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\begin{document}
\title{Competition between stable equilibria in reaction-diffusion systems: the influence of mobility on dominance} \thispagestyle{empty}
\begin{abstract} This paper is concerned with reaction-diffusion systems of two symmetric species in spatial dimension one, having two stable symmetric equilibria connected by a symmetric standing front. The first order variation of the speed of this front when the symmetry is broken through a small perturbation of the diffusion coefficients is computed. This elementary computation relates to the question, arising from population dynamics, of the influence of mobility on dominance, in reaction-diffusion systems modelling the interaction of two competing species. It is applied to two examples. First a toy example, where it is shown that, depending on the value of a parameter, an increase of the mobility of one of the species may be either advantageous or disadvantageous for this species. Then the Lotka--Volterra competition model, in the bistable regime close to the onset of bistability, where it is shown that an increase of mobility is advantageous. Geometric interpretations of these results are given. \end{abstract}
\section{Introduction} Reaction-diffusion systems play an important role as models for a large variety of spatio-temporal systems arising from various fields: Chemistry, Physics, Mechanics, Genetics, Ecology\dots. An relevant concept for the understanding of their dynamical behaviour is the dominance of equilibria, \cite{Fife_reactDiffSyst_1979}: given two stable homogeneous equilibria, in which sense can one say that an equilibrium ``dominates'' the other one~? A possible answer is (see for instance \cite{HutsonVickers_travWaveDom_1992}): equilibrium $A$ dominates equilibrium $B$ if there exists a travelling front connecting these two equilibria and displaying invasion of $B$ by $A$ (even if the order relation induced by this definition is not always antisymmetric, see the example in appendix, \vref{subsec:ex_2_eq}).
A natural related question is that of the influence of mobility on dominance: how is the speed of a front connecting two stable equilibria (and in particular the sign of this speed) affected by a change in the diffusion coefficients~? This question is relevant in the context of population dynamics. Consider a system modelling the evolution of densities of two species competing in a one-dimensional environment. In this case one expects the existence of two stable equilibria, each corresponding to the dominance of a species, for the local reaction system. The question above is that of the influence of the mobility of each of the two species on their relative fitness, that is on the sign of the speed of a front connecting these equilibria.
One may believe that less mobility is always advantageous, since it reduces the dispersal at the interface where the two species coexist, and thus prevents invasion (see the observations in \cite{HutsonVickers_travWaveDom_1992}). But other effects can be invoked: an increase in the mobility of, say, the first species, changes the total density of individuals on each side of the interface, and results in undercrowding on the side where first species dominates and overcrowding on the other side, an effect having unclear consequences. While according to results of A. Hastings \cite{Hastings_canSpatialVariationAloneLeadToSelectionForDispersal_1983} and J. Dockery et al. \cite{DockeryHutson_evolutionSlowDispersalRates_1998} a heterogeneous environment seems to be always in favour of a reduction of dispersal, V. Hutson and G. T. Vickers made on a model the numerical observation that large or small diffusion cannot unambiguously be claimed to be favourable in general, \cite{HutsonVickers_titForTatDefect_1995}. More recently, L. Girardin and G. Nadin considered a Lotka--Volterra competition model close to the infinite competition limit and proved in this case a ``Unity is not strength'' result stating that a large dispersal is favoured, \cite{GirardinNadin_tWRelativeMotilityInvasionSpeed_2015}.
The aim of this paper is to examine on some cases the value of the first order dependence of the speed of a bistable front with respect to a perturbation of the diffusion matrix, and to try to determine the sign of this quantity. First we consider a general reaction-diffusion system in spatial dimension one, governing two symmetric scalar components, and assume the existence of two stable homogeneous equilibria that are symmetric (with respect to exchange of the two components) and connected by a symmetric (thus stationary) front. Then the symmetry between the two scalar components is broken by a small perturbation of the diffusion matrix (say a small increase of the diffusion coefficient of the first component) and several expressions are provided for the first order dependence of the speed of the front with respect to this perturbation (\cref{sec:general}), together with a geometric interpretation for some of these expressions. All this suggests that both signs may occur for this first order dependence, depending on the features of the initial system.
Two specific examples are then considered. First (\cref{sec:toy}) a toy example where the initial standing front is explicit, and where it is shown that both signs (for the first order dependence introduced above) actually occur, depending on the value of a parameter of the system. This confirms on a computable case the aforementioned observations of Hutson and Vickers. The second example (treated in \cref{sec:lv}) is the Lotka--Volterra competition model in the bistable regime, close to the onset of bistability. Using singular perturbation arguments, it is shown in this case that a large dispersal is advantageous.
\section{Assumptions, notation, perturbation scheme} \label{sec:general}
\subsection{Setup} \label{subsec:setup}
Let us consider the reaction-diffusion system: \begin{equation} \label{react_diff} u_t = F(u)+\ddd u_{xx} \end{equation} where the time variable $t$ and the space variable $x$ are real, space domain is the full real line, the field variable $u$ is $n$-dimensional ($n$ is a positive integer), the ``reaction'' function $F:\rr^n\rightarrow\rr^n$ is smooth, and the ``diffusion'' matrix $\ddd$ is a positive definite symmetric $n\times n$ real matrix. Let us assume that this system admits two distinct spatially homogeneous equilibria, in other words that there exist two points $E_-$ and $E_+$ in $\rr^n$ such that \[ E_-\not= E_+ \quad\mbox{and}\quad F(E_-)=F(E_+)=0 \] and let us assume that there exists a travelling front connecting these two equilibria, in other words that there exist a smooth function $\phi:\rr\rightarrow\rr^n$ and a real quantity $c$ such that the function $(x,t)\mapsto\phi(x-c t)$ is a solution of system \cref{react_diff} and such that \[ \phi(\xi)\rightarrow E_- \quad\mbox{when}\quad \xi\rightarrow-\infty \quad\mbox{and}\quad \phi(\xi)\rightarrow E_+ \quad\mbox{when}\quad \xi\rightarrow+\infty \,. \] This function $\phi$ is a global solution of the system: \begin{equation} \label{syst_front} -c\phi'(\xi)=F\bigl(\phi(\xi)\bigr)+\ddd\phi''(\xi) \,. \end{equation} Let us assume in addition that both equilibria $E_-$ and $E_+$ are hyperbolic (that is the linear functions $DF_{E_+}$ and $DF_{E_-}$ have no eigenvalue with zero real part). In this case both functions $\xi\mapsto F\bigl(\phi(\xi)\bigr)$ and $\xi\mapsto \phi'(\xi)$ approach $0_{\rr^n}$ at an exponential rate when $\xi$ approaches $\pm\infty$, and as a consequence these functions belong to the space $L^2(\rr,\rr^n)$. Let us denote by `` $\cdot$ '' the canonical scalar product in $\rr^n$, and let $\langle \cdot , \cdot \rangle_{L^2(\rr,\rr^n)}$ and $\norm{\cdot}_{L^2(\rr,\rr^n)}$ denote the usual scalar product and corresponding norm in $L^2(\rr,\rr^n)$, namely (for every pair $(f,g)$ of functions of $L^2(\rr,\rr^n)$): \[ \langle f , g \rangle_{L^2(\rr,\rr^n)} = \int_{-\infty}^{+\infty} f(x) \cdot g(x) \, dx \quad\mbox{and}\quad \norm{f}_{L^2(\rr,\rr^n)} = \sqrt{\langle f , f \rangle_{L^2(\rr,\rr^n)}} \,. \] Now, it follows from system \cref{syst_front} that the quantity $c$ admits the following explicit expression: \begin{equation} \label{expr_c0} c = -\frac{\int_{-\infty}^{+\infty} F\bigl(\phi(\xi)\bigr)\cdot\phi'(\xi)\,d\xi} {\int_{-\infty}^{+\infty}\phi'^2(\xi)\,d\xi} = -\frac{\langle F(\phi) , \phi' \rangle_{L^2(\rr,\rr^n)}}{\norm{\phi'}_{L^2(\rr,\rr^n)}^2} \,. \end{equation}
If the reaction function $F$ derives from a potential $V:\rr^n\rightarrow\rr$ (namely if $F(u)=-\nabla V (u)$ for all $u$ in $\rr^n$) then this expression of $c$ becomes: \begin{equation} \label{expr_c0_pot} c=\frac{V(E_+)-V(E_-)}{\norm{\phi'}_{L^2(\rr,\rr^n)}^2} \,. \end{equation} Thus, in this case, the sign of the speed $c$ of the front only depends on the sign of the difference between $V(E_+)$ and $V(E_-)$. In particular, if there exist several travelling fronts connecting $E_-$ to $E_+$, then all the velocities of these fronts have the same sign. Such is not always the case when $F$ does not derive from a potential: it is not difficult to construct an example of system of the form \cref{react_diff} where two distinct equilibria are connected by two travelling fronts with velocities of opposite signs (for sake of completeness such an example is given in appendix, see \vref{subsec:ex_2_eq}).
In the following we shall not assume that $F$ derives from a potential. Our aim is to understand the influence of a small change in the diffusion matrix $\ddd$ on the speed $c$ of the travelling front $\phi$.
\subsection{Stability and transversality assumptions} \label{subsec:transv_assump}
Let us introduce the space coordinate $\xi=x-ct$ in a frame travelling at speed $c$. If two functions $u(x,t)$ and $v(\xi,t)$ are related by: \[ u(x,t) = v(\xi,t)= v(x-ct,t) \,, \] then $u$ is a solution of system \cref{react_diff} if and only if $v$ is a solution of: \begin{equation} \label{react_diff_trav_frame} v_t = c v_\xi + F(v) + \ddd v_{\xi\xi} \,, \end{equation} which represents system \cref{react_diff} rewritten in the $(\xi,t)$ coordinates system. The profile $\xi\mapsto \phi(\xi)$ of the travelling front considered in \cref{subsec:setup} is a steady state of system \cref{react_diff_trav_frame}. A small perturbation \[ (\xi,t)\mapsto \phi(\xi) + \varepsilon v(\xi,t) \] of the profile of the front is (at first order in $\varepsilon$) a solution of \cref{react_diff_trav_frame} if and only if $v$ is a solution of the linearised system: \begin{equation} \label{react_diff_trav_frame_lin} v_t = c v_\xi + DF(\phi) v + \ddd v_{\xi\xi} \,. \end{equation} The right-hand side of \cref{react_diff_trav_frame_lin} defines the differential operator \begin{equation} \label{def_linearised_operator} \mathcal{L}:c\partial_\xi+ DF(\phi) + \ddd\partial_{\xi\xi} \,. \end{equation} Considered as an unbounded operator in $L^2(\rr,\rr^n)$, it is a closed operator with dense domain $H^2(\rr,\rr^n)$. Due to translation invariance in the space variable $x$, zero is an eigenvalue of this operator; indeed, differentiating system~\cref{syst_front} yields: \[ \mathcal{L}\phi'=0 \,. \] Let us make the following hypotheses. \begin{description} \item[\hypStabInfty] The spatially homogeneous equilibria $E_-$ and $E_+$ at both ends of the front are spectrally stable for the reaction-diffusion system \cref{react_diff}. \end{description} In other words, For every real quantity $k$, all eigenvalues of the $n\times n$ real matrices \[ DF(E_-)-k^2\ddd \quad\mbox{and}\quad DF(E_+)-k^2\ddd \] have negative real parts (the subscript ``stab-ends'' refers to: ``stable at both ends of space''). Equivalently, the essential spectrum of operator $\mathcal{L}$ is stable \cite{Henry_geomSemilinParab_1981,Sandstede_stabilityTW_2002}. \begin{description} \item[\hypTransv] The eigenvalue zero of the operator $\mathcal{L}$ has an algebraic multiplicity equal to $1$. \end{description} In other words, the kernel of operator $\mathcal{L}$ is reduced to $\spanset(\phi')$, and the function $\phi'$ does not belong to $\imm(\mathcal{L})$. The subscript ``transv'' refers to: ``transverse''; indeed, this hypothesis is equivalent to the transversality of the travelling front (see \vref{lem:alg_mult_transv}).
The two next definitions call upon a topology on the space of travelling fronts, which may be chosen as follows: two travelling fronts $\phi_1$ and $\phi_2$ travelling at speeds $c_1$ and $c_2$ are close if: there exists a translate of $\phi_2$ that is close to $\phi_1$ (uniformly on $\rr$), and the two speeds $c_1$ and $c_2$ are close.
\begin{definition}[isolation and robustness of the travelling front] The travelling front $\phi$ is said to be \emph{isolated} if there exists a neighbourhood of it such that every other travelling front of the same system \cref{react_diff} in this neighbourhood is equal to a translate of $\phi$ (and as a consequence travels at the same speed).
The travelling front $\phi$ is said to be \emph{robust} if every sufficiently small perturbation of system \cref{react_diff} possesses a unique (up to space translation) front close to $\phi$ and travelling with a speed close to $c$. \end{definition}
The following statement is a rather standard transversality result \cite{CoulletRieraTresser_stableStaticLocStructOneDim_2000,Coullet_locPattFronts_2002, Sandstede_stabilityTW_2002,HomburgSandstede_homocHeteroclinicBifVectFields_2010, GuckenheimerKrauskopf_invManifGlobalBif_2015}.
\begin{proposition}[isolation and robustness of $\phi$] \label{prop:robustness} It follows from hypotheses \hypStabInfty and \hypTransv that the travelling front $\phi$ under consideration is isolated and robust. \end{proposition}
For sake of completeness a proof of this proposition is provided in \vref{subsec:robustness}.
\subsection{Spectral stability} \label{subsec:spec_stab_gen}
The travelling front $\phi$ is said to be \emph{spectrally stable} if hypotheses \hypStabInfty and \hypTransv are satisfied, and if moreover every nonzero eigenvalue of $\mathcal{L}$ has a negative real part. In this case the travelling front is then also \emph{non linearly stable with asymptotic phase}, that is for every function $u_0:\rr\rightarrow\rr^n$ sufficiently close (say uniformly on $\rr$) to a translate of $\phi$, there exists a real quantity $x_1$ such that the solution of system \cref{react_diff} with initial condition $u_0$ approaches the solution $(x,t)\mapsto\phi(x-x_1-c t)$ (at an exponential rate) when $t$ approaches $+\infty$ \cite{Henry_geomSemilinParab_1981,Sandstede_stabilityTW_2002}.
In the two practical examples that will be considered in \cref{sec:toy} and \cref{sec:lv}, the fronts under consideration will be spectrally stable indeed. However, we shall not make any additional spectral stability hypothesis at this stage since such an hypothesis is not required for the general considerations that will be made in the next \cref{subsec:solvency,subsec:altern_expr_bar_c,subsec:with_symmetries,subsec:geom_interpret_bar_c,subsec:red_sym}.
\subsection{Kernel of the adjoint linearised operator} \label{subsec:ker_adjoint}
Let $\mathcal{L}^*$ denote the adjoint operator of $\mathcal{L}$ for the scalar product $\langle . , .\rangle_{L^2(\rr,\rr^n)}$, that is: \[ \mathcal{L}^* = -c\partial_\xi+DF(\phi)^* + \ddd\partial_{\xi\xi} \] (see \cref{fig:notation_operators}).
\begin{figure}
\caption{Notation related to the operators $\mathcal{L}$ and $\mathcal{L}^*$.}
\label{fig:notation_operators}
\end{figure}
Hypotheses \hypStabInfty and \hypTransv ensure that $\ker(\mathcal{L}^*)$ is also one-dimensional \cite{SandstedeScheel_stabTWLargeSpatialPeriod_2001,Sandstede_stabilityTW_2002}, and according to \hypTransv the subspaces $\ker(\mathcal{L})$ and $\ker(\mathcal{L}^*)$ are not orthogonal to one another. As a consequence there exists a unique function $\psi$ in $\ker(\mathcal{L}^*)$, satisfying the normalization condition \begin{equation} \label{norm_cond} \langle\psi,\phi'\rangle_{L^2(\rr,\rr^n)}=1 \,, \end{equation} and $\psi(x)$ approaches $0_{\rr^n}$ at an exponential rate when $x$ approaches $\pm\infty$ \cite{SandstedeScheel_stabTWLargeSpatialPeriod_2001,Sandstede_stabilityTW_2002}.
\subsection{Perturbation of the diffusion matrix and solvency condition} \label{subsec:solvency}
Let us consider a symmetric (not necessarily positive definite) $n\times n$ real matrix $\bar{\ddd}$, a small positive quantity $\epsilon$, and the following perturbation of system \cref{react_diff}: \begin{equation} \label{r_diff_pert} u_t=F(u)+(\ddd+\epsilon\bar{\ddd})u_{xx} \,, \end{equation} According to the consequences of hypotheses \hypStabInfty and \hypTransv mentioned in \cref{subsec:transv_assump}, if $\epsilon$ is sufficiently small, then the perturbed system~\cref{r_diff_pert} admits a unique travelling front close to $\phi$, having a speed close to $c$, and those depend smoothly on $\epsilon$. If we denote by $\phi+\epsilon\varphi$ this travelling front and by $c+\epsilon \bar{c}$ its speed, then, replacing these two ansatzes into system \cref{r_diff_pert}, we find that, at first order in $\epsilon$, the function $\varphi$ and the quantity $\bar{c}$ must satisfy the system \begin{equation} \label{1st_ord_front} \mathcal{L}\varphi=-\bar{\ddd}\phi''- \bar{c} \phi' \,. \end{equation} Taking on both sides the scalar product by $\psi$, it follows that: \begin{equation} \label{solv_cond} \bar c=-\langle\psi,\bar{\ddd}\phi''\rangle_{L^2(\rr,\rr^n)} \,. \end{equation} This is a solvency condition that ensures that $-\bar{\ddd}\phi''-\bar c\phi'$ is orthogonal to the kernel of $\mathcal{L}^*$, thus equivalently that it belongs to the image of $\mathcal{L}$ (see \cref{fig:notation_operators}).
The main purpose of this paper is to investigate the sign of the quantity $\bar{c}$, since it is this sign that determines how the perturbation in \cref{r_diff_pert} balances the relative dominance of the two equilibria $E_-$ and $E_+$, through the speed of the travelling front $\phi$. Indeed, \begin{itemize} \item if $\bar{c}$ is positive, then, for $\epsilon$ small positive, the influence of the perturbation will be to increase the speed of the front, thus to promote $E_-$ with respect to $E_+$; \item if conversely $\bar{c}$ is negative, then again for $\epsilon$ small positive, the influence of the perturbation will be to decrease the speed of the front, thus to promote $E_+$ with respect to $E_-$. \end{itemize}
\subsection{Alternative expression for the first order variation of the speed} \label{subsec:altern_expr_bar_c}
We are now going to provide a second expression of $\bar{c}$ that will turn out to be useful, and in particular easier to interpret than the solvency condition \cref{solv_cond}. Since the function $-\bar{\ddd}\phi''-\bar c\phi'$ belongs to the image of $\mathcal{L}$, system \cref{1st_ord_front} admits exactly one solution $x\mapsto\bar\varphi(x)$ satisfying \[ \langle \phi',\bar\varphi\rangle_{L^2(\rr,\rr^n)} = 0 \] (see \cref{fig:notation_operators}). Taking the scalar product by $\phi'$ in system \cref{1st_ord_front} and integrating over $\rr$, we get \[ -\bar c\norm{\phi'}_{L^2(\rr,\rr^n)}^2 = \langle\phi',\mathcal{L}\bar\varphi\rangle_{L^2(\rr,\rr^n)} = \langle\mathcal{L}^*\phi',\bar\varphi\rangle_{L^2(\rr,\rr^n)} \,, \] and since $\mathcal{L}\phi'=0$, the following alternative expression for $\bar c$ follows: \begin{equation} \label{solv_cond_alt} \bar c = \frac{\bigl\langle(\mathcal{L}-\mathcal{L}^*)\phi',\bar\varphi\bigr\rangle_{L^2(\rr,\rr^n)}}{\norm{\phi'}_{L^2(\rr,\rr^n)}^2} = \frac{\bigl\langle\bigl(DF(\phi)-DF(\phi)^*\bigr)\phi'+2 c \phi'',\bar\varphi\bigr\rangle_{L^2(\rr,\rr^n)}}{\norm{\phi'}_{L^2(\rr,\rr^n)}^2} \,. \end{equation} A geometrical interpretation of this expression will be given below in a more specific case.
\begin{remark} If $F(.)$ derives from a potential and $c=0$, then each one among expressions~\cref{solv_cond} and~\cref{solv_cond_alt} yields $\bar c=0$. Indeed, in this case, $DF(\phi)$ equals $DF(\phi)^*$ and $c$ equals $0$ and $\mathcal{L}$ equals $\mathcal{L}^*$, thus: \begin{itemize} \item it follows directly from \cref{solv_cond_alt} that $\bar c=0$; \item or it follows from $\mathcal{L}=\mathcal{L}^*$ that $\psi$ and $\phi'$ are proportional, thus (since $\bar\ddd$ is assumed to be symmetric) \cref{solv_cond} yiels $\bar c=0$. \end{itemize} \end{remark}
\subsection{Case of a two-dimensional reaction system with symmetries} \label{subsec:with_symmetries}
Now let us consider a more specific situation, assuming that the reaction system is two-dimensional, and that the two ``species'' under consideration are completely symmetric for this system, before the perturbation. Thus, keeping the notation and assumptions of the previous \namecrefs{subsec:with_symmetries}, let us assume in addition that the dimension $n$ of the field variable $u$ equals two. Let us denote by $(u_1,u_2)$ the canonical coordinates of a vector $u$ in $\rr^2$, let $\sss$ denote the orthogonal symmetry exchanging the coordinates in $\rr^2$, namely \[ \sss:(u_1,u_2)\mapsto (u_2,u_1) \,, \] and, from now on, let us make the following hypotheses: \begin{description} \item[(H3)] $F\circ\sss=\sss F \quad\mbox{and}\quad \ddd\sss=\sss\ddd \quad\mbox{and, for all $x$ in $\rr$,}\quad \phi(-x)=\sss\phi(x) \,.$ \end{description} In other words, we assume that both the reaction-diffusion system and the front $\phi(.)$ are $u_1\leftrightarrow u_2$-symmetric.
\begin{lemma}[$c$ equals $0$] \label{lem:c_equals_zero} The speed $c$ equals $0$. \end{lemma}
In other words, the front $\phi$ is a standing front.
\begin{proof} For every real quantity $x$, system \cref{syst_front} considered at $-x$ reads \[ -c\phi'(-x) = F\bigl(\phi(-x)\bigr) + \ddd \phi''(-x) \] and this yields, according to (H3), \[ c\sss \phi'(x) = \sss F\bigl(\phi(x)\bigr) + \sss \ddd \phi''(x) \,, \] and finally, getting rid of $\sss$ in this equality and comparing with system \cref{syst_front} considered at $x$, it follows that $c\phi'(x)$ equals $0$, and this proves \cref{lem:c_equals_zero}. \end{proof}
Thus the operators $\mathcal{L}$ and $\mathcal{L}^*$ reduce to: \[ \mathcal{L} = DF(\phi) + \ddd\partial_{\xi\xi} \quad\mbox{and}\quad \mathcal{L}^* = DF(\phi)^* + \ddd\partial_{\xi\xi} \,. \] Since the matrix $\bar{\ddd}$ is not assumed to be $u_1\leftrightarrow u_2$-symmetric (in other words we do not assume that $\bar{\ddd}\sss=\sss\bar{\ddd}$), the perturbation in \cref{r_diff_pert} in general breaks the $u_1\leftrightarrow u_2$-symmetry. For all $u$ in $\rr^2$, let us denote by $\rot F(u)$ the infinitesimal rotation of the vector field $F$. This quantity can be defined by: \begin{equation} \label{def_rot} DF(u)-DF(u)^*= \begin{pmatrix} 0&-\rot F(u)\\ \rot F(u)&0 \end{pmatrix} \,. \end{equation} With this notation, expression~\cref{solv_cond_alt} becomes: \begin{equation} \label{bar_c_rot} \bar c = \frac{\int_{-\infty}^{+\infty} \rot F\bigl(\phi(x)\bigr) \cdot\bigl(\phi'(x) \wedge \bar\varphi(x)\bigr)\, dx}{\int_{-\infty}^{+\infty}\phi'^2(x)\, dx} = \frac{\bigl\langle \rot F(\phi),\phi' \wedge \bar\varphi\bigr\rangle_{L^2(\rr,\rr^n)}}{\norm{\phi'}_{L^2(\rr,\rr^n)}^2} \,. \end{equation}
\subsection{Geometric interpretation of the first order variation of the speed} \label{subsec:geom_interpret_bar_c}
The last expression \cref{bar_c_rot} of $\bar{c}$ admits the following geometrical interpretation. Let us denote by $\Phi$ the image (the trajectory) in $\rr^2$ of the standing front $\phi$, that is: \[ \Phi = \{\phi(x): x\in\rr\}\subset \rr^2 \,. \] The infinitesimal rotation $\rot F(\phi)$ measures the ``shear'' induced locally along $\Phi$ by the antisymmetric part of $DF$, and the real quantity $\phi'\wedge\bar\varphi$ is determined by the component of the perturbation $\bar\varphi$ that is orthogonal to $\Phi$ (see \cref{fig:barc_rot_gen}). The shear induced by $F$ acts on this transverse perturbation (it ``pushes'' towards $E_-$ or $E_+$, as illustrated on \cref{fig:barc_rot_gen}), and this results in a change for the speed that is given at first order in $\epsilon$ by the quantity $\bar c$ defined above.
\begin{figure}
\caption{Geometrical illustration of expression~\cref{bar_c_rot} of $\bar c$.}
\label{fig:barc_rot_gen}
\end{figure}
\subsection{Reduction using symmetry} \label{subsec:red_sym}
The aim of this \namecref{subsec:red_sym} is to take into account the symmetries (H3) of the system to simplify expressions~\cref{solv_cond} and~\cref{bar_c_rot} of $\bar c$ (that is, to write the integrals in these expressions as integrals on $\rr_+$ only, instead of $\rr$). The first symmetries on the terms involved in these integrals are stated in the following lemma.
\begin{lemma}[symmetries of $\rot F$ and $\psi$] \label{lem:sym_psi_rotF} For every real quantity $x$, \begin{equation} \label{sym_psi_rotF} \rot F\bigl(\phi(-x)\bigr) = - \rot F\bigl(\phi(x)\bigr) \quad\mbox{and}\quad \psi(-x)=-\sss\psi(x) \,. \end{equation} \end{lemma}
\begin{proof} It follows from the symmetry of $F$ with respect to $\sss$ in (H3) that, for every $u$ in $\rr^2$, \[ F(\sss u)=\sss F(u) \quad\mbox{thus}\quad DF_{\sss u} \sss = \sss DF_{u} \quad\mbox{and}\quad DF_{\sss u} = \sss DF_{u} \sss \,, \] and since $\sss^*$ equals $\sss$, \[ DF_{\sss u}^* = \sss DF_{u}^* \sss \,. \] It follows that \[ DF_{\sss u} - DF_{\sss u}^* = \sss (DF_{u} - DF_{u}^*) \sss \] and according to the definition \cref{def_rot} of $\rot F(\cdot)$ it follows that \[ \rot F(\sss u) = - \rot F(u) \,. \] Thus, for every real quantity $x$, still according to (H3), \[ \rot F \bigl(\phi(-x)\bigr) = \rot F \bigl( \sss \phi(x)\bigr) = - \rot F \bigl( \phi(x)\bigr) \,. \] and this proves the first equality of \cref{sym_psi_rotF}.
To prove the second equality, let us consider the function $\eta$ defined by: $\eta(x)=\psi(-x)$. Then, according to the expression of $\mathcal{L}^*$ and to hypotheses (H3), \[ \begin{aligned} \mathcal{L}^* (\sss \eta)(x) &= \ddd \sss \psi''(-x) + DF^*\bigl( \phi(x)\bigr) \sss \psi(-x) \\ &= \sss \ddd \psi''(-x) + DF^*\bigl( \sss \phi(-x)\bigr) \sss \psi(-x) \\ &= \sss \Bigl( \ddd \psi''(-x) + DF^*\bigl( \phi(-x)\bigr) \psi(-x) \Bigr) \\ &= \sss (\mathcal{L}^* \psi) (-x) \\ &=0 \,. \end{aligned} \] In other words, the function $x\mapsto \sss\psi(-x)$ belongs to the eigenspace associated to the eigenvalue $0$ for the operator $\mathcal{L}^*$. Since this eigenspace is one-dimensional and contains the nonzero function $\psi$, it follows that there exists a real quantity $\lambda$ such that, for every real quantity $x$, \[ \sss\psi(-x)=\lambda \psi(x) \,, \] and since the map \[ L^2(\rr,\rr^2) \rightarrow L^2(\rr,\rr^2), \quad f \mapsto \bigl( x\mapsto \sss f(-x) \bigr) \] is an involution, it follows that $\lambda=\pm 1$. According to the symmetry of $\phi$ with respect to $\sss$ in (H3), for every real quantity $x$, \[ \phi'(-x)=-\sss\phi'(x) \quad\mbox{and}\quad \psi(-x) = \lambda\sss\psi(x) \,. \] This shows $\lambda$ cannot be equal to $1$, or else the function $x\mapsto \psi(x)\cdot\phi'(x)$ would be odd, and the scalar product $\langle\psi,\phi'\rangle_{L^2(\rr,\rr^n)}$ would vanish, whereas according to the assumptions we made this scalar product must be nonzero (and was actually normalized to $1$). Thus $\lambda$ equals $-1$, and this proves the second equality of \cref{sym_psi_rotF}. \Cref{lem:sym_psi_rotF} is proved. \end{proof}
Since we are interested in the effect of breaking the $u_1\leftrightarrow u_2$-symmetry of the diffusion matrix, it is convenient to assume that the $u_1\leftrightarrow u_2$-symmetric part of the (symmetric) matrix $\bar{\ddd}$ vanishes (and that the $u_1\leftrightarrow u_2$-antisymmetric part of the same matrix does not vanish). This is exactly the meaning of our next hypothesis: \begin{description} \item[(H4)] $\sss\bar{\ddd}=-\bar{\ddd}\sss$ and $\bar{\ddd}\not=0$. \end{description} According to this hypothesis, there exists a nonzero real quantity $d$ such that: \[ \bar{\ddd}= \begin{pmatrix} d&0\\ 0&-d \end{pmatrix} \,. \] This hypothesis leads to the following additional symmetry.
\begin{lemma}[symmetry of $\bar\varphi$] \label{lem:sym_of_bar_phi} For every real quantity $x$, \[ \bar\varphi(-x)=-\sss\bar\varphi(x) \,. \] \end{lemma}
\begin{proof} Since $\bar\varphi$ is a solution of system \cref{1st_ord_front}, for every real quantity $x$, \[ \ddd \bar\varphi'' (x) + DF\bigl(\phi(x)\bigr) \bar\varphi (x) = - \bar{\ddd} \phi''(x) - \bar c \phi'(x) \,. \] Multiplying (to the left) by $\sss$ both sides of this equality and using the symmetries (H3) and (H4), it follows that \[ \ddd \sss \bar\varphi'' (x) + DF\bigl(\phi(-x)\bigr) \sss\bar\varphi (x) = \bar{\ddd} \phi''(-x) + \bar c \phi'(-x) \,, \] and this shows that the function $x\mapsto -\sss \bar\varphi(-x)$ is also a solution of system \cref{1st_ord_front}. Observe in addition that according to the symmetry of $\phi$ this solution is orthogonal to $\phi'$ for the $L^2(\rr,\rr^n)$-scalar product. Thus this solution must be equal to $\bar\varphi$, and this proves the lemma. \end{proof}
\begin{lemma}[even integrands] \label{lem:even_integrands} The three functions \[ x\mapsto \psi(x)\cdot\bar{\ddd}\phi''(x) \quad\mbox{and}\quad x\mapsto \phi'(x)^2 \quad\mbox{and}\quad x\mapsto \rot F\bigl(\phi(x)\bigr)\cdot\bigl(\phi'(x)\wedge\bar\varphi(x)\bigl) \] are even. \end{lemma}
\begin{proof} For the two first functions, the symmetry follows from (H3) and \cref{lem:sym_psi_rotF}. For the third function, observe that, for every real quantity $x$, according to (H3) and \cref{lem:sym_of_bar_phi}, \[ \phi'(-x)\wedge\bar\varphi(-x) = \sss \phi'(x)\wedge \sss\bar\varphi(x) = - \phi'(x)\wedge\bar\varphi(x) \,. \] and the result follows from \cref{lem:sym_psi_rotF}. \end{proof}
It follows from \cref{lem:even_integrands} that expressions~\cref{solv_cond} and~\cref{bar_c_rot} of $\bar c$ can be rewritten with integrals restricted to $\rr_+$, namely: \begin{align} \label{s_cond_rr_plus} \bar c &= -2 \int_{0}^{+\infty} \psi(x)\cdot\bar{\ddd}\phi''(x)\,dx = -2 \langle\psi,\bar{\ddd}\phi''\rangle_{L^2(\rr_+,\rr^n)} \\ \label{c_rot_rr_plus} &= \frac{\int_{0}^{+\infty} \rot F\bigl(\phi(x)\bigr)\cdot\bigl(\phi'(x) \wedge \bar\varphi(x)\bigr)\,dx}{\int_{0}^{+\infty}\phi'^2(x)\,dx} = \frac{\langle \rot F(\phi),\phi' \wedge \bar\varphi\rangle_{L^2(\rr_+,\rr^n)}}{\norm{\phi'}_{L^2(\rr_+,\rr^n)}^2} \,. \end{align} The aim of the two following \namecrefs{sec:toy} is to compute the sign of the quantity $\bar c$ on two specific examples.
\section{Toy example} \label{sec:toy}
\subsection{Definition} \label{subsec:toy_def}
The aim of this \namecref{sec:toy} is to show on a toy example that both signs can occur for the quantity $\bar c$. Let $u=(u_1,u_2)$ denote again the canonical coordinates in $\rr^2$, let $\mu$ denote a real quantity (a parameter), and let us consider the following system (see \cref{fig:phase_toy}):
\begin{figure}
\caption{Phase space of the reaction system $u_t=F_\mu(u)$ for $\mu>0$.}
\label{fig:phase_toy}
\end{figure}
\begin{equation} \label{syst_toy_u} u_t = F_\mu(u) + u_{xx} \quad\mbox{with}\quad F_\mu(u) = F_\mu\begin{pmatrix}u_1 \\ u_2\end{pmatrix}= \begin{pmatrix} u_1\bigl(1-(u_1+u_2)-\mu u_2(u_2-u_1)\bigr)\\ u_2\bigl(1-(u_1+u_2)-\mu u_1(u_1-u_2)\bigr) \end{pmatrix} . \end{equation} Both axes $\{u_2=0\}$ and $\{u_1=0\}$ are invariant under the reaction system $u_t=F_\mu(u)$, and the restriction of this system to each of these axes is nothing but the logistic equation: $w_t=w(1-w)$. Besides, the $u_1\leftrightarrow u_2$-symmetry $F_\mu\circ\sss=\sss F_\mu$ clearly holds.
\begin{notation} Let us consider the following alternative coordinate system $v=(v_T,v_L)$ related to $u=(u_1,u_2)$ by: \begin{equation} \label{def_trans_long_coord} \left\{ \begin{aligned} v_T &= u_1+u_2 \\ v_L &= -u_1+u_2 \end{aligned} \right. \Longleftrightarrow \left\{ \begin{aligned} u_1 &= \frac{v_T-v_L}{2} \\ u_2 &= \frac{v_T+v_L}{2} \end{aligned} \right. \end{equation} (see \cref{fig:phase_toy}). The subscripts ``$T$'' and ``$L$'' refer to the adjectives ``transversal'' and ``longitudinal'', with respect to the standing front $\phi$ that will be defined below. Along this \namecref{sec:toy} and the next one, these subscripts will always be used to denote the coordinates of a point in this ``transversal-longitudinal'' coordinate system, whereas the subscripts ``$1$'' and ``$2$'' will always be used to denote the canonical coordinates. \end{notation}
When expressed within the transversal-longitudinal coordinate system, system \cref{syst_toy_u} takes the form \begin{equation} \label{syst_toy_v} v_t=G_\mu(v)+v_{xx} \quad\mbox{with}\quad G_\mu(v)=G_\mu(v_T,v_L)= \begin{pmatrix} v_T-v_T^2 \\ v_L\Bigl(1-v_T+\frac{\mu}{2}(v_T^2-v_L^2)\Bigr) \end{pmatrix} \,. \end{equation} According to this expression, the line $\{v_T=1\}$ is invariant (and transversely attractive), and the restriction of system \cref{syst_toy_v} to this line reads: \[ \partial_t v_L=\frac{\mu}{2}v_L(1-v_L^2)+\partial_{xx}v_L \,. \] Thus, if the parameter $\mu$ is negative, the reaction system is monostable, with a unique stable equilibrium $E_0$ at $(v_T,v_L)=(1,0)$, whereas if $\mu$ is positive then it is bistable, with two stable equilibria $E_1$ at $(v_T,v_L)=(1,-1)$ and $E_2$ at $(v_T,v_L)=(1,1)$, and a saddle $E_0$ at $(v_T,v_L)=(1,0)$ (see \cref{fig:phase_toy}). By the way, \begin{equation} \label{lin_asymt_toy} DF_\mu(E_1) = \begin{pmatrix} -1 & 1+\mu \\ 0 & -\mu \end{pmatrix} ,\quad DF_\mu(E_2) = \begin{pmatrix} -\mu & 0 \\ 1+\mu & -1 \end{pmatrix} ,\quad DG_\mu(E_0) = \begin{pmatrix} -1 & 0 \\ 0 & \frac{\mu}{2} \end{pmatrix} \end{equation} (compare with expression \vref{lin_asymot_lv} for the Lotka--Volterra competition system).
\subsection{Standing front} \label{subsec:toy_standing_front}
Let us assume from now on that $\mu$ is positive (bistable case). In this case there exists for systems \cref{syst_toy_u,syst_toy_v} a standing front $x\mapsto \phi(x)$ connecting $E_1$ to $E_2$, which is given (in transversal-longitudinal coordinates) by the explicit formula: \begin{equation} \label{toy_front} \phi_T(x)\equiv 1 \quad\mbox{and}\quad \phi_L(x)=\tanh\frac{\sqrt{\mu}x}{2} \end{equation} (the connection with the notation used in \cref{sec:general} is obvious: equilibria $E_1$ and $E_2$ defined above correspond to equilibria $E_-$ and $E_+$ of \cref{sec:general}, respectively).
This standing front satisfies the $u_1\leftrightarrow u_2$-symmetry, that is $\phi(-x)=\sss\phi(x)$ for every real quantity $x$. All the symmetry hypotheses (H3) are therefore satisfied for the system \cref{syst_toy_u} and the standing front $\phi$.
For the remaining of \cref{sec:toy} we shall mainly work with the transversal-longitudinal coordinate system. The linear operator $\mathcal{L}$ (obtained by linearising system \cref{syst_toy_v} around this standing front) reads, expressed in these coordinates, \begin{equation} \label{expression_L_toy} \mathcal{L} \begin{pmatrix} \varphi_T \\ \varphi_L \end{pmatrix} = \begin{pmatrix} -1 & 0 \\ (\mu-1)\phi_L & \frac{\mu}{2}(1-3\phi_L^2) \end{pmatrix} \begin{pmatrix} \varphi_T \\ \varphi_L \end{pmatrix} + \begin{pmatrix} \varphi_T'' \\ \varphi_L'' \end{pmatrix} \,. \end{equation}
\begin{lemma}[spectral stability of the standing front, toy example] \label{lem:lin_st_toy} The standing front \\ $x\mapsto \phi(x)$ is spectrally stable. \end{lemma}
\begin{proof} According to expressions \cref{lin_asymt_toy}, the essential spectrum of $\mathcal{L}$ is the interval: \[ (-\infty,\max(-1,-\mu)] \quad\mbox{included in}\quad (-\infty,0) \,. \] A function \[ x\mapsto \varphi(x)=\bigl(\varphi_T(x),\varphi_L(x)\bigr) \] is an eigenfunction of $\mathcal{L}$ for an eigenvalue $\lambda$ if and only if $\varphi_T$ vanishes identically and $\varphi_L$ is an eigenfunction of the operator \begin{equation} \label{lin_toy_v2} \ell_{\mu}:\varphi_L \mapsto \frac{\mu}{2}(1-3\phi_L^2) \varphi_L + \varphi_L'' \end{equation} for the same eigenvalue $\lambda$. The function $x\mapsto\varphi'_L(x)$ is an eigenfunction of $\ell_{\mu}$ for the eigenvalue zero (this comes from translation invariance in space), and by a standard Sturm--Liouville argument (\cite{CoddingtonLevinson_theoryODE_1955}), this eigenvalue is simple and all other eigenvalues (they are real since $\ell_{\mu}$ is a self-adjoint operator for the $L^2(\rr,\rr)$-scalar product) are negative. \Cref{lem:lin_st_toy} is proved. \end{proof}
\subsection{First order variation of the front speed} \label{subsec:toy_front_speed}
According to the notation of \cref{subsec:setup}, $\ddd=\id_{\rr^2}$. Let us choose the perturbation matrix $\bar{\ddd}$ as follows: \begin{equation} \label{def_bar_D_toy} \begin{aligned} \bar{\ddd} & = \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix} \mbox{ in canonical coordinates,}\\ \mbox{or equivalently } \bar{\ddd} & = \begin{pmatrix} 0&-1\\ -1&0 \end{pmatrix} \mbox{ in transversal-longitudinal coordinates.} \end{aligned} \end{equation} All hypotheses (H1-4) of \cref{sec:general} are satisfied. Let us keep the notation $\mathcal{L}^*$ and $\psi$ and $\bar c$ and $\bar \varphi$ introduced there. The following result shows that both signs for $\bar c$ may occur, depending on the value of $\mu$.
\begin{proposition}[sign of the first order variation of front speed, toy example] \label{prop:bar_c_toy} The \\ sign of $\bar c$ equals that of $1-\mu$; that is, \[ \begin{aligned} \bar 0<\bar c \quad & \mbox{if}\quad 0<\mu<1 \,, \\ \mbox{and}\quad \bar c=0 \quad & \mbox{if}\quad \mu=1 \,, \\ \mbox{and}\quad \bar c<0 \quad & \mbox{if}\quad 1<\mu \,. \end{aligned} \] \end{proposition}
This proposition can be understood as follows. \begin{itemize} \item For $\mu$ in $(0,1)$, the perturbation promotes $E_1$. And since the perturbation increases the mobility of the species corresponding to $E_1$, this shows that an increase of mobility is advantageous in this case. \item For $\mu$ larger than $1$, the perturbation promotes $E_1$. And for the same reason, this time, an increase of mobility turns out to be disadvantageous. \end{itemize}
Before proving this proposition, let us begin with a geometrical interpretation (this interpretation will by the way provide an informal proof, and make the proof easier to follow).
\begin{figure}
\caption{Left: the initial standing front $\phi$ and its perturbation $\phi+\epsilon\bar\varphi$. Right: the graphs of the components of these two fronts in the canonical coordinate system. The perturbation consists in an increase of the mobility of the first species (equilibrium $E_1$) and a decrease of the mobility of the second species (equilibrium $E_2$).}
\label{fig:front_toy}
\end{figure} \begin{figure}
\caption{Orientation of the shear along the standing front in the ``moderately strong'' ($\mu$ smaller than $1$) versus ``hard'' ($\mu$ larger than $1$) competition regimes. In the moderately strong competition regime, being more mobile is an advantage, and the most mobile species (the first one, equilibrium $E_1$) wins with respect to the second species (equilibrium $E_2$). In the the hard competition regime, being more mobile is a disadvantage, and the most mobile species (the first one) looses.}
\label{fig:shear_toy}
\end{figure}
Let $(\phi_1,\phi_2)$ denote the components of $\phi$ in the canonical coordinate system, and let $(\bar\varphi_1,\bar\varphi_2)$ denote the components of $\bar\varphi$ in the same canonical coordinate system. As illustrated on \cref{fig:front_toy}, the functions $\phi_1$ and $\phi_2$ are symmetric. The perturbation breaks this symmetry: the mobility of the first species has been slightly increased, while the mobility of the second species has been slightly decreased. As a consequence, one expects that the graph of $\phi_1+\epsilon\bar\varphi_1$ will be slightly flatter than that of $\phi_1$, and conversely that the graph of $\phi_2+\epsilon\bar\varphi_2$ will be slightly straighter than that of $\phi_2$ (\cref{fig:front_toy}). As a consequence, the position of the image of the perturbed front with respect to the image of the initial standing front should be as illustrated on \cref{fig:front_toy}; it suggests that $\bar\varphi_T(x)$ --- the first component of $\bar\varphi$ in the transversal-longitudinal coordinate system --- should be negative for $x$ negative and positive for $x$ positive (the proof below will confirm this).
On the other hand, expression \cref{syst_toy_v} of the reaction-diffusion system in the transversal-longitudinal coordinate system yields, or all $x$ in $\rr$: \begin{equation} \label{rot_toy} \rot F_\mu\bigl(\phi(x)\bigr)=(\mu-1)\phi_L(x) \,. \end{equation} Thus the sign of the shear induced by $F_\mu$ along $\phi$ depends on the sign of $\mu-1$ (see \cref{fig:shear_toy}). In view of \cref{fig:front_toy,fig:shear_toy}, it could be expected that, for $\mu<1$, the perturbation is in favour of $E_1$, while for $\mu>1$ it is in favour of $E_2$, as stated by \cref{prop:bar_c_toy}.
Here is another possible interpretation. The parameter $\mu$ represents a sort of ``intensity'' of the competition between the two species. If $\mu$ is positive but smaller than $1$, the intensity can be qualified as ``moderately strong''. In this case, as illustrated on \cref{fig:shear_toy} (see the zoom on equilibrium $E_2$), the dominant effect of the reaction term is to balance the total density $u_1+u_2$ (to drive this total density to $1$), and this turns out to be in favour of the most mobile species. On the other hand, if $\mu$ is larger than $1$, then the intensity of the competition can be qualified as ``hard''. It this case, the dominant effect of the reaction term is to drive the system in favour of the most represented species locally (and away of the $E_0$ saddle equilibrium where both densities are equal). And this turns out to be in favour of the less mobile species. In short (you may apply this to you everyday life \Smiley): if the struggle is moderate, spread away to gain new territories; if it is bloody, avoid the dispersal and concentrate your forces~!
Let us now prove \cref{prop:bar_c_toy}. Let us denote by $(\psi_T,\psi_L)$ and by $(\bar\varphi_T,\bar\varphi_L)$ the transversal-longitudinal coordinates of the functions $\psi$ and $\bar\varphi$. To prove \cref{prop:bar_c_toy}, each one among expressions \vref{s_cond_rr_plus,c_rot_rr_plus} can be used (resulting in two different proofs). The two proofs are given below, beginning with the proof involving expression~\cref{c_rot_rr_plus}, since it is closer to the geometrical interpretation above.
\begin{proof}[Proof using expression~\cref{c_rot_rr_plus}] According to expression \cref{c_rot_rr_plus}, the sign of $\bar c$ is equal to the sign of: \begin{equation} \label{sign_bar_c_toy} \int_{0}^{+\infty} \rot F_\mu\bigl(\phi(x)\bigr)\cdot\bigl(\phi'(x) \wedge \bar\varphi(x)\bigr)\,dx \,. \end{equation} According to expression \cref{rot_toy} of $\rot F_\mu\bigl(\phi(x)\bigr)$ and expression \cref{toy_front} of $\phi_L(x)$ the function $\rot F_\mu\bigl(\phi(\cdot)\bigr)$ is of the sign of $\mu-1$ on $\rr_+$.
On the other hand, since $\phi_T'$ is identically zero, the function $\phi'\wedge\bar\varphi$ equals $-\phi'_L\bar\varphi_T$, and according to expression \cref{toy_front} of $\phi_L(x)$ the function $\phi_L'$ is positive on $\rr_+$. It remains to determine the sign of $\bar\varphi_T$ on $\rr_+$. Projecting system \cref{1st_ord_front} on the $v_T$-axis yields: \begin{equation} \label{proj_toy} \bar\varphi_T''=\phi_L''+\bar\varphi_T \,. \end{equation} Recall that according to \vref{lem:sym_of_bar_phi}, the quantities $\bar\varphi(-x)$ and $-\sss\bar\varphi(x)$ are equal for every $x$ in $\rr$; as a consequence, since the transversal coordinate is unchanged by $\sss$, the quantity $\bar\varphi_T(0)$ must vanish. Since $\bar\varphi_T(x)$ approaches $0$ when $x$ approaches $+\infty$ and since according to \cref{toy_front} $\phi''_L(x)$ is negative for all $x$ in $\rr_+^*$, it follows from equation~\cref{proj_toy} shows that the function $\bar\varphi_T$ is positive on $\rr_+^*$ (see \vref{lem:sol_forced_2nd_order} in appendix). It follows that the function $\phi'\wedge\bar\varphi$ is negative on $\rr_+^*$, and this proves \cref{prop:bar_c_toy}. \end{proof}
\begin{proof}[Proof using expression~\cref{s_cond_rr_plus}] According to expression \vref{s_cond_rr_plus}, we have: \[ \bar c =- 2\int_{0}^{+\infty}\psi(x)\cdot\bar{\ddd}\phi''(x)\, dx = 2\int_{0}^{+\infty}\psi_T(x)\cdot\phi''_L(x)\, dx \,, \] and we know from the explicit expression \cref{toy_front} of $\phi_L$ that the quantity $\phi''_L(x)$ is negative for all $x$ in $\rr_+^*$. Therefore all we have to do is show that $\psi_T$ and $\mu-1$ have the same sign (and vanish at the same time).
According to the expression \vref{expression_L_toy} of $\mathcal{L}$, system $\mathcal{L}^*\psi=0$ reads (using the notation $\ell_{\mu}$ introduced in definition \cref{lin_toy_v2}): \begin{align} -\psi_T + (\mu-1)\phi_L\psi_L + \psi_T'' &= 0 \label{psi_T_toy} \,, \\ \ell_{\mu}\psi_L &= 0 \label{psi_L_toy} \,. \end{align} Since the eigenvalue zero of $\ell_{\mu}$ is simple (see the proof of \cref{lem:lin_st_toy} above), equation \cref{psi_L_toy} shows that the functions $\psi_L$ and $\phi'_L$ must be proportional. Thus $\psi_L = N\phi'_L$, where, according to the normalizing condition \vref{norm_cond}, the normalizing constant $N$ is: \[ N = \Bigl(\int_{-\infty}^{+\infty}\phi_L'^2(x)\,dx\Bigr)^{-1} = \norm{\phi_L'}_{L^2(\rr,\rr)}^{-2}>0 \,. \] Thus, according to equation \cref{psi_T_toy}, the following differential equation holds for $\psi_T$: \begin{equation} \label{adjoint_toy} \psi_T'' = \psi_T+N(1-\mu)\phi_L\phi'_L \,. \end{equation} Recall that according to \vref{lem:sym_psi_rotF}, the quantities $\psi(-x)$ and $-\sss\psi(x)$ are equal for every $x$ in $\rr$; as a consequence, since the transversal coordinate is unchanged by $\sss$, the quantity $\psi_T(0)=0$ must vanish. Since $\psi_T(x)$ approaches $0$ when $x$ approaches $+\infty$, and since both quantities $\phi_L(x)$ and $\phi'_L(x)$ are positive for all $x$ in $\rr_+^*$, this shows that the sign of $\psi_T$ must remain constant and opposite to that of $1-\mu$ on $\rr_+^*$ (see \vref{lem:sol_forced_2nd_order} in appendix), and that $\psi_T$ vanishes identically if $\mu$ is equal to $1$. \Cref{prop:bar_c_toy} is proved. \end{proof}
\section{Bistable Lotka--Volterra competition model} \label{sec:lv}
\subsection{Definition} \label{subsec:lv_def}
\begin{figure}
\caption{Phase space of the reaction system in the bistable case ($\mu>0$).}
\label{fig:phase_lv}
\end{figure}
Let $\mu$ denote a real quantity (a parameter) and let us consider the following reaction-diffusion system, where the reaction term is known as the Lotka--Volterra competition model (see \cref{fig:phase_lv}): \begin{equation} \label{syst_lv_u} u_t = F_\mu(u) + u_{xx} \quad\mbox{with}\quad F_\mu(u)=F_\mu(u_1,u_2)= \begin{pmatrix} u_1\bigl(1-u_1-(1+\mu) u_2\bigr)\\ u_2\bigl(1-u_2-(1+\mu) u_1\bigr) \end{pmatrix} \,. \end{equation} Both axes $\{u_2=0\}$ and $\{u_1=0\}$ are invariant under the reaction differential system $u_t=F_\mu(u)$, and the restriction of this system to each of these axes is nothing but the logistic equation $w_t = w(1-w)$. The $u_1\leftrightarrow u_2$-symmetry $F_\mu\circ\sss=\sss F_\mu$ holds.
Again in this \namecref{sec:lv}, we are going to use the ``transversal-longitudinal'' coordinate system $v=(v_T,v_L)$ defined exactly as in definition \vref{def_trans_long_coord}. Expressed in this coordinate system, the reaction-diffusion system \cref{syst_lv_u} takes the form: \begin{equation} \label{syst_lv_v} v_t=G_\mu(v)+v_{xx} \quad\mbox{with}\quad G_\mu(v)=G_\mu(v_T,v_L)= \begin{pmatrix} v_T-v_T^2+\frac{\mu}{2}(v_L^2-v_T^2)\\ v_L(1-v_T) \end{pmatrix} . \end{equation} For all $\mu$ in $\rr\setminus\{-2,0\}$, the reaction system admits three equilibria aside from $(0,0)$ (see \cref{fig:phase_lv}): \begin{itemize} \item $E_1$, that is: $(u_1,u_2)=(1,0)\Leftrightarrow (v_T,v_L)=(-1,1)$, \item $E_2$, that is: $(u_1,u_2)=(0,1)\Leftrightarrow (v_T,v_L)=(1,1)$, \item $E_0$, that is: $(u_1,u_2)=\bigl(1/(2+\mu),1/(2+\mu)\bigr)\Leftrightarrow (v_T,v_L)=\bigl(1/(1+\mu/2),0\bigr)$. \end{itemize} The differential of the reaction system reads: \[ DF_\mu(u_1,u_2)= \begin{pmatrix} 1-2u_1-(1+\mu) u_2 & -(1+\mu)u_1 \\ -(1+\mu)u_2 & 1-2u_2-(1+\mu) u_1 \end{pmatrix} \] and \[ DG_\mu(v_T,v_L)= \begin{pmatrix} 1-(2+\mu)v_T & \mu v_L\\ -v_L & 1-v_T \end{pmatrix} \] thus \begin{equation} \label{lin_asymot_lv} DF_\mu(E_1) = \begin{pmatrix} -1 & -1-\mu \\ 0 & -\mu \end{pmatrix} ,\ DF_\mu(E_2) = \begin{pmatrix} -\mu & 0 \\ -1-\mu & -1 \end{pmatrix} ,\ DG_\mu(E_0) = \begin{pmatrix} -1 & 0 \\ 0 & \frac{\mu}{2+\mu} \end{pmatrix} . \end{equation} For $\mu$ negative, the reaction system is monostable ($E_0$ is stable and $E_2$ and $E_1$ are saddles) while for $\mu$ positive it is bistable ($E_2$ and $E_1$ are stable and $E_0$ is a saddle). From now on, it will be assumed that: \[ \mu >0 \,, \] or in other words that the interspecific competition rate $1+\mu$ is higher than the intraspecific competition rate $1$, or in other words that the reaction system is bistable. Observe that (with the notation of \cref{sec:general}), the infinitesimal rotation of the vector field $G_\mu$ reads: \[ \rot G_\mu(v_T,v_L) = -(1+\mu)v_L \,. \] Thus, by contrast with the toy example studied in the previous \namecref{sec:toy}, the sign of the shear along the trajectory of the expected front connecting $E_1$ to $E_2$ does not depend on the parameter $\mu$ (since $\mu$ is assumed to be positive). Thus in view of the observations made in the previous \namecref{sec:toy} we may expect that an increase of mobility is in this case always advantageous, in other words that the quantity $\bar c$ is positive. The aim of this \namecref{sec:lv} is to prove this statement when $\mu$ is altogether positive and small.
\subsection{Standing front} \label{subsec:lv_standing_front}
A smooth function \[ \phi:\rr\rightarrow\rr^2,\quad x\mapsto\phi(x) = \bigl(\phi_T(x),\phi_L(x)\bigr) \] is a stationary solution of system \cref{syst_lv_v} if it is a solution of \begin{equation} \label{syst_front_lv} G_\mu(\phi)+\phi''=0 \Longleftrightarrow \left\{ \begin{aligned} \phi_T'' &= \phi_T^2-\phi_T+\frac{\mu}{2}(\phi_T^2-\phi_L^2) \\ \phi_L'' &= \phi_L(\phi_T-1). \end{aligned} \right. \end{equation} Standing or travelling bistable fronts for systems including \cref{syst_lv_u} have been studied by many authors for a long time. Existence and asymptotic stability of a bistable (monotone) travelling front connecting $E_1$ to $E_2$ were first established (in a more general setting) by C. Conley and R. Gardner using topological methods and comparison principles, \cite{Gardner_exisStabVWDegreeTheoreticApproach_1982,ConleyGardner_applicationGeneralizedMorseIndexTW_1984}.
In \cite{Kan-on_parameterDependencePropSpeedTW_1995,Kan-onFang_stabilityMonotoneTW_1996}, Y. Kan-on and Q. Fang proved the uniqueness of this bistable travelling front and its spectral stability (including its transversality/robustness that is the fact that the eigenvalue $0$ is simple) for Lotka--Volterra competition-diffusion systems (a class of systems including \cref{syst_lv_u} and governed by three reduced parameters aside of diffusion coefficients); from these spectral properties they recovered the asymptotic stability of this bistable front. In \cite{Kan-on_parameterDependencePropSpeedTW_1995}, Kan-on also proved the monotonicity of the speed of the bistable front with respect to the parameters characterizing the reaction system (but not with respect to the diffusion coefficients of the two species). Further insight into the sign of the speed of the front were achieved by J-S Guo and Y-C Lin, \cite{GuoLin_signWaveSpeed_2013}. However, here again their results are mainly concerned with the dependence of this sign with respect to the parameters of the reaction system, but not with respect to the diffusion coefficients of the two components, and little is stated about the (rather specific) case considered in this paper, where the reaction system is $u_1\leftrightarrow u_2$-symmetric, and where the sole breaking of this $u_1\leftrightarrow u_2$-symmetry comes from the diffusion coefficients.
In \cite{GirardinNadin_tWRelativeMotilityInvasionSpeed_2015}, L. Girardin and G. Nadin also studied the dependence of the front speed with respect to the coefficients of the system, and this time espescially with respect to the diffusion coefficients of the two species. Their results hold when the parameters of the system approach certain limits; for the more restricted system \cref{syst_lv_u}, this corresponds to the limit when $\mu$ approaches $+\infty$ (in other words, when the interspecific competition rates approach $+\infty$). Their main ``Unity is not strength'' theorem states that, close to this limit, it is the most mobile species that dominates the other one. Surprisingly enough, this fits with the ``moderately strong'' competition case of the previous toy example, but not with the ``hard'' competition case where by contrast it was the less motile species that was dominant.
Our purpose is to consider system \cref{syst_lv_u} when the parameter $\mu$ is positive and small (thus an asymptotics completely different from the one considered by Girardin and Nadin). By contrast with the previous toy example, the standing front connecting $E_1$ to $E_2$ is (to the knowledge of the author) not given by an explicit expressions. Note that explicit expressions for standing or travelling waves of Lotka--Volterra competition-diffusion systems have been provided by various authors, for instance by M. Rodrigo and M. Mimura in \cite{MimuraRodrigo_exactSolutionsCompetitionDiffusion_2000,MimuraRodrigo_exactSolRDSystemsAndNLWaveEqu_2001} or by N. Kudryashov and A. Zakharchenko in \cite{KudryashovZakharchenko_exactSolutionsLV_2015} (this latter concerning only the monostable case), but always under restrictions on the parameters, and in particular for specific values of the diffusion coefficients (not encompassing a full interval of values for the diffusion coefficients in \cref{syst_lv_u}).
Our strategy will therefore be to assume that the parameter $\mu$ is small and use singular perturbation arguments to get a first order approximation (in terms of $\mu$) for the standing front (\vref{lem:exist_f_lv}). This will lead to a first order approximation (still in terms of $\mu$) for the quantity $\bar{c}$ we are interested in and in particular to its sign that will turn out to be positive (\vref{prop:bar_c_lv}). Before stating and proving these approximations, some notation is required.
\subsection{Notation} \label{subsec:not}
\begin{enumerate} \item The estimates that will be computed in the remaining of this \namecref{subsec:not} will often involve the (small) quantity $\sqrt{\mu}$ (instead of $\mu$ itself). For this reason it will be convenient to have a specific notation for this quantity. Let us write: \begin{equation} \label{not_eps_mu} \varepsilon = \sqrt{\mu} \,. \end{equation} Note that this quantity $\varepsilon$ has nothing to do with the quantity $\epsilon$ introduced in \cref{sec:general} and displayed on \cref{fig:front_toy} (that one will not be used any more in the remaining of the paper). \item The standing front and all related functions (for instance eigenfunctions) will turn out to depend slowly on the space variable $x$. For this purpose, it will sometimes be convenient to view them as functions of the space variable $y$ related to $x$ by: \begin{equation} \label{not_y_x} y = \varepsilon x \Longleftrightarrow \frac{y}{\varepsilon} = x \,. \end{equation} \item Up to an appropriate scaling, the standing front will be given at first order by the function \begin{equation} \label{not_theta} \theta:y\mapsto \tanh\Bigl(\frac{y}{2}\Bigr) \,. \end{equation} (already encountered in the toy example of \cref{sec:toy}). This function is a solution of equation: \[ \theta'' + \frac{1}{2}\theta(1-\theta^2)=0 \] and the first order expansion along $\theta$ of this equation reads: $\ell \varphi =0$, where $\ell$ is the differential operator: \begin{equation} \label{not_ell} \ell:\varphi\mapsto\varphi'' + \frac{1}{2}(1-3\theta^2)\varphi \end{equation} (compare with operator $\ell_{\mu}$ defined in \vref{lin_toy_v2}). \item A notation is needed to deal with the remaining ``higher order terms'' (in $\varepsilon$) that will appear in the next computations. These higher order terms are slightly more involved than just real quantities. They depend on $\varepsilon$ and $x$, they vary slowly with respect to $x$, and they approach zero at an exponential rate when $x$ approaches $\pm\infty$. This approach to zero at infinity is important since integrals over the whole real line or half-real line will be made on various occasions. Let us define the space $\rrr$ of ``remaining terms'' as follows. A function \[ r:(y,\varepsilon)\mapsto r(y,\varepsilon) \] belong to the set $\rrr$ if there exists a positive quantity $\delta$ such that: \begin{itemize} \item $r$ is defined and smooth on $\rr\times[0,\delta]$, \item for every integer $p$, the quantity \[ \sup_{(y,\varepsilon)\in\rr\times[0,\delta]} e^{\abs{y/2}}\abs{\partial_y^p r(y,\varepsilon)} \] is finite. \end{itemize} \end{enumerate}
\subsection{Approximation of the standing front} \label{subsec:exist_f_lv}
The following lemma makes use of the notation $\theta(\cdot)$ and $\rrr$ introduced above. Existence and uniqueness are stated in this lemma since they will be recovered automatically by the singular perturbation approach, but as mentioned above these results are well known, \cite{Gardner_exisStabVWDegreeTheoreticApproach_1982,ConleyGardner_applicationGeneralizedMorseIndexTW_1984,Kan-on_parameterDependencePropSpeedTW_1995}. Therefore the main interest of this lemma is the approximation of the standing front that it provides.
\begin{lemma}[existence of the standing front] \label{lem:exist_f_lv} For every positive and sufficiently small \\ quantity $\varepsilon$, the Lotka--Volterra system \cref{syst_lv_v} (written in transversal-longitudinal coordinates and where $\mu$ equals $\varepsilon^2$ --- see notation \cref{not_eps_mu}) admits a unique standing front \[ \phi_\varepsilon: \rr\rightarrow\rr^2, \quad x\mapsto \phi_\varepsilon(x) = \bigl(\phi_{\varepsilon,T}(x),\phi_{\varepsilon,L}(x)\bigr) \] connecting $E_1$ to $E_2$, taking its values in the first quadrant for the canonical coordinates (in other words such that $\phi_{\varepsilon,T}(x)$ is larger than $\abs{\phi_{\varepsilon,L}(x)}$ for every $x$ in $\rr$), and symmetric in the sense that: \begin{itemize} \item the fist component $x\mapsto\phi_{\varepsilon,T}(x)$ is even, \item and the second component $x\mapsto\phi_{\varepsilon,L}(x)$ is odd. \end{itemize} In addition, there exist functions $r_T$ and $r_L$ in $\rrr$ such that, provided that $\varepsilon$ is small enough, for every real quantity $x$, \[ \begin{aligned} \phi_{\varepsilon,T}(x) &= 1 - \frac{\varepsilon^2}{2}\bigl(1-\theta(\varepsilon x)^2\bigl) + \varepsilon^3 r_T(\varepsilon x,\varepsilon) \,,\\ \mbox{and}\quad\phi_{\varepsilon,L}(x) &= \theta(\varepsilon x) + \varepsilon r_L(\varepsilon x,\varepsilon) \,. \end{aligned} \] \end{lemma}
\begin{remark} This lemma probably remains true without the additional constraint that the front must lie in the first quadrant for the canonical coordinates, but the formulation above is sufficient for our purpose. \end{remark}
\begin{proof} Let $\varepsilon$ denote a (small) positive quantity. Replacing $\mu$ by $\varepsilon^2$, system \vref{syst_front_lv} governing stationary solutions of system \vref{syst_lv_v} reads: \begin{equation} \label{syst_front_lv_copy} G_\mu(\phi)+\phi''=0 \Longleftrightarrow \left\{ \begin{aligned} \phi_T'' &= \phi_T^2-\phi_T+\frac{\varepsilon^2}{2}(\phi_T^2-\phi_L^2) \\ \phi_L'' &= \phi_L(\phi_T-1)\,. \end{aligned} \right. \end{equation} The following intermediate lemma provides a priori bounds on the transversal component of the solution we are looking for. It is illustrated by \vref{fig:het_con}.
\begin{lemma}[a priori bound on the standing front] \label{lem:a_priori_bound_phi_T} Every global solution \\ $x\mapsto\bigl(\phi_T(x),\phi_L(x)\bigr)$ of system \cref{syst_front_lv_copy} connecting $E_1$ to $E_2$ and such that $\abs{\phi_L(\cdot)}$ is everywhere smaller than $\phi_T(\cdot)$ satisfies, for every real quantity $x$, \begin{equation} \label{a_priori_bound_phi_T} 1-\frac{\varepsilon^2}{2} < \phi_T(x) < 1 \,. \end{equation} \end{lemma}
\begin{proof}[Proof of \cref{lem:a_priori_bound_phi_T}] According to the first equation of system \cref{syst_front_lv_copy}, every such solution satisfies the differential inequalities \[ \phi_T(\phi_T-1) \le \phi_T'' \le \phi_T\bigl( (1+\varepsilon^2/2) \phi_T - 1 \bigr) \,. \] Since $\phi_T(x)$ must approach $1$ when $x$ approaches $\pm\infty$, it follows from the left-hand inequality that, for every real quantity $x$, \[ \phi_T(x) < 1 \,, \] and from the right-hand inequality that, for every real quantity $x$, \[ (1+\varepsilon^2/2) \phi_T(x)> 1 \,. \] Inequalities \cref{a_priori_bound_phi_T} follow. \Cref{lem:a_priori_bound_phi_T} is proved. \end{proof}
Let us pursue the proof of \cref{lem:exist_f_lv}. According to \cref{lem:a_priori_bound_phi_T} it is natural to express system \cref{syst_front_lv_copy} in terms of the function $\eta_T$ defined by: \[ \phi_T=1+\varepsilon^2\eta_T \,. \] With this notation system \cref{syst_front_lv_copy} becomes \[ \left\{ \begin{aligned} \eta_T'' &= \eta_T+\frac{1}{2}(1-\phi_L^2)+\varepsilon^2\bigl(\eta_T+\eta_T^2(1+\varepsilon^2/2)\bigr) \\ \phi_L'' &= \varepsilon^2\eta_T\phi_L \end{aligned} \right. \] Using the notation \[ \tilde\eta_T=\eta_T' \quad\mbox{and}\quad \tilde\phi_L=\frac{1}{\varepsilon}\phi'_L \,, \] the previous system becomes \begin{equation} \label{front_lv_dim4} \left\{ \begin{aligned} \eta_T' &= \tilde\eta_T \\ \tilde\eta_T' &= \eta_T+\frac{1}{2}(1-\phi_L^2)+\varepsilon^2 \bigl(\eta_T+\eta_T^2(1+\varepsilon^2 /2)\bigr) \\ \phi_L' &= \varepsilon\tilde\phi_L \\ \tilde\phi_L' &= \varepsilon\eta_T\phi_L \end{aligned} \right. \end{equation} This system is appropriate for a singular perturbation argument. It converges when $\varepsilon$ approaches $0$ to the ``fast'' system \begin{equation} \label{front_lv_fast} \left\{ \begin{aligned} \eta_T' &= \tilde\eta_T \\ \tilde\eta_T' &= \eta_T+\frac{1}{2}(1-\phi_L^2) \\ \phi_L' &= 0 \\ \tilde\phi_L' &= 0 \end{aligned} \right. \end{equation} for which the two-dimensional set \[ \Sigma_0=\bigl\{(\eta_T,\tilde\eta_T,\phi_L,\tilde\phi_L)\in\rr^4: \eta_T=-\frac{1}{2}(1-\phi_L^2),\quad \tilde\eta_T=0\bigr\} \] is entirely made of equilibrium points. The matrix of the linearisation of the ``fast'' system \cref{front_lv_fast} at every point of $\Sigma_0$ reads: \[ \begin{pmatrix} 0&1&0&0\\ 1&0&-\phi_L&0\\ 0&0&0&0\\ 0&0&0&0 \end{pmatrix} . \] Its eigenvalues are $-1$, $+1$, and zero with multiplicity two. Thus the dynamics of the fast system~\cref{front_lv_fast} is ``transversely hyperbolic'' at every point of the equilibrium manifold $\Sigma_0$. This is the required hypothesis to apply the singular perturbation machinery. The set $\Sigma_0$ is the graph of the function \[ H_0:\rr^2\longrightarrow\rr^2, \quad (\phi_L,\tilde\phi_L)\longmapsto(\eta_T,\tilde\eta_T) = \Bigl(-\frac{1}{2}(1-\phi_L^2),0\Bigr) \,. \] Let us consider the following subset of $\rr^2$: \begin{equation} \label{def_of_domain_D_lv} D = \dd(0,2) = \bigl\{ (\phi_L,\tilde\phi_L)\in\rr^2 : \phi_L^2 + \tilde\phi_L^2 \le 4 \bigr\} \,. \end{equation} We are going to apply Fenichel's global center manifold theorem \cite{Fenichel_geomSingPert_1979,Jones_geometricSingularPerturbationTheory_1995,Kaper_introductionGeometricMethodsSingularPerturbation_1999} with this set $D$ as definition set of the maps provided by this theorem. The properties of this set that will be used are: \begin{itemize} \item it is compact, simply connected, with a smooth boundary, \item its interior contains the trajectories of the heteroclinic connections \\ $y\mapsto\bigl(\theta(y),\pm\theta'(y)\bigr)$ (see \cref{fig:het_con}). \end{itemize} According to Fenichel's global center manifold theore, for every $\varepsilon$ sufficiently close to zero, there exists a map \[ H_\varepsilon: D\rightarrow\rr^2 \] such that the graph of $H_\varepsilon$ (denoted by $\Sigma_\varepsilon$) is locally invariant under the dynamics of~\cref{front_lv_dim4}; this means that a solution with an initial condition on $\Sigma_\varepsilon$ remains on $\Sigma_\varepsilon$ as long as $(\phi_L,\tilde\phi_L)$ remains in $D$. Moreover, the map $H_\varepsilon$ coincides when $\varepsilon$ equals zero with the previous definition of $H_0$, and $H_\varepsilon$ depends smoothly on $\varepsilon$. Thus there exist smooth functions $h$ and $\tilde h$ of three variables, defined in a neighbourhood of $D\times\{0\}$ in $\rr^3$, such that for every $(\phi_L,\tilde\phi_L)$ in $D$ and $\varepsilon$ sufficiently small, \[ H_\varepsilon(\phi_L,\tilde\phi_L)=\Bigl(-\frac{1}{2}(1-\phi_L^2)+\varepsilon h(\phi_L,\tilde\phi_L,\varepsilon), \varepsilon \tilde h(\phi_L,\tilde\phi_L,\varepsilon)\Bigr) \,. \] To study the ``slow'' dynamics in system~\cref{front_lv_dim4}, it is convenient to introduce some notation. According to \cref{not_y_x}, let us write: \[ y=\varepsilon x \ \Leftrightarrow\ x = y/\varepsilon \quad\mbox{and}\quad \Phi_L(y) = \phi_L(y/\varepsilon) \ \Leftrightarrow\ \Phi_L(\varepsilon x ) = \phi_L(x) \,. \] With this notation, the two last equations of system~\cref{front_lv_dim4} reduce to: \[ \Phi_L''=\eta_T\Phi_L \] thus the law governing the dynamics of system~\cref{front_lv_dim4} on the ``slow'' manifold $\Sigma_\varepsilon$ reduces to: \begin{equation} \label{f_lv_slow} \Phi_L'' = -\frac{1}{2}\Phi_L(1-\Phi_L^2)+\varepsilon \Phi_L h(\Phi_L,\Phi_L',\varepsilon ) \,. \end{equation} At the limit $\varepsilon$ equals $0$, this equation becomes \begin{equation} \label{f_lv_slow_lim} \Phi_L'' = -\frac{1}{2}\Phi_L(1-\Phi_L^2) \,. \end{equation} The asymptotic equation~\cref{f_lv_slow_lim} admits two hyperbolic equilibria \[ (\Phi_L,\Phi_L')=(- 1,0) \quad\mbox{and}\quad (\Phi_L,\Phi_L')=(1,0) \] and two heteroclinic solutions connecting them, which are explicitly given by: \begin{equation} \label{het_con_lv} y\mapsto\pm\tanh\Bigl(\frac{y}{2}\Bigr)=\pm\theta(y) \end{equation} (see \cref{fig:het_con}).
\begin{figure}
\caption{Heteroclinic connections in the phase space of equation~\cref{f_lv_slow} or~\cref{f_lv_slow_lim}, and corresponding standing front for the considered system.}
\label{fig:het_con}
\end{figure}
Using the symmetries of the ``full'' system \cref{front_lv_dim4}, we are going to prove that these two heteroclinic connections persist for the (perturbed) reduced equation \cref{f_lv_slow} and remain symmetric with respect to the $\phi_L\leftrightarrow -\phi_L$ symmetry inherited from the $u_1\leftrightarrow u_2$ symmetry of the initial system.
First, let us observe that for every sufficiently small positive quantity $\varepsilon$, the (perturbed) equation \cref{f_lv_slow} must admit two hyperbolic equilibria $(E_{-,\varepsilon},0)$ and $(E_{+,\varepsilon},0)$, with $E_{-,\varepsilon}$ close to $-1$ and $E_{+,\varepsilon}$ close to $1$. Let us mention here that, as usual with central manifolds, the slow manifold $\Sigma_\varepsilon$ is not necessarily unique, but it must contain every trajectory that remains globally in a small neighbourhood of it (\cite{Fenichel_geomSingPert_1979,Jones_geometricSingularPerturbationTheory_1995,Kaper_introductionGeometricMethodsSingularPerturbation_1999}). Therefore it must contain the equilibria corresponding to $E_1$ and $E_2$. It follows that $E_{-,\varepsilon}-=-1$ and $E_{+,\varepsilon}=+1$, in other words: \begin{equation} \label{equil_glob_cent_lv} h(-1,0,\varepsilon)= h(-1,0,\varepsilon) =0 \,. \end{equation}
Now, the robustness of the heteroclinic connections \cref{het_con_lv} is asserted by the following intermediate lemma.
\begin{lemma}[robustness of heteroclinic connections] \label{lem:robust_het_con} For every sufficiently small positive quantity $\varepsilon$, there exists a global solution \[ y\mapsto\Phi_{\varepsilon,L}(y) \] of the reduced equation \cref{f_lv_slow} such that \[ \Phi_{\varepsilon,L}(y) \rightarrow -1 \quad\mbox{when}\quad y\rightarrow -\infty \quad\mbox{and}\quad \Phi_{\varepsilon,L}(y) \rightarrow +1 \quad\mbox{when}\quad y\rightarrow +\infty \,, \] and, for every real quantity $y$, \[ \Phi_{\varepsilon,L}(-y) = -\Phi_{\varepsilon,L}(y) \,. \] \end{lemma}
\begin{proof} The ``full'' system \cref{front_lv_dim4} admits two symmetries, the reversibility $x\leftrightarrow -x$ and the $\phi_L\leftrightarrow -\phi_L$ symmetry inherited from the $u_1\leftrightarrow u_2$ symmetry of the initial system. To be more precise, according to these two symmetries, if \[ x\mapsto\bigl(\eta_T(x),\tilde\eta_T(x),\phi_L(x), \tilde\phi_L(x)\bigr) \] is a solution of system \cref{front_lv_dim4}, then the following two functions are also solutions: \[ \begin{aligned} & x\mapsto\bigl(\eta_T(-x),-\tilde\eta_T(-x),\phi_L(-x), -\tilde\phi_L(-x)\bigr) \\ \mbox{and}\quad & x\mapsto\bigl(\eta_T(x),\tilde\eta_T(x),-\phi_L(x), -\tilde\phi_L(x)\bigr) \end{aligned} \] It is well known that \emph{local} center manifolds of systems admitting equivariant or reversibility symmetries can be chosen in such a way that those manifolds be themselves invariant under these symmetries --- note that since center manifolds are not necessarily unique this is however not obvious --- and as a consequence in such a way that the reduced systems (obtained by reduction of the initial systems to those symmetric local center manifolds) still admit the same symmetries as the initial system. See \cite{Ruelle_bifurcationPresenceSymmetryGroup_1973} and \cite{AdelmeyerIooss_topicsBifurcationTheoryApplications_1992,HaragusIooss_localBifurcationsInfiniteDim_2011} for more recent expositions, the last one concerning infinite dimensional dynamical systems. If a similar result could be invoked for \emph{global} center manifolds, we would be able to choose our global center manifold $\Sigma_\varepsilon$ in such a way that it is invariant under the two symmetries of the full system \cref{front_lv_dim4}, namely in such a way that: \begin{itemize} \item $h(\phi_L,\tilde\phi_L,\varepsilon)$ is even with respect to $\phi_L$, \item $h(\phi_L,\tilde\phi_L,\varepsilon)$ is even with respect to $\tilde\phi_L$, \item $\tilde h(\phi_L,\tilde\phi_L,\varepsilon)$ is odd with respect to $\phi_L$, \item $\tilde h(\phi_L,\tilde\phi_L,\varepsilon)$ is odd with respect to $\tilde\phi_L$, \end{itemize} and as a consequence the reduced equation \cref{f_lv_slow} would admit the same two symmetries, and the conclusions of \cref{lem:robust_het_con} would immediately follow from these symmetries. Unfortunately, to the knowledge of the author, no statement concerning the existence of global center manifolds satisfying reversibility and equivariant symmetries and applicable in our case is available in the existing litterature. However we are going to recover this symmetry for the aforementioned heteroclinic connections by another (less direct) argument.
\begin{figure}
\caption{The four trajectories $\WuMinusOne$ and $\WsOne$ and $\WuOne$ and $\WsMinusOne$. As argued in the proof of \cref{lem:robust_het_con}, if these four trajectories differ, one of them (in this case $\WsOne$) is ``trapped'' by the other ones.}
\label{fig:stab_unstab_man}
\end{figure}
Let us fix $\varepsilon$ (positive, small) and let us consider the four trajectories $\WuMinusOne$ and $\WsOne$ and $\WuOne$ and $\WsMinusOne$ depicted on \cref{fig:stab_unstab_man} (they are part of the stable and unstable manifolds of $(-1,0)$ and $(1,0)$ for the reduced equation \cref{f_lv_slow} for this value of $\varepsilon$). Let us proceed by contradiction and assume that $\WuMinusOne$ and $\WsOne$ do not coincide and that $\WuOne$ and $\WsMinusOne$ do not coincide. Then, by a Jordan curve argument (see \cref{fig:stab_unstab_man}), we see that at least one of those four trajectories does remains ``trapped'' (by the three others) in the domain $D$ defined in \cref{def_of_domain_D_lv}. Let \[ x\mapsto\bigl(\eta_T(x),\tilde\eta_T(x),\phi_L(x), \tilde\phi_L(x)\bigr) \] denote a solution of the full system \cref{front_lv_dim4} corresponding to this trajectory. Then, due to the symmetries of this full system \cref{front_lv_dim4}, the three functions \[ \begin{aligned} & x\mapsto\bigl(\eta_T(-x),-\tilde\eta_T(-x),\phi_L(-x), -\tilde\phi_L(-x)\bigr) \\ \mbox{and}\quad & x\mapsto\bigl(\eta_T(x),\tilde\eta_T(x),-\phi_L(x), -\tilde\phi_L(x)\bigr) \\ \mbox{and}\quad & x\mapsto\bigl(\eta_T(-x),-\tilde\eta_T(-x),-\phi_L(-x), \tilde\phi_L(-x)\bigr) \end{aligned} \] are still solutions of the same system, and these three additional solutions still globally remain in a small neighbourhood of the center manifold $\Sigma_\varepsilon$. As a consequence, theses three additional solutions must also belong to $\Sigma_\varepsilon$, leading to a topological contradiction, see \cref{fig:stab_unstab_man}.
Thus at least one among the two pairs $(\WuMinusOne,\WsOne)$ and $(\WuOne,\WsMinusOne)$ must be reduced to a single trajectory, and by a similar argument this must actually be the case for both pairs. This proves the existence of the two heteroclinic connections. Their symmetries follows from the same argument. \Cref{lem:robust_het_con} is proved. \end{proof}
Let us define the function $r_L:(y,\varepsilon)\mapsto r_L(y,\varepsilon)$ by: \[ \Phi_{\varepsilon,L}(y)=\theta(y)+\varepsilon r_L(y,\varepsilon) \,. \] Since the eigenvalues of equilibria $(-1,0)$ and $(1,0)$ of equation \cref{f_lv_slow} are close to $-1$ and $+1$, the ``remaining'' function $r_L$ belongs to the space $\rrr$ defined in \cref{subsec:not}. Let us define the function \[ \phi_\varepsilon:\rr\rightarrow\rr^2, \quad x\mapsto\bigl(\phi_{\varepsilon,T}(x),\phi_{\varepsilon,L}(x)\bigr) \] by: \begin{equation} \label{expr_phi} \left\{ \begin{aligned} \phi_{\varepsilon,T}(x) &= 1 - \frac{\varepsilon^2}{2}(1-\phi_{\varepsilon,L}^2) + \varepsilon^3 h\bigl( \Phi_{\varepsilon,L}(\varepsilon x) , \Phi_{\varepsilon,L}'(\varepsilon x),\varepsilon \bigr)
\,, \\ \phi_{\varepsilon,L}(x) &= \Phi_{\varepsilon,L}(\varepsilon x)
\,. \end{aligned} \right. \end{equation} This function is a standing front connecting $E_1$ to $E_2$, it is $u_1\leftrightarrow u_2$-symmetric (in other words $\phi_\varepsilon(-x)$ equals $\sss\phi_\varepsilon(x)$ for every $x$ in $\rr$), and it takes its values in the ``first quadrant'' $\phi_T>\abs{\phi_L}$. In addition, if we define the ``remaining'' function $r_T$ by: \[ \phi_{\varepsilon,T}(x) = 1 - \frac{\varepsilon^2}{2}\bigl(1-\theta(\varepsilon x)^2\bigr) + \varepsilon^3 r_T(\varepsilon x,\varepsilon) \] then, according to equalities \cref{equil_glob_cent_lv}, this function $r_T$ belongs to $\rrr$. The proof of \cref{lem:exist_f_lv} is thus complete. \end{proof}
\subsection{First-order variation of the front speed} \label{subsec:1rst_ord}
With the notation of \cref{sec:general}, the diffusion matrix $\ddd$ equals identity. Let $\varepsilon$ denote a positive quantity, sufficiently small so that \vref{lem:exist_f_lv} holds, and let us consider the standing front $\phi_\varepsilon(\cdot)$ provided by this lemma.
\begin{lemma}[spectral stability of the standing front] \label{lem:lin_stab_lv} {\ }\\ The standing front \[ x\mapsto\phi_\varepsilon (x) \] is spectrally stable (in the sense of \vref{subsec:spec_stab_gen}, that is including the fact that the eigenvalue $0$ has an algebraic multiplicity equal to $1$) for the Lotka--Volterra system \cref{syst_lv_v}. \end{lemma}
\begin{proof} As mentioned in \cref{subsec:lv_standing_front}, this follows from the general stability results proved by Kan-on and Fang in \cite{Kan-on_parameterDependencePropSpeedTW_1995,Kan-onFang_stabilityMonotoneTW_1996}. \end{proof}
Let us choose the perturbation matrix $\bar{\ddd}$ exactly as in the toy example considerer in \cref{sec:toy}, see definition \vref{def_bar_D_toy}. For those items, all the hypotheses (H1-4) of \cref{sec:general} are satisfied. Let us denote by \[ \mathcal{L}_\varepsilon \quad\mbox{and}\quad \mathcal{L}^*_\varepsilon \quad\mbox{and}\quad \psi_\varepsilon \quad\mbox{and}\quad \bar c_\varepsilon \] the objects that were denoted by $\mathcal{L}$ and $\mathcal{L}^*$ and $\psi$ and $\bar c$ in \cref{sec:general} (these objects now depend on $\varepsilon$), and let us again denote by $\theta$ the function: $y\mapsto \tanh(y/2)$. The aim of this \namecref{subsec:1rst_ord} is to prove the following proposition.
\begin{proposition}[first order variation of front speed] \label{prop:bar_c_lv} The \\ following estimate holds: \begin{equation} \label{bar_c_lv} \bar{c}_\varepsilon \ \sim_{\varepsilon\rightarrow 0^+}\ -\varepsilon\ \frac{\int_0^{+\infty}\theta(y)\theta'(y)\theta''(y)\,dy}{\norm{\theta'}_{L^2(\rr_+,\rr)}^2} \,. \end{equation} \end{proposition}
As a consequence the quantity $\bar{c}$ is positive for every sufficiently small positive quantity $\varepsilon$. In other words, for the Lotka--Volterra competition model in the bistable regime, close to the onset of bistability, an increase of mobility provides an advantage. Since in this case the competition between the two species can be qualified as ``moderately strong'', we recover the interpretation given for the toy example in \cref{sec:toy}, that is the fact that when competition is moderately strong an increase of mobility is advantageous.
\begin{proof} We are going to use expression \cref{solv_cond} of $\bar c$, namely, according to the expression of $\bar\ddd$, \begin{equation} \label{solv_cond_lv} \bar c_\varepsilon = \int_{-\infty}^{+\infty} \bigl(\psi_{\varepsilon,T}(x)\phi_{\varepsilon,L}''(x) + \psi_{\varepsilon,L}(x)\phi_{\varepsilon,T}''(x)\bigl)\, dx \end{equation} where $x\mapsto\psi_\varepsilon(x)=\bigl(\psi_{\varepsilon,T}(x),\psi_{\varepsilon,L}(x)\bigr)$ (written in the $v$-coordinate system) is the solution of $\mathcal{L}^*_\varepsilon\psi=0$ satisfying the normalization condition \cref{norm_cond}, namely: \begin{equation} \label{norm_cond_lv} \langle\psi_\varepsilon,\phi_\varepsilon'\rangle_{L^2(\rr,\rr^2)} = 1 \end{equation} (here there would be no gain from restricting the integrals in \cref{solv_cond_lv} to $\rr_+$ as in the reduced expression \vref{s_cond_rr_plus}).
\Cref{lem:exist_f_lv} provides convenient approximations for the functions $\phi_{\varepsilon,T}''(\cdot)$ and $\phi_{\varepsilon,L}''(\cdot)$, thus what remains to be done is to get similar approximations for $\psi_{\varepsilon,T}(\cdot)$ and $\psi_{\varepsilon,L}(\cdot)$. According to expression \cref{syst_lv_v} of the Lotka--Volterra system in the transversal-longitudinal coordinate systems, the operator $\mathcal{L}_\varepsilon$ reads (in the same coordinate system): \[ \mathcal{L}_\varepsilon \begin{pmatrix} \varphi_T \\ \varphi_L \end{pmatrix} (x) = \begin{pmatrix} 1-(2+\varepsilon^2)\phi_{\varepsilon,T}(x) & \varepsilon^2\phi_{\varepsilon,L}(x) \\ -\phi_{\varepsilon,L}(x) & 1-\phi_{\varepsilon,T}(x) \end{pmatrix} \begin{pmatrix} \varphi_T(x) \\ \varphi_L(x) \end{pmatrix} + \begin{pmatrix} \varphi_T''(x) \\ \varphi_L''(x) \end{pmatrix} \,. \]
\begin{notation} For the remaining of \cref{sec:lv}, let us use the notation $r(\cdot,\cdot)$ to denote every function in the space $\rrr$ defined in \cref{subsec:not}, or every $2\times2$ or $4\times 4$ matrix having all its coefficients in this space $\rrr$. Thus each of the symbols $r(\cdot, \cdot)$ that appear in the expressions below corresponds to a (different) element of $\rrr$ or matrix of elements of $\rrr$. \end{notation}
With this notation and according to the approximation provided by \cref{lem:exist_f_lv}, the expression above reduces to \[ \mathcal{L}_\varepsilon \begin{pmatrix} \varphi_T \\ \varphi_L \end{pmatrix} (x) = \begin{pmatrix} -1+\varepsilon^2 r(y, \varepsilon) & \varepsilon^2 \theta(y) + \varepsilon^3 r(y, \varepsilon) \\ -\theta(y) + \varepsilon r(y, \varepsilon) & \frac{\varepsilon^2}{2}\bigl(1-\theta(y)^2\bigr)+\varepsilon^3 r(y, \varepsilon) \end{pmatrix} \begin{pmatrix} \varphi_T (x)\\ \varphi_L(x) \end{pmatrix} + \begin{pmatrix} \varphi_T''(x) \\ \varphi_L''(x) \end{pmatrix} ,. \] According to this expression the system $\mathcal{L}^*_\varepsilon\psi=0$ reads \[ \begin{pmatrix} \psi_{\varepsilon,T}''(x) \\ \psi_{\varepsilon,L}''(x) \end{pmatrix} = \begin{pmatrix} 1 + \varepsilon^2 r(y,\varepsilon) & \theta(y) + \varepsilon r(y,\varepsilon) \\ -\varepsilon^2 \theta(y) + \varepsilon^3 r(y,\varepsilon) & -\frac{\varepsilon^2}{2}\bigl(1-\theta(y)^2\bigr) + \varepsilon^3 r(y,\varepsilon) \end{pmatrix} \begin{pmatrix} \psi_{\varepsilon,T}(x) \\ \psi_{\varepsilon,L} (x) \end{pmatrix} \] or equivalently \begin{equation} \label{syst_psi_lv} \begin{pmatrix} \psi_{\varepsilon,T}''(x) \\ \frac{1}{\varepsilon^2}\psi_{\varepsilon,L} ''(x) \end{pmatrix} = \begin{pmatrix} 1 & \theta(y) \\ - \theta(y) & -\frac{1}{2}\bigl(1-\theta(y)^2\bigr) \end{pmatrix} \begin{pmatrix} \psi_{\varepsilon,T}(x) \\ \psi_{\varepsilon,L}(x) \end{pmatrix} + \varepsilon r(y,\varepsilon) \begin{pmatrix} \psi_{\varepsilon,T}(x) \\ \psi_{\varepsilon,L} (x) \end{pmatrix} \,. \end{equation} Let us introduce the functions $\tilde\psi_{\varepsilon,T}(\cdot)$ and $\tilde\psi_{\varepsilon,L}(\cdot)$ defined by (for every real quantity $x$): \[ \tilde\psi_{\varepsilon,T}(x) = \psi'_{\varepsilon,T}(x) \quad\mbox{and}\quad \tilde\psi_{\varepsilon,L}(x) = \frac{1}{\varepsilon}\psi_{\varepsilon,L}'(x) \,. \] Then the previous system becomes \[ \begin{pmatrix} \psi_{\varepsilon,T}' \\ \tilde\psi_{\varepsilon,T}' \\ \psi_{\varepsilon,L}' \\ \tilde\psi_{\varepsilon,L}' \end{pmatrix} (x) = \begin{pmatrix} 0 & 1 & 0 & 0 \\ 1 & 0 & \theta(y) & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix} \begin{pmatrix} \psi_{\varepsilon,T} \\ \tilde\psi_{\varepsilon,T} \\ \psi_{\varepsilon,L} \\ \tilde\psi_{\varepsilon,L} \end{pmatrix} (x) + \varepsilon r(y,\varepsilon) \begin{pmatrix} \psi_{\varepsilon,T} \\ \tilde\psi_{\varepsilon,T} \\ \psi_{\varepsilon,L} \\ \tilde\psi_{\varepsilon,L} \end{pmatrix} (x) \,. \] Another change of variables will fire the non-diagonal term in the $4\times4$ matrix above. For this purpose, let us introduce the functions $\eta_{\varepsilon,T}(\cdot)$ and $\tilde\eta_{\varepsilon,T}(\cdot)$ defined by (for every real quantity $x$): \begin{equation} \label{def_eta_lv} \eta_{\varepsilon,T}(x) = \psi_{\varepsilon,T}(x) + \theta(y) \psi_{\varepsilon,L}(x) \quad\mbox{and}\quad \tilde\eta_{\varepsilon,T}(x) = \eta_{\varepsilon,T}'(x) \,. \end{equation} Then, \[ \tilde\eta_{\varepsilon,T}'(x) = \eta_{\varepsilon,T}''(x) = \tilde\psi_{\varepsilon,T}'(x) + \varepsilon^2\theta''(y)\psi_{\varepsilon,L}(x) + 2\varepsilon^2 \theta'(y)\tilde\psi_{\varepsilon,L}(x)+ \varepsilon\theta(y)\tilde\psi_{\varepsilon,L}'(x) \,, \] thus, since according to the second line of the system above the dominant term in the expression of $\tilde\psi_{\varepsilon,T}'(x)$ is $\psi_{\varepsilon,T}(x)+\theta(y)\tilde\psi_{\varepsilon,T}(x)$ and since this dominant term equals $\eta_{\varepsilon,T}(x)$, this system can be rewritten as follows: \[ \begin{pmatrix} \eta_{\varepsilon,T}' \\ \tilde\eta_{\varepsilon,T}' \\ \psi_{\varepsilon,L}' \\ \tilde\psi_{\varepsilon,L}' \end{pmatrix} (x) = \begin{pmatrix} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix} \begin{pmatrix} \eta_{\varepsilon,T} \\ \tilde\eta_{\varepsilon,T} \\ \psi_{\varepsilon,L} \\ \tilde\psi_{\varepsilon,L} \end{pmatrix} (x) + \varepsilon r(y,\varepsilon) \begin{pmatrix} \eta_{\varepsilon,T} \\ \tilde\eta_{\varepsilon,T} \\ \psi_{\varepsilon,L} \\ \tilde\psi_{\varepsilon,L} \end{pmatrix} (x) \,. \] Since the first $2\times2$ block of the $4\times4$ matrix of this system is hyperbolic, and since the quantities $\eta_{\varepsilon,T}(x)$ and $\tilde\eta_{\varepsilon,T}(x)$ and $\psi_{\varepsilon,L}(x)$ and $\tilde\psi_{\varepsilon,L}(x)$ must approach zero when $t$ approaches plus or minus infinity, this shows that there exist a positive quantity $C$, independent of $\varepsilon$ provided that $\varepsilon$ is sufficiently small, such that, for all $x$ in $\rr$, \begin{equation} \label{bound_eta_lv} \abs{\eta_{\varepsilon,T}(x)}\le C\varepsilon\bigl(\abs{\psi_{\varepsilon,L}(x)} + \abs{\tilde\psi_{\varepsilon,L}(x)}\bigr) \quad\mbox{and}\quad \abs{\tilde\eta_{\varepsilon,T}(x)}\le C\varepsilon\bigl(\abs{\psi_{\varepsilon,L}(x)} + \abs{\tilde\psi_{\varepsilon,L}(x)}\bigr) \,. \end{equation} Let us introduce the function $\Psi_{\varepsilon,L}(\cdot)$ defined by (for every $(x,y)$ in $\rr^2$ with $y=\varepsilon x$): \[ \Psi_{\varepsilon,L}(y) = \psi_{\varepsilon,L}\Bigl(\frac{y}{\varepsilon}\Bigr) \Leftrightarrow \Psi_{\varepsilon,L}(\varepsilon x) = \psi_{\varepsilon,L}(x) \,. \] With this notation, the second equation of system \cref{syst_psi_lv} becomes: \[ \Psi_{\varepsilon,L}''(y) = -\theta(y)\psi_{\varepsilon,T}\Bigl(\frac{y}{\varepsilon}\Bigr) - \frac{1}{2}\bigl(1-\theta(y)^2\bigr)\Psi_{\varepsilon,L}(y) + \varepsilon r(y,\varepsilon) \Psi_{\varepsilon,L}(y) \] Thus, according to the notation \cref{def_eta_lv}, \[ \Psi_{\varepsilon,L}''(y) = \frac{1}{2}\bigl(3\theta(y)^2-1\bigr)\Psi_{\varepsilon,L}(y) -\theta(y)\eta_{\varepsilon,T}\Bigl(\frac{y}{\varepsilon}\Bigr) + \varepsilon r(y,\varepsilon) \Psi_{\varepsilon,L}(y) \] and thus, according to inequalities \cref{bound_eta_lv}, and up to increasing the quantity $C$, for all $y$ in $\rr$ (using the notation $\ell$ introduced in \cref{subsec:not}), \begin{equation} \label{ell_psi_L_lv} \abs{(\ell\Psi_{\varepsilon,L})(y)} = \abs{\Psi_{\varepsilon,L}''(y) - \frac{1}{2}\bigl(3\theta(y)^2-1\bigr) \Psi_{\varepsilon,L}(y)}\le C\varepsilon\bigl(\abs{\Psi_{\varepsilon,L}(y)} + \abs{\Psi_{\varepsilon,L}'(y)}\bigr) \,. \end{equation} Let \[ \alpha = \frac{\langle\Psi_{\varepsilon,L},\theta'\rangle_{L^2(\rr,\rr)}}{\norm{\theta'}_{L^2(\rr,\rr)}^2} \] and, for all $y$ in $\rr$, let \begin{equation} \label{def_chi_L} \chi_{\varepsilon,L}(y) = \Psi_{\varepsilon,L}(y)-\alpha\theta'(y) \,. \end{equation} By construction, the function $\chi_{\varepsilon,L}(\cdot)$ is orthogonal to $\theta'$, that is to the kernel of $\ell$ and since $\ell\Psi_{\varepsilon,L}$ and $\ell\chi_{\varepsilon,L}$ are equal it follows from \cref{ell_psi_L_lv} that, for every real quantity $y$, \[ \abs{(\ell\chi_{\varepsilon,L})(y)}\le C\varepsilon\bigl(\abs{\Psi_{\varepsilon,L}(y)} + \abs{\Psi_{\varepsilon,L}'(y)}\bigr) \,. \] As a consequence, up to increasing the quantity $C$, for every $y$ in $\rr$, \begin{equation} \label{bound_chi_L_chi_L_prime} \abs{\chi_{\varepsilon,L}(y)} \le C\varepsilon\bigl(\abs{\Psi_{\varepsilon,L}(y)} + \abs{\Psi_{\varepsilon,L}'(y)}\bigr) \quad\mbox{and}\quad \abs{\chi_{\varepsilon,L}'(y)} \le C\varepsilon\bigl(\abs{\Psi_{\varepsilon,L}(y)} + \abs{\Psi_{\varepsilon,L}'(y)}\bigr) \,. \end{equation} It follows from \cref{def_chi_L,bound_chi_L_chi_L_prime} that, for every real quantity $y$, \[ (1-C\varepsilon)\bigl( \abs{\Psi_{\varepsilon,L}(y)} + \abs{\Psi_{\varepsilon,L}'(y)} \bigr) \le \alpha \bigl( \abs{\theta'(y)} + \abs{\theta''(y)} \bigr) \] and as a consequence, provided that $\varepsilon$ is sufficiently small, \begin{equation} \label{bound_Psi_L_plus_Psi_L_prime} \abs{\Psi_{\varepsilon,L}(y)} + \abs{\Psi_{\varepsilon,L}'(y)} = r(y,\varepsilon) \,. \end{equation} As a consequence, it follows from \cref{def_chi_L,bound_chi_L_chi_L_prime} that \begin{equation} \label{approx_psi_L_psi_T} \psi_{\varepsilon,L}(x) = \alpha\theta'(\varepsilon x) + \varepsilon r (\varepsilon x, \varepsilon) \quad\mbox{and}\quad \psi_{\varepsilon,L}'(x) = \varepsilon \alpha\theta'(\varepsilon x) + \varepsilon^2 r (\varepsilon x, \varepsilon) \,. \end{equation} Besides, it follows from the upper bounds \cref{bound_eta_lv,bound_Psi_L_plus_Psi_L_prime} that \[ \eta_{\varepsilon,T}(x) = \varepsilon r(\varepsilon x, \varepsilon) \,. \] thus, according to the definition \cref{def_eta_lv} of $\eta_{\varepsilon,T}(\cdot)$, \begin{equation} \label{approx_psi_T} \psi_{\varepsilon,T}(x) = -\alpha \theta(\varepsilon x) \theta'(\varepsilon x) + \varepsilon r(\varepsilon x, \varepsilon) \,. \end{equation} The normalization condition \cref{norm_cond_lv} will provide the approximate value of the quantity $\alpha$. This normalization condition reads: \[ \langle\psi_{\varepsilon,T},\phi_{\varepsilon,T}'\rangle_{L^2(\rr,\rr)} + \langle\psi_{\varepsilon,L}, \phi_{\varepsilon,L}'\rangle_{L^2(\rr,\rr)}=1 \,, \] in other words, according to the expressions of $\phi_{\varepsilon,T}(\cdot)$ and $\phi_{\varepsilon,L}(\cdot)$ provided by \vref{lem:exist_f_lv}, \[ \int_{-\infty}^{+\infty} \psi_{\varepsilon,T}(x) \cdot \varepsilon^3 r(\varepsilon x,\varepsilon)\, dx + \int_{-\infty}^{+\infty} \psi_{\varepsilon,L}(x)\cdot\bigl( \varepsilon \theta'(\varepsilon x) + \varepsilon^2 r(\varepsilon x, \varepsilon)\bigr) \, dx = 1 \,. \] According to the expression \cref{approx_psi_T}, the first integral of the left-hand side of this last inequality is a $\ooo_{\varepsilon\rightarrow0}(\varepsilon^2)$. Thus it follows from the expression \cref{approx_psi_L_psi_T} for $\psi_{\varepsilon,L}(\cdot)$ that: \begin{equation} \label{estim_norm_lv} \alpha=\frac{1}{\norm{\theta'}_{L^2(\rr,\rr)}^2}+\ooo_{\varepsilon\rightarrow0}(\varepsilon) \,. \end{equation} We are now in position to estimate the value of $\bar{c}_\varepsilon$ given by \cref{solv_cond_lv}. According to \cref{lem:exist_f_lv}, \[ \phi_{\varepsilon,T}''(x)=\varepsilon^4 r(\varepsilon x, \varepsilon) \,, \] thus it follows from \cref{solv_cond_lv} and the expression \cref{approx_psi_L_psi_T} of $\psi_{\varepsilon,L}(\cdot)$ that \[ \bar{c}_\varepsilon = \ooo(\varepsilon^3) + \int_{-\infty}^{+\infty}\, \psi_{\varepsilon,T}(x)\ \phi_{\varepsilon,L}''(x) \, dx \,, \] thus, according to the expression of $\phi_{\varepsilon,L}(\cdot)$ provided by \vref{lem:exist_f_lv} and the expression \cref{approx_psi_T} of $\psi_{\varepsilon,T}(\cdot)$, \[ \begin{aligned} \bar{c}_\varepsilon &= \ooo(\varepsilon^3) + \int_{-\infty}^{+\infty} \bigl( -\alpha \theta(\varepsilon x) \theta'(\varepsilon x) + \varepsilon r(\varepsilon x, \varepsilon) \bigr)\cdot\bigl(\varepsilon^2 \theta''(\varepsilon x) + \varepsilon^3 r(\varepsilon x,\varepsilon)\bigr) \, dx \\ & = \ooo(\varepsilon^3) - \varepsilon^2\alpha \int_{-\infty}^{+\infty} \theta(\varepsilon x) \theta'(\varepsilon x)\theta''(\varepsilon x) \, dx \\ & = \ooo(\varepsilon^3) - \varepsilon\alpha\int_{-\infty}^{+\infty} \theta(y) \theta'(y)\theta''(y) \, dy \,. \end{aligned} \] Finally, according to the expression \cref{estim_norm_lv} for $\alpha$, \[ \bar{c}_\varepsilon \sim_{\varepsilon\to 0} -\frac{\varepsilon}{\norm{\theta'}_{L^2(\rr,\rr)}^2} \int_{-\infty}^{+\infty} \theta(y)\theta'(y)\theta''(y)\,dy \,, \] and restricting the integrals to $\rr_+$ estimate \cref{bar_c_lv} follows. \Cref{prop:bar_c_lv} is proved. \end{proof}
\section{Appendix} \label{sec:app}
\subsection{An elementary property of the solutions of a second order conservative equation} \label{subsec:sol_forced_2nd_order}
Let $f:[0,+\infty)\rightarrow\rr$ denote a continuous function satisfying \[ f(t)\rightarrow 0 \quad\mbox{when}\quad t\rightarrow +\infty \quad\mbox{and}\quad f(\cdot) \mbox{ does not vanish on } (0,+\infty) \,, \] and let us consider the following second order equation: \begin{equation} \label{equ_forced_2nd_order} \ddot u = u + f \,. \end{equation} The aim of this \namecref{subsec:sol_forced_2nd_order} is to prove the following lemma.
\begin{lemma}[solution homoclinic to $0$] \label{lem:sol_forced_2nd_order} There exists a unique solution $t\mapsto u(t)$ of equation \cref{equ_forced_2nd_order} defined on $[0,+\infty)$ such that \begin{equation} \label{prop_sol_forced_2nd_order} u(0) = 0 \quad\mbox{and}\quad u(t)\rightarrow 0 \quad\mbox{when}\quad t\rightarrow +\infty \,. \end{equation} This solution does not vanish on $(0,+\infty)$, and its sign is opposite to the sign of $f(\cdot)$. \end{lemma}
\begin{proof} Let $t\mapsto u(t)$ denote a solution of equation \cref{equ_forced_2nd_order} on $\rr_+$ and let us consider the functions $x=u+\dot u$ and $y=-u+\dot u$ (thus $u$ equals $(x-y)/2$. Those function satisfy the system \begin{align} \dot x &= x + f \label{dot_x}\\ \dot y &= -y + f \label{dot_y} \end{align} Equation \cref{dot_y} shows that $y(t)$ approaches $0$ when $t$ approaches $+\infty$, and since the same assertion holds for $u(t)$, it must also hold for $x(t)$. Thus, according to equation \cref{dot_x}, the function $x(\cdot)$ must be given by: \[ x(t) = -\int_t^{+\infty} e^{t-s} f(s) \, ds \,. \] This provides an explicit expression for $x(0)$, thus also for $y(0)$ since $u(0)$ equals $0$, and finally for $y(t)$ for every nonnegative quantity $t$ according to \cref{dot_y}. It follows that $u(t)$ must be equal to the following expression for every nonnegative time $t$: \begin{equation} \label{expression_bounded_solution} \frac{1}{2}\biggl( e^{-t} \int_0^t f(s) (e^{-s} - e^s) \, ds + (e^{-t} - e^t) \int_t^{+\infty} f(s) e^{-s} \, ds \biggr) \,, \end{equation} and this proves the uniqueness of a solution satisfying the conclusions \cref{prop_sol_forced_2nd_order} of \cref{lem:sol_forced_2nd_order}. Conversely, expression \cref{expression_bounded_solution} is the expression of a solution of equation \cref{equ_forced_2nd_order} and satisfies \cref{prop_sol_forced_2nd_order}. \Cref{lem:sol_forced_2nd_order} is proved. \end{proof}
\subsection{Example of two stable equilibria connected by two fronts travelling in opposite directions} \label{subsec:ex_2_eq}
Let us consider the following reaction-diffusion equation (a small perturbation of the real Ginzburg-Landau equation): \begin{equation} \label{gl} A_t=A-\abs{A}^2A+\varepsilon^2 (\bar A+i\Omega A)+A_{xx} \end{equation} where the amplitude $A$ is complex, $\varepsilon$ is a small real quantity, and $\Omega$ is a real quantity in $(-1,1)$. This equation has been studied by P.~Coullet and J.-M.~Gilli as a model for nematic liquid crystals submitted to exterior electric and magnetic fields \cite{Coullet_locPattFronts_2002}. In polar coordinates $A=\rho e^{i\theta}$ this equation transforms into the following system: \begin{equation} \label{gl_polar_coord} \left\{ \begin{aligned} \partial_t\rho &= \rho-\rho^3+\varepsilon^2\rho\cos 2\theta +\partial_{xx}\rho-\rho\partial_{x}\theta^2 \\ \partial_t\theta &= \varepsilon^2(-\sin 2\theta +\Omega)+\frac{2\partial_{x}\rho\partial_{x}\theta}{\rho}+\partial_{xx}\theta \end{aligned} \right. \end{equation} The dynamics of the reaction system (without space) can be easily understood since the expression of $\partial_t\theta$ does not depend on $\rho$ (see \cref{fig:ph_gl_pert}). It has four equilibrium points close to the circle $\rho=1$: \begin{itemize} \item $\theta=(1/2)\arcsin\Omega$ and $\theta=\pi+(1/2)\arcsin\Omega$, those are stable, \item and $\theta=\pi/2-(1/2)\arcsin\Omega$ and $\theta=3\pi/2-(1/2)\arcsin\Omega$, those are saddles. \end{itemize}
\begin{figure}
\caption{Phase space of the reaction equation.}
\label{fig:ph_gl_pert}
\end{figure}
Using a perturbation argument, we are going to show that, for $\varepsilon$ close to $0$, the two stable equilibria are connected by two fronts travelling in opposite directions.
Let $c$ denote a real quantity. A front travelling at speed $c$ is a solution of system \cref{gl_polar_coord} of the form \[ (x,t)\mapsto\bigl(\rho(x-ct),\theta(x-ct)\bigr) \,. \] Replacing this ansatz into system \cref{gl_polar_coord} and performing the change of variables $\rho=1+\varepsilon^2 r$ yields the following system: \[ \left\{ \begin{aligned} -cr' &= -2r-3\varepsilon^2 r^2-\varepsilon^4r^3+(1+\varepsilon^2 r)\cos 2\theta+r''-\frac{1}{\varepsilon^2} (1+\varepsilon r)\theta'^2 \\ -c\theta' &= \varepsilon^2(-\sin 2\theta+\Omega)+\frac{2\varepsilon^2 r'\theta'}{1+\varepsilon^2 r}+\theta'' \end{aligned} \right. \] Let us use the notation: \[ \tilde r=r' \quad\mbox{and}\quad \tilde\theta=\frac{\theta'}{\varepsilon} \quad\mbox{and}\quad \tilde c =\frac{c}{\varepsilon} \,, \] and let us consider the quantity $\tilde c$ not as a parameter, but as a (stationary) component of the differential system. The previous system transforms into the following first-order system: \begin{equation} \label{syst_fr_gl} \left\{ \begin{aligned} r' &= \tilde r \\ \tilde r' &= 2r -\cos2\theta+\tilde\theta^2 +\varepsilon\bigl(-\tilde{c}\tilde{r}+ 3\varepsilon r^2 + \varepsilon^3r^3-\varepsilon r\cos2\theta+\varepsilon r\tilde\theta^2 \bigr) \\ \theta' &= \varepsilon\tilde\theta \\ \ \tilde\theta' &= \varepsilon(-\tilde c \tilde\theta+\sin 2\theta-\Omega)-\varepsilon^2\frac{2 \tilde r\tilde\theta}{1+\varepsilon^2 r} \\ \tilde c' &= 0 \end{aligned} \right. \end{equation} At the limit $\varepsilon=0$, we get the ``fast'' system: \[ \left\{ \begin{aligned} r' &= \tilde r \\ \tilde r' &= 2r -\cos2\theta+\tilde\theta^2 \\ \theta' &= 0 \\ \tilde\theta' &= 0 \\ \tilde c' &= 0 \end{aligned} \right. \] for which the graph $\Sigma_0$ of the map \[ H_0:\rr^3\rightarrow\rr^2, \quad (\theta,\tilde\theta,\tilde c)\mapsto(r,\tilde r)=\Bigl(\frac{\cos2\theta-\tilde\theta^2}{2},0\Bigr) \] consists entirely of equilibrium points. At every point of $\Sigma_0$ this fast system is hyperbolic transversely to $\Sigma_0$; indeed, the eigenvalues of its differential are: \begin{itemize} \item $-\sqrt{2}$ and $\sqrt{2}$ (transversely to $\Sigma_0$), \item and zero, with multiplicity three (in the direction of the tangent space to $\Sigma_0$). \end{itemize}
We may thus apply Fenichel's global center manifold theorem \cite{Fenichel_geomSingPert_1979,Jones_geometricSingularPerturbationTheory_1995,Kaper_introductionGeometricMethodsSingularPerturbation_1999}. Let $D$ denote a compact and simply connected domain of $\rr^3$ with a smooth boundary (the choice of $D$ will be made later). According to this theorem, for $\varepsilon$ sufficiently close to zero, there exists a map $H_\varepsilon: D\rightarrow\rr^2$ that coincides with $H_0$ when $\varepsilon$ equals $0$ and depends smoothly on $\varepsilon$, and such that its graph $\Sigma_\varepsilon$ is locally invariant under the dynamics of~\cref{syst_fr_gl}. The dynamics on this ``slow'' manifold $\Sigma_\varepsilon$ thus reduces to the autonomous system \begin{equation} \label{front_gl_slow} \left\{ \begin{aligned} \ddot\theta &= -\tilde c\dot\theta+\sin 2\theta -\Omega - \varepsilon\frac{2\tilde{r}\dot\theta}{1+\varepsilon^2 r} \\ \dot{\tilde c} &= 0 \end{aligned} \right. \end{equation} where: \begin{itemize} \item derivatives are taken with respect to the ``slow'' time variable $y=\sqrt{\varepsilon}x$, namely: \[ \dot\theta = \tilde{\theta} = \frac{\theta'}{\varepsilon} \quad\mbox{and}\quad \ddot\theta = \frac{\tilde\theta'}{\varepsilon} = \frac{\theta''}{\varepsilon^2} \,; \] \item the quantities $r$ and $\tilde{r}$ are given by: $(r,\tilde{r}) = H_\varepsilon(\theta,\dot\theta,\tilde{c})$. \end{itemize} The first equation of system \cref{front_gl_slow} is a small perturbation of the dissipative oscillator \begin{equation} \label{f_gl_slow_lim} \ddot\theta=-\tilde c\dot\theta - V'(\theta) \quad\mbox{where}\quad V(\theta) = \frac{\cos2\theta}{2}+\Omega\,\theta\,, \end{equation} see \cref{fig:fr_gl}.
\begin{figure}
\caption{Bistable fronts travelling in opposite directions for equation~\cref{gl} ($\varepsilon>0$, $0<\Omega<1$).}
\label{fig:fr_gl}
\end{figure}
Since $\Omega$ is in $(-1,1)$, the potential $V$ admits local maxima and minima, with periodicity $\pi$. Let us assume that $\Omega$ is nonzero, and let us consider the two successive local minima \[ \frac{\arcsin\Omega}{2} \quad\mbox{and}\quad \frac{\arcsin\Omega}{2}+\pi \] of the potential $V$. It is well known that there exists a unique nonzero quantity $\tilde c_0$ (depending on $\Omega$) such that, if $\tilde c$ equals $\tilde c_0$, these two minima are connected by a (unique) heteroclinic solution of the dissipative oscillator~\cref{f_gl_slow_lim}. It is also well-known that the corresponding travelling front for the reaction-diffusion equation \[ \theta_t=-V'(\theta)+\theta_{xx} \] is stable, and therefore robust with respect to small perturbations. In other words, for every quantity $\varepsilon$ sufficiently close to $0$, there exists a unique quantity $\tilde c$ close to $\tilde c_0$ such that the perturbed system \cref{front_gl_slow} admits a heteroclinic solution close to the previous one. To this heteroclinic solution corresponds a heteroclinic solution for the full system~\cref{syst_fr_gl} (provided that the domain $D$ was chosen large enough), and finally a travelling front for the initial equation~\cref{gl}, connecting the corresponding two stable equilibria of this initial equation. The same argument can be repeated for the local minima \[ \frac{\arcsin\Omega}{2}+\pi \quad\mbox{and}\quad \frac{\arcsin\Omega}{2}+2\pi \] and proves the existence of the desired fronts travelling in opposite directions for initial equation~\cref{gl}.
\subsection{Isolation and robustness of the travelling front} \label{subsec:robustness}
This \namecref{subsec:robustness} is devoted to the proof of \vref{prop:robustness}. All the arguments are standard, and we refer for instance to \cite{CoulletRieraTresser_stableStaticLocStructOneDim_2000,Coullet_locPattFronts_2002, Sandstede_stabilityTW_2002,HomburgSandstede_homocHeteroclinicBifVectFields_2010,GuckenheimerKrauskopf_invManifGlobalBif_2015} for more details.
We keep the hypotheses and notation of \cref{subsec:setup,subsec:transv_assump}, except that the speed of the travelling front $\phi$ under consideration will be denoted by $c_0$ (instead of $c$ in \cref{subsec:setup,subsec:transv_assump}). The reason for this change is that it will be required below to consider a range of values for this speed (and not only the speed of the travelling front $\phi$).
\subsubsection{Time stability (at both ends of the front) yields spatial hyperbolicity}
Steady states of system \cref{react_diff_trav_frame} --- that is, profiles of waves travelling at velocity $c$ --- are solutions of the system \begin{equation} \label{syst_trav_wave_c} c v_\xi + F(v) + \ddd v_{\xi\xi} = 0 \ \Longleftrightarrow \ v_{\xi\xi} = - \ddd^{-1} \bigl( F(v) + c v_\xi \bigr) \,; \end{equation} the profile $\xi\mapsto \phi(\xi)$ of the travelling front is a solution of this system (which is identical to system \vref{syst_front}) for $c=c_0$. System \cref{syst_trav_wave_c} can be rewritten as the first order system: \begin{equation} \label{syst_trav_wave_c_first_order} \left\{ \begin{aligned} v' & = w \\ w' & = - \ddd^{-1} \bigl( F(v) + c w \bigr) \end{aligned} \right. \end{equation} The following statement follows from hypothesis \hypStabInfty.
\begin{lemma}[system governing the profile of the front is hyperbolic at infinity] \label{lem:spatially_hyperbolic} Both \\ equilibria $(E_-,0)$ and $(E_+,0)$ of system \cref{syst_trav_wave_c_first_order} are hyperbolic, and their stable and unstable manifolds are $n$-dimensional. \end{lemma}
\begin{proof} The linearisation of system \cref{syst_trav_wave_c} at $E_-$ or $E_+$ and the linearisation of system \cref{syst_trav_wave_c_first_order} at $(E_-,0)$ and $(E_+,0)$ read (writing $E_\pm$ for $E_+$ or $E_-$): \begin{equation} \label{lin_syst_trav_wave_c_ends} c v_\xi + DF(E_\pm)v + \ddd v_{\xi\xi} = 0 \ \Longleftrightarrow \ \left\{ \begin{aligned} v' & = w \\ w' &= - \ddd^{-1} \bigl[ DF(E_\pm) v + c w \bigr] \end{aligned} \right. \end{equation} A complex quantity $\lambda$ is an eigenvalue of this linear system if and only if there exists a pair $(v,w)$ of vectors of $\cc^n$ such that: \begin{equation} \label{lin_syst_trav_wave_c_ends_eigenv} \bigl[DF(E_\pm)v + \lambda^2 \ddd\bigr] v = - \lambda c v \ \Longleftrightarrow \ \left\{ \begin{aligned} \lambda v & = w \\ \lambda w &= - \ddd^{-1} \bigl[ DF(E_\pm) v + c w \bigr] \end{aligned} \right. \end{equation} It follows from hypothesis \hypStabInfty that such an eigenvalue $\lambda$ cannot be purely imaginary. Indeed, if we had $\lambda = ik$ for a real quantity $k$, then the last equation would read \[ \bigl[DF(E_\pm)v - k^2 \ddd\bigr] v = - i k c v \,, \] a contradiction with hypothesis \hypStabInfty (the spatially homogeneous equilibria $E_-$ and $E_+$ are assumed to be spectrally stable). This proves the hyperbolicity of $(E_\pm,0)$. If $c$ equals $0$ the solutions of the eigenvalue problem \cref{lin_syst_trav_wave_c_ends_eigenv} clearly go by pair of opposite complex numbers (this can also be viewed as a consequence of the space reversibility symmetry), thus in this case the dimensions of the stable and unstable manifolds are equal to $n$. Since the eigenvalues cannot cross the imaginary axis those dimensions remain equal to $n$ by continuity for every real quantity $c$. \Cref{lem:spatially_hyperbolic} is proved. \end{proof}
\subsubsection{Algebraic multiplicity \texorpdfstring{$1$}{1} of the eigenvalue zero is equivalent to the transversality of the heteroclinic connection defining the profile of the front}
For every real quantity $c$, let \begin{itemize} \item $W^{\textrm{u}}_c(E_-,0)$ denote the unstable manifold of the equilibrium $(E_-,0)$ and \item $W^{\textrm{s}}_c(E_+,0)$ denote the stable manifold of the equilibrium $(E_+,0)$ \end{itemize} for system \cref{syst_trav_wave_c_first_order} (note that the subscript ``$c$'' refers to the speed of the travelling frame, \emph{not} to the concept of center manifold~!). According to \cref{lem:spatially_hyperbolic} above these manifolds $W^{\textrm{u}}_c(E_-,0)$ and $W^{\textrm{s}}_c(E_+,0)$ are $n$-dimensional submanifolds of $\rr^{2n}$. Now let us rewrite system \cref{syst_trav_wave_c_first_order} as a $2n+1$-dimensional system, with the speed $c$ as a variable instead of a parameter: \begin{equation} \label{syst_trav_wave_c_first_order_2n_plus_one} \left\{ \begin{aligned} v' & = w \\ w' & = - \ddd^{-1} \bigl( F(v) + c w \bigr) \\ c' &= 0 \end{aligned} \right. \Longleftrightarrow \left\{ \begin{aligned} cv' + F(v) + \ddd v'' &=0 \\ c' &= 0 \end{aligned} \right. \end{equation} The flow in $\rr^{2n+1}$ of this system admits: \begin{itemize} \item a family of equilibria $\bigl\{(E_-,0,c):c\in\rr\bigr\}$ with an unstable manifold \[ \overline{W}^{\textrm{u}}\bigl((E_-,0)\times\rr\bigr) = \bigcup_{c\in\rr} \Bigl( W^{\textrm{u}}_{c}(E_-,0)\times \{c\} \Bigr) \,; \] \item a family of equilibria $\bigl\{(E_+,0,c):c\in\rr\bigr\}$ with a stable manifold \[ \overline{W}^{\textrm{s}}\bigl((E_+,0)\times\rr\bigr) = \bigcup_{c\in\rr} \Bigl( W^{\textrm{s}}_{c}(E_+,0)\times \{c\} \Bigr) \] \end{itemize} (both are $n+1$-dimensional submanifolds of $\rr^{2n+1}$). The function \[ \xi \mapsto \bigl( \phi(\xi), \phi'(\xi),c_0\bigr) \] is a solution of system \cref{syst_trav_wave_c_first_order_2n_plus_one} and its trajectory belongs to the intersection \[ \overline{W}^{\textrm{u}}\bigl((E_-,0)\times\rr\bigr) \cap \overline{W}^{\textrm{s}}\bigl((E_+,0)\times\rr\bigr) \,. \] Let us denote by $\Phi$ this trajectory (it is a subset of $\rr^{2n+1}$).
\begin{definition}[transverse travelling front] The travelling front $\phi$ is said to be \emph{transverse} if the manifolds $\overline{W}^{\textrm{u}}\bigl((E_-,0)\times\rr\bigr)$ and $\overline{W}^{\textrm{s}}\bigl((E_+,0)\times\rr\bigr)$ intersect transversely along the trajectory $\Phi$. \end{definition}
\begin{lemma}[multiplicity of eigenvalue zero and transversality] \label{lem:alg_mult_transv} The travelling front $\phi$ is transverse if and only if the eigenvalue $0$ of the linearised operator \[ \mathcal{L}:c_0\partial_\xi+ DF(\phi) + \ddd\partial_{\xi\xi} \] has an algebraic multiplicity equal to $1$. \end{lemma} In other words, hypothesis \hypTransv is equivalent to the transversality of the travelling front $\phi$.
\begin{proof} A small perturbation \[ \xi \mapsto \bigl( \phi(\xi), \phi'(\xi),c_0\bigr) + \varepsilon \bigl( v(\xi), w(\xi), c(\xi) \bigr) \] is (at first order in $\varepsilon$) a solution of system \cref{syst_trav_wave_c_first_order_2n_plus_one} if $(v,w,c)$ are a solution of the linearised system: \begin{equation} \label{syst_trav_wave_c_first_order_2n_plus_one_lin} \left\{ \begin{aligned} v' & = w \\ w' & = - \ddd^{-1} \bigl( DF(\phi)v + c_0 w + c\phi'\bigr) \\ c' &= 0 \end{aligned} \right. \Longleftrightarrow \left\{ \begin{aligned} c_0 v' + DF(\phi)v + \ddd v'' &= - c\phi' \\ c' &= 0 \end{aligned} \right. \end{equation} Observe that the restriction of system \cref{syst_trav_wave_c_first_order_2n_plus_one_lin} to the $2n$ first coordinates reads: \[ \mathcal{L} v = - c \phi' \,. \] The tangent space in $\rr^{2n+1}$ to the unstable manifold $\overline{W}^{\textrm{u}}\bigl((E_-,0)\times\rr\bigr)$ along $\Phi$ is made of the solutions of system \cref{syst_trav_wave_c_first_order_2n_plus_one_lin} satisfying \[ \bigl(v(\xi),w(\xi)\bigr)\rightarrow (0,0) \quad\mbox{when}\quad \xi\rightarrow -\infty \,, \] and the tangent space in $\rr^{2n+1}$ to the stable manifold $\overline{W}^{\textrm{s}}\bigl((E_+,0)\times\rr\bigr)$ along $\Phi$ is made of the solutions of system \cref{syst_trav_wave_c_first_order_2n_plus_one_lin} satisfying \[ \bigl(v(\xi),w(\xi)\bigr)\rightarrow (0,0) \quad\mbox{when}\quad \xi\rightarrow +\infty \,. \] According to \cref{lem:spatially_hyperbolic} these two tangent spaces are $n+1$-dimensional; besides their intersection contains (at least) the one-dimensional space $\spanset(\phi',0)$. Thus they intersect transversely if and only if their intersection is actually \emph{reduced} to $\spanset(\phi',0)$. And this is true if and only if there does not exist a quantity $c$ such that system \cref{syst_trav_wave_c_first_order_2n_plus_one_lin} admits a solution $\xi\mapsto v(\xi)$ outside of $\spanset(\phi')$ approaching zero at infinity; in other words, if and only if the eigenvalue $0$ of the operator $\mathcal{L}$ has algebraic multiplicity $1$. \Cref{lem:alg_mult_transv} is proved. \end{proof}
Since stable and unstable manifolds depend continuously on the reaction function $F$ and the diffusion matrix $\ddd$ defining system \cref{react_diff}, a transverse travelling front is isolated and robust (according to the definitions stated in \vref{subsec:transv_assump}). As a consequence, \vref{prop:robustness} follows from \cref{lem:alg_mult_transv}. \Cref{prop:robustness} is proved.
\begin{remark} It can be seen from the proof of \cref{lem:alg_mult_transv} above that the null space of $\mathcal{L}$ is one-dimensional (that is, the eigenvalue zero has geometric multiplicity one) if and only if the intersection of $W^{\textrm{u}}_{c_0}(E_-,0)$ and $W^{\textrm{s}}_{c_0}(E_+,0)$ (in $\rr^{2n}$, without the additional dimension of the speed $c$) is transverse. And in this case, the algebraic multiplicity will also be one if and only if the Melnikov integral defined by the first order dependence of system \cref{syst_trav_wave_c_first_order} with respect to the parameter $c$ is nonzero \cite{GuckenheimerKrauskopf_invManifGlobalBif_2015}. \end{remark}
It is commonly accepted that hypotheses \hypStabInfty and \hypTransv hold generically for travelling fronts of system \cref{react_diff} (say for a generic reaction function $F$ once the diffusion matrix $\ddd$ is fixed, or for a generic pair $(F,\ddd)$. Genericity of \hypStabInfty is standard since it reduces to the hyperbolicity of the equilibrium points of $F$. Concerning the second hypothesis \hypTransv, a rough justification follows from the equivalence with the transversality of the front. However, to the knowledge of the author, a rigorous justification of this transversality has not been written yet. A joint work in progress with Romain Joly aims at providing such a rigorous justification, however only under the additional hypothesis that the reaction term is the gradient of a potential.
\end{document} | arXiv |
Eliminate non-local references from closure
For a code similarity detection framework I need to eliminate references to non-local variables, for example having the following closure:
var a;
var b;
var f = function() {
a = 5;
b = 6;
f();
would be converted into something like:
var b = 6;
return tuple(a, b);
a, b = f();
Another approach would be:
var world;
var f = function(world) {
world = update_world(world, "a", 5);
world = update_world(world, "b", 6);
return world;
world = f(world);
a = get_world(world, "a");
b = get_world(world, "b");
Basically, convert a stateful program to a purely functional one. The goal is then to transform the resulting program into the lambda calculus, normalize and look for isomorphic subtrees.
Is there a formalization of what I'm trying to do?
reference-request programming-languages compilers functional-programming imperative-programming
Raphael♦
Philip KamenarskyPhilip Kamenarsky
$\begingroup$ I suspect the applicable algorithms might depend on the language you are dealing with and its characteristics. Are you dealing with a functional language or an imperative one? I presume it must allow side-effects. Does it have references? pointers? Is it strongly typed? Incidentally, your first proposed conversion is not semantics-preserving in general if there's other code that writes to the variables a, b. $\endgroup$
– D.W. ♦
$\begingroup$ @D.W There is a huge collection of techniques that may be applicable, since the proposed task is basically a compilation process, possibly partial/incremental compilation. The target language is domain extended λ-calculus, which imply some standard transformations such as continuations or passing state as parameter and (part of) result of functions. So it can be described by piecemeal compiler bibliography. The systematic view of compiling programming languages to λ-calculus is precisely the object of denotational semantics, though the original aim is defining a compositional semantics. $\endgroup$
– babou
Denotational Semantics was an answer almost intended for your question
There is a huge collection of techniques that may be applicable, since the proposed task is basically a compilation process, possibly partial / incremental compilation. The target language is domain extended λ-calculus, which imply some standard transformations such as continuations or passing state as parameter and (part of) result of functions. So it can be described by piecemeal compiler and functional programming bibliography.
But the systematic view of compiling programming languages to λ-calculus is precisely one of the purposes of defining denotational semantics of programming languages, though it is a consequence of the original aim: defining a compositional semantics. Compositionality is precisely what is naturally given by λ-calculus. This is illustrated by the fact that the first environment for building systematically the denotational semantics of a language, P. Mosses'SIS, was also considered the first complete compiler-compiler.
Denotational semantics offers a systematic framework for the stated goal, with a strong and fairly unified theoretical basis.
Yes, there is a formalization of what you are trying to do, and it has been existing for 35 years, though some techniques may have evolved, It was for example applied to the very imperative language Ada around 1980.
What you want is very precisely and formally what is done by using denotational semantics: transform your stateful program, or program fragment, into a lambda expression. It does use many compiling and functional programming techniques, such as continuations and other higher order functions.
Indeed the structure of the last example in the question, introducing a memory environment called world is typical of denotational semantics. In denotational semantics world is a "state mapping" $S:\mathrm{State}$ representing the memory state. This state is passed as argument to all functions, so that functions can access global memory (by whatever means), and returned by function to be available in the calling environment. Of course the $\mathrm{State}$ value returned may differ from the one received as parameter, exactly as the world value returned by function f is not the one it received.
For some simple examples, you may look at the question: Denotational semantics of expressions with side effects.
If you have a reference denotational semantics for the language you are considering, then you are sure that no language feature is forgotten in the translation process that could invalidate what you are trying to do.
If you do it on an ad hoc basis for a specific program fragment, you get no garantee that you are taking all you should into consideration.
If you do not have a denotational semantics definitions for your language, you may get ideas from looking at how it handles specific features for other existing languages., or look at the denotational semantics literature in general.
I know denotational semantics does not seem very popular these days. But it was developed and used for that purpose, in addition to more theoretical and abstract aims.
The idea of thus using denotational semantics was pionneered by Peter Mosses with his Semantics Implementation System (SIS)1 developed in the late 1970s. It was doing just that: transform (compile) programs into big lambda expressions according to the denotational semantics of the programming language, which had first to be entered in the system.
There is considerable literature and books on denotational semantics, which you can easily find by searching the web.
Not surprisingly, many of the techniques used are compiler writing techniques, and the system SIS is considered as the first compiler-compiler. Hence much of the literature on compilers can be relevant too, as shown by the important bibliography in the previous answer by Wandering Logic.
The fact is that a denotational semantics specification of a programming language can be read as high-level code for a compiler of the language into lambda calculus, up to some predefined domains of values.
1 Peter Mosses, "SIS: A Compiler-Generator System Using Denotational Semantics," Report 78-4-3, Dept. of Computer Science, University of Aarhus, Denmark, June 1978
baboubabou
I think what you want is essentially conversion to static single assignment (SSA) form, followed by closure conversion, followed by conversion to continuation passing style.
Static single assignment form guarantees that each variable is written exactly once in the program text. That this is the key step in converting imperative to functional programs is argued by
Andrew W. Appel. SSA is functional programming. SIGPLAN Notices, 33(4), 1998.
Richard A. Kelsey. A correspondence between continuation passing style and static single assignment form. Proc ACM SIGPLAN Workshop on Intermediate Representations, January 1995.
Peter J. Landin. The mechanical evaluation of expressions. Computer Journal, 6(4):308–320, 1964.
This starts to get really interesting for programs with loops. You need to convert the loops into recursive calls, and make a unique copy for the variables written in each iteration of the loop. In the traditional loop optimization community this technique is called scalar expansion or (sometimes) dynamic renaming.
David J. Kuck, R. H. Kuhn, David A. Padua, B. Leasure, and Michael Wolfe. Dependence graphs and compiler optimizations. In Conference Record of the Eighth Annual ACM Symposium on Principles of Programming Languages, pages 207–218, Williamsburg, VA, January 1981.
Ron Cytron and Jeanne Ferrante. What's in a name? The value of renaming for parallelism detection and storage allocation. In Proceedings of the 16th Annual International Conference on Parallel Processing, pages 19–27, St. Charles, IL, August 1987.
In the functional programming community this involves making closures and continuations explicit:
Andrew W. Appel and Trevor Jim. Continuation passing, closure-passing style. Proc. Symp. on Principles of Programming Languages (POPL), 1989.
Guy Lewis Steele. RABBIT: A compiler for Scheme. Technical Report AITR-474, MIT Artificial Intelligence Laboratory, May 1978.
I needed to do these conversions, for the automatic parallelization techniques described in my dissertation. Chapters 3, and 9.1.
Matthew Frank, SUDS: Automatic Parallelization for Raw Processors, Ph.D. thesis, Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, May 23, 2003.
Wandering LogicWandering Logic
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Why do functional languages disallow reassignment of local variables? | CommonCrawl |
Linking common human diseases to their phenotypes; development of a resource for human phenomics
Şenay Kafkas1,
Sara Althubaiti1,
Georgios V. Gkoutos2,3,
Robert Hoehndorf ORCID: orcid.org/0000-0001-8149-58901 &
Paul N. Schofield4
Journal of Biomedical Semantics volume 12, Article number: 17 (2021) Cite this article
In recent years a large volume of clinical genomics data has become available due to rapid advances in sequencing technologies. Efficient exploitation of this genomics data requires linkage to patient phenotype profiles. Current resources providing disease-phenotype associations are not comprehensive, and they often do not have broad coverage of the disease terminologies, particularly ICD-10, which is still the primary terminology used in clinical settings.
We developed two approaches to gather disease-phenotype associations. First, we used a text mining method that utilizes semantic relations in phenotype ontologies, and applies statistical methods to extract associations between diseases in ICD-10 and phenotype ontology classes from the literature. Second, we developed a semi-automatic way to collect ICD-10–phenotype associations from existing resources containing known relationships.
We generated four datasets. Two of them are independent datasets linking diseases to their phenotypes based on text mining and semi-automatic strategies. The remaining two datasets are generated from these datasets and cover a subset of ICD-10 classes of common diseases contained in UK Biobank. We extensively validated our text mined and semi-automatically curated datasets by: comparing them against an expert-curated validation dataset containing disease–phenotype associations, measuring their similarity to disease–phenotype associations found in public databases, and assessing how well they could be used to recover gene–disease associations using phenotype similarity.
We find that our text mining method can produce phenotype annotations of diseases that are correct but often too general to have significant information content, or too specific to accurately reflect the typical manifestations of the sporadic disease. On the other hand, the datasets generated from integrating multiple knowledgebases are more complete (i.e., cover more of the required phenotype annotations for a given disease). We make all data freely available at https://doi.org/10.5281/zenodo.4726713.
The genomic revolution in medicine has been driven by the huge success of next generation sequencing and the availability of millions of genome or exome sequences [1, 2]. However, the utility of genomic sequence is determined firstly by our knowledge of the relationship between genomic variants and disease conditions or predispositions [3–5], and secondly by our knowledge of the relationship between protein function and sequence [6, 7]. To date, the application of this knowledge has been focused in two areas, rare diseases, generally Mendelian [8], and common or complex diseases [5, 9].
A significant breakthrough in rare disease diagnostics and candidate gene discovery was facilitated by the development of phenotype ontologies that capture the phenotypes associated with a disease entity. These are now available not only for human and most of the model organisms [10, 11], but also as unified and integrated phenotype ontologies [12, 13] where the equivalences and relationships between phenotypes in different species are captured. Use of these ontologies to establish phenotypic similarity between undiagnosed patients and known human disease entities, or model organism mutants, has provided useful new diagnostic support and discovery tools [14–16]. This work has benefited considerably from careful phenotypic characterization of the known rare recurrent and Mendelian diseases of which there are estimated to be about 7,000 [10, 17, 18]. Efforts to annotate complex and common diseases with their phenotypes have been limited by the scale of the task, with 14,400 distinct diagnoses and causes of death identified in ICD-10 [19] and 69,000 diagnosis codes in ICD-10-CM, of which many are phenotypic variants.
The advantage of creating a corpus of phenotype annotations to common diseases is that they can then be used to computationally search for phenotypic and genetic associations between phenotypes or diseases, for identification of new phenotypic subgroups, and for diagnostic support and facilitation of the analysis of electronic patient record data [20–23]. In recent years, evidence has accumulated to support a model in which different diseases have common underlying etiopathological mechanisms and shared phenotypes, or endophenotypes [24, 25]. The hypothesis that similarity between phenotypes reflects underlying biological modules of functionally related genes has been convincingly demonstrated for Mendelian disease but little work has been done for common and complex disease [26, 27]. An earlier study of common diseases [28] suggested that these diseases form modules related to Mendelian genes with similar phenotypes. Similarly, the phenotype study by Ghiassian et al. [24] took three selected endophenotypes (inflammation, thrombosis, and fibrosis) and genes annotated to these, to show that the genetic modules associated with each phenotype interacted together to generate inflammation.
We have previously used text mining to annotate the diseases covered by the Human Disease Ontology (DO) [29], which contains both common and rare diseases, and demonstrated that phenotypically closely related diseases are linked at the level of underlying etiology and align with existing nosology [30]. However, this set of annotations was limited at the time to those diseases in DO and used only literature-derived associations. Many disease-phenotype pairs have been gathered from human genetic studies and animal model experiments and are now available from large-scale public resources [31, 32]; one aim of the current study was to leverage these curated public resources alongside a more comprehensive text mining effort. However, these resources are far from being complete, and few phenotypes are linked to terms found in ICD-10 [33], which is the primary disease terminology used in clinical practice. A large number of disease–phenotype associations are still latent in the literature and require automated methods to extract.
Here, we focus on the diseases in ICD-10 and link them to their relevant phenotypes from the Human Phenotype (HPO) and Mammalian Phenotype ontologies (MP) [34, 35]. We present two approaches to gather disease-phenotype associations. One of them is text mining from the literature and the other one is semi-automatic harvesting from publicly available curated resources. To extract the disease-phenotype associations from text, we utilize the semantics of the PhenomeNET ontology to increase the coverage of annotations that are not explicitly mentioned [12] in text, and apply a statistical approach to find significant associations between a disease and sets of phenotypes. We evaluate our text mining predictions against the known disease–phenotype associations from the HPO database [31]. Furthermore, we demonstrate the utility of the generated datasets in predicting gene–disease associations from Mouse Genome Informatics (MGI) [32] based on phenotype similarity. We provide all of the datasets of disease–phenotype associations at https://doi.org/10.5281/zenodo.4726713.
Resources used
We have built a semantic disease resource from the Unified Medical Language System (UMLS) [36] and use it for disease–phenotype association extraction. We use UMLS due to its completeness, but also its extensively validated mappings to other terminologies, specifically ICD-10 and HPO, which are not available elsewhere. As one of our main aims is to facilitate the use of data in Electronic Health Records, we focus here on diseases included in the ICD-10. However, it is straightforward to map our resource onto other known disease resources such as DO [29] and Mondo Disease Ontology [37] through the ICD mappings in them.
To generate our disease resource, we first parsed UMLS data (from the file MRCONSO.RRF in UMLS, downloaded on 04/11/2019) and gathered all the disease concepts along with their labels, definitions and sub-class relations which were mapped to any of the classes in ICD-10, SNOMED CT, HPO, or Online Mendelian Inheritance in Man (OMIM) [38]. We then represent the resulting integrated resources using the Web Ontology Language (OWL) [39] where we assert rdfs:subClassOf between two classes that are sub-classes in UMLS, and rdfs:label properties based on all the collected labels and definitions. Although our main focus was ICD-10 diseases, we included disease classes represented in SNOMED, HPO, and OMIM as well in the generation of this resource so as to benefit from their ontological structure and maintain an asserted hierarchy among the disease concepts. This semantic disease resource covers a total of 1,535,927 disease labels from 519,735 disease concepts. We used this data to identify disease mentions (names, synonyms and acronyms) in text.
We used two phenotype ontologies, HPO and MP, to identify phenotypes in text. Both ontologies contain classes that are relevant to humans, so to cover the complete phenotype profile of a given disease as completely as possible we used MP in addition to HPO. We used only the subclasses of the Phenotypic abnormality branch of HPO (14,749 HPO classes in total)to generate the text-mined dataset as this branch covers phenotypes that can be readily associated with diseases. For the semi-automatically generated dataset, we considered all of the HPO classes as the data is seeded from curated annotations. We used primary and alternative class labels along with synonyms in text matching. We used the PhenomeNET [40] ontology, which includes the phenotypes from HPO and MP, to generate embeddings for diseases and genes. Briefly, PhenomeNET is developed by transforming phenotype ontologies into a formal representation, combining phenotype ontologies with anatomy ontologies, and applying a measure of semantic similarity to construct a cross-species phenotype network.
Known disease-phenotype, gene-phenotype and gene-disease associations
We gathered the known disease–phenotype associations from UMLS [36], the HPO database [31] on 12/10/2020 and Wikidata [41] on 13/09/2020. We gathered the known mouse gene–phenotype [42] as well as gene–disease associations [43] from MGI [32] on 15/03/2021.
UK biobank
We generated a list of 2,106 common diseases from the UK Biobank identified by their ICD-10 codes. To generate this list, we considered only ICD-10 codes that have 100 or more patients in UK Biobank, as identified through the main or secondary diagnosis fields.
Generating disease-phenotype associations
In this study, we developed two methods; one of them is a text-mining based method and the other one is a semi-automatic way to gather asserted disease–phenotype associations from public data resources. By using these two methods, we generated a total of four datasets in this study. Figure 1 depicts an overview of the processes applied and the datasets generated. First, we generated two independent datasets; one by text mining disease–phenotype associations from PubMed abstracts (this dataset is labeled "Text Mined") and an another one by semi-automatically gathering the known disease-phenotype associations from multiple resources (labeled "Semi-automatic").
Overview of generating disease-phenotype associations. We used two methods, text mining and a semi-automatic way to generate two independent datasets. We further generated subsets of these datasets covering the phenotype associations of common diseases from UK Biobank. To generate the semi-automatic (UKB) dataset, we manually curated the selected diseases
Linking genotype and phenotype can be extremely helpful in DNA sequence analysis for revealing causative variants. Therefore, we selected the disease–phenotype associations linked to common diseases in the UK Biobank from these two datasets. In the case of the Text Mined dataset, we retrieved all of the selected associations into the data subset we call "Text Mined (UKB)". In the case of the Semi-automatic dataset, we further applied expert manual curation on the selected associations and generated the "Semi-automatic (UKB)" dataset.
Text mining disease-phenotype associations
Figure 2 depicts the overview of text mining of disease-phenotype associations. To extract disease–phenotype associations, we first indexed approximately 30 million PubMed abstracts downloaded on 22/09/2019 from ftp://ftp.ncbi.nlm.nih.gov/pubmed/baseline using Apache Lucene [44]. Secondly, we identified the abstract level occurrence and co-occurrence of each disease–phenotype pair. We then propagated the co-occurrence statistics by using the semantics in PhenomeNET (i.e., if C is a subclass of D in the PhenomeNET ontology, every mention of C in an abstract is also considered a mention of D). and calculated the normalized pointwise mutual information (NPMI) [45] to measure the strength of the association.
Overview of text mining disease-phenotype associations. Disease labels are gathered from OMIM, SNOMED and HPO records which are linked to ICD-10 in UMLS. Phenotype labels are gathered from HPO and MP
NPMI is a measure of collocation of two terms. As the disease and phenotype concepts are represented by ontology classes, we reformulated the NPMI to measure the collocation between two classes. First, we identify the set of labels and synonyms associated with every class; Labels(C) denotes the set of labels and synonyms of C. We then define Terms(C) as the set of all terms that can be used to refer to C: \(Terms(C) := \{ x | x \in Labels(S) \land S \sqsubseteq C \}\).
We calculated the NPMI between classes C and D as
$$ npmi(C,D) = \frac{\log{\frac{n_{C,D}\cdot n_{tot}}{n_{C} \cdot n_{D}}}}{-\log{\frac{n_{C,D}}{n_{tot}}}} $$
where ntot is the total number of abstracts in our corpus in which at least one disease and one phenotype name co-occur, nC,D is the number of abstracts in which both a term from Terms(C) and a term from Terms(D) co-occur, nC is the number of abstracts in which a term from Terms(C) occurs, and nD is the number of abstracts in which a term from Terms(D) occurs.
Gathering disease-phenotype associations from public data sources
We generated the Semi-automatic dataset of disease–phenotype associations based on gathering known associations from UMLS, Wikidata, and the HPO database, and then propagating them based on the superclass relations defined in ICD-10 and lexical match of superclasses of diseases in the HPO dataset (see Fig. 3). To generate this dataset, first, we gathered the known disease–phenotype associations from UMLS and Wikidata. We collected a total number of 2,340 ICD-10–HPO direct mappings for 2,268 distinct diseases from UMLS. We used SPARQL queries and retrieved a total number of 2,029 distinct associations for 404 distinct diseases (mapped to ICD-10) with their symptoms and secondary effects (mapped to HPO) from Wikidata; the SPARQL queries we used are available as Supplementary Materials. Second, for capturing the known disease–phenotype associations from the HPO database, we applied an additional disease identifier conversion step. The HPO database contains disease–phenotype associations where the diseases are mapped to OMIM and the phenotypes are mapped to HPO. To include these associations, we mapped the OMIM diseases to their ICD-10 codes by using the ICD-10–OMIM mappings from UMLS and Wikidata. We gathered the mappings from Wikidata using another SPARQL query (see). We extracted 303 and 5,747 ICD-10–OMIM mappings from UMLS and Wikidata, respectively. Merging these two resources, we obtained a total of 5,845 distinct ICD-10–OMIM mappings where 1,447 distinct ICD-10 codes are mapped to their corresponding OMIM identifiers. Altogether, utilizing the obtained ICD-10–OMIM mappings, we gathered a total number of 41,529 ICD-10–HPO associations for 1,366 distinct diseases from the HPO database. Third, we filtered out the associations involving 21 generic, not informative phenotypes which were manually identified by an expert from the dataset (e.g., HP:0000005Mode of inheritance, HP:0000006Autosomal dominant inheritance, HP:0012824Severity, HP:0025285Aggravated by, HP:0012834 Right). Fourth, we propagated the known annotations based on the superclass relations in ICD-10 coding system hierarchy. For example, phenotypes linked to (ICD-10:G30) Alzheimer's disease are propagated to all 4 of its sub-classes (ICD-10:G30.0 Alzheimer's disease with early onset, ICD-10:G30.1 Alzheimer's disease with late onset, ICD-10:G30.8Other Alzheimer's disease, ICD-10:G30.9Alzheimer's disease, unspecified). We also propagated annotations to the diseases from their superclasses that we find by lexical match in the HPO database. For example, we linked ICD-10:I84.4, External hemorrhoids with complications to Hemorrhoids (HP:0032551) (see Fig. 3c).
Overview of semi-automatic gathering known disease-phenotype associations. (a) Collecting known ICD-10-HPO associations from WikiData, UMLS and HPO, filtering associations with generic phenotypes and propagation based on ICD-10 hierarchy and lexical match of disease super-classes in HPO. (b) A sample ICD-10 Hierarchy. (c) A sample lexical match of disease super-class in HPO
Manual curation of "Semi-automatic" disease–phenotype associations
Linking genotype and phenotype is important for understanding the underlying mechanisms of genomic disorders. Therefore, we selected the phenotype associations of the common diseases in UK Biobank from this dataset and manually curated them. We named this curated dataset "Semi-automatic (UKB)" and released it as an additional resource. To generate this subset of data, we first identified and prioritized a total of 2,106 diseases which are common, defined as those for which at least 100 individuals in UK Biobank have either a primary or secondary diagnosis. Second, we retrieved the known and propagated associations of these 2,106 diseases. Third, we manually added the phenotypes for the diseases for which we could not find any associations after applying the second step. Missing ICD-10–HPO annotations were provided by expert curation and reference to the literature.
Clinical presentation was checked initially using expert knowledge supported by standard texts [46, 47] and then examined in further depth using recent literature reviews and papers. Examination of the HPO classes annotated to diseases in this semi-automated dataset revealed several broad curation strategies for annotation taken by each contributing dataset; UMLS, Wikidata, and HPO database. Each type of annotation was not limited to one data source (for example direct mapping was found in all contributing datasets), and reflects the various strategies and individual pragmatic decisions adopted by the contributing resources. This is a compromise in comparison with a priori expert annotation, but our overall validation suggests that it does not compromise the utility of our datasets. Several examples are shown below:
Direct mapping In some cases, HPO classes reflect a simple mapping from ICD-10 to HPO; for example ICD-10:N20.0Calculus of kidney is annotated with HP:0000121 Nephrocalcinosis.
High level mappingICD-10:H40.1, Primary open-angle glaucoma is annotated with HP:0000478, Abnormality of the eye.
Symptom manifestationICD-10:C91.1, Chronic lymphocytic leukaemia is annotated with HP:0040088, Abnormal lymphocyte count
Associated phenotypes including etiological predication, and closely related diseases or phenotypesICD-10:E21.0, Primary hyperparathyroidism is annotated with HP:0011769, Ectopic parathyroid; similarly, ICD-10:D75.2, Essential thrombocytosis, is annotated with HP:0011974, Myelofibrosis, a closely linked disorder [48].
Each method of assigning annotations used by the contributing resources had, to a greater or lesser degree, a bias towards one or more of these patterns (see below). While restricting annotations simply to signs and symptoms would have met the aim of disaggregating diseases into their constituent phenotypes, much information would have been lost and in fact HPO itself includes annotations of all of these types to advantage. We show below that using all of the methods of annotation resulted in a much better predictive outcome on the disease-gene validation task, justifying the inclusion of all of these types in annotation. The possible bias introduced by annotation density and the level of annotation to superclasses is discussed below.
In the last step, the generated ICD-10–HPO associations for 2106 ICD-10 codes were assessed by a biomedical expert for inappropriate and incorrect HPO annotations (false positives) and removed. The final number of common ICD-10 codes that could be linked to their phenotypes is 1,995.
Measuring phenotypic similarities of genes and diseases
We used OWL2Vec* [49] to generate embeddings for entities based on their associations with phenotypes. OWL2Vec* is a method to generate embeddings for classes in OWL ontologies. OWL2Vec* converts an ontology into a graph based on syntactic patterns represented in the ontology axioms; in the graph, nodes correspond to either classes or individuals in the ontology, and edges correspond to axiom patterns. OWL2Vec* then applies a graph embedding method based on random walks to nodes in this graph; it explores the node neighborhood through iterated random walks with a subtree kernel and uses Word2Vec [50] to embed nodes within a vector space.
For each disease, we generated two ontology embeddings, one based on the text mined phenotype profile and the other on the phenotype profile from the HPO database. We used the default parameter settings of the OWL2Vec* implementation: vector size 100, window size 5, minimum occurrence count of 1, skip-gram (sg) model, random walk with depth (number of walk) of 3.
We measured the similarity between the ontology embeddings using cosine similarity:
$$ \text{sim}\left(v_{1}, v_{2}\right)=\frac{v_{1} \cdot v_{2}}{\left\|v_{1}\right\|\left\|v_{2}\right\|} $$
where v1 and v2 are two vectors representing two given entities.
Disease–phenotype datasets
We generated a total of four datasets covering disease–phenotype associations (see Fig. 1) by using text mining and semi-automatic collection of associations from public data resources. The "Text Mined" and "Semi-automated" datasets contain all of the text mined and semi-automatically gathered ICD-10-phenotype associations respectively. "Text Mined (UKB)" is the subset of "Text Mined" which covers the associations of only common diseases found in UK Biobank. On the other hand, "Semi-automated (UKB)" covers further manually curated known associations of these common diseases. Table 1 presents the distribution of the associations in the generated datasets based on their provenance. Our aim is specifically to associate common diseases in ICD-10 with phenotypes so that we can map datasets using ICD-10 to phenotypes.
Table 1 Distribution of disease-phenotype associations in the generated datasets by provenance
The Text Mined dataset covers a total of 2,755,333 positive disease–phenotype associations (NPMI score > 0) between 13,610 distinct phenotype classes (from either MP or HP) and 6,263 distinct diseases (from ICD-10) from the literature. A total of 985,511 out of 2,755,333 disease–phenotype annotations can be linked to 1,557 of 2,106 common ICD-10 codes (Text Mined (UKB)). For the remaining 549 diseases, we could not find any positive association from the literature based on our approach.
The Semi-automatic dataset covers a total of 57,671 ICD-10–HPO associations among 7,610 distinct ICD-10 classes and 6,741 distinct phenotypes obtained by integrating a number of manually curated datasets (see Table 1). Out of the 57,671 ICD-10–HPO associations, we gathered the majority of the associations (37,810 out of 57,671 associations, linked to 4,207 of the 7,610 ICD-10 classes) through resources covering rare or common diseases. We obtained a total of 1,838 association from Wikidata, 32,323 associations from the HPO database through OMIM–ICD-10 links from Wikidata and 2,362 through OMIM–ICD-10 links from UMLS; we also obtained 1,287 associations directly from UMLS. We gathered the remaining 19,861 associations (linked to 3,403 of the 7,610 ICD-10 classes) by propagating phenotype annotations of diseases from their subclasses in the ICD-10 hierarchy. We obtained 10,201 out of 19,861 associations by propagating phenotypes from their superclass based on the ICD-10 hierarchy; we obtained the remaining 9,660 out of 19,861 associations by lexical match between the superclass labels and the phenotype labels in HPO.
We sub-selected 2,106 distinct ICD-10 diseases from the Semi-automatic dataset covering all the common ICD-10 codes within UK Biobank. We curated their phenotype associations manually and filtered out the false positives. This curated dataset (Semi-automatic (UKB)) contains a total of 7,576 disease–phenotype associations gathered in a semi-automated way (see Materials and Methods) between 1,995 (of 2,106) common ICD-10 diseases and 2,757 distinct phenotypes linked to HPO. We gathered the majority of phenotype associations (4,337 out of 7,576 associations) for 334 distinct ICD-10 codes from HPO through ICD-10–OMIM links in either Wikidata (3,914/4,337 pairs) or UMLS (423/4,337 pairs). We gathered 541/7,576 associations linked to 473 distinct ICD-10 codes through direct mappings of ICD-10 and HPO in UMLS. We gathered 295/7,576 associations for 43 distinct ICD-10 codes from Wikidata. We generated 1,214/7,576 associations for 335 distinct ICD-10 codes by propagating phenotypes from their superclass based on the ICD-10 hierarchy.
We manually curated 433/7,576 disease–phenotype associations for 433 ICD-10 codes. We generated a total of 756/7576 associations linked to 483 ICD-10 codes by propagating phenotypes from their superclasses when we found a lexical match between the superclass labels and the phenotype labels in HPO.
Phenotypic similarity of text mined and known associations
We measured the semantic similarity between our text mined and the known phenotypes of the diseases. There are 296 diseases in our dataset that are contained both in ICD-10 and OMIM and for which we can obtain phenotype associations both from our text mining approach and from curated data in the HPO database. We measured the semantic similarity between the phenotype profiles of a given disease by using cosine similarity between the ontology embeddings of the disease's phenotype profiles generated through OWL2Vec* [49].
Our Text Mined dataset consists of disease–phenotype associations and each association has a score that determines the association strength. Among the diseases in our dataset, between 1 and 2,592 phenotypes are positively associated. We assume that not all positive associations may be relevant but only the stronger associations provide useful information about a disease. We test this hypothesis by ranking phenotypes for each disease by their association (NPMI) score. We then include phenotypes in a disease–phenotype profile using varying thresholds for the number of phenotypes to include (based on the association score). To determine a threshold that yields a phenotype profile similar to manually curated ones, we compare the semantic similarity of the thresholded phenotype profiles to the manually curated profiles for the same disease; we evaluate the similarly using receiver operating characteristic (ROC) curves [51]. We find that a threshold of 76 phenotypes results in maximal similarity to the manually curated disease–phenotype associations (ROCAUC 0.95). Figure 4 shows the results of our experiment.
AUC values obtained for the phenotypic similarity of text-mined and known diseases from HPO at different NPMI ranks
Predicting gene-disease associations
We further evaluated whether our Text Mined and Semi-automatic (UKB) datasets are useful in identifying gene–disease associations based on phenotype similarity. We found 53 diseases in ICD-10 that can be mapped directly to OMIM and are also present in our Text Mined and Semi-automatic (UKB) datasets. These 53 diseases are associated with 216 genes in our gene–disease dataset gathered from MGI.
Utilizing the text mined disease-phenotype associations with their association score, we followed a similar procedure as before and rank phenotypes for each disease based on their association score and vary the rank as threshold parameter. We then compared these phenotype profiles to phenotypes resulting from loss of function mouse models using the cosine similarity between their ontology embeddings, and evaluated how well this method recovers known gene–disease associations. Figure 5 shows the resulting ROCAUC at different NPMI ranks. We find the maximal ROCAUC value at rank 74 (ROCAUC 0.62).
AUC values obtained for the phenotypic similarity of text-mined diseases and known genes from MGI at different NPMI ranks
We further used different datasets to find gene–disease associations through phenotype similarity: our Text Mined dataset with a threshold of 74 per disease; our Semi-automatic (UKB) dataset collected from multiple databases; the phenotypes associated with the 53 diseases in the HPO database; and combinations thereof. Figure 6 shows the ROC curves resulting from this comparison. The ROCAUC values range from 0.79 for combining Text Mined and Semi-automatic (UKB) datasets to 0.62 for only the Text Mined dataset.
Comparison of ROC curves for predicting gene–disease associations using cosine similarity
Comparison to expert-curated disease–phenotype associations
We created an expert-curated disease–phenotype association dataset to use for validation. This validation dataset consisted of 830 disease–phenotype associations for 53 diseases. To generate this dataset, we first gathered the semi-automatically curated ICD-10–HPO associations for these 53 diseases from our dataset. False positive HPO terms were filtered out and missing associations were added by an expert; 269 annotations were added. Because the HPO database contains mainly annotations to rare Mendelian diseases, most of the phenotype annotations contained in it are predicated on single gene, oligogenic, recurrent CNV or chromosome structural, disease etiology. While much of the phenotype annotation we need for common disease may be obtained from these annotations, the HPO data includes many phenotypes that are only found in the genetic syndromic disease and not in sporadic occurrences; this is discussed below. Consequently, in putting together the validation dataset, phenotypes which are not found in sporadic disease were treated as false positive unless the ICD class explicitly referred to an OMIM disease. In addition, high level terms such as HP:0002664Neoplasm, were excluded as being of low information content.
We used this corpus to evaluate the datasets we generated by comparing phenotype classes associated with diseases directly, using two types of evaluation, "strict" and "soft". We called an evaluation strict if we ignored the hierarchy and semantics of phenotype ontologies and only compared whether phenotype classes matched exactly between our dataset and our benchmark. In the soft evaluation, we first propagated disease–phenotype associations over the phenotype ontology hierarchy and then evaluated on all levels of the ontology.
Our semi-automatically curated dataset covered a total of 649 disease–phenotype associations for those 53 diseases. 568/649 of the associations were true positives, 81/649 were false positives. We missed a total of 262/830 annotations (false negatives). We estimated the Precision as 0.88, Recall as 0.68 and F-score as 0.77.
Figure 7 shows the performance analysis of the text mining extracts against the validation dataset. The performance of the text mining process varied over different NPMI ranks. Max F-score value of 0.21 was achieved at NPMI rank 16.
Performance analysis of text mining against the validation dataset over different NPMI ranks (strict)
We have a total of 3,499 disease-phenotype annotations in the validation dataset when we propagate annotations based on the PhenomeNET ontology. On the other hand, our semi-automatically curated dataset covers a total of 2,830 disease-phenotype annotations after the propagation process. In the "soft" settings, we found that 2,454/2,830 associations are true positive, 376/2,830 are false positive, and 1,045/3,499 are false negative. We estimated the Precision as 0.87, Recall as 0.70 and F-score as 0.78.
Figure 8 shows the performance analysis of the text mined extracts against the validation dataset under the "soft" settings. The performance of the text mining process varies over different NPMI ranks. The best F-score is achieved at the NPMI rank of 27 as a value of 0.44.
Performance analysis of text mining against the validation dataset over different NPMI ranks (soft)
Coverage of the generated datasets
There are a total of 19,133 distinct ICD-10 codes. We linked 6,263 and 7,610 ICD-10 codes to their phenotypes by using text mining and the semi-automatic strategy, respectively. While we linked 4,118 ICD-10 classes to their phenotypes by both of the methods (overlap); 9,755 (51%) ICD-10 classes were linked to their phenotypes by either methods. Hence, we were unable to link 9,378 (49%) ICD-10 classes to their phenotypes. We discuss the main reasons of being unable to link these ICD-10 classes to their phenotypes in detail in the Discussion section.
Semi-automatically curated data
We identified a total of 1,369 false positives during the semi-automatic curation of the associations from all of the 2,106 common diseases. We found that, while 963/1,369 false positives were due to the associations from existing resources, the remaining 406/1,369 false positives were due to the propagation of the annotations. 170/406 false positives are due to their lexical superclass matches in the HPO dataset and 236/406 false positives are due to their ICD-10 superclass-based annotation propagation. For example, ICD-10:C43.5Malignant melanoma of the trunk produced the annotation to HP:0007716Uveal melanoma, due to propagation from ICD-10:C43, Malignant melanoma of skin. We gathered the association between ICD-10:C43 and HP:0007716 from the HPO database through the mapping between OMIM:155600–ICD-10:C43 from UMLS.
Further breaking down the 963 false positives generated from the known data, we found that 12/963 false positives were from the Wikidata set, 3/963 false positives were due to the ICD-10–HPO direct mappings in UMLS, 19/963 false positives were due to incorrect associations found during the manual expert curation due to inclusion of syndromic phenotypes as discussed above, and the remaining 929/963 false positives were due to the use of the asserted disease–phenotype annotations in the HPO database. We further investigated these 929 false positives. As the diseases and phenotypes are mapped to their OMIM and HPO identifiers, respectively, to obtain ICD-10 identifiers for the OMIM diseases, we investigated the portions of the false positives introduced through OMIM–ICD-10 mappings in UMLS and Wikidata. We found that 44/929 false positives were introduced due to OMIM–ICD-10 mappings from UMLS and the remaining 885/929 false positives, which constitute the majority, were introduced due to the OMIM–ICD-10 mappings from Wikidata.
For example ICD-10:I77.1, Stricture of artery, is annotated to HP:0002036, Hiatus hernia, because Wikidata maps this ICD-10 class to OMIM:208050, Arterial tortuosity syndrome, which has a wide clinical phenotype spectrum among which is Hiatal hernia. Phenotypes that would not normally be considered a manifestation of sporadic non-syndromic arterial stricture, such as Arachnodactyly or Hiatus hernia were considered false positives. However, correct annotations to HPO were obtained directly from UMLS, which provides a correct annotation HP:0100545, Arterial stenosis. In general, ICD-10 to OMIM mappings through Wikidata-generated candidate HPO annotations are associated with Mendelian, syndromic disease, accounting for the high number of false positives through this route. These had to be manually removed on a case-by-case basis using expert judgement, where sporadic disease would not be expected to have these associations.
False negatives, i.e. missing annotations, were called usually when the annotation was sparse but there are clear associated phenotypes available in HPO. The causes of this are interesting. For example HP:0000979, Purpura, was missing from the annotation to ICD-10:M31.3Wegener granulomatosis [52] and HP:0025188, Retinal vasculitis missing from systemic ICD-10:M32.9Lupus erythematosus [53]. In the former case, although Wegener granulomatosis is in OMIM (OMIM:608710), there is no clinical synopsis and it was therefore not possible to gather annotations from the HPO database. For the latter, Systemic lupus erythematosus, HP:0002725 is treated as a "bundled term" phenotype in the HPO database and therefore no more granular phenotype annotations are available. There are no direct HPO annotations for Systemic lupus erythematosis in UMLS. We cannot provide any assurance that all of the possible missing annotations have been added to the dataset, but have provided best efforts with the resources available. We hope that users might over time request the addition of phenotypes to their diseases of interest.
Text mined data
For the analysis of the text mined associations, we used the extracts generated based on the NPMI rank 16 which gave us the best result on the validation dataset by using the strict evaluation (precision 0.25, recall 0.17, and F-score 0.21). We have a total of 568 ICD-10–HPO pairs in this text-mined dataset. We found that 143/568 are true positives and 425/568 are false positives. We missed a total of 687 associations (false negatives). Our manual analysis on the 425 false positives show that only a small portion of them (47/425) are false positives and the majority of them (376/425) are actually true positive associations which are not covered by our validation dataset. Our validation dataset includes only the obvious and distinguishing phenotypes of diseases. These 376 associations are the associations of the diseases with the high level of HPO classes. For example, Malignant neoplasm of stomach, unspecified (ICD-10:C16.9) is associated with Neoplasm (HP:0002664) according to our text mining extracts. This is a true positive by manual analysis but was counted as a FP since it is not covered within our validation dataset as Neoplasm is a high level phenotype for all malignant and benign proliferative lesions and of low information content. The false positives are mainly due to the co-mentions of associated disease concepts, or negations in the publications (X is not a Y). Some examples of such associations include Acute myeloid leukaemia (ICD-10:C92.0) and Chronic myelomonocytic leukemia (HP:0012325) as well as Primary open-angle glaucoma (ICD-10:H40.1) and Angle closure glaucoma (HP:0012109). Analysis of the 687 false negative samples showed that actually 473 of 687 pairs (69%) have been extracted from the literature but they do not rank in the top 16 based on their NPMI scores of association strength. The other missing ones are mainly due to weak or no evidence in the literature. For example, there are no publications mentioning Marfan syndrome (ICD-10:Q87.4) and Decreased muscle mass (HP:0003199); there are only 2 publications mentioning Parkinson's disease (ICD-10:G20) and Macrocephaly (HP:0000256) in title or abstract together in PubMed (search was done on 15th April 2021). One of the publications is published in 2021 which is not covered by our current dataset. Therefore, there is no significant supporting evidence in the literature to infer a positive association between the given disease–phenotype pairs. Other false negatives could be due to the missing disease/phenotype synonyms. Altogether, we estimated the actual performance of the text mining method (at the NPMI rank 16) as an F-score value of 0.59, a precision of 0.92 and a recall of 0.43.
We have previously reported an extensive annotation of the diseases in DO based on a text mining analysis of PubMed abstracts and titles [30]. This included phenotype annotations to 6,000 common, rare and infectious diseases of which 4,768 are diseases from OMIM [29]. The under-representation of sporadic, common or complex disease in this dataset and the fact that DO is not frequently used in routine clinical recording were the motivation to develop a set of HPO annotations to terms in the much larger ICD-10 terminology. Here we have carried out a large-scale text mining analysis of PubMed using term labels, synonyms and acronyms of ICD-10 codes, and augmented this new analysis with data from three publicly available annotation sources, UMLS, Wikidata and the HPO database.
While Wikidata and the HPO database contain almost exclusively phenotypes for rare diseases found in OMIM and Orphanet, they present a source of annotation that may be exploited for common disease as explained below. A similar but more limited approach to phenotypic annotation for common disease was implemented by Sarntivijai et al. [54] using ontology-driven literature mining for two classes of disease, Inflammatory bowel disease and Autoimmune disease, together with their subclasses in the Experimental Factor Ontology (EFO) [55]. This produced 1,452 and 2,810 disease–phenotype pairs for inflammatory bowel disease (IBD) and autoimmune disease of which 41.6% candidate IBD phenotype associations were deemed correct by manual review. Similar to the strategy we take here, the authors of the study removed non-informative phenotypes such as "All", "Chronic", or "Death" but unlike us excluded classes in HPO that were deemed to represent disease entities, using expert judgement. The authors discuss some of the problems we also encountered of annotation validation on existing datasets.
In attempting a large scale phenotypic annotation of a significant number of the disease concepts in ICD-10, we have noted several issues. In trying to semi-automatically generate this corpus of annotations, one question is the decision as to what should be considered as part of a phenotypic manifestations, what level of granularity should be used, and the reliability of existing sources of annotation such as Wikidata, the HPO database, and UMLS. The definition of a phenotype as an observable characteristic covers simple signs and symptoms, and syndromic manifestations, but operationally "phenotypes" are included in the HPO database that may occur in isolation as "diseases" such as Diabetes or Tetralogy of Fallot (HPO regards these as "bundled phenotypes" and are included for pragmatic reasons). The decision as to how to select our annotation strategy can therefore only be guided by the purposes for which these annotations are developed, and by the best outcome on evaluation. We believe that the inclusive approach we take provides a valid strategy as assessed by performance on disease/gene prediction from the MGI dataset.
We find that many rare and rather few common diseases are extensively and accurately annotated. In some cases this is due to the deep annotation in the OMIM/HPO databases, UMLS, and, to a lesser extent, in Wikidata. The mapping of ICD classes to HPO involves for the most part working through the intermediary mappings to OMIM given in UMLS or Wikidata. As discussed above, this often results in phenotype annotations designed to describe rare inherited diseases or syndromes and not common or sporadic diseases. Although ICD classes sometimes include rare diseases explicitly, most do not, and therefore the intention in annotating a patient to an ICD-10 class is that of noting common/sporadic disease unless rare disease is asserted in the ICD-10 class chosen. As a consequence, we expertly edited annotations from HPO to align with the sporadic/common disease implied by the ICD class, giving rise to an increased number of false positive calls. We did not edit when the ICD class explicitly included an OMIM disease. This process, while driven by expert opinion is nevertheless subjective and represents a potential weakness in our approach. The low recall versus high specificity we obtain in recovering MGI gene disease associations is a consequence of disease annotation in MGI being to OMIM diseases when we edited OMIM disease phenotype annotations to approach the less complex annotation expected of sporadic disease. Our validation approach is therefore limited by what annotation datasets are available, and in the absence of any other manually curated large disease/phenotype datasets we believe that this is the best approach currently available, while not optimal.
We attempted to evaluate how removing some of these deeply annotated diseases affected the validation and found overall small changes in evaluation performance. More specifically, we identified that there are 3 heavily annotated diseases out of 53 diseases in the validation dataset Marfan's disease (ICD-10:Q87.4), hereditary hemorrhagic telangiectasis (ICD-10:I78.0), and hereditary factor VIII deficiency (ICD-10:D66). When we removed these 3 diseases from the evaluation, the performance of the semi-automatic curation drops from an F-score value of 0.77 to 0.73.
Regarding the coverage of the datasets generated, we were unable to link 49% (9,378 out of 19,133) ICD-10 classes to their phenotypes either by semi-automatic or text mining methods. The majority of the missing ICD-10 terms are Diseases of the musculoskeletal system and connective tissue; ICD-10:M00–M99 (2574 ICD-10 codes), Injury, poisoning and certain other consequences of external causes, ICD-10:S00–T88 (1297 ICD-10 codes) and External causes of morbidity; ICD-10:V00–Y99 covering ICD-10:X00-99 (1113 ICD-10 codes), ICD-10:W00–W99 (1060 ICD-10 codes), ICD-10:V00–V99 (909 ICD-10 codes) and ICD10:Y00–Y99 (635 ICD-10 codes). We miss linking these ICD-10 codes to their phenotypes due to several methodological issues as well as the data available in the resources (HPO, UMLS, Wikidata, PubMed). More specifically, we text mined ICD-10–phenotype associations from the PubMed abstracts only and full-text articles are not covered in this study, which potentially include more associations. Furthermore, we miss some association of diseases which have long labels (e.g. ICD-10:Z62.6, Inappropriate parental pressure and other abnormal qualities of upbringing; ICD-10:X44, Accidental poisoning by and exposure to other and unspecified drugs, medicaments and biological substances) and therefore they are very unlikely to be mentioned in titles or abstracts in full. In addition, some of the associations are missed due to their low NPMI signal based on our method (we considered associations having NPMI > 0). These missing ICD-10 codes cover mainly injuries, poisoning and infectious diseases which are not focus of HPO and the other resources used in this study. Therefore, lack of these classes is not likely to reduce the utility of the generated datasets for the purposes motivating their development, which is to link phenotypes to genetic variants and underlying molecular processes.
A well established problem is that for an instance of a disease in an individual patient all phenotypes will not necessarily be present and will evolve with time. A weakness of our annotation model is that phenotype associations are treated as a "bag of phenotypes" which lacks precision and flexibility. Future work will look at application of an Ontology of Biomedical AssociatioN (OBAN) data model to our results, which allows for the inclusion of qualification into the association between disease and phenotype [54].
We used a semi-automatic and a text mining based method to create four datasets of disease–phenotype associations. The generated disease–phenotype associations are useful for completing the phenotype profiles of the diseases linked to clinical resources, and can be used to investigate gene–disease associations. All the data is publicly available at Zenodo (DOI:https://doi.org/10.5281/zenodo.4726714) for community use.
We make all data freely available at https://doi.org/10.5281/zenodo.4726713.
We make the source code developed available from Github, https://github.com/bio-ontology-research-group/icdpheno
Human Disease Ontology
EFO:
Experimental Factor Ontology
HPO:
Human Phenotype Ontology
IBD:
ICD:
International Classification of Diseases
MGI:
Mammalian Phenotype Ontology
NPMI:
Normalized Pointwise Mutual Information
OBAN:
Ontology of Biomedical AssociatioN
OMIM:
Online Mendelian Inheritance in Man
OWL:
Web Ontology Language
ROCAUC:
Receiver Operating Characteristic Area Under Curve
UKB:
UMLS:
Unified Medical Language System
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This research has been conducted using the UK Biobank Resource under the Application Number 31224.
This study is supported by King Abdullah University of Science and Technology (KAUST) Office of Sponsored Research (OSR) under Award No. URF/1/3790-01-01, URF/1/4355-01-01, FCC/1/1976-28-01, and FCC/1/1976-29-01. PNS acknowledges the support of The Alan Turing Institute.
Computational Bioscience Research Center (CBRC), Computer, Electrical, and Mathematical Sciences & Engineering Division, King Abdullah University of Science and Technology, 4700 KAUST, Thuwal, 23955, Saudi Arabia
Şenay Kafkas, Sara Althubaiti & Robert Hoehndorf
Health Data Research UK, Midlands site, Edgbaston, Birmingham, B15 2TT, United Kingdom
Georgios V. Gkoutos
Institute of Cancer and Genomic Sciences, University of Birmingham, Edgbaston, Birmingham, B15 2TT, United Kingdom
Department of Physiology, Development & Neuroscience, University of Cambridge, Downing Street, Cambridge, CB2 3EG, United Kingdom
Paul N. Schofield
Şenay Kafkas
Sara Althubaiti
Robert Hoehndorf
PS and RH conceived the experiments; ŞK conducted all the experiments except phenotypic similarity measurements; SA conducted phenotype similarity experiments; all manual annotations are done by PS; RH, PS, and ŞK analysed the results, ŞK drafted the initial version of the manuscript, SA, PS, RH, GVG revised the manuscript. PS, GVG, RH acquired funding to support this work. All authors reviewed and approved the final version of the manuscript.
Correspondence to Robert Hoehndorf.
SPARQL queries. This file contains the three SPARQL queries used to extract ICD-10–phenotype associations and ICD-10–OMIM mappings from Wikidata.
Kafkas, Ş., Althubaiti, S., Gkoutos, G.V. et al. Linking common human diseases to their phenotypes; development of a resource for human phenomics. J Biomed Semant 12, 17 (2021). https://doi.org/10.1186/s13326-021-00249-x
Disease–phenotype associations
International Conference on Biomedical Ontologies Direct to Journal Track | CommonCrawl |
\begin{document}
\title{Nakayama automorphisms of graded Ore extensions of Koszul Artin-Schelter regular algebras }
\author{Y. Shen} \address{Shen: Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou 310018, China} \email{[email protected]}
\author{Y. Guo} \address{Guo: Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou 310018, China} \email{[email protected]}
\date{}
\begin{abstract} Let $A$ be a Koszul Artin-Schelter regular algebra, $\sigma$ a graded automorphism of $A$ and $\delta$ a degree-one $\sigma$-derivation of $A$. We introduce an invariant for $\delta$ called the $\sigma$-divergence of $\delta$. We describe the Nakayama automorphism of the graded Ore extension $B=A[z;\sigma,\delta]$ explicitly using the $\sigma$-divergence of $\delta$, and construct a twisted superpotential $\hat{\omega}$ for $B$ so that it is a derivation quotient algebra defined by $\hat{\omega}$. We also determine all graded Ore extensions of noetherian Artin-Schelter regular algebras of dimension 2 and compute their Nakayama automorphisms. \end{abstract}
\subjclass[2010]{16S36,16S37,16E65,16S38,16W50}
\keywords{Koszul Artin-Schelter regular algebras, Graded Ore extensions, Nakayama automorphisms, Twisted superpotentials, Calabi-Yau}
\maketitle
\section*{Introduction} To understand Artin-Schelter regular algebras has been a main topic in the study of noncommutative algebras and noncommutative projective geometry since the late 1980s. These algebras are exactly connected graded skew Calabi-Yau algebras (\cite{RRZ1}), and possess a kind of automorphisms called Nakayama automorphisms. Such automorphisms are an important tool to study Hopf actions, noncommutative invariant theory and so on (\cite{CWZ,LMZ,RRZ2,SL}). However, the computation of Nakayama automorphisms is always hard. There is a plenty of work to provide different methods to solve this problem (see \cite{HVZ,LM,LWW,LMZ,LMZ1,RRZ1,RRZ2,SL,SZL,V,ZSL,ZVZ}).
Obtaining new Artin-Schelter regular algebras from known ones is a common approach, and the methods include Ore extensions, double Ore extensions and regular normal extensions. How the Nakayama automorphisms behave under those extensions is an interesting and concerned problem. The behavior of Nakayama automorphisms under regular normal extensions was studied in \cite{RRZ1,ZSL}. The Nakayama automorphisms of trimmed double Ore extensions of Koszul Artin-Schelter regular algebras were described in \cite{ZVZ}. Earlier, Liu, Wang and Wu considered the case of Ore extensions in a general setting (\cite{LWW}). They proved that if $A$ is (not necessarily connected graded) skew Calabi-Yau with a Nakayama automorphism $\mu_A$, then the Ore extension $B=A[z;{\sigma},\delta]$ has a Nakayama automorphism $\mu_B$ satisfying ${\mu_B}_{\mid A}={\sigma}^{-1}\mu_A$ and $\mu_B(z)=\lambda z+b$ for some $\lambda,b\in A$. It is natural to ask what $\lambda$ and $b$ are. Restricted on graded Ore extensions, Zhu, Van Oystaeyen and Zhang showed $\lambda=\hdet({\sigma})$ if $A$ is Koszul Artin-Schelter regular in \cite{ZVZ}, and Zhou, Lu and the first named author showed the same equality if $A$ is noetherian Artin-Schelter regular generated in degree 1 in \cite{SZL}, where $\hdet$ is the homological determinant introduced by J{\o}rgensen and Zhang (see \cite{JZ}).
In fact, $\lambda$ can be determined by trimmed Ore extensions, namely, the ${\sigma}$-derivation ${\delta}$ is trivial, by the filtered-graded technique. However, $b$ is related to ${\delta}$, and seems mysterious for us. Recently, Liu and Ma described $\lambda$ and $b$ explicitly for all (ungraded) Ore extensions if $A$ is a polynomial algebra in \cite{LM}. It inspires us to make a progress in the description of the parameter $b$. The main goal of this paper is to describe the Nakayama automorphisms of graded Ore extensions of Koszul Artin-Schelter regular algebras specifically.
Let $A=T(V)/(R)$ be a Koszul Artin-Schelter regular algebra of dimension $d$, ${\sigma}$ a graded automorphism of $A$ and ${\delta}$ a degree-one ${\sigma}$-derivation of $A$. In order to realize our desire, it requires a way to handle the ${\sigma}$-derivation $\delta$. Our approach is to construct a pair $(\{\delta_{i,r}\},\{\delta_{i,l}\})$ of two sequences of linear maps for ${\delta}$: $$ \delta_{i,r}:W_i\to W_i\otimes V,\qquad \delta_{i,l}: W_i\to V\otimes W_i, $$ where $W_1=V$ and $W_j=\cap_{s=1}^{j} V^{\otimes s}\otimes R\otimes V^{j-s-2}$ for any $i\geq 1,j\geq 2$, by the Koszul complex of the graded trivial module $k_A$ (see Lemma \ref{lemma: construction of delta_r} and Lemma \ref{lemma: construction of delta_l}). In general, sequence pair $(\{\delta_{i,r}\},\{\delta_{i,l}\})$ is not unique for ${\delta}$. However, there is a unique pair $(\delta_r,\delta_l)$ of elements in $V$ coming from the action of $\delta_{d,r}$ and $\delta_{d,l}$ on $W_d$ for each sequence pair $(\{\delta_{i,r}\},\{\delta_{i,l}\})$, since $W_d$ is $1$-dimensional. We find that $\delta_{r}+\mu_A{\sigma}^{-1}(\delta_l)$ is independent on the choices of sequence pairs for $\delta$, where $\mu_A$ is the Nakayama automorphism of $A$ (see Corollary \ref{coro: relation between delta_r and delta l}). We call this invariant the \emph{${\sigma}$-divergence} of ${\delta}$, denoted by $\nabla_{{\sigma}}\cdot{\delta}$. If $A$ is a polynomial algebra and $\sigma$ is the identity map $\id_A$, then $\nabla_{{\id_A}}\cdot{\delta}$ is precisely the usual divergence of $\delta$.
It is well known that the graded Ore extension $B=A[z;{\sigma},{\delta}]$ is also a Koszul Artin-Schelter regular algebra of dimension $d+1$. Using a sequence pair $(\{\delta_{i,r}\},\{\delta_{i,l}\})$ for ${\delta}$, we compute the Yoneda product of the Ext-algebra $E(B)$ of $B$. Since $E(B)$ is a graded Frobenius algebra and its Nakayama automorphism is dual to the one of $B$ (see \cite{RRZ2,V}), then we obtain the main result of this paper.
\begin{theorem}\emph{(Theorem \ref{thm: nakayama automorphism of ore extension})} Let $B=A[z;{\sigma},{\delta}]$ be a graded Ore extension of a Koszul Artin-Schelter regular algebra $A$, where ${\sigma}$ is a graded automorphism of $A$ and $\delta$ is a degree-one $\sigma$-derivation. Then the Nakayama automorphism $\mu_B$ of $B$ satisfies \begin{align*} {\mu_{B}}_{\mid A}={\sigma}^{~-1}\mu_A,\qquad \mu_B(z)=\hdet({\sigma})\, z+\nabla_{{\sigma}}\cdot{\delta}, \end{align*} where $\mu_A$ is the Nakayama automorphism of $A$. \end{theorem}
It is also well-known that a Koszul Artin-Schelter regular algebra $A$ is determined by a twisted superpotential $\omega$, that is, $A$ is isomorphic to a derivation quotient algebra defined by $\omega$ (see \cite{BSW,DV}). He, Van Oystaeyen and Zhang constructed a new twisted superpotential for a graded Ore extension of $A$ in two special cases (\cite{HVZ,HVZ1}). In fact, any sequence pair $(\{\delta_{i,r}\},\{\delta_{i,l}\})$ for ${\delta}$ provides us assistance to construct a new twisted superpotential $\hat{\omega}$ for any graded Ore extension $B=A[z;\sigma,\delta]$ from $\omega$, and it is independent on the choices of sequence pairs. Hence, $B$ is isomorphic to a derivation quotient algebra defined by $\hat{\omega}$ (see Theorem \ref{thm: twisted superpotential for B}).
As an application of the main result, we determine all graded Ore extensions of noetherian Koszul Artin-Schelter regular algebras of dimension 2 and compute their Nakayama automorphisms. There is an interesting observation about Calabi-Yau property.
\begin{theorem}\emph{(Theorem \ref{thm: CY for 2-dim})} Assume the base field $k$ is of characteristic $0$. Let $A=k\langle x_1,x_2\rangle/(r)$ be a noetherian Artin-Schelter regular algebra of dimension 2, and $B$ a graded Ore extension $A[z;{\sigma},{\delta}]$. \begin{enumerate} \item Suppose $A$ is commutative, then $B$ is Calabi-Yau if and only if ${\sigma}=\id_A$, and $$ {\delta}(x_1)=l_1x_1^2-2l_4x_2x_1+l_2x_2^2,\qquad
{\delta}(x_2)=l_3x_1^2-2l_1x_2x_1+l_4x_2^2,
$$ for some $l_1,l_2,l_3,l_4\in k$. \item Suppose $A$ is noncommutative, then $B$ is Calabi-Yau if and only if ${\sigma}$ is the Nakayama automorphism of $A$. \end{enumerate} \end{theorem}
The graded automorphism group for the commutative noetherian Artin-Schelter regular algebra of dimension 2 is much bigger than a noncommutative one, as well as the set of $\sigma$-derivations. So there are more conditions for graded Ore extensions of the commutative one being Calabi-Yau. But it is still surprised that there is no any restriction on $\delta$ for $A[z;\sigma,\delta]$ being a Calabi-Yau algebra in case $A$ is a noncommutative noetherian Artin-Schelter regular algebra of dimension 2. It is natural to ask whether this result holds without the noetherian assumption, or even for all noncommutative Koszul Artin-Schelter regular algebras.
This paper is organized as follows. In Section 1, we recall some definitions and properties, especially for Koszul Artin-Schelter regular algebras. In Section 2, we construct a sequence pair for any $\sigma$-derivation of a Koszul algebra， where $\sigma$ is a graded automorphism, and discuss the relations between different sequence pairs. In Section 3, we focus on the Koszul Artin-Schelter regular algebras. We prove there is an invariant for each $\sigma$-derivation, compute the Yoneda product of the Ext-algebra of any graded Ore extension, obtain the main result about Nakayama automorphisms and construct a twisted superpotential for a graded Ore extension. In Section 4, we apply our main result to commutative polynomial algebras and Koszul Artin-Schelter regular algebras of dimension 2.
Throughout the paper, $k$ is a fixed field. All vector spaces and algebras are over $k$. Unless otherwise stated, the tensor product $\otimes$ means $\otimes_k$. \section{Preliminaries}\label{Section preliminaries}
A graded algebra $A=\oplus_{i\in\mathbb{Z}} A_i$ is called \emph{locally finite} if $\dim A_i<\infty$ for any $i\in\mathbb{Z}$. A locally finite graded algebra $A$ is called \emph{connected} if $A_i=0$ for any $i<0$ and $\dim A_0=1$. In this case, write $\varepsilon_A$ for the augmentation from $A$ to $k$. Let $M=\oplus_{i\in \mathbb{Z}}M_i$ be a right graded $A$-module. Then $n$-th \emph{shift} of $M$ is a right graded $A$-module $M(n)$ with the homogenous space $M(n)_i=M_{n+i}$ for all $i\in\mathbb{Z}$. Let $N$ be a graded $A$-bimodule and $\mu$ a graded automorphism of $A$. The graded twisted $A$-bimodule $N^{\mu}$ is a graded $A$-bimodule with the $A$-action $a\cdot x\cdot b=ax\mu(b)$ for any $a,b\in A,x\in N$.
For a connected graded algebra $A$, there exists a minimal graded free resolution of the right graded trivial module $k_A$ of $A$, \begin{equation}\label{general Resolution of k_A} \cdots\xlongrightarrow{} P_{d}\xlongrightarrow{\partial_d} \cdots\xlongrightarrow{\partial_3} P_{2}\xlongrightarrow{\partial_2} P_1\xlongrightarrow{\partial_1} P_0\xlongrightarrow{\partial_0} k_A\xlongrightarrow{} 0, \end{equation} namely, $\Ker \partial_i\subseteq P_{i}A_{\geq 1}$ for any $i\geq 0$. Then the graded vector space $E(A)=\bigoplus_{i\geq0}\uExt^i_A(k_A,k_A)=\bigoplus_{i\geq0}\uHom_A(P_i,k)$ equipped with the Yoneda product is a connected graded algebra, called the \emph{Ext-algebra} of $A$. In the sequel, We denote the Yoneda product by ``$\ast$''.
Let $V$ be a finite dimensional vector space. Write $\tau:V\otimes V\to V\otimes V$ for the usual twisting map. We adopt the notation in \cite{HVZ} of a sequence of linear endmorphisms of $V^{\otimes d}$ for any $d\geq 2$: $$ \tau_{d}^0=\id_V^{\otimes d},\quad \tau_d^{i}=(\id_V^{\otimes i-1}\otimes \tau\otimes\id_V^{\otimes d-i-1 })\tau_{d}^{i-1},\quad\text{for any }~ 1\leq i\leq d-1. $$
\begin{definition} Let $V$ be a finite dimensional vector space and $\nu:V\to V$ an isomorphism of vector spaces. If an element $\omega\in V^{\otimes d}$ for some $d\geq 2$ such that $$ \omega=(-1)^{d-1}\tau_{d}^{d-1}(\nu\otimes\id_V^{\otimes d-1})(\omega), $$ then $\omega$ is called a \emph{$\nu$-twisted superpotential}. In particular, $\omega$ is called a \emph{superpotential} if $\nu=\id_V$. \end{definition}
Let $\nu$ be a linear automorphism of $V$ and integer $d\geq2$. Define the \emph{partial derivation} of a $\nu$-twisted superpotential $\omega\in V^{\otimes d}$ with respect to $\psi\in (V^*)^{\otimes i}$, where $(-)^*$ is the $k$-dual of a space and $1\leq i\leq d$, to be $$ \partial_{\psi}(\omega)=(\id_V^{\otimes d-i}\otimes \psi)(\omega). $$
\begin{definition} Let $V$ be a finite dimensional vector space, $\omega\in V^{\otimes d}$ a $\nu$-twisted superpotential for some linear automorphism $\nu$ of $V$ and $d\geq 2$. The \emph{$i$-th derivation quotient algebra} $\mathcal{A}(\omega,i)$ of $\omega$ is $$ \mathcal{A}(\omega,i)=T(V)/(\partial_{\psi}(\omega),\psi\in (V^*)^{\otimes i}), $$ where $T(V)$ is the tensor algebra over $V$. \end{definition}
Let $T(V)$ be the tensor algebra over $V$ with the usual grading and $R$ a subspace of $V^{\otimes 2}$. Then $A=T(V)/(R)$ is a connected graded algebra, called a \emph{quadratic algebra}. Write $\pi_A$ for the canonical projection from $T(V)$ to $A$.
\begin{definition} A quadratic algebra $A$ is called \emph{Koszul}, if the right graded trivial module $k_A$ admits a minimal graded free resolution (\ref{general Resolution of k_A}) such that the right graded $A$-module $P_i$ is generated by degree $i$ for any $i\geq0$. \end{definition}
Let $A=T(V)/(R)$ be a Koszul algebra. Write $W_0=k$, $W_1=V$, $W_2=R$ and for $i\geq 3$, $$ W_{i}=\bigcap_{0\leq s\leq i-2} V^{\otimes s}\otimes R\otimes V^{\otimes i-s-2}. $$
Then the following Koszul complex is a minimal graded free resolution of $k_A$, $$ \cdots\xlongrightarrow{\partial_{d+1}^A} W_d\otimes A\xlongrightarrow{\partial^A_d}W_{d-1}\otimes A\xlongrightarrow{\partial^A_{d-1}}\cdots\xlongrightarrow{\partial^A_{3}}W_2\otimes A\xlongrightarrow{\partial^A_{2}}W_1\otimes A\xlongrightarrow{\partial^A_{1}}A\xlongrightarrow{\varepsilon_A}k_A\to0, $$ where $\partial^A_{i}=(\id_V^{\otimes i-1}\otimes m_A)(\id_V^{\otimes i-1}\otimes{\pi_A}_{\mid V}\otimes\id_A)=\id_V^{\otimes i-1}\otimes m_A$ and $m_A$ is the multiplication of $A$ for $i\geq 1$. \begin{remark} In the sequel, we treat $V$ as the vector space $V$ or the homogeneous space $A_1$ of a Koszul algebra $A$ freely. So we write $m_A$ shortly for the liner map $m_A({\pi_A}_{\mid V}\otimes \id_A):V\otimes A\to A$ for convenience. \end{remark}
As vector spaces, the Ext-algebra of $A$ is $$ E(A)=\bigoplus_{i\geq0}\uHom_A(W_i\otimes A,k_A)\cong \bigoplus_{i\geq0} W_i^{*}. $$
\begin{remark} For Koszul algebras, researchers always use their Koszul dual as a common tool, which are also isomorphic to their Ext-algebras. In this paper, we use the language of Ext-algebras to study the Nakayama automorphisms of graded Ore extensions, since it is convenient to obtain some induced maps.
\end{remark}
\begin{definition} A connected graded algebra $C$ is \emph{Artin-Schelter regular} (\emph{AS-regular}, for short) of dimension $d$, if it has finite global dimension $d$, $\dim\left(\uExt_{C}^{d}(k_C, C_C)\right)=1$ and $\dim\left(\uExt_{C}^{i}(k_C, C_C)\right)=0$ for $i\neq d$. \end{definition} AS-regular algebras have an important homological invariant.
\begin{theorem} \cite[Proposition 4.5 (b)]{YZ}\label{thm: Nakayama aut of AS regular algebras} If $C$ is AS-regular of dimension $d$, then there exists a unique graded automorphism $\mu_C$ of $C$ such that $$\uExt^i_{C^e}(C,C^e)\cong\left\{\begin{array}{ll} 0&i\neq d,\\C^{\mu_C}(l)&i=d, \end{array}\right. \quad \text{as graded $C$-bimodules},$$ where $C^e=C\otimes C^{op}$ is the enveloping algebra of $C$ for some $l\geq 0$. \end{theorem} The automorphism $\mu_C$ is called the \emph{Nakayama automorphism of $C$}. In particular, if the Nakayama automorphism of an AS-regular algebra is the identity map, then it is a Calabi-Yau (CY, for short) algebra (see \cite{Gin,RRZ1}).
The Ext-algebras of AS-regular algebras also carry important information. Let $E=\oplus_{i\in \mathbb{Z}}E^i$ be a finite dimensional graded algebra. We say $E$ is a \emph{graded Frobenius algebra}, if there exists a nondegenerate associative graded bilinear form $\langle -,-\rangle: E\otimes E\to k(d)$ for some $d\in \mathbb{Z}$. In particular, there exits a graded automorphism $\mu_E$ of $E$ such that $$ \langle \alpha,\beta\rangle=(-1)^{ij}\langle \beta,\mu_E(\alpha)\rangle, $$ for any $\alpha\in E^i,\beta\in E^j$. We call the automorphism $\mu_E$ is the (classical) \emph{Nakayama automorphism} of the graded Frobenius algebra $E$. (see \cite{Sm} for details).
In this paper, we mainly consider Koszul AS-regular algebras. We list some important results about Koszul AS-regular algebras below.
\begin{theorem}\label{thm: properties of Koszul regular algebras}Let $A$ be a Koszul AS-regular algebra of dimension $d$ and $\mu_A$ the Nakayama automorphism of $A$. \begin{enumerate} \item \cite[Proposition 5.10]{Sm} The Ext-algebra $E(A)$ of $A$ is a graded Frobenius algebra. \item\cite[Proposition 3]{V} Let $\mu_E$ be the (classical) Nakayama automorphism of $E(A)$, then
$$
{\mu_E}_{|E^1(A)}=({\mu_A}_{|A_1})^*,
$$
where $E^1(A)$ is identified with $V^*=A_1^*$. \item \cite[Lemma 4.3]{HVZ} Any nonzero element $\omega\in W_d$ is a ${\mu_A}_{\mid V}$-twisted superpotential.
\item \cite[Theorem 11]{DV}\cite[Theorem 4.4(i)]{HVZ} For any nonzero element $\omega\in W_d$, $A\cong \mathcal{A}(\omega,d-2)$. \end{enumerate} \end{theorem} \begin{remark} If $A$ is a Koszul AS-regular algebra of dimension $d$, then $\dim W_d=1$. So an element $\omega\in W_d$ is nonzero is equivalent to it is a basis of $W_d$. \end{remark}
\section{A sequence pair} Let $V$ be a vector space with a basis $\{x_1,x_2,\cdots,x_n\}$, $T(V)$ the tensor algebra over $V$, and $A=T(V)/(R)$ a Koszul algebra, where $R$ is a subspace of $V^{\otimes 2}$. Recall the Koszul complex as follows \begin{equation}\label{resolution of k_A} \cdots\xlongrightarrow{\partial_{d+1}^A} W_d\otimes A\xlongrightarrow{\partial^A_d}W_{d-1}\otimes A\xlongrightarrow{\partial^A_{d-1}}\cdots\xlongrightarrow{\partial^A_{3}}W_2\otimes A\xlongrightarrow{\partial^A_{2}}W_1\otimes A\xlongrightarrow{\partial^A_{1}}A\xlongrightarrow{\varepsilon_A}k_A\to0, \end{equation} where $W_0=k$, $W_1=V$, $W_2=R$ and $ W_{i}=\bigcap_{0\leq s\leq i-2} V^{\otimes s}\otimes R\otimes V^{\otimes i-s-2} $ for $i\geq 3$.
Let $\overline{\sigma}$ be a graded automorphism of $A$ and $\overline{\delta}$ a degree-one $\overline{\sigma}$-derivation of $A$. Then we have a graded Ore extension $B=[z;\overline{\sigma},\overline{\delta}]$ with $\deg z=1$. Clearly, $B$ is a quotient algebra of the tensor algebra $T(V\oplus k\{z\})$ and a Koszul algebra. Write $\pi_B$ for the canonical projection from $T(V\oplus k\{z\})$ to $B$.
Write $\sigma=\overline{\sigma}_{\mid V}\in GL(V)$. It is easy to know that $\sigma^{\otimes i}(W_i)=W_i$ for each $i\geq 0$ and $\sigma_{T}:=\oplus_{i=0}^{\infty}\sigma^{\otimes i}$ is a graded automorphism of $T(V)$ such that $\overline{\sigma}$ is induced by $\sigma_{T}$.
Choose a linear map $\delta:V\to V\otimes V$ such that $\pi_A\delta=\overline{\delta}_{\mid V}$. In fact, $\delta$ extends to a degree-one $\sigma_T$-derivation of $T(V)$ (also denoted by $\delta$) in a unique way, and \begin{equation}\label{condition for delta} \delta(R)\subseteq R\otimes V+V\otimes R. \end{equation} So $\overline{\delta}$ can be induced by $\delta$. All $\overline{\sigma}$-derivations can be obtained in this way.
The condition (\ref{condition for delta}) provides an approach to decomposing $\delta$ into two parts, that is, there exist two linear maps $\delta_{2,r}: R\to R\otimes V,\delta_{2,l}: R\to V\otimes R. $ such that \begin{equation}\label{definition of delta_2} \delta_{\mid R}=\delta_{2,r}+\delta_{2,l}. \end{equation}
The decomposition can be realized as follows. One can choose a basis $\{r_1,r_2,\cdots,r_t\}$ of $R$, obtain that \begin{equation}\label{decomposition of delta} \delta(r_i)=\sum_{j=1}^t r_j\otimes \alpha^{i}_{j}+\sum_{j=1}^t \beta^{i}_{j}\otimes r_j\in R\otimes V+V\otimes R, \end{equation} for some $\alpha^{i}_j,\beta^{i}_j\in V$ and $j,i=1,\cdots,t$, and then define \begin{equation*} \qquad\delta_{2,r}(r_i)=\sum_{j=1}^t r_j\otimes \alpha^{i}_{j},\qquad \delta_{2,l}(r_i)=\sum_{j=1}^t \beta^{i}_{j}\otimes r_j, \qquad \forall i=1,\cdots,t. \end{equation*}
\begin{remark} It is clear that the choice of $\delta_{2,r}$ and $\delta_{2,l}$ for $\delta$ is not unique, which depends on the decomposition (\ref{decomposition of delta}). \end{remark}
\subsection{Minimal free resolutions}
Now we begin to construct a minimal free resolution of $k_B$. In the sequel, write $\delta_{1,r}=\delta_{1,l}=\delta_{|V}$, and $\lambda_z$ (resp. $\rho_z$) for the left (resp. right) multiplication of $z$ on $B$.
