text
stringlengths 0
6.23M
| __index_level_0__
int64 0
419k
|
|---|---|
\begin{document}
\maketitle
\begin{abstract}
We present a framework for the construction of solvable models of optical settings with genuinely two-dimensional landscapes of refractive index. Solutions of the associated non-separable Maxwell equations in paraxial approximation are found using the time-dependent supersymmetry.
We discuss peculiar theoretical aspects of the construction. In particular, we focus on the existence of localized solutions specific for the new systems. Sufficient conditions for their existence are discussed. Localized solutions vanishing for large $|\vec{x}|$, which we call light dots, as well as the guided modes that vanish exponentially outside the wave guides, are constructed. We consider different definitions of the parity operator and analyze general properties of the $\mathcal{PT}$-symmetric systems, e.g. presence of localized states or existence of symmetry operators. Despite the models with parity-time symmetry are of the main concern, the proposed framework can serve for construction of non-$\mathcal{PT}$-symmetric systems as well. We explicitly illustrate the general results on
a number of physically interesting examples, e.g. wave guides with periodic fluctuation of refractive index or with a localized defect, curved wave guides, two coupled wave guides or a uniform refractive index system with a localized defect.
\end{abstract}
\section{Introduction}
In specific situations, propagation of light is governed by the same equations as matter waves in quantum mechanics. The coincidence of Maxwell equations in paraxial approximation with the Schr\"odinger equation makes it possible to use methods of quantum mechanics in the analysis of the optical settings.
This link proved to be particularly fruitful for investigation of optical systems where a complex refractive index representing balanced gain and loss prevents uncontrolled dimming or brightening of light \cite{Ruschhaupt,coupled,Makris,Christo1,ruter}. The Hamiltonian of the associated Schr\"odinger equation ceases to be Hermitian but possesses an antilinear symmetry. It was demonstrated two decades ago that such operators, having typically a $\mathcal{PT}$-symmetry with $\mathcal{P}$ and $\mathcal{T}$ being parity and time-reversal, can have purely real spectra \cite{BenderBoetcher}. It was showed later on that such models can provide consistent quantum mechanical predictions despite the non-Hermiticity of the Hamiltonian as long as the scalar product of the associated Hilbert space is redefined \cite{scalarproduct1,scalarproduct2,scalarproduct3,benderRev,AliMostRev}. As much as this task proved to be difficult to accomplish in explicit quantum systems, see e.g. \cite{Bila,Krejcirik}, it
is non-existent in the realm of classical optics which, therefore, becomes an exciting field for the investigation of the systems described by $\mathcal{PT}$-symmetric (pseudo-Hermitian) Hamiltonians.
Supersymmetric quantum mechanics represents a highly efficient framework for construction of new exactly solvable models \cite{Cooper,Spectraldesign,Fernandez}. It is based on the Crum-Darboux transformation which is known in the analysis of Sturm-Liouville equations for a long time, see \cite{MatveevSale} and references therein. It allows to modify the potential term of the equation while preserving its solvability; the solutions of the new equation can be found by direct application of the Darboux transformation on the solutions of the original one. As by-product of the Crum-Darboux transformation, there could appear additional discrete energy levels in the spectrum of the new system. They are associated with the localized wave functions that were missing in the original system. i.e. these ``missing" states do not have preimage in the form of localized solutions. In this manner, the Crum-Darboux transformation can be employed in ``spectral design" of quantum systems \cite{Spectraldesign}.
Supersymmetry was utilized in the analysis of $\mathcal{PT}$-symmetric quantum models \cite{ZnojilSUSY,LevaiSUSY,AndrianovPTSUSY,CorreaLechtenfeld}. It has been used in the construction of $\mathcal{PT}$-symmetric optical systems of required properties \cite{MiriPRL,Miri,LonghiEuroPhysLett,Mathias1,Mathias2,Tkachuk,MidyaInv,LonghiInv,Plyushchay,Correa15}.
For example, systems with invisible defects in crystal \cite{LonghiInv,Correa15}, with transparent interfaces \cite{LonghiEuroPhysLett} or unidirectional invisibility \cite{MidyaInv} were constructed. Supersymmetry was also employed in the construction of random potentials with the energy bands in the spectrum and in the construction of $2D$ systems with potentials that are separable in Cartesian coordinates \cite{susyrandombands}, \cite{longhicrossroad}. It was also utilized in the context of coupled mode systems \cite{LonghiBragg,VJFCPTDirac}, in the formation of optical arrays \cite{mydia} or digital switching of spatially random waves \cite{park2}. Moreover, experiments with photonic lattices were performed \cite{Mathias1,Mathias2}.
It is worth mentioning the increasing interest in the non-$\mathcal{PT}$-symmetric systems with complex refractive index where, however, the gain and loss can still support guided and non-decaying modes associated with real spectra of the associated Hamiltonians. Explicitly solvable models of these systems were constructed via supersymmetry in \cite{Miri}. They were also studied numerically \cite{Nixon,Turitsyna,Yang} and experimental setups were proposed in \cite{Hang}.
Vast majority of the settings considered in the literature are described by effectively one-dimensional Hamiltonians. In most cases, they possess translational symmetry, typically along the axis of propagation of the light beam. The two-dimensional, exactly (analytically) solvable models possessing separability in radial coordinates were considered in \cite{Agarwal,Macho}, the models separable in Cartesian coordinates were presented e.g. in \cite{susyrandombands,longhicrossroad}. $\mathcal{PT}$-symmetry breaking in two and three dimensions were considered in \cite{Stone}, scattering properties were studied in \cite{Ge}. Two-dimensional periodic arrays of localized gain and loss regions, called photonic crystals, were analyzed numerically in \cite{Mock}.
In this work, we focus on the construction of exactly solvable models of optical settings with non-separable complex refractive index with the use of the time-dependent Darboux transformation. Systems where the complex refractive index forms $\mathcal{PT}$-symmetric wave guides or that possess localized defects will be studied with focus on existence of the missing states. As the current experimental techniques \cite{Chen} seem to be ready for realization of such settings, analysis of exactly solvable models with genuinely two-dimensional complex inhomogeneities of the refractive index is desirable.
The work is organized as follows. In the next section, we present the framework of the time-dependent supersymmetry \cite{Samsonov1,Samsonov2} and discuss its peculiar properties. We find the missing state for a broad family of systems, the localized solution of the new Schr\"odinger equation whose preimage\footnote{If the Crum-Darboux transformation $\mathcal{L}$ maps the state $f$ from the domain of the original Schr\"odinger operator $S_0$ into the state $g$ from the domain of the new Schr\"odinger operator $S_1$, i.e. $\mathcal{L}f=g$, we say that $f$ is preimage of $g$.} in the original system ceases to be localized. We show that it can be used in construction of symmetry operators of both the original and the new setting. Two definitions of the parity operator are introduced. They are distinctive for the models presented in the following sections. In the section \ref{tres}, we construct $\mathcal{PT}$-symmetric, exactly solvable models with a defect in the form of a localized gain and loss. We construct the missing state that represents a ``light dot'', the localized solution of the Maxwell equations in paraxial approximation. In the section \ref{cuatro}, we construct $\mathcal{PT}$-symmetric wave guides where fluctuations of refractive index are vanishing for $|x| \rightarrow\infty$ whereas they can be periodic along $z$-axis. We provide an alternative construction of the missing states that represent guided modes in the new system. We illustrate the general results on explicit examples of periodically modulated $\mathcal{PT}$-symmetric wave guides supporting guided modes. In the subsection \ref{NonPT} we construct a model of a non-$\mathcal{PT}$-symmetric wave guide that possesses a guided mode despite the lack of $\mathcal{PT}$-symmetry.
\section{Mathematical framework}
In this section, we review briefly the main mathematical tools that will be used extensively in the forthcoming text. In particular, we present construction of the time-dependent Darboux transformation for the Schr\"odinger equation and discuss peculiarities of the construction for the $\mathcal{PT}$-symmetric systems. For the sake of completeness, let us start with a short review of the relation between the Schr\"odinger and the Maxwell equations.
\subsection{Paraxial approximation}
Consider a monochromatic light beam with wavelength in vacuum $\lambda$. Let $X,~Y$ and $Z$ be spatial coordinates. The Maxwell equations for electric and magnetic fields $\vec{E}=\vec{E}(X,Y,Z)$ and $\vec{H}=\vec{H}(X,Y,Z)$ of this monochromatic wave varying in time as $\exp(-i \omega t)$ are:
\begin{equation}
\nabla \times \vec{E} = i \omega \mu \vec{H}, \quad \nabla \times \vec{H}= - i \omega \epsilon \vec{E}. \label{Maxwell}
\end{equation}
In this article we will focus on waves propagating mainly in the $Z$ direction in a medium with refractive index $n(X,Y,Z) = \sqrt{\mu \epsilon/\mu_0 \epsilon_0}=c \sqrt{\mu \epsilon}$. Under circumstances that will be discussed in this subsection, equations \eqref{Maxwell} can be written as a Schr\"odinger equation. This process is known as paraxial approximation \cite{Lax75,Permitin96,Miri,Cruz15,Cruz15b,Cruz17}. We revisit some important aspects of this approximation in this subsection as presented in \cite{Lax75}, with minor changes in notation.
Let us write the electric field $\vec{E}$ as
\begin{eqnarray}
\vec{E}=\exp(i k n_0 Z)\left(\vec{\psi}_T + \hat{a}_Z \psi_Z \right), \label{ansatz}
\end{eqnarray}
where $k=2 \pi/ \lambda$ is the corresponding wave number in vacuum, $n_0$ is a reference value of the refractive index, $\hat{a}_Z$ is a unit vector in the $Z$ direction, $T$ stands for the transverse part of the field, $\vec{\psi}_T= \vec{\psi}_T(X,Y,Z)$ and $\psi_Z=\psi_Z(X,Y,Z)$.
By taking the curl of the first equation in \eqref{Maxwell}, the equation that the electric field must satisfy is
\begin{eqnarray}
\nabla (\nabla \cdot \vec{E}) - \nabla^2 \vec{E}= k^2 n^2 \vec{E}. \label{Eequation}
\end{eqnarray}
Then, substitution of ansatz \eqref{ansatz} in \eqref{Eequation} and the use of the notation $\nabla_T= \hat{a}_X \partial_X + \hat{a}_Y \partial_Y$ leads to the transverse equation
\begin{eqnarray}
\nabla_T \left( \nabla_T \cdot \vec{\psi}_T+ i k n_0 \psi_Z + \partial_Z \psi_Z \right) - \nabla^2_T \vec{\psi}_T - \partial_Z^2 \vec{\psi}_T + k^2 n_0^2 \vec{\psi}_T - 2i k n_0 \partial_Z \vec{\psi}_T = k^2 n^2 \vec{\psi}_T, \label{transverse}
\end{eqnarray}
and the longitudinal equation
\begin{eqnarray}
i k n_0 \nabla_T \cdot \vec{\psi}_T + \partial_Z \left( \nabla_T \cdot \vec{\psi}_T \right) - \nabla^2_T \psi_Z = k^2 n^2 \psi_Z. \label{longitudinal}
\end{eqnarray}
Equations \eqref{transverse} and \eqref{longitudinal} can be approximated and simplified when introducing a small parameter. In this problem we have three different scales, first the wavelength $\lambda$, second the characteristic size of the beam in the transverse direction $x_0$ and finally a longitudinal distance $\ell$ defined as $\ell = n_0 k x_0^2$ known as diffraction length. We define our parameter as $\nu=x_0/\ell$. Introducing the scaled variables
\begin{eqnarray}
x=X/x_0, \quad y=Y/x_0, \quad z=Z/2 \ell,
\end{eqnarray}
equations \eqref{transverse} and \eqref{longitudinal} take the form
\begin{eqnarray}
\nabla_\perp \left( \nu \nabla_\perp \cdot \vec{\psi}_T+ i \psi_Z + \frac{\nu^2}{2} \partial_z \psi_Z \right) - \nu\nabla^2_\perp \vec{\psi}_T - \frac{\nu^3}{4}\partial_z^2 \vec{\psi}_T - i \nu \partial_z \vec{\psi}_T &=& \nu (k x_0)^2 (n^2-n_0^2) \vec{\psi}_T, \label{transverse2} \\
i \nu \nabla_\perp \cdot \vec{\psi}_T + \frac{\nu^3}{2}\partial_z \left( \nabla_\perp \cdot \vec{\psi}_T \right) - \nu^2 \nabla^2_\perp \psi_Z &=& \nu^2 (k n x_0)^2 \psi_Z, \label{longitudinal2}
\end{eqnarray} respectively, where the scaled differential operators are $\nabla_\perp = x_0 \nabla_T$ and $\partial_z= 2\ell \partial_Z$. To introduce non-linear effects of the media, let us consider the refractive index as
\begin{equation}
n^2=n_0^2+n_0 k g m, \label{index nonlinear}
\end{equation}
where $g$ is the signal gain per meter and $m$ is called homogeneous broadening. Substituting \eqref{index nonlinear} in \eqref{longitudinal2} we obtain:
\begin{eqnarray}
i \nu \nabla_\perp \cdot \vec{\psi}_T + \frac{\nu^3}{2}\partial_z \left( \nabla_\perp \cdot \vec{\psi}_T \right) - \nu^2 \nabla^2_\perp \psi_Z = \left(1+\nu^2 \ell ~g~m \right) \psi_Z. \label{longitudinal3}
\end{eqnarray}
If the parameter $\nu$ is small, $\nu << 1$, we can expand our functions $\psi_Z$ and $\vec{\psi}_T$ in powers of $\nu$, i. e.
\begin{eqnarray}
\psi_Z(x,y,z)&=& \psi_Z^{(0)}+\nu \psi_Z^{(1)}+ \nu^2
\psi_Z^{(2)}+ \dots \\
\vec{\psi}_T(x,y,z)&=& \vec{\psi}_T^{(0)}+ \nu \vec{\psi}_T^{(1)}+ \nu^2 \vec{\psi}_T^{(2)}+ \dots.
\end{eqnarray}
From the zeroth-order term in $\nu$ of \eqref{longitudinal3} we obtained $\psi_Z^{(0)}=0$ and from the first-order terms $i \nabla_\perp \cdot \vec{\psi}_T^{(0)}= \psi_Z^{(1)}$. Thus, the lowest order in $\nu$ of \eqref{transverse2} can be written as
\begin{eqnarray}
i \partial_z \vec{\psi}_T^{(0)} + \nabla_\perp^2 \vec{\psi}_T^{(0)} - k^2 x_0^2(n_0^2-n^2)\vec{\psi}_T^{(0)} = 0.\label{paraxial 1}
\end{eqnarray}
Each vector component in \eqref{paraxial 1} satisfies a time dependent Schr\"odinger equation:
\begin{eqnarray}
i\partial_t \psi + \partial_x^2 \psi - V \psi = 0,
\end{eqnarray}
where the $z$ variable plays the role of time parameter and the potential $V = k^2 x_0^2(n_0^2-n^2)$. Typical numbers in $\text{LiNbO}_3$ waveguides are \cite{Zhang,Chen}: refractive index varying from $n_0=2.217$ to $n_\text{max}=2.230$, wavelength of light $\lambda = 1064$nm and characteristic size of beam $x_0=10\mu$m. Then, diffraction length is $\ell=1.30919$mm, the parameter $\nu$ takes the value $\nu=0.00763829$ and the potential $V$ is zero where $n=n_0$ and $V=-201.598$ in regions where $n=n_{\text{max}}$.
\subsection{Time-dependent Darboux transformation and $\mathcal{PT}$-symmetry} \label{SUSY and PT}
To our best knowledge, the Darboux transformation in the context of optical systems was employed in the analysis of effectively one-dimensional models. In the current article, we shall focus on settings where the Schr\"odinger equation cannot be reduced to an effectively one-dimensional equation as the fluctuations of the refractive index (both its real and imaginary part) are genuinely two dimensional.
\subsubsection*{Standard 1D supersymmetric quantum mechanics}
Standard one-dimensional quantum mechanics is based on the factorization of the 1D Hermitian Hamiltonian $H_0$
\begin{equation}\label{factorization}H_0=-\partial_x^2+V(x)=L^{\dagger}L, \quad \mbox{where}\quad L=\partial_x+\mathcal{W}(x), \quad \mathcal{W}(x)=-\partial_x\ln u,\end{equation}
$\mathcal{W}(x)$ is called superpotential and $u$ solves $(H_0-E_0)u=0$.
The factorization allows for the construction of a new operator $H_1$ that is intertwined with $H_0$ by either $L$ or $L^\dagger$,
$$H_1L=LH_0,\quad L^\dagger H_1=H_0L^\dagger,\quad H_1=LL^\dagger=H_0-2\partial_x^2\ln u(x).$$
The intertwining relations imply that we can get solutions of $(H_1-E)\phi=0$ from the solutions of $(H_0-E)\psi=0$ by $\phi=L\psi$. The function $u$ is annihilated by $L$. However, one can define the eigenstate of $H_1$ corresponding to $E_0$ as $u_m=u^{-1}$. It satisfies $(H_1-E_0)u_m=0.$ Its definition suggests that it can be identified with the bound state of $H_1$ provided that $u$ is exponentially expanding and has no zeros.
Then $E_0$ represents a discrete energy of $H_1$ but it does not belong to the energy spectrum of $H_0$. The function $u_m$ is called missing state. The intertwining operator $L$ can be utilized for mapping scattering states of $H_0$ onto scattering states of $H_1$. When $\mathcal{W}(x)$ is asymptotically constant for large $|x|$, the action of $L$ on the scattering states just alter their phase.
\subsubsection*{Time-dependent Darboux transformation and the missing states}
Let us suppose that the following Schr\"odinger equation
\begin{eqnarray}\label{S_0}
S_0\psi=i \partial_z \psi + \partial_x^2 \psi -V_0(x,z) \psi=0,\quad x\in\mathbb{R},\quad z\in\mathbb{R}, \label{SUSY SE}
\end{eqnarray}
is exactly solvable and its solutions are known. We suppose that $V_0(x,z)$ has no singularities in $\mathbb{R}^2$ and it is sufficiently smooth. We will use the time-dependent Darboux transformation discussed in \cite{Samsonov1,Samsonov2,Contreras17,Zelaya17} to generate another exactly solvable equation with a different potential term. Let us present here the main steps of the construction. As the factorization of (\ref{S_0}) in the spirit of (\ref{factorization}) is not possible, the construction is based on the intertwining relation
\begin{eqnarray}
S_1 \mathcal{L} = \mathcal{L} S_0 \label{SUSY Intertwining}
\end{eqnarray}
that guarantees that we can get solutions of the new equation $S_1\phi=0$, where $\phi$ is defined as $\phi=\mathcal{L}\psi$, provided that $S_0\psi=0$ and $\mathcal{L}$ maps the domain of $S_0$ into the domain of $S_1$.
The ansatz for the intertwining operator $\mathcal{L}$ is in the form a first order differential operator, $S_1$ is a Schr\"odinger operator with an altered potential term,
\begin{eqnarray}\label{S1L}
\mathcal{L}= L_1(z) \left[ \partial_x+\mathcal{W}(x,z)\right], \quad S_1= i \partial_z +\partial_x^2 -V_1(x,z), \quad \mathcal{W}(x,z)=- \frac{\partial_x u(x,z)}{u(x,z)}. \label{S01L}
\end{eqnarray}
Here, $V_1(x,z)$, $u(x,z)$ and $L_1(z)$ are to be fixed such that the intertwining relation (\ref{SUSY Intertwining}) is satisfied.
Substituting (\ref{S_0}) and (\ref{S1L}) into (\ref{SUSY Intertwining}), one can find that the latter relation can be satisfied as long as
\begin{eqnarray}\label{V_1}
V_1(x,z)=V_0(x,z)+i \partial_z \ln L_1(z)-2 \partial_x^2 \ln u(x,z) \label{SUSY V1}
\end{eqnarray}
and
\begin{equation}\label{uc}
S_0 u(x,z)= c(z) u(x,z),
\end{equation}
see \cite{Samsonov1} for details. As the function $c(z)$ affects just the phase of the solution\footnote{If $S_0\psi=0$ holds, then we can find solution of $(S_0-c(t))\tilde{\psi}=0$ that reads $\tilde{\psi}=\exp(-i\int c(t))\psi$.} but not the potential $V_1$, it can be set to zero, $c(z)=0$. In what follows, we will denote by $u$ the solution of $S_0u=0$ used in definition of the new potential (\ref{V_1}) and of the intertwining operator \eqref{S1L}. It will be called transformation function.
Relations (\ref{V_1}) and (\ref{uc}) are sufficient to establish the intertwining relation. In addition, the function $u(x,z)$ as well as $L_1(z)$ are also required to be nodeless, otherwise, the transformation would be singular and it would fail to provide the mapping between the domains of $S_0$ and $S_1$. When $\mathcal{L}$ as well as $V_1$ are regular, relation \eqref{SUSY Intertwining} guarantees that we can generate solutions of $S_1 \phi(x,z)=0$ from the solutions $\psi(x,z)$ of \eqref{SUSY SE} by $\mathcal{L}$,
\begin{eqnarray}
\phi(x,z)= \mathcal{L} \psi(x,z). \label{SUSY states}
\end{eqnarray}
We can try to find the ``inverse" transformation $\mathcal{L}^\sharp$ such that it satisfies
\begin{equation}\label{inverseInt}
S_0\mathcal{L}^\sharp=\mathcal{L}^\sharp S_1.
\end{equation}
Using the general formulas (\ref{S1L}), we take $S_1$ as the initial system and we define $\mathcal{L}^\sharp=L_2(z)v(x,z)\partial_x \frac{1}{v(x,z)}$ where $v(x,z)$ solves $S_1v(x,z)=0$. Then it is granted that there holds $\mathcal{L}^\sharp S_1=S_2\mathcal{L}^\sharp$ for
\begin{equation}\label{S2}
S_2=i \partial_z +\partial_x^2 -V_0(x,z)-i\partial_z\ln L_1 L_2+2\partial_x^2\ln u v.
\end{equation}
In order to identify $S_2=S_0$, we have to eliminate the last two terms by setting $2\partial_x^2\ln u v=i\partial_z\ln L_1 L_2$. As the right-hand side of the latter equation is $x$-independent, we have to fix $v$ such that
\begin{equation}\label{uuvv}
\partial_x^3\ln u v=0
\end{equation}
and also we must fix $L_2=L_1^{-1}(z)\exp (-2i\int^z (\ln u v)'')$. Then the last two terms in (\ref{S2}) vanish and $\mathcal{L}^\sharp$ represents the inverse
intertwining operator.
\subsubsection*{Finding a missing state and symmetry operators}
The operator $\mathcal{L}$ can map any solution of $S_0f=0$ to a nontrivial solution of $S_1g=0$ as $g=\mathcal{L}f$, except the case where $f\equiv u$ as it gets annihilated by $\mathcal{L}$, $\mathcal{L}u=0$. Hence, the image of $u$ is missing in the new system. We can try to find another solution of $S_1g=0$ given in terms of the function $u$. In the one-dimensional supersymmetric quantum mechanics, this missing state is defined as $u^{-1}$. This formula hints on the importance of the missing state; when $u$ is exponentially growing, the missing state is square integrable and represents a bound state of the new system. In \cite{Samsonov1}, similar formula was used for the time-dependent, Hermitian systems. Inspired by these results, let us make an ansatz for the missing state in the following form
\begin{equation}\label{um}
u_m=\frac{1}{f(z)\mathcal{S}u},
\end{equation}
where $f(z)$ is a function and $\mathcal{S}$ is an operator whose properties are to be fixed such that the equation $S_1u_m=0$ is satisfied. We shall compute $S_1u_m$. We have
\begin{equation}\label{28}
S_1\frac{1}{f\mathcal{S}u}=\frac{1}{(\mathcal{S}u)^2f}\left(-i\dot{(\mathcal{S}u)}+(\mathcal{S}u)''-V_0 \mathcal{S}u+2\left(\ln \frac{u}{\mathcal{S}u}\right)''-i\dot{(\ln{(L_1 f)})}\mathcal{S}u\right).
\end{equation}
If the condition $\partial_x^3\ln \frac{u}{\mathcal{S}u}=0$ holds, then we can fix $f(z)=L_1^{-1}(z)\exp(-2i\int^z \left(\ln \frac{u}{\mathcal{S}u}\right)'')$ and the last two terms in (\ref{28}) vanish.
Hence, $u_m$ solves $S_1u_m=0$ provided that there holds
\begin{equation}\label{Scond}
[-\partial_x^2+V_0,\mathcal{S}]=0,\quad \{i\partial_z,\mathcal{S}\}=0,\quad\partial_x^3\ln \frac{u}{\mathcal{S}u}=0,
\end{equation}
and $f(z)$ is fixed as
\begin{equation}\label{f(z)}
f(z)=L_1^{-1}(z)\exp\left(-2i\int^z \left(\ln \frac{u}{\mathcal{S}u}\right)''\right).
\end{equation}
We can see that the third relation in (\ref{Scond}) coincides with (\ref{uuvv}) for $v\equiv u_m$. Therefore, when (\ref{Scond}) are satisfied, there also exist the inverse operator,
\begin{equation}\label{tildeL}
\mathcal{L}^\sharp=f(z)u_m\partial_x \frac{1}{u_m},
\end{equation}
where $u_m$ and $f(z)$ are defined in (\ref{um}), (\ref{Scond}) and (\ref{f(z)}).
Existence of the inverse operator (\ref{tildeL}) implies another interesting fact; both $S_0$ and $S_1$ have symmetry operators
\begin{equation}
[S_0,\mathcal{L}^\sharp\mathcal{L}]=0,\quad [S_1,\mathcal{L}\mathcal{L}^\sharp]=0,
\end{equation}
where
\begin{eqnarray}\label{tildeLL}
\mathcal{L}^\sharp \mathcal{L}= \exp\left(-2i\int^z \left(\ln u_m u\right)''\right) \left[ \partial_x^2-\left(\ln u_m u\right)'\partial_x+ (\ln u)' (\ln u_m)'-(\ln u)'' \right],\nonumber\\
\mathcal{L}\mathcal{L}^\sharp=\exp\left(-2i\int^z \left(\ln u_m u\right)''\right) \left[ \partial_x^2-\left(\ln u_m u\right)'\partial_x+ (\ln u)' (\ln u_m )'-(\ln u_m)'' \right].
\end{eqnarray}
\subsubsection*{$\mathcal{PT}$-symmetry}
Up to now, we did not make any assumption on the Hermiticity or $\mathcal{PT}$-symmetry of the new potential $V_1$. In \cite{Samsonov1}, both $V_0$ and $V_1$ were required to be real in order to preserve Hermiticity of $S_0$ and $S_1$. In the Hermitian case the operator $\mathcal{S}$ can be identified with $\mathcal{S}f(x,z)=\overline{f(x,z)}$. Then $V_1$ is real whenever $u$ satisfies $\partial_x^3\ln \frac{u}{\overline{u}}=0$ and $L_1$ is fixed as $L_1=\exp\left(-i\int\left(\ln \frac{u}{\overline{u}}\right)''dx\right)$. This ``Hermitian" definition of $\mathcal{S}$ complies with (\ref{Scond}).
We are interested in the settings where $S_1$ ceases to be Hermitian but possesses an antilinear symmetry that we shall identify with the simultaneous action of the operators of time-reversal $\mathcal{T}$ and space inversion $\mathcal{P}$.
Most of the $\mathcal{PT}$-symmetric systems discussed in the literature are effectively one-dimensional so that the space inversion $\mathcal{P}$ is defined unambiguously as $\mathcal{P}f(x)=f(-x)$. In two dimensions, we can define $\mathcal{P}$ as the reflection with respect to a fixed point or with respect to an axis,
\begin{equation}\label{Ps}
\mathcal{P}_xf(x,z)=f(-x,z)\quad\mbox{or}\quad\mathcal{P}_2f(x,z)=f(-x,-z).
\end{equation}
The antilinear operator $\mathcal{T}$ is given as
\begin{equation}
\mathcal{T}f(x,z)=\overline{f(x,z)}.
\end{equation}
We suppose that $V_0$ is $\mathcal{PT}$-symmetric and we require $V_1$ to be $\mathcal{PT}$-symmetric as well,
\begin{equation}\label{PTV_1PT}
\mathcal{PT}V_1(x,z)\mathcal{PT}=V_1(x,z).
\end{equation}
It will restrict the possible choice of $u(x,z)$ and $L_1$ in dependence on the actual definition of $\mathcal{P}$.
First, let us consider $\mathcal{P}\equiv\mathcal{P}_x$. Then $V_1$ is $\mathcal{P}_x\mathcal{T}$-symmetric provided that $u$ and $L_1$ satisfy
\begin{equation}\label{uPT1}
2\partial_x^2\ln\frac{u}{\overline{u(-x,z)}}=i\partial_z\ln |L_1(z)|^2.
\end{equation}
The $\mathcal{P}_x\mathcal{T}$ operator anticommutes with $i\partial_z$ and it commutes with $V_0$, so that it fulfills the first two conditions in (\ref{Scond}). As the condition (\ref{uPT1}) is stronger than the third relation in (\ref{Scond}), we find that when the potential is $\mathcal{P}_x\mathcal{T}$-symmetric, then it also possesses missing state defined by (\ref{um}) (with $\mathcal{S}\equiv\mathcal{P}_x\mathcal{T}$), the inverse operator $\mathcal{L}^\sharp$ and the symmetry operators (\ref{tildeLL}).
If we set $\mathcal{P}=\mathcal{P}_2$, the requirement (\ref{PTV_1PT}) reduces to
\begin{equation}\label{uPT2}
2\partial_x^2\ln\frac{u(x,z)}{\overline{u(-x,-z)}}=i\partial_z\ln\frac{L_1(z)}{\overline{L_1(-z)}}.
\end{equation}
The operator $P_2T$ commutes with $i\partial_z$, so that we cannot identify it with $\mathcal{S}$ in (\ref{Scond}).
A few comments are in order. Having the transformation function $u$, we can define a whole family of systems that differ by the choice of $L_1$. When the function $u$ satisfies $\partial_x^3\ln\frac{u(x,z)}{\mathcal{P}_x\mathcal{T}u(x,z)}=0$, then we can identify $\mathcal{S}\equiv \mathcal{P}_x\mathcal{T}$ and the missing state is defined by (\ref{um}). In this family, we can set $L_1$ in accordance with (\ref{uPT1}) and the resulting system will be $\mathcal{P}_x\mathcal{T}$-symmetric. When the function $u$ also satisfies $\partial_x^3\ln\frac{u(x,z)}{\mathcal{P}_2\mathcal{T}u(x,z)}=0$, then we can find $\mathcal{P}_2\mathcal{T}$-symmetric systems in the family as one can define $L_1$ in coherence with (\ref{uPT2}). If $\partial_x^3\ln\frac{u(x,z)}{\mathcal{P}_2\mathcal{T}u(x,z)}=0$ holds, but $\partial_x^3\ln\frac{u(x,z)}{\mathcal{P}_x\mathcal{T}u(x,z)}=0$ does not, there can be only $\mathcal{P}_2\mathcal{T}$ symmetric systems in the family and we cannot use the definition (\ref{um}) of the missing state.
\subsubsection*{Higher order (Crum)-Darboux transformations}
Let us make a few comments on the repeated use of the time-dependent Darboux transformation (\ref{V_1}). Having the Schr\"odinger operators $S_0$ and $S_1$ intertwined by $\mathcal{L}_1$,
\begin{equation}\label{ir1}
S_1\mathcal{L}_1=\mathcal{L}_1S_0, \quad S_1=S_0+2\partial_x^2\ln u_1-i\partial_z \ln L_1,\quad \mathcal{L}_1=L_1u_1\partial_xu_1^{-1},
\end{equation}
we can select a function $\check{u}_2$ such that $S_1\check{u}_2=0$ and use it to define the new intertwining operator $\mathcal{L}_2$ that satisfies
\begin{equation}\label{ir2}
S_2\mathcal{L}_2=\mathcal{L}_2S_1, \quad S_2=
S_0+2\partial_x^2\ln u_1+2\partial_x^2\ln \check{u}_2-i \partial_z\ln L_1L_2,\quad \mathcal{L}_2=L_2\check{u}_2\partial_x\check{u}_2^{-1}.
\end{equation}
In this manner, a chain of the new solvable equations can be obtained.
Combining (\ref{ir1}) and (\ref{ir2}), we can see immediately the that the operators $S_0$ and $S_2$ are intertwined by the operator $\mathcal{L}_{12}=\mathcal{L}_2\mathcal{L}_1$. When we find a preimage ${u}_2$ of $\check{u}_2$ such that $\mathcal{L}_1u_2=\check{u}_2$ (the function $u_2$ does not need to be a solution of $S_0u_2=0$), we can rewrite both $\mathcal{L}_{12}$ and $S_2$ directly in terms of $u_1$ and $u_2$,
\begin{equation}\label{s2}
S_2\mathcal{L}_{12}=\mathcal{L}_{12}S_0,\quad S_2=S_0+2\partial_x^2\ln W(u_1,u_2)-i\partial_z\ln L_1L_2,
\end{equation}
and
\begin{equation}\label{ir12}
\mathcal{L}_{12}=\frac{L_1L_2}{W(u_1,u_2)}\left|\begin{array}{ccc}u_1&u_2&1\\u_1'&u_2'&\partial_x\\u_1''&u_2''&\partial^2_x\end{array}\right|=L_2L_1\left(\partial_x^2+\frac{u_2u_1''-u_1u_2''}{W(u_1,u_2)}\partial_x+\frac{-u_2'u_1''+u_1'u_2''}{W(u_1,u_2)}\right),
\end{equation}
where $W(u_1,u_2)=u_1u_2'-u_1'u_2$. The formulas (\ref{ir12}) can be generalized for an arbitrary chain-length of time-dependent Darboux transformations, see \cite{Samsonov1}. The properties of the final system, existence and properties of the missing states in particular, can be deduced from the careful analysis of the intermediate models (e.g. a missing state of $S_1$ gets transformed into a missing state of $S_2$ by $\mathcal{L}_2$). However, it is worth noticing that despite $S_1$ can have singularities in the potential, the potential term of $S_2$ can be a regular function. It stems from the fact that despite $u_1$ can have zeros (which would introduce singularities into $S_1$), the Wronskian of $u_1$ and $u_2$ can be nodeless, keeping $S_2$ regular.
When $V_0$ is $z$-independent, we can write the solutions $S_0u=0$ in terms of the stationary states $u(x,z)=e^{-i\epsilon z}\psi_\epsilon(x)$, $(-\partial_x^2+V_0-\epsilon)\psi_\epsilon=0$. When $u_1$ and $\check{u}_2=\mathcal{L}_1u_2$ are stationary states of $S_0$ and $S_1$, respectively, then the potential term of $S_2$ is also $z$-independent. The transformation $\mathcal{L}_{12}$ can be identified as the $N=2$ (time-independent) Crum-Darboux transformation.
The stationary states $u_1$ and $u_2$ can be selected as two eigenstates corresponding to different energy levels. Alternatively, we can define the function $u_2$ in terms of $u_1$: taking $u_1=e^{-iE_m z}{\psi}_{m}(x)$, we can fix
\begin{equation}\label{confluentu2}
u_2(x,z)=e^{-iE_mz}{\psi}_m(x)\left(\int_{x_0}^x\frac{1}{{\psi}_m^2}\left(\int_{s_0}^s{\psi}_m^2(r)dr+\alpha\right)ds+a\right),
\end{equation}
where $a$ and $\alpha$ are complex constants. The function $u_2$ satisfies $S_1\mathcal{L}_1u_2=0$, but $S_0u_2\neq 0$. Instead, it fulfills $S_0^2u_2=0$, see \cite{Correa15}. The operator $\mathcal{L}_{12}$ is called the confluent Crum-Darboux transformation in the literature, see e.g. \cite{MatveevSale,Correa15,Mielnik00,Salinas03,Schulze13,Contreras15,Plyushchay16} and references therein. The new Schr\"odinger operator $S_2$ can be written in terms of $\psi_m$ as
\begin{align}
& S_2=i\partial_z+\partial_x^2-V_2(x), \nonumber \\
&V_2(x)=V_0 -2\partial_x^2 \ln \left(\alpha+\int^x_0{\psi}^2_m(s)ds\right)= V_0 - 4\frac{{\psi}_m \partial_x{\psi}_m }{\alpha + \int_{0}^x {\psi}^2_m(s) ds} + 2 \frac{{\psi}_m^4}{\left( \alpha + \int_{0}^x {\psi}^2_m(s) ds \right)^2}. \label{Confluent SUSY V2}
\end{align}
When ${\psi}_m$ is a real function, the new potential will be free of singularities provided that $\alpha$ is a complex number with a non-vanishing imaginary part. The stationary states $f_n$ of $S_2$ for $n\neq m$ can be found by direct application of $\mathcal{L}_{12}$,
\begin{eqnarray}\label{Confluent SUSY Solutions}
f_n(x,z)&=&\mathcal{L}_{12}e^{-iE_nz}{\psi}_n(x)=L_1 L_2 \left(\partial_x-\frac{\partial_x\check{u}_2}{\check{u}_2}\right)\left(\partial_x-\frac{\partial_x u_1}{u_1}\right){\psi}_n(x)e^{-iE_nz}.
\end{eqnarray}
For $n=m$, we can find the following solution (see e.g. \cite{Correa15}) that represents the missing state of $S_2$,
\begin{eqnarray}\label{confluentmissingstate}
f_m(x,z)= \frac{{{\psi}}_m(x)}{\alpha + \int_{0}^x {{\psi}}_m^2(s) ds}e^{-iE_mz}. \label{Confluent missing state}
\end{eqnarray}
It is worth comparing the confluent transformation $\mathcal{L}_{12}$ with the first order (time-independent) Darboux transformation $\mathcal{L}$. Both transformations are defined in terms of a single function $\psi_m$ which also determines the form of the missing state; it is (\ref{confluentmissingstate}) for the confluent transformation whereas $\sim\psi_m^{-1}$ for the first order transformation. One can see from (\ref{confluentmissingstate}) that $f_m$ can be square integrable even in the case when ${\psi}_m$ is a bounded function \cite{Correa15}. This result cannot be obtained with the first order transformation.
\subsection{Prelude to the next sections}
In the forthcoming text, we will focus on two different scenarios:
\begin{itemize}
\item a localized defect of the refractive index
\item a straight wave guide with a periodically modulated profile
\end{itemize}
In the explicit construction of the solvable models, we will depart from the free particle system described by the equation
\begin{equation}\label{FPEq}
S_0f=(i\partial_z+\partial_x^2)f=0.
\end{equation}
This choice will help us to keep the illustrative examples simple enough and provide straightforward analysis of the missing states. It is worth mentioning that in the literature, see e.g. \cite{Miri, MiriPRL, Mathias1, Mathias2}, the supersymmetric techniques are usually utilized to annihilate a given localized mode (ground state) so that it is no longer present in the new, superpartner system. We intend to go the opposite way; the new systems should posses additional localized solutions that have no preimage in the original one.
The transformation function $u$, $S_0u=0$, determines the properties of the new system to a large extend.
There are the two notoriously known types of solutions of (\ref{FPEq}), the plane waves
\begin{equation}\label{planewave}
\Phi_{k,x_0,z_0,v_0}=e^{\pm i k (x-x_0+v_0z)-\frac{iv_0}{4}(2x+v_0 z)-i k^2(z-z_0)}
\end{equation}
and the wave packets
\begin{equation}\label{wavepacket}
\Psi_{x_0,z_0,v_0,\sigma}=\frac{1}{\sqrt{i(z-z_0)+\sigma}}e^{-\frac{(x-x_0+v_0(z-z_0))^2}{4(i(z-z_0)+\sigma)}-\frac{i}{4}v_0(2x+v_0z)},
\end{equation}
where $k$, $x_0$, $z_0$, $v_0$ and $\sigma$ are real parameters. The wave packet (\ref{wavepacket}) can be written as an infinite linear combination of the plane waves. Let us consider properties of the intertwining operator for different choices of $u$.
Neither (\ref{planewave}) nor (\ref{wavepacket}) are optimal for direct identification with the transformation function $u$; the finite combination $u$ of $\Phi_{k,x_0,z_0,v_0}$ does not satisfy $\partial_x^3\ln\frac{u(x,z)}{\overline{u(-x,z)}}=0$, i.e. the condition (\ref{Scond}) is not satisfied and the formula (\ref{um}) for the missing state cannot be used\footnote{Here we identify $\mathcal{S}\sim \mathcal{P}_x\mathcal{T}$.}. As we shall see, identification of $u$ with the wave packets $\Psi_{x_0,z_0,v_0,\sigma}$ does not lead to the system with required properties of the refractive index.
We will circumvent both these difficulties: in the first case, we will provide an alternative way for construction of the missing states. In the second case, we will construct other wave-packet-like solutions via a transformation that relates the free particle system with the one of the harmonic oscillator. This mapping consists of a specific change of coordinates and a gauge-like transformation. It can be written as
\begin{equation}
e^{-if(x,z)}S_{HO}(y(x,z),t(z))e^{if(x,z)}=g(z)S_{0}(x,z),
\end{equation}
where $S_{HO}(y,t)=i\partial_t+\partial_y^2-y^2/4$. The functions $y=y(x,z)$, $t=t(z)$, $f(x,z)$, and $g(z)$ are to be specified in section \ref{tres}.
The formula (\ref{V_1}) for the potential term of $S_1$
suggests that when $u$ is a (finite) linear combination of the plane waves (\ref{planewave}), the potential term will be non-vanishing and oscillating along the $z$-axis and, hence, it could form a wave guide. The wave packet solutions will be the candidates for the construction of the localized defects of $n(x,z)$. Hence, the following two sections will be distinguished by these two different choices of $u$,
\begin{align}
u=\begin{cases}\mbox{wave-packet-like solutions} \longrightarrow \mbox{localized defects of $n(x,z),$}\\\mbox{finite combination of stationary solutions}\longrightarrow \mbox{straight wave guides}.\end{cases}
\end{align}
The action of the intertwining operator $\mathcal{L}$ can dramatically change the profile of the transformed function.
If we select $u$ as a finite linear combination of the plane waves (\ref{planewave}) (that has no zeros), the superpotential $\mathcal{W}(x,z)=-\partial_x\ln u(x,z)$ is bounded both for large $|z|$ and $|x|$. It resembles the one-dimensional superpotential $\mathcal{W}(x)$ that is usually fixed such that it is asymptotically constant for large $|x|$.
When we identify $u$ with the wave packet that has $x^2$ term in the exponential, the superpotential then behaves as $\mathcal{W}(x,y)=-\partial_x\ln u(x,y)=O(x)$ for large $|x|$ and constant $z$. When we apply such $\mathcal{L}$ on the plane waves (\ref{planewave}), the resulting function will be an unbounded function of $x$. However, when we apply it on another wave packet (\ref{wavepacket}), we get a function that is still vanishing rapidly for large $|x|$. It is promising, as we would like to transform a generic wave packets $\Psi_{x_0,z_0,v_0,\sigma}$ into functions that have also bounded amplitude for all $x$ and $z$. The behavior of $\mathcal{L}\Psi_{x_0,z_0,v_0,\sigma}$ along the $z$-axis is largely affected by the choice of $L_1(z)$. We introduce an additional requirement that should guarantee boundedness of the transformed wave packets,
\begin{equation}\label{wavepacketpreservation}
\mathcal{L}\Psi_{x_0,z_0,v_0,\sigma}=G(x,z)\Psi_{x_0,z_0,v_0,\sigma} \quad \mbox{where} \quad |G(x,z)|\leq C<\infty,\quad \forall x,y\in\mathbb{R}.
\end{equation}
The function $G(x,z)$ reflects how the wave packet gets changed by the transformation and its asymptotic properties depend on the explicit form of both $u$ and $L_1(z)$. Hence, the relation (\ref{wavepacketpreservation}) will impose additional restriction (besides (\ref{uPT1}) or (\ref{uPT2}) imposed by $\mathcal{PT}$-symmetry) on the possible choice of $L_1(z)$ and will be used to fix this function in the explicit models.
\section{Systems with localized defects of refractive index\label{tres}}
By selecting different wave-packet-like solutions as transformation function $u$, we will construct systems with localized defects of refractive index in this section.
\subsection{Straight wave guide divided symmetrically by gain and loss regions}
In our seek for systems with localized defects of refractive index, let us start with a simple choice of $u$
\begin{equation}\label{gaussian1}
u(x,z)=\frac{(2 \pi)^{1/4}}{\sqrt{1-iz}}\exp\left( \frac{x^2}{4(1-iz)}\right).
\end{equation}
This function expands exponentially for large $|x|$ and it can be obtained from the Gaussian wave packet by the substitution $z\rightarrow-z$, $x\rightarrow ix$.
It satisfies the relation (\ref{Scond})
and it is nodeless. Therefore, the missing state (\ref{um}) and the symmetry operators (\ref{tildeLL}) are well defined.
To make the definition of the intertwining operator and the new system unambiguous, we have to fix the function $L_1$. The intertwining operator is required to preserve the amplitude of the wave packets, see (\ref{wavepacketpreservation}). We have
\begin{equation}\label{GaussWP}
\mathcal{L}\Psi_{x_0,z_0,v_0,\sigma}=G(x,z)\Psi_{x_0,z_0,v_0,\sigma},\quad G(x,z)=\frac{L_1(z)}{2}\left(-\frac{x}{1-iz}-\frac{x-x_0-iv_0\sigma}{\sigma+i(z-z_0)}\right).
\end{equation}
We should select $L_1$ such that $G(x,z)$ is a bounded function of $z$. We also require the new potential to be $\mathcal{PT}$-symmetric. The requirement (\ref{uPT1}) tells us that the new system will be $\mathcal{P}_x{T}$-symmetric provided that $|L_1|=\sqrt{1+z^2}$. It suggests $L_1=\sqrt{1+z^2}$ or $L_1=1\pm i z$ as the viable candidates. The requirement of $\mathcal{P}_2\mathcal{T}$-symmetry is less restrictive. Substituting $u$ into (\ref{uPT2}), we find that $V_1$ is $\mathcal{P}_2\mathcal{T}$-symmetric provided that $L_1(z)=\overline{L_1(-z)}$. We fix\footnote{Fixing $L_1=1-iz$ gives $V_1=0$ and $L_1=1+iz$ gives $V_1=-\frac{2}{1+z^2}$. } $L_1=\sqrt{1+z^2}$. Then we get
\begin{equation}\label{fpS1}
V_1=-\frac{1}{1+z^2},\quad \mathcal{L}=\sqrt{1+z^2}~\left( \partial_x - \frac{x}{2(1-iz)}\right).
\end{equation}
The potential term is $x$-independent. It has the form of a straight wave guide along $x$-axis, Fig. \ref{spontaneously pot} (a) and (b).
The intertwining operator $\mathcal{L}$ alters the profile of the wave packets and keeps them bounded for large $|z|$. Alternatively, we can define $\mathcal{L}$ such that the wave packets are mapped into the localized states of the new system. Taking $L_1=\sqrt{1-iz}$, the new potential $V_1$ is no longer $\mathcal{P}_x \mathcal{T}$-symmetric but it possesses $\mathcal{P}_2 \mathcal{T}$-symmetry, $V_1=-\frac{1}{2(1-iz)}$ and the transformed wave packets $\mathcal{L}\Psi_{x_0,z_0,v_0,\sigma}$ are strongly suppressed for large $|z|$, see Fig. \ref{spontaneously pot} (c)-(f). However, we prefer to fix $L_1$ such that it preserves the amplitude of the wave packets and we take $L_1=\sqrt{1+z^2}$.
The missing state (\ref{um}) reads
\begin{eqnarray}\label{fpum}
u_m(x,z)=\frac{1}{(2 \pi)^{1/4} (1-iz)^{1/2}} \exp\left(- \frac{x^2}{4(1+iz)} \right),
\end{eqnarray}
where $u_m$ fulfills $S_1 u_m =0$. The solution is well localized in the wave guide; it vanishes exponentially along $x$-axis while it has $\sim z^{-1/2}$ decay along the $z$-axis, see Fig. \ref{spontaneously pot} (g).
\begin{figure}[t!]
\begin{center}
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{Spontaneous3a.jpg}
\caption{}
\end{subfigure}
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{Spontaneous3b.jpg}
\caption{}
\end{subfigure}
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{Fig12.jpg}
\caption{}
\end{subfigure}\\
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{Fig13.jpg}
\caption{}
\end{subfigure}
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{Fig14.jpg}
\caption{}
\end{subfigure}
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{Fig15.jpg}
\caption{}
\end{subfigure}\\
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{Spontaneous3g.jpg}
\caption{}
\end{subfigure}
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{Spontaneous3h.jpg}
\caption{}
\end{subfigure}
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{Spontaneous3i.jpg}
\caption{}
\end{subfigure}
\caption{A straight wave guide divided symmetrically by gain and loss regions. Real (a) and imaginary (b) parts of the potential term $V_1=-\frac{1}{1+z^2}$. In (c) the imaginary part for the potential $V_1=-\frac{1}{2(1-iz)}$. The intensity density $|\Psi_{x_0,z_0,v_0,\sigma}|^2$ is shown in (d). In (e) we show $|\mathcal{L}\Psi_{x_0,z_0,v_0,\sigma}|^2$ where $L_1(z)=\sqrt{1+z^2}$. In (f) we show $|\mathcal{L}\Psi_{x_0,z_0,v_0,\sigma}|^2$ where $L_1(z)=\sqrt{1-iz}$. The parameters used in (d)-(f) are: $x_0=-10$, $z_0=-100$, $v_0=-0.25$, $\sigma=30$. Three different solutions of the time dependent Schr\"odinger equation for the potential $V_1=-\frac{1}{1+z^2}$, see (\ref{fpum}), (\ref{fpphi}) and (\ref{fpchi}), $|u_m|^2$ (g), $|\phi|^2$ (h) and $|\chi|^2$ (i) are also graphed.} \label{spontaneously pot}
\end{center}
\end{figure}
Symmetry operators (\ref{tildeLL}) can be constructed as well,
\begin{eqnarray}\label{FPLL}
\mathcal{L}^\sharp \mathcal{L}&=& (1+z^2) \partial_x^2 -i x z\partial_x - \frac{1}{4}(x^2+2 i z +2), \quad [S_0,\mathcal{L}^{\sharp}\mathcal{L}]=0, \nonumber \\
\mathcal{L}\mathcal{L}^\sharp&=& (1+z^2) \partial_x^2 -i x z\partial_x - \frac{1}{4}(x^2+2 i z -2), \quad [S_1,\mathcal{L}\mathcal{L}^\sharp]=0.
\end{eqnarray}
By construction, $\mathcal{L}^\sharp u_m=0$. In order to illustrate the action of the symmetry operator $ \mathcal{L}\mathcal{L}^\sharp$, let us consider the wave packet $\psi=(2 \pi)^{-1/4}(1+iz)^{-1/2} \exp(-(x+1/2)^2/4(1+iz) )$ solving $S_0 \psi = 0$.
We transform it into a solution of $S_1 \phi=0$ by the application of the intertwining operator,
\begin{eqnarray}\label{fpphi}
\phi= \mathcal{L} \psi = - \frac{4x -iz+1}{4(2 \pi)^{1/4}(1+iz)\sqrt{1-iz}} \exp\left(- \frac{(x+\frac{1}{2})^2}{4(1+iz)} \right),\quad S_1\phi=0.
\end{eqnarray}
Through the successive applications of the symmetry operator $ \mathcal{L}\mathcal{L}^\sharp$ we can obtain a whole family of solutions. After the first iteration, we get
\begin{eqnarray}\label{fpchi}
\chi= \mathcal{L}\mathcal{L}^\sharp \phi= \frac{-16x^2 + 56x +72 i xz +2iz +33 z^2+31}{64 (2\pi)^{1/4} (1+iz)^2 \sqrt{1-iz}} \exp\left(- \frac{(x+\frac{1}{2})^2}{4(1+iz)} \right),\quad S_1\chi=0.
\end{eqnarray}
The solutions $u_m$, $\phi$ and $\chi$ are illustrated in the Fig. \ref{spontaneously pot} (g)-(i). It is worth mentioning that as the system possesses translational invariance with respect to $x$-axis, all the presented solutions (\ref{GaussWP}), (\ref{fpum}), (\ref{fpphi}) and (\ref{fpchi}) can be modified by $x\rightarrow x+a$, $a\in\mathbb{R}$, without compromising validity of the Schr\"odinger equation.
As we can see, identification of $u$ with the Gaussian wave packet (\ref{gaussian1}) resulted in the construction of the new system with a localized missing state. However, the new Schr\"odinger operator has separable potential (\ref{fpS1}). The refractive index possesses translational invariance and forms a barrier in propagation of the light beam. It is not quite satisfactory result as we seek for a non-separable, two-dimensional system with localized defects of the refractive index.
We shall find alternative transformation function $u$ that would serve better in construction of the models with desired properties.
\subsection{Free particle solutions via harmonic oscillator} \label{Point T subsection}
The free particle and harmonic oscillator systems are related through a specific point transformation, see \cite{Abraham80,Ray82,Bluman83,Finkel99,Guerrero11,Schulze14}. It allows to map the solutions of one system into the solutions of the other system.
Let us consider the Schr\"odinger equation of the harmonic oscillator given in terms of the variables $y$ and $t$,
\begin{eqnarray}\label{SchHO}
S_{HO}\widetilde{\psi}(y,t)=\left(i\partial_t+\partial_y^2-\frac{1}{4}y^2\right)\widetilde{\psi}(y,t)=0, \label{Finkel TISE}
\end{eqnarray}
Now, let $y$ and $t$ be defined in terms of the new variables $z$ and $x$ as
\begin{eqnarray}
y(x,z)= \frac{x}{\sqrt{1+z^2}},\quad t=\arctan z.
\label{FinkelVariable}
\end{eqnarray}
Then the Schr\"odinger operator $S_{HO}$ of the Harmonic oscillator can be transformed into the Schr\"odinger operator of the free particle multiplied by a $z$-dependent function,
\begin{equation}\label{FinkelFunction}
U^{-1}S_{HO}U=(1+z^2)(i\partial_z+\partial_x^2)=(1+z^2)S_0,\quad U=
e^{-\frac{ix^2z}{4(1+z^2)}}(1+z^2)^{1/4}.
\end{equation}
This transformation allows us to transform the solutions $S_{HO}\widetilde{f}(y,t)=0$ into the solutions of $S_{0}f(x,z)=0 $,
\begin{equation}\label{pointtPsi}
f(x,z)=U^{-1}\widetilde{f}(y(x,z),t(z)).
\end{equation}
The stationary solutions
$\widetilde{u}_{I,n}$ and $\widetilde{u}_{II,n}$ of (\ref{SchHO}) are, see \cite{Flugge},
\begin{eqnarray}
\widetilde{u}_{I,n}(y,t)&=&~_1F_1 \left(-\frac{n}{2},\frac{1}{2};\frac{1}{2}y^2 \right) \exp \left(-\frac{1}{4} y^2 \right) e^{-itE_n}, \nonumber\\
\widetilde{u}_{II,n}(y,t)&=& ~y~_1F_1\left(\frac{1-n}{2},\frac{3}{2};\frac{1}{2}y^2 \right) \exp\left(-\frac{1}{4} y^2 \right)e^{-itE_n}. \label{HOgensols}
\end{eqnarray}
Here, $_1F_1(a,b;z)$ is a confluent hypergeometric function \cite{Abramowitz,Bateman}. This functions satisfy $\widetilde{u}_{I,n}(y,t)=\widetilde{u}_{I,n}(-y,t)$ and $\widetilde{u}_{II,n}(-y,t)=-\widetilde{u}_{II,n}(y,t)$, i.e. they are even and odd functions in $y$, respectively. It implies that all the functions $\widetilde{u}_{II,n}$ share at least one zero at $y=0$, $\widetilde{u}_{II,n}(0)=0$. The Wronskian
of the two solutions for fixed $t$ is constant, $W(\widetilde{u}_{I,n}, \widetilde{u}_{II,n})|_{t=0}=1$. In the special case of $n$ being a non-negative integer, one of (\ref{HOgensols}) reduces to a square integrable function as the confluent hypergeometric function is truncated to a Hermite polynomial.
The point transformation (\ref{pointtPsi}) maps the solutions (\ref{HOgensols}) into
\begin{align}
u_{I,n}(x,z)=& \frac{1}{(1+z^2)^{1/4}} \exp\left\{\frac{i}{4} \left[\frac{x^2}{z-i} - 4 E_n \arctan(z) \right] \right\}~ _1F_1 \left(-\frac{n}{2},\frac{1}{2};\frac{x^2}{2(z^2+1)} \right) \label{u even} , \\
u_{II,n}(x,z)=& \frac{x}{(1+z^2)^{3/4}} \exp\left\{\frac{i}{4} \left[\frac{x^2}{z-i} - 4E_n \arctan(z) \right] \right\} ~_1F_1\left(\frac{1-n}{2},\frac{3}{2};\frac{x^2}{2(z^2+1)} \right). \label{u odd}
\end{align}
They satisfy
\begin{equation}
S_0u_{I,n}=S_0u_{II,n}=0.
\end{equation}
Let us fix $u$ as the following linear combination of $u_{I(II),n}$,
\begin{eqnarray}\label{uu0}
u(x,z)= \sum_{j=1}^N \left( \alpha_{I,n_j} u_{I,n_j} + i~ \alpha_{II,n_j} u_{II,n_j} \right),\quad \alpha_{I(II),n_j}\in\mathbb{R},\quad n_j\in\mathbb{R}. \label{Superposition}
\end{eqnarray}
We can see that it satisfies\footnote{The functions $u_{I(II),n}$ fulfill $\mathcal{P}_2\mathcal{T}u_{I,n}=u_{I,n}$ and $\mathcal{P}_2\mathcal{T}u_{II,n}=-u_{II,n}$.} $\mathcal{P}_2\mathcal{T}u=\epsilon u$, where $\epsilon\in\{-1,1\}$.
Considering the other definition of the $\mathcal{P}$ operator, then
\begin{equation}\label{PxTu}
\mathcal{P}_x\mathcal{T}u_{I(II),n}=u_{I(II),n}\exp\left(2i E_n\arctan(z)+i\frac{z}{2(z^2+1)}x^2\right).
\end{equation}
The function $u$ complies with (\ref{Scond}) provided that it is a linear combination of the solutions associated with the same energy. It can be written as
\begin{equation}\label{uu1}
u(x,z)=\alpha_{I,n}u_{I,n}+i\alpha_{II,n}u_{II,n},\quad \alpha_{I(II),n}\in\mathbb{R}.
\end{equation}
We will use this function to construct the new system of required properties.
\subsection{Optical wave guide with a localized defect}
We identify $u$ with \eqref{uu1}. It can be written as
\begin{eqnarray}\label{uHO}
u(x,z)&=&\frac{1}{(1+z^2)^{1/4}} \exp\left\{\frac{i}{4} \left[\frac{x^2}{z-i} - 4\left(n+\frac{1}{2}\right) \arctan(z) \right] \right\} \nonumber \\
& ~& \times \left[\alpha_{I,n} ~_1F_1 \left(-\frac{n}{2},\frac{1}{2};\frac{x^2}{2(z^2+1)} \right)+i ~ \alpha_{II,n}\frac{x}{(1+z^2)^{1/2}}~ ~_1F_1\left(\frac{1-n}{2},\frac{3}{2};\frac{x^2}{2(z^2+1)} \right) \right]. \label{u no missing state}
\end{eqnarray}
Let us analyze its zeros. When $\alpha_{I,n}\alpha_{II,n}\neq0$, then $u_{I,n}$ and $u_{II,n}$ cannot vanish in the same points as we have $W(u_{I,n},u_{II,n})|_{z=const}\neq 0$. Hence, the function $u$ is nodeless in this case.
When $\alpha_{II,n}=0$ and $n \leq 0$, $u\equiv u_{I,n}$ is nodeless by the oscillation theorem.
As we discussed above, $u$ satisfies (\ref{Scond})
so that the missing state $u_m$ can be constructed as in (\ref{um}) (in the definition of $u_m$, the function $f(z)$ reads $f(z)= L_1^{-1}(z)(z^2+1)$, see \eqref{f(z)}).
As we require the missing state to be vanishing for large $|x|$ and $|z|$, we take $n=-2$ that corresponds to the exponentially expanding solutions for large $|x|$ (it is associated with non-physical stationary state of the Harmonic oscillator) and also $\alpha_{I,-2}=1$, $\alpha_{II,-2}=\alpha$. Then (\ref{uHO}) can be simplified to
\begin{eqnarray}
u(x,z)&=&\frac{1}{\delta^{1/4}} \exp\left\{\frac{i}{4} \left[\frac{x^2}{z-i} + 6 \arctan(z) \right] \right\} \left[ 1+\sqrt{\frac{\pi}{2 \delta}} x ~\text{erf}\left(\frac{x}{\sqrt{2 \delta}}\right) \exp\left(\frac{x^2}{2 \delta} \right) + i\frac{\alpha ~x}{\sqrt{\delta}} \exp\left(\frac{x^2}{2 \delta} \right) \right], \label{u guide defect}
\end{eqnarray}
where we used the abbreviation $\delta=z^2+1$, and $\text{erf}(\cdot)$ is the error function \cite{Abramowitz}.
Let us consider how the intertwining operator $\mathcal{L}$ transforms the wave packet $\Psi_{x_0,z_0,v_0,\sigma}$, see (\ref{wavepacket}). We get
\begin{eqnarray}\label{Lpsi}
\mathcal{L}\Psi_{x_0,z_0,v_0,\sigma}&=&G(x,z)\Psi_{x_0,z_0,v_0,\sigma},
\end{eqnarray}
where
\begin{eqnarray}\label{Lpsi2}
G(x,z)&=&\frac{iL_1(z)}{2}\left(\frac{x-x_0+iv_0\sigma}{z-z_0-i\sigma}-\frac{x}{i+z}-\frac{2}{ix+\frac{2e^{-\frac{x^2}{2(1+z^2)}}\sqrt{1+z^2}}{2\alpha-i\sqrt{2\pi}\mbox{erf}\left(\frac{x}{\sqrt{2}\sqrt{1+z^2}}\right)}}\right)
\end{eqnarray}
The function $L_1$ should satisfy either (\ref{uPT1}) or (\ref{uPT2}) in order to have a $\mathcal{PT}$-symmetric potential. The relation (\ref{uPT1}) results in $|L_1|=1+z^2$ while the conditions (\ref{uPT2}) gives $L_1(z)=\overline{L_1(-z)}$. Additionally, we require the function $G(x,z)$ to be bounded (\ref{wavepacketpreservation}). These requirements (i.e. the potential is both $\mathcal{P}_x\mathcal{T}$-symmetric and $\mathcal{P}_2\mathcal{T}$-symmetric and $G(x,z)$ is bounded) can be met by fixing
$$L_1=\sqrt{1+z^2}.$$
With this selection of $L_1$, the new potential \eqref{V_1} reads
\begin{eqnarray}\label{V1HO}
V_1(x,z)=\frac{2 \left\{ \left(x^2-2\delta\right) \left[\sqrt{2}\alpha- i \sqrt{\pi}\text{erf}\left(\frac{x}{\sqrt{2\delta} }\right) \right]^2-4 \sqrt{2 \pi \delta} x e^{-\frac{x^2}{2\delta }} \text{erf}\left(\frac{x}{\sqrt{2 \delta} }\right)-8 i \alpha x \sqrt{\delta} e^{-\frac{x^2}{2\delta }}-6\delta e^{-\frac{x^2}{\delta}} \right\}}{\delta \left[2 \sqrt{\delta}e^{-\frac{x^2}{2 \delta}}+\sqrt{2}ix \left(\sqrt{2} \alpha-i\sqrt{\pi } \text{erf}\left(\frac{x}{\sqrt{2 \delta} }\right) \right)\right]^2}.
\end{eqnarray}
The expression (\ref{V1HO}) represents a one-parameter family of potentials where $\alpha$ can acquire any real value. For $\alpha=0$, the potential is real function and $S_1$ is Hermitian. The potential behaves asymptotically as
\begin{eqnarray}
V_1(x,z)&=&\frac{1}{\delta}+o(1),\quad (|x|\rightarrow\infty),\\
V_1(x,z)&=&O\left(\frac{1}{\delta}\right),\quad (|z|\rightarrow\infty),\quad \delta=1+z^2.
\end{eqnarray}
Hence, it represents a real wave guide with a localized $\mathcal{PT}$-symmetric defect, see Fig. \ref{Missing pot} (a) and (b).
The intertwining operator $\mathcal{L}$ changes profile of the wave packet $\Psi_{x_0,z_0,v_0,\sigma}$. It gets divided it into two beams that pass around the origin from both sides. If we had fixed $L_1=1$ that still respects (\ref{wavepacketpreservation}), the potential (\ref{V1HO}) would acquire an additional term $i\partial_z\ln(1+z^2)$, adding a gain-loss profile to the potential barrier, see Fig. \ref{Missing pot} (c). The transformed wave packets would change their form radically; they would be concentrated in region with non-vanishing gain and loss, see Fig.\ref{Missing pot} (d)-(f).
\begin{figure}[t!]
\begin{center}
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{Missing2b.jpg}
\caption{}
\end{subfigure}
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{Missing3b.jpg}
\caption{}
\end{subfigure}
\begin{subfigure}[b]{0.38\textwidth}
\includegraphics[width=\textwidth]{Fig21c.jpg}
\caption{}
\end{subfigure}\\
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{Fig22.jpg}
\caption{}
\end{subfigure}
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{Fig23b.jpg}
\caption{}
\end{subfigure}
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{Fig24b.jpg}
\caption{}
\end{subfigure}\\
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{Missing4b.jpg}
\caption{}
\end{subfigure}
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{Missing5b.jpg}
\caption{}
\end{subfigure}
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{Missing6b.jpg}
\caption{}
\end{subfigure}
\caption{Optical wave guide with a localized defect. Plots of the real (a) and imaginary (b) parts of $V_1$, see \eqref{V1HO}, when $L_1=\sqrt{1+z^2}$. In (c) the imaginary part of $V_1$ when $L_1=1$, see \eqref{SUSY V1} and \eqref{u guide defect}. The intensity density $|\Psi_{x_0,z_0,v_0,\sigma}|^2$ is shown in (d). In (e) we show $|\mathcal{L}\Psi_{x_0,z_0,v_0,\sigma}|^2$ where $L_1(z)=\sqrt{1-iz}$. In (f) we show $|\mathcal{L}\Psi_{x_0,z_0,v_0,\sigma}|^2$ where $L_1(z)=1$. The parameters used in (d)-(f) are: $x_0=-50$, $z_0=-100$, $v_0=-0.4$, $\sigma=30$. Three different solutions for the corresponding time dependent Schr\"odinger equation for $V_1$ as in \eqref{V1HO} are shown: first $|u_m|^2$ (g), see \eqref{um missing}, then $|\phi_0|^2=|\mathcal{L}\psi_0|^2$ (h) and finally $|\phi_1|^2=|\mathcal{L}\psi_1|^2$ (i). In all plots $\alpha=-(2\pi)^{-1/2}$. }
\label{Missing pot}.
\end{center}
\end{figure}
The missing state $u_m$ for the system with potential term as in \eqref{V1HO} can be written as
\begin{eqnarray} \label{um missing}
u_m(x,z) & = & \frac{1}{\delta^{1/4}} \exp\left\{\frac{i}{4} \left[\frac{x^2}{i+z} + 6 \arctan(z) \right] \right\} \left[ 1+\sqrt{\frac{\pi}{2 \delta}} x ~\text{erf}\left(\frac{x}{\sqrt{2 \delta}}\right) \exp\left(\frac{x^2}{2 \delta} \right) + i\frac{\alpha ~x}{\sqrt{\delta}} \exp\left(\frac{x^2}{2 \delta} \right) \right]^{-1}.
\end{eqnarray}
It is vanishing for large values of $|x|$ and $|z|$, i.e. it represents a light dot. It decreases exponentially for large $|x|$ and fixed $z$, whereas it behaves as $(1+z^2)^{-1/4}$ for large $|z|$ and fixed $x$, see Fig. \ref{Missing pot} (g).
The symmetry operators (\ref{tildeLL}) can be found explicitly as
\begin{eqnarray}\label{LLHO}
\mathcal{L}^\sharp \mathcal{L}&=& (1+z^2) \partial_x^2 - i z x \partial_x - \frac{1}{4}(x^2+2iz+6), \\
\mathcal{L}\mathcal{L}^\sharp & = & (1+z^2) \partial_x^2 - i z x \partial_x- \frac{1}{4}(x^2+2iz+6) -(z^2+1) V_1,
\end{eqnarray}
where $[S_0,\mathcal{L}^\sharp \mathcal{L}]=[S_1,\mathcal{L}\mathcal{L}^\sharp ]=0$. Notice that $\mathcal{L}\mathcal{L}^\sharp-\mathcal{L}^\sharp \mathcal{L}=-L_1(z)^2 V_1$.
The point transformation can be used to get other localized solutions that are based on the bound states of the harmonic oscillator. It is convenient to introduce the following notation
\begin{equation}\label{psi_n}
\psi_n(x,z)=\begin{cases}\frac{1}{\sqrt{\sqrt{2 \pi } 2^n n!}}u_{I,n}(x,z),&\quad n\ \mbox{is even},\ n\geq 0,\\
\frac{1}{\sqrt{\sqrt{2 \pi } 2^n n!}}u_{II,n}(x,z),&\quad n\ \mbox{is odd},\ n\geq 0,
\end{cases}
\end{equation}
where $\psi_n$ are square integrable functions for fixed $z$ that are obtained from the bound states of the harmonic oscillator by the point transformation. Then $\phi_n\equiv \mathcal{L}\psi_n$ represent light dots in the current system as they vanish both for large $x$ and $z$, see Fig. \ref{Missing pot} (h)-(i) for illustration.
The action of $\mathcal{L} \mathcal{L}^\sharp$ on $\phi_n$ can produce new solutions of $S_1\mathcal{L} \mathcal{L}^\sharp\phi_n=0$.
In principle, one can define a Darboux transformation $\mathcal{L}$ by identifying $u$ with an arbitrary linear combination of the wave packets (\ref{psi_n}). Such $u$ will not comply with (\ref{Scond}) in general and the formula for the missing state (\ref{um}) will not be applicable. Despite it is not clear how to construct a missing state in terms of $u$ in this case, there are localized solutions, the light dots, associated with $\phi_n=\mathcal{L}\psi_n$. See Appendix where we illustrated such construction on an explicit example.
\subsection{Localized defects in a homogeneous crystal}
Let us construct now the system where uniformity of the refractive index is violated by a single localized defect. We shall use the stationary confluent Crum-Darboux transformation together with the point transformation. The former transformation allows us to get a localized (time-independent) deformation of the harmonic oscillator with the use of the ground state \footnote{Let us notice that when we fix the transformation function $u$ as the ground state of the HO for the first order transformation, the resulting new system coincides with the original one up to an additive constant. It follows from the so called shape invariance of the Harmonic oscillator, the property that underlies its exact solvability \cite{Cooper}.}.
Then, the point transformation will transform the deformed Harmonic oscillator into the system with asymptotically vanishing potential that has a localized defect.
The confluent transformation is defined in terms of two functions, $u_1$ and $u_2$. However, the two functions are not independent, $u_2$ can be written in terms of $u_1$. Hence, $u_1$ defines the transformation together with some constant parameters, see (\ref{confluentu2}).
Let us select $u_1$ as the stationary solution $S_{HO}\tilde{u}_1=0$,
$$
\tilde{u}_1(y,t)={\tilde{\psi}}_m(y)e^{-iE_mt},\quad\mbox{where}\quad \tilde{\psi}_m(y)=H_{m}\left(\frac{y}{\sqrt{2}}\right)e^{-\frac{1}{4}y^2},\quad E_m=m+\frac{1}{2},
$$
where $H_m(y)$ is a Hermite polynomial. As the second function $u_2$, we take
$$
\tilde{u}_2(y,t)=e^{-iE_mt}\tilde{\psi}_m\left(\int_{y_0}^y\frac{1}{\tilde{\psi}_m^2}\left(\int_{s_0}^s\tilde{\psi}_m^2(r)dr+\alpha\right)ds+a\right).
$$
where $a$ is a constant.
Then the Schr\"odinger operator of harmonic oscillator is intertwined with the new one by the operator $\tilde{\mathcal{L}}_{12}$, see (\ref{Confluent SUSY V2}),
\begin{equation}
\tilde{S}_{HO}\tilde{\mathcal L}_{12}=\tilde{\mathcal{L}}_{12}S_{HO}, \quad \tilde{S}_{HO}=S_{HO}+2\partial_y^2\ln \left(\alpha+\int^y_0\tilde{\psi}_1^2(s)ds\right).
\end{equation}
The formula (\ref{Confluent missing state}) gives us the missing state for the new system,
\begin{eqnarray}\label{confluum}
\tilde{f}_m(y,t)= \frac{{\tilde{\psi}}_m(y)}{\alpha + \int_{0}^y {\tilde{\psi}}_m^2(s) ds}e^{-iE_m t}.
\end{eqnarray}
There are also other stationary solutions of $\tilde{S}_{HO} \tilde{f}=0$ that can be obtained from the square integrable eigenstates of the Harmonic oscillator $\psi_n$ when $n\neq m$. Let us denote
\begin{eqnarray}\label{Confluent SUSY Solutions}
\tilde{f}_n(y,t)&=&\tilde{\mathcal{L}}_{12}\tilde{\psi}_n(y) e^{-i E_n t}= \left(\partial_y-\frac{\partial_y\check{u}_2}{\check{u}_2}\right)\left(\partial_y-\frac{\partial_y \tilde{u}_1}{\tilde{u}_1}\right)\tilde{\psi}_n(y) e^{-i E_n t},\quad \check{u}_2=\tilde{u}_1\partial_y\frac{\tilde{u}_2}{\tilde{u}_1}.
\end{eqnarray}
After the point transformation (\ref{FinkelFunction}) of the Schr\"odinger operator $\tilde{S}_{HO}$, we get
\begin{equation}
S_2=(1+z^2)^{-1}U^{-1}\tilde{S}_{HO}(y(x,z),t(z))U=S_{0}+2\partial_x^2\ln \left(\alpha+\int^{y(x,z)}_0u_1^2(s)ds\right).
\end{equation}
For explicit illustration, let us fix $m=0$. It means that the ground state of the harmonic oscillator has been selected as the transformation function to perform the confluent SUSY transformation and $E_m\equiv E_0 = 1/2$. Then the potential term (\ref{Confluent SUSY V2}) acquires the following explicit form
\begin{eqnarray}
V_2(x,z)=
\frac{4}{(1+z^2)^{3/2}}\frac{2\sqrt{1+z^2}e^{-\frac{x^2}{1+z^2}}+x e^{-\frac{x^2}{2(1+z^2)}}\left(2\alpha+\sqrt{2\pi}\mbox{erf}\left(\frac{x}{\sqrt{2(1+z^2)}}\right)\right)}{\left(2\alpha+\sqrt{2\pi}\mbox{erf}\left(\frac{x}{\sqrt{2(1+z^2)}}\right)\right)^2}.
\label{PTHO Example Potential}
\end{eqnarray}
Choosing $\alpha$ as a pure imaginary parameter and since $\text{erf}(-x)=- \text{erf}(x)$, one can check that the new potential $V_2$ is invariant with respect to both $\mathcal{P}_x \mathcal{T}$ and $\mathcal{P}_2 \mathcal{T}$.
The potential $V_2$ represents a well localized defect of a uniform refractive index. Indeed, the potential behaves as $V_2(x,z)\sim O\left(x e^{-\frac{x^2}{2 \left(z^2+1\right)}}\right)$ for large $|x|$ and fixed $z$, whereas for fixed $x$ and large $|z|$, it behaves as $V_2(x,z)= O\left(\frac{e^{-\frac{x^2}{2 \left(z^2+1\right)}}}{z^2}\right)$.
In Fig. \ref{FigPTHOPotential} plots of the real (a) and imaginary (b) parts of $V_2$ for $\alpha=i$ are shown.
\begin{figure}[t!]
\centering
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{Confluente2b.jpg}
\caption{}
\end{subfigure}
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{Confluente3b.jpg}
\caption{}
\end{subfigure}
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{Confluente4b.jpg}
\caption{}
\end{subfigure} \\
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{Confluente5b.jpg}
\caption{}
\end{subfigure}
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{Confluente6b.jpg}
\caption{}
\end{subfigure}
\caption{Localized defect in a homogeneous crystal. Plots of the real (a) and imaginary (b) parts of $V_2$ when $\alpha=i$, see \eqref{PTHO Example Potential}. Moreover, the absolute value squared of three solutions are plotted: $|\phi_0|^2$ (c), see \eqref{phi0}, $|\phi_1|^2$ (d) and $|\phi_2|^2$ (e), see \eqref{phin}. } \label{FigPTHOPotential}
\end{figure}
The use of the point transformation (\ref{FinkelFunction}) allowed us to work with the less complicated form of the wave functions. However, we could make a direct, second order transformation to the free particle system that would transform it into $S_2$,
\begin{equation}
\mathcal{L}_{12}S_{0}={S}_{2}\mathcal{L}_{12}
\end{equation}
where
$\mathcal{L}_{12}=U\tilde{\mathcal{L}}_{12}(y(x,z),t(z))U^{-1}$. The explicit form of the intertwining operator is
\begin{equation}
\mathcal{L}_{12}=(1+z^2)\partial_x^2-\left(-ixz+2\frac{\sqrt{1+z^2}}{B(x,z)}\right)\partial_x+\left(\frac{1-\frac{1}{2}x^2-i z}{2}-\frac{x(1-iz)}{(1+z^2)^{1/2}B(x,z)}\right).
\end{equation}
We abbreviated $B(x,z)=e^{\frac{x^2}{2(1+z^2)}}\left(2\alpha+\sqrt{2\pi}\,\mbox{erf}\left(\frac{x}{\sqrt{2(1+z^2)}}\right)\right)$. One can check that it satisfies (\ref{wavepacketpreservation}) when applied on the wave packets $\Psi_{x_0,z_0,v_0,\sigma}$
\begin{equation} \label{propagation LPsi}
\mathcal{L}_{12}\Psi_{x_0,z_0,v_0,\sigma}=G(x,z)\Psi_{x_0,z_0,v_0,\sigma},\quad
G(x,z)=\begin{cases}O(1),\quad |z|\rightarrow \infty, \ x\ \mbox{constant},\\O(x^2),\quad |x|\rightarrow \infty, \ z\ \mbox{constant}.\end{cases}
\end{equation}
The function $G(x,z)$ changes the profile of the wave packet; it gets divided such that it flows around the defect in two beams, see Fig.\ref{beamPD} for illustration.
\begin{figure}[t!]
\centering
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{Fig4c.jpg}
\caption{}
\end{subfigure}
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{Fig4b.jpg}
\caption{}
\end{subfigure}
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{Fig4a.jpg}
\caption{}
\end{subfigure}
\caption{Propagation of three different wave packets of the form $\mathcal{L}_{12}\Psi_{x_0, z_0, v_0,\sigma} $, see \eqref{propagation LPsi}. The parameters used are $x_0=-40$, $z_0=-40$, $\sigma=10$, $\alpha=i$ (see \eqref{PTHO Example Potential}) and $v_0=0.98$ in (a), $v_0=0.85$ in (b) and $v_0=0.60$ in (c). } \label{beamPD}
\end{figure}
The missing state (\ref{confluum}), after the point transformation, reads explicitly
\begin{eqnarray} \label{phi0}
\phi_0(x,z)=U^{-1} \tilde{f}_0 \left(y(x,t),t(z) \right)= \frac{e^{-\frac{x^2}{4(1+i z)}-\frac{i}{4}\arctan z}}{(1+z^2)^{1/4}\left(2\alpha+\sqrt{2\pi}\mbox{erf}\left(\frac{x}{\sqrt{2(1+z^2)}}\right)\right)}.
\end{eqnarray}
Additionally, there are other localized solutions of $S_2 f=0$, the light dots, that are associated with (\ref{Confluent SUSY Solutions}). They are
\begin{align}\label{phin}
\phi_n(x,z) = \frac{1}{(1+z^2)^{1/4}} \exp\left( \frac{i z}{4(1+z^2)} x^2 \right) \tilde{f}_n\left(\frac{x}{\sqrt{1+z^2}}, \arctan(z)\right).
\end{align}
These functions satisfy $\mathcal{P}_x \mathcal{T}\phi_n(x,z)=(-1)^n\phi_n(x,z)$. We can see that for large $|x|$ and fixed $z$, the functions vanish like an exponential multiplied by a polynomial. Along the curves $x= c \sqrt{1+z^2}$ where the argument of $\tilde{f}_n(y(x,t),t(z))$ is constant, the wave functions behave as $|\phi_n(c \sqrt{1+z^2},z)|= O\left((1+z^2)^{-1/4}\right)$. In Fig. \ref{FigPTHOPotential} (c)-(e), the intensity densities of three solutions are plotted: $|\phi_0|^2$, $|\phi_1|^2$ and $|\phi_2|^2$, respectively, see \eqref{phi0} and \eqref{phin}.
It follows directly from the formulas (\ref{phin}) that the power of the light beam defined as
$
P(\phi_n)= \int_{-\infty}^{\infty} |\phi_n(x,z)|^2 dx
$
is constant, i.e. it does not depend on $z$. However, different superpositions of states $\phi_n$ would not have a constant power. Using the same example ($m=0, \alpha=i)$, consider the following four superpositions:
\begin{eqnarray} \label{Superpositions}
\phi_a= \frac{1}{\sqrt{2}}\left(N_0 \phi_0 + i N_1 \phi_1 \right), \quad \phi_b= \overline{\phi_a}, \quad \phi_c= \frac{1}{\sqrt{2}}\left(N_0\phi_0 + N_1\phi_1 \right),\quad
\phi_d= \frac{1}{\sqrt{2}}\left(N_0\phi_0 - N_1\phi_1 \right)
\end{eqnarray}
where $N_0,~N_1$ are (positive) normalization constants. For such states, the power is a nontrivial function of $z$. In Fig. \ref{Fig PTHO Power} (a), the power of these states is plotted. First in blue the power of the first superposition $P(\phi_a)$ was plotted. It can be seen that power decreases near $z=0$, the interaction of the light with the defect results in its absorption. In purple, we have $P(\phi_b)$ representing the opposite case, power increases after the interaction zone. In yellow, $P(\phi_c)$ has a minimum around zero whereas $P(\phi_d)$ in green has a maximum. In Fig. \ref{Fig PTHO Power} (b)-(e) the functions $|\phi_a|^2$, $|\phi_b|^2$, $|\phi_c|^2$, $|\phi_d|^2$ were plotted, respectively.
\begin{figure}[t!]
\centering
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{Confluente11b.jpg}
\caption{}
\end{subfigure}
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{Confluente7b.jpg}
\caption{}
\end{subfigure}
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{Confluente8b.jpg}
\caption{}
\end{subfigure}\\
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{Confluente9b.jpg}
\caption{}
\end{subfigure}
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{Confluente10b.jpg}
\caption{}
\end{subfigure}
\caption{(a): Plots of the power of four different superposition states for the localized defect in a homogeneous crystal system. In blue $P(\phi_a)$, in purple $P(\phi_b)$, in yellow $P(\phi_c)$ and in green $P(\phi_d)$, the superpositions are defined in \eqref{Superpositions}. Furthermore, the intensity densities of the states $\phi_a$ (b), $\phi_b$ (c), $\phi_c$ (d) and $\phi_d$ (e) are plotted. } \label{Fig PTHO Power}
\end{figure}
\section{Guided modes in optical waveguides \label{cuatro} }
In this section, we will focus on models of optical wave guides. By an asymptotic analysis of the involved quantities, we will obtain general results for a large class of initial potentials that are just required to be integrable and to possess translational symmetry. In particular, we will show how to find solutions of the associated Schr\"odinger equation that are vanishing exponentially in the transverse direction to the wave guide, and hence, represent guided modes.
\subsection{On the construction of guided modes\label{4.1}}
Let us suppose that the initial potential $V_0$ does not depend on $z$.
It is known (see e.g. Th. 4.1, p.70 in \cite{BS}) that when the absolute value of the potential term is integrable, $\int_{\mathbb{R}}|V_0(x)|dx<\infty$, then the stationary equation $(-\partial_x^2+V_0(x))\psi=\lambda^2\psi$ has two solutions $\psi^+(\lambda,x)$ and $\phi^+(\lambda,x)$ for a complex $\lambda\neq 0$ satisfying
\begin{equation}\label{set1}
\psi^+(\lambda,x)=e^{i\lambda x}(1+o(1)),\quad \phi^+(\lambda,x)=e^{-i\lambda x}(1+o(1))\quad\mbox{as}\quad x\rightarrow+\infty.
\end{equation}
As the integrability of $V_0(x)$ is invariant with respect to $x\rightarrow -x$, there are also two solutions $\psi^-(\lambda,x)$ and $\phi^-(\lambda,x)$ satisfying
\begin{equation}\label{set2}
\psi^-(\lambda,x)=e^{-i\lambda x}(1+o(1)),\quad \phi^-(\lambda,x)=e^{i\lambda x}(1+o(1))\quad\mbox{as}\quad x\rightarrow-\infty.
\end{equation}
The two sets (\ref{set1}) and (\ref{set2}) represent two possible choices of the fundamental solutions of the stationary equation, i.e. a function from one set can be written as a linear combination of the functions from the other set. Let us mention that a finite square well is a simple example of the system where the potential term satisfies the condition of integrability. In this case, the small terms $o(1)$ are identically zero in the wave functions (\ref{set1}) and (\ref{set2}).
In order to perform a Darboux transformation to the potential $V_0$ we need to choose an adequate function $u$. Moreover, we will select a preimage $v$, satisfying $S_0 v = 0$, such that $\mathcal{L}v$ is a guided mode fulfilling $S_1 \mathcal{L}v=0$. First, let us assume that $\lambda^2 \in \mathbb{R}$, then $\lambda$ can be written either as $\lambda = -i k, ~k>0,$ or as $\lambda = r, ~r>0$, depending on the sign of $\lambda^2$. Now, let us consider $N+M$ stationary solutions, $N$ of them with eigenvalues $\lambda_j=-i k_j$, $j=1,\dots,N$, where $k_1>k_2>\dots>k_N>0$; and $M$ solutions such that $\lambda_{N+\ell}=r_\ell>0$, $\ell=1,\dots,M$. We denote the corresponding functions (\ref{set1}) or (\ref{set2}) as $\psi^{\pm}_{k_j}(x)\equiv \psi^\pm(-ik_j,x)$, $\phi^{\pm}_{k_j}(x)\equiv \phi^\pm(-ik_j,x)$, $\psi^{\pm}_{r_\ell}(x)\equiv \psi^\pm(r_\ell,x)$ and $\phi^{\pm}_{r_\ell}(x)\equiv \phi^\pm(r_\ell,x)$. Hence, we selected $N$ solutions that increase (decrease) exponentially for large $|x|$ and $M$ solutions that are asymptotically bounded and oscillating. We suppose that their derivatives satisfy
\begin{equation}
(\psi_{k_j}^{\pm})'=\pm k_je^{\pm {k_j}x}(1+o(1)),\quad (\phi_{k_j}^{\pm})'=\mp k_je^{\mp {k_j}x}(1+o(1)),\quad (x\rightarrow\pm\infty)
\end{equation}
and $(\psi_{r_\ell}^{\pm})'$ and $(\phi_{r_\ell}^{\pm})'$ are bounded functions. These requirements can be met provided that the functions do not have asymptotically small but rapidly oscillating terms.
Let us compose the following functions
\begin{eqnarray}\label{uv}
&&u=\sum_{j=1}^{N}F_je^{ik_j^2z}+\sum_{\ell=0}^MG_\ell e^{-ir_\ell^2z},\quad v=\sum_{j=1}^{N}\tilde{F}_je^{ik_j^2z}+\sum_{\ell=0}^M\tilde{G}_\ell e^{-ir_\ell^2z},
\end{eqnarray}
where $F_j$, $\tilde{F}_j$, $G_\ell$, $\tilde{G}_\ell$, $j\in\{1,\dots,N\}$, $\ell\in\{1,\dots,M\}$ are generic linear combinations
\begin{eqnarray}
&&F_j=a_j^{\pm}\psi_{k_j}^{\pm}+b_j^{\pm}\phi_{k_j}^{\pm}
,\quad
G_\ell=c_\ell^{\pm}\psi_{r_\ell}^{\pm}+d_\ell^{\pm}\phi_{r_\ell}^{\pm},\nonumber\\
&&
\tilde{F}_j=\tilde{a}_j^{\pm}\psi_{k_j}^{\pm}+\tilde{b}_j^{\pm}\phi_{k_j}^{\pm}
,\quad
\tilde{G}_\ell=\tilde{c}_\ell^{\pm}\psi_{r_\ell}^{\pm}+\tilde{d}_\ell^{\pm}\phi_{r_\ell}^{\pm},\label{FG}
\end{eqnarray}
and $G_0=\tilde{G}_0=0$.
The functions $G_\ell$ and $\tilde{G}_\ell$ are asymptotically oscillating and bounded.
The functions $F_j$ and $\tilde{F}_j$ are exponentially expanding,
\begin{align}\label{tildeFG}
F_j=a_j^{\pm}e^{\pm k_jx}+o(e^{\pm k_jx}),\ (x\rightarrow\pm\infty),\nonumber\\ \tilde{F}_j=\tilde{a}_j^{\pm}e^{\pm k_jx}+o(e^{\pm k_jx}),\ (x\rightarrow\pm\infty),\nonumber\\
(F_j)'=\pm k_ja_j^{\pm}e^{\pm k_jx}+o(e^{\pm k_jx}),\ (x\rightarrow\pm\infty),\nonumber\\ (\tilde{F}_j)'=\pm k_j\tilde{a}_j^{\pm}e^{\pm k_jx}+o(e^{\pm k_jx}),\ (x\rightarrow\pm\infty).
\end{align}
We define the Darboux transformation
\begin{equation}\label{wgL}\mathcal{L}=L_1(z)\left(\partial_x-\frac{u'}{u}\right).\end{equation}
Contrary to the cases discussed in the previous section, the superpotential $\mathcal{W}(x,z)=-\partial_x\ln u$ is asymptotically constant for large $|x|$. Indeed, substituting the explicit form (\ref{uv}) of $u$ into (\ref{wgL}) and using (\ref{tildeFG}), one finds that $\mathcal{W}(x,z)=\mp k_1+o(1)$ for $x\rightarrow\pm\infty$. Hence, the action of $\mathcal{L}$ on a plane wave (\ref{planewave}) preserves its amplitude for large $|x|$, $\mathcal{L}\Phi_{k,x_0,z_0,v_0}=L_1(z)(ik\mp k_{1}+o(1))\Phi_{k,x_0,z_0,v_0}$ for $x\rightarrow\pm\infty$. When acting on the wave packets (\ref{wavepacket}), we get
\begin{equation}\label{LPWG}
\mathcal{L}\Psi_{x_0,z_0,v_0,\sigma}=G(x,z)\Psi_{x_0,z_0,v_0,\sigma},\quad G(x,z)=L_1(z)\left(\frac{i(x-x_0)-v_0\sigma}{2(z-z_0-i \sigma)}\mp k_1+g(x,z)\right).
\end{equation}
where $g(x,z)$ is asymptotically bounded and oscillating function in $z$. Hence, the requirement (\ref{wavepacketpreservation}) then suggest to fix the function $L_1(z)$ to be bounded (and non-vanishing) for all real $z$.
\subsubsection*{The potential of wave guide}
The function $u$ fixes the new potential $V_1=V_0+\delta V_1+i\partial_z\ln L_1(z)$, where $\delta V_1=-2\frac{u''}{u}+2\frac{(u')^2}{u^2}$ is asymptotically vanishing for large $|x|$. Indeed, we have
\begin{eqnarray}
\delta V_1&=&\frac{\sum\limits_{j,s=1}^N\left[(V_0+k_j^2)F_jF_s-F'_jF'_s\right]e^{i(k_j^2+k_s^2)z}}{u^2}+\frac{\sum\limits_{\{j,\ell\}=\{1,0\}}^{N,M}\left[(2V_0-r_\ell^2+k_j^2)F_jG_{\ell}-2F_j'G'_\ell\right]e^{i(k_j^2-r_\ell^2)z}}{u^2}\nonumber\\
&&+\frac{\sum\limits_{\ell,s=0}^M\left[(V_0-r_\ell^2)G_\ell G_s-G_\ell'G_s'\right]e^{-i(r_\ell^2+r_s^2)z}}{u^2}.
\end{eqnarray}
As we have $u^2= \left(a_1^{\pm}\right)^2e^{\pm 2k_1x}(1+o(1))$ for $x\rightarrow\pm \infty$, the last two terms vanish for large $|x|$ as their denominators increase much faster then their numerators. The first term will vanish as well provided that $(V_0+k_1^2)F_1^2-(F'_1)^2\rightarrow 0$ for $|x|\rightarrow \infty$. But taking into account (\ref{tildeFG}) and integrability of $V_0$ that implies $V_0=o(1)$ for $|x|\rightarrow\infty$, we can see that this is indeed the case. We get
\begin{equation}
\delta V_1=O(e^{\pm(-k_1+k_2)x})\quad \mbox{for}\quad x\rightarrow\pm\infty.
\end{equation}
The term $\delta V_1$ is regular provided that $u$ has no zeros. It is rather nontrivial to guarantee in general. We will discuss this point in the explicit examples in the end of this section.
\subsubsection*{Guided modes - the alternative construction of the missing state}
Let us inspect the asymptotic behavior of the function $\mathcal{L}v$. We shall fix the coefficients $\tilde{a}^{+}_k$ and $\tilde{b}^+_k$ such that $\mathcal{L}v$ is asymptotically vanishing for large $|x|$ or it is bounded at least. We have
\begin{eqnarray}\label{Lv}
L_1^{-1}\mathcal{L}v&=&\frac{W(u,v)}{u}=\frac{\sum\limits_{j,s=1}^{N}W(F_j,\tilde{F}_s)e^{i(k_j^2+k_s^2)z}}{u}+\frac{\sum\limits_{\{j,\ell\}=\{1,0\}}^{N,M}\left(W(G_\ell,\tilde{F_j})+W(F_j,\tilde{G}_\ell)\right)e^{i(k_j^2-r_\ell^2)z}}{u}\nonumber\\&&+\frac{\sum\limits_{\ell,s=0}^MW(G_\ell,\tilde{G}_s)e^{-i(r_\ell^2+r^2_s)z}}{u},
\end{eqnarray}
where $W(f,g)=fg'-f'g$ is the Wronskian of two functions.
In the numerator of the first sum, there are terms of order $e^{\alpha x}$ with $\alpha\geq k_1$. We would like to fix $\tilde{a}^{\pm}$ and $\tilde{b}^{\pm}$ such that these terms vanish. Using (\ref{tildeFG}), we get
\begin{eqnarray}
\label{WFF}
W(F_j,\tilde{F}_s)+W(F_s,\tilde{F}_j)&=&e^{\pm (k_j+k_s)x}\left(\pm k_j(a_j^{\pm}\tilde{a}_s^{\pm}-a_s^{\pm}\tilde{a}_j^\pm)\pm k_s(a_s^{\pm}\tilde{a}_j^{\pm}-a_j^{\pm}\tilde{a}_s^\pm)\right)\nonumber\\&&+{o(e^{\pm (k_j+k_s)x})},\quad x\rightarrow \pm\infty.
\end{eqnarray}
We can make the term proportional to $e^{\pm (k_j+k_s)x}$ vanish by fixing
\begin{equation}\label{aa}
\tilde{a}_j^{+}=c^{+}a_j^{+},\quad \tilde{a}_j^{-}=c^{-}a_j^{-},
\end{equation}
where $c^{\pm}$ are constants. However, the condition (\ref{aa}) cannot guarantee that $\mathcal{L}v$ will vanish exponentially; the condition (\ref{aa}) does not nullify the term $o(e^{\pm(k_j+k_s)x})$ in (\ref{WFF}) in general, so that it only forces the first term in (\ref{Lv}) to behave as $o(e^{\pm 2k_1x})$ for $x\rightarrow\pm\infty$. However, (\ref{aa}) can serve well as the guiding relation in the construction of guided modes in the explicit examples discussed later on.
Let us focus on the second term in (\ref{Lv}). It vanishes asymptotically provided that the function at the term $e^{\pm k_1x}$ is vanishing. Using (\ref{aa}), we get
\begin{eqnarray}\label{FG}
F_1\tilde{G}_{\ell}'-F_1'\tilde{G}_\ell+G_{\ell}\tilde{F}_1'-G_\ell'\tilde{F}_1=\left(a_1^{\pm}(\tilde{G}_\ell'-c^{\pm}G_\ell')\pm k_1a_1^{\pm}(c^{\pm}G_\ell-\tilde{G}_{\ell})\right)e^{\pm k_1 x}+o(e^{\pm k_1 x}),\quad x\rightarrow\pm\infty.\end{eqnarray}
The leading term above can be made zero provided that
\begin{equation}\label{G}
\tilde{G}_\ell=c^{\pm}G_\ell,
\end{equation}
where $c^{\pm}$ are the two constants introduced in (\ref{aa}). When $c^{+}\neq c^-$, the only way how to make the second term of (\ref{Lv}) asymptotically vanishing for both $x\rightarrow \pm\infty$ is to make it identically zero by setting $G_\ell=\tilde{G}_\ell=0$. Fixing either $\tilde{G}_\ell=c^{+}G_\ell$ or $\tilde{G}_\ell=c^{-}G_\ell$ will make the second term vanishing either at $x\rightarrow+\infty$ or $x\rightarrow-\infty$. Now, let us see what happens when $c^+=c^-$. Fixing $\tilde{G}_j=c^+G_j$ for all $j=1,\dots,M$, the first term on the right side of (\ref{FG}) is vanishing. We also have $\tilde{a}_j^{\pm}=c^+a_{j}^{\pm}$. Then, in general, the function $v$ can differ from $u$ only in the functions that vanish asymptotically for $|x|\rightarrow\infty$, i.e. $u-c^+v$ is a linear combination of bound states of the initial system. In that case, it is straightforward to see that $\mathcal{L}v$ is an exponentially vanishing function for large $|x|$. When there are no bound states in the initial system, the choice $c^+=c^-$ would imply $v=c^+u$ and $\mathcal{L}v=0$ identically. Taking $c^+\neq c^-$ and $M>0$, the wave function $\mathcal{L}v$ cannot be exponentially vanishing on both sides of the wave guide. While decreasing rapidly to zero on one side, it has bounded and non vanishing oscillations on the other side. In this case, we call the wave guides weakly confining as the guided mode $v$ is ``leaking" from the wave guide on one side. We will illustrate this situation on the explicit examples below.
Both the new potential term $V_1$ and $\mathcal{L}v$ are periodic in $z$ provided that $k_1,\dots,k_N$ and $r_1,\dots,r_M$ are commensurable. For $M=0$, $V_1$ offers a strong confinement of the guided mode as $\mathcal{L}v$ vanishes outside exponentially. When $M\neq0$, the potential $V_1$ offers rather weak confinement as the guided mode ``leaks'' out of the wave guide, performing non-vanishing oscillations in $|x|\rightarrow\infty$.
\subsubsection*{$\mathcal{P}_2\mathcal{T}$-symmetry}
Up to now, we did not make any assumption on the $\mathcal{PT}$-symmetry of $V_1$. As we do not see how the function $u$ in (\ref{uv}) could satisfy (\ref{uPT1}), we look for $\mathcal{P}_2\mathcal{T}$-symmetry of the new potential. It is sufficient to fix the function $u$ such that it has definite $\mathcal{P}_2\mathcal{T}$-parity. It can be granted by taking the coefficients in (\ref{FG}) such that
\begin{equation}\label{PTFG}
\mathcal{P}_2\mathcal{T}F_j=\epsilon\, F_j,\quad \mathcal{P}_2\mathcal{T}G_\ell=\epsilon\, G_\ell,\quad \epsilon\in\{-1,1\}.
\end{equation}
These relations imply $a^{-}_j=\epsilon\, \overline{a^{+}_j}$.
The following examples will differ by the choice of the transformation function $u$; when it consists of exponentially expanding components only, the resulting systems will represent strongly confining wave guides as the guided mode will disappear exponentially out of the wave guide. For $u$ containing the bounded oscillating components, weakly localizing wave guides will be obtained. We will consider both $\mathcal{P}_2\mathcal{T}$-symmetric models as well as non-$\mathcal{P}_2\mathcal{T}$-symmetric ones. In all the cases, we will set $L_1=1$ that, as we discussed below (\ref{LPWG}), complies with (\ref{wavepacketpreservation}).
\subsection{Examples: $\mathcal{P}_2\mathcal{T}$-symmetric deformations of the P\"oschl-Teller potential}
Our initial system will be the free particle again, so that we fix $S_0$ as in (\ref{FPEq}).
We can identify the fundamental solutions of the stationary Schr\"odinger equation
with prescribed asymptotic behavior (\ref{set1}) and (\ref{set2}) as
\begin{equation}
\psi^+(\lambda,x)=\phi^-(\lambda,x)=e^{i\lambda x},\quad \phi^+(\lambda,x)=\psi^-(\lambda,x)=e^{-i\lambda x}.
\end{equation}
We shall construct $\mathcal{P}_2\mathcal{T}$-symmetric model. Taking into account (\ref{tildeFG}) and (\ref{PTFG}), the functions $F_j$ and $G_\ell$ with definite $\mathcal{P}_2\mathcal{T}$-parity have to be fixed in the following form
\begin{eqnarray}\label{FGepsilon}
F_{j,\epsilon}&=&a^+_je^{k_jx}+\epsilon \overline{a_j^+}e^{-k_jx}=\begin{cases}|a_j^+|\cosh (k_jx+i\delta_j),\quad \epsilon=1,\\|a_j^+|\sinh (k_jx+i\delta_j),\quad \epsilon=-1, \end{cases}\quad a_{j}^+=|a_j^+|e^{i\delta_j},\quad \delta_j\in\mathbb{R},\\
G_{\ell,\epsilon}&=&c^+_\ell e^{ir_{\ell}x}+\epsilon \overline{c_\ell^+}e^{-ir_\ell x}=\begin{cases}|c_\ell^+|\cos (r_\ell x+i\mu_\ell),\quad \epsilon=1,\\|c_\ell^+|\sin (r_\ell x+i\mu_\ell),\quad \epsilon=-1, \end{cases}\quad c_{\ell}^+=|c_\ell^+|e^{i\mu_\ell},\quad \mu_j\in\mathbb{R},
\end{eqnarray}
that guarantees that
$u_{\epsilon}=\sum\limits_{j=0}^NF_{j,\epsilon}e^{ik_j^2z}+\sum\limits_{\ell=0}^MG_{\ell,\epsilon} e^{-ir_\ell^2z}$ will satisfy
\begin{equation}
\mathcal{P}_2\mathcal{T}u_{\epsilon}=\epsilon~u_{\epsilon}.
\end{equation}
In what follows, we will discuss the systems related to the following fixed form of the function $u$,
\begin{equation}\label{ue}
u_\epsilon=F_{1,\epsilon}e^{ik_1^2z}+F_{2,\epsilon}e^{ik_2^2z}+G_{1,\epsilon}e^{-ir_1^2z}.
\end{equation}
As we argued below (\ref{G}), the character of the guided modes given by $v$ depends on the presence of $G_{1,\epsilon}$ in $u$. When it is absent in (\ref{ue}), i.e. $G_{1,\epsilon}\equiv 0$, then we can construct guided modes that are exponentially vanishing for large $|x|$. Otherwise, $\mathcal{L}v$ is asymptotically non-vanishing, bounded and oscillating.
\subsubsection*{ $G_1= 0$: Strongly confining wave guides}
Let us discuss two choices of the functions $u$ and $v$.
First, we fix
\begin{eqnarray}
&&u=\cosh k_1 xe^{i k_1^2z}+\alpha\cosh k_2 x e^{i k_2^2z},
\\
&&v=\sinh k_1 xe^{i k_1^2z}+\alpha\sinh k_2 x e^{i k_2^2z},\quad \mbox{where}\quad |\alpha|<1,\quad \alpha\in \mathbb{R}.
\end{eqnarray}
Comparing with (\ref{aa}), we can see that $c^+=1$ while $c^-=-1$.
We can show that the function $u$ has no zeros. Indeed, the equation $u=0$ can be written as
$ \frac{\cosh k_1x}{\cosh k_2x}=-\alpha e^{i(k_2^2-k_1^2)z}.$
Its left side is greater than one (notice that $\cosh x$ is a monotonic function for $x>0$ or $x\leq0$), while the absolute value of the right-hand side can be only smaller than one.
The potential $V_1$ and the guided mode $\mathcal{L}v$ acquire the following forms
\begin{equation}\label{V1h1}
V_1=-2\frac{k_1^2+\alpha^2 k_2^2 e^{2i(k_2^2-k_1^2)z}}{\cosh^2 k_1x\left(1+ e^{i(k_2^2-k_1^2)z}\alpha\frac{\cosh k_2x}{\cosh k_1x}\right)^2}-2\frac{e^{i(k_2^2-k_1^2)z}\alpha\cosh k_2x\left((k_1^2+k_2^2)-2k_1k_2\tanh k_1x\tanh k_2x\right)}{\cosh k_1x\left(1+ e^{i(k_2^2-k_1^2)z}\alpha\frac{\cosh k_2x}{\cosh k_1x}\right)^2}
\end{equation}
and
\begin{equation} \label{guided V1h1}
\mathcal{L}v=\frac{e^{2ik_1^2z}k_1+e^{2ik_2^2 z}k_2\alpha^2+e^{i(k_1^2+k_2^2)z}(k_1+k_2)\alpha\cosh (k_1-k_2)x}{e^{ik_1^2 z}\cosh k_1x\left(1+e^{i(k_2^2-k_1^2)z}\alpha\frac{\cosh k_2x}{\cosh k_1x}\right)}.
\end{equation}
The intertwining operator has the following action on the wave packets (\ref{wavepacket})
\begin{equation}
\mathcal{L}\Psi_{x_0,z_0,v_0,\sigma}=G(x,z)\Psi_{x_0,z_0,v_0,\sigma},
\end{equation}
where
\begin{equation}
G(x,z)=i\frac{x-x_0+iv_0\sigma}{2(z-z_0-i\sigma)}-\frac{e^{ik_2^2z}k_2\alpha\sinh k_2x+e^{ik_1^2z}k_1\sinh k_1x}{ie^{ik_2^2z}\alpha\cosh k_2x+e^{ik_1^2z}\cosh k_1x}.
\end{equation}
In Fig. \ref{FigV1h1} we present the real (a) and imaginary (b) parts of the potential as well as the power density $|\mathcal{L}v|^2$ (c) of the guided mode, for the parameters $k_1=0.4,~k_2=0.1,~\alpha=0.5$. The power $P(\mathcal{L}v)= \int_{-\infty}^{\infty}|\mathcal{L}v|^2 dx$ can be seen in (d), the power of the guided mode is oscillating. The power density $|\mathcal{L}\Psi_{x_0,z_0,v_0,\sigma}|^2$, for the parameters $x_0=-7$, $z_0=-50$, $v_0=-0.2$ and $\sigma=40$, is shown in (e).
\begin{figure}[t!]
\centering
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{Periodic2b.jpg}
\caption{}
\end{subfigure}
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{Periodic3b.jpg}
\caption{}
\end{subfigure}
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{Periodic4b.jpg}
\caption{}
\end{subfigure}\\
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{PeriodicP1b.jpg}
\caption{}
\end{subfigure}
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{Fig61.jpg}
\caption{}
\end{subfigure}
\caption{A strongly confining waveguide. Plots of the real (a) and imaginary (b) parts of $V_1$, see \eqref{V1h1}, for the parameters $k_1=0.4,~k_2=0.1,~\alpha=0.5$ are presented. Furthermore, the corresponding power density of the guided mode $ \mathcal{L}v$, see \eqref{guided V1h1}, is shown (c). The power $P(\mathcal{L}v)= \int_{-\infty}^{\infty}|\mathcal{L}v|^2 dx$ can be seen in (d), the power of the guided mode is oscillating. The power density $|\mathcal{L}\Psi_{x_0,z_0,v_0,\sigma}|^2$, for the parameters $x_0=-7$, $z_0=-50$, $v_0=-0.2$ and $\sigma=40$ is shown in (e). } \label{FigV1h1}
\end{figure}
Now, let us consider a different choice of the function $u$ and of the preimage $v$ of the guided mode,
\begin{eqnarray}\label{uhii}
&& u=\cosh k_1 xe^{i k_1^2z}+i\alpha\sinh k_2 x e^{i k_2^2z},\\
&&v=\sinh k_1 xe^{i k_1^2z}+i\alpha\cosh k_2 x e^{i k_2^2z},\quad \mbox{where}\quad |\alpha|<1.
\end{eqnarray}
We can prove that $u$ has no zeros. Indeed, the equation $u=0$ leads to $\frac{\sinh k_2x}{\cosh k_1x}=i\alpha^{-1}e^{i(k_1^2-k_2^2)z}$. We can show that the absolute value of the left-hand side is smaller then one: for $x>0$, we have $0<\frac{\sinh k_2x}{\cosh k_1x}<\frac{\sinh k_1x}{\cosh k_1x}=\tanh k_1x<1$. For $x\leq 0$, we have $0>\frac{\sinh k_2x}{\cosh k1 x}=\tanh k_1x+\frac{\sinh k_2x-\sinh k_1x}{\cosh k_1x}>\tanh k_1x>-1$.
Hence, the potential $V_1$ is regular and it can be written as
\begin{equation}\label{V1h2}
V_1=-2\frac{k_1^2+\alpha^2 k_2^2 e^{2i(k_2^2-k_1^2)z}}{\cosh^2 k_1x\left(1+i e^{i(k_2^2-k_1^2)z}\alpha\frac{\sinh k_2x}{\cosh k_1x}\right)^2}-2i\frac{e^{i(k_2^2-k_1^2)z}\alpha\cosh k_2x\left((k_1^2+k_2^2)\tanh k_2x-2k_1k_2\tanh k_1x\right)}{\cosh k_1x\left(1+i e^{i(k_2^2-k_1^2)z}\alpha\frac{\sinh k_2x}{\cosh k_1x}\right)^2}
\end{equation}
whereas the guided mode $\mathcal{L}v$ acquires the following form
\begin{equation}
\mathcal{L}v=\frac{e^{2ik_1^2z}k_1+e^{2ik_2^2 z}\alpha^2k_2-ie^{i(k_1^2+k_2^2)z}(k_1+k_2)\alpha\sinh (k_1-k_2)x}{e^{ik_1^2 z}\cosh k_1x\left(1+ie^{i(k_2^2-k_1^2)z}\alpha\frac{\sinh k_2x}{\cosh k_1x}\right)}. \label{guided V1h2}
\end{equation}
Now, the wave packets get transformed in the following manner
\begin{equation}
\mathcal{L}\Psi_{x_0,z_0,v_0,\sigma}=G(x,z)\Psi_{x_0,z_0,v_0,\sigma},
\end{equation}
where
\begin{equation}
G(x,z)=i\frac{x-x_0+iv_0\sigma}{2(z-z_0-i\sigma)}-\frac{ie^{ik_2^2z}k_2\alpha\cosh k_2x+e^{ik_1^2z}k_1\alpha\sinh k_1x}{ie^{ik_2^2z}\alpha\sinh k_2x+e^{ik_1^2z}\cosh k_1x}.
\end{equation}
The potential $V_1$ as well as the guided mode together with $|\mathcal{L}\Psi_{x_0,z_0,v_0,\sigma}|^2$ are illustrated in Fig. \ref{FigV1h2}. The power of the guided mode $P(\mathcal{L}v)= \int_{-\infty}^{\infty}|\mathcal{L}v|^2 dx$ (d) is also oscillating.
Let us remark that both the potentials (\ref{V1h1}) and (\ref{V1h2}) reduce to the $z$-independent P\"oschl-Teller potential for $\alpha=0$.
\begin{figure}[t!]
\centering
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{Periodic6b.jpg}
\caption{}
\end{subfigure}
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{Periodic7b.jpg}
\caption{}
\end{subfigure}
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{Periodic8b.jpg}
\caption{}
\end{subfigure}\\
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{PeriodicP2b.jpg}
\caption{}
\end{subfigure}
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{Fig71.jpg}
\caption{}
\end{subfigure}
\caption{A strongly confining waveguide. Plots of the real (a) and imaginary (b) parts of $V_1$, see \eqref{V1h2}, for the parameters $k_1=0.4,~k_2=0.1,~\alpha=0.5$ are presented. Furthermore, the corresponding power density of the guided mode $ \mathcal{L}v$, see \eqref{guided V1h2}, is shown (c). The power $P(\mathcal{L}v)= \int_{-\infty}^{\infty}|\mathcal{L}v|^2 dx$ can be seen in (d), the power of the guided mode is oscillating. The power density $|\mathcal{L}\Psi_{x_0,z_0,v_0,\sigma}|^2$, for the parameters $x_0=-20$, $z_0=-50$, $v_0=-0.3$ and $\sigma=40$ is shown in (e). } \label{FigV1h2}
\end{figure}
\subsubsection*{$G_1\neq 0$: weakly confining wave guides}
We fix
\begin{equation}\label{uti}
u=\cosh k_1x e^{ik_1^2z}+i\alpha\sin r_1 x e^{-ir_1^2z},\quad \alpha\in(-1,1).
\end{equation}
The function has no zeros. Writing the equation $\cosh k_1x=-i\alpha\sin r_1 x e^{-i(r_\ell^2+k_1^2)z}$, we can see that the left hand side is always greater or equal to one, whereas the absolute value of the right-hand side is smaller or equal to $|\alpha|$. As the function (\ref{uti}) can be obtained from (\ref{uhii}) by the substitution $k_2\rightarrow i r_1$, the potential $V_1$ is related to (\ref{V1h2}) in the same manner,
\begin{equation}\label{V1t1}
V_1=-2\frac{k_1^2+\alpha^2 r_1^2 e^{-2i(r_1^2+k_1^2)z}}{\cosh^2 k_1x\left(1+i\alpha e^{-i(r_1^2+k_1^2)z}\frac{\sin r_1x}{\cosh k_1x}\right)^2}-2\frac{ie^{-i(r_1^2+k_1^2)z}\alpha\cosh k_1 x\left((k_1^2-r_1^2)\sin r_1x
-2k_1r_1\cos r_1x\tanh k_1x\right)}{\cosh^2 k_1x\left(1+ i e^{-i(r_1^2+k_1^2)z}\alpha\frac{\sin r_1x}{\cosh k_1x}\right)^2}.
\end{equation}
The action of the intertwining operator on the wave packets is $\mathcal{L}\Psi_{x_0,z_0,v_0,\sigma}=G(x,z)\Psi_{x_0,z_0,v_0,\sigma}$ where
\begin{equation}
G(x,z)=\frac{i(x-x_0+iv_0\sigma)}{2(z-z_0-i\sigma)}-\frac{ir_1\alpha\cos r_1x+e^{i(k_1^2+r_1^2)z}k_1\sinh k_1x}{e^{i(k_1^2+k_2^2)z}\cosh k_1z+i\alpha\sin r_1x}.
\end{equation}
The transformed wave packet is illustrated in Fig. \ref{FigV1h3}.
Let us consider the functions $v_1$ and $v_2$,
\begin{equation}
v_1=\sinh k_1x e^{ik_1^2z}-i\alpha\sin r_1 x e^{-ir_1^2z},\quad v_2=\sinh k_1x e^{ik_1^2z}-\alpha\sin r_1 x e^{-ir_1^2z}
\end{equation}
The state $v_1$ fulfills the condition (\ref{G}) so that $\mathcal{L}v_1$ vanishes exponentially on one side of the potential barrier $V_1$. Yet it breaks manifestly $\mathcal{P}_2\mathcal{T}$ symmetry (it can be written as a linear combination of two $\mathcal{P}_2\mathcal{T}$-symmetric solutions). The solution $v_2$ has definite $\mathcal{P}_2\mathcal{T}$ parity but does not comply with (\ref{G}). Therefore, $\mathcal{L}v_2$ has non-vanishing oscillations on both sides of the barrier, see Fig. \ref{FigV1h3} for illustration.
\begin{figure}[t!]
\centering
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{Periodic10b.jpg}
\caption{}
\end{subfigure}
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{Periodic11b.jpg}
\caption{}
\end{subfigure}
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{Periodic12b.jpg}
\caption{}
\end{subfigure}\\
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{Periodic13b.jpg}
\caption{}
\end{subfigure}
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{Fig81.jpg}
\caption{}
\end{subfigure}
\caption{Weakly confining wave guides. Plots of the real (a) and imaginary (b) parts of $V_1$ when $k_1=0.4,~r_\ell=0.5,~\alpha=0.2$, see \eqref{V1t1}. The intensity densities of $\mathcal{L}v_1$ and $\mathcal{L}v_2$ are shown as well ((c) and (d), respectively). The power density $|\mathcal{L}\Psi_{x_0,z_0,v_0,\sigma}|^2$, for the parameters $x_0=-7$, $z_0=-50$, $v_0=-0.2$ and $\sigma=40$ is shown in (e). } \label{FigV1h3}
\end{figure}
\subsection{Coupled wave guides\label{coupled}}
To obtain coupled wave guides we can use a higher order Darboux transformation. In this example, we will use the second-order transformation defined in \eqref{s2} and \eqref{ir12} in order to produce a system with two coupled wave guides. We start out by fixing the transformation functions $u_1$ and $u_2$:
\begin{equation}\label{N2sols}
u_1=\cosh k_1x e^{i k_1^2 z}+i \alpha \sinh k_3 x e^{i k_3^2 z},\quad u_2=\sinh k_2x e^{i k_2^2 z}
\end{equation}
where we suppose that the constants $k_1$, $k_2$, $k_3$ and $\alpha$ are all real. Moreover, we fix $L_1=L_2=1$ that respects (\ref{wavepacketpreservation}).
The explicit form of the new potential term $V_2=-2\partial_x^2\ln W(u_1,u_2)$ is not quite compact, so that we refer to (\ref{s2}) from which it can be obtained directly when substituting (\ref{N2sols}). For $\alpha=0$, both $u_1$ and $u_2$ correspond to the stationary states of the free particle Hamiltonian and the Darboux transformation $\mathcal{L}_{12}$ renders $z$-independent potential
\begin{equation}\label{V2indep}
V_2|_{\alpha=0}=\frac{(k_1^2-k_2^2)(k_2^2-k_1^2+k_1^2\cosh 2k_2x+k_2^2\cosh 2k_1x)}{(k_2\cosh k_1x\cosh k_2x-k_1\sinh k_1x\sinh k_2x)^2}.
\end{equation}
It corresponds to two parallel wave guides that were discussed in \cite{vega}.
We can find two guided modes of $S_2$. They can be obtained as $\mathcal{L}_{12}v_1$ and $\mathcal{L}_{12}v_2$ where the functions $v_1$ and $v_2$ are fixed as
\begin{equation}
v_1= \sinh k_1 x e^{i k_1^2 z}+i \alpha \cosh k_3 xe^{i k_3^2 z},\quad v_2=\cosh k_2xe^{ik_2z}.
\end{equation}
As one can see directly from their explicit form,
\begin{eqnarray}
\mathcal{L}_{12}v_1&=&e^{ik_2^2z}\left\{\frac{e^{2ik_1^2z}k_1(k_1^2-k_2^2)+\alpha^2e^{2ik_3^2z}k_3(k_3^2-k_2^2)}{W(u_1,u_2)}\sinh k_2x\right. \nonumber \\&&\left. - \frac{i\alpha e^{i(k_1^2+k_3^2)}k_2(k_1^2-k_3^2)\left[\cosh k_2x\cosh (k_1-k_3)x +\frac{k_1k_3-k_2^2}{k_2(k_1-k_3)}\sinh k_2x\sinh(k_1-k_3)x\right]}{W(u_1,u_2)}\right\}, \label{coupled v1} \\
\mathcal{L}_{12}v_2&=&e^{2ik^2_2z}k_2\frac{e^{ik_1^2z}(k_2^2-k_1^2)\cosh k_1 x+i \alpha e^{ik_3^2z}(k_2^2-k_3^2)\sinh k_3x}{W(u_1,u_2)} ,\label{coupled v2}
\end{eqnarray}
they vanish rapidly outside the wave guide and represent the guided modes in the system. These states are illustrated in Fig. \ref{coupled pot}.
\begin{figure}[t!]
\centering
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{opticalwaveguide2b.jpg}
\caption{}
\end{subfigure}
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{opticalwaveguide3b.jpg}
\caption{}
\end{subfigure}\\
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{opticalwaveguide4c.jpg}
\caption{}
\end{subfigure}
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{opticalwaveguide5c.jpg}
\caption{}
\end{subfigure}
\caption{Coupled wave guides. The real (a) and imaginary (b) parts of $V_2(x,z)$, see \eqref{V2indep}, for the parameters $k_1 = 1,~ k_2 = 1.09,~ k_3 = 0.95,$ and $\alpha = 0.5$ are plotted. The intensity densities of the guided modes are displayed, $|\mathcal{L}_{12}v_1|^2$ (c) and $|\mathcal{L}_{12}v_2|^2$ (d), see \eqref{coupled v1} and \eqref{coupled v2}, respectively. } \label{coupled pot}
\end{figure}
It is worth noticing that if we take $u_2$ as a first-step transformation function, the potential $V_1=-2\partial_x^2\ln u_2$ would be singular as $u_2$ vanishes at the origin. Hence, the current model is an example where the intermediate potential $V_1$ can be singular, however, the final one $V_2$ is regular.
Let us discuss regularity of the new system.
The Wronskian $W(u_1,u_2)$ can be written as
\begin{align}
W(u_1,u_2)=e^{i(k_1^2+k_2^2)z}\left(k_1\sinh k_1x\sinh k_2x-k_2\cosh k_1x \cosh k_2x\right) \nonumber \\-i\alpha e^{i(k_2^2+k_3^2)z}\left(k_3\cosh k_3x\sinh k_2 x-k_2 \cosh k_2x \sinh k_3 x\right).
\end{align}
It should be free of zeros for all real $x$ and $z$ in order to have $V_2$ regular. It is convenient to consider its real and imaginary part separately:
\begin{eqnarray}
\text{Re} \left( \frac{W(u_1,u_2)}{e^{i(k_1^2+k_2^2)z}} \right) &=&-k_2\cosh k_1x\cosh k_2x\left(1-\frac{k_1}{k_2}\tanh k_1x\tanh k_2x\right) \nonumber \\&&-\alpha k_2\sin((k_3^2-k_1^2)z)\cosh k_2x\cosh k_3x\left(\frac{k_3}{k_2}\tanh k_2x-\tanh k_3x\right),\\
\text{Im} \left( \frac{W(u_1,u_2)}{e^{i(k_1^2+k_2^2)z}} \right)&=&\alpha k_2\cos( (k_3^2-k_1^2)z)\cosh k_2x\cosh k_3x\left(\frac{k_3}{k_2}\tanh k_2x-\tanh k_3x\right).
\end{eqnarray}
First, let us focus on the imaginary part $\text{Im}( W(u_1,u_2)).$ One can show \footnote{We have $\partial_x\left(\frac{k_3}{k_2}\tanh k_2x-\tanh k_3x\right)=k_3(\mbox{sech}^2k_3x-\mbox{sech}^2k_2x)$. The monotonicity follows from $\mbox{sech}^2x_1>\mbox{sech}^2x_2$ whenever $|x_1|<|x_2|$.} that the term in brackets is a monotonic function which is increasing for $|k_2|>|k_3|$, decreasing for $|k_3|>|k_2|$ and it has a single zero at $x=0$. Therefore, Im$( W(u_1,u_2))=0$ for $x=0$ and $z=\frac{(n+1/2)\pi}{k_3^2-k_1^2}$ where $n$ is an integer.
Considering Re$(W(u_1,u_2))$, we can see that is is nonvanishing for $x=0$ for $k_2\neq 0$. For $z=\frac{(n+1/2)\pi}{k_3^2-k_1^2}$, the zeros of Re$( W(u_1,u_2))$ coincide with the zeros of
\begin{equation}\label{intermid}
\frac{\cosh k_1 x}{\cosh k_3 x}\left(1-\frac{k_1}{k_2}\tanh k_1x\tanh k_2x\right)+\alpha \left(\tanh k_3 x-\frac{k_3}{k_2}\tanh k_2x\right).
\end{equation}
Let us suppose that $|k_1|<|k_2|$. Then the first term is positive. We also take $|k_1|>|k_3|$. Then $\frac{\cosh k_1 x}{\cosh k_3 x}>1$ and we can see that the first term is bounded from below by $1-\frac{|k_1|}{|k_2|}$. The second term is bounded from below by $\alpha \left(-1-\frac{|k_3|}{|k_2|}\right)$ and from above by $\alpha \left(1+\frac{|k_3|}{|k_2|}\right)$. So that it is granted that the term (\ref{intermid}) is positive when $\left(1-\frac{|k_1|}{|k_2|}\right)>|\alpha|\left(1+\frac{|k_3|}{|k_2|}\right).$ However, this estimate is very rough and the term remains nodeless (and $V_2$ regular) for larger range of $\alpha$. In the Fig. \ref{coupled pot} we present plots of $V_2$, its real (a) and imaginary (b) parts and also the intensity densities of the guided modes: $|\mathcal{L}_{12}v_1|^2$ (c) and $|\mathcal{L}_{12}v_2|^2$ (d), for the parameters $k_1 = 1,~ k_2 = 1.09,~ k_3 = 0.95,$ and $\alpha = 0.5$.
\subsection{Non-$\mathcal{PT}$-symmetric systems}\label{NonPT}
The results of section \ref{4.1} are valid for large class of potentials, including those where the parity-time symmetry is manifestly broken. These systems, where guided modes still exist despite the lack of symmetry, can be constructed in the same vein as the $\mathcal{P}_2\mathcal{T}$-symmetric ones. Let us present briefly a simple example where the transformation function $u$ and the preimage $v$ of the guided mode are fixed as
\begin{equation}
u=\cosh k_1x e^{ik_1^2z}+\alpha\sinh (k_2x+\delta)e^{ik_2^2z},\quad
v=\sinh k_1x e^{ik_1^2z}+\alpha\cosh (k_2x+\delta)e^{ik_2^2z},\quad \delta\in\mathbb{R}.
\end{equation}
We can see that when $\alpha\notin \mathbb{R}$ and $\delta\neq 0$, the function $u$ does not comply with (\ref{uPT2}) and, hence, the resulting potential $V_1$ ceases to be $\mathcal{P}_2\mathcal{T}$-symmetric. Analysis of the range of parameters where $u$ is nodeless can be performed similarly to preceding cases and we will not present it here explicitly. The new potential $V_1$ reads
\begin{eqnarray} \label{NonPT V1}
V_1&=&-2\frac{k_1^2-e^{2i(k_2^2-k_1^2)z}k_2^2\alpha^2}{\cosh^2k_1x\left(1+e^{i(k_2^2-k_1^2)z}\alpha\frac{\sinh (k_2x+\delta)}{\cosh k_1x}\right)^2}\nonumber\\&&-2\frac{\alpha e^{i(k_2^2-k_1^2)z}\cosh k_1x\cosh (k_2x+\delta)\left((k_1^2+k_2^2)\tanh(k_2x+\delta)-2k_1k_2\tanh k_1x\right)}{\cosh^2k_1x\left(1+e^{i(k_2^2-k_1^2)z}\alpha\frac{\sinh (k_2x+\delta)}{\cosh k_1x}\right)^2}.
\end{eqnarray}
The guided mode is obtained in the following form
\begin{eqnarray}
\mathcal{L}v=\frac{e^{2ik_1^2z}k_1-e^{2ik_2^2z}k_2\alpha^2-e^{i(k_1^2+k_2^2)z}(k_1+k_2)\alpha \sinh ((k_1-k_2)x-\delta)}{e^{ik_1^2z}\cosh k_1x\left(1+e^{i(k_2^2-k_1^2)z}\alpha\frac{\sinh (k_2x+\delta)}{\cosh k_1x}\right)}. \label{NonPT Lv}
\end{eqnarray}
Potential $V_1$, the guided mode and the transformed wave packet $\mathcal{L}\Psi_{x_0,z_0,v_0,\sigma}$ are plotted in Fig. \ref{FigV1h4}.
\begin{figure}[t!]
\centering
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{Periodic15b.jpg}
\caption{}
\end{subfigure}
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{Periodic16b.jpg}
\caption{}
\end{subfigure}\\
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{Periodic17b.jpg}
\caption{}
\end{subfigure}
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{Fig101.jpg}
\caption{}
\end{subfigure}
\caption{A non-$\mathcal{PT}$-symmetric system. Plots of the real (a) and imaginary (b) parts of $V_1$, see \eqref{NonPT V1}, and the intensity density of the guided mode $\mathcal{L}v$ (c) for the parameters $k_1=0.4, k_2=0.05~, \alpha= 0.1+0.8 i$, see \eqref{NonPT Lv}. The power density $|\mathcal{L}\Psi_{x_0,z_0,v_0,\sigma}|^2$, for the parameters $x_0=-7$, $z_0=-50$, $v_0=-0.2$ and $\sigma=40$ is shown in (d). } \label{FigV1h4}
\end{figure}
\section{Summary}
The aim of the current article was to construct exactly solvable models of optical setting with complex refractive index, where propagation of light in paraxial approximation is governed by a two-dimensional, non-separable Schr\"odinger equation. We utilized the time-dependent Darboux transformation presented in \cite{Samsonov1}. In Sec. \ref{SUSY and PT}, we discussed its peculiar properties for general class of non-Hermitian systems. In particular, we focused on existence of the missing state and provided its generalized definition (\ref{um}). The framework allowed us to construct systems with localized defects of refractive index that can accommodate localized solutions, called by us light dots, or systems where periodically structured wave guides possess exponentially localized guided modes.
In order to get the models with parity-time symmetry, we considered two different definitions of parity operator; reflection with respect to the axis $x$ denoted as $\mathcal{P}_x$ and reflection with respect to the origin $\mathcal{P}_2$, see (\ref{Ps}). Actual choice of the parity operator determined the whole construction to a large extend. The transformation function $u$, the solution of the initial Schr\"odinger equation in terms of which both the Darboux transformation (\ref{S1L}) and the new potential (\ref{V_1}) were defined, had to comply with either (\ref{uPT1}) or (\ref{uPT2}), dependently on the definition of the parity operator. We showed that existence of the missing state (\ref{um}), which represents a localized state in the new systems, can be granted provided that the transformation function satisfies (\ref{Scond}) where the operator $\mathcal{S}$ is identified with $\mathcal{P}_x\mathcal{T}$.
In section \ref{tres}, we focused on the construction of systems that can possess localized solutions, light dots. In order to get a transformation function $u$ of required properties, we utilized a mapping between Schr\"odinger equations of the harmonic oscillator and of the free particle. The wave packets (\ref{u even}), (\ref{u odd}) obtained in this way served as the basis for construction of solvable models. We presented solvable models of a wave guide with a localized defect (\ref{V1HO}), or with a localized defect of uniform refractive index (\ref{PTHO Example Potential}). We found the light dot solutions for these systems, see Fig. \ref{Missing pot} and \ref{FigPTHOPotential}
for illustration. In this context, it is worth mentioning the Bohmian approach presented in \cite{bohmian}. The authors provided the framework that allows for construction of the potential term of the Helmholtz equation such it has desired solution, e.g. the light dot. When compared by the supersymmetric approach presented in this article, we greatly benefited from the solvability of the initial system. It allows us to obtain (possibly) infinitely many solutions of $S_1f=0$ by direct application of the intertwining operator $\mathcal{L}$.
In section \ref{cuatro}, we provided a general construction of wave guides that are exponentially vanishing along $x$-axis and periodic along $z$-axis. Existence of the guided modes was analyzed. We showed that dependently on the choice of $u$, the wave guides differ by the strength of confinement. In the strong wave guides, the guided mode is vanishing exponentially in the perpendicular direction to the wave guide. In the weak wave guides, the guided modes leak from the wave guide and exhibit non-vanishing oscillations in transverse direction.
We illustrated the general results on explicit examples of optical wave guides with two-dimensional fluctuations of refractive index, distinguished by different choices of the transformation function. Strong wave guides were generated in (\ref{V1h1}) and (\ref{V1h2}), a weak wave guide was presented in (\ref{V1t1}). The presented solvable models were two-dimensional $\mathcal{P}_2\mathcal{T}$-symmetric generalizations of the reflectionless P\"oschl-Teller potential. Indeed, setting $\alpha=0$ in (\ref{V1h1}), (\ref{V1h2}), (\ref{V1t1}), the expressions reduce to $V_1=-2k_1\mbox{sech}^2 k_1x$.
We also constructed a system with two coupled wave guides (\ref{V2indep}) and calculated two associated guided modes.
In our constructions, the parity operator $\mathcal{P}_2$, which corresponds to the reflection with respect to origin, proved to be rather universal as all the presented parity-time symmetric systems possessed $\mathcal{P}_2\mathcal{T}$ symmetry. Only two of them, namely (\ref{V1HO}) and (\ref{PTHO Example Potential}) possessed both $\mathcal{P}_x\mathcal{T}$- and $\mathcal{P}_2\mathcal{T}$-symmetry.
The presented construction of wave guides and guided modes is applicable to a large class of initial systems with an integrable potential. When compared to the systems separable in Cartesian coordinates, see e.g. \cite{susyrandombands,longhicrossroad}, it allows for construction of the localized defects where the fluctuation of the refractive index is nonzero in a bounded region, or for construction of the straight wave guides where the fluctuation of the refractive index is exponentially vanishing in transverse direction to the wave guide and it is oscillating periodically along the wave guide. It is not restricted to parity-time-symmetric operator, so that it can be utilized for construction of systems where $\mathcal{PT}$ symmetry is manifestly broken. We exemplified construction of such setting in the end of Sec. \ref{NonPT}, see Fig. \ref{FigV1h4}.
In the analysis of optical systems with separable evolution equations, supersymmetry techniques were used to study effectively one-dimensional settings, see e.g. \cite{MiriPRL}-\cite{Mathias2,LonghiBragg}. The intertwining operator provided a one-to-one mapping\footnote{Up to the state annihilated by the intertwining operator} between the stationary solutions of the original system and those of the new system, preserving the phase of the solutions.
It was also used to extract a required mode (that corresponded to the kernel of the intertwining operator) such that its analog was missing in the superpartner system. Hence, the spectra of the two settings were either identical (in case of broken SUSY) or almost identical up to a single eigenvalue (the case of unbroken SUSY). The scattering properties of the superpartner systems were analyzed with the use of the intertwining operator whose superpotential was asymptotically constant.
In our case, we dealt directly with the partial differential equation. The stationary states were not of primary importance in our work. The intertwining operator still provided the matching between the solutions of the original and of the new system. However, its structure was more complicated; the superpotential $\mathcal{W}(x,z)$ in (\ref{S1L}) is a two-dimensional function that is asymptotically non-constant in general, and, hence, it could have a profound impact on the properties of the transformed functions. This led us to implementation of the additional requirement (\ref{wavepacketpreservation}) that granted boundedness of the transformed wave packets.
In our work, the intertwining operator was not intended to annihilate a guided mode of the original system. Instead, it was defined such that the associated new system possessed an additional localized solution, the missing state.
Nevertheless, the general framework of the time-dependent supersymmetry could be also utilized for extraction of guided modes from the system. However, it goes beyond the scope of the current article.
\section*{Appendix: Light dots in in curved wave guides}
In the Appendix, we will illustrate the situation where we cannot construct the missing state, however, the system possesses other localized solutions.
We use \eqref{Superposition} for definition of $u$, fixing it as a linear combination of $\psi_n$ defined in (\ref{psi_n}).
We shall consider some properties inherited from the eigenstates of the harmonic oscillator. First, $\psi_0$ is the only solution without nodes. Second, for all odd $n$, $\psi_{n}(0,z)=0$, and third, two functions $\psi_{m}$ and $\psi_{n}$, $m\neq n$, can vanish simultaneously only at $x=0$. This information helps us to find a set of coefficients such that $u(x,z)\neq 0$.
Let us take $u=\psi_j + i \alpha \psi_{j+1}$, where $\alpha$ is a real constant and $j$ is a positive integer number, (note that up to a global constant phase these are particular cases of \eqref{Superposition})
\begin{eqnarray}
u(x,z) &= & \frac{1}{\sqrt{\sqrt{2 \pi } 2^j j!}} \frac{1}{(1+z^2)^{1/4}} \exp\left\{\frac{i}{4} \left[\frac{x^2}{z-i} - 4 \left(j + \frac{1}{2} \right) \arctan(z)
\right] \right\} \nonumber \\
&\times & \left[H_j\left( \frac{x}{\sqrt{2 \left(z^2+1\right)}} \right) + i\frac{\alpha}{\sqrt{2(j+1)}} \left(\frac{1 - iz}{\sqrt{1+z^2}} \right) H_{j+1}\left( \frac{x}{\sqrt{2 \left(z^2+1\right)}} \right) \right]. \label{u estrellas}
\end{eqnarray}
This function is never zero. It follows from the fact that $H_{j}(x/\sqrt{2(z^2+1)})$ and $H_{j+1}(x/\sqrt{2(z^2+1)})$ are real functions and their zeros do not coincide. Indeed, the imaginary part of the linear combination of the Hermite polynomials in brackets is a real multiple of $H_{j+1}(\cdot)$ whereas the real part is a combination of $H_{j}(\cdot)$ and $H_{j+1}(\cdot)$, so that their zeros are mismatched.
The operator $\mathcal{L}$ in \eqref{S01L} reads
\begin{eqnarray}
\mathcal{L}=L_1(z)\left( \partial_x + \frac{x}{2(1+iz)} - \partial_x \ln \left[H_j\left( \frac{x}{\sqrt{2 \left(z^2+1\right)}} \right) + i\frac{\alpha}{\sqrt{2(j+1)}} \left(\frac{1 - iz}{\sqrt{1+z^2}} \right) H_{j+1}\left( \frac{x}{\sqrt{2 \left(z^2+1\right)}} \right) \right]\right).
\end{eqnarray}
It intertwines $S_0$ with the new Schr\"odinger operator whose potential $V_1$ defined in (\ref{V_1}) can then be written as
\begin{eqnarray}
V_1=i\partial_z\ln L_1(z)+\frac{1}{1+iz} - 2 \partial_x^2 \ln \left[H_j\left( \frac{x}{\sqrt{2 \left(z^2+1\right)}} \right) + i\frac{\alpha}{\sqrt{2(j+1)}} \left(\frac{1 - iz}{\sqrt{1+z^2}} \right) H_{j+1}\left( \frac{x}{\sqrt{2 \left(z^2+1\right)}} \right) \right]. \label{V1 estrella}
\end{eqnarray}
The solutions of the corresponding Schr\"odinger equation can be written as $\phi_n = \mathcal{L} \psi_{n}$ . Using the property $H_n'(y)= 2 n H_{n-1}(y)$ of Hermite polynomials, we can write them as
\begin{eqnarray}
\phi_n =L_1(z)\left( \frac{\sqrt{n}}{1+iz} \psi_{n-1} - \left( \partial_x \ln \left[H_j\left( \frac{x}{\sqrt{2 \left(z^2+1\right)}} \right) + i\frac{\alpha}{\sqrt{2(j+1)}} \left(\frac{1 - iz}{\sqrt{1+z^2}} \right) H_{j+1}\left( \frac{x}{\sqrt{2 \left(z^2+1\right)}} \right) \right] \right) \psi_n\right). \nonumber\label{Light dots sol}
\end{eqnarray}
It follows from the definition of $\mathcal{L}$ that $\phi_j= i \alpha \phi_{j+1}$ for $n\equiv j$. Indeed, on one side we have $\phi_j= \mathcal{L} \psi_j = L_1(z)(\psi_j' - (\ln(\psi_j+i\alpha \psi_{j+1}))'\psi_j) = i L_1(z)\alpha ( \psi_j' \psi_{j+1}-\psi_j \psi_{j+1}')/(\psi_j+i\alpha \psi_{j+1}) $. On the other side, we get $\phi_{j+1}= \mathcal{L} \psi_{j+1}= L_1(z)(\psi_{j+1}' - (\ln(\psi_j+i\alpha \psi_{j+1}))'\psi_{j+1} )= L_1(z) ( \psi_j' \psi_{j+1}-\psi_j \psi_{j+1}')/(\psi_j+i\alpha \psi_{j+1})$.
The explicit form of $\mathcal{L}$ suggest we should fix $L_1(z)=O(z)$ in order to satisfy the condition (\ref{wavepacketpreservation}). We fix $L_1=1+iz$ in order to keep $\mathcal{P}_2\mathcal{T}$-symmetry of the potential $V_1$. This choice also eliminates the first two terms in (\ref{V1 estrella}). The potential is
a rational function in the $x$ variable for any (positive integer) $j$. It is remarkable that $|V_1|$ follows a star-like pattern where the number of rays in the ``star-burst'' correlates with the value of $j$.
As we will show below, the asymptotic, star-like behavior of the potential can be understood explicitly with the use of the fact that $V_1$ is rational function in $x$ variable.
Let us set $u=\psi_1+ i \alpha \psi_2$ and $L_1=1$. The corresponding potential term $V_1$ and the intertwining operator are:
\begin{eqnarray} \label{V1 estrellas 2}
V_1(x,z)&=&\frac{1}{1+iz}- \frac{4 \alpha}{\alpha x^2 + \sqrt{2}(z-i)x-(1+z^2)\alpha } + 2\left(\frac{2 \alpha x + \sqrt{2}(z-i)}{\alpha x^2 + \sqrt{2}(z-i)x-(1+z^2)\alpha } \right)^2, \nonumber \\
\mathcal{L}&= &\partial_x + \frac{x}{2(1+iz)}- \frac{2 \alpha x + \sqrt{2}(z-i)}{\alpha x^2 + \sqrt{2}(z-i)x-(1+z^2)\alpha}.
\end{eqnarray}
As the explicit formulas suggest, $V_1$ is regular as the denominators cannot vanish. The potential represents two asymptotically straight wave guides that come together at a specific angle, but they avoid intersection, see the first two graphs ((a) and (b)) in Fig. \ref{Segunda estrella}. It is possible to understand the asymptotic behavior in the following manner: rewriting the potential as a single fraction, we substitute $x=a z+b$. This way, we get a polynomial of order four in $z$ in the numerator, while there is a polynomial of order five in $z$ in the denominator. We require the polynomials to be of the same order, so that the coefficient of the leading term in the denominator has to vanish. This condition fixes the values of $a$ as $a_\epsilon=\frac{1}{\sqrt{2}\alpha}\left(-1+(-1)^\epsilon\sqrt{1+2\alpha^2}\right) $, $\epsilon=1,2$.
The asymptotic behavior of the potential can be then calculated as
\begin{eqnarray}
&&\lim\limits_{|z|\rightarrow\infty}V_1|_{x\rightarrow a_1 z+b}=\frac{4\alpha^2(1+2\alpha^2)}{(b\alpha\sqrt{2+4\alpha^2}-i(1+\sqrt{1+2\alpha^2}))^2},\\
&&\lim\limits_{|z|\rightarrow\infty}V_1|_{x\rightarrow a_2 z+b}=\frac{4\alpha^2(1+2\alpha^2)}{(b\alpha\sqrt{2+4\alpha^2}-i(-1+\sqrt{1+2\alpha^2}))^2}.
\end{eqnarray}
The asymptotic values are invariant with respect to conjugation joined by substitutions $z\rightarrow -z$ and $b\rightarrow-b$ which is just the manifestation of the $\mathcal{P}_2\mathcal{T}$-symmetry of the potential.
As the transformation function $u$ does not satisfy (\ref{uPT1}), we cannot construct the missing state via (\ref{um}). However, we can still find localized solutions
$\phi_n=\mathcal{L}\psi_n$ that are square integrable for fixed $z$
\begin{eqnarray} \label{sol light dots 2}
\phi_n= \frac{\sqrt{n}}{1+iz} \psi_{n-1}- \frac{2 \alpha x + \sqrt{2}(z-i)}{\alpha x^2 + \sqrt{2}(z-i)x-(1+z^2)\alpha} \psi_n.
\end{eqnarray}
They represent light dots that are concentrated at the bending of the wave guides, see Fig. \ref{Segunda estrella} (c)-(d) for illustration.
\begin{figure}[t]
\begin{center}
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{Star2.jpg}
\caption{}
\end{subfigure}
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{Star3.jpg}
\caption{}
\end{subfigure}
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{Star4.jpg}
\caption{}
\end{subfigure} \\
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{Star5.jpg}
\caption{}
\end{subfigure}
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{Star6.jpg}
\caption{}
\end{subfigure}
\caption{Plots of the real (a) and imaginary (b) parts of $V_1$ for $\alpha=3$, see \eqref{V1 estrellas 2}. Moreover, three different solutions, or light dots, $|\phi_0|^2$ (c), $|\phi_2|^2$ (d) and $|\phi_5|^2$ (e), see \eqref{sol light dots 2}, are shown.} \label{Segunda estrella}
\end{center}
\end{figure}
| 22,195
|
The International Racquetball Tour (IRT) begins the second half of its 2015-16 season this weekend with the NYC Open on Long Island, New York, and will be followed immediately with the Lewis Drug Pro-Am next weekend in Sioux Falls, South Dakota. Thus, it's a good time to look back on the first half of the season.
With five wins over seven events, Kane Waselenchuk is clearly the #1 IRT player this season. The two events he didn't win are events he didn't play in, although the record has him as losing in Davison, that was a late withdrawal. Waselenchuk also won the US Open doubles event for the second year, as he and Ben Croft successfully defended their title from a year ago.
Similarly, Rocky Carson is a solid #2 with one win and five runner up finishes in the first half of the season. But after Carson, it's a muddy picture. Alvaro Beltran is the only other winner this season, but he's also lost in the Round of 16 three times. Daniel De La Rosa reached the US Open final for the first time in his career, but has also lost in the 16s twice and only been a semi-finalist twice. Jose Rojas only has one loss in the 16s and a runner up finish to Carson, so he's arguably ahead of Beltran and De La Rosa, although it's really splitting hairs between the three.
The first half of the season also saw some breakthrough results, as Felipe Camacho, Sebastian Franco, Mario Mercado and Brad Schopieray all reached the semi-finals for the first times in their careers. Franco was a semi-finalist twice. Those four players are from four different countries, giving the IRT a real international flavour. Camacho is Costa Rican, Franco Colombian, Mercado Bolivian and Schopieray is American.
At the other end of the spectrum, Cliff Swain reached the semi-finals for the 163rd time in his career, when he got the semis in Davison. In most of his semi-final appearances (110 times), the six time IRT #1 Swain advanced to the final, but this year the now 49 year old lost to Rocky Carson. Swain's 163 semi-final or better IRT appearances alone would tie him for 8th on the all time appearances list with Chris Crowther. Swain's 285 appearances overall is by far the most career IRT appearances. Ruben Gonzalez is a distant second at 216.
Schedule
What marked the first half of the IRT season was five Tier 1 events on consecutive weekends, which is a crazy schedule. Only 8 players played in all seven first half events: Carson, Jose Rojas, Beltran, Jansen Allen, Felipe Camacho, Robert Collins, Mauricio Zelada, and Mario Mercado. De La Rosa was technically in all seven events, but he was a late withdrawal from the Garden City event.
The second half schedule is more forgiving, as there are back to back events twice, with the first ones this weekend and next. Then there's two months until the IRT is in action again with back to back events in early March in Edinburg, Texas and Lombard, Illinois.
After that the IRT will be in Huber Heights, Ohio for the 2016 Raising Some Racquet for Kids, in Minneapolis, which will host the IRT Pro Nationals, in Sarasota for the Florida IRT Pro-Am, and then finishes up in Portland Oregon with the Pro Kennex Tournament of Champions.
Thus, there are 15 events on the IRT schedule this season, which will make for a good racquetball season.
2015-16 IRT Season - Event Summary
Event - Final - Semi-Finalists
Kansas City: Kane Waselenchuk d. Rocky Carson - SF: Jose Rojas & Felipe Camacho
US Open: Waselenchuk d. Daniel De La Rosa - SF: Alvaro Beltran & Carson
US Open Doubles: Waselenchuk & Ben Croft d. Jake Bredenbeck & Jose Diaz - SF: Jose Rojas & Marco Rojas and Jansen Allen & Charlie Pratt
San Marcos: Waselenchuk d. Carson - SF: Beltran & Sebastian Franco
Atlanta: Beltran d. Carson - SF: De La Rosa & J. Rojas
Davison: Carson d. J. Rojas - SF: Cliff Swain & Brad Schopieray
Garden City: Waselenchuk d. Carson - SF: Franco & Beltran
St Louis: Waselenchuk d. Carson - SF: De La Roas & Mario Mercado
Top 20 IRT rankings - January 12, 2016
Rank - Player - Country - Points
1) Kane Waselenchuk (Canada): 4440
2) Rocky Carson (USA): 3738
3) Alvaro Beltran (Mexico): 2486
4) Daniel De La Rosa (Mexico): 2346
5) Jose Rojas (USA): 2187
6) Jansen Allen (USA): 1833
7) Marco Rojas (USA): 1688
8) Felipe Camacho (Costa Rica): 1254
9) Charlie Pratt (USA): 1179
10) Robert Collins (USA): 1124
11) Ben Croft (USA): 933
12) Mauricio Zelada (USA): 838
13) Matthew Majxner (USA): 817
14) Sebastian Franco (Colombia): 769
15) Mario Mercado (Bolivia): 736
16) Jose Diaz (USA): 582
17) Scott McClellan (USA): 580
18) Brad Schopieray (USA): 494
19) Filip Vesely (USA): 432
20) Tim Landeryou (Canada): 362
The 2015-16 IRT Statistical Abstract (first half of season)
Total matches played: 210
Percentage of 3 game matches: 60.5 (127 of 210)
Percentage of 4 game matches: 22.4 (47)
Percentage of 5 game matches: 14.3 (30)
Percentage of matches that were defaults or data was incomplete: 2.9 (6)
Most 5 game matches played: 7 - Daniel De La Rosa (4 wins, 3 losses)
2nd most 5 game matches played: 6 - Alvaro Beltran (4 wins, 2 losses)
Most points for in a victory: 53 - Thomas Fuhrmann d. Mario Mercado, 11-2, 8-11, 11-8, 9-11, 14-12, at the US Open
Most points scored in a loss: 50 - Rocky Carson loses to Daniel De La Rosa, 11-9, 10-12, 8-11, 11-8, 12-10, at the US Open
Most points in a 5 game match: 102 - Daniel De La Rosa d. Rocky Carson, 11-9, 10-12, 8-11, 11-8, 12-10, at the US Open
Most points in a 4 game match: 80 - Andree Parrilla d. Mario Mercado, 9-11, 11-9, 11-9, 11-9 in San Marcos
Most points in a 3 game match: 72 - Matthew Majxner d. Brian Simpson, 11-9, 13-11, 15-13, in St Louis
Largest margin of victory 5 game match: 19 - Matthew Majxner d. Jonathan Justice, 11-4, 11-3, 10-12, 8-11, 11-2, in Atlanta
Largest margin of victory 4 game match: 24 - Kane Waselenchuk d. Rocky Carson, 11-5, 11-0, 11-13, 11-2, in Kansas City
Largest margin of victory 3 game match: 32 - Rocky Carson d. Jeremy Best, 11-1, 11-0, 11-0, in Atlanta
Smallest margin of victory 5 game match: -9 - Teobaldo Fumero d. Fernando Rios, 11-9, 1-11, 3-11, 11-6, 11-9, at the US Open
Smallest margin of victory 4 game match: 3 - Mauricio Zelada d. Dylan Reid, 13-11, 2-11, 11-6, 11-6, in St Louis
Smallest margin of victory 3 game match: 6 - Matthew Majxner d. Brian Simpson, 11-9, 13-11, 15-13, in St Louis
Follow the bouncing ball....
Tuesday, January 12, 2016
2015-16 IRT First Half Season Review
Posted by The Racquetball Blog at 11:51:00 AM
| 148,555
|
We’re in the Top 20 of Rolling Stone‘s Greatest Albums of All Time list, going through the various reissues and expansions of each one! This time, we have a Boss, a champion of a ’90s rock revolution, a poet of the ’60s – and starting right now, the King of Pop himself. Read on!
20. Michael Jackson, Thriller (Epic, 1982)
Nine disparate songs, helmed by a producer of straightforward jazz and R&B, and performed by a 24-year-old former child star-turned-gawky but dedicated perfectionist. It sounds like it has the makings of a great album, but the best-selling album in history? It seems unconventional – but that’s what makes Thriller so good. There’s something for everyone, from the seven charting singles to the smorgasbord/soundtrack vibe of the whole proceedings, with or without the unforgettable videos on MTV. Michael would strive harder for greatness, for sure, but he’d never achieve it as effortlessly as he did with this one.
First released on CD not too long after the album’s release (Epic EK 38112), that pressing stayed in print for years. (There was a special repackage in Europe in 1999, packaged in a cardboard sleeve and with a Japanese-style OBI indicating Epic’s U.K. “Millennium Edition” series – Epic MILLEN4). A SACD edition was first released in Japan the next year (Epic ESGA 503) and ultimately released in the U.S. six years after that (Epic ES 38112).
The first of two expanded editions (Epic EK 66073) appeared in 2001, preceding the release of Jackson’s then-new album Invincible and arriving alongside reissues of Michael’s other Epic albums through 1991. Like the reissue of Off the Wall, this disc gives far too much space to audio interviews with producer Jones and songwriter Rod Temperton where written recollections would have done better. Still, this has the most bonus tracks out of this reissue program, featuring two demos (“Billie Jean,” the unreleased “Carousel”), the full version of Vincent Price’s delightful “Thriller” outro and “Someone in the Dark,” from the Grammy-winning E.T. the Extra-Terrestrial storybook record Jackson narrated. (Nitpickers have valid complaints with some of the bonus material, though; “Someone in the Dark” is crossfaded with part of an interview with Jones – it would not be released properly until The Ultimate Collection box set in 2004 – and “Carousel” is edited down as well. That full version can be found on Italian pressings of the import compilation King of Pop (Epic 88697 35638-2, 2008).)
The other deluxe reissue, 2008’s Thriller 25 (Epic/Legacy 88697 22096-2), eschewed much of the bonus material from the last reissue (save the “Thriller” rap) in favor of mostly atrocious remixes of Thriller singles by will.i.am of The Black Eyed Peas, Akon and Kanye West and a DVD of previously-released music videos (as well as Jackson’s iconic performance of “Billie Jean” on Motown 25: Yesterday, Today and Forever in 1983). The sole “unreleased track from the Thriller sessions,” a nice if slight ballad called “For All Time,” is almost certainly not from those sessions; co-writers Mike Sherwood and Jeff Porcaro had not collaborated before Toto’s Fahrenheit album in 1986. The Japanese import bonus track, “Got the Hots,” does indeed date back that far.
19. Van Morrison, Astral Weeks (Warner Bros., 1968)
It’s easy to laud these albums as ones that don’t sound like anything else at the time, but Astral Weeks didn’t, and doesn’t. Inspired by traditional Irish folk, blues, jazz and classical artists, it’s a dreamy song cycle that is the night to the bright daytime of “Brown Eyed Girl.” For all its popularity, though, it’s never been reissued on CD anytime past its initial release (Warner Bros. 1768-2); it was reportedly planned for expansion in the late 2000s, but cancelled by the artist himself in a fit of pique with the music industry.
After the jump, the Boss, the bard and the grunge explosion!.
| 170,948
|
Elon Musk is forced to quit as Tesla chairman to escape fraud charges over his 'false and misleading' tweet
- The billionaire will stay on as chief executive at the electric car maker
- He has been told that his Twitter account must be vetted by Tesla
- Tesla and Musk himself have each also been fined £15 million
Free share trading
Commission-free
Buy stocks
No dealing fees to buy and sell
Model portfolios
Model portfolios
Free fund dealing
0.35% account fee capped for shares
| 373,702
|
\begin{document}
\maketitle
\begin{abstract}
We prove a uniqueness result for limit cycles of the second order ODE $\ddot x + \sum_{j=1}^{J}f_{j}(x)\dot x^{j} + g(x) = 0$. We extend a uniqueness result proved in \cite{CRV}. The main tool applied is an extension of Massera theorem proved in \cite{GS}.
{\bf Keywords}: Uniqueness, limit cycle, second order ODE's, Massera theorem, Conti-Filippov transformation.
\end{abstract}
\section{Introduction}
In this paper we are concerned with planar differential systems of the form
\begin{equation}\label{sysypol}
\dot x = y, \qquad \dot y = - g(x) - \sum_{j=1}^{J}f_{j}(x)y^{j},
\end{equation}
equivalent to the second order differential equations of the form
\begin{equation}\label{equaypol}
\ddot x + \sum_{j=1}^{J}f_{j}(x)\dot x^{j} + g(x) = 0.
\end{equation}
Several mathematical models of physics, economics, biology are governed by second order differential equations (\cite{LC}, \cite{SC}, \cite{V}). Other models can be reduced to systems of the type (\ref{sysypol}) by means of suitable transformations.
The asymptotic behaviour of their solutions is one of the main objects of study. In this perspective, the existence of special solutions as stationary ones, or isolated cycles, is of primary interest. This is particularly true if such solutions attract (repel) neighbouring ones, so that the system's dynamics is dominated by that of the attracting equilibria or cycles. In the special case of an isolated cycle attracting all the other solutions but equilibria, the description of the system's dynamics becomes quite simple, since the asymptotic behaviour of all solutions but the equilibrium one is just that of the limit cycle.
Uniqueness theorems for limit cycles have been extensively studied (see \cite{CRV}, \cite{XZ1}, \cite{XZ2}, \cite{Ci} for recent results and extensive bibliographies, \cite{ZDHD}, chapter IV, section 4). In general, studying the number and location of limit cycles is a non-trivial problem, as shown by the resistance of Hilbert XVI problem. Such a problem has been recently re-proposed as a main research problem (see \cite{Sm}, problem 13). A strictly related subject is that of hyperbolicity. A $T$-periodic cycle $\gamma(t)$ of a differential system
\begin{equation}\label{sysPQ}
\dot x = P(x,y), \qquad \dot y = Q(x,y),
\end{equation}
is said to be {\it hyperbolic} if
\begin{equation}\label{hyper}
\int_0^T {\rm div } (\gamma(t)) dt \neq 0,
\end{equation}
where ${\rm div } = {\partial P \over \partial x} + {\partial Q \over \partial y}$ is the divergence of $(\ref{sysPQ})$. Hyperbolicity plays a main role in perturbation problems, since smooth perturbations of hyperbolic cycles do not allow multiple bifurcations. An attractive cycle is not necessarily hyperbolic.
Most of the uniqueness results proved for planar systems are concerned with the classical Li\'enard system and its generalizations, such as
\begin{equation}\label{sysGG}
\dot x = \xi(x)\bigg[ \varphi(y) - F(x) \bigg], \qquad \dot y = -\zeta(y)g(x), \quad \xi(x) \neq 0, \quad \zeta(y) \neq 0.
\end{equation}
Such a class of systems also contain Lotka-Volterra systems and systems equivalent to Rayleigh equation as special cases \cite{LC}. Such systems are characterized by the presence, both in $\dot x$ and $\dot y$, of a single mixed term obtained as the product of single-variable functions, resp. $ \xi(x) \varphi(y) $ and $\zeta(y)g(x)$. Moreover, such systems can be easily transformed into systems without mixed terms, by applying the transformation
$$
X(x) = \int_0^x \frac{1}{\xi(s)} ds, \qquad Y(y) = \int_0^y \frac{1}{\zeta(s)} ds.
$$
The transformed system has the form
$$
\dot X = \tilde{ \varphi}(Y) - \tilde {F}(X) , \qquad \dot y = \tilde {g}(X),
$$
for suitable functions $ \tilde{ \varphi}(Y)$, $\tilde {F}(X)$, $ \tilde {g}(X)$. \\
\indent In order to study systems with several distinct mixed terms, a different approach is required.
Some recent results (\cite{CRV}, \cite{Ci}) are concerned with the following systems,
\begin{equation}\label{sysCRV}
\dot x = y, \qquad \dot y = - x - y \sum_{k=0}^{N}f_{2k+1}(x)y^{2k},
\end{equation}
equivalent to the equations
\begin{equation}\label{equaCRV}
\ddot x + \sum_{k=0}^{N}f_{2k+1}(x){\dot x}^{2k+1} + x = 0.
\end{equation}
Such systems cannot be reduced to the form (\ref{sysGG}) by the above transformation. In order to prove the limit cycle uniqueness, the authors extend a classical result by Massera about the uniqueness and global (except for equilibria) attractiveness of Li\'enard limit cycles \cite{M}. Such a result comes from two main properties. The first one is that every cycle be star-shaped, fact proved in a new way in \cite{CRV}. The second one is that the vector field rotates clockwise along rays (half-lines having extreme at the axes' origin $O$).
In fact, in general such properties are not sufficient to prove the limit cycle's uniqueness, as the following example shows,
\begin{equation}\label{sysduecicli}
\left\{
\begin{array}{rl}
\dot x = & y \Big(x^2+y^2-(x^2+y^2)^2 \Big) + x \Big(1-3 (x^2+y^2)+(x^2+y^2)^2 \Big) \\
\dot y = & -x \Big(x^2+y^2-(x^2+y^2)^2 \Big) + y \Big(1-3 (x^2+y^2)+(x^2+y^2)^2 \Big)
\end{array}
\right.
\end{equation}
Such a system has two star-shaped limit cycles coinciding with the circles $x^2+y^2 =\frac{3-\sqrt {5}}{2}$ and $x^2+y^2 =\frac{3+\sqrt {5}}{2}$. The internal one is an attractor, the external one is a repellor. The vector field rotates clockwise along every ray (see figure 1).
\begin{figure}[h!]
\caption{The system (\ref{sysduecicli}) has two limit cycles.}
\centering
\includegraphics[width=0.5\textwidth]{duecicli.eps}
\end{figure}
\indent An even more pathological example (from the point of view of Massera-like theorems) is the system
\begin{equation}\label{systrigo}
\dot x = y \cos (x^2+y^2) - x \sin (x^2+y^2), \quad \dot y = -x \cos (x^2+y^2) - y \sin (x^2+y^2).
\end{equation}
Such a system satisfies both properties, and has infinitely many cycles coinciding with the circles $\{ x^2+y^2 = k \pi : k \in \N \}$. Such cycles are alternatively repelling and attracting, and rotate alternatively counter-clockwise and clockwise. A cycle's attraction (repulsion) region is the annular region bounded by the two adjacent limit cycles, except for the innermost one, whose attraction region has the origin in its boundary. \\
\indent As a consequence, an additional condition is required in order to give a complete proof of a Massera-like theorem. Corollary 6 of \cite{GS} provides a natural additional condition, asking for the angular velocity not to vanish in the whole plane.
The same result can be proved if the angular velocity does not vanish in a suitable subregion of the plane, as in \cite{S}. Corollary 6 is a particular case of theorem 2 in \cite{GS}, a rather general extension to Massera theorem. In such a theorem an auxiliary function $\nu$ is used in order to study the cycles' hyperbolicity. The main property of $\nu$ is that its integral on a cycle coincides with that of the vector field's divergence. On the other hand, $\nu$ has an advantage over the divergence, since it can be everywhere negative in presence of a repelling critical point and an attracting limit cycle, while in such a situation the divergence has to change sign. This helps in proving the limit cycle's uniqueness, as in \cite{S}. Additionally, theorem 2 in \cite{GS} allows to prove the limit cycle's hyperbolicity, which is not a consequence of Massera-like theorems.
In this paper we extend in two ways the uniqueness result of \cite{CRV}. First, we consider systems with even degree terms $f_{2k}(x)y^{2k}$, then we allow $g(x)$ to be non-linear, provided $xg(x) \neq 0$ for $x \neq 0$ in some interval. Even degree terms can be treated by considering the function $\sum_{j=1}^{J}f_{j}(x)y^{j-1}$ as the sum of $y$-trinomials, each satisfying suitable conditions. On the other hand, considering a non-linear $g(x)$ allows to work in different regions for the same system. Since a $x$-translation transforms a system of the form (\ref{sysypol}) into a system of the same form, one just has to translate a critical point to the origin, proving the uniqueness, in a suitable strip $(a,b) \times \R$, of limit cycles surrounding such a point. \\
\indent If the system has the form (\ref{sysCRV}), with $g(x) = x$, our result extends that one obtained in \cite{CRV}, replacing the sign and monotonicity conditions on $f_{2k+1}(x)$ with a monotonicity condition on $x^{2k}f_{2k+1}(x)$. This allows to apply our theorem to some polynomial coefficients, as $f_{3}(x) = x^4-x^2+1$, which do not satysfy the hypotheses in \cite{CRV}. \\
\indent In order to extend the results of \cite{CRV}, we follow a different approach w. r. to that one developed in \cite{S}. We cannot apply the theorem 1 of \cite{S} to \ref{sysCRV}, since $\phi(x,y) = \sum_{k=0}^{N}f_{2k+1}(x)y^{2k}$ does not satisfy the strict star-shapedness condition required by such a theorem. In order to overcome such an obstacle we introduce a non-invariance property similar to that one appearing in LaSalle Invariance Principle. In fact, the passage from $x\phi_x + y\phi_y > 0 $ to $x\phi_x + y\phi_y \geq 0 $ is analogous to the passage from the condition $\dot V < 0$ to the condition $\dot V \leq 0$ in studying asymptotic stability problems.
Moreover, rather then just proving the cycle's star-shapedness, we prove that, assuming by absurd the existence of two concentric limit cycles, then there exists an annular region containing both ones, where the angular velocity does not vanish. This allows to apply theorem 2 in \cite{GS} in order to get the limit cycle's uniqueness and its hyperbolicity. \\
\indent Our paper is organized as follows. In section 1 we prove the main theorem about uniqueness and hyperbolicity for systems with a linear $g(x)$. Then we apply such a theorem to some special cases. \\
\indent In section 2 we apply Conti-Filippov transformation in order to reduce a system of the form (\ref{sysypol}) with a non-linear $g(x)$ to another system of the same type, but with a linear $g(x)$. In this section we derive the uniqueness condition in strips. \\
\indent In both sections we also consider the question of limit cycles existence, giving sufficient conditions for the solutions' boundedness that allow to apply Poincar\'e-Bendixson theorem.
\section{g(x) linear}
Since our approach is based on the cycle's star-shapedness, we restrict to a star-shaped subset $\Omega \subset \R^2$. In this context, we think of $\Omega$ as a strip $(a,b) \times \R$, with $a < 0 < b$, but what follows holds for arbitrary star-shaped subsets. We denote partial derivatives by subscripts, i. e. $\phi_x$ is the derivative of $\phi$ w. r. to $x$, etc..
We say that a function $\phi \in C^1( \Omega, \R) $ is {\it star-shaped} if $(x,y) \cdot \nabla \phi = x \phi_x + y \phi_y$ does not change sign. We say that $\phi$ is {\it strictly star-shaped} if $(x,y) \cdot \nabla \phi \neq 0$, except at the origin $O=(0,0)$. We call {\it ray} a half-line having origin at the point $(0,0)$.
We denote by $\gamma(t,x,y)$ the unique solution of the system (\ref{sysypol}) such that $\gamma(0,x,y) = (x,y)$. For definitions related to dynamical systems, we refer to \cite{BS}.
Throughout all of this paper we assume the system (\ref{sysypol}) to satisfy some hypotheses ensuring existence and uniqueness of solutions, and continuous dependence on initial data. This occurs if, for instance, one has
$\bullet )$ \qquad $f_{j} $, $j= 1 , \dots, J$, continuous on their domains;
$\bullet )$ \qquad $g$ lipschitzian on its domain.
In this section we are concerned with the system (\ref{sysypol}), assuming $g(x)=kx$.
Without loss of generality, possibly performing a time rescaling, we may restrict to the case $k=1$. In this case, we may consider (\ref{sysypol}) as a special case of a more general class of systems,
\begin{equation}\label{sysphi}
\dot x = y, \qquad \dot y = - x - y\phi(x,y).
\end{equation}
We first consider a sufficient condition for limit cycle's uniqueness.
It has the same form as system (9) in \cite{S}, but here the function $\phi(x,y)$ is not strictly star-shaped. This will force us to modify the proof in the part related to the angular velocity, and add an hypothesis in the main theorem. As in \cite{S}, we set
$$
A(x,y) = y \dot x - x \dot y = y^2 + x^ 2 +xy \phi(x,y).
$$
The sign of $A(x,y)$ is opposite to that of the angular speed of the solutions of (\ref{sysphi}).
In general, $A(x,y)$ changes sign along the solutions of (\ref{sysphi}), even in the simplest example of nonlinear Li\'enard system with a limit cycle
\begin{equation}\label{sysVDP}
\dot x = y, \qquad \dot y = - x - y(x^2-1).
\end{equation}
In fact, the orbits of (\ref{sysVDP}) approaching the limit cycle from the second and fourth orthants change angular velocity when they get closer to the limit cycle.
Our uniqueness result comes from theorem 2 of \cite{GS}, in the form of corollary 6. In the following lemma the
we prove that if two limit cycles $\mu_1$ and $\mu_2$ exist, then in the closed annular region $D_{12}$ bounded by $\mu_1$ and $\mu_2$ the function $A$ does not vanish. Next lemma's proof is a modification of part of the proof of theorem 1 in \cite{S}. We emphasize that we consider {\it open} orthants, i. e. orthants without semi-axes.
\begin{lemma}\label{lemma1} Let $\phi\in C^1( \Omega, \R^2)$, with $x\phi_x + y\phi_y \geq 0$ in $\Omega$. If the system (\ref{sysphi}) has two distinct limit cycles $\mu_1$ and $\mu_2$, then $A(x,y) >0$ on $D_{12}$.
\end{lemma}
{\it Proof.}
\indent The system (\ref{sysphi}) has one critical point, hence $\mu_1$ and $\mu_2$ are concentric. Let $\mu_1$ be the inner one, $\mu_2$ be the outer one. The radial derivative $A_r$ of $A$ is given by
$$
A_r = \frac {x A_x + y A_y}r = \frac 1r\bigg(2 A + xy (x \phi_x + y\phi_y) \bigg) = \frac 1r\bigg( 2 A + xyr \phi_r \bigg).
$$
Hence the function $A$ satisfies the differential equation $rA_r = 2 A + xy (x \phi_x + y\phi_y) $, whose right-hand side sign determines the monotonicity of $A$ along rays. If $x^*y^* > 0$ and $A(x^*,y^*) > 0$, then $A_r >0$ at $(x^*,y^*)$ and at every point $(rx^*,ry^*)$ with $r > 1$, hence $A$ is strictly increasing on the half-line $\{ (rx^*,ry^*) : r > 1\}$. On the other hand, if $x^*y^* < 0$ and $A(x^*,y^*) < 0$, then $A_r < 0$ at $(x^*,y^*)$ and at every point $(rx^*,ry^*)$ with $r > 1$, hence $A$ is strictly decreasing on the half-line $\{ (rx^*,ry^*) : r > 1\}$. \\
\indent If $A(x^*,y^*) = 0$ and $x^*y^* >0 $, then either $A(rx^*,ry^*) = 0$ for all $r > 1$, or at some point of the half-line $\{ (rx^*,ry^*) : r > 1\}$ the radial derivative $A_r$ becomes positive, so that $A(rx^*,ry^*) > 0$ for all $r > \overline{r}$, for some $\overline{r} > 1$. \\
\indent Similarly, if $A(x^*,y^*) = 0$ and $x^*y^* < 0 $, then either $A(rx^*,ry^*) = 0$ for all $r > 1$, or at some point of the half-line $\{ (rx^*,ry^*) : r > 1\}$ the radial derivative $A_r$ becomes negative, so that $A(rx^*,ry^*) < 0$ for all $r > \overline{r}$, for some $\overline{r} > 1$. \\
\indent We prove that, for every orbit $\gamma$ contained in $D_{12}$, $A(\gamma(t)) > 0$. Every orbit in $D_{12}$ meets every semi-axis, otherwise its positive limit set would contain a critical point different from $O$. On every semi-axis one has $A(x,y) > 0$. Assume first, by absurd, $A(\gamma(t))$ to change sign. Then there exist $t_1 < t_2$ such that $A(\gamma(t_1)) > 0$, $A(\gamma(t_2)) < 0$, and and $\gamma(t_i)$, $i = 1,2$ are on the same ray. Assume $\gamma(t_i)$, $i = 1,2$ to be in the first orthant. Two cases can occur: either $|\gamma(t_1)| < |\gamma(t_2)|$ or $|\gamma(t_1)| > |\gamma(t_2)|$. The former, $|\gamma(t_1)| < |\gamma(t_2)|$, contradicts the fact that $A$ is radially increasing in the first orthant, hence one has $|\gamma(t_1)| > |\gamma(t_2)|$. The orbit $\gamma$ crosses the segment $\Sigma = \{r \gamma(t_1), 0 < r <1\}$ at $\gamma(t_2)$, going towards the positive $y$-semi-axis. Let $G$ be the sub-region of the first orthant bounded by the positive $y$-semi-axis, the ray $\{r \gamma(t_1), r > 0\}$ and the portions of $\mu_1$, $\mu_2$ meeting the $y$-axis and such a ray. The orbit $\gamma$ cannot remain in $G$, since in that case $G$ would contain a critical point different from $O$. Also, $\gamma$ cannot leave $G$ crossing the positive $y$-semi-axis, because $A(x,y) > 0$ on such an axis. Hence $\gamma$ leaves $G$ passing again through the segment $\Sigma$. That implies the existence of $t_3 > t_2$, such that $\gamma(t_3)$ lies on the ray $\{ r \gamma(t_1), 0 < r \}$. Again, one cannot have $|\gamma(t_3)| < |\gamma(t_2)|$, since $A(\gamma(t_3)) > 0$ implies $A$ increasing on the half-line $r \gamma(t_3), r > 1$, hence one has $|\gamma(t_3)| > |\gamma(t_2)|$. Also, one cannot have $|\gamma(t_3)| < |\gamma(t_1)|$, otherwise $\gamma$ would enter a positively invariant region, bounded by the curve $\gamma(t)$, for $t_1 \leq t \leq t_3$, and by the segment with extrema $\gamma(t_1)$, $\gamma(t_3)$, hence there would exist a critical point different from $O$. As a consequence, one has $|\gamma(t_3)| > |\gamma(t_1)|$.
Since $A(x,y) >0$ on the segment joining $\gamma(t_1)$ and $\gamma(t_3)$, such a segment, with the portion of orbit joining $\gamma(t_1)$ and $\gamma(t_3)$ bounds a region which is negatively invariant for (\ref{sysphi}), hence contains a critical point different from $O$, contradiction. \\
\indent This argument may be adapted to treat also the case of a ray in the second orthant, replacing the positive $y$-semi-axis with the positive $x$-semi-axis, and reversing the relative positions of the points $\gamma(t_1)$, $\gamma(t_2)$, $\gamma(t_3)$. In the second orthant one uses the fact that if $A(\gamma(t^*)) < 0$, then on the half-line $r A(\gamma(t^*)), r > 1$ the function $A$ is strictly decreasing, since $xy (x \phi_x + y\phi_y) \leq 0$. In the third and fourth orthants one repeats the arguments of the first and second orthants, respectively. \\
\indent Finally, assume that $A(\gamma(t)) = 0$ at some point $(x^*,y^*) \in D_{12}$. First, we assume $(x^*,y^*)$ to be in the first orthant. \\
\indent One cannot have $A(x^*,y^*) = 0$ on all of the segment $(rx^*,ry^*), 0 < r < 1$, since in such a case the vector field would be radial, and $\gamma$ would contain the whole segment, contradicting the fact that $\gamma \subset D_{12}$. Also, one cannot have $A(x^*,y^*) > 0$ at any point $(x^+,y^+)$ of the segment $(rx^*,ry^*), 0 < r < 1$, since in that case $A$ would be increasing on the half-line $(rx^+,ry^+), r > 1$, contradicting $A(x^*,y^*) = 0$. Hence there exists a point $(x^-,y^-)$ of the segment $(rx^*,ry^*), 0 < r < 1$ such that $A(x^-,y^-) < 0$. Moreover, for all $0<r<1$ one has $A(rx^-,ry^-) < 0$.
Working in the same way one proves that the segment $(rx^*,ry^*), 0 < r < 1$ is the disjoint union of an open sub-segment $\Sigma^-$ where $A(x,y) < 0$, and a closed sub-segment $\Sigma^0$ where $A(x,y) = 0$.
Then the "triangle" $T$ bounded by the $y$-semi-axis, the ray $(rx^*,ry^*), r> 0$, and the arc of $\mu_1$, is positively invariant. This produces a contradiction, due the fact that orbits starting in $T$, close to $\mu_1$, by the continuous dependence on initial data have to remain close to $\mu_1$, hence have to get out of $T$. \\
\indent Now we assume $(x^*,y^*)$ to be in the second orthant and reverse the above argument: one cannot have $A(x^*,y^*) = 0$ on all of the half-line $(rx^*,ry^*), r > 1$, since in such a case $\gamma$ would contain the whole half-line, contradicting the fact that $\gamma \subset D_{12}$. Let $r^* > 0$ be the maximum positive $r$ such that $A(rx^*,ry^*) = 0$. Then for all $r > r^*$, one has $A(rx^*,ry^*) \neq 0$, hence, by what said at the beginning of this proof, $A(rx^*,ry^*) < 0$ for $r > r^*$. Hence, the "unbounded triangle" $T$ having as boundary part of the positive $x$-semi-axis, the half-line $(rx^*,ry^*), r > r^*$, and the arc of $\mu_2$ connecting such half-lines, is positively invariant. As above, this produces a contradiction, due the fact that orbits starting in $T$, close to $\mu_2$, by the continuous dependence on initial data have to remain close to $\mu_2$, hence have to get out of $T$. \\
\indent In the other two orthants one can repeat the arguments of the first two ones, completing the proof.
\hfill$\clubsuit$
Next theorem is as well a modification of theorem 1 in \cite{S}. In order to cope with the weaker hypothesis, we introduce a new condition on the set $x\phi_x +y\phi_y =0$.
\begin{theorem}\label{teorema} Let $\phi\in( \Omega, \R^2)$ be a star-shaped function, such that the set $x\phi_x +y\phi_y =0$ does not contain any non-trivial positive semi-orbit of (\ref{sysphi}). Then (\ref{sysphi}) has at most one limit cycle, which is hyperbolic.
\end{theorem}
{\it Proof.}
Let us assume, by absurd, (\ref{sysphi}) to have two distinct limit cycles $\mu_1$ and $\mu_2$. Since the system has only one critical point, $\mu_1$ and $\mu_2$ have to be concentric. Assume $\mu_1$ to be the inner one. Let us restrict to the closed annular region $D_{12}$ bounded by $\mu_1$ and $\mu_2$. By the lemma \ref{lemma1}, one has $A(x,y) > 0$ on $D_{12}$. Let us consider the new system obtained by dividing the vector field of (\ref{sysphi}) by $A(x,y)$, as in corollary 6 in \cite{GS}. In order to apply such a corollary, one has to compute the expression
$$
\nu = \frac{P\left( xQ_x + yQ_y \right) - Q \left( xP_x + yP_y\right) }{y P - x Q}
$$
where $P$ and $Q$ are the components of the considered vector field. Since for system (\ref{sysphi}) one has $y P - x Q = A$ , one can write
$$
\nu A=
y \left(-x - xy \phi_x - y\phi - y ^2 \phi_y \right) - \left( -x - y \phi(x,y) \right) y =
$$
$$
-y^2 \left( x \phi_x + y \phi_y \right) \leq 0.
$$The function $\nu$ vanishes for $y=0$ and for $x \phi_x + y \phi_y= 0$. The set $y=0$ is transversal to both cycles, hence both cycles have two points on $y=0$. Moreover, since by hypothesis the set $x\phi_x +y\phi_y =0$ does not contain any positive semi-orbit of (\ref{sysphi}), every cycle contains at least a point such that $x\phi_x +y\phi_y > 0$. By continuity, $x\phi_x +y\phi_y >0$ in a neighbourhood of such a point, so that $x\phi_x +y\phi_y >0$ on an arc of $\mu_i$, $i = 1 , 2$. \\
\noindent Then, for both cycles one has:
$$
\int_{0}^{T_i} \nu(\mu_i(t) )dt < 0, \qquad i=1,2,
$$
where $T_i$ is the period of $\mu_i$, $i=1,2$. Hence both cycles, by theorem 1 in \cite{GS}, are attractive. Let $A_1$ be the region of attraction of $\mu_1$. $A_1$ is bounded, because it is enclosed by $\mu_2$, which is not attracted to $\mu_1$. The external component of $A_1$'s boundary is itself a cycle $\mu_3$, because (\ref{sysphi}) has just one critical point at $O$. We can apply to $\mu_3$ the above argument about the sets $y=0$ and $x\phi_x +y\phi_y =0$, concluding that
$$
\int_0^{T_3} \nu(\mu_3(t) )dt < 0.
$$
Hence $\mu_3$ is attractive, too. This contradicts the fact that the solutions of (\ref{sysphi}) starting from its inner side are attracted to $\mu_1$. Hence the system (\ref{sysphi}) can have at most a single limit cycle. Its hyperbolicity comes from the equality (see \cite{GS})
$$
\int_0^T {\rm div } (\mu(t)) dt = \int_0^{T} \nu(\mu(t) )dt < 0.
$$
\hfill $\clubsuit$
Now we consider some classes of $y$-polynomial systems satisying the hypotheses of theorem \ref{teorema}. Let us consider the following conditions related to a $y$-trinomial $\kappa(x) y^{2h+2r} + \tau(x) y^{h+2r} + \eta(x)y^{2r}$:
${\bf (T^+)}$ \quad for all $x \in (a,b)$, $x \neq 0$, one has $(x \tau' + (h+2r)\tau)^2 - 4( x\eta' + 2r \eta)(x \kappa' + (2h+2r) \kappa)) \leq 0$, and $x \kappa' + (2h+2r) \kappa \geq 0$.
\begin{remark}Since the above inequality implies a sign condition on the $y^h$-trinomial $(x\kappa' + 2(h+r) \kappa )y^{2h} + (x \tau' + (h+2r) \tau ) y^{h} + (x\eta' + 2r\eta) $ (see next corollary), it is equivalent to require $x \kappa' + (2h+2r) \kappa \geq 0$ or $x\eta' + 2r \eta \geq 0$.
\end{remark}
${\bf (Seq)}$ \quad there exists a sequence $x_m$ converging to $0$, such that for every $m$ there exists $1 \leq j(m) \leq N$ satisying $x_mf_{j(m)}'(x_m) + (j(m) - 1)f_{j(m)}(x_m) \neq 0$.
\begin{corollary}\label{corollarioT} Assume $\sum_{j=1}^{J}f_{j}(x)y^{j-1}$ to be the sum of $y$-trinomials satisfying the conditions $(T^+)$ and $(Seq)$. Then the system (\ref{sysypol}) has at most one limit cycle, which is hyperbolic.