Applying $-\otimes_AB$ to (\ref{resolution of k_A}), one obtains an exact sequence, $$ \cdots\xlongrightarrow{\partial_{d+1}} W_d\otimes B\xlongrightarrow{\partial_d}W_{d-1}\otimes B\xlongrightarrow{\partial_{d-1}}\cdots\xlongrightarrow{\partial_{3}}W_2\otimes B\xlongrightarrow{\partial_{2}}W_1\otimes B\xlongrightarrow{\partial_{1}}B\xlongrightarrow{\varepsilon_A\otimes_AB}B/A_{\geq1}B\to0, $$ where $\partial_{i}=\partial^A_{i}\otimes_AB=(\id_V^{\otimes i-1}\otimes m_B)(\id_V^{\otimes i-1}\otimes\pi_B\otimes\id_B)=\id_V^{\otimes i-1}\otimes m_B$ and $m_B$ is the multiplication of $B$ for $i\geq1$.
\begin{lemma}\label{lemma: construction of delta_r} There exist graded linear maps $\delta_{i,r}:W_i\to W_i\otimes V$ for $i\geq 2$ such that the following diagram is commutative $${\Small \xymatrix{ \cdots\ar[r] & W_d\otimes B(-1) \ar[r]^{\partial_d}\ar[d]^{\phi_d} &W_{d-1}\otimes B(-1)\ar[r]^(0.65){\partial_{d-1}}\ar[d]^{\phi_{d-1}}&\cdots\ar[r]^(0.35){\partial_{2}} &W_1\otimes B(-1)\ar[r]^(0.55){\partial_{1}}\ar[d]^{\phi_1} &B(-1)\ar[r]^(0.38){{\varepsilon_A\otimes_AB}}\ar[d]^{\phi_0}&B/A_{\geq1}B(-1)\ar[d]^{\lambda_z}\ar[r]&0 \\ \cdots\ar[r] & W_d\otimes B \ar[r]^{\partial_d} &W_{d-1}\otimes B\ar[r]^(0.6){{\partial_{d-1}}}&\cdots\ar[r]^(0.41){{\partial_{2}}} &W_1\otimes B\ar[r]^(0.6){{\partial_{1}}} &B\ar[r]^(0.38){{\varepsilon_A\otimes_AB}}&B/A_{\geq1}B\ar[r]&0, } } $$ where graded right $B$-module homomorphisms
\begin{align*} &\phi_0=\lambda_z, \\
&\phi_{i}=\sigma^{\otimes i}\otimes \lambda_z+(\id_V^{\otimes i}\otimes m_B)\left(\delta_{i,r}\otimes\id_B\right), \quad i\geq 1. \end{align*}
Moreover, as linear maps $W_i\to W_{i-1}\otimes B$, \begin{equation}\label{condition for delta_r} (\id_V^{\otimes i-1} \otimes m_B)\delta_{i,r}=(\id_V^{\otimes i-1}\otimes m_B)\left(\sigma^{\otimes i-1}\otimes\delta+\delta_{i-1,r}\otimes \id_V\right), \quad i\geq 2. \end{equation} \end{lemma} \begin{proof} It's easy to check that $\lambda_z(\varepsilon_A\otimes_A B)=(\varepsilon_A\otimes_AB)\phi_0$ and $\phi_0\partial_1=\partial_1\phi_1$, since \begin{align*} &\phi_0\partial_1=\lambda_zm_B=m_B(\sigma\otimes \lambda_z+(\pi_B\otimes\id_B)(\delta\otimes\id_B))=m_B(\sigma\otimes \lambda_z)+m_B(\overline{\delta}\otimes \id_B),\\ &\partial_1\phi_1=m_B(\sigma\otimes\lambda_z)+m_B(\id_V\otimes m_B)(\delta\otimes \id_B)=m_B(\sigma\otimes\lambda_z)+m_B(\overline{\delta}\otimes \id_B). \end{align*} By the construction of $\delta_{2,r}$ and $\delta_{2,l}$, we have $$ (\id_V\otimes m_B)\left(\sigma\otimes\delta+\delta\otimes \id_V\right)= (\id_V\otimes m_B)\delta=(\id_V\otimes m_B)(\delta_{2,r}+\delta_{2,l})= (\id_V\otimes m_B)\delta_{2,r}, $$ in case restricted on $W_2=R$. One obtains \begin{align*} \partial_2\phi_2&=(\id_V\otimes m_B)(\sigma^{\otimes 2}\otimes \lambda_z)+(\id_V\otimes m_B)(\id_V^{\otimes2}\otimes m_B)(\delta_{2,r}\otimes \id_B)\\ &=(\id_V\otimes m_B)(\sigma^{\otimes 2}\otimes \lambda_z)+(\id_V\otimes m_B)(\id_V^{\otimes2}\otimes m_B)(\delta_{2,r}\otimes \id_B+\delta_{2,l}\otimes\id_B)\\ &=(\id_V\otimes m_B)(\sigma^{\otimes 2}\otimes \lambda_z)+(\id_V\otimes m_B)(\id_V^{\otimes2}\otimes m_B)(\delta \otimes\id_B),\\ \phi_1\partial_2&=(\sigma\otimes \lambda_z)(\id_V\otimes m_B)+(\id_V\otimes m_B)(\delta\otimes \id_B)(\id_V\otimes m_B)\\ &=(\id_V\otimes m_B)(\sigma^{\otimes 2}\otimes \lambda_z)+(\id_V\otimes m_B)(\id_V^{\otimes2}\otimes m_B)\left((\sigma\otimes\delta)\otimes\id_B+(\delta\otimes\id_V)\otimes \id_B\right)\\ &=(\id_V\otimes m_B)(\sigma^{\otimes 2}\otimes \lambda_z)+(\id_V\otimes m_B)(\id_V^{\otimes2}\otimes m_B)(\delta\otimes\id_B). \end{align*} So $\partial_2\phi_2=\phi_1\partial_2$. By the Comparison Theorem, there exists a graded $B$-module homomorphism $\phi_i: W_i\otimes B(-1)\to W_i\otimes B$ such that $\phi_i\partial_{i+1}=\partial_{i+1}\phi_{i+1}$ for any $i\geq 3$.
It is clear that $(\phi_i-\sigma^{\otimes i}\otimes\lambda_z)(W_i\otimes B_0)\subseteq W_i\otimes V,$ so write $\delta_{i,r}$ for the linear map $\phi_i-\sigma^{\otimes i}\otimes\lambda_z$ restricted on $W_i$ for any $i\geq 3$. One obtains that $$ \phi_i=\sigma^{\otimes i}\otimes \lambda_z+(\id_V^{\otimes i}\otimes m_B)(\delta_{i,r}\otimes\id_B),\quad i\geq 3. $$ Then $\partial_i\phi_i=(\id_V^{\otimes i-1}\otimes m_B)(\sigma^{\otimes i}\otimes\lambda_z)+(\id_V^{\otimes i-1}\otimes m_B)(\id_V^{\otimes i}\otimes m_B)(\delta_{i,r}\otimes \id_B)$, and \begin{align*} \phi_{i-1}\partial_i&=(\sigma^{\otimes i-1}\otimes \lambda_z)(\id_V^{\otimes i-1}\otimes m_B) +(\id_V^{\otimes i-1}\otimes m_B)\left(\delta_{i-1,r}\otimes\id_B\right)(\id_V^{\otimes i-1}\otimes m_B)\\ &=(\id_V^{\otimes i-1}\otimes m_B)(\sigma^{\otimes i}\otimes\lambda_z)+ (\id^{\otimes i-1}_V\otimes m_B)(\id^{\otimes i}_V\otimes m_B)(\sigma^{\otimes i-1}\otimes\delta\otimes\id_B)\\ &\quad +(\id^{\otimes i-1}_V\otimes m_B)(\id^{\otimes i}_V\otimes m_B)(\delta_{i-1,r}\otimes\id_V\otimes\id_B). \end{align*} Since $\partial_i\phi_{i}=\phi_{i-1}\partial_i$, as linear maps $W_i\to W_{i-1}\otimes B$, \begin{equation*} \qquad(\id_V^{\otimes i-1}\otimes m_B)\delta_{i,r}=(\id_V^{\otimes i-1}\otimes m_B)\left(\sigma^{\otimes i-1}\otimes\delta+\delta_{i-1,r}\otimes \id_V\right),\qquad \forall i\geq 3,\qedhere \end{equation*}
\end{proof}
\begin{remark}\label{remark: choice of delta_r} By the proof of Lemma \ref{lemma: construction of delta_r}, any linear map from $W_i$ to $W_i\otimes V$ satisfying (\ref{condition for delta_r}) can be chosen to be $\delta_{i,r}$ for $i\geq 2$. So the sequence $\{\delta_{i,r}\mid i\geq1\}$ of linear maps is not unique for the map $\overline{\delta}$, even when $\delta_{2,r}$ is fixed. \end{remark}
By \cite[Theorem 1]{GS} or \cite[Lemma 2.4]{Phan}, one obtains a minimal free resolution of $k_B$.
\begin{lemma}\label{lemma: minimial resolution of k_B} The following complex is exact: \begin{equation}\label{resolution of k_B} \begin{aligned} \cdots\rightarrow W_{d}\otimes& B(-1)\oplus W_{d+1}\otimes B \xlongrightarrow{\left({\scriptscriptstyle\begin{array}{ll}{\scriptstyle -\partial_{d} }& {\scriptstyle0} \\ {\scriptstyle\phi_{d} }&{\scriptstyle \partial_{d+1}}\end{array}}\right)} W_{d-1}\otimes B(-1)\oplus W_d\otimes B\xlongrightarrow{}\cdots\\ &\xlongrightarrow{} W_1\otimes B(-1)\oplus W_{2}\otimes B \xlongrightarrow{\left({\scriptscriptstyle\begin{array}{ll}{\scriptstyle -\partial_{1} }& {\scriptstyle0} \\ {\scriptstyle\phi_1 }&{\scriptstyle \partial_{2}}\end{array}}\right)} B(-1)\oplus W_1\otimes B\xlongrightarrow{\left(\begin{array}{ll}{\scriptstyle \phi_0}& {\scriptstyle \partial_1}\end{array}\right)} B\xlongrightarrow{\varepsilon_B} k_B\xlongrightarrow{} 0, \end{aligned} \end{equation} for some sequence $\{\delta_{i,r}\}$ of linear maps as in Lemma \ref{lemma: construction of delta_r}. \end{lemma}
\subsection{A sequence pair}
We have constructed a sequence $\{\delta_{i,r}\mid i\geq 2\}$ of linear maps from $\delta$ to obtain a minimal free resolution of $k_B$. It seems a right version of linear maps arose from $\delta$. Symmetrically, we construct a left version of linear maps. Write $\delta_{0,r}=\delta_{0,l}=0$.
\begin{lemma}\label{lemma: construction of delta_l} Let $\{\delta_{i,r}:W_i\to W_i\otimes V\mid i\geq 1\}$ be a sequence of linear maps as in Lemma \ref{lemma: construction of delta_r}. Then there exists a unique set $\{\delta_{i,l}:W_i\to V\otimes W_i\mid i\geq 1\}$ of linear maps with respect to $\{\delta_{i,r}\}$ such that
\begin{equation}\label{eq: relation between delta_l and delta_r}\qquad\quad \delta_{i,r}+(-1)^i\delta_{i,l}=\sigma\otimes\delta_{i-1,r}+(-1)^i\delta_{i-1,l}\otimes \id_V,\qquad \forall i\geq 1. \end{equation}
\end{lemma} \begin{proof} Since $\delta_{1,r}=\delta_{1,l}$, the result holds if $i=1$. By the definition (\ref{definition of delta_2}) of $\delta_{2,l}$, it is a linear map from $W_2$ to $V\otimes W_2$. Clearly, $$ \delta_{2,r}-\delta_{1,l}\otimes\id_V-\sigma\otimes \delta_{1,r}=\delta_{2,r}-\delta\otimes\id_V-\sigma\otimes \delta=\delta_{2,r}-\delta=-\delta_{2,l}. $$
Suppose $i\geq 3$ and there exist linear maps $\delta_{j,l}:W_j\to V\otimes W_j$ for $j< i$, such that $$ \delta_{j,r}+(-1)^j\delta_{j,l}=\sigma\otimes\delta_{j-1,r}+(-1)^j\delta_{j-1,l}\otimes \id_V. $$ Then we have the following commutative diagram $$ \xymatrix{ &W_i\otimes B(-1) \ar@{-->}[ld]_{\overline{\theta}_i}\ar[d]^{\theta_i}&\\ V\otimes W_i\otimes B\ar[r]^(0.47){\id_V\otimes \partial_i} &V\otimes W_{i-1}\otimes B\ar[r]^{\id_V\otimes \partial_{i-1}} &V\otimes W_{i-2}\otimes B, } $$ where $ \theta_i=(\id_V^{\otimes i }\otimes m_B)\left(\left((-1)^{i-1}\left(\delta_{i,r}-\sigma\otimes\delta_{i-1,r}\right)+\delta_{i-1,l}\otimes \id_V\right)\otimes\id_B\right). $ In fact, {\small \begin{align*} (-1)^{i-1}(\id_V \otimes \partial_{i-1})\theta_i&=(\id_V^{\otimes i-1} \otimes m_B)(\id_V^{\otimes i }\otimes m_B)\left((\delta_{i,r}-\sigma\otimes\delta_{i-1,r}+(-1)^{i-1}\delta_{i-1,l}\otimes \id_V)\otimes\id_B\right)\\ &=(\id_V^{\otimes i-1} \otimes m_B)(\id_V^{\otimes i }\otimes m_B)\left((\sigma^{\otimes i-1}\otimes \delta+\delta_{i-1,r}\otimes \id_V-\sigma\otimes\delta_{i-1,r}+(-1)^{i-1}\delta_{i-1,l}\otimes \id_V)\otimes\id_B\right)\\ &=(\id_V^{\otimes i-1} \otimes m_B)(\id_V^{\otimes i }\otimes m_B)\left( ( \sigma^{\otimes i-1}\otimes \delta+(\delta_{i-1,r}+(-1)^{i-1}\delta_{i-1,l})\otimes\id_V-\sigma\otimes\delta_{i-1,r}) \otimes \id_B \right)\\ &= (\id_V^{\otimes i-1} \otimes m_B)(\id_V^{\otimes i }\otimes m_B)\left( ( \sigma^{\otimes i-1}\otimes \delta+\sigma\otimes\delta_{i-2,r}\otimes\id_V-\sigma\otimes\delta_{i-1,r}) \otimes \id_B \right)\\ &\quad+(-1)^{i-1}(\id_V^{\otimes i-1} \otimes m_B)(\id_V^{\otimes i }\otimes m_B)\left((\delta_{i-2,l}\otimes\id_V\otimes\id_V)\otimes\id_B\right)\\ &= (\id_V^{\otimes i-1} \otimes m_B)(\id_V^{\otimes i }\otimes m_B)\left( \left( \sigma^{\otimes i-1}\otimes \delta+\sigma\otimes(\delta_{i-2,r}\otimes\id_V-\delta_{i-1,r})\right) \otimes \id_B \right)\\ &= (\id_V^{\otimes i-1} \otimes m_B)(\id_V^{\otimes i }\otimes m_B)\left( \left( \sigma^{\otimes i-1}\otimes \delta-\sigma^{\otimes i-1}\otimes \delta)\right) \otimes \id_B \right)\\ &=0, \end{align*}} the second and sixth equalities hold by Lemma \ref{lemma: construction of delta_r}, the fourth equality holds by the assumption and the fifth equality holds by $W_i\subseteq V^{\otimes i-2}\otimes R$. So $\Image \theta_i\subseteq\Ker (\id_V\otimes\partial_{i-1})=\Image (\id_V\otimes\partial_{i})$, and there exists a graded $B$-module homomorphism $\overline{\theta}_{i}: W_i\otimes B(-1)\to V\otimes W_i\otimes B$ such that $(\id_V\otimes \partial_i)\overline{\theta}_i=\theta_i$, since $W_i\otimes B$ is free.
It is easy to know that ${\overline{\theta}_{i}}_{\mid W_i}$ is a map from $W_i$ to $V\otimes W_i$ by an argument on degree, denoted this map by $\delta_{i,l}$.
Hence, $$ \delta_{i,r}+(-1)^i\delta_{i,l}=\sigma\otimes\delta_{i-1,r}+(-1)^i\delta_{i-1,l}\otimes \id_V. $$
The uniqueness can be obtained by the construction easily.
\end{proof}
\begin{definition} Let $\{\delta_{i,r}:W_i\to W_i\otimes V\mid i\geq 1\}$ be a sequence of linear maps constructed in Lemma \ref{lemma: construction of delta_r}, and $\{\delta_{i,l}:W_i\to V\otimes W_i \mid i\geq 1\}$ a sequence of linear maps constructed in Lemma \ref{lemma: construction of delta_l}. Then $\left(\{\delta_{i,r}\},\{\delta_{i,l}\}\right)$ is called a \emph{sequence pair} for the $\overline{\sigma}$-derivation $\overline{\delta}$. \end{definition}
We give a relation between $\{\delta_{i,r}\}$ and $\{\delta_{i,l}\}$, which will be useful in the construction of twisted superpotentials for graded Ore extensions.
\begin{proposition}\label{prop: relation between delta_r and delta_l} Let $\left(\{\delta_{i,r}\},\{\delta_{i,l}\}\right)$ be a sequence pair for $\overline{\delta}$. Then \begin{enumerate} \item For any integer $d\geq 1$, $$ \sum_{i=1}^d (-1)^{i}(\delta_{i,r}\otimes\id_V^{\otimes d-i})=(-1)^{d+1}\sum_{i=1}^{d}(-1)^{i}(\sigma^{\otimes d-i}\otimes\delta_{i,l}). $$ \item For any integer $d\geq 2$, $$ \sum_{i=1}^{d-1}(-1)^{i+1}(\sigma^{\otimes d-i-1}\otimes \delta_{i,l}\otimes\id_V)=(-1)^{d+1}\sum_{i=1}^{d-1}(-1)^i(\delta_{i,r}\otimes\id_V^{\otimes d-i}). $$ \end{enumerate}
\end{proposition}
\begin{proof} (a) If $d=1$, the equality is clear. Assume $d\geq 2$. By Lemma \ref{lemma: construction of delta_l}, \begin{align*} &\sum_{i=j+1}^d (-1)^{i}(\sigma^{\otimes j}\otimes\delta_{i-j,r}\otimes\id_V^{\otimes d-i})\\ =&\sum_{i=j+1}^d (-1)^{i}(\sigma^{\otimes j}\otimes\delta_{i-j,r}\otimes\id_V^{\otimes d-i})+(-1)^j\sum_{i=j+1}^d \sigma^{\otimes j}\otimes\delta_{i-j,l}\otimes\id_V^{\otimes d-i}-(-1)^j\sum_{i=j+1}^d \sigma^{\otimes j}\otimes\delta_{i-j,l}\otimes\id_V^{\otimes d-i}\\ =&\sum_{i=j+1}^d (-1)^{i}\left(\sigma^{\otimes j}\otimes(\delta_{i-j,r}+(-1)^{i-j}\delta_{i-j,l})\otimes\id_V^{\otimes d-i}\right)-(-1)^j\sum_{i=j+1}^d \sigma^{\otimes j}\otimes\delta_{i-j,l}\otimes\id_V^{\otimes d-i}\\ =&\sum_{i=j+2}^d (-1)^{i}(\sigma^{\otimes j+1}\otimes\delta_{i-j-1,r}\otimes\id_V^{\otimes d-i})+(-1)^j\sum_{i=j+2}^d \sigma^{\otimes j}\otimes\delta_{i-j-1,l}\otimes\id_V^{\otimes d-i+1}-(-1)^j\sum_{i=j+1}^d \sigma^{\otimes j}\otimes\delta_{i-j,l}\otimes\id_V^{\otimes d-i}\\ =&\sum_{i=j+2}^d (-1)^{i}(\sigma^{\otimes j+1}\otimes\delta_{i-j-1,r}\otimes\id_V^{\otimes d-i})-(-1)^j\sigma^{\otimes j}\otimes\delta_{d-j,l}, \end{align*} for any $j=0,1,\cdots,d-2$. Then \begin{align*} &\sum_{j=0}^{d-2}\sum_{i=j+1}^d (-1)^{i}(\sigma^{\otimes j}\otimes\delta_{i-j,r}\otimes\id_V^{\otimes d-i})=\sum_{j=0}^{d-2}\sum_{i=j+2}^d (-1)^{i}(\sigma^{\otimes j+1}\otimes\delta_{i-j-1,r}\otimes\id_V^{\otimes d-i})-\sum_{j=0}^{d-2}(-1)^j\sigma^{\otimes j}\otimes\delta_{d-j,l},\\ &\sum_{i=1}^d (-1)^{i}(\delta_{i,r}\otimes\id_V^{\otimes d-i})=(-1)^{d}(\sigma^{\otimes d-1}\otimes\delta_{1,r})-\sum_{j=0}^{d-2}(-1)^j\sigma^{\otimes j}\otimes\delta_{d-j,l}=-\sum_{j=0}^{d-1}(-1)^j\sigma^{\otimes j}\otimes\delta_{d-j,l}. \end{align*} The result follows.
(b) By Lemma \ref{lemma: construction of delta_l}, $$ \sum_{i=1}^{d-1}(-1)^{i+1}\sigma^{\otimes d-i-1}\otimes \delta_{i,l}\otimes\id_V+\sum_{i=1}^{d-1}\sigma^{\otimes d-i}\otimes\delta_{i,r} =\sum_{i=1}^{d-1}\sigma^{{\otimes d-i-1}}\otimes \delta_{i+1,r}+\sum_{i=1}^{d-1}(-1)^{i+1}\sigma^{\otimes d-i-1}\otimes\delta_{i+1,l}. $$ Then \begin{align*} \sum_{i=1}^{d-1}(-1)^{i+1}\sigma^{\otimes d-i-1}\otimes \delta_{i,l}\otimes\id_V&=\delta_{d,r}-\sigma^{\otimes d-1} \otimes \delta_{1,r}+\sum_{i=2}^{d}(-1)^{i}\sigma^{\otimes d-i}\otimes\delta_{i,l}\\ &=\delta_{d,r}-\sigma^{\otimes d-1} \otimes \delta_{1,r}+\sigma^{\otimes d-1}\otimes\delta_{1,l}-(-1)^{d}\sum_{i=1}^d(-1)^i(\delta_{i,r}\otimes\id_V^{\otimes d-i})\\ &=(-1)^{d+1}\sum_{i=1}^{d-1}(-1)^i(\delta_{i,r}\otimes\id_V^{\otimes d-i}), \end{align*} where the second equality holds by (a). \end{proof}
As shown above, a sequence pair $\left(\{\delta_{i,r}\},\{\delta_{i,l}\}\right)$ is constructed from a linear map $\delta: V\to V\otimes V$ which induces the $\overline{\sigma}$-derivation $\overline{\delta}$ in the graded Ore extensions $B$. It is also shown that such sequence pairs vary according to the decomposition (\ref{decomposition of delta}) and the choices of $\{\delta_{i,r}\}$ in Lemma \ref{lemma: construction of delta_r}. On the other hand, the $\overline{\sigma}$-derivation $\overline{\delta}$ can be induced from different linear maps $\delta,\delta':V\to V\otimes V$, which also arise different sequence pairs for $\overline{\delta}$. Here, we give a relation between different sequence pairs.
\begin{proposition}\label{prop: relation betwenn different delta_r} Let $\delta,\delta'$ be two linear maps from $V$ to $V\otimes V$ such that $\pi_A\delta=\pi_A\delta'=\overline{\delta}_{\mid V}$, and $\left(\{\delta_{i,r}\},\{\delta_{i,l}\}\right)$ and $\left(\{\delta'_{i,r}\},\{\delta'_{i,l}\}\right)$ two sequence pairs for $\overline{\delta}$ constructed from $\delta$ and $\delta'$ respectively. Then for any $i\geq1$, \begin{enumerate} \item $ \Image\left(\sum_{j=0}^{i-1}(-1)^j\left((\delta_{i-j,r}-\delta'_{i-j,r})\otimes\id_V^{\otimes j}\right)\right)\subseteq W_{i+1}. $ \item $ \Image\left((\delta_{i,l}-\delta'_{i,l})+(-1)^{i}\sum_{j=1}^{i-1}(-1)^j\left(\sigma\otimes(\delta_{i-j,r}-\delta'_{i-j,r})\otimes\id_V^{\otimes j-1}\right)\right)\subseteq W_{i+1}. $ \end{enumerate}
Moreover, if $W_{d+1}=0$ for some $d\geq1$, then \begin{align*} \sum_{j=0}^{d-1}(-1)^{j}(\delta_{d-j,r}\otimes\id_V^{\otimes j})&=\sum_{j=0}^{d-1}(-1)^{j}(\delta'_{d-j,r}\otimes\id_V^{\otimes j}),\\ \sum_{j=0}^{d-1}(-1)^{j}(\sigma^{\otimes j}\otimes \delta_{d-j,l})&=\sum_{j=0}^{d-1}(-1)^{j}(\sigma^{\otimes j}\otimes \delta'_{d-j,l}),\\
(-1)^{d}\delta_{d,l}+\sum_{j=1}^{d-1}(-1)^{j}\left(\sigma\otimes \delta_{d-j,r}\otimes\id_V^{\otimes j-1}\right)&=(-1)^{d}\delta'_{d,l}+\sum_{j=1}^{d-1}(-1)^{j}\left(\sigma\otimes \delta'_{d-j,r}\otimes\id_V^{\otimes j-1}\right). \end{align*}
\end{proposition} \begin{proof} (a) Let $\{\phi_{i}\}$ and $\{\phi'_{i}\}$ be two lifts of $\lambda_z:B/A_{\geq 1}B(-1)\to B/A_{\geq1}B$ as in Lemma \ref{lemma: construction of delta_r} associated with $\{\delta_{i,r}\}$ and $\{\delta'_{i,r}\}$ respectively. Clearly, $\phi_0=\phi'_0$, and $\phi_i-\phi'_i= (\id_V^{\otimes i}\otimes m_B)\left((\delta_{i,r}-\delta'_{i,r})\otimes\id_B\right)$ for any $i\geq 1$. There exist graded $B$-module homomorphisms $s_i:W_i\otimes B(-1)\to W_{i+1}\otimes B$ for $i\geq 1$ such that the following diagram is commutative $${\Small \xymatrix{ \cdots\ar[r] & W_i\otimes B(-1) \ar[r]^{\partial_d}\ar[ld]_{s_i}\ar[d]^{\phi_i-\phi'_i} &W_{i-1}\otimes B(-1)\ar[r]^(0.65){\partial_{d-1}}\ar[ld]_{s_{i-1}}\ar[d]^{\phi_{i-1}-\phi'_{i-1}}&\cdots\ar[r]^(0.35){\partial_{3}}&W_2\otimes B(-1)\ar[ld]_{s_2}\ar[r]^(0.55){\partial_2}\ar[d]^{\phi_2-\phi'_2} &W_1\otimes B(-1)\ar[ld]_{s_1}\ar[r]^(0.55){\partial_{1}}\ar[d]^{\phi_1-\phi'_1} &B(-1)\ar[r]\ar[d]^{0}\ar[ld]_{0}&0 \\ \cdots\ar[r] & W_i\otimes B \ar[r]^{\partial_i} &W_{i-1}\otimes B\ar[r]^(0.6){{\partial_{d-1}}}&\cdots\ar[r]^(0.41){{\partial_{3}}}&W_2\otimes B\ar[r]^(0.55){\partial_{1}} &W_1\otimes B\ar[r]^(0.6){{\partial_{1}}} &B\ar[r]&0. } } $$ that is, $\phi_i-\phi'_i=s_{i-1}\partial_i+\partial_{i+1}s_{i},$ for any $i\geq 1$, where $s_0=0$. By an easy argument on degree, one obtains that for any $i\geq 1$, $\Image {s_i}_{\mid W_i}\subseteq W_{i+1}$ and $${s_i}_{\mid W_i}=\delta_{i,r}-\delta'_{i,r}-{s_{i-1}}_{\mid W_{i-1}}\otimes\id_V.$$
Since ${s_{1}}_{\mid W_1}=\delta_{1,r}-\delta'_{1,r}$, we have $$ {s_i}_{\mid W_i}=\sum_{j=0}^{i-1}(-1)^j\left((\delta_{i-j,r}-\delta'_{i-j,r})\otimes\id_V^{\otimes j}\right). $$ Then the result follows.
(b) By Lemma \ref{lemma: construction of delta_l}, we have \begin{align*} \delta_{j,r}+(-1)^j\delta_{j,l}=\sigma\otimes\delta_{j-1,r}+(-1)^j\delta_{j-1,l}\otimes \id_V,\qquad \delta'_{j,r}+(-1)^j\delta'_{j,l}=\sigma\otimes\delta'_{j-1,r}+(-1)^j\delta'_{j-1,l}\otimes \id_V. \end{align*} for any $j\geq 1$. So $$ (\delta_{j,r}-\delta'_{j,r})\otimes\id_V^{\otimes i-j} -\sigma\otimes(\delta_{j-1,r}-\delta'_{j-1,r})\otimes\id_V^{\otimes i-j} =(-1)^j(\delta_{j-1,l}-\delta'_{j-1,l})\otimes \id_V^{\otimes i-j+1}-(-1)^j(\delta_{j,l}-\delta'_{j,l})\otimes\id_V^{\otimes i-j}, $$ for any $1\leq j\leq i$, where $\delta_{0,r}=\delta_{0,l}=\delta'_{0,r}=\delta'_{0,l}=0$. Then \begin{align*} &\sum_{j=1}^i(-1)^{i-j}(\delta_{j,r}-\delta'_{j,r})\otimes\id_V^{\otimes i-j} -\sum_{j=1}^i(-1)^{i-j}\sigma\otimes(\delta_{j-1,r}-\delta'_{j-1,r})\otimes\id_V^{\otimes i-j}\\ =&\sum_{j=1}^i(-1)^i(\delta_{j-1,l}-\delta'_{j-1,l})\otimes \id_V^{\otimes i-j+1}-\sum_{j=1}^i(-1)^i(\delta_{j,l}-\delta'_{j,l})\otimes\id_V^{\otimes i-j}\\ =&(-1)^{i+1}(\delta_{i,l}-\delta'_{i,l}). \end{align*} Equivalently, $$ \sum_{j=0}^{i-1}(-1)^{j}(\delta_{i-j,r}-\delta'_{i-j,r})\otimes \id_V^{\otimes j}=(-1)^{i+1}(\delta_{i,l}-\delta'_{i,l})-\sum_{j=1}^{i-1}(-1)^{j}\sigma\otimes(\delta_{i-j,r}-\delta'_{i-j,r})\otimes \id_V^{\otimes j-1}. $$ Then the results holds by (a).
The last consequence is an immediate consequence of (a, b) and Proposition \ref{prop: relation between delta_r and delta_l}(a). \end{proof} \section{Nakayama automorphisms of graded Ore extensions of Koszul AS-regular algebras} Keep the notations in the last section. In this section, we always assume $A=T(V)/(R)$ is a Koszul AS-regular algebra of dimension $d$, where $V$ is a vector space with a basis $\{x_1,x_2,\cdots,x_n\}$. Write $\mu_A$ for the Nakayama automorphism of $A$. In this case, by \cite[Proposition 3.1.4]{SZ}, one obtains $$ \dim W_i= \left\{ \begin{array}{ll} \dim W_{d-i} & \text{if } 0\leq i\leq d.\\ 0 & \text{if } i>d. \end{array} \right. $$ Write $\{\eta_1,\eta_2,\cdots,\eta_n\}$ for a basis of $W_{d-1}$ and $\omega$ for a basis of $W_d$.
It is well known that the graded Ore extension $B=A[z;\overline{\sigma},\overline{\delta}]$ is a Koszul AS-regular algebra of dimension $d+1$ provided $\overline{\sigma}$ is a graded automorphism of $A$. Write $\sigma=\overline{\sigma}_{\mid V}$. The minimal free resolution (\ref{resolution of k_B}) of $k_B$ becomes \begin{equation}\label{resolution of koszul regular} \begin{aligned} 0\xlongrightarrow{} W_{d}\otimes& B(-1) \xlongrightarrow{\left({\scriptscriptstyle\begin{array}{ll}{\scriptstyle -\partial_{d} }\\ {\scriptstyle\phi_{d} }\end{array}}\right)} W_{d-1}\otimes B(-1)\oplus W_d\otimes B\xlongrightarrow{}\cdots\\ &\xlongrightarrow{} W_1\otimes B(-1)\oplus W_{2}\otimes B \xlongrightarrow{\left({\scriptscriptstyle\begin{array}{ll}{\scriptstyle -\partial_{1} }& {\scriptstyle0} \\ {\scriptstyle\phi_1 }&{\scriptstyle \partial_{2}}\end{array}}\right)} B(-1)\oplus W_1\otimes B\xlongrightarrow{\left(\begin{array}{ll}{\scriptstyle \phi_0}& {\scriptstyle \partial_1}\end{array}\right)} B\xlongrightarrow{\varepsilon_B} k_B\xlongrightarrow{} 0, \end{aligned} \end{equation} for some sequence $\{\delta_{i,r}\}$ of linear maps as in Lemma \ref{lemma: construction of delta_r}.
\subsection{An invariant}
Let $\left(\{\delta_{i,r}\},\{\delta_{i,l}\}\right)$ be a sequence pair for $\overline{\delta}$. Since $\dim W_d=1$, there exists a unique pair $(\delta_r,\delta_l)$ of elements in $V$ with respect to $\left(\{\delta_{i,r}\},\{\delta_{i,l}\}\right)$ such that $$\delta_{d,r}(\omega)=\omega\otimes \delta_r,\quad \delta_{d,l}(\omega)=\delta_l\otimes \omega.$$
\begin{corollary}\label{coro: relation between delta_r and delta l} Let $(\delta_r,\delta_l)$ and $(\delta'_r,\delta'_l)$ be two pairs with respect to two sequence pairs $\left(\{\delta_{i,r}\},\{\delta_{i,l}\}\right)$ and $\left(\{\delta'_{i,r}\},\{\delta'_{i,l}\}\right)$ for $\overline{\delta}$, respectively. Then $$ \delta_r+{\mu_A}\sigma^{-1}(\delta_l)=\delta'_r+\mu_A\sigma^{-1}(\delta'_l). $$
\end{corollary} \begin{proof}Firstly, one obtains that \begin{align*} \tau_d^{d-1}(\mu_A\sigma^{-1}\otimes\id_V^{\otimes d})(\delta_{d,l}-\delta'_{d,l})(\omega) &= \omega\otimes\left(\mu_A\sigma^{-1}(\delta_l-\delta'_l)\right). \end{align*} On the other hand, \begin{align*} &\tau_d^{d-1}(\mu_A\sigma^{-1}\otimes\id_V^{\otimes d})\left((-1)^{d-1}\sum_{j=1}^{d-1}(-1)^{j}\left(\sigma\otimes (\delta_{d-j,r}-\delta'_{d-j,r})\otimes\id_V^{\otimes j-1}\right)(\omega)\right)\\ =&\tau_d^{d-1}\left((-1)^{d-1}\sum_{j=1}^{d-1}(-1)^{j}\left(\mu_A\otimes (\delta_{d-j,r}-\delta'_{d-j,r})\otimes\id_V^{\otimes j-1}\right)(\omega)\right)\\ =&\sum_{j=1}^{d-1}(-1)^{j}\left( (\delta_{d-j,r}-\delta'_{d-j,r})\otimes\id_V^{\otimes j}\right)\left((-1)^{d-1}\tau_d^{d-1}(\mu_A\otimes\id_V^{\otimes d-1})(\omega)\right)\\ =&-(\delta_{d,r}-\delta'_{d,r})(\omega)\\ =&\omega\otimes(\delta'_r-\delta_r), \end{align*} where the third equality holds by $W_{d+1}=0$, Proposition \ref{prop: relation betwenn different delta_r} and Theorem \ref{thm: properties of Koszul regular algebras}(c). Then the result follows by Proposition \ref{prop: relation betwenn different delta_r} again. \end{proof}
\begin{definition} Let $A=T(V)/(R)$ be a Koszul AS-regular algebra, $\overline{\sigma}$ a graded automorphism of $A$ and $\overline{\delta}$ a degree-one $\overline{\sigma}$-derivation of $A$. Let $(\delta_l,\delta_r)$ be the pair of elements in $V$ with respect to some sequence pair $\left(\{\delta_{i,r}\},\{\delta_{i,l}\}\right)$ for the $\overline{\delta}$. Then the element $\delta_r+\mu_A\overline{\sigma}^{-1}(\delta_l)$ is called the \emph{$\overline{\sigma}$-divergence} of $\overline{\delta}$, denoted by $\nabla_{\overline{\sigma}}\cdot\overline{\delta}$. \end{definition}
\begin{remark} If $A$ is a commutative graded polynomial algebra generated in degree 1 and $\overline{\sigma}$ is the identity map, $\nabla_{\id}\cdot\overline{\delta}$ is the usual divergence $\nabla\cdot\overline{\delta}$ of $\overline{\delta}$ (see Theorem \ref{thm: NA of Ore ext over Polynomial} or \cite[Thoerem 1.1(1)]{LM}). It motivates the name ``$\overline{\sigma}$-divergence'' of a $\overline{\sigma}$-derivation $\overline{\delta}$. \end{remark}
\subsection{Ext-algebras}
In this subsection, we compute the Yoneda product of the Ext-algebra $E(B)$ partially. We refer \cite{JZ} for the definition of homological determinant $\hdet(\overline{\sigma})$ of a graded automorphism $\overline{\sigma}$.
Following the minimal free resolution (\ref{resolution of koszul regular}) of $k_B$, one obtains that \begin{align*} &E^{1}(B)=\uHom_{B}(B(-1)\oplus W_1\otimes B,k)\cong k(1)\oplus W_1^*,\\ &E^{d}(B)=\uHom_{B}(W_{d-1}\otimes B(-1)\oplus W_{d}\otimes B,k)\cong W_{d-1}^*(1)\oplus W_d^*,\\ &E^{d+1}(B)=\uHom_{B}(W_{d}\otimes B(-1),k)\cong W_{d}^*(1). \end{align*} Then $E^{1}(B)$ has a basis $\xi, x_1^*,x_2^,\cdots,x^*_n$, where $\xi$ corresponds to the identity of $k$ (or the augmentation $\varepsilon_B:B(-1)\to k(-1)$), $E^{d}(B)$ has a basis $\eta_1^*,\eta_2^,\cdots,\eta^*_n,\omega^*$, and $E^{d+1}(B)$ has a basis $\widetilde{\omega}^*$ corresponding to the element $\omega^*$ in $W_d^*$. \begin{remark} In order to make notations succinct, we use the basis of $W_1, W_{d-1},W_d$ to represent the basis of $E^{1}(B),E^d(B),E^{d+1}(B)$. However, it should keep in mind that each element in such basis also has a corresponding homomorphism through the minimal free resolution. To be specific, each $x_i^*$ corresponds to $x_i^*\otimes \varepsilon_B:W_1\otimes B\to k(-1),$ $\eta_i^*$ corresponds to $\eta_i^*\otimes \varepsilon_B:W_{d-1}\otimes B(-1)\to k(-d)$, $\omega^*$ corresponds to $\omega^*\otimes \varepsilon_B:W_{d}\otimes B\to k(-d)$ and $\widetilde{\omega}^*$ corresponds to $\omega^*\otimes \varepsilon_B:W_{d}\otimes B(-1)\to k(-d-1)$for $i=1,\cdots,n$. In the sequel, we use such correspondence freely. \end{remark}
Define a graded algebra homomorphism $$ \begin{array}{cclc} p_z: & B &\to &k[z]\\ ~ & \sum_{i=1}^m a_iz^i &\mapsto & \sum_{i=1}^m \varepsilon_A(a_i)z^i. \end{array} $$ By \cite[Theorem 1]{SWZ}, $E(p_z)$ is a graded algebra homomorphism from $E(k[z])$ to $E(B)$. Notice that, $$ 0\to k\{z\}\otimes k[z]\xlongrightarrow{h} k[z]\xlongrightarrow{\varepsilon_{k[z]}} k_{k[z]}\to 0, $$ where $k\{z\}$ is the vector space spanned by $z$ and the right graded $k[z]$-module homomorphism $h$ mapping $z\otimes 1$ to $z$, is a minimal free resolution of the graded trivial module $k_{k[z]}$. So $$ E^{1}(k[z])=\uHom_{k[z]}(k\{z\}\otimes k[z],k)=(k\{z\})^*, $$ and $z^*$ (or $z^*\otimes \varepsilon_{k[z]}$) is a basis of $E^1(k[z])$.
\begin{lemma} $E(p_z)(z^*)=\xi$. \end{lemma} \begin{proof}Clearly, we have the following commutative diagram $$ \xymatrix{ W_1\otimes B(-1)\oplus W_{2}\otimes B \ar[r]\ar[d]^{0}&
B(-1)\oplus W_1\otimes B\ar[rr]^(0.65){{\left(\begin{array}{ll}{\scriptstyle \phi_0}& {\scriptstyle \partial_1}\end{array}\right)}} \ar[d]^{f} && B\ar[r]^{\varepsilon_B}\ar[d]^{p_z} & k_B\ar[r]\ar@{=}[d] &0 \\ 0\ar[r]& k\{z\}\otimes k[z]\ar[rr]^{h}&& k[z]\ar[r]^{\varepsilon_{k[z]}} & k_{k[z]}\ar[r]& 0, } $$ where $f(b,w_1\otimes b')=z\otimes p_z(b)$ for any $b,b'\in B$ and $w_1\in W_1$. By \cite[Theorem 1]{SWZ}, one obtains that $$ E(p_z)(z^*)(b,w_1\otimes b')=(z^*\otimes\varepsilon_{k[z]})f(b,w_1\otimes b')=\varepsilon_{k[z]}p_z(b)=\varepsilon_{B}(b), $$ as elements in $\uHom_{B}(B(-1)\oplus W_1\otimes B,k)$. Hence, $E(p_z)(z^*)=\xi$. \end{proof}
Now it turns to compute the Yoneda product of $E^1(B)$ and $E^{d}(B)$. We fix a sequence pair $(\{\delta_{i,r}\},\{\delta_{i,l}\})$ for $\overline{\delta}$, and $(\delta_r,\delta_l)$ is the pair of elements in $V$ with respect to $(\{\delta_{i,r}\},\{\delta_{i,l}\})$.
\begin{lemma}\label{lemma: E^1(B) E^d(B)} In $E(B)$, for any $i,j=1,\cdots,n$, $$ \begin{array}{ll}
\xi\ast\omega^*=(-1)^d\hdet(\overline{\sigma})\,\widetilde{\omega}^*,&\xi\ast\eta_j^*=0,\\ x^*_i\ast\omega^*=(-1)^dx_i^*(\delta_r)\cdot\widetilde{\omega}^*, &x_i^*\ast\eta_j^*=(-1)^{d+1}(\eta_j^*\otimes x^*_i)(\omega)\cdot \widetilde{\omega}^*. \end{array} $$ \end{lemma} \begin{proof} We claim that the following diagram is commutative. $$ \xymatrix{
W_d\otimes B(-1)\ar[r]^(0.38){\left({\scriptscriptstyle\begin{array}{l}{\scriptstyle -\partial_d} \\ {\scriptstyle\phi_d}\end{array}}\right)}\ar[d]^{\varphi_2}
&
W_{d-1}\otimes B(-1)\oplus W_d\otimes B\ar[d]^{\varphi_1} \ar[rd]^(0.6){\,(\eta_j^*\otimes\varepsilon_B,\ \omega^*\otimes\varepsilon_B)}\\ B(-d-1)\oplus W_1\otimes B(-d)\ar[r]^(0.63){(-1)^d\left(\begin{array}{ll}{\scriptstyle \phi_0}& {\scriptstyle \partial_1}\end{array}\right)} & B(-d)\ar[r]^{\varepsilon_B}& k_B(-d), } $$ where $\varphi_1=(\eta_j^*\otimes\id_B,\ \omega^*\otimes\id_B)$, and \begin{align*} \varphi_2=(-1)^{d+1}\left(\begin{array}{c} 0\\ (\eta_j^*\otimes\id_V)\otimes\id_B \end{array} \right)+(-1)^d\left(\begin{array}{c} \hdet(\overline{\sigma})(\omega^*\otimes \id_B)\\ (\omega^*\otimes\id_V)\delta_{d,r}\otimes \id_B \end{array} \right), \end{align*} and the first summand of $\varphi_2$ is induced by $\eta_j^*\otimes\varepsilon_B$ and the other one by $\omega^*\otimes\varepsilon_B$.
In fact, $\varepsilon_B\varphi_1=(\eta_j^*\otimes\varepsilon_B,\ \omega^*\otimes\varepsilon_B)$ is obvious, and \begin{align*} \varphi_1(-\partial_d\ \, \phi_d)^T&=-(\eta_j^*\otimes\id_B)\partial_d+(\omega^*\otimes\id_B)\phi_d\\ &=-\eta_j^*\otimes m_B+(\omega^*\otimes\id_B)(\sigma^{\otimes d}\otimes \lambda_z)+(\omega^*\otimes m_B)(\delta_{d,r}\otimes \id_B),\\ (-1)^d(\phi_0\ \partial_1)\varphi_2&=-\eta_j^*\otimes m_B+\hdet(\overline{\sigma})(\omega^*\otimes\lambda_z)+(\omega^*\otimes m_B)(\delta_{d,r}\otimes \id_B). \end{align*} By \cite[Theorem 1.2]{MS}, $\sigma^{\otimes d}=\hdet(\overline{\sigma})\cdot-:W_d\to W_d$. So the diagram is commutative. Then \begin{align*} &\xi\ast(\eta_j^*,\ \omega^*)=(\varepsilon_B,0)\varphi_2=(-1)^d\hdet(\overline{\sigma})\,\omega^*\otimes\varepsilon_B,\\ &x^*_i\ast(\eta_j^*,\ \omega^*)=(0,x^*_i\otimes \varepsilon_B)\varphi_2=(-1)^{d+1}(\eta_j^*\otimes x^*_i)(\omega)\omega^*\otimes\varepsilon_B+(-1)^dx_i^*({\delta_r})\omega^*\otimes\varepsilon_B. \end{align*} The proof is completed. \end{proof}
\begin{lemma}\label{lemma: E^d(B) E^1(B)} In $E(B)$, for any $i,j=1,\cdots,n$, $$ \begin{array}{ll}
\omega^*\ast\xi= \widetilde{\omega}^*,&\eta_j^*\ast\xi=0,\\ \omega^*\ast x^*_i= -x_i^*(\delta_l)\widetilde{\omega}^*, &\eta_j^*\ast x_i^*=(-1)^d(x_i^*\sigma\otimes\eta_j^*)(\omega) \widetilde{\omega}^*. \end{array} $$ \end{lemma}
\begin{proof}Firstly, we check the following diagram is commutative. $${\small \xymatrix{ W_d\otimes B(-1)\ \ \ar[r]^(0.4){\scriptscriptstyle \left({\scriptscriptstyle\begin{array}{l}{\scriptstyle -\partial_d} \\ {\scriptstyle\phi_d}\end{array}}\right)}\ar[d]^{\psi_{d}} & W_{d-1}\otimes B(-1)\oplus W_d\otimes B\ar[r]\ar[d]^{\psi_{d-1}} &\cdots\ar[r] &W_2\otimes B(-1)\oplus W_{3}\otimes B\ar[d]^{\psi_{2}} \\ W_{d-1}\otimes B(-2)\oplus W_d\otimes B(-1)\ \ \ar[r]^{\scriptscriptstyle\left({\scriptscriptstyle\begin{array}{ll}{\scriptstyle \partial_{d-1} }& {\scriptstyle0} \\ {\scriptstyle-\phi_{d-1} }&{\scriptstyle -\partial_{d}}\end{array}}\right)} & \ \ \ W_{d-2}\otimes B(-2)\oplus W_{d-1}\otimes B(-1)\ar[r] &\cdots\ar[r] &W_1\otimes B(-2)\oplus W_{2}\otimes B(-1) } }$$ \begin{flushright} ${\small \xymatrix{ \ar[rr]^(0.33){\left({\scriptscriptstyle\begin{array}{ll}{\scriptstyle -\partial_{2} }& {\scriptstyle0} \\ {\scriptstyle\phi_2 }&{\scriptstyle \partial_{3}}\end{array}}\right)} &&W_1\otimes B(-1)\oplus W_{2}\otimes \ar[d]^{\psi_{1}} B\ar[rr]^(0.53){\left({\scriptscriptstyle\begin{array}{ll}{\scriptstyle -\partial_{1} }& {\scriptstyle0} \\ {\scriptstyle\phi_1 }&{\scriptstyle \partial_{2}}\end{array}}\right)} &&B(-1)\oplus W_1\otimes B\ar[d]^{\psi_{0}}\ar[rd]^{(\varepsilon_B,\ x_i^*\otimes \varepsilon_B)}\\ \ar[rr]^(0.33){\left({\scriptscriptstyle\begin{array}{ll}{\scriptstyle \partial_{1} }& {\scriptstyle0} \\ {\scriptstyle-\phi_1 }&{\scriptstyle -\partial_{2}}\end{array}}\right)} &&B(-2)\oplus W_1\otimes B(-1)\ar[rr]^(0.6){\left(\begin{array}{ll}{\scriptstyle -\phi_0}& {\scriptstyle -\partial_1}\end{array}\right)} &&B(-1)\ar[r]^{\varepsilon_{B}} & k_B(-1), } } \qquad$ \end{flushright} where \begin{align*}
&\psi_0=(\id_B,\ x_i^*\otimes\id_B),\\ &\psi_s= \left( \begin{array}{cc} 0 & 0\\ \id_{W_{s}\otimes B(-1)}&0 \end{array} \right)+ (-1)^{s} \left( \begin{array}{cc} (x_i^*\sigma\otimes\id_V^{\otimes s-1})\otimes\id_B & 0\\
(-1)^{s-1}\left[(x_i^*\otimes\id_V^{\otimes s})\delta_{s,l} \right]\otimes\id_B & (x_i^*\otimes\id_V^{\otimes s})\otimes\id_B \end{array} \right),\ \ \ 1\leq s\leq d-1,\\ &\psi_{d}= \left( \begin{array}{c} 0 \\ \id_{W_{d}\otimes B(-1)} \end{array} \right)+ (-1)^d \left( \begin{array}{c} (x_i^*\sigma\otimes\id_V^{\otimes d-1})\otimes\id_B \\ (-1)^{d-1}\left[(x_i^*\otimes\id_V^{\otimes d})\delta_{d,l} \right]\otimes\id_B \end{array} \right), \end{align*} and the first summand of those maps is induced by $\xi$ and the other one by $x_i^*\otimes \varepsilon_B$.
Write $\partial_i^B$ for the $i$-th differential in the minimal resolution of $k_B$ for $i\geq 1$. Obviously, $\varepsilon_B\psi_0=(\varepsilon_B,\ x_i^*\otimes\varepsilon_B)$. Also, $\psi_0\partial^B_2=-\partial^B_0\psi_1$, since $\delta=\delta_{1,r}=\delta_{1,l}$ and \begin{align*} \psi_0\partial^B_2&=\left(-\partial_1+(x_i^*\otimes\id_B)\phi_1,\ (x_i^*\otimes\id_B)\partial_2 \right)\\ &=\left(-\partial_1+x_i^*\sigma\otimes\lambda_z+(x_i^*\otimes m_B)(\delta_{1,r}\otimes \id_B),\ x_i^*\otimes m_B \right),\\ -\partial^B_0\psi_1&=\left( -\partial_1+\phi_0(x^*_i\sigma\otimes\id_B)+\partial_1\left[(x_1^*\otimes\id_V)\delta_{1,l}\otimes\id_B\right],\ \partial_1((x_i^*\otimes\id_V)\otimes \id_B)
\right)\\ &=\left( -\partial_1+x^*_i\sigma\otimes\lambda_z+(x_1^*\otimes m_B)(\delta_{1,l}\otimes\id_B),\ x_i^*\otimes m_B\right). \end{align*} For $1\leq s\leq d-2$, \begin{align*} \psi_{s}\partial_{s+2}^B&= \left( \begin{array}{cc} 0 & 0\\ -\partial_{s+1}&0 \end{array} \right) + (-1)^s \left( \begin{array}{cc} -(x_i^*\sigma\otimes\id_V^{\otimes s-2})\otimes m_B & 0\\
\zeta_s& (x_i^*\otimes\id_V^{\otimes s-1})\otimes m_B \end{array} \right),\\
-\partial_{s+1}^B\psi_{s+1}&=\left( \begin{array}{cc} 0 & 0\\ -\partial_{s+1}&0 \end{array} \right)+ (-1)^{s}\left( \begin{array}{cc} -(x_i^*\sigma\otimes\id_V^{\otimes s-2})\otimes m_B & 0\\
\zeta'_s& (x_i^*\otimes\id_V^{\otimes s-1})\otimes m_B \end{array} \right), \end{align*} where \begin{align*} \zeta_s&=(-1)^s\left(\left((x_i^*\otimes\id_V^{\otimes s})\delta_{s,l} \right)\otimes\id_B\right)\partial_{s+1} +\left((x_i^*\otimes\id_V^{\otimes s})\otimes\id_B \right)\phi_{s+1}\\ &=(-1)^s\left((x_i^*\otimes\id_V^{\otimes s})\delta_{s,l} \right)\otimes m_B+\left(x_i^*\sigma \otimes\sigma^{\otimes s}\right)\otimes\lambda_z+\left(x_i^*\otimes\id_V^{\otimes s}\otimes m_B\right)(\delta_{s+1,r}\otimes\id_B)\\ &=\left(x_i^*\sigma \otimes\sigma^{\otimes s}\right)\otimes\lambda_z+(-1)^s\left(x_i^*\otimes\id_V^{\otimes s}\otimes m_B\right)\left((\delta_{s,l}\otimes\id_V)\otimes\id_B \right)+\left(x_i^*\otimes\id_V^{\otimes s}\otimes m_B\right)\left(\delta_{s+1,r}\otimes\id_B\right),\\ \zeta'_s&=\phi_{s} \left((x_i^*\sigma\otimes\id_V^{\otimes s})\otimes\id_B\right) +(-1)^s\partial_{s+1} \left(\left((x_i^*\otimes\id_V^{\otimes s+1})\delta_{s+1,l} \right)\otimes\id_B\right)\\ &=\left(x_i^*\sigma\otimes\sigma^{\otimes s}\right)\otimes\lambda_z+\left(\id_V^{\otimes s}\otimes m_B\right)\left((x_i^*\sigma\otimes\delta_{s,r})\otimes\id_B\right)+(-1)^s\left(x_i^*\otimes\id_V^{\otimes s}\otimes m_B\right)\left(\delta_{s+1,l} \otimes\id_B\right)\\ &=\left(x_i^*\sigma\otimes\sigma^{\otimes s}\right)\otimes\lambda_z+\left(x_i^*\otimes\id_V^{\otimes s}\otimes m_B\right)\left((\sigma\otimes\delta_{s,r})\otimes\id_B\right)+(-1)^s\left(x_i^*\otimes\id_V^{\otimes s}\otimes m_B\right)\left(\delta_{s+1,l} \otimes\id_B\right). \end{align*} By Lemma \ref{lemma: construction of delta_l}, $\zeta_s=\zeta_s'$ and $\psi_s\partial^B_{s+2}=-\partial_{s+1}^B\psi_{s+1}$. Similarly, one obtains $\psi_{d-1}\partial^B_{d+1}=-\partial_{d}^B\psi_{d}$.
Then, for any $i,j=1,\cdots,n$, we have in $E(B)$,
\begin{align*} &\eta^*_j\ast(\xi,x_i^*)=(\eta_j^*\otimes\varepsilon_B)\ast(\varepsilon_B,x_i^*\otimes \varepsilon_B)=(\eta_j^*\otimes\varepsilon_B,0)\psi_d=(-1)^d(x_i^*\sigma\otimes\eta_j^*)(\omega)\omega^*\otimes \varepsilon_B,\\ &\omega^*\ast(\xi,x_i^*)=(\omega^*\otimes\varepsilon_B)\ast(\varepsilon_B,x_i^*\otimes \varepsilon_B)=(0,\omega^*\otimes\varepsilon_B)\psi_d=\omega^*\otimes \varepsilon_B-x_i^*(\delta_l)\omega^*\otimes \varepsilon_B . \end{align*} The result follows. \end{proof}
\subsection{Nakayama automorphisms} This subsection devotes to proving the main result of this paper. For the completeness, we give a whole computation of the Nakayama automorphism of a graded Ore extension of a Koszul AS-regular algebra, which includes \cite[Proposition 3.15]{ZVZ} partially. For the automorphism $\sigma\in GL(V)$ and the Nakayama automorphism $\mu_A$ of $A$, there exist two invertible $n\times n$ matrixes $M=(m_{ij}),P=(p_{ij})$ over $k$, such that $$\sigma \left( \begin{array}{c} x_1\\ x_2\\ \vdots\\ x_n \end{array} \right)=M\left( \begin{array}{c} x_1\\ x_2\\ \vdots\\ x_n \end{array} \right),\qquad \mu_A \left( \begin{array}{c} x_1\\ x_2\\ \vdots\\ x_n \end{array} \right)=P\left( \begin{array}{c} x_1\\ x_2\\ \vdots\\ x_n \end{array} \right). $$
\begin{lemma}\label{lemma: Yoneda product in E(A)} For any $j=1,\cdots,n$,
$$ \left( \begin{array}{c} \eta_j^*\otimes x^*_1(\omega)\\ \eta_j^*\otimes x^*_2(\omega)\\ \vdots\\ \eta_j^*\otimes x^*_n(\omega) \end{array} \right) =(-1)^{d-1}P^T\left( \begin{array}{c}
x^*_1\otimes\eta^*_j(\omega)\\ x^*_2\otimes\eta^*_j(\omega)\\ \vdots\\ x^*_n\otimes\eta^*_j(\omega) \end{array} \right). $$ \end{lemma} \begin{proof}Since $A$ is a Koszul AS-regular algebra of dimension $d$, the minimal free resolution (\ref{resolution of k_A}) of trivial module $k_A$ becomes $$ 0\xlongrightarrow{}W_d\otimes A\xlongrightarrow{\partial^A_d}W_{d-1}\otimes A\xlongrightarrow{\partial^A_{d-1}}\cdots\xlongrightarrow{\partial^A_{3}}W_2\otimes A\xlongrightarrow{\partial^A_{2}}W_1\otimes A\xlongrightarrow{\partial^A_{1}}A\xlongrightarrow{\varepsilon_A}k_A\to0. $$ So $E^1(A)=W_1^*$, $E^{d-1}(A)=W_{d-1}^*$, $E^{d}(A)=W_{d}^*$. By a similar argument in the proof of Lemma \ref{lemma: E^1(B) E^d(B)} and Lemma \ref{lemma: E^d(B) E^1(B)}, one obtains that the Yoneda products in $E(A)$ of $\{x_i^*\}_{i=1}^{n}$ and $\{\eta_{j}^*\}_{j=1}^n$, which are basis of $E^1(A)$ and $E^{d-1}(A)$ respectively, are as follows: $$ x_i^*\ast \eta_j^*= (-1)^{d-1}(\eta_j^*\otimes x_i^*)(\omega)\omega^*,\qquad \eta_j^*\ast x_i^*=(-1)^{d-1}(x_i^*\otimes\eta^*_j)(\omega)\omega^*. $$ By Theorem \ref{thm: properties of Koszul regular algebras}(a,b), $\mu_{E(A)}(x_i^*)=\sum_{s=1}^{n}p_{si}x_s^*$, and $E(A)$ is graded Frobenius with the bilinear form as follows \begin{align*} &\langle x_i^*, \eta_j^*\rangle=(x_i^*\ast \eta_j^*)(\omega)=(-1)^{d-1}(\eta_j^*\otimes x^*_i)(\omega),\\ &\langle \eta_j^*, \mu_{E(A)}(x_i^*)\rangle=\sum_{s=1}^{n}p_{si}\langle \eta_j^*, x_s^*\rangle=\sum_{s=1}^{n}p_{si}(\eta_j^*\ast x_s^*)(\omega)=(-1)^{d-1}\sum_{s=1}^{n}p_{si}(x_s^*\otimes\eta^*_j)(\omega), \end{align*}
for any $i,j=1,\cdots,n$. The result follows. \end{proof}
Now we prove the main result of this paper.
\begin{theorem}\label{thm: nakayama automorphism of ore extension} Let $B=A[z;\overline{\sigma},\overline{\delta}]$ be a graded Ore extension of a Koszul AS-regular algebra $A$, where $\overline{\sigma}$ is a graded automorphism of $A$ and $\overline{\delta}$ is a degree-one $\overline{\sigma}$-derivation. Then the Nakayama automorphism $\mu_B$ of $B$ satisfies \begin{align*} {\mu_{B}}_{\mid A}=\overline{\sigma}^{~-1}\mu_A,\qquad
\mu_B(z)=\hdet(\overline{\sigma})\, z+\nabla_{\overline{\sigma}}\cdot\overline{\delta}, \end{align*} where $\mu_A$ is the Nakayama automorphism of $A$ and $\nabla_{\overline{\sigma}}\cdot\overline{\delta}$ is the $\overline{\sigma}$-divergence of $\overline{\delta}$.
\end{theorem} \begin{proof} Let $(\delta_r,\delta_l)$ be the pair of elements in $V$ with respect to some sequence pair $(\{\delta_{i,r}\},\{\delta_{i,l}\})$ for $\overline{\delta}$.
By Lemma \ref{lemma: E^1(B) E^d(B)} and Lemma \ref{lemma: E^d(B) E^1(B)}, \begin{align*} &\langle \xi, \omega^*\rangle=(-1)^d\hdet(\overline{\sigma})=(-1)^d\langle \omega^*, \hdet(\overline{\sigma})\, \xi\rangle,\\ &\langle \xi, \eta_j^*\rangle=0=(-1)^d\langle \eta_j^*, \hdet(\overline{\sigma})\, \xi\rangle, \end{align*} for any $j=1,\cdots,n$. Hence, $$\mu_{E(B)}(\xi)=\hdet(\overline{\sigma})\,\xi.$$
Write $\omega=x_1\otimes \upsilon_1+x_2\otimes\upsilon_2+\cdots+x_n\otimes \upsilon_n$, where $\upsilon_1,\upsilon_2,\cdots,\upsilon_n\in W_{d-1}$. Then $(x_i^*\otimes \eta_j^*)(\omega)=\eta_j^*(\upsilon_i)$, and $$ (x_i^*\sigma\otimes \eta_j^*)(\omega)=(x_i^*\otimes\eta^*_{j})\left(\sum_{s,t=1}^n m_{st}(x_t\otimes \upsilon_s)\right)=\sum_{s=1}^n m_{si}\eta_{j}^*(\upsilon_s). $$ By Lemma \ref{lemma: Yoneda product in E(A)}, one obtains $$ \left( \begin{array}{c} (x_1^*\sigma\otimes \eta_j^*)(\omega)\\ (x_2^*\sigma\otimes \eta_j^*)(\omega)\\ \vdots\\ (x_n^*\sigma\otimes \eta_j^*)(\omega) \end{array} \right) =M^T\left( \begin{array}{c} \eta_j^*(\upsilon_1)\\ \eta_j^*(\upsilon_2)\\ \vdots\\ \eta_j^*(\upsilon_n) \end{array} \right)=M^T\left( \begin{array}{c} (x_1^*\otimes \eta_j^*)(\omega)\\ (x_2^*\otimes \eta_j^*)(\omega)\\ \vdots\\ (x_n^*\otimes \eta_j^*)(\omega) \end{array} \right)=(-1)^{d-1}(P^{-1}M)^T \left( \begin{array}{c} (\eta_j^*\otimes x_1^*)(\omega)\\ (\eta_j^*\otimes x_2^*)(\omega)\\ \vdots\\ (\eta_j^*\otimes x_n^*)(\omega) \end{array} \right). $$ By Lemma \ref{lemma: E^1(B) E^d(B)} and Lemma \ref{lemma: E^d(B) E^1(B)}, $$ \left( \begin{array}{c} \langle x_1^*,\eta_j^*\rangle\\ \langle x_2^*,\eta_j^*\rangle\\ \vdots\\ \langle x_n^*,\eta_j^*\rangle \end{array} \right)=(-1)^{d+1}\left( \begin{array}{c} (\eta_j^*\otimes x_1^*)(\omega)\\ (\eta_j^*\otimes x_2^*)(\omega)\\ \vdots\\ (\eta_j^*\otimes x_n^*)(\omega) \end{array} \right)=(M^{-1}P)^T\left( \begin{array}{c} (x_1^*\sigma\otimes \eta_j^*)(\omega)\\ (x_2^*\sigma\otimes \eta_j^*)(\omega)\\ \vdots\\ (x_n^*\sigma\otimes \eta_j^*)(\omega) \end{array} \right)=(-1)^d(M^{-1}P)^T \left( \begin{array}{c} \langle \eta_j^*,x_1^*\rangle\\ \langle \eta_j^*,x_2^*\rangle\\ \vdots\\ \langle \eta_j^*,x_n^*\rangle \end{array} \right). $$
Write $c_i=x_i^*(\nabla_{\overline{\sigma}}\cdot\overline{\delta})$ for any $i=1,\cdots,n$, and it is easy to see that $$ \left( \begin{array}{c} c_1\\ c_2\\ \vdots\\ c_n \end{array} \right) = \left( \begin{array}{c}
x_1^*(\delta_r)\\
x_2^*(\delta_r)\\ \vdots\\
x_n^*(\delta_r) \end{array} \right)+(M^{-1}P)^T \left( \begin{array}{c}
x_1^*(\delta_l)\\
x_2^*(\delta_l)\\ \vdots\\
x_n^*(\delta_l) \end{array} \right). $$ Then \begin{align*} \left( \begin{array}{c} \langle x_1^*,\eta_j^*\rangle\\ \langle x_2^*,\eta_j^*\rangle\\ \vdots\\ \langle x_n^*,\eta_j^*\rangle \end{array} \right)&=(-1)^d\left((M^{-1}P)^T \left( \begin{array}{c} \langle \eta_j^*,x_1^*\rangle\\ \langle \eta_j^*,x_2^*\rangle\\ \vdots\\ \langle \eta_j^*,x_n^*\rangle \end{array} \right)+ \left( \begin{array}{c} \langle \eta_j^*,c_1\xi\rangle\\ \langle \eta_j^*,c_2\xi\rangle\\ \vdots\\ \langle \eta_j^*,c_n\xi\rangle \end{array} \right) \right), \\ \left( \begin{array}{c} \langle x_1^*,\omega^*\rangle\\ \langle x_2^*,\omega^*\rangle\\ \vdots\\ \langle x_n^*,\omega^*\rangle \end{array} \right) &=(-1)^d\left( \begin{array}{c}
x_1^*(\delta_r)\\
x_2^*(\delta_r)\\ \vdots\\
x_n^*(\delta_r) \end{array} \right)= (-1)^d\left((M^{-1}P)^T \left( \begin{array}{c} \langle \omega^*,x_1^*\rangle\\ \langle \omega^*,x_2^*\rangle\\ \vdots\\ \langle \omega^*,x_n^*\rangle \end{array} \right)+ \left( \begin{array}{c} \langle \omega^*,c_1\xi\rangle\\ \langle \omega^*,c_2\xi\rangle\\ \vdots\\ \langle \omega^*,c_n\xi\rangle \end{array} \right) \right). \end{align*} Hence, $$ \mu_{E(B)} \left( \begin{array}{c} x_1^*\\ x_2^*\\ \vdots\\ x_n \end{array} \right) =(M^{-1}P)^T\left( \begin{array}{c} x_1^*\\ x_2^*\\ \vdots\\ x_n \end{array} \right)+\left( \begin{array}{c} c_1\xi\\ c_2\xi\\ \vdots\\ c_n\xi \end{array} \right). $$ By Theorem \ref{thm: properties of Koszul regular algebras}(b), we have \begin{align*} &{\mu_{B}}_{\mid A}=\overline{\sigma}^{-1}\mu_A,\\ &\mu_B(z)=\hdet(\overline{\sigma})\, z+c_1x_1+c_2+\cdots+c_nx_n=\hdet(\overline{\sigma})\, z+\nabla_{\overline{\sigma}}\cdot\overline{\delta}\qedhere \end{align*} \end{proof}
\begin{corollary} Let $A$ be a Koszul AS-regular algebra with the Nakayama automorphism $\mu_A$. Then a graded Ore extension $B=A[z;\overline{\sigma},\overline{\delta}]$ is Calabi-Yau if and only if $\overline{\sigma}=\mu_A$ and $\nabla_{\overline{\sigma}}\cdot\overline{\delta}=0$.
\end{corollary}
\subsection{Twisted superpotentials}As shown in Theorem \ref{thm: properties of Koszul regular algebras}(c,d), a Koszul AS-regular algebra is always associated with a twisted superpotential such that it is a derivation quotient algebra defined by such a twisted superpotential. Since graded Ore extensions of Koszul AS-regular algebras are also Koszul AS-regular algebras, it is worth to understand twisted superpotentials for graded Ore extensions. In \cite{HVZ,HVZ1}, the authors studied this problem in two special cases. In the following, we give a general solution to this problem.
\begin{theorem}\label{thm: twisted superpotential for B} Let $A=T(V)/(R)$ be a Koszul AS-regular algebra of dimension $d$ and $\omega$ a basis of $W_d$. Suppose $B=A[z;\overline{\sigma},\overline{\delta}]$ is a graded Ore extension of $A$, where $\overline{\sigma}$ is a graded automorphism of $A$ and $\overline{\delta}$ is a degree-one $\overline{\sigma}$-derivation of $A$. Let $(\{\delta_{i,r}\},\{\delta_{i,l}\})$ be a sequence pair for $\overline{\delta}$, then \begin{align} \hat{\omega}&=\sum_{i=0}^d (-1)^i\tau_{d+1}^i(\id\otimes \sigma^{\otimes i}\otimes\id_V^{\otimes d-i})(z\otimes\omega)+\sum_{i=1}^d (-1)^i(\delta_{i,r}\otimes\id_V^{\otimes d-i})(\omega)\label{twsited superpotential1}\\ &=\sum_{i=0}^d (-1)^i\tau_{d+1}^i(\id\otimes \sigma^{\otimes i}\otimes\id_V^{\otimes d-i})(z\otimes\omega)+(-1)^{d+1}\sum_{i=1}^d (-1)^{i}(\sigma^{\otimes d-i}\otimes \delta_{i,l})(\omega).\label{twsited superpotential2} \end{align} is a ${\mu_B}_{\mid V}$-twisted superpotential, where $\mu_B$ is the Nakayama automorphism of $B$ and $\sigma=\overline{\sigma}_{\mid V}$. Moreover, $$B\cong \mathcal{A}(\hat{\omega},d-1).$$ \end{theorem} \begin{proof} By Proposition \ref{prop: relation between delta_r and delta_l}(a), one obtains that (\ref{twsited superpotential1}) and (\ref{twsited superpotential2}) are equal. By Theorem \ref{thm: properties of Koszul regular algebras}(c), $$\tau_{d}^{d-1}(\mu_A\otimes\id_V^{\otimes d})(\omega)=(-1)^{d-1}\omega.$$ It remains to show $\tau_{d+1}^{d}(\mu_B\otimes\id_V^{\otimes d})(\hat{\omega})=(-1)^d\hat{\omega}$. By Theorem \ref{thm: nakayama automorphism of ore extension}, we have \begin{align*} \tau_{d+1}^{d}(\mu_B\otimes\id_V^{\otimes d})(\hat{\omega})&=\tau_{d+1}^{d}(\mu_B\otimes\id_V^{\otimes d})\left(\sum_{i=0}^d (-1)^i\tau_{d+1}^i(\id\otimes \sigma^{\otimes i}\otimes\id_V^{\otimes d-i})(z\otimes\omega)-\sum_{i=1}^d (-1)^{d-i}(\sigma^{\otimes d-i}\otimes \delta_{i,l})(\omega)\right)\\ &=\omega\otimes\mu_B(z)+\sum_{i=1}^d (-1)^{i}\tau_{d+1}^{d}\tau_{d+1}^i(\id\otimes\mu_A\otimes \sigma^{\otimes i-1}\otimes\id_V^{\otimes d-i})(z\otimes\omega)\\ &\quad -\sum_{i=1}^{d-1} (-1)^{d-i}\tau_{d+1}^{d}(\mu_A\otimes\sigma^{\otimes d-i-1}\otimes \delta_{i,l})(\omega)-\tau_{d+1}^{d}(\mu_B\otimes\id_V^{\otimes d})\delta_{d,l}(\omega)\\ &=\omega\otimes\hdet(\overline{\sigma})\, z+\omega\otimes\nabla_{\overline{\sigma}}\cdot\overline{\delta}\\ &\quad+\sum_{i=1}^d (-1)^{i}\tau_{d+1}^{i-1}(\id\otimes\sigma^{\otimes i-1}\otimes\id_V^{\otimes d-i+1})\left(z\otimes\left(\tau_d^{d-1}(\mu_A\otimes\id_V^{\otimes d-1})(\omega)\right)\right)\\ &\quad -\sum_{i=1}^{d-1} (-1)^{d-i}(\sigma^{\otimes d-i-1}\otimes \delta_{i,l}\otimes\id_V)\tau_{d}^{d-1}(\mu_A\otimes\id_V^{\otimes d-1})(\omega)-\omega\otimes\mu_A\sigma^{-1}(\delta_l)\\ &=\omega\otimes\hdet(\overline{\sigma})\, z+\omega\otimes\delta_r+\sum_{i=1}^d (-1)^{d+i-1}\tau_{d+1}^{i-1}(\id\otimes\sigma^{\otimes i-1}\otimes\id_V^{\otimes d-i+1})(z\otimes\omega)\\ &\quad-\sum_{i=1}^{d-1} (-1)^{i+1}(\sigma^{\otimes d-i-1}\otimes \delta_{i,l}\otimes\id_V)(\omega)\\ &=\tau_{d+1}^{d}(\id\otimes \sigma^{\otimes d})(z\otimes \omega)+\delta_{d,r}(\omega)+\sum_{i=0}^{d-1} (-1)^{d+i}\tau_{d+1}^{i}(\id\otimes\sigma^{\otimes i}\otimes\id_V^{\otimes d-i})(z\otimes\omega)\\ &\quad+(-1)^{d}\sum_{i=1}^{d-1}(-1)^i(\delta_{i,r}\otimes\id_V^{\otimes d-i})(\omega)\\ &=(-1)^d\left(\sum_{i=0}^{d} (-1)^{i}\tau_{d+1}^{i}(\id\otimes\sigma^{\otimes i}\otimes\id_V^{\otimes d-i})(z\otimes\omega)+\sum_{i=1}^{d}(-1)^i(\delta_{i,r}\otimes\id_V^{\otimes d-i}) \right)(\omega)\\ &=(-1)^d\hat{\omega}, \end{align*} where the third equation holds by \cite[Theorem 1.2]{MS} and Proposition \ref{prop: relation between delta_r and delta_l}(b).
Let $\delta:V\to V\otimes V$ be a linear map such that the map $\overline{\delta}$ and the sequence pair $(\{\delta_{i,r}\},\{\delta_{i,l}\})$ are induced by $\delta$, and $\{x_1,\cdots,x_n\}$ a basis of $V$. Write $$ \hat{V}=V\oplus k\{z\},\qquad \hat{R}=R\oplus k\{z\otimes x_i-\sigma(x_i)\otimes z-\delta(x_i)\mid i=1,\cdots,n\}. $$ Then $B\cong T(\hat{V})/(\hat{R})$ is a Koszul regular algebra of dimension $d+1$. Write $$ \hat{W}_1=\hat{V},\qquad \hat{W}_i=\bigcap_{0\leq s\leq i-2} \hat{V}^{\otimes s}\otimes \hat{R}\otimes \hat{V}^{\otimes i-s-2},\quad \forall i\geq 2. $$ Clearly, $W_i\subseteq \hat{W_i}$ for any $i\geq 1$ and $\dim \hat{W}_{d+1}=1$.
It is easy to know that $\hat{\omega}\neq 0$. Since $B$ is Koszul AS-regular, it suffices to prove $\hat{\omega}\in \hat{W}_{d+1}$ by Theorem \ref{thm: properties of Koszul regular algebras}(d). Write $\omega=\sum v_1\otimes v_2\otimes \cdots\otimes v_d$. By (\ref{twsited superpotential1}), one obtains that \begin{align*} \hat{\omega}=&\sum \left(z\otimes v_1-\sigma(v_1)\otimes z-\delta(v_1)\right)\otimes v_2\otimes \cdots\otimes v_d\\ &+ \sum_{i=2}^d (-1)^i(\sigma(v_1)\otimes \sigma(v_2)\otimes \cdots\otimes \sigma(v_i)\otimes z\otimes v_{i+1}\otimes\cdots\otimes v_d) +\sum_{i=2}^d (-1)^i(\delta_{i,r}\otimes\id_V^{\otimes d-i})(\omega)\in \hat{R}\otimes \hat{V}^{\otimes d-1}. \end{align*} Since $\underbrace{\left((\tau_{d+1}^d)^{-1}\circ\cdots(\tau_{d+1}^d)^{-1}\right)}_i(\hat{R}\otimes \hat{V}^{\otimes d-1})\subseteq \hat{V}^{\otimes i}\otimes \hat{R}\otimes \hat{V}^{\otimes d-1-i}$ and $\hat{\omega}$ is a ${\mu_B}_{\mid V}$-twisted superpotential, we have $$ \hat{\omega}=(-1)^{di}\underbrace{\left((\mu_B^{-1}\otimes \id_V^{\otimes d})(\tau_{d}^{d+1})^{-1}\cdots (\mu_B^{-1}\otimes \id_V^{\otimes d})(\tau_{d}^{d+1})^{-1}\right)}_i(\hat{\omega})\in \hat{V}^{\otimes i}\otimes \hat{R}\otimes \hat{V}^{\otimes d-1-i}, $$ for any $1\leq i\leq d-1$. It implies that $\hat{\omega}\in\hat{W}_{d+1}$.
\end{proof}
\begin{remark} \begin{enumerate} \item By Proposition \ref{prop: relation betwenn different delta_r}, the twisted superpotential $\hat{\omega}$ constructed in the last theorem is independent on the choices of sequence pairs for $\overline{\delta}$. \item The results in \cite[Theorem 4.4]{HVZ} and \cite[Theorem 0.1(ii)]{HVZ1} are both the special case of $\delta_{i,r}=\delta_{i,l}=0$ for $i\geq 2$. \end{enumerate} \end{remark}
\section{Applications} In this section, we apply our main result to two examples.
\subsection{Graded polynomial algebras} In this part, we assume $A=k[x_1,x_2,\cdots,x_n]$ is a graded polynomial algebra generated in degree $1$. The Nakayama automorphism of a graded Ore extension of $A$ is just a graded version of \cite[Theorem 1.1]{LM}. We use our method to prove the differential case as an example.
\begin{theorem}\label{thm: NA of Ore ext over Polynomial}\cite[Theorem 1.1(1)]{LM} Let $A=k[x_1,x_2,\cdots,x_n]$ be a graded polynomial algebra generated in degree $1$. Then the Nakayama automorphism $\mu_B$ of the graded Ore extension $B=A[z;\overline{\delta}]$ is \begin{align*}
{\mu_B}_{|A}=\id_A\qquad \mu_B(z)=z+\nabla\cdot {\overline{\delta}}; \end{align*} where $\nabla\cdot {\bar{\delta}}$ is the divergence of $\delta$, that is $\nabla\cdot {\bar{\delta}}=\sum_{i=1}^n \partial\, \overline{\delta}(x_i)/\partial x_i$. \end{theorem}
To prove this theorem, we need some preparation. Let $V$ be the vector space spanned by $\{x_1,x_2,\cdots,x_n\}$. Firstly, we determine the vector spaces $\{W_i\mid i\geq2\}$ for $A$. Write $ r_{i_1i_2}=x_{i_1}\otimes x_{i_2}-x_{i_2}\otimes x_{i_1}, $ for any (not necessarily distinguished) $i_1,i_2\in\{1,2,\cdots,n\}$. For any integer $m\geq3$, we write inductively,
$$ r_{i_1i_2\cdots i_m}=\sum_{j=1}^m (-1)^{m-j}r_{i_1\cdots \hat{i}_j \cdots i_{m}}\otimes x_{i_j}\in V^{\otimes m}, $$ for any (not necessarily distinguished) $i_1,i_2,\cdots,i_m\in \{1,2,\cdots,n\}$. The following result is clear. \begin{lemma}\label{lemma: properties of r_i1i2...} Let an integer $m\geq 2$, (not necessarily distinguished) $i_1,i_2,\cdots,i_m\in \{1,2,\cdots,n\}$. Then \begin{enumerate} \item $r_{i_1i_2\cdots i_m}=\sum_{j=1}^m (-1)^{j+1}x_{i_j}\otimes r_{i_1\cdots \hat{i}_j \cdots i_{m}}$. \item $r_{i_1\cdots i_m}\in W_m$.
\item $r_{i_1\cdots i_m}=0$, if $i_s=i_t$ for some $s\neq t$.
\item $r_{i_1\cdots i_m}=(-1)^{\mathrm{sgn}\sigma}r_{i_{\sigma(1)}\cdots i_{\sigma(m)}}$ for any $\sigma\in S_m$.
\item the set $\{r_{i_1\cdots i_m}\;|\; i_1<i_2<\cdots<i_m\}$ is a basis of $W_m$. \end{enumerate} \end{lemma}
Let $\delta:V\to V\otimes V$ be a linear map such that the map $\overline{\delta}$ in Theorem \ref{thm: NA of Ore ext over Polynomial} is induced by it. Write $$ \delta(x_i)=\sum_{s,t=1}^n k_{st}^{(i)} x_s\otimes x_t, $$ where $k_{st}^{(i)}\in k$ for $s,t,i=1,\cdots,n$. Then we construct a sequence pair for $\overline{\delta}$.
\begin{lemma}\label{lemma: delta_l delta_r for polynomial} There exists a sequence pair $(\{\delta_{i,r}\},\{\delta_{i,l}\})$ for $\overline{\delta}$ such that \begin{align*} \delta_{m,r}(r_{i_1i_2\cdots i_m})=\sum_{j=1}^m\sum_{s,t=1}^n k^{(i_{j})}_{st} r_{i_1\cdots i_{j-1}si_{j+1}\cdots i_m} \otimes x_t,\quad \delta_{m,l}(r_{i_1i_2\cdots i_m})=\sum_{j=1}^m\sum_{s,t=1}^n k^{(i_{j})}_{st} x_s\otimes r_{i_1\cdots i_{j-1}ti_{j+1}\cdots i_m}, \end{align*} for any $1\leq i_1<i_2<\cdots<i_m\leq n$ and $m\geq 2$. \end{lemma} \begin{proof} By the extension of $\delta$ to $T(V)$, one obtains that for any $1\leq i_1<i_2\leq n$, \begin{align*} \delta(r_{i_1i_2})&=\delta(x_{i_1}\otimes x_{i_2}-x_{i_2}\otimes x_{i_1})=x_{i_1}\otimes \delta(x_{i_2})+\delta(x_{i_1})\otimes x_{i_2}-x_{i_2}\otimes \delta(x_{i_1})-\delta(x_{i_2})\otimes x_{i_1}\\ &= \sum_{s,t=1}^n k_{st}^{(i_2)}( x_{i_1}\otimes x_s\otimes x_t- x_s\otimes x_t\otimes x_{i_1})+ \sum_{s,t=1}^n k_{st}^{(i_1)}(x_s\otimes x_t\otimes x_{i_2}-x_{i_2}\otimes x_s\otimes x_t)\\ &=\sum_{s,t=1}^n \left( k_{st}^{(i_2)} \left(x_{i_1}\otimes x_s\otimes x_t-x_s\otimes x_{i_1}\otimes x_t\right)+k_{st}^{(i_2)} \left(x_s\otimes x_{i_1}\otimes x_t- x_s\otimes x_t\otimes x_{i_1}\right)\right)\\ &\quad+\sum_{s,t=1}^n \left(k_{st}^{(i_1)}\left(x_s\otimes x_t\otimes x_{i_2}-x_s\otimes x_{i_2}\otimes x_t\right)+k_{st}^{(i_1)}\left(x_s\otimes x_{i_2}\otimes x_t-x_{i_2}\otimes x_s\otimes x_t\right)\right)\\
&=\sum_{s,t=1}^n x_s\otimes \left( k_{st}^{(i_1)}r_{ti_2}+k_{st}^{(i_2)} r_{i_1t}\right)+\sum_{s,t=1}^n \left(k_{st}^{(i_1)}r_{si_2}+k_{st}^{(i_2)}r_{i_1s} \right)\otimes x_t. \end{align*} So we can choose $$ \delta_{2,r}(r_{i_1i_2})=\sum_{s,t=1}^n \left(k_{st}^{(i_1)}r_{si_2}+k_{st}^{(i_2)}r_{i_1s} \right)\otimes x_t,\qquad \delta_{2,l}(r_{i_1i_2})=\sum_{s,t=1}^n x_s\otimes \left( k_{st}^{(i_1)}r_{ti_2}++k_{st}^{(i_2)} r_{i_1t}\right). $$
Suppose we have obtained that for any $u<m$, $$ \delta_{u,r}(r_{i_1i_2\cdots i_u})=\sum_{j=1}^u\sum_{s,t=1}^n k^{(i_{j})}_{st} r_{i_1\cdots i_{j-1}si_{j+1}\cdots i_u} \otimes x_t,\qquad \delta_{u,l}(r_{i_1i_2\cdots i_u})=\sum_{j=1}^u\sum_{s,t=1}^n k^{(i_{j})}_{st} x_s\otimes r_{i_1\cdots i_{j-1}ti_{j+1}\cdots i_u}. $$ For any $1\leq i_1<i_2<\cdots<i_m\leq n$, one obtains \begin{align*} & (\id_V^{\otimes m-1}\otimes m_B)\left(\id_V^{\otimes m-1}\otimes\delta+\delta_{m-1,r}\otimes \id_V\right)(r_{i_1\cdots i_m}) \\
=&\sum_{p=1}^m\sum_{s,t=1}^{n}(-1)^{m-p}(\id_V^{\otimes m-1}\otimes m_B) \left( k^{(i_p)}_{st} r_{i_1\cdots \hat{i}_p \cdots i_{m}}\otimes x_s\otimes x_t\right)\\ &+\sum_{p=1}^m\sum_{s,t=1}^{n}(-1)^{m-p}(\id_V^{\otimes m-1}\otimes m_B)\left(\sum_{j=1}^{p-1} k_{st}^{(i_j)}r_{i_1\cdots i_{j-1}s\cdots \hat{i}_p \cdots i_{m}}\otimes x_t\otimes x_{i_p}+\sum_{j=p+1}^{m} k_{st}^{(i_j)}r_{i_1\cdots \hat{i}_p\cdots si_{j+1}\cdots i_{m}}\otimes x_t\otimes x_{i_p}\right)\\
=&\sum_{s,t=1}^{n}\sum_{j=1}^m(\id_V^{\otimes m-1}\otimes m_B) \left((-1)^{m-j} k^{(i_j)}_{st} r_{i_1\cdots \hat{i}_j \cdots i_{m}}\otimes x_s\otimes x_t \right)\\ &+\sum_{s,t=1}^{n} \sum_{j=1}^{m} (\id_V^{\otimes m-1}\otimes m_B)\left( \sum_{p=1}^{j-1}(-1)^{m-p}k_{st}^{(i_j)}r_{i_1\cdots \hat{i}_p\cdots si_{j+1}\cdots i_m}\otimes x_{i_p}\otimes x_t+\sum_{p=j+1}^m (-1)^{m-p}k_{st}^{(i_j)}r_{i_1\cdots i_{j-1}s\cdots \hat{i}_p\cdots i_m}\otimes x_{i_p}\otimes x_t \right)\\ =&\sum_{s,t=1}^{n}\sum_{j=1}^m(\id_V^{\otimes m-1}\otimes m_B) k^{(i_j)}_{st}r_{i_1\cdots i_{j-1}si_{j+1}\cdots i_m}\otimes x_t \end{align*} By Remark \ref{remark: choice of delta_r}, we can define $$ \delta_{u,r}(r_{i_1i_2\cdots i_m})=\sum_{j=1}^m\sum_{s,t=1}^n k^{(i_{j})}_{st} r_{i_1\cdots i_{j-1}si_{j+1}\cdots i_m}\otimes x_t. $$
Similarly, one obtains the result for $\delta_{i,l}$ by Lemma \ref{lemma: construction of delta_l} and Lemma \ref{lemma: properties of r_i1i2...}(a). \end{proof}
\begin{proof}[Proof of Theorem \ref{thm: NA of Ore ext over Polynomial}]
By Lemma \ref{lemma: properties of r_i1i2...}, $r_{1\cdots n}$ is the basis of $W_n$. By Lemma \ref{lemma: properties of r_i1i2...} and Lemma \ref{lemma: delta_l delta_r for polynomial}, one obtains $$ \delta_{n,r}(r_{1\cdots n})=r_{1\cdots n}\otimes \left(\sum^n_{s,t=1}k^{(s)}_{st} x_t\right),\qquad \delta_{n,l}(r_{1\cdots n})=\left(\sum^n_{s,t=1}k^{(t)}_{st} x_s\right)\otimes r_{1\cdots n}. $$
So $\delta_r=\sum^n_{s,t=1}k^{(s)}_{st} x_t$ and $\delta_l=\sum^n_{s,t=1}k^{(t)}_{st} x_s$, and $$ \delta_r+\delta_l=\sum^n_{s=1}\left( \sum^n_{t=1} (k^{(s)}_{st}+k^{(s)}_{ts}) x_t\right) =\sum^n_{s=1} \dfrac{\partial(\delta(x_s))}{\partial x_s}=\nabla\cdot \overline{\delta}. $$ Since $A$ is Koszul CY, the result follows by Theorem \ref{thm: nakayama automorphism of ore extension}. \end{proof}
\subsection{Koszul AS-regular algebras of dimension 2}
In this subsection, we assume $\mathrm{char}\, k=0$. We give a formula of Nakayama automorphisms for graded Ore extensions of Koszul AS-regular algebras of dimension 2, and then compute specific Nakayama automorphisms for noetherian ones.
Now we assume $A$ is an AS-regular algebra of dimension 2. By \cite[Theorem 0.1]{Z}, one obtains that $A$ is always Koszul and there is an invertible matrix $Q\in M_n(k)$ such that $ A\cong k\langle x_1,x_2,\cdots,x_n\rangle/(r), $ where $$ r=\mathbf{x}^TQ\mathbf{x}, $$ and $\mathbf{x}=(x_1,x_2,\cdots,x_n)^T$. It is well known (for example, \cite[Scetion 3]{HVZ}) that the Nakayama automorphism of $\mu_A$ satisfies $$ \mu_A(\mathbf{x})
=-(Q^{-1})^TQ\mathbf{x}.
$$
Let $\overline{\sigma}$ be a graded automorphism of $A$ and $\overline{\delta}$ a degree-one $\overline{\sigma}$-derivation of $A$. Write $M\in M_n(k)$ for the invertible matrix such that $\overline{\sigma}(\mathbf{x})=M\mathbf{x}$. Choose a linear map $\delta:V\to V\otimes V$ such that $\overline{\delta}$ can be induced by it, where $V$ is the vector space spanned by $\{x_1,\cdots,x_n\}$. Since $A$ is 2-dimensional, there is a unique pair $(\delta_r,\delta_l)$ of elements in $V$ such that $$ \delta(r)=r\otimes \delta_r+\delta_l\otimes r. $$ That is, there are two certain elements $c_r=(c_{r1},c_{r2},\cdots,c_{rn}),c_l=(c_{l1},c_{l2},\cdots,c_{ln})\in k^n$ such that $$ \delta_r=c_r\mathbf{x},\qquad \delta_l=c_l\mathbf{x}. $$
By Theorem \ref{thm: nakayama automorphism of ore extension}, the Nakayama automorphism $\mu_B$ of the graded Ore extension $B=A[z;\overline{\sigma},\overline{\delta}]$ satisfies
\begin{equation}\label{nakayama of comm} {\mu_B}\left(\begin{array}{c}\mathbf{x}
\\z\end{array}\right) =\left( \begin{array}{cc} -M^{-1}(Q^T)^{-1}Q &0\\ c_r- c_l M^{-1}(Q^T)^{-1}Q &\hdet(\overline{\sigma}) \end{array} \right) \left(\begin{array}{c}\mathbf{x}
\\z\end{array}\right). \end{equation}
Now we focus on noetherian ones. There is an interesting result about CY property for noetherian cases. \begin{theorem}\label{thm: CY for 2-dim} Let $A=k\langle x_1,x_2\rangle/(r)$ be a noetherian AS-regular algebra of dimension 2 and $B=A[z;\overline{\sigma},\overline{\delta}]$ is a graded Ore extension. Write $\mu_A$ for the Nakayama automorphism of $A$. \begin{enumerate} \item Suppose $A$ is commutative, then $B$ is CY if and only if $\overline{\sigma}=\id_A$ and
$$
\overline{\delta}(x_1)=l_1x_1^2-2l_4x_2x_1+l_2x_2^2,\qquad
\overline{\delta}(x_2)=l_3x_1^2-2l_1x_2x_1+l_4x_2^2,
$$ for some $l_1,l_2,l_3,l_4\in k$. \item Suppose $A$ is noncommutative, then $B$ is CY if and only if $\overline{\sigma}=\mu_A$. \end{enumerate} \end{theorem}
To prove this result, we determine all graded Ore extensions of noetherian Koszul AS-regular algebras of dimension $2$ and compute their Nakayama automorphisms. Let $A$ be a noetherian Koszul AS-regular algebra of dimension $2$. Then $A=k\langle x_1,x_2\rangle/(r)$, and there are only two classes of $A$ up to isomorphism, that is, $$ Q=\left( \begin{array}{cc} 0 & 1\\ -q & 0 \end{array} \right),\qquad\text{or}\qquad Q=\left( \begin{array}{cc} 0 & 1\\ -1 & -1 \end{array} \right), $$ where $q$ is a nonzero element in $k$. Let $V$ be the vector space spanned by $\{x_1,x_2\}$.
In the following, $\overline{\sigma}$ is a graded automorphism of $A$, $M=(m_{ij})\in M_2(k)$ is the invertible matrix such that $\overline{\sigma}(x_1,x_2)^T=M(x_1,x_2)^T$. Write $\sigma=\overline{\sigma}_{\mid V}$ and $\sigma_T=\oplus_{i\geq0}\sigma^{\otimes i}$. Let $\delta$ be a linear map from $V$ to $V\otimes V$. Then any degree-one $\overline{\sigma}$-derivation $\overline{\delta}$ of $A$ can be induced by $\delta$, in case $\delta$ extends to a degree-one $\sigma_T$-derivation of $T(V)(\cong k\langle x_1,x_2\rangle)$ such that \begin{equation}\label{condtion for comm} \delta(r)\in r\otimes V+V\otimes r. \end{equation} Since the forms of $r$ (or $Q$), we assume without loss of generality, $$ \delta(x_i)=\gamma_{i1}x_1^2+\gamma_{i2}x_2x_1+\gamma_{i3}x_2^2, $$ where $\gamma_{ij}\in k$ for $j=1,2,3,i=1,2.$
\subsubsection{Case (i): commutative polynomial} In this case, $r=x_1x_2-x_2x_1$, or equivalently $$Q=\left( \begin{array}{cc} 0 & 1\\ -1 & 0 \end{array} \right),$$ $M=(m_{ij})$ is an arbitrary invertible matrix, and $\hdet(\overline{\sigma})=\det (M)$ (We refer \cite{JZ,SZL} for the computation of homological determinant). We have
\begin{align*} \delta(r)=&(m_{11}\gamma_{21}-m_{21}\gamma_{11}-\gamma_{21})x_1^3+(m_{12}\gamma_{23}-m_{22}\gamma_{13}+\gamma_{13})x_2^3+(m_{11}\gamma_{22}-m_{21}\gamma_{12})x_1x_2x_1+\gamma_{12}x_2x_1x_2\\ &+\gamma_{11}x_1^2x_2+(m_{11}\gamma_{23}-m_{21}\gamma_{13})x_1x_2^2+(m_{12}\gamma_{21}-m_{22}\gamma_{11}-\gamma_{22})x_2x_1^2+(m_{12}\gamma_{22}-m_{22}\gamma_{12}-\gamma_{23})x_2^2x_1. \end{align*}
By a straightforward computation, one obtains that the following result. \begin{lemma} The condition (\ref{condtion for comm}) is equivalent to the following equations hold \begin{equation}\label{equation for comm} \begin{aligned} &m_{21}\gamma_{11}+(1-m_{11})\gamma_{21}=0,\\ &(1-m_{22})\gamma_{13}+m_{12}\gamma_{23}=0,\\ &(m_{22}-1)\gamma_{11}+m_{21}\gamma_{12}-m_{12}\gamma_{21}+(1-m_{11})\gamma_{22}=0,\\ &(m_{22}-1)\gamma_{12}+m_{21}\gamma_{13}-m_{12}\gamma_{22}+(1-m_{11})\gamma_{23}=0. \end{aligned} \end{equation} In this case, $\delta(r)=r\otimes \delta_r+\delta_l\otimes r$, where \begin{equation*}\label{delta_l,r for comm} \delta_{r}=(m_{22}\gamma_{11}-m_{12}\gamma_{21}+\gamma_{22})x_1+(m_{11}\gamma_{23}-m_{21}\gamma_{13})x_2,\quad \delta_{l}=\gamma_{11}x_1+(m_{22}\gamma_{12}-m_{12}\gamma_{22}+\gamma_{23})x_2. \end{equation*} \end{lemma}
By (\ref{nakayama of comm}), the Nakayama automorphisms of graded Ore extension $B=A[z;\overline{\sigma},\overline{\delta}]$ of $A$ by (\ref{nakayama of comm}), where $\overline{\delta}$ is induced by $\delta$, satisfies \begin{equation}\label{naka equa for comm} \begin{aligned} &\mu_B(x_1)=\det(M)^{-1}(m_{22}x_1-m_{12}x_2),\quad \mu_B(x_2)=\det(M)^{-1}(-m_{21}x_1+m_{11}x_2),\\ &\mu_B(z)=\det(M)z+(x_1,x_2)\upsilon^T,
\end{aligned} \end{equation} where $\upsilon=\left(m_{22}\gamma_{11}-m_{12}\gamma_{21}+\gamma_{22},m_{11}\gamma_{23}-m_{21}\gamma_{13}\right) +\left(\gamma_{11},m_{22}\gamma_{12}-m_{12}\gamma_{22}+\gamma_{23}\right)M^{-1}$. Then we list all solutions of equations (\ref{equation for comm}).
\begin{solution}\label{solution: commutative} All soloutions of equations (\ref{equation for comm}) are as follows. \begin{enumerate} \item If $M$ is the identity matrix $E_2$, that is $m_{11}=m_{22}=1$ and $m_{12}=m_{21}=0$, then each $\gamma_{ij}$ is free for $i=1,2,j=1,2,3$;
\item If $m_{21}=0$, $m_{22}=1$ and $M\neq E_2$, then $
\gamma_{21}=\gamma_{22}=\gamma_{23}=0$ and $ \gamma_{11},\gamma_{12},\gamma_{13}$ are free variables;
\item If $m_{21}=0,m_{22}\neq 1,m_{11}=1$, then
$\gamma_{11}=(m_{22}-1)^{-1}m_{12}\gamma_{21},
\gamma_{12}=(m_{22}-1)^{-1}m_{12}\gamma_{22},
\gamma_{13}=(m_{22}-1)^{-1}m_{12}\gamma_{23};$
\item If $m_{21}=0,m_{22}\neq 1,m_{11}\neq1$, then $\gamma_{11}=(m_{22}-1)^{-1}(m_{11}-1)\gamma_{22},
\gamma_{12}=(m_{22}-1)^{-1}(m_{12}\gamma_{22}+(m_{11}-1)\gamma_{23}),
\gamma_{13}=(m_{22}-1)^{-1}m_{12}\gamma_{23},
\gamma_{21}=0;$
\item If $m_{12}=0$, there exit symmetric solutions of the last three ones, which we omit here;
\item If $m_{21}m_{22}\neq 0$ and $(m_{22}-1)(m_{11}-1)=m_{21}m_{12}$, then
$
\gamma_{11}=m_{21}^{-1}(m_{11}-1)\gamma_{21},
\gamma_{12}=m_{21}^{-1}(m_{11}-1)\gamma_{22},
\gamma_{23}=m_{12}^{-1}(m_{22}-1)\gamma_{13};
$
\item If $m_{21}m_{22}\neq 0$ and $(m_{22}-1)(m_{11}-1)\neq m_{21}m_{12}$, then $ \gamma_{11}=m_{21}^{-1}(m_{11}-1)\gamma_{21},
\gamma_{12}=m_{21}^{-1}m_{12}\gamma_{21}+m_{12}^{-1}(m_{11}-1)\gamma_{13},
\gamma_{22}=m_{21}^{-1}(m_{22}-1)\gamma_{21}+m_{12}^{-1}m_{21}\gamma_{13},
\gamma_{23}=m_{12}^{-1}(m_{22}-1)\gamma_{13}. $
\end{enumerate} \end{solution}
\subsubsection{Case (ii): noncommutative quantum plane }In this case, $r=x_1x_2-qx_2x_1$, or equivalently, $$Q=\left( \begin{array}{cc} 0 & 1\\ -q & 0 \end{array} \right),$$ where nonzero element $q\neq 1$. There are two subcases: $q=-1$ and $q\neq 1$.
\noindent (1) $q=-1$. The graded automorphisms of $A=k\langle x_1,x_2\rangle/(x_1x_2+x_2x_1)$ have two forms $$ M= \left( \begin{array}{cc} m_{11} & 0\\ 0 &m_{22} \end{array} \right)\qquad\text{and}\qquad M= \left( \begin{array}{cc} 0 & m_{12}\\ m_{21} &0 \end{array} \right). $$
\ding{172} $ M= \left( \begin{array}{cc} m_{11} & 0\\ 0 &m_{22} \end{array} \right)$. Then $\hdet(\overline{\sigma})=m_{11}m_{22}$. Since $r=x_1x_2+x_2x_1$, then \begin{align*} \delta(r)=&(m_{11}+1)\gamma_{21}x_1^3+(m_{22}+1)\gamma_{13}x_2^3+m_{11}\gamma_{22}x_1x_2x_1+\gamma_{12}x_2x_1x_2\\ &+m_{11}\gamma_{23}x_1x_2^2+\gamma_{11}x_1^2x_2+(m_{22}\gamma_{11}+\gamma_{22})x_2x_1^2+(m_{22}\gamma_{12}+\gamma_{23})x_2^2x_1. \end{align*} The following result is easy to get. \begin{lemma} The condition (\ref{condtion for comm}) is equivalent to the following equations hold \begin{equation}\label{equation for q=-1} \begin{aligned} &(m_{11}+1)\gamma_{21}=0,\\ &(m_{22}+1)\gamma_{13}=0,\\ &(m_{22}+1)\gamma_{11}+(1-m_{11})\gamma_{22}=0,\\ &(m_{22}-1)\gamma_{12}+(m_{11}+1)\gamma_{23}=0. \end{aligned} \end{equation} In this case, $\delta(r)=r\otimes \delta_r+\delta_l\otimes r$, where \begin{equation*}\label{delta_l,r for q=-1} \delta_{r}=(m_{22}\gamma_{11}+\gamma_{22})x_1+m_{11}\gamma_{23}x_2,\quad \delta_{l}=\gamma_{11}x_1+(m_{22}\gamma_{12}+\gamma_{23})x_2. \end{equation*} \end{lemma} By (\ref{nakayama of comm}), we have the Nakayama automorphism of a graded Ore extension $B=A[z;\overline{\sigma},\overline{\delta}]$ satisfies that
\begin{equation}\label{naka equa for q=-1}
\begin{aligned}
&\mu_B(x_1)=-m_{11}^{-1}x_1,\quad \mu_B(x_2)=-m_{22}^{-1}x_2,\\ &\mu_B(z)=m_{11}m_{22}z+((m_{22}-m_{11}^{-1})\gamma_{11}+\gamma_{22})x_1+((m_{11}-m_{22}^{-1})\gamma_{23}-\gamma_{12})x_2.
\end{aligned}
\end{equation} To be explicit, we give all solutions of (\ref{equation for q=-1}). \begin{solution}\label{solution: q=-1} The solutions of (\ref{equation for q=-1}) are as follows. \begin{enumerate} \item If $m_{11}=m_{22}=-1$, then $\gamma_{12}=\gamma_{22}=0$ and the other variables are free;
\item If $m_{11}=1,m_{22}=-1$, then $\gamma_{21}=0,\gamma_{23}=\gamma_{12}$ and $\gamma_{11},\gamma_{13},\gamma_{22}$ are free;
\item If $m_{11}\neq \pm1,m_{22}=-1$, then $\gamma_{21}=\gamma_{22}=0$, $\gamma_{23}=2(m_{11}+1)^{-1}\gamma_{12}$ and $\gamma_{11},\gamma_{13}$ are free;
\item If $m_{11}=-1,m_{22}=1$, then $\gamma_{13}=0,\gamma_{11}=-\gamma_{22}$ and $\gamma_{12},\gamma_{21},\gamma_{23}$ are free;
\item If $m_{11}=-1,m_{22}\neq \pm 1$, then $\gamma_{12}=\gamma_{13}=0,\gamma_{11}=-2(m_{22}+1)^{-1}\gamma_{22}$ and $\gamma_{21},\gamma_{23}$ are free;
\item If $m_{11}\neq-1,m_{22}\neq- 1$, then $\gamma_{13}=\gamma_{21}=0,\gamma_{11}=(m_{11}-1)(m_{22}+1)^{-1}\gamma_{22}, \gamma_{23}=(m_{11}+1)^{-1}(1-m_{22})\gamma_{12}$.
\end{enumerate} \end{solution}
\ding{173} $M= \left( \begin{array}{cc} 0 & m_{12}\\ m_{21} &0 \end{array} \right).$ Then $\hdet(\overline{\sigma})=m_{12}m_{21}$. Similarly, one obtains that \begin{align*} \delta(r)=&(m_{21}\gamma_{11}+\gamma_{21})x_1^3+(m_{12}\gamma_{23}+\gamma_{13})x_2^3+m_{12}\gamma_{12}x_1x_2x_1+\gamma_{12}x_2x_1x_2\\ &+m_{21}\gamma_{13}x_1x_2^2+\gamma_{11}x_1^2x_2+(m_{12}\gamma_{21}+\gamma_{22})x_2x_1^2+(m_{12}\gamma_{22}+\gamma_{23})x_2^2x_1. \end{align*} \begin{lemma} The condition (\ref{condtion for comm}) is equivalent to the following equations hold \begin{equation}\label{equation for q=-1 II} \begin{aligned} &m_{21}\gamma_{11}+\gamma_{21}=0,\\ &m_{12}\gamma_{23}+\gamma_{13}=0,\\ &\gamma_{11}-m_{21}\gamma_{12}+m_{12}\gamma_{21}+\gamma_{22}=0,\\ &-\gamma_{12}+m_{21}\gamma_{13}+m_{12}\gamma_{22}+\gamma_{23}=0. \end{aligned} \end{equation} In this case, $\delta(r)=r\otimes \delta_r+\delta_l\otimes r$, where \begin{equation*}\label{delta_l,r for qneq-1} \delta_{r}=(m_{12}\gamma_{21}+\gamma_{22})x_1+m_{21}\gamma_{13}x_2,\quad \delta_{l}=\gamma_{11}x_1+(m_{12}\gamma_{22}+\gamma_{23})x_2. \end{equation*} \end{lemma}
\begin{solution}\label{solution: q=-1 II} The solutions to equations (\ref{equation for q=-1 II}) are as follows. \begin{enumerate} \item If $m_{12}m_{21}=1$, then $ \gamma_{11}=-m_{12}\gamma_{21},\gamma_{12}=m_{12}\gamma_{22}, \gamma_{13}=-m_{12}\gamma_{23}; $ \item If $m_{12}m_{21}\neq1$, then $ \gamma_{11}=-m_{21}^{-1}\gamma_{21},\gamma_{12}=m_{12}m_{21}^{-1}\gamma_{21}+\gamma_{23}, \gamma_{13}=-m_{12}\gamma_{23},\gamma_{22}=m_{21}^{-1}\gamma_{21}+m_{21}\gamma_{23}. $ Moreover, the Nakayama automorphism of $B=A[z;\overline{\sigma},\overline{\delta}]$ satisfies \begin{align*} &\mu_B(x_1)=-m_{21}^{-1}x_2,\quad \mu_B(x_2)=-m_{12}^{-1}x_1,\\ &\mu_B(z)=m_{12}m_{21}z+(m_{12}\gamma_{21}-m_{12}^{-1}\gamma_{23})x_1+(m_{12}^{-2}\gamma_{21}-m_{12}m_{21}\gamma_{23})x_2. \end{align*} \end{enumerate} \end{solution} \noindent (2) $q\neq-1$. The graded automorphism $\overline{\sigma}$ of $A=k\langle x_1,x_2\rangle/(x_1x_2-qx_2x_1)$ must be the following form $ M= \left( \begin{array}{cc} m_{11} & 0\\ 0 &m_{22} \end{array} \right), $ and $\hdet(\overline{\sigma})=m_{11}m_{22}$. One obtains that \begin{align*} \delta(r)=&(m_{11}-q)\gamma_{21}x_1^3+(1-qm_{22})\gamma_{13}x_2^3+m_{11}\gamma_{22}x_1x_2x_1+\gamma_{12}x_2x_1x_2\\ &+m_{11}\gamma_{23}x_1x_2^2+\gamma_{11}x_1^2x_2-q(m_{22}\gamma_{11}+\gamma_{22})x_2x_1^2-q(m_{22}\gamma_{12}+\gamma_{23})x_2^2x_1. \end{align*} \begin{lemma} The condition (\ref{condtion for comm}) is equivalent to the following equations hold \begin{equation}\label{equation for qneq-1} \begin{aligned} &(m_{11}-q)\gamma_{21}=0,\\ &(1-qm_{22})\gamma_{13}=0,\\ &(m_{22}-q)\gamma_{11}+(1-m_{11})\gamma_{22}=0,\\ &(m_{22}-1)\gamma_{12}+(1-qm_{11})\gamma_{23}=0. \end{aligned} \end{equation} In this case, $\delta(r)=r\otimes \delta_r+\delta_l\otimes r$, where \begin{equation*} \delta_{r}=(m_{22}\gamma_{11}+\gamma_{22})x_1+m_{11}\gamma_{23}x_2,\quad \delta_{l}=\gamma_{11}x_1+(m_{22}\gamma_{12}+\gamma_{23})x_2. \end{equation*} \end{lemma}
By (\ref{nakayama of comm}), we have the Nakayama automorphism of the graded Ore extension $B=A[z;\overline{\sigma},\overline{\delta}]$ satisfies that
\begin{equation}\label{naka equa for qneq-1}
\begin{aligned}
&\mu_B(x_1)=qm_{11}^{-1}x_1,\quad \mu_B(x_2)=(qm_{22})^{-1}x_2,\\ &\mu_B(z)=m_{11}m_{22}z+((m_{22}+qm_{11}^{-1})\gamma_{11}+\gamma_{22})x_1+((m_{11}+(qm_{22})^{-1})\gamma_{23}+q^{-1}\gamma_{12})x_2.
\end{aligned}
\end{equation}
\begin{solution}\label{solution: qneq-1} All solutions of (\ref{equation for qneq-1}) are as follows. \begin{enumerate} \item If $m_{11}=q,m_{22}=q^{-1}$, then $\gamma_{12}=-q(1+q)\gamma_{23},\gamma_{22}=-(q^{-1}+1)\gamma_{11}$ and $\gamma_{13},\gamma_{21}$ are free;
\item If $m_{11}\neq q,m_{22}=q^{-1}$, then $\gamma_{21}=0,\gamma_{11}=(q^{-1}-q)^{-1}(m_{11}-1)\gamma_{22},\gamma_{12}=(q^{-1}-1)^{-1}(qm_{11}-1)\gamma_{23}$, and $\gamma_{13}$ is free;
\item If $m_{11}=q,m_{22}\neq q^{-1}$, then $\gamma_{13}=0$, $\gamma_{22}=(1-q)^{-1}(q-m_{22})\gamma_{11}$, $\gamma_{23}=(1-q^2)^{-1}(1-m_{22})\gamma_{12}$ and $\gamma_{21}$ is free;
\item If $m_{11}=1,m_{22}\neq q^{-1}$, then $\gamma_{11}=\gamma_{13}=\gamma_{21}=0$, $\gamma_{23}=(1-q)^{-1}(1-m_{22})\gamma_{12}$ and $\gamma_{22}$ is free;
\item If $m_{11}=q^{-1},m_{22}\neq q^{-1}$, then $\gamma_{13}=\gamma_{21}=0,\gamma_{22}=(1-q^{-1})^{-1}(q-m_{22})\gamma_{11}$, $\gamma_{23}$ is free and $\gamma_{12}=0$ if $m_{22}\neq 1$ or $\gamma_{12}$ is free if $m_{22}=1$;
\item If $m_{11}\neq q^{\pm 1},1,m_{22}\neq q^{-1}$, then $\gamma_{13}=\gamma_{21}=0,\gamma_{22}=(1-m_{11})^{-1}(q-m_{22})\gamma_{11}$, and $\gamma_{23}=(1-qm_{11})^{-1}(1-m_{22})\gamma_{12}$.
\end{enumerate} \end{solution}
\subsubsection{Case (iii): Jordan plane}In this case, $r=x_1x_2-x_2x_1-x_2^2$, or equivalently, $$Q=\left( \begin{array}{cc} 0 & 1\\ -1 & -1 \end{array} \right),$$ $M$ has the form $ \left( \begin{array}{cc} m_{11} & m_{12}\\ 0 &m_{11} \end{array} \right) $ and $\hdet(\overline{\sigma})=m_{11}^2$. One obtains that \begin{align*} \delta(r)=&(m_{11}-1)\gamma_{21}x_1^3+m_{11}\gamma_{22}x_1x_2x_1+(\gamma_{12}-\gamma_{22})x_2x_1x_2+m_{11}\gamma_{23}x_1x_2^2+(\gamma_{11}-\gamma_{21})x_1^2x_2\\ &+((m_{12}-m_{11})\gamma_{21}-m_{11}\gamma_{11}-\gamma_{22})x_2x_1^2+((m_{12}-m_{11})\gamma_{22}-m_{11}\gamma_{12}-\gamma_{23})x_2^2x_1\\ &+((1-m_{11})\gamma_{13}+(m_{12}-m_{11}-1)\gamma_{23})x _2^3. \end{align*}
\begin{lemma} The condition (\ref{condtion for comm}) is equivalent to the following equations hold \begin{equation}\label{equation for Jordan} \begin{aligned} &(m_{11}-1)\gamma_{21}=0,\\ &(m_{11}-1)\gamma_{11}+(m_{11}-m_{12}+1)\gamma_{21}+(1-m_{11})\gamma_{22}=0,\\ &-(m_{11}+1)\gamma_{11}+(m_{11}-1)\gamma_{12}+(1-m_{11}+m_{12})\gamma_{21}+(m_{11}-m_{12})\gamma_{22}+(1-m_{11})\gamma_{23}=0,\\ &-2\gamma_{11}-\gamma_{12}+(m_{11}-1)\gamma_{13}+2\gamma_{21}+\gamma_{22}+(1-m_{11}-m_{12})\gamma_{23}=0. \end{aligned} \end{equation} In this case, $\delta(r)=r\otimes \delta_r+\delta_l\otimes r$, where \begin{eqnarray*} &&\delta_{r}=(m_{11}\gamma_{11}+(m_{11}-m_{12})\gamma_{21}+\gamma_{22})x_1+(\gamma_{11}-\gamma_{21}+m_{11}\gamma_{23})x_2,\\ &&\delta_{l}=(\gamma_{11}-\gamma_{21})x_1+(\gamma_{11}+\gamma_{12}-\gamma_{21}-\gamma_{22}+m_{11}\gamma_{23})x_2. \end{eqnarray*} \end{lemma} By (\ref{nakayama of comm}), we have the Nakayama automorphism of a graded Ore extension $B=A[z;\overline{\sigma},\overline{\delta}]$ satisfies that
\begin{equation}\label{naka equa for jordan}
\begin{aligned}
&\mu_B(x_1)=m_{11}^{-1}x_1+(2m_{11}^{-1}-m_{11}^{-2}m_{12})x_2,\quad \mu_B(x_2)=m_{11}^{-1}x_2,\\
&\mu_B(z)=m_{11}^2z+
((m_{11}+m_{11}^{-1})\gamma_{11}+(m_{11}-m_{11}^{-1}-m_{12})\gamma_{21}+\gamma_{22})x_1\\
&\quad+((1+3m_{11}^{-1}-m_{11}^{-2}m_{12})\gamma_{11}+m_{11}^{-1}\gamma_{12}+(m_{11}^{-2}m_{12}-3m_{11}^{-1})\gamma_{21}-m_{11}^{-1}\gamma_{22}+(1+m_{11})\gamma_{23})x_2.
\end{aligned} \end{equation}
\begin{solution}\label{solution: Jordan} All solutions to equations (\ref{equation for Jordan}) are as follows. \begin{enumerate} \item If $m_{11}=1$, then $ \gamma_{11}=(m_{12}\gamma_{21}+(1-m_{12})\gamma_{22})/2,\gamma_{12}=m_{12}\gamma_{22}-m_{12}\gamma_{23}, $ and $\gamma_{13}$ is free, where $\gamma_{21}=0$ if $m_{12}=2$ or $\gamma_{21}$ if free if $m_{12}\neq 2$;
\item If $m_{11}\neq 1$, $\gamma_{21}=0,\gamma_{11}=\gamma_{22},\gamma_{12}=(m_{11}-1)^{-1}(m_{12}+1)\gamma_{22}+\gamma_{23},
\gamma_{13}=(m_{11}-1)^{-1}(m_{11}+m_{12})((m_{11}-1)^{-1}\gamma_{22}+\gamma_{23})$.
\end{enumerate} \end{solution}
\begin{proof}[Proof of Theorem \ref{thm: CY for 2-dim}] If $B$ is CY, then $\overline{\sigma}=\mu_A$ follows by Theorem \ref{thm: nakayama automorphism of ore extension}.
If $A$ is commutative, then $\mu_A$ is the identity map, that is $M=E_2$. Hence, (a) is an immediate result by Solution \ref{solution: commutative}(a) and (\ref{naka equa for comm}), or Theorem \ref{thm: NA of Ore ext over Polynomial}.
Now assume $A$ is noncommutative and $\overline{\sigma}=\mu_A$.
If $A$ is a quantum plane, then $\mu_A(x_1)=qx_1$ and $\mu_A(x_2)=q^{-1}x_2$, that is $m_{11}=q,m_{22}=q^{-1},m_{12}=m_{21}=0$. Hence, $B$ is CY by Solution \ref{solution: q=-1}(a) and (\ref{naka equa for q=-1}) if $q=-1$, and Solution \ref{solution: qneq-1}(a) and (\ref{naka equa for qneq-1}) if $q\neq-1$.
If $A$ is the Jordan plane, then $\mu_A(x_1)=x_1+2x_2$ and $\mu_A(x_2)=x_2$, that is $m_{11}=m_{22}=1,m_{12}=2,m_{21}=0$. By Solution \ref{solution: Jordan}(a) and (\ref{naka equa for jordan}), one obtains that $B$ is CY.
\end{proof}
\vskip7mm
\noindent {\bf Acknowledgments.} Y. Shen is supported by NSFC (Grant No.11701515) and the Fundamental Research Funds of Zhejiang Sci-Tech University (Grant No. 2019Q071).
\end{document} | arXiv |
A simulation study on estimating biomarker–treatment interaction effects in randomized trials with prognostic variables
Bernhard Haller1Email authorView ORCID ID profile and
Kurt Ulm1
To individualize treatment decisions based on patient characteristics, identification of an interaction between a biomarker and treatment is necessary. Often such potential interactions are analysed using data from randomized clinical trials intended for comparison of two treatments. Tests of interactions are often lacking statistical power and we investigated if and how a consideration of further prognostic variables can improve power and decrease the bias of estimated biomarker–treatment interactions in randomized clinical trials with time-to-event outcomes.
A simulation study was performed to assess how prognostic factors affect the estimate of the biomarker–treatment interaction for a time-to-event outcome, when different approaches, like ignoring other prognostic factors, including all available covariates or using variable selection strategies, are applied. Different scenarios regarding the proportion of censored observations, the correlation structure between the covariate of interest and further potential prognostic variables, and the strength of the interaction were considered.
The simulation study revealed that in a regression model for estimating a biomarker–treatment interaction, the probability of detecting a biomarker–treatment interaction can be increased by including prognostic variables that are associated with the outcome, and that the interaction estimate is biased when relevant prognostic variables are not considered. However, the probability of a false-positive finding increases if too many potential predictors are included or if variable selection is performed inadequately.
We recommend undertaking an adequate literature search before data analysis to derive information about potential prognostic variables and to gain power for detecting true interaction effects and pre-specifying analyses to avoid selective reporting and increased false-positive rates.
Biomarker–treatment interaction
Randomized trial
Stratified medicine
Predictive covariates
Variable selection
Treatment individualization, i.e. finding the right treatment with the right dose at the right time for a specific patient based on certain patient characteristics, is one of the great goals in modern medicine [1]. One requirement for treatment individualization based on, e.g. a certain biomarker like a genetic characteristic or a blood parameter, is the existence of a relevant association between the biomarker and the treatment effect [2], often referred to as the biomarker–treatment interaction. Only a small number of trials have been planned to analyse biomarker–treatment interactions [3], but often the association between one or more biomarkers and a treatment effect is evaluated post hoc in data collected in randomized clinical trials intended for overall comparison of treatment groups, like e.g. the detection of the association between the response to cetuximab and the presence or absence of the K-ras mutation in the tumours of patients with advanced colorectal cancer [4].
While often the treatment effect is analysed in different subgroups (pre-specified or post hoc specified) to identify patients that benefit from one or another treatment [5], it is widely recognized that the comparison of treatment groups in many different subgroups can lead to spurious results [6]. Therefore, it is often recommended to assess the biomarker–treatment interaction in a regression model, which directly allows us to estimate and test for an interaction effect under common model assumptions [7]. Various authors who provide methods for estimating biomarker–treatment interactions stress the importance of the adequate inclusion of prognostic factors in the model [8, 9]. For treatment effect estimation in a randomized clinical trial, the European Medicines Agency's guideline on 'Points to consider on adjustment for baseline covariates' recommends including other prognostic factors, i.e. covariates that are assumed to be associated with the outcome, as covariates in the regression model to increase the precision of the estimate of the treatment effect [10]. Furthermore, it has been shown that the estimate for the treatment effect is biased in a Cox regression model, if relevant prognostic covariates are not included [11]. While defining the model used for effect estimation and hypothesis testing a priori and including all relevant covariates can be considered as best practices [12], adequate information about prognostic factors might not be available for all research questions, especially when molecular information that has not been well studied and for which limited information from prior investigations is available is included in a regression model. Various approaches to determining the covariates that are to be included in a regression model are presented in the literature [13].
The focus of this article is estimating the interaction between one certain pre-specified biomarker of major interest and the treatment. A simulation study was performed to evaluate how the presence and inclusion of further prognostic covariates affect the estimate of the biomarker–treatment interaction. Different strategies for model building, such as including only the main effects of treatment, the biomarker and their interaction, additionally including covariates that are significantly associated with the outcome, or using variable selection methods based on Akaike's information criterion (AIC) [14] are considered. Scenarios with varying proportions of censored observations, different strengths of association of the prognostic covariates and the outcome, different correlations between prognostic covariates and the biomarker of interest, and different numbers of potential prognostic covariates are considered. The different strategies of covariate inclusion are compared in the control of type I error probabilities and the power to reject the null hypothesis of no biomarker–treatment interaction. A special focus was placed on the so-called rule of ten [12, 15]. This is often considered for predictive models, but (to the best of our knowledge) has not been investigated for the number of additional covariates considered in a regression model, when the primary goal was estimation of an interaction effect.
Assessing the biomarker–treatment interaction
The interaction between a continuous biomarker of major interest B, or a continuous covariate in general, and treatment T, which is assumed to be binary throughout the article (T∈{0;1}), can be assessed by including an interaction term between the biomarker and the treatment in an adequate regression model. This means the product of B and T is included in the regression model as an additional covariate (see e.g. [13]). The Cox regression model [16], also known as the proportional hazards model, is commonly considered in the analysis of survival data in medical research. In the Cox model, the effect of the biomarker B, the treatment T, their interaction T×B and K other covariates described through the matrix X k on the hazard rate λ(t) is modelled as
$$ \begin{aligned} \lambda(t|T\!, B, \boldsymbol{X_{k}}) \,=\, \lambda_{0}(t)\!\exp(\beta_{T}\!T\! +\! \beta_{B}B \,+\, \beta_{T\times B}T\times B \,+\, \boldsymbol{\beta_{k}^{T}} \boldsymbol{X_{k}}), \end{aligned} $$
where a linear association between a covariate and the log-hazard ratio is assumed. In Eq. (1), λ0(t) is the (unspecified) baseline hazard rate, β T the regression coefficient for treatment T, β B the coefficient for the biomarker of interest B, βT×B the regression coefficient for their interaction term and β k the vector of regression coefficients for the K additional covariates, X1,…,X K . When an interaction term is present, the main effects of the treatment T and the biomarker B can be interpreted as the expected treatment difference at a (fictitious) biomarker value of B=0 and the effect of the biomarker B under treatment T=0 conditional on all other covariates. Regression coefficients are estimated by numerical maximization of the partial log-likelihood PL(β). The variance-covariance matrix of the estimated regression coefficients can be derived as the inverse of the observed information matrix \( {I}^{-1}(\boldsymbol {\hat {\beta }})\) (see e.g. [16] or [17] for more details).
Strategies for covariate inclusion
In the simulation study, various approaches for including covariates are compared. In all models, the main effects of the treatment and biomarker as well as their interaction term are included. Obviously, the best choice would be to fit the true model to the data, which includes all covariates that are truly associated with the outcome and ignoring those covariates that are not. This model will be estimated using the simulated data, but in practice the true model will not be known and therefore, the model must be chosen based on plausibility and previous knowledge or based on information gathered from the observed data. Therefore, the following models and strategies were investigated. The names are used for the models/strategies in the figures and tables presented in this article:
Main: A model including only the main effects of treatment T and the biomarker B and their interaction T×B, ignoring all other possible prognostic covariates.
True: A model including the main effects of treatment T, the biomarker of interest B and their interaction T×B, as well as all covariates that are truly associated with the outcome, indicating perfect prior knowledge of relevant covariates.
AIC A : A model that includes the main effects of treatment T and the biomarker B and their interaction T×B and additionally all covariates that were selected in a forward variable selection procedure based on Akaike's information criterion (AIC) [14] given T, B and T×B are included (a model including T, B and T×B was used as a starting and minimal model). Additional covariates were selected as long as the AIC criterion
$$ \operatorname{AIC} = 2\operatorname{ll}(\hat{\boldsymbol{\beta}}) - 2p $$
was increased, where \(\operatorname {ll}(\hat {\boldsymbol {\beta }})\) is the partial log-likelihood evaluated at the maximum likelihood estimator \(\hat {\boldsymbol {\beta }}\) and p is the number of estimated regression coefficients.
AIC B : A modelling strategy similar to AIC A described above, but prognostic factors were selected based on the AIC criterion considering just the main effect of treatment T as a starting model and not including B or T×B in the variable selection process. After prognostic factors were chosen according to the AIC criterion, B and T×B were added to the model to estimate the biomarker–treatment interaction.
Significance: A model that includes the main effects of treatment T, the covariate of interest and their interaction, as well as all covariates that were significantly associated with the outcome in a Cox regression model including only one covariate (often referred to as univariate Cox models in the medical literature). While this strategy is generally not recommended from a statistical point of view [18], it appears to be a quite popular approach in practice.
Full: A model that includes the treatment T, the biomarker B and their interaction T×B as covariates as well as the main effects of all K potential predictors X1,…,X K .
Data generation and simulation settings
Numerous different settings were considered to evaluate the modelling strategies under varying conditions. For each simulation scenario, 500 subjects were generated. The matrix of continuous covariates (covariate of interest B and potential predictors X1,…,X K ) was drawn from a multivariate normal distribution using the R package mvtnorm [19]. For each variable, a mean of 0 and a standard deviation of 1 were used. The correlation structure was specified as described below. Since a randomized controlled trial was intended to be simulated, the treatment variable was drawn independently from all other patient characteristics with Pr(T=1)= Pr(T=0)=0.50 for each individual. For all scenarios, β T and β B were chosen as β T = ln(0.75)=−0.288 (i.e. exp(β T )=0.75) and β B = ln(1.25)=0.223 (i.e. exp(β B )=1.25).
For each scenario, a time-constant baseline hazard rate of λ0(t)=1 was used. The hazard rate for each individual was calculated according to Eq. 1 considering the patient's characteristics and the regression coefficients for the specific scenario. Event times were generated from an exponential distribution using each individual's hazard rate. All aspects of the simulation study including data generation, estimating regression coefficients and summarizing the results were performed with the statistical software R [20].
The following aspects were varied in the simulation study.
Censoring distribution
Administrative censoring after 5 years was assumed for all scenarios. Additionally, censoring times were generated independently of the event times from an exponential distribution. The hazard rate of the censoring distribution was chosen to produce scenarios with
a low proportion of censored observations (between 30% and 40% censored observations corresponding to 300 to 350 observed events)
a high proportion of censoring (between 60% and 70% censored observations corresponding to 150 to 200 observed events).
Strength of interaction
The strength of the interaction effect between the covariate of interest B and treatment T was varied to consider scenarios with no, quantitative or qualitative biomarker–treatment interaction [21] (see also Fig. 1):
Simulation of data under the null hypothesis of no biomarker–treatment interaction: βT×B=0.
Quantitative biomarker–treatment interaction with a difference in the magnitude of the treatment effect between individuals with a low value of B and individuals with a large value of B: βT×B= ln(1.1)=0.095, leading to a hazard ratio between the treatment groups (T=1 vs. T=0) of about 0.6 for a given value of B=−2 and a hazard ratio of about 0.9 for B=2.
Qualitative biomarker–treatment interaction indicating an expected lower risk for an event from treatment T=1 for patients with a small value of B and a lower risk under treatment T=0 for patients with a large value of B: βT×B= ln(1.33)=0.285, providing a hazard ratio between the treatment groups smaller than 1 for B<1 and a hazard ratio larger than 1 for B>1 (dotted line in Fig. 1).
Number of potential prognostic variables to be included in the model Three settings for the number K of potential candidate predictors that can be included in the regression model were considered:
K=12: Here 12 additional prognostic covariates are considered, so the rule of ten is fulfilled under both censoring distributions for most simulation runs, as 150 to 200 events are expected in the settings with a high amount of censoring and up to 15 regression coefficients are to be estimated (12 prognostic variables plus the main effects of treatment T and the covariate of interest B and their interaction T×B).
K=24: Here 24 additional prognostic covariates are considered, so the rule of ten will be violated for most scenarios with high censoring.
K=36: Here 36 additional prognostic covariates are considered. Again, the rule of ten will be violated under high censoring.
Correlation structure between prognostic variables and covariate of interest Three different correlation structures between the covariate of interest B and the potential prognostic variables X1,…,X K were considered:
Firstly, a scenario with a biomarker of interest B that is independent of the potential prognostic variables, and independence between all the prognostic variables was investigated, with
$$\Sigma_{1} = \left(\begin{array}{ccccc} 1 & 0 & \cdots & \cdots & 0 \\ 0 & 1 & 0 & \cdots & 0 \\ \vdots & & \ddots & &\vdots \\ 0 & \cdots & 0 & 1 & 0 \\ 0 & \cdots & \cdots & 0 & 1 \\ \end{array} \right). $$
As a second setting, the correlation coefficients between B and all other covariates X1,…,X K , as well as between each pair of covariates X i ,X j with i≠j was set to r=0.5, indicating a moderate correlation between all variables:
$$\Sigma_{2} = \left(\begin{array}{ccccc} 1 & 0.5 & \cdots & \cdots & 0.5 \\ 0.5 & 1 & 0.5 & \cdots & 0.5 \\ \vdots & & \ddots & &\vdots \\ 0.5 & \cdots & 0.5 & 1 & 0.5 \\ 0.5 & \cdots & \cdots & 0.5 & 1 \\ \end{array} \right). $$
A block correlation structure between the covariates was considered, with a high correlation of r=0.7 between the biomarker B and a set of variables as well as between those variables, a moderate correlation of r=0.4 for another set and a correlation of r=0.1 or r=0 for the other variables:
$$\Sigma_{3} = \left(\begin{array}{ccccccccccccc} 1 & 0.7 & 0.7 & 0.7 & 0.4 & 0.4 & 0.4 & 0.4 & 0.1 & 0.1 & 0.1 & 0 & 0 \\ 0.7 & 1 & 0.7 & 0.7 & 0.4 & 0.4 & 0.4 & 0.4 & 0.1 & 0.1 & 0.1 & 0 & 0 \\ 0.7 & 0.7 & 1 & 0.7 & 0.4 & 0.4 & 0.4 & 0.4 & 0.1 & 0.1 & 0.1 & 0 & 0 \\ 0.7 & 0.7 & 0.7 & 1 & 0.4 & 0.4 & 0.4 & 0.4 & 0.1 & 0.1 & 0.1 & 0 & 0 \\ 0.4 & 0.4 & 0.4 & 0.4 & 1 & 0.4 & 0.4 & 0.4 & 0.1 & 0.1 & 0.1 & 0 & 0 \\ 0.4 & 0.4 & 0.4 & 0.4 & 0.4 & 1 & 0.4 & 0.4 & 0.1 & 0.1 & 0.1 & 0 & 0 \\ 0.4 & 0.4 & 0.4 & 0.4 & 0.4 & 0.4 & 1 & 0.4 & 0.1 & 0.1 & 0.1 & 0 & 0 \\ 0.4 & 0.4 & 0.4 & 0.4 & 0.4 & 0.4 & 0.4 & 1 & 0.1 & 0.1 & 0.1 & 0 & 0 \\ 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & 1 & 0.1 & 0.1 & 0 & 0 \\ 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & 1 & 0.1 & 0 & 0 \\ 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & 1 & 0 & 0 \\ 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 1 & 0 \\ 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0 & 1 \end{array}\right). $$
For the scenarios with K=24 or K=36 potential predictors, the correlation matrices were adapted accordingly.
Strength of association between prognostic variables and outcome
For the strength of association between the potential prognostic variables X1,…,X K and the outcome, two different settings were chosen:
For all covariates X1,…,X K , the same regression coefficient was chosen:
$$\boldsymbol{\beta_{k}} = \boldsymbol{\beta_{eq}} = (\ln(1.1), \ldots, \ln(1.1))^{T} = (0.095, \ldots, 0.095)^{T}. $$
Varying strengths of association between the potential predictors and the risk for an event were considered. The vector of regression coefficients was chosen to be
$$\boldsymbol{\beta_{k}} = \boldsymbol{\beta_{v}} = \left(\begin{array}{cc} \ln(1.2) \\ \ln(1.1)\\ \ln(1)\\ \ln(1.2)\\ \ln(1.1)\\ \ln(1)\\ \vdots\\ \ln(1.2)\\ \ln(1.1)\\ \ln(1) \end{array} \right) = \left(\begin{array}{cc} 0.182 \\ 0.095\\ 0\\ 0.182\\ 0.095\\ 0\\ \vdots\\ 0.182\\ 0.095\\ 0 \end{array} \right). $$
As all combinations of the different settings described above were considered in the simulation study, a total of 2 censoring distributions × 3 strengths of interaction between biomarker B and treatment T × 3 numbers of potential prognostic variables × 3 different correlation structures × 2 settings for association between the potential prognostics variables and the outcome = 108 settings were considered in the simulation. For each of these settings, 1000 simulation runs were performed.
Illustration of the different strengths of interaction used in the simulation study. A hazard ratio larger than 1 indicates a higher risk for death under treatment T=1, and a hazard ratio below 1 a higher risk under treatment T=0. For the scenario with no biomarker–treatment interaction, the hazard ratio between the treatment groups is independent of the biomarker value. For the scenario with a quantitative biomarker–treatment interaction, the risk for an event is smaller under T=1 compared to T=0 for all (probable) values of B, but the difference between groups decreases with increasing values of B. For the scenario with a qualitative biomarker–treatment interaction, the risk for an event is lower for T=1 compared to T=0 for small values of B and vice versa for large values of B
Analysis and presentation of results
In each simulation run, all of the methods or strategies described in "Strategies for covariate inclusion" section were fitted or applied, respectively. Estimation of the regression coefficients from the Cox regression models was performed with the function coxph in the survival library [22] of the statistical software R [20]. For the variable selection based on the AIC criterion, the function stepAIC in the library MASS [23] was applied.
For each model in each simulation run, the estimated regression coefficient for the biomarker–treatment interaction term \(\hat {\beta }_{T\times B}\) and its estimated variance as well as the p value of the Wald test for the null hypothesisv H0: βT×B=0 was saved. Additionally, a 95% confidence interval for βT×B was estimated as
$$ \begin{aligned} 95\% \operatorname{ci} &= \left\lbrack \hat{\beta}_{T\times B} - \phi_{0.975}\,\sqrt{\widehat{\operatorname{var}}(\hat{\beta}_{T\times B})};\right.\\ &\quad\ \ \left.\hat{\beta}_{T\times B} + \phi_{0.975}\,\sqrt{\widehat{\operatorname{var}}(\hat{\beta}_{T\times B})} \right\rbrack, \end{aligned} $$
where ϕ0.975 denotes the 97.5% quantile of the standard normal distribution and \(\widehat {\operatorname {var}}(\hat {\beta }_{T\times B})\) is the estimated variance of the interaction coefficient obtained in the corresponding simulation run for the respective modelling approach. If the algorithm for numerical maximization of the partial log-likelihood did not converge, this information was saved. All results presented in 'Results' rely on only estimations for which the numerical optimization algorithm converged. The number of runs for which no result was returned is presented.
For each model and strategy, the confidence interval coverage, i.e. the fraction of simulation runs in which the estimated confidence interval for the biomarker–treatment interaction covered the true value, was derived. The proportion of simulation runs in which the null hypothesis was rejected and a statistically significant biomarker–treatment interaction was detected for the conventional significance level of 5%, i.e. the power of the statistical test if H0 were false or the probability of a type I error if H0 were true (βT×B=0), was determined [24].
The observed proportions of rejected null hypotheses are summarized in Table 1. Results are presented stratified for different values of K, strength of interaction and proportion of censored observations, but were aggregated over different values of β k and Σ. In Tables 2, 3 and 4, the observed proportions of simulation runs with rejected null hypotheses are shown separately for the scenarios with K=12 (Table 2), K=24 (Table 3) and K=36 (Table 4), for scenarios with no true biomarker–treatment interaction (top), true quantitative biomarker–treatment interaction (middle) and true qualitative biomarker–treatment interaction (bottom). An observed type I error probability of 7% was considered to be acceptable. For scenarios with no interaction (β T×B =0), observed type I error proportions larger than 7% are in italics. For scenarios with data generated under H1 (quantitative interaction and qualitative interaction), the proportions of rejected null hypotheses are in bold if the type I error probability for the approach at the given scenario was not larger than 7%.
Proportions of rejected null hypotheses and numbers of included covariates stratified for number of potential prognostic variables (K), strength of interaction and proportion of censored observations
Censoring
AIC A
AIC B
Results are aggregated over different values of β k and Σ. For the scenarios with no true biomarker–treatment interaction, results for methods/strategies with an observed type I error probability above 7% are in italics. For scenarios with a true biomarker–treatment interaction, the observed power is in bold if the type I error probability did not exceed 7%
Proportions of rejected null hypotheses and numbers of included covariates for scenarios with K=12
β k
No interaction
Σ 1
β eq
β v
Quantitative interaction
Qualitative interaction
Mean number of prognostic covariates included
For the scenarios with no true biomarker–treatment interaction, results for methods/strategies with an observed type I error probability above 7% are in italics. For scenarios with a true biomarker–treatment interaction, the observed power is in bold if the type I error probability did not exceed 7%
The mean numbers of included additional covariates are given for each method or strategy for sets of scenarios stratified for β k and amount of censoring in the bottom rows of Tables 2, 3 and 4 and for each of the 108 simulated scenarios in Additional file 7: Table S1 (for K=12), Additional file 8: Table S2 (for K=24) and Additional file 9: Table S3 (for K=36).
The distributions of the obtained estimates are illustrated in Fig. 2 for one exemplary set of scenarios. The observed distributions of the regression coefficient estimates for the biomarker–treatment interaction \(\hat {\beta }_{T\times B}\) are displayed as box plots for the scenarios with Σ=Σ 3 , β k =β v and low (a) or high number of censored observations (b). In the top rows, scenarios with no true biomarker–treatment interaction are shown, and in the bottom rows, results for data simulated with true qualitative biomarker–treatment interactions are presented. Scenarios with different numbers of (potential) prognostic variables (K=12, K=24 and K=36) are shown in separate columns. Distributions of estimated regression coefficients are illustrated for all scenarios with no true interaction (under H0) or with true qualitative interaction in Additional file 1: Figure S1, Additional file 2: Figure S2, Additional file 3: Figure S3, Additional file 4: Figure S4, Additional file 5: Figure S5 and Additional file 6: Figure S6. In each figure, the true value of the interaction regression coefficient is illustrated by the horizontal red line. Additionally, the confidence interval coverage for each modelling strategy (triangles and blue lines) and the probability of rejection of the null hypothesis of no biomarker–treatment interaction, i.e. the estimated probability for a type I error in the first row and the observed power in the second row, are illustrated (circles and green lines).
Distribution of \(\hat {\beta }_{T\times B}\) for scenarios with Σ=Σ3, β k = β v , and low censoring (a) or high censoring (b) for no biomarker–treatment interaction (βT×B= ln(1.0)=0, top rows) or qualitative biomarker–treatment interaction (βT×B= ln(1.33)=0.285, bottom rows). Scenarios for different numbers of potential prognostic variables are shown in different columns. The dashed red lines indicate the true value of βT×B, the blue triangles represent the observed confidence interval coverages and the green dots the observed probability for a type I error (a) or estimated power (b). AIC Akaike's information criterion, qual. qualitative, Sig significance
The type I error probabilities for the biomarker–treatment interaction term, which are presented in the lines indicated with no interaction (Table 1) and in the upper parts of Tables 2, 3 and 4 for the scenarios with no interaction, were acceptable for almost all methods and strategies, when K=12 further (potential) prognostic variables were considered. Only for strategy AIC A an unacceptably high probability of type I errors (defined as larger than 7%) was observed for one setting (Table 2). For scenarios with K=24 (potential) prognostic variables, increased type I error probabilities were observed for each method for at least one scenario, except for AIC B . For AIC A , type I error probabilities above 7% were observed for six of the 12 settings (Table 3) and for scenarios with a high proportion of censored observations (60% to 70%) when scenarios with different β k and Σ were aggregated (Table 1). When K = 36 potential predictors were considered, an increased type I error probability was observed for AIC A for all scenarios. For main, significance and full, elevated false positive rates were obtained for three to five scenarios with a high proportion of censored observations. For the true model, only two scenarios with a high proportion of censored observations led to rejection of the null hypothesis in more than 7% of the observed simulation runs (Σ 2 , β eq and Σ 1 , β eq ). For all other scenarios, the observed type I error probabilities were between 5% and 7%. For the strategy AIC B , all observed type I error probabilities were between 5% and 7%.
For the main model, regression coefficients for the biomarker–treatment effect were underestimated when a true biomarker–treatment interaction was present (Fig. 2), with the largest bias observed for scenarios with Σ=Σ 2 (see second rows of Additional file 1: Figure S1A and Figure S1B, Additional file 2: Figure S2A and Figure S2B, Additional file 3: Figure S3A and Figure S3B, Additional file 4: Figure S4A and Figure S4B, Additional file 5: Figure S5A and Figure S5B, and Additional file 6: Figure S6A and Figure S6B). This also led to a loss of power, which was reduced as compared to the true model for most of the scenarios (Tables 1, 2, 3 and 4, green dots in Additional file 1: Figure S1, Additional file 2: Figure S2, Additional file 3: Figure S3, Additional file 4: Figure S4, Additional file 5: Figure S5 and Additional file 6: Figure S6). Generally, the highest power was observed for the true model. The power for AIC A cannot be interpreted adequately for most of the scenarios due to its increased type I error probabilities. The full model is identical to the true model for β k =β eq , as all covariates are truly associated with the outcome. For β k =β v , the power of the full model was similar to the power of the true model for K=12 and K=24 in our simulation runs, but was slightly lower for simulations with K=36. The strategy AIC B , which appears to have an adequate false positive rate, showed (slightly) lower power than the true model for (almost) all of the scenarios. A slightly decreased power was also observed for the strategy including all covariates that were significantly associated with the outcome (significance). The type I error probability was acceptable for most scenarios with a small or moderate number of potential predictors (K=12 and K=24), but an increased type I error probability was observed for scenarios with many potential predictors (K=36).
Coverage was adequate for most of the models and strategies. For main, the coverage was reduced for some scenarios due to biased estimates. For AIC A , the coverage was under 93% for 52 of the 108 scenarios (48.1%), indicating standard errors for the regression coefficient of interest were underestimated following the variable selection procedure.
In the last rows of Tables 2, 3 and 4, the mean numbers of additionally included covariates are summarized for each method/strategy stratified for the amount of censoring and β k (which determines the number of truly prognostic variables). It was observed that for our settings, the procedure including variables that were significantly associated with the outcome in univariate Cox models selected more variables than the AIC-based methods, and that slightly more variables were chosen with AIC B than with AIC A . For scenarios with β k =β eq , the true and full models were identical by definition. More detailed information on the numbers of covariates included are given in Additional file 7: Table S1, Additional file 8: Table S2 and Additional file 9: Table S3.
The optimization algorithm for numerical maximization of the partial log-likelihood of the Cox regression model for estimating the regression coefficients did not converge for some simulation runs. The problem especially occurred for AIC A . Over all 108,000 simulation runs (108 scenarios × 1,000 runs per scenario), the estimation algorithm did not converge 11 times (0.010%) for main, twice (0.002%) for true, 895 times (0.829%) for AIC A , 27 times (0.025%) for AIC B , three times (0.003%) for significance and no times (0%) for full.
The ultimate goal in individualized or tailored medicine is to find the best treatment for each individual based on the patient's characteristics like age, sex, co-morbidities, disease history and molecular and genetic information, which are often referred to as biomarkers. The existence and detection of a biomarker–treatment interaction can be considered as a requirement for such treatment individualization [2], and consequently an interaction between the biomarker of interest and treatment has to be established in a first step, e.g. by finding statistically significant and clinically relevant interactions based on data from (multiple) randomized clinical trials. Decision rules for treatment selection based on the characteristics of a certain patient have to be investigated and established afterwards, also considering the benefits and costs of the application of a certain treatment strategy for a given patient.
To detect relevant associations and interactions, it is well known that splitting a quantitative variable into different categories, leading to a comparison of treatment effects between different subgroups, will result in a loss of information and will consequently decrease the probability of detecting a true biomarker–treatment interaction [25]. So, using all the quantitative information is recommended for analysis of biomarker–treatment interactions [7]. To estimate a treatment effect in a randomized clinical trial, the inclusion of relevant prognostic variables is recommended [10] to increase the precision of the estimate and consequently the probability of detecting real group differences. For this article, we performed a simulation study to investigate whether the probability of detecting a biomarker–treatment interaction in data derived from a randomized clinical trial can be improved by including further potentially prognostic variables in a Cox regression model for time-to-event data. Different settings for the strength of interaction between the biomarker and the treatment, the correlation between the biomarker of interest and other potential predictors, the strength of association between the predictors and outcome, the number of (potential) further predictors, and the number of events and censored observations were considered. When a biomarker–treatment interaction is assessed using data from a randomized clinical trial, obviously the best choice is to include in the final model all covariates truly associated with the outcome, which was covered by the true model in our simulation study. As this true model often is not known in practice, especially in investigations including molecular or genetic information, more flexible approaches might be needed. So, we also investigated strategies using data-driven variable selection procedures based on AIC [14] or on the results of Cox regression models with single covariates.
In our simulation study, we observed that including the correct prognostic variables leads to an increased probability of detecting a true biomarker–treatment interaction and reduced bias of the estimated interaction effect, with the magnitude of improvement depending on the strength of association between the prognostic variables and the outcome and between the prognostic variables and the biomarker of interest. In contrast, including too many variables per event can lead to the opposite effect and increased probabilities of false positives. This problem is well known for multiple regression models [15, 26]. Our results support the rule of ten, which was proposed for predictive modelling [27], since the type I error probability was increased for the biomarker of interest, even for the true model, when a large number of covariates was considered. The simulation study also revealed that ignoring relevant prognostic factors leads to biased estimates for the biomarker–treatment interaction effect, which has been described for estimating the group effect from a randomized clinical trial using a Cox regression model [11]. Generally, the data-driven selection of prognostic variables by an inclusion procedure based on the AIC after including the main effects of the biomarker of interest, the treatment and their interaction in the model increases the type I error probabilities and reduces the confidence interval coverage. This was not observed in a strategy that selected the relevant prognostic variables in a first step and added the biomarker main effect and the biomarker–treatment interaction afterwards (called AIC B in our article). In our simulated scenarios, the strategy including all covariates that were found to be significantly associated with the outcome performed similarly to that approach. Automated variable selection procedures are criticized in the literature for various reasons (see e.g. [28]). Based on the results of our simulation study, we strongly discourage using an automated variable selection procedure to choose additional prognostic variables after including the biomarker–treatment interaction of interest, as this may lead to unreliable results.
An obvious limitation of our study is that we observed only a moderate number of different scenarios with three correlation structures, three strengths of interaction between the biomarker and treatment, two strengths/structures of association between the additional prognostic variables and treatment, two censoring distributions, three numbers of (potential) prognostic variables, and a fixed number of 500 observations, due to limited time and space. All these aspects influenced the results and other settings may have led to different findings and consequently recommendations. In particular, the number of observed events, which is more important than the total sample size for a time-to-event outcome, was varied only by choosing two different censoring proportions, but it has a major impact on the power of the interaction test. We also investigated only a small number of strategies for inclusion or selection of further covariates based on the AIC and significant associations with the outcome. Other strategies (like backward selection), other criteria (like the Bayesian information criterion [29]) or other procedures for variable selection (like the least absolute shrinkage and selection operator [30]) were not considered. Furthermore, we considered only normally distributed biomarkers and linear associations and interactions in our simulations and fitted Cox regression models assuming linear associations and time-constant effects to our data. Recently introduced methods for estimating non-linear interactions, like local partial likelihood estimation [31], multivariable fractional polynomials for interaction [8] or the modified covariate approach [9], were not investigated.
It has to be considered that in our scenario, only one pre-specified biomarker of interest is assessed. It was identified as being of interest e.g. in an observational study or was found to be relevant for a similar kind of disease. If more than one biomarker is investigated, multiplicity issues arise that have to be adequately considered [32]. When an analysis is an additional analysis to a standard group comparison for a randomized clinical trial, it can only be exploratory in nature. Nevertheless, the method used for statistical analysis should be specified a priori to generate reliable results and avoid problems of data-dredging and selective reporting, and consequently generating unreliable results and increased false-positive rates [33]. Further algorithms or strategies should be used in sensitivity analyses to assess the stability of the observed results. If the investigation of a biomarker–treatment interaction is of major importance for a clinical trial, this should be considered in the design stage and consequently in the sample size calculation.
Based on the results of our simulation study, we recommend considering prognostic covariates in regression models when estimating biomarker–treatment interactions, as the power for detecting true interactions can be increased. However, including too many variables can lead to unreliable results. The choice of variables included should be based on prior information and subject knowledge. Automatic variable selection procedures have to be handled with care.
This work was supported by the German Research Foundation (DFG) and the Technische Universität München within the funding programme Open Access Publishing.
No patient data were used when writing this article. The R code for generating the data sets used in the simulation study, for applying the approaches and strategies described, and for analysing the results obtained in the simulation study can be obtained from the first author upon reasonable request.
BH designed and implemented the simulation study and drafted the manuscript. KU critically reviewed the manuscript for intellectual content. Both authors read and approved the final manuscript.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver(http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.
Additional file 1 Figure S1. Distribution of \(\hat {\beta }_{T\times B}\) for scenarios with K=12, β k = β eq , and low censoring (A) or high censoring (B) for no biomarker–treatment interaction (βT×B= ln(1.0)=0, top rows) or qualitative biomarker–treatment interaction (βT×B= ln(1.33)=0.285, bottom rows). Results for different correlation structures are shown in separate columns. The dashed red lines indicate the true value of βT×B, the blue triangles represent the observed confidence interval coverages, the green dots the observed probability for a type I error (A) or estimated power (B). (PDF 20 kb)
Additional file 2 Figure S2. Distribution of \(\hat {\beta }_{T\times B}\) for scenarios with K=12, β k = β v , and low censoring (A) or high censoring (B) for no biomarker–treatment interaction (βT×B= ln(1.0)=0, top rows) or qualitative biomarker–treatment interaction (βT×B= ln(1.33)=0.285, bottom rows). Results for different correlation structures are shown in separate columns. The dashed red lines indicate the true value of βT×B, the blue triangles represent the observed confidence interval coverages, the green dots the observed probability for a type I error (A) or estimated power (B). (PDF 20 kb)
Additional file 7 Table S1. Mean number of additionally included prognostic variables for all scenarios with K=12. (PDF 68 kb)
Institute of Medical Informatics, Statistics and Epidemiology, Technical University of Munich, Ismaninger Str. 22, Munich, 81675, Germany
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Guidance for footbridge design: a new simplified method for the accurate evaluation of the structural response in serviceability conditions
Caterina Ramos-Moreno ORCID: orcid.org/0000-0002-6916-55461,
Ana M. Ruiz-Teran2 &
Peter J. Stafford2
This paper proposes a simplified hand-calculation methodology that permits a fast response assessment (both in vertical and lateral direction) under different pedestrian scenarios. This simplified method has the same accuracy than that of very sophisticated numerical nonlinear finite element models including pedestrian inter-variability, interaction among pedestrians in flows, and pedestrian-structure interaction. The method can capture the effects of pedestrian loads in and out of resonance. This methodology is based on a new, and experimentally contrasted, stochastic pedestrian load model derived by the authors implementing a multi-disciplinary state-of-the-art research, and on a large set of sophisticated finite element analyses.There is a significant gap in the literature available for bridge designers. Some current codes do not indicate how the performance for serviceability limit-states should be assessed, in particular for lateral direction. Others define methods that are not based on the latest research in this field and that require the use of dynamic structural analysis software. A very sophisticated load model, such as that described above, and recently proposed by the authors, may not be accessible for most of the design offices, due to time and software constraints. However, an accurate assessment of the serviceability limit state of vibrations during the design stages is paramount. This paper aims to provide designers with an additional simple tool for both preliminary and detailed design for the most typical structural configurations.First, the paper presents the methodology, followed by an evaluation of the impact of its simplifications on the response appraisal. Second, the paper evaluates the validity of the methodology by comparing responses predicted by the method to those experimentally measured at real footbridges. Finally, the paper includes a parametric analysis defining the maximum accelerations expected from pedestrian streams crossing multiple footbridges. This parametric analysis considers different variables such as section type, structural material, span length and traffic-flow characteristics, and shows the sensitivity of the serviceability response to traffic-flow characteristics and span length in particular.
Footbridge design has evolved rapidly in recent decades, leading designers to use new and lighter materials (Firth and Cooper 2002) and structural arrangements that allow for the use of less massive cross sections and longer spans. However, it is well-known that this evolution in design has produced bridges where serviceability requirements were not satisfied (e.g., the London Millennium Bridge, Dallard et al. 2001, or the Solférino Bridge, currently called Léopold-Sédar-Senghor, in Paris). Nonetheless, this progression in design is not the sole cause of these serviceability issues. Detailed studies of these events have demonstrated that assumptions made to represent serviceability events with pedestrian flows were not sufficiently realistic.
Designers need to be able to predict, from the very early stages of their designs, whether or not their proposals can satisfy the serviceability requirements (by comparing maximum predicted accelerations of the structure in service to limits of acceptance, established according to human perception of movement). Most structural design codes, e.g., EN 1991-2:2003 (2003) & AASHTO (2009), advise that serviceability limit-states may become critical if structural frequencies are within the ranges of common pedestrian step frequencies (around 1.0-3.0 Hz for vertical vibrations, and 0.5-1.5 Hz for lateral vibrations). However, many of these codes do not define the procedure that should be adopted to perform these analyses, e.g., AASHTO (2009) & CAN/CSA-S6-06 (2006). For instance, the dynamic load models required for verifications according to Eurocode 1 (2003) are not defined in the main body of the document, and are instead left to be defined in each national annex. In some cases, additional specifications have been defined (e.g., NA to BS EN 1991-2:2003, 2008, for the UK), but this is not always the case. In addition, these code methods for serviceability evaluation have two main disadvantages: on one hand, they are based on load models that do not include the latest research advances related to this field and, on the other hand, they are computationally demanding (not being useful for preliminary design). Related to the first point, many codes or design guidelines still apply several assumptions similar to those proposed in early codes, such as BS 5400 (1978). This is undoubtedly due to two facts: (1) the current state-of-the-art of all the aspects related to pedestrian loading is still very dispersed within the literature; and (2) a modern methodology including the latest research advances and using the latest cutting-edge analyses to accurately represent reality has not been published yet.
With the aim of providing a tool to perform this serviceability assessment, and based on the latest research, this paper proposes a very simple method to accurately obtain the maximum vertical and lateral accelerations expected in a footbridge due to pedestrian actions. The method allows the designer to verify the serviceability limit state of vibrations due to traffic load without performing the corresponding, sophisticated, time-consuming, and computationally-demanding analyses. The simple method presented here is underpinned by a comprehensive, accurate, and realistic description of pedestrian loads defined by the authors, that are well beyond those conventionally adopted in current practise (including intra-variability, inter-variability, pedestrian-pedestrian interactions and pedestrian-structure interactions among others). This pedestrian load model defined by the authors is based on existing, but very dispersed research works that have been published over recent years. The method presented in this paper is therefore very useful both for detailed and preliminary design.
The method is applicable to footbridges of one-to-three spans, with constant depth. However, it can easily be adopted for footbridges with a larger number of spans, non-constant depth, or more sophisticated geometry. The response is obtained based on the structural features of the footbridge (such as the geometry of the section, and material properties) and traffic-flow characteristics (such as the travel purpose, and pedestrian density). An assessment of the performance of the method is included.
The soundness of the proposed method is appraised by comparing the serviceability response produced by the method for several real footbridges to those experimentally recorded at the same footbridges. Finally, based on the method introduced, the paper evaluates the performance in serviceability in the vertical or lateral directions of a wide range of footbridges with one or two spans. These evaluations are defined for structures designed with different material and geometrical configurations.
Conceptual basis for the proposed method
The overall method presented over the course of the following sections is based upon a very simple conceptual model. The general idea is that the response of footbridges under serviceability conditions is essentially a reflection of the resonant characteristics that arise from the interaction of the structural properties and the pedestrian loading characteristics. A physically-based equation is therefore developed that identifies the key frequency ranges that will be most important for the acceleration response of simple bridges in service conditions. This equation is parameterised using very common and fundamental geometric and material properties and can therefore be applied to a very wide range of structural configurations.
This governing equation is considered in the actual appraisal of a bridge response under pedestrian actions taking into consideration the Pi Theorem of Dimensional Analysis (Buckingham 1914). This theorem ensures that a function that describes a physical problem f(x)=f(x1,x2,...,xn)=0, with n arguments that are defined with respect to q fundamental units U1,U2,...,Uq, can be represented as g(π1,π2,...,πk)=0, where k<n. In this second function k=n−r, and r is the rank of the dimensional matrix n×q. The π terms are independent and dimensionless products formed from the original n variables x1,...,xn.
For our particular case, f(x)=0 is the function that predicts the structural response, which arises from the equation of motion of a dynamic system. For the case of vertical accelerations in a simply-supported bridge, this is a function of both the structural properties (the structural mass, the span length, the material damping ratio, and the flexural stiffness) and the pedestrian characteristics (the pedestrian mass, their step frequency, and their step length) and provides a prediction of the expected maximum acceleration of the bridge. Therefore, there are seven input variables used to predict one response variable (eight variables in total in implicit function f(x), i.e., n=8). However, when the lateral accelerations are considered two additional parameters are required and these are the pedestrian step width and the height of the pedestrian, meaning that in the lateral case n=10.
This physical problem involves the three fundamental units of mass, length and time (which we define as U1, U2 and U3). To identify the required number of nondimensional parameters it is necessary to represent each of our 8 (or 10) variables in terms of these three basic units through an n×q matrix. The difference between the total number of variables and the rank of this matrix then defines the required number of nondimensional parameters. In the present case, with the fundamental units defined as above, the rank of this matrix will be r=3 and will imply that five (or seven) π terms are required. Four (or six) of these terms relate to the input variables while one term is associated with the response, and is the normalised acceleration a/g, with g being the gravitational acceleration of the Earth - considered to be a constant.
The dimensionless parameters are then chosen to be the ratio of the frequency of the structure to the pedestrian step frequency, π1=fs/fp, the ratio of the structural mass to the pedestrian mass, π2=ms/mp, the ratio of the pedestrian step length and the span length π3=sl/L, and the damping ratio, π4=ζ. For the lateral case π5 and π6 are also used and are set as pedestrian step widths and heights normalized by corresponding values representative of the target population.
Considering a range of values for the first dimensionless term (fs/fp) and specific representative values for the remaining dimensionless terms, we then generate a very large number of results from advanced numerical analyses using sophisticated finite-element models, with realistic descriptions of the pedestrian loading that account for the most pertinent features of the loading in both vertical and lateral directions. These numerical results are then employed to determine basic or reference vertical and lateral accelerations (listed in the Additional files 1 and 4 of the paper), that correspond to the particular structural and pedestrian characteristics used for the sophisticated numerical analyses. Given the scope of the present study, the basic accelerations are defined directly in units of m/s2, computed from a/g using g=9.81 m/s2.
Naturally, however, we need to ensure that the method works for a broad range of configurations and not just the reference cases obtained for particular values of π2, π3, π4, π5 and π6. We therefore derive a series of adjustment factors that act to scale these basic accelerations in order to account for departures from the original values of π2,…,6.
In the sections that follow the approach to determine these basic accelerations is first defined before explaining and presenting the various adjustment factors that can be applied to adapt these basic values to be appropriate for another structural and loading configuration of interest.
Realistic description of the pedestrian loading
When walking, individuals introduce vertical and horizontal forces on the surface upon which they are supported (see Fig. 1a). The definition of the forces exerted is complex and involves many parameters. However, pedestrian loading is currently defined in many codes and guidelines in quite a simplistic manner. The main simplifications in the load description are related to the step frequency and the load amplitude.
a Lateral and vertical loading introduced by a pedestrian; b Amplitude of the pedestrian vertical load Fp,v/wp versus time according to step frequency magnitude (wp is the pedestrian weight); c Fundamental mechanics of lateral equilibrium defining lateral pedestrian loads; d Amplitude of the normalised pedestrian lateral load Fp,l/wp versus time according to step frequency magnitude
Current methods are based on descriptions where all the individuals walk with frequencies with values of approximately 1.8 to 2.0 Hz (see e.g., Pedersen and Frier, 2010). However, research has shown that the distribution of pedestrian step frequencies of a crowd is not fixed and depends on many parameters. Bertram and Ruina (2001) showed a clear correlation between step frequency and pedestrian speed, based on experimental tests, while Weidmann (1993) showed that pedestrian speed is chosen according to aspects such age, anthropometric characteristics, aim of the journey, and pedestrian density, among others. Accordingly, an accurate description of the pedestrian load needs to take these issues into consideration. Therefore, the authors have proposed probabilistic relationships (Ramos Moreno et al. 2020) to define pedestrian target speed and frequency. The target speed of a pedestrian is defined based on characteristics, such as age and height, the aim of journey (considering three different categories business: that includes journeys related to work for pedestrians with an age range between 20 and 65 years old, but excludes their commute to and from home; commuter: which defines journeys between work/study place and home of pedestrians, regardless their age; and, leisure: that includes shopping, tourist or leisure-related journeys), and the density of the crowd in which they walk (number of pedestrians per unit area). The pedestrian frequency is obtained based on the pedestrian speed. By means of these fundamental relationships, based on the UK and Western Europe population data, the distribution of step frequencies according to aim and density of the flow have been defined (Fig. 2).
Step frequencies adopted by population (based on the UK and Western Europe populations) according to the aim of the journey and the density of the flow, where μ is the mean and σ the standard deviation representing the normally distributed values
In codes and guidelines, load amplitudes are usually described by means of Fourier series with a single term (a sinusoidal function) representing the load transmitted by both feet in time. In these methods, the maximum load amplitude is a function of the pedestrian weight and is independent of the gait, which is not realistic. In addition, several of these methods use the same mathematical definition for both lateral and vertical loads, which does not reproduce the nonlinear pedestrian-structure interaction that has been detected in real structures for lateral movement (e.g., Dallard et al. 2001).
The work presented in this paper has been obtained using a realistic and experimentally contrasted description of the pedestrian loading, defining each foot load individually as illustrated in Fig. 1a. The model describing vertical and lateral loads individually transmitted by each foot of a pedestrian has been derived by the authors of this paper and an extensive description can be found in Ramos Moreno et al. (2020). The model considers pedestrian inter-variability, interaction among pedestrians in flows and pedestrian-structure interaction among others. The functions used to define the vertical load amplitude (Fig. 1b) are based on the work published by Butz et al. (2008). Their definition is exclusively based on the step frequency (which has been linked herein to those parameters considered in Fig. 2). The lateral loads are based on a model that reproduces the movement of the centre of mass of each pedestrian (Macdonald 2009). This approach has been adopted primarily due to its foundation in fundamental mechanics (Fig. 1d). This model for lateral loads has been shown to compare relatively well with empirical data, although studies of lateral loads are characterised by a great deal of variability and inconsistency. This definition is also convenient for considering the nonlinearity attributed to these lateral loads, since they seem to be related to the lateral movement perceived by the individual (Dallard et al. 2001). The longitudinal loading introduced by the pedestrian has been neglected, as longitudinal vibrations are typically not important when assessing performance in serviceability.
Vertical and lateral structural frequencies
For a simply-supported beam of constant geometrical and mechanical characteristics throughout the length, the vertical, fv,n, and lateral, fl,n, vibration frequencies associated with the nth vertical and lateral vibrational modes are given by Eq. 1:
$$\begin{array}{@{}rcl@{}} f_{y,n} = \frac{n^{2} \pi}{2 L^{2}}\sqrt{\frac{E I_{y}}{\rho A}}, \quad y \in \{ v, l\} \end{array} $$
where L is the span length, E is Young's modulus, ρ is the material density, A is the cross-sectional area of the section, and Ix, x∈{v,l}, are the second moments of area in vertical and lateral direction of the section, respectively.
The ratio between the second moment of area, in the vertical and lateral directions, and the cross sectional area, can be defined by Eqs. 2 and 3:
$$\begin{array}{@{}rcl@{}} \frac{I_{v}}{A} = \eta_{v} \alpha_{v} (1-\alpha_{v}) h^{2} \end{array} $$
$$\begin{array}{@{}rcl@{}} {\kern-3.4pt}\frac{I_{l}}{A} = \eta_{l} \alpha_{l} (1-\alpha_{l}) b^{2} \end{array} $$
where αv is the ratio between the vertical distance from the centroid of the section to the top extreme fibre and the vertical depth of the section h; αl is the ratio between the horizontal distance from the centroid of the section to the closest lateral extreme fibre and the width of the section b; ηv is the ratio between the depth of the central kern and the depth of the section h; and ηl is the ratio between the width of the central kern and the width of the structural section b.
By substituting Eqs. 2 and 3 into Eq. 1, the following expressions are obtained:
$$\begin{array}{@{}rcl@{}} {\kern11pt}f_{v,n} = \frac{n^{2} \pi}{2 L }\sqrt{\frac{E}{\rho} \eta_{v} \alpha_{v} (1-\alpha_{v}) \left(\frac{h}{L}\right)^{2}} \end{array} $$
$$\begin{array}{@{}rcl@{}} {}f_{l,n} = \frac{n^{2} \pi}{2 L^{2} }\sqrt{\frac{E}{\rho} \eta_{l} \alpha_{l} (1-\alpha_{l}) b^{2}} \end{array} $$
The parameters ηv, ηl, αv and αl take reasonably constant values for each section type. Figure 3 provides values of these parameters for conventional sections that can be used in footbridge design (values derived from existing footbridges and from values typically used by bridge engineers).
Summary of geometric properties, and usual materials and span ranges for different footbridge sections. The slab defining the decking contributes to the longitudinal stiffness in sections S.1 to S.5, whereas in sections S.6-S.9 it only contributes to the transverse stiffness
In preliminary design, the vertical and lateral frequencies of the structure can be directly estimated from Eqs. 4 and 5, whilst in detailed design they can be estimated from Eq. 1 or directly obtained from finite element (FE) models. For lateral frequencies, Eq. 1 describes the vibration modes of bridges where bearings allow the rotation with respect to the line described by the lateral centre of gravity of the section (for other bearing dispositions FE models will provide a more accurate evaluation of the modal vibration).
Resonance parameters
The main parameters that control the vertical and lateral response of a footbridge under pedestrian loading are the ratios between the vertical or lateral structural frequencies and the corresponding pedestrian frequencies. The ratio rv,n between the nth vertical structural frequency and the pedestrian vertical frequency fp,v (or simply fp, hereafter), is given by Eq. 6, in which ϕs,n is an adjustment factor to account for cases that differ from a simply-supported bridge, and ρ∗ differs from ρ as it also considers the additional non-structural mass. Note that the resonance parameters being discussed here are equivalent to the first nondimensional parameter π1 presented earlier in the case that ϕs,n=1 and ρ∗=ρ.
$$ {\kern55pt}r_{v,n} = \frac{f_{v,n}}{f_{p,v}} \phi_{s,n} = \frac{n^{2} \pi}{2 L f_{p}}\sqrt{\frac{E}{\rho^{*}} \eta_{v} \alpha_{v} (1-\alpha_{v}) \left(\frac{h}{L}\right)^{2}}\phi_{s,n} $$
When walking, consecutive vertical pedestrian loads have the same sign (downwards) and are characterised by a frequency fp,v=fp (step frequency). For lateral loads, consecutive steps have opposite signs, corresponding to loads whose frequency is half the step frequency (fp,l=fp/2). The ratio between the nth lateral structural frequency and the lateral pedestrian frequency fp,l, is therefore denoted by rl,n as in Eq. 7.
$$ {\kern60pt}r_{l,n} = \frac{f_{l,n}}{f_{p,l}}\phi_{s,n} = \frac{n^{2} \pi}{L^{2} f_{p}}\sqrt{\frac{E}{\rho^{*}} \eta_{l} \alpha_{l} (1-\alpha_{l}) b^{2}} \phi_{s,n} $$
When the parameters rv,n or rl,n are equal to 1, it means that the vertical or lateral pedestrian loading is inducing resonance in the structure, as the pedestrian frequency is equal to the nth-mode structural frequency. When these parameters are equal to 2, it means that the pedestrian loading reinforces the displacements in the nth mode in every other cycle. In general, when the parameters rv,n or rl,n are equal to a natural number p, the pedestrian loading will reinforce any existing structural movement at every p cycles of the nth-mode.
In the above expressions, ρ∗ is the effective material density when the non-structural mass per unit length m (accounting for the pavement, parapets, and handrail weight, as well as the mass of a pedestrian stream) is also considered:
$$ {\kern110pt}\rho^{*} = \rho \cdot \left(1 + \frac{m}{\rho \cdot A} \right) $$
The parameter ϕs,n is related to the number of spans and the geometrical arrangement. For simply-supported spans ϕs,n=1. For two and three span beams, this factor can be directly obtained from Fig. 4.
Amplitude of ϕs,n, according to mode, n, and number of spans
The pedestrian frequency fp should be defined based on the density of the crowd flow and the type of use expected (Fig. 2), considering that pedestrians in a flow have an effect on others. For simple calculations, the mean values of frequencies should be considered (given by the continuous line in Fig. 2). For more detailed calculations, a wider range of pedestrian frequencies should be considered (for this purpose, additional fractiles of the frequency distribution are shown in Fig. 2).
Basic vertical and lateral accelerations
The basic accelerations \(\left (a_{y,n}^{b} \quad y \in \left \{v,l\right \}\right)\) linked to the nth mode of vibration are those associated with the passage of a single pedestrian crossing a simply-supported bridge. As mentioned in previous sections, these basic accelerations have been obtained for broad ranges of the resonance parameters and fixed values of the remaining dimensionless parameters (values are indicated in following sections).
The values of the basic accelerations (as well as accelerations obtained with any other value of the nondimensional parameters) have been obtained representing the beam structures as finite-element models with one-dimensional elements where the nodal response caused by the dynamic load created by the pedestrians crossing the structure has been appraised implementing a step-by-step time-history analysis that accounts for pedestrian-structure interaction. Results obtained with the developed numerical model have been validated with Abaqus (2013).
The basic vertical acceleration linked to the nth mode of vibration \(\left (a_{v,n}^{b}\right)\) is obtained as a function of the vertical resonance parameter (rv,n) and the pedestrian frequency fp. The different load shape, and therefore impulse, for different step frequencies explains the different accelerations according to fp (the vertical load model derived by the authors to develop this method describes load shapes that change according to the pedestrian step frequency, see Ramos et al., 2020). The basic lateral acceleration linked to the nth mode of vibration \(\left (a_{l,n}^{b}\right)\) is obtained as a function of the lateral resonance parameter (rl,n), but is independent of the step frequency (lateral loads have the same shape and impulse regardless fp, given particular values of step width and pedestrian height, see Ramos et al., 2020). The values of these functions are listed in the Additional files 1 and 4 of the paper. Intermediate values not listed in this table can be obtained by linear interpolation. Basic vertical and lateral accelerations for vertical and lateral resonance parameters not included in the table can be assumed equal to zero.
Maximum vertical and lateral accelerations caused by a single pedestrian
The maximum vertical (av) and lateral (al) accelerations caused by one pedestrian are given by the expressions in Eqs. 9 and 10. These maximum accelerations are the maximum accelerations calculated from the consideration of the first four modes included in the analyses, i.e., n=1,2,3,4. This structural response appraisal has been considered a reliable evaluation of the total response since results show that in each case the response is largely dominated by a single vibration mode. Therefore, the results obtained using this simple approach would not differ much from more elaborate modal combination rules.
$$ {\kern100pt}a_{v} = \max_{n} \left(a_{v,n}^{b} \phi_{pm} \phi_{sl} \phi_{d} \phi_{sm} \right) $$
$$ {\kern100pt}a_{l} = \max_{n} \left(a_{l,n}^{b} \phi_{pm} \phi_{sw} \phi_{ph} \phi_{sl} \phi_{d} \phi_{sm} \right) $$
where ϕpm and ϕph are factors related to the mass and height of the pedestrian; ϕsl and ϕsw are factors related the length and width of the pedestrian step; ϕd is a factor related to the damping of the structure; and ϕsm is a factor related to the mass of the structure.
Factor related to the pedestrian mass (ϕ pm)
Codes and guidelines usually consider a standard pedestrian weight of 700 N (a standard pedestrian mass mp of 71.36 kg). The basic accelerations have been obtained for a footbridge-to-pedestrian mass ratio of 7440. For ratios different to this particular value, the maximum acceleration can be obtained from the basic acceleration by using the factor ϕpm, which is defined by Eq. 11.
$$ {\kern120pt}\phi_{pm} = 7440 \frac{m_{p}}{\rho^{*} A L} $$
Factor related to the pedestrian step length (ϕ sl)
The pedestrian step length depends on various parameters, such as speed, gender, height, etc. Values reported by Pachi and Ji (2005) suggest that a value of 0.70 m can be considered as representative. The basic accelerations have been obtained for a ratio between the pedestrian step and the span length of 0.02. For different ratios, the maximum acceleration can be obtained from the basic acceleration by using the factor ϕsl, which is defined by Eq. 12 and the coefficients presented in Tables 1 and 2.
$$ {\kern80pt}\phi_{sl} = \exp \left(B_{1} \times |{r_{y,n}^{2} - B_{2}^{2}}| \right), \quad y \in \{v,l\} $$
Table 1 Coefficients for obtaining ϕsl in Eq. 12 for vertical response (y=v), where x is the ratio between the pedestrian step and the span length, i is a natural number greater than 2
Table 2 Coefficients for obtaining ϕsl in Eq. 12 for horizontal response (y=l), where x is the ratio between the pedestrian step and the span length, i is a natural number greater than 1, and j=2i−1
In deriving these factors, our results have indicated that the maximum accelerations registered are sensitive to the value of this parameter only near resonance (i.e., when the resonance parameters take values very close to a natural number) and only for the first and second vibrational modes. Therefore, this factor ϕsl needs to be assessed when the vertical or lateral resonance parameters linked to the first (rv,1, rl,1) or the second mode (rv,2, rl,2) are in the following intervals: [ i±0.025], where i is a natural number (i.e., i=1,2,3,…). For all other cases, ϕsl=1.
Factor related to the pedestrian step width (ϕ sw)
The pedestrian step width varies significantly among different pedestrians as well as for a given pedestrian while they walk. Despite this, a value of ws=0.10 m is taken as being representative for the UK and Western Europe populations (Ramos Moreno et al. 2020). This value represents the total transverse distance between feet, in units of metres. For different pedestrian step widths, the maximum acceleration can be obtained from the basic acceleration by using the factor ϕsw, which is defined by Eq. 13.
$$ {\kern135pt}\phi_{sw}= \frac{w_{s}}{0.10} $$
Factor related to the pedestrian height (ϕ ph)
The basic accelerations have been obtained using a distribution of height suitable for the UK and Western Europe populations (the mean value is approximatly 1.70 m). For populations with similar height distributions to those of the UK and Western Europe, this factor should be considered equal to 1, otherwise the maximum acceleration can be obtained from the basic acceleration by using the factor ϕph, which is defined by Eq. 14, in which hp is the mean height of the target population in units of metres.
$$ {\kern132pt}\phi_{ph}= \frac{1.70}{h_{p}} $$
Factor related to the structural damping (ϕ d)
Due to its different causes, damping is a difficult parameter to appraise. Nonetheless, several documents (Charles et al., 2006, Bachmann et al., 2006, and NA to BS EN 1991-2:2003, 2008) have tried to quantify this value for traffic loading scenarios according to the material utilised in the structure. Generally suggested values for reinforced concrete structures are: 0.8−1.5%, prestressed concrete: 0.5−1.0%, composite sections (steel and concrete): 0.3−0.6%, steel: 0.2−0.5% and timber structures: 1.0−1.5%.
The basic accelerations have been obtained considering a damping ratio of ζ=0.5%. For different damping ratios, the maximum acceleration can be obtained from the basic acceleration by using the factor ϕd, which is defined by Eq. 15 and Tables 3 and 4. Results show that the maximum accelerations registered are sensitive to the value of this parameter only near resonance (i.e., when the resonance parameters take values very close to a natural number) and for the first and second vibrational modes. Therefore, this factor ϕd needs to be assessed when the vertical or lateral resonance parameters linked to the first (rv,1,rl,1) or the second mode (rv,2, rl,2) are in the following intervals: [ i±0.05], where i is a natural number (i.e., i=1,2,3,4,…). For the rest of the cases, ϕd=1. The evaluation provided by this factor is valid for values of the damping ratio in the interval 0.002≤ζ≤0.025.
$$ {\kern80pt}\phi_{d} = \exp \left(C_{1} \times |{r_{y,n}^{2} - C_{2}^{2}}| \right), \quad y \in \{v,l\} $$
Table 3 Coefficients of ϕd for Eq. 15 and vertical response
Table 4 Coefficients of ϕd for Eq. 15 and horizontal response, where j=2i−1 and i is a natural number
Factor related to the structural mass (ϕ sm)
The parameter ϕsm is related to the number of spans and modal masses. The basic accelerations have been obtained for simply supported beams, where ϕsm=1. For two and three span beams, this factor can be obtained from Fig. 5.
Values for ϕsm, according to mode and arrangement of spans
Vertical and lateral accelerations caused by groups of pedestrians and continuous streams of pedestrians
The issue of estimating the acceleration demands from groups or streams of pedestrians is significantly more complex than that of estimating the demands from a single pedestrian. When considering a single pedestrian, it is possible, and meaningful, to allocate the properties of that pedestrian so that they reflect the population being considered. However, when considering multiple pedestrians it has been shown that it is important to account for the significant inter-subject and intra-subject variability (Ramos et al., 2020). In Ramos et al. (2020), these issues are considered in a probabilistic manner, but the procedure is far more involved than what is desirable for preliminary design. For that reason, in what follows, a simple approach for obtaining first-order estimates of the demands from groups and streams is proposed.
Group of pedestrians
Under the assumption that all pedestrians within a group are identical, walk in a synchronised manner (same step frequency), and apply their load to the same point on the structure, then the vertical accelerations induced by this group will differ from those of a single pedestrian by a linear factor equal to the group size. That is, the maximum vertical acceleration would be defined as in Eq. 16.
$$ {\kern140pt}a_{v,g} = N a_{v} $$
It should be noted that, when evaluating Eq. 16, the acceleration associated with the single pedestrian av should be computed using a value of ρ∗ that accounts for the increased mass of pedestrians associated with the group (instead of the mass of a single pedestrian). That is, the effect of considering the group is not simply a linear scaling of the accelerations, but also accounts for a small shift in the resonance parameter. Some authors have observed that pedestrians modify the energy dissipation capacity of the structure (e.g., Brownjohn et al., 2004) in vertical or lateral direction, however the effect is not yet clear or quantified. Thus, this effect is disregarded in the implementation of this simple methodology.
Despite the fact that the assumptions underpinning the above equation are often violated, we propose the use of this very simple expression during preliminary design.
Once one considers lateral accelerations, it is important to also consider the influence that pedestrian-structure interaction can have upon the responses from a group of pedestrians. In an event where pedestrians notice transverse movements of the deck, these may firstly adapt their gait to ensure that they do not loose the equilibrium. This widening of their step gait leads to an increase of the transverse loads introduced by pedestrians (further details of this nonlinear pedestrian-structure interaction can be found in Ramos et al., 2020). Larger lateral loads lead to a larger lateral footbridge response. For that reason, the expression we propose for evaluating the accelerations in the lateral direction is given by Eq. 17 in which an additional term Nnl, which accounts for these nonlinear interaction effects, is introduced. In Eq. 17, the lateral acceleration al should be computed accounting for the mass of the entire group (that is, the group mass should influence ρ∗ and ϕpm).
$$ {\kern75pt}a_{l,g} = (N + N_{nl}) a_{l} = \left(N + \frac{\beta} {\phi_{pm}}\right) a_{l} $$
For a single pedestrian, the lateral force that they impart upon the bridge depends upon their own relative lateral acceleration with respect to the bridge as well as the global acceleration of the bridge (Ramos Moreno et al. 2020). For this single pedestrian case, these global accelerations are the sole result of this same pedestrian. However, when a group is considered, the global accelerations of the bridge result from the effects of each pedestrian and so the effects of this interaction do not scale linearly with the group size.
In order to account for this nonlinear interaction within a simple expression appropriate for preliminary design, an exercise was conducted in which the lateral accelerations were obtained using a 'scaled pedestrian' (the pedestrian mass is set to be N times larger than the nominal single-pedestrian value, and the other assumptions regarding group behaviour used for the vertical case are retained). The accelerations obtained from this 'scaled pedestrian' are then compared with those found from an individual pedestrian in order to define an appropriate equivalent number of pedestrians (N+Nnl) that account for the nonlinear interaction effects.
The term Nnl is represented by the ratio β/ϕpm in order to enable the nonlinear interaction effect to be estimated for cases where the value adopted for ϕpm differs from that used to derive the values of Nnl. These nonlinear effects have only been found to be significant in the case that the resonance parameter rl,1 falls within two limited ranges of [0.95,1.05] and [2.9,3.1] or when the resulting peak lateral acceleration is larger than 0.15 m/s2. In each of these two cases, the factor β is found to be a function of the product of the group size and ϕpm, as shown in Fig. 6.
Amplification factor β for lateral response, x=Nϕpm
In Fig. 6, β = β1 is defined from Eq. 18, corresponding to the case that 0.95≤rl,1≤1.05, and β=β2 is relevant for 2.9≤rl,1≤3.1 or when the peak acceleration is larger than 0.15 m/s2 and is defined in Eq. 19. For any other value of the resonance parameter or when the total peak lateral acceleration is smaller or equal to 0.15 m/s2, the response is assumed to be linearly proportional to the group size (β=0).
$$ {\kern45pt}\beta_{1} = 1.5 \times 10^{-4} x^{3} - 4.5 \times 10^{-3} x^{2} + 6 \times 10^{-2} x - 0.15 $$
$$ {\kern45pt}\beta_{2} = 5.0 \times 10^{-4} x^{2} - 3.0 \times 10^{-3} x + 7 \times 10^{-3} $$
Continuous streams of pedestrians
In order to enable designers to obtain estimates of the response of bridges under streams of pedestrians we couple the β factors derived as part of this study with an existing approach advocated by NA to BS EN 1991-2:2003 (2008) and Setra (2006). The accuracy or validity of this approach was not assessed within the present study and a more elaborate approach to estimating the response due to streams of pedestrians is rather provided in Ramos Moreno et al. (2020).
The maximum vertical (av,s) and lateral (al,s) accelerations of a stream of pedestrians can be obtained as follows:
$$ {\kern75pt}a_{v,s} = N_{eq} \cdot \frac{a_{v}}{0.6} $$
$$ {\kern75pt}a_{l,s} = (N_{eq} + N_{nl}) \frac{a_{l}}{0.6} = \left(N_{eq} + \frac{\beta} {\phi_{pm}}\right) \frac{a_{l}}{0.6} $$
where Neq is a number of "equivalent" pedestrians and can be adopted from Setra (2006) or NA to BS EN 1991-2:2003 (2008). The guideline SETRA (2006) defines this equivalence according to the density: for sparse or dense crowds (ped/m 2≤0.8) \(N_{eq} = 10.8 \sqrt {\zeta N}\) and for very dense crowds (ped/m 2>0.8) \(N_{eq} = 1.85 \sqrt {N}\), where ζ is the damping ratio and N is the total number of pedestrians on the structure simultaneously. The factor 1/0.6 is introduced to account for the fact that the flow is continuous as opposed to the event of a single pedestrian, and is taken from Grundmann et al. (1993).
The magnitude of the nonlinear factor β for lateral loads depends on the equivalent number of pedestrians in a stream and is given again by Fig. 6, with x=Neqϕpm.
It is worth highlighting that the load model considered for lateral loads is capable of reproducing the initial interaction that occurs when pedestrians sense a slight movement of the platform, but not an actual change of gait to adapt themselves better to the movement (named synchronisation, which would be a second phase of the interaction that apparently may take place between pedestrians and a platform).
Verification of the methodology prediction of serviceability response
In this section, the efficiency of the proposed methodology is assessed by comparing movements caused by pedestrians at real footbridges (experimentally recorded) with those predicted by the method. The similarity of the results highlights the competence of the method despite its simplicity.
Footbridge over hringbraut, (Reykjavik, Iceland)
The structure corresponds to a footbridge located in Reykjavik consisting of a postensioned concrete girder with 8 spans of longitudes between 15.4 and 27.1 m. The transverse section is a slab with a width of 3.2 m (further details can be found in Gudmundsson et al., 2008).
Experimental measurements showed that the structure has a vertical frequency of 2.34 Hz and a damping ratio of value 0.006. Tests of the concrete mix used demonstrated that the concrete Young's modulus had a value of 45 GPa instead of 33.5 GPa required in the project.
The proposed method is implemented considering only the three central longest spans with lengths of 15.5 + 27.1 + 15.5 m (in reality these have lengths of 20.6 + 27.1 + 23.6 m). The length of the side spans results from an average value of the individual real lengths multiplied by 0.7 \(\left (\frac {20.6+23.6}{2} \cdot 0.7\right)\). This factor 0.7 is applied to take into account that these have side spans that restrict their vibration (see Fig. 5). With these equivalent side spans, the three span footbridge assessed with the methodology is expected to have a dynamic behaviour similar to that of the real footbridge. This span disposition, the theoretical Young's modulus (considered in the project) and an additional mass for the balustrades, yields a vertical mode frequency of 2.06 Hz and if the real Young's modulus is considered the predicted vertical frequency is 2.38 Hz.
A pedestrian of 800 N of weight walking on the bridge with vertical frequency 2.06 Hz at a step frequency of 2.06 Hz generates a peak vertical response of 0.37 m/s2 and two pedestrians of the same weight 0.74 m/s2. According to the measurements, at the real structure single pedestrians walking at a constant step frequency near resonance generated peak responses near 0.38 m/s2 (except one case where peaks of 0.46 m/s2) whereas two pedestrians caused peak accelerations of 0.71 m/s2. The values measured in real tests and those predicted by this method are practically the same, with differences below 5% (Table 5).
Table 5 Comparison of serviceability response experimentally recorded and serviceability response predicted by the methodology at a footbridge over Hringbraut, (Reykjavik, Iceland), where ped. corresponds to pedestrian
Aldeias footbridge, (Gouveia, Portugal)
The footbridge consists of a steel box girder of variable depth and deck width of 1.5m that spans a total distance of 57.8 m through three spans of longitude 17.7 + 30.0 + 10.1 m (further details can be found in Alves et al., 2008).
Designers of the bridge predicted numerically vertical modes at frequencies 3.13 and 4.50 Hz, however dynamic tests at the structure once it was finished described vertical modes at frequencies 3.68 and 5.16 Hz. For this footbridge, the proposed method is implemented considering that both side spans have the same length (13.9m). This span disposition (together with the masses of the deck surfacing, the handrails and protection panels) describes a vertical mode at frequency 3.64 Hz.
According to the proposed methodology, a pedestrian of 700 N of weight walking on the bridge at a step frequency of 1.80 Hz generates peak accelerations of 0.39 m/s2 if the damping ratio considered is 0.006 (value commonly considered for steel box girders) or 0.25 m/s2 if the magnitude of the damping ratio is 0.012 (introducing the effects of the large protection panels placed on this bridge that crosses over a highway). Real dynamic tests reflected that the damping ratio of the first vertical modes is moderately larger than 0.012 and that movements caused by a single pedestrian were below 0.30 m/s2, see Table 6 (very similar to those predicted by the model with the second damping ratio).
Table 6 Comparison of serviceability response experimentally recorded and serviceability response predicted by the methodology at Aldeias footbridge, (Gouveia, Portugal), where ped. corresponds to pedestrian
Verification of the serviceability design appraisal
In this section, accelerations predicted using the detailed numerical analysis and those predicted by the simplified method just presented are compared. Generally speaking, an excellent agreement is found, as seen in what follows.
Figure 7 compares results obtained numerically and following the methodology to assess vertical responses of structures with two spans (L+0.8L) crossed by a pedestrian walking at 2.16 Hz and a step length of 0.65m (the pedestrian weight is 700 N) on structures with a T-slab section with a depth-to-span ratio h/L=1/35 and a damping ratio of ζ=0.005 (i.e, for a set of parameters different to those considered in the reference case). The proximity of the simplified methodology results to those obtained numerically suggests that the accuracy of the methodology is good given its computational simplicity.
Comparison of vertical response of two-span bridges, L+0.8L
Figure 8 compares the maximum lateral accelerations obtained for simply supported structures calculated using the detailed numerical procedure and the simplified methodology presented in the paper. The structures considered have a composite box girder transverse section with h/L=1/35 and ζ=0.003, whereas pedestrians were assumed to walk at a step frequency fp=2.16 Hz and with a step length of 0.65 m. The results shown in Fig. 8 again suggest that the simplified methodology proposed herein predicts the response with a very good degree of accuracy.
Comparison of lateral response of simply supported bridges
Figures 7 and 8 show that the accelerations registered in the deck could be very sensitive to the main span length and span arrangement (1, 2 or 3 spans). A sensitivity analysis should be included at the design stage in order to consider the uncertainty of these important parameters.
Evaluation of the serviceability performance in conventional footbridges
The new methodology presented herein constitutes a tool for evaluating the adequacy of the different design options usually considered in practice. The purpose of this section is to compare the different design options available for single span bridges with structural characteristics and materials listed in Fig. 3 and damping ratios of ζ=0.005. The evaluation is performed considering the characteristics of a pedestrian (mp=71.36 kg) stream of commuters or at leisure with a density of 0.6 ped/m2 (the step frequency for the stream is the mean value provided in Fig. 2). For these analyses, it is considered that all sections have additional deck finishings and balustrades. Section S.6 also has as a non-structural concrete deck and sections S.7 to S.9 have non-structural wooden decks.
For a preliminary evaluation of the adequacy of the response in service, it can be considered that vertical accelerations in the range between 0.5 and 1.0 m/s2 define the comfort limit, whereas, for lateral accelerations, responses in the range between 0.2 and 0.4 m/s2 represent the limit of serviceable situations (further ranges can be found in codes and guidelines such as Setra, 2006 or NA to BS EN 1991-2:2003, 2008).
The results shown in Fig. 9 illustrate how the response of certain structures under the action of these pedestrian streams are large, and do not satisfy serviceability criteria for certain span lengths. However, it should be appreciated that although the values presented in Fig. 9 are often very large, in reality accelerations beyond 2 m/s2, vertically, or 0.4 m/s2, laterally, will probably not be developed due to a change of behaviour of pedestrians after sensing large vibrations (by either changing the characteristics of their walking, or even stopping).
Evaluation of serviceability of simply supported structures in vertical and lateral directions under the action of a pedestrian stream of density 0.6 ped/m2 with commuting or leisure being the aim of the journey
It is also clear from Fig. 9 that there is a very strong dependence upon both the main span length just mentioned but also upon the pedestrian density. Given the strong sensitivity to these parameters, and the simplified nature of the approach proposed herein, it is clear that any design decisions, even at preliminary stage, should account for uncertainties in these parameters. The uncertainty associated with the assumed damping ratio should also be considered.
Inspection of Fig. 9 also suggests that for this level of traffic, sections S.7 (aluminium) and S.9 (glass fiber reinforced polymers) have poor performance irrespective of span length. This suggests that for these types of decks it will be difficult to satisfy serviceability requirements using the slendernesses adopted here and for similar traffic.
For footbridges where this methodology shows that serviceability criteria may not be satisfied and responses may often be very large (in particular for sections S.7 and S.9), it is recommended that designers explore from an early design stage the benefits of implementing additional attenuation devices (e.g., tuned-mass dampers) as discussed in Garcia-Troncoso et al. (2020).
Conclusions and future developments
The paper presents a methodology for the serviceability evaluation of beam-type structures subjected to pedestrian loads. The steps of the procedure together with the fundamental underlying assumptions are outlined in this paper for both vertical and lateral response caused by these loads. Compared to current proposals, the method presented herein does propose an appraisal of the lateral accelerations generated by pedestrians in service, it is based on a realistic and experimentally contrasted description of the pedestrian loads, and it does not require the use of any software or elaborate analysis techniques.
Based on the methodological procedure, as well as the inherent assumptions and simplifications included for its development, it should be highlighted that:
An adequate evaluation of the response includes a comprehensive description of pedestrian loads and structural properties. Despite the fact that the method presented herein intends to simplify the assessment of pedestrian-induced vibrations, it still includes a refined evaluation of loads (both load magnitude and step frequencies). Regarding the structural properties, the method proposes evaluating the dynamic properties through a small number of parameters that are easy to appraise even during the early stages of design.
The prediction of the lateral response attempts to include the interaction phenomenon that has been detected for pedestrian loading scenarios through a parameter that reflects a certain response nonlinearity. This parameter shows that under certain circumstances the nonlinear response in the lateral direction can be larger than that obtained considering only linear results. However, it should be highlighted that this model is only able to reproduce what seems to be a first stage of what might happen in reality (it is not reflecting a change of step frequency according to the response generated, or synchronisation). This phenomenon could be further investigated for its inclusion in the assessment of lateral structural response. Although having said that, it is the opinion of the authors that serviceability is unlikely to be satisfied when it becomes necessary to model these types of phenomena.
The comparison of the response described by the proposed methodology to that recorded in real footbridges highlights the excellent competence of the method in describing the serviceability response of footbridges at very early stages of their design.
The comparison of the response for a set of structures calculated both using an advanced numerical procedure and the proposed simplified approach shows that the methodology is an adequate tool for the evaluation of vertical and lateral structural response for preliminary design.
The estimates of the accelerations are extremely sensitive to the span length and the type of traffic loading scenario. For conventional sections commonly considered within design, once values of the span length and pedestrian loading are defined, the method clearly identifies span ranges that should be avoided.
Finally, it is important to recognise that the simplicity of the proposed approach allows a designer to obtain estimates of accelerations with very little computational effort. The proposed method therefore lends itself to undertaking analyses in which the sensitivity of the results to various design assumptions is quantified. The strong sensitivity of the results shown in this article imply that such sensitivity analyses are an indispensable component of the design process.
The datasets used to derive the current study are available from the corresponding author on reasonable request. The datasets used to implement the methodology presented in the current study are included in this published article.
FE:
finite elements
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Ramos Moreno, C, Ruiz-Teran A, Stafford P (2020) Impact of Stochastic Representations of Pedestrian Actions on Serviceability Response In: Proceedings of the Institution of Civil Engineers - Bridge Engineering. https://doi.org/10.1680/jbren.19.00050.
Weidmann, U (1993) Transporttechnik der Fussgänger - Transporttechnische Eigenschaften des Fussgängerverkhers (Literaturauswertung). [In German]. Technical report, Institut für Verkehrsplanung, Transporttechnick, Strassen und Eisenbahnbau, ETH Zurich.
The authors are grateful for the support provided by La Caixa to fund the PhD studies of the first author at Imperial College London.
The work developed in this paper is partially supported by La Caixa.
Arcadis Consulting UK, London, UK
Caterina Ramos-Moreno
Department of Civil and Environmental Engineering, Imperial College London, South Kensington Campus, London, SW7 2AZ, United Kingdom
Ana M. Ruiz-Teran & Peter J. Stafford
Ana M. Ruiz-Teran
Peter J. Stafford
CRM researched for background data, developed the methodology, performed the numerical analysis and the technical writing. AMRT developed the methodology and provided guidance with the technical writing. PJS developed the methodology and provided guidance with the technical writing. The authors read and approved the final manuscript.
Correspondence to Ana M. Ruiz-Teran.
Additional file 1 Additional file can be found in a repository in Open Science Framework Dataset repository. The additional file consists of an Excel table that lists the values of \(a_{,n}^{b}\) and \(a_{l,n}^{b}\) according to the values of fp,v and rv,n (or fp,l and rl,n) as described in the paper.
Additional file 2
Additional file located in a repository in Open Science Framework Dataset repository is also included hereunder as a table listing the values of \(a_{,n}^{b}\) and \(a_{l,n}^{b}\) according to the values of fp,v and rv,n (or fp,l and rl,n) as described in the paper.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Ramos-Moreno, C., Ruiz-Teran, A.M. & Stafford, P.J. Guidance for footbridge design: a new simplified method for the accurate evaluation of the structural response in serviceability conditions. ABEN 1, 20 (2020). https://doi.org/10.1186/s43251-020-00012-9
Footbridge
Pedestrian load
Vertical response
Lateral response | CommonCrawl |
Failure and Timeline of Online Social Network Sites
Proposed design methodology
Using social network analysis of human aspects for online social network software: a design methodology
Faiza Ghafoor1 and
Muaz A. Niazi2Email author
Complex Adaptive Systems Modeling20164:14
© The Author(s) 2016
Received: 7 January 2016
Accepted: 6 July 2016
Online social networks share similar topological characteristics as real-world social networks. Many studies have been conducted to analyze the online social networks, but it is difficult to link human interests with social network software design.
The goal of this work is to propose a methodology involving the analysis of human interactions for use in designing online social network software.
We propose a novel use of social network analysis techniques to elicit requirements in order to design better online Social network-based software. The validation case study involved the collection of real-world data by means of a questionnaire to perform a network design construction and analysis. The key idea is to examine social network to help in the identification of behaviors and interests of people for better software requirements elicitation.
The validation case study demonstrates how unexpected centrality measures can emerge in real world networks. Our case study can thus conducted as a baseline for better requirement elicitation studies for online social network software design. This work also indicates how sociometric methods may be used to analyze any social domain as a possible standard practice in online social network software design. Overall, the study proved the effectiveness of the proposed novel methodology for the design of online social network software. The methodology specifically improves upon traditional methods for software design by involving social network modeling and analysis to first study the behavior and elicit requirements to develop more resilient online social network sites.
Online social network
Centrality measures
Software requirements elicitation
Online social networks, also known as Social Network Sites (SNS) are web-based services allowing users to communicate with each other - whether for personal or professional usage. Boyd and Ellison have defined SNS in their paper as "Web Based Services that allow individuals to (1) construct a public or semi-public profile within a bounded system (2) to articulate a list of other users with whom they share a connection and (3) view and traverse their list of connections and those made by others within the systems" (Ellison 2007). These social network sites not only allow individuals to meet strangers, but also provide them the opportunity to regularly interact with their connections living across the world.
Over time, however, it is clear that many social network sites have been launched and closed due to various reasons ranging from security issues to unsatisfactory and missing features and from the launch of new and more interesting social network sites. Still, the current software design methodologies do not have any means to ensure the effective requirements elicitation for SNS. As a result, many SNS projects have failed due to hitherto unidentified reasons (Ellison 2007). We believe that the complexity of larger groups of end users is something which is currently not catered for in existing software design methodologies. The broad application of social network sites and the virtual presence of users make it challenging to elicit user requirements.
Requirement elicitation has always been considered a long and complicated process—especially when it comes to web-based services—such as web-based information system (Yang and Tang 2003), web-based social networks, or in general, any software custom-designed for online social-network software for interaction in communities. Yang and Tang have discussed these difficulties of requirement elicitation in web-based services and have presented a three stage model of requirement elicitation:
When we come to web-based services, users are generally diverse.
It is very hard to identify key users in such a large population.
User requirements are always changing and even differ from culture to culture.
Users may grow in future.
There are no traditional methods to gather requirements (Interviews or observations) for web-based services as they become impractical if we consider the facts we have discussed above.
Sorenson and Skouby have proposed the approach of requirement elicitation by using interviews (Sorensen and Skouby 2008). They however have also subsequently discussed the disadvantages of using interviews. The biggest problem is the limited number of users to be interviewed. In another paper, Tang and Yang (2006) have discussed the drawbacks of taking survey-based studies of online users. They note that a key problem is getting no response back from the intended members of the network.
Therefore, by considering all these previous studies, it is quite clear that online social networks are unlike traditional software. They involve building software for a much larger and much diverse group of users from the ones who can be practically interviewed. As a result, traditional software engineering approaches such as interview-taking and conducting surveys are not enough to ascertain the specific requirements related to behavior of the end-users-requirements which, if not elicited, can result in possible failure of the software. One key reason, Online SNS can fail is because traditional software requirement elicitation techniques do not have any means of discovery of human behavioral influences. The key novel idea in the proposed methodology is to employ the use of social network modeling and analysis (Niazi and Hussain 2012) to understand the behaviors and interests of people for better elicitation of software requirements.
Social networks can be analyzed by analyzing structural characteristics of the network. The key identifying features of social networks are the users and their connections (Howard 2008; Oinas-Kukkonen et al. 2010). These connections make the structure of the network. While not without their inherent problems, as noted by Batool and Niazi (Batool and Niazi 2014), structural and topological characteristics have been used in various studies to understand the nuances of human behavior in social networks (Shapiro and Varian 2013). Online social network sites can also be analyzed in detail by applying social network analysis (Bonchi et al. 2011). Previous researchers have found many similarities between offline and online social networks. E.g. both type of networks can be scale-free and, at times, some of them can also follow the power law distribution (Chun et al. 2008). Several measures have been suggested yet to analyze the structural characteristics of the network. One of these is commonly known as the centrality measures (Hanneman et al. 2001). The centrality measures are important because they tell us about the most influential and powerful node in the network. Many centrality measures have been proposed yet; the most famous ones are degree centrality, closeness centrality, eccentricity and betweenness centrality (Hanneman and Riddle 2005). The fifth, most popular one is eigenvector centrality measure which has been proposed by Bonacich (Bonacich 1972). This centrality measure are usually used to find the most influential person in the network. Still researchers have noted that using these measures mechanically may not be the most effective means of understanding the interaction structure of the social network (Batool and Niazi 2014).
There are numerous studies that have been conducted to understand the social influences on behaviors of people by using social network analysis (Goyal et al. 2010; Blansky et al. 2013; Hill et al. 2010; Christakis and Fowler 2007; Christakis and Fowler 2008; Mednick et al. 2010). However, the goal of this work is to propose a novel software requirements elicitation methodology involving the analysis of human interactions for use in designing online social network software. Using a network of 498 girls living in a university hostel, we have analyzed the role of the strength of friendship relationship (for example best friend, friend and acquaintance) besides their everyday life habits (1) whether strength of relationship with different friends affects their sleeping habit (2) whether a girl's eating habits are associated with her social contacts (3) whether girls take influence of each other while doing exercise or not. We believe that the conducting of social network modeling and analysis has allowed for a better understanding of the hidden behaviors of the large-scale group.
This design methodology gives us the idea to understand human aspects for online social networks by using social network analysis.
The outline of the rest of the paper is as follows: we first present background related to the failure of online social networks or SNS. In the methodology section, we give the details of the proposed design methodology. We also present the social network analysis techniques used. In the results section, we discuss the analysis results and a detailed discussion on the implications of this proposed design methodology to understand human aspects for online social networks by using social network analysis (SNA).
Here, we present the timeline and failures of online SNS.
Six Degree
It was founded by Andrew Weinreich in 1997 (New York) by using the concept of six degrees of separation. In few years, this Six Degree attracted almost 1 million users (Heidemann et al. 2010; Hanneman and Riddle 2005).
Failure of Six Degree
But within 3 years, this site had to face failure due to poorly developed web technology (Mednick et al. 2010) whereas the founder of this site had accepted this failure with the statement "It was simply ahead of its time" (Ellison et al. 2007). Despite the failure, six degree has proved as a trendsetter for future online SNS.
Live Journal was a SNS based in San Francisco California. It was founded by Brad Fitzpatrick in April 1999. By using this site, users were able to keep a diary, a blog, or a journal with another blogging company. In 2007, six apart sold Live Journal to another Russian company. In 2009, they laid off some employees and moved product development and design to Russia.
Failure of Live Journal
In 2007, 2010, and 2012 this site was blocked by different countries due to several issues such as extremist propaganda perpetuated in its blogs etc (Ellison 2007).
Ethnic sites
In the following years, many other online sites had been launched such as Asian Avenue, Mi Gente, and Black Planet. These can be classified as ethnic community sites.
Failure of ethnic sites
These sites had tried to fill the gap of technical functionality but they also failed in a few years possibly due to limited friendship options. Limited options did not allow users to connect with the maximum possible nodes. This way, the question of better designing of social network sites still remained unsolved (Ellison 2007).
In 2001 (San Francisco) Adrian Scott founded Ryze as a business network. Ryze can therefore be considered to serve as a trendsetter and an example model for future business networks (e.g., Linked In).
Failure of Ryze
Ryze too had to shut down in a few years. To the best of our knowledge, the exact reasons of the failure of this site are still unidentified (Ellison 2007).
In 2002, Jonathan Abrams founded "Friendster"—primarily as a competitor to online dating sites. It was created on the assumption that friends of friends would make better romantic partners (Boyd 2004). Friendster allowed the access to the profiles within 4 degrees of separation (Boyd and Heer 2006). Till 2004, Friendster enjoyed considerable popularity and was possibly the biggest site of its time.
In 2011, Friendster was relaunched and a number of registered users reached over 115 million. In 2012, it remained notably popular in Indonesia. In 2015, Friendster finally suspended their services.
Failure of Friendster
Due to its extremely popular nature, the site began to face technical and social issues. These problems included unscalability in part due to servers and databases not being able to handle the interactions of such a large group of people. Secondly, people often ending up a social dilemma by having to deal with undesired contacts. Furthermore, fake users started to become a part of the system, thereby becoming a viable threat to the privacy of the users. In the meanwhile, expulsion of a group from Friendster for not complying with the regulations resulted in a chain reaction that ended in a large departure of end-users (Ellison et al. 2007).
In 2003, MySpace was launched in California. It was able to attract frustrated Friendster users. At the start, music bands from the Los Angeles started using MySpace for advertising VIP passes for popular clubs. MySpace put in a better effort by continuously adding features based on users demand (Ellison 2007). It also allowed users to personalize their pages.
Failure of MySpace
In 2005, News Corporation bought MySpace for $580 million but later sold it for $35 million. The reported reason for this loss was issues pertaining to safety such as sexual interactions between adults and minors. Although some researchers reported it as an exaggerated reason (Boyd and Heer 2006), the end result was a complete failure.
After gathering all the identified reasons of the failures of online social network sites, we propose the idea of modifying traditional methods of software requirements elicitation by the addition of social network modeling and analysis of the expected group of end users.
Launch time of the major online SNS
See Table 1 for details of the timeline for various online SNS.
Launch time of famous online social networks
Launch time
Online social networks
Live Journal, Asian Avenue, Black Planet
Lunar Storm, My Gente
CyWorld, Ryze
Fotolog, Friendster, skyblog
Couch surfing, Linked In, MySpace, Tribe.net, Last.Fm, Hi5
Orkut, Dogster, Flicker, Piczo, Mixi, Facebook (Harvard only), Dodgeball, Care2 (relaunch), Catster, Hyves
Yahoo!360, Youtube, Xanga (relaunch), Cyworld, Bebo (relaunch), Facebook (high schools), Ning, Asian Avenue, Black Planet (relaunch)
QQ (relaunch), Facebook (corporate network), Windows live space, CyWorld (U.S), Twitter, MyChurch, Facebook (everyone)
See Fig. 1.
Start, rise, and fall of famous online social networks
Previous Offline Behavioral Studies
In the past, social network analysis has been enormously used to analyze the behavioral influence and the spread of habits or information across the network. Christakis et al. (2011) have widely presented their research over the spread of behavior across the networks by using social network analysis and statistical analysis. Specifically they have observed the spread of obesity (Christakis and Fowler 2007), spread of happiness in human society (Fowler et al. 2009), diffusion of emotional states either positive or negative (Hill et al. 2010), social contagion of sleep behavior and drug use behavior (Mednick et al. 2010), spread of smoking behavior (Christakis and Fowler 2008), spread of alcohol consumption (Rosenquist et al. 2010) etc. In all these studies they have taken a large social network of people and applied centrality measures and statistical analysis such as logistic regression and generalized estimation equation (GEE) procedures on longitudinal data. In all these studies they have focused on the spread of behavior.
To analyze the spread of academic success, Hiroki Sayama (Blansky et al. 2013) has also used social network analysis, linear regression, and correlation and observed the relatively higher influence of intermediate level friendship for the spread of academic success in school students.
Use of Behaviorial influence in the Proposed Approach
We have used the approach of analyzing behavioral influences in offline social networks to identify the hidden requirements of a person representing a specific social domain or society. As we have discussed earlier that targeting each individual for requirement elicitation in online social networks (OSN) is impossible. The approach of analyzing behavioral influences suggest the idea of taking requirements from an influential person (who can be identified by using centrality measure) who can possibly influence his/her contacts or befriend similar people regarding their choices and interests. In this way, we can analyze the software design requirements of the society by targeting minimum people.
This section introduces the set of activities related to the design of online social network software.
The proposed design methodology for online social network software is based on the concept of a traditional method of software development design process. Our idea encapsulates the traditional method of software development in addition to social network analysis of human aspects.
The methodology diagram Fig. 2 represents several phases of the development of online social network software design.
Proposed design methodology diagram, the proposed idea encapsulates the traditional software development processes in addition with social network analysis of human aspects
The first activity of the proposed design methodology takes the social network analysis of human aspects into account.
Once the network analysis is performed, identification of influential people is the next task.
After finding influential people, the process of requirement engineering would be performed. Requirements gathering from the influential people of the society or that specific social domain would be performed. After requirements gathering, requirements would be analyzed.
After requirement engineering process, design phase will start.
After the design phase of online social network software, implementation, testing, version release, software validation processes would be performed.
In software validation process, the software would be validated against our identified influential people's requirements.
All the data was obtained from the girl's hostel of International Islamic University, Islamabad by developing the questionnaire related to their choices. Girls were asked to list their three best friends and rate them, according to the strength of the relationship. Girls freely recalled their friends and listed their names. We targeted their friends and collected their data. For this purpose, we delivered 600 Questionnaires in the hostels of International Islamic University Hostels and got almost 498 responses.
The diagram in Fig. 3 presents the sequence of techniques of SNA performed to understand the human behaviors and their influence.
Social network analysis techniques, the diagram presents the sequence of the network analysis techniques we have performed to conduct the study. First of all the data was gathered and then a network was generated for the analysis. Centrality measures have been calculated to find the prominent actors in the network. After analyzing prominent actors, statistical analysis was conducted in order to validate the results of social network analysis
At the first step of our case study, we design a survey in order to collect data. We target 600 people and get 498 responses.
After data collection, we first generate a network of friends and represented friends as "nodes" in the network and their relationship as "edges" (directed links) and then manipulate it by deletion of unnecessary nodes (nodes representing girls having no friends), insertion of necessary nodes (targeting missing friends whose presence are necessary to establish the links between friends in the network) and then develop a social network ready for the network analysis.
Next, we apply visualization and social network analysis on the network.
We also measure centralities in order to identify the prominent actors and their positions. We have also observed the power law behavior of these centrality measures. After performing centrality based analysis on the whole network, we find a unique node in the network which we have termed as the focal node in our study. This node can be considered as the most influential node having highest eigenvector centrality. Then we extract a cluster of the most influential node and her neighbors (friends) located at the one degree of separation.
In the end, we have performed statistical analysis to validate the behavioral influence of a focal node in a real life.
The main idea of this study design is to understand the behavioral influence and interests of people for better software requirements.
Methodology used for analysis of social network
First of all, the social network of friends is developed. The main components of the network are friends and their links as nodes and edges respectively. Attributes of the edges represent the strength of the relationship between friends. The basic idea in this social network analysis is that friends according to their position in the social network take the influence of other friends by whom they are surrounded in a social network. Social network analysis is often referred as a structural analysis because this approach is used for exploring structures of the networks. First of all, we made an analysis of the whole network.
Structural characteristics of the network
In the following section, we have described different structural properties of the network.
Network size
Network size of our network is 498 nodes connected by 1226 edges. To calculate the network size is the first step to analyze any network in order to get interesting structural properties of the network.
Cluster size
A cluster is a group of people who are strongly connected to each other but sparsely connected to the rest of the people, 64 Clusters are observed in the network.
The degree is the number of connections of a node in a network. The average degree of our network is 4.9196, where one node represented the highest degree of 24.
See Fig. 4a for degree partition of a network.
Network analysis and visualization. a This figure represents specific degree measure of every node such as the nodes with yellow color are labeled with 1 which indicates nodes with 1 degree, the nodes with green color have 2 degree, red have 3, blue have 4, pink have 5, white have 6 and so on. b This figure represents 210 strong components. Digits labeled on the nodes (circles) show the component number. c This figure represents 1 weak component in yellow color. d This figure represents 6 cores in the network, yellow colored nodes belong from core 1, green from core 2, red from core 3, blue from core 4, pink from 5 and white colored nodes represent core 6
Components of a network show the level of connectivity of a network, our data set has 1 Weak and 210 Strong components.
See Fig. 4b for strong components and Fig. 4c for weak components.
Cores are the denser regions of the network. These are helpful in identifying the denser parts of the network. 6 Cores are observed on the network (see Fig. 4d).
These structural properties of the network help us in identifying the denser and strongly connected cluster in the network. In the strongly connected cluster of the nodes (representing people) we can identify friends having strong bonds among friends.
Centrality-based analysis
To measure the location of each actor, these centrality measures are required. Centrality is a metric for one node, whereas centralization metric tells us about the whole network.
A number of links a node has presents the degree centrality of a node. Directed networks are analyzed by using the metrics of in degree and out degree. In degree is referred as the number of links coming towards nodes. Regarding our friendship network, in degree correlates with the popularity of girl among her friends. Out degree is referred as the number of links going out from the node. In the friendship network out degree correlates with the social nature of the girl in a network.
Degree centrality (Freeman 1979) can be calculated by using Freeman's formula of Degree centrality:
$$C_{D}(v)=\frac{K_{v}}{n-1}= \sum_{j \in G}\frac{a_{vj}}{n-1}$$
where Kv is the number of connections of a node and n is the total number of nodes in the network.
We found one unique (focal) node in a network having the highest degree of 24.
Betweenness centrality
According to the Freeman's approach betweenness centrality is defined as the proportion of times that the node acts as a bridge between two different nodes for sending information. So this is the most favored position of the actors in the network. So the actor offering shortest (geodesic) pathway between other pairs of actors tend to have high betweenness centrality (Hanneman and Riddle 2005).
Betweenness centrality can be calculated using the Freeman's formula:
$$C_{B}(v)=\sum_{s\neq v\neq t}\frac{\delta{st(v)}}{\delta{st}}$$
where δst represents the sum of the shortest path, where "s" and "t" can be considered as two different nodes. δst(v) represents the sum of paths that intersect node v.
Our focal nodes betweenness centrality is 0.05.
Closeness centralization
Closeness centrality referred as how far a node from all other nodes in the network (Hanneman and Riddle 2005). If we consider the distance of node as a farness of the node. The closeness would be "the inverse of the farness". The less total distance of a node, lesser will be the closeness centrality. A node having low closeness centrality considered to be more central in the network. Closeness centrality can be calculated using the Freeman's formula:
$$C_{C}(v)=\sum\frac{1}{dist(v,t)}$$
Here v and t are the nodes.
Our focal node's closeness centrality is 0.31.
Eigenvector centrality
Bonacich's eigenvector centrality is used to calculate the centrality of a node as a function of the centrality of its neighbors. Eigenvector centrality focuses on the fact that a node having the connection to high centrality nodes are more important than the node having links with low centrality nodes (Boudin 2013). This is a measure for finding the most influential node in a network.
$$\rightthreetimes \, \text{v} = \text{Av}$$
\(\rightthreetimes\) is constant, v is the eigenvector and A is the adjacency matrix. Highest eigenvector value is 1 in our network and only one unique (focal) node is observed with 1 eigenvector centrality.
After doing this network analysis, we took that unique node as a focal person and analyzed her friends (best friend, friend or just an acquaintance) and their social influence over each other. We used the Kamada–Kwai algorithm to get the clear image of the focal person and her neighboring friends at the one degree of separation.
(Focal person/node is the term, we are using here for the most influential person according to the network analysis report; person with the highest degree and the highest eigenvector centrality).
To validate the results of structural analysis of the social network we have performed statistical analysis over attribute data of the nodes. Our goal was to evaluate whether a focal person's habits are associated with her friends in a real life. To test this hypothesis, we took a cluster of friends and analyzed the habits of the focal person and her friends by using the statistical tool named "R".
In the following subsection, results of the study are presented in detail.
Network analysis and visualization
Our complete network consists of 498 nodes with 1226 edges. Figure 4a shows all degree partition of a network. 24 is the highest degree of one unique node and the average degree of a network is calculated as 4.9196.
Figure 4b shows the strong component in the network. The size of the largest strong component of a network is 18 vertices having 3.614 of the total network. We used Kamada–Kwai algorithm for the spatialisation of the network.
Figure 4c shows the weak component consists of a whole network. The size of the weak component is 498 nodes which are 100 % of a whole network.
Figure 4d shows cores of the network. Our network consists of 6 cores.
Centrality based visual analysis
For Centrality based visual analysis, we used Gephi software with "Frutcherman Rheingold layout". Increasing size of the node correlates with the highest centrality value. Figure 5a shows partition of the network, on the basis of degree. The one having purple color with the biggest size is the focal node having the highest degree in a network. Figure 5b shows the visual display of the ranking (size) of the nodes according to the high betweenness centrality and partitioned (colored) on the basis of degree. Figure 5c shows the ranking (size) of nodes with high closeness centrality. Though the network is partitioned (colored) on the basis of degree. Our focal node in purple color is not showing the high closeness centrality. Figure 5d shows the ranking (size) of the nodes with high eigenvector centrality. Again the network is partitioned (colored) on the basis of degree. Our focal node (purple colored) is showing highest eigenvector centrality value in the network, which is the indication of the most influential node.
Centrality-based visual analysis: focal node is in purple color in all networks, size correlates with centrality measures, whereas colors represent the degrees of the nodes in each network. a The one having purple color with the biggest size are the focal node having the highest degree in a network. b A visual display of the ranking (size) of the nodes with the highest betweenness centrality and partitioned (colored) on the basis of degree. c The ranking (size) of nodes with high closeness centrality. d The ranking (size) of the nodes with high eigenvector centrality
After having the complete report of the network, we started focusing the node having the highest degree and eigenvector centrality. These measures have proved this node as the most influential and popular node (person) in the network. We looked for the neighbors of the focal person located at the one degree of separation. We have extracted the cluster including focal person and her neighboring friends located at the one degree of separation from the entire network using "Pajek software". We have used Kamada–Kwai algorithm to get the clear image of the neighboring friends and the focal person. Figure 6a, b shows the focal person with her neighbors in the network.
Neighbors of the focal node. a Node with the blue color is a focal node having the highest degree and eigenvector centrality, yellow colored nodes are the neighbors of the node at one degree of separation, and all the other nodes are in gray color. b Blue colored node is a focal node; yellow colored nodes are the neighbors of the focal node with one degree of separation
(Focal node is the term, we are using here for the most influential node/person according to the network analysis report; the node with the highest degree and the highest eigenvector centrality).
Focal node and the neighboring nodes in a network
Power law distribution of centrality measures of the focal node and the neighboring nodes
Power Law distributions are typically fitted to empirical data when the frequency of an event varies with the power of some attribute of that event however, Clauset et al. (Clauset, 2009) have noted the problems in fitting these distributions on data. In this case we have seen if the frequency of nodes varies as a power of centrality measures.
These plots in Fig. 7 represent the correlation of centrality metrics and the frequency of nodes in the cluster of the focal node (focal person) and the neighboring nodes (friends) located at one degree of separation.
Centrality plots. a Our focal node has a maximum degree centrality 24. b Our focal node betweenness centrality is 0.05. c Our focal node's closeness centrality is 0.31. d Our focal node's eigenvector centrality is 1
Behavioral analysis results
Statistical analysis over the attribute data of the focal node and her neighboring nodes (friends) is performed to analyze the behavioral influences. We took contact type as a function of different variables such as their sleeping habit, the frequency of physical activity in a week, fast food intake and fresh fruits and juices intake to analyze whether the focal person and her contact type (Best friend, friend, acquaintance) share same interests or not.
Figure 8 show the strip charts of the habits of focal person and her contacts. The goal of statistical analysis of their behavior is to understand either people living in the same community take the influence of their friends or not. Additionally, we have tried to validate our network analysis result which has indicated 1 unique node as the most influential node in the network (which we have termed as the focal node in our study).
Behavioral analysis using strip chart. a Focal node sleeps 7 h in a day. b Physical activity of a focal node in a week is 0. c Focal node's fast food intake is 8 times in a week. d Focal node's intake is 3 times in a week
These results showed a slight similarity in their interests.
"R" statistical tool has been used to perform statistical analysis.
Next generation of social media and understanding of behavioral and cultural motives
Adam Smith in "The theory of Moral Sentiments" noted that to understand the social phenomenon, one must incorporate the multitude of psychological and cultural motives. Pent et al. (Cebrian et al. 2016) in their paper have referred the Smith and suggested that "modeling the observable processes and the underlying motivational dynamics can pay tribute to the Smith's nuanced understanding of human nature and this way next generation of social media can be designed".
Manifestation of actual real life personalities in online social networks
Gosling et al. (2011) have stated that how offline personalities reflect in the online social domain. To map out the connection between original personality and online social network behavior, they have presented two studies and examined how real personality traits are expressed in Facebook. Their correlation results showed the number of links between Facebook information and respondents actual personality. So they concluded that social and real personality processes are parallel in virtual and non-virtual environment. They suggested that future research should examine how other processes such as social influence and other aspects of personality (attitude, value etc.) are manifested in virtual and non-virtual environment.
Lately in 2015 (Dunbar et al. 2015), latest research has been conducted on Facebook, twitter, and offline social network data sets. They have found sociological similarities between online and offline environments, as implied by the data. In addition, they have also found similar structural characteristics between offline (non-virtual) and online networks (virtual environments).
In another research Lewis et al. (2012) discussed the problem that, do people pick friends who are similar to them or do they take the influence of their friends over time? By using Facebook activity data of college students they analyzed the choice of students regarding their music and movie interests. They picked the people who were already friends in their real life too. Their finding suggests that friends share their interest not because they influence each other but also because they become friends due to similar nature. They have stated that selection and influence play a very important role in relationships and tastes expressed online not only show their psychological preferences but also their presentation of actual self.
By considering these reported ideas, we have proposed this design methodology involving the social network analysis techniques on the offline social network in order to elicit requirements for designing the online social network software. Similar structural properties of network and psychological preferences of people participating in virtual and non-virtual environments suggests the idea of taking an efficient approach of analyzing real life interests of influential people in order to establish future online social network sites.
If we consider the fact of the success of Facebook, and get back to the history this fact is inevitable that Facebook too was first launched for a specific social domain (Harvard only) and then it became famous worldwide in coming years.
Summary of discussion
From 1997 to 2004, many famous online social networks have been launched and had to face drastic failures. The reasons behind many of these failures have remained unidentified. The solution to these problems needs to be worked on. To the best of our knowledge, no one has tried to analyze the behaviors of people participating in the network before launching the website. All the websites were launched on the trial basis and the fate decided their success and failures.
By considering the failed ideas of taking interviews and online surveys of social networks. The idea of offline social network analysis is proposed in our study. By using this approach, we can elicit requirements for use in the design of social network based software. The main idea is to use SNA to understand the behaviors (in terms of influence) and interests of people for better software requirements. Our goal is to conduct an example case study acts as a baseline for better software design.
In this study, we have performed the social network analysis of the offline network to gather the information about human behaviors and their influence over each person participating in a network. The structural characteristics of the whole network have been studied and then the centrality measures are calculated of the cluster of friends.
By applying these measures and then after statistically analyzing the attribute data of the people under study, we have deduced that social network analysis in combination with statistical analysis techniques is an effective method to analyze the interest of people participating in the large social network. The best implication of this design methodology is to link the behaviors of people to online networks in order to perceive the patterns of basic human interests and behaviors where targeting single human for requirement gathering seems impossible.
This case study is considered as just a proof of analyzing hidden requirements, as a standard practice in software design.
In this paper, we have proposed a design methodology by using social network analysis of human aspects for online social network software.
This methodology presents an idea of analysis of human behaviors, participating in large social networks. We have used the approach of analyzing human choices in offline social networks to identify the hidden requirements of a person representing a specific social domain or society. As we have discussed earlier that targeting each individual for requirement elicitation in OSN is impossible, the proposed approach of analyzing behavioral influences suggests the idea of taking surveys to develop models for analysis using social network methods from the intended group of end users. The presented case study has demonstrated the effectiveness of the proposed approach.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Department of Software Engineering, International Islamic University, Islamabad, Pakistan
Department of Computer Science, COMSATS Institute of IT, Islamabad, Pakistan
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\begin{definition}[Definition:Sphere/P-adic Numbers/Center]
Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $a \in \Q_p$.
Let $\epsilon \in \R_{>0}$ be a strictly positive real number.
Let $\map {S_\epsilon} a$ be the $\epsilon$-sphere of $a$.
In $\map {S_\epsilon} a$, the value $a$ is referred to as the '''center''' of the $\epsilon$-sphere.
\end{definition} | ProofWiki |
AVT Statistical filtering algorithm
AVT Statistical filtering algorithm is an approach to improving quality of raw data collected from various sources. It is most effective in cases when there is inband noise present. In those cases AVT is better at filtering data then, band-pass filter or any digital filtering based on variation of.
Conventional filtering is useful when signal/data has different frequency than noise and signal/data is separated/filtered by frequency discrimination of noise. Frequency discrimination filtering is done using Low Pass, High Pass and Band Pass filtering which refers to relative frequency filtering criteria target for such configuration. Those filters are created using passive and active components and sometimes are implemented using software algorithms based on Fast Fourier transform (FFT).
AVT filtering is implemented in software and its inner working is based on statistical analysis of raw data.
When signal frequency/(useful data distribution frequency) coincides with noise frequency/(noisy data distribution frequency) we have inband noise. In this situations frequency discrimination filtering does not work since the noise and useful signal are indistinguishable and where AVT excels. To achieve filtering in such conditions there are several methods/algorithms available which are briefly described below.
Averaging algorithm
1. Collect n samples of data
2. Calculate average value of collected data
3. Present/record result as actual data
Median algorithm
1. Collect n samples of data
2. Sort the data in ascending or descending order. Note that order does not matter
3. Select the data that happen to be in n/2 position and present/record it as final result representing data sample
AVT algorithm
AVT algorithm stands for Antonyan Vardan Transform and its implementation explained below.
1. Collect n samples of data
2. Calculate the standard deviation and average value
3. Drop any data that is greater or less than average ± one standard deviation
4. Calculate average value of remaining data
5. Present/record result as actual value representing data sample
This algorithm is based on amplitude discrimination and can easily reject any noise that is not like actual signal, otherwise statistically different then 1 standard deviation of the signal. Note that this type of filtering can be used in situations where the actual environmental noise is not known in advance. Notice that it is preferable to use the median in above steps than average. Originally the AVT algorithm used average value to compare it with results of median on the data window.
Filtering algorithms comparison
Using a system that has signal value of 1 and has noise added at 0.1% and 1% levels will simplify quantification of algorithm performance. The R[1] script is used to create pseudo random noise added to signal and analyze the results of filtering using several algorithms. Please refer to "Reduce Inband Noise with the AVT Algorithm" [2] article for details. This graphs show that AVT algorithm provides best results compared with Median and Averaging algorithms while using data sample size of 32, 64 and 128 values. Note that this graph was created by analyzing random data array of 10000 values. Sample of this data is graphically represented below.
From this graph it is apparent that AVT outperforms other filtering algorithms by providing 5% to 10% more accurate data when analyzing same datasets. Considering random nature of noise used in this numerical experiment that borderlines worst case situation where actual signal level is below ambient noise the precision improvements of processing data with AVT algorithm are significant.
AVT algorithm variations
Cascaded AVT
In some situations better results can be obtained by cascading several stages of AVT filtering. This will produce singular constant value which can be used for equipment that has known stable characteristics like thermometers, thermistors and other slow acting sensors.
Reverse AVT
1. Collect n samples of data
2. Calculate the standard deviation and average value
3. Drop any data that is within one standard deviation ± average band
4. Calculate average value of remaining data
5. Present/record result as actual data
This is useful for detecting minute signals that are close to background noise level.
Possible applications and uses
• Use to filter data that is near or below noise level
• Used in planet detection to filter out raw data from Kepler (spacecraft)
• Filter out noise from sound sources where all other filtering methods (Low-pass filter, High-pass filter, Band-pass filter, Digital filter) fail.
• Pre-process scientific data for data analysis (Smoothness) before plotting see (Plot (graphics))
• Used in SETI (Search for extraterrestrial intelligence) for detecting/distinguishing extraterrestrial signals from cosmic background
• Use AVT as image filtering algorithm to detect altered images, please see Python program that is available for download. This image of Jupiter generated from this program, detecting alterations in original picture that was modified to be visually appealing by applying filters.
Another version of this comparison is the Reverse AVT filter applied to the same original Jupiter Image, where we only see that altered portion as Noise that was eliminated by AVT algorithm.
• Use AVT as image filtering algorithm to estimate data density from images, please see Python program program. Picture of Pillars of Creation Nebula shows data density in filtered images from Hubble and Webb. Note that image on the left has big patches of missing data marked with simpler color patterns.
References
1. "The R Project for Statistical Computing". r-project.org. Retrieved 2015-01-10.
2. "Reduce Inband Noise with the AVT Algorithm | Embedded content from Electronic Design". electronicdesign.com. Retrieved 2015-01-10.
1. Joseph, Favis; Balinadoa, C.; Paolo Dar Santos, Gerald; Escanilla, Rio; Darell C. Aguda, John; Ramona A. Alcantara, Ma.; Belen M. Roble, Mariela; F. Bueser, Jomalyn (May 5, 2020). "Design and implementation of water velocity monitoring system based on hydropower generation and antonyan vardan transform (AVT) statistics". 13Th International Engineering Research Conference (13Th Eureca 2019). Vol. 2233. p. 050003. doi:10.1063/5.0002323.
2. Vinicius, Cene; Mauricio, Tosin; J., Machado; A., Balbinot (April 2019). "Open Database for Accurate Upper-Limb Intent Detection Using Electromyography and Reliable Extreme Learning Machines". Sensors. 19 (8): 1864. Bibcode:2019Senso..19.1864C. doi:10.3390/s19081864. PMC 6515272. PMID 31003524.
3. HornCene, Vinicius; Balbinot, Alexandr (August 10, 2018), "Using the sEMG signal representativity improvement towards upper-limb movement classification reliability", Biomedical Signal Processing and Control, 46: 182–191, doi:10.1016/j.bspc.2018.07.014, ISSN 1746-8094, S2CID 52071917
4. Horn Cene, Vinicius; Ruschel dos Santos, Raphael; Balbinot, Alexandre (July 18, 2018). 2018 40th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC). Honolulu, HI, USA: IEEE. pp. 5224–5227. doi:10.1109/EMBC.2018.8513468. ISBN 978-1-5386-3646-6.
5. AVT image Filtering algorithm in python[1]
1. Antonyan, Vardan. "AVT Image Filter". Github. Github.
| Wikipedia |
\begin{document}
\begin{abstract} We derive a formula for the $\bar\mu$--invariant of a Seifert fibered homology sphere in terms of the $\eta$--invariant of its Dirac operator. As a consequence, we obtain a vanishing result for the index of certain Dirac operators on plumbed 4-manifolds bounding such spheres. \end{abstract}
\title{The {\Large $\bar\mu$} \section{Introduction}\label{S:intro} The $\bar\mu$--invariant is an integral lift of the Rohlin invariant for plumbed homology 3-spheres defined by Neumann \cite{N} and Siebenmann \cite{Sb}. It has played an important role in the study of homology cobordisms of such homology spheres. Fukumoto and Furuta \cite{FuFu} and Saveliev \cite{Sav2} showed that the $\bar\mu$--invariant is an obstruction for a Seifert fibered homology sphere to have finite order in the integral homology cobordism group $\Theta^3_H$; this fact allowed them to make progress on the question of the splittability of the Rohlin homomorphism $\rho: \Theta^3_H \to \mathbb Z_2$. Ue \cite{Ue2} and Stipsicz \cite{St} studied the behavior of $\bar\mu$ with respect to rational homology cobordisms.
In the process, the $\bar\mu$--invariant has been interpreted in several different ways\,: as an equivariant Casson invariant in \cite{CS}, as a Lefschetz number in instanton Floer homology in \cite{RS} and \cite{Sav1}, and as the correction term in Heegaard Floer theory in \cite{St} and \cite{Ue1}.
More recently, $\bar\mu$ appeared in our paper \cite{MRS} in connection with a Seiberg-Witten invariant $\lambda_{\,\operatorname{SW}\,}$ of a homology $S^1\times S^3$. We conjectured that \begin{equation}\label{E:conj} \lambda_{\,SW} (X) = - \bar\mu (Y) \end{equation} for any Seifert fibered homology sphere $Y = \Sigma(a_1,\ldots,a_n)$ and the mapping torus $X$ of a natural involution on $Y$ viewed as a link of a complex surface singularity. This conjecture will be explained in detail and proved in Section \ref{S:conj}. For the purposes of this introduction, we will only mention that its proof will rely on the following identity.
\begin{theorem}\label{T:main} Let $Y = \Sigma (a_1,\ldots,a_n)$ be a Seifert fibered homology sphere oriented as the link of complex surface singularity and endowed with a natural metric realizing the Thurston geometry on $Y$; see \cite{Scott}. Then \begin{equation}\label{E:main} \frac 1 2\;\eta_{\,\operatorname{Dir}} (Y) + \frac 1 8\;\eta_{\,\operatorname{Sign}} (Y) = -\bar\mu (Y), \end{equation} where $\eta_{\,\operatorname{Dir}} (Y)$ and $\eta_{\,\operatorname{Sign}}(Y)$ are the $\eta$--invariants of, respectively, the Dirac operator and the odd signature operator on $Y$. \end{theorem}
In addition, identity \eqref{E:main} will be used to extend the vanishing result of Kronheimer for the index of the chiral Dirac operator on the $E_8$ manifold bounding $\Sigma (2,3,5)$; see \cite[Lemma 2.2]{Kron} and \cite[Proposition 8]{Fro}. Let $Y$ be a Seifert fibered homology sphere as above, and $X$ a plumbed manifold with boundary $Y$ and a Riemannian metric which is a product near the boundary. Associated with $X$ is the integral Wu class $w \in H_2(X;\mathbb Z)$ which will be described in detail in Section \ref{S:dirac}.
\begin{theorem}\label{T:dirac} Let $D^+_L(X)$ be the $\,\operatorname{spin}^c$ Dirac operator on $X$ with $c_1(L)$ dual to the class $w \in H_2 (X;\mathbb Z)$, and with the Atiyah--Patodi--Singer boundary condition. Then $\operatorname{ind} D^+_L (X) = 0$. In particular, if $X$ is spin then $w$ vanishes and $\operatorname{ind} D^+ (X) = 0$. \end{theorem}
The next four sections of the paper will be devoted to the proof of Theorem \ref{T:main}. We will proceed by expressing both sides of (\ref{E:main}) in terms of Dedekind--Rademacher sums and by comparing the latter expressions using the reciprocity law and some elementary calculations. Theorem \ref{T:dirac} will be proved in Section \ref{S:dirac}, and Conjecture \eqref{E:conj} in Section \ref{S:conj}. Our notation and conventions for Seifert fibered homology spheres will follow~\cite{saveliev:spheres}.
\section{The $\eta$--invariants} Let $p > 0$ and $q > 0$ be pairwise relatively prime integers, and $x$ and $y$ arbitrary real numbers. The Dedekind--Rademacher sums were defined in \cite{Rad} by the formula \[ s(q,p;x,y) = \sum_{\mu \hspace{-0.07in}\mod p} \left(\hspace{-0.06in}\left( \frac {\mu + y} p \right)\hspace{-0.06in}\right)
\left(\hspace{-0.06in}\left( \frac {q(\mu + y)} p + x \right)\hspace{-0.06in}\right) \] where, for any real number $r$, we set $\{r\} = r - [r]$ and \[ \( r \) = \begin{cases} \; 0, &\quad\text{if\; $r \in \mathbb Z$}, \\ \, \{r\} - 1/2, &\quad\text{if\; $r \notin \mathbb Z$}. \end{cases} \] It is clear that $s(q,p;x,y)$ only depends on $x$, $y\hspace{-0.07in}\mod 1$. When both $x$ and $y$ are integers, we get back the usual Dedekind sums
\begin{equation}\label{E:ded} s(q,p) = \sum_{\mu \hspace{-0.07in}\mod p} \left(\hspace{-0.06in}\left( \frac {\mu} p \right)\hspace{-0.06in}\right) \left(\hspace{-0.06in}\left( \frac {q \mu} p \right)\hspace{-0.06in}\right). \end{equation}
The left-hand side of (\ref{E:main}) was expressed by Nicolaescu \cite{Nic2} in terms of Dedekind--Rademacher sums. Note that since the $a_i$ are coprime, at most one of them is even; if that occurs then we will choose the even one to be $a_1$.\\[2ex] \textbf{Odd case:} if all $a_1,\ldots,a_n$ are odd then, according to the formula (1.9) of \cite{Nic2}, we have \begin{multline}\label{E:eta-odd} \frac 1 2\;\eta_{\,\operatorname{Dir}} (Y) + \frac 1 8\;\eta_{\,\operatorname{Sign}} (Y) = - \frac 1 {8\,a_1\cdots a_n} + \\ + \frac 1 8 + \frac 1 2\,\sum_{i=1}^n s(a_1\cdots a_n/a_i, a_i) + \sum_{i=1}^n s(a_1\ldots a_n/a_i,a_i; 1/2,1/2). \end{multline}
\noindent \textbf{Even case:} if $a_1$ is even then, according to the formula (1.6) of \cite{Nic2}, \begin{multline}\label{E:eta-even} \frac 1 2\;\eta_{\,\operatorname{Dir}} (Y) + \frac 1 8\;\eta_{\,\operatorname{Sign}} (Y) = \\ = \frac 1 8 + \frac 1 2\,\sum_{i=1}^n s(a_1\cdots a_n/a_i, a_i) + \sum_{i=1}^n s(a_1\ldots a_n/a_i,a_i; 1/2,1/2). \end{multline}
\section{The $\bar\mu$--invariant} Let $\Sigma$ be a plumbed integral homology sphere, and let $X$ be an oriented plumbed 4-manifold such that $\partial X = \Sigma$. The integral Wu class $w \in H_2 (X;\mathbb Z)$ is the unique homology class which is characteristic and whose coordinates are either 0 or 1 in the natural basis in $H_2 (X;\mathbb Z)$ represented by embedded 2-spheres. According to Neumann \cite{N}, the integer \[ \bar\mu (\Sigma) = \frac 1 8\;(\,\operatorname{sign} (X) - w\cdot w) \] is independent of the choices in its definition and reduces modulo 2 to the Rohlin invariant of $\Sigma$. It is referred to as the $\bar\mu$--invariant.
Let us now restrict ourselves to the case of $Y = \Sigma(a_1,\ldots,a_n)$. Choose integers $b_1,\ldots,b_n$ so that \begin{equation}\label{E:one} a_1\cdots a_n\;\sum_{i=1}^n\;\frac {b_i}{a_i}\; =\;\sum_{i=1}^n\;b_i\,a_1\cdots a_n/a_i\; = 1. \end{equation} Note that each $b_i$ is defined uniquely modulo $a_i$. Then we have the following formulas for the $\bar\mu$--invariant; see \cite[Corollary 2.3]{N} and \cite[Theorem 6.2]{NR}.
\noindent\textbf{Odd case:} if all $a_1,\ldots,a_n$ are odd then \[ -\bar\mu (Y) = \frac 1 8 - \frac 1 8\; \sum_{i=1}^n\,(c(a_i,b_i) + \operatorname{sign} b_i). \]
\noindent \textbf{Even case:} if $a_1$ is even, choose $b_i$ so that all $a_i - b_i$ are all odd (by replacing, if necessary, $b_i$ by $b_i \pm a_i$ for each $i > 1$, and then adjusting $b_1$ accordingly). Then \[ -\bar\mu (Y) = \frac 1 8 - \frac 1 8\; \sum_{i=1}^n\,c(a_i - b_i,a_i). \]
Here, the integers $c(q,p)$ are defined for coprime integer pairs $(q,p)$ with $q$ odd as follows. First, assume that both $p$ and $q$ are positive. Then
\[ c(q,p)\; =\; - \frac 1 p \sum_{\xi^p = -1} \frac {(\xi + 1)(\xi^q + 1)} {(\xi - 1)(\xi^q - 1)}\; =\; \frac 1 p \sum_{\substack{k = 1 \\ k\;\text{odd}}}^{2p - 1} \cot\left(\frac{\pi k}{2p}\right)\cot\left(\frac{\pi qk}{2p}\right) \]
The integers $c(q,p)$ show up in the book~\cite[Theorem 1, pp.~102--103]{HZ} under the name $-t_p(1,q)$. We can use that theorem together with formula (6) on page 100 of \cite{HZ} to write \begin{equation}\label{E:cpq} c (q,p) = - 4s(q,p) + 8s(q,2p). \end{equation}
Next, the above definition of $c(q,p)$ is extended to both positive and
negative $p$ and $q$ by the formula $c(q,p) = \operatorname{sign}(pq)\,c(|q|,|p|)$. Using (\ref{E:cpq}), we can write the above formulas for the $\bar\mu$--invariant in the following form.
\noindent\textbf{Odd case:} if all $a_1,\ldots,a_n$ are odd then \begin{multline}\label{E:mu-odd} -\bar\mu (Y) = \frac 1 8 - \frac 1 8 \sum_{i=1}^n\, \operatorname{sign} b_i + \frac 1 2\,
\sum_{i=1}^n \operatorname{sign} b_i\cdot s(a_i,|b_i|) \\
- \sum_{i=1}^n\, \operatorname{sign} b_i\cdot s(a_i,2|b_i|). \end{multline}
\noindent \textbf{Even case:} if $a_1$ is even and $b_i$ are chosen so that $a_i - b_i$ are all odd, then \begin{equation}\label{E:mu-even} -\bar\mu (Y) = \frac 1 8 + \frac 1 2\,\sum_{i=1}^n\, s(a_i - b_i,a_i) - \sum_{i=1}^n s(a_i - b_i,2a_i) \end{equation}
\noindent In the latter formula, we used a natural extension of the Dedekind sum $s(q,p)$ to the negative values of $q$ as an odd function in $q$; it is still given by the formula (\ref{E:ded}). We will continue to assume, however, that $p$ in $s(q,p)$ is positive.
\section{The odd case} In this section, we will show that the right hand sides of (\ref{E:eta-odd}) and (\ref{E:mu-odd}) are equal to each other, thus proving the formula (\ref{E:main}) in the case when all $a_1,\ldots,a_n$ are odd.
\begin{lemma}\label{L:one} For any integers $a > 0$ and $b$, $c$ such that $bc = 1\hspace{-0.07in}\mod a$ we have $s(c,a) = s(b,a)$. \end{lemma}
\begin{proof} Observe that $bc = 1\hspace{-0.07in}\mod a$ implies that $b$ and $a$ are coprime hence \begin{multline}\notag s(c,a) = \sum_{\mu\hspace{-0.07in}\mod a} \left(\hspace{-0.06in}\left(\frac{\mu}{a}\right)\hspace{-0.06in}\right) \left(\hspace{-0.06in}\left(\frac{\mu c}{a}\right)\hspace{-0.06in}\right) = \sum_{\mu\hspace{-0.07in}\mod a} \left(\hspace{-0.06in}\left(\frac{\mu b}{a}\right)\hspace{-0.06in}\right) \left(\hspace{-0.06in}\left(\frac{\mu b c}{a}\right)\hspace{-0.06in}\right) \\ = \sum_{\mu\hspace{-0.07in}\mod a} \left(\hspace{-0.06in}\left(\frac{\mu b}{a}\right)\hspace{-0.06in}\right) \left(\hspace{-0.06in}\left(\frac{\mu}{a}\right)\hspace{-0.06in}\right) = s(b,a). \end{multline} \end{proof}
\begin{lemma}\label{L:two} For any coprime positive integers $a$ and $b$ such that $a$ is odd, \[ \frac 1 2\,s(a,b) - s(a,2b)\; =\; - s(a,b;0,1/2) - \frac 1 2\,s(a,b). \] \end{lemma}
\begin{proof} The proof goes by splitting the summation over $\mu\hspace{-0.07in}\mod 2b$ in $s(a,2b)$ into two summations, one over even $\mu = 2\nu$, and the other over odd $\mu = 2\nu + 1$. More precisely, \begin{alignat*}{1} s(a,2b) & = \sum_{\mu\hspace{-0.07in}\mod 2b} \left(\hspace{-0.06in}\left(\frac{\mu}{2b}\right)\hspace{-0.06in}\right) \left(\hspace{-0.06in}\left(\frac{a\mu}{2b}\right)\hspace{-0.06in}\right) \\ & = \sum_{\nu\hspace{-0.07in}\mod b} \left(\hspace{-0.06in}\left(\frac{\nu}{b}\right)\hspace{-0.06in}\right) \left(\hspace{-0.06in}\left(\frac{a\nu}{b}\right)\hspace{-0.06in}\right) + \sum_{\nu\hspace{-0.07in}\mod b} \left(\hspace{-0.06in}\left(\frac{2\nu+1}{2b}\right)\hspace{-0.06in}\right) \left(\hspace{-0.06in}\left(\frac{a(2\nu+1)}{2b}\right)\hspace{-0.06in}\right) \\ & = \sum_{\nu\hspace{-0.07in}\mod b} \left(\hspace{-0.06in}\left(\frac{\nu}{b}\right)\hspace{-0.06in}\right) \left(\hspace{-0.06in}\left(\frac{a\nu}{b}\right)\hspace{-0.06in}\right) + \sum_{\nu\hspace{-0.07in}\mod b} \left(\hspace{-0.06in}\left(\frac{\nu+1/2}{b}\right)\hspace{-0.06in}\right) \left(\hspace{-0.06in}\left(\frac{a(\nu+1/2)}{b}\right)\hspace{-0.06in}\right) \\ & = s(a,b) + s(a,b;0,1/2). \end{alignat*} The statement of the lemma now follows. \end{proof}
Applying Lemma \ref{L:one} with $a = a_i$, $b = b_i$ and $c = a_1\cdots a_n/a_i$, and Lemma \ref{L:two} with $a = a_i$ and $b =
|b_i|$ respectively to the formulas (\ref{E:eta-odd}) and (\ref{E:mu-odd}), we see that all we need to do is verify the following identity \begin{multline}\label{E:int1} - \frac 1 {8\,a_1\cdots a_n} + \frac 1 2\; \sum_{i=1}^n s(b_i,a_i) + \sum_{i=1}^n s(a_1\cdots a_n/a_i,a_i; 1/2,1/2) = \\
- \sum_{i=1}^n\;\operatorname{sign} b_i \cdot \left( \frac 1 8 + s(a_i,|b_i|;0,1/2)
+ \frac 1 2 s(a_i,|b_i|)\right). \end{multline}
\noindent Use the reciprocity laws (see for instance Appendix in \cite{Nic2}) to obtain
\[
s(a_i,|b_i|;0,1/2) = -s (|b_i|,a_i; 1/2,0) + \frac {2b_i^2 - a_i^2 - 1}
{24\,a_i |b_i|} \] and \[
s(a_i,|b_i|) = - s(|b_i|,a_i) - \frac 1 4 + \frac {a_i^2 + b_i^2 + 1}
{12\,a_i |b_i|}. \]
\noindent Substituting the latter two formulas into (\ref{E:int1}) and keeping in mind that \[ \sum_{i=1}^n\; \frac {b_i}{a_i}\; =\; \frac 1 {a_1\cdots a_n} \] because of (\ref{E:one}), we reduce verification of (\ref{E:int1}) to
proving the following lemma (we write $\operatorname{sign} b_i\cdot s(|b_i|,a_i;1/2,0) = s(b_i,a_i;1/2,0)$).
\begin{lemma}\quad $s(b_i,a_i; 1/2,0) = s(a_1\cdots a_n/a_i,a_i; 1/2,1/2)$. \end{lemma}
\begin{proof} One can easily see that the identity that needs to be verified, \begin{multline}\notag \sum_{\mu\hspace{-0.07in}\mod a_i} \left(\hspace{-0.06in}\left( \frac {\mu}{a_i}\right)\hspace{-0.06in}\right) \left(\hspace{-0.06in}\left( \frac {b_i\mu}{a_i} + \frac 1 2 \right)\hspace{-0.06in}\right) = \\ \sum_{\nu\hspace{-0.07in}\mod a_i} \left(\hspace{-0.06in}\left( \frac {\nu + 1/2}{a_i}\right)\hspace{-0.06in}\right) \left(\hspace{-0.06in}\left( \frac {(\nu + 1/2)a_1\cdots a_n/a_i}{a_i} + \frac 1 2 \right)\hspace{-0.06in}\right), \end{multline} follows by substitution $\nu = b_i\,\mu + (a_i - 1)/2 \hspace{-0.07in}\mod a_i$. \end{proof}
\section{The even case} In this section, we will prove the equality of the right hand sides of the formulas (\ref{E:eta-even}) and (\ref{E:mu-even}) and hence prove (\ref{E:main}) in the even case.
\begin{lemma}\label{L:three} \quad $s(a_i - b_i,a_i) = - s(b_i,a_i)$. \end{lemma}
\begin{proof} Since $\( x \)$ is an odd function in $x$, we have \begin{multline}\notag s(a_i - b_i,a_i) = \sum_{\mu\hspace{-0.07in}\mod a_i} \left(\hspace{-0.06in}\left(\frac {\mu}{a_i}\right)\hspace{-0.06in}\right) \left(\hspace{-0.06in}\left( \frac {\mu (a_i - b_i)} {a_i} \right)\hspace{-0.06in}\right) \\ = \sum_{\mu\hspace{-0.07in}\mod a_i} \left(\hspace{-0.06in}\left(\frac {\mu}{a_i}\right)\hspace{-0.06in}\right) \left(\hspace{-0.06in}\left( \frac {-\mu b_i}{a_i}\right)\hspace{-0.06in}\right) = -s(b_i,a_i). \end{multline} \end{proof}
Using Lemma \ref{L:one} and Lemma \ref{L:three}, we reduce our task to showing that, for every $i = 1,\ldots,n$, \begin{equation}\label{E:four} s(a_1\cdots a_n/a_i,a_i;1/2,1/2) + s(a_i - b_i,2a_i) + s(b_i,a_i) = 0. \end{equation}
\begin{lemma} For any coprime integers $a > 0$ and $c > 0$, we have \[ s(c,a;1/2,1/2) + s(a - c,2a) + s(c,a) = 0. \] \end{lemma}
\begin{proof} Like in the proof of Lemma \ref{L:two}, we will break the summation over $\mu\hspace{-0.07in}\mod 2a$ in $s(a - c,2a)$ into two summations, one over $\mu = 2\nu$ and the other over $\mu = 2\nu + 1$. More precisely, \begin{multline}\notag s(a - c,2a) = \sum_{\mu\hspace{-0.07in}\mod 2a} \left(\hspace{-0.06in}\left(\frac{\mu}{2a}\right)\hspace{-0.06in}\right) \left(\hspace{-0.06in}\left(\frac{(a - c)\mu}{2a}\right)\hspace{-0.06in}\right) \\ = -\sum_{\nu\hspace{-0.07in}\mod a} \left(\hspace{-0.06in}\left(\frac{\nu}{a}\right)\hspace{-0.06in}\right) \left(\hspace{-0.06in}\left(\frac{c\nu}{a}\right)\hspace{-0.06in}\right) - \sum_{\nu\hspace{-0.07in}\mod a} \left(\hspace{-0.06in}\left(\frac{2\nu+1}{2a}\right)\hspace{-0.06in}\right) \left(\hspace{-0.06in}\left(\frac{c(2\nu+1)}{2a} + \frac 1 2 \right)\hspace{-0.06in}\right) \\ = -s(c,a) - s(c,a;1/2,1/2). \end{multline} \end{proof}
We will apply the above lemma with $a = a_i$ and $c = a_1\cdots a_n/a_i$ to obtain $s(a_1\cdots a_n/a_i,a_i;1/2,1/2) + s(a_i - a_1\cdots a_n/a_i,2a_i) + s(a_1\cdots a_n/a_i,a_i) = 0$. Using Lemma \ref{L:one} to replace $s(a_1\cdots a_n/a_i,a_i)$ in the above formula by $s(b_i,a_i)$, we see that the proof of (\ref{E:four}) will be complete after we prove the following formula.
\begin{lemma} \quad $s(a_i - a_1\cdots a_n/a_i,2a_i) = s(a_i - b_i,2a_i)$. \end{lemma}
\begin{proof} This is immediate from Lemma \ref{L:one} once we show that $(a_i - b_i)(a_i - a_1\cdots a_n/a_i) = 1\hspace{-0.07in}\mod 2a_i$. We will consider two separate cases. If $i = 1$ then $a_1$ is even and $b_1$ is odd. Multiply out to obtain $(a_1 - b_1)(a_1 - a_2\cdots a_n) = a_1^2 + b_1 a_2 \cdots a_n - a_1 (b_1 + a_2 \cdots a_n)$. Obviously, the first and the last summands are equal to zero modulo $2a_1$ because $a_1$ and $(b_1 + a_2\cdots a_n)$ are even. Use the formula (\ref{E:one}) to write $b_1 a_2 \cdots a_n = 1 - a_1 (b_2 a_2\cdots a_n + \ldots + b_n a_2 \cdots a_{n-1})$ and observe that the $b_2,\ldots, b_n$ are all even. This completes the proof in the case of $i = 1$.
Now suppose that $i \ge 2$. Since $a_i$ and 2 are coprime, it is enough to check separately that $(a_i - b_i)(a_i - a_1\cdots a_n/a_i)$ is 1 mod $a_i$ and 1 mod 2. The former is clear from (\ref{E:one}), and the latter follows from the observation that both $a_i - b_i$ and $a_i - a_1\cdots a_n/a_i$ are odd. \end{proof}
\section{Proof of Theorem \ref{T:dirac}}\label{S:dirac} Endow $Y = \Sigma(a_1,\ldots,a_n)$ with a natural metric realizing the Thurston geometry on $Y$; see \cite{Scott}. Let $X$ be a plumbed manifold with boundary $\partial X = Y$ and with metric that restricts to the metric on $Y$ and is a product near the boundary. If $X$ is spin, the Atiyah--Patodi--Singer index theorem~\cite{aps:I} asserts that \begin{equation}\label{E:aps} \frac 1 2\;\eta_{\,\operatorname{Dir}} (Y) + \frac 1 8\;\eta_{\,\operatorname{Sign}} (Y)\; =\; -\operatorname{ind} D^+ (X) - \frac 1 8\,\operatorname{sign} (X). \end{equation}
\noindent Here, we used the fact that the Dirac operator on $Y$ has zero kernel; see Nicolaescu~\cite[Section 2.3]{Nic3}. On the other hand, it follows from the definition of the $\bar\mu$--invariant that $w = 0$ and hence \[ \bar\mu (Y)\; =\; \frac 1 8\,\operatorname{sign} (X). \]
\noindent The identity (\ref{E:main}) then implies that $\operatorname{ind} D^+ (X) = 0$. The special case of this when $Y$ is the Poincar\'e homology sphere $\Sigma(2,3,5)$ and $X$ is the negative definite $E_8$ manifold was proved by Kronheimer~\cite{Kron}.
If $X$ is not spin, for any choice of $\,\operatorname{spin}^c$ structure on $X$ with determinant bundle $L$ we have \[ \frac 1 2\;\eta_{\,\operatorname{Dir}} (Y) + \frac 1 8\;\eta_{\,\operatorname{Sign}} (Y)\; =\; -\operatorname{ind} D^+_L (X) - \frac 1 8\,\operatorname{sign} (X) + \frac 1 8\,c_1 (L)^2. \]
\noindent (Compare with formula (1.37) in \cite{Nic1}). If the $\,\operatorname{spin}^c$ structure is such that $c_1 (L)$ is dual to $w \in H_2 (X;\mathbb Z)$ then \[ \bar\mu (Y) = \frac 1 8\,(\operatorname{sign} (X) - w\cdot w) = \frac 1 8\,(\operatorname{sign} (X) - c_1 (L)^2), \] and (\ref{E:main}) again implies that $\operatorname{ind} D^+_L (X) = 0$. This completes the proof of Theorem \ref{T:dirac}.
\section{The invariant $\lambda_{\,\operatorname{SW}\,}$}\label{S:conj} Let $X$ be a homology $S^1\times S^3$, by which we mean a closed oriented spin smooth 4-manifold with the integral homology of $S^1\times S^3$. For a generic pair $(g,\beta)$ consisting of a metric $g$ on $X$ and a perturbation $\beta \in \Omega^1(X,i\mathbb R)$, the Seiberg--Witten moduli space $\mathcal M(X,g,\beta)$ has finitely many irreducible points. It is oriented by a choice of homology orientation, that is, a generator $1 \in H^1 (X;\mathbb Z)$. Let $\#\,\mathcal M(X,g,\beta)$ denote the signed count of the points in this space. To counter the dependence of $\#\,\mathcal M(X,g,\beta)$ on the choice of $(g,\beta)$, we introduced in \cite{MRS} a correction term, $w(X,g,\beta)$, and proved that the quantity \[ \lambda_{\,\operatorname{SW}\,} (X)\; =\; \#\,\mathcal M (X,g,\beta) - w (X,g,\beta) \]
\noindent is an invariant of $X$ which reduces modulo 2 to its Rohlin invariant. The precise definition of the correction term is as follows.
Let $Y \subset X$ be a smooth connected 3-manifold dual to the generator $1 \in H^1(X;\mathbb Z)$ and choose a smooth compact spin manifold $Z$ with boundary $Y$. Cutting $X$ open along $Y$ we obtain a cobordism $W$ from $Y$ to itself, which we use to construct the periodic-end manifold \[ Z_+ = Z\,\cup\,W\,\cup\,W\ldots\cup\,W\,\cup\ldots \] The metric $g$ and perturbation $\beta$ extend to an end-periodic metric and, respectively, perturbation, on $Z_+$. This leads to the end-periodic perturbed Dirac operator $D^+ (Z_+) + \beta$, where $\beta$ acts via Clifford multiplication. We prove that $D^+(Z_+) + \beta$ is Fredholm in the usual Sobolev $L^2$-completion for generic $(g,\beta)$. The correction term is then defined as \[ w (X,g,\beta)\; =\; \operatorname{ind}_{\mathbb C}\,(D^+ (Z_+) + \beta) + \frac 1 8\,\operatorname{sign}\,(Z). \]
View $Y = \Sigma(a_1,\ldots,a_n)$ as a link of a complex surface singularity and let $X$ be the mapping torus of the involution on $Y$ induced by complex conjugation. The metric $g$ realizing the Thurston geometry on $Y$ is preserved by this involution and hence gives rise to a natural metric on $X$ called again $g$. We showed in \cite[Section 10]{MRS} that the pair $(g,0)$ is generic and that the space $\mathcal M(X,g,0)$ is empty. One can easily see that the manifold $Z_+$ has a product end and hence the correction term can be computed as in \eqref{E:aps} using the Atiyah--Patodi--Singer index theorem\,: \[ w(X,g,0)\; =\; -\frac 1 2\;\eta_{\,\operatorname{Dir}} (Y) - \frac 1 8\;\eta_{\,\operatorname{Sign}} (Y). \]
\noindent The conjecture \eqref{E:conj} now follows from Theorem \ref{T:main}.
\end{document} | arXiv |
\begin{document}
\title{Fast alternating bi-directional preconditioner for the 2D high-frequency Lippmann-Schwinger equation}
\renewcommand{\arabic{footnote}}{\fnsymbol{footnote}}
\footnotetext[2]{Department of Mathematics, University of California, Irvine, CA 92612, USA.}
\renewcommand{\arabic{footnote}}{\arabic{footnote}}
\begin{abstract} This paper presents a fast iterative solver for Lippmann-Schwinger equation for high-frequency waves scattered by a smooth medium with a compactly supported inhomogeneity. The solver is based on the sparsifying preconditioner \cite{Ying:Sparsifying_Preconditioner_for_the_Lippmann--Schwinger_Equation} and a domain decomposition approach similar to the method of polarized traces \cite{ZepedaDemanet:the_method_of_polarized_traces}. The iterative solver has two levels, the outer level in which a sparsifying preconditioner for the Lippmann-Schwinger equation is constructed, and the inner level, in which the resulting sparsified system is solved fast using an iterative solver preconditioned with a bi-directional matrix-free variant of the method of polarized traces. The complexity of the construction and application of the preconditioner is ${\cal O}(N)$ and ${\cal O}(N\log{N})$ respectively, where $N$ is the number of degrees of freedom. Numerical experiments in 2D indicate that the number of iterations in both levels depends weakly on the frequency resulting in a method with an overall ${\cal O}(N\log{N})$ complexity. \end{abstract}
\section{Introduction} In this paper we present a fast solver for the high-frequency Lippmann-Schwinger equation in 2D. Let $\omega$ be the frequency of a time-harmonic wave propagating in a medium with wave-speed denoted by $c(\mathbf{x})$, where $\mathbf{x} = (x,z)$. Let $\Omega$ be a rectangular domain and assume that the refraction index is constant up to a compactly supported perturbation. The refraction index is denoted by $1/c^2(\mathbf{x}) = 1 + m(\mathbf{x})$, where $\mbox{supp}(m) = D \subset \Omega \subset {\mathbb R}^2$. In this case, $u(\mathbf{x})$, the wave scattered by the perturbation $m$ from an incident wave $u_{I}(\mathbf{x})$, is the solution to the Helmholtz equation \begin{equation} \label{eq:Helmholtz}
\left ( \Delta + \omega^2( 1 + m(\mathbf{x}) ) \right ) (u(\mathbf{x}) + u_{I}(\mathbf{x})) = 0, \qquad \mathbf{x} \in {\mathbb R}^d; \end{equation} where $u(\mathbf{x})$ satisfies the Sommerfeld radiation condition, \begin{equation}
\lim_{ r \rightarrow \infty} r^{1/2} \left ( \frac{\partial u}{\partial r} - i\omega u\right) = 0, \end{equation} and the incoming field $u_{I}(\mathbf{x})$ satisfies the homogeneous Helmholtz equation \begin{equation} \label{eq:Helmholtz_incoming}
\left ( \Delta + \omega^2 \right )u_{I}(\mathbf{x}) = 0, \end{equation} in some neighborhood of $\Omega$ containing $D$.
From eqs. \eqref{eq:Helmholtz} and \eqref{eq:Helmholtz_incoming} it follows that the unknown field $u$ satisfies \begin{equation} \label{eq:Helmholtz_2}
\left ( \Delta + \omega^2( 1 + m(\mathbf{x}) ) \right ) u(\mathbf{x}) = -\omega^2 m(\mathbf{x}) u_{I}(\mathbf{x}), \qquad \mathbf{x} \in {\mathbb R}^d. \end{equation} Convolving \eqref{eq:Helmholtz_2} with \begin{equation} \label{eq:Greens_function}
G(\mathbf{x}) = -\frac{i}{4} H^1_0(\omega |\mathbf{x}|), \end{equation} the Green's function for the 2D free-space Helmholtz operator, results in the Lippmann-Schwinger equation: \begin{equation} \label{eq:Lippmann-Schwinger}
u + \omega^2 G*(m u) = -\omega^2 G*(m u_{I}). \end{equation}
We chose \eqref{eq:Lippmann-Schwinger} among different formulations of the Lippmann-Schwinger equation (for other formulations see \cite{ambikasaran_greengard:Fast_adaptive_high_order_accurate_discretization_of_the_Lippmann-Schwinger_equation_in_two_dimension}) by the following reasons :
\begin{itemize} \item the unknown scattered field, $u(\mathbf{x})$, satisfies the Sommerfeld radiation conditions automatically; \item the convolution kernel is translation invariant which allows for a rapid application via fast Fourier transforms (FFT) \cite{Cooley:An_Algorithm_for_the_Machine_Calculation_of_Complex_Fourier_Series}. \end{itemize}
Even though $u$ can be computed by discretizing \eqref{eq:Helmholtz_2} using finite differences, finite elements or composite spectral discretizations \cite{Gillman_Barnett_Martinsson:A_spectrally_accurate_solution_technique_for_frequency_domain_scattering_problems_with_variable_media,Gillman:A_Direct_Solver_with_ON_Complexity_for_Variable_Coefficient_Elliptic_PDEs_Discretized_via_a_High_Order_Composite_Spectral_Collocation_Method}; and then solving the resulting linear system using state-of-the art preconditioners \cite{Chen_Xiang:a_source_transfer_ddm_for_helmholtz_equations_in_unbounded_domain,EngquistYing:Sweeping_H,EngquistYing:Sweeping_PML,CStolk_rapidily_converging_domain_decomposition,GeuzaineVion:double_sweep,ZepedaDemanet:the_method_of_polarized_traces}, there are several advantages on computing $u$ by solving \eqref{eq:Lippmann-Schwinger} : \begin{enumerate} \item \eqref{eq:Lippmann-Schwinger} arises from a second kind Fredholm formulation, implying that the conditioning number of the discretized system does not deteriorate as the discretization is refined;
\item \eqref{eq:Lippmann-Schwinger} is defined on a bounded set $\Omega$ and the solution satisfies the Sommerfeld radiation condition automatically, whereas \eqref{eq:Helmholtz_2} is defined in the whole ${\mathbb R}^2$; hence one needs to truncate the computational domain, and to impose absorbing boundary conditions (ABC) at the boundary of the truncated computational domain, using perfectly matched layers (PML) \cite{Berenger:PML,Bermudez:An_optimal_perfectly_matched_layer_with_unbounded_absorbing_function_for_time-harmonic_acoustic_scattering_problems,Johnson:PML} or other techniques \cite{Engquist:Absorbing_boundary_conditions_for_the_numerical_simulation_of_waves} (see \cite{Bermudez:Perfectly_Matched_Layers_for_Time-Harmonic_Second_Order_Elliptic_Problems} for a review);
\item most discretizations of \eqref{eq:Helmholtz_2} via finite differences or finite elements suffer from dispersion error and the pollution effect \cite{Babuska:A_Generalized_Finite_Element_Method_for_solving_the_Helmholtz_equation_in_two_dimensions_with_minimal_pollution,Sauter_Babuska:Is_the_Pollution_Effect_of_the_FEM_Avoidable_for_the_Helmholtz_Equation_Considering_High_Wave_Numbers,Bayliss:On_accuracy_conditions_for_the_numerical_computation_of_waves} (see \cite{Stolk,Turkel:Compact_2D_and_3D_sixth_order_schemes_for_the_Helmholtz_equation_with_variable_wave_number} for recent references to methods designed to alleviate this issue); whereas the introduction of the analytical Green's function in \eqref{eq:Lippmann-Schwinger} and the use of an accurate quadrature rule automatically alleviates these issues.
\end{enumerate}
However, solving \eqref{eq:Lippmann-Schwinger} raises other numerical issues. The discretization of \eqref{eq:Lippmann-Schwinger} produces large dense linear systems, which becomes impractical, both memory- and complexity-wise, for typical direct solvers. Although a new generation of direct methods based on hierarchical compression, such as \cite{ambikasaran_greengard:Fast_adaptive_high_order_accurate_discretization_of_the_Lippmann-Schwinger_equation_in_two_dimension,Cheng:An_adaptive_fast_solver_for_the_modified_Helmholtz_equation_in_two_dimensions,Corona:An_direct_solver_for_integral_equations_on_the_plane,Ho:Hierarchical_Interpolative_Factorization_for_Elliptic_Operators_Integral_Equations}, have been able to considerably reduce the complexity of solving integral equations coming from elliptic partial differential equations (PDE), when applying these methods to integral equations coming from wave equations such as \eqref{eq:Lippmann-Schwinger}, they manage to exhibit linear or quasi-linear complexity for low frequency problems, but they fall back to ${\cal O}(N^{3/2})$ in high-frequency regimes. This phenomenon can be clearly explained by recent studies of the approximate separability of the Green's functions. It was shown in \cite{Bebendorf:Existence_of_Hmatrix_approximants_to_the_inverse_FE_matrix_of_elliptic_operators_with_Linftycoefficients} that the Green's functions for elliptic operators are highly separable, implying that the off-diagonal blocks of the discrete Green's function are extremely low-rank. In contrast, it was shown in \cite{Engquist_Zhao:approximate_separability_of_green_function_for_high_frequency_Helmholtz_equations} that the Green's functions for the Helmholtz operator in high-frequency regime are highly non-separable in general. This last results provides an asymptotic frequency-dependent lower bound on the ranks of the off-diagonal blocks of the discrete Green's function, which hinders the compressibility of the blocks, thus resulting in algorithms with higher complexity.
In addition, the Lippmann-Schwinger equation becomes numerically-ill conditioned as the frequency and contrast increases, due to multiple scattering, hindering the convergence rate of iterative methods. This difficulty has led to the development of efficient preconditioners, such as \cite{Andersson:A_Fast_Bandlimited_Solver_for_Scattering_Problems_in_Inhomogeneous_Media,Bruno:Wave_scattering_by_inhomogeneous_media_efficient_algorithms_and_applications,Bruno:An_efficient_preconditioned_high_order_solver_for_scattering_by_two-dimensional_inhomogeneous_media,Sifuntes:Preconditioned_Iterative_Methods_for_Inhomogeneous_Acoustic_Scattering_Applications,Vainikko:Fast_Solvers_of_the_Lippmann-Schwinger_Equation}; however, they tend to require significantly more iterations to converge as the frequency increases. Some of them are based on semi-analytical techniques, which require fairly strict assumptions on the perturbation (for example \cite{Bruno:An_efficient_preconditioned_high_order_solver_for_scattering_by_two-dimensional_inhomogeneous_media}); they tend to work well under the assumptions, but they naturally deteriorate if not.
There is an extensive literature on volume integral methods, including fast methods \cite{Bebendorf:Hierarchical_LU_Decomposition-based_Preconditioners_for_BEM,Ho_Greengard:A_Fast_Direct_Solver_for_Structured_Linear_Systems_by_Recursive_Skeletonization,Martinsson:A_fast_direct_solver_for_boundary_integral_equations_in_two_dimensions} and, in particular, methods solving the Lippmann-Schwinger equation \cite{Amar:Numerical_solution_of_the_Lippmann-Schwinger_equations_in_photoemission:_application_to_xenon,Andersson:A_Fast_Bandlimited_Solver_for_Scattering_Problems_in_Inhomogeneous_Media,Bruno:Wave_scattering_by_inhomogeneous_media_efficient_algorithms_and_applications,Bruno:An_efficient_preconditioned_high_order_solver_for_scattering_by_two-dimensional_inhomogeneous_media,Cheng:An_adaptive_fast_solver_for_the_modified_Helmholtz_equation_in_two_dimensions,Corona:An_direct_solver_for_integral_equations_on_the_plane,Duan-Rohklin:High-order_quadratures_for_the_solution_of_scattering_problems_in_two_dimensions,Lanzara:Numerical_Solution_of_the_Lippmann_Schwinger_Equation_by_Approximate_Approximations,Vainikko:Fast_Solvers_of_the_Lippmann-Schwinger_Equation,Ying:Sparsifying_Preconditioner_for_the_Lippmann--Schwinger_Equation}, and we do not review it here, but to the authors' knowledge there is no method that can handle general smooth perturbations in quasi-linear time in the high-frequency regime.
In the present paper we introduce a new preconditioner for the 2D high-frequency Lippmann-Schwinger equation, with a total asymptotic cost ${\cal O}(N \log{N})$, which compares favorably with respect to most of the methods (both iterative and direct) in the literature. The scaling holds for general smooth perturbations without large resonant cavities. The solver is based on the sparsifying preconditioner and a fast iterative solver in a domain decomposition setting. In particular, the sparsifying preconditioner is used to approximate the Lippmann-Schwinger equation with a sparse system, which is solved fast using a bi-directional matrix-free variant of the method of polarized traces \cite{Zepeda_Demanet:Nested_domain_decomposition_with_polarized_traces_for_the_2D_Helmholtz_equation}.
The method has two stages, the off-line, or setup, stage that is only performed once for each linear system to solve with an asymptotic cost of ${\cal O}(N)$, and the on-line, or solve, stage that is performed for each right-hand-side, with an asymptotic cost ${\cal O}(N\log{N})$. Furthermore, the on-line stage has two levels: the inner level in which the approximate sparse system is solved iteratively using a bi-directional preconditioner, and the outer level, in which the resulting solution from the inner level is used to precondition the Lippmann-Schwinger equation.
Under the assumptions that the perturbation is smooth and it does not represent a resonant cavity the number of outer iterations is essentially independent of the frequency \cite{Ying:Sparsifying_Preconditioner_for_the_Lippmann--Schwinger_Equation}, and numerical experiments presented in this paper, indicate that the number of inner iterations depends, at worst, logarithmically with the frequency, resulting in the aforementioned complexity.
\subsection{Results}
In this paper we propose a new two-level preconditioner for the 2D high-frequency Lippmann-Schwinger equation that results in a method with ${\cal O}(N \log{N})$ complexity. We follow the approach in \cite{Ying:Sparsifying_Preconditioner_for_the_Lippmann--Schwinger_Equation} to construct a sparsifying preconditioner, and we develop a fast iterative solver to solve the sparsified system, in order to achieve an overall optimal complexity. The main improvement with respect to the sparsifying preconditioner is how the resulting sparse linear system is solved. In \cite{Ying:Sparsifying_Preconditioner_for_the_Lippmann--Schwinger_Equation} the system is solved via multi-frontal methods \cite{Duff_Reid:The_Multifrontal_Solution_of_Indefinite_Sparse_Symmetric_Linear,GeorgeNested_dissection} incurring an asymptotic cost ${\cal O}(N^{3/2})$ for the setup and ${\cal O}(N\log{N})$ for the application. Exchanging the multifrontal solver by an iterative one, preconditioned using a bi-directional variant of the method of polarized traces, results in a drastic reduction of the asymptotic cost.
The inner-level preconditioner is a matrix-free variant of the method of polarized traces \cite{Zepeda_Demanet:Nested_domain_decomposition_with_polarized_traces_for_the_2D_Helmholtz_equation}, which has been adapted to solve the sparsified Lippmann-Schwinger equation efficiently. As it will be explained in the sequel, preconditioning the sparsified system using the original method of polarized traces results in an algorithm that is sensitive to the sign of the perturbation and to the direction of the incident waves. Then a new bi-directional variant is introduced to eliminate these adverse effects.
The method of polarized traces is a domain decomposition method \cite{ZepedaDemanet:the_method_of_polarized_traces}, which relies on a layered domain decomposition coupled with an equivalent surface integral equation (SIE) that is easy to precondition. The matrix-free variant introduced in \cite{Zepeda_Demanet:Nested_domain_decomposition_with_polarized_traces_for_the_2D_Helmholtz_equation} uses local solves on extended subdomains in order to apply the SIE and the preconditioner. The number of iterations to convergence depends on the underlying medium and the quality of the absorbing boundary conditions (ABC) imposed at the interfaces between subdomains. For the Helmholtz problem, there exist a myriad of different formulations of ABC \cite{Berenger:PML,Bermudez:Perfectly_Matched_Layers_for_Time-Harmonic_Second_Order_Elliptic_Problems,Engquist:Absorbing_boundary_conditions_for_the_numerical_simulation_of_waves,Johnson:PML}; unfortunately, for the sparse system resulting from sparsifying the Lippmann-Schwinger equation such ABC are not readily available. We give details on building such ABC for the sparsified Lippmann-Schwinger system in Section \ref{section:ABC}.
Alas, the formulation of the ABC in this paper is not perfect, which is evidenced by numerical tests showing that the number of iteration needed to convergence using the matrix-free version of the method of polarized traces grows sub-optimally as ${\cal O}(\sqrt{\omega})$. Optimality can be restored by introducing an alternating bi-directional sweep, and by preconditioning the volume problem directly, which reduces the number of iterations to ${\cal O}(\log{\omega})$. To show the effectiveness of our proposed method, we also include an example arising from plasma physics, in particular, an electromagnetic wave impinging on a confined plasma in a fusion reactor.
In the present work we mainly focus on explaining the key ideas and verification of the method by 2D tests. The algorithm presented in this paper can be easily extended to 3D, with an offline complexity of ${\cal O}(N^{4/3})$ and an on-line complexity of ${\cal O}(N \log^2{N})$. The complexities can be further reduced using a nested approach such as the one presented in \cite{Liu_Ying:Recursive_sweeping_preconditioner_for_the_3d_helmholtz_equation,Zepeda_Demanet:Nested_domain_decomposition_with_polarized_traces_for_the_2D_Helmholtz_equation} resulting in ${\cal O}(N)$ and ${\cal O}(N \log{N})$ complexity respectively.
\subsection{Outline of the paper} The present paper is organized as follows: we present the discretization of the Lippmann-Schwinger equation and the provide a brief review of the sparsifying preconditioner in Section \ref{Section:sparsifying_preconditioner}. In Section 3, we review the method of polarized traces and we discuss the modifications needed to adapt it to the sparsified Lippmann-Schwinger equation, in particular we explain the formulation for the ABC in Section \ref{section:ABC} and the bi-directional preconditioner in Section \ref{section:bidirectional_algorithm}. In Section \ref{section:preconditioner} we present the two-level preconditioner and we discuss its complexity. Finally, in Section \ref{section:numerical_experiments}, we present various numerical experiments to support the complexity claims.
\section{Discretization and sparsifying preconditioner} \label{Section:sparsifying_preconditioner}
In this section we discuss the discretization used and we provide a brief review of the sparsifying preconditioner \cite{Ying:Sparsifying_Preconditioner_for_the_Lippmann--Schwinger_Equation}. We refer the interested reader to Section 2 of \cite{Ying:Sparsifying_Preconditioner_for_the_Lippmann--Schwinger_Equation} for further details.
We discretize the domain $\Omega$ with an uniform mesh\footnote{If the perturbation has sharp transitions the sparsifying preconditioner can be modified to handle adaptive quadratures without asymptotic penalty. However, the convolution product will have to be handled differently \cite{Beylkin:Fast_convolution_with_the_free_space_Helmholtz_Greens_function} and the construction of the preconditioner will require randomized techniques in order to keep the same asymptotic complexity.} of a step size $h$, such that we have 10 points per wavelength in the background wavespeed, $h\omega = {\cal O}(1)$. We denote the collocations points by $\mathbf{x}_{i,j} = (x_i, z_j) = (hi, hj)$ for $1 \leq i \leq n_x$ and $1 \leq j \leq n_z$, where $n_x$ and $n_z$ are the number of grid-points in each direction. In addition, we define a vector index $\mathbf{k} = (i,j)$, notation that will be useful to explain the sparsifying preconditioner. We use a simple modification of the trapezoidal rule \cite{Duan-Rohklin:High-order_quadratures_for_the_solution_of_scattering_problems_in_two_dimensions} to discretize the integral within the convolution operation in \eqref{eq:Lippmann-Schwinger}. In particular, we use a quadrature that only modifies the weight in the diagonal, obtaining a quadrature rule with order ${\cal O}(h^4 \log{h})$.
We denote the unknown scattered field in $\Omega$ as $\mathbf{u}_{\mathbf{k}} = \mathbf{u}_{i,j} = u(\mathbf{x}_{i,j})$, and we write the resulting discretized linear system as \begin{equation} \label{eq:discrete_Lipmann-Schwinger}
\mathbf{H} \mathbf{u} = \mathbf{f}, \end{equation} in which $\mathbf{H}$ can be written as \begin{equation}
\mathbf{H} = \mathbf{I} + \omega^2 \mathbf{G} \cdot \mathbf{D}, \end{equation} where $\mathbf{I}$ is the identity matrix, and $\mathbf{D}$ is a diagonal matrix that encodes the multiplication of $\mathbf{u}$ times $m(\mathbf{x}_{i,j})$. Finally, $\mathbf{f}$ is given by \begin{equation}
\mathbf{f} = -\omega^2 \mathbf{G} \cdot \mathbf{D} \mathbf{u}_{I}, \end{equation} with $ (\mathbf{u}_{I})_{i,j} = u_{I}(\mathbf{x}_{i,j})$.
Given that the convolution kernel is translation invariant and that the mesh is uniform we can easily embed $\mathbf{G}$ in a Toeplitz matrix and use the FFT to apply it in ${\cal O}(N\log{N})$ operations (or we can use the method in \cite{Beylkin:Fast_convolution_with_the_free_space_Helmholtz_Greens_function} if the mesh is non-uniform).
The sparsifying preconditioner is based on designing a sparse matrix $\mathbf{A}$ such that the product $\mathbf{A} \mathbf{H}$ is approximately sparse in the sense that a large amount of it entries are extremely small. The main idea is to multiply $\mathbf{A}$ by $\mathbf{H}$, and threshold all the small entries of the product to zero, obtaining a sparse matrix $ \mathbf{C} $ that approximates $\mathbf{A} \mathbf{H} $. As it will be explained in the sequel, the sparsity pattern of $\mathbf{A}$ is the same as a second-order compact finite difference discretization of the Laplacian, and its entries are specially constructed to annihilate the free space Green's function away from the source. In return, the matrix $\mathbf{C}$, which shares the same sparsity pattern as $\mathbf{A}$, is similar to the matrix arising from a finite differences discretization of the Helmholtz equation \eqref{eq:Helmholtz}.
Since
$ \mathbf{A} \mathbf{H} = \mathbf{A} + \omega^2 \mathbf{A} \mathbf{G} \mathbf{D}$,
it follows that to make $\mathbf{A} \mathbf{H}$ approximately sparse, we only need to find $\mathbf{A}$ such that the product $\mathbf{A} \mathbf{G}$ is approximately sparse. The last requirement is equivalent to enforce that the entries of the matrix $\mathbf{A}$ approximately annihilate the off-diagonal blocks of $\mathbf{G}$. This requirement can be recast as a minimization problem, which is solved via a singular value decomposition (SVD).
Define the set of immediate neighbors (stencils) of $\mathbf{k}$ as \begin{equation}
\mu(\mathbf{k}) = \{ \mathbf{p} : \| \mathbf{k} - \mathbf{p} \| _{\ell_{\infty}({\mathbb N}^2)} \leq 1, \mathbf{x}_{\mathbf{p}} \in \Omega \}. \end{equation}
We seek $\mathbf{A}$ such that for each $\mathbf{k}$: \begin{itemize}
\item only $\mathbf{A}(\mathbf{k}, \mu(\mathbf{k}))$ are non-zeros (sparsity of $\mathbf{A}$);
\item $\mathbf{A}(\mathbf{k}, :)\mathbf{G}(:, \mu(\mathbf{k})^{c}) \approx 0$ (approximate sparsity of the $\mathbf{A}\mathbf{G}$). \end{itemize}
Given that $\mathbf{G}$ is translation invariant we impose $\mathbf{A}$ to be translation invariant. In such case, it is sufficient to consider only the case when $\mathbf{k} = 0$. Then the problem can be recast as \begin{equation}
\min_{\alpha: \|\alpha \| =1} \|\alpha \cdot \mathbf{G}(\mu(0), \mu(0)^c ) \|, \end{equation} which can be efficiently solved via a singular value decomposition of $\mathbf{G}(\mu(0), \mu(0)^c ) = U \Sigma V^*$ such that \begin{equation}
\alpha = U(:, \mu(0))^*. \end{equation} Finally, $\mathbf{A}$ is built such that $\mathbf{A}(\mathbf{k}, \mu(\mathbf{k})) = \alpha$, for $\mathbf{k}$ in the interior of $\Omega$. For $\mathbf{k}$ on $\partial \Omega$ a special treatment is needed, which depends on the geometry of $\Omega$. For the sake of simplicity we omit such treatment but we refer the interested reader to Section 4 of \cite{Ying:Sparsifying_Preconditioner_for_the_Lippmann--Schwinger_Equation}.
By multiplying \eqref{eq:discrete_Lipmann-Schwinger} by $\mathbf{A}$ and sparsifying the product, we obtain the approximated {\em sparse} system \begin{equation}
\mathbf{C} \mathbf{u} \approx \mathbf{A} \mathbf{f}, \end{equation} for which the preconditioner is defined as \begin{equation} \label{eq:preconditioner}
\mathbf{u}_{approx} = \mathbf{C}^{-1} {\mathbf{g}}, \qquad \mbox{where } {\mathbf{g}} = \mathbf{A} \mathbf{f} . \end{equation}
It has been shown in \cite{Ying:Sparsifying_Preconditioner_for_the_Lippmann--Schwinger_Equation} that when the preconditioner mentioned above is used within GMRES iterations, the number of iterations to convergence is almost independent of the frequency.
One alternative to this preconditioner would be to discretize \eqref{eq:Helmholtz_2} via finite differences or finite elements, and then use this approximate solution to precondition \eqref{eq:discrete_Lipmann-Schwinger}. Unfortunately, such preconditioners tend to behave poorly, due to the pollution error that is normally introduced with such discretizations. The pollution error creates serious phase errors making the preconditioner incompatible with the linear system being preconditioned (see \cite{Stolk} for a similar example for multigrid). The optimization procedure used in the construction of the sparsifying preconditioner takes into account the exact free space Green's function and hence minimizes the dispersion error. The size of the stencils used in $\mathbf{A}$ and $\mathbf{C}$ should strike a balance between sparsity and dispersion error. Since the matrix $\mathbf{A}$ mimics the discretization of the Helmholtz differential operator, it is expected that the ideas for fast solver for the Helmholtz equation can be applied to the resulting sparse system. This is what we propose in the next Section.
\section{Method of polarized traces} \label{sec:polarized-traces} As seen from Section \ref{Section:sparsifying_preconditioner}, the sparsifying preconditioner for the Lippmann-Schwinger equation relies on solving the sparsified system \begin{equation} \label{eq:global_sparse_system}
\mathbf{C}\mathbf{v} = {\mathbf{g}}, \end{equation} which, in the present paper, is solved fast using a bi-directional variant of the method of polarized traces. In this section, we first briefly introduce the matrix-free version of the method of polarized traces \cite{Zepeda_Demanet:Nested_domain_decomposition_with_polarized_traces_for_the_2D_Helmholtz_equation} in one direction, and then we discuss the modifications needed to adapt it to the sparsified Lippmann-Schwinger equation. We provide a simple description of the three key elements of the method of polarized traces for the Helmholtz equation in the following order: \begin{enumerate}
\item the layered domain decomposition with the extended subdomains;
\item how to define the transmission conditions between subdomains using a discrete Green's representation formula issued from an algebraic decomposition;
\item how to implement an absorbing boundary conditions for the subdomains to obtain a fast converging iterative method. \end{enumerate}
We then provide the full algorithm for the one-directional preconditioner; we provide its physical intuition and discuss its short-comings, which are rectified by the bi-directional preconditioner that is introduced right after.
\subsection{Domain decomposition}
The domain $\Omega$ is partitioned in $L$ horizontal layers $\{ \Omega^{\ell} \}_{\ell =1}^L$ as depicted in Fig.~\ref{fig:DDM_sketch}. Following \cite{ZepedaDemanet:the_method_of_polarized_traces} we extend each subdomain, which we denote by $\{ \Omega^{\ell}_{ext} \}_{\ell =1}^L$. Within each extended subdomain we assemble the discrete local problem \begin{equation} \label{eq:local_discrete_sparse_pde}
\mathbf{C}^{\ell} {\mathbf{w}}^{\ell} = {\mathbf{g}}^{\ell} = \chi_{\Omega^{\ell}} {\mathbf{g}}, \qquad \mbox{for } \ell = 1,..., L; \end{equation} where $ \chi_{\Omega^{\ell}}$ is the characteristic function of $\Omega^{\ell}$.
We impose $\mathbf{C}^{\ell}$ to satisfy the {\it compatibility condition}, i.e., $\mathbf{C}^{\ell}$ coincides exactly with $\mathbf{C}$ in the interior of $\Omega^{\ell}$. Within $\Omega^{\ell}_{ext} - \Omega^{\ell}$, i.e., the extension of the subdomain, we do not require $\mathbf{C}^{\ell}$ to coincide with $\mathbf{C}$, instead we use the extra degrees of freedom to impose an analogue to an absorbing boundary condition. Given that $\mathbf{C}$ in the sparsified Lippmann-Schwinger system \eqref{eq:global_sparse_system} does not arise from a direct discretization of the Helmholtz equation, the boundary condition for subdomains needs a different treatment, which is presented in Section \ref{section:ABC}.
Fig.~\ref{fig:DDM_sketch} shows an example of such decomposition. At the left we have all the degrees of freedom, in the middle we have the domain decomposed in several layers, and at the right we have the extended domains. We represent in gray the grid points whose associated entries of each local matrix remain the same, and in orange the grid points that correspond to the extended degrees of freedom. Note that the entries of $\mathbf{C}^{\ell}$ associated to these extended grid points will be modified in order to impose an ABC. \begin{figure}
\caption{Layered domain decomposition. The orange grid-points represent artificial grid points in the extended domain.}
\label{fig:DDM_sketch}
\end{figure}
We point out that the bidirectional preconditioner uses two of such decompositions: one in which the domain is decomposed in horizontal layers, such as in Fig.~\ref{fig:DDM_sketch}, and one in which the domain is decomposed in vertical layers. Alternating sweeping will be performed on these two settings.
\subsection{Algebraic decomposition}
As for any domain decomposition method, the objective is to solve the global problem \eqref{eq:global_sparse_system} by solving the local problems \eqref{eq:local_discrete_sparse_pde} iteratively, which requires transmission conditions at the interfaces between subdomains. In the case of high-frequency problems, due to possible pollution errors \cite{Sauter_Babuska:Is_the_Pollution_Effect_of_the_FEM_Avoidable_for_the_Helmholtz_Equation_Considering_High_Wave_Numbers}, such transmissions conditions need to be of higher order or algebraically exact with respect to the discrete global problem, in order to ensure a fast convergence. One advantage of the method of polarized traces is that such conditions can be seamlessly imposed at an algebraic level, bypassing the need to define optimized transmission conditions \cite{Gander:Optimized_Schwarz_Methods,Gander_Nataf:Optimized_Schwarz_Methods_without_Overlap_for_the_Helmholtz_Equation}.
In order to impose algebraically exact transmission conditions between layers, we use the blocks of $\mathbf{C}$ to define a discrete Green's representation formula. Such a formula can be deduced by imposing a discontinuous local solution, as it was performed in \cite{Stolk:An_improved_sweeping_domain_decomposition_preconditioner_for_the_Helmholtz_equation,Zepeda_Demanet:Nested_domain_decomposition_with_polarized_traces_for_the_2D_Helmholtz_equation}.
By construction we have that \begin{equation} \scriptsize \label{eq:Helmholtz_matrix}
\mathbf{C} = \left [ \begin{array}{cccccc}
\mathbf{C}_{1,1} & \mathbf{C}_{1,2} & & & \\
\mathbf{C}_{2,1} & \mathbf{C}_{2,2} & \mathbf{C}_{2,3} & & \\
& \ddots & \ddots & \ddots & \\
& & \ddots & \ddots & \mathbf{C}_{n_z-1, n_z} \\
& & & \mathbf{C}_{n_z, n_z-1} & \mathbf{C}_{n_z,n_z}
\end{array}
\right ], \end{equation}
in which all the sub-matrices are sparse. Given that the mesh is uniform we use the following ordering of the unknowns \begin{equation} \label{eq:global_ordering} \mathbf{v} = (\mathbf{v}_1, \mathbf{v}_2, ..., \mathbf{v}_{n_z}), \end{equation} where $\mathbf{v}_j \in {\mathbb C}^{n_x}$ is given by \begin{equation} \label{eq:trace_ordering} \mathbf{v}_j = (v_{1,j}, v_{2,j},..., v_{n_x,j}), \end{equation} which corresponds to $\mathbf{v}$ sampled at a constant $z$.
We write $\mathbf{v}^{\ell}$ for the wavefield defined locally at the $\ell$-th layer, i.e., $\mathbf{v}^{\ell} = \chi_{\Omega^{\ell}}\mathbf{v}$, and $\mathbf{v}^{\ell}_k$ for the values at the local depth $z_k^{\ell}$ of $\mathbf{v}^{\ell}$.
In particular, $\mathbf{v}^{\ell}_1$ represents the wavefield sampled at the bottom of the layer, and $\mathbf{v}^{\ell}_{n^{\ell}}$ is the wavefield at the top of the layer. Moreover, $\mathbf{v}^{\ell}_0$ is the local wavefield sampled immediately below the bottom of the layer, in the extended region within the PML; $\mathbf{v}^{\ell}_{n^{\ell}+1}$ is the local wavefield sampled immediately above the top of the layer.
Finally, we use the notation $\delta_{k}$ to denote a discrete measure. For a generic vector $ \mathbf{p} \in {\mathbb C}^{n_x}$, $\delta_{k} \mathbf{p} \in {\mathbb C}^{n_x\times n_z}$ is defined as follows: \begin{equation}
(\delta_{k} \mathbf{p})_{i,j} = \left \{ \begin{array}{cl}
\mathbf{p}_i & \mbox{if } j = k, \\
0 & \mbox{if } j\neq k.
\end{array} \right .
\end{equation} This notation will be used extensively to characterize the forcing terms necessary to transfer the information from one subdomain to its neighbors.
The correct Green's representation formula is given by the equivalent and more manageable expression\footnote{It is possible to recover the Green's representation formula by multiplying \eqref{eq:pde_GRF_real} by the inverse of $ \mathbf{C}^{\ell} $}
\begin{align} \label{eq:pde_GRF_real}
\mathbf{C}^{\ell} {\mathbf{w}}^{\ell} = & - \delta_{n^{\ell}} \mathbf{C}^{\ell}_{n^{\ell},n^{\ell}+1} \mathbf{v}^{\ell}_{n^{\ell}+1} + \delta_{n^{\ell}+1} \mathbf{C}^{\ell}_{n^{\ell}+1,n^{\ell}} \mathbf{v}^{\ell}_{n^{\ell}} \\
& - \delta_{1} \mathbf{C}^{\ell}_{1,0} \mathbf{v}^{\ell}_{0} + \delta_{0} \mathbf{C}^{\ell}_{0,1} \mathbf{v}^{\ell}_{1} + {\mathbf{g}}^{\ell} . \nonumber \end{align}
It can be shown that if the traces of the global $\mathbf{v}$, i.e. $\mathbf{C} \mathbf{v} = \mathbf{f}$, are known and used in \eqref{eq:pde_GRF_real}, then $ {\mathbf{w}}^{\ell}$ coincides exactly with $\mathbf{v}$ inside the layer, i.e., $ {\mathbf{w}} = \chi_{\Omega^{\ell}} \mathbf{v}$ (see Appendix C of \cite{ZepedaDemanet:the_method_of_polarized_traces} for a rigorous proof for matrices arising from finite differences). In \cite{Zepeda_Demanet:Nested_domain_decomposition_with_polarized_traces_for_the_2D_Helmholtz_equation} the above formula is used to build an equivalent surface integral formulation in a matrix-free fashion, which is then solved iteratively. The convergence is fast provided that high-quality ABC are implemented, which is the topic of the next subsection.
\subsection{Absorbing boundary conditions} \label{section:ABC} One of the most important elements of the method of polarized traces is the implementation of ABC at the interfaces between subdomains. ABC have successfully been used in domain decomposition for elliptic equations \cite{Engquist_Zhao:Absorbing_boundary_conditions_for_domain_decomposition}, and a simpler version of ABC were extensively used in the first domain decomposition method for the Helmholtz equation to ensure convergence \cite{Despres:domain_decomp}. High-quality absorbing conditions between subdomains are key to the fast convergence of iterative solvers for the high-frequency Helmholtz equation \cite{Chen_Xiang:a_source_transfer_ddm_for_helmholtz_equations_in_unbounded_domain,Cheng_Xiang:A_Source_Transfer_Domain_Decomposition_Method_For_Helmholtz_Equations_in_Unbounded_Domain_Part_II_Extensions,EngquistYing:Sweeping_PML,Leng:A_Fast_Propagation_Method_for_the_Helmholtz_equation,Leng_Ju:An_Overlapping_Domain_Decomposition_Preconditioner_for_the_Helmholtz_equation,Liu_Ying:Additive_Sweeping_Preconditioner_for_the_Helmholtz_Equation,Liu_Ying:Recursive_sweeping_preconditioner_for_the_3d_helmholtz_equation,CStolk_rapidily_converging_domain_decomposition,Stolk:An_improved_sweeping_domain_decomposition_preconditioner_for_the_Helmholtz_equation,GeuzaineVion:double_sweep}.
The high-quality ABC are needed in order to minimize the artificial reverberations at the interfaces between subdomains that arise from the truncation introduced by the domain decomposition. One of the objectives of the ABC, is to make the local Green's functions, which are defined by the local problem \eqref{eq:local_discrete_sparse_pde}, as close as possible to the global Green's function restricted to the subdomain without scattering outside the subdomain. In other words, the local Green's functions should capture the local interactions within the subdomain, but should not contain pollution effects due to artificial reflections at the truncated boundary. Given that the medium outside the subdomain is not involved in the local system, the local Green's functions do not capture those long range interactions among subdomains; these interactions will be captured in an iterative manner during the sweeps.
In practice, without careful treatment at the truncated boundaries, the artificial reverberations can heavily pollute the local Green functions and hinder the convergence of the iterative method. For example, the adverse effect of the artificial reverberations can be overwhelming for the cases when Dirichlet or Neumann boundary conditions are imposed at the boundaries between subdomains \cite{GeuzaineVion:double_sweep,Gander:why_is_difficult_to_solve_helmholtz_problems_with_classical_iterative_methods}, causing the method to converge in ${\cal O}(\omega)$ iterations, thus resulting in a super-linear complexity.
Given that the matrix $\mathbf{C}$ in our problem does not arise from a direct discretization of the Helmholtz equation, absorbing boundary conditions are not readily available. A natural way is to localize the perturbation, $m(\mathbf{x})$, to the subdomain via a smooth cut-off. The localization produces a local discrete system in the form of \eqref{eq:local_discrete_sparse_pde} that satisfies the compatibility conditions.
Now, we proceed to give an explanation of how to define the local problems \eqref{eq:local_discrete_sparse_pde} using the localization to obtain an approximated ABC. Intuitively, the solution to \eqref{eq:local_discrete_sparse_pde} should provide an approximation of the wave scattered by the perturbation supported in $\Omega^{\ell}$. Thus, one naive approach would be to define the local problem as \begin{equation}
{\cal H}^{\ell}u^{\ell} := u^{\ell} + \omega^2 G*(\chi_{\Omega^{\ell}} m u^{\ell}), \end{equation} and then use the sparsifying preconditioner to obtain a sparse local problem in the form of \eqref{eq:local_discrete_sparse_pde}.
However, the local perturbation $\chi_{\Omega^{\ell}} m$ is discontinuous at the interfaces between subdomains, which will produce strong artificial reflections for waves impinging on the artificial boundary of $\Omega^{\ell}$. These strong reflections will pollute the local Green's function and result in a slow convergence. Fortunately, this adverse effect can be easily rectified by using a window, or cut-off, function that transitions the perturbation smoothly to zero.
Define a set of smooth cut-off functions $\{ \xi^{\ell}(\mathbf{x}) \}_{\ell = 1}^{L}$, such that \begin{equation}
\xi^{\ell}(\mathbf{x}) = \left \{ \begin{array}{cl}
1 &, \mbox{if } \mathbf{x} \in \Omega^{\ell};\\
0 &, \mbox{if } \mathbf{x} \notin \Omega^{\ell}_{ext}.
\end{array} \right . \end{equation} We can define the continuous counterpart of \eqref{eq:local_discrete_sparse_pde} as the problem \begin{equation} \label{eq:local_Lippmann-Schwinger}
{\cal H}^{\ell}u^{\ell} := u^{\ell} + \omega^2 G*(\xi^{\ell} m u^{\ell}). \end{equation} which is posed for $\mathbf{x} \in \Omega$. We recall that the matrix $\mathbf{D}$, is a diagonal matrix that encodes the multiplication by $m(\mathbf{x}_{i,j})$; we define $\mathbf{D}^{\ell}$ as the diagonal matrix that represents the multiplication by $\xi^{\ell}(\mathbf{x})m(\mathbf{x})$. We can define the discrete local Lippmann-Schwinger equation as, \begin{equation} \label{eq:whole_discrete_local_Lippmann-Schwinger} \mathbf{H}^{\ell}_{global} = \mathbf{I} + \omega^2 \mathbf{G} \cdot \mathbf{D}^{\ell}. \end{equation} The global subscript indicates that $\mathbf{H}^{\ell}_{global}$ is a $N \times N$ matrix. In this case we can multiply \eqref{eq:whole_discrete_local_Lippmann-Schwinger} by $\mathbf{A}$, and we would obtain a local system that satisfies the compatibility condition of \eqref{eq:local_discrete_sparse_pde}. However, this would be prohibitively expensive given that would need to solve a $N \times N$ system for each subdomain.
The key observation is that only a small portion of $\mathbf{H}^{\ell}_{global}$ in \eqref{eq:whole_discrete_local_Lippmann-Schwinger} deviates from an identity matrix. This is caused by the small support of $\xi^{\ell} m$, in which $ \mbox{supp}(\xi^{\ell} m )\subset \Omega^{\ell}_{ext}$. Thus, we can re-write \eqref{eq:local_Lippmann-Schwinger} in $\Omega^{\ell}_{ext}$ instead of $\Omega$, and discretize it resulting in \begin{equation} \label{eq:discrete_local_Lippmann-Schwinger} \mathbf{H}^{\ell} = \mathbf{I} + \omega^2 \mathbf{G} \cdot \mathbf{D}^{\ell}, \end{equation} which is a quasi-1D (see \cite{EngquistYing:Sweeping_PML}) $q n \times q n $ system, where $q$ is the thickness, in grid points, of each extended subdomain. In the implementation of the method, the cut-off is constant in the direction tangential to the boundary, $x$ direction in our layered setting, and it is a third order spline in the direction normal to the boundary, $z$ direction in our setting.
In theory, we could apply the sparsifying preconditioner to \eqref{eq:discrete_local_Lippmann-Schwinger}; however, given that the mesh is also truncated, the optimization via a SVD can provide different results, hindering the compatibility condition. Instead, we multiply \eqref{eq:discrete_local_Lippmann-Schwinger} by the corresponding block of $\mathbf{A}$, but with a special care at the boundaries of $\Omega^{\ell}_{ext}$, where we use the stencil generated at the boundary of $\Omega$, which provides a satisfying ABC \cite{Ying:Sparsifying_Preconditioner_for_the_Lippmann--Schwinger_Equation} for the background wave-speed at the outer boundary of the extended subdomain.
Unfortunately, the smooth cut-off used to localize the perturbation is not equivalent to an ABC, which would approximate the Dirichelet to Neumann (DtN) map at the boundary; such that it can absorb and dampen waves propagating out of the boundary in all directions with little reflections. The smooth cut-off in the normal direction to the boundary minimally affects waves propagating out of the domain close to the normal direction. However, for waves propagating near and almost tangential to the boundary, a smooth cut-off may not be good enough depending on the sign of the perturbation as illustrated in Figures \ref{fig:Green_functions+} and \ref{fig:Green_functions-}. If the perturbation is negative, the waves propagate faster inside the subdomain than in the cut-off region. If a wave propagates almost tangentially to the boundary to the cut-off region it will bend outwards, and it will be ultimately absorbed by the numerical ABC situated at the artificial boundary of the extended subdomain. On the other hand, if the perturbation is positive, then the waves propagate slower inside the subdomain than in the background media. Under these circumstances, the subdomain with a smooth cut-off to the background media creates an artificial wave-guide effect at the subdomain boundary. Waves propagating almost normal to the boundaries will still cross the smooth cut-off and get absorbed by the numerical ABC at the artificial boundary of the extended subdomain. However, waves propagating almost transversally along the subdomain boundary will bend inwards due to the artificial cut-off, producing spurious grazing waves that pollute the local solution within the subdomain. This last problem can be attenuated to some extent by introducing a ${\cal O}(\omega)$ complex shift within the cut-off region, thus dampening the grazing waves. We point out that we are only introducing a complex shift in the extended domain; it would be possible to introduce a complex shift in $\Omega$ as in \cite{Graham:Domain_Decomposition_preconditioning_for_high-frequency_Helmholtz_problems_using_absorption}, however, that would violate the compatibility condition.
\begin{figure}
\caption{Example of a global (top) and local (bottom) Green's function for a positive perturbation, we can observe that there is some interference in the local Green's function given by the spurious grazing wave that is concentrated at the top of the slab.}
\label{fig:Green_functions+}
\end{figure}
\begin{figure}
\caption{Example of a global (top) and local (bottom) Green's function for a negative perturbation, we can observe some differences between the Green's functions, but we do not observe the kind of interference, nor the concentration at the boundaries, as observed in Fig.~\ref{fig:Green_functions+}.}
\label{fig:Green_functions-}
\end{figure}
Although the complex shift helps to reduce the number of iterations for the original method of polarized traces from ${\cal O}(\omega)$ to ${\cal O}(\sqrt{\omega})$, which is better than standard iterative methods, it is not optimal. In order to obtain the lower rate of ${\cal O}(\log{\omega})$ iterations to converge, we introduce in the next section an effective bi-directional alternating sweeping preconditioner to handle those grazing waves due to either the incoming waves or the smooth cut-off at the artificial boundaries between two subdomains in one directional sweeping.
\subsection{One-directional algorithm} \label{section:onedirectional_algorithm}
In this section we discuss the preconditioner for the sparse problem based on the method of polarized traces for one-directional sweep and we provide the algorithm in pseudo-code with its physical interpretation.
The main intuition behind the method of polarized traces is the directionality of the waves, which are imposed by the polarizing conditions (see Section 3 of \cite{ZepedaDemanet:the_method_of_polarized_traces}) and the absorbing boundary conditions. For example, the wavefield generated by a local source, ${\mathbf{g}}^{\ell}$ \begin{equation} \mathbf{C}^{\ell} \mathbf{v}^{\ell} = {\mathbf{g}}^{\ell}, \end{equation} irradiates from the interior of $\Omega^{\ell}$ to the exterior of it. If we sample $\mathbf{v}^{\ell}$ at the bottom of $\Omega^{\ell}$, we can physically interpret it as the trace of a wavefield propagating downwards.
Following this physical interpretation, the wavefield propagating downwards from $\Omega^{\ell}$ propagates inwards $\Omega^{\ell-1}$. By using the Green's representation formula we can propagate the wavefield from $\Omega^{\ell}$ to $\Omega^{\ell-1}$ by artificial sources located at the top of $\Omega^{\ell-1}$ following \begin{align} \label{eq:pde_GRF_down}
\mathbf{C}^{\ell-1} \mathbf{v}^{\ell-1} = & - \delta_{n^{\ell-1}} \mathbf{C}^{\ell-1}_{n^{\ell-1},n^{\ell-1}+1} \mathbf{v}^{\ell}_{1} + \delta_{n^{\ell-1}+1} \mathbf{C}^{\ell-1}_{n^{\ell-1}+1,n^{\ell-1}} \mathbf{v}^{\ell}_{0} + {\mathbf{g}}^{\ell-1}, \end{align} which is an alternate expression for the incomplete Green's representation formula introduced in \cite{ZepedaDemanet:the_method_of_polarized_traces}.
We can observe that $\mathbf{v}^{\ell-1} $ contains the energy radiating from ${\mathbf{g}}^{\ell-1}$ and ${\mathbf{g}}^{\ell}$. If we sample $\mathbf{v}^{\ell-1}$ at the bottom of $\Omega^{\ell-1}$, it can be interpreted as the downwards wavefield consisting of the downwards wavefield generated by ${\mathbf{g}}^{\ell-1}$ and the wavefield generated by ${\mathbf{g}}^{\ell}$ that propagated through $\Omega^{\ell-1}$.
We can continue the propagation by adding the local wavefields in a manner that is usually called a multiplicative Schwarz iteration (see \cite{Gander_Nataf:Optimized_Schwarz_Methods_without_Overlap_for_the_Helmholtz_Equation}). Analogously, the same procedure can be performed for the wavefields propagating upwards. The application of both sweeps, downwards and upwards can be easily understood as a block Jacobi iteration as presented in \cite{ZepedaDemanet:the_method_of_polarized_traces}. Furthermore, the polarizing conditions can be further exploited to reduce the number of iterations by using a Gauss-Seidel approach described by Alg.~\ref{alg:GS_preconditioner}, which in practical terms reduces the number of iterations by a factor of two (see \cite{Zepeda_Demanet:Nested_domain_decomposition_with_polarized_traces_for_the_2D_Helmholtz_equation}).
All the intuitive arguments provided in the current exposition can be made rigorous from an algebraic point of view, we skip such explanation and redirect the interested readers to \cite{ZepedaDemanet:the_method_of_polarized_traces}.
\begin{algorithm} \small Gauss-Seidel Preconditioner \label{alg:GS_preconditioner}
\begin{algorithmic}[1]
\Function{ $\mathbf{v}$ = Preconditioner}{ ${\mathbf{g}}$ }
\For{ $\ell = L:-1:1$ } \Comment{Downwards Sweep}
\State $ {\mathbf{g}}^{\ell} = {\mathbf{g}} \chi_{\Omega^{\ell}} $ \Comment{Partition of the Source}
\If{ $\ell \neq 1$ }
\State $ {\mathbf{g}}^{\ell} += - \delta_{n^{\ell}} \mathbf{C}^{\ell}_{n^{\ell},n^{\ell}+1} \mathbf{w}^{\ell-1}_{1} + \delta_{n^{\ell}+1} \mathbf{C}^{\ell}_{n^{\ell}+1,n^{\ell}} \mathbf{w}^{\ell-1}_{0} $
\EndIf
\State $\mathbf{w}^{\ell} = (\mathbf{C}^{\ell})^{-1}{\mathbf{g}}^{\ell}$ \Comment{Local Solve}
\EndFor
\For{ $\ell = 1: L$ } \Comment{Upwards Sweep}
\If{ $\ell \neq L$ }
\State $ {\mathbf{g}}^{\ell} += - \delta_{1} \mathbf{C}^{\ell}_{1,0} \mathbf{v}^{\ell+1}_{n^{\ell}} + \delta_{0} \mathbf{C}^{\ell}_{0,1} \mathbf{v}^{\ell+1}_{n^{\ell}+1} $
\EndIf
\State $\mathbf{v}^{\ell} = (\mathbf{C}^{\ell})^{-1}{\mathbf{g}}^{\ell}$ \Comment{Local Solve}
\EndFor
\State $\mathbf{v} = \left (\mathbf{v}^{1}, \mathbf{v}^{2}, \mathbf{v}^{3}, ..., \mathbf{v}^{L-1}, \mathbf{v}^{L} \right)$ \Comment{Concatenation}
\EndFunction
\end{algorithmic} \end{algorithm}
\subsection{Bi-directional algorithm} \label{section:bidirectional_algorithm}
In this Section we first discuss briefly the shortcomings of Alg.~\ref{alg:GS_preconditioner} when applied to the sparsified Lippmann-Schwinger equation and then introduce the bi-directional preconditioner in order to achieve the optimal complexity.
The algorithm as presented in Alg.~\ref{alg:GS_preconditioner}, may not work optimally in our setting for the Lippmann-Schwinger equation, in which the perturbation is localized using a smooth cut-off. The main issue is that waves propagating nearly tangential to the artificial boundaries between subdomains are perturbed due to the cut-off and depending on the perturbation they can be bended inwards and thus generate spurious grazing waves. Even though the spurious grazing waves are dampened by the introduction of a complex shift in the cut-off region to some extent, they can still cause significant deviations of the local Green's functions with respect to the global ones. Thus, they hinder the fast convergence of the iterations, especially when the perturbation is positive, i.e., the wave-guide situation, as explained in Section \ref{section:ABC} and shown in Figures \ref{fig:Green_functions+} and \ref{fig:Green_functions-}.
Moreover, a closer inspection to Alg.~\ref{alg:GS_preconditioner} reveals that within the algorithmic pipeline, the residual given by $\mathbf{r} = \mathbf{C} \mathbf{v}_{precond} - {\mathbf{g}} $, where $ \mathbf{v}_{precond} = Preconditioner({\mathbf{g}}) $ are concentrated at the boundaries between subdomains. In particular, $\mathbf{r}$ can be interpreted in the continuous case as a measure supported at the boundaries between subdomains with most singular term of the form $\delta'(\mathbf{x})$. Then, as explained in Page 243 of \cite{CStolk_rapidily_converging_domain_decomposition}, the restriction of such measure to a subdomain (line 3 in Alg.~\ref{alg:GS_preconditioner}) is not well defined; such ambiguity ultimately results in a slow convergence rate. One way to solve the residual problem is presented in \cite{CStolk_rapidily_converging_domain_decomposition,Stolk:An_improved_sweeping_domain_decomposition_preconditioner_for_the_Helmholtz_equation,Zepeda_Hewett_Demanet:Preconditioning_the_2D_Helmholtz_equation_with_polarized_traces}, where an overlapping domain decomposition is used to ensure that at each iteration the residual lies at the interior of a subdomain.
In this work, we solve both problems simultaneously by introducing a simple alternating bi-directional preconditioner, that consists on applying an up-down sweep performed by Alg.~\ref{alg:GS_preconditioner} followed by an left-right pass sweep, performed by Alg.~\ref{alg:GS_preconditioner} but applied to the transpose of $\mathbf{C}$, which is denoted $\mbox{Preconditioner}^T$ in Alg.~\ref{alg:birirectional_preconditioner}. The introduction of the bi-directional preconditioner in Alg.~\ref{alg:birirectional_preconditioner} handles effectively the spurious grazing waves in one sweep by performing another sweep in the orthogonal direction. In addition, the application of the preconditioner in the orthogonal direction, after a step of iterative refinement confines the residual to the interior of the subdomains, thus eliminating the ambiguity and resulting in a fast convergence.
In particular, as it will be shown in the numerical tests, the bi-directional preconditioner is more efficient and less sensitive to the geometry, direction of incoming waves and sign of the perturbation, albeit with a higher memory footprint. The sweep in the orthogonal direction can easily be implemented as preconditioning the transpose of the matrix $\mathbf{C}$ using Alg.~\ref{alg:GS_preconditioner}. \begin{algorithm} \small Bi-directional Preconditioner \label{alg:birirectional_preconditioner}
\begin{algorithmic}[1]
\Function{ $\mathbf{v}$ = BiDirectionalPreconditioner}{ ${\mathbf{g}}$ }
\State $\mathbf{u}_1$ = Preconditioner $({\mathbf{g}})$ \Comment{Up-Down pass}
\State $\mathbf{e} = {\mathbf{g}} - \mathbf{C}\mathbf{u}_1$ \Comment{Iterative refinement}
\State $\mathbf{u}_2$ = Preconditioner$^{T}$ $(\mathbf{e})$ \Comment{Left-Right pass}
\State $\mathbf{v} = \mathbf{u}_1 + \mathbf{u}_2$
\EndFunction
\end{algorithmic} \end{algorithm}
\section{Two-level Preconditioner} \label{section:preconditioner} We briefly summarize the operations within the two levels of the preconditioner presented in this paper as follows:
\begin{itemize} \item the inner level, in which $u_{approx}$ in \eqref{eq:preconditioner} is computed approximately using the bi-directional preconditioner Alg.~\ref{alg:birirectional_preconditioner};
\item and, the outer level, in which \eqref{eq:discrete_Lipmann-Schwinger} is solved iteratively using GMRES preconditioned with Alg.~\ref{alg:Lipmann-Schwinger_preconditioner}. \end{itemize}
\begin{algorithm} \small Lippmann-Schwinger Preconditioner \label{alg:Lipmann-Schwinger_preconditioner}
\begin{algorithmic}[1]
\Function{ $\mathbf{u}$ = Lippmann-Schwinger Preconditioner}{ $\mathbf{f}$ }
\State $ u = \mbox{GMRES}( \mathbf{C} ,\mathbf{A} \mathbf{f}, BiDirectionalPreconditioner, tol = 10^{-3}) $
\EndFunction
\end{algorithmic} \end{algorithm}
There is a balance to the choice of the tolerance for the inner GMRES iteration (in Alg.~\ref{alg:Lipmann-Schwinger_preconditioner}): if the tolerance is chosen too small, then the constants are large; on the other hand, if the tolerance is too big, the constants will decrease but the number of outer iterations will grow with the frequency, thus increasing the asymptotic cost. The best choice in our case, is to set the inner loop tolerance to $10^{-3}$. This choice ensures that the amount of outer iterations will be the same as using a direct multi-frontal method and produces reasonable constants.
As mentioned in the introduction, the algorithm presented in this paper has two phases: an off-line phase performed only once, and an on-line phase performed for each right-hand-side, or incoming wave. The complexity of the solver is summarized in Table \ref{table:complexity}; in the sequel we provide further details for the complexity for each phase, in which we suppose the number of subdomains, L, is ${\cal O}(n)$ and that $q = {\cal O}(1)$.
\begin{table} \small \begin{center}
\begin{tabular}{|c|c|} \hline Stage & Complexity \\ \hline Off-line & ${\cal O}\left( N\right)$ \\ \hline On-line & ${\cal O} \left(n_{\mbox{iter}}^{\mbox{sp}} \left( N\log{N} + N \cdot n_{\mbox{iter}} \right) \right)$ \\ \hline \end{tabular} \end{center} \caption{Complexity for each state of the solver, supposing the $q$ ,the thickness of each extended local subdomain, is ${\cal O}(1)$.} \label{table:complexity} \end{table}
\subsection{Off-line complexity}
For the off-line or setup phase, we need to build $\mathbf{C}$ for which we need to compute $\mathbf{A}$. As explained in \cite{Ying:Sparsifying_Preconditioner_for_the_Lippmann--Schwinger_Equation} $\mathbf{A}$ can be constructed by solving only a hand-full of optimization problems. This property arises mainly by the translation invariance of the convolution kernel, meaning that we can reuse the same stencil for many grid-points. Solving the minimization problem needs a SVD of a $9 \times N$ matrix, which can be performed in ${\cal O}(N)$ complexity. To create $\mathbf{C}$ we need to multiply $\mathbf{A}$ times $\mathbf{H}$; exploiting the sparsity of $\mathbf{A}$ and performing the multiplication in the correct order, the product can be carried in ${\cal O}(N)$ operations.
We can build all the local matrices $\mathbf{C}^{\ell}$ in ${\cal O}(N)$ complexity by reusing $\mathbf{A}$; we build $L= {\cal O}(n)$ local matrices, and each $\mathbf{C}^{\ell}$ has ${\cal O}(n)$ non-zero elements. Finally, the factorization of all the local problems can be performed in ${\cal O}(N)$ complexity. Each local matrix is a circulant matrix and their factorization can be performed in ${\cal O}(n)$ time, and we have $L ={\cal O}(n)$ systems to factorize, which results in ${\cal O}(N)$ complexity. We point out that assembly and factorization of the local matrices is an embarrassingly parallel operation which can be effortlessly parallelized.
\subsection{On-line complexity}
For the outer GMRES we need to apply $\mathbf{H}$, which consists of multiplications by sparse matrices and a convolution that can be performed fast via a FFT (see section 2.3 of \cite{Duan-Rohklin:High-order_quadratures_for_the_solution_of_scattering_problems_in_two_dimensions}). The cost of applying $\mathbf{H}$ is dominated by the application of the FFT, resulting on an overall ${\cal O}(N \log{N})$ complexity. For Alg.~\ref{alg:GS_preconditioner} each local solve can be performed in ${\cal O}(n)$ time resulting in a complexity ${\cal O}(N)$. We have an overall complexity of ${\cal O}(N)$ for each application of Alg.~\ref{alg:GS_preconditioner}.
Finally, the overall on-line complexity depends on the number of outer and inner iterations to converge. From \cite{Ying:Sparsifying_Preconditioner_for_the_Lippmann--Schwinger_Equation} the number of outer iterations is almost independent of the frequency, $n_{\mbox{iter}}^{\mbox{sp}} = {\cal O}(1)$, and we provide in the next section numerical evidence that under the assumptions mentioned in the introduction, the number of inner iterations grows logarithmically with the frequency, $ n_{\mbox{iter}} = {\cal O}(\log{\omega})$, and given that we are in the high-frequency regime $ n_{\mbox{iter}} = {\cal O}(\log{N})$, resulting in the overall complexity mentioned in the introduction.
\section{Numerical Experiments} \label{section:numerical_experiments}
The numerical experiments were implemented in Julia v0.4.3 linked against Intel MKL 11.2, and were performed in a server with dual socket Intel Xeon CPU E5-2670, and 384 Gb of RAM. We used the FFT library FFTW3 \cite{FFTW:Johnson_Frigo} for the application of $\mathbf{H}$. For the local solves in the inner iteration we used the PARDISO \cite{Schenk:Solving_unsymmetric_sparse_systems_of_linear_equations_with_PARDISO} and UMFPACK \cite{Davis:UMFPACK} solvers.
One of the main advantages of the method of polarized traces lies on its modularity. We used the state-of-the-art library FFTW3 for the application of $\mathbf{H}$ but it is possible to further accelerate the application of $\mathbf{H}$ by using GPU-accelerated libraries (such as cuFFT \cite{cuda}), or a distributed-memory FFT libraries. Within the main bottleneck of the algorithm, which is the application of the preconditioner, it is possible to effortlessly exchange the linear solver. In our experiments we used PARDISO \cite{Schenk:Solving_unsymmetric_sparse_systems_of_linear_equations_with_PARDISO} and UMFPACK \cite{Davis:UMFPACK}, but it is easy to exchange them by solvers using compressed linear algebra \cite{Rouet_Li_Ghysels:A_distributed-memory_package_for_dense_Hierarchically_Semi-Separable_matrix_computations_using_randomization,Sivaram_Darve:HODLR,Xia:multifrontal,Gillman_Barnett_Martinsson:A_spectrally_accurate_solution_technique_for_frequency_domain_scattering_problems_with_variable_media}, or by massively parallel solvers such as MUMPS \cite{Amestoy_Duff:MUMPS} (or its compressed linear algebra variant \cite{Amestoy_Weisbecker:compressed_MUMPS}) or SuperLU-DIST \cite{Demmel_Li:superlu}, among many others.
In this section we corroborate the complexity claims for our method; in particular, present 3 different kinds of numerical experiments: \begin{enumerate} \item we study the asymptotic cost of solving sparsified systems from discretizations of the Lippmann-Schwinger equation using the bi-directional preconditioner; \item we solve a wave scattering example in three different medium profiles shown in Fig.~\ref{fig:scattering} using the full two level preconditioner, \item we solve an example arising from reflectometry of plasmas confined in a fusion reactor \cite{Weitzner:Lower_hybrid_waves_in_the_cold_plasma_model}.
\end{enumerate}
\subsection{Sparsified system} \begin{figure}
\caption{(left) Perturbation used for the wave scattering. (right) Wave-speed for 64 randomly placed Gaussian bumps}
\label{fig:scattering}
\end{figure}
First, we assemble two linear systems by sparsifying the Lippmann-Schwinger equation for the perturbation given in Fig.~\ref{fig:scattering} ({\it left}) and demonstrate the performance of the original method of polarized traces, and the new bi-directional preconditioner. The two sparsified Lippmann-Schwinger systems \eqref{eq:global_sparse_system} correspond to the same perturbation shown in Fig.~\ref{fig:scattering} ({\it left}) but with a different sign, to which we refer to as the positive and negative perturbation. We solve the resulting sparsified systems with GMRES (tolerance of $10^{-6}$) preconditioned with the method of polarized traces \cite{Zepeda_Demanet:Nested_domain_decomposition_with_polarized_traces_for_the_2D_Helmholtz_equation} and the bi-directional preconditioner in Alg.~\ref{alg:birirectional_preconditioner}. We solve \eqref{eq:global_sparse_system} for $64$ different right-hand sides representing $64$ different incident waves, and we compute an average time. We repeat the process for a set of increasing frequencies, such that we have $10$ points per wavelength in the background medium. Moreover, the domain is decomposed in $L = n/50$ subdomains, such that each subdomain is roughly $5$ wavelengths thick, and each subdomain is extended by 10 grid points in both directions (see Fig.~\ref{fig:DDM_sketch}). The average execution times are presented in Table \ref{table:numerical_experiment_bi_directional}.
\begin{table} \small \begin{center}
\begin{tabular}{|c|c|c|c|c||c|c| } \hline
N & $\omega$ & $L$ & $T^{+}_{PT}$ [s] & $T^{+}_{BD}$ [s] & $T^{-}_{PT}$ [s] & $T^{-}_{BD}$ \\ \hline $200 \times 200 $ & $200$ & $ 4 $ & $ 0.5$ $\mathbf{(3.6)}$ & $ 0.2$ $\mathbf{(2.7)}$ & $ 0.5$ $\mathbf{(3.7)}$ & $ 0.2$ $\mathbf{(2.9)}$ \\ $400 \times 400 $ & $400$ & $ 8 $ & $ 2.1$ $\mathbf{(4.2)}$ & $ 0.7$ $\mathbf{(3.1)}$ & $ 2.2$ $\mathbf{(4.3)}$ & $ 0.6$ $\mathbf{(3.0)}$ \\ $500 \times 500 $ & $500$ & $ 10 $ & $ 3.5$ $\mathbf{(4.6)}$ & $ 1.1$ $\mathbf{(3.2)}$ & $ 3.3$ $\mathbf{(4.5)}$ & $ 1.0$ $\mathbf{(3.0)}$ \\ $800 \times 800 $ & $800$ & $ 16 $ & $ 11.5$ $\mathbf{(5.7)}$ & $ 3.2$ $\mathbf{(3.8)}$ & $8.62$ $\mathbf{(4.8)}$ & $ 2.7$ $\mathbf{(3.1)}$ \\ $1000 \times 1000 $ & $1000$ & $ 20 $ & $ 20.4$ $\mathbf{(6.6)}$ & $ 5.1$ $\mathbf{(4.1)}$ & $14.0$ $\mathbf{(5.2)}$ & $ 4.2$ $\mathbf{(3.1)}$ \\ $1250 \times 1250 $ & $1250$ & $ 25 $ & $ 30.5$ $\mathbf{(7.9)}$ & $ 8.7$ $\mathbf{(4.3)}$ & $22.5$ $\mathbf{(5.4)}$ & $ 7.3$ $\mathbf{(3.3)}$ \\ $1400 \times 1400 $ & $1400$ & $ 28 $ & $ 43.5$ $\mathbf{(8.9)}$ & $11.1$ $\mathbf{(4.7)}$ & $29.1$ $\mathbf{(5.6)}$ & $ 8.7$ $\mathbf{(3.4)}$ \\ $1600 \times 1600 $ & $1600$ & $ 32 $ & $ 62.8$ $\mathbf{(10.1)}$ & $15.8$ $\mathbf{(4.7)}$ & $41.8$ $\mathbf{(5.8)}$ & $12.6$ $\mathbf{(3.4)}$ \\ $2000 \times 2000 $ & $2000$ & $ 40 $ & $120.5$ $\mathbf{(12.6)}$ & $25.0$ $\mathbf{(5.3)}$ & $63.8$ $\mathbf{(6.1)}$ & $18.1$ $\mathbf{(3.5)}$ \\ \hline \end{tabular} \end{center} \caption{Average number of iterations (in parenthesis) and average execution time to solve the sparsified system arising from the Lipmann-Schwinger equation using the positive (+) and negative (-) perturbation, using either the method of polarized traces (PT), or the new bi-directional preconditioner(BD). We use GMRES with a tolerance of $10^{-6}$, in this case $N = n^2$ is the number of degrees of freedom, $\omega$ is the frequency and $L$ is the number of subdomains.} \label{table:numerical_experiment_bi_directional}
\end{table}
Table \ref{table:numerical_experiment_bi_directional} clearly depicts the different behavior between the matrix-free version of the method of polarized traces, and its bi-directional variant. We can clearly observe that the execution time increases faster for the original method of polarized traces, which is mainly due to the increase of the number of iteration to convergence. This trend is more marked in the case of the positive perturbation as explained in Section \ref{section:ABC}. In particular, we observe that the execution time (due to an increase in the number of iterations) grows as ${\cal O}(N \omega^{1/2})= {\cal O}(N^{5/4} )$ instead of ${\cal O}(N\log{N})$ and thus increasing overall the complexity of the global algorithm. In addition, we can observe that the constants are lower for the bi-directional preconditioner. We point out that the bottleneck of both preconditioners is the local solves, which are performed using a multifrontal solver; the main difference between both methods is the number of such solves at each iteration: for the method of polarized traces we need $7$ solves\footnote{The larger amount of solves needed for the matrix-free version of the method of polarized traces is due to the application of the integral operator using local solves at each GMRES iteration, in addition to the solves required for the application of the preconditioner.} per layer and per iteration, whereas the bi-directional preconditioner needs only 4; in addition, the bi-directional preconditioner converges in a lower number of iterations.
\begin{figure}
\caption{{\it Left:} execution time of the off-line stage; {\it Right:} execution time of the on-line stage of solving the Sparsified Lippmann-Schwinger equation using GMRES preconditioned with Alg.~\ref{alg:birirectional_preconditioner}. The tolerance was fixed to $10^{-6}$, and $\omega h = {\cal O}(1)$.}
\label{fig:scaling_sparsified_lippmann_schwinger}
\end{figure}
Fig.~\ref{fig:scaling_sparsified_lippmann_schwinger} ({\it left}), shows the scaling for the factorization of the local problems and the assembly of the sparsified system. In addition, Fig.~\ref{fig:scaling_sparsified_lippmann_schwinger} ({\it right}), shows the execution time for the solution of the sparsified system using the bi-directional algorithm as reported in Table \ref{table:numerical_experiment_bi_directional} but in logarithmic scale. We can observe from Fig.~\ref{fig:scaling_sparsified_lippmann_schwinger} that the assembly and factorization cost is ${\cal O}(N)$ and that the online cost of one inner iteration is ${\cal O}( N \log{N})$, and, in particular, $n_{\mbox{iter}} = {\cal O}(\log{\omega})$.
\subsection{Wave scattering}
\begin{figure}
\caption{Real part of the total wavefield generated by the scattering of a plane wave impinged on the negative perturbation ({\it left}) and on the positive perturbation ({\it right}), in both examples we have $\omega = 500$.}
\label{fig:wavefields1}
\end{figure}
We test the performance of the two-level preconditioner by solving the Lippmann-Schwinger equation \eqref{eq:discrete_Lipmann-Schwinger} associated to three problems in wave scattering. We use three different perturbation profiles, the negative smooth perturbation, the positive smooth perturbation and Gaussian bumps as shown in Fig.~\ref{fig:scattering} ({\it left}) and ({\it right}). The profiles were chosen to depict the behavior of the method in three generic settings for wave propagation: a defocussing medium (negative perturbation), a focusing medium (positive perturbation) and a medium with multiple scattering (Gaussian bumps). We can observe the total wavefield scattered by a plane-wave impinging on each perturbation in Fig.~\ref{fig:wavefields1} ({\it left}), Fig.~\ref{fig:wavefields1} ({\it right}) and Fig.~\ref{fig:wavefields2} ({\it left}), respectively.
\begin{figure}
\caption{ Average execution time for the on-line stage of solving the high-frequency Lippmann-Schwinger equation with GMRES (tolerance of $10^{-10}$) preconditioned with Alg.~\ref{alg:Lipmann-Schwinger_preconditioner} with an inner tolerance of $10^{-3}$, for the three different perturbation profiles; $\omega h = {\cal O}(1)$.}
\label{fig:scaling2}
\end{figure}
For each perturbation profile we solve a system of increasing size, keeping $\omega h$ constant. Each system is solved using GMRES with a tolerance of $10^{-10}$, preconditioned with Alg.~\ref{alg:Lipmann-Schwinger_preconditioner} with a inner tolerance of $10^{-3}$; the domain is decomposed in $L = n/50$ subdomains, and each subdomain is extended 10 grid points in each direction (see Fig.~\ref{fig:DDM_sketch}). At each frequency, we compute the wavefield generated by a set of 64 different incident waves impinging on the perturbation by solving the Lippmann-Schwinger equation, and we report the average execution time in Fig.~\ref{fig:scaling2}. From Fig.~\ref{fig:scaling2} we can observe that the execution time scales as ${\cal O}(N \log N)$ for each perturbation profile, which corroborates the claims made in the introduction. For the off-line stage we obtain the same execution times as in Fig.~\ref{fig:scaling_sparsified_lippmann_schwinger} ({\it left}).
\begin{figure}
\caption{ Left: average execution time for the on-line stage of solving the Lippmann-Schwinger with the Gaussian bumps perturbation with the original sparsifying preconditioner, i.e., using multifrontal methods, and using the two-level preconditioner with $\omega h = {\cal O}(1)$; Right: execution time of the off-line stage for the two-level preconditioner and the original sparsifying preconditioner.}
\label{fig:scaling3}
\end{figure}
Moreover, for the sake of comparison, we time the execution time of solving the Lippmann-Schwinger equation using the original sparsifying preconditioner \cite{Ying:Sparsifying_Preconditioner_for_the_Lippmann--Schwinger_Equation} (i.e., solving \eqref{eq:global_sparse_system} via a multifrontal solver \cite{Davis:UMFPACK}). For the comparison we used the multiple scattering perturbation in Fig.~\ref{fig:scattering} ({\it right}), with the same set of frequencies and incoming waves as before. We use the implementation of the sparsifying preconditioner in \cite{Zepeda:Lippmann_Schwinger}. The average execution times of the on-line stage are reported in Fig.~\ref{fig:scaling3} ({\it left}) together with the averages times for the two-level preconditioner. From Fig.~\ref{fig:scaling3} ({\it left}) we can observe that, unsurprisingly, the original sparsifying preconditioner has the same asymptotic on-line cost as the two-level preconditioner presented in this paper; however, the main difference lies in the off-line stage, in which the multifrontal solver has a complexity of ${\cal O}(N^{3/2})$ compared to ${\cal O}(N)$ for the two-level preconditioner as shown in Fig.~\ref{fig:scaling3} ({\it right}).
\subsection{Plasma physics}
\begin{figure}
\caption{Perturbation for the plasma physics example (taken from \cite{ambikasaran_greengard:Fast_adaptive_high_order_accurate_discretization_of_the_Lippmann-Schwinger_equation_in_two_dimension}) we can observe the at the interior the perturbation becomes negative, thus yielding a strongly elliptic problem locally, thus becoming an evanescent medium.}
\label{fig:plasma}
\end{figure}
\begin{figure}
\caption{Real part of the total wavefield generated by the scattering of a plane wave in the Gaussian bumps model (left) and in the plasma model (right). In both examples $\omega = 500$.}
\label{fig:wavefields2}
\end{figure}
\begin{figure}
\caption{Average execution time for solving the Lippmann-Schwinger equation using the plasma perturbation profile ({\it left}), and the average number of outer iterations needed for convergence ({\it right}). The problem was solved with GMRES (tolerance of $10^{-10}$) preconditioned with the two-level preconditioner with an inner tolerance of $10^{-3}$ and $\omega h = {\cal O}(1)$. The extra logarithmic factor comes from the increasing number of outer iterations.}
\label{fig:scalingplasma}
\end{figure}
Finally, we test the performance of the two-level preconditioner using a problem arising from plasma physics, in particular, a radio-frequency (RF) wave propagating in a fusion reactor. Under some simplifications, the propagation of a RF wave can be described using a cold plasma model \cite{Weitzner:Lower_hybrid_waves_in_the_cold_plasma_model}. We borrow from \cite{ambikasaran_greengard:Fast_adaptive_high_order_accurate_discretization_of_the_Lippmann-Schwinger_equation_in_two_dimension} a model\footnote{We refer the interested reader to Example 4.3 of \cite{ambikasaran_greengard:Fast_adaptive_high_order_accurate_discretization_of_the_Lippmann-Schwinger_equation_in_two_dimension} for further details.} corresponding to an idealized plasma density profile shown in Fig.~\ref{fig:plasma}. The physics in this case are different, an incoming wave propagates until it hits a cut-off region in which the problem becomes strictly elliptic or evanescent, thus causing the waves to reflect, as depicted in Fig.~\ref{fig:wavefields2} ({\it right}).
We solve a system of increasing size, keeping $\omega h$ constant; we decompose the domain such that we have roughly $5$ wavelengths inside each subdomain, and each subdomain is extended 10 grid points in each direction. At each frequency, we compute the wavefield generated by a set of 64 different incident waves impinging on the perturbation by solving the Lippmann-Schwinger equation and we report the average execution times in Fig.~\ref{fig:scalingplasma} ({\it left}).
In the case of the plasma example, we observe from Fig.~\ref{fig:scalingplasma} ({\it left}) a slightly bigger scaling. The main reason for this scaling is the fact that the number of outer iteration grows logarithmically, as shown in Fig.~\ref{fig:scalingplasma} ({\it right}), which is presumably caused by the transition of the physics at the cut-off region. Nonetheless, the number of inner iterations grows like ${\cal O}(\log{N})$, the same as in the wave scattering case, resulting in an overall complexity of ${\cal O}(N \log^2{N})$, which, to the our knowledge, compares favorably to most of current solvers.
\section{Conclusions} We have presented a new fast iterative method for the high-frequency Lippmann-Schwinger equation in 2D. The method has an asymptotic cost of ${\cal O}(N \log{N})$ for the wave scattering problems presented in this paper.
Some future work and extensions are: the design of more efficient ABC, in order to use only one directional preconditioner, thus decreasing the memory footprint; extensions to 3D, in which a nested approach can be used to obtain low complexity algorithms; parallelization of the solvers using state-of-the-art linear solvers; reduction of the constant via compressed linear algebra, which can provide extra savings in the plasma example, where the local ellipticity of the problem can be exploited.
Another line of research is to further reduce the on-line complexity by using the original reduction to a SIE of the method of polarized traces, coupled to the precomputation of integral operators and their compression.
We point out that it is envisageable to change the outer GMRES loop with the flexible GMRES \cite{Saad:A_Flexible_Inner-Outer_Preconditioned_GMRES_Algorithm} algorithm, and reduce the inner-level tolerance to accelerate the algorithm, without a penalty on the asymptotic complexity.
\section{Acknowledgments}
We would like to thank Lexing Ying, and Leslie Greengard for fruitful discussions; Antoine Cerfon for his help with the plasma example, and Laurent Demanet for computational resources.
\end{document} | arXiv |
# Arrays and their role in sorting and searching
Arrays are an essential data structure in programming. They allow us to store multiple values of the same type in a single variable. In the context of sorting and searching algorithms, arrays play a crucial role as they provide a way to store and manipulate the data we want to sort or search.
An array is a collection of elements that are stored in contiguous memory locations. Each element in the array has an index, which represents its position in the array. The index starts at 0 for the first element and increments by 1 for each subsequent element.
Arrays can be used to store various types of data, such as numbers, characters, or even objects. They provide random access to elements, which means we can directly access any element in the array using its index.
Arrays are particularly useful when implementing sorting and searching algorithms because they allow us to efficiently access and manipulate the data. By using arrays, we can easily compare elements, swap their positions, and perform other operations required by these algorithms.
Suppose we have an array of integers called `numbers`:
```cpp
int numbers[] = {5, 2, 8, 1, 9};
```
We can access individual elements of the array using their index. For example, to access the first element, we use `numbers[0]`. To access the third element, we use `numbers[2]`. We can also modify the elements of the array by assigning new values to them.
## Exercise
Consider the following array:
```cpp
char letters[] = {'a', 'b', 'c', 'd', 'e'};
```
What is the index of the letter 'c' in the array?
### Solution
The index of the letter 'c' in the array is 2.
# Common data types used in C++
Before we dive deeper into sorting and searching algorithms, let's briefly discuss some common data types used in C++. Understanding these data types is essential for implementing and working with algorithms effectively.
1. Integer (`int`): This data type is used to store whole numbers. It can represent both positive and negative values.
2. Floating-point (`float` and `double`): These data types are used to store decimal numbers. The `float` type provides single-precision floating-point numbers, while the `double` type provides double-precision floating-point numbers with higher precision.
3. Character (`char`): This data type is used to store individual characters. Characters are enclosed in single quotes, such as 'a' or '5'.
4. Boolean (`bool`): This data type is used to represent logical values. It can have two possible values: `true` or `false`.
5. String (`std::string`): This data type is used to store sequences of characters. Strings are enclosed in double quotes, such as "Hello, world!".
These are just a few of the common data types used in C++. There are many more data types available, including arrays, structures, and classes, which provide more complex ways to store and organize data.
Here are some examples of variables declared using different data types:
```cpp
int age = 25;
float pi = 3.14;
char grade = 'A';
bool isStudent = true;
std::string name = "John Doe";
```
In the examples above, we declare variables of different data types and assign them values. These variables can be used in algorithms to store and manipulate data.
## Exercise
Declare a variable called `temperature` of type `double` and assign it the value 98.6.
### Solution
```cpp
double temperature = 98.6;
```
# Efficiency analysis of algorithms
Efficiency analysis is an essential aspect of designing and implementing algorithms. It allows us to evaluate the performance of different algorithms and choose the most efficient one for a given problem. In the context of sorting and searching algorithms, efficiency analysis helps us understand how these algorithms perform as the size of the input data increases.
There are several factors to consider when analyzing the efficiency of an algorithm:
1. Time complexity: This measures the amount of time an algorithm takes to run as a function of the input size. It helps us understand how the algorithm's performance scales with larger inputs. Common notations used to represent time complexity include Big O notation (e.g., O(n), O(n^2), O(log n)).
2. Space complexity: This measures the amount of memory an algorithm uses as a function of the input size. It helps us understand how the algorithm's memory usage scales with larger inputs. Space complexity is also represented using Big O notation.
3. Best case, worst case, and average case analysis: Algorithms can have different performance characteristics depending on the input data. Analyzing the best case, worst case, and average case scenarios helps us understand the algorithm's behavior in different situations.
Efficiency analysis allows us to compare different algorithms and choose the most suitable one for a specific problem. It helps us optimize our code and ensure that it can handle large datasets efficiently.
Let's compare the time complexity of two sorting algorithms: bubble sort and merge sort.
Bubble sort has a worst-case time complexity of O(n^2), which means that its performance degrades quadratically as the input size increases. In the best case scenario, when the input is already sorted, bubble sort has a time complexity of O(n). The average case time complexity is also O(n^2).
Merge sort, on the other hand, has a consistent time complexity of O(n log n) in all scenarios. This makes it more efficient than bubble sort for large datasets.
## Exercise
Which sorting algorithm would you choose for sorting a large dataset: bubble sort or merge sort? Explain your choice.
### Solution
I would choose merge sort for sorting a large dataset because it has a consistent time complexity of O(n log n) in all scenarios. This means that its performance remains efficient even as the input size increases. Bubble sort, on the other hand, has a time complexity of O(n^2) in the worst case, which can be very slow for large datasets.
# Pointers and their use in sorting and searching
Pointers are a powerful feature of the C++ programming language. They allow us to store and manipulate memory addresses, which can be used to access and modify data indirectly. Pointers are particularly useful in sorting and searching algorithms as they provide a way to efficiently manipulate arrays and other data structures.
In C++, a pointer is a variable that stores the memory address of another variable. By using pointers, we can directly access and modify the value of a variable stored in memory. This is especially useful when working with large datasets or when we need to swap elements during sorting.
Sorting and searching algorithms often require swapping elements in an array. Pointers make it easy to swap elements by directly accessing their memory addresses. This avoids the need to create temporary variables or perform complex calculations.
Pointers can also be used to pass arrays and other data structures to functions efficiently. Instead of passing the entire data structure, we can pass a pointer to it, reducing the amount of memory required and improving performance.
Here's an example that demonstrates the use of pointers in swapping elements of an array:
```cpp
void swap(int* a, int* b) {
int temp = *a;
*a = *b;
*b = temp;
}
int main() {
int numbers[] = {5, 2, 8, 1, 9};
int size = sizeof(numbers) / sizeof(numbers[0]);
// Swap the first and last elements
swap(&numbers[0], &numbers[size - 1]);
// Print the modified array
for (int i = 0; i < size; i++) {
cout << numbers[i] << " ";
}
return 0;
}
```
In this example, the `swap` function takes two pointers as arguments and swaps the values they point to. We pass the memory addresses of the first and last elements of the `numbers` array to the `swap` function, effectively swapping their values.
## Exercise
Write a function called `findMax` that takes an array of integers and its size as arguments. The function should return a pointer to the maximum element in the array.
### Solution
```cpp
int* findMax(int* array, int size) {
int* max = array;
for (int i = 1; i < size; i++) {
if (array[i] > *max) {
max = &array[i];
}
}
return max;
}
```
# Recursive algorithms and their applications
Recursive algorithms are algorithms that solve a problem by breaking it down into smaller subproblems of the same type. These smaller subproblems are then solved recursively until a base case is reached. Recursive algorithms are often used in sorting and searching because they provide an elegant and efficient way to solve complex problems.
One common example of a recursive algorithm is the factorial function. The factorial of a non-negative integer n is the product of all positive integers less than or equal to n. The factorial function can be defined recursively as follows:
$$
\text{factorial}(n) =
\begin{cases}
1 & \text{if } n = 0 \\
n \times \text{factorial}(n-1) & \text{otherwise}
\end{cases}
$$
This recursive definition breaks the problem of computing the factorial of n into the smaller subproblem of computing the factorial of n-1. The base case is when n is 0, in which case the factorial is defined to be 1.
Recursive algorithms can also be used to solve problems that can be naturally divided into smaller subproblems. For example, the merge sort algorithm, which we will discuss later, uses recursion to sort an array by dividing it into two halves, sorting each half recursively, and then merging the sorted halves.
Recursive algorithms can be elegant and concise, but they can also be difficult to understand and debug. It is important to define the base case correctly and ensure that the recursive calls make progress towards the base case to avoid infinite recursion.
Here's an example of a recursive algorithm to compute the sum of all elements in an array:
```cpp
int sumArray(int* array, int size) {
if (size == 0) {
return 0;
} else {
return array[size - 1] + sumArray(array, size - 1);
}
}
int main() {
int numbers[] = {1, 2, 3, 4, 5};
int size = sizeof(numbers) / sizeof(numbers[0]);
int sum = sumArray(numbers, size);
cout << "Sum: " << sum << endl;
return 0;
}
```
In this example, the `sumArray` function takes an array and its size as arguments. If the size is 0, indicating an empty array, the function returns 0. Otherwise, it adds the last element of the array to the sum of the rest of the array, computed recursively by calling `sumArray` with a smaller size.
## Exercise
Write a recursive function called `binarySearch` that takes a sorted array of integers, a target value, and the indices of the first and last elements in the array as arguments. The function should return the index of the target value in the array, or -1 if the target value is not found.
Hint: Use the binary search algorithm, which works by repeatedly dividing the search interval in half until the target value is found or the interval is empty.
### Solution
```cpp
int binarySearch(int* array, int target, int first, int last) {
if (first > last) {
return -1;
} else {
int middle = (first + last) / 2;
if (array[middle] == target) {
return middle;
} else if (array[middle] > target) {
return binarySearch(array, target, first, middle - 1);
} else {
return binarySearch(array, target, middle + 1, last);
}
}
}
```
# Selection sort algorithm
The selection sort algorithm is a simple and intuitive sorting algorithm. It works by repeatedly finding the minimum element from the unsorted part of the array and putting it at the beginning. This process is repeated until the entire array is sorted.
The selection sort algorithm can be implemented in C++ as follows:
```cpp
void selectionSort(int* array, int size) {
for (int i = 0; i < size - 1; i++) {
int minIndex = i;
for (int j = i + 1; j < size; j++) {
if (array[j] < array[minIndex]) {
minIndex = j;
}
}
swap(array[i], array[minIndex]);
}
}
```
In this implementation, the outer loop iterates through the array from the first element to the second-to-last element. In each iteration, it finds the minimum element from the remaining unsorted part of the array by comparing each element with the current minimum. If a smaller element is found, its index is updated. After the inner loop completes, the minimum element is swapped with the first element of the unsorted part, effectively putting it in its correct position. This process is repeated until the entire array is sorted.
The time complexity of the selection sort algorithm is O(n^2), where n is the number of elements in the array. This makes it inefficient for large arrays, but it can be useful for small arrays or for partially sorted arrays where the number of swaps is minimized.
Let's consider an example to illustrate how the selection sort algorithm works. Suppose we have the following array:
```
[5, 3, 8, 2, 1]
```
In the first iteration, the minimum element is 1, so it is swapped with the first element:
```
[1, 3, 8, 2, 5]
```
In the second iteration, the minimum element is 2, so it is swapped with the second element:
```
[1, 2, 8, 3, 5]
```
In the third iteration, the minimum element is 3, so it is swapped with the third element:
```
[1, 2, 3, 8, 5]
```
In the fourth iteration, the minimum element is 5, so it is swapped with the fourth element:
```
[1, 2, 3, 5, 8]
```
After the fifth iteration, the array is fully sorted.
## Exercise
Write a C++ program that uses the selection sort algorithm to sort an array of integers in ascending order. The program should take the size of the array and the elements of the array as input, and output the sorted array.
For example, if the input is:
```
5
5 3 8 2 1
```
The output should be:
```
1 2 3 5 8
```
### Solution
```cpp
#include <iostream>
using namespace std;
void selectionSort(int* array, int size) {
for (int i = 0; i < size - 1; i++) {
int minIndex = i;
for (int j = i + 1; j < size; j++) {
if (array[j] < array[minIndex]) {
minIndex = j;
}
}
swap(array[i], array[minIndex]);
}
}
int main() {
int size;
cin >> size;
int* array = new int[size];
for (int i = 0; i < size; i++) {
cin >> array[i];
}
selectionSort(array, size);
for (int i = 0; i < size; i++) {
cout << array[i] << " ";
}
cout << endl;
delete[] array;
return 0;
}
```
This program reads the size of the array from the user and dynamically allocates an array of that size. It then reads the elements of the array from the user. After that, it calls the `selectionSort` function to sort the array in ascending order. Finally, it prints the sorted array.
# Insertion sort algorithm
The insertion sort algorithm is another simple and intuitive sorting algorithm. It works by dividing the array into a sorted and an unsorted part. Initially, the sorted part contains only the first element, and the unsorted part contains the remaining elements. The algorithm then repeatedly takes an element from the unsorted part and inserts it into its correct position in the sorted part. This process is repeated until the entire array is sorted.
The insertion sort algorithm can be implemented in C++ as follows:
```cpp
void insertionSort(int* array, int size) {
for (int i = 1; i < size; i++) {
int key = array[i];
int j = i - 1;
while (j >= 0 && array[j] > key) {
array[j + 1] = array[j];
j--;
}
array[j + 1] = key;
}
}
```
In this implementation, the outer loop iterates through the array from the second element to the last element. In each iteration, the current element is stored in a variable called `key`, and a variable `j` is initialized to the index of the previous element. The inner loop compares the `key` with each element in the sorted part of the array, moving elements greater than the `key` one position to the right. Finally, the `key` is inserted into its correct position in the sorted part of the array.
The time complexity of the insertion sort algorithm is O(n^2), where n is the number of elements in the array. Like the selection sort algorithm, this makes it inefficient for large arrays, but it can be useful for small arrays or for partially sorted arrays where the number of comparisons and swaps is minimized.
Let's consider an example to illustrate how the insertion sort algorithm works. Suppose we have the following array:
```
[5, 3, 8, 2, 1]
```
In the first iteration, the `key` is 3. Since 5 is greater than 3, 5 is moved one position to the right:
```
[3, 5, 8, 2, 1]
```
In the second iteration, the `key` is 8. Since 5 is not greater than 8, no elements are moved:
```
[3, 5, 8, 2, 1]
```
In the third iteration, the `key` is 2. Since 8 is greater than 2, 8 is moved one position to the right. Similarly, 5 is moved one position to the right:
```
[3, 5, 2, 8, 1]
```
Finally, the `key` is inserted into its correct position:
```
[3, 2, 5, 8, 1]
```
After the remaining iterations, the array is fully sorted.
## Exercise
Write a C++ program that uses the insertion sort algorithm to sort an array of integers in ascending order. The program should take the size of the array and the elements of the array as input, and output the sorted array.
For example, if the input is:
```
5
5 3 8 2 1
```
The output should be:
```
1 2 3 5 8
```
### Solution
```cpp
#include <iostream>
using namespace std;
void insertionSort(int* array, int size) {
for (int i = 1; i < size; i++) {
int key = array[i];
int j = i - 1;
while (j >= 0 && array[j] > key) {
array[j + 1] = array[j];
j--;
}
array[j + 1] = key;
}
}
int main() {
int size;
cin >> size;
int* array = new int[size];
for (int i = 0; i < size; i++) {
cin >> array[i];
}
insertionSort(array, size);
for (int i = 0; i < size; i++) {
cout << array[i] << " ";
}
cout << endl;
delete[] array;
return 0;
}
```
This program reads the size of the array from the user and dynamically allocates an array of that size. It then reads the elements of the array from the user. After that, it calls the `insertionSort` function to sort the array in ascending order. Finally, it prints the sorted array.
# Bubble sort algorithm
The bubble sort algorithm is a simple sorting algorithm that works by repeatedly swapping adjacent elements if they are in the wrong order. It gets its name from the way smaller elements "bubble" to the top of the list.
The bubble sort algorithm can be implemented in C++ as follows:
```cpp
void bubbleSort(int* array, int size) {
for (int i = 0; i < size - 1; i++) {
for (int j = 0; j < size - i - 1; j++) {
if (array[j] > array[j + 1]) {
int temp = array[j];
array[j] = array[j + 1];
array[j + 1] = temp;
}
}
}
}
```
In this implementation, the outer loop iterates through the array from the first element to the second-to-last element. The inner loop compares adjacent elements and swaps them if they are in the wrong order. After each iteration of the outer loop, the largest element is guaranteed to be in its correct position at the end of the array.
The time complexity of the bubble sort algorithm is O(n^2), where n is the number of elements in the array. This makes it inefficient for large arrays, but it can be useful for small arrays or for partially sorted arrays where the number of comparisons and swaps is minimized.
Let's consider an example to illustrate how the bubble sort algorithm works. Suppose we have the following array:
```
[5, 3, 8, 2, 1]
```
In the first iteration of the outer loop, the largest element (8) "bubbles" to the end of the array:
```
[3, 5, 2, 1, 8]
```
In the second iteration, the second largest element (5) "bubbles" to the second-to-last position:
```
[3, 2, 1, 5, 8]
```
After the remaining iterations, the array is fully sorted:
```
[1, 2, 3, 5, 8]
```
## Exercise
Write a C++ program that uses the bubble sort algorithm to sort an array of integers in ascending order. The program should take the size of the array and the elements of the array as input, and output the sorted array.
For example, if the input is:
```
5
5 3 8 2 1
```
The output should be:
```
1 2 3 5 8
```
### Solution
```cpp
#include <iostream>
using namespace std;
void bubbleSort(int* array, int size) {
for (int i = 0; i < size - 1; i++) {
for (int j = 0; j < size - i - 1; j++) {
if (array[j] > array[j + 1]) {
int temp = array[j];
array[j] = array[j + 1];
array[j + 1] = temp;
}
}
}
}
int main() {
int size;
cin >> size;
int* array = new int[size];
for (int i = 0; i < size; i++) {
cin >> array[i];
}
bubbleSort(array, size);
for (int i = 0; i < size; i++) {
cout << array[i] << " ";
}
cout << endl;
delete[] array;
return 0;
}
```
This program reads the size of the array from the user and dynamically allocates an array of that size. It then reads the elements of the array from the user. After that, it calls the `bubbleSort` function to sort the array in ascending order. Finally, it prints the sorted array.
# Merge sort algorithm
# Quick sort algorithm
The quick sort algorithm is another efficient sorting algorithm. It is a divide-and-conquer algorithm that works by selecting a pivot element from the array and partitioning the other elements into two sub-arrays, according to whether they are less than or greater than the pivot. The sub-arrays are then recursively sorted.
The basic steps of the quick sort algorithm are as follows:
1. Choose a pivot element from the array. This can be any element, but for simplicity, we'll choose the last element in the array.
2. Partition the array into two sub-arrays: one with elements less than the pivot and one with elements greater than the pivot. This is done by iterating through the array and swapping elements as necessary.
3. Recursively apply the quick sort algorithm to the sub-arrays. This means repeating steps 1 and 2 for each sub-array until the entire array is sorted.
4. Combine the sorted sub-arrays to get the final sorted array.
The efficiency of the quick sort algorithm depends on the choice of pivot element. In the worst case, if the pivot is consistently chosen as the smallest or largest element, the algorithm can have a time complexity of O(n^2). However, on average, the quick sort algorithm has a time complexity of O(n log n), making it one of the fastest sorting algorithms.
Let's go through an example to see how the quick sort algorithm works. Consider the following array:
```
[10, 14, 19, 26, 27, 31, 33, 35, 42, 44]
```
We'll choose the last element, 44, as the pivot. We'll then partition the array into two sub-arrays:
```
[10, 14, 19, 26, 27, 31, 33, 35] and [42]
```
Next, we'll recursively apply the quick sort algorithm to each sub-array. For the first sub-array, we'll choose 35 as the pivot and partition it into:
```
[10, 14, 19, 26, 27, 31, 33] and [35]
```
We'll continue this process until we have sub-arrays with only one element. Then, we'll combine the sorted sub-arrays to get the final sorted array:
```
[10, 14, 19, 26, 27, 31, 33, 35, 42, 44]
```
## Exercise
Sort the following array using the quick sort algorithm: [31, 15, 7, 22, 18, 29, 5, 11]
### Solution
The sorted array is: [5, 7, 11, 15, 18, 22, 29, 31]
# Binary search algorithm
The binary search algorithm is a commonly used algorithm for searching for an element in a sorted array. It works by repeatedly dividing the search interval in half until the target element is found.
The basic steps of the binary search algorithm are as follows:
1. Start with the entire sorted array.
2. Set the lower bound to the first index of the array and the upper bound to the last index of the array.
3. While the lower bound is less than or equal to the upper bound, do the following:
- Calculate the mid point by taking the average of the lower and upper bounds.
- If the element at the mid point is equal to the target element, return the mid point as the index of the target element.
- If the element at the mid point is less than the target element, set the lower bound to mid point + 1.
- If the element at the mid point is greater than the target element, set the upper bound to mid point - 1.
4. If the target element is not found after the loop, return -1 to indicate that the element is not in the array.
The binary search algorithm is efficient because it reduces the search interval by half with each iteration. It has a time complexity of O(log n), where n is the number of elements in the array.
Let's go through an example to see how the binary search algorithm works. Consider the following sorted array:
```
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
```
We want to search for the element 5. We start with the entire array and set the lower bound to 0 and the upper bound to 9.
In the first iteration, we calculate the mid point as (0 + 9) / 2 = 4. The element at the mid point is 4, which is less than the target element 5. Therefore, we update the lower bound to mid point + 1, which is 5.
In the second iteration, we calculate the mid point as (5 + 9) / 2 = 7. The element at the mid point is 7, which is greater than the target element 5. Therefore, we update the upper bound to mid point - 1, which is 6.
In the third iteration, we calculate the mid point as (5 + 6) / 2 = 5. The element at the mid point is 5, which is equal to the target element 5. Therefore, we return the mid point as the index of the target element.
The binary search algorithm found the target element 5 at index 5.
## Exercise
Using the binary search algorithm, find the index of the element 23 in the following sorted array: [10, 14, 19, 23, 26, 31, 33, 35, 42, 44]
### Solution
The index of the element 23 is 3. | Textbooks |
•https://doi.org/10.1364/OE.435163
Few pulses femtosecond laser exposure for high efficiency 3D glass micromachining
Enrico Casamenti, Sacha Pollonghini, and Yves Bellouard
Enrico Casamenti,* Sacha Pollonghini, and Yves Bellouard
Galatea Laboratory, IEM/STI, Ecole Polytechnique Fédérale de Lausanne (EPFL), Rue de la Maladière 71b, CH - 2000 Neuchâtel, Switzerland
*Corresponding author: [email protected]
Enrico Casamenti https://orcid.org/0000-0002-2384-8106
Yves Bellouard https://orcid.org/0000-0002-8409-4678
E Casamenti
S Pollonghini
Y Bellouard
Enrico Casamenti, Sacha Pollonghini, and Yves Bellouard, "Few pulses femtosecond laser exposure for high efficiency 3D glass micromachining," Opt. Express 29, 35054-35066 (2021)
Spotlight Summary
Spotlight Summary by Sebastian Simeth and Martin Reininghaus
New manufacturing technologies are capable of generating arbitrary shapes and high-aspect-ratios, including undercuts or even in-volume fluidic channel structures at high μm-precision. Amongst other approaches, 3D micromachining of transparent glass compounds using femtosecond laser inscription and subsequent wet chemical etching is one of these novel manufacturing technologies. For example, this method opens completely new manufacturing capabilities for complex new optics or highly integrated temperature resistant workpieces and enables completely new applications in the consumer or electronics sector.
Casamenti et al. present the pathway towards an elaborated 3D glass micromachining method by demonstrating improved aspect ratios, a higher process efficiency and faster processing times. Therefore, the authors identify key parameters in both the laser inscription and chemical etching process and investigate their correlation. By changing the exposure dose and the polarization of the femtosecond laser radiation, different regimes in the process windows are identified. The variation between a hydrofluoric-acid based and two hydroxide-based etchants reveals different etching mechanisms. Subsequent annealing after the laser inscription reveals thermal relaxation affecting the etching mechanism. The investigations executed reveal deeper insights into the selective laser-induced etching process (SLE) upon the influence of laser-induced defects in the glass matrix on the etching selectivity.
Article Reference
Opt. Express 29(22) 35054-35066 (2021) View: Abstract | HTML | PDF
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Tailored surface birefringence by femtosecond laser assisted wet etching
Rokas Drevinskas, et al.
Opt. Express 23(2) 1428-1437 (2015)
Chemical etching of fused silica after modification with two-pulse bursts of femtosecond laser
Valdemar Stankevič, et al.
Towards fast femtosecond laser micromachining of fused silica: The effect of deposited energy.
Sheeba Rajesh, et al.
Laser Machining and Material Processing
Femtosecond lasers
Femtosecond pulses
Laser materials processing
Laser matter interactions
Nonlinear effects
Original Manuscript: June 24, 2021
Revised Manuscript: September 17, 2021
Manuscript Accepted: September 20, 2021
December 1, 2021 Spotlight on Optics
Advanced three-dimensional manufacturing techniques are triggering new paradigms in the way we design and produce sophisticated parts on demand. Yet, to fully unravel its potential, a few limitations have to be overcome, one of them being the realization of high-aspect-ratio structures of arbitrary shapes at sufficiently high resolution and scalability. Among the most promising advanced manufacturing methods that emerged recently is the use of optical non-linear absorption effects, and in particular, its implementation in 3D printing of glass based on femtosecond laser exposure combined with chemical etching. Here, we optimize both laser and chemical processes to achieve unprecedented aspect ratio levels. We further show how the formation of pre-cursor laser-induced defects in the glass matrix plays a key role in etching selectivity. In particular, we demonstrate that there is an optimal energy dose, an order of magnitude smaller than the currently used ones, yielding to higher process efficiency and lower processing time. This research, in addition to a conspicuous technological advancement, unravels key mechanisms in laser-matter interactions essential in chemically-based glass manufacturing and offers an environmentally-friendly pathway through the use of less-dangerous etchants, replacing the commonly used hydrofluoric acid.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Laser-based method for fabricating three-dimensional structures at micro/nano-scales has been a long time endeavor embraced by researchers since the 80s. From advanced stereolithographic concepts [1,2] to the reporting of sub-laser wavelength resolution in polymers [3,4], it rapidly expanded towards glass materials at the turn of the millennium, with the demonstration of micron to sub-micron femtosecond laser processing of glass [5–8]. This two-step process consists of first, exposing a substrate to femtosecond laser irradiation to define patterns of arbitrarily shapes throughout the material volume, and second, placing the substrate in an etchant that dissolves laser-exposed volumes. The laser does not remove any material but instead locally modifies it, introducing self-organized nanostructures [9] consisting in a series of nano-planes parallel one to another that are preferentially etched according to their orientation [10]. Contrary to laser induced photo-polymerization that defines the final shape of the object by direct-laser writing, the glass-based exposure-etching method is a negative process as the laser-exposed volume is removed, and what is left is the non-exposed region. As a result, while sub-wavelength resolution exposure patterns can be imprinted using just-above-the-threshold pulse energies, the ultimate feature size and process capability depend not only on the chemical etching selectivity between laser-affected zones and non-affected ones, but also on the etchant diffusion process since its concentration is depleted as it infiltrates the material through the laser-exposed volumes. Achieving the highest possible aspect ratio with the smallest possible feature sizes is therefore directly correlated to the ability to produce laser-modified zones exhibiting the highest possible etching rate enhancement with respect to the etching rate of the pristine material, and yet, with the lowest possible pulse energies so that the non-linear affected zone remains the smallest possible.
To date, two chemical solutions have been reportedly used. The first one consists of using a low-concentration (typically 2.5 to 5 v%), room-temperature-bath of hydrofluoric acid (HF) [6], while the second is to use a high molar concentration solution (8 to 10 M) of potassium hydroxide (KOH) brought at a temperature between 80 and 90 °C [11]. While HF achieves etching contrast in the order of 1:50 to 1:100, KOH exhibits significantly higher values [12]. Both methods pose difficulties in terms of implementation, not only for the operators' safety but also in terms of environmental impact. HF acid is one of the most hazardous acids to manipulate and requires specific recycling procedures. KOH, as all alkali metal hydroxides, remains highly corrosive and hazardous when used at high concentrations.
Here, we investigate sodium hydroxide (NaOH) as an etching solution. Our observations demonstrate an etching contrast, four and two times higher than HF and KOH, respectively. The highest etching rate with NaOH is also observed for very low net exposure doses (∼ 1.5 J/mm2), an order of magnitude lower than the ones conventionally used. This specific observation enables ten-fold accelerated laser-exposure velocity than the current state of the art [13] as fewer overlapping pulses are required. We further demonstrate that at least in the very low exposure dose regime, etching enhancement mechanisms by HF, NaOH, and KOH are driven by the presence of defects such as non-bridging oxygen hole (NBOHC) and oxygen deficiency centers (ODC), and not primarily by the presence of nanogratings [14].
2. Experiments
2.1 Protocol
Fused silica substrates (Corning 7980 0F, OH 1000 ppm, 1 mm-thick, and 25 mm-square) are exposed to a femtosecond laser in a regime where no ablation occurs. In practice, we use an Ytterbium-fiber amplifier laser (Yuzu from Amplitude), emitting 270 fs-pulses at a wavelength of 1030 nm and a constant repetition rate of 333 kHz, chosen far-away from the regime where thermal accumulation is observed (∼ 1 MHz). Laser-exposure patterns are written by translating the substrate with linear stages (PI-Micos, UPS 150) under a 0.4-numerical aperture microscope objective that focuses the beam down to a measured optical waist of 1.94 µm. The etching rate efficiency is assessed by measuring the etchant progression in straight line patterns passing through the entire specimen and buried under the material surface at a fixed depth of 70 microns. A digital microscope (Hirox KH-8700) is used to measure the length of the etched patterns after an etching time of four hours for the three etchants considered here.
Lines-patterns are written in back and forth directions, and under two different linear polarization states, aligned and perpendicular to the writing direction, respectively. The pattern lengths are chosen to exceed the actual specimen size to exclude acceleration and deceleration phases of the moving stages, ensuring a constant cruising speed and hence, a constant exposure dose throughout the specimen.
Although near the absorption threshold (found at pulse energies of ∼ 160 nJ), the laser-pattern width at the focal point can effectively be smaller than the spot size itself (due to non-linear absorption effects), in what follows, for comparative purpose, we use the optical beam waist as the metric for calculating the net laser exposure dose (or deposited energy) according to the following formula [13]:
$$\Phi = \frac{{4{E_p}f}}{{\pi wv}}$$
Where Φ is the exposure dose (in J/mm2), f the repetition rate of the laser (in Hz), w is the optical beam waist (in mm), v is the velocity of translation of the substrate (in mm/s) and Ep the pulse energy (in J).
A quantitative comparison between hydrofluoric acid (HF, 2.5 v%), potassium hydroxide (KOH, 45 wt%), and sodium hydroxide (NaOH, 5 wt%) as chemical etchants is performed. HF etching is carried at room temperature, while KOH and NaOH etching are performed at 90 °C. Six different pulse energies are explored from 160 to 260 nJ (with a step of 20 nJ), spanning across a region where the laser modified zone stretches from ∼ 6 µm to ∼ 17 µm along the laser propagation axis. These pulse energy conditions correspond to a pulse-irradiance $\psi = {E_p}/(\Delta t \cdot \pi {w^2}/4)$, where Δt is the pulse duration (in s), of ∼ 200 to ∼ 326 W/mm2.
For each pulse energy, the exposure dose is varied 30 times between ∼ 0.5 and ∼ 100 J/mm2 by tuning the laser-scanning speed between 0.5 and 85 mm/s, while keeping the repetition rate constant (333 kHz). The samples are cut into two perpendicularly to the sets of lines to expose the laser affected zones directly to the etchant, avoiding the beam-clipping effects that occur on the edges of the specimens. In this fashion, two samples are produced for each specific combination of parameters for statistical purposes.
2.2 Effect of pulse energy
Figure 1 shows the etching rate comparison of the three etchants for a given set of pulse energies and as a function of the exposure dose for a fixed polarization, chosen perpendicular to the writing direction. This polarization is reportedly the one that leads to the highest etching rates [10].
Fig. 1. Etching rate versus exposure doses for three different etchants: HF 2.5% vol, KOH 45% wt, and NaOH 5% wt. Six different pulse energy levels are explored: 160 nJ (A), 180 nJ (B), 200 nJ (C), 220 nJ (D), 240 nJ (E), and 260 nJ (F). In each graph, the squares represent the mean value and the bars the standard deviation for the two measurements performed. See Data File 1 for underlying values.
The three etchants follow a similar overall trend, with noticeable and significant differences in the moderate pulse energy regime, found around 240 nJ. For the near-absorption threshold pulse energy (∼160 nJ), a single peak of etching efficiency is observed at doses around 10 J/mm2, followed by a decay. There, the three etchants display a remarkably identical etching efficiency, with NaOH decaying at a faster rate towards higher doses. This trend - a sharp increase followed by a decay - is consistent with what we reported in [13].
As the pulse energy is further increased, we observe the gradual appearance of a second peak at significantly lower doses, centered on 1.5 J/mm2. While this peak eventually reaches the same amplitude that the one at 10 J/mm2 for HF, it is double in magnitude for the bases, KOH and NaOH. In general, NaOH and KOH etching efficiencies are higher than HF for nearly all the exposure doses, with NaOH reaching the highest recorded etching rate value of >300 µm/h, about four times more than for HF and twice more than KOH under similar exposure conditions. Fig. S1 in the Supplemental document shows the sets of lines written with 260 nJ of pulse energy after etching.
A salient feature, common to all three etchants, is the presence of a valley, where the etching rate efficiency significantly drops before recovering to higher values. This region is consistently found around 3-4 J/mm2 and defines a transition zone, signaling a dramatic change in the mechanism promoting an accelerated etching in laser-affected zones between low (1.5 J/mm2) and high (10 J/mm2) exposure doses. It is remarkable to observe that the highest etching rate is observed at the lowest doses, where only a few pulses overlap, in a number not sufficient for the formation of clear distinguishable nanogratings to be observed [15,16].
Figure 2(A) shows the lines aspect ratio, i.e. the ratio between the measured etched length and the width at the entrance point of the etched tunnel (see inset of Fig. 2(C)), for the two characteristics etching rate peaks (∼ 1.5 J/mm2 and 10 J/mm2, respectively) defining local maxima, for the three etchants and for each pulse energy. While HF seems to show a weak dependence on the pulse energy, both KOH and NaOH show a clear maximum around 220-240 nJ. This maximum is particularly pronounced for NaOH - where an impressive aspect ratio approaching 400 is observed, while it does not exceed 100 and 50 for KOH and HF respectively (consistently with what previously reported in [11,13]). To visualize the difference between the etchants, grid structures are laser-inscribed and etched for around 18 hours and shown in Fig. 2(B). Here the access for the etchant is defined by a vertical plane that intersects the lines. It is clear that NaOH and KOH etch the laser-modified lines faster and are more affected by the laser polarization than HF and, at the same time, NaOH is the most selective of the three.
Fig. 2. (A) Aspect ratio of the etched lines at two characteristic exposure doses, for the three etchants and six different pulse energies. (B) Visual comparison of the etching progress for different laser parameters depending on the etchant used after 18 hours. (C) Comparison of the aspect ratio obtainable with various subtractive glass micromachining techniques. In the inset, a schematic of the aspect ratio definition used.
Each etchant naturally etches pristine fused silica, albeit at different rates. For similar etching conditions as in here, it is typically about ∼ 3 µm/h for HF [8], it is about three times less for KOH (∼ 0.9 µm/h) [17] and only ∼ 0.5 µm/h for NaOH. To explore the effects of etching in terms of surface quality, a comparative study is performed and is provided in the Supplemental document of this manuscript. In summary, HF leads to the least rough surfaces (Ra ∼ 80 nm), while NaOH behaves similarly to KOH and results in a roughness around 50% worse than the HF one (Ra ∼ 120 nm). Finally, in Fig. 2(C) the performance of the most common subtractive glass micro-manufacturing methods are compared, taking as reference the highest aspect ratio values reported in [11,13,18–26]. The presented version of femtosecond laser assisted chemical etching with NaOH clearly stands out in terms of aspect ratio and unlike all the other techniques mentioned, enables the manufacturing of arbitrary 3D geometries, and this, independently from the etchant chosen.
2.3 Effect of polarization
The polarization has a direct effect on the orientation of the nanostructures [9] and is known to affect the etching rate both for HF [10] and KOH [11]. Figure 3 shows the etching polarization contrast, defined as the difference between etching rates for the two orthogonal linear polarization directions, defined as parallel and perpendicular to the writing direction. Like the other etchants, NaOH turns out to be also polarization sensitive. Interestingly, the polarization dependence drops to near zero (or even inverted) for doses corresponding to the low-etching-rate valley (∼ 3-4 J/mm2, see Fig. 1), where the material removal rate drops to a few tens of micron per hour. At very low doses, the difference between the two polarization states is the most pronounced and culminates at 300 µm/h for NaOH, for pulses of 240 nJ and doses of ∼ 1 J/mm2, value at which the etching rate for parallel polarization is near zero.
Fig. 3. Etching contrast between perpendicular and parallel laser polarization (defined with respect to the writing direction) versus exposure doses for different pulse energies and for the three etchants: (A) HF, (B) KOH, and (C) NaOH. For visual purposes, the lines represent the average trends based on the many experimental points. The solid lines show the etching contrast, while the dotted ones give the etching rate for parallel polarization. See Data File 1 for underlying values.
The etching rate contrast is actually the highest at very low doses despite the fact that only a handful of pulses do overlap, not allowing for a clear formation of self-organized nanogratings as visible through structural changes (see [15,16] and further in this manuscript). An important consequence of this observation is that there is an inherent anisotropy in the laser-affected zones already from the very first pulses, and well before nanogratings are formed.
The large polarization contrast has a direct implication on the process implementation as it requires ensuring that the polarization has consistently the same orientation with respect to the writing direction, which can for instance be achieved by mounting a half-wave plate in the beam path and rotating it to follow an arbitrary trajectory. A similar study is performed on the effect of the writing direction and is reported in the Supplemental document. In particular, the writing direction differential remains between +/- 25 µm/h for HF, but reaches up to 100 µm/h for KOH and 150 µm/h for NaOH. This effect decreases for increasing deposited energy and might be due to the presence of a pulse-front-tilt in the laser beam.
2.4 Cross-section analysis of laser-written patterns
To further understand the nature of the material changes under varying pulse exposure conditions and to correlate it with etching observations, one sample with the same set of parameters tested is finely polished and etched in HF for three minutes to reveal the cross-sectional view of the laser written lines.
Pictures of the specimen are taken using a scanning electron microscopy (Zeiss SEM Gemini 450 operated at 5.0 kV and 100 pA) and assembled in Fig. 4 to investigate morphological correlations between exposure dose and pulse energies with the etched patterns. In Fig. 4, the red-highlighted values correspond to the largest etching rate region for each pulse energy, while the blue-highlighted values to the energy deposited that leads to the formation of nanogratings, whether it is partial or complete. Interestingly, the highest etching rate does not match with the formation of well-structured nanogratings for pulse energies equal to or above 200 nJ. The first etching rate peak (found around 1.5 J/mm2) has no direct correlation with morphological traits of nanogratings or possible micro-cracks. Interestingly, at very few overlapping pulses, the connection between porous zones is poor and does not form a continuous pattern as highlighted elsewhere [27,28]. Yet, the etching rate is among the highest observed suggesting an etching mechanism unrelated to the completeness of porous structures, interconnected or not.
Fig. 4. Scanning electron microscope pictures of the cross-section of laser written lines for different pulse energy and energy deposited values. The squares represent the energy levels tested, with the green ones corresponding to the indicated images. The parameters highlighted in blue led to the formation of clearly visible nanogratings. The ones highlighted in red correspond to the largest etching rate. The scale bar is 1 µm. See Dataset 1 [47] for underlying micrographs.
2.5 Effect of annealing on etching rates
Non-ablative femtosecond laser interaction with fused silica leads to permanent defects in the glass-matrix when the material comes at rest. At a molecular level, oxygen deficiency centers (ODCs), non-bridging-oxygen-hole-centers (NBOHC), color centers (E' centers) and others [29] form and are considered precursors for the formation of free molecular oxygen trapped in nano-pores [30–32].
On one hand, nanogratings (forming 'microscopic morphological defects') resist to extreme annealing temperatures (above 1000°C) [33,34] (even though they undergo a slight degradation above 400°C [35]), while other localized laser-induced structural changes such as laser-induced densification, vanish around 900°C [36,37]. On the other hand, defects at the atomic glass matrix level that require much less energy to form, can be suppressed at 300°C [36] and be fully annealed at 500°C [30].
We take advantage of these observations to differentiate between morphological/structural versus glass-matrix defects as potential contributors for the observed accelerated etching, in particular in the low-dose exposure regimes. In practice, three specimens with a set of lines exploring different laser parameters were prepared as in the previous experiment, but considering only two pulse energies (200 and 240 nJ) and maintaining a polarization state perpendicular to the writing direction for optimal etching. The specimens were annealed at 300°C for 10 h, with a 1 °C/minute heating/cooling rate constant to limit the creation of defects during cooling [38]. Substrates were then etched for two hours in different etchants and the etching rate measured according to the same principle described in the previous paragraphs. Polishing was then performed to remove the portion of material containing etched lines. This procedure is repeated four times in order to test the effect of annealing at 300, 500, 700, and 900°C on the etching rate. The results are shown in Fig. 5.
Fig. 5. Comparison of etching rates versus energy deposited before and after annealing for a fixed pulse energy of 200 nJ (A) and 240 nJ (B), for the three etchants (HF, KOH, and NaOH), and for the three distinct zones of interest defined in Fig. 1. The etching rate is normalized with respect to the etching rate before annealing. See Data File 2 for underlying values.
Let us first consider the effect of the annealing on the HF etching rate (Fig. 5(A)-(B), blue bars). The annealing up to 500°C – reported to quench defects [30,36] – reduces the etching rate only in the case of lower exposure doses and pulse energies (region I for 200 nJ), but does not decrease nor the higher-energy deposited peak nor the etching rate for a pulse energy of 240 nJ.
In fact, in both cases, we assist to a fictitious increase of the rate (only in region III for 200 nJ while in both region I and III for 240 nJ), which is due to a capillarity-driven degradation in etching selectivity of HF [11] and the decreased etching time from 4 h to 2 h for the measurements done before and after annealing, respectively. Such discrepancy among 200 and 240 nJ can be attributed to the creation of a larger amount of defects when higher pulse energy (i.e. field strength) is used, which can also explain the sharp decrease of noise in the etching rate measurements of Fig. 1 for increasing pulse energy and is in line with the stochastic essence of non-linear absorption phenomena. In other words, we assume that the decrease in defects' density, after annealing at 300°C and 500°C for 240 nJ of pulse energy, increases the mean distance between neighboring defects to a value that impacts minimally the HF.
The annealing temperature is further increased to values at which possible localized densification effects on the silica matrix are relaxed into larger and more stable ring structures (i.e. 700 and 900 °C) [34–37]. At this point, the HF etching performance degrades down to 50 and 30% of the original etching speed before annealing. Such behavior hints that the HF etching mechanism is driven mainly by porosity and associated localized densification of the glass structure, at least in regions II and III. Indeed, structural changes locally cause a sharp increase in active surface and densification, suggested by the presence of small member rings, which augments the chemical reactivity of the Si-O bonds [39–41].
KOH (orange bars in Fig. 5) and NaOH (green bars in Fig. 5) show radically different behavior. At pulse energies of 200 nJ, the first annealing at 300°C almost completely cancels the low-energy deposited etching rate peak, while it halves it in the case of 240 nJ. As before, we interpret the different behaviors between pulse energies as due to the initial density of defects, larger for higher pulse energy, and assume that the annealing is not enough to cancel out all the defects for 240 nJ. Although to a lower extent, the etching peak corresponding to the highest energy doses (∼ 10 J/mm2) starts decreasing already at 300°C. When the annealing temperature is further increased, the decrease in etching rate continues smoothly for KOH and sharply for NaOH. Those trends indicate that the etching mechanism of KOH and NaOH is different from the one of HF and driven mainly by the presence of defects in the matrix and only partially by the presence of porosity or associated densification effects. In this context, NaOH tends to be the most selective and the most affected by the laser polarization state, the writing direction (see Fig. S3), or the annealing conditions. Finally, it should be noticed that even if mainly defects-driven the etching rate of NaOH and KOH is highly dependent on the laser polarization (as shown in Fig. 3(B)-(C)), which indicates that a self-organized structural configuration of the defects do exist at the very first stage of the laser interaction, and before the occurrence of pores and nanogratings forming a pathway for the etchant to progress.
We interpret these observations as follows. For low exposure dose (region I in Fig. 6), the etching enhancement is due to the presence of defects localized within the laser-affected-zone. With the pulse duration considered here, it is known that defects mostly consist of NBOHCs and E' centers (see in particular [42]). Both the cross-sectional analysis (Fig. 4) and the annealing study (especially for 200 nJ of pulse energy) supports this hypothesis. Further insights could be obtained from defects density estimation through electron spin resonance (ESR) spectroscopy [43,44]. The lower (or none in case of HF) decrease in etching rate after 300°C and 500°C annealing of modifications with 240 nJ of pulse energy is linked to the density of defects created versus the amount canceled by the annealing. We assume that the mean-free(of defects)-path is reduced to a value that starts to influence the etching rate of KOH and NaOH, but not the one of HF, due to their difference in etching rate of pristine silica (see vectors of Fig. 6).
Fig. 6. Schematic interpretation of the etching mechanism in the region of interest defined above. Top to bottom: the graph shows a qualitative representation of the variation in defects' and pores' quantity depending on exposure dose and pulse energy; the three panels provide a visualization of the modifications induced in each regime with a fixed vertical polarization; and the colored vectors display the etching rate of dense of defects versus pristine SiO2 for HF, KOH, and NaOH.
Further increasing the exposure dose (region II in Fig. 6) leads to a sharp decrease in etching rate which we attribute to the transition from defects to pores. A possible explanation is that higher exposure doses result in the quenching of defects, suppressing the etching enhancement effect. This interpretation remains speculative and further analysis is required to unravel the cause behind the dip in etching rate for region II. Finally, in the high exposure dose (region III in Fig. 6) the nano-pores start coalescing and forming coherent networks with a preferential orientation and ultimately the so-called nanogratings. In this fashion, an accelerated etching is recovered, but this time is driven mainly by the inherent porosity of the material and to a lesser extent by possible additional stress-induced etching-enhancement effects within the laser modified regions [13,45,46]. Due to the difference in etching pristine material, in the case of hydroxide-based etchants, this second etching mechanism is significantly less favorable than the one driven by oriented defects, while it results in a similar enhancement for HF-based etchant.
3. Summary and conclusion
Let us summarize the main findings of this study:
• For the three etchants investigated, a similar phenomenological behavior is observed: a first regime in the low-exposure doses corresponding to around ten overlapping pulses (∼ 1.5 J/mm2), followed by a second regime defined with a local minimum in etching rate and then a third regime, at higher dose, where an etching maximum is recovered.
• For the three-etchant investigated, the local etching maximum corresponding to the low exposure doses (∼ 1.5 J/mm2) with few pulses overlap, the so-called nanogratings are not yet distinctly observed. This first exposure regime at low doses is observed only above a given pulse energy threshold (here at ∼ 180 nJ), higher than the one for observing modifications. Despite the absence of a visible self-organized pattern after etching, depending on the linear polarization orientation a strongly anisotropic etching behavior is observed. It is particularly pronounced in the case of NaOH, where an anisotropic ratio of ∼ 1200 to 1 µm is observed after only 4 hours.
• HF acid-based etchant displays a different behavior than hydroxides-based etchants. While KOH and NaOH highest etching rates are observed for the low-exposure dose regime, for HF similar local maxima in etching rate are found in both the low- and high-exposure dose. Annealing below 500°C has a dramatic effect on the low-exposure dose etching behavior, eventually completely canceling the etching enhancement for NaOH and KOH at low pulse energy, while it has limited effects on the HF etching behavior.
• Sodium hydroxide (NaOH) as an etchant for femtosecond laser-assisted 3D micro-manufacturing shows superior performances compared to known alternative methods, based on HF and KOH solutions. It reaches an unprecedented etching rate of 300 µm/h, twice more than KOH and near four times more than HF. As the natural etching rate of silica by NaOH is very low, the contrast between exposed and unexposed etching rate yields an extreme aspect ratio approaching 1 to 400, for tunnel-like patterns.
In conclusion, from a technological point of view, the observation of an optimal etching rate surpassing the previously reported ones and this, at very low doses, offers a boost in achievable laser-exposure writing speeds larger than one order of magnitude. Moreover, while maintaining an optimal etching rate and selectivity, the aspect ratio obtainable is significantly increased, which is of practical importance for fabricating slender and denser structures.
In this work, EC designed and carried out the experiments reported here. SP performed the initial preparatory experiments. YB and EC discussed and interpreted the results. YB conceived the research. The article text was mainly written by EC and YB. All authors contributed to revisions. The Galatea Laboratory acknowledges the sponsoring of Richemont International.
See Supplement 1 for supporting content. Data underlying the results presented in this paper are available in Data File 1, Data File 2, and Dataset 1 [47].
Supplemental document
See Supplement 1 for supporting content.
1. H. Kodama, "Automatic method for fabricating a three-dimensional plastic model with photo-hardening polymer," Rev. Sci. Instrum. 52(11), 1770–1773 (1981). [CrossRef]
2. K. Ikuta and K. Hirowatari, "Real three dimensional micro fabrication using stereo lithography and metal molding," in Proceedings IEEE Micro Electro Mechanical SystemsIEEE (1993), pp. 42–47.
3. S. Maruo, O. Nakamura, and S. Kawata, "Three-dimensional microfabrication with two-photon-absorbed photopolymerization," Opt. Lett. 22(2), 132–134 (1997). [CrossRef]
4. S. Kawata, H.-B. Sun, T. Tanaka, and K. Takada, "Finer features for functional microdevices," Nature 412(6848), 697–698 (2001). [CrossRef]
5. Y. Kondo, J. Qiu, T. Mitsuyu, K. Hirao, and T. Yoko, "Three-dimensional microdrilling of glass by multiphoton process and chemical etching," Jpn. J. Appl. Phys. 38(Part 2, No. 10A), L1146–L1148 (1999). [CrossRef]
6. A. Marcinkevičius, S. Juodkazis, M. Watanabe, M. Miwa, S. Matsuo, H. Misawa, and J. Nishii, "Femtosecond laser-assisted three-dimensional microfabrication in silica," Opt. Lett. 26(5), 277 (2001). [CrossRef]
7. Y. Cheng, K. Sugioka, K. Midorikawa, M. Masuda, K. Toyoda, M. Kawachi, and K. Shihoyama, "Three-dimensional micro-optical components embedded in photosensitive glass by a femtosecond laser," Opt. Lett. 28(13), 1144 (2003). [CrossRef]
8. Y. Bellouard, A. Said, M. Dugan, and P. Bado, "Fabrication of high-aspect ratio, micro-fluidic channels and tunnels using femtosecond laser pulses and chemical etching," Opt. Express 12(10), 2120 (2004). [CrossRef]
9. Y. Shimotsuma, P. G. Kazansky, J. Qiu, and K. Hirao, "Self-organized nanogratings in glass irradiated by ultrashort light pulses," Phys. Rev. Lett. 91(24), 247405 (2003). [CrossRef]
10. C. Hnatovsky, R. S. Taylor, E. Simova, V. R. Bhardwaj, D. M. Rayner, and P. B. Corkum, "Polarization-selective etching in femtosecond laser-assisted microfluidic channel fabrication in fused silica," Opt. Lett. 30(14), 1867 (2005). [CrossRef]
11. S. Kiyama, S. Matsuo, S. Hashimoto, and Y. Morihira, "Examination of etching agent and etching mechanism on femotosecond laser microfabrication of channels inside vitreous silica substrates," J. Phys. Chem. C 113(27), 11560–11566 (2009). [CrossRef]
12. C. A. Ross, D. G. MacLachlan, D. Choudhury, and R. R. Thomson, "Optimisation of ultrafast laser assisted etching in fused silica," Opt. Express 26(19), 24343 (2018). [CrossRef]
13. S. Rajesh and Y. Bellouard, "Towards fast femtosecond laser micromachining of fused silica: the effect of deposited energy," Opt. Express 18(20), 21490 (2010). [CrossRef]
14. C. Hnatovsky, R. S. Taylor, E. Simova, P. P. Rajeev, D. M. Rayner, V. R. Bhardwaj, and P. B. Corkum, "Fabrication of microchannels in glass using focused femtosecond laser radiation and selective chemical etching," Appl. Phys. A 84(1-2), 47–61 (2006). [CrossRef]
15. R. Taylor, C. Hnatovsky, and E. Simova, "Applications of femtosecond laser induced self-organized planar nanocracks inside fused silica glass," Laser & Photon. Rev. 2(1-2), 26–46 (2008). [CrossRef]
16. S. Richter, M. Heinrich, S. Döring, A. Tünnermann, S. Nolte, and U. Peschel, "Nanogratings in fused silica: formation, control, and applications," J. Laser Appl. 24(4), 042008 (2012). [CrossRef]
17. H. Seidel, L. Csepregi, A. Heuberger, and H. Baumgärtel, "Anisotropic etching of crystalline silicon in alkaline solutions: i . orientation dependence and behavior of passivation layers," J. Electrochem. Soc. 137(11), 3612–3626 (1990). [CrossRef]
18. X. Li, T. Abe, and M. Esashi, "Deep reactive ion etching of Pyrex glass using SF6 plasma," Sensors and Actuators A: Physical 87(3), 139–145 (2001). [CrossRef]
19. B. J. Park, Y. J. Choi, and C. N. Chu, "Prevention of exit crack in micro drilling of soda-lime glass," CIRP Ann. 51(1), 347–350 (2002). [CrossRef]
20. B. R. Sarkar, B. Doloi, and B. Bhattacharyya, "Parametric analysis on electrochemical discharge machining of silicon nitride ceramics," Int. J. Adv. Manuf. Technol. 28(9-10), 873–881 (2006). [CrossRef]
21. M. K. Bhuyan, F. Courvoisier, P.-A. Lacourt, M. Jacquot, L. Furfaro, M. J. Withford, and J. M. Dudley, "High aspect ratio taper-free microchannel fabrication using femtosecond Bessel beams," Opt. Express 18(2), 566 (2010). [CrossRef]
22. L. Jiang, P. Liu, X. Yan, N. Leng, C. Xu, H. Xiao, and Y. Lu, "High-throughput rear-surface drilling of microchannels in glass based on electron dynamics control using femtosecond pulse trains," Opt. Lett. 37(14), 2781 (2012). [CrossRef]
23. L. Brusberg, M. Queisser, C. Gentsch, H. Schröder, and K.-D. Lang, "Advances in CO2-laser drilling of glass substrates," Phys. Procedia 39, 548–555 (2012). [CrossRef]
24. S. Karimelahi, L. Abolghasemi, and P. R. Herman, "Rapid micromachining of high aspect ratio holes in fused silica glass by high repetition rate picosecond laser," Appl. Phys. A 114(1), 91–111 (2014). [CrossRef]
25. L. Hof and J. Abou Ziki, "Micro-hole drilling on glass substrates—a review," Micromachines 8(2), 53 (2017). [CrossRef]
26. A. A. Tseng, Y.-T. Chen, and K.-J. Ma, "Fabrication of high-aspect-ratio microstructures using excimer laser," Optics and Lasers in Engineering 41(6), 827–847 (2004). [CrossRef]
27. S. Richter, A. Plech, M. Steinert, M. Heinrich, S. Döring, F. Zimmermann, U. Peschel, E. B. Kley, A. Tünnermann, and S. Nolte, "On the fundamental structure of femtosecond laser-induced nanogratings," Laser & Photon. Rev. 6(6), 787–792 (2012). [CrossRef]
28. F. Zimmermann, A. Plech, S. Richter, S. Döring, A. Tünnermann, and S. Nolte, "Structural evolution of nanopores and cracks as fundamental constituents of ultrashort pulse-induced nanogratings," Appl. Phys. A 114(1), 75–79 (2014). [CrossRef]
29. L. Skuja, "Optically active oxygen-deficiency-related centers in amorphous silicon dioxide," J. Non-Cryst. Solids 239(1-3), 16–48 (1998). [CrossRef]
30. H.-B. Sun, S. Juodkazis, M. Watanabe, S. Matsuo, H. Misawa, and J. Nishii, "Generation and recombination of defects in vitreous silica induced by irradiation with a near-infrared femtosecond laser," J. Phys. Chem. B 104(15), 3450–3455 (2000). [CrossRef]
31. M. Lancry, F. Zimmerman, R. Desmarchelier, J. Tian, F. Brisset, S. Nolte, and B. Poumellec, "Nanogratings formation in multicomponent silicate glasses," Appl. Phys. B 122(3), 66 (2016). [CrossRef]
32. S. Richter, M. Heinrich, S. Döring, A. Tünnermann, and S. Nolte, "Formation of femtosecond laser-induced nanogratings at high repetition rates," Appl. Phys. A 104(2), 503–507 (2011). [CrossRef]
33. E. Bricchi and P. G. Kazansky, "Extraordinary stability of anisotropic femtosecond direct-written structures embedded in silica glass," Appl. Phys. Lett. 88(11), 111119 (2006). [CrossRef]
34. M. Cavillon, Y. Wang, B. Poumellec, F. Brisset, and M. Lancry, "Erasure of nanopores in silicate glasses induced by femtosecond laser irradiation in the Type II regime," Appl. Phys. A 126(11), 876 (2020). [CrossRef]
35. S.-E. Wei, Y. Wang, H. Yao, M. Cavillon, B. Poumellec, G.-D. Peng, and M. Lancry, "Thermal stability of type II modifications by IR femtosecond laser in silica-based glasses," Sensors 20(3), 762 (2020). [CrossRef]
36. J. J. Witcher, W. J. Reichman, L. B. Fletcher, N. W. Troy, and D. M. Krol, "Thermal annealing of femtosecond laser written structures in silica glass," Opt. Mater. Express 3(4), 502 (2013). [CrossRef]
37. Y. Li, M. Yang, D. N. Wang, J. Lu, T. Sun, and K. T. Grattan, "Fiber Bragg gratings with enhanced thermal stability by residual stress relaxation," Opt. Express 17(22), 19785 (2009). [CrossRef]
38. Y. Su, H. Yin, X. He, and Y. Zhou, "Influence of annealing on the structure of silica glass," J. Wuhan Univ. Technol.-Mat. Sci. Edit. 28(5), 902–906 (2013). [CrossRef]
39. C. Fiori and R. A. B. Devine, "Evidence for a wide continuum of polymorphs in a-SiO2," Phys. Rev. B 33(4), 2972–2974 (1986). [CrossRef]
40. A. Agarwal and M. Tomozawa, "Correlation of silica glass properties with the infrared spectra," J. Non-Cryst. Solids 209(1-2), 166–174 (1997). [CrossRef]
41. K. Awazu and H. Kawazoe, "Strained Si–O–Si bonds in amorphous SiO2 materials: A family member of active centers in radio, photo, and chemical responses," J. Appl. Phys. 94(10), 6243–6262 (2003). [CrossRef]
42. M. Sakakura, Y. Lei, L. Wang, Y.-H. Yu, and P. G. Kazansky, "Ultralow-loss geometric phase and polarization shaping by ultrafast laser writing in silica glass," Light Sci Appl 9(1), 15 (2020). [CrossRef]
43. K. M. Davis, K. Miura, N. Sugimoto, and K. Hirao, "Writing waveguides in glass with a femtosecond laser," Opt. Lett. 21(21), 1729 (1996). [CrossRef]
44. X. W. Wang, R. Buividas, F. Funabiki, P. R. Stoddart, H. Hosono, and S. Juodkazis, "Analysis of defects patterned by femtosecond pulses inside KBr and SiO2 glass," Appl. Phys. A 122(3), 194 (2016). [CrossRef]
45. S. Juodkazis, K. Yamasaki, V. Mizeikis, S. Matsuo, and H. Misawa, "Formation of embedded patterns in glasses using femtosecond irradiation," Appl. Phys. A 79(4-6), 1549–1553 (2004). [CrossRef]
46. A. Champion and Y. Bellouard, "Direct volume variation measurements in fused silica specimens exposed to femtosecond laser," Opt. Mater. Express 2(6), 789 (2012). [CrossRef]
47. E. Casamenti, S. Pollonghini, and Y. Bellouard, "Few pulses femtosecond laser exposure for high efficiency 3D glass micromachining: data," figshare, 2021, https://doi.org/10.6084/m9.figshare.16635142.
H. Kodama, "Automatic method for fabricating a three-dimensional plastic model with photo-hardening polymer," Rev. Sci. Instrum. 52(11), 1770–1773 (1981).
K. Ikuta and K. Hirowatari, "Real three dimensional micro fabrication using stereo lithography and metal molding," in Proceedings IEEE Micro Electro Mechanical SystemsIEEE (1993), pp. 42–47.
S. Maruo, O. Nakamura, and S. Kawata, "Three-dimensional microfabrication with two-photon-absorbed photopolymerization," Opt. Lett. 22(2), 132–134 (1997).
S. Kawata, H.-B. Sun, T. Tanaka, and K. Takada, "Finer features for functional microdevices," Nature 412(6848), 697–698 (2001).
Y. Kondo, J. Qiu, T. Mitsuyu, K. Hirao, and T. Yoko, "Three-dimensional microdrilling of glass by multiphoton process and chemical etching," Jpn. J. Appl. Phys. 38(Part 2, No. 10A), L1146–L1148 (1999).
A. Marcinkevičius, S. Juodkazis, M. Watanabe, M. Miwa, S. Matsuo, H. Misawa, and J. Nishii, "Femtosecond laser-assisted three-dimensional microfabrication in silica," Opt. Lett. 26(5), 277 (2001).
Y. Cheng, K. Sugioka, K. Midorikawa, M. Masuda, K. Toyoda, M. Kawachi, and K. Shihoyama, "Three-dimensional micro-optical components embedded in photosensitive glass by a femtosecond laser," Opt. Lett. 28(13), 1144 (2003).
Y. Bellouard, A. Said, M. Dugan, and P. Bado, "Fabrication of high-aspect ratio, micro-fluidic channels and tunnels using femtosecond laser pulses and chemical etching," Opt. Express 12(10), 2120 (2004).
Y. Shimotsuma, P. G. Kazansky, J. Qiu, and K. Hirao, "Self-organized nanogratings in glass irradiated by ultrashort light pulses," Phys. Rev. Lett. 91(24), 247405 (2003).
C. Hnatovsky, R. S. Taylor, E. Simova, V. R. Bhardwaj, D. M. Rayner, and P. B. Corkum, "Polarization-selective etching in femtosecond laser-assisted microfluidic channel fabrication in fused silica," Opt. Lett. 30(14), 1867 (2005).
S. Kiyama, S. Matsuo, S. Hashimoto, and Y. Morihira, "Examination of etching agent and etching mechanism on femotosecond laser microfabrication of channels inside vitreous silica substrates," J. Phys. Chem. C 113(27), 11560–11566 (2009).
C. A. Ross, D. G. MacLachlan, D. Choudhury, and R. R. Thomson, "Optimisation of ultrafast laser assisted etching in fused silica," Opt. Express 26(19), 24343 (2018).
S. Rajesh and Y. Bellouard, "Towards fast femtosecond laser micromachining of fused silica: the effect of deposited energy," Opt. Express 18(20), 21490 (2010).
C. Hnatovsky, R. S. Taylor, E. Simova, P. P. Rajeev, D. M. Rayner, V. R. Bhardwaj, and P. B. Corkum, "Fabrication of microchannels in glass using focused femtosecond laser radiation and selective chemical etching," Appl. Phys. A 84(1-2), 47–61 (2006).
R. Taylor, C. Hnatovsky, and E. Simova, "Applications of femtosecond laser induced self-organized planar nanocracks inside fused silica glass," Laser & Photon. Rev. 2(1-2), 26–46 (2008).
S. Richter, M. Heinrich, S. Döring, A. Tünnermann, S. Nolte, and U. Peschel, "Nanogratings in fused silica: formation, control, and applications," J. Laser Appl. 24(4), 042008 (2012).
H. Seidel, L. Csepregi, A. Heuberger, and H. Baumgärtel, "Anisotropic etching of crystalline silicon in alkaline solutions: i . orientation dependence and behavior of passivation layers," J. Electrochem. Soc. 137(11), 3612–3626 (1990).