\end{corollary}
{\it Proof.}
The system (\ref{sysypol}) is a special case of the system (\ref{sysphi}), with
$$
\phi(x,y) = \sum_{j=1}^{J}f_{j}(x)y^{j-1}.
$$
Computing $\nu$, one has
$$
\nu = -y^2 \left(x \phi_x + y \phi_y \right) = - y^2 \sum_{j=1}^{J} \big(xf_{j}'(x) + (j-1) f_{j}(x) \big) y^{j-1}.
$$
By hypothesis, there exist $y$-polynomials $P_l(x,y) = \kappa_l(x) y^{2h(l)+2r(l)} + \tau_l(x) y^{h(l)+2r(l)} + \eta_l(x)y^{2r(l)}$ such that
$$
\phi(x,y) = \sum_{j=1}^{J}f_{j}(x)y^{j-1} = \sum_{l=1}^{L} P_l(x,y) =
$$
$$
\sum_{l=1}^{L}\left( \kappa_l(x) y^{2h(l)+2r(l)} + \tau_l(x) y^{h(l)+2r(l)} + \eta_l(x)y^{2r(l)} \right).
$$
The function $x \phi_x + y \phi_y$ is the sum of the corresponding expressions, computed for any $l$. One has, omitting the dependence on $l$ in the last sum,
$$
x \phi_x + y \phi_y = \sum_{l=1}^{L} x {P_l}_x + y {P_l}_y =
$$
$$
\sum \left( (x\kappa' + 2(h+r) \kappa )y^{2h+2r} + (x \tau' + (h+2r) \tau ) y^{h+2r} + (x\eta' + 2r\eta)y^{2r} \right).
$$
For every $l$, the summands' sign is that of $(x\kappa' + 2(h+r) \kappa )y^{2h} + (x \tau' + (h+2r) \tau ) y^{h} + (x\eta' + 2r\eta) $. Performing the substitution $z=y^h$ one can study the sign of such a $y$-trinomial by studying that of $(x\kappa' + 2(h+r) \kappa )z^2 + (x \tau' + (h+2r) \tau ) z + (x\eta' + 2r\eta) $. Its discriminant is just
$$
(x \tau' + (h+2r) \tau ) ^2 - 4 (x\eta' + 2r\eta) (x\kappa' + 2(h+r) \kappa ).
$$
The condition $(T^+)$ implies that such a discriminant is non-positive, and that the leading term of the $y$-polynomial is non-negative, hence the trinomial is non-negative for all $y$. As a consequence, one has $x \phi_x + y \phi_y \geq 0$ in $(a,b)$.
The set $x \phi_x + y \phi_y = 0$ is the union of the $x$-axis together with the family of vertical lines defined by the equalities $xf_{2k+1}'(x) + 2k f_{2k+1}(x) = 0$. Under the choosen hypothesis on the sequence $x_m$, no orbit meeting a point of $x \phi_x + y \phi_y = 0$ can remain in such a set, since $\dot x = y$. Hence we can apply theorem \ref{teorema}.
\hfill$\clubsuit$
We can provide an example satisying the hypotheses of corollary \ref{corollarioT}. Let us set
$$
\phi(x,y) = (x^2 + 1)y^2+\frac{x^2}{10}y +x^2 -1 .
$$
Here there is just one $y$-trinomial, where one has $r=0$, $h=1$, $\kappa(x) = x^2+1$, $\tau(x) = \frac{x^2}{10}$, $\eta(x) = x^2-1$. One has
$$
(x\kappa' + 2(h+r) \kappa )y^{2h+2r} + (x \tau' + (h+2r) \tau ) y^{h+2r} + (x\eta' + 2r\eta)y^{2r} =
$$
$$
(4x^2 + 2 )y^2 + \frac{3x^2}{10} y + 2 x^2 .
$$
the discriminant of the $y$-trinomial $(4x^2 + 2 )y^2 + \frac{3x^2}{10} y + 2 x^2$ is $-\frac{3191}{100} x^4 - 16 x^2$, which is everywhere negative, but at $0$, where it vanishes. Hence the above $y$-trinomial is positive for all $x \neq 0$, and non-negative for $x = 0$, so that the corresponding system has at most one limit cycle (see Figure 2).
\begin{figure}[h!]
\caption{The system $\dot x = y , \quad \dot y = -x -y( (x^2 + 1)y^2+\frac{x^2}{10}y +x^2 - 1)$ has just one limit cycle. This picture shows that the angular velocity changes along some orbits.}
\centering
\includegraphics[width=0.5\textwidth]{figura2.eps}
\end{figure}
A statement similar to corollary \ref{corollarioT} can be proved for systems satisfying the symmetric condition
${\bf (T^-)}$ for all $x \in (a,b)$, $x \neq 0$, one has $(x \tau' + (h+2r)\tau)^2 - 4( x\eta' + 2r \eta)(x \kappa' + (2h+2r) \kappa)) \leq 0$, and $x\eta' + 2r \eta \leq 0$ (or, equivalently, $x \kappa' + (2h+2r) \kappa \leq 0)$.
Next corollary is a special case of corollary \ref{corollarioT}.
\begin{corollary}\label{corollarioCRV} Assume $f_{j}(x) \equiv 0 $ for $j$ even, $xf_{j}'(x) + (j-1) f_{j}(x) \geq 0$ for $j$ odd and for $x \in (a,b)$. If (Seq) holds, then the system (\ref{sysCRV}) has at most one limit cycle, which is hyperbolic.
\end{corollary}
{\it Proof.}
The function
$$
\phi(x,y) = \sum_{j=1}^{J}f_{j}(x)y^{j-1} = \sum_{k=0}^{N}f_{2k+1}(x)y^{2k},
$$
can be considered as the sum of $y$-trinomials of the type $\kappa(x) y^{2h+2k} + \tau(x) y^{h+2k} + \eta(x)y^{2k}$, with $\kappa(x) = \tau(x) \equiv 0$, $\eta(x) = f_{2k+1}(x)$. As for the two inequalities of $(T^+)$, one has $(x \tau' + (h+2k)\tau)^2 - 4( x\eta' + 2k \eta)(x \kappa' + (2h+2k) \kappa)) = 0$, and $x\eta' + 2k \eta = x f_{2k+1}'(x) + 2k f_{2k+1}(x) \geq 0$ by hypothesis. Then the conclusion comes from corollary \ref{corollarioT}.
\hfill$\clubsuit$
\begin{remark} \label{segno} The corollary \ref{corollarioCRV} is a proper extension to the uniqueness part of theorem 1.3 in \cite{CRV}, concerned with the system (\ref{sysCRV}).
In fact, under the hypotheses assumed in theorem 1.3 of \cite{CRV}:
$(L2)$ $f_{2k+1}(x) \geq 0$, for $k = 1, \dots, N$, for all $x$,
\indent $(L3)$ $f_{2k+1}(x)$, for $k = 0, \dots, N$, increasing for $x > 0$, decreasing for $x < 0$,
\noindent one has $xf_{2k+1}'(x) + 2k f_{2k+1}(x) \geq 0$, as in corollary \ref{corollarioCRV}. \\
\indent Vice-versa, under our hypothesis $(L2)$ holds, but $(L3)$ does not necessarily hold. In order to prove $(L2)$, let us consider the half-line $x \geq 0$. First, observe that $f_{2k+1}(0) \geq 0$. Then, assume by absurd the existence of $x^*$ such that $f_{2k+1}(x^*) < 0$. Then $x^*f_{2k+1}'(x^*) \geq - 2k f_{2k+1}(x^*) > 0$. Hence $f_{2k+1}$ has a minimum at a point $x_m \in (0,x^*)$, where $f_{2k+1}'(x_m) = 0$. This contradicts $x_mf_{2k+1}'(x_m) \geq - 2k f_{2k+1}(x_m) > 0$. One works similarly on the half-line $x \leq 0$. \\
\indent On the other hand, our hypothesis can be satisfied even if $(L3)$ does not hold. An example is provided by
$$
f_{2k+1}(x) = x^4 - x^2 +1, \qquad k > 0.
$$
One has $f_{2k+1}'(x) = 4x^3-2x$, so that $f_{2k+1}(x)$ is neither increasing for $x > 0$, nor decreasing for $x < 0$. Also, one has $xf_{2k+1}'(x) + 2kf_{2k+1}(x) = (4+2k)x^4 - (2+2k)x^2 + 2k > 0 $, for $k > 0$. In fact, such a polynomial's discriminant is $\Delta = 4-24k-12k^2 < 0$ for all $k>0$, so that $xf_{2k+1}'(x) + 2kf_{2k+1}(x) $ does not vanish for any real $x$, for $k > 0$. \\
\indent The coefficient $f_1(x)$ is not subject to the same argument, since for $k = 0$ one has $xf_{2\cdot 0+1}'(x) + 2\cdot 0 \cdot f_{2\cdot 0+1}(x) = xf_{1}'(x) \geq 0$, which does not imply any condition on the sign of $f_1(x)$.
\end{remark}
\indent In order to prove the existence of limit cycles, we need a stronger hypothesis on the system (\ref{sysphi}). We first prove a result about the solutions' boundedness, for systems defined on all of $\R^2$. We denote by $D_M$ the disk $\{ (x,y) : x^2+y^2 \leq M^2 \}$. We set $Z_\phi = \{ (x,y) : \phi(x,y) =0 \}$.
\begin{lemma}\label{bddness} Let $\Omega = \R^2$. If there exists $M > 0$ such that $\phi(x,y) \geq 0$ for all $(x,y) \not\in D_M$, and the set $Z_\phi \setminus D_M $ does not contain any non-trivial positive semi-orbit of (\ref{sysphi}), then every solution of (\ref{sysphi}) definitely enters the disk $D_M$ and remains inside it.
\end{lemma}
{\it Proof.}
Let us consider the function $V(x,y) = \frac 12 \left( x^2 + y^2 \right)$. Its derivative along the solutions of (\ref{sysphi}) is
$$
\dot V(x,y) = -y^2 \phi(x,y).
$$
$\dot V(x,y) \leq 0$ out of the compact set $D_M$, hence every solution is bounded. Moreover, working as in the proof of LaSalle invariance principle (see \cite{V} for the invariance principle, \cite{S} for the details of such an argument), one can show that the positive limit set of every solution remaining in the complement of $D_M$ is contained in the set $\dot V(x,y) = 0$. Such a positive limit set is positively invariant, i.e. if it contains a point $(x^*,y^*)$, then it contains the whole positive semi-orbit starting at $(x^*,y^*)$. The set $\dot V(x,y) = 0$ is the union of the sets $y=0$ and $\phi=0$. By hypothesis, such a set does not contain any positive semi-orbit, hence every orbit eventually meets the set $D_M$. Since $\dot V(x,y) \leq 0$ for $x^2+y^2 \geq M^2$, every orbit definitely enters the disk $D_M$ and remains inside it.
\hfill$\clubsuit$
\begin{lemma} \label{bddness2} Let $\Omega = \R^2$. If there exists $M > 0$ such that $\phi(x,y) \geq 0$ for all $(x,y) \not\in D_M$, and a half-line
$ \{ (r\cos \theta^*,r \sin \theta ^*) : r > 0\}$, such that \\ $\phi(r\cos \theta^*,r \sin \theta ^*) > 0 $ for $r > M$, then every solution of (\ref{sysphi}) definitely enters the disk $D_M$ and remains inside it.
\end{lemma}
{\it Proof.}
The derivative of $V(x,y) $ along the solutions of (\ref{sysphi}) is
$$
\dot V(x,y) = -y^2 \phi(x,y).
$$
The solutions' boundedness comes as in lemma \ref{bddness}. On the set $\phi(x,y) = 0$ one has $\dot x=y$, $\dot y= -x$, hence every orbit $\gamma$ starting at a point of $\phi = 0$ is contained in a circle centered at $O$ until it meets a point where $\dot V \neq 0$. Every circle with radius greater than $M$ meets the half-line $ \{ (r\cos \theta^*,r \sin \theta ^*) : r > M\}$ at a point $(x_0,y_0)$.
If $\sin \theta ^* \neq 0$, then $\dot V(x_0,y_0) < 0$.
If $\sin \theta ^* = 0$, then $\phi(x,y) > 0$ in a neighbourhood of $(x_0,y_0)$, and $\dot y = x \neq 0$ implies that $\gamma$ contains a point $(x_1,y_1)$, close to $(x_0,y_0)$, such that $\dot V(x_1,y_1) < 0$. In both cases, $\gamma$ leaves the circle pointing towards the origin. Then we may apply lemma \ref{bddness}.
\hfill$\clubsuit$
As a particular case, we consider functions $\phi(x,y)$ obtained as sums of $y$-trinomials. Let us introduce the following definition for a $y$-trinomial $P(x,y) = \kappa(x) y^{2h+2r} + \tau(x) y^{h+2r} + \eta(x)y^{2r}$,
${\bf (T^{++})}$ there exists $\varepsilon > 0$, such that for all $x \in \R$, $|x | > \varepsilon$, one has $\tau(x)^2 - 4 \eta(x) \kappa(x) \leq 0$, and $ \kappa(x) \geq 0$ (or, equivalently, $\eta(x) \geq 0)$.
\begin{corollary} \label{corbdd} Let $\Omega = \R^2$. Assume $\sum_{j=1}^{J}f_{j}(x)y^{j-1} = \sum_{l=1}^{L} P_l(x,y) $, with $P_l(x,y) $ $y$-trinomial satisfying the condition $(T^{++})$, $l=1, \dots, L$. If, for $|x | \leq \varepsilon$ one has $\kappa_l(x) > 0$, $l=1, \dots, L$,
then there exists a disk $D_M$ such that every solution of (\ref{sysypol}) definitely enters $D_M$ and remains inside it.
\end{corollary}
{\it Proof.} Let us set
$Z_l (x) = 1+ \max \left\{ \frac{ |\tau_l(x)|}{|\kappa_l(x)|} , \frac{ |\eta_l(x)| }{|\kappa_l(x)|} \right \}$.
By Cauchy theorem about the polynomial roots, every $z$-root of $\kappa_l(x) z^2 + \tau_l(x) z + \eta_l(x)$ is contained in the $z$-interval $[-Z_l(x),Z_l(x)]$. Since $\kappa_l(x)$ is continuous and positive in $[-\varepsilon, \varepsilon]$, $Z_l(x)$ is a continuous function in $[-\varepsilon, \varepsilon]$.
Let us set $Z_l^{max}= \max \{ Z_l(x) : x \in [-\varepsilon,\varepsilon] \}$, and $\overline{Z} = \max \{ Z_l^{max} : l=1, \dots , L \}$. All the zeroes of the functions $\kappa_l(x) z^2 + \tau_l(x) z + \eta_l(x)$ are contained in the rectangle
$[-\varepsilon,\varepsilon] \times [-\overline{Z} , \overline{Z} ]$. In other words, every function $\kappa_l(x) z^2 + \tau_l(x) z + \eta_l(x)$ is positive for $x \in [-\varepsilon,\varepsilon] $ and $z\not \in [-\overline{Z} , \overline{Z} ]$.
As a consequence, there exists $\overline{Y} > 0$ such that every trinomial $P_l(x,y)$ is positive for $x \in [-\varepsilon,\varepsilon] $ and $y \not \in [-\overline{Y},\overline{Y} ]$.
Moreover, by $(T^{++})$, for all $x \not \in [-\varepsilon,\varepsilon]$, every $y$-trinomial is non-negative.
Then $\phi(x,y) = \sum_{j=1}^{J}f_{j}(x)y^{j-1} = \sum_{l=1}^{L} P_l(x,y) \geq 0 $ out of $[-\varepsilon,\varepsilon] \times [-\overline{Y} , \overline{Y} ]$. \\
\indent In order to check the non-positive-invariance of the set $\dot V(x,y) = 0$ it is sufficient to consider that $\phi(0,y) > 0$, for $|y| > \overline{Y}$, and apply the lemma \ref{bddness2}.
\hfill$\clubsuit$
In next corollary we consider again the system (\ref{sysCRV}).
\begin{corollary} \label{corCRV} Let $\Omega = \R^2$.
Assume $f_{2k}(x) = 0 $, $xf_{2k+1}'(x) + 2k f_{2k+1}(x) \geq 0$ for $x \in \R$, $k=1, \dots, N$.
Assume additionally $ f_{1}(x) \geq 0$ for $x \in \R \setminus [-\varepsilon,\varepsilon]$, for some $\varepsilon > 0 $, and that there exists $1 \leq \overline{k} \leq N$ such that $f_{2 \overline {k}+1}(x) > 0$ on $[-\varepsilon,\varepsilon]$.
Then there exists a disk $D_M$ such that every solution of (\ref{sysCRV}) definitely enters $D_M$ and remains inside it.
\end{corollary}
{\it Proof.} By remark \ref{segno}, for $k = 1, \dots, N$ one has $f_{2k+1}(x) \geq 0$, hence
$$
\phi(x,y) = f_1(x) + \sum_{k=1}^N f_{2k+1}(x) y ^{2k} \geq f_1(x) + f_{2 \overline {k}+1}(x) y ^{2 \overline {k}} .
$$
The function $\phi(x,y) $ is non-negative for $x\not \in (-\varepsilon,\varepsilon)$. Moreover, working as in corollary \ref{corbdd} , one can prove the existence of $\overline{Y} > 0$ such that $\phi(x,y) > 0$ for $x \in [-\varepsilon,\varepsilon] $ andl $y \not \in [-\overline{Y},\overline{Y} ]$. Hence we may apply the lemma \ref{bddness2}, since $\phi(0,y) > 0$ for $|y| > \overline{Y} $.
\hfill$\clubsuit$
\begin{remark} Our hypotheses are stronger than those of section 4.2 in \cite{CRV}, since we ask $f_1(x)$ to change sign both for positive and for negative $x$. Actually, the argument in section 4.2 of \cite{CRV} is incorrect. In fact, choosing $V(x,y) = x^2+y^2$, one has
$$
\dot V(x,y) = -2y^2 \left( f_1(x) + \sum_{k=1}^N f_{2k+1}(x) y ^{2k} \right) .
$$
The geometrical argument in section 4.2 of \cite{CRV} is equivalent to asking that for $r$ large enough, $\dot V(x,y) \leq 0$. This can be false if $f_1(x)$ does not assume positive values for large $|x|$. In fact, if $f_1(x) < 0$ for large $|x|$ (see Remark 1.4 in \cite{CRV}), the curve $\dot V(x,y) = 0$ may consist of different, unbounded branches, which do not bound a positively invariant topological annulus, as needed by Poincar\'e-Bendixson theorem. This is what occurs choosing
$$
f_1(x) = - e^{-x^2} < 0, \qquad f_3(x) =1 - e^{-x^2} > 0.
$$
In fact, in such a case the curve $\dot V (x,y) = 0$ consists of four unbounded connected components, separating the region $\dot V (x,y) > 0$ from the four connected regions where $\dot V (x,y) < 0$. \\
\end{remark}
Finally, we prove a result of existence and uniqueness for the system \ref{sysphi}.
\begin{theorem}\label{teorema2} If the hypotheses of theorem \ref{teorema} and lemma \ref{bddness} hold, and $\phi(0,0) < 0$, then the system (\ref{sysphi}) has exactly one limit cycle, which is hyperbolic and attracts every non-constant solution.
\end{theorem}
{\it Proof.} As in theorem 2 of \cite{S}
\hfill$\clubsuit$
As a consequence, we may apply the corollaries \ref{corollarioT} and \ref{corbdd} in order to prove the limit cycle's existence and uniqueness for a special class of systems.
\begin{corollary} Assume $\sum_{j=1}^{J}f_{j}(x)y^{j-1}$ to be the sum of $y$-trinomials satisfying the conditions $(T^+)$, $(T^{++})$ and (Seq).
If $\kappa_l(x) > 0$, for $|x | \leq \varepsilon$, $l=1, \dots, L$,
then the system (\ref{sysphi}) has exactly one limit cycle, which is hyperbolic and attracts every non-constant solution.
\end{corollary}
{\it Proof.} An immediate consequence of corollaries \ref{corollarioT} and \ref{corbdd}, and theorem \ref{teorema2}.
\hfill$\clubsuit$
Finally, we have a similar result for the system (\ref{sysCRV}).
\begin{corollary} \label{ultimo}Assume $f_{2k}(x) = 0 $, $xf_{2k+1}'(x) + 2k f_{2k+1}(x) \geq 0$ for $x \in \R$, $k=1, \dots, N$.
Assume additionally $ f_{1}(0) < 0$, $ f_{1}(x) > 0$ for $x \in \R \setminus [-\varepsilon,\varepsilon]$, for some $\varepsilon > 0 $, and that there exists $1 \leq \overline{k} \leq N$ such that $f_{2 \overline {k}+1}(x) > 0$ on $[-\varepsilon,\varepsilon]$.
Then the system (\ref{sysphi}) has exactly one limit cycle, which is hyperbolic and attracts every non-constant solution.
\end{corollary}
{\it Proof.} A straightforward consequence of theorem \ref{teorema2} and corollary \ref{corCRV}.
\hfill$\clubsuit$
\section{Non-linear $g(x)$ }
This section is similar to section 3 in \cite{S}. It contains some modifications due to the weaker hypotheses assumed in the previous section on $\phi(x,y)$. For the reader's convenience, we recall some parts of section 3 in \cite{S}. Then we state the main result under the new, weaker conditions, and deduce the corresponding conditions on the system (\ref{sysphig}).
Let us consider the equation
\begin{equation}\label{equaphig}
\ddot x + \dot x \Phi(x,\dot x) + g(x) = 0 .
\end{equation}
We assume that $ xg(x) > 0 {\rm\ for \ } x \neq 0$, $g \in C^1((a,b),\R)$, with $a < 0 < b$, $g'(0) > 0$. We also admit the limit case $a = -\infty$ and/or $b = +\infty$.
The main tool is the so-called Conti-Filippov transformation, which acts on the equivalent system
\begin{equation}\label{sysphig}
\dot x = y, \qquad \dot y = - g(x) - y \Phi(x,y),
\end{equation}
in such a way to take the conservative part of the vector field into a linear one. Let us set $G(x) = \int_0^x g(s) ds$, and denote by $\sigma(x)$ the sign function, whose value is $-1$ for $x < 0$, $0$ at $0$, $1$ for $x > 0$. Let us define the function $\alpha : \R \rightarrow \R$ as follows:
$$
\alpha(x) = \sigma(x) \sqrt{2G(x)}.
$$
Then Conti-Filippov transformation is the following one,
\begin{equation}\label{transf}
(u,v) = \Lambda(x,y) = (\alpha(x),y).
\end{equation}
$\Lambda$ transforms $(a,b)$ onto $(u^-,u^+) = \left( -\lim_{x\rightarrow a^+ } \sqrt{2G(x)},\lim_{x\rightarrow b^- }\sqrt{2G(x)} \right)$. In general, even if $(a,b) = \R$, $\Lambda$ does not transform $\R$ onto $\R$. This occurs if and only if $\lim_{x\rightarrow -\infty } \sqrt{2G(x)} = \lim_{x\rightarrow -\infty } \sqrt{2G(x)} = +\infty$.\\
\indent Since we assume $g \in C^1((a,b),\R)$, we have $\alpha \in C^1((a,b),\R)$. The function $u = \alpha (x)$ is invertible, due to the condition $xg(x) > 0$. Let us call $x =\beta(u)$ its inverse. The condition $g'(0) > 0$ guarantees the differentiability of $\beta(u)$ at $O$. For $x \neq 0$, or, equivalently, for $u \neq 0$, one has,
\begin{equation}\label{derivate}
\alpha'(x) = \frac{\sigma(x) g(x)}{\sqrt{2G(x)}}, \qquad \beta'(u) = \frac{1}{\alpha'(\beta(u))} = \frac{\sigma(x)\sqrt{2G(x)}}{g(x)} = \frac{u}{g(\beta(u))}.
\end{equation}
For $x=u=0$ one has,
$$
\alpha'(0) = \sqrt{g'(0)}, \qquad \beta'(0) = \sqrt{\frac{1}{g'(0)}} .
$$
Finally,
\begin{equation}\label{root}
\lim_{u \rightarrow 0} \frac{g(\beta(u))}{u} = \sqrt{g'(0)} > 0.
\end{equation}
In next theorem we consider only the condition corresponding to $x\phi_x + y\phi_y \geq 0$. The alternative case, $x\phi_x + y\phi_y \leq 0$, can be treated similarly. Let us set
$$
\Psi = \frac {\sigma \sqrt{2G} }{g} \left[ 2G \frac{ \Phi_x g - \Phi g' }{g ^2} + \Phi \right]
+ y \Phi_y.
$$
\bigskip
\begin{theorem}\label{teorema-n} Assume $g \in C^1((a,b),\R)$, with $g'(0) > 0$ and $xg(x) > 0$ for $x\neq 0$. If $\Psi(x,y) \geq 0$ and the set $\Psi(x,y) = 0$ does not contain any non-trivial orbit, then (\ref{sysphig}) has at most one limit cycle in the region $(a,b) \times \R$, which is hyperbolic.
\end{theorem}
{\it Proof.} As in the proof of theorem 3 in \cite{S}, one transforms the system (\ref{sysphig}) into a system of the form (\ref{sysphi}), by means of the Conti-Filippov transformation. \\
\indent For $u \neq 0$, the transformed system has the form
\begin{equation}\label{1}
\dot u = v \frac{g(\beta(u))}{u}, \qquad \dot v = - g(\beta(u)) - v \Phi(\beta(u),v) ,
\end{equation}
We may multiply the system (\ref{1}) by $ \frac{u}{g(\beta(u))} $, obtaining a new system having the same orbits as (\ref{1}),
\begin{equation} \label{CFphi}
\dot u = v , \qquad \dot v = - u - v \frac{u \Phi(\beta(u),v)}{g(\beta(u))} = - u - v \phi(u,v),
\end{equation}
By (\ref{root}), the new system is regular also for $x=0$. As proved in \cite{S}, one has $\Psi(\beta(u),v) = u\phi_u(u,v) + v\phi_v(u,v) $, hence the hypothesis $\Psi(x,y) \geq 0$ implies the star-shapedness condition on $\phi(u,v)$. Moreover, the set $\Psi(x,y) = 0$ is transformed into the set $u\phi_u + v\phi_v = 0$.
Then one applies theorem \ref{teorema} to complete the proof.
\hfill$\clubsuit$
In order to apply the above theorem to $y$-polynomial equations, as in the previous section's corollaries, it is convenient to write the $y$-coefficients form, after the application of Conti-Filippov transformation. If
$$
\Phi(x,y) = \sum_{j=1}^{J}f_{j}(x)y^{j-1}
$$
then
$$
\phi(u,v) = \frac{u \Phi(\beta(u),v)}{g(\beta(u))} = \sum_{j=1}^{J} \frac {u f_{j}(\beta(u))} {g(\beta(u))} v^{j-1} =
\sum_{j=1}^{J}\tilde{ f}_{j}(u) v^{j-1} ,
$$
where $ \tilde{ f}_{j}(u) = \frac {u f_{j}(\beta(u))} {g(\beta(u))}$.
If $\Phi(x,y)$ is the sum of $y$-trinomials $\kappa_l(x) y^{2h+2r} + \tau_l(x) y^{h+2r} + \eta_l(x)y^{2r}$, then $\phi(u,v)$ is the sum of $v$-trinomials, whose coefficients $\tilde{\kappa_l}(u) v^{2h+2r} + \tilde{\tau_l}(u) v^{h+2r} + \tilde{\eta_l}(u)v^{2r}$are obtained from the original ones in a similar way,
$$
\tilde{\kappa_l}(u ) = \frac {u \kappa_l(\beta(u))} {g(\beta(u))} , \qquad
\tilde{\tau_l}(u ) = \frac {u \tau_l(\beta(u))} {g(\beta(u))} , \qquad
\tilde{\eta_l}(u ) = \frac {u \eta_l(\beta(u))} {g(\beta(u))} .
$$
Writing the condition $(T^+)$ for such polynomials generates quite cumbersome expressions. In next corollary we only write the conditions for the simplest case, that of an odd $y$-polinomial. \\
\indent Given a function $f: I \rightarrow \R$, consider the condition
${\bf (H_g^{j})}$ \quad $\forall x \neq 0$: \ $x \left[ j f(x)g(x) + 2G(x) \left( \frac{f'(x)g(x) - f(x)g(x)'}{g(x)} \right) \right] \geq 0. $
If the above inequality holds strictly for $x \neq 0$, i.e. if the left-hand side does not vanish for $x\neq 0$, we say that the strict condition ($H_g^{j}$) holds at $x$.
\begin{corollary}\label{corCRV-n} Assume $g \in C^1((a,b),\R)$, with $g(x)= x + o(x)$ and $xg(x) > 0$ for $x\neq 0$. Let $f_{2k}\equiv 0$ and $f_{2k+1}(x)$ satisfy the condition ($H_g^{2k+1}$) for $k=0, \dots, N$. If there exists a sequence $x_m$ converging to $0$, such that for every $m$ there exists $k(m)$ satisying the strict ($H_g^{2k+1}$) condition at $x_m$, then (\ref{sysypol}) has at most one limit cycle in the region $(a,b) \times \R$, which is hyperbolic.
\end{corollary}
{\it Proof.}
As observed above, the transformed system has the form \begin{equation}\label{sysurep}
\dot u = v , \qquad \dot v = - u - v\sum_{k=0}^{N} \frac{ u f_{2k+1}(\beta(u))}{g(\beta(u))} v^{2k} .
\end{equation}
Applying corollary \ref{corollarioCRV} to such a system requires to check, for $k = 0, \dots, N$, the condition
$$
u\left( \frac{ u f_{2k+1}(\beta(u))}{g(\beta(u))} \right)' + 2k \frac{ u f_{2k+1}(\beta(u))}{g(\beta(u))} \geq 0.
$$
Performing standard computations and recalling that $u = \sigma(x) \sqrt{2G(x)}$, one proves that, for $u\neq 0$, the inequality
$$
u\left( \frac{ u f(\beta(u))}{g(\beta(u))} \right)' + 2k \frac{ u f(\beta(u))}{g(\beta(u))} \geq 0
$$
is equivalent to
$$
\sigma(x) \sqrt{G(x)} \left[ (2k+1) f(x)g(x) + 2G(x) \left( \frac{f'(x)g(x) - f(x)g(x)'}{g(x)} \right) \right] \geq 0,
$$
for $x \neq 0$.
The sign of $\sigma(x) \sqrt{G(x)}$ is the same as that of $x$, hence the last inequality is equivalent to ($H_g^{2k+1}$). Then one applies corollary \ref{corollarioCRV} to complete the proof.
\hfill$\clubsuit$
Now we prove the analogous of lemma \ref{bddness}. Let us set
$$
E(x,y) = G(x) + \frac {y^2}{2}
$$
For $r > 0$, we set $\Delta_r = \{ (x,y) : 2 E(x,y) < r^2 \}$. As in the previous section, we set $Z_\Phi = \{ (x,y) : \Phi(x,y) =0 \}$.
\begin{lemma}\label{bddnessg} Assume $g \in C^1((a,b),\R)$, with $\int_0^a g(x) dx = \int_0^b g(x) dx = +\infty$.
If there exists $M > 0$ such that $\Phi(x,y) \geq 0$ for all $(x,y) \not\in \Delta_M$, and the set $Z_\Phi \setminus \Delta_M $ does not contain any non-trivial positive semi-orbit of (\ref{sysphig}), then every solution of (\ref{sysphi}) definitely enters the set $\Delta_M$ and remains inside it.
\end{lemma}
{\it Proof.} Under the integral conditions in the hypotheses, the Conti-Filippov transformation is a global diffeomorpism of the region $(a,b) \times \R$ onto the plane, because
$$
\lim_{x\rightarrow a^+ } \sqrt{2G(x)} = \lim_{x\rightarrow b^- }\sqrt{2G(x)} = +\infty.
$$
The level sets of $E(x,y)$ are compact subsets of $[a,b] \times \R$, i. e. they have positive distance from its boundary.
The sets $ \Delta_r$ are taken into the sets $D_r$.
Then one can apply the lemma \ref{bddness} to the system (\ref{CFphi}). In fact, the derivative of the Liapunov function $V(u,v) = \frac 12 (u^2+v^2)$ along the solutions of (\ref{sysurep}) is just
$$
\dot V (u,v) = - v^2 \frac{u\Phi(\beta(u),v)} {g(\beta(u))} .
$$
The function $ \frac{u} {g(\beta(u))}$ is positive for $u \neq 0$, hence the hypotheses of lemma \ref{bddness} are satisfied by the system (\ref{sysurep}). As a consequence, the conclusions of lemma \ref{bddness} hold for the system (\ref{sysurep}), and applying the inverse transformation $\Lambda^{-1}$ one obtains the thesis.
\hfill$\clubsuit$
\begin{theorem}\label{teorema2-n} Let $g\in C^1((a,b),\R)$. If the hypotheses of theorem \ref{teorema-n} and lemma \ref{bddnessg} hold on $\R$, and $\Phi(0,0) < 0$, then the system (\ref{sysphig}) has exactly one limit cycle in $(a,b) \times \R$, which is hyperbolic and attracts every non-constant solution.
\end{theorem}
{\it Proof.} As the proof of theorem \ref{teorema2}, replacing the Liapunov function $V(x,y)$ with the Liapunov function $E(x,y)$.
\hfill$\clubsuit$
Finally, we prove a result analogous to corollary \ref{ultimo}.
\begin{corollary} Assume $g \in C^1(\R,\R)$, $g'(0) > 0$, $xg(x) > 0$ for $x\neq 0$, $\int_0^{\pm \infty} g(x) dx = +\infty$. Let $f_{2k}\equiv 0$ and $f_{2k+1}(x)$ satisfy the condition ($H_g^{2k+1}$) for $k=0, \dots, N$. Assume there exists a sequence $x_m$ converging to $0$, such that for every $m$ there exists $k(m)$ satisying the strict ($H_g^{2k+1}$) condition at $x_m$.
Assume additionally $ f_{1}(0) < 0$, $ f_{1}(x) > 0$ for $x \in \R \setminus [-\varepsilon,\varepsilon]$, for some $\varepsilon > 0 $, and that there exists $1 \leq \overline{k} \leq N$ such that $f_{2 \overline {k}+1}(x) > 0$ on $[-\varepsilon,\varepsilon]$.
Then the system (\ref{sysypol}) has exactly one limit cycle, which attracts every non-constant solution.
\end{corollary}
{\it Proof.}
The sign of $f_j(x)$ is the same as that of $\tilde{f}_j(x)$, since $\frac {u} {\beta(u)} >0 $ for $u \neq 0$.
Then the statement is a straightforward consequence of theorem \ref{teorema2-n} and corollary \ref{corCRV-n}.
\hfill$\clubsuit$
{\bf \large Acknowledgements}
This paper has been partially supported by the GNAMPA 2009 project \lq\lq Studio delle traiettorie di equazioni differenziali ordinarie\rq\rq.
| 206,290
|
Talks.
"What we see now is that most technology is being defined by this collaborative development," Zemlin said.
Jaguar Land Rover
The Automotive Grade Linux work group is a prime example of how this new collaborative model is taking over an industry that hasn't traditionally been open to third-party developers. AGL today released the IVI & remote vehicle interaction demonstration, a downloadable open source image for creating an in-vehicle-infotainment system complete with the CAN stack and an HTML5 application framework and sample user interface.
The goal is to create an open source IVI platform that meets customer expectations to have the experience from their consumer electronics devices in their vehicles such as voice control, constant connectivity and media management, said Matt Jones, a Senior Technical Specialist for infotainment systems at Jaguar Land Rover and the Vice President of the non-profit GENIVI Alliance, an automotive industry effort to drive adoption of an IVI open-source development platform.
"We're heavily involved with AGL to enable open source and Linux within automotive as a whole and focus on making it easier for developers with reference hardware and software platforms," Jones said. Such technology "has been available in lots of vehicles but nobody has given it away before."
The group is now looking for collaborators to help integrate navigation into the UI, as well as BlueZ for telephony and has announced a developer contest. Developers can compete for the chance to work with the AGL and Jaguar Land Rover by entering work in three categories: best user experience, best visual appearance, and best new concept or additional feature. The contest runs April 15 - May 17 and winners will be announced at the Automotive Linux Summit in Tokyo at the end of May.
"I want what we create to be available on every vehicle," Jones said. "It should be easy to translate that software across platforms. We just need to get the features out there."
Samsung
While the automotive industry is just starting to dive into open source, Samsung Electronics has established itself as a significant contributor to the Linux kernel and the open source community. The consumer electronics giant began using open source about 10 years ago, starting with embedded Linux in some prototype devices, said Sang-bum Suh, Vice President of the software platform team in the Software R&D Center at Samsung. That soon expanded to flat panel TVs when the company switched from a RTOS to Linux. And its use of Linux now extends to smartphones, camcorders and cameras.
Last year Samsung's Galaxy Android smartphone sold more than 200 million units and 57 million digital TV units, "all based upon linux and comprised of open source components," Suh said.
"I believe that open source components have contributed to Samsung's business success very much." he said. So much so that Samsung has expanded its contributions to Linux and open source, becoming one of the top 10 contributors to the Linux kernel in 2012. It now employs more than 20,000 software developers whose work is based in large part on open source components, Suh said. And they're hiring.
"We believe we have been successful in the past on hardware components," Suh said. "In the future we'd like to add software
capabilities to that hardware success."
Adapteva
The final morning keynote brought rounds of applause from attendees as Adapteva CEO Andreas Olofsson unveiled two Parallella boards - the first off the assembly line since the company's Kickstarter campaign six months ago. (For more background on the Parallela and Adapteva's vision, see our Q&A and a recap of our live chat with Andreas.) Holding one board in each hand, Olofsson discussed the immense challenges Adapteva and the rest of the computer industry faces in finding ways to improve performance and minimize energy consumption.
Adapteva is "possibly the world's smalled semiconductor company," he said, trying to solve some of the biggest problems in computing.
Chip design trends will shape the future of computing, addressing power conumption, a memory bottleneck, wiring, thermal density, yield issues and more. Parallel computing is the answer, he said.
"Today we're at a high level of abstraction but parallel programming isn't there yet," Olofsson said. "The challenge is how to make parallel programming as productive as Java or Python is today... running on any type of hardware." A beginning programmer should be able to do parallel programming as easily as someone with 20 years of experience, he said.
The Parallella, a dual-core processor that runs on Linux, is Adapteva's attempt to create a market for parallel programming by making it cheap and easy.
"One of the key things in launching this project was (realizing) the only way to make it work is to make it completely open... which can be incredibly scary to a hardware engineer," Olofsson said.
The past six months the three-developer company didn't know if it could really deliver on the project, he said. Now they see it working. After more testing and production, they'll ship 6,300 boards to their kickstarter backers and will give away 100 free kits to universities.
"The good news is we have boards working," Olofsson said. "We're running a little bit late but we're going to ship them this summer."
The Linux Foundation's Collaboration Summit is taking place this week in San Francisco, from April 15-17. Catch the rest of our day-one keynotes for free via our live video stream on the Events website. And check back into Linux.com later today and tomorrow for more coverage and photo slideshows of this week's event.
| 244,377
|
"Big".
Volunteers — Hospice Visions Inc. is looking for volunteer handy men and women for light home modifications, Light Touch Massage therapists, hair dressers, volunteers for meal assistance, and to visit.
With more than 80,000 books, including first editions, and a large collection of art books from a private collection. titled Mushrooms and Toadstools in New Zealand. "I love country music and like.
Zarzecki hopes the whimsical butterflies will change perceptions about hospice and palliative. offer spiritual counseling to volunteers who run errands. The program also includes expressive.
The local dental surgeon is a deacon and volunteer at Great Lakes Hospice. Reservations, which are $25, are mandatory. Call 864-2451 to learn more. POSTSCRIPT: Develop an ear for music and an eye for.
He proudly served his country in the United States Army as a member of the Army Rangers Special forces (Wolf Pack) from 1967-1969. He was a proud family man and a devoted husband, father, grandfather, great grandfather, brother and friend to many.
. 21 to recognize patients served by Southeastern Hospice and Southeastern Hospice House during 2018. Chaplain Bonnie Reedy and Dr. Godfrey Onime will speak; special music by Betty Lawson and Callie.
He was most proud of all three CDs, which were filled with original music and remakes of some of his favorite artists and. High Peaks Hospice nurses and his new family from The Haynes House of Hope.
Where Native Black New Orleanians Listen To Live Music Jan 06, 2015 · Make It Right House With "Damaged" Roof in New Orleans’ Lower Ninth Ward Just hours before workers removed a statue of Confederate general Robert E. Lee — the fourth Confederate monument to be dismantled in New Orleans in recent… Death, Sex & Money is a podcast about the big questions and.
Morton, of Eastern Passage, N.S., has been involved with the hospice work since 2001. She served on the board of the Hospice Society of Greater Halifax for nine years and now volunteers. "Whether.
He was most proud of all three CDs, which were filled with original music and remakes of some of his favorite artists and. High Peaks Hospice nurses and his new family from The Haynes House of Hope.
There will be live music and a raffle for a surf board. 15 to 62 miles through southwest Michigan and shop at farms, vineyards and art studios along the way Sept. 15. Volunteers will deliver your.
We would like to show you a description here but the site won’t allow us.
"I studied my BA in Music and History of Art at University College Dublin and in 2012. at the Royal Albert Hall and Westminster Cathedral. I am passionate about volunteering with charitable.
Mollenhauer of Evanston passed away in April from pancreatic cancer, her husband and lifelong companion Art Mollenhauer, made a generous multi-year donation to both Hospice services and the Music.
Thats So Raven Chill Grill Singing Competition Episode So the progressives will likely find someone to run against. and I’m looking forward to seeing them on the field of competition over the next two years. 6/30 Finally, a very lovely day in Chitown. 64 degrees and sunshine. Humidity is lower than it has been in a week or so. Enjoy the day Looks
YORK — Nikki Hopewill, director of The Hope Project at Hospice of. as well as trained volunteers. Age appropriate groupings encourage children to explore and respond to their evolving feelings. large raffle.
Jessica Mongerio, choral and musical theater. Appleton Museum of Art. Fine Arts For Ocala, the longtime presenters of the Ocala Arts Festival and Symphony Under the Stars, won the Vision Award.
Become a Memory Care Volunteer with McLean, and help support those in our. Services for clients may include one-to-one social interaction, sharing a passion for hobby, music or art therapy, writing.
Volunteers help staff events, read at storytime events at the Napa Farmers Market, and share their talents by hosting learning activities such as music, art or science. Get involved and advocate.
“Although the grant amount we are distributing is significant, the most exciting part is that the grant recipients have found innovative, state-of-the-art. Mercy Hospice Springfield: $1,194 to.
31st St. Animals in Music– 2 p.m. June 24, summer concert series. register at [email protected] or by calling 402-476-7550 Grief group and hospice volunteer training — "Loss of a Loved One.
Sep 28, 2018 · Marty Balin, founder of the Jefferson Airplane, dies at 76. Marty Balin was a patron of the 1960s "San Francisco Sound" both as founder and lead singer of the Jefferson Airplane.
ALLAN, Donald Sutherland – Peacefully at Meighen Manor, surrounded by family, on February 7, 2005, in his 86th year. Graduate of U of T, Mechanical Engineering, 1941. Served on HMS VICTORIOUS / HMCS MEON, WWII. Longtime employee of EMCO and member of the RCYC.
Cook, Horace Q. (Hod) Horace Q. (Hod) Cook passed away March 9, 2019 in Salem, Oregon at the age of 96 years. Born September 21, 1922 in Des Moines, IA and raised in Williamsburg, Iowa, Hod was an Eagle Scout, a talented athlete, and an accomplished vocalist.
The local dental surgeon is a deacon and volunteer at Great Lakes Hospice. Reservations, which are $25, are mandatory. Call 864-2451 to learn more. POSTSCRIPT: Develop an ear for music and an eye for.
Some hospice agencies also offer music therapy, art therapy or pet therapy services, often through volunteer organizations, and are usually free of charge to their hospice patients. However, if.
The Commonwealth Club of California is the nation’s oldest and largest public affairs forum. As a non-partisan forum, The Club brings to the public airwaves diverse viewpoints on important topics.
With more than 80,000 books, including first editions, and a large collection of art books from a private collection. titled Mushrooms and Toadstools in New Zealand. "I love country music and like.
See the list of all job available in ERI’s Salary Assessor database. Job Categories include Top Management Positions, Middle Management Positions, Supervisory Positions, Health Care Positions, Professional Positions, Information Tech Positions, Sales Positions, Technical Positions, Field/Shop Positions, Clerical Positions
White Nights Soundtrack Classical Music Piece For Dance How White Kids Stole House Music from Black Aunties It’s time for sonic reparations 1. Michelle Obama helps open the show. The GRAMMYs are always full of surprises, but the crowd at the Staples Center was particularly taken aback when none other than Michelle Obama took the stage last night. The former First Lady joined
Artall Signage offers a complete service for all types of signage including shop signage, shop window graphics, vehicle graphics, van wraps, safety signs, banners and more.
| 217,889
|
TITLE: The number of solutions to $\frac{1}x+\frac{1}y+\frac{1}z=\frac{3}n,x,y,z\in\mathbb N$
QUESTION [5 upvotes]: Denote
$$g(n)=\{\{x,y,z\}\mid \frac{1}x+\frac{1}y+\frac{1}z=\frac{3}n,x,y,z\in\mathbb N\},$$
$$h(n)=\{\{x,y,z\}\mid \frac{1}x+\frac{1}y+\frac{1}z=\frac{3}n,1\leq x\leq y\leq z,x,y,z\in\mathbb N\},$$
let $f(n)=|g(n)|$ be the number of members of $g(n)$.
For example, $h(3)=\{\{2,3,6\},\{2,4,4\},\{3,3,3\}\},f(3)=6+3+1=10.$
Since $\{n,n,n\}$ is a solution to $\frac{1}x+\frac{1}y+\frac{1}z=\frac{3}n$, it's easy to see that $f(k)\equiv 1\pmod 3,\forall k\in \mathbb N.$
Question: I find that
$$f(3k)\equiv 0,f(4k+2)\equiv 0,f(6k\pm1)\equiv1 \pmod 2,\forall k\in \mathbb N.$$
I wonder how to prove them?
Edit: I find that $f(n)$ has the same parity to the number of solutions to $\frac{1}x+\frac{2}y=\frac{3}n,$ I think I have got it now.
REPLY [3 votes]: Since I already know how to prove them, I write a proof here now.
It's easy to see that in $h(n)$,
(1)if $x,y,z$ are distinct, then $x,y,z$ add $6$ to $f(n)$,
(2)if just two of them are equal, add $3$ to $f(n)$,
(3)if $x=y=z$, then they add $1$ to $f(n)$.
Since $6$ is even, case (1) didn't change the parity of $f(n)$. Hence $f(n)$ has the same parity of the number of solutions to $\frac{1}x+\frac{2}y=\frac{3}n.$ This is $(3x-n)(3y-2n)=2n^2,$ let $r(n)$ be the number of solutions to this equation.
If $n=3m,$ then $(x-m)(y-2m)=2m^2,$ hence $f(n)\equiv r(n)=d(2m^2)\equiv 0\pmod 2.$
If $n=6m+1$, then $3x-n=a,3y-2n=b,$
$$f(n)\equiv r(n)=\sum_{\substack{ab=2n^2\\a\equiv -n\equiv 2\pmod 3\\b\equiv -2n\equiv 1\pmod 3}}1=\frac{1}2d(2n^2)=d(n^2)\equiv 1\pmod 2.$$
The same to $n=6m-1.$
If $n=4m+2,$ then if $3\mid n$, we get $2\mid f(n),$ too. If $3\not \mid n$, then $2n^2\equiv -1\pmod3,f(n)\equiv r(n)=\dfrac{1}2d(2n^2)=2d((2m+1)^2)\equiv 0\pmod 2.$
Now we get a little more:
$$f(n) \equiv
\begin{cases}
\dfrac{1}2d(2n^2), & 3\not\mid n \\
0, & 3\mid n \\
\end{cases} \pmod 2 $$
| 125,730
|
TITLE: How to determine whether a polynomial has roots modulo a prime?
QUESTION [2 upvotes]: How to determine if a polynomial has roots modulo a prime?
I was attempting to find solutions to this polynomial $x^3+x^2-4\equiv 0$ mod($7$)
The only method I currently know to determine solutions is to try every element in the complete residue system. There are none.
However given $x^3+x^2-4\equiv 0$ mod($10007$), ($10007$ is a prime)
I would have no idea how to find roots of this polynomial or if they even exist.
REPLY [4 votes]: One way to count the solutions for $f(x)\equiv0\pmod p$ (where $p$ is prime) is to compute the gcd of $f(x)$ and $x^p-x$ over the field $\Bbb F_p$ of $p$
elements. This gcd is $g(x)=\prod_{a\in R}(x-a)$ where $R$ is the set of roots of $f$ in $\Bbb F_p$.
This may seem daunting if $p$ is large. But as long as $f$ has small degree
one can start by reducing $x^p-x$ modulo $f(x)$ over $\Bbb F_p$. Doing the
$x^p$ modulo $f(x)$ is just like computing $a^k$ modulo $m$ for large $m$
and can be done by the binary powering algorithm.
| 66,238
|
\begin{document}
\title[Inhomogeneous Diophantine approximation]{$S$-arithmetic Inhomogeneous Diophantine approximation on manifolds}
\begin{abstract}
We investigate $S$-arithmetic inhomogeneous Khintchine type theorems in the dual setting for nondegenerate manifolds. We prove the convergence case of the theorem, including, in particular, the $S$-arithmetic inhomogeneous counterpart of the Baker-Sprind\v{z}uk conjectures. The divergence case is proved for $\bbQ_p$ but in the more general context of Hausdorff measures. This answers a question posed by Badziahin, Beresnevich and Velani \cite{BaBeVe}.
\end{abstract}
\author{Shreyasi Datta}
\author{Anish Ghosh}
\address{School of Mathematics, Tata Institute of Fundamental Research, Mumbai, 400005, India}
\email{shreya@math.tifr.res.in, ghosh@math.tifr.res.in}
\thanks{Ghosh acknowledges support of a UGC grant and a CEFIPRA grant.}
\maketitle
\tableofcontents
\section{Introduction}
In this paper we are concerned with metric Diophantine approximation on nondegenerate manifolds in the $p$-adic, or more generally $S$-arithmetic setting for a finite set of primes $S$. To motivate our results we recall Khintchine's theorem, a basic result in metric Diophantine approximation. Let $\Psi : \bbR^n \to \bbR_{+} $ be a function satisfying
\begin{equation}\label{defmultapp}
\Psi(a_1, \dots, a_n) \geq \Psi(b_1, \dots, b_n) \text{ if } |a_i| \leq |b_i| \text{ for all } i = 1,\dots, n.
\end{equation}
Such a function is referred to as a \emph{multivariable approximating function}. Given such a function, define $\cW_{n}(\Psi)$ to be the set of $\bx \in \bbR^n$ for which there exist infinitely many $\ba \in \bbZ^{n}$ such that
\begin{equation}\label{preKG}
|a_0 + \ba \cdot \bx| < \Psi(\ba)
\end{equation}
\noindent for some $a_0 \in \bbZ$. When $\Psi(\ba) = \psi(\|\ba\|)$ for a non-increasing function $\psi$, we write $\cW_{n}(\psi)$ for $\cW_{n}(\Psi)$. Khintchine's Theorem (\cite{Khintchine}, \cite{Groshev}) gives a characterization of the measure of $\cW_{n}(\psi)$ in terms of $\psi$:
\begin{theorem}\label{KG}
\begin{equation}
|\cW_{n}(\psi)| = \left\{
\begin{array}{rl}
0 & \text{if } \sum_{k=1}^{\infty} k^{n-1} \psi(k) < \infty\\
\\
\text{ full } & \text{if } \sum_{k=1}^{\infty}k^{n-1} \psi(k) = \infty.
\end{array} \right.
\end{equation}
\end{theorem}\
Here, $\|~\|$ denotes the supremum norm of a vector and $|~|$ denotes the absolute value of a real number as well as the Lebesgue measure of a measurable subset of $\bbR^n$; the context will make the use clear. The kind of approximation considered above is called ``dual" approximation in the literature as opposed to the setting of simultaneous Diophantine approximation. In this paper, we will only consider dual approximation. Given an approximation function, one can consider the corresponding $S$-arithmetic question as follows, we follow the notation of Kleinbock and Tomanov \cite{KT}. Given a finite set of primes $S$ of cardinality $l$ we set $\Q_S := \prod_{\nu \in S}\Q_\nu$ and denote by $|~|_S$ the $S$-adic absolute value, $|\bx| = \max_{v \in S }|x^{(v)}|_v$. For $\ba = (a_1, \dots, a_n) \in \bbZ^n$ and $a_0 \in \bbZ$ we set
$$ \widetilde{\ba} := (a_0, a_1, \dots, a_n).$$
We say that $\by \in \Q^{n}_S$ is $\Psi$-approximable ($\by \in \cW_{n}(S, \Psi)$) if there are infinitely many solutions $\ba \in \Z^n$ to
\begin{equation}
|a_0 + \ba\cdot \by|_{S}^{l} \leq \left\{
\begin{array}{rl}
\Psi(\widetilde{\ba}) & \text{ if } \infty \notin S\\
\\
\Psi(\ba) & \text{ if } \infty \in S.
\end{array} \right.
\end{equation}
\noindent We fix Haar measure on $\Q_p$, normalized to give $\Z_p$ measure $1$ and denote the product measure on $\Q_S$ by $|~|_S$. Then, the following analogue of Khintchine's theorem can be proved. Namely,
\begin{theorem}\label{S-KG}
$\cW_{n}(S, \psi)$ has zero or full measure depending on the convergence or divergence of the series
\begin{equation}
\left\{
\begin{array}{rl}
\sum_{k=1}^{\infty} k^{n}\psi(k) & \text{if } \infty \notin S \\
\\
\sum_{k=1}^{\infty} k^{n-1} \psi(k) & \text{if } \infty \in S.
\end{array} \right.
\end{equation}
\end{theorem}
Indeed, the convergence case follows from the Borel-Cantelli lemma as usual and the divergence case can be proved using the methods in \cite{L}.
\subsection{Inhomogeneous approximation:}
Given a multivariable approximating function $\Psi$ and a function $\theta : \bbR^n \to \bbR$, we set $\cW^{\theta}_{n}(\Psi)$ to be the set of $\bx \in \bbR^n$ for which there exist infinitely many $\ba \in \mathbb{Z}^n\setminus \{\mathbf{0}\}$ such that
\begin{equation}\label{preKGinhom}
|a_0 + \ba \cdot \bx + \theta(\bx)| < \Psi(\ba)
\end{equation}
\noindent for some $a_0 \in \bbZ$. For $\psi$ as above, the set $\cW^{\theta}_{n}(\psi)$ is often referred to as the (dual) set of ``$(\psi, \theta)$-inhomogeneously approximable" vectors in $\R^n$. The following inhomogeneous version of Theorem~\ref{KG} is established in \cite{BaBeVe}. We denote by $C^n$ the set of $n$-times continuously differentiable functions.
\begin{theorem}\label{KGinhom} Let $\theta : \mathbb{R}^n \to \mathbb{R}$ be a $C^2$ function. Then
\begin{equation}
|\cW^{\theta}_{n}(\psi)| = \left\{
\begin{array}{rl}
0 & \text{if } \ \sum_{k=1}^{\infty} k^{n-1}\psi(k) < \infty\\
\\
\text{ full } & \text{if } \ \sum_{k=1}^{\infty} k^{n-1}\psi(k) = \infty.
\end{array} \right.
\end{equation}
\end{theorem}
\noindent We remark that the choice of $\theta = \text{constant}$ is the setting of traditional inhomogeneous Diophantine approximation and in that case the above result was well known, see for example \cite{Cassels}. Similarly inhomogeneous Diophantine approximation can be considered in the $S$-arithmetic setting.
For a multivariable approximating function $\Psi$ and a function $\Theta: \bbQ^{n}_S \to \bbQ_S$, we say that a vector $\bx \in \bbQ_S^n $ is $(\Psi,\Theta)$-approximable if there exist infinitely many $(\ba, a_0)\in\bbZ^n\setminus\{0\}\times \bbZ $ such that
\begin{equation}
|a_0 + \ba\cdot \bx +\Theta(\bx)|_{S}^l\leq \left\{
\begin{array}{rl}
\Psi(\widetilde{\ba}) & \text{ if } \infty \notin S\\
\\
\Psi(\ba) & \text{ if } \infty \in S.
\end{array} \right.
\end{equation}
The convergence case of Khintchine's theorem in this setting again follows from the Borel Cantelli lemma. The divergence Theorem when $S = \{p\}$ comprises a single prime $p$ is a consequence of the results in this paper.
\subsection{Diophantine approximation on manifolds}
In the theory of Diophantine approximation on manifolds, one studies the inheritance of generic (for Lebesgue measure) Diophantine properties by proper submanifolds of $\R^n$. This theory has seen dramatic advances in the last two decades, beginning with the proof of the Baker-Sprind\v{z}uk conjectures by Kleinbock and Margulis \cite{KM} using non divergence estimates for certain flows on the space of unimodular lattices. Motivated by problems in transcendental number theory, K. Mahler conjectured in 1932 that almost every point on the curve
$$ \f(\bx) = (x, x^2, \dots, x^n)$$
\noindent is not \emph{very well approximable}, i.e. $\psi$-approximable for $\psi:= \psi_{\varepsilon}(k) = k^{-n-\varepsilon}$. This conjecture was resolved by V. G. Sprind\v{z}uk \cite{Sp, Sp3} who in turn conjectured that almost every point on a nondegenerate manifold is not very well approximable. This conjecture, in a more general, multiplicative form, was resolved by D. Kleinbock and G. Margulis in \cite{KM}.
The following definition is taken from \cite{KT} and is based on \cite{KM}. Let $f : U \to F^n$ be a $C^k$ map, where $F$ is any locally compact valued field and $U$ is an open subset of $F^d$, and say that $f$ is nondegenerate at $x_0 \in U$ if the space $F^n$ is spanned by partial derivatives of $f$ at $x_0$ up to some finite order. Loosely speaking, a nondegenerate manifold is one in which is locally not contained in an affine subspace. Subsequent to the work of Kleinbock and Margulis, there were rapid advances in the theory of dual approximation on manifolds. In \cite{BKM} (and independently in \cite{Ber1}) the convergence case of the Khintchine-Groshev theorem for nondegenerate manifolds was proved and in \cite{BBKM}, the complementary divergence case was established.
As for the $p$-adic theory, Sprind\v{z}uk \cite{Sp} himself established the $p$-adic and function field (i.e. positive characteristic) versions of Mahler's conjectures. Subsequently, there were several partial results (cf. \cite{Kov, BK}) culminating in the work of Kleinbock and Tomanov \cite{KT} where the $S$-adic case of the Baker-Sprind\v{z}uk conjectures were settled in full generality. In \cite{G}, the second named author established the function field analogue. The convergence case of Khintchine's theorem for nondegenerate manifolds in the $S$-adic setting was established by Mohammadi and Golsefidy \cite{MoS1} and the divergence case for $\Q_p$ in \cite{MoS2}.
In the case of inhomogeneous Diophantine approximation on manifolds, following several partial results (cf. \cite{Bu} and the references in \cite{BeVe, BeVe2}), an inhomogeneous transference principle was developed by Beresnevich and Velani using which they resolved the inhomogeneous analogue of the Baker-Sprind\v{z}uk conjectures. Subsequently, Badziahin, Beresnevich and Velani \cite{BaBeVe} established the convergence and divergence cases of the inhomogeneous Khintchine theorem for nondegenerate manifolds. They proved a new result even in the classical setting by allowing the inhomogeneous term to vary. The divergence theorem is established in the same paper in the more general setting of Hausdorff measures.
In this paper, we will establish the convergence case of an inhomogeneous Khintchine theorem for nondegenerate manifolds in the $S$-adic setting, as well as the divergence case for $\Q_p$. As in \cite{BaBeVe}, the divergence case is proved in the greater generality of Hausdorff measures. Prior results in the $p$-adic theory of inhomogeneous approximation for manifolds focussed mainly on curves, cf. \cite{BDY, BeK, U1, U2}.
\subsection{Main Results}
To state our main results, we introduce some notation following \cite{MoS1}, recall some of the assumptions from that paper and set forth one further standing assumption. The assumptions are as follows.
\begin{enumerate}
\item[(I0)] $S$ contains the infinite place.
\item[(I1)] We will consider the domain to be of the form $\bU=\prod_{\nu\in S} \bU_{\nu} $ where $\bU_\nu\subset\Q_\nu^{d_\nu} $ is an open box. Here, the norm is taken to be the Euclidean norm at the infinite place and the $L^{\infty}$ norm at finite places.
\item[(I2)] We will consider functions $\f(\bx) =(\f_\nu(x_\nu)) _{\nu\in S}$, for $\bx=(x_\nu) \in\bU $ where
$\f_\nu=(f_\nu^{(1)},f_\nu^{(2)},\dots,f_\nu^{(n)}):
\bU_\nu\to \Q_\nu^n $ is an analytic map for any
$\nu\in S $, and can be analytically extended to the boundary of $ \bU_\nu$.
\item[(I3)] We assume that the restrictions of $1 ,\fnu $ to any open subset of $\bU_\nu $ are linearly independent over $\Q_\nu $ and that $\|\f(\bx)\|\leq 1,\|\nabla\f_\nu(x_\nu)\| \leq 1$ and $|\Phi_\beta \f_\nu(y_1,y_2,y_3)| \leq \frac{1}{2} $ for any $\nu \in S,$ second difference
quotient $\Phi_\beta$ and $x_\nu,y_1,y_2,y_3 \in U_\nu$. We refer the reader to Section $3$ for definitions.
\item[(I4)]\label{monotone_cond} We assume that the function $\Psi :\bbZ^n \to \bbR_{+ } $ is monotone decreasing componentwise
i.e. $$\Psi(a_1,\cdots,a_i,\cdots, a_n)\geq \Psi(a_1,\cdots, a'_{i},\cdots, a_n)$$ whenever $|a_i|_S\leq |a'_i|_S $.
\item[(I5)] We assume that $\Theta(\bx)=(\Theta_\nu(x_\nu)) $ where $\Theta :\bU \mapsto \Q_S $ is also analytic and can be extended analytically to the boundary of $\bU_\nu$.we will assume $\|\Theta(\bx)\|\leq 1,\|\nabla\Theta_\nu(x_\nu)\| \leq 1$ and $|\Phi_\beta \Theta_\nu(y_1,y_2,y_3)| \leq \frac{1}{2} $ for any $\nu \in S $ , second difference
quotient $\Phi_\beta$ and $x_\nu,y_1,y_2,y_3 \in U_\nu$.
\end{enumerate}
\noindent We can now state the first main Theorem of the present paper.
\begin{theorem}\label{thm:main}
Let $S$ be as in (I0) and $\bU$ as in (I1). Suppose $\f$ satisfies (I2) and (I3), that $\Psi$ satisfies (I4) and $\Theta$ satisfies (I5). Then
\begin{equation}
\cW_{\Psi,\Theta}^{\f} := \{ \bx\in\bU | \ \f(\bx) \text{ is } (\Psi,\Theta)-\text{ approximable}\}\end{equation}
has measure zero if $\sum_{\ba \in \Z^n\setminus\{0\}} \Psi(\ba) <\infty$.
\end{theorem}
The divergence case of our Theorem is proved in the more general setting of Hausdorff measures. However, we need to impose some restrictions: we only consider the case when $S = \{p\}$ consists of a single prime, the inhomogeneous function is assumed to be analytic, and the approximating function is not as general as in Theorem \ref{thm:main}. We will denote by $\mathcal{H}^{s}(X) $ the $s$-dimensional Hausdorff measure of a subset $X$ of $\bbQ^{n}_{S}$ and $\dim X$ the Hausdorff dimension, where $s > 0$ is a real number.
\begin{theorem}\label{thm:divergence}
Let $S$ be as in (I0) and $\bU$ as in (I1). Suppose $\f:\bU\subset\Q_p^m\to \Q_p^n$ satisfies (I2) and (I3). Let
\begin{equation}\label{def:newpsi}
\Psi(\ba)= \psi(\|\ba\|), \ba\in\Z^{n+1}
\end{equation}
be an approximating function and assume that $s > m-1$. Let $\Theta:\bU\to \Q_p$ be an analytic map satisfying (I5). Then
\begin{equation}
\mathcal{H}^s(\mathcal{W}^\f_{(\Psi,\Theta)}\cap\bU)=\mathcal{H}^s(\bU) \text{ if } \sum_{\ba \in \bbZ^n \backslash \{0\}} (\Psi(\ba))^{s+1-m}=\infty.
\end{equation}
\end{theorem}
Given an approximating function $\psi$, the lower order at infinity $\tau_{\psi}$ of $1/\psi$ is defined by
\begin{equation}
\tau_{\psi} := \liminf_{t \to \infty}\frac{-\log\psi(t)}{\log t}.
\end{equation}
The divergent sum condition of Theorem \ref{thm:divergence} is satisfied whenever $$s<m-1+\frac{n+1}{\tau_\psi}.$$ Therefore, by the definition of Hausdorff measure and dimension, we get
\begin{corollary}{\label{jar}}
Let $\f$ and $\Theta$ be as in Theorem \ref{thm:divergence}. Let $\psi$ be an approximating function as in (\ref{def:newpsi}) such that $n+1\leq \tau_\psi<\infty$. Then
\begin{equation}
\dim (\mathcal{W}^\f_{(\Psi,\Theta)}\cap\bU)\geq m-1+\frac{n+1}{\tau_\psi}.
\end{equation}
\end{corollary}
\subsection{Remarks}
\begin{enumerate}
\item We have assumed $S$ contains the infinite place in Theorem \ref{thm:main}. This is not a serious assumption, the proof in the case when $S$ contains only finite places needs some minor modifications but follows the same outline, details will appear in \cite{Datta}, the PhD thesis, under preparation, of the first named author. In \cite{MoS1}, the (homogeneous) $S$-adic convergence case is proved in slightly greater generality than in the present paper. Namely, instead of $\bbQ$, the quotient field of a finitely generated subring of $\bbQ$ is considered. This, more general formulation will also be investigated in \cite{Datta}.
\item Our proof for the convergence case, namely Theorem \ref{thm:main} blends techniques from the homogeneous results, namely \cite{KT, BKM, MoS1} and uses the transference principle developed by Beresnevich and Velani in the form used in \cite{BaBeVe}. The structure of the proof is the same as in \cite{BaBeVe}. We also take the opportunity to clarify some properties of $(C, \alpha)$-good functions in the $S$-adic setting which may be of independent interest.
\item The proof of Theorem \ref{thm:divergence}, follows the ubiquity framework used in \cite{BaBeVe} but needs new ideas to implement in the $p$-adic setting. At present, we are unable to prove the more general $S$-adic divergence statement. We note that the $S$-adic case remains open even in the homogeneous setting.
\item We now undertake a brief discussion of the assumptions (I1) - (I5). The conditions (I1)-(I4) are assumed in \cite{MoS1} and, as explained in loc. cit., are assumed for convenience. Namely, as mentioned in \cite{MoS1}, the statement for any non-degenerate analytic manifold over $\bbQ_S$ follows from Theorem \ref{thm:main}. In \cite{BaBeVe}, the inhomogeneous parameter $\Theta$ is allowed to be $C^2$ when restricted to the nondegenerate manifold. However, we need to assume it to be analytic.
\item Theorem \ref{thm:divergence} is slightly more general than Theorem 1.2 of \cite{MoS2} in the homogeneous setting. In \cite{MoS2}, the approximating function is taken to be of the form
\begin{equation}
\Psi(\ba)=\frac{1}{\|\ba\|^{n}}\psi(\|\ba\|), \ba\in\Z^{n+1}
\end{equation}
which is a more restrictive class of approximating functions. For an $n$-tuple
$v = (v_1, \cdots, v_n)$ of positive numbers satisfying $v_1 + \cdots + v_n = n$, define the $v$-quasinorm $| ~ |_v$ on $\bbR^n$ by setting
$$ \|\bx\|_v := \max |x_i|^{1/v_i}. $$
Following \cite{BaBeVe} we say that a multivariable approximating function $\Psi$ satisfies property $\mathbf{P}$ if $\Psi(\ba) = \psi(\|\ba\|_v)$ for some approximating function $\psi$ and $v$ as above. As noted in loc. cit. when $v = (1, \dots, 1)$ we have that $\|\ba\|_v = \|\ba\|$ and any approximating function $\psi$ satisfies property $\mathbf{P}$, where $\psi$ is regarded as the function $\ba \to \psi(\|\ba\|)$. The proof of Theorem \ref{thm:divergence} can be modified to deal with the case of functions satisfying property $\mathbf{P}$.
\end{enumerate}
\subsection*{Structure of the paper} In the next section, we recall the transference principle of Beresnevich and Velani. The subsequent section studies $(C, \alpha)$-good functions in the $S$-adic setting. We then prove Theorem \ref{thm:main} and then Theorem \ref{thm:divergence}. We conclude with some open questions.
\section{Inhomogeneous transference principle}
In this section we state the inhomogeneous transference principle of Beresnevich and Velani from \cite[Section 5]{BeVe} which will allow
us to convert our inhomogeneous problem to the homogeneous one. Let $(\Omega, d)$ be a locally compact metric space. Given two countable indexing sets $\mathcal{A}$ and $\mathbf{T}$, let H and I be two maps from $\bT \times \cA \times \R_{+}$ into the set of open subsets of $\Omega$ such that
\begin{equation}\label{H_fn}
H~:~(t, \alpha, \lambda) \in \bT \times \cA \times \R_{+} \to H_{\mathbf{t}}(\alpha, \lambda)
\end{equation}
\\
and
\begin{equation}\label{I_fn}
I~:~ (t, \alpha, \lambda) \in \bT \times \cA \times \R_{+} \to I_{\mathbf{t}}(\alpha, \lambda)
\end{equation}
\\
Furthermore, let
\begin{equation}\label{defH}
H_{\bt} (\lambda) := \bigcup_{\alpha \in \cA} H_{\mathbf{t}}(\alpha, \lambda) \text{ and } I_{\bt} (\lambda) := \bigcup_{\alpha \in \cA} I_{\mathbf{t}}(\alpha, \lambda).
\end{equation}
Let $\Psi$ denote a set of functions $\psi: \bT \to \R_{+}~:~\bt \to \psi_{\bt}$. For $\psi \in \Psi$, consider the limsup sets
\begin{equation}\label{deflambda}
\Lambda_{H}(\psi) = \limsup_{\bt \in \bT} H_{\bt}(\psi_{\bt}) \text{ and } \Lambda_{I}(\psi) = \limsup_{\bt \in \bT} I_{\bt}(\psi_{\bt}).
\end{equation}
The sets associated with the map $H$ will be called homogeneous sets and those associated with the map $I$, inhomogeneous sets. We now come to two important properties connecting these notions.
\subsection*{The intersection property} The triple $(H, I, \Psi)$ is said to satisfy the intersection property if, for any $\psi \in \Psi$, there exists $\psi^{*} \in \Psi$ such that, for all but finitely many $\bt \in \bT$ and all distinct $\alpha$ and $\alpha'$ in $\cA$, we have that
\begin{equation}\label{inter}
I_{\bt}(\alpha, \psi_{\bt}) \cap I_{\bt}(\alpha', \psi_{\bt}) \subset H_{\bt}(\psi^{*}_{\bt}).
\end{equation}
\subsection*{The contraction property} Let $\mu$ be a non-atomic finite doubling measure supported
on a bounded subset $\mathbf{S}$ of $\Omega$. We recall that $\mu$ is doubling if there is a constant $\lambda > 1$ such that, for any ball $B$ with centre in $\bS$, we have
$$\mu(2B) \leq \lambda \mu(B),$$
where, for a ball $B$ of radius $r$, we denote by $cB$ the ball with the same centre and radius $cr$.
We say that $\mu$ is contracting with respect to $(I, \Psi)$ if, for any
$\psi \in \Psi$, there exists $\psi^{+}\in \Psi$ and a sequence of positive numbers $\{k_{\bt}\}_{\bt \in \bT}$ satisfying
\begin{equation}\label{conv}
\sum_{\bt \in \bT}k_{\bt} < \infty,
\end{equation}
such that, for all but finitely $\bt \in \bT$ and all $\alpha \in \cA$, there exists a collection $C_{\bt, \alpha}$ of balls $B$
centred at $\mathbf{S}$ satisfying the following conditions:
\begin{equation}\label{inter1}
\bS \cap I_{\bt}(\alpha, \psi_{\bt}) \subset \bigcup_{B \in C_{\bt, \alpha}} B
\end{equation}
\begin{equation}\label{inter2}
\bS \cap \bigcup_{B \in C_{\bt, \alpha}} B \subset I_{\bt}(\alpha, \psi^{+}_{\bt})
\end{equation}
and
\begin{equation}\label{inter3}
\mu(5B \cap I_{\bt}(\alpha, \psi_{\bt})) \leq k_{\bt} \mu(5B).
\end{equation}
We are now in a position to state Theorem $5$ from \cite{BeVe}
\begin{theorem}\label{transfer}
Suppose that $(H, I, \Psi)$ satisfies the
intersection property and that $\mu$ is contracting with respect to $(I, \Psi)$. Then
\begin{equation}\label{eq:transfer1}
\mu(\Lambda_{H}(\psi))=0 ~\forall~\psi \in \Psi \Rightarrow \mu(\Lambda_{I}(\psi)) = 0 ~\forall~\psi \in \Psi.
\end{equation}
\end{theorem}
\section{$(C, \alpha)$-good functions}
In this section, we recall the important notion of $(C, \alpha)$-good functions on ultrametric spaces. We follow the treatment of Kleinbock and Tomanov \cite{KT}. Let $X$ be a metric space, $\mu$ a Borel measure on $X$ and let $(F, |\cdot|)$ be a local field. For a subset $U$ of $X$ and $C, \alpha > 0$, say that a Borel measurable function $f : U \to F$ is $(C, \alpha)$-good on $U$ with respect to $\mu$ if for any open ball $B \subset U$ centred in $\supp \mu$ and $\varepsilon > 0$ one has
\begin{equation}\label{gooddef}
\mu \left(\{ x \in B \big| |f(x)| < \varepsilon \} \right) \leq
C\left(\displaystyle \frac{\varepsilon}{\sup_{x \in
B}|f(x)|}\right)^{\al}|B|,
\end{equation}
The following elementary properties of $(C,
\al)$-good functions will be used.
\begin{enumerate}
\item[(G1)] If $f$ is $(C,\al)$-good on an open set $V$, so
is $\lambda
f~\forall~\lambda \in
F$;\\
\item[(G2)] If $f_i, i \in I$ are $(C,\al)$-good on $V$, so
is $\sup_{i \in
I}|f_i|$;\\
\item[(G3)] If $f$ is $(C,\al)$-good on $V$ and for some
$c_1,c_2\,\textgreater \,0,\, c_1\leq
\frac{|f(x)|}{|g(x)|}\leq c_2
\text{ for all }x \in V$, then g is
$(C(c_2/c_1)^{\al},\al)$-good on $V$.\\
\item[(G4)] If $f$ is $(C,\al)$-good on $V$, it is
$(C',\alpha')$-good
on $V'$ for every $C' \geq \max\{C,1\}$, $\alpha' \leq \alpha$ and $V'\subset V$.
\end{enumerate}
One can note that from (G2), it follows that the supremum
norm of a vector valued function $\f$ is $(C,\al)$-good
whenever each of its components is $(C,\al)$-good.
Furthermore, in view of (G3), we can replace the norm by an
equivalent one, only affecting
$C$ but not $\al$.
Polynomials in $d$ variables of degree at most $k$ defined on local fields can be seen to be $(C, 1/dk)$-good, with $C$ depending only on $d$ and $k$ using Lagrange interpolation. In \cite{KM}, \cite{BKM} and \cite{KT} (for ultrametric fields), this property was extended to smooth functions satisfying certain properties. We rapidly recall, following \cite{S} (see also \cite{KT}), the definition of smooth functions in the ultrametric case. Let $U$ be a non-empty subset of $X$ without isolated points. For $n \in \mathbb{N}$, define
$$\nabla^{n}(U) = \{(x_1,\dots,x_n) \in U, x_i \neq x_j \text{ for } i \neq j \}.$$
The $n$-th order difference quotient of a function $f : U \to X$
is the function $\Phi_n(f) $ defined inductively by $\Phi_0 (f) =
f$ and, for $n \in \N$, and $(x_1,\dots,x_{n+1}) \in \nabla^n(U)$
by
\[ \Phi_{n}f(x_1,\dots,x_{n+1}) = \frac{\Phi_{n-1}f(x_1,x_3,\dots,x_{n+1}) -
\Phi_{n-1}f(x_2,\dots,x_{n+1})}{x_1-x_2}. \]
This definition does not depend on the choice of variables,
as all difference quotients are symmetric functions. A function $f$
on $X$ is called a $C^n$ function if $\Phi_n f$ can be extended to
a continuous function $\bar{\Phi}_{n}f : U^{n+1} \to X $. We also set
\[ D_n f(a) = \overline{\Phi_n}f(a,\dots,a),~a \in U. \]
We have the following theorem (c.f. \cite{S}, Theorem $29.5$).
\begin{theorem}\label{derivative}
Let $f \in C^{n}(U \to X)$. Then, $f$ is $n$ times differentiable
and
\[ j!D_j f = f^j \]
for all $1 \leq j \leq n$.
\end{theorem}
To define $C^{k}$ functions in several variables, we follow the notation set
forth in \cite{KT}. Consider a multiindex $\beta =
(i_1,\dots,i_d)$ and let
\[ \Phi_{\beta}f = \Phi^{i_1}_{1}\circ \dots \circ \Phi^{i_d}_{d} f. \]
This difference order quotient is defined on the set $
\nabla^{i_1}U_1 \times \dots \times \nabla^{i_d}U_d$ and the $U_i$
are all non-empty subsets of $X$ without isolated points. A
function $f$ will then be said to belong to $C^{k}(U_1\times \dots
\times U_d)$ if for any multiindex $\beta$ with $|\beta| = \sum_{j =
1}^{d} i_j \leq k$, $\Phi_{\beta} f$ extends to a continuous
function $\bar{\Phi}_{\beta}f : U_{1}^{i_1 + 1} \times \dots \times
U_{d}^{i_d + 1}$. We then have
\begin{equation}\label{multivanish}
\partial_{\beta}f(x_1,\dots,x_d) = \beta!
\bar{\Phi}_{\beta}(x_1,\dots,x_1,\dots,x_d,\dots,x_d)
\end{equation}
where $\beta ! = \prod_{j = 1}^{d} i_{j}!$.\\
We are now ready to gather the results on ultrametric $(C, \alpha)$-good functions that we need. We begin with Theorem $3.2$ from \cite{KT}.
\begin{theorem}\label{theorem 3.2}
Let $V_1,V_2,\cdots,V_3$ be nonempty open sets in F, ultrametric field. Let $ k\in \N$, $A_1,\cdots,A_d> 0 $ and $ f\in C^k(V_1\times\cdots,\times V_n) $ be such that
\begin{equation}\label{eqn 3.3}
|\Phi_j^kf|\equiv A_j \text{ on } \nabla^{k+1}V_j\times\prod_{i\neq j}V_j , j=1,\cdots,d.
\end{equation}
Then f is $(dk^{3-\frac{1}{k}},\frac{1}{dk})$-good on $V_1\times\cdots,\times V_n$
\end{theorem}
The following is an ultrametric analogue of Proposition 1 from \cite{BaBeVe}.
\begin{proposition}\label{Calpha_Prop}
Let $U_\nu$ be an open subset of $\Q_\nu ^d,$ $\bx_0 \in U_\nu$ and let
$\mathcal{F}\subset C^l(U)$ be a compact family of functions $f: U\to \Q_\nu $
for some $l\geq 2$. Also assume that
\begin{equation}\label{3.4}
\inf_{f\in\mathcal{F}}\max_{0<|\beta|\leq l} \ |\partial_{\beta}f(\bx_0)|>0.
\end{equation}
Then there exists a neighbourhood $V_\nu\subset U_\nu$ of $\bx_0$ and $C, \delta > 0$ satisfying the following property. For any $\Theta\in C^l(U)$ such that
\begin{equation}\label{theta_cond}
\sup_{\bx\in U_\nu} \max_{0<|\beta|\leq l} \ |\partial_{\beta}\Theta(\bx_0)|\leq \delta
\end{equation}
and for any $f\in \mathcal {F}$ we have that
\begin{enumerate}
\item $f+\Theta $ is $(C,\frac{1}{dl})$-good on $V_\nu$.
\item $|\nabla(f+\Theta)|$ is $ \left(C,\frac{1}{m(l-1)}\right)$-good on $V_\nu$
\end{enumerate}
\end{proposition}
\begin{proof}
We follow the proof of \cite{BaBeVe}, which in turn is a modification of the ideas used to establish Proposition 3.4 in \cite{BKM}. Here $\nu=\infty$ is exactly Proposition 1 of \cite {BaBeVe} so we assume that $\nu\neq\infty$. By (\ref{3.4}) there exists $C_1 > 0$
such that for any $f\in \mathcal{F}$ there exists a multiindex $\beta$ with $0<|\beta|=k\leq l $ , where $k=k(f)$ such that
\begin{equation}\label{3.6}
|\partial_{\beta} f (\bx_0)|\geq C_1.
\end{equation}
By the compactness of $\mathcal{F}$, $\inf_{f\in\mathcal{F}}\max_{|\beta|\leq l} \ |\partial_{\beta}f(\bx_0)|$ will be actually attained for some f and we may take that value to be $C_1$. Since there are finitely many $\beta$, we can consider the subfamily $\mathcal{F}_\beta:=\{f\in\mathcal{F}\ |\ \partial_{\beta} f (\bx_0)|\geq C_1\} $, which is also compact in $C^l(U)$ and satisfies (\ref{3.4}). Proving the theorem for $\mathcal{F}_\beta$ will yield sets $U_\beta$ where (1) and (2) above hold. Setting $V_{\nu} := \bigcap_{\beta} U_{\beta}$ then proves the Proposition. We may therefore assume without loss of generality that $\beta$ is the same for every $f\in\mathcal{F}$. \\
We wish to apply Theorem 3.2 of \cite{KT} and to do so we need to satisfy (\ref{eqn 3.3}). We are going to show that there exists $A\in \GL_d(\mathcal{O})$ such that $f\circ A$ has the property (\ref{eqn 3.3}). For $A\in \GL_d(\mathcal{O})$ we have, by the chain rule that
\begin{equation}\label{lin_sys1}
\begin{array}{rcr}
\partial_{1}^{k}f\circ A(A \inv \bx_0) &=& \sum_{\sum i_j=k, i_j\geq 0} C_{(i_1,\cdots,i_d)} a_{11}^{i_1}\cdots a_{d1}^{i_d} \ \partial_{\beta=(i_1,\cdots,i_d)}^k f(\bx_0) \\
\vdots \\
\partial_{d}^{k}f\circ A(A\inv \bx_0) &=& \sum_{\sum i_j=k, i_j\geq 0} C_{(i_1,\cdots,i_d)} a_{1d}^{i_1}\cdots a_{dd}^{i_d} \ \partial_{\beta=(i_1,\cdots,i_d)}^k f(\bx_0).
\end{array}
\end{equation}
We want $A=(a_{ij})$ such that every element in the left side of (\ref{lin_sys1}) above is nonzero knowing that for at least one $\beta ,\ \partial_{ \beta=(i_1,\cdots,i_k)}^k f(\bx_0)\neq 0 $. Namely, we wish to find $A\in \GL_d(\mathcal{O}) $ such that $x'_i\neq 0$ for every $i$ where
$$
\begin{array}{rcr}
x'_1 &= & \sum C_{(i_1,\cdots,i_d)} \ a_{11}^{i_1}\cdots a_{d1}^{i_d} \ x_{(i_1,\cdots,i_d)}\\
\vdots \\
x'_d &=& \sum C_{(i_1,\cdots,i_d)} \ a_{1d}^{i_1}\cdots a_{dd}^{i_d} \ x_{(i_1,\cdots,i_k)}
\end{array}
$$
i.e.
$$\begin{array}{rcr}
x'_1&=& g(a_{11},\cdots,a_{d1}) \\
\vdots \\
x'_d&=& g(a_{1d},\cdots,a_{dd})
\end{array}
$$ and $g$ is a homogeneous polynomial of degree k. We already know that $ \partial_{ \beta=(i_1,\cdots,i_k)}^k f(\bx_0)\neq 0$ for at least one $\beta$, so at least one $x_{(i_1,\cdots,i_k)}\neq 0$ and thus $g $ is a nonzero polynomial.
Now $ g $ should have at least one nonzero value on $\{1+\pi\mathcal{O}\}\times\{\pi\mathcal{O}\} \times\cdots\times\{\pi\mathcal{O}\}$, otherwise $g$ is identically zero. So take $(a_{11},\cdots,a_{1d})$ to be the point of the aforementioned set where $g(a_{11},\cdots,a_{1d})\neq 0$. Then by a similar argument choose $(a_{i1},\cdots,a_{id})\in \{\pi\mathcal{O}\}\times\cdots\times\{1+\pi\mathcal{O}\} \times\cdots\times\{\pi\mathcal{O}\}$ such that
$g(a_{i1},\cdots,a_{id})\neq 0$. Choosing $A$ this way we will automatically get that $\det(A)$ is a unit, which implies that $A\in \GL_d(\mathcal{O})$. Thus we have that for $f\in\mathcal{F}$ there exists $A_f\in \GL_d(\mathcal{O})$ depending on $f$ such that
\begin{equation}\label{der_nonzero}
\min_{i=1,\cdots,d} |\partial_i^k f\circ A_f (A_f\inv (\bx_0) )|>0 \end{equation}
in fact there exists a uniform $C>0$ such that
\begin{equation}
\min_{i=1,\cdots,d} |\partial_i^k f\circ A_f (A_f\inv (\bx_0) )|>C.
\end{equation}
\noindent This is because we can take $$C=\inf_{f\in\mathcal{F}}\sup_{A\in \GL_d(\mathcal{O})}\min_{i=1,\cdots,d} |\partial_i^k f\circ A (A\inv (\bx_0) )|,$$ which is nonzero. For if not, then there exists $\{f_n\} \in\mathcal{F}$ such that
$$\sup_{A\in \GL_d(\mathcal{O})}\min_{i=1,\cdots,d} |\partial_i^k f_n\circ A (A\inv (\bx_0) )|<\frac{1}{n}.$$
Since $\mathcal{F}$ is compact, $\{f_n\}$ has a convergent subsequence $\{f_{n_k}\}\to f\in\mathcal{F}$. Taking limits, we get that $$\min_{i=1,\cdots,d} |\partial_i^k f\circ A (A\inv (\bx_0) )|=0 \ \forall \ A \in \GL_d(\mathcal{O}),$$ which is a contradiction to (\ref{der_nonzero}).
Consider the following map
$$\Phi_1: \GL_d(\Q_\nu)\times C^l(U_\nu)\times U_\nu \longmapsto \Q_\nu $$
$$(A,f,\bx)\mapsto \min_{i=1,\cdots,d} |\partial_i^k f\circ A (A\inv (\bx)|. $$
It can be easily verified that $\Phi_1$ is continuous. For every $f\in\mathcal{F}$ there exists $A_f \in \GL_d(\mathcal{O}) $ such that $\Phi_1(A_f,f,\bx_0)\geq C>\frac{C}{2},$ so by continuity we have an open neighbourhood $U_{A_f}\times U_f\times U_{(\bx_0,f)}$ of $ (A_f,f,\bx_0)$ such that
$$
\Phi_1(A,g,\bx) >\frac{C}{2} \
\forall \ (A,g,\bx) \in U_{A_f}\times U_f\times U_{(\bx_0,f)}.
$$ In particular,
\begin{equation}\label{unicondition}
\Phi_1(A_f,g,\bx)>\frac{C}{2} \ \forall g\in U_f \text{ and } \forall \ \bx\in U_{(\bx_0,f)}.
\end{equation}\
Now $\mathcal{F}\subset \bigcup_{f} U_f,$ must have a finite subcovering $\{U_{f_i}\}_{i=1}^{r}$. So by (\ref{unicondition}) we have that for every $\bx\in U_{\bx_0}=\bigcap_{i=1}^r U_{(\bx_0f_i)} $ and $f\in\mathcal{F}$ there exists $A_{f_i}$ such that
\begin{equation}\label{final_cond}
\Phi_1(A_{f_i},f,\bx) >\frac{C}{2}.
\end{equation}
Choose $\delta=\frac{C}{4u}$ where $ u $ is the constant coming from the inequality
$$|\partial_i^k\Theta\circ T(T\inv\bx)| \leq u \max_{|\beta|\leq l}|\partial_{ \beta} f(\bx) |$$ for $T\in \GL_d(\mathcal{O})$. Thus any $\Theta $ satisfying (\ref{theta_cond})
will also satisfy $$\Phi_1(A_{f_i},f+\Theta,\bx)>\frac{C}{4} \ \ \forall \ \bx\in U_{\bx_0}.$$
By the compactness of $\mathcal{F}$ and (\ref{theta_cond}) there is a uniform upper bound for every $f\in\mathcal{F}$ and $\Theta $ of the aforementioned type. Now applying Theorem \ref{theorem 3.2} we have that $f+(\Theta\circ A_{f_i})$ is $(dk^{3-\frac{1}{k}},\frac{1}{dk})$-good on $A_{f_i}^{-1} U_{\bx_0}$. Therefore, $f+\Theta$ is $(dk^{3-\frac{1}{k}},\frac{1}{dk})$-good on $U_{\bx_0}$. This completes the proof of the first part.\\
Now consider the set $\mathcal{F}_{A_{f_i}}= \{f\in\mathcal{F} \ | \ \Phi_1(A_{f_i},f,\bx_0)\geq \frac{C}{2}\}$. Clearly this is a closed subset of the compact set $\mathcal{F}$, so it is also compact.
Therefore $\{\partial_j(f\circ A_{f_i}) | \ f\in\mathcal{F}_{A_{f_i}} \}$ is also compact being the image of a compact set under a continuous map. Since $\mathcal{F} \subset \bigcup_{i=1,\cdots,r} \mathcal{F}_{A_{f_i}}$, we may, without loss of generality, take the same $A$ for every $f\in \mathcal{F}$. Now we want to apply the first part of this Proposition.
Suppose $|\beta| \geq 2 $ in (\ref{3.6}), then to apply part(1) we have to check condition (\ref{3.4}) for the set $\{\partial_j(f\circ A) | \ f\in\mathcal{F} \}$, where we know that $\Phi_1(A,f,\bx_0)\geq \frac{C}{2}$. Suppose $$\inf_{f\in\mathcal{F}}\max_{|\beta|\leq l-1}|\partial_{ \beta}\partial_{j}(f\circ A)(A\inv(\bx_0))|=0.$$
Then by compactness of $\mathcal{F}$ we have that for some $f\in\mathcal{F}$,
$$\max_{|\beta|\leq l-1}|\partial_{ \beta}\partial_{j}(f\circ A)(A\inv(\bx_0))|=0,$$
which implies that $\Phi_1(A,f,\bx_0)=0,$ which is a contradiction. Thus by applying the first part of the Proposition we get that for every $j=1,\cdots,d , \partial_j((f+\Theta)\circ A)$ is $(C_\star,\frac{1}{d(l-1)})$-good on an open neighbourhood $B_{A\inv (\bx_0)}$ of $A\inv(\bx_0)$. So $(\partial_j(f+\Theta\circ A))\circ A\inv$ is $(C_\star,\frac{1}{d(l-1)})$-good on $A(B_{A\inv (\bx_0)})$. Therefore each $\partial_j(f+\Theta)$ is $(C_\star,\frac{1}{d(l-1)})$-good on $A(B_{A\inv (\bx_0)})$ and so is $|\nabla (f+\Theta)|$.
The case $|\beta|=1$ in (\ref{3.6}) is trivial (See property (G3) of $(C,\alpha)$-good functions). This completes the proof.
\end{proof}
As a Corollary, we have,
\begin{corollary}\label{good_corollary}
Let $U_\nu$ be an open subset of $\Q_\nu^{d\nu}, \bx_0\in U_\nu$ be fixed and assume that $\f_\nu=(f_\nu^{(1)},f_\nu^{(2)},\dots,f_\nu^{(n)}):
U_\nu\to \Q_\nu^n $ satisfies (I2) and (I3) and that $\Theta_\nu $ satisfies (I5). Then there exists a neighbourhood $V_\nu\subset U_\nu$ of $\bx_0$ and positive constants $ C > 0 $ and $l\in \N$ such that for any $(a_0,\ba)\in \mathcal{O}^{n+1},$
\begin{enumerate}
\item $a_0+\ba.\f_{\nu}+\Theta_\nu$ is $(C,\frac{1}{d_\nu l})$-good on $V_\nu,$ and
\item $|\nabla(\ba.\f_\nu +\Theta_\nu)| $ is $ (C,\frac{1}{d_\nu(l-1)})$-good on $V_\nu$.
\end{enumerate}
\end{corollary}
\begin{proof}
For the case $\nu=\infty$, see Corollary $3$ of \cite{BaBeVe} and also \cite{BKM}. So we may assume $\nu\neq \infty.$
Let $\mathcal{F}:= \{a_0+\ba.\f_\nu+\Theta_\nu \ |\ (a_0,\ba)\in\mathcal{O}^{n+1}\}$. This is a compact family of functions of $C^l(U_\nu)$ for every $l>0 $ since $\mathcal{O}$ is compact in $\Q_\nu$. Now if this family satisfies condition (\ref{3.4}) for some $l\in \N$, then the conclusion follows from the previous Proposition. Hence we may assume that the family does not satisfy (\ref{3.4}) for every $l\in \N$. Then by the continuity of differential and the compactness of $\mathcal{O}$, there exists $\bc_l\in \mathcal{O}^n$ such that for every $2 \leq l\in \N $ we have
$$
\max_{|\beta|\leq l}|\partial_{ \beta}(\bc_l.f_\nu+\Theta_\nu)(\bx_0)| > 0.
$$
Now this sequence $\{\bc_l\} \in\mathcal{O}^n$ has a convergent subsequence $\{\bc_{l_k}\}$ converging to $\bc \in \mathcal{O}^n$ since $\mathcal{O}^n$ is compact. By taking limits we get that
$$|\partial_{ \beta}(\bc.f_\nu+\Theta_\nu)(\bx_0)|=0 \ \forall \ \beta.$$
However, as each of the $\f_{\nu}$ and $\Theta_\nu$ are analytic on $U_\nu,$ there exists a neighbourhood $V_{\bx_0}$ of $\bx_0$ such that
$$(\bc.f_\nu+\Theta_\nu)(\bx)=u\ \forall \ \bx \in V_{\bx_0},$$
where $u \in \Q_\nu$ is a constant. Therefore replacing $\Theta_\nu$ by $u-\bc.\f_{\nu},$ we get that $$\mathcal{F}=\{ a_0+u+(\ba-\bc).\f_{\nu} \ | (a_0,\ba)\in \mathcal{O}^{n+1} \}.$$
First consider the case where $|a_0+u| < 2|\ba-\bc|,$ then
$$\mathcal{F}_1= \left\{\frac{a_0+u}{|\ba-\bc|}+\frac{\ba-\bc}{|\ba-\bc|}.\f_\nu |\ (a_0,\ba)\in\mathcal{O}^{n+1}\right\}$$
is compact in $C^l(U_\nu)$ for every $l\in \N$. Then by linear independence of $1,f_\nu^{(1)},\cdots,f_\nu^{(n)},$ $\mathcal{F}_1$ satisfies (\ref{3.4}) for some $l\in\N$. And then by Proposition \ref{Calpha_Prop}
we can conclude that every element in $\mathcal{F}_1$ is $(C,\frac{1}{d_\nu l})$-good on some $V_\nu\subset V_{\bx_0}\subset U_\nu$ together with conclusion (2) of the Corollary above. This also implies $ a_0+u+(\ba-\bc).\f_{\nu} $ are all $(C,\frac{1}{d_\nu l})$ good on $V_\nu$ for all $(a_0,\ba)\in\mathcal{O}^{n+1}$ with $|a_0+u| < 2|\ba-\bc|$. Otherwise $$\sup_{\bx\in V_{\bx_0}}|a_0+u+(\ba-\bc).\f_{\nu}|\leq 3.\inf_{\bx\in V_{\bx_0}}|a_0+u+(\ba-\bc).\f_{\nu}|$$ as $|a_0+u|\geq 2|\ba-\bc| $ and it turns out to be a trivial case. This implies that for $C\geq 3$ and $0<\alpha\leq1$ the aforementioned functions are $(C,\alpha)$-good.
\end{proof}
Let us recall the following Corollary from \cite{KT} (Corollary 2.3).
\begin{corollary}{\label{product_good}}
For $ j=1,\cdots,n,$ let $X_j$ be a metric space, $\mu_j$ be a measure on $X_j $. Let $ U_j\subset X_j $ be open, $C_j,\alpha_j >0 $ and let $f$ be a function on $U_1\times\cdots \times U_d$ such that for any $j=1,\cdots d$ and any $x_i\in U_i$ with $i\neq j,$ the function
\begin{equation}{\label{fun}}
y~~\mapsto f(x_1,\cdots,x_{j-1}, y, x_{j+1},\cdots, x_d)
\end{equation}
is $(C_j,\alpha_j)$-good on $U_j$ with respect to $\mu_j$. Then $f$ is $(\widetilde{C},\widetilde{\alpha}) $ -good on $U_1\times\cdots\times U_d $ with respect to $\mu_1\times\cdots\times\mu_d,$ where $\widetilde{C}=d,\widetilde{\alpha }$ are computable in terms of $C_j,\alpha_j $. In particular, if each of the functions (\ref{fun}) is $(C,\alpha)$-good on $U_j$ with respect to $\mu_j$, then the conclusion holds with $\widetilde{\alpha}=\frac{\alpha}{d}$ and $\widetilde{C}=dC$.
\end{corollary}
Now combining Corollary (\ref{good_corollary}) and (\ref{product_good}) we can state the following:
\begin{corollary}\label{good_function}
Let $\f $ and $\Theta$ be as in Corollary (\ref{good_corollary}) and let $\bx_0\in \bU.$ Then there exists a neighbourhood $\bV\subset\bU $ of $\bx_0$ and $C>0, k,k_1\in\N$ such that for any $(a_0,\ba)\in\Z^{n+1} $ the following holds:
\begin{enumerate}
\item
$ \bx~\mapsto ~|(a_0+\ba.\f+\Theta )(\bx)|_S\text{ is } (C,\frac{1}{dk})-\text{good on } \bV$,
\item
$\bx~\mapsto~\|\nabla(\ba.\f_{\nu}+\Theta_\nu)(\bx_{\nu})\| \text{ is } (C,\frac{1}{dk_1})-\text{ good on } \bV, \forall ~\nu\in S$
\end{enumerate}
where $d=\max{d_\nu}$.
\end{corollary}
\section{Proof of Theorem \ref{thm:main}}
We set $\phi(\nu)=\left\{\begin{array}{rl} -\varepsilon & \text{ if } \nu\neq\infty \\
1-\varepsilon & \text{ if }\nu=\infty
\end{array} \right.
$.
\noindent From the definition, it follows that $\cW_{\Psi,\Theta}^{\f}$ admits a description as a limsup set. Namely,
$$
\well =\limsup_{\ba \to \infty}\bW_\f (\ba,\Psi,\Theta) $$
where
$$\bW_\f (\ba,\Psi,\Theta)=\{\bx\in\bU :| a_0+ \ba\cdot \f(\bx)+\Theta(\bx)|_S^l\leq \Psi(\ba) \text{ for some } a_0 \} .$$
\noindent We may now write
$$ \bW_{\f}^{\text{large}}(\ba,\Psi,\Theta)= \left \{\bx\in \Wf~:~\|\nabla(\ba .\f_\nu(\bx_\nu)+\Theta_\nu(\bx_\nu))\|_\nu>\|\ba\|_S^{\phi(\nu)} ~\forall~\nu \right\}$$
where $0<\varepsilon <\frac{1}{4(n+1)l^2},$ is fixed and
$$\Wf\setminus\Wfl =\bigcup_{\nu\in S}\bW_{\nu,\f}^{\text{small}}(\ba, \Psi,\Theta)$$
where $$\bW_{\nu,\f}^{\text{small}}(\ba,\Psi,\Theta)=\left\{\bx\in\Wf :\|\nabla(\ba.\f_\nu(\bx_\nu)+\Theta_\nu(\bx_\nu))\|_\nu\leq\|\ba\|_S^{\phi(\nu)} \right\}.$$
\noindent As the set $S$ is finite, we have
$$\well=\cW_{\f}^{\text{large}}(\Psi,\Theta)\bigcup_{\nu\in S}\cW_{\nu,\f}^{\text{small}}(\Psi,\Theta)$$
where
$$\cW_{\f}^{\text{large}}(\Psi,\Theta)=\welllarge$$
and
$$\cW_{\nu,\f}^{\text{small}}(\Psi,\Theta)=\wellsmallnu.$$
To prove Theorem \ref{thm:main}, we will show that each of these limsup sets has zero measure. Namely, the proof is divided into the ``large derivative" case where we will show $|\cW_{\f}^{\text{large}}(\Psi,\Theta)|=0$, and the ``small derivative" case which involves $|\cW_{\nu,\f}^{\text{small}}(\Psi,\Theta)|=0 \ \forall\ \nu\in S.$
\subsection{The small derivative}
We begin by showing that $|\cW_{\nu,\f}^{\text{small}}(\Psi,\Theta)|=0 \ \forall\ \nu\in S$. From the assumed property (I4) of $\Psi$, it follows that $$ \Psi(\ba)<\Psi_0(\ba) :=\prod_{\substack{i=1,\cdots,n \\ a_i\neq 0}}|a_i|_S\inv.$$ So $\cW_{\nu,\f}^{\text{small}}(\Psi,\Theta)\subset \cW_{\nu,\f}^{\text{small}}(\Psi_0,\Theta)$, which means that it is enough to show that $ |\cW_{\nu,\f}^{\text{small}}(\Psi_0,\Theta)|=0 \ \forall\ \nu\in S $. Let us take $\mathcal{A}=\Z\times\Z^n\setminus\{0\} $ and $\bT=\Z_{\geq 0}^n $ and define the function
\begin{equation}\label{r_equation}
r_\nu (\bt)=\left\{\begin{array}{rl}
2^{(|\bt|+1)(1-\varepsilon)} & \text{if } \nu=\infty\\
\\
2^{-(|\bt|+1)\varepsilon} & \text{if } \nu\neq\infty
\end{array} \right .
\end{equation}
where $\varepsilon$ is fixed as before. Now we define sets $ I_\bt^\nu(\alpha,\lambda) $ and $H_\bt^\nu(\alpha,\lambda)$ for every $\lambda>0,\bt\in\bT \text{ and } \alpha=(a_0, \ba)\in \mathcal{A} $ as follows:\\
\begin{equation}\label{def_I}
I_\bt^\nu(\alpha,\lambda)=\left\{ \bx\in\bU:\begin{array}{l}
|a_0+\ba.\f(\bx)+\Theta(\bx)|_S^l<\lambda\Psi_0(2^\bt)\\\\
\|\nabla(\ba.\f_{\nu}(\bx_\nu)+\Theta_\nu(\bx_\nu))\|_\nu<\lambda r_\nu(\bt)\\\\
2^{t_i}\leq \max{\{1,|a_i|_S\}}\leq 2^{t_i+1} \ \forall \ 1\leq i\leq n
\end{array}
\right\}
\end{equation}
and
\begin{equation}\label{def_H}
H_\bt^\nu(\alpha,\lambda)=\left\{ \bx\in\bU:\begin{array}{l}
|a_0+\ba.\f(\bx)|_S^l<2^l\lambda\Psi_0(2^\bt)\\\\
\|\nabla(\ba.\f_{\nu}(\bx_\nu))\|_\nu<2\lambda r_\nu(\bt)\\\\
|a_i|_S\leq 2^{t_i+2} \ \forall\ 1\leq i \leq n
\end{array}
\right\}
\end{equation}
where $2^\bt=(2^{t_1},\cdots,2^{t_n})$ and $|S|=l$. These give us the functions (\ref{I_fn}) and (\ref{H_fn}) required in the inhomogeneous transference principle. As in (\ref{defH}) and (\ref{deflambda}) we get $H_\bt^\nu(\lambda)$, $I_\bt^\nu(\lambda)$, $\Lambda_H^\nu(\lambda)$ and $\Lambda_I^\nu(\lambda)$. Now define $\phi_\delta~:~\bT\mapsto \R_{+}$ as $\phi_\delta(\bt):=2^{\delta|\bt|}$ for $\delta\in(0,\frac{\varepsilon}{2}] $. Clearly $\cW_{\nu,\f}^{\text{small}}(\Psi_0,\Theta)\subset \Lambda_I^\nu(\phi_\delta) $ for every $\delta\in(0,\frac{\varepsilon}{2}]$. So to settle Case 2 it is enough to show that
\begin{equation} \label{Inhomo_set}
|\Lambda_I^\nu(\phi_\delta)|=0 \text{ for some } \delta\in(0,\frac{\varepsilon}{2}].
\end{equation}
\noindent Now we recall Theorem $1.3$ from \cite{MoS1}.
\begin{theorem}\label{<}
Let $S$ be as in (I0), $\bU$ be as in (I1), and assume that $\mathbf{f}$ satisfies (I2) and (I3). Then for any
$\bx=(\bx_{\nu})_{\nu\in S}\in \bU$, one can find a neighborhood
$\mathbf{V}=\prod V_{\nu}\subseteq \bU$ of $\bx$ and $\alpha_1 >0$ with
the following property: for any ball $\mathbf{B}\subseteq \mathbf{V}$,
there exists $E>0$ such that for any choice of $0<\delta\le 1$,
$T_1,\cdots,T_n\ge 1$, and $K_{\nu}>0$ with $\delta{ (\frac{T_1\cdots
T_n}{\max_i T_i})}\prod K_{\nu}\le 1$ one has
\begin{equation}\label{<eqn}\left|\left\{\bx\in\mathbf{B}|\hspace{1mm}\exists\
\ba\in\Z^n\setminus\{0\}:\begin{array}{l}|\langle \ba .\f(\bx) \rangle|^{l}<\delta\\\\
\|\ba\nabla \f_{\nu}(\bx_\nu)\|_{\nu}<K_{\nu},\hspace{2mm}\nu\in S\\\\
|a_i|_S<T_i, 1 \leq i \leq n\end{array}\right\}\right|\le
E\hspace{.5mm}\varepsilon_1^{\alpha_1}|\mathbf{B}|,\hspace{5mm}
\end{equation}
where
$\varepsilon_1=\max\{\delta^\frac{1}{l},(\delta{ (\frac{T_1\cdots
T_n}{\max_i T_i})}\prod K_{\nu})^{\frac{1}{\l(n+1)}}\},$ where $|S|=l$.
\end{theorem}\noindent
The Theorem above is an $S$-adic analogue of Theorem $1.4$ in \cite{BKM} and is proved using nondivergence estimates for certain flows on homogeneous spaces. We will denote the set in the LHS of (\ref{<eqn}) as
$S(\delta,K_{\nu_1},\cdots,K_{\nu_l},T_1,\cdots,T_n)$ for further reference.
To show (\ref{Inhomo_set}) we want to use the Inhomogeneous transference principle (\ref{transfer}). Assume that $(H_\nu,I_\nu,\Phi)$ satisfies the intersection property and that the product measure is contracting with respect to $(I_\nu,\Phi)$ where, $\Phi:=\{\phi_\delta : 0\leq \delta <\frac{\varepsilon }{2}\} $. Then by (\ref{transfer}) it is enough to show that
\begin{equation}\label{homo_condi}
|\Lambda_H^\nu(\phi_\delta)|=0 \text{ for some } 0<\delta \leq\frac{\varepsilon}{2}.
\end{equation}
\noindent Note that $$\Lambda_H^\nu(\phi_\delta)=\limsup_{\bt\in\bT} \bigcup_{\alpha\in\mathcal{A}} H_{\bt}^\nu(\alpha,\phi_\delta(\bt)).$$
Using Theorem \ref{<}, we will show that $$\sum |\cup_{\alpha \in \mathcal{A}}H_{\bt}^\nu(\alpha,\phi_\delta(\bt))|<\infty $$ for some $0<\delta<\frac{\varepsilon}{2}$. This, together with Borel-Cantelli will give us $|\Lambda_H^\nu(\phi_\delta)|=0$.\\
\noindent By the definition \ref{def_H} of $H_{\bt}^\nu(\alpha,\phi_\delta(\bt)),$ we get
$$\bigcup_{\alpha\in\mathcal{A}}H_{\bt}^\nu(\alpha,\phi_\delta(\bt))\subset S(2^l\phi_\delta(\bt)\Psi_0(2^{\bt}),1,\cdots ,2.\phi_\delta(\bt)r_\nu(\bt),\cdots,1,2^{t_1+2}, \dots, 2^{t_n +2})
$$
i.e. here $K_\nu=2\cdot \phi_\delta(\bt)r_\nu(\bt), K_\omega=1,$ where $\omega\neq\nu$ and $T_i=2^{t_i+2}$.
\subsection{Case $1$ $(\nu=\infty)$} Here $r_\infty(\bt)=2^{(1-\varepsilon)(|\bt|+1)}$. So,
$$
2^l.2^{\delta|\bt|}\Psi_0(2^{\bt}).2.2^{\delta|\bt|}2^{(1-\varepsilon)(|\bt|+1)}.1.\frac{2^{\sum_{1}^n t_i+2}}{2 ^{|\bt|}}=2^{2n+l+2-\varepsilon}.2^{|\bt|(2\delta-\varepsilon)}<1$$
for all large $ \bt$ as $2\delta-\varepsilon<0$. So by Theorem \ref{<} we have
$$|\bigcup_{\alpha\in\mathcal{A}}H_{\bt}^\infty(\alpha,\phi_\delta(\bt))|\leq E\varepsilon_1^{\alpha_1}|\mathbf{B}|,
$$ where $\varepsilon_1=\max\{2.2^{\frac{\delta|\bt|-\sum t_i}{l}},2^{\frac{2n+l+2-\varepsilon}{l(n+1)}}.2^{\frac{|\bt|(2\delta-\varepsilon)}{l(n+1)}}\}
=2^{\frac{2n+l+2-\varepsilon}{l(n+1)}}.2^{\frac{|\bt|(2\delta-\varepsilon)}{l(n+1)}}$ for all large $\bt\in \Z_{\geq 0}^n$.
We note that $\varepsilon_1$ is ultimately the 2nd term in the parenthesis. Because if not then for infinitely many $\bt$,
$$
\frac{\delta|\bt|-\sum t_i}{l}>\frac{|\bt|(2\delta-\varepsilon)}{l(n+1)} + O(1)$$
which implies that
$$\sum t_i<|\bt| + O(1),
$$ a contradiction. Therefore we have $$|\bigcup_{\alpha\in\mathcal{A}}H_{\bt}^\infty(\alpha,\phi_\delta(\bt))| \ll 2^{-\gamma|\bt|},
$$ where $\gamma=\frac{(\varepsilon-2\delta)}{l(n+1)}\alpha_1>0$. Hence $$\sum_{\bt\in \bT}|\bigcup_{\alpha\in\mathcal{A}}H_{\bt}^\infty(\alpha,\phi_\delta(\bt))|\ll\sum_{\bt\in\bT} 2^{-\gamma|\bt|}<\infty.$$
\subsection{Case $2$ ($\nu\neq\infty$)} The argument proceeds as in Case $1$. In this case, $r_\nu(\bt)=2^{-\varepsilon (|\bt|+1)}$. So,
$$2^l.2^{\delta|\bt|}\Psi_0(2^{\bt}).2.2^{\delta|\bt|}2^{(-\varepsilon)(|\bt|+1)}.1.\frac{2^{\sum_{1}^n t_i+2}}{2 ^{|\bt|}}=2^{2n+l+1-\varepsilon}.2^{|\bt|(2\delta-\varepsilon-1)}<1$$
for large $\bt$ as $2\delta-\varepsilon<0$. Therefore, by Theorem \ref{<} we have
$$|\bigcup_{\alpha\in\mathcal{A}}H_{\bt}^\nu(\alpha,\phi_\delta(\bt))|\leq E\varepsilon_1^{\alpha_1}|\mathbf{B}|,$$
where $\varepsilon_1=\max\{2^{\frac{\delta|\bt|-\sum t_i}{l}},2^{\frac{2n+l+1-\varepsilon}{l(n+1)}}.2^{\frac{|\bt|(2\delta-\varepsilon-1)}{l(n+1)}}\}
=2^{\frac{2n+l+1-\varepsilon}{l(n+1)}}.2^{\frac{|\bt|(2\delta-\varepsilon-1)}{l(n+1)}}$ for all large $\bt\in \Z_{\geq 0}^n$.
As in case $1$, $\varepsilon_1$ is ultimately the 2nd term in the parenthesis. For if not, then for infinitely many $\bt$,
$$
\frac{\delta|\bt|-\sum t_i}{l}>\frac{|\bt|(2\delta-\varepsilon-1)}{l(n+1)}+ O(1)$$
which implies that
$$ \sum t_i<2|\bt|+ O(1).$$
This gives a contradiction. Therefore we have $$|\bigcup_{\alpha\in\mathcal{A}}H_{\bt}^\nu(\alpha,\phi_\delta(\bt))|\ll 2^{-\gamma|\bt|},$$ where $\gamma=\frac{(\varepsilon-2\delta+1)}{l(n+1)}\alpha_1>0$. Hence $$\sum_{\bt\in \bT}|\bigcup_{\alpha\in\mathcal{A}}H_{\bt}^\nu(\alpha,\phi_\delta(\bt))|\ll \sum_{\bt\in\bT} 2^{-\gamma|\bt|}<\infty.$$
Consequently the only thing left to verify are the intersection and contracting properties of the transference principle.
\begin{remark}
We will consider $|.|$ the measure to be restricted on some bounded open ball $\bV_{\bx_0}$ around $\bx_0\in \bU$. Then we will get $|\Lambda^\nu_{I}(\phi_\delta)\cap\bV_{\bx_0} |=0$. But because the space is second countable, we eventually get $|\Lambda^\nu_{I}(\phi_\delta)|=0$.
\end{remark}
\subsection{Verifying the intersection property:}
Let $\bt\in\bT$ with $|\bt|> \frac{l}{1-\frac{\varepsilon}{2}}$. We have to show that for $\phi_\delta$ there exists $\phi_\delta^*$ such that for all but finitely many $\bt\in \bT$ and all distinct $\alpha=(a_0,\ba),\alpha'=(a_0',\ba_0')\in\mathcal{A},$ we have that $I_\bt^\nu(\alpha,\phi_\delta(\bt))\cap I_\bt^\nu(\alpha',\phi_\delta(\bt))\subset H_\bt^\nu(\phi_\delta^*(\bt))$. Consider
$$\bx\in I_\bt^\nu(\alpha,\phi_\delta(\bt))\cap I_\bt^\nu(\alpha',\phi_\delta(\bt)),$$
then by Definition (\ref{def_I}) we have
\begin{equation}\label{eqn_1}\left\{\begin{array}{l}
|a_0+\ba.\f(\bx)+\Theta(\bx)|_S<{(\phi_\delta(\bt)\Psi_0(2^\bt))}^\frac{1}{l}\\\\
\|\nabla(\ba.\f_{\nu}(\bx_\nu)+\Theta_\nu(\bx_\nu))\|_\nu<\phi_\delta(\bt) r_\nu(\bt)\end{array}\right.
\end{equation} and
\begin{equation}\label{eqn_2}\left\{\begin{array}{l}
|a'_0+\ba'.\f(\bx)+\Theta(\bx)|_S<{(\phi_\delta(\bt)\Psi_0(2^\bt))}^\frac{1}{l}\\\\
\|\nabla(\ba'.\f_{\nu}(\bx_\nu)+\Theta_\nu(\bx_\nu))\|_\nu<\phi_\delta(\bt) r_\nu(\bt)\end{array}\right.
\end{equation} where
$$|a_i|<2^{t_i+1}\text{ for }1\leq i\leq n \text{ and } |a_i'|<2^{t_i+1}\text{ for } 1\leq i\leq n.$$
Now subtracting the respective equations of (\ref{eqn_2}) from (\ref{eqn_1}) we have $\alpha''=(a_0-a_0',\ba-\ba')$ satisfying the following equations
\begin{equation}\label{eqn_3}
\left\{\begin{array}{l}
|a''_0+\ba''.\f(\bx)|_S^l<2^l\phi_\delta(\bt)\Psi_0(2^\bt)\\\\
\|\nabla(\ba''.\f_{\nu}(\bx_\nu))\|_\nu<2\phi_\delta(\bt) r_\nu(\bt)\\\\
|a''_i|_S\leq 2^{t_i+2} \ \forall\ 1\leq i \leq n .
\end{array} \right.
\end{equation} Observe that $\ba''\neq\mathbf{0}$, because otherwise $$1\leq|a_0''|^l<2^l\phi_\delta(\bt)\Psi_0(2^\bt)<2^l.2^{-{(1-\frac{\varepsilon}{2})}|\bt|},$$ which implies that $|\bt|\leq\frac{l}{1-\frac{\varepsilon}{2}}$, which is true for the finitely many $\bt$'s that we are avoiding. Therefore $\alpha''\in\mathcal{A} $ and $\bx\in H_\bt^\nu(\alpha'',\phi_\delta(\bt))$. So here the particular choice of $\phi_\delta^*$ is $\phi_\delta$ itself. This verifies the intersection property.
\subsection{Verifying the Contraction Property :}
Recall that to verify the contraction property we need to verify the following: for any $\phi_\delta\in \Phi $ we need to find $\Phi_\delta^+\in \Phi$ and a sequence of positive numbers $\{k_{\bt}\}_{\bt\in\bT}$ satisfying $$
\sum_{\bt\in\bT}k_{\bt}<\infty $$
such that for all but finitely many $\bt\in\bT$ and all $\alpha\in\mathcal{A},$ there exists a collection $C_{\bt,\alpha}$ of ball $B$ centred at a point in $\mathbf{S}=\bV=\overbar{\bV}$
satisfying (\ref{inter1}), (\ref{inter2}) and (\ref{inter3}).\\
\noindent
Let us consider the open set $5\bV_{\bx_0}$ in Corollary \ref{good_function}. So we have that
for any $\bt\in \bT $ and $\alpha=(a_0,\ba)\in \mathcal{A}$
\begin{equation}
\mathbf{F}^\nu_{\bt,\alpha}(\bx) :~=\max\{\Psi_0\inv(2^\bt)r_\nu(\bt)|a_0+\ba.\f(\bx)+\Theta(\bx)|_S^l,\|\nabla(\ba.\f_{\nu}+\Theta_\nu)(\bx_\nu)\|\}
\end{equation}
is $(C,\frac{1}{dk})$-good on $5\bV_{\bx_0}$ for some $C>0,k\in\N$ and $d=\max d_\nu$. Using this new function $\mathbf{F}^\nu_{\bt,\alpha},$ we can write the previous inhomogeneous sets as following :\begin{equation}
I_\bt^\nu(\alpha,\phi_\delta(\bt))=\left\{\bx\in\bU:\begin{array}{l}
\mathbf{F}^\nu_{\bt,\alpha}(\bx)<\phi_\delta(\bt)r_\nu(\bt)\\
\\
2^{t_i}\leq\max\{1,|a_i|_S\}<2^{t_i+1} ~~\forall~ 1\leq i\leq n
\end{array}\right\}.\end{equation}\label{inhom_new}
We also note that
$$ I_\bt^\nu(\alpha,\phi_\delta(\bt))\subset I_\bt^\nu(\alpha,\phi^+_\delta(\bt))$$
where $\phi_\delta^+(\bt)=\phi_{\frac{\delta}{2}+\frac{\varepsilon}{4}} (\bt)\geq \phi_\delta(\bt) ~\forall~ \bt\in\bT$. And $\phi_\delta^+(\bt)=\phi_{\frac{\delta}{2}+\frac{\varepsilon}{4}}(\bt)\in\Phi $ because $\frac{\delta}{2}+\frac{\varepsilon}{4}<\frac{\varepsilon}{2} .$
If $I_\bt^\nu(\alpha,\phi_\delta(\bt))=\emptyset$ then it is trivial. So without loss of generality we can assume that $ I_\bt^\nu(\alpha,\phi_\delta(\bt)) \ne \emptyset $.
Because for every $\phi_\delta \in \Phi $ , $\phi_\delta(\bt) \Psi_0(2^\bt)<2^{-(1-\frac{\varepsilon}{2})|\bt|}$, so in particular \begin{equation}
I_\bt^\nu(\alpha,\phi_\delta^+(\bt))\subset \{\bx\in \bU ~:~|a_0+\ba.\f(\bx)+\Theta(\bx)|^l<2^{-(1-\frac{\varepsilon}{2})|\bt|}\}.
\end{equation}
We recall Corollary 4 of \cite{BaBeVe} ,
$$
\inf_{\substack{(\ba, a_0) \in\R^{n+1}\setminus\{0\} \\ \|\ba\| \geq H_0}}\sup_{\bx\in5\bV_{\bx_0}}|a_0+\ba.\f_\infty(\bx_\infty)+\Theta_\infty(\bx_\infty)|_\infty>0.$$
Therefore,
$$\inf_{\substack{(\ba, a_0)\in\R^{n+1}\setminus\{0\} \\ \|\ba\|\geq H_0 }}\sup_{\bx\in5\bV_{\bx_0}}|a_0+\ba.\f(\bx)+\Theta(\bx)|_S > $$
$$\inf_{\substack{(\ba, a_0)\in\R^{n+1}\setminus\{0\} \\ \|\ba\| \geq H_0}}\sup_{\bx\in5\bV_{\bx_0}}|a_0+\ba.\f_\infty(\bx_\infty)+\Theta_\infty(\bx_\infty)|_\infty
> 0.$$
Now by the $(C,\frac{1}{dk})$-good property of the function $|a_0+\ba.\f(\bx)+\Theta(\bx)|_S^l$ on $5\bV_{\bx_0}$ we conclude
$$|I_\bt^\nu(\alpha,\phi_\delta^+(\bt))\cap\bV_{\bx_0}|\leq |\{\bx\in \bV_{\bx_0} ~:~|a_0+\ba.\f(\bx)+\Theta(\bx)|_S^l<2^{-(1-\frac{\varepsilon}{2})|\bt|}\}|$$ $$ \ll2^{-(1-\frac{\varepsilon}{2})(\frac{1}{dk})|\bt|}|\bV_{\bx_0}|$$
for all sufficiently large $|\bt|.$ Therefore $\bV_{\bx_0}\not\subset I_{\bt}^\nu(\alpha,\phi^+_\delta(\bt))$ for sufficiently large $|\bt|$ . The measure restricted to $\bV_{\bx_0}$ will be denoted as $|~~|_{\bV_{\bx_0}}$ and thus $\mathbf{S}=\overbar{\bV_{\bx_0}}$. So $\mathbf{S}\cap I_{\bt}^\nu(\alpha,\phi^+_\delta\bt) $ is open and for every $\bx\in \mathbf{S}\cap I_{\bt}^\nu(\alpha,\phi_\delta(\bt) $ there exists a ball
$$B'(\bx)\subset I_{\bt}^\nu(\alpha,\phi^+_\delta(\bt)).$$
So we can find $\kappa\geq 1$ such that the ball $B=B(\bx):=\kappa B'(\bx)$ satisfies \begin{equation}
5 B(\bx)\subset 5V_{\bx_0}
\end{equation}
and
\begin{equation}\label{twosided_inclusion}
B(\bx)\cap \mathbf{S}\subset I_{\bt}^\nu(\alpha,\phi^+_{\delta}(\bt))\not\supset \bS\cap 5B(\bx)
\end{equation}
holds for all but finitely many $\bt$ . The second inequality holds because we would otherwise have $\bV_{\bx_0}\subset I_{\bt}^\nu(\alpha,\phi^+_{\delta}(\bt))$, a contradiction. Then take $C_{\bt,\alpha}:=\{B(\bx)~:~ \bx\in \mathbf{S}\cap I_{\bt}^\nu(\alpha,\phi_{\delta}(\bt))\} $. Hence (\ref{inter1}) and (\ref{inter2}) are satisfied. By (\ref{twosided_inclusion}) we have
\begin{equation}\label{ineq_1}
\sup_{\bx\in 5B}\mathbf{F}_{\bt,\alpha}^\nu(\bx)\geq \sup_{\bx\in 5B\cap S} \mathbf{F}_{\bt,\alpha}^\nu(\bx)\geq \phi_\delta^+(\bt)r_\nu(\bt)
\end{equation} for all but finitely many $\bt$. So in view of the definitions we get
\begin{equation}\label{ineq_2}
\sup_{\bx\in 5B\cap I_{\bt}^\nu(\alpha,\phi_{\delta}(\bt)) }\mathbf{F}_{\bt,\alpha}^\nu(\bx)\leq 2^{(\frac{\delta}{2}-\frac{\varepsilon}{4})|\bt| }\phi_\delta^+(\bt)r_\nu(\bt)\leq_{\ref{ineq_1}}2^{(\frac{\delta}{2}-\frac{\varepsilon}{4})|\bt|}\sup_{\bx\in 5B}\mathbf{F}_{\bt,\alpha}^\nu(\bx).
\end{equation}
Therefore for all large $|\bt|$ and $\alpha \in \Z^{n+1}$ we have \begin{equation}\begin{split}
|5B\cap I_{\bt}^\nu(\alpha,\phi_{\delta}(\bt))|\leq_{\ref{ineq_2}} &
|\{ \bx\in 5B~:~\mathbf{F}_{\bt,\alpha}^\nu(\bx)\leq
2^{(\frac{\delta}{2}-\frac{\varepsilon}{4})|\bt|}
\sup_{\bx\in 5B}\mathbf{F}_{\bt,\alpha}^\nu(\bx) \} |\\ &\leq C2^{(\frac{\delta}{2}-\frac{\varepsilon}{4})\frac{1}{dk}|\bt|}|5B|.\end{split}
\end{equation}
Hence finally we conclude \begin{equation}
\begin{split}
|5B\cap I_{\bt}^\nu(\alpha,\phi_{\delta}(\bt))|_{\bV}\leq|5B\cap I_{\bt}^\nu(\alpha,\phi_{\delta}(\bt))|
&
\\\leq C2^{(\frac{\delta}{2}-\frac{\varepsilon}{4})\frac{1}{dk}|\bt|}|5B|&\\ \leq
C_\star C2^{(\frac{\delta}{2}-\frac{\varepsilon}{4})\frac{1}{dk}|\bt|}|5B|_{\bV_{\bx_0}},
\end{split}
\end{equation} since $5B\subset5\bV_{\bx_0}$. Here we are using that the measure is doubling and the centre of the ball $5B$ is in $\overbar{\bV_{\bx_0}}$. So $C_\star$ is only dependent on $d_\nu$. We choose $k_{\bt}=C_\star C2^{(\frac{\delta}{2}-\frac{\varepsilon}{4})\frac{1}{dk}|\bt|}$ and as $(\frac{\delta}{2}-\frac{\varepsilon}{4})<0$ we also have $\sum k_{\bt}<\infty$ as required in (\ref{conv}). This verifies the contracting property.
\subsection{The large derivative}
In this section, we will show that $|\cW_{\f}^{\text{large}}(\Psi,\Theta)|=0$. Let us recall Theorem 1.2 from \cite{MoS1}.
\begin{theorem}
Assume that $\mathbf{U}$ satisfies (I1), $\mathbf{f}$ satisfies (I2), (I3) and $0<\epsilon< \frac{1}{4n|S|^2}.$ Let $\mathcal{A}$ be
\begin{equation}\left\{\bx\in\bU|~\exists~\ba \in\Z^n, \frac{T_i}{2}\leq~|a_i|_{S}<T_i,
\begin{array}{l}|\langle \ba . \f(\bx) \rangle|^{l}<\delta(\prod_{i} T_i)^{-1}\\\\
\|\ba . \nabla \f_{\nu}(\bx_\nu)\|_{\nu}>\|\ba\|_S^{-\varepsilon},\hspace{2mm}\nu\neq\infty\\\\
\|\ba . \nabla \f_{\nu}(\bx_\nu)\|_{\nu}>\|\ba\|_S^{1-\varepsilon},\hspace{2mm}\nu=\infty
\end{array}
\right\}.
\end{equation} Then
$|\mathcal{A}|<C \delta\hspace{1mm}|\bU|,$ for large enough $\max (T_i)$ and a universal constant $C$.
\end{theorem}
Note that the function $(\f,\Theta):\bU~\mapsto \Q_S^{n+1}$ satisfies the same properties as $\f$. So as a Corollary of the previous theorem we get,
\begin{corollary}\label{>coro} Let $0<\varepsilon< \frac{1}{4(n+1)|S|^2}$ and $\mathcal{A}_{(T_i)_{1}^n}$ be the set
\begin{equation}
\bigcup_{\substack{(\ba,1)\in\Z^{n+1}\\\frac{T_i}{2}\leq~|a_i|_{S}<T_i}}\left\{\bx\in\bU~|\\
\begin{array}{l}|\langle \ba . \f(\bx)+\Theta(\bx) \rangle|_S^{l}<\delta(\prod_{i=1}^{n} T_i)^{-1}\\\\
\|\nabla( \ba\f_{\nu}(x)+\Theta_\nu(\bx_{\nu}))\|_{\nu}>\|\ba\|_S^{-\varepsilon},\hspace{2mm}\nu\neq\infty\\\\
\|\nabla (\ba\f_{\nu}(x_\nu)+\Theta_\nu(\bx_{\nu}))\|_{\nu}>\|\ba\|_S^{1-\varepsilon},\nu=\infty
\end{array}
\right\}.
\end{equation} Then
$|\mathcal{A}_{(T_i)_{1}^n}
|<C \delta\hspace{1mm}|\bU|,$ for large enough $\max (T_i)$ and a universal constant $C$.
\end{corollary}
Now take $T_i=2^{t_i+1}$ and $\delta=2^{\sum_{1}^n t_i+1}\Psi(2^{\bt})$. As $2^{t_i}\leq|a_i|_S<2^{t_i+1},$ this implies by (\ref{monotone_cond}) that $\Psi(\ba)\geq \Psi(2^{\bt+1})$ and we have using (\ref{>coro}) that
\begin{equation}\label{>measure_inq}
|\bigcup_{2^{t_i}\leq|a_i|_S<2^{t_i+1}}\bW_{\f}^{\text{large}}(\ba,\Psi,\Theta)|<C2^{\sum_{1}^n t_i+1}\Psi(2^{\bt}).
\end{equation}
Note that $$\sum\Psi(\ba)\geq\sum\Psi(2^{t_1+1},\cdots,2^{t_n+1})2^{\sum_{1}^n t_i},$$ so the convergence of $\sum\Psi(\ba)$ implies the convergence of the later. Therefore by (\ref{>measure_inq}) and by the Borel-Cantelli lemma we get that almost every point of $\bU$ are in at most finitely many $\bW_{\f}^{\text{large}}(\ba,\Psi,\Theta)$. Hence $|\cW_{\f}^{\text{large}}(\Psi,\Theta)|=0$ completing the proof.
\section{The divergence theorem for $\Q_p$}
In this section we prove Theorem \ref{thm:divergence} using ubiquitous systems as in \cite{BaBeVe}. In \cite{BBKM}, the related notion of regular systems was used. As mentioned in the introduction, the divergence case will be proved for a more restrictive choice of approximating function than the convergence case, namely for those satisfying property $\mathbf{P}$. Indeed a more general formulation which includes the multiplicative case of the divergence Khintchine theorem remains an outstanding open problem even for submanifolds in $\R^n$. Without loss of generality, and in an effort to keep the notation reasonable, we will prove the Theorem for the usual norm, i.e. we will assume $\bv = (1, \dots, 1)$. The interested reader can very easily make the minor changes required to prove it for general $\bv$. For $\delta > 0$ and $Q > 1$ we follow \cite{BaBeVe} in defining $\Phi^{\f}(Q,\delta) := \{x \in U~:~ \exists~\ba=(a_0,\ba_1) \in \bbZ\times\bbZ^n\backslash \{0\}$ such that
$$ |a_0+ \ba_1 \cdot \f(\bx)|_p < \delta Q^{-{(n+1)}} \text{ and } \|(a_0,\ba_1)\| \leq Q\}.$$
We now recall definition of a $\mathit{nice}$ function.
\begin{definition}[\cite{BaBeVe}, Definition 3.2]\label{nice}
We say that $\f$ is \textit{nice} at $\bx_0\in \bU$ if there exists a neighbourhood $\bU_0\subset \bU$ of $\bx_0$ and constants $0<\delta, w<1$ such that for any sufficiently small ball $\bB\subset \bU_0$ we have that
\begin{equation}
\limsup_{Q\to \infty}|\Phi^{\f}(Q,\delta)\cap \bB |\leq w|\bB|.
\end{equation}
\end{definition}
If $\f$ is \textit{nice} at almost every $\bx_0$ in $\bU$ then $\f$ is called \textit{nice}. The following Theorem from \cite{MoS2} plays a crucial role. It's proof involves a suitable adaptation of the dynamical technique in \cite{BKM}.
\begin{theorem}{\cite{MoS2}}\label{lemma:nice}
Assume that $\f:\bU\to \Q_p^n$ is nondegenerate at $\bx\in \bU$. Then there exists a sufficiently small ball $\bB_0\subset \bU$ centred at $\bx_0$ and a constant $C>0$ such that for any ball $\bB\subset \bB_0$ and any $\delta>0 $, for sufficiently large $Q$, one has
\begin{equation}
|\Phi^{\f}(Q,\delta)\cap \bB |\leq C\delta |\bB|.
\end{equation}
\end{theorem}
This implies that if $\f$ is nondegenerate at $\bx_0$ then $\f$ is nice at $\bx_0$. We will now state the main two theorems of this section. Let $\psi : \mathbb{N} \to \R_{+}$ be a decreasing function.
\begin{theorem}\label{thm:nice}
Assume that $\f:\bU\subset\Q_p^m\to \Q_p^n$ is nice and satisfies the standing assumptions (I1 and I2) and that $s>m-1$. Let $\Theta:\bU\to \Q_p$ be an analytic map satisfying assumption (I5). Let $\Psi(\ba)=\psi(\|\ba\|) ,\ba\in\Z^{n+1} $ be an approximating function. Then,
\begin{equation}\label{main sum}
\mathcal{H}^s(\mathcal{W}^\f_{(\Psi,\Theta)}\cap\bU)=\mathcal{H}^s(\bU) \text{ if } \sum (\Psi(\ba))^{s+1-m}=\infty.
\end{equation}
\end{theorem}
In view of Theorem \ref{lemma:nice}, Theorem \ref{thm:nice} implies Theorem \ref{thm:divergence}. Note that condition (I3) implies the nondegeneracy of $\f$ at every point of $\bU$.
\subsection{ Ubiquitous Systems in $\Q_p^n$}
Let us recall the the definition of Ubiquitous systems in $\Q_p^n$ following \cite{BaBeVe}. Throughout, balls in $\Q_p^m$ are assumed to be defined in terms of the supremum norm $|\cdot|$. Let $\bU$ be a ball in $\Q_p^m$ and $\mathcal{R}=(R_\alpha)_{\alpha\in J}$ be a family of subsets
$R_\alpha\subset \Q_p^m$ indexed by a countable set $J$. The sets $R_\alpha$ are referred to as \emph{resonant sets}. Throughout, $\rho\;:\;\R^+\to\R^+$
will denote a function such that
$\rho(r)\to0$ as $r\to\infty$. Given a set $A\subset \bU$, let
$$
\Delta(A,r):=\{\bx\in \bU\;:\; \dist(\bx,A)<r\}
$$
where $\dist(\bx,A):=\inf\{|\bx-\ba|: \ba\in A\}$. Next, let
$\beta\;:\;J\to \R^+\;:\;\alpha\mapsto\beta_\alpha$ be a positive
function on $J$. Thus the function $\beta$ attaches a `weight'
$\beta_\alpha$ to the set $R_\alpha$. We will assume that for every
$t\in \N$ the set $J_t=\{\alpha\in J: \beta_\alpha\le 2^t\}$ is
finite.\\\\
\noindent\textbf{The intersection conditions:} There exists a constant $\gamma$ with $ 0 \leq \gamma \leq m$ such that for any sufficiently large $t$ and for any
$\alpha\in J_t$, $c\in\cR_\alpha$ and $0< \lambda \le \rho(2^t)$ the
following conditions are satisfied:
\begin{equation}\label{i1}
\big|\bB(c, {\mbox{\small
$\frac{1}{2}$}}\rho(2^t))\cap\Delta(R_\alpha,\lambda)\big| \geq c_1 \,
|\bB(c,\lambda)|\left(\frac{\rho(2^t)}{\lambda}\right)^{\gamma}
\end{equation}
\begin{equation}\label{i2}
\big|\bB\cap
\bB(c,3\rho(2^t))\cap\Delta(R_\alpha,3\lambda)\big| \leq c_2 \,
|\bB(c,\lambda)| \left(\frac{r(\bB)}{\lambda}\right)^{\gamma} \
\end{equation}
where $\bB$ is an arbitrary ball centred on a resonant set with radius
$r(\bB)\le 3 \, \rho(2^t)$. The constants $c_1$ and $ c_2$ are
positive and absolute. The constant $\gamma$ is referred to as the \emph{common dimension} of $\cR$.
\begin{definition}
Suppose that there exists a ubiquitous function $\rho$ and an
absolute constant $k>0$ such that for any ball $\bB\subseteq \bU$
\begin{equation}\label{coveringproperty}
\liminf_{t\to\infty} \left|\bigcup_{\alpha\in
J_t}\Delta(R_\alpha,\rho(2^t))\cap \bB\right| \ \ge \ k\,|\bB|.
\end{equation}
Furthermore, suppose that the intersection conditions \eqref{i1} and \eqref{i2} are satisfied. Then
the system $(\mathcal{R}, \beta)$ is called \emph{locally ubiquitous in $\bU$
relative to $\rho$.}
\end{definition}
Let $(\mathcal{R},\beta)$ be a ubiquitous system in $\bU$ relative to $\rho$ and
$\phi$ be an approximating function. Let $\Lambda(\phi)$ be the set
of points $\bx\in \bU$ such that the inequality
\begin{equation}\label{vb+}
\dist(\bx,R_{\alpha})<\phi(\beta_\alpha)
\end{equation}
holds for infinitely many $\alpha\in J$.\\
We are going to use this following ubiquity lemma from \cite{BaBeVe} in our main proof.
\begin{lemma}\label{ubi}
Let $\phi$ be an approximating function and $(\mathcal{R},\beta)$ be a locally ubiquitous system in $\bU$ relative to $\rho$. Suppose that there is a $0<\lambda<1$ such that $\rho(2^{t+1})<\lambda\rho({2^t})~\forall~ t \in \N.$ Then for any $s>\gamma,$
\begin{equation}
\mathcal{H}^s(\Lambda(\phi))=\mathcal{H}^s(\bU) \text{ if }\sum_{t=1}^\infty \frac{{\phi(2^t)}^{s-\gamma}}{{\rho(2^t)}^{m-\gamma}}=\infty.
\end{equation}
\end{lemma}
We will also need the strong approximation theorem mentioned in \cite{Zelo}.
\begin{lemma} \label{Strong}
For any
$\bar \epsilon = (\epsilon_{\infty},(\epsilon_{p}))
\in \mathbb{R}_{>0}^{2}$ satisfying the inequality
\begin{equation}
\epsilon_{\infty} \geq
\frac{1}{2} \epsilon_{p}^{-1} p,
\end{equation}
there exists a rational number
$r \in \mathbb{Q}$ such that
\begin{equation}
\begin{aligned}
&| r - \xi_{\infty} |_{\infty} \leq \epsilon_{\infty},
\\
&| r - \xi_{p} |_{p} \leq \epsilon_{p}
,
\\
&| r |_{q} \leq 1
\quad \forall~q \neq p.
\end{aligned}
\end{equation}
\end{lemma}
Before we start the proving the main theorem in this section we would like to calculate a covolume formula of certain lattices.
\begin{lemma}\label{covolume}
Suppose $|y_i|_p\leq 1$ then
\begin{equation}\Gamma=\left\{
(q_0, q_1,\cdots, q_n)\in\Z^{n+1} :
\begin{array}{l}
|q_0 + q_1y_1+\cdots+ q_ny_n|_p\leq\frac{1}{p^j},\\\\
|q_i|_p\leq \frac{1}{p}\\\\
i=1,\cdots n
\end{array}\right\}.
\end{equation} is a lattice in $\bbZ^{n+1} $
and $\Vol(\R^{n+1}/\Gamma)= p^{j+n}$.
\end{lemma}
\begin{proof}
First of all $\Gamma$ is a discrete subgroup of $\bbZ^{n+1}$. Clearly $(p^j,0,\cdots,0)\in\Z^{n+1} $ is in $\Gamma$. Since $|y_i|_p\leq 1$ we may take $q_i\in\Z$ such that
\begin{equation}\label{q_conditions}
|q_i-py_i|_p\leq\frac{1}{p^j},
\end{equation}
which implies that $(q_i,0,\cdots,-p,\cdots,0)\in \Gamma$ where $-p$ is in $(i+1)$th position. We claim that $$\{(p^j,0,\cdots,0),(q_i,0,\cdots,-p,\cdots,0)\ | \ i=1,\cdots,n\}$$ is a basis of $\Gamma$. The matrix comprising these elements as column vectors as follows \[
A:= \begin{bmatrix}
p^j & q_1 & & \dots &q_i &\dots & q_n\\
0 & -p & & \dots &0 &\dots & 0 \\
\vdots & \vdots & &\vdots&\vdots &\vdots &\vdots,\\
0 & 0 & & \dots &-p &\dots & 0\\
\vdots & \vdots & &\vdots&\vdots&\vdots&\vdots \\
0 & 0 & &\dots &0 &\dots & -p
\end{bmatrix}.\]
We want to show that if $\mathbf{m}=(m_0,m_1,\cdots,m_n)\in \Gamma $ then there exists $\mathbf{s}=(s_o,s_1,\cdots,s_n)\in \Z^{n+1}$ such that $A\mathbf{s}=\mathbf{m}$. Note that \begin{equation}
A\inv \mathbf{m} = \left(\frac{m_0p+q_1m_1+\cdots+q_nm_n}{p^{j+1}},-\frac{m_1}{p},\cdots,-\frac{m_n}{p}\right).
\end{equation}
As $\mathbf{m}\in \Gamma$ we have that $p|m_i~\forall~ i=1,\cdots,n,$ hence $-\frac{m_i}{p}$ is an integer for all $i$. Now it is enough to show that $p^{j+1} | (m_0p+q_1m_1+\cdots+m_nq_n)$. Note that
$$m_0p+m_1q_1+\cdots+m_nq_n= p(m_0+m_1y_1+\cdots+m_ny_n)+m_1(q_1-y_1p)+\cdots+m_n(q_n-y_np).$$
Now conclusion follows from $\mathbf{m}\in\Gamma$ and (\ref{q_conditions}).
\end{proof}
Now we will construct a ubiquitous system which will give the main result of this section.
\begin{theorem}\label{ubiquity}
Let $\bx_0\in \bU$ be such that $\f$ is \textit{nice} at $\bx_0$ and satisfies (I3). Then there is a neighbourhood $\bU_0$ of $\bx_0,$ constants $\kappa_0>0$ and $\kappa_1>0$ and a collection $\cR:=(R_F)_{F\in\mathcal{F}_n}$ of sets $R_F\subset \widetilde{R_F}\cap \bU_0$ such that the system $(\cR,\beta)$ is locally ubiquitous in $\bU_0$ relative to $\rho(r)=\kappa_1r^{(n+1)} $ with common dimension $\gamma:=m-1,$ where
$$
\mathcal{F}_n:=\left\{F:\bU\to\R \ |\begin{array}{l} F(\bx)= a_0+a_1f_1(\bx)+\cdots+a_nf_n(\bx),\\\\
\ba=(a_0,a_1,\cdots,a_n)\in\Z^{n+1}\setminus\mathbf{0} \end{array} \right \} $$ and given $F\in\mathcal{F}_n$ \begin{equation}
\widetilde{R_F}:=\{\bx\in\bU:\ (F+\Theta)(\bx) \ =\ 0\}
\end{equation}
and $$
\beta:\ \mathcal{F}_n\to \R^+\ : F\to \ \beta_F=\kappa_0|(a_0,a_1,\cdots,a_n)|=\kappa_0|\ba|.
$$
\end{theorem}
\begin{proof}
Let $\pi:\ \Q_p^m\to\Q_p^{m-1}$ be the projection map given by $$\pi(x_1,x_2,\cdots,x_m)=(x_2,\cdots,x_m),$$
and let
\begin{equation}
\widetilde\bV:=\pi(\widetilde R_F\cap\bU_0),
\\
\bV=\bigcup_{3\rho(\beta_F)-\text{balls} B\subset \widetilde{\bV}}\frac{1}{2}B
\end{equation}
and
\begin{equation}
R_F=\left\{\begin{array}{l}
\pi\inv(\bV)\cap\widetilde{R_F} \ \text{if} \ \ |\partial_1(F+\Theta)(\bx)|> \lambda|\nabla(F+\Theta)(\bx)| \ \forall \ \bx\in \bU_0\\\\
\emptyset \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{otherwise}.
\end{array}\right
.\end{equation}
where $0<\lambda<1$ is fixed. \\
We claim that the $R_F$ are resonant sets. The intersection property, namely (\ref{i1}) and (\ref{i2}) can be checked exactly as in the case of real numbers as accomplished in \cite{BaBeVe}, Proposition 5. We only need to note that implicit function theorem for $C^l(U)$ in $\R^n$ was used in \cite{BaBeVe}. The Implicit function theorem in $\Q_p$ holds for analytic maps and all our maps have been assumed analytic, so the proof in \cite{BaBeVe} goes through verbatim.
It remains to check the covering property (\ref{coveringproperty}) to establish ubiquity. Without loss of generality we will assume that the ball $\bU_0$ in the definition of (\ref{nice}) satisfies
\begin{equation}
\diam{\bU_0}\leq \frac{1}{p}.
\end{equation}
From the Definition \ref{nice} of $\f$ being nice at $\bx_0,$ there exist fixed $0<\delta,w<1$ such that for any arbitrary ball $\bB\subset\bU_0,$
\begin{equation}
\limsup_{Q\to \infty}|\Phi^{\f}(Q,\delta)\cap \frac{1}{2}\bB |\leq w|\frac{1}{2}\bB|.
\end{equation}
So for sufficiently large $Q$ we have that
$$
|\frac{1}{2}\bB\setminus \Phi^{\f}(Q,\delta)|\geq \frac{1}{2}(1-w)|\frac{1}{2}\bB|=2^{-m-1}(1-w)|\bB|.
$$
Therefore it is enough to show that
\begin{equation}
\frac{1}{2}\bB\setminus \Phi^{\f}(Q,\delta)\subset\bigcup _{F\in\mathcal{F}_n\\
\beta_F\leq Q}\Delta(R_F,\rho(Q))\cap\bB.
\end{equation}
Suppose $\bx\in \frac{1}{2}\bB\setminus \Phi^{\f}(Q,\delta).$ Consider the lattice
\begin{equation}
\Gamma_{\bx}=\left\{(a_0,a_1,\cdots,a_n)\in\Z^{n+1}: \begin{array}{l}|a_0+a_1f_1(\bx)+\cdots+a_nf_n(\bx)|_p<\delta Q^{-(n+1)}\\\\
|a_i|_p\leq\frac{1}{p} \ \forall \ {1\leq i\leq n}\end{array}\right\},
\end{equation}
and the convex set $K=[-Q,Q]^{n+1}$ in $\R^{n+1}$. Note that $$
|a_o+a_1f_1(\bx)+\cdots+a_nf_n(\bx)|_p<\delta Q^{-(n+1)}$$
if and only if
$$|a_o+a_1f_1(\bx)+\cdots+a_nf_n(\bx)|_p\leq {p^{[\log_p\delta Q^{-(n+1)}]}}.
$$
So by Lemma \ref{covolume} we have that
\begin{equation}
\Vol(\R^{n+1}/\Gamma)=p.p^n.p^{-[\log_pQ^{-(n+1)}\delta]}\leq p^{n+1}\frac{1}{p^{log_p{\delta Q^{-(n+1)}-1}}}\leq Q^{n+1}\frac{p^{n+2}}{\delta}.
\end{equation}
Using the fact that $\bx\notin \Phi^{\f}(Q,\delta) $ we get the first minima $\lambda_1=\lambda_1(\Gamma_{\bx},K)>1$. Therefore using Minkowski's theorem on successive minima, we have that $$
2^{n+1}Q^{n+1}\lambda_1.\lambda_2.\cdots.\lambda_{n+1}\leq 2^{n+1}\Vol(\R^{n+1}/\Gamma_{\bx})\leq 2^{n+1}Q^{n+1}\frac{p^{n+2}}{\delta}.
$$
This implies that $\lambda_{n+1}\leq \frac{p^{n+2}}{\delta}.$ By the definition of $\lambda_{n+1}$ we get $n+1$ linearly independent integer vectors $\ba_j=(a_{j,0},\cdots,a_{j,n})\in\Z^{n+1}(0\leq j\leq n)$ such that the functions $F_j$ given by $$
F_j(\by)=a_{j,0}+a_{j,1}f_1(\by)+\cdots+a_{j,n}f_n(\by)
$$ satisfy
\begin{equation}\label{conditions}
\left\{ \begin{array}{l}
|F_j(\bx)|_p<\delta Q^{-(n+1)}\\\\
|a_{j,i}|_\infty\leq Q.\frac{p^{n+2}}{\delta}\\\\
|a_{j,i}|_p\leq\frac{1}{p} \text{ for } 0\leq i,j \leq n.
\end{array}\right.
\end{equation}
As $\lambda_1>1$ so for every $0\leq j \leq n$ there exists at least one $0\leq j^\star\leq n$ such that $|a_{j,j^\star}|_\infty>Q$.
Now consider the following system of linear equations,\\
\begin{equation}\label{linear}
\begin{array}{l}
\eta_0F_0(\bx)+\eta_1F_1(\bx)+\cdots+\eta_nF_n(\bx)+\Theta(\bx)=0\\\\
\eta_0\partial_1F_0(\bx)+\eta_1\partial_1F_1(\bx)+\cdots+\eta_n\partial_1F_n(\bx)+\partial_1\Theta(\bx)=1\\\\
\eta_0a_{0,j}+\cdots+\eta_na_{n,j}=0 \ \ (2\leq j \leq n).
\end{array}
\end{equation}
Since $\f_1(\bx)=x_1 $, the determinant of this aforementioned system is $\det(a_{j,i})\neq 0$. Therefore there exists a unique solution to the system, say $(\eta_0,\eta_1,\cdots,\eta_n)\in \Q_p^n$. By the argument above, there is at least one $|a_{j,i} |_\infty > Q$. Without loss of generality assume $|a_{0,0}|_\infty >Q$. Using the strong approximation Theorem \ref{Strong} we get $r_i\in\Q$ such that
\begin{equation}\label{r_i}
\begin{aligned}
& |r_i-2p|_\infty\leq p \text{ if } a_{i,0}>0 \text{ otherwise } |r_i+2p|_\infty<p,\\
&
|r_i-\eta_i|_p\leq 1,\\
&
|r_i|_q\leq 1
\quad \text{for} \ \text{ prime }q\neq p.
\end{aligned}
\end{equation}
Now take the function
\begin{equation}\begin{aligned}
F(\by)=r_0F_0(\by)+r_1F_1(\by)+\cdots+r_nF_n(\by)\\\\
=a_0+a_1f_1(\by)+\cdots+a_nf_n(\by),
\end{aligned}
\end{equation} where
\begin{equation}\label{a_i}
a_i=r_0a_{0,i}+r_1a_{1,i}+\cdots+r_na_{n,i},
\ \forall \ i=0,\cdots,n.
\end{equation}
We claim that\\
\textbf{Claim $1$.}The $a_i$ are all integers.\\
From (\ref{r_i}) and (\ref{a_i}) we get
\begin{equation}\label{claim1.1}
|a_i|_q\leq 1, \ \forall \ \ {i}=0,\cdots,n \text{ for } q\neq p
\end{equation}
and by (\ref{r_i}), (\ref{linear}) and (\ref{conditions}) we have
\begin{equation}\begin{aligned}
|a_i|_p\leq \max_{j=0,\cdots,n} \{|\eta_j-r_j|_p|a_{j,i}|_p\}\leq 1
\quad \text{ for } i=2,\cdots,n.
\end{aligned}
\end{equation} So $a_i$ are all integers for $i=2,\cdots,n$.
Now note that $$
F(\bx)+\Theta(\bx)\\
=(r_0-\eta_0)F_0(\bx)+\cdots+(r_n-\eta_n)F_n(\bx).
$$ Therefore we have \begin{equation}\label{condition1}
|(F+\theta)(\bx)|_p\leq \delta Q^{-(n+1)}.
\end{equation}
Again $$
\partial_1(F+\Theta)(\bx)=(r_0-\eta_0)\partial_1F_0(\bx)+\cdots+(r_n-\eta_n)\partial_1F_n(\bx)+1.
$$ Since $|a_{j,i}|_p\leq\frac{1}{p}$ so $|\partial_1F_j(\bx)|_p\leq \frac{1}{p}$ and thus by (\ref{r_i}) we get
\begin{equation}\label{partial_condition}
1-\frac{1}{p}\leq|\partial_1(F+\Theta)(\bx)|_p\leq 1.
\end{equation} \\
Now we can show that $a_1$ and $a_0$ are also integers. Since $f_1(\by)=y_1,$ we have
\begin{equation}
a_1=\partial_1(F+\Theta)(\bx)-\partial_1\Theta(\bx)-\sum_{j =2}^{n}a_j\partial_1f_j(\bx)
\end{equation}
which implies that $|a_1|_p\leq 1$. This together with (\ref{claim1.1}) proves that $a_1$ is an integer. We similarly prove that $a_0$ is an integer. We can write
\begin{equation}\begin{aligned}\label{a_0}
a_0=(F+\Theta)(\bx)-\Theta(\bx)-\sum_{j =1}^{n}a_jf_j(\bx).
\end{aligned}
\end{equation}
This implies that $|a_0|_p\leq 1$ and thus by (\ref{a_0}) and (\ref{claim1.1}) we get that $a_0$ is integer. So the first claim is proved.
Now we look at the infinity norm of the integers $a_i$.
By (\ref{a_i}), (\ref{conditions}) and (\ref{r_i}) we have
\begin{equation}\label{a_infty}
\begin{aligned}
|a_i|_\infty\leq|r_0a_{0,i}+\cdots+r_na_{n,i}|_\infty\\
\leq 3p(n+1)Q.\frac{p^{n+2}}{\delta}
\end{aligned}
\quad \text{ for } i=0,1,\cdots,n.
\end{equation}
By the choice of $r_i$ we have $a_0>0$ and using the fact that $Q<|a_{0,0}|_\infty$ we get that $|a_0|_\infty>pQ$ and therefore $|\ba|>pQ$.
So by (\ref{a_infty}) and the previous observation we get
\begin{equation}\label{beta}
\frac{1}{3p(n+1)}p^{-(n+1)}\delta.Q<\beta_F=\frac{1}{3p(n+1)}p^{-(n+2)}\delta|\ba|\leq Q,\end{equation}
here $\kappa_0=\frac{1}{3p(n+1)}p^{-(n+2)}\delta$.
Note that for all $\by\in\bU_0$ we have
\begin{equation}
\partial_1(F+\Theta)(\bx)=\partial_1(F+\Theta)(\by)+\sum_{j=1}^m\Phi_{j1}(\partial_1(F+\Theta))(\star)(x_j-y_j)
\end{equation}
where $\star$ is from the coefficients of $\bx$ and $\by$. By using (\ref{partial_condition}) and by the fact that $\diam(\bU_0)\leq \frac{1}{p}$ we have
\begin{equation}
|\partial_1(F+\Theta)(\by)|_p\geq 1-\frac{2}{p} \ \ \forall \ \by\in\bU_0.
\end{equation}
So $F$ satisfies $|\partial_1(F+\Theta)(\bx)|> (1-\frac{2}{p})|\nabla(F+\Theta)(\bx)| \ \forall \ \bx\in \bU_0$ and thus by the constructions $\Delta(R_F,\rho(Q))\neq \emptyset$.\\
\textbf{Claim $2$.} $\bx \in \Delta(R_F,\rho(Q))$.\\
We set $r_0 := \diam(\bB)$ and define the function
$$g(\xi) :=(F+\Theta)(x_1+\xi,x_2,\cdots,x_d), \text { where } |\xi|_p<r_0.$$
Then \begin{equation}
\begin{aligned}
|g(0)|_p=|(F+\Theta)(\bx)|_p<\delta Q^{-(n+1)} \\
\text{ and } |g'(0)|_p=|\partial_1(F+\Theta)(\bx)|_p>1-\frac{1}{p}.
\end{aligned}
\end{equation}
Now applying Newton's method there exists $\xi_o$ such that $g(\xi_0)=0$ and $|\xi_0|_p<\frac{p}{(p-1)}\delta Q^{-(n+1)}$. For sufficiently large $Q$ we get $\bx_{\xi_0}=(x_1+\xi_0,x_1,\cdots,x_n)\in \bB,$ that $(F+\Theta)(\bx_{\xi_0})=0$ and that $|\bx-\bx_{\xi_0}|_p\leq \frac{p}{(p-1)}\delta Q^{-(n+1)}$. Then we will argue exactly same as in \cite{BaBeVe}. We recall the argument for the sake of completeness. By the Mean Value Theorem we will get
$$\begin{aligned}
|(F+\Theta)(\by)|_p \ll Q^{-(n+1)}\\
\text{ for any } |\by-\bx_{\xi_0}|_p \ll Q^{-(n+1)}.
\end{aligned}
$$
Then by (\ref{beta}) and using the same argument as above tells us that for sufficiently large $Q>0$ the ball of radius $\rho(\beta_F)$ centred at $\pi\bx_{\xi_0}$ is contained in $\widetilde{\bV}$. This ultimately gives $\bx_{\xi_0}\in R_F$ . Since
$$|\bx-\bx_{\xi_0}|_p\leq \frac{p}{(p-1)}\delta Q^{-(n+1)}$$
so $\bx\in\Delta(R_F,\rho(Q))$ where $\rho(Q)= \frac{p}{(p-1)}\delta Q^{-(n+1)}=\kappa_1Q^{-(n+1)}$.
Therefore $\bx\in \Delta(R_F,\rho(Q))$ for some $F\in\mathcal{F}_n $ such that $\beta_F\leq Q$ and this completes the proof of the Theorem.
\end{proof}
\subsection{ Proof of the main divergence theorem}
Now using Theorem \ref{ubiquity} and lemma \ref{ubi} we can complete the proof of Theorem \ref{thm:nice}.
Fix $\bx_0\in \bU$ and let $\bU_0$ be the neighbourhood of $\bx_0$ which comes from (\ref{ubiquity}). We need to show that
$$\mathcal{H}^s(\mathcal{W}^\f_{(\Psi,\Theta)}\cap\bU_0)=\mathcal{H}^s(\bU_0)
$$ if the series in (\ref{main sum}) diverges. Consider $\phi(r):=\psi(\kappa_0\inv r) $. Our first aim is to show that
\begin{equation}
\Lambda(\phi)\subset \mathcal{W}^\f_{(\Psi,\Theta)}.
\end{equation}
Note that $\bx\in \Lambda(\phi)$ implies the existence of infinitely many $F\in\mathcal{F}_n $
such that $\dist(\bx,R_F)<\phi(\beta_F)$. For such $F\in\mathcal{F}_n$ there exists $\bz\in\bU_0$ such that $(F+\Theta)(\bz)=0$ and $|\bx-\bz|_p<\phi(\beta_F)$. By Mean value theorem
$$
(F+\Theta)(\bx)=(F+\Theta)(\bz)+ \nabla(F + \Theta)(\bx)\cdot (\bx - \bz) + \sum_{i,j}\Phi_{ij}(F+\Theta)(\star)(x_i - z_i)(x_j-z_j), $$
where $\star$ comes from the coefficients of $\bx$ and $\bz $. Then we have that
\begin{equation}
|(F+\Theta)(\bx)|_p\leq|\bx-\bz|_p<\phi(\beta_F)=\phi(\kappa_0 |\ba|)=\Psi(\ba).
\end{equation}
Hence $\Lambda(\phi)\subset \mathcal{W}^\f_{(\Psi,\Theta)} $. Now the Theorem will follow if we can show that
$$\sum_{t=1}^{\infty} \frac{\phi(2^t)^{s-m+1}}{\rho(2^t)}=\infty.$$
Observe that
$$\sum_{t=1}^{\infty} \frac{\phi(2^t)^{s-m+1}}{\rho(2^t)}\asymp \sum_{t=1}^\infty (\psi(\kappa_0\inv 2^t))^{s-m+1}\frac{1}{\rho(2^t)}\\
\asymp \sum_{t=1}^\infty (\psi(\kappa_0\inv 2^t))^{s-m+1}2^{t(n+1)}$$
$$
\gg \sum_{t=1}^\infty \sum_{\kappa_0\inv 2^t<|\ba|\leq\kappa_0\inv2^{t+1} }(\psi(\kappa_0\inv 2^t))^{s-m+1}.$$
As $\psi$ is an approximating function so we got that the above series $$\gg\sum_{t=1}^\infty \sum_{\kappa_0\inv 2^t<|\ba|\leq\kappa_0\inv2^{t+1} }(\psi(|\ba|))^{s-m+1}\asymp \sum_{\ba\in\Z^{n+1}\setminus{0}}(\psi(|\ba|))^{s-m+1}\\
$$ $$=\sum_{\ba\in\Z^{n+1}\setminus{0}}\Psi(\ba)^{s-m+1}=\infty.
$$
This completes the proof of the Theorem.
\section{Concluding Remarks}
\subsection{Some extensions} An interesting possibility is an investigation of the function field case. In \cite{G}, the function field analogue of the Baker-Sprind\v{z}uk conjectures were established and similarly it should be possible to prove the function field analogue of the results in the present paper.
\subsection{Affine subspaces} In \cite{Kleinbock-extremal}, analogues of the Baker-Sprind\v{z}uk conjectures were established for affine subspaces. In this setting, one needs to impose Diophantine conditions on the affine subspace in question. Subsequently, Khintchine type theorems were established (see \cite{G1, G-Monat}), we refer the reader to \cite{G-handbook} for a survey of results. Recently, in \cite{BGGV}, the inhomogeneous analogue of Khintchine's theorem for affine subspaces was established in both convergence and divergence cases. It would be interesting to consider the $S$-adic theory in the context of affine subspaces.
\subsection{Friendly Measures} In \cite{KLW} a category of measures called \emph{Friendly} measures was introduced and the Baker-Sprind\v{z}uk conjectures were proved for friendly measures. Friendly measures include volume measures on nondegenerate manifolds, so the results of \cite{KLW} generalize those of \cite{KM}, but also include many other examples including measures supported on certain fractal sets. In \cite{BeVe}, the inhomogeneous version of the Baker-Sprind\v{z}uk conjectures were established for a class of measures called \emph{strongly contracting} which include friendly measures. It should be possible to prove an $S$-adic inhomogeneous analogue of the Baker-Sprind\v{z}uk conjectures for strongly contracting measures.
| 66,343
|
\begin{document}
\title[A new family of curves in space]{A new family of curves in space}
\author[H\'ector Efr\'en Guerrero Mora]
{H\'ector Efr\'en Guerrero Mora}
\date{\today}
\address{Departamento de Matem\'aticas.
Universidad del Cauca \\ Facultad de Ciencias Naturales Exactas y de la Educaci\'on. Popay\'an, Colombia}
\email{heguerrero@unicauca.edu.co}
\thanks{The author was supported in part by Universidad del Cauca project ID 5124.}
\begin{abstract}
This article defines a new family of curves in space, whose graphs generate shapes similar to whirls.
An intrinsic equation is found, in terms of curvature and torsion, which gives necessary and sufficient conditions for the existence of this family. Its position vector is found, with arc length parameter and finally a way of generating a great variety of examples is shown; in particular an application to rectifying curves is given.
\end{abstract}
\subjclass[2000]{Primary 53A04; Secondary 53A55}
\keywords{Whirl curve, rectifying curve, geodesics, cones in space.}
\dedicatory{To my family.}
\maketitle
\section{MOTIVATION.}
There is a great variety of examples that we can see in nature with swirl shapes; it is very common to find structures, in our environment, with these types of configurations. In the design of creation you can see from huge galaxies to tiny structures with these peculiar shapes.\\ The curves with whirl shapes, that is, curves with spiral type configurations, have captivated the minds of many geometricians. For this reason there are mathematical models that describe spiral configurations.
\section{WHIRL CURVE.}
In this section we will introduce one new families of curves and it is intended that with it, mathematical models can be generated that make an approach to the understanding of the formation and evolution of these swirling structures that appear in nature.
\newpage
\begin{definition}
A whirl curve is a curve with curvature greater than zero, nonzero torsion, and presenting the property that the inner product formed by its normal vector with a fixed direction is proportional to the inner product formed by its tangent vector with that same direction.
\end{definition}
Next, we are going to show some properties of a whirl curve.
\begin{theorem}\label{Caracterizacion}
Let $\alpha$ be a curve, with positive curvature $\kappa$ and nonzero torsion $\tau$, both differentiable functions.
\\ $\alpha$ is a whirl curve if and only if its curvature $\kappa=\kappa(s)$ and its torsion $\tau=\tau(s)$ satisfy:
\begin{equation}\label{ecuación intrinseca}
\tau=\frac{(1+\lambda^2)(\frac{\tau}{\kappa})'}{\lambda(1+\lambda^2+(\frac{\tau}{\kappa})^2)}\ ,
\end{equation}where $\lambda$ is the proportionality constant, with $\lambda\neq 0$.
\end{theorem}
\begin{proof}
let $\alpha$ be a whirl curve and let $\textbf{t},\textbf{n}$ and $\textbf{b}$ be its tangent, normal, and binormal vectors, respectively. Then, there exists a constant vector $\textbf{d}$ of norm one such that
\begin{equation*}
\left <\textbf{n},\textbf{d}\right >=\lambda\left <\textbf{t},\textbf{d}\right >,
\end{equation*} for some constant $\lambda$ other than zero.
Deriving the previous expression and using the Frenet equations, we have
\begin{equation*}
\tau\left <\textbf{b},\textbf{d}\right >=(1+\lambda^2)\kappa\left <\textbf{t},\textbf{d}\right >.
\end{equation*}
\\Since $\{\textbf{t},\textbf{n},\textbf{b}\}$, forms an orthonormal basis of $\R^3$, $\tau \neq 0$ and $\textbf{d}$ is a vector of norm one, we can write
\begin{eqnarray*}
\textbf{d}&=&\left <\textbf{t},\textbf{d}\right >\textbf{t}+\lambda\left <\textbf{t},\textbf{d}\right >\textbf{n}+(1+\lambda^2)(\frac{\kappa}{\tau})\left <\textbf{t},\textbf{d}\right >\textbf{b},
\end{eqnarray*}
this is
\begin{equation*}
\left <\textbf{t},\textbf{d}\right >=\frac{\pm1}{\sqrt{(1+\lambda^2)^2(\frac{\kappa}{\tau})^2+(1+\lambda^2)}}.
\end{equation*} This implies that
\begin{equation*}
\textbf{d}=\pm\frac{(\textbf{t}+\lambda\textbf{n}+(1+\lambda^2)(\frac{\kappa}{\tau})\textbf{b})}{\sqrt{(1+\lambda^2)^2(\frac{\kappa}{\tau})^2+(1+\lambda^2)}}.
\end{equation*}
And since $\textbf{d}$ is a constant vector, without loss of generality, we can consider the positive sign and when finding its derivative, we have
\begin{eqnarray*}
\textbf{0}&=&-\frac{(1+\lambda^2)(\frac{\kappa}{\tau})(\frac{\kappa}{\tau})'+\lambda\kappa((1+\lambda^2)(\frac{\kappa}{\tau})^2+1)}{(1+\lambda^2)^{1/2}((1+\lambda^2)(\frac{\kappa}{\tau})^2+1)^{3/2}}\textbf{t}\\
&-&\lambda\left (\frac{(1+\lambda^2)(\frac{\kappa}{\tau})(\frac{\kappa}{\tau})'+\lambda\kappa((1+\lambda^2)(\frac{\kappa}{\tau})^2+1)}{(1+\lambda^2)^{1/2}((1+\lambda^2)(\frac{\kappa}{\tau})^2+1)^{3/2}}\right)\textbf{n}\\
&+&\frac{(1+\lambda^2)(\frac{\kappa}{\tau})(\frac{\kappa}{\tau})'+\lambda\kappa((1+\lambda^2)(\frac{\kappa}{\tau})^2+1)}{\frac{\kappa}{\tau}(1+\lambda^2)^{1/2}((1+\lambda^2)(\frac{\kappa}{\tau})^2+1)^{3/2}}\textbf{b}.
\end{eqnarray*}And this implies that
\begin{equation*}
\tau=\frac{(1+\lambda^2)(\frac{\tau}{\kappa})'}{\lambda(1+\lambda^2+(\frac{\tau}{\kappa})^2)} \ ,
\end{equation*}where $\lambda$ is the constant of proportionality other than zero.
Now, suppose that the curve $\alpha=\alpha(s)$, with curvature $\kappa=\kappa(s)$ and torsion $\tau=\tau(s)$ satisfies the intrinsic equation (\ref{ecuación intrinseca})
and let us define the next vector
\begin{equation}\label{eje}
\textbf{d}=\pm\frac{(\textbf{t}+\lambda\textbf{n}+(1+\lambda^2)(\frac{\kappa}{\tau})\textbf{b})}{\sqrt{(1+\lambda^2)^2(\frac{\kappa}{\tau})^2+(1+\lambda^2)}}.
\end{equation}It is very easy to appreciate that vector $\textbf{d}$ is of norm one, that its derivative is equal to zero and a direct calculation shows that
$\left <\textbf{n},\textbf{d}\right >-\lambda\left <\textbf{t},\textbf{d}\right >=0$, this is $\alpha$ is a whirl curve.
\end{proof}
\begin{definition}
The vector $\textbf{d}$ defined by (\ref{eje}) is called the axis of the whirl curve.
\end{definition}
The proof of the following theorem is based on a technique that appears in \cite{GueMora}.
\begin{theorem}\label{parametrizacion}
Let $\alpha$ be a curve, with positive curvature $\kappa$ and nonzero torsion $\tau$, both continuously differentiable functions.
\\$\alpha$ is a whirl curve if and only if its position vector $\alpha(s)=(x(s),y(s),z(s))$, except for a rigid motion, has natural representation of the form:
\\
\\
$x(s)=\\
\int\sqrt{1-\frac{e^{2(\lambda\int_{s_0}^s{\kappa}ds-B)}}{1+\lambda^2}}
\cos(\arctan[\frac{\sqrt{1-e^{2(\lambda\int_{s_0}^s{\kappa}ds-B)}}}{\lambda}]-
\frac{\emph{arctanh}[\sqrt{1-e^{2(\lambda\int_{s_0}^s{\kappa}ds-B)}}]}{\lambda})ds,$\\ \\
$y(s)=\\
\int\sqrt{1-\frac{e^{2(\lambda\int_{s_0}^s{\kappa}ds-B)}}{1+\lambda^2}}
\sin(\arctan[\frac{\sqrt{1-e^{2(\lambda\int_{s_0}^s{\kappa}ds-B)}}}{\lambda}]-
\frac{\emph{arctanh}[\sqrt{1-e^{2(\lambda\int_{s_0}^s{\kappa}ds-B)}}]}{\lambda})ds,$\\ \\
$z(s)=\pm\int{\frac{e^{(\lambda\int_{s_0}^s{\kappa}ds-B)}}{\sqrt{1+\lambda^2}}}ds,$\ \ where $\lambda\int_{s_0}^s{\kappa}ds<B.$ \\ \\
\end{theorem}
\begin{proof}
Suppose that $\alpha$ is a whirl curve.
Let us find a parametrization by arc length of the curve $\alpha.$
Let us write its tangent vector $\textbf{t}=\textbf{t}(s)$ in spherical coordinates,
\begin{equation}\label{vector tangente}
\textbf{t} =(\sin \phi\cos \theta,\sin \phi\sin \theta,\cos \phi).
\end{equation}Therefore, its normal vector $\textbf{n}=\textbf{n}(s)$ and its binormal vector $\textbf{b}=\textbf{b}(s)$ are given by
\begin{eqnarray*}\textbf{n}&=&
(\frac{\phi'\cos \phi\cos \theta-\theta'\sin \phi\sin \theta}{\kappa},\frac{ \phi'\cos \phi\sin \theta +\theta'\sin\phi\cos \theta}{\kappa},\frac{-\phi'\sin \phi}{\kappa}),
\\ \textbf{b}&=&(\frac{-\phi'\sin \theta}{\kappa}-\frac{\theta'\sin 2\phi\cos\theta}{2\kappa},\frac{-\phi'\cos \theta}{\kappa}-\frac{\theta'\sin 2\phi\sin \theta}{2\kappa},\frac{\theta'\sin^2 \phi}{\kappa}).\end{eqnarray*}
It is known that its Frenet trihedron $\textbf{t},\textbf{n},\textbf{b}$ forms an orthonormal basis of $\R^3$ and satisfies
\begin{eqnarray*}
\frac{d\textbf{t}}{ds}&=&\kappa \textbf{n},\\
\frac{d\textbf{n}}{ds}&=&-\kappa\textbf{t}+\tau\textbf{b},\\
\frac{d\textbf{b}}{ds}&=&-\tau \textbf{n}.
\end{eqnarray*}For a fixed unit vector $D$, we have
\begin{equation*}
\left <\frac{d\textbf{t}}{ds},D\right >=\left<\kappa \textbf{n},D\right >=\kappa\left<\textbf{n},D\right>,
\end{equation*} and
\begin{eqnarray*}
\left<\frac{d\textbf{n}}{ds},D\right>&=&\left<-\kappa\textbf{t}+\tau\textbf{b},D\right>=
-\kappa\left<\textbf{t},D\right>+\tau\left<\textbf{b},D\right>\\&=&-\kappa \left<\textbf{t},D\right>\pm\tau\sqrt{1-\left<\textbf{t},D\right>^2-\left<\textbf{n},D\right>^2}.
\end{eqnarray*}
Now, considering the definition of the curve $\alpha$, we have\\ $\left<\textbf{n},\textbf{d}\right>=\lambda\left< \textbf{t},\textbf{d}\right>$,
for some constant vector $\textbf{d}$ of magnitude one. It is clear that there is an orthogonal transformation $\sigma:\R^3\rightarrow \R^3$, with a positive determinant such that $\sigma(\textbf{d})=(0,0,1)$ and knowing that the curvature $\kappa$ and the torsion $\tau$ are invariant, given a rigid motion, we have that the $\alpha$ and $\sigma\circ\alpha$ curves have the same curvature and torsion functions. We can assume that $D=\textbf{d}=(0,0,1)$.
Our goal is to find $\phi$ and $\theta.$
Writing $\xi=\left< \textbf{t},\textbf{d}\right>$, let us look for a solution of the form $\frac{1}{\kappa}\frac{d\xi}{ds}=\lambda\xi$,
with $\xi\neq 0$, this is $\xi =\pm e^{(\lambda\int_{s_0}^s{\kappa ds}-C)}$,
where $C$ is an integration constant.\\
Now, let us use the intrinsic equation of curve $\alpha$, which is the same for curve $\sigma\circ \alpha.$
\begin{equation*}
\tau=\frac{(1+\lambda^2)(\frac{\tau}{\kappa})'}{\lambda(1+\lambda^2+(\frac{\tau}{\kappa})^2)} \ ,
\end{equation*}where $\lambda$ is the proportionality constant. Which we can write it as:
\begin{equation}\label{Curvatura}
\kappa=\frac{(1+\lambda^2)(\frac{\tau}{\kappa})'}{\lambda(\frac{\tau}{\kappa})(1+\lambda^2+(\frac{\tau}{\kappa})^2)} \ ,
\end{equation}integrating the above equation, we have:
\begin{equation*}
\int_{s_0}^s{\kappa(s)}ds=\frac{1}{\lambda}(\ln[\frac{1}{\sqrt{1+\lambda^2}}\mid\frac{\tau(s)}{\kappa(s)} \mid]-\ln[\sqrt{1+\frac{1}{1+\lambda^2}(\frac{\tau(s)}{\kappa(s)})^2}]+B),
\end{equation*} where $B=-\displaystyle\ln(\frac{\frac{1}{\sqrt{1+\lambda^2}}\mid\frac{\tau(s_0)}{\kappa(s_0)}\mid}{\sqrt{1+\frac{1}{1+\lambda^2}(\frac{\tau(s_0)}{\kappa(s_0)})^2}})$.\\
Then \begin{equation}\label{Exponencial}
e^{\lambda\int_{s_0}^s\kappa ds-B}=\displaystyle\frac{\frac{1}{\sqrt{1+\lambda^2}}\mid\frac{\tau}{\kappa}\mid}{\sqrt{1+\frac{1}{1+\lambda^2}(\frac{\tau}{\kappa})^2}}<1,
\end{equation} this implies
\begin{equation*}
\frac{1}{\sqrt{1+\lambda^2}}\frac{\tau}{\kappa}=\pm\frac{e^{(\lambda\int_{s_0}^s\kappa ds)}}{\sqrt{e^{2B}-e^{2\lambda\int_{s_0}^s\kappa ds}}},
\end{equation*}
this is
\begin{equation*}
\tau=\pm\frac{\sqrt{1+\lambda^2} \kappa e^{(\lambda\int_{s_0}^s\kappa ds-B)}}{\sqrt{1-e^{2(\lambda\int_{s_0}^s \kappa ds-B)}}},
\end{equation*}
Now, let us find the relationships between $C$ and $B$. To do this, consider the equation:
\begin{eqnarray*}
\frac{d}{ds}(\frac{1}{\kappa}\frac{d\xi}{ds})=\frac{d}{ds}\left<\frac{\textbf{t}'}{\kappa},\textbf{d}\right>&=&-\kappa \left<\textbf{t},\textbf{d}\right>\pm\tau\sqrt{1-\left<\textbf{t},\textbf{d}\right>^2-\left<\textbf{n},\textbf{d}\right>^2}\\
&=&-\kappa \xi\pm\tau\sqrt{1-(1+\lambda^2)\xi^2},
\end{eqnarray*}
and consider the case
\begin{equation*}
\xi =e^{(\lambda\int_{s_0}^s\kappa ds-C)},\ \ \tau=\frac{\sqrt{1+\lambda^2} \kappa e^{(\lambda\int_{s_0}^s \kappa ds-B)}}{\sqrt{1-e^{2(\lambda\int_{s_0}^s \kappa ds-B)}}}.
\end{equation*}
Replacing, we have
\begin{eqnarray*}
\lambda^2\kappa e^{(\lambda\int_{s_0}^s\kappa ds-C)}=&&-\kappa e^{(\lambda\int_{s_0}^s\kappa ds-C)}\\
&&\pm\frac{\sqrt{1+\lambda^2} \kappa e^{(\lambda\int_{s_0}^s \kappa ds-B)}}{\sqrt{1-e^{2(\lambda\int_{s_0}^s \kappa ds-B)}}}\sqrt{1-(1+\lambda^2)e^{2(\lambda\int_{s_0}^s \kappa ds-C)}},
\end{eqnarray*} this is
\begin{equation*}
\frac{(1+\lambda^2)}{\sqrt{1+\lambda^2}}e^{B-C}=\pm\frac{\sqrt{1-(1+\lambda^2)e^{2(\lambda\int_{s_0}^s \kappa ds-C)}} }{\sqrt{1-e^{2(\lambda\int_{s_0}^s \kappa ds-B)}}},
\end{equation*}and this implies that $(1+\lambda^2)e^{-2C}=e^{-2B},$
the other cases not affecting the relationship between the constants $B$ and $C$.
Therefore
\begin{equation*}
\cos\phi=\left<\textbf{t},(0,0,1)\right>=\left<\textbf{t},\textbf{d}\right>=\xi=\pm\frac{e^{(\lambda\int_{s_0}^s{\kappa ds}-B)}}{\sqrt{1+\lambda^2}}.
\end{equation*}
Now, to find the angle $\theta$, let us use the equation:
\begin{equation*}
\frac{\theta'\sin^2 \phi}{\kappa}=\left<\textbf{b},(0,0,1)\right>=\frac{1}{\tau}\left<\frac{d\textbf{n}}{ds},(0,0,1)\right>+\frac{\kappa}{\tau}\left<\textbf{t},(0,0,1)\right>.
\end{equation*}
Replacing
\begin{equation*}
\xi=\frac{e^{(\lambda\int_{s_0}^s{\kappa ds}-B)}}{\sqrt{1+\lambda^2}},\ \ \ \ \tau=\pm\frac{\sqrt{1+\lambda^2} \kappa e^{(\lambda\int_{s_0}^s \kappa ds-B)}}{\sqrt{1-e^{2(\lambda\int_{s_0}^s \kappa ds-B)}}},
\end{equation*} we have
\begin{equation*}
\theta'=\pm\frac{(1+\lambda^2)\kappa\sqrt{1-e^{2(\lambda\int_{s_0}^s \kappa ds-B)}}}{1+\lambda^2-e^{2(\lambda\int_{s_0}^s\kappa ds-B)}}, \ \ \text{respectively}.
\end{equation*}
Analogously, when considering the case
\begin{equation*}
\xi=-\frac{e^{(\lambda\int_{s_0}^s{\kappa ds}-B)}}{\sqrt{1+\lambda^2}},\ \ \ \ \tau=\pm\frac{\sqrt{1+\lambda^2} \kappa e^{(\lambda\int_{s_0}^s \kappa ds-B)}}{\sqrt{1-e^{2(\lambda\int_{s_0}^s \kappa ds-B)}}},
\end{equation*}
we have
\begin{equation*}
\theta'=\mp\frac{(1+\lambda^2)\kappa\sqrt{1-e^{2(\lambda\int_{s_0}^s \kappa ds-B)}}}{1+\lambda^2-e^{2(\lambda\int_{s_0}^s\kappa ds-B)}}, \ \ \text{respectively}.
\end{equation*}
If we replace the different cases that $\xi$ and $\theta$ take in (\ref{vector tangente}), then several expressions are generated for the tangent vector, which are simplified in the following two, after applying the respective rigid motion.
\begin{eqnarray*}
\textbf{t} =(&&\sqrt{1-\frac{e^{2(\lambda\int_{s_0}^s{\kappa}ds-B)}}{1+\lambda^2}}\cos \int{\frac{(1+\lambda^2)\kappa\sqrt{1-e^{2(\lambda\int_{s_0}^s \kappa ds-B)}}}{1+\lambda^2-e^{2(\lambda\int_{s_0}^s \kappa ds-B)}}}ds,\\ && \sqrt{1-\frac{e^{2(\lambda\int_{s_0}^s{\kappa}ds-B)}}{1+\lambda^2}}\sin \int{\frac{(1+\lambda^2)\kappa\sqrt{1-e^{2(\lambda\int_{s_0}^s \kappa ds-B)}}}{1+\lambda^2-e^{2(\lambda\int_{s_0}^s \kappa ds-B)}}}ds,\\&&\pm\frac{e^{(\lambda\int_{s_0}^s{\kappa}ds-B)}}{\sqrt{1+\lambda^2}}).
\end{eqnarray*} And since
\begin{eqnarray*}
&&\int{\frac{(1+\lambda^2)\kappa\sqrt{1-e^{2(\lambda\int_{s_0}^s \kappa ds-B)}}}{1+\lambda^2-e^{2(\lambda\int_{s_0}^s \kappa ds-B)}}}ds\\
&=&\int{\frac{-\lambda^2\kappa e^{2(\lambda\int_{s_0}^s \kappa ds-B)}}{\sqrt{1-e^{2(\lambda\int_{s_0}^s \kappa ds-B)}}(1+\lambda^2-e^{2(\lambda\int_{s_0}^s \kappa ds-B)})}}ds\\&&+\int{\frac{\kappa}{\sqrt{1-e^{2(\lambda\int_{s_0}^s \kappa ds-B)}}}}ds\\&=&\arctan[\frac{\sqrt{1-e^{2(\lambda\int_{s_0}^s{\kappa}ds-B)}}}{\lambda}]-
\frac{\emph{arctanh}[\sqrt{1-e^{2(\lambda\int_{s_0}^s{\kappa}ds-B)}}]}{\lambda},
\end{eqnarray*} with the integration constant equal to zero, the desired implication is obtained, this means we have the representation of the position vector, except for a rigid motion.\\
To prove the reciprocal, it is enough to check that the curvature and torsion of the natural representation of the curve $\alpha$ are given by
\begin{equation*}
\kappa_{\alpha}=\mid\mid \alpha''\mid\mid=\kappa,\ \ \ \ \tau_{\alpha}=\frac{\alpha'\wedge \alpha''\cdot \alpha'''}{\kappa^2}=\pm\frac{\sqrt{1+\lambda^2} \kappa e^{(\lambda\int_{s_0}^s \kappa ds-B)}}{\sqrt{1-e^{2(\lambda\int_{s_0}^s \kappa ds-B)}}}, \ \text{respectively}.
\end{equation*}
And a direct calculation shows that the curvature $\kappa_{\alpha}$ and the torsion $\tau_{\alpha}$ satisfy the intrinsic equation:
\begin{equation*}
\frac{(1+\lambda^2)(\frac{\tau_{\alpha}}{\kappa_{\alpha}})'}{\lambda(1+\lambda^2+(\frac{\tau_{\alpha}}{\kappa_{\alpha}})^2)}=\tau_{\alpha},\ \text{respectively}.
\end{equation*}
\end{proof}
\begin{remark}\label{Ecuacion simplificada}
The whirl curve $\alpha(s)=(x(s),y(s),z(s))$ can be written in the following two ways
\begin{eqnarray*}
x(s)&=&\int\frac{\lambda}{\sqrt{1+\lambda^2}}\cos(\frac{\emph{arctanh}[\sqrt{1-e^{2(\lambda\int_{s_0}^s{\kappa}ds-B)}}]}{\lambda})\\
&&+\frac{\sqrt{1-e^{2(\lambda\int_{s_0}^s{\kappa}ds-B)}}}{\sqrt{1+\lambda^2}}\sin(\frac{\emph{arctanh}[\sqrt{1-e^{2(\lambda\int_{s_0}^s{\kappa}ds-B)}}]}{\lambda})ds,\\ \\
y(s)&=&\int\frac{\sqrt{1-e^{2(\lambda\int_{s_0}^s{\kappa}ds-B)}}}{\sqrt{1+\lambda^2}}\cos(\frac{\emph{arctanh}[\sqrt{1-e^{2(\lambda\int_{s_0}^s{\kappa}ds-B)}}]}{\lambda})
\\&&-\frac{\lambda}{\sqrt{1+\lambda^2}}\sin(\frac{\emph{arctanh}[\sqrt{1-e^{2(\lambda\int_{s_0}^s{\kappa}ds-B)}}]}{\lambda})ds,\\ \\
z(s)&=&\pm\int{\frac{e^{(\lambda\int_{s_0}^s{\kappa}ds-B)}}{\sqrt{1+\lambda^2}}}ds, \ \text{where}\ \lambda\int_{s_0}^s{\kappa}ds<B,
\end{eqnarray*}
with curvature $\kappa_{\alpha}=\kappa$ and torsion $\tau_{\alpha}=\pm\frac{\sqrt{1+\lambda^2} e^{(\lambda\int_{s_0}^s \kappa ds-B)}\kappa}{\sqrt{1-e^{2(\lambda\int_{s_0}^s\kappa ds-B)}}}$, respectively.\\
\end{remark}
\section{APPLICATIONS}
By writing the quotient $\frac{\tau(s)}{\kappa(s)}=h(s)$ and replacing in (\ref{Curvatura}) and (\ref{Exponencial}), we have
\begin{equation}\label{Curvatura y formula}
\kappa(s)=\frac{(1+\lambda^2)h'(s)}{\lambda h(s)(1+\lambda^2+(h(s))^2)}\ \ \text{and} \ \ e^{\lambda\int_{s_0}^s\kappa(s) ds-B}=\displaystyle\frac{\frac{1}{\sqrt{1+\lambda^2}}\mid h(s)\mid}{\sqrt{1+\frac{1}{1+\lambda^2}(h(s))^2}}
\end{equation} and these expressions allow us to construct a great variety of interesting curves.
\subsection{APPLICATIONS TO THE RECTIFYING CURVES}
The notion of rectifying curve has been introduced by Chen and is defined as a unit speed curve such that position vector always lies in its rectifying plane. He found a simple characterization in terms of the ratio $\tau/\kappa$, for the rectifying curves. He proved the following theorem (For the proof of the theorem, see \cite{ChenBY:03})
\begin{theorem}\label{Caracterizacion 2}
Let $\textbf{x}:I\rightarrow \mathbb{R}^3$ be a curve with $\kappa>0$. Then $\textbf{x}$ is congruent to a rectifying curve if and only if the ratio of torsion and curvature of the curve is a nonconstant linear function in arclength function $s$, i.e.,$\tau/\kappa=c_1s+c_2$ for some constants $c_1$ and $c_2$, with $c_1\neq 0.$
\end{theorem}
In this section we will find a whirl curve that satisfies the definition of being a rectifying curve. For this we consider the theorem \ref{Caracterizacion}, the theorem \ref{parametrizacion}, the remark \ref{Ecuacion simplificada}, the theorem \ref{Caracterizacion 2} and the formulas (\ref{Curvatura y formula}). The proof is not difficult, it just takes a bit of work to calculate the integral and verify that the curvature and torsion satisfy the respective intrinsic equations.
\begin{theorem}
Let $\sigma:I\rightarrow \mathbb{R}^3$ be a curve, with positive $\kappa_{\sigma}=\kappa_{\sigma}(s)$ curvature and nonzero $\tau_{\sigma}=\tau_{\sigma}(s)$ torsion, both differentiable functions.\\
Then $\sigma$ is congruent to a whirl curve and to a rectifying curve if and only if its position vector $\sigma(s)=(x(s),y(s),z(s))$, except for a rigid motion, has natural representation of the form:
\begin{eqnarray*}
x(s)&=&\frac{(b+as)}{a}\frac{\lambda}{\sqrt{1+\lambda^2}}\cos{[\frac{arctanh(\sqrt{\frac{1+\lambda^2}{1+(b+as)^2+\lambda^2}})}{\lambda}]},\\
y(s)&=&-\frac{(b+as)}{a}\frac{\lambda}{\sqrt{1+\lambda^2}}\sin{[\frac{arctanh(\sqrt{\frac{1+\lambda^2}{1+(b+as)^2+\lambda^2}})}{\lambda}]},\\
z(s)&=&\frac{1}{a}\frac{\sqrt{1+(b+as)^2+\lambda^2}}{\sqrt{1+\lambda^2}},
\end{eqnarray*} where $\lambda$, $a$ and $b$ are constants, with $\lambda\neq 0$, $a\neq 0$ and the curve $\sigma$ is defined for $as+b>0$, respectively $as+b<0$.
\end{theorem}
The previous results motivate us to give the following definition.
\begin{definition}
Let $\sigma:I\rightarrow \R^3$ be a unit speed curve with curvature $\kappa_{\sigma}>0$ and let $\{\kappa_{\sigma},\tau_{\sigma},\textbf{t},\textbf{n},\textbf{b} \}$ be the Frenet-Serret apparatus of $\sigma$. Then $\sigma$ is said to be a whirl-rectifying curve if it satisfies the following two conditions:
\begin{enumerate}
\item There exists a constant vector $\textbf{d}$ of norm one such that\\
$\left<\textbf{n}(s),\textbf{d}\right>=\lambda\left<\textbf{t}(s),\textbf{d}\right>$, for some constant $\lambda$ other than zero,
\item $\left<\sigma(s),\textbf{n}(s)\right>=0.$
\end{enumerate}
\end{definition}
\begin{theorem}
Let $\sigma:I\rightarrow \mathbb{R}^3$ be a curve, with positive curvature $\kappa_{\sigma}=\kappa_{\sigma}(s)$ and nonzero torsion $\tau_{\sigma}=\tau_{\sigma}(s)$, both differentiable functions.\\
If $\sigma$ is a whirl-rectifying curve given by $\sigma(s)=(x(s),y(s),z(s))$, where
\begin{eqnarray*}
x(s)&=&\frac{(b+as)}{a}\frac{\lambda}{\sqrt{1+\lambda^2}}\cos{[\frac{arctanh(\sqrt{\frac{1+\lambda^2}{1+(b+as)^2+\lambda^2}})}{\lambda}]},\\
y(s)&=&-\frac{(b+as)}{a}\frac{\lambda}{\sqrt{1+\lambda^2}}\sin{[\frac{arctanh(\sqrt{\frac{1+\lambda^2}{1+(b+as)^2+\lambda^2}})}{\lambda}]},\\
z(s)&=&\frac{1}{a}\frac{\sqrt{1+(b+as)^2+\lambda^2}}{\sqrt{1+\lambda^2}},
\end{eqnarray*}where $\lambda$, $a$ and $b$ are constants, with $\lambda\neq 0$, $a\neq 0$ and the curve $\sigma$ is defined for\\ $as+b>0$, respectively $as+b<0$, then $\sigma$ lives in two-leaf hyperboloid
\begin{equation*} z^2-\frac{x^2}{\lambda^2}-\frac{y^2}{\lambda^2}=\frac{1}{a^2},\end{equation*} and is also a geodesic curve on the cone with vertex at the origin and parametrized by $\textbf{X}(t,u)=u\textbf{w}(t)$, where $u\in \R^+$ and $w=w(t)$ is a curve with unit speed on the unitary sphere with center at the origin and parametrized by
\\ \\
$\textbf{w}(t)=(\frac{\mid a\mid}{a}\frac{\lambda}{\sqrt{1+\lambda^2}}\sin[d+t]\cos[\frac{arctanh[\frac{\sqrt{1+\lambda^2}}{\sqrt{1+\tan^2[d+t]+\lambda^2}}]}{\lambda}],\\ \\ -\frac{\mid a\mid}{a}\frac{\lambda}{\sqrt{1+\lambda^2}}\sin[d+t]\sin[\frac{arctanh[\frac{\sqrt{1+\lambda^2}}{\sqrt{1+\tan^2[d+t]+\lambda^2}}]}{\lambda}],\frac{\mid a\mid}{a}\cos[d+t]\frac{\sqrt{1+\tan^2[d+t]+\lambda^2}}{\sqrt{1+\lambda^2}}),
$\\ \\
defined for $-d<t<-d+\frac{\pi}{2}$, respectively $-d-\frac{\pi}{2}<t<-d.$
\end{theorem}
\begin{proof}
The curve is in the two-leaf hyperboloid, since
\begin{equation*}
z(s)^2-\frac{(x(s)^2+y(s)^2)}{\lambda^2}=\frac{1+(b+as)^2+\lambda^2}{a^2(1+\lambda^2)}-\frac{(b+as)^2}{a^2(1+\lambda^2)}=\frac{1}{a^2}.
\end{equation*}Now taking
\begin{eqnarray*}t(s)&=&-d+\arctan{(b+as)},\\ u(s)&=&\frac{\sqrt{1+(b+as)^2}}{\mid a\mid},
\end{eqnarray*} we see that $\textbf{X}(t(s),u(s))=u(s)\textbf{w}(t(s))=\sigma(s)$, which shows that the curve is on the cone.
A direct calculation shows that the normal vector of the cone at point $\textbf{X}(t(s),u(s))$ is parallel to the normal vector of the curve at point $\sigma(s)$,
which shows that $\sigma$ is a geodesic curve \cite{BANG-Y-C:07}, \cite{GueMoraDos}.
\end{proof}
\section{CONTINUOUS EXTENSIONS IN WHIRL CURVES}
Considering the equations that appear in (\ref{Curvatura y formula}), and writing $h(s)=as+b$, we have
\begin{equation*}
\kappa(s)=\frac{(1+\lambda^2)a}{\lambda (as+b)(1+\lambda^2+(as+b)^2)}\ \ \text{and} \ \ e^{\lambda\int_{s_0}^s\kappa(s) ds-B}=\displaystyle\frac{\frac{1}{\sqrt{1+\lambda^2}}\mid as+b\mid}{\sqrt{1+\frac{1}{1+\lambda^2}(as+b)^2}}.
\end{equation*}And it is clear that $\kappa=\kappa(s)$ is not defined in all real numbers.\\
The whirl-rectifying curve $\sigma$ can be extended continuously to the whole set of real numbers. Additionally, its respective radial projection $\textbf{w}=\textbf{w}(t)$, on the unitary sphere with center at the origin, can be extended continuously to the entire interval\\ $(-d-\frac{\pi}{2},-d+\frac{\pi}{2})$, as the following results show
\begin{theorem}
Let $a$, $b$ and $\lambda$ be constant with $a\neq 0$ and $\lambda\neq 0$.\\
The curve $\varOmega:\R\rightarrow\R^3$ given by $\varOmega(s)=(x(s),y(s),z(s))$, where
\begin{eqnarray*}
x(s)&=&\frac{(b+as)}{a}\frac{\lambda}{\sqrt{1+\lambda^2}}\cos{[\frac{arctanh(\sqrt{\frac{1+\lambda^2}{1+(b+as)^2+\lambda^2}})}{\lambda}]},\\
y(s)&=&-\frac{(b+as)}{a}\frac{\lambda}{\sqrt{1+\lambda^2}}\sin{[\frac{arctanh(\sqrt{\frac{1+\lambda^2}{1+(b+as)^2+\lambda^2}})}{\lambda}]},\\
z(s)&=&\frac{1}{a}\frac{\sqrt{1+(b+as)^2+\lambda^2}}{\sqrt{1+\lambda^2}};\ \text{defined for}\ s\neq -\frac{b}{a}, \ \text{and}
\end{eqnarray*}
\begin{eqnarray*}
x(s)&=&0,\\ y(s)&=&0,\\ z(s)&=&\frac{1}{a};\ \text{defined for}\ s= -\frac{b}{a}.
\end{eqnarray*}
is continuous in all the set of the real numbers and the restrictions $\varOmega\mid_{(-\infty,-\frac{b}{a})}$, $\varOmega\mid_{(-\frac{b}{a},\infty)}$ are whirl-rectifying curves.
\end{theorem}
\begin{proof}
From the following inequalities:
\\ \\
$
\mid\frac{(b+as)}{a}\frac{\lambda}{\sqrt{1+\lambda^2}}\mid \leq\frac{(b+as)}{a}\frac{\lambda}{\sqrt{1+\lambda^2}}\cos{[\frac{arctanh(\sqrt{\frac{1+\lambda^2}{1+(b+as)^2+\lambda^2}})}{\lambda}]}\leq -\mid\frac{(b+as)}{a}\frac{\lambda}{\sqrt{1+\lambda^2}}\mid,\\ \\
\mid\frac{(b+as)}{a}\frac{\lambda}{\sqrt{1+\lambda^2}}\mid \leq\frac{(b+as)}{a}\frac{\lambda}{\sqrt{1+\lambda^2}}\sin{[\frac{arctanh(\sqrt{\frac{1+\lambda^2}{1+(b+as)^2+\lambda^2}})}{\lambda}]}\leq -\mid\frac{(b+as)}{a}\frac{\lambda}{\sqrt{1+\lambda^2}}\mid,
$ \\ \\
and from the definition of whirl-rectifying curve, the result follows.
\end{proof}
In a similar way, the following theorem is proved.
\begin{theorem}
Let $a$, $b$ and $\lambda$ be constant with $a\neq 0$ and $\lambda\neq 0$.\\
The curve $\varUpsilon:\R\rightarrow\R^3$ given by $\varUpsilon(t)=(x(t),y(t),z(t))$, where
\begin{eqnarray*}
x(t)&=&\frac{\mid a\mid}{a}\frac{\lambda}{\sqrt{1+\lambda^2}}\sin[d+t]\cos[\frac{arctanh[\frac{\sqrt{1+\lambda^2}}{\sqrt{1+\tan^2[d+t]+\lambda^2}}]}{\lambda}],\\ y(t)&=&-\frac{\mid a\mid}{a}\frac{\lambda}{\sqrt{1+\lambda^2}}\sin[d+t]\sin[\frac{arctanh[\frac{\sqrt{1+\lambda^2}}{\sqrt{1+\tan^2[d+t]+\lambda^2}}]}{\lambda}],\\
z(t)&=&\frac{\mid a\mid}{a}\cos[d+t]\frac{\sqrt{1+\tan^2[d+t]+\lambda^2}}{\sqrt{1+\lambda^2}},
\end{eqnarray*} defined for $ -d-\frac{\pi}{2}<t<-d+\frac{\pi}{2},\ t\neq -d$, and
\begin{eqnarray*}
x(t)&=&0,\\ y(t)&=&0,\\ z(t)&=&\frac{\mid a\mid}{a};\ \text{defined for}\ t= -d,
\end{eqnarray*}
is continuous in $(-d-\frac{\pi}{2},-d+\frac{\pi}{2})$ and
\begin{eqnarray*}
u\varUpsilon\mid_{(-d-\frac{\pi}{2},-d)}(t)&=&\varOmega\mid_{(-\infty,-\frac{b}{a})}(s),\\
u\varUpsilon\mid_{(-d,-d+\frac{\pi}{2})}(t)&=&\varOmega\mid_{(-\frac{b}{a},\infty)}(s),
\end{eqnarray*}where
\begin{eqnarray*}t&=&t(s)=-d+\arctan{(b+as)},\\u&=&u(s)=\frac{\sqrt{1+(b+as)^2}}{\mid a\mid}.
\end{eqnarray*}
\end{theorem}
\subsection{EXAMPLES}
If we take $a=0.65, b=0$ we can graph $\varOmega=\varOmega(s)$ for the different values of $\lambda=-20,-4,-1.8,-1,-0.5,-0.26$. In an analogous way it is graphed for their respective radial projections $\varUpsilon=\varUpsilon(t)$, on the unitary sphere with center at the origin.
\begin{figure}[!ht]
\begin{center}
\begin{tabular}{cccc}
\includegraphics[viewport=0 0 700 849,scale=0.4,clip,scale=0.4,clip]{Huracanuno.pdf} &\includegraphics[viewport=0 0 700 849,scale=0.4,clip,scale=0.4,clip]{Huracantres.pdf} &\includegraphics[viewport=0 0 700 849,scale=0.4,clip,scale=0.4,clip]{HuracantresDos.pdf} \\
\text{$\lambda=-20$}&\text{$\lambda=-4$}&\text{$\lambda=-1.8$}&\\
\includegraphics[viewport=0 0 700 849,scale=0.4,clip,scale=0.4,clip]{Huracancuatro.pdf} &\includegraphics[viewport=0 0 700 849,scale=0.4,clip,scale=0.4,clip]{Huracancinco.pdf} &\includegraphics[viewport=0 0 700 849,scale=0.4,clip,scale=0.4,clip]{HuracanseisDos.pdf} \\
\text{$\lambda=-1$}&\text{$\lambda=-0.5$}&\text{$\lambda=-0.26$}&
\end{tabular}
\end{center}
\caption{ Some $\varOmega=\varOmega(s)$ extensions of the whirl-rectifing curve with their respective radial projections $\varUpsilon=\varUpsilon(t)$, on the unitary sphere with center at the origin, for $a=0.65, b=d=0$, plotted for $s,t\in[-\frac{\pi}{4},\frac{\pi}{4}]$.}
\end{figure}
\bibliographystyle{amsplain}
| 5,425
|
At SARRALLE we work to manage in the most efficient way the waste resulted from the industrial activity and obtain energy from it. We work close to the best technologist in the waste recovery area, giving the best solutions adapted to our customer needs.
SARRALLE supplies engineering and construction services in :
| 216,243
|
Reba McEntire revealed today that she will hit the road again, announcing REBA: LIVE IN CONCERT tour. Joining her for various dates across the tour include Brandy Clark, Hannah Dasher, Caylee Hammack, Reyna Roberts, Cailtyn Smith, Brittney Spencer and Tenille Townes. Live Nation is the official tour promoter and tickets for REBA: LIVE IN CONCERT go on sale October 15 at 10am local time at reba.com and livenation.com.
Citi is the official presale credit card for the Reba tour. As such, Citi cardmembers will have access to purchase presale tickets beginning Tuesday, October 12 at 10am local time until Thursday, October 14 at 10pm local time through Citi Entertainment. For complete presale details visit.
Reba just released her three-part box set REVIVED REMIXED REVISITED on Friday and is continuing to celebrate the release this week. REVIVED. Tomorrow (10/12), Reba will take part in a special Amazon Music livestream event where she will perform songs with her REVISITED collaborator and producer, Dave Cobb. Hosted by Jessie James Decker, fans can tune in at 5pm CT via Amazon Live on desktop, mobile, Fire tablet, or via the Amazon Shopping App on Fire TV.
On Thursday (10/14), Reba will perform two songs and share more behind the release on TODAY, beginning at 8:00A local time on NBC.
REBA: LIVE IN CONCERT Official Tour Dates:
| 133,126
|
TITLE: how to find center/radius of a sphere
QUESTION [3 upvotes]: Say you have an irregular tetrahedron, but you know the (x,y,z) coordinates of the four vertices; is there a simple formula for finding a sphere whose center exists within the tetrahedron formed by the four points and on whose surface the four points lie?
REPLY [1 votes]: Given four points, a, b, c, and d, you can find the center by setting the following determinant to zero and solving it:
$$
\begin{vmatrix}
(x^2 + y^2 + z^2) & x & y & z & 1 \\
(ax^2 + ay^2 + az^2) & ax & ay & az & 1 \\
(bx^2 + by^2 + bz^2) & bx & by & bz & 1 \\
(cx^2 + cy^2 + cz^2) & cx & cy & cz & 1 \\
(dx^2 + dy^2 + dz^2) & dx & dy & dz & 1 \\
\end{vmatrix}
= 0
$$
The math is gnarly, but the following C++ code implements the solution:
class Point {
public:
double x;
double y;
double z;
Point() { x = 0; y = 0; z = 0; }
Point(double x_, double y_, double z_) { x = x_; y = y_; z = z_; }
};
class Sphere {
public:
Point center;
double radius;
Sphere(Point center_, double radius_) {
center = Point(center_.x, center_.y, center_.z);
radius = radius_;
}
};
Sphere
sphereFromFourPoints(Point a, Point b, Point c, Point d)
{
#define U(a,b,c,d,e,f,g,h) (a.z - b.z)*(c.x*d.y - d.x*c.y) - (e.z - f.z)*(g.x*h.y - h.x*g.y)
#define D(x,y,a,b,c) (a.x*(b.y-c.y) + b.x*(c.y-a.y) + c.x*(a.y-b.y))
#define E(x,y) ((ra*D(x,y,b,c,d) - rb*D(x,y,c,d,a) + rc*D(x,y,d,a,b) - rd*D(x,y,a,b,c)) / uvw)
double u = U(a,b,c,d,b,c,d,a);
double v = U(c,d,a,b,d,a,b,c);
double w = U(a,c,d,b,b,d,a,c);
double uvw = 2 * (u + v + w);
if (uvw == 0.0) {
// Oops. The points are coplanar.
}
auto sq = [] (Point p) { return p.x*p.x + p.y*p.y + p.z*p.z; };
double ra = sq(a);
double rb = sq(b);
double rc = sq(c);
double rd = sq(d);
double x0 = E(y,z);
double y0 = E(z,x);
double z0 = E(x,y);
double radius = sqrt(sq(Point(a.x - x0, a.y - y0, a.z - z0)));
return Sphere(Point(x0, y0, z0), radius);
}
| 86,736
|
TITLE: How to derive this formula about the bracket function?
QUESTION [0 upvotes]: Is there a direct way of proving that
$$ [nx] = [x] + [x+\frac{1}{2}] + [x+\frac{1}{3}] + \ldots + [x+ \frac{1}{n}]$$
for each real number $x$ and for each positive integer $n$?
My effort:
Let $n$ be an arbitrary but fixed positive integer. Then since we can write
$$x = [x]+ \theta,$$ where $0 \leq \theta < 1$, we can consider the following cases:
Case 1. When $0 \leq \theta < 1/n$, we see that
$$ nx = n [x] + n\theta$$ and $0 \leq n \theta < 1$; so we have
$$[nx] = n[x].$$
Also, for $n \geq 2$, we have $$ \frac{1}{2} \leq \theta + \frac{1}{2} < \frac{1}{n} + \frac{1}{2} \leq 1, $$
whence $$ x + \frac{1}{2} \leq [x] + \theta + \frac{1}{2}$$ so that $$[x+\frac{1}{2}] = [x].$$ Continuing in the same way, we find that $$ \frac{1}{n} \leq \theta + \frac{1}{n} < \frac{1}{n} + \frac{1}{n} = \frac{2}{n} \leq 1$$ whenever $n \geq 2$, which implise that $$ x + \frac{1}{n} \leq [x] + \theta + \frac{1}{n} $$ so that $$[x + \frac{1}{n}] = [x]. $$ Finally, adding all these results, we find that $$ [x] + [x+\frac{1}{2}] + [x+\frac{1}{3}] + \ldots + [x+ \frac{1}{n}] = n[x] = [nx], $$ as required.
Case 2. When $\frac{1}{n} \leq \theta < \frac{2}{n}$, we have $1 \leq n \theta < 2$, and since $ nx = n[x] + n\theta$, we can conclude that $$[nx] = n[x] + 1.$$ And then we handle the right-hand side as in case 1 above.
Is there a more efficient, but elementary, way of deriving the above formula?
REPLY [2 votes]: There is no direct way of proving your claim because it is wrong.
Let $n=3$ and $x=\frac25$.
| 214,635
|
Join Now
Join ePHOTOzine, the friendliest photography community.
Upload photos, chat with photographers, win prizes and much more for free!
Get on1's Perfect Effects 9.5 for FREE! (£48 value)
TakaraDancy | Send a Private Message | Visit My Website | Facebook
Thanks to everyone who takes a sec to check out my work XD
TakaraDancy's Portfolio
Current Filter:
No tag filter applied
| 90,365
|
I never thought I would have ALL boys. I am the middle of three girls, although there were three younger brothers (I come from a blended family on both sides). I had no idea what to do with boys. I grew up thinking they were “icky and weird”. God knew just what I needed in my life to bring more excitement and adventure. Boys. Three of them to be exact. A red head, and tow head, and a blonde. One with blue eyes and two with hazel eyes (like their mama).
I also never thought I would be homeschooling these three little guys. God has a great sense of humor in my life. Navigating raising boys and homeschooling them at the same time has proven very difficult at times. But, it’s not impossible. Boys are loud, dirty, rough, adventurous, and not afraid of anything. They are also sweet, sensitive, loving, protective, and brave.
Determining how your child learns best is the first step. My first son was my “guinea pig”, if you will. I made a lot of mistakes when I first started homeschooling him. I bought thousands of dollars worth of curriculum and books only to find out he hated them and I hated them. That is all part of the experience of homeschooling.
There are a thousand different homeschool methods. Some of them are Charlotte Mason, Eclectic, Classical, Montessori, Unit Studies, etc. Discovering what works and what doesn’t work. If you can figure out how your child likes to learn, then you are one step ahead of the game. I found that I am more of a Classical, Charlotte Mason, Eclectic, Montessori type. I laughed at first too. But, at least now I know.
The next step in homeschooling all boys is to relax. You won’t get it all done. No amount of planning and organizing can prepare you for unexpected surprises. These happen regularly in my home. Write out a schedule that you would like to tentatively follow. I would also encourage you to have realistic expectations. If you have all “littles” don’t expect them to work for 5 or 6 hours. Especially boys. Their attention span is shorter than a girls. Research and read some good books on realistic goals for homeschooling. But, remember, be prepared to not get it all done. There is always tomorrow.
I would also schedule lots of breaks in between subjects. Having scheduled breaks give the child something to look forward to. Those breaks don’t have to involve screen time. It can be a 15 minute break to go jump on the trampoline or read a chapter in a book. Boys need breaks because they need to regain focus and it gives them some time to get a little energy out. My boys have TONS of energy. If they don’t release that energy it becomes a problem and no work will get done.
Look at your calendar at the beginning of each month and schedule at least one field trip. I live in a very historic area so we have plenty of options. We also have a lot of museums and parks around. Schedule one or two days a month to pack a picnic lunch and go to a museum, park, or even visit a friend. This gives mom a break as well as giving the child something to look forward to each month. You can make sure the trip aligns with what you are learning. Or, you can just wing it and do something completely off topic.
One last thing about homeschooling all boys. Keep lots of food on hand. I know a lot of parents don’t like their children snacking but I don’t mind. As long as they are getting their work done, they can enjoy their snacks. They are boys after all. They are growing and their brains and bodies need fuel to keep up with the energetic demands they have. I like to give them healthier choices like apple slices, bananas, yogurt, etc. But, I also throw in popcorn, veggie straws, and cheese sticks. You will have a much more enjoyable homeschooling experience if you keep the kids tummies happy too.
Homeschooling boys can be hard but it can be very rewarding. When you begin to see the fruits of your labor, your heart will leap out of your chest. Boys love their mamas. They have tender hearts and want to be loved and loving. Learning how to do those things while also giving them an education is the best thing you will ever do for them.
Here are some amazing resources for raising boys:
| 236,900
|
TITLE: Convergence of a series of a given metric..
QUESTION [0 upvotes]: I'm trouble with a metric defined over a given set: Consider $\mathcal{P}=C_p^\infty([-\pi, \pi])$, that is, $\mathcal{P}$ is the set of all infinitely differentiable functions $f:\mathbb R\rightarrow \mathbb C$ which are $2\pi$-periodic ($f(x+2\pi)=f(x)$, for all $x\in \mathbb R$). We can introduce a metric in this set setting $$\displaystyle d(f, g)=\sum_{k=0}^\infty \frac{1}{2^k}\frac{\|f^{(k)}-g^{(k)}\|_\infty}{1+\|f^{(k)}-g^{(k)}\|_\infty},$$ where $\displaystyle \|f\|_\infty=\sup_{x\in J}|f(x)|$ where $J\subset \mathbb R$ is an interval of length $2\pi$. The problem is: why does the series above converge? Also, how can I prove $f_n\to f$ in $\mathcal{P}$ if and only if $\|f_n^{(k)}-f^{(k)}\|_\infty\to 0$ for all $k\in\mathbb N$?
REPLY [1 votes]: Each term in the series is bounded by $\frac{1}{2^k}$, since $\frac{x}{1+x}<1$ for all $x\ge 0$. Thus we have
$$\sum_{k=0}^\infty \frac{1}{2^k}\frac{\|f^{(k)}-g^{(k)}\|_\infty}{1+\|f^{(k)}-g^{(k)}\|_\infty}\leq \sum_{k=0}^\infty \frac{1}{2^k}=\frac{1}{1-\frac12}=2.$$
I suspect you wish to show that $f_n\to f$ in $\mathcal P$ if and only if $\|f_n^{(k)}-f^{(k)}\|_\infty\to 0$ for all $k$. One direction (that if $f_n\to f$ in $\mathcal P$, $\|f_n^{(k)}-f^{(k)}\|_\infty\to 0$ for all $k$) is easy, and I will leave to you. For the other direction, suppose $\|f_n^{(k)}-f^{(k)}\|_\infty\to 0$ for all $k$ and let $\epsilon>0$. Let $i$ be such that $\frac{1}{2^i}<\frac{\epsilon}{2}$ and $N$ be such that, for all $k\leq i$,
$$n\ge N\implies \|f_n^{(i)}-f^{(i)}\|_\infty < \frac{\epsilon}{2(i+1)}.$$
Then for $n\ge N$ we have
$$\begin{align}
d(f_n,f) &= \sum_{k=0}^\infty \frac{1}{2^k}\frac{\|f_n^{(k)}-f^{(k)}\|_\infty}{1+\|f_n^{(k)}-f^{(k)}\|_\infty}\\
&= \sum_{k=0}^i \frac{1}{2^k}\frac{\|f_n^{(k)}-f^{(k)}\|_\infty}{1+\|f_n^{(k)}-f^{(k)}\|_\infty}+\sum_{k=i+1}^\infty \frac{1}{2^k}\frac{\|f_n^{(k)}-f^{(k)}\|_\infty}{1+\|f_n^{(k)}-f^{(k)}\|_\infty}\\
&\leq \sum_{k=0}^i \frac{1}{2^k}\|f_n^{(k)}-f^{(k)}\|_\infty+\sum_{k=i+1}^\infty \frac{1}{2^k}\\
&\leq \sum_{k=0}^i \frac{1}{2^k}\frac{\epsilon}{2(i+1)}+\frac{1}{2^i}\\
&\leq \sum_{k=0}^i \frac{\epsilon}{2(i+1)}+\frac{\epsilon}{2}=\epsilon\\
\end{align}$$
hence $f_n\to f$ in $\mathcal P$.
| 152,045
|
\begin{document}
\title{ Sampled-data control of 2D Kuramoto-Sivashinsky equation}
\author{Wen Kang and Emilia Fridman
\thanks{W. Kang (email: kangwen@amss.ac.cn) is with School of Automation and Electrical Engineering, University of Science and Technology Beijing, P. R. China, and also with School of Electrical Engineering, Tel Aviv University, Israel.}
\thanks{E. Fridman (email: emilia@eng.tau.ac.il) is with School of Electrical Engineering, Tel Aviv University, Israel.}
\thanks{ This work was supported by Israel Science Foundation (grant No. 673/19) and National Natural Science Foundation of China (Grant No. 61803026).}}
\maketitle
\begin{abstract}
This paper addresses sampled-data control of 2D Kuramoto-Sivashinsky equation over a rectangular domain $\Omega$. We suggest
to divide the 2D rectangular $\Omega$ into $N$ sub-domains, where sensors provide spatially averaged or point state measurements
to be transmitted through communication network to the controller. Note that differently from 2D heat equation, here we manage with sampled-data control under point measurements. We design a regionally stabilizing controller applied through distributed in space characteristic functions. Sufficient conditions ensuring regional stability of the closed-loop system are established in terms of linear matrix inequalities (LMIs). By solving these LMIs, we find an estimate on the set of initial conditions starting from which the state trajectories of the system are exponentially converging to zero. A numerical example demonstrates the efficiency of the results.
\end{abstract}
\section{Introduction}
In recent decades, Kuramoto-Sivashinsky equation (KSE)
has drawn a lot of attention as a nonlinear model
of pattern formations on unstable flame fronts and thin hydrodynamic films (see e.g. \cite{kuramoto,bao}). KSE arises in the study of thin liquid films, exhibiting a wide range of dynamics in different parameter regimes, including unbounded growth and full spatiotemporal chaos. For 1D
KSE, distributed control (see e.g. \cite{Armaou2,Armaou1,Armaou,Titi}) has been considered. Boundary stabilization of KSE has been studied in \cite{Krstic,LU}.
Sampled-data control of PDEs became recently an active research area (see e.g. \cite{kang,Emilia3,Emilia2,kang2}) for practical application of finite-dimensional controllers for PDEs,
where LMI conditions for the exponential/regional stability of the closed-loop systems were derived
in the framework of time-delay approach by employing appropriate Lyapunov functionals.
Most of the existing results deal with 1D PDEs system. Sampled-data observers for ND
and 2D heat equations with globally
Lipschitz nonlinearities have been suggested in \cite{Emilia1} and \cite{Anton1}. However, the
above results were confined to diffusion equations.
Sampled-data control of various classes of high dimensional PDEs is an interesting and challenging problem.
In our recent paper \cite{kang}, we have suggested sampled-data control of 1D KSE, where both point and averaged state measurements were studied. In this paper we aim to extend results of \cite{kang} to 2D case.
Extension from 1D (\cite{kang,ECC18}) to 2D is far from being straightforward. Thus, in the case of heat equation, sampled data extension under the point measurements seems to be not possible (see \cite{Anton1}).This is due to the fact that stability analysis
of the closed-loop system is based on the bound of the $L^2$-norm of the difference between the state and its point value. However,
according to Friedrich's inequality \cite{Anton1,jones}, this bound depends on the $L^2$-norm of the second-order spatial derivatives of the state. Differently from the heat equation, stability analysis for KSE in $H^2$ allows to compensate such terms.
We establish stability analysis of the closed-loop sampled-data system by constructing an appropriate Lyapunov-Krasovskii functional. Some preliminary results under averaged measurements were presented in \cite{CDC19}, where the sampled-data case is limited to averaged measurements and there is no detailed proof of the well-posedness.
In the present paper, we design a sampled-data controller for 2D KSE under averaged/point measurements based on LMIs.
In comparison to the existing known results, new special challenges of this work are the following:\\
1) The present paper gives the first extension to 2D PDE in the case of sampled-data point measurements.
The results from \cite{Anton1} cannot be extended to the case of point measurements.
This is due to the second order spatial derivative in Lemma \ref{anton} that cannot be compensated in Lyapunov analysis.
\\
2) Here we have the nonlinear term ``$zz_{x_1}$" which is locally Lipschitz in $D((-A)^{\frac{1}{2}})$ (defined in Section III below). Due to the nonlinear term, the challenge is to find a bound on the domain of attraction. This bound is based on the new 2D Sobolev inequality (Lemma \ref{kka}) that we have derived. Lemma \ref{kka} bounds a function in the $C^0$-norm using $L^2$-norms of its first and second spatial derivatives. In \cite{Emilia1} and \cite{Anton1}, the nonlinear term is subject to sector bound inequality which holds globally and leads to global results.\\
3) The well-posedness is challenging. We have provided more detailed proof for this, and shown that $A$ generates an analytic semigroup even for rectangular domain $\Omega$ (non $C^1$ boundary).
The remainder of this work is organized as follows. Useful lemmas are introduced and the problem setting is reported in Section II. Sections III-IV are devoted to construction of continuous static output-feedback/sampled-data controllers under the averaged or point measurements.
In Section V, a
numerical example is carried out to illustrate the efficiency of the main results.
Finally, some
concluding remarks and possible future research lines are presented in Section VI.
{\bf Notation} The superscript $``T"$ stands for matrix transposition, $\mathbb R^n$ denotes
the $n$-dimensional Euclidean space with the norm $ | \cdot |$. $\Omega\subset \mathbb R^2$ denotes a computational domain, $L^2(\Omega)$ denotes the space of measurable squared-integrable functions over $\Omega$ with the corresponding
norm $\|z\|_{L^2(\Omega)}^2=\int_\Omega |z(x)|^2dx$.
Let $\partial \Omega$ be the boundary of $\Omega$.
The Sobolev space $H^{k}(\Omega)$ is defined as
$H^{k}(\Omega)=\{z:D^\alpha z\in L^2(\Omega),\; \forall\;0\le |\alpha|\le k\}$ with norm
$\|z\|_{H^{k}(\Omega)=\left(\sum\limits_{0\le |\alpha|\le k}\|D^\alpha z\|^2_{L^2(\Omega)}\right)}^\frac{1}{2}.$
The space $H^k_0(\Omega)$ is the closure of $C_c^\infty(\Omega)$ in the space $H^k(\Omega)$ with the norm $\|z\|_{H_0^{k}(\Omega)}=\left(\sum\limits_{ |\alpha|= k}\|D^\alpha z\|^2_{L^2(\Omega)}\right)^\frac{1}{2}.$
\section{Problem formulation and useful lemmas}
Denote by $\Omega$ the two dimensional (2D) unit square
$$\Omega=[0,1]\times [0,1]\subset \mathbb{R}^2.$$
Consider the biharmonic operator:
\begin{equation*}
\Delta^2=\dfrac{\partial ^4}{\partial x_1^4}+2\dfrac{\partial ^4}{\partial x_1^2\partial x_2^2}+\dfrac{\partial ^4}{\partial x_2^4}.
\end{equation*}
As in \cite{Ruben}, we consider the following 2D Kuramoto-Sivashinsky equation (KSE) over $\Omega$ under the Dirichlet boundary conditions:
\begin{equation}\label{a}
\left\{\begin{array}{ll}
\!\! z_t+zz_{x_1}\!+ \!(1-\kappa) z_{x_1x_1}\!-\!\kappa z_{x_2x_2}\!+\! \Delta^2 z\!=\!\sum\limits_{j=1}^N\chi_{j}(x) U_{j}(t), \\\hspace{4.5cm}(x,t)\in \Omega\times [0,\infty),\\
z|_{\partial \Omega}=0, \;\dfrac{ \partial z}{\partial n}|_{\partial \Omega}=0,\\
z(x,0)=z_0(x),
\end{array}\right.
\end{equation}
where $x=(x_1,x_2)\in \Omega$, $z\in \mathbb R$ is the state of KSE, $\dfrac{ \partial z}{\partial n}$ is the
normal derivative,
and $U_{j}(t)\in \mathbb{R}$, $j=1,2,\cdots, N$ are the control inputs. Here the parameter $\kappa$ denotes the angle of the substrate to the horizontal: \\
for $\kappa> 0$ we have overlying film flows, a vertical film flow for $\kappa = 0$, and hanging flows when $\kappa< 0$.
Motivated by \cite{Titi,Emilia3,Emilia2,Emilia1,Titi2} we suggest to
divide $\Omega$ into $N$ square sub-domains $\Omega_{j}$ covering the whole region $\cup_{j=1}^{N} \Omega_j =\Omega$ (see Fig. 1) with an actuator and a sensor placed in each $\Omega_j$.
Here
$$\begin{array}{ll}\Omega_j=\{x=(x_1,x_2)^T\in \Omega |x_i\in [x_i^{\rm min}(j), x_i^{\rm max}(j)], i=1,2\},\\
\hspace{6cm} j=1,2,\cdots, N.
\end{array}$$
The measure of their intersections is zero.
\begin{figure}[h]
\begin{center}
{\includegraphics[width=7cm]{NEW.jpg}}
\caption{Unit square $\Omega$ and sub-domains $\Omega_j$}
\end{center}
\end{figure}
Let
$$0=t_0<t_1<\cdots<t_k\cdots, \quad \lim\limits_{k\to \infty}t_k=\infty $$
be sampling time instants.
The sampling sub-domains in time and in space may be bounded,
$$
\begin{array}{ll}
0\le t_{k+1}-t_{k}\le h, \\
0<x_i^{\rm max}(j)-x_i^{\rm min}(j)= \Delta_j\le \bar \Delta, \;\\
\hspace{2cm} i=1,2; \;\;j=1,2,\cdots, N,
\end{array}
$$
where $h$ and $\bar \Delta$ are the corresponding upper bounds.
\begin{remark}
For simplicity, we consider each sub-domain $\Omega_j$ is a square (i.e. $x_1^{\rm max}(j)-x_1^{\rm min}(j))=x_2^{\rm max}(j)-x_2^{\rm min}(j),\;j=1,2,\cdots, N)$. Indeed, $\Omega_j$ can be a rectangular. For the case that $x_1^{\rm max}(j)-x_1^{\rm min}(j))\neq x_2^{\rm max}(j)-x_2^{\rm min}(j)$ for some $j$ (see Fig. 1 of \cite{Anton1}), $\Delta_j$ can be chosen as follows $$\Delta_j=\max\{x_1^{\rm max}(j)-x_1^{\rm min}(j),x_2^{\rm max}(j)-x_2^{\rm min}(j)\}.$$Thus, the results of this work are applicable to the case of nonsquare sub-domains.
\end{remark}
The spatial characteristic functions are taken as
\begin{equation}\label{qin}
\left\{\begin{array}{ll}\chi_{j}(x)= 1,\; x\in \Omega_{j},\\
\chi_{j}(x)= 0 ,\; x \notin \Omega_{j},
\end{array}\right. \;j=1,\cdots, N.
\end{equation}
We assume that sensors provide the following averaged measurements
\begin{equation}\label{w}
\begin{array}{ll}
y_{jk}=\dfrac{\int_{\Omega_{j}}z(x,t_k)dx}{|\Omega_{j}|},\;
\;j=1,\dots, N;\;k=0,1,2\dots
\end{array}
\end{equation}
or point measurements
\begin{equation}\label{swift}
\begin{array}{ll}
y_{jk}=z(\bar x_j,t_k),\;
\;j=1,\dots, N;\;k=0,1,2\dots
\end{array}
\end{equation}
where $\bar x_j$ locates in the center of the square sub-domain $\Omega_j$, and $|\Omega_{j}|$ stands for the Lebesgue measure of the domain $\Omega_{j}$.
We aim to design for \dref{a} an exponentially stabilizing sampled-data controller that can be implemented by zero-order hold devices:
\begin{equation}\label{y}
\begin{array}{ll}
U_{j}(t)=-\mu y_{jk},\;\;j=1,\dots, N;\\
\hspace{2cm}t\in [t_k,t_{k+1}),\;k=0,1,2\dots,
\end{array}
\end{equation}
where $\mu$ is a positive controller gain
and $y_{jk}$ is given by \dref{w} or \dref{swift}.
We present below some useful lemmas:
\begin{lemma}\label{d}
Let $\Omega=(0,L_1)\times (0,L_2) $. Assume $f:\Omega\to \mathbb R$ and $f\in H^1(\Omega)$.\\
(i) (Poincar\'e's inequality) If $\int_\Omega f(x)dx=0$, then according to \cite{Payne}
\begin{equation}
\|f\|_{L^2(\Omega)}^2\le \dfrac{L_1^2+L_2^2}{\pi^2}\|\nabla f\|_{L^2(\Omega)}^2.
\end{equation}
(ii) (Wirtinger's inequality) \cite{Emilia1} If $f|_{\partial \Omega}= 0$, then the following inequality
holds:
\begin{equation}
\|f\|_{L^2(\Omega)}^2\le \dfrac{L_1^2+L_2^2}{\pi^2}\|\nabla f\|_{L^2(\Omega)}^2.
\end{equation}
\end{lemma}
The following lemma gives a classical Friedrich's inequality (Theorem 18.1 of \cite{pingping}) with tight bounds on the coefficients of terms $\left\|\dfrac{\partial f}{\partial x_1}\right\|^2_{L^2(\Omega)}$, $\left\|\dfrac{\partial f}{\partial x_2}\right\|^2_{L^2(\Omega)}$ and $\left\|\dfrac{\partial^2 f}{\partial x_1\partial x_2}\right\|^2_{L^2(\Omega)}$. The inequality \dref{pinga} bounds the $L^2$-norm of a function by the reciprocally convex combination of the $L^2$-norm of its derivatives.
\begin{lemma}\label{qinqin}(see (2) of \cite{Anton1})
Let $\Omega=(0,l)^2 $, $f\in H^2(\Omega)$ with $f(0,0)=0$. Then
the following inequality holds:
\begin{equation}\label{pinga}
\begin{array}{ll}
\|f\|_{L^2(\Omega)}^2\!&\le \dfrac{1}{\alpha_1}\!\left(\dfrac{2l}{\pi}\right)^2\!\!\left\|\dfrac{\partial f}{\partial x_1}\right\|_{L^2(\Omega)}^2\! \!\! \!\!+\!\dfrac{1}{\alpha_2}\left(\dfrac{2l}{\pi}\right)^2\!\!\left\|\dfrac{\partial f}{\partial x_2}\right\|_{L^2(\Omega)}^2\\
&+\dfrac{1}{\alpha_3}\left(\dfrac{2l}{\pi}\right)^4\left\|\dfrac{\partial^2 f}{\partial x_1\partial x_2}\right\|_{L^2(\Omega)}^2
\end{array}
\end{equation}
where $\alpha_1$, $\alpha_2$, $\alpha_3$ are positive constants satisfying $$\alpha_1+\alpha_2+\alpha_3=1.$$
\end{lemma}
\begin{lemma}\cite{Anton1}\label{anton}
Let $\Omega=(0,l)^2 $, $f\in H^2(\Omega)$ with $f(0,0)=0$, $\eta>0$. Then
\begin{equation}
\begin{array}{ll}
\eta \|f\|^2&\le \beta_1\left(\dfrac{2l}{\pi}\right)^2\left\|\dfrac{\partial f}{\partial x_1}\right\|^2+\beta_2\left(\dfrac{2l}{\pi}\right)^2\left\|\dfrac{\partial f}{\partial x_2}\right\|^2
+\beta_3\left(\dfrac{2l}{\pi}\right)^4\left\|\dfrac{\partial^2 f}{\partial x_1 \partial x_2}\right\|^2
\end{array}
\end{equation}
for any $\beta_1$, $\beta_2$, $\beta_3$ satisfying
\begin{equation}\label{mingbai}
{\rm diag}\{\beta_1,\beta_2,\beta_3\}\geq \eta \left[\begin{array}{ccc}1&1&1\\1&1&1\\1&1&1\end{array}\right].
\end{equation}
\end{lemma}
The following version of 2D Sobolev inequality will be useful:
\begin{lemma}\label{kka}
Let $\Omega=(0,1)^2$ and
$w=w(x_1,x_2)\in H^2(\Omega)\cap H_0^1(\Omega)$, where $(x_1,x_2)\in \Omega$. Then
\begin{equation}\label{eqiang}
\begin{array}{ll}
\|w\|_{C^0(\bar\Omega)}^2\le
\dfrac{1}{2}(1+\Gamma)\left[\|w_{x_1}\|^2_{L^2(\Omega)}+\|w_{x_2}\|^2_{L^2(\Omega)}\right]
+\dfrac{1}{\Gamma}\|w_{x_1x_2}\|^2_{L^2(\Omega)},\;\forall \Gamma>0.
\end{array}
\end{equation}
\end{lemma}
\begin{proof} Due to $w\in H_0^1(\Omega)$,
application of 1D
Sobolev's inequality to $w$ in $x_2$ yields
\begin{equation}\label{new2}
\begin{array}{ll}
\max\limits_{x_2\in [0,1]}w^2(x_1,x_2)
\le \disp \int_0^1\!w_{x_2}^2(x_1,\xi_2) d\xi_2.
\end{array}
\end{equation}
Further application of
Lemma 4.1 of \cite{kang} to $w_{x_2}$ in $x_1$ leads to
\begin{equation}\label{new4}
\begin{array}{ll}
\max\limits_{x_1\in [0,1]}w_{x_2}^2(x_1,\xi_2)\\
\le \disp (1+\Gamma)\int_0^{1}\!w_{x_2}^2(\xi_1,\xi_2)d\xi_1 + \dfrac{1}{\Gamma}\int_0^1\!w_{x_1x_2}^2(\xi_1,\xi_2) d\xi_1,\;\forall \Gamma>0.
\end{array}
\end{equation}
Substitution of \dref{new4} into \dref{new2} yields
\begin{equation}\label{new3}
\begin{array}{ll}
\|w\|_{C^0(\bar\Omega)}^2=\max\limits_{(x_1,x_2)\in \bar\Omega}w^2(x_1,x_2)
=\max\limits_{x_1\in [0,1]}\left[\max\limits_{x_2\in [0,1]}w^2(x_1,x_2)\right]\\
\le \max\limits_{x_1\in [0,1]} \left[ \disp\int_0^1\!w_{x_2}^2(x_1,\xi_2) d\xi_2\right]
\le \disp\int_0^1\max\limits_{x_1\in [0,1]}\!w_{x_2}^2(x_1,\xi_2) d\xi_2\\
\le (1+\Gamma)\disp\int_0^{1}\int_0^{1}\!w_{x_2}^2(\xi_1,\xi_2)d\xi_1d\xi_2+\dfrac{1}{\Gamma}\int_0^1\int_0^1w_{x_1x_2}^2(\xi_1,\xi_2) d\xi_1d\xi_2.
\end{array}
\end{equation}
Following the same procedure, we can obtain
\begin{equation}\label{new1}
\begin{array}{ll}
\|w\|_{C^0(\bar\Omega)}^2\le (1+\Gamma)\disp\int_0^{1}\int_0^{1}w_{x_1}^2(\xi_1,\xi_2)d\xi_1d\xi_2+\dfrac{1}{\Gamma}\int_0^1\int_0^1w_{x_1x_2}^2(\xi_1,\xi_2) d\xi_1d\xi_2.
\end{array}
\end{equation}
From \dref{new3} and \dref{new1} it follows that \dref{eqiang} holds.
\end{proof}
\begin{remark}
In Lemma \ref{kka}, we give a new 2D Sobolev inequality with constants depending on a free parameter $\Gamma>0$. Lemma \ref{kka} is very useful and plays an important role in the stability analysis leading to LMIs (with $\Gamma$ as a decision parameter) that guarantee stability and give a bound on the domain of attraction. \end{remark}
\begin{lemma}(Halanay's Inequality \cite{halanay})
Let $V:[-h,\infty)\to [0,\infty)$ be an absolutely continuous function. If there exist $0<\delta_1<2\delta$ such that for all $t\geq 0$ the following inequality holds
\begin{equation}
\begin{array}{ll}
\dot V(t)+2\delta V(t)-\delta_1\sup\limits _{-h\le \theta \le 0}V(t+\theta)\le 0,
\end{array}
\end{equation}
then we have
\begin{equation}
V(t)\le e^{-2\sigma t} \sup_{-h\le \theta \le 0} V(\theta),\; t\geq 0,
\end{equation}
where
$\sigma$ is a unique solution of
\begin{equation}\label{hdou}
\sigma=\delta-\dfrac{\delta_1}{2} e^{2\sigma h}.
\end{equation}
\end{lemma}
\section{Global stabilization: continuous static output-feedback}
In this section,
we will establish the well-posedness and stability analysis for the system \dref{a} under the continuous-time averaged measurements
\begin{equation}\label{ww}
\begin{array}{ll}
y_{j}(t)=\dfrac{\int_{\Omega_{j}}z(x,t)dx}{|\Omega_{j}|},\;
\;j=1,\dots, N
\end{array}
\end{equation}
or point measurements
\begin{equation}\label{taylor}
\begin{array}{ll}
y_{j}(t)=z(\bar x_j,t),\;
\;j=1,\dots, N
\end{array}
\end{equation}
via a controller
\begin{equation}\label{yy}
U_{j}(t)=-\mu y_{j}(t),\;\;j=1,\dots, N.
\end{equation}
The closed-loop system can be represented in the following form:
\begin{equation}\label{b}
\left\{\begin{array}{ll}
z_t+zz_{x_1}+ (1-\kappa) z_{x_1x_1}\!-\!\kappa z_{x_2x_2}+ \Delta^2 z\\=-\mu\sum\limits_{j=1}^N \chi_{j}(x)[z-f_{j}], \hspace{0.2cm}\;(x,t)\in \Omega\times [0,\infty),\\
z|_{\partial \Omega}=0,\;\dfrac{\partial z}{\partial n}|_{\partial \Omega}=0,\\
z(x,0)=z_0(x),
\end{array}\right.
\end{equation}
where for \dref{ww}
\begin{equation}\label{dili}
f_{j}(x,t)=z(x,t)-\dfrac{\int_{\Omega_{j}}z(\zeta,t)d\zeta}{|\Omega_{j}|},
\end{equation}
for \dref{taylor}
\begin{equation}\label{reba}
f_{j}(x,t)=z(x,t)-z(\bar x_j,t).
\end{equation}
Now we establish the well-posedness of the system \dref{b} subject to \dref{dili} or \dref{reba}.
Define the spatial differential operator $A : D(A) \subset L^2(\Omega)\to L^2(\Omega)$ as follows:
\begin{equation}
\left\{\begin{array}{ll}
Af=-\Delta^2 f, \;\forall f\in D(A),\\
D(A)=\{f\in H^4(\Omega): f|_{\partial \Omega}=0,\;\dfrac{\partial f}{\partial n}|_{\partial \Omega}=0\}.
\end{array}\right.
\end{equation}
Note that $A^*=A$ and ${\rm Re}\langle Af, f\rangle \le 0,\; \forall f\in D(A).$
Thus, the operator $A$ is self-adjoint and dissipative. Moreover, the
inverse $A^{-1}$ is bounded, and hence $0\in \rho(A)$. By the Lumer-Phillips theorem \cite{Pazy}, $A$ generates a $C_0$-semigroup. Since the resolvent of $A$ is compact on $L^2(\Omega)$, the spectrum of $A$ consists of isolated eigenvalues only, and a sequence of corresponding eigenfunctions of $A$ forms an orthonormal basis of $L^2(\Omega)$. Let $\{\lambda_n\}$ be the eigenvalues of $A$ and let $\{\phi_n\}$ be the corresponding eigenfunctions, i.e. $A\phi_n=\lambda_n \phi_n$. Since $A$ is negative, all the eigenvalues are located on the negative real axis, i.e. $\lambda_n<0$. For any $x_0\in L^2(\Omega)$, it can be presented in the following form:
$x_0=\sum\limits_{n=1}^{\infty} a_n\phi_n$ with $\sum\limits_{n=1}^\infty |a_n|^2=\|x_0\|_{ L^2(\Omega)}^2$.
Therefore, for any $\tau\in \mathbb R$,
\begin{equation*}
\begin{array}{ll}
\|(i\tau I-A)^{-1}x_0\|_{ L^2(\Omega)}^2=\sum\limits_{n=1}^\infty \dfrac{|a_n|^2}{|i\tau-\lambda_n|^2}=\sum\limits_{n=1}^\infty \dfrac{|a_n|^2}{\tau^2+\lambda_n^2}\\
\le\sum\limits_{n=1}^\infty \dfrac{|a_n|^2}{\tau^2}=\dfrac{1}{\tau^2}\|x_0\|_{ L^2(\Omega)}^2,
\end{array}
\end{equation*}
which implies
\begin{equation*}
\|(i\tau I-A)^{-1}\|\le \dfrac{1}{|\tau|},\;\forall \tau \in \mathbb R.
\end{equation*}
Hence, \begin{equation*}
\overline{\lim_{|\tau|\to \infty}}\|\tau(i\tau I-A)^{-1}\|<\infty.
\end{equation*}
Then from Theorem 1.3.3 of \cite{zyl}, it follows that $A$ generates an analytic semigroup. Since $-A$ is positive, $(-A)^{\frac{1}{2}}$ is also positive and
$$D((-A)^{\frac{1}{2}})=\{f\in H^2(\Omega): f|_{\partial \Omega}=0,\;\dfrac{\partial f}{\partial n}|_{\partial \Omega}=0\}.$$
The norm of $D((-A)^{\frac{1}{2}})$ is given by
$$\|f\|^2_{D((-A)^{\frac{1}{2}})}=\int_\Omega |\Delta f|^2dx.$$
Throughout the paper, we assume that $z_0\in D((-A)^{\frac{1}{2}})$. We can rewrite the system \dref{b} subject to \dref{dili} or \dref{reba} as the evolution equation:
\begin{equation}\label{meme}
\left\{\begin{array}{ll}
\dfrac{d}{dt}z(\cdot, t)=Az(\cdot,t)+F(z(\cdot,t)),\\
z(\cdot,0)=z_0(\cdot)
\end{array}\right.
\end{equation}
subject to
\begin{equation*}
\begin{array}{ll}
F(z(\cdot,t))&=-z(x,t)z_{x_1}(x,t)-(1-\kappa)z_{x_1x_1}(x,t)\\&+\kappa z_{x_2x_2}(x,t)
-\mu\sum\limits_{j=1}^N \chi_{j}(x)[z(x,t)-f_{j}(x,t)].
\end{array}
\end{equation*}
Note that the nonlinear term $F$ is locally Lipschitz continuous, that is, there exists a positive constant $l(C)$ such that the following inequality holds:
$$\|F(z_1)-F(z_2)\|_{L^2(\Omega)}\le l(C)\|z_1-z_2\|_{D((-A)^{\frac{1}{2}})}$$
for any $z_1$, $z_2\in D((-A)^{\frac{1}{2}})$ with $\|z_1\|_{D((-A)^{\frac{1}{2}})}\le C$, $\|z_2\|_{D((-A)^{\frac{1}{2}})}\le C$.
Here we prove the nonlinear term $F$ is locally Lipschitz continuous for the case of point measurements. From the expression of $F(z(\cdot,t))$, using Minkowskii's inequality we have
$$\begin{array}{ll}\|F(z_1)-F(z_2)\|_{L^2(\Omega)}\le \|z_1z_{1x_1}-z_2z_{2x_1}\|_{L^2(\Omega)}
+|1-\kappa|\cdot\|z_{1x_1x_1}-z_{2x_1x_1}\|_{L^2(\Omega)}\\
\hspace{1.2cm}+|\kappa|\cdot\|z_{1x_2x_2}-z_{2x_2x_2}\|_{L^2(\Omega)}+\mu \|z_1-z_2\|_{L^2(\Omega)}\\
\hspace{1.2cm}+\mu \sum \limits_{j=1}^N\|\int_{\bar x_j}^x z_{1\xi}(\xi,t)-z_{2\xi}(\xi,t)d\xi\|_{L^2(\Omega_j)}.
\end{array}\eqno(*)$$
Note that $D((-A)^{\frac{1}{2}})=\{f\in H^2(\Omega): f|_{\partial \Omega}=0,\;\dfrac{\partial f}{\partial n}|_{\partial \Omega}=0\}$. Since $z_1$, $z_2 \in D((-A)^{\frac{1}{2}})$ with $\|z_1\|_{D((-A)^{\frac{1}{2}})}\le C$, $\|z_2\|_{D((-A)^{\frac{1}{2}})}\le C$, we obtain that
$z_1-z_2 \in D((-A)^{\frac{1}{2}})$, and there exist some positive constants $M_1$ and $M_2$ such that
$$
\left\{\begin{array}{ll}\|z_1\|_{L^2(\Omega)}\le M_1\left[\|z_{1x_1}\|_{L^2(\Omega)}+\|z_{1x_2}\|_{L^2(\Omega)}\right], \\
\|z_{1x_1}\|_{L^2(\Omega)}+\|z_{1x_2}\|_{L^2(\Omega)}\le M_2\|z_1\|_{D((-A)^{\frac{1}{2}})},\\
|z_2\|_{L^2(\Omega)}\le M_1\left[\|z_{2x_1}\|_{L^2(\Omega)}+\|z_{2x_2}\|_{L^2(\Omega)}\right], \\
\|z_{2x_1}\|_{L^2(\Omega)}+\|z_{2x_2}\|_{L^2(\Omega)}\le M_2\|z_2\|_{D((-A)^{\frac{1}{2}})},\\
\|z_1\!-\!z_2\|_{L^2(\Omega)}\!\!\le \! M_1\!\left[\|z_{1x_1}\!-\!z_{2x_1}\|_{L^2(\Omega)}\!+\!\|z_{1x_2}\!-\!z_{2x_2}\|_{L^2(\Omega)}\right], \\
\|z_{1x_1}\!\!-\!\!z_{2x_1}\!\|_{L^2(\Omega)}\!+\!\|z_{1x_2}\!\!-\!\!z_{2x_2}\|_{L^2(\Omega)}\!\!\le \!M_2\|z_1\!-\!z_2\|_{D((-A)^{\frac{1}{2}})}.
\end{array}\right.
$$
Hence, the following holds:
\begin{equation*}
\begin{array}{ll}
\|z_1z_{1x_1}-z_2z_{2x_1}\|_{L^2(\Omega)}
=\|z_1z_{1x_1}-z_2z_{1x_1}+z_2z_{1x_1}-z_2z_{2x_1}\|_{L^2(\Omega)}\\
\le \|z_1-z_2\|_{L^2(\Omega)}\|z_{1x_1}\|_{L^2(\Omega)}\!+\!\|z_2\|_{L^2(\Omega)}\|z_{1x_1}-z_{2x_1}\|_{L^2(\Omega)}\\
\le M_1M_2\|z_1-z_2\|_{D((-A)^{\frac{1}{2}}} M_2\|z_1\|_{D((-A)^{\frac{1}{2}}} \\+M_1M_2\|z_2\|_{D((-A)^{\frac{1}{2}}}M_2\|z_1-z_2\|_{D((-A)^{\frac{1}{2}}}\\
\le 2M_1M_2^2C\|z_1-z_2\|_{D((-A)^{\frac{1}{2}}},
\end{array}
\end{equation*}
\begin{equation*}
\begin{array}{ll}
|1-\kappa|\!\cdot\!\|z_{1x_1x_1}-z_{2x_1x_1}\|_{L^2(\Omega)}\!+\!|\kappa|\!\cdot\!\|z_{1x_2x_2}-z_{2x_2x_2}\|_{L^2(\Omega)}\\
\le \max\left\{|1-\kappa|, |\kappa|\right\}\|z_1-z_2\|_{D((-A)^{\frac{1}{2}}},
\end{array}
\end{equation*}
\begin{equation*}
\mu \|z_1-z_2\|_{L^2(\Omega)}\le \mu M_1M_2\|z_1-z_2\|_{D((-A)^{\frac{1}{2}}}.
\end{equation*}
Moreover, Lemma 3 implies that there exist some positive constants $M_3$, $M_4$ and $M_5$ such that
\begin{equation*}
\begin{array}{ll}
\mu \sum \limits_{j=1}^N\|\int_{\bar x_j}^x z_{1\xi}(\xi,t)-z_{2\xi}(\xi,t)d\xi\|_{L^2(\Omega_j)}\\
\le \mu \left[M_3\|z_{1x_1}-z_{2x_1}\|_{L^2(\Omega)}\!+\!M_4\|z_{1x_2}-z_{2x_2}\|_{L^2(\Omega)}\!+\!M_5\|z_{1x_1x_2}-z_{2x_1x_2}\|_{L^2(\Omega)}\right]\\
\le \mu M_3M_2\|z_1-z_2\|_{D((-A)^{\frac{1}{2}}}\!+ \!\mu M_4M_2\|z_1-z_2\|_{D((-A)^{\frac{1}{2}}}\!+ \!\mu M_5 \|z_1-z_2\|_{D((-A)^{\frac{1}{2}}}.
\end{array}
\end{equation*}
Substitution of the above inequalities into the right-hand side of (*) yields
$$\|F(z_1)-F(z_2)\|_{L^2(\Omega)}\le l(C)\|z_1-z_2\|_{D(-A)^{\frac{1}{2}}},$$
where
$$\begin{array}{ll}l(C)=2M_1M_2^2C+\max\left\{|1-\kappa|, |\kappa|\right\} +\mu (M_1M_2+M_3M_2+M_4M_2+M_5 ).\end{array}$$
From Theorem 6.3.1 of \cite{Pazy}, it follows that the system \dref{b} subject to \dref{dili} or \dref{reba} has a unique local classical solution $z\in C([0,T), L^2(\Omega))\cap C^1((0,T), L^2(\Omega))$
for any initial function $z_0\in D((-A)^{\frac{1}{2}})$.
\begin{remark} The above mentioned well-posedness result is dependent on the initial condition $z_0\in D((-A)^{\frac{1}{2}})$. If the initial function $z_0\in L^2(\Omega)$, the solution of the system \dref{b} subject to \dref{dili} or \dref{reba} may become a mild solution or weak solution.
\end{remark}
\subsection{Distributed controller under averaged measurements }
\begin{proposition}
Consider the closed-loop system \dref{b} subject to \dref{dili}.
Given positive scalars $\bar \Delta$, if there exist $\delta>0$, $\mu>0$ and
$\lambda_i\geq 0$ $(i=1, 2)$
such that the following LMI holds:
\begin{equation}\label{wo}
\begin{array}{ll}
\!\Upsilon& \triangleq \left[ \begin{array}{ccc}-2\mu+2\delta-\lambda_1\dfrac{\pi^2}{2} &\hspace{0.5cm}\Upsilon_{12}&\hspace{0.5cm}\mu \\
\ast &-2&\hspace{0.5cm}0\\
\ast&\ast&\hspace{0.5cm}-\lambda_2\end{array}\right]\le 0
\end{array}
\end{equation}
where $$\Upsilon_{12}=-\dfrac{\lambda_1}{2}-\lambda_2\dfrac{\bar \Delta^2}{\pi^2}-(1-\kappa),$$
then the
closed-loop system is globally exponentially stable in the $L^2$-sense:
\begin{equation}\label{kk}
\int_\Omega z^2(x,t)dx\le e^{-2\delta t }\int_\Omega z^2(x,0)dx,\; \forall t\geq 0.
\end{equation}
Furthermore, if the strict LMI \dref{wo} is feasible for $\delta=0$, then the closed-loop system is exponentially stable with a small enough decay rate.
\end{proposition}
\begin{proof}
The proof is divided into three parts.\\
Step 1: We have shown that there exists a local clasical solution to \dref{b} subject to \dref{dili}, where $T=T(z_0)$. By Theorem 6.23.5 of \cite{kran}, we obtain that the solution exists for any $T>0$ if this solution admits a priori estimate. In Step 3, it will be shown that the feasibility of LMI \dref{wo} guarantees that the solution of \dref{b} subject to \dref{dili} admits a priori bound, which can further guarantee the existence of the solution for all $t\geq 0$.
Step 2: Assume formally that there exists a classical solution of \dref{b} subject to \dref{dili} for all $t\geq 0$. We consider
the following Lyapunov-Krasovskii functional:
\begin{equation}\label{k}
V(t)=\| z(\cdot,t)\|_{L^2(\Omega)}^2.
\end{equation}
Since $z|_{\partial \Omega}=0$ and $\dfrac{\partial z}{\partial n}|_{\partial \Omega}=0$, integration by parts leads to
\begin{equation}\label{tongtong}
\int_{\Omega} z^2z_{x_1}dx=0,
\end{equation}
\begin{equation}\label{jin}
\begin{array}{ll}
\int_{\Omega} z_{x_1x_2}^2dx=\int_\Omega z_{x_1x_1}z_{x_2x_2}dx.
\end{array}
\end{equation}
Furthermore, from \dref{jin} it follows that
\begin{equation}\label{jin2}
\begin{array}{ll}
-2 \int_{\Omega} [z_{x_1x_1}^2+z_{x_2x_2}^2+2z_{x_1x_2}^2]dx
=-2 \int_{\Omega} [z_{x_1x_1}+z_{x_2x_2}]^2dx
=-2 \int_{\Omega}|\Delta z|^2dx.
\end{array}
\end{equation}
Differentiating \dref{k} along \dref{b}, integrating by part and using \dref{tongtong}, \dref{jin2} we obtain
\begin{equation}\label{henghengshu}
\begin{array}{ll}
\dot V(t)+2\delta V(t)=2\int_\Omega zz_tdx+2\delta \int_\Omega z^2dx\\
=2(1-\kappa)\int_\Omega z_{x_1}^2dx-2\kappa \int_\Omega z_{x_2}^2dx-2\int_\Omega |\Delta z|^2dx\\
-(2\mu-2\delta) \int_\Omega z^2dx+2\mu \sum\limits_{j=1}^N \int_{\Omega_{j} }zf_{j} dx.
\end{array}
\end{equation}
From Lemma \ref{d}, the Wirtinger's inequality yields
\begin{equation}\label{kk1}
\begin{array}{ll}
\lambda_1\left [\int_\Omega \nabla z^T\nabla zdx -\frac{\pi^2}{2}\int_\Omega z^2 dx\right]\geq 0,
\end{array}
\end{equation}
where $\lambda_1\geq 0$.\\
For the case of averaged measurements, $f_j$ is given by \dref{dili}.
Since $\int_{\Omega_{j}}f_{j}(x,t) dx=0$, from Lemma \ref{d}, the Poincar\' e inequality leads to
\begin{equation*}
\begin{array}{ll}
\int_{ \Omega_{j}}f_{j}^2dx \le \frac{2\bar \Delta^2}{\pi^2}\int_{ \Omega_{j}} \nabla z^T \nabla z dx.
\end{array}
\end{equation*}
Hence,
\begin{equation}\label{mm}
\begin{array}{ll}
\lambda_2 \sum\limits_{j=1}^N\left[\frac{2\bar \Delta^2}{\pi^2}\int_{ \Omega_{j}}\nabla z^T \nabla z dx- \int_{ \Omega_{j}} f_{j}^2dx\right]\geq0,
\end{array}
\end{equation}
where $\lambda_2\geq 0$.\\
Integration by parts yields
\begin{equation}\label{ee}
\begin{array}{ll}
-\int_{\Omega}\nabla z^T\nabla z dx=\int_{\Omega}z\Delta zdx.
\end{array}
\end{equation}
Applying S-procedure \cite{Yakubovich}, we add to $\dot V(t)+2\delta V(t)$ the left-hand side of \dref{kk1}, \dref{mm} and use \dref{ee}. Then it follows that
\begin{equation*}
\begin{array}{ll}
\dot V(t)+2\delta V(t)
\le \dot V(t)+2\delta V(t)+\lambda_1\left [\int_{\Omega}\nabla z^T\nabla zdx -\frac{\pi^2}{2}\int_\Omega z^2 dx \right]\\
+\lambda_2\left[ \frac{2\bar \Delta^2}{\pi^2}\int_{\Omega}\nabla z^T \nabla z dx- \sum\limits_{j=1}^N\int_{\Omega_j}f_{j}^2dx\right]\\
\le 2(1-\kappa)\int_\Omega z_{x_1}^2dx-2\kappa \int_\Omega z_{x_2}^2dx-2\int_\Omega |\Delta z|^2dx\\
-(2\mu-2\delta+\lambda_1\frac{\pi^2}{2}) \int_\Omega z^2dx+2\mu \sum\limits_{j=1}^N \int_{\Omega_{j} }zf_{j} dx\\
-(\lambda_1+\lambda_2\frac{2\bar \Delta^2}{\pi^2})\int_\Omega z\Delta z dx-\lambda_2\sum\limits_{j=1}^N\int_{\Omega_{j}} f_{j}^2dx.
\end{array}
\end{equation*}
Note that
\begin{equation}\label{rami1}
\begin{array}{ll}
2(1-\kappa)\int_\Omega z_{x_1}^2dx-2\kappa \int_\Omega z_{x_2}^2dx
= 2(1-\kappa)\int_\Omega \nabla z^T \nabla z dx -2 \int_\Omega z_{x_2}^2dx\\
\le 2(1-\kappa)\int_\Omega \nabla z^T \nabla z dx.
\end{array}
\end{equation}
Then substitution of \dref{ee} into \dref{rami1} yields
\begin{equation}
\begin{array}{ll}
2(1-\kappa)\int_\Omega z_{x_1}^2dx-2\kappa \int_\Omega z_{x_2}^2dx
\le -2(1-\kappa)\int_\Omega z\Delta z dx.
\end{array}
\end{equation}
Hence,
\begin{equation*}
\dot V(t)+2\delta V(t)\le \sum_{j=1}^N \int_{\Omega_j}\! \left[ \begin{array}{ccc}z&\Delta z&f_j \end{array}\right] \!\Upsilon \!\left[ \begin{array}{c}z\\ \Delta z\\f_j \end{array}\right] dx\le 0
\end{equation*}
if $\Upsilon\le 0$ holds.
Therefore, $$V(t) \le e^{-2\delta t} V(0),\; \forall t\geq 0.$$
Note that the feasibility of the strict LMI \dref{wo} with $\delta = 0$ implies its feasibility with a small enough $\delta_0 > 0$. Therefore, if the strict LMI \dref{wo} holds for $\delta = 0$, then the closed-loop system is exponentially stable with a small decay rate $\delta_0 > 0$.
Step 3: The feasibility of LMI \dref{wo} yields that the solution of \dref{b} subject to \dref{dili} admits a priori estimate
$V(t) \le e^{-2\delta t} V(0)$. By Theorem 6.23.5 of \cite{kran}, continuation of this solution under a priori bound to entire interval $[0,\infty)$.
\end{proof}
\subsection{Distributed controller under point measurements }
\begin{proposition}
Consider the closed-loop system \dref{b} subject to \dref{reba}.
Given positive scalars $\bar \Delta$, if there exist $\delta>0$, $\mu>0$, $\eta>0$, $\lambda_1\geq 0$, $\lambda_2\in \mathbb R$ and
$\beta_i>0$ $(i=1, 2,3) $
such that \dref{mingbai} is satisfied and the following LMIs hold:
\begin{equation}\label{weiweihe}
2(1-\kappa)+\beta_1\left(\!\frac{\bar \Delta}{\pi}\right)^2+\lambda_1-\lambda_2\le 0,
\end{equation}
\begin{equation}
-2\kappa+\beta_2\left(\!\frac{\bar \Delta}{\pi}\right)^2+\lambda_1-\lambda_2\le 0,
\end{equation}
\begin{equation}\label{woa}
\Lambda\!\!=\!\!\left[\begin{array}{cccc}-2\mu+2\delta-\lambda_1\dfrac{\pi^2}{2}&-\dfrac{\lambda_2}{2}&-\dfrac{\lambda_2}{2}&\mu\\
\ast&-2 &-2+\dfrac{\beta_3}{2}\left(\!\frac{\bar \Delta}{\pi}\right)^4&0\\
\ast&\ast&-2&0\\
\ast&\ast&\ast&-\eta \end{array}\right]\!\!\le \! 0,
\end{equation}
the closed-loop system is globally exponentially stable satisfying \dref{kk}.
Furthermore, if the strict LMI \dref{woa} is feasible for $\delta=0$, then the closed-loop system is exponentially stable with a small enough decay rate.
\end{proposition}
\begin{proof}
Step 1: We have shown that there exists a local clasical solution to \dref{b} subject to \dref{reba}, where $T=T(z_0)$. By Theorem 6.23.5 of \cite{kran}, we obtain that the solution exists for any $T>0$ if this solution admits a priori estimate. In Step 3, it will be shown that the feasibility of LMIs \dref{weiweihe}-\dref{woa} guarantees that the solution of \dref{b} subject to \dref{reba} admits a priori bound, which can further guarantee the existence of the solution for all $t\geq 0$.
Step 2: Assume formally that there exists a classical solution of \dref{b} subject to \dref{reba} for all $t\geq 0$. Consider $V$ given by \dref{k}. Differentiating $V$ along \dref{b} and integrating by parts, we have \dref{henghengshu}. \\
For the case of point measurements, $f_j$ is given by \dref{reba}.
From Lemma \ref{anton}, we have
\begin{equation*}
\begin{array}{ll}
\eta\|f_j\|_{L^2(\Omega_j)}^2\!\!& \le \beta_1\left(\!\frac{\bar \Delta}{\pi}\right)^2\!\!\|z_{ x_1}\|_{L^2(\Omega_j)}^2\! +\!\beta_2\left(\!\frac{\bar \Delta}{\pi}\right)^2\!\|z_{ x_2}\|_{L^2(\Omega_j)}^2
\!+\!\beta_3\left(\frac{\bar \Delta}{\pi}\right)^4\!\|z_{x_1 x_2}\|_{L^2(\Omega_j)}^2
\end{array}
\end{equation*}
for any scalars $\beta_1$, $\beta_2$, $\beta_3$ such that \dref{mingbai} holds.\\
Hence,
\begin{equation}\label{yingying}
\begin{array}{ll}
\sum\limits_{j=1}^{N}&\left[\beta_1\left(\!\frac{\bar \Delta}{\pi}\right)^2\!\!\!\|z_{ x_1}\|_{L^2(\Omega_j)}^2\! +\beta_2\left(\!\frac{\bar \Delta}{\pi}\right)^2\!\|z_{ x_2}\|_{L^2(\Omega_j)}^2\right.\\&\left.
+\beta_3\left(\frac{\bar \Delta}{\pi}\right)^4\!\!\|z_{x_1 x_2}\|_{L^2(\Omega_j)}^2-\eta\|f_j\|_{L^2(\Omega_j)}^2\right]\geq 0.
\end{array}
\end{equation}
From \dref{ee}, for any $\lambda_2\in \mathbb R$ we have
\begin{equation}
\lambda_2\left[\int_{\Omega}\nabla z^T\nabla z dx+\int_{\Omega}z\Delta zdx\right]=0.
\end{equation}
Similarly, we add to $\dot V(t)+2\delta V(t)$ the left-hand side of \dref{kk1} and \dref{yingying}. Then by taking into account \dref{henghengshu}, we obtain
\begin{equation*}
\begin{array}{ll}
\hspace{0.28cm}\dot V(t)+2\delta V(t)\\
\le \dot V(t)+2\delta V(t)+\lambda_1\left [\int_{\Omega}\nabla z^T\nabla zdx -\frac{\pi^2}{2}\int_\Omega z^2 dx \right]\\
+\lambda_2\left[-\int_{\Omega}\nabla z^T\nabla z dx-\int_{\Omega}z\Delta zdx\right]\\
+\left[\beta_1\left(\frac{\bar \Delta}{\pi}\right)^2\|z_{ x_1}\|_{L^2(\Omega_j)}^2+\beta_2\left(\frac{\bar \Delta}{\pi}\right)^2\|z_{ x_2}\|_{L^2(\Omega_j)}^2\right.\\\left.
+\beta_3\left(\frac{\bar \Delta}{\pi}\right)^4\|z_{x_1 x_2}\|_{L^2(\Omega_j)}^2-\eta\|f_j\|_{L^2(\Omega_j)}^2\right]\\
\le \left[2(1-\kappa)+\beta_1\left(\frac{\bar \Delta}{\pi}\right)^2+\lambda_1-\lambda_2\right]\int_\Omega z_{x_1}^2dx\\
+\left[-2\kappa+\beta_2\left(\frac{\bar \Delta}{\pi}\right)^2+\lambda_1-\lambda_2\right] \int_\Omega z_{x_2}^2dx\\
+\beta_3\left(\!\frac{\bar \Delta}{\pi}\right)^4\int_\Omega z_{x_1x_2}^2dx-2\int_\Omega |\Delta z|^2dx\\
-(2\mu-2\delta+\lambda_1\frac{\pi^2}{2}) \int_\Omega z^2dx+2\mu \sum\limits_{j=1}^N \int_{\Omega_{j} }zf_{j} dx\\
-\lambda_2\int_{\Omega}z\Delta zdx-\eta\sum\limits_{j=1}^N\int_{\Omega_{j}} f_{j}^2dx.
\end{array}
\end{equation*}
By using \dref{jin} and \dref{jin2}, the following inequality holds for all $t\geq 0$
\begin{equation}
\dot V(t)+2\delta V(t) \le \sum\limits_{j=1}^N\int_{\Omega_{j}}\psi^T(x,t)\Lambda \psi(x,t) dx
\end{equation}
where
$$\psi(x,t)={\rm col}\{z, z_{x_1x_1},z_{x_2x_2},f_j \}.$$
Therefore, the LMIs \dref{weiweihe}-\dref{woa} yield \dref{kk}.
Step 3: The feasibility of LMIs \dref{weiweihe}-\dref{woa} yields that the solution of \dref{b} subject to \dref{reba} admits a priori estimate
$V(t) \le e^{-2\delta t} V(0)$. By Theorem 6.23.5 of \cite{kran}, continuation of this solution under a priori bound to entire interval $[0,\infty)$.
\end{proof}
\section{Sampled-data regional stabilization}
\subsection{Sampled-data control under averaged measurements}
For $j= 1, \cdots , N$; $k = 0, 1,\cdots$ we consider the quantities
\begin{equation}\label{gaozhong}
f_{j}(x,t)=z(x,t)-\dfrac{\int_{\Omega_{j}}z(\zeta,t)d\zeta}{|\Omega_{j}|},
\end{equation}
\begin{equation}\label{chuzhong}
g_j(t)=\dfrac{1}{t-t_k}\dfrac{\int_{\Omega_{j}}\int_{t_k}^tz_s(\zeta,s)ds d\zeta}{|\Omega_{j}|}.
\end{equation}
Then the controller \dref{y} subject to \dref{w} leads to the closed-loop
system
\begin{equation}\label{niu}
\left\{\begin{array}{ll}
z_t\!+zz_{x_1}\!+ (1-\kappa) z_{x_1x_1}\!-\!\kappa z_{x_2x_2}+ \Delta^2 z
=-\mu\sum\limits_{j=1}^N \chi_{j}(x)[z-f_{j}-(t-t_k)g_j], \\
\hspace{8cm}(x,t)\in \Omega\times [t_k,t_{k+1}),\\
z|_{\partial \Omega}=0,\;\dfrac{\partial z}{\partial n}|_{\partial \Omega}=0.
\end{array}\right.
\end{equation}
Now we use the step method (see e.g. \cite{Bellman,Emilia4}) to establish the proof of the well-posedness for system \dref{niu}.
For $t\in [t_0,t_1]$, we consider the following equation:
\begin{equation}\label{t0}
\left\{\begin{array}{ll}
z_t+zz_{x_1}+ (1-\kappa) z_{x_1x_1}\!-\!\kappa z_{x_2x_2}+ \Delta^2 z
=-\mu\sum\limits_{j=1}^N \chi_{j}(x)\dfrac{\int_{\Omega_j}z_0(x)dx}{|\Omega_j|}, \\
z|_{\partial \Omega}=0,\;\dfrac{\partial z}{\partial n}|_{\partial \Omega}=0.
\end{array}\right.
\end{equation}
Then system \dref{t0} can be represented as an evolution equation \dref{meme} subject to
\begin{equation*}
\begin{array}{ll}
\!F(z(\cdot,t))\!&\!=-z(x,t)z_{x_1}(x,t)\!-(1-\kappa)z_{x_1x_1}(x,t)\!+\kappa z_{x_2x_2}(x,t)\\&-\mu\sum\limits_{j=1}^N \chi_{j}(x)\dfrac{\int_{\Omega_j}z_0(x)dx}{|\Omega_j|}.
\end{array}
\end{equation*}
Note that the nonlinearity $F(z(\cdot,t))$ is locally Lipschitz continuous. From Theorem 3.3.3 of \cite{Henry}, it follows that
there exists a unique local strong solution $z(\cdot,t)\in C([0,T];D((-A)^{\frac{1}{2}}))\cap C^1((0,T];D(A))$ of \dref{t0} initialized with $z_0\in D((-A)^{\frac{1}{2}})$
on some interval $[0,T]\subset [0,t_1]$, where $T=T(z_0)>0$. By Theorem 6.23.5 of \cite{kran}, we obtain that if this solution admits a priori estimate, then the solution exists on the entire $[0,t_1]$. The priori estimate on the solutions starting from the domain of attraction
will be guaranteed by the stability conditions that we will provide (see Theorem \ref{beibei}).
Then we apply the same line of reasoning step-by-step to the time segments $[t_1,t_2]$, $[t_2,t_3]$, $\cdots$. Following this procedure, we find that the strong solution exists for all $t\geq 0$.
In order to derive the stability conditions for \dref{niu} we employ
the following Lyapunov-Krasovskii functional
\begin{equation}\label{jingjing}
\begin{array}{ll}
V_1(t)&=p_1\|z\|^2_{L^2(\Omega)} +p_2\|\Delta z\|^2_{L^2(\Omega)}\\
&+r(t_{k+1}-t) \int_{\Omega} \int_{t_k}^t e^{2\delta (s-t)}z_s^2(x,s)dsdx,\\
&t\in [t_k,t_{k+1}),\; p_1>0,\; p_2>0,\; r>0.
\end{array}
\end{equation}
\begin{remark}
Note that without delay/sampling behavior, the energy norm is usually used. In the present work, due to the sampling terms, we need to use Lyapunov-Krasovskii functionals (see e.g. \cite{Emilia2,emilia5}). Therefore, additionally to the energy norm $p_1\|z\|^2_{L^2(\Omega)} +p_2\|\Delta z\|^2_{L^2(\Omega)}$, we employ the term $r(t_{k+1}-t) \int_{\Omega} \int_{t_k}^t e^{2\delta (s-t)}z_s^2(x,s)dsdx$ to deal with the sampling.
\end{remark}
For convenience we define $$V=D((-A)^{\frac{1}{2}})$$ with the norm
\begin{equation*}
\|z\|_V^2=p_1\|z\|^2_{L^2(\Omega)} +p_2\|\Delta z\|^2_{L^2(\Omega)}.
\end{equation*}
Here $p_1$ and $p_2$ are positive constants that are related to the Lyapunov-Krasovskii
functional \dref{jingjing}. By using Lyapunov-Krasovskii functional \dref{jingjing}, in
Theorem \ref{beibei} we provide LMI conditions for regional exponential
stability of \dref{niu} and for a bound on the domain of attraction.
\begin{theorem}\label{beibei}
Consider the closed-loop system \dref{niu}. Given positive scalars $C$, $h$, $\mu$, $\bar \Delta$ and $\delta$, let there exist
scalars $r>0$, $\Gamma>0$, $p_1>0$, $p_2>0$, $\lambda_i\geq 0$ $(i=1,2)$ and $\lambda_3\in \mathbb R$ satisfy the linear matrix inequalities:
\begin{equation}\label{yangyang}
\Xi_i|_{z=C}<0,\; \Xi_i|_{z=-C}<0,\; i=1,2
\end{equation}
\begin{equation}\label{ninian}
\left[\begin{array}{cc} p_2-(1+\Gamma)\dfrac{1}{\pi^2}&\hspace{0.3cm} \sqrt{\dfrac{1}{2}} \\ \ast&\hspace{0.3cm} \Gamma \end{array}\right]> 0,
\end{equation}
where
\begin{equation}\label{uu}
\Xi_1=\begin{pmat}[{......|}]
&&&&&&& -r h z\cr
&&&&&&&0\cr
&&&&{ {\Phi_1}}&&&-(1-\kappa)rh\cr
&&&&&&&\kappa rh\cr
&&&&&&& -rh\cr
&&&&&&& -\mu rh\cr
&&&&&&& \mu rh\cr\-
&&&&{{\ast} } & && -rh\cr
\end{pmat},
\end{equation}
\begin{equation}\label{qq}
\Xi_2=\begin{pmat}[{......|}]
&&&&&&& -r h z\cr
&&&&&&&0\cr
&&&&{ {\Phi_2}}&&&-(1-\kappa)rh\cr
&&&&&&&\kappa rh\cr
&&&&&&& -rh\cr
&&&&&&& -\mu rh\cr
&&&&&&& \mu rh\cr
&&&&&&& \mu rh^2 \cr\-
&&&&{{\ast} } & && -rh\cr
\end{pmat},
\end{equation}
$\Phi_1=\{\phi_{ij}\}$ is a
symmetric matrix composed from
$$
\begin{array}{ll}
\phi_{11}=2p_1(1-\kappa)+\lambda_1+\dfrac{2\bar \Delta^2}{\pi^2}\lambda_2-\lambda_3,\;
\phi_{15}=-p_2z,\\
\phi_{22}=-2p_1\kappa+\lambda_1+\dfrac{2\bar \Delta^2}{\pi^2}\lambda_2-\lambda_3,\\
\phi_{33}=-2p_1+2\delta p_2,\; \phi_{34}=2\delta p_2,\\ \phi_{35}=-p_2(1-\kappa),\phi_{36}=-\dfrac{\lambda_3}{2},\\
\phi_{44}=-2p_1+2\delta p_2,\; \phi_{45}=p_2\kappa, \; \phi_{46}=-\dfrac{\lambda_3}{2},\\
\phi_{55}=-2p_2,\;\phi_{56}=-p_2\mu,\; \phi_{57}=p_2\mu,\\
\phi_{66}=-2p_1\mu+2\delta p_1-\dfrac{\pi^2}{2}\lambda_1, \;
\phi_{67}=p_1\mu,\\
\phi_{77}=-\lambda_2,
\end{array}
$$
\begin{equation}
\Phi_2=\begin{pmat}[{......|}]
&&&&{ {\Phi_1}} & &&\ast \cr\-
0&0&0&0&p_2\mu h&p_1\mu h&0&-rhe^{-2\delta h}\cr
\end{pmat}.
\end{equation}
Then for any initial state $z_0\in V$ satisfying $\|z_0\|_V^2<C^2$, a unique solution of \dref{niu} exists and satisfies
$$\begin{array}{ll} p_1\|z\|^2_{L^2(\Omega)} +p_2\|\Delta z\|^2_{L^2(\Omega)}\\\le e^{-2\delta t} [p_1\|z_0\|^2_{L^2(\Omega)} +p_2\|\Delta z_0\|^2_{L^2(\Omega)}],\; , t\ge 0.
\end{array}$$
Furthermore, if the strict LMI \dref{yangyang} is feasible for $\delta= 0$, then the closed-loop system is exponentially stable with a small enough decay rate.
\end{theorem}
\begin{proof}
The proof is divided into three parts.\\
Step 1: We have shown that there exists a unique local strong solution to \dref{niu} on some interval
$[0,T]\subset [0,t_1]$. By Theorem 6.23.5 of \cite{kran}, we obtain that the solution exists on the entire interval $[0,t_1]$ if this solution admits a priori estimate. In Step 3, it will be shown that the feasibility of LMIs \dref{yangyang}, \dref{ninian} guarantees that the solution of \dref{niu} admits a priori bound, which can further guarantee the existence of the solution for all $t\geq 0$.
Step 2: Assume formally that there exists a solution of \dref{niu} for all $t\geq 0$. Differentiating $V_1$ along \dref{niu}, we have
\begin{equation}\label{yanzi}
\begin{array}{ll}
\dot V_1(t)+2\delta V_1(t)
&=2p_1\int_\Omega zz_tdx+2p_2\int_\Omega \Delta z \Delta z_tdx\\
&-r\int_{\Omega} \int_{t_k}^t e^{2\delta (s-t)}z_s^2(x,s)dsdx \\
&+r(t_{k+1}-t)\int_{\Omega}z_t^2(x,t)dx\\
&+2\delta p_1 \int_\Omega z^2dx+2\delta p_2 \int_\Omega |\Delta z|^2dx.
\end{array}
\end{equation}
Substitution of $z_t$
from \dref{niu} leads to
\begin{equation}\label{kkw1}
\begin{array}{ll}
r(t_{k+1}-t)\int_{\Omega}z_t^2(x,t)dx\\
=r(t_{k+1}\!-t)\sum\limits_{j=1}^N\int_{\Omega_j}[-zz_{x_1}-(1-\kappa) z_{x_1x_1}+\kappa z_{x_2x_2}\\
- \Delta^2 z-\mu z+\mu f_j+ \mu (t-t_k)g_j]^2dx\\
\le rh\sum\limits_{j=1}^N\int_{\Omega_j} [-zz_{x_1}-\!(1-\kappa) z_{x_1x_1}\!+\kappa z_{x_2x_2}- \Delta^2 z\\
-\mu z+\mu f_j+ \mu (t-t_k)g_j]^2dx.
\end{array}
\end{equation}
Integration by parts yields
\begin{equation}\label{erb}
\begin{array}{ll}
2p_1\int_\Omega zz_tdx
=2p_1(1-\kappa)\int_\Omega z_{x_1}^2dx-2p_1\kappa \int_\Omega z_{x_2}^2dx-2p_1\int_\Omega |\Delta z|^2dx\\
-2p_1\mu \int_\Omega z^2dx+2p_1\mu \sum\limits_{j=1}^N \int_{\Omega_{j} }z[f_{j}+ (t-t_k)g_j]dx,
\end{array}
\end{equation}
and
\begin{equation}
\begin{array}{ll}
2p_2\int_\Omega \Delta z \Delta z_tdx
=2p_2\int_\Omega \Delta^2 z \cdot z_tdx\\
=2p_2\int_\Omega \Delta ^2 z [-zz_{x_1} -(1-\kappa)z_{x_1x_1}+\kappa z_{x_2x_2}-\Delta^2 z]dx\\
-2p_2\mu \int_\Omega\Delta ^2 z [z-f_j-(t-t_k)g_j]dx.
\end{array}
\end{equation}
The Jensen inequality leads to
\begin{equation}
\begin{array}{ll}
-r\int_{\Omega} \int_{t_k}^t e^{2\delta (s-t)}z_s^2(x,s)dsdx
\le -r e^{-2\delta h}\int_{\Omega}\frac{1}{t-t_k}\left[\int_{t_k}^{t} z_s(x,s)ds\right]^2dx\\
\le -r (t-t_k)e^{-2\delta h} \sum\limits_{j=1}^N \int_{\Omega_j} g_j^2dx.
\end{array}
\end{equation}
From \dref{ee}, we obtain
\begin{equation}\label{een}
\begin{array}{ll}
\lambda_3\left[-\int_{\Omega}\nabla z^T \nabla z dx-\int_{\Omega}z\Delta zdx\right]=0,
\end{array}
\end{equation}
where $\lambda_3\in \mathbb R$.\\
Set $$\begin{array}{ll}
\eta_1 ={\rm col}\{z_{x_1}, z_{x_2}, z_{x_1x_1}, z_{x_2x_2}, \Delta^2 z, z, f_j\},\\
\eta_2 ={\rm col}\{ \eta_1,g_j\},\\
\beta \triangleq\left[\begin{array}{ccccccccc} -z&0& -(1-\kappa) & \kappa &-1&-\mu&\mu& \mu(t-t_k) \end{array}\right].
\end{array}
$$
Then
\begin{equation}\label{kkw3}
\begin{array}{ll}
[-zz_{x_1}-\!(1-\kappa) z_{x_1x_1}\!+\kappa z_{x_2x_2}- \Delta^2 z
-\mu z+\mu f_j+ \mu (t-t_k)g_j]^2
= \eta_2^T\beta^T\beta\eta_2.
\end{array}
\end{equation}
From \dref{kkw1} and \dref{kkw3} we have
\begin{equation}\label{ppy}
\begin{array}{ll}
r(t_{k+1}-t)\int_{\Omega}z_t^2(x,t)dx\le rh \sum\limits_{j=1}^N \disp\int_{\Omega_j}\eta_2^T\beta^T\beta\eta_2dx.
\end{array}
\end{equation}
Applying S-procedure, we add to $\dot V_1(t)+2\delta V_1(t)$ the left-hand side of \dref{kk1}, \dref{mm}, \dref{een}. Then,
\begin{equation}\label{sihuanghu}
\begin{array}{ll}
\dot V_1(t)+2\delta V_1(t)\\
\le \dot V_1(t)+2\delta V_1(t)+\lambda_1\left [\int_{\Omega} \nabla z^T \nabla z dx -\frac{\pi^2}{2}\int_\Omega z^2 dx\right]\\
+\lambda_2\left[ \frac{2\bar \Delta^2}{\pi^2}\int_{\Omega}\nabla z^T\nabla z dx- \sum\limits_{j=1}^N\int_{\Omega_{j}}f_{j}^2dx\right]\\
+\lambda_3\left[-\int_{\Omega}\nabla z^T \nabla z dx-\int_{\Omega}z\Delta zdx\right]\\
\le \sum\limits_{j=1}^N \disp\int_{\Omega_j} \dfrac{h-t+t_k}{h}\eta_1^T\Phi_1 \eta_1+\dfrac{t-t_k}{h}\eta_2^T\Phi_2 \eta_2 dx\\
+rh\sum\limits_{j=1}^N \disp\int_{\Omega_j}\eta_2^T\beta^T\beta\eta_2dx
-(4p_1-4\delta p_2)\int_{\Omega} z_{x_1x_2}^2dx.
\end{array}
\end{equation}
Note that LMIs \dref{yangyang} imply that $\phi_{33}<0$, i.e. $p_1> \delta p_2$.
As in \cite{Anton}, first we assume that
\begin{equation}\label{lele}
\|z(\cdot,t)\|_{C^0(\bar\Omega)}<C,\; \forall t\geq 0.
\end{equation}
Note that $\dfrac{h-t+t_k}{h}+\dfrac{t-t_k}{h} =1$ and $t-t_k\le h$.
Under the assumption \dref{lele},
applying Schur complement to \dref{ppy}, from \dref{kkw3}-\dref{sihuanghu} we obtain
\begin{equation}\label{rami}
\begin{array}{ll}
\dot V_1(t)+2\delta V_1(t)\\
\le\sum\limits_{j=1}^N \disp\int_{\Omega_j} \dfrac{h-t+t_k}{ h} \left[\begin{array}{cc}\eta_1^T& 1\end{array}\right]\Xi_1\left[\begin{array}{c}\eta_1
\\1\end{array}\right]dx\\+\sum\limits_{j=1}^N \disp\int_{\Omega_j}\dfrac{t-t_k}{ h}
[\begin{array}{cc}\eta_2^T& 1\end{array}]\Xi_2\left[\begin{array}{c}\eta_2\\1\end{array}\right]dx
\le 0
\end{array}
\end{equation}
if $\Xi_1<0$, $\Xi_2<0$ for all $z\in (-C,C)$.\\
Matrices $\Xi_1$ and $\Xi_2$ given by \dref{uu}, \dref{qq} are affine in $z$.
Hence, $ \Xi_1< 0$ and $ \Xi_2< 0$ for all $z \in (-C, C)$ if these inequalities
hold in the vertices $z=\pm C$ hold, i.e. if LMIs \dref{yangyang} are feasible.
We prove next that \dref{lele} holds. Lemma \ref{kka} and Schur complement theorem lead to
\begin{equation}\label{tingtingmeng}
\begin{array}{ll}
\!\!\|z\|_{C^0(\bar\Omega)}^2\!\!\le \!\dfrac{1}{2}(1+\Gamma)\!\!\left[\|z_{x_1}\|^2_{L^2(\Omega)}\!
+\!\|z_{x_2}\|^2_{L^2(\Omega)}\right]
\!\!+\!\dfrac{1}{\Gamma}\!\|z_{x_1x_2}\|^2_{L^2(\Omega)}\\
\le \left[(1+\Gamma)\dfrac{1}{\pi^2} +\dfrac{1}{2\Gamma}\right]\|\Delta z\|^2_{L^2(\Omega)}
\le V_1(t).
\end{array}
\end{equation}
The last inequality in \dref{tingtingmeng} follows from \dref{ninian}, and for the second inequality in \dref{tingtingmeng} we use
the Wirtinger's inequality, \dref{jin} and \dref{jin2}.
Therefore, it is
sufficient to show that
\begin{equation}\label{qqp}
V_1(t)<C^2,\;\forall t\geq 0.
\end{equation}
Indeed, for $t = 0$, the inequality \dref{qqp} holds. Let \dref{qqp} be false
for some $t_1$. Then $V_1(t_1)\geq C^2>V_1(0)$. Since $V_1$ is continuous in time, there must exist $t^*\in (0,t_1]$ such that
\begin{equation}\label{vv}
V_1(t)<C^2\;\forall t\in [0,t^*)\; and \; V_1(t^*)=C^2.
\end{equation}
The first relation of \dref{vv}, together with the feasibility of \dref{yangyang}, guarantees that $\dot V_1(t)+2\delta V_1(t)\le 0$
on $[0,t^*)$. Therefore, $V_1(t^*) \le V_1(0) <C^2$. This
contradicts the second relation of \dref{vv}. Thus, \dref{qqp} and consequently, \dref{rami} is true, which implies provided that $\|z_0\|_V <C$. \\
Note that the feasibility of LMI \dref{yangyang} with $\delta= 0$ implies its feasibility with a small enough $\delta_0> 0$. Therefore, if LMI \dref{yangyang} holds for $\delta = 0$, then the closed-loop system is exponentially stable with a small decay rate.
Step 3: The feasibility of LMIs \dref{yangyang}, \dref{ninian} yields that the solution of \dref{niu} admits a priori estimate
$V_1(t) \le e^{-2\delta t} V_1(0)$. By Theorem 6.23.5 of \cite{kran}, this solution (under a priori bound) can be continued to entire interval $[0,\infty)$.
\end{proof}
\subsection{Sampled-data control under point measurements}
Under the controller \dref{y} subject to \dref{swift}, the closed-loop
system becomes
\begin{equation}\label{YAZI}
\left\{\begin{array}{ll}
z_t+zz_{x_1}+ (1-\kappa) z_{x_1x_1}\!-\!\kappa z_{x_2x_2}+ \Delta^2 z\\
=-\mu\sum\limits_{j=1}^N \chi_{j}(x)z(\bar x_j,t_k), \;
(x,t)\in \Omega\times [t_k,t_{k+1}),\\
z|_{\partial \Omega}=0,\;\dfrac{\partial z}{\partial n}|_{\partial \Omega}=0.
\end{array}\right.
\end{equation}
\begin{theorem}\label{woy}
Consider the closed-loop system \dref{YAZI}. Given positive scalars $C$, $h$, $\mu$, $\bar \Delta$ and $\delta_1<2\delta$, let there exist
scalars $r>0$, $\Gamma>0$, $p_1>0$, $p_2>0$, $\eta>0$, $\lambda_i\geq 0$ $(i=1,2)$, $\lambda_3\in \mathbb R$ and
$\beta_i>0$ $(i=1, 2,3) $
such that \dref{mingbai} is satisfied and the following LMIs hold:
\begin{equation}\label{jinhan}
-2\delta_1p_2+\beta_3\left(\frac{\bar\Delta}{\pi}\right)^4\le 0,
\end{equation}
\begin{equation}
\bar \Theta=\left[\begin{array}{ccc} -\delta_1p_2&-\dfrac{\beta_1}{2}\left(\frac{\bar \Delta}{\pi}\right)^2&-\dfrac{\beta_2}{2}\left(\frac{\bar \Delta}{\pi}\right)^2\\
\ast&-\delta_1p_2&0\\
\ast&\ast&-\delta_1p_2\end{array}\right]\le 0,
\end{equation}
\begin{equation}\label{hans}
\left[\begin{array}{cc} p_2-(1+\Gamma)\dfrac{1}{\pi^2}&\hspace{0.3cm} \sqrt{\dfrac{1}{2}} \\ \ast&\hspace{0.3cm} \Gamma \end{array}\right]> 0,
\end{equation}
\begin{equation}\label{kkw}
\Lambda_i|_{z=C}<0,\; \Lambda_i|_{z=-C}<0,\; i=1,2
\end{equation}
where
\begin{equation*}
\Lambda_1=\begin{pmat}[{|.}]
{ {\Theta_0}}&\Theta_1\cr\-
{{\ast} } & -rh\cr
\end{pmat},
\end{equation*}
\begin{equation*}
\Lambda_2=\begin{pmat}[{|.}]
{ {\Theta_2}}&{\Theta_3}\cr\-
{{\ast}}& {-rh}\cr
\end{pmat},
\end{equation*}
$\Theta_0=\{\theta_{ij}\}$ is a symmetric matrix composed from
\begin{equation*}
\begin{array}{ll}
\theta_{11}=2p_1(1-\kappa)+\lambda_1-\lambda_2,\;
\theta_{22}=-2p_1\kappa+\lambda_1-\lambda_2,\\
\theta_{33}=-2p_1+2\delta p_2,\;
\theta_{35}=-p_2(1-\kappa),\;
\theta_{36}=-\dfrac{\lambda_2}{2},\\
\theta_{44}=-2p_1+2\delta p_2,\;
\theta_{45}=p_2\kappa,\;
\theta_{46}=-\dfrac{\lambda_2}{2},\\
\theta_{55}=-2p_2,\;
\theta_{56}=-p_2\mu,\;
\theta_{57}=p_2\mu,\\
\theta_{66}=-2p_1\mu+2\delta p_1-\dfrac{\pi^2}{2}\lambda_1,\;
\theta_{67}=p_1\mu,\\
\theta_{77}=-\eta,
\end{array}
\end{equation*}
\begin{equation}
\Theta_1=[\begin{array}{ccccccc}-r h z &0&-(1-\kappa)rh&\kappa rh& -rh&-\mu rh&\mu rh\end{array}]^T
\end{equation}
\begin{equation}
\Theta_2=\begin{pmat}[{......|}]
&&&&{ {\Theta_0}} & &&\ast \cr\-
0&0&0&0&p_2\mu h&p_1\mu h&0&-rhe^{-2\delta h}\cr
\end{pmat},
\end{equation}
\begin{equation}
\Theta_3=[\begin{array}{cc}\Theta_1&\mu rh^2\end{array}]^T
\end{equation}
Then for any initial state $z_0\in V$ satisfying $\|z_0\|_V^2<C^2$, a unique solution of \dref{YAZI} exists and satisfies
$$\begin{array}{ll} p_1\|z\|^2_{L^2(\Omega)} +p_2\|\Delta z\|^2_{L^2(\Omega)}\le e^{-2\sigma t} [p_1\|z_0\|^2_{L^2(\Omega)} +p_2\|\Delta z_0\|^2_{L^2(\Omega)}],\;t\ge 0,
\end{array}$$
where $\sigma$ is a unique positive solution of \dref{hdou}.
\end{theorem}
\begin{proof}
See Appendix.
\end{proof}
\section{Numerical example}
Consider the system \dref{a} under the sampled-data control law \dref{y} with the averaged measurements \dref{w}. Here we choose
$\mu=0.95$.
By verifying LMI conditions of Theorem \ref{beibei} with $\delta=0.1$, $\kappa=-0.5$,
$\bar \Delta=1/4$, $C=2$, we find that
the closed-loop system \dref{niu} preserves the exponential stability within
a given domain of initial conditions $\|z_0\|_V^2<4$ for $t_{k+1}-t_k \le h\le 0.39$. Note that $\bar \Delta=1/4$ corresponds to $N=16$ square subdomains with the sides length $1/4$.
The feasible solutions of LMIs with $h=0.35$ are given as follows:
$p_1=80.6354$, $p_2=5.145$.
We compute the solution of the closed-loop system \dref{niu} numerically via finite element method.
Let $\xi=0.1$ and $M=1/\xi=10$. Define $(x_{1}^i,x_2^j)=(i\xi,j\xi)$, $i,j=0,1,2,\dots, M$.
We divide $\Omega$ on $M^2=100$ squares $R_{ij}$ defined by
$$R_{ij}\triangleq \{(x_1,x_2)\in \Omega |x_{1}^i\le x_1\le x_{1}^{i+1}, x_{2}^j\le x_2\le x_{2}^{j+1}\}.$$
On the node $(x_{1}^i,x_2^j)$, four finite element basis functions are selected as
\begin{equation*}
\begin{array}{ll}
N_1^{i,j}(x)=\left\{\begin{array}{ll}\!\!\left(1-\frac{x_1-x_1^i}{\xi}\right)\left(\frac{x_2-x_2^j}{\xi}\right),\; &x\in R_{ij},\\
0,\; &otherwise\end{array}\right.\\
N_2^{i,j}(x)=\left\{\begin{array}{ll}\!\!\left(1-\frac{x_1-x_1^i}{\xi}\right)\left(1-\frac{x_2-x_2^j}{\xi}\right),\; &x\in R_{ij},\\
0,\; &otherwise\end{array}\right.\\
N_3^{i,j}(x)=\left\{\begin{array}{ll}\!\!\left(\frac{x_1-x_1^i}{\xi}\right)\left(1-\frac{x_2-x_2^j}{\xi}\right),\; &x\in R_{ij},\\
0,\; &otherwise\end{array}\right.\\
N_4^{i,j}(x)=\left\{\begin{array}{ll}\!\!\left(\frac{x_1-x_1^i}{\xi}\right)\left(\frac{x_2-x_2^j}{\xi}\right),\; &x\in R_{ij},\\
0,\; &otherwise\end{array}\right.\\
\end{array}
\end{equation*}
We consider the Galerkin approximation solution of the closed-loop system in finite dimensional space
generated by these basis functions, which takes the form
$$z^{M}(x,t)=\sum_{k=1}^4\sum_{i,j=1}^M m_k^{i,j}(t)N_k^{i,j}(x),$$
where $m_k^{i,j}(t)$ are determined by standard finite element Galerkin method to satisfy some ODEs.
Fig. 2 shows snapshots of the state $z(x,t)=z(x_1,x_2,t)$ at different times for the closed-loop system \dref{niu} with $t_{k+1}-t_k=0.35$, $N=16$ and initial condition $z(x_1,x_2, 0)\! =\!0.236\! \sin (\pi x_1)\!\sin (\pi x_2)$,\; $(x_1,x_2)\!\in \Omega=(0,1)^2$.
It is seen that the closed-loop system is stable.
Fig. 3 demonstrates the time evolution of $V_1(t)$ via the finite difference method, where the steps of space and time are taken
as $1/4$ and $0.00025$, respectively.
By verifying the LMI conditions of Theorem 1, we obtain the maximum value $h=0.39$ that preserves the exponential stability. By simulation of the solution to the closed-loop system starting from the same initial condition, we find that stability is preserved for essentially larger values of $h$ till approximately $h=2.45$.
\begin{figure}[h]
\begin{center}
\subfigure
{\includegraphics[width=0.7\linewidth]{h.pdf}}
\caption{Snapshots of state $z(x_1,x_2,t)$ at different time $t\in \{0,1.4,14\}$}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\subfigure
{\includegraphics[width=0.7\linewidth]{2DSMALLKSE.png}}
\caption{Lyapunov function $V_1(t)$}
\end{center}
\end{figure}
For the sampled-data controller \dref{y} under the point
measurement \dref{swift}, by choosing $\mu=0.95$,
and using Yalmip we verify LMI conditions of Theorem \ref{woy} with $\delta=0.2$, $\delta_1=0.15$, $\kappa=-0.5$, $\bar \Delta=1/4$, $C=2$. Then we find that the resulting closed-loop system is exponentially stable for $t_{k+1}-t_k \le h\le 0.37$ for any initial values satisfying $ \| z_0\|_{L^2(0,1)} < 1$.
Since point measurements use less information
on the state, the point measurements allow smaller
sampling intervals than averaged measurements.
Simulations of the solutions to the closed-loop system under the point measurements confirm the theoretical results.
\section{Conclusion}
The present paper discusses sampled-data control of 2D KSE
under the spatially distributed averaged or point measurements. Sufficient LMI conditions have been investigated such that the regional stability of the closed-loop system is guaranteed.
Our results are applicable to sampled-data controller design of high dimensional distributed parameter systems. Our next step places its main focus on
$H_\infty$ filtering problem of high dimensional coupled ODE-PDE/PDE-PDE system.
\appendix
\renewcommand\thesection{\appendixname~\Alph{section}}
\renewcommand\theequation{\Alph{section}.\arabic{equation}}
\renewcommand\thelemma{\Alph{section}.\arabic{lemma}}
\renewcommand\thetheorem{\Alph{section}.\arabic{theorem}}
\begin{center}
Proof of Theorem \ref{woy}
\end{center}
Step 1: We have shown that there exists a unique local strong solution to \dref{YAZI} on some interval
$[0,T]\subset [0,t_1]$. By Theorem 6.23.5 of \cite{kran}, we obtain that the solution exists on the entire interval $[0,t_1]$ if this solution admits a priori estimate. In Step 3, it will be shown that the feasibility of LMIs \dref{yangyang}, \dref{ninian} guarantees that the solution of \dref{YAZI} admits a priori bound, which can further guarantees the existence of the solution for all $t\geq 0$.
Step 2: Assume formally that there exists a strong solution of \dref{YAZI} starting from $\|z_0\|_V<C$ for all $t\geq 0$.
Differentiating $V_1$ along \dref{YAZI}, we obtain the inequality \dref{yanzi}.
Denote
\begin{equation}
f_j(x,t)=z(x,t)-z(\bar x_j,t)=\int_{\bar x_j}^x z_\xi(\xi,t)d\xi,
\end{equation}
From Lemma \ref{anton}, we have
\begin{equation}
\begin{array}{ll}
\eta\|f_j\|_{L^2(\Omega_j)}^2\!\!& \le\! \beta_1\left(\!\frac{\bar \Delta}{\pi}\right)^2\!\!\!\|z_{ x_1}\|_{L^2(\Omega_j)}^2\! +\beta_2\left(\!\frac{\bar \Delta}{\pi}\right)^2\!\!\!\|z_{ x_2}\|_{L^2(\Omega_j)}^2\\&
+\beta_3\left(\frac{\bar \Delta}{\pi}\right)^4\!\!\|z_{x_1 x_2}\|_{L^2(\Omega_j)}^2
\end{array}
\end{equation}
for any scalars $\beta_1$, $\beta_2$, $\beta_3$ such that \dref{mingbai} holds.\\
Hence,
\begin{equation}\label{any}
\begin{array}{ll}
\sum\limits_{j=1}^{N}&\left[\beta_1\left(\frac{\bar \Delta}{\pi}\right)^2\|z_{ x_1}\|_{L^2(\Omega_j)}^2 +\beta_2\left(\frac{\bar \Delta}{\pi}\right)^2\!\|z_{ x_2}\|_{L^2(\Omega_j)}^2\right.\\&\left.
+\beta_3\left(\frac{\bar \Delta}{\pi}\right)^4\|z_{x_1 x_2}\|_{L^2(\Omega_j)}^2-\eta\|f_j\|_{L^2(\Omega_j)}^2\right]\geq 0.
\end{array}
\end{equation}
Denote
\begin{equation*}
\rho(x,t)\triangleq\frac{1}{t-t_k}\int_{t_k}^t z_s(x,s)ds.
\end{equation*}
Then, we have
\begin{equation*}
z(x,t)=z(x,t_k)+(t-t_k)\rho(x,t).
\end{equation*}
By using Jensen's inequality, we have
\begin{equation}\label{mimi}
\begin{array}{ll}
-r \int_{ \Omega}\int_{t_k}^t e^{2\delta (s-t)}z_s^2(x,s)ds dx
\le -r e^{-2\delta h}(t-t_k) \int_{ \Omega} \rho ^2(x,t)dx.
\end{array}
\end{equation}
Integration by parts leads to
\begin{equation}\label{gigi}
\begin{array}{ll}
2p_1\int_{ \Omega}zz_tdx
=2p_1(1-\kappa)\int_\Omega z_{x_1}^2dx-2p_1\kappa \int_\Omega z_{x_2}^2dx-2p_1\int_\Omega |\Delta z|^2dx\\
+2p_1\mu \sum\limits_{j=1}^N \int_{\Omega_{j} }z\left[(t-t_k)\rho(x,t)+f_j(x,t_k)\right]dx\\
-2p_1\mu \int_\Omega z^2dx,
\end{array}
\end{equation}
\begin{equation}\label{kiki}
\begin{array}{ll}
2p_2\int_\Omega \Delta z \Delta z_tdx
=2p_2\int_\Omega \Delta^2 z \cdot z_tdx\\
=2p_2\int_\Omega \Delta ^2 z [-zz_{x_1} -(1-\kappa)z_{x_1x_1}+\kappa z_{x_2x_2}-\Delta^2 z]dx\\
-2p_2\mu \sum\limits_{j=1}^N\int_{\Omega_j}\Delta ^2 z [z(x,t)-(t-t_k)\rho(x,t)-f_j(x,t_k)]dx.
\end{array}
\end{equation}
Then from \dref{mimi}-\dref{kiki}, adding \dref{any} into $\dot V_1+2\delta V_1$ we obtain
\begin{equation}\label{juan}
\begin{array}{ll}
&\dot V_1(t)+2\delta V_1(t)-\delta_1\sup\limits_{\theta\in [-h,0]}V_1(t+\theta)
\le \dot V_1(t)+2\delta V_1(t)-\delta_1 V_1(t_k)\\
&+\lambda_1\left [\int_{\Omega} \nabla z^T \nabla z dx -\frac{\pi^2}{2}\int_\Omega z^2 dx\right]\\
&+\lambda_2\left[-\int_{\Omega}\nabla z^T \nabla z dx-\int_{\Omega}z\Delta zdx\right]\\
&+\left[\beta_1\left(\!\frac{\bar \Delta}{\pi}\right)^2\int_{\Omega}z_{ x_1}^2(x,t_k)dx +\beta_2\left(\!\frac{\bar \Delta}{\pi}\right)^2\int_{ \Omega}z_{ x_2}^2(x,t_k)dx\right.\\
&\left.+\beta_3\left(\frac{\bar \Delta}{\pi}\right)^4\int_{\Omega}z_{x_1 x_2}^2(x,t_k)dx-\eta \sum\limits_{j=1}^N\int_{\Omega_j}f_j^2(x,t_k)dx\right]\\
&\le 2p_1(1-\kappa)\int_\Omega z_{x_1}^2dx-2p_1\kappa \int_\Omega z_{x_2}^2dx-2p_1\int_\Omega |\Delta z|^2dx\\
&+2p_1\mu \sum\limits_{j=1}^N \int_{\Omega_{j} } z\left[(t-t_k)\rho+f_j(x,t_k)\right]dx-2p_1\mu \int_\Omega z^2dx\\
&+2p_2\int_\Omega \Delta ^2 z [-zz_{x_1} -(1-\kappa)z_{x_1x_1}+\kappa z_{x_2x_2}-\Delta^2 z]dx\\
&-2p_2\mu \int_\Omega\Delta ^2 z [z-(t-t_k)\rho-f_j(x,t_k)d\xi]dx\\
&-r e^{2\delta h}(t-t_k) \int_{ \Omega} \rho ^2(x,t)dx
+r(t_{k+1}-t)\int_{\Omega}z_t^2(x,t)dx\\
&+2\delta p_1 \int_\Omega z^2dx+2\delta p_2 \int_\Omega |\Delta z|^2dx
-\delta_1 p_1 \int_\Omega z^2(x,t_k)dx\\
&-\delta_1 p_2 \int_\Omega |\Delta z(x,t_k)|^2dx+(\lambda_1-\lambda_2)\int_{ \Omega}[z_{x_1}^2+z_{x_2}^2]dx\\
&-\lambda_1\frac{\pi^2}{2}\int_\Omega z^2dx-\lambda_2\int_\Omega z\Delta zdx+\beta_1\left(\frac{\bar \Delta}{\pi}\right)^2\int_{\Omega}z_{ x_1}^2(x,t_k)dx \\&+\beta_2\left(\frac{\bar \Delta}{\pi}\right)^2\int_{ \Omega}z_{ x_2}^2(x,t_k)dx
+\beta_3\left(\frac{\bar \Delta}{\pi}\right)^4\int_{\Omega}z_{x_1 x_2}^2(x,t_k)dx
-\eta \sum\limits_{j=1}^N\int_{\Omega_j}f_j^2(x,t_k)dx.
\end{array}
\end{equation}
Using \dref{jin2}, we have
\begin{equation}\label{sun}
\begin{array}{ll}
&-2p_1\int_{ \Omega}|\Delta z|^2 dx\\
&=-2p_1 \int_{\Omega} [z_{x_1x_1}^2(x,t)+z_{x_2x_2}^2(x,t)+2z_{x_1x_2}^2(x,t)]dx,
\end{array}
\end{equation}
\begin{equation}\label{anzui}
\begin{array}{ll}
&-\delta_1p_2 \int_{\Omega}|\Delta z(x,t_k)|^2dx\\
&=-\delta_1p_2 \int_{\Omega} [z_{x_1x_1}^2(x,t_k)+z_{x_2x_2}^2(x,t_k)+2z_{x_1x_2}^2(x,t_k)]dx.
\end{array}
\end{equation}
Set
\begin{equation*}
\begin{array}{ll}
\eta_0=&\{z_{x_1}(x,t),z_{x_2}(x,t),z_{x_1x_1}(x,t),z_{x_2x_2}(x,t),\Delta^2 z(x,t),\\&z(x,t),f_j(x,t_k)\},\\
\eta_1=&\{z_{x_1}(x,t),z_{x_2}(x,t),z_{x_1x_1}(x,t),z_{x_2x_2}(x,t),\Delta^2 z(x,t),\\&z(x,t),f_j(x,t_k),\rho(x,t)\},\\
\bar \eta=&\{z(x,t_k),z_{x_1x_1}(x,t_k),z_{x_2x_2}(x,t_k)\}
\end{array}
\end{equation*}
Substituting \dref{sun}, \dref{anzui} into \dref{juan}, we obtain
\begin{equation}\label{shu}
\begin{array}{ll}
&\dot V_1(t)+2\delta V_1(t)-\delta_1\sup\limits_{\theta\in [-h,0]}V_1(t+\theta)\\
&\le \sum\limits_{j=1}^N\disp \int_{\Omega_j} \dfrac{h-t+t_k}{h}\eta_0^T\Theta_0\eta_0 +\dfrac{t-t_k}{h}
\eta_1^T\Theta_1\eta_1 dx\\
&+\int_{ \Omega}\bar\eta^T\bar \Theta \bar \eta dx
+r(t_{k+1}-t)\int_{ \Omega}z_t^2(x,t)dx\\
&-(4p_1-4\delta p_2)\int_{ \Omega}z_{x_1x_2}^2(x,t)dx\\
&-[2\delta_1p_2-\beta_3(\frac{\bar\Delta}{\pi})^4]\int_{\Omega}z_{x_1 x_2}^2(x,t_k)dx.
\end{array}
\end{equation}
Substitution of $z_t$ from \dref{YAZI} yields
\begin{equation}\label{airen}
\begin{array}{ll}
r(t_{k+1}-t)\int_{\Omega}z_t^2(x,t)dx\\
\le rh\sum\limits_{j=1}^N\int_{\Omega_j} [-zz_{x_1}-\!(1-\kappa) z_{x_1x_1}\!+\kappa z_{x_2x_2}- \Delta^2 z\\
-\mu z+ \mu (t-t_k)\rho+\mu f_j(x,t_k)]^2dx.
\end{array}
\end{equation}
Set $\bar\psi={\rm col}\{z_{x_1}(x,t), z_{x_1x_1}(x,t), z_{x_2x_2}(x,t), \Delta^2 z(x,t), \\
z(x,t), f_j(x,t_k), \rho(x,t)\}$ and
\begin{equation*}
\beta \triangleq\left[\begin{array}{cccccccc} -z& -(1-\kappa) & \kappa &-1&-\mu&\mu & \mu(t-t_k)\end{array}\right].
\end{equation*}
Then
\begin{equation}\label{mami}
\begin{array}{ll}
[-zz_{x_1}-\!(1-\kappa) z_{x_1x_1}\!+\kappa z_{x_2x_2}- \Delta^2 z
-\mu z+\mu f_j(x,t_k)\\+ \mu (t-t_k)\rho]^2
= \bar\psi^T\beta^T\beta\bar \psi.
\end{array}
\end{equation}
Application of Schur complement theorem to \dref{mami}, together with \dref{shu} and \dref{airen}, implies
\begin{equation*}
\dot V_1(t)+2\delta V_1(t)-\delta_1\sup\limits_{\theta\in [-h,0]}V_1(t+\theta)\le 0
\end{equation*}
if \dref{jinhan}-\dref{hans} are satisfied, and $\Lambda_1<0$, $\Lambda_2<0$ hold for all $z\in (-C,C)$.
Similar to Theorem \ref{beibei}, LMI \dref{kkw} imply $\Lambda_1<0$, $\Lambda_2<0$ for all $z\in (-C,C)$. Thus the result is established via Halanay's inequality.
Step 3: The feasibility of LMIs \dref{jinhan}-\dref{kkw} yields that the solution of \dref{YAZI} admits a priori estimate
$V_1(t)\le e^{-2\sigma t} \sup\limits_{-h\le \theta \le 0} V_1(\theta)$. By Theorem 6.23.5 of \cite{kran}, continuation of this solution under a priori bound to entire interval $[0,\infty)$.
| 52,903
|
TITLE: Variance of exponentially distributed variable
QUESTION [0 upvotes]: For $X$~$\text{Exp}(1)$ (exponential distribution with parameter $1$), show that $$\text{Var}[(x-1)^2]=8$$
I know I have to calculate pdf, $E(X)$ and $E(X^2)$, but I what really confuses me is the brackets in variance $(x-1)^2$. Any suggestions?
REPLY [0 votes]: Let's find the raw moments: $$\operatorname{E}[X^k] = \int_{x=0}^\infty x^k e^{-x} \, dx = k!, \quad k = 1, 2, \ldots,$$ which is a consequence of the definition of the gamma function.
Then the rest is elementary algebra and using the linearity of expectation: $$\begin{align*}\operatorname{Var}[(X-1)^2] &= \operatorname{E}[(X-1)^4] - \operatorname{E}[(X-1)^2]^2 \\
&= \operatorname{E}[X^4 - 4X^3 + 6X^2 - 4X + 1] - (\operatorname{E}[X^2 - 2X + 1])^2 \\
&= \operatorname{E}[X^4] - 4 \operatorname{E}[X^3] + 6 \operatorname{E}[X^2] - 4 \operatorname{E}[X] + 1 - (\operatorname{E}[X^2] - 2 \operatorname{E}[X] + 1)^2 \\
&= 4! - 4(3!) + 6(2!) - 4(1!) + 1 - (2! - 2(1!) + 1)^2.
\end{align*}$$
| 65,328
|
Welcome to CrossFitABF!
We’re excited to have you come check out our team and our facilities. Please click the Schedule Now link to pick a day that works best for you. Your session is completely FREE of charge and it will give you a great introduction into CrossFitABF, our culture and our facilities. We’re positive you’re going to love the group interaction, we’re certain you’ll make new friends and we KNOW you’re going to love what we have to offer.
During your introductory session, we will answer any questions you have about CrossFitABF.
- Show up 15 minutes before class starts to sign in.
- Wear anything you’re comfortable sweating in! No Dress Code here!
- Bring water! You’re going to be thirsty after, if not during your class.
Feel free to call or email us with any questions you have about CrossFit or your FREE Introductory Session. Otherwise, click the link to see our schedule.
| 257,390
|
Celebrity makeup artist and Lorac Cosmetics Founder Carol Shaw worked with Hunger Games star Stef Dawson for last night's Hunger Games: Mockingjay, Part 1 premiere.
Carol gave Stef a modern, romantic look with a shimmering defined eye, glowing skin and a rosy-nude matte lip.
Wanna know how to steal this pretty red carpet look for yourself? Check out all the details below on how to recreate Dawson's look:
Complexion
Eyes
Cheeks
Brows
Loving the look? Let us know with a note below!
See Now: 100 Most Beautiful Women In The World
© 2018 BeautyWorldNews.com All rights reserved. Do not reproduce without permission.
|Beauty Buzz|
|Beauty Buzz|
|BEAUTY|
| 15,954
|
WEEKEND SPORT: Gaelic football and Tour de France
Fachtna Kelly looks at the main sporting events this weekend including the big Gaelic football clashes and the final hours of the Tour de France.
Fachtna Kelly looks at the main sporting events this weekend including the clash of old rivals Dublin and Meath in the Leinster championship and Meath and Tyrone in the qualifiers, as well as the final hours of the Tour de France.
Are Dublin confident of victory? Well, they may be making all the usual noises about respecting the opposition and all that, but they did name their team awfully early. Perhaps the biggest
| 30,944
|
Dissertation on motivation
Brandy cronométrico ENFACE, popular persuasive essay topics for middle school their mortal grump rogues renegotiation. Sheffy interneural divorced and his geologizing hiccup inexpiably applicator and siding. Frederich petticoated dark and deranged metabolize your time corella or alphanumeric form. Derek dyable stangs its circulating offside and underlapped! scorifying opportunistic Ruddy, his very convexly reflux. ladylike slogging Augustin, his forced circumnutates unscrupulous awards. Useful Tabbie together, their anticlimax rematches. Clayborn nudist achieved its systematizes and blotted with repentance! dissertation on motivation Kalvin bearable sips his phoneme disbowelling. sanctifies evil Finley, its digest very scurrilously. Harvey thermolabile subentire and denitrification his phosphorite depersonalize or scutters prodigiously. scleroid improve creative writing Levi shrugs, she gets very stern. Hercules yesíferos unwigged bitchier and its bias ferrets and gradually loosening. Myron sick beats, its contents mockingly. Flint kidnapped restore its downgraded week. adust compare contrast essay romeo juliet and Scottish Roderick rehearsing his hole or made fascinating. Christie skimp renaissance humanism as an intellectual movement wall, hydroxides coruscate osmotically persists. Holly canorous shimmers his cudgel inclusive. Gerold condensed and non-inverted x-Heights Waylay their keyboards and allusive rabbits. Peter Siles confervoid she grew to rely unwisely? Renard aidful literalise departure and tunings almost! epicene and Willmott separated naphthalized its sublease clover or chastely disturbances. Pattie reproductive thickened and interspersed persuasive speech rubric high school his rattler exaggerate wrinkles and drolly. unposted open sports history essays eluding peerless? Hamlen mythologic spend their ballast secularises unsteadfastly? anhedonic and conjunctiva Bernard flews beautification or malignant sixth. sleazy Clark inshrine that BABIRUSA exciting scientific essay biology jumps. Efren bulbiferous definition, his benevolence duel indeterminably replicas. Derek goose disorderly and steric their Moonlights scrumps and cadged stunned. Scarface unenlightened incarnadined, their tabards Research paper tips for kids touses sincerely hamstring. euhemerise famous Cobbie, his short cloak Rosily monkeys. clumsy and dumb as a jackass Kalle bluing your dissertation on motivation blinders and prolapses graphitizes negligently. Humphrey solid state power their redesigns helplessly. Toddy dissertation on motivation afferent stinks, your titularly unregister. anti-American and Constantinos immunization dissertation on motivation of their inurements prods and hits glidingly crab. Salem psychoneurotic slashed and lattices their spring Caliços or cold welding axiomatically. chug extenuative to mutualise compunctiously? Michale shame covered his near disbelief. Raymundo diphtheroid detected, your very selfish racket.
| 252,488
|
Grafton, C(ornelius) W(arren) 1909–1982
GRAFTON, C(ornelius) W(arren) 1909–1982
PERSONAL: Born 1909, in China; died 1982; children: Sue. Education: Earned degrees in journalism and law.
CAREER: Attorney in Louisville, KY.
AWARDS, HONORS: Mary Roberts Rinehart award, 1943.
WRITINGS:
NOVELS
The Rat Began to Gnaw the Rope (crime), Farrar & Rinehart (New York, NY), 1943.
The Rope Began to Hang the Butcher (crime), Farrar & Rinehart (New York, NY), 1944.
My Name Is Christopher Nagel, Rinehart (New York, NY), 1947.
Beyond a Reasonable Doubt (crime), Rinehart (New York, NY), 1950.
SIDELIGHTS: Attorney C. W. Grafton, the father of bestselling author Sue Grafton, only wrote a handful of novels in his lifetime, but his three mysteries, The Rat Began to Gnaw the Rope, The Rope Began to Hang the Butcher, and Beyond a Reasonable Doubt have been compared to the works of such writers as Erle Stanley Gardner, Rex Stout, and Raymond Chandler. Grafton took his first two titles from a nursery rhyme, and had he continued to write more in the series, books such as "The Water Began to Quench the Fire" and "The Stick Began to Beat the Dog" would have been next. As it is, Grafton's lawyer protagonist Gilmore Henry of Calhoun County, Kentucky, appears only in The Rat Began to Gnaw the Rope and The Rope Began to Hang the Butcher.
These books combine fast-paced plotting and a breezy style with the author's thorough knowledge of the law, as well as of the business world. His invention of crimes with their roots in events decades in the past foreshadows some of the techniques of John D. MacDonald and Ross Macdonald. More determined than some writers to root his fiction in a specific time, Grafton also conveys a strong sense of immediate pre-World War II America with war clouds conspicuous on the horizon. At times, he throws in as many topical references (prices and products; song titles; names of radio and movie stars, politicians, and sports heroes) as would someone writing a historical novel about the time.
The Rat Began to Gnaw the Rope is a complex tale about a stock manipulation scheme. Henry, a short, chubby, but likable character, is introduced as a sleuth not quite in the same mold as his predecessors. The wounds—mostly physical—he suffers during his investigations affect him more personally than the average hard-boiled hero, and he consequently comes off as a more sensitive character. In The Rope Began to Hang the Butcher there is more court action than in the previous Henry book; the portrait of a backwoodsy Kentucky court where the judge wanders around the room during the trial, challenging out-of-towners to tell him from advocates or spectators, is particularly unique. This time, Henry must unravel a scheme involving insurance and real estate, and he manages to do so with the all the perspicuity of the fictional Perry Mason.
Beyond a Reasonable Doubt, Grafton's return to the genre without his series character, is his best-known book. An unusual and suspenseful courtroom novel, the story reveals early on that lawyer Jess London is guilty of the not-unjustified murder of his brother-in-law, Mitchell Sothern. The novel draws its suspense from the question of whether and how he will manage to escape punishment. Tried for the crime after recanting an earlier confession, Jess acts as his own attorney, surviving some of the narrowest escapes in courtroom fiction.
BIOGRAPHICAL AND CRITICAL SOURCES:
PERIODICALS
New York Times Book Review, February 8, 1981, review of Beyond a Reasonable Doubt, p. 31; September 4, 1983, review of The Rat Began to Gnaw the Rope, p. 19.
| 51,195
|
DNR Filmclub | Man Bites Dog (Rémy Belvaux, André Bonzel and Benoît Poelvoorde, 1992)
Let op!De webshop sluit 1 uur voor de voorstelling. Neem contact op met de kassa via 070-2119988
Mockumentary films between reality and fiction
Controversial winner of the International Critics’ Prize at the 1992 Cannes Film Festival.
- In French with English subtitles.
| 159,067
|
King County employees now have access to a new wellness tool. Omada® is a program to support you and eligible covered family members in achieving your goals toward healthy weight and reducing your risk of certain chronic diseases. Omada is available at no cost to you, if you’re eligible. This is another way that King County is Investing in YOU.
You can find out if you or your family members qualify by taking a 1-minute risk screener. You’ll receive the program at no additional cost if you or your family members are enrolled in the King County medical plan (Kaiser Permanente or Regence BlueShield), are at risk for type 2 diabetes, are 18 years or older, and are accepted to the program.
Omada is an innovative digital lifestyle change program that surrounds you with the tools and support you need to lose weight and reduce your risk for certain chronic diseases. Omada is a “CDC-recognized” program, meaning that it’s backed by research and has met the high-quality standards set by the US Centers for Disease Control and Prevention. You can be sure that the work you put in will pay off.
Omada guides you step-by-step to better health. You’ll get:
- A professional health coach to keep you on track – on your best days and your worst.
- Smart technology to track your progress, and reveal what is (and isn’t) working for you.
- Each week, you’ll learn healthy tips for better eating, fitness, sleep, and stress management.
- The support of a small group of peers just like you for encouragement at every step.
Learn more and see if you’re eligible for Omada. We’re excited to start the journey toward a healthy lifestyle together with this great program.
Omada celebrates the diversity that each participant brings to the program, including, but not limited to, race, ethnicity, sex, gender identity, sexual orientation, socioeconomic background, origin, ability and disability, and religion. Learn more.
Contact Balanced You at 206.263.9626 or via email.
| 65,869
|
Tasty, Fairly Easy, and Keeps Well{!}:
(c) 2014, Davd
This recipe will be of most interest to readers who live near the sea. It’s tough enough to buy any fresh saltwater fish in say, Winnipeg or Kansas City—but mackerel? They don’t keep as well as many other fish—which is why the {!} after “keeps well” in the subtitle. (It’s true, too—vinegar is a good preservative, and this recipe seems to include enough vinegar to preserve the pickled fish for a week or two in the ‘fridge.) Salmon and cod and sole—often haddock, “Alaska pollock”, and some exotic farmed fish like ‘basa’—can often be bought “fresh frozen”; but if you find mackerel, or herring, inland, it’s almost certain to be smoked or pickled.
Living near the Gulf of St. Lawrence, i found mackerel to be one of the few species of fish that are available for ordinary people to catch in quantity for food. Salmon and sea bass are very restricted, i’m not sure if the public may angle for cod or haddock at all… but mackerel, you may catch all you can. Late summer and early fall are the easiest time to catch mackerel fairly near shore—so here, if you have a place nearby to fish for them, or somebody gave you a few, is something to do with mackerel that will make them a treat, and give some enthusiasm to your thanks for the next ones.
If you start with “round” (whole) mackerel, my first advice is: Fillet them soon! If somebody brings me a few mackerel and i don’t have time to fillet them that day—make it, within three hours—i put them in the freezer. Since they fillet rather well half-thawed, that’s a practical way to organize the work. I suggest peeling the thin, plastic-like skin from the fillets. (If you have a dog or cat, then when you have cooked the fillets,, pour the “juice” from the container that held them into the cooking pan, add the skin, and cook slowly a few minutes for your pet. My dog likes this treat and giving it to him helps him to let me eat the fillets in peace.)
If somebody gives you mackerel fillets, thank him [or her] enthusiastically. They’re well worth having if you know how to pickle them quickly [or smoke them]; and if you fillet them yourself, you’ll have the task of disposing of the rest of the fish. It has a fairly strong odor and will attract cats and larger carnivores1.
In Europe, i had eaten smoked mackerel that were delicious. Here—well, i won’t name the “brand” i found to be fit to eat but not nearly as appealing as the smoked mackerel i had met in Finnish (and if i recall correctly, Swedish and German) stores when on sabbatical. I didn’t find any processed mackerel that appealed to me as those in the Baltic countries had—and i had learnt to appreciate mackerel over there.
Reading several Internet recipes, and Lapointe’s Poissonerie, i found these seasoning patterns repeatedly:
— bay, onion, and pepper
— mustard and butter
— onion, vinegar, and mixed pickling spice
Dill and parsley were fairly often specified.
So, given what was in the house, including the fillets of two mackerel, i tried: Half a cup of white vinegar,
— 10 peppercorns, 1 bay leaf, a teaspoon of dill seed,
—and a generous amount of chive [cut no longer than ¼”—about half a centimetre—long],
… simmered in a small stainless steel pan. (If you don’t grow chives, i suggest you start with a rounded tablespoon of finely chopped onion [bulb or greens], and then adjust how much onion you use, according to your liking.)
When the liquid had simmered for 3-5 minutes—and it must simmer very gently or the pan will boil dry or too near to dry—i poached 4 fillets [2 fish] in that liquid, which required me to cut the fillets to fit in the pan, and cook them in 2-3 “batches”. Each pan-full took less than five minutes to cook, turning the fillets once, with the vinegar just barely boiling.
The mackerel came out delicious, not rank at all—and rank taste can be a problem with mackerel. It was good plain and with mustard..
I usually eat mackerel with potatoes or rye bread. I also tried some pickled mackerel with rye porridge for breakfast, just to see—and that was pretty good, too.
Take some time, i suggest, with the first batch you make: Be sure the vinegar doesn’t boil hard2, let the dried herbs simmer slowly so they can have 4-5 minutes to flavour the vinegar, and cook the mackerel at “barely a boil”. If you pickle mackerel often, you’ll get used to the timing, and then the work will not need your full attention.
It’s worthwhile making up plenty of this, and freezing it in small amounts to take out now and then over the winter and spring for a treat, if you have the time and the mackerel. If so, you can double, triple, even more than triple the amounts of vinegar and herbs. (You can also use this recipe to cook thawed fillets during the winter. I wouldn’t keep mackerel frozen all the way through spring.)
Like white fish poached in salsa picante, this can become something that distinguishes you as a cook. It’s distinctive, most men like it [and fairly many children and women, too] and it “repeats well”: A guest (or a host to whom you take it as a contribution) will be glad to see and taste it again.
Notes:
1. I recently filleted two mackerel, and buried the carcases about a foot deep, covered with some crab shells, then dirt, then a layer of dog dung, then more dirt. Two days later i saw that a passing fox, or perhaps a raccoon or cat, had started to dig for the fish and stopped at the level of the dung.
2. If much of the volume boiled away, you’d have to add water, and guessing how much water is difficult. Vinegar contains much more water than acetic acid, and the boiling point of acetic acid is hotter than that of water, so the little bit that is lost by gentle simmering causes no trouble. day, June 5th, Moncton, New Brunswick was shut down: No bus service, schools and stores closed, residents of one section of town ordered to lock their houses and hide in their basements. In the first minutes of June 6th, the man “everyone” said had done it, was captured.
Justin Bourque, the radio news told me June 5th, had recently become a survivalist, moved out of his parents’ home [at age 23 or 24], and was reputed to have taken up drugs more worrisome than ‘recreational pot’. He expressed hatred for police and government control. A gun dealer whose shop had survivalist clients and staff, stated publicly that Bourque had not bought weapons nor ammunition from it, though he was socially linked to staff. (When Calgary’s Earl Silverman, frustrated for many years in his efforts to gather support for a safe house for men violently abused by women, committed suicide, his friends remembered him kindly1. People who were linked to Bourque that i heard of, were “distancing themselves”.)
Then, once Bourque had been captured by the RCMP, the news mentioned a court appearance and charges of murder, but no more about his background and character, only that his father had expressed concern to the authorities and been told nothing could be done if Justin had not yet committed a crime..
Several years ago now, in Mayerthorpe, Alberta, another lone gunman killed four Mounties. I did hear his name mentioned once on the radio this past week, in connection with the problem of identifying potential multiple murderers before they kill. (Will Bourque’s name be shoved in our ears every June? I doubt it. The name of the Mayerthorpe killer hasn’t been. Marc Lepine’s name is shoved in our ears by ideological Feminist “vested interests.” The normal reaction to a despicable deed by a probable madman is to let its memory fade with time, which is what happened after the Mayerthorpe murders.)
Bourque’s was not a “men’s rights” nor even misogynist action. One of the two wounded police was a woman. The other wounded constable, and the three who were killed, were men. Given that women make up a minority of police, it seems evident that Bourque’s hatred was directed at police, not women.
As an act in support of anarchy or even increased liberty, it was a dismal failure: The evening of June 6th, after Bourque’s capture, people gathered at the Moncton police station in a vigil of sympathy and support. Before and after, flowers and candles were left at the station in memory of the officers who had been killed and sympathy for the wounded. For six days sympathy and public support for the RCMP dominated the CBC News—the shooting dominated the news for longer than the attacks of September 11, 2001, in which over 3,000 people were killed, dominated the US news2.
Far from bringing the police or the law into disrepute or even doubt, the attacks increased public support dramatically. Those who agree, somewhat or greatly, with survivalism, or oppose the increased regimentation of these times (as compared, for instance, to the times when i was a young man and a boy) are going to be less rather than more likely to express their anti-authoritarianism—for fear of being associated with Bourque, and because of the increased sympathy for “authority”.
If he thought he was some kind of Robin-Hood, he was very much mistaken. Robin-Hood was one of a band of two dozen or more “outlaws”, men who the Ruling Class of the time had denied the normal protections of the law. Robin-Hood, Little John, Alan A’Dale, Friar Tuck, and the rest, robbed rich people who passed through Sherwood Forest—and gave much of what they took, to the poor. They occasionally committed, and much more often threatened, violence—but what they did about wealth was much more like the teachings of Jesus, Moses and Muhammad than like self-serving bandits or pirates… and they were a community, living an alternative to the legal and social-class arrangements of the time, and linked socially to the common people around them.
As for Western Canada’s Louis Riél, he had more men—and women, and children—with him, by far, than Robin-Hood. The Prairie Métis had a working society in much more harmony than conflict with the “First Nations” around them—and even a Provisional Government—they weren’t rebels; they were defending themselves. Like the First Nations, they were defeated by a colonial army with more weapons and troops than they had.
Justin Bourque did himself and his cause, much more harm than if he had gone to the police station or the courthouse, doused himself with gasoline, and lit it on fire; or hanged himself in some out-of-the-way place, in either case, without hurting anyone else. Tom Ball “set himself on fire in front of the Cheshire County Court House,” (in the State of New Hampshire, USA, as reported in a local newspaper); and his Last Words are respected by many men’s interest advocates3. Earl Silverman hanged himself in his garage shortly before the sale of his house closed; he had struggled for two decades to gain some recognition and social support for men who had been violently abused by women, being himself such a man. His story is respected, and if we can get his Last Statement, we intend to provide it also.
In other words, by killing three police and wounding two others, Bourque made himself more of a loser, not less. Or as my Christian faith teaches, seek fellowship, seek refuge, but don’t seek vengeance.
I believe we do live in an overly authoritarian society. How valid Bourque’s grievance against police and government was, what it was even, i can’t say; and unlike Earl Silverman’s last words, i’m not looking for Bourque’s. He tainted them too badly with his last acts. I can mourn Earl Silverman, and wish he had accepted my invitation for a retreat. I can mourn Tom Ball likewise, though i learned of his life only from his death, too late to think about inviting him. Like most sane men whose work is not “doing psychiatry”, i don’t want to meet, much less invite, the sort who might go on a murderous rampage.
Notice that i did not recommend suicide. I respect it as a better alternative to futile violence; i also regard it as an inferior alternative to getting together.
If anyone reading this has strong negative attitudes or feelings—antipathies, is the Latin word—toward some social wrong, and has come to the end of his [or her] patience, my advice is, to repeat: Seek fellowship, seek refuge, but don’t seek vengeance. Even better, don’t wait until you’re bitter—seek fellowship, and if you don’t have a good home, seek refuge with other like-minded men.
If there’s a man like Earl Silverman out there who doesn’t have the fellowship or the refuge he should have (maybe simply because where he lives, shelter is so expensive, as it was in Earl’s Calgary) there’s an e-mail address at everyman dot ca called replies—human readers should be able to put that together, spam robots, i hope, can’t—and i check it every week or so. Maybe this week i’ll try to check it oftener. I have two spare bedrooms4, over 90 acres of mostly forest, and a good prayer garden; and i’m interested in corresponding with men who are getting together in other places.
I expect to give a little priority here, to those who like ecoforestry and horticulture, who write well, and to fellow Christians; but Buddhists, Jews, Muslims, and men of smaller well-disciplined faiths like Sikhs and Native American spiritualities, need not be shy.
Notes:
1. Silverman’s faults were mentioned; they were faults many men and women have. He was honoured not by pretending the faults weren’t there, but for who he was even with them.
2. This is not to imply that the Moncton shootings will claim as much public attention in future, as the destruction of the World Trade Center has repeatedly claimed since 2001. (The damage to the Pentagon building has had far less subsequent attention.) I happen to live a few hours’ journey from Moncton. What this is meant to assert, not merely imply, is that public support for the RCMP and for “authority” was strikingly increased by the shootings.
Since the state funeral for the constables who were killed, there have been many local stories about memorials to them and for other police killed on duty, about grief counselling for their families and colleagues, and about charitable fund raising for the families they left behind.
3. Not all of us who respect the statement overall, advocate the use of Molotov cocktails. It is worth remembering that the USA began with a Revolutionary War in which, Tom Ball wrote, his ancestor Elijah served—also as a sergeant.
4. The house is plain and awkwardly designed—but i could afford to pay for it and the land “in full” and have some savings left. A handful of good men could build a much better house to live in, and this one could become the guest house.
| 376,207
|
April 24, 2012
The fourth Discipline we will deal with is Journaling
JOURNALING: Letting the Soul out to Dry
On the second to last day of the Denver Urban Semester we had a time when we exchange gifts that we had been challenged to make out of things we found around the apartments or the city. I made a bird out of macaroni and cheese (my meal for the summer) and wrote scripture on it, I was surprised it made it home. The gift I received was a small journal with a passage from 2 Corinthians 12 and a very encouraging letter from a the young woman who had drawn my name for the gift exchange. The first morning I wrote in that journal was the morning after I received it, I didn’t write in it again for two weeks.
At that point Journaling was still a very new concept for me, I always thought that journaling was like keeping a diary. To me that meant keeping secrets from people and as my sisters probably did in their diaries “talk about boys.” On the other side of the coin it seemed that journaling was just too time consuming and I thought I was doing good enough just by getting into regularity with my quiet time schedule. Why should I add journaling, didn’t I write enough on this blog enough, why add another discipline.
Discovering God’s Vulnerability…
Christian George starts off his chapter on journaling by reminding us of a very simple and easily forgotten truth, that Jesus stepped out of the glory of heaven and took on the skin of mankind (Godology Christian George, 52). That the Son of God in His infinite wisdom decided to set aside the riches and glory of heaven and step into man’s skin to discover what it was like just to be like us.
When you look at it this way it’s easy to see how Jesus was vulnerable during his earthly ministry. For anyone of power to step down into the trenches with the people who they call workers or brothers opens the door for a lot of trouble. Look at the CBS show “Undercover Boss” where CEO’s and Executive’s from large corporation like NASCAR and White Castle disguise themselves as employees so they can better serve their employees.
In a more perfect and better way this is what Christ did when He came down from heaven. He walked alongside us, hung out with the less desirable members of society and ticked off the richer upper echelon who thought they had it all figured out. Then, because of what Jesus taught a change came to that system, but for that change to happen Christ had to be stripped of all glory and crucified, be buried and raised to life. He had to become vulnerable and submissive, even to death on a cross (Phil 2:11).
…And Encountering Our Own
How often do we actually want to admit that we are vulnerable? Think about it, we live in a culture that teaches us to be strong and to build walls on independence. Then when we think we have it figured out someone comes along and tells us we need to tone it down. These mixed messages often just make us angry and so we slip into a deadly apathy thinking we are not allowed to be anything.
But then we hide that feeling too.
I have discovered though, that journaling provides us a way to be vulnerable as well as a way to celebrate God’s vulnerability. We show our vulnerability by pouring out our hearts and our souls to Christ and writing down prayers. I often use Journaling as a chance to write down what God is revealing to me in scripture, hardly to record the day’s events, but mostly to praise my creator. Journaling becomes a way of getting our emotions out, but it’s also good practice for actually showing that vulnerability within our friendships and other relationships as well.
Celebrating the Incarnation
George has the following to say concerning this discipline
Journaling is an inward practice that reminds us of an upward reality-that God glued Himself to our planet. By inscribing our thoughts and prayers on paper, we appreciate Christ’s condescension, journaling is a celebration of the incarnation (55)
So journaling isn’t just therapeutic, it is, like Art, a way to celebrate the incarnation and bring glory to God. When we acknowledge God in our private lives it becomes easier to acknowledge Him in our public lives. It gives us a chance to reflect on who God is and what exactly it is that He’s doing.
If we are to seek out God on this level we may find that we will fall deeper and deeper in love with who He is and have a clearer view of who we are.
– – –
So why is Journalism a Discipline? Well we’ve seen how prayer, obedience and Art allow us to focus on God, and that’s what the discipline of Journalism does. It allows us to pray, write out what God is asking us to do, or even draw something. It also becomes a very good tool for looking back on what we were struggling with or what we were rejoicing in two or three days, weeks or months earlier.
So go and grab a notebook and pick up a pen and take the time to write down what you are starting to work through spiritually. Write out your prayers, or a scripture you are trying to memorize or an experience or whatever might be on your heart at the given time. Then sit back and reflect on what you’ve just written and see how deeply you start to fall in love with a God who is madly in love with you.
LET YOUR SOUL OUT TO DRY, GO, WRITE & REFLECT!
God Bless You
Jonathan David Faulkner
Also Available from the Good Discipline Series
Also Available from Jonathan Faulkner
Available titles from the 10:31 Life Ministries Writing Team
A Chosen Generation (1 Peter 2:9): Mediocre Christianity by Angel Edwards
College Commitment: Those Puzzled Athenians-and Us. by Rev. David Faulkner
Confessions of a College Freshmen: On Buying Your Own Groceries by Amy Faulkner
To contact or support 10:31 Life Ministries email us at: hi1031.ministries@yahoo.com
To contact Jonathan Faulkner email him at: Jonathan@altrocklive.com
| 143,373
|
TITLE: Let $a$ be a real number and $\mid x-a \mid < \frac{\mid a \mid}{2}$, then $\mid x \mid > \frac{\mid a \mid}{2}.$
QUESTION [0 upvotes]: Here's what I got, but I feel like I might not be going in the right direction with this. Can someone guide me, please?
We will prove this directly. That is, we will assume $a$ is a real number and $\mid x-a \mid < \frac{\mid a \mid}{2}$, and we will show $\mid x \mid > \frac{\mid a \mid}{2}$.
We will start with $\mid x-a \mid < \frac{\mid a \mid}{2}$. We will then divide both sides of the inequality by $\mid a \mid$ to get $\frac{x-a}{\mid a \mid}=\frac{1}{2}$. We can then simplify this to $ \mid 1-\frac{x}{a} \mid <\frac{1}{2}$. We then add the additive inverse of 1 from both sides to get $ \mid \frac{-x}{ a } \mid <-\frac{1}{2}$. We then add the additive inverse of $-\frac{1}{2}$ and the additive inverse of $\frac{-x}{\mid a \mid}$ to both sides to get $\frac{1}{2} < \mid \frac{x}{a} \mid$. We can then multiply both sides by $\mid a \mid$ to get $\frac{\mid a \mid}{2} < x$. We can then rewrite this as $x < \frac{\mid a \mid}{2}$ and we can then get $\mid x \mid < \frac{\mid a \mid}{2}$ which completes this proof.
REPLY [2 votes]: Can you use the triangle inequality? You have $|a| - |x| \le |x-a| < \frac{|a|}{2}$ so $-|x| < \frac{|a|}{2} - |a| = -\frac{|a|}{2} \Rightarrow |x| > \frac{|a|}{2}$.
| 120,417
|
\begin{document}
\hspace*{2.5 in}CUQM-121, HEPHY-PUB 840/07\\
\vspace*{0.4 in}
\title{Ultrarelativistic $N$-boson systems}
\author{Richard L. Hall}
\address{Department of Mathematics and Statistics, Concordia University,
1455 de Maisonneuve Boulevard West, Montr\'eal,
Qu\'ebec, Canada H3G 1M8}
\author{Wolfgang Lucha}
\address{Institute for High Energy Physics, Austrian Academy of Sciences,
Nikolsdorfergasse 18, A-1050 Vienna, Austria}
\eads{\mailto {rhall@mathstat.concordia.ca}, \mailto {wolfgang.lucha@oeaw.ac.at}}
\begin{abstract}General analytic energy bounds are derived for $N$-boson systems governed by
ultrarelativistic Hamiltonians of the~form
$$H=\sum_{i=1}^N\|{\bf p}_i\|+\sum_{1=i<j}^NV(r_{ij}),$$
where $V(r)$ is a static attractive pair potential.
It is proved that a translation-invariant model Hamiltonian $H_c$ provides a lower bound to $H$ for all $N \ge 2.$ This result was conjectured in an earlier paper but proved only for $N = 2,3,4.$ As an example, the energy in the case of the linear potential $V(r) = r$ is determined with error less than $0.55\%$ for all $N\ge2.$
\end{abstract}
\pacs{03.65.Ge, 03.65.Pm\hfil\break
{\it Keywords\/}:
Semirelativistic Hamiltonians, Salpeter Hamiltonians, boson systems}
\vskip0.2in
\maketitle
\section{Introduction}
We consider first the semirelativistic $N$-body Hamiltonian $H$ given by
$$H=\sum_{i=1}^N\sqrt{\|{\mathbf p}_i\|^2+m^2}+\sum_{1=i<j}^NV(r_{ij}),\eqno{(1)}$$
and the following model Hamiltonian $H_c$
$$H_c= \sum_{1=i<j}^N\left[{\gamma}^{-1}\sqrt{\gamma\|{\mathbf p}_i-{\mathbf p}_j\|^2+ (mN)^2}~+~ V(r_{ij}) \right],\eqno{(2)}$$
where $\gamma = {N\choose 2} = \half N(N-1).$
If $\Psi(\rho_2,\rho_3,\dots,\rho_N)$ is the lowest boson eigenstate of $H$ expressed in terms of Jacobi relative coordinates, then it was proved in Ref.~\cite{HallLucha2007} that the model facilitates a `reduction' $\langle H_c\rangle = \langle{\mathcal H}\rangle$ to the expectation of a one-body Hamiltonian ${\mathcal H}$ given by
$${\cal H} = N\sqrt{\lambda p^2+ m^2}~~+~\gamma V(r),\quad \lambda ={{2(N-1)}\over{N}}. \eqno{(3)}$$
The question remains as to the relation between $H$ and the model $H_c.$ It is known from earlier work (discussed in~\cite{HallLucha2007}) that the {\bf lower bound conjecture}
$$\langle H\rangle \geq \langle H_c\rangle\eqno{(4)}$$
is true for the following cases: for the armonic oscillator $V(r) = vr^2,$ for all attractive $V(r)$ in the nonrelativistic large-$m$ limit, and for static gravity $V(r) = -v/r$. This list was augmented in Ref.~\cite{HallLucha2007} by the following cases: in general for $N = 3$, and, if $m = 0,$ for $N = 4.$ The purpose of the present article is to extend this list to include the ultrarelativistic cases $m = 0$ for all $N\ge 2$ and arbitrary attractive $V(r).$
\section{The general lower bound for $m = 0.$}
It was shown in Ref.~\cite{HallLucha2007} that the non-negativity of the expectation $\langle\delta(m,N)\rangle$ is sufficient to establish the validity of the conjecture (4), where
$$
\delta(m, N) = \sum_{i = 1}^{N}\sqrt{\|{\mathbf p}_i\|^2 + m^2}\ -\ {{2}\over{N-1}}\sum_{1=i<j}^{N}
\sqrt{{{N-1}\over{2N}}\|{\mathbf p}_i -{\mathbf p}_j\|^2+ m^2}.\eqno{(5)}
$$
Thus for the new cases we are now able to treat we must consider $\langle\delta(0,N)\rangle$. By using the necessary boson permutation symmetry of $\Psi$, the expectation value we need to study is reduced to
$$
\langle\delta(0, N)\rangle = N\left\langle\|{\mathbf p}_1\|\ -\ \sqrt{{{N-1}\over{2N}}}\|{\mathbf p}_1 -{\mathbf p}_2\|\right\rangle.\eqno{(6)}
$$
\clearpage
The principal result of this paper, the lower bound for $m = 0$ and all $N\ge2$, is an immediate consequence of the following:
\nll{\bf Theorem~1}~~~$\langle\delta(0, N)\rangle= 0$.
\nll{\bf Proof of Theorem~1}
\nll Without loss of generality we adopt in momentum space a coordinate origin such
that $\sum_{i=1}^N{\mathbf p}_i := {\mathbf p} = {\mathbf 0}.$
We define the mean lengths
$$\langle ||{\mathbf p}_1||\rangle := k \quad {\rm and} \quad \langle ||{\mathbf p}_1-{\mathbf p}_2||\rangle := d.\eqno{(7)}$$
We wish to make a correspondence between mean lengths such as $k$ and $d$ and the sides of triangles that can be constructed with these lengths. We consider the triangle formed by the three vectors
$\{{\mathbf p}_1, {\mathbf p}_2,~ {\mathbf p}_1-{\mathbf p}_2\}.$ We suppose that the corresponding angles in this triangle are $\{\phi_{12}, \theta_1,\theta_2\}$ (the same notation is used for other similar triples). We now consider projections of one side on a unit vector along an adjacent side and define the mean angles $\phi$ and $\theta$ by the relations
$$\langle \|{\mathbf p}_1\|\cos(\phi_{12})\rangle := \langle \|{\mathbf p}_1\|\rangle\cos(\phi)$$
and
$$\langle\|{\mathbf p}_1-{\mathbf p}_2\|\cos(\theta_1)\rangle := \langle\|{\mathbf p}_1-{\mathbf p}_2\|\rangle\cos(\theta).$$
Thus, on the average, this triangle is isosceles with one angle $\phi$ and the other two angles $\theta.$ Since ${\mathbf p} = 0,$ we have $\langle {\mathbf p}_1\cdot {\mathbf p}\rangle = 0.$ Hence
$$||{\mathbf p}_1||^2 + \sum_{i = 2}^N||{\mathbf p}_1||||{\mathbf p}_i||\cos(\phi_{1i})= 0.$$
Thus, by dividing by $||{\mathbf p}_1||$ and using boson symmetry, we find
$$\langle\left(||{\mathbf p}_1|| + (N-1)||{\mathbf p}_2||\cos(\phi_{12})\right)\rangle = \langle||{\mathbf p}_1||\left(1 + (N-1)\cos(\phi_{12})\right)\rangle= 0. $$
We therefore conclude that $k(1+(N-1)\cos(\phi)) = 0,$ that is to say
$$\cos(\phi) = -\frac{1}{N-1}.$$
We now consider again the triangle formed by the three vectors $\{{\mathbf p}_1, {\mathbf p}_2,~{\mathbf p}_1-{\mathbf p}_2\}.$ We have immediately from the dot product ${\mathbf p}_1\cdot ({\mathbf p}_1-{\mathbf p_2})$
$$\|{\mathbf p}_1\| \|{\mathbf p}_1-{\mathbf p_2}\|\cos(\theta_1) = \|{\mathbf p}_1\|(\|{\mathbf p}_1\| - \|{\mathbf p}_2\|\cos(\phi_{12})).$$
By dividing by $\|{\mathbf p}_1\|$ and taking means we obtain
$$d\cos(\theta) = k(1-\cos(\phi)).$$
But $\theta = (\pi/2 - \phi/2)$ and $\cos(\phi) = -1/(N-1).$ Hence we conclude
$$\frac{k}{d} = \left(\frac{N-1}{2N}\right)^{\half}.$$
This equality establishes Theorem~1.\qed
\section{The linear potential}
We apply the new bound to the case of the linear potential $V(r) = r.$ The weaker $N/2$ lower bound (discussed in Ref.~\cite{HallLucha2007}) is always available, but, up to now, we knew no way of obtaining tight bounds for this problem. For a comparison upper bound, we use a Gaussian trial function $\Phi$ and the original Hamiltonian $H$ to obtain a scale-optimized variational upper bound $E \leq E_g = (\Phi, H\Phi).$
As we showed in Ref.~\cite{HallLucha2007}, for the linear potential $V(r) = r$ in three spatial dimensions, the conjecture (now proven) implies that the $N$-body bounds are given for $N \ge 2$ by
$$ N\left(\frac{(N-1)^3}{2N}\right)^{1\over 4}e = E_c^L \leq E \leq E_g^U = 4N\left(\frac{(N-1)^3}{2N\pi^2}\right)^{\frac{1}{4}},\eqno{(8)}$$
where $e \approx 2.2322 $ is the bottom of the spectrum~\cite{Boukraa89} of the one-body problem $h = \|{\mathbf p}\| + r$. From (8) we see that the ratio $R = E_g/E_c = 4/(\pi^{\half}e) \approx 1.011.$
The energy of the ultrarelativistic many-body system with linear pair potentials is therefore determined by these inequalities with error less than 0.55\% for all $N\ge 2.$ Earlier we were able to obtain such close bounds for all $N$ only for the harmonic oscillator~\cite{Hall04}.
\section{Conclusion}
We have enlarged the number of semirelativistic problems that satisfy the lower-bound conjecture $\langle H\rangle \geq \langle H_c\rangle$ to include all problems with $m = 0$ and $N\ge 2.$ An extension of the geometric reasoning used in Ref.~\cite{HallLucha2007} from pyramids to more general simplices would perhaps have provided an alternative proof. However, the more algebraic approach adopted here, relying in the end on mean angles in a triangle, seemed to provide a more independent and robust approach.
\section*{Acknowledgement}
One of us (RLH) gratefully acknowledges both partial
financial support of his research under Grant No.~GP3438 from~the Natural
Sciences and Engineering Research Council of Canada and hospitality of the
Institute for High Energy Physics of the Austrian Academy of Sciences in
Vienna.
\medskip
| 123,799
|
Council chiefs have been urged to remove a glass ceiling holding back black and Asian social care workers from promotion after it emerged just 2.5% of social services directors are from ethnic minorities.
Just four of the 152 directors of adult social services in England are believed to be from ethnic minority backgrounds, compared to 17% of registered social workers, according to Association of Directors of Social Services statistics. The Association of Directors of Children’s Services said it kept no data on the issue.
People from ethnic minorities make up 12% of the population in England, according to the Office of National Statistics, showing how frontline social work is over-represented by people from non-white backgrounds.
But the Race Equality Foundation, which works to promote racial equality in health and social care, said the under-representation at senior management level dated back to the 1970s.
Despite efforts over the last 10-15 years, many councils have lost the impetus to promote equality through senior appointments, according to Jabeer Butt, deputy chief executive of the foundation.
“Councillors and directors need to change their mindsets when hiring people to senior positions,” he added.
A national programme supporting ethnic minority staff in social care to gain promotions, Get Ahead, delivered by the Improvement and Development Agency, closed in 2010 after four years when funding ran out.
Former social care director Roy Taylor, who is leading Adass’s response to the issue, admitted local authorities were in danger of appearing out of touch with service users from different cultures. He is planning an event later this year to discuss solutions, bringing together recruitment agencies and managers.
“If you have a senior management team which is overwhelmingly white, BME service users will not feel full of confidence that their needs will be met,” he said.
Butt added that a lack of leadership in promoting diversity in social care was resulting in a poor standard of care for some people from ethnic minorities. “You get pockets of good practice but this is dependent on individuals and teams, rather than across the local authority – but this picture was identified as far back as 1977.”
Barriers for progression include diversity falling off the agenda in the current round of public spending cuts, and a lack of “champions” raising awareness of the issue, according to Taylor.
Butt added that BME managers were being excluded from networks of senior councillors and executives responsible for appointments.
To read about Neelam Bhardwaja’s career journey to become director of social services at Cardiff Council visit the Social Work Blog
What do you think? Have your say on career progression barriers among ethnic minority managers on CareSpace
Keep up to date with the latest developments in social care. Sign up to our daily and weekly emails
Related articles
Special report on social work and ethnic minority issues
Why are so few BME social workers promoted to management?
| 348,861
|
\begin{document}
\begin{abstract}
Let $A$ be a basic finite-dimensional $k$-algebra standardly stratified for a partial order $\leqslant$ and $\Delta$ be the direct sum of all standard modules. In this paper we study the extension algebra $\Gamma = \text{Ext} _A^{\ast} (\Delta, \Delta)$ of standard modules, characterize the stratification property of $\Gamma$ for $\leqslant$ and $\leqslant ^{op}$, and obtain a sufficient condition for $\Gamma$ to be a generalized Koszul algebra (in a sense which we define).
\end{abstract}
\keywords{Extension algebras, standardly stratified, Koszul, Quasi-hereditary.}
\subjclass[2000]{16G10, 16E40.}
\maketitle
Let $A$ be a basic finite-dimensional $k$-algebra standardly stratified with respect to a poset $(\Lambda, \leqslant)$ indexing all simple modules (up to isomorphism), $\Delta$ be the direct sum of all standard modules, and $\mathcal{F} (\Delta)$ be the category of finitely generated $A$-modules with $\Delta$-filtrations. That is, for each $M \in \mathcal{F} (\Delta)$, there is a chain $0 = M_0 \subseteq M_1 \subseteq \ldots \subseteq M_n = M$ such that $M_i / M_{i-1}$ is isomorphic to an indecomposable summand of $\Delta$, $1 \leqslant i \leqslant n$. Since standard modules of $A$ are relative simple in $\mathcal{F} (\Delta)$, we are motivated to exploit the extension algebra $\Gamma = \text{Ext} _A^{\ast} (\Delta, \Delta)$ of standard modules. These extension algebras were studied in \cite{Abe, Drozd, Klamt, Mazorchuk1, Miemietz}. In this paper, we are interested in the stratification property of $\Gamma$ with respect to $(\Lambda, \leqslant)$ and $(\Lambda, \leqslant ^{op})$, and its Koszul property since $\Gamma$ has a natural grading. A particular question is that in which case it is a \textit{generalized Koszul algebra}, i.e., $\Gamma_0$ has a linear projective resolution.
By Gabriel's construction, we associate a locally finite $k$-linear category $\mathcal{E}$ to the extension algebra $\Gamma$ such that the category $\Gamma$-mod of finitely generated left $\Gamma$-modules is equivalent to the category of finitely generated $k$-linear representations of $\mathcal{E}$. We show that the category $\mathcal{E}$ is a \textit{directed category} with respect to $\leqslant$. That is, the morphism space $\mathcal{E} (x,y) = 0$ whenever $x \nleqslant y$. With this terminology, we have:
\begin{theorem}
If $A$ is standardly stratified for $(\Lambda, \leqslant)$, then $\mathcal{E}$ is a directed category with respect to $\leqslant$ and is standardly stratified for $\leqslant ^{op}$. Moreover, $\mathcal{E}$ is standardly stratified for $\leqslant$ if and only if for all $\lambda, \mu \in \Lambda$ and $s \geqslant 0$, Ext$ _A^s (\Delta_{\lambda}, \Delta_{\mu})$ is a projective End$ _A (\Delta_{\mu})$-module.
\end{theorem}
In particular, if $A$ is a quasi-hereditary algebra, then $\Gamma$ is quasi-hereditary with respect to both $\leqslant$ and $\leqslant ^{op}$. We also generalize the above theorem to abstract \textit{stratifying systems} and \textit{Ext-projective stratifying systems} (EPSS) described in \cite{Erdmann, Marcos1, Marcos2, Webb}.
In the case that the standardly stratified algebra $A$ is a graded algebra and $A_0$ is semisimple, its Koszul duality has been studied in \cite{Agoston1, Agoston2, Mazorchuk2, Mazorchuk3, Mazorchuk4}. Since the extension algebra $\Gamma = \text{Ext} _A^{\ast} (\Delta, \Delta)$ has a natural grading, and $\Gamma_0 = \text{End} _A(\Delta)$ in general is not semisimple, we study the general Koszul property of $\Gamma$ by using some generalized Koszul theories developed in \cite{Green, Li1, Madsen2, Madsen3}, where the degree 0 parts of graded algebras are not required to be semisimple.
Take a fixed EPSS $(\underline {\Theta}, \underline{Q})$. As an analogue to linear modules of graded algebras, we define \textit{linearly filtered modules} in this system. With this terminology, a sufficient condition can be obtained for $\Gamma$ to be a generalized Koszul algebra.
\begin{theorem}
Let $(\underline {\Theta}, \underline{Q})$ be an EPSS indexed by a finite poset $(\Lambda, \leqslant)$ such that Ext$_A ^i (Q, \Theta) =0$ for all $i \geqslant 1$ and Hom$ _A (Q, \Theta) \cong \text{Hom} _A (\Theta, \Theta)$. Suppose that all $\Theta_{\lambda}$ are linearly filtered for $\lambda \in \Lambda$. If $M \in \mathcal{F} (\Theta)$ is linearly filtered, then the graded $\Gamma$-module $\text{Ext} _A^{\ast} (M, \Theta)$ has a linear projective resolution. In particular, $\Gamma = \text{Ext} _A^{\ast} (\Theta, \Theta)$ is a generalized Koszul algebra.
\end{theorem}
The paper is organized as follows: in section 1 we characterize the stratification property of $\Gamma$; in section 2 we define linearly filtered modules, study their basic properties and prove the second theorem.
Throughout this paper $A$ is a finite-dimensional basic associative $k$-algebra with identity 1, where $k$ is algebraically closed. We only consider finitely generated modules and denote by $A$-mod the category of finitely generated left $A$-modules. Maps and morphisms are composed from right to left.
\section{Stratification property of extension algebras}
Let $(\Lambda, \leqslant)$ be a finite preordered set parameterizing all simple $A$-modules $S_{\lambda}$ (up to isomorphism). This preordered set also parameterizes all indecomposable projective $A$-modules $P_{\lambda}$ (up to isomorphism). According to \cite{Cline}, the algebra $A$ is standardly-stratified with respect to $(\Lambda, \leqslant)$ if there exist modules $\Delta_{\lambda}$, $\lambda \in \Lambda$, such that the following conditions hold:
\begin{enumerate}
\item the composition factor multiplicity $[\Delta_{\lambda} : S_{\mu}] = 0$ whenever $\mu \nleqslant \lambda$;
and
\item for every $\lambda \in \Lambda$ there is a short exact sequence $0 \rightarrow K_{\lambda} \rightarrow P_{\lambda} \rightarrow \Delta_{\lambda} \rightarrow 0$ such that $K_{\lambda}$ has a filtration with factors $\Delta_{\mu}$ where $\mu > \lambda$.
\end{enumerate}
In some literatures the preordered set $(\Lambda, \leqslant)$ is supposed to be a poset (\cite{Dlab}) or even a linearly ordered set (\cite{Agoston1, Agoston2}). Algebras standardly stratified in this sense are called \textit{strongly standardly stratified} (\cite{Frisk1, Frisk2}). In this paper $(\Lambda, \leqslant)$ is supposed to be a poset.
If $A$ is standardly stratified, then \textit{standard modules} can be defined as \footnote{In \cite{Agoston1, Agoston2} standard modules are defined as $\Delta_{\lambda} = P_{\lambda} / \sum _{\mu > \lambda} \text{tr} _{P_{\mu}} (P_{\lambda})$. Note that in their setup $\leqslant$ is a linear order, so this description of standard modules coincides with ours.}:
\begin{equation*}
\Delta_{\lambda} = P_{\lambda} / \sum _{\mu \nleqslant \lambda} \text{tr} _{P_{\mu}} (P_{\lambda}),
\end{equation*}
where tr$_{P_{\mu}} (P_{\lambda})$ is the trace of $P_{\mu}$ in $P _{\lambda}$. See \cite{Dlab, Webb} for more details. Let $\Delta$ be the direct sum of all standard modules and $\mathcal{F} (\Delta)$ be the full subcategory of $A$-mod such that each object in $\mathcal{F} (\Delta)$ has a filtration by standard modules. Clearly, since $A$ is standardly stratified for $\leqslant$, $_AA \in \mathcal{F} (\Delta)$, or equivalently, every indecomposable projective $A$-module has a filtration by standard modules.
Throughout this section we suppose that $A$ is standardly stratified with respect to $\leqslant$ if it is not specified. We also remind the reader that $\mathcal{F} (\Delta)$ is closed under extensions, kernels of epimorphisms, and direct summands, but is not closed under cokernels of monomorphisms.
Given $M \in \mathcal{F} (\Delta)$ and a fixed filtration $0 = M_0 \subseteq M_1 \subseteq \ldots \subseteq M_n =M$, we define the \textit{filtration multiplicity} $m_{\lambda} = [ M: \Delta_{\lambda}]$ to be the number of factors isomorphic to $\Delta_{\lambda}$ in this filtration. By Lemma 1.4 of \cite{Erdmann}, The filtration multiplicities defined above are independent of the choice of a particular filtration. Moreover, since each standard module has finite projective dimension, we deduce that every $A$-module contained in $\mathcal{F} (\Delta)$ has finite projective dimension. Therefore, the extension algebra $\Gamma = \text{Ext} _A^{\ast} (\Delta, \Delta)$ is finite-dimensional.
\begin{lemma}
Let $\Delta_{\lambda}$, $\Delta_{\mu}$ be standard modules. Then Ext$ _A^n (\Delta _{\lambda}, \Delta_{\mu}) =0$ if $\lambda \nleqslant \mu$ for all $n \geqslant 0$.
\end{lemma}
\begin{proof}
First, we claim $[\Omega^i (\Delta_{\lambda}) : \Delta_{\nu}] = 0$ whenever $\lambda \nleqslant \nu$ for all $i \geqslant 0$, where $\Omega$ is the Heller operator. Indeed, for $i =0$ the conclusion holds clearly. Suppose that it is true for all $i \leqslant n$ and consider $\Omega ^{n+1} (\Delta _{\lambda})$. We have the following exact sequence:
\begin{equation*}
\xymatrix {0 \ar[r] & \Omega ^{n+1} (\Delta _{\lambda}) \ar[r] & P \ar[r] & \Omega^n (\Delta _{\lambda}) \ar[r] & 0.}
\end{equation*}
By the induction hypothesis, $[\Omega ^n (\Delta _{\lambda}) : \Delta_{\nu}] = 0$ whenever $\lambda \nleqslant \nu$. Therefore, $[P : \Delta_{\nu}] = 0$ whenever $\lambda \nleqslant \nu$, and hence $[\Omega ^{n+1} (\Delta _{\lambda}) : \Delta_{\nu}] = 0$ whenever $\lambda \nleqslant \nu$. The claim is proved by induction.
The above short exact sequence induces a surjection Hom$ _A (\Omega ^n (\Delta _{\lambda}), \Delta_{\mu}) \rightarrow \text{Ext} ^n_A (\Delta _{\lambda}, \Delta_{\mu})$. Thus it suffices to show Hom$ _A (\Omega ^n (\Delta _{\lambda}), \Delta_{\mu}) =0$ for all $n \geqslant 0$ if $\lambda \nleqslant \mu$. By the above claim, all filtration factors $\Delta_{\nu}$ of $\Omega ^n (\Delta _{\lambda})$ satisfy $\nu \geqslant \lambda$, and hence $\nu \nleqslant \mu$. But Hom$ _{A} (\Delta_{\nu}, \Delta_{\mu}) = 0$ whenever $\nu \nleqslant \mu$. The conclusion follows.
\end{proof}
Gabriel's construction gives rise to a bijective correspondence between finite-dimensional algebras and locally finite $k$-linear categories with finitely many objects. Explicitly, To each finite-dimensional $k$-algebra $A$ with a chosen set of orthogonal primitive idempotents $\{ e_{\lambda} \} _{\lambda \in \Lambda}$ satisfying $\sum _{\lambda \in \Lambda} e_{\lambda} =1$ we define a $k$-linear category $\mathcal{A}$ with $\Ob \mathcal{A} = \{ e_{\lambda} \} _{\lambda \in \Lambda}$ and $\mathcal{A} (e_{\lambda}, e_{\mu}) = e_{\mu} A e_{\lambda} \cong \text{Hom} _{A} (A e_{\mu}, A e_{\lambda})$. Conversely, given a locally finite $k$-linear category $\mathcal{A}$ with finitely many objects, we define $A = \bigoplus _{x, y \in \text{Ob } \mathcal{A}} \mathcal{A} (x,y)$, and the multiplication in $A$ is induces by the composition of morphisms in $\mathcal{A}$ in an obvious way. Clearly, $A$-mod is Morita equivalent to the category of all finite-dimensional $k$-linear representations of $\mathcal{A}$. We then call $\mathcal{A}$ the \textit{associated category} of $A$ and $A$ the \textit{associated algebra} of $\mathcal{A}$. The category $\mathcal{A}$ is called \textit{directed} if there is a partial order $\preccurlyeq$ on $\Ob \mathcal{A}$ such that $\mathcal{A} (x, y) = 0$ unless $x \preccurlyeq y$.
Now let $\Gamma = \text{Ext} _A^{\ast} (\Delta, \Delta)$. This is a graded finite-dimensional algebra equipped with a natural grading. In particular, $\Gamma_0 = \text{End} _A (\Delta)$. For each $\lambda \in \Lambda$, $\Delta_{\lambda}$ is an indecomposable $A$-module. Therefore, up to isomorphism, the indecomposable projective $\Gamma$-modules are exactly those $\text{Ext} _A^{\ast} (\Delta_{\lambda}, \Delta)$, $\lambda \in \Lambda$.
The associated $k$-linear category $\mathcal{E}$ of $\Gamma$ has the following structure: $\Ob \mathcal{E} = \{ \Delta _{\lambda} \} _{\lambda \in \Lambda}$; the morphism space $\mathcal{E} (\Delta _{\lambda}, \Delta_{\mu}) = \text{Ext} _A^{\ast} (\Delta_{\lambda}, \Delta_{\mu})$. The partial order $\leqslant$ induces a partial order on $\Ob \mathcal{E}$ which we still denote by $\leqslant$, namely, $\Delta _{\lambda} \leqslant \Delta_{\mu}$ if and only if $\lambda \leqslant \mu$.
\begin{proposition}
The associated category $\mathcal{E}$ of $\Gamma$ is directed with respect to $\leqslant$. In particular, $\Gamma$ is standardly stratified with respect to $\leqslant ^{op}$ and all standard modules are projective.
\end{proposition}
\begin{proof}
The first statement follows from the previous lemma. The second statement is also clear. Indeed, since $\Gamma$ is directed with respect to $\leqslant$, $e_{\mu} \Gamma e_{\lambda} \cong \text{Hom} _{\Gamma} (Q_{\mu}, Q_{\lambda}) = 0$ if $\mu \ngeqslant \lambda$, where $Q_{\mu}, Q_{\lambda}$ are projective $\Gamma$-modules. Thus tr$ _{Q_{\mu}} (Q_{\lambda}) = 0$ whenever $\mu \ngeqslant \lambda$, or equivalently, tr$ _{Q_{\mu}} (Q_{\lambda}) = 0$ whenever $\mu \nleqslant ^{op} \lambda$. Therefore, all standard modules with respect to $\leqslant ^{op}$ are projective.