X. Li, T. Abe, and M. Esashi, "Deep reactive ion etching of Pyrex glass using SF6 plasma," Sensors and Actuators A: Physical 87(3), 139–145 (2001).
B. J. Park, Y. J. Choi, and C. N. Chu, "Prevention of exit crack in micro drilling of soda-lime glass," CIRP Ann. 51(1), 347–350 (2002).
B. R. Sarkar, B. Doloi, and B. Bhattacharyya, "Parametric analysis on electrochemical discharge machining of silicon nitride ceramics," Int. J. Adv. Manuf. Technol. 28(9-10), 873–881 (2006).
M. K. Bhuyan, F. Courvoisier, P.-A. Lacourt, M. Jacquot, L. Furfaro, M. J. Withford, and J. M. Dudley, "High aspect ratio taper-free microchannel fabrication using femtosecond Bessel beams," Opt. Express 18(2), 566 (2010).
L. Jiang, P. Liu, X. Yan, N. Leng, C. Xu, H. Xiao, and Y. Lu, "High-throughput rear-surface drilling of microchannels in glass based on electron dynamics control using femtosecond pulse trains," Opt. Lett. 37(14), 2781 (2012).
L. Brusberg, M. Queisser, C. Gentsch, H. Schröder, and K.-D. Lang, "Advances in CO2-laser drilling of glass substrates," Phys. Procedia 39, 548–555 (2012).
S. Karimelahi, L. Abolghasemi, and P. R. Herman, "Rapid micromachining of high aspect ratio holes in fused silica glass by high repetition rate picosecond laser," Appl. Phys. A 114(1), 91–111 (2014).
L. Hof and J. Abou Ziki, "Micro-hole drilling on glass substrates—a review," Micromachines 8(2), 53 (2017).
A. A. Tseng, Y.-T. Chen, and K.-J. Ma, "Fabrication of high-aspect-ratio microstructures using excimer laser," Optics and Lasers in Engineering 41(6), 827–847 (2004).
S. Richter, A. Plech, M. Steinert, M. Heinrich, S. Döring, F. Zimmermann, U. Peschel, E. B. Kley, A. Tünnermann, and S. Nolte, "On the fundamental structure of femtosecond laser-induced nanogratings," Laser & Photon. Rev. 6(6), 787–792 (2012).
F. Zimmermann, A. Plech, S. Richter, S. Döring, A. Tünnermann, and S. Nolte, "Structural evolution of nanopores and cracks as fundamental constituents of ultrashort pulse-induced nanogratings," Appl. Phys. A 114(1), 75–79 (2014).
L. Skuja, "Optically active oxygen-deficiency-related centers in amorphous silicon dioxide," J. Non-Cryst. Solids 239(1-3), 16–48 (1998).
H.-B. Sun, S. Juodkazis, M. Watanabe, S. Matsuo, H. Misawa, and J. Nishii, "Generation and recombination of defects in vitreous silica induced by irradiation with a near-infrared femtosecond laser," J. Phys. Chem. B 104(15), 3450–3455 (2000).
M. Lancry, F. Zimmerman, R. Desmarchelier, J. Tian, F. Brisset, S. Nolte, and B. Poumellec, "Nanogratings formation in multicomponent silicate glasses," Appl. Phys. B 122(3), 66 (2016).
S. Richter, M. Heinrich, S. Döring, A. Tünnermann, and S. Nolte, "Formation of femtosecond laser-induced nanogratings at high repetition rates," Appl. Phys. A 104(2), 503–507 (2011).
E. Bricchi and P. G. Kazansky, "Extraordinary stability of anisotropic femtosecond direct-written structures embedded in silica glass," Appl. Phys. Lett. 88(11), 111119 (2006).
M. Cavillon, Y. Wang, B. Poumellec, F. Brisset, and M. Lancry, "Erasure of nanopores in silicate glasses induced by femtosecond laser irradiation in the Type II regime," Appl. Phys. A 126(11), 876 (2020).
S.-E. Wei, Y. Wang, H. Yao, M. Cavillon, B. Poumellec, G.-D. Peng, and M. Lancry, "Thermal stability of type II modifications by IR femtosecond laser in silica-based glasses," Sensors 20(3), 762 (2020).
J. J. Witcher, W. J. Reichman, L. B. Fletcher, N. W. Troy, and D. M. Krol, "Thermal annealing of femtosecond laser written structures in silica glass," Opt. Mater. Express 3(4), 502 (2013).
Y. Li, M. Yang, D. N. Wang, J. Lu, T. Sun, and K. T. Grattan, "Fiber Bragg gratings with enhanced thermal stability by residual stress relaxation," Opt. Express 17(22), 19785 (2009).
Y. Su, H. Yin, X. He, and Y. Zhou, "Influence of annealing on the structure of silica glass," J. Wuhan Univ. Technol.-Mat. Sci. Edit. 28(5), 902–906 (2013).
C. Fiori and R. A. B. Devine, "Evidence for a wide continuum of polymorphs in a-SiO2," Phys. Rev. B 33(4), 2972–2974 (1986).
A. Agarwal and M. Tomozawa, "Correlation of silica glass properties with the infrared spectra," J. Non-Cryst. Solids 209(1-2), 166–174 (1997).
K. Awazu and H. Kawazoe, "Strained Si–O–Si bonds in amorphous SiO2 materials: A family member of active centers in radio, photo, and chemical responses," J. Appl. Phys. 94(10), 6243–6262 (2003).
M. Sakakura, Y. Lei, L. Wang, Y.-H. Yu, and P. G. Kazansky, "Ultralow-loss geometric phase and polarization shaping by ultrafast laser writing in silica glass," Light Sci Appl 9(1), 15 (2020).
K. M. Davis, K. Miura, N. Sugimoto, and K. Hirao, "Writing waveguides in glass with a femtosecond laser," Opt. Lett. 21(21), 1729 (1996).
X. W. Wang, R. Buividas, F. Funabiki, P. R. Stoddart, H. Hosono, and S. Juodkazis, "Analysis of defects patterned by femtosecond pulses inside KBr and SiO2 glass," Appl. Phys. A 122(3), 194 (2016).
S. Juodkazis, K. Yamasaki, V. Mizeikis, S. Matsuo, and H. Misawa, "Formation of embedded patterns in glasses using femtosecond irradiation," Appl. Phys. A 79(4-6), 1549–1553 (2004).
A. Champion and Y. Bellouard, "Direct volume variation measurements in fused silica specimens exposed to femtosecond laser," Opt. Mater. Express 2(6), 789 (2012).
E. Casamenti, S. Pollonghini, and Y. Bellouard, "Few pulses femtosecond laser exposure for high efficiency 3D glass micromachining: data," figshare, 2021, https://doi.org/10.6084/m9.figshare.16635142.
Abe, T.
Abolghasemi, L.
Abou Ziki, J.
Agarwal, A.
Awazu, K.
Bado, P.
Baumgärtel, H.
Bellouard, Y.
Bhardwaj, V. R.
Bhattacharyya, B.
Bhuyan, M. K.
Bricchi, E.
Brisset, F.
Brusberg, L.
Buividas, R.
Cavillon, M.
Champion, A.
Chen, Y.-T.
Choi, Y. J.
Choudhury, D.
Chu, C. N.
Corkum, P. B.
Courvoisier, F.
Csepregi, L.
Davis, K. M.
Desmarchelier, R.
Devine, R. A. B.
Doloi, B.
Döring, S.
Dudley, J. M.
Dugan, M.
Esashi, M.
Fiori, C.
Fletcher, L. B.
Funabiki, F.
Furfaro, L.
Gentsch, C.
Grattan, K. T.
Hashimoto, S.
He, X.
Heinrich, M.
Herman, P. R.
Heuberger, A.
Hirao, K.
Hirowatari, K.
Hnatovsky, C.
Hof, L.
Hosono, H.
Ikuta, K.
Jacquot, M.
Jiang, L.
Juodkazis, S.
Karimelahi, S.
Kawachi, M.
Kawata, S.
Kawazoe, H.
Kazansky, P. G.
Kiyama, S.
Kley, E. B.
Kodama, H.
Kondo, Y.
Krol, D. M.
Lacourt, P.-A.
Lancry, M.
Lang, K.-D.
Lei, Y.
Leng, N.
Li, X.
Li, Y.
Liu, P.
Lu, J.
Lu, Y.
Ma, K.-J.
MacLachlan, D. G.
Marcinkevicius, A.
Maruo, S.
Masuda, M.
Matsuo, S.
Midorikawa, K.
Misawa, H.
Mitsuyu, T.
Miura, K.
Miwa, M.
Mizeikis, V.
Morihira, Y.
Nakamura, O.
Nishii, J.
Nolte, S.
Park, B. J.
Peng, G.-D.
Peschel, U.
Plech, A.
Poumellec, B.
Qiu, J.
Queisser, M.
Rajeev, P. P.
Rajesh, S.
Rayner, D. M.
Reichman, W. J.
Richter, S.
Ross, C. A.
Said, A.
Sakakura, M.
Sarkar, B. R.
Schröder, H.
Seidel, H.
Shihoyama, K.
Shimotsuma, Y.
Simova, E.
Skuja, L.
Steinert, M.
Stoddart, P. R.
Su, Y.
Sugimoto, N.
Sugioka, K.
Sun, H.-B.
Sun, T.
Takada, K.
Tanaka, T.
Taylor, R.
Taylor, R. S.
Thomson, R. R.
Tian, J.
Tomozawa, M.
Toyoda, K.
Troy, N. W.
Tseng, A. A.
Tünnermann, A.
Wang, D. N.
Wang, L.
Wang, X. W.
Watanabe, M.
Wei, S.-E.
Witcher, J. J.
Withford, M. J.
Xiao, H.
Xu, C.
Yamasaki, K.
Yan, X.
Yang, M.
Yao, H.
Yin, H.
Yoko, T.
Yu, Y.-H.
Zhou, Y.
Zimmerman, F.
Zimmermann, F.
Appl. Phys. A (7)
Appl. Phys. B (1)
CIRP Ann. (1)
Int. J. Adv. Manuf. Technol. (1)
J. Appl. Phys. (1)
J. Electrochem. Soc. (1)
J. Laser Appl. (1)
J. Non-Cryst. Solids (2)
J. Phys. Chem. B (1)
J. Phys. Chem. C (1)
J. Wuhan Univ. Technol.-Mat. Sci. Edit. (1)
Laser & Photon. Rev. (2)
Light Sci Appl (1)
Micromachines (1)
Opt. Mater. Express (2)
Optics and Lasers in Engineering (1)
Phys. Procedia (1)
Phys. Rev. B (1)
Phys. Rev. Lett. (1)
Rev. Sci. Instrum. (1)
Sensors and Actuators A: Physical (1)
Data File 1 Data set of etching rates of fused silica after femtosecond laser exposure for 6 sets of pulse energies and different laser exposure doses after etching with HF, KOH, and NaOH.
Data File 2 Data set of etching rates of fused silica after femtosecond laser exposure for 2 sets of pulse energies and different laser exposure doses after etching with HF, KOH, and NaOH and their variation after annealing at different temperatures.
Dataset 1 Cross-section SEM micrographs. Each figure contains the laser affected zone for opposite laser writing direction. Deposited energy increasing left to right. Polarization always perpendicular to the writing direction.
Supplement 1 Overview of etched lines, roughness study after etching, and effect of writing direction on etching rate..
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\begin{document}
\title[Ricci-Yamabe maps for Riemannian flows]{Ricci-Yamabe maps for Riemannian flows and their volume variation and volume entropy}
\author{Mircea Crasmareanu} \address{Faculty of Mathematics \\ University "Al. I.Cuza" \\ Iasi, 700506 \\ Rom\^{a}nia\newline
http://www.math.uaic.ro/$\sim$mcrasm}
\email{[email protected]}
\thanks{The second author was supported by The Scientific and Technological Research Council of Turkey (T\"UB{\.I}TAK) Grant No.1059B141600696.}
\author{S{\.I}nem G\"uler}
\address{Department of Mathematics \\ Ayaza\u ga Campus, Faculty of Science and Letters, \\ Istanbul Technical University \\ 34469 Maslak, Istanbul \\ Turkey}
\email{[email protected]}
\begin{abstract} The aim of this short note is to produce new examples of geometrical flows associated to a given Riemannian flow $g(t)$. The considered flow in covariant symmetric $2$-tensor fields will be called Ricci-Yamabe map since it involves a scalar combination of Ricci tensor and scalar curvature of $g(t)$. Due to the signs of considered scalars the Ricci-Yamabe flow can be also a Riemannian or semi-Riemannian or singular Riemannian flow. We study the associated function of volume variation as well as the volume entropy. Finally, since the two-dimensional case was the most handled situation we express the Ricci flow equation in all four orthogonal separable coordinate systems of the plane. \end{abstract}
\subjclass[2010]{53C44, 53A05}
\keywords{Riemannian flow, Ricci-Yamabe map, volume variation, volume entropy}
\maketitle
\section*{Introduction}
The recent applications of the Hamilton-Ricci flow (\cite{c:ln}) in Perelman's proof of the geometrization conjecture and Brendle-Schoen's proof of the differentiable sphere theorem attract the attention to the geometry of Riemannian flows. These flows and other general geometric flows have been applied to several topological, geometrical and physical problems and we cite only some of them: the mean curvature flow, the K\"ahler-Ricci, Calabi and Yamabe flows, the curve shortening flows and so on.
The present paper aims to introduce a scalar combination of Ricci and Yamabe flow under the name of {\it Ricci-Yamabe map}. Due to the signs of involved scalars ($\alpha $ and $\beta $) the Ricci-Yamabe map can be also a Riemannian or semi-Riemannian or singular Riemannian flow. This kind of multiple choices can be useful in some geometrical or physical (e.g. relativistic) theories. Another strong motivation for our map is the fact that, although the Ricci and Yamabe flow are identical in dimension two, they are essentially different in higher dimensions. Let us remark that an interpolation flow between Ricci and Yamabe flows is considered in \cite{c:c} under the name {\it Ricci-Bourguignon flow} but it depends on a single scalar, see also the pages 79-80 of the book \cite{c:g}. A normalized version of Ricci-Bourguignon-Yamabe flow is studied in \cite{m:t} from the point of view of spectral geometry.
In the first section we introduce and discuss five examples of Ricci-Yamabe maps: conformal (particular cone), convex-Euclidean, generalized Poincar\'e, generalized cigar flow and time dependent 2D warped metrics; a sixth example, consisting in gradient flows introduced by Min-Oo and Ruh in \cite{mo:r}, is very briefly recalled due to its relationship with the Ricci flow. For some of them the general dimension $n$ is too complicated from the point of view of their Ricci tensor and scalar curvature and then the case $n=2$ is particularly analyzed with respect to the (global sometimes) coordinates $(u, v)$; on this way the present study can be useful in the shape analysis of surfaces following the path of \cite{z:g}. Also, for an $(\alpha , \beta )$-Ricci-Yamabe flow we express the variation of some geometrical quantities: the Christoffel symbols, the scalar curvature and the volume form.
Another main topic studied for the introduced Ricci-Yamabe map is the volume variation to which the second section is devoted. The given Riemannian flow $g(t)$ will be called $(\alpha , \beta)$-{\it RY-expanding} or {\it steady} or ${\it shrinking}$ if this variation is positive, zero or negative, respectively. For the example of generalized cigar flow with an exponential potential we derive all three possibilities: expanding, steady and shrinking. Also, we get two examples of $(\alpha , \beta)$-RY-expanding flows which are uniform i.e. they depend only on the time variable $t$ and not on the coordinates of $M$. We finish the second section with a study of the volume entropy for an $(\alpha , \beta)$-Ricci-Yamabe flow following the technique of \cite{v:c}.
In the last section we study a generalization of the conformal example. More precisely, we allow a general conformal transformation $f(t, x, y)(dx^2+dy^2)$ of the Euclidean plane metric and its corresponding Ricci flow equation. The previous coordinate system $(x, y)$ being provided by the Euclidean geometry can be without a geometrical or physical significance for a different geometry. But it has been known that for some constant curvature spaces there exist {\it orthogonal separable coordinate systems}, namely coordinate systems for which a given Hamiltonian system in classical me\-cha\-nics and Schr\"odinger equation of the quantum mechanics admit solutions via separation of variables; hence these coordinate systems are related to the superintegrability problem, \cite{b:cp}. For example, in the real 2D space there are four such systems, for the complex plane there are six while for 2D sphere there are only two. We derive the expression of the Ricci flow equation in all 2D coordinates systems and also for their solitonic analogs. On this way, we add to the PDE flavor of the topic of Ricci flow three new equations: the polar, the parabolic and the elliptic Ricci flow equation.
\section{The Ricci-Yamabe map of a Riemannian flow}
For a smooth $n$-dimensional manifold $M^n$ let $T^s_2(M)$ be the linear space of its symmetric tensor fields of $(0, 2)$-type and $Riem(M)\subsetneqq T^s_2(M)$ be the infinite space of its Riemannian metrics. Following the general theory of geometric flows we introduce:
\begin{definition}\label{def:1.1} A {\it Riemannian flow} on $M$ is a smooth map: \begin{equation} \label{1.1} g:I\subseteq \mathbb{R}\rightarrow Riem(M) \end{equation} where $I$ is a given open interval. We can call it also as {\it time-dependent} (or {\it non-stationary}) {\it Riemannian metric}. \end{definition}
Throughout this work we fix a Riemannian flow $g(\cdot )$ and one denotes by $Ric(t)$ the Ricci tensor field of $g(t)$ and by $R(t)$ the corresponding scalar curvature. Let also be given $\alpha $ and $\beta $ some scalars.
\begin{definition}\label{def:1.2}
The map $RY^{(\alpha , \beta , g)}:I\rightarrow T^s_2(M)$ given by: \begin{equation} \label{1.2} RY^{(\alpha, \beta ,g)}(t):=\frac{\partial g}{\partial t}(t)+2\alpha Ric(t)+\beta R(t)g(t) \end{equation} is called {\it the $(\alpha , \beta)$-Ricci-Yamabe map} of the Riemannian flow $(M, g)$. If $RY^{(\alpha, \beta ,g)}\equiv 0$ then $g(\cdot )$ will be called {\it an $(\alpha , \beta)$-Ricci-Yamabe flow}.
\end{definition}
\begin{Particular Case}\label{part case:1.3} We have: \begin{enumerate} \item[(1)] the equation $RY^{(1, 0, g)}\equiv 0$ is the Ricci flow while the equation $RY^{(0, 1, g)}\equiv 0$ is the Yamabe flow (\cite[p. 520]{c:ln}). The equation of an $(\alpha /2, \beta )$-Ricci-Yamabe flow appears at page 50 in \cite{b:u} where the Ricci curvature is interpreted as a vector field on the space of metrics. \item[(2)] for $n=2$ since $Ric=\frac{R}{2}g=Kg$ with $K$ being the Gaussian curvature we get: \begin{equation} \label{1.3} RY^{(\alpha , \beta , g)}(t)=\frac{\partial g}{\partial t}(t)+2(\alpha +\beta )K(t)g(t). \end{equation} Hence the Ricci and Yamabe flows coincide on surfaces. \item[(3)] Due to the signs of $\alpha $ and $\beta $ the Ricci-Yamabe map can be also a Riemannian or semi-Riemannian or singular (degenerate) Riemannian flow and this kind of freedom can be useful in some geometrical or physical (e.g. relativistic) theories. For example, a recent bi-metric approach of spacetime geometry appears in \cite{a:ks} and \cite{b:c}. \end{enumerate}
\end{Particular Case}
\begin{example}\label{exp:1.4.}
$(1)$ (\textit{Cone and conformal flow}) Fix $g\in Riem(M)$ and a smooth $f:I\rightarrow \mathbb{R}^*_{+}=(0, +\infty )$. We define $g(t):=f(t)g$ which we call {\it conformal flow}. In particular, for $I=(0, +\infty )$ and $f(t)=t$ we obtain {\it the cone flow}. Applying Exercise 1.11 of \cite[p. 6]{c:ln} one have: \begin{equation} \label{1.4} Ric(t)=Ric(g), \quad R(t)=\frac{1}{f(t)}R(g) \end{equation} and then the $(\alpha , \beta )$-{\it Ricci-Yamabe-conformal map} is: \begin{equation} \label{1.5} RY^{(\alpha , \beta , g)}(t)=[f^{\prime }(t)+\beta R(g)]g+2\alpha Ric(g), \end{equation} which becomes constant for the cone flow. In particular, if $g$ is an Einstein metric, from: \begin{equation} \label{1.6} Ric(g)=\frac{R(g)}{n}g, \end{equation} we get: \begin{equation} \label{1.7} RY^{(\alpha , \beta , g)}(t)=[f^{\prime }(t)+(\beta +\frac{2\alpha }{n})R(g)]g, \end{equation} which is a time-dependent Einstein metric. \\
$(2)$ A fixed Riemannian geometry $(M, g)$ is called {\it convex-Euclidean} (\cite{b:cp}) if it supports a Riemannian flow: \begin{equation} \label{1.8} g(t)=(1-t)g+tI, \end{equation} with $t\in [0, 1]$ and $I$ the covariant version of the Kronecker tensor field. For example, every pa\-ral\-le\-li\-za\-ble manifold, in particular any Lie group, is a convex-Euclidean one. In the following, due to complicated equations on the general case we restrict to $n=2$ and suppose that $g$ is isothermal: $g(u, v):=E(u, v)I$ with the smooth function $E>0$ on the surface $M$. Then all terms of the convex-Euclidean flow are isothermal: \begin{equation} \label{1.9} g(t)=[(1-t)E+t]I. \end{equation} The expression of the Gaussian curvature for isothermal metrics is well-known and hence: \begin{equation} \label{1.10} K(t)=-\frac{1}{2[(1-t)E+t]}\Delta _{u, v}(\ln [(1-t)E+t]), \end{equation} where $\Delta _{u, v}$ is the usual 2D Laplacian: $\Delta _{u, v}=\partial ^2_{uu}+\partial ^2_{vv}$. In conclusion: \begin{equation} \label{1.11} RY^{(\alpha , \beta , g)}(t)=[1-E-(\alpha +\beta )\Delta _{u, v}(\ln [(1-t)E+t])]I, \end{equation} which is a time-dependent isothermal (semi-, singular) metric. \\
$(3)$ (\textit{Poincar\'e flow}) For the Poincar\'e half space model of hyperbolic geometry in $\mathbb{R}^n_{+}=\{(x^1,...,x^n)\in \mathbb{R}^n; x^n>0\}$ (\cite[p. 135]{p:p}) we define {\it the Poincar\'e flow} on $I=\mathbb{R}$ as: \begin{equation} \label{1.12} g(t)=\frac{1}{(x^n)^t}\left[d(x^1)^2+...+d(x^n)^2\right]=\frac{1}{(x^n)^t}g^n_e, \end{equation} which satisfies $\partial _tg(t)=(-\ln x^n)\cdot g(t)$ and can be considered as a conformal flow for the Euclidean metric $g^n_e$ on $\mathbb{R}^n_{+}$. Applying the formulae of \cite[p. 35]{c:ln} we have: \begin{equation} \label{1.13} Ric(t)=\frac{(2-n)t^2-2t}{4(x^n)^2}g_e^{n-1}+\frac{(1-n)t}{2(x^n)^2}d(x^n)^2, \quad R(t)=\frac{1-n}{(x^n)^{2-t}}\left[t+\frac{(n-2)t^2}{4}\right] \end{equation} and then: \begin{align} \label{1.14} RY^{(\alpha , \beta , g)}(t)\cdot (x^n)^2=&\{(1-n)\beta \left[t+\frac{(n-2)t^2}{4}\right]-(x^n)^{2-t}\ln x^n\}g^n_e\\ \notag +&[(1-\frac{n}{2})t^2-t]\alpha g_e^{n-1}+(1-n)\alpha td(x^n)^2. \end{align} For $n=2$ we obtain: \begin{equation} \label{1.15} RY^{(\alpha , \beta , g)}(t)=\frac{-(x^2)^{2-t}\ln x^2-(\alpha +\beta )t}{(x^2)^2}g^2_e. \end{equation}
$(4)$ (\textit{Generalized cigar flow}) In \cite[p. 154]{c:ln} it is given the Hamilton cigar $2D$ metric on $M=\mathbb{R}^2$ as a steady Ricci soliton: \begin{equation} \label{1.16} g_c(u, v)=\frac{1}{1+u^2+v^2}I \end{equation} and then we introduce {\it the generalized cigar flow}: \begin{equation} \label{1.17} g^f_c(t)=\frac{1}{f(t)+u^2+v^2}I \end{equation} for a smooth $f:\mathbb{R}\rightarrow \mathbb{R}^{*}_+$ with $f(0)=1$. Its Ricci-Yamabe map is: \begin{equation} \label{1.18} RY^{(\alpha , \beta , g^f)}(t)=\frac{4(\alpha +\beta )f(t)-f^{\prime }(t)}{(f+u^2+v^2)^2}I \end{equation} and then $RY^{(\alpha , \beta , g^f)}\equiv 0$ if and only if $f(t)=e^{4(\alpha +\beta )t}$. Recalling that for $\alpha =1$ and $\beta =0$ we have a Ricci flow we re-obtain the result of \cite{c:ln} that $(1.17)$ is a Ricci flow for $f(t)=e^{4t}$. \\
$(5)$ (\textit{Time dependent warped metrics}) Let the 2D Riemannian flow: \begin{equation} \label{1.19} g(t)=du^2+f(t)G(u)dv^2 \end{equation} again for a smooth $f:\mathbb{R}\rightarrow \mathbb{R}^{*}_+$ with $f(0)=1$. Its $(\alpha , \beta )$-Ricci-Yamabe map is: \begin{equation} \label{1.20} RY^{(\alpha , \beta , g)}(t)=f^{\prime }(t)G(u)dv^2-\frac{\alpha +\beta }{\sqrt{f(t)G}}\Delta \ln G\cdot g(t). \end{equation} For $G(u)=u^2$ and $f(t)=t^2$ which do not satisfies $f(0)=1$ the metric $\eqref{1.19}$ appears in the Exercise 1.6.7 of \cite[p. 33]{p:p}. With $G(u)=sn_k^2(u)$ of \cite[p. 117]{p:p} we obtain the time dependent rotationally symmetric metrics generalizing the constant curvature ($=k$) metrics. Hence this flow $\eqref{1.19}$ has the $(\alpha , \beta )$-Ricci-Yamabe map: \begin{equation} \label{1.21} RY^{(\alpha , \beta , g)}(t)=f^{\prime }(t)sn_k^2(u)dv^2-\frac{2k(\alpha +\beta )}{\sqrt{f(t)}}\cdot g(t). \end{equation} Let us note that Ricci flows on general warped product metrics are studied in \cite{wj:l}. \\
$(6)$ (\textit{Gauge flows of Min-Oo-Ruh type}) Starting with a given Riemannian metric $g$, a gradient flow is considered in \cite{mo:r} by means of a family $\theta _t$ of tensor fields of $(1, 1)$-type: \begin{equation} \label{1.22} g(t)=g(\theta _t \cdot , \theta _t\cdot ) \end{equation}
with $\theta _0=I$. The infinitesimal gauge transformation $\dot{\theta }:=\frac{d}{dt}\theta _t|_{t=0}$ is used to express the infinitesimal change of the Levi-Civita connection.
The choice $\dot{\theta }=-Ric$ of $(1, 1)$-type yields exactly the Hamilton's Ricci flow and the Lagrangian of the gradient flow then becomes a Yang-Mills functional $D\rightarrow \int _M\|F^D\|^2$ on the space of Cartan connections of hyperbolic type.
The choice $\dot{\theta }=-\alpha Ric-\frac{\beta }{2}RI$ yields the $(\alpha , \beta )$-Ricci-Yamabe flow. Moreover, given the vector field $\xi $ and $\lambda \in \mathbb{R}$ the choice $Ric=-\nabla ^g\xi -\lambda I$ with $\nabla ^g\xi $ the covariant derivative of $\xi $ with respect to the fixed metric $g$ yields {\it the Ricci soliton} $(g, \xi , \lambda )$ on $M$ i.e. a self-similar solution of the Ricci flow, \cite{c:ln}. \quad $\Box $
\end{example}
We finish this section with the variation of some geometrical objects along an $(\alpha , \beta)$-Ricci-Yamabe flow:
\begin{proposition}\label{prop:1.5} Let $g(t)$ be an $(\alpha , \beta)$-Ricci-Yamabe flow. Then: \begin{itemize} \item[(1)] {\it the variation of the Christoffel symbols is}: \begin{align} \label{1.23} \partial _t\Gamma ^k_{ij}(t)=&\alpha g^{kl}\left(\nabla _lR_{ij}-\nabla _iR_{jl}-\nabla_jR_{il}\right)(t)\\ \notag -&\frac{\beta }{2}\left[(\nabla _iR)(t)\delta ^k_j+(\nabla _jR)(t)\delta ^k_i-(\nabla ^kR)(t)g_{ij}(t)\right], \end{align} \item[(2)] {\it the variation of the scalar curvature is}: \begin{equation} \label{1.24}
\partial _tR(t)=[\alpha +(n-1)\beta ]\Delta _{g(t)}R(t)+2\alpha \|Ric(t)\|_{g(t)}^2+\beta R^2(t), \end{equation} {\it with $\Delta _{g(t)}$ the Laplacian with respect to $g(t)$; for $n=2$ this relation becomes}: \begin{equation} \label{1.25} \left\{ \begin{array}{ll} \partial _tR(t)=(\alpha +\beta )\left[\Delta _{g(t)}R(t)+R^2(t)\right], \\ \partial _tK(t)=(\alpha +\beta )\left[\Delta _{g(t)}K(t)+2K^2(t)\right],\\ \end{array} \right. \end{equation} \item[(3)] the variation of the volume form is: \begin{equation} \label{1.26} \partial _td\mu (t)=-(\alpha +\frac{n}{2}\beta )R(t)d\mu (t). \end{equation}
It follows that if $2\alpha +n\beta =0$, then the volume is constant and the variation of the scalar curvature is: \begin{equation} \label{1.27}
\frac{1}{\alpha}\partial _tR(t)=\frac{2-n}{n}\Delta _{g(t)}R(t)+2\|Ric(t)\|_{g(t)}^2-\frac{2}{n}R^2(t), \end{equation} provided $\alpha \neq 0$; particularly is zero in dimension $n=2$. For an $(\alpha , \beta)$-Ricci-Yamabe flow on a closed Riemannian surface $M^2$ with $\alpha +\beta \geq 1$ and $t\in (0, T]$, we have the next global inequality on $M$: \begin{equation} \label{1.28} K(t)\geq -\frac{1}{2(\alpha +\beta )t}. \end{equation} \end{itemize} \end{proposition}
\begin{proof} We apply the formulae of \cite{c:ln} with the variation tensor field: \begin{equation} \label{1.29} v(t)=-2\alpha Ric(t)-\beta R(t)g(t) \end{equation} and the corresponding variation function: \begin{equation} \label{1.30} V(t)=-(2\alpha +n\beta )R(t). \end{equation} We have $div\ v=-(\alpha +\beta )dR$ and $div(div\ v)=-(\alpha +\beta )\Delta R$. For the item (3) at the Ricci flow $\alpha =1$ we derive the corresponding {\it normalized Ricci flow} of \cite[p. 128]{c:ln} with the associated scalar $\beta =-\frac{2}{n}$ multiplied with the average scalar curvature. The inequality $(1.28)$ follows from the maximum principle. \end{proof}
\begin{Particular Case}\label{particular case:1.6}
Suppose that the given Riemannian flow $g(t)$ is {\it Ricci-recurrent} i.e. for a fixed time-dependent $1$-form $\eta $ we have: \begin{equation} \label{1.31} \nabla _l(t)R_{ij}(t)=\eta _l(t)R_{ij}(t). \end{equation} Then $\eqref{1.23}$ becomes, with $\eta ^k=g^{ka}\eta _a$: \begin{align} \label{1.32} \partial _t\Gamma ^k_{ij}(t)=&\alpha \left(\eta ^k(t)R_{ij}(t)-\eta _i(t)R_{j}^k(t)-\eta _j(t)R_{i}^k(t)\right)\\ \notag -&\frac{\beta }{2}\left(\eta _i(t)\delta ^k_j+\eta _j(t)\delta ^k_i-g_{ij}\eta ^k(t)\right)R(t), \end{align} since $\nabla _l(t)R(t)=\eta _lR(t)$. The relation $\eqref{1.25}$ becomes: \begin{equation} \label{1.33}
\partial _tR(t)=(\alpha +\beta )\left[(div _{g(t)}\eta (t)+\|\eta (t)\|^2_{g(t)})R(t)+R^2(t)\right]. \end{equation} In fact, in dimension $2$ if $K(t)>0$, then $\eqref{1.31}$ holds for: \begin{equation} \label{1.34} \eta =d(\ln K) \end{equation} and hence $\eqref{1.32}$ reads: \begin{equation} \label{1.35}
\partial _tK(t)=(\alpha +\beta )\left[(\Delta _{g(t)}\ln K(t)+\|\nabla \ln K(t)\|^2_{g(t)})K(t)+2K^2(t)\right]. \end{equation}
\end{Particular Case}
\section{The volume variation of Ricci-Yamabe maps and volume entropy of a Ricci-Yamabe flow}
Inspired by \cite{m:c} and \cite{c:k} we introduce:
\begin{definition}\label{def:2.1}
The {\it $(\alpha , \beta )$-Ricci-Yamabe volume variation} for the Riemannian flow $g(\cdot )$ is the smooth function $V^{(\alpha , \beta , g)}:M\times I\rightarrow M\times I$ given by: \begin{equation} \label{2.1} \partial _tV^{(\alpha , \beta , g)}(t):=Tr _{g(t)}RY^{(\alpha , \beta , g)}(t)=\sum _{i, j=1}^ng^{ij}(t){RY}^{(\alpha , \beta , g)}_{ij}(t) \end{equation} where, as usual, $g^{ij}(t)$ are the components of inverse $g^{-1}(t)$. The flow will be called $(\alpha , \beta)$-{\it RY-expanding} or {\it steady} or ${\it shrinking}$ if this variation is positive, zero or negative, respectively. Moreover, if this variation depends only on $t$, then we say that the flow is {\it uniform}. \end{definition}
\begin{example}\label{exp:2.2}
$(1)$ For conformal flows we have from $\eqref{1.5}$: \begin{equation} \label{2.2} \partial _tV^{(\alpha , \beta , g)}(t)=\frac{nf^{\prime }(t)+(2\alpha +n\beta )R(g)}{f(t)} \end{equation} which yields: \begin{equation} \label{2.3} V(t)^{(\alpha , \beta , g)}=n\ln f(t)+(2\alpha +n\beta )R(g)F(t) \end{equation} with $F$ an anti-derivative for $\frac{1}{f}$. In particular, for $2\alpha +n\beta =0$ or flat metric $g$, the conformal flow is uniform $(\alpha , \beta )-RY-$expanding. \\
$(2)$ For the two-dimensional Poincar\'e flow by using $\eqref{1.15}$ we get: \begin{equation} \label{2.4} V(t)^{(\alpha , \beta , g)}=-\frac{\alpha +\beta}{2}t^2+(x^2)^{2-t}. \end{equation} Hence, on the horizontal lines $x^2=constant$ this flow is $(\alpha , \beta )$-RY-expanding, steady or shrinking according to $\alpha +\beta $ being a negative, null or positive number. The existence of a preferred direction in the physical space is discussed in \cite[p. 50]{b:l}. \\
$(3)$ For the generalized cigar flow its variation of volume is: \begin{equation} \label{2.5} \partial _tV^{(\alpha , \beta , g)}(t)=\frac{4(\alpha +\beta )f-f^{\prime }}{f+u^2+v^2}. \end{equation} For example, choosing an exponential potential $f(t)=e^{ct}$ we derive that the flow is $(\alpha , \beta)$-RY-expanding, steady or shrinking according to $\alpha +\beta $ being greater, equal or lower than $\frac{c}{4}$. \\
$(4)$ For the time dependent warped metrics $\eqref{1.19}$ with $G(u)=sn_k^2(u)$ from $\eqref{1.21}$ we have: \begin{equation} \label{2.6} V^{(\alpha , \beta , g)}(t)=\ln f(t)-4k(\alpha +\beta )F(t) \end{equation} with $F$ and anti-derivative for $\frac{1}{\sqrt{f}}$. In particular, for $\alpha +\beta =0$ or flat metric $g$, this flow is uniform $(\alpha , \beta )-RY-$expanding. \quad $\Box $
\end{example}
More generally, the steady case is characterized by:
\begin{proposition}\label{2.3}
The Riemannian flow $g(\cdot )$ is $(\alpha , \beta)$-RY-steady if and only if its scalar curvature is: \begin{equation} \label{2.7} (2\alpha +n\beta )R(t)=-Tr_{g(t)}(\partial _tg(t)). \end{equation} \end{proposition}
\begin{proof} From $\eqref{2.1}$ we derive that the steady case is characterized by: \begin{equation} \label{2.8} n\beta R(t)=-Tr_{g(t)}(\partial _tg(t)+2\alpha Ric(t)). \end{equation} Since $Tr$ is a linear operator we derive immediately the conclusion. \end{proof}
We finish this section with a study of the volume entropy for an $(\alpha , \beta)$-Ricci-Yamabe flow supposing that $M$ is compact and with $n\geq 3$. Let $\pi : \tilde{M}\rightarrow M$ be the universal cover of $M$ and $\omega (M, g, r>0)=\frac{1}{r}\ln V_{\tilde{g}}(\tilde{B}(r))$ be {\it the volume growth function} of the Riemannian metric $g$; here $V_{\tilde{g}}$ is the volume with respect to the cover metric $\tilde{g}$. Then {\it the volume entropy} of $(M, g)$ is: $h(M, g)=\lim _{r\rightarrow \infty }\omega (M, g, r)$.
We have a result similar to Proposition 2.1 of \cite{v:c}:
\begin{proposition}\label{prop:2.4}
Let $g(t)$ be an $(\alpha , \beta)$-Ricci-Yamabe flow on the compact $M$ satisfying:
\begin{itemize} \item[(i)] {\it the injectivity radius $i(M, g(t))\geq i\in \mathbb{R}$ uniformly in time}; {\it in particular $g(t)$ has the sectional curvature uniformly bounded from above by a positive constant and an uniform lower bound of the lengths of closed geodesics}, \item[(ii)] $n\geq 3$ {\it and} $2\alpha +n\beta >0$. \end{itemize}
Then the volume entropy $h(M, g(t))$ is nondecreasing. \end{proposition}
\begin{proof}
With a computation similar to that of \cite{v:c} and using $\eqref{1.26}$ we obtain: \begin{equation} \label{2.9} \partial _t \omega (M, g(t), r)=-\frac{2\alpha +n\beta }{2rV_{\tilde{g}(t)}(\tilde{B}(r))}\int _{\tilde{B}(r)}\tilde{R}(t)d\tilde{ \mu }(t). \end{equation} The hypothesis (i) assures the existence of a constant $C>0$ such that: \begin{equation} \label{2.10} \lim _{r\rightarrow \infty}\frac{1}{V_{\tilde{g}(t)}(\tilde{B}(r))}\int _{\tilde{B}(r)}\tilde{R}(t)d\tilde{ \mu }(t)\leq n(n-1)C. \end{equation} Hence, with (ii) we get $\partial _th(M, g(t))\geq 0$ which means the conclusion.
\end{proof}
\section{The orthogonal companions of the 2D Ricci flow equation and the associated solitonic PDEs}
In this section we start with the general conformal-Euclidean 2D flow on an open subset of the plane, \cite[p. 286]{ci:co}: \begin{equation} \label{3.1} g(t)=\exp(h(x, y, t))[dx^2+dy^2] \end{equation} for which the Ricci flow equation is: \begin{equation} \label{3.2} (\exp h)_t=\Delta h \end{equation} with $\Delta $ the Euclidean Laplacian: $\Delta h=h_{xx}+h_{yy}$.
In the following, we consider the Ricci flow equation $\eqref{3.2}$ in the separable coordinate systems of $\mathbb{R}^2$ having the model as the paper \cite{b:cp}. \\
$(1)$ The Cartesian coordinates $(x, y)$ form the first separable coordinate system of the plane. With the separation of variables $h=f(t, x)g(t, y)$ we have: \begin{equation} \label{3.3} (f_tg+fg_t)\exp (fg)=f_{xx}g+fg_{yy}, \end{equation} while the separation of variables $h=f(t, x)+g(t, y)$ yields the equation: \begin{equation} \label{3.4} (f_t+g_t)\exp (f+g)=f_{xx}+g_{yy}. \end{equation}
The following three systems are: \\
$(2)$ Polar coordinates: $u=x\cos y$, $v=x\sin y$. The equation $\eqref{3.2}$ becomes {\it the polar Ricci flow equation}: \begin{align} \label{3.5} h_{uu}(\cos ^2y+x^2\sin ^2y)+&h_{vv}(\sin ^2y+x^2\cos ^2y)\\ \notag +&2h_{uv}\sin y\cos y(1-x^2)=(\exp h)_t. \end{align} \begin{itemize} \item[(2.1)] By searching for $h$ of the form $h=\varphi (t, w=u+\alpha v)$ we get the {\it polar solitonic Ricci flow equation}: \begin{align} \label{3.6} \varphi _{ww}[\cos ^2y+&x^2\sin ^2y+\alpha ^2(\sin ^2y+x^2\cos ^2y)\\ \notag +&2\alpha \sin y\cos y(1-x^2)]=\varphi _t\exp \varphi . \end{align} \item[(2.2)] By searching for $h$ of the form $h=f(t, u)g(t, v)$ we obtain: \begin{align} \label{3.7} f_{uu}g(\cos ^2y+x^2\sin ^2y)+&fg_{vv}(\sin ^2y+x^2\cos ^2y)\\ \notag +&2f_{u}g_{v}\sin y\cos y(1-x^2)=(f_tg+fg_t)\exp(fg). \end{align} \item[(2.3)] By searching for $h$ of the form $h=f(t, u)+g(t, v)$ we derive: \begin{equation} \label{3.8} f_{uu}(\cos ^2y+x^2\sin ^2y)+g_{vv}(\sin ^2y+x^2\cos ^2y)=(f_t+g_t)\exp(fg). \end{equation} \end{itemize}
$(3)$ Parabolic coordinates: $\xi =\frac{1}{2}(u^2-v^2)$, $\eta =uv$. The Euclidean metric $g=d\xi ^2+d\eta ^2$ becomes a Liouville one: $g=(u^2+v^2)(du^2+dv^2)$ and the Ricci flow equation is now {\it the parabolic Ricci flow equation}: \begin{equation} \label{3.9} 2\sqrt{\xi ^2+\eta ^2}(h_{\xi \xi }+h_{\eta \eta })=(\exp h)_t. \end{equation} \begin{itemize} \item[(3.1)] With $h=\varphi (t, w=\xi +\alpha \eta )$, we derive {\it the parabolic solitonic Ricci flow equation}: \begin{equation} \label{3.10} 2\sqrt{\xi ^2+\eta ^2}(1+\alpha ^2)\varphi _{ww}=\varphi _t(\exp \varphi ). \end{equation} \item[(3.2)] For $h=f(t, \xi )g(t, \eta )$ we get: \begin{equation} \label{3.11} 2\sqrt{\xi ^2+\eta ^2}(f_{\xi \xi }g+fg_{\eta \eta })=(f_tg+fg_t)(\exp fg). \end{equation} \item[(3.3)] For $h=f(t, \xi )+g(t, \eta )$ we obtain: \begin{equation} \label{3.12} 2\sqrt{\xi ^2+\eta ^2}(f_{\xi \xi }+g_{\eta \eta })=(f_t+g_t)\exp (f+g). \end{equation} \end{itemize}
$(4)$ Elliptic coordinates: $x^2=c^2(u-1)(v-1)$, $y^2=-c^2uv$. The Euclidean metric $g=dx^2+dy^2$ becomes a Lorentzian one: $g=\frac{c^2(v-u)}{4}\left(\frac{du^2}{u(u-1)}-\frac{dv^2}{v(v-1)}\right)$ and the Ricci flow equation is now {\it the elliptic Ricci flow equation}: \begin{align} \label{3.13} (\exp h)_t=&h_x\Delta _{u, v}x+h_y\Delta _{u, v}y\\ \notag +&\frac{c^2}{4}\left[h_{xx}\frac{v-u}{u(u-1)}+2h_{xy}\left(\frac{v(1-v)}{u(u-1)}+\frac{u(1-u)}{v(v-1)}\right)+h_{yy}\frac{u-v}{v(v-1)}\right], \end{align} where $\Delta _{u, v}$ is the Euclidean Laplacian in the variables $(u, v)$; supposing $c>0$ we have: \begin{equation} \label{3.14} \left\{ \begin{array}{ll} \Delta _{u, v}x=-\frac{c}{4}\left(\frac{1}{u-1}\sqrt{\frac{v-1}{u-1}}+\frac{1}{v-1}\sqrt{\frac{u-1}{v-1}}\right), \\ \\ \Delta _{u, v}y=-\frac{c}{4}\left(\frac{1}{u}\sqrt{\frac{-v}{u}}+\frac{1}{v}\sqrt{\frac{-u}{v}}\right). \end{array} \right. \end{equation} \begin{itemize} \item[(4.1)] With $h=\varphi (t, w=x+\alpha y)$ we obtain {\it the elliptic solitonic Ricci flow equation}: \begin{align} \label{3.15} \varphi _t\exp \varphi=&\varphi _w(\Delta _{u, v}x+\alpha h_y\Delta _{u, v}y)\\ \notag +&\varphi _{ww}[\frac{v-u}{u(u-1)}+2\alpha \left(\frac{v(1-v)}{u(u-1)}+\frac{u(1-u)}{v(v-1)}\right)+\alpha ^2\frac{u-v}{v(v-1)}]. \end{align} \item[(4.2)] With $h=f(t, x)g(t, y)$ we get: \begin{align} \label{3.16} (f_tg+fg_t)(\exp fg)=&f_xg\Delta _{u, v}x+fg_y\Delta _{u, v}y+f_{xx}g\frac{v-u}{u(u-1)}\\ \notag +&2f_{x}g_{y}\left(\frac{v(1-v)}{u(u-1)}+\frac{u(1-u)}{v(v-1)}\right)+fg_{yy}\frac{u-v}{v(v-1)}. \end{align} \item[(4.3)] With $h=f(t, x)+g(t, y)$ we obtain: \begin{align} \label{3.17} (f_t+g_t)\exp (f+g)=&f_x\Delta _{u, v}x+g_y\Delta _{u, v}y\\ \notag +&f_{xx}\frac{v-u}{u(u-1)}+g_{yy}\frac{u-v}{v(v-1)}. \end{align} \end{itemize} On this way, we add to the PDE flavor of the topic of Ricci flow three new equations: the polar, the parabolic and the elliptic Ricci flow equation, as well as their solitonic companions.
\end{document} | arXiv |
Mathematical Biosciences and Engineering, 2016, 13(1): 209-225. doi: 10.3934/mbe.2016.13.209.
Primary: 92B05, 34D23; Secondary: 34D20.
Global analysis on a class of multi-group SEIR model with latency and relapse
Jinliang Wang, Hongying Shu
1. School of Mathematical Science, Heilongjiang University, Harbin 150080
2. Department of Mathematics, Tongji University, Shanghai 200092
Received: , Published:
Abstract Related pages
In this paper, we investigate the global dynamics of a multi-group SEIR epidemic model,allowing heterogeneity of the host population, delay in latency and delay due torelapse distribution for the human population.Our results indicate that when certain restrictions on nonlinear growth rate and incidence are fulfilled, the basic reproduction number $\mathfrak{R}_0$ plays the key role of a global threshold parameter in the sense that the long-time behaviors of the model depend only on $\mathfrak{R}_0$. The proofs of the main results utilize the persistence theory indynamical systems, Lyapunov functionals guided by graph-theoretical approach.
Keywords: relapse; Lyapunov functional.; global stability; latency; Multi-group model
Citation: Jinliang Wang, Hongying Shu. Global analysis on a class of multi-group SEIR model with latency and relapse. Mathematical Biosciences and Engineering, 2016, 13(1): 209-225. doi: 10.3934/mbe.2016.13.209
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Let $x,$ $y,$ and $z$ be positive real numbers such that $xyz = 1.$ Find the minimum value of
\[(x + 2y)(y + 2z)(xz + 1).\]
By AM-GM,
\begin{align*}
x + 2y &\ge 2 \sqrt{2xy}, \\
y + 2z &\ge 2 \sqrt{2yz}, \\
xz + 1 &\ge 2 \sqrt{xz},
\end{align*}so
\[(x + 2y)(y + 2z)(xz + 1) \ge (2 \sqrt{2xy})(2 \sqrt{2yz})(2 \sqrt{xz}) = 16xyz = 16.\]Equality occurs when $x = 2y,$ $y = 2z,$ and $xz = 1.$ We can solve to get $x = 2,$ $y = 1,$ and $z = \frac{1}{2},$ so the minimum value is $\boxed{16}.$ | Math Dataset |
\begin{document}
\title{Quantum superpositions of ``common-cause'' and ``direct-cause'' causal structures} \author{Adrien Feix} \author{{\v C}aslav Brukner} \affiliation{Faculty of Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria} \affiliation{Institute for Quantum Optics and Quantum Information (IQOQI), Boltzmanngasse 3, 1090 Vienna, Austria} \date{\today} \begin{abstract} The constraints arising for a general set of causal relations, both classically and quantumly, are still poorly understood. As a step in exploring this question, we consider a \emph{coherently controlled superposition} of ``direct-cause'' and ``common-cause'' relationships between two events. We propose an implementation involving the spatial superposition of a mass and general relativistic time dilation. Finally, we develop a computationally efficient method to distinguish such genuinely quantum causal structures from classical (incoherent) mixtures of causal structures and show how to design experimental verifications of the nonclassicality of a causal structure. \end{abstract} \maketitle \section{Introduction}\label{sec:introduction} The deeply rooted intuition that the basic building blocks of the world are \emph{cause-effect-relations} goes back over a thousand years~\cite{aristotle_metaphysics_1933,hume_enquiry_1975,reichenbach_direction_1956} and yet still puzzles philosophers and scientists alike.
In physics, general relativity provides a theoretic account of the causal relations that describe which events in spacetime can influence which other events. For two (infinitesimally close) events separated by a time-like or light-like interval, one event is in the future light cone of the other, such that there could be a direct cause-effect relationship between them. When a space-like interval separates two events, no event can influence the other. The causal relations in general relativity are \emph{dynamical}, since they are imposed by the dynamical light cone structure~\cite{brown_behaviour_2009}.
Incorporating the concept of causal structure in the quantum framework leads to novelties: it is expected that such a notion will be both \emph{dynamical}, as in general relativity, as well as \emph{indefinite}, due to quantum theory~\cite{hardy_quantum_2007}. One might then expect indefiniteness with respect to the question of whether an interval between two events is time-like or space-like, or even whether event $A$ is prior to or after event $B$ for time-like separated events. Yet, finding a unified framework for the two theories is notoriously difficult and the candidate models still need to overcome technical and conceptual problems.
One possibility to separate conceptual from technical issues is to consider more general, \emph{theory-independent} notions of causality. The \emph{causal model} formalism~\cite{spirtes_causation_1993,pearl_causality:_2000} is such an approach, which has found applications in areas as diverse as medicine, social sciences and machine learning~\cite{illari_causality_2011}. The study of its quantum extension, allowing for non-local correlations~\cite{wood_lesson_2015,cavalcanti_modifications_2014,henson_theory-independent_2014,fritz_beyond_2016} or including new information-theoretic principles~\cite{pienaar_graphseparation_2015,chaves_informationtheoretic_2015,costa_quantum_2015} might provide intuitions and insights that are currently missing from the theory-laden take at combining quantum mechanics with general relativity.
Recently, it was found that it is possible to formulate quantum mechanics without any reference to a global causal structure~\cite{oreshkov_quantum_2012}. The resulting framework---the \emph{process matrix formalism}---allows for processes which are incompatible with any definite order between operations. One particular case of such a process is the ``quantum switch'', where an auxiliary quantum system can coherently control the order in which operations are applied~\cite{chiribella_quantum_2013}. This results in a quantum controlled superposition of the processes ``$A$ causing $B$'' and ``$B$ causing $A$''. The quantum switch can also be realized through a preparation of a massive system in a superposition of two distinct states, each yielding a different but definite causal structure for future events~\cite{zych_quantum_2015,zych_bell_????}. Furthermore, it provides computational~\cite{araujo_computational_2014a} and communication~\cite{feix_quantum_2015,allard_guerin_exponential_2016} advantages over standard protocols with a fixed order of events. The first experimental proof-of-principle demonstration of the switch has been reported recently~\cite{procopio_experimental_2015}.
Given that one can implement superpositions of two different causal orders, one may ask if and how one could realize situations in which two events are in superpositions of being in ``common-cause'' ($A$ does not cause $B$ directly) and ``direct-cause'' ($A$ and $B$ share no common cause) relationships. Here we show that such superpositions exist and how to verify them.
We develop a framework for the computationally efficient verification of \emph{coherent superpositions} of ``direct-cause'' and ``common-cause'' causal structures. We propose a natural physical realization of a quantum causal structure with the spatial superposition of a mass and general relativistic time dilation using the approach developed in Refs.~\cite{zych_bell_????,zych_quantum_2015}. Finally, using the process matrix formalism, we define a degree of ``nonclassicality of causal structures'' and show how to design \emph{experimental verifications} thereof using a semidefinite program~\cite{nesterov_interior_1987}.
\section{Quantum causal models} \label{sec:causalstructures} To formalize the pre-theoretic notion of causality, the standard approach is to use \emph{causal models}~\cite{spirtes_causation_1993,pearl_causality:_2000}, consisting of (i) a causal network and (ii) model parameters. The \emph{causal network} is represented by a \emph{directed graph}, whose nodes are variables and whose directed edges represent causal influences between variables. The causal influence from $A$ to $B$ is identified with the possibility of \emph{signaling} from $A$ to $B$. To exclude the possibility of causal loops, one imposes the condition that the graph should be \emph{acyclic} (a ``DAG''), which induces a \emph{partial order} (``causal order'') over the variables. The \emph{model parameters} then determine how the probability distribution of each variable or set of variables is to be computed as a function of the value of its parent nodes.
Fully characterizing the causal model requires information which is available only through ``interventions'', where the value of one or more variables is \emph{set} to take a specific value, independently of the values of the rest of the variables. In the resulting causal network, the connections from all its parents are eliminated. Intervening on all relevant variables is sufficient to completely reconstruct the full causal model~\cite{pearl_causality:_2000}. Since this is often practically impossible, it is crucial to investigate the possibilities of causal inference from a \emph{limited} set of interventions.
Moving to \emph{quantum} causal models, we will define variables as results of generalized quantum operations applied to incoming quantum systems (``local operation''). Formally, a local operation $\mathcal{M}_A: A_I \to A_O$ is a map from a density matrix $\rho_{A_I} \in A_I$ to $\rho_{A_O} \in A_O$ (where $A_I$ ($A_O$) denotes the space of linear operators on the Hilbert space $\mathcal H^{A_I}$ ($\mathcal H^{A_O}$)). The Choi-Jamio\l{}kowski (CJ) isomorphism~\cite{choi_completely_1975,jamiolkowski_linear_1972} provides a convenient representation of the local map as a positive operator $M_A \in A_I \otimes A_O$ (the explicit definition is given in Appendix~\ref{app:cj}).
The \emph{quantum causal structure}, which is the quantum analogue of the classical causal network, maps the aforementioned local operations to a probability distribution. It can be thought of as a \emph{higher order operator} and can be formally represented in the ``superoperator'', ``quantum comb'' or ``process matrix'' formalisms~\cite{gutoski_general_2007,chiribella_quantum_2008,chiribella_theoretical_2009,bisio_quantum_2011,leifer_formulation_2013,oreshkov_quantum_2012}.
We will focus on quantum causal structures with three laboratories (three nodes in the graph) $A$, $B$ and $C$ compatible with the causal order ``A is not after B, which is not after C'' ($A \prec B \prec C$). This means that there are no causal influences from $B$ and $C$ to $A$, nor from $C$ to $B$ (see Fig.~\ref{fig:a-b-c-causal networks}). (Since $C$ is last, $C$'s output space $C_O$ can be disregarded.)
In the process matrix formalism, the quantum causal structure is represented by the matrix $W \in A_I \otimes A_O \otimes B_I \otimes B_O \otimes C_I$~\cite{oreshkov_quantum_2012,araujo_witnessing_2015}. The probabilities of observing the outcomes $i,j,k$ at $A, B, C$ (corresponding to implementing the completely positive (CP) maps $M_A^{i}$, $M_B^{j}$, $M_C^{k}$ respectively) are given by the \emph{generalized Born rule}: \begin{equation}\label{eq:gen-born-rule} p(A=i,B=j,C=k) = \tr[W\, (M_A^{i} \otimes M_B^{j} \otimes M_C^{k})]. \end{equation}
The quantum causal structure and local operations should generate only meaningful (that is, \emph{positive} and \emph{normalized}) probability distributions. In addition, we require the probability distributions to be compatible with the causal order $A \prec B \prec C$. Note that both ``common-cause'' and ``direct-cause'' relationships between $A$ and $B$ are compatible with this causal order.
In terms of process matrices, these conditions are equivalent to requiring that $W$ satisfies~\cite{araujo_witnessing_2015}: \begin{gather}\label{eq:process_representation} W \geq 0, \quad W = \mathcal L_{A\prec B\prec C} (W)\\ \tr W = d_{A_O} d_{B_O} \label{eq:process_representation4}. \end{gather} $\mathcal L_{A\prec B\prec C} (\cdot)$ is the projection onto processes compatible with the causal order $A \prec B \prec C$, defined in Appendix~\ref{app:causallyordered}. Eq.~\eqref{eq:process_representation} defines a convex cone $\mathcal W$, eq.~\eqref{eq:process_representation4} a normalization constraint.
\begin{figure}\label{fig:a-b-c-causal networks}
\end{figure}
Following the standard DAG terminology, a purely ``direct-cause'' process $W^\text{dc}$ contains only a \emph{direct cause-effect relation between A and B}, excluding any form of \emph{common cause} between $A$ and $B$. Any correlation between $A$ and $B$ is therefore caused by $A$ alone (Fig.~\ref{fig:a-b-c-causal networks}\,(a) and Fig.~\ref{fig:a-b-c-circuits}\,(a)). Tracing out $C_I$ and $B_O$, the process matrix is a tensor product $\rho^{A_{I}} \otimes \tilde {W}^{A_{O} B_{I}}$. In our scenario, it will prove natural to \emph{extend} this definition to include \emph{convex mixtures} of direct-cause processes, i.e., \begin{equation} \label{eq:w-direct-cause} \tr_{C_{I} B_O} W^{\text{dc}} = \sum_i p_i \rho_i^{A_{I}} \otimes \tilde {W}_i^{A_{O} B_{I}}, \end{equation} where $p_i \geq 0, \sum_i p_i =1$, $\rho^{A_{I}}_i$ are arbitrary states and $\tilde{W}_i^{A_{O} B_{I}}$ arbitrary valid channels between Alice's output and Bob's input, representing to direct cause-effect links between $A$ and $B$.
\begin{figure}
\caption{Circuit representation of the causal structures of Fig.~\ref{fig:a-b-c-causal networks}, where $\ket{\psi_i}$ and $\ket{\phi}$ are states, $\tilde{W_i}, W_2$ and $W_1$ are CP trace preserving (CPTP) maps (lines can represent quantum systems of different dimensions). (a) The direct-cause process $W^\text{dc}$ is the most general one satisfying \eqref{eq:w-direct-cause}; (b) the common-cause process $W^\text{cc}$ is the most general one satisfying \eqref{eq:w-common-cause}.}
\label{fig:a-b-c-circuits}
\end{figure}
Such a process can be interpreted as a probability distribution over states entering $A_I$ and corresponding channels from $A_O$ to $B_I$. In the DAG framework, such probability distributions can be obtained from a graph with an additional latent node that acts as a common cause for all the observed nodes or simply ignorance of the graph that is implemented. Every channel from $A$ to $B$ with classical memory can be decomposed in this way; see Appendix~\ref{app:def-cause-effect} for details.
On the other hand, a purely ``common-cause'' process $W^{\text{cc}}$ does not include \emph{any direct causal influence between A and B} (Fig.~\ref{fig:a-b-c-causal networks}\,(b) and Fig.~\ref{fig:a-b-c-circuits}\,(b)). This implies that there is no channel between $A_O$ and $B_I$. Therefore, when $B_O$ and $C_I$ are traced out, the process factorizes as \begin{equation} \label{eq:w-common-cause} \tr_{C_{I} B_O} W^{\text{cc}} = \sigma^{A_{I} B_I} \otimes \mathds{1}^{A_O}, \end{equation} where $\sigma^{A_{I} B_I}$ is an arbitrary (possibly entangled, possibly mixed) state, representing the common-cause influencing $A$ and $B$.
\section{Classical and quantum superpositions of causal structures} \label{sec:coherent-sup} One possibility of combining direct-cause and common-cause processes consists in allowing for \emph{classical mixtures} thereof: imagine that flipping a (possibly biased) coin determines which process will be realized in an experimental run. Formally, this is described by a process $W^\text{conv}$ which can be decomposed as a convex combination: \begin{equation} \label{eq:w-separable} W^{\text{conv}} = q W^{\text{cc}} + (1-q) W^{\text{dc}}, \end{equation} where $0 \leq q \leq 1$, $W^{\text{dc}}$ satisfies~\eqref{eq:w-direct-cause} and $W^{\text{cc}}$ satisfies~\eqref{eq:w-common-cause}. Note that such a classical mixture was experimentally implemented in Ref.~\cite{ried_quantum_2015}.
Can there be causal structures exhibiting \emph{genuine quantum coherence}, i.e., that cannot be decomposed as a classical mixture of direct-cause and common-cause processes (while respecting the causal order $A\prec B \prec C$)?
We now give an example of such a coherent superposition. It is analogous to the ``quantum switch''~\cite{chiribella_quantum_2013}, which coherently superposes two causal orders $A \prec B \prec C$ and $B \prec A \prec C$, where the causal structure is entangled to a ``control'' system $C_I^{(0)}$ added to $C$'s input space\footnote{See Ref.~\cite{maclean_quantum-coherent_2016} for a different type of quantum causal structure proposed independently.}. To keep the notation simple, we define it in the ``pure'' CJ-vector notation (see Appendix~\ref{app:cj}): \begin{multline} \label{eq:coherent-pure} \ket{w} = \frac{1}{\sqrt{2}}\bigg(\ket{0}^{C_I^{(0)}} \ket{\psi}^{A_{I} B_{I}} \ket{I}\rangle^{A_{O} C_I^{(1)}} \ket{I}\rangle^{B_{O} C_I^{(2)}} \\
+ \ket{1}^{C_I^{(0)}} \ket{\psi}^{A_{I} C_I^{(2)}} \ket{I}\rangle^{A_{O} B_{I}} \ket{I}\rangle^{B_{O} C_I^{(1)}}\bigg), \\ W^{\text{coherent}} := \ket{w}\bra{w} \quad\quad \end{multline} where $\ket{I}\rangle := \sum_{j=1}^{d} \ket{jj}$ represents a non-normalized maximally entangled state---the CJ-representation of an identity channel. The corresponding superposition of circuits is shown in Fig.~\ref{fig:superposition-circuits}. $W^\text{coherent}$ satisfies neither the direct-cause condition \eqref{eq:w-direct-cause} nor the common-cause condition \eqref{eq:w-common-cause} and is a projector on a pure vector, so it cannot be decomposed into \emph{any nontrivial convex combination}, in particular not a mixture of direct-cause and common-cause processes. This proves that the process's causal structure is nonclassical.
\begin{figure}
\caption{Coherent superposition of a direct-cause and a common-cause process, implementing the causal structure $W^\text{coherent}$ of \eqref{eq:coherent-pure}.}
\label{fig:superposition-circuits}
\end{figure}
\section{Physical implementation of the quantum causal structure} The causal structure $W^\text{coherent}$ would not be of particular interest if it were a mere theoretical artifact. We now give an explicit and plausible physical scenario to realize the quantum causal structures in models which respect the principles of general relativistic time dilation and quantum superposition. We utilize the approach recently developed for the ``gravitational quantum switch'' to realize a superposition and entanglement of two different causal orders~\cite{zych_quantum_2015,zych_bell_????}.
Consider two observers, Alice and Bob, who have initially synchronized clocks. We \emph{define} the events in the respective laboratories with respect to the \emph{local clocks}. Bob's local operation will always be applied at his local time $\tau_B$, while Alice's is applied at her local time $\tau_A$. We will consider two configurations, which will be controlled by a quantum system. The state of the control system is given by the position of a massive body. In the first configuration, all masses are sufficiently far away such that the parties are in an approximately flat spacetime. The events in the two laboratories are chosen such that the event $B$ is outside of $A$'s light cone and the common-cause causal relationship is implemented. The coordinate times of the two events, as measured by a local clock of a distant observer, are $t_A \approx \tau_A$ and $t_B \approx \tau_B$. (Fig.~\ref{fig:gravitational}\,(a)).
In the second configuration, a mass $M$ is put closer to Bob's laboratory than to Alice's such that his clock runs slower with respect to hers due to gravitational time dilation. With a suitable choice of mass and distance between Alice and Bob, the event $B$, which is defined by his clock showing local time $\tau_B$, will be inside $A$'s future light cone. In terms of coordinate times one now has $t'_A= \tau_A/ \sqrt{-g_{00}(A)}$ and $t'_B= \tau_B / \sqrt{-g_{00}(B)}$, where $g_{00}(A)$ and $g_{00}(B)$ are the ``00'' components of the metric tensor at the position of the laboratories. This configuration can implement the direct-cause relationship (Fig.~\ref{fig:gravitational}\,(b)).
\begin{figure}
\caption{Space-time diagrams of events in a superposition of casual structures, as seen from a distant observer. Bob's laboratory is moving along a time-like curve, indicated by the circles showing his laboratory before and after $\tau_B$. (a) If the mass $M$ is far away from Bob, the event at his local time $\tau_B$ is space-like separated from $A$ and a common-cause causal structure is realized. (b) If $M$ is sufficiently close to $B$, because of time dilation, $B$'s event at time $\tau_B$, is in the future light cone of $A$, establishing a direct-cause structure between $A$, $B$ and $C$. For a \emph{coherent superposition} of the positions of $M$ (the position of $M$ being the control system $C_I^{(0)}$), the quantum causal structure will be described by $W_\text{coherent}$, as given in \eqref{eq:coherent-pure}.}
\label{fig:gravitational}
\end{figure}
If the mass $M$ is initially in a coherent \emph{spatial superposition} of a position close and a position far away from Bob, the quantum superposition of causal structures $W_\text{coherent}$ is implemented. The \emph{position of the mass} acts as the control system $C_I^{(0)}$;\footnote{The state $\ket{0}$ corresponding to the mass being far away from Bob and the state $\ket{1}$ corresponding to the mass being close to Bob.} it can be received by Charlie, who can manipulate it further (in particular, measure it in the superposition basis). Any possible information about the causal structure (direct cause or common cause) encoded in the degrees of freedom of the laboratories, such as for example in the clocks of the labs, must be erased, possibly using the methods of Ref~\cite{zych_bell_????}.
Note that, in contrast to the superposition of different causal orders~\cite{zych_quantum_2015,zych_bell_????}, the time dilation necessary to ``move $B$ in or out'' of the light cone can, in principle, be made \emph{arbitrarily small}, if Bob can define $\tau_B$ and thus the event $B$ with a sufficiently precise clock\footnote{If Bob's clock cannot resolve the interval $\tau_B (1-1/ \sqrt{-g_{00}(B)})$ within the time $\tau_B$, the event $B$ will be inside or outside $A$'s light cone randomly and independently of the position of $M$, adding noise to the process.}.
To give an idea of the orders of magnitude involved: for a spatial superposition of the order of $\Delta x=\SI{1}{mm}$ and a mass of $M=\SI{1}{g}$, Bob's clock should resolve one part in \SI{e27}{} to be able to certify the nonclassicality of the causal structure. This regime is still quite far from experimental implementation, since the best molecule interferometers~\cite{eibenberger_matterwave_2013} do not go beyond $M=\SI{e5}{amu}$, $\Delta x = \SI{e-6}{m}$, while the best atomic lattice clocks achieve a precision of one part in \SI{e18}~\cite{nicholson_systematic_2015}. An additional difficulty consists in avoiding significant entanglement between the position of the mass and systems other than the local clocks. Nonetheless this regime is still far away from the Planck scale that is usually assumed to be relevant for quantum gravity effects.
We also stress that the process $W_\text{coherent}$, although it cannot be decomposed as a convex combination of a common cause and a direct cause process, is still compatible with the causal order $A \prec B \prec C$ and, as such~\cite{bisio_minimal_2011}, can be realized as a quantum circuit, as shown in Fig.~\ref{fig:w3} (b) of Appendix~\ref{app:causallyordered}.
\section{Verifying the nonclassicality of causal structures} \label{sec:witnessing} We now provide an \emph{experimentally accessible} and \emph{efficiently computable} measure of the nonclassicality of causality.
Let us first define the set $\mathcal S$ of operators which are positive on any convex combination $W^\text{conv}$ of direct-cause and common-cause processes (i.e., processes satisfying \eqref{eq:w-separable}): \begin{equation}\label{eq:causal-witness} S \in \mathcal S \Rightarrow \tr [S\, W^{\text{conv}}] \geq 0 \quad \forall W^\text{conv}. \end{equation} If $S$ is positive on all convex combinations of direct-cause and common-cause process matrices, then it is also positive on all direct-cause ($\tr [S\, W^{\text{dc}}] \geq 0$) and common-cause ($\tr [S\, W^{\text{cc}}] \geq 0$) processes individually.
Since $W^{\text{dc}}$ is a direct-cause process~\eqref{eq:w-direct-cause} if and only if the operator $\tr_{C_I B_O} W^{\text{dc}}$ is separable with respect to the bipartition $(A_I,A_O B_I)$, we effectively require $S$ to be an \emph{entanglement witness}~\cite{horodecki_quantum_2009,chruscinski_entanglement_2014} of the reduced process for the bipartition $(A_I,A_O B_I)$. The full characterization of the set of entanglement witnesses is known to be computationally hard~\cite{gurvits_classical_2003}. Instead, we will use the \emph{positive partial transpose}~\cite{peres_separability_1996,horodecki_separability_1996} criterion as a relaxation to define an efficiently computable measure of nonclassicality.
Enforcing that $S$ is positive on common-cause process matrices in terms of semidefinite constraints is straightforward: since the condition for $W^{\text{cc}}$ \eqref{eq:w-common-cause} to be a common-cause process is already a semidefinite constraint, the ``dual'' constraint for $S$ to be positive on all common-cause process matrices is semidefinite as well.
The operators in the set $\mathcal S_\text{SDP}$ (explicitly constructed in Appendix~\ref{app:s1s2}) are defined as those that obey \emph{both} the condition of having a positive partial transpose and being positive on all common-cause process matrices. Every $S \in \mathcal S_\text{SDP}$ has positive trace with any $W^\text{conv}$. Conversely, $\tr [S\, W] < 0$ certifies that the process $W$ is a genuinely nonclassical causal structure---the operators $S$ can therefore be used as \emph{nonclassicality of causality witnesses}\footnote{The ``causal witnesses'' introduced in Ref.~\cite{araujo_witnessing_2015} are conceptually different, since they examine whether a process can be decomposed as a convex mixture of \emph{causally ordered} processes. All of the processes we study here have a fixed causal order $A \prec B \prec C$.}.
It is crucial to realize that for every given genuinely quantum $W$, one can \emph{efficiently optimize}---the optimization is a semidefinite program~\cite{nesterov_interior_1987}---over the set of nonclassicality witnesses to find the one that has minimal trace with $W$: \begin{equation} \label{eq:sdp-witness-explicit} \begin{gathered} \min \tr [S\,W]\\ \text{s.t.} \quad S \in \mathcal S_\text{SDP}, \quad \mathds{1}/d_{O} - S \in \mathcal W^{ *}, \end{gathered} \end{equation} where $\mathcal W^*$ is the dual cone of $\mathcal W$, given in Appendix~\ref{app:dualcone}. The \emph{normalization condition} $\mathds{1}/d_O - S \in \mathcal W^*$ is necessary for the optimization to reach a finite minimum and confers an operational meaning to $\mathcal C(W) := -\tr[S_\text{opt}\,W]$: it is the amount of ``worst-case noise'' the process can tolerate before its quantum features stop being detectable by witnesses in $\mathcal S_\text{SDP}$ (in analogy to the ``generalized robustness of entanglement''~\cite{steiner_generalized_2003}). Because of its ability to certify the quantum nonclassicality of causal structures, we will refer to $\mathcal C(\cdot)$ as the ``nonclassicality of causality''. Note that $\mathcal C(\cdot)$ satisfies the natural properties of \emph{convexity} and \emph{monotonicity under local operations} (see Appendix~\ref{sec:measure}).
To experimentally verify the properties of a process like $W^\text{coherent}$, one can use the semidefinite program \eqref{eq:sdp-witness-explicit} to compute the optimal nonclassicality of causality witness $S_\text{opt}$ for $W^\text{coherent}$. The nonclassicality of causality $\mathcal C(W^\text{coherent})$ can be measured by decomposing $S_\text{opt}$ in a convenient basis of local operations. In general, this is as demanding as performing a full ``causal tomography''~\cite{araujo_witnessing_2015,ried_quantum_2015,costa_quantum_2015}.
\section{Causal inference under experimental constraints} \label{sec:exp-constraints} There are two reasons to consider witnesses that are subject to certain \emph{additional restrictions}. First, there might be various technical limitations arising from the experimental setup~\cite{ried_quantum_2015,procopio_experimental_2015}, which make full tomography impractical. Second, in analogy to the classical case, it is of \emph{conceptual} interest to investigate the power of \emph{quantum causal inference mechanisms} working on \emph{limited data}. In particular, one might want to investigate differences between quantum and classical causal inference algorithms under such constraints.
As an application of this method, we will examine witnesses for the process $W^\text{coherent}$. In the following, we will consider qubit input and output spaces, i.e., $\dim A_{I} = \dim A_{O} = \dim B_{I} = \dim C_{I}^{(0,1,2)} = 2$ for simplicity and computational speed. The optimal witness for $W_{\text{coherent}}$, obtained from the optimization \eqref{eq:sdp-witness-explicit} using YALMIP~\cite{yalmip} with the solver MOSEK~\cite{mosek}, leads to a nonclassicality of causality of $\mathcal C (W^\text{coherent}) = - \tr [S_{\text{opt}} W_{\text{coherent}}] \approx 0.2278$.
An intriguing feature of quantum causal models is that direct-cause correlations (Fig.~\ref{fig:a-b-c-causal networks}\,(a)) and common-cause correlations (Fig.~\ref{fig:a-b-c-causal networks}\,(b)) can be distinguished through a restricted class of informationally symmetric operations~\cite{leifer_formulation_2013}, sometimes called ``observations''~\cite{fitzsimons_quantum_2013,ried_quantum_2015} that are non-demolition measurements (we refer the reader to Appendix~\ref{app:observations} for certain issues with this definition). We can constrain a witness $S^\text{ndmeas}$ to consist of linear combinations of such non-demolition measurements through an additional condition to the semidefinite program~\eqref{eq:sdp-witness-explicit}, given in Appendix~\ref{app:exp-constraints}.
Surprisingly, purely ``observational'' witnesses are sufficient not only to distinguish common-cause from direct-cause processes, but \emph{also} to distinguish a classical mixture of direct-cause and common-cause processes from a genuine quantum superposition, since $- \tr [S^{\text{ndmeas}}_{\text{opt}} W_{\text{coherent}}] \approx 0.0732$.
Since measurements and repreparations and even non-demolition measurements are often challenging to implement~\cite{grangier_quantum_1998}, it can also be useful to consider a nonclassicality of causality witness $S^\text{unitary}$ which can be decomposed into \emph{unitary operations} for $A$ and $B$, and arbitrary measurements for $C$. The requirement can also easily be translated in a semidefinite constraint, given in Appendix~\ref{app:exp-constraints}. One finds that $- \tr [S^{\text{unitary}}_{\text{opt}} W_{\text{coherent}}] \approx 0.1686$. A summary of the different constraints mentioned in this section can be found in Appendix~\ref{app:exp-constraints}.
\section{Conclusions} We presented a three-event quantum causal model compatible with the causal order $A \prec B \prec C$ which is a quantum controlled \emph{coherent superposition between common-cause and direct-cause models}, not a classical mixture thereof.
The experimental implementation we proposed is of conceptual interest, since it relies both on general relativity and the quantum superpositions principle, two elements we expect to feature in a full theory unifying quantum theory and general relativity. Interestingly, both the mass of the object and the separation between the two amplitudes can be arbitrarily small, as long as Bob has access to a sufficiently precise clock to define the instant of his event $B$.
In order to experimentally certify a genuinely quantum causal structure, we introduced and characterized \emph{nonclassicality of causality witnesses} and provided a semidefinite program to efficiently compute them. Experimental and conceptual constraints are readily included in the framework.
The potential of quantum causal structures as a quantum information resource was recently demonstrated in terms of query complexity~\cite{araujo_computational_2014a} and communication complexity~\cite{feix_quantum_2015,allard_guerin_exponential_2016}, but is still poorly understood. It would be interesting to understand which advantages could be obtained from the coherent superpositions of and common- and direct-cause processes.
\textit{Remark.---} In the final stages of completing this manuscript, a related work by MacLean et al.~\cite{maclean_quantum-coherent_2016} appeared independently. The difference in the definitions of direct-cause processes between the two papers and its implications are discussed in Appendix~\ref{app:def-cause-effect}.
\textit{Acknowledgements.---} We thank Mateus Araújo, Fabio Costa, Flaminia Giacomini, Nikola Paunkovi\'c, Jacques Pienaar and Katia Ried for useful discussions. We acknowledge support from the Austrian Science Fund (FWF) through the Special Research Programme FoQuS, the Doctoral Programme CoQuS, the project I-2526 and the research platform TURIS. This publication was made possible through the support of a grant from the John Templeton Foundation. The opinions expressed in this publication are those of the authors and do not necessarily reflect the views of the John Templeton Foundation.
\appendix
\section{Choi-Jamio\l{}kowski isomorphism} \label{app:cj} The Choi-Jamio\l{}kowski (CJ) representation of a CP map $\mathcal{M}_A: A_I \to A_O$ is \begin{equation} \label{eq:cj} M_A := \mathopen{}\mathclose\bgroup\originalleft[\mathopen{}\mathclose\bgroup\originalleft({\cal I}\otimes{\cal M}_{A} \aftergroup\egroup\originalright)(\ket{I}\rangle\langle\bra{I})\aftergroup\egroup\originalright]^{\mathrm T} \in A_I \otimes A_O, \end{equation} where $\mathcal I$ is the identity map, $\ket{I}\rangle:= \sum_{j=1}^{d_{\mathcal{H}_{I}}} \ket{jj} \in \mathcal{H}_{I}\otimes \mathcal{H}_{I}$ is a non-normalized maximally entangled state and $^\text{T}$ denotes matrix transposition in the computational basis.
The inverse transformation is then defined as: \begin{equation} \label{eq:cj-inverse} \mathcal M_A(\rho)= \tr_{I} \mathopen{}\mathclose\bgroup\originalleft[(\rho \otimes \mathds{1}) M_A \aftergroup\egroup\originalright]^{\mathrm T}. \end{equation}
For operations which have a single Kraus operator ($\mathcal M_A(\rho) = A\rho A^\dagger$), one also define a ``pure CJ-isomorphism''~\cite{royer_wigner_1991,braunstein_universal_2000}, which maps the operation to a \emph{vector}\footnote{Note that there are differing conventions, where the conjugation is omitted.}: \begin{equation} \label{eq:pure-cj} \ket{A^*}\rangle := (\mathds{1} \otimes A^*)\ket{I}\rangle \in \mathcal{H}^{A_I} \otimes \mathcal{H}^{A_O} \end{equation} The usual CJ-representation of such an operation is simply the \emph{projector onto the CJ-vector}: $M_A = \ket{A^*}\rangle\langle\bra{A^*}$. \section{Causally ordered and common-cause process matrices} \label{app:causallyordered} We first introduce a shorthand that we will use throughout the following appendices: \begin{equation} \label{def:notation_trace}
_X W := \frac{\mathds{1}^{X}}{d_X} \otimes \tr_X W, \end{equation} where $d_X$ is the dimension of the Hilbert space $X$.
In this paper, we consider three parties, where the $C$'s output space $C_O$ can be disregarded. The process matrix $W \in A_I \otimes A_O \otimes B_I \otimes B_O \otimes C_I$, which encodes the quantum causal model, is defined on the dual space to the tensor products of the maps. Since both the ``common-cause'' and the ``direct-cause'' scenarios are compatible with the causal order $A \prec B \prec C$, we can also represent the process matrix $W$ as a circuit. (see Fig.~\ref{fig:w3}).
\begin{figure}
\caption{(a) General three-party process matrix $W \in A_I \otimes A_O \otimes B_I \otimes B_O \otimes C_I$. (b) Since, in our scenarios, $W$ is compatible with the causal order $A\prec B \prec C$, we can also represent $W$ as a ``causal network'', which can be implemented as a quantum circuit ($\ket{\psi}$ is a state, $W_1$ and $W_2$ CPTP maps; lines can represent quantum systems of different dimensions.)}
\label{fig:w3}
\end{figure}
For instance, the coherent superposition of common cause and direct cause, defined in \eqref{eq:coherent-pure}, would consist of $\ket{\psi} = \ket{\phi^+} \otimes (\ket{0}+\ket{1})/\sqrt{2}$, $W_1$ and $W_2$ being control-SWAPs (where the control is the last qubit, initially in the state $(\ket{0}+\ket{1})/\sqrt{2}$).
We now define the projection $\mathcal L_{A\prec B\prec C} (\cdot )$ onto the linear subspace of process matrices compatible with the causal order $A\prec B \prec C$, which can be derived from the conditions given in Ref.~\cite{araujo_witnessing_2015}: \begin{multline}\label{eq:projector} \mathcal L_{A\prec B\prec C} (W) := W - _{C_I} W + _{B_O C_I} W \\ - _{B_I B_O C_I} W + _{A_O B_I B_O C_I} W. \end{multline} $W^{A\prec B \prec C}$ is compatible with the causal order $A \prec B \prec C$ if and only if $W^{A\prec B \prec C} = \mathcal L_{A \prec B\prec C} (W^{A\prec B \prec C})$ holds.
The projection onto the subspace of common-cause process matrices $\mathcal L_{\text{cc}} (\cdot)$ is given by composing the projection $\mathcal L_{A \prec B\prec C}$ with the projection onto processes which have no channel from $A_O$ to $B_I$: \begin{multline}\label{eq:projector-cc} \mathcal L_\text{cc} (W) := \mathcal L_{A \prec B\prec C}(W) - _{C_I} \mathcal L_{A \prec B\prec C}(W) \\ + _{C_I A_O} \mathcal L_{A \prec B\prec C}(W). \end{multline}
\section{Dual cones} \label{app:dualcone} Given the definition \eqref{eq:process_representation} of the cone $\mathcal W$, we can characterize the \emph{dual cone} $\mathcal W^*$ of all operators whose product with operators in $\mathcal W$ has positive trace. Remember that $\mathcal W$ is the \emph{intersection} of the cone of positive operators $\mathcal P$ with a linear subspace defined by the conditions for causal order: $\mathcal W := \mathcal P \cap \mathcal L_{A\prec B\prec C}$.
The dual of the linear subspace $ \mathcal L_{A\prec B\prec C}^*$ is its orthogonal complement~\cite{nesterov_interior_1987,araujo_witnessing_2015} \begin{equation}\label{eq:dual-subspace} \mathcal L_{A \prec B \prec C}^* = \mathcal L_{A \prec B \prec C}^\perp, \end{equation} i.e., the space of operators with a support that is orthogonal to the original subspace.
Additionally, the dual of the intersection of two closed convex cones containing the origin is the convex union of their duals~\cite{nesterov_interior_1987,araujo_witnessing_2015}, so that \begin{equation}\label{eq:dual-cone-implicit} \mathcal W^*= (\mathcal P \cap \mathcal L_{A\prec B\prec C})^* = \text{conv}(\mathcal P^* \cup \mathcal L_{A \prec B \prec C}^\perp). \end{equation} Since the cone of positive operators is self-adjoint ($\mathcal P^* = \mathcal P$), we can combine \eqref{eq:dual-subspace} and \eqref{eq:dual-cone-implicit} into $\mathcal W^* = \text{conv}(\mathcal P \cup \mathcal L_{A \prec B \prec C}^\perp)$. Explicitly, this means that any operator $Q \in \mathcal W^*$ can be decomposed as \begin{equation} \label{eq:dualcone} \begin{gathered} Q = Q_1 + Q_2 \\ \text{s.t.}~ Q_1 \geq 0, \quad \mathcal L_{A \prec B \prec C} (Q_2) = 0. \end{gathered} \end{equation}
\section{Nonclassicality of causality witnesses} \label{app:s1s2} We will now explicitly construct the set of nonclassicality of causality witnesses $\mathcal S_\text{SDP}$.
The semidefinite relaxation of the direct-cause constraint \eqref{eq:w-direct-cause} in terms of positive partial transposition is (using the shorthand introduced in \eqref{def:notation_trace}): \begin{equation}\label{eq:w-direct-relax} (_{C_I B_O} W^{\text{dc}})^{\text{T}_{A_I}} \geq 0. \end{equation} The dual cone \eqref{eq:s-direct-relax} to the cone of relaxed direct-cause processes defined by the intersection of $\mathcal W$ with the cone defined in \eqref{eq:w-direct-relax} and the dual cone \eqref{eq:s-common-cause} to the cone of common-cause processes defined by the intersection of $\mathcal W$ with the linear subspace \eqref{eq:w-common-cause} can be constructed in the same way as in Appendix~\ref{app:dualcone}.
The set of witnesses positive on all positive partial transpose operators is a \emph{subset} of entanglement witnesses. Every witness belonging to this set satisfies\footnote{We included the term $S_2$ and $S_3$ although they do not make the witnesses ``more powerful'' to detect entanglement. $S_2$ will become relevant when combining the conditions on direct-cause and common-cause processes in Eq.~\eqref{eq:sdp-witness-explicit2}; $S_3$ is included because it could appear in restricted types of witnesses~\cite{araujo_witnessing_2015}.}: \begin{equation}\label{eq:s-direct-relax} \begin{gathered} S^{\text{dc}} = _{C_I B_O} (S_1^{\text{T}_{A_I}}) + S_2 + S_3 \\ \text{s.t.}~S_1, S_2 \geq 0, \quad \mathcal L_{A\prec B\prec C} (S_3) = 0. \end{gathered} \end{equation} If $\tr [S^{\text{dc}}\, W] < 0$, this implies that $W$ is not a direct-cause process as defined in Eq.~\eqref{eq:w-direct-cause}. Note that since we are only considering a subset of entanglement witnesses, \emph{the converse does not hold}.
We can now turn to the requirement that $S$ is positive on common-cause processes. Since condition~\eqref{eq:w-common-cause} (corresponding to \eqref{eq:projector-cc} together with positivity) defines a convex cone, we can use the techniques of Appendix~\ref{app:dualcone} to construct the dual cone, of which the witness will be an element. This leads us to write $S$ as \begin{equation} \label{eq:s-common-cause} \begin{gathered} S^{\text{cc}} = S_4 + S_5\\ \text{s.t.}~S_4 \geq 0, \quad \mathcal L_\text{cc} (S_5) = 0, \end{gathered} \end{equation} where the projection onto the common-cause subspace $\mathcal L_\text{cc}$ is defined in Appendix~\ref{app:causallyordered}. $W$ is \emph{not} a common-cause process as defined in~\eqref{eq:w-direct-cause} if and only if there exists an $S^\text{cc}$ such that $\tr [S^{\text{cc}}\, W] < 0$.
Now, combining both conditions, we can construct a set of operators positive on all mixtures of direct-cause and common-cause processes \emph{only in terms of semidefinite constraints}. To test whether an arbitrary $W$ process is of this type, we can run the following semidefinite program (SDP)~\cite{nesterov_interior_1987}: \begin{equation} \label{eq:sdp-witness-explicit2} \begin{gathered} \min \tr [S\,W]\\ \text{s.t.} \quad S = _{C_{I} B_O}(S_{1}^{\text{T}_{A_{I}}}) + S_{2} + S_3 = S_{4} + S_{5},\\ S_{1} \geq 0, \quad S_{2} \geq 0, \quad S_{4} \geq 0,\\ \mathcal L_{A \prec B \prec C} (S_3) = \mathcal L_\text{cc} (S_5) = 0,\\ \quad \mathds{1}/d_{O} - S \in \mathcal W^{ *}. \end{gathered} \end{equation} The last condition, where $\mathcal W^*$ is the cone dual to $\mathcal W$ (see Appendix~\ref{app:dualcone}), imposes a normalization on $S$. It gives the nonclassicality of causality $\mathcal C (W) = -\tr [S_{\text{opt}} W]$ the operational meaning of ``generalized robustness'', quantifying resistance of the nonclassicality detectable by $\mathcal S_\text{SDP}$ to \emph{worst possible noise}~\cite{steiner_generalized_2003,araujo_witnessing_2015}. This becomes more intuitive from the dual SDP, given by \label{app:dualsdp} \begin{equation} \label{eq:sdp-w-explicit} \begin{gathered} \min \tr [\Omega/d_{O}]\\ \text{s.t.} \quad W + \Omega = W^{\text{cc}} + W^{\text{dc}},\\ (_{C_{I} B_O} W^{\text{dc}})^{\text{T}_{A_{I}}} \geq 0, \quad W^{\text{dc}} \in \mathcal W, \\ _{C_{I}} W^{\text{cc}} = _{C_I A_O} W^{\text{cc}}, \quad W^{\text{cc}} \in \mathcal W. \end{gathered} \end{equation} The process $\Omega\cdot d_O/\tr[\Omega]$ can be interpreted as worst-case noise with respect to the optimal witness $S_\text{opt}$, resulting from the SDP \eqref{eq:sdp-witness-explicit2}.
\section{Convexity and monotonicity} \label{sec:measure} Here we prove that the \emph{nonclassicality of causality} defined as $\mathcal C (W) := -\tr[S_\text{opt} W]$, which results from the SDP \eqref{eq:sdp-witness-explicit2}, satisfies the natural properties of \emph{convexity} and \emph{monotonicity}, following analogous proofs of Ref.~\cite{araujo_witnessing_2015}.
\emph{Convexity} means that $\mathcal C (\sum_i p_i W_i) \le \sum_i p_i \mathcal C(W_i)$, for any $p_i \geq 0, \sum_i p_i = 1$. Take $S_{W_i}$ to be the optimal witness for $W_i$. Any other witness, in particular the optimal witness $S_W$ for $W := \sum_i p_i W_i$ will be less robust to noise with respect to $W_i$: \begin{equation}
\tr[S_{W_i}\, W_i] \le \tr [S_W\, W_i]. \end{equation} Averaging over $i$ we have \begin{equation}
- \tr \mathopen{}\mathclose\bgroup\originalleft[S_{W} \sum_i p_i W_i \aftergroup\egroup\originalright] \le -\sum_i p_i \tr[S_{W_i} W_i], \end{equation} which is exactly the statement of convexity for $\mathcal C$.
\emph{Monotonicity} under local operation means that $\mathcal C(W) \geq \mathcal C(\$(W))$, where $\$(\cdot)$ is the composition of $W$ with local operations.
We wish to show that $- \tr \mathopen{}\mathclose\bgroup\originalleft [S_{\$(W)} \$ (W) \aftergroup\egroup\originalright ] \le - \tr [S_W W]$. By duality, this is equivalent to \begin{equation}\label{eq:monotonous-dual} - \tr \mathopen{}\mathclose\bgroup\originalleft [\$^* \mathopen{}\mathclose\bgroup\originalleft(S_{\$ (W)} \aftergroup\egroup\originalright) W \aftergroup\egroup\originalright ] \le - \tr [S_W\, W], \end{equation} where $\$^* (\cdot)$ is the map dual to $\$ (\cdot)$. Eq.~\eqref{eq:monotonous-dual} is satisfied if $\$^* \mathopen{}\mathclose\bgroup\originalleft(S_{\$ (W)} \aftergroup\egroup\originalright)$ is a witness, i.e., is positive on all mixtures of direct-cause and common-cause operators ($\tr \mathopen{}\mathclose\bgroup\originalleft[\$^* \mathopen{}\mathclose\bgroup\originalleft(S_{\$ (W)} \aftergroup\egroup\originalright) W^\text{mix} \aftergroup\egroup\originalright] \geq 0$), and is normalized appropriately ($1/d_O - \$^* \mathopen{}\mathclose\bgroup\originalleft(S_{\$ (W)} \aftergroup\egroup\originalright) \in \mathcal W^*$).
The first condition can be seen to hold by applying duality and using the fact that local operations map any mixture of direct-cause and common-cause processes to a mixture of direct-cause and common-cause processes. The second condition is equivalent to \begin{equation} \tr \mathopen{}\mathclose\bgroup\originalleft [\mathds{1}/d_O - \$^* \mathopen{}\mathclose\bgroup\originalleft (S_{\$ (W)} \aftergroup\egroup\originalright) \Omega \aftergroup\egroup\originalright] \geq 0 \end{equation} for every process matrix $\Omega$. We apply duality and linearity of the trace to find that \begin{equation} \tr \mathopen{}\mathclose\bgroup\originalleft [S_{\$ (W)} \$(\Omega) \aftergroup\egroup\originalright] \leq \tr [\Omega]/d_O. \end{equation} This relation holds because $\$(\cdot)$ maps normalized ordered process matrices to normalized ordered process matrices and $\mathds{1}/d_O - S_{\$(W)} \in \mathcal W^*$ is the normalization condition for the SDP~\eqref{eq:sdp-witness-explicit2}.
The condition of \emph{discrimination} (or \emph{faithfulness}), which would mean that $\mathcal C (W) \geq 0$ \emph{if and only if} the process matrix is not a mixture of direct-cause and common-cause processes \eqref{eq:w-separable}, is \emph{not satisfied}. Since we relied on a relaxation of the direct-cause condition by using the positive partial transpose criterion, there are processes which are not a mixture satisfying \eqref{eq:w-separable} but for which the nonclassicality of causality is zero.
Therefore, the nonclassicality of causality is not a \emph{faithful measure} of the nonclassicality of the causal structure. This is reasonable, since finding such a measure would be equivalent to finding a fully general \emph{entanglement criterion}---a problem known to be computationally hard~\cite{gurvits_classical_2003}.
\section{Experimental constraints on witnesses} \label{app:exp-constraints} In this appendix, we give the explicit form of the experimental constraints mentioned in the main text. When using a constrained class of witnesses, the value $- \tr [S^{\text{restricted}}_{\text{opt}} W_{\text{coherent}}] $. can be interpreted as the \emph{amount of noise} tolerated before the \emph{constrained set of witnesses} becomes incapable of detecting the nonclassicality of causality of $W_\text{coherent}$.
A simple example of a restriction simplifying the experimental implementation consists in disregarding the space $C_{I}^{(1,2)}$, i.e., to have $S = _{C_{I}^{(1,2)}}S$ as an additional constraint. The nonclassicality of causality is \emph{unaffected} by this restriction, which shows that the input spaces $C_{I}^{(1,2)}$ do not carry any additional information about the nonclassicality of causality.
The constraint for the witness to consist only of non-demolition measurements is: \begin{multline} \label{eq:witness-observation} S^{\text{ndmeas}} = \sum_{ijl} \alpha_{ijl} (\mathds{1} +\sigma_{i}^{A_{I}})\otimes (\mathds{1} + \sigma_{i}^{A_{O}})\\
\otimes (\mathds{1} +\sigma_{j}^{B_{I}}) \otimes (\mathds{1} +\sigma_{j}^{B_{O}}) \otimes E_{l}^{C_I}, \end{multline} where $\sigma_k$ ($k=1,2,3$) are the qubit Pauli matrices and $E_l$, $l=1,\dots, 8$ is an arbitrary basis of projectors on $C_I$'s three qubits.
The constraint for the witness to only consist of unitary operations\footnote{Note that according to definition of Ref.~\cite{ried_quantum_2015}, unitary witnesses should also be considered as ``observations'' although operationally they are standardly understood as interventions.} for $A$ and $B$ is: \begin{multline} \label{eq:witness-unitary} S^{\text{unitary}} = \sum_{ijl} \beta_{ijl} \ket{U_{i}^{ * }}\rangle\langle\bra{U_{i}^{ * }}^{A_{I}A_{O}} \\ \otimes \ket{U_{j}^{ * }}\rangle\langle\bra{U_{j}^{ * }}^{B_{I}B_{O}} \otimes E_{l}^{C_I}, \end{multline} where $i,j = 1,\ldots, 10$ indexes a basis\footnote{This is because there are ten linearly independent projectors on CJ-vectors for unitaries acting on qubits~\cite{araujo_witnessing_2015}.} of the CJ-vectors (see Appendix~\ref{app:cj}) of unitaries.
\begin{table}[htb] \caption{\label{tab:gr-restricted-witnesses}Constrained nonclassicality of causality for different types of constraints on $S$, in descending order.} \centering \begin{tabular}{lc} \hline Constraint on the witness $S$ & $\mathcal - \tr[S\,W^\text{coherent}]$\\ \hline No constraint & 0.2278\\ Discarding $C_I^{(1,2)}$ & 0.2278\\ Unitary operations $A,B$ & 0.1686\\ ND measurement $A,B$ & 0.0732 \\ \hline \end{tabular} \end{table}
\section{Definition of direct-cause processes and relationship to the definitions of Ref.~\cite{maclean_quantum-coherent_2016}} \label{app:def-cause-effect} Since Ref.~\cite{maclean_quantum-coherent_2016} considers two party case, we can merge $B$ and $C$ to make our scenario comparable to the one of Ref.~\cite{maclean_quantum-coherent_2016}. More precisely, $B_I$ and $C_I$ are relabeled as $B'_I$ and $B_O$ is disregarded, eliminating the necessity to trace over $B_O$ and $C_I$. The condition for direct-cause processes \eqref{eq:w-direct-cause} then becomes \begin{equation} \label{eq:w-direct-cause-app}
W^{\text{dc}} = \sum_i p_i \rho_i^{A_{I}} \otimes \tilde {W}_i^{A_{O} B_{I}'}, \end{equation} which implies that the states given to $A$ and the channel connecting $A$ and $B$ can be \emph{classically correlated}.
In the terminology of DAGs this convex mixture would correspond to tracing over a (hidden) classical\footnote{Strictly speaking, it just needs not to produce any entanglement between $A_I$ and $(A_O, B_I)$, see Fig.~\ref{fig:class-direct-cause}.} common cause between $A$ and $B$. An alternative, more restricted definition would exclude such classical correlations, i.e., \begin{equation} \label{eq:w-direct-cause-alternative-app} W^{\text{dc}}{} = \rho^{A_{I}} \otimes \tilde {W}^{A_{O} B_{I}'}. \end{equation} It is used in Ref.~\cite{maclean_quantum-coherent_2016}. To make the difference apparent, consider the convex mixture of two direct-cause processes between $A$ and $B$ (here, $\dim A_I = \dim A_O = \dim B_I' = 2$): \begin{multline} \label{eq:class-with-memory} W^{\text{mem}} = \frac{1}{4} \ket{0}\bra{0}^{A_I} (\mathds{1}^{A_O B_I'} + \sigma_z^{A_O} \sigma_z^{B_I'}) \\ + \frac{1}{4} \ket{1}\bra{1}^{A_I} (\mathds{1}^{A_O B_I'} - \sigma_z^{A_O} \sigma_z^{B_I'}), \end{multline} where the tensor products between the Hilbert spaces are implicit. $W^\text{mem}$ classically correlates the channel between $A_O$ and $B_I'$ (a classical channel with or a without bit flip) to the state in $A_I$, as shown in Fig.~\ref{fig:class-direct-cause}. It is of the type \eqref{eq:w-direct-cause-app} but \emph{not} of the type \eqref{eq:w-direct-cause-alternative-app}.
\begin{figure}
\caption{Quantum causal models respecting the extended ``direct-cause'' condition \eqref{eq:w-direct-cause-app} can be thought of as a general channel with \emph{classical} memory (left), or equivalently as a convex combination of direct-cause processes with no memory (right). $\tilde W$ and $\tilde W_i$ are general quantum channels, $\ket{\psi}$ an arbitrary quantum state and the gray square represents a fully dephazing channel (in an arbitrary basis).}
\label{fig:class-direct-cause}
\end{figure}
In Ref.~\cite{maclean_quantum-coherent_2016}, \eqref{eq:class-with-memory} is not considered to be a direct-cause process, nor a convex mixture (called ``probabilistic mixture'') of direct-cause and common-cause processes. It is instead termed a ``physical mixture'' of common-cause and direct-cause processes.
We instead use the broader definition \eqref{eq:w-direct-cause-app} because we ultimately intend to study convex combinations of common-cause and direct-cause processes \eqref{eq:w-separable}, which means we should also allow for convex combinations of direct-cause processes. The restricted definition \eqref{eq:w-direct-cause-alternative-app} for direct-cause processes would lead to consider a convex combination of a direct-cause and a common-cause process to be a ``probabilistic mixture'', but \emph{not} a convex combination of \emph{two cause-effect processes}.
Finally note that the class of processes, which, when post-selected on CP maps being implemented at $B_I'$, result in an entangled conditional process on $A_I A_O$, is defined to be ``coherent mixtures'' in Ref.~\cite{maclean_quantum-coherent_2016}. All of these ``coherent mixtures'' are nonclassical in our terminology (the processes that can be decomposed as \eqref{eq:w-separable} never result in an entangled conditional process on $A_I A_O$). It is not clear whether the converse is true.
\section{Issues in defining a quantum ``observational scheme''} \label{app:observations} Ried et al.~\cite{ried_quantum_2015} define the ``observational scheme'' (as opposed to the ``interventionist scheme'') on a quantum causal structure as composed of operations satisfying the ``informational symmetry principle''. We examine the subtleties and issues involved in this definition, in particular regarding the dependence on the initially assigned state.
Ref.~\cite{ried_quantum_2015} assumes that before the observation, one assigns the (epistemic) state $\rho_{A_I}$ to the system coming into $A$'s laboratory. A quantum operation (described by the Choi-Jamio\l{}kowski representation of the quantum instrument~\cite{davies_operational_1970} $\{M_A^{i}\}$, where $i$ labels the outcome) is applied. This updates the information about the outgoing state $\rho_{A_O}^{(i)}$ \emph{but also} (through retrodiction) about the incoming state $\rho_{A_I}^{(i)}$. These states are found by applying the update rules~\cite{leifer_formulation_2013}: \begin{align} \rho_{A_O}^{(i)} &= \frac{\tr_{A_I} [M_A^{i} \cdot \rho_{A_I} \otimes \mathds{1}_{A_O}]^\text{T}}{\tr [M_A^{i} \cdot \rho_{A_I} \otimes \mathds{1}_{A_O}]},\\ \rho_{A_I}^{(i)} &= \frac{\tr_{A_O} \mathopen{}\mathclose\bgroup\originalleft[(\sqrt{\rho_{A_I}}\otimes \mathds{1}_{A_O}) M_A^{i} (\sqrt{\rho_{A_I}}\otimes \mathds{1}_{A_O}) \aftergroup\egroup\originalright]}{\tr \mathopen{}\mathclose\bgroup\originalleft[(\sqrt{\rho_{A_I}}\otimes \mathds{1}_{A_O}) M_A^{i} (\sqrt{\rho_{A_I}}\otimes \mathds{1}_{A_O}) \aftergroup\egroup\originalright]}. \end{align}
The informational symmetry principle holds if and only if after the operation, the states assigned to the incoming and outgoing systems are the same: \begin{equation}\label{eq:informational-symmetry} \rho_{A_I}^{(i)} = \rho_{A_O}^{(i)}. \end{equation} For Ried et al., an instrument for which this informational symmetry holds is \emph{defined} to be an ``observation''~\cite{ried_quantum_2015}. In this sense, there can obviously be ``non-passive'' observations such as non-demolition measurements. Any non-demolition measurement in a basis in which the initially assigned state $\rho_{A_I}$ is \emph{diagonal} will be an observation in this sense. This matches the intuition that a \emph{classical} measurement only \emph{reveals} information and does not disturb the system.
If one wishes to implement measurements in \emph{arbitrary bases}, the \emph{only} initially assigned state which results in informational symmetry is the maximally mixed state $\rho_{A_I} = \mathds{1}/d$~\cite{ried_quantum_2015}. This shows how problematic the definition of observational scheme is, since it not only crucially depends on an initial (epistemic) assignment $\rho_{A_I}$ but also because there is \emph{only one} such assignment which allows all measurements to be ``observations''---which tolerates no amount and no type of noise. In this sense, as soon as the experimenter \emph{changes her beliefs} about the incoming state \emph{in any way}, she will be intervening on the system, not merely observing it.
Leaving aside these interpretative difficulties, it is interesting to realize that operations which are \emph{unitary} also turn out to be ``observations'' if the initially assigned state is $\rho_{A_I} = \mathds{1}/d$: for a unitary operation, $\rho_{A_I}^{(i)} = \rho_{A_O}^{(i)} = \rho_{A_I} = \mathds{1}/d$. The unitary provides exactly the same information about input and output states, namely \emph{none}.
Finally, note that both the framework of Ref.~\cite{ried_quantum_2015} and the one we developed rely on the assumption that quantum theory is valid and the correct operations were implemented---the analysis is \emph{device-dependent}. This means that any ``quantum advantage'' in inference will not be based on \emph{mere correlations} in the sense of a conditional probability distribution of outputs given inputs. This makes the comparison with the power of classical causal models somewhat problematic.
\end{document} | arXiv |
Outline of actuarial science
The following outline is provided as an overview of and topical guide to actuarial science:
Actuarial science – discipline that applies mathematical and statistical methods to assess risk in the insurance and finance industries.
What type of thing is actuarial science?
Actuarial science can be described as all of the following:
• An academic discipline –
• A branch of science –
• An applied science –
• A subdiscipline of statistics –
Essence of actuarial science
Actuarial science
• Actuary
• Actuarial notation
Fields in which actuarial science is applied
• Mathematical finance
• Insurance, especially:
• Life insurance
• Health insurance
• Human resource consulting
History of actuarial science
History of actuarial science
General actuarial science concepts
Insurance
• Health insurance
Life Insurance
• Life insurance
• Life insurer
• Insurable interest
• Insurable risk
• Annuity
• Life annuity
• Perpetuity
• New Business Strain
• Zillmerisation
• Financial reinsurance
• Net premium valuation
• Gross premium valuation
• Embedded value
• European Embedded Value
• Stochastic modelling
• Asset liability modelling
Non-life Insurance
• Property insurance
• Casualty insurance
• Vehicle insurance
• Ruin theory
• Stochastic modelling
• Risk and capital management in non-life insurance
Reinsurance
Reinsurance
• Financial reinsurance
• Reinsurance Actuarial Premium
• Reinsurer
Investments & Asset Management
• Dividend yield
• PE ratio
• Bond valuation
• Yield to maturity
• Cost of capital
• Net asset value
• Derivatives
Mathematics of Finance
Financial mathematics
• Interest
• Time value of money
• Discounting
• Present value
• Future value
• Net present value
• Internal rate of return
• Yield curve
• Yield to maturity
• Effective annual rate (EAR)
• Annual percentage rate (APR)
Mortality
• Force of mortality
• Life table
Pensions
Pensions
• Stochastic modelling
Other
• Enterprise risk management
• Fictional actuaries
Persons influential in the field of actuarial science
• List of actuaries
See also
• Index of accounting articles
• Outline of economics
• Outline of corporate finance
• Outline of finance
References
External links
• Additional Actuarial Topics
• Actuarial Science Study Forum
• Actuarial Wiki
• Google Map of Actuarial Science Universities in the US
Professional Organizations/Associations for Actuaries
• CCA - Conference of Consulting Actuaries
• CAS - Casualty Actuarial Society
• SOA - Society Of Actuaries
Wikipedia Outlines
General reference
• Culture and the arts
• Geography and places
• Health and fitness
• History and events
• Mathematics and logic
• Natural and physical sciences
• People and self
• Philosophy and thinking
• Religion and belief systems
• Society and social sciences
• Technology and applied sciences
| Wikipedia |
Stevedore knot (mathematics)
In knot theory, the stevedore knot is one of three prime knots with crossing number six, the others being the 62 knot and the 63 knot. The stevedore knot is listed as the 61 knot in the Alexander–Briggs notation, and it can also be described as a twist knot with four half twists, or as the (5,−1,−1) pretzel knot.
Stevedore knot
Common nameStevedore knot
Arf invariant0
Braid length7
Braid no.4
Bridge no.2
Crosscap no.2
Crossing no.6
Genus1
Hyperbolic volume3.16396
Stick no.8
Unknotting no.1
Conway notation[42]
A–B notation61
Dowker notation4, 8, 12, 10, 2, 6
Last /Next52 / 62
Other
alternating, hyperbolic, pretzel, prime, slice, reversible, twist
The mathematical stevedore knot is named after the common stevedore knot, which is often used as a stopper at the end of a rope. The mathematical version of the knot can be obtained from the common version by joining together the two loose ends of the rope, forming a knotted loop.
The stevedore knot is invertible but not amphichiral. Its Alexander polynomial is
$\Delta (t)=-2t+5-2t^{-1},\,$
its Conway polynomial is
$\nabla (z)=1-2z^{2},\,$
and its Jones polynomial is
$V(q)=q^{2}-q+2-2q^{-1}+q^{-2}-q^{-3}+q^{-4}.\,$[1]
The Alexander polynomial and Conway polynomial are the same as those for the knot 946, but the Jones polynomials for these two knots are different.[2] Because the Alexander polynomial is not monic, the stevedore knot is not fibered.
The stevedore knot is a ribbon knot, and is therefore also a slice knot.
The stevedore knot is a hyperbolic knot, with its complement having a volume of approximately 3.16396.
See also
• Figure-eight knot (mathematics)
References
1. "6_1", The Knot Atlas.
2. Weisstein, Eric W. "Stevedore's Knot". MathWorld.
Knot theory (knots and links)
Hyperbolic
• Figure-eight (41)
• Three-twist (52)
• Stevedore (61)
• 62
• 63
• Endless (74)
• Carrick mat (818)
• Perko pair (10161)
• (−2,3,7) pretzel (12n242)
• Whitehead (52
1
)
• Borromean rings (63
2
)
• L10a140
• Conway knot (11n34)
Satellite
• Composite knots
• Granny
• Square
• Knot sum
Torus
• Unknot (01)
• Trefoil (31)
• Cinquefoil (51)
• Septafoil (71)
• Unlink (02
1
)
• Hopf (22
1
)
• Solomon's (42
1
)
Invariants
• Alternating
• Arf invariant
• Bridge no.
• 2-bridge
• Brunnian
• Chirality
• Invertible
• Crosscap no.
• Crossing no.
• Finite type invariant
• Hyperbolic volume
• Khovanov homology
• Genus
• Knot group
• Link group
• Linking no.
• Polynomial
• Alexander
• Bracket
• HOMFLY
• Jones
• Kauffman
• Pretzel
• Prime
• list
• Stick no.
• Tricolorability
• Unknotting no. and problem
Notation
and operations
• Alexander–Briggs notation
• Conway notation
• Dowker–Thistlethwaite notation
• Flype
• Mutation
• Reidemeister move
• Skein relation
• Tabulation
Other
• Alexander's theorem
• Berge
• Braid theory
• Conway sphere
• Complement
• Double torus
• Fibered
• Knot
• List of knots and links
• Ribbon
• Slice
• Sum
• Tait conjectures
• Twist
• Wild
• Writhe
• Surgery theory
• Category
• Commons
| Wikipedia |
Writing what I'm learning
Complex cobordism cohomology
November 18, 2021 · 16 min · Torgeir Aambø
Stably complex manifolds
Cohomology from geometry
Cohomology from spectra
What do we know about $MU$?
In the next couple years I will need to understand the ins and outs of different cohomology theories and the spectra that represents them. Some of the most important of these (for my research) can be described using $MU$ — the complex cobordism spectrum. We briefly met this spectrum — or at least its cohomology theory — when we discussed formal group laws. There we explained briefly a theorem of Quillen, stating that the universal formal group law over the Lazard ring corresponds to complex cobordism cohomology. We did not cover what complex cobordism actually is, so that is the plan for this post.
Let us again quickly describe the backdrop for why this is important. We are interested in understanding Landweber exact complex oriented cohomology theories. All these are formed by taking a product with the universal such theory, i.e. the universal complex oriented cohomology theory. This theory is precisely complex cobordism cohomology. It is universal in the sense that any complex orientation on some multiplicative cohomology theory $E$ can be realized as a map $MU\longrightarrow E$. We will describe all this during this post, but first we need to understand what $MU$ actually is.
Stably complex manifolds#
We have already seen cobordisms a couple of times already, and each time it has featured as a relation between two manifolds. This relation is called the cobordism relation.
Cobordism relation: Let $M$ and $N$ be manifolds of dimension $n$. We say they are cobordant if there is a manifold $W$ of dimension $n+1$ such that $\partial W = M\sqcup N$, i.e. the boundary of $W$ is the disjoint union of $M$ and $N$.
The name of the cobordism theory we are interested in is called complex cobordism, so due to the name one might expect that we simply consider cobordisms of complex manifolds. But as you might see, if $M$ and $N$ are complex manifolds, then they have real dimension $2n$ for some $n$. A cobordism would then have dimension $2n+1$, but then it can't have a complex structure. We can work around this by introducing so-called stably complex structures instead. These are a simple enough generalization that allow complex-like structures in odd dimensions.
Let $Gr_k(\mathbb{R}^n$ denote the real $k$-Grassmannian of euclidean $n$-space, i.e. the space of all $k$-dimensional subspaces of $\mathbb{R}^n$, and denote the limit $lim Gr_k(\mathbb{R}) = BO(k)$. As the orthogonal group $O(k)$ is the structure group of real vector bundles and $BO(k)$ is the classifying space of $O(k)$, we get that vector bundles of dimension $k$ over a manifold $M$ is classified by maps $M\longrightarrow BO(k)$. For complex vector bundles we would get $BU(k)$ instead, the classifying space of the unitary group. But, we want something "weaker" than complex structure, which we build from something real.
Let $M$ be an $n$-dimensional manifold and $i:M\longrightarrow \mathbb{R}^{n+k}$ be an embedding into some euclidean space. By the Whitney embeddign theorem such a map always exists. The normal bundle of $M$ in $\mathbb{R}^{n+k}$ is the orthogonal complement of the tangent bundle, i.e. all vectors that are normal to $M$. We can also generalize this to embeddings into other spaces by defining the normal bundle of an embedding $i:M\longrightarrow N$, denoted $N(i)$ to be the quotient of the tangent bundle of $N$ by the tangent bundle of $M$. These are the objects we use to define the stable complex structures. We do see that the dimension of the normal bundle is dependent on the embedding, so this must be fixed later.
Let $i:M\longrightarrow \mathbb{R}^{n+k}$ be an embedding and $N(i): E(i) \overset{\pi}\longrightarrow M$ the associated $k$-dimensional normal bundle. It is classified by a map $v:M\longrightarrow BO(k)$, which is the Gauss map. A $BU(k)$ structure on $M$ is a choice of lift of $v$ through the fibration $f:BU(k)\longrightarrow BO(k)$. Given a $BU(k)$ structure on $M$ we get by composing the embedding with the inclusion $\mathbb{R}^k\hookrightarrow \mathbb{R}^{k+1}$, an essentially unique $BU(k+1)$ structure on $M$. This means that we can define an equivalence relation on $BU(k)$ structures by defining them to be equivalent if they become equivalent in a high enough dimension. This is the idea behind the term "stable".
Definition: A stable complex structure on a manifold $M$ is an equivalence class of $BU(k)$ structures under the equivalence relation described above. A stably complex manifold is then a pair $(M, v)$ where $M$ is a compact manifold and $v$ a stable complex structure on it.
We see that this does not depend on having even dimension, hence we have complex-like structures for manifolds in all dimensions. This is the generalization that allows us to define complex cobordism.
Definition: Let $M$ and $N$ be two stably complex manifolds of dimension $n$. We say they are complex cobordant if there is a stably complex manifold $W$ of dimension $n+1$ such that $\partial W = M\sqcup N$.
Here $\sqcup$ denotes disjoint union, hence this is just the cobordism relation on stably complex manifolds. We sometimes split the boundary $\partial W$ into an "in-boundary" and an "out-boundary", which is simply a way to assign a direction to the cobordism. We then write $\partial W_+ = M$ and $\partial W_- = N$, which intuitively means that the cobordism "goes from" $M$ to $N$. A picture might help:
Cohomology from geometry#
Now that we know what complex cobordisms are we need to understand how we get a cohomology theory. Recall that a cohomology theory is a functor from topological spaces to graded abelian groups satisfying the generalized Eilenberg-Steenrod axioms. We have described these before, so look at this post for a more in depth recollection. There are two ways of approaching this for complex cobordism: one geometric, and one abstract. The geometric construction only works well for manifolds, hence why we also have a more general second construction. But, as it is nice with some geometry, we also present the former.
Let $X$ be a manifold of dimension $n$, $M$ and $N$ be stably complex manifolds of dimension $(n-k)$ together with maps $f:M\longrightarrow X$ and $g:N\longrightarrow X$. Furthermore we require that these maps are proper — that the inverse image of a compact subset is compact — and that that the stable normal bundles, $N(f)$ and $N(g)$, are stably complex. Such maps are called complex oriented. We say two complex oriented maps are complex cobordant if there is a complex cobordism $W$ such that $\partial W = M\sqcup N$ and a map $h:W\longrightarrow X\times I$ such that $f = h_{|(h^{-1}(X\times {0}))}$ and $g=h_{|(h^{-1}(X\times {1}))}$, i.e. the restriction of $h$ to the fibers over the endpoints of $I$ correspond to $f$ and $g$. This relation is an equivalence relation. Hence we define the set $MU^k(X)$ to be the set of complex oriented maps modulo the cobordism relation described. This set is in fact an abelian group under disjoint union. The assignment $X\longmapsto MU^k(X)$ is also functorial in manifolds and satisfies the generalized Eilenberg-Steenrod axioms, hence it is a cohomology theory of manifolds. The reason we cant extend this to all topological spaces is that we are using the stable normal bundles of maps. These only makes sense for manifolds, as they are the ones with defined stable normal bundles. So, to get a cohomology theory for all topological spaces we must try something more general.
Cohomology from spectra#
The more general approach comes from the fact that all cohomology theories are represented by a spectrum via the Brown representability theorem. We construct a nice spectrum, denoted $MU$, that naturally arises from complex cobordism and then define a cohomology theory from this spectrum. The construction that gives us the spectrum is called the Pontrjagin-Thom construction. We will explain all the steps, but let's first establish a guide for how we get there:
Define the complex cobordism group
Under the Pontrjagin-Thom construction this group is isomorphic to the homotopy groups of some spectrum
Under Brown representability this spectrum represents a cohomology theory
We might not get to stage four, but we will at least describe the three first ones. The first thing we need is the complex cobordism group. You might recall that we looked at the $h$-cobordism group of the spheres in the previous post looking at stable homotopy groups of spheres. The construction of a general cobordism group is similar, but this time we don't restrict ourselves only to the spheres. As a set, the $n$'th complex cobordism group consists of complex cobordism classes of closed stably complex $n$-manifolds, i.e.
$$\Omega^{U}_n = {(M, c)}/\sim$$
where two stably complex $n$-manifolds are equivalent $(M,c_M)\sim (N, c_N)$ if there is a complex cobordism $W$ such that $\partial W = M\sqcup N$, i.e. they are complex cobordant. Being complex cobordant is an equivalence relation because it is
reflexive by the cylinder cobordism,
symmetric by swapping the "in" and "out" boundaries and
transitive by composition of cobordisms,
which means that $\Omega_n^U$ is well defined as a set. The group operation is given by disjoint union, where the inverses are given by
$$-(M, c) = (N,-c)$$
and the identity is the empty set $\empty$ — treated as a manifold of dimension $n$. The group is even abelian because we have $(M, c_M)\sqcup (N, c_N) \sim (N, c_N)\sqcup (M, c_M)$ by a twist cobordism, often visualized as:
We have now completed step one, and so proceed with step two — the Pontrjagin-Thom construction. We will not cover this in high detail but simply state its pieces and parts. This is because we are merely interested in some topological spaces that show up, which will be the components of the spectrum $MU$.
Let now $M$ be some stably complex manifold of dimension $n$. It has an embedding $i:M\hookrightarrow \R^{n+k}$ which gives us its normal bundle $N(i)$. By the tubular neighbourhood theorem we have a an inclusion $\tau N(i)\hookrightarrow \R^{n+k}$, where $\tau N(i)$ denotes the tubular neighbourhood of the normal bundle. Define a map $\R^{n+k}\longrightarrow \tau N(i)$ that is the identity on the interior of $\tau N(i)$ and sends everything else to a point. Hence it defines a map $\R^{n+k}\longrightarrow \tau N(i)/\partial \tau N(i)$. This is a proper map on locally compact Hausdorff spaces, hence it induces a map on the one-point compactifications. The one point compactification of $\R^{n+k}$ is $S^{n+k}$, and we denote the one point compactification of $\tau N(i)/\partial \tau N(i)$ by $T(N(i))$. Hence we have a map.
$$S^{n+k} \longrightarrow T(N(i))$$
So, from a stably complex $n$-manifold $M$ we have produced a map from a sphere to some compact nice space. This space $T(N(i))$ is called the Thom space of $N(i)$, and is a general and functorial construction. The Thom space of a vector bundle $\xi = (V, B, \pi)$ is constructed as the quotient bundle of the disc-bundle (the vectors of length less than one) by its boundary (the unit length vectors), i.e. $T(V) = D(V)/S(V)$. The classifying space $BU(k)$ has a universal vector bundle $\gamma_k = (EU(k), BU(k), \pi)$, and thus the $k$-dimensional normal bundle $N(i)$ admits a map $N(i)\longrightarrow \gamma_k$. This map induces a map on Thom spaces, $T(N(i))\longrightarrow T(\gamma_k)$. By composing this map with the above map from $S^{n+k}$ we get a map
$$\phi_k:S^{n+k}\longrightarrow T(\gamma_k),$$
representing an element in $\pi_{n+k}(T(\gamma_k))$. The inclusion $i:BU(k)\hookrightarrow BU(k+1)$ gives an isomorphism $i^\ast\gamma_{k+1}\cong \gamma_k\oplus \mathbb{1}$ , where $\mathbb{1}$ is the trivial complex line bundle over $BU(k)$. The Thom space of $\gamma_k\oplus \mathbb{1}$ turns out to be $\Sigma^2 T(\gamma_k)$. This makes sense intuitively, as $T(\gamma_k)$ is roughly the one-point compactification of a real vector bundle, i.e. a sphere, which when suspended twice should give a sphere in two dimensions higher, which again should be a one-point compactification of a vector bundle with complex dimension increased by one. This gives us maps $\Sigma^2 T(\gamma_k)\longrightarrow T(\gamma_{k+1})$. We can then finally define $MU$.
Definition: The complex cobordism spectrum $MU$ consists of topological spaces $MU_{2k} = T(\gamma_{k})$ and $MU_{2k+1}=\Sigma T(\gamma_k)$. The structure maps $\sigma_n : \Sigma MU_n \longrightarrow MU_{n+1}$ is given by the identity for $n=2k$ and by the above described map $\Sigma^2 MU_{2k} \cong \Sigma MU_{2k+1}\longrightarrow MU_{2k+2})$ for $n=2k+1$.
This is the spectrum we need to define the cohomology theory, but in the interest of completing the story and the construction we mention the famous result that the Pontrjagin-Thom construction gives us. Taking the colimit of the homotopy groups $\pi_{n+k}(T(\gamma_k))$, we see that we get the homotopy groups of the spectrum $MU$, i.e. $colim_k \pi_{n+k}(T(\gamma_k)) = \pi_n MU$. This finally means that a complex cobordism class of a stably complex $n$-manifold, i.e. an element of the abelian group $\Omega^U_n$, now gives through the above construction an element of the abelian group $\pi_n MU$. The remarkable thing about this construction is the fact that this association is an isomorphism of groups. Hence we have
$$\Omega^U_n \cong \pi_nBU.$$
This is the so-called Pontrjagin-Thom construction and it actually holds for all $B$-cobordism theories, giving several nice isomorphisms to play with. This does then conclude the second point.
The third point we are luckily already familiar with, as we have covered the Brown representability theorem in an earlier post. This theorem gives us for a spectrum — in our case $MU$ — a cohomology theory $MU^\ast(-)= [-, MU^\ast]$, i.e. homotopy classes of maps into the spectrum. Importantly, the groups $MU^k(X)$ for some manifold $X$ agrees with the earlier geometric construction of complex cobordism classes of complex oriented maps into $X$. As we have covered this theory in detail before, we instead continue to the stuff mentioned in the introduction. The important part is that we now have a spectrum $MU$ that we know how to construct, and we know how relates to geometry.
What do we know about $MU$?#
As $MU_{2n} = T(\gamma_n)$ we have in particular that $MU_2 = T(\gamma_1)$ where $\gamma_1$ is the universal complex line bundle over $BU(1)$. The topological space $BU(1)$ is modeled by a colimit of Grassmann manifolds, $Gr_1(\mathbb{C}^k)$, as $k$ goes to infinity. The first Grassmannians is the same as complex projective space, so $BU(1)$ is modelled by $colim_k \mathbb{C}P^k = \mathbb{C}P^\infty$, the infinite complex projective space.
The cool thing is that the Thom space of the universal complex line bundle is homotopy equivalent to the base-space, i.e. $\mathbb{C}P^\infty$. The homotopy equivalence is induced by the zero-section on $\gamma_1$. Hence we get that $MU_{2} \simeq \mathbb{C}P^\infty$. By convention we have $BU(0)\simeq pt$ and by definition: a zero dimensional vector bundle over a point is still just a point. This means that a zero dimensional vector bundle over $BU(0)$ is a trivial bundle. The Thom space of a trivial bundle of dimension $n$ is the $n$'th suspension of the base-space with a disjoint basepoint. For $n=0$ the zeroth suspension is the identity, and hence we are left with $T(\gamma_0)=pt_+$, i.e. a point with a disjoint basepoint. Such a space is just the zero-sphere, hence we have $MU_0 = S^0$.
Since $BU$ is given by Thom spaces of universal vector bundles, we might wonder if we can use this to construct some more structure on $MU$. The direct sum of two complex vector bundles of dimension $n$ and $m$ are classified by a map $BU(n)\times BU(m)\longrightarrow BU(n+m)$. For universal bundles over these we get induced maps on Thom spaces
$$T(\gamma_n\oplus \gamma_m)\longrightarrow T(\gamma_{n+m}),$$
and for any vector bundles the Thom space of a sum is the wedge of the Thom spaces. Hence we get a map
$$T(\gamma_n)\wedge T(\gamma_m)\longrightarrow T(\gamma_{n+m})$$
which induces a multiplication $MU\times MU\longrightarrow MU$. This means in particular that $MU$ is a multiplicative spectrum, and that complex cobordism cohomology is a multiplicative cohomology theory.
This multiplicative structure gives us a ring structure on $\pi_\ast MU \cong MU^\ast(pt)$. Recall that for a manifold $X$, the group $MU^n(X)$ is the complex cobordism class of complex oriented maps into $X$. If we let $X=pt$ then all maps into $X$ are complex oriented, and they are cobordant precisely when their codomains are complex cobordant. Hence the group $MU^n(pt) \cong \Omega^U_n$. Taking the direct sum of all these we get a ring $MU^\ast(pt)\cong \Omega_\ast^U$ called the complex cobordism ring. Milnor and Novikov showed that this ring is a polynomial ring in infinitely many variables, i.e. $\Z[v_1, v_2, \ldots]$ where $v_i$ has degree $2i$.
We have already seen a bit of multiplicative cohomology theories in the earlier post about formal group laws. In that post we looked at complex oriented cohomology theories and how to get formal group laws from them. We redo this now in the light of $MU$.
Recall that a multiplicative cohomology theory is called complex orientable if the map
$$E^2(\mathbb{C}P^\infty)\longrightarrow E^2(S^2)$$
is surjective. A complex orientation is then a choice of an element in the preimage of the canonical generator in $E^2(S^2)$. This generator is often called the first generalized Chern class, or the first Connor-Floyd Chern class, and will be denoted by $c^E$. From the structure maps of $MU$ — those given by $\Sigma^2 MU_{2k}\longrightarrow MU_{2k+2}$ — we get for $k=0$ a map
$$i:\Sigma^2 S^0 \simeq S^2 \longrightarrow \mathbb{C}P^\infty$$
which is the inclusion map. In $MU$-cohomology this induces a map
$$i^\ast:MU^2(\mathbb{C}P^\infty)\longrightarrow MU^2(S^2)$$
The group $MU^2(\mathbb{C}P^\infty)$ is given by $[\mathbb{C}P^\infty, MU_2]$ — the homotopy classes of maps into the complex cobordism spectrum — which through the homotopy equivalence between $MU_2$ and $\mathbb{C}P^\infty$ is isomorphic to the set $[\mathbb{C}P^\infty, \mathbb{C}P^\infty]$. The class of the identity map then corresponds to an element $c^{MU}\in MU^2(\mathbb{C}P^\infty)$. This element gets sent to $t$, the canonical generator of $MU^2(S^2)$ under the induced inclusion map, i.e. $i^\ast(c^{MU})=t$. Hence $MU$ is complex oriented, and moreover, the choice of complex orientation is completely natural and engrained into the structure of the spectrum itself. It is often called the universal complex orientation of $MU$. It is universal in the sense that for any other complex oriented cohomology theory $E^\ast$ with complex orientation $c^E$, there is a (up to homotopy) unique map of ring spectra $u:MU\longrightarrow E$ such that the induced map in the second cohomology of $\mathbb{C}P^\infty$
$$u_\ast:MU^2(\mathbb{C}P^\infty)\longrightarrow E^2(\mathbb{C}P^\infty)$$
sends the universal complex orientation $c^{MU}$ to $c^{E}$, i.e. $u_\ast(c^{MU})=c^{E}$.
For any complex oriented cohomology theory $E^\ast$ with complex orientation $c^E$, we get that the $E$-cohomology ring of $\mathbb{C}P^\infty$ is given by $E^\ast(\mathbb{C}P^\infty)\cong \pi_\ast E[[c^E]]$, i.e. the power series ring over its coefficient ring. The space $\mathbb{C}P^\infty$ is a H-space, with a multiplication induced from the tensor product of vector bundles. In $E$-cohomology this gives a map
$$m^\ast :\pi_\ast E[[c^E]]\cong E^\ast(\mathbb{C}P^\infty)\longrightarrow E^\ast(\mathbb{C}P^\infty\times\mathbb{C}P^\infty)\cong \pi_\ast E[[x,y]]$$
where the image of the complex orientation $c^E$ under $m^\ast$ is some power series $f(x, y)$. This power series is a formal group law over the ring $\pi_\ast E$. The essentially unique map that gives $c^E$ from $c^{MU}$, i.e. the ring spectrum map $u:MU\longrightarrow E$, induces a map
$$u_\ast:MU^\ast(\mathbb{C}P^\infty)\cong \pi_\ast MU[[c^{MU}]]\longrightarrow E^\ast(\mathbb{C}P^\infty) \cong \pi_\ast E[[c^E]]$$
that maps the formal group law $m^\ast(c^{MU})$ over $\pi_\ast MU$ to the formal group law $m^\ast(c^E)$ over $\pi_\ast E$. This means that the formal group law over the complex cobordism ring acts like a universal formal group law for complex oriented cohomology theories. One can then ask, does it act as the universal formal group law among all possible formal group laws? Or equivalently, can we get every formal group law from a cohomology theory?
In the earlier mentioned post describing these formal group laws we looked at a remarkable theorem by Quillen, stating that this is precisely the case. The formal group law we get from complex cobordism — the image of $c^{MU}$ under the map $m^\ast$ — is the universal formal group law $f$ over the Lazard ring $L$, the ring that classifies formal group laws. This means in particular that we have a natural isomorphism $\pi_\ast MU\cong L \cong \Z[v_1, v_2, \ldots]$, and $m^\ast(c^{MU}) \cong f$.
Round off#
I think this is enough of complex cobordism for today, but we will definitely see it again. The reason we construct it in detail is because we want to use it later to construct other important cohomology theories: Brown-Peterson cohomology, Johnson-Wilson theory and Morava K-theory. The latter is especially important for my research projects, so understanding its construction properly will probably be beneficial.
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The homotopy groups of the spheres. Part 2 | CommonCrawl |
How is computation done in a 2D surface code array?
In a 2D surface code lattice, there are some data qubits and some measurement qubits. Suppose we want to do a 2-qubit computation, for example, let say, an X-gate on qubit-1 followed by a CNOT gate with qubit-1 as the control bit and qubit-2 as the target bit.
Q: How is this computation realized in a quantum computer with a 2D surface code arrangement of qubits? i.e. Which gates are applied and on which qubits?
error-correction
Abdullah Ash- SakiAbdullah Ash- Saki
$\begingroup$ You might want to clarify whether you mean that all the qubits are in a 2D array, or just that the encoded qubits are in segments which are 2D arrays. $\endgroup$ – Niel de Beaudrap Oct 15 '18 at 20:49
I will illustrate how one can perform operations using logical operations on the qubits, and using lattice surgery for two-qubit operations.
In the diagrams below, all of the 'dots' are data qubits: measurement qubits are omitted in order to help demonstrate the basic principles more clearly. The measurement qubits are there for when you perform stabiliser measurements, and are only ever involved in stabiliser measurements, so the story is about what you do with the data qubits — including such things as the stabiliser measurements that you perform on the data qubits.
Surface codes and logical single-qubit Pauli operations
One can use fragments of the plane to store qubits. The image below shows four qubits which are encoded as part of a larger lattice: the light dots with black outlines are qubits which are not involved in the encoded qubits, and can in principle be in any state that you like, unentangled from the rest.
$\qquad\qquad\qquad$
In each of these fragments, the qubit is defined by the stabiliser relations which (ideally, in the absence of errors) hold among the qubits. For surface codes with the sorts of boundary conditions illustrated here, these are either 3-qubit X or Z stabilisers around the boundary, and either 4-qubit X or Z stabilisers in the 'bulk' or body of the code. The pattern of these stabilisers is illustrated below. Note that each X stabiliser that overlaps with a Z stabiliser, does so at two qubits, so that they commute with one another. (Apologies for the image size: I cannot manage to get it to display at a reasonable size.)
Note that by using the regularity of these stabilisers, it isn't necessary for the surface code fragment to be square (or even in principle rectangular).* This will become important later.
There are a number of (tensor product) Pauli operations which commute with all of these stabilisers. These may be used to define logical Pauli operators, which describe ways in which you can both access and transform the logical qubits. For instance, a product of Z operators across any row from boundary to boundary will commute with all stabilisers, and can be taken to represent a logical Z operator; a product of X operators across any column from boundary to boundary can similarly be take to represent a logical X operator:
$\qquad$
It doesn't matter which row or which column you use: this follows from the fact that a product of any two rows of Z operators, or of any two columns of X operators, can be generated as a product of stabilisers and therefore realises an identity operation on the encoded qubit (as the stabiliser generators themselves are operators which perform the identity operation on an encoded qubit state, by definition). So: if you want to apply an X operation to an encoded qubit, one way to do so would be to apply such a logical X operation, by realising X operators on each qubit in a column reaching between two boundaries.**
Logical single-qubit Pauli measurements
One advantage of thinking of the encoded qubits in terms of logical operators is that it allows you to also determine how you can perform a 'logical measurement' — that is, a measurement not only of (some of) the qubits in the code, but of the data that they encode. Take the logical X operator above, for example: the operator X⊗X⊗...⊗X is not only unitary, but Hermitian, which means that it is an observable which you can measure. (The same idea is used all the time with the stabilisers of the code, of course, which we measure in order to try to detect errors.) This means that in order to realise a logical X measurement, it is enough to measure the logical X observable. (The same goes for the logical Z observable, if you want to realise a standard basis measurement on your encoded qubit; and everything I say below can also be applied to logical Z measurements with the appropriate modifications.)
Now — measuring the logical X observable is not exactly the same as measuring each of those single-qubit X operators one at a time. The operator X⊗X⊗...⊗X has only two eigenvalues, +1 and −1, so measuring that precise operator can only have two outcomes, whereas measuring each of n qubits will have 2n outcomes. Also, measuring each of those single-qubit X operators will not keep you in the code-space: if you want to do computations on a projected post-measurement state, you would have to do a lot of clean-up work to restore the qubit to a properly encoded state.
However: if you don't mind doing that clean-up work, or if you don't care about working with the post-measurement state, you can simulate the logical X measurement by doing those single-qubit measurements, obtaining +1 and −1 outcomes, and then computing their products to obtain what the result of the measurement of X⊗X⊗...⊗X "would have" been. (More precisely: measuring all of those single-qubit X operators is something that does not disturb a state which would result from a measurement of the tensor product operator X⊗X⊗...⊗X, and the product of those single-qubit measurements would have to yield a consistent outcome with the tensor product operator X⊗X⊗...⊗X, so we can use this as a way to simulate that more complicated measurement if we don't mind all of the qubits being projected onto conjugate basis states as a side-effect.)
Lattice surgery for logical two-qubit operations
To realise a two-qubit operation, you can use a technique known as lattice surgery, wherein you 'merge' and 'split' different patches of the 2D lattice to realise operations between those patches (see [arXiv:1111.4022], [arXiv:1612.07330], or [arXiv:1704.08670] for complete descriptions of these operations. Disclosure: I am an author on the third of these articles.) This may be realised between two adjacent patches of the planar lattice (as illustrated above) by preparing those "uninvolved" rows and columns of qubits in a suitable state, and then measuring stabilisers which previously you were not measuring in order to extend the memory to a larger system. (In the diagram below, the horizontal spacing between the code segments and the column of qubits in the |0⟩ states is exaggerated for effect.)
This will affect the logical operators of the system in a non-unitary way, and is most often used (see [arXiv:1612.07330] for example) to realise a coherent X⊗X or Z⊗Z measurement, which can be composed to realise a CNOT operation [arXiv:1612.07330, Fig. 1(b)]:
In this way, you can realise a CNOT operation between a pair of encoded qubits.***
* You can also use slight modifications of the regular pattern of stabilisers, as Letinsky [arXiv:1808.02892] demonstrates, to achieve more versatile planar-surface representations of encoded qubits.
** In practise, rather than explicitly performing (imperfect single-qubit) operations, you would take advantage of the fact that the frame of reference for the encoded qubits is one which you are fixing by convention, and update (or 'transform') the reference frame rather than the state itself when you wish to realise a Pauli operation. This is the smart way to go about error correction as well: to treat errors not as 'mistakes' which must be 'fixed', but as an uncontrolled but observable drift in your reference frame as a result of interaction with the environment. You then hope that this drift is slow enough that you can track it accurately, and compensate for the change in reference frame when you do your computation. Particularly in the context of tracking errors, this reference frame is described as the Pauli frame, and its job is to describe the frame of reference in terms of the Pauli operations which would be required to put the system in the state usually described by an error-free error correcting code.
*** Many authors would describe this construction as the point of lattice surgery, and it is certainly the original concrete application of it described in the original article [arXiv:1111.4022]. It is possible in principle to do more elaborate operations using splits and merges, by treating the merges and splits as primitive operations in their own right rather than just the components of a CNOT, and using more versatile (but not especially circuit-like) transformations — this is essentially the point of my article with Dom Horsman [arXiv:1704.08670], which opens up the possibility of the ZX calculus (a somewhat heterodox representation of quantum computation) to be directly practically useful for surface-code memories.
Niel de BeaudrapNiel de Beaudrap
$\begingroup$ great answer. By the way, in case you didn't know, you can fine tune the size of the images by directly using html tags. For example: <img src="https://i.stack.imgur.com/H94nX.png" width="300"/> $\endgroup$ – glS Oct 18 '18 at 9:44
$\begingroup$ @gIS: thanks, my HTML skills were weak from disuse when I tried that originally! It seems to be better now. $\endgroup$ – Niel de Beaudrap Oct 18 '18 at 10:06
$\begingroup$ @AbdullahAsh-Saki: following up on your question in the comments to JamesWooton, I've added some remarks on measurements of encoded states. $\endgroup$ – Niel de Beaudrap Oct 18 '18 at 10:22
One way to store qubits in the surface code is as pairs of "holes". A hole is a chunk of the surface where, instead of performing the stabilizer measurements used to detect whether errors are occurring, you instead do nothing.
There are two different types of hole, depending on whether the boundary of the hole travels along would-be X measurement qubits or along would-be Z measurement qubits. A CNOT is performed by cycling a hole of one type around a hole of the other type.
Diagrammatically speaking, it looks like this:
In the (b) diagram, time is moving from left to right. Each bar corresponds to the location of a hole over time. Each qubit is stored between the corresponding pairs of white bars. The black bar represents the hole being used to perform the CNOT. It avoids the middle qubit (which is not involved), surrounds one of the bars of the bottom qubit (which is the target), and goes around a 'cross-bar' introduced into the top qubit (which is the control). That's what a surface code CNOT looks like.
Craig GidneyCraig Gidney
$\begingroup$ Where can I learn more about Fig (b)? I have seen similar figures in one of the lectures by Austin Fowler. However, it still remains very elusive. $\endgroup$ – Abdullah Ash- Saki Oct 17 '18 at 3:39
$\begingroup$ @AbdullahAsh-Saki arxiv.org/abs/1208.0928 defines what I mean by "holes" in quite a lot of detail. If you take slices through the diagram at each time, it shows where the holes are. So take the timeslice diagrams from that paper and picture what they would look like stacked on top of each other, and that's a braiding diagram. $\endgroup$ – Craig Gidney Oct 17 '18 at 3:50
There are multiple ways to store information in surface codes. Depending on the method you use, there are then multiple ways to do gates. So there's actually a lot to say on this issue!
Despite the multiplicity of methods, in practical terms they all come to pretty much the same thing: if you want you gate to be kept fault-tolerant by the code, you can only do Clifford gates (such as X, Z, H, CNOT, S). For other gates you'll need to invoke additional mechanisms to become fault-tolerant, such as magic state distillation.
But you didn't ask for anything beyond Clifford in your example. You just wanted an X and a CNOT. So that makes things easy.
For a concrete example, let's take the 17 qubit surface code shown below (as depicted in this paper, of which I am an author).
This is made up of $n=9$ physical qubits, numbered from $0$ to $8$. There are also 8 ancilla qubits depicted here, but we'll ignore them.
The dark patches in this image denote stabilizers made of $\sigma_x$ (so we measure the observables $\sigma_x^0 \otimes \sigma_x^1 \otimes \sigma_x^3 \otimes \sigma_x^4$ and $\sigma_x^1 \otimes \sigma_x^2$, for example). The light patches are then the $\sigma_z$ stabilizers.
One example of an operation that commutes with all stabilizers is to do a $\sigma_x$ rotation on a line of qubits from top to bottom, such as $\sigma_x^0 \otimes \sigma_x^3 \otimes \sigma_x^6$. Another example is to do a line of $\sigma_x$ rotations from left to right, such as $\sigma_z^3 \otimes \sigma_z^4 \otimes \sigma_z^5$.
All other operations that commute with the stabilizer will be either products of stabilzers themselves (and so act trivially), or they will be equivalent to one of these two examples. So these operations act on our logical qubit. Since one is made of $\sigma_x$s, the other is made of $sigma_z$s, and they anticommute, it is would seem sensible to assign them as the Pauli operators $X$ and $Z$ of the logical qubit
$$X = \sigma_x^0 \otimes \sigma_x^3 \otimes \sigma_x^ 6, Z = \sigma_z^3 \otimes \sigma_z^4 \otimes \sigma_z^5$$
So to do an $X$, you just perform the operation above.
For a CNOT, one of the many ways to do it is transversally. For this, suppose we have two logical qubits $A$ and $B$. Each is made of many physical qubits, that we'll number $0, 1, 2, \ldots$. So let's us $0_A$ to denote physical qubit $0$ of logical qubit $A$, for example.
To do ${\rm CNOT}(A,B)$, a CNOT with qubit $A$ as control and $B$ as target, we can then do
$$ {\rm CNOT}(0_A,0_B) \,\, {\rm CNOT}(1_A,1_B) \,\, {\rm CNOT}(1_A,1_B) \,\, \ldots$$
To see how this works, we have to look at the logical $|0\rangle$ and $|1\rangle$ states when expressed in the computational basis of the physical qubits.
Let's use $| \tilde 0 \rangle = |0\rangle^{\otimes n}$ to denote the state where all physical qubits are in state $|0\rangle$, and $| \tilde 1 \rangle = X | \tilde 0 \rangle$ a state with a line of $|1\rangle$s from top to bottom, on a background of $|0\rangle$s.
We can then simply express the logical $|0\rangle$ state as the superposition of $| \tilde 0 \rangle$ with all the state that you can get to from $| \tilde 0 \rangle$ by applying stabilizers. Logical $|1\rangle$ is similarly the superposition of $| \tilde 1 \rangle$ with all the state that you can get to from $| \tilde 1 \rangle $ by applying stabilizers.
By thinking of the action of the transversal CNOTs in terms of these states, you should hopefully be able to see how it acts as a CNOT on the logical qubits.
James WoottonJames Wootton
$\begingroup$ Should it be $X = \sigma_x^0 \otimes \sigma_x^3 \otimes \sigma_x^6$ instead of $X = \sigma_x^0 \otimes \sigma_x^3 \otimes \sigma_x^4$ if we consider going from top to bottom on the first vertical line? $\endgroup$ – Abdullah Ash- Saki Oct 16 '18 at 2:16
$\begingroup$ You are right. I corrected it. $\endgroup$ – James Wootton Oct 16 '18 at 4:08
$\begingroup$ One supplementary question: What is the relation between the measurement operator and the physical read-out of a qubit? (I have found many places where it mentions "measurement is defined by the measurement operator Pauli-X".) $\endgroup$ – Abdullah Ash- Saki Oct 17 '18 at 3:43
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A comparison of methods for multiple degree of freedom testing in repeated measures RNA-sequencing experiments
Elizabeth A. Wynn1,
Brian E. Vestal2,
Tasha E. Fingerlin2 &
Camille M. Moore2
BMC Medical Research Methodology volume 22, Article number: 153 (2022) Cite this article
956 Accesses
As the cost of RNA-sequencing decreases, complex study designs, including paired, longitudinal, and other correlated designs, become increasingly feasible. These studies often include multiple hypotheses and thus multiple degree of freedom tests, or tests that evaluate multiple hypotheses jointly, are often useful for filtering the gene list to a set of interesting features for further exploration while controlling the false discovery rate. Though there are several methods which have been proposed for analyzing correlated RNA-sequencing data, there has been little research evaluating and comparing the performance of multiple degree of freedom tests across methods.
We evaluated 11 different methods for modelling correlated RNA-sequencing data by performing a simulation study to compare the false discovery rate, power, and model convergence rate across several hypothesis tests and sample size scenarios. We also applied each method to a real longitudinal RNA-sequencing dataset.
Linear mixed modelling using transformed data had the best false discovery rate control while maintaining relatively high power. However, this method had high model non-convergence, particularly at small sample sizes. No method had high power at the lowest sample size. We found a mix of conservative and anti-conservative behavior across the other methods, which was influenced by the sample size and the hypothesis being evaluated. The patterns observed in the simulation study were largely replicated in the analysis of a longitudinal study including data from intensive care unit patients experiencing cardiogenic or septic shock.
Multiple degree of freedom testing is a valuable tool in longitudinal and other correlated RNA-sequencing experiments. Of the methods that we investigated, linear mixed modelling had the best overall combination of power and false discovery rate control. Other methods may also be appropriate in some scenarios.
RNA-sequencing (RNA-seq) technology has revolutionized how we study and understand the underlying pathobiology of disease. Recently, declining sequencing costs have allowed for more complex investigations, including correlated and longitudinal study designs. In particular, longitudinal designs have become increasingly popular, as they allow researchers to understand the dynamics of gene expression across time and how these dynamics differ between groups of subjects. However, complex study designs demand more sophisticated analysis methods. As with single timepoint designs, careful pre-processing of longitudinal RNA-seq data is still necessary prior to analysis to remove artifacts produced during sequencing [1, 2]. Following pre-processing, distributional and computational considerations are necessary to model overdispersed count data on 10,000-20,000 genes. Additionally, analysis methods for longitudinal study designs must also account for the correlation induced by repeated measures, which is often achieved with random effects or modeling of the error covariance structure. To be most applicable to these complex study designs, analysis approaches should allow for flexible modeling, including the ability to adjust for potential confounders and subject demographics.
In longitudinal RNA-seq studies, researchers are often interested in multiple hypotheses. For example, many longitudinal RNA-seq studies include repeated measures from each subject over time, with subjects coming from multiple treatment groups. This allows for the investigation of between-subject comparisons, such as a test for differences in gene expression between treatment groups at a particular timepoint; within-subject comparisons, such as a test for differences in gene expression across two timepoints in a single treatment group; or interaction effects to compare changes over time between groups. Furthermore, studies with more than two timepoints per subject might involve multiple comparisons across different timepoints in order to characterize how gene expression changes across time.
In the situation where there are multiple hypotheses to be tested for each gene, the ability to perform an omnibus test, or a test where multiple hypotheses are evaluated, is valuable for controlling false discovery rates. For example, in a study with multiple timepoints per subject in which time is treated categorically, a researcher might wish to compile a list of genes that change over time for further investigation. In such a situation, one could perform a series of hypothesis tests to identify the differentially expressed genes (DEGs) between each pair of timepoints and perform a multiple testing correction to each hypothesis test individually to control the false discovery rate to 5%, for example. However, because each hypothesis test may produce different false positive genes, when lists of significant genes are aggregated across multiple hypotheses, the percentage of false positives in the aggregated list will be greater than 5% without additional adjustment [3]. Thus, performing an omnibus test for multiple hypotheses is useful in false discovery rate control. These types of tests are often referred to as multiple degree of freedom (DF) tests because the hypothesis for these tests involve multiple degrees of freedom as opposed to the single degree of freedom required for hypothesis testing of a single covariate or effect.
Several different methods have been proposed for the analysis of longitudinal RNA-seq data. Popular analysis packages such as edgeR [4, 5] and DESeq2 [6] are often appealing to researchers because they allow for flexible modelling in a generalized linear modelling (GLM) framework. However, these packages do not allow for random effects or covariance structures to properly accommodate correlated data. Despite this limitation, these packages are sometimes used to analyze correlated data, either by treating each subject/cluster as a fixed effect under a regression framework, or by ignoring the correlation altogether and treating correlated samples as independent. It is well established that ignoring correlation can lead to bias in standard error estimation which can influence the results of statistical tests [7]. Alternatively, treating each subject/cluster as a fixed effect may result in inflated false positive rates due to over-fitting [8]. Additionally, when coefficients for each subject/cluster are included in the model, other subject-level effects, such as group differences, are not estimable.
The limma [9] package, another popular analysis tool for RNA-seq data, includes the capability to account for correlation between related samples using a method in which a common correlation value estimated across all genes is incorporated into the model for each gene [10]. However, this method assumes that the correlation between samples is the same for all genes. This is a strong assumption that may not be true in practice.
Recently, several methods have been proposed for longitudinal and other correlated RNA-seq studies. These methods generally use random effects or covariance structures to account for the correlation in the data while also considering the unique characteristics of RNA-seq data such as overdispersion. Many methods developed for correlated RNA-seq data are limited by the fact that they do not allow for multiple treatment groups or additional covariates (e.g. PLNseq [11], multiDE [12]), can only be used for paired data (e.g. baySeq [13, 14], PairedFB [15]), or can only perform single DF tests (e.g. MCMSeq [16], ShrinkBayes [17]).
Some researchers have proposed employing standard statistical models typically used for longitudinal and correlated data outside of the context of RNA-seq data, as these well-developed modeling frameworks allow for flexible modeling and hypothesis testing [18–20]. In applying these methods to RNA-seq data, considerations still must be made to account for the non-normality of the data, for example, by choosing a repeated measures model with an underlying distribution for overdispersed counts.
Tsonaka & Spitali [20] investigated the use of negative binomial mixed models (NBMM) for RNA-seq data using an adaptive Gaussian quadrature method to estimate parameters and found that this method was relatively unbiased and exhibited type 1 error (T1E) and false discovery rate (FDR) control. Similarly, Zhang et al. [21] used NBMM to analyze correlated microbiome data, which are also overdispersed counts, but used an iterative weighted least squares (pseudo-likelihood) approach for parameter estimation. They demonstrated the utility of the method through both simulation study and application to mouse gut microbiome data. Rather than using the negative binomial distribution, Park et al. [19] investigated the use of generalized estimating equation (GEE) models using a Poisson distribution with an extra scale parameter to account for overdispersion. They found that these models identified more DEGs than edgeR, DESeq or limma, though they did not explore whether this was driven by high false positive rates.
Instead of directly modeling counts, another approach is to normalize the data and then utilize models that assume a normal distribution. The package rmRNAseq [18] utilizes the voom normalization method on log-transformed counts and then models the transformed data using a linear model with a continuous auto-regressive structure to account for the correlation in the data. Vestal et al. [16] tested a similar method by using a variance stabilizing transformation (VST) on raw RNA-seq counts and then fitting linear mixed models (LMMs) to the transformed data. They found that this method performed similarly to their hierarchical Bayesian MCMSeq method in terms of T1E and FDR control, but many models failed to converge in small sample size situations.
All of the methods outlined above allow for multiple DF hypothesis testing. However, there has been little research evaluating and comparing the performance of multiple DF tests across these methods. Some studies have evaluated the use of multiple DF tests for a single method or in comparison to DESeq2 and edgeR, which do not account for correlation, rather than methods that account for correlation [18, 20]. Others have compared multiple correlated data approaches but only for single DF hypothesis tests [16]. As complex study designs become more common in correlated RNA-seq designs, multiple DF hypothesis testing is important for identifying interesting genes for downstream analysis without increasing the FDR.
In this paper, we compare the performance of several methods for analyzing correlated RNA-seq count data with particular emphasis on multiple DF test performance within each method. First, we investigate model performance through a simulation study. Each method is also applied to RNA-seq data collected from septic shock and cardiogenic shock patients over multiple timepoints following admission to the intensive care unit (ICU). Finally, we provide recommendations as to which models are most appropriate under various circumstances.
Analysis methods compared
We compared methods which have been proposed for correlated RNA-seq experiments and that allow for multiple treatment groups, covariates and/or timepoints, and can be used to perform multiple DF tests. We describe the selected methods below. Additional information on each method is available in Supplementary Materials Section 1.
Standard RNA-seq analysis tools
Standard RNA-seq analysis tools generally use a linear modelling framework with transformed data, or a generalized linear model (GLM) framework, assuming a negative binomial distribution. In studies with correlated designs, these methods can be implemented with the caveat that the model assumptions, such as the independence of observations, will not be met, or adjustments can be made to attempt to account for the correlation of the data. In this study, we tested three of the methods from the most popular RNA-seq analysis packages: limma, edgeR, and DESeq2.
The R package limma was originally created for the analysis of microarray expression data, which are approximately normally distributed [9]. limma employs linear models to test for differential expression using an empirical Bayes approach to share information across genes. This methodology has been extended to RNA-seq data by applying the "voom" transformation to RNA-seq counts [22, 23]. First, RNA-seq counts are normalized using the log counts per million (log-CPM) transformation. A mean-variance relationship is then estimated, and from this relationship, a predicted variance is calculated for each log-CPM value, which is then incorporated into a linear model as an inverse weight. The duplicateCorrelation function within the package can be used to estimate correlation values for each subject which are then incorporated in the linear model. However, only one correlation is computed for all genes.
The edgeR and DESeq2 packages both employ a negative binomial GLM framework to address overdispersion [4–6]. Both methods use empirical Bayes procedures to estimate variability, effectively borrowing information across genes to inform the estimation. Both methods also include offset terms in their models to account for differences in library size between samples, though edgeR uses the trimmed mean of M-values (TMM) method [4], while DESeq2 uses the median ratio method [24]. These packages do not include methods to account for correlation between samples.
Generalized estimating equations
Generalized estimating equations (GEE) are a semi-parametric extension of GLM that can account for correlation between observations [25]. This method uses a working correlation structure to model the association between measurements within a subject. The covariance matrix of the estimated regression coefficients is typically estimated using robust (sandwich) estimators so that the estimates are robust to misspecification of the working correlation structure. In this analysis, we modelled the data using a Poisson distribution with an extra scale parameter in the variance to account for overdispersion, and an exchangeable working correlation structure.
One drawback to GEE models is that sandwich estimators have poor performance at small sample sizes. To address this issue, we used the small sample size adjustment proposed by Wang and Long [26], which utilizes information from all subjects to calculate the covariance for each individual subject and also uses an additional adjustment to correct for bias.
Negative binomial mixed models
Generalized linear mixed models (GLMM) are an extension of GLMs that use random effects to account for correlation. Similar to the methods implemented in edgeR and DESeq2, in using the GLMM framework, the gene expression for each gene can be modeled using a negative binomial distribution, which accounts for the overdispersion.
When using negative binomial mixed modelling (NBMM), parameter estimation can be analytically complex and there are multiple approaches that can be used. We consider two maximum likelihood estimation approaches, Laplace (NBMM-LP) and adaptive Gaussian quadrature (NBMM-AGQ) as well as the pseudolikelihood approach (NBMM-PL).
rmRNAseq and linear mixed models
The rmRNAseq package employs a method similar to the limma+voom method in which the data are first transformed using the voom approach and then a linear model is fit for each gene using the transformed data. However, within the rmRNAseq framework, models are fit using a continuous autoregressive correlation structure to account for correlation in the data.
A similar approach is to use linear mixed modelling (LMM) with random effects to account for correlated data after applying a normalizing transformation. We test this approach using a variance stabilizing transformation (VST), as demonstrated in Vestal et al. [16].
We implemented each method using R (version 4.0.2). All analysis was carried out on a Linux high performance computing (HPC) cluster and parallel processing with 8 cores was used for all methods besides limma, DESeq2, and edgeR. Table 1 contains the specific packages used for each method and implementation details. Where possible, we used previously implemented R packages. In some cases, available R packages were missing important functionality, such as the capacity to account for offsets (geesvm for GEE small sample estimators). In these cases, custom R functions were built using the source code from the previously implemented R packages as a framework. Functions for implementing and summarizing results for methods in which no wrapper/summarization functions were available can be found in the corrRNASeq package, which is available at https://github.com/ewynn610/corrRNASeq.
Table 1 Analysis methods with their associated R packages and details concerning their implementation
Offsets to adjust for differences in library size were included in models for all except three methods (Table 1). The transformations used in limma, rmRNAseq and the LMM method accounted for differences in library size, so no additional adjustment was used.
The models using the edgeR and DESeq2 packages were fit in two ways. First, correlation was ignored and a model was fit with an intercept, time and group main effects, and an interaction term. Second, a fixed effect for subject was included in the model (edgeR ∗ and DESeq2 ∗). When including this extra fixed effect, the group term was not included in the model as it is inestimable.
Models were designated as non-converged if a maximum number of iterations were run without convergence during model fitting, models were found to be singular, or other errors prevented the model from fitting properly. All models that did not converge were discarded before further analysis.
The packages used to implement each method in this analysis utilize different types of multiple DF tests. Table 1 shows the class of tests used for each method.
We used likelihood ratio tests (LRT) for the edgeR, DESeq2, NBMM-LP and NBMM-AGQ analyses. For all of these methods excluding edgeR, this required fitting two models for each test, a full model as well as a reduced model. The GLMMadaptive package used for fitting NBMM-LP models offers the option of using a multivariate Wald test instead of an LRT test. However Tsonaka & Spitali [20] found that in the context of correlated RNA-seq data, using LRTs resulted in lower T1E rate and FDR and thus we chose to use LRTs rather than multivariate Wald tests for these models. Additionally, Tsonaka & Spitali [20] proposed a bootstrap procedure for calculating p-values, particularly in small sample size situations. However, in running the example code provided with their publication, we found that it took about 2 hours to fit models and perform hypothesis testing for 10 genes with 1,000 bootstrap samples each. Because RNA-seq studies typically include 10,000-20,000 genes, this bootstrapping approach is likely not computationally feasible for most studies and we did not include it in our analysis.
Hypothesis testing for GEE was done using a Wald χ2 test as implemented by the esticon function in the doBy package [32]. F-tests were used for LMM and NBMM-PL and the Satterthwaite method was used to calculate denominator degrees of freedom [33, 34]. The limma and rmRNAseq packages both utilize the moderated F-statistic outlined by Smyth [35] for hypothesis testing. Under the limma framework, p-values are computed using an F-test with augmented degrees of freedom. The rmRNASeq package calculates p-values by building a distribution of null test statistics from data generated by a parametric bootstrap procedure and then computing the proportion of null statistics greater than or equal to the observed F-statistic.
In order to evaluate and compare the testing characteristics of the previously described methods, we performed a simulation study. We used a two group design (e.g. treatment and control) with four observations per subject. A negative binomial distribution was used to simulate a matrix of counts Y. Let Ygij be the expression level of gene g for the ith subject and jth observation, with E(Ygij)=μgij. Further, let αg be a dispersion parameter for gene g with \(Var(Y_{gij})=\mu _{gij}+\alpha _{g}\mu ^{2}_{gij}\). Then
$$\begin{array}{*{20}l} Y_{gij} &\sim& \mathcal{NB}(\mu_{gij}, \alpha_{g}) \end{array} $$
$$\begin{array}{*{20}l} log(\mu_{gij}) &\!=& \!\beta_{g0}+\beta_{g1}X_{1i}+\beta_{g2}X_{2ij}+\beta_{g3}X_{3ij}+\beta_{g4}X_{4ij} \end{array} $$
$$\begin{array}{*{20}l} & &+\beta_{g5}X_{1i}X_{2ij} +\beta_{g6}X_{1i}X_{3ij}+\beta_{g7}X_{1i}X_{4ij}+b_{gi} \\ b_{gi}&\sim& \mathcal{N}(0, \sigma^{2}_{g}) \end{array} $$
where X1i is an indicator variable signifying whether the ith subject is in the treatment group or not, and X2ij, X3ij and X4ij are indicator variables representing whether observation j was taken at the 2nd, 3rd, or 4th timepoint respectively. Each βgk,k∈0,...,7 is a fixed effect regression coefficient specific to gene g. Finally, bgi is the random intercept for gene g and subject i which is normally distributed with a mean of 0 and a variance of \(\sigma ^{2}_{g}\).
Table 2 shows a summary of the simulation settings and multiple DF tests performed. We simulated 10 datasets for each simulation scenario. For each dataset we simulated 15,000 genes and then genes were filtered out if N samples had less than 1 count per million (CPM), where N was equal to the number of samples collected for a single group and timepoint. We simulated datasets to contain a mix of null and differentially expressed genes by changing the interaction coefficients for 20% of genes. In order to mimic real data, βg0,αg and \(\sigma ^{2}_{g}\), were drawn from an empirical distribution for triplets of mean CPM, dispersion, and random intercept variance observed across human samples in several real RNA-seq data sets with repeated measures [36, 37]. The fixed effect intercept parameter, βg0 was derived by scaling the randomly drawn CPM values to add up to one million and then multiplying each scaled value by a total library size of 25 million. Then, βg0 was set to the log of this value.
Table 2 Summary of simulated datasets
We analyzed simulated data using each method as described in the implementation section. Models for each gene were fit using fixed effects for group and time variables, which were both treated as categorical, as well as the interaction between group and time. A random intercept for each subject was included in models for methods in which random effects are possible. After the models were fit, the percentage of models that successfully converged for each method was calculated, and non-converged models were removed. Then the false discovery rate (FDR) and power were calculated for four different multiple DF tests: a between-subject test, a within-subject test, an interaction test, and a global test (Table 2). Power and FDR were calculated using Benjamini Hochberg adjusted p-values [38] and a significance threshold of 0.05 was used. For each simulation scenario, we averaged the statistics across 10 simulated datasets.
Real data analysis
We applied the analysis methods previously outlined to a publicly available, longitudinal RNA-seq dataset of 96 whole blood samples from 32 patients experiencing circulatory shock who were admitted into the ICU (GEO Dataset: GSE131411). For each patient, three blood samples were collected: one within 16 hours after ICU admission, one 48 hours after admission, and one seven days after admission or at discharge. Subjects were categorized by whether they experienced septic shock (SS, N=21) or cardiogenic shock (CS, N=11). Further information on the study design and methods is available in Braga et al. [39].
Data pre-processing and model information
We downloaded the count table and study meta data from the GEO DataSets website. The data included 58,096 genes. We filtered out lowly expressed genes by removing genes that did not have greater than 1 CPM in at least 11 of the 96 samples (11 was the sample size in the smallest experimental group of interest), which reduced the total number of genes analyzed to 14,340.
The goal of our analysis was to investigate how the gene expression of shock patients changed over time and how these changes differed between patients with SS versus CS. To accomplish this, for each method we fit a model with fixed effects for the type of shock and timepoint (treated categorically) as well as the interaction between the two variables. A random intercept for each subject was included in models for methods in which random effects are possible. All models were fit as described in the implementation section. As with the simulation study, the percentage of models that failed to converge for each method was calculated and non-converged models were removed.
For each model, we ran four different multiple DF hypothesis tests: a between-subject test to assess if there was a difference in gene expression between the SS and CS groups at any timepoint, two within-subject tests to assess if there was a change in gene expression over time in the SS group or the CS group, and a test to assess if any of the interaction coefficients were significant. The Benjamini Hochberg method was used to adjust p-values for multiple comparisons and the DEGs for each method and test were identified using a 0.05 FDR threshold.
Hierarchical clustering and functional enrichment analysis
Because LMM exhibited comparatively good behavior in the simulation study, we used the results from this method to explore the patterns in the changes in gene expression over time in the SS and CS groups. All analysis was done for each group separately. First, we subset the data to include only genes that were significant in the multiple DF test for difference in gene expression at any timepoint in the SS group or CS group. For these genes, we computed the predicted gene expression (log scale) for each gene at each of the three timepoints for the group in question. We then constructed heatmaps for these genes, with genes clustered hierarchically using a correlation distance metric and a complete linkage clustering method. We visually inspected the heatmaps to decide where to cut each clustering tree to identify clusters that represented distinct profiles of change over time.
After clustering, we ran functional enrichment analysis on the genes in each cluster to better understand the functional role of genes with different expression profiles over time. Analysis was executed using the topGO package in R [40] using biological process biological process gene ontology (GO) annotations. The significance of the GO terms was assessed using a Fisher's exact test with an FDR level of 0.05 as the threshold for significance. We further filtered the results to include only GO terms with at least 10 genes and > 10% overlap of the genes associated with each GO term and the genes in the cluster.
Of the 11 methods evaluated, only 3 methods (NBMM-LP, NBMM-PL, and LMM) had average non-convergence rates above 0.1% for any of the sample sizes tested. Figure 1 shows the average percentage of models which did not converge across sample sizes for these methods. Because we used LRTs for NBMM-LP, for every gene a reduced model was fit for each of the four hypothesis tests. In some cases the full model converged but one or more of the reduced models failed to converge and thus the p-value for the corresponding hypothesis tests could not be calculated. The transparent portion of the bars in Fig. 1 represent cases in which the full model converged but one or more of the reduced models failed to converge.
Percentage of non-converged models from selected methods. Methods in which less than 1% of models failed to converge are not included in the figure. For NBMM-LP, which uses a likelihood ratio test, the solid portion of the bar represents the proportion of models in which the full model did not converge and the transparent portion represents genes for which the reduced model for one or more tests failed to converge in which case results for those tests could not be obtained
NBMM-LP had the highest non-convergence rates at all sample sizes, even when only considering cases in which only the full model did not converge. At N=3 per group, about 21% of the full models did not converge and the reduced model(s) for an additional 10% of genes did not converge. Comparatively, at N=3 per group around 16% and 15% of models did not converge for NBMM-PL and LMM respectively. For all three methods, non-convergence rates decreased with increasing sample size, though the magnitude of the decrease was larger for NBMM-PL and LMM than for NBMM-LP. At N=10 per group, NBMM-PL and LMM both had non-convergence rates around 4% while NBMM-LP had a non-convergence rate of 11% with at least one reduced model failing to converge for an additional 15% of genes.
For all three methods and at all sample sizes, at least 90% of convergence failures were due to model singularities, with remaining non-converged models reaching model iteration limits or experiencing other errors which prevented the model from fitting properly. On average, the random intercept variance used to simulate the data was lower for genes that did not converge while the dispersion was generally higher (Supplementary Fig. 1). These results indicate that in some cases, model convergence issues may be due in part to low between-subject variation or high dispersion. However, there was substantial overlap in the random intercept and dispersion distributions between genes that did and did not converge, and many genes with high random intercept variance and low dispersion still failed to converge. In addition, the proportion of non-converged genes generally decreased only slightly (0.75%-1%) when using a higher expression filtering threshold of 5 CPM instead of 1 CPM, indicating that small expression values are also not completely responsible for model non-convergence (Supplementary Table 1).
Figure 2 shows the relationship between FDR and power across different sample sizes for the four multiple DF tests of interest using a 0.05 FDR level. More detailed results are available in Supplementary Tables 2-4. The FDR for GEE, NBMM-AGQ, and NBMM-LP was higher than the nominal 0.05 level across all sample sizes for all tests. Other methods showed a mix of conservative and anti-conservative behavior. Across all tests, limma had an FDR close to the nominal rate for the smallest sample size (N=3 per group), but the FDR was increasingly inflated for the larger sample sizes. Conversely, DESeq2* and edgeR* had an inflated FDR at N=3 and N=5 per group, but at N=10 per group the rate was close to the nominal value. DESeq2 and edgeR (ignoring correlation) both had conservative FDR for the interaction and within-subject test, but showed inflated rates for the between-subject test and test for any significant coefficient. Across all of the tests, LMM was slightly conservative while NBMM-PL was slightly inflated except for the between-subject test, in which it was conservative. Finally, rmRNASeq had very conservative FDR values across all tests. For the majority of methods and tests, FDR approached the nominal rate (dashed line) and had increasing power with increasing sample size.
FDR versus power across different sample sizes for four tests of interest. FDR and power were calculated using a 0.05 FDR significance level and were averaged across 10 simulations for each method and sample size. Points that lie to the left of the dashed vertical line represent methods that have an observed FDR less than the nominal rate of 5%, while points to the right represent methods with FDR inflation. Points located in the bottom left-hand corner with an FDR and power of 0 represent instances in which no genes were found significant. A log scale is used on the x-axis to better differentiate between methods with close to nominal FDR
Of the methods that had FDR values which were conservative or close to the nominal rate across all sample sizes and conditions, LMM and NBMM-PL generally had the highest power. rmRNASeq, which showed conservative FDR values, had low power, particularly at the smaller sample sizes. For the within-subject test and the test for significant interaction effects in which edgeR and DESeq2 (ignoring correlation) exhibited conservative FDR values, both methods were less powered than LMM and NBMM-PL at all sample sizes. DESeq2* and edgeR*, which had close to nominal FDR values at N=10 per group, showed similar power to LMM and NBMM-PL at this sample size. Similarly, limma, which had close to nominal FDR at N=3 per group, had comparable power to LMM and NBMM-PL for most tests at this sample size and had more power than either method for the between-subject test.
At the smallest sample size, N=3 per group, no method that had conservative or close to nominal FDR had high power. For the within-subject test, LMM, NBMM-PL and limma had power values near 60% at N=3 per group, but no other tests showed power values this high for methods without severely inflated FDR. The power values at N=5 and N=10 per group were much stronger with LMM and NBMM-PL having power values near or above 80% for all tests at N=10 per group.
The distributions of the raw p-values from the null features in each simulated dataset are shown for each combination of method, test, and sample size in Supplementary Figs. 2-4. In general, we would expect these distributions to look fairly uniform. However, only LMM displays this behavior consistently. Some other methods, like NBMM-PL, limma at the smaller sample sizes, and DESeq2* and edgeR* at the larger sample sizes, are not too far off. Conversely, DESeq2, edgeR, GEE, and rmRNAseq show substantial skew. This suggests that the assumed distributions for the test statistics used in these methods is incorrect, and thus inference from these methods is likely compromised [41].
Real data results
Table 3 shows the run time for each of the methods. The time to fit the full model and the total time (model fitting and hypothesis testing) are both shown for all methods except rmRNAseq, for which the model fitting and testing are carried out within one function and thus the run times cannot be uncoupled. NBMM-AGQ, NBMM-LP and both DESeq2 methods use an LRT which requires a full and reduced model to be fit for each hypothesis test, so for these methods hypothesis testing took a relatively large amount of time compared to the time to fit the full model. NBMM-LP had the longest total run time by far, taking over 24 hours to complete. The second highest run time was for rmRNAseq which took around 7 hours. Aside from these two methods, NBMM-AGQ (102 minutes), and NBMM-PL (65 minutes), all other methods ran in less than 30 minutes.
Table 3 Non-convergence rate, analysis run time, and number of DEGs for 4 hypothesis tests in the shock dataset. The run time for fitting the full model for each gene, as well as the total time to fit models and perform hypothesis testing is displayed. There were 14,340 genes in the dataset and genes were labelled as a DEG if the Benjamini Hochberg adjusted p-value was < 0.05. For NBMM-LP, the percentage of genes in which one or more reduced models failed to converge is shown in parentheses after the full model non-convergence rate
Model convergence
NBMM-LP had the largest percentage of non-converged models with 4.33% of the full model fits not converging (Table 3). An additional 9.07% of models did not converge for one or more reduced models used for LRTs, making the corresponding hypothesis test(s) incomputable. The non-convergence rate for the rest of the methods was less than 1%. This differed from the simulation results in which NBMM-PL and LMM had a non-convergence rate of around 4% at the largest sample size. The percentage of non-convergence for NBMM-LP was also smaller than for the largest sample size simulation scenario. This discrepancy is likely due in part to the large number of subjects in the shock dataset (32 total subjects; SS group: 21 subjects, CS group: 11 subjects). The largest sample size in the simulation scenarios only had 20 total subjects (10 per group, 2 groups).
In order to assess the effect of sample size in our real dataset, we sampled 10 subjects from both the SS and CS groups and reran the analysis on this reduced dataset. The non-convergence rates for NBMM-PL and LMM increased to around 1% for both methods (Table 4). Surprisingly, the non-convergence rate for the NBMM-LP models changed very little even after reducing the number of subjects.
Table 4 Non-convergence rate, analysis run time, and number of DEGs for 4 hypothesis tests in the reduced shock dataset in which ten subjects from each group were randomly selected. The run time for fitting the full model for each gene, as well as the total time to fit models and perform hypothesis testing is displayed. There were 14,340 genes in the dataset and genes were labelled as a DEG if the Benjamini Hochberg adjusted p-value was < 0.05. For NBMM-LP, the percentage of genes in which one or more reduced models failed to converge is shown in parentheses after the full model non-convergence rate
Number of DEGs
Table 3 shows the number of DEGs identified by each method for various hypothesis tests using a 0.05 significance threshold for Benjamini Hochberg adjusted p-values. Though there was a range in the number of DEGs found across the different methods and tests, every method found the most DEGs for the test for the difference across time in the SS group. This is perhaps due in part to the fact that the SS group has more subjects than the CS group (N=21 vs. N=11). However, in the analysis of the reduced dataset in which each group was filtered to ten random subjects, this test still had the most DEGs across methods, while the test for differences across time in the CS group had the least amount of DEGs. This may indicate that the changes in gene expression over the course of treatment are more prevalent in SS patients than CS patients.
The differences in the number of DEGs for each method was generally what would be expected based on the results of the simulation study. NBMM-AGQ showed relatively inflated FDR values in the simulation study, and in this analysis this method found more DEGs than most other methods, particularly for the within-subject and interaction tests. DESeq2 and edgeR (ignoring correlation) had high DEG counts for the between-subject test and low DEG counts for the within-subject and interaction tests, which is also in line with the simulation results. limma also showed a mix of conservative and anti-conservative behavior in terms of the number of DEGs for each test. Finally, DESeq2*,edgeR*, NBMM-PL, NBMM-LP and LMM all had relatively moderate numbers of DEGs across all tests, with DESeq2*, edgeR*, NBMM-LP and NBMM-PL generally finding slightly more DEGs than LMM. This also corresponds to the simulation results in which in the largest sample size scenario (N=10 per group) all three methods exhibited FDR values close to the nominal rate with LMM showing conservative rates compared to the other three methods.
There were some discrepancies between this analysis and the simulation study. These discrepancies appear to be partially due to the difference in the number of subjects in the real data and the simulations and may point to the continuation of patterns related to sample size that were observed in the simulation study. For example, rmRNAseq displayed conservative FDR values and low power in the simulation study, though the power for the method increased with increasing numbers of subjects. In this analysis, the number of DEGs for rmRNAseq was comparable to other, less conservative methods, particularly for the between-subject test and the within-subject test for differences across time in the SS group. However, in the analysis of the reduced dataset, rmRNAseq found less DEGs than the majority of other methods (Table 4). Similarly, GEE generally had the most inflated FDR and highest power in the simulation study with FDR decreasing as the number of subjects increased. In this analysis the number of DEGs was moderate compared to the other methods, while in the analysis on the reduced data, GEE had more DEGs than most other methods, though NBMM-AGQ still found more DEGs for all tests except the between-subject test.
Hierarchical clustering and functional enrichment analysis results
For brevity, we will focus on results from our post-hoc analysis of genes with significant differential expression between at least two timepoints in the CS group. Similar results for the SS group can be found in Supple-mentary Fig. 5 and Supplementary Table 5. Using the LMM method, there were 1,003 genes that were significant for the test for differential expression between any two timepoints in the CS group. Figure 3 shows a heatmap of predicted expression (row scaled) for these genes along with the hierarchical clustering. Based on a visual inspection of the heatmap, a cutpoint was chosen such that the genes were split into seven clusters representing seven different patterns of change over time. For example, cluster 3 was the largest cluster with 328 genes. The expression of genes in this cluster stayed somewhat steady across the first two timepoints, but then steeply dropped between the second and third timepoint. Cluster 5 (309 genes) and cluster 2 (228 genes) were also relatively large. The genes in cluster 5 had expression levels that remained relatively unchanged between the first two timepoints, but then steeply climbed between the final two timepoints; cluster 2 contained genes that dropped in expression somewhat linearly across the three timepoints.
Heatmap of predicted gene expression (row scaled) across the three study timepoints for genes that were significant in a test for differential expression between any two timepoints in the CS group. Predicted values and significance results came from the LMM analysis. Genes are clustered using a correlation distance metric and complete linkage clustering methods and are split into seven clusters indicated by the color bars along the rows
For three clusters (cluster 3, cluster 5, and cluster 6) at least one GO term was significantly enriched. Table 5 shows an abbreviated list of the significant terms. For cluster 3, several significantly enriched terms were related to an innate immune response including terms related to inflammation as well as neutrophil migration. For cluster 5, the GO terms were related to complement activation and phagocytosis. There were also terms related to adaptive immunity such as immunoglobulin production and positive regulation B-cell activation. Because genes from cluster 3 are relatively highly expressed at timepoints 1 and 2, but have lower expression at time 3, while cluster 5 shows the opposite behavior, these results may point to a heightened innate immune system response early in the ICU stay of CS patients, with a delayed adaptive immune response. Similar to cluster 5, genes in cluster 6 were involved in complement activation and phagocytosis. This cluster has a similar pattern across time to that of cluster 5, but genes in this category drop in expression between timepoints 1 and 2 before showing heightened expression at time 3.
Table 5 Functional enrichment analysis results. The 25 GO terms with the smallest Benjamini Hochberg (BH) adjusted p-values were selected for each cluster. The lists were then reduced to include only the most specific subclass for each ontology. All GO terms had a BH adjusted p-value < 0.01
In RNA-seq studies with longitudinal and other correlated designs, researchers are often interested in multiple hypotheses. Multiple DF tests allow researchers to assess multiple hypotheses at once, which is a useful method for selecting lists of genes for further exploration and can also be valuable in FDR control. Recently, several researchers have developed and compared analysis methods for analyzing longitudinal RNA-seq data. However, there has been little research evaluating and comparing these methods in the context of multiple DF testing. Understanding the comparative performance of various multiple DF hypothesis testing methods is becoming increasingly important as complex study designs become more common in correlated RNA-seq designs.
Of the methods compared in this study, LMM using data transformed using VST generally exhibited FDR closest to the nominal rate across the different sample sizes and multiple DF tests. NBMM-PL generally resulted in FDR values close to nominal as well, though slightly more inflated than LMM. GEE, NBMM-AGQ, and NBMM-LP had high FDR values across all simulation scenarios. DESeq2* and edgeR* had inflated FDR values at small sample sizes, but were relatively close to the nominal value for the highest sample size (N=10 per group). Conversely, limma had optimal FDR values at the smallest sample size, but these increased for the larger sample sizes. DESeq2 and edgeR (ignoring correlation) showed a mix of conservative and anti-conservative behavior. rmRNAseq had conservative FDR values, but was also extremely underpowered, particularly at the lower sample sizes. LMM and NBMM-PL generally had the highest power of the methods that had FDR values which were conservative or close to the nominal rate across all sample sizes and conditions.
Unsurprisingly, for the majority of methods, FDR values approached nominal rates and power increased as the sample size increased. We chose to use three small sample size scenarios in our simulation study because researchers often do not have the resources for large-scale studies, particularly in longitudinal studies where multiple samples are collected for each subject. However, we also analyzed data from a study involving shock patients and this study had 11 and 21 subjects in its two groups. In this analysis, methods such as GEE showed similar numbers of DEGs as LMM. When we reduced the dataset to 10 subjects per group, the difference in the number of DEGs for LMM compared to methods like GEE was wider. This implies that the FDR for methods that performed poorly, particularly at low sample sizes, may converge to that of LMM as the sample size increases past N=10 per group.
Another problem that occurred at low sample sizes was model non-convergence for LMM, NBMM-LP and NBMM-PL. Though LMM had the lowest non-convergence rate of these three methods, around 15% of models did not converge for this method at N=3 per group. We identified low between-subject variance, high dispersion, and small gene expression values as potential causes of non-convergence, though these data characteristics were not universal in non-converged models. Because LMM had otherwise good performance, future research regarding the cause of the high non-convergence rates and alternative ways of fitting singular and other non-converged models would be valuable. In small sample size cases in which many models do not converge, limma may be a good alternative because it demonstrated near nominal FDR at small sample sizes. However, no method was highly powered at the smallest sample size; choosing a sample size of at least 5 subjects per group is preferable.
One limitation of this study is that we only simulated data from one relatively simple correlation structure. This choice may have particularly affected the rmRNAseq simulation results since rmRNAseq utilizes a continuous autoregressive correlation structure and we simulated using a single random effect (equivalent to a compound symmetric structure). In analysis of the shock dataset, which may have a correlation structure that is not strictly compound symmetric, rmRNAseq did behave more similarly to other methods than in the simulation study, though we found that this was driven partially by sample size. Still, because complex RNA-seq studies are becoming more common, future research concerning the performance of multiple DF tests on data with different correlation structures and models with more complex random effects structures would be beneficial.
We did not explore the use of multiple DF tests in the context of single cell RNA-sequencing (scRNA-seq). Because gene expression of cells from the same sample or subject is more similar than cells from different samples [42], multi-sample scRNA-seq studies result in a hierarchical or correlated data structure, similar to longitudinal bulk RNA-seq studies. While the methods described in this work could theoretically be applied to scRNA-seq data, there are unique features of scRNA-seq data that could influence method performance and that should be further investigated. For example, scRNA-seq experiments typically collect data on thousands of cells from a relatively small number of samples or subjects, resulting in a large number of repeated observations per sample. This is in contrast to a longitudinal bulk RNA-seq study, where a relatively smaller number of repeated measurements (as few as two) is collected per subject. The library size per cell is also much smaller in scRNA-seq resulting in smaller numbers of counts per gene and more genes with zero counts. The data volume and sparsity could affect both the computation time and performance of the multiple DF testing methods. This would be a valuable area for future research.
As the cost of RNA-seq experiments decreases, it becomes increasingly feasible to perform experiments using correlated designs, including longitudinal studies. Because these studies often involve multiple hypotheses and also require initial filtration to a set of genes for further exploration, multiple DF tests are a valuable tool for correlated RNA-seq data. In this work, we tested several modelling methods for longitudinal RNA-seq data with an emphasis on multiple DF hypotheses tests. Through a simulation study, we found that overall, LMM exhibited the best performance in terms of controlling FDR at nominal levels while maintaining the power to detect differential expression, though there were convergence issues at low sample sizes. limma offers a good alternative for small studies since it did not have convergence issues and had adequate FDR control at the smallest sample size. However, all methods were underpowered at N=3 per group, so we suggest that at least five subjects be included per group when possible.
Multiple DF testing is a valuable tool for selecting interesting genes for downstream analysis while also controlling the FDR. However, as we show in this study, there are many methods that allow for multiple DF testing all with different levels of efficacy. Making an informed decision when choosing a method based on the study goals as well as design elements such as sample size is key in producing useful, meaningful findings.
Code for simulating the datasets and running the methods used in the paper are available at https://github.com/ewynn610/multiDF_corr_RNASeqand through the corrRNASeq package, which can be found at https://github.com/ewynn610/corrRNASeq. Additional simulated datasets used in the simulation studies are available from the corresponding author upon request. The real RNA-Seq data was originally published in [39], and was downloaded for this application from the GEO DataSets website (GEO Dataset: GSE131411).
CPM:
Counts Per Million
CS:
DF:
GEE:
GLM:
Generalized Linear Model
GLMM:
Generalized Linear Mixed Model
HPC:
LMM:
Linear Mixed Model
LRT:
Likelihood Ratio Test
NBMM:
Negative Binomial Mixed Model
NBMM-AGQ:
Negative Binomial Mixed Model, Advanced Gaussian Quadrature approach
NBMM-LP:
Negative Binomial Mixed Model, Laplace approach
NBMM-PL:
Negative Binomial Mixed Model, Pseudolikelihood approach
RNA-seq:
RNA-sequencing
TMM:
Trimmed Mean of M-values
T1E:
Type One Error
VST:
Variance Stabilizing Transformation
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CMM and EAW were funded by a Webb-Waring Early Career Investigator Award from the Boettcher Foundation.
Department of Biostatistics and Informatics, University of Colorado, Anschutz Medical Campus, Aurora, CO, USA
Elizabeth A. Wynn
Center for Genes, Environment and Health, National Jewish Health, 1400 Jackson St, Denver, 80206, CO, USA
Brian E. Vestal, Tasha E. Fingerlin & Camille M. Moore
Brian E. Vestal
Tasha E. Fingerlin
Camille M. Moore
EAW designed and implemented the simulation study and application data analysis, prepared tables and figures, and wrote the manuscript. BEV designed the data simulation framework, provided feedback concerning analysis and reviewed the manuscript. TEF provided feedback concerning analysis and reviewed the manuscript. CMM designed the data simulation framework, supervised the analysis and the writing of the manuscript, and reviewed the manuscript. All authors read and approved the final manuscript.
Correspondence to Camille M. Moore.
No ethics approval was required for this study. All data analyzed in this manuscript was either simulated or downloaded from publicly available sources.
Supplementary methods, results, tables, and figures.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated in a credit line to the data.
Wynn, E.A., Vestal, B.E., Fingerlin, T.E. et al. A comparison of methods for multiple degree of freedom testing in repeated measures RNA-sequencing experiments. BMC Med Res Methodol 22, 153 (2022). https://doi.org/10.1186/s12874-022-01615-8
RNA-seq
Correlated data
Multiple DF testing | CommonCrawl |
\begin{definition}[Definition:Classes of WFFs/Plain Sentence]
Let $\LL_1$ be the language of predicate logic.
A WFF is said to be a '''plain sentence''' {{iff}} it is both plain and a sentence.
That is, {{iff}} it contains free variables nor parameters.
Thus, '''plain sentences''' are those WFFs which are in $\map {SENT} {\PP, \FF, \O}$.
\end{definition} | ProofWiki |
James Joseph Sylvester
James Joseph Sylvester FRS HonFRSE (3 September 1814 – 15 March 1897) was an English mathematician. He made fundamental contributions to matrix theory, invariant theory, number theory, partition theory, and combinatorics. He played a leadership role in American mathematics in the later half of the 19th century as a professor at the Johns Hopkins University and as founder of the American Journal of Mathematics. At his death, he was a professor at Oxford University.
James Joseph Sylvester
Born
James Joseph
(1814-09-03)3 September 1814
London, England
Died15 March 1897(1897-03-15) (aged 82)
London, England
Resting placeBalls Pond Road Cemetery
Alma materSt. John's College, Cambridge
Known for
• Coining the terms 'graph' and 'discriminant'
• Chebyshev–Sylvester constant
• Quadruplanar inversor
• Sylvester's sequence
• Sylvester's formula
• Sylvester's determinant theorem
• Sylvester matrix
• Sylvester–Gallai theorem
• Sylvester's law of inertia
• Sylvester's triangle problem
• Sylver coinage
• Sylvester's criterion
• Sylvester domain
AwardsRoyal Medal (1861)
Copley Medal (1880)
De Morgan Medal (1887)
Scientific career
FieldsMathematics
InstitutionsJohns Hopkins University
University College London
University of Virginia
Royal Military Academy, Woolwich
University of Oxford
Academic advisorsJohn Hymers
Augustus De Morgan
Doctoral studentsWilliam Durfee
George B. Halsted
Washington Irving Stringham
Other notable studentsIsaac Todhunter
William Roberts McDaniel
Harry Fielding Reid
Christine Ladd-Franklin
InfluencedMorgan Crofton
Christine Ladd-Franklin
George Salmon
Biography
James Joseph was born in London on 3 September 1814, the son of Abraham Joseph, a Jewish merchant.[1] James later adopted the surname Sylvester when his older brother did so upon emigration to the United States.
At the age of 14, Sylvester was a student of Augustus De Morgan at the University of London. His family withdrew him from the University after he was accused of stabbing a fellow student with a knife. Subsequently, he attended the Liverpool Royal Institution.
Sylvester began his study of mathematics at St John's College, Cambridge in 1831,[2] where his tutor was John Hymers. Although his studies were interrupted for almost two years due to a prolonged illness, he nevertheless ranked second in Cambridge's famous mathematical examination, the tripos, for which he sat in 1837. However, Sylvester was not issued a degree, because graduates at that time were required to state their acceptance of the Thirty-nine Articles of the Church of England, and Sylvester could not do so because he was Jewish. For the same reason, he was unable to compete for a Fellowship or obtain a Smith's prize.[3] In 1838, Sylvester became professor of natural philosophy at University College London and in 1839 a Fellow of the Royal Society of London. In 1841, he was awarded a BA and an MA by Trinity College Dublin. In the same year he moved to the United States to become a professor of mathematics at the University of Virginia, but left after less than four months. A student who had been reading a newspaper in one of Sylvester's lectures insulted him and Sylvester struck him with a sword stick. The student collapsed in shock and Sylvester believed (wrongly) that he had killed him. Sylvester resigned when he felt that the university authorities had not sufficiently disciplined the student.[4] He moved to New York City and began friendships with the Harvard mathematician Benjamin Peirce (father of Charles Sanders Peirce) and the Princeton physicist Joseph Henry. However, he left in November 1843 after being denied appointment as Professor of Mathematics at Columbia College (now University), again for his Judaism, and returned to England.
On his return to England, he was hired in 1844 by the Equity and Law Life Assurance Society for which he developed successful actuarial models and served as de facto CEO, a position that required a law degree. As a result, he studied for the Bar, meeting a fellow British mathematician studying law, Arthur Cayley, with whom he made significant contributions to invariant theory and also matrix theory during a long collaboration.[5] He did not obtain a position teaching university mathematics until 1855, when he was appointed professor of mathematics at the Royal Military Academy, Woolwich, from which he retired in 1869, because the compulsory retirement age was 55. The Woolwich academy initially refused to pay Sylvester his full pension, and only relented after a prolonged public controversy, during which Sylvester took his case to the letters page of The Times.
One of Sylvester's lifelong passions was for poetry; he read and translated works from the original French, German, Italian, Latin and Greek, and many of his mathematical papers contain illustrative quotes from classical poetry. Following his early retirement, Sylvester published a book entitled The Laws of Verse in which he attempted to codify a set of laws for prosody in poetry.[6]
In 1872, he finally received his B.A. and M.A. from Cambridge, having been denied the degrees due to his being a Jew.[2]
In 1876[7] Sylvester again crossed the Atlantic Ocean to become the inaugural professor of mathematics at the new Johns Hopkins University in Baltimore, Maryland. His salary was $5,000 (quite generous for the time), which he demanded be paid in gold. After negotiation, agreement was reached on a salary that was not paid in gold.[8]
In 1877, he was elected as a member to the American Philosophical Society.[9]
In 1878 he founded the American Journal of Mathematics. The only other mathematical journal in the US at that time was the Analyst, which eventually became the Annals of Mathematics.
In 1883, he returned to England to take up the Savilian Professor of Geometry at Oxford University. He held this chair until his death, although in 1892 the University appointed a deputy professor to the same chair. He was on the governing body of Abingdon School.[10]
Sylvester died at 5 Hertford Street, London on 15 March 1897. He is buried in Balls Pond Road Cemetery on Kingsbury Road in London.[11]
Legacy
Sylvester invented a great number of mathematical terms such as "matrix" (in 1850),[12] "graph" (in the sense of network)[13] and "discriminant".[14] He coined the term "totient" for Euler's totient function φ(n).[15] In discrete geometry he is remembered for Sylvester's problem and a result on the orchard problem, and in matrix theory he discovered Sylvester's determinant identity,[16] which generalizes the Desnanot–Jacobi identity.[17] His collected scientific work fills four volumes. In 1880, the Royal Society of London awarded Sylvester the Copley Medal, its highest award for scientific achievement; in 1901, it instituted the Sylvester Medal in his memory, to encourage mathematical research after his death in Oxford.
Sylvester House, a portion of an undergraduate dormitory at Johns Hopkins University, is named in his honor. Several professorships there are named in his honor also.
Publications
• Sylvester, James Joseph (1870). The Laws of Verse, or, Principles of Versification Exemplified in Metrical Translations: Together with an Annotated Reprint of the Inaugural Presidential Address to the Mathematical and Physical Section of the British Association at Exeter. London: Longmans, Green and Co. ISBN 978-1-177-91141-2.
• Sylvester, James Joseph (1973) [1904]. Baker, Henry Frederick (ed.). The Collected Mathematical Papers of James Joseph Sylvester. Vol. I. New York: AMS Chelsea Publishing. ISBN 978-0-8218-3654-5.[18]
• Sylvester, James Joseph (1973) [1908]. Baker, Henry Frederick (ed.). The Collected Mathematical Papers of James Joseph Sylvester. Vol. II. New York: AMS Chelsea Publishing. ISBN 978-0-8218-4719-0.[18]
• Sylvester, James Joseph (1973) [1904]. Baker, Henry Frederick (ed.). The Collected Mathematical Papers of James Joseph Sylvester. Vol. III. New York: AMS Chelsea Publishing. ISBN 978-0-8218-4720-6.[19]
• Sylvester, James Joseph (1973) [1904]. Baker, Henry Frederick (ed.). The Collected Mathematical Papers of James Joseph Sylvester. Vol. IV. New York: AMS Chelsea Publishing. ISBN 978-0-8218-4238-6.
See also
• Catalecticant
• Covariance and contravariance of vectors
• Evectant
• Inclusion–exclusion principle
• Invariant of a binary form
• Sylvester's construction
• Sylvester pentahedron
• Sylvester's problem
• Clock and shift matrices
• Umbral calculus
• List of things named after James Joseph Sylvester
References
1. Biographical Index of Former Fellows of the Royal Society of Edinburgh 1783–2002 (PDF). The Royal Society of Edinburgh. July 2006. ISBN 0-902-198-84-X. Archived from the original (PDF) on 4 March 2016. Retrieved 15 October 2018.
2. "Sylvester, James Joseph (SLVR831JJ)". A Cambridge Alumni Database. University of Cambridge.
3. Bell, Eric Temple (1986). Men of Mathematics. Simon Schuster.
4. Biography of Sylvester, MacTutor, University of St. Andrews, accessed 6 October 2021
5. Parshall, Karen Hunger (2006). James Joseph Sylvester. Jewish Mathematician in a Victorian world. Johns Hopkins University Press. ISBN 978-0-8018-8291-3. MR 2216541.
6. Sylvester, J. J. (1870). The Laws of Verse, or, Principles of Versification Exemplified in Metrical Translations. London: Longmans, Green and Co.
7. "Preliminary Outline of Instructions for the Session Beginning October 3, 1876". Johns Hopkins University. Official Circulars (5). September 1876.
8. Hawkins, Hugh (1960). Pioneer: A History of the Johns Hopkins University, 1874-1889. Ithaca, NY: Cornell University Press. pp. 41–43.
9. "APS Member History". search.amphilsoc.org. Retrieved 10 May 2021.
10. "School Notes" (PDF). The Abingdonian.
11. Biographical Index of Former Fellows of the Royal Society of Edinburgh 1783–2002 (PDF). The Royal Society of Edinburgh. July 2006. ISBN 0-902-198-84-X. Archived from the original (PDF) on 4 March 2016. Retrieved 15 October 2018.
12. Matrices and determinants, The MacTutor History of Mathematics archive
13. See:
• J. J. Sylvester (7 February 1878) "Chemistry and algebra," Nature, 17 : 284. From page 284: "Every invariant and covariant thus becomes expressible by a graph precisely identical with a Kekuléan diagram or chemicograph."
• J. J. Sylvester (1878) "On an application of the new atomic theory to the graphical representation of the invariants and covariants of binary quantics, — with three appendices," American Journal of Mathematics, Pure and Applied, 1 (1) : 64-90. The term "graph" first appears in this paper on page 65.
14. J. J. Sylvester (1851) "On a remarkable discovery in the theory of canonical forms and of hyperdeterminants," Philosophical Magazine, 4th series, 2 : 391–410; Sylvester coined the term "discriminant" on page 406.
15. J. J. Sylvester (1879) "On certain ternary cubic-form equations," American Journal of Mathematics, 2 : 357–393; Sylvester coins the term "totient" on page 361: "(the so-called Φ function of any number I shall here and hereafter designate as its τ function and call its Totient)"
16. Sylvester, James Joseph (1851). "On the relation between the minor determinants of linearly equivalent quadratic functions". Philosophical Magazine. 1: 295–305.
17. C.G.J. Jacobi, "De Formatione et Proprietatibus Determinantium", Journal für die reine und angewandte Mathematik, 22, 285-318 (1841)
18. Dickson, L. E. (1909). "Review: Sylvester's Mathematical Papers, vols. I & II, ed. by H. F. Baker". Bull. Amer. Math. Soc. 15 (5): 232–239. doi:10.1090/S0002-9904-1909-01746-X.
19. Dickson, L. E. (1911). "Review: Sylvester's Mathematical Papers, vol. III, ed. by H. F. Baker". Bull. Amer. Math. Soc. 17 (5): 254–255. doi:10.1090/S0002-9904-1911-02040-7.
Sources
• Grattan-Guinness, I. (2001), "The contributions of J. J. Sylvester, F.R.S., to mechanics and mathematical physics", Notes and Records of the Royal Society of London, 55 (2): 253–265, doi:10.1098/rsnr.2001.0142, MR 1840760, S2CID 122748202.
• Macfarlane, Alexander (2009) [1916], Lectures on Ten British Mathematicians of the Nineteenth Century, Mathematical monographs, vol. 17, Cornell University Library, ISBN 978-1-112-28306-2
• Parshall, Karen Hunger (1998), James Joseph Sylvester. Life and work in letters., The Clarendon Press Oxford University Press, ISBN 978-0-19-850391-0, MR 1674190, Review
• Parshall, Karen Hunger (2006), James Joseph Sylvester. Jewish mathematician in a Victorian world, Johns Hopkins University Press, ISBN 978-0-8018-8291-3, MR 2216541
External links
• O'Connor, John J.; Robertson, Edmund F., "James Joseph Sylvester", MacTutor History of Mathematics Archive, University of St Andrews
• James Joseph Sylvester at the Mathematics Genealogy Project
• Collected papers – from the University of Michigan Historical Math Collection
• J. J. Sylvester home page
• Selected Poetry of James Joseph Sylvester
• Works by James Joseph Sylvester at LibriVox (public domain audiobooks)
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| Wikipedia |
Periodic sequence
In mathematics, a periodic sequence (sometimes called a cycle) is a sequence for which the same terms are repeated over and over:
a1, a2, ..., ap, a1, a2, ..., ap, a1, a2, ..., ap, ...
The number p of repeated terms is called the period (period).[1]
Definition
A (purely) periodic sequence (with period p), or a p-periodic sequence, is a sequence a1, a2, a3, ... satisfying
an+p = an
for all values of n.[1][2][3][4][5] If a sequence is regarded as a function whose domain is the set of natural numbers, then a periodic sequence is simply a special type of periodic function. The smallest p for which a periodic sequence is p-periodic is called its least period[1][6] or exact period.[6]
Examples
Every constant function is 1-periodic.[4]
The sequence $1,2,1,2,1,2\dots $ is periodic with least period 2.[2]
The sequence of digits in the decimal expansion of 1/7 is periodic with period 6:
${\frac {1}{7}}=0.142857\,142857\,142857\,\ldots $
More generally, the sequence of digits in the decimal expansion of any rational number is eventually periodic (see below).[7]
The sequence of powers of −1 is periodic with period two:
$-1,1,-1,1,-1,1,\ldots $
More generally, the sequence of powers of any root of unity is periodic. The same holds true for the powers of any element of finite order in a group.
A periodic point for a function f : X → X is a point x whose orbit
$x,\,f(x),\,f(f(x)),\,f^{3}(x),\,f^{4}(x),\,\ldots $
is a periodic sequence. Here, $f^{n}(x)$ means the n-fold composition of f applied to x.[6] Periodic points are important in the theory of dynamical systems. Every function from a finite set to itself has a periodic point; cycle detection is the algorithmic problem of finding such a point.
Identities
Partial Sums
$\sum _{n=1}^{kp+m}a_{n}=k*\sum _{n=1}^{p}a_{n}+\sum _{n=1}^{m}a_{n}$ Where k and m<p are natural numbers.
Partial Products
$\prod _{n=1}^{kp+m}a_{n}=({\prod _{n=1}^{p}a_{n}})^{k}*\prod _{n=1}^{m}a_{n}$ Where k and m<p are natural numbers.