\end{proof}
The following proposition characterizes the stratification property of a $k$-linear category directed with respect to $\leqslant$.
\begin{proposition}
Let $\mathcal{C}$ be a locally finite $k$-linear category directed with respect to a partial order $\leqslant$ on $\Ob \mathcal{C}$. Then it is stratified for this order. The standard modules are isomorphic to indecomposable summands of $\bigoplus _{x \in \text{Ob } \mathcal{C}} \mathcal{C} (x, x)$. Moreover, this stratification is standard if and only if for each pair of objects $x, y \in \Ob \mathcal{C}$, $\mathcal{C} (x, y)$ is a projective $\mathcal{C} (y,y)$-module.
\end{proposition}
\begin{proof}
This is just a collection of results in \cite{Li1}. The first statement is Corollary 5.4; the second statement comes from Proposition 5.5; and the last statement is Theorem 5.7.
\end{proof}
Now we restate and prove the first theorem.
\begin{theorem}
If $A$ is standardly stratified for $(\Lambda, \leqslant)$, then $\mathcal{E}$ is a directed category with respect to $\leqslant$ and is standardly stratified for $\leqslant ^{op}$. Moreover, $\mathcal{E}$ is standardly stratified for $\leqslant$ if and only if for all $\lambda, \mu \in \Lambda$ and $s \geqslant 0$, Ext$ _A^s (\Delta_{\lambda}, \Delta_{\mu})$ is a projective End$ _A (\Delta_{\mu})$-module.
\end{theorem}
\begin{proof}
The first statement follows from Proposition 1.2 and the second statement follows from Proposition 1.3.
\end{proof}
In the case that $A$ is quasi-hereditary, we have:
\begin{corollary}
If $A$ is a quasi-hereditary algebra with respect to $\leqslant$, then $\Gamma$ is quasi-hereditary with respect to both $\leqslant$ and $\leqslant ^{op}$.
\end{corollary}
\begin{proof}
We have showed that $\Gamma$ is standardly stratified with respect to $\leqslant ^{op}$ and the corresponding standard modules $_{\Gamma} \Delta _{\lambda} \cong \Gamma 1_{\lambda}$ for $\lambda \in \Lambda$. Therefore,
\begin{equation*}
\text{End} _{\Gamma} (_{\Gamma} \Delta _{\lambda}) = \text{End} _{\Gamma} (\Gamma 1_{\lambda}) \cong 1_{\lambda} \Gamma 1_{\lambda} = \text{Ext} _A^{\ast} (\Delta_{\lambda}, \Delta_{\lambda}) = \text{End}_A (\Delta_{\lambda}) \cong k
\end{equation*}
since $A$ is quasi-hereditary. So $\Gamma$ is also quasi-hereditary with respect to $\leqslant^{op}$.
Now consider the stratification property of $\Gamma$ with respect to $\leqslant$. The associated category $\mathcal{E}$ is directed with respect to $\leqslant$. Since $\text{Ext} _A^{\ast} (\Delta_{\mu}, \Delta_{\mu}) = \text{End} _A (\Delta_{\mu}) \cong k$ for all $\mu \in \Lambda$, $\mathcal{E} (\Delta_{\lambda}, \Delta_{\mu}) = \text{Ext} _A^{\ast} (\Delta_{\lambda}, \Delta_{\mu})$ is a projective $k$-module for each pair $\lambda, \mu \in \Lambda$. Therefore, $\mathcal{E}$ is standardly stratified for $\leqslant$ by the previous theorem. Moreover, by Proposition 1.3, the standard modules of $\mathcal{E}$ (or the standard modules of $\Gamma$) are precisely indecomposable summands of $\bigoplus _{\lambda \in \Lambda} \text{Ext} _A^{\ast} (\Delta_{\lambda}, \Delta_{\lambda}) \cong \bigoplus _{\lambda \in \Lambda} k_{\lambda}$. Clearly, for $\lambda \in \Lambda$, End$ _{\Gamma} (k_{\lambda}, k_{\lambda}) \cong k$, so $\Gamma$ is quasi-hereditary with respect to $\leqslant$.
\end{proof}
The following example from 8.2 in \cite{Frisk2} illustrates why we should assume that $\leqslant$ is a partial order rather than a preorder. Indeed, in a preordered set $(\Lambda, \leqslant)$ we cannot deduce $x = y$ if $x \leqslant y$ and $y \leqslant x$.
\begin{example}
Let $A$ be the path algebra of the following quiver with relations $\alpha_1 \beta_1 = \alpha_2 \beta_2 = \alpha_2 \alpha_1 = \beta_1 \beta_2 =0$. Define a preorder $\leqslant$ by letting $x \leqslant y < z$ and $y \leqslant x < z$.
\begin{equation*}
\xymatrix{x \ar@/^/[r] ^{\alpha_1} & y \ar@/^/[r] ^{\alpha_2} \ar@/^/[l] ^{\beta_1} & z \ar@/^/[l] ^{\beta_2}}.
\end{equation*}
Projective modules and standard modules are described as follows:
\begin{equation*}
P_x \cong \Delta_x = \begin{matrix} x \\ y \\ x \end{matrix} \qquad P_y = \begin{matrix} & y & \\ x & & z \\ & & y \end{matrix} \qquad \Delta_y = \begin{matrix} y \\ x \end{matrix} \qquad P_z \cong \Delta_z = \begin{matrix} z \\ y \end{matrix}
\end{equation*}
Then the associated category $\mathcal{E}$ of $\Gamma = \text{Ext} _A^{\ast} (\Delta, \Delta)$ is not a directed category since both Hom$_A (\Delta_x, \Delta_y)$ and Hom$ _A (\Delta_y, \Delta_x)$ are nonzero.
\end{example}
Now we generalize the above results to \textit{Ext-Projective Stratifying Systems} (EPSS). From now on the algebra $A$ is finite-dimensional and basic, but we do not assume that it is standardly stratified for some partial order, as we did before. The EPSS we describe in this paper is indexed by a finite poset $(\Lambda, \leqslant)$ rather than a linearly ordered set as in \cite{Marcos1, Marcos2}. However, this difference is not essential and all properties described in \cite{Marcos1, Marcos2} can be applied to our situation with suitable modifications.
\begin{definition}
(Definition 2.1 in \cite{Marcos2}) Let $\underline{\Theta} = \{ \Theta_{\lambda} \}_{\lambda \in \Lambda}$ be a set of nonzero $A$-modules and $\underline{Q} = \{ Q_{\lambda} \} _{\lambda \in \Lambda}$ be a set of indecomposable $A$-modules, both of which are indexed by a finite poset $(\Lambda, \leqslant)$. We call $(\underline {\Theta}, \underline{Q})$ an EPSS if the following conditions are satisfied:
\begin{enumerate}
\item Hom$ _A (\Theta_{\lambda}, \Theta_{\mu}) =0$ if $\lambda \nleqslant \mu$;
\item for each $\lambda \in \Lambda$, there is an exact sequence $0 \rightarrow K_{\lambda} \rightarrow Q_{\lambda} \rightarrow \Theta_{\lambda} \rightarrow 0$ such that $K_{\lambda}$ has a filtration only with factors isomorphic to $\Theta_{\mu}$ satisfying $\mu > \lambda$;
\item for every $A$-module $M \in \mathcal{F} (\underline {\Theta})$ and $\lambda \in \Lambda$, Ext$ _A^1 (Q_{\lambda}, M) =0$.
\end{enumerate}
\end{definition}
We denote $\Theta$ and $Q$ the direct sums of all $\Theta _{\lambda}$'s and $Q_{\lambda}$'s respectively, $\lambda \in \Lambda$.
Given an EPSS $(\underline {\Theta}, \underline{Q})$ indexed by $(\Lambda, \leqslant)$, $(\underline {\Theta}, \leqslant)$ is a \textit{stratifying system} (SS): Hom$ _A (\Theta_{\lambda}, \Theta_{\mu}) =0$ if $\lambda \nleqslant \mu$, and Ext$ _A^1 (\Theta_{\lambda}, \Theta_{\mu}) =0$ if $\lambda \nless \mu$. Conversely, given a stratifying system $(\underline {\Theta}, \leqslant)$, we can construct an EPSS $(\underline {\Theta}, \underline{Q})$ unique up to isomorphism. See \cite{Marcos2} for more details. Moreover, as described in \cite{Marcos2}, the algebra $B = \text{End} _A (Q) ^{op}$ is standardly stratified, and the functor $e_Q = \text{Hom} _A (Q, -)$ gives an equivalence of exact categories between $\mathcal{F} (\Theta)$ and $\mathcal{F} (_B \Delta)$.
To study the extension algebra $\Gamma = \text{Ext} _A^{\ast} (\Theta, \Theta)$, one may want to use projective resolutions of $\Theta$. However, different from the situation of standardly stratified algebras, the regular module $_AA$ in general might not be contained in $\mathcal{F} (\Theta)$. If we suppose that $_AA$ is contained in $\mathcal{F} (\Theta)$ (in this case the stratifying system $(\underline {\Theta}, \leqslant)$ is said to be \textit{standard}) and $\mathcal{F} (\Theta)$ is closed under the kernels of surjections, then by Theorem 2.6 in \cite{Marcos1} $A$ is standardly stratified for $\leqslant$ and those $\Theta_{\lambda}$'s coincide with standard modules of $A$. This situation has been completely discussed previously. Alternately, we use the \textit{relative projective resolutions} whose existence is guaranteed by the following proposition.
\begin{proposition}
(Corollary 2.11 in \cite{Marcos2}) Let $(\underline {\Theta}, \underline{Q})$ be an EPSS indexed by a finite poset $(\Lambda, \leqslant)$. Then for each $M \in \mathcal{F} (\Theta)$, there is a finite resolution
\begin{equation*}
\xymatrix {0 \ar[r] & Q^d \ar[r] & \ldots \ar[r] & Q^0 \ar[r] & M \ar[r] & 0}
\end{equation*}
such that each kernel is contained in $\mathcal{F} (\Theta)$, where $0 \neq Q^i \in \text{add} (Q)$ for $0 \leqslant i \leqslant d$.
\end{proposition}
The number $d$ in this resolution is called the \textit{relative projective dimension} of $M$.
\begin{proposition}
Let $(\underline {\Theta}, \underline{Q})$ be an EPSS indexed by a finite poset $(\Lambda, \leqslant)$ and $d$ be the relative projective dimension of $\Theta$. If Ext$ _A^s (Q, \Theta) = 0$ for all $s \geqslant 1$, then for $M, N \in \mathcal{F} (\Theta)$ and $s > d$, Ext$ _A^s (M, N) =0$.
\end{proposition}
\begin{proof}
Since both $M$ and $N$ are contained in $\mathcal{F} (\Theta)$, it is enough to show that Ext$ _A^s (\Theta, \Theta) = 0$ for all $s > d$. If $d =0$, then $Q = \Theta$ and the conclusion holds trivially. So we suppose $d \geqslant 1$. Applying the functor Hom$ _A (-, \Theta)$ to the exact sequence
\begin{equation*}
\xymatrix {0 \ar[r] & K_1 \ar[r] & Q \ar[r] & \Theta \ar[r] & 0}
\end{equation*}
we get a long exact sequence. In particular, from the segment
\begin{equation*}
\xymatrix {\text{Ext} _A^{s-1} (Q, \Theta) \ar[r] & \text{Ext} _A^{s-1} (K_1, \Theta) \ar[r] & \text{Ext} _A^s (\Theta, \Theta) \ar[r] & \text{Ext} _A^s (Q, \Theta)}
\end{equation*}
of this long exact sequence we deduce that $\text{Ext} _A^s (\Theta, \Theta) \cong \text{Ext} _A^{s-1} (K_1, \Theta)$ since the first and last terms are 0. Now applying Hom$ _A (-, \Theta)$ to the exact sequence
\begin{equation*}
\xymatrix {0 \ar[r] & K_2 \ar[r] & Q^1 \ar[r] & K_1 \ar[r] & 0}
\end{equation*}
we get $\text{Ext} _A^{s-1} (K_1, \Theta) \cong \text{Ext} _A^{s-2} (K_2, \Theta)$. Thus $\text{Ext} _A^s (\Theta, \Theta) \cong \text{Ext} _A^{s-d} (K_d, \Theta)$ by induction. But $K_d \cong Q^d \in \text{add} (Q)$. The conclusion follows.
\end{proof}
Thus $\Gamma = \text{Ext} _A^{\ast} (\Theta, \Theta)$ is a finite-dimensional algebra under the given assumption.
There is a natural partition on the finite poset $(\Lambda, \leqslant)$ as follows: let $\Lambda_1$ be the subset of all minimal elements in $\Lambda$, $\Lambda_2$ be the subset of all minimal elements in $\Lambda \setminus \Lambda_1$, and so on. Then $\Lambda = \sqcup_{i \geqslant 1} \Lambda_i$. With this partition, we can introduce a \textit{height function} $h: \Lambda \rightarrow \mathbb{N}$ in the following way: for $\lambda \in \Lambda_i \subseteq \Lambda$, $i \geqslant 1$, we define $h(\lambda) = i$.
For each $M \in \mathcal{F} (\Theta)$, we define supp$(M)$ to be the set of elements $\lambda \in \Lambda$ such that $M$ has a $\Theta$-filtration in which there is a factor isomorphic to $\Theta_{\lambda}$. For example, supp$ (\Theta_{\lambda}) = \{ \lambda \}$. By Lemma 2.6 in \cite{Marcos2}, the multiplicities of factors of $M$ is independent of the choice of a particular $\Theta$-filtration. Therefore, supp$(M)$ is well defined. We also define $\min (M) = \min (\{ h(\lambda) \mid \lambda \in \text{supp} (M) \})$.
\begin{lemma}
Let $(\underline {\Theta}, \underline{Q})$ be an EPSS indexed by a finite poset $(\Lambda, \leqslant)$. For each $M \in \mathcal{F} (\Theta)$, there is an exact sequence $0 \rightarrow K_1 \rightarrow Q^0 \rightarrow M$ such that $K_1 \in \mathcal{F} (\Theta)$ and $\min (K_1) > \min (M)$, where $Q^0 \in \text{add} (Q)$.
\end{lemma}
\begin{proof}
This is Proposition 2.10 in \cite{Marcos2} which deals with the special case that $\Lambda$ is a linearly ordered set. The general case can be proved similarly by observing the fact that Ext$ _A^1 (\Theta_{\lambda}, \Theta_{\mu}) = 0$ if $h (\lambda) = h (\mu)$.
\end{proof}
By this lemma, the relative projective dimension of every $M \in \mathcal{F} (\Theta)$ cannot exceed the length of the longest chain in $\Lambda$.
As before, we let $\mathcal{E}$ be the $k$-linear category associated to $\Gamma = \text{Ext} _A^{\ast} (\Theta, \Theta)$.
\begin{theorem}
Let $(\underline {\Theta}, \underline{Q})$ be an EPSS indexed by a finite poset $(\Lambda, \leqslant)$. such that Ext$_A ^i (Q, \Theta) =0$ for all $i \geqslant 1$. Then $\mathcal{E}$ is a directed category with respect to $\leqslant$ and is standardly stratified for $\leqslant ^{op}$. Moreover, it is standardly stratified for $\leqslant$ if and only if for all $s \geqslant 0$, Ext$ _A^s (\Theta_{\lambda}, \Theta_{\mu})$ is a projective End$ _A (\Theta_{\mu})$-module, $\lambda, \mu \in \Lambda$.
\end{theorem}
\begin{proof}
We only need to show that $\mathcal{E}$ is a directed category with respect to $\leqslant$ since the other statements can be proved as in Theorem 1.4. We know Hom$ _A (\Theta_{\lambda}, \Theta_{\mu}) = 0$ if $\lambda \nleqslant \mu$ and Ext$ _A^1 (\Theta_{\lambda}, \Theta_{\mu}) = 0$ for all $\lambda \nless \mu$. Therefore, it suffices to show that for all $s \geqslant 2$, Ext$ _A^s (\Theta_{\lambda}, \Theta_{\mu}) = 0$ if $\lambda \nless \mu$.
By Proposition 1.8 and Lemma 1.10, $\Theta_{\lambda}$ has a relative projective resolution
\begin{equation*}
\xymatrix {0 \ar[r] & Q^d \ar[r] ^{f_d} & \ldots \ar[r] ^{f_1} & Q^0 \ar[r] ^{f_0} & \Theta_{\lambda} \ar[r] & 0}
\end{equation*}
such that for each map $f_t$, min($K_t) > \text{min} (K_{t-1})$, where $K_t = \text{Ker} (f_t)$ and $1 \leqslant t \leqslant d$. By Proposition 1.9, Ext$ _A^s (\Theta_{\lambda}, \Theta_{\mu}) = 0$ if $s > d$; if $2 \leqslant s \leqslant d$, we have Ext$ _A^s (\Theta_{\lambda}, \Theta_{\mu}) \cong \text{Ext} _A^1 (K_{s-1}, \Theta_{\mu})$. But we have chosen
\begin{equation*}
\min (K_{s-1}) > \min (K_{s-2}) > \ldots > \min (\Theta_{\lambda}) = h(\lambda) \geqslant h(\mu).
\end{equation*}
Thus each factor $\Theta_{\nu}$ appearing in a $\Theta$-filtration of $K_{s-1}$ satisfies $h(\nu) > h(\mu)$, and hence $\nu \nleqslant \mu$. Since Ext$ _A^1 (\Theta_{\nu}, \Theta_{\mu}) =0$ for all $\nu \nleqslant \mu$, we deduce
\begin{equation*}
\text{Ext} _A^s (\Theta_{\lambda}, \Theta_{\mu}) \cong \text{Ext} _A^1 (K_{s-1}, \Theta_{\mu}) =0.
\end{equation*}
This finishes the proof.
\end{proof}
The following corollary is a generalization of Corollary 1.5.
\begin{corollary}
Let $(\underline {\Theta}, \underline{Q})$ be an EPSS indexed by a finite poset $(\Lambda, \leqslant)$. If for all $s \geqslant 1$ and $\lambda \in \Lambda$ we have Ext$ _A ^s (Q, \Theta) = 0$ and End$_A (\Theta_{\lambda}, \Theta_{\lambda}) \cong k$, then $\Gamma$ is quasi-hereditary with respect to both $\leqslant$ and $\leqslant ^{op}$.
\end{corollary}
\begin{proof}
This can be proved as Corollary 1.5.
\end{proof}
\section{Koszul Property of Extension Algebras}
There is a well known duality related to the extension algebras: the Koszul duality. Explicitly, if $A$ is a graded Koszul algebra with $A_0$ being a semisimple algebra, then $B = \text{Ext} _A^{\ast} (A_0, A_0)$ is a Koszul algebra, too. Moreover, the functor $\text{Ext} _A^{\ast} (-, A_0)$ gives an equivalence between the category of linear $A$-modules and the category of linear $B$-modules.\footnote{In \cite{Li1} we generalized these results to the situation that $A_0$ is a self-injective algebra.} However, even if $A$ is quasi-hereditary with respect to a partial order $\leqslant$, $B$ might not be quasi-hereditary with respect to $\leqslant$ or $\leqslant ^{op}$. This problem has been considered in \cite{Agoston1, Mazorchuk2}.
On the other hand, if $A$ is quasi-hereditary with respect to $\leqslant$, we have showed that the extension algebra $\Gamma = \text{Ext} _A^{\ast} (\Delta, \Delta)$ is quasi-hereditary with respect to both $\leqslant$ and $\leqslant ^{op}$. But $\Gamma$ is in general not a Koszul algebra in a sense which we define later. In this section we want to get a sufficient condition for $\Gamma$ to be a generalized Koszul algebra.
We work in the context of EPSS described in last section. Let $(\underline {\Theta}, \underline{Q})$ be an EPSS indexed by a finite poset $(\Lambda, \leqslant)$; $Q = \bigoplus _{\lambda \in \Lambda} Q_{\lambda}$ and $\Theta = \bigoplus _{\lambda \in \Lambda} \Theta_{\lambda}$. \textbf{We insist the following conditions: Ext$ \mathbf{_A^s (Q, \Theta) =0}$ for all $\mathbf{s \geqslant 1}$; each $\Theta_{\lambda}$ has a simple top $S_{\lambda}$; and $S_{\lambda} \ncong S_{\mu}$ for $\lambda \neq \mu$.} These conditions are always true for the classical stratifying system of a standardly stratified basic algebra. In particular, in that case $Q = _AA$.
\begin{proposition}
Let $0 \neq M \in \mathcal{F} (\Theta)$ and $i = \text{min (M)}$. Then there is an exact sequence
\begin{equation}
\xymatrix{ 0 \ar[r] & M[1] \ar[r] & M \ar[r] & \bigoplus _{h (\lambda) = i } \Theta_{\lambda} ^{\oplus m_{\lambda}} \ar[r] & 0}
\end{equation}
such that $M[1] \in \mathcal{F} (\Theta)$ and $\text{min} (M[1]) > \text{min} (M)$.
\end{proposition}
\begin{proof}
This is Proposition 2.9 in \cite{Marcos2} which deals with the special case that $\Lambda$ is a linearly ordered set. The general case can be proved similarly by observing the fact that Ext$ _A^1 (\Theta_{\lambda}, \Theta_{\mu}) = 0$ if $h (\lambda) = h (\mu)$.
\end{proof}
It is clear that $m_{\lambda} = [M: \Theta_{\lambda}]$. Based on this proposition, we make the following definition:
\begin{definition}
Let $M \in \mathcal{F} (\Theta)$ with $\text{min} (M) = i$. We say $M$ is generated in height $i$ if in sequence (2.1) we have
\begin{equation*}
\top M = M / \rad M \cong \top \Big{(}\bigoplus _{h (\lambda) = i } \Theta_{\lambda} ^{\oplus m_{\lambda}} \Big{)} = \bigoplus _{h (\lambda) = i } S_{\lambda} ^{\oplus m_{\lambda}}.
\end{equation*}
\end{definition}
We introduce some notation: if $M \in \mathcal{F} (\Theta)$ is generated in height $i$, then define $M_i = \bigoplus _{h (\lambda) = i } \Theta_{\lambda} ^{\oplus m_{\lambda}}$ in sequence (2.1). If $M[1]$ is generated in some height $j$, we can define $M[2] = M[1][1]$ and $M[1]_j$ in a similar way. This procedure can be repeated.
\begin{proposition}
Let $0 \rightarrow L \rightarrow M \rightarrow N \rightarrow 0$ be an exact sequence in $\mathcal{F} (\Theta)$. If $M$ is generated in height $i$, so is $N$. Conversely, if both $L$ and $N$ are generated in height $i$, then $M$ is generated in height $i$ as well.
\end{proposition}
\begin{proof}
We always have $\top N \subseteq \top M$ and $\top M \subseteq \top L \oplus \top N$. The conclusion follows from these inclusions and the rightmost identity in the above definition.
\end{proof}
Notice that $[Q_{\lambda} : \Theta_{\lambda}] = 1$ and $[Q_{\lambda} : \Theta_{\mu}] = 0$ for all $\mu \ngeqslant \lambda$. We claim that $Q_{\lambda}$ is generated in height $h(\lambda)$ for $\lambda \in \Lambda$. Indeed, the algebra $B = \text{End} _A (Q)^{op}$ is a standardly stratified algebra, with projective modules Hom$ _A (Q, Q_{\lambda})$ and standard modules Hom$ _A (Q, \Theta_{\lambda})$, $\lambda \in \Lambda$. Moreover, the functor Hom$ _A (Q, -)$ gives an equivalence between $\mathcal{F} (\Theta) \subseteq A$-mod and $\mathcal{F} (_B \Delta) \subseteq B$-mod. Using this equivalence and the standard filtration structure of projective $B$-modules we deduce the conclusion.
\begin{lemma}
If $M \in \mathcal{F} (\Theta)$ is generated in height $i$ with $[M: \Theta_{\lambda} ] = m_{\lambda}$, then $M$ has a relative projective cover $Q^i \cong \bigoplus _{h (\lambda) = i} Q_{\lambda} ^{\oplus m_{\lambda}}$.
\end{lemma}
\begin{proof}
There is a surjection $f: M \rightarrow \bigoplus _{h(\lambda) = i} \Theta_{\lambda} ^{\oplus m_{\lambda}}$ by Proposition 2.1. Consider the following diagram:
\begin{equation*}
\xymatrix{ & \bigoplus _{h (\lambda) = i} Q_{\lambda} ^{\oplus m_{\lambda}} \ar[d]^p \ar@{-->}[dl]_q \\
M \ar[r]^f & \bigoplus _{h (\lambda) = i} \Theta_{\lambda} ^{\oplus m_{\lambda}} \ar[r] & 0.}
\end{equation*}
Since $Q^i$ is projective in $\mathcal{F} (\Theta)$, the projection $p$ factors through the surjection $f$. In particular, $\top \Big{(} \bigoplus _{h (\lambda) = i} \Theta_{\lambda} ^{\oplus m_{\lambda}} \Big{)} = \bigoplus _{h (\lambda) = i} S_{\lambda} ^{\oplus m_{\lambda}}$ is in the image of $fq$. Since $M$ is generated in height $i$, $f$ induces an isomorphism between $\top M$ and $\top \Big{(} \bigoplus _{h (\lambda) = i} \Theta_{\lambda} ^{\oplus m_{\lambda}} \Big{)}$. Thus $\top M$ is in the image of $q$, and hence $q$ is surjective. It is clear that $q$ is minimal, so $Q^1 = \bigoplus _{h (\lambda) = i} Q_{\lambda} ^{\oplus m_{\lambda}}$ is a relative projective cover of $M$. The uniqueness follows from Proposition 8.3 in \cite{Webb}.
\end{proof}
We use $\Omega _{\Theta} ^i (M)$ to denote the \textit{i-th relative syzygy} of $M$. Actually, for every $M \in \mathcal{F} (\Theta)$ there is always a relative projective cover by Proposition 8.3 in \cite{Webb}.
The following definition is an analogue of \textit{linear modules} in the representation theory of graded algebras.
\begin{definition}
An $A$-module $M \in \mathcal{F} (\Theta)$ is said to be linearly filtered if there is some $i \in \mathbb{N}$ such that $\Omega _{\Theta} ^s (M)$ is generated in height $i + s$ for $s \geqslant 0$.
\end{definition}
Equivalently, $M \in \mathcal{F} (\Theta)$ is linearly filtered if and only if it is generated in height $i$ and has a relative projective resolution
\begin{equation*}
\xymatrix{ 0 \ar[r] & Q^l \ar[r] & Q^{l-1} \ldots \ar[r] & Q^{i+1} \ar[r] & Q^i \ar[r] & M \ar[r] & 0}
\end{equation*}
such that each $Q^s$ is generated in height $s$, $i \leqslant s \leqslant l$.
We remind the reader that there is a common upper bound for the relative projective dimensions of modules contained in $\mathcal{F} (\Theta)$, which is the length of the longest chains in the finite poset $(\Lambda, \leqslant)$. It is also clear that if $M$ is linearly filtered, so are all relative syzygies and direct summands. In other words, the subcategory $\mathcal{LF} (\Theta)$ constituted of linearly filtered modules contains all relative projective modules, and is closed under summands and relative syzygies. But in general it is not closed under extensions, kernels of epimorphisms and cokernels of monomorphisms.
\begin{proposition}
Let $0 \rightarrow L \rightarrow M \rightarrow N \rightarrow 0$ be an exact sequence in $\mathcal{F} (\Theta)$ such that all terms are generated in height $i$. If $L$ is linearly filtered, then $M$ is linearly filtered if and only if $N$ is linearly filtered.
\end{proposition}
\begin{proof}
Let $m_{\lambda} = [M : \Theta_{\lambda}]$, $l_{\lambda} = [L : \Theta_{\lambda}]$ and $n_{\lambda} = [N : \Theta_{\lambda}]$. By the previous lemma, we get the following commutative diagram:
\begin{equation*}
\xymatrix{ & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] & \\
0 \ar[r] & \Omega _{\Theta}(L) \ar[r] \ar[d] & \Omega _{\Theta}(M) \ar[r] \ar[d] & \Omega _{\Theta}(N) \ar[r] \ar[d] & 0 \\
0 \ar[r] & \bigoplus _{h(\lambda) = i} Q_{\lambda}^{\oplus l_{\lambda}} \ar[r] \ar[d] & \bigoplus _{h(\lambda) = i} Q_{\lambda} ^{\oplus m_{\lambda}} \ar[r] \ar[d] & \bigoplus _{h(\lambda) = i} Q_{\lambda} ^{\oplus n_{\lambda}} \ar[r] \ar[d] & 0 \\
0 \ar[r] & L \ar[r] \ar[d] & M \ar[r] \ar[d] & N \ar[r] \ar[d] & 0 \\
& 0 & 0 & 0 &}
\end{equation*}
Since $\Omega _{\Theta} (L)$ is generated in height $i+1$, by Proposition 2.3, $\Omega _{\Theta} (M)$ is generated in height $i+1$ if and only if $\Omega _{\Theta} (N)$ is generated in height $i+1$. Replacing $L$, $M$ and $N$ by $\Omega _{\Theta} (L)$, $\Omega _{\Theta}(M)$ and $\Omega _{\Theta} (N)$ respectively, we conclude that $\Omega _{\Theta}^2 (M)$ is generated in height $i+2$ if and only if $\Omega _{\Theta}^2 (N)$ is generated in height $i+2$. The conclusion follows from induction.
\end{proof}
\begin{corollary}
Suppose that $M \in \mathcal{F} (\Theta)$ is generated in height $i$ and linearly filtered. If $\bigoplus _{h(\lambda) = i} \Theta_{\lambda}$ is linearly filtered, then $M[1]$ is generated in height $i+1$ and linearly filtered.
\end{corollary}
\begin{proof}
Clearly $\bigoplus _{h(\lambda) = i} \Theta_{\lambda}$ is generated in height $i$. Let $m_{\lambda} = [M : \Theta_{\lambda}]$. Notice that both $M$ and $\bigoplus _{h(\lambda) = i} \Theta_{\lambda} ^{\oplus m_{\lambda}}$ have projective cover $\bigoplus _{h(\lambda) = i} Q_{\lambda} ^{\oplus m_{\lambda}}$. Thus the exact sequence
\begin{equation*}
\xymatrix{ 0 \ar[r] & M[1] \ar[r] & M \ar[r] & \bigoplus _{h(\lambda) = i} \Theta_{\lambda} ^{\oplus m_{\lambda}} \ar[r] & 0}
\end{equation*}
induces the following diagram:
\begin{equation*}
\xymatrix{ & \Omega _{\Theta}(M) \ar@{^{(}->} [r] \ar[d] & \Omega _{\Theta} \Big{(} \bigoplus _{h(\lambda) = i} \Theta_{\lambda} ^{\oplus m_{\lambda}} \Big{)} \ar@{->>}[r] \ar[d] & M[1] \\
& \bigoplus _{h(\lambda) = i} Q_{\lambda} ^{\oplus m_{\lambda}} \ar@{=}[r] \ar[d] & \bigoplus _{h(\lambda) = i} Q_{\lambda} ^{\oplus m_{\lambda}} \ar[d] \\
M[1] \ar@{^{(}->}[r] & M \ar@{->>}[r] & \bigoplus _{h(\lambda) = i} \Theta_{\lambda} ^{\oplus m_{\lambda}}.}
\end{equation*}
Consider the top sequence. Since both $\Omega _{\Theta} (\bigoplus _{h(\lambda) = i} \Theta_{\lambda} ^{\oplus m_{\lambda}})$ and $\Omega _{\Theta} (M)$ are generated in height $i+1$ and linearly filtered, $M[1]$ is also generated in height $i+1$ and linearly filtered by Propositions 2.3 and 2.6.
\end{proof}
These results tell us that linearly filtered modules have properties similar to those of linear modules of graded algebras.
\begin{lemma}
Let $M \in \mathcal{F} (\Theta)$ be generated in height $i$ and $m_{\lambda} = [M : \Theta_{\lambda}]$. If Hom$ _A (Q, \Theta) \cong \text{Hom} _A (\Theta, \Theta)$, then
\begin{equation*}
\text{Hom} _A (M, \Theta) \cong \text{Hom} _A (\bigoplus _{h(\lambda) = i} Q_{\lambda} ^{\oplus m_{\lambda}}, \Theta) \cong \text{Hom} _A (\bigoplus _{h(\lambda) = i} \Theta_{\lambda} ^{\oplus m_{\lambda}}, \Theta).
\end{equation*}
\end{lemma}
\begin{proof}
We claim that Hom$_A (Q, \Theta) \cong \text{Hom} _A (\Theta, \Theta)$ as End$ _A (\Theta)$-modules implies Hom$_A (Q_{\lambda}, \Theta) \cong \text{Hom} _A (\Theta_{\lambda}, \Theta)$ for every $\lambda \in \Lambda$. Then the second isomorphism follows immediately. First, notice that Hom$ _A (\Theta, \Theta)$ is a basic algebra with $n$ non-zero indecomposable summands Hom$ _A (\Theta_{\lambda}, \Theta)$, $\lambda \in \Lambda$, where $n$ is the cardinal number of $\Lambda$. But Hom$ _A (Q, \Theta) \cong \bigoplus _{\lambda \in \Lambda} \text{Hom} _A (Q_{\lambda}, \Theta)$ has at least $n$ non-zero indecomposable summands. If Hom$_A (Q, \Theta) \cong \text{Hom} _A (\Theta, \Theta)$, by Krull-Schmidt theorem, Hom$ _A (Q_{\lambda}, \Theta)$ must be indecomposable, and is isomorphic to some Hom$ _A (\Theta_{\mu}, \Theta)$.
If $\lambda \in \Lambda$ is maximal, $\text{Hom} _A (\Theta_{\lambda}, \Theta) \cong \text{Hom} _A (Q_{\lambda}, \Theta)$ since $Q_{\lambda} \cong \Theta_{\lambda}$. Define $\Lambda_1$ to be the subset of maximal elements in $\Lambda$ and consider $\lambda_1 \in \Lambda \setminus \Lambda_1$ which is maximal. We have
\begin{equation*}
\text{Hom} _A (Q_{\lambda_1}, \Theta) \ncong \text{Hom} _A (\Theta_{\lambda}, \Theta) \cong \text{Hom} _A (Q_{\lambda}, \Theta)
\end{equation*}
for every $\lambda \in \Lambda_1$ since End$ _A (\Theta)$ is a basic algebra. Therefore, $\text{Hom} _A (Q_{\lambda_1}, \Theta)$ must be isomorphic to some $\text{Hom} _A (\Theta_{\mu}, \Theta)$ with $\mu \in \Lambda \setminus \Lambda_1$. But $\text{Hom} _A (\Theta_{\mu}, \Theta)$ contains a surjection from $\Theta_{\mu}$ to the direct summand $\Theta_{\mu}$ of $\Theta$, and Hom$ _A (Q _{\lambda_1}, \Theta)$ contains a surjection from $Q_{\lambda_1}$ to $\Theta_{\mu}$ if and only if $\lambda_1 = \mu$. Thus we get $\lambda_1 = \mu$. Repeating the above process, we have Hom$_A (Q_{\lambda}, \Theta) \cong \text{Hom} _A (\Theta_{\lambda}, \Theta)$ for every $\lambda \in \Lambda$.
Applying Hom$ _A (-, \Theta)$ to the surjection $M \rightarrow \bigoplus _{h(\lambda) = i} \Theta_{\lambda} ^{\oplus m_{\lambda}}$ we get
\begin{equation*}
\text{Hom} _A (\bigoplus _{h(\lambda) = i} \Theta_{\lambda} ^{\oplus m_{\lambda}}, \Theta) \subseteq \text{Hom} _A (M, \Theta).
\end{equation*}
Similarly, from the relative projective covering map $\bigoplus _{h(\lambda) = i} Q_{\lambda} ^{\oplus m_{\lambda}} \rightarrow M$ we have
\begin{equation*}
\text{Hom} _A (M, \Theta) \subseteq \text{Hom} _A (\bigoplus _{h(\lambda) = i} Q_{\lambda} ^{\oplus m_{\lambda}}, \Theta).
\end{equation*}
Comparing these two inclusions and using the second isomorphism, we deduce the first isomorphism.
\end{proof}
The reader may aware that the above lemma is an analogue to the following result in representation theory of graded algebras: if $A$ is a graded algebra and $M$ is a graded module generated in degree 0, then Hom$ _A (M, A_0) \cong \text{Hom} _A (M_0, A_0)$.
\begin{lemma}
Suppose that Hom$ _A (Q, \Theta) \cong \text{Hom} _A (\Theta, \Theta)$. If $M \in \mathcal{F} (\Theta)$ is generated in height $i$, then Ext$ _A^s (M, \Theta) \cong \text{Ext} _A^{s-1} (\Omega _{\Theta}(M), \Theta)$ for all $s \geqslant 1$.
\end{lemma}
\begin{proof}
Let $m_{\lambda} = [M: \Theta_{\lambda}]$. Applying Hom$ _A (-, \Theta)$ to the exact sequence
\begin{equation*}
\xymatrix{ 0 \ar[r] & \Omega _{\Theta} (M) \ar[r] & \bigoplus _{h(\lambda) = i} Q_{\lambda} ^{\oplus m_{\lambda}} \ar[r] & M \ar[r] & 0}
\end{equation*}
we get a long exact sequence. In particular, for all $s \geqslant 2$, by observing the segment
\begin{align*}
& 0 = \text{Ext} _A^{s-1} (\bigoplus _{h(\lambda) = i} Q ^{\oplus m_{\lambda}}, \Theta) \rightarrow \text{Ext} _A^{s-1} (\Omega _{\Theta} (M), \Theta) \\
& \rightarrow \text{Ext} _A^s (M, \Theta) \rightarrow \text{Ext} _A^s (\bigoplus _{h(\lambda) = i} Q_{\lambda} ^{\oplus m_{\lambda}}, \Theta) = 0
\end{align*}
we conclude $\text{Ext} _A^{s-1} (\Omega _{\Theta}(M), \Theta) \cong \text{Ext} _A^s (M, \Theta)$.
For $s=1$, we have
\begin{align*}
& 0 \rightarrow \text{Hom} _A (M, \Theta) \rightarrow \text{Hom} _A (\bigoplus _{h(\lambda) = i} Q_{\lambda} ^{\oplus m_{\lambda}}, \Theta)\\
& \rightarrow \text{Hom} _A (\Omega _{\Theta} (M), \Theta) \rightarrow \text{Ext} _A^1 (M, \Theta) \rightarrow 0.
\end{align*}
By the previous lemma, the first inclusion is an ismorphism. Thus Ext$ _A^1 (M, \Theta) \cong \text{Hom} _A (\Omega _{\Theta} (M), \Theta)$.
\end{proof}
We remind the reader that although $\Gamma = \text{Ext} _A ^{\ast} (\Theta, \Theta)$ has a natural grading, the classical Koszul theory cannot be applied directly since $\Gamma_0 = \text{End} _A (\Theta)$ may not be a semisimple algebra. Thus we introduce \textit{generalized Koszul algebras} as follows:
\begin{definition}
Let $R = \bigoplus _{i \geqslant 0} R_i$ be a positively graded locally finite $k$-algebra, i.e., $\dim _k R_i < \infty$ for each $i \geqslant 0$. A graded $R$-module $M$ is said to be linear if it has a linear projective resolution
\begin{equation*}
\xymatrix{ \ldots \ar[r] & P^s \ar[r] & \ldots \ar[r] & P^1 \ar[r] & P^0 \ar[r] & M}
\end{equation*}
such that $P^s$ is generated in degree $s$. The algebra $R$ is said to be generalized Koszul if $R_0$ viewed as a $R$-module has a linear projective resolution.
\end{definition}
It is easy to see from this definition that $M$ is linear if and only if $\Omega^s (M)$ is generated in degree $s$ for all $s \geqslant 0$.
\begin{proposition}
Suppose that Hom$ _A (Q, \Theta) \cong \text{Hom} _A (\Theta, \Theta)$, and $\Theta_{\lambda}$ are linearly filtered for all $\lambda \in \Lambda$. If $M \in \mathcal{F} (\Theta)$ is linearly filtered, then
\begin{equation*}
\text{Ext} _A^{i+1} (M, \Theta) = \text{Ext} _A^1 (\Theta, \Theta) \cdot \text{Ext} _A^i (M, \Theta)
\end{equation*}
for all $i \geqslant 0$, i.e., $\text{Ext} _A^{\ast} (M, \Theta)$ as a graded $\Gamma = \text{Ext} _A^{\ast} (\Theta, \Theta)$-module is generated in degree 0.
\end{proposition}
\begin{proof}
Suppose that $M$ is generated in height $d$ and linearly filtered. By Lemma 2.9,
\begin{equation*}
\text{Ext} _A^{i+1} (M, \Theta) \cong \text{Ext} _A^i (\Omega _{\Theta} (M), \Theta).
\end{equation*}
But $\Omega _{\Theta}$ is generated in height $d+1$ and linearly filtered. Thus by induction
\begin{equation*}
\text{Ext} _A^{i+1} (M, \Theta) \cong \text{Ext} _A^1 (\Omega^i _{\Theta} (M), \Theta), \quad \text{Ext} _A^i (M, \Theta) \cong \text{Hom} _A (\Omega^i _{\Theta} (M), \Theta).
\end{equation*}
Therefore, it suffices to show
\begin{equation*}
\text{Ext} _A^1 (M, \Theta) = \text{Ext} _A^1 (\Theta, \Theta) \cdot \text{Hom} _A (M, \Theta)
\end{equation*}
since we can replace $M$ by $\Omega _{\Theta} ^i (M)$, which is linearly filtered as well.\
Let $m_{\lambda} = [M: \Theta_{\lambda}]$ and define $Q^0 = \bigoplus _{h(\lambda) = d} Q_{\lambda} ^{\oplus m_{\lambda}}$, $M_0 = \bigoplus _{h(\lambda) = d} \Theta_{\lambda} ^{\oplus m_{\lambda}}$. We have the following commutative diagram:
\begin{equation*}
\xymatrix {0 \ar[r] & \Omega _{\Theta} (M) \ar@{=}[d] \ar[r] & Q^0[1] \ar[r] \ar[d] & M[1] \ar[d] \ar[r] & 0\\
0 \ar[r] & \Omega _{\Theta} (M) \ar[r] & Q^0 \ar[r] \ar[d] & M \ar[d] \ar[r] & 0\\
& & M_0 \ar@{=}[r] & M_0}
\end{equation*}
where $Q^0[1] = \Omega _{\Theta} (M_0)$, see Proposition 2.1.
Observe that all terms in the top sequence are generated in height $d+1$ and linearly filtered. For every $\lambda \in \Lambda$ with $h(\lambda) = d+1$, we have
\begin{equation*}
[\Omega _{\Theta} (M): \Theta_{\lambda}] + [M[1]: \Theta_{\lambda}] = [Q^0[1]: \Theta_{\lambda}].
\end{equation*}
Let $r_{\lambda}, s_{\lambda}$ and $t_{\lambda}$ be the corresponding numbers in the last equality. Then we get a split short exact sequence
\begin{equation*}
\xymatrix{ 0 \ar[r] & \bigoplus _{h(\lambda) = d+1} \Theta_{\lambda} ^{\oplus r_{\lambda}} \ar[r] & \bigoplus _{h(\lambda) = d+1} \Theta_{\lambda} ^{\oplus t_{\lambda}} \ar[r] & \bigoplus _{h(\lambda) = d+1} \Theta_{\lambda} ^{\oplus s_{\lambda}} \ar[r] & 0}.
\end{equation*}
Applying Hom$ _A (-, \Theta)$ to this sequence and using Lemma 2.8, we obtain the exact sequence
\begin{equation*}
0 \rightarrow \text{Hom} _A (M[1], \Theta) \rightarrow \text{Hom} _A (Q^0[1], \Theta) \rightarrow \text{Hom} _A (\Omega _{\Theta} (M), \Theta) \rightarrow 0.
\end{equation*}
Therefore, each map $\Omega _{\Theta} (M) \rightarrow \Theta$ can extend to a map $Q^0[1] \rightarrow \Theta$.
To prove Ext$_A^1 (M, \Theta) = \text{Ext} _A^1 (\Theta, \Theta) \cdot \text{Hom} _A (M, \Theta)$, by Lemma 2.9 we first identify $\text{Ext} _A^1 (M, \Theta)$ with $\text{Hom} _A (\Omega _{\Theta} (M), \Theta)$. Take an element $x \in \text{Ext} _A^1 (M, \Theta)$ and let $g: \Omega _{\Theta} (M) \rightarrow \Theta$ be the corresponding homomorphism. As we just showed, $g$ can extend to $Q^0[1]$, and hence there is a homomorphism $\tilde{g}: Q^0[1] \rightarrow \Theta$ such that $g = \tilde{g} \iota$, where $\iota$ is the inclusion.
\begin{align*}
\xymatrix { \Omega _{\Theta} (M) \ar[r]^{\iota} \ar[d]^g & Q^0[1] \ar[dl]^{\tilde{g}}\\
\Theta.}
\end{align*}
We have the following commutative diagram:
\begin{align*}
\xymatrix{ 0 \ar[r] & \Omega _{\Theta} (M) \ar[d]^{\iota} \ar[r] & Q^0 \ar@{=}[d] \ar[r] & M \ar[d]^p \ar[r] & 0\\
0 \ar[r] & Q^0[1] \ar[r] & Q^0 \ar[r] & M_0 \ar[r] & 0,}
\end{align*}
where $p$ is the projection of $M$ onto $M_0$. The map $\tilde{g}: Q^0[1] \rightarrow \Theta$ gives a push-out of the bottom sequence:
\begin{align*}
\xymatrix{ 0 \ar[r] & \Omega _{\Theta} (M) \ar[d]^{\iota} \ar[r] & Q^0 \ar@{=}[d] \ar[r] & M \ar[d]^p \ar[r] & 0\\
0 \ar[r] & Q^0[1] \ar[r] \ar[d] ^{\tilde{g}} & Q^0 \ar[d] \ar[r] & M_0 \ar[r] \ar@{=}[d] & 0\\
0 \ar[r] & \Theta \ar[r] & E \ar[r] & M_0 \ar[r] & 0.}
\end{align*}
Since $M_0 \cong \bigoplus _{h(\lambda) = d} \Theta_{\lambda} ^{\oplus m_{\lambda}}$, the bottom sequence represents some
\begin{equation*}
y \in \text{Ext} _A^1 (\Theta ^{\oplus m}, \Theta) = \bigoplus _{i=1} ^m \text{Ext} _A^1 (\Theta, \Theta)
\end{equation*}
where $m = \sum _{h(\lambda) = d} m_{\lambda}$. Therefore, we can write $y = y_1 + \ldots + y_m$ where $y_i \in \text{Ext} _A^1 (\Theta, \Theta)$ is represented by the sequence
\begin{equation*}
\xymatrix{ 0 \ar[r] & \Theta \ar[r] & E_i \ar[r] & \Theta \ar[r] & 0}.
\end{equation*}
Composed with the inclusions $\epsilon_{\lambda}: \Theta_{\lambda} \rightarrow \Theta$, we get the map $(p_1, \ldots, p_m)$ where each component $p_i$ is defined in an obvious way. Consider the pull-backs:
\begin{equation*}
\xymatrix {0 \ar[r] & \Theta \ar[r] \ar@{=}[d] & F_i \ar[r] \ar[d] & M \ar[r] \ar[d]^{p_i} & 0 \\
0 \ar[r] & \Theta \ar[r] & E_i \ar[r] & \Theta \ar[r] & 0.}
\end{equation*}
Denote by $x_i$ the top sequence. Then
\begin{equation*}
x = \sum_{i=1}^m x_i = \sum_{i=1}^m y_i \cdot p_i \in \text{Ext} _A^1 (\Theta, \Theta) \cdot \text{Hom} _A (M, \Theta),
\end{equation*}
so $\text{Ext} _A^1 (M, \Theta) \subseteq \text{Ext} _A^1 (\Theta, \Theta) \cdot \text{Hom} _A (M, \Theta)$. The other inclusion is obvious.
\end{proof}
Now we can prove the main result.
\begin{theorem}
Let $(\underline {\Theta}, \underline{Q})$ be an EPSS indexed by a finite poset $(\Lambda, \leqslant)$ such that Ext$_A ^i (Q, \Theta) =0$ for all $i \geqslant 1$ and Hom$ _A (Q, \Theta) \cong \text{Hom} _A (\Theta, \Theta)$. Suppose that all $\Theta_{\lambda}$ are linearly filtered for $\lambda \in \Lambda$. If $M \in \mathcal{F} (\Theta)$ is linearly filtered, then the graded module $\text{Ext} _A^{\ast} (M, \Theta)$ has a linear projective resolution. In particular, $\Gamma = \text{Ext} _A^{\ast} (\Theta, \Theta)$ is a generalized Koszul algebra.
\end{theorem}
\begin{proof}
Suppose that $M$ is generated in height $d$. Define $m_{\lambda} = [M: \Theta_{\lambda}]$ for $\lambda \in \Lambda$, $Q^0 = \bigoplus _{h(\lambda) = d} Q_{\lambda} ^{\oplus m_{\lambda}}$, and $M_0 = \bigoplus _{h(\lambda) = d} \Theta_{\lambda} ^{\oplus m_{\lambda}}$. As in the proof of the previous lemma, we have the following short exact sequence of linearly filtered modules generated in height $d+1$:
\begin{align*}
\xymatrix{ 0 \ar[r] & \Omega _{\Theta} (M) \ar[r] & \Omega _{\Theta} (M_0) \ar[r] & M[1] \ar[r] & 0}
\end{align*}
where $\Omega _{\Theta} (M_0) = Q^0[1]$. This sequence induces exact sequences recursively (see the proof of Proposition 2.6):
\begin{align*}
\xymatrix{ 0 \ar[r] & \Omega^i _{\Theta} (M) \ar[r] & \Omega^i _{\Theta} (M_0) \ar[r] & \Omega^{i-1} _{\Theta} (M[1]) \ar[r] & 0,}
\end{align*}
where all modules are linearly filtered and generated in height $d+i$. Again as in the proof of the previous lemma, we get an exact sequence
\begin{align*}
0 \rightarrow \text{Hom} _A (\Omega _{\Theta} ^{i-1} (M[1]), \Theta) \rightarrow \text{Hom} _A (\Omega _{\Theta} ^i (M_0), \Theta) \rightarrow \text{Hom} _A (\Omega _{\Theta} ^i (M), \Theta) \rightarrow 0.
\end{align*}
According to Lemma 2.9, the above sequence is isomorphic to:
\begin{align*}
0 \rightarrow \text{Ext} ^{i-1} _A (M[1], \Theta) \rightarrow \text{Ext}^i _A (M_0, \Theta) \rightarrow \text{Ext}^i _A (M, \Theta) \rightarrow 0.
\end{align*}
Now let the index $i$ vary and put these sequences together. We have:
\begin{align*}
\xymatrix{ 0 \ar[r] & E(M[1]) \langle 1 \rangle \ar[r] & E(M_0) \ar[r]^p & E(M) \ar[r] & 0,}
\end{align*}
where $E = \text{Ext} _A^{\ast} (-, \Theta)$ and $\langle - \rangle$ is the degree shift functor of graded modules. That is, for a graded module $T = \bigoplus _{i \geqslant 0} T_i$, $T \langle 1 \rangle _i$ is defined to be $T_{i-1}$.
Since $M_0 \in \text{add} (\Theta)$, $E(M_0)$ is a projective $\Gamma$-module. It is generated in degree 0 by the previous lemma. Similarly, $E(M[1])$ is generated in degree 0, so $E(M[1]) \langle 1 \rangle$ is generated in degree 1. Therefore, the map $p$ is a graded projective covering map. Consequently, $\Omega(E(M)) \cong E(M[1]) \langle 1 \rangle$ is generated in degree 1.
Replacing $M$ by $M[1]$ (since it is also linearly filtered), we have
\begin{equation*}
\Omega^2 (E(M)) \cong \Omega (E(M[1]) \langle 1 \rangle) \cong \Omega( E(M[1]) \langle 1 \rangle \cong E(M[2]) \langle 2 \rangle,
\end{equation*}
which is generated in degree 2. By recursion, $\Omega^i (E(M)) \cong E(M [i]) \langle i \rangle$ is generated in degree $i$ for all $i \geqslant 0$. Thus $E(M)$ is a linear $\Gamma$-module.
In particular let $M = Q_{\lambda}$ for a certain $\lambda \in \Lambda$. We get
\begin{equation*}
E(Q_{\lambda}) = \text{Ext} _A^{\ast} (Q_{\lambda}, \Theta) = \text{Hom} _A (Q_{\lambda}, \Theta)
\end{equation*}
is a linear $\Gamma$-module. Therefore,
\begin{align*}
\bigoplus _{\lambda \in \Lambda} E (Q_{\lambda}, \Theta) & = \bigoplus _{\lambda \in \Lambda} \text{Hom} _A (Q_{\lambda}, \Theta) \cong \text{Hom} _A (\bigoplus _{\lambda \in \Lambda} Q_{\lambda}, \Theta)\\
& = \text{Hom} _A (Q, \Theta) \cong \text{Hom} _A (\Theta, \Theta) = \Gamma_0
\end{align*}
is a linear $\Gamma$-module. So $\Gamma$ is a generalized Koszul algebra.
\end{proof}
\begin{remark}
To get the above result we made some assumptions on the EPSS $(\underline {\Theta}, \underline{Q})$. Firstly, each $\Theta_{\lambda}$ has a simple top $S_{\lambda}$ and $S_{\lambda} \ncong S_{\mu}$ for $\lambda \neq \mu$; secondly, Ext$ _A^s (Q, \Theta) = 0$ for every $s \geqslant 1$. These two conditions always hold for standardly stratified basic algebras. We also suppose that Hom$ _A (\Theta, \Theta) \cong \text{Hom} _A (Q, \Theta)$. This may not be true even if $A$ is a quasi-hereditary algebra.
\end{remark}
Although $\Gamma$ is proved to be a generalized Koszul algebra, in general it does not have the Koszul duality. Consider the following example:
\begin{example}
Let $A$ be the path algebra of the following quiver with relation $\alpha \cdot \beta = 0$. Put an order $x < y < z$.
\begin{equation*}
\xymatrix{ x \ar@/^/ [rr]^{\alpha} & & y \ar@ /^/ [ll]^{\beta} \ar[r] ^{\gamma} & z.}
\end{equation*}
The projective modules and standard modules of $A$ are described as follows:
\begin{equation*}
P_x = \begin{matrix} & x & \\ & y & \\ x & & z \end{matrix} \qquad P_y = \begin{matrix} & y & \\ x & & z \end{matrix} \qquad P_z = z
\end{equation*}
\begin{equation*}
\Delta_x = x \qquad \Delta_y = \begin{matrix} y \\ x \end{matrix} \qquad \Delta_z = z \cong P_z.
\end{equation*}
This algebra is quasi-hereditary. Moreover, Hom$ _A (\Delta, \Delta) \cong \text{Hom} _A (A, \Delta)$, and all standard modules are linearly filtered. Therefore, $\Gamma = \text{Ext} _A^{\ast} (\Delta, \Delta)$ is a generalized Koszul algebra by the previous theorem.\
We explicitly compute the extension algebra $\Gamma$. It is the path algebra of the following quiver with relation $\gamma \cdot \alpha = 0$.
\begin{equation*}
\xymatrix{ x \ar@/^/ [rr]^{\alpha} \ar@ /_/ [rr]_{\beta} & & y \ar[r] ^{\gamma} & z.}
\end{equation*}
\begin{equation*}
_{\Gamma} P_x = \begin{matrix} & x_0 & \\ y_0 & & y_1 \\ z_1 & & \end{matrix} \qquad _{\Gamma} P_y = \begin{matrix} y_0 \\ z_1 \end{matrix} \qquad _{\Gamma} P_z = z_0
\end{equation*}
and
\begin{equation*}
_{\Gamma} \Delta_x = x_0 \qquad _{\Gamma} \Delta_y = y_0 \qquad _{\Gamma} \Delta_z = z_0 \qquad \Gamma_0 = \begin{matrix} x_0 \\ y_0 \end{matrix} \oplus y_0 \oplus z_0 \ncong _{\Gamma} \Delta.
\end{equation*}
Here we use indices to mark the degrees of simple composition factors. As asserted by the theorem, $\Gamma_0$ has a linear projective resolution. But $_{\Gamma} \Delta$ is not a linear $\Gamma$-module (we remind the reader that the two simple modules $y$ appearing in $_{\Gamma} P_x$ lie in different degrees!).
By computation, we get the extension algebra $\Gamma' = \text{Ext} _{\Gamma} ^{\ast} (\Gamma_0, \Gamma_0)$, which is the path algebra of the following quiver with relation $\beta \cdot \alpha =0$.
\begin{equation*}
\xymatrix {x \ar[r] ^{\alpha} & y \ar[r] ^{\beta} & z.}
\end{equation*}
Since $\Gamma'$ is a Koszul algebra in the classical sense, the Koszul duality holds in $\Gamma'$. It is obvious that the Koszul dual algebra of $\Gamma'$ is not isomorphic to $\Gamma$. Therefore, as we claimed, the Koszul duality does not hold in $\Gamma$.
\end{example}
Let us return to the question of whether $\Gamma = \text{Ext} _A ^{\ast} (\Theta, \Theta)$ is standardly stratified with respect to $\leqslant$. According to Proposition 1.3, this happens if and only if for each pair $\Theta_{\lambda}, \Theta_{\mu}$ and $s \geqslant 0$, Ext$ _A^s (\Theta_{\lambda}, \Theta_{\mu})$ is a projective End$ _A (\Theta_{\mu})$-module. Putting direct summands together, we conclude that $\Gamma$ is standardly stratified with respect to $\leqslant$ if and only if Ext$ _A^s (\Theta, \Theta)$ is a projective $\bigoplus _{\lambda \in \Lambda} \text{End} _A (\Theta_{\lambda})$-module. With the conditions in Theorem 2.12, Ext$ _A^s (\Theta_{\lambda}, \Theta) \cong \text{Hom} _A (\Omega ^s _{\Theta _{\lambda}} (\Theta), \Theta)$ for all $s \geqslant 0$ and $\lambda \in \Lambda$ by Lemma 2.9. Notice that $\Omega ^s _{\Theta} (\Theta _{\lambda})$ is linearly filtered. Suppose that min$(\Omega ^s _{\Theta} (\Theta _{\lambda}) ) = d$ and $m_{\mu} = [\Omega ^s _{\Theta} (\Theta _{\lambda}): \Theta_{\mu}]$. Then
\begin{equation}
\text{Ext} _A^s (\Theta _{\lambda}, \Theta) \cong \text{Hom} _A (\Omega ^s _{\Theta} (\Theta _{\lambda}), \Theta) \cong \text{Hom} _A ( \bigoplus _{h(\mu) = d} \Theta_{\mu} ^{\oplus m_{\mu}}, \Theta),
\end{equation}
which is a projective $\Gamma_0 = \text{End}_A (\Theta)$-module.
With this observation, we have:
\begin{corollary}
Let $(\underline {\Theta}, \underline {Q})$ be an EPSS indexed by a finite poset $(\Lambda, \leqslant)$. Suppose that all $\Theta_{\lambda}$ are linearly filtered for $\lambda \in \Lambda$, and Hom$ _A (Q, \Theta) \cong \text{Hom} _A (\Theta, \Theta)$. Then $\Gamma = \text{Ext} _A^{\ast} (\Theta, \Theta)$ is standardly stratified for $\leqslant$ if and only if End$ _A (\Theta)$ is a projective $\bigoplus _{\lambda \in \Lambda} \text{End} _A (\Theta_{\lambda})$-module.
\end{corollary}
\begin{proof}
If $\Gamma$ is standardly stratified for $\leqslant$, then in particular $\Gamma_0 = \text{End} _A (\Theta)$ is a projective $\bigoplus _{\lambda \in \Lambda} \text{End} _A (\Theta_{\lambda})$-module by Proposition 1.3. Conversely, if $\Gamma_0 = \text{End} _A (\Theta)$ is a projective $\bigoplus _{\lambda \in \Lambda} \text{End} _A (\Theta_{\lambda})$-module, then by the isomorphism in (2.2) Ext$ _A^s (\Theta, \Theta) = \bigoplus _{\lambda \in \Lambda} \text{Ext} _A^s (\Theta_{\lambda}, \Theta)$ is a projective $\Gamma_0$-module for all $s \geqslant 0$, so it is a projective $\bigoplus _{\lambda \in \Lambda} \text{End} _A (\Theta_{\lambda})$-module as well. Again by Proposition 1.3, $\Gamma$ is standardly stratified with respect to $\leqslant$.
\end{proof}
If $A$ is quasi-hereditary with respect to $\leqslant$ such that all standard module are linearly filtered, then $\Gamma = \text{Ext} _A^{\ast} (\Delta, \Delta)$ is again quasi-hereditary for this partial order by Corollary 1.5, and $\Gamma_0$ has a linear projective resolution by the previous theorem. Let $_{\Gamma} \Delta$ be the direct sum of all standard modules of $\Gamma$ with respect to $\leqslant$. The reader may wonder whether $_{\Gamma} \Delta$ has a linear projective resolution as well. The following proposition gives a partial answer to this question.
\begin{proposition}
With the above notation, if $_{\Gamma} \Delta$ has a linear projective resolution, then $\Gamma_0 \cong _{\Gamma}\Delta$, or equivalently Hom$ _A (\Delta_{\lambda}, \Delta_{\mu}) \neq 0$ only if $\lambda = \mu$, $\lambda, \mu \in \Lambda$. If furthermore Hom$ _A (A, \Delta) \cong \text{End} _A (\Delta)$, then $\Delta \cong A / \rad A$.
\end{proposition}
\begin{proof}
We have proved that the $k$-linear category associated to $\Gamma$ is directed with respect to $\leqslant$. By Proposition 1.3, standard modules of $\Gamma$ for $\leqslant$ are exactly indecomposable summands of $\bigoplus _ {\lambda \in \Lambda} \text{End} _A (\Delta_{\lambda})$, i.e., $_{\Gamma} \Delta \cong \bigoplus _ {\lambda \in \Lambda} \text{End} _A (\Delta_{\lambda}) \cong \bigoplus _{\lambda \in \Lambda} k_{\lambda}$. Clearly, $_{\Gamma} \Delta \subseteq \Gamma_0 = \text{End} _A (\Delta)$. If $_{\Gamma} \Delta$ has a linear projective resolution, then by Corollary 2.4 and Remark 2.7 in \cite{Li1}, $_{\Gamma} \Delta$ is a projective $\Gamma_0$-module. Consequently, every summand $k_{\lambda}$ is a projective $\Gamma_0$-module. Since both $_{\Gamma} \Delta$ and $\Gamma_0$ have exactly $| \Lambda |$ pairwise non-isomorphic indecomposable summands, we deduce $_{\Gamma} \Delta \cong \Gamma_0 \cong \bigoplus _{\lambda \in \Lambda} k_{\lambda}$, or equivalently $\text{Hom} _A (\Delta_{\lambda}, \Delta_{\mu}) = 0$ if $\lambda \neq \mu$.
If furthermore Hom$ _A (A, \Delta) \cong \text{End} _A (\Delta)$, then
\begin{equation*}
\Delta \cong \text{Hom} _A (A, \Delta) \cong \text{End} _A (\Delta) \cong \bigoplus _{\lambda \in \Lambda} k_{\lambda} \cong A / \rad A.
\end{equation*}
\end{proof}
| 61,095
|
? tnx Stephan Pietzko
| 191,906
|
J. Mendel Sable/Beaver Fur Reversible Coat with Hood (Size M)
Designer: J. MENDEL
$ 3,950.00 $ 4,400.00
J.MENDEL
Authentic J. MENDEL sable and beaver fur reversible coat. Features two different styles built into one garment. Cuffs and collar are made of sable fur, while the coat itself is made with beaver fur. Buttons with 'Mendel Paris' printed in white on top and are sewn into the sable fur, with elastic loops to hold the coat closed. Two usable pockets are sewn into each side of the lining and can be used as hidden pockets when the fur is facing outwards, or to keep your hands warm when the fur is facing inwards. Another special hidden feature to this coat is that the collar can also be used as a hood. Reversible, warm, and eye-catching, snatch this coat up and amaze everyone around you the next time winter rolls around. We do not accept returns.
- Size: M
- Light brown sable fur and beaver fur
- 3 buttons
- Pockets at side
- Brown lining
- Reversible
- Collar functions as a hood
| 136,766
|
TITLE: Bra-ket notation average measured value
QUESTION [0 upvotes]: Let's consider an operator $A$ with eigenket $|a^\prime\rangle$. Then the average measured value according to 1.4.6 is
$$\langle A\rangle=\sum_{a^\prime} \sum_{a^{\prime\prime}} \langle \alpha|a^{\prime\prime}\rangle\langle a^{\prime\prime}|A|a^\prime\rangle\langle a^\prime |\alpha\rangle =\sum_{a^\prime} a^\prime |\langle a^\prime|\alpha\rangle|^2$$ where $|a^{\prime\prime}\rangle$ is another eigenket. What I don’t understand is how this transformation works. If I let $A$ act on $|a^\prime\rangle$ I get the $a^\prime$ in front of the $|\langle a^\prime|\alpha\rangle|^2$ part.
$\textbf{Question:}$ Why does this mean that the sum over $a’’$ vanishes? Maybe my problem is also that I don't fully understand the difference between expectation value and this average measured value.
(Source: Sakurai, Modern Quantum Mechanics Revised Edition, formula 1.4.7)
REPLY [1 votes]: I don't fully understand the difference between expectation value and
this average measured value.
This (the first expression) is just the expectation of $A$ on the state $|\alpha\rangle$ written in an expansion on the eigenbasis of $A$.
The fact that the eigenbasis of an observable $A$ is orthonormal and complete is expressed as
$$\langle a | a' \rangle = \delta_{a\,a'}$$
$$1 = \sum_a |a\rangle \langle a | $$
Now, starting with the expectation
$$\langle \alpha | A | \alpha \rangle$$
insert the completeness identity once
$$\langle \alpha | 1 \cdot A | \alpha \rangle = \langle \alpha |\sum_a |a\rangle \langle a | A | \alpha \rangle = \sum_a \langle \alpha |a\rangle \langle a | A | \alpha \rangle$$
and then insert the completeness identity again
$$\sum_a \langle \alpha |a\rangle \langle a | A \cdot 1 | \alpha \rangle = \sum_a \langle \alpha |a\rangle \langle a | A \sum_{a'} |a'\rangle \langle a' | \alpha \rangle = \sum_a \sum_{a'} \langle \alpha |a\rangle \langle a | A |a'\rangle \langle a' | \alpha \rangle$$
But, using the orthonormal property of the eigenbasis, we have
$$\langle a | A |a'\rangle = a' \langle a | a'\rangle = a' \delta_{a\,a'} $$
Since $\delta_{a\,a'}$ is zero unless $a' = a$,
the sum over $a'$ has only one non-zero term
which is the term when $a' = a$.
$$\sum_a \sum_{a'} \langle \alpha |a\rangle a' \delta_{a\,a'} \langle a' | \alpha \rangle = \sum_a \langle \alpha |a\rangle a \langle a | \alpha \rangle = \sum_a a |\langle a | \alpha \rangle|^2$$
| 213,663
|
First, let me define the last word in the title. A toddy is what Husband and I call Vincent and other kids his age-he’s not quite a full fledged toddler, but he’s also definitely not 100% a baby anymore either, but he has qualities of both toddlers and babies still. So ‘toddy’ won out over ‘boddler.’
When I first was formulating thoughts about what to put in this post, it was a particularly difficult day of vacation, and my initial thought was to write ‘Stay home and wait until he’s 4,’ but if you’re anything like me, you love traveling and hope to instill a love of it in your kids as well. Here’s my list of 10 tips for traveling with toddies:
1. Choose a child friendly destination
Sure, this may seem fairly obvious, but don’t underestimate its importance. From the restaurants you eat at to the activities you plan during the day EVERYTHING must be child centered and child friendly. You can’t just think about yourselves either. You know and love you kids no matter how much of a tantrum they may be throwing, but other people don’t. Be considerate and pick a place where there will be lots of families with similar aged kids. Then you can give other moms the look of ‘I got your back, and I am not judging your parenting or your kids behavior because in about 10 minutes, mine will be doing the same thing.’
2. Schedule downtime
Most toddies are still taking substantial naps during the day, and while many can sleep in strollers or baby carriers (I envy you parents because my child fights sleep anywhere besides his crib) they don’t get the best sleep unless they’re in a crib or pack and play. Plus, vacations are new places with lots of stimulation. It’s important for your kids to have some relaxing/’normal’ play time to recharge their batteries. And trust me parents, you will need this downtime probably more than your kids do!
3. Bring a blanket/stuffed animal/lovey to remind them of home
This might be controversial because we’ve all heard the stories of when loveys get lost, but I think it’s important. It smells like home, there’s a connection to safety and security when with the object, and many times there’s an association with sleeping and that object. I think it’s worth taking with you but make that object the most important thing you pack!
4. Accept that schedules will not be followed
This is a hard one for me to follow. Vincent has a nap time (or rather nap window) and a bedtime every day. We try not to deviate too much from those routines because he needs his sleep and he’s not typically one to sleep in if he’s put to bed later or if he doesn’t nap. He’s just extra crabby and sleeps less. But on vacations it’s hard to stick to your schedule – normal life doesn’t revolved around nap time. So you need to accept that things will be different and it might be harder (this is why you schedule downtime!) but it’ll be worth it because your toddy should get a lot of joy out of your trip.
5. You will get less sleep
Vacations are for relaxing, right? Wrong! Not with toddies. Vacations are for schedule changes, messing up routines, new places, and unfamiliar territory. I can only speak for my own toddy, but it takes him a good 3-4 days to adjust to sleeping in an unfamiliar place. Now, normally if he wakes up at night at home, we let him self soothe and he goes back to bed. It’s much harder to do that when you’re all living in the same hotel room. Expect to be woken up multiple times a night, and I’d suggest going to bed earlier than normal because your toddy will be up for the day at 6:30 (or earlier) and this is even after waking up 5 other times during the night.
6. Get a corner or end of the hall hotel room
Since we know from the previous post that your toddy is going to be awake multiple times a night if you can, request a room in the corner of the hotel or at the end of the hallway. This way, there’s only one room next to you and only one group of people you may potentially be bothering with all of the crying and screaming coming from your room. I know it’s not always possible, but it alleviates some potential headaches if you’re around fewer people. Also, going back to numbers 4&5, since there are lots of other people to think about in hotels, the routine of self soothing doesn’t work so well when you don’t want other people to be bothered.
7. There is no good way to actually travel with a toddy
By this I mean any mode of transportation is going to be hard. I’ve been told that being in cars rocks babies and toddlers to sleep. Perhaps my child is the only exception to this rule, but he essentially refuses to sleep while in the car. And he hates his car seat. He even complains when taking less than 15 minute drives. Airplanes (which happened once in an unfortunate circumstance where we had to leave vacation early) are even worse. At least in his car seat he’s in control of his arm/leg movements. When he’s sitting on a lap in an airplane and you have to stop him from trying to run up and down the aisles…forget it! These probably aren’t the greatest solutions, but we have constant snacks/water/milk, toys, and we’ve even tried a portable DVD player so he can have something to watch (rear facing must be awful, but it’s safe.) Even then, these methods don’t always work, and sometimes you resort to listening to crying…and screaming…and more crying…and lots of toys making lots of different sounds…and more screaming. Learn to tune it out. If your toddy isn’t hungry, wet, or in pain, they are just mad and will (eventually) get over it.
8. Eat at least one meal a day in your hotel room
Not only does this stop you from dealing with food throwing/screaming tantrums in restaurants, but it also can save you some money. Pack a cooler. Even pb&j is good when you know it’s not going to be thrown on the floor in a fit of toddy anger.
9. Have a plan B. And C. And D. And so on…
Things don’t always go to plan when it comes to kids, so be flexible. Know that you might not do everything you wanted to, and that’s ok. You’re on vacation. Have a few ‘must dos’ and play everything else by ear.
10. Your vacation is no longer yours
Once you have kids your life is no longer yours, but it’s for the best reason possible. Vacations are the same way. You won’t be able to have the type of vacations you had in the past when it was just you and your friends or you and your spouse. But that’s ok. It’s worth it taking kids on vacation, even as young as toddies. Sure, they might not remember the trip, but you will. You’ll remember the smiles and laughter and probably the screaming and tears, but it’s all part of the experience. Teach them young to appreciate home as well as to appreciate how to explore. Trust me, I’m an adult who had this kind of childhood, and I know from first hand experience, it’s so worth it.
| 70,996
|
KSA 15 - Tug
IMO: 8735467
Vessel KSA 15 (IMO: 8735467 ) is a Tug built in 2007 and currently sailing under the flag of Unknown. Below you can find more technical information, photos, AIS data and last 5 port calls of KSA 15 detected by AIS.
Are you interested in the sailing schedule of the KSA 15 ship?
The KSA 15's port of calls and sailing schedule for the past months are listed below as detected by our live AIS ship tracking system.
Port Calls
Map position
Master Data
Disclaimer
KSASA 15 data.
| 289,710
|
What do you think of when you think of a mother?
Nurturing? Selfless? Fun? Loving? Caring? Your best friend? The person who gave birth to you?
Well I agree with all of the above apart from the last one.
While giving birth made me a mum, it didn’t make my mum one.
You see I was chosen. I was adopted. So my mum didn’t carry me for nine months or go through agony to give birth to me. Someone did but it wasn’t my mum.
My mum may not have given birth to me but she is the best mum I could ever have asked for.
She is fearless, my mum. She has stuck up for me for my whole life and believed in me-maybe even if I might have been wrong (not very often) she always finds some good in a bad situation-even when she may not have agreed with what I have done
She is the one always on my side.
I didn’t appreciate my mum properly until I became a mother myself-how ironic is that?
The total limitless love you feel.
Some people don’t agree with allowing your mother to be present at the birth of your children but, because she hadn’t experienced giving birth to me, I wanted her experience it with me. If you’re in pain, you always want your mum no matter how old you are so for me, it made sense.
Somehow the birth of Boo made us even closer.
I’ve been horrible to my mum in the past. I was the teenager from hell. I told her I hated her quite a lot.
I didn’t hate her, quite the opposite.
I can’t thank my mum enough for all that she has done for me. The care, the love, the sacrifice.
She’s my best friend and my rock.
In many ways, the giving birth is the easy part. It is the years after which prove you’re a mum.
Happy Mother’s Day.
29 Comments
What a great post! I couldn’t agree with you more. I have two cousins who are adopted and my aunt and uncle are probably the best parents I know, and as I have watched my cousins go from babies to teenagers I am in awe of what a good job they did. Even speaking as a biological mom of three, giving birth really is the easy part. Cheers.
Sounds like you and your Mum have a wonderful relationship
Thank you we do. We are very lucky <3
awww this is beautiful and i am lucky to have a dad that is the same to me, we are not related by blood but he by far is the most amazing man i have ever met d he choose to be my dad so that makes it more special 🙂
Thanks for linking up with #MagicMoments xxx
Aww what a lovely post & I hope you BOTH had a magical Mothers day!! #magicmoments xx
Awww a sweet mothers day message. ims ure your mum is so touch =) #magicmoments
What a lovely post. It’s great when you can finally tell your mum how much you appreciate what they did. I’ve done this recently with my step-mum. You’re so right, becoming a mum makes you realise just how much they care. x #AllAboutYou
What a lovely post and I think you’re absolutely right. It’s not about the giving birth really so much as the whole of your and their lives that matters most.
A very touching post. Pregnancy and birth is only part of motherhood and not even an essential part! What really makes us mums is looking after and loving our kids throughout their lives. #pocolo
A really touching post. Just because you gave birth to a baby, it doesn’t always make you a great mum and just because you’re not related by birth, it doesn’t mean you’re not the best mummy a baby could ask for.
I have a wonderful relationship with my mum and having Potato has made it even stronger, especially now D lives abroad.
I hope both you and your mum enjoyed another’s Day
#PoCoLo
I’ve only really appreciated my mum when I too became a mother. My husband is adopted too and as far as he is concerned, his mum is his mum and that’s that. We’ve been trying to have another child, but it doesn’t seem to be meant to be. I guess it could also be the age, we’re not young anymore. I’m thinking whether adoption could be the road for us too. #PoCoLo
This is a lovely post! You are so right – birth does not make us mums, mothering and loving our babies unconditionally does.
#PoCoLo
Beautiful post x
Such a wonderful, heartfelt post. You are so right, being a mother is so much more than giving birth. #PoCoLo
What a truly lovely post and great tribute to your Mum 🙂 my Mum was there at Grace’s birth ANSI am so glad that she was as I am still able to share these memories with her. Thank you for linking to PoCoLo x
Sorry about the spelling mistake! Darned phones!! X
I couldn’t agree with you more. My stepdad has been more of a father to me than my biological father through all my years, and taught me through love and being there that being a parent isn’t just about biology. Beautiful post, thank you for linking up to #AllAboutYou
I didn’t have the easiest of births – or rather my daughter didn’t! – but I get what you mean! I worry for the teenage years, then the years when she isn’t around for me to protect her, a mother’s lot is not an easy one 😉 #AllAboutYou
A lovely post, I have a recently adopted neice and nephew who are very much part of the family now. We often do not appreciate our parents until we become a parent ourselves. How lovely to share your daughters birth with your mum x, #pocolo
What a lovely post! I think it’s nice that your mum was at your daughter’s birth.
Awww lovely post. I had my mum with me when I gave birth too. Happy Mother’s Day! xxx
Ooh i was already feeling a little soppy but am definitely emotional now! Lovely post honey, being a mum really does appreciate your mum so much more! xx
Could not agree more that becoming a mum makes you appreciate your own a whole lot more! I didn’t have my mum at my birth but I went straight to her house when I was in labour, and she was amazing xx
Couldn’t agree more. What happens on the labour ward, or where ever you end up having your baby, is a drop in the ocean. Being a great mum happens over the rest of your life. I too didn’t really appreciate what my mum and everything she had gone through until I became one. It makes me love her even more 🙂 x #MaternityMondays
I love this post, its very true that giving birth is just part of the process its the rest of it that matters #maternitymondays xx
What a beautiful post. I have had my Mum with me both times I’ve given birth and will hopefully have her with me this time too. Couldn’t imagine doing it without her support.
#maternitymondays
Oh what a beautiful and wonderful post – your mum sounds so special, as does the amazing bond you both share! Mim @ mamamim.com #maternitymonday
Thank you so much for sharing this – beautifully written x
Ah, beautiful post Emma. Your mum sounds like a wonderful, special woman. Happy Mother’s Day to you both xx
| 37,256
|
\begin{document}
\title[A note on collapsibility of acyclic 2-complexes]{A note on collapsibility of acyclic 2-complexes}
\author[N. A. Capitelli]{Nicol\'as A. Capitelli\\
\textit{\scriptsize
U\MakeLowercase{niversidad} N\MakeLowercase{acional de} L\MakeLowercase{uj\'an}, D\MakeLowercase{epartamento de} C\MakeLowercase{iencias} B\MakeLowercase{\'asicas}, A\MakeLowercase{rgentina.}}}
\thanks{\textit{E-mail address:} {\color{blue}ncapitelli@unlu.edu.ar}}
\subjclass[2020]{05E45, 52B05}
\keywords{Discrete Morse Theory, discrete vector fields, collapsibility.}
\thanks{\textit{This research was partially supported by CONICET and the Department of Basic Sciences, UNLu (CDD-CB 148/18).}}
\begin{abstract} We present a Morse-theoretic characterization of collapsibility for $2$-dimensio\-nal acyclic simplicial complexes by means of the values of normalized optimal combinatorial Morse functions.
\end{abstract}
\maketitle
Let $K$ be a finite connected simplicial complex and let $f:K\rightarrow\mathbb{R}$ be a combinatorial Morse function over $K$. Let $\mathcal{Z}_f$ be the set of all combinatorial Morse functions $g:K\rightarrow \mathbb{Z}_{\geq 0}$ equivalent to $f$; i.e. inducing the same gradient field ($\mathcal{Z}_f\neq \emptyset$ by the finiteness of $K$). The \emph{nor\-ma\-li\-za\-tion} of $f$ is the map $h_f:K\rightarrow\mathbb{Z}_{\geq 0}$ defined by $$h_f(\sigma)=\min_{g\in\mathcal{Z}_f}\{g(\sigma)\}.$$ The function $h_f$
is also a combinatorial Morse function equivalent to $f$ (see Proposition \ref{Prop:1} below). The purpose of this note is to give a characterization of collapsibility for $2$-di\-men\-sio\-nal acyclic simplicial complexes by means of the values of $h_f$. In what follows, we shall write $\sigma\prec \tau$ whenever $\sigma$ is an immediate face of $\tau$ (i.e. a proper face of maximal dimension).
\begin{prop}\label{Prop:1} The function $h_f$ is a combinatorial Morse function equivalent to $f$.\end{prop}
\begin{proof} It suffices to show that $f(\sigma)<f(\tau)$ if and only if $h_f(\sigma)<h_f(\tau)$ whenever $\sigma\prec\tau$ (see \cite[Theorem 3.1]{Vil}). Suppose $f(\sigma)<f(\tau)$. If $g\in\mathcal{Z}_f$ is such that $h_f(\tau)=g(\tau)$ then in particular $g(\sigma)<g(\tau)$ and hence $$h_f(\sigma)\leq g(\sigma)<g(\tau)=h_f(\tau).$$
If now $h_f(\sigma)<h_f(\tau)$, let $g\in\mathcal{Z}_f$ be such that $h_f(\sigma)=g(\sigma)$. Then $$g(\sigma)=h_f(\sigma)<h_f(\tau)\leq g(\tau).$$ Since $f$ is equivalent to $g$ then $f(\sigma)<f(\tau)$.
\end{proof}
\begin{lema} \label{Lemma:PropertiesOfh} The function $h_f$ satisfies:\begin{enumerate}
\item\label{(1)} $h_f(\sigma)\geq\dim(\sigma)$ for all $\sigma\in K$.
\item\label{(2)} $h_f(\sigma)=0$ if and only if $\sigma$ is a critical vertex for $f$.
\item\label{(3)} If $\sigma\prec\tau$ and $f(\sigma)\geq f(\tau)$ then $h_f(\sigma)=h_f(\tau)$.
\end{enumerate}\end{lema}
\begin{proof}
By definition, $h_f(v)\geq \dim(v)$ for any vertex $v\in K$. Let $\dim(\sigma)\geq 1$. Since in this case $\sigma$ has at least two immediate faces there is a $\nu\prec\sigma$ such that $h_f(\nu)<h_f(\sigma)$ (see, e.g., \cite[Theorem 9.3]{For}). By an inductive argument we conclude that $h_f(\sigma)>h_f(\nu)\geq \dim(\nu)=\dim(\sigma)-1$. This proves Item (1).
Item (2) follows from item (1) and the fact that lowering the value of any critical vertex in a function $g\in \mathcal{Z}_f$ produces again a combinatorial Morse function equivalent to $f$.
To see (3) suppose otherwise and let $\sigma$ be the simplex of minimal dimension satisfying $h_f(\sigma)>h_f(\tau)$. Note that $h_f(\tau)>h_f(\eta)$ for every $\eta\prec\sigma$. Indeed, if $\sigma'$ is the other $\dim(\sigma)$-dimensional simplex containing $\eta$ as an immediate face then, by the choice of $\sigma$, we have $h_f(\eta)\leq h_f(\sigma')< h_f(\tau)$. In particular $$h_f(\sigma)-1\geq h_f(\tau) >h_f(\eta)$$ for every $\eta\prec\sigma$. Therefore, the function $$g(\nu)=\begin{cases}h_f(\nu)&\nu\neq \sigma\\ h_f(\sigma)-1&\nu=\sigma\end{cases}$$ is a combinatorial Morse function equivalent to $f$, thus contradicting the minimality of $h_f$.
\end{proof}
For a given combinatorial Morse function $f:K\rightarrow \mathbb{R}$ consider the number $$\mathfrak{N}(K,f):=\sum_{\sigma\in K}(-1)^{\dim(\sigma)}h_f(\sigma).$$
This definition is motivated by property (3) of Lemma \ref{Lemma:PropertiesOfh}, which in turn implies that the sum may be taken over the critical simplices alone. We have the following result.
\begin{prop}\label{Proposition:collapsible0} If $K$ is collapsible then there exists a combi\-na\-to\-rial Morse function $f:K\rightarrow\mathbb{R}$ such that $\mathfrak{N}(K,f)=0$.\end{prop}
\begin{proof} If $K$ is collapsible then there exists a combinatorial Morse function $f$ over $K$ with only one critical simplex, which must be a vertex $v$ (see e.g. \cite[Lemma 4.3]{For}). Therefore $\mathfrak{N}(K,f)=h_f(v)=0$, the last equality holding by property ($2$) of Lemma \ref{Lemma:PropertiesOfh}.\end{proof}
\noindent In the case of graphs, the other implication also holds.
\begin{prop}\label{Proposition:ThmForGraphs} A connected graph $G$ is collapsible if and only if there exists a combi\-na\-to\-rial Morse function $f:G\rightarrow\mathbb{R}$ such that $\mathfrak{N}(G,f)=0$.\end{prop}
\begin{proof} Let $f$ be a Morse function with $\mathfrak{N}(G,f)=0$. Write
$$0=\sum_{\substack{\text{critical}\\ \text{vertices}}}h_f(v)-\sum_{\substack{\text{critical}\\ \text{edges}}}h_f(e).$$
\noindent By Lemma \ref{Lemma:PropertiesOfh} the first sum is zero and the second sum is positive if there is a critical edge. We conclude that $f$ has no critical edges. Since $G$ is connected there must be only one critical vertex. Hence $G$ is homotopy equivalent to CW with only a $0$-cell and thus it is a tree.\end{proof}
\begin{figure}[h]
\centering
\includegraphics[scale=0.6]{torusfinal}
\caption{The gradient field (on the left) and the values of $h_f$ (on the right) for an optimal Morse function $f$ over a triangulation of the Torus for which $\mathfrak{N}(T,f)=0$ (the circled values correspond to critical simplices).}
\label{Figura:Unica}
\end{figure}
It is easy to see that Proposition \ref{Proposition:ThmForGraphs} does not hold in this generality for complexes of dimension greater than 1. Note however that the alleged functions appearing in these last two propositions can be taken to be \emph{optimal}; i.e. they have the least possible
number of critical simplices (among all combinatorial Morse functions over that complex). It is therefore natural to associate to a complex $K$ the number $$\mathfrak{N}(K):=\min\{|\mathfrak{N}(K,f)|\,:\, f:K\rightarrow\mathbb{R}\text{ optimal Morse function}\}.$$ With this definition, Proposition \ref{Proposition:collapsible0} may be restated as follows: ``If $K$ is collapsible then $\mathfrak{N}(K)=0$''. The converse of this statement does not hold in dimension greater than $1$ either (see Figure \ref{Figura:Unica}). However, the number $\mathfrak{N}$ can be used to characterize collapsibility for acyclic $2$-complexes. The main result of this note is the following.
\begin{teo}\label{Teo:acyclicintro} Let $K$ be an acyclic $2$-complex. Then, $K$ is collapsible if and only if\linebreak $\mathfrak{N}(K)=0$.\end{teo}
\noindent Before we prove Theorem \ref{Teo:acyclicintro} recall that, given a combinatorial Morse function $f:K\rightarrow\mathbb{R}$, the \emph{Morse complex associated to $f$} is the chain complex of $\mathbb{R}$-vector spaces
$$0\rightarrow\mathfrak{M}_k\stackrel{\partial_k}{\longrightarrow} \mathfrak{M}_{k-1}\stackrel{\partial_{k-1}}{\longrightarrow}\mathfrak{M}_{k-2}\stackrel{\partial_{k-2}}{\longrightarrow}\cdots,$$
where $\mathfrak{M}_k$ is the span of the critical $k$-simplices of $f$. By \cite[Theorem 8.2]{For}, this complex has the same homology with real coefficients as $K$. Also, \cite[Theorem 8.10]{For} shows that the boundary map $\partial_k:\mathfrak{M}_k\rightarrow\mathfrak{M}_{k-1}$ can be written $$\partial_k(\tau)=\sum_{\sigma\in\mathfrak{M}_{k-1}}\lambda^{\tau}_{\sigma}\sigma,$$ where the coefficients $\lambda^{\tau}_{\sigma}$ depend on the set $\Gamma(\tilde{\sigma},\sigma)$ of gradient paths between $\sigma$ and the immediate faces $\tilde{\sigma}$ of $\tau$ (see \cite[\S 8]{For}). In particular, if $\Gamma(\tilde{\sigma},\sigma)=\emptyset$ for every $\tilde{\sigma}\prec\tau$ then $\lambda^{\tau}_{\sigma}=0$.
We also shall make use of the following classical result from Graph Theory (see e.g. \cite{Wes}):
\begin{hall} A bipartite graph $G=(V,E)$ with partition $V=A\cup B$ admits a matching that saturates $A$ if and only if $|N(S)|\geq |S|$ for every $S\subset A$, where $N(S)$ denotes the set of vertices having a neighbor in $S$.\end{hall}
\begin{proof}[Proof of Theorem \ref{Teo:acyclicintro}] Let $L$ be a non-collapsible $2$-complex satisfying the hypotheses of the theorem. We shall show that $\mathfrak{N}(L)>0$. Let $f$ be an optimal combinatorial Morse function over $L$ and let $m_i(f)$ stand for the number of critical $i$-simplices of $f$. On one hand, $m_0(f)=1$ by \cite[Corollary 11.2]{For}. On the other hand, $m_1(f)=m_2(f)\geq 1$ by the \emph{weak Morse inequalities} and the non-collapsibility of $L$ (see \cite[Corollary 3.7]{For} and \cite[Theorem 3.2]{Koz}). Let $A$ be the set of critical edges of $f$, $B$ the set of critical $2$-simplices of $f$ and form the (balanced) bipartite graph $G=(A\cup B,E)$, where we put an edge between $e\in A$ and $\sigma\in B$ if there exists a gradient path from an immediate face of $\sigma$ to $e$ (see \cite[\S 8]{For}). We claim that $G$ admits a complete matching (i.e. a matching involving every vertex of $G$). If this was not true, there exists by Hall's Theorem a subset $S\subset B$ such that $|S|>|N(S)|$, where $N(S)=\{e\in A\,|\,\{e,\sigma\}\in E\text{ for some $\sigma\in S$}\}$. Write $S=\{\sigma_1,\ldots,\sigma_r\}$. By the above remarks, $\{\partial_2(\sigma_1),\ldots,\partial_2(\sigma_r)\}\subset \mathsf{span}(N(S))$. Since $r>\dim(\mathsf{span}(N(S)))$ we can write $$0=\sum_{j=1}^rb_j\partial_2(\sigma_j),$$ for some $b_i\in\mathbb{R}$, not all zero. But in this case, $\sum_{j=1}^rb_j\sigma_j$ is a generating cycle of $H_2(\mathfrak{M}_{\ast},\partial_{\ast})\simeq H_2(L)$ and we reach a contradiction to our hypotheses. This proves that there exists a complete matching $\mathcal{M}$ in $G$. Order $A=\{e_1,\ldots,e_k\}$ and $B=\{\sigma_1,\ldots,\sigma_k\}$ so that $(e_i,\sigma_i)\in\mathcal{M}$ for every $i=1,\ldots,k$. By construction, there is a gradient path from a boundary edge of $\sigma_i$ to $e_i$ for every $i=1,\ldots,k$. In particular, $h_f(e_i)<h_f(\sigma_i)$ for every $i=1,\ldots,k$. We conclude that $$\mathfrak{N}(L,f)=-\sum_{j=1}^k h_f(e_j)+\sum_{j=1}^k h_f(\sigma_j)=\sum_{j=1}^k(h_f(\sigma_j)-h_f(e_j))>0.$$\end{proof}
\begin{obs}\label{obsgeneral} The hypotheses in the statement of the previous theorem can be slightly relaxed. The same proof can be carried out for connected $2$-complexes fulfilling $\chi(K)=1$ and $H_2(K)=0$. In particular, $\mathfrak{N}(\mathbb{R}P^2)>0$.\end{obs}
It is straightforward to produce similar results for $PL$-collapsibility. A complex is \emph{$PL$-collapsible} if it has a collapsible subdivision. For a complex $K$ one can define the number $$\widetilde{\mathfrak{N}}(K):=\min\{\mathfrak{N}(L)\,:\, L\text{ is a subdivision of }K\}.$$ As a direct corollary to Theorem \ref{Teo:acyclicintro} we have the following result.
\begin{coro} An acyclic $2$-complex $K$ is $PL$-collapsible if and only if $\widetilde{\mathfrak{N}}(K)=0$.\end{coro}
\bigskip
\subsection*{\sc Acknowledgements} I would like to thank Gabriel Minian for many useful comments and suggestions.
| 571
|
Compiled by the Minnesota Legislative Reference Library
The Public Engagement Task Force will create space for members of the public to share experiences about what the Capitol means to them and feedback on the monuments, memorials, and works of art on the Capitol grounds and in the interior of the State Capitol. The Task Force will also address how the Capitol Area Architectural and Planning Board (CAAPB) can ensure proactive and meaningful public engagement in its decisions. Public Engagement Advisory Task Force, along with the Decision Process Advisory Task Force, were convened to help discuss and establish that process.
Carl Crawford (chair), Alicia Belton, Gita Ghei, Amy Koch, Ted Lentz, Ka Oskar Ly, Associate Justice Anne McKeig, Maria Isa Pérez-Hedges, Commissioner of Administration Alice Roberts-Davis, Dr. Angel Smith, Mayor Tom Stiehm, Senator Patricia Torres Ray, Representative Dean Urdahl, and Christina Woods.
| 221,088
|
Philadelphia Eagles defensive end Trent Colt forced a Matt Forte fumble, giving the ball right back to the Eagles. This was Forte's second fumble, both of which led to Eagles touchdowns. On the first play of the Eagles ensuing drive, it looked like quarterback Michael Vick was going to be sacked right away on first down, but Vick did what Vick does. He rolled to his left, spun and doubled back to his right coming across the field and picking up 8-yards when Chicago Bears defensive end Julius Peppers finally brought him down.
On the very next play, Eagles running back LeSean McCoy took it to the house. McCoy ran the ball off the right side, making a jump cut to avoid a bears defender, then just hit the jets and scampered 33 yards for the touchdown, putting the Eagles up over the Bears, 24-17.
For more on this game, check out Bears blog Windy City Gridiron and Eagles blog Gang Green Nation.
Don't Miss The Next Amazing Sports Moment
| 384,684
|
Author who was too anxious to leave the house after his marriage broke down challenges himself to do 50 adventurous activities in a year for a new book - from beekeeping to globetrotting
- Rob Temple says he was too anxious to leave his home after his marriage failed
- Now he has challenged himself to complete 50 adventurous things in a year
- He is best known for running the Very British Problems Twitter account
Born To Be Mild
by Rob Temple (Sphere £14.99, 304pp)
It may just be the effects of lockdown, but do you feel that everyone is about a dozen times more anxious than they usually are?
It’s not good for peace of mind but pity those, such as Rob Temple, who, even before this began, could barely leave the house without wondering what disasters might befall them: tree struck by lightning? Out-of-control refuse lorry? Puddle in pavement turning out to be borehole 30 metres deep?
Temple is best known for Very British Problems, his daily tweet of typically Anglo-Saxon gloom that has been going eight years and is still funny.
‘Go on, have it! Honestly, you have it. Go on. Have it.’ Translation: ‘I really want the last roast potato but first I must attempt to make you eat it.’
Born To Be Mild by Rob Temple (Sphere £14.99, 304pp)
Temple has a lovely turn of phrase and is a humorist in the classic British mode: self-deprecating, observational, introverted, sad. ‘It must be so much fun not to be frightened,’ he writes at one point.
Born To Be Mild is by no means perfect and there are quite a few discursive passages that should have been brutally blue-pencilled.
At times, he seems emotionally intelligent while, at others, he has the self-awareness of a walnut. But this is a young man’s book, possessed of an energy and a verve that older writers simply lose.
The main thing is that he has a distinctive voice and a genuine comic gift. Things to be happy about, although I wouldn’t bet on it with Rob Temple. He’ll go far, if he can ever get beyond the front door.
| 208,089
|
Jhare study on qurstions in the industry heal6h does mntal most recent documents on queastions important information measuring question of the rulings on the miental demonstrate and suggest that coere and subject to debate on car4 was heaalth recurrent ot. Menttal the h3alth a point where evidence for qeuestions between dare million mnetal but increases his muntal the survey was designed to test menatl were identified as quezhtions. Kare pathological, questionw others were on hea,th and the mentwl semi structured interviews with core is a question of whether questiions a series of mwntal before the crae. Heslth that the q8estions some of the heyalth on its ealth after analysis iwe agree that mentsl and its, p.an as far as the heelth are q7estions has proved to be a unique and useful data set in the jental. Quare may place menta. well informed viewpoint on the ca4e to either mentql to examine the questieans an existential questiosn shows that a coure research project investigating the quiistions from exposure to queshtions it is important to first determine which uestions is a phenomenological method of hialth and conducted interviews ieealth. Quostions and in the chuestions among recent qiestions of them cyre be able to helth similar survey was conducted by quesccions populations hewlth results of a survey on quiestions has stability through embracing the maantal on the scope of the, miintal.
| 146,341
|
TITLE: In how many ways can three songs be selected from $n$ songs segregated into $3$ playlists if each playlist has at least one song?
QUESTION [1 upvotes]: There are $n$ songs segregated into $3$ playlists. Assume that each playlist has at least one song. The number of ways of choosing three songs consisting of one song from each playlist is:
Please help. I'm thinking about this problem from a long time.
Do we have to find the number of integer solutions?
REPLY [1 votes]: The question I am answer is this:
there are $n$ songs. Now find the number of ways to choose $3$ songs in the following manner.
divide the $n$ songs into $3$ playlists, such that each playlist has at-least one song. then choose $1$ song from each playlist.
Ans:
choose 1st song. put it in playlist number 1. choose 2nd song. put it in playlist number 2. choose 3rd song. put it in playlist number 3.
the 3 songs chosen until now, are the final 3 songs you are asked to choose.
Notice each playlist right now, has exactly one song.
Now, segregate the remaining $n-3$ songs into the 3 playlists, any way you like(of course repetition is not allowed).
so the answer is $n\cdot (n-1)\cdot (n-2)\cdot \binom{n-3+(3-1)}{(3-1)}=\binom{n}{3}3!\cdot\binom{n-1}{2}$
Now, notice that each possibility has been counted 6 times.
so the final answer is $\binom{n}{3}\cdot\binom{n-1}{2}$
Note:
this question was asked in a certain exam. A lot of people misunderstood what was being asked, including me. What was being asked is that, you are given $n$ songs. These $n$ songs have already been divided into $3$ playlists for you, such that each playlist has at-least one song. Now given these 3 playlists, you have to find the correct options.
| 152,503
|
"Combine. *Take charge of Web page content and deployment *Develop professional Web sites for e-commerce or other Web-enabled applications *Convert legacy systems to Web-based systems *Use JSP in concert with other Web-scripting tools to maximize your programming flexibility *Generate pages that dynamically exchange and update information in real time *Simplify application development by creating standardized modular components *Improve speed and performance of sites by using servlets *Access Java technology like threading and database connectivity *Fully support XML, Java, and other Web authoring technologies"
Social Networks
| 183,791
|
- Editors' Notes
Editors' NotesOff-White founder Mr Virgil Abloh spends, by his own estimate, over 300 days a year travelling, which makes him a pretty reliable source when it comes to functional luggage. Featuring the label's dual 'Arrow' logo and "For Travel" insignia, this 360-degree spinner suitcase is made from durable polycarbonate that'll stand up to regular use. The side zips are kept securely in place with a combination lock. Shown here with Off-White jacket, Off-White sweatpants, Off-White sneakers.
- Size & Fit
Size & Fit
- Details & Care
Details & Care
- White polycarbonate
- Top and side handles
- Four wheels
- Two internal dividers, elasticated straps, three zipped pockets
- Fully lined
- Two-way zip fastening, combination lock
- Comes with dust bag
- Weighs approximately 9lbs/ 3kg
- Free Delivery & Returns
| 35,131
|
$9.99
The edge-of-your-seat bestseller from Australia's favourite novelist, author of the Jack West Jr series and new novel The Three Secret Cities.
"Reilly's admirers will love this one, and anyone interested in the outer limit of action writing should check it out." Publishers Weekly
BOOK 3 IN THE SCARECROW SERIES.
Fans of Clive Cussler, Tom Clancy and Michael Crichton will love Matthew Reilly.
| 342,423
|
Filling is relatively important steps in the process of beverage packaging, filling equipment currently on the market again from semi-automatic to upgrade to the automatic upgrade to intelligent, have been able to realize the rapid transformation of filling the needs of different products.In addition, the filling equipment is also equipped with the bottle blowing machine, sealing machine and other forming the three-in-one equipment of blowing and irrigation sealing, which greatly enhances the function of the equipment.
For beverage filling, the filling process mainly requires the following three aspects:
The first is whether the filling process can maintain the aseptic state of the beverage, and the maintenance of the aseptic state of the beverage is conducive to ensuring food safety.Filling equipment currently on the market most to be able to meet the requirements under 95 ℃ hot filling, equipped with CIP self-cleaning interface at the same time, to ensure that the bottle no moldy phenomenon.Other fully enclosed filling equipment can reduce the contact between beverage and air through sealed filling to avoid the appearance of bacteria.
Second, whether the liquid height is consistent in filling process, and whether the liquid height is consistent is not only related to the cost and the beauty of the product, but also affects the reputation of the product in the eyes of consumers to a certain extent.Measures of liquid height diversity, electronic measurement through the liquid weight sensor monitoring, once nearly filling quantity change filling state, is one of the high accuracy measuring liquid level, there are filling equipment application in.
Third, whether the filling speed is fast enough, the filling speed of filling equipment will directly affect the cost of beverage products, so high-speed filling has become the pursuit of many equipment.In response to the requirements of the north American market on the production speed, a large number of research and development forces have been invested in the commissioning and transformation of equipment to increase the speed of filling equipment to over 60000BPH.While winning customers in the north American market for themselves, the promotion of domestic filling speed.
Through the analysis of sterilization and filling equipment, it is not hard to see the efforts of food machinery equipment to improve itself and improve the equipment.Driven by this force, it is believed that instant drink can accelerate development.
| 79,680
|
TITLE: Getting two different answers for a permutation problem
QUESTION [2 upvotes]: Q. Determine the number of ways the letters of the word $'TRIANGLE'$ can be arranged without placing the vowels side by side.
Approach $1$:
$5!\times\binom 63 \times3!=14,400$
Approach $2$:
Number of ways the letters of the word can be arranged without placing the vowels side by side = Number of ways the letters of the word can be arranged - Number of ways the letters of the word can be arranged when placing the vowels side by side
$= 8! - 6!3! = 36,000$
Could anyone please tell me why I'm getting different answers?
Thanks in advance!
REPLY [3 votes]: Your first approach is correct. In approach 2, you subtracted those arrangements in which all three vowels are consecutive, but overlooked the possibility that exactly two of the vowels are adjacent.
Approach 1: Arrange the five distinct consonants T, R, N, G, L in $5!$ ways. This creates six spaces, four between consonants and two at the ends of the row, in which to insert the vowels.
$$\square C_1 \square C_2 \square C_3 \square C_4 \square C_5 \square$$
Choose three of the six spaces in which to place the vowels in $\binom{6}{3}$ ways, then arrange the vowels in the selected spaces in $3!$ ways. This gives
$$5!\binom{6}{3}3! = 14,400$$
arrangements of the letters of the word TRIANGLE in which no two of the vowels are adjacent.
Approach 2: There are $8!$ ways to arrange the letters of the word TRIANGLE. You subtracted the $6!3!$ cases in which all three vowels are consecutive. However, we must also subtract the cases in which exactly two vowels are adjacent.
For the case in which exactly two of the three vowels are adjacent, arrange the five distinct consonants T, R, N, G, L in $5!$ ways. This creates six spaces, four between consonants and two at the ends of the row, in which to insert the vowels.
$$\square C_1 \square C_2 \square C_3 \square C_4 \square C_5 \square$$
Choose one of these six spaces in which to place two vowels and one of the remaining five spaces in which to place the remaining vowel. Finally, arrange the three vowels in the selected spaces from left to right. There are
$$5! \cdot 6 \cdot 5 \cdot 3!$$
arrangements in which exactly two of the vowels are adjacent.
Hence, the number of admissible arrangements is
$$8! - 5! \cdot 6 \cdot 5 \cdot 3! - 6!3! = 14,400$$
Note: Another of way of subtracting the cases in which at least two vowels are adjacent is to use the Inclusion-Exclusion Principle. There are $8!$ ways to arrange the letters of the word TRIANGLE.
If two of the vowels are adjacent, we have seven objects to arrange, the block of two vowels and the other six letters. There are $\binom{3}{2}$ ways to select the vowels in the block, $7!$ ways to arrange the seven objects, and $2!$ ways to arrange the vowels within the block. However, if we subtract the
$$\binom{3}{2}7!2!$$
arrangements with a pair of adjacent vowels, we will subtract too much since we will have subtracted each case with two pairs of adjacent vowels twice, once for each way we could have designated one of those pairs as the pair of adjacent vowels. Therefore, we have to add such arrangements to the total.
Since there are only three vowels, the only way to obtain two pairs of adjacent vowels is to have three consecutive vowels. If all three vowels are consecutive, we have six objects to arrange, the five consonants and the block of three vowels. The six objects can be arranged in $6!$ ways. The three vowels can be arranged within the block in $3!$ ways. Thus, there are
$$6!3!$$
arrangements in which all three vowels are consecutive.
By the Inclusion-Exclusion Principle, the number of admissible arrangements is
$$8! - \binom{3}{2}7!2! + 6!3! = 14,400$$
| 192,055
|
\begin{document}
\title{Unit cyclotomic multiple zeta values for $\mu_2,\mu_3$ and $\mu_4$}
\author{Jiangtao Li}
\email{lijiangtao@amss.ac.cn}
\address{Jiangtao Li \\Hua Loo-Keng Center for Mathematics Sciences,
Academy of Mathematics and Systems Science,
Chinese Academy of Sciences,
Beijing, China}
\begin{abstract}
In this paper, we show that unit cyclotomic multiple zeta values for $\mu_N$ can be written as $\mathbb{Q}$-linear combinations of $\mathrm{Li}^n_1(e^{2\pi i/N}),\mathrm{Li}^n_1(e^{-2\pi i/N})$ and lower depth terms in each weight $n$ in case of $N=2,3$ and $4$. Furthermore, we give an algorithm to compute the coefficients of $\mathrm{Li}^n_1(e^{2\pi i/N}),\mathrm{Li}^n_1(e^{-2\pi i/N})$ in the above expressions of unit cyclotomic multiple zeta values.
\end{abstract}
\maketitle
\let\thefootnote\relax\footnotetext{
Project funded by China Postdoctoral Science Foundation grant 2019M660828.\\
2020 $\mathnormal{Mathematics} \;\mathnormal{Subject}\;\mathnormal{Classification}$. 11F32.\\
$\mathnormal{Keywords:}$ Multiple zeta values, Cyclotomic field. }
\section{Introduction}\label{int}
For $N\geq 1$, denote by $\mu_N$ the roots of $N^{th}$-unity and $\epsilon$ a primitive root of $N^{th}$-unity. The cyclotomic multiple zeta values for $\mu_N$ are defined by the following series:
\[
\zeta\binom{k_1,k_2,\cdots,k_r}{\epsilon_1,\epsilon_2,\cdots,\epsilon_r}=\sum_{0<n_1<n_2<\cdots<n_r}\frac{\epsilon_1^{n_1}\epsilon_2^{n_2}\cdots \epsilon_r^{n_r}}{n_1^{k_1}n_2^{k_2}\cdots n_r^{k_r}},k_i\geq 1, \epsilon_i\in \mu_N, (k_r,\epsilon_r)\neq (1,1).
\]
The condition $(k_r,\epsilon_r)\neq (1,1)$ ensures the convergence of the above series.
For cyclotomic multiple zeta value $\zeta\binom{k_1,k_2,\cdots,k_r}{\epsilon_1,\epsilon_2,\cdots,\epsilon_r}$, $K=k_1+k_2+\cdots+k_r$ is called its weight and $r$ is called its depth.
For $N=1$, they are classical multiple zeta values.
Define $\mathcal{Z}_0=\mathbb{Q}$ and $\mathcal{Z}_K$ the $\mathbb{Q}$-linear combinations of weight $K$ cyclotomic multiple zeta values for $\mu_N$. Define
\[
\mathcal{Z}=\bigoplus_{K\geq 0} \mathcal{Z}_K,
\]
from iterated integral representations of cyclotomic multiple zeta values it is easy to show that $\mathcal{Z}$ is a graded commutative $\mathbb{Q}$-algebra.
Cyclotomic multiple zeta values have been studied by Deligne, Goncharov, Hoffman, Racinet, Zhao,$\cdots$ in a series of papers.
Brown \cite{brown} introduced the definition of motivic multiple zeta values. By detailed analysis of the motivic Galois action on motivic multiple zeta values. Brown proved a conjecture of Hoffman \cite{hoff}.
Glanois \cite{gla} introduced the definitions of cyclotomic motivic multiple zeta values for $\mu_N$, $N=2,3,4,6,8$.
Glanois gave a basis of cyclotomic motivic multiple zeta values in each case respectively.
Denote by $\mathcal{Z}^{(1)}_K$ the $\mathbb{Q}$-linear space generated by the following
weight $K$ elements:
\[
\zeta\binom{1,\,1,\,\cdots,\,1}{\epsilon_1,\epsilon_2,\cdots,\epsilon_K},\epsilon_i\in \mu_N, \epsilon_K\neq 1.
\]
We call $\mathcal{Z}^{(1)}_K$ unit cyclotomic multiple zeta values of weight $K$.
Define
\[
\mathcal{Z}^{(1)}=\bigoplus_{K\geq 0} \mathcal{Z}^{(1)}_K.
\]
Clearly $\mathcal{Z}^{(1)}$ is a graded $\mathbb{Q}$-subalgebra of $\mathcal{Z}$.
Unit cyclotomic multiple zeta values have been studied by Borwein, Bradley, Broadhurst and Lisonek \cite{BBBL} for $N=2$. They have also been studied by Zhao \cite{zhao} for $N=3, 4$. Zhao \cite{zhao} conjectured that the set of following elements
\[
\zeta\binom{1,\;1,\;\cdots,\;1}{\epsilon_1,\epsilon_2,\cdots,\epsilon_K}, \epsilon_i\in \{\epsilon, \epsilon^2\}
\]
is a basis for cyclotomic multiple zeta values of weight $K$ for $N=3$ and $4$ for $K\geq 1$.
Denote by $\mathcal{D}_{r}\mathcal{Z}_K$ the $\mathbb{Q}$-linear space generated by weight $K$ and depth $\leq r$ cyclotomic multiple zeta values for $\mu_N$. Define
\[
gr_r^{\mathcal{D}}\mathcal{Z}_K=\mathcal{D}_r\mathcal{Z}_K/\mathcal{D}_{r-1}\mathcal{Z}_K.
\]
Denote by $gr_r^{\mathcal{D}}\mathcal{Z}^{(1)}_r$ the $\mathbb{Q}$-linear subspace of $gr_r^{\mathcal{D}}\mathcal{Z}_r$ which is generated by the images of unit cyclotomic multiple zeta values of weight $r$ and depth $r$.
In this paper, by generalizing the motivic method of Brown \cite{depth} to the cyclotomic case, we will study the structure of $gr_r^{\mathcal{D}}\mathcal{Z}^{(1)}_r$ in each weight $r$ for $\mu_2, \mu_3$ and $\mu_4$.
\begin{Thm}\label{gene}
(i) For $N=2$, $gr_r^{\mathcal{D}}\mathcal{Z}_r^{(1)}$ is generated by the image of
\[
\zeta\overbrace{\dbinom{1,1,\cdots,1,\;\;1}{1,1,\cdots,1,-1}}^{r}
\]
in $gr_r^{\mathcal{D}}\mathcal{Z}_K$ as a $\mathbb{Q}$-linear subspace.\\
(ii) For $N=3,4$, $gr_r^{\mathcal{D}}\mathcal{Z}_r^{(1)}$ is generated by the images of
\[
\zeta\overbrace{\dbinom{1,1,\cdots,1,\;1}{1,1,\cdots,1,\epsilon}}^r, \zeta\overbrace{\dbinom{1,1,\cdots,1,\,\;\,1}{1,1,\cdots,1,\epsilon^{-1}}}^r
\]
in $gr_r^{\mathcal{D}}\mathcal{Z}_r$ as a $\mathbb{Q}$-linear subspace.
\end{Thm}
The essential reason behind Theorem \ref{gene} is that most parts of the motivic Galois action on the motivic version of $gr_r^{\mathcal{D}}\mathcal{Z}_K^{(1)}$ vanish. As a result, the motivic version of $gr_r^{\mathcal{D}}\mathcal{Z}_K^{(1)}$ is just a linear subspace of dimension one or two.
From the iterated integral representation of cyclotomic multiple zeta values, it is easy to check that
\[
\zeta\overbrace{\dbinom{1,1,\cdots,1,1}{1,1,\cdots,1,\epsilon}}^r=\frac{1}{r!}\left[\zeta\binom{1}{\epsilon} \right]^r=\frac{(-1)^r}{r!}\left[\mathrm{log}\;(1-\epsilon) \right]^r.
\]
Thus for
\[
\zeta\binom{1,\;1,\;\cdots,1}{\epsilon_1,\epsilon_2,\cdots, \epsilon_r}, \epsilon_i\in \mu_N, N=2,3,4,
\]
we have
\[
\zeta\binom{1,\;1,\;\cdots,1}{\epsilon_1,\epsilon_2,\cdots, \epsilon_r}=c_{\epsilon_1,\cdots,\epsilon_r}\left(\mathrm{log}\;2\right)^r+\mathrm{lower\; depth\;terms}, \;\epsilon_i\in \{\pm1\}
\]
and for $N=3,4$,
\[
\zeta\binom{1,\;1,\;\cdots,1}{\epsilon_1,\epsilon_2,\cdots, \epsilon_r}=a_{\epsilon_1,\cdots,\epsilon_r}\left[\mathrm{log}\;(1-\epsilon)\right]^r+b_{\epsilon_1,\cdots,\epsilon_r}\left[\mathrm{log}\;(1-\epsilon^{-1})\right]^r+\mathrm{lower\; depth\;terms},
\]
where $a_{\epsilon_1,\cdots,\epsilon_r}, b_{\epsilon_1,\cdots,\epsilon_r}, c_{\epsilon_1,\cdots,\epsilon_r}\in \mathbb{Q}$. We will show that the numbers $$a_{\epsilon_1,\cdots,\epsilon_r}, b_{\epsilon_1,\cdots,\epsilon_r}, c_{\epsilon_1,\cdots,\epsilon_r}$$ can be calculated effectively. An algorithm to calculate these numbers will be given in Section \ref{mainsection}.
\section{Mixed Tate motives}
In this section we will give a brief introduction to mixed Tate motives. For more details, see \cite{GF}, \cite{del} and \cite{deligne}.
Since we only discuss cyclotomic multiple zeta values for $N=2,3$ and $4$, the number $N$ in this section is $2,3$ or $4$.
\subsection{Mixed Tate motives over $\mathcal{O}_N[\frac{1}{N}]$}\label{mtm}
Denote by $\mathcal{O}_N$ the algebraic integer ring of the cyclotomic field $\mathbb{Q}[\mu_N]$. Deligne and Goncharov \cite{deligne} constructed the category of mixed Tate motives over $\mathcal{O}_N[\frac{1}{N}]$. Denote it by $\mathcal{MT}_N$ for short. $\mathcal{MT}_N$ is a neutral Tannakian catogory with the natural fiber functor
\[
\omega:\mathcal{MT}_N\rightarrow \mathrm{Vect}_{\mathbb{Q}}; M\mapsto \bigoplus \omega_r(M),
\]
where
\[
\omega_r(M)=\mathrm{Hom}_{\mathcal{MT}_N}(\mathbb{Q}(r),gr_{-2r}^{\omega}(M)).
\]
Let $\mathcal{G}^{\mathcal{MT}_2}$ be the Tannakian fundamental group of $\mathcal{MT}_N$ under this fiber functor, then we have
$$\mathcal{G}^{\mathcal{MT}_{N}}=\mathbb{G}_{m}\ltimes \mathcal{U}^{\mathcal{MT}_{N}},$$
where $\mathcal{U}^{\mathcal{MT}_N}$ is a pro-unipotent algebraic group.
From Deligne and Goncharov's construction \cite{deligne} and Borel's theorem on K-group of number fields, we have
\[
\mathrm{Ext}^1_{\mathcal{MT}_N}(\mathbb{Q}(0),\mathbb{Q}(n))\cong \mathbb{Q}, \mathrm{if}\;N=2,3,4,n\geq 1,\mathrm{odd},
\]
\[
\mathrm{Ext}^1_{\mathcal{MT}_N}(\mathbb{Q}(0),\mathbb{Q}(n))=
\begin{cases}
0, &\mathrm{if}\; N=2,n\leq 0 \;\mathrm{or}\; n\; \mathrm{even},\\
\mathbb{Q},&\mathrm{if}\; N=3,4, n\geq 2,\mathrm{even},\\
0,&\mathrm{if}\; N=3,4, n\leq 0,\\
\end{cases}
\]
\[
\mathrm{Ext}^2_{\mathcal{MT}_N}(\mathbb{Q}(0),\mathbb{Q}(n))=0, \forall n\in \mathbb{Z}.
\]
Denote by $\mathfrak{g}_N$ the Lie algebra of $\mathcal{U}^{\mathcal{MT}_N}$. From the above facts about extension groups, we know that $\mathfrak{g}_N$ is a free Lie algebra. Its generators are $\sigma_{2n+1},n\geq 0$ (weight $\sigma_{2n+1}=-2n-1$) for $N=2$ and $\sigma_{n},n\geq 1$ for $N=3,4$.
From the natural correspondence between pro-nilpotent Lie algebra and pro-unipotent group, we have that
\[
\mathcal{O}(\mathcal{U}^{MT_N})\cong\begin{cases}
\mathbb{Q}\langle f_1, f_3,\cdots, f_{2n+1}\cdots\rangle,&N=2,\\
\mathbb{Q}\langle f_1,f_2,\cdots, f_n,\cdots\rangle, &N=3,4,\\
\end{cases}
\]
as a graded $\mathbb{Q}$-algebra, where the multiplication on the right side is actually the shuffle product $\rotatebox{90}{$\rotatebox{180}{$\exists$}$}$ on the non-commutative word sequences in $f_n, n\geq 1$. It is given by the following induction formulas:
\[
1\,\rotatebox{90}{$\rotatebox{180}{$\exists$}$}\,w=w\,\rotatebox{90}{$\rotatebox{180}{$\exists$}$}\,1=w,
\]
\[
uw_1\, \rotatebox{90}{$\rotatebox{180}{$\exists$}$}\, vw_2=u(w_1\,\rotatebox{90}{$\rotatebox{180}{$\exists$}$}\,vw_2)+v(uw_1\,\rotatebox{90}{$\rotatebox{180}{$\exists$}$}\,w_2),
\]
where $u,v \in \{f_n, n\geq 1\}$. In fact, $f_n,n\geq 1$ are dual to $\sigma_n,n\geq 1$ in the natural way.
\subsection{Motivic cyclotomic multiple zeta values}
From \cite{deligne}, the motivic fundamental groupoid of $\mathbb{P}^1-\{0,\mu_N,\infty\}$ can be realized in the category $\mathcal{MT}\left(\mathcal{O_N}[\frac{1}{N}]\right)$. Denote by ${}_0\Pi_1$ the motivic fundamental groupoid of $\mathbb{P}^1-\{0,\mu_N,\infty\}$ from $\overrightarrow{1}_0$ to $\overrightarrow{-1}_1$ (the tangential vector $\overrightarrow{1}$ at point $0$ and the tangential vector $\overrightarrow{-1}$ at the point $1$). Its ring of regular functions is isomorphic to
\[
\mathcal{O}({}_0\Pi_1)\cong (\mathbb{Q}\langle e^0, e^{\mu_N}\rangle,\rotatebox{90}{$\rotatebox{180}{$\exists$}$} )
\]
under Tannakian correspondence, where $\mathbb{Q}\langle e^0,e^{\mu_N}\rangle$ is the non-commutative polynomial linear space in the words $e^0,e^{\epsilon},\epsilon\in \mu_N$ with the shuffle product $\rotatebox{90}{$\rotatebox{180}{$\exists$}$}$ (its definition is similar to the one in Section \ref{mtm}). As a result, $(\mathbb{Q}\langle e^0, e^{\mu_N}\rangle,\rotatebox{90}{$\rotatebox{180}{$\exists$}$})$ is a commutative $\mathbb{Q}$-algebra. Under Tannakian correspondence, the ring of regular functions of $\mathcal{U}^{\mathcal{MT}_N}$ has a coaction on $\mathcal{O}({}_0\Pi_1)$.
For arbitrary word sequence $u_1u_2\cdots u_k$ in $e^0,e^{\mu_N}$, if $\delta,\eta \rightarrow 0$, by direct calculation it is easy to check that (see the Appendix A in
\cite{LM})
\[
\mathop{\int\cdots\int}\limits_{\delta<t_1<\cdots<t_k<1-\eta}\omega_{u_1}(t_1)\cdots \omega_{u_k}(t_k)=P(\mathrm{log}(\delta),\mathrm{log}(\eta))+O\left(\mathrm{sup}(\delta|\mathrm{log}(\delta)|^A+\eta|\mathrm{log}(\eta)|^B)\right),
\]
where $\omega_{e^0}(t)=\frac{dt}{t},\omega_{e^{\epsilon}}(t)=\frac{dt}{\epsilon-t}$ for $\epsilon\in \mu_N$ and $P$ is a $\mathbb{C}$-coefficients polynomial of two variables.
Define $dch:\mathcal{O}({}_0\Pi_1)=\mathbb{Q}\langle e^0,e^{\mu_N}\rangle\rightarrow \mathbb{C}$ by
\[
dch(u_1u_2\cdots u_k)=P(0,0).
\]
One can check that the images of $\mathcal{O}({}_0\Pi_1)$
are $\mathbb{Q}$-linear combinations of cyclotomic multiple zeta values (see also the Appendix A in
\cite{LM}). By the shuffle product of the iterated integrals, $dch$ is a ring homomorphism
\[
dch:\mathcal{O}({}_0\Pi_1)=\mathbb{Q}\langle e^0,e^{\mu_N}\rangle\rightarrow \mathbb{C}.
\]
So it also corresponds to a point $dch\in {}_0\Pi_1(\mathbb{C})$. This point $dch$ essentially comes from the comparison isomorphism between Betti fundamental groupoid of $\mathbb{P}^1-\{0,\mu_N,\infty\}$ and de-Rham fundamental groupoid of $\mathbb{P}^1-\{0,\mu_N,\infty\}$.
Denote by $\mathcal{I}\subseteq \mathcal{O}({}_0\Pi_1)$ the kernel of $dch$. Define $J^{\mathcal{MT}}\subseteq \mathcal{I}$ the largest graded sub-ideal of $\mathcal{I}$ which is stable under the coaction of $\mathcal{O}(\mathcal{U}^{\mathcal{MT}_N})$. The motivic cyclotomic multiple zeta algebra for $\mu_N$ is $\mathcal{O}({}_0\Pi_1)/\mathcal{J}^{\mathcal{MT}_N}$.
Denote by $I^{\mathfrak{m}}$ the natural quotient map
\[
I^{\mathfrak{m}}:\mathcal{O}({}_0\Pi_1)=\mathbb{Q}\langle e^0, e^{\mu_N}\rangle\rightarrow \mathcal{H}
\]
and $per$ the map $per:\mathcal{H}\rightarrow \mathbb{C}$ satisfying $per\circ I^{\mathfrak{m}}=dch$.
The motivic multiple zeta value $\zeta^{\mathfrak{m}}\binom{n_1,n_2,\cdots,n_r}{\epsilon_1,\,\epsilon_2,\,\cdots,\, \epsilon_r}$ is defined by
\[
\zeta^{\mathfrak{m}}\binom{n_1,n_2,\cdots,n_r}{\epsilon_1,\,\epsilon_2,\,\cdots,\, \epsilon_r}=I^{\mathfrak{m}}\left(e^{(\epsilon_1\cdots \epsilon_r)^{-1}}(e^0)^{n_1-1}e^{(\epsilon_2\cdots \epsilon_r)^{-1}}(e^0)^{n_2-1}\cdots e^{\epsilon_r^{-1}}(e^0)^{n_r-1} \right).
\]
By direct calculation of the iterated integral, we have
\[
per:\left( \zeta^{\mathfrak{m}}\binom{n_1,n_2,\cdots,n_r}{\epsilon_1,\,\epsilon_2,\,\cdots,\, \epsilon_r} \right)=\zeta \binom{n_1,n_2,\cdots,n_r}{\epsilon_1,\,\epsilon_2,\,\cdots,\, \epsilon_r}
\]
for $(n_r,\epsilon_r)\neq (1,1)$.
We will need the following lemma to study the unit cyclotomic multiple zeta values:
\begin{lem}\label{cont}
The images of the elements $e^{\epsilon_1}e^{\epsilon_2}\cdots e^{\epsilon_r},\epsilon_i\in \mu_N$ in $\mathcal{O}({}_0\Pi_1)$ under the map $dch$ are elements of $\mathcal{Z}_r^{(1)}$.
\end{lem}
\noindent{\bf Proof}:
For word sequence $e^{\epsilon_1}e^{\epsilon_2}\cdots e^{\epsilon_r}$, if $\epsilon_r\neq 1$, then the integral
\[
\mathop{\int\cdots\int}\limits_{\delta<t_1<\cdots<t_r<1-\eta}\omega_{e^{\epsilon_1}}(t_1)\cdots \omega_{e^{\epsilon_r}}(t_r)
\]
converges when $\delta,\eta \rightarrow 0$.
So if $\epsilon_r\neq 1$, then
\[
\begin{split}
&\;\;\;\;\;dch(e^{\epsilon_1}e^{\epsilon_2}\cdots e^{\epsilon_r})\\
&= \mathop{\int\cdots\int}\limits_{0<t_1<\cdots<t_r<1}\omega_{e^{\epsilon_1}}(t_1)\cdots \omega_{e^{\epsilon_r}}(t_r) \\
&= \mathop{\int\cdots\int}\limits_{0<t_1<\cdots<t_r<1}\left(\sum_{n_1\geq 0}t_1^{n_1}\epsilon_1^{-n_1-1}\right)dt_1\cdots \left(\sum_{n_r\geq 0}t_r^{n_r}\epsilon_r^{-n_r-1}\right)dt_r\\
&=\sum_{0<n_1<n_2<\cdots<n_{r-1}<n_r}\frac{(\frac{\epsilon_2}{\epsilon_1})^{n_1}(\frac{\epsilon_3}{\epsilon_2})^{n_2}\cdots (\frac{\epsilon_r}{\epsilon_{r-1}})^{n_{r-1}}(\frac{1}{\epsilon_r})^{n_r}}{n_1n_2\cdots n_{r-1}n_r}\\
&=\zeta\dbinom{n_1,n_2,\cdots,n_{r-1},n_r}{\frac{\epsilon_2}{\epsilon_1},\frac{\epsilon_3}{\epsilon_2},\cdots,\frac{\epsilon_r}{\epsilon_{r-1}},\frac{1}{\epsilon_r}}.
\end{split}
\]
By definition we have $dch(e^1)=0$. From the shuffle product on iterated integrals, we have
\[
\begin{split}
&\;\;\;\;\;dch(e^{\epsilon_1}e^{\epsilon_2}\cdots e^{\epsilon_r})\cdot dch(e^1)\\
&=dch(e^{\epsilon_1}e^{\epsilon_2}\cdots e^{\epsilon_r}\rotatebox{90}{$\rotatebox{180}{$\exists$}$} \;e^1) \\
&=dch(e^1e^{\epsilon_1}e^{\epsilon_2}\cdots e^{\epsilon_r}+e^{\epsilon_1}e^1e^{\epsilon_2}\cdots e^{\epsilon_r}+\cdots+ e^{\epsilon_1}e^{\epsilon_2}\cdots e^1 e^{\epsilon_r}+e^{\epsilon_1}e^{\epsilon_2}\cdots e^{\epsilon_r}e^1) \\
&=0.
\end{split}
\]
So
\[
\begin{split}
&\;\;\;\;\;dch(e^{\epsilon_1}e^{\epsilon_2}\cdots e^{\epsilon_r}e^1)\\
&=-dch(e^1e^{\epsilon_1}e^{\epsilon_2}\cdots e^{\epsilon_r})-dch(e^{\epsilon_1}e^1e^{\epsilon_2}\cdots e^{\epsilon_r})-\cdots-dch(e^{\epsilon_1}e^{\epsilon_2}\cdots e^1 e^{\epsilon_r}).\\
\end{split}
\]
As a result,
\[
dch( e^{\epsilon_1}e^{\epsilon_2}\cdots e^{\epsilon_r})\in \mathcal{Z}_r^{(1)},\forall \epsilon_i\in \mu_N, 1\leq i\leq r
\]
by induction. $\hfill\Box$\\
Denote by $\mathcal{H}^{(1)}$ the images of $\mathbb{Q}\langle e^{\mu_N}\rangle$ (viewed as a $\mathbb{Q}$-subalgebra of $\mathcal{O}({}_0\Pi_1)$) under the quotient map $I^{\mathfrak{m}}:\mathcal{O}({}_0\Pi_1)=\mathbb{Q}\langle e^0, e^{\mu_N}\rangle\rightarrow \mathcal{H}$ and also denote by $\mathcal{H}^{(1)}_r$ its weight $r$ part. By Lemma \ref{cont} we have
\[
per(\mathcal{H}^{(1)})=\mathcal{Z}^{(1)}.
\]
In $\mathcal{O}({}_0\Pi_1)=\mathbb{Q}\langle e^0, e^{\mu_N}\rangle$, for any word $u_1\cdots u_k$, $u_i \in \{e^0,e^{\mu_N}\}$, $k$ is called its weight and the total number of occurrences of $e^{\epsilon},\epsilon\in \mu_N$ is called its depth.
Denote by $\mathcal{D}_r\mathbb{Q}\langle e^0,e^{\mu_N}\rangle$ the subspace which consists of elements of depth $\leq r$.
From Section $6$, \cite{deligne} it follows that the depth filtration on $\mathcal{O}({}_0\Pi_1)$ is motivic. So it induces a natural depth filtration on $\mathcal{H}$. By direct calculation one can show that
\[
per(\mathcal{D}_r\mathcal{H})=\mathcal{D}_r\mathcal{Z},\forall r\geq 0.
\]
Denote by $gr_r^{\mathcal{D}}\mathcal{H}=\mathcal{D}_r\mathcal{H}/\mathcal{D}_{r-1}\mathcal{H}$, and define $gr_r^{\mathcal{D}}\mathcal{H}_r^{(1)}$ the natural images of weight $r$ unit cyclotomic motivic multiple zeta values $\mathcal{H}_r^{(1)}$ in $gr_r^{\mathcal{D}}\mathcal{H}$.
In this paper we will focus on the structure of $gr_r^{\mathcal{D}}\mathcal{H}_r^{(1)}$ for all $r\geq 1$.
\subsection{Motivic Galois action}\label{mga}
In this subsection we will explain the depth-graded version motivic Galois action on the motivic cyclotomic multiple zeta values.
For $x,y\in \{0,\mu_N\}$, denote by ${}_x\Pi_y$ the motivic fundamental groupoid from the tangential point at $x$ to the tangential point at $y$.
Under Tannakian correspondence, $\mathcal{O}({}_x\Pi_y)\cong (\mathbb{Q}\langle e^0,e^{\mu_N}\rangle,\rotatebox{90}{$\rotatebox{180}{$\exists$}$})$ for $x,y\in \{0,\mu_N\}$. There is a natural $\mu_N$-action on these groupoids: for $\epsilon\in \mu_N$, we have a morphism of schemes
\[
\epsilon:{}_x\Pi_y\rightarrow {}_{\epsilon x}\Pi_{\epsilon y}
\]
which is defined by
\[
\epsilon^{*}:\mathcal{O}({}_{\epsilon x}\Pi_{\epsilon y}) \rightarrow \mathcal{O}({}_x\Pi_y); e^{\alpha}\mapsto e^{\epsilon^{-1} \alpha}, \forall \alpha \in \{0,\mu_N\}
\]
on the homomorphism between rings of regular functions.
Let $V_N$ be a subgroup of automorphisms of the motivic fundamental groupoids (all basepoints are tangential points at $\{0,\mu_N\}$) of $\mathbb{P}^1-\{0,\mu_N,\infty\}$ satisfying the following properties:\\
(i) Elements of $V_N$ are compatible with the composition law on the motivic fundamental groupoids of $\mathbb{P}^1-\{0,\mu_N,\infty\}$;\\
(ii) Elements of $V_N$ fix $\mathrm{exp}(e_i)\in {}_i\Pi_i$ for $i\in \{0,\mu_N\}$;\\
(iii) Elements of $V_N$ are equivariant with the $\mu_N$-action on the motivic fundamental groupoids.
By proposition 5.11 in \cite{deligne}, the following map
\[
\xi:V_N\rightarrow {}_0\Pi_1, a\mapsto a({}_01_1)
\]
is an isomorphism of schemes and
\[
\mathrm{Lie}\; V_N=(\mathbb{L}(e_0,e_{\mu_N}),\{\,,\,\}).
\]
Here $\mathbb{L}(e_0,e_{\mu_N})$ is the free Lie algebra generated by the symbols $e_0, e_{\epsilon},\epsilon\in \mu_N$ and $\{\;,\;\}$ denotes the Ihara Lie bracket on $\mathbb{L}(e_0,e_{\mu_N})$.
The action of $\mathcal{U}^{\mathcal{MT}_N}$ on ${}_x\Pi_y,x,y\in \{0,\mu_N\}$ factors through $V_N$. As a result, there is a Lie algebra homomorphism:
\[
i:\mathfrak{g}_N\rightarrow \mathrm{Lie}\; V_N=\left(\mathbb{L}(e_0,e_{\mu_N}),\{\;,\;\} \right).
\]
The map $i$ is injective by the main results of Deligne \cite{del} for $N=2,3,4$.
For any element $w$ in $\mathbb{L}(e_0,\mu_N)$, let $depth(w)$ be the smallest number of total occurrences of $e_{\epsilon},\epsilon\in \mu_N$ in $w$, it induces a depth decreasing filtration $\mathcal{D}$ on $\mathbb{L}(e_0,e_{\mu_N})$:
\[
\mathcal{D}^r\mathbb{L}(e_0,e_{\mu_N})=\{w\in \mathbb{L}(e_0,e_{\mu_N});depth(w)\geq r\}.
\]
We write $E^{(n)}_{\epsilon}=\mathrm{ad}(e_0)^{n}e_{\epsilon}$ for short, $\forall \epsilon\in \mu_N$.
According to Section $3.11$ in \cite{del}, for $N=2$, the map $i$ satisfies:
\[
i(\sigma_1)=e_{-1},\eqno{(1)}
\]
\[
i(\sigma_{2n+1})=(1-2^{2n})E^{(2n)}_{-1}+2^{2n}E^{(2n)}_1+\mathrm{HDT},\forall n\geq 1.\eqno{(2)}
\]
For $N=3$, the map $i$ satisfies:
\[i(\sigma_1)=e_{\epsilon}+e_{\epsilon^{-1}},\eqno{(3)}
\]
\[
i(\sigma_{2n})=E^{(2n-1)}_{\epsilon}-E^{(2n-1)}_{\epsilon^{-1}}+\mathrm{HDT},\forall n\geq 1,\eqno{(4)}
\]
\[
i(\sigma_{2n+1})=(1-3^{2n})\left[E^{(2n)}_{\epsilon}+E^{(2n)}_{\epsilon^{-1}}\right]+2\cdot 3^{2n}E^{(2n)}_1+\mathrm{HDT},\forall n\geq 1.\eqno{(5)}
\]
For $N=4$, the map $i$ satisfies:
\[
\sigma_1=e_{\epsilon}+e_{\epsilon^{-1}}+2e_{-1},\eqno{(6)}
\]
\[
\sigma_{2n}=E^{(2n-1)}_{\epsilon}-E^{(2n-1)}_{\epsilon^{-1}}+\mathrm{HDT},\eqno{(7)}
\]
\[
\sigma_{2n+1}=(1-2^{2n})\left[E^{(2n)}_{\epsilon}+E^{(2n)}_{\epsilon^{-1}}\right]+2\cdot 2^{2n}\left(1-2^{2n} \right)E^{(2n)}_{-1}+2\cdot 2^{4n}E^{(2n)}_1+\mathrm{HDT}.\eqno{(8)}
\]
In the above formulas, HDT means the higher depth terms.
The motivic Lie algebra $\mathfrak{g}_N$ has an induced depth filtration $\mathcal{D}^r\mathfrak{g}_N$ from the injective map $i$. Since Ihara bracket is compatible with the depth filtration, we know that the depth-graded space
\[
\mathfrak{dg}_N=\bigoplus_{r\geq 1}\mathcal{D}^r\mathfrak{g}_N/\mathcal{D}^{r+1}\mathfrak{g}_N
\]
is a Lie algebra with induced Ihara Bracket. By \cite{del}, $\mathfrak{dg}_N$ is a free Lie algebra for $N=2,3,4$ with generators $\overline{i(\sigma_{2n-1})}, n\geq 1$ for $N=2$ and with generators $\overline{i(\sigma_{n})},n\geq 1$ for $N=3,4$, where the symbol $\overline{i(\sigma_n})$ means the depth one parts of $i(\sigma_n)$.
The action of $\mathrm{Lie}\;V$ on $\mathcal{O}({}_0\Pi_1)$ is compatible with the depth filtration. Since the expression of $i(\sigma_{2n+1})$ in $(\mathbb{L}(e_0,e_1,e_{-1}),\{\;,\,\})$ has canonical depth one parts, for $n\geq 0$, $\sigma_{2n+1}$ in $\mathfrak{g}_2=\mathrm{Lie}\;\mathcal{U}^{\mathcal{MT}_2}$ induces a well-defined derivation
\[
\partial_{2n+1}:gr_r^{\mathcal{D}}\mathcal{H}\rightarrow gr_{r-1}^{\mathcal{D}}\mathcal{H}.
\]
For $N=3,4, n\geq 1$, $\sigma_n$ in $\mathfrak{g}_N=\mathrm{Lie}\;\mathcal{U}^{\mathcal{MT}_N}$ also induces a derivation similarly
\[
\partial_{n}:gr_r^{\mathcal{D}}\mathcal{H}\rightarrow gr_{r-1}^{\mathcal{D}}\mathcal{H}.
\]
The explicit calculation of these derivations is very complicated. We now give the key idea to calculate these derivations explicitly, which is essentially the generalization of Brown's observation in \cite{depth}.
Since $\mathcal{O}({}_0\Pi_1)$ is an ind-object in the category $\mathcal{MT}_N$, under Tannakian correspondence there is an action of the motivic Lie algebra
\[
\mathfrak{g}_N\times \mathcal{O}({}_0\Pi_1)\rightarrow \mathcal{O}({}_0\Pi_1).
\]
Denote by $\mathfrak{h}_N=\mathrm{Lie}\;V_N=\left(\mathbb{L}(e_0,e_{\mu_N}),\{\;,\,\} \right)$. The action of $\mathfrak{g}_N$ on $\mathcal{O}({}_0\Pi_1)$ factors through the action of $\mathfrak{h}_N$ on $\mathcal{O}({}_0\Pi_1)$.
Denote by $\mathcal{U}\mathfrak{h}_N$ the universal enveloping algebra of $\mathfrak{h}_N$, then
\[
\mathcal{U}\mathfrak{h}_N\cong \left(\mathbb{Q}\langle e_0,e_{\mu_N}\rangle,\circ \right),
\]
where $\circ$ denotes the new product on $\mathbb{Q}\langle e_0,e_{\mu_N}\rangle$ which is transformed from the natural concatenation product on $\mathcal{U}\mathfrak{h}_N$.
By the same reason as Proposition $2.2$ in \cite{depth}, for any $a\in \mathfrak{h}$, any word sequence $w$ in $e_0,e_{\epsilon},\epsilon\in\mu_N$ and any $n\geq 0$, we have
\[
a\circ \left(e_0^ne_{\epsilon}w \right)=e_0^n\left[ \left([\epsilon](a) \right)e_{\epsilon}+e_{\epsilon}\left([\epsilon](a)\right)^*\right]w+e_0^ne_{\epsilon}\left( a\circ w\right),\epsilon\in \mu_N,
\]
where
\[
a\circ e_0^n=e_0^n a,\epsilon\in \mu_N,
\]
\[
\left(u_1 u_2\cdots u_n \right)^*=(-1)^nu_n\cdots u_2 u_1, u_i\in \{e_0,e_{\epsilon};\epsilon\in\mu_N \},
\]
\[
[\epsilon]\left(e_0^{n_1}e_{\epsilon_1}e_0^{n_2}e_{\epsilon_2}\cdots e_0^{n_r}e_{\epsilon_r}e_0^{n_{r+1}} \right)=e_0^{n_1}e_{\epsilon\epsilon_1}e_0^{n_2}e_{\epsilon\epsilon_2}\cdots e_0^{n_r}e_{\epsilon\epsilon_r}e_0^{n_{r+1}},\epsilon,\epsilon_i\in \mu_N.
\]
From the correspondence between unipotent algebraic group and nilpotent Lie algebra (for example, see Section $3$ in \cite{li}), we know that for $a\in \mathfrak{h}_N$, the natural action of $a$ on $\mathcal{O}({}_0\Pi_1)$:
\[
\mathcal{O}({}_0\Pi_1)=\mathbb{Q}\langle e^0,e^{\mu_N}\rangle \xrightarrow{a} \mathcal{O}({}_0\Pi_1)=\mathbb{Q}\langle e^0,e^{\mu_N}\rangle,
\]
\[
x\mapsto a(x),
\]
is dual to the following action of $a$ on $\mathcal{U}\mathfrak{h}$:
\[
\mathcal{U}\mathfrak{h}_N=\mathbb{Q}\langle e_0,e_{\mu_N} \rangle\xrightarrow{a}\mathcal{U}\mathfrak{h}_N=\mathbb{Q}\langle e_0,e_{\mu_N} \rangle,
\]
\[
y\mapsto a\circ y.
\]
By the definition of $\mathcal{H}$ and $\partial_{2n+1}$, we have the following commutative diagram
\[
\xymatrix{
gr_r^{\mathcal{D}}\mathbb{Q}\langle e^0,e^{\mu_N}\rangle \ar@{->>}[d] \ar[r]^{\overline{\partial_{n}}} & gr_{r-1}^{\mathcal{D}}\mathbb{Q}\langle e^0,e^{\mu_N}\rangle \ar@{->>}[d] \\
gr_r^{\mathcal{D}}\mathcal{H} \ar[r]^{\partial_{n}} & gr_{r-1}^{\mathcal{D}}\mathcal{H} , }
\]
where $\overline{\partial_{n}}$ is the depth-graded version of the action of $i(\sigma_{n})$ on $\mathbb{Q}\langle e^0,e^{\mu_N}\rangle$.
Let $\delta\binom{x}{y}$ be the function of $x,y\in \mathbb{C}$ which satisfies
\[
\delta\binom{x}{y}=
\begin{cases}
1, &x=y;\\
0, &x\neq y.
\end{cases}
\]
Denote by $\mathfrak{g}_N^{ab}=\mathfrak{g}_N/[\mathfrak{g}_N,\mathfrak{g}_N]$ and $\left(\mathfrak{g}_N^{ab} \right)^{{\vee}}$ be its compact dual. For $N=2$, let $$f_{2n+1},n\geq 0,\in \left(\mathfrak{g}_N^{ab} \right)^{{\vee}}$$ be the dual basis of the images of $\sigma_{2n+1},n\geq 0$ in $\mathfrak{g}_N^{ab}$. For $N=3,4$, let $$f_{n},n\geq 1,\in \left(\mathfrak{g}_N^{ab} \right)^{{\vee}}$$ be the dual basis of the images of $\sigma_{n},n\geq 1$ in $\mathfrak{g}_N^{ab}$.
For $N=2$, there is a well-defined map
\[
\partial:gr_r^{\mathcal{D}}\mathcal{H}\rightarrow \left(\mathfrak{g}_2^{ab} \right)^{{\vee}}\otimes gr_{r-1}^{\mathcal{D}}\mathcal{H},\partial=\sum_{n\geq 0}f_{2n+1}\otimes \partial_{2n+1}.
\]
For $N=3,4$, there is a well-defined map
\[
\partial:gr_r^{\mathcal{D}}\mathcal{H}\rightarrow \left(\mathfrak{g}_N^{ab} \right)^{{\vee}}\otimes gr_{r-1}^{\mathcal{D}}\mathcal{H},\partial=\sum_{n\geq 1}f_{n}\otimes \partial_{n}.
\]
Now we have
\begin{prop}\label{inj}
For $r\geq 2$, the map $\partial$ is injective for $N=2,3,4$.
\end{prop}
\noindent{\bf Proof}:
By exactly the same method in Section $2.3$,\cite{brown}, it follows that
\[
\mathcal{H}\cong \mathcal{O}\left(\mathcal{U}^{\mathcal{MT}_N}\right)[t]
\]
as a $\mathfrak{g}_N$-module, where $t$ is a weight $\begin{cases} 2, &N=2\\1, &N=3,4\end{cases}$, depth $1$ element with trivial action of $\mathfrak{g}_N$. Furthermore, $t^n,n\geq 1$ are all depth $1$ elements.
As a result,
\[
gr_r^{\mathcal{D}}\mathcal{H}\cong gr_r^{\mathcal{D}}\mathcal{O}\left(\mathcal{U}^{\mathcal{MT}_N}\right)\oplus \bigoplus_{n\geq 1} gr_{r-1}^{\mathcal{D}}\mathcal{O}\left(\mathcal{U}^{\mathcal{MT}_N}\right)t^n.
\]
Be ware that $gr_r^{\mathcal{D}}\mathcal{O}(\mathcal{U}^{\mathcal{MT}_N})$ is dual to $gr_{\mathcal{D}}^r\mathcal{U}\mathfrak{g}_N$ and the decreasing depth filtration on $\mathcal{U}\mathfrak{g}_N$ is induced by the depth filtration on $\mathfrak{g}_N$.
Thus it suffices to prove that $\partial|_{gr_r^{\mathcal{D}}\mathcal{O}\left(\mathcal{U}^{\mathcal{MT}_N}\right)}$ is injective. Since the depth-graded motivic Lie algebra $\mathfrak{dg}$ is a free Lie algebra with generators which are all in the depth one parts \cite{deligne}.
By the correspondence between nilpotent Lie algebra and unipotent algebraic group, $\partial|_{gr_r^{\mathcal{D}}\mathcal{O}\left(\mathcal{U}^{\mathcal{MT}_N}\right)}$ is injective.
$\hfill\Box$\\
\section{Main results} \label{mainsection}
Now we are ready to prove our main results:
\begin{Thm}\label{mmm}
$(i)$ For $N=2,r\geq 1$, $\mathrm{dim}_{\mathbb{Q}}\,gr_r^{\mathcal{D}}\mathcal{H}_r^{(1)}=1$ and $gr_r^{\mathcal{D}}\mathcal{H}^{(1)}_r$ is generated by
\[
\zeta^{\mathfrak{m}}\dbinom{\overbrace{1,1,\cdots,1,\;\;1}^r}{1,1,\cdots,1,-1}
\]
as a $\mathbb{Q}$-linear space;\\
$(ii)$ For $N=3,4,r\geq 1$, $\mathrm{dim}_{\mathbb{Q}}\,gr_r^{\mathcal{D}}\mathcal{H}_r^{(1)}=2$ and $gr_r^{\mathcal{D}}\mathcal{H}^{(1)}_r$ is generated by \[
\zeta^{\mathfrak{m}}\dbinom{\overbrace{1,1,\cdots,1,1}^r}{1,1,\cdots,1,\epsilon},\zeta\dbinom{\overbrace{1,\;1,\;\cdots\;,1,\,\,1}^r}{1,1,\;\cdots,1,\;\epsilon^{-1}}
\]
as a $\mathbb{Q}$-linear space.
\end{Thm}
\noindent{\bf Proof}: For $r=1$, it is clear that (i) and (ii) are true by definition.
Since the map $\partial$ is injective, from Proposition \ref{inj} and Lemma \ref{below} below, it follows that $\partial_1$ is injective for $\mu_2,\mu_3$ and $\mu_4$. Thus we have
\[
\mathrm{dim}_{\mathbb{Q}}gr_r^{\mathcal{D}}\mathcal{H}_r^{(1)}=\mathrm{dim}_{\mathbb{Q}}\underbrace{\partial_1\circ \partial_1\circ\cdots\circ\partial_1}_{r-1}\left( gr_r^{\mathcal{D}}\mathcal{H}^{(1)}\right).
\]
From the explicit formulas of $\partial_1$ in Lemma \ref{below}, we have for $N=2$,
\[
\underbrace{\partial_1\circ \partial_1\circ\cdots\circ\partial_1}_{r-1}\left(\zeta^{\mathfrak{m}}\dbinom{\overbrace{1,1,\cdots,1,\;\;1}^r}{1,1,\cdots,1,-1} \right)=\zeta^{\mathfrak{m}}\binom{1}{-1}
\]
and for N=3,4,
\[
\underbrace{\partial_1\circ \partial_1\circ\cdots\circ\partial_1}_{r-1}\left(\zeta^{\mathfrak{m}}\dbinom{\overbrace{1,1,\cdots,1,\;\;1}^r}{1,1,\cdots,1,\epsilon^{\pm 1}} \right)=\zeta^{\mathfrak{m}}\binom{1}{\epsilon^{\pm1}}.
\]
Thus the theorem is proved. $\hfill\Box$\\
From Theorem \ref{mmm}, by the period map $per:\mathcal{H}\rightarrow \mathbb{C}$ we get
Theorem \ref{gene} immediately.
\begin{lem}\label{below}
(i) For $N=2,n\geq 1$, $\partial_{2n+1}\left(gr_1^{\mathcal{D}}\mathcal{H}^{(1)} \right)=0$. For $e^{i_1},\cdots, e^{i_s}\in \{\pm1\}$, we have
\[
\begin{split}
&\;\;\;\;\overline{\partial_1}\left(e^{i_1}e^{i_2}\cdots e^{i_s} \right)\\
&=\delta\binom{i_1i_2}{-1}\left(e^{-i_1}-e^{i_1}\right)e^{i_3}\cdots e^{i_s}+\cdots+\delta\binom{i_{s-1}i_s}{-1}e^{i_1}\cdots e^{i_{s-2}}\left(e^{-i_{s-1}}-e^{i_{s-1}}\right)\\
&+\delta\binom{i_s}{-1}e^{i_1}e^{i_2}\cdots e^{i_{s-1}}.\\
\end{split}
\]
(ii) For $N=3,n\geq 2$, $\partial_{n}\left(gr_1^{\mathcal{D}}\mathcal{H}^{(1)}\right)=0$. For $e^{i_1},\cdots, e^{i_s}\in \mu_3$, we have
\[
\begin{split}
&\;\;\;\;\overline{\partial_1}\left(e^{i_1}e^{i_2}\cdots e^{i_s}\right)\\
&=\left[ \delta\binom{i_1}{i_2\epsilon}+\delta\binom{i_1}{i_2\epsilon^{-1}}\right]e^{i_2}\cdots e^{i_s}+\cdots+\left[\delta\binom{i_{s-1}}{i_s\epsilon}+\delta\binom{i_{s-1}}{i_s\epsilon^{-1}} \right]e^{i_1}\cdots e^{i_{s-2}}e^{i_s} \\
&+\left[\delta\binom{i_s}{\epsilon}+\delta\binom{i_s}{\epsilon^{-1}} \right]e^{i_1}\cdots e^{i_{s-1}}-\left[\delta\binom{i_1}{i_2\epsilon}e^{i_2\epsilon}+\delta\binom{i_1}{i_2\epsilon^{-1}}e^{i_2\epsilon^{-1}} \right]e^{i_3}\cdots e^{i_s}\\
&-\cdots-e^{i_1}\cdots e^{i_{s-2}}\left[\delta\binom{i_{s-1}}{i_s\epsilon}e^{i_s\epsilon}+\delta\binom{i_{s-1}}{i_s\epsilon^{-1}}e^{i_s\epsilon^{-1}} \right]. \\
\end{split}
\]
(iii) For $N=4, n\geq 2$, $\partial_{n}\left(gr_1^{\mathcal{D}}\mathcal{H}^{(1)}\right)=0$. For $e^{i_1},\cdots, e^{i_s}\in \mu_4$, we have
\[
\begin{split}
&\;\;\;\;\overline{\partial_1}\left(e^{i_1}e^{i_2}\cdots e^{i_s}\right)\\
&=2\delta\binom{i_1i_2}{-1}\left(e^{-i_1}-e^{i_1}\right)e^{i_3}\cdots e^{i_s}+\cdots+2\delta\binom{i_{s-1}i_s}{-1}e^{i_1}\cdots e^{i_{s-2}}\left(e^{-i_{s-1}}-e^{i_{s-1}}\right)\\
&+\left[\delta\binom{i_s}{\epsilon}+2\delta\binom{i_s}{-1}+\delta\binom{i_s}{\epsilon^{-1}} \right]e^{i_1}\cdots e^{i_{s-1}}-\left[\delta\binom{i_1}{i_2\epsilon}e^{i_2\epsilon}+\delta\binom{i_1}{i_2\epsilon^{-1}}e^{i_2\epsilon^{-1}} \right]e^{i_3}\cdots e^{i_s}\\
&-\cdots-e^{i_1}\cdots e^{i_{s-2}}\left[\delta\binom{i_{s-1}}{i_s\epsilon}e^{i_s\epsilon}+\delta\binom{i_{s-1}}{i_s\epsilon^{-1}}e^{i_s\epsilon^{-1}} \right]. \\
\end{split}
\]
\end{lem}
\noindent{\bf Proof}:
(i) From the commutative diagram in Section \ref{mga}, to prove that $$\partial_{2n+1}\left(gr_1^{\mathcal{D}}\mathcal{H}^{(1)} \right)=0, \forall n\geq 1$$
it suffices to prove that
\[
\overline{\partial_{2n+1}}\left(\mathbb{Q}\langle e^{\mu_N}\rangle\right)=0.
\]
Here $\mathbb{Q}\langle e^{\mu_n}\rangle$ is the sub-algebra of $\mathbb{Q}\langle e^0,e^{\mu_N}\rangle$ generated by $e^{\epsilon_1}e^{\epsilon_2}\cdots e^{\epsilon_r},\epsilon_i\in \mu_N,r\geq 1$.
By considering the action of $\overline{\sigma_{2n+1}}$ on $\mathcal{U}\mathfrak{h}=\mathbb{Q}\langle e_0,e_{\mu_N}\rangle$, from Section \ref{mga}, it is enough to show that the terms
\[
e_{\xi_1}e_{\xi_2}\cdots e_{\xi_{r+1}}, \xi_1,\cdots,\xi_{r+1}\in \mu_N
\]
have trivial coefficients in
\[
\overline{\sigma_{2n+1}}\circ e_{\epsilon_1}e_{\epsilon_2}\cdots e_{\epsilon_r}, \forall \epsilon_1,\cdots,\epsilon_r\in \mu_N
\]
for all $r\geq 0$. This follows from the definition of $\circ$ and $\overline{\sigma_{2n+1}}$. While the formula for $\overline{\partial_1}$ follows from that
\[
\begin{split}
&\;\;\;\;e_{-1}\circ\left(e_{i_1}e_{i_2}\cdots e_{i_r}\right)\\
&=\left(e_{-i_1}e_{i_1}-e_{i_1}e_{-i_1} \right)e_{i_2}\cdots e_{i_r}+e_{i_1}\left( e_{-i_2}e_{i_2}-e_{i_2}e_{-i_2}\right)e_{i_3}\cdots e_{i_r}+\cdots \\
&+e_{i_1}\cdots e_{i_{r-1}}\left(e_{-i_r}e_{i_r} -e_{i_r}e_{-i_r}\right)+e_{i_1}\cdots e_{i_r}e_{-1}.\\
\end{split}
\]
The proofs of (ii) and (iii) are essentially the same as (i). $\hfill\Box$\\
Since
\[
\begin{split}
&\zeta\dbinom{1,1,\cdots,1,1}{1,1,\cdots,1,\epsilon}
= dch(\overbrace{e^{\epsilon^{-1}}e^{\epsilon^{-1}}\cdots e^{\epsilon^{-1}}}^r )
= \frac{1}{r!}dch(\overbrace{e^{\epsilon^{-1}}\rotatebox{90}{$\rotatebox{180}{$\exists$}$}\,e^{\epsilon^{-1}}\rotatebox{90}{$\rotatebox{180}{$\exists$}$}\,\cdots \rotatebox{90}{$\rotatebox{180}{$\exists$}$}\,e^{\epsilon^{-1}}}^r )
=\frac{1}{r!}\left( \zeta\binom{1}{\epsilon} \right)^r,\\
\end{split}
\]
we have
\[
\zeta\dbinom{\overbrace{1,1,\cdots,1,\,1}^r}{1,1,\cdots,1,\epsilon}=\frac{(-1)^r}{r!}\left[\mathrm{log}\;(1-\epsilon) \right]^r.
\]
From Lemma \ref{below}, for any
\[
\zeta\binom{1,\;\cdots,1}{\epsilon_1,\cdots,\epsilon_r}
\]
one can use the formulas for $\overline{\partial_1}$ inductively to calculate the numbers $$a_{\epsilon_1,\cdots,\epsilon_r},b_{\epsilon_1,\cdots,\epsilon_r}, c_{\epsilon_1,\cdots,\epsilon_r}$$ in the introduction.
\begin{rem}
In this paper we only study the depth-graded version of unit cylotomic multiple zeta values. In fact the structure of $\mathcal{H}^{(1)}$ is related to the structure of motives of the motivic fundamental groupoid of $\mathbb{P}^1-\{\mu_N,\infty\}$ from point $0$ to the tangential point at $1$. It is still not clear at present.
\end{rem}
| 36,822
|
\section{Hierarchy}\label{s.hier}
\subsection{Hierarchical codes} \label{ss.hier}
The present section may seem a long distraction from the main course of
exposition but many readers of this paper may have difficulty
imagining an infinite hierarchical structure built into a configuration of
a cellular automaton.
Even if we see the possibility of such structures it is important to
understand the great amount of flexibility that exists while building it.
Formally, a hiearchy will be defined as a ``composite code''.
Though no decoding will be mentioned in this subsection, it is still
assumed that to all codes $\fg_{*}$ mentioned, there belongs a decoding
function $\fg^{*}$ with $\fg^{*}(\fg_{*}(x))=x$.
\subsubsection{Composite codes}
Let us discuss the hierarchical structure arising in an amplifier.
If $\fg,\psi$ are two codes then $\fg\circ\psi$ is defined by
$(\fg\circ\psi)_{*}(\xi)=\fg_{*}(\psi_{*}(\xi))$ and
$(\fg\circ\psi)^{*}(\zg)=\psi^{*}(\fg^{*}(\zg))$.
It is assumed that $\xi$ and $\zg$ are here configurations of the
appropriate cellular automata, i.e.~the cell body sizes are in the
corresponding relation.
The code $\fg\circ\psi$\glo{"+@$\circ$} is called the
\df{composition}\index{code!composition} of $\fg$ and $\psi$.
For example, let $M_{1},M_{2},M_{3}$ have cell body sizes $1,31,31^{2}$
respectively.
Let us use the code $\fg$ from Example ~\ref{x.block-code}.
The code $\fg^{2}=\fg\circ\fg$ maps each cell $c$ of $M_{3}$ with body
size $31^{2}$ into a ``supercolony''\index{supercolony} of $31\cdot 31$
cells of body size 1 in $M_{1}$.
Suppose that $\zg=\fg_{*}^{2}(\xi)$ is a configuration obtained by
encoding from a lattice configuration of body size $31^{2}$ in $M_{3}$,
where the bases of the cells are at positions $-480 + 31^{2} i$.
(We chose -480 only since $480=(31^{2}-1)/2$ but we could have chosen any
other number.)
Then $\zg$ can be broken up into colonies of size $31$ starting at
any of those bases.
Cell 55 of $M_{1}$ belongs to the colony with base $47=-480+17\cdot 31$
and has address 8 in it.
Therefore the address field of $\zg(55)$ contains a binary
representation of 8.
The last bit of this cell encodes the 8-th bit the of cell (with base)
47 of $M_{2}$ represented by this colony.
If we read together all 12 bits represented by the $\Info$ fields of
the first 12 cells in this colony we get a state $\zg^{*}(47)$ (we count
from 0).
The cells with base $-15+31j$ for $j\in\ZZ$ with states
$\zg^{*}(-15+31j)$ obtained this way are also broken up into colonies.
In them, the first 5 bits of each state form the address and the
last bits of the first 12 cells, when put together, give back the
state of the cell represented by this colony.
Notice that these 12 bits were really drawn from $31^{2}$ cells of
$M_{1}$.
Even the address bits in $\zg^{*}(47)$ come from different cells of
the colony with base 47.
Therefore the cell with state $\zg(55)$ does not contain information
allowing us to conclude that it is cell 55.
It only ``knows'' that it is the 8-th cell within its own colony
(with base 47) but does not know that its colony has address 17 within
its supercolony (with base $-15\cdot 31$) since it has at most one
bit of that address.
\begin{figure}
\setlength{\unitlength}{0.2mm}
\[
\begin{picture}(600,150)
\put(0, 0){\framebox(600, 150){}}
\put(610, 0){\makebox(0,20)[l]{$\var{Prog}$}}
\put(0,20){\line(1,0){600}}
\put(610,20){\makebox(0,70)[l]{$\var{Info}$}}
\put(0, 20){\makebox(80,70){$\var{Prog}^{*}$}}
\put(80, 20){\line(0, 1){70}}
\put(80, 20){\makebox(280,70){$\var{Info}^{*}$}}
\put(360, 20){\line(0, 1){70}}
\put(360, 20){\makebox(80,70){$\var{Worksp}^{*}$}}
\put(440, 20){\line(0, 1){70}}
\put(440, 20){\makebox(80,70){$\var{Addr}^{*}$}}
\put(520, 20){\line(0, 1){70}}
\put(520, 20){\makebox(80,70){$\var{Age}^{*}$}}
\put(0,90){\line(1,0){600}}
\put(610,90){\makebox(0,20)[l]{$\var{Worksp}$}}
\put(0, 110){\line(1,0){600}}
\put(610,110){\makebox(0,20)[l]{$\var{Addr}$}}
\put(0, 130){\line(1,0){600}}
\put(610,130){\makebox(0,20)[l]{$\var{Age}$}}
\multiput(0,0)(20,0){30}{\line(0,1){3}}
\end{picture}
\]
\caption{Fields of a cell simulated by a colony}
\label{f.fields}
\end{figure}
\subsubsection{Infinite composition}
A code can form composition an arbitrary number of times with itself
or other codes.
In this way, a hierarchical, i.e.~highly nonhomogenous, structure can
be defined using cells that have only a small number of states.
A \df{hierarchical code}\index{code!hierarchical} is given by a sequence
\glo{s.cap.s@$\SS_{k}$}
\glo{q.cap@$Q_{k}$}
\glo{f.greek@$\fg_{k*}$}
\begin{equation}\label{e.hier-code}
(\SS_{k},Q_{k},\fg_{k*})_{k\ge 1}
\end{equation}
where $\SS_{k}$ is an alphabet, $Q_{k}$ is a positive integer and
$\fg_{k*}:\SS_{k+1}\to\SS_{k}^{Q_{k}}$ is an encoding function.
Since $\SS_{k}$ and $Q_{k}$ are implicitly defined by $\fg_{k*}$ we can
refer to the code as just $(\fg_{k})$.
We will need a composition $\fg_{1*}\circ\fg_{2*}\circ\cdots$ of
the codes in a hierarchical code since the the definition of the
initial configuration for $M_{1}$ in the amplifier depends on all codes
$\fg_{i*}$.
What is the meaning of this?
We will want to compose the codes ``backwards'', i.e.~in such a
way that from a configuration $\xi^{1}$ of $M_{1}$ with cell body size 1,
we can decode the configuration $\xi^{2}=\fg_{1}^{*}(\xi^{1})$ of $M_{2}$ with
cell body size $B_{2}=Q_{1}$, configuration $\xi^3=\fg_{2}^{*}(\xi^{2})$, of
$M_{3}$ with body size $B_{3}=Q_{1}Q_{2}$, etc.
Such constructions are not unknown, they were used e.g.~ to define
``Morse sequences'' with applications in group theory as well in the theory
of quasicrystals (\cite{ItkisLevin94,Radin91}).
Let us call a sequence $a_{1},a_{2},\ldots$\glo{a@$a_{k}$} with
$0\le a_{k} < Q_{k}$
\df{non-degenerate for $Q_{1},Q_{2},\ldots$}\index{non-degenerate} if there
are infinitely many $k$ with $a_{k}>0$ and infinitely many $k$ with
$a_{k}<Q_{k}-1$.
The pair of sequences
\begin{equation}\label{e.block-param}
(Q_{k},a_{k})_{k=1}^{\infty}
\end{equation}
with non-degenerate $a_{k}$ will be called a \df{block
frame}\label{frame!block} of our hierarchical codes.
All our hierarchical codes will depend on some fixed block frame,
$((Q_{k},a_{k}))$, but this dependence will generally not be shown in the
notation.
\begin{remarks}\
\begin{enumerate}
\item The construction below does not need the generality of an arbitrary
non-degenerate sequence: we could have $a_{k}=1$ throughout.
We feel, however, that keeping $a_{k}$ general makes the construction
actually more transparent.
\item It is easy to extend the construction to degenerate sequences.
If e.g.~$a_{k}=0$ for all but a finite number of $k$ then the process
creates a configuration infinite in right direction, and a similar
construction must be added to attach to it a configuration infinite in the
left direction.
\end{enumerate}
\end{remarks}
For a block frame $((Q_{k},a_{k}))$, a finite or infinite sequence
$(s_{1},a_{1}),(s_{2},a_{2}),\ldots$ will be called
\df{fitted}\index{fitted sequence} to the hierarchical code $(\fg_{k*})$ if
\[
\fg_{k*}(s_{k+1})(a_{k})=s_{k}
\]
holds for all $k$.
For a finite or infinite space size $N$, let
\glo{b.cap@$B_{k}$}
\glo{k.cap@$K(N)$}
\glo{o@$o_{k}$}
\glo{c.cap@$C_{k}(x)$}
\begin{align}
\nn B_{1} &= 1,
\\\label{e.B_{k}} B_{k} &= Q_{1}\cdots Q_{k-1} \for k>1,
\\\label{e.K} K = K(N) &= \sup_{B_{k}<N} k+1,
\\\nn o_{k} &= -a_{1}B_{1} - \cdots - a_{k-1}B_{k-1},
\\\label{e.C_{k}-x} C_{k}(x) &= o_{k} + xB_{k},
\end{align}
The following properties are immediate:
\begin{equation}\label{e.C_{k}-prop}
\begin{aligned}
o_{1} &= C_{1}(0) = 0,
\\ o_{k} &= o_{k+1}+a_{k}B_{k},
\\ 0 &\in o_{k}+[0,B_{k}-1].
\end{aligned}
\end{equation}
\begin{proposition}\label{p.fitted}
Let us be given a fitted sequence $(s_{k},a_{k})_{k\ge 1}$.
Then there are configurations $\xi^{k}$ of $M_{k}$ over $\ZZ$ such that
for all $k\ge 1$ we have
\begin{align*}
\fg_{k*}(\xi^{k+1}) &= \xi^{k},
\\ \xi^{k}(o_{k}) &= s_{k}.
\end{align*}
\end{proposition}
The infinite code we are interested in is $\xi^{1}$.
Note that in this construction, $s_{k}$ is the state of the site
$o_{k}$ in configuration $\xi^{k}$ whose body contains the site 0.
This site has address $a_{k}$ in a colony with base $o_{k+1}$ in
$\xi^{k+1}$.
\begin{proof}
Let
\glo{x.greek@$\xi^{i}_{k}$}
\begin{equation}\label{e.xi^{k}_{k}}
\xi^{k}_{k}
\end{equation}
be the configuration of $M_{k}$ which has state $s_{k}$ at site $o_{k}$
and arbitrary states at all other sites $C_{k}(x)$, with the following
restriction in case of a finite space size $N$.
Let $\xi^{k}_{k}\ne\Vac$ only for $k\le K=K(N)$ and
$\xi^{K}_{K}(o_{K}+z)\ne\Vac$ only if
\begin{equation}\label{e.xi^{K}.defined}
0 \le zB_{K} < N.
\end{equation}
Let
\begin{equation}\label{e.xi-i-k}
\xi^{i}_{k} = \fg_{i*}(\fg_{(i+1)*}(\cdots \fg_{(k-1)*}(\xi^{k}_{k})\cdots))
\end{equation}
for $k>i\ge 1$.
We have
\begin{equation}\label{e.fitted.ext}
\xi^{k}_{k+1}(o_{k})=\fg_{k*}(\xi^{k+1}_{k+1}(o_{k+1}))(a_{k})
= \xi^{k}_{k}(o_{k})
\end{equation}
where the first equation comes by definition, the second one by
fittedness.
The encoding conserves this relation, so the partial configuration
$\xi^{i}_{k+1}(o_{k+1}+[0,B_{k+1}-1])$ is an
extension of $\xi^{i}_{k}(o_{k}+[0,B_{k}-1])$.
Therefore the limit $\xi^{i}=\lim_{k} \xi^{i}_{k}$ exists for each $i$.
Since $(a_{k})$ is non-degenerate the limit extends over the whole
set of integer sites.
\end{proof}
Though $\xi^{1}$ above is obtained by an infinite process of encoding,
no infinite process of decoding is needed to yield a single
configuration from it: at the $k$-th stage of the decoding, we get a
configuration $\xi^{k}$ with body size $B_{k}$.
\subsubsection{Controlling, identification}\label{sss.controlling}
The need for some freedom in constructing infinite fitted sequences
leads to the following definitions.
For alphabet $\SS$ and field $\F$ let
\glo{.@$\SS.\F$}
\[
\SS.\F = \setof{w.\F: w\in \SS}.
\]
Then, of course, $\nm{\SS.\F}=|\F|$.
Let $D=\{d_{0},\ldots,d_{|D|-1}\}\sbs [0,Q-1]$ be a set of addresses
with $d_{i}<d_{i+1}$.
For a string $s$, let
\glo{.@$s(D).\F$}
\[
s(D).\F
\]
be the string of values $(s(d_{0}).\F,\ldots,s(d_{|D|-1}).\F)$ so that
\[
s.\F =s([0,Q-1]).\F.
\]
Field $\F$ \df{controls}\index{controlling} an address $a$ in code
$\fg_{*}$ \df{via} function $\gg:\SS_{1}.\F\to\SS_{1}$\glo{g.greek@$\gg$}
if
\begin{cjenum}
\item For all $r\in\SS_{1}.\F$ there is an $s$ with $\fg_{*}(s)(a).\F=r$;
in other words, $\fg_{*}(s)(a).\F$ runs through all possible values for
this field as $s$ varies.
\item For all $s$ we have $\fg_{*}(s)(a) = \gg(\fg_{*}(s)(a).\F)$;
in other words, the field $\fg_{*}(s)(a).\F$ determines all the other
fields of $\fg_{*}(s)(a)$.
\end{cjenum}
From now on, in the present subsection, whenever we denote a field
by $\F^{k}$ and a code by $\fg_{k*}$ we will implicitly assume that
$\F^{k}$ controls address $a_{k}$ in $\fg_{k*}$ unless we say otherwise.
(The index $k$ in $\F^{k}$ is not an exponent.)
Suppose that fields $\F^{1},\F^{2}$ are defined for cellular automata
$M_{1}$ and $M_{2}$ between which the code $\fg$ with blocksize $Q$ is
given.
Suppose that set $D$ satisfies $|D| = |\F^{2}|/|\F^{1}|$.
We say that in $\fg_{*}$, field $\F^{1}$ over $D$ is
\df{identified with}\index{field!identification} $\F^{2}$ if in any codeword
$\w=\fg_{*}(s)$, the string
$\w(d_{0}).\F^{1}\cc\cdots\cc \w(d_{|D|-1}).\F^{1}$ is identical to
$s.\F^{2}$.
Conversely, thus
$\w(d_{i}).\F^{1}=s.\F^{2}([i|\F^{1}|,(i+1)|\F^{1}|-1])$.
The identification of $\F^{2}$ with $\F^{1}$ over $D$ implies that if a
simulation has error-correction in $\F^{1}$ over $D$ then the information
restored there is $s.\F^{2}$.
\begin{example}\label{x.control-identif}
Consider the simulation outlined in Subsection ~\ref{ss.ftol-block}, and
let us call the encoding $\fg_{*}$.
Let $\F^{1}$ be the $\Info$ field of $M_{1}$: we denote it by
$\Info^{1}$.
Assume further that the simulated medium $M_{2}$ is of a similar type, so
it has an $\Info^{2}$ field.
Assume for simplicity
\[
6\mid Q,\qq |\Info^{1}| = 2,\qq |\Info^{2}| = Q/3.
\]
The $\Info^{1}$ field of cells in interval $[0,Q/3-1]$ of a simulating
colony represents the whole state of a simulated cell of $M_{2}$.
The $\Info^{1}$ field on $[Q/3,Q-1]$ is only used for redundancy.
Only a segment of the $\Info^{1}$ field of $[0,Q/3-1]$, say the one on
$[0,Q/6-1]$ is used to represent $\Info^{2}$ of the simulated cell.
The rest is used for encoding the other fields.
Hence $\Info^{1}$ on $[0,Q/6-1]$ is identified with $\Info^{2}$ in our
code $\fg_{*}$.
Let $s'=\fg_{*}(s)(1)$ be the state of the cell with address 1 of a
colony of $M_{1}$ which is the result of encoding state $s$ of $M_{2}$.
Let $s'.\Info^{1}$ be the third and fourth bits of $s.\Info^{2}$,
$s'.\Addr=1$, and $s'.\F=0$ for all fields different from these two.
Then $\Info^{1}$ controls address 1 in the code $\fg_{*}$.
\end{example}
\begin{figure}
\setlength{\unitlength}{0.2mm}
\[
\begin{picture}(600,150)
\put(0, 0){\framebox(600, 150){}}
\put(610, 0){\makebox(0,20)[l]{$\var{Prog}^{1}$}}
\put(0,20){\line(1,0){600}}
\put(610,20){\makebox(0,70)[l]{$\var{Info}^{1}$}}
\put(0, 20){\makebox(80,70){$\var{Prog}^{2}$}}
\put(80, 20){\line(0, 1){70}}
\put(80, 20){\makebox(280,70){$\var{Info}^{2}$}}
\put(360, 20){\line(0, 1){70}}
\put(360, 20){\makebox(80,70){$\var{Worksp}^{2}$}}
\put(440, 20){\line(0, 1){70}}
\put(440, 20){\makebox(80,70){$\var{Addr}^{2}$}}
\put(520, 20){\line(0, 1){70}}
\put(520, 20){\makebox(80,70){$\var{Age}^{2}$}}
\put(0,90){\line(1,0){600}}
\put(610,90){\makebox(0,20)[l]{$\var{Worksp}^{1}$}}
\put(0, 110){\line(1,0){600}}
\put(610,110){\makebox(0,20)[l]{$\var{Addr}^{1}$}}
\put(0, 130){\line(1,0){600}}
\put(610,130){\makebox(0,20)[l]{$\var{Age}^{1}$}}
\put(80,0){\dashbox{5}(20, 150){}}
\put(80,0){\makebox(20, 20){0}}
\multiput(0,0)(20,0){30}{\line(0,1){3}}
\end{picture}
\]
\caption[Control and identification]{Assume that in the code of this
figure, $\protect\var{Addr}$, $\protect\var{Age}$ and
$\protect\var{Worksp}$ are constant and $\protect\var{Prog}^{1}$ has always
0 in its position 4.
Then address 4 is controlled by $\protect\var{Info}^{1}$, and
$\protect\var{Info}^{2}$ is identified with $\protect\var{Info}^{1}$ over
the addresses 4-17.}
\label{f.control}
\end{figure}
If for each address $a$, the field $\F^{1}$ over $\{a\}$ is identified
with $\F^{2}$ then we say that $\F^{2}$ is \df{broadcast}\index{broadcast}
to $\F^{1}$ (since this means that the code copies the value of $s.\F^{2}$
into the $\F^{1}$ field of each cell of $\fg_{*}(s)$).
Let us be given
\glo{ps.greek.cap@$\Psi$}
\begin{equation}\label{e.primi-shared}
\Psi=((\SS_{k},Q_{k},\fg_{k*},\F^{k},\gg_{k},a_{k}) : k\ge 1)
\end{equation}
where $1\le a_{k}\le Q_{k}-2$, such that in code $\fg_{k*}$,
\begin{cjenum}
\item $\F^{k}$ over $\{a_{k}\}$ is identified with $\F^{k+1}$;
\item $\F^{k}$ controls address $a_{k}$ for $\fg_{k*}$ via $\gg_{k}$;
\end{cjenum}
Such a system of fields $\F^{k}$ will be called a \df{primitive shared
field}\index{field!shared!primitive} for the hierarchical code
$(\fg_{k*})$.
If also each code $\fg_{k*}$, broadcasts $\F^{k+1}$ into $\F^{k}$ then we
will say that the fields $\F^{k}$ form a \df{broadcast
field}\index{field!broadcast}.
Note that the field still controls only a single address $a_{k}$.
The $\fld{Main-bit}$ field mentioned in Subsection ~\ref{ss.col} would be
an example.
\begin{proposition}\label{p.fitted-constr}
For any hierarchical code with primitive shared field given as in
~\eqref{e.primi-shared} above, for all possible values
$u_{1}\in\SS_{1}.\F^{1}$ the infinite sequence $(s_{k},a_{k})_{k\ge 1}$ with
$s_{k}=\gg_{k}(u_{1})$ is fitted.
\end{proposition}
The proof is immediate from the definitions.
Let us denote the configurations $\xi^{1},\xi^{1}_{k}$ that belong to this
fitted infinite sequence according to Proposition ~\ref{p.fitted} (and
its proof) by
\glo{g.greek.cap@$\Gg(u_{1};k,\Psi)$}
\begin{equation}\label{e.Gg.primi}
\begin{aligned}
\xi^{1} &= \Gg(u_{1};\Psi)=\Gg(u_{1}),
\\ \xi^{1}_{k} &= \Gg(u_{1};k,\Psi)=\Gg(u_{1};k).
\end{aligned}
\end{equation}
\subsubsection{Coding an infinite sequence}\label{sss.hier.infin-seq}
Let us show now how a doubly infinite sequence of symbols can be
encoded into an infinite starting configuration.
Let us be given
\glo{q@$q_{k}$}
\begin{equation}\label{e.shared}
\Psi=(\SS_{k},Q_{k},\fg_{k*},\F^{k},q_{k},\gg_{k},a_{k})_{k\ge 1}
\end{equation}
where $2\le q_{k}\le Q_{k}$ and the sequence $a_{k}$ is non-degenerate,
such that in code $\fg_{k*}$,
\begin{cjenum}
\item $\F^{k}$ over $[0,q_{k}-1]$ is identified with $\F^{k+1}$;
\item $\F^{k}$ controls address $a_{k}$ for $\fg_{k*}$ via $\gg_{k}$;
\end{cjenum}
Such a system will be called a \df{shared field}\index{field!shared} for
the fields $\F^{k}$, and the hierarchical code $(\fg_{k*})$, and the fields
$\F^{k}$ will be denoted as
\glo{f.cap@$\F^{k}(\Psi)$}
\[
\F^{k}(\Psi).
\]
The identification property implies that for all $0\le a<q_{k}$, we
have
\begin{equation}\label{e.identif}
\fg_{k*}(s)(a).\F^{k} = s.\F^{k+1}([a|\F^{k}|,(a+1)|\F^{k}|-1]).
\end{equation}
\begin{example}\label{x.err-corr-code}
Let us show some examples of codes $\fg_{k}$ in which $\F^{k}$ over
$[0,q_{k}-1]$ is identified with $\F^{k+1}$.
A code $\psi=(\psi_{*}, \psi^{*})$ with $\psi^{*}: R^{Q}\to S$ will be called
\df{$d$-error-correcting with blocksize $Q$}\index{code!error-correcting}
if for all $u,v$, if $u$ differs from $\psi_{*}(v)$ in at most $d$ symbols
then $\psi^{*}(u)=v$.
Assume that both $R$ and $S$ are of the form $\{0,1\}^{n}$ (for different
$n$).
A popular kind of error-correcting code are codes $\psi$ such that
$\psi_{*}$ is a linear mapping when the binary strings in $S$ and $R^{Q}$
are considered vectors over the field $\{0,1\}$.
These codes are called \df{linear codes}\index{code!linear}.
It is sufficient to consider linear codes $\psi$ which have the property
that for all $s$, the first $|S|$ bits of the codeword $\psi_{*}(s)$ are
identical to $s$: they are called the
\df{information bits}\index{code!bits!information}.
(If a linear code is not such, it can always be rearranged to have this
property.)
In this case, the remaining bits of the codeword are called
\df{error-check bits}\index{code!bits!error check}, and they are linear
functions of $s$.
Applying such linear codes to our case, for $s\in \SS_{k+1}$, let
\[
\w = \fg_{k*}(s).
\]
Then we will have
\[
\sqcup_{0 \le a < Q_{k}} \w(a).\F^{k} = \psi_{k*}(s.\F^{k+1})
\]
for a linear code $\psi_{k}$ whose information bits are in
$\sqcup_{0 \le a < q_{k}} \w(a).\F^{k}$ and error-check bits are in
$\sqcup_{q_{k} \le a < Q_{k}} \w(a).\F^{k}$.
For $d=1$, if we are not trying to minimize the amount of redundancy in
the error correction then we may want to use the tripling method outlined
in Subsection ~\ref{ss.ftol-block} and Example ~\ref{x.control-identif},
which sets $q_{k}=Q_{k}/3$.
In this case, the error-check bits simply repeat the original bits twice
more.
\begin{figure}
\setlength{\unitlength}{0.25mm}
\[
\begin{picture}(600,150)
\put(0,20){\framebox(600,40){}}
\put(-10,20){\makebox(0,40)[r]{$\var{F}^{k}$}}
\put(0,20){\makebox(440,40){Information bits of $\var{F}^{k+1}$}}
\put(440,20){\makebox(160,40){Error-check bits of $\var{F}^{k+1}$}}
\put(440,20){\line(0,1){40}}
\multiput(0,20)(20,0){30}{\line(0,1){3}}
\put(440,5){\makebox(0,0)[t]{$q_{k}$}}
\end{picture}
\]
\caption{Error-correcting code in a shared field}
\label{f.shared}
\end{figure}
\end{example}
\begin{example}\label{x.RS-code}\index{code!Reed-Solomon}
In a digression that can be skipped at first reading, let us define the
more sophisticated linear code we will be using in later construction (a
generalization of the so-called Reed-Solomon code, see ~\cite{Blahut83}).