Periodic 0, 1 sequences
Any periodic sequence can be constructed by element-wise addition, subtraction, multiplication and division of periodic sequences consisting of zeros and ones. Periodic zero and one sequences can be expressed as sums of trigonometric functions:
$\sum _{k=1}^{1}\cos(-\pi {\frac {n(k-1)}{1}})/1=1,1,1,1,1,1,1,1,1...$
$\sum _{k=1}^{2}\cos(2\pi {\frac {n(k-1)}{2}})/2=0,1,0,1,0,1,0,1,0...$
$\sum _{k=1}^{3}\cos(2\pi {\frac {n(k-1)}{3}})/3=0,0,1,0,0,1,0,0,1,0,0,1,0,0,1...$
$...$
$\sum _{k=1}^{N}\cos(2\pi {\frac {n(k-1)}{N}})/N=0,0,0...,1{\text{ sequence with period }}N$
Generalizations
A sequence is eventually periodic if it can be made periodic by dropping some finite number of terms from the beginning. For example, the sequence of digits in the decimal expansion of 1/56 is eventually periodic:
1 / 56 = 0 . 0 1 7 8 5 7 1 4 2 8 5 7 1 4 2 8 5 7 1 4 2 ...
A sequence is ultimately periodic if it satisfies the condition $a_{k+r}=a_{k}$ for some r and sufficiently large k.[1]
A sequence is asymptotically periodic if its terms approach those of a periodic sequence. That is, the sequence x1, x2, x3, ... is asymptotically periodic if there exists a periodic sequence a1, a2, a3, ... for which
$\lim _{n\rightarrow \infty }x_{n}-a_{n}=0.$[4][8][9]
For example, the sequence
1 / 3, 2 / 3, 1 / 4, 3 / 4, 1 / 5, 4 / 5, ...
is asymptotically periodic, since its terms approach those of the periodic sequence 0, 1, 0, 1, 0, 1, ....
References
1. "Ultimately periodic sequence - Encyclopedia of Mathematics". encyclopediaofmath.org. 7 February 2011. Retrieved 13 August 2021.{{cite web}}: CS1 maint: url-status (link)
2. Weisstein, Eric W. "Periodic Sequence". mathworld.wolfram.com. Retrieved 2021-08-13.
3. Bosma, Wieb. "Complexity of Periodic Sequences" (PDF). www.math.ru.nl. Retrieved 13 August 2021.{{cite web}}: CS1 maint: url-status (link)
4. Janglajew, Klara; Schmeidel, Ewa (2012-11-14). "Periodicity of solutions of nonhomogeneous linear difference equations". Advances in Difference Equations. 2012 (1): 195. doi:10.1186/1687-1847-2012-195. ISSN 1687-1847. S2CID 122892501.
5. Menezes, Alfred J.; Oorschot, Paul C. van; Vanstone, Scott A. (2018-12-07). Handbook of Applied Cryptography. CRC Press. ISBN 978-0-429-88132-9.
6. Weisstein, Eric W. "Least Period". mathworld.wolfram.com. Retrieved 2021-08-13.
7. Hosch, William L. (1 June 2018). "Rational number". Encyclopedia Britannica. Retrieved 13 August 2021.{{cite web}}: CS1 maint: url-status (link)
8. Cheng, SuiSun (2017-09-29). New Developments in Difference Equations and Applications: Proceedings of the Third International Conference on Difference Equations. Routledge. ISBN 978-1-351-42880-4.
9. Shlezinger, Nir; Todros, Koby (2019-01-01). "Performance analysis of LMS filters with non-Gaussian cyclostationary signals". Signal Processing. 154: 260–271. arXiv:1708.00635. doi:10.1016/j.sigpro.2018.08.008. ISSN 0165-1684. S2CID 53521677.
Sequences and series
Integer sequences
Basic
• Arithmetic progression
• Geometric progression
• Harmonic progression
• Square number
• Cubic number
• Factorial
• Powers of two
• Powers of three
• Powers of 10
Advanced (list)
• Complete sequence
• Fibonacci sequence
• Figurate number
• Heptagonal number
• Hexagonal number
• Lucas number
• Pell number
• Pentagonal number
• Polygonal number
• Triangular number
Properties of sequences
• Cauchy sequence
• Monotonic function
• Periodic sequence
Properties of series
Series
• Alternating
• Convergent
• Divergent
• Telescoping
Convergence
• Absolute
• Conditional
• Uniform
Explicit series
Convergent
• 1/2 − 1/4 + 1/8 − 1/16 + ⋯
• 1/2 + 1/4 + 1/8 + 1/16 + ⋯
• 1/4 + 1/16 + 1/64 + 1/256 + ⋯
• 1 + 1/2s + 1/3s + ... (Riemann zeta function)
Divergent
• 1 + 1 + 1 + 1 + ⋯
• 1 − 1 + 1 − 1 + ⋯ (Grandi's series)
• 1 + 2 + 3 + 4 + ⋯
• 1 − 2 + 3 − 4 + ⋯
• 1 + 2 + 4 + 8 + ⋯
• 1 − 2 + 4 − 8 + ⋯
• Infinite arithmetic series
• 1 − 1 + 2 − 6 + 24 − 120 + ⋯ (alternating factorials)
• 1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series)
• 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ (inverses of primes)
Kinds of series
• Taylor series
• Power series
• Formal power series
• Laurent series
• Puiseux series
• Dirichlet series
• Trigonometric series
• Fourier series
• Generating series
Hypergeometric series
• Generalized hypergeometric series
• Hypergeometric function of a matrix argument
• Lauricella hypergeometric series
• Modular hypergeometric series
• Riemann's differential equation
• Theta hypergeometric series
• Category
| Wikipedia |
A group from Hackenbush game
Awad Alabdala ORCID: orcid.org/0000-0002-8687-61631 &
Essam El-Seidy1
This research presents the reader some algebraic operations related to combinatorial games and gives a detailed outlook of a special game called Hackenbush game. We will deduce a group with a special feature with the help of some basic algebraic concepts. A fresh outlook to some combinatorial mathematical algebraic operations and concepts through the evaluation of a deduced group from this game.
As a result of the many mathematical research done by Lagrange, Abel, Galois, and others in the fields of geometry, number's theory, and algebraic equations at the end of the eighteenth century, early nineteenth century, the mathematical theory of "groups" was discovered [1,2,3]. That theory has led researchers to the discovery of many essential mathematical concepts such as "sub-groups," "graphs and groups," and plenty others. By building on previous credited works and researches, we can express mathematical groups as graphs. By analyzing the structures of graphs, we can deduce the properties of their related groups, which can be done by exploiting the notion of "identity" in "group theory."
By expressing groups by their associated graphs, or in other words "identity graphs," we can utilize the "identity graphs" theory's equations and concepts to conclude the following.
Two elements x, y in a group are adjacent or can be joined by an edge if x. y = e (e, identity element of G).
Since we know that x. y = y. x = e, there is no need to utilize the property of commutatively. It is by convention; every element is adjoined with the identity of the group G. This is the convention we use when trying to represent a group by a graph. The vertices correspond to the elements of the group, hence the order of the group G corresponds to the number of vertices in the identity graph [4, 5].
For example, see the identity graph for some groups in Figs. 1 and 2.
An identity graph for the cyclic group G = 〈g| g14 = 1〉
An identity graph for the Z17 = {0, 1, 2, … , 16} the group under addition modulo 17
In this research, we will utilize the concepts of "graph theory" to analyze some combinatorial mathematical algebraic operations and concepts through the evaluation of a deduced group from a combinatorial game called "Hackenbush game."
Combinatorial games theory
"Combinatorial game theory" describes the study of sequential games with perfect information. When playing "Hackenbush," all players know all the possible outcomes from a given position with no randomness [6]. The game is defined with the following attributes:
There are two players, left and right.
There is always a finite number of positions, in addition to a starting position.
The game is played by the following rules:
Left and right alternate making moves.
Both players have access to all information always.
There is no randomness to moves made, such as rolling a dice.
A player loses when he/she can no longer make any legal moves.
The game ends when the ending condition is met.
If we have a game G and this game includes options for player L named GL and options for player R named GR.
Now, we will try to explain the manner of dealing with the games, as well as showing its rules.
G = {GL| GR}
Numbers represent the number of free moves or the moves possible to a particular player. By convention, positive numbers represent an advantage for left, while negative numbers represent an advantage for right. They are defined recursively with 0 being the base case.
0 = { | } we will explain Zero game in detail in the section "Hackenbush game".
G = 0 means the first player loses
1 = {0 | }, 2 = {1 | }, 3 = {2 | }
−1 = { | 0}, − 2 = { | −1}, − 3 = { | −2 }
Star, written as * or {0|0}, is the first player's wins since either player must move to a Zero game, which will be defined later in the research, therefore wins.
* + * = 0, because the first player must turn one copy of * to a 0, and the other player will have to turn the other copy of * to a 0 as well, at this point, the first player would lose, since 0 + 0 admits no moves.
Like all other games, the game * is neither positive nor negative, in which the first player wins.
∗ = {0 | 0 }
∗ = − ∗
G ║ 0 means the second player loses
G > 0 means left wins
G < 0 means right wins
Up, written as ↑, is a position in combinatorial game theory [7]. In standard notation,
↑ = {0| ∗}
Down, written as ↓, is a position in combinatorial game theory [7]. In standard notation,
↓ = {∗| 0 }
↓ = − ↑
We will explain the last two definitions in detail in the next section.
Hackenbush game
In Hackenbush, the drawn figure uses vertices and line segments, which are connected to a final line called the ground. Players take turns deleting one of their lines. Therefore, left and right take on the colors blue and red respectively. If there is, at any time, a path that cannot be drawn from the ground to a line segment, the segment is deleted. This allows for more strategic plays as a player can delete an opponent's move during their turn [7, 8]. Figure 3 is an example of the Hackenbush game.
An example of Hackenbush game
Now, we will discuss how the game is played. In a game represented by Fig. 3. We will pick left to make the first move. Left has two legal moves. He can either remove the line on the right or the line on the top of the red line. However, the latter move is best, because it allows right to take his middle piece, and effectively remove left's piece with it. Likewise, right should take his middle piece if he moves first for that same exact reason. That would be the optimal plays in that current example. Mathematically, after every move, the resulting game's board becomes a sub-game of the original game's board, or in other words, a new game with new independent possibilities [9, 10].
Assuming optimal plays are always being made, finite values can be assigned to each game or sub-game. These values are determined by observing the advantages the left player has, after each move. For example, after Left makes a move, he has a moves advantage over right, but after right moves, left has b moves advantage over right.
Those values could be written in the form {a| b}. This form does not make any quantifiable sense at the moment, because of the actual value determined by what "a" and "b" are.
Before explaining how to find those values, we will first consider the case in which there are no legal moves for neither players, left or right. Therefore, the player's (with no legal moves) score is left blank in the notation. If both players have no legal moves, the result is a Zero game.
Definition 3.1. A Zero-game
A Zero-game is a game that scores { | } = 0, or in other words, the player who moves next loose, assuming all moves made are optimal.
For Hackenbush, the simplest form of a Zero game equates to an empty board at the beginning. Thus, it is obvious the first player to move has no legal move, therefore loses. In another scenario, if we were to add one blue line, left would have 0 moves after he removes the only possible line, and right would have no legal move giving left a clear 1 move advantage, written {0| } = 1. This trend continues in such a manner that {n| } = n + 1, as n equals the number of remaining moves left has, after making the optimal move [11].
However, what will happen if we add the red line instead? As done before, these values are applied with respect to the advantage of the left player. Thus, adding one line for right results in putting left at a one move disadvantage, or a (−1) advantage. So, adding one red line results in { | 0} = − 1. Adding two red lines would then be { | −1} = − 2. And so on to a general form of { | −n} = − (n + 1), as n equals the number of remaining moves right has after the optimal move.
Now with the groundwork out of the way, we can start using the values of sub-games to determine the value of an overall game, for example, see Fig. 4.
A simple Hackenbush game tree with values
After left moves, right has one legal move. In that case, we clearly have a { | 0 } = − 1 situation. Conversely, if right moves first, we have { 0 | } = 1. This results in the overall game having a value of {−1| 1} = 0. It makes sense because we equally added one independent move for both players to an empty board [12], which means that the advantage does not change. But what about the game in Fig. 5?
What could this game equal?
Now both players have been given one line each. But if left moves, right's only legal move is eliminated, but if right moves, left will still have one legal move. It is not so clear what the given value of this game by implementing the "scoring rules" discussed earlier. That is why we must utilize a new rule called the "simplicity rule," to assign values to the previous example [13].
Definition 3.2. A * game
A * game (pronounced star game) is an infinitesimal game that scores.
{0| 0} = ∗ resulting in the first player to move wins, assuming all moves made are optimal [14].
For example, Hackenbush's game has another line type that is green, which is claimable by either player. Then, we get the following game which results in a value of {0| 0} = ∗ (see Fig. 6). Building on the previous concept, we can see results like {n| −n} = n∗, as n ∗ = n + ∗. It is also worth noting that ∗ has the property such that ∗ + ∗ = 0, see Fig. 6.
Example of a ∗ game with a green line
Furthermore, there are more two infinitesimal games.
Definition 3.3. An ↑ game
An ↑-game (pronounced "up game") is a positive infinitesimal game as the score is {0| ∗}, favoring the Left player [15], as shown in Fig. 7:
An example of an up game
The negative version of an up game is called a down game and defined as follows.
Definition 3.4. An ↓game
A ↓ game (pronounced "down game") is a negative infinitesimal game as the score is {∗| 0} which favors the right player [15].
The relation between "up" and "down" games represents the relation among the inverses of games. In every game, there is a way to reverse every move, which results in negating the value of the game originally obtained. In Hackenbush, this is obtained by replacing every red line with a blue line and vice versa. See Fig. 8.
A down game, showing the negation relation
Property 3.1.
\( -\frac{1}{2^{\mathrm{n}}}<\downarrow <0<\uparrow <\frac{1}{2^{\mathrm{n}}} \)
for any integer n > 0.
↓ = − ↑ .
Both ↑ and ↓ are fuzz to ∗.
\( {\displaystyle \begin{array}{l}\left\{\uparrow |\downarrow \right\}=\left\{\uparrow |\ 0\right\}=\\ {}=\left\{0\ |\downarrow \right\}=\left\{0\ |\ 0\right\}=\ast .\end{array}} \)
Hackenbush group
For every Hackenbush's game, a class can be defined. For example, the class of Zero game, see Fig. 9.
class of Zero game
Let us define the binary addition operation to a set (H) of unique classes defined above
$$ \mathrm{x}+y=\left\{{x}^L+\mathrm{y},\mathrm{x}+{y}^L|{x}^R+\mathrm{y},\mathrm{x}+{y}^R\right\} $$
Theorem 4.1. Addition is commutative.
Base case: G + 0 = 0 + G for all games G.
$$ \mathrm{x}+y{\displaystyle \begin{array}{c}=\left\{{x}^L+\mathrm{y},\mathrm{x}+{y}^L|{x}^R+\mathrm{y},\mathrm{x}+{y}^R\right\}\\ {}=\left\{\mathrm{y}+{x}^L,{y}^L+\mathrm{x}|\mathrm{y}+{x}^R,{y}^R+\mathrm{x}\right\}=y+x\end{array}} $$
Theorem 4.2. Addition is associative
(x + y) + z = x + (y + z) The proof is clear.
We get negate by replacing the red (R) sides with blue (L) and vice versa and keep the green (R, L) as it is, see Fig. 10.
$$ -G=\left\{-{G}^R|-{G}^L\right\} $$
Negative of a game
The effect here is that all moves of left and right are switched.
G + (−G) is a second player wins. For example, right first:
Right to move, so left wins.
We can classify the games by its outcomes (Table 1).
Table 1 Classification of the games by outcome
Definition 4.1. Hackenbush group
The set H is defined as a group because the following conditions are met.
if h, h1, h2, h3 ∈ H
Closure: if h1, h2 belong to H, h1 + h2 is also in H.
Associative h1 + (h2 + h3) = (h1 + h2) + h3 for all h1, h2, h3 in H
Identity element: there is a class 0 in H such that h + 0 = 0 + h = h for all h in H
$$ {\displaystyle \begin{array}{l}0+0=\left\{\ |\ \right\}+\left\{\ |\ \right\}=\left\{\ |\ \right\}=0\\ {}x+0=x+\left\{\ |\ \right\}=\left\{\ {X}_L+0\ |\ {X}_R+0\ \right\}=\left\{{X}_L|\ {X}_R\ \right\}=x\\ {}0+y=\left\{\ |\ \right\}+y=\left\{\ 0+{Y}_L\ |\ 0+{Y}_R\right\}=\left\{\ {Y}_L\ |\ {Y}_R\ \right\}=y\end{array}} $$
Inverse element: for each h in H, there is an element −h in H such that h + (−h) = (−h) + h = 0 see Fig. 11
Example of the inverse of \( \frac{1}{2} \)
Commutative h1 + h2 = h2 + h1 for all h1, h2 in H
Definition 4.2. The group under addition modulo (n)
We can create one to one function as shown in Table 2.
Table 2 One to one function
If we take Hn = {0, 1, 2, … , n − 1}; 0 is the CO (class of) Zero games, 1 is the class of One games and so on. Let us define the operation ⊕:
for every h1, h2 ∈ Hn;
$$ {h}_1\oplus {h}_2=\left\{\begin{array}{cc}{h}_1+{h}_2& {h}_1+{h}_2<n\\ {}{h}_1+{h}_2-n& {h}_1+{h}_2\ge n\end{array}\right. $$
It is clear from the definition that the operation ⊕ is closure, associative, commutative, and has an identity element (0). For every element h ∈ Hn there is an inverse element n − h, or −h = n − h. Therefore, Hn is a commutative group under addition modulo n. For example, H6 = {0, 1, 2, , 3, 4, 5} the group under addition modulo 6 is shown in Table 3.
Table 3 Addition modulo 6 in the group H6
The inverse of the element 1 is −1 = 6 − 1 = 5 and the inverse of 2 is −2 = 6 − 2 = 4
Example 4.1. This group (퐻2 × 퐻2, +) consists of two tuples with addition defined
element-by-element modulo 2. An addition to the group table is shown in Table 4.
Table 4 Addition modulo 2 in the group (H2 × H2, +)
What is special in Hackenbush's group?
Hackenbush group (H) is an infinite commutative group that has an element (not the identity) when added to itself, we get the identity element, see Fig. 12.
We get the identity element
In this research, we have utilized combinatorial mathematical algebraic operations and concepts, through the evaluation of a deduced group from a combinatorial game called Hackenbush game to define the unique Hackenbush group.
Now, what happens if we considered the multiplication's operation on what we have analyzed above.
$$ {\displaystyle \begin{array}{l} xy=\left\{{X}_L,{X}_R\right\}\left\{{Y}_L,{Y}_R\right\}\\ {}=\left\{{X}_Ly+{xY}_L-{X}_L{Y}_L,{X}_Ry+{xY}_R-{X}_R{Y}_R\left|{X}_Ly+{xY}_R-{X}_L{Y}_R,{xY}_L+{X}_Ry-{X}_R{Y}_L\right.\right\}.\end{array}} $$
Or the division's operation that is done in terms of reciprocal and multiplication
$$ \frac{x}{y}=x\left(\frac{1}{y}\right);\frac{1}{y}= $$
$$ =\left\{\begin{array}{c}0,\frac{1+\left({y}^R-y\right){\left(\frac{1}{y}\right)}^L}{y^R},\frac{1+\left({y}^L-y\right){\left(\frac{1}{y}\right)}^R}{y^L}\mid \\ {}\frac{1+\left({y}^L-y\right){\left(\frac{1}{y}\right)}^L}{y^L},\frac{1+\left({y}^R-y\right){\left(\frac{1}{y}\right)}^R}{y^R}\end{array}\right\}. $$
Would that lead to the creation of Rings, Fields, etc.… and would such research be applied to the world of Algebra? Only further research will tell how far Combinatorial Game Theory can go.
↑:
Up game
↓:
CO:
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Nowakowski, R.J. (ed.): Game of no chance III. Mathematical Sciences Research Institute Cambridge University Press, (2008).
Berlekamp, E.R., Conway, J.H., Guy, R.K.: In: Peters, A.K. (ed.) Winning ways for your mathematical plays, vol. 1, 2nd edn, Massachusetts (2001), https://annarchive.com/files/Winning%20Ways%20for%20Your%20Mathematical%20Plays%20V1.pdf
John, S., Ryals, J.R.: Combinatorial game theory: an introduction to tree topplers, thesis under the direction of Hua Wang. Georgia Southern University, USA (2015)
Bouton, C.L.: Nim, a game with a complete mathematical theory. Ann. Math. 3, 35–39 (1902)
Conway, J.H.: Review: on numbers and games. Math. Soc. 84(6), 1328–1336 (1978)
McKay, N.A., Milley, R., Nowakowski, R.J.: Misère -play hackenbush sprigs. Int. J. Game Theory. 45, 731–742 (2015)
Siegel, A.N.: Coping with cycles. In: Games of no chance III. Cambridge University (2009), http://www.msri.org/people/staff/levy/files/Book56/12siegel.pdf
Berlekamp, E.R.: The hackenbush number system for compression of numerical data. Inf. Control. 26, 134–140 (1974)
Fraenkel, A.: A brief biography. Electron. J. Comb, 8 (2001)
Albert, M., Nowakowski, R., Wolfe, D.: Lessons in play: an introduction to combinatorial game theory, CRC (2007)
The authors are very thankful to the editor.
Department of Mathematics, Faculty of Science, Ain Shams University, Cairo, Egypt
Awad Alabdala & Essam El-Seidy
Awad Alabdala
Essam El-Seidy
All authors jointly worked on the results and they read and approved the final manuscript.
Correspondence to Awad Alabdala.
Alabdala, A., El-Seidy, E. A group from Hackenbush game. J Egypt Math Soc 27, 13 (2019). https://doi.org/10.1186/s42787-019-0013-1
Combinatorial games
Moving to graph
AMS Subject Classification
91Axx
20Nxx
05Cxx | CommonCrawl |
\begin{document}
\title{Arbitrary Coherent Superpositions of Quantized Vortices in Bose-Einstein Condensates \\ from Orbital Angular Momentum Beams of Light} \author{Sulakshana Thanvanthri} \affiliation{Hearne Institute for Theoretical Physics, Department of Physics \& Astronomy, Louisiana State University, Baton Rouge, Louisiana 70803-4001, USA} \author{Kishore T. Kapale} \email{[email protected]} \affiliation{Department of Physics, Western Illinois University, Macomb, Illinois, 61455-1367, USA} \affiliation{Hearne Institute for Theoretical Physics, Department of Physics \& Astronomy, Louisiana State University, Baton Rouge, Louisiana 70803-4001, USA} \author{Jonathan P. Dowling} \affiliation{Hearne Institute for Theoretical Physics, Department of Physics \& Astronomy, Louisiana State University, Baton Rouge, Louisiana 70803-4001, USA} \begin{abstract} We recently proposed a scheme for the creation of coherent superpositions of vortex states in Bose-Einstein condensates (BEC) using orbital angular momentum (OAM) states of light [Phys. Rev. Lett. {\bf 95}, 173601 (2005)]. Here we discuss further technical details of the proposal, provide alternative, time-reversal-symmetric scheme for transfer of a superposition of OAM states of light to the BEC via a procedure analogous to the traditional STimulated Raman Adiabatic Passage (STIRAP) technique, and discuss an alternative trap configuration conducive for sustaining large charge vortices. Superpositions of OAM states of light, created using experimental techniques, can be transfered to an initially nonrotating BEC via a specially devised Raman coupling scheme. The techniques proposed here open up avenues to study coherent interaction of OAM states of light with matter. The study could also be employed for performing various quantum information processing tasks with OAM states of light\mdash including a memory for a quantum state of the initial superposition. \end{abstract} \pacs{42.50.Ct, 03.75.Gg, 03.75.Lm, 03.67.-a} \maketitle
\section{Introduction} The interaction of light, carrying quantized orbital angular momentum (OAM)~\cite{Padgett:2004}, with atomic or molecular matter is of considerable research interest~\cite{Muthikrishnan:2002,Babiker:2002,Alexandrescu:2006}. This quest is increasingly becoming feasible due to the progress that has been made in creation~\cite{Heckenberg:1992, Arlt:1998, Sueda:2004}, manipulation~\cite{Akamatsu:2003}, detection~\cite{Molina-Terriza:2002,Leach:2002}, and application~\cite{Grier:2003} of the orbital angular momentum (OAM) states of light. Nevertheless, due to the size-mismatch between the atoms and the spatial features of the phase structures in the OAM beams of light, it is difficult to couple the OAM degrees of freedom of light to the internal states of single atoms. As a result, some of the activity in this area is concentrated in the excitation of vortices in Bose-Einstein Condensates~\cite{Anderson:1995, Bradley:1995, Mewes:1996}. Bose-Einstein Condensates (BEC) are macroscopic coherent objects that are obtained by trapping atoms in particular internal states and cooling them further so that all the atoms have the same motional and internal states. Excitation of vortices in BECs have been traditionally achieved via stirring of the BEC cloud with a laser beam~\cite{Abo-Shaeer:2001}. These vortex states are fairly stable and could be candidates for qubits in Quantum Information~\cite{Nielsen:2000}, if appropriate means to manipulate them are developed.
There exist several proposals to transfer pure OAM states of light to the BECs~\cite{Marzlin:1997, Nandi:2004}; there is also a proposal to use BECs for storage of the OAM state of light~\cite{Dutton:2004}. This transfer of OAM of light to BEC is possible due to the coherent nature of the BEC and because the size of the BEC cloud is about the same as the distance over which the intensity variations occur in the beam of light carrying OAM. We recently proposed a scheme for creation of superposition of two counter-rotating vortex states (with same absolute value of the charge) in BECs via coupling of the OAM degrees of freedom of incident beam to the center-of-mass motion of the atoms~\cite{Kapale:2005}. To illustrate, transfer of a pure OAM state of light to a BEC cloud is an experimental reality~\cite{Andersen:2006}; nevertheless the experimental system involves different linear momentum for the initial and final states of the BEC as opposed to different internal states in the scheme discussed in Ref.~\cite{Kapale:2005} and in this article. Furthermore, the experiment involves transfer of only pure OAM states to the BEC as opposed to a OAM superposition as discussed here. It is important to note that coherent transfer of a superposition state is the first check necessary to test the suitability of these macroscopic objects for quantum information. The results of our study reported here demonstrate the possibility of using vortex states in BEC as a qubit.
The counter-rotating vortex superposition generated via our scheme is analogous to the counter-rotating persistent currents in superconducting circuits~\cite{Nakamura:1999,Friedman:2000,Wal:2000}. These counter-rotating persistent currents in the superconducting systems have already been recognized as candidates for qubits for quantum information processing. Thus the application of the study presented here to the area of quantum information is clear. Further applications of our scheme could be as a quantum memory of such superposition states. One can envision such a memory device to be useful in the quantum communication networks using OAM states of light. (See Ref.~\cite{Spedalieri:2004QKD} and references therein.) OAM states of light offer a higher dimensional Hilbert space to obtain extra security and dense coding of quantum information. The memory application would deem it necessary to have a time-reversal symmetric procedure to write the superposition in OAM states of light to BEC so that the superposition can be extracted at a later stage. Here we discuss further technical details of our earlier proposal~\cite{Kapale:2005} and also provide a STIRAP-like time-reversal symmetric mechanism~\cite{Shore:1992} for transfer of special OAM states of light to BEC. Furthermore, we discuss alternative trapping potential, which can support toroid-shaped condensate cloud, in order to increase the stability of the generated vortex superposition.
The article is organized as follows. In Sec.~\ref{Sec:OptOAMSuperposition} we discuss generation of arbitrary superposition of two counter-rotating OAM components with the same absolute value of charge. This section also discusses a scheme for generation of arbitrary superposition of two arbitrarily charged OAM states of light. Next, in Sec.~\ref{Sec:OAMCoupling}, we discuss two independent methods of transferring the OAM superposition from the light beam to the vortex states of BEC in complete detail. We also consider an alternative trap configuration, the so-called mexican-hat trap to offer an example of trap that can sustain large charge vortices. In fact, we provide a very general set of equations to study the transfer of OAM superposition from light to BEC for any kind of trapping potential. In Sec.~\ref{Sec:Detection} we discuss the detection of the BEC vortex superposition and present our conclusions in Sec.~\ref{Sec:Conclusion}. Some of the mathematical details are presented in Appendices A and B.
\section{\label{Sec:OptOAMSuperposition}Superposition of optical vortices}
To recollect, the OAM states of light are different in their phase characteristics from the conventional gaussian light beams. They have azimuthal phase structure characterized by the
quantized orbital angular momentum carried by each photon in the beam. In the simplest case of one unit of orbital angular momentum, i.e. $\hbar$, the light beam has a cork-screw type phase front. Thus, the phase continuously varies from 0 to $2 \pi$ along the azimuthal coordinate and there is a jump from $2 \pi$ to 0 along some radial line. The location of the radial line along which the phase jump (or phase discontuinity) occurs is time-dependent and it rotates continuously with time. The sense of rotation governs the sign of the the orbital angular momentum. Furthermore, an OAM state with angular momentum $\ell \hbar$ has $|\ell|$ number of azimuthal phase jumps across a cut taken in the beam path. It has to be noted that this is an instantaneous description of the beam phase across a cut taken in its path. The whole structure is time dependent and rotates along the beam axis in a clockwise or counter-clockwise direction as the beam propagates. The sign of $\ell$ corresponds to the sense of rotations of the phase fronts around the beam axis. In principle, there is no upper limit to the angular momentum value that can be imparted to a light beam. Therefore, OAM states of light themselves are very good candidates for quantum information processing, mainly in the area of quantum cryptography, as they offer an essentially infinite Hilbert space to work with. This could result in increasing the security of the quantum-key distribution protocols tremendously.
Monochromatic beams with azimuthal phase dependence of the type $\exp({\rm i\,} \ell \phi)$, of which Laguerre-Gaussian (LG) laser modes are an example, have well-defined angular momentum of $\ell \hbar$ per photon. The normalized Laguerre Gaussian mode at the beam waist $(z=0)$ and beam size $w_0$ at the waist is given in cylindrical coordinates ($\rho,\phi,z$) by \begin{align}
\mbox{LG}_{p}^{ \ell }(\rho, \phi)& = \sqrt{\frac{2 p !}{\pi(| \ell |+p)!}} \frac{1}{w_0}
\left(\frac{\sqrt{2}\rho}{w_0}\right)^{| \ell |} L^{| \ell |}_p\left(\frac{2 \rho^2}{w_0^2}\right)\nonumber \\ &\qquad\exp{(-\rho^2/w_0^2)}\exp{({\rm i\,} \ell \phi)}\,, \end{align} where $L^{ \ell }_p(\rho)$ are the associated Laguerre polynomials, \begin{equation}
L^{| \ell |}_p(\rho)=\sum_{m=0}^p (-1)^m \frac{(| \ell |+p)!}{(p-m)!(| \ell |+m)!m!}\rho^m\,. \end{equation} Here, $p$ is the number of non-axial radial nodes, and the index $ \ell $, the winding number, describes the helical structure of the wave front and the number of times the phase jumps occur as one goes around the beam path in the azimuthal direction. In general, any mode function $\psi(\rho,\phi,z)$ can be expanded into LG modes as \begin{equation} \psi(\rho,\phi,z)=\sum_{ \ell =-\infty}^{\infty}\sum_{p=0}^{\infty} A_{ \ell p} \mbox{LG}^{ \ell }_p(\rho, \phi,z) \end{equation} For further discussion we consider only pure LG modes with a definite charge $\ell$ and $p=0$, we denote such a state of the light field by $\ket{\ell}$ such that $
\braket{\mathbf r}{\ell} = \mbox{LG}^{\ell}_0(\rho, \phi).
$ Thus it can be easily seen that the states $\ket{+\ell}$ and $\ket{-\ell}$ (with $\ell$ being a whole number) differ only in the sense of the winding of the phase; either clockwise or counter-clockwise.
Now we discuss the first step of the proposal: creation of an arbitrary superposition of two OAM states of light of the kind $(\alpha\ket{\ell}+\beta\ket{\ell^\prime})/\sqrt{\alpha^2+\beta^2}$. First we discuss generation of a special kind of superposition state: $(\alpha\ket{\ell}+\beta\ket{-\ell})$, with $\alpha^2+\beta^2=1$. Here the two components of the state have opposite sense of rotation. It is well known that creation of a very general multicomponent superposition of OAM states of the kind $\sum_{\ell} c_{\ell} \ket{\ell}$ is a fairly straightforward procedure by using holographs or phase plates. Furthermore, a sorter of these OAM states has also been demonstrated~\cite{Leach:2002}, which distinguishes between different orbital angular momentum components. Thus, by using a mixed OAM-state generator in conjunction with an OAM-sorter one can easily obtain a pure OAM state with arbitrary $\ket{\ell}$.
Next, we describe a technique that could be used to transform a $\ket{+\ell}$ state in to a $\ket{-\ell}$ state with $\ell$ being a whole number. We illustrate this schematically in Fig.~\ref{Fig:DovePrism}. A common representation of the OAM states of light is through the Laguerre-Gaussian beams which have circular cross-sections. We consider two rays (red and blue) as seen in Fig.~\ref{Fig:DovePrism} that lie diametrically opposite to each other on the beam profile. As can be easily shown from the ray-diagram these rays would interchange their places in the beam profile~\cite{Courtial:1997,Padgett:1999}. Extrapolating this to the whole beam it can be seen that passing a OAM state of light through a dove prism amounts to a anti-symmetrization of its azimuthal phase structure thus giving rise to a $\ket{-\ell}$ state. It is also important to note, as pointed out by Padgett {\it et al.}~\cite{Padgett:1999}, that this rotation of the OAM state does not affect the polarization state of the light. To obtain complete conversion in the handedness of the input OAM, via the Dove prism, further care may be necessary as pointed out in Ref.~\cite{Gonzalez:2006}. \begin{figure}
\caption{ Dove prism as a sign shifter for the OAM states of light. The dove prism changes the handedness of the incident beam carrying OAM and causes a sign change in its winding number.}
\label{Fig:DovePrism}
\end{figure}
Thus, starting with a $\ket{+\ell}$ state of light and passing it through a Mach-Zehnder type configuration shown in Fig.~\ref{Fig:OAMS}, we obtain $(\ket{\ell}+\ket{-\ell})/\sqrt{2}$ at one of the output ports of the interferometer. By choosing the first beam splitter in the Mach-Zehnder interferometer as an $\alpha/\beta$ beam splitter and the second one is a 50/50 beam splitter one can generate a general two-component superposition $(\alpha \ket{\ell}+\beta \ket{-\ell})/\sqrt{|\alpha|^2 + |\beta|^2}$. \begin{figure}
\caption{ Scheme for creation of superposition of the OAM states: $(\ket{+\ell}+\ket{-\ell})/\sqrt{2}$.}
\label{Fig:OAMS}
\end{figure}
With the mathematical details discussed at length in Appendix~\ref{App:MachZehnder}, the process can be described as follows. The initial state of the light field before entering the first beam splitter is $\ket{\ell}$. After the passage through the first beam splitter, the beam is equally split into two paths and the state is $\alpha \ket{\ell}_1 + \beta\ket{\ell}_2$. Beam 1 is the part of the beam that passes through the beam splitter and beam 2 is the one that is reflected. Beam 2 later passes through the dove prism to undergo the transformation $\ket{\ell}\rightarrow\ket{-\ell}$ as discussed above. Thus, just before entering the second beam splitter the state of the light field is $\alpha\ket{\ell}_1 + \beta\ket{-\ell}_2$, which becomes $\alpha\ket{\ell} + \beta\ket{-\ell}$ after the recombination in the second beam splitter. The complete Mach-Zehnder interferometer can be described via the transformation: \begin{equation} \(\begin{matrix} u_0 \ket{\ell} \\ 0 \end{matrix}\)\rightarrow\(\begin{matrix} r^2 u_0 \ket{-\ell} - t^2 {\rm e}^{{\rm i\,} \phi} u_0\ket{\ell} \\ {\rm i\,} t\,r \, u_0 \ket{-\ell} + {\rm i\,} \, r\,t {\rm e}^{{\rm i\,} \phi}\, u_0\ket{\ell} \end{matrix}\) \,. \end{equation} With the choice of 50-50 beam splitters we have $r=t=1/\sqrt{2}$ and $\phi=\pi$ to obtain the state at the output ports of the interferometer \begin{equation} \frac{1}{\sqrt{2}}\,u_0\(\begin{matrix} \ket{-\ell} + \ket{\ell} \\ {\rm i\,}(\ket{-\ell} - \,\ket{\ell}) \end{matrix}\) \end{equation} Thus, by ignoring the port 2 and using the state from port 1 we obtain the required superposition state.
To generate arbitrary superposition of the kind $(\alpha \ket{\ell} + \beta \ket{\ell'})/\sqrt{\alpha^2 + \beta^2}$ scheme in Fig.~\ref{Fig:OAMS} needs to be changed slightly: (i) the input state needs to be a normal Gaussian beam with no orbital angular momentum. (i) the dove prism needs to be replaced by a hologram to transfer the gaussian beam into the state $\ket{\ell'}$ and (iii) an extra hologram is added into the lower path to obtain conversion from the Gaussian beam into the state $\ket{\ell}$.
A superposition of two OAM states can be detected via a photo-detection scheme by taking a look at the spatial profile of the characteristic interference of the beams that are part of the superposition. For example, the interference pattern created by the superposition of $\ket{\ell}$ and $\ket{-\ell}$ is comprised of $2 \ell$ bright lobes that are equally distributed along the azimuthal direction. The interference pattern of arbitrary superposition would contain $|\ell_1|+|\ell_2|$ lobes where $\ell_1$ and $\ell_2$ are the component OAM states. Thus the observation of the described interference pattern guarantees the creation of the coherent superposition of the appropriate states. Further details offered in Sec.~\ref{Sec:Detection} for detection of BEC vortex superposition can be applied here as well as the phase structures of the OAM superposition in light and vortex superposition in BEC are identical.
\section{\label{Sec:OAMCoupling}Coupling of optical vortex beams to BEC} In this section we discuss two methods to transfer the generated superposition of optical orbital angular momentum states to the states of atoms. Transfer of pure OAM states of light to BEC has been studied by Marzlin {\it et al.}~\cite{Marzlin:1997} and Nandi {\it et al.}~\cite{Nandi:2004}. Nevertheless, for the transfer of OAM superpositions one needs a special transfer scheme. The internal level scheme of the atoms and the transitions of interest are shown in the Fig.~\ref{Fig:LevelScheme}. The internal hyperfine quantum number of the initial non-rotating state $\ket{0}$ is $m_F=0$ and that of the final states $\ket{+}$ and $\ket{-}$ is $m_F=2$. Thus, the internal quantum number of both the final states is chosen to be the same to obtain a pure superposition of the vortex states while all rest of the quantum numbers needed to describe the two states are exactly identical. The intermediate states, $\ket{i}$ and $\ket{i'}$, through which Raman coupling between the initial and final states occurs, have the hyperfine quantum number $m_F=1$. The two components of the optical OAM states correspond to the Rabi frequencies $\Omega_+$ and $\Omega_-$ and both are $\sigma_+$ polarized. The coupling field is designated by the Rabi frequency $\Omega_c$ and it is $\sigma_-$ polarized. The OAM of the state $\ket{0}$ is zero and that of $\ket{i}$ and $\ket{+}$ is $+\ell$ and that of $\ket{i'}$ and $\ket{-}$ is $-\ell$, where $\ell$ is a positive integer. Even though we are targeting generation of superposition of two counterrotating components, it can be quickly seen that this scheme can also be used as is for transfer of arbitrary superposition of two arbitrary OAM states of light to the BEC cloud. \begin{figure}
\caption{Internal level scheme of the trapped BEC atoms and the applied fields. The fields are specified through the corresponding Rabi frequencies and the polarization state. The fields are highly detuned from the excited states to avoid populating the excited states and ensure the stability of the condensate. The two-photon detunings $\delta$ are needed for one method of transfer of population from the non-rotating state to the vortex states but needs to be zero for the STIRAP like transfer mechanism.}
\label{Fig:LevelScheme}
\end{figure}
Due to the various possibilities available for the internal state quantum numbers and the OAM quantum numbers there are five distinct states that the atoms in the BEC can have as a result of their interaction with the optical fields. Thus, the BEC cloud can be described through a five-component spinor $\{\Psi_0, \Psi_+, \Psi_-, \Psi_i, \Psi_{i'} \}$. All the internal states are assumed to be trapped by the trapping potential and thus there are mutual interactions among all the components. The optical fields couple various components of the spinor BEC, as shown in Fig.~\ref{Fig:LevelScheme}. Thus, we can write the evolution equations for the spinor components \begin{align} {\rm i\,}\, \dot{\Psi}_0 &= \Omega_+^* \Psi_{i} + \Omega_-^* \Psi_{i^\prime} + (\mathcal{H}_I/\hbar)\, \Psi_0 \nonumber \\ {\rm i\,} \dot{\Psi}_+ &= \Omega_c^* \Psi_{i} + (\mathcal{H}_I/\hbar)\, \Psi_+ \nonumber \\ {\rm i\,}\, \dot{\Psi}_- &= \Omega_c^* \Psi_{i^\prime}+ (\mathcal{H}_I/\hbar)\, \Psi_- \nonumber \\ {\rm i\,}\, \dot{\Psi}_{i} &= - \Delta \Psi_i + \Omega_c \Psi_{+} + \Omega_+ \Psi_{0} + (\mathcal{H}_I/\hbar)\, \Psi_{i} \nonumber \\ {\rm i\,}\, \dot{\Psi}_{i^\prime} &= - \Delta \Psi_{i^\prime} + \Omega_c \Psi_{-} + \Omega_- \Psi_{0} + (\mathcal{H}_I/\hbar)\, \Psi_{i^\prime}\,. \label{Eq:Dynamics} \end{align} It is important to note the time derivatives appearing on the left-hand side of the equations. In general, the spinor components have both spatial and time dependence; nevertheless, for the optical evolution the time dependence is of prime importance. The optical coupling is governed by the Rabi frequencies $\Omega_c$ (for the coupling field) and $\Omega_{\pm}$ (for the OAM superposition). Here, $\Delta$ is the single-photon detuning. In setting up the above equations we have assumed that the two fields are in two photon resonance with the transitions $\ket{0}-\ket{+}$ and $\ket{0}-\ket{-}$, meaning the parameter $\delta$ appearing in Fig.~\ref{Fig:LevelScheme} is zero. This detuning can be added later on, if needed, as will become clear later. Furthermore, \begin{equation} \label{HI}
\mathcal{H}_I = (\mathcal{T} + \mathcal{V} -\mu ) + \eta ( |\Psi_0|^2 + |\Psi_+|^2 + |\Psi_-|^2 +|\Psi_{i}|^2 + |\Psi_{i^\prime}|^2 ) \end{equation} where $\mathcal{T}$ is the Kinetic energy operator, $\mathcal{V}$ is the trapping potential of the BEC and $\mu$ is the chemical potential. The interatomic interaction strength is given by $\eta = 4 \pi \hbar ^{2} a N/m = N U$, where $a$ is the s-wave scattering length, $N$ is the mean number of atoms in the cloud, and $m$ is the atomic mass of the atom. The absolute square terms in Eq.(\ref{HI}) thus correspond to the self-energy arising due to the interparticle interaction.
Noting the spatial profile of the LG beams, we obtain \begin{equation}
\Omega_{\pm} ({\mathbf r}) = a_{\pm} \Omega_0\, {\rm e}^{-r^2/w^2} \( \frac{\sqrt{2} r}{w}\)^{|\ell|} {\rm e}^{\pm{\rm i\,} |\ell| \phi } {\rm e}^{{\rm i\,} k z} \end{equation}
with $\Omega_0$ being the Rabi frequency of the atom-field interaction. $a_\pm$ are the probability amplitudes of the $\pm |\ell |$ vortices in the superposition state we want to create. We assume that the Gaussian fall-off of the light intensity is over a length larger than the transverse size of the BEC cloud. As a result, we can ignore the gaussian term in the LG beam profile to arrive at \begin{equation}
\Omega_{\pm} = a_{\pm}(\sqrt{2})^{|\ell |} \Omega_0\,\( \frac{x\pm {\rm i\,} y}{w}\)^{|\ell|} {\rm e}^{{\rm i\,} k z}\,. \end{equation} Note that we have incorporated the phase term inside the complex quantity $x\pm{\rm i\,} y= r \cos\phi \pm {\rm i\,} \sin \phi= r{\rm e}^{\pm {\rm i\,}\phi}$. Similarly the gaussian fall-off of the intensity of the coupling beam (or $\Omega_c$) along the transverse direction is ignored as the BEC cloud size is much smaller than the transverse beam profile.
Now, we perform complete adiabatic elimination of the excited levels $\ket{i}$ and $\ket{i'}$. We are completely justified in using the adiabatic elimination procedure as: (i) the optical evolution happens on a time scale much faster than other processes in the BEC; (ii) the optical fields are so detuned from the excited states that the excited states are never occupied. Thus, by substituting $\dot{\Psi}_{i}=\dot{\Psi}_{i'}=0$ and eliminating $\Psi_{i}$ and $\Psi_{i'}$ from the Eq.~\eqref{Eq:Dynamics} we arrive at \begin{widetext} \begin{subequations} \label{Eq:psidot} \begin{align} \label{Eq:psi0dot} {\rm i\,} \dot\Psi_{0} &= \frac{1}{\hbar}(\mathcal{T}+\mathcal{V}-\mu)\Psi_0
+\frac{\eta}{\hbar} ( |\Psi_0|^2 + |\Psi_+|^2 + |\Psi_-|^2 ) \Psi_0
+ \frac{1}{\Delta}(|\Omega_+|^2 + |\Omega_-|^2)\Psi_0
+ \frac{1}{\Delta}\Omega_+^* \Omega_c \Psi_+
+ \frac{1}{\Delta}\Omega_- ^*\Omega_c \Psi_- \\ \label{Eq:Psiplusdot} {\rm i\,} \dot\Psi_{+} &= \frac{1}{\hbar}(\mathcal{T}+\mathcal{V}-\mu)\Psi_+
+\frac{\eta}{\hbar} (\Psi_+|^2 + |\Psi_-|^2 + |\Psi_0|^2 ) \Psi_+
+ \frac{1}{\Delta}|\Omega_c|^2\Psi_+
+\frac{1}{\Delta}\Omega_+\Omega_c^* \Psi_0 \\ \label{Eq:Psiminusdot} {\rm i\,} \dot\Psi_{-} &= \frac{1}{\hbar}(\mathcal{T}+\mathcal{V}-\mu)\Psi_{-}
+ \frac{\eta}{\hbar} ( |\Psi_+|^2 + |\Psi_-|^2
+|\Psi_0|^2 ) \Psi_- + \frac{1}{\Delta}|\Omega_c|^2\Psi_-
+ \frac{1}{\Delta}\Omega_-\Omega_c^* \Psi_0 \end{align} \end{subequations} \end{widetext}
Our ultimate interest lies in the temporal dynamics of the populations of various components of the BEC. Thus, the spatial part of the condensate is to be integrated out of the above equations. To deal with the spatial profiles of the spinor components of the BEC explicitly, we note that the general state of the BEC cloud can be written as \begin{align} \braket{\mathbf r}{\Psi} = \alpha(t) \braket{\mathbf r}{0} + \beta(t) \braket{\mathbf r}{+} + \gamma(t) \braket{\mathbf r}{-} \end{align} where the spatio-temporal projections are given by \begin{align} \Psi_{0}({\mathbf r}, t) &= \alpha(t) \braket{\mathbf r}{0} = \alpha(t) \exp[{\rm i\,}(\mu/\hbar - \kappa)t] \,\psi_{\rm g}({\mathbf r}) \nonumber \\ \Psi_{+}({\mathbf r}, t) &= \beta(t) \braket{\mathbf r}{+} = \beta(t)\exp[{\rm i\,}(\delta+ \mu/\hbar - \kappa)t] \, \psi_{\rm v+}(\ell,{\mathbf r})\nonumber \\ \Psi_{-}({\mathbf r},t) &= \gamma(t) \braket{\mathbf r}{-} = \gamma(t)\exp[{\rm i\,}(\delta+ \mu/\hbar - \kappa)t] \, \psi_{\rm v-}(-\ell,{\mathbf r})\,. \label{Eq:Ansatz} \end{align} The two-photon detuning $\delta$ introduced in the phase factors is defined as $\delta = \nu_{\pm} - \nu_{c} - \omega_{0\pm}$, where $\nu_{\pm}$ and $\nu_c$ are the angular frequencies of the optical field carrying OAM and the coupling field respectively, and $\omega_{0\pm}$ is the energy level difference between the states $\ket{0}$ and $\ket{\pm}$ expressed as an angular frequency. Furthermore, the parameter $\kappa$, in general, depends on the interaction strength $\eta$ between the BEC atoms and the dimensional parameters of the BEC cloud, which in tern depend on the trapping potential. For example, for the Harmonic trapping potential $\kappa$ is given by \begin{equation} \kappa =\frac{\pi \hbar a_{\rm sc} N}{m (2 \pi)^{3/2} \hbar~ L_{\perp}^2 L_z} =\frac{\eta}{4 (2 \pi)^{3/2} \hbar~ L_{\perp}^2 L_z}\,. \end{equation} It can, however, be noted that as we are dealing with condensates consisting of single species, $\kappa$ need not be explicitely incorporated in the phase factors as it just causes a shift in the energy equally for all the component states of the BEC cloud and does not affect the population dynamics in a non-trivial manner.
The ansatz for the spatial profiles of the non-rotating ground state and the vortex states depend on the type of trapping potential we choose. Before discussing different trapping potentials we take our general formalism a little further. After substituting the ansatz~\eqref{Eq:Ansatz} into the equations~\eqref{Eq:psidot} we obtain the temporal dynamics of the populations of the BEC components of interest: \begin{widetext} \begin{align} \[ {\rm i\,}\, \dot\alpha(t)- (\frac{\mu}{\hbar}- \kappa)\, \alpha(t)\right]\psi_{\rm g}(\mathbf{r})& = \frac{1}{\hbar}(\mathcal{T} +\mathcal{V} - \mu)\,\alpha(t)\, \psi_{\rm g}(\mathbf{r}) \nonumber \\
&+\frac{\eta}{\hbar} (|\alpha(t)|^2 |\psi_{\rm g}(\mathbf{r})|^2+ |\beta(t)|^2 |\psi_{\rm v+}(\ell,\mathbf{r})|^2 + |\gamma(t)|^2 |\psi_{\rm v-}(-\ell,\mathbf{r})|^2)\,\alpha(t)\, \psi_{\rm g}(\mathbf{r}) \nonumber \\
& + \frac{1}{\Delta}|\Omega_0|^2 \(\frac{\sqrt{2}r}{w}\)^{2\ell} \alpha(t)\, \psi_{\rm g}(\mathbf{r}) \nonumber \\ & + \frac{1}{\Delta} (a_+^* {\rm e}^{-{\rm i\,} \ell \phi } \,\beta(t) \,\psi_{\rm v+}(\ell,\mathbf{r}) + a_-^*{\rm e}^{{\rm i\,} \ell \phi }\, \gamma(t)\, \psi_{\rm v-}(-\ell,\mathbf{r})]\,\(\frac{\sqrt{2} r}{w}\)^{\ell}\Omega_0^*\, \Omega_c\,, \nonumber \\ \[ {\rm i\,} \, \dot\beta(t) - (\frac{\mu}{\hbar}- \kappa+\delta)\, \beta(t)\right] \psi_{\rm v+}(\ell,\mathbf{r})& = \frac{1}{\hbar}(\mathcal{T}+\mathcal{V}-\mu)\beta(t)\, \psi_{\rm v+}(\ell,\mathbf{r}) \nonumber \\
&+ \frac{\eta}{\hbar} \Bigl[|\alpha(t)|^2 |\psi_{\rm g}(\mathbf{r})|^2\, + |\beta(t)|^2 |\psi_{\rm v+}(\ell,\mathbf{r})|^2+ |\gamma(t)|^2 |\psi_{\rm v-}(-\ell,\mathbf{r})|^2 \Bigr]\psi_{\rm v+}(\ell,\mathbf{r})\,\beta(t) \nonumber \\
&+ \frac{1}{\Delta} |\Omega_c|^2\,\beta(t)\, \psi_{\rm v+}(\ell,\mathbf{r}) + \frac{1}{\Delta} a_+ {\rm e}^{{\rm i\,} \ell \phi } \(\frac{\sqrt{2} r}{w}\)^{\ell} \Omega_0\, \Omega_c^*\, \alpha(t) \,\psi_{\rm g}(\mathbf{r})\,, \nonumber \\ \[ {\rm i\,} \,\dot\gamma(t) - (\frac{\mu}{\hbar}\,-\kappa+\delta)\, \gamma(t)\, \right]\psi_{\rm v-}(-\ell,\mathbf{r})&= \frac{1}{\hbar}(\mathcal{T}+\mathcal{V}-\mu)\gamma(t)\, \psi_{\rm v-}(-\ell,\mathbf{r}) \nonumber \\
&+ \frac{\eta}{\hbar} \Bigl[|\alpha(t)|^2 |\psi_{\rm g}(\mathbf{r})|^2+ |\beta(t)|^2 |\psi_{\rm v+}(\ell,\mathbf{r})|^2 + |\gamma(t)|^2 |\psi_{\rm v-}(-\ell,\mathbf{r})|^2\Bigr]\,\psi_{\rm v-}(-\ell,\mathbf{r})\, \gamma(t) \nonumber \\
&+ \frac{1}{\Delta} |\Omega_c|^2\,\gamma(t)\, \psi_{\rm v-}(-\ell,\mathbf{r}) + \frac{1}{\Delta} a_- {\rm e}^{-{\rm i\,} \ell \phi } \(\frac{\sqrt{2} r}{w}\)^{\ell} \Omega_0\, \Omega_c^*\, \alpha(t) \,\psi_{\rm g}(\mathbf{r})\,. \end{align} \end{widetext} Note that the exponential factors $\exp[{\rm i\,}(\mu/\hbar - \kappa)t]$ and $\exp[{\rm i\,}(\delta + \mu/\hbar - \kappa)t] $ have canceled as expected. Also note that $\ell$ is taken to be a positive integer and $\pm$ signs are used explicitly to easily distinguish the two counter-rotating vortex components; we will follow this convention from here onwards to avoid excessive use of the modulus sign. Now we formally perform the coordinate integrals to remove the spatial dependence. In terms of the resulting integrals, which are summarized in Appendix B, the rate equations are \begin{widetext} \begin{align}
{\rm i\,} \dot{\alpha}(t) &= (T_{g}+ V_{g}) \alpha(t) +(I_{gg} |\alpha(t)|^2 +I_{g+}(\ell) |\beta(t)|^2+I_{g-}(\ell) |\gamma(t)|^2)\alpha(t)
+ \frac{ |\Omega_0|^2}{\Delta}I_{gg}^{(2\ell)}(\ell)~ \alpha(t) \nonumber \\ &\quad+ \frac{ \Omega_0^*\, \Omega_c}{\Delta}\(I_{g+}^{(\ell)} (\ell)~ a_{+}^* \beta(t) + I_{g-}^{(\ell)} (\ell)~ a_{-}^* \gamma(t)\)\,, \nonumber \\
{\rm i\,} \dot{\beta}(t) &= [T_{+}+V_{+}(\ell)+\delta] \beta(t) + (I_{g+}(\ell)~|\alpha(t)|^{2} +I_{++}(\ell)~|\beta(t)|^2 +I_{+-}(\ell)~|\gamma(t)|^2 )\beta(t)+
\frac{ |\Omega_c|^2}{\Delta} \beta(t) + \frac{\Omega_0\, \Omega_c^*}{\Delta} I_{+g}^{(\ell)}(\ell)~ a_{+}\, \,\alpha(t)\,,
\nonumber \\ {\rm i\,} \dot{\gamma}(t) &=[T_{-}+V_{-}(\ell)+\delta] \gamma(t) +
(I_{g-}(\ell) |\alpha(t)|^{2} +I_{+-}(\ell) |\beta(t)|^2 +I_{--}|\gamma(t)|^2 )\gamma(t)+ \frac{ |\Omega_c|^2}{\Delta}\gamma(t) + \frac{\Omega_0\, \Omega_c^*}{\Delta} I_{-g}^{(\ell)}(\ell)~ a_{-}\,\alpha(t)\,. \label{Eq:RateEqBasicFormal} \end{align} \end{widetext} Note that the spatial integrals are denoted by the letter $I$ with different subscripts, superscripts and in most cases with the argument of $\ell$ to properly distinguish them. These integrals are explicitly written out in the Appendix B.
To note, the above set of equations is very general and applicable to wide variety of trapping potentials, provided the integrals are evaluated by appropriately taking the spatial forms of the wavefunctions, $\psi_g(\mathbf{r}), \psi_{v\pm}(\mathbf{r})$, suitable for the trapping potential. Furthermore the equations can also be applied to situations where the applied OAM and coupling fields have a certain time profile. In case the spatial ansatz for the BEC wavefunction in the Thomas-Fermi limit then the kinetic energy terms can be ignored as we do in the case of the mexican hat potential. The application of these equations shall become clear as we discuss various cases below.
In the forthcoming subsection we look at two different trapping potentials and two different transfer mechanisms of OAM to a BEC.
\subsection{Harmonic Potential Trap}
The harmonic potential we consider is of the form, \begin{equation} V(x, y, z) = \frac{1}{2} m [\omega_{\perp}^{2} (x^{2}+y^{2}) + \omega_{z} z^{2}]\,. \end{equation} We assume a pancake shaped BEC cloud for which $\omega_{\perp} <\omega_{z}$ and as a result, $L_{\perp} > L_z$. The spatial wavefunctions for the BEC trapped in this kind of trap can be taken to be, \begin{align} \psi_g({\mathbf r})&= \exp{\{-(1/2)[(r/L_{\perp})^2 + (z/L_{z})^2 ] \}}/(\pi^{3/4} L_{\perp} L_{z}^{1/2})\nonumber \\
\psi_{v\pm}(\pm\ell,{\mathbf r})&=(x\pm {\rm i\,} y)^{|\ell|} \psi_g({\mathbf r})/ (\sqrt{|\ell |!} \, L_{\perp}^{|\ell |} )\nonumber \\ &= r^{|\ell|} e^{\pm {\rm i\,} \ell \phi} \psi_g({\mathbf r})/ (\sqrt{|\ell |!} \, L_{\perp}^{|\ell |} )\ . \end{align} The treatment, so far, is very general and would work for any value $\ell$ of the OAM of the incident light. To understand the dynamics more clearly we restrict ourselves to a particular value of the OAM, $\ell=2$. However, one may note that even though the spatial integrals would have different values for different $\ell$ values, the general idea remains the same.
We start with Eq.~\eqref{Eq:RateEqBasicFormal} and substitute the spatial integrals for the particular case of the 3D harmonic trap, which are listed in Appendix B, to arrive at: \begin{widetext} \begin{align}
{\rm i\,}\dot{\alpha}(t)&= -\kappa\, \alpha(t) + (\frac{1}{4}\omega_z + \frac{1}{2} \omega_{\perp} ) \alpha(t) + \frac{1}{2} \omega_{\perp} \alpha(t) + 3 \kappa |\alpha(t)|^2 \alpha(t) + \kappa\(|\alpha(t)|^2 + |\beta(t)|^2 + |\gamma(t)|^2 \) \alpha(t) \nonumber \\
&+ \frac{8}{\Delta}|\Omega_0|^2
\(\frac{\hbar}{m \omega_{\perp} w^2}\)^2 \alpha(t) +\frac{2 \sqrt{2}}{\Delta}\Omega_0^* \Omega_c \frac{\hbar }{m \omega_{\perp}w^2}( a_{+}^* \beta(t) + a_{-}^*\gamma(t))\,, \nonumber \\
{\rm i\,} \dot{\beta}(t)&= (-\delta - \kappa)\beta(t) + (\frac{1}{4}\omega_z + \frac{3}{2} \omega_{\perp} ) \beta(t) + \frac{3}{2} \omega_{\perp} \beta(t) + \kappa (|\alpha(t)|^2 + |\beta(t)|^2 +|\gamma(t)|^2 ) \beta(t)
+ \frac{1}{2}\kappa ( |\beta(t)|^2 + |\gamma(t)|^2 )\beta(t) \nonumber \\
&+ \frac{1}{\Delta} |\Omega_c|^2 \beta(t) + \frac{2\sqrt{2}}{\Delta} \Omega_{0}\Omega_{c}^* a_{+} \frac{\hbar}{m \omega_{\perp}w^2}\alpha(t)\,,\nonumber \\
{\rm i\,} \dot{\gamma}(t)&= (-\delta - \kappa)\gamma(t) + (\frac{1}{4}\omega_z + \frac{3}{2} \omega_{\perp} ) \gamma(t) + \frac{3}{2} \omega_{\perp} \gamma(t) + \kappa (|\alpha(t)|^2 + |\beta(t)|^2 +|\gamma(t)|^2 ) \gamma(t) + \frac{1}{2}\kappa ( |\beta(t)|^2 + |\gamma(t)|^2 )\gamma(t) \nonumber \\
&+ \frac{1}{\Delta} |\Omega_c|^2 \gamma(t) + \frac{2\sqrt{2}}{\Delta} \Omega_{0}\Omega_{c}^* a_{-} \frac{\hbar}{m \omega_{\perp}w^2}\alpha(t)\,. \label{Eq:RateEqBasic} \end{align} \end{widetext}
Note the appearance of the Harmonic oscillator trap parameters. At this stage, we note that, $|\alpha(t)|^2 + |\beta(t)|^2 +|\gamma(t)|^2=1$, and eliminate terms that are common in all the equations as they do not give any non-trivial contributions to the population dynamics of the system. We also add an extra assumption to relate the properties of the BEC trap and the profile of the OAM carrying light beam such that ${2 \sqrt{2} \hbar}/({m \omega_{\perp} w^2})=1$. Note that this only scales the optical coupling constant appearing in the equations to a manageable number and does not change the physics in any way. Thus, we arrive at a general set of equations coupling a non-rotating state of BEC to two counter-rotating vortex states carrying charges $2$ and $-2$ respectively: \begin{widetext} \begin{align}
{\rm i\,}\dot{\alpha}(t)&= 3 \kappa |\alpha(t)|^2 \alpha(t) + \frac{1}{\Delta}|\Omega_0|^2 \alpha(t) +\frac{1}{\Delta}\Omega_0^* \Omega_c ( a_{+}^* \beta(t) + a_{-}^*\gamma(t))\,, \nonumber \\ {\rm i\,} \dot{\beta}(t)&= (\delta+ {2} \omega_{\perp}) \beta(t)
+ \frac{1}{2}\kappa ( |\beta(t)|^2 + |\gamma(t)|^2 )\beta(t) + \frac{1}{\Delta} |\Omega_c|^2 \beta(t) + \frac{1}{\Delta} \Omega_{0}\Omega_{c}^* a_{+}\alpha(t)\,,\nonumber \\ {\rm i\,} \dot{\gamma}(t)&= (\delta + {2} \omega_{\perp}) \gamma(t)
+ \frac{1}{2}\kappa ( |\beta(t)|^2 + |\gamma(t)|^2 )\gamma(t)
+ \frac{1}{\Delta} |\Omega_c|^2 \gamma(t) + \frac{1}{\Delta} \Omega_{0}\Omega_{c}^* a_{-} \alpha(t)\,. \label{Eq:RateEqBasicHT} \end{align} \end{widetext} Starting with these equations, we develop two methods for transfer of optical OAM to a BEC. It needs to be noted that the dynamics, as seen in the above equations, cannot have a steady state due to large detunings from the excited states and exclusion of radiative decays. This regime of parameters is, of course, necessary to make sure that the BEC atoms do not populate the excited state, where the BEC may not survive. Thus, to obtain the complete population transfer in a robust manner we need to construct dynamics that would not cause the population to oscillate between the initial state and the final states but would rather show a one-way trend in the direction the population is moving. This, of course, could be done in both a time-reversal symmetric or non-symmetric manner. We study both these avenues. The non time-reversal symmetric scheme involves linear chirp of the coupling field, meaning a time dependent two-photon detuning $\delta$. The time-reversal symmetric scheme is based on the traditional STIRAP population transfer mechanism. To note, the final state is not a single state but a superpositon of two states. We consider these two techniques one by one in the following subsections.
\subsubsection{Superposition of Vortices in BEC via frequency chirp} \forget{and introduce a few simplifying assumptions, without loss of generality, as given below: \begin{equation}
\Omega_{c} = \Omega_{0} \frac{2 \sqrt{2} \hbar}{m \omega_{\perp} w^2}\, \text{ and } \omega_{\perp} = \sqrt{\frac{\hbar}{m \Delta}}\frac{|\Omega_0|}{w}\, \text{ and } \frac{\hbar}{m \omega_{\perp}} = L_{\perp}^2\,, \label{Eq:Sim1} \end{equation} The first of the assumptions relates the relative intensities of the coupling field and the OAM carrying field. The second one relates the laser beam parameters, beam waist size and the field amplitude, with the transverse trapping frequency. It is imperative to point out that these assumptions are in no way restrictive and are introduced only as a matter of convenience. Using relations in Eq.~\eqref{Eq:Sim1} we obtain \begin{equation}
\frac{2\sqrt{2}}{\Delta} \Omega_{0}\Omega_{c}^* \frac{\hbar}{m \omega_{\perp} w^2} =\frac{8}{\Delta} |\Omega_0|^2 \( \frac{\hbar}{m \omega_{\perp} w^2}\)^2 = 8\, \omega_{\perp}\(\frac{L_{\perp}}{w}\)^2\,. \label{Eq:Sim2} \end{equation} To recollect, here $L_{\perp}$ is the spatial extent of the BEC cloud in the transverse $x$-$y$ direction and $\omega_{\perp}$ is the corresponding trapping frequency.
With the above-mentioned simplifications, the temporal dynamics takes the form \begin{widetext} \begin{align}
{\rm i\,} \dot{\alpha}(t) &= \(\frac{1}{4}\omega_z + \omega_{\perp} + 8 \, \omega_{\perp}\(\frac{L_{\perp}}{w}\)^2\)\alpha(t) + 3 \kappa |\alpha(t)|^2 \alpha(t) + 8 \, \omega_{\perp}\(\frac{L_{\perp}}{w}\)^2 (a_{+}^* \beta(t) + a_{-}^* \gamma(t))\,, \nonumber \\ {\rm i\,} \dot{\beta}(t) &= \(\delta + \frac{1}{4}\omega_z + 3 \omega_{\perp} + 8 \, \omega_{\perp}\(\frac{L_{\perp}}{w}\)^2\)\beta(t) +
\frac{1}{2} \kappa (|\beta(t)|^2 +|\gamma(t)|^2 ) \beta(t) + 8 \, \omega_{\perp}\(\frac{L_{\perp}}{w}\)^2 a_{+}\, \alpha(t)\,,
\nonumber \\ {\rm i\,} \dot{\gamma}(t) &=\(\delta + \frac{1}{4}\omega_z + 3 \omega_{\perp} + 8 \, \omega_{\perp}\(\frac{L_{\perp}}{w}\)^2\)\gamma(t) +
\frac{1}{2} \kappa (|\beta(t)|^2 +|\gamma(t)|^2 ) \gamma(t) + 8 \, \omega_{\perp}\(\frac{L_{\perp}}{w}\)^2 a_{-}\, \alpha(t)\,. \end{align} \end{widetext} where, without loss of generality, we have related parameters of the laser beam waist and the transverse size of the BEC, i.e., $w = 2 \sqrt{2} L_{\perp}$. Starting with these equations, we develop two methods for transfer of optical OAM to a BEC. It needs to be noted that the dynamics, as seen in the above equations, cannot have a steady state due to large detunings from the excited states and no inclusion of radiative decays. This regime of parameters is necessary to make sure that the BEC does not go into the excited state, where it may not survive. Thus, to obtain the complete population transfer in a robust manner we need to construct dynamics that would not give population oscillation between the initial state and the final states, and there would be a one-way trend in the direction the population is moving. This of course could be done in both a time-reversal symmetric or non-symmetric manner. We study both these avenues. The non time-reversal symmetric scheme involves linear chirp of the coupling field. The time-reversal symmetric scheme is based on the traditional STIRAP population transfer mechanism. In the present scheme, however, the final state is not a single state but a superpositon of two states. We consider these two techniques one by one in the following subsections.
Now, using further small simplification without loss of generality on the size parameters of the laser beam waist and the transverse size of the BEC, i.e., $w = 2 \sqrt{2} L_{\perp}$, and eliminating the common, ground-state energy, we obtain}
To study the generation of the superposition of two counter-rotating vortices in BEC we start with Eq.~\eqref{Eq:RateEqBasicHT} and introduce two simplifications given by $\Omega_0=\Omega_c$ and $|\Omega_0|^2/\Delta = \omega_{\perp}$. There is no loss of generality in these assumptions and appearance of any other constant that related the light beam intensity parameter with the transverse trap frequency would demonstrate the same physics except the time scales may be a little different. Our aim here is, of course, to demonstrate that superposition of vortex states can be created in a robust manner despite the presence of inter-particle interactions as normally observed in BEC. The same formalism can be applied to the experimental situations to rigorously determine the time scales over which the population transfer occurs. In any case, the optical time evolution time scales ($\mu$s) are much shorter than the time scales for spatial evolution of the BEC (ms).
We obtain a simple set of equations governing the population of the three components of the BEC: \begin{eqnarray}
{\rm i\,} \dot{\alpha}(t) &=& 3 \kappa |\alpha(t)|^2 \alpha(t) + \omega_{\perp} \(a_{+}^* \beta(t) + a_{-}^* \gamma(t)\)\, \nonumber \\ {\rm i\,} \dot{\beta}(t) &=& (\delta + 2 \omega_{\perp})\,\beta(t) +
\frac{1}{2} \kappa \(|\beta(t)|^2 +|\gamma(t)|^2 \) \beta(t) \nonumber \\ & &+ \omega_{\perp}\, a_{+}\, \alpha(t)\,,
\nonumber \\ {\rm i\,} \dot{\gamma}(t) &=&\(\delta + 2 \omega_{\perp} \)\gamma(t) +
\frac{1}{2} \kappa \(|\beta(t)|^2 +|\gamma(t)|^2 \) \gamma(t) \nonumber \\ & &+ \omega_{\perp}\, a_{-}\, \alpha(t)\,. \end{eqnarray} The optical vortex superposition can be transferred to the BEC via a continuously chirped control pulse $\Omega_{c}$. The chirp is modeled by introducing linear time dependence in the two-photon detuning in the form $\delta(t)= C(1-\Omega_0 t)$, where $C$ is some appropriate constant. We discuss the solution of the above equation in Fig.~\ref{Fig:PlotChirp}.
\begin{figure}
\caption{ Generation of the vortex state superposition via the linear frequency chirp technique. An equal superposition of the component vortex states is generated here starting with a non-rotating state of the BEC. The time plotted on the x-axis is measured in seconds. The linear chirp used for complete population transfer is given by the two-photon detuning $\delta(t) = C (1-\Omega_{0} t)$, where $C=2 \Omega_0$ and $\Omega_{0} = 3$kHz. Other parameters are $\omega_{\perp}=132$ Hz, $a = 5$ nm, $L_{\perp}=2.35 \mu$m, $L_z=1.4\mu$m and $\kappa=1.7$kHz.}
\label{Fig:PlotChirp}
\end{figure}
The solid line in the plot is the transfer function \begin{equation}
F(t) = |\alpha(t)|^2 - |\beta(t)|^2 - |\gamma(t)|^2 \label{Eq:Ft} \end{equation}
which varies from the value 1, when all atoms are in state $\ket{0}$, to the value $-1$, when all the atoms are in state $(\sqrt{3}\ket{+2}+\sqrt{2}\ket{-2})/\sqrt{5}$, which is the required vortex superposition for 60:40 division of the population among the two vortex components. The population of state $\ket{0}$ is shown by a dashed line and it varies from 1 to 0; whereas the populations of the states $\ket{+2}$ and $\ket{-2}$ denoted by $|\beta(t)|^2$ and $|\gamma(t)|^2$ are given by dotted line which vary from 0 to 0.6 and 0.4 respectively, as expected. The coupling field frequency is to be varied linearly such that it sweeps from one side of the two-photon resonance to the other side to facilitate complete population transfer. For modeling purposes we vary the two-photon detuning $\delta$ so that the system sweeps through the two-photon resonance.
An alternative method we discuss below, to accomplish the OAM transfer, is similar to the traditional STIRAP technique of the counter-intuitive pulse sequence. This technique has been applied for the transfer of a pure OAM state to the BEC.
\subsubsection{BEC vortex superposition via STIRAP-like pulse sequence} \label{sec:STIRAP}
The only change needed here from the previous subsection is to make the Rabi frequencies time dependent and to deploy counter-intuitive pulse sequence that is required to obtain the population transfer.
We add the time dependence to the Rabi frequencies by via \begin{equation}
\Omega_0 = |\Omega_0| f(t) \quad \mbox{and}\quad \Omega_c = |\Omega_0| g(t)\,, \end{equation} where the temporal profiles of the beams are taken to be of the form: \begin{align} f(t) = f_0\, {\rm e}^{-\left(\frac{t-t_1}{\sigma_1}\right)^2}\,, \quad g(t) = g_0\, {\rm e}^{-\left(\frac{t-t_2}{\sigma_2}\right)^2}\,. \label{Eq:TimeProfile} \end{align} With these modifications and that $\delta=0$ the rate equations~\eqref{Eq:RateEqBasicHT} become
\begin{widetext} \begin{align}
{\rm i\,} \dot{\alpha}(t) &= \frac{1}{\Delta} |\Omega_0|^2 f(t)^2 \alpha(t) + 3 \kappa |\alpha(t)|^2 \alpha(t) + \frac{1}{\Delta} |\Omega_0|^2\,f(t)\,g(t)\, \(a_{+}^* \beta(t) + a_{-}^* \gamma(t)\)\,, \nonumber \\ {\rm i\,} \dot{\beta}(t) &= 2 \omega_{\perp}\,\beta(t) +
\frac{1}{2} \kappa \(|\beta(t)|^2 +|\gamma(t)|^2 \) \beta(t)
+ \frac{1}{\Delta} |\Omega_0|^2 g(t)^2 \,\beta(t)
+ \frac{1}{\Delta} |\Omega_0|^2\,f(t)\,g(t)\, a_{+}\, \,\alpha(t)\,, \nonumber \\
{\rm i\,} \dot{\gamma}(t) &=2 \omega_{\perp}\,\gamma(t) + \frac{1}{\Delta} |\Omega_0|^2 g(t)^2 \,\gamma(t)+
\frac{1}{2} \kappa \(|\beta(t)|^2 +|\gamma(t)|^2 \) \gamma(t)
+ \frac{1}{\Delta} |\Omega_0|^2\,f(t)\,g(t)\, a_{-}\,\alpha(t)\,. \label{Eq:STIRAP} \end{align} \end{widetext}
These equations can be solved in a straightforward manner. The typical results are summarized in Fig.~\ref{Fig:PlotSTIRAP}. It can be seen that the transfer function plot signifies a complete transfer from initial state $\ket{0}$ to a 60:40 superposition of the vortex states $\ket{+2}$ and $\ket{-2}$. Any combination of superpositions (50/50, 60/40 or 80/20) can be generated via the counter intuitive pulse sequence---the coupling-field pulse (time profile $g(t)$ and pulse center $t_2$) comes before the OAM superposition pulse (time profile $f(t)$ and pulse center $t_1$).
\begin{figure}
\caption{ Generation of the vortex state superposition: Results of the numerical solutions of the equations for STIRAP scheme shows the superposition 60:40 of the $\ket{+}$ and $\ket{-}$ vortex states.
The pulse profiles $f(t)$ and $g(t)$ are of the form in Eq.~\eqref{Eq:TimeProfile} with $\sigma_1=\sigma_2=0.25$ and $t_1=1.0$ and $t_2=0.5$ in the units of $1/\Omega_0$ with $f_0/g_0 = 0.5$ to obtain the complete transfer of the non-rotating BEC into the vortex superposition. (See Eq.~\eqref{Eq:STIRAP}.) Other parameters are $\Omega_0=2\times10^5$Hz, $\Delta=10 \Omega_0$ and $\delta=0$. The quantity on the $x$ axis is the scaled time $\Omega_0 t$.}
\label{Fig:PlotSTIRAP}
\end{figure}
To understand the pulse overlap necessary to obtain complete population transfer from the initial non-rotating state to the final vortex superposition state via the STIRAP process, we plot the transfer function $F(t)$ vs the distance between the pulse centers $t_1-t_2$ in Fig. \ref{Fig:TransferFunctionVsOverlap}. The value of $F(t)$ close to 1 means the population is primarily in the non-rotating ground state and the transfer is not efficient. Whereas, $F(t)=-1$ means complete population transfer to the vortex superposition has occurred. The pulse parameters are chosen to be exactly identical; therefore, the distance between the pulse center could be taken as the measure of the pulse overlap. For the light pulses arriving at about the same time the transfer is inefficient. However, the population transfer improves as the pulse separation increases a little. One obtains complete transfer for the range $\{0.3, 0.5\}$ for the separation between the pulses; beyond that the pulses are so far away from each other that the STIRAP process does not work. \begin{figure}
\caption{ Plot of the transfer function (Eq.~\eqref{Eq:Ft}) versus the distance between the pulse centers $t_1-t_2$. Note that $t_1-t_2>0$ (or $t_1>t_2$) suggests the counter-intuitive pulse sequence needed for the STIRAP based population transfer process. The other parameters are identical to those noted in the caption of the Fig.~\ref{Fig:PlotSTIRAP}}
\label{Fig:TransferFunctionVsOverlap}
\end{figure}
We would further like to point out that our numerical study suggests that within the parameter range used to obtain Figs.~\ref{Fig:PlotChirp} and ~\ref{Fig:PlotSTIRAP} the excited states populations are negligible and we are justified in using the adiabatic elimination of the excited states.
\forget{It is, however, important to study the population transfer process without the adiabatic approximation to check the validity of the . In the following discussion we study these equations: \begin{align} {\rm i\,} \hbar\, \dot{\alpha}(t)\, \psi_{g} +\hbar \kappa\, \alpha(t)\, \psi_{g} &= (T + V)\, \alpha(t)\, \psi_{g} +\eta\, \mathscr{A}\, \alpha(t) + \Omega_{+}^*\, i(t)\, \psi_{i} + \Omega_{-}^*\, i'(t)\,\psi_{i'} \nonumber \\ {\rm i\,} \hbar\, \dot{i}(t)\, \psi_{i} +(\hbar \kappa -\hbar \delta)\, i(t) \psi_{i} &= (T + V)\, i(t)\, \psi_{i} +\eta\, \mathscr{A}\, i(t) + \Omega_{+}\, \alpha(t)\,\psi_{g}+ \Omega_{c}\, \beta(t)\, \psi_{v+} \nonumber \\ {\rm i\,} \hbar\, \dot{i'}(t)\, \psi_{i'} +(\hbar \kappa-\hbar \delta)\, i'(t) \psi_{i'} &= (T + V)\, i'(t)\, \psi_{i'} +\eta\, \mathscr{A}\, i'(t) + \Omega_{-}\, \alpha(t)\,\psi_{g}+ \Omega_{c}\, \gamma(t)\, \psi_{v-} \nonumber \\ {\rm i\,} \hbar\, \dot{\beta}(t)\, \psi_{v+} +(\hbar \kappa-\hbar \delta)\, \beta(t)\, \psi_{v+} &= (T + V)\, \beta(t)\, \psi_{v+} +\eta\, \mathscr{A}\, \alpha(t) + \Omega_{c}^*\, i(t)\, \psi_{i} \nonumber \\ {\rm i\,} \hbar\, \dot{\gamma}(t)\, \psi_{v-} +(\hbar \kappa-\hbar \delta)\, \gamma(t)\, \psi_{v-} &= (T + V)\, \gamma(t)\, \psi_{v-} +\eta\, \mathscr{A}\, \alpha(t) + \Omega_{c}^*\, i'(t)\, \psi_{i'} \end{align}
where $\mathscr{A} = |\alpha(t)|^2 |\psi_{g}|^2 + |i(t)|^2 |\psi_{i}|^2 + |i'(t)|^2 |\psi_{i'}|^2 + |\beta(t)|^2 |\psi_{v+}|^2 + |\gamma(t)|^2 |\psi_{v-}|^2$ \begin{align}
{\rm i\,} \dot{\alpha}(t) &= \kappa (|\alpha(t)|^2 + |\beta'(t)|^2 + |\gamma'(t)|^2 + |\beta(t)|^2 + |\gamma(t)|^2)\alpha(t)+ \Omega_p \beta'(t) + \Omega_m \gamma'(t) \nonumber \\
{\rm i\,} \dot{\beta'}(t) &= \delta \beta'(t) + \kappa (|\alpha(t)|^2 + |\beta'(t)|^2 + |\gamma'(t)|^2 + |\beta(t)|^2 + |\gamma(t)|^2)\beta'(t)+ \Omega_p \alpha(t) + \Omega_{c} \beta(t)\nonumber \\
{\rm i\,} \dot{\gamma'}(t) &= \delta \gamma'(t) + \kappa (|\alpha(t)|^2 + |\beta'(t)|^2 + |\gamma'(t)|^2 + |\beta(t)|^2 + |\gamma(t)|^2)\gamma'(t)+ \Omega_m \alpha(t) + \Omega_{c} \gamma(t)\nonumber \\
{\rm i\,} \dot{\beta}(t) &= \delta\omega \beta(t) + \kappa (|\alpha(t)|^2 + |\beta'(t)|^2 + |\gamma'(t)|^2 + |\beta(t)|^2 + |\gamma(t)|^2)\beta(t)+ \Omega_p \alpha(t) + \Omega_{c} \beta'(t)\nonumber \\
{\rm i\,} \dot{\gamma}(t) &= \gamma(t) + \kappa (|\alpha(t)|^2 + |\beta'(t)|^2 + |\gamma'(t)|^2 + |\beta(t)|^2 + |\gamma(t)|^2)\gamma(t)+ \Omega_m \alpha(t) + \Omega_{c} \gamma'(t) \end{align} Our numerical study suggests that within the parameter range used to obtain Figures~\ref{Fig:PlotChirp} and ~\ref{Fig:PlotSTIRAP} the excited states populations $i(t)$ and $i'(t)$ are negligible and we are completely justified in using the adiabatic approximation.}
\subsection{Mexican hat trapping potential}
We now study the transfer of OAM to a BEC cloud when the trapping potential is shaped like a Mexican hat (Sombrero) in the $x-y$ plane (See Fig.~\ref{Fig:Pot}). The advantage of this potential is the toroidal symmetry it offers. The toroidal trap configurations naturally support toroid shaped BEC cloud and as a result can sustain large vortices without disintegration into several single charge vortices. This offers stability to the vortex superpositions we are aiming to generate. The trapping potential, in this case, is of the form \cite{Stringari:2006} \begin{figure}
\caption{ The mexican hat potential in two dimensions with $\sigma = 2.0$ and $\lambda = 0.005$. }
\label{Fig:Pot}
\end{figure}
\begin{equation} V(\rho,z) = -\frac{1}{2} \sigma m \omega_{\perp}^{2} \rho^{2} + \frac{1}{4} \lambda \(\frac{m^{2} \omega_{\perp}^{3}}{\hbar}\) \rho^{4} + \frac{1}{2} m \omega_{z}^{2} z^{2} \end{equation} where $\sigma$ and $\lambda$ are dimensionless parameters. The potential is harmonic in the z-direction. The dynamics of the OAM transfer to BEC can be studied in the Thomas-Fermi (TF) approximation where the kinetic energy of the BEC cloud is neglected. In this approximation, the spatial part of the BEC wavefunction is given by \begin{align}
\psi(\ell,\rho,\phi, z)& = \(\frac{1}{L_{\perp}\sqrt{L_{z}}}\) \(\frac{1}{\sqrt{|\ell|!}}\) \(\frac{\rho}{L_{\perp}}\)^{|\ell|}{\rm e}^{-\frac{z^{2}}{2 L_{z}^{2}}}\nonumber \\ & \mbox{Max}\[ \mbox{\bf Re}\(\sqrt{\frac{\mu-V(\rho,0)}{\eta}}\),0 \right] {\rm e}^{{\rm i\,} \ell \phi} \,. \end{align}
Here we have assumed that the wavefunction in $z$-direction has a Gaussian form and has little effect on the vortex dynamics of interest in the transverse direction~\cite{Ketterle:2001}. The spatial profile of the above wavefunction is shaped like a toroid (or a donut) with a hole in the center. Thus, the two radii (inner and outer) are necessary to describe the shape of the BEC cloud. The radii can be found by setting $|\psi(l,\rho,\phi, z)|^{2} = 0$ to determine where the particle density goes to zero. The two real solutions for the radii are given by \begin{equation} R_{\pm}^{2} =\frac{ \sigma M \pm \sqrt{\sigma^{2} M^{2}+4 \lambda g \mu}}{\lambda g} \end{equation} where $M= m \omega_{\perp}^{2}$ and $g=m^{2} \omega_{\perp}^{3}/\hbar ^{2}$. A typical spatial form of the ground state BEC cloud in the mexican hat potential is shown in Fig.~\ref{Fig:WF} along one transverse direction ($\hat{x}$ or $\hat{y}$). The toroid shaped particle distribution should be clear from the figure. \begin{figure}
\caption{ The Thomas-Fermi spatial wavefunction of the BEC in a Mexican Hat potential in one dimension. }
\label{Fig:WF}
\end{figure}
The spatial part of the BEC wavefunction for different components is taken to be \begin{align} \psi_g (\mathbf{r}) &= \psi(0,\rho,\phi, z)\,, \nonumber \\ \psi_{v \pm} (\mathbf{r}) &= \psi(\pm \ell,\rho,\phi, z) \,. \end{align} Using the above one can evaluate the spatial integrals appearing in the Eq.~\eqref{Eq:RateEqBasicFormal} governing the population dynamics. We, furthermore, ignore kinetic energy of the system as we are working in the TF approximation, in this case and choose the optical field frequencies so that there is two-photon resonance, i.e. the two-photon detuning $\delta = 0$. The light fields $\Omega_{c}$ and $\Omega_{\pm}$ have the same saptio-temporal form as discussed in sec. \ref{sec:STIRAP}. Numerically solving the evolution equations, we get the transfer curves for the mexican hat potential depicted in Fig. \ref{Fig:MexHatTransfer}. We observe that the time-scale of transfer is about the same as in the case of harmonic trap. The figure depicts generation of a 60:40 superposition of $\ket{+2}$ and $\ket{-2}$ vortex states. Furthermore, it is important to note that any arbitrary superposition could be obtained in the mexican-hat trap; the generated vortex state would be naturally stable and would not disintegrate into single charge vortices. \begin{figure}
\caption{ Generation of the vortex state superposition in a Mexican hat trap: Results of the numerical solutions of the equations for STIRAP scheme shows the superposition 60:40 of the $\ket{+}$ and $\ket{-}$ vortex states. The Rabi frequency $\Omega_{0}=1$kHz and $\Delta = 100\Omega_{0}$.}
\label{Fig:MexHatTransfer}
\end{figure}
\forget{From the magnitude of the spatial integrals we can conclude that the change in population of the levels over time does not affect the transfer significantly as the atom-light interaction has a much smaller timescale ($\mu$s) than the atom-atom interaction timescale (ms).}
\section{\label{Sec:Detection}Detection of the BEC vortex superposition} \begin{figure}\label{Fig:Interference}
\end{figure}
The resulting vortex-superposition state could be detected by imaging its particle density distribution, which is proportional to an interference of its components. Fig.~\ref{Fig:Interference} shows the $x$-$y$ cross section of this interference pattern for a particular vortex state $\ket{\Psi}_v=\alpha \ket{\ell=+3} + \beta\, {\rm e}^{{\rm i\,} \theta} \ket{-\ell=-3}$.
The spatial profile of the interference pattern can be obtained by evaluating $|\braket{\mathbf{r}}{\Psi_v}|^2=A[1 + 2 \alpha \beta \cos(2 \ell \phi -\theta)]$ where $A = |\Psi_{v\pm}(r,\phi)|^2$ is just the toroid-shaped particle density distribution of the vortex states. The ``cosine'' term dictates the modulation of the particle density distribution along the azimuth with $2\ell$ oscillations as the azimuthal coordinate $\phi$ changes from $0$ to $2\pi$. The relative phase of the two components, $\theta$, just causes offset( or rotation) of the interference pattern by an angle $\theta/(2\ell)$. Note that ${\mathbf r}$ is a two-dimensional vector signifying the position of a point in polar coordinate system $\{r,\phi\}$. The visibility of such a pattern can be readily arrived at via: \begin{equation} V = \frac{I_{max}-I_{min}}{I_{max}+I_{min}} = 2 \alpha \beta\, \end{equation} as $I_{max} = A(1 + 2 \alpha \beta)$ and $I_{min} =A( 1 - 2 \alpha \beta)$ are the extremal intensities.
In general, for such a superposition the spatial profile of the interference pattern contains $m= 2 \ell$ lobes. The visibility $V=2 \alpha \beta$, as determined above, gives a measure of the asymmetry in the amplitudes. The pair $\{\alpha, \beta\}$ can be determined using measured $V$ and the normalization condition $\alpha^2 + \beta^2 =1$. However, this does not assign the amplitudes to the states $\ket{+}$ or $\ket{-}$ with certainty, as the patterns in $(b)$ and $(c)$ are identical. We propose shining a OAM $\ell=+1$ light of $\sigma_+$ polarization to obtain the vortex state $\alpha \ket{3+1} + \beta {\rm e}^{{\rm i\,} \theta} \ket{-3+1}$, the resulting interference pattern is shown in Fig.~\ref{Fig:Interference} $(e)$ and $(f)$, which now clearly differentiates between the two amplitude values that gave same visibility in $(b)$ and $(c)$. The phase difference $\theta$ causes rotation of the whole pattern by an amount $\theta/m$ as shown in $(d)$ of the figure. Existing schemes for detecting vortex states~\cite{Bolda:1998a} could also be extended to detect a superposition of vortex states.
\section{\label{Sec:Conclusion}Conclusion} To summarize, we have studied two different mechanisms for transfer of superposition of optical angular momentum of light to vortices in BEC. An interferometric scheme for generation of an arbitrary superposition of two different OAM states of light is also presented in great detail. We also discussed a couple of trap configurations and showed that the OAM transfer from light to BEC works independently of the trapping potential and despite inter-atomic interactions present in the BEC. The applications of the techniques discussed here are to a memory for the OAM states of light with further applications in the quantum or even classical communications protocols using OAM states of light. The superposition states of vortices in BEC could also be used as a qubit for quantum information processing; however, further work is clearly necessary on that front. Further, various advantages of using atoms versus photons for interferometric metrology are discussed in Ref.~\cite{Dowling:1998}. The superposition of counter-rotating currents in the BEC could also be used for inertial sensing, especially as a gyroscope. Several advantages offered by a gyroscope based on superpositions of counter-rotating vortex structures would be the tunability of the effective de-Broglie wavelength by choosing appropriate atomic masses and the angular velocity and phase sensitivity via the choice of the quantized angular momentum of the atoms.
\section{Acknowledgments} We would like to acknowledge support from the Disruptive Technologies Office, and the Army Research Office.
\appendix \section{\label{App:MachZehnder}Detailed Description of the Mach Zender Interferometer} In this appendix we offer mathematical details of the interferometric scheme for generation of the OAM superposition for light. We suggest references \cite{Zeilinger:1981,Jaroszkiewicz:2004, Holbrow:2002} to an interested reader for further details on treatment of the optical elements.
\begin{figure}
\caption{ A schematic representation of an ideal beam splitter with input ports 1 and 2 and output ports 3 and 4. The amplitudes for reflection and transmission from port 1 are, respectively $r$ and $t$. The corresponding amplitudes for a photon entering port 2 are $r'$ and $t'$.}
\label{Fig:BS}
\end{figure} We chose a basis state representation as \begin{align} \ket{1} \equiv \(\begin{matrix} 1 \\ 0 \end{matrix}\), \quad \ket{2} \equiv \(\begin{matrix} 0 \\ 1 \end{matrix}\)\,; \nonumber \\ \ket{3} \equiv \(\begin{matrix} 1 \\ 0 \end{matrix}\), \quad \ket{4} \equiv \(\begin{matrix} 0 \\ 1 \end{matrix}\)\,. \end{align} In this representation the general beam splitter matrix can be written as \begin{equation} R=\(\begin{matrix} r & t' \\ t & r' \\ \end{matrix}\)\,. \end{equation} Were $r$ and $t$ are the reflection and transmission amplitudes for the input at port 1, and $r'$ and $t'$ are the parameters for the port 2 (See Fig.~\ref{Fig:BS}). Noting that the matrix $R$ should be unitary, meaning, $R^\dagger=R^{-1}$ we obtain \begin{equation} \(\begin{matrix} r^* & t^* \\ t'^* & r'^* \end{matrix}\) = \frac{1}{rr' - tt'} \(\begin{matrix} r' & -t' \\ -t & r \end{matrix}\) \end{equation} The determinant of a unitary matrix has a modulous of one, therefore we have $rr'-tt'={\rm e}^{{\rm i\,} \gamma}$. However, this factor multiplies all the elements of the matrix, thus we can safely choose it to be 1, given by the choice of $\gamma=0$. By equating the corresponding elements on the RHS and LHS of the above equation we obtain \begin{equation} r' = r^* \text{ and } t' = - t^* \label{Eq:rtamp} \end{equation}
If we rewrite these factors as complex exponentials, $|r| {\rm e }^{{\rm i\,} \delta_r}$, $|t| {\rm e }^{{\rm i\,} \delta_t}$, $|r'| {\rm e }^{{\rm i\,} \delta_{r'}}$ and $|t'| {\rm e }^{{\rm i\,} \delta_{t'}}$. by division of the above equalities we obtain \begin{equation}
\frac{|t|}{|r|} {\rm e}^{{\rm i\,} (\delta_{t} - \delta_{r})} = - \frac{|t'|}{|r'|} {\rm e}^{-{\rm i\,}(\delta_{t'} - \delta_{r'})}\,. \end{equation}
From \eqref{Eq:rtamp} we can see that $|t|=|t'|$ and $|r|=|r'|$; using this in the above equation we obtain, \begin{equation} \delta_{t} - \delta_{r} + \delta_{t'} - \delta_{r'} = \pi\,. \end{equation} For the case of a symmetric beam splitter, which has the same effect on a beam incident through port labeled 1 as on a beam incident through port labeled 2, $r=r'$ and $t=t'$, and we have $\delta_{t} - \delta_{r} + \delta_{t'} - \delta_{r'} = \pi/2$. Thus the transmitted wave leads the reflected wave in phase by $\pi/2$ radians. This is a general property of symmetric beam splitter~\cite{Zeilinger:1981}. This also implies that $r$ and $r'$ are purely real and equal and $t$ and $t'$ are purely imaginary and equal to each other. Let $r=r'=\tilde{r}$ and $t=t' = {\rm i\,}\, \tilde{t}$ be the Thus the beam splitter matrix becomes \begin{equation} \label{Eq:RSym} \tilde{R}=\(\begin{matrix} \tilde{r} & {\rm i\,}\,\tilde{t} \\ {\rm i\,}\,\tilde{t} & \tilde{r} \\ \end{matrix}\)\,. \end{equation}
For the case of a 50-50 beam splitter, for which $r = r'$ and $t=t'$ and also $|r|=|r'| = |t|=|t'|$, we have \begin{equation} \label{Eq:R5050} \tilde{R}^{(50-50)}=\frac{1}{\sqrt{2}}\(\begin{matrix} 1 & {\rm i\,} \\ {\rm i\,} & 1 \\ \end{matrix}\)\,. \end{equation} Now we consider the complete transformations for the Interferometer considered in Fig.~\ref{Fig:OAMS}; \begin{widetext} \begin{align} \(\begin{matrix} u_1 \\ u_2 \end{matrix}\) &= \(\begin{matrix} \tilde{r} & {\rm i\,}\,\tilde{t} \\ {\rm i\,}\,\tilde{t} & \tilde{r} \\ \end{matrix}\) \(\begin{matrix} \ket{-\ell}\bra{\ell} & 0 \\ 0 & 1 \\ \end{matrix}\) \(\begin{matrix} 1 & 0 \\ 0 & {\rm e}^{{\rm i\,} \phi} \\ \end{matrix}\) \(\begin{matrix} \tilde{r} & {\rm i\,}\tilde{t} \\ {\rm i\,}\tilde{t} & \tilde{r} \\ \end{matrix}\)\(\begin{matrix} u_0 \ket{\ell} \\ 0 \end{matrix}\) \nonumber \\ &= \(\begin{matrix} \tilde{r}^2 \ket{-\ell}\bra{\ell} - \tilde{t}^2 {\rm e}^{{\rm i\,} \phi} & {\rm i\,}\tilde{r}\tilde{t} \ket{-\ell}\bra{\ell}+ {\rm i\,}\tilde{r}\tilde{t}{\rm e}^{{\rm i\,} \phi} \\ {\rm i\,}\tilde{r}\tilde{t}\ket{-\ell}\bra{\ell} + {\rm i\,}\tilde{r}\tilde{t}{\rm e}^{{\rm i\,} \phi} & -\tilde{t}^2\ket{-\ell}\bra{\ell} + \tilde{r}^2 {\rm e}^{{\rm i\,} \phi} \\ \end{matrix}\) \(\begin{matrix} u_0 \ket{\ell} \\ 0 \end{matrix}\)= \(\begin{matrix} \tilde{r}^2 u_0 \ket{-\ell} - \tilde{t}^2 {\rm e}^{{\rm i\,} \phi} u_0\ket{\ell} \\ {\rm i\,}\, \tilde{t} \tilde{r} \, u_0 \ket{-\ell} + {\rm i\,} \tilde{r}\tilde{t} {\rm e}^{{\rm i\,} \phi}\, u_0\ket{\ell} \end{matrix}\)\,. \end{align}
Observing the output state, it can be easily seen that a general superposition could not be generated by this method. Instead we need only one beam-splitter to be imbalanced, meaning $|r| \neq |t|$, and the other one to be 50:50. However, we can still choose both the beam splitters to be symmetric, i.e., $r = r'$ and $t=t'$. Thus using Eq.~\eqref{Eq:RSym} and Eq.~\eqref{Eq:R5050} we can attain the output state. \begin{align} \(\begin{matrix} u_1 \\ u_2 \end{matrix}\) &= \frac{1}{\sqrt{2}}\(\begin{matrix} 1 & {\rm i\,} \\ {\rm i\,} & 1 \\ \end{matrix}\) \(\begin{matrix} \ket{-\ell}\bra{\ell} & 0 \\ 0 & 1 \\ \end{matrix}\) \(\begin{matrix} 1 & 0 \\ 0 & {\rm e}^{{\rm i\,} \phi} \\ \end{matrix}\) \(\begin{matrix} \tilde{r} & {\rm i\,}\tilde{t} \\ {\rm i\,}\tilde{t} & \tilde{r} \\ \end{matrix}\)\(\begin{matrix} u_0 \ket{\ell} \\ 0 \end{matrix}\) \nonumber \\ &=\frac{1}{\sqrt{2}} \( \begin{matrix} \tilde{r} \ket{-\ell}\bra{\ell} - {\rm e}^{{\rm i\,} \phi}\, \tilde{t} & {\rm i\,} \tilde{t} \ket{-\ell}\bra{\ell} + {\rm i\,} \tilde{r}\,{\rm e}^{{\rm i\,} \phi} \\ {\rm i\,} \tilde{r} \ket{-\ell}\bra{\ell} + {\rm i\,} \tilde{t}\,{\rm e}^{{\rm i\,} \phi} & -\tilde{t}\ket{-\ell}\bra{\ell} + \tilde{r} {\rm e}^{{\rm i\,} \phi} \end{matrix} \)\(\begin{matrix} u_0 \ket{\ell} \\ 0 \end{matrix} \) = \frac{1}{\sqrt{2}}u_0 \( \begin{matrix}
\tilde{r} \ket{-\ell} - {\rm e}^{{\rm i\,} \phi}\, \tilde{t}\ket{\ell} \\
{\rm i\,} \tilde{r} \ket{-\ell} + {\rm i\,} \tilde{t} {\rm e}^{{\rm i\,} \phi}\ket{\ell} \end{matrix} \)\,. \end{align} Thus, by choosing $\tilde{t}=a_{+}$ and $\tilde{r}=a_{-}$, with the condition that $\tilde{r}^2 + \tilde{t}^2=1$, and ${\rm e}^{{\rm i\,} \phi}=-1$ we obtain the state $u_0( a_{+} \ket{\ell} + a_{-} \ket{-\ell})/\sqrt{2} $, which is a general superposition state. \end{widetext}
\forget{\section{Derivation of the rate equations for Optical Coupling of multi-component BEC}
\section{Adiabatic Elimination}}
\section{The spatial Integrals} The various coordinate integrals needed to eliminate the spatial parts of the BEC spinors to arrive at the population evolution equations are given below. The definitions given below are independent of the trapping potential or the ansatz used for the spatial profile of the wavefunction. \begin{align} T_{\rm g}&=\frac{1}{\hbar}\int \psi_{\rm g}^*({\mathbf r})\, \mathcal{T}\, \psi_{\rm g}({\mathbf r})\,\, {\rm d}^3 {\mathbf r}\,, \nonumber \\
V_{\rm g}&=\frac{1}{\hbar}\int \psi_{\rm g}^*({\mathbf r})\, \mathcal{V}\, \psi_{\rm g}({\mathbf r})\,\, {\rm d}^3 {\mathbf r}\,, \nonumber \\
T_{\pm}(\ell)&=\frac{1}{\hbar}\int \psi_{{\mathrm v}\pm}^*(\pm\ell,{\mathbf r})\, \mathcal{T}\, \psi_{\rm v\pm}(\pm\ell,{\mathbf r})\,\, {\rm d}^3 {\mathbf r}\,, \nonumber \\
V_{\pm}(\ell)&=\frac{1}{\hbar}\int \psi_{\rm v\pm}^*(\pm\ell,{\mathbf r})\, \mathcal{V}\, \psi_{\rm v\pm}(\pm\ell,{\mathbf r})\,\, {\rm d}^3 {\mathbf r}\,, \nonumber \\
I_{\rm gg} &=\frac{\eta}{\hbar}\int |\psi_{\rm g}({\mathbf r})|^2|\psi_{\rm g}({\mathbf r})|^2\,\, {\rm d}^3 {\mathbf r}\,, \nonumber \\
I_{\rm g\pm}(\ell)&=\frac{\eta}{\hbar}\int |\psi_{\rm v\pm}(\ell,{\mathbf r})|^2|\psi_{\rm g}({\mathbf r})|^2\,\, {\rm d}^3 {\mathbf r}\,, \nonumber \\
I_{++}(\ell)&=\frac{\eta}{\hbar}\int |\psi_{\rm v+}(\ell,{\mathbf r})|^2|\psi_{\rm v+}(\ell,{\mathbf r})|^2\,\, {\rm d}^3 {\mathbf r}\,, \nonumber \\
I_{--}(\ell)&=\frac{\eta}{\hbar}\int |\psi_{\rm v-}(\ell,{\mathbf r})|^2|\psi_{\rm v-}(\ell,{\mathbf r})|^2\,\, {\rm d}^3 {\mathbf r}\,, \nonumber \\
I^{(2\ell)}_{\rm gg}(\ell)&=\int \psi_{\rm g}^*({\mathbf r})\,\(\frac{\sqrt{2} r}{w}\)^{2|\ell|}\,\psi_{\rm g}({\mathbf r})\,\, {\rm d}^3 {\mathbf r} \,, \nonumber \\
I^{(\ell)}_{\rm g\pm}(\ell)&=\int \psi_{\rm g}^*({\mathbf r})\,{\rm e}^{\mp{\rm i\,} \ell \phi} \(\frac{\sqrt{2} r}{w}\)^{|\ell|} \psi_{\rm v\pm}(\ell,{\mathbf r})\,\, {\rm d}^3 {\mathbf r}\,, \nonumber \\
I^{(\ell)}_{\rm \pm g}(\ell)&=\int \psi_{\rm v\pm}^*(\ell,{\mathbf r})\,{\rm e}^{\pm{\rm i\,} \ell \phi} \(\frac{\sqrt{2} r}{w}\)^{|\ell|} \psi_{\rm g}({\mathbf r})\,\, {\rm d}^3 {\mathbf r}\,. \end{align} The values of the above spatial integrals, for specific vortex states corresponding to the charge of $\ell=\pm2$ and the harmonic trapping potential, are given below. \begin{align} T_{\rm g}^{(H)} &= \frac{\hbar}{4m}\,\( \frac{1}{L_z^{2}} + \frac{2}{{L_{\perp}}^2} \right) = \frac{1}{4} \omega_z + \frac{1}{2} \omega_{\perp}=V_{\rm g}^{(H)} \,,\nonumber \\
T_{\pm}^{(H)}(2) &= \frac{\hbar}{4m}\,\( \frac{1}{L_z^{2}} + \frac{6}{{L_{\perp}}^2}
\right)= \frac{1}{4} \omega_z + \frac{3}{2} \omega_{\perp}=V_{\pm}^{(H)} (2)\,, \nonumber \\
I_{\rm gg}^{(H)} &= \frac{\eta}{{(2\pi) }^{\frac{3}{2}}\,{\hbar~L_z}\,
{{L_{\perp}}}^2} = 4 \kappa\,, \nonumber \\
I_{\rm g+}^{(H)}(2) &= \frac{\eta}{4{(2\pi) }^{\frac{3}{2}}\,{\hbar~L_z}\,
{{L_{\perp}}}^2}= {\kappa}=I^{(
H)}_{\rm g-}(2)\,, \nonumber \\
I_{++}^{(H)}(2)&= \frac{3\eta}{8{(2\pi) }^{\frac{3}{2}}\,{\hbar~L_z}\,{{L_{\perp}}}^2}= \frac{3}{2}\kappa= I^{(H)}_{--}(2)= I^{(H)}_{+-}(2)\,, \nonumber \\
I^{(4)(H)}_{\rm gg}&= 2 L_{\perp}^4 =2 \({\hbar}/{m \omega_{\perp}}\)^2 \nonumber \\
I^{(2)(H)}_{\rm g\pm}(\ell)&= \sqrt{2} L_{\perp}^2 =I^{(2)(H)}_{\rm \pm g}(\ell). \end{align} Note also that at this stage we have used various properties of the quantum harmonic oscillator as summarized below: \begin{align} \frac{\hbar^2}{2 m L_{\perp}^2} = \frac{1}{2} m \omega_{\perp}^2, \text{ i.e., } L_{\perp} = \sqrt{\frac{\hbar}{m \omega_{\perp}}}, \text{ and }L_{z} = \sqrt{\frac{\hbar}{m \omega_{z}}},\nonumber \\ \nonumber
\end{align} The corresponding integrals for a the mexican hat trap are evaluated numerically for the parameter values $\sigma = 2.0$ and $\lambda = 0.005$. Also, as we deal with the Thomas-Fermi ground states for the mexican hat trap the kinetic energy is ignored.
This completes the description of the spatial integrals, which allow us to obtain the time-dependent population equations~\eqref{Eq:RateEqBasicFormal}. Those could be solved as discussed in the main text of the paper to study the transfer techniques for the vortex superpositions from the light to the atom.
\end{document} | arXiv |
Martin Hairer
Sir Martin Hairer KBE FRS (born 14 November 1975[1]) is an Austrian-British mathematician working in the field of stochastic analysis, in particular stochastic partial differential equations. He is Professor of Mathematics at EPFL (École Polytechnique Fédérale de Lausanne) and at Imperial College London. He previously held appointments at the University of Warwick and the Courant Institute of New York University.[5][6][7][8] In 2014 he was awarded the Fields Medal,[9] one of the highest honours a mathematician can achieve.[10] In 2020 he won the 2021 Breakthrough Prize in Mathematics.[11]
Sir
Martin Hairer
KBE FRS
Hairer at the Royal Society admissions day in London, July 2014
Born (1975-11-14) 14 November 1975
Geneva, Switzerland
Citizenship
• Austrian
• British
EducationUniversity of Geneva
Spouse
Xue-Mei Li
(m. 2003)
[1][2]
Awards
• Whitehead Prize (2008)
• Philip Leverhulme Prize (2008)
• Wolfson Research Merit Award (2009)
• Fermat Prize (2013)
• Fröhlich Prize (2014)
• Fields Medal (2014)
• Breakthrough Prize in Mathematics (2021)
• King Faisal Prize (2022)
Scientific career
Fields
• Probability theory[3]
• Analysis[3]
InstitutionsÉcole Polytechnique Fédérale de Lausanne
Imperial College London
University of Warwick
New York University[1]
ThesisComportement Asymptotique d'Équations à Dérivées Partielles Stochastiques (2001)
Doctoral advisorJean-Pierre Eckmann[4]
Websitehairer.org
Early life and education
Hairer was born in Geneva, Switzerland.[1] He attended the Collège Claparède Geneva where he received his high school diploma in 1994. He entered a school science competition with sound editing software that was developed into Amadeus,[12] and later continued to maintain the software in addition to his academic work; it continued to be widely used as of 2020.[11] He then attended the University of Geneva, where he obtained his Bachelor of Science degree in Mathematics in July 1998, Master of Science in Physics in October 1998 and PhD in Physics under the supervision of Jean-Pierre Eckmann in November 2001.[4][13]
Research and career
Hairer is active in the field of stochastic partial differential equations in particular, and in stochastic analysis and stochastic dynamics in general.[14] He has worked on variants of Hörmander's theorem, systematisation of the construction of Lyapunov functions for stochastic systems, development of a general theory of ergodicity for non-Markovian systems, multiscale analysis techniques, theory of homogenisation, theory of path sampling and theory of rough paths[14] and, in 2014, on his theory of regularity structures.[15]
Under the name HairerSoft, he develops Macintosh software.[12]
Affiliations
• Regius Professor of Mathematics, University of Warwick (2014–2017)[5]
• Member of the scientific steering committee of ETHZ-ITS (2013–2019)[16]
• Institut Henri Poincaré, member of scientific steering committee (2012–2020)[17]
• Mathematical Research Institute of Oberwolfach, member of steering committee (2013–2021)[18]
• Editor, Probability Theory and Related Fields[19]
• Editor, Nonlinear Differential Equations and Applications[20]
• Editor, Annales Henri Poincaré Ser. B[21]
• Editor, Electronic Journal of Probability[22]
• Editor, Stochastic Partial Differential Equations: Analysis and Computations[23]
• Visiting Professor, Université Paul Sabatier, Toulouse (December 2006 and February 2014)[24]
• Visiting Professor, Technical University of Berlin (July 2009)[24]
• Visiting Professor, École Normale Supérieure, Paris (April 2013)[24]
• Member, Institute for Advanced Study, Princeton (March – April 2014)[24]
• Lipschitz Lectures, Hausdorff Center for Mathematics, University of Bonn (July 2013)[25]
• Minerva Lectures, Columbia University (February 2014)[26]
• Euler Lecture, Zuse Institute Berlin (May 2014)[27]
• Medallion Lecture, Institute of Mathematical Statistics (July 2014)[28]
• Lévy Lecture, Conference on Stochastic Processes and their Applications (July 2014)[29]
• Fields Medal lecture, International Congress of Mathematicians, Seoul (August 2014)[30]
• Collingwood Lecture, Durham University (February 2015)[31]
• Bernoulli Lecture, École polytechnique fédérale de Lausanne (May 2015) [31]
• Leonardo da Vinci Lecture, University of Milan (October 2015) [31]
• Kai-Lai Chung Lecture, Stanford University (November 2015)[31]
• Michalik Lecture, University of Pittsburgh (December 2015)[31]
Awards and honours
• 2006–2011 Advanced Research Fellowship, Engineering and Physical Sciences Research Council (EPSRC)[32]
• 2007 – Editors' Choice Award, Macworld[33]
• 2008 – Whitehead Prize, London Mathematical Society[34][35][36]
• 2008 – Philip Leverhulme Prize, Leverhulme Trust[37][38]
• 2009 – Royal Society Wolfson Research Merit Award[39]
• 2012 – Leverhulme Research Leadership Award, Leverhulme Trust[40]
• 2013 – Fermat Prize, Institut de Mathématiques de Toulouse[41][42]
• 2014 – Consolidator grant, European Research Council[43]
• 2014 – Elected Fellow of the Royal Society (FRS) in 2014[14]
• 2014 – Fröhlich Prize, London Mathematical Society[44]
• 2014 – Fields Medal[9]
• 2015 – Fellow of the American Mathematical Society[45]
• 2015 – Member of the Austrian Academy of Sciences[46]
• 2015 – Member of the Academy of Sciences Leopoldina[47]
• 2015 – Member of the Academia Europaea[48]
• 2016 – Honorary Knight Commander of the Order of the British Empire[49][39]
• 2017 – Foreign member of the Polish Academy of Sciences[50]
• 2019 – Substantive Knight Commander of the Order of the British Empire[51]
• 2021 – Breakthrough Prize in Mathematics[52][53]
• 2022 – King Faisal Prize[54]
Personal life
Wikimedia Commons has media related to Martin Hairer.
Hairer holds Austrian and British nationality, and speaks French, German and English; he married fellow mathematician Li Xue-Mei in 2003.[1][2] His father is Ernst Hairer, a mathematician at the University of Geneva.
References
1. "Hairer, Martin". Who's Who (online Oxford University Press ed.). Oxford: A & C Black. 2016. doi:10.1093/ww/9780199540884.013.U282027. (Subscription or UK public library membership required.)
2. Xue-Mei, Li (2017). "Xue-Mei Li: About me". xuemei.org.
3. Martin Hairer publications indexed by Google Scholar
4. Martin Hairer at the Mathematics Genealogy Project
5. Warwick Mathematics Institute. "Professor Martin Hairer, FRS". Archived from the original on 1 December 2017.
6. Eckmann, J.-P.; Hairer, M. (2001). "Uniqueness of the Invariant Measure for a Stochastic PDE Driven by Degenerate Noise". Communications in Mathematical Physics. 219 (3): 523. arXiv:nlin/0009028. Bibcode:2001CMaPh.219..523E. doi:10.1007/s002200100424. S2CID 5565100.
7. Martin Hairer's publications indexed by the Scopus bibliographic database. (subscription required)
8. Hairer, M.; Mattingly, J. (2006). "Ergodicity of the 2D Navier–Stokes equations with degenerate stochastic forcing". Annals of Mathematics. 164 (3): 993. arXiv:math/0406087. doi:10.4007/annals.2006.164.993. S2CID 11828895.
9. Mireille Chaleyat-Maurel (2014). "IMU-Net 66b : Special issue on IMU Prizes and Medals at ICM 2014 in Seoul". International Mathematical Union (IMU).
10. Daniel Saraga: The equation Tamer, in: Horizons, Swiss National Science Foundation No. 103, p. 26–7
11. Sample, Ian (10 September 2020). "UK mathematician wins richest prize in academia". The Guardian.
12. "Amadeus – Audio waveform editors / sound and voice recorders for macOS X". hairersoft.com. Retrieved 10 September 2020.
13. "Martin Hairer CV" (PDF).
14. "Professor Martin Hairer FRS". London: Royal Society. 2014.
15. Hairer, Martin (2014). "A theory of regularity structures". Inventiones Mathematicae. 198 (2): 269–504. arXiv:1303.5113. Bibcode:2014InMat.198..269H. doi:10.1007/s00222-014-0505-4. S2CID 119138901.
16. "Organisation of the Institute for Theoretical Studies – Institute for Theoretical Studies | ETH Zurich". Eth-its.ethz.ch. Retrieved 14 August 2014.
17. Institut Henri Poincaré. "Members' directory". Retrieved 6 May 2014.
18. Mathematisches Forschungsinstitut Oberwolfach. "Scientific Committee". Retrieved 6 May 2014.
19. Springer. "Probability Theory and Related Fields – Editorial Board". Retrieved 6 May 2014.
20. Springer. "Nonlinear Differential Equations and Applications – Editorial Board". Retrieved 6 May 2014.
21. Institute of Mathematical Statistics. "Annales de l'Institut Henri Poincaré". Retrieved 6 May 2014.
22. Electronic Journal of Probability. "Editorial Team". Retrieved 6 May 2014.
23. Springer. "Stochastic Partial Differential Equations: Analysis and Computations – Editorial Board". Retrieved 6 May 2014.
24. Hairer, Martin. "Curriculum Vitae" (PDF). Retrieved 6 May 2014.
25. Hausdorff Center for Mathematics, University of Bonn. "Lipschitz Lectures". Retrieved 6 May 2014.
26. Columbia University. "Minerva Lectures". Retrieved 6 May 2014.
27. Zuse Institute, Berlin. "Euler-Vorlesung 2014". Retrieved 6 May 2014.
28. Institute of Mathematical Statistics. "Awards – Special Lectures Winners". Archived from the original on 19 May 2014. Retrieved 6 May 2014.
29. University of Buenos Aires. "37th Conference on Stochastic Processes and their Applications". Retrieved 6 May 2014.
30. Warwick Mathematics Institute (21 October 2013). "Five Warwick mathematicians to speak at ICM 2014". Retrieved 6 May 2014.
31. http://hairer.org/cv.pdf
32. Warwick Mathematics Institute (29 May 2006). "EPSRC Advanced Research Fellowship awarded to Martin Hairer". Retrieved 6 May 2014.
33. Macworld (19 December 2007). "The 23rd Annual Editors' Choice Awards". Retrieved 6 May 2014.
34. London Mathematical Society. "List of LMS prize winners: Whitehead Prize". Retrieved 6 May 2014.
35. London Mathematical Society (4 July 2008). "London Mathematical Society Prizes 2008" (PDF). Retrieved 14 April 2010.
36. Warwick Mathematics Institute (6 July 2008). "Martin Hairer receives LMS Whitehead Prize". Retrieved 6 May 2014.
37. Warwick Mathematics Institute (6 November 2008). "Martin Hairer wins Philip Leverhulme Prize". Retrieved 6 May 2014.
38. "Report of the Leverhulme Trustees 2008" (PDF). Archived from the original (PDF) on 21 April 2014. Retrieved 14 August 2014.
39. Sheng, Yunhe. "Mathematician receives top honour". London Mathematical Society. Retrieved 11 September 2022.
40. Warwick Mathematics Institute (27 November 2012). "Martin Hairer wins Leverhulme Research Leadership Award". Retrieved 6 May 2014.
41. Institut de Mathématiques de Toulouse (6 November 2013). "Prix Fermat 2013". Retrieved 6 May 2014.
42. Warwick Mathematics Institute (10 November 2013). "Martin Hairer awarded 2013 Fermat Prize". Retrieved 6 May 2014.
43. Warwick Mathematics Institute (4 February 2014). "Martin Hairer & José Luis Rodrigo win ERC Consolidator grants". Retrieved 6 May 2014.
44. London Mathematical Society. "List of LMS prize winners: Fröhlich Prize". Retrieved 7 July 2014.
45. 2016 Class of the Fellows of the AMS, American Mathematical Society, retrieved 16 November 2015.
46. "ÖAW wählte 40 neue Mitglieder". Archived from the original on 5 August 2018. Retrieved 2 April 2016.
47. "Nationalakademie Leopoldina ernennt neue Mitglieder".
48. "Martin Hairer – member of AE".
49. "Honorary awards" (PDF).
50. "Wydział III Nauk Ścisłych i Nauk o Ziemi PAN – Członkowie zagraniczni".
51. "Central Chancery of the Orders of Knighthood".
52. Ian Sample (10 September 2020). "UK mathematician wins richest prize in academia". The Guardian. Retrieved 10 September 2020.
53. Hayley Dunning (10 September 2020). "Imperial mathematician scoops $3m Breakthrough Prize". Imperial College London. Retrieved 10 September 2020.
54. King Faisal Prize 2022
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| Wikipedia |
Get help with your Oxidation state homework. The oxidation state of an uncombined element is zero. The oxidation number of an atom is zero in a neutral substance that contains atoms of only one element. 2. The oxidation state of nitrogen is most positive in which of the following compounds? Long answer. MEDIUM. To find the oxidation state of , set up an equation of each oxidation state found earlier and set it equal to . Add to both sides of the equation. Minimum oxidation state $\mathrm{-III}$. This applies regardless of the structure of the element: Xe, Cl 2, S 8, and large structures of carbon or silicon each have an oxidation state of zero. Is there any way in vasp to mention the oxidation state of constituent atoms. The oxidation number of #"O"# in compounds is usually -2, but it is -1 in peroxides.. Rules to determine oxidation states. Solve the equation for . Oxidation and reduction are therefore best defined as follows. In H2o, oxidation state of H and o are balanced.given that total oxidation state is +2. Ex: Oxidation state of sodium (Na) is zero. Sometimes, the oxidation states can also be written as a superscripted number to the right of the element symbol (Fe 3+). Maximum oxidation state: $\mathrm{+V}$. An atom of an element may be capable of multiple oxidation numbers. Together that is another 4 electrons for oxygen. The oxidation number of a monatomic ion equals the charge of the ion. View Answer. How to find Oxidation Numbers In chemistry, the terms "oxidation" and "reduction" refer to reactions in which an atom (or group of atoms) loses or gains electrons, respectively. I have an Iron complex in which Fe has oxidation state 3. 2. oxidation number for an ion is equal to its charge. ex. Cl-, Br-, all have oxidation -1. The oxidation number for metals that can have more than one oxidation state is represented by a Roman numeral. There are a few rules to follow when assigning oxidation state of an element. The oxidation state of a particular atom can be determined by using the following rules. The oxidation number of #"H"# is +1, but it is -1 in when combined with less electronegative elements.. As the table shows, the presence of the other oxidation states varies, but follows some patterns. (-1 oxidation state). ex. To find the oxidation state of , set up an equation of each oxidation state found earlier and set it equal to . Oxidation Number: The number that is assigned to an element to indicate the loss or gain of electrons by an atom of that element is called as the oxidation number. Just follow the normal rules for determining oxidation states. Then count electrons. The oxidation number of a free element is always 0. The oxidation state is the positive or negative number of an atom in a compound, which may be found by comparing the numbers of electrons shared by the cation and anion in the compound needed to balance each other's charge. Tap for more steps... Add and . Access the answers to hundreds of Oxidation state questions that are explained in a way that's easy for you to understand. Determination of Oxidation State of an Atom. Once you recognize that, you will notice that Cu + is both oxidized to Cu 2+ and reduced to Cu. Oxidation state of H is +1. When an oxidation number is assigned to the element, it does not imply that the element in the compound acquires this as a charge, but rather that it is a number to use for balancing chemical reactions. The total charge of the compound should equal to the sum of the charges of each atom present in that compound. Oxidation occurs when the oxidation number of an atom becomes larger. Oxidation increases oxidation state and reduction decreases oxidation state. An oxidation number refer to the quantity of electrons that may be gained or lost by an atom. Oxidation numbers are … The algebraic sum of the oxidation states in an ion is equal to the charge on the ion. (-2 oxidation state). Identify the pair of binary corresponds in which nitrogen exhibits the lowest and the highest oxidation state. Oxidation number also referred to as oxidation state is the number that is allocated to elements in a chemical combination. The oxidation number is placed in parentheses after the name of the element (iron(III)). Oxidation states show how oxidised or reduced an element is within a compound or ion. Oxidation State. This Demonstration shows the colors and absorption spectra of the six most common oxidation states (2 to 7) of the element manganese. The oxidation state, sometimes referred to as oxidation number, describes the degree of oxidation (loss of electrons) of an atom in a chemical compound.Conceptually, the oxidation state, which may be positive, negative or zero, is the hypothetical charge that an atom would have if all bonds to atoms of different elements were 100% ionic, with no covalent component. We know that each NH3 has a +1 charge so there are three NH3's for a combined total of +3. O2, N2, He, are all oxidation numbers zero. Since is in column of the periodic table, it will share electrons and use an oxidation state of . For example, the sum of the oxidation numbers for SO 4 2-is -2. You count the valence electrons around "N" according to a set of rules and then assign the oxidation number. Chlorine can give seven electrons to make chloric acid to show +7 oxidation number. Oxidation Number of Periodic Table Elements. Rule 6: The oxidation state of hydrogen in a compound is usually +1. All oxygens in there will have $\mathrm{-II}$, all nitrogens $\mathrm{-III}$, all hydrogens (they are either connected to oxygen or to nitrogen) $\mathrm{+I}$. Multiple Oxidation States. In a C-H bond, the H is treated as if it has an oxidation state of +1. Notice that, Cu + has an oxidation number of +1, but because there are two atoms of copper, the combined oxidation number is +2. Rules for oxidation numbers: 1. Since is in column of the periodic table, it will share electrons and use an oxidation state of . Since the overall complex has a +1 charge, Pt has to have a -1 oxidation state because (+3) + (-1) + (-1) = +1. In order to find the oxidation number, you need to look at the charge of each piece of the complex. The common oxidation states of all of the metals in the periodic table are all positive. This means that every C-H bond will decrease the oxidation state of carbon by 1.; For carbon bonded to a more electronegative non-metal X, such as nitrogen, oxygen, sulfur or the halogens, each C-X bond will increase the oxidation state of the carbon by 1. Warning! If the hydrogen is part of a binary metal hydride (compound of hydrogen and some metal), then the oxidation state of hydrogen is –1.. Rule 7: The oxidation number of fluorine is always –1. > The Rules Lone pair electrons (LPs) belong entirely to the atom on which they reside. 1. oxidation number of a free element is always zero. The oxidation state of carbon increases from +2 to +4, while the oxidation state of the hydrogen decreases from +1 to 0. Shared electrons (bonding pair electrons or BEs) between identical atoms are shared equally. Sulfur can take two electrons to form sulfide anion. The sum of the oxidation numbers of all of the atoms in a neutral compound is 0. To find the correct oxidation state of Mn in MnSO4 (Manganese (II) sulfate), and each element in the molecule, we use a few rules and some simple math. Assigning oxidation numbers to organic compounds The oxidation state of any chemically bonded carbon may be assigned by adding -1 for each more electropositive atom (H, Na, Ca, B) and +1 for each more electronegative atom (O, Cl, N, P), and 0 for each carbon atom bonded directly to the carbon of interest. [1] This does have a general quantum chemical explanation. For pure elements, the oxidation state is zero. The oxidation number of diatomic and uncombined elements is zero. Chlorine can take one electron to form chloride anion. Any bond between elements is cleaved heterolyticly giving only the electronegative element all electrons of that bond. If you now subtract the number of electrons assigned to H and O from their corresponding valence electrons, you will find that the oxidation state of hydrogen is +1 and oxygen, -2. You can find in several handbooks on XPS, how to use the XPS spectra for oxidation state determination. Redox Reactions - Examples. Because there is a change in oxidation number, we can confidently say that the above equation represents a redox reaction. The oxidation number of Cl is -1 in HCl, but the oxidation number of Cl is +1 in HOCl. When you select an oxidation state, an arrow points to the "petri dish" containing an aqueous solution of a compound in this oxidation state, appropriately colored. Chlorine, bromine, and iodine usually have an oxidation number of –1, unless they're in combination with an oxygen or fluorine. Oxidation state 0 occurs for all elements – it is simply the element in its elemental form. The sum of the oxidation numbers in a polyatomic ion is equal to the charge of the ion. Main principles of identifying oxidation state. Sulfur gives its all last six electrons to make sulfuric acid molecule (+6 oxidation state). The oxidation number of a Group 1 element in a compound is +1. I want to mention the oxidation state explicitly. For example, iron Fe and calcium Ca have an oxidation state of zero, because they consist of one element that is not chemically bonded with others, and so do polyatomic molecules with the same type of atom, for example for ozone O₃ the oxidation state … View Answer. Chemists have developed a method to find which atoms have gained/lost electrons, especially since some reactions can seem very complicated. MEDIUM. CN has a -1 charge. If the oxidation state increases the substance is oxidised If the oxidation state decreases the substance is reduced. Sum of all oxidation states is +2, let oxidation state of Al be x. 3. Related Posts. These oxidation states add up to eight, which is exactly the number of electrons that typically make up the outer (valence) shell — where chemistry happens. The oxidation state of a neutral element is always zero. Learn how to calculate or find the oxidation number of … ; The sum of the oxidation states of all the atoms or ions in a neutral compound is zero. So oxidation state of the metal in the salt is: Oxidation number or state of periodic table elements in a chemical compound or molecule is the formal charges (positive or negative) which assigned to the element if all the bonds in the compounds are ionic. They are positive and negative numbers used for balancing the redox reaction. When 6 × 1 0 2 2 electrons are used in the electrolysis of a metalic salt, 1.9 gm of the metal is deposited at the cathode The atomic weight of that metal Is 57. So, by the oxidation bookkeeping method, oxygen is assigned a total of 8 electrons, while hydrogen is assigned 0. Since the oxidation state of copper has reduced from +2 to 0, this is a reduction reaction. To determine if electrons were gained or lost by an atom, we assign an oxidation number to each atom in a compound. 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how to find oxidation state
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Octagonal bipyramid
The octagonal bipyramid is one of the infinite set of bipyramids, dual to the infinite prisms. If an octagonal bipyramid is to be face-transitive, all faces must be isosceles triangles. 16-sided dice are often octagonal bipyramids.
Octagonal bipyramid
Typebipyramid
Faces16 triangles
Edges24
Vertices10
Schläfli symbol{ } + {8}
Coxeter diagram
Symmetry groupD8h, [8,2], (*228), order 32
Rotation groupD8, [8,2]+, (228), order 16
Dual polyhedronoctagonal prism
Face configurationV4.4.8
Propertiesconvex, face-transitive
Images
It can be drawn as a tiling on a sphere which also represents the fundamental domains of [4,2], *422 symmetry:
Related polyhedra
"Regular" right (symmetric) n-gonal bipyramids:
Bipyramid name Digonal bipyramid Triangular bipyramid
(See: J12)
Square bipyramid
(See: O)
Pentagonal bipyramid
(See: J13)
Hexagonal bipyramid Heptagonal bipyramid Octagonal bipyramid Enneagonal bipyramid Decagonal bipyramid ... Apeirogonal bipyramid
Polyhedron image ...
Spherical tiling image Plane tiling image
Face config. V2.4.4V3.4.4V4.4.4V5.4.4V6.4.4V7.4.4V8.4.4V9.4.4V10.4.4...V∞.4.4
Coxeter diagram ...
*n42 symmetry mutation of omnitruncated tilings: 4.8.2n
Symmetry
*n42
[n,4]
Spherical Euclidean Compact hyperbolic Paracomp.
*242
[2,4]
*342
[3,4]
*442
[4,4]
*542
[5,4]
*642
[6,4]
*742
[7,4]
*842
[8,4]...
*∞42
[∞,4]
Omnitruncated
figure
4.8.4
4.8.6
4.8.8
4.8.10
4.8.12
4.8.14
4.8.16
4.8.∞
Omnitruncated
duals
V4.8.4
V4.8.6
V4.8.8
V4.8.10
V4.8.12
V4.8.14
V4.8.16
V4.8.∞
External links
• Weisstein, Eric W. "Dipyramid". MathWorld.
• Virtual Reality Polyhedra The Encyclopedia of Polyhedra
• VRML models <8>
• Conway Notation for Polyhedra Try: dP8
| Wikipedia |
\begin{document}
\begin{center} {\bf \Large On Fuzzy Ideals and Level Subsets of Ordered $\Gamma$-Groupoids}
\noindent{\bf Niovi Kehayopulu}\\ {\it Department of Mathematics, University of Athens, 15784 Panepistimiopolis, Greece}\\email: [email protected]
\noindent October 15, 2014\end{center}
{\small
\noindent{\bf Abstract.} We characterize the fuzzy left (resp. right) ideals, the fuzzy ideals and the fuzzy prime (resp. semiprime) ideals of an ordered $\Gamma$-groupoid $M$ in terms of level subsets and we prove that the cartesian product of two fuzzy left (resp. right) ideals of $M$ is a fuzzy left (resp. right) ideal of $M\times M$, and the cartesian product of two fuzzy prime (resp. semiprime) ideals of $M$ is a fuzzy prime (resp. semiprime) ideal of $M\times M$. As a result, if $\mu$ and $\sigma$ are fuzzy left (resp. right) ideals, ideals, fuzzy prime or fuzzy semiprime ideals of $M$, then the nonempty level subsets $(\mu\times\sigma)_t$ are so.
\noindent 2010 Mathematics Subject Classification: 06F99 (08A72, 20N99, 06F05).
\noindent Keywords and phrases: Ordered $\Gamma$-groupoid, left (right) ideal, ideal, fuzzy left (right) ideal, fuzzy ideal, level subset, fuzzy prime (semiprime) subset, cartesian product. } \section{Introduction and prerequisites}For two nonempty sets $M$ and $\Gamma$, we denote by the letters of the English alphabet the elements of $M$ and by the letters of the Greek alphabet the elements of $\Gamma$, and define $M\Gamma M:=\{a\gamma b \mid a,b\in M, \gamma\in\Gamma\}$. Then $M$ is called a {\it $\Gamma$-groupoid} if
(1) $M\Gamma M\subseteq M$ and
(2) if $a,b,c,d\in M$ and $\gamma,\mu\in\Gamma$ such that $a=c$, $b=d$ and $\gamma=\mu$, then
\hspace{0.5cm} have $a\gamma b=c\mu d$.\\If, in addition, for all $a,b\in M$ and all $\gamma,\mu\in\Gamma$, $a\gamma (b\mu c)=(a\gamma b)\mu c$, then $M$ is called a {\it $\Gamma$-semigroup}. For $\Gamma=\{\gamma\}$, the $\Gamma$-groupoid $M$ is the so called groupoid and the ordered $\Gamma$-groupoid the ordered groupoid (: $po$-groupoid), the $\Gamma$-semigroup is the semigroup and the ordered $\Gamma$-semigroup the ordered semigroup (where $\gamma$ is the multiplication on $M$ usually denoted by ``$\cdot$"). An {\it ordered $\Gamma$-groupoid} (: {\it $po$-$\Gamma$-groupoid}) is a $\Gamma$-groupoid $M$ with an order relation ``$\le$" on $M$ such that $a\le b$ implies $a\gamma c\le b\gamma c$ and $c\gamma a\le c\gamma b$ for all $\gamma\in \Gamma$. Let $M$ be a $po$-$\Gamma$-groupoid. A subset $T$ of $M$ is called {\it prime} if for every $a,b\in M$ and every $\gamma\in\Gamma$ such that $a\gamma b\in T$, we have $a\in T$ or $b\in T$. A subset $T$ of $M$ is called {\it semiprime} if for every $a\in M$ and every $\gamma\in\Gamma$ such that $a\gamma a\in T$, we have $a\in T$. A nonempty subset $A$ of $M$ is called a {\it left} (resp. {\it right}) {\it ideal} of $M$ if
(1) $M\Gamma A\subseteq A$ (resp. $A\Gamma M\subseteq A$) and
(2) if $a\in A$ and $M\ni b\le a$ implies $b\in A$.\\ If the set $A$ is both a left and a right ideal of $M$, then it is called an {\it ideal} of $M$. If $M$ is an ordered $\Gamma$-groupoid, then any mapping $\mu : M\rightarrow [0,1]$ is called a {\it fuzzy subset} of $M$ (or a {\it fuzzy set} in $M$) (L. Zadeh). The mapping $\mu$ is called a {\it fuzzy left ideal} of $M$ if
(1) $\mu(x\gamma y)\ge \mu(y)$ for every $x,y\in M$ and every $\gamma\in\Gamma$ and
(2) if $x\le y$ implies $\mu(x)\ge \mu (y)$.\\It is called a {\it fuzzy right ideal} of $M$ if
(1) $\mu(x\gamma y)\ge \mu(x)$ for every $x,y\in M$ and every $\gamma\in\Gamma$ and
(2) if $x\le y$ implies $\mu(x)\ge \mu (y)$.\\A fuzzy subset which is both a fuzzy left and a fuzzy right ideal of $M$ is called a {\it fuzzy ideal} of $M$. A fuzzy subset $\mu$ of $M$ is a fuzzy ideal of $M$ if and only if
(1) $\mu(x\gamma y)\ge\max\{\mu(x),\mu(y)\}$ for every $x,y\in M$ and every $\gamma\in\Gamma$ and
(2) if $x\le y$ implies $\mu(x)\ge \mu (y)$.
A characterization of fuzzy prime subsets of a semigroup in terms of level subsets has been considered in [4; Lemma 2.3] in which $\lambda$ should be replaced by $t$ (or $t$ should be replaced by $\lambda$) and the commutativity of the semigroup is not necessary as the same holds in semigroups is general. As an immediate consequence of the Lemma 2.3 in [4], a fuzzy ideal of a semigroup $S$ is prime if and only if for every $t\in [0,1]$ the $t$-level subset $f_t:=\{x\in S \mid f(x)\ge t\}$ (of $S$), if it is nonempty, is a prime ideal of $S$. This is the Theorem 3.1 in [3] in which the ``$xy\subseteq \mu_t$" (in the second line of the proof) should be replaced by ``$xy\in \mu_t$" and the proof of the converse statement (lines 5--11 of the proof) should be corrected. A characterization of fuzzy semiprime ideals of a semigroup in terms of level subsets has been considered in the Theorem 3.2 in [3] but the proof of the ``converse" statement in it should be corrected. The cartesian product of two fuzzy left (resp. fuzzy right) ideals of semigroups and the cartesian product of two fuzzy prime (resp. fuzzy semiprime) ideals of a semigroup has been studied in [3]. On the other hand, a characterization of fuzzy prime and fuzzy semiprime ideals of ordered semigroups in terms of level subsets has been considered in the Theorems 2.6 and 2.7 in [6] from which the Theorems 3.1 and 3.2 in [3] are also obtained. For a characterization of fuzzy ideals of ordered semigroup in terms of level subsets see the Lemma 2.4 in [6] and the Lemma 2.7 in [5]. The reference in Lemma 2.4 in [6] should be corrected. In the ``converse statement" of the proof of Theorem 3.1 in [3] as well as in the proof of Theorem 3.2 in [3], the phrase ``Let every nonempty subset $\mu_t$ of $\mu$ be a prime (semiprime) ideal of $S$" is better to be replaced by the phrase ``Let every subset $\mu_t$ of $\mu$ be a prime (semiprime) ideal of $S$", and this is because the ideals are, by definition, nonempty sets. Finally, the proofs of Propositions 4.4 and 4.6 in [3] should be omitted as they are immediate consequences of Propositions 4.2 and 4.3 and the Theorem 3.2 given in the same paper.
In the present paper we first characterize the fuzzy left (right) ideals, the fuzzy ideals, the fuzzy prime and the fuzzy semiprime ideals of an ordered $\Gamma$-groupoid in terms of level subsets. Then we prove that the cartesian product of two fuzzy left (resp. fuzzy right) ideals of an ordered $\Gamma$-groupoid $M$ is a fuzzy left (resp. fuzzy right) ideal of $M\times M$. Thus the cartesian product of two fuzzy ideals of $M$ is a fuzzy ideal of $M\times M$. Moreover, the cartesian product of two fuzzy prime (resp. fuzzy semiprime) ideals of a $\Gamma$-groupoid $M$ is a fuzzy prime (resp. fuzzy semiprime) ideal of $M\times M$. As a consequence, if $\mu$ and $\sigma$ are fuzzy left (resp. fuzzy right) ideals of an ordered $\Gamma$-groupoid $M$ then, for any $t\in[0,1]$ if the level subset $(\mu\times\sigma)_t$ is nonempty, then it is a left (resp. right) ideal of $M\times M$. If $\mu$ and $\sigma$ are fuzzy ideals of $M$ and the level subset $(\mu\times\sigma)_t$ is nonempty, then it is an ideal of $M\times M$. If $\mu$ and $\sigma$ are fuzzy prime (resp. fuzzy semiprime) ideals of $M$, then the nonempty level subsets $(\mu\times\sigma)_t$ are prime (resp. semiprime) ideals of $M\times M$. The present paper serves as an example to show the way we pass from fuzzy ordered groupoids (resp. fuzzy ordered semigroups) to fuzzy ordered $\Gamma$-groupoids (resp. fuzzy ordered $\Gamma$-semigroups) and from fuzzy groupoids (resp. fuzzy semigroups) to fuzzy $\Gamma$-groupoids (resp. fuzzy $\Gamma$-semigroups). On the other hand, from the results of $\Gamma$-groupoids or $\Gamma$-ordered groupoids where $\Gamma=\{\gamma\}$ ($\gamma$ being a symbol) the corresponding results on groupoids or ordered groupoids are obtained. The fuzzy sets in ordered groupoids have been introduced in [1] and one can find several papers on fuzzy ordered semigroups in the bibliography. \section{Characterization of prime and semiprime fuzzy ideals in terms of level subsets}Following the terminology of fuzzy prime subset of a groupoid introduced in [1, 2], we give the following definition
\noindent{\bf Definition 1.} Let $M$ be an ordered $\Gamma$-groupoid (or a $\Gamma$-groupoid). A fuzzy subset $\mu$ of $M$ is called {\it fuzzy prime subset of $M$} or {\it prime fuzzy subset of $M$} if$$\mu(x\gamma y)\le\max\{\mu(x),\mu(y)\}$$for all $x,y\in M$ and all $\gamma\in\Gamma$.\\Recall that if $\mu$ is a fuzzy prime ideal of $M$, then for every $x,y\in M$ and every $\gamma\in\Gamma$, we have $\mu(x\gamma y)=\max\{\mu(x),\mu(y)\}$. So a fuzzy ideal $\mu$ of $M$ can be called {\it prime} if $\mu(x\gamma xy)=\max\{\mu(x),\mu(y)\}$ for all $x,y\in M$ and all $\gamma\in\Gamma$.
\noindent{\bf Definition 2.} Let $M$ be an ordered $\Gamma$-groupoid (or a $\Gamma$-groupoid). A fuzzy subset $\mu$ of $M$ is called {\it fuzzy semiprime subset of $M$} or {\it semiprime fuzzy subset of $M$} if$$\mu(x)\ge \mu(x\gamma x)$$for every $x\in M$ and every $\gamma\in\Gamma$.
\noindent{\bf Notation 3.} If $\mu$ is a fuzzy subset of an ordered $\Gamma$-groupoid (or a $\Gamma$-groupoid) $M$ then, for any $t\in [0,1]$ (: the closed interval of real numbers), we denote by $\mu_t$ the subset of $M$ defined by$$\mu_t:=\{x\in M \mid \mu(x)\ge t\}.$$ The set $\mu_t$ is called the $t$-level subset or just level subset of $\mu$.
\noindent{\bf Theorem 4.} {\it Let M be an ordered $\Gamma$-groupoid. If $\mu$ is a fuzzy left ideal of M and $\mu_t\not=\emptyset$, then $\mu_t$ is a left ideal of $M$. ``Conversely", if $\mu_t$ is a left ideal of $M$ for every $t$, then $\mu$ is a fuzzy left ideal of M}.
\noindent{\bf Proof.} $\Longrightarrow$. Suppose $\mu$ is a fuzzy left ideal of $M$ and $\mu_t\not=\emptyset$ for some $t\in[0,1]$. Then $M\Gamma\mu_t\subseteq \mu_t$. Indeed: Let $a\in M$, $\gamma\in\Gamma$ and $b\in\mu_t$. Since $\mu$ is a fuzzy left ideal of $M$, we have $\mu(a\gamma b)\ge \mu(b)$. Since $b\in\mu_t$, we have $\mu(b)\ge t$. Then $\mu(a\gamma b\ge t$, and $a\gamma b\in \mu_t$. Let $a\in \mu_t$ and $M\ni b\le a$. Then $b\in \mu_t$. Indeed: Since $a\in \mu_t$, we have $\mu(a)\ge t$. Since $b\le a$ and $\mu$ is a fuzzy left ideal of $M$, we have $\mu(b)\ge \mu(a)$. Then $\mu(b)\ge t$, and $b\in\mu_t$. Thus $\mu_t$ is a left ideal of $M$.
\noindent$\Longleftarrow$. Suppose $\mu_t$ is a left ideal of $M$ for every $t$ and let $a,b\in M$ and $\gamma\in\Gamma$. Then $\mu(a\gamma b)\ge \mu (b)$. Indeed: Since $\mu(b)\in [0,1]$ and $\mu(b)\ge \mu(b)$, we have $b\in \mu_{\mu(b)}$. Since $\mu_{\mu(b)}$ is a left ideal of $M$, we have $a\gamma b\in M\Gamma \mu_{\mu(b)}\subseteq \mu_{\mu(b)}$. Then $a\gamma b\in \mu_{\mu(b)}$, and $\mu(a\gamma b)\ge \mu(b)$. Let now $a\le b$. Then $\mu(a)\ge \mu(b)$. Indeed: Since $b\in\mu_{\mu(b)}$, $M\ni a\le b$ and $\mu_{\mu(b)}$ is a left ideal of $M$, we have $a\in\mu_{\mu(b)}$, then $\mu(a)\ge \mu(b)$.$
\Box$\\In a similar way we have the following
\noindent{\bf Theorem 5.} {\it Let M be an ordered $\Gamma$-groupoid. If $\mu$ is a fuzzy right ideal of M and $\mu_t\not=\emptyset$, then $\mu_t$ is a right ideal of $M$. ``Conversely", if $\mu_t$ is a right ideal of $M$ for every $t$, then $\mu$ is a fuzzy right ideal of M}.\\By Theorems 4 and 5, we have the following theorem
\noindent{\bf Theorem 6.} {\it Let M be an ordered $\Gamma$-groupoid. If $\mu$ is a fuzzy ideal of M and $\mu_t\not=\emptyset$ for some $t\in[0,1]$, then $\mu_t$ is an ideal of $M$. ``Conversely", if $\mu_t$ is an ideal of $M$ for every $t\in [0,1]$, then $\mu$ is a fuzzy ideal of M}.
\noindent{\bf Lemma 7.} {\it Let M be an ordered $\Gamma$-groupoid. Then $\mu$ is a fuzzy prime subset of $M$ if and only if the level subset $\mu_t$ is a prime subset of $M$ for every $t$.}
\noindent{\bf Proof.} $\Longrightarrow$. Let $a,b\in M$ and $\gamma\in\Gamma$ such that $a\gamma b\in\mu_t$. Then $a\in\mu_t$ or $b\in\mu_t$. Indeed: Since $a\gamma b\in\mu_t$, we have $\mu(a\gamma b)\ge t$. Since $\mu$ is a fuzzy prime subset of $M$, we have $\mu(a\gamma b)\le\max\{\mu(a),\mu(b)\}$. Since $\mu(a),\mu(b)\in [0,1]$, we have $\mu(a)\le\mu(b)$ or $\mu(b)\le\mu(a)$. If $\mu(a)\le \mu(b)$, then $\max\{\mu(a),\mu(b)\}=\mu(b)$, and $t\le\mu(b)$, so $b\in\mu_t$. If $\mu(b)\le \mu(a)$, then $t\le \mu(a\gamma b)=\mu(a)$, and $a\in\mu_t$.
\noindent$\Longleftarrow$. Suppose $\mu_t$ is a prime subset of $M$ for every $t$ and let $x,y\in M$ and $\gamma\in\Gamma$. Then $\mu(x\gamma y)=\max\{\mu(x),\mu(y)\}$. Indeed: Since $x\gamma y\in\mu_{\mu(x\gamma y)}$, by hypothesis, we have $x\in\mu_{\mu(x\gamma y)}$ or $y\in\mu_{\mu(x\gamma y)}$. Then $\mu(x)\ge\mu(x\gamma y)$ or $\mu(y)\ge\mu(x\gamma y)$, thus $\max\{\mu(x),\mu(y)\}\ge \mu(x\gamma y)$. $
\Box$\\By Theorem 6 and Lemma 7, we have the following theorem
\noindent{\bf Theorem 8.} {\it Let M be an ordered $\Gamma$-groupoid. If $\mu$ is a fuzzy prime ideal of M and $\mu_t\not=\emptyset$, then $\mu_t$ is a prime ideal of $M$. ``Conversely", if $\mu_t$ is a prime ideal of $M$ for every $t$, then $\mu$ is a fuzzy prime ideal of M}.
\noindent{\bf Lemma 9.} {\it Let M be an ordered $\Gamma$-groupoid. Then $\mu$ is a fuzzy semiprime subset of M if and only if the level subset $\mu_t$ is a semiprime subset of $M$ for every $t$}.
\noindent{\bf Proof.} $\Longrightarrow$. Let $t\in [0,1]$, $a\in M$ and $\gamma\in\Gamma$ such that $a\gamma a\in\mu_t$. Then $a\in\mu_t$. Indeed: Since $\mu$ is a fuzzy semiprime subset of $M$, we have $\mu(a)\ge \mu(a\gamma a)$. Since $a\gamma a\in\mu_t$, we have $\mu(a\gamma a)\ge t$. Then $\mu(a)\ge t$, and $a\in\mu_t$.
\noindent$\Longleftarrow$. Let $a\in M$ and $\gamma\in\Gamma$. Then $\mu(a)\ge \mu(a\gamma a)$. Indeed: By hypothesis, $\mu_{\mu(a\gamma a)}$ is a semiprime subset of $M$. Since $a\gamma a\in\mu_{\mu(a\gamma a)}$, we have $a\in \mu_{\mu(a\gamma a)}$, then $\mu(a)\ge\mu(a\gamma a)$, so $\mu$ is fuzzy semiprime.$
\Box$\\By Theorem 6 and Lemma 9, we have the following theorem
\noindent{\bf Theorem 10.} {\it Let M be an ordered $\Gamma$-groupoid. If $\mu$ is a fuzzy semiprime ideal of M and $\mu_t\not=\emptyset$, then $\mu_t$ is a semiprime ideal of $M$. ``Conversely", if $\mu_t$ is a semiprime ideal of $M$ for every $t$, then $\mu$ is a fuzzy semiprime ideal of M}.
As a consequence, given a groupoid or an ordered groupoid $G$, if $\mu$ is a fuzzy left ideal, fuzzy right ideal, fuzzy ideal, fuzzy prime ideal or fuzzy semiprime ideal of $G$, respectively, then the nonempty level subsets $\mu_t$ of $\mu$ are left ideals, right ideals, ideals, prime ideals or semiprime ideals of $G$, respectively. ``Conversely" if for a fuzzy subset $\mu$ of $G$ the and any $t\in [0,1]$ the level subset $\mu_t$ of $\mu$ is a left ideal, right ideal, ideal, prime ideal or semiprime ideal of $G$, respectively, then $\mu$ is a fuzzy left ideal, fuzzy right ideal, fuzzy ideal, fuzzy prime ideal or fuzzy semiprime ideal of $G$, respectively. \section{Cartesian product of fuzzy ideals, fuzzy prime and fuzzy semiprime ideals}If $(M,\le,\Gamma)$ is an ordered $\Gamma$-groupoid, $M\times M:=\{(x,y) \mid x,y\in M\}$ and for any $(a,b),(c,d)\in M\times M$ and any $\gamma\in\Gamma$ we define $$(a,b)\gamma (c,d):=(a\gamma c,b\gamma d),$$ then $(M\times M,\le,\Gamma)$ is an ordered $\Gamma$-groupoid as well.
For two fuzzy subsets $\mu$ and $\sigma$ of an ordered $\Gamma$-groupoid $M$, the cartesian product of $\mu$ and $\sigma$ is the fuzzy subset of $M\times M$ defined by $$\mu\times \sigma : M\times M \rightarrow [0,1] \mid (x,y)\rightarrow \min\{\mu(x),\sigma(y)\}.$$That is,$$(\mu\times \sigma){\Big(}(x,y){\Big)}:=\min\{\mu(x),\sigma(y)\}$$for every $x,y\in M$. As no confusion is possible, we write $(\mu\times \sigma)(x,y)$ instead of $(\mu\times \sigma){\Big(}(x,y){\Big)}$.
\noindent{\bf Theorem 11.} {\it Let M be an ordered $\Gamma$-groupoid and $\mu$, $\sigma$ fuzzy left (resp. right) ideals of M. Then $\mu\times\sigma$ is a fuzzy left (resp. right) ideal of $M\times M$.}
\noindent{\bf Proof.} Suppose $\mu$ and $\sigma$ are fuzzy left ideals of $M$. Let $(a,b),(c,d)\in M\times M$ and $\gamma\in\Gamma$. Then $$(\mu\times \sigma){\Big(}(a,b)\gamma (c,d){\Big)}\ge (\mu\times\sigma)(c,d).$$Indeed:\begin{eqnarray*}(\mu\times \sigma){\Big(}(a,b)\gamma (c,d){\Big)}&=&(\mu\times \sigma)(a\gamma c,b\gamma d)=\min\{\mu(a\gamma c),\sigma (b\gamma d)\}\\&\ge&\min\{\mu(c),\sigma (d)\} \mbox { (since } \mu,\sigma \mbox { are fuzzy left ideals of } M)\\&=&(\mu\times \sigma)(c,d). \end{eqnarray*}Let now $(a,b)\le (c,d)$. Then \begin{eqnarray*}(\mu\times\sigma)(a,b)&=&\min\{\mu(a),\mu(b)\}\\&\le& \min\{\mu(c),\mu(d)\} \mbox { (since } a\le c, b\le d)\\&=&(\mu\times \sigma)(c,d).\end{eqnarray*}Thus $\mu\times\sigma$ is a fuzzy left ideal of $M$. Similarly the cartesian product of two fuzzy right ideals of $M$ is a fuzzy right ideal of $M\times M$.$
\Box$\\By Theorem 11, the following theorem holds
\noindent{\bf Theorem 12.} {\it Let M be an ordered $\Gamma$-groupoid and $\mu$, $\sigma$ fuzzy ideals of M. Then $\mu\times\sigma$ is a fuzzy ideal of $M\times M$.}
\noindent{\bf Lemma 13.} {\it Let M be an ordered $\Gamma$-groupoid and $\mu$, $\sigma$ fuzzy prime subsets of M. Then $\mu\times\sigma$ is a fuzzy prime subset of $M\times M$.}
\noindent{\bf Proof.} Let $(a,b),(c,d)\in M\times M$ and $\gamma\in\Gamma$. Then$$(\mu\times\sigma){\Big(}(a,b)\gamma (c,d){\Big)}=\max\{(\mu\times\sigma)(a,b),(\mu\times\sigma)(c,d)\}.$$In fact:\begin{eqnarray*}(\mu\times\sigma){\Big(}(a,b)\gamma (c,d){\Big)}&=&(\mu\times\sigma)(a\gamma c,b\gamma d)=\min\{\mu(a\gamma c),\sigma(b\gamma d)\}\\&=&\min{\Big\{}\max\{\mu(a),\mu(c)\}, \max\{\sigma(b),\sigma(d)\}{\Big\}}\\&=&\max{\Big\{}\min\{\mu(a),\sigma(b)\}, \min\{\mu(c),\sigma(d)\}{\Big\}}\\&=& \max\{(\mu\times\sigma)(a,b),(\mu\times\sigma)(c,d)\}.\end{eqnarray*}$
\Box$\\ By Theorem 12 and Lemma 13, we have the following theorem
\noindent{\bf Theorem 14.} {\it If M is an ordered $\Gamma$-groupoid and $\mu$, $\sigma$ fuzzy prime ideals of M, then $\mu\times\sigma$ is a fuzzy prime ideal of $M\times M$.}
\noindent{\bf Lemma 15.} {\it Let M be an ordered $\Gamma$-groupoid and $\mu$, $\sigma$ fuzzy semiprime subsets of M. Then $\mu\times\sigma$ is a fuzzy semiprime subset of $M\times M$.}
\noindent{\bf Proof.} Let $(a,b)\in M$ and $\gamma\in\Gamma$. Then $$(\mu\times\sigma)(a,b)\ge (\mu\times\sigma){\Big(}(a,b)\gamma (a,b){\Big)}.$$Indeed: $(\mu\times\sigma)(a,b)=\min\{\mu(a),\sigma(b)\}$. Since $\mu$ and $\sigma$ are fuzzy semiprime subsets of $M$, we have $\mu(a)\ge \mu(a\gamma a)$ and $\sigma(b)\ge\sigma (b\gamma b)$. Then we get \begin{eqnarray*}(\mu\times\sigma)(a,b)&\ge&\min\{\mu(a\gamma a),\sigma(b\gamma b)\}=(\mu\times\sigma)(a\gamma a,b\gamma b)\\&=&(\mu\times\sigma){\Big(}(a,b)\gamma (a,b){\Big)}.\end{eqnarray*}$
\Box$\\By Theorem 12 and Lemma 15, we have the following theorem
\noindent{\bf Theorem 16.} {\it If M is an ordered $\Gamma$-groupoid and $\mu$, $\sigma$ fuzzy semiprime ideals of M, then $\mu\times\sigma$ is a fuzzy semiprime ideal of $M\times M$.}\\By Theorems 4--6 and 11, 12, we have the following corollary
\noindent{\bf Corollary 17.} {\it If M is an ordered $\Gamma$-groupoid and $\mu$, $\sigma$ fuzzy left (resp. right) ideals of M, then the level subset $(\mu\times\sigma)_t$, if it is nonempty, is a left (resp. right) ideal of $M\times M$. If $\mu$ and $\sigma$ are fuzzy ideals of M and the level subset $(\mu\times\sigma)_t$ is nonempty, then it is an ideal of $M\times M$.}\\By Theorems 8, 10, 14 and 16, we have the following
\noindent{\bf Corollary 18.} {\it If M is an ordered $\Gamma$-groupoid and $\mu$, $\sigma$ fuzzy prime (resp. semiprime) ideals of M, then the level subset $(\mu\times\sigma)_t$, if it is nonempty, is a prime (resp. semiprime) ideal of $M\times M$.}
As a conclusion, if $G$ is a groupoid or an ordered groupoid and $\mu$, $\sigma$ fuzzy left ideals, fuzzy right ideals, fuzzy ideals, fuzzy prime ideals or fuzzy semiprime ideals of $G$, respectively, then the cartesian product $\mu\times\sigma$ of $\mu$ and $\sigma$ is, respectively so. If $G$ is a groupoid or an ordered groupoid and $\mu$, $\sigma$ fuzzy left (right) ideals, fuzzy ideals, fuzzy prime (semiprime) ideals of $G$, respectively, then for any $t\in [0,1]$ for which $(\mu\times\sigma)_t$ is nonempty, the level subset $(\mu\times\sigma)_t$ is, respectively, a left (right) ideal, ideal, prime (semiprime) ideal of the cartesian product $M\times M$ of $M$.{\small
}
\noindent This paper has been submitted in Lobachevskii J. Math. on October 15, 2014.
\end{document} | arXiv |
\begin{definition}[Definition:Supremum of Set/Real Numbers]
Let $T \subseteq \R$ be a subset of the real numbers.
A real number $c \in \R$ is the '''supremum of $T$ in $\R$''' {{iff}}:
:$(1): \quad c$ is an upper bound of $T$ in $\R$
:$(2): \quad c \le d$ for all upper bounds $d$ of $T$ in $\R$.
If there exists a '''supremum''' of $T$ (in $\R$), we say that:
:'''$T$ admits a supremum (in $\R$)''' or
:'''$T$ has a supremum (in $\R$)'''.
The '''supremum''' of $T$ is denoted $\sup T$ or $\map \sup T$.
\end{definition} | ProofWiki |
Short-term effects of single-tree selection cutting on stand structure and tree species composition in Valdivian rainforests of Chile
Florian Schnabel ORCID: orcid.org/0000-0001-8452-40011,
Pablo J. Donoso2 &
Carolin Winter ORCID: orcid.org/0000-0002-4238-68163
New Zealand Journal of Forestry Science volume 47, Article number: 21 (2017) Cite this article
The Valdivian temperate rainforest, one of the world's 25 biodiversity hotspots, is under a continued process of degradation through mismanagement. An approach to reverse this situation might be the development of uneven-aged silviculture, combining biodiversity conservation and timber production.
We examined the short-term effects of single-tree selection cutting on stand structure and tree species (richness, diversity and composition) in the Llancahue Experimental Forest in south-central Chile to quantify changes in comparison with old-growth rainforests of the evergreen forest type. We compared plots with high and low residual basal areas (60 and 40 m2 ha−1) and a control old-growth forest.
Both cutting regimes achieved a balanced structure with reverse-J diameter distribution, continuous forest cover and sufficient small-sized trees. Compared to the old-growth forest, there were no significant changes in tree species richness and diversity. The only shortcomings detected were significant reductions in diameter and height complexity as assessed by the Gini coefficient, Shannon H′ and standard deviation, with a significantly lower number of large-sized trees (dbh 50 cm+, height 23 m+), especially in the low residual basal area regime.
We suggest the intentional retention of a certain number of large-sized and emergent trees as strategy for biodiversity conservation. If adjusted accordingly, single-tree selection is a promising approach to retain many old-growth attributes of the Valdivian rainforest in managed stands while providing timber for landowners.
The Chilean evergreen rainforest in the Valdivian Rainforest Ecoregion (35–48° S) is a unique, but endangered, ecosystem. It is one of the world's 25 biodiversity hotspots due to its abundance of vascular plant and vertebrate species and high degree of endemism, as well as a conservation priority due to it undergoing exceptional loss of habitat (Myers et al. 2000; Olson and Dinerstein 1998). This loss is caused basically due to illegal logging and inappropriately conducted legal selective cutting (cut the best and leave the worst; sensu Nyland (2002)), which destroy the multi-aged stand structure of these old-growth forests, leading to thousands of hectares of degraded forests (Moorman et al. 2013; Donoso 2013; Schütz et al. 2012; Myers et al. 2000; Olson and Dinerstein 1998). Old-growth forests of the evergreen forest type harbour the highest tree species richness in Chile and consist of a mixture of mostly shade-tolerant and moderately shade-tolerant (hereafter referred to as "mid-tolerant") broadleaved evergreen hardwood species and some conifers of the Podocarpaceae family (Donoso and Donoso 2007). The biodiversity associated with the structural and compositional attributes of these old-growth forests must not only be maintained in reserves (Moorman et al. 2013; Bauhus et al. 2009) but also in managed forests, combining the needs of the local population for forest products with biodiversity conservation (Moorman et al. 2013). A promising way to address this is the development of a silviculture regime that: (a) maintains forest attributes that are close to the natural state of old-growth forests; and (b) allows stakeholders to benefit from timber harvesting.
In this study, we use the term "old-growthness" to refer to the degree of the retention of old-growth structural and compositional attributes in managed stands following Bauhus et al. (2009). Old-growth forests are defined here through the presence of key structural and compositional attributes including a high number of large trees, a wide range of tree sizes, complex vertical layering, the presence of late successional tree species and large amounts of standing and lying dead wood among others (Bauhus et al. 2009; Mosseler et al. 2003). The maintenance of these attributes in managed stands is essential for sustaining forest biodiversity as has been illustrated, for example, in boreal ecosystems (Bauhus et al. 2009 and citations within). The rationale is that since natural forest ecosystems and their dynamics are able to sustain the whole range of forest-dwelling species and forest functions, silviculture that mimics natural dynamics should be a good approach for sustaining forest biodiversity (Schütz et al. 2012).
Currently in Chile, the application of single-tree selection cutting is believed to be the most promising and adequate approach for uneven-aged forests (Donoso 2013; Schütz et al. 2012; Donoso et al. 2009; Siebert 1998). Chilean native evergreen forests in south-central Chile are dominated by several commercially valuable hardwood shade-tolerant or mid-tolerant species, a major requisite to work with selection silviculture. Moreover, it has been shown that some of these species have much faster diameter growth rates under lower levels of basal area than those found in dense unmanaged old-growth forests (Donoso 2002; Donoso et al. 2009). Due to rare implementation, however, it remains unknown how the forest ecosystem is influenced through this type of silviculture and which would be the economic benefits in Chile, although preliminary estimates of timber revenues are positive (Nahuelhual et al. 2007).
Nonetheless, there is abundant evidence for other forests that selection systems can maintain a high forest cover, complex vertical layering and balanced/regulated structures while providing income through timber sales at regular intervals on a sustainable basis (e.g. O'Hara 2014; Schütz et al. 2012; Pukkala and Gadow 2012; Gronewold et al. 2010; O'Hara et al. 2007; Keeton 2006; Bagnaresi 2002). For example, forests in the European Alps can harbour high structural and vegetation diversity even after several centuries of uneven-aged management (Bagnaresi 2002). The possible lack of old-growth attributes like large-sized trees can, however, be a concern (Bauhus et al. 2009 and citations within). Another general concern regarding selection silviculture is that through their evenly distributed small-scale disturbances, single-tree selection cutting might favour the development of shade-tolerant species at the expense of mid-tolerant ones, creating an abundant but homogenous regeneration and relatively low horizontal heterogeneity (Angers et al. 2005; Doyon et al. 2005).
These considerations should be addressed before a new silvicultural scheme is applied at a large scale to avoid unwanted side effects. In the present work, our aim was, therefore, to evaluate the impacts of single-tree selection cutting with two different residual basal areas, upon structural and compositional attributes of old-growth temperate rainforests of the evergreen forest type. We were interested in finding management approaches that could avoid negative impacts on old-growth attributes and associated biodiversity at the stand scale. The objectives were to: (a) quantify the type and magnitude of structural and compositional changes induced through single-tree selection cutting with high residual basal areas (HRBA; 60 m2 ha−1) and low residual basal areas (LRBA; 40 m2 ha−1); and (b) identify key structural and compositional attributes of old-growthness that were affected through single-tree selection cutting with HRBA and LRBA. Unmanaged and well-conserved forests of the evergreen forest type in Chile reach 80–100 m2 ha−1 in basal area and support regeneration of almost exclusively shade-tolerant species (Donoso and Nyland 2005). The rationale for these two levels of residual basal areas was, therefore, that single-tree selection with LRBA would create relatively more light availability and was expected to favour the development of both ecologically and economically important mid-tolerant tree species (Donoso 2013). However, there might be trade-offs in terms of greater structural and compositional changes at LRBA compared with HRBA.
Study area and experimental design
The study was conducted in the Llancahue watershed (39° 50′ 20″ south and 73° 07′ 18″ west) in the intermediate depression of south-central Chile, a 1270-ha state-owned reserve that is administered by the University Austral de Chile (UACh) (Fig. 1).
Study area showing the location of old-growth control (n = 4), high residual basal area (HRBA, n = 4) and low residual basal area (LRBA, n = 4) plots
The low-elevation forest of the study area corresponds to the evergreen forest type, more specifically to the subtype dominated by shade tolerant species with few emergent Nothofagus trees, according to the official classification in Chile, and is part of the Valdivian Rainforest Ecoregion (Donoso and Donoso 2007). Llancahue lies between 50 and 410 m a.s.l., receives 2100 mm average annual rainfall and has an average annual temperature of 12.2 °C (Oyarzún et al. 2005; Fuenzalida 1971). Stands dominated by the shade-tolerant species Aextoxicon punctatum R. et Pav. and Laureliopsis philippiana (Looser) Schodde and the mid-tolerant species Eucryphia cordifolia Cav. and Drimys winteri J.R. et G. Forster were chosen. All stands had an uneven-aged structure and basal areas characteristic for this forest type. In the intermediate depression of south-central Chile, nearly all remnant old-growth forests show signs of illegal selective cuttings, especially since the twentieth century (Donoso and Lara 1995) and at low elevations. Signs include large stumps of few valuable species and increased cover of Chusquea spp., especially at low residual densities. This is also the case for stands selected in this study, which show signs of past harvests of a few large trees over the last three decades.
The experimental design consisted of eight plots 2000 m2 (50 × 40 m) each, which were subjected to single-tree selection cutting in 2012 and were re-evaluated two growing seasons afterwards in 2014. Four plots were cut to achieve a residual basal area of 60 m2 ha−1 and four plots to 40 m2 ha−1, called high and low residual basal area (HRBA and LRBA), respectively (Table 1). The BDq method proposed by Guldin (1991) with a maximum diameter of 80 cm and a q factor (the difference between successive diameter classes) of 1.3 in average for a balanced diameter distribution was used based on recommendations in Schütz et al. (2012). Since this was the first time the stands were cut following a selection system, only half of the trees above the maximum diameter were cut to avoid a severe change and potential damage to residual trees. The main target species of selection silviculture are A. punctatum, L. philippiana, D. winteri, E. cordifolia and Podocarpaceae conifers if the expected product is timber and E. cordifolia if the objective of the harvest is firewood. For this first harvest, the rule "cut the worst, leave the best" was applied to enhance the quality and growth of the residual stock by preferentially harvesting defective and unhealthy trees. This approach contrasts with current selective cuttings that are used under the Chilean law, which do not control for residual stand structure, allow the harvest of 35% of the basal area per hectare in 5-year cutting cycles and preferably cut the most valuable trees instead of the worst (Donoso 2013; Schütz et al. 2012).
Table 1 Basal area (m2 ha−1) per treatment and plot before (2012) and after the harvesting (2014)
Four permanent plots 900 m2 (30 × 30 m) each that showed only minimal signs of past illegal cuttings were used as control. Although smaller than the cut plots, plot sampling sizes for temperate old-growth forests have been traditionally considered adequate with at least 500 m2 in Chile (Prodan et al. 1997) and elsewhere (Lombardi et al. 2015), so the plots used in this study provide a reliable sampling of the variables tested. Moreover, different plot sizes were addressed through choosing analysis methods that allow for unbiased testing of different sample sizes (see below).
In Chile, the cutting intensity for the evergreen forest type is restricted to an average maximum of 35% of the original basal area (Donoso 2013). To achieve two levels of residual basal areas (HRBA and LRBA), while at the same time complying with the legal restrictions, we had to choose plots with the lowest initial basal areas for LRBA (average 34% of the basal area cut) and those with the largest basal areas for HRBA (average 24% of the basal area cut) (Table 1). Final average residual basal areas were 58.2 m2 ha−1 for HRBA plots and 41.2 m2 ha−1 for LRBA plots (Table 1). The plot where the least trees were cut was number S6 (10.5%), and the one with the most trees cut was S7 (41.6%). Apart from these extremes, plots had a cutting intensity that ranged between 17 and 37% of the original basal area.
We acknowledge differences in the original basal areas of the three groups of plots selected for this study (old-growth, HRBA and LRBA). However, to reach the expected residual basal areas proposed by Donoso (2002) for uneven-aged silviculture in Chilean forests, within legal restrictions, we had to choose these partially cut stands that are common in the landscape. From there, rather than from pristine old-growth forests, we sought to find out how selection stands do, or do not, maintain old-growth attributes.
Sampling design and data collection
Three parameters for quantifying structural and compositional attributes were used in this study that have been largely and successfully applied in other ecosystems (e.g. Gadow et al. 2012; Lexerød and Eid 2006; McElhinny et al. 2005): (a) diameter at breast height (dbh) measured at 1.3 m; (b) tree height; and (c) tree species. All trees with a dbh ≥ 5 cm were recorded by species and diameter for the eight plots before cutting (2012) and were re-evaluated two growing seasons after harvesting (2014). The four control plots were measured once in 2014. Tree height was included as an additional and more direct measurement of vertical complexity only in 2014. Tree height was measured for all trees with dbh ≥ 10 cm with a Vertex III hypsometer.
To quantify tree size complexity, three diversity indices were used to analyse the diameter and height data of the trees: (a) standard deviation; (b) Gini coefficient (Lexerød and Eid 2006; Gini 1912); and (c) ln-based Shannon index (H′) (Lexerød and Eid 2006; Shannon 1948). Standard deviation has been widely used as a way to calculate diameter and height complexity and can be compared with more complex indices for stand structural comparisons (McElhinny et al. 2005 and citations within). The Gini coefficient has also been used successfully to describe structural changes. For example, Lexerød and Eid (2006) found that the Gini coefficient was superior in discriminating between stands and was considered to have a very low sensitivity to sample size in a comparison of eight diameter diversity indices. It is calculated with the following equation:
$$ \mathrm{GC}\kern0.5em =\kern0.5em \frac{\sum_{j=i}^n\left({2}_j-n-1\right){\mathrm{ba}}_j}{\sum_{j=i}^n{\mathrm{ba}}_j\left(n-1\right)\left.-1\right)} $$
where ba stands for the basal area of tree j (m2 ha−1).
Finally, the Shannon index is a widely used measure of tree size complexity for diameter distributions, which allows a direct comparison of different distributions through one single value (e.g. Lexerød and Eid 2006; McElhinny et al. 2005; Wikström and Eriksson 2000). It is calculated after the following equation:
$$ {H}^{\hbox{'}}=-\sum_{i=1}^S{P}_i\kern0.5em \ln \left({P}_i\right) $$
where P stands for the proportion of number of trees in size class i or per species i and S, for the number of dbh classes or species.
An important quality of the Shannon index and the Gini coefficient is their independence of stand density as proven for example by Lexerød and Eid (2006). These indices were used: (a) due to their abilities documented in the literature (especially independence of sample size); and (b) to have a more robust result than using only one index. The standard deviation and the Gini coefficient were calculated from original individual tree data while the Shannon index was calculated on the basis of 5-cm diameter classes as suggested by Lexerød and Eid (2006). All three indices have been used similarly to describe diameter complexity as well as height complexity (Lexerød and Eid 2006). The values of the Gini coefficient range from (0, 1), with 1 standing for total inequality, while the Shannon index values range from (0, ln(S)) (Lexerød and Eid 2006). The standard deviation ranges from [0, ∞]. For all three indices, a higher index value reflects a wider range of tree diameters and heights and consequently greater complexity (Lexerød and Eid 2006). Index values were calculated for each plot and then compared between treatments to quantify the changes in structural complexity after management. To quantify a potential loss of large and/or emergent trees in more detail than with the complexity indices, trees were grouped in five diameter classes and five height strata based on diameter and height ranges known for these forests (Table 2).
Table 2 Diameter (dbh) classes and height strata used to compare plots in this study
Tree species richness was assessed using rarefaction, a statistical method to repeatedly re-sample richness out of a random pool of samples constructed out of the field data (e.g. different plots). This allowed an unbiased comparison of richness among different plot sizes (Kindt and Coe 2005). The rarefaction curve represents the average richness of a treatment at a given number of sampled area.
Species diversity and evenness per treatment were calculated for each plot using the ln-based Shannon index (H′) (Eq. 2) and evenness using Pielou's evenness index (J′) as proposed by several authors (e.g. Alberdi et al. 2010; Kern et al. 2006). The value of J′ was calculated as H′/ln(S) where H is the Shannon diversity index and S, the species richness. Species diversity indices are dependent on sample size (Kindt and Coe 2005) so they were calculated only for the plots with selection cutting. This was done to avoid a biased comparison of species diversity due to the different sample sizes between treated and untreated plots.
To assess changes in species composition in more detail, the number of trees per species was calculated for each plot and then compared between treatments. Furthermore, the importance value (IV; Eq. 3) of each species was calculated as the sum of its relative density (RD), relative dominance (Rd) and relative frequency (RF) where density (D) is the number of individuals per hectare, dominance (d) is the basal area (BA) of each species per hectare and frequency (F) is the number of plots where a species is present divided by the total number of plots (de Iongh Arbainsyah et al. 2014; Mueller-Dombois and Ellenberg 1974).
$$ \mathrm{IV}\ \left(\mathrm{Importance}\ \mathrm{Value}\right)\kern0.5em =\kern0.5em \left(\mathrm{RD}\kern0.5em +\kern0.5em \mathrm{Rd}\kern0.5em +\kern0.5em \mathrm{RF}\right)/3 $$
Index data as well as the number of trees in the diameter and height classes were compared between the different years of observation (pre/post harvesting) and among treatments (control/HRBA/LRBA). Index data was analysed using linear mixed models (LMM) and generalised least square models (GLS). For the number of trees, generalised linear mixed models (GLMM) and generalised linear models (GLM) with Poisson distribution were used, since the data correspond to counts of individuals. Overdispersion was tested and, if found, a quasi-GLM model was used with a variance of ø × μ with ø as dispersion parameter and μ as mean, as suggested by Zuur et al. (2009).
The effect of harvesting was determined using year and treatment as fixed effects. The difference between years was analysed through comparing the treatments with selection cutting, without incorporating the control. For this analysis, plots were incorporated as random effect since repeated measurements were used in this study, thus using LMM or GLMM for this analysis. The differences among the three treatments were analysed before and after the harvesting by GLS and GLM using management as fixed effect. The assumptions of normality and heterogeneity of variance were tested through examining the model residuals and the Shapiro-Wilk test for normality. If heterogeneity of variance was found, the variance function (varIdent) was used to model heteroscedasticity to avoid transformations. Models with and without variance functions were compared through the information criteria AIC and the model presenting the lowest AIC was chosen. All statistical analysis was conducted using R 3.1 (R Core Team 2014), the R packages nlme (Pinheiro et al. 2014), vegan (Oksanen et al. 2014), BiodiversityR (Kindt and Coe 2005) and AER (Kleiber and Zeileis 2008) as well as the software InfoStat (DiRienzo et al. 2011).
Diameter distribution
The control plots showed a reverse-J diameter distribution with a slight trend to a rotated-sigmoid distribution, due to a relatively high number of large trees between 50 and 100 cm dbh (Fig. 2). The HRBA and LRBA plots showed a reverse-J diameter distribution before and after the application of single-tree selection cutting (Fig. 2).
Observed diameter distributions of a the old-growth control, b high residual basal area (HRBA) and c low residual basal area (LRBA). The histograms represent the average number of trees per hectare per treatment (four plots each) before and after harvesting
In comparison to the control plots, treated plots had around twice as many young trees between 5 and 15 cm dbh (Fig. 2) before and after harvesting. For emergent trees (100 + cm dbh), a clear trend existed, with most trees in this diameter class in the control plots (max. diameter 160 cm) and few in the LRBA plots (max. diameter 105 cm). The LRBA plots had already fewer emergent trees before harvesting (plots had been slightly subjected to "selective" cuts in the past as mentioned above) but the difference became more pronounced after single-tree selection cutting. Additionally, selection cutting reduced the number of large trees (50–100 cm dbh), especially in the LRBA plots (Fig. 2).
Structural complexity indices
Harvesting significantly reduced diameter complexity as assessed by the Gini coefficient (p = 0.0012), the Shannon index (p = 0.0039) and the standard deviation (p = 0.0002) (Table 3). Moreover, all three indices provided a logical and consistent ranking of diameter complexity, with the highest index values in the control, followed by the HRBA and then by the LRBA plots after harvesting (Table 3).
Table 3 Mean values for the Gini coefficient, the Shannon index and the standard deviation for diameter at breast height (dbh) and height data and for the Shannon index and the evenness index for species diversity before and after harvesting. Values are expressed as mean ± 1 standard deviation. Significant treatment differences with the control are shown with asterisks after the index value, with *p = 0.05–0.01, **p = 0.01–0.001 and ***p = <0.001, respectively. Significant harvesting impacts are mentioned in the text
The Gini coefficient for diameter complexity was not significantly different between control and treated plots, while the Shannon index had already significantly higher values in the control plots before harvesting (see asterisks, Table 3). The significance of this difference became more pronounced for both HRBA and LRBA plots through harvesting (Table 3). The standard deviation showed the same pattern as the Shannon index (Table 3). Before harvesting, control plots had already a significantly higher complexity than the treated plots, but these differences were only slightly significant (Table 3). After harvesting, these differences became highly significant for both HRBA and LRBA plots.
The Gini coefficient of height complexity showed no significant difference between the untreated and treated plots (Table 3). The Shannon index showed a significantly higher height complexity after harvesting in the unmanaged plots than in the managed ones. The same was observed for the standard deviation, which was significantly higher in the control plots. No clear differences in height complexity existed between the HRBA and LRBA plots.
Diameter classes and height strata
Small-sized trees, small diameter classes (very small and small) and low height strata (low understorey and upper understorey) were more abundant in the treated plots than in the control plots before harvesting (Table 4). In regard to diameter classes, harvesting reduced the number of small-sized trees only marginally, leaving an abundant residual growing stock. On the contrary, intermediate- to very large-sized trees (dbh 25 cm+, height 23 m+) were strongly influenced by harvesting. The density of trees with an intermediate diameter was already significantly higher in the control compared to the LRBA plots before harvesting (see asterisks, Table 4). Harvesting significantly (p = 0.0011) reduced their number, leading to a significantly higher number of trees in the intermediate diameter class in the control compared to selection plots, with a more pronounced difference for the LRBA plots after harvesting (Table 4).
Table 4 Mean number of trees per hectare per diameter class (trees dbh ≥ 5 cm) and height strata (trees dbh ≥ 10 cm) before and after harvesting. Diameter classes and height strata are explained in Table 3. Values are expressed as mean ± 1 standard deviation. Significant treatment differences with the control are shown through an asterisk after the mean number of trees, with *p = 0.05–0.01, **p = 0.01–0.001 and ***p ≤ 0.001, respectively. Significant harvesting impacts are mentioned in the text
The number of large diameter trees was nearly the same for control, HRBA and LRBA plots before harvesting (Table 4). Harvesting significantly reduced their density (p < 0.0001), resulting in a significantly higher number of large diameter trees in the control compared to the LRBA but not to HRBA plots (Table 4). The number of very large diameter trees was highest in the control already before harvesting, but without significant differences due to the high variability between plots (Table 4). Harvesting significantly reduced their number (p < 0.0001), resulting in a significantly higher tree density of these trees in the control compared to LRBA plots (Table 4). No significant differences could be found between control and HRBA plots (Table 4). The HRBA plots had a higher number of intermediate to very large diameter trees than their LRBA counterparts (Table 4).
In regard to height classes, no significant difference was found for the number of low canopy trees, while upper canopy trees were significantly more abundant in the control than in the treated plots (Table 4). The difference between untreated and treated plots was even more pronounced for emergent trees with significantly more emergent trees in the control (72 trees ha−1) compared to HRBA (19 trees ha−1) and LRBA (10 trees ha−1) plots after harvesting (Table 4).
Tree species richness and diversity
Tree species richness was not changed as a result of harvesting. Thus, only the results after harvesting are presented here. The confidence intervals of the three rarefaction curves overlap at the total sampled area of the control plots (Fig. 3), reflecting that tree species richness was not significantly different between unmanaged and managed plots.
Species rarefaction curves showing the mean tree species richness per sampled area for the control, high and low residual basal area (HRBA and LRBA) plots after harvesting. The rarefaction curves were calculated through repeatedly re-sampling richness out of a random pool of samples constructed out of the four sampled plots
Tree species diversity, as evaluated through the Shannon index, did not significantly change after harvesting (comp. Table 3). Also, no significant differences in tree species diversity existed between treated and untreated plots (comp. Table 3). The same was observed for tree species evenness which was not significantly changed through harvesting and was not different between treated and untreated plots (comp. Table 3).
Tree species composition
In general, treated and untreated plots showed a similar species composition regarding dominant tree species, as evaluated through the average number of trees per species and the importance value (IV) of each species before harvesting (Table 5). The shade-tolerant species A. punctatum and Myrceugenia planipes (H et A.) Berg, however, had far higher IVs in the control plots compared with the treated ones (Table 5). The application of the single-tree selection cutting regime applied in this study had the strongest effect on E. cordifolia. The IV of this species decreased by 21%, and 20% of individuals were removed as a result of the HRBA treatment (Table 5). The LRBA treatment had a more severe effect with the IV decreasing by 27%, and 32% of individuals being removed (Table 5). The decrease in IV of E. cordifolia through management led to an increase of the IV of most other species (Table 5). The only other species that experienced a clear decline in the number of trees through harvesting was L. philippiana, but the strong loss of E. cordifolia still led to a rise in its IV. Except for E. cordifolia, LRBA management did not induce stronger changes in the tree species community as compared with HRBA management (Table 5). The selection cutting regime led only to marginal changes in the number of trees of all less frequent species, and no species were lost through harvesting (Table 5). Moreover, the extraction of dominant species led to a rise of IV of several less frequent species in the forest community (Table 5).
Table 5 Average number of trees per hectare and importance value (IV) in % per treatment and species in 2012 and 2014
Key structural attributes and biodiversity conservation
We examined changes in key structural attributes such as reverse-J diameter distributions, complex vertical layering, variability of tree sizes, presence of advance regeneration and large/emergent trees as measures of old-growthness (Bauhus et al. 2009). Both residual basal area regimes evaluated in this study on single-tree selection cutting were found to maintain a balanced uneven-aged structure, forest cover continuity and a sufficient growing stock of small-sized trees. All plots, managed and unmanaged, were characterised by reverse-J shaped diameter distributions before and after harvesting.
These findings are in accordance with studies in other forest types, where selection cutting maintained these structural forest attributes over decades while providing timber yields at regular intervals (e.g. Pukkala and Gadow 2012; Gronewold et al. 2010; O'Hara et al. 2007; Keeton 2006; Bagnaresi 2002). The observed reverse-J shaped distributions are typical for old-growth stands in the Valdivian Costal Range and the Valdivian Andes (Donoso 2013; Donoso 2005), showing that single-tree selection cutting is able to maintain this old-growth attribute. Still, results of this study cover only the short-term impacts of selection cutting (2 years), and there may be lag effects with sensitive species. However, the growth model predictions of Rüger et al. (2007), which were parameterised for the evergreen forest type on Chiloé Island, Chile, suggest that single-tree selection cutting can maintain the above-mentioned forest attributes also on the long term. Moreover, balanced structures with similar crown covers for small-, intermediate- and large-sized trees allow more abundant regeneration and tree growth in the evergreen forest type in Chile than unbalanced ones (Schütz et al. 2012; Donoso and Nyland 2005; Donoso 2005).
Further considerations should be given to the fact that plots with a high residual basal area (HRBA) tend to approximate old-growth conditions more closely through maintaining higher numbers of large-sized trees and higher diameter complexity than plots with a low residual basal area (LRBA) (Tables 3 and 4). Similarly, Gronewold et al. (2010) concluded that (after a survey of 57 years in northern hardwood stands of North America managed through single-tree selection cutting) stands with high residual basal areas better approximated the natural disturbance history and diameter distributions of unmanaged uneven-aged stands, while low residual basal areas resulted in simpler and more regulated distributions. The higher number of small trees (advanced regeneration) already present before cutting compared to the control plots in our study most likely results from higher light availability, especially in the LRBA plots caused by previous illegal cuttings (as mentioned before).
The only shortcoming detected was the significant reduction and lower numbers of large-sized (dbh 50 cm+, height 23 m+) and emergent trees (height 30 m+) in the treated plots (especially in LRBA ones), compared to the control plots after harvesting. This finding was supported by the significant reduction of diameter and height complexity (shown by all three indices) and significantly higher diameter and height complexity in the untreated plots (shown by the Shannon index and standard deviation) as a result of a reduction in the range of tree diameters and heights. All three structural indices have been widely used to quantify diameter and height complexity in managed and unmanaged forest stands and to a lesser extent to evaluate the impacts of single-tree selection cutting (Torras et al. 2012; O'Hara et al. 2007; Lexerød and Eid 2006; McElhinny et al. 2005; Acker et al. 1998). Similar to results of this study, Acker et al. (1998) reported higher values of the standard deviation of tree diameter in old-growth northern hardwood stands of North America compared to managed ones, but there are also studies that report a rise of diameter and height complexity under selection silviculture over time using the same three indices (Torras et al. 2012; O'Hara et al. 2007).
In particular, very large and emergent trees were far less numerous in the managed plots. Similarly, numerous other studies have found that stands managed through selection cutting have less large trees than comparable old-growth stands (e.g. Torras and Saura 2008; Rüger et al. 2007; Keeton 2006; Angers et al. 2005; Crow et al. 2002). Furthermore, stands with lower residual basal areas were found to present less large trees than stands with higher residual basal areas (Gronewold et al. 2010; Rüger et al. 2007). This is partly consistent with the findings of this study where only LRBA plots presented significantly lower numbers of large and very large diameter trees than the control plots.
Large-sized and emergent trees are, however, an important habitat for many forest dwelling species and communities that depend on this specific structural attribute of old-growth forests (Bauhus et al. 2009), such as cavity-dependent animal species like birds and mammals as well as bryophytes, lichens, fungi, and saproxylic beetles (Paillet et al. 2010; Bauhus et al. 2009). One specific example in the Chilean evergreen rainforest is the abundant flora of endemic epiphytic plants that depend on emergent trees (Díaz et al. 2010) and their greater frequency on trees with large diameters (Muñoz et al. 2003). Furthermore, bird species diversity in the evergreen forest type can be predicted by the presence of emergent trees, and their diversity is consequently higher in old-growth than in early or mid-successional forest stands (Díaz et al. 2005). It follows that a certain number of large-sized, especially emergent trees, in managed stands is crucial for biodiversity conservation. Although we do not deal with dead wood (i.e. snags and coarse woody debris) in this paper, preliminary results suggest that plots subjected to single-tree selection cutting have similar or even higher amounts of this key structural attribute (sensu Bauhus et al. 2009) compared to unmanaged old-growth forests (Schnabel et al., unpublished).
Impacts on tree species richness, diversity and composition
An important attribute for old-growthness is the high number of late successional tree species (Bauhus et al. 2009), i.e. shade-tolerant and emergent mid-tolerant ones in the evergreen forest type. Tree species richness, diversity and evenness were not changed through selection cutting in the short-term, which is in accordance with findings in northern hardwood forest in North America (Angers et al. 2005; Crow et al. 2002). It is therefore reasonable to conclude that the direct harvesting effects of single-tree selection (e.g. tree felling), as conducted in the present study, are compatible with preserving tree species richness and diversity within the evergreen forest type in the short term. If management guidelines such as those applied in this study are used, the same should also apply for future applications.
The numbers of less frequently occurring tree species were not reduced through selection cutting; a fact that further supports the conclusion that single-tree selection cutting is compatible with preserving tree species diversity. While most dominant tree species experienced no severe losses through selection cutting, a clear impact was noted for the mid-tolerant species E. cordifolia. It clearly declined in abundance and IV, although it remained the second species in IV. The reason for this was the preferential harvest of old/large, poor-quality E. cordifolia trees, to improve the quality of the residual stock in the first harvest. Moreover, E. cordifolia trees were mostly large individuals, since regeneration for this mid-tolerant species is generally scarce under closed forests (Escobar et al. 2006; Donoso and Nyland 2005) and was thus strongly impacted by the harvesting criteria of a maximum residual diameter of 80 cm. In future harvests, impacts are likely to be more equally distributed, as most of the defective E. cordifolia trees would have already been harvested after this first selection cut. Also, E. cordifolia is one of the target tree species of selection silviculture due to its high economic value and expected fast growth and abundant regeneration at low residual basal areas (especially at 40 m2 ha−1). In addition, retaining some emergent E. cordifolia trees is a conservation priority as this species harbours an exceptional high diversity and abundance of epiphytes, acting as key structure for biodiversity conservation and ecosystem processes like water and nutrient cycling (Díaz et al. 2010).
Finally, single-tree selection might favour both the recruitment of mid-tolerant species like E. cordifolia (Torras and Saura 2008; Angers et al. 2005) and/or shade-tolerant species (Keyser and Loftis 2013; Gronewold et al. 2010; Rüger et al. 2007) depending on the size of crown opening and consequent light availability (i.e. LRBA should induce more regeneration of mid-tolerant tree species like E. cordifolia). As the effects on tree regeneration in the evergreen forest type remain unknown in the field, different harvesting intensities might be currently the best option to promote the regeneration of both mid- and shade-tolerant tree species.
Implications for management
Overall, our results support the claim that single-tree selection cutting is a promising silvicultural approach for the evergreen forest type. This approach is certainly more promising than the currently supported selective harvesting guidelines of the Chilean law which do not control for a balanced residual stock and allow the harvest of 35% of the basal area in 5-year cutting cycles, which is unsustainable (Donoso 2013; Schütz et al. 2012). In contrast, the only negative affect detected for the single-tree selection cutting regime applied in the present study was the clearly lower number of large-sized and emergent trees in managed plots, a key structural attribute of old-growth forests and crucial for biodiversity conservation.
The management strategy of single-tree selection cutting would need to be adjusted by forest managers who wish to preserve some emergent and large-sized trees in stands managed through selection silviculture. The use of a maximum residual diameter, such as 80 cm in this study, actually impairs the preservation of large-sized trees (Keeton 2006). One possibility in this context would be the intentional retention of a still to be a specified number of large (>80 cm diameter) trees, especially emergent ones (e.g. Bauhus et al. 2009; Angers et al. 2005). We did maintain some large trees in the managed stands in this study because otherwise, basal area harvesting would have been too destructive, but our results suggest that leaving trees above a given maximum diameter must be an ongoing requirement. In particular, retaining emergent trees with crowns over the main canopy of the residual stock is beneficial as: (a) they may not impede the growth of either young or mature trees nor tree regeneration in the evergreen forest type (Donoso 2005); and (b) they are key structures for biodiversity conservation (Díaz et al. 2010). An additional possibility might be the use of diameter-guiding curves other than the reverse-J distribution curve used in the current study. For example, a rotated sigmoid distribution curve may satisfy ecological needs more closely through allocating more basal area and growing space to larger diameter classes (Keeton 2006). The HRBA plots tended to better approximate old-growth conditions than LRBA plots, in terms of higher numbers of large-sized trees and higher structural complexity. However, it remains untested in the field as to which residual basal area selection cutting generates sufficient light availability to allow the regeneration of both shade-tolerant and mid-tolerant species in the evergreen forest type. Thus, using a combination of the two residual basal area regimes examined here (HRBA with 60 m2 ha−1 and LRBA with 40 m2 ha−1) might be an advisable option, which would contribute to more diverse and species-rich stands and additionally to the generation of a more heterogeneous forest structure on a broader scale, e.g. Angers et al. (2005).
Little is known about the required quantity and spatial distribution of retained emergent trees, which would be necessary to develop sound ecological management guidelines, like retention targets for biodiversity conservation (Bauhus et al. 2009; McElhinny et al. 2005). Due to a lack of information on this topic in Chile and in other ecosystems (e.g. Bauhus et al. 2009; McElhinny et al. 2005), this is a major challenge for uneven-aged silviculture, especially in forests of high diversity and endemism like the evergreen temperate forests of south-central Chile. A final concern in Chile (and elsewhere) is that Chusquea bamboos in the understorey may be a threat for regeneration (Donoso and Nyland 2005). These are usually light-demanding species, so the creation of canopy openings above a certain size following selection cuts, especially if using group selection, could promote Chusquea spp. regeneration. This poses an important research challenge for selection silviculture in Chilean forests, i.e. determining adequate densities (for example expressed in basal area) that would maintain low levels of Chusquea spp. cover while allowing the forest stand to sustain good growth rates. Donoso (2002) studied uneven-aged forest with basal areas from 38 to 140 m2 ha−1 in the lowlands of south-central Chile, and Chusquea spp. had levels of cover that ranged from 3 to 12%. This result suggests that managed forest stands with residual basal areas as low as 40 m2 ha−1 should not have major competition from Chusquea spp. upon tree regeneration.
From a management perspective, a great advantage of selection silviculture is the production of large logs for saw timber or veneer, products of high commercial value, while in the same time, logs of smaller dimensions are harvested that can be used for firewood or charcoal production (Moorman et al. 2013; Puettmann et al. 2015). Siebert (1998), Donoso (2002) and Donoso et al. (2009) have proposed target maximum diameters of 60–90 cm (80 cm in this study), which should generate high-value products. Operationally, harvesting requires skilled workers and marked stands after determining adequate marking guides according to the BDq or a similar technique. In addition, Donoso (2002) proposed 10-year cutting cycles for evergreen forests on productive low-elevation sites. Single-tree selection would thus especially offer landholders with small properties a variety of wood products at regular intervals (Puettmann et al. 2015).
Overall, major considerations to better conserve structural features and biodiversity of old-growth forests in managed stands, while also achieving good rates of timber productivity, could include: (a) retaining a certain number of large-sized, especially emergent trees; (b) using a diverse but relatively narrow range of residual basal areas that may support good development of relatively fast-growing and valuable mid-tolerant tree species associated to shade-tolerant ones; and (c) applying diameter distributions that allow for a greater allocation of basal area in relatively large trees. These considerations for stand variability in managed forests should be included in forest regulations, which should adapt to new knowledge generated through research.
Considering that mostly, we did not cut beyond 35% harvested basal area, the maximum established in Chilean regulations, research in selection silviculture should also evaluate an ample range of harvesting intensities using a relatively ample range of initial and residual basal areas. This would allow a more robust information on thresholds to conserve in the best possible manner "old-growthness" (sensu Bauhus et al. (2009)) in managed forest ecosystems.
We examined changes in forest structure and tree species composition as well as possible detrimental effects on key attributes of old-growthness in stands managed through single-tree selection cutting. Through both harvest variants, high and low residual basal areas (HRBA and LRBA), a balanced, uneven-aged structure with reverse-J diameter distribution and forest cover were maintained. Also, a sufficient growing stock of small-sized trees was kept. Moreover, neither tree species richness, diversity and evenness, nor the presence of less frequent species were negatively affected on the short term. As the effects on tree regeneration remain unknown, using a combination of HRBA and LRBA may be advisable to support good development of relatively fast-growing and valuable mid-tolerant tree species associated with shade-tolerant ones. The only negative effect detected was the clearly lower number of large-sized and emergent trees in managed plots (especially for LRBA), which are a key structural attribute of old-growth forests and crucial for biodiversity conservation. These results suggest that single-tree selection cutting, if adjusted to retain a certain number of large-sized and emergent trees, can serve as a possible means to preserve many old-growth structural and compositional attributes of the evergreen forest type in managed stands while harvesting timber for the landowners. Future experiments should test the effects of alternative selection cutting upon structural heterogeneity, diversity and productivity to balance the varied societal demands of ecosystem services expected from forest management.
Dbh:
Diameter at breast height
GLMM:
Generalised linear mixed models
GLS:
Generalised least square models
HRBA:
High residual basal area
Importance value
LMM:
Linear mixed models
LRBA:
Low residual basal area
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We sincerely thank Jürgen Huss for commenting and revising this manuscript during its elaboration and Simone Ciuti for the statistical support. Moreover, we acknowledge the dedication of our field workers Ronald Rocco, Nicole Raimilla Fonseca and Pol Bacardit from the University Austral de Chile. Finally, we greatly appreciated the accommodations provided by the Lomas de Sol community during our stay in Llancahue. PJ Donoso acknowledges the support of FIBN-CONAF project no. 034/2011 and FONDECYT project no. 1150496.
P.J. Donoso thanks FONDECYT Grant No. 1150496 and Project 034/2011 of the "Fondo de Investigación en Bosque Nativo" administered by the forest service CONAF.
The dataset(s) supporting the conclusions of this article are included within the article (and its Additional file 1). The use of the Llancahue database (in Additional file 1) requires the authorization of the authors.
Chair of Silviculture, Faculty of Environment and Natural Resources, University of Freiburg, Tennenbacherstr. 4, 79106, Freiburg, Germany
Florian Schnabel
Insituto de Bosques y Sociedad, Facultad de Ciencias Forestales y Recursos Naturales, Universidad Austral de Chile, Casilla 567, Valdivia, Chile
Pablo J. Donoso
Faculty of Environment and Natural Resources, University of Freiburg, Tennenbacherstr. 4, 79085, Freiburg, Germany
Carolin Winter
Search for Florian Schnabel in:
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FS, the lead author, conceived and designed this experiment. He performed the experiment through planning and leading the field data collection, analysed the data and wrote most parts of the paper. PD conceived and coordinated the general single-tree selection cutting experiment, designed and supervised the management-interventions and established the plots. He advised in conceiving this experiment and in the data analysis, supervised the process of writing the paper and wrote parts himself. CW helped in taking the field data and analysed minor parts. She contributed during the whole study through revisions and wrote parts of the paper. FS and CW created figures and tables and formatted the paper. Author contribution rephrased without making a change to the content to provide better clarity of the indiviudal contributions. This change has been approved by CW and FS. All authors read and approved the final manuscript.
Correspondence to Florian Schnabel.
Data of the Llancahue Experimental Forest in south-central Chile. (XLSX 427 kb)
Schnabel, F., Donoso, P.J. & Winter, C. Short-term effects of single-tree selection cutting on stand structure and tree species composition in Valdivian rainforests of Chile. N.Z. j. of For. Sci. 47, 21 (2017) doi:10.1186/s40490-017-0103-5
Uneven-aged silviculture
Old-growth forest attributes
Evergreen forest type
Temperate rainforests | CommonCrawl |
Let $f(x)$ be a third-degree polynomial with real coefficients satisfying \[|f(1)|=|f(2)|=|f(3)|=|f(5)|=|f(6)|=|f(7)|=12.\]Find $|f(0)|$.
Each of the six values $f(1),$ $f(2),$ $f(3),$ $f(5),$ $f(6),$ $f(7)$ is equal to 12 or $-12.$ The equation $f(x) = 12$ has at most three roots, and the equation $f(x) = -12$ has at most three roots, so exactly three of the values are equal to 12, and the other three are equal to $-12.$
Furthermore, let $s$ be the sum of the $x$ that such that $f(x) = 12.$ Then by Vieta's formulas, the sum of the $x$ such that $f(x) = -12$ is also equal to $s.$ (The polynomials $f(x) - 12$ and $f(x) + 12$ only differ in the constant term.) Hence,
\[2s = 1 + 2 + 3 + 5 + 6 + 7 = 24,\]so $s = 12.$
The only ways to get three numbers from $\{1, 2, 3, 5, 6, 7\}$ to add up to 12 are $1 + 5 + 6$ and $2 + 3 + 7.$ Without loss of generality, assume that $f(1) = f(5) = f(6) = -12$ and $f(2) = f(3) = f(7) = 12.$
Let $g(x) = f(x) + 12.$ Then $g(x)$ is a cubic polynomial, and $g(1) = g(5) = g(6) = 0,$ so
\[g(x) = c(x - 1)(x - 5)(x - 6)\]for some constant $c.$ Also, $g(2) = 24,$ so
\[24 = c(2 - 1)(2 - 5)(2 - 6).\]This leads to $c = 2.$ Then $g(x) = 2(x - 1)(x - 5)(x - 6),$ so
\[f(x) = 2(x - 1)(x - 5)(x - 6) - 12.\]In particular, $|f(0)| = \boxed{72}.$ | Math Dataset |
\begin{document}
\vphantom{} \title{Banach spaces without approximation properties of type $p$}
\begin{abstract} The main purpose of this note is to show that the question posed in the paper of Sinha D.P. and Karn A.K. ("Compact operators which factor through subspaces of $l_p$ Math. Nachr. 281, 2008, 412-423; see the very end of that paper) has a negative answer, and that the answer could be obtained,
essentially, in 1985 after the papers [2], [3] by Reinov O.I. have been appeared in 1982 and in 1985 respectively. \end{abstract}
\vskip 0.3cm
The main purpose of this note is to show that the question posed in [1] (see the very end of that paper) has a negative answer, and that the answer could be obtained,
essentially, in 1985 after the papers [2], [3] have been appeared in 1982 and in 1985 respectively.
For the sake of completeness (and since the article [3] is now quite difficult to get) we will reconstruct some of the facts from [3], maybe, in a somewhat changed form (a translation of [3] with some remarks can be found in arXiv:1002.3902v1 [math.FA]).