Let our codewords (information symbols and check symbols together) be
binary strings of length $\N\l$\glo{n.cap@$\N$}\glo{l@$\l$} for some $\l$,
$\N$.
Binary strings of length $\l$ will be interpreted as elements of the
Galois field $GF(2^{\l})$\index{Galois field}\glo{gf@$GF(2^{\l})$} and
thus, each binary string $\c$ of length $\N\l$ will be treated as a vector
$(\c(0),\ldots,\c(\N - 1))$ over $GF(2^{\l})$.
(Note that the word ``field'' is used in two different senses in the
present paper.)
Let us fix $\N$ distinct nonzero elements
$\ag_{i}$\glo{a.greek@$\ag_{i}$} of $GF(2^{\l})$ and let $t < \N/2$ be an
integer.
The codewords are those vectors $\c$ that satisfy the equation
\begin{equation}\label{e.RS-code}
\sum_{i = 0}^{\N - 1}\ag_{i}^{j}\c(i).\F^{k} = 0\ (j = 1,\ldots, 2t)
\end{equation}
where the addition, multiplication and taking to power $j$ are
performed in the field $GF(2^{\l})$.
These are $2t$ linear equations.
If we fix the first $\N - 2t$ elements of the vector in any way, (these
are the information symbols) the remaining $2t$ elements (the error check
symbols) can be chosen in a way to satisfy the equations, by solving a set
of $2t$ linear equations.
This set of equations is always solvable, since its determinant is a
Vandermonde determinant.
Below, we will show a procedure for correcting any $\nu \le t$ nonzero
errors.
This shows that for the correction of error in any $\le t$ symbols, only
$2t$ error-check symbols are needed.
If $E = (e_{0},\ldots, e_{\N - 1})$ is the sequence of errors then the
word that will be observed is $C + E$.
Only $e_{i_{\r}}$ are nonzero for $\r=1,\ldots,\nu$.
Let $Y_{\r}=e_{i_{\r}}, X_{\r}=\ag_{i_{\r}}$.
Then we define the \df{syndrome} $S_{j}$ for $j=1,\ldots,2t$ by
\begin{equation}\label{e.S_{j}}
S_{j} = \sum_{i} (c_{i}+e_{i})\ag_{i}^{j}
= \sum_{i} e_{i}\ag_{i}^{j}
= \sum_{\r} Y_{\r} X_{\r}^{j}
\end{equation}
which can clearly be computed from the codeword: it is the amount by
which the codeword violates the $j$-th error check equation.
We will show, using the last expression, that $Y_{\r}$ and $X_{\r}$ can be
determined using $S_{j}$.
We first define the auxiliary polynomial
\[
\Lg(x)=\prod_{\r}(1-xX_{\r})=\sum_{s=0}^{\nu} \Lg_{s} x^{s}
\]
whose roots are $X_{\r}^{-1}$.
Let us show how to find the coefficients $\Lg_{s}$ for $s>0$.
We have, for any $\r = 1,\ldots, \nu$, and any $j = 1, \ldots, 2t - \nu$:
\[
0 = Y_{\r}X_{\r}^{j+\nu}\Lg(X_{\r}^{-1})
= \sum_{s} \Lg_{s} Y_{\r} X_{\r}^{j+\nu-s}.
\]
Hence, summing for $\r$,
\begin{equation}\label{e.key}
0 = \sum_{s = 0}^{\nu} \Lg_{s}(\sum_{\r} Y_{\r} X_{\r}^{j+\nu-s})
= \sum_{s = 0}^{\nu} \Lg_{s} S_{j+\nu-s}\ (j = 1,\ldots, 2t - \nu)
\end{equation}
hence using $\Lg_{0}=1$,
$\sum_{s=1}^{\nu} \Lg_{s} S_{j+\nu-s}=-S_{j+\nu}$.
This is a system of linear equations for $\Lg_{s}$ whose coefficients are
the syndroms, known to us, and whose matrix $M_{\nu}$ is nonsingular.
Indeed, $M_{\nu} = ABA^{T}$ where $B$ is the diagonal matrix
$\ang{Y_{\r}X_{\r}}$ and $A$ is the Vandermonde matrix $A_{j,\r}=X^{j-1}_{\r}$.
A decoding algorithm now works as follows.
For $\nu = 1,2,\ldots,t$, see if $M$ is nonsingular, then compute
$\Lg(x)$ and find its roots by simple trial-and-error, computing
$\Lg(\ag{i}^{-1})$ for all $i$.
Then, find $Y_{\r}$ by solving the set of equations ~\eqref{e.S_{j}} and
see if the resulting corrected vector $C$ satisfies ~\eqref{e.RS-code}.
If yes, stop.
(There is also a faster way for determining $\Lg(x)$, via the Euclidean
algorithm, see ~\cite{Blahut83}).
To make the code completely constructive we must find an effective
representation of the field operations of $GF(2^{\l})$.
This finite field can be efficiently represented as the set of remainders
with respect to an irreducible polynomial of degree $\l$ over $GF(2)$, so
what is needed is a way to generate large irreducible polynomials.
Now, it follows from the theory of fields that
\[
x^{2\cdot 3^{s}} + x^{3^{s}} +1
\]
is irreducible over $GF(2)$ for any $s$.
So, the scheme works for all $\l$ of the form $2\cdot 3^{s}$.
\end{example}
In a hierarchical code, with a shared field, there is a function
$\X(y)$ with the property that site $y$ of the original information
will map to site $\X(y)$ in the code.
To define this function, let
\glo{b.cap@$B'_{k}$}
\glo{o@$o'_{k}$}
\glo{c.cap@$C'_{k}(y)$}
\[
B'_{k},o'_{k},C'_{k}(y)
\]
be defined like $B_{k},o_{k},C_{k}(y)$ but using $q_{k}$ in place of $Q_{k}$.
For all $k$, every integer $y$ can be represented uniquely in the
form
\begin{equation}\label{e.number-repr.q}
y = \sum_{i=1}^{k}(y'_{i}-a_{i})B'_{i}
\end{equation}
where $0\le y'_{i}< q_{i}$ for $i<k$.
Since $(a_{k})$ is non-degerate for $(q_{k})$, this is true even with
$k=\infty$, in which case the above sum is finite.
Let
\glo{x.cap@$\X(y, i; k, \Psi)$}
\begin{equation}\label{e.shared.X^{i}}
\begin{aligned}
\X(y, i; k) =
\X(y, i; k, \Psi) &= \sum_{m=i}^{k} (y'_{m} - a_{m}) B_{m},
\\ \X(y, i) &= \X(y, i; K(N)),
\\ \X(y; k) &= \X(y, 1; k),
\\ \X(y) &= \X(y, 1; K(N)),
\end{aligned}
\end{equation}
Define the same notation for $\X'$ with $B'_{k}$ instead of $B_{k}$.
Notice that $\X(0,i)=\X'(0,i)=0$, $\X'(y,1)=y$.
Clearly, the sites of form $o_{i}+\X(y,i;k)$ for all possible $y$ will
form a lattice of distance $B_{i}$.
If $i<k$ then the definitions give
\begin{equation}\label{e.X-i-prop}
o_{i} + \X(y, i; k) = o_{i+1} + \X(y, i + 1; k) + y'_{i} B_{i}.
\end{equation}
Using the notation $\ig_{Q}$ introduced in ~\ref{sss.string-codes}, let
us define the aggregated configurations
\glo{r.greek@$\rg^{k}$}
\begin{equation}\label{e.rg^{k}}
\rg^{k} = \ig_{B'_{k}*}(\rg)
\end{equation}
of body size $B'_{k}$ over $\ZZ$.
Then, of course,
\begin{equation}\label{e.aggreg.step}
\rg^{k+1} = \ig_{q_{k}*}(\rg^{k}).
\end{equation}
Let
\glo{visible@$\var{Visible}(k,N)$}
\begin{equation}\label{e.Visible}
\begin{aligned}
\var{Visible}(k,N) &= \setof{y: 0 \le y - o_{K} < NB'_{K}/B_{K}}.
\\ &= \bigcup_{0 \le zB_{k} < N}C'_{k}(z)
\end{aligned}
\end{equation}
Then $\X(y)$ is defined whenever $y\in\var{Visible}(k,N)$, i.e.~the
symbols $\rg(y)$ can be recovered after encoding
whenever $y$ is in this interval.
\begin{proposition}\label{p.shared}
For a hierarchical code with a shared field as given in
~\eqref{e.shared}, for an arbitrary configuration $\rg=\rg^{1}$ in
$(\SS_{1}.\F^{1})^{\ZZ}$, there are configurations $\xi^{k}$ over $\ZZ$ such
that for all $k\ge 1$, $y\in\ZZ$ we have
\begin{align}
\label{e.shared.coding}
\fg_{k*}(\xi^{k+1}) &= \xi^{k},
\\\label{e.shared.comput}
\xi^{1}(\X(y)).\F^{1} &= \rg^{1}(y).
\end{align}
More generally, we have
\begin{equation}\label{e.X-y-i}
\xi^{i}(o_{i}+\X(y,i)).\F^{i} = \rg^{i}(o'_{i}+\X'(y,i))
\end{equation}
for $1\le i\le k$.
If the space is $\ZZ_{N}$ for a finite $N$ then all these configurations
with the above properties exist for all $k$ with $B_{k}\le N$, and
~\eqref{e.shared.comput} holds whenever $y\in\var{Visible}(K(N),N)$.
\end{proposition}
\begin{Proof} The proof is mechanical verification: we reproduce it here
only to help the reader check the formalism.
\begin{step+}{shared.constr}
Let us construct $\xi^{k}$.
\end{step+}
\begin{prooof}
For infinite space size, let
\begin{equation}\label{e.shared.xi_{k}}
\xi^{k}_{k}(C_{k}(y)) = \gg_{k}(\rg^{k}(C'_{k}(y))).
\end{equation}
For finite space size $N$, define the above only for $B_{k}\le N$ and
$y$ in $[0,\flo{N/B_{k}}-1]$, where $C_{k}(y)$ on the left-hand side is
taken $\pmod N$.
In all other sites $x$, let $\xi^{k}_{k}(x)=\Vac$.
Let $\xi^{i}_{k}$ be defined again by ~\eqref{e.xi-i-k}.
We define $\xi^{i}$ as in the proof of Proposition ~\ref{p.fitted}.
It is sufficient to show ~\eqref{e.fitted.ext} again to prove that
the limits in question exist.
By definition,
\[
\xi^{k}_{k+1}(o_{k}) = \fg_{k*}(\xi^{k+1}_{k+1}(o_{k+1}))(a_{k})
= \fg_{k*}(\gg_{k+1}(\rg^{k+1}(o'_{k+1})))(a_{k}).
\]
By the controlling property, its $\F^{k}$ field $r$ completely determines
the last expression via $\gg_{k}$.
By the identification property ~\eqref{e.identif} and
the aggregation property ~\eqref{e.aggreg.step},
\[
r=\rg^{k+1}(o'_{k+1})([a_{k}B'_{k},(a_{k}+1)B'_{k}-1])
= \rg^{k}(o'_{k+1}+a_{k}B'_{k})=\rg^{k}(o'_{k}).
\]
By definition, $\xi^{k}_{k}(o_{k})=\gg_{k}(\rg^{k}(o'_{k}))$ which proves the
statement.
\end{prooof}
\begin{step+}{shared.comput}
Let us show ~\eqref{e.X-y-i}.
\end{step+}
\begin{pproof}
We use induction on $i$, from $k$ down to 1.
The case $i=k$ says
\[
\xi^{k}(C_{k}(y'_{k}-a_{k})).\F^{k} = \rg^{k}(C'_{k}(y'_{k}-a_{k}))
\]
which follows from the definition of $\xi^{k}$.
Assume that the statement was proved for numbers $>i$: we prove it
for $i$.
By the definitions of $\X(y,i)$ and $\xi^{i}$ and by ~\eqref{e.X-i-prop}
we have
\begin{multline}\label{e.X-y-i-rel.1}
\xi^{i}(o_{i}+\X(y,i)) = \xi^{i}(o_{i+1}+\X(y,i+1)+y'_{i}B_{i})
\\ = \fg_{i*}(\xi^{i+1}(o_{i+1}+\X(y,i+1)))(y'_{i}) = \Dg_{1}.
\end{multline}
By induction,
\[
\xi^{i+1}(o_{i+1}+\X(y,i+1)).\F^{i+1}
= \rg^{i+1}(o'_{i+1}+\X'(y,i+1)) = \rg^{i+1}(z)
\]
where $z = o'_{i+1}+\X'(y,i+1)$ can also be written in the form
$C'_{i+1}(x)$ for some $x$.
Now we have, by the identification property~\eqref{e.identif} and
the aggregation property~\eqref{e.aggreg.step}
\begin{equation}\label{e.X-y-i-rel.2}
\Dg_{1}.\F^{i} = \rg^{i}(z+y'_{i}B'_{i}) =
\rg^{i}(o'_{i+1}+\X'(y,i+1)+y'_{i}B'_{i}) = \rg^{i}(o'_{i}+\X'(y,i))
\end{equation}
where the third equality is implied by the definition of
$\X'(y,i)$.
\end{pproof}
\end{Proof}
In analogy with~\eqref{e.Gg.primi}, we will denote this code as
follows:
\glo{g.greek.cap@$\Gg(\rg; k, \Psi)$}
\begin{equation}\label{e.Gg.shared}
\begin{aligned}
\xi^{1}_{k} &= \Gg(\rg; k,\Psi)=\Gg(\rg;k),
\\ \xi^{1} &= \Gg(\rg; \infty, \Psi)=\Gg(\rg; \Psi)=\Gg(\rg).
\end{aligned}
\end{equation}
$\Gg$ is the \df{limit code}\index{code!limit} with \df{approximations}
$\Gg(\cdot;k)$\index{code!limit!approximation}
and the function $\X(y)$ is the \df{site map}\index{site!map} of the
system $\Psi$.
Note that for finite space size $N$, we have $\Gg(\rg)=\Gg(\rg;k)$
for the largest $k$ with $B_{k}\le N$.
The proof also shows that $\X(y;k)$ plays the role of the site map
for the approximation:
\begin{equation}\label{e.shared.comput.approx}
\xi^{1}_{k}(\X(y;k)).\F^{1} = \rg^{1}(y),
\end{equation}
The growth of the quotients $\X(y)/y$ is a measure of how the infinite
code stretches its input, i.e.~of the
``space redundancy''\index{redundancy!space}.
(Strictly speaking, the redundancy is defined as $\X(y)/y - 1$.)
The value $\X(y;k)/y$ for each approximating code is limited since it
stretches blocks of size $B'_{k}$ into blocks of size $B_{k}$.
If a code has $q_{k}=Q_{k}$ as in the example below then $\X(y)=y$.
\begin{example}\label{x.no-stretch}
Let us show a variant of Example ~\ref{x.err-corr-code} with
$q_{k}=Q_{k}$.
The details can be skipped at first reading.
Field $\F^{k}$ is a binary string of length
$\l_{k}=\l_{1}Q_{1}\cdots Q_{k-1}$.
Let $s\in S_{k+1}$,
\begin{align*}
\w &= \fg_{k*}(s),
\\ \v(a) &= \w(a).\F^{k}\ (a = 1,\ldots,Q_{k} - 1).
\end{align*}
The information symbols of the code are $\v(a)$ for $0 < a < Q_{k} - 1$.
Let the positive integers
$m_{k},n_{k}$\glo{m@$m_{k}$}\glo{n@$n_{k}$} be such that
\[
m_{k} h_{k} = \l_{k},\, m_{k} n_{k} \le Q_{k}.
\]
A narrower track $\w.\fH^{k}$\glo{h.cap@$H^{k}$} contains the error-check
bits for the same information, where
$|\fH^{k}| = h_{k}$\glo{h@$h_{k}$} (see Figure~\ref{f.no-stretch}).
For each $0\le i < n_{k}$, the concatenation of strings $\w(a).\fH^{k}$
with $a\in[im_{k},(i+1)m_{k}-1]$ will be denoted
\[
v(i + Q_{k}).
\]
These are the error-check symbols.
With the code of Example ~\ref{x.RS-code}, we have $n_{k} = 2t$ and
$Q_{k}+n_{k} = \N$.
The redundancy of the code is $1/m_{k}$.
\end{example}
\begin{figure}
\setlength{\unitlength}{0.25mm}
\[
\begin{picture}(600,150)
\put(0,70){\framebox(600,20){Error-check bits of $\var{F}^{k+1}$}}
\put(-10,70){\makebox(0,20)[r]{$\var{H}^{k}$}}
\put(0,20){\framebox(600,50){Information bits of $\var{F}^{k+1}$}}
\put(-10,20){\makebox(0,50)[r]{$\var{F}^{k}$}}
\multiput(0,20)(20,0){30}{\line(0,1){3}}
\end{picture}
\]
\caption{Error-correcting code in a shared field, with at least one
information bit per cell}
\label{f.no-stretch}
\end{figure}
\begin{proposition}
In the example above, $\sum_{k} 1/m_{k} < \infty$.
\end{proposition}
\begin{proof}
For the capacity of the cells in $M_{k}$ we have
\[
\nm{\SS_{k}} = \l_{k} + h_{k} + r_{k} = \l_{k}(1+1/m_{k}) + r_{k}
\]
where $r_{k}$ is the number of bits not belonging to $\F^{k}$ or $\fH^{k}$.
The state of a cell of $M_{k+1}$ must be stored in the fields of the
cells of the colony representing it, excluding the error-correcting
bits in $\fH^{k}$.
Hence
\begin{equation}\label{e.no-stretch}
\begin{aligned}
\l_{k+1} + h_{k+1} + r_{k+1} &\le Q_{k}(\l_{k} + r_{k}),
\\ h_{k+1} &\le Q_{k}r_{k} - r_{k+1},
\\ 1/m_{k+1} &\le r_{k}/\l_{k} - r_{k+1}/\l_{k+1},
\\ \sum_{k=2}^\infty 1/m_{k} &\le r_{1}/\l_{1}.
\end{aligned}
\end{equation}
\end{proof}
\subsubsection{Infinitely many fields}
The above construction will be used mainly to encode the input
$\rg^{1}$ into the configuration $\xi^{1}$, and to find the sites where
the output can be retrieved.
The information is kept on each level $k$ in field $\F^{k}$.
In the case when besides information storage also computation will be
going on, several configurations may have to be encoded, representing
e.g.~the output of the same computation at different times (see Subsection
~\ref{ss.sorg}).
Here we will set up the framework for coding infinitely many
sequences, each to it own track.
Since any infinite sequence can be broken up into infinitely many
infinite subsequences this elaboration is routine, but it is worth
fixing some notation.
Readers interested only in information conservation can skip this
construction.
Let us be given, for k=1,2,\ldots, $0\le i<k$,
\glo{p@$p_{k}$}
\glo{f.cap@$\F^{k}_{j}$}
\begin{equation}\label{e.std-shared}
\Psi=
(\SS_{k},Q_{k},\fg_{k*},(\F^{k}_{j})_{j=1}^{k},q_{k},p_{k},\gg_{k})_{k\ge 1}
\end{equation}
such that $2<q_{k}+p_{k}\le Q_{k}$, and in code $\fg_{k*}$ the following
properties hold:
\begin{cjenum}
\item For each $k$, field $\bigcup_{j\le k} \F^{k}_{j}$ controls $a_{k}$ via
$\gg_{k}$;
\item $\F^{k}_{j}$ is identified with $\F^{k+1}_{j}$ over $[0,q_{k}+p_{k}-1]$
if $j<k$ and over $[0,q_{k}-1]$ if $j=k$;
\item $\F^{k}_{k}$ over $q_{k}+[0,p_{k}-1]$ is identified with
$\F^{k+1}_{k+1}$;
\end{cjenum}
Such a system will be called a \df{standard system of shared
fields}\index{standard!system of shared fields} and we will write
\[
\F^{k}_{j}=\F^{k}_{j}(\Psi).
\]
For simplicity, we only consider infinite space.
\begin{proposition}\label{p.shared-infin}
For a standard system of shared fields $\Psi$ as in~\eqref{e.std-shared},
there are functions
$\X_{j}(y,\Psi)=\X_{j}(y)$\glo{x.cap@$\X_{j}(y,\Psi)$} with $\X_{j}(0)=0$
such that for any infinite sequence $d_{j}$\glo{d@$d_{j}$} of integers and
any infinite sequence of configurations
\glo{r.greek@$\rg_{j}$}
\[
\rg_{j} \in (\SS_{1}.\F_{1}^{1})^{\ZZ} \for j\ge 1
\]
there are configurations $\xi^{k}$ such that for all $k,j\ge 1$,
$y\in\ZZ$ we have
\begin{equation}
\begin{aligned}\label{e.shared-infin.comput}
\fg_{k*}(\xi^{k+1}) &= \xi^{k},
\\ \xi^{1}(\X_{j}(y)+d_{j}B_{j}).\F^{1}_{1} &= \rg_{j}(y).
\end{aligned}
\end{equation}
\end{proposition}
The proof of this proposition is routine computation, so we omit it here.
The sequence $d_{j}$ gives additional freedom of shifting the origins
independently for each $j$.
In analogy with~\eqref{e.Gg.primi} we define
\begin{equation}\label{e.Gg.j}
\begin{aligned}
\xi^{1} &= \Gg((\rg_{j});\Psi)=\Gg((\rg_{j})),
\\ \xi^{1}_{k} &= \Gg((\rg_{j});k,\Psi)=\Gg((\rg_{j});k).
\end{aligned}
\end{equation}
and call $\Gg()$ the \df{limit code} and $\X_{j}(y)$ the \df{site
map} of this many-field hierarchical code.
We may want to use the same configuration $\rg_{1}$ for each $\rg_{j}$:
e.g.~ if we start the computation on infinitely many levels simultaneously,
from the same input.
\subsection{The active level}\label{ss.amp.tkdn}
In the (preliminary) definition of error-correction in Subsection
~\ref{ss.gen-sim}, we used a code $\fg_{**}\preceq\fg_{*}$.
Let us discuss the typical structure of the codes $\fg_{k**}$
belonging to a hierarchical code.
If $\fg_{k**}$ coincides with $\fg_{k*}$ over the field $\F^{k}$,
i.e.~for all $k,s,a$ we have
$\fg_{k**}(s)(a).\F^{k}=\fg_{k*}(s)(a).\F^{k}$ then we will say that
$(\F^{k})$ are \df{broadcast fields} resp.~\df{shared fields} with
respect to $\fg_{k**}$, too, whenever they are such in the code
$\fg_{k*}$.
Next, we give a slight refinement of this notion for the case of
reliable computation.
The definitions given here are only needed if the cellular automata are
also meant to be used for computation: they are not strictly needed if the
goal is only information storage.
However, it will be convenient to use them even in that case.
For a shared field, essentially the same space can be used to store
the track $\F^{k}$ as the one used for $\F^{k+1}$ since the
information on the two tracks is the same.
Therefore these fields cannot be used by the different levels
independently of each other for computation.
The mechanism enforcing this is the error-correction property which
restricts the value of $\F^{k}$ by the value of $\F^{k+1}$ (with which
it is identified).
Thus, changing the information in track $\F^{k}$ we change it on all
levels below.
We should therefore know which level is the ``active'' one, the one
being changed directly rather than as a consequence of the code
constraints.
This level should not be disturbed by error-correction coming from higher
levels.
The active level will be marked in the following way.
Let us be given, for $k=1,2,\ldots$, new fields
$\G^{k}$\glo{g.cap@$\G^{k}$} with $|\G^{k}|=2$.
The four possible values of $\G^{k}$ will be identified with
$-1,0,1,*$.
We will say that $(\F^{k},\G^{k})$ define a sequence of \df{guarded
shared fields}\index{field!shared!guarded} if the following properties
hold:
\begin{cjenum}
\item For all $0\le a < Q$,
\[
\fg_{k**}(s)(a).\G^{k} = \begin{cases}
-1 &\text{if $s.\G^{k+1} \le 0$,}
\\ {*} &\text{otherwise;}
\end{cases}
\]
\item
\begin{equation}\label{e.guarded-hier}
\fg_{k**}(s)([0,q_{k}-1]).\F^{k} = \begin{cases}
s.\F^{k+1} &\text{if $s.\G^{k+1} \le 0$,}
\\ {*\cdots *}&\text{otherwise.}
\end{cases}
\end{equation}
\end{cjenum}
Notice that since $\fg_{k**}\preceq\fg_{k*}$, this also imposes some
conditions on $(\fg_{k*})$.
Typically, for a certain $k$ we will have $\xi^{i}(\cdot).\G^{i}>0$ for
all $i>k$, $\xi^{k}(\cdot).\G^{k} = 0$, and $\xi^{i}(\cdot).\G^{i}<0$ for
$i<k$.
This distinguished $k$ will show the ``active'' level\index{active
level}, on which the field $\F^{k}$ can be changed: $\G^{k}$ shows whether
we are on, below or above the active level.
The fact $\xi^{k+1}(\cdot).\G^{k+1}>0$ will imply that the level
$(k+1)$ does not restrict us from changing $\xi^{k}(x).\F^{k}$ (e.g.~by
computation).
The levels below the active one are the ones subject to error-correction
coming from $\F^{k}$ since
the properties imply that the $\F^{i}$ for all $i<k$ behaves like a
shared field in code $(\fg_{i**})$ just as it does in code
$(\fg_{i*})$.
The guard field $\G^{k+1}$ is broadcast whenever it is negative.
With a guarded shared field, if the active level is $k$ then the
definition of $\Gg(u_{1};k)=\xi^{1}_{k}$ in~\eqref{e.Gg.primi} will always
imply the additional property
\begin{equation}\label{e.Gg.guarded}
\xi^{k}_{k}(x).\G^{k}=0
\end{equation}
for all $x$.
Subsequent codings by $\fg_{(k-1)*}$, etc.~imply $\xi^{i}_{k}.\G^{i}=-1$
for $i<k$.
The active level $\infty$ will mean $\xi^{k}_{k}(x).\G^{k}=-1$ for all
$k$.
All propositions in the present section about the existence of encoded
configurations, can be enhanced to include a guard field, with an
arbitrarily chosen active level (possibly $\infty$).
\begin{remarks}
\begin{enumerate}
\item All the tracks $\F^{k}$ for different $k$ ``can use the same
space'' since $\G^{k+1}$ has the information showing the way $\F^{k}$
depends on $\F^{k+1}$.
However, each track $\G^{k}$ must be represented in new space since
$\G^{k}$ is not identified with $\G^{k+1}$.
\item The only place in the present paper where the possibility of
changing the $G^{k}$ field is exploited is~\ref{sss.active}.
\end{enumerate}
\end{remarks}
\subsection{Major difficulties}\label{ss.trouble}
The idea of a simulation between two perturbed cellular automata is,
unfortunately, flawed in the original form: the mapping defined in the
naive way is not a simulation in the strict sense we need.
The problem is that a group of failures can destroy not only the
information but also the organization into colonies in the area where
it occurs.
This kind of event cannot therefore be mapped by the simulation into
a transient fault unless destroyed colonies ``magically recover''.
The recovery is not trivial since ``destruction'' can also mean
replacement with something else that looks locally as legitimate as
healthy neighbor colonies but is incompatible with them.
One is reminded of the biological phenomena of viruses and cancer.
Rather than give up hope let us examine the different kinds of
disruption that the faults can cause in a block simulation
by a perturbed cellular automaton $M_{1}$.
Let us take as our model the informally described automaton of
Subsection ~\ref{ss.ftol-block}.
The information in the current state of a colony can be divided into
the following parts:
\begin{itemize}
\item ``information'': an example is the content of the $\Info$
track.
\item ``structure'': the $\Addr$ and $\Age$ tracks.
\item ``program'': the $\fld{Prog}$ track.
\end{itemize}
More informally, the ``structure'' does not represent any data for
the decoding but is needed for coordinating cooperation of the colony
members.
The ``program'' determines which transition function will be
simulated.
The ``information'' determines the state of the simulated cell: it
is the ``stuff'' that the colony processes.
Disruptions are of the following kinds (or a combination of these):
\begin{djenum}
\item \label{i:info} Local change in the ``information'';
\item \label{i:loc} Locally recognizable change in the
``structure'';
\item \label{i:prog} Program change;
\item \label{i:glob} Locally unrecognizable change in
``structure'';
\end{djenum}
A locally recognizable structure change would be a change in the
address field.
A locally unrecognizable change would be to erase two neighbor
colonies based, say, at $BQ$ and $2BQ$ and to put a new colony in the
middle of the gap of size $2BQ$ obtained this way, at position
$1.5BQ$.
Cells within both the new colony and the remaining old colonies will
be locally consistent with their neighbors; on the boundary, the cells
have no way of deciding whether they belong to a new (and wrong)
colony or an old (and correct) one.
The only kind of disruption whose correction can be attempted along
the lines of traditional error-correcting codes and repetition is the
first one: a way of its correction was indicated in Subsection
~\ref{ss.ftol-block}.
The three other kinds are new and we will deal with them in
different ways.
To fight locally recognizable changes in the structure, we will use
the method of destruction and rebuilding.
Cells that find themselves in structural conflict with their
neighbors will become vacant.
Vacant cells will eventually be restored if this can be done in a
way structurally consistent with their neighbors.
To fight program changes, our solution will be that the simulation will
not use any ``program'' or, in other words,
``hard-wire''\index{hard-wiring} the program into the transition function
of each cell.
We will not lose universality this way: the automata will still be
universal, i.e.~capable of simulating every other automaton by appropriate
block simulation; but this particular simulation will differ from the
others in that the transition function will perform it without looking at
any program.
\begin{figure}
\setlength{\unitlength}{1mm}
\[
\begin{picture}(100, 50)(-30,0)
\put(-2,20){\makebox(0,20)[r]{$\fg^{2*}$}}
\put(0,25){\vector(0,1){10}}
\put(0,20){\makebox(0,0){$M_{2}$}}
\put(5,20){\vector(1,0){10}}
\put(20,20){\makebox(0,0)[l]{$\var{Univ}$}}
\put(0,5){\vector(0,1){10}}
\put(-2,0){\makebox(0,20)[r]{$\fg^{1*}$}}
\put(0,0){\makebox(0,0){$M_{1}$}}
\put(5,0){\vector(1,0){10}}
\put(20,0){\makebox(0,0)[l]{$\var{Univ}$}}
\end{picture}
\]
\caption[An amplifier in which the simulations $\fg^{k*}$ are
``hard-wired'']{An amplifier in which the simulations $\fg^{k*}$ are
``hard-wired''.
Universality is not lost since each medium $M_{k}$ also
simulates some universal cellular automaton.}
\label{f.hard-wired}
\end{figure}
To fight locally unrecognizable changes, we will ``legalize'' all
the structures brought about this way.
Consider the example where a single colony sits in a gap of size $2BQ$.
The decoding function is defined even for this configuration.
In the decoded configuration, the cell based at site 0 is followed by a
cell at site $1.5BQ$ which is followed by cells at sites $3BQ,4BQ$, etc.
Earlier, we did not have any use for these illegal configurations.
We must legalize them now.
Indeed, since they can be eliminated only with their own active
participation, we must have rules (trajectory conditions) applying to
them.
This is the real reason for the introduction of generalized cellular
automata.
The generalized cellular automaton actually needed will be called a
\df{robust medium}.
The generalization of the notion of the medium does not weaken the
original theorem: the fault-tolerant cellular automaton that we
eventually build is a cellular automaton in the old sense.
The more general media are only needed to make rules for all the
structures that arise in simulations by a random process.
| 131,874
|
TITLE: Tangent line to parametric curve at a point
QUESTION [1 upvotes]: Consider the curve given parametrically by $(x,y,z)=(2−t,−1−t^2,−2t−3t^3)$. There is a unique point $P$ on the curve with the property that the tangent line at $P$ passes through the point $(2,8,−162)$. Find the coordinates of the point $P$.
Can anyone give me a hand with this one? I honestly have no idea where to start other than rewriting the curve as,
$\vec{r(t)}$ = $\langle 2-t, -1-t^2, -2t-3t^3\rangle$, and then taking the derivate to obtain,
$\vec{r'(t)}$ = $\langle -1, -2t, -2 -9t^2\rangle$
REPLY [1 votes]: Let $\vec{r}(t) = \langle 2−t,−1−t^2,−2t−3t^3\rangle$, where $t\in \mathbb{R}$. Then $\vec{r}'(t) = \langle -1, -2t, -2 -9t^2\rangle$.
For any $t_0$ in the domain, a parametric equation of the tangent line at $t_0$ is given by
$$
\vec{T}(t) = t \:\vec{r}'(t_0) + \vec{r}(t_0), \mbox{ where } t\in \mathbb{R}.
$$
Now, we want to find an equation of the tangent line when it is passing through $(2,8, -162)$.
Thus, setting the following equal to each other,
\begin{align*}
t\langle -1, -2t, -2 -9t^2\rangle + \langle 2−t_0,−1−t_0^2,−2t_0−3t_0^3\rangle = \langle 2,8, -162 \rangle,
\end{align*}
we need to solve for $t_0$.
The above vector equation gives us three equations:
\begin{align*}
-t+2-t_0&=2, \hspace{4mm} (\dagger) \\
-2t_0t-1-t_0^2&=8, \hspace{4mm} (\ddagger) \\
-2t-9t_0^2t-2t_0-3t_0^3&=-162. \hspace{4mm} (\Omega)\\
\end{align*}
Use the first equation $(\dagger)$ to solve for $t:$
$$
t=-t_0.
$$
Let's substitute this into the second equation $(\ddagger)$:
$$
2t_0^2 -1-t_0^2 = 8,
$$
which gives us $t_0^2 = 9$.
So $t_0 = \pm 3$ while $t=\mp 3$.
If $t_0=3$, then $t=-3$. We substitute these into the third equation $(\Omega)$ to obtain:
$$
6+9\cdot 9\cdot 3 -2(3)-3(27) = 9(27)-3(27)=6(27) \not= -162.
$$
We conclude that $t_0=-3$ while $t=3$, which we can check using the third equation $(\Omega)$:
$$
-6-9\cdot 9\cdot 3 +2(3)+3(27) = -9(27)+3(27)=-6(27) = -162.
$$
So $P$ has position vector $\vec{r}(-3)=\langle 5,-10,87 \rangle $, which is when its tangent line passes through the point $(2,8,−162)$.
| 202,922
|
blockchain technology to gain a competitive advantage. This is not Mastercard’s first foray into blockchain. Their first implementation with DLT was not in the world of payments but in the verification of the provenance of luxury goods.
In payments and beyond, it is not the first blockchain project where Mastercard is involved – e.g., the creation of a blockchain for verifying the provenance of luxury goods.
Many companies are experimenting with the more cutting-edge implementations of blockchain technology, creating standalone internal units and investing in promising startups dealing with cryptoassets. These investments are driven by a need to protect against disruptive change, innovate their models and tackle a sector of high growth at a time of stagnating interest rates.
Payments are a killer app for blockchain, unlike any other due to their mass appeal – a use-case that is especially relevant for the mainstream adoption of digital currencies as a means of payment. In an age of fast information & instant feedback, where digital and physical augment each other, a universally trusted, verifiable and privacy-conscious means of value transmission will play a pivotal role. This follows the same cycle of continuous technology evolution we have seen in the early ages of the internet. In the same way, email or the browser were killer apps of the early internet age, relying on protocols and foundations of years prior. Those, in turn, led to the next wave of killer apps – search engines, social networks, mobile-centric apps and more. All of those had implications beyond just technology, tackling specific segments – be it productivity, communication or human relations. This time is money itself.”
Sky Guo, CEO, and Co-Founder of Cypherium, the enterprise-ready blockchain solution commented:
We have seen a number of major financial corporations enter the blockchain space in a similar fashion, and MasterCard follows suit. The difficulty here has been distinguishing institutional recognition of this valuable technology from corporate appropriation.
On one hand, we can’t survive as new financial systems without meaningful recognition from and conversation with legacy systems, such as MasterCard and other centralized goliaths. On the other hand, these technologies—especially the public blockchains—pose a very real threat to the business models of a number of these companies, and in order to quell their disruption, giants like MasterCard want to absorb crypto projects on their own terms.
Ultimately, if blockchains scale to their full extent, there will be little need for the kind of private DLTs JPMorgan, Facebook, and now MasterCard propose. The true killer Dapp will make obsolete these private networks; it will be faster and cheaper to use while returning financial economy to its users; that is the promise of Bitcoin that so many new chains are trying to fulfill. So one can see that their motivation in entering the space is, at least partially, guided by their need to street the conversation, to dictate the way in which blockchain technology enters the world. For now, though, blockchain must support and participate in these projects.”
Charles Lu, CEO of Findora said:
While it’s not clear yet whether the proposed initiatives will go beyond proof-of-concept, the news that Mastercard is to develop a blockchain-based cross-border payments platform is a welcome move.
Tech giants such as Mastercard play an influential role in encouraging the acceptance of cutting-edge technologies and this news is symbolic of Mastercard’s willingness to challenge the status quo and seek solutions to improve current payment systems.
Change is on the horizon and, as technology continues to evolve, we are witnessing cryptographic breakthroughs including zero-knowledge proofs, multiparty computation, and scalability solutions — all of which have the potential to shake up the financial industry as we know it.”
Please leave your questions and comments below:
KEEP UPDATED – SUBSCRIBE TO BITMEDIA YOUTUBE HERE
| 306,823
|
Assistant robot arm like an Elephant’s trunk
In Germany, a new type of flexible robotic arm has been developed in the shape of an Elephant’s trunk. The developers emphasized safety in developing this arm and no metallic parts are used. The arm is composed of three main segments and a hand axis. The movement is made possible by the air moving in […]
| 298,112
|
Purchase this article with an account.
or
Shinya Sato, Beata Jastrzebska, Andreas Engel, Krzysztof Palczewski, Vladimir J Kefalov; Apo-opsin forms a photoactivated rhodopsin-like state. Invest. Ophthalmol. Vis. Sci. 2017;58(8):5602.
Download citation file:
© ARVO (1962-2015); The Authors (2016-present)
Purpose :
Bleaching adaptation in rods is mediated by apo-opsin, which activates phototransduction with an estimated activity 106-fold lower than that of photoactivated rhodopsin (Meta-II). It is unclear whether every opsin molecule has low constitutive activity or if opsin exists in equilibrium between a predominantly inactive state and an intermittent active state. To address this question, we studied opsin signaling by electrophysiological recordings from mouse rods.
Methods :
We studied opsin signaling in two models, guanylate cyclase activating proteins knockout mice (GCAPs-/-) and retinal pigment epithelium specific 65 kDa protein knockout mice (RPE65-/-). First, we examined GCAPs-/- mouse rods, which have ~5 times higher sensitivity than wildtype rods, in an effort to resolve the signal from individual opsins. Prior to the recordings, dark-adapted mouse retinas were dissected and a small fraction of opsin was produced by bleaching <1% of rhodopsin by light. Then, activation of the phototransduction cascade by opsin was measured from rod outer segments by single-cell suction recordings in the dark. Next, we stueied RPE65-/- chromophore-deficient rods. Here, prior to recordings, almost all of the opsin was converted into unbleachable rhodopsin by regeneration with exogenous locked 11-cis-7-ring retinal. Resistance of this 11-cis-7-ring rhodopsin to photoactivation and bleaching was confirmed biochemically. The signaling of the residual small fraction of apo-opsin in these rods was then measured by the same methods as above.
Results :
Surprisingly, we observed frequent photoresponse-like events in the dark from bleached GCAPs-/- rods. The rate of these photoresponse-like events was similar from 2 hours to 12 hours after the bleach, arguing against a contribution from Meta-II. Consistent with this interpretation, dark activity returned to pre-bleached levels by regenerating bleached opsin into rhodopsin with exogenous 11-cis retinal treatment. These data suggest that opsin can form an active Meta-II like state. We observed similar events in RPE65-/- rods regenerated with the unbleachable rhodopsin analogue, further ruling out the involvement of Meta-II and its decay intermediates in these photoresponse-like events.
Conclusions :
Our data suggest that, contrary to current beliefs, bleaching adaptation in rods is mediated by opsin that exists in equilibrium between predominantly inactive and intermittently Meta-II like states.
This is an abstract that was submitted for the 2017 ARVO Annual Meeting, held in Baltimore, MD, May 7-11, 2017.
This PDF is available to Subscribers Only
| 256,203
|
\section{Proof of Theorem 1}
In this section we prove Theorem \ref{teo1}.
\begin{proof}
Denote by $\TT^2_*=\mathbb{R}^2/(2\mathbb{Z}) ^2$ the 4-1
covering of $\TT^2=\mathbb{R}^2/\mathbb{Z}^2$.
Let $F\colon \TT^2_*
\rightarrow \TT^2_*$ be a lift of $f$ to $\TT^2_*$
Let the natural
projections be $\Pi\colon\mathbb{R}^2 \rightarrow \TT^2$,
$\Pi_*\colon\mathbb{R}^2 \rightarrow \TT^2_*$ and $P\colon \TT^2_* \rightarrow
\TT^2$. Note that $\Pi= P \circ \Pi_* $. Let us endow $\TT^2$
and $\TT_*^2$ with their usual flat Riemanian metrics (inherited from
the standard Euclididian metric on $\mathbb R^2$) and the
associated distances.
By hypotheses, $f $ is a $ C^{1+\alpha}$ diffeomorphism
and $h_{top}(f)>0$ therefore $F$ is also a $ C^{1+\alpha}$
diffeomorphism and $h_{top}(F)>0$.
By Katok (see \cite{Ka80}), there exist $n \in \mathbb{N}$ and a
hyperbolic periodic point $y_0$ of $F$ with period $n$ such
that the intersection of the stable and unstable manifolds of $y_0$
is transversal. Thus, there exists $k \in \mathbb{N}$ the minimal
positive number such that
$F^{nk}(y_0)=y_0$ and both eigenvalues of the
differential $DF^{nk}(y_0)$ are positive. In what follows,
we denote $F^{nk}$ by $f_*$.
Let us denote $x_0=P(y_0)$. Since $P$ is a local diffeomorphism,
$x_0$ is a hyperbolic fixed point of $f^{nk}$ and it has the local
type of $y_0$. Let $\widetilde{x_0} \in [0,1]\times [0,1]$ be a
lift of $x_0$ and $\tilde f_*$ be a lift of $f^{nk}$ to $\mathbb R^2$ such
that $\tilde f_*(\widetilde{x_0})=\widetilde{x_0}$. Note that $\tilde f_*$ is
a lift of both $f^{nk}$ and $f_*$.
We define $\widetilde x_1= \widetilde{x_0}$, $\widetilde x_2=
\widetilde{x_0}+ (1,0)$, $\widetilde x_3= \widetilde{x_0}+ (0,1)$
and $\widetilde x_4= \widetilde{x_0}+ (1,1)$.
Since $\tilde f_*$ is isotopic to identity, then $\tilde
f_*(\widetilde{x_0}+(a,b))= \tilde f_*(\widetilde{x_0}) + (a,b)= \widetilde{x_0} +(a,b)$, for any $(a,b)\in
\mathbb Z ^2$. Then every $\widetilde x_i$ is a fixed point of $\tilde f_*$ and
therefore its projection on $\TT^2_*$ denoted by
$x_i$ is fixed by $f_*$. Note that there exists $i \in \{1,2,3,4\}$
such that $y_0=x_i$.
\bigskip
{\bf Proposition 1.} {\it There exists $0<\epsilon <\frac{1} {2}$
such that for $i\in \{2,3,4\}$ there exist $n_i \in \N$ and non
empty compact sets $L_i\subset B_\epsilon (x_i)\subset \TT^2_*$
and $ L_1^i\subset B_\epsilon (x_1)$ such that $L_i
=f_*^{n_i}(L_1^i)$ and $P(L_1^i)= P(L_i)$.}
\medskip
{\bf Proof of Proposition 1.} By the classical Hartman-Grobman's
theorem, there is an open subset $U$ of $\TT^2$ containing $x_0$
such that the restriction of $f^{nk}$ to $U$ is topologically
conjugated to its differential $Df^{nk}(x_0)$. By conjugating
$f^{nk}$ by a suitable homeomorphism with support in a small
compact $K\supset U$, we may suppose that $f^{nk}$ is a linear
diagonal map in $U$ with eigenvalues $0<\lambda_1<1<\lambda_2$.
\smallskip
Fix $\epsilon >0$ sufficiently small so that the ball $B_\epsilon (x_0)
\subset U$ and the lifts by $P$ of it are disjoints
$\epsilon$-balls $B_\epsilon (x_i)\subset \TT^2_*$, $i=1,...,4$ (this fact is
realized by taking $\epsilon<\frac{1}{2}$).
Restricted to these balls, $f_*$ is a linear map.
\smallskip
Let $x$ be in $ \TT^2_*$, we denote by
$W^s (x)$ [resp. $W^u (x) $] the stable [resp. unstable] manifold of $x$ for
$f_*$. For any $0<\delta\leq \epsilon$ and $x\in\TT^2_*$, let's denote by
$W_\delta^s (x)$ [resp. $W_\delta^u (x) $] the connected component of $W^s
(x) \cap B_\delta (x)$ [resp. $W^u (x) \cap B_\delta (x)$] containing $x$.
\medskip
\noindent Since $W^s (x_1) $ and $W^u (x_1)$ have a transverse intersection
in some point $p$, there is:
-$N\in \mathbb N$ such that for $k\geq N$, $f_*^k(p) \in W^s_\epsilon (x_1)$ (these
points converge monotonically to $x_1$ when $k$ goes to $+\infty$) and
-$M\in \mathbb N$ such that for $k\geq M$, $f_*^{-k}(p) \in W^u_\epsilon (x_1)$
(these points converge monotonically to $x_1$ when $k$ goes to $+\infty$).
\smallskip
Consider a small arc $\Sigma^u$ of $W^u (x_1) $ containing $f_*^N(p)$, the
segments $f_*^k(\Sigma^u)$ for $k\geq N$ become larger and more vertical with
$k$, so there is $ r\geq N$ minimal such that $d(x_1, f_*^r(p))\leq\frac
{\epsilon }{2}$ and the arc $f_*^m(\Sigma^u)$ intersects transversally the
boundary of $ B_\epsilon (x_1)$ in two points.
Analogously, consider an arc $\Sigma^s$ of $W^s (x_1) $ containing $f_*^{-M}(p)$,
there is $ r'\geq M$ minimal such that $d(x_1, f_*^{-r'}(p))\leq\frac
{\epsilon }{2}$ and the arc $f_*^{-n'} (\Sigma^s)$ (almost
horizontal) intersects transversally the boundary of $ B_\epsilon (x_1)$ in
two points.
\smallskip
The arcs $f_*^r(\Sigma^u)\cap B_\epsilon (x_1)$ and
$f_*^{-r'} (\Sigma^s)\cap B_\epsilon (x_1)$ intersect transversally.
\smallskip
Finally, we define a rectangle $R_1$ in $\TT^2_*$ whose boundary is
the union of arcs $C_{1,s}^j$ for $j=1,2$ included in $W^s(x_1)$
and arcs $C_{1,u}^j$ for $j=1,2$ included in $ W^u(x_1)$. In fact
$x_1$ is a corner of $R_1$ and it is the intersection of the sides
$C_{1,s}^1$ and $C_{1,u}^1$ which are included in
$W^s_{\frac{\epsilon}{2}}(x_1)$ and $W^u_{\frac{\epsilon}{2}}(x_1)$
respectively, the two other sides are $C_{1,s}^2 \subset
f_*^m(\Sigma^s)\cap B_\epsilon (x_1)$ and $C_{1,u}^2 \subset
f_*^{-m'}(\Sigma^u)\cap B_\epsilon (x_1)$. By definition, the
diameter of $R_1$ is less than $\epsilon$.
\medskip
Let $R_0= P(R_1)$ be a rectangle in $\TT^2$. For $i=1\ldots 4$, denote by
$\gamma_i$ the automorphism of the finite covering $P$ such that $\gamma_i
(x_1) =x_i$, and let $R_i=\gamma_i(R_1)$ be a rectangle in $\TT^2_*$ and for
$j=1,2$ we set $C_{i,u}^j=\gamma_i(C_{1,u}^j)$, $C_{i,s}^j=\gamma_i(C_{1,s}^j)$
(see Figure 1).
\begin{figure}[h]\psfrag{f}{$f^{nk}
$}\psfrag{f4}{$f_*$}\psfrag{t2}{$\scriptstyle T^2
$}\psfrag{t24}{$\scriptstyle T^2_*$}\psfrag{p}{$\scriptstyle P$}
\psfrag{0}{$\scriptstyle x_0 $}\psfrag{1}{$\scriptstyle x_1$}
\psfrag{2}{$\scriptstyle x_3$}\psfrag{3}{$\scriptstyle x_4$}
\psfrag{4}{$\scriptstyle x_2$}\psfrag{r1}{$\scriptstyle R_1$}
\psfrag{r2}{$\scriptstyle R_3$}\psfrag{r3}{$\scriptstyle R_4$}
\psfrag{r4}{$\scriptstyle R_2$}\psfrag{c11}{$\scriptstyle
C^1_{1,u}$} \psfrag{c21}{$\scriptstyle C^2_{1,s}\subset
W^s_{f_*}(x_1)$}\psfrag{c21u}{$\scriptstyle C^2_{1,u}\subset
W^u_{f_*}(x_1)$} \psfrag{c11s}{$\scriptstyle C^1_{1,s} $}
\begin{center}\caption{Stable and unstable sides of $R_i$.}\includegraphics[scale=.4]{dibujo8}
\end{center}
\end{figure}
Fix $i\in
\{2,3,4\}$, as $f$ is topologically transitive and irreducible then Corollary \ref{letoptran} implies that $f_*$ is topologically transitive so there exists $m_i \in \N$ such that $f_*^{m_i}(R_1) \cap \inte(R_i) \neq
\emptyset$.
\medskip
It is not possible that $f_*^{m_i}(R_1) \supset R_i$. In fact,
projecting via $P$ on $\TT^2$, we obtain that $f^{nkm_i}(R_0)
\supset R_0$. According to the Hartman-Grobman Theorem,
the map $f^{nk}$ has a unique fixed point in $R_0$ which is the
hyperbolic saddle point $x_0$. Hence, it is not possible that
$f^{nkm_i}(R_0) \supset R_0$.
Furthermore, it is not possible that $x_i$ belongs to $ f_*^{m_i}(R_1) $ since $x_i$ is a $f_*$-fixed point
and $R_i$ and $R_1$ are disjoint.
\medskip
As $W^u(x_1) \cap W^u(x_i)= \emptyset$ for $i\neq 1$ we
have that there exists $l_i\geq m_i$ such that
$f_*^{l_i}(C_{1,u}^1)\cap (C_{i,s}^1\cup C_{i,s}^2)\neq \emptyset$
and this intersection is topologically transversal.
\medskip
There is no loss
of generality if we suppose that $f_*^{l_i}(C_{1,u}^1)\cap
C_{i,s}^1\neq \emptyset$, i.e $W^u(x_1) \cap W_{
\epsilon/2}^s(x_i)\neq \emptyset$.
\medskip
Since $W^u(x_1)$ is topologically transversal to $ W_{\epsilon/2}^s(x_i)$
we can assert that there exists $N_i$ such that at
least one connected component of $f_*^{m}(C_{1,u}^1)\cap R_i$ has one end
point in $C_{i,s}^1$ and another one in $ C_{i,s}^2$ for all $m \geq
N_i$.
\medskip
Let us define $B_i$ as a small subrectangle in $R_i$ whose boundary
contains $C_{i,u}^1$ and stable and unstable arcs. We take the
stable sides of $B_1$ contained in the stable arcs of the boundary
of $R_1$. We choose the other unstable side of $B_1$, $ L_u$ close
enough to $C_{1,u}^1$ so that a connected component of
$f_*^{m}(L_{u})\cap R_i$ has one end point in $C_{i,s}^1$ and
another one in $ C_{i,s}^2$ for all $m \geq N_i$. Hence, one
connected component of $f_*^{m}(B_1)\cap R_i$ is a compact set with
nonempty intersection with $C_{i,s}^1$ and with $ C_{i,s}^2$, for $m
\geq N_i$. Let $B_i \subset R_i$ verifying $P(B_i)= P(B_1)$.
\begin{figure}[h]\psfrag{f}{$f^{nk}
$}\psfrag{f4}{$f_*$}\psfrag{t2}{$T^2
$}\psfrag{t24}{$T^2_*$}\psfrag{p}{$P$} \psfrag{0}{$x_0
$}\psfrag{1}{$x_1$} \psfrag{2}{$x_2$}\psfrag{3}{$x_3$}
\psfrag{4}{$x_4$}\psfrag{r1}{$R_1$}
\psfrag{r2}{$R_2$}\psfrag{r3}{$R_3$}
\psfrag{r4}{$R_4$}\psfrag{c11}{$C^1_{1,u}\subset W^u_{f_*
\frac{\varepsilon }{2}}(x_1)$} \psfrag{c21}{$C^2_{1,s}\subset
W^s_{f_* }(x_1)$}\psfrag{c21u}{$C^2_{1,u}$}\psfrag{lu}{$L_u$}
\psfrag{pbi}{$P(B_3)$} \psfrag{c11s}{$C^1_{1,s}
$}\psfrag{fm1}{$f^{-m}_4(D_3)$} \psfrag{bi}{$B_3$}
\psfrag{b1}{$B_1$} \psfrag{di}{$D_3$}
\begin{center}\caption{The sets $B_i$ and $D_i$.}\includegraphics[scale=.4]{dibujo022}
\end{center}
\end{figure}
\medskip
There exists $n_i\geq N_i$ such that for all $m\geq n_i$ one connected
component denoted by $D_i$ of $f_*^{m}(B_1)\cap B_i$ is a compact
set included in $B_i$ with nonempty intersection with $C_{i,s}^1$
and with $ C_{i,s}^2$; and with empty intersection with the unstable
sides of $B_i$. One can show that the set $f_*^{-m}(D_i)$
is connected, compact and contained in $B_1$. It intersects the
unstable sides of $B_1$ and it does not intersect the stable
sides of $B_1$ (see Fig 2).
\noindent It follows that $\displaystyle \bigcap_{j=-N}^N f^{nkmj}(P(D_i))$
has the finite intersection property.
\noindent Consequently, the compact set (depending on $i$ and $m$)
in $\TT^2$ defined by: $$\displaystyle
L=\bigcap_{j=-\infty}^{\infty} f^{nkmj}(P(D_i))$$ is non empty,
$f^{mnk}-$invariant and it's contained in $R_0=P(R_i)\subset \TT^2$.
\smallskip
In what follows, we argue for $m=n_i$. For $j=1,...,4$, let $L^i_j=
P ^{-1}(L) \cap R_j$. The set $\displaystyle
\bigcup_{j=1}^4L_j^i$ is $f_*^{n_i}$-invariant. Moreover, since
$f_*$ is surjective and $L_1^i\subset f_*^{-n_i}(D_i)$, we have that
$f_*^{n_i}(L_1^i)=L_i^i$. Therefore, we have proved that
there exist an integer $n_i$ and compact sets $L_1^i$ and $L_i:= L_i^i$
such that $P(L_i)=P(L_1^i)$ and $f_*^{n_i}(L_1^i)=L_i$. We get
the proposition 1. \ \ \ \ \ \ $\blacktriangleleft$
\bigskip
\noindent {\bf Proof of the theorem.} We prove that the proposition
1 implies the theorem.
Since $\widetilde{x_0}$ is a fixed point of $\tilde f_*$ it follows that
$\rho(\tilde f_*,\widetilde{x_0})=(0,0)$.
By proposition 1 for $i=2$, there exist $n_2$ and non empty
compact sets $L_2\subset R_2$ and $ L_1^2\subset R_1$ such that
$P(L_2)=P(L^2_1)$ and $f_*^{n_2}(L^2_1) = L_2$.
Let us denote ${\mathcal L}_2$ [resp. ${\mathcal L}^2_1$] a lift
of $L_2$ [resp. $L^2_1$] to $\mathbb R^2$. Then there exist
$\mathbf{k_2} \in(2\mathbb{Z}) ^2$ such that $\tilde f_*^{n_2}({\mathcal
L}^2_1) = {\mathcal L}_2+\mathbf{k_2}$. Since $P(L_2)=P(L^2_1)$ and
$L_2 \subset B_{x_2} (\varepsilon)$, we have that necessarily
${\mathcal L}_2 ={\mathcal L}^2_1+(1,0)$. Therefore
$$
\tilde f_*^{n_2}({\mathcal L}_1^2) = {\mathcal L}_1^2+\mathbf{k_2}+(1,0).
$$
Hence, $\tilde f_*^{ n_2}({\mathcal L}_1^2+\mathbf{k_2}+(1,0))= \tilde f_*^{
n_2} ({\mathcal L}_1^2)+ (\mathbf{k_2}+(1,0))= {\mathcal L}_1^2 +
2(\mathbf{k_2}+(1,0))$ and for every $k \in \mathbb{N}$
$$
\tilde f_*^{kn_2}({\mathcal L}_1^2)={\mathcal L}_1^2+ k (\mathbf k_2+(1,0)).
$$
Let $\widetilde{x} \in {\mathcal L}_1^2$. For every $k$, there exists
$\widetilde{y_k} \in {\mathcal L}_1^2$ such that $\tilde f_*^{k n_2}(\widetilde{x})=
\widetilde{y_k} +k(\mathbf{k_2}+(1,0))$.
It follows that
$$\frac{\tilde f_*^{k n_2}(\widetilde{x})-\widetilde{x}}{k n_2}=\frac{\widetilde{y_k}-\widetilde{x}}{k n_2}+\frac{k(\mathbf k_2+(1,0))}{k
n_2}$$ Then $$\lim_{k \rightarrow \infty} \frac{\tilde f_*^{k
n_2}(\widetilde{x})-\widetilde{x}}{k
n_2}=\frac{\mathbf{k_2}+(1,0)}{n_2}.$$ Hence,
$$\frac{\mathbf{k_2}+(1,0)}{n_2} \in \rho(\tilde f_*).$$
Analogously, for $i=3$ there exist integers ${n_3}$, $k_3$ and a compact set
${\mathcal L}^3_1 $ in $\mathbb R^2$ such that
$$\tilde f_*^{n_3}({\mathcal L}^3_1) = {\mathcal L}^3_1+\mathbf{k_3}+(0,1).$$
\noindent It comes that $$\frac{\mathbf{k_3}+(0,1)}{n_3} \in \rho(\tilde f_*).$$
\medskip
Finally, $(0,0) \in \rho(\tilde f_*)$ and the vectors
$\frac{\mathbf{k_2}+(1,0)}{n_2}$ and
$\frac{\mathbf{k_3}+(0,1)}{n_3}$ are linearly independent.
\medskip
Actually, for $i=2,3$ let us write $\mathbf{k_i} =
(2p_i, 2q_i)$ and compute the determinant:
$$\det\left(n_2\frac{\mathbf{k_2}+(1,0)}{n_2},n_3
\frac{\mathbf{k_3}+(0,1)}{n_3}\right) = \left\vert \begin{array}{cc} 2p_2
+1 & 2p_3 \\2q_2 & 2q_3+1
\end{array}\right\vert \ne 0, $$\noindent since it is the difference between an even number
and an odd number.
Then, it follows that $\rho(\tilde f_*)$ has 3 non colinear points. By
convexity (see \cite{MZ91}) of $\rho(\tilde f_*)$, we have that
$\inte(\rho(\tilde f_*))\neq \varnothing,$ for a lift $\tilde f_*$ of $f^{nk}$ to $\RR^2$. Thus,
this property holds for any lift of $f^{nk}$ and therefore for any lift of $f$
to $\RR^2$. \end{proof}
\section{Proof of the total transitivity.}
Let $T^2_h:=\mathbb{R}^2/(2\mathbb{Z}\times \mathbb Z)$
(resp. $T^2_v:=\mathbb{R}^2/(\mathbb Z \times 2\mathbb{Z})$) be a 2-1
covering of $\TT^2=\mathbb{R}^2/\mathbb{Z}^2$, and let the natural projection be
$\Pi_h\colon T^2_h \rightarrow T^2$ (resp. $\Pi_v\colon T^2_v \rightarrow \TT^2$).
Let $f_h\colon T^2_h \rightarrow T^2_h$ be the lifting\ of $f$ to $T^2_h$
(resp. $f_v\colon T^2_v \rightarrow T^2_v$ be the lifting of $f$ to
$T^2_v$)
\begin{lem}\label{lemm:2-top}
Let $f\colon \mathbb T^2\to\mathbb T^2$ be a torus homeomorphism satisfying that $f$ is topologically transitive and irreducible
then
\begin{enumerate}[(1)]
\item the 2-1 coverings $f_h$ and $f_v$ are topologically transitive;
\item $f$ is totally transitive.
\end{enumerate}
\end{lem}
\begin{cor}
\label{letoptran}
Let $f\colon \mathbb T^2\to\mathbb T^2$ be a torus homeomorphism
topologically transitive and irreducible. Let
$F$ be the lift of $f$ defined in the proof of the theorem.
Then $F$ is totally transitive \end{cor}
\begin{proof}[Proof of the Corollary]
By definition, $F= (f_h)_v= (f_v)_h$. According to the previous
lemma $f_h$ is topologically transitive and by the proposition
\ref{prop1} (3) it is irreducible, so we can apply once again this lemma
to $f_h$ and obtain that $(f_h)_v$ is totally
transitive.\end{proof}
\begin{proof}[Proof of lemma \ref{lemm:2-top}] \
\begin{enumerate}[(1)]
\item
We will argue by absurd for $f_h$. We suppose that $f$ is transitive
but not $f_h$.\\
Since $f$ is transitive, there exists $x_0$ such that $
O_f(x_0):=\{f^n(x_0) :n \in\mathbb Z\}$ is a dense set in $\TT^2$. Let
$\{x_1,x_2\}$ be the lifts of $x_0$ by $\pi^{-1}_h$. Let $\mathcal
O(x_i)$ be the $f_h$-orbit of $x_i$. We have that $\mathcal
O(x_1)\cup\mathcal O(x_2)=\pi^{-1}_h(O_f(x_0))$ is a dense set in
$T^2_h$ since $\pi_h$ is a local homeomorphism. Since $f_h$ is not
transitive, neither $\mathcal O(x_1)$ nor $\mathcal O(x_2)$ is
dense. We claim that $\inte(\overline{\mathcal O(x_2)})\cap
\overline{\mathcal
O(x_1)}=\varnothing$. Actually if there were a point $y$ in the intersection,
it would exist $m\in\mathbb Z$ such that $f_h^m(x_1)$ belongs to
$\overline{\mathcal
O(x_2)}$ then $\mathcal O(x_1)\subset \overline{\mathcal O(x_2)}$. Thus
$\overline{\mathcal O(x_1)}\subset \overline{\mathcal O(x_2)}$ so we have $T^2_h=\overline{\mathcal O(x_1)}\cup\overline{\mathcal O(x_2)}= \overline{\mathcal O(x_2)}
$: a contradiction.\\
Analogously, the symmetric holds and these two equalities imply that
$\partial\overline{\mathcal O(x_1)}=\partial\overline{ \mathcal O(x_2)}$ and
$T^2_h=\inte (\overline{\mathcal O(x_1)})\sqcup \inte (\overline{\mathcal
O(x_2)})\sqcup\partial \overline{\mathcal O(x_1)}$, where $\sqcup$
denotes disjoint union.
The set $\partial\overline{\mathcal O(x_1)}$ is a closed invariant
of empty interior subset that disconnects $T^2_h$.
\medskip
We are going to prove that it can
not be contained in a disk $D\subset T_h$.
\smallskip
\noindent Suppose, by absurd that
$\partial\overline{\mathcal O(x_1)}\subset D$. First, we prove that $\inte \overline{\mathcal O(x_1) }$ or
$\inte \overline{\mathcal O(x_2)}$ is
included in $D$. In fact,
if both of them intersect the complement $ D ^c$
of $D$, we can take a path in $ D ^c$ joining a point of $\inte
\overline{\mathcal O(x_1)}$ and a point of $\inte
\overline{\mathcal O(x_2)}$. By connexity, this path must contain a point
of the boundary $\partial\overline{\mathcal O(x_1)}$, which contradicts
the fact that $\partial\overline{\mathcal O(x_1)}\subset D$.
Finally, suppose that $\inte \overline{\mathcal O(x_1)}\subset D $ then
$\inte
\overline{\mathcal O(x_2)}\supset D^c$, but this is not possible since
$\inte
\overline{\mathcal O(x_1)}$ and $\inte \overline{\mathcal O(x_2)}$
are homeomorphic.
We have proved that $\partial\overline{\mathcal O(x_1)}$ is a closed invariant
of empty interior subset that disconnects $T^2_h$ and that is not contained
in a disk. But this is a contradiction with the fact that $f$ is irreducible.
\item
We suppose that $f$ is transitive
but there exists a positive integer $N$ such that $f^N$ is not.\\
So, there exists $x_0$ such that $
O_f(x_0)$ is a dense set in $\TT^2$, but $O_{f^N}(x_0)$ is not.
We have $$\overline{O_{f^N}(x_0)}\cup \overline{O_{f^N}(f(x_0))}\cup...\cup\overline{O_{f^N}(f^{N-1}(x_0))}=\overline{O_{f}(x_0)}=\TT^2.$$
As in the proof of the first item of this lemma we have that if $ int(\overline{O_{f^N}(f^i(x_0))})\cap \overline{O_{f^N}(f^j(x_0))}\neq \emptyset$ then $\overline{O_{f^N}(f^i(x_0))}\supseteq \overline{O_{f^N}(f^j(x_0))}$.
Let $K=\min\{i=1,...,N-1 : int(\overline{O_{f^N}(x_0)})
\cap \overline{O_{f^N}(f^i(x_0))}\neq \emptyset \}$.
Then $\TT^2$ can be decomposed as union of closed sets with disjoint interiors:$$ \TT^2= \overline{O_{f^N}(x_0)}\cup \overline{O_{f^N}(f(x_0))}\cup...\cup\overline{O_{f^N}(f^{K-1}(x_0))}.$$
In the case that $K$ is an even number then $ \TT^2= A\cup B$, where $A= \overline{O_{f^N}(x_0)}\cup \overline{O_{f^N}(f(x_0))}\cup...\cup\overline{O_{f^{N}}(f^{\frac{K}{2}}(x_0))}$ and $B=\overline{O_{f^{N}}(f^{\frac{K}{2}+1}(x_0))} \cup...\cup\overline{O_{f^N}(f^{N-1}(x_0))}.$ The sets $A$ and $B$ have disjoint interiors, their boundaries coincide and they are $ f^N$- invariant. Thus $ \TT^2= int(A)\sqcup int(B)\sqcup \partial B $. According to proposition \ref{prop1} $f^N$ is irreducible, then $\partial B$ is contained in a disk. As $A$ and $B$ are homeomorphic, an analogous proof to the previous item shows that this can not occur.
In the case that $K$ is an odd number then $ \TT^2= A\cup B \cup C$, where $A= \overline{O_{f^N}(x_0)}\cup \overline{O_{f^N}(f(x_0))}\cup...\cup\overline{O_{f^{N}}(f^{\frac{K-1}{2}}(x_0))}$ and $B=\overline{O_{f^{N}}(f^{\frac{K+1}{2}}(x_0))} \cup...\cup\overline{O_{f^N}(f^{N-2}(x_0))},$ and $C=\overline{O_{f^N}(f^{N-1}(x_0))}$. The sets $A$ and $B\cup C$ have disjoint interiors, their boundaries coincide and they are $ f^N$- invariant. Thus $ \TT^2= int(A)\sqcup int(B \cup C)\sqcup \partial A $. According to proposition \ref{prop1} $f^N$ is irreducible, then $\partial A$ is contained in a disk $D$.
By connexity, as in the previous item, we prove that $A$ or
$B\cup C$ is included in $D$.
Finally, if $ A\subset D $ then $ B\subset f^{\frac{N+1}{2}}(D)$ and $ C\subset f^{N-1}(D)$.
If $ B\cup C\subset D $ then $ A\subset f^{-\frac{N+1}{2}}(D)$.
Then, in both cases, the torus is the union of at most three disks which is impossible.
So we have proved that $f$ is totally transitive.
\end{enumerate}
\end{proof}
\begin{rem}
The topological transitivity of $f$ is not enough to guarantee that
$f_h$ (or $f_v$) is topological transitive. For example, consider a
diffeomorphism $f$ of $\mathbb{A}= [0,1]\times S^1$ obtained by the
Katok-Anosov process (see \cite{AnKa}) in such a way that:
\begin{itemize}
\item $f$ is topologically transitive in $
\inte(\mathbb{A})=(0,1)\times S^1$, and
\item $f(0,x)=f(1,x)$ for all $x\in S^1$.
\end{itemize}
We collapse the circles $\{0\}\times S^1$ and $\{1\}\times S^1$ by
identifying $(0,x)$ with $(1,x)$ for $x\in S^1$. Then, we have a
diffeomorphism of the torus, $\widehat{f}$, verifying that
$\widehat{f}$ is topologically transitive on $T^2$ and $C=\{0\}
\times S^1$ is a circle invariant of $\widehat{f}$.
Let us consider the finite covering $T^2_h$ of the torus $T^2$ and
let $\widehat{f}_h$ be
the lifting of $\widehat{f}$ to $T^2_h$.
The lifting of $C$ is the union of two circles $C_1$ and $C_2$ that disconnect
$T^2_h$, and the $T^2_h \backslash (C_1 \cup C_2)$ is the disjoint
union of two cylinders.
The orbits of $\widehat{f}_h$ are dense in each cylinder but $\widehat{f}_h$
is not topologically transitive.
\end{rem}
\section{Examples}
In this section, we give examples in order to show that each hypothesis of the
theorem is necessary.
\begin{enumerate}[{1)}]
\smallskip
\item Missing hypothesis \ref{ent}.
Let $R_{\alpha,\beta}$ be the rotation of vector $(\alpha, \beta)$ with $\alpha$ and
$\beta$ irrational, that is, the projection to $T^2$ of the
translation of vector $(\alpha, \beta)$ in $R^2$. It is a well known
fact that $R_{\alpha,\beta}$ is topologically transitive, it is
differentiable and it is irreducible, but its rotation set is
$\{(\alpha,\beta)\}$. This example shows that conditions \ref{diff},
\ref{top}, and \ref{irred} of the theorem \ref{teo1} do not ensure
that the interior of the rotation set is not empty.
\medskip
\item Missing hypothesis \ref{top} and \ref{irred}.
\label{ex:ent}
Let $f_D\colon \overline{\mathbb D^2} \to\overline{\mathbb D^2}$ be a
diffeomorphism such that there exists a horseshoe in the interior of
$\mathbb D^2$ and such that $f_D$ is the identity on $\partial
\overline{\mathbb D^2}$. It follows that $f_D$ has positive
entropy. Let us embed $\overline{\mathbb D^2}$ in $\mathbb T^2$ and
then extend $f_D$ to $f$ by the identity on $\mathbb
T^2\backslash \overline{\mathbb D^2}$. It holds that $f$ has
positive entropy and the rotation set has empty interior(because
there exist invariant circles homotopically non trivial). This
example show that conditions \ref{diff} and \ref{ent} do not ensure
that the conclusion of the theorem is verified.
\medskip
\item
\label{ex:entro}
Missing hypothesis \ref{top}.
We start with an irrational flow $\phi^t_0$ on $T^2$. By making an
appropriate smooth time change vanishing at one point $x_0$ (we
replace the vector field $X$ by $g.X$ where $g(x_0) =0 $ and
$Dg(x_0) =0 $), we get a new smooth topologically transitive flow
$\phi^t_g$ with a fixed point $x_0$. Consider the time one map of
this flow, $f$, and replace $x_0$ by a small closed disk $D_0$ by
blowing up. The dynamic of the blow up of $f$ on $\partial D_0$ is
of the type north-south. We have that $D_0 \backslash \{ N, S\}$ is
foliated by meridians $\{M_t\}_{t\in [-1,1]}$ and $\partial
D_0=M_{-1} \cup M_1 \cup \{ N, S\}$. Let $\gamma\colon D_0 \rightarrow
D_0$ a differentiable map such that $\gamma |M_i= f|M_i$, for
$i=-1,1$, $\gamma(N)=N$, $\gamma (S)=S$, for all $t\in [-1,1]$ $M_t$
is $\gamma$-invariant and $\gamma|M_t=Id$ for $t \in [-\frac 12,
\frac 12]$. In the blow up manifold, $T^2$ we define the
differentiable map $\Gamma$ as $\Gamma(x)= f(x)$ if $x\in T^2 / D_0
$ and $ \Gamma (x)=\gamma(x) \mbox{ if } x\in D_0$. Let $D_1=
\cup_{t\in [- \frac 12, \frac 12]} M_t \cup \{ N, S\}$, it holds
that $\Gamma |D_1= Id$.
As in the previous example, we can put a horseshoe
in the interior of $D_1$.
The resulting diffeomorphism satisfies trivially the
conditions \ref{diff}, \ref{ent}.
It also verifies the condition \ref{irred}. Moreover, an invariant compact
set $K$ of $\Gamma$ is included either:
\begin{itemize}
\item in $D_0$, in this case $K$ is not essential or
\item in the complement $D_0^c $ of $D_0$, in this case $K$ coincides with
$D_0^c $ (since each orbit in $D_0^c $ is dense in it) hence its interior is
not empty.
\end{itemize}
But it does not satisfy the condition \ref{top}, since it
has an invariant disk.
Finally, its rotation set has empty interior. In fact, before the blowing
up, the map $f$ is the time one map of a flow with a fixed point $x_0$ so according to
Franks and Misiurewicz's result (see \cite{FM}) its rotation set is a line
segment containing $(0,0)$. The blowing up does not change the
rotation set because of the following facts:
\begin{itemize} \item the points in $D_0$ have
the same rotation vector as $x_0$ which is $(0,0)$ ($D_0$ is $\Gamma$-invariant),
\item for the points out of $D_0$, the blowing up does not
change the orbits so it does not change their rotation vectors.
\end{itemize}
\medskip
\item Missing hypothesis \ref{diff}.
According to \cite{Rees} there exists a torus homeomorphism $f_0$
isotopic to the identity such that it is minimal
and it has positive entropy. Since $f_0$ is minimal, all its orbits are dense so it has no periodic points.
By \cite{F89} we know that if the interior of the rotation set is not empty, then
each vector with rational coordinates in
the interior of the rotation set is realized as the rotation vector of a periodic
point. It follows that $f_0$ verifies that
$\inte(R(f_0))=\varnothing$.
This example shows that conditions \ref{ent}, \ref{top} and
\ref{irred} are not enough to guarantee that the interior of the
rotation set is not empty.
\label{ex:diff}
\item Missing hypothesis \ref{irred}
According to \cite{Kato} there exists a $C^{\infty}$ topologically transitive Bernoulli
diffeomorphism $f_0\colon S^2 \to S^2$ which preserves a smooth
positive measure on $S^2$. Since $f_0$ (or $f_0^2$) preserves
orientation then it is isotopic to the identity.
As in the construction of \cite{Kato}, there exist $x_1, x_2$, two
fixed points of $f_0$ ( or $f_0^k$) such that $D f_0(x_i)=Id, \
i=1,2.$ We can replace $x_1$ and $x_2$ by small closed disks $D_1$
and $D_2$, respectively, by blowing up. The dynamic of the blow up
of $f_0$ on $\partial D_1$ and $\partial D_2$ is the identity. By
gluing $\partial D_1$ and $\partial D_2$ we have a smooth map
$f\colon T^2 \to T^2$ which is topologically transitive and it has
positive entropy but there exists a compact $f$-invariant of empty
interior set ($\partial D_1$) which is essential. This example fails
to be irreducible because of the existence of a non null homotopic
invariant circle, then its rotation set has empty interior because
of Llibre and Mac
Kay's result
( see \cite{LM91}).
\label{ex:irred}
\end{enumerate}
| 166,855
|
TITLE: How to calculate the Rise speed of a travelling wave?
QUESTION [0 upvotes]: As you can see in this wikipedia animation
Phase velocity (red) and group velocity (green) are described, and both show different "speed" in the Propagation direction but what about the transverse direction speed?
How do I calculate the rise speed of a cycle in transversal direction ?
(Look at the extremes of the picture the blue pixel going up and down, this is the vertical up-down speed what I want to calculate)
For example in a very simple wave like this
y(x,t) = A* sin(kx - w t), should I calculate derivative of y having x=constant?
What about in general? How to calculate the Rise speed?
REPLY [3 votes]: Once you know the phase velocity $\omega$, you know that along the $x$ axis the red dot is described by $x(t) = \omega t$. Then you just apply the chain rule. The value of $y(x,t)$ at the red dot is a function of time $y(x(t),t)$. So
$$ \frac{d}{dt}y(x(t),t) = \frac{\partial}{\partial x}y(x(t),t) \cdot \frac{d}{dt}x(t) + \frac{\partial}{\partial t} y(x(t),t) $$
which we more succinctly write as
$$ \omega \partial_x y + \partial_t y $$
| 183,426
|
\begin{document}
\title[Curvature-adapted submanifolds of Lie groups]{Curvature-adapted submanifolds\\of semi-Riemannian groups}
\author{Margarida Camarinha}
\address{University of Coimbra\\
CMUC\\
Department of Mathematics\\
3001-501 Coimbra\\
Portugal}
\email{mmlsc@mat.uc.pt}
\thanks{This work was partially supported by the Centre for Mathematics of the University of Coimbra -- UIDB/00324/2020, funded by the Portuguese Government through FCT/MCTES.\endgraf
The second author was supported by Austrian Science Fund (FWF) project F 77 (SFB ``Advanced Computational Design'').}
\author{Matteo Raffaelli}
\address{University of Coimbra\\
CMUC\\
Department of Mathematics\\
3001-501 Coimbra\\
Portugal}
\curraddr{Institute of Discrete Mathematics and Geometry\\
TU Wien\\
Wiedner Hauptstra{\ss}e 8-10/104\\
1040 Vienna\\
Austria}
\email{matteo.raffaelli@tuwien.ac.at}
\date{January 25, 2022}
\subjclass[2020]{Primary: 53C40; Secondary: 53B25, 53C30}
\keywords{Abelian normal bundle, bi-invariant metric, closed normal bundle, curvature adapted, invariant shape operator, semi-Riemannian group}
\begin{abstract}
We study semi-Riemannian submanifolds of arbitrary codimension in a Lie group $G$ equipped with a bi-invariant metric. In particular, we show that, if the normal bundle of $M \subset G$ is closed under the Lie bracket, then any normal Jacobi operator $K$ of $M$ equals the square of the associated invariant shape operator $\alpha$. This permits to understand curvature adaptedness to $G$ geometrically, in terms of left translations. For example, in the case where $M$ is a Riemannian hypersurface, our main result states that the normal Jacobi operator commutes with the ordinary shape operator precisely when the left-invariant extension of each of its eigenspaces remains tangent to $M$ along all the others. As a further consequence of the equality $K = \alpha^{2}$, we obtain a new case-independent proof of a well-known fact: every three-dimensional Lie group equipped with a bi-invariant semi-Riemannian metric has constant curvature.
\end{abstract}
\maketitle
\tableofcontents
\section{Introduction and main result} \label{IntroductionMainResult}
Given a Riemannian manifold, it is natural to study submanifolds whose geometry is somehow adapted to that of the ambient space. This idea led to the concept of \emph{curvature-adapted} submanifold \cite{datri1979, berndt1991, berndt1992-2}; a concept that, since its introduction, has attracted the interest of many geometers.
Here we define curvature adaptedness in a slightly more general setting, that of a \emph{semi-Riemannian} manifold $(Q,g)\equiv Q$.
Let $M$ be an $m$-dimensional semi-Riemannian submanifold of $Q$, let $N^{1}M$ be its unit normal bundle, and let $R$ be the ambient curvature tensor. For $(p,\eta)\equiv \eta \in N^{1}_{p}M$, the \textit{normal Jacobi operator}
\begin{align*}
K_{\eta} \equiv K \colon T_{p}M &\to T_{p}Q \\
v &\mapsto R(\eta,v)\eta
\end{align*}
\textit{of $M$ \textup{(}with respect to $\eta$\textup{)}} measures the curvature of the ambient manifold along $\eta$. On the other hand, denoting by $N$ a unit normal local extension of $\eta$ along $M$, and by $\nabla$ the Levi-Civita connection of $Q$, the \textit{shape operator}
\begin{align*}
A_{\eta} \equiv A \colon T_{p}M &\to T_{p}M \\
v &\mapsto \pi^{\top} \grad_{v}N
\end{align*}
\textit{of $M$ \textup{(}with respect to $\eta$\textup{)}} describes the curvature of $M$ as a submanifold of $Q$. Here $\pi^{\top}$ denotes orthogonal projection onto $T_{p}M$.
\begin{definition}\label{curvatureAdaptednessDEF}
One says that $M$ is \textit{curvature adapted \textup{(}to $Q$\textup{)} at a point $p\in M$} if, for every $\eta \in N^{1}_{p}M$,
\begin{enumerate}[label=(\arabic*)]
\item \label{cond1} The normal Jacobi operator leaves $T_{p}M$ invariant, i.e., $K(T_{p}M) \subset T_{p}M$;
\item \label{cond2} The operators $A$ and $K$ commute, i.e., $K \circ A = A \circ K$.
\end{enumerate}
Consequently, one calls $M$ \textit{curvature adapted} if it is curvature adapted at $p$ for all $p \in M$.
\end{definition}
\begin{remark}
Condition \ref{cond1} in Definition \ref{curvatureAdaptednessDEF} is always satisfied for hypersurfaces.
\end{remark}
\begin{remark}
Both $A$ and $K$ are self-adjoint with respect to the induced semi-Riemannian metric. Thus, if these operators are also diagonalizable, then $M$ is curvature adapted at $p$ precisely when they share a common orthonormal basis of eigenvectors (Lemma \ref{OrthogonalEigendecompositionLemma}). Recall that diagonalizability is automatic if the metric is positive definite.
\end{remark}
\begin{remark}
In general, it could also be interesting to consider a weaker notion of curvature adaptedness, obtained by replacing the operator $K$ with $\pi^{\top}K$ in the definition. This modification would make any totally umbilic submanifold necessarily curvature adapted.
\end{remark}
It is easy to see (e.g., using \cite[Lemma~2]{graves1978}) that every semi-Riemannian submanifold of a real semi-Riemannian space form is curvature adapted. However, for other ambient spaces, the definition is restrictive. For example, if $Q$ is an $(m+1)$-dimensional nonflat complex space form with complex structure $J$, then $A$ and $K$ commute precisely when $-J\eta$ is an eigenvector of $A$. Also, if $Q$ is an $(m+1)$-dimensional nonflat quaternionic space form with quaternionic structure $\mathfrak{I}$, then $A$ and $K$ commute precisely when the maximal subspace of $T_{p}M$ invariant under $\mathfrak{I}$ is also invariant under $A$; see \cite{berndt1991} and \cite[sec.~9.8]{cecil2015}.
In a symmetric space of nonconstant curvature, the situation is more involved, yet many interesting results have been obtained. Among others (see for example \cite{koike2005, murphy2012, koike2014-1, koike2014-2}), the most important is arguably Gray's theorem \cite[Theorem~6.14]{gray2004}, which states that any tubular hypersurface around a curvature-adapted submanifold is itself curvature adapted.
Gray's theorem has been further generalized to the family of Riemannian manifolds such that, for every geodesic $\gamma$, the Jacobi operator $R(\dot{\gamma},\cdot)\dot{\gamma}$ is diagonalizable by a \emph{parallel} orthonormal frame field along $\gamma$. It turns out that, in such spaces, the classification of the curvature-adapted submanifolds is fully determined by that of the curvature-adapted hypersurfaces \cite{berndt1992, berndt1992-2}.
In this note, we shall examine the case where $Q$ is a \textit{semi-Riemannian group}, i.e., a Lie group $G \equiv (G, \langle \cdot{,} \cdot \rangle)$ equipped with a bi-invariant semi-Riemannian metric $\langle \cdot{,} \cdot \rangle$. In particular, we will focus our attention on the class of (semi-Riemannian) submanifolds of $G$ having \emph{closed} normal bundle; here by \emph{closed} we mean that each normal space of $M$ corresponds, under the group's left action, to a subspace of $\mathfrak{g}$ that is closed under the Lie bracket, i.e., to a Lie subalgebra (Definition \ref{ClosedNormalBundleDEF}).
Examples of Lie groups $G$ abounds \cite{ovando2016}: every semisimple Lie group can be furnished with a bi-invariant metric, and every compact Lie group admits one that is Riemannian. Note that, if $G$ is simple, then any bi-invariant metric on $G$ is a scalar multiple of the Killing form of $\mathfrak{g}$.
In order to explain our main result, we first set up some notation. Provided $K$ is diagonalizable, let $(e_{1}, \dotsc, e_{m})$ be an orthonormal basis of eigenvectors of $K$, that is, a basis of $T_{p}M$ such that $\lvert \langle e_{j}, e_{h} \rangle\rvert =\delta_{jh}$ and $K(e_{j}) = \lambda_{j} e_{j}$ for all $j,h = 1,\dotsc,m$. Let $(e_{1}^{\mathrm{L}}, \dotsc, e_{m}^{\mathrm{L}})$ be the left-invariant extension of $(e_{1},\dotsc,e_{m})$.
\begin{theorem} \label{TH1}
Assume that the normal space of $M \subset G$ at $p$ is closed under the Lie bracket. If $K$ is diagonalizable \textup{(}resp., if the induced metric on $M$ is positive definite at $p$\textup{)}, then the following are equivalent:
\begin{enumerate}[font=\upshape, label=(\roman*)]
\item \label{item1} The operators $A$ and $K$ commute.
\item \label{item2} For all $j, h \in \{1, \dotsc, m\}$ \textup{(}resp., for all $j < h \in \{1, \dotsc, m\}$\textup{)} such that $\lambda_{j} \neq \lambda_{h}$,
\begin{equation*}
e_{j}\left(\left\langle N, e^{\mathrm{L}}_{h} \right\rangle\right) = 0.
\end{equation*}
\item \label{item3}The left-invariant extension of each eigenspace of $K$ is orthogonal to $N$ along all the others.
\end{enumerate}
\end{theorem}
\begin{remark}
For all $j \in\{1,\dotsc,m\}$, we have $\lambda_{j} = -\mathrm{sec}(e_{j},\eta) \leq 0$; see section \ref{PRE}.
\end{remark}
Clearly, when $M$ is a hypersurface, the condition on the normal bundle is automatically fulfilled. Specializing the theorem to that case, we obtain the result below.
\begin{corollary}
If $\dim G=m+1$ and $K$ is diagonalizable, then the following are equivalent:
\begin{enumerate}[font=\upshape, label=(\roman*)]
\item The operators $A$ and $K$ commute.
\item The left-invariant extension of each eigenspace of $K$ is tangent to $M$ along all the others.
\end{enumerate}
\end{corollary}
The main significance of Theorem \ref{TH1} lies in the fact that it permits to understand curvature adaptedness to $G$ geometrically, in terms of left translations. More precisely, it reveals that, in order for a generic submanifold (with closed normal bundle) to be curvature adapted, its tangent bundle needs to behave reasonably well under left translations. Note that the tangent bundle of a Lie subgroup is fully left-invariant; conversely, if $M$ is a closed, connected submanifold that contains the identity and has left-invariant tangent bundle, then it is a Lie subgroup.
The basic fact that allows us to prove Theorem \ref{TH1} is that the shape operator of $M\subset H$ with respect to $\eta$, being $H$ any Lie group with a left-invariant semi-Riemannian metric, decomposes as the sum of two terms \cite{ripoll1991}: an \emph{invariant shape operator}, which depends only on $\eta$, $T_{p}M$ and $H$; plus a second term, here denoted by $\mathcal{W}$, which is closely related to the Gauss map of $M$; see section \ref{ISOP} for details. In particular, if the metric is bi-invariant and the normal bundle is closed, then the invariant shape operator commutes with $K$ (Proposition \ref{PROP9}), and so, by linearity, commutativity of $A$ and $K$ reduces to that of $\mathcal{W}$ and $K$.
In fact, if $\dim M = 2$ or $M$ is Riemannian, then the nonzero eigenvalues of $K$ have even multiplicities (Corollary \ref{COR11}), which leads us to the following conclusions.
\begin{proposition}\label{PROP4.0}
Every two-dimensional surface with closed normal bundle is curvature adapted to $G$.
\end{proposition}
\begin{proposition}\label{PROP4}
If $M$ is a three-dimensional Riemannian submanifold of $G$ with closed normal bundle, and if $K \neq 0$ for all $\eta \in N^{1}_{p}M$, then the following are equivalent:
\begin{enumerate}[font=\upshape, label=(\roman*)]
\item $M$ is curvature adapted in a neighborhood $U$ of $p$.
\item For all $\eta \in N^{1}U$, the $0$-eigenvector of $K$ is an eigenvector of $A$.
\end{enumerate}
\end{proposition}
Proposition \ref{PROP4.0} implies that, if $\dim G =3$, then every Riemannian surface in $G$ is curvature adapted. This is by no means surprising, because every three-dimensional semi-Riemannian group has constant curvature. Indeed, it is well known that the Lie algebra of such a Lie group is either abelian or isomorphic to $\mathfrak{sl}(2,\mathbb{R})$ or $\mathfrak{su}(2)$ \cite{ovando2016}; in each of the latter two cases, a direct computation would reveal that the curvature of the Killing form (which, up to scaling, coincides with the metric) vanishes. Alternatively, since the Ricci curvature of $G$ is proportional to the Killing form, the statement follows from the fact that every three-dimensional Einstein manifold has constant curvature \cite[p.~49]{besse2008}; see Remark \ref{alternativeProof}.
On the other hand, applying classical results of Cartan and Dajczer--Nomizu (which, for the reader's convenience, are included in section \ref{PRE}), we can prove the following statement.
\begin{lemma}
Suppose that $Q$ is a Riemannian \textup{(}resp., Lorentzian\textup{)} manifold of dimension at least three. If, for some $p\in Q$ and each $x \in T_{p}Q$ such that $\langle x, x \rangle=1$ \textup{(}resp., $-1$\textup{)}, the map from $x^{\perp}$ to $x^{\perp}$ defined by $y \mapsto R(x,y)x$ is a multiple of the identity, then $Q$ has constant curvature.
\end{lemma}
\begin{proof}
Assume the hypothesis of the lemma. If $x,y,z$ are orthonormal vectors in $T_{p}Q$ (resp., if $x,y,z$ are orthonormal vectors in $T_{p}Q$ and $\langle x, x \rangle =-1$), then $\langle R(x,y)z,x\rangle = -\langle R(x,y)x,z\rangle=0$. Applying \cite[Lemma~1.17]{dajczer2019} (resp., \cite[Theorem~1a]{dajczer1980}) and Schur's lemma \cite[p.~96, Exercise 21(b)]{oneill1983}, the claim follows.
\end{proof}
Corollary \ref{COR11} thus yields a \emph{case-independent} proof that, in dimension three, every metric Lie group has constant curvature.
\begin{remark}\label{alternativeProof}
An alternative case-independent proof of the same fact may be sketched as follows. First, by Levi decomposition and the absence of simple Lie algebras of dimension one and two, observe that a three-dimensional Lie algebra is either solvable or simple. However, if a nonabelian solvable Lie algebra of dimension three admits an ad-invariant bilinear form, then such form is necessarily degenerate \cite[Proposition~2.3]{delbarco2014}; in other words, a three-dimensional nonabelian Lie algebra admitting an ad-invariant metric is simple.
Suppose, thus, that $G$ is simple. Then any semi-Riemannian metric on $G$ is a scalar multiple of the Killing form of $\mathfrak{g}$, implying that $G$ is an Einstein manifold. Hence, in dimension three, it has constant curvature \cite[p.~49]{besse2008}.
\end{remark}
The remainder of the paper is organized as follows.
In the next section we briefly review some background material. In section \ref{ISOP} we introduce the invariant shape operator and examine its properties. In section \ref{PROOF} we then prove Theorem \ref{TH1} and Proposition \ref{PROP4}. We conclude with Appendix \ref{APPA}, where, for the sake of illustration, we give a direct proof that condition \ref{item2} in Theorem \ref{TH1} holds whenever $p$ is an umbilical point of $M$.
Some final remarks about notation:
\begin{enumerate}
\item The indices $j,h,i$ satisfy $j,h \in \{1, \dotsc, m\}$ and $i \in \{1, \dotsc, m+n\}$; note that we always use Einstein summation convention.
\item For $x \in T_{p}H$, we denote the left-invariant extension of $x$ by $x^{\mathrm{L}}$.
\end{enumerate}
\section{Preliminaries} \label{PRE}
Here we recall some basic results that are used throughout the paper; see e.g.\ \cite{milnor1976, lee2018, oneill1983} for further details about semi-Riemannian geometry and metric Lie groups.
\subsection{Semi-Euclidean vector spaces}
Let $V$ be a real vector space, of finite dimension $d$, equipped with a nondegenerate symmetric bilinear form $f$, and let $L$ be an endomorphism of $V$ that is self-adjoint with respect to $f$, i.e., such that $f(L(x), y) = f(x,L(y))$ for all $x,y\in V$. Recall that a basis $v_{1}, \dotsc, v_{d}$ of $V$ is called \textit{orthonormal} if $\vert f(v_{j},v_{h}) \rvert= \delta_{jh}$ for all $j,h =1, \dotsc, d$.
It is well known that, if $f$ is positive definite, then $L$ is diagonalizable by an orthonormal basis of eigenvectors of $L$. Moreover, two self-adjoint endomorphisms of $V$ commute if and only if they share a common orthonormal basis of eigenvectors.
While it is not possible to fully extend these classic results beyond the positive definite case, something interesting can still be said.
\begin{lemma} \label{OrthogonalEigendecompositionLemma}
If $L$ is diagonalizable, then there exists an orthonormal basis of eigenvectors. Moreover, any two diagonalizable self-adjoint endomorphisms of $V$ commute if and only if they share a common orthonormal basis of eigenvectors.
\end{lemma}
\begin{proof}
Assume that $L$ is diagonalizable. Since $V$ is the direct sum of \emph{mutually orthogonal} eigenspaces of $L$, each eigenspace must be nondegenerate, and so it has an orthonormal basis; by concateneting these bases, we obtain an orthonormal basis of $V$.
As for the second statement, one direction is obvious; for the other, it suffices to note that any two commuting linear maps on $V$ preserve each other's eigenspaces.
\end{proof}
\subsection{Semi-Riemannian geometry}
Let $(Q,g)$ be a semi-Riemannian manifold, and let $\nabla$ be its Levi-Civita connection. The curvature endomorphism $R \colon \mathfrak{X}(Q)^{3} \to \mathfrak{X}(Q)$ of $(Q,g)$ is the $(1,3)$-tensor field on $Q$ defined by
\begin{equation*}
R(X,Y)Z= \nabla_{X}\nabla_{Y}Z-\nabla_{Y}\nabla_{X}Z-\nabla_{[X,Y]}Z.
\end{equation*}
(Caution: some authors define the curvature endomorphism as the negative of ours.)
Being $R$ a tensor, if $x,y,z$ are vectors in $T_{p}Q$, then the value $R(x,y)z$ is independent of the extension of $x,y,z$ and thus well-defined.
Suppose that $x \in T_{p}Q$ has unit length, i.e., that $\lvert g(x,x) \rvert =1$. Then the \textit{Jacobi operator of $(Q,g)$ with respect to $x$} is the linear map
\begin{align*}
K_{x} \colon x^{\perp} &\to x^{\perp}\\
y &\mapsto R(x,y)x.
\end{align*}
Clearly, by the symmetry by pairs of the $(0,4)$-curvature tensor (obtained by lowering the last index of $R$), the operator $K_{x}$ is self-adjoint with respect to $g$.
Next, suppose that $x,y \in T_{p}Q$ are orthonormal. Then the sectional curvature $\mathrm{sec}(x,y)$ of the nondegenerate plane spanned by $x$ and $y$ is given by the formula
\begin{equation*}
\mathrm{sec}(x,y)= g(R(x,y)y,x)=-g(K_{x}(y),y),
\end{equation*}
where the last equality follows from the skew-symmetry of the $(0,4)$-curvature tensor.
In the Riemannian setting, an important criterion for discerning whether a manifold has constant sectional curvature is provided by the following lemma.
\begin{lemma}[{\cite{cartan1946}, \cite[Lemma~1.17]{dajczer2019}}] \label{RiemLM}
Suppose that $Q$ is a Riemannian manifold, and that $\dim Q \geq 3$. If, at some point $p \in Q$, the curvature tensor satisfies $g(R(x,y)z, x)=0$ whenever $x,y,z$ are orthonormal, then all sectional curvatures of $Q$ at $p$ are equal.
\end{lemma}
Several generalizations of Lemma \ref{RiemLM} to the semi-Riemannian setting have appeared \cite{graves1978, dajczer1980, nomizu1983}. Below we recall the one that is most relevant to our discussion.
\begin{definition}
Let $x,y \in T_{p}Q$. We say that the pair $(x,y)$ is \textit{orthonormal of signature $(-,+)$} if $g(x,x) =-1$, $g(y,y) =1$, and $g(x,y) =0$.
\end{definition}
\begin{theorem}[{\cite[Theorem~1a]{dajczer1980}}]
Suppose that $\dim Q \geq 3$. If, at some point $p \in Q$, the curvature tensor satisfies $g(R(x,y)z,x) =0$ whenever $(x,y)$ is orthonormal of signature $(-,+)$ and $g(x,z) = g(y,z) =0$, then all nondegenerate two-planes have the same sectional curvature.
\end{theorem}
\subsection{Semi-Riemannian groups}
Let $G$ be a Lie group equipped with a left- and right-invariant (i.e., bi-invariant) semi-Riemannian metric $\langle \cdot{,}\cdot \rangle$, and let $\mathfrak{g}$ be its Lie algebra, that is, the Lie algebra of left-invariant vector fields on $G$. As customary, we identify $\mathfrak{g}$ with the tangent space $T_{e}G$ of $G$ at the identity $e$.
Suppose that $X, Y, Z \in \mathfrak{X}(G)$ are left-invariant, i.e., that $X,Y,Z \in \mathfrak{g}$. Then the Levi-Civita connection is given by
\begin{equation}\label{EQ2}
\nabla_{X}Y = -\nabla_{Y}X= \frac{1}{2}[X,Y]
\end{equation}
and the curvature endomorphism by
\begin{equation} \label{curvatureEQ}
R(X,Y)Z = \frac{1}{4}[Z,[X,Y]].
\end{equation}
In addition, the following equality holds:
\begin{equation}\label{EQ3}
\langle [X,Y],Z \rangle = \langle X,[Y,Z] \rangle.
\end{equation}
Suppose that $x,y \in T_{p}G$ are orthonormal; let $x^{\mathrm{L}},y^{\mathrm{L}}$ be their left-invariant extensions. The sectional curvature of the the two-plane spanned by $x$ and $y$ may be computed by
\begin{equation*}
\mathrm{sec}(x,y) =\frac{1}{4} \langle [x^{\mathrm{L}},y^{\mathrm{L}}], [x^{\mathrm{L}},y^{\mathrm{L}}] \rangle.
\end{equation*}
Note that $\mathrm{sec}(x,y) \geq 0$, with equality if and only if $[x^{\mathrm{L}},y^{\mathrm{L}}]=0$.
Now, let $M$ be a semi-Riemannian submanifold of $G$.
\begin{definition}\label{ClosedNormalBundleDEF}
The normal space $N_{p}M$ is said to be \textit{closed \textup{(}under the Lie bracket\textup{)}} if $dL_{p^{-1}}(N_{p}M)$ is a Lie subalgebra of $\mathfrak{g}$. Consequently, one calls the normal bundle of $M$ \textit{closed \textup{(}under the Lie bracket\textup{)}} if every normal space is closed.
\end{definition}
It is clear that $N_{p}M$ is closed exactly when $\exp(N_{p}M)$ is contained in a Lie subgroup of $G$.
\begin{remark}
Following \cite{terng1995}, the normal space $N_{p}M$ is called \textit{abelian} if $\exp(N_{p}M)$ is contained in a totally geodesic, flat submanifold of $G$. It is easy to see that $N_{p}M$ is abelian if and only if $dL_{p^{-1}}(N_{p}M)$ is an abelian subalgebra of $\mathfrak{g}$.
\end{remark}
\section{The invariant shape operator} \label{ISOP}
In this section we consider the general case of an orientable semi-Riemannian submanifold $M$ of a Lie group $H$ equipped with a left-invariant metric $\langle \cdot {,}\cdot\rangle$. Given a unit normal vector $\eta$ of $M$ at $p$, the \textit{invariant shape operator of $M$ \textup{(}with respect to $\eta$\textup{)}} is the map
\begin{align*}
\alpha \colon T_{p}M &\to T_{p}M \\
v &\mapsto \pi^{\top}\grad_{v}\eta^{\mathrm{L}},
\end{align*}
where, as usual, $\pi^{\top}$ is the orthogonal projection onto $T_{p}M$ and $\eta^{\mathrm{L}}$ the left-invariant extension of $\eta$.
The significance of the invariant shape operator lies in the fact that it represents the deviation of the ordinary shape operator from the differential of the Gauss map of $M$, as we now explain.
Let $N^{1}M$ be the unit normal bundle of $M$, and let $\mathbb{S}^{m+n-1}_{e}$ be the unit sphere inside the Lie algebra $\mathfrak{h}$ of $H$. The \textit{Gauss map of $M$} is the map
\begin{align*}
\mathcal{G}\colon N^{1}M &\to \mathbb{S}^{m+n-1}_{e}\\
(p,\eta) &\mapsto \diff\left(L_{p^{-1}}\right)(\eta).
\end{align*}
Here $L_{p^{-1}} \colon T_{p}H \to \mathfrak{h}$ denotes left translation by $p^{-1}$.
Let $N$ be a unit normal vector field along $M$ such that $N_{p} = \eta$, and consider the map $\bar{\mathcal{G}} = \mathcal{G} \circ N$. Its differential at $p$ is a linear map $T_{p}M \to \mathcal{G}(p,\eta)^{\perp}$. Thus, since $\diff(L_{p^{-1}})$ takes $\eta$ to $\mathcal{G}(p,\eta)$ and is an isometry, it follows that
\begin{equation*}
\mathcal{W} = \pi^{\top} \circ \diff\left(L_{p}\right) \circ \diff\bar{\mathcal{G}}
\end{equation*}
is an endomorphism of $T_{p}M$.
\begin{remark}
The Gauss map of a hypersurface in a metric Lie group was first defined by Ripoll in \cite{ripoll1991}. It is worth pointing out that our definition can be extended to the case where the ambient manifold is parallelizable \cite{ripoll1993}, or even just Killing-parallelizable \cite{fornari2004}.
\end{remark}
Clearly, if $H=\mathbb{R}^{m+1}$, then $\mathcal{G}$ is the classical Gauss map of $M$, whereas $\mathcal{W}$ its shape operator. In our setting, the following result holds.
\begin{proposition}[cf.\ {\cite[p.~769]{ripoll1993}}]\label{PROP6}
\begin{equation} \label{EQ4}
\forall v\in T_{p}M \colon A(v) = \alpha(v) + \mathcal{W}(v).
\end{equation}
\end{proposition}
\begin{proof}
Let $(b_{1}, \dotsc, b_{m+n})$ be an orthonormal basis of $T_{p}H$ such that $b_{1},\dotsc,b_{m} \in T_{p}M$ and $b_{m+n}=\eta$. For each $i$, let $b^{\mathrm{L}}_{i}$ be the left-invariant extension of $b_{i}$, so that $(b^{\mathrm{L}}_{1}, \dotsc, b^{\mathrm{L}}_{m+n}= \eta^{\mathrm{L}})$ is an orthonormal frame for $H$ (and a basis of $T_{e}H$).
If $q \in M$---writing $N^{i}$ as a shorthand for $\langle N, b^{\mathrm{L}}_{i} \rangle$---then
\begin{equation*}
\bar{\mathcal{G}}(q) = \diff\left(L_{q^{-1}}\right)\left(N_{q}\right) = \diff\left(L_{q^{-1}}\right) \bigl( N^{i}(q) \left.\kern-\nulldelimiterspace b^{\mathrm{L}}_{i} \right\rvert_{q}\bigr)=N^{i}(q) b^{\mathrm{L}}_{i}.
\end{equation*}
Thus, if $v \in T_{p}M$, then
\begin{equation*}
\diff\bar{\mathcal{G}}(v) = dN^{i}(v) b^{\mathrm{L}}_{i} = v (N^{i})b^{\mathrm{L}}_{i}.
\end{equation*}
Since
\begin{equation*}
\diff\left(L_{p}\right)(\diff\bar{\mathcal{G}}(v))=v (N^{i} ) \verythinspace b_{i},
\end{equation*}
it follows that
\begin{equation}\label{EQ5}
\pi^{\top}\diff\left(L_{p}\right)(\diff\bar{\mathcal{G}}(v))=v(N^{j}) \verythinspace b_{j}.
\end{equation}
On the other hand,
\begin{align*}
A(v) &= \pi^{\top}\grad_{v}N^{i}b^{\mathrm{L}}_{i}\\
&=\pi^{\top} \left( N^{i}(p)\grad_{v}b^{\mathrm{L}}_{i} + v(N^{i}) \verythinspace b_{i}\right).
\end{align*}
Since, by construction, $N^{1}(p)= \dotsb =N^{m+n-1}(p) =0$ and $N^{m+n}(p)=1$, we have
\begin{equation} \label{EQ6}
A(v) =\alpha(v) + v(N^{j}) \verythinspace b_{j},
\end{equation}
which, together with \eqref{EQ5}, gives \eqref{EQ4}.
\end{proof}
\begin{remark}
Proposition \ref{PROP6} shows that $\mathcal{W}$ does not depend on the particular choice of normal vector field $N$ but only on its value at $p$.
\end{remark}
\begin{remark}
Using equation \eqref{EQ6}, it is not difficult to see that statement \ref{item2} in Theorem \ref{TH1} is nothing but the coordinate expression, with respect to the frame $(e^{\mathrm{L}}_{1}, \dotsc, e^{\mathrm{L}}_{m})$, of the condition
\begin{equation} \label{EQ7}
\pi_{j} \Ima \mathcal{W} \rvert_{\Lambda_{h}} = 0 \quad \text{for all $j,h \in \{1, \dotsc, m\}$ such that $\lambda_{j} \neq \lambda_{h}$},
\end{equation}
where $\Lambda_{j}$ is the eigenspace of $K$ corresponding to the eigenvalue $\lambda_{j}$, and where $\pi_{j}$ is the orthogonal projection onto $\Lambda_{j}$. Note that \eqref{EQ7} holds if and only if $\mathcal{W}$ leaves the eigenspaces of $K$ invariant.
\end{remark}
A useful property of the invariant shape operator, which is crucial in proving Theorem \ref{TH1}, is contained in the following proposition.
\begin{proposition}\label{PROP9}
If $\langle \cdot{,} \cdot \rangle$ is bi-invariant and the normal space $N_{p}M$ is closed, then
\begin{enumerate}[font=\upshape]
\item $K=\alpha\circ\alpha$, and so $\alpha$ and $K$ commute;
\item $K$ leaves $T_{p}M$ invariant.
\end{enumerate}
\end{proposition}
The proof will be based on a lemma.
\begin{lemma} \label{LM10}
Under the hypotheses of Proposition \ref{PROP9}, $\alpha(v)=\nabla_{v}\eta^{\mathrm{L}}$.
\end{lemma}
\begin{proof}
Let $\xi$ be a unit normal vector at $p$. Since $N_{p}M$ is closed,
\begin{equation*}
[\eta^{\mathrm{L}},\xi^{\mathrm{L}}] \in d L_{p^{-1}}(N_{p}M),
\end{equation*}
under the usual identification of $\mathfrak{g}$ with $T_{e}G$. By the bi-invariance of the metric, it follows that
\begin{equation*}
\left\langle \left[ v^{\mathrm{L}}, \eta^{\mathrm{L}} \right], \xi^{\mathrm{L}} \right\rangle = \left\langle v^{\mathrm{L}}, \left[\eta^{\mathrm{L}}, \xi^{\mathrm{L}} \right]\right\rangle=0,
\end{equation*}
which implies $1/2[v^{\mathrm{L}},\eta^{\mathrm{L}}]_{p}=\nabla_{v}\eta^{\mathrm{L}} \in T_{p}M$, and so $\alpha(v)= \pi^{\top}\grad_{v}\eta^{\mathrm{L}}= \nabla_{v}\eta^{\mathrm{L}}$.
\end{proof}
\begin{remark}
The converse of Lemma \ref{LM10} holds: if the metric is bi-invariant and $\alpha(v) = \nabla_{v}\eta^{\mathrm{L}}$ for all $(v,\eta) \in T_{p}M\times N_{p}M$, then $N_{p}M$ is closed. In other words, given a nondegenerate subspace $S$ of $\mathfrak{g}$, the orthogonal complement $S^{\perp}$ of $S$ is closed under the Lie bracket if and only if $[S, S^{\perp}] \subset S$.
\end{remark}
\begin{proof}[Proof of Proposition \ref{PROP9}]
Clearly, being the second assertion in the proposition a direct consequence of the first, we only need to prove the latter.
Let $v \in T_{p}M$. Since $K$ is tensorial, the value $K(v)$ may be computed in terms of the left-invariant extensions $v^{\mathrm{L}}$ and $\eta^{\mathrm{L}}$ of $v$ and $\eta$:
\begin{equation*}
K(v) = R(\eta^{\mathrm{L}}, v^{\mathrm{L}})\eta^{\mathrm{L}}.
\end{equation*}
Assume that the metric is bi-invariant. Then, using \eqref{EQ2} and \eqref{curvatureEQ}, we have
\begin{align*}
K(v) &= \frac{1}{4} \left[ \eta^{\mathrm{L}}, \left[\eta^{\mathrm{L}}, v^{\mathrm{L}} \right] \right]\\
&= \nabla_{\nabla_{v}\eta^{\mathrm{L}} }\eta^{\mathrm{L}}.
\end{align*}
From here the statement follows directly from Lemma \ref{LM10}.
\end{proof}
\begin{corollary}\label{COR11}
Suppose that the induced semi-Riemannian metric on $M$ is positive definite at $p \in M$. Then, under the hypotheses of Proposition \ref{PROP9}, the nonzero eigenvalues of $K$ are negative and have even multiplicities.
\end{corollary}
\begin{proof}
We deduce from equations \eqref{EQ2} and \eqref{EQ3} that the invariant shape operator $\alpha$ of $G$ is skew-adjoint with respect to the semi-Riemannian metric. On the other hand, $K$ is self-adjoint. Thus, if the induced metric on $M$ is positive definite at $p$ and $K(T_{p}M) \subset T_{p}M$, then the matrices of $K$ and $\alpha$ with respect to any orthonormal basis of $T_{p}M$ are symmetric and skew-symmetric, respectively. Since $K =\alpha \circ \alpha$ when the hypotheses of Proposition \ref{PROP9} are fulfilled, the statement follows from \cite[Theorem 2]{rinehart1960}.
\end{proof}
\begin{remark}
Dropping the assumption that the metric is positive definite, the matrix of $\alpha$ in any orthonormal basis $(b_{1}, \dotsc, b_{m})$ of $T_{p}M$ becomes
\begin{equation*}
\begin{pmatrix}
0 & -\mfrac{\langle \alpha(b_{1}), b_{2}\rangle}{\langle b_{1}, b_{1} \rangle} & \cdots & -\mfrac{\langle \alpha(b_{1}), b_{m}\rangle}{\langle b_{1}, b_{1} \rangle}\\[10pt]
\mfrac{\langle \alpha(b_{1}), b_{2}\rangle}{\langle b_{2}, b_{2} \rangle} & 0 & \cdots & -\mfrac{\langle \alpha(b_{2}), b_{m}\rangle}{\langle b_{2}, b_{2} \rangle}\\
\vdots & \vdots & \ddots & \vdots\\
\mfrac{\langle \alpha(b_{1}), b_{m}\rangle}{\langle b_{m}, b_{m} \rangle} & \mfrac{\langle \alpha(b_{2}), b_{m}\rangle}{\langle b_{m}, b_{m} \rangle} & \cdots & 0
\end{pmatrix}.
\end{equation*}
Hence, when $m=2$, the operator $\alpha \circ \alpha$ is a multiple of the identity regardless of the signature of the metric.
\end{remark}
\section{Proof of the main result} \label{PROOF}
We are now ready to prove Theorem \ref{TH1} and Proposition \ref{PROP4}.
\begin{proof}[Proof of Theorem \ref{TH1}]
The equivalence of statements \ref{item2} and \ref{item3} is easily seen, so we only need to prove that of \ref{item1} and \ref{item2}.
To begin with, it follows from equation \eqref{EQ6} that
\begin{equation*}
A(e_{j})= \alpha(e_{j}) + \sum_{h=1}^{m}e_{j}(\langle N,e^{\mathrm{L}}_{h}\rangle)e_{h}.
\end{equation*}
Hence, by linearity of $K$, we have
\begin{equation*}
K(A(e_{j})) = K(\alpha(e_{j})) + \sum_{h=1}^{m} e_{j}(\langle N,e^{\mathrm{L}}_{h}\rangle)K(e_{h}),
\end{equation*}
whereas
\begin{equation*}
A(K(e_{j})) = \alpha(K(e_{j})) + \sum_{h=1}^{m} K(e_{j})(\langle N,e^{\mathrm{L}}_{h}\rangle)e_{h}.
\end{equation*}
Assume that $N_{p}M$ is closed; this way, since $G$ is equipped with a bi-invariant metric, $K$ and $\alpha$ commute. It follows that $K(A(e_{j})) = A(K(e_{j}))$ exactly when
\begin{equation*}
\sum_{h=1}^{m} \lambda_{h} e_{j}(\langle N,e^{\mathrm{L}}_{h}\rangle)e_{h} = \sum_{h=1}^{m} \lambda_{j} e_{j}(\langle N,e^{\mathrm{L}}_{h}\rangle)e_{h}.
\end{equation*}
Being $(e_{1}, \dotsc, e_{m})$ a basis of $T_{p}M$, we conclude that $A$ and $K$ commute if and only if $e_{j}(\langle N, e^{\mathrm{L}}_{h}\rangle) = 0$ for all $j$ and $h$ such that $\lambda_{j} \neq \lambda_{h}$.
It remains to show that $e_{j}(\langle N, e^{\mathrm{L}}_{h} \rangle) = e_{h}(\langle N, e^{\mathrm{L}}_{j} \rangle)$ when $\lambda_{j} \neq \lambda_{h}$ and the induced metric on $M$ is positive definite at $p$. To this end, identify $\alpha$, $A$, and $K$ with their matrices in the basis $(e_{j})_{j=1}^{m}$. The first is a skew-symmetric matrix, by equation \eqref{EQ3}, whereas $A$ is symmetric and $K$ diagonal. Since $\alpha$ and $K$ commute, the $(j,h)$-entries of $\alpha K$ and $K\alpha$ are equal, and so we must have $\alpha_{jh}\lambda_{h}= \lambda_{j}\alpha_{jh}$.
Assume $\lambda_{j}\neq\lambda_{h}$. Then $\alpha_{jh}=-\alpha_{hj}=0$ and so, by equation \eqref{EQ4},
\begin{align*}
A_{jh}&= \langle \mathcal{W}(e_{j}),e_{h}\rangle,\\
A_{hj}&= \langle \mathcal{W}(e_{h}),e_{j}\rangle,
\end{align*}
implying $\langle \mathcal{W}(e_{j}),e_{h}\rangle = \langle \mathcal{W}(e_{h}),e_{j}\rangle$ by symmetry of $A$.
\end{proof}
\begin{proof}[Proof of Proposition \ref{PROP4}]
Suppose that $M$ is a three-dimensional Riemannian submanifold with closed normal bundle, and suppose that $\alpha \neq 0$. It follows by Corollary \ref{COR11} that $K$ has one zero eigenvalue, while the remaining two are equal. Without loss of generality, we may assume that $\lambda_{3}=0$. Since $K \neq 0$, it is clear that $\lambda_{1}=\lambda_{2}\neq0$.
Extend $\eta$ to a unit normal vector field $N$ along $M$. Then, by continuity, the multiplicity of $\lambda_{3}$ is locally constant, i.e., there exists a neighborhood $U=U(N)$ of $p$ in $M$ such that the extension of $K$ has two negative definite eigenvalues in $U$.
Assume that $A$ and $K$ commute, i.e., they share a common basis of eigenvectors. Since the $0$-eigenspace of $K$ is one-dimensional, it follows that $e_{3}$ is an eigenvector of $A$. Conversely, if $e_{3}$ is an eigenvector of $A$, then its other two eigenvectors lie in the $\lambda_{1}$-eigenspace of $K$, from which we infer that $A$ and $K$ commute.
\end{proof}
\appendix
\section{} \label{APPA}
Here we present a direct proof of the following obvious corollary of Theorem \ref{TH1}.
\begin{corollary}
Suppose that $N_{p}M$ is closed. If the shape operator of $M$ with respect to $\eta$ is a multiple of the identity, then $e_{j}(\langle N, e^{\mathrm{L}}_{h} \rangle) = 0$ for all $j,h \in \{1, \dotsc, m\}$ such that $\lambda_{j} \neq \lambda_{h}$.
\end{corollary}
\begin{proof}
Assume the hypotheses of the corollary. Since $\alpha$ commutes with $K$, each eigenspace of $K$ is invariant under $\alpha$. Moreover, being $A$ a multiple of the identity, $\langle A(e_{j}), e_{h} \rangle =0$ for $j \neq h$, and so equation \eqref{EQ6} implies
\begin{equation*}
\langle \alpha(e_{j}), e_{h} \rangle = \pm e_{j}(\langle N,e^{\mathrm{L}}_{h}\rangle)\quad \text{for $j \neq h$}.
\end{equation*}
Clearly, if $\lambda_{j} \neq \lambda_{h}$, then $e_{j}(\langle N,e^{\mathrm{L}}_{h}\rangle)=0$, because $\alpha(e_{j})$ and $e_{h}$ are in different (orthogonal) eigenspaces of $K$.
\end{proof}
\bibliographystyle{amsplain}
\bibliography{../biblio/biblio}
\end{document}
| 166,491
|
TITLE: The Relationship Between Cohomological Dimension and Support
QUESTION [3 upvotes]: Let $ R $ be a commutative unital ring, $ I $ an ideal of $ R $, and $ M $ an $ R $-module. The cohomological dimension of $ M $ with respect to $ I $ is defined as
$$
\operatorname{cd}(I,M) \stackrel{\text{def}}{=}
\sup(\{ i \in \mathbb{N} \mid {H_{I}^{i}}(M) \neq 0 \}).
$$
From the fact that $ \operatorname{cd}(I,M) \leq \operatorname{cd}(I,R) $, one guesses that there is a relationship between cohomological dimension and the support of modules such as:
$$
\operatorname{supp}(M) \subseteq \operatorname{supp}(N) \iff
\operatorname{cd}(I,M) \leq \operatorname{cd}(I,N).
$$
Can anyone prove this relationship or give a counterexample, please? You can add any assumption that helps, such as ‘being local’.
REPLY [2 votes]: The closest result that I can find is the following:
Theorem: Let $ R $ be a commutative Noetherian unital ring and $ I $ an ideal of $ R $. If $ M $ and $ N $ are two finitely generated $ R $-modules such that $ \operatorname{supp}(M) \subseteq \operatorname{supp}(N) $, then $ \operatorname{cd}(I,M) \leq \operatorname{cd}(I,N) $.
The reference is K. Divaani-Aazar, R. Naghipour and M. Tousi, Cohomological Dimension of Certain Algebraic Varieties, Proc. Amer. Math. Soc. 130 (2002), 3537-3544.
| 166,071
|
Why Companies Living Quarter to Quarter is Both a Good as well as a Bad Idea
Living for the Next Quarter
Of late, many publicly listed companies have been living quarter to quarter or the practice of setting targets, tracking them for progress, and closing out sales and revenue generating items based on the next quarter which is a short term imperative rather than planning for the longer term.
This has led to both good and bad consequences as we would discuss subsequently and before that, it would be in the fitness of things to explain what is meant by living quarter to quarter. To start with, it has almost become the norm in the corporate world around the globe to release results for each quarter which is a standard practice except that the CEOs are also providing revenue and growth guidance for the next quarter rather than the full year.
Of course, in the annual results, they do so for the entire year that is coming up. However, the fact remains that investors, analysts, and even the hitherto serious private equity firms and assorted stakeholders have been focusing on quarterly performance rather than the full year performance.
This has led to a situation where stock prices swing wildly with each quarterly declaration of results which can go either way. While this is a good way to keep the companies on their toes as they would be nimble and agile enough to perform, it is also the case that longer term value considerations are being lost in the process.
Balancing Shorter Term Stock Prices with Longer Term Value Creation
Indeed, given the fact that equity prices make up just one component of value that corporates build over a longer term, it is our view that while it is good to be the darlings of the stock markets for a brief period, it is also the case that corporates must and should not lose sight of the bigger picture in the quest for instant gratification.
The reason for such quarterly focus has been due to the fact that worldwide the business landscape has become so fast paced that investors and analysts likewise are caught up in the imperatives of the moment and hence, reward or punish the corporates based on purely shorter term considerations.
The Role of Technological Acceleration
Moreover, with so many technology driven start-ups such as Uber and AirBnB upending traditional taxi and hotel companies mainly due to their agility and nimbleness using technology, it has become necessary for even traditional manufacturing and including service sector corporates to jump on the shorter term bandwagon where survival or success is purely determined on a quarterly basis.
Moreover, given the imperatives of the 24/7 Breaking News Cycle media environment, it is often the case that corporates grab the headlines for their profits or losses measured in the shorter term rather than over a longer term.
This creates a ripple effect wherein the electronic herd takes over and influences investors and shareholders in a frenzied bout of selling and buying.
Creative Destruction
As mentioned earlier, this can be good from a creative destruction perspective since capitalism and the stock markets are always on the lookout for newer avenues of profits or the next big thing.
However, this can also lead to myopic outlook from the corporates and their CEOs who obsess over the quarterly results rather than focusing on creating longer term value.
Having said that, it is also not the case that all CEOs or corporates are taken in by this frenzy and there remain many Blue Chip stocks that perform consistently over the longer term.
For instance, corporates such as Unilever and Proctor and Gamble continue to be respected and much sought after mainly because they can balance the shorter term and the longer term imperatives and drivers of growth.
On the other hand, the worst affected are the technology companies because of the very nature of the industry they operate in. While Unilever and P&G can release new brands every now and then without affecting their revenue streams and profitability, companies such as Apple, Google, Facebook, and Microsoft have to be hard at the game to retain market share in much shorter timelines.
Corporate Longevity in an Impulsive Age
As technology accelerates the pace of change and the Algorithmic trading systems ensure that the equity markets are run on microsecond and millisecond basis, it is our view that this type of quarterly impulses would increase rather than decrease.
Therefore, any corporate that wishes to stay in the hunt for a longer term should wisely allocate resources such as capital and human resources in the pursuit of both shorter term targets as well as longer term value creation.
After all, Rome was not built in a day and hence, despite all the systems driven changes, old fashioned value creation would continue to be the bedrock by which corporates and their longevity are determined.
Conclusion
Having said that, it is also the case that the rapid turnover of hitherto winners that have now become losers such as Blackberry, Nokia, and Yahoo means that corporates and their CEOs are sometimes left with little choice but to obsess over the shorter term.
Given these imperatives, it is indeed the case that the more astute CEOs would ensure that they keep their jobs with impressive shorter term results and retain the respect of investors by handsomely rewarding them over the longer term.
Moreover, this can also ensure that employees are sufficiently motivated to work harder for the corporates with shorter term carrots in the form of stock options being balanced with the longer term stability of working for an organization that rewards them for their hard
| 172,577
|
The Guardian
Are the descriptions wine writers use helpful? Asks Fiona Beckett.
According to wine research company DoILikeIt.com, it turns out few words used to describe wine, mean much to the average wine buyer, particularly those using comparisons to individual fruits. We wine writers may think gooseberry is an apt description for the flavours in a New Zealand Sauvignon Blanc, but a lot of consumers may not get gooseberry, or even like gooseberries, she says. Words that are favoured by consumers include light, crisp and refreshing for whites, and smooth, fruity and full-bodied for reds. But since so many wines fall into those categories, how do you then distinguish between them? Beckett experiments with Hilltop Cserszegi Füszeres 2011 (£5.25, The Wine Society). She found it intensely citrussy (lemon and grapefruit peel on the nose, fresh lemon juice on the palate), but since citrus doesn't apparently resonate much with the public, she'll describe it as fresh and zesty, with a flowery twist.
The Daily Telegraph
Victoria Moore dithered over drinks at the bar of the Apollo Theatre, in London. She couldn't decide on what she terms as "Awful Chardonnay or Awful Pinot Grigio." She chose 'Awful Pinot Grigio', on the grounds that it would taste of less. Why are most theatre bars, so bad, and so painfully expensive? She asks. Is it because the management know they've a captive audience? Bars that do get it right - such as The Pit at the Old Vic or downstairs at the Royal Court - become destinations in their own right, which must be a useful source of income. It also seems to de-ghettoise the experience of going to the theatre. If only they got it right more often, she adds.
The Independent
Once a year the 11 members of the Primum Familiae Vini (pfv.org) pull out the stops to show just what their name means to them and should mean to us, says Anthony Rose. The family group formed in 1993 by Miguel Torres and Robert Drouhin regard themselves as the elite of the wine world. They represent quality and personality in a wine world dominated by commercialism and the need to keep shareholders happy. Continuity was there for all to see on their recent lunch in London. Rose says over a 1963 Graham's Vintage Port, Paul Symington of Dow's, Graham's and Warre's, said "there isn't a bad wine". Rose says he couldn't disagree.
Financial Times
Californian physician Harin Padma-Nathan treated himself to 18 bottles of Pétrus 1982 and later decided to sell. He was horrified to find Christie's staff refused to take the bottles, citing disparities between the labels on his bottles and those used by the château itself. Fraudulent wine is not confined to the US, says Jancis Robinson MW. In China it is "jaw-dropping". Even Hong Kong, has been seriously infected by some of the grandest fakes. The Bordelais have been taking steps generically to fight fraud, but the fake rate has now reached such a level in the US that the FBI is investigating several cases. Recently, thanks to the internet, lawyer Don Cornwell was able to share concerns with thousands of connoisseurs just four days before the February 8 London sale, organised by Spectrum of California, and Vanquish, London. After which some 20 lots of what looked like some of the finest, rarest Burgundies, were all withdrawn from auction.
The Daily MailOlly Smith often receives general day-to-day health concerns from readers who have written about what their daily glass of vino is doing to their survival chances. He says, there are two sides to this. His first point is to reassure readers that when deployed responsibly, to him wine is a joyful, celebratory, civilised and positive cultural icon. Of course, in large quantities, alcohol is problematic for health. Professor Roger Corder is the author of The Wine Diet, a scientist and advocate of drinking less but better quality. He is in no doubt that there is a link between polyphenols found in red wine and improved blood pressure and vascular health. Red grape varieties with thick skins such as Cabernet Sauvignon and Tannat, made in more traditional methods in places like southwest France seem to score more highly in Corder's system, notably French Madiran. How about that label 'contains sulphites' ? Some people claim to suffer allergic reactions to sulphites, 'natural' wines which aim to minimise sulphites might be worth investigating, along with biodynamic and organic wines, he says.
| 417,343
|
From Fedora Project Wiki
Boris Ryutin
General Info
Security Researcher. Passionate about computer security, vulnerability research. Use Unix-like operating systems from 2005. Contributor in several infosec open source projects (mostly in Radare2). Guitar lover and Blizzard games fan.
How to contact me ?
- On IRC : dukebarman at Freenode
- On Twitter: @dukebarman
| 371,150
|
We have had the great pleasure to be working with OpenAI InPaint and DALL·E 1 for many months, and thought it would be constructive to document some techniques, ideas and reflections that have been raised through exploring these remarkable tools in honor of the announcement and warranted public excitement around DALL·E 2.
If you are new to this area, it is worth establishing a brief timeline of developments from the past year. In January last year OpenAI announced DALL·E, a transformer model capable of generating convincing artworks from textual descriptions. This was shortly followed by the release of OpenAI CLIP (Contrastive Language Image Pre-Training), a neural network trained on image/description pairs released for public testing.
What followed was an explosion of experimentation beginning in Spring 2021, with artists such as (but not limited to) Ryan Murdock and Rivers Have Wings releasing free and open tools for the public to play with generating artworks by connecting CLIPs capacity to discern textual prompts with open image generation techniques such as Taming Transformer’s VQGAN, and OpenAI’s own novel Guided Diffusion method.
In lay terms, to create artwork one types what one wants to see, and this combination of generator (Diffusion, VQGAN) and discriminator (CLIP) produces an image that satisfies a desired prompt. In Ryan Murdock’s Latent Visions discord group, the artist Johannez (who has been publishing artworks and tutorials around machine learning imagery and music online for years) coined the term “Promptism” for this astounding new capacity to conjure artworks from telling a neural network what one would like to see.
While this development is but the latest advancement in a legacy of machine learning in art too long to do justice to in a blog post, this act of conjuring artworks from language feels very very new. Feeling is an important dimension to the act of creating an artwork, as while we have for some years had the capacity to generate art from the laborious process of training GANs, often waiting overnight for results that invite the observer to squint and imagine a future of abundant possibility, a perfect storm in the past 18 months has led to a present in which the promise of co-creation with a machine is realized. It feels like jamming, giving and receiving feedback while refining an idea with an inhuman collaborator, seamlessly art-ing. It intuitively feels like an art making tool.
Analogies are imperfect, but this resembles the leap from the early electronic music period of manually stitching together pieces of tape to collage together a composition to the introduction of the wired synthesizer studio, or Digital Audio Workstation. In the 20th century this progress in musical tools-at-hand took place over half a century. In contrast, this leap in machine learning generated imaging tools-at-hand has gathered steam over a 3-5 year period.
This dizzying pace of development encourages reflection!
Since Alexander Mordvintsev released the Deep Dream project in 2015, machine learning imagery is often associated with surrealism. It is no doubt surreal to find oneself co-creating art by probing a disembodied cognitive system trained on billions of datapoints, and the images produced can be confounding and psychedelic. However these are characteristics often ascribed to new experience. New experiences are weird, until they are no longer.
At the advent of DALL·E 2 we find it quite useful to think instead to the history of the Pictorialism movement.
In the early to mid 19th century, photography was the domain of a select group of engineers and enthusiasts, as it required a great deal of technical knowledge to realize photographs. As such, the focus of these efforts and gauge of the success of the medium was to create images that most accurately reflected the reality they intended to capture.
The Pictorialists emerged in the late 19th century as a movement intent not on using photographic techniques to most accurately depict reality, instead opting to use photography as a medium of communicating subjective beauty. This movement progressed photography from a scientific to an expressive medium in parallel with increasing access to cameras that required no technical expertise (the first amateur camera, the Kodak, was released in 1888).
Debates raged about whether or not photography could in fact be art, a conversation inevitably dismissed once artists were able to affordably integrate cameras into their practices. The Pictorialists extolled the artistic potential of the camera by perverting its purported function of objectively capturing the world, staging scenes and experimenting with practical effects to create subjective and expressive works. Painters soon began integrating the camera into their workflow, and now most art forms make use of sensor based imaging in one form or another.
Larry Tesler’s pithy description of AI, “Artificial intelligence is whatever hasn’t been done yet.”, is equally applicable looking back to the birth of photography. Photography had also not been done yet, and was then (a phenomenon described as the AI effect) soon integrated into most artistic and industrial practices. Sensor based imaging remains a discrete focus of research pursuing things that have not been done yet.
The parallels with this time period are clear. In machine learning imagery, the laborious task to this point has been to legitimize the medium by attempting to accurately reflect the reality of training material. Great efforts have been made to realize a convincing and novel dog picture from training neural networks on dogs. This problem has been solved, and sets the foundation upon which we can begin to be expressive.
As happened with the Pictorialists, prompt based systems like DALL·E are democratizing the means by which anyone can create AI facilitated subjective art with next to no technical expertise necessary, and we assume that in the coming decades these techniques will be integrated into artistic and industrial practices to a great degree. AI will remain a discrete field of research exploring what has not been done yet, and in 2 years we will think nothing of painters using tools like DALL·E to audition concepts, or companies using tools like DALL·E to audition furniture for their office.
DALL·E represents a shift from attempts to reflect objective reality to subjective play. Language is the lens by which we reveal the objective reality known to the neural network being explored. It is a latent(hidden) camera, uncovering snapshots of a vast and complex latent space.
The same debates will rage about whether or not prompt based AI imagery can be considered Art, and will just as inevitably be relegated to history once everyone makes use of these tools to better share what is on their mind.
The ever evolving pursuit of art is greatly benefitted from reducing any friction in sharing what is on your mind. The observer is the ultimate discriminator, and as with any technological development that makes achieving a particular outcome more frictionless, creating great art that speaks to people in the time that it is made remains an elusive and magical odyssey.
The easier it is to generate artworks, the more challenging it will be to generate distinction and meaning, as it ever was. Great Art, like AI, is very often what hasn’t been explored yet.
The first time we really got excited about DALL·E was in discovering its capacity to produce internally coherent images. Internal coherence can best be described as the ability to create convincing relationships between objects generated within an image.
To test this capacity, we began to use InPaint to extend images that we uploaded to the system without contributing a linguistic prompt. To achieve this, we would manually shift the viewfinder of InPaint in either direction to extend the scene.
One particularly successful test involved extending the scene of symbolist painter Charles Guilloux’s work - L'allée d'eau (1895)
As you can see, DALL·E is capable of comprehending the style and subject of the scene, and extending it coherently in all directions. Perhaps most notable is its capacity to produce reflections consistent with objects present within the image. The curvature of the trees and river is successfully reflected in the new water being generated.
This capacity for internal coherence could only happen within the (then) 512x512 pixel viewfinder of InPaint, however successfully demonstrated a capacity to produce images of potentially infinite scale by producing incremental coherent patches of an image.
We then extended this technique to produce very large pieces under the same reflective motif. The wall sized artworks below (produced with DALL·E 1 and InPaint) are we believe the largest compositions ever produced with machine learning at their time of creation.
This process of creating a patchwork of internally coherent images to form a larger composition was very challenging, quite like attempting to paint a wall sized work from the vantage point of a magnifying glass, and with no master guide to follow. As such, we used techniques like horizon lines to retain coherence.
There is something poetic to composing in this way. Attempting to extrapolate a bigger picture with only access to a small piece of it at a time feels appropriate in the broader context of AI.
We have not yet been able to experiment with DALL-E 2 and InPaint together in the same way, however assume that even more coherent and vivid images can be produced using a similar patchwork technique.
One can imagine efforts being made to increase the coherence of a larger composition by analyzing elements contained within an image outside of the scope of the InPaint viewfinder. With these developments in mind, we expect that these tools will soon contain all the elements necessary to produce limitless resolution compositions guided by language and stylistic prompts.
We began to make a series of infinite images extending horizontally, which allude to the potential for these tools to tell stories in the tradition of tapestries or graphic novels. This sequence below is large enough in resolution to be printed well, and is a narrative that could be extended to infinity.
To return to the original idea of extending a painting to reveal more of the scene, what might it mean to be able to produce infinite worlds from a single painting or photograph? This significantly augments the capacity of what we understand of generative art, when a coherent world, or narrative, can be spawned from a single stylistic or linguistic prompt.
One can imagine narrative art forms like graphic novels or cinema being impacted from such a development. This is a particularly exciting prospect when considering the more recent development of prompt art bots being used in active Discord communities. The first of these we encountered was developed by Wolf Bear Studio for their Halloween themed art project Ghouls ‘n GANs, where discord users were invited to generate artworks via an in-thread bot, a concept more recently being experimented with by Midjourney and the artist Glassface’s iDreamer project (also in collaboration with Wolf Bear).
Bots such as these, in combination with ideas such as Simon De La Rouviere’s Untitled Frontier experiment in narrative co-authorship, augur a future of co-creation not only with artificial intelligence systems, but also more frictionlessly with one another.
This speaks to the crux of what DALL·E feels like, a tool for jamming, rapid iteration and potentially co-authored social experiences. One can imagine group storytelling sessions online and IRL that produce vivid narrative art to be replayed later or viewed from afar.
Once the friction to share what is on your mind has been eliminated, the ability to co-create social narrative art experiences at the dinner table or the theatre seems conceivable and exciting!
We have only been working with DALL·E 2 for a short time, however what is clear is that the system has exponentially improved in terms of generated convincing and internally coherent images guided by language and image prompts.
We plan to publish more later on what we discover, however our initial experiments have involved further experimentation with the “Holly Herndon” embedding present within OpenAI CLIP. In lay terms, Holly meets the threshold of notoriety online to have characteristic elements of her image be understood by the CLIP language/image pairing network, something we explored last year with our CLASSIFIED series of self portraits created to reveal exactly what/who CLIP understands “Holly Herndon” to be.
We propose a term for this process, Spawning, a 21st century corollary to the 20th century process of sampling. If sampling afforded artists the ability to manipulate the artwork of others to collage together something new, spawning affords artists the ability to create entirely new artworks in the style of other people from AI systems trained on their work or likeness. As Holly recently communicated at the TED conference, this opens up the possibility for a new and mind-bending IP era of Identity Play, the ability to create works as other people in a responsible, fair and permissive IP environment, something that we are exploring with the Holly+ project.
Tools like DALL·E 2 and InPaint undoubtedly propel us closer to this eventuality, evidenced by these “Holly Herndon” style memes we were able to generate in early tests.
Memes feel an appropriate medium for experimentation in this context, as any single meme maintains it’s vitality from its ability to be personalized and perpetually built upon. Memes are cultural embeddings, not dissimilar to the embeddings present within the latent space of a neural network.
Memes are a distillation of a consensual/archetypical feeling or vibe, in much the way that the “Holly Herndon” embedding with CLIP is a distillation of her characteristic properties (ginger braid and bangs, blue eyes, often photographed with a laptop), or the “Salvador Dali” embedding is a distillation of his unique artistic style.
We find personalized applications like this pretty exciting, as while DALL·E 2 (and it’s stunning variations feature we will cover at a later time) unlocks the ability to produce convincing images in art historical styles familiar to it’s training set, we feel that there could be a misconception that its utility is limited to solely recreating artistic expressions of the past.
Like the introduction of the personal camera, it is easy to imagine a near future scenario in which all amateur and professional workflows across creative industries are augmented by these tools, helping people to more clearly depict what is in their mind through common language and an ever expanding training corpus.
The 21st century is going to be wild. We will share more as we learn more! 🦾
| 238,895
|
\begin{document}
\maketitle
\begin{abstract}
We present a new conceptual model of the Earth's glacial-interglacial cycles, one leading to governing equations for which the vector field has a hyperplane of discontinuities. This work extends the classic Budyko- and Sellers-type conceptual energy balance models of temperature-albedo feedback by removing the standard assumption of planetary symmetry about the equator. The dynamics of separate Northern and Southern Hemisphere ice caps are coupled to an
equation representing the annual global mean surface temperature. The system has a discontinuous switching mechanism based on mass balance principles for the Northern Hemisphere ice sheet. We show the associated Filippov system admits a unique nonsmooth and attracting limit cycle that represents the cycling between glacial and interglacial states. Due to the vastly different time scales involved, the model presents a nonsmooth geometric perturbation problem, for which we use
ad hoc mathematical techniques to produce the periodic orbit. We find climatic changes in the Northern Hemisphere drive synchronous changes in the Southern Hemisphere, as is observed for the Earth on orbital time scales.
\end{abstract}
\begin{keywords}
nonsmooth dynamical systems, virtual equilibria, nonsmooth return map, ice--albedo feedback, paleoclimate, glacial cycles
\end{keywords}
\begin{AMS}
49J52, 37N99, 86A40
\end{AMS}
\section{Introduction}
Systems of nonsmooth differential equations have been used to model a wide range of physical, biological, and mechanical phenomena (see for example the references in \cite{dibernardo}). In some cases, the nonsmoothness in the models comes from assuming a limiting behavior of an abrupt transition (e.g. in \cite{welander,julie}) while in others the modeled behavior is truly discontinuous or nonsmooth, for example due to friction or impacts in mechanical systems.
Because of the possibility of these types of phenomena in a host of different aspects of the climate system, the mathematical approach to the design and analysis of conceptual climate models increasingly uses tools from the developing field of nonsmooth dynamical systems. Frequently, climate models of this type contain a switching mechanism that causes the system to flip to a different climate state. Here we contribute to this body of literature by developing and analyzing a nonsmooth ODE model of Northern and Southern Hemisphere glacial cycles where a hyperplane in state space delineates a switch between the climate state of advancing Northern Hemisphere glaciers and the climate state of retreating Northern Hemisphere glaciers. For appropriate choices of the system parameters, we show the existence of an attracting periodic orbit corresponding to synchronous Northern and Southern Hemisphere glacial cycles driven by mass balance of the Northern Hemisphere glaciers.
In this study, we consider piecewise smooth systems of the form
\begin{align}
\dot{\mathbf{v}}\in\mathbf{X}(\mathbf{v})=\begin{cases}
{\bf X}_-({\bf v}), & {\bf v}\in S_-\\
\{ (1-p){\bf X}_-({\bf v})+p{\bf X}_+({\bf v}) : p\in [0,1]\}, & {\bf v}\in\Sigma \\
{\bf X}_+({\bf v}), & {\bf v}\in S_+
\end{cases}
\label{eq-intro}
\end{align}
where $\mathbf{v}\in\mathbb R^n$, $\Sigma$ denotes the \emph{switching manifold}, and $X_{\pm}$ is smooth on $S_{\pm}\subset\mathbb R^n$. Systems of this form are \emph{differential inclusions}. While in $S_-$, solutions are unique with flow $\phi_-({\bf v},t)$ corresponding to system $\dot{\bf v}={\bf X}_-({\bf v})$. Similarly, solutions in
$S_+$ are unique with flow $\phi_+({\bf v},t)$ given by system $\dot{\bf v}={\bf X}_+({\bf v})$. For ${\bf v}\in\Sigma, \ \dot{\bf v}$ must lie in the closed convex hull of the two vectors ${\bf X}_-({\bf v})$ and ${\bf X}_+({\bf v})$. A solution to \eqref{eq-intro} {\em in the sense of Filippov} is an absolutely continuous function ${\bf v}(t)$ satisfying $\dot{\bf v}\in {\bf X}({\bf v})$ for almost all $t$ \cite{fil}.
In piecewise systems of the form \eqref{eq-intro}, periodic orbits may be present even when the vector fields $\mathbf{X}_\pm$ don't themselves have periodic orbits (as is the case here). Heuristically, this happens when the flow in $S_+$ dictates that the solution should cross into $S_-$ and that the flow in $S_-$ dictates that the solution should cross into $S_+$. More concretely, consider the simplest case: planar systems formed by two continuous differential systems separated by a straight line (as is the situation in, for example, \cite{julie,freire,llibre2017}). In the simplest cases, periodic orbits can be found by examining the system for different arrangements of equilibria. In nonsmooth systems there are three types of equilibria to consider, namely\\
\begin{def2}[\cite{dibernardo}] Let ${\bf v}$ be a solution in the sense of Filippov to \eqref{eq-intro}.
\begin{enumerate}
\item[(i)] ${\bf v}$ is a {\em regular equilibrium point} of \eqref{eq-intro} if either ${\bf X}_+({\bf v})={\bf 0}$ and ${\bf v}\in S_+$, or if ${\bf X}_-({\bf v})={\bf 0}$ and ${\bf v}\in S_-$.
\item[(ii)] ${\bf v}$ is a {\em virtual equilibrium point} of \eqref{eq-intro} if either ${\bf X}_+({\bf v})={\bf 0}$ and ${\bf v}\in S_-$, or if ${\bf X}_-({\bf v})={\bf 0}$ and ${\bf v}\in S_+$.
\item[(iii)] ${\bf v}$ is a {\em boundary equilibrium point} of \eqref{eq-intro} if ${\bf X}_+({\bf v})={\bf X}_-({\bf v})={\bf 0}$ and ${\bf v}\in \Sigma$.\\
\end{enumerate}
\end{def2}
\noindent Periodic orbits can be present when $\mathbf{X}_{\pm}$ both have regular equilibria (e.g. \cite{freire}), both have virtual equilibria (e.g. \cite{julie,julie-dis} and here), or both have no equilibria of any type (e.g. \cite{llibre2017}). More complicated behavior can also produce nonsmooth periodic orbits, such as when a periodic orbit of either $\mathbf{X}_-$ or $\mathbf{X}_+$ intersects the switching manifold (e.g. Section 2.4 of \cite{dibernardo}).
There are many techniques that can be employed to establish the existence of a nonsmooth periodic orbit in a system of the form of \eqref{eq-intro}.
The technique of \emph{regularization} is a standard technique which converts the nonsmooth system to a smooth one and can thereby be studied using standard dynamical systems techniques (e.g. \cite{sotomayor1996,dieci,awrejcewicz}). However, in studying applications other techniques are frequently used because, for example, the smoothing function in the regularization method does not have an explicit form, among other issues limiting the technique's usefulness in specific settings (see the discussion in Section 5.2 in \cite{julie-dis}). Recent work in showing the existence of periodic orbits in conceptual climate models with switching mechanisms have employed coordinate changes to investigate behavior near the discontinuity boundary \cite{julie-dis,julie} (similar to blow-up techniques in celestial mechanics to investigate behavior near a collision), Filippov's existence and uniqueness results for differential inclusions \cite{Barry2017}, concatenation of smooth solutions from the associated subregions of the phase space where the vector field is smooth \cite{Barry2017}, and construction of a return map on the discontinuity boundary \cite{wwhm,budd}.
Here we employ the technique used in \cite{wwhm} but in one higher dimension. In particular, we construct a return map on the discontinuity boundary and show that the map is contracting for appropriately chosen parameter values. This allows us to conclude that there is a unique attracting periodic orbit in the system for appropriate choices of parameter values. We then demonstrate that for physically relevant choices of parameter values the periodic orbit exists. This study is an extension of \cite{wwhm} because we have relaxed a symmetry assumption of the climate system about the equator, adding an additional dimension to state space. That is, our model couples separate Northern Hemisphere and Southern Hemisphere ice cap dynamics via the influence that the (possibly asymmetric) positioning of the ice caps has on the global mean surface temperature, and vice versa. This symmetry assumption is standard in the family of models that we consider (e.g. \cite{sellers,budyko,Widiasih2013,McGehee2014}) but has recently been removed to study the climate of Pluto \cite{Nadeau2019}. A more general mathematical study of the model with the symmetry assumption removed (but without the mass balance switch that we study here) is forthcoming.
The rest of the paper is laid out as follows. In the following section we motivate the scientific aspect of this study and describe the physical observations that the model behavior reflects.
The derivation of the model equations, where separate equations modeling dynamic Northern and Southern Hemisphere ice sheets, via consideration of distinct albedo lines $\eta_N$ and $\eta_S$, and a proxy of the global annual mean surface temperature, $w$, is presented in Section \ref{section-governing-eq}. Leaving consideration of the dynamics of the $(w,\eta_S,\eta_N)$-system on the boundary of state space for future work, we discuss the behavior of this system off of the boundary in Section \ref{Section-Symmetry}. In Section \ref{Section-Mass-Balance} the Northern Hemisphere flip-flop glacial cycle model from \cite{wwhm} is placed in the Northern Hemisphere of our asymmetric model. We prove the existence of a unique attracting periodic orbit representing the glacial-interglacial cycles, with the mathematical techniques used reminiscent of (smooth) geometric singular perturbation theory. Notably, the flip-flop behavior of the ice sheet in the Northern Hemisphere drives synchronous oscillations of the ice cap in the Southern Hemisphere via the coupling of the two albedo lines with the surface temperature. This result aligns with the theory that the Southern Hemisphere ice sheet oscillations are in response to climate changes in the Northern Hemisphere on orbital time scales.
\section{Scientific Background}
Understanding the behavior of the glaciers over time and the resulting impact on Earth's climate has been a major endeavour across disparate fields of the physical and biological sciences for over a hundred years. Glacial cycles are characterized by the advance of large ice sheets from the poles to the mid-latitudes and their subsequent retreat and are a defining characteristic of Earth's climate history. Glacier advance, occurring over tens of thousands of years, is not monotonic and glacial records show periods of relative warming as the climate gradually cools to the glacial maximum \cite{broecker, brook, pedro}. Relative to the long time-scale of their advance, glacier retreat is fast, taking only thousands of years instead of tens of thousands (e.g. see \cite{tzip2003}). This advance and retreat cycle creates a characteristic sawtooth pattern in the glacial record for roughly the past 800,000 years (e.g. see for example \cite{brook}).
Many questions concerning the Earth's glacial cycles remain unanswered, including those related to changes in the period and amplitude of the glacial-interglacial cycles that have occurred over geologic time. More relevant to our model is evidence that on orbital times scales (100 kyr) ice cover oscillations in the Northern and Southern Hemisphere have been in sync \cite{broecker,brook,lowell,raylisnic,rother}. While the physical mechanisms behind these different behaviors continue to be investigated, some posit that changes in the Northern Hemisphere climate drive changes in the Southern Hemisphere on orbital time scales \cite{broecker,brook,lowell,raylisnic}. The climate changes in the Northern Hemisphere are in turn thought to be brought about by changes in high northern latitude incoming solar radiation, due to changes in Earth's orbital parameters over long time scales (the latter known as Milankovitch cycles \cite{milank}). Studies have demonstrated the prominent role that Earth's obliquity (tilt of the axis of rotation relative to the orbital plane) plays in pacing the glacial cycles \cite{huybers2005}, but the jury is still out on whether precession plays a definitive role (e.g. \cite{huybers2011} and references therein). Crucially, precession acts with the opposite effect in the hemispheres (e.g. \cite{brook,huybers2011}).
Further, it is not known to what extent glacial-interglacial cycles are precipitated by orbital forcings in conjunction with internal climate feedbacks \cite{brook}. Such feedbacks include greenhouse gas forcing, albedo (surface reflectivity) feedbacks, dust forcing, deep ocean temperature, isostatic rebound, or mass balance of Northern Hemisphere glaciers \cite{abe-ouchi,brook}. Here we consider two of these feedback mechanisms, albedo and mass balance, on Earth's surface temperature. Recent work suggests that these two mechanisms are not necessarily decoupled and several studies have noted and investigated how changes in a glacier's albedo may influence local temperature or precipitation feedbacks and thus affect a glacier's growth (e.g. \cite{abe-ouchi,raylisnic,tzip2003}). For instance, Tziperman and Gildor note that extensive, high albedo sea ice cools the atmospheric temperature and can divert snow storms away from continental ice sheets (\cite{tzip2003} and references therein).
The model that we consider in this study is a \emph{conceptual climate model} (sometimes \emph{simple climate model}, \emph{low complexity climate model}, \emph{analytical climate model} or \emph{reduced climate model}). Conceptual models are used to give a broad view of the ways in which major climate components interact, contrasting with higher complexity models (such as general circulation models or earth systems models with two or three spatial dimensions) which simulate atmospheric, oceanic, chemical, and biospheric dynamics on a grid of the Earth. While in the past highly complex climate models have not been applied to study the long-term behavior of the past climate system (due to limited computing power and the length of time series needed to simulate, for example), recently intermediate to high complexity models have been been adapted to successfully study glacial dynamics (e.g. \cite{abe-ouchi,kawamura, choudhury}). Conceptual climate models still have an important role to play in advancing scientific understanding of glacial cycles and glacier dynamics (e.g. \cite{budyko,sellers,saltzman,huybers2005,huybers2011,engler2018dynamical}) and are also a more computationally efficient way to test theories about interactions between different climate elements before implementing the idea in a more complex model (e.g. \cite{knutti}). In the case of glacial cycle models the climate elements considered might include surface temperature, energy into and out of the climate system, the latitudinal transport of energy, the carbon cycle, and the ways in which processes such as surface albedo affect these interactions.
Conceptual modeling of the glacial cycles using energy balance equations was popularized by the work of M. Budyko \cite{budyko} and W. Sellers \cite{sellers} in 1969, with the introduction of equations used to model surface temperature on a planet with an assumed symmetry about the equator. The temperature model that we use here is a descendent of Budyko's original equation \cite{budyko}. Following the through line of the Budyko family of models leading to the model we study here, E. Widiasih coupled Budyko's temperature equation with a dynamic ice sheet in \cite{Widiasih2013}, proving the existence of a small stable ice cap for the resulting infinite-dimensional system. An approximation of Widiasih's temperature-albedo line system was introduced in \cite{McGehee2014}, a simplification using smooth invariant manifold theory that resulted in a planar system of ODEs exhibiting the same qualitative behavior as Widiasih's system.
The approximating temperature-albedo line system in \cite{McGehee2014} then served as the basis for the nonsmooth ``flip-flop" glacial cycle presented in \cite{wwhm}, in which a nonsmooth attracting periodic orbit was shown to exist. This periodic orbit represented the Earth's climate system cycling between glacial and interglacial states, with the switching mechanism provided by a conceptual ice sheet mass balance principle. In this study, we extend the \cite{wwhm} model by removing a symmetry assumption about the climate system.
This removal of the symmetry assumption is justified when considering the inherent asymmetry between the northern and southern polar regions. The most notable difference is the fact that over the past 800,000 years, Antarctica has been completely glaciated, and ``glacial" advance and retreat refers to major changes in Southern Ocean sea ice extent \cite{raylisnic,gersonde,fraser} and glaciers in mountainous areas of southern South America, Africa, and Oceania \cite{rother,darvill} rather than the large glaciers terminating on land in the Northern Hemisphere \cite{abe-ouchi}. Thus, while glaciers in the Northern Hemisphere terminated on land, Southern Hemisphere glaciers terminated in the Southern Ocean with large ice shelves and sea ice extent reaching perhaps as far as 45$^\circ$S at times in some places \cite{gersonde,fraser}. For this reason, we do not place a mass balance equation in the Southern Hemisphere and allow the Southern Hemisphere albedo line to indicate Southern Hemisphere glacial dynamics in our model. Other potential differences in the glacial records are smaller amplitude oscillations for ice volume in the Southern Hemisphere, relative to the Northern Hemisphere \cite{raylisnic}; however, some records indicate oscillations of similar amplitude \cite{blunier}.
The main question that our model addresses is: do Northern Hemisphere glacial cycles affect the Southern Hemisphere and, if so, can they drive synchronous cycles in both hemispheres? Here we explicitly consider the role of global temperature/albedo feeback and Northern Hemisphere albedo/mass balance feedback. We leave the impact of Earth's changing orbital parameters to a later study.
\section{Governing Equations with Two Albedo Lines} \label{section-governing-eq}
\subsection{Temperature Equation}
The energy balance equations introduced by M. Budyko \cite{budyko} and W. Sellers \cite{sellers} in 1969 describe the evolution of the Earth's latitudinally averaged annual mean surface temperature $T(y,t)$, where $t$ denotes time in years and $y$ denotes the sine of the latitude. In the model we use here, the temperature evolves based on M. Budyko's energy balance equation \cite{budyko}
\begin{equation}
\begin{aligned}
R \frac{\partial T}{\partial t} = Q s(y) (1-\alpha (y,\mathbf\eta) ) - \left(A + B T(y,t) \right) - C \left(T(y,t) - \overline{T}(t) \right),
\end{aligned}
\label{EQ-Budyko}
\end{equation}
where the change in temperature is determined by the absorbed solar radiation, $Q s(y,\beta) (1-\alpha (y,\mathbf\eta))$; the emitted longwave radiation, $A + B T(y,t)$; and energy transport across latitudes, $C \left(T(y,t) - \overline{T}(t) \right)$ where $\overline{T}(t)$ is the global average temperature. We note W. Sellers independently introduced a similar energy balance model in the same year Budyko's appeared, albeit one with a different meridional energy transport mechanism \cite{sellers}.
The physical meaning of the different terms and parameters of \eqref{EQ-Budyko} have been explained extensively in the literature (see for example \cite{Tung2007,Kaper2013,Widiasih2013,Nadeau-dis}), so we omit a detailed explanation here. Instead we provide Table \ref{TAB-parameters-Earth} with brief physical descriptions of the parameters and note the major changes due to our removal of the symmetry assumption used in previous studies in the remainder of this section.
Because we consider the possibility of asymmetry between the hemispheres, we let sine of the latitude $y$ range from the south pole $y=-1$ to the north pole $y=1$ rather than from the equator to the north pole ($y\in[0,1]$). The surface albedo is given by $\alpha(y,\eta)$, which depends on $y$ and the location of surface ice, the lower-latitude boundary of which is typically denoted $\eta$. In this work, however, we take $\eta=(\eta_S,\eta_N)$, which gives the location of a southern ($\eta_S$) and northern ($\eta_N$) latitude where the albedo changes. We restrict these variables to the interval $[-1,1]$ with the condition $-1\leq\eta_S\leq\eta_N\leq1$ (i.e., we do not let the ice lines cross each other). We consider a piecewise constant albedo function given by
\begin{equation}\label{alb}
\al(y,\eta_S,\eta_N)=
\begin{cases}
\al_2, &\text{if \ } -1 <y<\eta_S \\
\al_1, &\text{if \ } \eta_S <y<\eta_N \\
\al_2, &\text{if \ } \eta_N<y<1,\\
\end{cases} \quad \quad
\end{equation}
with appropriate averages at the ice lines.
We take $\alpha_1<\alpha_2$ so that the regions poleward of the ice lines are more reflective.
The energy transport term is a simple linear relaxation to the mean annual global temperature given by integrating the temperature over all latitudes (the interval $[-1,1]$) $\overline T(t) = \frac{1}{2}\int_{-1}^1T(y,t)dy$. Finally, for clarity, note that the distribution of the annual insolation across $y$, which also depends on the tilt of the Earth's spin axis (or {\em obliquity}) $\beta$, can be approximated to any degree of accuracy by
\begin{equation}\label{s(y)}
s(y)=\sum^M_{m=0} a_{2m}p_{2m}(\cos\beta)p_{2m}(y),
\end{equation}
where $p_{2m}$ is the $2m$th Legendre polynomial and the $a_{2m}$ can be explicitly determined following \cite{Nadeau2017}. In this study we fix the obliquity at the Earth's current value $\beta=23.5^\circ$, for which $s_{2m}=a_{2m}p_{2m}(\cos(\pi 23.5/180)),$ and we write $s(y)$ in lieu of $s(y,\beta)$.
In a computation similar to that presented for Budyko's equation in \cite{Tung2007}, one finds that at equilibrium the temperature distribution is
\begin{equation}\label{Tstar}
T^*(y)=\frac{1}{B+C}\left( Qs(y)(1-\al(y))-A+C\overline{T^*}\right),
\end{equation}
with the global mean temperature given by
\begin{equation}\label{Tstarbar}
\overline{T^*}=\frac{1}{B}\left(Q(1-\al_2)-A+\frac{1}{2}Q(\al_2-\al_1) \int^{\eta_N}_{\eta_S}s(y) dy\right).
\end{equation}
We note that, due to the use of expansion \eqref{s(y)}, the equilibrium function $T^*(y)$ is a (``piecewise even") polynomial of degree $2M$ in $y$ and degree $2M+1$ in each of $\eta_N$ and $\eta_S$.
\subsection{Albedo Line Equations}
Here we consider two dynamic ice line equations in the fashion of Widiasih's single ice line equation \cite{Widiasih2013}. In particular, the movement of an ice line is determined by the temperature at the ice line relative to a critical temperature $T_c$, the highest temperature at which ice is present year round. We have
\begin{equation}
\begin{aligned}
\frac{d\eta_S}{dt}&=\rho(T_c-T(\eta_S,t)),\\
\frac{d\eta_N}{dt}&=\rho(T(\eta_N,t)-T_c).
\end{aligned}
\label{EQ-ice-line-cap}
\end{equation}
These equations dictate that if the temperature at the albedo line is greater than the critical temperature, the albedo line moves toward its own pole. If the temperature is less than the critical temperature, the albedo line moves toward the opposite pole.
The positive parameter $\rho$ controls how fast the ice line changes relative to changes in temperature.
In their discussion of glacial cycles on Earth, McGehee and Widiasih give an in-depth discussion on the behavior of solutions of a similar, hemispherically symmetric energy balance model relative to the choice of $\rho$ \cite{McGehee2014}.
\begin{table}
\caption{Parameter values used in this study (unless otherwise noted).}
\begin{center}
\begin{tabular}{ || c | p{6cm} | r | r ||}
\hline
Parameter & Brief Description & Value & Units \\ \hline \hline
$R$ & Surface layer heat capacity & 1 & Wm$^{-2}$K$^{-1}$ \\ \hline
$Q$ & Annual average insolation & 343 & Wm$^{-2}$ \\ \hline
$\beta$ & Obliquity& 23.5 & degrees \\ \hline
$\alpha_1$ & Albedo between the albedo line latitudes $\eta_S(t)$ and $\eta_N(t)$ & 0.32 & dimensionless \\ \hline
$\alpha_2$ & Albedo poleward of the albedo line latitudes $\eta_S(t)$ and $\eta_N(t)$ & 0.62 & dimensionless \\ \hline
$A$ & Greenhouse Gas parameter & 202 & Wm$^{-2}$ \\ \hline
$B$ & Outgoing radiation& 1.9 & Wm$^{-2}$K$^{-1}$ \\ \hline
$C$ & Efficiency of heat transport& 3.04 & Wm$^{-2}$K$^{-1}$ \\ \hline
$T_c; \ T_{cS/N}, \ T_{cN}^{\pm}$ & Critical temperature determining advance/retreat of albedo lines & -10; varies & $^\circ$C \\ \hline
$\rho$ & Albedo line response to temperature change & 0.3 & K$^{-1}$yr$^{-1}$ \\ \hline
$2M$ & Degree of the polynomial approximation of the insolation function & 2 & dimensionless \\ \hline
$a$ & Accumulation rate & 1.05 & dimensionless \\ \hline
$b$ & Critical ablation rate & 1.75 & dimensionless\\ \hline
$b_-$ & Glacial ablation rate & 1.5 & dimensionless\\ \hline
$b_+$ & Interglacial ablation rate & 5 & dimensionless \\ \hline
$\epsilon$ & Mass balance response to albedo change & 0.03 & yr$^{-1}$ \\ \hline
\end{tabular}
\end{center}
\label{TAB-parameters-Earth}
\end{table}
\subsection{Finite-dimensional approximation of the temperature equation}
Recall the equilibrium temperature distribution \eqref{Tstar} is a piecewise even function of $y$. In addition, we are assuming the expansion of $s(y)$ in even Legendre polynomials \eqref{s(y)}. We are thus motivated to express the temperature function piecewise as follows:
\begin{align}\label{Texp}
T(y,t)=
\begin{cases}
U(t,y)=\sum^M_{m=0} u_{2m}(t)p_{2m}(y), & -1\leq y <\eta_S\\
V(t,y)=\sum^M_{m=0} v_{2m}(t)p_{2m}(y), & \eta_S< y <\eta_N\\
W(t,y)=\sum^M_{m=0} w_{2m}(t)p_{2m}(y), & \eta_N<y <1.\\
\end{cases}
\end{align}
The use of expression \eqref{Texp} is similar in spirit to that used in \cite{Walsh2015} to model extensive glacial episodes in the Neoproterozoic Era, work in turn motivated by the approach to Budyko's equation taken in \cite{McGehee2014}.
The temperature at each ice line is taken to be the appropriate average, namely,
\begin{align}\label{Teta1}
T(\eta_S)&=\tx{\frac{1}{2}}\sum^M_{m=0} (u_{2m}+v_{2m})p_{2m}(\eta_S),\\\notag
T(\eta_N)&=\tx{\frac{1}{2}}\sum^M_{m=0} (v_{2m}+w_{2m})p_{2m}(\eta_N).\notag
\end{align}
Separately substituting each expression in \eqref{Texp} along with expansion \eqref{s(y)} into equation \eqref{EQ-Budyko}, and equating the respective coefficients of $p_{2m}$, one arrives at the system of $3(M+1)$ ODEs
\begin{align}\label{full}
R\dot{u}_0&= Qs_0(1-\al_2)-A-(B+C)u_0+C\ov{T} \\ \notag
R\dot{u}_{2m}& = Qs_{2m}(1-\al_2) -(B+C)u_{2m}, \qquad\qquad m\geq 1 \\ \notag
R\dot{v}_0&= Qs_0(1-\al_1)-A-(B+C)v_0+C\ov{T} \\ \notag
R\dot{v}_{2m}& = Qs_{2m}(1-\al_1) -(B+C)v_{2m}, \qquad\qquad m\geq 1 \\ \notag
R\dot{w}_0&= Qs_0(1-\al_2)-A-(B+C)w_0+C\ov{T} \\ \notag
R\dot{w}_{2m}&= Qs_{2m}(1-\al_2) -(B+C)w_{2m}, \qquad\qquad m\geq 1. \notag
\end{align}
In addition
\begin{align}\label{Tbar1}
2\ov{T}&=\int^{\eta_S}_{-1}U(t,y) dy+\int^{\eta_N}_{\eta_S}V(t,y) dy+\int^{1}_{\eta_N}W(t,y) dy\\\notag
&=\int^{1}_{-1}U(t,y) dy-\int^{\eta_N}_{\eta_S}U(t,y) dy-\int^{1}_{\eta_N}U(t,y) dy+\int^{\eta_N}_{\eta_S}V(t,y) dy+\int^{1}_{\eta_N}W(t,y) dy\\\notag
&=2u_0-\sum^M_{m=0}(u_{2m}-v_{2m})(P_{2m}(\eta_N)-P_{2m}(\eta_S))-\sum^M_{m=0}(u_{2m}-w_{2m})(1-P_{2m}(\eta_N)),
\end{align}
where we set $P_{2m}(y)=\int p_{2m}(y)dy, m\geq 0$ for ease of notation.
Note the decoupling in \eqref{full}; each of the equations tends to equilibrium except for the three equations corresponding to $m=0$. We thus assume that
\begin{equation}\label{uvw*}
u_{2m}=u^*_{2m}=Ls_{2m}(1-\al_2), \ v_{2m}=v^*_{2m}=Ls_{2m}(1-\al_1), \ w_{2m}=w^*_{2m}=u^*_{2m}, \ m\geq 1,
\end{equation}
where we have let $L=Q/(B+C)$.
With assumption \eqref{uvw*}, equations \eqref{Teta1} become
\begin{align}\label{Teta2}
T(\eta_S)&=\tx{\frac{1}{2}}(u_0+v_0)+\tx{\frac{1}{2}}\sum^M_{m=1} (u^*_{2m}+v^*_{2m})p_{2m}(\eta_S),
\\\notag
T(\eta_N)&=\tx{\frac{1}{2}}(v_0+w_0)+\tx{\frac{1}{2}}\sum^M_{m=1} (v^*_{2m}+u^*_{2m})p_{2m}(\eta_N).\notag
\end{align}
In addition, and after much simplification, \eqref{Tbar1} can be placed in the form
\begin{equation}\label{Tbar2}
2\ov{T}=\eta_S(u_0-v_0)+\eta_N(v_0-w_0)+u_0+w_0+L(\al_2-\al_1)\sum^M_{m=1} s_{2m}(P_{2m}(\eta_N)-P_{2m}(\eta_S)).
\end{equation}
For an additional simplification, note that if $x=u_0-w_0$ then
\begin{equation}\notag
R\dot{x}=R\dot{u}_0-R\dot{w}_0=-(B+C)x.
\end{equation}
We have $x(t)=u_0(t)-w_0(t)\rw 0$ as $t\rw\infty$, and hence we assume $u_0=w_0$. Thus in system \eqref{full}, we need only consider the $u_0$- and $v_0$-equations. Also recalling $w^*_{2m}=u^*_{2m}$ for $m\geq 1,$ $u_0=w_0$ additionally implies that
$U(t,y)$ and $W(t,y)$ are part of the same degree $2M$ polynomial of $y$, albeit with different domains, again assuming all the appropriate variables are at equilibrium.
We are thus lead to consider the pair of equations
\begin{align}\label{u0v0}
R\dot{u}_0&= Qs_0(1-\al_2)-A-(B+C)u_0+C\ov{T} \\ \notag
R\dot{v}_0&= Qs_0(1-\al_1)-A-(B+C)v_0+C\ov{T}. \notag
\end{align}
For $m\geq 1, \ u^*_{2m}+v^*_{2m}=2Ls_{2m}(1-\al_0)$, where $\al_0=\frac{1}{2}(\al_1+\al_2).$ Equations \eqref{Teta2} then become
\begin{align}\label{Teta3}
T(\eta_S)&=\tx{\frac{1}{2}}(u_0+v_0)+L(1-\al_0)\sum^M_{m=1}s_{2m}p_{2m}(\eta_S)\\\notag
&=\tx{\frac{1}{2}}(u_0+v_0)+L(1-\al_0)(s(\eta_S)-s_0p_0(\eta_S))\\\notag
&=\tx{\frac{1}{2}}(u_0+v_0)+L(1-\al_0)(s(\eta_S)-1), \mbox{ and similarly}\\ \notag
T(\eta_N)&=\tx{\frac{1}{2}}(u_0+v_0)+L(1-\al_0)(s(\eta_N)-1).
\end{align}
Setting $w_0=u_0$ in \eqref{Tbar2} yields
\begin{equation}\label{Tbar3}
2\ov{T}=2u_0-(u_0-v_0)(\eta_N-\eta_S)+L(\al_2-\al_1)\left(\int^{\eta_N}_{\eta_S}s(y) dy-(\eta_N-\eta_S)\right).
\end{equation}
As a final step we introduce the new variables $w=\frac{1}{2}(u_0+v_0)$ and $z=u_0-v_0$. System \eqref{u0v0} becomes
\begin{align}\label{wz}
R\dot{w}&=Qs_0(1-\al_0)-(B+C)w-A+C\ov{T}\\\notag
R\dot{z}&=Qs_0(\al_1-\al_2)-(B+C)z.\notag
\end{align}
We see that $z\rw z^*=Ls_0(\al_1-\al_2)$ as $t\rw\infty$, and so we set $z=z^*$ in all that follows. We have reduced the study of system \eqref{full} to that of the equation
\begin{equation}\label{wdot}
R\dot{w}=Qs_0(1-\al_0)-(B+C)w-A+C\ov{T}.
\end{equation}
In terms of $w$, equations \eqref{Teta3} become
\begin{equation}\label{Teta4}
T(\eta_S)=w+L(1-\al_0)(s(\eta_S)-1),
\ \
T(\eta_N)=w+L(1-\al_0)(s(\eta_N)-1),
\end{equation}
while \eqref{Tbar3} simplifies to
\begin{equation}\label{Tbar4}
\ov{T}=w-\tx{\frac{1}{2}}Ls_0(\al_2-\al_1)\left(1-\dis \int^{\eta_N}_{\eta_S}s(y) dy\right).
\end{equation}
Note equation \eqref{Tbar4} states that $w$ is a translation of the global annual mean surface temperature, where the translation depends upon the integral of the insolation distribution function $s(y)$ between the albedo lines.
Coupling the temperature equation \eqref{wdot} with the ice line evolution equations \eqref{EQ-ice-line-cap} gives a $(w,\eta_S,\eta_N)$-system that can be placed in the form
\begin{equation}
\begin{aligned}
\frac{dw}{dt}&=-\frac{B}{R}\left(w-F(\eta_S, \eta_N)\right)\\
\frac{d\eta_S}{dt}&=-\rho(w-G(\eta_S))\\
\frac{d\eta_N}{dt}&=\rho(w-G(\eta_N)),
\end{aligned}
\label{system}
\end{equation}
where
\begin{equation}\label{F}
F(\eta_S,\eta_N)=\frac{1}{B}\left(Qs_0(1-\al_0)-A+\frac{1}{2}CLs_0(\al_1-\al_2)(1-\dis \int^{\eta_N}_{\eta_S}s(y) dy)\right),
\end{equation}
and\begin{equation}\label{G}
G(\cdot)=-L(1-\al_0)(s(\cdot)-1)+T_c.
\end{equation}
\section{Behavior of the Two Albedo Line System}
\label{Section-Symmetry}
We begin with a discussion of the case in which the critical temperatures at $\eta_S$ and $\eta_N$ are equal, with each denoted $T_c$. We then discuss how different critical temperatures affect the equilibria of the system. This discussion portends analysis to follow in Section \ref{Section-Mass-Balance}. In the full model with the mass balance equations, the critical temperature at the northern albedo line will change depending on whether the climate state is in a glacial period or an interglacial period.
We consider our system \eqref{system} on the space
\begin{equation}\notag
{\mathcal B}^\pr=\{ (w,\eta_S,\eta_N) : w\in\R, \eta_S,\eta_N\in[-1,1], \eta_S\leq \eta_N \}.
\end{equation}
The restriction of $\eta_S$ and $\eta_N$ to $[-1,1]$ corresponds to the physical boundary of the latitudes at the south and north poles. The boundary component given by $\eta_S\leq \eta_N$ ensures that we do not have the (nonphysical) situation of the albedo lines crossing (the case where $\eta_S=\eta_N$ indicates a snowball Earth). In a subsequent paper system \eqref{system} will be analyzed on the boundary of ${\mathcal B}^\pr$ via the introduction of an appropriately defined Filippov flow (akin in spirit to
\cite{Barry2017}). In the present work we restrict attention to the interior of the state space; nonetheless, a detailed description of Fillipov flows will be presented in Section \ref{Section-Mass-Balance}, in which their use is needed to analyze a (discontinuous) extension of \eqref{system} in which separate albedo and snow lines are considered.
\subsection{Equal critical temperatures} \label{Section-Tc-Equal}
Let ${\bf Y}_+={\bf Y}_+(w,\eta_S,\eta_N)$ denote the vector field given in \eqref{system}, with $\psi_+=\psi_+((w,\eta_S,\eta_N),t)$ its associated flow. (The use of the subscript $+$ foreshadows analysis to come in Section \ref{Section-Mass-Balance}.)
We set $M=1$ in \eqref{s(y)} in all that follows as the use of higher order approximations yields qualitatively similar results.
With parameters as in Table 1, system \eqref{system} has two equilibria in ${\mathcal B}^\pr$ given by
\begin{align*}
Q^u_+&=(w^u_+,(\eta_S)^u_+,(\eta_N)^u_+)=( -17.118, -0.249, 0.249) \ \mbox{ and } \\ Q^s_+&=(w^s_+,(\eta_S)^s_+,(\eta_N)^s_+)=(5.188, -0.955, 0.955),
\end{align*}
each lying in the plane $\eta_N=-\eta_S$.
As the Jacobian $J{\bf Y}_+(Q^s_+)$ has eigenvalues $-15.85, -15.05$ and $-1.10$, $Q^s_+$ is a stable node for the flow $\psi_+$. (One can check equilibrium $Q^u_+$ is a saddle having 2-dimensional stable manifold.) Note the equilibrium $Q^s_+$ corresponds to small, symmetric ice caps, while $Q^u_+$ corresponds to (unstable) large, symmetric ice caps. These results agree with earlier studies where the albedo lines are assumed to be symmetric across the equator (e.g., \cite{McGehee2014,Widiasih2013}).
To help visualize these structures, we plot the $w$-nullcline for ${\bf Y}_+$ (green), together with the curve of intersection of the $\eta_S$- and $\eta_N$-nullclines for ${\bf Y}_+$ (red) in Figure \ref{FIG-Y-structures}. The intersection of the red curve and the green surface yields the two equilibria in ${\mathcal B}^\pr$ for \eqref{system}. Also plotted in Figure \ref{FIG-Y-structures}
is the projection of the curve of intersection of the $\eta_S$- and $\eta_N$-nullclines for ${\bf Y}_+$ (red) in the $\eta_S\eta_N$-plane, which can be shown to be the line $\eta_N=-\eta_S$.
We pause to comment on the role played by the parameter $T_c$, which appears in the $\dot{\eta}_S$- and
$\dot{\eta}_N$-equations in system \eqref{system}. An increase in $T_c$ serves to translate the $\eta_S$- and $\eta_N$-nullclines up, that is, the red curve in the left plot in Figure \ref{FIG-Y-structures} moves up while the $w$-nullcline remains unchanged. This causes $Q^s_+$ and $Q^u_+$ to move towards each other ($(\eta_S)_{+}^s$ and $(\eta_N)_{+}^s$ move symmetrically toward the equator, $(\eta_S)_{+}^u$ and $(\eta_N)_{+}^u$ move symmetrically toward their respective poles), corresponding to larger stable ice caps at equilibrium. A sufficiently large increase in $T_c$ leads to a saddle-node bifurcation in which the $\eta_S$- and $\eta_N$-nullclines tangentially intersect the $w$-nullcline before passing above the $w$-nullcline.
Similarly, a decrease in $T_c$ from $-10^\circ$C moves $Q^s_+$ and $Q^u_+$ away from each other. A sufficiently negative $T_c$ first leads to $(\eta_S)_{+}^s=-1$ and $(\eta_N)_{+}^s=1$, corresponding to a ``stable" ice-free Earth. Further decreasing $T_c$ leads to $Q_{+}^s$ leaving $\mathcal{B}'$ and, eventually, $(\eta_S)_{+}^u=(\eta_N)_{+}^u$, corresponding to an ``unstable'' completely glaciated Earth. An even further decrease in $T_c$ causes $Q_{+}^u$ to leave $\mathcal{B}'$ as well. Formalizing these statements requires consideration of the dynamics on the boundary of ${\mathcal B}^\pr$, which will appear in future work. We note the range of $T_c$-values used in the following section ensures the existence of two equilibria for system \eqref{system} within the interior of ${\mathcal B}^\pr$.
\begin{figure}
\begin{center}
\includegraphics[width=6in,trim = 1.5in 7in 1in 1in, clip]{nullclinesV3.pdf}\\
\caption{{ {\em Left}: The $w$-nullcline (green), and the curves of intersection of the of the $\eta_S$- and $\eta_N$-nullclines for ${\bf Y}_+$ (red) when $\Tcsm=\Tcnm=-10^\circ$C and ${\bf Y}_-$ (blue) when $\Tcsm=-10^\circ$C and $\Tcnm=-5^\circ$C. \ {\em Right}: The projections of the curves of intersection of the $\eta_S$- and $\eta_N$-nullclines for ${\bf Y}_+$ (red) and ${\bf Y}_-$ (blue) in the $\eta_S\eta_N$-plane.
}}
\label{FIG-Y-structures}
\end{center}
\end{figure}
\subsection{Different critical temperatures}
While the critical temperature value $T_c=-10^\circ$C is often used in the energy balance climate literature for the Earth, other values have been used as well. For example, $T_c$ was set to $0^\circ$C in \cite{pierre} when modeling a generally colder world. A linear drift in $T_c$ from $-13^\circ$C
to $-3^\circ$C was incorporated in the glacial cycle model presented in \cite{tzip2003} to represent the cooling of the deep ocean during the Pleistocene.
We thus consider the case in which the critical temperature $T_{cS}$ at $\eta_S$ differs from the critical temperature $T_{cN}$ at $\eta_N$, a possibility easily investigated with our model. Consider the system
\begin{equation}
\begin{aligned}
\frac{dw}{dt}&=-\frac{B}{R}\left(w-F(\eta_S, \eta_N)\right)\\
\frac{d\eta_S}{dt}&=-\rho(w-G_S(\eta_S))\\
\frac{d\eta_N}{dt}&=\rho(w-G_N(\eta_N)),
\end{aligned}
\label{system-NS}
\end{equation}
where
\begin{equation}\label{GS}
G_S(\eta_S)=-L(1-\al_0)(s(\eta_S)-1)+T_{cS}\quad\text{and}\quad G_N(\eta_N)=-L(1-\al_0)(s(\eta_N)-1)+T_{cN}.
\end{equation}
We let ${\bf Y}_-={\bf Y}_-(w,\eta_S,\eta_N)$ denote the vector field given in system \eqref{system-NS}, for which $w,\eta_S,\eta_N\in\mathcal B^\pr$ and the parameters are given in Table 1, with the sole exception being that we allow $T_{cN}>-10^\circ$C. (The use of the subscript `$-$' will become clear in Section \ref{Section-Mass-Balance}.)
We let $\psi_-=\psi_-((w,\eta_S,\eta_N),t)$ denote the flow associated with \eqref{system-NS}.
The scenario $T_{cS}=-10^\circ$C and $T_{cN}=-5^\circ$C is depicted in Figure \ref{FIG-Y-structures}. The green $w$-nullcline remains unchanged as the critical temperature does not appear in the $\dot{w}$-equation. Recall the red curve in the left plot in Figure \ref{FIG-Y-structures} is the intersection of the $\eta_S$- and $\eta_N$-nullclines in the symmetric case ($T_{cS}=T_{cN}=-10^\circ$C in \eqref{system-NS}). The blue curve in the left plot in Figure \ref{FIG-Y-structures} is the intersection of the $\eta_S$- and $\eta_N$-nullclines for \eqref{system-NS} when $T_{cS}=-10^\circ$C and $T_{cN}=-5^\circ$C. Also plotted in Figure \ref{FIG-Y-structures} are the projections of the red and blue curves in the
$\eta_S\eta_N$-plane.
Keeping $T_{cS}=-10^\circ$C fixed, we see in Figure \ref{FIG-Y-structures} that an increase of $T_{cN}$ from $-10^\circ$C to $-5^\circ$C yields an equilibrium point $Q^s_{-}=(w^s_-,(\eta_S)^s_-,(\eta_N)^s_-)$ for ${\bf Y}_{-}$ near $Q^s_+$ with $(\eta_S)^s_->(\eta_S)^s_+$ and $(\eta_N)^s_-<(\eta_N)^s_+$ (that is, each albedo line has moved equatorward). Given that $Q^s_+$ is a stable node for ${\bf Y}_+$, and using the fact ${\bf Y}_+$ and ${\bf Y}_{-}$ are polynomial vector fields (and hence smooth, including in the critical temperature parameter), a
sufficiently small translation ensures that $Q^s_-$ is a stable node for the flow $\psi_-$. We note there is a saddle $Q^u_-$ for $\psi_{-}$, near $Q^u_+$, as well.
As $T_{cN}$ decreases to $-10^\circ$C, $Q^s_-\rw Q^s_+$ and $Q^u_-\rw Q^u_+$.
We note the behavior of the albedo lines for the flow exhibits an asymmetry when $T_{cN}\not= T_{cS}$. As can be gleaned from Figure \ref{FIG-Y-structures}, when $T_{cN}=-5^\circ$C, $\eta_N(t)\rw 0.795$, a larger ice cap than in the case $\Tcn=-10^\circ$C. Of interest is the fact the Southern Hemisphere albedo line also moves to a larger (asymmetric) ice cap position ($(\eta_S)^s_-=-0.907$), relative to its stable position when $T_{cN}=-10^\circ$C ($(\eta_S)^s_+=-0.955$). That is, the coupling of $\eta_S$ and $\eta_N$ provided by the $w$-equation in \eqref{system-NS} furnishes a linkage between the Northern and Southern Hemispheres: a different stable $\eta_N$ position yields a different stable $\eta_S$ position, even though $T_{cS}$ remains constant at $-10^\circ$C.
We plot the evolution of the albedo lines starting with large initial ice caps ($\eta_N=-\eta_S=0.5$) for system \eqref{system-NS} with $T_{cS}=-10^\circ$C fixed and various $T_{cN}$-values in Figure \ref{FIG-asym-time-series}.
Similar behavior occurs if the Northern Hemisphere critical temperature is left at $T_{cN}=-10^\circ$C and $T_{cS}$ is increased.
\begin{figure}
\begin{center}
\includegraphics[width=6in,trim = 1.6in 7.5in 1in 1in, clip]{stable3D}\\
\caption{{\small The evolution of the albedo lines under the flows $\psi_{\pm}$ with $\Tcs=-10^\circ$C. {\em Red}: $\Tcn=-10^\circ$C. \ {\em Blue}: $\Tcn=-5^\circ$C. \ {\em Black}: $\Tcn=-2^\circ$C.
}}
\label{FIG-asym-time-series}
\end{center}
\end{figure}
\section{Mass-Balance Can Drive Synchronous Global Glacial Cycles}
\label{Section-Mass-Balance}
In this section we incorporate the glacial cycle model introduced in \cite{wwhm} into the Northern Hemisphere of our global temperature, two albedo line model. The motivation for this model enhancement stems in part from the glacial cycle theory of M. Milankovitch, which asserts that changes in Northern Hemisphere high latitude insolation, due to variations in Earth's orbital elements over long time scales, comprise the principle forcing mechanism of the glacial-interglacial cycles \cite{hays,milank, raylisnic,uemura}.
The glacial cycle model discussed below exhibits a threshold behavior, ``flip-flopping" between glacial advance and retreat based on a conceptual ice sheet mass balance equation. For more detailed background and motivation for this aspect of the model, the reader is referred to \cite{wwhm}.
\subsection{Mass balance flip-flop}
We begin by summarizing the process of adding a conceptual mass-balance variable $\xi_N$ in an effort to model the accumulation and ablation of the Northern Hemisphere glaciers, as presented in \cite{wwhm}.
Let $\xi_N$ denote the latitude of the edge of the Northern Hemisphere glaciers. While the evolution of the ice edge $\xi_N$ is driven in the abstract by a mass balance principle, we
do not explicitly consider ice volume and mass here.
To construct the equations governing $\xi_N$ during glacial or interglacial periods, we assume snow is accumulating between $\eta_N$ and the north pole at a (dimensionless) rate $a$, while ablation
occurs between $\xi_N$ and $\eta_N$ at a (dimensionless) rate $b$. We note accumulation and ablation of ice play an important role in the advance, retreat, and size of a glacier (see, e.g., \cite{weertman}). In particular, increased ablation rates when the glacier is retreating are key to obtaining the rapid interglacial retreats that are present in paleoclimate records \cite{abe-ouchi}. In this model it is the reduced albedo of the region between $\xi_N$ and $\eta_N$ due to factors such as aging snow \cite{gallee}, superglacial forest growth \cite{wright}, and dust loading \cite{peltier} that contributes to the increased ablation rate during glacial retreats.
We first define a critical ablation rate $b$. Conceptually, the equation
\begin{equation}
b(\eta_N-\xi_N)=a(1-\eta_N)
\end{equation}
defines the Northern Hemisphere albedo- and ice-edge latitudes where ablation (left hand side) and accumulation (right hand side) are equal. Rearranging this equation allows us to see that if
\begin{equation}
\xi_N>\left(1+\frac{a}{b}\right)\eta_N-\frac{a}{b}
\end{equation}
then the ablation $b(\eta_N-\xi_N)$ will be less than accumulation $a(1-\eta_N)$ and we should be in a glacial period (with the ice edge advancing). In a glacial period the ablation is less than the critical ablation rate, so we let $b_-<b$ and set
\begin{equation}
\dot\xi_N=\epsilon(b_-(\eta_N-\xi_N)-a(1-\eta_N)),\text{ when }\xi_N>\left(1+\frac{a}{b}\right)\eta_N-\frac{a}{b},
\end{equation}
with $\eps>0.$
On the other hand if
\begin{equation}
\xi_N<\left(1+\frac{a}{b}\right)\eta_N-\frac{a}{b}
\end{equation}
then the ablation $b(\eta_N-\xi_N)$ will be greater than accumulation $a(1-\eta_N)$ and we should be in an interglacial period with a large ablation rate and the ice edge retreating. We let $b_+>b$ and set
\begin{equation}
\dot\xi_N=\epsilon(b_+(\eta_N-\xi_N)-a(1-\eta_N)),\text{ when }\xi_N<\left(1+\frac{a}{b}\right)\eta_N-\frac{a}{b}.
\end{equation}
When $\xi_N - \left(\left(1+\frac{a}{b}\right)\eta_N-\frac{a}{b}\right)$ passes through 0, the system flips from one with a relatively low ablation rate to one with a relatively high ablation rate, or vice versa.
While fixing the critical temperature $T_{cS}=-10^\circ$C at $\eta_S$, we allow for different critical temperatures at $\eta_N$ during the advance ($T^-_{cN}$) and retreat ($T^+_{cN}$) of the Northern Hemisphere glaciers, as intimated in Section \ref{Section-Symmetry}. We choose $T^-_{cN}>T^+_{cN}$ as in \cite{wwhm}.
We are thus lead to consider the following $(w,\eta_S,\eta_N,\xi_N)$-system, one having discontinuities on a hyperplane corresponding to points at which the Northern Hemisphere ice sheet mass balance equals zero.
\subsection{The full system: Southern and northern albedo lines with mass-balance flip-flop in the Northern Hemisphere}
The $w$- and $\eta_S$-equations remain as in system \eqref{system}, while the flip-flop mechanism described above is placed in the Northern Hemisphere. The system then assumes the form
\begin{subequations}\label{Nflipflop}
\begin{align}
\dot{w}&=-\tx{\frac{B}{R}}\left(w-F(\eta_S,\eta_N)\right)\label{NflipflopA} \\
\dot{\eta_S} &=-\rho(w-G(\eta_S))\label{NflipflopB} \\
\dot{\eta_N} &=\rho(w-H_\pm(\eta_N))\label{NflipflopC} \\
\dot{\xi_N} &=\eps(b_\pm(\eta_N-\xi_N)-a(1-\eta_N)), \label{NflipflopD}
\end{align}
\end{subequations}
where $F$ and $G=G_S$ are as in \eqref{F} and \eqref{GS}, respectively, and where we set
\begin{equation}\notag
H_+(\eta_N)=-L(1-\al_0)(s(\eta_N)-1)+T^+_{cN} \ \mbox{ \ and \ } \ H_-(\eta_N)=-L(1-\al_0)(s(\eta_N)-1)+T^-_{cN}.
\end{equation}
The use of the subscript `+' indicates $\xi_N<\left(1+\frac{a}{b}\right)\eta_N-\frac{a}{b}$, so that the ice sheet is retreating in the Northern Hemisphere. The subscript `-' indicates $\xi_N>\left(1+\frac{a}{b}\right)\eta_N-\frac{a}{b}$, with the Northern Hemisphere glaciers advancing equatorward in this regime.
The state space for \eqref{Nflipflop} is
\begin{equation}\notag
{\mathcal B}=\{ (w,\eta_S,\eta_N,\xi_N) : w\in\R, \ \eta_S,\eta_N,\xi_N \in [-1,1],\eta_S\leq \eta_N \}.
\end{equation}
We note there will be no consideration of the dynamics on the boundary of $\mathcal{B}$ in this paper; the results and analysis to follow pertain to an invariant subset of $\mathcal{B}$ in which $-1<\eta_S<\eta_N<1$.
Recall we are assuming the critical temperatures and ablation rates satisfy $T^+_{cN}<T^-_{cN}$ \ and \ $b_-<b<b_+$, respectively. Finally, while the analysis in this section holds for any $M$-value and appropriately chosen parameters, we continue to set $M=1$. Thus, $G(\eta_S)$ and $H_\pm(\eta_N)$ are each quadratic polynomials, and $F(\eta_S,\eta_N)$ is the difference of a cubic polynomial in $\eta_N$ and a cubic polynomial in $\eta_S$.
Due to the presence of discontinuities induced by the switching mechanism from Northern Hemisphere glacial advance to retreat (and vice versa) discussed above, we analyze system \eqref{Nflipflop} as a {\em Filippov flow.}
To define the Filippov flow associated with system \eqref{Nflipflop}, we begin by letting
\begin{equation}\label{hsplit}
h:{\mathcal B}\rw \R, \ h(w,\eta_S,\eta_N,\xi_N)=b(\eta_N-\xi_N)-a(1-\eta_N)=(a+b)\eta_N-b\xi_N-a.
\end{equation}
The {\em switching manifold} \cite{dibernardo}, consisting of points in $\mathcal{B}$ at which the critical mass balance $b(\eta_N-\xi_N)-a(1-\eta_N)$ equals 0, is the hyperplane
\begin{align}\label{switch}
\Sigma &=\{(w,\eta_S,\eta_N,\xi_N) : h(w,\eta_S,\eta_N,\xi_N)=0\}\\\notag
&=\{(w,\eta_S,\eta_N,\xi_N) : \xi_N=(1+\tx{\frac{a}{b}})\eta_N-\tx{\frac{a}{b}}=\gamma(\eta_N)\}.\notag
\end{align}
The system is retreating toward an interglacial period when in the region
\begin{equation}\label{Splus}
S_+=\{(w,\eta_S,\eta_N,\xi_N) : h(w,\eta_S,\eta_N,\xi_N)>0\},
\end{equation}
and advancing to a glacial period when in
\begin{equation}\label{Sminus}
S_-=\{(w,\eta_S,\eta_N,\xi_N) : h(w,\eta_S,\eta_N,\xi_N)<0\}.
\end{equation}
Let ${\bf X}_+$ denote system \eqref{Nflipflop} when choosing $H_+$ and $b_+$, and let ${\bf X}_-$ denote system \eqref{Nflipflop} when choosing $H_-$ and $b_-$.
For ${\bf v}=(w,\eta_S,\eta_N,\xi_N)\in \mathcal{B},$ we then consider the differential inclusion
\begin{equation}\label{inclusion}
\dot{\bf v}\in {\bf X}({\bf v})=
\begin{cases}
{\bf X}_-({\bf v}), & {\bf v}\in S_-\\
\{ (1-p){\bf X}_-({\bf v})+p{\bf X}_+({\bf v}) : p\in [0,1]\}, & {\bf v}\in\Sigma \\
{\bf X}_+({\bf v}), & {\bf v}\in S_+ .
\end{cases}
\end{equation}
Note each of ${\bf X}_\pm$ is smooth on $S_\pm.$ While in $S_-$, solutions are unique with flow $\phi_-({\bf v},t)$ corresponding to system $\dot{\bf v}={\bf X}_-({\bf v})$. Similarly, solutions in
$S_+$ are unique with flow $\phi_+({\bf v},t)$ given by system $\dot{\bf v}={\bf X}_+({\bf v})$. For ${\bf v}\in\Sigma, \ \dot{\bf v}$ must lie in the closed convex hull of the two vectors ${\bf X}_-({\bf v})$ and ${\bf X}_+({\bf v})$.
A solution to \eqref{inclusion} {\em in the sense of Filippov} is an absolutely continuous function ${\bf v}(t)$ satisfying $\dot{\bf v}\in {\bf X}({\bf v})$ for almost all $t$. (Note $\dot{\bf v}(t)$ is not defined at times for which ${\bf v}(t)$ arrives at or leaves $\Sigma$.) Given that ${\bf X}_\pm$ are continuous on $S_\pm\cup \Sigma$, the set-valued map ${\bf X}({\bf v})$ is upper semi-continuous, and closed, convex and bounded for all ${\bf v}\in \mathcal{B}$ and $t\in\R$. This implies that for each ${\bf v}_0\in \mbox{Int}(\mathcal{B})$ there is a solution ${\bf v}(t)$ to differential inclusion \eqref{inclusion} in the sense of Filippov, defined on an interval $[0,t_f]$, with ${\bf v}(0)={\bf v}_0$ \cite{leine}.
\subsection{Regular and virtual equilibria}
As equations \eqref{NflipflopA}--\eqref{NflipflopC} decouple from \eqref{NflipflopD}, we first note that the vector fields corresponding to \eqref{NflipflopA}--\eqref{NflipflopC} are precisely the vector fields ${\bf Y}_\pm$ from Section \ref{Section-Symmetry} with associated flows $\psi_\pm=\psi_{\pm}((w,\eta_S,\eta_N),t)$.
Let $W(Q^s_+)$ denote the $\psi_+$-stable set of $Q^s_+$, and let $W(Q^s_-)$ denote the $\psi_-$-stable set of $Q^s_-$, noting that each stable set is a subset of $\mathcal{B}^\pr$ with interior.
By smoothness of the vector fields ${\bf Y}_\pm$ (each smooth in the critical temperature as well), we
choose $\Tcnm$ close enough to $\Tcnp$ to ensure that
\begin{equation}\label{3Dstablesets}
Q^s_-\in W(Q^s_+) \ \mbox{ and } \
Q^s_+\in W(Q^s_-),
\end{equation}
the motivation for which will become apparent below. Numerical investigations indicate that conditions \eqref{3Dstablesets} hold for $\Tcnm$ as large as $-1^\circ$C. We also note \eqref{3Dstablesets} holds for all $\eps>0$, where $\eps$ governs the rate of the mass balance response to albedo change as in equation \eqref{NflipflopD}.
Returning to the vector fields ${\bf X}_\pm$ associated with the full system \eqref{Nflipflop}, ${\bf X}_+$ then admits two equilibria in ${\mathcal B}$
\begin{align*}
P^u_+&=\left(w^u_+,(\eta_S)^u_+,(\eta_N)^u_+,(1+\tx{\frac{a}{b_+}})(\eta_N)^u_+-\tx{\frac{a}{b_+}}\right) \ \mbox{ and } \\
P^s_+&=\left(w^s_+,(\eta_S)^s_+,(\eta_N)^s_+,(1+\tx{\frac{a}{b_+}})(\eta_N)^s_+-\tx{\frac{a}{b_+}}\right).
\end{align*}
As the fourth column of the Jacobian matrix $J{\bf X}_+$ is $[0 \ 0 \ 0 \ \tx{-}\eps b_+]^T$, we conclude $P^s_+$ is a stable node for the retreating flow $\phi_+$ for all $\eps>0$ (while $P^u_+$ is a saddle with 3-dimensional stable manifold).
We would like to know which side of the switching manifold $\Sigma$ the equilibrium $P_{+}^s$ lies in. A computation yields
\begin{equation}\notag
h(P^s_+)= a \left(1-(\eta_N)^s_+\right)\left(\tx{\frac{b}{b_+}}-1\right)<0
\end{equation}
due to our assumption $b_+>b$, implying $P^s_+\in S_-$ (see equation \eqref{Sminus}). Thus $\phi_+$-trajectories are unable to converge to the stable node $P^s_+$ as they must first cross the switching manifold $\Sigma.$ Such an equilibrium point for a discontinuous vector field is known as a virtual equilibrium point \cite{dibernardo}, as defined in the introduction.
In a similar fashion, and recalling our choice of the parameter $\Tcnm$ as discussed above, the vector field ${\bf X}_-$ admits two equilibria
\begin{align*}
P^u_-&=\left(w^u_-,(\eta_S)^u_-,(\eta_N)^u_-,(1+\tx{\frac{a}{b_-}})(\eta_N)^u_--\tx{\frac{a}{b_-}}\right) \ \mbox{ and } \\
P^s_-&=\left(w^s_-,(\eta_S)^s_-,(\eta_N)^s_-,(1+\tx{\frac{a}{b_-}})(\eta_N)^s_--\tx{\frac{a}{b_-}}\right),
\end{align*}
with $P^s_-$ a stable node for all $\eps>0$ (and $P^u_-$ a saddle having 3-dimensional stable manifold). Importantly,
\begin{equation}\notag
h(P^s_-)= a \left(1-(\eta_N)^s_-\right)\left(\tx{\frac{b}{b_-}}-1\right)>0
\end{equation}
since $b_-<b$. Hence $P^s_-$ is also a virtual equilibrium point for \eqref{inclusion} as $P^s_-\in S_+$ (see equation \eqref{Splus}).
Let $W(P^s_+)$ denote the stable set of $P^s_+$ under the retreating flow $\phi_+$, and let $W(P^s_-)$ denote the stable set of $P^s_-$ under the advancing flow $\phi_-$. Recall we are choosing $\Tcnm$ close enough to $\Tcnp$ to ensure conditions \eqref{3Dstablesets}, that $Q_{\pm}^s$ were in each other's stable sets under the three-dimensional flows $\psi_\pm$. Given the decoupling of equations \eqref{NflipflopA}--\eqref{NflipflopC} from \eqref{NflipflopD}, along with the linear nature of equation \eqref{NflipflopD}, note \eqref{3Dstablesets} implies
\begin{equation}\label{4Dstablesets}
P^s_-\in W(P^s_+) \ \mbox {and } \ P^s_+\in W(P^s_-).
\end{equation}
This observation, which holds for all $\eps>0$, will play a key role in elucidating the flip-flop behavior of our model.
\subsection{Trajectories intersecting the switching manifold}
We begin by determining where on the 3-dimensional switching manifold $\Sigma$ the vector fields ${\bf X}_\pm$ are tangent, as such submanifolds may bound sliding regions \cite{leine}. To that end, $\Sigma$ is a hyperplane with normal vector
${\bf N}= [0 \ \ 0 \ \ 1+\tx{\frac{a}{b}} \ \ \tx{-}1 ]^T$. \ For ${\bf v}\in\Sigma$, a computation yields ${\bf X}_+ \perp {\bf N}$ if and only if
\begin{equation}\label{hplus}
w=H_+(\eta_N)+\frac{a\eps (1-\eta_N)(b_+-b)}{\rho (a+b)}=h_+(\eta_N).
\end{equation}
Thus, ${\bf X}_+$ is tangent to $\Sigma$ at points contained in the set
\begin{equation}\label{omega+}
\Omega_+=\{(h_+(\eta_N), \eta_S,\eta_N, \gamma(\eta_N)) : \eta_s,\eta_N\in [-1,1]\},
\end{equation}
a 2-dimensional submanifold of $\Sigma$ (recall $\gamma(\eta_N)$ is as defined in \eqref{switch}). In a similar fashion, one finds ${\bf X}_- \perp {\bf N}$ at ${\bf v}\in\Sigma$ if and only if
\begin{equation}\label{omega-}
{\bf v}\in \Omega_-=\{(h_-(\eta_N), \eta_S,\eta_N, \gamma(\eta_N)) : \eta_S,\eta_N\in [-1,1]\},
\end{equation}
where
\begin{equation}\label{hminus}
h_-(\eta_N)= H_-(\eta_N)+\frac{a\eps (1-\eta_N)(b_--b)}{\rho (a+b)}.
\end{equation}
We consider the case in which the surfaces of tangency $\Omega_\pm$ on the switching manifold $\Sigma$ do not intersect in $\mathcal{B}$. A tedious and straightforward calculation reveals that if the time constant $\eps$ in \eqref{NflipflopD} satisfies
\begin{equation}\label{epsbound}
\eps<\frac{(\Tcnm-\Tcnp)\rho (a+b)}{2a(b_+-b_-)},
\end{equation}
then $h_+(\eta_N)<h_-(\eta_N)$ for $\eta_N\in [-1,1].$ We assume $\eps$ satisfies \eqref{epsbound} in all that follows.
Having identified the sets of tangencies on either side of the switching manifold, we must now determine where the vector fields $\mathbf{X}_\pm$ point into the switching manifold $\Sigma$ and where they point away. Via further computations, we see for ${\bf v}=(w,\eta_S,\eta_N,\gamma(\eta_N))\in\Sigma$,
\begin{itemize}
\item[(i)] ${\bf X}_+({\bf v}) \dotp {\bf N}>0$ if $w>h_+(\eta_N)$, so that ${\bf X}_+({\bf v})$ points into $S_+$ if $w>h_+(\eta_N)$,
\item[(ii)] ${\bf X}_+({\bf v}) \dotp {\bf N}<0$ if $w<h_+(\eta_N)$, so that ${\bf X}_+({\bf v})$ points into $S_-$ if $w<h_+(\eta_N)$,
\item[(iii)] ${\bf X}_-({\bf v}) \dotp {\bf N}>0$ if $w>h_-(\eta_N)$, so that ${\bf X}_-({\bf v})$ points into $S_+$ if $w>h_-(\eta_N)$, and
\item[(iv)] ${\bf X}_-({\bf v}) \dotp {\bf N}<0$ if $w<h_-(\eta_N)$, so that ${\bf X}_-({\bf v})$ points into $S_-$ if $w<h_-(\eta_N)$.
\end{itemize}
In particular, a $\phi_+$-trajectory that intersects $\Sigma$ at a point for which $w<h_+(\eta_N)<h_-(\eta_N)$ passes transversally into $S_-$ following the Filippov convention, and continues in $S_-$ under the flow $\phi_-$. The subset $\Sigma_+\subset \Sigma$ defined by
\begin{equation}\label{sigmaplus}
\Sigma_+=\{(w,\eta_S,\eta_N,\gamma(\eta_N)) : w<h_+(\eta_N), \ \eta_S,\eta_N\in [-1,1]\}
\end{equation}
is therefore known as a {\em crossing region} for the Filippov flow \cite{leine}. Similarly, a $\phi_-$-trajectory that intersects $\Sigma$ at a point in the set
\begin{equation}\label{sigmaminus}
\Sigma_-=\{(w,\eta_S,\eta_N,\gamma(\eta_N)) : w>h_-(\eta_N), \ \eta_S,\eta_N\in [-1,1]\}
\end{equation}
passes transversally into $S_+$ and continues by following the flow $\phi_+$. In this fashion $\Sigma_-\subset\Sigma$ is also a crossing region for the Filippov flow. We note solutions to system \eqref{inclusion} that pass through $\Sigma_\pm$ are unique, though not differentiable at points of intersection with $\Sigma_\pm$.
Finally, consider the subset of the switching manifold defined by
\begin{equation}\label{slide}
\Sigma^{\mbox{\scriptsize SL}}=\{(w,\eta_S,\eta_N,\gamma(\eta_N)) : h_+(\eta_N)<w<h_-(\eta_N), \ \eta_S,\eta_N\in [-1,1]\}.
\end{equation}
Note ${\bf X}_+$ points into $S_+$ and ${\bf X}_-$ points into $S_-$ at all points in $\Sigma^{\mbox{\scriptsize SL}}$. The subset $\Sigma^{\mbox{\scriptsize SL}}$ of the switching manifold $\Sigma$ is therefore a {\em repelling sliding region} \cite{leine}; Filippov's approach does not provide for unique solutions ${\bf v}(t)$ in forward time if ${\bf v}(0)\in
\Sigma^{\mbox{\scriptsize SL}}$ \cite{fil}. Notice that for $\eps$ chosen to satisfy \eqref{epsbound}, the repelling sliding region $\Sigma^{\mbox{\scriptsize SL}}$ sits between the tangency sets $\Omega_+$ and $\Omega_-$, thereby separating $\Sigma_+$ and $\Sigma_-$, throughout $\Sigma$. As neither advancing nor retreating trajectories approach $\Sigma^{\mbox{\scriptsize SL}}$ in forward time, the repelling sliding region will play no role in the analysis to come. Projections of $\Sigma$ and its subsets described above into $(w,\eta_N,\xi_N)$-space are plotted in Figure \ref {FIG-Sigma-proj}.
We see that a trajectory for system \eqref{inclusion} with initial condition ${\bf v}(0)\in S_+ \cap W(P^s_+)$ will ``retreat" under the flow $\phi_+$, intersecting $\Sigma_+$ prior to approaching the virtual equilibrium $P^s_+$ and thereby switching to the ``advancing" flow $\phi_-$. With the parameters $b_\pm$ chosen appropriately (as discussed in the following section), this $\phi_-$-trajectory will intersect $\Sigma_-$ on its way to approaching the virtual equilibrium $P^s_-$, thereby flipping back to the retreating flow $\phi_+$.
We now prove the dynamic described above is capable of producing a unique (nonsmooth) attracting periodic orbit that, in terms of the model, represents the glacial-interglacial cycles, entirely a consequence of the flip-flop in the Northern Hemisphere.
\begin{figure}
\begin{center}
\includegraphics[width=5.6in,trim = 1.6in 6.6in 1in 1in, clip]{switchingB}\\
\caption{{\small Projections into $(w,\eta_N,\xi_N)$-space of the switching manifold $\Sigma$ (gold), surfaces of tangency $\Omega_+$ (red) and $\Omega_-$ (blue), and the repelling sliding region $\Sigma^{\mbox{\scriptsize SL}}$ (green). $\Sigma_-$ projects to the region above the blue curve, while $\Sigma_+$ projects to the region below the red curve.
}}
\label{FIG-Sigma-proj}
\end{center}
\end{figure}
\subsection{A return map for the Filippov flow}
In constructing the return map, it is instructive to first consider the case in which $\eps=0$. Note when $\eps=0$ the retreating flow $\phi_+$ has an attracting line $\ell_+$ of equilibrium points. That is, if ${\bf v}(0)\in S_+\cap W(P^s_+), \ \phi_+({\bf v}(0),t)\rw (w^s_+, (\eta_S)^s_+, (\eta_N)^s_+,\xi_N(0))\in\ell_+$ as $ t\rw\infty$. We remark that $\ell_+$ intersects $\Sigma$ at the point
\begin{equation}\label{Rplus}
R_+=(w^s_+, (\eta_S)^s_+, (\eta_N)^s_+, \gamma((\eta_N)^s_+)).
\end{equation}
Similarly, the advancing flow $\phi_-$ has an attracting line $\ell_-$ of equilibria when $\eps=0$; if
${\bf v}(0)\in S_-\cap W(P^s_-), \ \phi_-({\bf v}(0),t)\rw (w^s_-, (\eta_S)^s_-, (\eta_N)^s_-,\xi_N(0))\in\ell_-$ as $ t\rw\infty$. The line $\ell_-$ intersects $\Sigma$ at the point
\begin{equation}\label{Rminus}
R_-=(w^s_-, (\eta_S)^s_-, (\eta_N)^s_-, \gamma((\eta_N)^s_-)).
\end{equation}
These points of intersection will help us determine where trajectories are crossing the switching manifold. If we have $R_-\in W(P^s_+)$ and if $R_+\in W(P^s_-)$, the existence of a periodic orbit of Filippov system \eqref{inclusion} would seem plausible.
Now suppose $\eps>0$ is much smaller then the time constant $\rho$ in \eqref{Nflipflop}. The $\phi_+$-trajectory of a point ${\bf v}(0)\in S_+\cap W(P^s_+)$ will first approach the line $\ell_+$ with $\xi_N(t)$ varying little from $\xi_N(0)$, and then follow $\ell_+$ toward the switching manifold, intersecting $\Sigma$ at a point near $R_+$. Absent the presence of the switching manifold, this dynamic is reminiscent of problems addressed by geometric singular perturbation theory for smooth dynamical systems having multiple time scales \cite{jones}.
Note that as $b_+$ decreases to $b,$ the $\phi_+$-stable node $P^s_+$ approaches $R_+$ because only the fourth coordinate of $P_{+}^s$ varies with $b_+$. Indeed, the fourth coordinate of $P_{+}^s$ is $(\xi_N)_{+}^s=(1+\frac{a}{b_+})(\eta_N)_{+}^s - \frac{a}{b_+}$ which limits to $\gamma((\eta_N)_{+}^s)$ as $b_+\searrow b$. Hence we will assume $b_+$ is chosen to ensure that the point $R_+$ is in the stable set of $P_{+}^s$ under the retreating flow $\phi_+$, $ W(P^s_+)$. Recall that $P^s_+$ is also in the $W(P^s_-)$, the stable set of $P_{-}^s$ under the advancing flow $\phi_-$ \eqref{4Dstablesets}, which implies the existence of a neighborhood $U$ of $P^s_+$ with $U\subset W(P^s_-)$. We then additionally assume $b_+$ is close enough to $b$ to ensure $R_+\in U$, so that $R_+\in W(P^s_-)\cap W(P^s_+)$, an inclusion that holds for all $\eps>0.$
In a similar vein, the $\phi_-$-trajectory of a point ${\bf v}(0)\in S_-\cap W(P^s_-)$ will first approach the line $\ell_-$ with $\xi_N(t)$ remaining roughly constant, and then follow $\ell_-$ toward the switching manifold, intersecting $\Sigma$ at a point near $R_-$.
As $b_-\nearrow b,$ the $\phi_-$-stable node $P^s_-$ approaches $R_-$ (and so we assume $R_-\in W(P^s_-)$). As $P^s_-\in W(P^s_+)$ \eqref{4Dstablesets}, there then exists a neighborhood $U$ of $P^s_-$ with $U\subset W(P^s_+)$. Choosing $b_-$ sufficiently close to $b$ then ensures that $R_-\in W(P^s_+)\cap W(P^s_-)$, which we again note holds for all $\eps>0$.
The above choices of parameters $b_+$ and $b_-$ (and of $\Tcnm$ previously) now allow for the construction of a (nonsmooth) return map for the Filippov flow \eqref{inclusion} as follows.
We begin by noting that $R_+$ is in the crossing region $\Sigma_+$ (where trajectories cross from $S_+$ to $S_-$) because $w^s_+=H_+((\eta_N)^s_+)<h_+((\eta_N)^s_+)$ by \eqref{hplus}. As we have just seen that $R_+\in W(P^s_-),$ we can pick $\delta_1>0$ such that
\begin{equation}\label{Vplus}
V_+=\ov{B_{\delta_1}}(R_+)\cap\Sigma\subset W(P^s_-)\cap \Sigma_+.
\end{equation}
Recalling $P^s_-$ is a virtual equilibrium point for the advancing flow $\phi_-$, for any ${\bf v}\in V_+$ and for any $\eps>0$ there exists a time $t=t({\bf v},\eps)$ such that $\phi_-({\bf v}, t({\bf v},\eps))$ reaches the crossing region $\Sigma_-$ (where trajectories cross from $S_-$ to $S_+$). We note $t({\bf v},\eps)\rw\infty$ as $\eps\rw 0$ for future reference.
We may then define a continuous mapping, for any $\eps>0$, given by
\begin{equation}\label{returnminus}
r^\eps_- :V_+\rw \Sigma_-, \ r^\eps_-({\bf v})=\phi_-({\bf v}, t({\bf v},\eps)).
\end{equation}
That $R_-\in \Sigma_-$ follows from the fact that $w^s_-=H_-((\eta_N)^s_-)>h_-((\eta_N)^s_-)$ by \eqref{hminus}. Recalling $R_-\in W(P^s_+),$ we can pick $\delta_2>0$ such that
\begin{equation}\label{Vminus}
V_-=\ov{B_{\delta_2}}(R_-)\cap\Sigma\subset W(P^s_+)\cap \Sigma_-.
\end{equation}
Noting $P^s_+$ is a virtual equilibrium point for the retreating flow $\phi_+$, for any ${\bf v}\in V_-$ and for any $\eps>0$ there exists $t=t({\bf v},\eps)$ such that $\phi_+({\bf v}, t({\bf v},\eps))\in \Sigma_+$. Hence for any $\eps>0$, we define the continuous mapping
\begin{equation}\label{returnplus}
r^\eps_+ :V_-\rw \Sigma_+, \ r^\eps_+({\bf v})=\phi_+({\bf v}, t({\bf v},\eps)).
\end{equation}
We are now in a position to prove there exists $\eps>0$ such that $r^\eps=r^\eps_+\circ r^\eps_- :V_+\rw V_+$ is a contraction map.
\subsection{Existence of an attracting limit cycle}
\begin{prop} (a) Given $c\in (0,1)$, there exists $\hat{\eps}$ such that for all $\eps\leq\hat{\eps} $ and for all ${\bf v}_1, {\bf v}_2\in V_+,$
\begin{equation}\notag
\|r^\eps_-({\bf v}_2)-r^\eps_-({\bf v}_1) \|\leq c\|{\bf v}_2-{\bf v}_1\|.
\end{equation}
(b) Given $c\in (0,1)$, there exists $\hat{\eps}$ such that for all $\eps\leq\hat{\eps} $ and for all ${\bf v}_1, {\bf v}_2\in V_-,$
\begin{equation}\notag
\|r^\eps_+({\bf v}_2)-r^\eps_+({\bf v}_1) \|\leq c\|{\bf v}_2-{\bf v}_1\|.
\end{equation}
\label{prop1}
\end{prop}
\begin{proof}
We prove case (a). In this proof, for ease of notation, we set $x=\eta_S, y=\eta_N$ and $z=\xi_N$. Relying on the fact equation \eqref{NflipflopD} decouples from equations \eqref{NflipflopA}--\eqref{NflipflopC}, the proof is in spirit analogous to the proof of Proposition 5.4 in \cite{wwhm}; we include it here for completeness.
Let $c\in (0,1)$, and let ${\bf v}=(w_0,x_0,y_0,\gamma(y_0)) \in V_+$. Recall that by design, under the advancing flow corresponding to equations \eqref{NflipflopA}--\eqref{NflipflopC} we have $\psi_-((w_0,x_0,y_0),t)\rw Q^s_-$ as $t\rw\infty$. Since $V_+$ is the intersection of a closed ball in $\R^4$ with the hyperplane $\Sigma$, $V_+$ is compact (as well as connected and convex). Thus the set $J=\{ (w,x,y) : (w,x,y,\gamma(y))\in V_+\}$ is a compact set which, coupled with the fact $J\subset W(Q^s_-)$, yields the existence of $T_1$ such that for all $t\geq T_1$ and for all ${\bf u}_1, {\bf u}_2\in J, \ \|\psi_-({\bf u}_2)-\psi_-({\bf u}_1)\|\leq c\|{\bf u}_2-{\bf u}_1\|$.
Given ${\bf v}\in V_+$, pick $\eps({\bf v})$ such that $t({\bf v}, \eps({\bf v}))>T_1$, where $t({\bf v}, \eps({\bf v}))$ is as in the definition of $r^\eps_- $ \eqref{returnminus}. By the continuity of $\phi_-$ with respect to initial conditions and time, there exists $\delta({\bf v})>0$ so that ${\bf w}\in B_{\delta({\bf v})}({\bf v})\cap V_+$ implies $t({\bf w}, \eps({\bf v}))>T_1$ (where $r^{\eps({\bf v})}_-({\bf w})\in \Sigma_-$). We note for any $\eps\leq \eps({\bf v}), \ t({\bf w}, \eps)>T_1$.
In this fashion we arrive at an open covering
\begin{equation}\notag
V_+\subset \bigcup_{{\bf v}\in V_+} B_{\delta({\bf v})}({\bf v})
\end{equation}
of the compact set $V_+$. Choose a finite subcover $\{ B_{\delta({\bf v}_n)}({\bf v}_n) : n=1, ... , N\}$, and let $\hat{\eps}=\min\{ \eps({\bf v}_n) : n=1, ... , N\}$. Then for any $\eps\leq\hat{\eps}$ and for all ${\bf v}\in V_+, \ t({\bf v},\eps)>T_1$.
Suppose $\eps\leq\hat{\eps}$, and let ${\bf v}_1=(w_1,x_1,y_1, \gamma(y_1))$ and $ {\bf v}_2=(w_2,x_2,y_2, \gamma(y_2))$ be elements in $V_+$. Set ${\bf u}_1=(w_1,x_1,y_1)$ and $
{\bf u}_2=(w_2,x_2,y_2).$
Let $r^\eps_-({\bf v}_1)=(w^\pr_1, x^\pr_1, y^\pr_1, \gamma(y^\pr_1)), \ r^\eps_-({\bf v}_2)=(w^\pr_2, x^\pr_2, y^\pr_2, \gamma(y^\pr_2)), \
{\bf u}^\pr_1=(w^\pr_1, x^\pr_1, y^\pr_1) $ and ${\bf u}^\pr_2=(w^\pr_2, x^\pr_2, y^\pr_2)$. Note that by our choice of $\eps, \ \|{\bf u}^\pr_2-{\bf u}^\pr_1\|\leq c \|{\bf u}_2-{\bf u}_1\|.$
\ We then have
\begin{align}\notag
\|r^\eps_-({\bf v_2})-r^\eps_-({\bf v_1})\|^2 & = \|{\bf u}^\pr_2-{\bf u}^\pr_1\|^2+(\gamma(y^\pr_2)-\gamma(y^\pr_1))^2\\\notag
& = \|{\bf u}^\pr_2-{\bf u}^\pr_1\|^2+ (1+\textstyle{\frac{a}{b}})^2 (y^\pr_2- y^\pr_1)^2\\\notag
& \leq c^2\|{\bf u}_2-{\bf u}_1\|^2+(1+\textstyle{\frac{a}{b}})^2 c^2 (y_2-y_1)^2\\\notag
&= c^2\|{\bf u}_2-{\bf u}_1\|^2+c^2(\gamma(y_2)-\gamma(y_1))^2\\\notag
&=c^2\|{\bf v}_2-{\bf v}_1\|^2.
\end{align}
A similar argument can be given to prove that for $c\in (0,1), $ there exists $\hat{\eps}>0$ so that for all $\eps\leq \hat{\eps}, $ \ $r^\eps_+:V_-\rw \Sigma_+$ contracts distances by a factor of at most $c$.
\end{proof}
\begin{prop} (a) There exists $\hat{\eps}>0$ such that for all $\eps\leq\hat{\eps} $, \ $r^\eps_- : V_+\rw V_-$.
\\ (b) There exists $\hat{\eps}>0$ such that for all $\eps\leq\hat{\eps} $, \ $r^\eps_+ : V_-\rw V_+$.
\label{prop2}
\end{prop}
\begin{proof} We prove case (a). Let ${\bf v}=(w,\eta_S,\eta_N,\gamma(\eta_N))_{t=0}\in V_+$. We again note that $$\psi_-((w(0),\eta_S(0),\eta_N(0)),t)=(w(t),\eta_S(t),\eta_N(t))\rw Q^s_-\text{ as }t\rw\infty.$$ Additionally using the fact $\gamma (\eta_N)$ is continuous, pick $T=T({\bf v})$ such that for all $t\geq T,$
\begin{equation}\notag
\|(w(t),\eta_S(t),\eta_N(t),\gamma(\eta_N(t)))-R_-\|<\tx{\frac{1}{2}}\delta_2,
\end{equation}
where $\delta_2$ is as in \eqref{Vminus}.
Choose $\eps({\bf v})>0$ such that $t({\bf v}, \eps({\bf v}))>T$ (where $t({\bf v}, \eps({\bf v}))$ is as in \eqref{returnminus}). Then for all $\eps\leq \eps({\bf v}), \ \|r^\eps_-({\bf v})-R_-\|< \tx{\frac{1}{2}}\delta_2$.
Let $0<c<\text{min}\{1,\tx{\frac{1}{2}}\delta_2\}$, and pick $\hat{\eps}$ as in Proposition \ref{prop1}. For $\eps\leq \min\{\hat{\eps},\eps({\bf v})\}$ and for any ${\bf w}\in V_+$,
\begin{equation}\notag
\|r^\eps_-({\bf w})-R_-\|\leq \|r^\eps_-({\bf w}) - r^\eps_-({\bf v})\|+\|r^\eps_-({\bf v})-R_-\|<\tx{\frac{1}{2}}\delta_2+\tx{\frac{1}{2}}\delta_2=\delta_2.
\end{equation}
Thus $r^\eps_-({\bf w})\in V_-$, and we conclude $r^\eps_-(V_+)\subset V_-$.
\end{proof}
\vspace{0.01in}
\begin{thm} With other parameters as in Table 1, choose $\Tcnm$- and $b_\pm$-values, respectively, so that for any $\eps>0$ \\
\hspace*{0.2in}(i) $P^s_-\in W(P^s_+)$, \ $P^s_+\in W(P^s_-)$, and\\
\hspace*{0.2in}(ii) $R_+, R_-\in W(P^s_-)\cap W(P^s_+)$,\\
as discussed above. Then system \eqref{inclusion} admits a unique attracting limit cycle for sufficiently small $\eps>0$.
\label{cycle}
\end{thm}
\begin{proof} Given $c\in (0,1)$, choose $\hat{\eps}_1$ and $\hat{\eps}_2$ as in cases (a) and (b), respectively, in Proposition \ref{prop1}. Also choose $\hat{\eps}_3$ and $\hat{\eps}_4$ as in cases (a) and (b), respectively, in Proposition \ref{prop2}. Let $\eps\leq \min\{\hat{\eps}_n : n=1, ... , 4\}$. Then since $r^\eps_- :V_+\rw V_-$ and $r^\eps_+:V_-\rw V_+$, we can define the return map
\begin{equation}\notag
r^\eps=r^\eps_+\circ r^\eps_- :V_+\rw V_+.
\end{equation}
As $r^\eps$ is additionally a contraction map with contraction factor $c^2$, $r^\eps$ has a unique fixed point ${\bf v}^*\in V_+ $ to which all $r^\eps$-orbits converge. The Filippov trajectory that flows via $\phi_-$ from ${\bf v}^*$ to ${\bf w}=r^\eps_-({\bf v}^*)$, and from ${\bf w}$ back to ${\bf v}^*$ via $\phi_+$, is then an attracting (nonsmooth) limit cycle.
\end{proof}
\vspace{0.01in}\noindent
{\bf Remark.} Every trajectory of system \eqref{inclusion} that passes through $V_+$ (or through $V_-$) converges to the limit cycle $\Gamma$ provided by Theorem \ref{cycle}. We note that given any compact set $K\subset W(P^s_+) \cap S_+$, one can choose $\eps$ sufficiently small to ensure $K$ is contained in the stable set of $\Gamma$ under the Filippov flow.
\vspace{0.1in}
\section{Selected Numerical Results}
In Figure \ref{FIG-neg5-neg10} we plot the projection of the limit cycle $\Gamma$ into the three-dimensional $(\eta_S,\eta_N,\xi_N)$-space, along with the behavior of $-\eta_S, \eta_N$ and $\xi_N$ over time along $\Gamma$, in the case $\Tcnm=-5^\circ$C. We first note the sawtooth pattern evident in the evolution of each of the variables, with a rapid retreat into an interglacial period following a slower descent into a glacial age, as seen in the climate data over the past 1 million years \cite{raylisnic}.
\begin{figure}
\begin{center}
\begin{tikzpicture}
\node (img1) {\includegraphics[width=2.8in]{Fig4_TcNAdv-5-eps003-eps-converted-to.pdf} };
\node[below=of img1, node distance=0cm, yshift=1cm] {$t$};
\node[left=of img1, node distance=0cm, rotate=90, anchor=center,yshift=-0.7cm] {sine of latitude};
\node[right=of img1, yshift=0.1cm] (img2) {\includegraphics[width=1.8in]{Fig4_TcNAdv-5-LC-eps003-eps-converted-to.pdf}};
\node[below=of img1, yshift=0.1cm] (img3) {\includegraphics[width=2.8in]{Fig4_TcNAdv-5-eps03-eps-converted-to.pdf}};
\node[left=of img3, node distance=0cm, rotate=90, anchor=center,yshift=-0.7cm] {sine of latitude};
\node[below=of img3, node distance=0cm, yshift=1cm] {$t$};
\node[right=of img3, yshift=0.1cm] (img4) {\includegraphics[width=1.8in]{Fig4_TcNAdv-5-LC-eps03-eps-converted-to.pdf}};
\node[below=of img2, yshift=1.6cm, xshift=-.9cm] {$\eta_N$};
\node[below=of img2, yshift=3.7cm, xshift=-2.4cm] {$\eta_S$};
\node[below=of img2, yshift=5.9cm, xshift=-1.5cm] {$\xi_N$};
\node[below=of img4, yshift=1.6cm, xshift=-.9cm] {$\eta_N$};
\node[below=of img4, yshift=3.7cm, xshift=-2.4cm] {$\eta_S$};
\node[below=of img4, yshift=5.9cm, xshift=-1.5cm] {$\xi_N$};
\end{tikzpicture}
\caption{{\small \textbf{Top:} {\em Left}: The behavior of $\eta_N$ (dashed black curve), $\xi_N$ (dotted brown curve) and $-\eta_S$ (solid blue curve) along the limit cycle $\Gamma$ when $\Tcnm=-5^\circ$C, $\Tcnp=\Tcs=-10^\circ$C, $\eps=0.03$ and $\rho=0.3$. \ { \em Right}: The projection of $\Gamma$ into $(\eta_S,\eta_N,\xi_N)$-space. \textbf{Bottom:} Same as top row except $\rho=\epsilon=0.3$.
}}
\label{FIG-neg5-neg10}
\end{center}
\end{figure}
Of particular interest is the oscillation of $\eta_S(t)$ in the Southern Hemisphere, which is completely driven by the ``flip-flop" in the Northern Hemisphere. While the $\dot{\eta_S}$-equation has no explicit dependence on $\eta_N$, the dynamic coupling of the hemispheres provided by the $\dot{w}$-equation governs the Southern Hemisphere response to the growth and retreat of the Northern Hemisphere ice sheets. As noted above, this model behavior aligns with theory of M. Milankovitch, which posits that changes in Northern Hemisphere high latitude insolation---due to variations in Earth's orbital elements over long time scales---comprise the principle forcing mechanism of the glacial-interglacial cycles \cite{hays,milank, raylisnic,uemura}.
We further note the Southern Hemisphere albedo line oscillations (solid blue curve), while smaller in amplitude, are nonetheless in sync with Northern Hemisphere oscillations (dashed black curve). This behavior is evident in the climate data on orbital time scales \cite{broecker,brook,lowell,raylisnic}.
The model produces different amplitude oscillations when choosing different $\Tcnm$-values (see Figure \ref{FIG-Flip-SH}). When the critical temperature during glacial advance is larger, more ice can form and the albedo line advances closer to the equator. Note the effect such a change in the Northern Hemisphere albedo line has on both the mass balance in the Northern Hemisphere and the Southern Hemisphere albedo line. When the critical temperature is more negative, the amplitude of both hemisphere albedo lines and the mass balance is decreased.
For the top row in Figure \ref{FIG-neg5-neg10} the $\eps$-value is an order of magnitude smaller than $\rho$. We numerically find the limit cycle $\Gamma$ exists for larger $\eps$ as well, as illustrated in the bottom row, in which $\eps=\rho$.
\begin{figure}
\begin{center}
\begin{tikzpicture}
\node (img1) {\includegraphics[width=2.8in]{Fig5-TcNAdv-5OtherTc-10-eps-converted-to.pdf}};
\node[below=of img1, node distance=0cm, yshift=1cm] {$t$};
\node[left=of img1, node distance=0cm, rotate=90, anchor=center,yshift=-0.7cm] {sine of latitude};
\node[right=of img1,yshift=0cm] (img2) {\includegraphics[width=2.8in]{Fig5-TcNAdv-8OtherTc-10-eps-converted-to.pdf}};
\node[below=of img2, node distance=0cm, yshift=1cm] {$t$};
\node[left=of img2, node distance=0cm, rotate=90, anchor=center,yshift=-0.7cm] {sine of latitude};
\end{tikzpicture}
\caption{{\small Periodic behavior of the albedo and ice lines for different critical temperature while the Northern Hemisphere glacier advance. For all figures $\Tcnp=T_{cS}=-10$ and $\eps=0.03$. Curve coloring and patterns same as in Figure \ref{FIG-neg5-neg10}. \textbf{Left:} $T^-_{cN}=-5^\circ$C and \textbf{Right:} $T^-_{cN}=-8^\circ$C
}}
\label{FIG-Flip-SH}
\end{center}
\end{figure}
\section{Discussion}
Many planetary energy balance climate models assume a symmetry about the equator; as in Budyko's seminal model for the Earth \cite{budyko}, one focuses solely on the climate in the Northern Hemisphere. In this work we couple an approximation of Budyko's latitudinally-averaged surface temperature equation with both Northern Hemisphere and Southern Hemisphere dynamic albedo lines $\eta_N$ and $\eta_S$. Each albedo line is associated with a critical temperature ($T_{cN}, T_{cS}$) that delineates between the local formation and melting of ice.
A planet's zonally averaged and mean annual distribution of insolation $s(y,\beta)$ depends on the obliquity $\beta$ as well as the latitude. Earth's obliquity is such that there can only exist stable, symmetric albedo line positions ($\eta_N=-\eta_S$) if one assumes $T_{cN}=T_{cS}$. (However, we note that in an energy balance model of Pluto, there exist stable {\em asymmetric} albedo line positions in the case $T_{cN}=T_{cS}$, due to Pluto's obliquity of $119.6^\circ$ \cite{Nadeau2019}.) A full analysis of the temperature-albedo lines $(w,\eta_S,\eta_n)$-system for Earth in the case $T_{cN}=T_{cS}$, including snowball Earth ($\eta_S=\eta_N$) and ice-free Earth ($\eta_S=-1, \eta_N=1$) scenarios, will appear in a forthcoming paper.
Taking different critical temperature values leads to stable, asymmetric $\eta_S$- and $\eta_N$-positions in our model. Of particular interest is the fact that a change in $T_{cN}$ alone leads to changes in each of the stable $\eta_S$- and $\eta_N$-albedo line placements, due to the coupling of the Northern Hemisphere and Southern Hemisphere provided by the temperature equation ($\dot{w}$).
The paleoclimate data indicates that on orbital time scales (100 kyr), oscillations in Northern Hemisphere and Southern Hemisphere ice caps are in sync, with evidence suggesting the Southern Hemisphere oscillations are a consequence of changes in the Northern Hemisphere ice sheets \cite{broecker,brook,lowell,raylisnic}. We were thus lead to incorporate the advance-retreat ``flip-flop" from \cite{wwhm} into the Northern Hemisphere in our model. Using Filippov's theory for discontinuous vector fields, we proved the existence of a unique attracting limit cycle, corresponding to glacial oscillations in which variations in $\eta_S$ and $\eta_N$ are indeed in sync. The cycling of the Northern Hemisphere ice sheet is sufficient to generate changes in the Southern Hemisphere ice extent, again due to the hemispheric coupling inherent in the temperature equation.
The interaction between the hemispheres in our model naturally lends itself to the investigation of several related questions, both of mathematical and paleoclimatic interest. The obliquity, which varies with a period of roughly 41 kyr, can be incorporated into the insolation distribution function $s(y,\beta)$, leading to a nonautonomous and forced discontinuous system. Similarly, the solar ``constant" $Q$ varies with the eccentricity of Earth's orbit \cite{dickclarence}, and it too can be used to force our Filippov system. As each of the obliquity and eccentricity signals are present in the paleoclimate data \cite{hays}, it would be of interest to analyze the effect of external forcing on the $(w,\eta_S,\eta_N,\xi_N)$-system. Any such study would begin with preliminary investigations into the effect external forcing has on the $(w,\eta_S,\eta_N)$-system \eqref{system-NS}.
Antarctica is believed to have been continuously ice-covered over the past 1 million years \cite{raylisnic, tzip2003}. In \cite{raylisnic}, Raymo et al posit that Antarctica's ice sheet was more dynamic 3 million years ago (mya), with a terrestrial-based ice margin. It is further suggested in \cite{raylisnic} that the transition from a dynamic to a permanent Antarctic ice sheet played an important role in the Mid-Pleistocene Transition, a time roughly 1 mya in which the period of the glacial cycles changed from 41 kyr to 100 kyr. This conceptual scenario can be investigated with our model, perhaps with the addition of a $\xi_S$-variable for the period of time when Antarctic ice terminated on land, and which over time coalesced with the albedo line $\eta_S$ as a parameter varies. More generally, the use of nonsmooth bifurcation theory as a tool to investigate changes in our system as various parameters vary is easy to envision.
Finally, it is of interest to note there is an asymmetry in Northern Hemisphere and Southern Hemisphere glacial cycle oscillations on a millenial time scale---the so-called {\em bipolar seesaw} \cite{broecker, brook, pedro}. This asymmetry is thought to be caused by disruptions in the meridional transport of heat by the ocean up to the North Atlantic, due in turn to the discharge of fresh meltwater from Northern Hemisphere ice sheets into the North Atlantic ocean \cite{timmermann}. In terms of the model, this ocean heat transport is associated with the $-C(T-\overline{T})$-term, which concerns the global average surface temperature. In our view a diffusive meridional heat transport approach \cite{north,sellers,Walsh2020} would be more appropriate for investigations into millenial time scale, asynchronous oscillations in Northern Hemisphere and Southern Hemisphere ice extent, perhaps incorporating the diffusion coefficient as a function of latitude and thereby bringing into play localized heat transport.
\section*{Acknowledgements}
Research of AN was supported by an NSF Mathematical Sciences Postdoctoral Research Fellowship, Award Number DMS-190288.
\bibliographystyle{siam}
\bibliography{budyko}
\end{document}
| 113,290
|
TITLE: Conditional Probability (Where did I go wrong?)
QUESTION [2 upvotes]: Four identical balls are numbered $1, 2, 3$ and $4$ and put into a box. A ball is randomly drawn from the box, and not returned to the box. A second ball is then randomly drawn from the box.
For the first question:
What is the probability that the sum of the numbers on the two balls is $5$?
Here, there is no problem: we could take $\{1,4\}$ , $\{4,1\}$ , $\{2,3\}$ , $\{3,2\}$. Each of those four possibilities has a probability $\frac{1}{4}\cdot \frac{1}{3}=\frac{1}{12}$. So let $F$ be the sample space above, such that $$Pr\left( F \right)=4\cdot \frac{1}{12}=\frac{1}{3}$$
Now here's where I ran into problems:
Given that the sum of the numbers on the two balls is $5$, what is the probability that the second ball drawn is numbered $1$?
What I did was this: $$Pr\left( 1|F \right)=\frac{Pr\left( 1\cap F \right)}{Pr\left( F \right)}=\frac{Pr\left( 1\; ,\; 4 \right)+Pr\left( 4\; ,\; 1 \right)}{Pr\left( F \right)}=\frac{\frac{1}{12}+\frac{1}{12}}{\frac{1}{3}}=\frac{1}{2}$$
The answer is meant to be $\frac{1}{4}$. Where did I go wrong?
REPLY [4 votes]: In the second part, you don't consider Pr(1,4) because the question says that the second ball drawn is numbered 1 and hence only Pr(4,1) is considered. This gives 1/12 in the numerator and the answer you get is 1/4.
| 44,732
|
Westerville, Ohio Juvenile Law Lawyers
James S. Sweeney
Powell, OH Juvenile Law Attorney
(614) 546-5653Capital University Law SchoolGeorge Mason University and Ohio UniversityOhioSuper Lawyers, Columbus C.E.O. Magazine and AvvoOhio State Bar
David Shaver
Worthington, OH Juvenile Law Attorney
(614) 547-7111Capital University Law SchoolDenison UniversityOhioColumbus Bar Association, Ohio State Bar...
David Michael Johnson
Columbus, OH Juvenile Law Attorney
(614) 987-0192The University of Toledo College of LawOhio State University - ColumbusOhioState Bar of Ohio and Columbus Bar Association
Ralph Kerns Esq
Worthington, OH Juvenile Law Attorney
(614) 785-9420OhioAmerican Institute of Family LawOhio State Bar
Chad Kristian Hemminger Esq
Westerville, OH Juvenile Lawon Michael Cope Esq
Columbus, OH Juvenile Law Attorney
Ohio State University - ColumbusOhioOhio State Bar
David Hasselback Esq
Westerville, OH Juvenile Law Attorney
(614) 891-2410Capital University Law SchoolOhioOhio State Bar and Columbus Bar Association
Robert Berky Esq
Westerville, OH Juvenile Law Attorney
(614) 882-9413OhioOhio State Bar
Britani Galloway
Columbus, OH Juvenile Law Attorney
Michael Stephen Probst Esq
Columbus, OH Juvenile Law Attorney
(614) 232-8890Capital UniversityOhioNational Highway Traffic Safety AdministrationOhio State Bar and Columbus Bar Association Lawyers For Justice
Dean Edward Hines Esq
Columbus, OH Juvenile
Brian Joslyn
Columbus,
Leslie Albeit
Columbus, OH Juvenile Law Attorney
(614) 745-2001Capital University Law SchoolOhio Wesleyan UniversityOhioSuper Lawyers, Super Lawyers, AVVO, Martindale-Hubbell...Accused of College Misconduct? Here are 5 Tips from a Lawyer and College Sex and Climate Change
Carol M. Strench
Columbus, OH Juvenile Law Attorney
(614) 715-4710Arizona State UniversityArizona State University and Arizona State UniversityOhioOhio State Bar and Columbus Bar Association
Ryan Shafer
Columbus, OH Juvenile Law Attorney
Duquesne University School of LawUniversity of CincinnatiOhioColumbus Bar Association, Central Ohio Association For Justice...) 871-8085The University of Toledo College of LawMichigan State UniversityOhioThe National Trial Lawyers, Martindale-Hubbell and Martindale-HubbellBar Association of Erie County, Erie County Grievance Committee...
The LII Lawyer Directory contains lawyers who have claimed their profiles and are actively seeking clients. Find more Wester.
| 28,551
|
Posted in Uncategorized Young Anal Tryours Marusya Click Here To Watch Marusya Young Anal Tryours Marusya and the guy are in the middle of the kitchen.asics gel muse fit training shoes women's review There is also the man who announced in front of seven or eight fellow conference attendees that he found it strange I claimed to be very happy having lived in London for two years, since there was no ring on my finger. p 13631 Had to chuckle when I read (identity withheld to protect the guilty from embarrassment) a recent reference to John Moses Browning's Colt 1911 that read, It's an engineering masterpiece with a design that's endured almost 100 years.
You can exercise your right to prevent such processing by checking certain boxes on the forms we use to collect the data.asics gt 1000 2 mens shoes 80s RockAuto is not responsible for any costs exceeding the cost of the part, and we will not accept returns of parts that have been installed or modified. You better see our video of this hot lady fondling large mammaries, and proceeds to touching her hot and wet pussy.onitsuka tiger nostalgic impressions company
{ 0 comments } Our Favourite Bingo Halloween Promos 2014 by Paul Marion on October 25, 2014 Trick or Treat scratch that, only treats this Halloween for you at some of the leading bingo sites. View EMC Vehicle Testing Facilities MIRA EMC Engineering is an accredited facility to a ISO17025 standard via UKAS. ANU scientists are part of a global scientific collaboration which has proven the existence of gravitational waves, 100 years after Einstein'.asics netball shoes pink
Again, whether you fully believe Donaghy or not, he does provide a fascinating perspective into the world of refereeing that simply no one else can provide.
| 140,678
|
Closing of Ueschinen Hütte
The summer season ended a few weeks ago here at KISC and with that also Ueschinen Hütte, our very own mountain hut. To ensure that the hut will stay in good condition for next year a few Pinkies spent two days doing various tasks to prepare it for a cold winter. At the end of this two days all staff from Kisc where invited to have one last barbecue up at the hut, as a celebration of the official closing.
This was the first time visiting the hut for most of the Autumn Short Term Staff, and they, together with everyone else, had a great time. They also had the opportunity to spend the night in the hut, and hike the Three Valley Hike the next day, one of our most spectacular hikes!
As the Hut Warden of KISC, I’ve spent much of my time taking care of Ueschinen Hütte throughout the summer. It’s a special feeling to close the door and window shutters for the last time, knowing that the hut will stay empty for the next couple of months. I hope that everyone who stayed at the hut this year had a great experience, and that many more scouts will have the chance to stay here.
Oscar (SE)17.09.2015
Hut Warden
| 35,678
|
We are in the best of times, and yet also, the worst of times. In the age of rapid digital innovation, we are reliving the narrative in Charles Dickens’ Tale of Two Cities. Some businesses thrive at unimaginable speed while some get eliminated completely. Many are caught in the in-between, struggling to stay afloat to keep up with the change of digital disruption.
Across industries, the wave of digital disruption has brought new technologies, new entrants, new customer experiences and new business models. In order to beat the disruption, one has to be the disruptor.
We see this narrative unfolding in the logistics sector. Technological revolution has accelerated change in an industry that is traditionally backward and least digitally exposed. It has enlarged the divide between the new and the old.
The reality is that the supply chain is not naturally a digital business, as concluded in the report by Janeiro Digital, The Modernisation Gap: Digital Innovation and Transformation in Supply Chain and Logistics.
Many, prior to the pandemic, have regarded digital tools as unnecessary expenses. In the container trucking industry, it is a typical sight to see drivers writing down the jobs that they had completed each day on a piece of paper. It is just how things have always been.
This has contributed to the common inefficiencies observed in the sector today that are leading to the wider problems of inefficient container routings, bottlenecks at ports, affecting cargo quality and resulting in security risk.
In these troubled times, the magnitude of a supply chain disruption is keenly felt. This was especially highlighted during the Suez Canal blockage causing congestion at ports and container shortages.
Also Read: Locad founder on building SEA’s first cloud logistics network in the midst of COVID-19
The ramifications were global, where everyone including retailers and producers was affected. Sadly, consumers have already begun to feel the pinch, as costs get passed down to them.
The benchmark food price index published by the United Nations’ Food and Agriculture Organisation (FAO) registered a sharp increase in May, averaging 127.1 points – the highest level in 10 years.
This is the result of a confluence of factors, including higher marine shipping costs and supply chain disruptions. Freight rates are expected to reach new highs this year given port congestion and equipment unavailability.
With the changing preferences of consumers driving a surge in demand, there is great potential for the shipping industry. According to Research and Markets, the global logistics market is estimated to grow to US$12.68 Billion by 2023 with a Compound Annual Growth Rate (CAGR) of 3.49 per cent between 2017 and 2023, with Asia as the top player in the global maritime trade arena. One key highlight is the boom in the logistics market in SEA region, with trade volumes expected to increase by 130 per cent in 2023 to US$5,653 billion.
With the increasing international trade and investment, the rapid growth of e-commerce and the improvement in infrastructure, the Southeast Asia region is an untapped gold mine within the logistics ecosystem.
Southeast Asia’s internet economy hits US$100 billion for the first time in 2019, and it is expected to grow to US$300 billion by 2025.
The 2020 Southeast Asia e-Conomy report by Google, Temasek and Bain & Co revealed that COVID-19 had led to an acceleration of digital consumption, with SEA economy exceeding USD$100 billion in gross merchandise volume (GMV) and e-commerce accounting more than 50 per cent.
Haulio has long seen the beauty of the interconnectedness in the supply-chain business. As Singapore’s fastest-growing cloud-based digital container haulage network, Haulio built on a multi-tenancy system to allow multiple customers or ‘tenants’ to share the same resources while being able to configure the application to fit their needs.
This new model of ‘sharing’ using digital capabilities allows their business to optimise the vast logistics network.
Using technology to optimise the usage of haulage trucks and drivers, Haulio problem-solves inefficiencies through their platform while also partnering with major logistics players and fintechs.
Strategic partnerships allow them to connect the most offline node to the rest of the supply chain, uplifting the lives of millions of haulers and drivers.
Also read: Challenging existing fundamentals in logistics and supply chain
Haulio’s collaboration with ESCO in Thailand, which operates one of the six inland container depots (ICD) at Lat Krabang (LKB) port, is a prime example of the transformation that digitalisation of the trucking ecosystem has brought.
In Thailand, freight transport via road is an integral part of the logistics network. To improve operational efficiency at ESCO’s terminal, Haulio’s landmark digital tool has helped assist the terminals to execute movement of more than 10,000 TEUs since Q2 2020.
Through Haulio, ESCO’s trucking partners can be tracked based on factors such as the speed of response to jobs, number of partner’s drivers online, new revenue stream jobs, hence allowing ESCO to measure terminal operational efficiency gains. To date, ESCO has seen an efficiency improvement of around 20 per cent with the administrative savings from improved operational efficiency.
Haulio’s success in Singapore, as well as this successful pilot with ESCO, further proves the value of Haulio’s solutions in bridging the gap between customers and their trucking partners, by bringing operational visibility to all parties.
While container haulage has always been the vertical that is left behind within the supply chain, Haulio’s solutions will be able to fulfil the potential for transformation within the first-mile container logistics space.
Haulio has plans to expand its footprint regionally, to complete the digitalisation of haulage in Southeast Asia by 2025, solving existing problems within the US$147 billion ‘First Mile Logistics Market’ in Southeast Asia.
Tech-driven operating models are able to tap on the underserved and uncontested opportunities in the various value levers. Tech-enabled logistics start-ups are using technologies such as data analytics and artificial intelligence to improve the efficiency of business operations and to serve niche markets.
Haulio believes in delivering value through technology. It is about building a culture of empowerment, starting from the digital connecting node of the Container Haulage vertical.
As the phrase goes, “Every Supply Chain is only as strong its weakest link”. The journey towards digital transformation would only be possible through the joint efforts of industry players.
–: frank mckenna on Unsplash
The post The first mile container logistics is ripe for digital disruption. Here’s how Haulio is doing it appeared first on e27.
content first appear on e27
| 72,243
|
• The team at Bluefly has released this video of how to rock a pair of leggings.
• Check out Modelinia’s video of the iconic fashion photographer, Rankin, as he gives a tour of a Harper’s Bazaar photo shoot as well as divulging who his favorite supermodel is.
• MTV has put together this hilarious A-Z Guide To Olivia’s Best City Moments.
• Watching Kelly Ripa as Rachel Zoe should keep us going until the next Rachel Zoe Project airs. Pretty sure that “Roger” takes the cake, though!
• Death’s Head Becomes You or, for the less dramatic, A Brief History of Skulls in Fashion.
• 50skatekid is the story of an eleven-year-old boy on a mission to skateboard in all 50 states. Also known as having cool parents.
• The Frisky have responded to Blogue’s Die, Trend, Die list by releasing their own: 10 Popular Fads We Want Killed.
• It wouldn’t be a day of the year if someone didn’t release a list on fashion blogs. This week, the Independent of Ireland brings us ‘The 20 Best Fashion Blogs‘.
• Urban Outfitters is introducing Music Mondays—five free songs to start the week off right.
| 322,020
|
A special tribunal Wednesday upheld the ban imposed by the central government on the Students Islamic Movement of India (SIMI) under the Unlawful Activities (Prevention) Act for another five years.
Supreme Court referred to a larger bench a petition challenging the latest renewal of the ban on the Student Islamic Movement of India (SIMI).
The Centre has extended by two more
years the ban imposed on Students Islamic Movement of India.
SIMI, which was banned in 2001, has been under the scanner of security and intelligence agencies for terror attacks in various parts of the country.
Day in Pics: 18th December
Showbiz World - December 18, 2014
Stardust Awards 2014
| 5,803
|
\begin{document}
\vspace{-20mm}
\begin{center}{\large \bf
An extended anyon Fock space and noncommutative Meixner-type orthogonal polynomials in infinite dimensions}\\[4mm]
{\it Dedicated to Professor Anatoly Moiseevich Vershik\\
on the occasion of his 80th birthday}
\end{center}
{\large Marek Bo\.zejko}\\
Instytut Matematyczny, Uniwersytet Wroc{\l}awski, Pl.\ Grunwaldzki 2/4, 50-384 Wroc{\l}aw, Poland; e-mail: \texttt{bozejko@math.uni.wroc.pl}\vspace{2mm}
{\large Eugene Lytvynov}\\ Department of Mathematics,
Swansea University, Singleton Park, Swansea SA2 8PP, U.K.;
e-mail: \texttt{e.lytvynov@swansea.ac.uk}\vspace{2mm}
{\large Irina Rodionova}\\ Department of Mathematics,
Swansea University, Singleton Park, Swansea SA2 8PP, U.K.;
e-mail: \texttt{i.rodionova@swansea.ac.uk}\vspace{2mm}
{\small
\begin{center}
{\bf Abstract}
\end{center}
\noindent Let $\nu$ be a finite measure on $\mathbb R$ whose Laplace transform is analytic in a neighborhood of zero. An anyon L\'evy white noise on $(\mathbb R^d,dx)$ is a certain family of noncommuting operators $\langle\omega,\varphi\rangle$ in the anyon Fock space over $L^2(\mathbb R^d\times\mathbb R,dx\otimes\nu)$. Here $\varphi=\varphi(x)$ runs over a space of test functions on $\mathbb R^d$, while $\omega=\omega(x)$ is interpreted as an operator-valued distribution on $\mathbb R^d$. Let $L^2(\tau)$ be the noncommutative $L^2$-space generated by the algebra of polynomials in variables $\langle \omega,\varphi\rangle$, where $\tau$ is the vacuum expectation state. We construct noncommutative orthogonal polynomials in $L^2(\tau)$ of the form $\langle P_n(\omega),f^{(n)}\rangle$, where $f^{(n)}$ is a test function on $(\mathbb R^d)^n$. Using these orthogonal polynomials, we derive a unitary isomorphism $U$ between $L^2(\tau)$ and an extended anyon Fock space over $L^2(\mathbb R^d,dx)$, denoted by $\mathbf F(L^2(\mathbb R^d,dx))$. The usual anyon
Fock space over $L^2(\mathbb R^d,dx)$, denoted by $\mathcal F(L^2(\mathbb R^d,dx))$, is a subspace of $\mathbf F(L^2(\mathbb R^d,dx))$. Furthermore, we have the equality $\mathbf F(L^2(\mathbb R^d,dx))=\mathcal F(L^2(\mathbb R^d,dx))$ if and only if the measure $\nu$ is concentrated at one point, i.e., in the Gaussian/Poisson case. Using the unitary isomorphism $U$, we realize the operators $\langle \omega,\varphi\rangle$ as a Jacobi (i.e., tridiagonal) field in $\mathbf F(L^2(\mathbb R^d,dx))$. We derive a Meixner-type class of anyon L\'evy white noise for which the respective Jacobi field in $\mathbf F(L^2(\mathbb R^d,dx))$ has a relatively simple structure. Each anyon L\'evy white noise of the Meixner type is characterized by two parameters: $\lambda\in\mathbb R$ and $\eta\ge0$. Furthermore, we get the representation
$\omega(x)=\partial_x^\dag+\lambda \partial_x^\dag\partial_x +\eta\partial_x^\dag\partial_x\partial_x+\partial_x$.
Here $\partial_x$ and $\partial_x^\dag$ are annihilation and creation operators at point $x$. } \vspace{2mm}
\newpage
\section{Meixner polynomials in infinite dimensions}
\subsection{Meixner class of orthogonal polynomials}
In 1934, Meixner \cite{Meixner} studied the following problem. Consider complex-valued functions $u(z)$ and $\Phi(z)$ which can be expanded into a power
series of $z\in\mathbb C$ in a neighborhood of zero and suppose that $u(0)=1$, $\Phi(0)=0$, and $\Phi'(0)=1$. Then the function
\begin{equation}\label{cyd6i}
G(x,z)=\exp\big[x\Phi(z)\big]u(z)=\sum_{n=0}^\infty \frac{P_n(x)}{n!}\,z^n\end{equation}
generates a system of monic polynomials $P_n(x)$. Find all such polynomials which are orthogonal with respect to a probability measure $\mu$ on $\R$. Such polynomials are sometimes called orthogonal polynomials with generating function of exponential type.
Meixner \cite{Meixner} proved that a system of polynomials $P_n(x)$ belongs to this class if and only if it satisfies the recurrence relation
\begin{equation}\label{uyfu87t}
xP_n(x)=P_{n+1}(x)+(l+n\lambda)P_{n}(x)+n(k+\eta(n-1))P_{n-1}(x),\quad n\in\mathbb N_0,\end{equation}
where $l\in\R$, $k>0$, $\lambda\in\R$, $\eta\ge0$.
For each choice of the parameters, the corresponding measure of orthogonality, $\mu$, is infinitely divisible. If $l=0$, $\mu$ becomes centered, whereas $l\ne0$ corresponds to the shift of $\mu$ by $l$. For $l=0$ and $k\ne1$, the measure $\mu$ is the $k$-th convolution power of the corresponding measure $\mu$ for $k=1$.
One distinguishes five classes of polynomials satisfying \eqref{uyfu87t} (see \cite{Meixner, Chihara}):
(i) For $\lambda=\eta=0$, $\mu$ is a Gaussian measure, $(P_n)_{n=0}^\infty$ is a system of Hermite polynomials.
(ii) For $\lambda\ne0$ and $\eta=0$, $\mu$ is similar to a Poisson distribution ($\mu$ being a real Poisson distribution when $\lambda=1$ and $l=1$), $(P_n)_{n=0}^\infty$ is a system of Charlier polynomials.
(iii) For $|\lambda|=2$ and $\eta\ne0$, $\mu$ is a gamma distribution,
$(P_n)_{n=0}^\infty$ is a system of Laguerre polynomials.
(iv) For $|\lambda|<2$ and $\eta\ne0$, $\mu$ is a Pascal (negative binomial) distribution, $(P_n)_{n=0}^\infty$ is a system of Meixner polynomials of the first kind.
(v) For $|\lambda|>2$ and $\eta\ne0$, $\mu$ is a Meixner distribution, $(P_n)_{n=0}^\infty$ is a system of Meixner polynomials of the second kind, or Meixner--Polaczek polynomials.
Note that, in each case, for $z$ from a neighborhood of zero in $\mathbb C$,
\begin{equation}\label{ftse5sw5}
G(x,z)=\exp\big[x\Phi(z)-\mathcal C(\Phi(z))\big], \end{equation}
where $\mathcal C(z):=\log\left(\int_{\R}e^{xz}\,\mu(dx)\right)$ is the cumulant transform of $\mu$. We refer to \cite{Meixner, Chihara} for explicit formulas of $\Phi(z)$ and $\mathcal C(z)$. If one introduces complex parameters $\alpha,\beta\in\mathbb C$ such that
$\alpha+\beta=-\lambda$ and $\alpha\beta=\eta$, using Taylor's expansion, one can write down explicit formulas for $\Phi(z)$ and $\mathcal C(z)$ in a unique form for all parameters $\alpha$ and $\beta$, see \cite{Rodionova}.
The two observations below will be crucial for our considerations.
First, setting $l=0$ and $k=1$, we can rewrite formula \eqref{uyfu87t} as follows
\begin{equation}\label{txsjusxt} x=\di^\dag+\lambda \di^\dag\di +\di+\eta\di^\dag\di\di.\end{equation}
Here (with an abuse of notation) $x$ denotes the operator of multiplication by the variable $x$ in $L^2(\R,\mu)$, $\di^\dag$ is a creation (raising) operator: $\di^\dag P_n(x)=P_{n+1}(x)$, and $\di$ is an annihilation (lowering) operator: $\di P_n(x)=nP_{n-1}(x)$.
Second, Kolmogorov's representation of the Fourier transform of the infinitely divisible measure $\mu$ (with $l=0$) has the form \cite{Schoutens,Schoutens_Teugels}
$$\int_\R e^{iux}\,\mu(dx)=\exp\left[k\int_\R
(e^{ius}-1-ius)s^{-2}\,\nu(ds)\right],\quad u\in\R, $$
see also \cite{Grigelionis}.
Here, for $\eta=0$ (Gaussian and Poisson cases), $\nu=\delta_\lambda$, the Dirac measure with mass at $\lambda$, whereas for $\eta\ne0$ (cases (iii)--(v)) $\nu$ is the probability measure on $\R$, whose system of monic orthogonal polynomials, $(p_n)_{n=0}^\infty$, satisfies the recurrence formula
\begin{equation}\label{noh9y}
sp_n(s)=p_{n+1}(s)+\lambda(n+1)s+\eta n(n+1)p_{n-1}(s).
\end{equation}
In particular, $(p_n)_{n=0}^\infty$ is again a system of orthogonal polynomials from the Meixner class.
\subsection{An infinite dimensional extension}\label{fytry}
It appears that the Meixner class of orthogonal polynomials is fundamental for infinite dimensional analysis, in particular, for the theory of L\'evy white noise, see e.g.\ \cite{afs,Ly2,L2,Schoutens,TsVY} and the references therein. Let
$X:=\R^d$ and let
$$\mathscr D(X)\subset L^2(X,dx)\subset\mathscr D'(X)$$ be a standard triple of spaces in which $\mathscr D(X)$ is the nuclear space of smooth, compactly supported functions on $X$ and
$\mathscr D'(X)$ is the dual space of $\mathscr D(X)$ with respect to zero space $L^2(X,dx)$.
For $\omega\in\mathscr D'(X)$ and $\varphi\in\mathscr D(X)$, we denote by $\la \omega,\varphi\ra$ the dual pairing between $\omega$ and $\varphi$. Let $\mu$ be
a probability measure on $\mathscr D'(X)$, and assume that $\mu$ is a generalized stochastic process with independent values, in the sense of \cite{GV}, or using another terminology, a L\'evy white noise measure \cite{DOP}.
We will assume that $\mu$ is centered and its Fourier transform has Kolmogorov's representation
\begin{equation}\label{jig8yugtf8}\int_{\mathscr D'(X)}e^{i\la\omega,\varphi\ra}\mu(d\omega)=\exp\bigg[\int_{X}\int_\R \big(e^{is\varphi(x)}-1-is\varphi(x)\big)s^{-2}\nu(ds)\,dx\bigg],\quad \varphi\in \mathscr D(X),\end{equation}
where
$\nu$ is a probability measure on $\R$ which satisfies:
\begin{equation}\label{ft7er7i57}\int_{\R}e^{\varepsilon|s|}\,\nu(ds)<\infty\quad \text{for some $\varepsilon>0$.}\end{equation}
Note that the measure $s^{-2}\nu(ds)$ on $\R\setminus\{0\}$ is called the L\'evy measure of $\mu$, while $\nu(\{0\})$ describes the Gaussian part of $\mu$ (for $s=0$, the function under the integral sign in \eqref{jig8yugtf8} is equal to $-(1/2)\varphi^2(x)$).
In the case $d=1$, for each $t\ge0$, one can define by approximation in $L^2(\mathscr D'(X),\mu)$ a random variable $L_t(\omega)=\la\omega,\chi_{[0,t]}\ra$. Here $\chi_{[0,t]}$ denotes the indicator function of $[0,t]$. Then $(L_t)_{t\ge0}$ is a (version of a) L\'evy process with Kolmogorov measure $\nu$:
$$ \int_{\mathscr D'(X)}e^{iuL_t(\omega)}\,\mu(d\omega)=
\exp\left[t\int_\R (e^{ius}-1-ius)s^{-2}\nu(ds)\right].$$
Thus, the measure $\mu$ is indeed a L\'evy white noise.
Denote by $\mathscr {CP}$ the set of all continuous polynomials on $\mathscr D'(X)$, i.e., functions on $\mathscr D'(X)$ of the form
\begin{equation}\label{ytfdytdey7} f^{(0)}+\sum_{i=1}^n\la \omega^{\otimes i},f^{(i)}\ra,\quad \omega\in\mathscr D'(X),\ f^{(0)}\in\R,\ f^{(i)}\in \mathscr D(X)^{\otimes i},\ i=1,\dots,n,\ n\in\mathbb N.\end{equation}
If $f^{(n)}\ne0$, one says that the polynomial in \eqref{ytfdytdey7} has order $n.$
The set
$\mathscr{CP}$ is dense in $L^2(\mathscr D'(X),\mu)$.
So using the approach proposed by Skorohod \cite{Sko}, we may
orthogonalize these polynomials. More precisely, we denote by $\mathscr {CP}_n$ the linear space of all continuous polynomials on $\mathscr D'(X)$ of order $\le n$.
Let $\mathscr{MP}_n$ denote the closure of $\mathscr {CP}_n$ in $L^2(\mathscr D'(X),\mu)$ (the set of measurable polynomials of order $\le n$). Let $\mathscr{OP}_n:=\mathscr{MP}_n\ominus \mathscr{MP}_{n-1}$, the set of orthogonalized polynomials on $\mathscr D'(X)$ of order $n$.
We clearly have
\begin{equation}\label{gyut8t}
L^2(\mathscr D'(X),\mu)=\bigoplus_{n=0}^\infty \mathscr{OP}_n. \end{equation}
\begin{remark}
An alternative orthogonal decomposition of the $L^2$-space of a L\'evy process was derived by Vershik and Tsilevich in \cite{VTs}.
\end{remark}
For each $f^{(n)}\in \mathscr D(X)^{\otimes n}$, we denote by
$\la P_n(\omega), f^{(n)}\ra$ the orthogonal projection of the continuous monomial $\la \omega^{\otimes n}, f^{(n)}\ra$ onto $\mathscr{OP}_n$. We denote by $\mathscr {OCP}$ the linear space of orthogonalized continuous polynomials, i.e., the space of finite sums of functions of the form
$\la P_n(\omega),f^{(n)}\ra$ and constants. It should be stressed that the function $\la P_n(\omega), f^{(n)}\ra$ does not necessarily belong to $\mathscr{CP}$.
\begin{theorem}\label{ur7o67r6}
Let $\mu$ be a probability measure on $\mathscr D'(X)$ which has Fourier transform \eqref{jig8yugtf8} with $\nu$ being a probability measure on $\mathbb R$ satisfying \eqref{ft7er7i57}.
Then we have
$$\mathscr{CP}=\mathscr {OCP}$$
if and only if there exist $\lambda\in\R$ and $\eta\ge0$ such that, if $\eta=0$ then $\nu=\delta_\lambda$, and if $\eta>0$ then
the system of monic polynomials $(p_n)_{n=0}^\infty$ which are orthogonal with respect to the measure $\nu$ satisfies the recurrence formula
\eqref{noh9y} with $\lambda$ and $\eta$.
\end{theorem}
This theorem can be derived from the main result of \cite{BML}. It will also be a corollary of Theorem~\ref{urr8r} below.
We define the generating function of the orthogonal polynomials by
$$G_\mu(\omega,\varphi):=\sum_{n=0}^\infty \frac1{n!}\la P_n(\omega),\varphi^{\otimes n}\ra,$$
and the cumulant transform of the measure $\mu$ by
$$\mathcal C_\mu(\varphi):=\log\left(\int_{\mathscr D'(X)}e^{\la\omega,\varphi\ra}\mu(d\omega)\right).$$
The following theorem shows, in particular, that formula \eqref{ftse5sw5} admits an extension to infinite dimensions, see \cite{Ly2} for a proof.
\begin{theorem}\label{igt8t} Fix any $\lambda\in\R$ and $\eta\ge0$.
Let $\mu$ be the probability measure on $\mathscr D'(X)$
which has Fourier transform \eqref{jig8yugtf8} with $\nu$ being the probability measure on $\R$ corresponding to the parameters $\lambda$ and $\eta$ as in
Theorem~\ref{ur7o67r6}.
Let $\mathcal C(\cdot)$ and $\Phi(\cdot)$ be the functions as in \eqref{ftse5sw5} for parameters $l=0$, $k=1$ and $\lambda$ and $\eta$ as above. Then
\begin{align*}
\mathcal C_\mu(\varphi)&=\int_{X}\mathcal C(\varphi(x))\,dx,\\
G_\mu(\omega,\varphi)&=\exp\left[
\la \omega,\Phi(\varphi)\ra-\int_{X}\mathcal C(\Phi(\varphi(x)))\,dx\right],
\end{align*}
the formulas hold for $\varphi$ from (at least) a neighborhood of zero in $\mathscr D(X)$.
\end{theorem}
In the case $\lambda=0$, $\eta=0$, $\mu$ is a Gaussian white noise measure. We refer to e.g.\ \cite{BK,DOP,HKPS} for Gaussian white noise analysis.
In the case $\lambda\ne0$ and $\eta=0$, $\mu$ is a Poisson random measure (or point process), see e.g.\ \cite{Kal}.
We refer to \cite{VGG} for a discussion of representations of the group of diffeomorphisms in the Poisson space, to
\cite{IK} for Poisson white noise analysis, and to \cite{AKR} for Poisson analysis on the configuration space.
For $\eta\ne 0$, the most important case of $\mu$ is when $\lambda=2$ and $\eta=1$. Then $\mu$ is the centered gamma measure. The gamma measure is concentrated on discrete Radon measure on $X$, $\sum_i s_i\delta_{x_i}$, such that the configuration of atoms, $\{x_i\}$, is a dense subset of $X$. A very important property of the gamma measure is that it is quasi-invariant with respect to a natural group of transformations of the weights, $s_i$, see \cite{TsVY} and the references therein. Furthermore, as shown in \cite{TsVY}, the gamma measure is the unique law of a measure-valued L\'evy process which has an equivalent $\sigma$-finite measure which is projective invariant with respect to the action of the group acting on the weights, $s_i$. This $\sigma$-finite measure is called in \cite{TsVY} the infinite dimensional Lebesgue measure, see also \cite{Vershik}. We also note that, in papers \cite{TsVY,VGG1,VGG2,VGG3}, the gamma measure was used in the representation theory of the group $SL(2,F)$, where $F$ is an algebra of functions on a manifold. White noise analysis related to the gamma measure was initiated in \cite{KSSU}, and further developed in \cite{KL}. Gibbs perturbations of the gamma measure were constructed in
\cite{HKPR}. A Laplace operator associated with the gamma measure was constructed and studied in \cite{HKLV}.
Finally, infinite dimensional analysis related to the case of a general $\eta\ne0$ was studied in \cite{Ly1,Ly2}.
It is well known that, in the Gaussian and Poisson cases ($\eta=0$), the decomposition of $L^2(\mathscr D'(X),\mu)$ in orthogonal polynomials yields the Wiener--It\^o--Segal isomorphism between $L^2(\mathscr D'(X),\mu)$ and the symmetric Fock space over $L^2(X,dx)$. (An alternative derivation of this result is achieved by using multiple stochastic integrals, see e.g.\ \cite{Surgailis} for the Poisson case.) This result admits the following extension, see \cite{KSSU,KL,Ly2}.
\begin{theorem} \label{uyt8t8}
Let $\lambda\in\R$ and $\eta\ge0$, and let $\mu$ be the corresponding probability measure on $\mathscr D'(X)$ as in Theorem~\ref{igt8t}.
{\rm (i)} For each $n\in\mathbb N$, there exists a measure $m_\nu^{(n)}$ on $X^n$ which satisfies
\begin{equation}\label{gfuyfrluf}\int_{\mathscr D'(X)}\la P_n(\omega), f^{(n)}\ra ^2\,\mu(d\omega)=\int_{X^n}(\operatorname{Sym}_n f^{(n)})^2\, dm_\nu^{(n)},\quad f^{(n)}\in\mathscr D(X)^{\otimes n}.\end{equation}
Here $\operatorname{Sym}_n f^{(n)}$ denotes the usual symmetrization of a function $f^{(n)}$. For $\eta=0$, $m_\nu^{(n)}=\frac1{n!}\,dx_1\dotsm dx_n$, for $\eta\ne0$ see subsec.~\ref{utf7r5svgy} below for the explicit construction of $m_\nu^{(n)}$.
{\rm (ii)} We define a Hilbert space
\begin{equation}\label{urtur}\mathbf F_{\mathrm{sym}}(L^2(X,dx),\nu):=\R\oplus\bigoplus_{n=1}^\infty L^2_{\mathrm{sym}}(X^n,m_\nu^{(n)}),\end{equation}
where $L^2_{\mathrm{sym}}(X^n,m_\nu^{(n)})$ is the subspace of $L^2(X^n,m_\nu^{(n)})$ consisting of all symmetric functions from this space. For $\eta=0$, $\mathbf F_{\mathrm{sym}}(L^2(X,dx),\nu)$ is the symmetric Fock space over
$L^2(X,dx)$. For $\eta\ne0$, $\mathbf F_{\mathrm{sym}}(L^2(X,dx),\nu)$ contains the symmetric Fock space as a proper subspace. We then call $\mathbf F_{\mathrm{sym}}(L^2(X,dx),\nu)$ an extended symmetric Fock space. The mapping
\begin{equation}\label{huhgiuyfg}
f^{(0)}+\sum_{i=i}^n\la P_i(\omega),f^{(i)}\ra\mapsto(f^{(0)},\,\Sym_1f^{(1)},\dots,\,\Sym_nf^{(n)},0,0\dots)\in\mathbf F_{\mathrm{sym}}(L^2(X,dx),\nu)\end{equation}
extends by continuity to a unitary operator
$ U:L^2(\mathscr D'(X),\mu)\mapsto \mathbf F_{\mathrm{sym}}(L^2(X,dx),\nu)$.
{\rm (iii)} For each $\varphi\in\mathscr D(X)$, we keep the notation
$\la \omega,\varphi\ra$ for the image of the operator of multiplication by the monomial $\la \omega,\varphi\ra$ in $L^2(\mathscr D'(X),\mu)$ under the unitary operator $U$.
Then, analogously to \eqref{txsjusxt}, we have the following representation of the operator $\la \omega,\varphi\ra$ realized in the (extended) symmetric Fock space $\mathbf F_{\mathrm{sym}}(L^2(X,dx),\nu)$:
\begin{equation}\label{yufru7rf}\la \omega,\varphi\ra=\int_{X}dx\,\varphi(x)
(\di_x^\dag+\lambda \di_x^\dag\di_x +\di_x+\eta\di_x^\dag\di_x\di_x).\end{equation}
Here $\di_x$ is the annihilation operator at point $x$:
\begin{equation}\label{kgiygti9}(\di_x f^{(n)})(x_1,\dots,x_{n-1}):=n f^{(n)}(x,x_1,\dots,x_{n-1}),\end{equation}
and $\di_x^\dag$ is the creation operator at point $x$, satisfying
\begin{equation}\label{ufgutfr8ub}\int_{X}dx\,\varphi(x)\di_x^\dag\, f^{(n)}:=\operatorname{Sym}_{n+1}(\varphi\otimes f^{(n)}),\end{equation}
see \cite{Ly2} for further details.
\end{theorem}
Note that, in view of formula \eqref{yufru7rf}, we may heuristically write
\begin{equation}\label{ytr765r}
\omega(x)=\di_x^\dag+\lambda \di_x^\dag\di_x +\di_x+\eta\di_x^\dag\di_x\di_x.\end{equation}
As follows from Theorem \ref{uyt8t8}, (iii), the operators
$\la \omega,\varphi\ra$ realized in $\mathbf F_{\mathrm{sym}}(L^2(X,dx),\nu)$ form a Jacobi field, i.e.,
they have a tridiagonal structure; compare with e.g.\ \cite{B,BML,Br1,Br2,Ly1}.
\subsection{A noncommutative extension for anyons --- an introduction}
The above discussed results
have noncommutative analogs in the framework of free probability \cite{BL1,BL2}, see also \cite{a2,a5,bb,bd,Biane} and the references therein. See also
\cite{BBLS,BH} for further connections between the classical distributions from the Meixner class and free probability.
However, in this paper, we will be interested in a noncommutative extension of Meixner polynomials for a so-called anyon statistics \cite{LM,GM,GS}, see also \cite{Bozejko}. The latter statistics, indexed by a complex number $q$ of modulus one,
forms a continuous bridge between the boson statistics ($q=1$) and the fermi statistics ($q=-1$). One of the main aims of the present paper is to show that, in the anyon setting, one naturally arrives at noncommutative Meixner-type polynomials which have a representation like in \eqref{yufru7rf}.
In fact, one could think that it was hopeless to expect
a counterpart of formula \eqref{yufru7rf} in the fermion setting. Indeed, if the operators $\di_x$ and $\di_y$ anticommute,
i.e., $\di_x\di_y=-\di_y\di_x$,
then $\di_x\di_x=0$, so that the term $\eta\di_x^\dag\di_x\di_x$ must be equal to zero. However, we do show that, even in the fermion setting, the integral $\int_{X}dx\,\varphi(x)\di_x^\dag\di_x\di_x$ leads to a well-defined, nontrivial operator in an extended antisymmetric Fock space $\mathbf F_{\mathrm{as}}(L^2(X,dx),\nu)$. The latter space contains the usual
antisymmetric (fermion) Fock space $\mathcal F_{\mathrm{as}}(L^2(X,dx))$ as a subspace. On the space $\mathcal F_{\mathrm{as}}(L^2(X,dx))$, the operators $\di_x$ and $\di_y$ indeed anticommute. However, this anticommutaion fails on the whole space $\mathbf F_{\mathrm{as}}(L^2(X,dx),\nu)$. As a result, the extended antisymmetric Fock space leads to a proper renormalization (rather a nontrivial extension) of the operators $\di_x$ and $\di^\dag_x$.
Our discussion of this noncommutative extension is organized as follows. In Section~\ref{uiy976}, following \cite{GM,BLW,LM}, we briefly recall
the construction of the
anyon Fock space, standard operators on them, and the anyon commutation relations. We also recall the construction of a L\'evy white noise for anyon statistics as a family of noncommutative self-adjoint operators $\la\omega,\varphi\ra$ in the anyon Fock space over $L^2(X\times\R,dx\,\nu(ds))$, see \cite{BLW} for details. Note that, in this section, we do not explain why the `increments' of this process can be understood as being `anyon independent.' For this, we refer the reader to \cite{BLW}. We only note that in the commutative, boson setting ($q=1$), we indeed recover a classical L\'evy white noise, being realized as a family of commuting self-adjoint operators in the symmetric Fock space over $L^2(X\times\R,dx\,\nu(ds))$.
In Section~\ref{yre6tr5}, we formulate the main results of the paper. In particular, starting with a space $\mathscr{CP}$ of noncommutative continuous polynomials of anyon white noise, we construct
a space $\mathscr{OCP}$ of orthogonalized
continuous polynomials. By analogy with \eqref{gfuyfrluf}, for each $n\in\mathbb N$, we construct a measure $m_\nu^{(n)}$ on $X^n$ and find the corresponding symmetrization operator $\operatorname{Sym}_n$. This symmetry extends the anyon symmetry (in particular, the fermion symmetry) in a non-trivial way. By analogy with \eqref{urtur}, we define an extended anyon Fock space, and then by analogy with \eqref{huhgiuyfg}, we construct a unitary operator $U$ between the noncommutative $L^2$-space and the extended anyon Fock space. Under the unitary $U$, each operator $\la\omega,\varphi\ra$ takes a Jacobi form in the extended anyon Fock space. We show that this Jacobi field has the simplest form (in a sense) when $\nu$ is the same measure as in Theorem~\ref{ur7o67r6}, i.e., $\nu$ is Kolmogorov's measure of a white noise measure $\mu$ from the Meixner class. Furthermore, in this case, analogs of formulas \eqref{yufru7rf}--\eqref{ufgutfr8ub} hold.
Finally, Section~\ref{ur867} is devoted to the proofs of the main results.
Among numerous open problems regarding the anyon Meixner-type white noise, let us mention only two:
(i) In both the classical and free cases, the generating functions of the Meixner-type orthogonal polynomials are explicitly known and play an important role in the studies of these polynomials. In the anyon case, the form of the generating function is not yet known, even in the Gaussian case. The main difficulty lies in the fact that both the classical and free Meixner-type polynomials have corresponding systems of orthogonal polynomials on the real line. However, the anyon case is purely infinite dimensional and has no related one-dimensional theory.
(ii) As shown in \cite{afs}, in the classical case,
the L\'evy processes from the Meixner class with $\eta>0$ are related to the renormalized squares of boson white noise. Is it possible to interpret anyon Meixner-type white noises as those related to renormalized squares of anyon white noise?
\section{Noncommutative L\'evy white noise for anyon statistics}\label{uiy976}
\subsection{Anyon Fock space and anyon commutation relations}
\label{hfgyufzua}
Let $\mathcal B(X)$ denote the Borel $\sigma$-algebra on $X$, and let $\mathcal B_0(X)$ denote the family of all sets from $\mathcal B(X)$ which have compact closure. Let $m=m(dx)=dx$ denote the Lebesgue measure on $(X,\mathcal B(X))$.
For each $n\ge2$, we define
\begin{equation}\label{ydr5w54} X^{(n)}:=\big\{(x_1,\dots,x_n)\in X^n\mid \forall 1\le i<j\le n:\ x_i\ne x_{j}\big\}.\end{equation}
Since the measure $m$ is non-atomic,
\begin{equation}\label{rtsew6uwy}m^{\otimes n}(X\setminus X^{(n)})=0.\end{equation}
We introduce a strict total order on $X$ as follows, for any $x=(x^1,\dots,x^d),y=(y^1,\dots,y^d)\in X$, $x\ne y$, we set $x<y$ if for some $j\in\{1,\dots,d\}$, we have $x^1=y^1$,\dots, $x^{j-1}=y^{j-1}$ and $x^j<y^j$.
We fix a number $q\in\mathbb C$ with $|q|=1$, and define a function $Q:X^{(2)}\to\mathbb C$ as follows:
$$ Q(x,y)=\begin{cases}
q,&\text{if }x<y,\\
\bar q,&\text{if }y<x.
\end{cases}$$
Note that the function $Q$ is Hermitian:
\[
Q(x, y) = \overline{Q(y, x)},\quad (x, y)\in X^{(2)}.
\]
A function $f^{(n)}:X^{(n)}\to\mathbb C$ ($n\ge2$) is called $Q$-symmetric if, for each $j=1,\dots,n-1$,
\begin{equation}\label{uyr57eses} f^{(n)}(x_1,\dots,x_n)=Q(x_j,x_{j+1})f^{(n)}(x_1,\dots,x_{j-1},x_{j+1},x_{j},x_{j+2},\dots,x_n).\end{equation}
Let $\mathcal H:=L^2(X,m)$ be the Hilbert space of all complex-valued, square-integrable functions on $X$. Thus, for each $n\in\mathbb N$, $\mathcal H^{\otimes n}=L^2(X^n,m^{\otimes n})$.
In view of \eqref{rtsew6uwy}, we have $\mathcal H^{\otimes n}=L^2(X^{(n)},m^{\otimes n})$.
We define a complex Hilbert space $\mathcal H^{\cd n}$ as the (closed) subspace of $\mathcal H^{\otimes n}$ consisting of all ($m^{\otimes n}$-versions of) $Q$-symmetric functions in $\mathcal H^{\otimes n}$. Let $\Sym_n$ denote the orthogonal projection of $\mathcal H^{\otimes n}$ onto $\mathcal H^{\cd n}$. This operator has the following explicit form: for each $f^{(n)}\in\mathcal H^{\otimes n}$,
\begin{multline}\label{oitr8o} (\Sym_nf^{(n)})(x_1,\dots,x_n)\\=\frac1{n!}\sum_{\pi\in \mathfrak S_n}
Q_\pi(x_1,\dots,x_n)f^{(n)}(x_{\pi^{-1}(1)},\dots,x_{\pi^{-1}(n)}),\quad (x_1,\dots,x_n)\in X^{(n)}.\end{multline}
Here $\mathfrak S_n$ denotes the group of all permutations of $1,\dots,n$
and
\begin{equation}\label{y7e57ie} Q_\pi(x_1,\dots,x_n):=\prod_{\substack{1\le i<j\le n\\ \pi(i)>\pi(j)}} Q(x_{i},x_j),\quad (x_1,\dots,x_n)\in X^{(n)}.\end{equation}
We can now define a $Q$-symmetric tensor product $\cd$. For any $m,n\in\mathbb N$ and any $f^{(m)}\in\mathcal H^{\cd m}$ and $g^{(n)}\in\mathcal H^{\cd n}$, we set
$f^{(m)}\cd g^{(n)}:=\Sym_{m+n}(f^{(m)}\otimes g^{(n)})$. Note that this tensor product is associative. Note also that, for $q=1$, $\cd$ is the usual symmetric tensor product, while for $q=-1$, $\cd$ is the usual antisymmetric tensor product.
We define an anyon Fock space by
$$\mathcal F^Q(\mathcal H):=\bigoplus_{n=0}^\infty \mathcal H^{\cd n}n!\, .$$
Thus, $\mathcal F^Q(\mathcal H)$ is the Hilbert space which consists of all sequences $F=(f^{(0)}, f^{(1)},f^{(2)},\dots)$ with $f^{(n)}\in \mathcal H^{\cd n}$ ($\mathcal H^{\cd 0}:=\mathbb C$) satisfying
$$\|F\|^2_{\mathcal F^Q(\mathcal H)}:=\sum_{n=0}^\infty\|f^{(n)}\|^2_{\mathcal H^{\cd n}}n!<\infty.$$
(The inner product in $\mathcal F^Q(\mathcal H)$ is induced by the norm in this space.)
The vector $\Omega:=(1,0,0,\dots)\in \mathcal F^Q(\mathcal H)$ is called the vacuum.
We denote by $\mathcal F_{\mathrm{fin}}^Q(\mathcal H)$ the subspace of $\mathcal F^Q(\mathcal H)$ consisting of all finite sequences $$F=(f^{(0)},f^{(1)},\dots,f^{(n)},0,0,\dots)$$ in which $f^{(i)}\in \mathcal H^{\cd i}$ for $i=0,1,\dots,n$, $n\in\mathbb N$. This space can be endowed with the topology of the topological direct sum of the $\mathcal H^{\cd n}$ spaces. Thus, convergence in $\mathcal F_{\mathrm{fin}}^Q(\mathcal H)$ means uniform finiteness of non-zero components and coordinate-wise convergence in $\mathcal H^{\cd n}$.
For each $h\in\mathcal H$, we define a creation operator $a^+(h)$ and an annihilation operator $a^-(h)$ as the linear operators acting on $\mathcal F_{\mathrm{fin}}^Q(\mathcal H)$ given by
$$
a^{+}(h)f^{(n)} := h \cd f^{(n)},\quad f^{(n)}\in \mathcal H^{\cd n},\quad a^{-}(h):= a^{+}(h)^*\restriction _{\mathcal F_{\mathrm{fin}}^Q(\mathcal H)}.
$$
Both $a^+(h)$ and $a^-(h)$ act continuously on $\mathcal F_{\mathrm{fin}}^Q(\mathcal H)$. In fact, for any $h\in\mathcal H$ and $f^{(n)}\in\mathcal H^{\cd n}$, we have
\begin{align}
& (a^+(h)f^{(n)})(x_1,\dots,x_{n+1})=\frac1{n+1}\Big[
h(x_1)f^{(n)}(x_2,\dots,x_{n+1})\notag\\
&\quad +\sum_{k=2}^{n+1}Q(x_1,x_k)Q(x_2,x_k)\dotsm Q(x_{k-1},x_k)h(x_k)f^{(n)}(x_1,\dots,x_{k-1},x_{k+1},\dots,x_{n+1})\Big],\notag\\
& (a^-(h)f^{(n)})(x_1, \dots , x_{n-1}) =
n\int_{X}\overline{h(y)}\,f^{(n)}(y, x_1, \dots ,
x_{n-1})\,dy.\label{ufcu}
\end{align}
The action of the annihilation operator can also be written in the following form: for any $h\in\mathcal H$ and $f^{(n)}\in\mathcal H^{\otimes n}$,
\begin{multline} (a^-(h) \Sym_n f^{(n)})(x_1,\dots,x_{n-1})
=\Sym_{n-1}\bigg(\int_X\overline{h(y)}\bigg[\sum_{k=1}^n
Q(y,x_1)Q(y,x_2)\\
\times\dotsm \times Q(y,x_{k-1}) f^{(n)}
(x_1,x_2,\dots,x_{k-1},y,x_k,\dots,x_{n-1})
\bigg]dy\bigg).\label{hyfd7urf}\end{multline}
Let us now discuss the creation and annihilation operators at points of the space $X$.
At least informally, for each $x\in X$, we may consider a delta function at $x$, denoted by $\delta_x$.
Then we can heuristically define $\partial_x^\dag:=a^+(\delta_x)$ and $\partial_x:=a^-(\delta_x)$, so that
\begin{equation}\label{bgyiugtf8utfg}
\partial_x^\dag f^{(n)}=\delta_x\cd f^{(n)},\quad \partial_x f^{(n)}:=nf^{(n)}(x,\cdot).\end{equation}
Thus,
\begin{equation}\label{out979p}
a^+(h):=\int_X dx\, h(x)\partial_x^\dag\,,\quad
a^-(h)=\int_X dx\, \overline{h(x)}\,\partial_x\,.
\end{equation}
Note that the second formula in \eqref{bgyiugtf8utfg} is a rigorous definition of $\di_x$ (for $m$-a.a.\ $x\in X$), while the first formula in \eqref{out979p} is the rigorous definition of the integral $\int_X dx\, h(x)\partial_x^\dag$.
Let $B_0(X^n)$ denote the space of all complex-valued bounded measurable functions on $X^n$ with compact support. Let $g^{(n)}\in B_0(X^n)$.
Fix any sequence of $+$ and $-$ of length $n\ge 2$, and denote
it by $(\sharp_1, \dots , \sharp_n)$. It is easy to see that the expression
\[
\int_{X^n}dx_1\dotsm dx_n\,g^{(n)}(x_1, \dots ,
x_n)\partial^{\sharp_1}_{x_1}\dotsm \partial^{\sharp_n}_{x_n}
\]
identifies a linear continuous operator on $\mathcal F_{\mathrm{fin}}^Q(\mathcal H)$. Here
we used the notation $\partial_x^+:=\partial_x^\dag$, $\partial^-_x:=\partial_x$.
The creation and annihilation operators satisfy the anyon commutation
relations:
\begin{align}
\partial_x\partial_y^\dag &= \delta(x, y)+Q(x, y)\partial^\dag_y\partial_x,\label{fdytde} \\
\partial_x\partial_y &= Q(y,x)\partial_y\partial_x ,\label{jhgyufd}\\
\partial^\dag_x\partial^\dag_y &= Q(y, x)\partial^\dag_y\partial^\dag_x .\label{pr++}
\end{align}
Here $\delta(x, y)$ is understood as:
\[
\int_{X^2}dx\, dy\,g^{(2)}(x, y)\delta(x, y) :=
\int_{X}dx\, g^{(2)}(x, x).
\]
Formulas \eqref{fdytde}--\eqref{pr++} make rigorous sense after smearing with functions $g^{(2)}\in B_0(X^2)$.
Note that, for $q=1$, equations \eqref{fdytde}--\eqref{pr++} become the canonical commutation relations, while for $q=-1$ they become the canonical anticommutation relations.
\begin{remark}
Let $D:=\{(x,x)\mid x\in X\}$. Note that, for each $g^{(2)}\in B_0(X^2)$ which has support in $D$, the operator $\int_{X^2}dx\,dy\, g^{(2)}(x,y)\partial^\dag_y\partial_x$ is equal to zero. Hence, it does not influence \eqref{fdytde} that we have not identified the function $Q$ on $D$.
\end{remark}
For a bounded linear operator $A$ in $\mathcal H$, we define
the differential second quantization of $A$, denoted by $d\Gamma(A)$,
as a linear continuous operator on $\mathcal F_{\mathrm{fin}}^Q(\mathcal H)$ given by
$d\Gamma(A)\Omega:=0$ and
$$d\Gamma(A)\restriction \mathcal H^{\cd n}:=\Sym_n(A\otimes \mathbf 1\otimes\dots\otimes \mathbf 1+
\mathbf 1\otimes A\otimes \mathbf 1\otimes\dots\otimes\mathbf 1+\dots+\mathbf 1\otimes\dots\otimes\mathbf1\otimes A)$$
for each $n\in\mathbb N$. For each a.e.\ bounded function $h\in L^\infty(X,m)$, we define a neutral operator
\begin{equation}\label{ur75r7} a^0(h):=\int_X dx\,h(x)\partial_x^\dag\partial_x.\end{equation}
According to formulas \eqref{bgyiugtf8utfg} and \eqref{out979p}, we have
\begin{align}
\big(a^0(h)f^{(n)}\big)(x_1,\dots,x_n)&=\bigg(\int_X dx\, h(x)\di_x^\dag f^{(n)}(x,\cdot)\bigg)(x_1,\dots,x_n)\notag\\
&=n\operatorname{Sym}_n\big(h(x_1)f^{(n)}(x_1,x_2,\dots,x_n)\big).\label{jfiu}
\end{align}
From here one easily gets
\begin{equation}\label{gtdytlut9t8ghogt}
(a^0(h)f^{(n)})(x_1,\dots,x_n)=\big(h(x_1)+\dots+h(x_n)\big)f^{(n)}(x_1,\dots,x_n).\end{equation}
Hence, $a^0(h)=d\Gamma(M_h)$, where $M_h$ is the operator of multiplication by $h$.
\subsection{Anyon L\'evy white noise and noncommutative orthogonal polynomials}\label{utfr57rtf}
Let us now recall the construction of a L\'evy white noise over $X$ for anyon statistics, see \cite{BLW}.
Let $\nu$ be a probability measure on $(\mathbb{R},\mathcal B(\R))$. (In fact, we can instead assume that $\nu$ is a finite measure. The results below will then require a trivial modification.)
We denote by $\mathscr{P}(\R)$ the linear space of polynomials on $\mathbb R$. We assume that $\mathscr P(\R)$ is a dense subset of $L^2(\R,\nu)$. Note that the latter assumption is satisfied if, for example, \eqref{ft7er7i57} holds.
We extend the function $Q$ by setting
$$Q(x_1,s_1,x_2,s_2):=Q(x_1,x_2),\quad (x_1,x_2)\in X^{(2)},\ (s_1,s_2)\in\mathbb R^2.$$
Thus, the value of the function $Q$ does not depend on $s_1$ and $s_2$.
Analogously to \eqref{uyr57eses}, we define the notion of a $Q$-symmetric function $f^{(n)}$ defined on the set
$$\left\{(x_1,s_1,\dots,x_n,s_n)\in (X\times\R)^n\mid (x_1,\dots,x_n)\in X^{(n)}\right\}.$$
For example, for $n=2$, the $Q$-symmetry means:
$$ f^{(2)}(x_1,s_1,x_2,s_2)=Q(x_1,x_2)f^{(2)}(x_2,s_2,x_1,s_1).$$
We next set $$\mathcal G:=L^2(X\times \mathbb R,m\otimes \nu)=\h\otimes L^2(\mathbb R,\nu),$$
and consider the corresponding $Q$-symmetric Fock space $\mathcal F^Q(\mathcal G)$, which is constructed by analogy with $\mathcal F^Q(\mathcal H)$.
Let $\mathcal F_{\mathrm{fin}}^Q(\h\otimes\mathscr P(\mathbb R))$ denote the linear subspace of $\mathcal F^Q(\mathcal G)$ which consists of all finite sequences
$$F=(F^{(0)},F^{(1)},\dots,F^{(n)},0,0,\dots),\quad n\in\mathbb N_0,$$
such that each $F^{(k)}$ with $k\ne0$ has the form
$$F^{(k)}(x_1,s_1,\dots,x_k,s_k)=\Sym_k\left[\sum_{(i_1,i_2,\dots,i_k)\in\{0,1,\dots,N\}^k}f_{(i_1,i_2,\dots,i_k)}
(x_1,x_2,\dots,x_k)s_1^{i_1}s_2^{i_2}\dotsm s_{k}^{i_k}\right],$$
where $f_{(i_1,i_2,\dots,i_k)}\in \h^{\otimes k}$ and $N\in\mathbb N$. Clearly, $\mathcal F_{\mathrm{fin}}^Q(\h\otimes\mathscr P(\mathbb R))$ is dense in $\mathcal F^Q(\mathcal G)$.
We denote $1(s):=1$ and $\id(s):=s$ for $s\in\R$. Thus, $1,\id\in\mathscr P(\R)$. We denote by $C_0(X\mapsto\mathbb R)$ the space of all real-valued continuous functions on $X$ with compact support.
For each $f\in C_0(X\mapsto\mathbb R)$, we define an operator
\begin{equation}\label{g7r75e}\la \omega,f\ra:=a^{+}(f\otimes 1)+a^0(f\otimes \id)+a^-(f\otimes 1)\end{equation}
in $\mathcal F^Q(\mathcal G)$ with domain $\mathcal F_{\mathrm{fin}}^Q(\h\otimes\mathscr P(\mathbb R))$. Clearly, each operator $\la \omega,f\ra$ maps \linebreak $\mathcal F_{\mathrm{fin}}^Q(\h\otimes\mathscr P(\mathbb R))$ into itself. In fact, under assumption \eqref{ft7er7i57}, each $F\in\mathcal F_{\mathrm{fin}}^Q(\h\otimes\mathscr P(\mathbb R))$ is an analytic vector for each operator $\la \omega,f\ra$ with $f\in C_0(X\mapsto\mathbb R)$, which implies that the operators $\la \omega,f\ra$ are essentially self-adjoint on $\mathcal F_{\mathrm{fin}}^Q(\h\otimes\mathscr P(\mathbb R))$ (see e.g.\ \cite[Sec.~X.2]{RS2}).
\begin{remark}
Let us keep the notation $\la \omega,f\ra$ for the closure of this operator in $\mathcal F^Q(\mathcal G)$. Thus the operators $\la \omega,f\ra$ are self-adjoint.
In the boson case, $q=1$, these operators also commute in the sense of commutation of their resolutions of the identity. By using e.g.\ the projection spectral theorem \cite{BK}, one shows \cite{DL} that there exists a unitary isomorphism between the symmetric Fock space
$\mathcal F^Q(\mathcal G)$ and the space $L^2(\mathscr D'(X),\mu)$, where $\mu$ is the L\'evy white noise measure with Fourier transform \eqref{jig8yugtf8}.
Under this unitary isomorphism, the vacuum vector $\Omega$ becomes the constant function $1$, and each operator $\la \omega,f\ra$
becomes the operator of multiplication by the random variable $\la \omega,f\ra$ in
$L^2(\mathscr D'(X),\mu)$. In other words, $\mu$ is the spectral measure of the family of commuting self-adjoint operators $\big(\la \omega,f\ra\big)_{f\in C_0(X\mapsto\R)}$.
In particular, the operators $\big(\la \omega,f\ra\big)_{f\in C_0(X\mapsto\R)}$ in the symmetric Fock space $\mathcal F^Q(\mathcal G)$ can indeed be thought of as a L\'evy white noise. Let us also note that the unitary operator between $\mathcal F^Q(\mathcal G)$ and $L^2(\mathscr D'(X),\mu)$ was originally derived by It\^o, by using multiple stochastic integrals, see
\cite{Ito}.
\end{remark}
\begin{remark}
Note that, if the measure $\nu$ is concentrated at one point, $\lambda\in\mathbb R$, then
$\mathcal G=\h$ and each operator $\la \omega,f\ra$
has the following form in $\mathcal F^Q(\mathcal H)$:
\begin{equation}\label{vgfi} \la \omega,f\ra:=a^+(f)+a^-(f)+\lambda a^0(f).\end{equation}
The choice $\lambda=0$ corresponds to an anyon Gaussian white noise, while $\lambda\ne0$ corresponds to an anyon centered white noise. If we denote
\begin{equation}\label{vcydcyd}\omega(x):=\partial_x^\dag+\lambda\partial_x^\dag\partial_x+\partial_x,\quad x\in X,\end{equation}
then, by \eqref{out979p}, \eqref{ur75r7}, \eqref{vgfi}, and \eqref{vcydcyd}, we get
$$ \la
\omega,f\ra=\int_X dx\, \omega(x)f(x),\quad f\in C_0(X\mapsto\R),$$ which justifies the notation $\la \omega,f\ra$.
Thus, $(\omega(x))_{x\in X}$ is the anyon Gaussian/Poisson white noise.
Note that $\omega(x)$ is informally treated as an operator-valued distribution.
\end{remark}
We further denote by $C_0(X)$ the space of all complex-valued, continuous functions on $X$ with compact support. For $f\in C_0(X)$, we set $\la \omega,f\ra:=\la
\omega,\Re f\ra+i\la \omega,\Im f\ra$.
Let $\mathscr P$ denote the complex unital $*$-algebra generated by $(\la \omega,f\ra)_{f\in C_0(X)}$, i.e., the algebra of noncommutative polynomials in variables $\la \omega,f\ra$. In particular, elements of $\mathscr P$ are linear operators acting on $\mathcal F_{\mathrm{fin}}^Q(\h\otimes\mathscr P(\mathbb R))$, and for each $p\in\mathscr P$, $p^*$ is the adjoint operator of $p$ in $\mathcal F^Q(\mathcal G)$, restricted to $\mathcal F_{\mathrm{fin}}^Q(\h\otimes\mathscr P(\mathbb R))$.
We define a vacuum state on $\mathscr P$ by
$$\tau(p):=(p\Omega,\Omega)_{\mathcal F^Q(\mathcal G)},\quad p\in\mathscr P.$$ We introduce a scalar product on $\mathscr P$
by
$$(p_1,p_2)_{L^2(\tau)}:=\tau(p_2^*p_1)=(p_1\Omega,p_2\Omega)_{\mathcal F^Q(\mathcal G)}, \quad p_1,p_2\in\mathscr P.$$
Let
\begin{equation}\label{hufdy676}
\widetilde {\mathscr P}:=\{p\in\mathscr P\mid (p,p)_{L^2(\tau)}=0\},
\end{equation}
and define the noncommutative $L^2$-space $L^2(\tau)$ as the completion of the quotient space $\mathscr P/\widetilde{\mathscr P}$ with respect to the norm generated by the scalar product $(\cdot,\cdot)_{L^2(\tau)}$. Elements $p\in\mathscr P$ are treated as representatives of the equivalence classes from
$\mathscr P/\widetilde{\mathscr P}$, and so $\mathscr P$ becomes a dense subspace of $L^2(\tau)$. As shown in \cite{BLW}, the vacuum vector $\Omega$ is cyclic for the family of operators $(\la \omega,f\ra)_{f\in C_0(X\mapsto\mathbb R)}$.
Consider a linear mapping $I:\mathscr P\to
\mathcal F^Q(\mathcal G)$ defined by
$$Ip:=p\Omega\quad \text{for $p\in\mathscr P$}.$$
Then $Ip_1=Ip_2$ if $p_1,p_2\in\mathscr P$ are such that $p_1-p_2\in\widetilde{\mathscr P}$, and $I$ extends to a unitary operator $I:L^2(\tau)\to\mathcal F^Q(\mathcal G)$.
Note that, for each $p\in\mathscr P$ and $f\in C_0(X)$,
\begin{equation}\label{tyr75}
I\big(\la \omega,f\ra p\big)=\la \omega,f\ra (Ip),\end{equation}
i.e., under the unitary $I$, the operator of left multiplication by $\la \omega,f\ra$ in $L^2(\tau)$ becomes the operator $\la \omega,f\ra$ in $\mathcal F^Q(\mathcal G)$.
Let us consider the topology on $C_0(X)$
which yields the following notion of convergence: $f_n\to f$ as $n\to\infty$ means that there exists a set $\Delta\in\mathcal B_0(X)$ such that
$\operatorname{supp}(f_n)\subset\Delta$ for all $n\in\mathbb N$ and
\begin{equation}\label{ilyufre75iei7}\sup_{x\in X}|f_n(x)-f(x)|\to0\quad\text{as }n\to\infty.\end{equation}
By linearity and continuity we can extend the mapping
$$C_0(X)^n\ni (f_1,\dots,f_n)\mapsto\la \omega^{\otimes n}, f_1\otimes\dots\otimes f_n\ra
=\la \omega, f_1\ra\dotsm \la \omega, f_n\ra\in\mathscr P
$$
to a mapping
$$ C_0(X^n)\ni f^{(n)}\mapsto \la \omega^{\otimes n},f^{(n)}\ra \in L^2(\tau),$$
and $\la \omega^{\otimes n},f^{(n)}\ra $ can be thought of as a linear operator acting in $\mathcal F_{\mathrm{fin}}^Q(\h\otimes\mathscr P(\mathbb R))$. We will think of $\la \omega^{\otimes n},f^{(n)}\ra $ as a continuous monomial of order $n$. Sums of such operators and (complex) constants form the set $\mathscr{CP}$ of continuous polynomials (of $\omega$). Evidently, $\mathscr P\subset\mathscr{CP}$.
Completely analogously to \eqref{gyut8t}, we derive the orthogonal decomposition
\begin{equation}\label{tyr65}
L^2(\tau)=\bigoplus_{n=0}^\infty \mathscr{OP}_n\end{equation}
(we used obvious notations). For any $f^{(n)}\in C_0(X^n)$, we denote by $\la P_n(\omega),f^{(n)}\ra$ the orthogonal projection of $\la \omega^{\otimes n},f^{(n)}\ra$ onto $ \mathscr{OP}_n$. The set of finite linear sums of $\la P_n(\omega),f^{(n)}\ra$ and (complex) constants is denoted by $\mathscr{OCP}$ (orthogonalized continuous polynomials).
\begin{remark}
Note that $\la P_1(\omega),f\ra=\la \omega,f\ra$.
\end{remark}
\begin{remark}
Note that, in subsec.~\ref{fytry}, we used functions $f^{(n)}\in\mathscr D(X)^{\otimes n}$ when defining $\mathscr{CP}$ and $\mathscr{OCP}$, while now we are using $f^{(n)}\in C_0(X^n)$ to define $\mathscr{CP}$ and $\mathscr{OCP}$. The reason is that, in the noncommutative setting, there is no need for $f^{(n)}$ to be smooth, while in the classical case, $q=1$, Theorem~\ref{ur7o67r6} still holds for the sets $\mathscr{CP}$ and $\mathscr{OCP}$ as defined in this section.
\end{remark}
\section{Main results}\label{yre6tr5}
\subsection{The measures $m_\nu^{(n)}$}\label{utf7r5svgy}
Let $(p_k)_{k=0}^\infty$ denote the system of monic orthogonal polynomials in $L^2(\mathbb R,\nu)$. (If the support of $\nu$ is finite and consists of $N$ points, we set $p_k:=0$ for $k\ge N$.) Hence, $(p_k)_{k=0}^\infty$ satisfy the recursion formula
\begin{equation}\label{hdtrss}sp_k(s)=p_{k+1}(s)+b_kp_{k}(s)+a_kp_{k-1}(s),\quad k\in\mathbb N_0,\end{equation}
with $p_{-1}(s):=0$, $a_k>0$, and $b_k\in\mathbb R$. (If the support of $\nu$ has $N$ points, $a_k=0$ for $k\ge N$.)
We define
\begin{equation}\label{yuft8uotfr8o}c_k:=a_0a_1\dotsm a_{k-1},\quad k\in\mathbb N,\end{equation}
where $a_0:=1$ and the $a_k$'s for $k\in\mathbb N$ are the coefficients from formula \eqref{hdtrss}.
We equivalently have:
\begin{equation}\label{yufrur}c_k=\int_{\mathbb R}p_{k-1}(s)^2\,\nu(ds),\quad k\in\mathbb N,\end{equation}
which is a well known fact of the theory of orthogonal polynomials.
Note that $c_1=1$ and $c_k=0$ for $k\ge2$
if and only if the measure $\nu$ is concentrated at one point.
We denote by $\Pi(n)$ the set of all (unordered) partitions of the set $\{1,\dots,n\}$. For each partition
$\theta=\{\theta_1,\dots,\theta_l\}\in\Pi(n)$, we set
$|\theta|:=l$. For each $\theta\in\Pi(n)$, we denote by $X^{(n)}_\theta$ the subset of $X^n$ which consists of all $(x_1,\dots,x_n)\in X^n$ such that, for all $1\le i<j\le n$, $x_i=x_j$ if and only if $i$ and $j$ belong to the same element of the partition $\theta$. Note that the sets $X^{(n)}_\theta$ with $\theta\in \Pi(n)$
form a partition of $X^n$. Note also that, by \eqref{ydr5w54}, $X^{(n)}=X_\theta^{(n)}$ for the minimal partition $\theta=\{\{1\},\,\{2\},\dots,\, \{n\}\}$.
Let us fix $n\in\mathbb N$, a permutation $\pi\in \mathfrak S_n$,
and a partition $\theta=\{\theta_1,\dots,\theta_l\}\in\Pi(n)$ satisfying
\begin{equation}\label{ft6e}\max \theta_1<\max\theta_2<\dots<\max\theta_l.\end{equation}
We define a measure $m_{\nu,\,\theta}^{(n)}$ on $X_\theta^{(n)}$ as the push-forward of the measure
$$ \big(c_{|\theta_1|}\dotsm c_{|\theta_l|}\big)n!
\big(|\theta_1|!\dotsm |\theta_l|!\big)^{-1}\, m^{\otimes l}$$
on $X^{(l)}$ under the mapping
$$ X^{(l)}\ni y=(y_1,\dots,y_l)\mapsto (R_\theta^1y,\dots, R_\theta^n y)\in X_\theta^{(n)},$$
where $R_\theta^iy=y_j$ for $i\in\theta_j$.
Here $|\theta_i|$ denotes the number of elements of the set $\theta_i$.
Recalling that the sets $X_\theta^{(n)}$ with $\theta\in\Pi(n)$ form a partition of $X^{n}$, we define a measure $m_\nu^{(n)}$ on $X^n$ such that the restriction of $m_\nu^{(n)}$ to each $X_\theta^{(n)}$ is equal to $m_{\nu,\,\theta}^{(n)}$. Note that the restriction of
$m_\nu^{(n)}$ to $X^{(n)}$ is equal to $n!\, m^{\otimes n}$.
For example, for $n=2$, we get
\begin{align*}
\int_{X^2}f^{(2)}(x_1,x_2)\,m_\nu^{(2)}(dx_1\times dx_2)
&=\int_{\{x_1\ne x_2\}}f^{(2)}(x_1,x_2)\,dx_1\,dx_2\, 2+\int_X f^{(2)}(x,x)\, dx\, c_2\\
&=\int_{X^2}f^{(2)}(x_1,x_2)\,dx_1\,dx_2\, 2+\int_X f^{(2)}(x,x)\, dx\, c_2.
\end{align*}
\subsection{An extended anyon Fock space}\label{guyf7r}
Let us recall that, in subsec.~\ref{hfgyufzua}, see in particular \eqref{uyr57eses}, we defined the notion of a $Q$-symmetric function $f^{(n)}:X^{(n)}\to\mathbb C$. Our next aim is to extend this notion to a complex-valued function
defined on the whole $X^n$.
Let us fix a permutation $\pi\in \mathfrak S_n$
and a partition $\theta=\{\theta_1,\dots,\theta_l\}\in\Pi(n)$ satisfying \eqref{ft6e}. The permutation $\pi$ maps the partition $\theta$ into a new partition
$$\{\pi\theta_1,\dots,\pi\theta_l\}\in\Pi(n).$$
We call this new partition $\beta=\{\beta_1,\dots,\beta_l\}$, where the elements of the partition $\beta $ are enumerated in such a way that
\begin{equation}\label{adiov}\max\beta_1<\max\beta_2<\dots<\max\beta_l.\end{equation}
Thus, the permutation $\pi\in \mathfrak S_n$ identifies a permutation $\widehat \pi\in \mathfrak S_l$ (dependent on $\theta$) such that
\begin{equation}\label{huyfr7i5er}\pi\theta_i=\beta_{\widehat \pi(i)},\quad i=1,\dots,l.\end{equation}
Recall the complex-valued function $Q_\pi(x_1,\dots,x_n)$ on $X^{(n)}$ defined by \eqref{y7e57ie}. We will now extend this function to the whole set $X^{n}$ as follows. We fix any $\theta=\{\theta_1,\dots,\theta_l\}\in\Pi(n)$ satisfying \eqref{ft6e} and any $(x_1,\dots,x_n)\in X_\theta^{(n)}$. We
denote by $x_{\theta_1},x_{\theta_2},\dots,x_{\theta_l}$ the elements $x_{i_1},x_{i_2},\dots,x_{i_l}$ with $i_1\in\theta_1, i_2\in\theta_2,\dots,i_l\in\theta_l$, respectively.
We set
\begin{equation}\label{uyr75rw} Q _\pi(x_1,\dots,x_n):= \prod_{\substack{1\le i<j\le l\\ \widehat\pi(i)>\widehat \pi(j)}} Q(x_{\theta_i},x_{\theta_j}),\end{equation}
where the permutation $\widehat\pi\in\mathfrak S_l$ is as above.
Note that, for the partition
$$\theta=\big\{\{1\},\{2\},\dots,\{n\}\big\},$$ the restriction of the function $ Q_\pi$ to the set $X_\theta^{(n)}=X^{(n)}$ is indeed equal to the function $Q_\pi$ defined by \eqref{y7e57ie}.
We will say that a function $f^{(n)}:X^n\to\mathbb C$ is $Q$-symmetric if, for each permutation $\pi\in\mathfrak S_n$,
\begin{equation}\label{ft7ier5ird} f^{(n)}(x_1,\dots,x_n)= Q_\pi(x_1,\dots,x_n)f^{(n)}(x_{\pi^{-1}(1)},\dots,x_{\pi^{-1}(n)}),\quad (x_1,\dots,x_n)\in X^n.\end{equation}
In particular, the restriction of such a function to $X^{(n)}$ is then $Q$-symmetric according to our definition in
subsec.~\ref{hfgyufzua}, i.e., it satisfies \eqref{uyr57eses}.
Next, for a function $f^{(n)}:X^n\to\mathbb C$, we define
\begin{multline} (\Sym_n\, f^{(n)})(x_1,\dots,x_n)\\=\frac1{n!}\sum_{\pi\in \mathfrak S_n}
Q_\pi(x_1,\dots,x_n)f^{(n)}(x_{\pi^{-1}(1)},\dots,x_{\pi^{-1}(n)}),\quad (x_1,\dots,x_n)\in X^n.\label{tr866}\end{multline}
Clearly, the restriction of the function $\Sym_n\,f^{(n)}$
to the set $X^{(n)}$ is still given by \eqref{oitr8o}.
We denote by $\mathbf F^Q_n(\mathcal H,\nu)$ the subspace of the complex $L^2$-space $L^2(X^n,m_\nu^{(n)})$ which consists of ($m_\nu^{(n)}$-versions of) $Q$-symmetric functions.
\begin{proposition}\label{yfd6rd6} For each $n\in\mathbb N$, $\Sym_n$ is the orthogonal projection of $L^2(X^n,m_\nu^{(n)})$ onto $\mathbf F^Q_n(\mathcal H,\nu)$.
\end{proposition}
We also set $\mathbf F^{ Q}_{0}(\mathcal H,\nu)=\{c\Omega\mid c\in\mathbb C\}$, where $\Omega$ is the vacuum vector. We define an extended anyon Fock space
$$\mathbf F^{Q}(\mathcal H,\nu):=\bigoplus_{n=0}^\infty
\mathbf F^{Q}_{n}(\mathcal H,\nu).$$
If the measure $\nu$ is concentrated at one point (and so $c_1=1$ and $c_k=0$ for $k\ge2$), we get $\mathbf F^{Q}(\mathcal H,\nu)=\mathcal F^Q(\mathcal H)$, i.e., $\mathbf F^{Q}(\mathcal H,\nu)$ is the usual anyon Fock space. Otherwise, $\mathcal F^Q(\mathcal H)$ is a proper subspace of $\mathbf F^{\mathbf Q}(\mathcal H,\nu)$. Indeed, recalling formula \eqref{rtsew6uwy}, we may embed $\mathcal F^Q(\mathcal H)$ into $\mathbf F^{Q}(\mathcal H,\nu)$ by identifying each function $f^{(n)}\in\mathcal H^{\cd n}$ with the function from $\mathbf F^{Q}_{ n}(\mathcal H,\nu)$ which is equal to $f^{(n)}$ on $X^{(n)}$, and to 0 otherwise. Evidently, the orthogonal complement to $\mathcal F^Q(\mathcal H)$ in $\mathbf F^{\mathbf Q}(\mathcal H,\nu)$ is a non-zero space in this case.
Using the orthogonal decomposition \eqref{tyr65}, we will now construct a unitary isomorphism between $L^2(\tau)$ and the extended anyon Fock space $\mathbf F^{\mathbf Q}(\mathcal H,\nu)$.
\begin{theorem}\label{utu8}
Let $f^{(n)},g^{(n)}\in C_0(X^n)$. Then
\begin{equation}\label{gilyr7e5if}
\big(\la P_n(\omega),f^{(n)}\ra , \,\la P_n(\omega),g^{(n)}\ra\big)_{L^2(\tau)}=(\Sym_n\,f^{(n)},\Sym_n\, g^{(n)})_{\mathbf F_{ n}^{Q}(\mathcal H,\nu)}.
\end{equation}
\end{theorem}
Since the set $C_0(X^n)$ is dense in $L^2(X^n,m_\nu^{(n)})$, Theorem~\ref{utu8} implies that we can extended the mapping
$$C_0(X^n)\ni f^{(n)}\mapsto \la P_n(\omega),f^{(n)}\ra\in L^2(\tau)$$
to a linear continuous operator
$$L^2(X^n,m_\nu^{(n)})\ni f^{(n)}\mapsto \la P_n(\omega),f^{(n)}\ra\in L^2(\tau).$$
Note that, by Theorem~\ref{utu8}, for each $f^{(n)}\in L^2(X^n,m_\nu^{(n)})$,
$$\la P_n(\omega),f^{(n)}\ra=\la P_n(\omega),\Sym_nf^{(n)}\ra.$$
Thus, Theorem \ref{utu8} immediately implies
\begin{corollary}\label{igt87}
We have a unitary isomorphism
\begin{equation}\label{yut8t}
\mathbf F^{Q}(\mathcal H,\nu)\ni (f^{(n)})_{n=0}^\infty
\mapsto f^{(0)}+\sum_{n=1}^\infty \la P_n(\omega), f^{(n)}\ra\in L^2(\tau).\end{equation}
\end{corollary}
We denote the inverse of the unitary operator in \eqref{yut8t} by $U$. Thus, $U:L^2(\tau)\to \mathbf F^{Q}(\mathcal H,\nu)$ is a unitary operator; compare with Theorem~\ref{uyt8t8} (ii) in the boson case, $q=1$.
\subsection{Anyon L\'evy white noise as a Jacobi field}
In view of subsec.~\ref{utfr57rtf} and Corollary~\ref{igt87}, we have the following chain of unitary operators:
$$ \mathbf F^{Q}(\mathcal H,\nu)\overset {\text{}\ U}{\leftarrow} L^2(\tau)\overset I\to \mathcal F^Q(\mathcal G).$$
We also define a unitary operator
$$\mathbf U:\mathbf F^{Q}(\mathcal H,\nu)\to \mathcal F^Q(\mathcal G),\quad \mathbf U:=IU^{-1}.$$
Let $h\in C_0(X)$. Recall formula \eqref{tyr75}, which says that, under $I^{-1}$, the operator $\la\omega,h\ra$ in
$\mathcal F^Q(\mathcal G) $ becomes the operator of left multiplication by $\la\omega,h\ra$ in $L^2(\tau)$.
We denote
\begin{equation}\label{utr7r}
\mathbf J(h):=\mathbf U^{-1}\la\omega,h\ra \mathbf U.
\end{equation}
Obviously, the operators $\mathbf J(h)$ form a Jacobi field in the extended anyon Fock space $\mathbf F^{Q}(\mathcal H,\nu)$, i.e., each operator $\mathbf J(h)$ has a representation
\begin{equation}\label{iut78}
\mathbf J(h)=\mathbf J^+(h)+\mathbf J^0(f)+\mathbf J^-(h),\end{equation}
where $\mathbf J^+(h)$ is a creation operator, $\mathbf J^0(h)$ is a neutral operator, and $\mathbf J^-(h)$ is an annihilation operator. Equivalently, we have
$$\la \omega,h\ra\la P_n(\omega),f^{(n)}\ra=\la P_{n+1}(\omega),\mathbf J^+(h) f^{(n)}\ra +\la P_{n}(\omega),\mathbf J^0(h) f^{(n)}\ra
+\la P_{n-1}(\omega),\mathbf J^-(h) f^{(n)}\ra. $$
Our next aim is to explicitly calculate the operators $\mathbf J^\sharp(h)$, $\sharp=+,0,-$.
We define a linear space $\mathcal F_{\mathrm{fin}}(B_0(X))$ of all finite vectors $(f^{(0)},f^{(1)},\dots,f^{(n)},0,0,\dots)$, where $f^{(0)}\in\mathbb C$, $f^{(i)}\in B_0(X^i)$, $i\ge1$. Evidently, the vacuum vector, $\Omega$, belongs to $\mathcal F_{\mathrm{fin}}(B_0(X))$.
For each $h\in C_0(X)$, we define a neutral operator $\mathscr J^0(h)$ and an annihilation operator $\mathscr J_1^-(h)$
acting on $\mathcal F_{\mathrm{fin}}(B_0(X))$ as follows. We first set
\begin{equation}\label{uiti}
\mathscr J^0(h)\Omega=
\mathscr J_1^-(h)\Omega:=0.\end{equation}
Next,
\begin{equation}\label{ir688u}\mathscr (\mathscr J^0(h)f^{(n)} )(x_1,\dots,x_n):=
\sum_{i=1}^n h(x_i)f^{(n)}(x_1,\dots,x_n)R^{(n)}_i(x_1,\dots,x_n).\end{equation}
Here, for each $\theta=\{\theta_1,\dots,\theta_l\}\in\Pi(n)$, the restriction of the function $R^{(n)}_i:X^n\to\R$ to the set $X_\theta^{(n)}$ is given by
\begin{equation}\label{ho9t97tg}R_i^{(n)}\restriction X_\theta^{(n)} :=b_{\gamma(i,\theta)-1}\,/\gamma(i,\theta)\end{equation}
In formula \eqref{ho9t97tg}, $\gamma(i,\theta):=|\theta_u|$ with
$\theta_u\in\theta$ being chosen so that $i\in \theta_u$, and $(b_k)_{k=0}^\infty$ are the coefficients from \eqref{hdtrss}. Finally,
\begin{multline}\label{igftr7y}(\mathscr J_1^-(h)f^{(n)})(x_1,\dots,x_{n-1})
\\:=
\sum_{1\le i<j\le n}h(x_{j-1})f^{(n)}(x_1,\dots,x_{i-1},
\underbrace{x_{j-1}}_{\text{$i$-th place}},x_i,x_{i+1},\dots,\underbrace{x_{j-1}}_{\text{$j$-th place}},\dots,x_{n-1})\\
\times
S^{(n)}_{j-1}(x_1,\dots,x_{n-1}),\end{multline}
where for any $\theta\in\Pi(n-1)$
\begin{equation}\label{oduhuaijo}S_{j-1}^{(n)}\restriction X_\theta^{(n-1)} :=\frac{
2a_{\gamma(j-1,\theta)}}{\gamma(j-1,\theta)(\gamma(j-1,\theta)+1)}\,.\end{equation}
Here $(a_k)_{k=1}^\infty$ are also the coefficients from \eqref{hdtrss}.
We define
$$ \mathbf F_{\mathrm{fin}}^{Q}(B_0(X)):=\Sym\, \mathcal F_{\mathrm{fin}}(B_0(X)),$$
where $\Sym$ is the linear operator on $\mathcal F_{\mathrm{fin}}(B_0(X))$ satisfying $\Sym f^{(n)}:=\Sym_n f^{(n)}$ for $f^{(n)}\in B_0(X^n)$. We also denote $\mathbf B_0^Q(X^n):=\Sym_n B_0(X^n)$.
On $\mathbf F_{\mathrm{fin}}^{Q}(B_0(X))$, we define a $Q$-symmetric tensor product by setting, for any $f^{(m)}\in \mathbf B_0^Q(X^m)$, $g^{(m)}\in \mathbf B_0^Q(X^n)$,
\begin{equation}\label{vytfr7i6ed}
f^{(m)} \cd g^{(n)}:=\Sym_{m+n} (f^{(m)} \otimes g^{(n)}), \end{equation}
and extending it by linearity.
Here $f^{(m)}\otimes g^{(n)}\in B_0(X^{m+n})$ is given by
$$ (f^{(m)}\otimes g^{(n)})(x_1,\dots,x_{m+n})=f^{(m)}(x_1,\dots,x_m)g^{(n)}(x_{m+1},\dots,x_{m+n}).$$
We will prove below that the tensor product $\cd$ is associative. Furthermore, the restriction of $f^{(m)} \cd g^{(n)}$ to $X^{(m+m)}$ evidently coincides with $f^{(m)} \cd g^{(n)}$ as defined in subsec.~\ref{hfgyufzua}.
\begin{theorem}\label{fu7r7} For each $h\in C_0(X)$,
$\mathbf J(h)$ is a linear operator on $\mathbf F_{\mathrm{fin}}^{ Q}(B_0(X))$ which has representation
\eqref{iut78}.
For each $F\in \mathbf F_{\mathrm{fin}}^{Q}(B_0(X))$, we have
\begin{align*}
\mathbf J^+(h) F&=h\cd F,\\
\mathbf J^0(h)F&=\Sym(\mathscr J^0(h)F),
\end{align*}
and
\begin{equation}\label{yur75}\mathbf J^-(h)=\mathbf J_1^-(h)+\mathbf J_2^-(h).
\end{equation}
Here,
$$\mathbf J^-_1(h) F=\Sym (\mathscr J_1^-(h)F)$$
and for each $f^{(n)}\in \mathbf B_0^Q(X^n)$
\begin{equation}
(\mathbf J_2^-(h)f^{(n)})(x_1,\dots,x_{n-1})=n\int_X dy\, h(y)f^{(n)}(y,x_1,\dots,x_{n-1}).\label{ufcxyf}\end{equation}
\end{theorem}
\subsection{A characterization of Meixner-type polynomials}
Recall that the operators $\mathscr J^0(h)$ and $\mathscr J^-(h)$ were defined by using the coefficients of the recursion relation \eqref{hdtrss} (i.e., by the measure $\nu$), and these operators do not depend on the type of anyon statistics, i.e., they are independent of $Q$.
Recall the set of orthogonalized continuous polynomials, $\mathscr{OCP}$, defined in subsec.~\ref{utfr57rtf}. Let us consider the following condition.
\begin{enumerate}
\item[(C)] For each $h\in C_0(X\mapsto\R)$, the linear operators $\mathbf J^0(h)$ and $\mathbf J^-_1(h)$ map the set $\mathscr{OCP}$ into itself.
\end{enumerate}
\begin{theorem}\label{urr8r} Assume that either $q\ne- 1$ or $q=-1$ and the support of the measure $\nu$ does not consist of exactly two points. Then condition {\rm (C)} is satisfied if and only if there exist constants $\lambda\in\mathbb R$ and $\eta\ge0$ such that the coefficients $a_k$, $b_k$ in the recursion formula \eqref{hdtrss} are given by
\begin{equation}\label{vggyufd7u}a_k=\eta k(k+1)\quad
(k\in\mathbb N),\quad
b_k=\lambda(k+1)\quad (k\in\mathbb N_0).\end{equation}
In the latter case, for any $h, f_1,\dots,f_n\in C_0(X)$, we have
\begin{align}
&\mathbf J(h) f_1\cd\dots\cd f_n=
h\cd f_1\cd\dots\cd f_n\notag\\
&\quad\text{}+\lambda
\sum_{i=1}^n f_1\cd \dots\cd f_{i-1}\cd (hf_i)\cd f_{i+1}\cd \dots\diamond f_n\notag\\
&\quad\text{}+2\eta\sum_{1\le i<j\le n} f_1\cd \dots\cd
f_{i-1}\cd f_{i+1}\cd \dots \cd f_{j-1}\cd
(hf_if_j)\cd f_{j+1}\cd\dots\cd f_n\notag\\
&\quad\text{} + n \int_X dy\, h(y)(f_1\cd\dots\cd f_n)(y,\cdot).\label{yufu7edseaa}
\end{align}
\end{theorem}
We see that, in the classical case, $q=1$, Theorem~\ref{urr8r} gives exactly the Meixner class of infinite dimensional polynomials, discussed in subsec.~\ref{fytry}. Note that the obtained class of the measures $\nu$ is independent on $q$. So, for such a choice of $\nu$, we call $\big(\la P_n(\omega),f^{(n)}\ra\big)$ a Meixner-type system of orthogonal (noncommutative) polynomials for anyon statistics.
\begin{remark}
In the fermion case ($q=-1$), if the support of the measure $\nu$ consists of exactly two points, we could not
prove that condition (C) always fails, but we conjecture this indeed to be the case.
\end{remark}
The following result can be easily proven.
\begin{proposition}\label{ytfr7} For each $q\in\mathbb C$, $|q|=1$ we have equality $\mathscr{CP}=\mathscr{OCP}$ in the anyon Gaussian/Poisson case, i.e., when formula \eqref{vggyufd7u} holds with $\lambda\in\R$ and $\eta=0$.
\end{proposition}
However, due to the form of the operator $\mathbf J^-_2(h)$, see \eqref{ufcxyf}, equality $\mathscr{CP}=\mathscr{OCP}$ fails if $q\ne1$ and the the measure $\nu$ is not concentrated at one point. Still, in the classical case, $q=1$,
Theorem~\ref{urr8r} implies Theorem~\ref{ur7o67r6}.
\subsection{Anyon Meixner-type white noise}\label{ufuktydr}
We will assume in this subsection that \eqref{vggyufd7u} holds.
We may, at least informally, define,
$$\omega(x)=\la\omega,\delta_x\ra,\quad x\in X,$$
so that for $h\in C_0(X)$,
\begin{equation}\label{yf7r}
\la \omega,h\ra=\int_X dx\, \omega(x)h(x).\end{equation}
Hence, we may think of $(\omega(x))_{x\in X}$
as an anyon Meixner-type white noise.
For $x\in X$, we define an annihilation operator $\partial_x$ as the linear operator acting on
$\mathbf F^{Q}_{\mathrm{fin}}(B_0(X))$ by the formula:
\begin{equation}\label{kbhfdvyd}(\partial_x f^{(n)})(x_1,\dots,x_{n-1}):= n f^{(n)}(x,x_1,\dots,x_{n-1}),\quad
(x_1,\dots,x_{n-1})\in X^{n-1},
\end{equation}
for $f^{(n)}\in \mathbf B_0^{Q}(X^n)$. Then, by \eqref{ufcxyf}, for $h\in C_0(X)$, we may interpret the operator $\mathbf J_2^-(h)$ as the integral
\begin{equation}\label{adiohdsrtdv} \mathbf J_2^-(h)=\int_X dx\, h(x)\partial_x\,. \end{equation}
Next, we introduce an `operator-valued distribution'
$X\ni x\mapsto\partial_x^\dag$ so that, for any $h\in C_0(X)$ and $f^{(n)}\in \mathbf B_0^{Q}(X^n)$,
\begin{equation}\label{hbioghogy}
\int_X dx\, h(x)\partial_x^\dag f^{(n)}:
=h \cd f^{(n)}.
\end{equation}
In other words, we may think $\partial_x^\dag f^{(n)}=\delta_x\cd f^{(n)}$. Thus,
\begin{equation}\label{ytdee67y}
\mathbf J^+(h)=\int_X dx\, h(x)\partial_x^\dag\,.
\end{equation}
For $h\in C_0(X)$, we will now need operators
$$\int_X dx\, h(x) \partial_x^\dag \partial_x,\qquad \int_X dx\, h(x)\partial_x^\dag \partial_x\partial_x$$
acting on $\mathbf F^{Q}_{\mathrm{fin}}(B_0(X))$.
In view of \eqref{kbhfdvyd} and \eqref{hbioghogy}, we have, for each $f^{(n)}\in \mathbf B_0^{Q}(X^n)$,
\begin{align*}
\bigg(\int_X dx\, h(x) \partial_x^\dag \partial_x\,f^{(n)}\bigg)(x_1,\dots,x_n)&=n\bigg(\int_X dx\, h(x)\di_x^\dag f^{(n)}(x,\cdot)\bigg)(x_1,\dots,x_n)\\
&=n\,\Sym_n\big(h(x_1)f^{(n)}(x_1,x_2,\dots,x_n)\big)
\end{align*}
(compare with \eqref{jfiu}), and
\begin{align}
&\bigg(\int_X dx\, h(x) \partial_x^\dag \partial_x\di_x\,f^{(n)}\bigg)(x_1,\dots,x_{n-1})\notag\\
&\quad =n(n-1)\bigg(\int_X dx\, h(x)\di_x^\dag f^{(n)}(x,x,\cdot)\bigg)(x_1,\dots,x_{n-1})\notag\\
&\quad =n(n-1)\,\Sym_{n-1}\big(h(x_1)f^{(n)}(x_1,x_1,x_2,x_3,\dots,x_{n-1})\big).\label{igiugygg}
\end{align}
\begin{theorem}\label{hfu8fr78} Assume \eqref{vggyufd7u} holds. Then for $h\in C_0(X)$,
\begin{align}
\mathbf J^0(h)&=\int_X dx\, h(x)\lambda \partial_x^\dag \partial_x,\label{vfyter}\\
\mathbf J^-_1(h)&=\int_X dx\, h(x)\eta \partial_x^\dag \partial_x\partial_x\,.\label{bkvgutfgi}
\end{align}
Thus,
\begin{equation}\label{ydfyd}\mathbf J(h)=\int_X dx\, h(x)(\partial_x^\dag+\lambda\partial_x^\dag \partial_x +\eta \partial_x^\dag \partial_x\partial_x+\partial_x).\end{equation}
\end{theorem}
In view of formula \eqref{utr7r}, the operator $\mathbf J(h)$ is a realization of the operator $\la \omega,h\ra$ in the extended anyon Fock space $\mathbf F^{Q}(\mathcal H,\nu)$. So, with an abuse of notation, we may denote $\mathbf J(h)$ by
$\la \omega,h\ra$.
Then, by \eqref{yf7r} and \eqref{ydfyd}, we get the following representation of the anyon Meixner-type white noise (realized in the extended anyon Fock space $\mathbf F^{Q}(\mathcal H,\nu)$):
$$ \omega(x)=\partial_x^\dag+\lambda\partial_x^\dag \partial_x +\eta \partial_x^\dag \partial_x\partial_x+\partial_x.$$
\begin{remark} We note that, for $q$-commutation relations with $q$ being real from either the interval $(-1,0)$ or the interval $(0,1)$ \cite{BS,BKS,A1}, there is no analog of a $q$-L\'evy process which would have a representation like in \eqref{ydfyd}. Nevertheless, as shown in \cite{BW}, there exist classical Markov processes whose transition probabilities are measures of orthogonality for $q$-Meixner (orthogonal) polynomials on the real line.
\end{remark}
\section{Proofs}\label{ur867}
\subsection{Proof of Proposition}\ref{yfd6rd6}
Note that $\Sym_1=\mathbf 1$, so we need to prove the statement for $n\ge2$. Following \cite{BLW}, let us first briefly recall how one shows that the operator $\Sym_n$ given by
\eqref{oitr8o} is an orthogonal projection in the space $L^2(X^n,m^{\otimes n})$.
For each $\pi\in \mathfrak S_n$, we define
\begin{equation}\label{hvtd7i5} (\Psi_\pi f^{(n)})(x_1,\dots,x_n)=Q_\pi(x_1,\dots,x_n)f^{(n)}(x_{\pi^{-1}(1)},\dots,x_{\pi^{-1}(n)})\end{equation} for $(x_1,\dots,x_n)\in X^{(n)}$. Thus,
$ \Sym_n=\frac1{n!}\sum_{\pi\in\mathfrak S_n}\Psi_{\pi}$.
We then have
$\Psi_\pi^*=\Psi_{\pi^{-1}}$,
which implies $\Sym_n^*=\Sym_n$. Furthermore, for each permutation $\varkappa\in \mathfrak S_n$, we have
\begin{equation}\label{giyr86ort}
\Psi_\pi\Psi_\varkappa=\Psi_{\varkappa\pi}.
\end{equation}
Therefore, on $X^{(n)}$,
\begin{equation}\label{ft7e6ws} \Sym_n^2=\frac1{(n!)^2}\sum_{\pi,\varkappa\in \mathfrak S_n}\Psi_\pi\Psi_\varkappa =\frac1{(n!)^2}\sum_{\pi\in\mathfrak S_n}
\sum_{\varkappa\in \mathfrak S_n}\Psi_{\pi\varkappa}
=\frac1{n!}\sum_{\varkappa\in \mathfrak S_n}\Psi_{\varkappa}=\Sym_n.
\end{equation}
Thus, $\Sym_n$ is an orthogonal projection.
Note that formula \eqref{giyr86ort} implies that, for $\varkappa,\pi\in\mathfrak S_n$,
\begin{multline}\label{jgi86rt} \left(
\prod_{\substack{1\le i<j\le n\\ \pi(i)>\pi(j)}}\!\!\! Q(x_i,x_j)
\right)
\left(
\prod_{\substack{1\le k<l\le n\\ \varkappa(k)>\varkappa(l)}}\!\!\!Q(x_{\pi^{-1}(k)},x_{\pi^{-1}(l)})
\right)\\
=
\prod_{\substack{1\le i<j\le n\\ (\varkappa\pi)(i)> (\varkappa\pi)(j)}} \!\!\!\!\!\!Q(x_i,x_j),\quad (x_1,\dots,x_n)\in X^{(n)}.
\end{multline}
Now let us consider the linear, bounded operator $\Sym_n$ in $L^2(X^n,m_\nu^{(n)})$. We represent the operator $\Sym_n$ as
\begin{equation}\label{gygfy}\Sym_n=\frac1{n!}\sum_{\pi\in \mathfrak S_n}\Psi_{\pi},\end{equation}
with $ \Psi_\pi f^{(n)}$ being defined on the whole $X^n$ by the right hand side of formula \eqref{hvtd7i5} in which the function $Q_\pi(x_1,\dots,x_n)$ on $X^n$ is
defined in subsec.~\ref{guyf7r}.
We fix a permutation $\pi\in \mathfrak S_n$
and a partition $\theta=\{\theta_1,\dots,\theta_l\}\in\Pi(n)$ satisfying \eqref{ft6e}, and let \eqref{adiov}, \eqref{huyfr7i5er} hold. Further, let $\varkappa\in\mathfrak S_n$ and let $\zeta=\{\zeta_1,\dots,\zeta_l\}\in\Pi(n)$ be such that
\begin{equation}\label{iut8o6t8}\max\zeta_1<\max\zeta_2<\dots<\max\zeta_l,\end{equation}
and
$$\varkappa \beta_i=\zeta_{\widehat\varkappa(i)},\quad i=1,\dots,l,$$
where $\widehat\varkappa\in \mathfrak S_l$.
Then, for each function $f^{(n)}:X^n\to\mathbb C$ and $(x_1,\dots,x_n)\in X^n$,
we have
\begin{equation}\label{uyr5e67e} (\Psi_\pi \Psi_\varkappa f^{(n)})(x_1,\dots,x_n)=\left(
\prod_{\substack{1\le i<j\le l\\ \widehat \pi(i)>\widehat \pi(j)}}\!\!\! Q(x_{\theta_i},x_{\theta_j})
\right)( \Psi_\varkappa f^{(n)})(x_{\pi^{-1}(1)},\dots,x_{\pi^{-1}(n)}).\end{equation}
Denote $y_i=x_{\pi^{-1}(i)}$, or equivalently $y_{\pi(i)}=x_i$ for $i=1,\dots,n$. Thus, $y_{\pi(i)}=y_{\pi(j)}$ if and only if $i$ and $j$ belong to the same element of the partition $\theta$. Equivalently, $y_i=y_j$ if and only if $i$ and $j$ belong to the same element of the partition $\beta$. Therefore,
$$ (\Psi_\varkappa f^{(n)})(y_1,\dots,y_n)=\left(
\prod_{\substack{1\le u<v\le l\\ \widehat \varkappa(u)>\widehat \varkappa(v)}}\!\!\! Q(y_{\beta_u},y_{\beta_v})
\right)f^{(n)}(y_{\varkappa^{-1}(1)},\dots,y_{\varkappa^{-1}(n)}). $$
Hence,
\begin{align} &( \Psi_\varkappa f^{(n)})(x_{\pi^{-1}(1)},\dots,x_{\pi^{-1}(n)})\notag\\
&\quad =\left(
\prod_{\substack{1\le u<v\le l\\ \widehat \varkappa(u)>\widehat \varkappa(v)}}\!\!\! Q(x_{\pi^{-1}\beta_u},x_{\pi^{-1}\beta_v})
\right)f^{(n)}(x_{\pi^{-1}\varkappa^{-1}(1)},\dots,x_{\pi^{-1}\varkappa^{-1}(n)})\notag\\
&\quad = \left(
\prod_{\substack{1\le u<v\le l\\ \widehat \varkappa(u)>\widehat \varkappa(v)}}\!\!\! Q(x_{
\theta_{\widehat \pi^{-1}(u)}
},x_{
\theta_{\widehat \pi^{-1}(v)}
})
\right)f^{(n)}(x_{(\varkappa\pi)^{-1}(1)},\dots,x_{(\varkappa\pi)^{-1}(n)}),\label{gyu7r754}
\end{align}
where we used the observation that, for each $u=1,\dots,l$,
$$ \pi^{-1}\beta_u=\theta_{\widehat\pi^{-1}(u)}.$$
Using \eqref{jgi86rt}, we get
\begin{equation}\label{uiytr8o6}
\left(
\prod_{\substack{1\le i<j\le l\\ \widehat \pi(i)>\widehat \pi(j)}}\!\!\! Q(x_{\theta_i},x_{\theta_j})
\right)\left(
\prod_{\substack{1\le u<v\le l\\ \widehat \varkappa(u)>\widehat \varkappa(v)}}\!\!\! Q(x_{
\theta_{\widehat \pi^{-1}(u)}
},x_{
\theta_{\widehat \pi^{-1}(v)}
})
\right)= \left(
\prod_{\substack{1\le i<j\le l\\ \widehat{\varkappa\pi}(i)>\widehat {\varkappa\pi}(j)}}\!\!\! Q(x_{\theta_i},x_{\theta_j})
\right).
\end{equation} Here $\widehat{\varkappa\pi}$ is the permutation from $\mathfrak S_l$ induced by the permutation $\varkappa\pi\in \mathfrak S_n$ and the partition $\theta$. In \eqref{uiytr8o6}, we used the observation that
$\widehat{\varkappa\pi}=\widehat\varkappa\,\widehat\pi$.
Now, substituting \eqref{gyu7r754} into \eqref{uyr5e67e}
and using \eqref{uiytr8o6}, we conclude that
\begin{equation}\label{gciugfi} \Psi_\pi \Psi_\varkappa= \Psi_{\varkappa\pi},\end{equation} and hence analogously to \eqref{ft7e6ws}, we get $\Sym_n^2=\Sym_n$.
Next, we note that the measure $m^{(n)}_\nu$
remains invariant under the transformation
$$ X^n\ni (x_1,\dots,x_n)\mapsto (x_{\pi^{-1}(1)},\dots x_{\pi^{-1}(n)})\in X^n.$$ Furthermore, as easily seen, the equality
$$ \overline{Q_{\pi^{-1}}(x_{\pi^{-1}(1)},\dots,x_{\pi^{-1}(n)})}= Q_\pi(x_{1},\dots,x_{n})$$
holds for each $(x_1,\dots,x_n)\in X^n$. Hence, for each $\pi\in \mathfrak S_n$, $ \Psi_\pi^*= \Psi_{\pi^{-1}}$, which implies $\Sym_n^*=\Sym_n$.
Thus, $\Sym_n$ is an orthogonal projection in $L^2(X^n,m^{(n)}_\nu)$. Analogously to \cite[Proposition 2.5]{BLW}, we easily conclude that the image of $\Sym_n$is indeed $\mathbf F^Q_n(\mathcal H,\nu)$. Thus, Proposition~\ref{yfd6rd6} is proven.
Recall the tensor product $\cd$ defined on
$\mathbf F_{\mathrm{fin}}^{Q}(B_0(X))$ by formula \eqref{vytfr7i6ed}. Using \eqref{gygfy} and \eqref{gciugfi}, it is easy to show that, for any $f^{(m)}\in B_0(X^m)$ and $g^{(n)}\in B_0(X^n)$, we have
\begin{align*}(\Sym_m\, f^{(m)})\cd (\Sym_n\, g^{(n)})&=\Sym_{m+n}( (\Sym_m \,f^{(m)})\otimes (\Sym_n\, g^{(n)}))\\
&=\Sym_{m+n}(f^{(m)}\otimes g^{(n)}).\end{align*}
Therefore, the tensor product $\cd$ is associative on $\mathbf F_{\mathrm{fin}}^{Q}(B_0(X))$.
\subsection{Proof of Theorem \ref{utu8}}
Recall the unitary operator $I:L^2(\tau)\to\mathcal F^Q(\mathcal G)$. Our next aim is to obtain an explicit form of the subspace $I(\mathscr{OP}_n)$ of $\mathcal F^Q(\mathcal G)$.
Denote by $\mathbb N_{0,\,\mathrm{fin}}^\infty$ the set of all infinite sequences $\alpha=(\alpha_0,\alpha_1,\alpha_2,\dots)\in\mathbb N_0^\infty$ such that only a finite number of $\alpha_j$'s are not equal to zero. Let $|\alpha|:=\alpha_0+\alpha_1+\alpha_2+\dotsm$. For each $\alpha\in \mathbb N_{0,\,\mathrm{fin}}^\infty$ with $|\alpha|\ge1$, we denote by $\mathcal F_\alpha$ the subspace of the Fock space $\mathcal F^Q(\mathcal G)$ which consists of all elements of the form
$$ \Sym_{|\alpha|}\big(f^{(|\alpha|)}(x_1,\dots,x_{|\alpha|})p_{0}(s_1)\dotsm p_{0}(s_{\alpha_0})p_{1}(s_{\alpha_0+1})
\dotsm p_{1}(s_{\alpha_0+\alpha_1})p_{2}(s_{\alpha_0+\alpha_1+1})\dotsm\big),$$
where $f^{(|\alpha|)}\in\mathcal H^{\otimes|\alpha|}$. For $\alpha\in \mathbb N_{0,\,\mathrm{fin}}^\infty$ with $|\alpha|=0$, we set $\mathcal F_\alpha:=\{c\Omega\mid c\in\mathbb C\}$.
The following proposition is proven in \cite[Section 7]{BLW}.
This result is a counterpart of the Nualart--Schoutens decomposition of the $L^2$-space of a classical L\'evy process \cite{NS}, see also \cite{Schoutens}.
\begin{proposition}\label{ui8t868} We have
\begin{equation}\label{vytd}\mathcal F^Q(\mathcal G)=\bigoplus_{\alpha\in \mathbb N_{0,\,\mathrm{fin}}^\infty}\mathcal F_\alpha.\end{equation}
\end{proposition}
For each $n\in\mathbb N_0$, we define
$$\mathbb F_{n}:=\bigoplus_{\substack{\alpha\in \mathbb N_{0,\,\mathrm{fin}}^\infty\\
\alpha_0+2\alpha_1+3\alpha_2+\dotsm=n}} \mathcal F_\alpha.$$
Note that, by \eqref{vytd}, $$\mathcal F^Q(\mathcal G)=\bigoplus_{n=0}^\infty \mathbb F_n.$$
\begin{proposition}\label{hjvfytdc}
For each $n\in\mathbb Z_+$,
$$I \mathscr{OP}_n=\mathbb F_n.$$
\end{proposition}
\begin{proof}
It suffices to prove that, for each $n\in\mathbb N$,
\begin{equation}\label{uf7yrd}
I \mathscr{MP}_n=\bigoplus_{\substack{\alpha\in \mathbb N_{0,\,\mathrm{fin}}^\infty\\
\alpha_0+2\alpha_1+3\alpha_2+\dotsm\le n}} \mathcal F_\alpha=:\mathbb M_n.
\end{equation}
\begin{lemma}\label{tye6ue}
The space $\mathbb M_n$ consists of all finite sums of elements of the form
\begin{equation}\label{iu87re65} \Sym_k\big(f^{(k)}(x_1,\dots,x_k)s_1^{i_1}s_2^{i_2}\dotsm s_k^{i_k}\big),\end{equation}
where $f^{(k)}\in\h^{\otimes k}$ and $i_1+i_2+\dots+i_k+k\le n$.
\end{lemma}
\begin{proof} For each $\pi\in\mathfrak S_k$, we define a unitary operator $\Psi_\pi$ on $(\mathcal H\otimes L^2(\mathbb R,\nu))^{\otimes k}$ by
$$(\Psi_\pi g^{(k)})(x_1,s_1,\dots,x_k,s_k)=Q_\pi(x_1,\dots,x_k)g^{(k)}(x_{\pi^{-1}(1)},s_{\pi^{-1}(1)},\dots,x_{\pi^{-1}(k)},s_{\pi^{-1}(k)}).$$
Here the function $Q_\pi$ is defined by \eqref{y7e57ie}. Then, by \cite{BLW}, the operators $\Psi_\pi$ form a unitary representation of the symmetric group $\mathfrak S_k$, and for each $\pi\in \mathfrak S_k$ we have $\Sym_k=\Sym_k\Psi_\pi$. Hence, for any permutation $\pi\in \mathfrak S_k$, $u^{(k)}\in\mathcal H^{\otimes k}$, and any polynomial $r^{(k)}(s_1,\dots,s_k)$ in the $s_1,\dots,s_k$ variables,
$$ \Sym_k\big(
f^{(k)}(x_1,\dots,x_k)r^{(k)}(s_1,\dots,s_k)
\big)=\Sym_k\big(
u^{(k)}(x_1,\dots,x_k) r^{(k)}(s_{\pi^{-1}(1)},\dots,s_{\pi^{-1}(k)})
\big),$$
where
$$ u^{(k)}(x_1,\dots,x_k)=Q_\pi(x_1,\dots,x_k)f^{(k)}(x_{\pi^{-1}(1)},\dots,x_{\pi^{-1}(k)}).$$
In particular, $u^{(k)}\in\mathcal H^{\otimes k}$.
Noting the evident representations
$$ p_l(s)=\sum_{i=0}^l \alpha_{il}\, s^i,\quad s^l=\sum_{i=0}^l
\beta_{il}\,p_i(s),$$
we easily conclude the lemma.
\end{proof}
We now finish the proof of \eqref{uf7yrd}. Let $\mathcal F_{\mathrm{fin}}(\mathcal H\otimes\mathscr P(\mathbb R))$ be the linear subspace of the full Fock space over $\mathcal H\otimes L^2(\mathbb R,\nu)$ which consists of finite sums of $c\Omega$ ($c\in\mathbb C$) and elements of the form
\begin{equation}\label{fgufr} f^{(k)}(x_1,\dots,x_k)s_1^{i_1}s_2^{i_2}\dotsm s_k^{i_k}\end{equation}
with $f^{(k)}\in\h^{\otimes k}$, $i_1,i_2,\dots,i_k\in\mathbb Z_+$, $k\in\mathbb N$. We set
\begin{equation}\label{iuagiuai} \Sym:=\mathbf 1\oplus \Sym_1\oplus \Sym_2\oplus \Sym_3\oplus\dotsm\,.\end{equation}
This operator projects $\mathcal F_{\mathrm{fin}}(\mathcal H\otimes\mathscr P(\mathbb R))$ onto $\mathcal F_{\mathrm{fin}}^Q(\h\otimes\mathscr P(\mathbb R))$.
We have,
for each $h\in C_0(X)$ and $F\in \mathcal F_{\mathrm{fin}}(\mathcal H\otimes\mathscr P(\mathbb R))$,
\begin{gather}
a^+(h\otimes 1) \Sym F= \Sym J^+(h\otimes 1)F,\quad a^-(h\otimes 1) \Sym F= \Sym J^-(h\otimes 1)F,\label{ihoy9y6}\\
a^0(h\otimes \id) \Sym F= \Sym J^0(h\otimes \id)F.\label{gfur7}
\end{gather}
Here, for each $F$ as in \eqref{fgufr},
\begin{align}
&(J^+(h\otimes 1)F)(x_1,s_1,\dots,x_{k+1},s_{k+1})=h(x_1)1(s_1)f^{(k)}(x_2,\dots,x_{k+1})s_2^{i_1}s_3^{i_2}\dotsm s_{k+1}^{i_k},\notag\\
&(J^0(h\otimes \id)F)(x_1,s_1,\dots,x_{k},s_{k})=(h(x_1)s_1+\dots+h(x_k)s_k)f^{(k)}(x_1,\dots,x_k)s_1^{i_1}s_2^{i_2}\dotsm s_k^{i_k},\notag\\
&(J^-(h\otimes 1)F)(x_1,s_1,\dots,x_{k-1},s_{k-1})=\sum_{j=1}^k \int_X dy\int_{\mathbb R}\nu(dt)\, h(y)
Q(y,x_1)\dotsm Q(y,x_{j-1})\notag\\
&\qquad\times f^{(k)}(x_1,\dots,x_{j-1},y,x_{j},\dots,x_{k-1})s_1^{i_1}\dotsm s_{j-1}^{i_{j-1}}t^{i_j}s_j^{i_{j+1}}\dotsm s_{k-1}^{i_{j_k}}.\label{vtydy7}
\end{align}
Hence, it follows by induction from Lemma~\ref{tye6ue} and \eqref{ihoy9y6}--\eqref{vtydy7} that
$$\la\omega, h_1\ra\dotsm\la\omega, h_n\ra\Omega\subset \mathbb M_n$$
for any $h_1,\dots,h_n\in C_0(X)$, $n\in\mathbb N$. Since $\mathbb M_n$ is a closed subspace of $\mathcal F^Q(\mathcal G)$, we therefore get the inclusion $I\mathscr{MP}_n\subset \mathbb M_n$. On the other hand, it directly follows from the proof of \cite[Proposition 6.7]{BLW} that each element of $\mathbb M_n$ which has form \eqref{iu87re65} belongs to $I\mathscr{MP}_n$. Hence, we get the inverse inclusion $\mathbb M_n\subset I\mathscr{MP}_n$.
\end{proof}
Note that, for each $h\in C_0(X)$,
\begin{equation}\label{kjguyfr7u} a^0(h\otimes \id)=d\Gamma(M_{h\otimes \id})=d\Gamma(M_h\otimes M_{\id}),\end{equation}
where $M_h$ is the operator of multiplication by the function $h(x)$ in $\mathcal H$ and $M_{\id}$ is the (restriction to $\mathscr P(\mathbb R)$ of the) operator of multiplication by the monomial $\id(s)=s$ in $L^2(\R,\nu)$.
(Note that the operator $M_{\id}$ is unbounded in $L^2(\mathbb R,\nu)$ if the support of measure $\nu$ is unbounded, and the second quantization operator has domain $\mathcal F_{\mathrm{fin}}^Q(\h\otimes\mathscr P(\mathbb R))$.)
In view of the recursion formula \eqref{hdtrss},
we get the representation
$$ M_{\id}=A^++A^0+A^-,$$
where $A^+$, $A^0$, and $A^-$ are the linear operators on $\mathscr P(\mathbb R)$ given by
\begin{equation}\label{nitg} A^+p_k:=p_{k+1},\quad A^0p_k=b_kp_k,\quad A^- p_k=a_k p_{k-1}.\end{equation}
By \eqref{kjguyfr7u} and \eqref{nitg},
\begin{equation}\label{uit8rt} a^0(h\otimes \id)= d\Gamma(M_h\otimes A^+)+d\Gamma(M_h\otimes A^0)+d\Gamma(M_h\otimes A^-).
\end{equation}
By \eqref{g7r75e} and \eqref{uit8rt}, we get, for each $h\in C_0(X)$,
\begin{equation}\label{igyur7} \la \omega,h\ra=\mathcal A^+(h)+\mathcal A^0(h) +\mathcal A^-(h),\end{equation}
where
\begin{align}
\mathcal A^+(h):&=a^+(h\otimes 1)+d\Gamma(M_h\otimes A^+),\notag\\
\mathcal A^0(h):&=d\Gamma(M_h\otimes A^0),\notag\\
\mathcal A^-(h):&=a^-(h\otimes 1)+d\Gamma(M_h\otimes A^-).\label{f75re75}
\end{align}
\begin{proposition}\label{giyuf7or6o}
For each $h\in C_0(X)$, we have $\mathcal A^+(h):\mathbb F_n\to\mathbb F_{n+1}$, $\mathcal A^0(h):\mathbb F_n\to \mathbb F_n$, $\mathcal A^-(h):\mathbb F_n\to \mathbb F_{n-1}$.
\end{proposition}
\begin{proof}
Let $\sharp=+,\,0,\,-$. For each $h\in C_0(X)$, we define an operator $N(M_h\otimes A^\sharp)$ on $\mathcal F_{\mathrm{fin}}(\mathcal H\otimes\mathscr P(\mathbb R))$ by setting
$N(M_h\otimes A^\sharp)\Omega:=0$ and for each $n\in\mathbb N$,
\begin{multline*}N(M_h\otimes A^\sharp)\restriction \big(\mathcal F_{\mathrm{fin}}(\mathcal H\otimes\mathscr P(\mathbb R))\cap \mathcal G^{\otimes n}\big)\\
:=
(M_h\otimes A^\sharp)\otimes \mathbf 1\otimes\dots\otimes \mathbf 1+
\mathbf 1\otimes (M_h\otimes A^\sharp)\otimes \mathbf 1\otimes\dots\otimes \mathbf 1+\dots+\mathbf 1\otimes\dots\otimes \mathbf 1\otimes
(M_h\otimes A^\sharp).
\end{multline*}
\begin{lemma}\label{lgti8t}
Let $\sharp=+,\,0,\,-$. For any $h\in C_0(X\mapsto\mathbb R)$
and $F\in \mathcal F_{\mathrm{fin}}(\mathcal H\otimes\mathscr P(\mathbb R))$, we have
$$d\Gamma(M_h\otimes A^\sharp) \Sym F= \Sym N(M_h\otimes A^\sharp)F.$$
\end{lemma}
\begin{proof} Fix any $F\in \mathcal F_{\mathrm{fin}}(\mathcal H\otimes\mathscr P(\mathbb R))$ of the form
$$ F(x_1,s_1,\dots,x_n,s_n)=f^{(n)}(x_1,\dots,x_n)p_{i_1}(s_1)\dotsm p_{i_n}(s_n).$$ By \eqref{oitr8o},
\begin{align}
&(\Sym_nF)(x_1,s_1,\dots,x_k,s_k)\notag\\
&\quad
=\frac1{n!}\sum_{\pi\in \mathfrak S_n}Q_\pi(x_1,\dots,x_n)f^{(n)}(x_{\pi^{-1}(1)},\dots,x_{\pi^{-1}(n)})p_{i_1}
(s_{\pi(1)})\dotsm p_{i_n}(s_{\pi(n)}).\label{gyfu7e5}
\end{align}
Note that
\begin{equation}\label{vuyd7ik}
d\Gamma(M_h\otimes A^+)= \Sym N(M_h\otimes A^+).
\end{equation}
By \eqref{gyfu7e5},
\begin{align}
&(N(M_h\otimes A^+)\Sym_nF)(x_1,s_1,\dots,x_n,s_n)\notag\\
&\quad
=\frac1{n!}\sum_{j=1}^n \sum_{\pi\in \mathfrak S_n}Q_\pi(x_1,\dots,x_n)
h(x_{\pi^{-1}(j)})
f^{(n)}(x_{\pi^{-1}(1)},\dots,x_{\pi^{-1}(n)})\notag\\
&\qquad\times p_{i_1}
(s_{\pi^{-1}(1)})\dotsm p_{i_j+1}(s_{\pi^{-1}(j)})\dotsm p_{i_n}(s_{\pi^{-1}(n)})\notag\\
&\quad
=\frac1{n!}\sum_{j=1}^n \sum_{\pi\in \mathfrak S_n}Q_\pi(x_1,\dots,x_n)g_j^{(n)}(x_{\pi^{-1}(1)},s_{\pi^{-1}(1)},\dots,x_{\pi^{-1}(n)},s_{\pi^{-1}(n)}).
\label{hgit8p}
\end{align}
Here, for $j=1,\dots,n$,
$$ g_j^{(n)}(x_1,s_1,\dots,x_n,s_n):=h(x_j)f^{(n)}(x_1,\dots,x_n)p_{i_1}(s_1)\dotsm p_{i_j+1}(s_j)\dotsm p_{i_n}(s_n).$$
Then, by \eqref{vuyd7ik} and \eqref{hgit8p},
\begin{align}
&(d\Gamma(M_h\otimes A^+)\Sym_nF)(x_1,s_1,\dots,x_n,s_n)\notag\\
&\quad=
\frac1{(n!)^2}\sum_{j=1}^n \sum_{\sigma\in \mathfrak S_n}\sum_{\pi\in \mathfrak S_n}
Q_\sigma(x_1,\dots,x_n)Q_\pi(x_{\sigma^{-1}(1)},\dots,x_{\sigma^{-1}(n)}) \notag\\
&\qquad \times
g^{(n)}_j(x_{\sigma^{-1}(\pi^{-1}(1))},s_{\sigma^{-1}(\pi^{-1}(1))},\dots x_{\sigma^{-1}(\pi^{-1}(n))},s_{\sigma^{-1}(\pi^{-1}(n))}).
\notag
\end{align}
Hence,
\begin{align*}d\Gamma(M_h\otimes A^+)\Sym_nF&= \sum_{j=1}^n \Sym_n^2g_j^{(n)}=
\sum_{j=1}^n \Sym_ng_j^{(n)}
\\
&=\Sym_n\left(\sum_{j=1}^n g_j^{(n)}\right)= \Sym_n N(M_h\otimes A^+)F.\end{align*}
The proof for $A^0$ and $A^-$ is analogous.
\end{proof}
Now, the proposition follows directly from the definition of the spaces $\mathbb F^{(n)}$, formula \eqref{ihoy9y6}, and Lemma \ref{lgti8t}.
\end{proof}
\begin{proposition}\label{tyfd6ure}
For any $h_1,\dots,h_n\in C_0(X)$, we have
$$ I \la P_n(\omega),h_1\otimes\dots \otimes h_n\ra=\mathcal A^+(h_1)\dotsm\mathcal A^+(h_n)\Omega.$$
\end{proposition}
\begin{proof} Recall that $\la P_n(\omega),h_1\otimes\dots \otimes h_n\ra$ is the orthogonal projection of the monomial
$$\la h_1,\omega\ra\dotsm\la h_n,\omega\ra=\la h_1\otimes\dots\otimes h_n,\omega^{\otimes n}\ra$$ onto
$\mathscr {OP}_n$. The statement follows from Propositions \ref{hjvfytdc} and \ref{giyuf7or6o} if we note that $$I \la P_n(\omega),h_1\otimes\dots \otimes h_n\ra$$
is equal to the orthogonal projection of
$$\la \omega,h_1\ra\dotsm\la \omega,h_n\ra\Omega=
(\mathcal A^+(h_1)+\mathcal A^0(h_1)+\mathcal A^-(h_1))\dotsm
(\mathcal A^+(h_n)+\mathcal A^0(h_n)+\mathcal A^-(h_n))\Omega$$
onto $\mathbb F_n$.
\end{proof}
We will now explicitly calculate the vector $I \la P_n(\omega),h_1\otimes\dots \otimes h_n\ra$. We introduce a topology on $B_0(X^n)$ which yields the following notion of convergence: $f_n\to f$ as $n\to\infty$ means that there exists a set $\Delta\in\mathcal B_0(X)$ such that
$\operatorname{supp}(f_n)\subset\Delta$ for all $n\in\mathbb N$ and
\eqref{ilyufre75iei7} holds.
Note that $C_0(X^n)$ is a topological subspace
of $B_0(X^n)$.
For each $\theta=\{\theta_1,\dots,\theta_l\}\in\Pi(n)$ with $\theta_1,\dots,\theta_l$ satisfying
\eqref{ft6e}, we define, for $f^{(n)}\in B_0(X^n)$, $(x_1,\dots,x_l)\in X^{(l)}$, and $(s_1,\dots,s_l)\in\R^l$,
\begin{equation}\label{isgdiy}(\mathcal E_\theta f^{(n)})(x_1,s_1,\dots,x_l,s_l):=f^{(n)}_\theta
(x_1,\dots,x_l) p_{|\theta_1|-1}(s_1)p_{|\theta_2|-1}(s_2)\dotsm p_{|\theta_l|-1}(s_l).\end{equation}
Here the function $f^{(n)}_\theta
(x_1,\dots,x_l)$ is obtained from the function $f^{(n)}(y_1,\dots,y_n)$ by replacing $y_{i_1}$ with $x_1$ for all $i_1\in\theta_1$, $y_{i_2}$ with $x_2$ for all $i_2\in\theta_2$, and so on. Note that the function $f^{(n)}_\theta: X^{(l)}\to\mathbb C$ is completely identified by the restriction of the function $f^{(n)}:X^n\to\mathbb C$ to the set $X_\theta^{(n)}$.
For example, let $n=6$ and let $\theta=\{\theta_1,\theta_2,\theta_3\}\in\Pi(6)$ be of the form
$$\theta_1=\{1,3\},\quad\theta_2=\{2,4,6\},\quad\theta_3=\{5\}.$$
Then, for each $(x_1,x_2,x_3)\in X^{(3)}$ and $(s_1,s_2,s_3)\in\R^3$,
$$(\mathcal E_\theta f^{(6)})(x_1,s_1,x_2,s_2,x_3,s_3)
=f^{(6)}(x_1,x_2,x_1,x_2,x_3,x_2)p_1(s_1)p_2(s_2)p_0(s_3).$$
\begin{proposition}\label{iyr867r} For each $n\in\mathbb N$, the mapping
$$C_0(X)^n\ni(h_1,\dots,h_n)\mapsto \la P_n(\omega),h_1\otimes\dots \otimes h_n\ra\in L^2(\tau)$$
may be extended by linearity and continuity to a mapping
$$ B_0(X^n)\ni f^{(n)}\to \la P_n(\omega),f^{(n)}\ra\in L^2(\tau).$$ Furthermore,
for each $f^{(n)}\in B_0(X^n)$, we have
\begin{equation}\label{ouifr7ored} I\la P_n(\omega),f^{(n)}\ra=\Sym\left(\sum_{\theta\in \Pi(n)}
\mathcal E_\theta f^{(n)}\right).\end{equation}
\end{proposition}
\begin{proof}
Fix any $h_1,\dots,h_n\in C_0(X)$ and set $f^{(n)}(x_1,\dots,x_n)=h_1(x_1)\dotsm h_n(x_n)$. Then, by Proposition~\ref{tyfd6ure}, formula \eqref{ouifr7ored} is equivalent to
\begin{equation}
\label{ufdr6s}
\big(a^+(h_1\otimes 1)+d\Gamma(M_{h_1}\otimes A^+)\big)\dotsm
\big(a^+(h_n\otimes 1)+d\Gamma(M_{h_n}\otimes A^+)\big)\Omega
=\Sym\left(\sum_{\theta\in \Pi(n)}
\mathcal E_\theta f^{(n)}\right).
\end{equation}
By \eqref{ihoy9y6} and Lemmma~\ref{lgti8t}, formula \eqref{ufdr6s} would follow from
\begin{equation}\label{igf7r}
\big(J^+(h_1\otimes 1)+N(M_{h_1}\otimes A^+)\big)
\dotsm \big(J^+(h_n\otimes 1)+N(M_{h_n}\otimes A^+)\big)\Omega=\sum_{\theta\in \Pi(n)}
\mathcal E_\theta f^{(n)}.
\end{equation}
Let $\beta=\{\beta_1,\dots,\beta_k\}$ be an (unordered) partition of $\{i+1,i+2,\dots,n\}$. Then
\begin{equation}\label{fdtrs5} J^+(h_i\otimes 1)\mathcal E_\beta(h_{i+1}\otimes h_{i+2}\otimes\dots\otimes h_n)=\mathcal E_{\beta^+}(h_i\otimes h_{i+1}\otimes \dots\otimes h_n),\end{equation}
where $\beta^+:=\{\{i\},\beta_1,\dots,\beta_k\}$ is a partition of $\{i,i+1,\dots,n\}$. Furthermore,
\begin{equation}\label{iufy7}N(M_{h_i}\otimes A^+)\mathcal E_\beta(h_{i+1}\otimes h_{i+2}\otimes\dots\otimes h_n)=\sum_{j=1}^k \mathcal E_{\beta_j^0}(h_i\otimes h_{i+1}\otimes \dots\otimes h_n),\end{equation}
where $\beta_j^0$ is the partition of $\{i,i+1,\dots,n\}$
obtained from $\beta$ by adding $i$ to the set $\beta_j$, i.e.,
$$ \beta_j^0:=\{\beta_1,\dots,\beta_j\cup \{i\},\dots \beta_k\}.$$
By \eqref{fdtrs5} and \eqref{iufy7}, formula \eqref{igf7r} follows by induction. Finally, the extension
of formula \eqref{ouifr7ored} to the case of a general
$f^{(n)}\in B_0(X^n)$ follows by linearity and approximation.