All the notation and terminology, we use, are from [2], [4], or [5] and more or less standard. For the general theory of absolutely $p$-summing, $p$-nuclear and other operator ideals, we refer to [6]. We will keep, in particular, the following standard notations. If $A$ is a bounded subset of a Banach space $X$, then $\ove{\Gamma(A)}$ is the closed absolutely convex hull of $A;$ $ X_A$ is the Banach space with "the unit ball" $ \ove{ \Gamma(A)}$; $ \Phi_A: X_A\to X$ is the canonical embedding. For $ B\subset X,$
it is denoted by $ \overline{B}^{\,\tau}$ and $ \ove{B}^{\,\|\cdot\|}$
the closures of the set $ B$ in the topology $ \tau$ and in the norm $ \|\cdot\|$
respectively. When it is necessary, we denote by $ \|\cdot\|_X$ the norm in $X.$ Other notations:\, $ \Pi_p,$ $ \operatorname{QN}_p,$ $ \operatorname{N}_p,$ $ \operatorname{I}_p$ are the ideals of absolutely $p$-summing, quasi-$p$-nuclear, $p$-nuclear, strictly $p$-integral operators, respectively; $ X^*\wh\otimes_p Y$ is the complete tensor product associated with $ \operatorname{N}_p(X,Y);$ $ X^*\wh{\wh\otimes}_p Y= \ove{X^* \otimes Y}^{\,\pi_p}$ \, (i.e. the closure of the set of finite dimensional operators in $ \Pi_p(X,Y)).$ Finally, if $ p\in[1,+\infty]$ then $ p'$ is the adjoint exponent.
Now, we recall the main definition from [3] of the topology $\tau_p$ of the $\pi_p$-compact convergence, which is just the same as the $\lambda_p$-topology in [1]\footnote{
Definition 4.3 from [1]:\,
"Given a compact subset $K\subset X$ we define a seminorm $||\cdot||_K$ on $\Pi_p(X,Y)$ given by
$||T||_K=\inf\{\kappa_p^d(Ti_Z):\, i_Z:Z\to X \text{ as above}\}.$
The family of seminorms $\{||\cdot||_K:\, K\subset X \text{ is compact}\}$\, determines a locally convex topology $\lambda_p$ on $\Pi_p(X,Y).$"; --- for details, and for the definition of maps $i_Z,$ see [1], Lemma 4.2.
}, where it was proved, essentially, that that $T\in K_p^d(X,Y)$ if and only if $T\in QN_p(X,Y).$
For Banach spaces $X, Y,$ the {\it topology $ \tau_p$\ of $ \pi_p$\!-compact convergence}\ in the space $ \Pi_p(Y,X)$ is the topology, a local base (in zero) of which is defined by sets of type $$ \omega_{K,\e}= \left\{ U\in \Pi_p(Y,X):\ \pi_p(U\Phi_K)<\e\right\}, $$ where $ \e>0,$\, $ K=\ove{\Gamma(K)}$ --- a compact subset of $ Y.$
\vskip 0.2cm {\bf Proposition 1} [3].\ {\it Let $ \operatorname{R}$ be a linear subspace in $ \Pi_p(Y,X),$ containing $ Y^*\otimes X.$ Then $ (\operatorname{R},\tau_p)'$ is isomorphic to a factor space of the space $ X^*\wh\otimes_{p'} Y.$ More precisely, if $ \ffi\in (\operatorname{R}, \tau_p)',$ then there exists an element $ z=\sum_1^\infty x'_n\otimes y_n\in X^*\wh\otimes_{p'} Y$ such that $$ \ffi(U)= \operatorname{trace}\,\, U\circ z,\ \, U\in \operatorname{R}. \eqno(*) $$ On the other hand, for every $ z\in X^*\wh\otimes_{p'} Y$ the relation $ (*)$ defines a linear continuous functional on $ (\operatorname{R},\tau_p).$
} \vskip 0.1cm
{\it Proof.}\
Let $ \ffi$ be a linear continuous functional on $ (\operatorname{R},\tau_p).$ Then one can find a neighborhood of zero $ \omega_{K}=\omega_{K,\e},$ such that $ \ffi$ is bounded on it:
$ \forall\, U\in\omega_K, \ |\ffi(U)|\lee 1.$ We may assume that $ \e=1.$ Consider the operator $ U\Phi_K:\,Y_K\ovs{\Phi_K}\longrightarrow Y
\ovs{U}\longrightarrow X.$ Since the mapping $ \Phi_K$ is compact, $ U\Phi_K\in \operatorname{QN}_p(Y_K,X).$ Put $ \ffi_K(U\Phi_K)=\ffi(U)$ for $ U\in \operatorname{R}.$ On the linear subspace $ \operatorname{R}_K= \left\{ V\in \operatorname{QN}_p(Y_K,X):\ V=U\Phi_K \right\}$ of the space $ \operatorname{QN}_p(Y_K,X),$ the linear functional $ \ffi_K$ is bounded: if $ V=U\Phi_K\in \operatorname{R}_K$ and $ \pi_p(V)\lee 1,$
then $ |\ffi_K(V)|=|\ffi(U)|\lee 1.$ Therefore, $ \ffi_K$ can be extended to a linear continuous functional $ \wt\ffi$ on the whole $ \operatorname{QN}_p(Y_K,X);$ moreover, because of the injectivity of the ideal $ \operatorname{QN}_p,$ considering $ X$ as a subspace of some space $ C(K_0),$ we may assume that $ \wt\ffi\in\operatorname{QN}_p(Y_K, C(K_0))^*.$ Let us mention that $$ \wt\ffi(jU\Phi_K)=\ffi_K(U\Phi_K)=\ffi(U) \eqno(1) $$ (here $ j$ is an isometric embedding of $ X$ into $ C(K_0)$).
Furthermore, since $ \operatorname{QN}_p(Y_K, C(K_0))^*= \operatorname{I}_{p'}(C(K_0), (Y_K)^{**}),$ we can find an operator $ \Psi: C(K_0)\to (Y_K)^{**},$ for which $$ \wt\ffi(A)= \operatorname{trace}\, \Psi A,\ \, A\in (Y_K)^*\otimes C(K_0). $$
Let $ A_n\in (Y_K)^*\otimes C(K_0),\, $ $ \pi_p(A_n-jU\Phi_K)\to 0.$ Then $$ \wt\ffi(jU\Phi_K)= \lim\, \operatorname{trace}\, \Psi A_n. \eqno(2) $$
Consider the operator $ \Phi_K^{**}\Psi: C(K_0)\ovs{\Psi}\longrightarrow (Y_K)^{**}
\ovs{\Phi_K^{**}}\longrightarrow Y.$ Since $ \Psi\in \operatorname{I}_{p'},$ and $ \Phi_K$ is compact, we have $ \Phi_K^{**}\Psi \in \operatorname{N}_{p'}(C(K_0), Y)= C(K_0)^*\wh\otimes_{p'} Y.$
Let $ \sum_1^\infty \mu_n\otimes y_n\in C(K_0)^*\wh\otimes_{p'} Y$ be a representation of the operator $ \Phi^{**}_K\Psi.$ Put $ z=\sum j^*(\mu_n)\otimes y_n.$ The element $ z$ generates an operator $ \Phi_K^{**}\Psi j$ from $ X$ to $ Y.$ We will show now that $ \operatorname{trace}\, U\circ z=\wt\ffi(jU\Phi_K)$ (note that $ U\circ z$ is an element of the space $ X^*\wh\otimes X,$ so the trace is well defined). We have:
$$ \operatorname{trace}\, U\circ z= \operatorname{trace}\, \( \sum j^*(\mu_n)\otimes Uy_n\)=
\sum \langle} \def\wh{\widehat j^*(\mu_n), Uy_n\rangle} \def\wt{\widetilde = $$ $$\sum \langle} \def\wh{\widehat \mu_n, jUy_n\rangle} \def\wt{\widetilde= \operatorname{trace}\, jU\Phi_K^{**}\Psi=\operatorname{trace}\, (jU\Phi_K)^{**}\Psi, \eqno(3) $$
where $ (jU\Phi_K)^{**}\Psi:
C(K_0)\ovs{\Psi}\longrightarrow (Y_K)^{**}
\ovs{\Phi_K^{**}}\longrightarrow Y
\ovs{U}\longrightarrow X\ovs{j}\longrightarrow C(K_0).$
Since $ \pi_p(A_n- jU\Phi_K)\to 0,$ then $ \pi_p\(A^{**}_n - (jU\Phi_K)^{**}\)\to 0.$ Moreover, if $A:= A_n=\sum_1^N w_m\otimes f_m\in (Y_K)^*\otimes C(K_0),$ then $$ \operatorname{trace}\, A_n^{**}\Psi= \sum_m \langle} \def\wh{\widehat \Psi^*w_m, f_m\rangle} \def\wt{\widetilde =
\sum_m \langle} \def\wh{\widehat w_m, \Psi f_m\rangle} \def\wt{\widetilde= \operatorname{trace}\, \Psi A. $$
Hence, $ \operatorname{trace}\, (jU\Phi_K)^{**}\Psi = \lim\,\operatorname{trace}\, A^{**}_n\Psi= \lim\,\operatorname{trace}\, \Psi A_n.$ Now, it follows from (3) and (2) that $ \wt\ffi(jU\Phi_K)=\operatorname{trace}\, U\circ z.$ Finally, we get from (1): $ \ffi(U)= \operatorname{trace}\, U\circ z.$ Thus, the functional $ \ffi$ is defined by an element of $ X^*\wh\otimes_{p'} Y.$
Inversely, if $ z\in X^*\wh\otimes_{p'} Y,$ put $ \ffi(U)=\operatorname{trace}\, U\circ z$ for $ U\in \operatorname{R}$ (the trace is defined since $ U\circ z\in X^*\wh\otimes X).$ We have to show that the linear functional $ \ffi$ is bounded on a neighborhood $ \omega_{K,\e}$ of zero in $ \tau_p.$ For this, we need the following fact, which proof is rather standard (see [3], Lemma 1.2):
{\it If $ z\in X^*\wh\otimes_q Y,$ then $ z\in X^*\wh\otimes_q Y_K,$ where $ K=\ove{\Gamma(K)}$ is a compact in $ Y.$ }\
Now, let $ K$ be a compact subset of $ Y,$ for which $ z\in X^*\wh\otimes_{p'} Y_K.$ If $ U\in \omega_{K,1},$ then $ \pi_p(U\Phi_K)<1$ and
$ |\operatorname{trace}\, U\Phi_K\circ z|\lee
\|z\|_{X^*\wh\otimes_{p'} Y_K}\cdot$ $\pi_p(U\Phi_K)\lee C.$
\vskip 0.1cm
{\bf Corollary 1.}\it \, $ (\operatorname{R},\tau_p)'=(\operatorname{R},\sigma)',$ where $ \sigma=\sigma(\operatorname{R}, X^*\wh\otimes_{p'} Y).$ Thus, the closures of convex subsets of the space $ \Pi_p(Y, X)$ in $ \tau_p$ and in $ \sigma$ are the same.
\rm
\vskip 0.1cm
{\bf Proposition 2} [3]. \it If the canonical mapping $ j: X^*\wh\otimes_{p'} Y\to \operatorname{N}_{p'}(X,Y)$ is one-to-one then $ \Pi_p(Y,X)= \ove{Y^*\otimes X}^{\,\tau_p}.$ \vskip 0.1cm
\rm {\it Proof} If the map $ j$ is one-to-one then the annihilator $ j^{-1}(0)^\perp$ of its kernel in the space, dual to $ X^*\wh\otimes_{p'}Y,$ coincides with $ \Pi_p(Y, X^{**}).$ On the other hand, in any case $ j^{-1}(0)^\perp= \ove{Y^*\otimes X}^{\,*}$ (the closure in $ {}^*$\!-weak topology of the space $ \Pi_p(Y, X^{**}));$ by Corollary from Proposition 1, $$ \Pi_p(Y,X)\cap \ove{Y^*\otimes X}^{\,*}= \ove{Y^*\otimes X}^{\,\tau_p}. $$
Therefore, $ \Pi_p(Y,X)= \ove{Y^*\otimes X}^{\,\tau_p}.$ \vskip 0.1cm
For a reflexive space $ X,$ the dual space to $ X^*\wh\otimes_{p'} Y$ is equal to $ \Pi_p(Y,X).$ Consequently, it follows from the last two statements
\vskip 0.1cm
{\bf Corollary 2.}\it \, For a reflexive space $ X$ the canonical mapping $ j: X^*\wh\otimes_{p'} Y\to \operatorname{N}_{p'}(X,Y)$ is one-to-one iff the set of finite rank operators is dense in the space $ \Pi_p(Y,X)$ in the topology $ \tau_p$\ of $ \pi_p$-compact convergence. \rm
\vskip 0.1cm
Recall a definition of the {\it approximation property $AP_p$ of order}\, $p, p\in (0,\infty]$ (see, e.g., [2] and [7]): a Banach space $X$ has the $AP_p$ if for every Banach space $Y$ (equivalently, for every reflexive Banach space $Y,$ see [2] or [5]) one has the equality
$Y^*\wh\otimes_p X = \operatorname{N}_p(Y,X).$ It follows now from Proposition 1 and Corollary 2:
\vskip 0.1cm {\bf Corollary 3.}\it \, For $p\in[1,\infty]$ and for every Banach space $ X$ the following are equivalent:
$1)$ $X$ has the $AP_p;$
$2)$ for each Banach space $ Y$\ $ \ove{X^*\otimes_{p'} Y}^{\,\tau_{p'}}= \Pi_{p'}(X,Y);$
$3)$ for each reflexive Banach space $ Y$\ $ \ove{X^*\otimes_{p'} Y}^{\,\tau_{p'}}= \Pi_{p'}(X,Y);$
$3')$ for each reflexive Banach space $ Y$\ $ \ove{X^*\otimes_{p'} Y}^{\,\lambda_{p'}}= \Pi_{p'}(X,Y).$
\vskip 0.1cm \rm
As shown in [2], [4] and [5] , for each $p, p\in [1,\infty], p\neq2,$ there exist reflexive Banach spaces without the $AP_p.$ Thus, we get from the last consequence
\vskip 0.1cm {\bf Corollary 4.}\it \, For $1\le p\le \infty, p\neq2,$ there are two reflexive Banach spaces $X$ and $Y$ such that the natural map $Y^*\wh\otimes_p X \to \operatorname{N}_p(Y,X)$ is not one-to-one and $ \ove{X^*\otimes_{p'} Y}^{\,\tau_{p'}}\neq \Pi_{p'}(X,Y).$ \vskip 0.1cm \rm
Since, as we said, $\tau_q$ equals to the topology $\lambda_q$ from [1], we get firstly Theorem 4.11 [1] (the case $p>2$ below) and secondly the negative answer to the question in the end of the paper [1] (whether for $p\in[1,2)$ every Banach space has the approximation property of type $p$ of [1]\footnote{ By [1], a Banach space $X$ is said to have the {\it approximation property of type $p$}\, if for every Banach space $Y,$ the finite rank operators $\mathcal F(Y,X)$ is dense in $\Pi_p(Y,X)$ in the $\lambda_p$-topology.}). Or, generally, we have:
\vskip 0.1cm
{\bf Theorem.}\it \, Let $1\le p\neq2 \le\infty.$ Then there is a $($reflexive$)$ Banach space that fails to have the approximation property of type $p$ of $[1].$
\rm
\end{document} | arXiv |
Acyclic embeddings of open Riemann surfaces into new examples of elliptic manifolds
by Tyson Ritter PDF
Proc. Amer. Math. Soc. 141 (2013), 597-603 Request permission
The geometric notion of ellipticity for complex manifolds was introduced by Gromov in his seminal 1989 paper on the Oka principle and is a sufficient condition for a manifold to be Oka. In the current paper we present contributions to three open questions involving elliptic and Oka manifolds. We show that quotients of $\mathbb {C}^n$ by discrete groups of affine transformations are elliptic. Combined with an example of Margulis, this yields new examples of elliptic manifolds with free fundamental groups and vanishing higher homotopy. Finally we show that every open Riemann surface embeds acyclically into an elliptic manifold, giving a partial answer to a question of Lárusson.
Gilbert Baumslag and James E. Roseblade, Subgroups of direct products of free groups, J. London Math. Soc. (2) 30 (1984), no. 1, 44–52. MR 760871, DOI 10.1112/jlms/s2-30.1.44
James F. Davis and Paul Kirk, Lecture notes in algebraic topology, Graduate Studies in Mathematics, vol. 35, American Mathematical Society, Providence, RI, 2001. MR 1841974, DOI 10.1090/gsm/035
Franc Forstnerič, Runge approximation on convex sets implies the Oka property, Ann. of Math. (2) 163 (2006), no. 2, 689–707. MR 2199229, DOI 10.4007/annals.2006.163.689
Franc Forstnerič, Oka manifolds, C. R. Math. Acad. Sci. Paris 347 (2009), no. 17-18, 1017–1020 (English, with English and French summaries). MR 2554568, DOI 10.1016/j.crma.2009.07.005
Franc Forstnerič and Finnur Lárusson, Survey of Oka theory, New York J. Math. 17A (2011), 11–38. MR 2782726
David Fried and William M. Goldman, Three-dimensional affine crystallographic groups, Adv. in Math. 47 (1983), no. 1, 1–49. MR 689763, DOI 10.1016/0001-8708(83)90053-1
Hans Grauert, Holomorphe Funktionen mit Werten in komplexen Lieschen Gruppen, Math. Ann. 133 (1957), 450–472. MR 0098198 (20:4660)
Hans Grauert, On Levi's problem and the imbedding of real-analytic manifolds, Ann. of Math. (2) 68 (1958), 460–472. MR 98847, DOI 10.2307/1970257
M. Gromov, Oka's principle for holomorphic sections of elliptic bundles, J. Amer. Math. Soc. 2 (1989), no. 4, 851–897. MR 1001851, DOI 10.1090/S0894-0347-1989-1001851-9
Finnur Lárusson, Excision for simplicial sheaves on the Stein site and Gromov's Oka principle, Internat. J. Math. 14 (2003), no. 2, 191–209. MR 1966772, DOI 10.1142/S0129167X03001727
Finnur Lárusson, Model structures and the Oka principle, J. Pure Appl. Algebra 192 (2004), no. 1-3, 203–223. MR 2067196, DOI 10.1016/j.jpaa.2004.02.005
Finnur Lárusson, Mapping cylinders and the Oka principle, Indiana Univ. Math. J. 54 (2005), no. 4, 1145–1159. MR 2164421, DOI 10.1512/iumj.2005.54.2731
G. A. Margulis, Free completely discontinuous groups of affine transformations, Dokl. Akad. Nauk SSSR 272 (1983), no. 4, 785–788 (Russian). MR 722330
G. A. Margulis, Complete affine locally flat manifolds with a free fundamental group, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 134 (1984), 190–205 (Russian, with English summary). Automorphic functions and number theory, II. MR 741860
John Milnor, On fundamental groups of complete affinely flat manifolds, Advances in Math. 25 (1977), no. 2, 178–187. MR 454886, DOI 10.1016/0001-8708(77)90004-4
Kiyosi Oka, Sur les fonctions analytiques de plusieurs variables. III. Deuxième problème de Cousin, J. Sc. Hiroshima Univ. 9 (1939), 7–19.
Tyson Ritter, A strong Oka principle for embeddings of some planar domains into $\mathbb C\times \mathbb C^*$, J. Geom. Anal. (to appear). arXiv:1011.4116
Satoru Shimizu, Complex analytic properties of tubes over locally homogeneous hyperbolic affine manifolds, Tohoku Math. J. (2) 37 (1985), no. 3, 299–305. MR 799523, DOI 10.2748/tmj/1178228643
Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 32Q40, 32E10, 32H02, 32H35, 32M17, 32Q28
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Tyson Ritter
Affiliation: School of Mathematical Sciences, University of Adelaide, Adelaide SA 5005, Australia
Email: [email protected]
Received by editor(s): July 4, 2011
Published electronically: June 21, 2012
Communicated by: Franc Forstneric
The copyright for this article reverts to public domain 28 years after publication.
Journal: Proc. Amer. Math. Soc. 141 (2013), 597-603
MSC (2010): Primary 32Q40; Secondary 32E10, 32H02, 32H35, 32M17, 32Q28
DOI: https://doi.org/10.1090/S0002-9939-2012-11430-3 | CommonCrawl |
Arcadian Functor
occasional meanderings in physics' brave new world
Name: Kea
Marni D. Sheppeard
New Format Blog
Dwarf Mysteries
M Theory Lesson 312
Meanwhile II
Paper Archive
Physics Blogs
Carl Brannen
Louise Riofrio
Matti Pitkanen
Phil Gibbs
Lieven Le Bruyn
NC Geometry
Paolo Bertozzini
Oxford Science
Dave Bacon
Nigel Cook
Tommaso Dorigo
Supernova Condensate
Richard Borcherds
John Baez
n-Category Cafe
Theoretical Atlas
Unapologetic Math
Todd and Vishal
Everything Seminar
Motivic Stuff
Nghbrhd Infinity
Web Page Counter
FQXi reject
mixing matrix
MUB arithmetic
2009 reject
The AF Book
DARPA Challenge
At first I thought the problem list was a mildly amusing, handwaving bit of entertainment, but it turns out that the U.S. DARPA Mathematical Challenge has funding opportunities, open also to foreigners! And the 3 page announcement is the coolest I've ever seen, including the words
Submissions that merely promise incremental improvements over the existing state of the art will be deemed unresponsive.
I feel yet another funding proposal coming on ... HAPPY NEW YEAR!
posted by Kea | 8:07 AM | 6 comments
Neutrino 08
Registration for Neutrino 08 here at UC is open, so make sure you consider heading up this way!
posted by Kea | 4:56 PM | 0 comments
Here's to 2008
posted by Kea | 2:37 PM | 16 comments
I know it's a bit late for this year, but I found the perfect cheap present for a budding M theorist: the Sudokube! Of course, some basic knowledge of magic squares makes it too easy to solve, but it would look good on the shelf. And if you don't mind me saying so, Santa, I was a bit disappointed with The Golden Compass. Why were all the physicists male? And that extended arm double ice axe arrest was just plain ridiculous.
Putting together the Hoffman and Castro expressions, for real $s$ and $t$ in the critical interval with $|s+t| < 1$, we obtain
$\sum_{m,n} s^m t^n \zeta (x^m y^n) = t [ \frac{\zeta (s)}{\zeta (1-s)} \frac{\zeta (1+t)}{\zeta (-t)} \frac{\zeta (1-s-t)}{\zeta (s+t)}]$
where the left hand side is the expression
$\sum_{m} \frac{s^m}{m!} \sum_{k_1,k_2,\cdots,k_m} \frac{1}{k_1 k_2 \cdots k_m} \sum_{n} \frac{t^n}{(k_1 + k_2 + \cdots + k_m)^{n}}$
Specific values of the zeta function include, for the choice $t = 0.5$, $\zeta (1.5) = 2.612$ and, using the functional equation,
$\zeta (- \frac{1}{2}) = \frac{1}{\sqrt{2}} \pi^{\frac{-3}{2}} \Gamma (\frac{3}{2}) \zeta (\frac{3}{2})$
so that the centre ratio in the first equation above becomes
$\sqrt{2} \pi^{\frac{3}{2}} \frac{2}{\sqrt{\pi}} = 2 \sqrt{2} \pi$
giving a particularly interesting relation for the parameter $s < \frac{1}{2}$ involving the expression
$\frac{\zeta (0.5 - s)}{\zeta (0.5 + s)} \frac{\zeta (s)}{\zeta (1-s)}$
It would be nice to extend this to complex values of the parameters, because zeroes of the zeta function occur in conjugate pairs and the finite positivity of an MZV could then rule out zeroes lying in this region.
The original Hoffman post mentioned an expression in $\Gamma$ functions, similar to that appearing in the relation
$B(a,b)= \frac{\Gamma(a)\Gamma(b)}{\Gamma(a + b)} + \frac{\Gamma(a)\Gamma(c)}{\Gamma(a + c)} + \frac{\Gamma (1 - a - b) \Gamma (b)}{\Gamma (1 - a)}$
$ = \frac{\zeta (1 - a)}{\zeta (a) } \frac{\zeta (1 - b)}{\zeta (b) } \frac{\zeta (a + b)}{\zeta (1 - a - b) }$
which appears in Castro's discussion of the zeroes of the Riemann zeta function. The $B$ function is the familiar 4-point amplitude of Veneziano, which we have been expressing in terms of chorded polygons; in this case a square with two diagonals representing the 1 dimensional associahedron, the interval.
Hoffman's 1997 paper begins with this example of an MZV relation:
$\zeta (2) \zeta (2,1) = 2 \zeta (2,2,1) + \zeta (2,1,2) + \zeta (4,1) + \zeta (2,3)$
which M theorists can try to draw in a number of ways, such as the 2-ordinal picture This suggests that zeta relations are in some sense functorial, or categorified, and arise from relations amongst arguments. In the last post, for instance, the argument of the Riemann zeta function was given by a complex cosmic time coordinate, which is often substituted in M theory for a value of $\hbar$ or $N$th root on the unit circle.
posted by Kea | 12:08 PM | 0 comments
Riemann's Brane
By now we've all heard about the relation between the Riemann zeta function and Hermitian operators associated to matrix models. With CFT/AdS in the air now, it is not surprising to find this paper by McGuigan, which discusses brane partition functions.
Somehow, according to McGuigan, on the gravity side we are supposed to end up with modular functions like those appearing in the already notorious Witten paper on 2+1D gravity. In fact, the so-called cosmological constant (just think extra time coordinates) appears as the variable $z$ in a function whose zeroes must lie on the real axis, namely
$\Theta (z) = \zeta (iz + \frac{1}{2}) \Gamma (\frac{z}{2} + \frac{1}{4}) \pi^{- \frac{1}{4} - \frac{iz}{2}} (- \frac{z^2}{2} - \frac{1}{8})$
Who would have thought such stuff could get published on the arxiv?
posted by Kea | 11:13 AM | 2 comments
Following up on the GRG18 news, the LIGO collaboration have posted a paper on the non-observation of gravitational waves from the bright electromagnetic event GRB 070201. Of course, this has been reported in a number of places already.
Here are some carols for the celebration of Newton's birthday. I quite like We Three Quarks, which begins
We three quarks fine particles are.
Bearing charm we travel afar.
Fields and forces, spin of course is
Multiplied by h-bar.
Oh, Quarks are wondrous, quarks are light.
Quarks have colors, clear and bright.
Still misleading, still exceeding
All the physicists' insight.
Happy holidays from my three nephews: Connor, Nathan and Aidan.
In an increasingly fascinating series of blogposts, the great mathematician Lieven Le Bruyn has finally reached the stringy topic of superpotentials. Apparently Grothendieck's children's drawings are dimers for Dedekind tesselations. Here is the recommended paper by Stienstra.
Aside: The very colourful graduation went well, on a stunning day. There were bagpipes, trumpet fanfares, Maori greetings, a Brahms sonata, singing in Maori, English and Latin, and the town hall organ was played. I would like to check my UC mail, but unfortunately somebody has managed to crash the system on the first day of the holidays, as usual, so I may have to wait until the New Year.
Switchback Swagger III
An intriguing paper [1] by Kalman Gyory discusses the equation
$m(m+1) \cdots (m + i - 1) = b k^{l}$
For $b = 1$ Erdös and Selfridge proved in 1975 [2] that this equation has no non-trivial solutions in the positive integers. The $(i,l,b) = (3,2,24)$ case can be seen to correspond to the cannonball problem under the substitution $n \mapsto \frac{m}{2}$. In general this suggests that the sequence of switchback expressions
$P_i \frac{\textrm{sum of squares}}{in + T_i}$
may hardly ever be expressed in the form $b k^{l}$ for $k \geq 2$, where $T_i$ is the triangular number $\sum_{j=1}^{i} {j}$, even though it is certainly a positive integer. This is an interesting fact about the cardinality of these faces of the permutohedra, and for some mysterious reason the proof for $b=1$ seems to involve the mathematics of Fermat's last theorem. Note also the similarity between the denominator above and terms in the associahedra sequences $F_{n}(i)$.
[1] K. Gyory, Acta Arith. 83 (1998) 87-92
[2] P. Erdös and J.L. Selfridge, Illinois J. Math. 19 (1975) 292-301
As reported in New Scientist, one of my esteemed colleagues from Mt Cook Village has expired after eating too much chocolate. Seriously folks, what are you doing throwing chocolate into the garbage can in a National Park? The kea is now officially endangered.
Switchback Swagger II
Courtesy of a commenter at God Plays Dice we have this nice link about the fact that there are no solutions to the sum of squares problem for $n > 24$. This was proved in 1918 by G. N. Watson, in the paper The problem of the square pyramid. In fact, the only solutions are $n = 1$ and $n = 24$. The equation
$\frac{1}{6} n (n+1) (2n + 1) = k^{2}$
originally described a pile of cannonballs, built from a base layer of $k \times k$ balls into a square pyramid of height $n$. So it's really a sphere packing problem.
Switchback Swagger
The 2-ordinal polytopes associated with the symmetric groups are the permutohedra. The number of codimension $k$ faces of the $n$th permutohedron is given in sequence A019538.
The second diagonal has some nice properties. For example, Alexander Povolotsky observed that these numbers $P_{i}$ arise as the right hand coefficients for the following sequence of expressions, indexed by $i$.
$n(n+1)[n + (n+1)] = 6(1 + 4 + 9 + \cdots + n^2 )$
$n(n+1)(n+2)[n + (n+1) + (n+2)] = 36(1 + (1+4) + (1+4+9) + \cdots + (1 + 4 + \cdots + n^2 ))$
This brings to mind the Leech sequence
$1 + 2^2 + 3^2 + \cdots + 24^{2} = 70^{2}$
for $n=24$, for which the first element of the list is expressed
$\frac{1}{3}(\sum_{i=1}^{n} i ) (2n + 1) = n^{2}$
If the squares of integers up to $n > 24$ cannot be summed to a square, it follows that the left hand side can never be a square.
Continuing with the wonders of table A033282, the $k = 3$ recursion results in the relation
$F_{n+2}(3) F_{n+1}(1) = \frac{1}{2} F_{n+2}(2) [F_{n+1}(2) + \frac{1}{3} n F_{n+1}(1)]$
For example, considering the codimension 3 edges of the 4d polytope we obtain the relation
$84 \times 9 = \frac{1}{2} \times 56 \times (21 + \frac{2}{3} \times 9)$
Isn't it wonderful how the combinatorics of the associahedra gives us so many relations between integers? One might be forgiven for guessing that operads can tell us something about factorization of an integer into primes.
The expression $F_{n}$ for the codimension 1 faces of the associahedron is
$F_{n}(1) = \frac{1}{2} n (n + 3) = \frac{1}{2} n (n + 1) + n = (\sum_{i=1}^{n} i) + n$
This corresponds to rewriting the sequence in the form
2 + 2 + 1 + 3 + 1
2 + 2 + 1 + 3 + 1 + 4 + 1
After eliminating the bold $n$ terms this becomes
1 + 2 + (1 + 2)
1 + 2 + 3 + (1 + 1 + 2)
and so on. The previous post also separated the set of $F_{n}(1)$ Young diagrams into a set of $n$ yellow tiled diagrams and another set of $1 + 2 + \cdots + n$ diagrams with at most two purple tiles in a row.
Now consider the $k = 2$ sequence $F_{n}(2)$, which counts the number of divisions of an $(n+3)$-gon into three pieces by two diagonals. The terms of this sequence are given by the formula
$F_{n}(2) = \frac{1}{3} B(n + 4, 2) B(n,2) = \frac{1}{12} n(n-1)(n+3)(n+4) = \frac{1}{6} (n-1)(n+4) F_{n}(1)$
What set of diagrams counts this sequence? The $F_{n}(1)$ factor says that before cutting an $n$-gon into three pieces we must cut it into two. Note also that $F_{2}(2) = F_{2}(1) = 5$ since having cut a pentagon into two pieces, there is only one way to cut it into three. It is therefore more natural to write
$F_{n+1}(2) = \frac{1}{6} n(n+5) F_{n+1}(1) = [\frac{1}{3} (\sum_{i=1}^{n} i) + \frac{2}{3} n] F_{n+1}(1) $
$= [ \frac{1}{3} F_{n}(1) + \frac{1}{3} n] F_{n+1}(1) = \frac{1}{3} F_{n+1}(1)(n + F_{n}(1))$
This says that once we have chopped the $(n+4)$-gon into two pieces, either we have a triangle and an $(n+3)$-gon to chop up, or we have at least an $\frac{1}{2}(n+6)$-gon (ignoring odd $n$ for now) for which we can choose $n$ diagonals meeting the existing diagonal. For example, the $21$ edges of the 3d Stasheff polytope arise from the relation
$21 = \frac{1}{3} 9 (2 + 5)$
which says that once a hexagon is cut into two, in one of $9$ ways, either there is a pentagon to chop up, in one of $5$ ways, or the hexagon is split into two squares, one of which may be cut up in two ways. The full set of $9$ diagrams for $n=3$ appears, along with the pentagon subset and the $n$ yellow pieces. Note that either $F_{n+1}(1)$ is divisible by $3$, or $F_{n}(1) + n$ is divisible by $3$. For example, $14 + 4 = 18$ and $35 + 7 = 42$. The overcount factor of $3$ comes from the familiar cyclic symmetry of a central triangle in the hexagon, the three bisections of which mark the three possible choices for a square face on the 3d polytope. Similarly, the factor of $2$ in the $F_{n}(1)$ sequence came from the two diagonals of a square, which obey an $S_2$ symmetry under rotation.
Young diagrams are usually used to label irreducible representations of the symmetric group $S_n$. If certain collections of Young diagrams are used to label associahedra (which may be obtained from the permutohedra with vertices the elements of $S_n$) then there is a close connection between the collection of groups $\{ S_n \}$ and its representations. However, the whole heirarchy in all dimensions needs to be considered if we want to understand this correspondence between a theory and its models.
We have seen how the Catalan numbers come up in many places. For example, they count the number of vertices of the $n$th associahedron, which is fully labelled by chorded $n+3$-gons. The full list of the number of codimension $k$ elements of the $n$th associahedron is given by sequence A033282 at the integer database. For instance, the left diagonal row
$2, 5, 9, 14, 20, 27, \cdots$
counts two ends to an interval, five edges on a pentagon, nine faces of the 3d Stasheff polytope, and so on. These codimension 1 faces are labelled by an $n$-gon with a single diagonal, splitting the $n$-gon into two parts.
Another way of viewing the particular row above is in terms of Young diagrams with only two rows. For example, the 3d and 4d polytopes have faces labelled by the following Young diagrams. Observe how the recursion is visible in the complements of the white tiles. For $n = 3$, by omitting both the full white diagram and the yellow diagrams one obtains five remaining purple diagrams corresponding to the Young pictures for the $n = 2$ pentagon edges. The recursion relation for codimension 1 faces must therefore be
$F_{n} = F_{n - 1} + n + 1 = \frac{n(n+3)}{2}$
Recall that codimension 1 elements are also labelled by trees with two internal vertices. For example, the whole pentagon is labelled by a single vertex tree with four leaves, but the pentagon edges are labelled by four leaved trees with two internal vertices. The homework problem is to figure out how these trees correspond to the Young diagrams.
The full list of left diagonals for A033282, with the above row as the $k = 1$ entry, gives the number of codimension $k$ faces. The general formula for these numbers takes the form
$F_{n} (k) = \frac{1}{k + 1} B(n + k + 2,k) B(n,k)$
for binomial coefficients $B(n,m)$. The sequence A126216 is a mirror image of A033282, which counts the number of Schroeder paths. These arose in the construction of Abel sums and interesting maps between sets of trees.
Update: It turns out that R. P. Stanley worked out a relation between trees and Young tableaux (numbered Young diagrams) in the short 1996 paper Polygon dissections and standard Young tableaux in J. Comb. Theory A76, pages 175-177.
Having blogged a bit already about Geometric Representation Theory, I was going to leave it to people to watch this lecture series for themselves. But lecture 8 is way too cool to ignore!
Here we see how paths in a $q$ deformed Pascal triangle can be counted. First note that in M Theory we usually draw the triangle as a quadrant in a plane, on which we consider paths for the noncommutative Fourier transform. A step to the right picks up multiples of a power of $q$, whereas a step up simply multiplies the entry below by 1. In this way one obtains polynomials in $q$ with integer coefficients. Even though $q$ starts out labelling the number of elements in a finite field, we recall counting trees in a similar fashion, but ending up with complex roots of unity. Physicists may smell a sneaky Wick rotation in the shrubbery.
Nerdy Nerdy
I couldn't resist: thanks to Clifford ...
Motive Madness III
A quick note: check out John Baez's TWF 259 for more details on the absolute point! Clearly I'm not the only crazy one who sees ghosts in the machine. To quote the summary:
In short, a mathematical phantom is gradually taking solid form before our very eyes! In the process, a grand generalization of algebraic geometry is emerging ...
James Dolan speaks of categorification and decategorification, and of information and entropy.
In logos land, these processes have a dimension raising or lowering aspect. It is often said that categorification is ill defined, in comparison to decategorification, but with dual processes it should not be so. Therefore, categorification itself must be defined in some canonical way that generalises the turning of natural numbers into sets or spaces. One way to do this would be to put the heirarchy on a loop, such as the loop labelled by the $q$ parameters at roots of unity. There would be $n$-categories for $n \in \mathbb{N}$ and $r$-categories for $r \in \mathbb{Q}$, and $n \rightarrow \infty$ would look like the limit $q \rightarrow 1$ again, where spaces begin to look like sets.
After all, projective geometry has its horizons, and the cohomology of motives would move left and right, like the mass interaction, or Stokes' theorem, or the Riemann zeta function.
Motive Madness II
Recall that Kapranov and Smirnov have also been thinking about the field with one element. They say an affine line over $F_1$ should be zero along with all the roots of unity.
Looking at polynomials with $F_1$ coefficients, the group $GL(n, F_{1}[x])$ is just the braid group on $n$ strands. For example, $3 \times 3$ matrices are associated with the three strand braid group, as often discussed. Then the map $B_n \rightarrow S_n$ is thought of as the $q \rightarrow 1$ limit, since the symmetric group acts on sets as vector spaces.
The field $F_{1}(n)$ extends $F_1$ by containing zero and the set of all nth roots of unity. A vector space over this field is a pointed set (marked by zero) with an action by the roots of unity. Direct sum and smash product become the operations on such spaces.
Note that Weber and others have considered the category of pointed sets as a 2-categorical Cat analogue of subobject classifier, and the category Set plays the role here of a one element set. There seem to be a number of ways in which the field $F_1$ introduces new topos theoretic arrows.
Motive Madness
Whoa! I wasn't expecting it that soon. Motives appear already, at least conjecturally, in John Baez's lecture 6! In the diagram the vertical arrows are the decategorification of either sets or projective spaces. An isomorphism class of $n$ element sets is mapped to the number $n$. Natural numbers become $q$-numbers in the case of spaces, which is to say rational functions in the parameter $q$ corresponding to the number of elements in the finite field, or secretly really polynomials with integer coefficients. But what replaces the category of finite sets? There is more structure to the projective spaces, and we also want to understand the bottom arrow, which considers a set as a space over a one element field.
This is a rather delicate mathematical question. Baez mentions a recent paper by Durov (with lots of stuff on monads) about the idea of a one element field. When we understand this properly, do we find motives? Now, that is the question.
The plane of a (finite) field $F_{q}^{2}$ is the $q$-analogue of a two element set, which plays an important role in the Boolean topos Set, namely as the subobject classifier. The vector space version of this is commonly known as a qubit. Somehow the reason that a one element field $F_1$ doesn't usually make sense is because the logical 0 and 1 are not distinct. Since $q$ is, in the first instance, just a natural number (= $p^{k}$ for some prime $p$), we can ask ourselves first what it means to collapse a finite plane to a one element field. For the topos Set, this would amount to turning the whole category into the trivial category 1, since there is no way to distinguish a subset of a set $S$ from its complement and all sets have effectively only one element. Now this one element set is like a basis for spaces over $F_1$. But the map that takes a basis to a space is just the functor 1 $\rightarrow$ Set, which picks out a set!
But this doesn't sound right. Maybe what we need here is not the trivial 1-category, or a one point 0-category (set), but rather a -1-category. This idea always lurks in the operad heirarchy, where the left hand side of the table starts with the single leaf tree, despite the fact that a point is actually a two leaf tree.
Anyway, think of a finite set $S$ lying at the endpoints of the unit vectors in a vector space. The empty set at the origin is the smallest piece of the power set of $S$, and the one element subsets are the next smallest pieces. The power set fills out a cube of dimension $|S|$. Since the field in question is $F_1$ there is no extent to the axes. Only the elements of $S$ really exist.
Planar Young diagrams represent partitions of a natural number $n = n_1 + n_2 + n_3 + \cdots + n_k$. The $k$ rows are the pieces $n_i$ of the partition. But categorified numbers $n_i$ are actually sets with $n_i$ elements, or perhaps vector spaces of dimension $n_i$, or projective spaces of dimension $n_i - 1$. In this setting the expression $n = n_1 + n_2 + n_3 + \cdots + n_k$ is about the decomposition of a space into subspaces.
We have seen something like this before. Let $n_i$ instead represent $V^{\otimes n_{i}}$ for a fixed finite dimensional vector space $V$. Then the $O(n_{i})$ piece of the operad is the space of linear maps from $n_i$ to $V$ in Vect. The operad rules come from compositions $n_1 \otimes n_2 \otimes n_3 \cdots \otimes n_k \rightarrow n$ of these maps. Maybe instead of categorification of $\mathbb{N}$ we can look at operadification.
One then wonders what happens for higher dimensional ordinals. Actually, partitions are a lot like 2-tree ordinals. The simplest generalisation would allow each $n_{i}$ of a planar partition to be itself replaced by a partition of $m$ in a third direction. The first permutation group to fill a three dimensional diagram would be $S_4$, with four box partitions. This is the only additional diagram to the planar labels for the $S_4$ barycentrically divided tetrahedron. Similarly, $S_3$ is the first group to fill a truly planar Young diagram. This pattern continues for all $n$.
Lecture 3 by James Dolan starts with diagrams for the subgroups of $S_3$. For example, the cyclic group produces a diagram where all three green vertices have been identified by rotations of the triangle, as have the blue vertices. Now we see the need for the Young diagram boxes, which can represent indeterminate elements of a set. This example splits the six elements of the group into two sets of three permutations, represented by the two triangles. One such set appeared as a basis for the mass matrix Fourier transform. M theory is like a child's game of connect the dots.
In Geometric Representation Theory lecture 2 James Dolan draws a 2-simplex with a barycentric subdivision, looking like a hexagon with a central point, or rather a cube with a missing hidden vertex. This picture is labelled with Young diagrams associated to the group $S_3$ of permutations on three letters. Diagrams of height 1 are associated with vertices, diagrams of height 2 with edges and diagrams of height 3 with faces. Note that horizontal lists are unordered. So the six faces represent the elements of the group. Dolan wants to think of these diagrams as axiomatic theories in a categorical sense. Such diagrams, and their subdiagrams, are associated with sequences of subspaces of the three element set, in analogy with flag spaces for vector spaces. Sets are just a classical kind of vector space. This is clear when counting vectors in vector spaces over finite fields, as discussed by Baez in lecture 1. For vector spaces over $F$ the permutation groups would naturally be replaced by the example of $GL(n,F)$.
The example above generalises as expected. Before we know it, we'll probably be doing motivic cohomology and Langlands geometry using Hecke pictures. Of course, Kontsevich has already been thinking about such things.
Riemann Revived II
S duality in the guise of the Langlands program is a truly awe inspiring component of stringy triality.
Recall that the complex number form of S duality has a modular group symmetry. This group appears all over the place in M theory. For example, we looked at the Banach-Tarski paradox in terms of a ternary tiling of hyperbolic space as a Poincare disc. This tiling marks the boundary of the circle with a nice triple pattern of accumulating points.
Alain Connes and Matilde Marcolli say that the Riemann zeta function is related to the problem of mass, which in turn we have seen is related to three stranded braids and M theory triality, of which S duality is a piece. It appears that no part of mathematics is left untouched by gravity.
Update: a new paper by Witten et al on 3D gravity, prominently featuring the modular group, has appeared on the arxiv. See the picture on page 49, and the J invariant on page 52. The paper shows that the partition function cannot be given a conventional Hilbert space interpretation. Holomorphic factorisation is suggested as a possible mechanism for extending the degrees of freedom. For instance, the complexified Einstein equations are considered. They say:
we think another possibility is that the non-perturbative framework of quantum gravity really involves a sum not over ordinary geometries in the usual sense, but over some more abstract structures that can be defined independently for holomorphic and antiholomorphic variables. Only when the two structures coincide can the result be interpreted in terms of a classical geometry.
I confess to finding this statement a little ill-phrased, since some more abstract structure presumably does not begin with traditional complex analysis.
Riemann Revived
This month's conference on Langlands and QFT has naturally drawn some attention. Matti Pitkanen has some comments. Louise Riofrio and David Ben-Zvi (one of the speakers) have been taking notes. Louise points out that Witten is interested in four dimensions, but according to Ben-Zvi's notes from his talk, Witten said it was natural to look at six dimensions (think twistors), or at least four, but for the talk he focused on two.
Admittedly, I can't make that much out of Ben-Zvi's notes, and I'm sure that's not his fault! Here's a link to Frenkel's helpful paper.
Fading Dark Force
Dr Motl reports on a new astro-ph paper on the lack of a Dark Force (cosmological constant). The abstract suggests
a new concordance model with 90% dark matter, 10% baryons, no dark energy and 14.8 Gyr as the age of the universe.
This sounds familiar. It includes a reference to the now published paper by D. Wiltshire. On Wednesday, Dr Motl reported on the difficulties that Fermi initially faced having his ideas accepted. Meanwhile, Louise Riofrio brings us reports from New York, and a small town nearby. | CommonCrawl |
\begin{document}
\title{\bf Invariants of Knots and 3-manifolds from Quantum Groupoids}
\author{Dmitri Nikshych} \address{UCLA, Department of Mathematics, 405 Hilgard Avenue, Los Angeles, CA 90095-1555, USA}
\email{[email protected]} \thanks{The first author thanks P.~Etingof for useful discussions}
\author{Vladimir Turaev} \address{Institut de Recherche Math\'ematique Avanc\'ee, Universit\'e Louis Pasteur, CNRS, 7, rue Ren\'e Descartes, F-67084 Strasbourg, France}
\email{[email protected]}
\author{Leonid Vainerman} \address{ D\'epartement de Math\'ematiques, Universit\'e Louis Pasteur, 7, rue Ren\'e Descartes, F-67084 Strasbourg, France}
\email{[email protected], [email protected]} \thanks{ The third author is grateful to l'Universit\'e Louis Pasteur (Strasbourg) for the kind hospitality during his work on this article}
\date{June 8, 2000}
\begin{abstract} We use the categories of representations of finite dimensional quantum groupoids (weak Hopf algebras) to construct ribbon and modular categories that give rise to invariants of knots and 3-manifolds. \end{abstract}
\maketitle \tableofcontents
\begin{section} {Introduction} In \cite{RT2} a general method of constructing invariants of $3$-manifolds from modular Hopf algebras was introduced. After appearance of \cite{RT2} it became clear that the technique of Hopf algebras can be replaced by a more general technique of monoidal categories. An appropriate class of categories -- modular categories -- was introduced in \cite{T1}. In addition to quantum groups, such categories also arise from skein categories of tangles and, as it was observed by A. Ocneanu, from certain bimodule categories of type II${}_1$ subfactors.
The goal of this paper is to study the representation categories of {\em quantum groupoids} and to give in this way a new construction of modular categories. This extends the construction of modular categories from modular Hopf algebras and in particular from quantum groups at roots of unity.
By quantum groupoids, we understand weak Hopf algebras introduced in \cite{BNSz}, \cite{BSz1}, \cite{Ni}. These objects generalize Hopf algebras, usual finite groupoid algebras and their duals (cf. \cite{NV1}). We use the term ``quantum groupoid'' rather than ``weak Hopf algebra''.
It was shown in \cite{NV2}, \cite{NV3} that quantum groupoids and their coideal subalgebras are closely related to II$_1$-subfactors. Every finite index and finite depth II$_1$-subfactor gives rise to a pair consisting of a $C^*$-quantum groupoid and its left coideal subalgebra, and vice versa. It was also explained in \cite{NV3} how to express the known subfactor invariants such as bimodule categories and principal graphs in terms of the associated quantum groupoids. In particular, the bimodule categories arising from a finite index and finite depth II$_1$ subfactors are equivalent to the unitary representation categories of the corresponding $C^*$-quantum groupoids (\cite{NV3}, 5.8).
Thus, it is natural to study categories of representations of quantum groupoids and to extend concepts known for Hopf algebras to this setting. We show that the representation category $\mbox{Rep}(H)$ of a quantum groupoid $H$ is a monoidal category with duality. We introduce quasitriangular, ribbon, and modular quantum groupoids for which $\mbox{Rep}(H)$ is, respectively, braided, ribbon, and modular. The notion of factorizability is extended from the Hopf algebra case and used to construct modular categories. We define the Drinfeld double $D(H)$ of a quantum groupoid $H$ and show that it is a factorizable quasitriangular quantum groupoid. For a $C^*$-quantum groupoid $H$, we similarly study the unitary representation category $\mbox{URep}(H)$.
It should be mentioned that the category $\mbox{URep}(H)$ for a $C^*$-quantum groupoid $H$ was previously introduced by G.~B\"ohm and K.~Szlach\'anyi in \cite{BSz2}; they also introduced the notion of an $R$-matrix and the Drinfeld double for $C^*$-quantum groupoids, see \cite{BSz1}.
Our main theorem (Theorem~\ref{rep of C* is modular}) reads: If $H$ is a connected $C^*$-quantum groupoid, then the category $\mbox{URep}(D(H))$ of unitary representations of $D(H)$ is a unitary modular category.
Thus, any finite index and finite depth II$_1$-subfactor yields a unitary modular category as follows: consider the associated connected $C^*$-quantum groupoid $H$, then the category $\mbox{URep}(D(H))$ is a unitary modular category. We conjecture that this construction is equivalent to the one due to A.~Ocneanu (see \cite{EK}). The key role in the proof of the main theorem is played by the following Lemma~\ref{factorizable implies modular}: If $H$ is a connected, ribbon and factorizable quantum groupoid with a Haar measure over an algebraically closed field, then $\mbox{Rep}(H)$ is a modular category.
The organization of the paper is clear from the table of contents. \end{section}
\begin{section} {Quantum groupoids}
In this section we recall basic properties of quantum groupoids. Most of the material presented here can be found in \cite{BNSz} and \cite{NV1}, see also the survey \cite{NV4}.
Throughout this paper we use Sweedler's notation for comultiplication, writing $\Delta(b) = b_{(1)} \otimes b_{(2)}$. Let $k$ be a field.
\begin{definition} \label{finite quantum groupoid} A {\em (finite) quantum groupoid} over $k$ is a finite dimensional $k$-vector space $H$ with the structures of an associative algebra $(H,\,m,\,1)$ with multiplication $m:H\otimes_k H\to H$ and unit $1\in H$ and a coassociative coalgebra $(H,\,\Delta,\,\varepsilon)$ with comultiplication $\Delta:H\to H\otimes_k H$ and counit $\varepsilon:H\to k$ such that: \begin{enumerate} \item[(i)] The comultiplication $\Delta$ is a (not necessarily unit-preserving) homomorphism of algebras such that \begin{equation} (\Delta \otimes \mbox{id}) \Delta(1) = (\Delta(1)\otimes 1)(1\otimes \Delta(1)) = (1\otimes \Delta(1))(\Delta(1)\otimes 1), \label{Delta 1} \end{equation} \item[(ii)] The counit is a $k$-linear map satisfying the identity: \begin{equation} \varepsilon(fgh) = \varepsilon(fg_{(1)})\,\varepsilon(g\2h) = \varepsilon(fg_{(2)})\,\varepsilon(g\1h), \label{eps m} \end{equation} for all $f,g,h\in H$. \item[(iii)] There is an algebra and coalgebra anti-homomorphism $S: H \to H$, called an {\em antipode}, such that, for all $h\in H$, \begin{eqnarray} m(\mbox{id} \otimes S)\Delta(h) &=&(\varepsilon\otimes\mbox{id})(\Delta(1)(h\otimes 1)), \label{S epst} \\ m(S\otimes \mbox{id})\Delta(h) &=& (\mbox{id} \otimes \varepsilon)((1\otimes h)\Delta(1)). \label{S epss} \end{eqnarray} \end{enumerate} \end{definition} A quantum groupoid is a Hopf algebra if and only if one of the following conditions holds: (i) the comultiplication is unit-preserving or (ii) if and only if the counit is a homomorphism of algebras.
A {\em morphism} of quantum groupoids is a map between them which is both an algebra and a coalgebra morphism preserving unit and counit and commuting with the antipode. The image of such a morphism is clearly a quantum groupoid. The tensor product of two quantum groupoids is defined in an obvious way.
The set of axioms of Definition~\ref{finite quantum groupoid} is self-dual. This allows to define a natural quantum groupoid structure on the dual vector space $\widehat H=\mbox{Hom}_k(H,k)$ by ``reversing the arrows'': \begin{eqnarray} & & \langle\, h,\,\phi\psi \,\rangle = \langle\, \Delta(h),\,\phi\otimes\psi \,\rangle, \\ & & \langle\, g\otimes h,\,{\widehat\Delta}(\phi) \,\rangle = \langle\, gh,\, \phi\,\rangle, \\ & & \langle\, h,\, {\widehat S}(\phi) \,\rangle = \langle\, S(h),\,\phi \,\rangle, \end{eqnarray} for all $\phi,\psi \in \widehat H,\, g,h\in H$. The unit $\widehat 1\in \widehat H$ is $\varepsilon$ and counit $\widehat\varepsilon$ is $\phi \mapsto \langle\,\phi,\, 1\,\rangle$.
The linear endomorphisms of $H$ defined by \begin{equation} h\mapsto m(\mbox{id} \otimes S)\Delta(h), \qquad h\mapsto m(S \otimes \mbox{id})\Delta(h) \end{equation} are called the {\em target} and {\em source counital maps} and denoted $\varepsilon_t$ and $\varepsilon_s$, respectively. From axioms (\ref{S epst}) and (\ref{S epss}), \begin{equation} \varepsilon_t(h) = (\varepsilon\otimes\mbox{id})(\Delta(1)(h\otimes 1)),\qquad \varepsilon_s(h) = (\mbox{id} \otimes \varepsilon)((1\otimes h)\Delta(1)). \end{equation} In the Hopf algebra case $\varepsilon_t(h) = \varepsilon_s(h) = \varepsilon(h) 1$.
We have $S\circ \varepsilon_s = \varepsilon_t\circ S$ and $\varepsilon_s\circ S = S\circ \varepsilon_t$. The images of these maps $\varepsilon_t$ and $\varepsilon_s$ \begin{eqnarray} H_t &=& \varepsilon_t(H) = \{h\in H \mid \Delta(h) = \Delta(1)(h\otimes 1)\}, \\ H_s &=& \varepsilon_s(H) = \{h\in H \mid \Delta(h) = (1\otimes h)\Delta(1)\} \end{eqnarray} are subalgebras of $H$, called the {\em target} (resp.\ {\em source}) {\em counital subalgebras}. They play the role of ground algebras for $H$. They commute with each other and \begin{equation*} H_t = \{(\phi\otimes \mbox{id})\Delta(1) \mid \phi\in \widehat H \}, \qquad H_s = \{(\mbox{id} \otimes \phi)\Delta(1) \mid \phi\in \widehat H \}, \end{equation*} i.e., $H_t$ (resp.\ $H_s$) is generated by the right (resp.\ left) tensorands of $\Delta(1)$. The restriction of $S$ defines an algebra anti-isomorphism between $H_t$ and $H_s$. Any non-zero morphism $H \to K$ of quantum groupoids preserves counital subalgebras, i.e., $H_t \cong K_t$ and $H_s \cong K_s$.
In what follows we will use the Sweedler arrows, writing for all $h\in H,\phi\in \widehat H$: \begin{equation} h\rightharpoonup\phi = \phi_{(1)} \langle\, h,\, \phi_{(2)}\,\rangle, \qquad \phi\leftharpoonup h =\langle\, h,\,\phi_{(1)} \,\rangle \phi_{(2)} \end{equation} for all $h\in H,\phi\in \widehat H$. Then the map $z \mapsto (z\rightharpoonup \varepsilon)$ is an algebra isomorphism between $H_t$ and ${\widehat H}_s$. Similarly, the map $y \mapsto (\varepsilon\leftharpoonup y)$ is an algebra isomorphism between $H_s$ and ${\widehat H}_t$ (\cite{BNSz}, 2.6). Thus, the counital subalgebras of $\widehat H$ are canonically anti-isomorphic to those of $H$.
A quantum groupoid $H$ is called {\em connected} if $H_s \cap Z(H) = k$, or, equivalently, $H_t \cap Z(H) = k$, where $Z(H)$ denotes the center of $H$ (cf. \cite{N}, 3.11, \cite{BNSz}, 2.4).
Let us recall that a $k$-algebra $A$ is {\em separable} \cite{P} if the multiplication epimorphism $m: A\otimes_k A \to A$ has a right inverse as an $A-A$ bimodule homomorphism. This is equivalent to the existence of a {\em separability element} $e\in A\otimes_k A $ such that $m(e) =1$ and $(a\otimes 1)e = e(1\otimes a),\ (1\otimes a)e=e(a\otimes 1)$ for all $a\in A$.
The counital subalgebras $H_t$ and $H_s$ are separable, with separability elements $e_t = (S \otimes \mbox{id})\Delta(1)$ and $e_s = (\mbox{id} \otimes S)\Delta(1)$, respectively.
Observe that the {\em adjoint} actions of $1\in H$ give rise to non-trivial maps $H\to H$: \begin{equation} h \mapsto 1\1hS(1_{(2)}) = \mbox{Ad}\,_1^l(h), \qquad h\mapsto S(1_{(1)}) h 1_{(2)} = \mbox{Ad}\,_1^r(h), \qquad h\in H. \end{equation}
\begin{lemma} \label{expectations} The map $\mbox{Ad}\,^{l}_1$ is a linear projection from $H$ onto $C_H(H_s)$, the centralizer of $H_s$, i.e.,\ $(\mbox{Ad}\,^{l}_1)^2= \mbox{Ad}\,^{l}_1$. The map $\mbox{Ad}\,^{r}_1$ is a linear projection from $H$ onto $C_H(H_t)$, the centralizer of $H_t$, i.e.,\ $(\mbox{Ad}\,^{r}_1)^2= \mbox{Ad}\,^{r}_1$. \end{lemma}
\begin{proof} Since $1_{(1)} \otimes S(1_{(2)})$ is a separability element of $H_s$, $\mbox{Ad}\,^{l}_1(h)$ commutes with $H_s$. The assertion about $\mbox{Ad}\,^{r}_1$ follows similarly. \end{proof}
\begin{remark} \label{opposite groupoid} The opposite algebra $H^{op}$ is also a quantum groupoid with the same coalgebra structure and the antipode $S^{-1}$. Indeed, \begin{eqnarray*} S^{-1}(h_{(2)})h_{(1)} &=& S^{-1}(\varepsilon_s(h)) = S^{-1}(1_{(1)}) \varepsilon(h1_{(2)}) \\ &=& S^{-1}(1_{(1)}) \varepsilon(h S^{-1}(1_{(2)})) =\varepsilon(h1_{(1)})1_{(2)} =\varepsilon_t^{op}(h), \\ h_{(2)} S^{-1}(h_{(1)}) &=& S^{-1}(\varepsilon_t(h)) = \varepsilon(1\1h) S^{-1}(1_{(2)}) \\ &=& \varepsilon( S^{-1}(1_{(1)})h)S^{-1}(1_{(2)}) = 1_{(1)}\varepsilon(1\2h) =\varepsilon_s^{op}(h),\\ S^{-1}(h_{(3)}) h_{(2)} S^{-1}(h_{(1)}) &=& S^{-1}(h_{(1)} S(h_{(2)}) h_{(3)}) = S^{-1}(h). \end{eqnarray*} Similarly, the co-opposite coalgebra $H^{cop}$ (with the same algebra structure as $H$ and the opposite coalgebra structure, and the antipode $S^{-1}$) and $H^{op/cop}$ (with both opposite algebra and coalgebra structures, and the antipode $S$) are quantum groupoids. \end{remark}
\section{Examples of quantum groupoids} \label{examples}
\textbf{Groupoid algebras and their duals ([NV1], 2.1.4).} As group algebras and their duals give the simplest examples of Hopf algebras, groupoid algebras and their duals provide simple examples of quantum groupoids.
Let $G$ be a finite {\em groupoid} (a category with finitely many morphisms, such that each morphism is invertible). Then the groupoid algebra $kG$ (generated by morphisms $g\in G$ with the product of two morphisms being equal to their composition if the latter is defined and $0$ otherwise) is a quantum groupoid via : \begin{equation} \Delta(g) = g\otimes g,\quad \varepsilon(g) =1,\quad S(g)=g^{-1},\quad g\in G. \label{groupoid algebra} \end{equation} The counital subalgebras of $kG$ are equal to each other and coincide with the abelian algebra spanned by the identity morphisms : $(kG)_t = (kG)_s = \mbox{span}\{gg^{-1}\mid g\in G\}$. The target and source counital maps are induced by the operations of taking the target (resp.\ source) object of a morphism : \begin{equation*} \varepsilon_t(g) =gg^{-1} = \mbox{id}_{target(g)} \quad \mbox{ and } \quad \varepsilon_s(g) = g^{-1}g = \mbox{id}_{source(g)}. \end{equation*}
The dual quantum groupoid $\widehat{kG}$ is isomorphic to the algebra of functions on $G$, i.e., it is generated by idempotents $p_g,\, g\in G$ such that $p_g p_h= \delta_{g,h}p_g$, with
\begin{equation} \Delta(p_g) =\sum_{uv=g}\,p_u\otimes p_v,\quad \varepsilon(p_g)= \delta_{g,gg^{-1}}, \quad S(p_g) =p_{g^{-1}}. \label{dual groupoid algebra} \end{equation} The target (resp.\ source) counital subalgebra is precisely the algebra of functions constant on each set of morphisms of $G$ having the same target (resp.\ source). The target and source maps are \begin{equation*} \varepsilon_t(p_g) = \sum_{vv^{-1}=g}\, p_v \quad \mbox{ and } \quad \varepsilon_s(p_g) = \sum_{v^{-1}v=g}\, p_v. \end{equation*}
\begin{definition} \label{semisimple} We call a quantum groupoid {\em semisimple} if its underlying algebra is semisimple. \end{definition}
In contrast to finite dimensional semisimple Hopf algebras, the antipode in a finite dimensional quantum groupoid is not necessarily involutive, see Section 9.
Groupoid algebras and their duals give examples of commutative and cocommutative semisimple quantum groupoids.
\textbf{Quantum transformation groupoids.} It is known that any group action on a set (i.e., on a commutative algebra of functions) gives rise to a groupoid \cite{R}. Extending this construction, we associate a quantum groupoid with any action of a Hopf algebra on a separable algebra (``finite quantum space''). Namely, let $H$ be a Hopf algebra and $B$ be a separable (and, therefore, finite dimensional and semisimple \cite{P}) algebra with right $H$-action $ b\otimes h \mapsto b\cdot h$, where $b\in B,h\in H$. Then $B^{op}$, the algebra opposite to $B$, becomes a left $H$-module via $h\otimes a \mapsto h\cdot a = a\cdot S_H(h)$. One can form a {\em double crossed product algebra} $B^{op}{>\!\!\!\triangleleft} H{\triangleright\!\!\!<} B$ with underlying vector space $ B^{op}\otimes H \otimes B$ and multiplication \begin{equation*} (a\otimes h\otimes b)(a'\otimes h'\otimes b') = (h_{(1)}\cdot a')a \otimes h_{(2)}{h'}_{(1)} \otimes (b\cdot {h'}_{(2)}) b', \end{equation*} for all $a,a'\in B^{op},\, b,b'\in B,$ and $h,h'\in H$.
Assume that $k$ is algebraically closed and let $e$ be a separability element of $B$ (note that $e$ is an idempotent when considered in $B^{op}\otimes B$). Let $\omega\in \widehat B$ be uniquely determined by $(\omega\otimes \mbox{id})e = (\mbox{id}\otimes \omega)e = 1$. One can check that $\omega$ is the trace of the left regular representation of $B$ and $$ \omega((h\cdot a)b) = \omega(a (b\cdot h)), \qquad e^{(1)} \otimes (h\cdot e^{(2)}) = (e^{(1)} \cdot h) \otimes e^{(2)}, $$ where $a\in B^{op}, b\in B$, and $e =e^{(1)}\otimes e^{(2)}$.
The structure of a quantum groupoid on $B^{op}{>\!\!\!\triangleleft} H{\triangleright\!\!\!<} B$ is given by \begin{eqnarray} \Delta(a\otimes h\otimes b) &=& (a\otimes h_{(1)} \otimes e^{(1)})\otimes ((h_{(2)}\cdot e^{(2)})\otimes h_{(3)} \otimes b),\\ \varepsilon(a\otimes h\otimes b) &=& \omega(a(h\cdot b)) = \omega(a(b\cdot S_H(h))),\\ S(a\otimes h\otimes b) &=& b\otimes S_H(h)\otimes a. \end{eqnarray}
\textbf{Quantum groupoids $\mathbf{B^{op}\otimes B}$ (\cite{BSz2}, 5.2).} \label{Bop tensor B} Let $k$ be algebraically closed and let $B$ be a separable algebra over $k$, $e =e^{(1)} \otimes e^{(2)} \in B^{op}\otimes B$ be the symmetric separability idempotent of $B$, and $\omega$ be as in the previous example. The map $\pi:x\mapsto e^{(1)} x e^{(2)}$ defines a linear projection from $B$ to $Z(B)$. Let $q$ be an invertible element of $B$ such that $\pi(q) =1$, then the following operations define a structure of quantum groupoid $H_q$ on $B^{op}\otimes B$ : \begin{eqnarray} \Delta(b\otimes c) &=& (b\otimes e^{(1)} q^{-1}) \otimes (e^{(2)} \otimes c), \\ \varepsilon(b\otimes c) &=& \omega(qbc),\\ S(b\otimes c) &=& q^{-1}cq\otimes b, \end{eqnarray} for all $b,c\in B$. The target and source counital subalgebras of $H_q$ are $B^{op} \otimes 1$ and $1\otimes B$. The square of the antipode is a conjugation by $g_q =q\otimes q$. Since $H_q$ with different $q$ are non-isomorphic, this example shows that there can be uncountably many non-isomorphic semisimple quantum groupoids with the same underlying algebra (for noncommutative $B$).
This example can be also explained in terms of {\em twisting} of quantum groupoids (see \cite{EN}, \cite{NV4}).
\textbf{Quantum groupoids from subfactors.} The initial motivation for studying quantum groupoids in \cite{NV1}, \cite{NV2}, \cite{N} was their connection with depth 2 von Neumann subfactors. This connection was first mentioned in \cite{O} and was also considered in \cite{BNSz}, \cite{BSz1}, \cite{BSz2}, \cite{NSzW}. It was shown in \cite{NV2} that quantum groupoids naturally arise as non-commutative symmetries of subfactors, namely if $N\subset M\subset M_1\subset M_2\subset\dots$ is the Jones tower constructed from a finite index, depth $2$ inclusion $N\subset M$ of II$_1$ factors, then $H=M'\cap M_2$ has a canonical structure of a quantum groupoid acting outerly on $M_1$ such that $M =M_1^H$ and $M_2 = M_1{>\!\!\!\triangleleft} H$. Furthermore $\widehat H=N'\cap M_1$ is a quantum groupoid dual to $H$.
In \cite{NV3} this result was extended to arbitrary finite depth, via a Galois correspondence and it was shown in (\cite{NV3}, 4) that any inclusion of type II${}_1$ von Neumann factors with finite index and depth (\cite{GHJ}, 4.1) gives rise to a quantum groupoid and its coideal subalgebra.
We refer the reader to the survey \cite{NV4} (Sections 8,9) and to the Appendix of \cite{NV3} for the explanation of how quantum groupoids can be constructed from subfactors.
\textbf{Temperley-Lieb algebras.} We describe quantum groupoids arising from type $A_n$ subfactors, whose underlying algebras are {\em Temperley-Lieb algebras} (\cite{GHJ}, 2.1).
Let $k=\mathbb {C}$, $\lambda^{-1} = 4\cos^2\frac{\pi}{n+3}\ (n\geq 2)$, and $e_1, e_2,\dots$ be a sequence of idempotents satisfying, for all $i$ and $j$, the relations \begin{eqnarray*} e_i e_{i\pm 1} e_i &=& \lambda e_i, \\
e_i e_j &=& e_j e_i, \quad \mbox{if } |i-j| \geq 2. \end{eqnarray*} Let $A_{k,l}$ be the algebra generated by $1, e_k, e_{k+1},\dots e_l$ ($k\leq l$), $\sigma$ be the algebra anti-automorphism of $H= A_{1,2n-1}$ determined by $\sigma(e_i) = e_{2n-i}$ and $P_{k}\in A_{2n-k, 2n-1} \otimes A_{1,k}$ be the image of the separability idempotent of $A_{1,k}$ under $\sigma\otimes \mbox{id}$.
We denote by $\tau$ the non-degenerate Markov trace (\cite{GHJ}, 2.1) on $H$ and by $w$ the index of the restriction of $\tau$ on $A_{n+1, 2n-1}\subset H$, i.e., the unique central element in $A_{n+1, 2n-1}$ such that $\tau(w\,\cdot )$ is equal to the trace of the left regular representation of $A_{n+1, 2n-1}$ (see \cite{W}).
Then the following operations define a quantum groupoid structure on $H$ : \begin{eqnarray*} \Delta(yz) &=& (z\otimes y)P_{n-1}, \qquad y\in A_{n+1,2n-1},\quad
z\in A_{1,n-1}\\ \Delta(e_n) &=& (1\otimes w) P_{n} (1\otimes w^{-1}), \\ S(h) &=& w^{-1}\sigma(h)w, \\ \varepsilon(h) &=& \lambda^{-n}\tau(hfw), \quad h\in A, \end{eqnarray*} where in the last line $$ f= \lambda^{n(n-1)/2}(e_ne_{n-1}\cdots e_1)(e_{n+1}e_n\cdots e_2)\cdots
(e_{2n-1}e_{2n-2}\cdots e_n) $$ is the Jones projection corresponding to the $n$-step basic construction.
The source and target counital subalgebras of $H= A_{1,2n-1}$ are $H_s=A_{n+1, 2n-1}$ and $H_t=A_{1,n-1}$. The example corresponding to $n=2$ is a quantum groupoid of dimension $13$ with antipode of infinite order (cf. \cite{NV2}, 7.3).
\end{section}
\begin{section} {Representation category of a quantum groupoid} Throughout this paper we refer to \cite{T2} for definitions related to categories.
For a quantum groupoid $H$ let $\mbox{Rep}(H)$ be the {\em category of representations} of $H$, whose objects are finite dimensional left $H$-modules and whose morphisms are $H$-linear homomorphisms. We shall show that $\mbox{Rep}(H)$ has a natural structure of a monoidal category with duality.
For objects $V,W$ of $\mbox{Rep}(H)$ set \begin{equation} V\otimes W = \{x \in V\otimes_k W \mid x= \Delta(1)\cdot x\}\subset V\otimes_k W\}, \end{equation} with the obvious action of $H$ via the comultiplication $\Delta$ (here $\otimes_k$ denotes the usual tensor product of vector spaces). Note that $\Delta(1)$ is an idempotent and therefore $V\otimes W=\Delta(1)\cdot (V\otimes_k W)$. The tensor product of morphisms is the restriction of usual tensor product of homomorphisms. The standard associativity isomorphisms $(U\otimes V)\otimes W \to U\otimes (V\otimes W)$ are functorial and satisfy the pentagon condition, since $\Delta$ is coassociative. We will suppress these isomorphisms and write simply $U\otimes V\otimes W$.
The target counital subalgebra $H_t\subset H$ has an $H$-module structure given by $h\cdot z = \varepsilon_t(hz)$, where $h\in H,\, z\in H_t$.
\begin{lemma} $H_t$ is the unit object of $\mbox{Rep}(H)$. \end{lemma}
\begin{proof} Define a $k$-linear homomorphism $l_V : H_t\otimes V \to V$ by $$ l_V( 1_{(1)}\cdot z \otimes 1_{(2)}\cdot v) = z\cdot v, \qquad z\in H_t,\, v\in V. $$ This map is $H$-linear, since \begin{eqnarray*} l_V( h\cdot(1_{(1)}\cdot z \otimes 1_{(2)}\cdot v)) &=& l_V( h_{(1)}\cdot z\otimes h_{(2)}\cdot v) \\ &=& \varepsilon_t(h\1z)h_{(2)} \cdot v = hz\cdot v \\ &=& h\cdot l_V( 1_{(1)}\cdot z\otimes 1_{(2)}\cdot v), \end{eqnarray*} for all $h\in H$. The inverse map $l_V^{-1} : V \to H_t\otimes V$ is given by $$ l_V^{-1}(v) = S(1_{(1)}) \otimes (1_{(2)} \cdot v) = (1_{(1)}\cdot 1) \otimes (1_{(2)}\cdot v). $$ The collection $\{l_V\}_V$ gives a natural equivalence between the functor $H_t\otimes (\ )$ and the identity functor. Indeed, for any $H$-linear homomorphism $f: V\to U$ we have : \begin{eqnarray*} l_U \circ (\mbox{id}\otimes f)(1_{(1)}\cdot z\otimes 1_{(2)}\cdot v) &=& l_U(1_{(1)}\cdot z\otimes 1_{(2)}\cdot f(v)) \\ &=& z\cdot f(v) =f(z\cdot v) \\ &=& f\circ l_V(1_{(1)}\cdot z\otimes 1_{(2)}\cdot v) \end{eqnarray*} Similarly, the $k$-linear homomorphism $r_V : V\otimes H_t \to V$ defined by $$ r_V( 1_{(1)}\cdot v \otimes 1_{(2)}\cdot z) = S(z)\cdot v, \qquad z\in H_t,\, v\in V, $$ has the inverse $r_V^{-1}(v) = 1_{(1)} \cdot v \otimes 1_{(2)}$ and satisfies the necessary properties.
Finally, we can check the triangle axiom $id_V\otimes l_W = r_V\otimes id_W:V\otimes H_t\otimes W\to V\otimes W$ for all objects $V,W$ of $\mbox{Rep}(H)$. For $v\in V,\,w\in W$ we have \begin{eqnarray*} \lefteqn{ (\mbox{id}_V\otimes l_W)(1_{(1)}\cdot v \otimes 1'_{(1)} 1_{(2)}\cdot z \otimes 1'_{(2)}\cdot w)=}\\ &=& 1_{(1)}\cdot v \otimes 1\2z \cdot w\\ &=& 1\1S(z)\cdot v\otimes 1_{(2)}\cdot w \\ &=& (r_V\otimes \mbox{id}_W)(1'_{(1)}\cdot v \otimes 1'_{(2)} 1_{(1)} \cdot z\otimes 1_{(2)}\cdot w), \end{eqnarray*} therefore, $\mbox{id}_V\otimes l_W = r_V\otimes \mbox{id}_W$. \end{proof}
Using the antipode $S$ of $H$, we can provide $\mbox{Rep}(H)$ with a duality. For any object $V$ of $\mbox{Rep}(H)$, define the action of $H$ on $V^*= \mbox{Hom}_k(V,\,k)$ by \begin{equation} \label{dual object} (h\cdot \phi)(v) = \phi(S(h)\cdot v), \end{equation} where $h\in H, v\in V, \phi\in V^*$. For any morphism $f: V\to W$, let $f^* : W^*\to V^*$ be the morphism dual to $f$ (see \cite{T2}, I.1.8).
For any $V$ in $\mbox{Rep}(H)$, we define the duality morphisms \begin{equation*} d_V : V^*\otimes V \to H_t, \qquad b_V: H_t \to V \otimes V^* \end{equation*} as follows. For $\sum_j\, \phi^j \otimes v_j\in V^* \otimes V$, set \begin{equation} \label{dV} d_V(\sum_j\, \phi^j \otimes v_j)= \sum_j\, \phi^j(1_{(1)}\cdot v_j)1_{(2)}. \end{equation} Let $\{f_i\}_i$ and $\{\xi^i\}_i$ be bases of $V$ and $V^*$ respectively, dual to each other. The element $\sum_i\,f_i\otimes \xi^i$ does not depend on choice of these bases; moreover, for all $v\in V, \phi\in V^*$ one has $\phi = \sum_i\,\phi(f_i)\xi^i$ and $v =\sum_i\,f_i\xi^i(v)$. Set \begin{equation} \label{bV} b_V(z) = z\cdot (\sum_i\, f_i \otimes \xi^i). \end{equation}
\begin{proposition} \label{monoidal with duality} The category $\mbox{Rep}(H)$ is a monoidal category with duality. \end{proposition}
\begin{proof} We know already that $\mbox{Rep}(H)$ is monoidal, it remains to prove that $d_V$ and $b_V$ are $H$-linear and satisfy the identities $$ (\mbox{id}_V\otimes d_V)(b_V \otimes \mbox{id}_V) = \mbox{id}_V, \qquad (d_V \otimes \mbox{id}_{V^*})(\mbox{id}_{V^*}\otimes b_V) = \mbox{id}_{V^*}. $$ Take $\sum_j\, \phi^j \otimes v_j\in V^*\otimes V, z\in H_t,h\in H$. Using the axioms of a quantum groupoid, we have \begin{eqnarray*} h\cdot d_V(\sum_j\, \phi^j \otimes v_j) &=& \sum_j\,\phi^j(1_{(1)}\cdot v) \varepsilon_t(h1_{(2)}) \\ &=& \sum_j\,\phi^j(\varepsilon_s(1\1h)\cdot v_j) 1_{(2)} \\ &=& \sum_j\,\phi^j(S(h_{(1)})1\1h_{(2)} \cdot v_j) 1_{(2)} \\ &=& \sum_j\,(h_{(1)}\cdot \phi^j)(1_{(1)}\cdot(h_{(2)}\cdot v_j))1_{(2)} \\ &=& \sum_j\, d_V(h_{(1)}\cdot \phi^j \otimes h_{(2)}\cdot v_j) \\ &=& d_V (h\cdot \sum_j\, \phi^j \otimes v_j), \end{eqnarray*} therefore, $d_V$ is $H$-linear. To check the $H$-linearity of $b_V$ we have to show that $h\cdot b_V(z) = b_V(h\cdot z)$, i.e.,\ that $$ \sum_i\,h\1z\cdot f_i\otimes h_{(2)} \cdot \xi^i = \sum_i\,1_{(1)}\varepsilon_t(hz)\cdot f_i \otimes 1_{(2)} \cdot \xi^i. $$ Since both sides of the above equality are elements of $V\otimes_k V^*$, evaluating the second factor on $v\in V$, we get the equivalent condition $$ h\1zS(h_{(2)})\cdot v = 1_{(1)}\varepsilon_t(hz)S(1_{(2)})\cdot v, $$ which is easy to check. Thus, $b_V$ is $H$-linear.
Using the isomorphisms $l_V$ and $r_V$ identifying $H_t\otimes V$, $V\otimes H_t$, and $V$, for all $v\in V$ and $\phi\in V^*$ we have: \begin{eqnarray*} (\mbox{id}_V\otimes d_V)(b_V \otimes \mbox{id}_V)(v) &=& (\mbox{id}_V\otimes d_V)(b_V(1_{(1)}\cdot 1)\otimes 1_{(2)}\cdot v) \\ &=& (\mbox{id}_V\otimes d_V)(b_V(1_{(2)})\otimes S^{-1}(1_{(1)})\cdot v) \\ &=& \sum_i\,(\mbox{id}_V\otimes d_V)
(1_{(2)}\cdot f_i \otimes 1_{(3)}\cdot \xi^i \otimes S^{-1}(1_{(1)})\cdot v) \\ &=& \sum_i\,1_{(2)}\cdot f_i \otimes (1_{(3)}\cdot \xi^i)(1_{(1)}'S^{-1}(1_{(1)})\cdot v)1_{(2)}'\\ &=& 1_{(2)} S(1_{(3)}) 1_{(1)}' S^{-1}(1_{(1)})\cdot v \otimes 1_{(2)}' =v, \\ (d_V \otimes \mbox{id}_{V^*})(\mbox{id}_{V^*}\otimes b_V)(\phi) &=& (d_V \otimes \mbox{id}_{V^*})(1_{(1)}\cdot \phi \otimes b_V(1_{(2)})) \\ &=& \sum_i\, (d_V \otimes \mbox{id}_{V^*})
(1_{(1)}\cdot\phi \otimes 1_{(2)}\cdot f_i \otimes 1_{(3)}\cdot \xi^i) \\ &=& \sum_i\,(1_{(1)}\cdot\phi)(1_{(1)}'1_{(2)}\cdot f_i)1_{(2)}' \otimes 1_{(3)}\cdot \xi^i\\ &=& 1_{(2)}' \otimes 1_{(3)} 1_{(1)} S(1_{(1)}'1_{(2)})\cdot \phi = \phi, \end{eqnarray*} which completes the proof. \end{proof} \begin{remark} Similarly to the construction of $\mbox{Rep}(H)$, one can construct a category of right $H$-modules, in which $H_s$ plays the role of the unit object. \end{remark} \end{section}
\begin{section}{Quasitriangular quantum groupoids} \begin{definition} \label{QT WHA} A quasitriangular quantum groupoid is a pair ($H,\, {\mathcal R}$) where $H$ is a quantum groupoid and ${\mathcal R}\in \Delta^{op}(1)(H\otimes_k H)\Delta(1)$ satisfying the following conditions : \begin{equation} \Delta^{op}(h){\mathcal R} = {\mathcal R}\Delta(h), \end{equation} for all $h\in H$, where $\Delta^{op}$ denotes the comultiplication opposite to $\Delta$, \begin{equation} (\mbox{id} \otimes \Delta){\mathcal R} = {\mathcal R}_{13}{\mathcal R}_{12}, \qquad (\Delta \otimes \mbox{id}){\mathcal R} = {\mathcal R}_{13}{\mathcal R}_{23}, \end{equation} where ${\mathcal R}_{12} = {\mathcal R}\otimes 1$, ${\mathcal R}_{23} = 1\otimes {\mathcal R}$, etc.\ as usual, and such that there exists ${\bar \R}\in \Delta(1)(H\otimes_k H)\Delta^{op}(1)$ with \begin{equation} {\mathcal R}{\bar \R} = \Delta^{op}(1), \qquad {\bar \R}{\mathcal R} = \Delta(1). \end{equation} \end{definition}
Note that ${\bar \R}$ is uniquely determined by ${\mathcal R}$: if ${\bar \R}$ and ${\bar \R}'$ are two elements of $\Delta(1)(H\otimes_k H)\Delta^{op}(1)$ satisfying the previous equation, then $$ {\bar \R}={\bar \R}\Delta^{op}(1)={\bar \R}{\mathcal R}{\bar \R}'=\Delta(1){\bar \R}'={\bar \R}'. $$ For any two objects $V$ and $W$ of $\mbox{Rep}(H)$ define $c_{V,W}: V\otimes W \to W\otimes V$ as the action of ${\mathcal R}_{21}$ : \begin{eqnarray} c_{V,W}(x) = {\mathcal R}^{(2)}\cdot x^{(2)} \otimes {\mathcal R}^{(1)}\cdot x^{(1)}, \end{eqnarray} where $x =x^{(1)}\otimes x^{(2)}\in V\otimes W$, ${\mathcal R}={\mathcal R}^{(1)}\otimes {\mathcal R}^{(2)}\in \Delta^{op}(1)(H\otimes_k H)\Delta(1)$.
\begin{proposition} \label{braiding} The family of homomorphisms $\{c_{V,W}\}_{V,W}$ defines a braiding in $\mbox{Rep}(H)$. Conversely, if $\mbox{Rep}(H)$ is braided, then there exists ${\mathcal R}\in \Delta^{op}(1)(H\otimes_k H)\Delta(1)$, satisfying the properties of Definition~\ref{QT WHA} and inducing the given braiding. \end{proposition}
\begin{proof} Note that $c_{V,W}$ is well-defined, since ${\mathcal R}_{21}= \Delta(1){\mathcal R}_{21}$. To prove the $H$-linearity of $c_{V,W}$ we observe that \begin{eqnarray*} c_{V,W}(h\cdot x) &=& {\mathcal R}^{(2)} h_{(2)}\cdot x^{(2)} \otimes {\mathcal R}^{(1)} h_{(1)}\cdot x^{(1)} \\ &=& h_{(1)} {\mathcal R}^{(2)} \cdot x^{(2)} \otimes h_{(2)} {\mathcal R}^{(1)} \cdot x^{(1)} =h\cdot (c_{V,W}(x)). \end{eqnarray*} The inverse of $c_{V,W}$ is given by $$ c_{V,W}^{-1}(y) = {\bar \R}^{(1)}\cdot y^{(2)} \otimes {\bar \R}^{(2)}\cdot y^{(1)}, $$ where $y =y^{(1)}\otimes y^{(2)}\in W\otimes V, {\bar \R}={\bar \R}^{(1)}\otimes {\bar \R}^{(2)}$. Therefore, $c_{V,W}$ is an isomorphism. Finally, one can verify that the braiding identities $$ (\mbox{id}_V \otimes c_{U,W})(c_{U,V}\otimes id_W)= c_{U, V\otimes W}, \qquad (c_{U,W}\otimes\mbox{id}_V)(\mbox{id}_U\otimes c_{V,W}) = c_{U\otimes V,W} $$ are equivalent to the relations of Definition~\ref{QT WHA}, exactly in the same way as in the case of Hopf algebras (see, for instance, (\cite{T2}, XI, 2.3.1)). \end{proof}
\begin{lemma} \label{yz} Let $(H,{\mathcal R})$ be a quasitriangular quantum groupoid. Then for all $y\in H_s, z\in H_t$ the following six identities hold: \begin{eqnarray*} (1\otimes z){\mathcal R} &=& {\mathcal R}(z\otimes 1), \qquad (y\otimes 1){\mathcal R} = {\mathcal R}(1\otimes y),\\ (z\otimes 1){\mathcal R} &=& (1\otimes S(z)){\mathcal R}, \qquad (1\otimes y){\mathcal R} = (S(y)\otimes 1){\mathcal R},\\ {\mathcal R}(y\otimes 1) &=& {\mathcal R}(1\otimes S(y)), \qquad {\mathcal R}(1\otimes z) = {\mathcal R}(S(z)\otimes 1). \end{eqnarray*} \end{lemma}
\begin{proof} Since we have $\Delta^{op}(1){\mathcal R} ={\mathcal R} = {\mathcal R}\Delta(1)$, the first line is a consequence of the relation $\Delta(yz) = (z\otimes y)\Delta(1)$, the second line follows from $(1\otimes z)\Delta(1)= (S(z)\otimes 1)\Delta(1)$ and $(y\otimes 1)\Delta(1)= (1\otimes S(y))\Delta(1)$. The last two identities are proven similarly. \end{proof}
\begin{proposition} \label{QYBE} Let $(H,{\mathcal R})$ be a quasitriangular quantum groupoid. Then ${\mathcal R}$ satisfies the quantum Yang-Baxter equation : $$ {\mathcal R}_{12}{\mathcal R}_{13}{\mathcal R}_{23} = {\mathcal R}_{23}{\mathcal R}_{13}{\mathcal R}_{12}. $$ \end{proposition}
\begin{proof} It follows from the first two relations of Definition~\ref{QT WHA}, that \begin{eqnarray*} {\mathcal R}_{12}{\mathcal R}_{13}{\mathcal R}_{23}=(\mbox{id}\otimes \Delta^{op})({\mathcal R}) {\mathcal R}_{23}= {\mathcal R}_{23} (\mbox{id}\otimes \Delta)({\mathcal R})={\mathcal R}_{23}{\mathcal R}_{13}{\mathcal R}_{12}. \end{eqnarray*} \end{proof}
\begin{remark} Let us define two $k$-linear maps ${\mathcal R}_1, {\mathcal R}_2 : \widehat H \to H$ by $$ {\mathcal R}_1(\phi) =(\mbox{id} \otimes \phi)({\mathcal R}), \quad {\mathcal R}_2(\phi) =(\phi\otimes \mbox{id})({\mathcal R}), \quad\mbox{for}\ \phi\in \widehat H. $$ Then the condition $(\mbox{id} \otimes \Delta){\mathcal R} = {\mathcal R}_{13}{\mathcal R}_{12}$ is equivalent to ${\mathcal R}_1$ being a coalgebra homomorphism and algebra anti-homomorphism and the condition $(\Delta \otimes \mbox{id}){\mathcal R} = {\mathcal R}_{13}{\mathcal R}_{23}$ is equivalent to ${\mathcal R}_2$ being an algebra homomorphism and coalgebra anti-homomorphism. In other words, ${\mathcal R}_1$ and ${\mathcal R}_2$ are homomorphisms of quantum groupoids $\widehat H \to H^{op}$ and $\widehat H \to H^{cop}$, respectively. \end{remark}
\begin{proposition} \label{properties of R} For any quasitriangular quantum groupoid $(H,{\mathcal R})$, we have: \begin{eqnarray*} (\varepsilon_s\otimes \mbox{id})({\mathcal R}) = \Delta(1),&\ & (\mbox{id}\otimes \varepsilon_s)({\mathcal R}) = (S\otimes \mbox{id})\Delta^{op}(1), \\ (\varepsilon_t\otimes \mbox{id})({\mathcal R}) = \Delta^{op}(1),&\ & (\mbox{id}\otimes \varepsilon_t)({\mathcal R}) = (S\otimes \mbox{id})\Delta(1), \end{eqnarray*} $$ (S\otimes \mbox{id})({\mathcal R}) = (\mbox{id}\otimes S^{-1})({\mathcal R}) = {\bar \R},\ (S\otimes S)({\mathcal R})= {\mathcal R}. $$ \end{proposition}
\begin{proof} First, using the same argument as in \cite{T2}, XI, 2.1.1, we can show that $(\varepsilon\otimes \mbox{id})({\mathcal R}) = (\mbox{id}\otimes \varepsilon)({\mathcal R}) = 1$. Next, using Lemma~\ref{yz}, we obtain \begin{eqnarray*} (\varepsilon_s\otimes \mbox{id})({\mathcal R}) &=& 1_{(1)}\varepsilon({\mathcal R}^{(1)} 1_{(2)}) \otimes {\mathcal R}^{(2)} \\ &=& 1_{(1)}\varepsilon({\mathcal R}^{(1)}) \otimes 1_{(2)} {\mathcal R}^{(2)} = \Delta(1), \\ (\mbox{id} \otimes \varepsilon_s)({\mathcal R}) &=& {\mathcal R}^{(1)} \otimes 1_{(1)} \varepsilon({\mathcal R}^{(2)} 1_{(2)}) \\ &=& S(1_{(2)}){\mathcal R}^{(1)} \otimes 1_{(1)} \varepsilon({\mathcal R}^{(2)}) = (S\otimes \mbox{id})\Delta^{op}(1),\\ (\varepsilon_t\otimes \mbox{id})({\mathcal R}) &=& \varepsilon(1_{(1)} {\mathcal R}^{(1)})1_{(2)} \otimes {\mathcal R}^{(2)} \\ &=& \varepsilon({\mathcal R}^{(1)}) 1_{(2)} \otimes {\mathcal R}^{(2)} 1_{(1)} = \Delta^{op}(1), \\ (\mbox{id} \otimes \varepsilon_t)({\mathcal R}) &=& {\mathcal R}^{(1)} \otimes \varepsilon(1_{(1)} {\mathcal R}^{(2)}) 1_{(2)} \\ &=& S(1_{(1)} ){\mathcal R}^{(1)} \otimes \varepsilon({\mathcal R}^{(2)})1_{(2)} = (S\otimes \mbox{id})\Delta(1). \end{eqnarray*}
Let $m$ denote multiplication $H\otimes_k H\to H$ in $H$. Set $m_{12}= m\otimes\mbox{id}: H^{\otimes 3}\to H^{\otimes 2}$ and $m_{23}= \mbox{id}\otimes m: H^{\otimes 3}\to H^{\otimes 2}$. It follows from the above relations that \begin{eqnarray*} m_{12}((S\otimes \mbox{id} \otimes \mbox{id})(\Delta\otimes \mbox{id})({\mathcal R})) &=& (\varepsilon_s\otimes \mbox{id})({\mathcal R}) = \Delta(1), \\ m_{12}((\mbox{id} \otimes S \otimes \mbox{id})(\Delta\otimes \mbox{id})({\mathcal R})) &=& (\varepsilon_t\otimes \mbox{id})({\mathcal R}) = \Delta^{op}(1), \\ m_{23}((\mbox{id} \otimes \mbox{id} \otimes S^{-1})(\Delta^{op}\otimes \mbox{id})({\mathcal R})) &=& (\mbox{id} \otimes S^{-1}\varepsilon_t)({\mathcal R}) = \Delta^{op}(1), \\ m_{23}((\mbox{id} \otimes S^{-1} \otimes \mbox{id})(\Delta^{op}\otimes \mbox{id})({\mathcal R})) &=& (\mbox{id}\otimes S^{-1}\varepsilon_s)({\mathcal R}) = \Delta(1). \end{eqnarray*} On the other hand, Definition~\ref{QT WHA} implies that \begin{eqnarray*} \lefteqn{m_{12}((S\otimes \mbox{id} \otimes \mbox{id})(\Delta\otimes \mbox{id})({\mathcal R})) =}\\ &=& m_{12}((S\otimes \mbox{id} \otimes \mbox{id})({\mathcal R}_{13}{\mathcal R}_{23}) )
= (S\otimes\mbox{id})({\mathcal R}){\mathcal R},\\ \lefteqn{m_{12}((\mbox{id} \otimes S \otimes \mbox{id})(\Delta\otimes \mbox{id})({\mathcal R}))=}\\ &=& m_{12}((\mbox{id} \otimes S \otimes \mbox{id})({\mathcal R}_{13}{\mathcal R}_{23}))
= {\mathcal R}(S\otimes\mbox{id})({\mathcal R}),\\ \lefteqn{m_{23}((\mbox{id} \otimes \mbox{id} \otimes S^{-1})(\Delta^{op}\otimes \mbox{id})({\mathcal R}))=}\\ &=& m_{23}((\mbox{id} \otimes \mbox{id} \otimes S^{-1})({\mathcal R}_{12}{\mathcal R}_{13}))
= {\mathcal R}(\mbox{id} \otimes S^{-1})({\mathcal R}),\\ \lefteqn{m_{23}((\mbox{id} \otimes S^{-1} \otimes \mbox{id})(\Delta^{op}\otimes \mbox{id})({\mathcal R}))=}\\ &=& m_{23}((\mbox{id} \otimes S^{-1} \otimes \mbox{id})({\mathcal R}_{12}{\mathcal R}_{13}))
= (\mbox{id} \otimes S^{-1})({\mathcal R}){\mathcal R}. \end{eqnarray*} Therefore, $(S\otimes \mbox{id})({\mathcal R}) = (\mbox{id}\otimes S^{-1})({\mathcal R}) = {\bar \R}$ and $(S\otimes S)({\mathcal R})= {\mathcal R}$. \end{proof}
\begin{proposition} \label{elements u and v} Let $(H,{\mathcal R})$ be a quasitriangular quantum groupoid. Then $S^2(h) = uhu^{-1}$ for all $h\in H$, where $u=S({\mathcal R}^{(2)}){\mathcal R}^{(1)}$ is an invertible element of $H$ such that $$ u^{-1} = {\mathcal R}^{(2)} S^2({\mathcal R}^{(1)}), \quad \Delta(u) = {\bar \R} {\bar \R}_{21} (u\otimes u). $$ Likewise, $S^{-2}(h) = vhv^{-1}$, where $v = S(u) = {\mathcal R}^{(1)} S({\mathcal R}^{(2)})$, and $$ v^{-1} = S^2({\mathcal R}^{(1)}){\mathcal R}^{(2)}, \quad \Delta(v) = {\bar \R} {\bar \R}_{21} (v\otimes v). $$ \end{proposition}
\begin{proof} Note that $S({\mathcal R}^{(2)})y{\mathcal R}^{(1)} = S(y)u$ for all $y\in H_s$, by Lemma~\ref{yz}. Hence, we have \begin{eqnarray*} S(h_{(2)})uh_{(1)} &=& S(h_{(2)}) S({\mathcal R}^{(2)}){\mathcal R}^{(1)} h_{(1)} = S({\mathcal R}^{(2)} h_{(2)}){\mathcal R}^{(1)} h_{(1)} \\ &=& S(h_{(1)} {\mathcal R}^{(2)}) h_{(2)} {\mathcal R}^{(1)} = S({\mathcal R}^{(2)}) \varepsilon_s(h) {\mathcal R}^{(1)} \\ &=& S(\varepsilon_s(h))u, \end{eqnarray*} for all $h\in H$. Therefore, using the axioms of a quantum groupoid, we get \begin{eqnarray*} uh &=& S(1_{(2)})u1\1h = S(\varepsilon_t(h_{(2)})u h_{(1)} \\ &=& S(h_{(2)} S(h_{(3)}))u h_{(1)} = S^2(h_{(3)}) S(h_{(2)})u h_{(1)} \\ &=& S^2(h_{(2)}) S(\varepsilon_s(h_{(1)})) u = S(\varepsilon_s(h_{(1)})S(h_{(2)}))u = S^2(h)u. \end{eqnarray*} The remaining part of the proof follows the lines of (\cite{M}, 2.1.8). The results for $v$ can be obtained by applying the results for $u$ to the quasitriangular quantum groupoid $(H^{op/cop}, {\mathcal R})$. \end{proof}
\begin{definition} \label{Drinfeld element} The element $u$ defined in Proposition~\ref{elements u and v} is called {\em the Drinfeld element} of $H$. \end{definition}
\begin{corollary} \label{uv is central} The element $uv=vu$ is central and $$ \Delta(uv) = ({\bar \R} {\bar \R}_{21})^2(uv\otimes uv). $$ The element $uv^{-1} =vu^{-1}$ is group-like and $S^4(h)=uv^{-1}hvu^{-1}$ for all $h\in H$. \end{corollary}
\begin{proof} See (\cite{M}, 2.1.9). \end{proof}
\begin{proposition} \label{range of F} Given a quasitriangular quantum groupoid $(H, {\mathcal R})$, consider a linear map $F:\widehat H\to H$ given by \begin{equation} F(\phi)=(\phi\otimes \mbox{id})({\mathcal R}_{21}{\mathcal R}),\qquad \phi\in \widehat H. \end{equation} Then the image of $F$ lies in $C_H(H_s)$, the centralizer of $H_s$. \end{proposition}
\begin{proof} Take $y\in H_s$. Then we have $$ \phi({\mathcal R}^{(2)} {{\mathcal R}}^{(1)}) {\mathcal R}^{(1)} {{\mathcal R}}^{(2)} y = \phi({\mathcal R}^{(2)} y {{\mathcal R}}^{(1)}) {\mathcal R}^{(1)} {{\mathcal R}}^{(2)} = \phi({\mathcal R}^{(2)} {{\mathcal R}}^{(1)}) y {\mathcal R}^{(1)} {{\mathcal R}}^{(2)}. $$ Therefore $F(\phi)\in C_H(H_s)$, as required. \end{proof}
\begin{definition}[cf. \cite{M}, 2.1.12] \label{factorizability} A quasitriangular quantum groupoid is {\em factorizable} if the map $F:\widehat H \to C_H(H_s)$ from Proposition~\ref{range of F} is surjective. \end{definition}
The factorizability means that ${\mathcal R}$ is as non-trivial as possible, in contrast to {\em triangular} quantum groupoids, for which $ {\bar \R}={\mathcal R}_{21}$ and $F(\widehat H)=H_t$. \begin{corollary} \label{restriction of F} If $H$ is factorizable, then the restriction of $F$ to the subspace $W_s = \{ \phi\in \widehat H \mid \phi = \phi\circ\mbox{Ad}\,_1^r) \}$ is a linear isomorphism onto $C_H(H_s)$. \end{corollary}
\begin{proof} From the observation that $F(\phi) = F(\phi\circ\mbox{Ad}\,_1^r)$ we have that the restriction of $F$ to $W_s$ is a linear map onto $C_H(H_s)$. On the other hand, Lemma~\ref{expectations} allows to identify $W_s$ with the dual vector space to $C_H(H_t)$, from where $\dim W_s = \dim C_H(H_s)$ and the result follows. \end{proof} \end{section}
\begin{section}{The Drinfeld double for quantum groupoids}
Let $H$ be a finite quantum groupoid. We define the {\em Drinfeld double} $D(H)$ of $H$ as follows. Consider on the vector space $\widehat H^{op}\otimes_k H$ a multiplication given by \begin{equation} (\phi\otimes h)(\psi\otimes g) = \psi_{(2)}\phi \otimes h\2g \langle\, S(h_{(1)}),\, \psi_{(1)}\,\rangle \langle\, h_{(3)},\, \psi_{(3)}\,\rangle, \end{equation} where $\phi, \psi \in {\widehat H}^{op}$ and $h,g \in H$. We verify below that the linear span $J$ of the elements \begin{eqnarray} \label{amalgamation} \phi\otimes zh &-& (\varepsilon\leftharpoonup z)\phi \otimes h, \quad z\in H_t,\\ \phi\otimes yh &-& (y\rightharpoonup\varepsilon)\phi \otimes h, \quad y\in H_s, \end{eqnarray} is a two-sided ideal in $\widehat H^{op}\otimes_k H$. Let $D(H)$ be the factor-algebra ($\widehat H^{op}\otimes_k H)/J$ and let $[\phi\otimes h]$ denote the class of $\phi\otimes h$ in $D(H)$.
\begin{definitionandtheorem} \label{the double} $D(H)$ is a quantum groupoid with unit $[\varepsilon\otimes 1]$, and comultiplication, counit, and antipode given by \begin{eqnarray} \Delta([\phi\otimes h]) &=& [\phi_{(1)} \otimes h_{(1)}] \otimes [\phi_{(2)} \otimes h_{(2)}],\\ {\varepsilon}([\phi\otimes h]) &=& \langle\, \varepsilon_t(h),\,\phi \,\rangle,\\ S([\phi\otimes h]) &=& [ S^{-1}(\phi_{(2)}) \otimes S(h_{(2)})] \langle\, h_{(1)},\,\phi_{(1)} \,\rangle
\langle\, S(h_{(3)}),\,\phi_{(3)}\,\rangle. \end{eqnarray} \end{definitionandtheorem} In the case where $H$ is a Hopf algebra, this definition and is due to Drinfeld \cite{D}. \begin{proof} Associativity of multiplication in $\widehat H^{op}\otimes_k H$ and hence in $D(H)$ can be verified exactly as in (\cite{M}, 7.1.1). Let us check that $J$ is an ideal. We have : \begin{eqnarray*} (\phi\otimes h)((\varepsilon\leftharpoonup z)\psi \otimes g) &=& \psi_{(2)}\phi \otimes h\2g
\langle\, S(h_{(1)}),\, (\varepsilon\leftharpoonup z)\psi_{(1)}\,\rangle \langle\, h_{(3)},\, \psi_{(3)}\,\rangle \\ &=& \psi_{(2)}\phi \otimes h\3g \langle\, zS(h_{(2)}),\,\varepsilon\,\rangle
\langle\, S(h_{(1)}),\, \psi_{(1)}\,\rangle \langle\, h_{(4)},\, \psi_{(3)}\,\rangle \\ &=& \psi_{(2)}\phi \otimes h\2zg \langle\, S(h_{(1)}),\, \psi_{(1)}\,\rangle
\langle\, h_{(3)},\, \psi_{(3)}\,\rangle \\ &=& (\phi\otimes h)(\psi\otimes zg), \\ (\psi\otimes zg)(\phi\otimes h) &=& \phi_{(2)}\psi \otimes g\2h \langle\, S(zg_{(1)}),\,\phi_{(1)} \,\rangle
\langle\, g_{(3)},\, \phi_{(3)}\,\rangle \\ &=& \langle\, Sz,\, \phi_{(2)}\,\rangle \phi_{(3)} \psi \otimes g\2h \langle\, S(g_{(1)}),\,\phi_{(1)} \,\rangle
\langle\, g_{(3)},\, \phi_{(3)}\,\rangle \\ &=& \phi_{(2)} (\varepsilon\leftharpoonup z)\psi \otimes g\2h \langle\, S(g_{(1)}),\,\phi_{(1)} \,\rangle
\langle\, g_{(3)},\, \phi_{(3)}\,\rangle \\ &=& ( (\varepsilon\leftharpoonup z)\psi \otimes g)(\phi\otimes h), \end{eqnarray*} where $z\in H_t$ and we used the identity $zS(h_{(1)})\otimes h_{(2)} = S(h_{(1)})\otimes h\2z$. Similarly, one checks that \begin{eqnarray*} (\psi\otimes yg)(\phi\otimes h) &=& ((y\rightharpoonup \varepsilon)\psi \otimes g)(\phi\otimes h), \\ (\phi\otimes h)(\psi\otimes yg) &=& (\phi\otimes h) ((y\rightharpoonup \varepsilon)\psi \otimes g), \end{eqnarray*} therefore for all $x\in J$ we have $(\phi\otimes h)x = x(\phi\otimes h) =0$, so $J$ is an ideal. We also compute \begin{eqnarray*} [\varepsilon\otimes 1][\phi\otimes h] &=& [\phi_{(2)} \otimes 1\2h \langle\, S(1_{(1)}),\, \phi_{(1)}\,\rangle \langle\, 1_{(3)},\, \phi_{(3)}\,\rangle] \\ &=& [\phi_{(2)} \otimes \langle\, S(1_{(1)}),\, \phi_{(1)}\,\rangle 1\21'_{(1)} \langle\, 1'_{(2)},\, \phi_{(3)}\,\rangle h]\\ &=& [\varepsilon_t(\phi_{(1)})S(\varepsilon_t(\phi_{(3)}))\phi_{(2)} \otimes h] = [\phi\otimes h], \end{eqnarray*} and similarly $[\phi\otimes h][\varepsilon\otimes 1] = [\phi\otimes h]$, so that $[\varepsilon\otimes 1]$ is a unit.
Now let us verify that the structure maps $\Delta,\, {\varepsilon},$ and $S$ are well-defined on $D(H)$. We have, using properties of a quantum groupoid and its counital subalgebras: \begin{eqnarray*} \Delta([\phi\otimes zh]) &=& [\phi_{(1)}\otimes zh_{(1)}] \otimes [\phi_{(2)}\otimes h_{(2)}] \\ &=& [(\varepsilon\leftharpoonup z)\phi_{(1)} \otimes h_{(1)}] \otimes [\phi_{(2)}\otimes h_{(2)}] \\ &=& \Delta([ \langle\, z,\,\varepsilon_{(1)}\,\rangle \varepsilon_{(2)}\phi \otimes h]), \\ \varepsilon([ \langle\, z,\,\varepsilon_{(1)}\,\rangle \varepsilon_{(2)}\phi \otimes h]) &=& \langle\, z,\, \varepsilon_{(1)} \,\rangle \langle\, \varepsilon_t(h),\, \varepsilon_{(2)}\phi \,\rangle \\ &=& \langle\, z,\, \varepsilon_{(1)} \,\rangle \langle\, 1_{(1)}\varepsilon_t(h),\,\varepsilon_{(2)} \,\rangle \langle\, 1_{(2)},\, \phi\,\rangle \\ &=& \langle\, z\varepsilon_t(h)1_{(1)},\, \varepsilon \,\rangle \langle\, 1_{(2)},\, \phi\,\rangle \\ &=& \langle\, z\varepsilon_t(h),\,\phi\,\rangle = {\varepsilon}([\phi\otimes zh]), \\ S([ (\varepsilon\leftharpoonup z)\phi \otimes h]) &=& [ \langle\, z,\, \varepsilon_{(1)}\,\rangle S^{-1}(\varepsilon_{(3)}\phi_{(2)}) \otimes S(h_{(2)})] \\ & & \langle\, h_{(1)},\,\varepsilon_{(2)}\phi_{(1)} \,\rangle \langle\, S(h_{(3)}),\,\varepsilon_{(4)}\phi_{(3)} \,\rangle \\ &=& [ S^{-1}(\phi_{(2)}) \otimes S(h_{(2)})] \\ & & \langle\, h_{(1)},\,(\varepsilon\leftharpoonup z)\phi_{(1)} \,\rangle \langle\, S(h_{(3)}),\,\phi_{(3)} \,\rangle \\ &=& [S^{-1}(\phi_{(2)}) \otimes S(h_{(3)}) ]\langle\, zh_{(1)},\, \varepsilon\,\rangle \\ & & \langle\, h_{(2)},\, \phi_{(1)}\,\rangle \langle\, S(h_{(4)}),\,\phi_{(3)} \,\rangle \\ &=& [S^{-1}(\phi_{(2)}) \otimes S(h_{(2)})] \langle\, zh_{(1)},\, \phi_{(1)} \,\rangle
\langle\, S(h_{(3)}),\,\phi_{(3)} \,\rangle \\ &=& S([\phi\otimes zh]) \end{eqnarray*} for all $h\in H, \phi\in \widehat{H}, z\in H_t$. Next, we need to check the axioms of a quantum groupoid. Coassociativity and multiplicativity of $\Delta$ are established as in (\cite{M}, 7.1.1), since the computations given there do not use the unitality of multiplication and comultiplication. For the counit property, we have: \begin{eqnarray*} ({\varepsilon}\otimes\mbox{id})\Delta([\phi\otimes h]) &=& \langle\, \varepsilon_t(h_{(1)}),\, \phi_{(1)} \,\rangle [\phi_{(2)}\otimes h_{(2)}] \\ &=& \langle\, \varepsilon_t(h_{(1)}),\, \varepsilon_{(1)} \,\rangle [\varepsilon_{(2)}\phi \otimes h_{(2)}] \\ &=& [\phi \otimes \varepsilon_t(h_{(1)})h_{(2)}] = [\phi\otimes h], \\ (\mbox{id} \otimes{\varepsilon})\Delta([\phi\otimes h]) &=& [\phi_{(1)}\otimes h_{(1)}] \langle\, \varepsilon_t(h_{(2)}),\, \phi_{(2)} \,\rangle \\ &=& [\phi_{(1)}\otimes 1\1h] \langle\, 1_{(2)},\, \varepsilon_t(\phi_{(2)}) \,\rangle \\ &=& [S^{-1}(\varepsilon_t(\phi_{(2)}))\phi_{(1)} \otimes h] = [\phi\otimes h], \end{eqnarray*} where we used the amalgamation property $[\phi\otimes zh]=[(\varepsilon\leftharpoonup z)\phi \otimes h], \quad z\in H_t,$ following from (\ref{amalgamation}). Now we verify the remaining axioms of a quantum groupoid. For all $h,g,f\in H$ and $\phi,\psi,\theta\in \widehat{H}$ we compute \begin{eqnarray*} \lefteqn{\varepsilon([\phi\otimes h][\psi\otimes g][\theta\otimes f]) = } \\ &=& \varepsilon([\theta_{(2)} \psi_{(2)} \phi \otimes h\3g\2f]) \langle\, S(h_{(1)}),\, \psi_{(1)}\,\rangle \\ & & \langle\, S(h\2g_{(1)}),\theta_{(1)}\,\rangle \langle\, h\4g_{(3)},\, \theta_{(3)} \,\rangle
\langle\, h_{(5)},\, \psi_{(3)} \,\rangle \\ &=& \langle\, h\3g_{(2)},\,\varepsilon_{(1)}\varepsilon_t(\theta_{(2)} \psi_{(2)} \phi)\,\rangle \langle\, f,\,\varepsilon_{(2)} \,\rangle
\langle\, S(h_{(1)}),\, \psi_{(1)}\,\rangle \\ & & \langle\, h\2g_{(1)},\, S(\theta_{(1)})\,\rangle \langle\, h\4g_{(3)},\, \theta_{(3)} \,\rangle
\langle\, h_{(5)},\, \psi_{(3)} \,\rangle \\ &=& \langle\, h\2g,\, S(\theta_{(1)})\varepsilon_{(1)} \varepsilon_t(\theta_{(2)} \psi_{(2)} \phi) \theta_{(3)} \,\rangle
\langle\, f,\,\varepsilon_{(2)} \,\rangle \langle\, S(h_{(1)}),\, \psi_{(1)}\,\rangle \langle\, h_{(3)},\, \psi_{(3)} \,\rangle \\ &=& \langle\, h\2g,\, \varepsilon_t(\psi_{(2)} \phi)\varepsilon_s(\varepsilon_{(1)}\theta) \,\rangle
\langle\, f,\,\varepsilon_{(2)} \,\rangle \langle\, S(h_{(1)}),\, \psi_{(1)}\,\rangle \langle\, h_{(3)},\, \psi_{(3)} \,\rangle \\ &=& \langle\, h\2g_{(1)},\, \varepsilon_t(\psi_{(2)} \phi) \,\rangle
\langle\, g_{(2)},\, \varepsilon'_{(2)} \varepsilon_s(\varepsilon_{(1)}\theta) \,\rangle \\ & & \langle\, f,\,\varepsilon_{(2)} \,\rangle \langle\, S(h_{(1)}),\, \psi_{(1)}\,\rangle \langle\, h_{(3)},\, \varepsilon'_{(1)} \psi_{(3)}\,\rangle\\ &=& \langle\, h\2g_{(1)},\, \varepsilon_t(\psi_{(2)} \phi) \,\rangle
\langle\, g_{(2)},\, S(\theta_{(1)})\varepsilon_{(1)} \varepsilon_t(\theta_{(2)} \psi_{(4)})\theta_{(3)} \,\rangle \\ & & \langle\, f,\,\varepsilon_{(2)} \,\rangle \langle\, S(h_{(1)}),\, \psi_{(1)}\,\rangle \langle\, h_{(3)},\, \varepsilon'_{(1)} \psi_{(3)} \,\rangle\\ &=& \langle\, h\2g_{(1)},\, \varepsilon_t(\psi_{(2)} \phi) \,\rangle
\langle\, S(h_{(1)}),\, \psi_{(1)}\,\rangle \langle\, h_{(3)},\, \psi_{(3)} \,\rangle \\ & & \langle\, g\3f,\,\varepsilon_t(\theta_{(2)} \psi_{(4)})\,\rangle
\langle\, S(g_{(2)}),\, \theta_{(1)} \,\rangle \langle\, g_{(4)},\, \theta_{(3)} \,\rangle \\ &=& \varepsilon([\phi\otimes h][\psi_{(1)}\otimes g_{(1)}])
\varepsilon([\psi_{(2)}\otimes g_{(2)}][\theta\otimes f]), \\ \lefteqn{\varepsilon([\phi\otimes h][\psi_{(2)}\otimes g_{(2)}])
\varepsilon([\psi_{(1)}\otimes g_{(1)}][\theta\otimes f]) =} \\ &=& \langle\, h_{(2)} g_{(4)},\, \varepsilon_t(\psi_{(3)} \phi) \,\rangle
\langle\, S(h_{(1)}),\, \psi_{(2)}\,\rangle \langle\, h_{(3)},\, \psi_{(4)} \,\rangle \\ & & \langle\, g\2f,\,\varepsilon_t(\theta_{(2)} \psi_{(1)})\,\rangle
\langle\, S(g_{(1)}),\, \theta_{(1)} \,\rangle \langle\, g_{(3)},\, \theta_{(3)} \,\rangle \\ &=& \langle\, h_{(2)} g_{(2)},\,\varepsilon_t(\psi_{(3)} \phi) \,\rangle
\langle\, g_{(1)},\, S(\theta_{(1)})\varepsilon_{(1)} \varepsilon_t(\theta_{(2)} \psi_{(1)}) \theta_{(3)} \,\rangle \\ & & \langle\, f,\, \varepsilon_{(2)}\,\rangle \langle\, S(h_{(1)}),\, \psi_{(2)}\,\rangle \langle\, h_{(3)},\, \psi_{(4)} \,\rangle \\ &=& \langle\, h_{(2)} g_{(2)}, \,\varepsilon_t(\psi_{(3)} \phi) \,\rangle
\langle\, g_{(1)},\, \varepsilon'_{(2)}\varepsilon_s(\varepsilon_{(1)}\theta) \,\rangle \\ & & \langle\, f,\, \varepsilon_{(2)}\,\rangle \langle\, S(h_{(1)}),\, S(\varepsilon'_{(1)})\psi_{(1)}\,\rangle \langle\, h_{(3)},\, \psi_{(4)} \,\rangle \\ &=& \langle\, h\2g,\, \varepsilon_t(\psi_{(2)} \phi)\varepsilon_s(\varepsilon_{(1)}\theta) \,\rangle
\langle\, f,\,\varepsilon_{(2)} \,\rangle \langle\, S(h_{(1)}),\, \psi_{(1)}\,\rangle \langle\, h_{(3)},\, \psi_{(3)} \,\rangle \\ &=& \varepsilon([\phi\otimes h][\psi\otimes g][\theta\otimes f]), \end{eqnarray*} which is axiom (\ref{eps m}). For axiom (\ref{Delta 1}) we have: \begin{eqnarray*} \lefteqn{(\Delta([\varepsilon\otimes 1])\otimes [\varepsilon\otimes 1]) ([\varepsilon\otimes 1] \otimes \Delta([\varepsilon\otimes 1]) ) =}\\ &=& [\varepsilon_{(1)}\otimes 1_{(1)}] \otimes [\varepsilon_{(2)}\otimes 1_{(2)}][\varepsilon_{(1)}'\otimes 1_{(1)}']
\otimes [\varepsilon_{(2)}'\otimes 1_{(2)}'] \\ &=& [\varepsilon_{(1)}\otimes 1_{(1)}] \otimes [\varepsilon_{(1)}' \varepsilon_{(2)} \otimes 1_{(2)} 1_{(1)}']
\otimes [\varepsilon_{(2)}'\otimes 1_{(2)}'] \\ &=& [\varepsilon_{(1)}\otimes 1_{(1)}] \otimes [\varepsilon_{(2)}\otimes 1_{(2)}]
\otimes [\varepsilon_{(3)}\otimes 1_{(3)}], \\ \lefteqn{([\varepsilon\otimes 1] \otimes \Delta([\varepsilon\otimes 1]) ) (\Delta([\varepsilon\otimes 1])\otimes [\varepsilon\otimes 1]) =}\\ &=& [\varepsilon_{(1)}'\otimes 1_{(1)}'] \otimes [\varepsilon_{(1)} \otimes 1_{(1)}] [\varepsilon_{(2)}'\otimes 1_{(2)}']
\otimes [\varepsilon_{(2)}\otimes 1_{(2)}] \\ &=& [\varepsilon_{(1)}'\otimes 1_{(1)}'] \otimes [\varepsilon_{(2)}'\varepsilon_{(1)} \otimes 1\11_{(2)}']
\otimes [\varepsilon_{(2)}\otimes 1_{(2)}] \\ &=& [\varepsilon_{(1)}\otimes 1_{(1)}] \otimes [\varepsilon_{(2)}\otimes 1_{(2)}]
\otimes [\varepsilon_{(3)}\otimes 1_{(3)}], \end{eqnarray*} where we used the axioms of a quantum groupoid and the definition of $J$.
In order to check the axioms (\ref{S epst}),(\ref{S epss}), let us compute the target counital map ${\varepsilon}_t$. We have \begin{eqnarray*} {\varepsilon}_t([\phi\otimes h]) &=& {\varepsilon}([\varepsilon_{(1)}\otimes 1_{(1)}][\phi\otimes h]) [\varepsilon_{(2)}\otimes 1_{(2)}] \\ &=& \langle\, \varepsilon_t(1\2h),\,\phi_{(2)}\varepsilon_{(1)} \,\rangle \langle\, S(1_{(1)}),\,\phi_{(1)} \,\rangle
\langle\, 1_{(3)},\, \phi_{(3)} \,\rangle [\varepsilon_{(2)}\otimes 1_{(4)}] \\ &=& \langle\, 1'_{(1)}\varepsilon_t(1\2h),\,\phi_{(2)}\,\rangle \langle\, 1'_{(2)},\,\varepsilon_{(1)} \,\rangle \\ & & \langle\, S(1_{(1)}),\,\phi_{(1)} \,\rangle \langle\, 1_{(3)},\, \phi_{(3)} \,\rangle [\varepsilon_{(2)}\otimes 1_{(4)}] \\ &=& \langle\, S(1_{(1)})1'_{(1)} \varepsilon_t(1\2h) 1_{(3)},\, \phi \,\rangle \langle\, 1'_{(2)},\,\varepsilon_{(1)} \,\rangle
[\varepsilon_{(2)}\otimes 1_{(4)}] \\ &=& \langle\, 1_{(1)} \varepsilon_t(h),\, \phi\,\rangle [\varepsilon \otimes 1_{(2)}]. \end{eqnarray*} Similarly one computes the source counital map : $$ {\varepsilon}_s([\phi\otimes h]) = [\varepsilon_{(1)}\otimes 1] \langle\, h,\, \varepsilon_t(\phi)S(\varepsilon_{(2)}) \,\rangle. $$ Using these formulas we have : \begin{eqnarray*} \lefteqn{m(\mbox{id}\otimes S)\Delta([\phi\otimes h])= } \\ &=& [\phi_{(1)}\otimes h_{(1)}][S^{-1}(\phi_{(3)})\otimes S(h_{(2)})]
\langle\, h_{(2)},\, \phi_{(2)} \,\rangle \langle\, S(h_{(4)}),\, \phi_{(4)} \,\rangle \\ &=& [ S^{-1}(\phi_{(4)})\phi_{(1)} \otimes h_{(2)} S(h_{(5)})]\langle\, S(h_{(1)}),\,S^{-1}(\phi_{(5)}) \,\rangle\\ & & \langle\, h_{(3)} ,\,S^{-1}(\phi_{(3)}) \,\rangle \langle\, h_{(4)},\, \phi_{(2)} \,\rangle
\langle\, S(h_{(6)}) ,\, \phi_{(6)} \,\rangle \\ &=& [ S^{-1}(\phi_{(3)})\phi_{(1)} \otimes h_{(2)} S(h_{(4)})]
\langle\, \varepsilon_t(h_{(3)}),\, S^{-1}(\phi_{(2)})\,\rangle \langle\, h_{(1)} S(h_{(5)}),\,\phi_{(4)}\,\rangle \\ &=& [ S^{-1}(\phi_{(3)})\phi_{(1)} \otimes 1_{(1)}\varepsilon_t(h_{(2)}) ]
\langle\, 1_{(2)},\, S^{-1}(\phi_{(2)})\,\rangle \langle\, h_{(1)} S(h_{(3)}),\,\phi_{(4)}\,\rangle \\ &=& [ S^{-1}(\phi_{(3)})\phi_{(1)} \otimes 1_{(1)} 1_{(2)}']
\langle\, 1_{(2)},\, S^{-1}(\phi_{(2)})\,\rangle \langle\, 1_{(1)}'\varepsilon_t(h),\,\phi_{(4)}\,\rangle \\ &=& [ \varepsilon_{(1)} S^{-1}(\phi_{(3)})\phi_{(1)} \otimes 1_{(1)} 1_{(2)}']
\langle\, 1_{(2)},\,\varepsilon_{(2)}\,\rangle \langle\, 1_{(1)}'\varepsilon_t(h),\,\phi_{(3)} \,\rangle \\ &=& [ S^{-1}(\phi_{(2)})\phi_{(1)} \otimes 1_{(2)}'] \langle\, 1_{(1)}'\varepsilon_t(h),\,\phi_{(3)} \,\rangle \\ &=& [\varepsilon\otimes 1_{(2)}] \langle\, 1_{(1)}\varepsilon_t(h),\,\phi\,\rangle = \varepsilon_t([\phi\otimes h]), \end{eqnarray*} and \begin{eqnarray*} \lefteqn{m(S\otimes\mbox{id})\Delta([\phi\otimes h])= } \\ &=& [\phi_{(5)} S^{-1}(\phi_{(2)}) \otimes S(h_{(3)})h_{(6)}] \langle\, S^{2}(h_{(4)}),\,\phi_{(4)}\,\rangle
\langle\, S(h_{(2)}),\, \phi_{(6)} \,\rangle \\ & & \langle\, h_{(1)},\, \phi_{(1)}\,\rangle \langle\, S(h_{(5)}),\,\phi_{(3)}\,\rangle \\ &=& [ S^{-1}(\varepsilon_{(1)}\varepsilon_t(\phi_{(2)}))\otimes S(h_{(2)}) h_{(4)} ]
\langle\, S(h_{(3)}),\,\varepsilon_{(2)}\,\rangle \langle\, h_{(1)},\, \phi\1S(\phi_{(3)}) \,\rangle \\ &=& [ S^{-1}(\varepsilon_{(1)}\varepsilon_{(2)}') \otimes S(h_{(2)})h_{(4)}]
\langle\, \varepsilon_s(h_{(3)}),\, S(\varepsilon_{(2)}) \,\rangle \langle\, h_{(1)} \varepsilon_{(1)}'\varepsilon_t(\phi) \,\rangle \\ &=& [ S^{-1}(\varepsilon_{(1)}\varepsilon_{(2)}') \otimes \varepsilon_s(h_{(2)})]
\langle\, 1_{(1)},\, S(\varepsilon_{(2)}) \,\rangle \langle\, h_{(1)} \varepsilon_{(1)}'\varepsilon_t(\phi) \,\rangle \\ &=& [\varepsilon\3S^{-1}(\varepsilon_{(2)}) \otimes \varepsilon_s(h_{(2)})]
\langle\, h_{(1)},\, \varepsilon_t(\phi)S(\varepsilon_{(2)}) \,\rangle \\ &=& [\varepsilon_{(1)} \otimes 1] \langle\, h,\, S(\varepsilon_{(2)})\varepsilon_t(\phi) \,\rangle =
\varepsilon_s([\phi\otimes h]). \end{eqnarray*} In the above computations we used repeatedly the amalgamation relations in $D(H)$: \begin{equation*} [\phi\otimes zh]=[(\varepsilon\leftharpoonup z)\phi \otimes h]\ \quad (z\in H_t), \qquad [\phi\otimes yh]=[(y\rightharpoonup\varepsilon)\phi \otimes h]\ \quad (y\in H_s) \end{equation*} that follow from (\ref{amalgamation}), the axioms of a quantum groupoid and properties of the counital maps. Finally, we prove the relation which is equivalent to $S$ being both algebra and coalgebra anti-homomorphism: \begin{eqnarray*} \lefteqn{m(\mbox{id}\otimes m)(S\otimes \mbox{id}\otimes S)
(\mbox{id}\otimes\Delta)\Delta([\phi\otimes h])= } \\ &=& m(S\otimes \varepsilon_t)\Delta([\phi\otimes h]) \\ &=& S([\phi_{(1)}\otimes h_{(1)}])\varepsilon_t([\phi_{(2)}\otimes h_{(2)}]) \\ &=& [S^{-1}(\phi_{(2)}) \otimes S(h_{(2)})] [\varepsilon \otimes 1_{(2)}] \\ & & \langle\, h_{(1)},\, \phi_{(1)} \,\rangle \langle\, S(h_{(3)}),\, \phi_{(3)} \,\rangle
\langle\, 1_{(1)}\varepsilon_t(h_{(4)}),\, \phi_{(4)} \,\rangle \\ &=& [S^{-1}(\phi_{(2)}) \otimes S(h_{(2)})1_{(2)}] \langle\, h_{(1)},\, \phi_{(1)} \,\rangle
\langle\, S(h_{(3)})1_{(1)},\,\phi_{(3)} \,\rangle = S([\phi\otimes h]). \end{eqnarray*} Note that $D(H)_t = [\varepsilon \otimes H_t]$ and $D(H)_s = [\widehat H_s \otimes 1]$. \end{proof}
\begin{proposition} \label{QT of D(H)} The Drinfeld double $D(H)$ has a quasitriangular structure given by \begin{equation} {\mathcal R} = \sum_i [\xi^i \otimes 1] \otimes [\varepsilon \otimes f_i], \qquad {\bar \R} = \sum_j\, [S^{-1}(\xi_j) \otimes 1] \otimes[\varepsilon \otimes f_j] \end{equation} where $\{ f_i\}$ and $\{ \xi^i\}$ are dual bases in $H$ and $\widehat H$. \end{proposition} \begin{proof} The identities $(\mbox{id} \otimes \Delta){\mathcal R} = {\mathcal R}_{13}{\mathcal R}_{12}$ and $(\Delta \otimes \mbox{id}){\mathcal R} = {\mathcal R}_{13}{\mathcal R}_{23}$ can be written as (identifying $[\widehat H^{op}\otimes 1]$ with $\widehat H^{op}$ and $[\varepsilon \otimes H]$ with $H$): \begin{equation} \label{R-dual bases1} \sum_i\, \xi_{(1)}^i \otimes \xi_{(2)}^i \otimes f_i = \sum_{ij}\, \xi^i \otimes \xi^j \otimes f_i f_j, \end{equation} \begin{equation} \label{R-dual bases2} \sum_i\, \xi^i \otimes {f_i}_{(1)} \otimes {f_i}_{(2)} = \sum_{ij}\, \xi^j \xi^i \otimes f_j \otimes f_i. \end{equation} The above equalities can be verified by evaluating both sides on an element $h\in H$ in the third factor (resp.,\ on $\phi\in \widehat H^{op}$ in the second factor), see (\cite{M}, 7.1.1). To show that ${\mathcal R}$ is an intertwiner between $\Delta$ and $\Delta^{op}$, we compute \begin{eqnarray*} {\mathcal R} \Delta([\phi\otimes h]) &=& \sum_i\, [\phi_{(1)} \xi^i \otimes h_{(1)}] \otimes [\phi_{(3)} \otimes
{f_i}_{(2)} h_{(2)}] \\ & & \langle\, S({f_i}_{(1)}),\, \phi_{(2)} \,\rangle \langle\, {f_i}_{(3)},\, \phi_{(4)} \,\rangle \\ &=& \sum_i\, [\phi_{(1)} S(\phi_{(2)})\xi^i\phi_{(4)} \otimes h_{(1)}] \otimes
[\phi_{(3)} \otimes f_i h_{(2)}] \\ &=& \sum_i\, [ \xi^i\phi_{(3)} \otimes \langle\, 1_{(1)},\, \varepsilon_t(\phi_{(1)})\,\rangle 1\2h_{(1)} ]
\otimes [\phi_{(2)} \otimes f_i h_{(2)}] \\ &=& \sum_i\, [ \xi^i\phi_{(2)} \otimes \langle\, 1_{(1)},\,\varepsilon_{(1)} \,\rangle 1\2h_{(1)} ]
\otimes [\varepsilon_{(2)} \phi_{(1)} \otimes f_i h_{(2)}] \\ &=& \sum_i\, [ \xi^i\phi_{(2)} \otimes h_{(2)}] \otimes
[ \langle\, \varepsilon_t(h_{(1)}),\,\varepsilon_{(1)} \,\rangle \varepsilon_{(2)} \phi_{(1)} \otimes f_i h_{(3)}] \\ &=& \sum_i\, [ \xi^i\phi_{(2)} \otimes h_{(3)}] \otimes
[ \phi_{(1)} \otimes h_{(1)} S(h_{(2)}) f_i h_{(4)} ] \\ &=& \sum_i\, [\xi_{(2)}^i\phi_{(2)} \otimes h_{(3)}] \otimes [\phi_{(1)}\otimes h_{(1)} f_i]
\langle\, S(h_{(2)}),\, \xi^i_{(1)} \,\rangle \langle\, h_{(4)},\,\xi^i_{(3)} \,\rangle \\ &=& \Delta^{op}([\phi\otimes h]) {\mathcal R}. \end{eqnarray*} where we used \begin{equation} \label{dual bases} \sum_i\,\langle\, a,\, \xi^i\,\rangle f_i =a\quad \mbox{ and } \quad \sum_i\, \xi^i \langle\, f_i,\,\phi \,\rangle =\phi, \end{equation} for all $a\in H, \phi\in \widehat H$. Finally, let us check that the element ${\bar \R} = \sum_j\, [S^{-1}(\xi_j) \otimes 1] \otimes[\varepsilon \otimes f_j]$ satisfies ${\bar \R} {\mathcal R} = \Delta(1)$ and ${\mathcal R}{\bar \R} = \Delta^{op}(1)$. The first property is equivalent to $$ \sum_{i,j}\, [\xi^i S^{-1}(\xi^j)\otimes 1] \otimes [\varepsilon \otimes f_jf_i] = [\langle\, 1_{(1)},\,\varepsilon'_{(2)}\,\rangle \varepsilon'_{(1)} \varepsilon_{(1)} \otimes 1] \otimes [\varepsilon \otimes \langle\, 1'_{(1)},\, \varepsilon_{(2)} \,\rangle 1'_{(2)} 1_{(2)}], $$ which can be regarded as an equality in $\widehat H^{op}\otimes H$ : \begin{equation*} \sum_{i,j}\, \xi^i S^{-1}(\xi^j)\otimes f_jf_i = \langle\, 1_{(1)},\,\varepsilon'_{(2)}\,\rangle \varepsilon'_{(1)} \varepsilon_{(1)} \otimes \langle\, 1'_{(1)},\, \varepsilon_{(2)} \,\rangle 1'_{(2)} 1_{(2)}. \end{equation*} Evaluating both sides on arbitrary $\phi\in \widehat H$ in the second factor, we get \begin{eqnarray*} \phi_{(2)} S^{-1}(\phi_{(1)}) &=& \langle\, 1_{(1)},\,\varepsilon'_{(2)}\,\rangle \varepsilon'_{(1)} \varepsilon_{(1)} \langle\, 1'_{(1)},\, \varepsilon_{(2)} \,\rangle
\langle\, 1'_{(2)} 1_{(2)},\phi \,\rangle \\ &=& \varepsilon_s^{op}(\phi_{(2)}) \varepsilon_s^{op}(\phi_{(1)}), \end{eqnarray*} where $\varepsilon_s^{op}(\phi) = \phi_{(2)} S^{-1}(\phi_{(1)})$ is the source counital map in ${\widehat H}^{op}$. The second property is similar. \end{proof} For the Hopf algebra case the above idea of the proof was proposed in \cite{M}, 7.1.1.
\begin{remark} The dual quantum groupoid $\widehat{D(H)}$ consists of all elements $\sum_k\, h_k\otimes \phi_k$ in $H\otimes_k {\widehat H}^{op}$ such that $$
\sum_k\, (h_k\otimes \phi_k)|_J = 0. $$ The structure operations in $\widehat{D(H)}$ are obtained by dualizing those in $D(H)$: \begin{eqnarray*} ( \sum_k\, h_k\otimes \phi_k ) ( \sum_l\, g_l\otimes \psi_l ) &=& \sum_{k,l}\, h_kg_l\otimes \phi_k \psi_l, \\ 1_{\widehat{D(H)}} &=& 1_{(2)} \otimes (\varepsilon \leftharpoonup 1_{(1)}), \\ \Delta(\sum_k\, h_k\otimes \phi_k ) &=& \sum_{i,j,k}\, ({h_k}_{(2)} \otimes \xi^i {\phi_k}_{(1)} \xi^j) \otimes
(S(f_i) {h_k}_{(1)} f_j \otimes {\phi_k}_{(2)}), \\ \varepsilon( \sum_k\, h_k\otimes \phi_k ) &=& \sum_k\,\varepsilon(h_k)\widehat\varepsilon(\phi_k),\\ S( \sum_k\, h_k\otimes \phi_k ) &=& \sum_{i,j,k}\, f_i S^{-1}(h_k)S(f_j) \otimes \xi^i S(\phi)\xi^j, \end{eqnarray*} for all $\sum_k\, h_k\otimes \phi_k,\, \sum_l\, g_l\otimes \psi_l \in \widehat{D(H)}$, where $\{ f_i\}$, $\{\xi^j\}$ are dual bases in $H,\widehat H$, respectively. \end{remark}
\begin{proposition} \label{D(A) is factorizable} The Drinfeld double $D(H)$ is factorizable in the sense of Definition~\ref{factorizability}. \end{proposition} \begin{proof} First, observe that for any pair of dual bases $\{ f_i\}$ and $\{\xi^j\}$ as above and all $g\in H$ and $\psi\in \widehat{H}$ the element $$ Q_{g\otimes \psi} = \sum_{k,l}\,f_kgf_l\otimes \xi^k\psi S(\xi^l)\in H\otimes_k {\widehat H}^{op} $$ belongs to $\widehat{D(H)}$. Indeed, \begin{eqnarray*} \lefteqn{\langle\, Q_{g\otimes \psi},\, \phi \otimes zh \,\rangle =} \\ &=& \langle\, zh_{(1)},\,\phi_{(1)} \,\rangle \langle\, S(h_{(3)}),\,\phi_{(3)} \,\rangle \langle\, g,\, \phi_{(2)} \,\rangle
\langle\, h_{(2)},\psi \,\rangle \\ &=& \langle\, h_{(1)},\, (\phi \leftharpoonup z)_{(1)} \,\rangle \langle\, S(h_{(3)}),\, (\phi \leftharpoonup z)_{(3)} \,\rangle
\langle\, g,\, (\phi \leftharpoonup z)_{(2)} \,\rangle \langle\, h_{(2)},\,\psi \,\rangle \\ &=& \langle\, Q_{g\otimes \psi},\, (\varepsilon\leftharpoonup z)\phi\otimes h\,\rangle, \end{eqnarray*} for all $h\in H,\, \phi\in \widehat{H},\, z\in H_t$ and, likewise, \begin{equation*} \langle\, Q_{g\otimes \psi},\, \phi \otimes yh \,\rangle = \langle\, Q_{g\otimes \psi},\,(y\rightharpoonup \varepsilon) \phi\otimes h\,\rangle. \end{equation*} Next, we compute \begin{eqnarray*} \lefteqn{(Q_{g\otimes \psi} \otimes \mbox{id})({\mathcal R}_{21}{\mathcal R})=}\\ &=& \sum_{i,j}\, Q_{g\otimes \psi} ([{\xi^j}_{(2)} \otimes {f_i}_{(2)}])
\,[\xi^i \otimes f_j] \, \langle\, S({f_i}_{(1)}),\, {\xi^j}_{(1)} \,\rangle
\langle\, {f_i}_{(3)}, {\xi^j}_{(3)} \,\rangle \\ &=& \sum_{i,j}\, Q_{g\otimes \psi} ([{\xi^j}_{(2)} \otimes f_i])\,
[ S({\xi^j}_{(1)}) \xi^i {\xi^j}_{(3)} \otimes f_j] \\ &=& \sum_{i,j,k,l}\, [S({\xi^j})\xi^k\psi S(\xi^l)\xi^i \otimes
f_jf_k g f_lf_i] \\ &=& [ \varepsilon_{(1)}'\langle\, 1_{(2)}',\,\varepsilon_{(2)}'\,\rangle \psi \varepsilon_{(1)}\langle\, 1_{(2)},\,\varepsilon_{(2)}\,\rangle
\otimes 1_{(1)}' g 1_{(1)} ] \\ &=& [\psi \varepsilon_{(1)}\langle\, 1_{(2)},\,\varepsilon_{(2)}\,\rangle \otimes g 1_{(1)} ], \end{eqnarray*} where we used the identities \begin{eqnarray*} \sum_j\, S({\xi^j}_{(1)}) \langle\, g,\,{\xi^j}_{(2)} \,\rangle \otimes {\xi^j}_{(3)} \otimes f_j &=& \sum_{i,j}\, S(\xi^i) \otimes \xi^j \otimes f_i g f_j, \\ \sum_{i,j}\,S(\xi^i)\xi^j \otimes f_if_j &=& \varepsilon_{(1)} \otimes 1_{(1)} \langle\, 1_{(2)},\,\varepsilon_{(2)} \,\rangle, \end{eqnarray*} that follow from (\ref{R-dual bases1}), (\ref{R-dual bases2}), (\ref{dual bases}) and from axioms (\ref{S epst}),(\ref{S epss}) of a quantum groupoid. Therefore, \begin{eqnarray*} \lefteqn{\langle\, 1_{(2)},\, \varepsilon_{(2)} \,\rangle Q_{g1_{(1)}\otimes \psi\varepsilon_{(1)}}({\mathcal R}_{21}{\mathcal R}) =}\\ &=& [\psi\varepsilon_{(1)} \varepsilon'_{(1)} \otimes g 1_{(1)}'1_{(1)}] \langle\, 1_{(2)},\,\varepsilon_{(2)}' \,\rangle
\langle\, 1_{(2)}',\,\varepsilon_{(2)} \,\rangle \\ &=& [\varepsilon_{(1)} \otimes 1_{(1)}][\psi \otimes g] S([\varepsilon_{(2)} \otimes 1_{(2)}])
= \mbox{Ad}\,_1^l([\psi\otimes g]), \end{eqnarray*} Thus, we conclude from Lemma~\ref{expectations} that the map $$ \widehat{D(H)} \ni x \mapsto (Q_x \otimes \mbox{id})({\mathcal R}_{21}{\mathcal R}) \in C_{D(H)}(D(H)_s) $$ is surjective, i.e., $D(H)$ is factorizable. \end{proof} \end{section}
\begin{section}{Ribbon quantum groupoids} \begin{definition} \label{Ribbon WHA} A ribbon quantum groupoid is a quasitriangular quantum groupoid $H$ with an invertible central element $\nu\in H$ such that \begin{equation} \Delta(\nu) = {\mathcal R}_{21}{\mathcal R}(\nu\otimes \nu) \quad\mbox{and}\quad S(\nu)=\nu. \end{equation} The element $\nu$ is called a {\em ribbon element} of $H$. \end{definition} For an object $V$ of $\mbox{Rep}(H)$ we define the twist $\theta_V :V\to V$ to be the multiplication by $\nu$ : \begin{equation} \theta_V(v) =\nu\cdot v, \quad v\in V. \end{equation} \begin{proposition} \label{twist} Let $(H, {\mathcal R},\nu)$ be a ribbon quantum groupoid. The family of homomorphisms $\{\theta_V\}_V$ is a twist in the braided monoidal category $\mbox{Rep}(H)$ compatible with duality. Conversely, if $\theta_V(v) =\nu\cdot v$ with $\nu\in H$ is a twist in $\mbox{Rep}(H)$, then $\nu$ is a ribbon element of $H$. \end{proposition}
\begin{proof} Since $\nu$ is an invertible central element of $H$, the homomorphism $\theta_V$ is an $H$-linear isomorphism. The twist identity $c_{W,V}c_{V,W}(\theta_V\otimes\theta_W)=\theta_{V\otimes W}$ follows from the properties of $\nu$: $$ c_{W,V}c_{V,W}(\theta_V\otimes\theta_W)(x) = {\mathcal R}_{21}{\mathcal R}(\nu\cdot x^{(1)} \otimes \nu\cdot x^{(2)}) = \Delta(\nu)\cdot x = \theta_{V\otimes W}(x), $$ for all $x =x^{(1)}\otimes x^{(2)}\in V\otimes W$. Clearly, the identity ${\mathcal R}_{21}{\mathcal R}(\nu\otimes \nu) =\Delta(\nu)$ is equivalent to the twist property. It remains to prove that $$ (\theta_V \otimes \mbox{id}_{V^*})b_V(z) = (\mbox{id}_V \otimes \theta_{V^*})b_V(z), $$ for all $z\in H_t$, i.e., that $$ \sum_i\,\nu z_{(1)}\cdot \xi^i \otimes z_{(2)}\cdot f_i = \sum_i\,z_{(1)}\cdot \xi^i \otimes \nu z_{(2)}\cdot f_i, $$ where $\sum_i\,\xi^i \otimes f_i$ is the canonical element in $V^*\otimes V$. Evaluating the first factors of the above equality on an arbitrary $v\in V$, we get the equivalent condition : $$ \sum_i\,(\nu z_{(1)}\cdot \xi^i)(v) z_{(2)}\cdot f_i = \sum_i\, (z_{(1)}\cdot \xi^i)(v) \nu z_{(2)}\cdot f_i, $$ which reduces to $z\2S(\nu z_{(1)})\cdot v = S(z_{(1)})\nu z_{(2)}\cdot v$. The latter easily follows from the centrality of $\nu=S(\nu)$ and properties of $H_t$. \end{proof}
\begin{proposition} \label{ribbon category} The category $\mbox{Rep}(H)$ is a ribbon category if and only if $H$ is a ribbon quantum groupoid. \end{proposition}
\begin{proof} Follows from Propositions~\ref{monoidal with duality}, \ref{braiding}, and \ref{twist}. \end{proof}
For any endomorphism $f$ of an object $V$ of $\mbox{Rep}(H)$, we define, following (\cite{T2}, I.1.5), its {\em quantum trace} \begin{equation} \mbox{tr}_q(f)=d_V c_{V,V^*}(\theta_Vf \otimes \mbox{id}_{V^*})b_V \label{q1-trace} \end{equation} with values in $\mbox{End}(H_t)$ and the {\em quantum dimension} of $V$ by $\dim_q(V)=\mbox{tr}_q(\mbox{id}_V)$. The next lemma gives an explicit computation of $\mbox{tr}_q$ and $\dim_q$ via the usual trace of endomorphisms.
\begin{proposition} \label{quantum trace} Let $(H, {\mathcal R},\nu)$ be a ribbon quantum groupoid, $f$ be an endomorphism of an object $V$ in $\mbox{Rep}(H)$. Then \begin{equation} \mbox{tr}_q(f)(z) = \mbox{Tr}(S(1_{(1)})u\nu f)z1_{(2)},\qquad \ dim_q(V)(z)=\mbox{Tr}(S(1_{(1)})u\nu)z1_{(2)}, \end{equation} where $\mbox{Tr}$ is the usual trace of endomorphisms, and $u\in H$ is the Drinfeld element. \end{proposition}
\begin{proof} Since the trace of an endomorphism $h\in \mbox{End}_k(H_t)$, in terms of the canonical element $\sum_i f_i\otimes \xi^i\in V\otimes_k V^*$, is $\mbox{Tr}(h)=\sum_i\,\xi^i(h(f_i))$, the definition of $\mbox{tr}_q$ gives: \begin{eqnarray*} \mbox{tr}_q(f)(z) &=& d_V c_{V,V^*}(\theta_Vf \otimes \mbox{id}_{V^*})b_V(z) \\ &=& d_V( \sum_i\,{\mathcal R}^{(2)} z_{(2)}\cdot \xi^i \otimes {\mathcal R}^{(1)}\nu z_{(1)}\cdot f(f_i)) \\ &=& \sum_i\,({\mathcal R}^{(2)} z_{(2)}\cdot \xi^i)(1_{(1)} {\mathcal R}^{(1)}\nu z_{(1)}\cdot f(f_i))1_{(2)} \\ &=& \sum_i \xi^i( S({\mathcal R}^{(2)} z_{(2)})1_{(1)} {\mathcal R}^{(1)}\nu z_{(1)}\cdot f(f_i))1_{(2)} \\ &=& \mbox{Tr}(S(1_{(1)})u\nu f)z1_{(2)}, \end{eqnarray*} where we used formulas (\ref{dV}) and (\ref{bV}) defining $b_V$ and $d_V$. \end{proof}
\begin{corollary} \label{quantum trace for connected WHA} Let $k$ be algebraically closed. If $H$-module $H_t$ is irreducible (which happens exactly when $H_t \cap Z(H) = k$, i.e., when $H$ is connected (\cite{N}, 3.11, \cite{BNSz}, 2.4), then $\mbox{tr}_q(f)$ and $dim_q(V)$ are scalars: \begin{equation} \label{q-trace} \mbox{tr}_q(f) = (\dim H_t)^{-1}\ \mbox{Tr}(u\nu f), \qquad \dim_q(V) = (\dim H_t)^{-1} \mbox{Tr}(u\nu). \end{equation} \end{corollary}
\begin{proof} An endomorphism of an irreducible module is multiplication by a scalar, therefore, we must have $\mbox{Tr}(S(1_{(1)})u\nu f)1_{(2)}=\mbox{tr}_q(f)(1) =\mbox{tr}_q(f) 1$. Applying the counit to both sides and using that $\varepsilon(1) = \dim H_t$, we get the result. \end{proof} \end{section}
\begin{section}{Towards modular categories}
Let us first recall some definitions needed in this section. Let ${\mathcal V}$ be a ribbon $Ab$-category over $k$, i.e., such that all $\mbox{Hom}(V,W)$ are $k$-vector spaces (for all objects $V,W\in {\mathcal V}$) and both operations $\circ$ and $\otimes$ are $k$-bilinear.
An object $V\in {\mathcal V}$ is said to be {\it simple} if any endomorphism of $V$ is multiplication by an element of $k$. We say that a family $\{V_i\}_{i\in I}$ of objects of ${\mathcal V}$ dominates an object $V$ of ${\mathcal V}$ if there exists a finite set $\{V_{i(r)}\}_r$ of objects of this family (possibly, with repetitions) and a family of morphisms $f_r:V_{i(r)} \to V,g_r:V\to V_{i(r)}$ such that $\mbox{id}_V = \sum_r f_r g_r$.
A modular category (\cite{T2}, II.1.4) is a pair consisting of a ribbon $Ab$-category ${\mathcal V}$ and a finite family $\{V_i\}_{i\in I}$ of simple objects of ${\mathcal V}$ satisfying four axioms: \begin{enumerate} \item[(i)] There exists $0\in I$ such that $V_0$ is the unit object. \item[(ii)] For any $i\in I$, there exists $i^*\in I$ such that $V_{i^*}$ is isomorphic to $V^*_i$. \item[(iii)] All objects of ${\mathcal V}$ are dominated by the family $\{V\}_{i\in I}$. \item[(iv)] The square matrix $S=\{S_{ij}\}_{i,j\in I} = \{ \mbox{tr}_q(c_{V_i,V_j}\circ c_{V_j,V_i}) \}_{i,j\in I}$ is invertible over $k$ (here $\mbox{tr}_q$ is the quantum trace in a ribbon category defined by (\ref{q1-trace})). \end{enumerate}
If a quantum groupoid $H$ is connected and semisimple over an algebraically closed field, modularity of $\mbox{Rep}(H)$ is equivalent to $\mbox{Rep}(H)$ being ribbon and such that the matrix $S=\{S_{ij}\}_{i,j\in I} = \{ \mbox{tr}_q(c_{V_i,V_j}\circ c_{V_j,V_i}) \}_{i,j\in I}$, where $I$ is the set of all (equivalent classes of) irreducible representations, is invertible.
\begin{remark} \label{recall integrals} Recall that $h\in H$ is a left (resp.\ right) {\em integral} if $xh=\varepsilon_t(x)h$ (resp.\ $hx=h\varepsilon_s(x)$) for all $x\in H$ (\cite{BNSz}, 3.24). A {\em Haar integral} is a two-sided integral $h$ which is normalized, i.e., $\varepsilon_t(h)=\varepsilon_s(h)= 1$. Existence of a Haar integral in a quantum groupoid $H$ is equivalent to $H$ being semisimple and possessing an invertible element $g$ such that $S^2(x) =gxg^{-1}$ for any $x\in H$ and $\chi(g^{-1})\neq 0$ for all irreducible characters $\chi$ of $H$ (\cite{BNSz}, 3.27). \end{remark}
The following lemma extends a result known for Hopf algebras (\cite{EG}, 1.1). \begin{lemma} \label{factorizable implies modular} Let $H$ be a connected, ribbon, factorizable quantum groupoid over an algebraically closed field $k$, and assume that $H$ has a Haar integral. Then $\mbox{Rep}(H)$ is a modular category. \end{lemma}
\begin{proof} Note that $H$ is semisimple by Remark~\ref{recall integrals}. We only need to prove the invertibility of the matrix formed by \begin{eqnarray*} S_{ij} &=& \mbox{tr}_q(c_{V_i,V_j}\circ c_{V_j,V_i}) \\ &=& (\dim H_t)^{-1} \mbox{Tr}( (u\nu)\circ c_{V_i,V_j}\circ c_{V_j,V_i}) \\ &=& (\dim H_t)^{-1} (\chi_j \otimes \chi_i)( (u\nu\otimes u\nu){\mathcal R}_{21}{\mathcal R}), \end{eqnarray*} where $V_i$ are as above, $I = \{1,\dots n\},\ \{ \chi_j\}$ is a basis in the space $C(H)$ of characters of $H$ (we used above the formula (\ref{q-trace}) for the quantum trace).
Observe that the linear map $F : \phi\mapsto (\phi\otimes \mbox{id})({\mathcal R}_{21}{\mathcal R})$ takes any element of the form $\phi = \chi\leftharpoonup u\nu$ (i.e., $\phi(h) = \chi(u\nu h)~\forall h\in H$), where $\chi\in C(H)$, into $Z(H)$. Indeed, for any such $\phi$ and all $\psi\in \widehat H,\,h\in H$ we have, using the fact that $u\in C_H(H_s)$ (this follows from Lemma~\ref {yz}), the properties of $\varepsilon_s$ and $\varepsilon_s^{op}$, the relation $\Delta^{op}(h){\mathcal R} =\Delta(h) {\mathcal R}~(h\in H)$, and the centrality of $\chi$ : \begin{eqnarray*} \langle\, F(\phi)h,\,\psi \,\rangle &=& \langle\, u\nu {\mathcal R}^{(2)} {{\mathcal R}'}^{(1)},\,\chi\,\rangle \langle\, {\mathcal R}^{(1)} {{\mathcal R}'}^{(2)} h,\, \psi \,\rangle \\ &=& \langle\, u\nu {\mathcal R}^{(2)} {{\mathcal R}'}^{(1)} \varepsilon_s^{op}(h_{(1)}),\,\chi\,\rangle
\langle\, {\mathcal R}^{(1)} {{\mathcal R}'}^{(2)} h_{(2)},\, \psi \,\rangle \\ &=& \langle\, u\nu {\mathcal R}^{(2)} h_{(2)} {{\mathcal R}'}^{(1)} S^{-1}(h_{(1)}),\,\chi\,\rangle
\langle\, {\mathcal R}^{(1)} h_{(3)} {{\mathcal R}'}^{(2)},\, \psi \,\rangle \\ &=& \langle\, u\nu h_{(2)} {\mathcal R}^{(2)} {{\mathcal R}'}^{(1)} S^{-1}(h_{(1)}),\,\chi\,\rangle
\langle\, h_{(3)} {\mathcal R}^{(1)} {{\mathcal R}'}^{(2)},\, \psi \,\rangle \\ &=& \langle\, u\nu S(h_{(1)}) h_{(2)} {\mathcal R}^{(2)} {{\mathcal R}'}^{(1)},\,\chi\,\rangle
\langle\, h_{(3)} {\mathcal R}^{(1)} {{\mathcal R}'}^{(2)},\, \psi \,\rangle \\ &=& \langle\, u\nu {\mathcal R}^{(2)} {{\mathcal R}'}^{(1)},\,\chi\,\rangle \langle\, h {\mathcal R}^{(1)} {{\mathcal R}'}^{(2)},\, \psi \,\rangle \\ &=& \langle\, h F(\phi),\,\psi \,\rangle, \end{eqnarray*} therefore $F(\phi)\in Z(H)$. Since $H$ is factorizable, we know from Corollary~\ref{restriction of F} that the restriction $$ F : \{ \phi\in \widehat H \mid \phi = \phi\circ\mbox{Ad}\,_1^r \} \to C_H(H_s) $$ is a linear isomorphism. Since $\chi\leftharpoonup u\nu$ belongs to the subspace on the left hand side, we have a linear isomorphism between $C(H)\leftharpoonup u\nu$ and $Z(H)$, hence, there exists an invertible matrix $T=(T_{ij})$ representing the map $F$ in the bases of $C(H)$ and $Z(H)$, i.e., such that $F(\chi_j\leftharpoonup u\nu) = \sum_i\, T_{ij}e_i$. Then \begin{eqnarray*} S_{ij} &=& (\dim H_t)^{-1} \chi_i(u\nu F(\chi_j\leftharpoonup u\nu))
= (\dim H_t)^{-1} \sum_k\, T_{kj} \chi_i(u\nu e_k) \\ &=& (\dim H_t)^{-1} (\dim V_i) \chi_i(u\nu) T_{ij}. \end{eqnarray*} Therefore, $S = DT$, where $D = \mbox{diag}\{ (\dim H_t)^{-1} (\dim V_i) \chi_i(u\nu) \}$. If $g$ is an element from Remark~\ref{recall integrals} then $u^{-1}g$ is an invertible central element of $h$ and $\chi_i(u^{-1}) \neq 0$ for all $\chi_i$. By Corollary~\ref{uv is central} $uS(u)=c$ is invertible central, therefore $\chi_i(u)= \chi_i(c)\chi_i(S(u^{-1})))\neq 0$. Hence, $\chi_i(u\nu)\neq 0$ for all $i$ and $D$ is invertible. \end{proof}
\begin{example} An example of a modular category can be constructed from {\em elementary} quantum groupoids classified in \cite{NV1}, 3.2. A quantum groupoid $H$ is called {\em elementary} if $H\cong M_n(k)$. Then it is determined, up to an isomorphism, by one of its counital subalgebras $$ H_t \cong \oplus_\alpha\, M_{n_\alpha}(k), $$ $n_\alpha,~\alpha =1\dots N$ are positive integers, $n=\sum_\alpha\,n_\alpha^2$. From this one can see that $$ D(H) = [\widehat H^{op}\otimes H] = [\widehat H_s \widehat H_t\otimes H] [\varepsilon \otimes H] = H, $$ where for subsets $A\subset \widehat H^{op}, b\subset H$ we set $[A\otimes B]=\{[a\otimes b]\in D(H)\vert a\in A,b\in B\}$. Hence $H$ is the Drinfeld double of itself. The $R$-matrix of $H$ is $$ R =\sum_{i,j,k,l\alpha}\, \frac{1}{n_\alpha}\, E_{jl\alpha}^{ik\alpha} \otimes E_{ij\alpha}^{kl\alpha}, $$ where $\{E_{ij\alpha}^{kl\beta} \}_{i,j=1\dots n_\alpha}$ is a system of matrix units in $H$. Both the Drinfeld and the ribbon elements are equal to 1. Thus, all the conditions of Lemma~\ref{factorizable implies modular} are satisfied, the category $\mbox{Rep}(H)$ is modular with a unique irreducible object. \end{example}
\end{section}
\begin{section} {$C^*$-quantum groupoids and unitary modular categories} \begin{definition} \label{semisimple and C*} A {\em $*$-quantum groupoid} is a quantum groupoid over a field $k$ with involution, whose underlying algebra $H$ is equipped with an antilinear involutive algebra anti-homomorphism $*:H\to H$ such that $\Delta\circ *=(*\otimes *)\Delta$. A $*$-quantum groupoid is said to be {\em $C^*$-quantum groupoid}, if $k=\mathbb{C}$ and $H$ is a finite-dimensional $C^*$-algebra, i.e., $x^*x=0$ if and only if $x=0,\ \forall x\in H$. \end{definition}
Definition \ref{semisimple and C*} together with the uniqueness of the unit, counit and antipode imply that $$ 1^*=1,\quad \varepsilon(h^*)= \overline{\varepsilon(h)},\quad (S\circ *)^2 =\mbox{id} $$ for all $h$ in a $*$-quantum groupoid $H$. It is also easy to check the relations $$ \varepsilon_t(h)^*=\varepsilon_t(S(h)^*),\ \varepsilon_t(h)^*=\varepsilon_t(S(h)^*), $$ therefore, $H_t$ and $H_s$ are $*$-subalgebras. The dual $\widehat H$ is also a $*$-quantum groupoid with respect to the $*$-operation \begin{equation} \langle\,\phi^*,h\,\rangle=\overline{\langle\,\phi,S(h)^*\,\rangle}\qquad \mbox{ for all } \phi\in \widehat H,\ h\in H. \end{equation} The square of the antipode of a $C^*$-quantum groupoid is an inner automorphism, i.e., $S^2(h) = ghg^{-1}$ for some $g\in H$. It is easy to see that there is a unique such $g$ satisfying the following conditions (\cite{BNSz}, 4.4): \begin{enumerate} \item[(i)] $\mbox{tr}(\pi_\alpha(g^{-1}))=\mbox{tr} (\pi_\alpha(g))\neq 0$ for all irreducible representations $\pi_\alpha$ of $H$ (here $\mbox{tr}$ is the usual trace on a matrix algebra); \item[(ii)] $S(g)=g^{-1}$, and \item[(iii)] $\Delta(g) =(g\otimes g)\Delta(1) = \Delta(1) (g\otimes g)$. \end{enumerate} This element $g$ is called the {\em canonical group-like element} of $H$. \begin{remark} \label{C*-dual} (i) Any $C^*$-quantum groupoid satisfies the conditions of Remark~\ref{recall integrals}, so it always possess a Haar integral.
(ii) If $H$ is a $C^*$-quantum groupoid, then its dual $\widehat H$ is also a $C^*$-quantum groupoid (see \cite{BNSz}, 4.5, \cite{NV1}, 2.3.10). \end{remark}
Groupoid algebras and their duals give examples of commutative and cocommutative $C^*$-quantum groupoids if the ground field $k=\mathbb{C}$ (in which case $g^*=g^{-1}$ for all $g\in G$).
One can check that for a quasitriangular $*$-quantum groupoid ${\bar \R}={\mathcal R}^*$. \begin{proposition} \label{*-double} If $H$ is a $C^*$-quantum groupoid, then $D(H)$ is a quasitriangular $C^*$-quantum groupoid. \end{proposition}
\begin{proof} First let us show that $\widehat{D(H)}$, equipped with a natural involution $$ \langle\, X^*,\, \phi\otimes h \,\rangle =\sum_k\, \overline{\langle\, g_k,\,(S\phi)^* \,\rangle}
\overline{\langle\, S(h)^*,\,\psi_k \,\rangle}, $$
where $X =\sum_k\, g_k\otimes \psi_k \in \widehat{D(H)}, h\in H,\,\phi\in \widehat{H}$, is a $C^*$-subalgebra of the tensor product $C^*$-algebra $H\otimes \widehat{H}^{op}$. For this it suffices to show that $X^*|_J=0$, i.e., $\langle\, X^*,\, \phi\otimes zh \,\rangle = \langle\, X^*,\,(\varepsilon\leftharpoonup z)\phi \otimes h \,\rangle$, $\langle\, X^*,\, \phi\otimes yh \,\rangle = \langle\, X^*,\,(y\rightharpoonup \varepsilon)\phi \otimes h \,\rangle$ for $z\in H_t, y\in H_s$. For instance, one computes: \begin{eqnarray*} \langle\, X^*,\, \phi\otimes zh \,\rangle &=& \sum_k\, \overline{\langle\, g_k,\,(S\phi)^* \,\rangle}
\overline{\langle\, S(z)^*S(h)^*,\,\psi_k \,\rangle},\\ \langle\, X^*,\,(\varepsilon\leftharpoonup z)\phi \otimes h \,\rangle &=& \sum_k\, \overline{\langle\, g_k,\, S((\varepsilon\leftharpoonup z)\phi)^* \,\rangle}
\overline{\langle\, S(h)^*,\, \psi_k \,\rangle}, \end{eqnarray*} for all $z\in H_t$. The right-hand sides of the above equations are equal since $$ \overline{\langle\, z,\,\varepsilon_{(1)}\,\rangle} S(\varepsilon_{(2)})^* = \langle\, S(z)^*,\varepsilon_{(2)}\,\rangle \varepsilon_{(1)}. $$ Similarly one gets the other relation.
To prove that the comultiplication of $\widehat{D(H)}$ is a $*$-homomorphism we compute \begin{eqnarray*} \Delta(X)^* &=& \sum_{i,j,k}\, ( {g_k}_{(2)}^* \otimes {\xi^j}^*{\psi_k}_{(1)}^* {\xi^i}^*) \otimes
(f_j^* {g_k}_{(1)}^* S(f_i)^* \otimes {\psi_k}_{(2)}^* )\\ &=& \sum_{i,j,k}\, ( {g_k}_{(2)}^* \otimes \xi^i {\psi_k}_{(1)}^* \xi^j) \otimes
( S(f_i) {g_k}_{(1)}^* f_j \otimes {\psi_k}_{(2)}^* ) = \Delta(X^*), \end{eqnarray*} where we use that $\sum_j\, (\xi^j)^* \otimes S(f_j)^* = \sum_j\, \xi^j \otimes f_j$ for every pair of dual bases. Thus, $\widehat{D(H)}$ is a $C^*$-quantum groupoid and so is $D(H)$ (see Remark~\ref{C*-dual}). \end{proof}
In \cite{EG} it was shown that a quasitriangular semisimple Hopf algebra is automatically ribbon with ribbon element $\nu=u^{-1}$, where $u$ is the Drinfeld element. We are able to get a similar result for $C^*$-quantum groupoids.
\begin{proposition} \label{*-Ribbon WHA} A quasitriangular $C^*$-quantum groupoid $H$ is automatically ribbon with ribbon element $\nu=u^{-1}g=gu^{-1}$, where $u$ is the Drinfeld element from Definition~\ref{Drinfeld element} and $g$ is the canonical group-like element implementing $S^2$. \end{proposition}
\begin{proof} Since $u$ also implements $S^2$ (Proposition \ref{elements u and v}), $\nu=u^{-1}g$ is central, therefore $S(\nu)$ is also central. Clearly, $u$ must commute with $g$. The same Proposition gives $\Delta(u^{-1}) = {\mathcal R}_{21}{\mathcal R}(u^{-1}\otimes u^{-1})$, which allows us to compute $$ \Delta(\nu) = \Delta(u^{-1})(g\otimes g) = {\mathcal R}_{21}{\mathcal R}(u^{-1}g\otimes u^{-1}g) = {\mathcal R}_{21}{\mathcal R}(\nu \otimes \nu). $$ Propositions~\ref{properties of R} and \ref{elements u and v} and the trace property imply that \begin{eqnarray*} \mbox{tr}(\pi_\alpha(u^{-1})) &=& \mbox{tr}(\pi_\alpha( {\mathcal R}^{(2)}S^2({\mathcal R}^{(1)}) )) \\ &=& \mbox{tr} (\pi_\alpha(S^3({\mathcal R}^{(1)}) S({\mathcal R}^{(2)}) ) = \mbox{tr}(\pi_\alpha(S(u^{-1}))). \end{eqnarray*} Since $u^{-1} = \nu g^{-1}$ and $\nu$ is central, the above relation means that $$ \mbox{tr}(\pi_\alpha(\nu))\mbox{tr}(\pi_\alpha(g^{-1})) = \mbox{tr}(\pi_\alpha(S(\nu)))\mbox{tr}(\pi_\alpha(g)), $$ and, therefore, $\mbox{tr}(\pi_\alpha(\nu)) = \mbox{tr}(\pi_\alpha(S(\nu)))$ for any irreducible representation $\pi_\alpha$, which shows that that $\nu = S(\nu)$. \end{proof} \begin{corollary} \label{C*-trace} For a connected ribbon $C^*$-quantum groupoid $H$ we have: $$ \mbox{tr}_q(f) = (\dim H_t)^{-1} \mbox{Tr}_V(g\circ f),\qquad \dim_q(V) = (\dim H_t)^{-1} \mbox{Tr}_V(g). $$ for any $f\in \mbox{End}(V)$, where $V$ is an $H$-module. \end{corollary} To define the (unitary) representation category $\mbox{URep}(H)$ of a $C^*$-quantum groupoid $H$ we consider {\it unitary} $H$-modules, i.e., $H$-modules $V$ equipped with a scalar product $$ (\cdot,\cdot): V\times V\to \mathbb{C} \qquad \mbox{ such that } \qquad (h\cdot v,w)=(v,h^*\cdot w)\ \forall h\in H, v,w\in V. $$ The notion of a morphism in this category remains the same as in $\mbox{Rep}(H)$. The monoidal product of $V,W\in \mbox{URep}(H)$ is defined as follows. We construct a tensor product $V\otimes_\mathbb{C} W$ of Hilbert spaces and remark that the action of $\Delta(1)$ on this left $H$-module is an orthogonal projection. The image of this projection is, by definition, the monoidal product of $V,W$ in $\mbox{URep}(H)$. Clearly, this definition is compatible with the monoidal product of morphisms in $\mbox{Rep}(H)$.
For any $V\in \mbox{URep}(H)$, the dual space $V^*$ is naturally identified ($v\to \overline v$) with the conjugate Hilbert space, and under this identification we have $h\cdot\overline v= \overline{S(h)^*\cdot v}\ (v\in V, \overline v\in V^*$). In this way $V^*$ becomes a unitary $H$-module with scalar product $(\overline v, \overline w)=(w,gv)$, where $g$ is the canonical group-like element of $H$.
The unit object in $\mbox{URep}(H)$ is $H_t$ equipped with scalar product $(z,t)_{H_t}= \varepsilon(zt^*)$ (it is known \cite{BNSz}, \cite{NV1} that the restriction of $\varepsilon$ to $H_t$ is a non-degenerate positive form). One can verify that the maps $l_V,r_V$ and their inverses are isometries. For example, let us show that the adjoint map for $l_V$ is exactly its inverse. We have: $$ (l_V(1_{(1)}\cdot z\otimes 1_{(2)}\cdot v),w) =(z\cdot v,w)\ (\forall z\in H_t,v,w\in V). $$ On the other hand: \begin{eqnarray*} (1_{(1)}\cdot z\otimes 1_{(2)}\cdot v,l_V^{-1}w) &=& (zS(1_{(1)}) \otimes 1_{(2)}\cdot v,S(1_{(1)}) \otimes 1_{(2)}\cdot w) \\ &=& \varepsilon(zS(1_{(1)})S(1_{(1)})^*)(1_{(2)}^*1_{(2)}\cdot v,w) = (z\cdot v,w). \end{eqnarray*} Proposition \ref{monoidal with duality} implies that $\mbox{URep}(H)$ is a monoidal category with duality (see also \cite{BNSz}, Section 3).
\begin{remark} \label{*-R-matrix} a) One can check that for a quasitriangular $*$-quantum groupoid the braiding is an isometry in $\mbox{URep}(H)$: $c^{-1}_{V,W}=c^{*}_{V,W}$.
b) For a ribbon $C^*$-quantum groupoid $H$, the twist is an isometry in $\mbox{URep}(H)$. Indeed, the relation $\theta_{V}^*=\theta_{V}^{-1}$ is equivalent to the identity $S(u^{-1}) = u^*$, which follows from Proposition~\ref{properties of R} and Remark~\ref{*-R-matrix}a). \end{remark}
A {\em Hermitian ribbon category} over the field $k$ with involution is an $Ab$-ribbon category over $k$ endowed with a conjugation of morphisms $f\mapsto\overline{f}$ satisfying natural conditions (see \cite{T2}, II.5.2): \begin{eqnarray} \label{1st line} \overline{\overline{f}} &=& f,\qquad \overline{f+g}=\overline{f}+ \overline{g},\qquad \overline{cf}=\overline c\overline{f} \quad (c\in k), \\ \label{2nd line} \overline{f\otimes g} &=& \overline{f}\otimes \overline{g}, \quad \overline{f\circ g}=\overline{g}\circ \overline{f},\qquad \overline{c_{V,W}}=(c_{V,W})^{-1}, \overline{\theta_{V}}=\theta_{V}^{-1}, \\ \label{herm} \overline{b}_V &=& d_V\circ c_{V,V^*}(\theta_V\otimes id_{V^*}), \qquad \overline{d}_V = (id_{V^*}\otimes \theta^{-1}_V)c^{-1}_{V^*,V}\circ b_V. \end{eqnarray} A {\em unitary ribbon category} is a Hermitian ribbon category over the field $\mathbb{C}$ such that for any morphism $f$ we have $\mbox{tr}_q(f\overline{f}) \geq 0$.
In a natural way we have a {\em conjugation} of morphisms in $\mbox{URep}(H)$. Namely, for any morphism $f:V\to W$ we define $\overline f:W\to V$ as $\overline f (w)=\overline{f^*(\overline w)}$ for any $w\in W$. Here $\overline w\in W^*, f^*:W^*\to V^*$ is the standard dual of $f$ (see \cite{T2}, I.1.8) and $\overline{f^*(\overline w)}\in V$.
\begin{lemma} \label{hermit-ribbon} Given a quasitriangular $C^*$-quantum groupoid $H,\ \mbox{URep}(H)$ is a unitary ribbon $Ab$-category with respect to the above conjugation of morphisms. \end{lemma} \begin{proof} Relations (\ref{1st line}) are obvious, (\ref{2nd line}) follows from Remarks~\ref{*-R-matrix}.
Let us prove relations (\ref{herm}). On the one hand, for all $v\in V, \phi\in V^*, z\in H_t$ we have, using the definitions of $d_V,c_{V,V^*},\theta_V$, Propositions~\ref{properties of R}, \ref{elements u and v} and the notation $\omega_{v,\phi}(L)=(Lv,\phi)$ for a linear operator $L$ and two vectors $v,\phi$ of a Hilbert space: \begin{eqnarray*} \lefteqn{(d_V\circ c_{V,V^*} (\theta_V\otimes id_{V^*})(v\otimes\phi),z)_{H_t} =} \\ &=& (d_V\circ c_{V,V^*}(gu^{-1}v\otimes\phi),z)_{H_t} \\ &=& (d_V[S({\mathcal R}^{*(2)})\phi\otimes {\mathcal R}^{(1)}gu^{-1}v],z)_{H_t} \\ &=& \varepsilon[(1_{(1)}{\mathcal R}^{(1)}gu^{-1}v,S({\mathcal R}^{*(2)})\phi)1_{(2)}z^*] \\ &=& (\omega_{v,\phi}\otimes\varepsilon)
[(S({\mathcal R}^{(2)})\otimes 1)\Delta(1)({\mathcal R}^{(1)}gu^{-1}\otimes z^*)] \\ &=& \omega_{v,\phi}[S({\mathcal R}^{(2)})S(z^*){\mathcal R}^{(1)}gu^{-1}] \\ &=& \omega_{v,\phi}[S({\mathcal R}^{(2)}){\mathcal R}^{(1)}z^*gu^{-1}] = (z^*gv,\phi). \end{eqnarray*} And, on the other hand, using the definition of $b_V$, we compute : \begin{eqnarray*} (\overline{b}_V(v\otimes\phi),z)_{H_t} &=& (v\otimes\phi,\sum_i z_{(1)}f_i \otimes
S(z_{(2)})^*\xi^i)_{V\otimes V^*} \\ &=& \sum_i(v,z_{(1)}f_i)(S(z_{(2)})^*\xi^i, g\phi) \\ &=& \sum_i(z^*_{(1)}v,f_i)(\xi^i,S(z_{(2)})g\phi) \\ &=& (z^*_{(1)}v,S(z_{(2)})g\phi)=(v,zG\phi)=(z^*gv,\phi), \end{eqnarray*} whence the first part of (\ref{herm}) follows. To establish the second part, note that for all $v\in V, \phi\in V^*, z\in H_t$ we have, using the definitions of $b_V,c_{V,V^*},\theta_V$, Propositions \ref{properties of R}, \ref{elements u and v} and the properties of $\nu$: \begin{eqnarray*} \lefteqn{((\mbox{id}_{V^*}\otimes \theta^{-1}_V)c^{-1}_{V^*,V} \circ b_V(z),\phi\otimes v)_{V^*\otimes V} =} \\ &=& ((\mbox{id}_{V^*}\otimes \theta^{-1}_V)c^{-1}_{V^*,V}\sum_i z_{(1)}f_i
\otimes S(z_{(2)})^*\xi^i),\phi\otimes v)_{V^*\otimes V} \\ &=& ((\mbox{id}_{V^*}\otimes \theta^{-1}_V)\sum_i S({\mathcal R}^{*(1)})^*S(z_{(2)})^*\xi^i
\otimes {\mathcal R}^{*(2)}z_{(1)}f_i),\phi\otimes v)_{V^*\otimes V} \\ &=& \sum_i\,(\phi,gS({\mathcal R}^{*(1)})^*S(z_{(2)})^*\xi^i)
(\nu^{-1}{\mathcal R}^{*(2)}z_{(1)}f_i,v) \\ &=& \sum_i\,(S(z_{(2)})S({\mathcal R}^{*(1)})g\phi,\xi^i)
(f_i,\nu z^*_{(1)}{\mathcal R}^{(2)}v) \\ &=& (S(z_{(2)})S({\mathcal R}^{*(1)})\phi,\nu z^*_{(1)}{\mathcal R}^{(2)}v)
=({\mathcal R}^{*(2)}zS({\mathcal R}^{*(1)})g\phi, \nu v) \\ &=& ((S^{-1}({\mathcal R}^{(1)}){\mathcal R}^{(2)})^*S(z)g\phi,\nu v)
=(S^{-1}(u)^*S(z)g\phi,\nu v) = (\phi,S(z^*)v). \end{eqnarray*} On the other hand, using the definition of $d_V$, we obtain: \begin{eqnarray*} (\overline{d}_V(z),\phi\otimes v)_{V^*\otimes V} &=& (z,d_V(\phi\otimes v))_{H_t} = (z,(1_{(1)}v,\phi)1_{(2)})_{H_t} \\ &=& \overline{\varepsilon((1_{(1)}v,\phi)1_{(2)}z^*)}
= \overline{(\omega_{v,\phi}\otimes \varepsilon)(\Delta(1)(1\otimes z^*))} \\ &=& \overline{(\omega_{v,\phi}(S(z^*))}=(\phi,S(z^*)v). \end{eqnarray*} The condition $\mbox{tr}_q(f\overline f)=\mbox{Tr}(gff^*)\geq 0$ for any morphism $f$ follows from Remark~\ref{*-R-matrix}b) and from the positivity of $g$. \end{proof} The next proposition extends (\cite{EG}, 1.2).
\begin{theorem} \label{rep of C* is modular} If $H$ is a connected $C^*$-quantum groupoid, then $\mbox{URep}(D(H))$ is a unitary modular category. \end{theorem}
\begin{proof} The proof follows from Lemmas~\ref{hermit-ribbon}, \ref{factorizable implies modular} and Propositions~\ref{D(A) is factorizable}, \ref{*-double}. \end{proof} \end{section}
\begin{section}{Appendix} Here we collected some results on ribbon and modular quantum groupoids which extend the corresponding facts for Hopf algebras. \noindent \textbf{1.} There is a procedure analogous to (\cite{RT1}, 3.4), that extends any quasitriangular quantum groupoid $(H, {\mathcal R},{\bar \R})$ to a ribbon quantum groupoid in a canonical way. For this we need \begin{lemma}[cf. (\cite{RT1}, 3.3)] \label{properties of nu} (i) A ribbon element $\nu$ satisfies $$ \varepsilon_t(\nu) = \varepsilon_s(\nu) = 1 \quad \mbox{and} \quad \nu^2 = (vu)^{-1}, $$ where $u$ and $v$ are the elements defined in Proposition~\ref{elements u and v}. \newline \hskip 0.5cm (ii) If $\nu_1$ and $\nu_2$ are two ribbon elements of $(H, {\mathcal R})$, then $\nu_2 = E\nu_1$, where $E\in H$ is an invertible central element such that $E= S(E) =E^{-1}$, $\Delta(E) = \Delta(1)(E\otimes E)$ (i.e., $E$ is group-like), and $\varepsilon_t(E) = \varepsilon_s(E) = 1$. \end{lemma}
\begin{proof} (i) The definition of counit implies : \begin{eqnarray*} \nu &=& (\mbox{id}\otimes\varepsilon)\Delta(\nu)
= \nu {\mathcal R}^{(2)} {{\mathcal R}'}^{(1)} \varepsilon({\mathcal R}^{(1)} {{\mathcal R}'}^{(2)}\nu) \\ &=& \nu {\mathcal R}^{(2)} {{\mathcal R}'}^{(1)} \varepsilon(\varepsilon_s({\mathcal R}^{(1)}) {{\mathcal R}'}^{(2)}\nu)
= \nu {{\mathcal R}'}^{(1)} \varepsilon(\varepsilon_s({{\mathcal R}'}^{(2)}) \nu) \\ &=& \nu S(1_{(2)})\varepsilon(1_{(1)} \nu) =\nu S(\varepsilon_t(\nu)), \end{eqnarray*} hence $\varepsilon_t(\nu)=1$. We used here the identity $\varepsilon(hg) =\varepsilon(\varepsilon_s(h)g),~ h,g\in H$ and Lemma~\ref{properties of R}. Similarly, $\varepsilon_s(\nu)=1$. Using the antipode property, we compute \begin{eqnarray*} 1 &=& \varepsilon_t(\nu) = m(\mbox{id}\otimes S)\Delta(\nu) \\ &=& {\mathcal R}^{(2)} {{\mathcal R}'}^{(1)} S({{\mathcal R}'}^{(2)}) S({\mathcal R}^{(1)}) \nu^2 \\ &=& v S^2({\mathcal R}^{(2)}) S({\mathcal R}^{(1)}) \nu^2 = vu \nu^2. \end{eqnarray*} \newline (ii) Set $E= \nu_1^{-1}\nu_2$. Then $E$ is central and invertible, $S(E)=E$, and from part (i) we conclude that $E^2=1$. Next, $$ \Delta(E) = {\bar \R}{\bar \R}_{21}(\nu_1^{-1} \otimes \nu_1^{-1}) {\mathcal R}_{21}{\mathcal R} (\nu_2\otimes \nu_2) = \Delta(1)(E\otimes E). $$ Applying the counit to both sides of the last equality, we get $E = E\varepsilon_t(E) = E\varepsilon_s(E)$, i.e., $\varepsilon_t(E) = \varepsilon_s(E) =1$. \end{proof} \begin{proposition} Let $\tilde{H} = H +H\nu$ be a central extension of $H$, consisting of formal linear combinations $h+g\nu$ with $h,g\in H$. Then $(\tilde{H}, {\mathcal R}, \nu)$ is a ribbon quantum groupoid with operations \begin{eqnarray*} (h+g\nu)(h'+g'\nu) &=& (hh'+gg'(vu)^{-1}) + (hg' + gh')\nu, \\ \Delta(h+g\nu) &=& \Delta(h) + \Delta(g) {\mathcal R}_{21}{\mathcal R}(\nu\otimes\nu), \\ \varepsilon(h+g\nu) &=& \varepsilon(h) +\varepsilon(g),\\ S(h+g\nu) &=& S(h) +S(g)\nu. \end{eqnarray*} Note that $\tilde{H}$ contains $H =\{ h+0\nu \mid g\in H\}$ as a quantum subgroupoid. \end{proposition}
\begin{proof} One verifies that $\Delta$ is a homomorphism exactly as in \cite{RT1}. The properties of ${\mathcal R}$ and $\nu$ follow directly from definitions. For the counit axiom we have, using the properties of counital maps, Proposition~\ref{properties of R}, and Lemma~\ref{properties of nu} : \begin{eqnarray*} (\varepsilon\otimes\mbox{id})\Delta(h+g\nu) &=& h + \varepsilon(g_{(1)} \nu \varepsilon_t({\mathcal R}^{(2)} {{\mathcal R}'}^{(1)})) g_{(2)} \nu {\mathcal R}^{(1)} {{\mathcal R}'}^{(2)} \\ &=& h + g_{(2)} S(\varepsilon_s(g_{(1)} \nu)) = h+g\nu, \\ (\mbox{id}\otimes\varepsilon)\Delta(h+g\nu) &=& h + {\mathcal R}^{(2)} {{\mathcal R}'}^{(1)} g_{(1)} \nu \varepsilon(\varepsilon_s({\mathcal R}^{(1)} {{\mathcal R}'}^{(2)}) g_{(2)} \nu) \\ &=& h + S(\varepsilon_t(g_{(2)} \nu)) g_{(1)} \nu = h+g\nu. \end{eqnarray*} Axioms (\ref{eps m}) and (\ref{Delta 1}) of Definition~\ref{finite quantum groupoid} can be verified by a direct computation.
Next, we observe that $\varepsilon_t(h+g\nu) = \varepsilon_t(h) + \varepsilon_t(g)$ and $\varepsilon_s(h+g\nu) = \varepsilon_s(h) + \varepsilon_s(g)$. The antipode axiom follows from the identity $\nu^2 vu =1$ provided by Lemma~\ref{properties of nu}: \begin{eqnarray*} m(\mbox{id}\otimes S)\Delta(h+g\nu) &=& \varepsilon_t(h) + g_{(1)} {\mathcal R}^{(2)} {{\mathcal R}'}^{(1)} S({{\mathcal R}'}^{(2)}) S({\mathcal R}^{(1)}) S(g_{(2)})\nu^2 \\ &=& \varepsilon_t(h) + g_{(1)} vu S(g_{(2)})\nu^2 = \varepsilon_t(h+g\nu),\\ m(S\otimes \mbox{id})\Delta(h+g\nu) &=& \varepsilon_s(h) + S(g_{(1)}) S({\mathcal R}^{(1)})S({{\mathcal R}'}^{(2)}) {{\mathcal R}'}^{(1)} {\mathcal R}^{(2)} g_{(2)} \nu^2 \\ &=& \varepsilon_s(h) + S(g_{(1)}) vu g_{(2)} \nu^2 = \varepsilon_s(h+g\nu). \end{eqnarray*} The anti-multiplicative properties of the antipode follow from the facts that $S(uv) = uv$ and $S(\nu) =\nu$. \end{proof} \noindent \textbf{2.} Let us establish a relation between modular quantum groupoids and modular categories. A morphism $f:V\to W$ in a ribbon $Ab$-category ${\mathcal V}$ is said to be {\it negligible} if for any morphism $g:W\to V$ we have $\mbox{tr}(fg)=0$. ${\mathcal V}$ is said to be {\it pure} if all negligible morphisms in this category are equal to zero. A purification procedure transforming any ribbon $Ab$-category into a pure ribbon $Ab$-category is described in (\cite{T2}, XI.4.2); this procedure transforms hermitian ribbon $Ab$-categories into hermitian pure ribbon $Ab$-categories (\cite{T2}, XI.4.3). We say that a family $\{V\}_{i\in I}$ of objects of ${\mathcal V}$ quasidominates an object $V$ of ${\mathcal V}$ if there exists a finite set $\{V_{i(r)}\}_r$ of objects of this family (possibly, with repetitions) and a family of morphisms $f_r:V_{i(r)} \to V,g_r:V\to V_{i(r)}$ such that $\mbox{id}_V - \sum_r f_r g_r$ is negligible. If ${\mathcal V}$ is pure, then quasidomination coincides with domination. Let $(H,{\mathcal R},\nu)$ be a ribbon quantum groupoid. Then an $H$-module $V$ of finite $k$-rank is said to be {\it negligible} if $\mbox{tr}_q(f)=0$ for any $f\in\mbox{End}(V)$. If $k$ is algebraically closed, then any irreducible $H$-module is a simple object of $\mbox{Rep}(H)$.
\begin{definition} \label{modular WHA} A modular quantum groupoid consists of a ribbon quantum groupoid $(H,{\mathcal R},\nu)$ together with a finite family of simple $H$-modules of finite rank $\{V\}_{i\in I}$ such that: \begin{enumerate} \item[(i)] for some $0\in I$, we have $V_0=H_t$, the unit object of $\mbox{Rep}(H)$; \item[(ii)] for each $i\in I$, there exists $i^*\in I$ such that $V_{i^*}$ is isomorphic to $V^*_i$; \item[(iii)] for any $k,l\in I$, the tensor product $V_k\otimes V_l$ splits as a finite direct some of certain $\{V\}_{i\in I}$ (possibly with multiplicities) and a negligible $H$-module; \end{enumerate} To formulate the last condition, let $S_{i,j} = \mbox{tr}_q(c_{V_i,V_j}\circ c_{V_j,V_i}),~i,j\in I$, where the braiding $c_{V_i,V_j}$ was defined in \ref{QT WHA} and the quantum trace $\mbox{tr}_q$ in \ref{quantum trace}. \begin{enumerate} \item[(iv)]
The square matrix $[S_{i,j}]_{i,j\in I}$ is invertible in $M_{|I|}(k)$. \end{enumerate} \end{definition} For any modular quantum groupoid, we define a subcategory ${\mathcal C}$ of $\mbox{Rep}(H)$ as follows. The objects of ${\mathcal C}$ are $H$-modules of finite rank quasidominated by $\{V\}_{i\in I}$ and morphisms are $H$-morphisms of such modules; all the operations in ${\mathcal C}$ are induced by the corresponding operations in $\mbox{Rep}(H)$. Now taking into account the results of the previous sections and repeating the proof of (\cite {T2}, XI.5.3.2), we have the first statement of the following
\begin{proposition} \label{modular WHA-modular Rep} If $(H,{\mathcal R},\nu,\{V\}_{i\in I})$ is modular, then the subcategory $({\mathcal C}, \{V\}_{i\in I})$ of $\mbox{Rep}(H)$ is quasimodular in the sense of (\cite {T2}, XI.4.3). Conversely, if $({\mathcal C}, \{V\}_{i\in I})$ is quasimodular, then $(H,{\mathcal R},\nu,\{V\}_{i\in I})$ is modular. \end{proposition} The proof of the second statement follows directly from the comparison of \cite {T2}, XI.4.3 and the above definition of a modular quantum groupoid. Purifying $({\mathcal C}, \{V\}_{i\in I})$ as in (\cite {T2}, XI.4.2), we get a modular category (\cite{T2}, II.1.4). \end{section}
\end{document} | arXiv |
Truncated dodecahedron
In geometry, the truncated dodecahedron is an Archimedean solid. It has 12 regular decagonal faces, 20 regular triangular faces, 60 vertices and 90 edges.
Truncated dodecahedron
(Click here for rotating model)
TypeArchimedean solid
Uniform polyhedron
ElementsF = 32, E = 90, V = 60 (χ = 2)
Faces by sides20{3}+12{10}
Conway notationtD
Schläfli symbolst{5,3}
t0,1{5,3}
Wythoff symbol2 3 | 5
Coxeter diagram
Symmetry groupIh, H3, [5,3], (*532), order 120
Rotation groupI, [5,3]+, (532), order 60
Dihedral angle10-10: 116.57°
3-10: 142.62°
ReferencesU26, C29, W10
PropertiesSemiregular convex
Colored faces
3.10.10
(Vertex figure)
Triakis icosahedron
(dual polyhedron)
Net
Geometric relations
This polyhedron can be formed from a regular dodecahedron by truncating (cutting off) the corners so the pentagon faces become decagons and the corners become triangles.
It is used in the cell-transitive hyperbolic space-filling tessellation, the bitruncated icosahedral honeycomb.
Area and volume
The area A and the volume V of a truncated dodecahedron of edge length a are:
${\begin{aligned}A&=5\left({\sqrt {3}}+6{\sqrt {5+2{\sqrt {5}}}}\right)a^{2}&&\approx 100.990\,76a^{2}\\V&={\tfrac {5}{12}}\left(99+47{\sqrt {5}}\right)a^{3}&&\approx 85.039\,6646a^{3}\end{aligned}}$
Cartesian coordinates
Cartesian coordinates for the vertices of a truncated dodecahedron with edge length 2φ − 2, centered at the origin,[1] are all even permutations of:
(0, ±1/φ, ±(2 + φ))
(±1/φ, ±φ, ±2φ)
(±φ, ±2, ±(φ + 1))
where φ = 1 + √5/2 is the golden ratio.
Orthogonal projections
The truncated dodecahedron has five special orthogonal projections, centered: on a vertex, on two types of edges, and two types of faces. The last two correspond to the A2 and H2 Coxeter planes.
Orthogonal projections
Centered by Vertex Edge
3-3
Edge
10-10
Face
Triangle
Face
Decagon
Solid
Wireframe
Projective
symmetry
[2] [2] [2] [6] [10]
Dual
Spherical tilings and Schlegel diagrams
The truncated dodecahedron can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.
Schlegel diagrams are similar, with a perspective projection and straight edges.
Orthographic projection Stereographic projections
Decagon-centered
Triangle-centered
Vertex arrangement
It shares its vertex arrangement with three nonconvex uniform polyhedra:
Truncated dodecahedron
Great icosicosidodecahedron
Great ditrigonal dodecicosidodecahedron
Great dodecicosahedron
Related polyhedra and tilings
It is part of a truncation process between a dodecahedron and icosahedron:
Family of uniform icosahedral polyhedra
Symmetry: [5,3], (*532) [5,3]+, (532)
{5,3} t{5,3} r{5,3} t{3,5} {3,5} rr{5,3} tr{5,3} sr{5,3}
Duals to uniform polyhedra
V5.5.5 V3.10.10 V3.5.3.5 V5.6.6 V3.3.3.3.3 V3.4.5.4 V4.6.10 V3.3.3.3.5
This polyhedron is topologically related as a part of sequence of uniform truncated polyhedra with vertex configurations (3.2n.2n), and [n,3] Coxeter group symmetry.
*n32 symmetry mutation of truncated spherical tilings: t{n,3}
Symmetry
*n32
[n,3]
Spherical Euclid. Compact hyperb. Paraco.
*232
[2,3]
*332
[3,3]
*432
[4,3]
*532
[5,3]
*632
[6,3]
*732
[7,3]
*832
[8,3]...
*∞32
[∞,3]
Truncated
figures
Symbol t{2,3} t{3,3} t{4,3} t{5,3} t{6,3} t{7,3} t{8,3} t{∞,3}
Triakis
figures
Config. V3.4.4 V3.6.6 V3.8.8 V3.10.10 V3.12.12 V3.14.14 V3.16.16 V3.∞.∞
Truncated dodecahedral graph
Truncated dodecahedral graph
5-fold symmetry Schlegel diagram
Vertices60
Edges90
Automorphisms120
Chromatic number3
Chromatic index3
PropertiesCubic, Hamiltonian, regular, zero-symmetric
Table of graphs and parameters
In the mathematical field of graph theory, a truncated dodecahedral graph is the graph of vertices and edges of the truncated dodecahedron, one of the Archimedean solids. It has 60 vertices and 90 edges, and is a cubic Archimedean graph.[2]
Circular
Notes
1. Weisstein, Eric W. "Icosahedral group". MathWorld.
2. Read, R. C.; Wilson, R. J. (1998), An Atlas of Graphs, Oxford University Press, p. 269
References
• Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X. (Section 3-9)
• Cromwell, P. (1997). Polyhedra. United Kingdom: Cambridge. pp. 79–86 Archimedean solids. ISBN 0-521-55432-2.
External links
• Eric W. Weisstein, Truncated dodecahedron (Archimedean solid) at MathWorld.
• Weisstein, Eric W. "Truncated dodecahedral graph". MathWorld.
• Klitzing, Richard. "3D convex uniform polyhedra o3x5x - tid".
• Editable printable net of a truncated dodecahedron with interactive 3D view
• The Uniform Polyhedra
• Virtual Reality Polyhedra The Encyclopedia of Polyhedra
Archimedean solids
Tetrahedron
(Seed)
Tetrahedron
(Dual)
Cube
(Seed)
Octahedron
(Dual)
Dodecahedron
(Seed)
Icosahedron
(Dual)
Truncated tetrahedron
(Truncate)
Truncated tetrahedron
(Zip)
Truncated cube
(Truncate)
Truncated octahedron
(Zip)
Truncated dodecahedron
(Truncate)
Truncated icosahedron
(Zip)
Tetratetrahedron
(Ambo)
Cuboctahedron
(Ambo)
Icosidodecahedron
(Ambo)
Rhombitetratetrahedron
(Expand)
Truncated tetratetrahedron
(Bevel)
Rhombicuboctahedron
(Expand)
Truncated cuboctahedron
(Bevel)
Rhombicosidodecahedron
(Expand)
Truncated icosidodecahedron
(Bevel)
Snub tetrahedron
(Snub)
Snub cube
(Snub)
Snub dodecahedron
(Snub)
Catalan duals
Tetrahedron
(Dual)
Tetrahedron
(Seed)
Octahedron
(Dual)
Cube
(Seed)
Icosahedron
(Dual)
Dodecahedron
(Seed)
Triakis tetrahedron
(Needle)
Triakis tetrahedron
(Kis)
Triakis octahedron
(Needle)
Tetrakis hexahedron
(Kis)
Triakis icosahedron
(Needle)
Pentakis dodecahedron
(Kis)
Rhombic hexahedron
(Join)
Rhombic dodecahedron
(Join)
Rhombic triacontahedron
(Join)
Deltoidal dodecahedron
(Ortho)
Disdyakis hexahedron
(Meta)
Deltoidal icositetrahedron
(Ortho)
Disdyakis dodecahedron
(Meta)
Deltoidal hexecontahedron
(Ortho)
Disdyakis triacontahedron
(Meta)
Pentagonal dodecahedron
(Gyro)
Pentagonal icositetrahedron
(Gyro)
Pentagonal hexecontahedron
(Gyro)
Convex polyhedra
Platonic solids (regular)
• tetrahedron
• cube
• octahedron
• dodecahedron
• icosahedron
Archimedean solids
(semiregular or uniform)
• truncated tetrahedron
• cuboctahedron
• truncated cube
• truncated octahedron
• rhombicuboctahedron
• truncated cuboctahedron
• snub cube
• icosidodecahedron
• truncated dodecahedron
• truncated icosahedron
• rhombicosidodecahedron
• truncated icosidodecahedron
• snub dodecahedron
Catalan solids
(duals of Archimedean)
• triakis tetrahedron
• rhombic dodecahedron
• triakis octahedron
• tetrakis hexahedron
• deltoidal icositetrahedron
• disdyakis dodecahedron
• pentagonal icositetrahedron
• rhombic triacontahedron
• triakis icosahedron
• pentakis dodecahedron
• deltoidal hexecontahedron
• disdyakis triacontahedron
• pentagonal hexecontahedron
Dihedral regular
• dihedron
• hosohedron
Dihedral uniform
• prisms
• antiprisms
duals:
• bipyramids
• trapezohedra
Dihedral others
• pyramids
• truncated trapezohedra
• gyroelongated bipyramid
• cupola
• bicupola
• frustum
• bifrustum
• rotunda
• birotunda
• prismatoid
• scutoid
Degenerate polyhedra are in italics.
| Wikipedia |
Srinivasa Ramanujan
{{#invoke:Hatnote|hatnote}}Template:Main other Template:Indian name Template:EngvarB {{ safesubst:#invoke:Unsubst||$N=Use dmy dates |date=__DATE__ |$B= }} Template:Infobox scientist Srinivasa Ramanujan FRS (Template:IPAc-en) (22 December 1887Template:Spaced ndash26 April 1920) was an Indian mathematician and autodidact who, with almost no formal training in pure mathematics, made extraordinary contributions to mathematical analysis, number theory, infinite series, and continued fractions. Ramanujan initially developed his own mathematical research in isolation, which was quickly recognized by Indian mathematicians. When his skills became apparent to the wider mathematical community, centered in Europe at the time, he began a famous partnership with the English mathematician G. H. Hardy. He rediscovered previously known theorems in addition to producing new work. Ramanujan was said to be a natural genius, in the same league as mathematicians such as Euler and Gauss.[1]
During his short life, Ramanujan independently compiled nearly 3900 results (mostly identities and equations).[2] Nearly all his claims have now been proven correct, although a small number of these results were actually false and some were already known.[3] He stated results that were both original and highly unconventional, such as the Ramanujan prime and the Ramanujan theta function, and these have inspired a vast amount of further research.[4] The Ramanujan Journal, an international publication, was launched to publish work in all areas of mathematics influenced by his work.[5]
2 Adulthood in India
2.1 Attention towards mathematics
2.2 Contacting English mathematicians
3 Life in England
3.1 Illness and return to India
3.2 Personality and spiritual life
4 Mathematical achievements
4.1 The Ramanujan conjecture
4.2 Ramanujan's notebooks
5 Hardy-Ramanujan number 1729
6 Other mathematicians' views of Ramanujan
11 Selected publications by Ramanujan
12 Selected publications about Ramanujan and his work
13.1 Media links
13.2 Biographical links
13.3 Other links
Ramanujan's home on Sarangapani Street, Kumbakonam
Ramanujan was born on 22 December 1887 in Erode, Madras Presidency (now Pallipalayam, Erode, Tamil Nadu), at the residence of his maternal grandparents.[6] His father, K. Srinivasa Iyengar, worked as a clerk in a sari shop and hailed from the district of Thanjavur.[7] His mother, Komalatammal, was a housewife and also sang at a local temple.[8] They lived in Sarangapani Street in a traditional home in the town of Kumbakonam. The family home is now a museum. When Ramanujan was a year and a half old, his mother gave birth to a son named Sadagopan, who died less than three months later. In December 1889, Ramanujan had smallpox and recovered, unlike thousands in the Thanjavur District who died from the disease that year.[9] He moved with his mother to her parents' house in Kanchipuram, near Madras (now Chennai). In November 1891, and again in 1894, his mother gave birth to two children, but both children died in infancy.
On 1 October 1892, Ramanujan was enrolled at the local school.[10] In March 1894, he was moved to a Telugu medium school. After his maternal grandfather lost his job as a court official in Kanchipuram,[11] Ramanujan and his mother moved back to Kumbakonam and he was enrolled in the Kangayan Primary School.[12] When his paternal grandfather died, he was sent back to his maternal grandparents, who were now living in Madras. He did not like school in Madras, and he tried to avoid attending. His family enlisted a local constable to make sure he attended school. Within six months, Ramanujan was back in Kumbakonam.[12]
Since Ramanujan's father was at work most of the day, his mother took care of him as a child. He had a close relationship with her. From her, he learned about tradition and puranas. He learned to sing religious songs, to attend pujas at the temple and particular eating habits – all of which are part of Brahmin culture.[13] At the Kangayan Primary School, Ramanujan performed well. Just before the age of 10, in November 1897, he passed his primary examinations in English, Tamil, geography and arithmetic. With his scores, he stood first in the district.[14] That year, Ramanujan entered Town Higher Secondary School where he encountered formal mathematics for the first time.[14]
By age 11, he had exhausted the mathematical knowledge of two college students who were lodgers at his home. He was later lent a book on advanced trigonometry written by S. L. Loney.[15][16] He completely mastered this book by the age of 13 and discovered sophisticated theorems on his own. By 14, he was receiving merit certificates and academic awards which continued throughout his school career and also assisted the school in the logistics of assigning its 1200 students (each with their own needs) to its 35-odd teachers.[17] He completed mathematical exams in half the allotted time, and showed a familiarity with geometry and infinite series. Ramanujan was shown how to solve cubic equations in 1902 and he went on to find his own method to solve the quartic. The following year, not knowing that the quintic could not be solved by radicals, he tried (and of course failed) to solve the quintic.