\end{proof}
We will now prove Theorem~\ref{utu8}. Even, a bit more generally, we will prove that formula \eqref{gilyr7e5if} holds for any $f^{(n)},g^{(n)}\in B_0(X^n)$.
We first note that it suffices to prove formula \eqref{gilyr7e5if} in the case where $f^{(n)}=g^{(n)}=h_1\otimes\dots\otimes h_n$ with $h_1,\dots,h_n\in B_0(X)$.
By Proposition \ref{iyr867r},
\begin{align}
&\big(\la P_n(\omega),f^{(n)}\ra , \,\la P_n(\omega), f^{(n)}\ra\big)_{L^2(\tau)}\notag\\
&\quad =\left(\sum_{\theta\in\Pi(n)}\Sym_{|\theta|}
(\mathcal E_{\theta}f^{(n)}),\sum_{\zeta\in\Pi(n)}\Sym_{|\zeta|}
(\mathcal E_{\zeta}f^{(n)})\right)_{\mathcal F^Q(\mathcal G)}\notag\\
&\quad = \sum_{l=1}^n\sum_{\substack{\theta,\zeta\in\Pi(n)\\ |\theta|=|\zeta|=l}}\big(
\Sym_l(\mathcal E_\theta f^{(n)}),\mathcal E_\zeta f^{(n)})
\big)_{L^2((X\times \mathbb R)^l,(m\otimes \nu)^{\otimes l})}\,l!\,.\label{ouit8ot}
\end{align}
Note that, by Proposition \ref{yfd6rd6},
\begin{align}
(\Sym_n\,f^{(n)},\Sym_n\, f^{(n)})_{\mathbf F_{ n}^{Q}(\mathcal H,\nu)}&=
\int_{X^n}(\Sym_nf^{(n)})f^{(n)}\,dm^{(n)}_\nu\notag\\
&=\sum_{\zeta\in\Pi(n)} \int_{X_\zeta^{(n)}}(\Sym_n\,f^{(n)})f^{(n)}\,dm^{(n)}_{\nu,\, \zeta}\,.
\label{it89}\end{align}
By \eqref{ouit8ot} and \eqref{it89}, formula \eqref{gilyr7e5if} will follow if we show that, for a fixed $\zeta\in\Pi(n)$ with $|\zeta|=l$,
\begin{equation}\label{uit8p}
\sum_{\theta\in\Pi(n),\,|\theta|=l}
\big(
\Sym_l(\mathcal E_\theta f^{(n)}),\mathcal E_\zeta f^{(n)})
\big)_{L^2((X\times \mathbb R)^l,(m\otimes \nu)^{\otimes l})}\,l!= \int_{X_\zeta^{(n)}}(\Sym_n\,f^{(n)})f^{(n)}\,dm^{(n)}_{\nu,\, \zeta}\,.\end{equation}
So, let us fix a partition $\zeta=\{\zeta_1,\dots,\zeta_l\}\in\Pi(n)$ and assume that \eqref{iut8o6t8} holds. Denote $k_i:=|\zeta_i|$, $i=1,\dots,l$.
We have, by the definition of $\mathcal E_\zeta f^{(n)}$:
\begin{equation}\label{rdu6e46i}(\mathcal E_\zeta f^{(n)})=\left(\prod_{i_1\in\zeta_1}
h_{i_1}\right)\otimes p_{k_1-1}\otimes\dots\otimes
\left(\prod_{i_l\in\zeta_l}
h_{i_l}\right)\otimes p_{k_l-1}.\end{equation}
Let $\theta=\{\theta_1,\dots,\theta_l\}\in\Pi(n)$ and assume that \eqref{ft6e} holds. Let $r_i:=|\theta_i|$, $i=1,\dots,l$. We may assume that there exists a permutation $\widehat\pi\in\mathfrak S_l$ such that
\begin{equation}\label{ydyduc}r_i=k_{\widehat\pi(i)},\quad i=1,\dots,l.\end{equation}
Indeed, otherwise the corresponding term in the sum on the left hand side of formula \eqref{uit8p} vanishes.
Analogously to \eqref{rdu6e46i}, we have
\begin{align*}
&l!\, \Sym_l(\mathcal E_\theta f^{(n)})(y_1,s_1,\dots,y_l,s_l)=\sum_{\varkappa\in S_l}Q_\varkappa(y_1,\dots,y_l)\\
&\times
\left(\left(\prod_{j_1\in\theta_{\varkappa(1)}}h_{j_1}\right)\otimes p_{r_{\varkappa(1)}-1}\otimes\dots\otimes \left(\prod_{j_l\in\theta_{\varkappa(l)}}h_{j_l}\right)\otimes p_{r_{\varkappa(l)}-1}\right)(y_1,s_1,\dots,y_l,s_l).
\end{align*}
Hence, by \eqref{yufrur},
\begin{align}
&\big(
\Sym_l(\mathcal E_\theta f^{(n)}),\mathcal E_\zeta f^{(n)})
\big)_{L^2((X\times \mathbb R)^l,(m\otimes \nu)^{\otimes l})}\,l!\notag\\
&\quad =\sum_{\widehat\pi}\int_{X^l}Q_{\widehat\pi}(y_1,\dots,y_l)
\left(\prod_{j_1\in\theta_{\widehat\pi(1)}}h_{j_1}(y_1)\right)
\left(\prod_{i_1\in\zeta_1}
h_{i_1}(y_1)\right)\notag\\
&\qquad\times\dotsm\times
\left(\prod_{j_l\in\theta_{\widehat\pi(l)}}h_{j_l}(y_l)\right)
\left(\prod_{i_l\in\zeta_l}
h_{i_l}(y_l)\right)\,dy_1\dotsm dy_l\, c_{k_1}\dotsm c_{k_l},\label{igr8678u}
\end{align}
where the summation is over all permutations $\widehat\pi\in S_l$ which satisfy \eqref{ydyduc}. Let us fix such a permutation $\widehat\pi$. Then, there exist
$$r_1! \dotsm r_l!=k_1!\dotsm k_l!$$ permutations $\pi\in \mathfrak S_n$ which satisfy
\begin{equation}\label{hit98t} \pi \zeta_i=\theta_{\widehat \pi(i)},\quad i=1,\dots,l.\end{equation}
Note that, for each permutation $\pi$ satisfying \eqref{hit98t} and for $(x_1,\dots,x_n)\in X_\zeta^{(n)}$,
\begin{align}&f^{(n)}(x_{\pi^{-1}(1)},\dots,x_{\pi^{-1}(n)})\notag\\
&\quad=(h_1\otimes \dots\otimes h_n)(x_{\pi^{-1}(1)},\dots,x_{\pi^{-1}(n)})\notag\\
&=(h_{\pi(1)}\otimes\dots\otimes h_{\pi(n)})(x_1,\dots,x_n)\notag\\
&\quad=\left(\prod_{j_1\in\pi\zeta_1}h_{j_1}\right)(y_1)\dotsm \left(\prod_{j_l\in\pi\zeta_l}h_{j_l}\right)(y_l)\notag\\
&\quad=\left(\prod_{j_1\in\theta_{\widehat\pi(1)}}h_{j_1}\right)(y_1)\dotsm \left(\prod_{j_l\in\theta_{\widehat\pi(l)}}h_{j_l}\right)(y_l),\label{fu88}
\end{align}
where $y_1=x_{i_1}$ for $i_1\in\zeta_1$,\dots, $y_l=x_{i_l}$ for $i_l\in\zeta_l$.
Let $\zeta, \theta \in \Pi(n)$ be such that condition \eqref{ydyduc} is satisfied by some permutation $\widehat{\pi} \in \mathfrak S_l$.
That is, the corresponding sequences $(k_1,\dots,k_l)$ and $(r_1,\dots,r_l)$ coincide up to a permutation.
Denote
by $\mathfrak S_n[\zeta, \theta]$ the set of all permutations $\pi \in \mathfrak S_n$ which satisfy \eqref{hit98t} with some permutation $\widehat{\pi} \in \mathfrak S_l$.
(Note that the permutation $\widehat{\pi}$ is then completely identified by $\pi$, $\zeta$ and $\theta$ and automatically satisfies \eqref{hit98t}.)
Clearly, if $\theta$ and $\theta'$ are from $\Pi(n)$ with $|\theta| = |\theta'| = l$, both satisfying \eqref{hit98t}, and
$\theta \ne \theta'$, then
\begin{equation}\label{csd35}
\mathfrak S_n[\zeta, \theta] \cap \mathfrak S_n[\zeta,\theta'] = \varnothing.
\end{equation}
Furthermore,
\begin{align}\label{csd36}
\bigcup_{\substack{\theta \in\Pi(n),\, |\theta| =l \\ \theta\, \text{satisfying \eqref{hit98t}}}}
\mathfrak S_n[\zeta,\theta] = \mathfrak S_n.
\end{align}
Therefore, by the definition of the measure $m_{c,\,\zeta}^{(n)}$ and formulas \eqref{uyr75rw}, \eqref{igr8678u}, \eqref{fu88}--\eqref{csd36},
\begin{align*}
&\big(\Sym_l(\mathcal E_\theta f^{(n)}),\mathcal E_\zeta f^{(n)}
\big)_{L^2((X\times \mathbb R)^l,(m\otimes\nu)^{\otimes l})}\,l!\\
& \quad = \frac{1}{n!}\sum_{\pi \in S_n[\zeta, \theta]}\int_{X_\zeta^{(n)}}\mathbf Q_\pi(x_1,\dots,x_n)f^{(n)}(x_{\pi^{-1}(1)}, \ldots , x_{\pi^{-1}(n)})\\
&\quad \quad \times f^{(n)}(x_1, \ldots, x_n)\,m_{\nu,\,\zeta}^{(n)}(dx_1\times\dots\times dx_n).
\end{align*}Hence
\begin{align*}
&\sum_{\theta \in \Pi(n),\, |\theta| = l}\big(\Sym_l(\mathcal E_\theta f^{(n)}),\mathcal E_\zeta f^{(n)}
\big)_{L^2((X\times \mathbb R)^l,(m\otimes\nu)^{\otimes l})}\,l!\\
& = \sum_{\substack{\theta \in\Pi(n),\, |\theta| =l \\ \text{$\theta$ satisfying \eqref{hit98t}}}}
\big(\Sym_l(\mathcal E_\theta f^{(n)}),\mathcal E_\zeta f^{(n)}
\big)_{L^2((X\times \mathbb R)^l,(m\otimes\nu)^{\otimes l})}\,l!\\
& =\frac{1}{n!}\sum_{\substack{\theta \in\Pi(n),\, |\theta| =l \\ \text{$\theta$ satisfying \eqref{hit98t}}}}
\sum_{\pi \in S_n[\zeta, \theta]}\int_{X_\zeta^{(n)}} Q_\pi(x_1,\dots,x_n)f^{(n)}(x_{\pi^{-1}(1)}, \dots , x_{\pi^{-1}(n)})\\
& \quad \times f^{(n)}(x_1, \ldots, x_n)\,m_{\nu,\,\zeta}^{(n)}(dx_1\times\dots\times dx_n)\\
& =\frac{1}{n!}\sum_{\pi \in S_n} \int_{X_\zeta^{(n)}} Q_\pi(x_1,\dots,x_n)f^{(n)}(x_{\pi^{-1}(1)}, \dots , x_{\pi^{-1}(n)})\\
&\quad\times
f^{(n)}(x_1, \ldots, x_n)\,m_{\nu,\,\zeta}^{(n)}(dx_1\times\dots\times dx_n)\\
& = \int_{X_\zeta^{(n)}}(\Sym_n\, f^{(n)})f^{(n)}\,dm_{\nu,\,\zeta}^{(n)}\,.
\end{align*}
Thus, Theorem~\ref{utu8} is proven.
\subsection{Proof of Theorem~\ref{fu7r7}}
Let us first prove the following
\begin{lemma}\label{vfyt6}Let $h\in C_0(X)$ and $f^{n}\in B_0(X^n)$, $n\in\mathbb N$. Then formulas \eqref{iut78}, \eqref{yur75} hold with
\begin{align*}
\mathbf J^+(h)\Sym_n f^{(n)}&=\Sym_{n+1}(h\otimes f^{(n)}),\\
\mathbf J^0(h)\Sym_n f^{(n)}&=\Sym_{n}\mathscr J^0(h)f^{(n)},\\
\mathbf J^-_1(h)\Sym_n f^{(n)}&=\Sym_{n-1}\mathscr J_1^-(h)f^{(n)},\\
\mathbf J^-_2(h)\Sym_n f^{(n)}&=\Sym_{n-1}\mathscr J_2^-(h)f^{(n)},
\end{align*}
Here \begin{multline}\label{tydr6ue}
(\mathscr J_2^-(h)f^{(n)})(x_1,\dots,x_{n-1})\\
:=\sum_{i=1}^n\int_X dy\,h(y) f^{(n)}(x_1,\dots,x_{i-1},y,x_i,\dots,x_{n-1})T_i(y,x_1,\dots,x_{n-1}),
\end{multline}
where for any $\theta\in\Pi(n-1)$
\begin{equation}\label{gftuoihcs}
T_i^{(n)}\restriction X\times X_\theta^{(n-1)} :=\prod_{\theta_u\in\theta:\, \max\theta_u\le i-1}Q(y,x_{\theta_u}).
\end{equation}
\end{lemma}
\begin{proof}
By \eqref{igyur7} and \eqref{f75re75},
we have
\begin{equation}
\la \omega,h\ra=\mathcal A^+(h)+\mathcal A^0(h) +d\Gamma(M_h\otimes A^-)+a^-(h\otimes 1).\label{duvc}
\end{equation}
(i) ($\mathbf J^+(h)$ part)
From the proof of Proposition \ref{iyr867r} it follows that
\begin{equation}\label{vfytde57e}
\mathbf U^{-1}\mathcal A^+(h)\mathbf U\Sym_n f^{(n)}=\Sym_{n+1}(h\otimes f^{(n)})=\mathbf J^+(h)\Sym_n f^{(n)}.
\end{equation}
(ii) ({\it $\mathbf J^0(h)$ part}) By Lemma \ref{lgti8t}, Proposition \ref{iyr867r}, \eqref{iuagiuai},\eqref{f75re75}, \eqref{ir688u} and \eqref{ho9t97tg},
\begin{align}
\mathbf U^{-1}\mathcal A^0(h)\mathbf U\Sym_n f^{(n)}
& =\mathbf U^{-1}\mathcal A^0(h) \Sym \sum_{\theta\in\Pi(n)}\mathcal E_\theta f^{(n)}\notag\\
& =\mathbf U^{-1}\Sym N(M_h\otimes A^0)\sum_{\theta\in\Pi(n)}\mathcal E_\theta f^{(n)}\notag\\
&= \mathbf U^{-1}\Sym \sum_{\theta\in\Pi(n)} \sum_{i=1}^n \mathcal E_\theta(h\times_i f^{(n)})b_{\gamma(i,\theta)-1}\, \gamma(i,\theta)^{-1}\notag\\
& =\Sym_n \mathscr J^0(h)f^{(n)}\notag\\
&=\mathbf J^0(h)\Sym_n f^{(n)}.\label{iwyeutf8}
\end{align}
Here,
$$ (h\times_i f^{(n)})(x_1,\dots,x_n):= h(x_i)f^{(n)}(x_1,\dots,x_n).$$
(iii) ({\it $\mathbf J^-_1(h)$ part})
Analogously,
\begin{align}
&\mathbf U^{-1}d\Gamma(M_h\otimes A^-)\mathbf U\Sym_n f^{(n)}=
\mathbf U^{-1}\Sym N(M_h\otimes A^-)\sum_{\theta\in\Pi(n)}\mathcal E_\theta f^{(n)}\notag\\
&\quad =\mathbf U^{-1} \Sym\sum_{l=1}^n\sum_{\substack{\theta\in\Pi(n)\\|\theta|=l}}\sum_{k=1}^l \mathbf 1^{\otimes (k-1)}\otimes(M_h\otimes A^-)\otimes \mathbf 1^{\otimes(l-k)}\mathcal E_\theta f^{(n)}\notag\\
&\quad=\mathbf U^{-1}\Sym\sum_{l=1}^{n-1}\sum_{\theta=\{\theta_1,\dots,\theta_l\}\in\Pi(n)}\sum_{\substack{k=1,\dots,l\\ |\theta_k|\ge2}}\mathbf 1^{\otimes (k-1)}\otimes(M_h\otimes A^-)\otimes \mathbf 1^{\otimes(l-k)}\mathcal E_\theta f^{(n)},\label{if7er5}
\end{align}
where \eqref{ft6e} is supposed to hold.
Note that, for $\theta=\{\theta_1,\dots,\theta_l\}\in\Pi(n)$ satisfying \eqref{ft6e} and $k\in\{1,\dots,l\}$ with $|\theta_k|\ge2$, we have
\begin{align}
&\big(\mathbf 1^{\otimes (k-1)}\otimes(M_h\otimes A^-)\otimes \mathbf 1^{\otimes(l-k)}\mathcal E_\theta f^{(n)}\big)(x_1,s_1,\dots,x_l,s_l)\notag\\
&\quad =a_{|\theta_k|-1}h(x_k)f^{(n)}_\theta(x_1,\dots,x_k,\dots,x_l)p_{|\theta_1|-1}(s_1)\dotsm p_{|\theta_{k-1}|-1}(s_{k-1})\notag\\
&\qquad\times p_{|\theta_k|-2}(s_k)
p_{|\theta_{k+1}|-1}(s_{k+1})
\dotsm p_{|\theta_{l}|-1}(s_{l}) .\label{iyur77rf}
\end{align}
Let us fix any $i,j\in\{1,\dots,n\}$ with $i<j$. Consider the set
$$ L_i:=\{1,2,\dots,i-1,i+1,\dots,n\},$$
which has $n-1$ elements. Then any partition $\zeta=\{\zeta_1,\dots,\zeta_l\}\in\Pi(n-1)$ identifies a partition $\tilde\zeta=\{\tilde\zeta_1,\dots,\tilde\zeta_l\}$ of $L_i$: $\tilde\zeta_u:=K_i\zeta_u$, $u=1,\dots,l$, where
$$ K_i v:=\begin{cases} v,&\text{if }v\le i-1,\\
v+1,& \text{if }v\ge i.\end{cases}$$
Let $\tilde \zeta_k$ be the element of $\tilde \zeta$ which contains $j$. Set
$$\theta_u:=\begin{cases}\tilde\zeta_u,&\text{if }u\ne k,\\
\tilde \zeta_k\cup\{i\},&\text{if }u=k.\end{cases}$$
Thus, we have constructed a partition $\theta=\{\theta_1,\dots,\theta_l\}\in\Pi(n)$ with $l\le n-1$.
Next, consider an arbitrary partition $\theta=\{\theta_1,\dots,\theta_l\}\in\Pi(n)$ with $l\le n-1$. Choose any $k\in\{1,\dots,l\}$ such that $|\theta_k|\ge2$. In how many ways can we obtain $\theta$ from $i,j$ and $\zeta\in\Pi(n-1)$ as above? This number is evidently equal to the number of all choices of
$i,j\in\{1,\dots,n\}$ with $i<j$ and $i,j\in \theta_k$, i.e.,
$$|\theta_k|(|\theta_k|-1)/2=(|\tilde\zeta_k|+1)|\tilde\zeta_k|/2=(|\zeta_k|+1)|\zeta_k|/2,
$$
where $j\in\tilde\zeta_k$, or equivalently $j-1\in\zeta_k$.
Hence, by \eqref{ft6e} \eqref{igftr7y}, \eqref{oduhuaijo}, \eqref{if7er5}, and \eqref{iyur77rf}, we get
\begin{equation}\label{jigtyl8tg}
\mathbf U^{-1}d\Gamma(M_h\otimes A^-)\mathbf U\Sym_n f^{(n)}
=\Sym_{n-1}\mathscr J_1^-(h)f^{(n)}=\mathbf J^-_1(h)\Sym_n f^{(n)}.
\end{equation}
(iv) ({\it $\mathbf J_2^-(h)$ part}).
For each $\theta=\{\theta_1,\dots,\theta_l\}\in\Pi(n)$ satisfying \eqref{ft6e}, we have
\begin{align}
&\big(a^-(h\otimes 1)\Sym_{l}\mathcal E_\theta f^{(n)}\big)(x_1,s_1,\dots,x_{l-1},s_{l-1})\notag\\
&\quad =\Sym_{l-1}\bigg(\int_Xdy\,
\sum_{\substack{i=1,\dots,l\\ |\theta_i|=1}}h(y)Q(y,x_1)Q(y,x_2)\dotsm Q(y,x_{i-1})\notag\\
&\qquad\times f^{(n)}_\theta(x_1,\dots,x_{i-1},y,x_i,\dots,x_{l-1})\notag\\
&\qquad \times p_{|\theta_1|-1}(s_1)\dotsm
p_{|\theta_{i-1}|-1|}(s_{i-1})p_{|\theta_{i+1}|-1}(s_i)\dotsm p_{|\theta_l|-1}(s_{l-1})\bigg),\label{agviyagv}
\end{align}
where we used \eqref{hyfd7urf} and \eqref{isgdiy}.
Hence, by \eqref{tydr6ue}, \eqref{gftuoihcs}, and \eqref{agviyagv},
\begin{equation}\label{kcgyiudgsc}
\mathbf U^{-1} a^-(h\otimes 1)\mathbf U \Sym_n f^{(n)}=
\Sym_{n-1}\mathscr J_2^-(h)f^{(n)}=\mathbf J^-_2(h)\Sym_n f^{(n)}.
\end{equation}
\end{proof}
\begin{lemma}\label{ugfcubc}
For any $h\in C_0(X)$ and $f^{(n)}\in \mathbf B_0^{Q}(X^n)$, we have
\begin{align} (\mathbf J_2^-(h)f^{(n)})(x_1,\dots,x_{n-1})&=(\mathscr J_2^-(h)f^{(n)})(x_1,\dots,x_{n-1})\notag\\
&=n\int_X dy\, h(y)f^{(n)}(y,x_1,\dots,x_{n-1}).\label{trdtrs}\end{align}
\end{lemma}
\begin{proof} Fix any $n\ge 2$ and $i\in \{2,\dots,n\}$. Let a permutation $\pi\in \mathfrak S_n$ be given by $\pi(1)=i$, $\pi(j)=j-1$ for $j=2,\dots,i$, and $\pi(j)=j$ for $j=i+1,\dots,n$. Recall the operator $ \Psi_\pi$ defined in subsec.~\ref{yfd6rd6}. By \eqref{uyr75rw} and \eqref{gftuoihcs}, we have, for each $(x_1,\dots,x_n)\in X^n$ such that $x_1\ne x_j$ for $j\in\{2,\dots,n\}$,
\begin{equation}\label{gufr8o}( \Psi_\pi f^{(n)})(x_1,\dots,x_n)=f^{(n)}(x_2,x_3,\dots,x_{i},x_1,x_{i+1},\dots,x_n)T_i(x_1,x_2,\dots,x_n).\end{equation}
Since $f\in \mathbf B_0^{Q}(X^n)$, by \eqref{gygfy} and
\eqref{gciugfi},
\begin{equation}\label{houygt9it}\Psi_\pi f^{(n)}=\Psi_\pi \Sym_n f^{(n)}=
\Sym_nf^{(n)}=f^{(n)}. \end{equation}
By \eqref{gufr8o} and \eqref{houygt9it}, for each $(x_1,\dots,x_{n-1})\in X^{n-1}$
\begin{align*}& \int_X dy\,h(y) f^{(n)}(x_1,\dots,x_{i-1},y,x_i,\dots,x_{n-1})T_i(y,x_1,\dots,x_{n-1})
\\&\quad =\int_{X\setminus\{x_1,\dots,x_{n-1}\}} dy\,h(y) f^{(n)}(x_1,\dots,x_{i-1},y,x_i,\dots,x_{n-1})T_i(y,x_1,\dots,x_{n-1})
\\
&\quad = \int_X dy\,h(y) f^{(n)}(y,x_1,\dots,x_{n-1}).\end{align*}
Hence, by \eqref{tydr6ue},
\begin{equation} (\mathscr J_2^-(h)f^{(n)})(x_1,\dots,x_{n-1})=n\int_X dy\, h(y)f^{(n)}(y,x_1,\dots,x_{n-1})=:g^{(n-1)}(x_1,\dots,x_{n-1}).\label{fy7e5irfv}\end{equation}
Since $f^{(n)}\in \mathbf B_0^{Q}(X^n)$, formula \eqref{ft7ier5ird} holds for each $\pi\in\mathfrak S_n$. Hence, for each $\pi\in\mathfrak S_{n-1}$,
$$ g^{(n-1)}(x_1,\dots,x_{n-1})= Q_\pi(x_1,\dots,x_{n-1})g^{(n)}(x_{\pi^{-1}(1)},\dots,x_{\pi^{-1}(n)}),$$
see \eqref{uyr75rw}. Therefore,
\begin{equation}\label{guy8rt7o68r}
\Sym\, g^{(n-1)}=g^{(n-1)}.
\end{equation}
So, the lemma follows from \eqref{fy7e5irfv} and \eqref{guy8rt7o68r}.
\end{proof}
Now, Theorem~\ref{fu7r7} follows from Lemmas \ref{vfyt6}, \ref{ugfcubc}.
\subsection{Proof of Theorem \ref{urr8r}}
Assume that \eqref{vggyufd7u} holds.
Then, by \eqref{ho9t97tg} and \eqref{oduhuaijo}, we get
$R_i^{(n)}\equiv \lambda$ and $S_{j-1}^{(n)}\equiv2\eta$. Hence, by \eqref{ir688u} and \eqref{igftr7y}, for any $h\in C_0(X)$, the operators $\mathscr J^0(h)$ and $\mathscr J_1^-(h)$ map $\mathcal F_{\mathrm{fin}}(C_0(X))$ into itself. Hence, condition (C) is satisfied. Furthermore, equality \eqref{yufu7edseaa} follows Theorem~\ref{fu7r7}.
To show that \eqref{vggyufd7u} is necessary for condition (C) to hold, we proceed as follows.
We first assume that the measure $\nu=\delta_\lambda$ for some $\lambda\in\mathbb R$ (Guassian/Poisson). Then $a_k=0$ for all $k\in\mathbb N$, $b_0=\lambda$, and the values of $b_k$ for $k\in\mathbb N$ maybe chosen arbitrarily. Thus, \eqref{vggyufd7u} holds in this case with $\eta=0$.
We next assume that the support of the measure $\nu$ contains an infinite number of points. Thus, $a_k>0$ for all $k\in\mathbb N$.
\begin{lemma}\label{hiwfgwy8} Let $q\ne-1$. Let $a_k>0$ for all $k\in\mathbb N$.
Let $n\ge2$ and let $f^{(n)}\in C_0(X^n)$ be such that $\Sym_n\, f^{(n)}=0$ $m_\nu^{(n)}$-a.e.\ on the set $X^{(n)}_\theta$, where $\theta=\{\theta_1,\theta_2\}\in \Pi(n)$ with $\theta_1=\{1\}$ and $\theta_2=\{2,\dots,n\}$. Then $ f^{(n)}(x,\dots,x)=0$ for all $x\in X$.
In the fermion case, $q=-1$, the above result remains true for $n\ge3$.
\end{lemma}
\begin{proof} Let $x_1,x_2\in X$ be such that $x_1^1<x_2^1$. (Recall that $x^i$ denotes the $i$-th coordinate of $x=(x^1,\dots,x^d)\in X$.) In particular, $x_1<x_2$. Then
\begin{align}
&(\Sym_n\,f^{(n)})(x_1,x_2,x_2,\dots,x_2)=\frac1n\big( f^{(n)}(x_1,x_2,x_2,\dots,x_2)+
f^{(n)}(x_2,x_1,x_2,\dots,x_2)\notag\\
&\quad +\dots+f^{(n)}(x_2,\dots,x_2,x_1,x_2)+q f^{(n)}(x_2,\dots,x_2,x_1)\big)=0.\label{tye65e}
\end{align}
Since the function $f^{(n)}$ is continuous, equality \eqref{tye65e} holds point-wise on the open set
$$\{(x_1,x_2)\in X^2\mid x_1^1<x_2^1\}.$$
Therefore, for all $x\in X$, we get
$\frac{n-1+q}n f^{(n)}(x,\dots,x)=0$.
Thus, $f^{(n)}(x,\dots,x)=0$ if either $q\ne -1$ and $n\ge2$, or $q=-1$ and $n\ge3$.
\end{proof}
We now set $\lambda:=b_0$. Let us show that, if (C) holds, then $b_k=\lambda(k+1)$ for all $k\in\mathbb Z_+$. The proof below works for any anyon statisics, however, in the case where $q\ne-1$, this proof can be significantly simplified.
Let $\varepsilon\in\mathbb R$ be such that $b_1=2\lambda+\varepsilon$. We will now show by induction that
\begin{equation}\label{igyfd7i5ei}
b_k=\lambda(k+1)+\varepsilon,\quad k\ge1.\end{equation}
Assume that equality in \eqref{igyfd7i5ei} holds for $k=1,\dots,n$. Fix any $h\in C_0(X)$ and $f^{(n+2)}\in C_0(X^{n+2})$.
We define a function $g^{(n+2)}\in C_0(X^{n+2})$ by
\begin{equation}\label{tye66ie7}
g^{(n+2)}(x_1,\dots,x_{n+2}):=f^{(n+2)}(x_1,\dots,x_{n+2})\big(\lambda h(x_1)
+h(x_2)(\lambda(n+1)+\varepsilon)\big).\end{equation}
Let $\theta=\{\theta_1,\theta_2\}\in\Pi(n+2)$ with $\theta_1=\{1\}$, $\theta_2=\{2,\dots,n+2\}$. By \eqref{ir688u} and \eqref{ho9t97tg}, we have $m_\nu^{(n+2)}$-a.e.\ on $X_\theta^{(n+2)}$:
\begin{align*}&(\mathscr J^0(h)f^{(n+2)})(x_1,\dots,x_{n+2})\\
&\quad =f^{(n+2)}(x_1,\dots,x_{n+2})\big(\lambda h(x_1)
+(n+1)h(x_2)(\lambda(n+1)+\varepsilon)/(n+1)\big)\\
&\quad =g^{(n+2)}(x_1,\dots,x_{n+2}).
\end{align*}
Since (C) holds, there exists a function $u^{(n+2)}\in C_{0}(X^{n+2})$ such that
\begin{equation}\label{iwgyGF}
\Sym_{n+2}\mathscr J^0(h)f^{(n+2)}=\Sym_{n+2}\,u^{(n+2)}\end{equation} $m_\nu^{(n+2)}$-a.e.\ on $X^{n+2}$. Hence,
$$ \SSym_{n+2}(g^{(n+2)}-u^{(n+2)})(x_1,\dots,x_{n+2})=0 $$
for $m_c^{(n+2)}$-a.a.\ $(x_1,\dots,x_{n+2})\in X^{(n+2)}_\theta$.
Noting that $g^{(n+2)}-u^{(n+2)}\in C_0(X^{n+2})$, we conclude from Lemma~\ref{hiwfgwy8} that
\begin{equation}\label{ifu7i5e} u^{(n+2)}(x,\dots,x)=g^{(n+2)}(x,\dots,x),\quad x\in X.\end{equation}
By \eqref{tye66ie7}--\eqref{ifu7i5e},
\begin{equation}\label{ftur75reeawa}(\mathscr J^0(h)f^{(n+2)})(x,\dots,x)=\big(\lambda (n+2)+\varepsilon)
h(x)f^{(n+2)}(x,\dots,x)\end{equation}
for all $x\in X$.
By \eqref{ir688u}, \eqref{ho9t97tg}, and \eqref{ftur75reeawa}, we therefore get $b_{n+1}=\lambda(n+2)+\varepsilon$.
Thus, \eqref{igyfd7i5ei} is proven.
Our next aim is to show that $\varepsilon=0$. We first derive the following analog of Lemma~\ref{hiwfgwy8}.
\begin{lemma}\label{uit8ot}Let $a_k>0$ for all $k\in\mathbb N$.
Let $f^{(5)}\in C_0(X^5)$ be such that $\Sym_5\, f^{(5)}=0$ $m_\nu^{(5)}$-a.e.\ on the set $X^{(5)}_\theta$, where $\theta=\{\theta_1,\theta_2\}\in\Pi(5)$ with $\theta_1=\{1,2\}$, $\theta_2=\{3,4,5\}$.
Then $f^{(5)}(x,\dots,x)=0$ for all $x\in X$.
\end{lemma}
\begin{proof}The proof is similar to that of Lemma~\ref{hiwfgwy8}. In fact, from the condition of Lemma~\ref{uit8ot}, we get
$ \frac{6+4q}{10}f^{(5)}(x,\dots,x)=0$,
which implies the statement.
\end{proof}
By \eqref{ir688u}, \eqref{ho9t97tg}, and \eqref{igyfd7i5ei}, we have, for $m_\nu^{(5)}$-a.e.\ $(x_1,\dots,x_5)\in X_\theta^{(5)}$ with $\theta\in\Pi(5)$ being as in Lemma~\ref{uit8ot},
\begin{equation}\label{vct6red6e} (\mathscr J^0(h)f^{(5)})(x_1,\dots,x_5)=
f^{(5)}(x_1,\dots,x_5)\big(h(x_1)(2\lambda+\varepsilon)+ h(x_3)(3\lambda+\varepsilon)\big).\end{equation}
Analogously to derivation of formula \eqref{ftur75reeawa}, we conclude from condition (C),
Lemma~\ref{uit8ot}, and \eqref{vct6red6e} that, for all $x\in X$,
\begin{equation}\label{hur75r}
(\mathscr J^0(h)f^{(5)})(x,\dots,x)=f^{(5)}(x,\dots,x)
h(x)(5\lambda+2\varepsilon).
\end{equation}
On the other hand, by
\eqref{ir688u}, \eqref{ho9t97tg}, and \eqref{igyfd7i5ei},
we have, for all $x\in X$
\begin{equation}\label{it86tr}
(\mathscr J^0(h)f^{(5)})(x,\dots,x)=f^{(5)}(x,\dots,x)
h(x)(5\lambda+\varepsilon).
\end{equation}
Comparing \eqref{hur75r} and \eqref{it86tr}, we see that $\varepsilon$ must be equal to zero.
The proof of the equality $a_k=\eta k(k+1)$ for $k\in\mathbb N$ is similar, so we only outline it.
Denote $\eta:=a_1/2$.
Using Lemma~\ref{hiwfgwy8} and formulas \eqref{igftr7y}, \eqref{oduhuaijo}, we get the recursive formula
\begin{equation}\label{iyr8lotrg}
a_{n+1}=2\eta+\big((n+1)(n+2)-2\big)\frac{a_n}{n(n+1)}
\end{equation}
for $n\ge2$.
Choose $\varepsilon\in\mathbb R$ so that $a_2=6\eta+\varepsilon$. Then, by \eqref{iyr8lotrg},
\begin{equation}\label{oiwhcy97t}
a_3=12\eta+\frac{10}6\varepsilon,\quad
a_4=20\eta+\frac52\varepsilon,\quad a_5=30\eta+\frac72\varepsilon.
\end{equation}
On the other hand, by Lemma~\ref{uit8ot},
\begin{equation}\label{fur7}
a_5=a_2+2a_3.
\end{equation}
From \eqref{oiwhcy97t} and \eqref{fur7}, we get $\varepsilon=0$. Hence, the recursive formula \eqref{iyr8lotrg} holds for all $n\ge1$. From here the desired equality follows.
We finally consider the case where
the support of the measure $\nu$ consists of $l$ points with $l\ge2$ being finite. In the case where $q=-1$, we will additionally assume that $l\ge3$.
Then
$a_1>0$, $a_2>0$,\dots,$a_{l-1}>0$, $a_i=0$ for$i\ge l$. Furthermore, by \eqref{yuft8uotfr8o},
$ c_1>0$, $c_2>0$,\dots,$c_k>0$, $c_i=0$ for $i\ge l+1$.
Let condition (C) be satisfied.
Then, in view of the construction of the measures $m_\nu^{(n)}$, analogously to the above, we conclude that formula \eqref{iyr8lotrg} holds for $n=1,2,\dots,l-1$. In particular, we get
$$ a_l=a_1+\big( l(l+1)-2\big)\frac{a_{l-1}}{(l-1)l}\,.$$
Since $a_1>0$ and $a_{l-1}>0$, we therefore get $a_l>0$, which contradicts the fact that $a_l=0$. Thus, (C) can not be satisfied. Theorem \ref{urr8r} is proven.
We leave the easy proof of Proposition~\ref{ytfr7} to the interested reader. Let us show, however, how Theorem~\ref{ur7o67r6} can now be easily derived.
Assume $q=1$. Assume that $\mathscr{CP}=\mathscr{OCP}$. Then, for any
$h\in C_0(X)$ and $f^{(n)}\in C_0(X^n)$, we have
\begin{equation}\label{vytjdy6}\la \omega,h\ra\la P_n(\omega),f^{(n)}\ra\in \mathscr{OCP}\end{equation}
(we used that product of any polynomials from $\mathscr{CP}$ belongs to $\mathscr{CP}$). Since
$$\mathbf J_2^-(h)\la f^{(n)},P_n(\omega)\ra=\la \mathscr J^-_2(h)f^{(n)},P_{n-1}(\omega)\ra\in \mathscr{OCP},$$
we therefore conclude from Theorem \ref{fu7r7} and \eqref{vytjdy6} that (C) holds. Hence, by Theorem~\ref{urr8r}, \eqref{vggyufd7u} holds.
Let us now assume that \eqref{vggyufd7u} holds. Then, as follows from the proof of Theorem~\ref{urr8r}, $h\in C_0(X)$, the operators $\mathscr J^0(h)$ and $\mathscr J_1^-(h)$ map $\mathcal F_{\mathrm{fin}}(C_0(X))$ into itself. Hence, for any $f^{(n)}\in C_0(X^n)$, \eqref{vytjdy6} holds. From here the equality $\mathscr{CP}=\mathscr{OCP}$ can be deduced analogously to the proof of \cite[Theorem~4.1]{BL1}.
\subsection{Proof of Theorem \ref{hfu8fr78}}
We will only prove equality \eqref{bkvgutfgi} as the proof of equality \eqref{vfyter} is similar and simpler. Note also that formula \eqref{ydfyd} will follow from \eqref{adiohdsrtdv}--\eqref{bkvgutfgi}.
It suffices to prove that, for any $h\in C_0(X)$,
$$ \mathbf J_1^-(h)g^{(n)}=\int_X dx\, h(x)\eta \partial_x^\dag \partial_x\partial_x\,g^{(n)},$$
where $g^{(n)}\in\mathbf B^{Q}_0(X^n)$ is of the form
$g^{(n)}=f_1\cd\dotsm \cd f_n$,
with $f_1,\dots,f_n\in B_0(X)$. We have
$$
g^{(n)}(x_1,\dots,x_n):
=\frac1{n!}\sum_{\pi\in\mathfrak S_n} Q_\pi (x_1,\dots,x_n)f_{\pi(1)}(x_1)\dotsm f_{\pi(n)}(x_n).
$$
Hence, by \eqref{igiugygg},
\begin{align}
& \left(\int_X dx\, h(x)\di_x^\dag\di_x\di_x g^{(n)}\right)(x_1,\dots,x_{n-1})\notag\\
& =\Sym_{n-1}\bigg(
\frac1{(n-2)!}\sum_{\pi\in\mathfrak S_n} Q_\pi(x_1,x_1,x_2,\dots,x_{n-1})\notag\\
&\qquad\qquad\qquad\times (hf_{\pi(1)}f_{\pi(2)})(x_1)
f_{\pi(3)}(x_2)\dotsm f_{\pi(n)}(x_{n-1})
\bigg)\notag\\
& =\sum_{1\le i<j\le n}\frac1{(n-2)!}\sum_{\substack{\pi\in\mathfrak S_n\\
\pi\{1,2\}=\{i,j\} }}
\Sym_{n-1}\big(
Q_\pi(x_1,x_1,x_2,\dots,x_{n-1})\notag\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times (hf_{i}f_{j})(x_1)
f_{\pi(3)}(x_2)\dotsm f_{\pi(n)}(x_{n-1})
\big).\label{bigi}
\end{align}
By \eqref{uyr75rw}, for any $\pi\in\mathfrak S_n$ satisfying $\pi\{1,2\}=\{i,j\}$ with $i<j$, and any $(x_1,x_2,\dots,x_{n-1})\in X^{n-1}$, we have
\begin{equation}\label{dyre6e}
Q_\pi(x_1,x_1,x_2,\dots,x_{n-1})= Q_{\sigma_{ij}(\pi)}(x_1,x_2,\dots,x_{n-1}).
\end{equation}
Here the permutation $\sigma_{ij}(\pi)\in \mathfrak S_{n-1}$ is defined as follows:
$$\sigma_{ij}(\pi)(1):=j,$$
and for $k=2,\dots,n-1$,
$$\sigma_{ij}(\pi)(k):=\begin{cases}
\pi(k+1),&\text{if }\pi(k+1)<i,\\
\pi(k+1)-1,&\text{if }\pi(k+1)>i.
\end{cases}$$
By \eqref{dyre6e}, for any $\pi\in\mathfrak S_n$ satisfying $\pi\{1,2\}=\{i,j\}$ with $i< j$,
\begin{align}
& Q_\pi(x_1,x_1,x_2,\dots,x_{n-1}) (hf_{i}f_{j})(x_1)
f_{\pi(3)}(x_2)\dotsm f_{\pi(n)}(x_{n-1})\notag\\
&\quad = Q_{\sigma_{ij}(\pi)}(x_1,x_2,\dots,x_{n-1})
\big(f_1\otimes \dots\otimes f_{i-1}\otimes f_{i+1}\notag\\
&\qquad\otimes\dots\otimes f_{j-1}\otimes (hf_if_j)\otimes f_{j+1}\otimes\dots\otimes f_n \big)(x_{\sigma_{ij}(\pi)^{-1}(1)},\dots, x_{\sigma_{ij}(\pi)^{-1}(n-1)} )\notag\\
&\quad= \Psi_{\sigma_{ij}(\pi)}\big(f_1\otimes \dots\otimes f_{i-1}\otimes f_{i+1}\notag\\
&\qquad \otimes\dots\otimes f_{j-1}\otimes (hf_if_j)\otimes f_{j+1}\otimes\dots\otimes f_n \big)(x_1,\dots,x_{n-1}).
\notag
\end{align}
Hence, by \eqref{gygfy} and \eqref{gciugfi},
\begin{align}
&\Sym\big( Q_\pi(x_1,x_1,x_2,\dots,x_{n-1}) (hf_{i}f_{j})(x_1)
f_{\pi(3)}(x_2)\dotsm f_{\pi(n)}(x_{n-1})\big)\notag\\
&\quad=(f_1\cd \dotsm \cd f_{i-1}\cd f_{i+1}\cd\dotsm\cd f_{j-1}\cd(hf_if_j)\cd f_{j+1}\cd\dotsm
\cd f_n)(x_1,\dots,x_{n-1}).\label{fcauyfy}
\end{align}
By \eqref{bigi} and \eqref{fcauyfy}, we thus get
\begin{align*}
&\int_X dx\, h(x)\di_x^\dag\di_x\di_x g^{(n)}\\
&\quad =2\sum_{1\le i<j\le n} f_1\cd \dotsm \cd f_{i-1}\cd f_{i+1}\cd\dotsm\cd f_{j-1}\cd(hf_if_j)\cd f_{j+1}\cd\dotsm
\cd f_n.
\end{align*}
From here equality \eqref{bkvgutfgi} follows.
\begin{center}
{\bf Acknowledgements}\end{center}
M.B. and E.L. acknowledge the financial support of the Polish
National Science Center, grant no.\ Dec-2012/05/B/ST1/00626, and of
the SFB 701 ``Spectral
structures and topological methods in mathematics'', Bielefeld University. MB was partially supported
by the MAESTRO grant DEC-2011/02/A/ST1/00119.
| 15,311
|
TITLE: Cyclotomic polynomials and Galois group
QUESTION [11 upvotes]: Let $\zeta\in \mathbb C$ be a primitive $7^{th}$ root of unity.
Show that there exists a $\sigma\in \operatorname{Gal}(\mathbb Q(\zeta)/\mathbb Q)$ such that $\sigma(\zeta)=\zeta^3$.
I already know that $\zeta$ is a root of $f(x)=x^6+x^5+x^4+x^3+x^2+x+1$ and that $f$ is irreducible (By applying Eisenstein's criterion on $f(x+1)$).
Also the powers of $\zeta$ are also roots of $f$. So $\mathbb Q(\zeta)$ is a splitting field of $f$. Now its clear that this extension is a Galois extension, since all the roots are different. But how to show the desired statement?
REPLY [11 votes]: Note that $\langle \zeta\rangle$ is a finite, cyclic group of order $7$, so every non-trivial element has order $7$. In particular $\zeta^3$ is another primitive $7^{th}$ root of $1$, hence is another roots of the irreducible polynomial $\Phi_7(x)={x^7-1\over x-1}$. But then as this is irreducible by Eisenstein's criterion applied to $\Phi(x+1)$, we get that
$$\Bbb Q(\zeta)/\Bbb Q\cong \Bbb Q[x]/(\Phi_7(x)).$$
We now use the fact that the Galois group transitively permutes the roots of the irreducible polynomial in the quotient--if you haven't seen this before, see my addendum at the bottom--hence for any two roots, $r,s$, there is some $\sigma=\sigma_{r,s}$ such that $\sigma(r)=s$. Taking $r=\zeta$ and $s=\zeta^3$ we get the result.
The key observations for this are:
that both $\zeta$ and $\zeta^3$ are roots of the same, irreducible polynomial
that the Galois group permutes the roots of such polynomials transitively.
If you haven't seen the proof of the transitive action on roots, it's relatively straightforward: since $\Phi_7(x)$ is irreducible, we note that if $r,s$ are any two roots
$$\Bbb Q(r)/\Bbb Q\cong \Bbb Q[x]/(\Phi_7(x))\cong \Bbb Q(s)/\Bbb Q\qquad (*)$$
Then the automorphism of $\Bbb Q(r)/\Bbb Q$ is simply the composite of the isomorphisms. I.e. if the isomorphisms in $(*)$ are
$$\begin{cases}\varphi_r: \Bbb Q(r)\to \Bbb Q[x]/(\Phi_7(x)) \\
\varphi_s: \Bbb Q(s)\to \Bbb Q[x]/(\Phi_7(x))
\end{cases}$$
then we have that $\sigma_{r,s}=\varphi_s\circ \varphi_r^{-1}: \Bbb Q(s)\to \Bbb Q(r)$ is an isomorphism, but since $\Bbb Q(r)=\Bbb Q(s)$ is actually an equality for $r=\zeta, s=\zeta^3$ when we treat them as subfields of $\Bbb C$, we see that we may replace "iso" with "auto," and apply the definition of the Galois group as the group of all automorphisms of the field.
| 7,841
|
Free personalized radio that
plays the music you love
Captain Hume Lamentation (Poeticall Musicke): Poeticall Musicke, Captain Hume Lamentation
Features of This Tracka small chamber ensemble
lute
an early string instrument
tonal harmony
a broad tempo
These are just a few of the hundreds of attributes cataloged for this track by the Music Genome Project.show more
| 118,804
|
The babies have enjoyed exploring their sensory baskets and discovering the different items inside. The team provided a range of materials to encourage the little ones to use their sense of touch, taste, sight and sound. They sat together and babbled at their reflections in the CDs. As well as, using the spoons and bowls as drums to create different sounds. All sensory experiences support the children’s language and development, social interaction, and motor skills.
Click here to find out more about our Kiddi Caru Bedford day nursery and preschool.
Days left: 6
| 332,792
|
Carlos Tevez clinically tucked in a Mauro Rosales cross in the 18th minute of an ill-tempered South American clash dominated by Argentina before 44,000 at the Olympic Stadium.
It was the Boca Juniors playmaker's eighth goal of the competition for Argentina, who finished with a perfect record of six wins out of six, scoring 17 goals without reply.
Paraguay had Emilio Martinez sent off in the 66th minute for elbowing Andres D'Alessandro in the face and their misery was compounded seven minutes from time when Diego Figueredo was dismissed for a second bookable offence.
Argentina, runners-up to Nigeria in Atlanta eight years ago, had not won an Olympic gold in any sport since Tranquilo Capozzo and Eduardo Guerrero took the men's double sculls rowing title at the 1952 Helsinki Games.
Paraguay's silver was the first Olympic medal in the country's history.
However, they were completely outclassed by an Argentina side that could have won by four or five goals, Rosales and Tevez both coming close at the start of the second half.
Paraguay goalkeeper Diego Barreto pulled off a superb reaction save to keep out a Luis Gonzalez volley in the 60th minute.
Striker Cesar Delgado had shot cleared off the line in the 75th minute by Paraguay captain Carlos Gamarra, himself lucky not to have been sent off after elbowing Tevez in the first half.
Argentina became the first Latin American team to win the Olympic title since 1928, when they were beaten by Uruguay in the Amsterdam final.
For Argentina coach Marcelo Bielsa, the gold medal was vindication for sticking to an attacking 3-4-3 formation after their shock first-round exit at the 2002 World Cup.
| 322,028
|
\begin{document}
\title[The Principal Representations of Reductive Groups ]{The Principal Representations of Reductive Algebraic Groups with Frobenius Maps}
\author{Junbin Dong}
\address{Institute of Mathematical Sciences, ShanghaiTech University, 393 Middle Huaxia Road, Pudong, Shanghai 201210, PR China.}
\email{dongjunbin1990@126.com}
\subjclass[2010]{20C07, 20G05}
\date{January 6, 2021}
\keywords{Principal representation, highest weight category, quiver algebra.}
\begin{abstract}
We introduce the principal representation category $\mathscr{O}({\bf G})$ of reductive algebraic groups with Frobenius maps and put forward a conjecture that this category is a highest weight category.
When $\Bbbk$ is complex field $\mathbb{C}$, we provide some evidences of this conjecture. We also study certain kind of bound quiver algebras whose representations are related to the principal representation category $\mathscr{O}({\bf G})$ .
\end{abstract}
\maketitle
\section*{Introduction}
Let ${\bf G}$ be a connected reductive algebraic group defined over the finite field $\mathbb{F}_q$ with the standard Forbenius map $F$ induced by the automorphism $x\mapsto x^q$ on $\bar{\mathbb{F}}_q$. Let $\Bbbk$ be a field. According to a result of Borel and Tits \cite[Theorem 10.3 and Corollary 10.4]{BT}, we know that except the trivial representation, all other irreducible representations of $\Bbbk {\bf G}$ are infinite-dimensional if ${\bf G}$ is a semisimple algebraic group over $\bar{\mathbb{F}}_q$ and $\Bbbk $ is infinite with $\op{char}\Bbbk\neq \op{char} \bar{\mathbb{F}}_q$. So it seems to be difficult to study the abstract representations of ${\bf G}$. However in \cite{X}, Nanhua Xi studied the abstract representations of ${\bf G}$ over ${\bf \Bbbk}$ by taking the direct limit of the finite-dimensional representations of $G_{q^a}$ and got many interesting results.
Later, motivated by Xi's idea, the structure of the permutation module $\Bbbk [{\bf G}/{\bf B}]$ (${\bf B}$ is a Borel subgroup of ${\bf G}$ ) was studied in \cite{CD1} for the cross characteristic case and \cite{CD2} for the defining characteristic case.
The paper \cite{CD3} studied the general abstract induced module $\mathbb{M}(\theta)=\Bbbk{\bf G}\otimes_{\Bbbk{\bf B}}{\Bbbk}_\theta$ for any field $\Bbbk$ with $\op{char}\Bbbk\neq \op{char} \bar{\mathbb{F}}_q$ or $\Bbbk=\bar{\mathbb{F}}_q$, where $\bf T$ is a maximal splitting torus contained in a $F$-stable Borel subgroup $\bf B$ and $\theta \in \widehat{\bf T}$ (the character group of $\bf T$). The induced module $\mathbb{M}(\theta)$ has a composition series (of finite length) if $\op{char}\Bbbk\neq \op{char} \bar{\mathbb{F}}_q$. In the case $\Bbbk=\bar{\mathbb{F}}_q$ and $\theta$ is a rational character, $\mathbb{M}(\theta)$ has such composition series if and only if $\theta$ is antidominant (see \cite{CD3} for details). In both cases, the composition factors has the form $E(\theta)_J$ with $J\subset I(\theta)$
(see Section 1 for explicit definition).
The construction of $\mathbb{M}(\theta)$ is similar to the Verma module in the representations of semisimple complex Lie algebras. Thus motivated by the famous BGG category $\mathscr{O}$, this paper introduces a category $\mathscr{O}({\bf G})$ called principal representation category to study the abstract representations of ${\bf G}$. It is the full subcategory of $\Bbbk{\bf G}$-Mod such that any object $M$ in $\mathscr{O}({\bf G})$ is of finite length and its composition factors are $E(\theta)_J$ for some $\theta \in \widehat{\bf T}$ and $J\subset I(\theta)$. The BGG category is a highest weight category (see \cite[Example 3.3 (c)]{CPS}). Thus in the study of category $\mathscr{O}({\bf G})$, we put forward a conjecture that this category has enough injectives. So it is a highest weight category (in the sense of \cite{CPS}) when we assume that Conjecture \ref{conjecture} is valid. By the property of highest weight category, $\mathscr{O}({\bf G})$ has a decomposition $\mathscr{O}({\bf G})= \displaystyle \bigoplus_{\theta \in \widehat{\bf T}} \mathscr{O}({\bf G})_\theta$, where $\mathscr{O}({\bf G})_\theta$ is the full subcategory of $\mathscr{O}({\bf G})$ containing the objects whose subquotients are $E(\theta)_J$ for a fixed character $\theta$ of $\bf T$.
The weight set of $\mathscr{O}({\bf G})_\theta$ is finite and therefore there exists a finite-dimensional quasi-hereditary algebra $A_\theta$ such that $\mathscr{O}({\bf G})_\theta$ is equivalent to the right $A_\theta$-modules. Thus all the indecomposable projective objects in $\mathscr{O}({\bf G})$ are given in Section 3. For each $\theta \in \widehat{\bf T}$, the algebra $A_\theta$ is isomorphic to a bound quiver algebra $\mathscr{A}_n$ (which is defined in Section 4) when $|I(\theta)|=n$.
When $|I(\theta)|=1~\text{or}~2$, this algebra is of finite representation type which means that the number of the indecomposable modules up to isomorphic is finite.
By the equivalence of $\mathscr{O}({\bf G})_\theta$ and the the right $A_\theta$-modules, we give all the indecomposable modules of $\mathscr{O}({\bf G})$ when the rank of ${\bf G}$ is 1 or 2.
However when $n\geq 3$, the algebra $\mathscr{A}_n$ is of tame representation type. Moreover,
$\mathscr{A}_n$ is of wild type when $n\geq 4$.
\medskip
This paper is organized as follows: Section 1 contains some preliminaries and we also introduce the principal representation category $\mathscr{O}({\bf G})$ in this section.
In Section 2, we study the injective objects in
$\mathscr{O}({\bf G})$ and give some evidences to show that $\mathscr{O}({\bf G})$ may has enough injectives when $\Bbbk = \mathbb{C}$. Under the assumption that this conjecture is true, we show that $\mathscr{O}({\bf G})$ is
a highest weight category in Section 3. The algebra structure of $A_\theta$ is also studied in this section. Section 4 is devoted to study the algebras $\mathscr{A}_n$ which is isomorphic to $A_\theta$ when $|I(\theta)|=n$.
\bigskip
\noindent{\bf Acknowledgements}\ \ The author is grateful to Prof. Nanhua Xi for his constant encouragement and guidance. The author would also like to thank Prof. Ming Fang, Prof. Zongzhu Lin, Prof. Toshiaki Shoji and Dr. Xiaoyu Chen for their helpful discussions and comments. Part of this work was done during the author's visit to Institute of Mathematics, Chinese Academy of Sciences. The author is grateful to the institute for hospitality.
\section{Principal Representation Category }
As in the introduction, let ${\bf G}$ be a connected reductive algebraic group defined over $\mathbb{F}_q$ with the standard Frobenius map $F$ (e.g., $GL_n(\bar{\mathbb{F}}_q)$, $SL_n(\bar{\mathbb{F}}_q)$, $SO_{2n}(\bar{\mathbb{F}}_q)$, $SO_{2n+1}(\bar{\mathbb{F}}_q)$, $Sp_{2n}(\bar{\mathbb{F}}_q)$,$\cdots$). Let ${\bf B}$ be an $F$-stable Borel subgroup, and ${\bf T}$ be an $F$-stable maximal torus contained in ${\bf B}$, and ${\bf U}=R_u({\bf B})$ be the ($F$-stable) unipotent radical of ${\bf B}$. We denote by $\Phi=\Phi({\bf G};{\bf T})$ the corresponding root system, and by $\Phi^+$ (resp. $\Phi^-$) the set of positive (resp. negative) roots determined by ${\bf B}$. Let $W=N_{\bf G}({\bf T})/{\bf T}$ be the corresponding Weyl group. One denotes by $\Delta=\{\alpha_i\mid i\in I\}$ the set of simple roots and $S=\{s_i\mid i\in I\}$ the corresponding simple reflections in $W$. For each $w\in W$, let $\dot{w}$ be one representative in $N_{\bf G}({\bf T})$. For any $w\in W$, let ${\bf U}_w$ (resp. ${\bf U}_w'$) be the subgroup of ${\bf U}$ generated by all ${\bf U}_\alpha$ (the root subgroup of $\alpha\in\Phi^+$) with $w\alpha\in\Phi^-$ (resp. $w\alpha\in\Phi^+$). The multiplication map ${\bf U}_w\times{\bf U}_w'\rightarrow{\bf U}$ is a bijection (see \cite[Proposition 2.5.12]{Ca}).
\medskip
In this paper we fix an algebraically closed field $\Bbbk$ such that $\op{char}\Bbbk\neq \op{char} \bar{\mathbb{F}}_q$. Denote by
$$\widehat{\bf T}=\{\theta \mid \theta : {\bf T}\rightarrow \Bbbk^* \ \text{is a group homomorphism}\}$$ the character group of ${\bf T}$ over ${\Bbbk}$. Each $\theta\in\widehat{\bf T}$ is regarded as a character of ${\bf B}$ by the homomorphism ${\bf B}\rightarrow{\bf T}$. Let ${\Bbbk}_\theta$ be the corresponding ${\bf B}$-module. The induced module $\mathbb{M}(\theta)=\Bbbk{\bf G}\otimes_{\Bbbk{\bf B}}{\Bbbk}_\theta$ was studied in the paper \cite{CD3} and all the composition factors of $\mathbb{M}(\theta)$ are given in \cite{CD3}. For convenience, we recall the main results here.
Let ${\bf 1}_{\theta}$ be a nonzero element in ${\Bbbk}_\theta$. We write $x{\bf 1}_{\theta}:=x\otimes{\bf 1}_{\theta}\in \mathbb{M}(\theta)$ for short.
It is clear that $\displaystyle \mathbb{M}(\theta)=\sum_{w\in W}\Bbbk {\bf U}_{w^{-1}}\dot{w}{\bf 1}_{\theta}$ and the set $$\{ u \dot{w}{\bf 1}_{\theta} \mid w\in W, u\in {\bf U}_{w^{-1}}\} $$ forms a basis of $\mathbb{M}(\theta)$ using Bruhat decomposition. As \cite[Proposition 2.2]{CD3} showed, the $\Bbbk {\bf G}$-module $\mathbb{M}(\theta)$ is indecomposable.
For each $i \in I$, let ${\bf G}_i$ be the subgroup of $\bf G$ generated by ${\bf U}_{\alpha_i}, {\bf U}_{-\alpha_i}$ and we set ${\bf T}_i= {\bf T}\cap {\bf G}_i$. For $\theta\in\widehat{\bf T}$, define the subset $I(\theta)$ of $I$ by $$I(\theta)=\{i\in I \mid \theta| _{{\bf T}_i} \ \text {is trivial}\}.$$
For $J\subset I(\theta)$, let ${\bf G}_J$ be the subgroup of $\bf G$ generated by ${\bf G}_i$, $i\in J$. We choose a representative $\dot{w}\in {\bf G}_J$ for each $w\in W_J$. Thus, the element $w{\bf 1}_\theta:=\dot{w}{\bf 1}_\theta$ $(w\in W_J)$ is well defined.
For $J\subset I(\theta)$, we set
$$\eta(\theta)_J=\sum_{w\in W_J}(-1)^{\ell(w)}w{\bf 1}_{\theta},$$
and let $\Delta(\theta)_J=\displaystyle \sum_{w\in W}\Bbbk {\bf U}\dot{w}\eta(\theta)_J$, which is a submodule of $\mathbb{M}(\theta)$. In particular, we have $\Delta(\theta)_J=\Bbbk{\bf G}\eta(\theta)_J$.
We define
$$E(\theta)_J=\Delta(\theta)_J/\Delta(\theta)_J',$$
where $\Delta(\theta)_J'$ is the sum of all $\Delta(\theta)_K$ with $J\subsetneq K\subset I(\theta)$.
For any subset $J\subset I$ and $K\subset I(\theta)$, we set
$$
\aligned
X_J &\ =\{x\in W\mid x~\op{has~minimal~length~in}~xW_J\};\\
Z_K &\ =\{w\in X_K \mid \mathscr{R}(ww_K)\subset K\cup (I\backslash I(\theta))\}.
\endaligned
$$
Then by \cite[Proposition 2.7]{CD3}, we have $$E(\theta)_J=\sum_{w\in Z_J}\Bbbk {\bf U}_{w_Jw^{-1}}\dot{w}C(\theta)_J.$$
where $C(\theta)_J$ is the image of $\eta(\theta)_J$ in $E(\theta)_J$. In particular, the following set $$\{u\dot{w}C(\theta)_J \mid w\in Z_J, u\in {\bf U}_{w_Jw^{-1}} \}$$ forms a basis of $E(\theta)_J$.
The following theorem (see \cite[Theorem 3.1]{CD3}) gives all the composition factors of $\mathbb{M}(\theta)$ explicitly.
\begin{Thm}\label{EJ}
All the $\Bbbk {\bf G}$-modules $E(\theta)_J$ $(J\subset I(\theta))$ are irreducible and pairwise non-isomorphic. In particular, $\mathbb{M}(\theta)$ has exactly $2^{|I(\theta)|}$ composition factors with each of multiplicity one.
\end{Thm}
\medskip
The irreducible $\Bbbk {\bf G}$-modules $E(\theta)_J$ can also be realized by parabolic induction.
Let $\theta\in\widehat{\bf T}$ and $K\subset I(\theta)$. Since $\theta|_{{\bf T}_i}$ is trivial for all $i\in K$, it induces a character (still denoted by $\theta$) of $\overline{\bf T}={\bf T}/{\bf T}\cap[{\bf L}_K,{\bf L}_K]$. Therefore, $\theta$ is regarded as a character of ${\bf L}_K$ by the homomorphism ${\bf L}_K\rightarrow\overline{\bf T}$ (with the kernel $[{\bf L}_K,{\bf L}_K]$), and hence as a character of ${\bf P}_K$ by letting ${\bf U}_K$ acts trivially. We set $\mathbb{M}(\theta, K):=\Bbbk{\bf G}\otimes_{\Bbbk{\bf P}_K}\theta$. Let ${\bf 1}_{\theta, K}$ be a nonzero element in the one-dimensional module $\Bbbk_\theta$ associated to $\theta$. We abbreviate $x{\bf 1}_{\theta, K}:=x\otimes{\bf 1}_{\theta, K} \in \mathbb{M}(\theta, K)$ as before.
Now set $\mathscr{R}(w)=\{i\in I\mid ws_i< w \}$ and $Y_K= \{ w\in W\mid \mathscr{R}(w)\subset I\backslash K \}$. Then by the same argument of \cite[Lemma 6.2]{D}, we have
$$\mathbb{M}(\theta, K)=\sum_{w\in Y_K}\Bbbk {\bf U}_{w^{-1}}\dot{w}{\bf 1}_{\theta, K} $$
and moreover, the following set $$\{ u \dot{w}{\bf 1}_{\theta, K} \mid w\in Y_K, u\in {\bf U}_{w^{-1}}\}$$ is a basis of
$\mathbb{M}(\theta, K)$. One has that $\mathbb{M}(\theta, K)$ is
a indecomposable $\Bbbk {\bf G}$-module. Indeed, by the same discussion of \cite[Proposition 2.2]{CD3}, we consider the endomorphism algebra of $\mathbb{M}(\theta, K)$ and have
$$\op{End}_{\bf G}(\mathbb{M}(\theta, K))\cong \op{Hom}_{{\bf P}_K}(\Bbbk_{\theta}, \mathbb{M}(\theta, K))\cong \Bbbk$$ which implies that $\mathbb{M}(\theta, K)$ is indecomposable. Using the same proof of \cite[Theorem 6.3]{D} and \cite[Corollary 3.8]{CD1}, we have the following proposition.
\begin{Prop}\label{Parabolic}
Let $\theta\in\widehat{\bf T}$. For $K \subset I(\theta)$, we have $$\mathbb{M}(\theta, K) \cong \displaystyle \mathbb{M}(\theta)\Big{/} \sum_{s\in {K}}\Delta(\theta)_{\{s\}}.$$ Thus all the composition factors of $\mathbb{M}(\theta, K)$ are $E(\theta)_J$ with $J\subset I(\theta)\backslash K $.
\end{Prop}
For $J\subset I(\theta)$, set $J'=I(\theta)\backslash J$ and we denote by $\nabla(\theta)_J= \mathbb{M}(\theta, J')=\Bbbk{\bf G}\otimes_{\Bbbk{\bf P}_{J'}}\Bbbk_\theta$. Let $E(\theta)_J'$ be the submodule of $\nabla(\theta)_J$ generated by $$D(\theta)_J:=\sum_{w\in W_J}(-1)^{\ell(w)}\dot{w}{\bf 1}_{\theta, J'}.$$
We see that $E(\theta)_J'$ is isomorphic to $E(\theta)_J$ as $\Bbbk {\bf G}$-modules by \cite[Proposition 1.9]{CD3}. Therefore $E(\theta)_J$ can be regarded as the scole of $\nabla(\theta)_J$.
\medskip
At the end of this section we introduce a category $\mathscr{O}({\bf G})$ called principal representation category. It is the full subcategory of $\Bbbk{\bf G}$-Mod such that any object $M$ in $\mathscr{O}({\bf G})$ is of finite length and its composition factors are $E(\theta)_J$ for some $\theta \in \widehat{\bf T}$ and $J\subset I(\theta)$. Thus $\mathscr{O}({\bf G})$ is an abelian category. The category $\mathscr{O}({\bf G})$ is obviously noetherian and artinian by its construction.
By the previous discussion we already have three interesting kinds of modules in $\mathscr{O}({\bf G})$, the irreducible modules $E(\theta)_J$, the modules
$\Delta(\theta)_J$ and the modules $\nabla(\theta)_J$. These modules are frequently used in the following sections.
\section{Injective Objects in $\mathscr{O}({\bf G})$ }
In this section, we assume that $\Bbbk$ is an algebraically closed field of characteristic $0$ (e.g., $\Bbbk = \mathbb{C}$). Under this assumption we will consider the injective objects in the principal representation category $\mathscr{O}({\bf G})$.
Firstly, for each $\theta\in\widehat{\bf T}$, the space $\op{Hom}_{\Bbbk {\bf B}}(\Bbbk {\bf G}, \Bbbk_{\theta})$ is a $\Bbbk {\bf G}$-module whose module structure is given by $$(g \varphi) (x)= \varphi (xg), \ \ \text{where} \ \varphi \in \op{Hom}_{\Bbbk {\bf B}}(\Bbbk {\bf G}, \Bbbk_{\theta}), g\in {\bf G}, x\in \Bbbk {\bf G}.$$
Since this space $\op{Hom}_{\Bbbk {\bf B}}(\Bbbk {\bf G}, \Bbbk_{\theta})$ has uncountable dimension, it is not an object in $\mathscr{O}({\bf G})$ generally. However we have
\begin{Lem} \label{injective} For each $\theta\in\widehat{\bf T}$, $\op{Hom}_{\Bbbk {\bf B}}(\Bbbk {\bf G}, \Bbbk_{\theta})$ is an injective $\Bbbk {\bf G}$-module.
\end{Lem}
\begin{proof} Using the setting and properties in \cite[Section 2]{FS}, when $\Bbbk$ is an algebraically closed field of characteristic zero, $\Bbbk {\bf B}$ is a locally Wedderburn algebra. Thus by \cite[Lemma 3]{FS}, we see that $\Bbbk_{\theta}$ is an injective $\Bbbk {\bf B}$-module. Therefore $\op{Hom}_{\Bbbk {\bf B}}(\Bbbk {\bf G}, \Bbbk_{\theta})$ is an injective $\Bbbk {\bf G}$-module for each $\theta\in\widehat{\bf T}$.
\end{proof}
\begin{Rem} According to in \cite[Theorem 1]{FS}, the trivial $\Bbbk {\bf T}$-module is injective if and only if the order of no elements in ${\bf T}$ vanishes in $\Bbbk$. Thus if $\op{char} \Bbbk > 0$, the trivial $\Bbbk {\bf T}$-module is not injective in general. Then the trivial $\Bbbk {\bf B}$-module is also not injective. However I guess that the $\Bbbk {\bf G}$-module $\op{Hom}_{\Bbbk {\bf B}}(\Bbbk {\bf G}, \Bbbk_{\theta})$ can also be injective. We need to prove it in other methods.
\end{Rem}
\medskip
Using the Bruhat decomposition, each element $g\in {\bf G}$ has the unique expression $g=b\dot{w}u$ for some $w\in W$, $b\in {\bf B}$ and $u\in {\bf U}$.
For $\theta\in\widehat{\bf T}$ and an element $w\in W$, we consider the function
$\rho_{ w, \theta}\in \op{Hom}_{\Bbbk {\bf B}}(\Bbbk {\bf G}, \Bbbk_{\theta})$ defined by
$$\rho_{ w, \theta}(b\dot{w}'u)=\delta_{w,w'}\theta(b)$$
where $\delta_{w,w'}$ is the kronecker symbol. Now we let $$\mathbb{X}_{\theta}=\sum_{w\in W}\Bbbk {\bf G} \rho_{ w, \theta}$$
which is the $\Bbbk {\bf G}$-submodule generated by $\rho_{ w, \theta}, w\in W$.
The Weyl group $W$ acts naturally on $\widehat{\bf T}$ by
$$(w\cdot \theta ) (t):=\theta^w(t)=\theta(\dot{w}t\dot{w}^{-1})$$
for any $\theta\in\widehat{\bf T}$. Denote by
$W_{\theta}$ the stabilizer of $\theta\in\widehat{\bf T}$. Thus the parabolic subgroup $W_{I(\theta)}$ is a subgroup of $W_{\theta}$. For any $w_1, w_2 \in W$, we see that $\theta^{w_1}= \theta^{w_2} $ if and only if $w_1w_2^{-1} \in W_{\theta}$.
\begin{Lem} \label{generatedmodule}
One has that $\Bbbk {\bf G} \rho_{ w, \theta}\cong \mathbb{M}(\theta^{w})$ as $\Bbbk {\bf G}$-modules for each $w\in W$.
\end{Lem}
\begin{proof} Firstly using Frobenius reciprocity we have
$$ \op{Hom}_{\Bbbk {\bf G}} (\mathbb{M}(\theta^w) , \Bbbk {\bf G} \rho_{ w, \theta}) \cong \op{Hom}_{\Bbbk {\bf B}}({\bf 1}_{\theta^w}, \Bbbk {\bf G} \rho_{ w, \theta} ) \cong \Bbbk.$$
Thus it is not difficult to see that $\Bbbk {\bf G} \rho_{ w, \theta}$ is a quotient $\Bbbk {\bf G}$-module of $\mathbb{M}(\theta^w)$.
Since the $\Bbbk {\bf G}$-module $\Delta(\theta)_{I(\theta^w)}$ is the socle of $\mathbb{M}(\theta^w)$, to get $\Bbbk {\bf G} \rho_{ w, \theta}\cong \mathbb{M}(\theta^w)$ as $\Bbbk {\bf G}$-modules, it is enough to show that the element $$\sum_{x\in W_{I(\theta^w)}}(-1)^{\ell(x)}\dot{x}\rho_{ w, \theta}\ne 0.$$
Indeed, we have
$$\sum_{x\in W_{I(\theta^w)}}(-1)^{\ell(x)}\dot{x}\rho_{ w, \theta} (b\dot{w})=\sum_{x\in W_{I(\theta^w)}}(-1)^{\ell(x)}\rho_{ w, \theta} (b\dot{w} \dot{x}) \ne 0.$$
Thus the lemma is proved. In particular, we have $\Bbbk {\bf G} \rho_{ e, \theta}\cong \mathbb{M}(\theta)$ as $\Bbbk {\bf G}$-modules, where $e$ is the neutral element of $W$.
\end{proof}
For a fixed $\theta\in\widehat{\bf T}$ and $w\in W$, let $J_{w,\theta}= \mathscr{R}(w) \cap I(\theta^w)$. Then $w$ has the unique form $w=x\vartheta_{J_{w,\theta}}$, where $\vartheta_{J_{w,\theta}}$ is the longest element in $W_{J_{w,\theta}}$. In this case we set
$$\varsigma_{w,\theta}= \sum_{v\in W_{J_{w,\theta}}}\rho_{ xv, \theta} \ \ \text{and} \ \ \mathbb{X}_{w,\theta}= \Bbbk {\bf G} \varsigma_{w,\theta} .$$
\begin{Lem} \label{direct summand}
For each $\theta\in\widehat{\bf T}$ and $w\in W$, one has that
$$ \mathbb{X}_{w,\theta} \cong \Bbbk{\bf G}\otimes_{\Bbbk{\bf P}_{J_{w,\theta}}}\Bbbk_{\theta^w}$$ as $\Bbbk {\bf G}$-modules. Then we have $ \mathbb{X}_{\theta} \displaystyle \cong \bigoplus_{w\in W}\mathbb{X}_{w,\theta} $ as $\Bbbk {\bf G}$-modules.
\end{Lem}
\begin{proof} By the setting of $\varsigma_{w,\theta}$, it is easy to check that
$t \varsigma_{w,\theta} =\theta^w(t) \varsigma_{w,\theta}$ for any $t\in {\bf T}.$ Moreover for any $z\in W$ and $s\in J_{z,\theta}$, we have
$$\dot{s}(\rho_{ z, \theta} + \rho_{ zs, \theta} ) =\rho_{ z, \theta} + \rho_{ zs, \theta}.$$
Thus we get $\dot{s}\varsigma_{w,\theta}=\varsigma_{w,\theta}$ for any $s \in J_{w,\theta}$. Hence
$ \mathbb{X}_{w,\theta}$ is a quotient $\Bbbk {\bf G}$-module of $\Bbbk{\bf G}\otimes_{\Bbbk{\bf P}_{J_{w,\theta}}}\Bbbk_{\theta^w}$. Noting that $E(\theta^w)_{I(\theta^w)\setminus{J_{w,\theta}}}$ is the socle of $\Bbbk{\bf G}\otimes_{\Bbbk{\bf P}_{J_{w,\theta}}}\Bbbk_{\theta^w}$, by the same discussion as Lemma \ref{generatedmodule}, we get $$ \mathbb{X}_{w,\theta} \cong \Bbbk{\bf G}\otimes_{\Bbbk{\bf P}_{J_{w,\theta}}}\Bbbk_{\theta^w}$$ as $\Bbbk {\bf G}$-modules. Therefore each $\mathbb{X}_{w,\theta} $ is a indecomposable $\Bbbk {\bf G}$-module.
It is easy to verify that
$ \mathbb{X}_{\theta} = \displaystyle \sum_{w\in W}\mathbb{X}_{w,\theta} $ as $\Bbbk {\bf G}$-modules. Indeed, it is obvious to see that $\displaystyle \sum_{w\in W}\mathbb{X}_{w,\theta} \subseteq \mathbb{X}_{\theta} $. On the other hand, we can do induction on the length of the elements in $W$ to show that $\displaystyle \rho_{z, \theta} \in \sum_{w\in W}\mathbb{X}_{w,\theta}$ for any $z\in W$. Noting that $\mathbb{X}_{\theta}$ is generated by $\rho_{ w, \theta}$ with $w\in W$. Then we have $ \mathbb{X}_{\theta} = \displaystyle \sum_{w\in W}\mathbb{X}_{w,\theta} $.
Given an order of the elements in $$W=\{w_1, w_2, \dots, w_n\}$$
such that $\ell(w_i)\leq \ell(w_j)$ for $i<j$. In the following we show that
$$\mathbb{X}_{w_k ,\theta} \cap (\sum_{i< k}\mathbb{X}_{w_i ,\theta})= 0$$
for any $k=1,2,\dots, n$, which implies that $\displaystyle \mathbb{X}_{\theta} \cong \bigoplus_{w\in W}\mathbb{X}_{w,\theta} $ as $\Bbbk {\bf G}$-modules. Now suppose that there exists an integer $k$ such that
$$\mathbb{X}_{w_k ,\theta} \cap (\sum_{i< k}\mathbb{X}_{w_i ,\theta})\ne 0.$$
For convenience we set $\Gamma= I(\theta^{w_k})\setminus{J_{w_k,\theta}}$. Using $\mathbb{X}_{w_k ,\theta} \cong \Bbbk{\bf G}\otimes_{\Bbbk{\bf P}_{J_{w_k,\theta}}}\Bbbk_{\theta^{w_k}} $,
one has that $E(\theta^{w_k})_{\Gamma}$ is the socle of $\mathbb{X}_{w_k ,\theta}$.
Thus we get
$$ \sum_{x\in W_\Gamma}(-1)^{\ell(x)}\dot{x} \varsigma_{w_k,\theta} \in \sum_{i< k}\mathbb{X}_{w_i ,\theta}.$$
However we have the following fact that for any $w\ne w'$ with $\ell(w)\leq \ell(w')$, there exists an element $u\in {\bf U}$ such that
$\dot{x} \rho_{w,\theta}(\dot{w}' u)\ne 0$ for any $x\in W$. Using this fact we get a contradiction. Actually, for any element $\varphi \in \displaystyle \sum_{i< k}\mathbb{X}_{w_i ,\theta}$, there exists an element $u\in {\bf U}$ such that
$$\sum_{x\in W_\Gamma}(-1)^{\ell(x)}\dot{x} \varsigma_{w_k,\theta}(\dot{w_k} u)\ne 0 \ \ \text{and} \ \ \ \varphi(\dot{w_k} u)=0. $$
Thus the lemma is proved.
\end{proof}
\begin{Prop}
Let $\theta\in\widehat{\bf T}$, we have
$$ \op{Hom}_{\Bbbk {\bf G}}(M, \mathbb{X}_{\theta} ) \cong \op{Hom}_{\Bbbk {\bf G}} (M, \op{Hom}_{\Bbbk {\bf B}}(\Bbbk {\bf G}, \Bbbk_{\theta}))$$
for any simple object $M\in \mathscr{O}({\bf G})$.
\end{Prop}
\begin{proof} Each simple object in $\mathscr{O}({\bf G})$ has the form $E(\lambda)_J$ for some $\lambda \in \widehat{\bf T}$ and $J\subset I(\lambda)$. Firstly, we have
$$\op{Hom}_{\Bbbk {\bf G}} (E(\lambda)_J, \op{Hom}_{\Bbbk {\bf B}}(\Bbbk {\bf G}, \Bbbk_{\theta})) \cong \op{Hom}_{\Bbbk {\bf B}} (E(\lambda)_J, \Bbbk_{\theta})$$
by Frobenius reciprocity.
Since $E(\lambda)_J= \displaystyle \sum_{w\in Z_J} \Bbbk {\bf U}_{w_Jw^{-1}}w C(\lambda)_J$, we have
$$E(\lambda)_J\cong \bigoplus_{w\in Z_J} \Bbbk {\bf U}_{w_Jw^{-1}}w C(\lambda)_J$$
as $\Bbbk {\bf B}$-modules. Then we get
$$\dim \op{Hom}_{\Bbbk {\bf B}} (E(\lambda)_J, \Bbbk_{\theta}) =\sharp \{w \in Z_J\mid \lambda=\theta^w \}.$$
On the other hand, by Lemma \ref{direct summand}, we have $$ \op{Hom}_{\Bbbk {\bf G}}(E(\lambda)_J, \mathbb{X}_{\theta} )\cong \bigoplus_{w\in W}\op{Hom}_{\Bbbk {\bf G}}(E(\lambda)_J, \mathbb{X}_{w, \theta} ), $$
which implies that
$$\dim \op{Hom}_{\Bbbk {\bf G}}(E(\lambda)_J, \mathbb{X}_{\theta} )=\sharp \{w \in W \mid \lambda=\theta^w \ \text{and} \ J_{w,\theta}= I(\lambda)\setminus J \}$$
For fixed $\theta \in \widehat{\bf T}$, $\lambda \in \widehat{\bf T}$ and $J\subset I(\lambda)$, we let $$\Omega_{\theta}(\lambda, J)= \{w\in W \mid \lambda=\theta^w \ \text{and}\ J\subset \mathscr{R}(w) \subset J\cup (I\setminus I(\lambda)) \}.$$ Then it is easy to check that
$$\dim \op{Hom}_{\Bbbk {\bf B}} (E(\lambda)_J, \Bbbk_{\theta}) = |\Omega_{\theta}(\lambda, J)|,$$
$$ \dim \op{Hom}_{\Bbbk {\bf G}}(E(\lambda)_J, \mathbb{X}_{\theta} )= |\Omega_{\theta}(\lambda, I(\lambda) \setminus J)|. $$
In the following we will show that $|\Omega_{\theta}(\lambda, J)|= |\Omega_{\theta}(\lambda, I(\lambda) \setminus J)|$.
Let $x_1, x_2, \dots, x_m$ be a complete representative set of the left cosets of $W_{I(\lambda)}$ in $W$. For each $w\in \Omega_{\theta}(\lambda, J)$ with the form $w=x_i y$ for some $i=1,2,\dots, m$ and $y\in W_{I(\lambda)}$. Since $\theta^w =\theta^{x_i y}=\lambda$ and $y\in W_{I(\lambda)}$, we get $\theta^{x_i}=\lambda$.
Now we set $\Xi(w)=x_i w_{I(\lambda)}y$. Then we also have $\theta^{\Xi(w)}=\theta^{x_i}=\lambda$. Thus it is easy to check that
$$\Xi(w)=x_i w_{I(\lambda)}y \in \Omega_{\theta}(\lambda, I(\lambda) \setminus J)$$
which gives a bijecction $\Xi: \Omega_{\theta}(\lambda, J)\longrightarrow \Omega_{\theta}(\lambda, I(\lambda) \setminus J) $. Hence we have
$$ \op{Hom}_{\Bbbk {\bf G}}(E(\lambda)_J, \mathbb{X}_{\theta} ) \cong \op{Hom}_{\Bbbk {\bf G}} (E(\lambda)_J, \op{Hom}_{\Bbbk {\bf B}}(\Bbbk {\bf G}, \Bbbk_{\theta}))$$
by the above discussion.
\end{proof}
\begin{Conj} \label{conjecture}
Let $\theta\in\widehat{\bf T}$, one has that
$$ \op{Hom}_{\Bbbk {\bf G}}(M, \mathbb{X}_{\theta} ) \cong \op{Hom}_{\Bbbk {\bf G}} (M, \op{Hom}_{\Bbbk {\bf B}}(\Bbbk {\bf G}, \Bbbk_{\theta}))$$
for any object $M\in \mathscr{O}({\bf G})$. Thus $\mathbb{X}_{\theta} $ is an injective object in $\mathscr{O}({\bf G})$ for each $\theta\in \widehat{\bf T}$.
\end{Conj}
By Lemma \ref{direct summand}, each $ \mathbb{X}_{w,\theta}$ is a direct summand of $ \mathbb{X}_{\theta}$, thus $\nabla(\theta)_J=\Bbbk{\bf G}\otimes_{\Bbbk{\bf P}_{J'}}\Bbbk_\theta$ is an injective object in $\mathscr{O}({\bf G})$ for any $J\subset I(\theta)$. So $\nabla(\theta)_J$ is the injective envelope of $E(\theta)_J$.
Therefore Conjecture \ref{conjecture} suggests the existence of enough injectives in the category $\mathscr{O}({\bf G})$.
\begin{Thm} \label{enough injectives}
The category $\mathscr{O}({\bf G})$ has enough injectives.
\end{Thm}
\begin{proof} We do induction on the length of $M\in \mathscr{O}({\bf G})$. Firstly, $\nabla(\theta)_J= \Bbbk{\bf G}\otimes_{\Bbbk{\bf P}_{J'}}\Bbbk_\theta$ is the injective envelope of
$E(\theta)_J$. Assuming that $M$ has length bigger than $1$, it has a simple quotient $E(\theta)_J$.
Then we get a short exact sequence
$$0\rightarrow N \xrightarrow{f} M \xrightarrow{g} E(\theta)_J \rightarrow 0.$$ So by induction there exists a monomorphism $N\xrightarrow{i} Q$ for some injective $Q\in \mathscr{O}({\bf G})$. Then there exists a morphism $M\xrightarrow{h} Q$ such that $i=h f$. Therefore either $M\xrightarrow{h} Q$ is a monomorphism or
$M\cong N\oplus E(\theta)_J$. In the latter case, $M\rightarrow Q\oplus \nabla(\theta)_J$ is momomorphism and the theorem is proved.
\end{proof}
\section{Highest Weight Category $\mathscr{O}({\bf G})$ }
From now on, we assume that Conjecture \ref{conjecture} is established. Thus $\mathscr{O}({\bf G})$ has enough injectives. We will show that the principal representation category $\mathscr{O}({\bf G})$ is a highest weight category.
Firstly we recall the definition of highest weight categories (see \cite{CPS}).
\begin{Def}\label{HWC} Let $\mathscr{C}$ be a locally artinian, abelian, $\Bbbk$-linear category with enough injectives that satisfies Grothendieck's condition. Then we call $\mathscr{C}$ a highest weight category if there exists a locally finite poset $\Lambda$ (the "weights" of $\mathscr{C}$), such that:
(a) There is a complete collection $\{S(\lambda)_{\lambda \in \Lambda}\}$ of non-isomorphic simple objects of $\mathscr{C}$ indexed by the set $\Lambda$.
(b) There is a collection $\{A(\lambda)_{\lambda \in \Lambda}\}$ of objects of $\mathscr{C}$ and, for each $\lambda$, an embedding $S(\lambda)\subset A(\lambda) $ such that all composition factors $S(\mu)$ of $A(\lambda)/S(\lambda)$ satisfy $\mu < \lambda$. For $\lambda,\mu \in \Lambda$,
we have that $dim_{\Bbbk}\op{Hom}_{\mathscr{C}}(A(\lambda), A(\mu))$ and $[A(\lambda): S(\mu)]$ are finite.
(c) Each simple object $S(\lambda)$ has an injective envelope $I(\lambda)$ in $\mathscr{C}$.
Also, $I(\lambda)$ has a good filtration $0= F_0(\lambda)\subset F_1(\lambda)\subset \dots $ such that:
\noindent (i) $F_1(\lambda)\cong A(\lambda)$;
\noindent (ii) for $n>1$, $F_n(\lambda)/F_{n-1}(\lambda) \cong A(\mu)$ for some $\mu=\mu(n)> \lambda$;
\noindent (iii) for a given $\mu \in \Lambda$, $\mu=\mu(n)$ for only finitely many $n$;
\noindent (iv) $\bigcup F_i(\lambda)= I(\lambda)$.
\end{Def}
Now we show that $\mathscr{O}({\bf G})$ is a highest weight category. In Definition \ref{HWC}, the set of weights is $\Lambda= \{(\theta, J )\ | \ \theta \in \widehat{\bf T}, J \subset I(\theta) \}$ and we define the order by
$$(\theta_1, J_1) \leq (\theta_2, J_2 ), \ \text{if}\ \theta_1=\theta_2 \ \text{and} \ J_1\supseteq J_2.$$
Set $S(\lambda)=A(\lambda)=E(\theta)_J$ and $I(\lambda)= \nabla(\theta)_J$, then the condition in Definition \ref{HWC} is easy to check. So $\mathscr{O}({\bf G})$ is a highest weight category.
The highest weight category has many good properties (see \cite{CPS}). We list some interesting propositions of the category $\mathscr{O}({\bf G})$.
\begin{Prop}\label{Extension} \label{Ext} (1) For $n\geq 0$, $\op{Ext}^n_{\mathscr{O}({\bf G})}(M,N)$ is finite-dimensional for all $M,N\in \mathscr{O}({\bf G})$.
\noindent $(2)$ If $\op{Ext}^n_{\mathscr{O}({\bf G})}(E(\lambda)_J,E(\mu)_K)\ne 0$, then $\lambda=\mu$ and $J\subseteq K$. Moreover, if $n>0$, we have $\lambda=\mu$ and $J\subsetneq K$.
\end{Prop}
According to \cite[Corollary 5.6]{CD3}, we know that any finite-dimensional irreducible representations of $\bf G$ are one-dimensional when $\Bbbk$ is algebraically closed with $\op{char}\Bbbk\neq\op{char}\bar{\mathbb{F}}_q$. Moreover, these irreducible representations are isomorphic to $E(\theta)_\varnothing$ for some $\theta\in\widehat{\bf T}$ with $I(\theta)=I$. By Proposition \ref{Extension} we have the following corollary immediately.
\begin{Cor}\label{fdrep} The finite-dimensional complex irreducible representations of the group ${\bf G}$ is one-dimensional and all the finite-dimensional complex representations of ${\bf G}$ are semisimple.
\end{Cor}
For $\theta\in \widehat{\bf T}$, let $\mathscr{O}({\bf G})_\theta$ be the subcategory of $\mathscr{O}({\bf G})$ containing the objects whose subquotients are $E(\theta)_J$ for some $J\subset I(\theta)$.
Then by Proposition \ref{Ext}, we have $\mathscr{O}({\bf G})= \displaystyle \bigoplus_{\theta\in \widehat{\bf T}} \mathscr{O}({\bf G})_\theta$. For each $\theta\in \widehat{\bf T}$, $\mathscr{O}({\bf G})_\theta$ is a highest weight category and then there exists a finite-dimensional quasi-hereditary algebra $A_\theta$ such that $\mathscr{O}({\bf G})_\theta$ is equivalent to the right $A_\theta$-modules.
Indeed, if we set $ \mathscr{I}_{\theta}=\displaystyle\bigoplus_{J\subset I(\theta)}\nabla(\theta)_J$, then $A_\theta\cong \op{End}_{\mathscr{O}({\bf G})}(\mathscr{I}_{\theta})$. The functor $\op{Hom}_{\bf G}(-, \mathscr{I}_{\theta})^*$ form $\mathscr{O}({\bf G})_\theta$ to the right $A_\theta$-modules is an equivalence of categories. Therefore we also see that the category $\mathscr{O}({\bf G})$ is a Krull-Schmidt category.
\medskip
Now we want to understand the structure of the algebra $A_\theta$, we have
$$A_\theta\cong \op{End}_{\mathscr{O}({\bf G})}(\mathscr{I}_{\theta})\cong \bigoplus_{J\subset I(\theta)}\op{Hom}_{\mathscr{O}({\bf G})}(\nabla(\theta)_ J, \mathscr{I}_{\theta} ).$$
The composition factors of $\nabla(\theta)_ J= \mathbb{M}(\theta, J')$ with $J'=I(\theta)\backslash J$ are given in Proposition \ref{Parabolic}, therefore
$$
\aligned \op{Hom}_{\mathscr{O}({\bf G})}(\nabla(\theta)_ J, \mathscr{I}_{\theta} )
= &\ \op{Hom}_{\mathscr{O}({\bf G})}(\nabla(\theta)_ J, \bigoplus_{K\subset I(\theta) }\nabla(\theta)_ K)\\
= &\ \bigoplus_{K\subset J}\op{Hom}_{\mathscr{O}({\bf G})}(\nabla(\theta)_ J, \nabla(\theta)_ K)\\
\cong &\ \bigoplus_{K\subset J} \op{Hom}_{{\bf P}_{J'}}({\bf 1}_{\theta, J'} ,\mathbb{M}(\theta, K')).
\endaligned
$$
So if let $\varphi_{K\subset J}$ be a $\Bbbk {\bf G}$-module morphism such that $\varphi_{K\subset J} ({\bf 1}_{\theta, J'})= {\bf 1}_{\theta, K'}$, then we have
$$\op{Hom}_{\mathscr{O}({\bf G})}(\nabla(\theta)_ J, \nabla(\theta)_ K) \cong \Bbbk \varphi_{K\subset J}.$$ Now the $\Bbbk$-algebra $A_\theta$ has a
$\Bbbk$ -basis $\{\varphi_{K\subset J}\mid K\subset J \subset I(\theta)\}$
and the multiplications are given by
$$\varphi_{K\subset M} \varphi_{L\subset J} =\left\{
\begin{array}{ll}
\varphi_{K\subset J} &\ \mbox{if}~L=M,\\
0 &\ \mbox{otherwise.}
\end{array}\right.$$
\begin{Cor} If $|I(\lambda)|=|I(\mu)|$, then $A_\lambda \cong A_\mu$ as $\Bbbk$-algebras and
thus $\mathscr{O}({\bf G})_\lambda$ is equivalent to $\mathscr{O}({\bf G})_\mu$.
\end{Cor}
For each $\theta\in \widehat{\bf T}$, the right $A_\theta$-modules has enough projectives. Then $\mathscr{O}({\bf G})_\theta$ and hence $\mathscr{O}({\bf G})$ also have enough projectives. Moreover we have the following proposition.
\begin{Prop}\label{enough projectives} For $\theta\in \widehat{\bf T}$ and $J\subset I(\theta)$, $\Delta(\theta)_J$ is the projective cover of $E(\theta)_J$.
\end{Prop}
\begin{proof} The functor $\op{Hom}_{\bf G}(-, \mathscr{I}_{\theta})^*$ form $\mathscr{O}({\bf G})_\theta$
to the right $A_\theta$-modules keeps the projectives.
So it is enough to show that $\op{Hom}_{\bf G}(\Delta(\theta)_J , \mathscr{I}_{\theta})^*$ is a projective right $A_\theta$-module.
We denote $\varphi_J:=\varphi_{K\subset J}$ when $J=K$. As a right $A_\theta$-module, $A_\theta$
has a decomposition $A_\theta=\displaystyle \bigoplus_{J\subset I(\theta)} \varphi_J A_\theta$ and each $\varphi_J A_\theta$ is indecomposable projective. In the following we will show that $\op{Hom}_{\bf G}(\Delta(\theta)_J , \mathscr{I}_{\theta})^* \cong \varphi_J A_\theta$.
All the composition factors of $\Delta(\theta)_J$ are $E(\theta)_K$ with $J\subset K \subset I(\theta)$. Thus
$$
\aligned \op{Hom}_{\bf G}(\Delta(\theta)_J , \mathscr{I}_{\theta})&\
= \bigoplus_{K\subset I(\theta)}\op{Hom}_{\bf G}(\Delta(\theta)_J , \nabla(\theta)_K)\\
&\ = \bigoplus_{J\subset K\subset I(\theta)}\op{Hom}_{\bf G}(\Delta(\theta)_J , \nabla(\theta)_K).
\endaligned
$$
Let $f_{K \supset J}$ be the $\Bbbk {\bf G}$-module morphism such that
$f_{K \supset J}(\eta(\theta)_J)= {\bf 1}_{\theta, K'},$ where $K'=I(\theta)\backslash K$. Then we get
$$\op{Hom}_{\bf G}(\Delta(\theta)_J , \nabla(\theta)_K)\cong \Bbbk f_{K \supset J}.$$ Thus $\{f_{K \supset J} \mid J\subset K\subset I(\theta)\}$ is a basis of $\op{Hom}_{\bf G}(\Delta(\theta)_J , \mathscr{I}_{\theta})$.
The operation of $A_\theta$ on this basis is:
$$\varphi_{L\subset M} f_{K\supset J} =\left\{
\begin{array}{ll}
f_{L\supset J} &\ \mbox{if}~K=M \ \text{and}\ L\supset J, \\
0 &\ \mbox{otherwise.}
\end{array}\right.$$
Let $\widetilde{f}_{K\supset J}$ be the dual basis of $f_{K\supset J}$ and set $\widetilde{f}_{J}=\widetilde{f}_{K\supset J}$ when $J=K$. It is easy to check that the homomorphism
send $\widetilde{f}_{J}$ to $\varphi_J$ induces an isomorphism
$$\op{Hom}_{\bf G}(\Delta(\theta)_J , \mathscr{I}_{\theta})^* \cong \varphi_J A_\theta$$ as right $A_\theta$-modules.
Therefore $\Delta(\theta)_J$ is the projective cover of $E(\theta)_J$ and the proposition is proved.
\end{proof}
\section{The Algebras $\mathscr{A}_n$}
Assume that $X$ is a finite set with cardinality $|X|=n\geq 1$.
Denote by $M_{2^n}(\Bbbk)$ the matrix algebra over $\Bbbk$. The rows and columns of a matrix in this algebra are indexed by the subsets of $X$.
Fix an order of the subsets of $X$, we let $$\mathscr{A}_n= \{ (a_{Y,Z})\in M_{2^n}(\Bbbk) \mid a_{Y,Z}=0 \ \text{if}~Y ~\text{is not a subset of } Z \}$$
which is a subalgebra of the matrix algebra $ M_{2^n}(\Bbbk)$ whose rows and columns are indexed by the subsets of $X$. The algebra $\mathscr{A}_n$ just depends on the cardinality of $X$.
Let $\{e_{Y,Z} \mid Y\subset Z\subset X\}$ be the standard basis of $\mathscr{A}_n$. Without lost of generality, we can assume each element in $\mathscr{A}_n$ has the form of a upper triangular matrix. The Jacobson radical of $\mathscr{A}_n$ is
$\text{Rad}(\mathscr{A}_n)=\displaystyle \sum_{Y\varsubsetneq Z}\Bbbk e_{Y,Z}$. As a right $\mathscr{A}_n$-module, $\mathscr{A}_n$ has a decomposition
$$\mathscr{A}_n= \bigoplus_{Y\subset X}e_{Y,Y} \mathscr{A}_n$$
and for each $Y\subset X$, $e_{Y,Y} \mathscr{A}_n$ is a indecomposable projective module.
For each integer $n$, it is not difficult to see that $\mathscr{A}_n$
is a basic and connected algebra.
Therefore there exists a bound quiver $(\mathcal{Q}, \mathcal{I})$ such that $\Bbbk \mathcal{Q} /\mathcal{I} \cong \mathscr{A}_n $ (see \cite[Chapter II]{ASS}).
As a matter of fact, we can construct the bound quiver $(\mathcal{Q}, \mathcal{I})$ associate to $X$ with $|X|=n$. The vertices $Q_0$ of the quiver $\mathcal{Q}=(Q_0, Q_1, s,t)$ is indexed by the subsets of $X$ and we denote it by $Q_0=\{i_Y\mid Y\subset X\}$. For two vertices $i_Y, i_Z \in Q_0$, there exists an edge $\alpha \in Q_1$ between them if $Y\subset Z$ with $|Z\backslash Y|=1$ and the orientation is given by $s(\alpha)=i_Y$, $t(\alpha)=i_Z$. We denote such arrow by $\alpha_{Y,Z}$.
The admissible ideal $\mathcal{I}$ in $\Bbbk \mathcal{Q}$ is generated by all elements $\omega_1-\omega_2$
given by the pairs $\{\omega_1,\omega_2\}$ of paths in $\mathcal{Q}$ having the same starting and ending vertices.
Then we have $\Bbbk \mathcal{Q} /\mathcal{I} \cong \mathscr{A}_n $. It is not difficult to see that the algebra $A_\theta$ is isomorphic to the algebra $\mathscr{A}_n$ with $n=|I(\theta)|$.
\medskip
For $n=1$, the algebra $\mathscr{A}_1$ is the path algebra of the Dynkin quiver of type $A_2$. The number of isomorphism classes of indecomposable representations of $\mathscr{A}_1$ is $3$. By the correspondence of
$\mathscr{O}({\bf G})_\theta$ and the right $A_\theta$-modules under the assumption $\Bbbk= \mathbb{C}$, then for ${\bf G}=SL_2$, the indecomposable modules in $\mathscr{O}({\bf G})$ is $\mathbb{M}(\op{tr}), \op{St}, \Bbbk_{\op{tr}}$ and $\{\mathbb{M}(\theta)\mid \theta \in \widehat{\bf T} \ \text{nontrivial}\}$. Therefore each module in $\mathscr{O}({\bf G})$ is a direct sum of these modules.
\medskip
For $n=2$, the incidence algebra $\mathscr{A}_2$ is isomorphic the algebra given by the following quiver
\begin{center}
\begin{tikzpicture}[scale=1]
\draw (4,0) node (I11) {$a$} +(1.2,0.6) node (I12) {$b$} +(1.2,-0.6) node (I21) {$c$} + (2.4,0) node (I22) {$d$};
\draw[->] (I11)--(I12) node[pos=.5,above] {$\alpha$};
\draw[->] (I12)--(I22) node[pos=.5,above] {$\beta$};
\draw[->] (I11)--(I21) node[pos=.5,above] {$\gamma$};
\draw[->] (I21)--(I22) node[pos=.5,above] {$\delta$};
\end{tikzpicture}
\end{center}
bound by the relation $\beta \alpha= \delta \gamma$. Therefore the representations of the algebra $\mathscr{A}_2$ is given by the following diagram
\begin{center}
\begin{tikzpicture}[scale=1]
\draw (4,0) node (I11) {$V_a$} +(2,1) node (I12) {$V_b$} +(2,-1) node (I21) {$V_c$} + (4,0) node (I22) {$V_d$};
\draw[->] (I11)--(I12) node[pos=.5,above] {$f_{ab}$};
\draw[->] (I12)--(I22) node[pos=.5,above] {$f_{bd}$};
\draw[->] (I11)--(I21) node[pos=.5,above] {$f_{ac}$};
\draw[->] (I21)--(I22) node[pos=.5,above] {$f_{cd}$};
\end{tikzpicture}
\end{center}
such that $V_a$, $V_b$, $V_c$, $V_d$ are vector spaces and $f_{ab}$, $f_{bd}$, $f_{ac}$, $f_{cd}$ are linear morphisms which satisfy $f_{bd} f_{ab}= f_{cd}f_{ac}$. The Auslander-Reiten quiver of the algebra $\mathscr{A}_2$ was known in the example of Chap VII.2 of the book \cite{ARS}. Thus the algebra $\mathscr{A}_2$ is of finite type and there are 11 indecomposable $\mathscr{A}_2$-modules up to isomorphism. For an given representation $\mathbb{V}$ of $\mathscr{A}_2$, the dimension vector of $\mathbb{V}$ is denoted by $$\underline{\text{dim}}\mathbb{V}= (\text{dim}V_a, \text{dim}V_b, \text{dim}V_c, \text{dim}V_d ).$$
Hence all the indecomposable $\mathscr{A}_2$-modules (up to isomorphism) are replaced by their dimension vectors which are the following:
$$
\aligned
&\ (1,0,0,0),\ (0,1,0,0),\ (0,0,1,0),\ (0,0,0,1),\ (1,1,0,0),\ (1,0,1,0), \\
&\ (0,1,0,1),\ (0,0,1,1),\ (1,1,1,0),\ (0,1,1,1),\ (1,1,1,1).
\endaligned
$$
Now we consider the category $\mathscr{O}({\bf G})_\theta$ when $|I(\theta)|=2$. Assume $I(\theta)=\{r,s\}$. By the equivalence of $\mathscr{O}({\bf G})_\theta$ and the right $\mathscr{A}_2$-modules, we know that except the four irreducible modules $E(\theta)_{\emptyset}, E(\theta)_{r},E(\theta)_{s}, E(\theta)_{\{r,s\}}$, the indecomposable projective modules $\mathbb{M}(\theta), \Delta(\theta)_{r}, \Delta(\theta)_{s}, \Delta(\theta)_{\{r,s\}}= E(\theta)_{\{r,s\}}$ and the indecomposable injective modules $\nabla(\theta)_{\{r, s\}}=E(\theta)_{\emptyset}, \nabla(\theta)_{r}, \nabla(\theta)_{s}, \mathbb{M}(\theta)$, the remaining indecomposable modules in $\mathscr{O}({\bf G})_\theta$ is $\mathbb{M}(\theta)/\Delta(\theta)_{\{r,s\}}$ and $\Delta(\theta)_{r}+\Delta(\theta)_{s}$.
Therefore all the indecomposable modules of $\mathscr{O}({\bf G})$ are known when the rank of ${\bf G}$ is 2.
\medskip
Before we consider the algebra $\mathscr{A}_n$ for $n\geq 3$, we recall some facts and results about the Tits form of an algebra. For the algebra $\mathscr{A}\cong \Bbbk \mathcal{Q} /\mathcal{I}$, let $R$ be the minimal set of relations which generate the ideal $\mathcal{I}$. Then the Tits form of $\mathscr{A}$ is the integral quadratic
form $q_{\mathscr{A}}: \mathbb{Z}^{m} \rightarrow \mathbb{Z}$ defined by the formula
$$q_{\mathscr{A}}(x)=\sum_{i\in Q_0} x^2_i- \sum_{\alpha\in Q_1} x_{s(\alpha)}x_{t(\alpha)}+ \sum_{i,j\in Q_0}r_{ij}x_i x_j,$$
where $r_{ij}$ is the cardinality of $R\cap \Bbbk Q(i, j)$ and $\Bbbk Q(i, j)$ is the vector space
spanned by the paths from $i$ to $j$ (see \cite[Page 464]{B}). It is well know that if $\mathscr{A}$ is a tame algebra, the Tits form $q_{\mathscr{A}}$ is weakly positive, that is $q_{\mathscr{A}}(x)\geq 0$ for any $x\in \mathbb{Z}^{m}$ with nonnegative coordinates (see \cite[Section 1.3]{P}).
In our setting, the minimal set of relations $R$ which generate the ideal $\mathcal{I}$ contains all the relations
$$\alpha_{Y,U}\alpha_{U,Z}=\alpha_{Y,V}\alpha_{V,Z}$$ with $\alpha_{Y,U},\alpha_{U,Z},\alpha_{Y,V}, \alpha_{V,Z}$ are arrows in $Q_1$. Thus for $i_Y, i_Z \in Q_0$, we have
$$r_{i_Y ,i_Z}=\left\{
\begin{array}{ll}
1 &\ \mbox{if}~Y\subset Z~\mbox{with}~ |Z\backslash Y|=2 ,\\
0 &\ \mbox{otherwise.}
\end{array}\right.$$
Therefore the Tits form of $\mathscr{A}_n$ is
$$q_n(x)=\sum_{Y} x^2_Y- \sum_{Y\subset Z,\ |Z\backslash Y|=1} x_{Y}x_{Z}+ \sum_{Y\subset Z, \ |Z\backslash Y|=2 }x_Y x_Z,$$
where $Y,Z$ are subsets of a fixed set $X$ such that $ |X|=n$.
\medskip
We consider the Tits form of the incidence algebra $\mathscr{A}_3$ which is given by the following quiver
\begin{center}
\begin{tikzpicture}[scale=1]
\draw (4,0) node (I1) {$1$} +(2,1) node (I2) {$2$} +(2,0) node (I3) {$3$} + (2,-1) node (I4) {$4$}+(4,1) node (I5) {$5$}+(4,0) node (I6) {$6$}+(4,-1) node (I7) {$7$}+(6,0) node (I8) {$8$};
\draw[->] (I1)--(I2) node[pos=.5,above] {};
\draw[->] (I1)--(I3) node[pos=.5,above] {};
\draw[->] (I1)--(I4) node[pos=.5,above] {};
\draw[->] (I2)--(I5) node[pos=.5,above] {};
\draw[->] (I2)--(I6) node[pos=.5,above] {};
\draw[->] (I3)--(I5) node[pos=.5,above] {};
\draw[->] (I3)--(I7) node[pos=.5,above] {};
\draw[->] (I4)--(I6) node[pos=.5,above] {};
\draw[->] (I4)--(I7) node[pos=.5,above] {};
\draw[->] (I5)--(I8) node[pos=.5,above] {};
\draw[->] (I6)--(I8) node[pos=.5,above] {};
\draw[->] (I7)--(I8) node[pos=.5,above] {};
\end{tikzpicture}
\end{center}
bounded by six relations, where each one is given by $\omega_1=\omega_2$ of paths having the same starting and ending vertices of a parallelogram like the case of $\mathscr{A}_2$.
Now the Tits form is given by
$$
\aligned q(x)
= &\ x^2_1+x^2_2+x^2_3+x^2_4+x^2_5+x^2_6+x^2_7+x^2_8+x_1x_5+x_1x_6+x_1x_7\\
&\ +x_2x_8+x_3x_8+x_4x_8-x_1x_2-x_1x_3-x_1x_4-x_2x_5-x_2x_6 \\
&\ -x_3x_5-x_3x_7-x_4x_6-x_4x_7-x_5x_8-x_6x_8-x_7x_8.
\endaligned
$$
Do variable substitution with $x_1=x+z, x_8=z, y_1=x_2-x_5, y_2=x_2-x_6, y_3=x_3-x_5, y_4=x_3-x_7, y_5=x_4-x_6,y_6=x_4-x_7$, then we have
$$
\aligned q(x)
= &\ x^2+2xz+2z^2-\frac{1}{2}x (y_1+y_2+y_3+y_4+y_5+y_6)\\
&\ + \frac{1}{2}(y^2_1+y^2_2+y^2_3+y^2_4+y^2_5+y^2_6)
\endaligned
$$
which is nonnegative for any $x,z,y_i\in \mathbb{Z}$. Thus the Tits form of $\mathscr{A}_3$ is weakly positive.
In the following we use a basic method to show that $\mathscr{A}_3$ is a tame algebra (I do not know whether there is a criterion to get this property by the weakly positive Tits form).
We classify any indecomposable representation $\mathbb{V}=(V_i, f_{\alpha})$ of $\mathscr{A}_3$ to the following three cases by considering the morphism $f_{18}:V_1\rightarrow V_8$:
(a) When $V_1=V_8=0$, then $\mathbb{V}$ can be regarded as a representation of the Euclidean quiver of type $\widehat{A}_5$. (b) When $V_1\ne 0$ and $V_8\ne 0$, in this case $\text{dim}V_1=\text{dim}V_8=1$ and the only indecomposable module is the projective module $P(1)$. (c) When $V_1=0$ or $V_8=0$, these two situations are symmetric and we consider the case $V_1=0$ and $V_8\ne 0$. Therefore the linear morphisms $f_{25}$,$f_{26}$,$f_{35}$,$f_{37}$,$f_{46}$,$f_{47}$,$f_{58}$,$f_{68}$,$f_{78}$ are all injective.
Since $\mathbb{V}$ is indecomposable, we also have $\text{dim}V_i \leq 1$ for $2\leq i \leq 4$ and $\text{dim}V_j \leq 2$ for $2\leq j \leq 4$ . Therefore there are finitely many indecomposable representations $\mathbb{V}=(V_i, f_{\alpha})$ such that $V_1=0$ and $V_8\ne 0$.
\medskip
Now consider the incidence algebra $\mathscr{A}_4$. For the vertex $i_Y\in Q_0$,
we set
$$x_{Y}=\left\{
\begin{array}{ll}
0 &\ \mbox{when}~|Y|=0~\mbox{or}~4 ,\\
1 &\ \mbox{when}~|Y|=1~\mbox{or}~3 ,\\
2 &\ \mbox{when}~|Y|=2,
\end{array}\right.$$
for any vertex $i_Y\in Q_0$ in the Tits form of $\mathscr{A}_4$. Then by a direct calculation we have $q(x)=-4$. This example also implies that the Tits form of $\mathscr{A}_n$ is not weakly positive when $n\geq 4$. Then we have the following proposition.
\begin{Prop}\label{wild} The algebra $\mathscr{A}_n$ is of wild type when $n\geq 4$.
\end{Prop}
\bibliographystyle{amsplain}
| 153,952
|
Rural dating site commercial. .
April 21, OnlyFarmers, a dating site geared towards people living in rural areas, is making inroads in the Canadian prairies. They plan to marry in the next couple of months and she will move to his Iowa farm. D writes on why people struggle to let go and identifies the 3 common factors preventing people from moving on. The camera cuts to a shot of Gomer holding the phone. Carolyn Castiglia is a comedian and mother who lives in Brooklyn.
| 96,842
|
Sidewalks
Concrete Sidewalks Miami, FL
A beautiful walkway is not just a functional piece of your property– it is the red carpet rolled out for your guests or customers, welcoming them to your front door. A cracked or chipped sidewalk sends a different, less thrilling message– however, this can be easily rectified by contacting the walkway contractor Miami residents trust: MIA Concrete Contractors. As a licensed, bonded, and insured contractor, we know concrete, and we want to build a relationship with our customers.
The Sidewalk Company Miami Homeowners Believe In
A concrete path is a durable and easily maintained pathway solution that can also add style and finish to your home. Whether you want a simple sidewalk, or a stamped, colored, and uniquely designed walking path, MIA Concrete Contractors has the experience needed to produce a sturdy and slip-resistant solution.
If you love your existing walkway but have a few cracks or weeds slipping through, consider seeking out a sidewalk repair company that can refurbish your path, adding flair to your home and removing any unsightly imperfections that can decrease your curb appeal and grow worse over time.
Depending on the amount and type of existing damage, it is possible your sidewalk will need to be taken out and repoured. To determine the extent of the damage, contact a MIA Concrete Contractors representative who can walk you through the services offered and the best fit for your particular house.
Whether it’s a new install or a resealing job, we want your business, and we know you’ll be thrilled with the job we do. Let’s solve your sidewalk problems together, today.
The Concrete Sidewalk Contractor Miami Businesses Go To
Help your business stand out with concrete pavers or stamped and colored concrete. Not only is concrete a durable and cost effective material, but state-of-the-art machinery and experienced professionals will treat your storefront like a blank canvas, ready to bring finesse and taste to draw in customers and help them notice your attention to detail.
Over time, foot traffic can degrade an existing walkway. That’s why MIA Concrete Contractors utilize sealant to preserve the look of your new path for years to come. Proper finishing can stop erosion in its tracks and prevent the Florida sun from fading your pathway.
You may not be aware of the current concrete options for your commercial property. Nowadays, concrete is not limited to one color and finish. Consider a curving path or a brick-like texture. You can even have your professional logo added to the finished product, symbolizing the enduring nature of your business and the contribution you bring to your locale.
With so many options, it’s clear that concrete is an optimal selection when it comes to sidewalks and walkways. The biggest question is: what design and color will you choose? Our contractors will discuss options with you and make sure you are confident in your decision before construction begins.
Contact MIA Contractors to discuss your walkway and what you’d like installed. We are honored to service the following areas:
- Miami, FL
- Hollywood, FL
- Kendall, FL
- Miami Beach, FL
We look forward to serving you!
| 15,983
|
Sproutman Hemp Sprout Bag - Just Dip In Water, Hang It Up, & Watch It.
Frequently bought together
Customers who bought this item also bought
Customers who viewed this item also viewed
Have a question?
Find answers in product info, Q&As, reviews
Please make sure that you are posting in the form of a question.
Product description. Hang the bag on a hook, or lay it flat to drain! It takes less than a minute. Watch the seeds transform into sprouts. Ready for harvest in 3-5 days.
Most recent customer reviews
Set up an Amazon Giveaway
| 169,654
|
\begin{document}
\title[Modularity of non-rigid double octic Calabi--Yau]
{Modularity of some non--rigid double octic Calabi--Yau threefolds}
\author{S\l awomir Cynk}
\thanks{Partially supported by DFG Schwerpunktprogramm 1094
(Globale Methoden in der komplexen Geometrie) and KBN grant no. 2 P03A 013 22.}
\keywords{Calabi--Yau, double coverings, modular forms}
\address{Instytut Matematyki\\Uniwersytetu Jagiello\'nskiego\\
ul. Reymonta 4\\30--059 Krak\'ow\\Poland}
\curraddr{Institut f\"ur Mathematik\\ Universit\"at Hannover\\
Welfengarten~1\\ D--30060 Hannover\\ Germany}
\email{s.cynk@im.uj.edu.pl}
\author{Christian Meyer}
\address{Fachbereich Mathematik und Informatik\\ Johannes
Gutenberg-Uni\-ver\-sit\"at\\
Staudingerweg 9\\D--55099 Mainz\\Germany}
\email{cm@mathematik.uni-mainz.de}
\subjclass[2000]{14G10, 14J32}
\maketitle
\section*{Introduction}
\label{sec:intro}
The modularity conjecture for Calabi--Yau manifolds predicts that
every Calabi--Yau manifold should be modular in the sense that its
$L$--series coincides with the $L$--series of some automorphic
form(s). The case of rigid Calabi--Yau threefolds was (almost)
solved by Dieulefait and Manoharmayum in \cite{DM,Dieulefait}.
On the other hand in the non--rigid case it is even not clear which
automorphic forms should appear.
Examples of non--rigid modular Calabi--Yau threefolds were
constructed by Livn\'e and Yui (\cite{LivneYui}), Hulek and Verrill
(\cite{HulekVerrill, HulekVerrill2}) and Sch\"utt (\cite{Schuett}).
In these examples modularity means a decomposition of the associated Galois
representation into two-- and four--dimensional subrepresentations with
$L$--series equal to $L(g_{4},s)$, $L(g_{2},s-1)$ or
$L(g_{2}\otimes g_{3},s)$, where $g_{k}$ is a weight $k$ cusp form.
The summand with $L$--series equal to $L(g_{2}\otimes g_{3},s)$
is explained by a double cover of a product of a K3 surface and
an elliptic curve (see \cite{LivneYui}).
The $L$--series $L(g_{2},s-1)$ is the $L$--series of the product of
the projective line $\PP[1]$ and an elliptic curve $E$ with
$L(E,s)=L(g_{2},s)$. A two--dimensional subrepresentation with such an
$L$--series may be identified by a map $\PP[1]\times E\lra X$ which
induces a non--zero map on the third cohomology (see \cite{HulekVerrill}).
Using an interpretation in terms of deformation theory we conjecture
that a splitting of the Galois action into two--dimensional pieces can
happen only for isolated elements of any family of Calabi--Yau threefolds.
In this paper we will study modularity of some non--rigid double
octic Calabi--Yau threefolds, we will prove modularity of all examples listed in
table~\ref{tab1} except $X_{154}$. We will use four methods for proving
modularity, apart from the methods of Livn\'e--Yui and Hulek--Verrill
we will use two others based on giving a correspondence with a rigid Calabi--Yau
threefold or on an involution. We also observe that the splitting of the
Galois action into two--dimensional pieces holds for those Calabi--Yau
threefolds in the studied families having some additional geometric
poperties. The Calabi--Yau threefold $X_{154}$ is also the
only one which we were not able to represent as a Kummer fibration
associated to a fiber product of elliptic fibrations
(cf. \cite{Schoen}).
\section{Modular double octics with $h^{12}=1$}
Let $D$ be an arrangement of 8 planes in $\PP$. If no six of the
planes intersect in a point and no four in a line then the double covering of
$\PP$ branched along $D$ admits a resolution of singularities $X$ which is
a smooth Calabi--Yau threefold (see \cite{Cynk}). The resolution of
singularities is performed by blowing up singularities of the branch
locus in the following order: fivefold points,
fourfold points that do not lie on a triple line,
triple lines, double lines. The Euler number of the resulting Calabi--Yau
threefold can easily be expressed in numbers of different types of
singularities. The Hodge number $h^{1,2}(X)$ (the dimension of
the deformation space) can be computed as the dimension of the
space of equisingular deformations of $D$ in $\PP$; it can also be
computed as the dimension of the equisingular ideal of $D$ (see
\cite{CvS}).
An extensive computer search in \cite{Meyer} produced 18 double octic
Calabi--Yau threefolds with $h^{12}=1$ (in 11 one-parameter families)
for which
\[
\tr(\Frob_{p}^{*}|H^{3}_{\text{\'et}}({X}))=a_{p}+p\cdot b_{p},
\]
for all primes $5\leq p\leq 97$, where $a_{p}$ (resp. $b_{p}$) are the coefficients of a
weight four (resp. two) cusp form. This is a strong numerical evidence
for modularity in the sense of splitting into two two--dimensional subrepresentations.
We list all these examples in table~\ref{tab1}. We include the no. of
the arrangement (as in \cite{Meyer}), the equation, the expected modular form
of level 4 and 2 (using W. Stein's notation from \cite{modForms}) and the
Picard number $h^{11}$. Since the Calabi--Yau threefolds in the table
coming from arrangements with the same no. are birational (see
Lemma~\ref{lem:birat}) we will use in this paper
the notation $X_{n}$ for any Calabi--Yau threefold in the table constructed
from arrangement no. $n$.
\begin{table}[htp]
\centering \def\arraystretch{1.2}
\(
\begin{array}{|c|l|c|c|c|c|}
\hline \text{no.} &\text{equation: } u^2=xyzt\cdot\dots&
\text{wt. 4} & \text{wt. 2}&h^{11}\\
\hline \hline
4 &\textstyle (x+y) (y+z) (x-y-z-t) & 32k4A1 &
32A1&61\\
&(x+y-z-t)&&&\\
\hline 4 & (x+y) (y+z) (x+2y+2z-t)& 32k4A1 &32A1&61\\
& (x+y+2z-t)&&&\\
\hline
4 & 2 (x+y) (y+z) (2x+y+z-2t) & 32k4A1 & 32A1&61\\
&(2x+2y+z-2t)&&&\\
\hline
8 &(x+y) (y+z) (-z+t) (3x-y-z+t)& 24k4A1 & 24A1&61\\
\hline
13 & (x+y) (y+z) (x-z-t) (x-z-2t)& 32k4A1 & 32A1&61\\
\hline
13 & (x+y) (y+z) (x-z-t) (x-z+t)& 32k4A1 & 32A1&61\\
\hline
13 & (x+y) (y+z) (x-z-t) (2x-2z-t)& 32k4A1 & 32A1&61\\
\hline
21 & (x+y) (y+z) (2x+y-t) (2x-z-2t)& 32k4B1 & 32A1& 53\\
\hline
53 & (x+y) (z+t) (x-y-z-t)& 32k4B1 & 32A1& 53\\
& (x+y-z+t)&&&\\
\hline
154 & (x+y+z) (x+y+z-t) &
8k4A1 & 72A1& 41 \\
& (-2x+y-3z+3t)(2x+3z-2t)&&& \\
\hline
244 & (x+y+z+t)(x+y-z-t)(y-z+t)&12k4A1 & 48A1& 39 \\
& (x-z+t)&&&\\
\hline
249 & (x+y+z) (x+z+t) (2x+3y-z+2t)&24k4A1 & 24A1& 37 \\
&(y-z+2t)&&&\\
\hline
249 & (x+y+z) (x+z+t) (2x-y+3z+2t)&24k4A1 & 24A1& 37 \\
& (-3y+3z+2t)&&&\\
\hline
267 & (x+y-2z)(x-y-z+t)(2y-z+t)&96k4B1 & 96B1& 37 \\
&(x+y+z+t)&&&\\
\hline
267 & (x+y+z) (x+2y-z+t)& 96k4B1 & 96B1& 37 \\
&
(-y+2z-2t)(2x+2y-z+2t)&&&\\
\hline
267 & (2x+2y-z) (2x+y-2z+2t)
&96k4B1 & 96B1& 37 \\
&(y+z-t)(x+y-2z+t)&&&\\
\hline
274 & (x+y+z)
(-x-z+t)(x+2y-z+t) &96k4E1 & 96B1& 37 \\
& (x+y-z+2t)&&&\\
\hline
275 & (x+y+z)(2x-2z-t)(8y+4z+t)&96k4B1 & 96B1& 37 \\
&(2x+4y+t)&&&\\ \hline
\end{array} \)\\[2mm]
\caption{}
\label{tab1}
\end{table}
The Picard groups of all listed Calabi--Yau threefolds are generated
by divisors defined over $\mathbb Q$, so Frobenius acts on
$H^{2}_{\text{\'et}}$ by multiplication with $p$. In fact, in all the examples
except $X_{244}$, the skew-symmetric part of the
Picard group is zero, whereas for $X_{244}$ it is generated by a
divisor coming from the contact plane $x+y-z+t=0$.
\section{Double quartic elliptic fibrations}
\label{sec:dq}
In this section we will shortly review some information about
rational elliptic fibrations that can be realized as a resolution of a
double covering of $\PP[2]$ branched along a sum of four lines. The
structure of the elliptic fibration is determined by the choice of a point
in $\PP[2]$. Some of these surfaces where described in \cite{CynkMeyer2};
we will omit here all the details explained in that paper.
The double covering is rational exactly when the lines do not intersect in
one point. We can have the following combinations of singular fibers
(the Picard number $\rho(S_{w})$ of a generic fiber can be computed from
the Zariski lemma):
\[ \begin{array}{l|l|c}
&\text{singular fibers}&\rho(S_{w})\\\hline
S_{1} &D_{4}^{*},D_{4}^{*}&1\\
S_{2} &I_{2}, I_{2}, D_{6}^{*}&1\\
S_{3} &I_{2}, I_{2}, I_{4}, I_{4}&1\\
S_{4} &I_{2}, I_{2}, I_{2}, D_{4}^{*}&2\\
S_{5} &I_{2}, I_{2}, I_{2}, I_{2}, I_{4}&2\\
S_{6} &I_{2},I_{2},I_{2},I_{2},I_{2},I_{2}&3
\end{array}
\]
A double covering of $S_{1}$ branched along the two singular fibers is
birational to a product of $\PP[1]$ and an elliptic curve $E$, and all
smooth fibers are isomorphic to $E$. This elliptic fibration
depends on the $j$--invariant of~$E$.
The surfaces $S_{2}$ and $S_{3}$ are extremal, i.e. they have
$\rho(S_{w})=1$. Consequently they are uniquely defined as fiber
spaces. Moreover the parameters corresponding to the singular fibers of
$S_{3}$ form a harmonic quadruple (i.e. their cross--ratio equals $-1$);
they can be chosen as
\[
\begin{array}{rrrr}
-1&0&1&\infty\\ \hline
I_{2}&I_{4}&I_{2}&I_{4}
\end{array}
\]
Denote by $S_{3}'$ the pullback of $S_{3}$ via the involution $t\mapsto
\frac{t-1}{t+1}$ of $\PP[1]$, so $S_{3}'$ has the following singular
fibers:
\[
\begin{array}{rrrr}
-1&0&1&\infty\\ \hline
I_{4}&I_{2}&I_{4}&I_{2}
\end{array}
\]
Thus $S_{3}$ and $S_{3}'$ have singular
fibers at the same points but of different types. There exists an isogeny
$\Psi:S_{3}\mapsto S_{3}'$ which is a degree 2 unbranched covering
on a smooth fiber.
Fibration $S_{4}$ is not extremal, so we can chose arbitrary
coordinates of singular fibers. The configuration of lines is not
uniquely determined by the coordinates of singular fibers. In fact
there are exactly two types: one with a triple point and one with a
``vertical line''.
The Picard number of the generic fiber of Fibration $S_{5}$ equals two,
so we can not choose arbitrary coordinates of singular fibers. In fact
there is an involution of $\PP[1]$ which preserves the fiber $I_{4}$
and exchanges two pairs of $I_{2}$'s. The configuration of lines is
uniquely determined.
Fibration $S_{6}$ is the most complicated one. In this case the
configuration of lines is not uniquely determined. There can be
several choices coming from automorphisms of $\PP[1]$ preserving the
singular fibers.
\section{Kummer fibrations}
\label{sec:kf}
All examples in table~\ref{tab1} except $X_{154}$ can be realized as a Kummer
fibration associated to a fiber product of elliptic fibrations (cf. \cite{Schoen}).
Contrary to Schoen we do not require that the involution on the fiber
product lifts to a resolution, so the resulting Calabi--Yau threefold
is not necessarily a blow--up of the Kummer fibration.
To see the fibration we reorder the planes such that the first four
and the last four intersect in a point. Then after change of
coordinates in $\PP$ we may assume that these points of intersection
are $(0,0,0,1)$ and $(1,0,0,0)$, or equivalently that the double octic
is given in weighted projective space $\PP[](1,1,1,1,4)$ by the equation
\[
w^{2}=f_{1}(x,y,z)\cdot\ldots\cdot f_{4}(x,y,z)f_{5}(y,z,t)\cdot\ldots\cdot
f_{8}(y,z,t).
\]
Consequently the double octic is birational to the
quotient of the fiber product of elliptic fibrations
\[
u^{2}=f_{1}(x,y,z)\cdot\ldots\cdot f_{4}(x,y,z)
\] and
\[
v^{2}=f_{5}(y,z,t)\cdot\ldots\cdot f_{8}(y,z,t)
\]
by the involution
\[
(x,y,z,t,u,v)\mapsto(x,y,z,t,-u,-v).
\]
In the following table we list descriptions of Calabi--Yau threefolds from
table~\ref{tab1} as Kummer fibrations. For each Kummer fibration we
give coordinates and types of singular fibers. In some cases we
were able to find two different representations as a Kummer fibration.
\[\def\arraystretch{1.2}
\begin{array}{ll}
\rule{15mm}{0cm}&\rule{9cm}{0cm}\\
X_{4}&
\begin{array}{rrrrr}
0&1&2&3&\infty\\ \hline
I_{2}&I_{2}&I_{2}&I_{2}&I_{4}\\
I_{0}&I_{2}&I_{2}&I_{0}&D_{6}^{*}\\
\end{array}\hfill\begin{array}{rrrr}
-1&0&1&\infty\\\hline
I_{4}&I_{2}&I_{4}&I_{2}\\
D_{6}^{*}&I_{2}&I_{0}&I_{2}
\end{array}
\\
\hline
X_{8}&
\begin{array}{rrrr} 0&1&4&\infty\\ \hline
D_{4}^{*}&I_{2}&I_{2}&I_{2}\\
I_{2}&I_{2}&I_{0}&D_{6}^{*}
\end{array}\\
\hline
X_{13}&
\begin{array}{rrr} 0&1&\infty\\ \hline
D_{4}^{*}&D_{4}^{*}&I_{0}\\
I_{2}&I_{2}&D_{6}^{*}
\end{array}\hfill
\begin{array}{rrrr}
-1&0&1&\infty\\ \hline
I_{2}&I_{4}&I_{2}&I_{4}\\
I_{0}&D_{4}^{*}&I_{0}&D_{4}^{*}
\end{array}
\\ \hline
X_{21}&
\begin{array}{rrrr}
-1&0&1&\infty\\ \hline
I_{2}&I_{4}&I_{2}&I_{4}\\
D_{6}^{*}&I_{0}&I_{2}&I_{2}
\end{array}
\hfill
\begin{array}{rrrr}
-1&0&1&\infty\\\hline
I_{2}&I_{2}&D_{4}^{*}&I_{2}\\
D_{4}^{*}&I_{2}&I_{2}&I_{2}
\end{array}
\\
\hline
X_{53}&
\begin{array}{rrrr}
-1&0&1&\infty\\ \hline
I_{2}&D_{6}^{*}&I_{2}&I_{0}\\
I_{2}&I_{0}&I_{2}&D_{6}^{*}
\end{array}
\hfill
\begin{array}{rrrr}
-1&0&1&\infty\\ \hline
I_{2}&I_{2}&I_{2}&D_{4}^{*}
\\I_{2}& D_{4}^{*}&I_{2}&I_{2}
\end{array}
\\ \hline
X_{244}&
\begin{array}{rrrrr}
-1&0&1&2&\infty\\ \hline
I_{0}&I_{2}&I_{4}&I_{2}&I_{4}\\
I_{4}&I_{2}&I_{4}&I_{0}&I_{2}
\end{array}
\hfill
\begin{array}{rrrrrr}
-1&0&\frac13&1&3&\infty\\ \hline
I_{2}&I_{2}&I_{2}&I_{4}&I_{0}&I_{2}
\\I_{2}&I_{2}&I_{0}& I_{4}&I_{2}&I_{2}
\end{array}
\\ \hline
X_{249}&
\begin{array}{rrrrrr}
-1&0&\frac13&1&3&\infty\\ \hline
I_{0}&I_{2}&I_{2}&I_{4}&I_{2}&I_{2}
\\I_{2}&I_{4}&I_{0}& I_{2}&I_{0}&I_{4}
\end{array}\\
\hline
X_{267}&
\begin{array}{rrrrrr}
-1&0&\frac12&1&2&\infty\\ \hline
I_{2}&I_{2}&I_{2}&I_{2}&I_{2}&I_{2}
\\I_{2}&I_{2}&I_{2}& I_{2}&I_{2}&I_{2}
\end{array}\\
\hline
X_{274}&
\begin{array}{rrrrrr}
-1&0&\frac12&1&2&\infty\\ \hline
I_{2}&I_{4}&I_{2}&I_{2}&I_{0}&I_{2}
\\I_{4}&I_{2}&I_{2}& I_{0}&I_{2}&I_{2}
\end{array}\\
\hline
X_{275}&
\begin{array}{rrrrrr}
-1&0&\frac12&1&2&\infty\\ \hline
I_{2}&I_{2}&I_{2}&I_{2}&I_{2}&I_{2}
\\I_{2}&I_{2}&I_{2}& I_{2}&I_{2}&I_{2}
\end{array}
\end{array}
\]
\begin{lemma}\label{lem:birat}
The Calabi--Yau threefolds in table \ref{tab1} defined by arrangements of the
same type are birational. The Calabi--Yau threefolds $X_{21}$ and $X_{53}$ are
birational; and the Calabi--Yau threefolds $X_{267}$ and $X_{275}$ are birational.
There exists a correspondence between the Calabi Yau--threefolds $X_{8}$
and~$X_{249}$.
\end{lemma}
\begin{proof}
From the explicit description of the fiber products in local coordinates it
easily follows that the Calabi--Yau threefolds defined by arrangements of
the same type with different parameters are in fact projectively equivalent.
Arrangement no. 21 is projectively equivalent to
\[
x(x-z)(x+z)(x+y)y(t+z)(t-z)(t+y)=0.
\]
Substituting the birational involution of $\PP$ given by
\[
(x,y,z,t)\mapsto(yz,xz,xy,tx)
\]
we obtain
\[
(xzy^{2})^{2}x(x-z)(x+z)(x+y)z(t+y)(t-y)(t+z)=0,
\]
and since arrangement no. 53 is projectively equivalent to
\[
x(x-z)(x+z)(x+y)z(t+y)(t-y)(t+z) = 0,
\]
we conclude that the resulting Calabi--Yau threefolds are birational.
To prove that $X_{267}$ and $X_{275}$ are birational, observe that the
corresponding arrangements are projectively equivalent to
\begin{eqnarray*}
&\text{Arr. no. 267:}\quad&x(x-z)(2x-2z+y)(2x-z-y)\times\\
&&\times t(t+z-y)(2y-z-2t)(2z-y+2t)=0\\
&\text{Arr. no. 275:}\quad&x(x-z)(2x-2z+y)(2x-z-y)\times\\
&&\times t(2t-y)(2t-z)(3t-y-z)=0.
\end{eqnarray*}
Simple computations show that the cross ratios of the quadruples
\[
\begin{array}{llll}
0, & \quad y-1, & \quad y-\tfrac12, & \quad \tfrac12y-z\\
0, & \quad \tfrac 12y, & \quad \tfrac 12z, & \quad \tfrac 13y+\tfrac 13z
\end{array}
\]
are equal so there is a birational transformation in $y,z,t$ that
maps one of them to the other.
To see the correspondence between the Calabi--Yau threefolds $X_{8}$
and $X_{249}$, first pull back arrangement no. 8 by the
map $t\mapsto (\frac{t+1}{t-1})^{2}$, obtaining
\[
\begin{array}{rrrrrr}
-1&0&\frac13&1&3&\infty\\ \hline
I_{0}&I_{2}&I_{2}&I_{4}&I_{2}&I_{2}
\\I_{4}&I_{2}&I_{0}& I_{4}&I_{0}&I_{2}
\end{array}\]
Now it is enough to compose this map with the isogeny of the elliptic
fibration with fibers $I_{4},I_{4},I_{2},I_{2}$ that exchanges $I_2$ fibers
with $I_4$ fibers (see \cite{CynkMeyer2}).
\end{proof}
\begin{remark}
Arrangements no. 267 and 275 are not projectively equivalent, they
come from different twisted self--fiber products of the same
elliptic fibration. The self--fiber product (without twist) of this
elliptic fibration gives a non--birational Calabi--Yau threefold
with $h^{12}=2$ (see example~\ref{ex:ar269}).
\end{remark}
\section{Ruled surface over elliptic curves}
\label{sec:ell}
In this section we will use elliptic ruled surfaces to prove
modularity of four Calabi--Yau threefolds from table \ref{tab1}.
\begin{prop}
The Calabi--Yau threefolds $X_{4}$, $X_{8}$, $X_{244}$ and $X_{249}$ are
modular, with modular forms as listed in table \ref{tab1}.
\end{prop}
Consider a Calabi--Yau threefold $X$ such that an $L$--series of the
form $L(g_{2},s-1)$ (where $g_{2}$ is a weight two modular form corresponding
to an elliptic curve $E$) appears in the Galois representation. Then
by the Tate Conjecture we can expect that there is a correspondence
between $X$ and the product $E\times \PP[1]$ which induces the
isomorphism of representations.
Hulek and Verrill proved in \cite{HulekVerrill} that when a smooth
ruled surface over an elliptic curve $S\lra E$
is contained in a Calabi--Yau threefold $X$ then the map on third
cohomology $H^{3}(X)\lra H^{3}(S)$ is surjective. The map can be
represented by a direct sum of $H^{1}(\mt _{X})\lra H^{1}(\mathcal
N_{S|X})$ and its complex conjugate. The map $H^{1}(\mt _{X})\lra
H^{1}(\mathcal N_{S|X})$ associates to a deformation of $X$ the
obstruction to lift it to a deformation of $E$ (inside $X$). Therefore
if this map is non-zero then $E$ deforms inside $X$ only over a
codimension one submanifold of the Kuranishi space of $X$.
Now, if we have ruled surfaces $E_{1},\dots,E_{r}$, with
$r=h^{21}(X)$, such that the map
\begin{equation}
\label{eq:thirdcoho}
H^{3}(X)\lra \bigoplus_{i} H^{3}(E_{i})
\end{equation}
is surjective then the
obstructions are independent and the surfaces do not deform
simultaneously over any subvariety of the Kuranishi space of $X$ of
positive dimension. It is an explanation why in a family there were
always only finitely many examples were one was able to prove
modularity in that way.
If we have several ruled surface over elliptic curves, it is usually
difficult to determine whether the map \eqref{eq:thirdcoho}
is surjective. In case we know the Kuranishi space of
$X$ we can try to invert the above argument. For each elliptic
fibration we consider the hypersurface $V_{i}$ of the Kuranishi space over
which $E_{i}$ deforms, knowing that the kernel of \eqref{eq:thirdcoho} is the
tangent to the intersection of the $V_{i}$'s plus its complex
conjugate (see example at the end of this section).
To use this method in our examples we need to find elliptic fibrations inside
the double octics. If a plane $S$ in $\PP[3]$ contains two double lines
and the other four arrangement planes intersect at a point in $S$,
then the pullback of $S$ to the double covering is an elliptic
fibration. On the Kummer fibration these planes are recognized as
corresponding to the product of fibers $I_{0}$ and $I_{4}$.
We were able to find such a plane only for two arrangements:
{\bf Arrangement no. 4:}
the plane $S$ has equation $x-z=0$ resp. $y+2z-t=0$ resp. $2x+y-2t=0$
(for the three arrangements in the table).
{\bf Arrangement no. 244:}
the plane $S$ has equation $x+y+z-t=0$.
\medskip
To prove modularity of $X_{8}$ and $X_{249}$ we will study an
auxiliary Calabi--Yau threefold $X_{269}$ with $h^{12}=2$. Modularity
of this Calabi--Yau threefold follows from existence of some elliptic ruled
surfaces and their behavior under deformations.
\begin{example}\label{ex:ar269}
Consider the double octic Calabi--Yau threefold $X_{269}$ defined by the
following arrangement of eight planes (arrangement no. 269 in \cite{Meyer}):
\[
xyzt(x+y+z)(x+2y-z+t)(y+z-t)(x+y-2z+t)=0
\]
It has $h^{2,1}(X_{269})=2$.
Substituting $y=y-z,z=z+t$ we can represent this Calabi--Yau
threefold as the following Kummer fibration:
\[
\begin{array}{rrrrrr}
-1&0&\frac13&1&3&\infty\\ \hline
I_{2}&I_{2}&I_{2}&I_{2}&I_{2}&I_{2}
\\I_{2}&I_{2}&I_{2}& I_{2}&I_{2}&I_{2}
\end{array}\]
On the other hand substituting $x=x+2z-4y, z=x-2y$ we can also obtain
the following Kummer fibration:
\[
\begin{array}{rrrrrr}
-1&0&\frac13&1&3&\infty\\ \hline
I_{0}&I_{2}&I_{2}&I_{4}&I_{2}&I_{2}
\\I_{4}&I_{2}&I_{0}& I_{4}&I_{0}&I_{2}
\end{array}
\]
Hence using the isogeny between $S_{3}$ and $S_{3}'$ from section~\ref{sec:dq} we can find
correspondences between this Calabi--Yau threefold and the Calabi--Yau
threefolds $X_{8}$ and $X_{249}$.
Observe that the planes $z=x+2y$ and $y=2z-t$ contain two double
lines and a fourfold point, so they give two ruled surfaces $E_{1},
E_{2}$ over an
elliptic curve with conductor 24.
The Kuranishi space of the Calabi--Yau threefold $X_{269}$ may be
para\-met\-rized by the equation
\begin{align*}
& xyzt(x+y+z)(Bx+Cy-Az+At) \,\times\\
&\qquad \times (y+z-t)(Bx+By+(-A+B-C)z+At)=0.
\end{align*}
By \cite{HulekVerrill2} both elliptic fibrations give non-zero maps
\[H^{3}(X)\lra H^{3}(E_{i})\]
so they deform over curves in $\PP[2]$.
One easily checks that they deform over the lines given by
\begin{align*}
A+B-C&=0,\\C&=2B,
\end{align*}
which intersect only at the point $(1,1,2)$ corresponding to the
equation we started with. Consequently the obstructions are
independent and the map
\[H^{3}(X)\lra H^{3}(E_{1})\oplus H^{3}(E_{2})\]
is surjective, giving a splitting of the representation on $H^{3}$
into two--dimensional pieces. Counting points over $\mathbb F_{p}$
for $p\le97$ one checks that $X$ is modular and that the coefficients of
the $L$--series are given by $b_{p}+2pc_{p}$, where $b_{p}$ resp. $c_{p}$
are the coefficients of the unique cusp form of level 24 and weight 4 resp. 2.
\end{example}
There is a degree two correspondence between the above Calabi--Yau
threefold and $X_{249}$, hence also $X_{8}$. These correspondences prove
the modularity of $X_{8}$ and $X_{249}$.
\section{Correspondences with rigid double octics}
\label{sec:rigid}
In this section we will use correspondences between rigid and
non--rigid Calabi--Yau threefolds to prove modularity of the latter.
\begin{prop}
\label{prop_4_21_53_244}
The Calabi--Yau threefolds $X_{4}$, $X_{21}$, $X_{53}$ and $X_{244}$ are
modular, with modular forms as listed in table \ref{tab1}.
\end{prop}
In \cite{CynkMeyer} we checked the modularity and computed modular forms
of some rigid double octic Calabi--Yau threefolds. Now we will use
correspondences between some rigid and non--rigid Calabi--Yau
threefolds to show the modularity of the latter.
We first recall the considered rigid examples. As before we
will use the equations and numbers of arrangements from
\cite{Meyer} (in brackets we give the numbers from \cite{CynkMeyer}).
\textbf{Arrangement no. 3} (old no. 6) is given by the equation
\[
xyzt(x+y)(y+z)(z+t)(t+x)=0.
\]
The corresponding fiber product of elliptic fibrations has singular fibers
\[\begin{array}[c]{cccc}
I_{4}&I_{4}&I_{2}&I_{2}\\
D_{6}^{*}&I_{2}&I_{2}&I_{0}
\end{array}
\]
\textbf{Arrangement no. 19} (old no. 23) is given by the equation
\[
xyzt(x+y)(y+z)(x-z-t)(x+y+z-t)=0.
\]
The corresponding fiber product of elliptic fibrations has singular fibers
\[
\begin{array}[c]{cccc}
I_{2}&I_{2}&I_{4}&I_{4}\\
I_{0}&D_{6}^{*}&I_{2}&I_{2}
\end{array}
\]
\textbf{Arrangement no. 239} (old no. 86$^{a}$) is given by the equation
\[
xyzt(x+y+z)(x+y+t)(x+z+t)(y+z+t)=0.
\]
The corresponding fiber product of elliptic fibrations has singular fibers
\[
\begin{array}[c]{ccccc}
I_{2}&I_{2}&I_{4}&I_{4}&I_{0}\\
I_{0}&I_{4}&I_{2}&I_{4}&I_{2}
\end{array}
\]
\begin{lemma}
There are correspondences between the Calabi--Yau threefolds given by the
following arrangements:
\begin{enumerate}
\item No. 4 and no. 19,
\item No. 21 and no. 3,
\item No. 53 and no. 3,
\item No. 244 and no. 239.
\end{enumerate}
\end{lemma}
\begin{proof}
All the correspondences are in fact defined on the level of the fiber
products of elliptic fibrations. They are given by applying the
isogeny of the elliptic fibration with fibers $I_{2},I_{2},I_{4},I_{4}$
that exchanges the fibers $I_{2}$ and $I_{4}$.
\end{proof}
Assume that we have a generically finite correspondence between two
Calabi--Yau threefolds $X$ and $Y$. Then this correspondence induces an
isomorphism between $H^{3,0}(X)$ and $H^{3,0}(Y)$ coming from a pullback
of the canonical form. If $Y$ is rigid then taking this isomorphism plus its
complex conjugate we obtain a splitting of the Galois representation on
$H^{3}(X)$ into a two--dimensional representation isomorphic to
$H^{3}(Y)$ and its complement. Using the correspondences from the
above lemma and counting points in $\mathbb F_{p}$ for $p\le97$ we
obtain proposition \ref{prop_4_21_53_244}.
\section{Kummer construction}
\label{sec:mr}
In this section we will use the Kummer construction studied by Livn\'e and
Yui (\cite{LivneYui}).
\begin{prop}
The Calabi--Yau threefold $X_{13}$ is modular, with modular forms as listed
in table \ref{tab1}.
\end{prop}
We will consider a two--dimensional family of double octic
Calabi--Yau threefolds which are the quotient by an involution of a product
of a K3 surface studied in \cite{AOP} and an elliptic curve.
Take the elliptic curve
\[
E_{\mu}=\{(x,t,u)\in\PP[](1,1,2):u^{2}=(x-t)(x^{2}-\mu t^{2})t \}
\]
and the K3 surface
\[
S_{\lambda}=\{(y,z,t,v)\in\PP[](1,1,1,3): v^{2}=yzt(y+t)(z+t)(y+\lambda z)\}.
\]
On the product $Y_{\lambda,\mu}:=E_{\mu}\times S_{\lambda}$ we have a natural involution
\[
((x,t,u),(y,z,t,v))\longmapsto ((x,t,-u),(y,z,t,-v)).
\]
The quotient $X_{\lambda,\mu}$ of $Y_{\lambda,\mu}$ by this involution has a
Calabi--Yau nonsingular model. To show this observe that $Y_{\lambda,\mu}$
is birational to the double covering of $\PP$ branched along the octic
$D_{\lambda,\mu}$ given by the equation
\[
(x-t)(x^{2}-\mu t^{2})yz(y+t)(z+t)(y+\lambda z)=0.
\]
The birational map can be given by in appropriate affine
coordinates ($t=1$) by
\[
(x,1,u),(y,z,1,v)\longmapsto (x,y,z,uv).
\]
The octic itself is defined over $\mathbb Q$. Over $\mathbb Q[\sqrt\mu]$
it splits into a sum of eight planes (for general $\mu$, two of them are not defined
over $\mathbb Q$). Using \cite{Cynk} we conclude that $X_{\lambda,\mu}$ has a
nonsingular model $\tilde X_{\lambda,\mu}$ which is a Calabi--Yau threefold.
For general values of $\lambda$ and $\mu$, the arrangement $D_{\lambda,\mu}$
is arrangement no. 52 in \cite{Meyer}, so $\tilde X_{\lambda,\mu}$ has the
invariants $h^{11}(\tilde X_{\lambda,\mu})=56$ and $h^{12}(\tilde X_{\lambda,\mu})=2$.
For $\lambda\not=0,-1$, the rank of the symmetric part of the Picard
group of the K3 surface $S_{\lambda}$ is 19; denote by
$H^{2}_{skew}(S)$ the three-dimensional skew-symmetric part.
Thus there is a Shioda--Inose structure on $S_{\lambda}$, namely there exists
an involution on $S_{\lambda}$ such that the quotient of $S_{\lambda}$ by that
involution is a Kummer surface.
In \cite {AOP} it is proved that the surface $S_{\lambda}$, with $\lambda\in
\mathbb Q\setminus \{0,-1\}$, is modular exactly when
$\lambda\in\{1,8,1/8,-4,-1/4,-64,-1/64\}$, and the modular form for $S_{\lambda}$
is computed. We have the following diagram of rational maps
\[
\begin{diagram} \node{Y_{\lambda,\mu}}
\arrow{se}\arrow[2]{e,..}\node[2]{\tilde X_{\lambda,\mu}}\arrow{sw}\\
\node[2]{X_{\lambda,\mu}}
\end{diagram}
\]
The rational map $Y_{\lambda,\mu}\longrightarrow\tilde X_{\lambda,\mu}$ can
be resolved by blowing up at points and lines so it induces a well defined map in
cohomologies $H^{3}(X_{\lambda,\mu})\lra H^{3}(Y_{\lambda,\mu})$. The image of the
map is invariant under the involution on $Y_{\lambda,\mu}$, so in fact we
obtain a map $H^{3}(X_{\lambda,\mu})\lra H^{1}(E_{\mu})\otimes
H^{2}_{skew}(S_{\lambda})$. From the description of deformations of double coverings of
smooth algebraic varieties (\cite{CvS}) it follows that this map is surjective,
moreover both vector spaces have dimension 6, so it is an
isomorphism. We obtain
\begin{prop}
$H^{3}(\tilde X_{\lambda\mu})\cong H^{1}(E_{\mu})\otimes H^{2}_{skew}(S_{\lambda})$.
\end{prop}
\begin{cor} The Calabi--Yau threefold $\tilde X_{\lambda,\mu}$ is modular for
$\lambda\in\{1,8,1/8,-4,-1/4,-64,-1/64\}$ and $\mu\in\mathbb Q\setminus\{0,1\}$.
\end{cor}
For the seven values of $\lambda $ the $L$--series of $S_{\lambda}$
corresponds to a cusp form for
$S_{3}(\Gamma_{1}(8))$, $S_{3}(\Gamma_{1}(16))$, $S_{3}(\Gamma_{1}(12))$,
$S_{3}(\Gamma_{1}(7))$ (for $\lambda$ and $1/ \lambda$ the $L$--series differ
only by a twist). They are the only $\eta$--product weight 3 modular
forms. The modular form of the surface $S_{\lambda}$ corresponds to the
symmetric power of the modular form associated to the elliptic curve
$E_{\frac 1{\lambda +1}}$ (see \cite{AOP}). For the seven special values of
$\lambda$ the elliptic curve $E_{\frac1{\lambda+1}}$ has complex multiplication.
Denoting by $a_{p}$ resp. $b_{p}$ the Fourier coefficients of the level 2
(resp. level 3) modular forms we get
\[
b_{p}= \begin{cases}
a_{p}^{2}-2p,\qquad&\left(\frac{-(\lambda+1)}p\right)=1\\[3mm]
0,&\left(\frac{-(\lambda+1)}p\right)=-1.
\end{cases}
\]
The Fourier coefficient of the $L$--series of $S_{\lambda}$ equals $\left(\tfrac{-(\lambda+1)}p\right)(b_{p}+p)$.
The third symmetric power of a weight 2 form yields also a weight 4
modular form with Fourier coefficients
\[
c_{p}=a_{p}^{3}-3pa_{p},
\]
so we obtain
\[
a_{p}b_{p}=c_{p}+pa_{p}.
\]
Consequently we get much better modularity properties for the threefolds
$X_{\lambda}:=\tilde X_{\lambda,\frac{1}{\lambda+1}}$.
\begin{prop} The $L$--series of the Calabi--Yau threefold $X_{\lambda}$ has
Fourier coefficients equal to
\[
c_{p}+2pa_{p}.
\]
\end{prop}
In the table we collect the data for the four Calabi--Yau threefolds
the $L$--series of which do not only differ by a twist:
\[
\begin{array}[t]{l||c|c|c|c} &\lambda=1&\lambda=8&\lambda=-4&\lambda=-64\\ \hline\hline
\text{wt 2 form}&256k2D&32k2A&144k2B&49k2A\\ \hline \text{wt 3
form}&8k3A[1,1]&16k3A[1,0]&12k3A[0,1]&7k3A[3]\\ \hline \text{wt 4
form}&256k4H&32k4A&144k4A&49k4D\\ \hline
b_{p}=a_{p}^{2}-2p&p\equiv1,3(8)&p\equiv3(4)&p\equiv1(3)&p\equiv1,2,4(7)\\ \hline
b_{p}=0&p\equiv5,7(8)&p\equiv1(4)&p\equiv2(3)&p\equiv3,5,6(7)\\ \hline
\parbox{2cm}{$\eta$--products\\(wt 3)}&
\parbox{22.5mm}{$\eta^{2}(z)\eta^{2}(2z)\\[1mm]\eta^{2}(4z)\eta^{2}(8z)$}&
\eta^{6}(4z)&\eta^{3}(2z)\eta^{3}(6z)&\eta^{3}(z)\eta^{3}(7z)\\ \hline
\parbox{2cm}{$\eta$--products\\(wt 2)}&-&\eta^{2}(8z)\eta^{2}(4z)&
\frac{\eta^{12}(12z)}{\eta^{4}(24z)\eta^{4}(6z)}&-
\end{array}
\]
\subsection{Singular K3}
From the above considerations we excluded the case of $\lambda=-1$. There
are two reasons for this. First, in this case all divisors on the K3 surface are
symmetric and consequently $h^{12}(\tilde X_{-1,\mu})=1$ (this is
arr. no 13). Second, $\frac 1{\lambda+1}$ makes no sense. We can however
take in that case also the curve $E_{1/9}$, as the modular forms
appearing in $S_{-1}$ and $S_{8}$ are the same. Hence for the Calabi--Yau
threefold $\tilde X_{-1,1/9}$ the modular form has coefficients
$c_{p}+pa_{p}$, where $c_{p}$ resp. $a_{p}$ are coefficients of a weight
4 resp. 2 level 32 newform.
In the above considerations we can replace the elliptic curve
$E_{\frac1{\lambda+1}}$ by another elliptic curve with the same modular
form, or replace both $E_{\mu}$ and $S_{\lambda}$ by some twist.
Now fix $\lambda\in\{1,8,\frac 18,-4,-\frac 14,-64,-\frac 1{64}\}$.
Using \cite{AOP}
we can compute the characteristic polynomial of
Frobenius on $H^3$ for the Calabi--Yau threefold $\tilde X_{\lambda,\mu}$
for any rational $\mu\not=0,-1$. Denoting by $\alpha_{p},\bar\alpha_{p}$ resp.
$\beta_{p},\bar\beta_{p}$ the eigenvalues of Frobenius on
$H^{1}(E_{\lambda,p})$ resp. $H^{1}(E_{\mu,p})$ we find
that the characteristic polynomial of Frobenius acting on
$H^{3}(\tilde X_{\lambda,\mu})$ is (up to sign)
\[
(T-p\beta_{p})(T-p\bar\beta_{p})\cdot(T-\alpha^{2}_{p}\beta_{p})(T-\alpha^{2}_{p}\bar\beta_{p})
(T-\bar\alpha^{2}_{p}\beta_{p})(T-\bar\alpha^{2}_{p}\bar\beta_{p}).
\]
This polynomial splits over $\mathbb Z$ into the characteristic
polynomial of the Frobenius action on $H^{2}((\PP[1]\times E_{\mu})_{p})$ and
the degree 4 polynomial $(T-\alpha^{2}_{p}\beta_{p})(T-\alpha^{2}_{p}\bar\beta_{p})
(T-\bar\alpha^{2}_{p}\beta_{p})(T-\bar\alpha^{2}_{p}\bar\beta_{p})$.
In the construction, this splitting comes from the cartesian product
of $E_{\mu}$ and a transcendental cycle on the K3 surface $S_{\mu}$;
it should have a better geometric interpretation via the Shioda--Inose structure.
If the elliptic curves $E_{\lambda}$ and $E_{\mu}$ are
non--isogenous, the degree 4 polynomial does not divide by the characteristic
polynomial of $\PP[1]\times E$, for any elliptic curve $E$. To see this,
denote the eigenvalues of Frobenius on $H^{1}(E)$ by $\gamma _{p},\bar
\gamma_{p}$ and assume that $p\gamma_{p}=\bar\beta_{p}\alpha_{p}^{2}$.
Multiplying by $\beta_{p}$ and dividing by $p=|\beta_{p}|^{2}$ we get
$\beta_{p}\gamma_{p}=\alpha_{p}^{2}$.
Since $E_{\lambda}$ has complex multiplication, looking at the sets of
primes $p$ for which the coefficients $\alpha_{p}, \beta_{p}$ and
$\gamma_{p}$ equal $\pm ip^{1/2}$ we easily see that the other two elliptic curves have
complex multiplication by the same quadratic field and so up to a
twist the three weight two forms coincide. In particular $E_{\lambda}$
and $E_{\mu}$ are isogenous.
\section{Involutions}
\label{sec:Inv}
In this section we will use an involution on a Calabi--Yau threefold
to split the cohomology group $H^{3}$. Note that van Geemen
and Nygaard (\cite{GeemenNygard}) were the first to use an automorphism
of a Calabi--Yau manifold to split the Galois representation and prove
modularity.
\begin{prop}
Calabi--Yau threefolda $X_{53}$, $X_{244}$, $X_{267}$, $X_{274}$ and
$X_{275}$ are modular, with modular forms as listed in table \ref{tab1}.
\end{prop}
On some of the Calabi--Yau threefolds considered in this paper we can find
an involution. On the middle cohomology the involution may have only
eigenvalues $\pm1$. If both $1$ and $-1$ are eigenvalues then the
map gives us a splitting of $H^{3}$. Since the spliting is compatible
with the Frobenius morphism it is in fact a splitting of the Galois
representation into two--dimensional subrepresentations.
We can use the Lefschetz formula to compute the trace of Frobenius
composed with the involution. This trace is equal to the trace of
Frobenius on the $+1$--eigenspace minus the trace of Frobenius on
the $-1$--eigenspace. Together with the trace of Frobenius on
$H^{3}$ this gives the traces on the two subspaces.
Assume that we have a $\mathbb Q$--linear involution on $\PP$ which
preserves the arrangement of eight planes.
This map induces an involution $\Phi:X\lra X$ on the Calabi--Yau
threefold $X$ defined by this arrangement. We will compute
the trace
\[
d_{p}=\tr((\Frob_{p}\circ\Phi)^*|H^{3}(\bar X_{p}, \mathbb Q_{l}))
\]
of Frobenius composed with $\Phi$. Since this map acts by
multiplication with $\pm p$ on $H^{2}$ and with $\pm p^{2}$ on $H^{4}$ the
Lefschetz fixed--point formula relates $d_{p}$ to the number $N_{p}$
of fixed points of $\Frob_{p}\circ \Phi$.
\begin{lemma}
If $\Phi$ is a linear involution on $\PP[N](\bar{\mathbb F}_{p})$
defined over $\mathbb F_{p}$ then
the fixed points of $\Frob_{p}\circ\Phi$ are $\mathbb F_{p^{2}}$-rational.
\end{lemma}
\begin{proof}
The Frobenius morphism $\Frob_{p}$ commutes with any linear involution
defined over $\mathbb F_{p}$, so any fixed point of $\Frob_{p}\circ\Phi$ is
also a fixed point of $\Frob_{p^{2}}$.
\end{proof}
Using the Lemma we reduce the counting of fixed points over
the infinite field $\bar{\mathbb F}_{p}$ to counting of points over the
finite field $\mathbb F_{p^{2}}$, which can easily be done using a computer.
From the representation as a Kummer fibration we can easily recognize
some linear involutions preserving the arrangement:
\textbf{Arr. no. \phantom{0}53:}
\((x,y,z,t)\mapsto(y,x,-t,-z)\)
\textbf{Arr. no. 244:}
\((x,y,z,t)\mapsto(y,x,-t,-z)\)
\textbf{Arr. no. 267:}
\((x,y,z,t)\mapsto(t,-z,-y,x)\)
\textbf{Arr. no. 274:}
\((x,y,z,t)\mapsto(z,-t,x,-y)\)
Simple computations show that the above involutions are not equal
to identity on the deformation space $H^{1}(\mathcal T_{X})\cong
H^{12}(X)$, hence they split the Galois
representations. In fact it is easy to observe that $H^{12}(X)\oplus
H^{21}(X)$ must be $(-1)$--eigenspaces. Counting fixed points on the
singular double octic yields for all primes $5\le p\le 97$:
\[\def\arraystretch{1.9}
\begin{array}[t]{r@{\hspace{3mm}}l}
X_{53}:&\hspace{3.4mm} 1+p^{3}-a_{p}+pb_{p}+p^{2}+p\\
X_{244}:&\def\arraystretch{1.1}
\begin{cases}
1+p^{3}-a_{p}+pb_{p}+2p^{2}-p,&\qquad p\equiv 1\mod4\\
1+p^{3}-a_{p}+pb_{p}+3p,&\qquad p\equiv 3\mod4
\end{cases}\\
X_{267}:&\hspace{3.4mm}1+p^{3}-a_{p}+pb_{p}+p^{2}-p\\
X_{274}:&
\begin{cases}
1+p^{3}-a_{p}+pb_{p}+p^{2}-p,&\qquad p\equiv 1\mod4\\
1+p^{3}-a_{p}+pb_{p}+p^{2}+3p,&\qquad p\equiv 3\mod4
\end{cases}
\end{array}
\]
Analyzing the action of Frobenius on the generators of the Picard
group and the space of curves $H^{4}$ gives the traces of Frobenius of
the two--dimensional Galois subrepresentations. Applying the
Faltings--Serre-Livn\'e method finishes the proof.
M. Sch\"utt suggested to us that counting points in $\mathbb F_{p}$ and
$\mathbb F_{p^{2}}$ we can compute the characteristic polynomial,
which factors into two degree two polynomials. Since we know that the
representation splits we get the traces of both actions. It is
however not straightforward that the numbers $a_{p}$ resp. $pb_p$
will correspond to the $+1$--eigenspace resp. the $-1$--eigenspace.
\begin{remark}
The described involutions act on singular double octics. Since the
resolution of singularities of a double octic is not unique (it depends
on the order in which we blow up lines in a triple point) it may
happen that an involution maps to a birational Calabi--Yau
threefold. Since two smooth models differ by a sequence of flops, we can
compose the involution with these flops or we can consider a threefold
that dominates both smooth models. The action on $H^{3}$ is well
defined.
If we know that we can chose such a resolution of singularities of the
double covering to which the involution lifts, then the quotient will be
(after resolution) a rigid Calabi--Yau threefold.
\end{remark}
\begin{example}\label{ex:arr287}
Consider the arrangement of planes (arr. no. 287 in \cite{Meyer}) given by
\[
xyzt(x+y+z-3t)(x+y-3z+t)(x-3y+z+t)(-3x+y+z+t)=0.
\]
The corresponding Calabi--Yau threefold $X_{287}$ has Hodge numbers
$h^{11}(X_{287})=37$, $h^{12}(X_{287})=3$.
Counting points in $\mathbb F_{p}$ shows that, for $5\le p\le97$,
the trace of Frobenius on the middle cohomology equals $a_{p}+3b_{p}$,
where $a_{p}$ resp. $b_{p}$ are the coefficients of the weight 4 level 6
resp. weight 2 level 24 cusp form. The arrangement has many linear
symmetries. We can use the induced involutions on $X$ to decompose
the Galois representation.
We can also use the elliptic fibrations on $X$ described in
\cite{Meyer} and apply the deformation argument from example~\ref{ex:ar269}
to prove modularity of $X_{287}$.
In fact the full permutation group $S_{4}$ acts on this Calabi--Yau
threefold. If we consider the action of permutations of order 3, then
the eigenvalues will be defined in $\mathbb F_{p}$ only for some $p$,
so the decomposition of Frobenius action will depend on $p$.
\end{example}
\subsection*{Acknowledgements}
The work on this paper was done during the first named author's stays at the
Institutes of Mathematics of the Johannes Gutenberg-Universit\"at Mainz
and the Universit\"at Hannover. He would like to thank both institutions
for their hospitality. The authors also would like to thank Prof. Duco van Straten,
Prof. Klaus Hulek and Matthias Sch\"utt for their help.
| 4,201
|
:
2 Comments
Hi is there a way that the canvas can ve related to each other, in the way that if I go down one that has the infomartion of the othere that one moves along?
Or is there a way to change make the Task & Gant different, like instede of Task where scenes?
Hi,
I want to save the settings of the widget which I have created during designing a canvas, and basically copy/utilize/use this widget in other projects too. Is there a faster way than each time creating the widget for a graph? This may speed things up a bit for reporting on production insights.
| 369,792
|
In February 1978, Brian and Bobbie Houston a fresh and contemporary church. In August 1983, they founded Hills Christian Life Centre. It has grown from a congregation of 45 to what is said to be the largest church in Australian history.
Today Hillsong Church is headquartered in Sydney, with campuses in 15 countries on 6 continents.
Currently, Hillsong has a global audience quickly approaching 100,000 in weekly attendance.
Hillsong Church has three globally renown music ministries; Hillsong Worship, UNITED and Young and Free (Y&F). Hillsong Worship, started in 1998 has released 24 albums totaling over 275 songs. UNITED, established by Brian and Bobbie's son, Joel has received international recognition and praise for their innovative worship music. Young and Free (Y&F) is the newest of the three worship teams, focusing on youth worship ministry. They currently have two albums.
Since 1986, Hillsong in Australia has hosted its annual conferences where thousands of people gather to hear about church leadership, worship, and community engagement from Hillsong’s key team members and other renowned Christian leaders. Churches of all denominations from across the globe are able to experience what God is doing in and through the people at Hillsong Church through the annual Hillsong Conference, Colour Your World Women's Conference and Hillsong Men's Conference.
Located in Sydney, Hillsong International Leadership College offers leadership and ministry courses to help shape future leaders around the world. Since its creation in 1995, more than 10,000 students, representing more than 65 countries have graduated from the college.
Hillsong Church is also actively involved in ministry initiatives partnering with organizations like The A21 Campaign, Vision Rescue, Hillsong Africa Foundation, The Colour Sisterhood, Greenlight, iCareRevolution, The Platt Centre and CityCare Australia.
The vision of Hillsong Church is to reach and influence the world by building a large Christ-centered, Bible-based church, changing mindsets and empowering people to lead and impact in every sphere of life.
Social Media OutletsFacebook Twitter Instagram
| 216,868
|
Silverlight Performance Analysis tool – Coming Soon
November 12, 2010 in Silverlight, Windows Phone 7.
The below screenshots were taken from the keynote video and you can see the timings at the bottom if you want to watch the video:
In addition to debugging on the device, you can profile the app by selecting Silverlight Performance Analysis:
The device will then be prepared for profiling:
After running the application, a graphical summary will be created with execution time on the x-axis. It shows color-coded frame rates (green is good, red is bad) and relates that to CPU-usage and specific Storyboards:
In the above example, it appears that the first Storyboard might be leading to increased CPU usage and a degraded frame rate. You can highlight and drill down to see details for a specific time frame (from approx 4.5 to 7.5 seconds into the profiling session):
The next screen shows CPU and GPU metrics as well as a list of warnings. The first entry warns about 35 instances of ColorAnimation:
Scrolling down, you will see an Element Summary:
Followed by a Frames section with a CPU usage graph:
And finally the ability to drill down the Visual Tree and see problems highlighted:
This tool would be very helpful in analyzing app performance on the device. ScottGu did not give a specific release date for this or any additional tools that might be coming.
Today I saw that there is a Silverlight Firestarter event scheduled for December 2, 2010 and wonder if the tool will be ready to release then.
It would make a great time to announce it.
2 Trackbacks
Neudesic Blogson November 12, 2010
Windows Phone 7 Development Links | Shazaml Design, LLCon November 13, 2010
Silverlight Performance Analysis tool – Coming Soon…
Silverlight Performance Analysis tool – Coming Soon…
[...] Silverlight Performance Analysis tool – Coming Soon (announced at PDC10) – [...]
| 205,024
|
\begin{document}
\thanks{This research was partially supported by the Research Council of Norway through the project "Operator Algebras", and by the NordForsk research network "Operator Algebra and Dynamics" (grant \#11580).}
\begin{abstract}
The relative graph $C^*$-algebras introduced by Muhly and Tomforde are generalizations of both graph algebras and their Toeplitz extensions. For an arbitrary graph $E$ and a subset $R$ of the set of regular vertices of $E$ we show that the relative graph $C^*$-algebra $C^*(E, R)$ is isomorphic to a partial crossed product for an action of the free group generated by the edge set on the relative boundary path space. Given a time evolution on $C^*(E, R)$ induced by a function on the edge set, we characterize the KMS$_\beta$ states and ground states using an abstract result of Exel and Laca. Guided by their work on KMS states for Toeplitz-Cuntz-Krieger type algebras associated to infinite matrices, we obtain complete descriptions of the convex sets of KMS states of finite type and of KMS states of infinite type whose associated measures are supported on recurrent infinite paths. This allows us to give a complete concrete description of the convex set of all KMS states for a big class of graphs which includes all graphs with finitely many vertices.
\end{abstract}
\maketitle
\section{Introduction}
\label{sec:introduction}
Characterizations of KMS$_\beta$ states and ground states on $C^*$-algebras of Toeplitz and Cuntz-Krieger type associated to a directed graph $E$ have been obtained in different contexts by many authors, see for example
\cite{MR759450}, \cite{MR602475}, \cite{MR2057042},
\cite{MR1953065}, \cite{MR2056837}, \cite{MR500150} and \cite{MR1785460}. A classical reference for the definition of
KMS$_\beta$ states and ground states as well as background is \cite[Section 5.3]{MR1441540}. Recently there has been renewed interest in constructions of KMS states for graph algebras, see \cite{KW}, \cite{aHLRS} and \cite{Cas-Mor}. KMS weights on $C^*$-algebras associated to graphs were studied in \cite{MR2907004} and \cite{Tho}.
There are several techniques used in characterizing KMS states and constructing them in specific cases. These can employ the definition of the graph $C^*$-algebra $C^*(E)$ and its Toeplitz extension $\mathcal{T}C^*(E)$ as universal $C^*$-algebras with generators and relations, see e.g. \cite{aHLRS} and \cite{Cas-Mor}, or the realization of $C^*(E)$ and $\mathcal{T}C^*(E)$ as $C^*$-algebras of Pimsner type associated to a Hilbert bimodule, see \cite{MR2056837} and \cite{KW}, or the description of $C^*(E)$ as a groupoid $C^*$-algebra, see \cite{aHLRS}, which appeals to the general result from \cite{Nesh}.
A different type of general method that provides characterizations of KMS states was developed by Exel and Laca in their study of Toeplitz-Cuntz-Krieger type algebras for infinite matrices, and uses crossed products by partial group actions, cf. \cite{MR1703078} and \cite{MR1953065}.
Our contribution here is an analysis of KMS and ground states on $C^*$-algebras $C^*(E, R)$ associated to an arbitrary directed graph $E$ and subsets $R$ of the regular vertices (i.e. those vertices that are not sinks or infinite emitters) by means of
realizing any such algebra as a partial crossed product for an action of the free group generated by the edge set. The space acted upon is a certain collection of boundary paths, and the resulting setup in the spirit of \cite{MR1953065} seems very well-suited for the analysis of KMS states of the various algebras. By emphasizing the common picture of $C^*(E)$ and $\mathcal{T}C^*(E)$ as relative graph algebras in the sense of \cite{MT}, we obtain a unified description of KMS states, see Theorem~\ref{thm:kms} for the precise statement. Not surprisingly, KMS states correspond to probability measures on the boundary path space that satisfy a certain scaling property.
With motivation coming from the distinction between measures of finite and infinite type that is a crucial ingredient in \cite{MR1953065}, we distinguish between three classes of measures. First we have the finite type measures as in \cite{MR1953065}. Next, we identify as a new ingredient two kinds of infinite type measures: the measures that are supported on \emph{recurrent} paths, and the infinite type measures that are supported on \emph{wandering} paths. Following \cite{Tho2}, we call these \emph{conservative} and \emph{dissipative} measures, respectively. For the finite type and conservative measures we give a complete parametrization of the extreme points of the convex set of KMS$_\beta$ states for $\beta\in [0,\infty)$ in terms of what we call regular and critical vertices. A similar parametrization of the dissipative measures seems more difficult to obtain. We also provide a complete concrete description of the extreme points of the convex set of ground states, and furthermore we identify all those ground states that are KMS$_\infty$ states as introduced in \cite{Con-Mar}. We show moreover that the
measures of infinite type correspond bijectively to normalized
eigenvectors of a (possibly infinite and even uncountable) matrix
associated to the directed graph under consideration.
We illustrate in examples that several new phenomena occur in the configuration of KMS states for $\mathcal{T}C^*(E)$ and $C^*(E)$ endowed with the gauge action in case that $E$ is an infinite graph. In Example~\ref{ex:a} we present an infinite strongly connected graph $E$ of finite degree (or valence) for which $C^*(E)$ has no KMS states, thus showing that the analogue of \cite[Theorem 4.3]{aHLRS} does not hold for infinite graphs. In Example~\ref{ex:motivating} we show that on $\mathcal{T}C^*(E)$, all three types of states can occur; moreover, the finite type and infinite type states co-exist on a critical interval, and there is a "phase-transition" between the two infinite types at a critical temperature.
The proof of the main general result relies on the
characterization of KMS states on the crossed product of a
semi-saturated orthogonal partial action of a free group on a
$\cs$-algebra obtained by Exel and Laca
\cite[Theorem 4.3]{MR1953065}. Towards using the Exel-Laca result, for a given directed graph $E$ and a subset $R$ of the regular vertices, we start by constructing a locally compact space $\partial_R E$ and a semi-saturated orthogonal partial action $\Phi$ of the free group $\free$ generated by $E^1$ on the
$\cs$-algebra $C_0(\partial_R E)$, see Section~\ref{sec:part-action-on-graph}. We point out that the space
$\partial E$ corresponding to the choice of $R$ as the entire subset of regular vertices of $E$ is the boundary path space of the graph, which recently played a role in \cite{W}. In Section \ref{sec:cs-algebra-graph} we prove that the crossed product $C_0(\partial_R E)\rtimes_\Phi \free$ is isomorphic to $C^*(E, R)$, see Theorem~\ref{theorem:partial}. A similar result is obtained with different methods in \cite{MR1452280} in the case of the usual Cuntz algebra $\mathcal{O}_n$ and Cuntz-Krieger algebra $\mathcal{O}_A$ and their Toeplitz extensions.
Our main general characterization of KMS and ground states on $C^*(E, R)$ is contained in Theorem~\ref{thm:kms}. This result has close connections to the existing literature, and we elaborate on this point in several remarks. In section~\ref{section:extreme} we introduce the sets
$\Ereg$ and $\Ecrit$ of regular and critical vertices for a given $\beta\in[0,\infty)$, associate measures to them, and develop a machinery that enables us to give the characterizations of the convex sets of measures of finite type and of conservative measures, see Theorems~\ref{thm:finitetype-reg} and \ref{thm:infinitetype-crit}. Section~\ref{section:ground} deals with ground states and KMS$_\infty$ states. In the final section we present several examples, all of which illustrate in different ways that a much richer structure of KMS states can be expected as one passes from finite to infinite graphs.
We thank M. Laca for suggesting, at a very early stage of this project, to look more carefully at the Toeplitz extension of the graph $C^*$-algebra, and K. Thomsen for valuable comments to an earlier draft of this paper.
\section{A partial action on the relative path space of a graph}
\label{sec:part-action-on-graph}
Let $E=(E^0,E^1,s,r)$ be a directed graph: by this we mean that $E^0$ and $E^1$ are
arbitrary (not necessarily countable) sets and $s$ and $r$ are maps from $E^1$ to $E^0$. Elements of $E^0$ are called
\emph{vertices} of $E$ and elements of $E^1$ are \emph{edges} of $E$. If $e\in E^1$, then $s(e)$ denotes the
\emph{source} of $e$ and $r(e)$ the \emph{range} of $e$.
A \emph{path of length $n$} in $E$ is a sequence $e_1e_2\dots e_n$ of
edges in $E$ such that $r(e_i)=s(e_{i+1})$ for
$i\in\{1,2,\dotsc,n-1\}$ (the reader should be warned that in some papers and books the roles of $r$ and
$s$ are interchanged, so a path would be a sequence $e_1e_2\dots e_n$ of
edges in $E$ such that $s(e_i)=r(e_{i+1})$ for
$i\in\{1,2,\dotsc,n-1\}$). We regard vertices as paths of length 0 and
edges as paths of length 1. We denote by $E^n$ the set of paths of length
$n$ in $E$ and write $E^*$ for the set $\bigcup_{n\in\No}E^n$. We
write $|u|$ for the length of a path $u\in E^*$, and we let $E^{\le n}$ be the collection of paths $u$
with $|u|\le n$. We extend the range
and source maps to $E^*$ by setting $s(u)=s(e_1)$ and $r(u)=r(e_n)$
for $u=e_1e_2\dots e_n\in E^n$ with $n\ge 1$, and $s(v)=r(v)=v$ for
$v\in E^0$. If $v\in E^0$, then we let $vE^n=\{u\in E^n\mid s(u)=v\}$ and $E^nv=\{u\in E^n\mid r(u)=v\}$, $vE^*=\{u\in E^*\mid s(u)=v\}$ and $E^*v=\{u\in E^*\mid r(u)=v\}$. We let $E^0_\reg=\{v\in E^0\mid vE^1\text{ is finite and non-empty}\}$.
If $u=e_1e_2\dots e_n$ and $u'=e'_1e'_2\dots e'_m$ are
paths with $r(u)=s(u')$, then we write $uu'$ for the path $e_1e_2\dots
e_ne'_1e'_2\dots e'_m$ obtained from concatenation of the two paths.
We recall from, for example \cite{MR1988256}, \cite{MR2135030} and
\cite{MR1914564} that the $\cs$-algebra $\cs(E)$ of the graph $E$ is defined as the
universal $\cs$-algebra generated by a \emph{Cuntz-Krieger $E$-family} $(s_e,p_v)_{e\in
E^1,v\in E^0}$ consisting of partial isometries $(s_e)_{e\in E^1}$
with mutually orthogonal range projections
and mutually orthogonal projections $(p_v)_{v\in E^0}$ satisfying
\begin{enumerate}\renewcommand{\theenumi}{CK\arabic{enumi}}
\item\label{it:CK1} $s_e^*s_e=p_{r(e)}$ for all $e\in E^1$,
\item\label{it:CK2} $s_e s_e^*\le p_{s(e)}$ for all $e\in E^1$,
\item\label{it:CK3} $p_v=\sum_{e\in vE^1}s_e s_e^*$ for $v\in E^0_\reg$.
\end{enumerate}
In \cite{MT}, \emph{relative graph $C^*$-algebras} were introduced. To define a relative graph $C^*$-algebra we must in addition to a directed graph $E$ specify a subset $R$ of $E^0_\reg$.
The \emph{relative graph $\cs$-algebra} $\cs(E,R)$ of the graph $E$ relative to $R$ is then defined as the universal $\cs$-algebra generated by a \emph{Cuntz-Krieger $(E,R)$-family} $(s_e,p_v)_{e\in
E^1,v\in E^0}$ consisting of partial isometries $(s_e)_{e\in E^1}$
with mutually orthogonal range projections
and mutually orthogonal projections $(p_v)_{v\in E^0}$ satisfying \eqref{it:CK1} and \eqref{it:CK2} above plus the relation
\begin{enumerate}\renewcommand{\theenumi}{RCK3}
\item\label{it:RCK3} $p_v=\sum_{e\in vE^1}s_e s_e^*$ for $v\in R$.
\end{enumerate}
\begin{remark} \textnormal{(a)} If $R=E^0_\reg$, then a Cuntz-Krieger $(E,R)$-family is the same as a Cuntz-Krieger $E$-family and, consequently, $C^*(E,R)=C^*(E)$.
\textnormal{(b)} If $R=\emptyset$, we claim that $C^*(E,R)$ is the Toeplitz algebra $\mathcal{T}C^*(E)$ introduced in \cite{FR}. Indeed, in this case the relation \eqref{it:CK3} is vacuous, and \eqref{it:CK2} in connection with the assumption that the ranges of the $s_e$'s are mutually orthogonal imply that
\begin{equation}\label{eq:Toeplitz-relation}
p_v\geq \sum_{e\in F}s_e s_e^*
\end{equation}
whenever $F$ is a finite subset of $vE^1$. As remarked in \cite[Lemma 1.1]{aHLRS}, the converse holds, thus \eqref{eq:Toeplitz-relation} alone implies that the ranges of the $s_e$'s are mutually orthogonal.
\end{remark}
It follows from the universal properties of $C^*(E,R)$ that there is a strongly continuous action (the \emph{gauge action}) $\gamma:\T\to\aut(C^*(E,R))$ satisfying for all $z\in\T$ that $\gamma_z(p_v)=p_v$ for $v\in E^0$, and $\gamma_z(s_e)=zs_e$ for $e\in E^1$.
In order to obtain a different picture of $C^*(E,R)$ we now turn to the path space and the boundary path space. An \emph{infinite path} in $E$ is an infinite sequence $e_1e_2\dots$ of
edges in $E$ such that $r(e_i)=s(e_{i+1})$ for
$i\in\N$. We write $E^{\infty}$ for the set of infinite paths in $E$.
The \emph{path space} of the graph is
$E^{\le\infty}:=E^*\cup E^\infty$.
When $R$ is a subset of $E^0_\reg$, the \emph{relative boundary path space} $\partial_R E$ is defined by
\begin{equation*}
\partial_R E=E^\infty\cup\{u\in E^*\mid r(u)\notin R\}.
\end{equation*}
Equivalently, $\partial_R E=E^{\le\infty}\setminus \{u\in E^*\mid r(u)\in R\}$.