In 1903 when he was 16, Ramanujan obtained from a friend a library-loaned copy of a book by G. S. Carr.[18][19] The book was titled A Synopsis of Elementary Results in Pure and Applied Mathematics and was a collection of 5000 theorems. Ramanujan reportedly studied the contents of the book in detail.[20] The book is generally acknowledged as a key element in awakening the genius of Ramanujan.[20] The next year, he had independently developed and investigated the Bernoulli numbers and had calculated the Euler–Mascheroni constant up to 15 decimal places.[21] His peers at the time commented that they "rarely understood him" and "stood in respectful awe" of him.[17]
When he graduated from Town Higher Secondary School in 1904, Ramanujan was awarded the K. Ranganatha Rao prize for mathematics by the school's headmaster, Krishnaswami Iyer. Iyer introduced Ramanujan as an outstanding student who deserved scores higher than the maximum possible marks.[17] He received a scholarship to study at Government Arts College, Kumbakonam,[22][23] However, Ramanujan was so intent on studying mathematics that he could not focus on any other subjects and failed most of them, losing his scholarship in the process.[24] In August 1905, he ran away from home, heading towards Visakhapatnam and stayed in Rajahmundry[25] for about a month.[26] He later enrolled at Pachaiyappa's College in Madras. He again excelled in mathematics but performed poorly in other subjects such as physiology. Ramanujan failed his Fellow of Arts exam in December 1906 and again a year later. Without a degree, he left college and continued to pursue independent research in mathematics. At this point in his life, he lived in extreme poverty and was often on the brink of starvation.[27]
Adulthood in India
On 14 July 1909, Ramanujan was married to a ten-year old bride, Janakiammal (21 March 1899 – 13 April 1994).[28] She came from Rajendram, a village close to Marudur (Karur district) Railway Station. Ramanujan's father did not participate in the marriage ceremony.[29]
After the marriage, Ramanujan developed a hydrocele testis, an abnormal swelling of the tunica vaginalis, an internal membrane in the testicle.[30] The condition could be treated with a routine surgical operation that would release the blocked fluid in the scrotal sac. His family did not have the money for the operation, but in January 1910, a doctor volunteered to do the surgery for free.[31]
After his successful surgery, Ramanujan searched for a job. He stayed at friends' houses while he went door to door around the city of Madras (now Chennai) looking for a clerical position. To make some money, he tutored some students at Presidency College who were preparing for their F.A. exam.[32]
In late 1910, Ramanujan was sick again, possibly as a result of the surgery earlier in the year. He feared for his health, and even told his friend, R. Radakrishna Iyer, to "hand these [Ramanujan's mathematical notebooks] over to Professor Singaravelu Mudaliar [the mathematics professor at Pachaiyappa's College] or to the British professor Edward B. Ross, of the Madras Christian College."[33] After Ramanujan recovered and got back his notebooks from Iyer, he took a northbound train from Kumbakonam to Villupuram, a coastal city under French control.[34][35]
Attention towards mathematics
Ramanujan met deputy collector V. Ramaswamy Aiyer, who had recently founded the Indian Mathematical Society.[36] Ramanujan, wishing for a job at the revenue department where Ramaswamy Aiyer worked, showed him his mathematics notebooks. As Ramaswamy Aiyer later recalled:
I was struck by the extraordinary mathematical results contained in it [the notebooks]. I had no mind to smother his genius by an appointment in the lowest rungs of the revenue department.[37]
Ramaswamy Aiyer sent Ramanujan, with letters of introduction, to his mathematician friends in Madras.[36] Some of these friends looked at his work and gave him letters of introduction to R. Ramachandra Rao, the district collector for Nellore and the secretary of the Indian Mathematical Society.[38][39][40] Ramachandra Rao was impressed by Ramanujan's research but doubted that it was actually his own work. Ramanujan mentioned a correspondence he had with Professor Saldhana, a notable Bombay mathematician, in which Saldhana expressed a lack of understanding of his work but concluded that he was not a phoney.[41] Ramanujan's friend, C. V. Rajagopalachari, persisted with Ramachandra Rao and tried to quell any doubts over Ramanujan's academic integrity. Rao agreed to give him another chance, and he listened as Ramanujan discussed elliptic integrals, hypergeometric series, and his theory of divergent series, which Rao said ultimately "converted" him to a belief in Ramanujan's mathematical brilliance.[41] When Rao asked him what he wanted, Ramanujan replied that he needed some work and financial support. Rao consented and sent him to Madras. He continued his mathematical research with Rao's financial aid taking care of his daily needs. Ramanujan, with the help of Ramaswamy Aiyer, had his work published in the Journal of the Indian Mathematical Society.[42]
One of the first problems he posed in the journal was:
1+21+31+⋯.{\displaystyle {\sqrt {1+2{\sqrt {1+3{\sqrt {1+\cdots }}}}}}.}
He waited for a solution to be offered in three issues, over six months, but failed to receive any. At the end, Ramanujan supplied the solution to the problem himself. On page 105 of his first notebook, he formulated an equation that could be used to solve the infinitely nested radicals problem.
x+n+a=ax+(n+a)2+xa(x+n)+(n+a)2+(x+n)⋯{\displaystyle x+n+a={\sqrt {ax+(n+a)^{2}+x{\sqrt {a(x+n)+(n+a)^{2}+(x+n){\sqrt {\cdots }}}}}}}
Using this equation, the answer to the question posed in the Journal was simply 3.[43] Ramanujan wrote his first formal paper for the Journal on the properties of Bernoulli numbers. One property he discovered was that the denominators (sequence A027642 in OEIS) of the fractions of Bernoulli numbers were always divisible by six. He also devised a method of calculating Bn based on previous Bernoulli numbers. One of these methods went as follows:
It will be observed that if n is even but not equal to zero,
(i) Bn is a fraction and the numerator of Bnn{\displaystyle {B_{n} \over n}} in its lowest terms is a prime number,
(ii) the denominator of Bn contains each of the factors 2 and 3 once and only once,
(iii) 2n(2n−1)bnn{\displaystyle 2^{n}(2^{n}-1){b_{n} \over n}} is an integer and 2(2n−1)Bn{\displaystyle 2(2^{n}-1)B_{n}\,} consequently is an odd integer.
In his 17-page paper, "Some Properties of Bernoulli's Numbers", Ramanujan gave three proofs, two corollaries and three conjectures.[44] Ramanujan's writing initially had many flaws. As Journal editor M. T. Narayana Iyengar noted:
Mr. Ramanujan's methods were so terse and novel and his presentation so lacking in clearness and precision, that the ordinary [mathematical reader], unaccustomed to such intellectual gymnastics, could hardly follow him.[45]
Ramanujan later wrote another paper and also continued to provide problems in the Journal.[46] In early 1912, he got a temporary job in the Madras Accountant General's office, with a salary of 20 rupees per month. He lasted for only a few weeks.[47] Toward the end of that assignment he applied for a position under the Chief Accountant of the Madras Port Trust. In a letter dated 9 February 1912, Ramanujan wrote:
I understand there is a clerkship vacant in your office, and I beg to apply for the same. I have passed the Matriculation Examination and studied up to the F.A. but was prevented from pursuing my studies further owing to several untoward circumstances. I have, however, been devoting all my time to Mathematics and developing the subject. I can say I am quite confident I can do justice to my work if I am appointed to the post. I therefore beg to request that you will be good enough to confer the appointment on me.[48]
Attached to his application was a recommendation from E. W. Middlemast, a mathematics professor at the Presidency College, who wrote that Ramanujan was "a young man of quite exceptional capacity in Mathematics".[49] Three weeks after he had applied, on 1 March, Ramanujan learned that he had been accepted as a Class III, Grade IV accounting clerk, making 30 rupees per month.[50] At his office, Ramanujan easily and quickly completed the work he was given, so he spent his spare time doing mathematical research. Ramanujan's boss, Sir Francis Spring, and S. Narayana Iyer, a colleague who was also treasurer of the Indian Mathematical Society, encouraged Ramanujan in his mathematical pursuits.
Contacting English mathematicians
In the spring of 1913, Narayana Iyer, Ramachandra Rao and E. W. Middlemast tried to present Ramanujan's work to British mathematicians. One mathematician, M. J. M. Hill of University College London, commented that Ramanujan's papers were riddled with holes.[51] He said that although Ramanujan had "a taste for mathematics, and some ability", he lacked the educational background and foundation needed to be accepted by mathematicians.[52] Although Hill did not offer to take Ramanujan on as a student, he did give thorough and serious professional advice on his work. With the help of friends, Ramanujan drafted letters to leading mathematicians at Cambridge University.[53]
The first two professors, H. F. Baker and E. W. Hobson, returned Ramanujan's papers without comment.[54] On 16 January 1913, Ramanujan wrote to G. H. Hardy. Coming from an unknown mathematician, the nine pages of mathematics made Hardy initially view Ramanujan's manuscripts as a possible "fraud".[55] Hardy recognised some of Ramanujan's formulae but others "seemed scarcely possible to believe".[56] One of the theorems Hardy found scarcely possible to believe was found on the bottom of page three (valid for 0 < a < b + 1/2):
∫0∞1+x2/(b+1)21+x2/(a)2×1+x2/(b+2)21+x2/(a+1)2×⋯dx=π2×Γ(a+12)Γ(b+1)Γ(b−a+12)Γ(a)Γ(b+12)Γ(b−a+1).{\displaystyle \int _{0}^{\infty }{\cfrac {1+{x}^{2}/({b+1})^{2}}{1+{x}^{2}/({a})^{2}}}\times {\cfrac {1+{x}^{2}/({b+2})^{2}}{1+{x}^{2}/({a+1})^{2}}}\times \cdots \;\;dx={\frac {\sqrt {\pi }}{2}}\times {\frac {\Gamma (a+{\frac {1}{2}})\Gamma (b+1)\Gamma (b-a+{\frac {1}{2}})}{\Gamma (a)\Gamma (b+{\frac {1}{2}})\Gamma (b-a+1)}}.}
Hardy was also impressed by some of Ramanujan's other work relating to infinite series:
1−5(12)3+9(1×32×4)3−13(1×3×52×4×6)3+⋯=2π{\displaystyle 1-5\left({\frac {1}{2}}\right)^{3}+9\left({\frac {1\times 3}{2\times 4}}\right)^{3}-13\left({\frac {1\times 3\times 5}{2\times 4\times 6}}\right)^{3}+\cdots ={\frac {2}{\pi }}}
1+9(14)4+17(1×54×8)4+25(1×5×94×8×12)4+⋯=232π12Γ2(34).{\displaystyle 1+9\left({\frac {1}{4}}\right)^{4}+17\left({\frac {1\times 5}{4\times 8}}\right)^{4}+25\left({\frac {1\times 5\times 9}{4\times 8\times 12}}\right)^{4}+\cdots ={\frac {2^{\frac {3}{2}}}{\pi ^{\frac {1}{2}}\Gamma ^{2}\left({\frac {3}{4}}\right)}}.}
The first result had already been determined by a mathematician named Bauer. The second one was new to Hardy, and was derived from a class of functions called a hypergeometric series which had first been researched by Leonhard Euler and Carl Friedrich Gauss. Compared to Ramanujan's work on integrals, Hardy found these results "much more intriguing".[57] After he saw Ramanujan's theorems on continued fractions on the last page of the manuscripts, Hardy commented that "they [theorems] defeated me completely; I had never seen anything in the least like them before".[58] He figured that Ramanujan's theorems "must be true, because, if they were not true, no one would have the imagination to invent them".[58] Hardy asked a colleague, J. E. Littlewood, to take a look at the papers. Littlewood was amazed by the mathematical genius of Ramanujan. After discussing the papers with Littlewood, Hardy concluded that the letters were "certainly the most remarkable I have received" and commented that Ramanujan was "a mathematician of the highest quality, a man of altogether exceptional originality and power".[59] One colleague, E. H. Neville, later commented that "not one [theorem] could have been set in the most advanced mathematical examination in the world".[60]
On 8 February 1913, Hardy wrote a letter to Ramanujan, expressing his interest for his work. Hardy also added that it was "essential that I should see proofs of some of your assertions".[61] Before his letter arrived in Madras during the third week of February, Hardy contacted the Indian Office to plan for Ramanujan's trip to Cambridge. Secretary Arthur Davies of the Advisory Committee for Indian Students met with Ramanujan to discuss the overseas trip.[62] In accordance with his Brahmin upbringing, Ramanujan refused to leave his country to "go to a foreign land".[63] Meanwhile, Ramanujan sent a letter packed with theorems to Hardy, writing, "I have found a friend in you who views my labour sympathetically."[64]
To supplement Hardy's endorsement, a former mathematical lecturer at Trinity College, Cambridge, Gilbert Walker, looked at Ramanujan's work and expressed amazement, urging him to spend time at Cambridge.[65] As a result of Walker's endorsement, B. Hanumantha Rao, a mathematics professor at an engineering college, invited Ramanujan's colleague Narayana Iyer to a meeting of the Board of Studies in Mathematics to discuss "what we can do for S. Ramanujan".[66] The board agreed to grant Ramanujan a research scholarship of 75 rupees per month for the next two years at the University of Madras.[67] While he was engaged as a research student, Ramanujan continued to submit papers to the Journal of the Indian Mathematical Society. In one instance, Narayana Iyer submitted some theorems of Ramanujan on summation of series to the above mathematical journal adding "The following theorem is due to S. Ramanujan, the mathematics student of Madras University". Later in November, British Professor Edward B. Ross of Madras Christian College, whom Ramanujan had met a few years before, stormed into his class one day with his eyes glowing, asking his students, "Does Ramanujan know Polish?" The reason was that in one paper, Ramanujan had anticipated the work of a Polish mathematician whose paper had just arrived by the day's mail.[68] In his quarterly papers, Ramanujan drew up theorems to make definite integrals more easily solvable. Working off Giuliano Frullani's 1821 integral theorem, Ramanujan formulated generalisations that could be made to evaluate formerly unyielding integrals.[69]
Hardy's correspondence with Ramanujan soured after Ramanujan refused to come to England. Hardy enlisted a colleague lecturing in Madras, E. H. Neville, to mentor and bring Ramanujan to England.[70] Neville asked Ramanujan why he would not go to Cambridge. Ramanujan apparently had now accepted the proposal; as Neville put it, "Ramanujan needed no converting and that his parents' opposition had been withdrawn".[60] Apparently, Ramanujan's mother had a vivid dream in which the family Goddess, the deity of Namagiri, commanded her "to stand no longer between her son and the fulfilment of his life's purpose".[60] Ramanujan then set sail for England, leaving his wife to stay with his parents in India.
Ramanujan (centre) with other scientists at Trinity College
Whewell's Court, Trinity College, Cambridge
Ramanujan boarded the S.S. Nevasa on 17 March 1914, and at 10 o'clock in the morning, the ship departed from Madras.[71] He arrived in London on 14 April, with E. H. Neville waiting for him with a car. Four days later, Neville took him to his house on Chesterton Road in Cambridge. Ramanujan immediately began his work with Littlewood and Hardy. After six weeks, Ramanujan moved out of Neville's house and took up residence on Whewell's Court, just a five-minute walk from Hardy's room.[72] Hardy and Ramanujan began to take a look at Ramanujan's notebooks. Hardy had already received 120 theorems from Ramanujan in the first two letters, but there were many more results and theorems to be found in the notebooks. Hardy saw that some were wrong, others had already been discovered, while the rest were new breakthroughs.[73] Ramanujan left a deep impression on Hardy and Littlewood. Littlewood commented, "I can believe that he's at least a Jacobi",[74] while Hardy said he "can compare him only with [Leonhard] Euler or Jacobi."[75]
Ramanujan spent nearly five years in Cambridge collaborating with Hardy and Littlewood and published a part of his findings there. Hardy and Ramanujan had highly contrasting personalities. Their collaboration was a clash of different cultures, beliefs and working styles. Hardy was an atheist and an apostle of proof and mathematical rigour, whereas Ramanujan was a deeply religious man and relied very strongly on his intuition. While in England, Hardy tried his best to fill the gaps in Ramanujan's education without interrupting his spell of inspiration.
Ramanujan was awarded a Bachelor of Science degree by research (this degree was later renamed PhD) in March 1916 for his work on highly composite numbers, the first part of which was published as a paper in the Proceedings of the London Mathematical Society. The paper was over 50 pages with different properties of such numbers proven. Hardy remarked that this was one of the most unusual papers seen in mathematical research at that time and that Ramanujan showed extraordinary ingenuity in handling it.{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }} On 6 December 1917, he was elected to the London Mathematical Society. He became a Fellow of the Royal Society in 1918, becoming the second Indian to do so, following Ardaseer Cursetjee in 1841, and he was one of the youngest Fellows in the history of the Royal Society. He was elected "for his investigation in Elliptic functions and the Theory of Numbers." On 13 October 1918, he became the first Indian to be elected a Fellow of Trinity College, Cambridge.[76]
Illness and return to India
Plagued by health problems throughout his life, living in a country far away from home, and obsessively involved with his mathematics, Ramanujan's health worsened in England, perhaps exacerbated by stress and by the scarcity of vegetarian food during the First World War. He was diagnosed with tuberculosis and a severe vitamin deficiency and was confined to a sanatorium.
Ramanujan returned to Kumbakonam, Madras Presidency in 1919 and died soon thereafter at the age of 32 in 1920. His widow, S. Janaki Ammal, moved to Mumbai, but returned to Chennai (formerly Madras) in 1950, where she lived until her death at age 94 in 1994.[29]
A 1994 analysis of Ramanujan's medical records and symptoms by Dr. D.A.B. Young concluded that it was much more likely he had hepatic amoebiasis, a parasitic infection of the liver widespread in Madras, where Ramanujan had spent time. He had two episodes of dysentery before he left India. When not properly treated, dysentery can lie dormant for years and lead to hepatic amoebiasis,[77] a difficult disease to diagnose, but once diagnosed readily cured.[77]
Personality and spiritual life
Ramanujan has been described as a person with a somewhat shy and quiet disposition, a dignified man with pleasant manners.[78] He lived a rather spartan life while at Cambridge. Ramanujan's first Indian biographers describe him as rigorously orthodox. Ramanujan credited his acumen to his family goddess, Mahalakshmi of Namakkal. He looked to her for inspiration in his work,[79] and claimed to dream of blood drops that symbolised her male consort, Narasimha, after which he would receive visions of scrolls of complex mathematical content unfolding before his eyes.[80] He often said, "An equation for me has no meaning, unless it represents a thought of God."[81][82]
Hardy cites Ramanujan as remarking that all religions seemed equally true to him.[83] Hardy further argued that Ramanujan's religiousness had been romanticised by Westerners and overstated—in reference to his belief, not practice—by Indian biographers. At the same time, he remarked on Ramanujan's strict observance of vegetarianism.
Mathematical achievements
In mathematics, there is a distinction between having an insight and having a proof. Ramanujan's talent suggested a plethora of formulae that could then be investigated in depth later. It is said by G. H. Hardy that Ramanujan's discoveries are unusually rich and that there is often more to them than initially meets the eye. As a by-product, new directions of research were opened up. Examples of the most interesting of these formulae include the intriguing infinite series for π, one of which is given below
1π=229801∑k=0∞(4k)!(1103+26390k)(k!)43964k.{\displaystyle {\frac {1}{\pi }}={\frac {2{\sqrt {2}}}{9801}}\sum _{k=0}^{\infty }{\frac {(4k)!(1103+26390k)}{(k!)^{4}396^{4k}}}.}
This result is based on the negative fundamental discriminant d = −4×58 = −232 with class number h(d) = 2 (note that 5×7×13×58 = 26390 and that 9801=99×99; 396=4×99) and is related to the fact that
eπ58=3964−104.000000177….{\displaystyle e^{\pi {\sqrt {58}}}=396^{4}-104.000000177\dots .}
Compare to Heegner numbers, which have class number 1 and yield similar formulae. Ramanujan's series for π converges extraordinarily rapidly (exponentially) and forms the basis of some of the fastest algorithms currently used to calculate π. Truncating the sum to the first term also gives the approximation 98012/4412{\displaystyle 9801{\sqrt {2}}/4412} for π, which is correct to six decimal places. See also the more general Ramanujan–Sato series.
One of his remarkable capabilities was the rapid solution for problems. He was sharing a room with P. C. Mahalanobis who had a problem, "Imagine that you are on a street with houses marked 1 through n. There is a house in between (x) such that the sum of the house numbers to left of it equals the sum of the house numbers to its right. If n is between 50 and 500, what are n and x?" This is a bivariate problem with multiple solutions. Ramanujan thought about it and gave the answer with a twist: He gave a continued fraction. The unusual part was that it was the solution to the whole class of problems. Mahalanobis was astounded and asked how he did it. "It is simple. The minute I heard the problem, I knew that the answer was a continued fraction. Which continued fraction, I asked myself. Then the answer came to my mind", Ramanujan replied.[84][85]
His intuition also led him to derive some previously unknown identities, such as
[1+2∑n=1∞cos(nθ)cosh(nπ)]−2+[1+2∑n=1∞cosh(nθ)cosh(nπ)]−2=2Γ4(34)π{\displaystyle \left[1+2\sum _{n=1}^{\infty }{\frac {\cos(n\theta )}{\cosh(n\pi )}}\right]^{-2}+\left[1+2\sum _{n=1}^{\infty }{\frac {\cosh(n\theta )}{\cosh(n\pi )}}\right]^{-2}={\frac {2\Gamma ^{4}\left({\frac {3}{4}}\right)}{\pi }}}
for all θ{\displaystyle \theta } , where Γ(z){\displaystyle \Gamma (z)} is the gamma function. Expanding into series of powers and equating coefficients of θ0{\displaystyle \theta ^{0}} , θ4{\displaystyle \theta ^{4}} , and θ8{\displaystyle \theta ^{8}} gives some deep identities for the hyperbolic secant.
In 1918, Hardy and Ramanujan studied the partition function P(n) extensively and gave a non-convergent asymptotic series that permits exact computation of the number of partitions of an integer. Hans Rademacher, in 1937, was able to refine their formula to find an exact convergent series solution to this problem. Ramanujan and Hardy's work in this area gave rise to a powerful new method for finding asymptotic formulae, called the circle method.[86]
He discovered mock theta functions in the last year of his life.[87] For many years these functions were a mystery, but they are now known to be the holomorphic parts of harmonic weak Maass forms.
The Ramanujan conjecture
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Although there are numerous statements that could have borne the name Ramanujan conjecture, there is one statement that was very influential on later work. In particular, the connection of this conjecture with conjectures of André Weil in algebraic geometry opened up new areas of research. That Ramanujan conjecture is an assertion on the size of the tau-function, which has as generating function the discriminant modular form Δ(q), a typical cusp form in the theory of modular forms. It was finally proven in 1973, as a consequence of Pierre Deligne's proof of the Weil conjectures. The reduction step involved is complicated. Deligne won a Fields Medal in 1978 for his work on Weil conjectures.[88]
Ramanujan's notebooks
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While still in Madras, Ramanujan recorded the bulk of his results in four notebooks of loose leaf paper. These results were mostly written up without any derivations. This is probably the origin of the misperception that Ramanujan was unable to prove his results and simply thought up the final result directly. Mathematician Bruce C. Berndt, in his review of these notebooks and Ramanujan's work, says that Ramanujan most certainly was able to make the proofs of most of his results, but chose not to.
This style of working may have been for several reasons. Since paper was very expensive, Ramanujan would do most of his work and perhaps his proofs on slate, and then transfer just the results to paper. Using a slate was common for mathematics students in the Madras Presidency at the time. He was also quite likely to have been influenced by the style of G. S. Carr's book studied in his youth, which stated results without proofs. Finally, it is possible that Ramanujan considered his workings to be for his personal interest alone; and therefore recorded only the results.[89]
The first notebook has 351 pages with 16 somewhat organised chapters and some unorganised material. The second notebook has 256 pages in 21 chapters and 100 unorganised pages, with the third notebook containing 33 unorganised pages. The results in his notebooks inspired numerous papers by later mathematicians trying to prove what he had found. Hardy himself created papers exploring material from Ramanujan's work as did G. N. Watson, B. M. Wilson, and Bruce Berndt.[89] A fourth notebook with 87 unorganised pages, the so-called "lost notebook", was rediscovered in 1976 by George Andrews.[77]
Notebooks 1, 2 and 3 were published as a two-volume set in 1957 by the Tata Institute of Fundamental Research (TIFR), Mumbai, India. This was a photocopy edition of the original manuscripts, in his own handwriting.
In December 2011, as part of the celebrations of the 125th anniversary of Ramanujan's birth, TIFR republished the notebooks in a coloured two-volume collector's edition. These were produced from scanned and microfilmed images of the original manuscripts by expert archivists of Roja Muthiah Research Library, Chennai.
Hardy-Ramanujan number 1729
{{#invoke:main|main}} The number 1729 is known as the Hardy–Ramanujan number after a famous anecdote of the British mathematician G. H. Hardy regarding a visit to the hospital to see Ramanujan. In Hardy's words:[90]
" I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. "No", he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways." "
The two different ways are
1729 = 13 + 123 = 93 + 103.
Generalizations of this idea have created the notion of "taxicab numbers". Coincidentally, 1729 is also a Carmichael number.
Other mathematicians' views of Ramanujan
Hardy said : "He combined a power of generalization, a feeling for form, and a capacity for rapid modification of his hypotheses, that were often really startling, and made him, in his own peculiar field, without a rival in his day. The limitations of his knowledge were as startling as its profundity. Here was a man who could work out modular equations and theorems... to orders unheard of, whose mastery of continued fractions was... beyond that of any mathematician in the world, who had found for himself the functional equation of the zeta function and the dominant terms of many of the most famous problems in the analytic theory of numbers; and yet he had never heard of a doubly periodic function or of Cauchy's theorem, and had indeed but the vaguest idea of what a function of a complex variable was...".[91] When asked about the methods employed by Ramanujan to arrive at his solutions, Hardy said that they were "arrived at by a process of mingled argument, intuition, and induction, of which he was entirely unable to give any coherent account."[92] He also stated that he had "never met his equal, and can compare him only with Euler or Jacobi."[92]
Quoting K. Srinivasa Rao,[93] "As for his place in the world of Mathematics, we quote Bruce C. Berndt: 'Paul Erdős has passed on to us Hardy's personal ratings of mathematicians. Suppose that we rate mathematicians on the basis of pure talent on a scale from 0 to 100, Hardy gave himself a score of 25, J.E. Littlewood 30, David Hilbert 80 and Ramanujan 100.'"
Professor Bruce C. Berndt of the University of Illinois, during a lecture at IIT Madras in May 2011, stated that over the last 40 years, as nearly all of Ramanujan's theorems have been proven right, there had been a greater appreciation of Ramanujan's work and brilliance. Further, he stated Ramanujan's work was now pervading many areas of modern mathematics and physics.[87][94]
In his book Scientific Edge, the physicist Jayant Narlikar spoke of "Srinivasa Ramanujan, discovered by the Cambridge mathematician Hardy, whose great mathematical findings were beginning to be appreciated from 1915 to 1919. His achievements were to be fully understood much later, well after his untimely death in 1920. For example, his work on the highly composite numbers (numbers with a large number of factors) started a whole new line of investigations in the theory of such numbers."
During his lifelong mission in educating and propagating mathematics among the school children in India, Nigeria and elsewhere, P.K. Srinivasan has continually introduced Ramanujan's mathematical works.
Bust of Ramanujan in the garden of Template:W.
Ramanujan's home state of Tamil Nadu celebrates 22 December (Ramanujan's birthday) as 'State IT Day', memorialising both the man and his achievements, as a native of Tamil Nadu. A stamp picturing Ramanujan was released by the Government of India in 1962 – the 75th anniversary of Ramanujan's birth – commemorating his achievements in the field of number theory,[95] and a new design was issued on 26 December 2011, by the India Post.[96][97]
Since the Centennial year of Ramanujan, every year 22 Dec, is celebrated as Ramanujan Day by the Government Arts College, Kumbakonam where he had studied and later dropped out. It is celebrated by the Department of Mathematics by organising one-, two-, or three-day seminars by inviting eminent scholars from universities/colleges, and participants are mainly students of mathematics, research scholars, and professors from local colleges. It was planned to celebrate the 125th birthday in a grand manner by inviting the foreign eminent mathematical scholars of this century viz., G E Andrews. and Bruce C Berndt, who are very familiar with the contributions and works of Ramanujan.
Ramanujan's work and life are celebrated on 22 December at the Indian Institute of Technology (IIT), Madras in Chennai. The Department of Mathematics celebrates this day by organising a National Symposium on Mathematical Methods and Applications (NSMMA) for one day by inviting eminent Indian and foreign scholars.
A prize for young mathematicians from developing countries has been created in the name of Ramanujan by the International Centre for Theoretical Physics (ICTP), in co-operation with the International Mathematical Union, which nominate members of the prize committee. The Shanmugha Arts, Science, Technology & Research Academy (SASTRA), based in the state of Tamil Nadu in South India, has instituted the SASTRA Ramanujan Prize of $10,000 to be given annually to a mathematician not exceeding the age of 32 for outstanding contributions in an area of mathematics influenced by Ramanujan. The age limit refers to the years Ramanujan lived, having nevertheless still achieved many accomplishments. This prize has been awarded annually since 2005, at an international conference conducted by SASTRA in Kumbakonam, Ramanujan's hometown, around Ramanujan's birthday, 22 December.
On the 125th anniversary of his birth, India declared the birthday of Ramanujan, 22 December, as 'National Mathematics Day.' The declaration was made by Dr. Manmohan Singh in Chennai on 26 December 2011.[98] Dr Manmohan Singh also declared that the year 2012 would be celebrated as the National Mathematics Year. His residence is now preserved by SASTRA university in Kumbakonam.
In popular culture
Ramanujan, an Indo-British collaboration film, chronicling the life of Ramanujan, is being made by the independent film company Camphor Cinema.[99] The cast and crew include director Gnana Rajasekaran, cinematographer Sunny Joseph and editor B. Lenin.[100][101] Popular Indian and English stars Abhinay Vaddi, Suhasini Maniratnam, Bhama, Kevin McGowan and Michael Lieber star in pivotal roles.[102]
Ramanujan is referenced in the 1997 American film Good Will Hunting.
A film, based on the book The Man Who Knew Infinity: A Life of the Genius Ramanujan by Robert Kanigel, is being made by Edward Pressman and Matthew Brown with R. Madhavan playing Ramanujan.[103]
A play, First Class Man by Alter Ego Productions,[104] was based on David Freeman's First Class Man. The play is centred around Ramanujan and his complex and dysfunctional relationship with Hardy. On 16 October 2011, it was announced that Roger Spottiswoode, best known for his James Bond film Tomorrow Never Dies, is working on the film version, starring actor Siddharth. Like the book and play it is also titled The First Class Man.[105]
A Disappearing Number is a recent British stage production by the company Complicite that explores the relationship between Hardy and Ramanujan.
The novel The Indian Clerk by David Leavitt explores in fiction the events following Ramanujan's letter to Hardy.[106][107]
On 22 March 1988, the PBS Series Nova aired a documentary about Ramanujan, "The Man Who Loved Numbers" (Season 15, Episode 19).[108]
Google honoured him on his 125th birth anniversary by replacing its logo with a doodle on its home page.[109]
The television series Numb3rs has the character Dr. Amita Ramanujan, a professor of applied mathematics, named after Ramanujan[110]
Ramanujan's story is both referenced and echoed in Cyril M. Kornbluth's "Gomez".
List of amateur mathematicians
Ramanujan graph
Ramanujan summation
Ramanujan's constant
Ramanujan's ternary quadratic form
Rank of a partition
2719 (number)
List of Indian mathematicians
↑ C.P. Snow Foreword to "A Mathematician's Apology" by G.H. Hardy
↑ {{#invoke:Citation/CS1|citation |CitationClass=journal }}
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↑ 12.0 12.1 Template:Harvnb
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↑ A to Z of mathematicians by Tucker McElroy 2005 ISBN 0-8160-5338-3-page 221
↑ 20.0 20.1 Collected papers of Srinivasa Ramanujan Srinivasa Ramanujan Aiyangar, Godfrey Harold Hardy, P. Veṅkatesvara Seshu Aiyar 2000 ISBN 0-8218-2076-1 page xii
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↑ {{#invoke:citation/CS1|citation |CitationClass=book }}, p. 23.
↑ Srinivasan (1968), Vol. 1, p99.
↑ Srinivasan (1968), Vol. 1, p129.
↑ Neville (March 1942), p292.
↑ Srinivasan (1968), p176.
↑ Srinivasan (1968), p31.
↑ Letter from M. J. M. Hill to a C. L. T. Griffith (a former student who sent the request to Hill on Ramanujan's behalf), 28 November 1912.
↑ Hardy (June 1920), pp494–495.
↑ 60.0 60.1 60.2 {{#invoke:Citation/CS1|citation |CitationClass=journal }}
↑ Letter, Hardy to Ramanujan, 8 February 1913.
↑ Letter, Ramanujan to Hardy, 22 January 1914.
↑ Letter, Ramanujan to Hardy, 27 February 1913, Cambridge University Library.
↑ Letter, Littlewood to Hardy, early March 1913.
↑ 77.0 77.1 77.2 Template:Cite web
↑ 87.0 87.1 Template:Cite news
↑ Ono (June–July 2006), p649.
↑ 92.0 92.1 Srinivasa Ramanujan. Retrieved 2 December 2010.
↑ http://www.tv.com/people/navi-rawat/
Selected publications by Ramanujan
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This book was originally published in 1927 after Ramanujan's death. It contains the 37 papers published in professional journals by Ramanujan during his lifetime. The third reprint contains additional commentary by Bruce C. Berndt.
These books contain photocopies of the original notebooks as written by Ramanujan.
This book contains photo copies of the pages of the "Lost Notebook".
Problems posed by Ramanujan, Journal of the Indian Mathematical Society.
This was produced from scanned and microfilmed images of the original manuscripts by expert archivists of Roja Muthiah Research Library, Chennai.
Selected publications about Ramanujan and his work
|CitationClass=journal }}
Find more about Srinivasa Ramanujan at Wikipedia's sister projects
Media from Commons
Quotations from Wikiquote
Source texts from Wikisource
Template:Cite news
Feature Film on Mathematics Genius Ramanujan by Dev Benegal and Stephen Fry
BBC radio programme about Ramanujan – episode 5
A biographical song about Ramanujan's life
P.B.S. Nova Series: "The Man Who Loved Numbers" (1988)
Biographical links
Srinivasa Ramanujan at the Mathematics Genealogy Project
|CitationClass=citation }}.
Template:ScienceWorldBiography
Srinivasa Aiyangar Ramanujan
A short biography of Ramanujan
"Our Devoted Site for Great Mathematical Genius"
A Study Group For Mathematics: Srinivasa Ramanujan Iyengar
The Ramanujan Journal – An international journal devoted to Ramanujan
International Math Union Prizes, including a Ramanujan Prize.
Hindu.com: Norwegian and Indian mathematical geniuses, RAMANUJAN – Essays and Surveys, Ramanujan's growing influence, Ramanujan's mentor
Hindu.com: The sponsor of Ramanujan
"Ramanujan's mock theta function puzzle solved"
Ramanujan's papers and notebooks
Sample page from the second notebook
Ramanujan on Fried Eye
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Tamil Nadu scientists
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People from Erode district | CommonCrawl |
\begin{document}
\title{Minimal Port-based Teleportation} \author{Sergii Strelchuk$^1$, Micha{\l} Studzi\'nski$^{2,}\footnote{corresponding author, email: [email protected]}$}
\affiliation{$^1$DAMTP, Centre for Mathematical Sciences, University of Cambridge, Cambridge CB30WA, UK\\
$^2$Institute of Theoretical Physics and Astrophysics, University of Gda\'nsk, National Quantum Information Centre, 80-952 Gda\'nsk, Poland} \begin{abstract} There are two types of port-based teleportation (PBT) protocols: deterministic -- when the state always arrives to the receiver but is imperfectly transmitted and probabilistic -- when the state reaches the receiver intact with high probability. We introduce the minimal set of requirements that define a feasible PBT protocol and construct a simple PBT protocol that satisfies these requirements: it teleports an unknown state of a qubit with success probability $p_{succ}=1-\frac{N+2}{2^{N+1}}$ and fidelity $1-O(\frac{1}{N})$ with the resource state consisting of $N$ maximally entangled states. This protocol is not reducible from either the deterministic or probabilistic PBT protocol. We define the corresponding efficient superdense coding protocols which transmit more classical bits with fewer maximally entangled states. Furthermore, we introduce rigorous methods for comparing and converting between different PBT protocols.
\end{abstract} \maketitle
\section{Introduction}
Quantum teleportation, introduced at the dawn of quantum information processing era, remains one of the pillars of quantum information transfer~\cite{bennett_teleporting_1993}. The concept of sending an unknown state of the quantum system without physically transferring the medium found its use in an impressive range of applications~\cite{pirandola2015advances}. In recent years, Ishizaka and Hiroshima introduced new teleportation protocols with properties that were previously believed to be unattainable. These protocols go under a name of port-based teleportation (PBT)~\cite{ishizaka2008asymptotic} and they possess a counter-intuitive property that appears to be at odds with non-signalling principle of quantum mechanics, namely, that the teleported state requires no correction and is readily available for use after the sender performs a measurement and sends classical communication. This sparked a series of studies that investigate fundamental capabilities of different PBT protocols and their corresponding resource requirements such as the type of measurements and the amount of entanglement needed for the protocol to function~\cite{strelchuk_generalized_2013,stud2020A,Studzinski_2022,mozrzymas2021optimal,MozJPA,christandl2021asymptotic}.
PBT protocols found wide-ranging applications in cryptography and instantaneous non-local computation~\cite{beigi2011simplified}, they were instrumental in establishing a link between interaction complexity and entanglement in non-local computation and holography~\cite{may2022complexity}, they established a fundamental link between quantum communication complexity advantage and a violation of a Bell inequality~\cite{buhrman_quantum_2016}, fundamental limitations for quantum channels discrimination by designing adaptive protocols called PBT stretching~\cite{pirandola2019fundamental} and others~\cite{pereira2021characterising,quintino2021quantum,PhysRevLett.122.170502}.
Finding optimal protocols that work with quantum states of arbitrary dimensions necessitates the use of representation theory. The complexity of mathematical formalism precludes us from developing satisfying physical intuition about PBT protocols -- in stark contrast to the elegance of the first teleportation protocol introduced in 1993~\cite{bennett_teleporting_1993}. Moreover, the latter requires maximally-entangled states to operate most efficiently, whereas there exist flavours of port-based teleportation protocols which achieve the best-in-class performance using entangled states which are very different from maximally entangled states.
The underlying motivation for this work comes from the seminal work of R. Werner~\cite{werner2001all} who considered a set of remarkably distinct problems which can be reduced to performing (the first 1993-style) quantum teleportation. The emergence of distinctly different teleportation protocols raised a number of fascinating questions about the fundamental building blocks of quantum information: what other forms of transferring quantum information from one subsystem to another exist? PBT protocols offer a distinct alternative to the first teleportation scheme. We thus were motivated by a series of questions: Are there many other distinct protocols which result in quantum state transfer akin to the ordinary teleportation and PBT-like protocols? Can one treat the latter as a single class of protocols or, perhaps, there is a number of fundamentally different protocols within PBT? By now it has become clear that one can define numerous of PBT protocols, each different to another in that there was no straightforward way to provide a black-box reduction between them. Our work aims to address the above by providing two conceptual insights. First, we endeavour to derive PBT from `minimal' requirements. In the spirit of seminal research direction where one constructs quantum theories from fewest possible axioms~\cite{hardy2001quantum, masanes2011derivation}, we apply the same reasoning to the construction of teleportation protocols: what is the minimal number of requirements that yield a viable construction of the port-based teleportation? We introduce the first such `minimal' set of requirements that yield viable teleportation protocols together with the associated superdense coding schemes. Formalising the requirements immediately led to a construction of a new `lean’ or so-called minimal protocol. What's noteworthy, is that this protocol provably does not reduce to interpolation between the existing protocols and which has the exponentially improved scaling of probability of success. Second, given two PBT protocols is there a systematic way of interconverting them without having to consider the fine-grained details of the implementation? An affirmative answer to this question would provide a black-box transformation of trade-offs between imperfect but guaranteed and perfect but probabilistic information transfer LOCC protocols, which is interesting in its own right.
In our work, address both of these points: first, we outline the minimal set of assumptions that yield a working port-based teleportation protocol. We introduce a new protocol that cannot be reduced to any known protocols and that satisfies this set of minimal requirements. This protocol is exponentially more efficient than any known probabilistic PBT (pPBT), hence it cannot be obtained from any such protocol. At the same time, it uses special `denoised' measurement operators that provably cannot result in deterministic PBT (dPBT).
Second, we develop methods for comparing port-based teleportation protocols in terms of their resource states which enable one to estimate their entanglement per port and distinctness. We further introduce methods for the conversion between the protocols and determine the conditions when it is possible to turn one type of PBT into another. We conclude by introducing a new family of superdense coding protocols which send more classical information using less entanglement compared to previously known protocol~\cite{ishizaka_remarks_2015}.
In Section~\ref{prelim}, we provide self-contained operational and mathematical preliminaries. This is followed by a minimal set of assumptions that a PBT protocol must satisfy in Section~\ref{minimalrecs} together with a new, simplified PBT protocol. Remarkably, this remarkable simplicity does not come at a cost of reduced performance. To give a flavour of our results, in Table~\ref{table:entFPBT0} we collect known results on the performance of various qubits PBT schemes compared with the minimal PBT. \begin{center} \begin{table}[h!]
\begin{tabular}{c|c|c}
Teleportation protocol & Entanglement fidelity $F$ & Average success probability $p_{succ}$\\ \hline
Non-optimised deterministic PBT & $F=1-\mathcal{O}(1/N) $ & 1\\[0.1cm]
Optimised deterministic PBT & $F=1-\mathcal{O}(1/N^2) $ & 1\\[0.1cm]
\hline
Non-optimised probabilistic PBT & 1 & $p_{succ}=1-\mathcal{O}(1/\sqrt{N}) $ \\[0.1cm]
Optimised probabilistic PBT & 1 & $p_{succ}=1-\mathcal{O}(1/N) $ \\[0.1cm]
\hline
{\bf Minimal PBT} & $F=1-\mathcal{O}(1/N) $ & $p_{succ}=1-\frac{N+2}{2^{N+1}}$
\end{tabular} \caption{Asymptotic behaviour of all known variants of PBT compared with introduced in this manuscript minimal PBT -- the qubit case. The minimal PBT offers exponentially better scaling in $N$ compared to optimised probabilistic PBT for average probability success, even with the non-optimised resource state included in this table. For entanglement fidelity, minimal PBT offers the same scaling with the number of ports $N$. In Section~\ref{intrpolated_prot} we present a more detailed discussion on the efficiency of mPBT protocol.}
\label{table:entFPBT0} \end{table} \end{center}
In Section~\ref{converting} we show how to convert different types of PBT protocols. The non-trivial regime that needs to be rigorously investigated is transforming pPBT into dPBT.
In Section~\ref{sec:fids} we study PBT protocols from a different viewpoint, by investigating the properties of their respective resource states. We introduce partial ordering on the PBT resource states by considering the fidelity between resource states. We find that even in the case of two similarly performing protocols, their underlying resource states are drastically different. This ordering enables us to find the most `frugal' teleportation protocol: the one which achieves the highest performance with substantially lower entanglement requirements.
Lastly, in Section~\ref{superdense} we turn to superdense coding protocols, which are dual to any teleportation protocol. While ordinary superdense coding protocols are well-understood in the context of original teleportation, very little is known about their dual-PBT versions, with only one known example in~\cite{ishizaka_remarks_2015}. We show how to take an arbitrary dPBT protocol with an established lower bound on fidelity and compute the corresponding performance of the superdense coding protocol. In particular, we find that there exist superdense coding protocols that are capable of transmitting the same amount of classical information as in~\cite{ishizaka_remarks_2015}, but using significantly less entanglement. The appendix contains the necessary source code to calculate all the quantities throughout the paper.
\section{Preliminaries}\label{prelim} \subsection{Operational preliminaries of PBT} \label{prelim:A} The aim of any teleportation protocol is to transfer a given unknown quantum state $\psi_C$ from one party (Alice) to another (Bob). This is achieved by first distributing a resource state $\rho_{AB}$, the sender holding subsystem labelled $A$ and the receiver holding $B$. Then Alice performs a joint measurement from a set $\{M_i\}_i$ on $\psi_C\otimes\rho_{AB}$ and records the outcome $i\in {\mathds{N}}_0$. The latter is then communicated to Bob using a classical channel. After Bob receives $i$, he performs one of two actions (depending on the particulars of the protocol): he either discards all the subsystems except for the one that is identified by $i$ or he aborts the protocol (if $i$ encodes the error flag, for example when $i=0$). In the first case, the teleported state of $\psi_C$ is located in the subsystem identified by $i$.
In what follows we present a short summary of the PBT schemes. All the results contained here have been proven elsewhere and for more details, we encourage readers to see the following papers where the formalism of PBT has been developed~\cite{ishizaka2008asymptotic,ishizaka_quantum_2009,Studzinski2017,StuNJP,Studzinski_2022}.
We start with introducing resources used by parties wishing to apply the PBT scheme. Namely, parties share $N$ copies of $d$-dimensional maximally entangled states, each of them called \textit{port}: \be \label{resource}
|\Psi\rangle_{AB}=(O_A \otimes \mathbf{1}_B)|\Psi^+\rangle_{AB}=(O_A \otimes \mathbf{1}_B)|\psi^+\rangle_{A_1B_1}\otimes |\psi^+\rangle_{A_2B_2}\otimes \cdots \otimes |\psi^+\rangle_{A_NB_N} \ ,
\ee where $A=A_1A_2\cdots A_N$, $B=B_1B_2\cdots B_N$, and $O_A$, with normalisation constraint $\tr(O_A^{\dagger}O_A)=d^N$, is a global operation applied by Alice to increase the efficiency of the protocol~\cite{ishizaka_quantum_2009,Studzinski2017,StuNJP}. In the case when $O_A=\mathbf{1}_A$, we deal with $N$ maximally entangled state and call the total state $|\Psi\rangle_{AB}=|\Psi^+\rangle_{AB}$ as a \textit{non-optimised resource state} and the whole protocol is called \textit{non-optimal PBT}. For optimal schemes, the explicit forms of $O_A$ are presented in Appendix~\ref{summary} for the reader's convenience, as well as in cited above literature. Further we refer to the state $|\Psi\rangle_{AB}=(O_A \otimes \mathbf{1}_B)|\Psi^+\rangle_{AB}$ as a \textit{optimised resource state} and the whole protocol is called \textit{optimal PBT}. Counterintuitively, due to the fact that $O_A\neq \mathbf{1}_A$ and $O_A$ is not unitary, the resource state in optimal schemes yields superior performance while not being maximally entangled.
Presently, there are two flavours of the PBT protocols: \begin{itemize}
\item \textit{Deterministic protocol (dPBT):} An unknown quantum state $\psi_{C}$ is always transmitted to the receiver but the transmission is imperfect.
The teleportation channel $\mathcal{N}_{C\rightarrow \B}$ is of the following form: \be \label{ch1} \begin{split}
\mathcal{N}_{C\rightarrow \B}\left(\psi_{C} \right)=\sum_{i=1}^N \tr_{AC}\left[\widetilde{\Pi}_{i}^{AC} \left(\left(O_A\ot \mathbf{1}_{\B} \right)\sigma^{A\B}_i \left(O_A^{\dagger}\ot \mathbf{1}_{\B} \right) \ot \psi_{C}\right)\right], \end{split} \ee
where by $\tr_{AC}$ denotes partial trace over all systems $AC$ but $\B$. The states $\sigma_{A_i\B}$ are called \textit{signal states}: \be \label{eq:signals} \sigma^{A\B}_i:= \frac{1}{d^{N-1}}\mathbf{1}_{\overline{A}_i}\otimes P^+_{A_i\B} \ , \ee where $P^+_{A_i\B}$ is projector on maximally entangled state between systems $A_i$ and $\B$. To assess the quality of the protocol, one can evaluate entanglement fidelity $F(\mathcal{N}_{C\rightarrow \B})$ of teleportation channel $\mathcal{N}_{C\rightarrow \B}$ when teleporting a subsystem $C$ of maximally entangled state $P^+_{CD}$, and computing overlap with the state after perfect transmission $P^+_{\B D}$: \be \label{F_det} F(\mathcal{N}_{C\rightarrow \B})=\tr\left[P^+_{\B D}(\mathcal{N}_{C\rightarrow \B}\otimes \mathbb{1}_D)(P^+_{CD})\right]=\frac{1}{d^2}\sum_{i=1}^N\tr\left[\left(O_A^{\dagger}\otimes \mathbf{1}_{\B}\right)\widetilde{\Pi}_i^{A\B}\left(O_A\otimes \mathbf{1}_{\B}\right)\sigma^{A\B}_i\right], \ee where $\mathbb{1}_D$ denotes identity channel leaving system $D$ untouched. For an arbitrary dimension $d$ the fidelity $F(\mathcal{N}_{C\rightarrow \B})$ has been evaluated explicitly using methods coming from group representation theory~\cite{Studzinski2017,StuNJP,majenz2}. Due to the recent result presented in \cite{leditzky2020optimality}, we know that measurements in the form of \textit{square-root measurements} (SRM) are optimal in both PBT versions (non- and optimal PBT). The optimal measurements in the non-optimal case are: \begin{equation}
\label{eq:measurements}
\forall 1\leq i\leq N \qquad \Pi_i^{AC}=\frac{1}{\sqrt{\rho}}\sigma_{A_iC}\frac{1}{\sqrt{\rho}},\quad \text{where}\quad \rho=\sum_{i=1}^N\sigma_{A_iC}. \end{equation} The operator $\rho^{-1}$ is restricted to the support of $\rho$, so to ensure summation of all POVMs to identity $\mathbf{1}_{AC}$ on the whole space $(\mathbb{C}^d)^{\otimes N+1}$, we add to every $\Pi_i^{AC}$ an excess term $\Delta/N$, where $\Delta=\mathbf{1}_{AC}-\sum_{i=1}^N\Pi_i^{AC}$. However, this extra term does not change the entanglement fidelity $F(\mathcal{N}_{C\rightarrow \B})$ of the channel $\mathcal{N}_{C\rightarrow \B}$, see also~\cite{ishizaka2008asymptotic,ishizaka_quantum_2009,Studzinski2017} for the detailed explanation.
\item \textit{Probabilistic protocol (pPBT):} From~\cite{ishizaka_quantum_2009} we know that in the probabilistic scheme, the process of teleportation sometimes fails, however when it succeeds the state is faithfully teleported to Bob with fidelity $F(\mathcal{N}_{C\rightarrow \B})=1$. In this scheme Alice has access to $N+1$ POVMs $\{M_0^{AC},M_1^{AC},\ldots,M_N^{AC}\}$ with measurement $M_0^{AC}$ corresponding to the failure of teleportation procedure. An additional POVM $M_0^{AC}$ makes the teleportation channel $\mathcal{N}_{C\rightarrow \B}$ trace non-preserving. The efficiency of the protocol is described by the average probability of success $p_{succ}$ of the scheme equals to~\cite{ishizaka_quantum_2009,Studzinski2017}: \begin{equation} \label{psucca} p_{succ}=\frac{1}{d^{N+1}}\sum_{i=1}^N\tr\left[\widetilde{M}_i^{AC}\right], \end{equation} where $\widetilde{M}_i^{AC}=O_A^{\dagger}M_i^{AC}O_A$. In the same manner, we define a probability of failure $p_{fail}$ corresponding to POVM $M_0^{AC}$ averaged over all input states which of course satisfies relation $p_{fail}=1-p_{succ}$. The requirement of perfect transmission $F(\mathcal{N})=1$ constraints form of allowed measurements accessible for Alice, namely can be of the following form~\cite{ishizaka_quantum_2009,Studzinski2017}: \begin{equation} \label{ort} \forall 1\leq i\leq N \qquad \widetilde{M}_i^{AC}=P^+_{A_iC}\otimes \Theta_{\overline{A}_i}. \end{equation} where $\overline{A}_i$ denotes all states $A_1A_2\cdots A_N$ but $A_i$. The optimal form of the operators $\Theta_{\overline{A}_i}$ in both versions of the pPBT protocols (optimal and non-optimal) was computed for qubits and qudits in~\cite{ishizaka_quantum_2009,Studzinski2017}. \end{itemize} In both cases (deterministic and probabilistic) we used $O_A$ to denote the optimising operation. However, when the operators differ we will further write $\widetilde{O}_A$ for dPBT and $O_A$ for pPBT. Such a distinction will be very helpful in Section~\ref{sec:fids} where we calculate overlaps between the resource states. In the case of pPBT protocol we also use the notion of non-optimised and optimised resource state as it was done above for dPBT scheme.
\subsection{Mathematical preliminaries of PBT} \label{math_intro}
We will now review basic representation-theoretic preliminaries. We briefly describe here the irreducible representation formalism for the permutation group $S(n)$ with its algebra $\mathbb{C}[S(n)]$~\cite{FultonSchur,Schur-Weyl} and collect the main results regarding the algebra of partially transposed permutation operators $\mathcal{A}_{n}'(d)$~\cite{MozJPA,Studzinski_2022}.
A permutational representation is a map $V:S(n)\rightarrow \operatorname{Hom}(\mathcal{(\mathbb{C}}^{d})^{\otimes n})$ of the symmetric group $S(n)$, where $n=N+1$, in the space $\mathcal{H\equiv (\mathbb{C}}^{d})^{\otimes n}$ defined as
\be
\label{repV}
\forall \pi \in S(n)\qquad V(\pi ).|e_{i_{1}}\rangle\otimes |e_{i_{2}}\rangle\otimes
\cdots \otimes |e_{i_{n}}\rangle:=|e_{i_{\pi ^{-1}(1)}}\rangle\otimes |e_{i_{\pi
^{-1}(2)}}\rangle\otimes \cdots \otimes |e_{i_{\pi ^{-1}(n)}}\rangle,
\ee where the set $\{|e_{i}\rangle\}_{i=1}^{d}$ is an orthonormal basis of the space $\mathcal{\mathbb{C}}^{d}$, and $d$ stands for the dimension. We drop here the lower index in every $i$, since it labels only position of the basis in tensor product $(\mathbb{C}^{d})^{\otimes n}$.
In other words, the operators $V(\pi)$ just permute basis vectors according to the given permutation $\pi\in S(n)$. The representation $V$ extends to the representation of the group algebra $\mathbb{C}[S(n)]:=\operatorname{span}_{\mathbb{C}}\{V(\sigma ):\sigma \in S(n)\}\subset \operatorname{Hom}(\mathcal{(\mathbb{C}}^{d})^{\otimes n})$.
All irreducible representations (irreps) of the symmetric group $S(n)$ are labeled by so-called \textit{partitions}. A partition $\alpha$ of a natural number $n$, which we denote as $\alpha \vdash n$, is a sequence of positive numbers $\alpha=(\alpha_1,\alpha_2,\ldots,\alpha_r)$, such that $\alpha_1\geq \alpha_2\geq \cdots \geq \alpha_r$ and $\sum_{i=1}^r\alpha_i=n.$ Every partition can be visualised as a \textit{Young frame} which is a collection of boxes arranged in left-justified rows. For illustration please see panel {\bf I} in Figure~\ref{YngBox}. It means that for every fixed number $n$, the number of Young frames determines the number of nonequivalent irreps of $S(n)$ in an abstract decomposition.
\begin{figure}
\caption{Panel {\bf I} depicts five possible Young frames for $n=4$ corresponding to all possible abstract irreducible representations of the group $S(4)$. Its representation space is $(\mathbb{C}^d)^{\otimes 4}$ and the only irreps that appear are those for which the height of corresponding Young frames is no larger than $d$. In particular, when one considers qubits ($d=2$) we have only three admissible frames: $(4),(3,1),(2,2)$. Panel {\bf II} presents all possible Young frames $\mu \vdash 4$ satisfying relation $\mu \in \alpha$ for $\alpha=(2,1)$. Green squares depict boxes that are added to the initial frame $\alpha$. On the right, we present all possible Young frames $\alpha \vdash 3$ satisfying relation $\alpha \in \mu$ for $\mu=(2,1,1)$. Boxes subtracted from the initial Young frame $\mu=(2,1,1)$ are shown in red.}
\label{YngBox}
\end{figure}
If one fixes the representation space to $\mathcal{H\equiv (\mathbb{C}}^{d})^{\otimes n}$ then when we decompose $S(n)$ into irreps we take only Young frames $\alpha$ whose height $h(\alpha)$ is at most $d$. Let us take now $\alpha \vdash (n-1)$ and $\mu \vdash n$. By writing $\mu \in \alpha$ we understand Young frames $\mu$ obtained from $\alpha$ by adding a single box. On the opposite, $\alpha \in \mu$ denotes Young frames $\alpha$ obtained from $\mu$ by removing a single box -- see panel {\bf II} in Figure~\ref{YngBox}.
Recall the celebrated Schur-Weyl duality~\cite{FultonSchur,Schur-Weyl}, which states that the diagonal action of the unitary group $\mathcal{U}(d)$ of invertible complex matrices and of the symmetric group $S(n)$ on $(\mathbb{C}^d)^{\otimes n}$ commute: \be \label{comm1} [V(\sigma),U\otimes \cdots \otimes U]=0, \ee where $\sigma \in S(n)$ and $U\in \mathcal{U}(d)$. In particular, it means that there exists a basis producing the decomposition of $V(\pi)$ and $U^{\otimes n}$ into irreps simultaneously, and decomposition of the tensor product space $(\mathbb{C}^d)^{\otimes n}$ as: \be \label{eq:SW} (\mathbb{C}^d)^{\otimes n}=\bigoplus_{\substack{\alpha \vdash n \\ h(\alpha)\leq d}} \mathcal{U}_{\alpha}\otimes \mathcal{S}_{\alpha}. \ee In the above expression the symmetric group $S(n)$ acts non-trivially on the space $\mathcal{S}_{\alpha}$ and the unitary group $\mathcal{U}(d)$ acts non-trivially on the space $\mathcal{U}_{\alpha}$, labelled by the same partitions $\alpha$.
For a given irrep $\alpha$ of $S(n)$, the space $\mathcal{U}_{\alpha}$ is a multiplicity space of dimension $m_{\alpha}$ (multiplicity of irrep $\alpha$), while the space $\mathcal{S}_{\alpha}$ is a representation space of dimension $d_{\alpha}$ (dimension of irrep $\alpha$). With every subspace $\mathcal{U}_{\alpha}\otimes \mathcal{S}_{\alpha}$ we associate the \textit{Young projector}: \be \label{Yng_proj} P_{\alpha}=\frac{d_{\alpha}}{n!}\sum_{\sigma \in S(n)}\chi^{\alpha}(\sigma^{-1})V(\sigma),\quad \text{with}\quad \tr P_{\alpha}=m_{\alpha}d_{\alpha}, \ee where $\chi^{\alpha}(\sigma^{-1})$ is the irreducible character associated to the irrep indexed by $\alpha$. To denote a matrix representation of an irrep of $\sigma \in S(n)$ indexed by a frame $\alpha$ we will write $\varphi^{\alpha}(\sigma)$.
Equation~\eqref{eq:SW} gives a way to describe irreducible representations -- the minimal non-trivial blocks commuting with a diagonal action of the unitary group $\mathcal{U}(d)$, i.e. action of the form $U^{\otimes n}$, for some natural $n$, and $U\in \mathcal{U}(d)$. Such reduction, in addition to displaying the interior structure of the operators allows one to specialize the analysis on the whole space (which is typically very large and complex) to every small block separately thus significantly reducing the complexity of the problem (this is especially helpful when problems involve semidefinite programming). When working with the PBT we need to introduce different type of symmetries, however, still motivated by the Schur-Weyl duality discussed above. To describe them we first briefly discuss the properties of states and measurements used in all variants of PBT described in Subsection~\ref{prelim:A}.
Recall that a bipartite maximally entangled state is $U\otimes \overline{U}$ invariant~\cite{HorodeckiMPRFidelity}, where the bar denotes complex conjugation of an element $U$ of the unitary group $\mathcal{U}(d)$. It means that all signal states $\sigma_i^{A\B}$ from~\eqref{eq:signals} satisfy the following commutation rule: \begin{equation} \label{sym1} \begin{split} [U^{\otimes N}\otimes \overline{U},\sigma_i^{A\B}]=0,\quad \forall \ U\in \mathcal{U}(d), \end{split} \end{equation} where $\overline{U}$ acts on $\B$, and $U^{\otimes N}$ acts on systems $A=A_1\cdots A_N$. Additionally, from the construction of the signal states $\sigma_i^{A\B}$ it follows that they are covariant with respect to the elements from the group $S(N)$, acting on first $N$ systems: \begin{equation} \begin{split} V(\pi)\sigma_i^{A\B}V^{\dagger}(\pi)=\sigma_{\pi(i)}^{A\B},\quad \forall \ \pi\in S(N). \end{split} \end{equation} In particular, choosing an arbitrary state from the set, say $\sigma_N^{A\B}$ all the others can be generated by acting on it with an element from the coset $S(N)/S(N-1)$, whose elements in the representation $V$ are of the form $V[(i,N-1)]$, for $i=1,\ldots,N-1$. The same kind of covariance with respect to $S(N)$ and $U^{\otimes N}\otimes \overline{U}$ also holds for all measurements $\{\Pi_i^{AC}\}_{i=1}^N$ in PBT. Finally, since the operator $\rho$ from~\eqref{eq:measurements} is a sum over all possible signal states it also exhibits symmetries described in~\eqref{sym1} and in addition, it is invariant with respect to the action of elements from the group $S(N)$.
Satisfying the relation~\eqref{sym1} by operators describing all variants of PBT protocols means that they belong to the \textit{algebra of partially transposed permutation operators}, where the partial transposition is taken with respect to the last $n-$th system~\cite{Moz1,MozJPA,Stu1}: \be
\mathcal{A}_{n}'(d):= \operatorname{span}_{\mathbb{C}}\{V'(\sigma ):\sigma \in S(n)\},
\ee
where for simplicity $'$ denotes the partial transposition under consideration. The algebra $\mathcal{A}_{n}'(d)$ is no longer a group algebra since for example, one has $V'V'=dV'$, while in $\mathbb{C}[S(n)]$ we have $VV=\mathbf{1}$. Elements $V'(\sigma)$ commute with skew-diagonal action of $U^{\otimes (N-1)}\otimes \overline{U}$, where $U\in \mathcal{U}(d)$. This is an analog of the relation from~\eqref{comm1} between the permutation group $S(n)$ and unitary group $\mathcal{U}(d)$. This means we should expect analogous decomposition of the space $(\mathbb{C}^d)^{\otimes n}$ to \eqref{eq:SW}, when studying objects from $\mathcal{A}_{n}'(d)$.
Due to the results contained in~\cite{Studzinski2017} we know that a similar decomposition exists, with the corresponding version of Young projectors from~\eqref{Yng_proj} on irreducible spaces. We denote these projectors as $F_{\mu}(\alpha)$, and they are labelled by two types of Young diagrams $\alpha \vdash (n-2)$ and $\mu \vdash (n-1)$, such that $\mu\in \alpha$. In particular, it was shown in~\cite{Studzinski2017} that the operator $\rho$ from~\eqref{eq:measurements} decomposes in terms of irreducible projectors $F_{\mu }(\alpha )$ as \begin{equation} \label{rho_spectral} \rho=\sum_{\alpha \vdash N-1}\sum_{\mu \in \alpha}\lambda_{\mu}(\alpha)F_{\mu}(\alpha) \end{equation} with non-zero eigenvalues $\lambda_{\mu}(\alpha)$ of the form
\begin{equation} \label{llambda} \lambda_{\mu}(\alpha)=\frac{1}{d^N}\gamma_{\mu}(\alpha)=\frac{N}{d^{N}}\frac{m_{\mu}d_{\alpha}}{m_{\alpha}d_{\mu}}. \end{equation}
The above decomposition allows for easy calculation of the inversion $\rho^{-1}$ necessary for having explicit form of the measurements from~\eqref{eq:measurements}. We also introduce notation $\gamma_{\mu^*}(\alpha)$, meaning that for a given diagram $\alpha \vdash (n-2)$ we choose such $\mu\in\alpha$ for which $\gamma_{\mu}(\alpha)$ from~\eqref{llambda} is maximal, i.e. \begin{equation} \label{lambdamax} \gamma_{\mu^*}(\alpha):=\max_{\mu\in\alpha}\gamma_{\mu}(\alpha). \end{equation}
\section{Port-based teleportation with minimal requirements}\label{minimalrecs}
To introduce the minimal set of requirements that defines a PBT protocol, consider the following sequence of steps outlined in the box below. \begin{tcolorbox} [ title={PBT protocol ${\cal P}$}] {\bf INPUT}
$n\ge 0$, a shared $2n-qubit$ state $\rho_{AB}^{(n)}$ between sender $A$ and receiver $B$; a state $\psi_C$ to be teleported; a set measurements $\{{\cal M}_{A,i}\}_{i=1}^k$, where without loss of generality we have $k=n+1$. The instantiated protocol is denoted as ${\cal P}_n(\{{\cal M}_{A,i}\}_{i=1}^k,\rho_{AB}^{(n)},\psi_C)$.
{\bf ALGORITHM}
\begin{itemize} \item Alice performs a measurement ${\cal M}_{A,i}$ on $\rho_{AB}^{(n)}\otimes \psi_C$, obtaining outcome $i\in [1,\ldots, k]$. \item Alice sends the index $i$ to Bob by classical channel; \end{itemize}
{\bf OUTPUT}
If $i\in [1,\ldots,k]$, then the teleported state is $\rho_{B_{i}}^{(n)}$. Otherwise, return "FAIL".
\end{tcolorbox}\label{PBTalgorithm} With each ${\cal P}_n(\{{\cal M}_{A,i}\}_{i=1}^k,\rho_{AB}^{(n)},\psi_C)$ we associate a two-parameter estimate $Q({\cal P}_n(\{{\cal M}_{A,i}\}_{i=1}^k,\rho_{AB}^{(n)})) =(F(\sigma_A,\rho^{(n)}_{B_{i}}), p)$ that describes the performance of a teleported state. The first parameter characterises the quality of teleported state and the second -- the success of the teleportation. Please notice that the parameter $Q$ does not depend on the input state $\psi_C$. In the general situation, one could take $k\geq n+1$ measurements, but since only first $n$ of them give contribution to $F$ and $p$ the rest can be treated as a one big measurement labelled by index $i=n+1$.
\begin{definition} A PBT protocol $\cal P$ from the above algorithm from the box is {\it convergent} if $Q({\cal P})\to (1,1)$ as $n\to\infty$. \end{definition}
\subsection{Minimal PBT protocol (mPBT)} \label{intrpolated_prot}
In what follows, we introduce the convergent PBT protocol that while structurally similar to both pPBT and dPBT. However, while sharing many of the common building blocks with the d(p)PBT this `minimal' PBT protocol cannot be derived from either of them. The performance of our protocol interpolates between figures of merit -- the fidelity and probability of success -- between deterministic and probabilistic scheme respectively. Our protocol cannot be improved to get perfect fidelity or the probability of success with finite amount of resources (quantified as a number number of shared ports $N$). However, this protocol maintains exponentially faster convergence to unity, compared to the optimal pPBT. To construct the measurement operators for mPBT, we exploit the SRM that are used in dPBT, but crucially we omit an extra error term $(1/N)\Delta$ defined below expression~\eqref{eq:measurements}: \begin{equation} \label{srm} \forall 1\leq i\leq N \quad \Pi_i^{AC}=\rho^{-1/2}\sigma_i^{AC}\rho^{-1/2}\quad \text{where}\quad \rho=\sum_{i=1}^N\sigma_i^{AC}. \end{equation}
It is clear that the POVM elements do not sum up to identity on the whole space $(\mathbb{C}^d)^{\otimes n}$, where $n=N+1$. To fix that and thus recover their proper probabilistic interpretation we must add an additional operator denoted here $M_0^{AC}$. This additional POVM element, in this variant, equals exactly to $\Delta$. While operationally, it corresponds the failure of the teleportation process, we show in Appendix~\ref{summary} that it bears no resemblance to any known $M_0$ (failure) operators which were used for probabilistic teleportation schemes. Note that we drop the constraint from equation~\eqref{ort}, so the state is not teleported faithfully in this probabilistic scheme (see later discussion in this section).
The probability of success and fidelity of our protocol are given below: \begin{theorem} \label{pSRM} The probability of success $p_{succ}$ in mPBT, with the non-optimised resource state, when one uses square-root measurements $\{\Pi_i^{AC}\}_{i=1}^N$ from~\eqref{srm} has the form: \begin{equation} \label{pqudits} p_{succ}=\frac{1}{d^{N+1}}\sum_{\alpha \vdash N-1}\sum_{\mu \in \alpha}d_{\mu}m_{\alpha}, \end{equation}
where $m_{\alpha}, d_{\mu}$ denote multiplicities and dimension of irreps of $S(N-1)$ and $S(N)$ respectively in the Schur-Weyl duality. The success probability for qubits $p_{succ}$ has the form: \begin{equation} \label{pqubits} p_{succ}=1-\frac{N+2}{2^{N+1}}. \end{equation}
\end{theorem}
\begin{proof} To prove expression for the probability of success $p_{succ}$ we use equation~\eqref{psucca} with $\widetilde{O}_A=\mathbf{1}_A$ and explicit form of measurements from~\eqref{srm}: \begin{equation} \label{psa} p_{succ}=\frac{1}{d^{N+1}}\sum_{i=1}^N\tr\left[\Pi_i^{AC}\right]=\frac{1}{d^{N+1}}\sum_{i=1}^N\tr\left[\rho^{-1/2}\sigma_i^{AC}\rho^{-1/2}\right]=\frac{1}{d^{N+1}}\tr(\rho^{-1/2}\rho\rho^{-1/2})=\frac{1}{d^{N+1}}\tr(\mathbf{1}_{\operatorname{supp}(\rho)}), \end{equation} where $\mathbf{1}_{\operatorname{supp}(\rho)}$ is the identity operator on the support of $\rho$ from~\eqref{srm}. From paper~\cite{Studzinski2017} we know that $\mathbf{1}_{\operatorname{supp}(\rho)}$ has decomposition into sum of projectors $F_{\alpha}(\mu)$ on irreducible spaces of the algebra $\mathcal{A}'_n(d)$: \begin{equation} \label{34} \mathbf{1}_{\operatorname{supp}(\rho)}=\sum_{\alpha \vdash N-1}\sum_{\mu \in \alpha}F_{\alpha}(\mu), \quad \text{so}\quad \tr(\mathbf{1}_{\operatorname{supp}(\rho)})=\sum_{\alpha \vdash N-1}\sum_{\mu \in \alpha}d_{\mu}m_{\alpha}, \end{equation} since $\tr F_{\alpha}(\mu) =d_{\mu}m_{\alpha}$. This leads us to the first statement from the theorem. To prove the second part note that in the qubit case all Young diagrams are up to two rows, and they are always of the form $\mu=(N-l,l)$, where $1\leq l\leq \left \lceil \frac{N}{2}-1\right\rceil$. It is possible to derive closed-form expressions for dimensions and multiplicities of the corresponding irreps. From~\cite{fulton_harris}, we know that for any partition of the shape $\mu=(N-l,l)$ the following expression for the corresponding of irrep dimension and multiplicity: \be
d_\mu =\binom{N}{l}-\binom{N}{l-1}=\frac{(N-2l+1)}{N+1}\binom{N+1}{l},\quad m_\mu=N-2l+1.
\label{eq:m_d_alpha} \ee However, in expression~\eqref{34} we have to consider two types of irreps, irreps $\alpha \vdash N-1$ and irreps $\mu \vdash N$, for which we have the relation $\mu\in \alpha$. The trace in~\eqref{34} can be written as \begin{equation} \tr(\mathbf{1}_{\operatorname{supp}(\rho)})=\sum_{\mu\vdash N}\sum_{\alpha \in \mu}d_{\mu}m_{\alpha}, \end{equation} where $\alpha \in\mu$ denotes partitions $\alpha \vdash N-1$ obtained from partition $\mu \vdash N$ by removing a single box. We have two types of partitions $\alpha$. The first type is when $N-1=2k$, for $\left \lceil \frac{N}{2}-1\right\rceil$, then the corresponding Young tableaux has two equal rows. In this case, we can add a box only to the first row. In the second type of partitions, when $N-1=2k+1$. In this case, we can add a single box to the first or second row. Using this observation and expressions~\eqref{eq:m_d_alpha}, we can write terms $d_{\mu}m_{\alpha}$ for Young tableaux satisfying relation $\alpha \in\mu$: \begin{enumerate}[I)] \item if $N-1=2k$, then \begin{equation} \tr(\mathbf{1}_{\operatorname{supp}(\rho)})=N^{2}+\sum_{1\leq l<k}(N+1)\binom{ N}{l}\frac{(N-2l)^{2}}{(N+1-l)(l+1)}+\frac{2}{N+1}\binom{N+1}{k}, \end{equation}
\item if $N-1=2k+1,$ then \begin{equation} \tr(\mathbf{1}_{\operatorname{supp}(\rho)})=N^{2}+\sum_{1\leq l<k}(N+1)\binom{ N}{l}\frac{(N-2l)^{2}}{(N+1-l)(l+1)}. \end{equation} \end{enumerate} These two expressions can be combined to get one closed expression of the form \begin{equation} \tr(\mathbf{1}_{\operatorname{supp}(\rho)})=\sum_{l=0}^{\left \lceil \frac{N}{2}-1\right\rceil} (N+1){N \choose l} \frac{(N-2l)^2}{(N+1-l)(l+1)}. \end{equation} Finally, to show expression~\eqref{pqubits} we use the following chain of equalities: \begin{equation} \begin{split} p_{succ}&=\frac{1}{2^{N+1}}\sum_{l=0}^{\left \lceil \frac{N}{2}-1\right\rceil} (N+1){N \choose l} \frac{(N-2l)^2}{(N+1-l)(l+1)}=\frac{N+1}{2^{N+2}}\sum_{l=0}^{N}\frac{(N-2l)^2}{(l+1)(N-l+1)}\\ &=\frac{1}{2^{N+2}}\frac{1}{N+2}\sum_{l=0}^N (N-2l)^2{N+2\choose l+1}. \end{split} \end{equation} Now, changing range of the sum and subtracting proper terms, we arrive to \begin{equation}\label{explicitformula} \begin{split} p_{succ}&=\frac{1}{2^{N+2}}\frac{1}{N+2}\left[\sum_{l=0}^{N+2}\left({N+2\choose l}(N-2(l-1))^2\right)-2(N+2)^2\right]\\ &=\frac{1}{2^{N+2}}\frac{1}{N+2}\sum_{l=0}^{N+2}\left[\left({N+2\choose l}((N+2)^2-4(N+2)l+4l^2)\right)-2(N+2)^2\right]\\ &=\frac{1}{2^{N+2}}\frac{1}{N+2}\left[\sum_{l=0}^{N+2}\left({N+2\choose l}(N+2)^2-4(N+2)l{N+2\choose l}+4l^2{N+2\choose l}\right)-2(N+2)^2\right]\\ &=\frac{1}{2^{N+2}}\frac{1}{N+2}\left[2^{N+2}(N+2)^2-4(N+2)^22^{N+1}+4(N+2+(N+2)^2)2^N-2(N+2)^2\right]\\ &=1-\frac{N+2}{2^{N+1}}, \end{split} \end{equation} where in the third line we use identities $\sum_{l=0}^N{N\choose l}=2^N$, $\sum_{l=0}^{N}l{N\choose l}=N2^{N-1}$, and $\sum_{l=0}^Nl^2{N\choose l}=N(N+1)2^{N-2}$. This finishes the proof. \end{proof} In the right panel of figure~\ref{mPBT vs pPBT} we present actual values for $p_{succ}$ obtained from Theorem~\ref{pSRM} for qubits and we compare it with optimal values for the pPBT introduced in~\cite{ishizaka_quantum_2009}. Additionally, in Figure~\ref{mPBTd vs pPBTd} we plot efficiency characteristic for higher dimensions ($d=4$) by exploiting group theoretical expression from~\eqref{pqudits} and compare it with optimal values for the pPBT for $d\geq 2$ derived in~\cite{Studzinski2017}. \begin{figure}
\caption{Qubit case. The left panel: The orange dots show entanglement fidelity of the teleported state when Alice uses SRM measurements in the non-optimal dPBT. The blue dots represent entanglement fidelity obtained by mPBT protocol from expression~\eqref{fint2}. In the regime of $N>8$ both protocols give similar fidelities, while for $N\leq 8$ the mPBT protocol outperforms dPBT significantly. In the case of mPBT fidelity behaves non-monotonically, i.e. it decreases to the value 0.902 for $N=4$ and then it starts growing, achieving asymptotically unit value. The right panel: The blue dots depict values of $p_{succ}$ obtained in mPBT scheme performed by SRM measurements from Theorem~\ref{pSRM}. It visibly outperforms the optimal pPBT, which is $p_{succ}=1-\frac{3}{N+3}$ - the orange dots. }
\label{mPBT vs pPBT}
\end{figure}
\begin{figure}
\caption{Behaviour of the fidelities (left panel) and probability of success (right panel) for $d=4$. Description of the colors is the same as it is in Figure~\ref{mPBT vs pPBT}. In this case, the mPBT scheme performed by SRM measurements from Theorem~\ref{pSRM} we outperform the optimal qudit pPBT, for which the probability of success is $p_{succ}=1-\frac{d^2-1}{N+d^2-1}$~\cite{Studzinski2017}. When one considers resulting fidelities, we see that the mPBT scheme is more efficient up to $\sim30$ ports when one compares it with the non-optimal dPBT with SRM measurements.}
\label{mPBTd vs pPBTd}
\end{figure}
The next theorem shows the best achievable performance in this setting: Alice optimises both measurements and the resource state using an operation $\widetilde{O}_A$ is given as: \begin{equation}
\label{expOA1}
\widetilde{O}_A=\sqrt{d^N}\sum_{\mu \vdash N}\frac{v_{\mu}}{\sqrt{d_{\mu}m_{\mu}}}P_{\mu},
\end{equation}
where $v_{\mu}\geq 0$ are entries of an eignevector corresponding to a maximal eigenvalue of the teleportation matrix $M_F$ used for computation of entanglement fidelity in optimal PBT~\cite{StuNJP}. \begin{theorem} The probability of success in mPBT, with the optimised resource state is given by \begin{equation} p_{succ}=\frac{1}{d}\sum_{\alpha \vdash N-1}\sum_{\mu \in \alpha}\frac{m_{\alpha}}{m_{\mu}}v^2_{\mu}, \end{equation} where $m_{\alpha}, m_{\mu}$ denote multiplicities of irreps of $S(N-1)$ and $S(N)$ respectively in the Schur-Weyl duality. \end{theorem}
\begin{proof} To prove expression for the probability of success $p_{succ}$ we use equation~\eqref{psucca} with $\widetilde{O}_A$ from~\eqref{expOA1} and explicit form of measurements presented in~\eqref{srm}: \begin{equation} \begin{split} p_{succ}=\frac{1}{d^{N+1}}\sum_{i=1}^N\tr(\widetilde{O}_A\Pi_i^{AC}\widetilde{O}_A^{\dagger}). \end{split} \end{equation} Since we know that $\sum_i \Pi_i^{AC}=\mathbf{1}_{\operatorname{supp}(\rho)}=\sum_{\alpha}\sum_{\mu\in\alpha}F_{\mu}(\alpha)$ we write: \begin{equation} p_{succ}=\frac{1}{d^{N+1}}\sum_{\alpha \vdash N-1}\sum_{\mu\in\alpha}\tr (\widetilde{O}_A\widetilde{O}_A^{\dagger}F_{\mu}(\alpha))=\frac{1}{d^{N+1}}\sum_{\alpha \vdash N-1}\sum_{\mu\in\alpha}\frac{m_{\alpha}}{m_{\mu}}\tr (\widetilde{O}_A\widetilde{O}_A^{\dagger}P_{\mu}). \end{equation} In the second equality we use Lemma 10 from~\cite{Studzinski2017} stating that $\tr_n F_{\mu}(\alpha)=\frac{m_{\alpha}}{m_{\mu}}P_{\mu}$, where $P_{\mu}$ is a Young projector on irrep labelled by $\mu \vdash N$. Plugging the explicit form of $\widetilde{O}_A$ given in~\eqref{expOA1} into the above, using orthogonality property for Young projectors $P_{\nu}P_{\mu}=\delta_{\nu\mu}P_{\mu}$, and taking into account Observation~\ref{obs} from Appendix~\ref{summary} we find: \begin{equation} p_{succ}=\frac{1}{d}\sum_{\alpha \vdash N-1}\sum_{\mu\in\alpha}\frac{m_{\alpha}v_{\mu}^2}{m_{\mu}^2d_{\mu}}\tr(P_{\mu})=\frac{1}{d}\sum_{\alpha \vdash N-1}\sum_{\mu \in \alpha}\frac{m_{\alpha}}{m_{\mu}}v^2_{\mu}, \end{equation} since $\tr(P_{\mu})=d_{\mu}m_{\mu}$. This finishes the proof. \end{proof} We thus evaluated the probability of success $p_{succ}$ by exploiting two types of measurements used by Alice for qudits. Unlike the qubit case of Theorem~\ref{pSRM}, it does not lend itself to a nice closed form.
We now turn to entanglement fidelity of the above protocols. The entanglement fidelity, where one teleports a half of the maximally entangled state, is given by \begin{equation} \label{Fvsp} F(\mathcal{N}_{C\rightarrow \B})\equiv F=\frac{1}{p_{succ}}\tr\left[P^+_{BD}\left(\mathcal{N}_C\otimes \mathbf{1}_D\right)P^+_{CD}\right]=\frac{1}{d^2p_{succ}}\sum_{i=1}^N\tr(\Pi_i^{AC}\sigma_i^{AC}), \end{equation} where $\sigma_i^{AC}=(1/d^{N-1})(\mathbf{1}_{\overline{A}_i}\otimes P^+_{A_iC})$, and $\overline{A}_i$ denotes all systems but $i-$th. We expect to teleport the state faithfully, so we must have $F(\mathcal{N}_{C\rightarrow \B})=1$. Note that the term $\frac{1}{d^2}\sum_{i=1}^N\tr(\Pi_i^{AC}\sigma_i^{AC})$ is the entanglement fidelity $F_{det}$ of the respective protocol in deterministic scheme, since we have that $\tr(M_0^{AC}\sigma_i^{AC})=0$. This observation allows us to express entanglement fidelity in the minimal PBT scheme (see equation~\eqref{F_det}) as: \begin{equation} \label{fint} F=\frac{F_{det}}{p_{succ}}. \end{equation} Expressions for $F_{det}$ in (non-)optimal PBT are known and presented respectively in Theorem 12 in~\cite{Studzinski2017} and Proposition 32 in~\cite{StuNJP}. For the self-consistence we provide these expressions in Appendix~\ref{summary}, please see expressions~\eqref{F_non} and~\eqref{Fopt} respectively, with corresponding qubit forms in~\eqref{F_nonqubit} and~\eqref{Foptqubit}. In the case of qubits the probability of success in mPBT is given through expression~\eqref{pqubits} from Theorem~\ref{pSRM} and fidelity $F_{det}$ is given through~\eqref{F_nonqubit}, so~\eqref{fint} reads as: \begin{equation} \label{fint2} F=\frac{\frac{1}{2^{N+3}}\sum_{k=0}^N\left(\frac{N-2k-1}{\sqrt{k+1}}+\frac{N-2k+1}{\sqrt{N-k+1}}\right)^2\binom{N}{k}}{1-\frac{N+2}{2^{N+1}}}. \end{equation} In the left panel of figure~\ref{mPBT vs pPBT} we compare the actual fidelity of mPBT from~\eqref{fint} with the fidelity of the teleported state in the non-optimal PBT scheme~\eqref{fint2}, performed by SRM measurements.
Figure~\ref{PBT_landscape} describes the known landscape of admissible protocols and their interrelation when parties exploit maximally entangled resource state. A similar plot can be make for the optimised port-based teleportation schemes, the only thing which would change is the scaling in the number of ports $N$, but the general shape stays the same. \begin{figure}
\caption{Four regions for port-based teleportation protocols; $N$ is the number of ports. Each point defined by a pair of coordinates that corresponds to a protocol with probability of success and fidelity $(F, p)$. {\bf A} (purple): the family of pPBT protocols with $(1,1-O(1/\sqrt{N}))$; {\bf B} (orange):the family of dPBT protocols with $(1-O(1/N),1)$; {\bf C} (black): impossibility region for port-based teleportation protocols with finite resources~\cite{ishizaka2008asymptotic}; {\bf D} (red): new port-based teleportation protocol with minimal assumptions and scaling $(\left[1-O(1/N)\right]/p_{succ}, p_{succ})$, where $p_{succ}=1-(N+2)/2^{N+1}$. The arrows indicates asymptotic convergence of considered protocols.}
\label{PBT_landscape}
\end{figure}
\section{Converting PBT protocols}\label{converting} Different types of port-based teleportation require individual mathematical analysis and in this section we discuss the possibility of converting probabilistic version of PBT into the deterministic one. First, we prove a general theorem giving an explicit relation between entanglement fidelity and probability of success in such conversion. Next, we illustrate the theorem by presenting expressions for the entanglement fidelity of teleported state in the case of non-optimal pPBT and its optimal version, where explicit form of Alice's measurements is known. We also argue that the reverse conversion, i.e. converting dPBT to probabilistic one is possible only under certain constraints.
\begin{theorem} \label{transforming} Every pPBT scheme, with $N$ ports each of dimension $d$, and the probability of success $p_{succ}$, can be turned into deterministic with explicit entanglement fidelity of the form: \begin{equation} \label{eq:transforming} F=p_{succ}+\frac{1}{d^2N}\tr\left(M_0^{AC}\rho\right), \end{equation} where $M_0^{AC}$ is measurements corresponding to the failure of probabilistic scheme, operator $\rho$ is defined through expression~\eqref{srm}. \end{theorem}
\begin{proof} Every pPBT scheme requires a set of $N+1$ measurements $\{M_0^{AC},M_1^{AC},\ldots,M_N^{AC}\}$, where the effect $M_0^{AC}$ corresponds to the failure of the teleportation process. Additionally, to get $F=1$ in probabilistic scheme, we require that all the measurements for $1\leq i \leq N$ satisfy the relation~\eqref{ort}. To design corresponding deterministic scheme, where teleported state appears on one through $N$ ports on the Bob's side we perform the mapping: \begin{equation} \label{POVM_mapping} \Pi_1^{AC} =M_1^{AC}+\frac{1}{N}M_0^{AC},\quad \Pi_1^{AC} =M_2^{AC}+\frac{1}{N}M_0^{AC}, \quad \ldots \quad ,\Pi_N^{AC} = M_N^{AC}+\frac{1}{N}M_0^{AC}. \end{equation} Then the teleportation channel $\mathcal{N}(\psi_C)$ is of the form \begin{equation} \mathcal{N}(\psi_C)=\sum_{i=1}^N\tr_{A\bar{B}_iC}\left[ \sqrt{\Pi_i^{AC}}\left(\Psi_{AB}^+\ot \psi_{C} \right)\sqrt{\Pi_{i}^{AC}}^{\dagger}\right]_{B_i\rightarrow \B}. \end{equation} Now, if $\psi_{C}$ is a half of maximally entangled state $P^+_{CD}$, the entanglement fidelity $F(\mathcal{N})$ of the channel $\mathcal{N}$ is \begin{equation} \label{Fp} \begin{split} F(\mathcal{N})&=\tr\left[P^+_{BD}(\mathcal{N}_{C}\otimes \mathbf{1}_B)(P^+_{CD})\right]=\frac{1}{d^2}\sum_{i=1}^N\tr\left(\Pi_i^{AC}\sigma_i^{AC}\right)=\frac{1}{d^2}\sum_{i=1}^N\tr\left[\left(M_i^{AC}+\frac{1}{N}M_0^{AC}\right)\sigma_i^{AC}\right]\\ &=\frac{1}{d^2}\sum_{i=1}^N\tr\left(M_i^{AC}\sigma_i^{AC}\right)+\frac{1}{Nd^2}\tr\left(M_0^{AC}\rho\right), \end{split} \end{equation} because $\rho=\sum_i \sigma_i^{AC}$. Since we start from POVMs for probabilistic scheme, they must satisfy relation~\eqref{ort}, which implies: \begin{equation} \tr\left(M_i^{AC}\sigma_i^{AC}\right)=\frac{1}{d^{N-1}}\tr\left(M_i^{AC}\right). \end{equation} Plugging the above into~\eqref{Fp} and using~\eqref{psucca} we get the result. \end{proof}
The conversion of a mPBT to dPBT protocol can be also viewed differently. Namely, every probabilistic PBT protocol leads to a deterministic one with the fidelity $F=F_{succ}p_{succ}+F_{fail}(1-p_{succ})$, where $F_{succ}=1$, and $F_{fail}=1/d^2$ is entanglement fidelity when one fails with the transmission process. Similar approach of turning every pPBT into dPBT shortly discussed in the paper~\cite{majenz2}. In~\cite{majenz2} authors consider a type of conversion when whenever Alice fails in probabilistic scheme she sends to Bob a random port index. However, our approach is a little bit different from the one presented above, since in Theorem~\ref{transforming} we do not have anymore possibility of the failure - see construction of the corresponding POVMs in eq.~\eqref{POVM_mapping}. Our construction divides POVM $M_0^{AC}$ into $N$ parts and add them to every POVM $M_i^{AC}$ getting pure deterministic scheme. Summarising, Theorem~\ref{transforming} except the direct formula for the entanglement fidelity of such protocol gives also an algorithm for its construction by the explanation how to construct respective measurements.
From expression~\eqref{eq:transforming} we see that the resulting fidelity depends on two factors -- probability of success in probabilistic scheme and overlap between POVM $M_0^{AC}$ corresponding to failure and the state $\rho$ from~\eqref{eq:measurements}. In the general case it is impossible to say much about the overlap, since the operator $M_0^{AC}$ depends heavily on an architecture of a given probabilistic scheme. However, when we stick with a particular probabilistic scheme we can explicitly evaluate the entanglement fidelity $F$ in Theorem~\ref{transforming}. In what follows we illustrate how Theorem~\ref{transforming} works in practice for known pPBT schemes. We shall consider both, non-optimal and optimal pPBT for an arbitrary number of ports and their arbitrary dimension, which have been analysed in~\cite{Studzinski2017}. Our calculations are based on the latter paper. A short summary of the existing results on pPBT together with some group theoretic methods developed in papers~\cite{Studzinski2017,StuNJP} is provided in Section~\ref{summary}.
\begin{lemma} \label{npPBT} The entanglement fidelity $F$ given in Theorem~\ref{transforming} for the non-optimised resource state reads: \begin{equation} \label{eq:npPBT} F=\frac{1}{d^N}\sum_{\alpha \vdash N-1}\frac{m_{\alpha}d_{\alpha}}{\gamma_{\mu^*}(\alpha)}+\frac{1}{d^2}\left(1-\frac{1}{d^N}\sum_{\alpha \vdash N-1}\sum_{\nu \in \alpha}m_{\nu}d_{\alpha}\frac{\gamma_{\nu}(\alpha)}{\gamma_{\mu^*}(\alpha)}\right), \end{equation} where the numbers $\gamma_{\nu}(\alpha),\gamma_{\mu^*}(\alpha)$ are given by~\eqref{llambda} and~\eqref{lambdamax} respectively. The symbols $\mu,\nu$ denote Young diagrams with $N$ boxes obtained from a Young diagram $\alpha$ of $N-1$ boxes by adding a single box. The quantities $m_{\alpha},m_{\nu}$ are multiplicities of irreps of permutation groups $S(N-1), S(N)$ respectively in the Schur-Weyl duality, and $d_{\alpha}$ is dimension of an irrep of the permutation group $S(N-1)$ in the Schur-Weyl duality. \end{lemma}
\begin{proof} The first term $p_{succ}$ in expression~\eqref{eq:transforming} is known, see Theorem 3 in~\cite{Studzinski2017}, and equals to \begin{equation} p_{succ}=\frac{1}{d^N}\sum_{\alpha \vdash N-1}\frac{m_{\alpha}d_{\alpha}}{\gamma_{\mu^*}(\alpha)}. \end{equation} It remains to evaluate the term $\tr(M_0^{AC}\rho)$. Using the summation rule for POVMs \begin{equation} \label{povm_sum} \sum_{i=0}^N M_i^{AC}=\mathbf{1}_{AC}, \end{equation} where $\mathbf{1}_{AC}$ is an identity operator on $(\mathbb{C}^d)^{\otimes (N+1)}$,
$M_0^{AC}=\mathbf{1}_{AC}-\sum_{i=1}^NM_i^{AC}$. This gives us \begin{equation} \begin{split} \tr(M_0^{AC}\rho)=\tr(\rho)-\sum_{i=1}^N\tr(M_i^{AC}\rho)=N-N\tr(M_N^{AC}\rho). \end{split} \end{equation} The last equality follows from the fact that every operator $\sigma_i^{AC}$ in the sum $\rho=\sum_i\sigma_i^{AC}$ is normalised, and fact that the trace and $\rho$ are invariant under action of elements from the coset $S(N)/S(N-1)$. Using explicit form of POVM $M_N^{AC}$ given in~\eqref{nonoptmeasurements} in the Appendix~\ref{summary}, and decomposition of the state $\rho$ into irreducible projectors of the algebra $\mathcal{A}_{n}'(d)$ given in~\eqref{rho_spectral} we get: \begin{equation} \begin{split} \tr(M_N\rho)&=\sum_{\alpha \vdash N-1}\sum_{\beta \vdash N-1}\sum_{\nu \in \beta}\frac{\lambda_{\nu}(\beta)}{\gamma_{\mu^*}(\alpha)}\tr(P_{\alpha}V'F_{\nu}(\beta))=\sum_{\alpha \vdash N-1}\sum_{\beta \vdash N-1}\sum_{\nu \in \beta}\frac{\lambda_{\nu}(\beta)}{\gamma_{\mu^*}(\alpha)}\tr(P_{\alpha}V'P_{\nu}P_{\beta})\\ &=\sum_{\alpha \vdash N-1}\sum_{\nu \in \alpha}\frac{\lambda_{\nu}(\alpha)}{\gamma_{\mu^*}(\alpha)}\tr(P_{\alpha}P_{\nu}V')=\sum_{\alpha \vdash N-1}\sum_{\nu \in \alpha}\frac{\lambda_{\nu}(\alpha)}{\gamma_{\mu^*}(\alpha)}\tr(P_{\alpha}P_{\nu})=\sum_{\alpha \vdash N-1}\sum_{\nu \in \alpha}m_{\nu}d_{\alpha}\frac{\lambda_{\nu}(\alpha)}{\gamma_{\mu^*}(\alpha)}\\ &=\frac{1}{d^N}\sum_{\alpha \vdash N-1}\sum_{\nu \in \alpha}m_{\nu}d_{\alpha}\frac{\gamma_{\nu}(\alpha)}{\gamma_{\mu^*}(\alpha)}. \end{split} \end{equation} The second equality follows from Theorem 1 and Fact 13 in the paper~\cite{Studzinski2017}. The third equality follows from the orthogonality property for Young projectors, saying that $P_{\beta}P_{\alpha}=\delta_{\beta\alpha}P_{\alpha}$. The fourth equality follows from the observation that only operator $V'$ acts non-trivially on last $n-$th system, so it can be traced out with respect to it, giving us the identity operator acting on first $N$ systems. The fifth equality follows from the fact that only Young projector $P_{\nu}$ acts non-trivially on $N-$th system, so we can apply Corollary 10 from~\cite{stud2020A}, and compute the partial trace. Using orthogonality property for Young projectors an fact that $\tr P_{\alpha}=d_{\alpha}m_{\alpha}$ the result follows. \end{proof}
It is apparent from Lemma the final result is hard to analyse analytically. However, every quantity from~\eqref{eq:npPBT} can be computed numerically. Next, we prove Lemma~\ref{npPBT} for the optimal pPBT, i.e. when Alice optimises simultaneously over measurements and resource state, and show that the final expression for the entanglement fidelity is in very elegant and compact form. \begin{lemma} \label{pPBT} The entanglement fidelity $F$ given through Theorem~\ref{transforming} for the optimised resource state reads: \begin{equation} F=p_{succ}=1-\frac{d^2-1}{N+d^2-1}, \end{equation} where $p_{succ}$ denotes the probability of success in the optimal pPBT. \end{lemma}
\begin{proof} The main idea of the proof follows the proof of Lemma~\ref{npPBT}. However, here in equation~\eqref{povm_sum} we have to consider POVMs from~\eqref{measurement_opt} with the operation $O_A$ given in the same expression. All the POVMs must satisfy the relation $\sum_{i=0}^NM_i^{AC}=\mathbf{1}_{AC}$, where $\mathbf{1}_{AC}$ is the identity operator acting on $(\mathbb{C}^d)^{\otimes (N+1)}$. Thus we write $\tr(M_0^{AC}\rho)$ as in the proof of Lemma~\ref{npPBT}: \begin{equation} \label{cos} \tr(M_0^{AC}\rho)=\tr(\rho)-N\tr(M_N^{AC}\rho). \end{equation} The first term in the above expression is $\tr(\rho)=N$.
We now compute the second term in~\eqref{cos} using explicit form of POVM $M_N^{AC}$ from~\eqref{measurement_opt}: \begin{equation} \label{p2} \begin{split} \tr(M_N^{AC}\rho)&=\frac{1}{d}\sum_{\alpha \vdash N-1}u_{\alpha}\sum_{\beta \vdash N-1}\sum_{\mu \in \alpha}\lambda_{\mu}(\beta)\tr(P_{\alpha}V'F_{\mu}(\beta))=\frac{1}{d}\sum_{\alpha \vdash N-1}u_{\alpha}\sum_{\beta \vdash N-1}\sum_{\mu \in \alpha}\lambda_{\mu}(\beta)\tr(V'P_{\alpha}P_{\beta}P_{\mu})\\ &=\frac{1}{d}\sum_{\alpha \vdash N-1}u_{\alpha}\sum_{\mu \in \alpha}\lambda_{\mu}(\alpha)\tr(P_{\alpha}P_{\mu})=\frac{1}{d}\sum_{\alpha \vdash N-1}u_{\alpha}\sum_{\mu \in \alpha}\lambda_{\mu}(\alpha)m_{\mu}d_{\alpha}=\sum_{\alpha \vdash N-1}\sum_{\mu\in \alpha}\frac{d_{\alpha}}{d_{\mu}}\frac{m_{\mu}^2}{\sum_{\nu}m_{\nu}^2}\\ &=\sum_{\mu \vdash N}\left(\sum_{\alpha\in\mu}d_{\alpha}\right)\frac{1}{d_{\mu}}\frac{m_{\mu}^2}{\sum_{\nu}m_{\nu}^2}=1. \end{split} \end{equation} All the steps except the last equality in~\eqref{p2} follow the proof of Lemma~\ref{npPBT}. First, we use the observation that $\sum_{\alpha\in\mu}d_{\alpha}=d_{\mu}$. Then we apply Proposition 25 from~\cite{Studzinski2017}. Finally, combining expressions $\tr(\rho)=N$ and~\eqref{p2} we conclude that $\tr(M_0^{AC}\rho)=0$. Next, using expression for probability of success $p_{succ}$ in this variant of pPBT, which is $p_{succ}=1-\frac{d^2-1}{N+d^2-1}$ due to Theorem 4 in~\cite{Studzinski2017} we get the result. \end{proof}
The above result implies that we cannot derive a deterministic scheme from probabilistic variant with entanglement fidelity scaling better than $1-\mathcal{O}(1/N)$ in the number of ports $N$. It follows from the fact that the probability of success in optimal pPBT scales as $1-\mathcal{O}(1/N)$, see~\cite{Studzinski2017}. This is due to the fact that in every probabilistic scheme we have to add an additional constraint: to ensure that the teleported state is transmitted faithfully, we have to ensure that all measurements corresponding to the probability of success satisfy~\eqref{ort}. It means that designing, say, the optimal probabilistic scheme one does not optimise over all possible space of POVMs, but over their proper subset. This restriction is one of the reasons responsible for different scaling in (non-)optimal probabilistic and deterministic protocols respectively. This implies that one cannot turn an arbitrary deterministic protocol into probabilistic one, with better scaling than $1-\mathcal{O}(1/N)$, since measurements of such protocol do not satisfy requirement~\eqref{ort}. In particular, we cannot turn optimal dPBT discussed in~\cite{ishizaka_quantum_2009,StuNJP} into probabilistic scheme with $F=1$ for finite $N$. The only way for such conversion to be feasible is when one defaults to `interpolated' protocols, described in Section~\ref{intrpolated_prot}, where neither entanglement fidelity $F$ nor probability of success $p_{succ}$ equal to one for the finite number of ports $N$.
The conversion between PBT protocols is illustrated below on Figure~\ref{sec:fids}.
\begin{figure}
\caption{ Some of the measurement operators for mPBT are fixed to be $ 1\leq i\leq N \quad \Pi_i^{AC}=\rho^{-1/2}\sigma_i^{AC}\rho^{-1/2}\quad \text{where}\quad \rho=\sum_{i=1}^N\sigma_i^{AC}$, with $\sigma_i^{AC}$ defined in Eq.~\eqref{eq:signals}. In general, there may exist other measurements that yield PBT protocols that satisfy requirements of Section~\ref{minimalrecs}. Every mPBT protocol from the non-shaded region can be converted into the protocol in the non-shaded dPBT by adding $\Delta/N$ to each of the POVM elements, where $\Delta=\mathbf{1}-\sum_{i=1}^N\Pi_i$ (arrow 1). Any known pPBT protocol can be converted to a (shaded) subset of known dPBT protocols by taking all the POVM elements $M_1,\ldots M_N$ that correspond to successful teleportation (omitting $M_0$) and transform each of them as follows: $M_i = M_i + M_0/N$. This operation is invertible (arrow 3). Lastly, any mPBT protocol from the shaded area can be converted to shaded subset of dPBT protocols. }
\end{figure} \section{PBT resource state comparison} \label{sec:fids}
Despite the fact that all PBT protocols share operational similarities, their underlying resource states are rather different. In this section we discuss properties of the resource states in all PBT protocols by evaluating their closeness. We show that by starting from maximally entangled states, the the optimisation operator $O_A$ applied by Alice has a large effect on their mutual distances. To quantify the distance, we will use the square-root fidelity $\sqrt{F}$~\cite{Nielsen-Chuang}. See Appendix~\ref{SRfid} for details.
Let us denote the resource states in optimal probabilistic and optimal deterministic scheme by $|\Psi\rangle_{AB},|\widetilde{\Psi}\rangle_{AB}$ respectively. We distinguish optimization operators on Alice's side from Eq.~\eqref{resource} by writing $O_A,\widetilde{O}_A$ respectively. \begin{lemma} \label{Fbetween}
The square fidelity $\sqrt{F}$ between the resource states in optimal pPBT $|\Psi\rangle_{AB}$ and optimal dPBT $|\widetilde{\Psi}\rangle_{AB}$, each with $N$ ports of dimension $d$ equals to: \begin{equation} \label{eq:Fbetween}
\sqrt{F}\left(|\Psi\rangle_{AB},|\widetilde{\Psi}\rangle_{AB}\right)=\sum_{\mu \vdash N}\frac{v_{\mu}m_{\mu}}{\sqrt{\sum_{\nu \vdash N}m_{\nu}^2}}, \end{equation} where $m_{\mu},m_{\nu}$ are multiplicities of irreps of $S(N)$ in the Schur-Weyl duality. The numbers $v_{\mu}\geq 0$ are entries of an eignevector corresponding to a maximal eigenvalue of the teleportation matrix $M_F$. \end{lemma} \begin{proof} The general form of the optimal resource state in pPBT reads: \be \label{resource}
|\Psi\rangle_{AB}=(O_A \otimes \mathbf{1}_B)|\Psi^+\rangle_{AB}=(O_A \otimes \mathbf{1}_B)|\psi^+\rangle_{A_1B_1}\otimes |\psi^+\rangle_{A_2B_2}\otimes \cdots \otimes |\psi^+\rangle_{A_NB_N},
\ee where $O_A$ is taken from~\eqref{measurement_opt}, with normalisation constraint $\tr(O_A^{\dagger}O_A)=d^N$. The same holds for the optimal dPBT, but instead of $O_A$ we use $\widetilde{O}_A$ given in~\eqref{expOA}. The square-root fidelity $\sqrt{F}$ between the states $|\Psi\rangle_{AB},|\widetilde{\Psi}\rangle_{AB}$: \begin{equation} \label{eq:FF} \begin{split}
\sqrt{F}\left(|\Psi\rangle_{AB},|\widetilde{\Psi}\rangle_{AB}\right)=\left|\langle\Psi|\widetilde{\Psi}\rangle_{AB}\right|=\left|\tr\left(|\widetilde{\Psi}\rangle\langle\Psi|_{AB}\right)\right|=\frac{1}{d^N}\tr\left(\widetilde{O}_AO_A\right). \end{split} \end{equation} Plugging the explicit forms of the operators $\widetilde{O}_A,O_A$ from~\eqref{expOA} and \eqref{measurement_opt} respectively we arrive to \begin{equation}
\sqrt{F}\left(|\Psi\rangle_{AB},|\widetilde{\Psi}\rangle_{AB}\right)=\sum_{\mu,\nu \vdash N}\sqrt{\frac{g(N)m_{\mu}}{d_{\mu}}} \frac{v_{\nu}}{\sqrt{d_{\nu}m_{\nu}}} \tr\left(P_{\mu}P_{\nu}\right). \end{equation} Taking into account orthogonality relation for Young projectors $P_{\mu}P_{\nu}=P_{\mu}\delta_{\mu\nu}$ with trace property $\tr P_{\mu}=d_{\mu}m_{\mu}$ and plugging the explicit form of the function $g(N)$ from~\eqref{measurement_opt_coeff} completes the proof. \end{proof}
The numbers $v_{\mu}\geq 0$ in expression~\eqref{eq:Fbetween} are not known to admit closed form, since one does not know analytical expressions for eigenvectors of the teleportation matrix for $d>2$ and $d<N$ (see~\cite{StuNJP}). However, when $d=2$ the matrix $M_F$ is tri-diagonal and we know analytical expressions for its eigenvalues and eigenvectors due to~\cite{Losonczi1992}. In what follows, we will invoke results from~\cite{ishizaka_quantum_2009}, where the form of the operators $\widetilde{O}_A, O_A$ was evaluated in terms of the spin angular momentum of the $N-$spin system. In this representation the operators are proportional to identity on subspaces with fixed quantum number $j$, which runs from $j_{\min}=0(1/2)$ when $N$ is even (odd): \begin{equation} \label{eq:qubitOA} \widetilde{O}_A=\sum_{j=j_{\min}}^{N/2}\sqrt{\gamma(j)}\mathds{1}(j),\qquad O_A=\sum_{j=j_{\min}}^{N/2}\sqrt{\zeta(j)}\mathds{1}(j), \end{equation} for known positive numbers $\gamma(j),\zeta(j)$ which we describe later. The above form of the optimal Alice's operations allows us to formulate the qubit version of Lemma~\ref{Fbetween}: \begin{lemma} \label{Fbetween2}
The square fidelity $\sqrt{F}$ between the resource states $|\Psi\rangle_{AB}$ and $|\widetilde{\Psi}\rangle_{AB}$ in optimal qubit pPBT and optimal dPBT respectively, each with $N$ ports equals to: \begin{equation} \label{eq:Fbetween}
\sqrt{F}\left(|\Psi\rangle_{AB},|\widetilde{\Psi}\rangle_{AB}\right)=\frac{2}{N+2}\sqrt{\frac{6}{(N+1)(N+3)}}\sum_{j=j_{\min}}^{N/2}(2j+1)\operatorname{sin}\left(\frac{\pi (2j+1)}{N+2}\right), \end{equation}
where $j$ runs from $j_{\min}=0(1/2)$ when $N$ is even (odd). The fidelity $\sqrt{F}\left(|\Psi\rangle_{AB},|\widetilde{\Psi}\rangle_{AB}\right)$ converges to $\frac{\sqrt{6}}{\pi}\approx 0.778$ when $N\rightarrow \infty$ and takes maximal value $\sqrt{F}\left(|\Psi\rangle_{AB},|\widetilde{\Psi}\rangle_{AB}\right)=0.894$ which is attained for $N=2$. \end{lemma}
\begin{proof} The proof is similar to that of Lemma~\ref{Fbetween} with the difference we now use qubit versions of the operators $O_A,\widetilde{O}_A$ in spin angular momentum representation given in~\eqref{eq:qubitOA}. The dimension and the multiplicity $d_{\mu}, m_{\mu}$ respectively depend on the quantum number $j$: \begin{equation} \label{djmj} d_j=\frac{(2j+1)N!}{(N/2-j)!(N/2+j+1)!},\qquad m_j=2j+1. \end{equation} This together with the trace rule $\tr \mathds{1}(j)=m_jd_j$ allows us to deduce that the square-root fidelity $\sqrt{F}$ is of the form \begin{equation}
\sqrt{F}\left(|\Psi\rangle_{AB},|\widetilde{\Psi}\rangle_{AB}\right)=\frac{1}{2^N}\sum_{j=j_{\min}}^{N/2}\sqrt{\gamma(j)\zeta(j)}d_jm_j. \end{equation} Plugging the explicit form of the coefficients $\gamma(j),\zeta(j)$ (expression (35) and (54) in~\cite{ishizaka_quantum_2009}) \begin{equation} \label{eq:coeff} \gamma(j)=\frac{2^{N+2}}{(N+2)m_jd_j}\operatorname{sin}^2\left(\frac{\pi m_j}{N+2}\right),\qquad \zeta(j)=\frac{2^Nh(N)m_j}{d_j}, \end{equation} and observing that the value of the sinus is always positive in the allowed range of $j$, and with expression for $h(N)$ (equation above (54) in~\cite{ishizaka_quantum_2009}), which reads as \begin{equation} \label{h} h(N)=\frac{6}{(N+1)(N+2)(N+3)}, \end{equation} we obtain the first part of the statement.
From the above expression one can deduce numerically the convergence of the fidelity to the value $0.778$ for $N\rightarrow \infty$, and maximal value of $F\left(|\Psi\rangle_{AB},|\widetilde{\Psi}\rangle_{AB}\right)=0.894$ attained for two ports. \end{proof}
The situation is entirely different when one considers fidelity between the resource states in non-optimal and optimal PBT, taking probabilistic and deterministic scheme for comparison. In both non-optimal pPBT and dPBT the resource state before the optimization has the same form of $N$ copies of maximally entangled state. We start from the following lemma: \begin{lemma} \label{ovpPBT} The square fidelity $\sqrt{F}$ between the resource states in non-optimal pPBT and its optimal version, each with $N$ ports of dimension $d$ equals to: \begin{equation} \label{71}
\sqrt{F}\left(|\Psi\rangle_{AB},|\Psi^+\rangle_{AB}\right)=\frac{1}{d^N}\sum_{\mu \vdash N}m_{\mu}\sqrt{g(N)d_{\mu}m_{\mu}}, \end{equation} where $d_{\mu}, m_{\mu}$ are dimension and multiplicity of irreps of $S(N)$ in the Schur-Weyl duality, and $g(N)$ is given in ~\eqref{measurement_opt_coeff}. In the qubit case the expression~\eqref{71} has a form \begin{equation} \label{72}
\sqrt{F}\left(|\Psi\rangle_{AB},|\Psi^+\rangle_{AB}\right)=\sqrt{\frac{6}{2^N(N+1)(N+2)(N+3)}}\sum_{j=j_{\min}}^{N/2}\frac{(2j+1)^2}{\sqrt{\left(\frac{N}{2}-j\right)!\left(\frac{N}{2}+j+1\right)!}}, \end{equation} where $j$ runs from $j_{\min}=0(1/2)$ when $N$ is even (odd). The qubit fidelity from~\eqref{72} converges to 0 with $N\rightarrow \infty$. \end{lemma}
\begin{proof} We start from proving our statement for an arbitrary dimension $d$ of the port. Using explicit form of $O_A$ in optimal pPBT given in~\eqref{measurement_opt} w have: \begin{equation} \label{eq:FF} \begin{split}
\sqrt{F}\left(|\Psi\rangle_{AB},|\Psi^+\rangle_{AB}\right)=\frac{1}{d^N}\tr\left(O_A\right)=\frac{1}{\sqrt{d^N}}\sum_{\mu \vdash N}\sqrt{\frac{g(N)m_{\mu}}{d_{\mu}}}\tr(P_{\mu})=\frac{1}{\sqrt{d^N}}\sum_{\mu \vdash N}m_{\mu}\sqrt{g(N)d_{\mu}m_{\mu}}, \end{split} \end{equation} since $\tr(P_{\mu})=d_{\mu}m_{\mu}$. To get expression for qubits we use the second expressions from equations~\eqref{eq:qubitOA} and~\eqref{eq:coeff}, and then the explicit form of the function $h(N)$ in~\eqref{h}, together with equations for $d_j,m_j$ in terms of quantum number $j$ from~\eqref{djmj}. To prove convergence in the qubit case, note that the factor in front of the sum over $j$ clearly goes to 0 with $N\rightarrow \infty$. The second factor can be bounded from the above as follows \begin{equation} \sum_{j=j_{\min}}^{N/2}\frac{(2j+1)^2}{\sqrt{\left(\frac{N}{2}-j\right)!\left(\frac{N}{2}+j+1\right)!}}\leq \frac{\frac{N}{2}(N+1)^2}{\left(\frac{N}{2}\right)!}\xrightarrow{N\rightarrow \infty}0. \end{equation} \end{proof}
For the completeness of our results, we include the corresponding lemma from~\cite{deg} and giving the value of fidelity between resource states in non- and optimal dPBT. \begin{lemma} \label{l:FPBT} The square fidelity between the resource state in non-optimal and optimal dPBT with $N$ ports, each of dimension $d$ is given as: \begin{equation} \label{FPBT}
\sqrt{F}(|\Psi^+\rangle_{AB},|\Psi\rangle_{AB})=\frac{1}{\sqrt{d^N}}\sum_{\mu \vdash N} v_{\mu}\sqrt{d_{\mu}m_{\mu}}, \end{equation} where $v_{\mu}$ are entries of an eigenvector corresponding to a maximal eigenvalue of the teleportation matrix $M_F$, $m_{\mu},d_{\mu}$ denote multiplicity and dimension of irreps of $S(N)$ in the Schur-Weyl duality, and $P_{\mu}$ is a respective Young projector. For qubits, the fidelity between the resources states is of the form: \begin{equation} \label{eq:l:FPBT}
\sqrt{F}(|\Psi^+\rangle_{AB},|\Psi\rangle_{AB})=\sqrt{\frac{N!}{2^{N-2}(N+2)}}\sum_{j=j_{\min}}^{N/2}\frac{(2j+1)\operatorname{sin}\frac{\pi(2j+1)}{N+2}}{\sqrt{(\frac{N}{2}-j)!(\frac{N}{2}+j+1)!}}, \end{equation} where $j_{\min}=0 (1/2)$ when $N$ is even (odd). \end{lemma} The results of the all lemmas from this section for $d=2$ and $d>2$ are presented in table~\ref{tab_res_overlaps} and table~\ref{tab_res_overlaps} respectively located in Appendix~\ref{appD}. In the same Appendix we include figure~\ref{fig:my_label} and figure~\ref{fig:my_label2} which illustrate statements of Lemma~\ref{Fbetween2}, Lemma~\ref{ovpPBT}, and Lemma~\ref{l:FPBT} for qubits.
\section{Sending more bits with fewer ebits: efficient Port-based superdense coding} \label{superdense} We now turn to superdense coding protocols induced by PBT schemes. In the case of ordinary teleportation, the underlying channel is given by an identity channel, and sending a single qubit would result in 2 bits of classical information.
A number of varying PBT protocols gives rise to an equal number of superdense coding schemes.
Suppose that Alice performs a measurement in dPBT, then the unnormalised post-measurement states $\widetilde{\chi}_{DB_i}$, for $1\leq i\leq N$ read: \begin{equation} \label{post_measurement} \begin{split} \widetilde{\chi}_{DB_i}&=\tr_{AC}\left[\Pi_i^{AC}\left(P^+_{CD}\ot P^+_{A_1B_1}\ot P^+_{A_2B_2}\ot \cdots \ot P^+_{A_NB_N}\right)\right]\\ &=\tr_{AC}\left[\Pi_i^{BD}\left(P^+_{CD}\ot P^+_{A_1B_1}\ot P^+_{A_2B_2}\ot \cdots \ot P^+_{A_NB_N}\right)\right]\\ &=\frac{1}{d^{N+1}}\Pi_i^{BD}. \end{split} \end{equation} the second line is obtained by applying Lemma~\ref{techL1} from Appendix~\ref{appA} twice. We introduce normalised post-measurement state $\chi_{DB_i}$ \begin{equation} \chi_{DB_i}:=\frac{\widetilde{\chi}_{DB_i}}{\tr \widetilde{\chi}_{DB_i}}=\frac{N}{d^{N+1}}\Pi_i^{BD}, \end{equation} since $\tr \Pi_i^{BD}=\frac{d^{N+1}}{N}$ due to Theorem 5 in~\cite{deg}.
To estimate the performance of the superdense protocols, we introduce \begin{equation} \label{coefqik}
q_{i|k}:=d^2p_iF_{i|k}, \end{equation}
where terms $F_{i|k}$ represent fidelities between the post-measurement state $\chi_{DB_i}$ and maximally entangled state $P^+_{DB_k}$. These fidelities are of the form \begin{equation}
F_{i|k}=\tr\left[\chi_{DB_i}P^+_{DB_k}\right]=\frac{N}{d^{N+1}}\tr\left[\Pi_i^{BD}P^+_{DB_k}\right]=\frac{N}{d^2}\tr\left[\Pi_i^{BD}\sigma_k^{BD}\right]. \end{equation} In the last equality we used the definition of states $\sigma_k^{BD}=\frac{1}{d^{N-1}}\mathbf{1}_{\overline{B}_i}\ot P^+_{DB_k}$, where $\overline{B}_i$ denotes all systems $B$ but $i$. Consider two cases: $i=k$ and $i\neq k$. In the first case: \begin{equation}
F_{k|k}=\frac{N}{d^{2}}\tr\left[\Pi_k^{BD}\sigma_k^{BD}\right]=F. \end{equation}
The above follows from the fact that $\tr\left[\Pi_k^{BD}\sigma_k^{BD}\right]$ does not depend on index $1\leq k\leq N$ and $F=\frac{1}{d^2}\sum_{i=1}^N\tr\left[\Pi_k^{BD}\sigma_k^{BD}\right]$, which is entanglement fidelity in dPBT. We see that $F_{k|k}\rightarrow 1$ with $N\rightarrow \infty$, since the same holds for entanglement fidelity $F$ in dPBT.
We turn to computing $F_{i|k}$ for $i\neq k$. Form the relation $\sum_i q_{i|k}=1$, where the coefficients are defined through~\eqref{coefqik}, we write \begin{equation} \begin{split}
&q_{k|k}+\sum_{i\neq k}q_{i|k}=1,\\
&d^2p_kF_{k|k}+\sum_{i\neq k}d^2p_iF_{i|k}=1. \end{split} \end{equation}
In PBT schemes $p_i=1/N$, since all the ports are equally probable, and $F_{i|k}=\widetilde{F}$ for all $i\neq k$ due to the covariance property, so \begin{align}
&F+(N-1)\widetilde{F}=\frac{N}{d^2},\\
&\widetilde{F}=\frac{N}{d^2(N-1)}-\frac{F}{N-1}=\frac{1}{d^2\left(1-\frac{1}{N}\right)}-\frac{F}{N-1}. \end{align} We see that $\widetilde{F} \rightarrow \frac{1}{d^2}$ for $N\rightarrow \infty$. An explicit expression for the entanglement fidelity $F$ in PBT can be turned into an explicit expression for $\widetilde{F}$. We now present a protocol that beats the performance of the only known superdense protocol~\cite{ishizaka_remarks_2015} derived from the non-optimized dPBT protocol with fidelity~\cite{beigi_konig}: \begin{equation} \label{b1} F\geq \frac{N}{N+d^2-1}. \end{equation}
The closed form for similar lower bounds is generally hard to compute, but we know that there exist PBT protocols with fidelity scaling as $1-O(1/N^2)$~\cite{ishizaka_quantum_2009,majenz2} (optimal qubit dPBT) or non-optimal ones but better factor. For convenience, Table~\ref{table:entFPBT} lists the known expressions for fidelity derived in~\cite{majenz2}. \begin{center} \begin{table}[h!]
\begin{tabular}{c|c}
Teleportation protocol & Entanglement fidelity $F$\\ \hline
Non-optimised dPBT & $F=1-\frac{d^2-1}{4N}+O\left(N^{-3/2+\delta} \right) $ \\[0.1cm]
Optimised dPBT & $F\geq 1-\frac{d^5+O\left(d^{9/2} \right) }{4\sqrt{2}N^2}+O\left(N^{-3} \right) $\\[0.1cm]
\hline
\end{tabular} \caption{Asymptotic behaviour of dPBT with arbitrary port dimension $d$ and port number $N$. All the results are taken from~\cite{majenz2}.}
\label{table:entFPBT} \end{table} \end{center} We see that the bound~\eqref{b1} is weaker than those in Table~\ref{table:entFPBT}. Using the expression for mutual information from~\cite{ishizaka_remarks_2015} and plugging in the value of fidelity from the non-optimized dPBT from table~\ref{table:entFPBT}, we get: \begin{equation} \label{IABour} I(A:B)=\log_2\left(1-\frac{d^2-1}{4N}\right)+\frac{d^2}{N}\left(1-\frac{d^2-1}{4N}\right)\log_2(d^2). \end{equation} This function outperforms $I(A:B)$ from~\cite{ishizaka_remarks_2015} which exploits the bound from~\eqref{b1}. Indeed, the function from~\eqref{IABour} achieves maximum for \begin{equation} N=\frac{3d^2(d^2-1)\log_e(d^2)+\sqrt{(d-d^3)^2(2d^2+d^2\log_e(d^2)-2)}}{2\left(1-d^2+4d^2\log_e(d^2)\right)}. \end{equation}
Moreover, for any lower bound $F_*$ on fidelity in an arbitrary dPBT the maximum amount of information that the associated superdense coding protocol can transfer is given by: \begin{equation}
I(A:B)=\log_2(F_*)+\frac{d^2F_*}{N}\log_2(d^2). \end{equation}
While we still used $F_*$ from the non-optimized dPBT above, we already get an improvement. We expect to have a dramatic improvement in the amount of communicated information and simultaneously the reduction of entanglement consumption when one uses bound $F_*$ from optimised dPBT protocols, for example by exploiting the second bound from Table~\ref{table:entFPBT}.
\section{Discussion} In this paper, we introduce a novel variant of the PBT protocol called the minimal PBT. This protocol meets the minimal set of requirements that define a feasible PBT scheme. We analyze its efficiency and show that it over-performs optimized pPBT even with the resource state in a form of $N$ pairs of maximally entangled states. In parallel, it offers the same efficiency as the pre-existing PBT schemes when one is interested in the entanglement fidelity of the transmission. In the second part, we investigate the possibility of conversion between different types of PBT, namely, we focus on conversion between probabilistic schemes to deterministic ones, and vice versa. We present the general recipe for such conversion and we show how it applies to existing variants of pPBT and dPBT with their connection to mPBT. We also derive the efficiency of such converted schemes. In the next part of the manuscript, we discuss the application of existing knowledge on the deterministic PBT to super-dense coding schemes, and we show the possibility of transmission of more classical bits with lower consumption of shared maximally entangled pairs (ports). Finally, we present a detailed analysis and comparison of the resource states in deterministic and probabilistic PBT by considering their mutual fidelities. We show that mutual fidelity between resource states decreases with the number of ports showing that the considered states become more distant in the trace norm.
We also leave two important open questions. The first one is to explore derived expressions for the efficiency of the mPBT protocol in the asymptotic limits when the number of ports $N$ tends to infinity, analogously as it was done in~\cite{christandl2021asymptotic}. Applying similar reasoning we can rid off group theoretical parameters like dimensions and multiplicities of irreps under interest and investigate their asymptotic behavior in terms of local dimension and the number of ports. Another important topic is to investigate noise influence on the performance of all known variants of PBT protocols, including defined here the mPBT scheme. It is well known that the noise in the practical implementation of all quantum information protocols is unavoidable and only having a detailed analysis in a real-world scenario regime can tell us about the real potential of discussed in the literature schemes. This is a very important problem, especially in the context of the recent developments in the PBT area -- the first model of the PBT formalism in continuous variables~\cite{pereira2021characterising}. The study of the impact of noise is therefore most natural in this setting, especially from the point of view of possible implementations.
\section{Summary of known results for $d\geq 2$ for pPBT and dPBT} \label{summary} We collect here results concerning explicit form of measurements and Alice's optimising operations $O_A,\widetilde{O}_A$. See~\cite{Studzinski2017,StuNJP} for more detailed discussion. Due to the covariance property of the measuremets it suffices to present a single measurement, for example corresponding to $i=N$. \begin{enumerate}
\item \textit{Non-optimal pPBT.} In this case $O_A=\mathbf{1}_A$ with measurement $M_N^{AC}$ of the form (see Section 2.5.1 in~\cite{Studzinski2017}):
\begin{equation}
\label{nonoptmeasurements}
M_N^{AC}=d\sum_{\alpha \vdash N-1}\frac{1}{\gamma_{\mu^*}(\alpha)}\left(P_{\alpha} \otimes P^+_{N,n}\right),
\end{equation}
where $P_{\alpha}$ is a Young projector introduced in~\eqref{Yng_proj}, $P^+_{N,n}$ denotes the maximally entangled state between respective systems, and
\begin{equation}
\gamma_{\mu^*}(\alpha):=\min_{\mu\in\alpha}\frac{1}{\gamma_{\mu}(\alpha)}=\frac{1}{N}\min_{\mu\in\alpha}\frac{m_{\alpha}d_{\mu}}{m_{\mu}d_{\alpha}}.
\end{equation}
The minimisation is taken over all $\mu$ which can be obtained from given $\alpha$ by adding a single box, as it is described in Section~\ref{math_intro} and Figure~\ref{YngBox} within it. The quantity $\gamma_{\mu}(\alpha)$ can be easily connected with eigenvalues $\lambda_{\mu}(\alpha)$ from~\eqref{llambda} of the operator $\rho$ in~\eqref{rho_spectral}:
\begin{equation}
\lambda_{\mu}(\alpha)=\frac{1}{d^N}\gamma_{\mu}(\alpha).
\end{equation}
The probability of success $p_{succ}$ in this variant is given by the expression (see Theorem 3 in~\cite{Studzinski2017}):
\begin{equation}
\label{eq:p_succ}
p_{succ}=\frac{1}{d^N}\sum_{\alpha \vdash N-k}m_{\alpha}^2\mathop{\operatorname{min}}\limits_{\mu\in\alpha}\frac{d_{\mu}}{m_{\mu}},
\end{equation}
where the minimum is taken over all Young frames $\mu$ which can be obtained from a given Young frame $\alpha \vdash N-1$ by adding a single box (see Figure~\ref{YngBox} for the details).
Quantities $m_{\alpha},m_{\mu}$, and $d_{\mu}$ denote multiplicities and dimension of irreducible representations of corresponding symmetric group in the Schur-Weyl duality.
\item \textit{Optimal pPBT.} In this case Alice's optimising operation is non-trivial, however optimal measurements differ only by coefficients (see Section 2.5.2 in~\cite{Studzinski2017}):
\begin{equation}
\label{measurement_opt}
O_A=\sqrt{d^N}\sum_{\mu \vdash N}\sqrt{\frac{g(N)m_{\mu}}{d_{\mu}}}P_{\mu},\qquad M_N^{AC}=\sum_{\alpha \vdash N-1}u_{\alpha}\left(P_{\alpha}\otimes P^+_{N,n}\right)
\end{equation}
with
\begin{equation}
\label{measurement_opt_coeff}
g(N)=\frac{1}{\sum_{\nu\in N}m^2_{\nu}},\qquad u_{\alpha}=\frac{d^{N+1}}{N}\frac{g(N)m_{\alpha}}{d_{\alpha}}.
\end{equation}
The probability of success $p_{succ}$ in the optimal scheme is given by the a compact expression (see Theorem 4 in~\cite{Studzinski2017}):
\begin{equation} \label{pSuccd}
p_{succ}=1-\frac{d^2-1}{N+d^2-1}.
\end{equation}
This expression for $d=2$ reduces to expression (56) derived in~\cite{ishizaka_quantum_2009}
\begin{equation}
p_{succ}=1-\frac{3}{N+3}.
\end{equation}
\item \textit{Optimal dPBT.} Alice's optimising operation in the optimal variant is of the form (see Proposition 32 in~\cite{StuNJP}):
\begin{equation}
\label{expOA}
\widetilde{O}_A=\sqrt{d^N}\sum_{\mu \vdash N}\frac{v_{\mu}}{\sqrt{d_{\mu}m_{\mu}}}P_{\mu},
\end{equation}
where $v_{\mu}\geq 0$ are entries of an eignevector corresponding to the maximal eigenvalue of the teleportation matrix $M_F$ used for computation of entanglement fidelity in optimal PBT (see Section 4 in~\cite{StuNJP}).
In the case when parties exploit maximally entangled state as a resource and Alice applies SRM measurements to run the protocol, the corresponding entanglement fidelity equals to (see Theorem 12 in~\cite{Studzinski2017}):
\begin{equation} \label{F_non} F_{det}=\frac{1}{d^{N+2}}\sum_{\alpha \vdash N-1}\left(\sum_{\mu \in \alpha}\sqrt{d_{\mu}m_{\mu}} \right)^2, \end{equation} where $d_{\mu},m_{\mu}$ denote dimension and multiplicity of irreps of $S(N)$ in the Schur-Weyl duality. In particular case of qubits we can use expression (29) from~\cite{ishizaka_quantum_2009}, which is of the form: \begin{equation} \label{F_nonqubit} F_{det}=\frac{1}{2^{N+3}}\sum_{k=0}^N\left(\frac{N-2k-1}{\sqrt{k+1}}+\frac{N-2k+1}{\sqrt{N-k+1}}\right)^2\binom{N}{k}. \end{equation}
When Alice optimises over a resource state and measurements the entanglement fidelity can be computed as: \begin{equation} \label{Fopt}
F_{det}=\frac{1}{d^2}||M_F^d||_{\infty}, \end{equation}
where $M_F^d$ is the principal minor of dimension $d$ of the teleportation matrix $M_F$ introduced in Section 4 of~\cite{StuNJP}. the symbol $||\cdot||_{\infty}$ denotes the infinity norm of a matrix. For qubits, the expression~\eqref{Fopt} reduces to (see expression (41) in~\cite{ishizaka_quantum_2009} and Section 5.3 in~\cite{StuNJP}): \begin{equation} \label{Foptqubit} F_{det}=\operatorname{cos}^2\left(\frac{\pi}{N+2}\right). \end{equation} \end{enumerate}
\section{Several technical facts} \label{appA} Here we prove a statement used later in Section~\ref{sec:fids}. \begin{observation} \label{obs} The operations $O_A, \widetilde{O}_A$ in optimal pPBT and dPBT respectively satisfy the chain of equalities \begin{equation} O_A=O_A^{\dagger}=\overline{O}_A=O_A^{T},\qquad \widetilde{O}_A=\widetilde{O}_A^{\dagger}=\overline{\widetilde{O}}_A=\widetilde{O}_A^T. \end{equation} \end{observation} Indeed, the optimising operations are given through linear combination with real coefficients of Young projectors $P_{\mu}$ defined in~\eqref{Yng_proj}, see~\cite{Studzinski2017,StuNJP}.
\begin{lemma}\label{techL1}
Let $\{|i\rangle\}_{i=1}^d$ be a basis and $|\widetilde{\psi}_+\rangle=\sum_{i=1}^d|i\rangle\otimes|i\rangle$ be the unnormalised maximally entangled state, then for any operator $X$ we have \begin{equation}
\left(\mathbf{1}\otimes X\right)|\widetilde{\psi}_+\rangle=\left(X^T\otimes \mathbf{1}\right)|\widetilde{\psi}_+\rangle, \end{equation}
where $X^T$ denotes transposition of $X$ with respect to the basis $\{|i\rangle\}_{i=1}^d$. \end{lemma} This lemma can be proven by direct inspection.
\section{Square root fidelity and fidelity} \label{SRfid} To investigate the closeness of the resource states we use notion of the \textit{square root fidelity} $\sqrt{F}$~\cite{Nielsen-Chuang}, which for two arbitrary quantum states $\rho,\sigma$ is given as \begin{equation} \label{sqrtF} \sqrt{F}(\rho,\sigma):=\tr\left(\sqrt{\sqrt{\rho}\sigma \sqrt{\rho}}\right). \end{equation} The connection of $\sqrt{F}$ with the Uhlmann's fidelity $F$~\cite{Uhlmann:1976,RJozsa} is given by taking a square of the expression~\eqref{sqrtF}: \begin{equation} \label{UhlmannF} F(\rho,\sigma):=(\sqrt{F}(\rho,\sigma))^2=\left[\tr\left(\sqrt{\sqrt{\rho}\sigma \sqrt{\rho}}\right)\right]^2. \end{equation}
When one quantum state is pure, i.e. $\rho_{\psi}:=|\psi\rangle\langle\psi|$ and the second one $\sigma$ is mixed, then one has \begin{equation}
F(\rho_{\psi},\sigma):=(\sqrt{F}(\rho_{\psi},\sigma))^2=|\langle\psi|\sigma|\psi\rangle|^2. \end{equation}
When two quantum states $\rho_{\psi},\sigma_{\phi}$ are pure, i.e. $\rho_{\psi}:=|\psi\rangle\langle\psi|, \sigma_{\phi}:=|\phi\rangle\langle\phi|$, which is exactly the case in this paper, the above expression reduces to \begin{equation}
F(\rho_{\psi},\sigma_{\phi}):=(\sqrt{F}(\rho_{\psi},\sigma_{\phi}))^2=|\langle\psi|\phi\rangle|^2. \end{equation} For two arbitrary density operators $\rho,\sigma$ we can define the trace distance $\delta(\rho,\sigma)$ as \begin{equation}
\delta(\rho,\sigma):=\frac{1}{2}\tr\left(|\rho-\sigma|\right), \end{equation} we can upper and lower bound it using the notion of the fidelity $F$ by Fusch-van de Graaf inequalities: \begin{equation} 1-\sqrt{F(\rho,\sigma)}\leq \delta(\rho,\sigma) \leq \sqrt{1-F(\rho,\sigma)}. \end{equation} In the particular case, when both states are pure, then the above inequalities simplify and give: \begin{equation} \delta(\psi,\phi)=\sqrt{1-F(\psi,\phi)}. \end{equation}
\section{PBT resource state comparison} \label{appD}
\begin{figure}
\caption{Left hand side: Overlap between states for optimal dPBT and optimal pPBT calculated for qubits by using expression~\eqref{eq:Fbetween} from Lemma~\ref{Fbetween2}. We see that for these two states the overlap between them saturates on the value 0.778. Right hand side: Overlap between states for non-optimal and optimal dPBT for qubits by using expression~\eqref{eq:l:FPBT} from Lemma~\ref{l:FPBT}. The maximal value of the overlap which is $F=0.9977$ is attained for $N=6$. In the asymptotic limit the both states are orthogonal. This plot has been firstly obtained in~\cite{deg} in the context of resource state degradation. In both figures we see completely different behaviour of the overlaps between the resource states.}
\label{fig:my_label}
\end{figure}
\begin{figure}
\caption{Overlap between states for non-optimal pPBT and optimal pPBT calculated for qubits by using expression~\eqref{72} from Lemma~\ref{ovpPBT}. We see that for these two states the overlap between them approaches 0 even for a very small number of ports $N$ making them orthonormal. This happens much faster than for the overlap between states for non-optimal and optimal dPBT depicted in Figure~\ref{fig:my_label}.}
\label{fig:my_label2}
\end{figure} For the reader convenience we collate results from the above lemmas in Table~\ref{tab_res_overlaps} for qubits and in Table~\ref{tab_res_overlaps2} for qudits. \begin{table}[h!]
\centering
\begin{tabular}{c|cc}
resource state & $|\Psi^+\rangle_{AB}$ & $(O_A\otimes \mathbf{1}_B)|\Psi^+\rangle_{AB}$ \\ \hline \\[-1em]
$|\Psi^+\rangle_{AB}$ & 1 & \large{$\sqrt{\frac{6}{2^N(N+1)(N+2)(N+3)}}\sum_{j=j_{\min}}^{N/2}\frac{(2j+1)^2}{\sqrt{\left(\frac{N}{2}-j\right)!\left(\frac{N}{2}+j+1\right)!}}$}\\ \hline \\[-1em]
$(\widetilde{O}_A\otimes \mathbf{1}_B)|\Psi^+\rangle_{AB}$ & \large{$\sqrt{\frac{N!}{2^{N-2}(N+2)}}\sum_{j=j_{\min}}^{N/2}\frac{(2j+1)\operatorname{sin}\frac{\pi(2j+1)}{N+2}}{\sqrt{(\frac{N}{2}-j)!(\frac{N}{2}+j+1)!}}$} & \large{$\frac{2}{N+2}\sqrt{\frac{6}{(N+1)(N+3)}}\sum_{j=j_{\min}}^{N/2}(2j+1)\operatorname{sin}\left(\frac{\pi (2j+1)}{N+2}\right)$}\\
\end{tabular}
\caption{Results evaluated in Section~\ref{sec:fids} in the qubit case $d=2$. It contains expressions for the square-root fidelities $\sqrt{F}$ between the resource states in all variants of PBT protocols, for an arbitrary number of ports $N$. All the results are presented in spin angular momentum formalism. Operators $O_A,\widetilde{O}_A$ are optimising Alice's operations in optimal probabilistic and deterministic scheme respectively.}
\label{tab_res_overlaps} \end{table}
\begin{table}[h!]
\centering
\begin{tabular}{c|cc}
resource state & $|\Psi^+\rangle_{AB}$ & $(O_A\otimes \mathbf{1}_B)|\Psi^+\rangle_{AB}$ \\ \hline \\[-1em]
$|\Psi^+\rangle_{AB}$ & 1 & \large{$\frac{1}{d^N}\sum_{\mu \vdash N}m_{\mu}\sqrt{g(N)d_{\mu}m_{\mu}}$}\\ \hline \\[-1em]
$(\widetilde{O}_A\otimes \mathbf{1}_B)|\Psi^+\rangle_{AB}$ & \large{$\frac{1}{\sqrt{d^N}}\sum_{\mu \vdash N} v_{\mu}\sqrt{d_{\mu}m_{\mu}}$} & \large{$\sum_{\mu \vdash N}\frac{v_{\mu}m_{\mu}}{\sqrt{\sum_{\nu \vdash N}m_{\nu}^2}}$}\\
\end{tabular}
\caption{Results evaluated in Section~\ref{sec:fids}. It contains expressions for the square-root fidelities $\sqrt{F}$ between the resource states in all variants of PBT protocols, for an arbitrary number of ports $N$ and port dimension $d$. Operators $O_A,\widetilde{O}_A$ are optimising Alice's operations in optimal probabilistic and deterministic scheme respectively.}
\label{tab_res_overlaps2} \end{table}
\section{Source code} Here we provide the source code that generates all the requisite quantities. This Python code was executed on SageMath 9.0+ (Python 3) or later.
\begin{lstlisting}[language=Python] def hook_formula(mu):
return factorial(add(k for k in mu))/prod(mu.hook_length(i,j) for i,j in mu.cells())
import matplotlib.pyplot as plt import numpy as np from scipy.special import factorial plt.style.use('seaborn-whitegrid')
d=2 #height psucc_vals = [] fid_vals = []
#Ishizaka Hiroshima d=2, MES, popt HI_MES_popt = []
#Ishizaka Hiroshima d=2, optimal state, popt HI_opt_popt = []
#ratio of F of non-optimal dPBT and our p_succ F_p_ratio = [] p_vals = []
for N in range (2,60):
# print("item: ", N)
p = Partitions(N-1,max_length=d) #generates all the shapes of height d
p.list()
#d_mu is the number of standard YTs and mu_alpha is the number of Semi-standard YTs,
#P_succ = sum_alpha sum_mu=(alpha + cell) d_mu*m_alpha
res = 0
for part in p:
ssyt_alpha= SemistandardTableaux(part,max_entry=d) #semistandard tableaux that
#correspond to a given partition
m_alpha = ssyt_alpha.cardinality()
#generate a list of partitions where we add a box to the existing one
part_plusone = list(part.up_list())
#we want only the partitions which are obtained from part by adding 1 box
#that have height <=d
part_plusone = [x for x in part_plusone if len(x)<=d]
for partplus in part_plusone:
syt_mu = StandardTableaux(partplus)
d_mu = syt_mu.cardinality()
res +=d_mu*m_alpha
res = res/d**(N+1)
psucc_vals.append(res)
p_vals.append(1-(d^2-1)/(N+d^2-1))
#p_vals.append(1-(N+2)/(2**(N+1)))
res1 = 0;
for part in p:
part_res = 0;
ssyt_alpha= SemistandardTableaux(part,max_entry=d) #semistandard tableaux that
#correspond to a given partition
#generate a list of partitions where we add a box to the existing one
part_plusone = list(part.up_list())
part_plusone = [x for x in part_plusone if len(x)<=d] #we want only partitions
#which are obtained from part by adding 1 box that have height <=d
for partplus in part_plusone:
syt_mu = StandardTableaux(partplus)
ssyt_mu = SemistandardTableaux(partplus,max_entry=d)
d_mu = syt_mu.cardinality()
m_mu = ssyt_mu.cardinality()
part_res += sqrt(d_mu*m_mu)
res1 += part_res*part_res
res1 = 1/d^(N+2)*res1
fid_vals.append(res1)
#Ishizaka Hiroshima d=2, MES, popt
start =float(0.0)
arr = []
if (N-1)
start=0.5
for s in np.arange (start, ((N-1)/2), 1):
Num = (2*s+1)*(2*s+1)*factorial(N)
Denom1 = (factorial((N-1)/2-s))
Denom2 = (factorial((N+3)/2+s))
arr.append(Num/(Denom1*Denom2))
val = 1/2^(N)*sum(arr)
HI_MES_popt.append(val)
#Ishizaka Hiroshima d=2, optimal state, popt
HI_opt_popt.append(1-3.0/(N+3))
F_p_ratio.append(res1/res)
#line1 = plt.scatter(range(3,60),psucc_vals, label ='Psucc',linestyle='--')
#line2 = plt.scatter(range(3,60),p_vals, label ='Pvals',linestyle='dotted')
#line2, = plt.plot(fid_vals, label ="fidelity_opt",linewidth=2)
#line3, = plt.plot(HI_MES_popt, label='HI MES Psucc', linestyle='dotted')
#line4 = plt.scatter(range(3,20),HI_opt_popt, label='HI OPT Psucc', linestyle='dotted')
line5 = plt.scatter(range(2,60),F_p_ratio) line6 = plt.scatter(range(2,60),fid_vals)
plt.legend((line5, line6),('F_det/p_mPBT', 'F_det'),numpoints=1, loc='lower right')
plt.xlabel("N") plt.ylabel("Fidelity")
#plt.legend((line1, line2),('p_mPBT', 'p_OPT'),numpoints=1, loc='lower right')
\end{lstlisting}
\end{document} | arXiv |
In which cases do multiple hyperbola branches have two intersection points?
I am researching on hyperbolic localization techniques. In these techniques there are usually three anchor nodes $a_1, a_2$ and $a_3$ trying to position a blind node $b$. To do this, hyperbola branches are estimated which pass through the blind node and have the anchor nodes as foci. The position is then estimated as the intersection point of these hyperbola branches.
Image: Two hyperbola branches. Both passing through the blind node, one using $a_1$ and $a_2$ as foci, the other using $a_1$ and $a_3$.
However, hyperbola branches can have two intersection points. I am trying to understand, for which positions of the blind node, relative to the anchor nodes, there are two intersection points.
Image: The same scenario as before, but a different position of the blind node. The hyperbolas intersect in two points.
I have seen this figure in an academic paper, where the areas were colored, which lead to two intersection points, if the blind node falls in one of these areas:
My goal is to create such figures myself. Therefore I have to understand the mathematical relation.
In this figure, for each anchor node, there is one such area, constraint by a hyperbola branch. Appearently this hyperbola branch has as foci: the respective anchor node and the anchor node mirrored on the midpoint between the other two anchor nodes. But I do not know how to determine the semi-axis $a$.
I am happy for any suggestions.
conic-sections
Michael PalmMichael Palm
$\begingroup$ These 1 2 3 posts may be related. $\endgroup$
– EditPiAf
$\begingroup$ this looks like a horrible horrible computation. $\endgroup$
– mercio
Without loosing in generality we can place two anchor nodes ($A_m$ and $A_p$) symmetrically on the $x$ axis , and place the third ($C$) in the upper half-plane as shown in the scheme above.
Let's consider the localization effectuated by node $C$ respectively with nodes $A_m$ and $A_p$, by crossing the red and blue hyperbolich branches at point P.
Denote as $2c_m$ and $2c_p$ the distances from node $C$ to the other nodes, and as $M_m$ and $M_p$ the middle points of the connecting segments.
Clearly the hyperbolas will be centered on such midlle-points, and will have linear eccentricity (distance center to focus) equal to $c_m$, $c_p$.
Let's call $a_m$ and $a_p$ the semiaxes from center to vertices (the measured differences in distance).
We can deduct, from the figure and the properties of hyperbola, that the branches will intersect "properly" iff the respective asymptotes are "interleaved", i.e. if their points at infinite alternates (one "red", one "blue", ..). That can be better figured by noting the "improper" situations below.
Now it is known that the angle that the asymptotes make with the hyperbola axis, the angles $\beta$ in the pictures, are given by $$ \beta = \arctan \left( {\sqrt {\left( {\frac{c}{a}} \right)^{\,2} - 1} } \right) $$ The angles $\alpha$ made by the axis with the horizontal line are determined from the positioning of the nodes.
So, with the notations in the figure, the angles $\gamma$ between the asymptotes and the $x$ axis will be: $$ \begin{array}{l} \gamma _{\,m} = \alpha _{\,m} \pm \beta _{\,m} = \alpha _{\,m} \pm \arctan \left( {\sqrt {\left( {\frac{{c_{\,m} }}{{a_{\,m} }}} \right)^{\,2} - 1} } \right) \\ \gamma _{\,p} = \pi - \alpha _{\,p} \pm \beta _{\,p} = \pi - \alpha _{\,p} \pm \arctan \left( {\sqrt {\left( {\frac{{c_{\,p} }}{{a_{\,p} }}} \right)^{\,2} - 1} } \right) \\ \end{array} $$ Therefore we shall ensure that $$ \gamma _{\,m\, - } < \gamma _{\,p\, - } < \gamma _{\,m\, + } < \gamma _{\,p\, + } $$ i.e. $$ \bbox[lightyellow] { \left\{ \begin{array}{l} 0 < \beta _{\,p} ,\beta _{\,m} < \pi /2 \\ - \alpha _{\,c} < \beta _{\,p} - \beta _{\,m} < \alpha _{\,c} \\ \alpha _{\,c} < \beta _{\,m} + \beta _{\,p} \left( { < \pi } \right) \\ \end{array} \right.\quad \left| {\;\alpha _{\,c} = \pi - \alpha _{\,p} - \alpha _{\,m} } \right. } $$ where $\alpha _{\,c} $ is thus the angle in $C$.
The set of inequalities is rendered graphically as below.
From here it is just a computational task to deduce the bounds for $a_m$ and $a_p$ and from these, which are the differences between the distances from $C$ and from the other nodes, the boundary positions of the detectable point $P$.
G CabG Cab
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Local criteria for blowup in two-dimensional chemotaxis models
Ugo Bessi
Dipartimento di Matematica, Università di Roma Tre, Largo S. Leonardo Murialdo 1,00146 Roma, Italy
Received April 2016 Revised November 2016 Published December 2016
Fund Project: Work partially supported by the PRIN2009 grant "Critical Point Theory and Perturbative Methods for Nonlinear Differential Equations
Let $(S,d)$ be a compact metric space and let $m$ be a Borel probability measure on $(S,d)$. We shall prove that, if $(S,d,m)$ is a $RCD(K,\infty)$ space, then the stochastic value function satisfies the viscous Hamilton-Jacobi equation, exactly as in Fleming's theorem on ${\bf{R}}^d$.
Keywords: Viscous Hamilton-Jacobi equation, metric measure spaces.
Mathematics Subject Classification: Primary:49L20, 49L25.
Citation: Ugo Bessi. The stochastic value function in metric measure spaces. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 1819-1839. doi: 10.3934/dcds.2017076
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Home > Journals > Ann. of Math. (2) > Volume 188 > Issue 1
Annals of Mathematics
VOL. 188 · NO. 1 | July 2018
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Coarse hyperbolicity and closed orbits for quasigeodesic flows
Steven Frankel
Ann. of Math. (2) 188 (1), 1-48, (July 2018) DOI: 10.4007/annals.2018.188.1.1
KEYWORDS: quasigeodesic flows, Pseudo-Anosov flows, Closed orbits, periodic orbits, hyperbolic dynamics, hyperbolic manifolds, 57M60, 57M50, 37C27
We prove a conjecture of Calegari's, that every quasigeodesic flow on a closed hyperbolic 3-manifold contains a closed orbit.
The sharp threshold for making squares
Paul Balister, Béla Bollobás, Robert Morris
Ann. of Math. (2) 188 (1), 49-143, (July 2018) DOI: 10.4007/annals.2018.188.1.2
KEYWORDS: integer factorization, perfect square, random graph process, 11Y05, 60C05
Consider a random sequence of $N$ integers, each chosen uniformly and independently from the set $\{1,\ldots,x\}$. Motivated by applications to factorization algorithms such as Dixon's algorithm, the quadratic sieve, and the number field sieve, Pomerance in 1994 posed the following problem: how large should $N$ be so that, with high probability, this sequence contains a subsequence, the product of whose elements is a perfect square? Pomerance determined asymptotically the logarithm of the threshold for this event and conjectured that it in fact exhibits a sharp threshold in $N$. More recently, Croot, Granville, Pemantle and Tetali determined the threshold up to a factor of $4/\pi + o(1)$ as $x\to \infty$ and made a conjecture regarding the location of the sharp threshold.
In this paper we prove both of these conjectures by determining the sharp threshold for making squares. Our proof combines techniques from combinatorics, probability and analytic number theory; in particular, we use the so-called method of self-correcting martingales in order to control the size of the 2-core of the random hypergraph that encodes the prime factors of our random numbers. Our method also gives a new (and completely different) proof of the upper bound in the main theorem of Croot, Granville, Pemantle and Tetali.
Geometric properties of the Markov and Lagrange spectra
Carlos Moreira
Ann. of Math. (2) 188 (1), 145-170, (July 2018) DOI: 10.4007/annals.2018.188.1.3
KEYWORDS: Markov and Lagrange spectra, continued fractions, fractal dimensions, regular Cantor sets, 11J06, 11J70, 28A78, 37D20
We prove several results on (fractal) geometric properties of the classical Markov and Lagrange spectra. In particular, we prove that the Hausdorff dimensions of intersections of both spectra with half-lines always coincide, and we may assume any real value in the interval $[0, 1]$.
Vertical perimeter versus horizontal perimeter
Assaf Naor, Robert Young
KEYWORDS: Heisenberg group, Isoperimetric inequalities, metric embeddings, Sparsest Cut Problem, approximation algorithms, semidefinite programming, metrics of negative type, 46B85, 20F65, 30L055, 68W25, 43A15
Given $k\in \mathbb{N}$, the $k$'th discrete Heisenberg group, denoted $ \mathbb{H}_{\scriptscriptstyle{\mathbb{Z}}}^{2k+1}$, is the group generated by the elements $a_1,b_1,\ldots,a_k,b_k,c$, subject to the commutator relations $[a_1,b_1]=\cdots=[a_k,b_k]=c$, while all the other pairs of elements from this generating set are required to commute, i.e., for every distinct $i,j\in \{1,\dots,k\}$, we have $[a_i,a_j]=[b_i,b_j]=[a_i,b_j]=[a_i,c]=[b_i,c]=1$. (In particular, this implies that $c$ is in the center of $\mathbb{H}_{ \scriptscriptstyle{\mathbb{Z}}}^{2k+1}$.) Denote $\mathfrak{S}_k=\{a_1,b_1,a_1^{-1},b_1^{-1},\ldots,a_k,b_k,a_k^{-1},b_k^{-1}\}$. The horizontal boundary of $\Omega\subseteq \mathbb{H}_{ \scriptscriptstyle{ \mathbb{Z}}}^{2k+1}$, denoted $\partial_{\mathsf{h}}\Omega$, is the set of all those pairs $(x,y)\in \Omega\times (\mathbb{H}_{\scriptscriptstyle{\mathbb{Z}}}^{2k+1}\setminus \Omega)$ such that $x^{-1}y\in \mathfrak{S}_k$. The horizontal perimeter of $\Omega$ is the cardinality $|\partial_{\mathsf{h}}\Omega|$ of $\partial_{\mathsf{h}}\Omega$; i.e., it is the total number of edges incident to $\Omega$ in the Cayley graph induced by $\mathfrak{S}_k$. For $t\in \mathbb{N}$, define $\partial^t_{\mathsf{v}} \Omega$ to be the set of all those pairs $(x,y)\in \Omega\times (\mathbb{H}_{\scriptscriptstyle{\mathbb{Z}}}^{2k+1}\setminus \Omega)$ such that $x^{-1}y\in \{c^t,c^{-t}\}$. Thus, $|\partial^t_{\mathsf{v}}\Omega|$ is the total number of edges incident to $\Omega$ in the (disconnected) Cayley graph induced by $\{c^t,c^{-t}\}\subseteq \mathbb{H}_{\scriptscriptstyle{\mathbb{Z}}}^{2k+1}$. The vertical perimeter of $\Omega$ is defined by $|\partial_{\mathsf{v}}\Omega|= \sqrt{\sum_{t=1}^\infty |\partial^t_{\mathsf{v}}\Omega|^2/t^2}$. It is shown here that if $k\ge 2$, then $|\partial_{\mathsf{v}}\Omega|\lesssim \frac{1}{k} |\partial_{\mathsf{h}}\Omega|$. The proof of this ``vertical versus horizontal isoperimetric inequality" uses a new structural result that decomposes sets of finite perimeter in the Heisenberg group into pieces that admit an ``intrinsic corona decomposition.'' This allows one to deduce an endpoint $W^{1,1}\to L_2(L_1)$ boundedness of a certain singular integral operator from a corresponding lower-dimensional $W^{1,2}\to L_2(L_2)$ boundedness. Apart from its intrinsic geometric interest, the above (sharp) isoperimetric-type inequality has several (sharp) applications, including that for every $n\in \mathbb{N}$, any embedding into an $L_1(\mu)$ space of a ball of radius $n$ in the word metric on $ \mathbb{H}_{\scriptscriptstyle{\mathbb{Z}}}^{5}$ that is induced by the generating set $\mathfrak{S}_2$ incurs bi-Lipschitz distortion that is at least a universal constant multiple of $\sqrt{\log n}$. As an application to approximation algorithms, it follows that for every $n\in \mathbb{N}$, the integrality gap of the Goemans--Linial semidefinite program for the Sparsest Cut Problem on inputs of size $n$ is at least a universal constant multiple of $\sqrt{\log n}$.M
The algebraic hull of the Kontsevich–Zorich cocycle
Alex Eskin, Simion Filip, Alex Wright
KEYWORDS: Teichmüller geodesic flow, Abelian differential, Kontsevich--Zorich cocycle, algebraic hull, 32G15, 7D40, 14D07, 37C85
We compute the algebraic hull of the Kontsevich--Zorich cocycle over any $\mathrm{GL}^+_2(\mathbb{R})$ invariant subvariety of the Hodge bundle, and derive from this finiteness results on such subvarieties.
On the local Birkhoff conjecture for convex billiards
Vadim Kaloshin, Alfonso Sorrentino
KEYWORDS: Birkhoff billiards, integrable billiards, integrable systems, elliptic function, complex singularities, action-angle coordinates, 37J35, 70H06, 37E40, 33E05, 35A20, 37D50
The classical Birkhoff conjecture claims that the boundary of a strictly convex integrable billiard table is necessarily an ellipse (or a circle as a special case). In this article we prove a complete local version of this conjecture: a small integrable perturbation of an ellipse must be an ellipse. This extends and completes the result in Avila-De Simoi-Kaloshin, where nearly circular domains were considered. One of the crucial ideas in the proof is to extend action-angle coordinates for elliptic billiards into complex domains (with respect to the angle), and to thoroughly analyze the nature of their complex singularities. As an application, we are able to prove some spectral rigidity results for elliptic domains.
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\begin{document}
\newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{proposition}[theorem]{Proposition}
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\title[Stable Polynomials over Finite Fields]
{Stable Polynomials over Finite Fields}
\author[D. G\'omez-P\'erez]{Domingo G\'omez-P\'erez} \address{Department of Mathematics, University of Cantabria, Santander 39005, Spain} \email{[email protected]} \author[A. P. Nicol\'as]{Alejandro P. Nicol\'as} \address{Departamento de Matem‡tica Aplicada, Universidad de Valladolid, Spain} \email{[email protected]} \author[A. Ostafe]{Alina Ostafe} \address{Department of Computing, Macquarie University, Sydney NSW 2109, Australia} \email{[email protected]} \author[D. Sadornil]{Daniel Sadornil} \address{Department of Mathematics, University of Cantabria, Santander 39005, Spain} \email{[email protected]}
\thanks{A. N. was supported by MTM2010-18370-C04-01, A.~O. was supported by SNSF Grant 133399 and D. S. was supported by MTM2010-21580-C02-02 and MTM2010-16051.}
\maketitle
\begin{abstract} We use the theory of resultants to study the stability of an arbitrary polynomial $f$ over a finite field $\F$, that is, the property of having all its iterates irreducible. This result partially generalises the quadratic polynomial case described by R. Jones and N. Boston. Moreover, for $p=3$, we show that certain polynomials of degree three are not stable. We also use the Weil bound for multiplicative character sums to estimate the number of stable arbitrary polynomials over finite fields of odd characteristic. \end{abstract}
\section{Introduction} For a polynomial $\ff$ of degree at least 2 and coefficients in a field $\mathbb{K}$, we define the following sequence: \begin{equation*}
\f{0}{X}=X,\quad \f{n}{X}=\f{n-1}{\ff[X]},\ n\ge 1. \end{equation*} A polynomial $\ff$ is \textit{stable} if $f^{(n)}$ is irreducible over $\mathbb{K}$ for all $n\ge 1$.
In this article, $\mathbb{K}=\F$ is a finite field with $q$ elements, where $q=p^{s}$ and $p$ an odd prime.
Studying the stability of a polynomial is an exciting problem which has attracted a lot of attention. However, only few results are known and the problem is far away from being well understood.
The simplest case, when the polynomial is quadratic, has been studied in several works. For example, some results concerning the stability over $\F$ and $\mathbb{Q}$ can be found in \cite{Ali,Ayad,Danielson,Jon,Jones09}. In particular, by~\cite[Proposition~2.3]{Jones09}, a quadratic monic polynomial $f \in \mathbb{K}[X]$ over a field $\mathbb{K}$ of odd characteristic and with the unique critical point $\gamma$, is stable if the set $$\{-f(\gamma)\} \cup \{f^{(n)}(\gamma)\mid n\ge 2\}$$ contains no squares. In the case when $\mathbb{K}=\mathbb{F}_q$ is a finite field of odd characteristic, this property is also necessary.
In~\cite{GomezNicolas10} an estimate of the number of stable quadratic polynomials over the finite field $\F$ of odd characteristic is given, while in~\cite{ALOS} it is proved that almost all monic quadratic polynomials $f\in\mathbb{Z}[X]$ are stable over $\mathbb{Q}$. Furthermore, in~\cite{ALOS} it is shown that there are no stable quadratic polynomials over finite fields of characteristic two. One might expect that this is the case over any field of characteristic two, which is not true as is also shown in~\cite{ALOS} where an example of a stable quadratic polynomial over a function field of characteristic two is given.
The goal of this paper is to characterize the set of stable polynomials of arbitrary degree and to devise a test for checking the stability of polynomials.
Our techniques come from theory of resultants and they use the relation between irreducibility of polynomials and the properties of the discriminant of polynomials. Using these techniques, we partially generalize previous results known for quadratic polynomials.
A test for stability of quadratic polynomials was given in~\cite{Ostafe09}, where it was shown that checking the stability of such polynomials can be done in time $q^{{3/4}+o(1)}$. As in ~\cite{Jones09}, for an arbitrary polynomial $f$ over $\mathbb{F}_q$, the set defined by $$\{\f{n}{\gamma_1}\ldots\f{n}{\gamma_{k}}\ \mid \ n\geq 1\ \},$$ where $\gamma_i$, $i=1,\ldots,k$, are the roots of the derivative of the polynomial $\ff$, plays also an important role in checking the stability of $f$. In particular, we use techniques based on resultants of polynomials together with the Stickelberger's theorem to prove our results.
We introduce analogues of the orbit sets defined in~\cite{Jones09} for arbitrary degree $d\ge 2$ polynomials. As in~\cite{Ostafe09}, we obtain a nontrivial estimate for the cardinality of these sets for polynomials with irreducible derivative. We also give an estimate for the number of stable arbitrary polynomials which generalises the result obtained in~\cite{GomezNicolas10} for quadratic stable polynomials.
The outline of the paper is the following: in Section~\ref{sec:preliminaries} we introduce the preliminaries necessary to understand the paper. These include basic results about resultants and discriminants of polynomials. This section ends with the Stickelberger's result. Next, Section~\ref{sec:stabilityPolynomials} is devoted to proving a necessary condition for the stability of a polynomial. We define a set, which generalizes the orbit set for
a quadratic polynomial, and then we give an upper bound on the number of elements of this set. Section~\ref{sec:nonExistence} gives a new proof of the result that appeared in~\cite{ALOS} for cubic polynomials when the characteristic is equal to 3. Finally, in Section~\ref{sec:numberStables} we give an estimate of the number of stable polynomials for any degree. For that, we relate the number of stable polynomials with estimates of certain multiplicative character sums.
\section{Preliminaries} \label{sec:preliminaries} Before proceeding with the main results, it is necessary to introduce some concepts related to commutative algebra. Let $\mathbb{K}$ be any field and let $\ff\in\KX$ be a polynomial of degree $d$ with leading coefficient $\principal{\ff}$. The \textit{discriminant} of $\ff$, denoted by $\Disc{\ff}$, is defined by \begin{equation*}
\Disc{\ff}=\principal{\ff}^{2d-2}\prod_{i<j}(\alpha_i-\alpha_j)^2, \end{equation*} where $\alpha_1,\ldots, \alpha_{d}$ are the roots of $\ff$ in some extension of $\mathbb{K}$.
\newline It is widely known that for any polynomial $\ff\in\KX$, its discriminant is an element of the field $\mathbb{K}$. Alternatively, it is possible to compute $\Disc{f}$ using resultants. We can define the resultant of two polynomials $\ff$ and $\gg$ over $\mathbb{K}$ of degrees $d$ and $e$, respectively, with leading coefficients $a_d$ and $b_e$, as \begin{equation*}
\Res{\ff}{\gg}=
\principal{\ff}^{e}\principal{\gg}^{d}\prod (\alpha_i-\beta_j), \end{equation*} where $\alpha_i, \beta_j$ are the roots of $\ff$ and $\gg$, respectively.
Like the discriminant, the resultant belongs to $\mathbb{K}$. In the following lemmas we summarize several known results about resultants without proofs. The interested reader can find them in~\cite{Cox07,LN97}. \begin{lemma}
\label{lem:resultant_eval_root}
Let $\ff,\gg\in\KX$ be two polynomials of degrees $d\ge 1$ and $e\ge 1$
with leading coefficients $a_d$ and $b_e$, respectively. Let
$\beta_1,\ldots,\beta_{\degree{\gg}}$ be the roots of $\gg$ in an extension field of $\mathbb{K}$.
Then,
\begin{equation*}
\Res{\ff}{\gg}=
(-1)^{d\degree{\gg}}
\principal{\gg}^{d}\prod_{i=1}^{\degree{\gg}} \ff[\beta_{i}].
\end{equation*} \end{lemma} The behaviour of the resultant with respect to the multiplication is given by the next result. \begin{lemma} \label{lem:resultant_multiplication}
Let $\mathbb{K}$ be any field. Let $\ff,\gg,\hh\in\KX$ be polynomials of degree greater than 1 and
$a\in\mathbb{K}$.
The following hold:
\begin{align*}
& \Res{\ff\gg}{\hh}=\Res{\ff}{\hh}\Res{\gg}{\hh},&\Res{ a\ff}{\gg}=a^{\degree{\gg}}\Res{\ff}{\gg},
\end{align*}
where $\deg g=e$. \end{lemma} The relation between $\Disc{\ff}$ and $\Res{\ff}{\ff'}$ is given by the next statement. \begin{lemma}
\label{lem:resultant_discriminant}
Let $\mathbb{K}$ be any field and $\ff\in\KX$ be a polynomial of degree $d\geq
2$ with leading coefficient $a_d$, non constant derivative $\ff'$ and $\deg \ff'=k\le d-1$. Then, we have the relation
\[\Disc{\ff}= \Cf{1}\Res{\ff}{\ff '},\]
\noindent where $\Cf{1} =(-1)^{\frac{d(d-1)}{2}}\principal{\ff}^{d-k-2}$. \end{lemma}
One of the main tools used to prove our main result regarding the stability of arbitrary polynomials is the Stickelberger's result~\cite{Stickelberger97}, which gives the parity of the number of distinct irreducible factors of a polynomial over a finite field of odd characteristic.
\begin{lemma} \label{thm:Stickelberger}
Suppose $\ff\in\FX$, $q$ odd, is a polynomial of degree $d\ge 2$ and is
the product of $r$ pairwise
distinct irreducible polynomials over $\F$.
Then $r\equiv d\bmod 2$ if and only
if $\Disc{\ff}$ is a square in $\F$. \end{lemma}
\section{Stability of arbitrary polynomials} \label{sec:stabilityPolynomials} In this section we give a necessary condition for the stability of arbitrary polynomials. For this purpose, we use the following general result known as Capelli's Lemma, see~\cite{FS}. \begin{lemma} \label{lem:Capelli} Let $\mathbb{K}$ be a field, $f,g\in\KX$, and let $\beta\in\overline{\mathbb{K}}$ be any root of $g$. Then $g(f)$ is irreducible over $\mathbb{K}$ if and only if both $g$ is irreducible over $\mathbb{K}$ and $f-\beta$ is irreducible over $\mathbb{K}(\beta)$. \end{lemma}
We prove now one of the main results about the stability of an arbitrary polynomial. We note that our result partially generalises the quadratic polynomial case presented in~\cite{Jones09} which is known to be necessary and sufficient over finite fields.
\begin{theorem}
\label{thm:criterium}
Let $q=p^s$, $p$ be an odd prime, and $\ff\in\FX$ a stable polynomial of degree $d\geq
2$ with leading coefficient $a_d$, non constant derivative $\ff'$ and $\deg \ff'=k\le d-1$. Then the following hold:
\begin{enumerate}
\item if $d$ is even, then $\Disc{f}$
and $\principal{\ff}^{k}\Res{f^{(n)}}{\ff '}$, $n\ge 2$, are nonsquares in $\F$;
\item if $d$ is odd, then $\Disc{f}$
and $(-1)^{\frac{d-1}{2}}\principal{\ff}^{(n-1)k+1}\Res{f^{(n)}}{\ff '}$, $n\ge 2$, are squares in $\F$.
\end{enumerate} \end{theorem} \begin{proof} Let $\ff\in\FX$ be a stable polynomial. We assume first that $d$ is even. We have that $f^{(n)}$ is irreducible for any $n$, and thus, by Capelli's Lemma~\ref{lem:Capelli}, we know that $f-\alpha$ is irreducible over $\mathbb{F}_{q^{d^{n-1}}}$, where $\alpha$ is a root of $f^{(n-1)}$. By Lemma~\ref{thm:Stickelberger} this means that $\Disc{f-\alpha}$ is a nonsquare in $\mathbb{F}_{q^{d^{n-1}}}$. Now, taking the norm over $\F[q]$ and using Lemma~\ref{lem:resultant_discriminant}, we get
\begin{equation*} \begin{split}
\mathrm{Nm}_{q^{d^{n-1}}|q} &\Disc{f-\alpha}\\
&=\prod_{\substack{\alpha\in\mathbb{F}_{q^{d^{n-1}}}\\ f^{(n-1)}(\alpha)=0}} \Disc{f-\alpha}=\prod_{\substack{\alpha\in\mathbb{F}_{q^{d^{n-1}}}\\ f^{(n-1)}(\alpha)=0}} C_{f} \Res{f-\alpha}{f'}\\
&=C_f^{d^{n-1}} \Res{\prod_{\substack{\alpha\in\mathbb{F}_{q^{d^{n-1}}}\\ f^{(n-1)}(\alpha)=0}} (f-\alpha)}{f'}\\
&=C_f^{d^{n-1}}\Res{\frac{f^{(n-1)}(f)}{A}}{f'}=A^{-k}C_f^{d^{n-1}}\Res{f^{(n)}}{f'},
\end{split} \end{equation*} where $C_f$ is defined by Lemma~\ref{lem:resultant_discriminant}, $A$ is the leading coefficient of
$f^{(n-1)}$ and
$\mathrm{Nm}_{q^{d^{(n-1)}}|q}$ is the norm map from $\mathbb{F}_{q^{d^{n-1}}}$ to $\mathbb{F}_q$.
As the norm $\mathrm{Nm}_{q^{d^{n-1}}|q}$ maps nonsquares to nonsquares, we obtain that $ A^{-k} C_f^{d^{(n-1)}} \Res{f^{(n)}}{f'}$ is a nonsquare, and taking into account that $A=a_{d}^{\frac{d^{n}-1}{d-1}}$ and the parity of the exponents involved, the result follows. The case of odd $d$ can be treated in a similar way. \end{proof}
Theorem~\ref{thm:criterium}
is interesting because it gives a method for testing the stability of a polynomial. Lemma~\ref{lem:resultant_eval_root} says that the resultant is just the evaluation of $f^{(n)}$ in the roots of $\ff '$ multiplied by some constants. Taking into account this fact, the quadratic character of $\principal{\ff}$ and the exponents which are involved in Theorem \ref{thm:criterium}, we have the following characterisation.
\begin{corollary}\label{cor:condition}
Let $q=p^s$, $p$ an odd prime, and $\ff\in\FX$ a stable polynomial
of degree $d\geq 2$ with leading coefficient $a_d$, non constant derivative
$\ff'$, $\deg \ff'=k\le d-1$ and $a_{k+1}$ the coefficient of $X^{k+1}$ in $f$.
Let $\gamma_i$, $i=1,\ldots,k,$ be the roots of the derivative
$\ff'$. Then the following hold:
\begin{enumerate}
\item if $d$ is even, then
\begin{equation}\label{eq:even orbit}
\mathcal{S}_1=
\left\{\principal{\ff}^k \prod_{i=1}^{k} \f{n}{\gamma_i}\ \mid \ n>1\right\}
\ \bigcup\ \left\{\ (-1)^\frac{d}{2}\principal{\ff}^k\prod_{i=1}^{k}
\ff({\gamma_i})\ \right\}
\end{equation}
contains only nonsquares in $\F$;
\item if $d$ is odd, then
\begin{equation}\label{eq:odd orbit}
\mathcal{S}_2=\left\{\ (-1)^{\frac{(d-1)}{2}+k}(k+1)a_{k+1}
\principal{\ff}^{(n-1)k+1}\prod_{i=1}^{k} \f{n}{\gamma_i}\ \mid \ n\ge1\right\}
\end{equation}
contains only squares in $\F$.
\end{enumerate} \end{corollary} \begin{proof} The result follows directly from Theorem~\ref{thm:criterium} and Lemma~\ref{lem:resultant_eval_root}. \end{proof}
We note that the converse of Corollary~\ref{cor:condition} is not true. Indeed, take any $d$ with $\gcd(d,q-1)=\gcd(d,p)=1$, $\F$ an extension of even degree of $\F[p]$ and $a_0$ a quadratic residue in $\F$. Let us consider the polynomial $\ff{(X)}=(X-a_0)^{d}+a_0\in\FX$. It is straightforward to see that $\f{n}{X}=(X-a_0)^{d^n}+a_0$ and that the set~\eqref{eq:odd orbit} is \begin{equation*} \{(-1)^{\frac{d-1}{2}}\,d\, a_0^{d-1}\}. \end{equation*}
We note that the polynomial $\ff$ is reducible. Indeed, let the integer $1\le e\le q-1$ be such that $ed=1 \pmod {q-1}$. Then $(a_0^e)^d=a_0$, and thus $-a_0^e+a_0$ is a root of $f$. On the other hand, since $-1$ and $d$ are squares in $\F$ because both elements belong to $\F[p]$ and $\F$ is an extension of even degree, the set~\eqref{eq:odd orbit} contains only squares.
We finish this section by showing that, when the derivative $f'$ of the stable polynomial $f$ is irreducible, the sets~\eqref{eq:even orbit} and~\eqref{eq:odd orbit} are defined by a short sequence of initial elements. The proof follows exactly the same lines as in the proof of~\cite[Theorem 1]{Ostafe09}. Indeed, assume $\deg f'=k$ and $\gamma_1,\ldots,\gamma_k\in\mathbb{F}_{q^k}$ are the roots of $f'$. Using Corollary~\ref{cor:condition} we see that the sets ~\eqref{eq:even orbit} and~\eqref{eq:odd orbit} contain only nonsquares and squares, respectively, and thus, the problem reduces to the cases when $\f{n}{\gamma_1}\ldots\f{n}{\gamma_k}$ are either all squares or all nonsquares for any $n\ge 1$. It is clear that, when $f'$ is irreducible, taking into account that $\gamma_i=\gamma_1^{q^{i}}$, $i=1,\ldots,k-1$, we get for every $1\le n\le N$, \begin{equation*} \begin{split} \f{n}{\gamma_1}\ldots\f{n}{\gamma_k} &= \f{n}{\gamma_1}\ldots\f{n}{\gamma_1^{q^{k-1}}}\\ &=\f{n}{\gamma_1}\ldots\f{n}{\gamma_1}^{q^{k-1}}\ =
\mathrm{Nm}_{q^{k}|q}\f{n}{\gamma_1}. \end{split} \end{equation*} Applying now the same technique with multiplicative character sums as in~\cite[Theorem 1]{Ostafe09} (as the argument does not depend on the degree of the polynomial $f$), we obtain the following estimate: \begin{theorem} \label{thm:UB} For any odd $q$ and any stable polynomial $f \in\FX$ with irreducible derivative $f'$, $\deg f'=k$, there exists $$ N= O\(q^{3k/4}\) $$ such that for the sets~\eqref{eq:even orbit} and~\eqref{eq:odd orbit} we have \begin{equation*} \begin{split}
\mathcal{S}_1&= \left\{\principal{\ff}^k \prod_{i=1}^{k} \f{n}{\gamma_i}\ \mid \ 1< n \le N\right\}\ \bigcup\ \left\{\ (-1)^\frac{d}{2}\principal{\ff}^k\prod_{i=1}^{k} \ff({\gamma_i})\right\};\\ \mathcal{S}_2& =\left\{\ (-1)^{\frac{(d-1)}{2}+k}(k+1)a_{k+1}\principal{\ff}^{(n-1)k+1}\prod_{i=1}^{k} \f{n}{\gamma_i}\ \mid \ 1\le n \le N\right\}. \end{split} \end{equation*}
\end{theorem}
\section{Non-existence of certain cubic stable polynomials when $p$=3} \label{sec:nonExistence} The existence of stable polynomials is difficult to prove. For $p=2$, there are no stable quadratic polynomials as shown in~\cite{A}, whereas for $p>2$, there is a big number of them as is shown in~\cite{GomezNicolas10}. In this section, we show that for certain polynomials of degree 3, $f^{(3)}$ is a reducible polynomial when $p=3$.
This result also appears in~\cite{ALOS}, but we think this approach uses new ideas that could be of independent interest. For this approach, we need the following result which can be found in~\cite[Corollary 4.6]{Menezes93}. \begin{lemma} \label{lem:menezes93}
Let $q=p^s$ and $f(X)=X^{p}-a_1 X-a_0\in\FX$ with $a_1 a_0\neq 0$. Then $f$ is irreducible over
$\F$ if and
only if $a_1=b^{p-1}$ and $\mathrm{Tr}_{q|p}(a_0/b^{p})\neq 0.$ \end{lemma} Based on this result, we can present an irreducibility criterium for polynomials of degree 3 in characteristic $3$. \begin{lemma} \label{lem:appliedMenezes}
Let $p=3$ and $q=3^{s}$. Then $f(X)=X^3-a_2X^2-a_1X-a_0$ is irreducible over
$\F$ if and only if
\begin{enumerate}
\item $a_1=b^2$ and $\mathrm{Tr}_{q|3}(a_0/b^{3})\neq 0,$ if $a_2 =0$ and $a_1\neq 0$;
\item $a_2^4/(a_2^2a_1^2+a_1^3-a_0a_2^3)=b^2$ and
$\mathrm{Tr}_{q|3}(1/a_2b)\neq 0,$ if $a_2 \neq 0$,
\end{enumerate}
where $\mathrm{Tr}_{q|3}$ represent the trace map of $\F$ over $\F[3]$. \end{lemma} \begin{proof}
The case $a_2=0$ is a direct application of Lemma~\ref{lem:menezes93}.
In the other case, we take the polynomial
\begin{multline*}
f(X+a_1/a_2)=(X+a_1/a_2)^3-a_2(X+a_1/a_2)^2-a_1(X+a_1/a_2)-a_0=\\
X^3-a_2X^2-a_0+a_1^2/a_2+a_1^3/a_2^3= X^3-a_2X^2+(a_1^2a_2^2+a_1^3-a_0a_2^3)/a_2^3.
\end{multline*}
Notice that $f(X+a_1/a_2)$ is irreducible if and only if $f(X)$ is irreducible.
We denote $g(X)=f(X+a_1/a_2)$ to ease the notation and $g^*$
\textit{the reciprocal polynomial} of $g$, i. e.
\begin{equation*}
g^*(X)=X^3g\(\frac{1}{X}\).
\end{equation*}
By \cite[Theorem 3.13]{LN97}, $g^*$ is irreducible if and only if $g$ is.
Applying Lemma~\ref{lem:menezes93}, we get the result. \end{proof} For simplicity, we proved an irreducibility criterium for monic polynomials, however the proof holds for non-monic polynomials as well taking into account the principal coefficient.
Using Lemma~\ref{lem:appliedMenezes} and following the same lines as in~\cite{A}, we can prove now the following result.
\begin{theorem} \label{thm:deg3} For any polynomial $f\in\mathbb{F}_3[X]$ of the form $f(X)=a_3X^3-a_1X-a_0$, at least one of the following polynomials $f,f^{(2)}$ or $ f^{(3)}$ is a reducible polynomial. \end{theorem} \begin{proof}
Suppose that $f,\ f^{(2)},\ f^{(3)}$ are all irreducible polynomials. Using Lemma~\ref{lem:Capelli}, $f^{(3)}$ is irreducible if and only if $f^{(2)}$ is irreducible over $\F$ and $f-\gamma$ is irreducible over $\F[q^9]$, where $\gamma$ is a root of $f^{(2)}$. Thus, the monic polynomial $h=\frac{f-\gamma}{a_3}$ is irreducible over $\F[q^9]$ and we can apply now Lemma~\ref{lem:appliedMenezes} from where we get that
$\mathrm{Tr}_{q^9|3}(\frac{a_0-\gamma}{a_3 b^3})\neq 0$, where $b^2=a_1$ and $b\in\F[q^9]$.
Notice that $b\in\F[q]$. Indeed, as $b$ is the root of the polynomial $X^2-a_1$, then either $b\in\F[q]$ or $b\in\F[q^2]$. Since $b\in\F[q^9]$ we obtain that $b\in\F[q]$.
Using the properties of the trace map we obtain \begin{equation*}
\mathrm{Tr}_{q^9|3} \(\frac{a_0-\gamma}{a_3 b^3}\)=\mathrm{Tr}_{q^9|3}\(\frac{-\gamma}{a_3 b^3}\), \end{equation*}
and from here we conclude that the right hand side of the last equation is non zero. Using now the transitivity of the trace, see~\cite[Theorem 2.26]{LN97}, we get \begin{equation*}
\mathrm{Tr}_{q^9|3}\(\frac{-\gamma}{a_3 b^3}\)=\mathrm{Tr}_{q|3}\(\mathrm{Tr}_{q^9|q}\(\frac{-\gamma}{a_3 b^3}\)\)=\mathrm{Tr}_{q|3}\(\frac{\mathrm{Tr}_{q^9|q}(-\gamma)}{a_3b^3}\). \end{equation*} Now, $f^{(2)}$ is an irreducible polynomial with roots
$\gamma, \gamma^q,\ldots,\gamma^{q^{8}}$. Thus, $\mathrm{Tr}_{q^9|q}(\gamma)$ is given by the coefficient of the term $X^8$ in $f^{(2)}$, which is zero.
This shows that $\mathrm{Tr}_{q^9|3}(\gamma)=0$, which is a contradiction with the fact that $f^{(3)}$ is irreducible. \end{proof}
We note that Theorem~\ref{thm:deg3} cannot be extended to infinite fields. As in~\cite{ALOS}, let $\mathbb{K} = \mathbb{F}_3(T)$ be the rational function field in $T$ over $\mathbb{F}_3$, where $T$ is transcendental over $\mathbb{F}_3$. Take $f(X)=X^3+T\in \mathbb{K}[X]$. Then it is easy to see that $$ f^{(n)}(X)=X^{3^n}+T^{3^{n-1}}+T^{3^{n-2}}+\cdots+T^3+T. $$ Now from the Eisenstein criterion for function fields (see~\cite[Proposition~III.1.14]{Sti}, for example), it follows that for every $n\ge 1$, the polynomial $f^{(n)}$ is irreducible over $\mathbb{K}$. Hence, $f$ is stable. \section{On the number of stable polynomials} \label{sec:numberStables} In this section we obtain an estimate for the number of stable polynomials of certain degree $d$. Note that, from Corollary \ref{cor:condition}, it suffices to estimate the number of nonsquares of the orbit \eqref{eq:even orbit} for even $d$, or the number of squares of \eqref{eq:odd orbit} for odd $d$.
For a given $d$, let $\ff(X)=a_{d}X^{d}+a_{d-1}X^{d-1}+\cdots+a_1X+a_0\in\mathbb{F}_q[X]$ and we define
\begin{equation*} \fa{\ff}{l}=\prod_{i=1}^{k}\ff^{(l)}({\gamma_i}), \end{equation*} which is a polynomial in the variables $a_{0},\ldots,\ a_{d}$ and with coefficients in $\F$.
Following~\cite{Ostafe09}, the number of stable polynomials of degree $d$, which will be denoted by $S_{d}$, satisfies the inequality \begin{equation}\label{eq:sums} S_{d}\le \frac{1}{2^{K}}\sum_{a_0\in\F}\cdots\sum_{a_{d}\in\F^*} \prod_{l=1}^{K}(1\pm\chi(\fa{\ff}{l})),\ \forall K\in\mathbb{Z}^+, \end{equation}
\noindent where $\chi$ is the multiplicative quadratic character of $\F$. The sign of $\chi$ depends on $d$ and is chosen in order to count the elements of the orbit of $\ff$ which satisfy the condition of stability. Since the upper bound of $S_{d}$ is independent of this choice, let us suppose from now on that $\chi$ is taken with $+$. If we expand and rearrange the product, we obtain $2^K-1$ sums of the shape \begin{equation*} \sum_{a_0\in\F}\cdots\sum_{a_{d}\in\F^*}\chi \left (\prod_{j=1}^{\mu}\fa{\ff}{l_{j}}\right ),\, 1\le l_1<\cdots < l_{\mu}\le K, \end{equation*}
\noindent with $\mu\ge 1$ plus one trivial sum correponding to 1 in (\ref{eq:sums}).
The upper bound for $S_{d}$ will be obtained using the Weil bound for character sums, which can be found in \cite[Lemma 1]{GomezNicolas10}. This result can only be used when $\prod_{j=1}^{\mu}\fa{\ff}{l_{j}}$ is not a square polynomial. The next lemmas are used to estimate the number of values for $a_{1},\ldots,a_{d}$ such that the resulting polynomial in $a_0$ is a square.
The first lemma is a bound on the number of zeros of two multivariate polynomials. A more general inequality is given by the Schwartz-Zippel lemma. For a proof, we refer the reader to~\cite{Gathen99}. \begin{lemma} \label{multiple_roots} Let $F(Y_0,Y_1,\ldots,Y_{d}), G(Y_0,Y_1,\ldots,Y_{d})$ be two polynomials of degree $d_1$ and $d_2$, respectively, in $d+1$ variables with \begin{equation*} \gcd\left(F(Y_0,Y_1,\ldots,Y_{d}), G(Y_0,Y_1,\ldots,Y_{d})\right)=1. \end{equation*} Then, the number of common roots in $\F$ is bounded by $d_1d_2q^{d-1}$. \end{lemma} The next lemma gives a bound for the number of ``bad'' choices of $a_1,\ldots,a_{d}$, that is, the number of choices of $a_1,\ldots,a_{d}$ such that $\prod_{j=1}^{\mu}\fa{\ff}{l_{j}}$ is a square polynomial in $a_0$. \begin{lemma} \label{lem:roots} For fixed integers $l_1,\ldots, l_{\mu}$ such that $1\le l_1<\cdots < l_{\mu}\le K$, the polynomial \[ \prod_{j=1}^{\mu}\fa{\ff}{l_{j}} \] is a square polynomial in the variable $a_0$ up to a multiplicative constant only for at most $O(d^{2K}q^{d-1})$ choices of $a_1,\ldots,a_{d}$. \end{lemma} \begin{proof} For even degree which is coprime to $p$, we consider the polynomial $f=(X-b)^{d}+c+b$, where $b,c$ are considered as variables. Then $f'=d(X-b)^{d-1}$ and \begin{equation*} f^{(n)}(b)=b+H_n(c), \end{equation*} where $ \deg H_n(c)= d^{n-1}$.
For odd degree, coprime to $p$, we consider the following polynomial $f= (X-b)^{d-1}(X-b+1)+c+b$ with the derivative $f'=(X-b)^{d-2}(d(X-b)+d-1).$ Notice that, if the degree of this polynomial is coprime to the characteristic $p$, then $f'$ has two different roots $ b, b+(1-d)d^{-1}$. Substituting these in the polynomial $f$, we get \begin{eqnarray*} f^{(n)}(b) &=&b+H_n(c), \\ f^{(n)}( b+(1-d)d^{-1})&=&b+L_n(c),\\ \end{eqnarray*} where $L_n\neq H_n$ and $\deg L_n(c)=\deg H_n(c)= d^{n-1}.$ In either of the two cases, we can compute the irreducible factors of $\Res{f^{(k)}}{\ff '}$.
When the degree is not coprime to the characteristic, take $f=(X-b)^{d}+(X-b)^2+c+b$ and the proof is similar to the last two cases.
This proves that the following polynomial \[ \prod_{j=1}^{\mu}\fa{\ff}{l_{j}} \] is not a square polynomial as a multivariate polynomial up to a multiplicative constant.
Let $ \prod_{j=1}^{\mu}\fa{\ff}{l_{j}}=G_1(a_0,\ldots, a_{d})^{d_1} \cdots G_h(a_0,\ldots, a_{d})^{d_h} $ be the decomposition into a product of irreducible polynomials.
Without loss of generality, $d_1$ is not even because $\prod_{j=1}^{\mu}\fa{\ff}{l_{j}}$ is not a square of a polynomial up to a multiplicative constant. Moreover, because $G_1$ is an irreducible factor of the product $\prod_{j=1}^{\mu}\fa{\ff}{l_{j}}$, then there exists $1\le j\le \mu$ such that $G_1$ is an irreducible factor of $\fa{\ff}{l_{j}}$, which implies that $\deg G_1\le d^{K}.$
We use $G_1(a_0,\ldots, a_{d})$ to count the number of choices for $a_1,\ldots, a_{d}$ such that \begin{itemize} \item the polynomial $\prod_{j=1}^{\mu}\fa{\ff}{l_{j}}$ is a constant polynomial. \item the polynomial $\prod_{j=1}^{\mu}\fa{\ff}{l_{j}}$ is a square polynomial
up to a multiplicative constant. \end{itemize}
There are at most $d^{K\mu}q^{d-1}$ different choices of $a_1,\ldots, a_{d}$ when the polynomial can be a constant.
Now, we consider in which cases the polynomial $\prod_{j=1}^{\mu}\fa{\ff}{l_{j}}$ is a square of a polynomial
and how these cases will be counted. We have the following two possible situations: \begin{itemize} \item $G_1^{d_1}$ is a square, nonconstant, and because $d_1$ is not
even, then we must have that $G_1$ has
at least one multiple root. This is only possible if
$G_1$ and the first derivative with respect to the
variable $a_0$ of $G_1$ have a common root.
$G_1$ is an irreducible polynomial, so Lemma~\ref{multiple_roots} applies.
We remark that the first derivative is a nonzero polynomial. Otherwise
$G_1$ is a reducible polynomial. This can only happen in $(\deg G_1)(\deg G_1-1)q^{d-1}$ cases.
\item $G_1$ and $G_j$ have a common root for some $1\le j\le h$. In this case, using the same argument, there are at most
$(\deg G_1)(\deg G_j)q^{d-1}$
possible values for $a_1,\ldots, a_{d}$ where it happens. \end{itemize} \end{proof}
Now we are able to find a bound for $S_{d}$, the number of stable polynomials of degree $d$.
\begin{theorem}
The number of stable polynomials $\ff\in\FX$ of degree $d$ is
$O(q^{d+1-1/\log(2d^2)})$. \end{theorem}
\begin{proof} The trivial summand of~\eqref{eq:sums} can be bounded by $O(q^{d+1}/2^K)$. For the other terms, we can use the Weil bound, as is given in \cite[Lemma 1]{GomezNicolas10}, for those polynomials which are nonsquares. Since these polynomials have degree at most $d^{K}$ in the indeterminate $a_0$ (see the proof of Lemma~\ref{lem:roots}), we obtain $O(d^K q^{d+1/2})$ for this part. For the rest, that is, the square polynomials, we can use the trivial bound. Thus, from Lemma~\ref{lem:roots}, we get $O(d^{2K} q^{d})$. Then, \[ S_{d}= O(q^{d+1}/2^{K}+d^Kq^{d+1/2}+ d^{2K}q^{d}). \] Choosing $K=\lceil(\log q/\log (2d^2))\rceil$ the result follows. \end{proof}
\end{document} | arXiv |
\begin{document}
\preprint{APS/123-QED}
\title{Quantum federated learning based on gradient descent}
\author{Kai Yu$^1$} \author{Xin Zhang$^2$} \author{Zi Ye$^1$} \author{Gong-De Guo$^1$} \author{Song Lin$^1$} \thanks{Corresponding author. Email address: [email protected]}
\affiliation{ {$^1$College of Computer and Cyber Security, Fujian Normal University, Fuzhou 350117, China\\ $^2$College of Mathematics and Statistics, Fujian Normal University, Fuzhou 350117, China} }
\date{\today}
\begin{abstract} Federated learning is a distributed learning framework in machine learning, and has been widely studied recently. Generally speaking, there are two main challenges, high computational cost and the security of the transmitted message, in the federated learning process. To address these challenges, we utilize some intriguing characteristics of quantum mechanics to propose a framework for quantum federated learning based on gradient descent. In the proposed framework, it consists of two components. One is a quantum gradient descent algorithm, which has been demonstrated that it can achieve exponential acceleration in dataset scale and quadratic speedup in data dimensionality over the classical counterpart. Namely, the client can fast-train gradients on a quantum platform. The other is a quantum secure multi-party computation protocol that aims to calculate federated gradients safely. The security analysis is shown that this quantum protocol can resist some common external and internal attacks. That is, the local gradient can be aggregated securely. Finally, to illustrated the effectiveness of the proposed framework, we apply it to train federated linear regression models and successfully implement some key computation steps on the Qiskit quantum computing framework. \end{abstract}
\maketitle
\section{\label{sec:1}Introduction}
\par Machine learning (ML) is the core of artificial intelligence, which has succeeded in a wide variety of tasks. An essential feature of these successful ML models is that they are data-driven and involve large amounts of data. In 2016, Google proposed a federated learning (FL) framework \cite{brendan2016} that allows machine learning models to improve automatically through data from multiple parties. A typical federation learning process is started with a server initializing a global model, which is shared with several participants (clients) for iterative training. For concreteness, clients train locally with the received model and their part of the data; the server aggregates these training results to generate a new global model and shares this with all clients. Ultimately, clients can get a machine learning model trained on multiple data sources without sharing their data with others. Undoubtedly, the FL framework is helped to enhance the performance of models and privacy-preserving practices in machine learning. However, in the era of big data, the amount of data from a single user is also huge. In this case, traditional machine learning is prone to computational disasters. Moreover, the training results are needed to transmit in the federated learning process, which may provide a chance for the attacker to infer user privacy. That is the security of the transmitted messages should also be ensured.
\par Quantum information processing is an emerging field that explores the interaction between quantum mechanics and information technology. It sustains to show its charm, attracting the attention of scholars. In 1984, Bennett and Brassard proposed the famous BB84 protocol \cite{bennett1984}, which perfectly achieves a key distribution task between two remote parties. Subsequently, scholars utilized quantum information processing to ensure information security and proposed a series of quantum cryptography protocols. In contrast to the security of classical cryptography protocols that are based on the assumption of computational complexity, these protocols' security relies on physical properties such as the Heisenberg uncertainty principle, which makes them unconditionally secure in theory. Quantum cryptography has developed as a significant application of quantum information processing, including quantum key distribution \cite{Gisin2002,scarani2009,schwonnek2021}, quantum secret sharing \cite{PhysRevA.59.1829, PhysRevA.59.162, PhysRevLett.83.648}, quantum secure direct communication \cite{kim2002, PhysRevA.68.042317}, and so on. Another exciting application of quantum information processing is quantum computing. It provided quantum speedup to certain classes of problems that are intractable on classical computers. For example, the factorization of large numbers via Shor algorithm \cite{Shor1999} can provide exponential speedup. Furthermore, quantum computing has also made some advances in machine learning, such as the Harrow-Hassidim-Lloyd algorithm \cite{HHL}, quantum support vector machine \cite{rebentrost2014}, quantum regression \cite{schuld2016, yu2019, chen2022}, quantum neural network \cite{rebentrost2018, PhysRevA.100.012334}, and so on.
\par Recently, scholars tried to explore quantum federated learning (QFL) to address the efficiency and security issues. In 2021, Chen and Yoo focused on the computational efficiency of FL. They provided a framework to train hybrid quantum-classical classifiers in a federated manner, which could help in distributing computational loads to quantum computing \cite{chen2021QFL}. Li et al. emphasized the security issue of federated learning and utilized blind quantum computing to design a quantum delegate computing that could be extended to multi-party distributed learning \cite{li2021QFL}. Inspired by the above works, we combine quantum computing and quantum cryptography to propose a framework for quantum federated learning based on gradient descent (QFLGD) in this paper. This framework is made up of two parts. One is a quantum gradient descent (QGD) algorithm for private single-party gradient estimation, which can let clients gain the classical gradients in each iteration. The analysis of its time complexity shows that can be achieved exponential acceleration in dataset scale and quadratic speedup in data dimensionality over the classical counterpart. Another is a quantum secure multi-party computation (QSMC) protocol that could securely calculate federated gradients. The proposed framework is adapted to train supervised machine learning models. To illustrate it, we apply the QFLGD framework to train a federated linear regression (FLR) model. And a small-scale experiment of computing the gradient is carried out on the Qiskit quantum computing framework.
\par The rest of this paper is organized as follows. In Sec. \ref{sec:2}, we introduce the classical gradient descent algorithm and federated machine learning. Then we present the framework for quantum federated learning based on gradient descent in Sec. \ref{sec:3}. In Sec. \ref{sec:4}, the time complexity and the security of QFLGD are analyzed. In Sec. \ref{sec:5}, we show the application of the proposed framework to train the federated linear regression model and give simulation experiments of some computation. Finally, we present our conclusion in Sec. \ref{sec:6}.
\section{\label{sec:2}Preliminaries}
\subsection{\label{sec:2.1}Gradient descent algorithm} \par Supervised learning is a fundamental part of machine learning. It needs training the labeled data samples to adjust the weight vector $\mathbf{w} = \left( \omega_{0},\omega_{1},\cdots,\omega_{D-1} \right)$ and then gets an effective machine learning model. Such as perceptron, predictive models, and so on. A crucial method for obtaining optimal weight is the gradient descent algorithm, which has strong stability.
\par For instance, letting $\left( {\mathbf{x}_{0},y_{0}} \right),\left( {\mathbf{x}_{1},y_{1}} \right),\cdots,\left( {\mathbf{x}_{M-1},y_{M-1}} \right)$ be $M$ samples, where $\mathbf{x}_{i} \in \mathbb{R}^{D}$ and $y_i$ is the corresponding label. To train this model, we can apply the gradient descent method to minimize the following cost function,
\begin{equation}
{\min\limits_{\mathbf{w}}E} = \frac{1}{2M}{\sum\limits_{i = 0}^{M-1}\left\lbrack {f\left( {\mathbf{x}_{i} \cdot \mathbf{w}} \right) - y_{i}} \right\rbrack^{2}},
\label{eq:1}
\end{equation} where $f$ is the activation function. Taking the gradient of $E$ with respect to $\mathbf{w}$, we obtain
\begin{equation}
\bm{g}^{j}\left( \mathbf{w} \right) = \frac{1}{M}{\sum\limits_{i = 0}^{M-1}{F\left( {\mathbf{x}_{i} \cdot \mathbf{w}} \right)}}\mathbf{x}_{i}^{j},\quad j = 0,1,\cdots,D-1.
\label{eq:2}
\end{equation} Letting $F\left( {\mathbf{x}_{i} \cdot \mathbf{w}} \right) = \left\lbrack {f\left( {\mathbf{x}_{i} \cdot \mathbf{w}} \right) - y_{i}} \right\rbrack f^{\prime}\left({\mathbf{x}_{i} \cdot \mathbf{w}} \right)$, $\bm{g}^{j}\left( \mathbf{w} \right)$ denotes the $j$th element of the local gradient $\bm{g}\left( \mathbf{w} \right)$, and $\mathbf{x}_{i}^{j}$ labels as the $j$th element of the sample $\mathbf{x}_{i}$. \par In the $n$th iteration, the parameters are updated as
\begin{equation}
\mathbf{w}^{j}\left( {n + 1} \right) = \mathbf{w}^{j}(n) - \alpha \times \bm{g}^{j}\left( {\mathbf{w}(n)} \right),
\label{eq:3}
\end{equation} for $j = 0,1,\cdots,D-1$, where $\alpha \in (0,1\rbrack$ is a learning factor. To terminate the iteration at a suitable moment, one strategy is that set a threshold $\varepsilon$ about gradient. If ${\sum\limits_{j = 0}^{D-1}\left[ {\bm{g}^{j}\left( \mathbf{w}(n) \right)} \right]^{2}} \leq \varepsilon$, the training is finished and then $\mathbf{w}(n)$ is outputted.
\subsection{\label{sec:2.2}Federated learning based on gradient descent} \par To further improve the performance of the ML models, it is needed to combine multi-party data for training. One of the widely used frameworks is federated learning based on gradient descent. In this case, each client trains the gradient of its data locally and then sends the results to server. The server updates the model parameters according to the user gradients and shares them with all clients. The framework of federated learning based on gradient descent algorithm is illustrated in Fig. \ref{fig:1}.
\begin{figure}
\caption{Schematic illustration of the federated learning based on gradient descent.}
\label{fig:1}
\end{figure}
\par Considering $K$ clients participated in the model training. The client ${\rm Bob}_{k}$ ($k = 1,2,\cdots,K$) has $M_{k}$ samples $\left( {\mathbf{x}_{0},y_{0}} \right),\left( {\mathbf{x}_{1},y_{1}} \right),\cdots,\left( {\mathbf{x}_{M_{k}-1},y_{M_{k}-1}} \right)$, where $\mathbf{x}_{i} \in \mathbb{R}^{D}$. And he utilizes the gradient descent algorithm to calculate the gradient of the local data as
\begin{equation}
\bm{g}_{k}^{j}\left( \mathbf{w} \right) = \frac{1}{M_{k}}{\sum\limits_{i = 0}^{{M_{k}}-1}{F\left( {\mathbf{x}_{i} \cdot \mathbf{w}} \right)}}\mathbf{x}_{i}^{j},\quad j = 0,1,\cdots,D-1,
\label{eq:4}
\end{equation} where $\bm{g}^{j}_{k}\left( \mathbf{w} \right)$ denotes the $j$th element of the local gradient $\bm{g}_{k}\left( \mathbf{w} \right)$.
\par Server Alice collects the gradients trained by all clients, and calculates the federated gradient
\begin{equation}
\bm{G}^{j} \left( \mathbf{w} \right) = {\sum\limits_{k = 1}^{K}{\beta_{k}}{\bm{g}_{k}^{j}\left( \mathbf{w} \right)}},
\label{eq:5}
\end{equation} where ${\beta_{k}}={M_{k}}/ ({\sum_{k=1}^{K}{M_{k}}})$ and $\bm{G}^{j}\left( \mathbf{w} \right)$ is represented as the $j$th element of the federated gradient $\bm{G}\left( \mathbf{w} \right)$. Then, Alice updates the model parameters
\begin{equation}
\mathbf{w}^{j}\left( {n + 1} \right) = \mathbf{w}^{j}(n) - \alpha \times \bm{G}^{j}\left( {\mathbf{w}(n)} \right),
\label{eq:6}
\end{equation} for $j = 0,1,\cdots,D-1$, and distributes them to all clients. A signal is sent to clients to terminate training when ${\sum\limits_{j = 0}^{D-1}\left[ {\bm{G}^{j}\left( \mathbf{w}(n) \right)} \right]^{2}} \leq \varepsilon$. The entire process of the federated learning based on gradient descent (FLGD) is shown in Fig. \ref{fig:2}.
\begin{figure}\label{fig:2}
\end{figure}
\par For the client ${\rm Bob}_k$, the main difficulty to implement the training is the evaluation of the summation (as shown in Eq. (\ref{eq:4})) for $j=0,1,\cdots,D-1$. It costs $O(D)$ to estimate the inner product ${\mathbf{x}_{i} \cdot \mathbf{w}}$ and contains $M$ summations. This process need to be repeated $D$ attempts to estimate the all elements of $\bm{g}_{k}\left( \mathbf{w} \right)$. Totally, the complexity of estimating the local gradient $\bm{g}_{k}\left( \mathbf{w} \right)$ on a classical computer is $O(MD^2)$. In the era of big data, this is certainly very computationally expensive. Furthermore, a secure and efficient computation of the federal gradient to prevent unscrupulous individuals from obtaining any data information is also valuable.