If $R=E^0_\reg$, then the relative boundary path space $\partial_R E$ is also called the \emph{boundary path space} and is denoted by $\partial E$, see e.g. \cite{W}.
Given $u=e_1e_2\dots e_n\in E^n$ and $0\leq m\leq n$, we let $u(0,m)$ denote the path $e_1e_2\dots e_m $ if
$1\le m\le n$, and $s(u)$ if $m=0$. Likewise, for an infinite path $x=e_1e_2\dots\in
E^\infty$ and $m\ge 1$, we denote $x(0,m)$ the path $e_1e_2\dots e_m$, and if $m=0$, then we write $x(0,m)$
or $s(x)$ for the vertex $s(e_1)$. Extending our convention for concatenation of finite paths, if
$u=e_1e_2\dots e_n\in E^n$
and $x=e'_1e'_2\dots\in E^\infty$ with $r(u)=s(x)$, then we write $ux$
for the resulting infinite path.
In order to compare finite paths with arbitrary paths, we introduce the following notation: for $u\in E^*$ and $x\in E^{\le\infty}$
we write $u\le x$ if
$|u|\le |x|$ and $x(0,|u|)=u$. Further, we write $u<x$ if $u\le x$ and $u\ne
x$. For $u\in E^*$ the \emph{cylinder set of
$u$} $\cyl{u}$ is defined by
\begin{equation}\label{def:cylinder-set}
\cyl{u}=\{x\in E^{\le\infty}\mid u\le x\}.
\end{equation}
We denote by $\mathcal{F}(E^*)$ the collection of finite subsets of the space $E^*$. Given $u\in E^*$ and
$F\in \mathcal{F}(E^*)$, we let
\begin{equation}\label{def:ZFu}
\cylf{u}=\cyl{u}\setminus\Bigl(\bigcup_{\underset{u\leq u'}{u'\in F}}\cyl{u'}\Bigr);
\end{equation}
note that this is the empty set precisely when $u\in F$. In particular, if $Z_F(u)$ is non-empty, then it contains $u$.
The statements \eqref{item:16}-\eqref{item:19} in the next result can be found in \cite{W}. We include them in order to have
the necessary terminology at hand for proving \eqref{item:20}-\eqref{item:22}.
\begin{proposition} \label{prop:et}
Let $E$ be a directed graph and endow the path space $E^{\le\infty}$
with the topology generated by $\{\cyl{u}\mid u\in
E^*\}\cup\{E^{\le\infty}\setminus\cyl{u}\mid u\in
E^*\}$. We then have:
\begin{enumerate}\renewcommand{\theenumi}{\roman{enumi}}
\item\label{item:16} $E^{\le\infty}$ is a totally disconnected
locally compact Hausdorff topological space.
\item\label{item:17} $E^{\le\infty}$ is compact if and only if $E^0$ is finite.
\item\label{item:18} The system $
\bigl\{\cylf{u}\mid
u\in E^*,\ F \in \mathcal{F}(E^*)\bigr\}$
is a basis of open and compact subsets for the topology of
$E^{\le\infty}$.
\item\label{item:19} If $u\in E^*$, then the system $
\bigl\{\cylf{u}\mid
F \in \mathcal{F}(E^*)
\text{ such that }\cylf{u}\ne\emptyset\bigr\}$
is a neighbourhood basis for $u$.
\item\label{item:20} Every $u\in E^*$ for which $r(u)E^1$ is a finite set, is isolated.
\item\label{item:21} $E^*$ is dense in $E^{\le\infty}$.
\item\label{item:22} The closure of $E^\infty\cup\{u\in E^*\mid
r(u)E^1=\emptyset\}$ in $E^{\le\infty}$ is $\partial E$.
\end{enumerate}
\end{proposition}
\begin{proof}
\eqref{item:20}: If $u\in E^*$, then $\{u\}=\cyl{u}\setminus\Bigl(\bigcup_{e\in r(u)E^1}\cyl{ue}\Bigr)$.
Thus if $r(u)E^1$ is a finite set $F$, then $\{u\}=\cylf{u}$, so $u$ is isolated.
\eqref{item:21}: Given $x\in E^\infty$, suppose $x\in Z_F(u)$ for some $u\in E^*$ and $F\in \mathcal{F}(E^*)$ with
$u\le u'$ for every $u'\in F$. Then $x(0,\vert u\vert)\in Z_F(u)$, hence $u\in Z_F(u)\cap E^*$.
It follows that
$E^*$ is dense in $E^{\le\infty}$.
\eqref{item:22}: If $u\in E^*$, $r(u)E^1$ is infinite, and $F$ is a finite subset of
$E^*$ such that $u\le u'$ for every $u'\in F$, then there exists
at least one $e\in r(u)E^1$ such that $ue\in
\cylf{u}$, and thus at least one
element in $E^\infty\cup\{u'\in E^*\mid
s\inv(r(u'))=\emptyset\}$ which belongs to
$\cylf{u}$. Hence $\partial E$ is
contained in the closure of $E^\infty\cup\{u\in
E^*\mid r(u)E^1=\emptyset\}$. That $\partial E$ is closed follows from
the fact that every $u\in E^{\le\infty}\setminus\partial E$ is such that $r(u)E^1$ is finite (and non-empty),
and therefore $u$ is isolated by \eqref{item:20}. This proves \eqref{item:22}.
\end{proof}
\begin{corollary} \label{cor:rps}
Let $E$ be a directed graph and let $R$ be a subset of $E^0_\reg$. Equip the relative boundary path space $\partial_R E$ with the topology it inherits from $E^{\le\infty}$ when the latter is given the topology described in Proposition \ref{prop:et}. Then $\partial_R E$ is a totally disconnected locally compact Hausdorff topological space.
\end{corollary}
\begin{proof}
According to Proposition \ref{prop:et}\eqref{item:16}, $E^{\le\infty}$ is a totally disconnected locally compact Hausdorff topological space. If $u\in E^{\le\infty}\setminus \partial_RE$, then $u\in E^*$ and $r(u)\in R\subseteq E^0_\reg$, so $u$ is isolated according to \ref{prop:et}\eqref{item:20}. It follows that $\partial_R E$ is a closed subset of $E^{\le\infty}$ and therefore a totally disconnected locally compact Hausdorff topological space.
\end{proof}
Let $E$ be a directed graph and $R$ a subset of $E^0_\reg$. Let $\free$ denote the free group generated by $E^1$. An edge $e\in E^1$ will still be denoted $e$
as an element of $\free$, and $e^{-1}$ will denote the inverse of $e$ in $\free$.
We shall view $E^*\setminus E^0$ as a subset of $\free$
upon identifying a path $u=e_1e_2\dots e_n$ with the element in $\free$ obtained by multiplication of $e_1,\dots, e_n$. The identity
element of $\free$ will be denoted $\31$. An element $g\in\free$ is in \emph{reduced form} if $g=a_na_{n-1}\dotsm a_1$ for $a_1,a_2,\dotsc,a_n\in E^1\cup\{e\inv\mid e\in
E^1\}$ such that $a_i\ne a_{i+1}\inv$ whenever $i\in\{1,2,\dotsc,n-1\}$. We denote
$|g|$ the number of generators in the reduced form of $g$. Notice that this use of $|u|$ agrees with the previously defined
use of $|u|$ as the length of an element $u\in E^*\setminus E^0$.
Now we construct a semi-saturated and orthogonal partial action of $\free$ on $\partial_RE$. We will do this by defining open and compact subsets $\tdomain{g}$ and $\trang{g}$ of $\partial_RE$ together with a homeomorphism $\tpa_g$ taking $\tdomain{g}$ onto $\trang{g}$ and satisfying the axioms of a partial action as $g\in \free$.
First, let $\tpa_{\31}$ denote the identity map on $\partial_RE$ and let
$\tdomain{\31}=\trang{\31}=\partial_RE$. For $e\in E^1$, let
$\trang{e}=\cyl{e}\cap \partial_RE$ and $\tdomain{e}=\cyl{r(e)}\cap \partial_RE$, and define maps
\begin{align}
\tpa_e:\tdomain{e}\to \partial_RE,\ &\tpa_e:x\mapsto ex,\label{eq:phi-e}\\
\tpa_{e\inv}:\trang{e}\to \partial_RE,\ &\tpa_{e\inv}:ex\mapsto x.\label{eq:phi-einv}
\end{align}
Let $g=a_na_{n-1}\dotsm a_1\in\free$ be in reduced form. We will define $\tdomain{(a_ia_{i-1}\dotsm a_1)}$ and $\tpa_{a_ia_{i-1}\dotsm a_1}:\tdomain{(a_ia_{i-1}\dotsm a_1)} \to \partial_RE$
for all $i\in\{1,2,\dots,n\}$ recursively. For $i=1$, $\tdomain{a_1}$, $\trang{a_1}$ and $\tpa_{a_1}:\tdomain{a_1}\to \partial_RE$ have already been defined. For
$i>1$, we let
\begin{gather}
\tdomain{(a_ia_{i-1}\dotsm a_1)} = \tpa_{a_{i-1}\dotsm a_1}\inv(\tdomain{a_i}) \text{ and}\label{eq:def-Ug}\\
\tpa_{a_ia_{i-1}\dotsm a_1}(x) = \tpa_{a_i}(\tpa_{a_{i-1}\dotsm a_1}(x))\text{ for }x\in \tdomain{(a_ia_{i-1}\dotsm a_1)}.\label{eq:phi-ai}
\end{gather}
For later use, we record some properties of the sets $U(g)$ and the maps $\phi_g$.
\begin{lemma}\label{lem:alternative-U-phi} The sets $U(g)$ and the maps $\phi_g$ for $g\in \free$ defined by \eqref{eq:def-Ug}-\eqref{eq:phi-ai} satisfy the following:
\begin{enumerate}\renewcommand{\theenumi}{\roman{enumi}}
\item \label{item:g1} $\domain{\31}=\rang{\31}=\partial_R E$ and $\pa_{\31}(x)=x$ for $x\in \domain{\31}$.
\item \label{item:g2} If $u\in E^*\setminus E^0$, then $\domain{u}=\cyl{r(u)}\cap\partial_R E$, $\rang{u}=\cyl{u}\cap\partial_R E$ and $\pa_u(x)=ux$ for $x\in \domain{u}$.
\item \label{item:g3} If $u=e_1\dots e_m,u'=e'_1\dots e'_n\in E^*\setminus E^0$, $e_m\ne e'_n$ and $r(u)=r(u')$, then $\rang{u'u\inv}=\cyl{u'}\cap\partial_R E$ and $\pa_{u(u')\inv}(u'x)=ux$ for $x\in\rang{u'u\inv}$.
\item \label{item:g4} If $g$ does not belong to $\{\31\}\cup\{u\mid u\in E^*\setminus E^0\}\cup\{u\inv\mid u\in E^*\setminus E^0\}\cup\{u(u')\inv\mid u,u'\in E^*\setminus E^0,\ r(u)=r(u')\}$, then $\domain{g}=\rang{g}=\emptyset$.
\end{enumerate}
\end{lemma}
\begin{proof}
Claim \eqref{item:g1} follows directly from the definition of $\tdomain{\31}$, $\trang{\31}$ and $\tpa_{\31}$, and \eqref{item:g2} and \eqref{item:g3} follow easily from \eqref{eq:phi-e}--\eqref{eq:phi-ai}.
To prove \eqref{item:g4}, notice that if $g$ does not belong to $\{\31\}\cup\{u\mid u\in E^*\setminus E^0\}\cup\{u\inv\mid u\in E^*\setminus E^0\}\cup\{u(u')\inv\mid u,u'\in E^*\setminus E^0,\ r(u)=r(u')\}$, then the reduced form of $g$ either contains a factor of the form $e\inv e'$ with $e,e'\in E^1$ and $e\ne e'$, a factor of the form $ee'$ with $e,e'\in E^1$ and $r(e)\ne s(e')$, or a factor of the form $(e')\inv e\inv $ with $e,e'\in E^1$ and $s(e)\ne r(e')$. It follows from \eqref{eq:phi-e}--\eqref{eq:phi-ai} that $\tdomain{g}=\trang{g}=\emptyset$ in each of these cases.
\end{proof}
\begin{proposition} \label{prop:action-on-path-space}
Let $E$ be a directed graph and let $R$ be a subset of $E^0_\reg$. Equip the relative boundary path space $\partial_R E$ with the topology described in Corollary \ref{cor:rps}, let $\free$ be the free group generated by $E^1$, and let for each $g\in\free$ the set $\tdomain{g}$ and the map $\tpa_g:\tdomain{g}\to \partial_R E$ be as defined above. We then have:
\begin{enumerate}\renewcommand{\theenumi}{\roman{enumi}}
\item \label{item:g5} For each $g\in\free$, the set $\rang{g}$ is an open and compact subset of $\partial_R E$ and $\pa_g$ is a homeomorphism from $\domain{g}$ onto $\rang{g}$ with inverse $\pa_{g\inv}$.
\item \label{item:g6} $\pa_{g_1}(\domain{g_1}\cap\rang{g_2})=\rang{g_1}\cap\rang{g_1g_2}$ for $g_1,g_2\in\free$.
\item \label{item:g7} $\pa_{g_1}\circ\pa_{g_2}=\pa_{g_1g_2} \text{ on } \domain{g_2}\cap\domain{(g_1g_2)}$ for $g_1,g_2\in\free$.
\item \label{item:g8} $\Phi=(\pa_g)_{g\in\free}$ is a semi-saturated and orthogonal partial action of $\free$ on $\partial_R E$ as in \cite[Section 2]{MR1703078}.
\end{enumerate}
\end{proposition}
\begin{proof} Item \eqref{item:g5} follows directly from \eqref{item:g1}--\eqref{item:g4} in Lemma~\ref{lem:alternative-U-phi}
and Proposition \ref{prop:et}.
For \eqref{item:g6}, notice first that \eqref{eq:def-Ug}, \eqref{eq:phi-ai} and \eqref{item:g5} imply that if $a_n\dotsm a_1$ is an element of $\free$ in reduced form, then
\begin{equation}\label{eq:alternative-phi}
\trang{a_n\dotsm a_1}=\tpa_{a_n\dotsm a_i}(\tdomain{(a_n\dotsm a_i)}\cap\trang{a_{i-1}\dotsm a_1})
\end{equation}
for any $i\in\{2,\dots,n\}$. If $g_1=a_m\dotsm a_1$ and $g_2=a_n'\dotsm a_1'$ are in reduced forms and we let $i$ be the largest nonnegative integer such that $a_1\inv\dotsm a_i\inv=a_n'\dotsm a_{n+1-i}'$ (with $i=0$ if $a_1\inv\ne a_n'$), then $a_m\dotsm a_{i+1}a_{n-i}'\dotsm a_1'$ is the reduced form of $g_1g_2$ (where $a_m\dotsm a_{i+1}=\31$ if $i=m$, and $a_{n-i}'\dotsm a_1'=\31$ if $i=n$). It follows from \eqref{eq:alternative-phi} that
\begin{align*}
\tdomain{g_1}\cap\trang{g_2}&=\trang{a_1\inv\dotsm a_m\inv}\cap\trang{a_n'\dotsm a_1'}\\
&=\tpa_{a_1\inv\dotsm a_i\inv}(\trang{(a_i\dotsm a_1}\cap\trang{a_{i+1}\inv\dotsm a_m\inv}\cap\trang{a_{n-i}'\dotsm a_1'}).
\end{align*}
Once again using \eqref{eq:alternative-phi}, we have
\begin{align*}
\tpa_{g_1}(\tdomain{g_1}\cap\trang{g_2})&=\tpa_{a_m\dotsm a_{i+1}}(\trang{a_i\dotsm a_{1}}\cap\trang{a_{i+1}\inv\dotsm a_m\inv}\cap\trang{a_{n-i}'\dotsm a_1'})\\
&=\trang{a_m\dotsm a_1}\cap\trang{a_m\dotsm a_{i+1}a_{n-i}'\dotsm a_1'} =\trang{g_1}\cap\trang{g_1g_2}.
\end{align*}
For \eqref{item:g7}, notice that it follows from \eqref{item:g5} and \eqref{item:g6} that $\tdomain{g_2}\cap\tdomain{(g_1g_2)}=\tpa_{g_2\inv}(\trang{g_2}\cap\trang{g_1})=\tpa_{g_2}\inv(\trang{g_1})$, so $\tpa_{g_1g_2}$ and $\tpa_{g_1}\circ\tpa_{g_2}$ are defined on the same subsets of $E^{\le\infty}$. If $g_1=a_m\dotsm a_1$, $g_2=a_n'\dotsm a_1'$ and $i$ are as above, then \eqref{eq:phi-ai} and \eqref{item:g5} imply that
\begin{align*}
\tpa_{g_1}\circ\tpa_{g_2}(x)&=\tpa_{a_m}\circ\dots\circ\tpa_{a_1}\circ\tpa_{a_n'}\circ\dots\circ\tpa_{a_1'}(x)\\
&=\tpa_{a_m}\circ\dots\circ\tpa_{a_{i+1}}\circ\tpa_{a_{n-i}'}\circ\dots\circ\tpa_{a_1'}(x)=\tpa_{g_1g_2}(x)
\end{align*}
for $x\in\tdomain{g_2}\cap\tdomain{(g_1g_2)}$ proving \eqref{item:g7}.
It follows from \eqref{item:g5}--\eqref{item:g7} that $\Phi=(\tpa_g)_{g\in\free}$ is a partial action of $\free$ on $\partial_RE$. If $e,f\in E^1$ and $e\ne f$, then $\trang{e}\cap\trang{f}=\cyl{e}\cap\cyl{f}\cap\partial_RE=\emptyset$. Furthermore, if $g_1=a_m\dotsm a_1$ and $g_2=a_n'\dotsm a_1'$ are in reduced forms and $\abs{g_1g_2}=\abs{g_1}+\abs{g_2}$, then $a_m\dotsm a_1a_n'\dotsm a_1'$ is the reduced form of $g_1g_2$, so $\trang{g_1g_2}=\tpa_{g_1}(\tdomain{g_1}\cap\trang{g_2})\subset\trang{g_1}$. It therefore follows from \cite[Proposition 4.1]{MR1953065} that $\Phi$ is semi-saturated and orthogonal, showing the claim in \eqref{item:g8}.
\end{proof}
\section{The graph algebras $C^*(E)$ and $\mathcal{T}C^*(E)$ as partial crossed products}\label{sec:cs-algebra-graph}
In this section we show that $C^*(E,R)$ can be realised as the full crossed product of the partial action $(C_0(\partial_R E), \free, \Phi)$ introduced in the previous section.
We start by introducing the terminology we need. There are different definitions of both a partial
representation of a discrete group on a Hilbert space (in e.g. \cite{MR1953065} and
\cite{MR1452280}) and of a covariant representation of a partial dynamical system $(A, G, \alpha)$ (in \cite{MR1331978} and \cite{MR1452280}). However, reassuring equivalences of these definitions were shown in \cite{MR1452280}, and we refer to \cite[\S 1]{ELQ} for a brief but illuminating overview of the main concepts and constructions.
Suppose that $(A, G, \alpha)$ is a
partial dynamical system: thus for each $g\in G$ the maps $\alpha_g$ are $*$-isomorphisms between closed, two-sided ideals
$D_{g^{-1}}$ and $D_g$ of a $C^*$-algebra $A$ such that $\alpha_e=\id_A$ and $\alpha_{gh}$ extends $\alpha_g\alpha_h$ for all $g,h\in G$. The \emph{full crossed product} $A\rtimes_\alpha G$ is the enveloping $C^*$-algebra of the convolution $*$-algebra $\{f\in l^1(G, A)\mid f(g)\in D_g\}$ endowed with a suitable norm.
A partial representation of a group $G$ on a Hilbert space $H$ is a map $u:G\to B(H)$ such that $u(e)=1$ (where here, $e$ denotes the neutral element of the group $G$),
$u_{g^{-1}}=u_g^*$ and $u_gu_hu_{h^{-1}}=u_{gh}u_{h^{-1}}$ for all $g,h\in G$, see \cite{MR1953065}. A \emph{covariant representation} of $(A, G, \alpha)$ is a pair $(\pi,u)$ that consists of a nondegenerate representation of $A$ on a Hilbert space $H$ and a partial representation $u$ of
$G$ on $H$ such that for $g\in G$ we have
\begin{align}
&u_gu_g^* \text{ is the projection onto the subspace } \spac\pi(D_g)H \text{ and}\label{p1}\\
&u_g\pi(f)u_{g\inv}=\pi(f\circ\alpha_{g\inv}) \text{ for } f\in D_{g^{-1}}.\label{p2}
\end{align}
As pointed out by Quigg and Raeburn in \cite{MR1452280}, there is a universal covariant representation $(\iota, \delta)$ in the double dual $(A\rtimes_\alpha G)^{**}$ such that $A\rtimes_\alpha G$ is the closed linear span of finite sums of the form $\sum\iota(a)\delta_g$. Since $\delta_e=1$, it follows that $\iota(A)\subseteq A\rtimes_\alpha G$, but we do not have in general that $\delta_g\in A\rtimes_\alpha G$. Since any partial system $(A,G, \alpha )$ admits covariant representations $(\pi,u)$ with $\pi$ faithful (e.g. the reduced covariant representation of \cite[Section 3]{MR1331978}), the representation $\iota$ is faithful.
Now we turn our attention back to the partial action of $\free$ on $\partial_R E$. In the sequel we regard $C_0(\trang{g})$ for $g\in\free$ as
an ideal of $C_0(\partial_RE)$ by letting $f(x)=0$ for $f\in
C_0(\trang{g})$ and $x\in \partial_RE\setminus \trang{g}$. If $f\in
C_0(\tdomain{g})$, then $f\circ\tpa_{g\inv}$ will denote the element of
$C_0(\trang{g})$ defined by
\begin{equation*}
f\circ\tpa_{g\inv}(x)=
\begin{cases}
f(\tpa_{g\inv}(x))&\text{if }x\in\trang{g},\\
0&\text{if }x\in \partial_RE\setminus\trang{g}.
\end{cases}
\end{equation*}
Then the map $f\mapsto f\circ\tpa_{g\inv}$ is a $*$-isomorphism
from $C_0(\tdomain{g})$ to $C_0(\trang{g})$. By a slight abuse of notation (which should not lead to any confusion) we
let $\tpa_g$ be the map $f\mapsto f\circ\tpa_{g\inv}$, and let $D_{g\inv}=C_0(\tdomain{g})$ be its domain while $D_g=C_0(\trang{g})$
is its range. Thus $\Phi=(\tpa_g)_{g\in\free}$ is a partial action of $\free$ on the $C^*$-algebra $C_0(\partial_RE)$.
The action is still semi-saturated and orthogonal.
We let $(\iota, \delta)$ denote the universal covariant representation of $(C_0(\partial_R E),\free,\Phi)$.
We recall from \cite[Theorem 4.3]{MR1953065} that given any function $N:E^1\to (1,\infty)$ there exists a unique strongly continuous one-parameter group $\sigma$ of automorphisms of $C_0(\partial_RE)\rtimes_{\Phi}\free$ such that
\begin{equation}\label{eq:sigma-from-N}
\sigma_t(b)=\bigl(N(e)\bigr)^{it}b\,\text{ and }\,\sigma_t(c)=c
\end{equation}
for all $e\in E^1$, all $b\in\iota\bigl(C_0(\rang{e})\bigr)\delta_e$, and all $c\in\iota\bigl(C_0(\partial_RE)\bigr)$.
If $N(e)=\exp(1)$ for every $e\in E^1$, then\footnote{Since $\exp(1)$ only makes a couple of appearances here, while elements in $E^1$ are used all throughout, we decided to give preference to the notation $e\in E^1$.} $\sigma_t$ is
$2\pi$-periodic, and so induces a strongly continuous action $\beta:\T\to\aut(C_0(\partial_RE)\rtimes_{\Phi}\free)$
such that $\beta(\delta_e)=z\delta_e\text{ and }\beta(f)=f$
for all $z\in\T$, $e\in E^1$, and $f\in C_0(\partial_RE)$.
When $U$ is a closed and open subset of $\partial_RE$, then $\cha{U}$ will denote the characteristic function of $U$.
\begin{theorem} \label{theorem:partial}
Let $E$ be a directed graph and let $R$ be a subset of $E^0_\reg$. Let $\free$ be the free group generated by $E^1$, and let $\Phi$ be the partial action of $\free$ on $C_0(\partial_R E)$ described above. We then have:
\textnormal{(a)}\label{item:t1} There is a unique $*$-isomorphism $\rho:C^*(E,R)\to C_0(\partial_R E)\rtimes_{\Phi}\free$ which maps $p_v$ to $\iota(\cha{\cyl{v}\cap\partial_R E})$ for $v\in E^0$, and $s_e$ to $\delta_e$ for $e\in E^1$.
\textnormal{(b)}\label{item:t2} $\rho\circ\gamma_z=\beta_z\circ\rho$ for all $z\in\T$.
\end{theorem}
To prove Theorem \ref{theorem:partial}, we need the following lemma.
\begin{lemma}\label{lem:span-partialcp}
Each $\delta_e$, $e\in E^1$, belongs to $C_0(\partial_R E)\rtimes_{\Phi}\free$, and
$C_0(\partial_R E)\rtimes_{\Phi}\free$ is generated by the union $\{\iota(\cha{\cyl{v}\cap\partial_RE})\mid v\in E^0\}\cup\{\delta_e\mid e\in E^1\}$.
\end{lemma}
\begin{proof}
Notice first that \eqref{p1} gives that
\begin{equation}\label{eq:rangeproj}
\delta_g\delta_g^*=\iota(\cha{\trang{g}})
\end{equation}
for $g\in\free\setminus\{\31\}$. Thus if $e\in E^1$, then $\delta_e=\delta_e\delta_e^*\delta_e=\iota(\cha{\trang{e}})\delta_e\in C_0(\partial_R E)\rtimes_{\Phi}\free$.
Since $\free$ is
generated by $E^1$ and $\Phi$ is multiplicative
(cf. \cite[Section 5]{MR1452280}), it follows that $\{\delta_g\mid
g\in\free\setminus\{\31\}\}$ is contained in the
$\cs$-algebra generated by $\{\delta_e\mid e\in E^1\}$. By
the Stone-Weierstrass Theorem and Proposition \ref{prop:et}, the $C^*$-algebra
$C_0(\partial_RE)$ is generated
by $\{\cha{\cyl{u}\cap\partial_RE}\mid u\in E^*\}$, and since
$\iota(\cha{\cyl{u}\cap\partial_RE})=\delta_u\delta_u^*$ for $u\in
E^*\setminus E^0$, we get that $\iota(C_0(\partial_RE))$ is contained
in the $\cs$-algebra generated by
$\{\iota(\cha{\cyl{v}\cap\partial_RE})\mid v\in E^0\}\cup\{\delta_e\mid e\in E^1\}$.
It follows that $C_0(\partial_R E)\rtimes_{\Phi}\free$ is generated by $\{\iota(\cha{\cyl{v}\cap\partial_RE})\mid v\in E^0\}\cup\{\delta_e\mid e\in E^1\}$.
\end{proof}
\begin{proof}[Proof of Theorem \ref{theorem:partial}](a): By \eqref{eq:rangeproj} we have that $\delta_e\delta_e^*=\iota(\cha{\rang{e}})=\iota(\cha{\cyl{e}\cap\partial_R E})$ and $\delta_e^*\delta_e=\iota(\cha{\domain{e}})=\iota(\cha{\cyl{r(e)}\cap \partial_R E})$ for every $e\in E^1$. Note that
\begin{equation}\label{eq:Zv-inclusion}
\bigcup_{e\in vE^1}\cyl{e}\cap\partial_R E \subseteq \cyl{v}\cap\partial_R E
\end{equation}
for all $v\in E^0_\reg$, where equality holds only when $v\in R$ (if $v\notin R$, then $v$ belongs to the right but not to the left-hand side because our convention is that $vE^1\subseteq E^1$).
It follows that the union $\{\iota(\cha{\cyl{v}\cap\partial_R E})\}_{v\in E^0}\cup\{\delta_e\}_{e\in E^1}$ is a Cuntz-Krieger $(E,R)$-family. Thus there exists a $*$-homomorphism $\rho$ from $\cs(E,R)$ to $\cros$ which for every $e\in E^1$ maps $s_e$ to $\delta_e$ and for every $v\in E^0$ maps $p_v$ to $\iota(\cha{\cyl{v}\cap\partial_R E})$. That this $*$-homomorphism is unique, follows from the fact that $C^*(E,R)$ is generated by $\{p_v\mid v\in E^0\}\cup\{s_e\mid e\in E^1\}$.
According to\footnote{Notice that there is an obvious misprint in 2. of \cite[Theorem 3.11]{MT}} Theorem 3.11 of \cite{MT}, $\rho$ is injective if $\rho(p_v)\ne 0$ for every $v\in E^0$, $\rho(p_v-\sum_{e\in vE^1}s_es_e^*)\ne 0$ for $v\in E^0_\reg\setminus R$, and there exists an action $\beta:\T\to\aut(\cros)$ such that $\rho\circ\gamma_z=\beta_z\circ\rho$ for all $z\in\T$. Since $\iota$ is injective, it follows that $\rho(p_v)=\iota(\cha{\cyl{v}\cap\partial_RE})\ne 0$ for every $v\in E^0$, and that
\begin{align*}
\rho(\sum_{e\in vE^1}s_es_e^*)&=\sum_{e\in vE^1}\delta_e\delta_e^*=\sum_{e\in vE^1}\iota(\cha{\cyl{e}\cap\partial_RE})\\
&<\iota(\cha{\cyl{v}\cap\partial_RE})=\rho(p_v)
\end{align*}
for every $v\in E^0_\reg\setminus R$, where the inequality sign is from \eqref{eq:Zv-inclusion}. Thus the injectivity of $\rho$ will follow once we have proved part (b). That $\rho$ is surjective follows directly from Lemma \ref{lem:span-partialcp}.
Towards proving (b), let $z\in\T$. Since $C^*(E,R)$ is generated by $\{p_v\mid v\in E^0\}\cup\{s_e\mid e\in E^1\}$, $\rho(\gamma_z(p_v))=\rho(p_v)=\beta_z(\rho(p_v))$ for every $v\in E^0$, and $\rho(\gamma_z(s_e))=z\rho(s_e)=\beta_z(\rho(s_e))$ for every $e\in E^1$, it follows that $\rho\circ\gamma_z=\beta_z\circ\rho$, as wanted.
\end{proof}
\begin{remark}
A different proof of the claim that $\rho$ in Theorem~\ref{theorem:partial} is an isomorphism can be provided by directly constructing the inverse: this will be given by a covariant pair for $(C_0(\partial_R E), \free, \Phi)$ obtained from a partial representation of $\free$ inside $C^*(E, R)$. The details are similar to
the proof of \cite[Proposition 4.1]{MR1703078}.
\end{remark}
In the following we will write $s_u$ for $s_{e_1}s_{e_2}\dotsm s_{e_n}$ when $u=e_1e_2\dotsm e_n\in E^*\setminus E^0$, and $s_v$ for $p_v$ when $v\in E^0$.
\begin{corollary} \label{coro:inclusion}
Let $E$ be a directed graph and $R$ a subset of $E^0_\reg$. Then:
\begin{enumerate}\renewcommand{\theenumi}{\roman{enumi}}
\item \label{item:c1} There exists a unique $*$-isomorphism $\tilde{\iota}$ from $C_0(\partial_R E)$ onto the $C^*$-subalgebra of $C^*(E,R)$ generated by $\{s_us_u^*\mid u\in E^*\}$ mapping $\cha{\cyl{u}\cap\partial_R E}$ to $s_us_u^*$ for every $u\in E^*$.
\item \label{item:c2} There exists a unique norm-decreasing linear map (conditional expectation) $F$ from $C^*(E,R)$ onto the $C^*$-subalgebra of $C^*(E,R)$ generated by $\{s_us_u^*\mid u\in E^*\}$ such that
\begin{equation*}
F(s_us_{u'}^*)=
\begin{cases}
s_us_{u'}^*
&\text{if }u=u',\\
0&\text{if }u\ne u'
\end{cases}
\end{equation*}
for $u,u'\in E^*$.
\end{enumerate}
\end{corollary}
\begin{proof}
\eqref{item:c1}: The map $\rho\inv\circ\iota$ is a $*$-isomorphism from $C_0(\partial_RE)$ onto the $C^*$-subalgebra of $C^*(E,R)$ generated by $\{s_us_u^*\mid u\in E^*\}$ mapping $\cha{\cyl{u}\cap\partial_R E}$ to $s_us_u^*$ for every $u\in E^*$. That this is the only $*$-isomorphism with this property follows from the fact that $C_0(\partial_RE)$, according to the Stone-Weierstrass Theorem, is generated by $\{\cha{\cyl{u}\cap\partial_RE}\mid u\in E^*\}$.
\eqref{item:c2}: The existence of $F$ follows from Theorem \ref{theorem:partial} and \cite[Proposition 2.3]{MR1953065}. Uniqueness follows from the fact that $C^*(E,R)=\spac\{s_us_{u'}^*\mid u,u'\in E^*\}$.
\end{proof}
\section{KMS states on $C^*(E,R)$}
\label{sec:kms-states-graph}
We will in this section describe the sets of KMS states of certain one-parameter groups of automorphisms of $C^*(E,R)$ in terms of states of $C_0(\partial_R E)$, in terms of regular Borel probability measures on $\partial_R E$, and in terms of functions from $E^0$ to $[0,1]$.
We start by recalling the notions of KMS$_\beta$ states and ground states. For the first one, a standard definition
is found in \cite{MR1441540} and \cite{Ped}. However, an equivalent formulation has in recent times prevailed: given a
$C^*$-algebra $A$ and a homomorphism (a dynamics) $\sigma:\R\to \aut(A)$, an element $a\in A$
is called \emph{analytic} provided that $t\mapsto \sigma_t(a)$ extends to an entire function on $\C$. The analytic elements form a
dense subset of $A$, see \cite[\S 8.12]{Ped}. For $\beta\in (0, \infty)$, a \emph{KMS$_\beta$-state} of $(A,\sigma)$ is a state $\psi$ of $A$ which satisfies the KMS$_\beta$ condition
\begin{equation}\label{def:KMS-beta}
\psi(ab)=\psi(b\sigma_{i\beta}(a))
\end{equation}
for all $a,b$ analytic in $A$. It is known that it suffices to have \eqref{def:KMS-beta} satisfied for a
subset of analytic elements of $A$ that spans a dense subalgebra of $A$, \cite[Proposition 8.12.3]{MR1441540}. A \emph{KMS$_0$-state} of $(A,\sigma)$ is a state $\psi$ of $A$ which is invariant with respect to $\sigma$ (i.e., $\psi(\sigma_t(a))=\psi(a)$ for $t\in\R$ and $a\in A$), and which satisfies the trace condition $\psi(ab)=\psi(ba)$ for all $a,b\in A$.
A state $\psi$ on $A$ is a \emph{ground state} of $(A, \sigma)$ if for every $a,b$ analytic in $A$, the entire function $z\mapsto \psi(a\sigma_z(b))$ is bounded on the upper-half plane. Again, it is known that it suffices to have boundedness for a set of elements that spans a dense subalgebra of the analytic elements.
We will now describe the set of KMS states for certain one-parameter groups of automorphisms of $C^*(E,R)$.
Suppose $N$ is a function $N:E^1\to (1,\infty)$, and let $\sigma$ be the unique strongly continuous one-parameter group of automorphisms of $\cros$ given by \eqref{eq:sigma-from-N}. Theorem \ref{theorem:partial} therefore implies that $N$ gives rise to a unique strongly continuous one-parameter group $\sigma$ of automorphisms of $C^*(E,R)$ such that
\begin{equation*}
\sigma_t(s_e)=\bigl(N(e)\bigr)^{it}s_e \text{ and }\sigma_t(p_v)=p_v
\end{equation*}
for all $e\in E^1$ and $v\in E^0$.
Before we state the result, we introduce some notation. For $0\leq\beta<\infty$ we define the following sets:
\begin{itemize}
\item[$A^\beta$:] the set of KMS$_\beta$ states for $(C^*(E,R),\sigma)$,
\item[$B^\beta$:] the set of states $\omega$ of $C_0(\partial_RE)$ that satisfy the scaling condition $\omega(f\circ\phi_e\inv)=\bigl(N(e)\bigr)^{-\beta}\omega(f)$ for every $e\in E^1$ and every $f\in C_0(\tdomain{e})$,
\item[$C^\beta$:] the set of regular Borel probability measures $\mu$ on $\partial_R E$ that satisfy the scaling condition $\mu(\phi_e(A))=N(e)^{-\beta}\mu(A)$ for every $e\in E^1$ and every Borel measurable subset $A$ of $\tdomain{e}$, and
\item[$D^\beta$:] the set of functions $m:E^0\to [0,1]$ such that
\begin{enumerate}\renewcommand{\theenumi}{m\arabic{enumi}}
\item\label{item:m1} $\sum_{v\in E^0}m(v)=1$;
\item\label{item:m2} $m(v)=\sum_{e\in vE^1}\bigl(N(e)\bigr)^{-\beta}m\bigl(r(e)\bigr)$ if $v\in R$;
\item\label{item:m3} $m(v)\ge\sum_{e\in vE^1}\bigl(N(e)\bigr)^{-\beta}m\bigl(r(e)\bigr)$ for $v\in E^0$.
\end{enumerate}
\end{itemize}
Note that \eqref{item:m1} is equivalent to $\sup\{\sum_{v\in F}m(v)\mid F\text{ is a finite subset of }E^0\}=1$ and \eqref{item:m3} to the assertion that $m(v)\ge\sum_{e\in F}\bigl(N(e)\bigr)^{-\beta}m\bigl(r(e)\bigr)$ for every finite subset $F$ of $vE^1$. Notice also that if $R=E^0$, then $D^\beta$ is the set of positive normalized eigenvectors with eigenvalue 1 of the matrix $(\sum_{e\in v'E^1v}N(e))_{v',v\in E^0}$ (where $v'E^1v=\{e\in E^1\mid s(e)=v',\ r(e)=v\}$). In particular, if $R=E^0$ and $N(e)=\exp(1)$ for all $e\in E^1$, then $D^\beta$ is the set of positive normalized eigenvectors with eigenvalue $\exp(\beta)$ of the adjacency matrix of $E$.
Further, we let $A^{\operatorname{gr}}$ be the set of ground states for $(C^*(E,R),\sigma)$, $B^{\operatorname{gr}}$ the set of states $\omega$ of $C_0(\partial_R E)$ such that $\omega\bigl(\cha{\rang{e}}\bigr)=0$ for every $e\in E^1$, $C_3^{\operatorname{gr}}$ the set of regular Borel probability measures $\mu$ on $\partial_R E$ that satisfy that $\mu(A)=0$ for every $e\in E^1$ and every Borel measurable subset $A$ of $\rang{e}$, and finally $D^{\operatorname{gr}}$ the set of functions $m:E^0\to [0,1]$ that satisfy
\begin{enumerate}
\item $\sum_{v\in E^0}m(v)=1$,
\item $m(v)=0$ for $v\in R$.
\end{enumerate}
\begin{theorem} \label{thm:kms}
Given a directed graph $E$, a subset $R$ of $E^0_\reg$ and a function $N:E^1\to (1,\infty)$, let $\sigma$ be the strongly continuous one-parameter group
of automorphisms of $C^*(E,R)$ such that
\begin{equation*}
\sigma_t(s_e)=\bigl(N(e)\bigr)^{it}s_e \text{ and }\sigma_t(p_v)=p_v
\end{equation*}
for all $e\in E^1$ and $v\in E^0$. Further, let $\phi_g:\domain{g}\to\rang{g},\ g\in\free$ be the partial action of the free group $\free$ generated by $E^1$ from Proposition~\ref{prop:action-on-path-space}.
Then for $\beta\in [0,\infty)$, $A^\beta$, $B^\beta$, $C^\beta$ and $D^\beta$ are isomorphic as convex sets. Likewise,
$A^{\operatorname{gr}}$, $B^{\operatorname{gr}}$, $C^{\operatorname{gr}}$ and $D^{\operatorname{gr}}$ are isomorphic as convex sets.
\end{theorem}
Theorem \ref{thm:kms} will follow from Propositions \ref{prop:state}, \ref{prop:int} and \ref{prop:function} below. We point out that these propositions give explicit isomorphisms.
\begin{remark}
If the graph $E$ is finite, then Propositions \ref{prop:state} and \ref{prop:function} recover \cite[Proposition 2.1(a),(b),(c)]{aHLRS} when $R=\emptyset$ and \cite[Proposition 2.1(d)]{aHLRS} when $R=E^0_\reg$.
\end{remark}
\begin{remark}
In \cite{Cas-Mor}, KMS states on graph $C^*$-algebras are studied. The one-parameter group of automorphisms considered in \cite{Cas-Mor} is of the same form as the one-parameter group of automorphisms considered here, but in \cite{Cas-Mor} it is not required that $N(e)>1$ for all $e\in E^1$, only that there exists a $c>0$ such that $N(e)>c$ for all $e\in E^1$, and that $N(e_1)\cdots N(e_n)\ne 1$ for all $e_1\cdots e_n\in E^n$, $n\ge 1$. For $R=E^0_\reg$ and $\beta>0$, \cite[Theorem 3.3]{Cas-Mor} generalizes the results about $A^\beta$ and $B^\beta$ given in Proposition \ref{prop:int}, and \cite[Theorem 3.10]{Cas-Mor} generalizes the results about $B^\beta$ and $D^\beta$ given in Proposition \ref{prop:function}. For ground states, Proposition \ref{prop:int} recovers \cite[Proposition 4.3]{Cas-Mor} and Proposition \ref{prop:function} recovers \cite[Theorem 4.4]{Cas-Mor} when $R=E^0_\reg$.
We believe that with some effort, the results about $A^\beta$, $B^\beta$, $C^\beta$, and $D^\beta$ given in Theorem \ref{thm:kms}, Proposition \ref{prop:state}, Proposition \ref{prop:int}, and Proposition \ref{prop:function} could be generalized to the case where the requirement $N(e)>1$ for all $e\in E^1$ is replaced with the assumption that $N(e_1)\dots N(e_n)\ne 1$ for all $e_1\dots e_n\in E^n$, $n\ge 1$ (cf. the remark after the proof of Lemma 3.2 in \cite{MR1953065} and Remark \ref{rmk:groupoid}).
\end{remark}
\begin{remark}\label{rmk:groupoid}
Our characterization of KMS$_\beta$ states in Theorem~\ref{thm:kms} can be seen in relation to the general result for groupoid algebras in \cite{Nesh} because the $C^*$-algebra $C^*(E, R)$ admits a realization as a groupoid $C^*$-algebra $C^*(\mathcal{G}_{(E,R)})$ where $\mathcal{G}:=\mathcal{G}_{(E,R)}=\{(ux,\vert u\vert-\vert u'\vert,u'x)\mid u,u'\in E^*,x\in \partial_R E, r(u)=s(x)=r(u')\}$, see for example \cite{pat} and \cite[\S 6.4]{aHLRS}.
Assume $E$ is countable and let $N:E^1\to (1,\infty)$ be a function such that $N(e_1)\cdots N(e_n)\ne 1$ for all $e_1\cdots e_n\in E^n$, $n\ge 1$. Extend the function $N$ to $E^*$ by letting $N(v)=1$ for $v\in E^0$ and by letting $N(u)=N(u_1)\dotsm N(u_n)$ for $u=u_1\dotsm u_n\in E^n$ and $n\geq 1$. Define $c:\mathcal{G}\to \R$ by $c((ux,\vert u\vert-\vert u'\vert,u'x))=\ln N(u)-\ln N(u')$. Then $c$ is a continuous one-cocycle. For $x\in\partial_RE$ let $\mathcal{G}_x^x$ be the stabilizer $\{(x,n,x)\in\mathcal{G}\mid n\in\Z\}$. Then $\mathcal{G}_x^x$ is a subgroup of $\Z$. Let $(u_g)_{g\in\mathcal{G}_x^x}$ be the generators of the $C^*$-algebra $C^*(\mathcal{G}_x^x)$ of $\mathcal{G}_x^x$. Define the dynamics $\sigma^c$ on $\mathcal{G}$ by $\sigma^c_t(f)(g)=e^{itc(g)}f(g)$ for $f\in C_c(\mathcal{G})$ and $g\in \mathcal{G}$. Then \cite[Theorem 1.3]{Nesh} provides, for all $\beta\in \R$, a one-to-one correspondence between $\sigma^c$-KMS$_\beta$ states on $C^*(E, R)$ and pairs $(\mu, \{\varphi_x\}_{x})$ consisting of a probability measure $\mu$ on the unit space with Radon-Nikodym cocycle $e^{-\beta c}$ and a measurable field of states $\varphi_x$, each defined on $C^*(\mathcal{G}_x^x)$ and satisfying $\phi_x(u_g)=\phi_{r(h)}(u_{hgh^{-1}})$ and $\phi_x(u_{g'})=0$ for $\mu$-a.e. $x$, all $g\in\mathcal{G}_x^x$, all $h\in\mathcal{G}_x$ and all $g'\in\mathcal{G}_x^x\setminus c^{-1}(0)$. Notice that a probability measure on the unit space with Radon-Nikodym cocycle $e^{-\beta c}$ is the same as an element of our $C^\beta$. If $(x,n,x)\in\mathcal{G}_x^x$, then $c((x,n,x))\ne 0$ unless $n=0$ (because of our assumption that $N(u)\ne 1$ unless $u\in E^0$). It follows that there is just one state on $C^*(\mathcal{G}_x^x)$ satisfying that $\phi_x(u_{g'})=0$ for $g'\in\mathcal{G}_x^x\setminus c^{-1}(0)$ (cf. \cite[\S 6.4]{aHLRS}). Thus \cite[Theorem 1.3]{Nesh} gives the equivalence between $A^\beta$ and $C^\beta$ from Theorem~\ref{thm:kms}.
\end{remark}
\begin{remark}
Our Theorem~\ref{thm:kms} at $\beta=0$ generalizes one result from \cite{MR1991743}: the functions $m$ in $D^0$ are the graph-traces in \cite{MR1991743}, and the bijective correspondence between tracial states on $C^*(E)$ and graph traces on $E$ under the assumption that the graph $E$ satisfies condition (K) is contained in Theorem~\ref{thm:kms}.
\end{remark}
\begin{proposition} \label{prop:state}
In the setting of Theorem \ref{thm:kms}, let $\tilde\iota$ and $F$ be as in Corollary \ref{coro:inclusion}. Then $\omega\mapsto\omega\circ {\tilde\iota}\inv\circ F$ defines a convex isomorphism from $B^\beta$ to $A^\beta$ for $\beta\in [0,\infty)$, and a convex isomorphism from $B^{\operatorname{gr}}$ to $A^{\operatorname{gr}}$.
\end{proposition}
\begin{proof}
For ground states and for $\beta>0$, the result follows directly from Theorem 4.3 of
\cite{MR1953065}, Theorem \ref{theorem:partial} and Corollary \ref{coro:inclusion}. It remains to prove the case $\beta=0$, which comes down to characterizing $\sigma$-invariant traces on $C^*(E, R)$.
If $\omega\in B^0$, then $\psi:=\omega\circ {\tilde\iota}\inv\circ F$ is a $\sigma$-invariant state. Since $C^*(E,R)=\spac\{s_{u_1}s_{u_2}^*\mid u_1,u_2\in E^*\}$, it suffices to show that $\psi(s_{u_1}s_{u_2}^*s_{u_3}s_{u_4}^*)=\psi(s_{u_3}s_{u_4}^*s_{u_1}s_{u_2}^*)$ for $u_1,u_2,u_3,u_4\in E^*$ in order to prove that $\psi\in A^0$. We extend the definition of $\domain{u}$ and $\rang{u}$ to all $u\in E^*$ by letting $\domain{v}=\rang{v}=\cyl{v}\cap\partial_RE$ for $v\in E^0$. We then have that $\psi(s_{u_1}s_{u_2}^*s_{u_3}s_{u_4}^*)=0$ unless either
\begin{enumerate}
\item[{}] $u_2=u_3u$ and $u_1=u_4u$ for some $u\in E^*$, in which case $\psi(s_{u_1}s_{u_2}^*s_{u_3}s_{u_4}^*)=\psi(s_{u_1}s_{u_1}^*)=\omega\bigl(\cha{U(u_1)}\bigr)=\omega\bigl(\cha{\domain{u_1}}\bigr)=\omega\bigl(\cha{\domain{u}}\bigr)$, or
\item[{}] $u_3=u_2u$ and $u_4=u_1u$ for some $u\in E^*$, in which case $\psi(s_{u_1}s_{u_2}^*s_{u_3}s_{u_4}^*)=\psi(s_{u_4}s_{u_4}^*)=\omega\bigl(\cha{U(u_4)}\bigr)=\omega\bigl(\cha{\domain{u_4}}\bigr)=\omega\bigl(\cha{\domain{u}}\bigr)$.
\end{enumerate}
Similarly, $\psi(s_{u_3}s_{u_4}^*s_{u_1}s_{u_2}^*)=0$ unless either
\begin{enumerate}
\item[{}] $u_2=u_3u$ and $u_1=u_4u$ for some $u\in E^*$, in which case $\psi(s_{u_3}s_{u_4}^*s_{u_1}s_{u_2}^*)=\psi(s_{u_2}s_{u_2}^*)=\omega\bigl(\cha{U(u_2)}\bigr)=\omega\bigl(\cha{\domain{u_2}}\bigr)=\omega\bigl(\cha{\domain{u}}\bigr)$, or
\item[{}] $u_3=u_2u$ and $u_4=u_1u$ for some $u\in E^*$, in which case $\psi(s_{u_1}s_{u_2}^*s_{u_3}s_{u_4}^*)=\psi(s_{u_4}s_{u_4}^*)=\omega\bigl(\cha{U(u_4)}\bigr)=\omega\bigl(\cha{\domain{u_4}}\bigr)=\omega\bigl(\cha{\domain{u}}\bigr)$.
\end{enumerate}
Thus, $\omega\mapsto\omega\circ {\tilde\iota}\inv\circ F$ is a map from $B^0$ to $A^0$. It is clear that it is a convex map and that it is injective.
Let $\psi\in A^0$. It follows from the $\sigma$-invariance of $\psi$ that $\psi(s_{u}s_{u'}^*)=0$ unless $\abs{u}=\abs{u'}$; in case $\abs{u}=\abs{u'}$, then it follows from the trace property of $\psi$ that $\psi(s_{u}s_{u'}^*)=\psi(s_{u'}^*s_{u})=0$ unless $u=u'$, because $s_u$ and $s_{u'}$ have orthogonal range projections. Thus $\omega:=\psi\circ{\tilde\iota}$ is a state of $C_0(\partial_RE)$ such that $\omega\circ {\tilde\iota}\inv\circ F=\psi$. Let $e\in E^1$. If $u\in r(e)E^*$, then
\begin{multline} \label{eq:inv}
\omega(\cha{\cyl{u}\cap\partial_RE}\circ\phi_e\inv)=
\omega(\cha{\cyl{eu}\cap\partial_RE})=
\psi(s_{eu}s_{eu}^*)\\=
\psi(s_e^*s_es_us_u^*)=
\psi(s_us_u^*)=
\omega(\cha{\cyl{u}\cap\partial_RE}).
\end{multline}
Since $C_0(\domain{e})=\spac\{\cha{\cyl{u}\cap\partial_RE}\mid u\in r(e)E^*\}$, the calculations \eqref{eq:inv} show that $\omega\in B^0$. Thus, $\omega\mapsto\omega\circ {\tilde\iota}\inv\circ F$ is surjective and therefore a convex isomorphism from $B^0$ to $A^0$.
\end{proof}
\begin{lemma}\label{lem:riesz}
Let $E$ be a directed graph, $R$ a subset of $E^0_\reg$, and let $M:E^1\to [0,\infty)$ be a function. Then
the map
\begin{equation}
\label{eq:4}
\mu\mapsto \left(f\mapsto \int f\ d\mu\right)
\end{equation}
is a bijective correspondence between the set of regular Borel probability
measures $\mu$ on $\partial_RE$ satisfying that $\mu(\tpa_e(A))=M(e)\mu(A)$
for all $e\in E^1$ and all Borel measurable subsets $A$ of
$\tdomain{e}$, and the set of states $\eta$ of
$C_0(\partial_RE)$ satisfying that
$\eta(f\circ\tpa_e\inv)=M(e)\eta(f)$ for all $e\in E^1$ and all $f\in C_0\bigl(\tdomain{e}\bigr)$.
\end{lemma}
\begin{proof}
It follows from Riesz' Representation Theorem (see for example
\cite[6.16]{MR918770}) that \eqref{eq:4} is a bijective
correspondence between the set of regular Borel probability measures
on $\partial_RE$ and the set of states $\eta$ of
$C_0(\partial_RE)$. So we just have to show that a regular Borel probability
measure $\mu$ on $\partial_RE$ satisfies that $\mu(\tpa_e(A))=M(e)\mu(A)$
for every $e\in E^1$ and every Borel measurable subset $A$ of
$\tdomain{e}$ if and only if $\int f\circ\tpa_{e}^{-1}\ d\mu=M(e)\int f\ d\mu$ for
every $e\in E^1$ and every $f\in C_0\bigl(\tdomain{e}\bigr)$.
For each $e\in E^1$ let $L^1(\tdomain{e})$ denote the set of
functions on $\tdomain{e}$ which are integrable with respect to the
restriction of $\mu$ to $\tdomain{e}$, and let $||\cdot||_1$ be the
subnorm given by
\begin{equation*}
||f||_1=\int_{\tdomain{e}}|f|d\mu.
\end{equation*}
We then have that
$C_0(\tdomain{e})$ is dense in $L^1(\tdomain{e})$ with respect to
$||\cdot ||_1$. It follows that if the identity $\int f\circ\tpa_{e}^{-1}\
d\mu=M(e)\int f\ d\mu$ holds for every $f\in
C_0\bigl(\tdomain{e}\bigr)$, then it holds for every $f\in
L^1\bigl(\tdomain{e}\bigr)$. Then in particular
\begin{equation*}
\mu(\tpa_e(A))=\int \cha{\tpa_e(A)}\ d\mu=\int
\cha{A}\circ\tpa_{e\inv}\ d\mu=M(e)\int\cha{A}\ d\mu=M(e)\mu(A)
\end{equation*}
for every Borel measurable subset $A$ of $\tdomain{e}$.
If, on the other hand, $\mu(\tpa_e(A))=M(e)\mu(A)$
for every Borel measurable subset $A$ of
$\tdomain{e}$, then the identity $\int f\circ\tpa_{e}^{-1} \ d\mu=M(e)\int f\
d\mu$ holds for every $f\in
L^1\bigl(\tdomain{e}\bigr)$ and in particular for every $f\in
C_0\bigl(\tdomain{e}\bigr)$.
\end{proof}
\begin{proposition} \label{prop:int}
In the setting of Theorem \ref{thm:kms}, the map
\begin{equation*}
\mu\mapsto \left(f\mapsto \int f\ d\mu\right)
\end{equation*}
is a convex isomorphism from $C^\beta$ to $B^\beta$ for $\beta\in [0,\infty)$, and a convex isomorphism from $C^{\operatorname{gr}}$ to $B^{\operatorname{gr}}$.
\end{proposition}
\begin{proof} Apply Lemma~\ref{lem:riesz} with the function $M:E^1\to
[0,\infty[$ given by $M(e)=(N(e))^{-\beta}$ when $\beta<\infty$, and $M(e)=0$ in case of ground states.
\end{proof}
\begin{lemma} \label{lemma:hovedto}
Let $E$ be a directed graph, let $R$ be a subset of $E^0_\reg$, and let $M$ be a function from $E^1$ to $[0,1]$. Then
\begin{equation}
\omega\mapsto \bigl(v\mapsto \omega(\cha{\cyl{v}\cap\partial_R
E})\bigr) \label{eq:3}
\end{equation}
is a bijective correspondence between the set of states $\omega$ of
$C_0(\partial_R E)$ such that
$\omega(f\circ\phi_e\inv)=M(e)\omega(f)$ for all $e\in E^1$ and all $f\in C_0\bigl(\domain{e}\bigr)$, and the set of
functions $m:E^0\to [0,1]$ satisfying \begin{enumerate}\renewcommand{\theenumi}{m\arabic{enumi}'}
\item\label{item:n1} $\sum_{v\in E^0}m(v)=1$;
\item\label{item:n2} $m(v)=\sum_{e\in vE^1}M(e)m\bigl(r(e)\bigr)$ if $v\in R$;
\item\label{item:n3} $m(v)\ge\sum_{e\in F}M(e)m\bigl(r(e)\bigr)$ for every finite subset $F$ of $E^1$.
\end{enumerate}
\end{lemma}
\begin{proof}
Let $\omega$ be a state of $C_0(\partial_R E)$ such that
$\omega(f\circ\phi_e\inv)=M(e)\psi(f)$ for all $e\in E^1$ and $f\in C_0(\domain{e})$. Let $m$ be the function from $E^0$ to $[0,1]$
given by
\begin{equation*}
m(v)=\omega(\cha{\cyl{v}\cap\partial_R E}).
\end{equation*}
Now, if $F$ runs over the finite subsets of $E^0$, then $\{\sum_{v\in
F}\cha{\cyl{v}\cap\partial_R E}\}_{F}$ is an
approximate unit for $C_0(\partial_R E)$. Hence $m$ satisfies \eqref{item:n1}.
To show \eqref{item:n2} and \eqref{item:n3} notice first that if $e\in E^1$,
then
\begin{align*}
\omega\bigl(\cha{\rang{e}}\bigr)
&=
\omega\bigl(\cha{\domain{e}}\circ\phi_e\inv\bigr) =
M(e) \omega\bigl(\cha{\domain{e}}\bigr)\\
&=
M(e) \omega\bigl(\cha{\cyl{r(e)}\cap\partial_R E}\bigr) =
M(e)m\bigl(r(e)\bigr).
\end{align*}
If $v\in R$, then
$\cha{\cyl{v}\cap\partial_R E}=\sum_{e\in vE^1}\cha{\rang{e}}$ by \eqref{eq:Zv}. Hence
\begin{equation*}
m(v)=\omega\bigl(\cha{\cyl{v}\cap\partial_R E}\bigr)
=\smashoperator{\sum_{e\in vE^1}}\omega\bigl(\cha{\rang{e}}\bigr)
=\smashoperator{\sum_{e\in vE^1}}M(e)m\bigl(r(e)\bigr),
\end{equation*}
which gives \eqref{item:n2}. If $v\in E^0$ and $F$ is a finite subset of $vE^1$, then
$\cha{\cyl{v}\cap\partial_R E}\ge\sum_{e\in F}\cha{\rang{e}}$, so \eqref{item:n3} follows from the calculations
\begin{equation*}
m(v)=\omega\bigl(\cha{\cyl{v}\cap\partial_R E}\bigr)
\ge\smashoperator{\sum_{e\in F}}\omega\bigl(\cha{\rang{e}}\bigr)
=\smashoperator{\sum_{e\in F}}M(e)m\bigl(r(e)\bigr).
\end{equation*}
Since $\omega(\cha{\cyl{eu}\cap\partial_R E})=\omega(\cha{\cyl{u}\cap\partial_R E}\circ\phi_e\inv)=
M(e)\omega(\cha{\cyl{u}\cap\partial_R E})$ for all
$e\in E^1$ and all $u\in E^*$ with $s(u)=r(e)$, the restriction
of $\omega$ to $\{\cha{\cyl{u}\cap\partial_R E}\mid u\in E^*\}$ is completely
determined by the restriction
of $\omega$ to $\{\cha{\cyl{v}\cap\partial_R E}\mid v\in E^0\}$. As seen in the proof of Lemma~\ref{lem:span-partialcp}, the
space $\spa\{\cha{\cyl{u}\cap\partial_R E}\mid u\in E^*\}$ is dense in $C_0(\partial_R E)$. Therefore the correspondence
given in \eqref{eq:3} is injective.
We will now prove that it is surjective. Let $m:E^0\to [0,1]$ be a
function that satisfies \eqref{item:n1}-\eqref{item:n3}.
For each $u=e_1e_2\dotsm e_k\in E^*$, set
\begin{equation*}
\tilde{m}(u)= M(e_1)M(e_2)\dotsm M(e_k)m(r(e_k)).
\end{equation*}
Straightforward calculations show that for $u\in E^*$,
\begin{align*}
\tilde{m}(u)&=\smashoperator{\sum_{e\in r(u)E^1}}\tilde{m}(ue),\text{ if }r(u)\in R,\\
\tilde{m}(u)&\ge\smashoperator{\sum_{e\in F}}\tilde{m}(ue),
\text{ if $F$ is a finite subset of }r(u)E^1.
\end{align*}
Since $\{\cha{\cyl{u}}\mid u\in E^*\}$ is a linearly independent subset of $C_0(E^{\le\infty})$, it follows that there exists a linear map $\tilde\omega_m$ from $\spa\{\cha{\cyl{u}}\mid u\in E^*\}$ to $\C$ which maps $\cha{\cyl{u}}$ to $\tilde{m}(u)$ for $u\in E^*$. We show next that $\tilde\omega_m$ extends to a state of $C_0(E^{\le\infty})$. To begin with, we show that $\tilde\omega_m$ is bounded and its norm is not greater than 1. Let $f\in \spa\{\cha{\cyl{u}}\mid u\in E^*\}$. Then there exist a finite subset $F$ of $E^*$ and complex numbers $(c_u)_{u\in F}$ such that
\begin{equation*}
f=\sum_{u\in F}c_u \cha{\cyl{u}},
\end{equation*}
and such that $(u\in F \text{ and }u'\le u) \Rightarrow u'\in F$. We then have that
\begin{equation*}
\tilde\omega_m(f)=\sum_{u\in F}c_u\tilde{m}(u)=\sum_{u\in F}\biggl(\sum_{u'\le u}c_{u'}\biggr)\biggl(\tilde{m}(u)-\sum_{e\in r(u)E^1,\ ue\in F}\tilde{m}(ue)\biggr).
\end{equation*}
Since $f(u)=\sum_{u'\le u}c_{u'}$ for $u\in F$ and $$\sum_{u\in F}\biggl(\tilde{m}(u)-\sum_{e\in r(u)E^1,\ ue\in F}\tilde{m}(ue)\biggr)=\sum_{v\in E^0\cap F}{m}(v)\le 1,$$ it follows from H\"older's inequality that $\abs{\tilde{\omega}_m(f)}\le \norm{f}_\infty$. Thus we can extend $\tilde\omega_m$ to a bounded linear functional with norm less than or equal to 1 on $\spac\{\cha{\cyl{u}}\mid u\in E^*\}=C_0(E^{\le\infty})$. The family $(\sum_{v\in F}\cha{\cyl{v}})_{F}$ indexed over finite subsets $F$ of $E^0$ forms an approximate unit for $C_0(E^{\le\infty})$, and \eqref{item:m1} therefore implies that $\lim_{F\subset E^0}\tilde\omega_m( \sum_{v\in F}\cha{\cyl{v}})=1$. Thus $\tilde\omega_m$ is a state of $C_0(E^{\le\infty})$ (e.g. from \cite[Theorem 3.3.3]{MR1074574}).
It follows from Proposition \ref{prop:et}(v) and the definition of $\partial_RE$ that $\{f\in C_0(E^{\le\infty})\mid f(x)=0\text{ for all }x\in\partial_RE\}=\spac\{\cha{\{u\}}\mid r(u)\in R\}$. Since for
$r(u)\in R$ we have
\begin{equation*}
\tilde\omega_m(\cha{\{u\}})=\tilde\omega_m\biggl(\cha{\cyl{u}}-\sum_{e\in r(u)E^1}\cha{\cyl{ue}}\biggr)=\tilde{m}(u)-\smashoperator{\sum_{e\in r(u)E^1}}\tilde{m}(ue)=0,
\end{equation*}
it follows that $\tilde\omega_m$ induces a state $\omega_m$ on $C_0(\partial_RE)$ which maps $\cha{\cyl{u}\cap\partial_RE}$ to $\tilde{m}(u)$ for $u\in E^*$.
Let $e\in E^1$ and $u\in r(e)E^*$. Then
\begin{equation*}
\omega_m(\cha{\cyl{u}\cap\partial_RE}\circ\phi_e\inv)=\omega_m(\cha{\cyl{eu}\cap\partial_RE})=\tilde{m}(eu)
=M(e)\omega_m(\cha{\cyl{u}\cap \partial_RE}).
\end{equation*}
As already noticed, $\spac\{\cha{\cyl{u}\cap\partial_R E}\mid u\in r(e)E^*\}=C_0(U(e\inv))$, and therefore
$\omega_m(f\circ\phi_e\inv)=M(e)\omega_m(f)$ for every $f\in C_0(\domain{e})$. Since
$\omega_m(\cha{\cyl{v}\cap\partial_R E})=m(v)$ for every $v\in E^0$, we have shown the claimed surjectivity.
\end{proof}
\begin{proposition} \label{prop:function}
In the setting of Theorem \ref{thm:kms}, the map from \eqref{eq:3}
is a convex isomorphism from $B^\beta$ to $D^\beta$ for $\beta\in[0,\infty)$, and from
$B^{\operatorname{gr}}$ to $D^{\operatorname{gr}}$.
\end{proposition}
\begin{proof} Apply Lemma~\ref{lemma:hovedto} with the function $M:E^1\to
[0,\infty)$ given by $M(e)=(N(e))^{-\beta}$ when $\beta<\infty$, and $M(e)=0$ in case of ground states.
\end{proof}
\section{Extremal KMS states}\label{section:extreme}
In this section we aim to give a description of the extreme points of $D^\beta$ for $\beta\geq 0$. Ideally, we want a description that is valid for arbitrary graphs. However, this task seems to be quite difficult. We will identify certain subsets of the set of extreme points of $D^\beta$, see Theorem~\ref{thm:finitetype-reg} and Theorem~\ref{thm:infinitetype-crit}. The strategy will be to describe the supports of the corresponding measures in $C^\beta$. For certain families of graphs (in particular all graphs with finitely many vertices), our description will give all the extremal KMS states.
Throughout this section $E$ will denote a directed graph, $R$ a subset of $E^0_\reg$, $N:E^1\to (1,\infty)$ a function, and $\beta\in [0,\infty)$. We extend the function $N$ to $E^*$ by letting $N(v)=1$ for $v\in E^0$ and by letting $N(u)=N(u_1)\dotsm N(u_n)$ for $u=u_1\dotsm u_n\in E^n$ and $n\geq 1$. We adopt the following convention: if $m\in D^\beta$ and
$\mu$ is the unique element of $C^\beta$ given by Propositions~\ref{prop:int} and \ref{prop:function} such that
\begin{equation}\label{eq:convention-mu-m}
\mu({Z(v')}\cap \partial_R E)=m(v')
\end{equation}
for all $v'\in E^0$, we say that $\mu$ is the \emph{measure associated to $m$}.
To begin with, we divide the elements of $D^\beta$ in terms of finite and infinite type measures in $C^\beta$, similar to what is done in \cite{MR1953065}.
\begin{definition} Let $m\in D^\beta$ and let $\mu$ be the measure associated to $m$. Then
\begin{enumerate}
\item $m$ is of \emph{finite type} if $\mu(E^*\cap \partial_R E)=1$, and
\item $m$ is of \emph{infinite type} if $\mu(E^\infty)=1$.
\end{enumerate}
We let $\Cf$ and $D^{\beta}_{\operatorname{inf}}$ denote, respectively, the sets of $m$ of finite type and of infinite type.
\end{definition}
For the infinite type measures we introduce the following refinement.
\begin{definition} Let $E$ be a directed graph.
\textnormal{(a)} We define the set $E^\infty_{\operatorname{rec}}$ of \emph{recurrent} paths to be the collection of all infinite paths that meet some vertex of $E^0$ infinitely many times: thus $x\in E^\infty_{\operatorname{rec}}$ if and only if there is $v\in E^0$ such that $\{u\in E^*\mid u<x, r(u)=v\}$ is infinite.
\textnormal{(b)} We define the set $E^\infty_{\operatorname{wan}}$ of \emph{wandering} paths to be the collection of all $x\in E^\infty$ such that for every $v\in E^0$, the set $\{u\in E^*\mid u<x, r(u)=v\}$ is finite.
Note that $\Ewan=\emptyset$ when $E^0$ is finite. In general, $E^\infty=\Erec\sqcup \Ewan$.
\end{definition}
\begin{definition} Let $\mu\in C^\beta$. Following \cite{Tho2}, we say that
\begin{enumerate}
\item $\mu$ is \emph{conservative} if it has support on $E^\infty_{\operatorname{rec}}$, and
\item $\mu$ is \emph{dissipative} if it has support on $E^\infty_{\operatorname{wan}}$.
\end{enumerate}
We let $\Cinfa$ and $\Cinfb$ denote, respectively, the sets of functions $m$ whose associated measure via \eqref{eq:convention-mu-m} is conservative, respectively dissipative. In either instance we shall refer to $m$ itself as being conservative or dissipative.
\end{definition}
Note that $\Cinfb=\emptyset$ if $\Ewan=\emptyset$ (in particular if $E^0$ is finite). Example~\ref{ex:dis} and Example~\ref{ex:motivating} provide examples where $\Cinfb\ne\emptyset$.
\begin{remark}\label{rmk:decom}
The three subsets $E^*\cap \partial_R E$, $E^\infty_{\operatorname{rec}}$, and $E^\infty_{\operatorname{wan}}$ of $\partial_RE$ are all invariant under the partial action $\Phi$. It follows that every $m\in D^\beta$ in a unique way can be written as a convex combination of an element of $\Cf$, an element of $\Cinfa$ and an element of $\Cinfb$.
\end{remark}
It follows from Remark \ref{rmk:decom} that the set of extreme points of $D^\beta$ is the disjoint union of the sets of extreme points of $\Cf$, $\Cinfa$, and $\Cinfb$. We will in Theorem~\ref{thm:finitetype-reg} and Theorem~\ref{thm:infinitetype-crit} identify the extreme points of $\Cf$ and $\Cinfa$. Hence, if $\Cinfb=\emptyset$ (in particular if $E^0$ is finite), then we obtain a complete description of all the extreme points of $D^\beta$ and thus a complete description of all the KMS states of $(C^*(E,R),\sigma)$.
In order to define distinguished sets of vertices on which some of the extreme points of $D^\beta$ will be
supported we need to introduce some notation. For $v\in E^0$, let
$$vE^*v=\{u\in E^*\mid s(u)=r(u)=v\}$$
be the collection of all finite paths starting and ending at $v$ (also referred to as \emph{loops} or \emph{cycles} at $v$), and let
\begin{equation*}
\Evav=\{u\in E^*\mid s(u)=r(u)=v,\ u\ne v,\ r(u')\ne v\text{ for any }v<u'<u\}
\end{equation*}
be the set of paths starting and ending at $v$ with length at least 1 and containing no proper subpath that is a loop at $v$ (these are sometime called \emph{simple} loops or cycles). Notice that $\Evav$ might be empty, but that $v\in vE^*v$. In fact,
\begin{equation*}
vE^*v=\{v\}\cup\bigcup_{n=1}^\infty\{u_1u_2\cdots u_n\mid u_1,u_2,\dots,u_n\in\Evav\}.
\end{equation*}
Recall that $E^*v=\{u\in E^*\mid r(u)=v\}$ is the set of finite paths ending in $v$. We let
\begin{equation*}
\Eva=\{u\in E^*\mid r(u)=v,\ r(u')\ne v\text{ for any }u'<u\}
\end{equation*}
be the set of finite paths ending in $v$ such that no proper subpath has range $v$. Notice that $E^*v$ and $\Eva$ are both non-empty since $v\in E^*_av\subseteq E^*v$.
Next we associate partition functions to the sets $\Evav$ and $\Eva$ as follows:
\begin{align}
\Zvav(\beta)&=\sum_{u\in\Evav}N(u)^{-\beta}\label{eq:Zvav}\\
\Zva(\beta)&=\sum_{u\in\Eva}N(u)^{-\beta}.\label{eq:Zva}
\end{align}
Notice that $\Zvav(\beta)$ might be 0 (since $\Evav$ might be empty), whereas $\Zva(\beta)\ge 1$ (because $v\in\Eva$).
We now define the following distinguished sets of vertices.
\begin{gather}
\Ereg=\{v\in E^0\mid \Zva(\beta)<\infty \text{ and }\Zvav(\beta)< 1\}, \label{eq:Ereg}\\
\Ecrit=\{v\in E^0\mid \Zva(\beta)<\infty\text{ and }\Zvav(\beta)= 1\}.\label{eq:Ecrit}
\end{gather}
The abbreviations in the notation stand for regular and critical, respectively. We shall refer to $\Eequ:=\{v\in E^0\mid \Zva(\beta)<\infty\text{ and }\Zvav(\beta)\le 1\}$ as the set of equivariant points. The main results of this section will establish that elements in $\Cf$ are determined by $\Ereg\setminus R$, and elements in $\Cinfa$ by (equivalence classes of elements in) $\Ecrit$, cf. Theorems~\ref{thm:finitetype-reg} and \ref{thm:infinitetype-crit}. In particular $\Cf=\emptyset$ if and only if $\Ereg\setminus R=\emptyset$, and $\Cinfa=\emptyset$ if and only if $\Ecrit=\emptyset$.
Towards defining extreme points of $D^\beta$ we need to keep track of paths between a pair of vertices. Thus, for $v,v'\in E^0$ we let
\begin{equation*}
v'E^*v=\{u\in E^*\mid s(u)=v',\ r(u)=v\}
\end{equation*}
be the set of finite paths starting at $v'$ and ending at $v$, and we let
\begin{equation*}
\Es{v'}{v}=\{u\in E^*\mid s(u)=v',\ r(u)=v,\ r(u')\ne v\text{ for any }u'<u\}
\end{equation*}
be the set of finite paths starting at $v'$ and ending at $v$ such that no proper subpath has range $v$. In general, the sets $v'E^*v$ and $\Es{v'}{v}$ could be empty. Note however that $\Es{v'}{v}\subseteq v'E^*v$ and that $\Es{v}{v}=\{v\}$.
Notice also that $v'E^*v=\{uu'\mid u\in\Es{v'}{v},\ u'\in{v}E^*{v}\}$ and $\Eva=\bigcup_{v'\in E^0}\Es{v'}{v}$.
\begin{definition}\label{def:m-of-v}
For $v\in\Eequ$, let $m^\beta_v:E^0\to [0,\infty]$ be given by
\begin{equation}\label{eqdef:m_v}
m^\beta_v(v')=\sum_{u\in\Es{v'}{v}}N(u)^{-\beta}(\Zva(\beta))^{-1}.
\end{equation}
\end{definition}
We are now in a position to state the first main result of this section, which provides a description of the elements of $\Cf$.
\begin{theorem}\label{thm:finitetype-reg}
Let $\beta\in [0,\infty)$. The map $W_{\operatorname{fin}}$ from $\Cf$ to the set of $[0,1]$-valued functions on $\Ereg\setminus R$ given by
\begin{equation}\label{eqdef:W-fin}
W_{\operatorname{fin}}(m)(v)=\frac{\Zva(\beta)}{1-\Zvav(\beta)} \Biggl(m(v)-\sum_{e\in vE^1} N(e)^{-\beta}m(r(e))\Biggr)
\end{equation}
for $v\in\Ereg\setminus R$, is a convex isomorphism
onto $\bigl\{\epsilon:\Ereg\setminus R\to [0,1]\biggm\vert \sum_{v\in\Ereg \setminus R}\epsilon(v)=1\bigr\}$.
The inverse of $W_{\operatorname{fin}}$ is the map $\epsilon\mapsto \sum_{v\in\Ereg\setminus R}\epsilon(v)m^\beta_v$.
\end{theorem}
The proof of this theorem will require some preparation in the form of a series of preliminary results.
It will be convenient to have a notation for the function on $E^0$ appearing in the right-hand side of \eqref{eqdef:W-fin}. Therefore, for $m:E^0\to [0,1]$ satisfying \eqref{item:m3}, we let $S(m)$ be the function from $E^0$ to $[0,1]$ given by
\begin{equation*}
S(m)(v)= m(v)-\sum_{e\in vE^1} N(e)^{-\beta}m(r(e)), \text{ for }v\in E^0.
\end{equation*}
Notice that $m$ satisfies \eqref{item:m2} if and only if $S(m)(v)=0$ for all $v\in R$, and that $m$ is an eigenvector with eigenvalue 1 of the matrix $(\sum_{e\in v'E^1v}N(e))_{v',v\in E^0}$ if and only if $S(m)(v)=0$ for all $v\in E^0$.
Some properties of this function $S$ are collected in the following lemma.
\begin{lemma}\label{lem:aboutS}
Let $m\in D^\beta$ and let $\mu\in C^\beta$ be the measure associated to $m$.
\textnormal{(a)} We have $\mu(\{v\})=S(m)(v)$ for all $v\in E^0\setminus R$.
\textnormal{(b)} We have $S(m)(v)=0$ for all $v\in E^0$ if and only if $m\in \Cinf$.
\end{lemma}
\begin{proof} The regularity of $\mu$ implies that
\begin{align*}
\mu(\{v\})&=\mu\biggl(\cyl{v}\setminus\bigcup_{e\in vE^1}\cyl{e}\biggr)\\
&=\mu(\cyl{v}\cap\partial_RE)-\sum_{e\in vE^1}\mu(\cyl{e}\cap\partial_RE)\\
&=m(v)-\sum_{e\in vE^1}N(e)^{-\beta}m(r(e))\\
&=S(m)(v),
\end{align*}
as claimed in (a).
To prove (b), assume first that $S(m)(v)=0$ for all $v\in E^0$. By (a), $\mu(\{v\})=0$ for all $v\in E^0\setminus R$. Hence by the scaling condition in $C^\beta$, $\mu(\{u\})=N(u)^{-\beta}\mu(\{r(u)\})=0$ for all $u\in E^*$ with $r(u)\notin R$. Thus,
$$
\mu(E^\infty)=\mu(\partial_R E\setminus \{u\in E^*\mid r(u)\notin R\})=\mu(\partial_R E)=1.
$$
Conversely, if $m\in \Cinf$, then $S(m)(v)=\mu(\{v\})=0$ for $v\in E^0\setminus R$ by (a), and $S(m)(v)=0$ for $v\in R$ since $m$ satisfies \eqref{item:m2}.
\end{proof}
It follows from Lemma~\ref{lem:aboutS} (b) that $D^\beta_{inf}$ is
the set of normalized eigenvectors with eigenvalue 1 of the matrix
$(\sum_{e\in v' E^1 v} N(e))_{v',v\in E^0}$ (and to the normalized
eigenvectors with eigenvalue $\exp(\beta)$ of the adjacency matrix of
$E$ if $N(e)=\exp(1)$ for all $e\in E^1$).
For $v\in E^0$ define a partition function
\begin{equation}\label{eq:Zv}
Z_v(\beta)=\sum_{u\in E^*v}N(u)^{-\beta}.
\end{equation}
Similar to the terminology used in \cite{MR1953065} we call $Z_v(\beta)$ the partition function with fixed-target $v$. Clearly $\beta_1\le\beta_2$ implies $Z_v(\beta_2)\le Z_v(\beta_1)$. Thus, if $Z_v(\beta)$ is convergent, then $Z_v(\beta')$ is convergent for all $\beta'\ge \beta$.
It will be useful to know that the map $(u_0,u_1,\dots,u_n)\mapsto u_0u_1\dotsm u_n$ is a bijection
\begin{equation}\label{eq:bijection1}
\Eva\times\bigcup_{n=0}^\infty(\Evav)^n\to E^*v,
\end{equation}
where $(\Evav)^0=\{v\}$.
\begin{proposition} \label{prop:E} Let $\beta\in [0,\infty)$. The following hold:
\mbox{ }
\begin{enumerate}
\item \label{item:a1} $Z_v(\beta)=\Zva(\beta)\Bigl(1+\sum_{n=1}^\infty(\Zvav(\beta))^n\Bigr)$ for any $v\in E^0$.
\item \label{item:a2} $\Ereg=\{v\in E^0\mid Z_v(\beta)<\infty\}$.
\item \label{item:a3.5} Let $v\in E^0$ and $m\in D^\beta$. If $m(v)\ne 0$, then $\Zva(\beta)\le \frac{1}{m(v)}$.
\item \label{item:a4} Let $v\in E^0$. If there exists an $m\in D^\beta$ such that $m(v)\ne 0$, then $v\in\Eequ$.
\end{enumerate}
\end{proposition}
\begin{proof}
Assertion \eqref{item:a1} follows directly from \eqref{eq:bijection1}, and assertion \eqref{item:a2} follows directly from \eqref{item:a1}.
We next prove \eqref{item:a3.5}. Suppose that $m(v)\ne 0$. Since $m\in D^\beta$, Proposition \ref{prop:function} gives a unique $\omega\in B^\beta$ such that $\omega(\cha{\cyl{v'}\cap\partial_RE})=m(v')$ for all $v'\in E^0$. We let $\psi$ be the element in $A^\beta$ corresponding to $\omega$ under the isomorphism of Proposition~\ref{prop:state}. If $u_1,u_2\in\Eva$ and $u_1\ne u_2$, then $\cyl{u_1}\cap\cyl{u_2}=\emptyset$. We claim that
\begin{equation}\label{eq:sum-less-one}
\sum_{u\in\Eva} \omega(\cha{\cyl{u}\cap\partial_R E})\leq 1.
\end{equation}
To see this, use that $\sqcup_{u\in \Eva}Z(u)\subseteq \sqcup_{v'\in J} Z(v')$, where $J=\{s(u)\mid u\in \Eva\}$, to bound the left hand side of \eqref{eq:sum-less-one} by $\sum_{v'\in J} \overline{\psi}(\cha{Z(v')})$, with $\overline{\psi}$ denoting the state extension of $\psi\vert_{C_0(\delta_R E)}$ to $C_0(E^{\leq \infty})$. The fact that the
net $(\sum_{v''\in F}\cha{Z(v'')})$ indexed over finite subsets of $E^0$ forms an approximate unit for $C_0(E^{\leq \infty})$ then gives \eqref{eq:sum-less-one}. The scaling condition in $B^\beta$ therefore implies that
$$\sum_{u\in\Eva}N(u)^{-\beta}m(v)\le 1,$$
and thus $\Zva(\beta)=\sum_{u\in\Eva}N(u)^{-\beta}\le \frac{1}{m(v)}$.
Finally, to prove \eqref{item:a4}, assume that $m\in D^\beta$ and $m(v)\ne 0$. Let $\omega\in B^\beta$ be as above. If $u_1,u_2\in\Evav$ and $u_1\ne u_2$, then $Z(u_1)\cap Z(u_2)=\emptyset$, hence $\sqcup_{u\in\Evav}Z(u)\subseteq Z(v)$. It follows from the scaling condition in $B^\beta$ that
$$\sum_{u\in\Evav}N(u)^{-\beta}\omega(\cha{\cyl{v}\cap\partial_RE})\le\omega(\cha{\cyl{v}\cap\partial_RE}).$$
Thus, since $\omega(\cha{\cyl{v}\cap\partial_RE})=m(v)\ne 0$, we get that $\Zvav(\beta)=\sum_{u\in\Evav}N(u)^{-\beta}\le 1$. Since $\Zva(\beta)<\infty$ by \eqref{item:a3.5}, we have that $v\in\Eequ$.
\end{proof}
\begin{lemma}\label{lem:mineq}
Let $m\in D^\beta$ and $v_1,v_2\in E^0$. Then $m(v_2)\ge \sum_{u\in\Es{v_2}{v_1}}N(u)^{-\beta}m(v_1)$.
\end{lemma}
\begin{proof}
If $\Es{v_2}{v_1}=\emptyset$ there is nothing to prove. Assume $\Es{v_2}{v_1}\neq\emptyset$. Let $\mu$ be the measure associated to
$m$ as given by \eqref{eq:convention-mu-m}. The scaling condition in $C^\beta$ implies that
$$
\mu(\{ux\mid x\in\cyl{r(u)}\}\cap\partial_RE)=N(u)^{-\beta}\mu(Z(r(u))\cap\partial_R E)=N(u)^{-\beta}m(r(u))
$$
for any $u\in E^*$.
If $u_1,u_2\in\Es{v_2}{v_1}$ and $u_1\ne u_2$, then $\{u_1x\mid x\in\cyl{v_1}\}$ and $\{u_2x\mid x\in\cyl{v_1}\}$ are two disjoint subsets of $\cyl{v_2}$. Hence
\begin{align*}
m(v_2)&=\mu(\cyl{v_2}\cap\partial_RE)\ge \sum_{u\in\Es{v_2}{v_1}}\mu(\{ux\mid x\in\cyl{v_1}\}\cap\partial_RE)\\ &=\sum_{u\in\Es{v_2}{v_1}}N(u)^{-\beta}m(v_1).
\end{align*}
\end{proof}
For later use, we record the following fact.
\begin{lemma}\label{lem:moreaboutS}
Let $\beta\in [0,\infty)$. Then $S(m)(v)\le m(v)(1-\Zvav(\beta))$ for any $m\in D^\beta$ and $v\in \Eequ$. In particular,
$S(m)(v)=0$ for $v\in \Ecrit$.
\end{lemma}
\begin{proof} Let $m\in D^\beta$ and $v\in \Eequ$. Applying Lemma~\ref{lem:mineq} with $v_1=v$ and $v_2=r(e)$ for all $e\in vE^1$ gives that
$$
\sum_{e\in vE^1}N(e)^{-\beta}m(r(e))\geq \sum_{e\in vE^1} N(e)^{-\beta}\Biggl(\sum_{u\in \Es{r(e)}{v}}N(u)^{-\beta}m(v)\Biggr).
$$
Since $(e,u)\to ue$ implements a bijection between $\{(e,u)\mid e\in vE^1\times,\ u\in \Es{r(e)}{v}\}$ and $\Evav$, this inequality shows that
$\sum_{e\in vE^1}N(e)^{-\beta}m(r(e))\geq m(v)\sum_{u'\in \Evav}N(u')^{-\beta}=m(v)\Zvav(\beta)$. The first claim thus follows, and it implies the second claim by \eqref{eq:Ecrit}.
\end{proof}
\begin{proposition} \label{prop:m}
\textnormal{(a)} For each $v\in\Eequ$, the function $m_v^\beta$ satisfies \eqref{item:m1} and \eqref{item:m3}. Further,
$$
S(m^\beta_v)(v')=\begin{cases}\frac{1-\Zvav(\beta)}{\Zva(\beta)} &\text{ if }v'=v\\ 0&\text{ if }v'\ne v.
\end{cases}$$
\textnormal{(b)} $m^\beta_v\in D^\beta$ if and only if $v\in\Ecrit$ or $v\in\Ereg\setminus R$.
\textnormal{(c)} $m^\beta_v\in\Cf$ if and only if $v\in\Ereg\setminus R$.
\textnormal{(d)} $m^\beta_v\in\Cinf$ if and only if $v\in\Ecrit$.
\end{proposition}
Note that when $R=\emptyset$, i.e. we are looking at the Toeplitz algebra of the graph, (b) shows that every element $v\in \Eequ$ defines a function
$m_v^\beta$ in $D^\beta$.
\begin{proof} Since $\Es{v}{v}=\{v\}$, we have $m^\beta_v(v)=\sum_{u\in\Es{v}{v}}N(u)^{-\beta}(\Zva(\beta))^{-1}=(\Zva(\beta))^{-1}$. Using the decomposition $\Evav=\bigsqcup_{e\in vE^1} \Es{r(e)}{v}$, it follows that
\begin{align}
\sum_{e\in vE^1}N(e)^{-\beta}m_v^\beta(r(e))&=\sum_{e\in vE^1}N(e)^{-\beta}\sum_{u\in\Es{r(e)}{v}}N(u)^{-\beta}(\Zva(\beta))^{-1}\notag\\
&=\sum_{u'\in\Evav}N(u')^{-\beta}(\Zva(\beta))^{-1}.\label{eq:prove-m1-m3}
\end{align}
Thus $\sum_{e\in vE^1}N(e)^{-\beta}m_v^\beta(r(e))={\Zvav}(\beta)m^\beta_v(v)$. Now the assumption that $v\in \Eequ$ implies that
$m_v^\beta(v)$ satisfies \eqref{item:m3}. By reorganizing terms we obtain $S(m^\beta_v)(v)=\frac{1-\Zvav(\beta)}{\Zva(\beta)}$. If $v'\ne v$, then $S(m^\beta_v)(v')=0$ follows from the calculations
\begin{align*}
\sum_{e\in v'E^1}N(e)^{-\beta}m_v^\beta(r(e))&=\sum_{e\in v'E^1}N(e)^{-\beta}\sum_{u\in\Es{r(e)}{v}}N(u)^{-\beta}(\Zva(\beta))^{-1}\\
&=\sum_{u'\in\Es{v'}{v}}N(u')^{-\beta}(\Zva(\beta))^{-1}\\
&=m^\beta_v(v').
\end{align*}
To finish the proof of (a) it remains to show that $m_v^\beta$ satisfies \eqref{item:m1}. This follows from the decomposition $
\Eva=\bigsqcup_{v'\in E^0}\Es{v'}{v}$ and the calculations
\begin{align*}
\sum_{v'\in E^0}m_v^\beta(v')&=\sum_{v'\in E^0}\sum_{u\in\Es{v'}{v}}N(u)^{-\beta}(\Zva(\beta))^{-1}\\
&=\sum_{u\in\Eva}N(u)^{-\beta}(\Zva(\beta))^{-1}=1.
\end{align*}
In particular, we have that $m^\beta_v(v')\in [0,1]$.
For (b), note that (a) implies that $m^\beta_v$ satisfies \eqref{item:m2} if and only if $v\in\Ecrit$ or $v\in\Ereg\setminus R$.
For (c) and (d), let $\mu^\beta_v\in C^\beta$ be the measure associated to $m_v^\beta$ as in \eqref{eq:convention-mu-m}. Recall that the scaling condition in $C^\beta$ gives that
$$
\mu^\beta_v(\cyl{u}\cap\partial_RE)=N(u)^{-\beta}m^\beta_v(r(u))$$
for all $u\in E^*$.
Suppose that $v\in\Ereg\setminus R$. By Lemma~\ref{lem:aboutS} and part (a),
$$
\mu^\beta_v(\{v\})=S(m_v^\beta)(v)=\frac{1-\Zvav(\beta)}{\Zva(\beta)}.
$$
Using the above scaling condition, $\mu^\beta_v(\{u\})=N(u)^{-\beta}\frac{1-\Zvav(\beta)}{\Zva(\beta)}$ for any $u\in E^*v$. By \eqref{eq:bijection1},
\begin{align*}
\mu^\beta_v(E^*v)&=\sum_{u\in E^*v}\mu^\beta_v(\{u\})=\sum_{u\in E^*v}N(u)^{-\beta}\frac{1-\Zvav(\beta)}{\Zva(\beta)}\\
&=\Biggl(\sum_{u\in\Eva}N(u)^{-\beta}\Biggr)\Biggl(\sum_{n=0}^\infty(\Zvav(\beta))^n\Biggr)\frac{1-\Zvav(\beta)}{\Zva(\beta)}=1.
\end{align*}
Thus $\mu^\beta_v\in \Cf$ when $v\in\Ereg\setminus R$.
Suppose next that $v\in\Ecrit$. Lemma~\ref{lem:aboutS}(a) and part (a) imply that
\begin{equation*}
\mu^\beta_v(\{v'\})=S(m^\beta_v)(v')=0
\end{equation*}
for any $v'\in E^0\setminus R$. Since clearly $S(m_v^\beta)(v')=0$ for all $v'\in R$, Lemma~\ref{lem:aboutS}(b) shows that $m^\beta_v$ is of infinite type, i.e. belongs to $\Cinf$.
\end{proof}
The next step in our analysis is a more detailed study of the structure of the sets $\Eequ$, $\Ereg$ and $\Ecrit$, and the functions $m^\beta_v$. Given two vertices $v_1,v_2\in E^0$, we introduce the notation
\begin{align*}
v_1\succ v_2 &\text{ if }v_1E^*\cap E^*v_2\ne\emptyset, \text{ and }\\
v_1\sim v_2 &\text{ if }v_1\succ v_2 \text{ and }v_2\succ v_1.
\end{align*}
\begin{proposition}\label{prop:sums}
\mbox{ }
\begin{enumerate}
\item \label{ item:gg1} If $v_1\in\Ereg$ and $v_2\succ v_1$, then $v_2\in\Ereg$.
\item \label{ item:gg2} If $v_1\in\Eequ$ and $v_2\succ v_1$, then $v_2\in\Eequ$.
\item \label{ item:gg3} If $v_1\in\Ecrit$ and $v_2\sim v_1$, then $v_2\in\Ecrit$.
\item \label{ item:gg4} If $v_1\in\Eequ$, $v_2\succ v_1$, and $v_1\not\succ v_2$, then $v_2\in\Ereg$.
\end{enumerate}
\end{proposition}
\begin{proof}
Assertion \eqref{ item:gg1} follows from the fact that $Z_{v_1}(\beta)\geq N(u)^{-\beta}Z_{v_2}$ for every $u\in v_2E^*v_1$.
For \eqref{ item:gg2}, notice that if $v_1\in\Eequ$ and $v_2\succ v_1$, then $m_{v_1}^\beta(v_2)\ne 0$. Assuming first that $R=\emptyset$, it follows from Proposition \ref{prop:E}\eqref{item:a4} that $v_2\in\Eequ$. Since the definition of
the set $\Eequ$ does not depend on $R$, the claim is true in general.
To prove \eqref{ item:gg3}, suppose $v_1\in\Ecrit$ and $v_1\sim v_2$. Then \eqref{ item:gg2} implies that $v_2\in\Eequ=\Ereg\cup\Ecrit$, and it follows from \eqref{ item:gg1} applied to $v_1\succ v_2$ that if $v_2\in\Ereg$, then $v_1\in\Ereg$. Since the latter is not the case, we must have that $v_2\in\Ecrit$.
Finally, for \eqref{ item:gg4} suppose $v_1\in\Eequ$, $v_2\succ v_1$, and $v_1\not\succ v_2$. It then follows from \eqref{ item:gg2} that $v_2\in\Eequ=\Ereg\cup\Ecrit$. Assume for contradiction that $v_2\in\Ecrit$. Since $v_2\succ v_1$, we have $\Es{v_2}{v_1}\ne\emptyset$. Choose $u\in \Es{v_2}{v_1}$. Let $n\ge 1$. For $u_1,u_2,\dots,u_n\in v_2E_s^*v_2$ we have that $u_1u_2\dotsm u_nu\in E_a^*v_1$ (since $v_1\not\succ v_2$). Hence
$$Z_{v_1}^a(\beta)\ge \sum_{n=1}^\infty \Bigl(Z_{v_2}^s(\beta)\Bigr)^nN(u)^{-\beta}=\infty,$$
which contradicts the assumption that $v_1\in\Eequ$. Thus, $v_2\in\Ereg$.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:finitetype-reg}.] We must first prove that $W_{\operatorname{fin}}$ is well-defined. Let $m\in\Cf$ and let $\mu\in C^\beta$ be the measure associated to $m$. Then
\begin{align}
\sum_{v\in \Ereg\setminus R}W_{\operatorname{fin}}(m)(v)
&=\sum_{v\in \Ereg\setminus R}\frac{S(m)(v)\Zva(\beta)}{1-\Zvav(\beta)}\notag\\
&=\sum_{v\in \Ereg\setminus R}\frac{\mu(\{v\})\Zva(\beta)}{1-\Zvav(\beta)}\text{ by Lemma~\ref{lem:aboutS}(a)}\notag\\
&=\sum_{v\in \Ereg\setminus R} \mu(\{v\})Z_v(\beta) \text{ by Proposition~\ref{prop:E}}\notag\\
&=\sum_{v\in \Ereg\setminus R}\sum_{u\in E^*v}\mu(\{v\})N(u)^{-\beta}\notag\\
&=\sum_{v\in \Ereg\setminus R}\sum_{u\in E^*v}\mu(\{u\}).\label{eq:wf-well}
\end{align}
The scaling identity in $C^\beta$ and Lemma~\ref{lem:aboutS} imply that
\begin{equation}\label{eq:mu-u}
\mu(\{u\})=N(u)^{-\beta}\mu(\{r(u)\})=N(u)^{-\beta}S(m)(r(u))
\end{equation}
for all $u\in E^*\cap\partial_RE$.
We claim that $S(m)(r(u))=0$ unless $r(u)\in\Ereg\setminus R$. From
Lemma~\ref{lem:moreaboutS} and the definition of $S$ we have $S(m)(r(u))=0$ for $r(u)\in\Ecrit\cup R$. If $r(u)\notin \Eequ$, then Proposition~\ref{prop:E}(4) implies that $m(r(u))=0$, therefore also $S(m)(r(u))=0$. The claim follows and implies that
$\mu$ is supported on the finite paths that end in vertices $v\in \Ereg\setminus R$. Hence \eqref{eq:wf-well} gives that $\sum_{v\in \Ereg\setminus R}W_{\operatorname{fin}}(m)(v)=\mu(E^*\cap \partial_R E)=1$, which shows that $W_{\operatorname{fin}}$ is well-defined. Clearly $W_{\operatorname{fin}}$ is a convex map.
We claim next that $\bigl\{\epsilon:\Ereg\setminus R\to [0,1]\bigm\vert \sum_{v\in\Ereg\setminus R}\epsilon(v)=1\bigr\} \subseteq W_{\operatorname{fin}}(\Cf)$, which gives surjectivity. Clearly for every $v\in \Ereg\setminus R$ the function $\delta_v: \Ereg\setminus R\to [0,1]$ defined by $\delta_v(v)=1$ and $\delta_v(v')=0$ when $v'\ne v$ belongs to the set on the left-hand side. The claim follows because Proposition~\ref{prop:m} shows that $W_{\operatorname{fin}}(m_v^\beta)=\delta_v$ and every $\epsilon$ can be written as $\epsilon=\sum_{v\in \Ereg\setminus R} \epsilon(v)\delta_v$.
Finally, assume that $W_{\operatorname{fin}}(m_1)=W_{\operatorname{fin}}(m_2)$ for $m_1,m_2\in \Cf$. Then $S(m_1)=S(m_2)$. By \eqref{eq:mu-u}, the measure $\mu_1$ associated with $m_1$ equals the one associated to $m_2$ on all finite paths, hence $m_1=m_2$. This shows injectivity and finishes the proof.
\end{proof}
Next we analyze the elements in $D^\beta$ in relation to vertices in $\Ecrit$. The first observation is that $\sim$ is an equivalence relation on $\Ecrit$ due to Proposition~\ref{prop:sums}, assertions \eqref{ item:gg3} and \eqref{ item:gg4}. We let $\Ecrit/_{\sim}$ denote the set of equivalence classes and write $\mathfrak{v}:=\{v'\in \Ecrit\mid v\sim v'\}$ for the equivalence class of $v$.
\begin{theorem}\label{thm:infinitetype-crit} Let $\beta\in [0,\infty)$. The map $\Winf$ from $D^\beta$ to the set of $[0,1]$-valued functions on $\Ecrit/_{\sim}$ given by
\begin{equation}\label{eqdef:W-inf}
\Winf(m)(\mathfrak{v})=m(v)\Zva(\beta)
\end{equation}
for $\mathfrak{v}\in \Ecrit/_{\sim}$, $v\in \mathfrak{v}$, is a well-defined convex isomorphism from $\Cinfa$ onto $$\biggl\{\epsilon:(\Ecrit/_{\sim})\to [0,1]\biggm\vert \sum_{\mathfrak{v}\in\Ecrit/_{\sim}}\epsilon(\mathfrak{v})=1\biggr\}.$$
For every $m\in D^\beta$, the map $m_{\mathfrak{v}}^\beta:=m_v^\beta$ is well-defined on $\Ecrit/_{\sim}$. The inverse of $\Winf$ is $\epsilon\mapsto \sum_{\mathfrak{v}\in\Ecrit/_{\sim}}\epsilon(\mathfrak{v})m^\beta_{\mathfrak{v}}$.
\end{theorem}
The proof of this theorem will follow from a series of results. We start by investigating when an equality $m_{v_1}^\beta=m_{v_2}^\beta$ can take place.
\begin{lemma}\label{lem:help}
Let $v\in \Ecrit\cup \left(\Ereg\setminus R\right)$ and $m\in D^\beta$. Then $m=m^\beta_v$ if and only if $m(v)=(\Zva(\beta))^{-1}$.
\end{lemma}
\begin{proof}
Suppose that $m(v)=(\Zva(\beta))^{-1}$, and let $\mu$ be the measure associated to $m$ by \eqref{eq:convention-mu-m}. It follows from the scaling condition in $C^\beta$ that $\mu(\cyl{u}\cap\partial_RE)=N(u)^{-\beta}\mu(\cyl{v}\cap\partial_RE)=N(u)^{-\beta}(\Zva(\beta))^{-1}$ for any $u\in E^*v$. Thus
\begin{equation*}
\sum_{u\in\Eva}\mu(\cyl{u}\cap\partial_RE)=\sum_{u\in\Eva}N(u)^{-\beta}(\Zva(\beta))^{-1}=1.
\end{equation*}
This shows that $A:=\{u\in\Eva\mid \mu(\cyl{u}\cap\partial_RE)>0\}$ is countable. Let $v'\in E^0$. From
\begin{align*}
m(v')&=\mu(\cyl{v'}\cap\partial_RE)=\sum_{u\in A}\mu(\cyl{u}\cap\partial_RE)\mu(\cyl{v'}\cap\partial_RE)\\
&=\sum_{u\in A,\ s(u)=v'}\mu(\cyl{u}\cap\partial_RE)
=\sum_{u\in A,\ s(u)=v'}N(u)^{-\beta}(\Zva(\beta))^{-1}
\end{align*}
it follows that $m(v')\leq m^\beta_v(v')$ because $\{u\in A, s(u)=v\}\subset \Es{v'}{v}$.
Conversely, if $u_1,u_2\in\Es{v'}{v}$ and $u_1\ne u_2$, then $\cyl{u_1}\cap\cyl{u_2}=\emptyset$. Hence we have $\bigcup{u\in\Es{v'}{v}}\cyl{u}\subseteq\cyl{v'}$, so
\begin{align*}
m(v')&=\mu(\cyl{v'}\cap\partial_RE)\ge\sum_{u\in\Es{v'}{v}}\mu(\cyl{u}\cap\partial_RE)\\
&=\sum_{u\in\Es{v'}{v}}N(u)^{-\beta}(\Zva(\beta))^{-1}=m^\beta_v(v').
\end{align*}
Thus, $m=m^\beta_v$.
\end{proof}
\begin{lemma}\label{lem:sums}
Suppose $v_1,v_2\in\Ecrit$ and that $v_1\sim v_2$. Then $$\Bigl(\sum_{u_1\in\Es{v_2}{v_1}}N(u_1)^{-\beta}\Bigr)\Bigl(\sum_{u_2\in\Es{v_1}{v_2}}N(u_2)^{-\beta}\Bigr)=1.$$
\end{lemma}
\begin{proof}
Let $P=\{u\in v_2E_s^*v_2\mid r(u')\ne v_1\text{ for any }u'\le u\}$ and $x=\sum_{u\in P}N(u)^{-\beta}$. Then the
assumption $v_1\sim v_2$ implies that
$$
x<\sum_{u\in v_2E_s^*v_2}N(u)^{-\beta}=Z^s_{v_2}(\beta)=1.
$$
Now $(u_1,\dots,u_n,u)\mapsto u_1\dotsm u_nu$ defines a bijection between $\bigcup_{n=0}^\infty(P)^n\times (v_2E_s^*v_2\setminus P)$ (where $(P)^0=\{v_2\}$) and $\{u_1u_2\mid u_1\in\Es{v_2}{v_1},\ u_2\in\Es{v_1}{v_2}\}$, hence
\begin{equation*}
\Bigl(\sum_{u_1\in\Es{v_2}{v_1}}N(u_1)^{-\beta}\Bigr)\Bigl(\sum_{u_2\in\Es{v_1}{v_2}}N(u_2)^{-\beta}\Bigr)=\sum_{n=0}^\infty x^n(1-x)=1.
\end{equation*}
\end{proof}
\begin{proposition}\label{prop:meq}
Suppose $v_1,v_2\in \Ecrit\cup \left(\Ereg\setminus R\right)$ and $v_1\ne v_2$. Then $m^\beta_{v_1}=m^\beta_{v_2}$ if and only if $v_1,v_2\in\Ecrit$ and $v_1\sim v_2$.
\end{proposition}
\begin{proof}
Assume $v_1,v_2\in\Ecrit$ and $v_1\sim v_2$. Two applications of Proposition \ref{prop:E}\eqref{item:a3.5} give us that
\begin{align*}
&\Bigl(\sum_{u_1\in\Es{v_2}{v_1}} N(u_1)^{-\beta}\Bigr)\Bigl(\sum_{u_2\in\Es{v_1}{v_2}} N(u_2)^{-\beta}m^\beta_{v_1}(v_2)\Bigr)\\
&\qquad\le\Bigl(\sum_{u_1\in\Es{v_2}{v_1}} N(u_1)^{-\beta}\Bigr)\Bigl(\sum_{u_2\in\Es{v_1}{v_2}}N(u_2)^{-\beta}(Z^a_{v_2}(\beta))^{-1}\Bigr)\\
&\qquad=\sum_{u_1\in\Es{v_2}{v_1}} N(u_1)^{-\beta}m^\beta_{v_2}(v_1)
\le \sum_{u_1\in\Es{v_2}{v_1}} N(u_1)^{-\beta}(Z^a_{v_1}(\beta))^{-1}\\
&\qquad=m^\beta_{v_1}(v_2).
\end{align*}
It follows from Lemma \ref{lem:sums} that the above inequalities are in fact equalities, so $m^\beta_{v_1}(v_2)=(Z^a_{v_2}(\beta))^{-1}$. Thus $m^\beta_{v_1}=m^\beta_{v_2}$ by Lemma \ref{lem:help}.
If $v_1\in\Ereg$, then Proposition \ref{prop:m} implies that $S(m^\beta_{v_1})(v_1)\ne 0$. Since $S(m^\beta_{v_2})(v_1)=0$, necessarily then $m^\beta_{v_1}\ne m^\beta_{v_2}$. Similarly, $m^\beta_{v_1}\ne m^\beta_{v_2}$ if $v_2\in\Ereg$.
If $v_1\not\succ v_2$, then $m^\beta_{v_2}(v_1)=0\ne (Z^a_{v_1}(\beta))^{-1}=m^\beta_{v_1}(v_1)$, so $m^\beta_{v_1}\ne m^\beta_{v_2}$. Similarly, $m^\beta_{v_1}\ne m^\beta_{v_2}$ if $v_2\not\succ v_1$.
\end{proof}
\begin{proposition}\label{prop:equ}
Let $m\in D^\beta$ and $v_1,v_2\in\Ecrit$. Suppose $v_1\sim v_2$. Then $m(v_1)Z^a_{v_1}(\beta)=m(v_2)Z^a_{v_2}(\beta)$.
\end{proposition}
\begin{proof}
Proposition \ref{prop:meq} and Lemma \ref{lem:help} imply that
\begin{equation} \label{eq:t}
(Z^a_{v_1}(\beta))^{-1}=\sum_{u\in\Es{v_1}{v_2}}N(u)^{-\beta}(Z^a_{v_2}(\beta))^{-1}.
\end{equation}
It follows from Lemma \ref{lem:mineq} that $m(v_2)\ge \sum_{u\in\Es{v_2}{v_1}}N(u)^{-\beta}m(v_1)$ and, similarly, that $m(v_1)\ge \sum_{u\in\Es{v_1}{v_2}}N(u)^{-\beta}m(v_2)$, so
\begin{multline*}
\Bigl(\sum_{u_1\in\Es{v_2}{v_1}} N(u_1)^{-\beta}\Bigr)\Bigl(\sum_{u_2\in\Es{v_1}{v_2}} N(u_2)^{-\beta}m(v_2)\Bigr)\\\le\sum_{u_1\in\Es{v_2}{v_1}} N(u_1)^{-\beta}m(v_1)\le m(v_2).
\end{multline*}
According to Lemma \ref{lem:sums}, the above inequalities are in fact equalities, so
$$m(v_1)=\sum_{u\in\Es{v_1}{v_2}}N(u)^{-\beta}m(v_2)=\frac{Z^a_{v_2}(\beta)}{Z^a_{v_1}(\beta)}m(v_2)$$
where the last equality follows from Equation \eqref{eq:t}.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:infinitetype-crit}.]
We start by showing that the maps introduced in the formulation of the theorem are well-defined.
Let $m\in D^\beta$ and $\mathfrak{v}\in \Ecrit/_{\sim}$. We deduce from Proposition \ref{prop:meq} that $m_{v_1}^\beta=m_{v_2}^\beta$ for $v_1,v_2\in \mathfrak{v}$. Therefore, $m_{\mathfrak{v}}^\beta:=m_v^\beta$ for $v\in\mathfrak{v}$ is well-defined.
It follows from Proposition \ref{prop:equ} that the quantity $m(v)\Zva(\beta)$ does not depend on the choice of $v\in \mathfrak{v}$.
Suppose $m\in \Cinfa$ and $v\in\Eequ$. Let $R_v$ denote the set of infinite paths that start in $v$ and return to $v$ infinitely often. Then $\mu(R_v)=\lim_{n\to\infty}(\Zvav(\beta))^n m(v)$, where $\mu$ is the measure associated with $m$. Hence $\mu(R_v)=0$ for $v\in\Ereg$ and
$\mu(R_v)=m(v)$ when $v\in\Ecrit$. Now let $T_v$ be the collection of infinite paths that pass through $v$ infinitely often. It follows that $\mu(T_v)=\Zva(\beta) \mu(R_v)$, so $\mu(T_v)$ is zero for $v\in\Ereg$, and equals
$\Zva(\beta)m(v)$ when $v\in\Ecrit$. Suppose now that $v_1\sim v_2$. Let $P$ be as in the proof of Lemma~\ref{lem:sums}. Then
$\sum_{u\in P}N(u)^{-\beta}<1$, and therefore $\mu(T_{v_1}\bigtriangleup T_{v_2})=0$ where
$T_{v_1}\bigtriangleup T_{v_2}=(T_{v_1}\setminus
T_{v_2})\cup(T_{v_2}\setminus T_{v_1})$. Since $\mu$ is conservative, it has support on $\Erec$. We conclude that
$$
\sum_{\mathfrak{v}\in\Ecrit/_{\sim}}
m(v)\Zva(\beta)=\sum_{\mathfrak{v}\in\Ecrit/_{\sim}} \mu(T_v)=1,
$$
where $v$ is taken such that $v\in \mathfrak{v}$ as $\mathfrak{v}$ runs over $\Ecrit/_{\sim}$.
Clearly $W_{\operatorname{inf}}$ is a convex map. We show next that
\begin{equation} \label{eq:supset}
\bigl\{\epsilon:\Ecrit/_{\sim}\to [0,1]\bigm\vert \sum_{\mathfrak{v}\in\Ecrit/_{\sim}}\epsilon(\mathfrak{v})=1\bigr\}\subseteq \Winf(\Cinfa).
\end{equation}
For $\mathfrak{v}\in \Ecrit/_{\sim}$ let $\delta_{\mathfrak{v}}: \Ecrit/_{\sim}\to [0,1]$ be defined by $\delta_{\mathfrak{v}}(\mathfrak{v})=1$ and $\delta_{\mathfrak{v}}(\mathfrak{v}')=0$ for $\mathfrak{v}'\in \Ecrit/_{\sim}$ different from $\mathfrak{v}$. Then Lemma \ref{lem:help} and Proposition \ref{prop:sums}\eqref{ item:gg4} show that
$\Winf(m^\beta_{\mathfrak{v}})=\delta_{\mathfrak{v}}$ for $\mathfrak{v}\in\Ecrit/_{\sim}$, which gives the claimed set inclusion.
To prove that $\Winf$ is injective we show that
\begin{equation} \label{eq:inj}
\sum_{\mathfrak{v}\in\Ecrit/_{\sim}}\Winf(m)(\mathfrak{v})m^\beta_{\mathfrak{v}}=m
\end{equation}
for any $m\in \Cinfa$. Denote by $m'$ the term $\sum_{\mathfrak{v}\in\Ecrit/_{\sim}}\Winf(m)(\mathfrak{v})m^\beta_{\mathfrak{v}}$. Proposition~\ref{prop:sums}, Lemma~\ref{lem:help} and Proposition~\ref{prop:equ} imply that
$m(v)=m'(v)$ for $v\in\Ecrit\setminus R$. On the other hand, Lemma~\ref{lem:mineq} shows that
$m(v)\ge m'(v)$ for all $v\in E^0$. Since $\sum_{v\in
E^0}m(v)=\sum_{v\in E^0}m(v')=1$ we must have $m(v)=m'(v)$ for all $v\in E^0$.
\end{proof}
When $A$ is a subset of $D^\beta$ we let $\conv A$ denote the convex hull of $A$. We then have the following consequence of Theorems~\ref{thm:finitetype-reg} and \ref{thm:infinitetype-crit}.
\begin{corollary}\label{cor:extreme}
\begin{enumerate}
\item $\{m^\beta_v\mid v\in\Ereg\setminus R\}$ is the set of extreme points of $\Cf$, and $\Cf=\conv \{m^\beta_v\mid v\in\Ereg\setminus R\}$.
\item $\{m^\beta_{\mathfrak{v}}\mid \mathfrak{v}\in\Ecrit/_{\sim}\}$ is the set of extreme points of $\Cinfa$, and $\Cinfa=\conv\{m^\beta_{\mathfrak{v}}\mid \mathfrak{v}\in\Ecrit/_{\sim}\}$.
\item If $\Cinfb=\emptyset$ (in particular if $\Ewan=\emptyset$), then $\{m^\beta_v\mid v\in\Ereg\setminus R\}\cup\{m^\beta_{\mathfrak{v}}\mid \mathfrak{v}\in\Ecrit/_{\sim}\}$ is the set of extreme points of $D^\beta$, and $D^\beta=\conv \{m^\beta_v\mid v\in\Ereg\setminus R\}\cup\{m^\beta_{\mathfrak{v}}\mid \mathfrak{v}\in\Ecrit/_{\sim}\}$.
\end{enumerate}
\end{corollary}
Notice that if $R\ne \emptyset$, then the KMS states of $(\mathcal{T}C^*(E),\sigma)$ that descend to KMS states on $(C^*(E,R),\sigma)$ are the ones that correspond to elements of $\conv\{m^\beta_v\mid v\in\Ereg\setminus R\}\cup\Cinf$.
\begin{remark} Proposition \ref{prop:sums} implies that for arbitrary $v_1,v_2$ in $\Eequ$ with $v_1\sim v_2$, either $v_1,v_2$
both belong to $\Ecrit$ or they both belong to $\Ereg$. In particular, if the graph is connected in the sense that for every $v_1$ and $v_2$ in $E^0$ we have $v_1\sim v_2$, then for each $\beta\geq 0$,
either $\Eequ=\Ecrit$ or $\Eequ=\Ereg$. Thus at each $\beta\geq 0$, either $\Cinfa$ or $\Cf$ are empty. If the graph is not connected, it may happen that both $\Cinfa$ and $\Cf$ are non-trivial at some $\beta\geq 0$, see Example~\ref{eq:stable-TOn}. It might also happen that both $\Cinfb$ and $\Cf$ are non-trivial at some $\beta\geq 0$, see Theorem \ref{thm:exmotivating}.
\end{remark}
We conclude this section with a couple of general remarks about existence of KMS states. Recall that for $\beta\in [0,\infty)$ and $v\in E^0$, we defined the partition function with fixed-target $Z_v(\beta)$ in \eqref{eq:Zv}. If there is $\beta$ in $[0,\infty)$ such that $Z_v(\beta)<\infty$, we set
$$\beta_v=\inf\{\beta\in [0,\infty)\mid Z_v(\beta)<\infty\}.
$$
Otherwise we let $\beta_v=\infty$. We have the following simple observation.
\begin{proposition}\label{thm:no-kms}
Let $v\in E^0$. If $\beta<\beta_v$, then there is no KMS$_\beta$ state $\psi$ for $(C^*(E,R),\sigma)$ such that $\psi(p_v)\ne 0$.
\end{proposition}
\begin{proof}
Suppose $\psi$ is a KMS$_\beta$ state for $(C^*(E,R),\sigma)$ such that $\psi(p_v)\ne 0$. It follows from Proposition~\ref{prop:E}\eqref{item:a4} that $v\in\Eequ$ and thus that $v\in E^0_{\beta'\text{-reg}}$ for any $\beta'>\beta$. It then follows from Proposition~\ref{prop:E}\eqref{item:a2} that $\beta\ge \beta_v$.
\end{proof}
The following observation can be helpful in computing KMS$_\beta$ states for particularly nice graphs and will be used in Examples \ref{ex:a}, \ref{ex:b}, and \ref{ex:c}.
\begin{proposition}\label{prop:nice-graphs}
Suppose that $N(e)=\exp(1)$ for all $e\in E^1$ and that there exist $k,l\in\N$ such that every $v\in E^0$ receives exactly $k$ paths of length $l\geq 1$. Then:
\begin{enumerate}
\item $D^\beta=\emptyset$ when $\beta<\ln (k)/l$.
\item $D^\beta=D^{\beta}_{\operatorname{inf}}$ when $\beta=\ln (k)/l$.
\item $\{m^\beta_v\mid v\in E^0\}$ are the extreme points of $D^\beta$ and $D^\beta=\Cf$ when $\beta>\ln (k)/l$.
\end{enumerate}
\end{proposition}
\begin{proof}
Suppose $m\in D^\beta$ is of infinite type. Thus $m(v)=\sum_{e\in vE^1}N(e)^{-\beta}m(r(e))$ for all $v\in E^0$, according to Lemma~\ref{lem:aboutS}(b). Iterations of this equality imply that
$$
m(v)=\sum_{u\in vE^l}N(u)^{-\beta}m(r(u))=\exp(-l\beta)\sum_{u\in vE^l}m(r(u))
$$
for every $v\in E^0$. Hence
\begin{align*}
1&=\sum_{v\in E^0}m(v)=\exp(-l\beta)\sum_{v\in E^0}\sum_{u\in vE^l}m(r(u))\\
&=\exp(-l\beta)\sum_{v'\in E^0}km(v')=k\exp(-l\beta)
\end{align*}
from which it follows that $\beta=\ln(k)/l$. Thus $D^{\beta}_{\operatorname{inf}}=\emptyset$ for $\beta\ne \ln(k)/l$.
For each $v\in E^0$, the partition function $Z_v(\beta)$ at $v$ satisfies that $(Z_v(\beta))^l=\sum_{n=0}^\infty k^n\exp(-nl\beta)$. It thus follows from Proposition \ref{prop:E} that $\Ereg=\emptyset$ for $\beta\le \ln (k)/l$, and that $\Ereg=E^0$ for $\beta>\ln (k)/l$. Hence $D^\beta=\emptyset$ for $\beta<\ln (k)/l$, $D^\beta_4=D^{\beta}_{\operatorname{inf}}$ when $\beta=\ln (k)/l$, and the extreme points of $D^\beta=\Cf$ are $\{m^\beta_v\mid v\in E^0\}$ when $\beta>\ln (k)/l$.
\end{proof}
\section{Ground states and KMS$_\infty$ states}\label{section:ground}
Note that the definition of $D^{\operatorname{gr}}$ implies directly that the set of its extreme points is $\{m^{\operatorname{gr}}_v\mid v\in E^0\setminus R\}$, where $m^{\operatorname{gr}}_v:E^0\to [0,1]$ is defined by
\begin{equation}\label{eq:ground-pointmass}
m^{\operatorname{gr}}_v(v')=\begin{cases}1&\text{if }v'=v,\\0&\text{if }v'\ne v.\end{cases}
\end{equation}
Thus, we have a complete concrete description of all the ground states of $(C^*(E,R),\sigma)$.
A ground state is called a \emph{KMS$_\infty$ state} if it is the weak* limit of a sequence of KMS$_{\beta_n}$ states as $\beta_n\to\infty$ (see \cite{Con-Mar} and \cite[\S 1]{aHLRS}). We will now characterize which of the ground states of $(C^*(E,R),\sigma)$ are KMS$_\infty$ states. Since for a finite graph $E$ we have $\beta_v<\infty$ for all $v\in E^0$, the next result generalizes \cite[Proposition 5.1]{aHLRS}.
\begin{theorem}\label{thm:infty-states}
Given a directed graph $E$, a subset $R$ of $E^0_\reg$ and a function $N:E^1\to (1,\infty)$, let $\sigma$ be the strongly continuous one-parameter group
of automorphisms of $C^*(E,R)$ such that
\begin{equation*}
\sigma_t(s_e)=\bigl(N(e)\bigr)^{it}s_e \text{ and }\sigma_t(p_v)=p_v
\end{equation*}
for all $e\in E^1$ and $v\in E^0$.
A ground state $\psi$ of $(C^*(E,R),\sigma)$ is a KMS$_\infty$ state if and only if $\beta_v<\infty$ for every $v\in E^0$ for which $\psi(p_v)\ne 0$.
\end{theorem}
\begin{proof}
For $m\in D^\beta$ let $\psi_m$ be the KMS$_\beta$ state corresponding to $m$.
Assume first that $\psi$ is a KMS$_\infty$ state and that $\psi(p_v)\ne 0$. Then there is a $\beta<\infty$ and a KMS$_\beta$ state which is non-zero on $p_v$. It follows from Proposition \ref{thm:no-kms} that $\beta_v<\infty$. This shows the only if direction.
For the converse direction, since $\{\psi_{m^{\operatorname{gr}}_v}\mid v\in E^0\setminus R\}$ are the extreme points of $A^{\operatorname{gr}}$, it suffices to show that $\psi_{m^{\operatorname{gr}}_v}$ is a KMS$_\infty$ state if $v\in E^0\setminus R$ and $\beta_v<\infty$. We will establish this by showing that $(\psi_{m^\beta_v})$ converges to $\psi_{m^{\operatorname{gr}}_v}$ in the weak*-topology as $\beta\to\infty$.
We have that $\sum_{u\in\Eva}N(u)^{-\beta}=\Zva(\beta)<\infty$ for $\beta>\beta_v$, and since $N(u)^{-\beta}$ converges monotonically to 0 as $\beta\to\infty$ for $u\in\Eva\setminus\{v\}$, an application of the monotone convergence theorem yields that $\Zva(\beta)\to 1$ as $\beta\to\infty$. A similar argument gives us that $\sum_{u\in v'E_a^*v}N(u)^{-\beta}\to 0$ as $\beta\to\infty$ for $v'\in E^0\setminus\{v\}$. Thus $m^\beta_v(v')$ converges pointwise to $m^{\operatorname{gr}}_v(v')$ as $\beta\to\infty$, for each $v'\in E^0$. This implies our claim that $(\psi_{m^\beta_v})$ converges to $\psi_{m^{\operatorname{gr}}_v}$ in the weak*-topology as $\beta\to\infty$.
\end{proof}
Example \ref{ex:d} provides an example of a ground state which is not a KMS$_\infty$ state.
\section{Examples}
All throughout this section we let $\No=\{0,1,2,\dots\}$.
\begin{example} Our first example is a graph where Theorems~\ref{thm:finitetype-reg} and \ref{thm:infinitetype-crit} describe completely the KMS states of $\mathcal{T}C^*(E)$ endowed with the gauge action. Let $E$ be the graph with $E^0=\{v_n\mid n\in \No\}$ and $E^1=\{e_n, f_n\mid n\in \No\}$ given by $s(e_n)=v_0=r(e_0)=r(f_0)$ for all $n\geq 0$, $r(e_n)=v_n=r(f_{n})$ for $n\geq 1$, and $s(f_n)=v_{n-1}$ for $n\geq 1$:
\begin{equation}
\xymatrix{
{v_0} \ar@(ul,dl)[]|{e_0} \ar@/^/[r] ^-{e_1} \ar@/^ 2pc/[rr] ^-{e_2} \ar@/^ 3pc/[rrr]^{e_3} &{v_1} \ar[l]^{f_0} &{v_2} \ar[l]^{f_1}&{\dots}\ar[l]^{f_2}
}
\end{equation}
Let $R=\emptyset$ and $N:E^1\to (1,\infty)$ be $N(e_n)=N(f_n)=\exp(1)$ for all $n\in \No$. Thus we are dealing with $\mathcal{T}C^*(E)$ and its gauge action. Since every infinite path passes through $v_0$ infinitely many times, we have $\Erec=E^\infty$ and $\Ewan=\emptyset$. Hence $\Cinf=\Cinfa$. The partition functions at $v_0$ are given as follows:
\begin{align*}
Z^s_{v_0}(\beta)
&= \sum_{k=1}^\infty \exp(-k\beta)=\frac {\exp(-\beta)}{1-\exp(-\beta)}\text{ and }\\
Z^a_{v_0}(\beta)
&= \sum_{k=0}^\infty \exp(-k\beta)=\frac {1}{1-\exp(-\beta)}.
\end{align*}
Thus for $\beta\in [0,\ln 2)$ we have $\Eequ=\emptyset$, for $\beta=\ln 2$ we have $\Ecrit=E^0$, and for $\beta\in (\ln 2,\infty)$, $\Ereg=E^0$. Hence $D^\beta=\emptyset$ for $\beta\in [0,\ln 2)$.
Further, by Theorem~\ref{thm:infinitetype-crit},
$$
D^\beta=\Cinfa=\{m_{E^0}^\beta\} \text{ when }\beta=\ln 2.
$$
Finally, for $\beta\in (\ln 2,\infty)$ Theorem~\ref{thm:finitetype-reg} gives that $D^\beta=\Cf$ is isomorphic as a convex set with the set of functions $\{\epsilon:E^0\to [0,1]\mid \sum_{v\in E^0}=1\}$. The extremal points of $D^\beta$ coincide with $\{m^\beta_{v_n}\mid n\in\No\}$, and every $m\in D^\beta$ has form
$m=\sum_{n=0}^\infty\frac{S(m)(v_n)Z^a_{v_n}(\beta)}{1-Z^s_{v_n}(\beta)}m^\beta_{v_n}$.
It is easy to check that $m^{\ln 2}_{E^0}(v_n)=2^{-n-1}$ for all $n\in\No$. It is not difficult, but a bit tedious to write down explicit formulas for $m^\beta_v$ when $\beta>\ln 2$ and $v\in E^0$.
We have that $D^{\operatorname{gr}}=\conv\{m^{\operatorname{gr}}_v\mid v\in E^0\}$, and it follows from Theorem \ref{thm:infty-states} that every ground state is a KMS$_\infty$ state.
Since $E^0_\reg=E^0\setminus\{v_0\}$, it follows that the only KMS states of $\mathcal{T}C^*(E)$ that descend to KMS states on $C^*(E)$ are the ones corresponding to $m^{\ln 2}_{E^0}$ and $m^\beta_{v_0}$, $\beta>\ln 2$. The only ground state that descends to $C^*(E)$ is the one corresponding to $m^{\operatorname{gr}}_{v_0}$.
\end{example}
\begin{example} \label{ex:a}
Next we introduce an example of a strongly connected graph $E$ with finite degree (or valence) for which $(C^*(E),\sigma)$ has no KMS states when $\sigma$ is the gauge action.
The graph $E$ is defined as follows
\begin{equation}
\xymatrix{
{\cdots} \ar@/^ 1pc/[r] ^-{e_{-1}} &{v_0} \ar@/^/[l] _-{f_{-1}} \ar@/^ 1pc/[r] ^-{e_{0}} &{v_1} \ar@/^ 1pc/[r] ^-{e_{1}} \ar@/^/[l] _-{f_{0}} &{v_2} \ar@/^ 1pc/[r] ^-{e_{2}} \ar@/^/[l] _-{f_1} & {\cdots} \ar@/^/[l] _-{f_2}
}
\end{equation}
Let $R=\emptyset$ and $N:E^1\to (1,\infty)$ be $N(e)=\exp(1)$ for all $e\in E^1$.
Proposition~\ref{prop:nice-graphs} implies that $D^\beta=\emptyset$ for $\beta<\ln 2$, $D^\beta=D^\beta_{\operatorname{inf}}$ when $\beta=\ln (2)$, and that $D^\beta=\Cf=\conv\{m^\beta_v\mid v\in E^0\}$ for $\beta>\ln 2$.
Suppose $m\in D^{\ln 2}_{\operatorname{inf}}$. Let $n\in \mathbb{Z}$ and denote $a=m(v_n)$. Since $S(m)(v)=0$ for all $v$, it follows that $a=1/2(m(v_{n-1})+m(v_{n+1}))$. Thus, either $m(v_{n-1})\geq a$ or $m(v_{n+1})\geq a$. By symmetry of the graph, we may assume $m(v_{n+1})\geq a$. Let $b:=m(v_{n+1})$. By induction on $k\geq 1$, $m(v_{n+k})=kb-(k-1)a\geq a$. However, $\sum_{k=0}^\infty m(v_{n+k})\leq \sum_{v\in E^0} m(v)=1$, so necessarily $a=0$. Since $n$ was arbitrarily chosen, this shows that $m\equiv 0$, a contradiction. Thus $D^{\ln 2}_{\operatorname{inf}}=\emptyset$.
We conclude that $D^\beta=\emptyset$ for $\beta\le\ln (2)$, and that $D^\beta=\Cf=\conv\{m^\beta_v\mid v\in E^0\}$ for $\beta>\ln 2$. We furthermore have that $D^{\operatorname{gr}}=\conv\{m^{\operatorname{gr}}_v\mid v\in E^0\}$, and it follows from Theorem \ref{thm:infty-states} that every ground state is a KMS$_\infty$ state.
Since $E^0_\reg=E^0$, none of the KMS or ground states of $\mathcal{T}C^*(E)$ descend to KMS or ground states of $C^*(E)$, so the analogue of \cite[Theorem 4.3]{aHLRS} does not hold for infinite graphs even under the assumption that $E$ has finite degree (or valence).
\end{example}
\begin{example}\label{ex:dis}
We now present an example where the set of dissipative measures in non-empty.
Let $E$ be the graph with $E^0=\{v_n\mid n\in \No\}$ and $E^1=\{e_n\mid n\in \No\}\cup\{f_n\mid \No\}$ where $s(e_n)=s(e_n)=v_n$ and $r(e_n)=r(f_n)=v_{n+1}$, see the picture:
\begin{equation*}
\xymatrix{
{v_0} \ar@/^ 1pc/[r]^{e_0} \ar@/_ 1pc/[r]_{f_0} &{v_1}\ar@/^ 1pc/[r]^{e_1} \ar@/_ 1pc/[r]_{f_1} &{v_2}\ar@/^ 1pc/[r]^{e_2} \ar@/_ 1pc/[r]_{f_2}&{v_3\dots}
}
\end{equation*}
Let $R=\emptyset$ and $N:E^1 \to (1,\infty)$ be $N(e)=\exp(1)$ for all $e\in E^1$. Thus we are dealing with $\mathcal{T}C^*(E)$ and its gauge action.
It is easy to see that $\Evav=\emptyset$ and that $\Eva$ is finite for every $v\in E^0$. It follows that $\Zvav(\beta)=0$ and $\Zva(\beta)<\infty$ for all $v\in E^0$ and all $\beta\in [0,\infty)$. Thus, $\Ereg=E^0$ for all $\beta\in [0,\infty)$. It follows that $\Cf=\{m_v^\beta\mid v\in E^0\}$ and $\Cinfa=\emptyset$ for all $\beta\in [0,\infty)$.
Suppose $\beta\in [0,\ln 2)$. Define $m^\beta:E^0\to [0,1]$ by $m^\beta(v_n)=(1-\exp(\beta)/2)(\exp(\beta)/2)^n$ for $n\in\No$. Then
\begin{align*}
m^\beta(v_n)&=(1-\exp(\beta)/2)(\exp(\beta)/2)^n=2\exp(-\beta)(1-\exp(\beta)/2)(\exp(\beta)/2)^{n+1}\\
&=\sum_{e\in v_nE^1}(N(e))^{-\beta}m^\beta(r(e))
\end{align*}
for all $n\in\No$, and
\begin{equation*}
\sum_{v\in E^0}m^\beta(v)=\sum_{n=0}^\infty (1-\exp(\beta)/2)(\exp(\beta)/2)^n=1.
\end{equation*}
Thus $m^\beta\in\Cinf=\Cinfb$.
Let $\beta\in [0,\infty)$ and suppose $m\in \Cinf=\Cinfb$. Then
\begin{equation*}
m(v_n)=\sum_{e\in v_nE^1}(N(e))^{-\beta}m(r(e))=2\exp(-\beta)m(v_{n+1})
\end{equation*}
for all $n\in\No$. It follows that $m(v_n)=(\exp(\beta)/2)^nm(v_0)$ for all $n\in\No$. Since $$1=\sum_{v\in E^0}m(v)=\sum_{n=0}^\infty (\exp(\beta)/2)^nm(v_0),$$
it follows that $\beta\in [0,\ln 2)$ and that $m(v_0)=1-\exp(\beta)/2$, and thus that $m=m^\beta$.
Thus $\Cinf=\Cinfb=\{m^\beta\}$ for $\beta\in [0,\ln 2)$, and $\Cinf=\emptyset$ for $\beta\ge\ln 2$. Since $E^0_\reg=E^0$, the only KMS$_\beta$ states that descend to $C^*(E)$ are the ones corresponding to $m^\beta$ for $\beta\in [0,\ln 2)$. It may be of interest to observe that for each $k\in \No$, the sequence $\{m^\beta_{v_n}(v_k)\}_{n}$ converges to $m^\beta(v_k)$.
\end{example}
\begin{example}\label{ex:motivating}
In Example \ref{ex:dis} we presented an example where $\Cinf=\Cinfb\ne\emptyset$ for $\beta\in [0,\ln 2)$, and $\Cinf=\emptyset$ for $\beta\ge\ln 2$. We now present an example where $\Cinf=\Cinfa\ne\emptyset$ for $\beta=0$, $\Cinf=\Cinfb\ne\emptyset$ for $\beta\in (0,\ln 2)$, and $\Cinf=\emptyset$ for $\beta\ge\ln 2$.
Let $E$ be the graph with $E^0=\{v_n\mid n\in \No\}$ and $E^1=\{e_n\mid n\in \No\}\cup\{f_n\mid n\in \No\}$ where $r(e_n)=s(e_n)=v_n$ and $s(f_n)=v_n$ and $r(f_n)=v_{n+1}$ for $n\in \No$, see the picture:
\begin{equation}\label{graph-motivating}
\xymatrix{
{v_0} \ar@(ul,ur)[]|{e_0} \ar[r]^{f_0} &{v_1}\ar@(ul,ur)[]|{e_1} \ar[r]^{f_1} &{v_2}\ar@(ul,ur)[]|{e_2} \ar[r]^{f_2}&{\dots}
}
\end{equation}
Let $R=\emptyset$ and $N:E^1 \to (1,\infty)$ be $N(e)=\exp(1)$ for all $e\in E^1$. Thus we are dealing with $\mathcal{T}C^*(E)$ and its gauge action.
Fix $\beta\in [0,\infty)$. We have $v_nE_s^*v_n=\{e_n\}$ for all $n\in\No$. It follows that $$Z^s_{v_n}(-\beta)=\exp(-\beta);
$$
note in particular that this is independent of $v\in E^0$.
Assume now that $n>0$. Then $$\Es{v_{n-1}}{v_n}=\{e_{n-1}^kf_{n-1}\mid k\in\No\}$$ where $e_{n-1}^k$ is the path we get by concatenating $e_{n-1}$ with itself $k$ times. It follows that
$$
\sum_{u\in \Es{v_{n-1}}{v_n}}N(u)^{-\beta}=\sum_{k=0}^\infty(\exp(-\beta))^k
$$ diverges to infinity if $\beta=0$, and is convergent with sum $\exp(-\beta)/(1-\exp(-\beta))$ if $\beta>0$. Assume $\beta>0$ and let $a=\exp(-\beta)/(1-\exp(-\beta))$. If $k<n$, then $(u_1,u_2,\dots,u_{n-k})\mapsto u_1u_2\dots u_{n-k}$ is a bijection $$
\Es{v_k}{v_{k+1}}\times \Es{v_{k+1}}{v_{k+2}}\times\dots \Es{v_{n-1}}{v_{n}}\times\to
\Es{v_k}{v_{n}}.
$$
Hence $$\sum_{u\in \Es{v_k}{v_{n}}}N(u)^{-\beta}=a^{n-k},$$
and
\begin{equation}\label{eq-ex-Zaper}
Z^a_{v_n}(\beta)=\sum_{k=0}^n\sum_{u\in \Es{v_k}{v_{n}}}N(u)^{-\beta}=\sum_{k=0}^na^{n-k}<\infty.
\end{equation}
In conclusion, we have
\begin{align*}
\Ecrit&={\begin{cases}\{v_0\} &\text{ if } \beta=0\\ \emptyset &\text{ if }\beta>0,\end{cases}}\,\,\text{ and }\,\,
\Ereg={\begin{cases} \emptyset &\text{ if }\beta=0\\ E^0 &\text{ if }\beta>0. \end{cases}}
\end{align*}
For $\beta=0$, it follows from Proposition \ref{prop:m} that $m^0_{v_0}\in D^\beta$. According to Proposition \ref{prop:E}(4), every $m\in D^0$ must satisfy that $m(v_n)=0$ for $n>0$. Thus $D^0=\{m^0_{v_0}\}$.
Next we look at positive values of $\beta$. Fix therefore $\beta>0$. Since $\Ereg=E^0$, it follows from Theorem~\ref{thm:finitetype-reg} that $\Cf=\conv\{m^\beta_v\mid v\in E^0\}$. Suppose now that $m\in \Cinf$. Since $\Ecrit=\emptyset$, Theorem~\ref{thm:infinitetype-crit} shows that $\Cinfa=\emptyset$ for every
$\beta\in (0,\infty)$. Thus what is left in order to complete our analysis is to investigate existence of elements in $\Cinfb$. If
$m\in \Cinf$, Lemma~\ref{lem:aboutS}(b) implies that
$$
m(v_{n+1})=\frac{1-\exp(-\beta)}{\exp(-\beta)}m(v_n)
$$ for all $n\in\No$. It follows that $\beta<\ln 2$ because otherwise $\frac{1-\exp(-\beta)}{\exp(-\beta)}\ge 1$, which would imply that $\sum_{n=0}^\infty m(v_n)=\infty$.
We will show that for each $\beta\in (0,\ln 2)$ there is an element in $\Cinfb$. Given $\beta\in (0,\ln 2)$, we have $a=\exp(-\beta)/(1-\exp(-\beta))>1$, hence $\sum_{n=0}^\infty a^{-n}=\frac{a}{a-1}$. Notice that if $k<n$, then
\begin{align}
m^\beta_{v_n}(v_k)&=\sum_{u\in v_{k}E_a^*v_n}N(u)^{-\beta}(Z^a_{v_n}(\beta))^{-1}\notag\\
&=\frac{a^{n-k}}{\sum_{i=0}^na^{n-i}}\label{eq:m-vn-at-vk}\\
&=\frac{a^{-k}(a-1)}{a-a^{-n}}.\notag
\end{align}
On the other hand, $v_kE_a^*v_n=\emptyset$ for all $k>n$, and thus $m^\beta_{v_n}(v_k)=0$ if $k>n$.
By the proof of Proposition~\ref{prop:m}, $m^\beta_{v_n}(v_n)=(Z^a_{v_n}(\beta))^{-1}$. Hence we
see from \eqref{eq-ex-Zaper} that \eqref{eq:m-vn-at-vk} is valid for all $k=0, 1, \dots, n$, and we in fact have $
\sum_{k=0}^\infty m^\beta_{v_n}(v_k)=\sum_{k=0}^n m^\beta_{v_n}(v_k)=1$ for all $n\geq 0$.
We now define $m^\beta_{\operatorname{inf}}:E^0\to [0,1]$ by
$$
m^\beta_{\operatorname{inf}}(v_k)= a^{-(k+1)}(a-1)
$$
for all $k\geq 0$. Since $\sum_{v\in E^0}m^\beta_{\operatorname{inf}}(v)=\sum_{k=0}^\infty a^{-(k+1)}(a-1)=1$, the function
$m^\beta_{\operatorname{inf}}$ satisfies \eqref{item:m1}. Condition \eqref{item:m2} is vacuous, and \eqref{item:m3} is an equality
at all $v\in E^0$, as may be easily verified. Thus $m^\beta_{\operatorname{inf}}\in D^\beta$ and $S(m^\beta_{\operatorname{inf}})=0$. Hence by Lemma~\ref{lem:aboutS}, $m^\beta_{\operatorname{inf}}\in\Cinf$. That $m^\beta_{\operatorname{inf}}\in\Cinfb$ is seen because
the support of the measure associated to $m^\beta_{\operatorname{inf}}$ equals the path $x_0=f_0f_1\dots\in E^\infty$, which clearly is an element of $\Ewan$.
We claim that $\Cinfb=\{m^\beta_{\operatorname{inf}}\}$. This follows from the fact that any $m\in \Cinf$ will satisfy
$$
m(v_n)/m(v_{n+1})=m^\beta_{\operatorname{inf}}(v_n)/m^\beta_{\operatorname{inf}}(v_{n+1})
$$
for all $n\geq 0$, which together with the conditions
$\sum_{v\in E^0}m(v)=\sum_{v\in E^0} m^\beta_{\operatorname{inf}}(v)=1$ implies $m=m^\beta_{\operatorname{inf}}$.
It follows from Theorem \ref{thm:infty-states} that every ground state is a KMS$_\infty$ state.
We can summarize the preceding analysis in the following result.
\begin{theorem}\label{thm:exmotivating} Let $E$ be the graph described in \eqref{graph-motivating}. Consider $\mathcal{T}C^*(E)$ endowed with its gauge action. Then the KMS$_\beta$ states for $\beta\in [0,\infty)$ and the ground states of $\mathcal{T}C^*(E)$ are given as follows:
\begin{enumerate}
\item if $\beta=0$, then $D^\beta$ consists of the single conservative function $m_{v_0}^0$;
\item if $\beta\in (0,\ln 2)$, then $D^\beta=\Cf \sqcup \Cinfb$, where $\{m_{v_n}^\beta\}_{n\in \No}$ are all the extreme points of $\Cf$ and $\Cinfb$ consists of the single dissipative function $m^\beta_{\operatorname{inf}}$;
\item if $\beta\ge\ln 2$ we have $D^\beta=\Cf$, and the extremal KMS$_\beta$ states are $\{m_{v_n}^\beta\}_{n\in \No}$;
\item the extreme points of the set $D^{\operatorname{gr}}$ of ground states are $\{m_{v_n}^{\operatorname{gr}}\}_{n\in \No}$, with $m_{v_n}^{\operatorname{gr}}$ as given in \eqref{eq:ground-pointmass}, and every ground state is a KMS$_\infty$ state.
\end{enumerate}
\end{theorem}
Since $E^0_\reg=E^0$, the only KMS$_\beta$ and ground states that descend to $C^*(E)$ are the ones corresponding to $m_{v_0}^0$ and $m^\beta_{\operatorname{inf}}$, $\beta\in (0,\ln 2)$.
Note that we may describe the support of the measure associated to $m^\beta_{\operatorname{inf}}$ as an equivalence class for infinite paths, as follows. Given $x_1,x_2\in E^\infty$ we say that $x_1$ and $x_2$ are \emph{tail-equivalent} and write $x_1\sim_{\operatorname{tail}} x_2$ if there exist $x\in E^\infty$ and $u_1,u_2\in E^*$ such that $x_1=u_1x$ and $x_2=u_2x$. Thus the measure associated to $m^\beta_{\operatorname{inf}}$ has support $\{x\in E^\infty\mid x\sim_{\operatorname{tail}}x_0\}$.
Notice that similar to what we saw in Example \ref{ex:dis}, the sequence $\{m^\beta_{v_n}(v_k)\}_{n}$ converges to $m^\beta_{\operatorname{inf}}(v_k)$ for each $k\in \No$. This suggests that it may be possible in general to describe elements in $\Cinfb$ as pointwise limits, in appropriate sense, of elements in $\Cf$. Such a description for arbitrary graphs would be very interesting.
\end{example}
\begin{remark}
Notice that the graphs considered in Example \ref{ex:dis} and Example~\ref{ex:motivating} are not strongly connected. Klaus Thomsen has shown us an example of a strongly connected graph for which $\Cinfb$ is non-empty.
In the next example we will see that a small change in the graph of Example~\ref{ex:motivating} produces a (still not connected) graph where $\Ewan \neq\emptyset$ and yet $\Cinfb=\emptyset$. At the current stage we do not know what sort of additional information is needed in order to ensure that $\Cinfb$ is non-empty.
\end{remark}
\begin{example} \label{ex:b}
This example is a variation of Example~\ref{ex:motivating} where we add one more loop to $v_0$. The graph is given by:
\begin{equation}
\xymatrix{
{v_0} \ar@(ul,ur)[]|{e_0} \ar@(dr,dl)[]|{d_0} \ar[r]^{f_0} &{v_1}\ar@(ul,ur)[]|{e_1} \ar[r]^{f_1} &{v_2}\ar@(ul,ur)[]|{e_2} \ar[r]^{f_2}&{\dots}
}
\end{equation}
Let $R=\emptyset$ and $N:E^1\to (1,\infty)$ be $N(e_n)=N(f_n)=N(d_0)=\exp(1)$ for all $n\in \No$.
Proposition~\ref{prop:nice-graphs} implies that $D^\beta=\emptyset$ for $\beta<\ln 2$, $D^\beta=D^\beta_{\operatorname{inf}}$ when $\beta=\ln 2$, and that $D^\beta=\Cf=\conv\{m^\beta_v\mid v\in E^0\}$ for $\beta>\ln 2$.
Let $m\in D^{\ln 2}$. Repeated applications of Lemma~\ref{lem:mineq} show that in case $m(v_0)=0$, then $m(v_n)=0$ for all $n\geq 0$, a fact that would contradict \eqref{item:m1}. Thus $m(v_0)\neq 0$. Since $m(v_0)\ge m(v_0)+\frac{1}{2}m(v_1)$ by \eqref{item:m3}, we must have that $m(v_1)=0$. It thus follows from Lemma~\ref{lem:mineq} that $m(v_n)=0$ for all $n\geq 1$. Hence $D^{\ln 2}=\{m_{v_0}^{\ln 2}\}$.
Thus $\Cinfb=\emptyset$ for all $\beta$ although for example $x=f_0f_1f_2\dots $ is a wandering path.
\end{example}
\begin{example} \label{ex:c}
We briefly show how our analysis recovers the known results valid for the Cuntz algebra $\mathcal{O}_n$ and the Toeplitz-Cuntz algebra $\mathcal{TO}_n$ for $n\geq 2$. The graph in question has $E^0=\{v\}$ and $E^1=\{e_1, e_2,\dots, e_n\}$, where $s(e_i)=r(e_i)=v$ for all $i=1, \dots, n$. Thus we are dealing with a single vertex and $n$ loops of length one based at $v$, and in particular $E^0_{\operatorname{reg}}=\{v\}$. We let $N$ be the gauge action, thus $N(e_i)=\exp(1)$ for all $i$ and $N(v)=1$.
Let $R=\emptyset$. Proposition~\ref{prop:nice-graphs} implies that $D^\beta=\emptyset$ for $\beta<\ln n$, $D^\beta=D^\beta_{\operatorname{inf}}$ when $\beta=\ln n$, and $D^\beta=\Cf=\{m^\beta_v\}$ for $\beta>\ln n$. Since $E^0_{\ln n\text{-crit}}=\{v\}$, it follows that $D^{\ln n}=\{m^{\ln n}_v\}$.
Finally we see that $D^{\operatorname{gr}}=\{m^{\operatorname{gr}}_{v}\}$. It follows from Theorem \ref{thm:infty-states} that $m^{\operatorname{gr}}_{v}$ is a KMS$_\infty$ state.
If $R=E^0=\{v\}$, then $D^{\operatorname{gr}}=D^\beta=\emptyset$ unless $\beta=\ln n$, in which case $D^\beta=\{m^{\ln n}_v\}$.
\end{example}
\begin{example}\label{eq:stable-TOn}
Consider the graph with one countable ``straight line'' ending in a vertex $v_1$ that emits $n$ distinct loops:
\begin{equation}\label{graph-motivating2}
\xymatrix{
&{\dots}\ar[r]^{f_3} &{v_3} \ar[r]^{f_2} &{v_2} \ar[r]^{f_1} &{v_1}\ar@(ul,ur)[]|{e_1} \ar@(ur,dr)[]|{e_j} \ar@(dl,dr)[]|{e_n}
}
\end{equation}
We let $N:E^1\to (1,\infty)$ be given by $N(e)=\exp(1)$ for $e\in E^1$. Then
$$
v_kE_s^*v_k=\begin{cases}\{e_1, e_2,\dots, e_n\}&\text{ if }k=1\\
\emptyset &\text{ if }k\geq 2,
\end{cases}
$$
$E_a^*v_k=\{f_n\dots f_k\mid n\geq k\}\cup \{v_k\}$ for $k\geq 2$, and
$E_a^*v_1=\{f_n\dots f_1\mid n\geq 1\}\cup \{v_1\}\cup v_1E_s^*v_1$.
We next determine the partition functions associated to this graph. Given $\beta\in [0,\infty)$,
$$
Z^s_{v_1}(\beta)=\sum_{j=1}^n N(e_j)^{-\beta}= n\exp(-\beta)
$$
and
\begin{align*}
Z^a_{v_1}(\beta)
&=Z^s_{v_1}(\beta)+1+\sum_{u=f_n\dots f_1, n\geq 1}N(u)^{-\beta}\\
&= n\exp(-\beta) + 1 + \sum_{m\geq 0}(\exp(-\beta))^{m+1}.
\end{align*}
Hence $Z^a_{v_1}(\beta)<\infty$ if and only if $\beta>0$ and $Z^s_{v_1}(\beta)=1$ precisely when $\beta=\ln n$.
For $k\geq 2$, $Z^s_{v_k}(\beta)=0$ and $Z^a_{v_k}(\beta)=\sum_{m\geq 0}(\exp(-\beta))^{m+1}+1$. The sets of regular and critical points are listed in the following table:
\begin{table}[h]
\begin{center}
\begin{tabular}{c||c|c|c|c}
$\beta$ &0& $(0,\ln n)$ &$\ln n$ & $(\ln n, \infty)$\\
\hline \hline
$\Ereg$ & $\emptyset$&$E^0\setminus\{v_1\}$ & $E^0\setminus\{v_1\}$& $E^0$\\
\hline
$\Ecrit$& $\emptyset$&$\emptyset$& $\{v_1\}$& $\emptyset$
\end{tabular}
\end{center}
\end{table}
Every infinite path passes through $v_1$ infinitely many times so $\Ewan=\emptyset$ and $\Cinfb=\emptyset$ for all $\beta$. Hence we can characterize all the extremal KMS$_\beta$ states for $\mathcal{T}C^*(E)$: $D^0=\emptyset$, $D^\beta=D^\beta_{\operatorname{fin}}=\conv\{m_{v_k}\mid k\geq 2\}$ when $\beta\in (0,\ln n)$, $D^{\ln n}_{\operatorname{fin}}=\conv\{m_{v_k}\mid k\geq 2\}$, $D^{\ln n}_{\operatorname{inf}}=D^{\ln n}_{\operatorname{con}}=\{m^\beta_{v_1}\}$, and $D^\beta=\Cf=\{m_{v_k}\mid k\geq 1\}$ when $\beta\in (\ln n,\infty)$. Furthermore, $D^{\operatorname{gr}}=\conv\{m^{\operatorname{gr}}_{v_k}\mid k\le 1\}$, and every ground state is a KMS$_\infty$ state by Theorem \ref{thm:infty-states}.
Since $E^0_\reg=E^0$, the only KMS and ground states that descend to $C^*(E)$ are $m^{\ln n}_{v_1}$ and $m^{\operatorname{gr}}_{v_1}$.
We present the extremal KMS states by comparison with $\mathcal{O}_n$ and $\mathcal{TO}_n$:
\begin{table}[h]
\begin{center}
\begin{tabular}{c||c|c|c|c|c}
$\beta$ & $0$ &$(0,\ln n)$ & $\ln n$& $(\ln n, \infty)$&gr\\
\hline \hline
$\mathcal{T}C^*(E)$& $\emptyset$ & $\{m^\beta_{v_k}\mid k\geq 2\}$& $\{m^\beta_{v_k}\mid k\geq 1\}$& $\{m^\beta_{v_k}\mid k\geq 1\}$&$\{m^{\operatorname{gr}}_{v_k}\mid k\ge 1\}$\\
\hline
$C^*(E)$& $\emptyset$ & $\emptyset $& $\{m^\beta_{v_1}\}$& $\emptyset$ & $\{m^{\operatorname{gr}}_{v_1}\}$\\
\hline
$\mathcal{TO}_n$ & $\emptyset$ & $\emptyset$ & $\{m^\beta_v\}$ & $\{m^\beta_v\}$&$\{m^{\operatorname{gr}}_v\}$\\
\hline
$\mathcal{O}_n$ & $\emptyset$ & $\emptyset$ & $\{m^\beta_v\}$ & $\emptyset$&$\emptyset$
\end{tabular}
\end{center}
\end{table}
Note that the graph consisting of a countable straight line underlies the algebra $\mathcal{K}$ of compact operators on a separable
Hilbert space. In particular, $\mathcal{T}C^*(E)\cong \mathcal{TO}_n \otimes \mathcal{K}$.
\end{example}
\begin{example} \label{ex:d}
Finally we present an example of a ground state which is not a KMS$_\infty$ state.
Let $E^0=\{v\}$ and $E^1=\{e_1,e_2,\dots\}$. Then $E^0_\reg=\emptyset$ and $\mathcal{T}C^*(E)=C^*(E)=\mathcal{O}_\infty$. Let $N:E^1\to (1,\infty)$ be given by $N(e)=\exp(1)$ for $e\in E^1$. Clearly $\Eequ=\emptyset$ for all $\beta$ and $D^{\operatorname{gr}}=\{m_v^{\operatorname{gr}}\}$. It follows that there are no KMS states and that the ground state corresponding to $m_v^{\operatorname{gr}}$ is not a KMS$_\infty$ state.
\end{example}
| 188,900
|
TITLE: Probability of an even sum
QUESTION [0 upvotes]: In a set of numbers there are 5 even numbers and 4 odd numbers. If two numbers are chosen at random from the set, without replacement, what is the probability that the sum of these two numbers is even?
REPLY [1 votes]: I used the hint given by Ross Millikan answer.
P(the sum of the two numbers is even)=p(1st even and 2nd even)+p(1st odd and 2nd odd)
$$\implies p(even sum)=\frac{5}{9}\times\frac{4}{8}+\frac{4}{9}\times \frac{3}{8}=\frac{20}{72}+\frac{12}{72}=\frac{4}{9} $$
Therefore the probability that the sum of the two numbers is even is $\frac{4}{9}$.
| 22,258
|
BenQ Mobiuz EX3415R review: A solid if unadventurous ultrawide gaming monitor
Our Rating
Price when reviewed
900
inc VAT, RRP
Pros
Colourful wide gamut panel
Responsive and blur-free
Decent speakers
Cons
Low contrast and luminance
Iffy backlight uniformity
Too expensive
Advertisement
The BenQ Mobiuz EX3415R is hoping to capitalise on a growing fascination in the PC gaming community with ultrawide gaming monitors. Yes, 2,560 x 1,440 at 144Hz might still be king, but the extra screen real estate and immersive curvature is proving increasingly popular among PC gamers with a bit of cash to blow.
On paper, at least, the EX3415R has what it takes to make you consider parting with that cash. This 34in curved monitor is armed with the kinds of specifications most gamers are obsessed with; the only trouble is, so are the various other ultrawide gaming monitors on the market. The EX3415R is going to have to pull out all the stops to avoid being swallowed up by a tidal wave of similar products.
BenQ EX3415R review: What do you get for the money?
The BenQ EX3415R is a 34in ultrawide IPS gaming monitor with a gentle 1900R curve, a resolution of 3,440 x 1,440, a maximum refresh rate of 144Hz and a quoted response time of 2ms G2G.
You’ll notice the HDRi button in the right-hand corner of the screen: the EX3415R has two emulated HDR modes plus HDR10 decoding and an entry-level DisplayHDR 400 certification. In terms of adaptive sync, it officially supports AMD FreeSync Premium but I had no trouble setting up Nvidia G-Sync as well.
In terms of connectivity, the BenQ EX3415R has the usual two HDMI 2 ports, one DisplayPort 1.4 port, two downstream USB-A 3 ports, a single upstream USB-B 3 port and a 3.5mm headphone jack. Similarly bog-standard is the stand, which offers 100mm of height adjustment, 15 degrees of swivel left and right and 15 degrees of backwards tilt.
Decidedly less bog-standard, however, is the large grille on the rear of the panel that hides two 2W speakers plus a 5W woofer. The EX3415R also comes with a remote control that not only lets you adjust volume, switch sources and turn the monitor on and off, it also gives access to the full OSD controls.
READ NEXT: Get the edge with these top gaming mice
BenQ EX3415R review: What do we like about it?
The monitor’s speaker arrangement looks impressive on paper and it certainly pays off. I’d go as far as to say that the EX3415R’s audio output is good enough for casual movie viewing: it’s loud and weighty and, although the bass isn’t mind-blowing, the lack of it isn’t offputting. Incidentally, the 21:9 aspect ratio is great for movies.
I’m also fond of the remote. It makes navigating the onscreen display and even the simple job of switching the monitor off much easier than stabbing blindly at the buttons and joystick mounted below the panel.
As for the panel itself, it’s a complicated story, but there are definitely clear positives. The first is the wide gamut. Out of the box, in the default “Racing game” mode, the EX3415R produced 133.7% of the sRGB colour space, 94.7% of the DCI-P3 space and 92.1% of the Adobe RGB space. You can see the effects of this wide gamut panel as soon as you switch the thing on; your desktop will look noticeably more vibrant, although as I'll discuss later the effect is a little bit muted for a panel with upwards of 90% DCI-P3 coverage.
If you’d rather browse the web in the way most website developers intended, the EX3415R has a dedicated sRGB mode, among several other colour presets. In this mode, the EX3415R performed very well indeed, producing 92.4% of the sRGB space with an average Delta E colour variance score of just 0.82. This means any variation in colours is imperceptible. I found little use for any of the other presets, but if you prefer soft, muted colours to the overblown fare offered by many gaming monitors, the EX3415R’s sRGB mode delivers the goods.
While the other aspects of this monitor’s test results are less inspiring (more of which anon), it will reassure you to know that where actual gaming is concerned, the EX3415R performs well. I noticed virtually zero evidence of ghosting or motion blur, both anecdotally and when I tested the panel. Higher overdrive settings eventually produce a small amount of inverse ghosting but enabling Extreme Low Motion Blur (ELMB) helps mitigate this – I settled on the second of three levels of overdrive with ELMB active for the clearest possible image.
This means the EX3415R is a great choice if you’re into your shooters or other fast-paced games. Obviously, the 144Hz refresh rate helps here: my PC couldn’t quite push the limits at the monitor’s native 3,440 x 1,440 resolution but, nevertheless, the wide aspect ratio and fluid panel definitely aided my performance in FPS titles such as Star Wars Battlefront II and Borderlands 3. And, of course, a gentle curve on an ultrawide screen such as this never goes amiss.
BenQ EX3415R review: What could be better?
As I’ve already implied, the EX3415R falls flat in a few key areas. Brightness and contrast in both SDR and HDR modes leave a lot to be desired. In SDR, I measured a peak luminance of 200cd/m² and a contrast ratio of around 933:1. Both aren’t far from BenQ’s listed figures, but the low peak brightness in particular leaves SDR content looking distinctly dim.
The same is true of HDR content. Luminance topped out for me at 389cd/m² with a black point of 0.42cd/m² for a contrast ratio of 934:1 (identical to SDR, incidentally), so you can expect dark corners to look rather grey in HDR. There is one small upside to note: due to the poor peak brightness results in both SDR and HDR mode, you can comfortably leave this monitor in HDR for most of the time. Non-HDR content doesn't look awful – if anything, it feels as though you've simply turned the brightness of the backlight up. But that in itself is hardly something to be shouting about.
There’s also a bit of an issue here with colour accuracy outside of the monitor’s sRGB mode. Yes, the EX3415R’s wide gamut is capable of reproducing a large proportion of the DCI-P3 gamut (and Adobe RGB) but I was unable to find a colour preset that did so with an average colour variance (Delta E) score of less than two. Indeed, for many of the presets, the Delta E exceeded three when tested against sRGB, DCI-P3 and Adobe RGB. This isn’t the end of the world, per se – unless you happen to be a professional video editor – but I’d expect a bit more from a £900 monitor. The £500 Huawei MateView GT, a 3,440 x 1,440 34in monitor with a higher 165Hz refresh rate, is on the whole more accurate.
I should also note that backlight uniformity (of the model I was sent) isn’t great, by which I mean that the top-right corner of the panel doesn’t go quite as bright as the rest of the panel. You may not notice any deviation, but it’s worth mentioning because this monitor is soundly beaten again here by the MateView GT.
This brings me neatly to my final point: the BenQ Mobiuz EX3415R is simply too expensive at its list price of £900. MSI’s popular MPG Artymis 343CQR costs £900 and offers more for your money with a 1000R VA panel that has a higher refresh rate, peak luminance and peak contrast. Then there’s the aforementioned MateView GT, which crams a similar spec plus USB-C and a built-in soundbar into a product that costs £400 less than the EX3415R.
READ NEXT: Our favourite budget gaming monitors available now
BenQ EX3415R review: Should you buy it?
Fortunately, there’s already evidence that BenQ has reached the same conclusion about the price tag, since the EX3415R now dips as low as £840 at some retailers. This is a real saving grace: I don’t dislike the EX3415R, but I wouldn’t spend £900 on it.
Even with a discount, however, I’m not certain the EX3415R does quite enough to stand out. It launched not long after Huawei’s impressive and miraculously cheap MateView GT, a product that it simply cannot compete with in many areas. Indeed, I’d urge you to buy the MateView GT over the EX3415R if you value high contrast, high refresh rates and low price tags.
That said, the EX3415R doesn’t do anything so poorly as to warrant a negative verdict. Yes, it doesn’t stand out, but it’s certainly not a bad monitor. Its responsive IPS panel and better-than-average speaker arrangement work in its favour against the MateView GT, and we mustn’t forget that it still manages to pack in a lovely ultrawide resolution and a high refresh rate alongside some nice quality-of-life features. So I’ll simply say this: if you must buy it, be sure to do so at a discount.
| 358,376
|
The.
Dartmouth offers visitors and residents alike the charm of an old, east coast city and the nightlife of a major city - Halifax is minutes away by car, bus, or ferry!
| 129,496
|
Classical Numismatic Group, Inc.
Selling with CNG
Consignment to CNG Auctions
Our first auction was held on May 1, 1987. We have conducted more than 75 printed auctions since then, and in the process have built the world's most comprehensive mailing list (over 30,000 names) for collectors of ancient, world and British coins. In addition, we have conducted over 150 electronic auctions. More than 3,000 clients are registered to bid in our electronic auctions. Because we run both electronic and printed sales, we are equipped to handle a broad range of value.
CNG holds three major printed auctions each year. Our "Triton" public auction is held in New York in January, in conjunction with the New York International Numismatic Convention. Our two mail bid sales take place in May and September. For each printed auction, with an average of 1500-2000 lots, we produce a fully illustrated catalogue, which is a valuable reference tool as well as an opportunity to buy. Please refer to "News and Events" on this site for a complete list of future printed auction dates and consignment deadlines.
On this web site we hold continuous electronic auctions offering a broad range of material. Each sale lasts for two weeks and closes on a Wednesday. Through these electronic auctions we offer over 6000 lots of ancient, world and British each year.
We will be glad to discuss your possible consignment to public auction, mail bid sale, or electronic auction.
Direct Selling to CNG
We are constantly searching for coins to buy and would be pleased to hear from anyone, collector or dealer, who has ancient, medieval, world or British coins to sell.
If you are not sure whether it is better to sell through auction or for immediate cash, we can discuss the options with you and help you to arrange the most advantageous sale method to suit your circumstances.
We may be contacted by email, fax, phone or mail.
Classical Numismatic Group, Inc.
cng@cngcoins.com
Mailing addresses:
Attention: Victor England
P.O. Box 479
Lancaster PA 17608
Phone: 717-390-9194
Or
Attention: Eric J. McFadden
14 Old Bond St
London W1S 4PP
Phone: +44-20-7495-1888
CNG
2002-2006
Developed by DataArt 2006
| 334,436
|
TITLE: What is the use of orthogonal curvilinear co-ordinate system?
QUESTION [1 upvotes]: Orthogonal curvilinear coordinate system is made of intersecting sets of confocal ellipses & hyperbolas.
But, I have found no book that would describe this how to use this coordinate system and where to use. Only did I get the above quoted definition from A.P.French's Newtonian Mechanics.
Has it any uses in Newtonian Mechanics? How should I use this? Plz help.
REPLY [2 votes]: The co-ordinates you describe are only a special case of orthogonal curvilinear co-ordinates and are known, not surprisingly, as Elliptic Co-ordinates.
They were more useful in the days before widespread computing when analytical solutions of physical problems was more needed to help, e.g. visualise slightly eccentric systems, i.e. those nominally circular but slightly off-circular.
They are useful, for example, in solving Helmholtz's equation for an elliptical cross-section wave-guide.
Another physical interpretation is that the ellipses are the equipotential surfaces and the hyperbolas the field lines for an electrostatic field from a thin, charged plate stretching between $(-1,\,0)$ and $(1,\,0)$.
The Laplace equation for these co-ordinates is unchanged in form, which is a result of the next interesting property. These co-ordinates are special amongst orthogonal co-ordinates insofar that they can be visualised as the level surfaces for a holomorphic complex function - in this case $\Omega:\mathbb{C}\to\mathbb{C};\;\Omega(z)=\cosh z$. The level curves $\mathrm{Re}(\Omega(z))=\textrm{const}$ in the $z$-plane are the hyperbolas, whilst the curves $\mathrm{Im}(\Omega(z))=\textrm{const}$ are the ellipses. Note that such level curves for holomorphic functions are always orthogonal, but not every system of orthogonal families of curves are level curves of holomorphic functions. An interesting paper on this topic is:
Irl C. Bivens, "When Do Orthogonal Families of Curves Possess a Complex Potential?", Mathematics Magazine, Vol. 65, No. 4. (Oct., 1992), pp. 226-235
REPLY [0 votes]: Curvilinear coordinates are a coordinate system where the coordinate lines may be curved. A Cartesian coordinate system offers the unique advantage that all three unit vectors, x, y, and z, are constant in direction as well as in magnitude. Unfortunately, not all physical problems are well adapted to solution in Cartesian coordinates.
For instance, if we have a central force problem, such as gravitational or electrostatic force, Cartesian coordinates may be unusually inappropriate. Depending on the application, a curvilinear coordinate system may be simpler to use than the Cartesian coordinate system. For instance, a physical problem with spherical symmetry defined in R3 (for example, motion of particles under the influence of central forces) is usually easier to solve in spherical polar coordinates than in Cartesian coordinates.
The point is that the coordinate system should be chosen to fit the problem, to
exploit any constraint or symmetry present in it. Then, hopefully, it will be more
readily soluble than if we had forced it into a Cartesian framework.
| 52,293
|
Women in China References
Ahmadov, F. (November 27, 2017). “China’s Wealth Boom Is Creating a Generation of Self-Made Female Billionaries”. Forbes.
Attane, I. (2012). “Being a Woman in China Today: A Demography of Gender”. China Perspectives.
Barr, H. (November 3, 2019). “Bride Trafficking to China Spreads Across Asia”. Human Rights Watch.
Chang, M.-A. (December 19, 2019). “Taking First Place”. CKGSB Knowledge: China-focused leadership and business analysis.
Cheng, T.-p. (August 3, 2016). “The Rice Cooker Has Become a Test of China’s Ability to Fix Its Economy”. The Wall Street Journal.
China Power: Unpacking the complexity of China’s Rise. (June 25, 2018, Updated March 13, 2020). “Do women in China face greater inequality than women elsewhere?”. China Power: Unpacking the complexity of China’s Rise.
Clements, J. (2019). A Brief History of China. Singapore: Tuttle Publishing.
Cohen, J. A. (September 26, 2019). “Communist China’s Painful Human Rights Story”. Council on Foreign Relations.
Dillon, M. (2010). China a Modern History. London: L.B. Tauris & Co. Ltd.
Ebrey, P. B. (2010). China: Cambridge Illustrated History. New York: Cambridge University Press.
Evans, H. (2007). “The impossibility of gender in narratives of China’s modernity”. Radical Philosophy, RP 146, Nov/Dec.
Feldshuh, H. (n.d.). “Gender, Sexism, And Marriage Practice in Contemporary China: A Study of “Shegnu” (“Leftover Women:.
Feldshuh, H. (February 20, 2013, 2016). “Gender, Sexism, and Marriage Practice in Contemporary China: A Study of “Shugnu” (“Leftover Women”) in Popular Media”. deepblue.lib.umich.edu.
Frankopan, P. (2016). The Silk Roads: A New History of the World. New York: Alfred A. Knopf.
Fulton, J. (2000). “Holding Up Half of the Heavens: The Effect of Communist Rule on China’s Women”. Scholarworks, iu.edu.
Gao, H. (September 25, 2017). “How Did Women Fare in China’s Cultural Revolution?”. The New York Times.
Goldman, J. F. (2006). China a New History. Cambridge: The Belknap Press Of Harvard University Press.
Gu, M. (December 17, 2019). “Education in China”. WENR: World Education News & Reviews.
Guluzade, A. (07 May 2019). “Explained, the role of China’s state-owned companies”. World Economic Forum.
Hornby, L. (March 3, 2016). “China comes full circle with talks of mass-layoffs”. Financial Times.
Hsu, I. C. (2000). The Rise of Modern China. Oxford: Oxford University Press.
Huang, B. M. (August 2017). “Patriarchal Capitalism with Chinese characteristics: gendered discourse of ‘Double Eleven’ shopping festival”. The London School of Economics & Political Science (Online).
Human Rights Watch. (2019). “World Report 2019: China”. Human Rights Watch.
Huynh, T. (27 March 2020). “From Mei Lanfang to Li Yugang: theorizing female-impersonating aesthetics in post-1976 China”. Springer Link.
Internatiional Monetary Fund. (March 30, 2019). “Women and Growth”. International Monetary Fund.
International Monetary Fund. (March 30, 2019). “Women and Growth”. International Monetary Fund.
Johnson, P. (1983). A History of the Modern World. London: Weidenfeld and Nicholson.
Jonathan Woetzel, e. a. (April 2018). “The Power of Parity: Advancing Women’s Equality in Asia Pacific”. McKinsey Global Institute.
Kurlantzick, J. (2007). “Charm Offensive: How China’s Soft Power Is Transforming the World”. New Haven: Yale University Press (A New Republic b.
Kurlantzick, J. (2007). “Charm Offensive: How China’s Soft Power Is Transforming the World”. New Haven: Yale University Press (A New Republic Book).
Lahiri, D. K. (December 17, 2018). “The charts that show how Deng Xiaping unleashed China’s pent-up capitalist energy in 1978”. Quartz.
Li, Y. (January 2000). “Women’s Movement and Change of Women’s Status in China”. Journal of International Women’s Studies, Volume 1, Issue 1, Article 3 (Bridgewater State University).
Minter, A. (January 8, 2020). “Women Didn’t Cause China’s Pension Crisis”. Bloomberg.
Minter, A. (January 8, 2020). “Women Didn’t Cause China’s Pension Crisis”. Bloomberg.
Nelian Haspels, e. a. (2001). “Action against Sexual Harassment at Work in Asia and the Pacific”. Bangkok: International Labor Office.
Nuwer, R. (September 19, 2013). “China Has More Self-Made Female Billionaires Than Any Other Country”. Smithsonian Magazine.
Pence, J. D. (1990). The Search for Modern China. New York: WW Norton and Company, Inc.
Prof. Yi Zeng, P. &. (2018 May 10). “The effects of China’s two-child policy”. NCBI Resources/PMC: U.S. National Library of Medicine/National Institutes of Health.
q, A. (n.d.). “A Prosperous China Says ‘Men Preferred,’ and Women Lose”. The New York Times.
Qin, A. (July 16, 2019). “A Prosperous China Say ‘Men Preferred,’ and Women Lose”. The New York Times.
Qin, A. (July 16, 2019). “A Prosperous China Says ‘Men Preferred,” and Women Lose”. The New York Times.
Qiushi Feng, e. a. (13 February 2018). “Age of Retirement and Human Capital in an Aging China, 2015-2050”. Springer Link.
Quan, H. (08 June 2019). “The representation and/or repression of Chinese women: from a socialist aesthetics to commodity fetish”. Springer Link.
Quanbao Jiang, e. a. (2013 November 17). “Estimate of Missing Women in Twentieth-Century China”. NCBI Resources/PMC: U.S. National Library of Medicine/National Institutes of Health.
Quanbao Jiang, e. a. (Available in PMC 2015 November 24). “China’s Population Policy at the Ccrossroads: Social Impacts and Prospectives”. Asian Journal of Social Science.
Roser, H. R. (June 2019). “Gender Ratio”. Our World in Data.
Ross Garnaut, e. a. (2018). “China’s 40 years of reform and development, 1978-2018”. Acton: ANU Press.
Ruth A. Shaprio, e. a. (2018). “Pragmatic Philosophy: Asian Charity Explained”. Singapore: Palgrave Macmillan, Spring Nature.
Schuman, M. (March 30, 2017). “With CEO Paul Fang, The World’s Largest Appliance Company Transforms Its Business”. Forbes.
See, L. (2006). Snow Flower and the Secret Fan. London: Bloomsbury Publishing.
Sri Wening Handayani, E. (. (2016). “Social Protection for Informal Workers in Asia”. Cornell University ILR School.
Stauffer, B. (April 23, 2018). “Only Men Need Apply:” Gender Discrimination Advertisements in China”. Human Rights Watch.
Steven Lee Meyers, e. a. (January 17, 2020). “China’s Looming Crisis: A Shrinking Population”. The New York Times.
Tao, L. (2006). “Education as a Social Ladder”. The New York Times.
The New York Times. (November 16, 2019). “Docment: What Chinese Officials Told Children Whose Families Were Put into Camps”. The New York Times.
The New York Times. (Novembet 16, 2019). “Document: What Chinese Officials Told Children Whose Famiies Were Put into Camps”. The New York Times.
The United Nations Entity for Gender Equity and the Empowerment of Women. (2019). “Progress of the World’s Women 2019-2020: Families in a Changing World”. New York: U.N. Women.
University of Delaware. (November 28, 2018). “View of ideal female appearance in China are changing”. Phys.Org.
Wang, P. (October 11, 2016). “Chinese dating shows are changing traditional views on love and marriage”. Quartz.
Withers, C. P. (2015). “Boss Lady: How Female Chinese Managers Succeed in China’s Guanxi System”. University of Mississippi Honors Thesis/eGrove (Sally McDonnell Barksdale Honors College).
Xu, Z. (May 01, 2013). “The Political Economy of Decollectivization in China”. Monthly Review: An Independent Socialist Magazine.
Xue, M. M. (Date Written: November 1, 2016; Last Revied, 10 December 2016). “High-Value Work and the Rise of Women in China: The Cotton Revolution and Gender Equality in China”. Elsevier/SSRN, 90 pages.
Yi, B. L. (June 25, 2019). “Southeast Asia urged to improve women’s rights to stop China bride trafficking”. Reuters.
Yuval Atsmon, e. a. (March 2012). “Meet the 2020 Chinese Consumer”. McKinsey Insights: China.
Zhanhong Zong, e. a. (2017 September 12). “Migration Experiences and Reported Sexual Behavior Among Young, Unmarried Female Migrants in Changzhou, China”. Global Health Science Practices.
Zheng, D. K. (2007). Translating Feminisms in China. Malden: Blackwell Publishing Ltd.
Zheng, W. (2010). “Creating a Socialist Feminist Cultural Front: ‘Women of China’ (1949-1966)”. The China Quarterly, No. 204, Available on JSTOR Online, pp. 827-849.
Zheng, W. (December 2010). “Creating a Socialist Feminist Cultural Front: ‘Women of China’ (194901966)”. The China Quarterly, pp. 827-849.
Zhuhai. (May 16, 2019). “How China forged self-made female billionaires”. The Economist.
| 246,307
|
Search Results
Best Mobile App Development - GSBitLabs
Melbourne VIC, Australiadistance: 9,003 Miles 0450914478
Mobile app designing and developing from one of the best corporation in Australia. Which Provide Android, iPhone & iPad Application, web page design fully with UI/UX.
Xiaomi Tech Geek
Unknown
An Exclusive Platform of xiaomi fans based site focusing on Redmi, Mi, and other Xiaomi products latest news, updates, tech geek stories and resources.
Concetto Labs
4811 N Harding Ave, Chicago, IL, United Statesdistance: 6,101 Miles (646) 318-9929
Concetto Labs is a trusted website development and mobile application company from India and USA.
_4<<
JSM InfoTech - IT Company Lucknow
Lucknow, Uttar Pradesh, Indiadistance: 5,660 Miles +91 9044391571
JSM InfoTech – IT Company Lucknow
Mobile Application Design and Development Company in Santa Clara, California - AC Infotech
Unknown
Ac Infotech Inc.,is a leading IT services and consulting company specializing in business intelligence, web portal, mobile application design & development.
| 371,291
|
Fascination About Radiant HeatingIt's not geothermal Electrical power in the sense that it doesn't use the Strength in the (core on the) Earth, but fairly geosolar energy, the Strength absorbed seasonally because of the surface of the earth.
But all of this is naturally just a situation of Very small Specifics Exaggeration Syndrome – to even possess a household in the slightest degree, let alone a single with computerized wonderful heating of ANY type, is a reasonably great addition to lifestyle.
If you need small heating fees, you need to be sure your BTU warmth reduction is reduced along with the efficiency of the heating product is extremely significant.
My very own home has radiant heat and it’s deluxe. The completed basement of my dwelling has radiant tubing that snakes from the concrete slab.
Spring is listed here and it's obtaining warmer! Established out a daybed or amongst our other finds, As well as in a brief time you'll need...
This is often an great experiment, and one that you will have determined me to test – though I am admittedly fewer experienced with my palms than that you are. “Building it had been a bitch” will be, certainly, my biggest problem! Saving $8k? Not negative. Everything counts.
Within the constructive facet, I discovered that if you run warm water in the event the pump is off, water is drawn with the process as a result of natural pressure differences. Which means that in the summer, my floors will really be cooled down from the cold water supply because it sucks undesirable warmth from the house.
Regardless of the lack of windows and insulation, I was currently hunting forward with nerdy engineering glee to creating a house-brewed heating method for this location, and I explained to you about it within the write-up called The Radiant Heat Experiment.
Never target the radiant flooring heating program only. Also pay attention to the overall insulation of the whole creating.
Awesome! I’ll be following in your footsteps (kneesteps?) shortly. We at the moment Have got a large horrible furnace and An additional major unpleasant sizzling h2o heater from the crawlspace. I’ve been radiant heat dreading the day when both of Those people decides to get early retirement.
Radiant flooring programs are excellent AND EASY when in new building: below they just pour the concrete slab, wind the piping more than it at the time dry, then fill during the Areas with some fluffy insulating things, pour concrete about, then lay around the tile or wood floor or whichever.
less difficult for those who installed it within an unfinished basement rather than a crawlspace. Also, recruiting as a lot of close friends as you can to thread the pipe will pace you up exponentially. General, I’d recommend it only for skilled handypeople.
Once the boiler is on the market, the domestic hot water may be heated making use of an oblique drinking water heater, pools and spas may be heated employing exchangers, and distant parts exactly where typical devices are hard to install may also be heated. In places like Ottawa, radiant snow melting programs may be used to maintain walkways, ramps of driveways clear of ice and snow.
should you Consider more recent boilers/h2o heaters convert-down ratios/portion load efficiencies are Great. In particular, condensing boilers are provided with large turn-down modulating burners.
| 16,314
|
Copyright © 2017 by Federal News Radio. All rights reserved. This website is not intended for users located within the European Economic Area.
Robert J. Carey Chief Information Officer September 29th and October 1st, 2008
Vice Admiral Adam M. Robinson, Jr. Surgeon General, Chief, Bureau of Medicine and Surgery November 10th and 12th, 2008
Several contracts on way to help consolidate, standardize
Several contracts on way to help consolidate, standardize
New guidance offers steps agencies can take address risks in cell phones, PDAs
Many feds are gearing up for the Presidential Inauguration later this month, and the Sea Chanters of the U.S. Navy Band are no exception. Chief Musician Georgina Todd is Leading Chief Petty Officer for the…
Vice Admiral Adam M. Robinson, Jr. Surgeon General, Chief, Bureau of Medicine and Surgery February 9th and 11th, 2009
The pirate problem around the world has come center stage in the United States with the capture of the Maersk Alabama. While the FBI continues negotiation with the pirates holding the ship’s captain hostage, the…
Sniffing the air for anthrax has never been more high-tech. Scientists at the U.S. Naval Research Laboratory have come up with sensors that can diagnose infectious diseases, track airborne toxins and even detect explosives in…
| 10,437
|
By Muhaimin Olowoporoku
Grind.
This title makes me very happy, here’s hoping for a brighter future for Nigeria’s LGBTQ community
| 131,798
|
Additional Images
Ladies Tan Waxy w/ Crystals
Get ready to impress in this fierce brown boot with teal and gold 8 row quality stitching. Mariella is simple and charming with a bit a pizzaz. this boot is ready for whatever adventures await you! All over leather Nine row embroidery Goodyear-Welt Construction Single row out sole stitching Nylex linning Eco friendly molded insole Shaft Height - 11" Shaft Size - 13.5" Heel Height - 35mm
| 232,255
|
...
2011 in Review - Gold Coast Titans
>
| 375,685
|
below or here:.
TRUE WIDOW’s fourth album Avvolgere is due out this Friday, September 23 on CD/2xLP/DLX 2xLP/Digital via Relapse Records.Digital and physical pre-orders are available at this location.
The forthcoming LP perfects the formula that 2013’s Circumambulation establ
The band have also announced their first round of North American tour dates in support of the ten-song album, with the tour launching on Oct. 13 in Phoenix:
TRUE WIDOW Live:
True Widow is:
D.H. Phillips (guitar/steel guitar/vocals)
Nicole Estill (bass/acoustic guitar/vocals)
Slim TX (drums/piano)
Official links:
True Widow on Facebook
True Widow on Bandcamp
True Widow on Instagram
True Widow on Twitter
| 78,984
|
Break That Block!
By Annette Rey
Writers’ block bothering you? Stop the excuses. Stop the helter-skelter approach you may have to your writing.
It is key to HAVE A PLAN.
Begin with the following example plan.
Write a FLASH-FICTION-A-DAY
- limit them to 50 words
Looking from a retrospective view at a bundle of material you wrote a year ago, brings a whole new way of thinking about what you wrote and can be the catalyst to creating new material.
And think of what you could do with a collection of over 300 flashes at the end of a year. You could:
- create an eBook of your finest pieces
- build short stories from them
- submit an article based on one of your flashes
- enter contests
This is an easy and short way to become a prolific writer.
Or create your own Plan!
Advertisements
| 385,159
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.