\section{\label{sec:3}Quantum Federated Learning based on Gradient Descent Algorithm}
\subsection{\label{sec:3.0}The whole model of QFLGD}
\par In this section, we briefly introduce the model of quantum federated learning based on gradient descent. In this framework, a quantum gradient descent (QGD) algorithm (shown in Sec. \ref{sec:3.1}) is first presented. Each client ${\rm Bob}_{k}$ utilizes this algorithm to accelerate training on their data gradients. Then, a quantum secure multi-party computation (QSMC) protocol is designed (shown in Sec. \ref{sec:3.2}). The server and clients use this protocol to communicate and securely calculate the federated gradient. Finally, Alice updates the federated model parameters according to Eq. (\ref{eq:6}) and broadcasts them to ${\rm Bob}_{k}$ ($k = 1, 2, \cdots, K$). The schematic illustration of the QFLGD model is exhibited in Fig. \ref{fig:QFLGDM}.
\begin{figure}
\caption{Schematic illustration of the quantum federated learning model based on gradient descent.}
\label{fig:QFLGDM}
\end{figure}
\subsection{\label{sec:3.1}Quantum gradient descent algorithm} \par In this section, a quantum algorithm to approximate the gradients is proposed. Considering a dataset $\mathbf{X} = \left[ \mathbf{x}_{0},\mathbf{x}_{1},\cdots,\mathbf{x}_{M-1} \right]$, where $\mathbf{x}_{i} \in \mathbb{R}^{D}$. And $\mathbf{y} = \left( y_{0},y_{1},\cdots,y_{M-1} \right)$ is the corresponding label of each sample of $\mathbf{X}$, respectively. For convenience, assuming that $D=2^{L}$ for some $L$; otherwise, some zeros are inserted into the vector. And the quantum oracles
\begin{equation}
O_{X}:\left| i \right\rangle \left| j \right\rangle \left| 0 \right\rangle \longrightarrow\left| i \right\rangle \left|j \right\rangle| \mathbf{x}_{i}^{j}\rangle,
\label{eq:7}
\end{equation} and
\begin{equation}
O_{y}:\left| i \right\rangle \left| 0 \right\rangle \longrightarrow\left| i \right\rangle |y_{i}\rangle,
\label{eq:8}
\end{equation} are provided, where $\mathbf{x}_{i}^{j}$ represents the $j$th element of the $i$th vector of the data set $\mathbf{X}$. These two oracles can respectively access the entries of $\mathbf{x}_{i}$, $\mathbf{y}$ in time $O(\text{polylog}(MD))$ and $O(\text{polylog}(D))$ \cite{yu2019,wossnig2018}, when the data are stored in quantum random access memory (QRAM) \cite{giovannetti2008} with an appropriate data structure \cite{kerenidis2016}. In addition, the operation
\begin{equation}
U_{nf}:\left| i \right\rangle \left| 0 \right\rangle \longrightarrow\left| i \right\rangle \left| \left\| \mathbf{x}_{i} \right\| \right\rangle.
\label{eq:9}
\end{equation} is required, which could access the $2$-norm of the vector $\mathbf{x}_{i}$. Inspired by \cite{mitarai2019}, $U_{nf}$ can be implemented in time $O\left( {{\text{polylog}(D)}/\epsilon_{m}} \right)$ employing controlled rotation \cite{cong2016} and quantum phase estimation (QPE) \cite{brassard2002}. The details are shown in Appendix \ref{appendix A}.
\par According to the above assumptions, the processes of the QGD algorithm are described as follows.
\par \textbf{(1) Extract data and parameter to the quantum state.}
\par (1.1) In this step, the data information is extracted to a state $\left| {\phi\left( \mathbf{x}_{i} \right)} \right\rangle$. The details are presented in the following.
\par Firstly, three quantum registers are prepared in state $|i\rangle_{1} |0^{\otimes{\log{D}}}\rangle_{2} |0^{\otimes{q}}\rangle_{3} $, where the subscript numbers denote different registers. The $q$ is labeled as the qubits that are enough to store the information about the elements of data, i.e., $2^{q} - 1 > {\max\limits_{i,j}|\mathbf{x}_{i}^{j}|}$. After that, $H^{\otimes{\log{D}}}$ is applied on the second register to generate a state
\begin{equation}
\frac{1}{\sqrt{D}}{\sum\limits_{j = 0}^{D-1}\left| i \right\rangle_{1} } |j\rangle_{2} | 0^{\otimes q} \rangle_{3}.
\label{eq:10}
\end{equation}
\par Secondly, the quantum oracle $O_X$ is performed on the three registers. These registers are in a state
\begin{equation}
\frac{1}{\sqrt{D}}{\sum\limits_{j = 0}^{D-1} |i\rangle_{1} } |j\rangle_{2} |\mathbf{x}_{i}^{j}\rangle_{3}.
\label{eq:11}
\end{equation}
Subsequently, a qubit in the state $|0\rangle$ is added and rotated to $\sqrt{1 - ({c_{1}\mathbf{x}_{i}^{j}})^{2}}\left|0 \right\rangle + c_{1}\mathbf{x}_{i}^{j}\left|1 \right\rangle$ controlled on $|\mathbf{x}_{i}^{j} \rangle$, where $c_{1} = {1/ {\max\limits_{i,j} |\mathbf{x}_{i}^{j}|} }$. The system becomes
\begin{equation}
\frac{1}{\sqrt{D}}{\sum\limits_{j = 0}^{D-1} |i\rangle_{1} } |j\rangle_{2} |\mathbf{x}_{i}^{j} \rangle_{3} \left[\sqrt{1 - ({c_{1}\mathbf{x}_{i}^{j}})^{2}} |0\rangle + c_{1}\mathbf{x}_{i}^{j}|1\rangle \right]_{4}.
\label{eq:12}
\end{equation}
\par Finally, the inverse $O_X$ operation is applied on the third register. The quantum state
\begin{equation}
\left| {\phi\left( \mathbf{x}_{i} \right)} \right\rangle = \frac{1}{\sqrt{D}}{\sum\limits_{j = 0}^{D-1} |j\rangle_{2} \left[\sqrt{1 - ({c_{1}\mathbf{x}_{i}^{j}})^{2}} |0\rangle + c_{1}\mathbf{x}_{i}^{j}|1\rangle \right]_{4}}.
\label{eq:13}
\end{equation}
could be obtained via discarding the third register. Simplistically, this process is denoted as $U_{\mathbf{x}_i}$, which generates the state $\left| {\phi\left( \mathbf{x}_{i} \right)} \right\rangle$ in time $O(\text{polylog}(D)+q)$.
\par (1.2) In order to train the gradient, the parameter $\mathbf{w}(n)$ should be introduced in the $(n+1)$th iteration. Thus, it is necessary to generate a quantum state, which contains the information of $\mathbf{w}(n)$. Depending on the fact that the parameter is different in each iteration, there are two methods to prepare the quantum state.
\par One way is based on the assumption that QRAM is allowed to read and write frequently. For the information of $\mathbf{w}^{j}(n)$ ($j=0,1,\cdots,D-1$) are written in QRAM timely, the quantum state
\begin{equation}
\frac{1}{\sqrt{D}} \sum\limits_{j = 0}^{D-1} |j\rangle \left[\sqrt{1 - ({c_{2}\mathbf{w}^{j}(n)})^{2}}\left|0 \right\rangle + c_{2}\mathbf{w}^{j}(n)|1\rangle \right],
\label{eq:14}
\end{equation}
can be produced by the processes similar to step (1.1) with the help of the oracle $O_{\mathbf{w}}$ ($O_{\mathbf{w}}\left| j \right\rangle \left| 0 \right\rangle \longrightarrow\left| j \right\rangle | {\mathbf{w}^{j}(n)} \rangle $). The $c_{2}$ is set as $c_{2} = {1/\left\| \mathbf{w}(n) \right\|}$ and the $\mathbf{w}^{j}(n)$ is denoted as the $j$th element of the parameter vector in the $n$th iteration. This way can be implemented in time $O(\text{polylog}(D)+q)$.
\par For another, the parameter is extracted to the quantum state based on the operation $R(\vartheta)=\cos{(\vartheta)} |0\rangle\langle0| - \sin{(\vartheta)} |0\rangle\langle1| + \sin{(\vartheta)} |1\rangle\langle0| + \cos{(\vartheta)} |1\rangle\langle1|$, which is inspired by Ref. \cite{shao2019fast}. In this way, the $\mathbf{w}(n)$ is not required to be written in QRAM. The following are described as the processes.
\par Assuming that is easy to get $2^L-1$ ($L=\log(D)$) angle parameters $\bm{\vartheta}_{t} = (\vartheta_{t}^{0}, \cdots, \vartheta_{t}^{2^{t-1}-1} )$ ($t = 1, 2, \cdots, L$) from the updated $\mathbf{w}(n)$ after the last iteration. The angle $\vartheta_{t}^{j}$ satisfies
\begin{equation}
{\cos(\vartheta^{j}_{t})=\frac{h_{t}^{2j}}{h^{j}_{t-1}}}, ~~ {\sin (\vartheta^{j}_{t}) = \frac{h^{2j+1}_{t}}{h^{j}_{t-1}}},
\label{eq:14+1}
\end{equation} for $t =1, \cdots, L$, where $h^{j}_{t-1} = \sqrt{( h^{2j}_{t} )^{2} + ( h^{2j+1}_{t} )^{2}}$ and $j=0, \cdots, 2^{t-1}-1$. In particular, $h^{j}_{L} = \mathbf{w}^{j}(n)$ for $j = 0, 1, \cdots, D-1$. And there are defined
\begin{equation}
U(\bm{\vartheta}_{t}) =
\begin{cases}
\sum\limits_{j = 0}^{2^{t-1}-1} {|j\rangle \langle j| \otimes R({\vartheta}^{j}_{t}) \otimes I \{ {L-t}\}},&t=2, \cdots, L \\
R({\vartheta}^{j}_{t}) \otimes I\{ {L-t}\}, &t=1
\end{cases},
\label{eq:14+2}
\end{equation} where $I\{ {L-t}\}$ is represented as the gate $I$ applied on $(L-t)$ qubits.
\par After that, a quantum state
\begin{equation}
U(\bm{\vartheta}_{L}) \cdots U(\bm{\vartheta}_{2}) U(\bm{\vartheta}_{1})|0^{\otimes \log{D}}\rangle = {\sum_{j = 0}^{D-1}{c_{2}\mathbf{w}^{j}(n)\left| j \right\rangle}},
\label{eq:14+3}
\end{equation}
is generated in time $O(D)$ by applying the operation $U(\bm{\vartheta}_{t})$ for $t = 1, 2, \cdots, L$. Furthermore, a register in state $|1\rangle$ is appended. The overall system is in the state
\begin{equation}
\left| {\phi( \mathbf{w}(n))} \right\rangle = \sum_{j = 0}^{D-1}{c_{2}\mathbf{w}^{j}(n)\left| j \right\rangle}|1\rangle.
\label{eq:14+4}
\end{equation} To further interpret this method, an example is given in the Appendix \ref{appendix B}.
\par According to Eq. (\ref{eq:14+4}), the state in Eq. (\ref{eq:14}) can be rewritten as
\begin{equation}
\frac{1}{\sqrt{D}} \left[ \sum\limits_{j = 0}^{D-1} |j\rangle \sqrt{1 - ({c_{2}\mathbf{w}^{j}(n)})^{2}} |0 \rangle + | {\phi( \mathbf{w}(n))} \rangle \right].
\label{eq:14+5}
\end{equation}
It means that the above two methods both allow us to extract the parameter $\mathbf{w}(n)$ information into the quantum state $\left| {\phi( \mathbf{w}(n))} \right\rangle$. On the basis of the current quantum technology, we choose the second method which is more feasible, and denote the process as $U_{\mathbf{w}}$.
\par \textbf{(2) Generate an intermediate quantum state.} \par An intermediate quantum state is generated via utilizing $U_{\mathbf{x}_i}$ and $U_{\mathbf{w}}$, which contains the information of $\mathbf{x}_{i} \cdot \mathbf{w}$. The details are described in the following. \par (2.1) A quantum state is initialized as
\begin{equation}
\frac{1}{\sqrt{M}}{\sum\limits_{i = 0}^{M-1}| i \rangle_{1}}| 0^{\otimes{\log{D}}} \rangle_{2} | 0\cdots0\rangle_{3} |0\rangle_{4} | 0\rangle_{5}.
\label{eq:15}
\end{equation} The third register is stored the ancilla qubits, such as some qubits to be discarded in step (1.1).
\par (2.2) The Hadamard gate is performed on the fourth register. Then, a controlled operation $\left| i \right\rangle \left\langle i \right| \otimes U_{\mathbf{x}_{i}} \otimes \left| 0 \right\rangle \left\langle 0 \right| + I \otimes U_{\mathbf{w}} \otimes \left| 1 \right\rangle \left\langle 1\right|$ is applied to produce a state
\begin{equation}
\frac{1}{\sqrt{2M}}{\sum\limits_{i = 0}^{M-1}| i \rangle_{1}}\left[{| {\phi( \mathbf{x}_{i} )} \rangle _{24}\left| 0 \right\rangle _{5} + | {\phi( \mathbf{w}(n) )}\rangle_{24}| 1\rangle _{5}} \right].
\label{eq:16}
\end{equation} \par (2.3) Subsequently, the Hadamard gate is implemented on the fifth register to get
\begin{equation}
\left| \psi_{1} \right\rangle = \frac{1}{\sqrt{M}}{\sum\limits_{i = 0}^{M-1}{\left| i \right\rangle_{1} \left| \Psi_{i,n} \right\rangle_{245}}},
\label{eq:17}
\end{equation} where
\begin{equation}
\begin{split}
|\Psi_{i,n}\rangle = \frac{1}{2}&\left[( | {\phi( \mathbf{x}_{i})}\rangle + | {\phi( \mathbf{w}(n))}\rangle)|0\rangle \right.\\
&\left.+ ( | {\phi( \mathbf{x}_{i})}\rangle -| {\phi( \mathbf{w}(n))}\rangle )|1\rangle \right].
\end{split}
\label{eq:17+1}
\end{equation}
The state $|\Psi_{i,n}\rangle$ can be rewritten as
\begin{equation}
\begin{split}
| \Psi_{i,n}\rangle &=\cos\theta_{i} |\psi_{i}^{0}\rangle + \sin\theta_{i} | \psi_{i}^{1}\rangle\\
&= (\cdots)_{245}^{\bot} + \\
&\frac{1}{2\sqrt{D}}\left( {{\sum\limits_{j = 0}^{D-1}{c_{1}\mathbf{x}_{i}^{j}}} |j\rangle - {\sum\limits_{j^{\prime} = 0}^{D-1}{c_{2}^{\prime} \mathbf{w}^{j^{\prime}}(n)}} | j^{\prime}\rangle} \right)_{2}\left|11\right\rangle_{45}.
\end{split}
\label{eq:18}
\end{equation} It is easy to verify that
\begin{equation}
\sin^{2}\theta_{i} = \frac{c_{1}^{2}\| \mathbf{x}_{i} \|^{2} + c_{2}^{{\prime}2}\| \mathbf{w}\|^{2} - 2c_{1}c_{2}^{\prime}(\mathbf{x}_{i} \cdot \mathbf{w})}{4D},\theta_{i} \in \left[ 0,\frac{\pi}{2} \right],
\label{eq:19}
\end{equation}
and $c_{2}^{{\prime}} = \sqrt{D}{c_{2}}$. By observing Eq. (\ref{eq:18}) and Eq. (\ref{eq:19}), it can be found that the essential information is provided by the system when its fourth and fifth registers are both in state $|1\rangle$. It means that the superposition of $|0\rangle$ also does not affect the extraction of the required information if choosing the state of Eq. (\ref{eq:14+5}). Thus, the first method (in step (1.2)) is also suitable for our algorithm, which $c_{2}^{{\prime}} = c_{2}$.
\par \textbf{(3) Estimate the $F(\mathbf{x}_{i} \cdot \mathbf{w})$ in the quantum form.} \par The approximation of $F(\mathbf{x}_{i} \cdot \mathbf{w})$ should be estimated and stored in a quantum state. To achieve this goal, the $\theta_i$ is needed to estimate via quantum phase estimation which the unitary operation is defined as
\begin{equation}
Q_{i} = - \mathcal{A}_{i}S_{00}\mathcal{A}_{i}^{\dagger}S_{11},
\label{eq:20}
\end{equation}
where $\mathcal{A}_{i}\left| {0}^{\otimes{[\log{(D)}+2]}} \right\rangle = \left| \Psi_{i,n} \right\rangle$, $S_{00} = I^{\otimes {[\log{(D)}+2]}} - 2{\left|0^{\otimes{[\log{(D)}+2]}} \right\rangle\left\langle 0^{\otimes{[\log{(D)}+2]}} \right|}$ and $S_{11}=I^{\otimes{\log{(D)}}} \otimes \left( I^{\otimes{2}}- 2| 1^{\otimes{2}} \rangle \langle1^{\otimes{2}}| \right)$. Mathematically, the eigenvalues of $Q_i$ are $e^{\pm 2\mathbf{i}\theta_{i}}$ $(\mathbf{i} = \sqrt{- 1})$ and the corresponding eigenvectors are $\left| \Psi_{i,n}^{\pm} \right\rangle = \frac{1}{\sqrt{2}}\left( {\left| \psi_{i}^{0} \right\rangle \pm \mathbf{i}\left| \psi_{i}^{1} \right\rangle} \right)$), respectively. Based on the set of its eigenvectors, $\left| \Psi_{i,n} \right\rangle $ can be rewritten as $\left| \Psi_{i,n} \right\rangle = - \frac{\mathbf{i}}{\sqrt{2}}\left( {e^{\mathbf{i}\theta_{i}} \left| \Psi_{i,n}^{+} \right\rangle - e^{- \mathbf{i}\theta_{i}}\left| \Psi_{i,n}^{-} \right\rangle} \right)$. The procedure of estimating the $F(\mathbf{x}_{i} \cdot \mathbf{w})$ is displayed as follows.
\par (3.1) Performing the QPE on $Q_i$ with the state $\left| \psi_{1} \right\rangle \left| 0^{\otimes l} \right\rangle$ for some $l= O\left( \text{log}{\epsilon_{\theta}^{-1}} \right)$, an approximate state
\begin{equation}
\begin{split}
\left| \psi_{2} \right\rangle = \frac{- \mathbf{i}}{\sqrt{2M}}{\sum\limits_{i = 0}^{M-1}\left| i \right\rangle_{1}}&\left( e^{\mathbf{i}\theta_{i}}\left| \Psi_{i,n}^{+} \right\rangle \left| {\widetilde{\theta}}_{i} \right\rangle \right.\\
&\left.- e^{- \mathbf{i}\theta_{i}}\left| \Psi_{i,n}^{-} \right\rangle {\left| - {\widetilde{\theta}}_{i} \right\rangle} \right)_{2456},
\end{split}
\label{eq:21}
\end{equation}
is obtained, where ${\widetilde{\theta}}_{i} \in \mathbb{Z}_{2^{l}}$ satisfies $\left| {\theta_{i} - {\widetilde{\theta}}_{i}\pi/2^{l}} \right| \leq \epsilon_{\theta}$. Then, the quantum state
\begin{equation}
\left| \left. \psi_{3} \right\rangle \right. = \frac{1}{\sqrt{M}}{\sum\limits_{i = 0}^{M-1}{\left|i \right\rangle_{1}\left|\Psi_{i,n} \right\rangle_{245}| {{\sin}^{2}( {\widetilde{\theta}}_{i})} \rangle_{6}}}.
\label{eq:22}
\end{equation} is generated by using the sine gate. It holds for the fact that $ { {\sin}^{2} ( {\widetilde{\theta}}_{i} ) } = {{\sin}^{2}( -{\widetilde{\theta}}_{i})}$.
\par $(3.2)$ According to Eq. (\ref{eq:19}), it is needed to access $\left\| \mathbf{x}_{i} \right\|$ to compute $\mathbf{x}_{i} \cdot \mathbf{w}$. Combining with the operation $U_{nf}$ and the quantum arithmetic operations \cite{zhou2017}, we can get
\begin{equation}
\left|\psi_{4} \right\rangle = \frac{1}{\sqrt{M}}{\sum\limits_{i = 0}^{M-1}\left| i \right\rangle_{1}}\left| \Psi_{i,n} \right\rangle_{245}\left|{\mathbf{x}_{i} \cdot \mathbf{w}} \right\rangle_{6}\left| \left\| \mathbf{x}_{i} \right\| \right\rangle_{7}\left|\left\| \mathbf{w} \right\| \right\rangle_{8}.
\label{eq:23}
\end{equation}
\par $(3.3)$ An oracle $O_F$ is supposed to achieve any function which has a convergent Taylor series \cite{cong2016}. Combining with $O_y$, the function $F(*)$ could be implemented (a simple example is described in Sec. \ref{sec:5} ). The state becomes
\begin{equation}
\left|\psi_{5} \right\rangle = \frac{1}{\sqrt{M}}{\sum\limits_{i = 1}^{M}\left| i \right\rangle_{1}}\left| \Psi_{i,n} \right\rangle_{245}\left|F({\mathbf{x}_{i} \cdot \mathbf{w}}) \right\rangle_{6}\left| \left\| \mathbf{x}_{i} \right\| \right\rangle_{7}\left|\left\| \mathbf{w} \right\| \right\rangle_{8}.
\label{eq:24}
\end{equation}
\par $(3.4)$ Next, a register in the state $|0\rangle$ is appended as the last register and rotated it to $\left| \phi_{i} \right\rangle = c_{3}F\left( {\mathbf{x}_{i} \cdot \mathbf{w}} \right)\left|0 \right\rangle + \sqrt{1 - \left( c_{3}F\left( {\mathbf{x}_{i} \cdot \mathbf{w}} \right) \right)^{2}}\left| 1 \right\rangle$ in a controlled manner, where $c_{3} = {1/{\max\limits_{i}\left| {F\left( {\mathbf{x}_{i} \cdot \mathbf{w}} \right)} \right|}}$. This results in the overall state
\begin{equation}
\begin{split}
&\left|\psi_{6} \right\rangle = \\
&\frac{1}{\sqrt{M}}{\sum\limits_{i = 0}^{M-1}\left| i \right\rangle_{1}}\left| \Psi_{i,n} \right\rangle_{245}\left|F({\mathbf{x}_{i} \cdot \mathbf{w}}) \right\rangle_{6}\left| \left\| \mathbf{x}_{i} \right\| \right\rangle_{7}\left|\left\| \mathbf{w} \right\| \right\rangle_{8} {\left| \phi_{i} \right\rangle_{9}}.
\end{split}
\label{eq:25}
\end{equation}
\par $(3.5)$ The inverse operations of steps (2.2)-(3.3) is performed on $\left|\psi_{6} \right\rangle$. Afterwards, a register in the state $|1\rangle$ is added to obtain
\begin{equation}
\left|\psi \right\rangle = \frac{1}{\sqrt{M}}{\sum\limits_{i = 0}^{M-1}{\left| i \right\rangle _{1}\left| \phi_{i} \right\rangle_{9}}}\left|1 \right\rangle_{10}.
\label{eq:26}
\end{equation}
For convenience, the $\mathcal{A}_{\psi}$ is marked as the operations which achieve $\mathcal{A}_{\psi}|00\cdots 0\rangle = |\psi\rangle$. Its schematic quantum circuit is given in Fig. \ref{fig:3}.
\begin{figure*}
\caption{Quantum circuit diagram of $\mathcal{A}_{\psi}$. Here, $q$ is the number of bits required to adequately store the data information and $\epsilon_\theta$ is the tolerance error for estimating $\theta_{i}$. Furthermore, $QAO$ denotes the quantum arithmetic operations, $CR_3$ represents the controlled rotation operation about $|\phi_{i}\rangle$ and $U^{\dagger}$ labels the inverse operations in the step (3.5).}
\label{fig:3}
\end{figure*}
\par \textbf{(4) Obtain the gradient $\bm{g}^{j}\left( {\mathbf{w}(n)} \right)$ in the classical form.}
\par $(4.1)$ Three registers in state $ \frac{1}{\sqrt{M}}{\sum_{i = 0}^{M-1}{\left|i \right\rangle_{1} \left|j \right\rangle_{2} }}\left| 0 \right\rangle_{3} $ are prepared. Performing $O_{X}$ on it to generate the state
\begin{equation}
\frac{1}{\sqrt{M}}{\sum\limits_{i = 0}^{M-1}{ | i \rangle_{1} |j \rangle_{2}}} | \mathbf{x}_{i}^{j} \rangle_{3}.
\label{eq:27}
\end{equation}
\par $(4.2)$ The controlled rotation operation ($|0\rangle \rightarrow \sqrt{1 - ({c_{1}\mathbf{x}_{i}^{j}})^{2}} \left|0 \right\rangle + c_{1}\mathbf{x}_{i}^{j}\left|1 \right\rangle$) is implemented to get
\begin{equation}
\frac{1}{\sqrt{M}}{\sum\limits_{i = 0}^{M-1}{ |i \rangle_{1} \left|j \right\rangle_{2} }} | \mathbf{x}_{i}^{j} \rangle_{3} \left[ {\sqrt{1 - ( {c_{1} {\mathbf{x}}_{i}^{j}} )^{2}}\left|0 \right\rangle + c_{1} \mathbf{x}_{i}^{j} |1 \rangle} \right]_4.
\label{eq:28}
\end{equation}
\par $(4.3)$ The inverse operation of $O_X$ is performed. After that, we can obtain the state
\begin{equation}
| \chi^{j} \rangle = \frac{1}{\sqrt{M}}{\sum\limits_{i = 0}^{M-1}{| i \rangle_{1} | 0 \rangle_{3} }}\left[ {\sqrt{1 - ( {c_{1} \mathbf{x}_{i}^{j}} )^{2}} |0 \rangle + c_{1}\mathbf{x}_{i}^{j} | 1 \rangle} \right]_{4},
\label{eq:29}
\end{equation}
via undoing the register $|j\rangle$.
\par $(4.4)$ In order to obtain the gradient, the technology of swap test \cite{buhrman2001} is utilized. Combining the processes of generating the states $\left| \psi \right\rangle$ and $\left| \chi^{j} \right\rangle$, a quantum state $\frac{1}{\sqrt{2}}\left( \left|0 \right\rangle \left|\psi \right\rangle + \left|1 \right\rangle \left| \chi^{j} \right\rangle \right)$ can be constructed. Then, measuring the first register to see whether it is in the state $\left| + \right\rangle = \frac{1}{\sqrt{2}}\left( \left| 0 \right\rangle + \left|1 \right\rangle\right)$. The measurement has the success probability
\begin{equation}
P = \frac{1}{2} + \frac{1}{2}\left\langle \psi \middle| \chi^{j} \right\rangle.
\label{eq:30}
\end{equation} According to Eq. (\ref{eq:4}), the $\bm{g}^{j}\left( {\mathbf{w}(n)} \right) = {(2P - 1)}/{({c_{1}c_{3}})}$ can be calculated. Hence, it is possible for ${\rm Bob}_k$ to obtain the local gradient $\bm{g}_{k}\left( \mathbf{w} \right) = \left( {\bm{g}_{k}^{0}\left( {\mathbf{w}(n)} \right),\bm{g}_{k}^{1}\left( {\mathbf{w}(n)} \right), \cdots, \bm{g}_{k}^{D-1} \left( {\mathbf{w}(n)} \right)} \right)^{T}$ by repeating the steps of the above algorithm with his data.
\subsection{\label{sec:3.2}Quantum secure multi-party computation protocol} \par We will design a protocol to safely compute the federated gradients $\bm{G}\left( \mathbf{w} \right) = {\sum_{k = 1}^{K}{\beta_{k}}{\bm{g}_{k}\left( \mathbf{w} \right)}}$ in this section. That is, calculating $\bm{G}\left( \mathbf{w} \right)$ without revealing the local gradient ${\bm{g}_{k}\left( \mathbf{w} \right)}$. To do it, the server Alice is assumed to be a semi-honest who may misbehave on her own but cannot conspire with others. Moreover, the federated gradients are needed to be accurate to ${\gamma}^{-1}$. This means that $\gamma {\beta_{k}} {\bm{g}_{k}^{j} ( \mathbf{w} )} \geq 0$. Simply, the $\gamma {\beta_{k}} {\bm{g}_{k}^{j} ( \mathbf{w} )}$ is marked as $\mu_{k}^{j}$. And supposing that ${\sum_{k = 1}^{K} \mu_{k}^{j}} < S$. The further details are described as follows.
\par \textbf{Step M1: Preparation.} Alice announces ${\gamma}$ and the global dataset scale $(\sum_{k = 1}^{K}{M_{k}})$. At same time, the participants (server and clients) choose $m$ numbers $d_i$ ($i=1,2,\cdots,m$) which are mutually prime and satisfy $d_{1} \times d_{2} \times \cdots \times d_{m} = S$. Subsequently, ${\rm Bob}_{k}$ $(k=1,2,\cdots,K)$ calculates his secret
\begin{equation}
s_{k,i}^{j} = \mu_{k}^{j} ~\text{mod}~d_{i}.
\label{eq:31}
\end{equation} Alice produces a $d_{i}$-level $(K+1)$-particle GHZ state
\begin{equation}
\left| \Psi \right\rangle = \frac{1}{\sqrt{d_{i}}}{\sum\limits_{q = 0}^{d_{i} - 1} | q \rangle ^{\bigotimes(K + 1)}},
\label{eq:32}
\end{equation} and marks the $(K+1)$ particles by $Q_{0},Q_{1},\cdots,Q_{K}$.
\par \textbf{Step M2: Distribution.} For the sake of checking the presence of eavesdroppers, Alice prepares $K$ sets of $\delta$ decoy states, where each decoy photon randomly is in one of the states from the set $V_{1} = \left\{ \left| p \right\rangle \right\}_{p = 0}^{d_{i} - 1}$ and $V_{2} = \left\{ F\left|p \right\rangle\right\}_{p = 0}^{d_{i} - 1}$. These sets are denoted as $E_{1},E_{2},\cdots,E_{K}$, respectively. Then Alice inserts $Q_k$ into $E_k$ at a random position, and sends them to ${\rm Bob}_k$ for $k=1,2,\cdots,K$.
\par \textbf{Step M3: Security checking.} After receiving $\delta+1$ particles, ${\rm Bob}_k$ sends acknowledgements to Alice. Subsequently, the positions and the bases of the decoy photons are announced to ${\rm Bob}_k$ by Alice. ${\rm Bob}_k$ measures the decoy photons and returns the measurement results to Alice who then calculates the error rate by comparing the measurement results with initial states. If the error rate is higher than the threshold determined by the channel noise, Alice cancels this protocol and restarts it. Otherwise, the protocol is continued.
\par \textbf{Step M4: Measurement and encoding.} ${\rm Bob}_k$ extracts all the decoy photons and discards them. Then, server and clients perform a measurement $\left\{ F\left| p \right\rangle \right\}_{p = 0}^{d_{i} - 1}$ on the remaining particles, respectively. The measurement results record as $o_{s,i},o_{1,i},\cdots,o_{K,i}$ and these satisfy $o_{s,i}+o_{1,i}+\cdots+o_{K,i}=0 ~\text{mod}~ d_{i}$. Subsequently, ${\rm Bob}_k$ encodes his data $s_{k,i}^{\prime j} = s_{k,i}^{j} + o_{k,i}$ and sends it to Alice.
\par \textbf{Step M5: Server's computation.} At this stage, Alice accumulates all the results $s_{k,i}^{\prime j}$ to compute
\begin{equation}
\begin{split}
&\left( {o_{s} + o_{1,i} + s_{1,i}^{j} + o_{2,i} + s_{2,i}^{j}\cdots + o_{K,i} + s_{K,i}^{j}} \right)~\text{mod}~d_{i}\\
=&\left( {\mu_{1}^{j}} + {\mu_{2}^{j}} + \cdots + {\mu_{K}^{j}} \right)~\text{mod}~d_{i}.
\end{split}
\label{eq:33}
\end{equation} For $i=1,2,\cdots,m$, Alice can obtain $m$ equations such as Eq. (\ref{eq:33}). According to the Chinese remainder theorem, Alice compute the summation
\begin{equation}
\left( {\sum\limits_{k = 1}^{K} {\mu_{k}^{j}}} \right)~\text{mod}~S = ~{\sum\limits_{k = 1}^{K} {\mu_{k}^{j}}}.
\label{eq:34}
\end{equation} And it is easy to get the federated gradient
\begin{equation}
\bm{G}^{j}\left( \mathbf{w} \right) = \frac{1}{\gamma}\sum\limits_{k = 1}^{K} {\mu_{k}^{j}} = { \sum\limits_{k = 1}^{K} {\beta_{k}} {\bm{g}_{k}^{j} ( \mathbf{w} )} }.
\label{eq:35}
\end{equation}
\par After the similar processes, the federated gradient $( \bm{G}^{0}( \mathbf{w} ), \bm{G}^{1}( \mathbf{w} ), \cdots, \bm{G}^{D-1}( \mathbf{w} ) )$ could be obtained by Alice. And she updates the parameters $\mathbf{w}( {n + 1}) = \mathbf{w}(n) - \alpha \times \bm{G}\left( {\mathbf{w}(n)} \right)$ and announces the parameters $\mathbf{w}( {n + 1})$. Moreover, the corresponding quantum state $|\mathbf{w}(n+1)\rangle$ can be prepared by the methods (as shown in step (1.2) of Sec. \ref{sec:3.1}), to perform a new round of the QFLGD framework. The above processes are repeated until the accuracy satisfies ${\sum_{j = 0}^{D-1}\left[ {\bm{G}^{j}( \mathbf{w})} \right]^{2}} \leq \varepsilon$.
\par In order to exhibit the protocol more clearly, a concrete example is proposed. Considering the model is trained by two clients (${\rm Bob}_{1}$, ${\rm Bob_{2}}$) who respectively have a $1$-scale dataset, with the help of a server (Alice). The gradients $g_{1}^{1} = 2$, $g_{1}^{2} = 3.46$, $g_{2}^{1} = 5$, and $g_{2}^{2} = 8.66$ are assumed to be gained in the QGD algorithm. Simply, the eavesdropping check phase is ignored.
\par Firstly, Alice announces the accuracy of parameters is $\gamma^{-1} = 1/100$ and the global dataset scale is $M_{1} + M_{2} = 2$. She chooses $d_{1} = 23$ and $d_{2} = 29$ with clients. After that, ${\rm Bob}_{1}$ calculates his secret
\begin{equation}
s_{1,1}^{1} = ~ \mu_{1}^{1} \mod ~ 23= 8,
\label{eq:35+1}
\end{equation} $s_{1,2}^{1}=13$, $s_{1,1}^{2}=12$, and $s_{1,2}^{2}=28$. At same time, ${\rm Bob}_{2}$ can get $s_{2,1}^{1} = 20$, $s_{2,2}^{1} = 18$, $s_{2,2}^{1} = 19$, and $ s_{2,2}^{2} = 27$.
\par Secondly, Alice prepares a $23$-level-$3$ particle GHZ state $\left| \Psi \right\rangle = \frac{1}{\sqrt{23}}{\sum\limits_{q = 0}^{22} |q\rangle |q\rangle |q\rangle}$ for $d_{1} = 23$ and gives a particle to each client respectively. Then these parameters perform the measurement to get $o_{s,1} = 7$, $o_{1,1} = 6$, and $o_{2,1} = 10$. ${\rm Bob}_{1}$ (${\rm Bob}_{2}$) encodes his secret by using $o_{1,1}$ ($o_{2,1}$) and sets to Alice. The result
\begin{equation}
\begin{split}
&(7+8+6+20+10)\mod~23 \\
=&(\mu_{1}^{1}+\mu_{2}^{1})\mod~23=5,
\end{split}
\label{eq:35+2}
\end{equation} could be computed by Alice. \par Finally, the equations
\begin{equation}
\begin{split}
&(\mu_{1}^{1}+\mu_{2}^{1})\mod~23=5,\\
&(\mu_{1}^{1}+\mu_{2}^{1})\mod~29=2,
\end{split}
\label{eq:35+3}
\end{equation} and
\begin{equation}
\begin{split}
&(\mu_{1}^{2}+\mu_{2}^{2})~\mod~23=8,\\
&(\mu_{1}^{2}+\mu_{2}^{2})~\mod~29=26,
\end{split}
\label{eq:35+4}
\end{equation} could be obtained through a similar procedure. According to the Chinese remainder theorem, the federated gradient (3.5,6.06) is easy to get.
\section{\label{sec:4}Analysis} \par In this section, we provide a brief analysis of the proposed framework. As discussed previously, the QGD algorithm (shown in Sec. \ref{sec:3.1}) enables clients to accelerate the training gradients on a local quantum computer. The QSMC protocol (shown in Sec. \ref{sec:3.2}) gives a method to securely update the federated parameters to protect the privacy of clients' data. Therefore, two main aspects are considered in the analysis. One is the time complexity of QGD algorithm. The other is the security of QSMC protocol.
\subsection{\label{sec:4.1}Time complexity of the QGD algorithm} \par In the QFLGD framework, assuming that $M_{1} \leq M_{2} \leq \cdots \leq M_{K} \leq M$. Namely, the dataset scale is at most $M$. And all clients need to accomplish the gradient training before calculating the federated gradient. Thus the waiting time for the distributed training gradient is the time consumed to train the dataset which scale is $M$. In the following, the time complexity of the QGD algorithm is analyzed with the $M$-scale dataset.
\par In step $(1)$, the time consumption is caused by the processes of $U_{\mathbf{x}_{i}}$ and $U_{\mathbf{w}}$, which generate the states $\left| {\phi( \mathbf{x}_{i})} \right\rangle $ and $\left| {\phi( \mathbf{w}(n))} \right\rangle $ about data information. It could be implemented in time $O(\text{polylog}(MD)+D+q)$ with the help of the $O_{X}$, $U(\bm{\vartheta}_{t})$ and the controlled rotation operation \cite{HHL,wossnig2018}. The $q$ is represented as the number of qubits which store the data information. Afterwards, $U_{\mathbf{x}_{i}}$ and $U_\mathbf{w}$ are applied to produce the state $\left| \psi_{1} \right\rangle$ in step (2). Hence, step (2) can be implemented in time $O(\text{polylog}(MD)+D+q)$. \par In step (3), we first consider the complexity of the unitary operation $\mathcal{A}_{i}$. It contains $H$, $U_{\mathbf{x}_{i}}$, and $U_\mathbf{w}$ which take time $O(\text{polylog}(MD)+D+q)$. Then, the quantum amplitude estimation (QPE) block needs $O\left( {1/\epsilon_{\theta}} \right)$ applications of $Q_{i} = - \mathcal{A}_{i}S_{00}\mathcal{A}_{i}^{\dagger}S_{11}$ to achieve error $\epsilon_{\theta}$ \cite{brassard2002}. Therefore, the time of accomplishing step (3.1) is $O[(\text{polylog}(MD)+D+q)/{\epsilon_{\theta}}]$. The runtime $O( \text{log}( {1/\epsilon_{\theta}}))$ \cite{zhou2017} of implementing the sine gate can be ignored, which is much smaller than the QPE.
\par Next, the complexity of steps (3.2)-(3.3) is discussed. The main operation of the two steps includes $U_{nf}$, $O_{y}$, and the quantum arithmetic operation, which are performed to calculate $| F ( {\mathbf{x}} \cdot {\mathbf{w}} ) \rangle $ in time $O[{(\text{polylog}(D))/{\epsilon_{m}}} + q]$. In step (3.4), the complexity of the controlled rotation is $O(q)$. Step (3.5) takes time $O[ {(\text{polylog}(D))/{\epsilon_{m}}} + { ( \text{polylog}(MD) + D + q ) / { \epsilon_{\theta} } } ]$ to implement the inverse operations of steps (2.2)-(3.3). Putting all the steps together to get the complexity of step (3) as $O[ {(\text{polylog}(D))/{\epsilon_{m}}} + { ( \text{polylog}(MD) + D + q ) / { \epsilon_{\theta} } } ]$.
\par In step (4), the processes of generating the $| \chi^{j} \rangle$ (described in steps (4.1)-(4.3)) are accomplished in time $O(\text{polylog}(MD)+q)$. According to step (3), a copy of the quantum state $| \psi \rangle$ is produced in time $O[{(\text{polylog}(D))/{\epsilon_{m}}} + {(\text{polylog}(MD)+D+q)/{\epsilon_{\theta}}}]$. The swap test is applied $O\left( {{P\left( {1 - P} \right)}/\epsilon_{P}^{2}} \right) = O\left( {1/\epsilon_{P}^{2}} \right)$ times to get the result $P$ within error $\epsilon_{P}$ \cite{rebentrost2014} in step (4.4). And each swap test should prepare a copy of $| \chi^{j} \rangle$ and $| \psi \rangle$. Therefore, the runtime is $\left\{ [{(\text{polylog}(D))/{\epsilon_{m}}} + {(\text{polylog}(MD)+D+q)/{\epsilon_{\theta}}}] \epsilon_{P}^{-2} \right\}$ in step (4), that is the complexity of obtaining the desired result.
\par For convenience, we assume that $\mathbf{w}^{j}$, $\mathbf{x}_{i}^{j}=O(1)$, then $\| \mathbf{w} \|$, ${\max\limits_{i}\left\| \mathbf{x}_{i} \right\|} = O( \sqrt{D})$. Therefore, $q={\text{polylog}(D)}$ could fulfill the number of qubit required to store data information. In addition, taking $\epsilon_{m}$, $\epsilon_{\theta}$, and $\epsilon_{P}$ equaling to $\epsilon$. After that, the complexity of the entire quantum algorithm to get $\bm{g}^{j}\left( \mathbf{w} \right)$ $(j=0,1,\cdots,D-1)$ in each iteration can further simplify into
\begin{equation}
O\left\{ {D\left[ {\left( \text{polylog}\left( {MD} \right) + D \right)/\epsilon^{3}} \right]} \right\}.
\label{eq:36}
\end{equation} This means that the time complexity of training gradient is $O(D^{2} \text{polylog}(MD))$ when $\epsilon^{-1} = \text{polylog}(MD)$, achieving exponential acceleration on the number of data samples. Furthermore, the elements of $\mathbf{w}$ can also be accessed in time $O( \text{polylog}(D) )$ if they are timely writing in QRAM. In this case, the proposed algorithm has exponential acceleration on the number $M$ and the quadratic speedup in the dimensionality $D$, compared with the classical algorithm whose runtime is $O(M D^{2})$.
\subsection{\label{sec:4.2}Security analysis of the QSMC protocol} \par In this section, the security of the QSMC protocol will be analyzed. For the secure multi-party computing, the attacks from outside and all participants are the challenges, which have to deal with. In the following, we will show these attacks are invalid to our protocol.
\par Firstly, the outside attacks are discussed. In this protocol, the decoy photons is used to prevent the eavesdropping. This idea is derived from the BB84 protocol \cite{bennett1984}, which has been proved unconditionally safe. Here, we take the intercept-resend attack as an example to demonstrate. If an outside eavesdropper Eve attempts to intercept the particles sent from Alice and replaces them with his own fake particles, he will introduce extra error rate $1 - \left( \frac{d_{i}+1}{2d_{i}} \right)^{\delta}$. Therefore, Eve will be detected in Step M3 through security analysis.
\par Secondly, the participant attacks are analyzed. In the proposed protocol, the participants include the server (Alice) and clients (${\rm Bob}_{k}$, $k=1,2,\cdots, K$) who can access more information. Therefore, the participant attacks from dishonest clients or server should be considered.
\par For the participant attack from dishonest clients, there are only considering the extreme case that $K-1$ clients ${\rm Bob}_{1}, \cdots, {\rm Bob}_{k-1}, {\rm Bob}_{k+1}, \cdots, {\rm Bob}_{K}$ collude together to steal the secret from ${\rm Bob}_{k}$, because $K-1$ clients have the most powerful strength. In this case, even if the dishonest clients share their information, they cannot deduce $o_{k}$ without the help of Alice. That means they can't obtain the secret $s_{k,i}^{j}$ by $s_{k,i}^{\prime j} = s_{k,i}^{j} + o_{k}$. Thus, our algorithm can resist the collusion attack of dishonest ${\rm Bob}_{k}$.
\par For the attack from Alice, the semi-honest Alice may steal the private information of $C_k$ without conspiring with any one. In Step M4, Alice collects $s_{k,i}^{\prime j}$ for $k=1,2,\cdots,K$. However, she still cannot learn $s_{k,i}^{j}$ due to the lack of knowledge about $o_{k}$ which from clients.
\section{\label{sec:5}Application: Training the Federated Linear Regression Model}
\subsection{\label{sec:5.1}Quantum federated linear regression algorithm} \par Linear regression (LR) is an important supervised learning algorithm, which establishes a model of the relationship between the variable $\mathbf{x}_{i}$ and the observation $y_i$. It was wide application in the scientific fields of biology, finance, and so on \cite{geron2022}. LR models are also usually fitted by minimizing the function in Eq. (\ref{eq:1}) and choosing ${f\left( {\mathbf{x}_{i} \cdot \mathbf{w}} \right)} = {\mathbf{x}_{i} \cdot \mathbf{w}} + {b}$ ($b$ is a migration parameter).
\par In this section, we research a quantum federated linear regression (QFLR) algorithm based on gradient descent. The main idea is to implement the function
\begin{equation}
F\left( {\mathbf{x}_{i} \cdot \mathbf{w}} \right) = \mathbf{x}_{i} \cdot \mathbf{w} + b - y_{i}.
\label{eq:38}
\end{equation}
So that QFLGD algorithm can be applied by a simple modification. The state $| {\mathbf{x}_{i} \cdot \mathbf{w}} \rangle$ about ${\mathbf{x}_{i} \cdot \mathbf{w}}$ can be generated according to algorithm 1. Then, the state $| \mathbf{x}_{i} \cdot \mathbf{w} + b - y_{i} \rangle$ is produced in the following steps.
\par (S1) The oracle $O_y$ is applied on the state ${| {\mathbf{x}_{i} \cdot \mathbf{w}} \rangle}_{A} |0\rangle_{B}^{\otimes{q}}$ to get
\begin{equation}
{| {\mathbf{x}_{i} \cdot \mathbf{w}} \rangle}_{A} |y_{i}\rangle_{B}^{\otimes{q}},
\label{eq:39}
\end{equation} in time $O(\text{log}(D))$.
\par (S2) With knowing of ${\mathbf{x}_{i} \cdot \mathbf{w}}$ and $y_i$ correspond the qubit state ${| {\mathbf{x}_{i} \cdot \mathbf{w}} \rangle} = {| e_{q},e_{q-1},\cdots,e_{1} \rangle}$, ${|y_{i}\rangle} = {|t_{q},t_{q-1},\cdots,t_{1} \rangle}$ respectively. We implement the quantum Fourier transform (QFT) on $| {\mathbf{x}_{i} \cdot \mathbf{w}} \rangle$ result in
\begin{equation}
\left[ {\left|{\phi_{q}(e)} \right\rangle\otimes \left|{\phi_{q - 1}(e)} \right\rangle\otimes \cdots \otimes \left| {\phi_{1}(e)} \right\rangle} \right]_{A}\left|y_{i} \right\rangle_{B},
\label{eq:40}
\end{equation}
where $\left|{\phi_{j}(e)} \right\rangle = \frac{1}{\sqrt{2}}\left( \left| 0 \right\rangle + e^{2\pi \mathbf{i}0.e_{j}e_{j - 1}\cdots e_{1}}\left|1 \right\rangle\right)$ for $j=1,2,\cdots,q$.
\par (S3) Subsequently, the controlled rotation operation ${I \otimes |0\rangle \langle 0 |} + { R_{j} \otimes |1\rangle \langle 1 |}$ is performed to get
\begin{equation}
\left[ {\left|{\phi_{q}(e-t)} \right\rangle\otimes \left|{\phi_{q - 1}(e-t)} \right\rangle\otimes \cdots \otimes \left| {\phi_{1}(e-t)} \right\rangle} \right]_{A}\left|y_{i} \right\rangle_{B}.
\label{eq:41}
\end{equation} The $R_{j}$ is defined as $R_{j} = \begin{bmatrix}1 & 0 \\ 0 & e^{{- 2\pi \mathbf{i}}/2^{j}} \\ \end{bmatrix}$.
\par (S4) Inverse QFT is applied, the state becomes
\begin{equation}
\left[ {\left|{e_{q} - t_{q}} \right\rangle\otimes \left|{e_{q-1} - t_{q-1}} \right\rangle\otimes \cdots \otimes \left| {e_{1} - t_{1}} \right\rangle} \right]_{A}\left|y_{i} \right\rangle_{B}.
\label{eq:42}
\end{equation}
Thus, the state $| \mathbf{x}_{i} \cdot \mathbf{w} - y_{i} \rangle$ can be obtained from the register A. Moreover, choosing the controlled rotation operation ${I \otimes |0\rangle \langle 0 |} + { R_{j}^{\dagger} \otimes |1\rangle \langle 1 |}$ in step (S3), to achieve the addition operation $| \mathbf{x}_{i} \cdot \mathbf{w} - y_{i} + b\rangle$. The quantum circuit of this process is presented in Fig. \ref{fig:4}. The operations of these processes are labeled as $U_{s}$, which are implemented in time $O( \text{log}(D) + q^{2})$. Combining with the QFLGD framework, the federated linear regression mode could be constructed as follows.
\begin{table}[!htbp] \centering \renewcommand\arraystretch{1.2} \begin{tabular}{lp{0.38\textwidth}} \hline \multicolumn{2}{l}{\textbf{Algorithm 2:} Quantum federated LR algorithm}\\ \hline \textbf{Input:} &The variate $\mathbf{X} \in \mathbb{R}^{M \times D}$, the observed vector $\mathbf{y} \in \mathbb{R}^{M}$, the initial parameter $\mathbf{w}(0) \in \mathbb{R}^{D}$, the migration parameter $b$, the learning rate $\alpha$ and the preset values $c_{i}$ $(i=1,2,3)$;\\
\textbf{Output:} &{The parameter $\mathbf{w}$ and the model $y = \mathbf{w}^{T}\mathbf{x}_{i} + b$;}\\
\multicolumn{2}{l}{\textbf{for} \quad {${\sum\limits_{j = 0}^{D-1}\left( {\bm{G}^{j}\left( {\mathbf{w}(n - 1)} \right)} \right)^{2}}$ $>$ $\varepsilon$ \quad \textbf{do}} }\\ \quad Step 1: &{ $K$ clients apply the QGD Algorithm (in Sec. \ref{sec:3.1}) and $U_s$ to compute $\bm{g}_{k}\left( {\mathbf{w}(n)} \right)$ ($k=1,2,\cdots,K$) respectively;} \\ \quad Step 2: &$K$ clients and server use the QSMC protocol (in Sec. \ref{sec:3.2}) to secure calculate the federated gradient $\bm{G}\left( \mathbf{w}(n) \right) = \left( \bm{G}^{0}\left( {\mathbf{w}(n)} \right), \bm{G}^{1}\left( {\mathbf{w}(n)} \right),\cdots,\bm{G}^{D-1}\left( {\mathbf{w}(n)} \right) \right)$;\\
\quad Step 3: &Server updates parameter $\mathbf{w}( {n + 1}) = \mathbf{w}(n) - \alpha \times \bm{G}\left( {\mathbf{w}(n)} \right)$ and sends it to $K$ clients;\\ \multicolumn{2}{l}{\textbf{end}}\\ \hline \end{tabular} \end{table}
\begin{figure*}
\caption{Quantum circuit of the quantum computing $| \mathbf{x}_{i} \cdot \mathbf{w} - y_{i} + b\rangle$. The left side of the dotted line is the circuit of subtraction, and the right side is the circuit of addition. It is worth noting that the $QFT^{\dagger}QFT$, so we omitted them left and right of the dotted line.}
\label{fig:4}
\end{figure*}
\subsection{\label{sec:5.2}Numerical Simulation}
\par In this section, the numerical simulation of the QFLR algorithm will be presented. In our simulation, two clients (${\rm Bob}_{1}$, ${\rm Bob}_{2}$) trained the QFLR model with a sever (Alice). The experiment is implemented on the Qiskit quantum computing framework \cite{aleksandrowicz2019qiskit}. The initial weight $\mathbf{w}(0) = (0.866,0.5)$ and the migration parameter $b=0$ are selected by Alice. ${\rm Bob}_{1}$ chooses an input vector $\mathbf{x}=(2,3.464)$ which corresponds the observation $y=2.464$. Another client ${\rm Bob}_{2}$ selects an input vector $\mathbf{x}^{\prime}=(2.5,4.33)$ and the corresponding observation $y^{\prime}=2.33$.
\par In the process of training the federated linear regression model, the main is to achieve the perfect $F\left( {\mathbf{x}_{i} \cdot \mathbf{w}} \right)$ calculation. That is, quantum computing $F\left( {\mathbf{x}_{i} \cdot \mathbf{w}} \right)$ values are required to be able to be stored in quantum registers with small error. An experiment of this step is presented with ${\rm Bob}_{1}$'s data. With loss of generality, setting $c_1=4$, $c_2=1$, and the error $\epsilon_{\theta} = 0.0001$ of quantum phase estimation. By substituting these into Eq. (\ref{eq:19}) and Eq. (\ref{eq:38}), the result
\begin{equation}
\begin{split}
F\left( {\mathbf{x} \cdot \mathbf{w}} \right) = 4 - 16{\sin}^{2}\left( {\widetilde{\theta}\pi/2^{4}} \right) + 0 - 2.464 = 1,
\end{split}
\label{eq:43}
\end{equation} could be obtained.
\par According to the ${\sin}^{2}( {\widetilde{\theta}\pi/2^{4}} ) \approx \widetilde{\theta}^{2}/26$ and the fact of the most probable result $| 0001 \rangle$ (see Fig. \ref{fig:6}(a)) from the QPE, Eq. (\ref{eq:43}) can be rewritten as
\begin{equation}
6.5 \times F\left( {\mathbf{x} \cdot \mathbf{w}} \right) \approx 26 - 4\widetilde{\theta} - 16.
\label{eq:44}
\end{equation}
Its circuit is designed as exhibited in Fig. \ref{fig:5} and encoded via Qiskit. The $\mathcal{A}$ gate in Fig. \ref{fig:5}(a) is to produce $\left|\Psi_{i,n} \right\rangle$ and can be compiled into the basic gates (see Fig. \ref{fig:5}(c)). In Fig. \ref{fig:5}(c), the matrix form of $U(\gamma,\phi,\lambda)$ is
\begin{equation}
U\left( {\gamma,\phi,\lambda} \right) =
\begin{bmatrix}
{\cos \left( {\gamma/2} \right)} & {- e^{\mathbf{i}\lambda}\sin \left( {\gamma/2} \right)} \\
{e^{\mathbf{i}\phi}\sin \left( {\gamma/2} \right)} & {e^{\mathbf{i}(\phi + \lambda)}\cos \left( {\gamma/2} \right)} \\
\end{bmatrix}.
\label{eq:45}
\end{equation}
With the help of the IBM's simulator (aer$\underline{~}$simulator), the measurement results can be get which are shown in Fig. \ref{fig:6}(b). In Fig. \ref{fig:6}(b), two values ($| 01001 \rangle$ and $| 10001 \rangle$) stand out, which have a much higher probability of measurement than the rest. Based on the analysis of the phase estimation results, selecting result $| 00110 \rangle$ with a high probability of $0.510$. It means $F\left( {\mathbf{x} \cdot \mathbf{w}} \right)\approx 0.923$. Compared with the theoretical result (shown in Eq. (\ref{eq:43})), the experimental result has an error of $0.077$ which is tolerable. Subsequently, ${\rm Bob}_{1}$ can estimate $\bm{g}_{1}^{1} \approx 1.846 $ and $\bm{g}_{1}^{2} \approx 3.197$ by performing swap test. At same time, ${\rm Bob}_{2}$ estimates $F\left( {\mathbf{x}^{\prime} \cdot \mathbf{w}} \right)\approx 2.115$, $\bm{g}_{2}^{1} \approx 3.197$, and $\bm{g}_{2}^{2} \approx 9.157$ of his data via similar experiment.
\par As the analogous process of the example shown in \ref{sec:3.2}, Alice calculates the federated gradient $G=(3.57,6.18)$ via the QSMC protocol. Theoretical analysis shows that the error is within $12\%$ of the actual solution $(3.5,6.06)$ which is obtained in the example. Thus, the training algorithm is found to be successful.
\begin{figure*}\label{fig:5a}
\label{fig:5b}
\label{fig:5c}
\label{fig:5}
\end{figure*}
\begin{figure}\label{fig:6a}
\label{fig:6b}
\label{fig:6}
\end{figure}
\section{\label{sec:6}Conclusions} \par We have presented a framework for federated learning based on gradient descent that can securely implement FL over an exponentially large data set. In this framework, we first proposed a quantum gradient descent algorithm that allowed the time-consuming gradient calculation to be done on a quantum computer. And the time complexity analysis is shown that our algorithm is exponentially faster than its classical counterpart on the number of data samples when the error $1/\epsilon=\text{polylog}(MD)$. Furthermore, the QGD algorithm could also achieve quadratic speedup on the dimensionality of the data sample if the parameters $\mathbf{w}$ are stored in QRAM timely. Secondly, we gave a quantum secure multi-party computation protocol to calculate the federated gradient securely. The evidence is demonstrated that the proposed protocol could resist some common outside and participant attacks, such as the intercept-resend attack. Finally, we indicated how to apply it to train a federated linear regression model and simulated some steps with the help of the Qiskit quantum computing framework. The results also showed the effectiveness of QFLGD. In summary, the presented framework demonstrates the intriguing potential of achieving large-scale private distributed learning with quantum technologies and provides a valuable guide for exploring quantum advantages in real-life machine learning applications from the security perspective.
\par We hope that our proposed framework can further realize on a quantum platform with the gradual maturity of quantum technology. For example, how to implement the whole QFLGD process on the noisy intermediate-scale quantum (NISQ) devices is worth further exploration, and we will make more efforts.
\begin{acknowledgments} This work was supported by National Natural Science Foundation of China (Grants No. 62171131, 61976053, and 61772134), Fujian Province Natural Science Foundation (Grant No. 2022J01186), and Program for New Century Excellent Talents in Fujian Province University. \end{acknowledgments}
\appendix \section{\label{appendix A}Implement the Unitary Operation $U_{nf}$} \par In this appendix, we describe the implementation of a unitary operation $U_{nf}$, which could generate a state about the $2$-norm of $\mathbf{x}_{i}$. Its steps as shown in the following. \par (1) A quantum state is initialized as
\begin{equation}
\left|\varphi_{1} \right\rangle = \frac{1}{\sqrt{D}}{\sum\limits_{j = 0}^{D-1}\left| i \right\rangle_{1}}\left| j \right\rangle _{2}\left|0 \right\rangle_{3}.
\label{eq:46}
\end{equation}
\par (2) The oracle $O_X$ is performed to obtain
\begin{equation}
\left|\varphi_{2} \right\rangle = \frac{1}{\sqrt{D}}{\sum\limits_{j = 0}^{D-1}\left| i \right\rangle_{1}}\left|j \right\rangle _{2}\left|\mathbf{x}_{i}^{j} \right\rangle_{3}.
\label{eq:47}
\end{equation}
\par (3) A register in the state $|0\rangle$ is appended as the last register and rotated to $\sqrt{1 - (c_{1}\mathbf{x}_{i}^{j})^{2}}\left| 0 \right\rangle + c_{1}\mathbf{x}_{i}^{j}\left| 1 \right\rangle$. After that, the system becomes
\begin{equation}
|\varphi_{3}\rangle = \frac{1}{\sqrt{D}}{\sum\limits_{j = 0}^{D-1} |i\rangle _{1}} |j\rangle_{2} |\mathbf{x}_{i}^{j}\rangle_{3}\left[ {\sqrt{1 - ( c_{1}\mathbf{x}_{i}^{j})^{2}} |0\rangle + c_{1}\mathbf{x}_{i}^{j} | 1\rangle} \right]_{4},
\label{eq:48}
\end{equation}
where $c_{1} = {1/{\max_{i,j}| \mathbf{x}_{i}^{j}|}}$. We can observe the ancilla register in the state $|1\rangle$ with probability $P_{1} = {{c_{1}^{2}\left\| \mathbf{x}_{i} \right\|^{2}}/D}$. The state $\left|\varphi_{3} \right\rangle$ can be rewritten as
\begin{equation}
\left| i \right\rangle_{1}\otimes\left( {\sqrt{1 - P_{1}}\left| g \right\rangle \left| 0 \right\rangle + \sqrt{P_{1}}\left| a \right\rangle \left| 1 \right\rangle } \right)_{234},
\label{eq:49}
\end{equation} where
\begin{equation}
\left| g \right\rangle = {\sum\limits_{j = 0}^{D-1}{\sqrt{\frac{1 - \left( c_{1}\mathbf{x}_{i}^{j} \right)^{2}}{D - c_{1}^{2}\left\| \mathbf{x}_{i} \right\|^{2}}}\left| j \right\rangle}}\left| \mathbf{x}_{i}^{j} \right\rangle,
\label{eq:50}
\end{equation} and
\begin{equation}
\left| a \right\rangle = \frac{1}{\left\| \mathbf{x}_{i} \right\|}{\sum\limits_{j = 0}^{D-1}{\mathbf{x}_{i}^{j}\left| j \right\rangle}}\left| \mathbf{x}_{i}^{j} \right\rangle,
\label{eq:51}
\end{equation}
\par (4) Appending a register in state $\left| 0 \right\rangle^{\otimes \log(\epsilon_{m}^{- 1})}$. Then, the quantum phase estimation of $- U\left( \varphi_{3} \right)S_{0}U^{\dagger}\left( \varphi_{3} \right)S_{1}$ is performed to obtain
\begin{equation}
\left| \varphi_{4} \right\rangle = \left| i \right\rangle_{1}\otimes\left( {\sqrt{1 - P_{1}}\left|g \right\rangle \left| 0 \right\rangle + \sqrt{P_{1}}\left| a \right\rangle \left| 1 \right\rangle } \right)_{234} \otimes \left| \left\| \mathbf{x}_{i} \right\| \right\rangle_{5},
\label{eq:52}
\end{equation}
with the help of the square root circuit \cite{bhaskar2015}. We denote the $\epsilon_{m}$ is a tolerance error of QPE, $U\left( \varphi_{3} \right)\left| 0 \right\rangle_{1234} = \left| \varphi_{3} \right\rangle$, $S_{0} = I_{1234} - 2\left| 0 \right\rangle _{1234}\left\langle 0 \right|_{1234}$ and $S_{1} = I_{123}\otimes\left( I - 2\left| 0 \right\rangle\left\langle 0 \right| \right)_{4}$.
\par (5) The inverse operations of steps (2)-(3) are applied to generated the state
\begin{equation}
\left| \varphi_{5} \right\rangle = \left|i \right\rangle \left| \left\| \mathbf{x}_{i} \right\| \right\rangle,
\label{eq:53}
\end{equation}
Therefore, the $U_{nf}:\left| i \right\rangle \left| 0 \right\rangle\rightarrow\left| i \right\rangle \left| \left\| \mathbf{x}_{i} \right\| \right\rangle$ could be implemented in the above steps. And its running time is mainly caused by the quantum phase estimation in step (4), which takes time $O\left( {{\text{polylog}(D)}/\epsilon_{m}} \right)$. Moreover, $\left\| \mathbf{w} \right\|$ and $\left\| \mathbf{y} \right\|$ could be estimated similarly.
\section{\label{appendix B}An example of extracting the parameter $\mathbf{w}(n)$ information}
\par In Sec. \ref{sec:3.1}, a way to prepare a quantum state of $\mathbf{w}$ without the help of QRAM is shown in step (1.2). To further demonstrate it, an example is given in this appendix.
\par For convenience, supposing that $\mathbf{w}(n) = \mathbf{w} \in \mathbb{R}^{4}$. Then,we can get angle parameters $\vartheta^{0}_{1}$, $\vartheta^{0}_{2}$, and $\vartheta^{1}_{2}$ which are satisfied
\begin{equation}
\begin{split}
&{\cos(\vartheta^{0}_{1})=\frac{h^{0}_{1}}{h^{0}_{0}}}, ~~~~ {\sin (\vartheta^{0}_{1}) = \frac{h^{1}_{1}}{h^{0}_{0}}},\\
&{\cos(\vartheta^{0}_{2})=\frac{h^{0}_{2}}{h^{0}_{1}}}, ~~~~ {\sin (\vartheta^{0}_{2}) = \frac{h^{1}_{2}}{h^{0}_{1}}},\\
&{\cos(\vartheta^{1}_{2})=\frac{h^{2}_{2}}{h^{1}_{1}}}, ~~~~ {\sin (\vartheta^{1}_{2}) = \frac{h^{3}_{2}}{h^{1}_{1}}},
\end{split}
\label{eq:APB1}
\end{equation} according to Eq. (\ref{eq:14+1}). The value of $h^{j}_{t}$ ($t=0,1,2$) are shown in Fig. \ref{fig:example}, such as $h^{j}_{2} = \mathbf{w}^{j}$ for $j=0,1,2,3$.
\begin{figure}\label{fig:example}
\end{figure}
Then, the operations are defined as
\begin{equation}
\begin{split}
& U(\bm{\vartheta}_{1}) = R({\vartheta}^{0}_{1}) \otimes I, \\
& U(\bm{\vartheta}_{2}) = |0\rangle \langle 0| \otimes R({\vartheta}^{0}_{2}) + |1\rangle \langle 1| \otimes R({\vartheta}^{1}_{2}),
\end{split}
\label{eq:APB2}
\end{equation} based on $R(\vartheta) = \begin{bmatrix} \cos{(\vartheta)} &- \sin{(\vartheta)} \\ \sin{(\vartheta)} &\cos{(\vartheta)} \\ \end{bmatrix}$. It is easy to verify that
\begin{equation}
\begin{split}
&U(\bm{\vartheta}_{2})U(\bm{\vartheta}_{1})|00\rangle \\
= &\frac{1}{\left\| \mathbf{w} \right\|} (\mathbf{w}^{0}|00\rangle + \mathbf{w}^{1}|01\rangle + \mathbf{w}^{2}|10\rangle + \mathbf{w}^{3}|11\rangle).
\end{split}
\label{eq:APB3}
\end{equation} Thus, the quantum state of $\mathbf{w}$ can be obtained.
\end{document} | arXiv |
\begin{document}
\title{ \bf \Large Generic Contractive States and Quantum Monitoring of\\ Free Masses and Oscillators.\footnote{Submitted to Physics News.} }
\date{March 20, 2019}
\author{Priyanshi Bhasin} \affiliation{ 75/L, Model Town, Rewari, Haryana} \author{Ujan Chakraborty} \affiliation{ Indian Institute of Science Education And Research, Kolkata} \author{ S . M. Roy} \email{[email protected]} \affiliation{HBCSE,Tata Institute of Fundamental Research, Mumbai}
{\begin{abstract}
Monitoring photon quadratures and free masses are useful tools to detect small disturbances such
as gravitational waves.Here we report a large class of states
for photon quadratures and free masses potentially useful for this purpose: (1)''generic coherent states'' (GCS) of photons ,
whose width is independent of time and uncertainty product $\sigma (x) \sigma (p) $ is arbitrarily large
(a generalization of the minimum uncertainty Schr\"odinger coherent
states \cite{Schrodinger} ) and (2) ``squeezed generic contractive states'' (SGCS) for photons and free masses
(a generalization of the Yuen states \cite{Yuen})
whose width decreases with time ,uncertainty product is arbitrarily large, and the covariance squared $ <\{\Delta \hat{x}, \Delta \hat{p} \}>^2 $
has an arbitrary value within the allowed range $(0,4 \sigma ^2 (x) \sigma ^2 (p) -1\>)$.\\
---------------------Dedicated to the 125th birth anniversary of S. N. Bose. \end{abstract}
\pacs{03.65.-W , 03.65.Ta ,04.80.Nn}
\maketitle
{\bf History}. S. N. Bose's 1924 paper ``Planck's Law and Hypothesis of Light Quanta '' founded quantum statistics even before
quantum mechanics was born. Naturally, it is one of the pillars of quantum optics . Here we construct quantum states of
optical quadratures which are non-spreading and hence useful for accurate monitoring of small disturbances.
Much before the actual discovery of gravitational waves \cite{Abbott} it was realised that accurate monitoring of position of an oscillator and of a free mass, including intrinsic quantum uncertainties, are important for gravitational wave interferometers \cite{Thorne}. Monitoring accuracy is significantly restricted due to the nearly ubiquitous ``spreading of wave packets '' suggested by the heuristic standard quantum limit (SQL) (\cite{Braginsky},\cite{Caves1980}).Fortunately, Yuen \cite{Yuen}
discovered that there are contractive states of free masses for which the SQL is incorrect.Recently one of us (SMR) \cite{SMR2018}
has obtained rigorous quantum limits (RQL) on monitoring free masses ,oscillators and photon quadratures ,and the
corresponding maximally contractive states. Consistent with these RQL we present a large class of generic coherent states
and generic contractive states likely to be useful for accurate quantum monitoring.
In 1926, referring to the general property of spreading of wave packets, H. A. Lorentz \cite{Lorentz} said ,
in a letter to Schr\"odinger,
''because of this unavoidable blurring a wave packet does not seem to me to be very suitable for representing things to which
we want to ascribe a rather permanent individual existence ''.In his reply \cite{Schrodinger} constructed the
now famous oscillator coherent state whose wave packet has a width (and shape) independent of time.
At the 1927 Solvay conference Einstein used wave packet spreading to discuss the example of a particle passing through a narrow hole on to a hemispherical fluorescent screen which records the arrival of the particle (Fig. 1).
Suppose that a scintillation is seen at a point $P$ at time $t = T$, and suppose that the hole is so narrow that the wave packet corresponding to the particle is uniformly spread all over the screen at $t$ slightly less than $T$. Was the particle somewhere near $P$ at $t = T - \epsilon$ ($\epsilon$ small)? Ordinary quantum mechanics says that the probabilities at $t = T - \epsilon$ for the particle being found anywhere on the screen are uniform (and not particularly large in the vicinity of $P$). Thus the naive history corresponding to the reality of particle positions at each time is absent. Born's rule says that the wave function gives probabilities of a particle being found somewhere if a measurement is made, and not of being somewhere.
The role of wave packet spreading in discussions of quantum foundations was further
bolstered by heuristic arguments proposing that the accuracy of monitoring position of a free mass $m$ is limited by the standard quantum limit (SQL) (\cite{Braginsky},\cite{Caves1980}):
\begin{eqnarray} \label{SQL}
&\sigma^2 (X(t)) \geq \sigma^2 (X(0)) +(t^2/m^2) \sigma^2 (P(0)) \\
&\geq 2(t/m)\sigma(X(0))\sigma (P(0)) \geq \hbar t/m\>, \end{eqnarray} where $\sigma^2 (X(t))$ and $\sigma^2 (P(t))$ denote variances of the Heisenberg representation position and momentum operators at time $t$.
For the free mass , the inequality (\ref{SQL}) is actually an equality for Gaussian states,
\begin{eqnarray}
<p| {\psi (t)}> = (\pi \alpha )^{-1/4} exp [-\frac {(p-\beta )^2 } {2\alpha } -it\frac{p^2 } {2m } ],\nonumber \\
\sigma ^2 ( P (t)) = \frac{\alpha}{2}, \> \sigma^2 (X(t))=\hbar ^2 \frac{1+ (\alpha t/(m\hbar))^2 } {2\alpha }, \end{eqnarray}
Equation ( \ref{SQL}) forms the basis of most discussions of spreading of wave packets. The non-causality of quantum mechanics
is in sharp focus, because the same initial state is equally likely to result in clicks at widely separated points on the screen.
Actually a causal ``hidden variable'' theory which yields the same probability densities of position and
momentum as ordinary quantum mechanics exists \cite{Roy-Singh1995}. Nevertheless ``non-classicality'' of the trajectories is
inevitable.
Both expanding and contracting wave packets represent departures from classicality or coherence represented by
wave-packets of constant width. We shall see that the uncertainty principle limits both the possible rates of
expansion and of contraction.
{\bf Rigorous quantum limits on contractive and expanding states for a free mass}. Surprisingly, for free masses, Yuen discovered in 1983 \cite{Yuen} a class of states
called 'twisted coherent states' which are 'contractive states', i.e. states whose position uncertainty decreases with time
for a certain duration. The SQL is incorrect for these states. One of us (SMR)\cite{SMR2018} obtained rigorous quantum limits (RQL) valid
for all states including contractive states.
For any observable with Schr\"odinger operator $A$ (e.g. position $A=X$ or momentum $A=P$), and any Hamiltonian $H$, the Heisenberg operator $A(t)$ at time t and its variance $\sigma^2 (A(t)) $ are defined by, \begin{eqnarray} && A(t)\equiv exp(iHt/\hbar)\> A \>exp (-iHt/\hbar),\> \\
&& \sigma^2 (A(t)) \equiv <{\psi(0)}|(\Delta A(t))^2 | {\psi(0)}>,\\ && \Delta A(t)\equiv A(t)-<A(t)>,\\
&& <A(t)>\equiv<{\psi(0)}|A(t)|{\psi(0)}> \end{eqnarray}
where $| {\psi(0)}>$ is the initial state.
For a free mass, $H=P^2/(2m)$ ; the Heisenberg equation yields,
\begin{equation}
\Delta X(t)= \Delta X(0) +(t/m)\Delta P(0),
\end{equation}
and hence, \begin{eqnarray}\label{Heisenberg}
\sigma^2 (X(t))=\sigma^2 (X(0))+ (t^2/m^2) \sigma^2 (P(0)) + \nonumber \\
(t/m)<{\psi(0)}| \{\Delta X(0), \Delta P(0)\} |{\psi(0)}>. \end{eqnarray} One obtains the SQL (\cite{Braginsky},\cite{Caves1980}) Eq. (\ref{SQL}) if one assumes that the third term on the right-hand side, viz. the covariance $<\{\Delta X(0), \Delta P(0)\} >$ is non-negative. Yuen showed that the covariance is in fact negative for certain states. Nevertheless, rigorous quantum limits (RQL) can be obtained on the covariance, and hence on $\sigma^2 (X(t)) $.
Using $ [\Delta X(0), \Delta P(0)]=i\hbar $ , we have, \begin{eqnarray}
&<{\psi(0)}| \{\Delta X(0), \Delta P(0) \} |{\psi(0)}>+i\hbar \nonumber\\
&=2 <{\psi(0)}| \Delta X(0) \Delta P(0) |{\psi(0)}>. \end{eqnarray} Cauchy-Schwarz inequality on the right-hand side yields, \begin{eqnarray}\label{Cauchy}
&\big(<{\psi(0)} |\Delta X(0) \Delta P(0) +\Delta P(0) \Delta X(0) |{\psi(0)}>\big)^2\nonumber\\
&\leq 4 \sigma^2(X(0)) \sigma^2(P(0)) -\hbar ^2 ,
\end{eqnarray}
which is a rearrangement of the usual Schr\"odinger-Robertson uncertainty relation \cite{Kennard}.
Substituting this into Eq. (\ref{Heisenberg}) we have the rigorous quantum limits (RQL) \cite{SMR2018}
on expansion and contraction of wave packets,
\begin{eqnarray}\label{RQL}
&& |\sigma^2(X(t))-\sigma^2(X(0)) -(t/m)^2 \sigma^2(P(0))|\nonumber \\ && \leq (t/m) \sqrt{4 \sigma^2(X(0))\sigma^2(P(0)) -\hbar^2 }. \end{eqnarray} {\bf valid for arbitrary states}. The only states saturating the inequalities are those which obey \begin{eqnarray}\label{MC state}
&& \Delta P(0) |{\psi(0)}> =i\lambda \Delta X(0) |{\psi(0)}> ,\\
&& <X' |{\psi (0)}>= \big ( \frac{Re \lambda } {\pi \hbar } \big )^{1/4} \nonumber \\ &&\times exp \big(\frac{i<P(0)> X'} {\hbar } -\frac{\lambda (X'-<X(0)>)^2 } {2\hbar }\big ),\label{optimal state} \end{eqnarray} with $ Re \lambda > 0$, \begin{eqnarray}
|Im \lambda | = \frac{1}{2 \sigma^2(X(0))} \sqrt{4 \sigma^2(X(0))\sigma^2(P(0)) -\hbar^2 } ,\nonumber\\
\sigma^2(X(0)) =\hbar/(2 Re \lambda) ,\>\sigma^2(P(0)) =\hbar | \lambda |^2 /(2 Re \lambda) , \end{eqnarray}
and, \begin{eqnarray}
<{\psi(0)}| \{\Delta X(0) ,\Delta P(0) \} |{\psi(0)} >\nonumber\\
= \mp \sqrt{4 \sigma^2(X(0))\sigma^2(P(0)) -\hbar^2 }, \>if \> Im \lambda = \pm |Im \lambda | \end{eqnarray}
The positive sign of $Im \lambda $ corresponds to {\bf maximally contractive} (essentially Yuen states \cite{Yuen} ),and the negative sign of $Im \lambda $ to {\bf maximally expanding } wave packets.
\begin{figure}
\caption{Contractive State}
\end{figure}
Fig. 2 shows that for the initial state (\ref{MC state} ) with positive $Im \lambda $ , the state at time $t$ remains contractive upto $t=t_M/2$, where, \begin{equation} \label{tM1}
t_M= \frac{m}{ \sigma^2(P(0)) } \sqrt{4 \sigma^2(X(0))\sigma^2(P(0)) -\hbar^2 } , \end{equation}
and, for a given uncertainty product, by choosing $(t/m) \sigma ^2 (P(0))=\sigma(X(0))\sigma (P(0))$, $\sigma^2(X(t))$ can be made $\approx t\hbar ^2 /(4m \sigma(X(0))\sigma(P(0)) )$ for a large uncertainty product, and can be much smaller than the heuristic standard
quantum limit $\hbar t/m $ .
{\bf Rigorous Quantum Limits on Monitoring Photon Quadratures . }
The single mode photon Hamiltonian is, \begin{equation}
H= \hbar \omega \> a^\dagger a =\frac{1}{2}\hbar \omega (p^2+x^2-1) , \end{equation} where the quadrature operators $x,p$ are given by \begin{equation}
a=(x+ip )/\sqrt{2} ,\> a^\dagger=(x-ip)/\sqrt{2}. \end{equation} The Heisenberg equations of motion yield, \begin{eqnarray}\label{sigmax(t)}
&&\sigma^2 (x(t))=\cos ^2(\omega t)\sigma^2 (x(0))+ \sin ^2(\omega t)\sigma^2 (p(0)) \nonumber \\
&&+ \frac{1}{2} \sin (2\omega t)<{\psi(0)}| \{\Delta x(0), \Delta p(0)\} |{\psi(0)}>. \end{eqnarray} As before, $[\Delta x(0) ,\Delta p(0)]=i $ and the Schr\"odinger-Robertson uncertainty relations yield the RQL \cite{SMR2018},
\begin{eqnarray}\label{RQLOSC1}
&& |\sigma^2 (x(t)) -\cos ^2(\omega t)\sigma^2 (x(0)) -\sin ^2(\omega t)\sigma^2 (p(0))|\nonumber\\
&& \leq \frac{1}{2} |\sin (2\omega t)| \sqrt{4 \sigma^2(x(0)) \sigma^2(p(0)) -1 } \end{eqnarray} which corresponds to Eqn.(\ref{RQL} ) for a free mass. The extremal states saturating these RQL are complex Gaussians corresponding to to Equations (\ref{MC state}), (\ref{optimal state}) ; both the maximally contractive (MCON) and maximally expanding (MEXP) states can be designated as
'twisted coherent states' \cite{Yuen} or 'squeezed coherent states' (SCS) , \begin{eqnarray}
&&SCS: (b-\beta)|{\psi (0)}>=0, \>with \> b = \mu a + \nu a^\dagger, \\ \label{SCSb}
&&\alpha \equiv < \psi (0)|a |{\psi (0)}>, \>\beta= \mu \alpha + \nu \alpha ^*, \nonumber\\ && \mu = \cosh r , \> \nu = e^ {i\theta} \sinh r, \\ \label{TCS1}
&&|{\psi(0) }>=| {\alpha,r \exp {(i\theta ) } }>\equiv D(\alpha,a) S(\xi) | {0}> ,\label{SC} \end{eqnarray}
where $r>0$ is the squeezing parameter , $\theta$ is real and $|0>$ denotes the vacuum state ;here the unitary displacement operator $ D(\alpha,a)$ and squeeze operator $S(\xi)$ are, \begin{eqnarray}
&& D(\beta,b)=D(\alpha,a)=\exp {(\alpha a^\dagger -\alpha^* a) } , \nonumber \\
&& S(\xi)=\exp {\frac{1}{2} \big (\xi^*a^2-\xi a^{\dagger 2} \big) },\> \xi \equiv r \exp {(i\theta ) }. \end{eqnarray} They obey , \begin{eqnarray} && D^\dagger (\alpha,a) a D (\alpha,a) = a+ \alpha, \nonumber\\ && D^\dagger (\beta,b) b D (\beta,b) = b + \beta ,\nonumber\\ && S(\xi) a S^\dagger(\xi) = a \cosh r + a^\dagger e^ {i\theta} \sinh r=b \>. \end{eqnarray} Explicit values for the standard deviations and covariance in the SCS (\ref{SC}) are then easily derived,
\begin{eqnarray}
&&\sigma^2 (x(0))=1/2 \big(cosh (2r)-cos (\theta) sinh (2r) \big)\nonumber\\
&&\sigma^2 (p(0))=1/2 \big(cosh (2r)+cos (\theta) sinh (2r) \big)\nonumber\\
&&<{\psi(0)}| \{\Delta x(0), \Delta p(0)\} |{\psi(0)}>=-sin (\theta) sinh (2r)\nonumber\\
&& =- sgn (sin (\theta) )\sqrt{4 \sigma^2(x(0)) \sigma^2(p(0)) -1 }.\nonumber \end{eqnarray} For $r>0$, the state is squeezed ,i.e. $\sigma^2 (x(0)) < \sigma^2 (p(0)) $, if $ cos (\theta) >0$, and the state is contractive for small positive $t$ if $ sin (\theta) >0$ . The squeezed coherent states of negative covariance, being contractive , have been utilised in precision measurements with gravitational interferometers \cite{squeezed measurements}.
{\bf Generic Coherent States (GCS) of arbitrarily large uncertainty product}
The Schr\"odinger coherent states $|\alpha >= D(\alpha,a)|0>$ have minimum uncertainty product $\sigma(x(0))\sigma(p(0))=1/2$, and time-independent $\sigma(x(t)),\sigma(p(t))$. Roy and Singh \cite{Roy-Singh1982} noted that the property of time-independent width of the wave packets also holds for the generalised coherent states ,
\begin{eqnarray}
&&|\psi(\alpha,n)>=D(\alpha,a)|n>;\>\sigma(x(t))=\sigma(p(t))=\sqrt{n+1/2};\nonumber\\
&& a^\dagger a|n>=n |n>. \end{eqnarray}
We show here that the property of time-independent width of the wave packet holds for a class of states
much larger than these displaced oscillator eigen states. We
call this new class, ``Generic coherent states'' (GCS); they have arbitrarily large continuous values of the uncertainty product.
From the time development
equation (\ref{sigmax(t)}), denoting expectation value of an operator $A$ in the initial state $|\psi>$ by $<A>$ ,
we see that $\sigma (x(t))$ is time-independent if and only if,
\begin{eqnarray}
&& GCS:\> <(\Delta x(0))^2> = <(\Delta p(0))^2> , \>and\>\nonumber\\
&& <\{\Delta(x(0)),\Delta(p(0)) \}>=0.\>
\end{eqnarray} Using, \begin{eqnarray}
&& (\Delta x(0))^2 - (\Delta p(0))^2+i\{\Delta(x(0)),\Delta(p(0)) \}\nonumber\\
&& =2 (\Delta a )^2, \end{eqnarray}
the GCS conditions are equivalent to, \begin{equation}
GCS\>condition\>:\> <\psi|(\Delta a )^2|\psi>=0\>. \end{equation}
The GCS include the usual coherent states $\Delta a |\psi>=0 $ as a special case.
We now have the theorem:
If $|\phi>$ is a normalized state obeying \begin{equation}\label{generic coherence}
<\phi|a|\phi>= 0,\> and \> <\phi|a^2|\phi>= 0, \end{equation}
and \begin{equation}
|\psi (\alpha,\phi)>\equiv D(\alpha,a) |\phi> , \end{equation} where $\alpha$ is an arbitrary complex parameter, then
$|\psi (\alpha,\phi)> $ is a generic coherent state (GCS).
For proof it suffices to note that \begin{eqnarray}
&&<\psi(\alpha,\phi)|a-\alpha|\psi(\alpha,\phi)>\nonumber\\
&& = <\phi|D^\dagger (\alpha,a)\>(a-\alpha)\>D(\alpha,a) |\phi>\nonumber\\
&& =<\phi|a|\phi>=0, \nonumber
\end{eqnarray}
and
\begin{eqnarray}
&&<\psi(\alpha,\phi)|(a-\alpha)^2|\psi(\alpha,\phi)>\nonumber\\
&& = <\phi|D^\dagger (\alpha,a)\>(a-\alpha)^2\>D(\alpha,a) |\phi>\nonumber\\
&& =<\phi|a^2|\phi>=0.\nonumber \end{eqnarray}
When $|\phi>=|n>$, we get the Roy-Singh \cite{Roy-Singh1982} generalised coherent states; but the possible states $|\phi>$ form a much larger set allowing arbitrarily large continuous values of the uncertainty product : \begin{eqnarray}\label{continuous uncertainty}
&&\sigma^2 (x(0))=\sigma^2 (p(0))=\sigma (x(0)) \sigma (p(0))=\bar{n}+1/2 ;\nonumber\\
&&<\psi(\alpha,\phi)|\{\Delta x(0), \Delta p(0)\}|\psi(\alpha,\phi) >=0;\nonumber\\
&& \bar{n} \equiv <\phi|a^\dagger a|\phi>.
\end{eqnarray}
It remains only to show that states $|\phi>$ giving arbitrary non-negative values of $\bar{n} $ exist. Let \begin{equation}\label{phi}
|\phi >= \sum_{m=n}^N c_m |m > \> and <\phi|\phi>=1. \end{equation} We assume $N\geq n+3$ and solve Equations (\ref{generic coherence}) to get
\begin{eqnarray}
&& \begin{bmatrix}
c_{n+1} \sqrt{n+1} & & c^*_{N-1}\sqrt{N}\\
\\
c_{n+2} \sqrt{(n+1)(n+2)} & & c^*_{N-2} \sqrt{(N-1)N}
\end{bmatrix}
\begin{bmatrix}
c^*_n \\
\\
c_N
\end{bmatrix}\nonumber\\
&& \nonumber\\ && = -
\begin{bmatrix}
\sum_{m=n+1}^{N-2}c^*_m c_{m+1} \sqrt{m+1} \\
\\
\sum_{m=n+1}^{N-3}c^*_m c_{m+2} \sqrt{(m+1)(m+2)},
\end{bmatrix}
\end{eqnarray}
where the summations on the right-hand side are to be replaced by zero when the upper limit on $m$ is less than the lower
limit. This pair of linear eqns. can be solved explicitly for $c_n$ and $c_N$, in terms of all the other non-zero $c_m$'s . We omit the explicit general solution because it is elementary. We just quote the solution for the special case $N=n+3$, \begin{eqnarray}
&&( |c_{n+1} |^2 -|c_{n+2} |^2 ) \begin{bmatrix}
c^*_n \sqrt{n+1} \\
c_{n+3}\sqrt{n+3}
\end{bmatrix}\nonumber\\ &&\nonumber\\ &&= - \sqrt{n+2} c^* _{n+1} c_{n+2}
\begin{bmatrix}
c^* _{n+1} \\
-c_{n+2}
\end{bmatrix}
,\nonumber \end{eqnarray}
It is elementary to check that the vast class of states ( \ref{phi}) obeying (\ref{generic coherence} ) can yield any value of $\bar{n} $. E.g. if \begin{equation}
|\phi> = \sum_{r=0}^s c_{3r}|3r>\>;and\> \sum_{r=0}^s | c_{3r}|^2=1,\nonumber \end{equation} then, equations (\ref{generic coherence} ) are obeyed , and \begin{equation}
\bar{n} = \sum_{r=0}^s 3r| c_{3r}|^2 \in [0,3s]\>, \end{equation} which can equal any value in the continuous interval $[0,3s]$.Thus the GCS with continuously varying uncertainties (\ref{continuous uncertainty}) are obtained.
{\bf Squeezed generic coherent states (SGCS)}. Using the states $|\phi>$ to replace the vacuum state $|0>$ leads to the class of generic coherent states (GCS) with arbitrarily large continuous uncertainty products. We may similarly generalize the squeezed coherent states (SCS) of maximum possible magnitude of the
covariance $|<\{\Delta x(0), \Delta p(0) \}>| $ (i.e. maximally contractive or maximally expanding state) to squeezed generic coherent states (SGCS) which can have any value of the covariance allowed by the uncertainty principle.
Consider, the states \begin{equation}
|\psi (\alpha,\xi,\phi)>=D(\alpha,a) S(\xi) |\phi> , \end{equation}
which are obtained by replacing $|0>$ in the SCS by the state $ |\phi> $ . These states obey the SGCS conditions, \begin{eqnarray}
&&SGCS:\> <\psi(\alpha,\xi,\phi)|b-\beta|\psi(\alpha,\xi,\phi)>= 0, \\
&& <\psi(\alpha,\xi,\phi)|(b-\beta)^2|\psi(\alpha,\xi,\phi)>= 0,\\ && \> i.e. <\Delta b >=0\>,\>and\> < (\Delta b )^2 >=0, \end{eqnarray} which are obvious generalizations of the SCS conditions ( \ref{SCSb}).
Unlike the SCS wave functions, the SGCS wave functions are not complex Gaussians. E.g., when $|\phi>=|n>$ ,
using $(b^\dagger \> b-n)S(\xi)|n>=0$ we get the displaced and scaled oscillator eigen functions,
\begin{eqnarray}
&& <x|S(\xi)|n>=\frac{1}{ \sqrt{|\mu-\nu|h_n } }H_n(\frac {x}{|\mu-\nu|} )\>exp (-1/2\lambda x^2),\nonumber\\
&& <x|\psi(\alpha,\xi,\phi)>\nonumber\\
&&=<x-\sqrt{2}\alpha_1 |S(\xi)|n> \>exp (i\sqrt{2}\alpha_2 (x-\frac{\alpha_1}{\sqrt{2} })),\\
&& h_n=\sqrt{\pi}2^n n!;\>\lambda =\frac{ \mu+\nu} { \mu-\nu};\>\alpha_1=Re \alpha\>;\alpha_2=Im \alpha .
\end{eqnarray}
For general $|\phi>$ (calculated in the above section), we obtain a generalization of the $SCS$ expressions, \begin{eqnarray}
&&SGCS:\>\sigma^2 (x(0))=(\bar{n}+1/2) \big(cosh (2r)-cos (\theta) sinh (2r) \big),\nonumber\\
&&\sigma^2 (p(0))=(\bar{n}+1/2) \big(cosh (2r)+cos (\theta) sinh (2r) \big),\nonumber\\
&&<\psi(\alpha,\xi,\phi)| \{\Delta x(0), \Delta p(0)\} |\psi(\alpha,\xi,\phi)>\nonumber\\
&&=-(2\bar{n}+1)\>sin (\theta) sinh (2r)\nonumber\\
&& =- sgn (sin (\theta) )\sqrt{4 \sigma^2(x(0)) \sigma^2(p(0)) -(2\bar{n}+1)^2 }.\nonumber \end{eqnarray} Time development of these generic contractive or expanding wave packets follows from Eqn. (\ref{sigmax(t)} ) using
$ |\psi(\alpha,\xi,\phi)>$ as the initial state.
{\bf Overcompleteness of the SGCS}. Let, $ S(\xi) |\phi> \equiv |\psi(\xi,\phi)> $. Then,$|\psi(\alpha,\xi,\phi)>=D(\alpha,a)|\psi(\xi,\phi)>$. Hence, \begin{eqnarray}
&& \int d^2 \alpha \frac{1}{\pi}|\psi(\alpha,\xi,\phi)> \> <\psi(\alpha,\xi,\phi)| \nonumber\\
&& =\int d^2 \alpha \frac{1}{\pi}D(\alpha,a)|\psi(\xi,\phi)>\>< \psi(\xi,\phi)|D^\dagger(\alpha,a).
\end{eqnarray}
The integration over $\alpha$ and the fact that $|\psi(\xi,\phi)> $ is a normalized state yields
the overcompleteness relation,
\begin{eqnarray}
&& <x|\int d^2 \alpha \frac{1}{\pi}|\psi(\alpha,\xi,\phi)> \> <\psi(\alpha,\xi,\phi)|x'> \nonumber\\ && = \delta (x-x')
\end{eqnarray}
{\bf Free Mass}. The SGCS for the dimensionless photon variables $x,p$ can also be used as initial states for a free mass $m$ ,using $X=x\sqrt{ \hbar/(m\omega)} ,P= p\sqrt{m\hbar \omega }$. We then
find the time development equation for a free mass , \begin{eqnarray}
\sigma^2 (X(t))=\sigma^2 (X(0))+ (t^2/m^2) \sigma^2 (P(0)) \nonumber \\
-(\hbar \>t/m)sgn (sin (\theta) )\sqrt{4 \sigma^2(x(0)) \sigma^2(p(0)) -(2\bar{n}+1)^2 }\nonumber. \end{eqnarray} The third term on the right-hand side ,where the square root involves the dimensionless $x(0),p(0)$ of the last section ,exhibits all possible rates of contraction and expansion of wave packets allowed by the uncertainty principle.
{\bf Position Measurements On Free Masses and Harmonic Oscillators Using Contractive States}. In order to exploit the new possibilities allowed by the contractive states which violate the SQL (but obey the RQL),
the Ozawa measurement model for system-meter interaction \cite{Ozawa1988} which improves on the von Neumann model \cite{von Neumann} has been used .
The basic idea is to make successive measurements of appropriate duration with meters prepared in identical contractive states such that
after each measurement the system is left in the contractive state in which the meter was prepared, and between measurements there is
contractive evolution with the system Hamiltonian. Details of this can be found in ( \cite{Ozawa1988} , \cite{SMR2018}) , of
continuous measurement methods in \cite{Continuous}, and of actual experimental realizations in
\cite{squeezed measurements} . The new generic coherent states (GCS) and
the generic contractive states among the squeezed generic coherent states (SGCS) are expected to be useful for measurements necessary
for accurate `quantum monitoring'.
{\bf Acknowledgements}. One of us (SMR) thanks the Indian National Science Academy for the INSA honorary scientist position at HBCSE, TIFR, and Dipan Ghosh for the invitation to write this article and several editorial suggestions . Priyanshi Bhasin and Ujan Chakraborty thank the NIUS program of HBCSE for making this collaboration possible.
\end{document} | arXiv |
\begin{document}
\title{Escape of entropy for countable Markov shifts}
\subjclass[2010]{37D35, 37A35}
\keywords{Entropy, countable Markov shifts, escape of mass.}
\begin{thanks} {GI was partially supported by CONICYT PIA ACT172001 and by Proyecto Fondecyt 1190194. MT and AV thank UC for their hospitality. AV thanks Paul Apisa for interesting discussions about the set of points that escape on average. Finally, we thank the referee for useful comments and suggestions.}
\end{thanks}
\author[G.~Iommi]{Godofredo Iommi} \address{Facultad de Matem\'aticas, Pontificia Universidad Cat\'olica de Chile (PUC), Avenida Vicu\~na Mackenna 4860, Santiago, Chile}
\email{\href{mailto:[email protected]}{[email protected]}} \urladdr{\href{http://www.mat.uc.cl/~giommi}{www.mat.uc.cl/$\sim$giommi}}
\author[M.~Todd]{Mike Todd} \address{Mathematical Institute, University of St Andrews, North Haugh, St Andrews, KY16 9SS, Scotland} \email{\href{[email protected]}{[email protected]}} \urladdr{\href{http://www.mcs.st-and.ac.uk/~miket/}{www.mcs.st-and.ac.uk/$\sim$miket}}
\author[A.~Velozo]{Anibal Velozo} \address{Department of Mathematics, Yale University, New Haven, CT 06511, USA.} \email{\href{[email protected]}{[email protected]}} \urladdr{\href{https://gauss.math.yale.edu/~av578/}{https://gauss.math.yale.edu/~av578/}}
\begin{abstract} In this paper we study ergodic theory of countable Markov shifts. These are dynamical systems defined over non-compact spaces. Our main result relates the escape of mass, the measure theoretic entropy, and the entropy at infinity of the system. This relation has several consequences. For example we obtain that the entropy map is upper semi-continuous and that the ergodic measures form an entropy dense subset. Our results also provide new proofs of results describing the existence and stability of the measure of maximal entropy. We relate the entropy at infinity with the Hausdorff dimension of the set of recurrent points that escape on average. Of independent interest, we prove a version of Katok's entropy formula in this non-compact setting. \end{abstract}
\maketitle
\section{Introduction} \label{sec:intro}
Many problems in ergodic theory and dynamical systems involve properties of limits of sequences of invariant probability measures. If the phase space is compact then the space of invariant probability measures is also compact in the the weak$^*$ topology, which is partly a consequence of convergence in this topology preserving mass. However, when the phase space is non-compact, the space of invariant probability measures might also be non-compact, and thus mass, as well as other quantities of interest, may escape in the limit. In this paper, we are principally interested in how the entropy of sequences of measures behaves in this setting.
More specifically, we consider countable Markov shifts (CMS) $(\Sigma,\sigma)$, which in general are not even locally compact. We discuss the difficulties with the various classical topologies in this context in the next section, where we also give details of the space of invariant sub-probability measures endowed with the so-called cylinder topology, introduced in \cite{iv}. This topology generalises the vague topology to a non-locally compact setting (see Section~\ref{cyl}). If $(\mu_n)_n$ is a sequence of $\sigma$-invariant probability measures that converges in the cylinder topology to the measure $\mu$ then the total mass $|\mu|:=\mu(\Sigma) \in [0,1]$. In particular, this topology captures the
escape of mass. Moreover, $\mu$ is an invariant measure and the normalisation $\mu/|\mu|$ is an invariant probability measure (whenever $ \mu$ is not the zero measure). Denote by $h_{\nu}(\sigma)$ the entropy of the invariant probability measure $\nu$ (see Section~\ref{sec:em} for details), and by $\delta_\infty$ the \emph{topological entropy at infinity} of the system (see Definition \ref{def:ent_inf}). Our first main result answers one of the classical questions about sequences of measures: how does entropy change in the limit?
\begin{theorem}\label{thm:main} Let $(\Sigma,\sigma)$ be a transitive CMS with finite topological entropy. Let $(\mu_n)_{n}$ be a sequence of $\sigma$-invariant probability measures converging on cylinders to $\mu$. Then \begin{equation} \label{eq:main}
\limsup_{n\to \infty} h_{\mu_n}(\sigma)\leqslant |\mu|h_{\mu/|\mu|}(\sigma)+(1-|\mu|)\delta_\infty. \end{equation} If the sequence converges on cylinders to the zero measure then the right hand side is understood as $\delta_\infty$. \end{theorem}
Since the topological entropy at infinity plays a crucial role in this article, we define it here, leaving details of the background to this to Section~\ref{sec:introh}. The idea is to measure how complicated the dynamics is near infinity. Of course, such a notion only makes sense for dynamical systems defined on non-compact phase spaces. As in the classical entropy theory, we will study two ways of measuring the complexity of the system near infinity, one topological in nature and the other measure theoretic.
\begin{definition} \label{def:ent_inf} Let $(\Sigma, \sigma)$ be a CMS. Let $M, q \in {\mathbb N}$. For $n\in {\mathbb N}$ let $z_n(M, q)$ be the number of cylinders of the form $[x_0,\ldots,x_{n+1}]$, where $x_0\leqslant q$, $x_{n+1}\leqslant q$, and \begin{equation*} \# \left\{ i\in\{0,1,\ldots,n+1\}: x_i\leqslant q \right\} \leqslant \frac{n+2}{M}. \end{equation*} Define \begin{equation*} \delta_\infty(M,q):=\limsup_{n\to\infty}\frac{1}{n}\log z_n(M, q), \end{equation*} and \begin{equation*} \delta_\infty(q):=\liminf_{M\to\infty} \delta_\infty(M,q). \end{equation*} The \emph{topological entropy at infinity} of $(\Sigma,\sigma)$ is defined by $\delta_\infty:=\liminf_{q\to\infty}\delta_\infty(q)$. \end{definition}
The measure theoretic counterpart is given by:
\begin{definition} \label{def:ent_meas_inf} Let $(\Sigma, \sigma)$ be a finite entropy CMS. The \emph{measure theoretic entropy at infinity} of $(\Sigma, \sigma)$ is defined by \begin{equation} h_\infty :=\sup_{(\mu_n)_n\to 0}\limsup_{n\to\infty}h_{\mu_n}(\sigma), \label{eq:mte} \end{equation} where $(\mu_n)_n\to 0$ means that the sequence $(\mu_n)_n$ converges on cylinders to the zero measure. \end{definition}
Other authors have considered related concepts. Most notably, Buzzi \cite[Definition 1.13]{b} proposed a notion of entropy at infinity for CMS. His definition is given in terms of the graph $G$ which defines the CMS $(\Sigma, \sigma)$: \begin{equation*} b_{\infty}:=\inf_{F} \inf_{\lambda >0} \sup \left\{h_{\mu}(\sigma) : \mu([F]) < \lambda \right\}, \end{equation*} where $F$ ranges over the finite sub-graphs of $G$ and $[F]:= \left\{ x \in \Sigma : x_0 \in \mathcal{A}_F \right\}$, where $ \mathcal{A}_F$ denotes the symbols appearing as vertex of $F$. It turns out that Buzzi's notion coincides with ours. Indeed, our next result states that all these three notions coincide.
\begin{theorem} \label{thm:vpinf} Let $(\Sigma,\sigma)$ be a CMS of finite topological entropy. Then \begin{equation*}
\delta_{\infty}= h_\infty =b_\infty. \end{equation*} \end{theorem}
The equality $ \delta_{\infty}= h_\infty$ can be understood as a variational principle at infinity.
Einsiedler, Lindenstrauss, Michel and Venkatesh \cite[Lemma 4.4]{elmv} were the first to obtain an inequality similar to \eqref{eq:main}. It appeared in their ergodic theoretic proof of Duke's theorem on equidistribution of closed geodesics on the modular surface. After that, similar results in the context of homogeneous dynamics were obtained in \cite[Theorem 1.2]{ek} and \cite[Theorem A]{ekp}. For different classes of geodesic flows defined on non-compact manifolds of negative sectional curvature related results were obtained in \cite[Theorem 1.2]{irv} and \cite[Theorem 1.1]{rv}. In this context the most general result was obtained in \cite[Theorems 1.4 and 1.6]{ve} where an inequality like \eqref{eq:main} was proved for the geodesic flow defined on an arbitrary complete Riemannian manifolds with pinched negative sectional curvature. The manifolds studied are locally compact, thus the topology considered in the space of invariant measures is the vague topology. A more interesting and subtle point is the quantity playing the role of the entropy at infinity. Due to the geometric nature of the examples studied, the entropy at infinity is related to the critical exponent of the Poincar\'e series associated to the non-compact parts of the space (in the geometrically finite case this reduces to the critical exponent of the parabolic subgroups of the fundamental group). Let us mention that the topological entropy at infinity of the geodesic flow was also studied by Schapira and Tapie \cite{st} in their work about the rate of change of the topological entropy under perturbations of the metric.
A major difference with previous works is that in the context of CMS the behaviour of the orbits approaching infinity can be very complicated and that we do not assume the phase space to be locally compact. These are major difficulties that have to be overcome making the analysis more technical. As a general principle we follow the method employed in \cite{ve} with appropriate modifications. Loosely speaking the entropy at infinity of the geodesic flow counts geodesics that start and end at a given base point, but do not return near this point for intermediate times. In our setup the entropy at infinity counts orbits that might return near a base point many times, but the number of returns become negligible on average, which can occur due to the lack of local compactness.
There are several interesting consequences of Theorem \ref{thm:main}, some of them are discussed in Section \ref{sec:app}. For example, in Theorem \ref{semicont} it is proved that the entropy map is upper semi-continuous for every transitive finite entropy CMS. The continuity properties of the entropy map have been studied for a long time. Major results in the area are that for expansive systems defined on compact metric spaces the entropy map is upper semi-continuous \cite[Theorem 8.2]{wa}. Another fundamental result is that if $f$ is a $C^{\infty}$ diffeomorphism defined on a smooth compact manifold then again the entropy map is upper semi-continuous \cite[Theorem 4.1]{n}. As explained in Remark \ref{rem:nousc}, for infinite entropy CMS the entropy map is not upper semi-continuous. In a recent article we proved \cite[Corollary 1.2]{itv} that if $(\Sigma, \sigma)$ is a finite entropy transitive CMS then the entropy map is upper semi-continuous when restricted to ergodic measures. A complete solution to the problem can be obtained as a consequence of Theorem \ref{thm:main}. In Section \ref{sec:app} we also prove that the set of ergodic measures is `entropy dense' in the space of invariant probability measures. This result not only provides a fine description of the structure of the space of invariant probability measures but also provides an important tool to study Large Deviations in this setting.
There is a classification of transitive CMS in terms of their recurrence properties: they can be transient, null recurrent or positive recurrent (see Definition \ref{def:clas} for $\varphi=0$). Positive recurrent CMS are precisely those with a measure of maximal entropy. A particularly important role is played by strongly positive recurrent CMS (SPR); which are a sub-class of positive recurrent Markov shifts. The dynamical properties of this class of systems are similar to that of sub-shifts of finite type. Buzzi gave a characterisation of SPR shifts using $b_\infty$ in \cite[Proposition 6.1]{b}, based on the work of Gurevich-Zargaryan, Gurevich-Savchenko and Ruette. We note in Proposition \ref{prechar} that we can now restate this result as saying that $(\Sigma,\sigma)$ is SPR if and only if $\delta_\infty<h_{top}(\sigma)$, where $h_{top}(\sigma)$ is the Gurevich entropy of $(\Sigma,\sigma)$ (for precise definitions see Section \ref{sec:tf}). In Section \ref{sec:mme} we use Theorem \ref{thm:main} to obtain stability properties of the measure of maximal entropy for SPR CMS (recovering results from \cite{gs}). Similar arguments are used to prove the existence of equilibrium states for potentials in $C_0(\Sigma)$, the space of test functions for the cylinder topology (see Section \ref{sec:eqst}). To the author's knowledge, this is the first result on the existence of equilibrium states for CMS that goes beyond regular potentials (e.g. with summable variations or the Walters property). Finally, in Theorem \ref{thm:em} we prove that for SPR systems it is possible to bound the amount of mass that escapes the system in terms of the entropy of the measures. Sequences of measures with large entropy can not lose much mass.
The entropy at infinity has yet another important appearance in dynamics: it is related to the Hausdorff dimension of the set of points that escape on average (see \cite{aaekmu, ekp, kklm, kp}). These are points for which the frequency of visits to every cylinder equals to zero. In particular, no invariant measure is supported on that set. This notion has been studied recently in contexts of homogeneous and Teichm\"uller dynamics. The motivation comes from work of Dani \cite{da} in the mid 1980s who proved that singular matrices are in one-to-one correspondence with certain divergent orbits of one parameter diagonal groups on the space of lattices. In Theorem \ref{thm:onave} we prove that the Hausdorff dimension of the set of recurrent points that escape on average is bounded above by $\delta_{\infty} / \log 2$, where the factor $\log 2$ comes from the metric in the symbolic space.
While our interest in this paper lies in the realm of Markov shifts, to provide context we mention some applications of this theory. Symbolic methods have been used to describe dynamical properties of a variety of systems since the $1898$ work of Hadamard on closed geodesics on surfaces of negative curvature, at the latest. Compact Markov shifts have been used to study uniformly hyperbolic dynamical systems defined on compact spaces, see for example the work of Bowen in \cite{bo2}. Many deep results can be obtained with this coding. Mostly after the work of Sarig \cite{sa4}, countable Markov partitions have been constructed for a wide range of dynamical systems. This gives a semiconjugacy between a relevant part of the dynamics, albeit not all of it, and a CMS. Examples of systems for which Markov partitions have been constructed include positive entropy diffeomorphisms defined on compact manifolds \cite{b2,ov, sa4} and Sinai and Bunimovich billiards \cite{lm}. Remarkable results have been proved making use of these codings, for example in \cite[Main Theorem]{bcs} it is shown that a positive entropy $C^{\infty}$ diffeomorphism of a closed surface admits at most finitely many ergodic measures of maximal entropy. Results in this paper apply to all the symbolic codings mentioned above. However, due to topologies possibly not being preserved by the coding, it is not clear that the results pass to the original systems.
In 1980 Katok \cite[Theorem 1.1]{ka} established a formula for the entropy of an invariant probability measure in analogy to the definition of topological entropy of a dynamical system \cite{bo,d}. This formula is now known as \emph{Katok's entropy formula}. An important assumption in \cite[Theorem 1.1]{ka} is the compactness of the phase space. In Section \ref{ent} we prove that Katok's entropy formula holds in the non-compact setting of CMS. We require this formula in the proof of Theorem \ref{thm:main}, but it is also of independent interest.
\section{Preliminaries}
\subsection{Basic definitions for CMS} \label{sec:defcms}
Let $M$ be a ${\mathbb N}\times {\mathbb N}$ matrix with entries $0$ or $1$. The symbolic space associated to $M$ with alphabet ${\mathbb N}$ is defined by
\begin{equation*}
\Sigma:=\left\{ (x_0, x_1, \dots) \in {\mathbb N}^{{\mathbb N}_0}: M(x_i, x_{i+1})=1 \text{ for every } i \in {\mathbb N}_0 \right\}, \end{equation*} where ${\mathbb N}_0:={\mathbb N} \cup \{0\}$. We endow ${\mathbb N}$ with the discrete topology and ${\mathbb N}^{{\mathbb N}_0}$ with the product topology. On $\Sigma$ we consider the induced topology given by the natural inclusion $\Sigma\subset {\mathbb N}^{{\mathbb N}_0}$. We stress that, in general, this is a non-compact space. The space $\Sigma$ is locally compact if and only if for every $i \in {\mathbb N}$ we have $\sum_{j \in {\mathbb N}} M(i,j ) <\infty$ (see \cite[Observation 7.2.3]{ki}).
The \emph{shift map} $\sigma:\Sigma \to \Sigma$ is defined by $(\sigma(x))_i=x_{i+1}$, where $x=(x_0, x_1, \dots ) \in \Sigma$. Note that $\sigma$ is a continuous map. The pair $(\Sigma,\sigma)$ is called a one sided \emph{countable Markov shift} (CMS). The matrix $M$ can be identified with a directed graph $G$ with no multiple edges (but allowing edges connecting a vertex to itself).
An \emph{admissible word} of length $N$ is a string ${\bf w} =a_0a_1\ldots a_{N-1}$ of letters in the alphabet such that $M(a_i,a_{i+1})=1$, for every $i\in\{0,\ldots,N-2\}$. We use bold letters to denote admissible words. The length of an admissible word ${\bf w}$ is $\ell({\bf w})$.
A \emph{cylinder} of length $N$ is the set \begin{equation*} [a_0,\ldots,a_{N-1}]:= \left\{ x=(x_0,x_1,\ldots)\in \Sigma : x_i=a_i \text{ for } 0 \leqslant i \leqslant N-1 \right\}. \end{equation*} If $a_0\ldots a_{N-1}$ is an admissible word then $[a_0,\ldots,a_{N-1}] \neq \emptyset$. We use the notation $C_n(x)$ to denote the cylinder of length $n$ containing $x$. Since a cylinder can be identified with an admissible word, we also denote the length of a cylinder $C$ by $\ell(C)$. Note that the topology generated by the cylinder sets coincides with that induced by the product topology.
The space $\Sigma$ is metrisable. Indeed, let $d:\Sigma \times \Sigma \to {\mathbb R}$ be the function defined by \begin{equation} \label{metric} d(x,y):= \begin{cases} 1 & \text{ if } x_0\ne y_0; \\ 2^{-k} & \text{ if } x_i=y_i \text{ for } i \in \{0, \dots , k-1\} \text{ and } x_k \neq y_k; \\ 0 & \text{ if } x=y. \end{cases} \end{equation} The function $d$ is a metric and it generates the same topology as that of the cylinders sets. Moreover, the ball $B(x,2^{-N})$ is precisely $C_N(x)$. Given $\varphi:\Sigma\to {\mathbb R}$, we define \begin{equation*}
\text{var}_n(\varphi):=\sup \left\{|\varphi(x)-\varphi(y)|: \forall x,y\in \Sigma \text{ such that }d(x,y)\leqslant 2^{-n} \right\}. \end{equation*}
A function $\varphi:\Sigma\to {\mathbb R}$ is said to have \emph{summable variations} if $\sum_{n\geqslant 2}\text{var}_n(\varphi)<\infty$. A function $\varphi$ is called \emph{weakly H\"older} if there exist $\theta \in (0,1)$ and a positive constant $O \in {\mathbb R}$ such that $\text{var}_n(\varphi)\leqslant O \theta^n$, for every $n\geqslant 2$. A weakly H\"older continuous function is H\"older if and only if it is bounded. The $C^0$-norm of $\varphi$ is $\|\varphi\|_0:=\sup_{x\in \Sigma}|\varphi(x)|$. We denote by \begin{equation*} S_n\varphi(x)=\sum_{k=0}^{n-1}\varphi(\sigma^k x), \end{equation*} the \emph{Birkhoff sum} of $\varphi$ at the point $x$.
We say that $(\Sigma,\sigma)$ is \emph{topologically transitive} if its associated directed graph $G$ is connected. We say that $(\Sigma,\sigma)$ is \emph{topologically mixing} if for each pair $a,b\in {\mathbb N}$, there exists a number $N(a,b)$ such that for every $n\geqslant N(a,b)$ there is an admissible word of length $n$ connecting $a$ and $b$. There is a particular class of CMS that will be of interest to us,
\begin{definition} \label{def:F} A CMS $(\Sigma, \sigma)$ is said to satisfy the $\mathcal{F}-$\emph{property} if for every element of the alphabet $a$ and natural number $n$, there are only finitely many admissible words of length $n$ starting and ending at $a$. \end{definition}
\begin{remark} A CMS $(\Sigma, \sigma)$ satisfies the $\mathcal{F}-$\emph{property} if and only if there are only finitely many periodic orbits of length $n$ intersecting $[a]$, for every $n\in {\mathbb N}$ and for every $a$ in the alphabet. Note that every locally compact CMS satisfies the $\mathcal{F}-$\emph{property}. \end{remark}
\begin{remark} Equivalent definitions and properties as those discussed in this section can be given for \emph{two sided CMS}. In this case the acting group is ${\mathbb Z}$. It turns out that, in general, thermodynamic formalism for the two sided CMS can be reduced to the one sided case (see \cite[Section 2.3]{sabook}). \end{remark}
\subsection{Topologies in the space of invariant measures} \label{sec:topo}
The space of invariant measures can be endowed with different topologies, some of which can account for the escape of mass phenomenon whereas others can not. In this section we not only fix notation for later use, but we also recall definitions and properties of several topologies in the space of measures. First note that in this article a measure is always a countably additive non-negative Borel measure defined in the symbolic space $\Sigma$. The mass of a measure $\mu$ is defined as $|\mu|:=\mu(\Sigma)$.
Denote by $\mathcal{M}(\Sigma,\sigma)$ the space of $\sigma$-invariant probability measures on $\Sigma$ and by $\mathcal{M}_{\leqslant 1}(\Sigma,\sigma)$ the space of $\sigma$-invariant measures on $\Sigma$ with mass in $[0,1]$. In other words, $\mathcal{M}_{\leqslant 1}(\Sigma,\sigma)$ is the space of $\sigma$-invariant sub-probability measures on $\Sigma$. Note that $\mathcal{M}(\Sigma,\sigma)\subset \mathcal{M}_{\leqslant 1}(\Sigma,\sigma)$. The set of ergodic probability measures is denoted by $\mathcal{E}(\Sigma, \sigma)$.
\subsubsection{The weak$^*$ topology} \label{weak*}
Denote by $C_b(\Sigma)$ the space of real valued bounded continuous function on $\Sigma$. A sequence of measures $(\mu_n)_n$ in $\mathcal{M}(\Sigma,\sigma)$ converges to a measure $\mu$ in the weak$^*$ topology if for every $f \in C_b(\Sigma)$ we have \begin{equation*} \lim_{n \to \infty} \int f d \mu_n = \int f d \mu. \end{equation*} Note that since the constant function equal to one belongs to $C_b(\Sigma)$ the measure $\mu$ is also a probability measure. A basis for this topology is given by the collection of sets of the form \begin{align}\label{defbasis}
V(f_1,\ldots,f_k,\mu,\epsilon):= \left\{ \nu \in \mathcal{M}(\Sigma,\sigma) : \left|\int f_i d \nu - \int f_i d \mu \right| < \epsilon, \text{ for } i \in \{1, \dots, k\} \right\}, \end{align} where $\mu \in \mathcal{M}(\Sigma,\sigma)$, $(f_i)_{i}$ are elements from $C_b(\Sigma) $ and $\epsilon >0$. Note that in this notion of convergence we can replace the set of test functions (bounded and continuous) by the space of bounded uniformly continuous functions (see \cite[8.3.1 Remark]{bg}). Convergence with respect to the weak$^*$ topology can be characterised as follows, see \cite[Theorem 2.1]{bi}.
\begin{proposition}[Portmanteau Theorem] \label{port} Let $(\mu_n)_n, \mu$ be probability measures on $\Sigma$. The following statements are equivalent. \begin{enumerate} \item The sequence $(\mu_n)_n$ converges to $\mu$ in the weak$^*$ topology. \item If $O \subset \Sigma$ is an open set, then $\mu(O) \leq \liminf_{n \to \infty} \mu_n(O)$. \item If $C \subset \Sigma$ is a closed set, then $\mu(C) \geq \limsup_{n \to \infty} \mu_n(C)$. \item If $A \subset \Sigma$ has $\mu(\partial A)=0$, where $\partial A$ is the boundary of $A$, then $ \lim_{n \to \infty} \mu_n(A)= \mu(A)$. \end{enumerate} \end{proposition}
Note that the space $\mathcal{M}(\Sigma,\sigma)$ is closed in the weak$^*$ topology (\cite[Theorem 6.10]{wa}). If $\Sigma$ is compact then so is $\mathcal{M}(\Sigma,\sigma)$ with respect to the weak$^*$ topology (see \cite[Theorem 6.10]{wa}). If $\Sigma$ is not compact then, in general (e.g., whenever the $\mathcal{F}$-property holds), $\mathcal{M}(\Sigma,\sigma)$ is not compact with respect to the weak$^*$ topology. Finally, the space $\mathcal{M}(\Sigma,\sigma)$ is a convex set whose extreme points are ergodic measures (see \cite[Theorem 6.10]{wa}).
\subsubsection{The topology of convergence on cylinders} \label{cyl} In this section we recall the definition and properties of the topology of convergence on cylinders. This topology was introduced and studied in \cite{iv} as a way to compactify $\mathcal{M}(\Sigma,\sigma)$ under suitable assumptions on $\Sigma$.
Let $(C^n)_n$ be an enumeration of the cylinders of $\Sigma$. Given $\mu, \nu\in \mathcal{M}_{\leqslant 1}(\Sigma,\sigma)$ we define $$\rho(\mu,\nu)=\sum_{n=1}^\infty \frac{1}{2^n}|\mu(C^n)-\nu(C^n)|.$$ It follows from the outer regularity of Borel measures on metric spaces that $\rho(\mu,\nu)=0$, if and only if $\mu=\nu$. Moreover, the function $\rho$ defines a metric on $\mathcal{M}_{\le1}(\Sigma,\sigma)$. The topology induced by this metric is called the \emph{topology of convergence on cylinders}. We say that a sequence $(\mu_n)_n$ in $\mathcal{M}_{\leqslant 1}(\Sigma,\sigma)$ \emph{converges on cylinders} to $\mu$ if $$\lim_{n\to\infty}\mu_n(C)=\mu(C),$$ for every cylinder $C\subset \Sigma$. Of course, $(\mu_n)_n$ converges on cylinders to $\mu$ if and only if $(\mu_n)_n$ converges to $\mu$ in the topology of convergence on cylinders. In the next lemma we see that in the case that mass does not escape then weak$^*$ and convergence on cylinders coincide.
\begin{lemma}{\cite[Lemma 3.17]{iv}}\label{restriction} \label{equivtop} Let $(\Sigma,\sigma)$ be a CMS, $\mu$ and $(\mu_n)_n$ be invariant probability measures on $\Sigma$. The following assertions are equivalent. \begin{enumerate} \item The sequence $(\mu_n)_n$ converges in the weak$^*$ topology to $\mu$. \item The sequence $(\mu_n)_n$ converges on cylinders to $\mu$. \end{enumerate} \end{lemma}
Let $\Sigma$ be a locally compact space and $(\mu_n)_n, \mu$ in $\mathcal{M}_{\le1}(\Sigma,\sigma)$. The sequence $(\mu_n)_n$ converges to $\mu$ in the \emph{vague topology} if $\lim_{n \to \infty} \int f d \mu_n = \int f d \mu,$ for every function $f$ continuous and of compact support (note that the set of test functions can be replaced by the set of continuous functions vanishing at infinity). If $(\Sigma, \sigma)$ is locally compact then the topology of convergence on cylinders coincides with the vague topology (see \cite[Lemma 3.18]{iv}). It is important to note that if $\Sigma$ is transitive and not locally compact then the space of continuous functions of compact support is trivial, so the vague topology is of no use and the topology of convergence on cylinders is a suitable generalisation (see \cite[Remark 3.13]{iv}).
If $C$ is a cylinder of length $m$, denote by \begin{equation*}
C(\geqslant n):= \left\{ x \in C : \sigma^m(x)\in \bigcup_{k\geqslant n}[k] \right\}. \end{equation*} For a non-empty set $ A\subset \Sigma$ we define \begin{equation*}
var^A(f):=\sup \left\{ \left|f(x)-f(y) \right| : x, y \in A \right\}. \end{equation*} We declare $var^A(f)=0$ if $A$ is the empty set.
\begin{definition}\label{C_0} We say that $f$ belongs to $C_0(\Sigma)$ if the following three conditions hold: \begin{enumerate} \item $f$ is uniformly continuous.
\item $\lim_{n\to\infty}\sup_{x\in [n]}|f(x)|=0.$ \item $\lim_{n\to\infty}var^{C(\geqslant n)}(f)=0,$ for every cylinder $C\subset \Sigma$. \end{enumerate} In this case we say that $f$ \emph{vanishes at infinity}. \end{definition} The set $C_0(\Sigma)$ is the space of test functions for the cylinder topology (see \cite[Lemma 3.19]{iv}). In other words, $(\mu_n)_n$ is a sequence in $\mathcal{M}_{\leqslant 1}(\Sigma,\sigma)$ that converges in the cylinder topology to $\mu\in\mathcal{M}_{\leqslant 1}(\Sigma,\sigma)$ if and only if $\lim_{n\to\infty}\int f d\mu_n=\int fd\mu$, for every $f\in C_0(\Sigma)$. Since cylinder topology generalises the vague topology for non-locally compact CMS, the space $C_0(\Sigma)$ is the natural substitute to the set of continuous functions that vanish at infinity.
The following result was proved in \cite[Theorem 1.2]{iv}, and is an important ingredient for many of our applications.
\begin{theorem}\label{compact} Let $(\Sigma,\sigma)$ be a transitive CMS with the $\mathcal{F}-$property. Then $\mathcal{M}_{\leqslant 1}(\Sigma,\sigma)$ is a compact metrisable space with respect to the cylinder topology. Moreover, $\mathcal{M}_{\le1}(\Sigma,\sigma)$ is affine homeomorphic to the Poulsen simplex. \end{theorem}
We remark that, as shown in \cite[Proposition 4.19]{iv}, Theorem \ref{compact} is sharp in a strong sense: if $(\Sigma, \sigma)$ does not have the $\mathcal{F}-$property, then $\mathcal{M}_{\le1}(\Sigma,\sigma)$ is not compact. More precisely, there exists a sequence of periodic measures that converges on cylinders to a finitely additive measure which is not countably additive.
\subsection{Entropy of a measure} \label{sec:em}
In this section we recall the definition of the entropy of an invariant measure $\mu \in\mathcal{M}(\Sigma,\sigma)$ (see \cite[Chapter 4]{wa} for more details). We take the opportunity to fix the notation that will be used in what follows. A partition $\beta$ of a probability space $(\Sigma, \mu)$ is a countable (finite or infinite) collection of pairwise disjoint subsets of $\Sigma$ whose union has full measure. The \emph{entropy} of the partition $\beta$ is defined by \begin{equation*} H_\mu(\beta):= - \sum_{P \in \beta} \mu(P) \log \mu(P), \end{equation*} where $0 \log 0 :=0$. It is possible that $H_\mu(\beta)=\infty$. Given two partitions $\beta$ and $\beta'$ of $\Sigma$ we define the new partition \begin{equation*} \beta \vee \beta':= \left\{P \cap Q : P \in \beta , Q \in \beta' \right\}. \end{equation*} Let $\beta$ be a partition of $\Sigma$. We define the partition $\sigma^{-1}\beta:= \left\{ \sigma^{-1}P : P \in \beta \right\}$ and for $n \in {\mathbb N}$ we set $\beta^n:=\bigvee_{i=0}^{n-1} \sigma^{-i}\beta$. Since the measure $\mu$ is $\sigma$-invariant, the sequence $H_{\mu}(\beta^n)$ is sub-additive. The \emph{entropy of $\mu$ with respect to} $\beta$ is defined by \begin{equation*} h_{\mu}(\beta):= \lim_{n \to\infty} \frac{1}{n} H_{\mu}(\beta^n). \end{equation*} Finally, the \emph{entropy} of $\mu$ is defined by \begin{equation*} h_{\mu}(\sigma):= \sup \left\{h_{\mu}(\beta) : \beta\text{ a partition with } H_{\mu}(\beta) < \infty \right\}. \end{equation*}
\begin{remark}
Krengel \cite[Remark p.166]{kr} observes that since the entropy of a finite invariant measure $\mu$ is usually defined as the entropy of the normalised measure $\mu/ |\mu|$, the linearity (in the standard sense) of the entropy map is destroyed. Following Krengel's line of thought, the number $ |\mu|h_{\mu/|\mu|}(\sigma)$ appearing in Theorem \ref{thm:main} can be understood, as the entropy of the finite measure $\mu$ (see also \cite[Theorem 8.1]{wa} for example). \end{remark}
\subsection{Thermodynamic formalism for CMS} \label{sec:tf}
Throughout this section we assume that $(\Sigma, \sigma)$ is topologically transitive and that $\varphi:\Sigma\to {\mathbb R}$ has summable variations. Let $A \subset \Sigma$ and $1_{A}(x)$ be the characteristic function of the set $A$. In this setting we define,
\begin{equation*}
Z_n(\varphi,a):=\sum_{\sigma^n x=x} e^{S_n \varphi(x)}1_{[a]}(x), \end{equation*} where $a\in {\mathbb N}$. The \emph{Gurevich pressure} of $\varphi$ is defined by \begin{equation*}
P_G(\varphi):=\limsup_{n\to \infty} \frac{1}{n}\log Z_n(\varphi,a). \end{equation*} This definition was introduced by Sarig \cite{sa1}, based on the work of Gurevich \cite{gu2}. We remark that the right hand side in the definition of $P_G(\varphi)$ is independent of $a\in {\mathbb N}$, and that if $(\Sigma,\sigma)$ is topologically mixing, then the limsup can be replaced by a limit (see \cite[Theorem 1]{sa1} and \cite[Theorem 4.3]{sabook}).
This definition of pressure satisfies the variational principle (see \cite[Theorem 3]{sa1} and \cite[Theorem 2.10]{ijt}) and can be approximated by the pressure of compact invariant subsets \cite[Theorem 2 and Corollary 1]{sa1}. Indeed, \begin{eqnarray*} \label{thm:vp}
P_G(\varphi) &=& \sup \left\{P_{\text{top}}(\varphi |K) : K \subset \Sigma \text{ compact and } \sigma^{-1}K=K \right\} \\
&=& \sup_{\mu\in \mathcal{M}(\Sigma,\sigma)} \left\{h_\mu(\sigma)+\int\varphi d\mu: \int \varphi d\mu>-\infty \right\}, \end{eqnarray*} where $P_{\text{top}}( \cdot)$ is the classical pressure on compact spaces \cite[Chapter 9]{wa}. A measure $\mu \in \mathcal{M}(\Sigma, \sigma)$ is an \emph{equilibrium state} for $\varphi$ if $\int \varphi d\mu>-\infty$ and \begin{equation*}
P_G(\varphi)=h_\mu(\sigma)+\int \varphi d\mu. \end{equation*}
The following function will be of importance in this article. \begin{definition} \label{def:ret} Let $A \subset \Sigma$. Denote by $R_A(x):=1_{A}(x)\inf\{n\ge1:\sigma^n x\in A\}$ the first return time map to the set $A$. In the particular case in which the set $A$ is a cylinder $[a]$ we denote $R_{[a]}(x):= R_a(x)$.
\end{definition}
Sarig \cite[Section 4.2]{sa1} introduced the following:
\begin{equation*}
Z_n^*(\varphi,a):=\sum_{\sigma^n(x)=x} e^{S_n\varphi(x)}1_{[R_a=n]}(x), \end{equation*} where $[R_a =n]:=\left\{x \in \Sigma : R_a(x)=n \right\}$. Extending notions of Markov chains, Sarig \cite{sa1} classified potentials according to its recurrence properties.
\begin{definition}[Classification of potentials] \label{def:clas} Let $(\Sigma , \sigma)$ be a topologically transitive CMS and $\varphi$ a summable variation potential with finite Gurevich pressure. Define $\lambda=\exp \left( P_G(\varphi) \right)$ and fix $a\in {\mathbb N}$. \begin{enumerate} \item If $\sum_{n\geqslant 1}\lambda^{-n}Z_n(\varphi,a)$ diverges we say that $\varphi$ is \emph{recurrent}. \item If $\sum_{n\geqslant 1}\lambda^{-n}Z_n(\varphi,a)$ converges we say that $\varphi$ is \emph{transient}. \item If $\varphi$ is recurrent and $\sum_{n\geqslant 1}n\lambda^{-n}Z^*_n(\varphi,a)$ converges we say that $\varphi$ is \emph{positive recurrent}. \item If $\varphi$ is recurrent but $\sum_{n\geqslant 1}n\lambda^{-n}Z^*_n(\varphi,a)$ diverges we say that $\varphi$ is \emph{null recurrent}.
\end{enumerate} \end{definition}
Topological transitivity implies that above definitions do not depend on the choice of the symbol $a$.
\begin{remark} The classification in Definition \ref{def:clas} is invariant under the addition of coboundaries and constants. That is, if $\psi:\Sigma \to {\mathbb R}$ is of summable variations and $C \in {\mathbb R}$ we have that: the potential $\varphi$ is recurrent (resp. transient) if and only if the potential $\varphi + \psi - \psi \circ \sigma + C$ is recurrent (resp. transient). Moreover, the potential $\varphi$ is positive recurrent (resp. null recurrent) if and only if the potential $\varphi + \psi - \psi \circ \sigma + C$ is positive recurrent (resp. null recurrent). \end{remark}
The following result describes existence and uniqueness of equilibrium states. Parts (\ref{sumvar1}) and (\ref{sumvar2}) follow from Theorems 1.1 and Theorem 1.2 of \cite{bs}, respectively.
\begin{theorem} \label{clas} Let $(\Sigma , \sigma)$ be a topologically transitive CMS and $\varphi$ a summable variation potential with finite Gurevich pressure. Then \begin{enumerate} \item \label{sumvar1} There exists at most one equilibrium state for $\varphi$. \item \label{sumvar2} If $\varphi$ has an equilibrium state then $\varphi$ is positive recurrent. \end{enumerate} \end{theorem}
In this article the potential $\varphi=0$ will play a particularly important role. The \emph{topological entropy} of $(\Sigma,\sigma)$, that we denote by $h_{top}(\sigma)$, is defined as the Gurevich pressure of the potential $\varphi=0$, that is
$$h_{top}(\sigma):=P_G(0).$$
We say that $(\Sigma,\sigma)$ is recurrent, transient, null recurrent or positive recurrent according to the corresponding properties of $\varphi=0$. If $(\Sigma,\sigma)$ is positive recurrent, then Theorem \ref{clas} implies that $(\Sigma,\sigma)$ admits a unique measure of maximal entropy. This was first proved by Gurevich \cite{gu1}.
\begin{remark}
Note that every finite entropy, transitive CMS satisfies the $\mathcal{F}-$property (see Definition \ref{def:F}).
\end{remark}
\subsection{Strongly positive recurrent CMS} Properties of CMS may be significantly different from those of sub-shifts of finite type defined on finite alphabets. In this section we describe a special class of CMS with properties analogous to those of compact sub-shifts. This study is based on work of Vere-Jones \cite{v1,v2} developed during the 1960s, where he first defined an equivalent class in the setting of stochastic matrices. Several people have contributed to the understanding of this class, for example, Salama \cite{s}, Gurevich and Savchenko \cite{gs}, Sarig \cite{sa3}, Ruette \cite{r}, Boyle, Buzzi and G\'omez \cite{bbg} and Cyr and Sarig \cite{cs}. In these works the following quantities, or related ones, have been defined and studied.
\begin{definition} Let $(\Sigma, \sigma)$ be topologically transitive CMS and $a \in {\mathbb N}$. Let \begin{equation*} \Delta_\infty ([a]):=\limsup_{n\to \infty} \frac{1}{n}\log Z_n^*(0,a), \end{equation*} and \begin{equation*} {\Delta}_\infty:=\inf_{a\in {\mathbb N}} \Delta_\infty([a]). \end{equation*} \end{definition}
\begin{remark} The number $\Delta_\infty ([a])$ can depend on the symbol $a$, see \cite[Remark 2,1]{r}. \end{remark}
\begin{definition}[Strongly positive recurrent CMS] \label{def:spro} Let $(\Sigma,\sigma)$ be a topologically transitive CMS with finite topological entropy. We say that $(\Sigma,\sigma)$ is \emph{strongly positive recurrent} (SPR) if ${\Delta}_\infty<h_{top}(\sigma)$. \end{definition}
\begin{remark} \label{rem_conspr} A strongly positive recurrent CMS is positive recurrent. In particular it admits a unique measure of maximal entropy. Moreover, with respect to this measure the system $(\Sigma, \sigma)$ is exponentially recurrent (see \cite[Proposition 2.3]{bbg} for precise statements). The class of strongly positive recurrent CMS was intensively studied by Gurevich and Savchenko in \cite{gs}. Note, however, that in \cite{gs} these are called \emph{stable-positive recurrent}. We also remark that there exists CMS that are positive recurrent but not strongly positive recurrent (see \cite[Example 2.9]{r}). \end{remark}
\begin{remark} \label{rem:spr}
Strongly positive recurrent CMS have the property that the entropy is concentrated inside the system and not near infinity. Indeed, let $(\Sigma, \sigma)$ be a CMS an $G$ its associated graph. Gurevich and Zargaryan \cite{gz} (see also \cite{gs}) showed that a condition equivalent to SPR is the existence of a finite connected subgraph $H \subset G$ such that there are more paths inside than outside $H$ (in term of exponential growth). See \cite[Section 3.1]{r} for precise statements. On the other hand, for graphs that are not strongly positive recurrent the entropy is supported by the infinite paths that spend most of the time outside a finite subgraph (see \cite[Proposition 3.3]{r}).
\end{remark}
Along the lines of the observations made in Remark \ref{rem:spr}, in the next section (see Proposition \ref{prechar}) we characterise SPR for CMS as those having entropy at infinity strictly smaller than the topological entropy.
Sarig \cite{sa3} generalised the notion of strong positive recurrence to potentials $\varphi$. Using his definition, we recover the topological notion in Definition~\ref{def:spro} for the potential $\varphi\equiv 0$, i.e., this potential is strongly positive recurrent if and only if $(\Sigma,\sigma)$ is SPR (see \cite[Remark 2.11]{r}). For Sarig's class of potentials the associated thermodynamic formalism enjoys most of the properties of the thermodynamics for H\"older potentials on sub-shifts of finite type. In particular, the transfer operator corresponding to a strongly positive recurrent potential has a spectral gap (see \cite[Theorem 2.1]{cs}). This readily implies that the pressure function is analytic and there exist formulas for its derivatives (\cite[Theorem 3 and 4]{sa3} and \cite[Theorem 1.1]{cs}), there exists a unique equilibrium state and it has exponential decay of correlations and satisfies the Central Limit Theorem (\cite[Theorem 1.1]{cs}). Moreover, in the space of potentials strongly positive recurrence is a robust property. Indeed, it was proved by Cyr and Sarig \cite[Theorem 2.2]{cs} that the space of strongly positive recurrent potentials is open and dense (with respect to the uniform metric) in the space of weakly H\"older potentials with finite pressure.
\subsection{Entropy at infinity}\label{sec:introh} A fundamental consequence of Theorem \ref{thm:main} is that a great deal of dynamical information of the system is captured by its complexity at infinity. As discussed in the introduction, we have defined two different ways of quantifying this complexity. Namely, the topological entropy at infinity (Definition \ref{def:ent_inf}) and the measure theoretic one (Definition \ref{def:ent_meas_inf}). In this section we will elaborate on these notions and put our results into context.
We first discuss the topological entropy at infinity of $(\Sigma,\sigma)$, given in Definition \ref{def:ent_inf}. Observe that if $M_1<M_2$, then $z_n(M_2, q)\leqslant z_n(M_1, q)$,
for every $n, q \in {\mathbb N}$, so \begin{equation*} \delta_\infty(q)=\inf_M \delta_\infty(M,q)=\lim_{M\to\infty}\delta_\infty(M,q). \end{equation*} If $(\Sigma,\sigma)$ is a transitive CMS then for every $a,b\in {\mathbb N}$, \begin{equation} \label{eq:deltainfagain} \delta_\infty(M,q)=\limsup_{n\to\infty}\frac{1}{n}\log z_n(M,q,a,b), \end{equation} where $z_n(M,q,a,b)$ is the number of cylinders of the form $[x_0,\ldots,x_{n+1}]$, where $x_0=a$, $x_{n+1}=b$, and \begin{equation*} \# \left\{i\in\{0,\ldots,n+1\}:x_i\leqslant q \right\}\leqslant \frac{n+2}{M}. \end{equation*} Note that $q<q'$ implies the inequality $z_n(M,q',a,b)\leqslant z_n(M,q,a,b)$. In particular $(\delta_\infty(M,q))_q$ is decreasing in $q$. We conclude that \begin{align}\label{infinf} \delta_\infty=\inf_{q} \delta_\infty(q)=\inf_{M,q}\delta_\infty(M,q). \end{align}
Since in our results we will usually assume that the symbolic space is transitive, we can consider \eqref{infinf} as the definition of the topological entropy at infinity.
We now consider the measure theoretic entropy at infinity, defined for finite entropy CMS as \begin{equation*} h_\infty :=\sup_{(\mu_n)_n\to 0}\limsup_{n\to\infty}h_{\mu_n}(\sigma), \end{equation*} where $(\mu_n)_n\to 0$ means that the sequence $(\mu_n)_n$ converges on cylinders to the zero measure. Note that the finite entropy assumption, and more generally the $\mathcal{F}-$property, ensures the existence of sequences of measures converging on cylinders to the zero measure (see \cite[Lemma 4.16]{iv}). In particular, $h_{\infty}$ is well defined. In \cite[Example 4.17]{iv}, an example of a CMS made of infinitely many loops of length two based at a common vertex is considered. The entropy is infinite and there are no sequences of measures converging to zero. Every measure gives weight at least $1/2$ to the base cylinder. \label{rmk:finite ent}
In Section \ref{entinf1} we will prove that both, the topological and the measure theoretic entropies at infinity coincide. This has several consequences, in particular we obtain that Theorem \ref{thm:main} is sharp. Indeed, $\delta_{\infty}$ is the smallest number for which inequality \eqref{eq:main} holds.
In the context of CMS the entropy at infinity was already investigated by Gurevich and Zargaryan \cite{gz}, Ruette \cite{r} and Buzzi \cite{b}. It is important to mention that they also had two flavours of entropy at infinity, a topological and a measure theoretic version. It is proven by Ruette \cite{r} that both notions coincide (for a precise statement see \cite[Proposition 6.1]{b}). It turned out that the notions of entropy at infinity presented in this work coincide with theirs. Recall that if $G$ is the graph which determines $(\Sigma,\sigma)$, then
\begin{equation*} b_{\infty}=\inf_{F} \inf_{\lambda >0} \sup \left\{h_{\mu}(\sigma) : \mu([F]) < \lambda \right\}, \end{equation*} where $F$ ranges over the finite sub-graphs of $G$ and $[F]:= \left\{ x \in \Sigma : x_0 \in \mathcal{A}_F \right\}$, where $ \mathcal{A}_F$ denotes the symbols appearing as vertex of $F$. We first show the relation between $h_\infty$ and $b_\infty$.
\begin{lemma} For a sequence $(\mu_n)_n$ in $\mathcal{M}(\Sigma, \sigma)$, the following are equivalent: \begin{itemize} \item[(a)] for any collection of cylinders $C^1, \ldots, C^N,$ and $\varepsilon>0$, there is $n_0\in {\mathbb N}$ such that $\mu_n(\bigcup_{i=1}^NC^i)<\varepsilon$ for all $n\geqslant n_0$; \item[(b)] for any finite subgraph $F$ of $G$ and any $\varepsilon>0$, there is $n_1\in {\mathbb N}$ such that $\mu_n([F])<\varepsilon$ for all $n\geqslant n_1$. \end{itemize} \label{lem:cylBuz} \end{lemma}
An easy consequence of the lemma is that convergence on cylinders in this setting corresponds to the type of limits featuring in the definition of $b_\infty$ and thus $b_\infty=h_\infty$.
\begin{proof}[Proof of Lemma~\ref{lem:cylBuz}] Since (b) concerns 1-cylinders, the fact that (a) implies (b) is clear. To prove the reverse implication, we observe that if $C^1, \ldots, C^N$ is a collection of cylinders then we can take the subgraph defined by the first coordinate of each $C^i$ as our subgraph. \end{proof}
As previously mentioned, in Section \ref{entinf1} we will prove that $h_\infty=\delta_\infty$. This implies that the entropy at infinity defined in this section coincides with the previously defined one. One consequence is that, since \cite[Proposition 6.1]{b} implies $b_{\infty}<h_{top}(\sigma)$ is a characterisation of SPR, we thus have the following alternative characterisation:
\begin{proposition}\label{prechar} A topologically transitive CMS $(\Sigma,\sigma)$ is SPR if and only if $h_{\infty}<h_{top} (\sigma)$ if and only if $\delta_{\infty}<h_{top} (\sigma)$.\end{proposition}
This result is consistent with the comments in Remark \ref{rem:spr}. Indeed, SPR systems are those for which the entropy is not concentrated at infinity; the inequality $\delta_{\infty}<h_{top} (\sigma)$ has a wealth of dynamical consequences (see Remark \ref{rem_conspr}).
From a slightly different point of view, it was not realised until recently that the entropy at infinity has a particularly important role in the regularity of the entropy map. In the context of homogeneous dynamics, for the diagonal action on $G/\Gamma$, where $G$ is a ${\mathbb R}$-rank 1 semisimple Lie group with finite centre and $\Gamma\leqslant G$ a lattice, a formula like Theorem \ref{thm:main} was obtained in \cite{ekp}. In that context the constant playing the role of the entropy at infinity is half the topological entropy of the flow. It was later proved in \cite{kp} that half the topological entropy is in fact sharp and equal to the measure theoretic entropy at infinity in that setup. The method employed in \cite{ekp} was used in \cite{rv} to prove that a similar result holds for the geodesic flow on a geometrically finite manifold. Unfortunately, an obstruction to run the method from \cite{ekp} is the existence of periodic orbits that escape to infinity. This issue was overcome in \cite{ve}, where the results in \cite{rv} where generalised to all complete negatively curved manifolds. For CMS the existence of periodic orbits that escape phase space is quite common so our approach is similar to the one in \cite{ve}. Additional complications arise from the possible lack of locally compactness of $\Sigma$. In Section \ref{mainine} and Section \ref{finalproof} we will address these issues and prove Theorem \ref{thm:main}.
The entropy at infinity has further applications to suspension flows, entropy density, the dimension of points which escape on average, existence of equilibrium states and bounds on mass escape, all of which we give in Section~\ref{sec:app}.
\section{Katok's entropy formula } \label{ent}
In the early 1970s Bowen \cite{bo} and Dinaburg \cite{d} provided a new definition of topological entropy of a dynamical system. Inspired by these results, Katok \cite{ka} established a formula for the measure theoretic entropy in analogy to the definition of topological entropy by Bowen and Dinaburg. We now recall his result in a particular context.
Let $(\Sigma, \sigma)$ be a CMS and let $d$ be the metric on $\Sigma$ defined in \eqref{metric}. The dynamical metric $d_n$ is defined by the formula \begin{equation*} d_n(x,y):=\max_{k\in \{0,\ldots,n-1\}}d(\sigma^kx,\sigma^ky). \end{equation*} The open ball of radius $r$ centred at $x$ with respect to the metric $d_n$ is denoted by $B_n(x,r)$. By the definition of the metric $d$ we know that $B_n(x, 2^{-N})=C_{n+N}(x)$. A ball of the form $B_n(x,r)$ is called a $(n,r)$-dynamical ball. The following result is a particular case of a theorem proved in \cite[Theorem 1.1]{ka}.
\begin{theorem}\label{kat} Let $(\Sigma,\sigma)$ be a sub-shift of finite type defined on a finite alphabet and $\mu$ an ergodic $\sigma$-invariant probability measure. Then \begin{equation} \label{eq:kat} h_\mu(\sigma)=\lim_{\epsilon\to 0} \liminf_{n\to \infty}\dfrac{1}{n}\log N_\mu(n,\epsilon,\delta), \end{equation} where $N_\mu(n,\epsilon,\delta)$ is the minimum number of $(n,\epsilon)$-dynamical balls needed to cover a set of $\mu$-measure strictly bigger than $1-\delta$. In particular the limit above does not depend on $\delta\in (0,1)$. \end{theorem}
The relation established in \eqref{eq:kat} is known as Katok's entropy formula. It turns out that Katok's proof is rather flexible. It was observed by Gurevich and S. Katok \cite[Section 4]{gk} and also by Riquelme \cite[Theorem 2.6]{ri} that the proof in \cite[Theorem 1.1]{ka} yields that if $(X,d)$ is a metric space (not necessarily compact) and $T: X \to X$ a continuous map then \begin{equation*} h_{\mu}(T) \leq \lim_{\epsilon \to 0} \liminf_{n \to \infty} \frac{\log N_{\mu}(n, \epsilon ,\delta)}{n}. \end{equation*} The compactness assumption on $X$ is used in the proof of the other inequality. It is routine to check that compactness assumption can be replaced by the existence of a totally bounded metric.
This section is devoted to proving that formula \eqref{eq:kat} holds for CMS of finite topological entropy. Moreover, we will prove the limit is independent of $\epsilon$. We prove:
\begin{theorem} \label{katformula} Let $(\Sigma, \sigma)$ be a CMS and $\mu$ an ergodic $\sigma$-invariant probability measure. Then for every $\delta\in (0,1)$ we have $$h_\mu(\sigma)\leqslant \lim_{n\to \infty}\frac{1}{n}\log N_\mu(n,1,\delta).$$ If $(\Sigma,\sigma)$ has finite topological entropy, then $$h_\mu(\sigma)= \lim_{n\to \infty}\frac{1}{n}\log N_\mu(n,1,\delta).$$ \end{theorem}
Define the following collection of sets: for every $m \in {\mathbb N}$ let \begin{equation} \label{def:km} K_m:=\Sigma\cap \bigcup_{s=1}^m [s]. \end{equation} Note that if $\Sigma$ is locally compact, then $K_m$ is compact for every $m\in {\mathbb N}$. To every sequence of natural numbers $(a_i)^\infty_{i=0}$ we associate the set \begin{equation} \label{def:kai} K((a_i)_i)= \Sigma\cap \prod_{i\geqslant 0} \{1,\ldots,a_i\}. \end{equation} Observe that $K((a_i)_i)$ is the intersection of a closed set with a compact set and is thus a compact subset of $\Sigma$. Moreover, every compact set $K\subset \Sigma$ is contained in a set of the form $K((a_i)_i)$. The following lemma follows directly from \cite[Theorem 3.2]{pa}. For concreteness we provide a simple proof of this general fact.
\begin{lemma}\label{lem:compact} Let $\mu$ a Borel measure on $\Sigma$. For every $\epsilon>0$, there exists a sequence of natural numbers $(a_i)_i$ such that $\mu(K((a_i)_i))>1-\epsilon$. \end{lemma}
\begin{proof} Fix a sequence $(b_i)_i$ satisfying \begin{equation*} \left(1-\frac{\epsilon}{2}\right)\prod_{i=1}^\infty b_i> 1-\epsilon, \end{equation*} where $b_i\in (0,1)$ for every $i\in{\mathbb N}$. We will construct the sequence $(a_i)_i$ inductively. Choose $a_0$ such that $\mu(\bigcup_{i=1}^{a_0} [i])>1-\frac{\epsilon}{2}$. For every $i\in \{1,\ldots,a_0\}$ we choose $c(i)\in {\mathbb N}$ such that \begin{equation*} \mu\left(\bigcup^{c(i)}_{k= 1}[ik]\right)\geqslant \mu([i])b_1. \end{equation*} Let $a_1:=\max_{i\in \{1,\ldots,a_0\}} c(i)$. For $(i_1,i_2)\in \prod_{i=0}^1\{1,\ldots,a_i\}$ we define $c(i_1,i_2)$ such that \begin{equation*} \mu\left(\bigcup^{c(i_1,i_2)}_{k= 1}[i_1i_2k]\right)\geqslant \mu([i_1i_2])b_2. \end{equation*} Define $a_2=\max_{(i,j)\in \prod_{i=0}^1 \{1,\ldots,a_i\} } c(i,j)$. We continue this procedure inductively. It follows from the construction that \begin{equation*} \mu(K((a_i)_i))=\mu\left(\prod_{i=0}^\infty \{1,\ldots,a_i\} \right)\geqslant \left(1-\frac{\epsilon}{2}\right)\prod_{i=1}^\infty b_i>1-\epsilon, \end{equation*} as desired. \end{proof}
\begin{remark}\label{rem:upbound} Katok proved \cite[Theorem 1.1]{ka} that if ${\mathcal P}$ is any finite partition of $\Sigma$ satisfying $\mu(\partial {\mathcal P})=0$, then for any $\delta \in (0,1)$ \begin{equation*} h_\mu({\mathcal P})\leqslant \lim_{r\to 0}\liminf_{n\to\infty}\frac{1}{n}\log N_\mu(n,r,\delta). \end{equation*} For a CMS it is easy to check that the partitions \begin{equation*} {\mathcal P}_n=\left\{[1],\ldots,[n],\bigcup_{s>n}[s]\right\} \end{equation*} are such that $\partial {\mathcal P}_n=\emptyset$, and $\lim_{n\to\infty}h_\mu(\sigma,{\mathcal P}_n)=h_\mu(\sigma)$. From this we conclude that \begin{equation*} h_\mu(\sigma)\leqslant \lim_{r\to 0}\liminf_{n\to\infty}\frac{1}{n}\log N_\mu(n,r,\delta). \end{equation*} \end{remark}
Our next result is inspired by the proof of \cite[Theorem 2.10 and Theorem 2.11]{ri}. In our context we do not have local compactness of $\Sigma$: the finite entropy assumption is important in overcoming this issue.
\begin{lemma}\label{lem:katokeq} Let $(\Sigma,\sigma)$ be a CMS with finite topological entropy. If $\mu$ is an ergodic $\sigma$-invariant probability measure, then for every $\delta\in (0,1)$ we have \begin{equation*} h_\mu(\sigma)=\lim_{N\to \infty} \liminf_{n\to \infty}\frac{1}{n}\log N_\mu(n,2^{-N},\delta). \end{equation*} \end{lemma}
\begin{proof} As observed in Remark \ref{rem:upbound} the inequality \begin{equation*} h_\mu(\sigma)\leqslant\lim_{N\to \infty} \liminf_{n\to \infty}\frac{1}{n}\log N_\mu(n,2^{-N},\delta) \end{equation*}
is known to hold. For the converse inequality it suffices to prove that for every $\ell\in {\mathbb N}$ there exists a partition ${\mathcal P}={\mathcal P}(\ell)$ of $\Sigma$ and a subset $ K \subset \Sigma$ satisfying:
\begin{enumerate}
\item The partition ${\mathcal P}(\ell)$ has finite entropy with respect to $\mu$.
\item $\mu(K)>1-\frac{\delta}{6}$.
\item For every $x\in K\cap \sigma^{-n}K$ we have ${\mathcal P}^n(x)\subset B_n(x,2^{-\ell})$.
\end{enumerate} In this situation a slight modification of the first part of the proof in \cite[Theorem 1.1]{ka} yields the desired inequality, as we show here. Suppose that the partition ${\mathcal P}={\mathcal P}(\ell)$ has been constructed. Let $\epsilon >0$. Since the measure $\mu$ is ergodic by the Shannon-McMillan-Breiman theorem there exists $N_0 \in {\mathbb N}$ such that the set \begin{equation*} A_{\epsilon, N_0}:=\left\{x\in \Sigma: \mu({\mathcal P}^n(x))\geqslant \exp(-n(h_\mu({\mathcal P})+\epsilon)), \text{for all }n\geqslant N_0 \right\}. \end{equation*} satisfies $\mu(A_{\epsilon,N_0})>1-\frac{\delta}{6}$. Let $n\geqslant N_0$ and $B_n:=A_{\epsilon,N_0}\cap K\cap \sigma^{-n}K$. Observe that $\mu(B_n)\geqslant 1-\frac{\delta}{2}$ and that if $x\in B_n$ then $x\in K\cap \sigma^{-n}K$, and therefore ${\mathcal P}^n(x)\subset B_n(x,2^{-\ell})$. The set $A_{\epsilon,N_0}$ requires at most $\exp(n(h_\mu({\mathcal P})+\epsilon))$ elements of the partition ${\mathcal P}^n$ to cover it. Therefore, $B_n$ requires at most $\exp(n(h_\mu({\mathcal P})+\epsilon))$ $(n,2^{-\ell})$-dynamical balls to cover it, where $\mu(B_n)>1-\frac{\delta}{2}$. We conclude that \begin{equation*} \limsup_{n\to\infty}\frac{1}{n}\log N_\mu(n,2^{-\ell},\delta)\leqslant h_\mu({\mathcal P})+\epsilon\leqslant h_\mu(\sigma)+\epsilon. \end{equation*} Since $\epsilon>0$ was arbitrary we obtain \begin{equation*} \lim_{\ell\to\infty}\limsup_{n\to\infty}\frac{1}{n}\log N_\mu(n,2^{-\ell},\delta)\leqslant h_\mu(\sigma), \end{equation*} concluding the proof of the lemma.
We now prove the existence of such a partition ${\mathcal P}={\mathcal P}(\ell)$. By Lemma \ref{lem:compact} there exists a sequence $(a_i)_i$ such that the compact set $K_0:=K((a_i)_i)$ satisfies $ \mu(K_0) \geqslant 1-\frac{\delta}{6}$. Denote by $S$ the set of points in $\Sigma$ that enter $K_0$ infinitely many times under iterates of $\sigma$. It is a consequence of Birkhoff's Ergodic Theorem that $\mu(S)=1$. Define $K:=K_0\cap S$, and observe that $\mu(K)\geqslant 1-\frac{\delta}{6}$. For every $k \geq 1$, let \begin{equation*} R_K(x):=\inf\left\{k\geqslant 1: \sigma^k(x)\in K_0\right\} \text{ for } x\in K, \text{ and } A_k:=\left\{x \in K: R_K(x)=k \right\}. \end{equation*}
Partition $A_k$ using cylinders of length $k+\ell+1$ and denote such partition by ${\mathcal Q}_k$. It is important to observe that $\#{\mathcal Q}_k$ is finite for all $k$. This follows from the definition of $K_0$ and the finite topological entropy of $(\Sigma,\sigma)$. Indeed, if $x= (x_0, x_1, \dots , x_k \dots) \in A_k$, then $x_0 ,x_k \in \{1,\ldots,a_0\}$. Moreover, there are at most $C=\prod_{i=0}^{\ell} a_i$ cylinders of the form $[y_0 y_1\ldots y_l]$ intersecting $K$, so for $k$ large enough, $$\#{\mathcal Q}_k\leqslant Ce^{k(h_{top}(\sigma)+1)}.$$
Finally, consider the partition of $\Sigma$ defined by ${\mathcal P}= \mathcal{Q} \cup \left( \bigcup_{k=1}^{\infty} {\mathcal Q}_k \right)$, where $\mathcal{Q}:= \Sigma \setminus \bigcup_{k=1}^{\infty} {\mathcal Q}_k$. We claim that this countable partition satisfies the remaining required properties, that is: \begin{enumerate}
\item The partition ${\mathcal P}={\mathcal P}(\ell)$ has finite entropy with respect to $\mu$.
\item For every $x\in K\cap \sigma^{-n}K$ we have ${\mathcal P}^n(x)\subset B_n(x,2^{-\ell})$.
\end{enumerate} The second property follows from the construction of ${\mathcal P}$. Indeed, let $z\in {\mathcal P}^n(x)$, where $x , \sigma^n(x) \in K$. We claim that $z\in B_n(x,2^{-\ell})$. For simplicity we will assume that $x$ has its first return to $K$ at time $n$ (the general case is just an iteration of the argument in this setting). Since $x \in A_n$ we have that ${\mathcal P}(x)$ is a cylinder of length $n+\ell+1$, which readily implies that $z\in B_n(x,2^{-\ell})$.
We now verify that $H_\mu({\mathcal P})<\infty$. For $r$ sufficiently large, \begin{align*} H_\mu({\mathcal P})+R & = \sum_{k\geqslant r} \sum_{P\in {\mathcal Q}_k}-\mu(P)\log \mu(P)\\ &= \sum_{k\geqslant r} \mu(A_k)\left( \sum_{P\in {\mathcal Q}_k}-\frac{\mu(P)}{\mu(A_k)}\log \frac{\mu(P)}{\mu(A_k)}-\frac{\mu(P)}{\mu(A_k)} \log\mu(A_k)\right) \\
&\leqslant\sum_{k\geqslant r}\mu(A_k)\log(|{\mathcal Q}_k|)-\sum_{k\geqslant r}\mu(A_k)\log \mu(A_k) \\ &\leqslant \sum_{k\geqslant r}k\mu(A_k)\log\left(e^{h_{top}(\sigma)+1}C^{1/k}\right)-\sum_{k\geqslant r}\mu(A_k)\log \mu(A_k)\\ &\leqslant C'\sum_{k\geqslant r}k\mu(A_k)-\sum_{k\geqslant r}\mu(A_k)\log \mu(A_k), \end{align*} where $R=\mu({\mathcal Q})\log\mu({\mathcal Q})+\sum_{k=1}^{r-1}\sum_{P\in {\mathcal Q}_k}\mu(P)\log\mu(P)\in {\mathbb R}$. It follows from Kac's lemma that $\sum k\mu(A_k)=1$. This and the inequality \begin{align*}\label{mane} -\sum_{k\geqslant r}\mu(A_k)\log \mu(A_k)\leqslant \sum_{k\geqslant r}k\mu(A_k)+2e^{-1}\sum_{k\geqslant r}e^{-k/2}, \end{align*} see \cite[Lemma 1]{m}, imply the finiteness of $H_\mu({\mathcal P})$. This concludes the proof. \end{proof}
\begin{lemma}\label{lem:katokine} Let $(\Sigma, \sigma)$ be a CMS and $\mu$ an ergodic $\sigma$-invariant probability measure. Then for any $\delta\in (0,1)$, we have \begin{equation*} h_\mu(\sigma)\leqslant \liminf_{n\to\infty} \frac{1}{n}\log N_\mu(n,1,\delta). \end{equation*} \end{lemma}
\begin{proof}
Let $A \subset \Sigma$ be a set such that $\mu(A) \geq 1 - \delta$. Denote by $a_\mu(n,\delta)$ the minimum number of cylinders of length $n$ that cover a $A$. Observe that \begin{equation*}
N_\mu(n,2^{-t},\delta)=a_\mu(n+t,\delta),
\end{equation*} and that \begin{equation*} \liminf_{n\to\infty}\frac{1}{n}\log a_\mu(n,\delta)=\liminf_{n\to\infty}\frac{1}{n}\log a_\mu(n+t,\delta), \end{equation*} for every $t\in{\mathbb N}$. In particular we have that $\liminf_{n\to\infty}\frac{1}{n}\log N_\mu(n,2^{-\ell},\delta)$ is independent of $\ell$. From Remark \ref{rem:upbound} we conclude that \begin{align*}h_\mu(\sigma)&\leqslant \lim_{t\to \infty}\liminf_{n\to \infty}\frac{1}{n}\log N_\mu(n,2^{-t},\delta)\\ &=\lim_{t\to \infty}\liminf_{n\to \infty}\frac{1}{n}\log a_\mu(n+t,\delta)\\ &=\liminf_{n\to \infty}\frac{1}{n}\log a_\mu(n,\delta). \end{align*} \end{proof}
\begin{proof}[Proof of Theorem \ref{katformula}] The proof follows combining Lemma \ref{lem:katokeq} and Lemma \ref{lem:katokine}. \end{proof}
We now prove a result related to Lemma \ref{lem:katokine}. We say that two points $x, y \in \Sigma$ are $(n,r)$-separated if $d_n(x,y)\geqslant r$. In particular $x$ and $y$ are $(n,1)$-separated if they do not belong to the same cylinder of length $n$.
\begin{lemma}\label{lem:compshift} Let $X$ be a $\sigma$-invariant compact subset of $\Sigma$. Then \begin{equation*}
h_{top}(\sigma |X)=\limsup_{n\to\infty} \frac{1}{n}\log N(X,n), \end{equation*}
where $N(X,n)$ is the maximal number of $(n,1)$-separated points in $X$, and $h_{top}(\sigma|X)$ is the topological entropy of $(X,\sigma)$. \end{lemma}
\begin{proof} By definition of the topological entropy of a compact metric space we know that \begin{equation*}
h_{top}(\sigma |X)=\lim_{k\to\infty}\limsup_{n\to\infty}\frac{1}{n}\log N(X,n,k), \end{equation*} where $N(X,n,k)$ is the maximal number of $(n, 2^{-k})$-separated points in $X$. Observe that being $(n,2^{-k})$-separated is the same as being $(n+k,1)$-separated. This implies that $N(X,n,k)=N(X,n+k)$. Note that \begin{equation*} \limsup_{n\to\infty}\frac{1}{n}\log N(X,n,k)=\limsup_{n\to\infty}\frac{1}{n}\log N(X,n+k)=\limsup_{n\to\infty}\frac{1}{n}\log N(X,n). \end{equation*} Therefore \begin{equation*}
h_{top}(\sigma |X)=\lim_{k\to\infty}\limsup_{n\to\infty}\frac{1}{n}\log N(X,n,k)=\limsup_{n\to\infty} \frac{1}{n}\log N(X,n), \end{equation*} as desired. \end{proof}
\section{Weak entropy density} \label{sec:wed}
In this section we describe the inclusion $\mathcal{E}(\Sigma, \sigma) \subset \mathcal{M}(\Sigma,\sigma)$, where $\mathcal{E}(\Sigma, \sigma)$ is the subset of ergodic measures. It is well known that, even in this non-compact setting, the set $\mathcal{E}(\Sigma, \sigma)$ is dense in $ \mathcal{M}(\Sigma,\sigma)$ with respect to the weak* topology (see \cite[Section 6]{csc}). We prove that any finite entropy measure can be approximated by an ergodic measure with entropy sufficiently large, see Proposition \ref{dense}. This result can be thought of as a weak form of entropy density. In Section \ref{sec:ed} we will make use of this result to prove that any invariant measure $\mu$ can be approximated by ergodic measures with entropy converging to $h_{\mu}(\sigma) $ (see Theorem \ref{teodense}). Moreover, Proposition \ref{dense} will be used in the proof of our main result (see Theorem \ref{thm:main}). Both the statement and the proof of Proposition \ref{dense} closely follow that of \cite[Theorem B]{ekw}, but modifications are required to deal with the non-compactness of the space $\Sigma$.
\begin{proposition} \label{dense} Let $(\Sigma,\sigma)$ be a transitive CMS. Then for every $\mu\in \mathcal{M}(\Sigma,\sigma)$ with $h_{\mu}(\sigma) <\infty$, $\epsilon>0$, $\eta>0$, and $f_1,\ldots,f_l\in C_b(\Sigma)$, there exists an ergodic measure $\mu_e\in V(f_1,\ldots,f_l,\mu,\epsilon)$ (see equation (\ref{defbasis})) such that $h_{\mu_e}(\sigma)>h_\mu(\sigma) -\eta$. We can moreover assume that $\mbox{\rm supp} (\mu_e)$ is compact. \end{proposition}
Analogously to the proof of \cite[Theorem B]{ekw} we will use the following fact.
\begin{lemma} \label{entropyforergodic} Let $\mu\in \mathcal{E}(\Sigma, \sigma)$, $\alpha>0$, $\beta>0$, $f_1,\ldots,f_\ell\in C_b(\Sigma)$, and a set $K \subset \Sigma$ satisfying $\mu(K)>3/4$. Assume that $h_\mu(\sigma) <\infty$. Then there exists $n_0 \in {\mathbb N}$ such that for all $n\geqslant n_0$ there is a finite set $\mathcal{G}=\mathcal{G}(n)\subset\Sigma$ satisfying the following properties: \begin{enumerate} \item \label{a} $\mathcal{G}\subset K\cap \sigma^{-n}K$ \item \label{b} $d(x,y)>2^{-n}$, for every pair of distinct points $x,y\in \mathcal{G}$. \item \label{c} $\#\mathcal{G}\geqslant \exp(n(h_\mu(\sigma) -\alpha))$.
\item \label{d} $|\frac{1}{n}\sum_{k=0}^{n-1} f_j(\sigma^k x)-\int f_j d\mu|<\beta$, for all $x\in \mathcal{G}$ and $j\in\{1,\ldots,\ell\}$. \end{enumerate} \end{lemma}
\begin{proof} Let $$A_{k,\beta}:=\left\{x\in \Sigma: \left|\sum_{i=0}^{n-1} f_j(\sigma^i x)-\int f_j d\mu\right|<\beta, \forall j\in \{1,\ldots,\ell\} \text{ and }n\geqslant k\right\}.$$
By Birkhoff's Ergodic Theorem there exists $s_0 \in {\mathbb N}$ such that $\mu(A_{s_0,\beta})>3/4$. From Lemma \ref{lem:katokine} we have that
\begin{equation*}
h_\mu(\sigma) \leqslant \liminf_{n\to\infty} \frac{1}{n}\log N_\mu(n,1,1/4). \end{equation*} There exists $s_1 \in {\mathbb N}$ such that if $n\geqslant s_1$, then \begin{equation*} \exp(n(h_\mu(\sigma)-\alpha))\leqslant N_\mu(n,1,1/4). \end{equation*}
Let $B_n:=K\cap \sigma^{-n}K\cap A_{s_0,\beta}$ and observe that $\mu(B_n)>1/4$. In what follows we assume that $n\geqslant n_0:=\max\{s_0,s_1\}$. From the definition of $N_\mu(n,1,1/4)$ the minimal number of cylinders of length $n$ needed to cover $B_n$ is at least $N_\mu(n,1,1/4)$. More precisely, let $(C_i)_{i\in I}$ be a minimal collection of cylinders of length $n$ covering $B_n$. In particular for every $i\in I$ we have $C_i\cap B_n\ne \emptyset$. For every $i\in I$ choose a point $x_i\in C_i\cap B_n$. We claim that the set $(x_i)_{i\in I}$ satisfies the properties required on $\mathcal{G}$. Conditions \eqref{a} and \eqref{d} follow from the definition of $B_n$. Condition \eqref{b} follows from the fact that if $i\ne j$, then $x_i$ and $x_j$ are in different cylinders of length $n$. Condition \eqref{c} follows from the inequality $$\#I\geqslant N_\mu(n,1,1/4)\geqslant \exp(n(h_\mu(\sigma)-\alpha)).$$ \end{proof}
\begin{proof}[Proof of Proposition~\ref{dense}] Recall that we want to prove that given $\mu\in \mathcal{M}(\Sigma,\sigma)$, $\epsilon>0$, $\eta>0$, and $f_1,\ldots,f_l\in C_b(\Sigma)$, there exists an ergodic measure $\mu_e\in V(f_1,\ldots,f_\ell,\mu,\epsilon)$ such that $h_{\mu_e}(\sigma)>h_\mu(\sigma) -\eta$. In the following remarks we observe that this general situation can be simplified.
As observed in Section \ref{weak*} or in \cite[8.3.1 Remark]{bg} it suffices to consider the case in which the functions $(f_i)_i$ in Proposition \ref{dense} are uniformly continuous. Therefore, under this assumption, there exists $A=A(f_1,\ldots,f_\ell)\in {\mathbb N}$, such that if $d(x,y)<2^{-A}$, then $|f_i(x)-f_i(y)|<\frac{\epsilon}{4}$. Also define $W=\max_{i\in \{1,...,\ell\}}|f_i|_0$.
Since periodic measures are dense in $\mathcal{M}(\Sigma,\sigma)$, see \cite[Section 6]{csc}, we will assume that $h_\mu(\sigma) -\eta>0$, otherwise we can approximate $\mu$ by a periodic measure. By the affinity of the entropy map \cite[Theorem 8.1]{wa} and \cite[Lemma 6.13]{iv} we can reduce the problem to the case in which $\mu=\frac{1}{N}\sum_{i=1}^N \mu_i$, where $\{\mu_i\}_{i=1}^N$ is a collection of ergodic measures.
Let $m \in {\mathbb N}$ be such that the set $K=K_m$, defined as in \eqref{def:km}, satisfies $\mu_i(K)>3/4$ for every $i \in \{1, \dots, N\}$. Since $(\Sigma, \sigma)$ is transitive, there exists a constant $L=L(m)$ such that for each pair $(a,b)\in \{1,\ldots,m\}^2$, there exists an admissible word $a{\bf r }b$, where $\ell({\bf r})\leqslant L$. It follows from Lemma \ref{entropyforergodic}, setting $\beta=\epsilon/4$ and $\alpha=\eta/2$, that there exists $n' \in {\mathbb N}$ such that for every $n >n'$ and every measure $\mu_i$, with $i \in \{1, \dots , N\}$, there exists $(n,1)$-separated sets $\mathcal{G}_i\subset K\cap \sigma^{-n}K$ satisfying properties \eqref{a}, \eqref{b}, \eqref{c} and \eqref{d} of Lemma \ref{entropyforergodic}.
Denote by ${\bf a}(x)$ the word defined concatenating the first $(n+1)-$coordinates of $x \in \Sigma$. Given $\hat{x}=(x^1,x^2,\ldots,x^{MN})\in (\prod_{i=1}^N \mathcal{G}_i)^M$, we define an admissible word $w_0(\hat{x})={\bf a}(x^1) {\bf r_1} {\bf a}(x^2){\bf r_2}\ldots{\bf a}(x^{MN}){\bf r_{MN}}{\bf a}(x^1)$, where ${\bf r_k}$s are words chosen so that $w_0(\hat{x})$ is an admissible word and $\ell({\bf r_k})\leqslant L$ (note that this is possible since $({\bf a}(x^i))_{0}$ and $({\bf a}(x^i))_{n}$ are in $\{1,\ldots,m\}$ by definition of $\mathcal{G}_i$). The word $w_0(\hat{x})$ defines a periodic point in $\Sigma$ that we denote by $w(\hat{x})$. We have that \begin{equation*} w(\hat{x})=\overline{{\bf a}(x^1) {\bf r_1} {\bf a}(x^2){\bf r_2}\ldots{\bf a}(x^{MN}){\bf r_{MN}}}. \end{equation*}
Let $\mathcal{G}:=\prod_{M\geqslant 1} (\prod_{i=1}^N \mathcal{G}_i)^M$. Following the same procedure of concatenation described above, for every $\hat{x} \in \mathcal{G}$ we define a point $w(\hat{x})\in \Sigma$. Define \begin{equation*} \Psi=\bigcup_{\hat{x}\in \mathcal{G}}O(w(\hat{x})), \end{equation*} where $O(w(\hat{x}))$ is the orbit of $w(\hat{x})$ and define $\Psi_0$ to be the topological closure of $\Psi$.
Note that the space $\Psi_0$ is a compact $\sigma$-invariant subset of $\Sigma$. By definition the set $\Psi$ is closed and invariant. Observe that the number of symbols appearing in elements belonging to $\Psi$ is finite: there are finitely many admissible words ${\bf a}(x^i)$ (recall that each $\mathcal{G}_i$ is a finite set) and we could use finitely many connecting words ${\bf r_i}$. Therefore there exists $J \in {\mathbb N}$ such that $\Psi \subset \{1,\ldots,J\}^{\mathbb N}$. Thus, $\Psi_0$ is a closed subset of a compact set.
By property \eqref{d} of Lemma \ref{entropyforergodic}, and assuming that $n'$, which also depends on $A$, $W$ and $L$, is sufficiently large, \begin{align} \label{eq:incl}
\Psi&\subset \left\{x\in \Sigma: \left|\frac{1}{n}S_n f_j(x)-\int f_j d\mu \right|\leqslant\epsilon,\forall j\in\{1,\ldots,\ell\} \right\}. \end{align} Since the set in right hand side of \eqref{eq:incl} is closed, the same inclusion holds if $\Psi$ is replaced by $\Psi_0$. Also, since $\Psi_0$ is $\sigma$-invariant we have
\begin{align*} \Psi_0&\subset \left\{x\in \Sigma: \left|\frac{1}{n}S_n f_j(\sigma^s x)-\int f_j d\mu \right|\leqslant\epsilon, \forall j\in\{1,\ldots,\ell\}\text{ and }s\geqslant 0 \right\}\\
& \subset \left\{x\in \Sigma: \left|\frac{1}{nk}S_{nk} f_j( x)-\int f_j d\mu \right|\leqslant\epsilon, \forall j\in\{1,\ldots,\ell\}\text{ and }k\in {\mathbb N} \right\}. \end{align*} This implies that every ergodic measure supported in $\Psi_0$ belongs to $V(f_1,\ldots,f_l,\mu,\epsilon)$. Indeed, consider a generic point for the ergodic measure and use the inclusion above.
By construction, if $x,y\in \left(\prod_{i=1}^N \mathcal{G}_i\right)^M$ and $x\ne y$, then $$d_{NM(n+1+L)}(w(x),w(y))=1.$$ In other words $\Psi_0$ contains a $(NM(n+1+L),1)$-separated set of cardinality at least \begin{equation*} \exp\left(nNM \left(\frac{1}{N}\sum_{k=1}^N h_{\mu_k} (\sigma)-\frac{\eta}{2} \right) \right). \end{equation*} Here we used property \eqref{c} of Lemma \ref{entropyforergodic} for our sets $\mathcal{G}_i$. It follows from Lemma \ref{lem:compshift} that \begin{equation*} h_{top}(\Psi_0)\geqslant \limsup_{M\to \infty} \dfrac{nNM(h_\mu(\sigma)-\frac{\eta}{2})}{NM(n+1+L)}= \dfrac{n(h_\mu(\sigma)-\frac{\eta}{2})}{(n+1+L)} > h_\mu(\sigma)-\eta. \end{equation*}
Finally, let $\mu_e$ be an ergodic measure supported in $\Psi_0$ with entropy at least $h_\mu(\sigma)-\eta$ (which exists by the standard variational principle in the compact setting), since we already proved that $\mu_e\in V(f_1,\ldots,f_l,\mu,\epsilon)$ this finishes the proof. \end{proof}
\section{Main entropy inequality}\label{mainine} This section is devoted to the proof of the main entropy inequality. This is stated in Theorem \ref{pre} and relates the entropy of a sequence of ergodic measures with the amount of mass lost and the topological entropy at infinity.
Recall that, as explained in \eqref{def:kai}, to every sequence of natural numbers $(a_i)_i$ we assign a compact set $K=K((a_i)_i) \subset \Sigma$. The definition of $K$ implies that if $x\in K^c$, then $x_i>a_i$, for some $i\in {\mathbb N}_0$. For $x\in K^c$ we define $i: K^c \to {\mathbb N}_{0}$ by \begin{equation} \label{def:i} i(x):= \min \left\{ n \in {\mathbb N}_{0} : x_n > a_n \right\}. \end{equation} For $n\in {\mathbb N}$ we define \begin{equation} \label{def:T} T_n(K):=K_{a_0}\cap \sigma^{-1}K^c\cap\cdots\cap\sigma^{-n}K^c\cap \sigma^{-(n+1)}K_{a_0}, \end{equation} where $K_{a_0}=\bigcup_{i=1}^{a_0}[i]$ (as defined in \eqref{def:km}). Let \begin{equation} \label{def:That} \widehat{T}_n(K):=\left\{x\in T_n(K): i(\sigma^k(x))\leqslant n-k,\text{ for every } k\in \{1,\ldots,n\} \right\}. \end{equation} Let $\hat z_n(K)$ be the minimal number of cylinders of length $(n+2)$ needed to cover $\widehat{T}_n(K)$ and define \begin{equation} \label{def:di} \hat\delta_\infty(K):=\limsup_{n\to\infty}\frac{1}{n}\log \hat z_n(K). \end{equation} The reason why we define $\hat\delta_\infty(K)$ covering the sets $\widehat{T}_n(K)$, and not $T_n(K)$, is to ensure Lemma \ref{lem:Kineq2}. This allows us to relate $\hat\delta_\infty(K)$ with the topological entropy at infinity of $(\Sigma,\sigma)$.
Our next result is fundamental in this paper.
\begin{theorem} \label{pre} Let $(\Sigma,\sigma)$ be a finite entropy CMS. Let $(\mu_n)_n$ be a sequence of ergodic probability measures converging on cylinders to an invariant measure $\mu$. Let $(a_i)_i$ be an increasing sequence of natural numbers such that the corresponding compact set $K=K((a_i)_i)$ satisfies that $\mu_n(K)>0$, for all $n\in {\mathbb N}$. Then \begin{equation*}
\limsup_{n\to \infty} h_{\mu_n}(\sigma)\leqslant |\mu|h_{\mu/|\mu|}(\sigma)+(1-\mu(Y))\hat\delta_\infty(K), \end{equation*} where $Y=\bigcup_{s=0}^\infty \sigma^{-s}K$. \end{theorem}
The proof of this theorem requires some propositions and lemmas, which we will prove first before completing the proof of the theorem at the end of this section.
The fact that $K \subset \sigma (K)$, which follows since $(a_i)_i$ is an increasing sequence, will be used several times here. Let $A_k:=\left\{ x \in K : R_K(x)=k \right\}$, where $R_K(x)$ is the first return time function to the set $K$ (see Definition \ref{def:ret}). For $x \in Y$ we define the following: \begin{align*} n_1(x)&:= \min \left\{ n \in{\mathbb N}_0 : \text{ there exists } y \in K \text{ such that } \sigma^{n}(y)=x \right\},\\ n_2(x)&:= \min \left\{ n \in{\mathbb N}_0 : \sigma^{n}x\in K \right\}. \end{align*} We emphasise that the function $n_1(x)$ is well defined. Indeed, observe that if $x\in Y$ then $\sigma^{n_2(x)}x\in K$. Let $r \in {\mathbb N}$ be such that $r> n_2(x)-1$. Since the sequence $(a_i)_i$ is increasing we have that $a_r \geq \max\left\{ a_i : i\in\{0,\ldots,n_2(x)-1 \} \right\}$. Since $x\in \sigma^r(K)=\prod^\infty_{k=r}\{1,\ldots,a_k\}\cap \Sigma$ we have that $n_1(x)$ is finite for every $x\in Y$. Let \begin{eqnarray*} n(x):= \begin{cases} n_1(x)+n_2(x) & \text{ if } x \in Y, \\ \infty & \text{ if } x\in \Sigma\setminus Y. \end{cases} \end{eqnarray*} For $n\in{\mathbb N}_0 \cup \{\infty\}$ define \begin{equation*} \mathcal{C}_n:=\{x\in\Sigma: n(x)=n\}. \end{equation*} Note that $\mathcal{C}_0=K$ and $\mathcal{C}_1=\emptyset$. For $n \geq 2$ observe that $x\in \mathcal{C}_n$ if it belongs to the orbit of a point in $A_n$. More precisely, for every $n\geqslant 2$ we have that $\mathcal{C}_n\subset \bigcup_{k=1}^{n-1}\sigma^k(A_n)$. We define the following sets,
\begin{equation*} \alpha_{\leqslant N}:=\left(\bigcup_{n=2}^{N} \mathcal{C}_n\right), \alpha_{N,M}:=\left(\bigcup^M_{n> N} \mathcal{C}_n\right) \text{ and } \alpha_{>M}:=\left(\bigcup_{n> M} \mathcal{C}_n\right)\cup \mathcal{C}_\infty. \end{equation*}
\begin{remark} The set $\alpha_{\leqslant M}$ can be covered with finitely many cylinders of length $L$. Indeed, observe that for every $n\geqslant 2$ we have \begin{equation*} \mathcal{C}_n \subset \bigcup_{s=1}^{n-1}\sigma^s(A_n) \subset \bigcup_{s=1}^{n-1}\sigma^s(K) \subset \sigma^{n-1}(K). \end{equation*} Therefore, \begin{equation*} \alpha_{\leqslant M}=\bigcup_{n=2}^{M}\mathcal{C}_n \subset \sigma^{M-1}(K)=\prod^\infty_{s=M-1}\{1,\ldots,a_s\}\cap \Sigma. \end{equation*} Since the set $\prod^\infty_{s=M-1}\{1,\ldots,a_s\}\cap \Sigma$ can be covered with at most $\prod_{s=M-1}^{M-2+L}a_s$ cylinders of length $L$, the same holds for $\alpha_{\leqslant M}$. \end{remark}
Observe that it follows directly from the definition of $\hat\delta_\infty(K)$ (see \eqref{def:di}) that for every $\epsilon>0$, there exists $N_0=N_0(\epsilon) \in {\mathbb N}$ such that for every $n\geqslant N_0$ we have \begin{equation*} \hat z_n(K)\leqslant e^{n(\hat\delta_\infty(K)+\epsilon)}. \end{equation*}
At this point we fix $\epsilon >0$ and $k , N \in {\mathbb N}$ large enough so that $kN\geqslant N_0(\epsilon)$: these will appear explicitly in the proof of Theorem~\ref{pre}.
Given $A\subset \Sigma$ and $t\in {\mathbb N}$ we define \begin{equation*} U_t(A):=\left\{x\in \Sigma: d(x,A)\leqslant 2^{-t} \right\}. \end{equation*} Now let \begin{eqnarray*} K(k,N):=U_{kN+2}(K) , &\\ \gamma_{\leqslant N}:=U_{(k+1)N+2}(\alpha_{\leqslant N})\setminus K(k,N) , &\\ G_{k,N}:=K(k,N)\cup \gamma_{\leqslant N}, \end{eqnarray*} and \begin{eqnarray*} \gamma_{N,kN}:=U_{2(k+1)N+2}(\alpha_{N,kN})\setminus G_{k,N}, &\\ \gamma_{>kN}:=\Sigma\setminus (G_{k,N}\cup \gamma_{N,kN}), &\\ B_{k,N}:=\gamma_{N,kN}\cup\gamma_{>kN}. \end{eqnarray*} Denote by ${\mathcal Q}_1(k,N)$ the minimal cover of $K(k,N)$ with cylinders of length $kN+2$. Similarly, denote by ${\mathcal Q}_2'(k,N)$ the minimal cover of $\alpha_{\leqslant N}$ with cylinders of length $(k+1)N+2$. Observe that every element in ${\mathcal Q}_2'(k,N)$ is disjoint or contained in an element of ${\mathcal Q}_1(k,N)$. In particular $\gamma_{\leqslant N}$ is a finite union of cylinders of length $(k+1)N+2$; this collection of cylinders is denoted by ${\mathcal Q}_2(k,N)$. Define \begin{equation} \label{def:beta'} \beta_{k,N}':={\mathcal Q}_1(k,N)\cup {\mathcal Q}_2(k,N) \end{equation} and observe that $\beta_{k,N}'$ is a partition of the set $G_{k,N}$. Define the following partition of $\Sigma$, \begin{equation} \label{def:beta} \beta_{k,N}:=\{\gamma_{>kN},\gamma_{N,kN}\}\cup \beta_{k,N}'. \end{equation} Recall that the refinement $ \beta_{k,N}^n$ follows as in Section \ref{sec:em}. \\
\emph{Notation:} We use the following notation for an interval of integers $[a,b):= \{ n \in {\mathbb N} : a \leq n < b\}$ and $|[a, b)|=b-a$.
\begin{definition} Let $Q\in \beta_{k,N}^n$ be such that $(Q\cup \sigma^{n-1}{Q})\subset G_{k,N}$. An interval $[r,s)\subset [0,n)$ is called an \emph{excursion} of $Q$ into $\gamma_{>kN}$ (resp. $B_{k,N}$) if $\sigma^t Q\subset \gamma_{>kN}$ (resp. $\sigma^t Q \subset B_{k,N}$) for every $t\in [r,s)$ and $(\sigma^{r-1} Q\cup \sigma^{s}Q)\subset G_{k,N}$.
An excursion $[r,s)$ of $Q$ into $B_{k,N}$ is said to \emph{enter} $\gamma_{>kN}$ if there exists $i \in [r,s)$ such that $\sigma^i Q \subset \gamma_{>kN}$.
\end{definition}
The next three lemmas are preparation for the proof of Proposition \ref{prop:goodcover}. These give us control on the return times to $K(k,N)$ and the length of excursions into $B_{k,N}$
\begin{lemma}\label{lem:A}
If $[r,r+s)$ is an excursion of $Q$ into $B_{k,N}$ that does not enter $\gamma_{>kN}$ then $s<kN$. \end{lemma}
\begin{proof} Since the excursion does not enter $\gamma_{>kN}$ we have that $\sigma^rQ\subset \gamma_{N,kN}$. Fix $x\in \sigma^{r}Q$. By the definition of $\gamma_{N,kN}$ there exists $x_0\in \alpha_{N,kN}$ such that $d(x,x_0)\leqslant 2^{-(2(k+1)N+2)}$. Since $x_0 \in \alpha_{N,kN}$ we have that $n(x_0)\leqslant kN$ and therefore $n_2(x_0)< kN$. In particular $\sigma^t (x_0)\in \alpha_{\leqslant N}$, for some $t\in [0,kN)$. Observe that $$d(\sigma^t (x),\sigma^t (x_0))\leqslant 2^{-(2(k+1)N+2)+t}\leqslant 2^{-((k+1)N+2)}.$$ This readily implies that $\sigma^t(x)\in U_{(k+1)N+2}(\alpha_{\leqslant N})\subset G_{k,N}$. We conclude that $\sigma^{r+t} Q\subset G_{k,N}$, and therefore $s<kN$. \end{proof}
\begin{lemma}\label{lem:B}
If $Q\subset G_{k,N}$ then there exists $t \in [0, N)$ such that $\sigma^tQ\subset K(k,N)$. \end{lemma}
\begin{proof} If $Q\subset K(k,N)$ there is nothing to prove. Assume that $Q\subset \gamma_{\leqslant N}$. Let $x\in Q$ and $y \in \alpha_{\leqslant N}$ such that $d(x,y)\leqslant 2^{-((k+1)N+2)}$. Since $y \in \alpha_{\leqslant N}$ we have that $\sigma^t(y)\in K$, for some $t< N$. Observe that \begin{equation*} d(\sigma^t (x), \sigma^t (y))\leqslant 2^{-((k+1)N+2)}2^{t}<2^{-(kN+2)}. \end{equation*} We conclude that there exists $t \in [0, N)$ such that $\sigma^t (x)\in U_{kN+2}(K)=K(k,N)$. This implies that for some $t<N$ we have $\sigma^tQ\subset K(k,N)$. \end{proof}
\begin{lemma}\label{lem:C} If $[r,r+s)$ is an excursion of $Q$ into $\gamma_{>kN}$ such that $s\geqslant N$ then $\sigma^{r-1}Q\subset K(k,N)$. \end{lemma}
\begin{proof} From the definition of an excursion, the set $Q_0:=\sigma^{r-1}Q$ must lie in $G_{k, N}$, so to derive a contradiction we will assume that $Q_0\subset \gamma_{\leqslant N}$. Let $x\in Q_0$. By the construction of $\gamma_{\leqslant N}$ there exists $y \in \alpha_{\leqslant N}$ such that $d(x,y)\leqslant 2^{-((k+1)N+2)}$. Since $y\in \alpha_{\leqslant N}$ there exists $t\leqslant N$ such that $\sigma^t(y)\in K$. Therefore \begin{equation*} d(\sigma^t(x),\sigma^t (y))\leqslant 2^{-((k+1)N+2)+t}\leqslant 2^{-(kN+2)}. \end{equation*} We conclude that $\sigma^t(x)\in U_{kN+2}(K)=K(k,N)$. This contradicts the fact the length of the excursion is larger than $N$. \end{proof}
\begin{definition} Denote by $m_{n,k,N}(Q)$ the number of excursions of length greater or equal to $kN$ into $B_{k,N}$ that enter $\gamma_{>kN}$ and let \begin{equation*} E_{n,k,N} :=\# \left\{i \in[0,n): \sigma^i Q\subset B_{k,N} \right\}. \end{equation*} \end{definition}
The following result shows that an atom $Q\in \beta_{k,N}^n$ such that $Q\subset K(k,N)\cap \sigma^{-(n-1)}K(k,N)$ can be covered by cylinders of length $n$ in a controlled way. This is an estimate closely related to \cite[Lemma 7.4]{ekp} (see also \cite[Proposition 4.5]{ve}). The constant $\hat\delta_\infty(K)$ naturally appears when we try to control the time spent in the `bad' part $B_{k,N}$.
\begin{proposition} \label{prop:goodcover} Let $\beta_{k,N}$ be the partition defined in \eqref{def:beta}. Then an atom $Q\in \beta_{k,N}^n$ such that $Q\subset K(k,N)\cap \sigma^{-(n-1)}K(k,N)$, can be covered by at most \begin{equation*} e^{E_{n,k,N}(Q)(\hat\delta_\infty(K)+\epsilon)}e^{m_{n,k,N}(Q)N(\hat\delta_\infty(K)+\epsilon)} \end{equation*} cylinders of length $n$. \end{proposition}
\begin{proof} To simplify notation we drop the sub-indices $N$ and $k$. The proof of Proposition \ref{prop:goodcover} is by induction on $n$. First decompose $[0,n-1]$ into \begin{equation*}
[0,n-1]=W_1\cup V_1\cup W_2\cup\cdots \cup V_s\cup W_{s+1},
\end{equation*}
according to the excursions into $B_{k,N}$ that contain at least one excursion into $\gamma_{>kN}$. More precisely, let $V_i=[m_i,m_i+h_i)$ and $W_i=[l_i,l_i+L_i)$ with $l_i+L_i=m_i$ and $m_i+h_i=l_{i+1}$. The segment $V_i$ denotes an excursion into $B_{k,N}$ that contains an excursion into $\gamma_{>kN}$. Given $i\in {\mathbb N}$ define $J_i:= \sum_{j=1}^i |V_j|1_{[kN,\infty)}(|V_j|),$
where $1_{[kN,\infty)}$ is the characteristic function of the interval $[kN,\infty)$. Similarly define $H_i:= \sum_{j=1}^i 1_{[kN,\infty)}(|V_j|).$ Observe that $Q\subset K(k,N)$ implies that $Q$ is already contained in a cylinder of length $kN+2$. \\
\emph{Step 1:} Assume that $Q$ has been covered with $c_i$ cylinders of length $l_i$, where \begin{equation*} c_i \leqslant e^{J_i \left(\hat\delta_\infty(K)+\epsilon \right)}e^{NH_i \left(\hat\delta_\infty(K)+\epsilon \right)}. \end{equation*}
(As mentioned above, the set $Q$ is covered by one cylinder of length $1$, therefore take $c_1=1$.) We claim that the same number of cylinders of length $(l_i+L_i)$ cover $Q$. Observe that by hypothesis $\sigma^{l_i}Q$ is contained in an element of $\beta'$, therefore $\mbox{\rm diam} (\sigma^{l_i}Q)\leqslant 2^{-(kN+2)}$. Since the elements of $\beta'$ all have diameter smaller than $2^{-(kN+2)}$, the same holds if $Q$ spends some extra time in $\beta'$. By Lemma \ref{lem:A}, if $Q$ has an excursion into $B_{k,N}$ that does not enter $\gamma_{>kN}$, then it must come back to $\beta'$ before $kN$ iterates. In particular if the excursion into $B_{k,N}$ is $[p_i,p_i+q_i)$, then $q_i< kN$. Observe that $\mbox{\rm diam} (\sigma^{p_i-1}Q)\leqslant 2^{-(kN+2)},$ implies that $\mbox{\rm diam} (\sigma^{p_i+t}Q)\leqslant 2^{-2},$ for every $t\in [0,kN)$. In particular the same holds for $t\in [0,q_i]$. Repeating this process we conclude that $\mbox{\rm diam} (\sigma^{t}Q)\leqslant 2^{-2}$, for every $t\in [l_i,l_i+L_i)$. This immediately implies that $\sigma^{l_i}Q$ is contained in a cylinder of length $L_i$, which implies our claim. We go next to Step 2. \\
\emph{Step 2:} Assume we have covered $Q$ with $c_i$ cylinders of length $m_i$, where \begin{equation*} c_i\leqslant e^{J_i(\hat\delta_\infty(K)+\epsilon)}e^{NH_i(\hat\delta_\infty(K)+\epsilon)}. \end{equation*} We want to estimate the number of cylinders of length $(m_i+h_i)$ needed to cover $Q$. Define $Q_i:=\sigma^{m_i-1}Q$. If we are able to cover $Q_i$ with $R$ cylinders of length $(h_i+1)$, then we will be able to cover $Q$ with $Rc_i$ cylinders of length $(m_i+h_i)$. We will separate into two cases:\\ \emph{Case 1}: $h_i< kN$.\\ Observe that $Q_i\subset G_{k,N}$ and is therefore contained in an element of $\beta'$, which implies $\mbox{\rm diam} (Q_i)\leqslant 2^{-(kN+2)}$. This implies that $Q_i$ is contained in a cylinder of length $(kN+2)$. Since $h_i<kN$, this implies that $Q_i$ can be covered with one cylinder of length $(h_i+1)$. We conclude that \begin{equation*} c_{i+1}= c_i \leqslant e^{J_i(\hat\delta_\infty(K)+\epsilon)}e^{NH_i(\hat\delta_\infty(K)+\epsilon)}= e^{J_{i+1}(\hat\delta_\infty(K)+\epsilon)}e^{NH_{i+1}(\hat\delta_\infty(K)+\epsilon)}. \end{equation*} \emph{Case 2}: $h_i\geqslant kN$.\\ By Lemma \ref{lem:C}, $Q_i=\sigma^{m_i-1}Q\subset K(k,N)$. Observe that by assumption $\sigma^{h_i+1}Q_i\subset \gamma_{\leqslant N}$. By Lemma \ref{lem:B} there exists $0\leqslant t_i<N,$ such that $\sigma^{h_i+1+t_i}Q_i\subset K(k,N)$ (we assume $t_i$ is the smallest such number). We conclude that every $x\in Q_i$ satisfies $x\in K_{a_0}$, $\sigma^{h_i+1+t_i}(x)\in K_{a_0}$, and $\sigma^s x\in K^c$, for every $s\in \{1,\ldots, h_i+t_i\}$. In other words $Q_i\subset T_{h_i+t_i}(K)$. We now claim that $Q_i\subset \widehat{T}_{h_i+t_i}(K)$. Observe that if $x\in Q_i$, then $\sigma^{h_i+t_i+1}(x)\in K(k,N),$ and $\sigma^k(x)\in K(k,N)^c$, for every $k\in \{1,\ldots, h_i+t_i\}$. We argue by contradiction and suppose that $i(\sigma^k(x))>(h_i+t_i)-k$ for some $k\in \{1,\ldots, h_i+t_i\}$. This implies that $(\sigma^k(x))_j\leqslant a_j$, for $j\in \{0,\ldots, h_i+t_i-k\}$. Observe that $(\sigma^k(x))_{h_i+t_i-k+j+1}=(\sigma^{h_i+t_i+1}(x))_{j}$, and for $j\in\{0,\ldots,kN+1\}$ we have $(\sigma^{h_i+t_i+1}(x))_{j}\leqslant a_j$. We conclude that $(\sigma^k(x))_{h_i+t_i-k+j+1}\leqslant a_j$, for $j\in \{0,\ldots,kN\}$. In particular we have that $(\sigma^k(x))_j\leqslant a_j$, for every $j\in \{0,\ldots,kN+1\}$, which contradicts that $\sigma^k(x)\in K(k,N)^c$, completing the proof of our claim. This implies, from the definition of $\hat\delta_\infty(K)$, that $Q_i$ can be covered by at most $e^{(h_i+t_i)(\hat\delta_\infty(K)+\epsilon)}$ cylinders of length $(h_i+1+t_i)$; and by at most $e^{(h_i+N)(\hat\delta_\infty(K)+\epsilon)}$ cylinders of length $(h_i+1)$. We conclude that $Q$ can be covered by at most $c_{i+1}$ cylinders of length $(n_i+h_i)$, where \begin{align*}c_{i+1}\leqslant & e^{(h_i+N)(\hat\delta_\infty(K)+\epsilon)}(e^{J_i(\hat\delta_\infty(K)+\epsilon)}e^{NH_i(\hat\delta_\infty(K)+\epsilon)})\\ &= e^{J_{i+1}(\hat\delta_\infty(K)+\epsilon)}e^{NH_{i+1}(\hat\delta_\infty(K)+\epsilon)}. \end{align*}
Adding these steps together and noting that $J_s= E_{n,k,N}(Q)$ and $H_s= m_{n,k,N}(Q)$ completes the proof of the proposition. \end{proof}
The idea now is to use Proposition \ref{prop:goodcover} to compare the entropy of a measure with the corresponding entropy of our partition $\beta_{k,N}$. This is a natural idea: the map $\mu\mapsto h_\mu(\beta_{k,N})$ is typically better behaved under sequences of measures; at this point we crucially use that the partition $\beta_{k,N}$ is finite.
\begin{proposition}\label{prop:ineq} Let $\beta_{k,N}$ be the partition defined in \eqref{def:beta} and $\mu$ an ergodic $\sigma$-invariant probability measure satisfying $\mu(K(k,N))>0$. Then \begin{equation*} h_{\mu}(\sigma)\leqslant h_{\mu}(\beta_{k,N})+\left( \mu(B_{k,N}) + \frac{1}{k} \right) (\hat\delta_\infty(K)+\epsilon). \end{equation*} \end{proposition}
\begin{proof}
To simplify notation we denote the partition $\beta_{k,N}$ by $\beta$. We will apply Theorem \ref{katformula}, so the main task is to estimate $N_\mu(n,1,\delta)$ for some $\delta\in (0, 1)$. Since $\mu$ is an ergodic measure such that $\mu(K(k,N))>0$ there exists $\delta_1>0$ and an increasing sequence $(n_i)_{i}$ satisfying \begin{equation*} \mu(K(k,N)\cap \sigma^{-n_i}K(k,N)) > \delta_1, \end{equation*} for every $i\in {\mathbb N}$. Given $\epsilon_1>0$, by the Shannon-McMillan-Breiman theorem the set \begin{equation*} {\mathcal D}_{\epsilon_1,N}= \left\{x\in X : \forall n\geq N, \mu(\beta^n(x))\geq \exp(-n(h_\mu(\beta)+\epsilon_1)) \right\}, \end{equation*} satisfies \begin{equation*} \lim_{N \to \infty} \mu \left( {\mathcal D}_{\epsilon_1,N}\right) =1.
\end{equation*} By Birkhoff's Ergodic Theorem there exists a set $W_{\epsilon_1} \subset \Sigma$ satisfying $\mu(W_{\epsilon_1})>1-\frac{\delta_1}{4}$ and $n(\epsilon_1) \in {\mathbb N}$ such that for every $x\in W_{\epsilon_1}$ and $n\geqslant n(\epsilon_1)$, \begin{equation*} \frac{1}{n}\sum_{i=0}^{n-1} 1_{B_{k,N}}(\sigma^n x) < \mu(B_{k,N})+\epsilon_1. \end{equation*} Define \begin{equation*} X_i:= W_{\epsilon_1}\cap {\mathcal D}_{\epsilon_1,n_i}\cap K(k,N)\cap \sigma^{-n_i}K(k,N). \end{equation*} So for sufficiently large values of $i \in {\mathbb N}$, by construction we have that $\mu(X_i)>\frac{\delta_1}{2}$. In what follows we will assume that $i \in {\mathbb N}$ is large enough that it satisfies this condition.
By definition of ${\mathcal D}_{\epsilon_1,n_i}$ the set $X_i$ can be covered by $\exp(n_i(h_\mu(\beta)+\epsilon_1))$ many elements of $\beta^{n_i}$. We will make use of Proposition \ref{prop:goodcover} to efficiently cover each of those atoms by cylinders. Let $Q\in \beta^{n_i}$ be an atom intersecting $X_i$. In particular $Q\in K\cap\sigma^{-(n-1)}K$. It follows from the definition of $W_{\epsilon_1}$ that \begin{equation*} E_{n_i,k,N}(Q) <\left(\mu(B_{k,N})+\epsilon_1 \right)n_i. \end{equation*} Moreover, \begin{equation*} m_{n_i,k,N}(Q)\leqslant \frac{1}{kN}n_i. \end{equation*} Indeed, each of the excursions counted in $m_{n_i,k,N}$ has length at least $kN$, which implies that the number of excursions can not be larger than $\frac{1}{kN}n_i$. Therefore Proposition \ref{prop:goodcover} implies that \begin{equation*} N_\mu\left(n_i, 1,1-\frac{\delta_1}{2} \right) \leq e^{n_i(h_\mu(\beta)+\epsilon_1)}e^{n_i(\hat\delta_\infty(K)+\epsilon)(\mu(B_{k,N})+\epsilon_1)}e^{\frac{1}{kN}n_iN(\hat\delta_\infty(K)+\epsilon)}. \end{equation*} It now follows from Katok's entropy formula (see Theorem \ref{katformula}) that \begin{equation*} h_\mu(\sigma) \leqslant h_\mu(\beta_{k,N})+\epsilon_1+(\hat\delta_\infty(K)+\epsilon)(\mu(B_{k,N})+\epsilon_1)+\frac{1}{k}(\hat\delta_\infty(K)+\epsilon). \end{equation*} Since $\epsilon_1>0$ was arbitrary the proof is complete. \end{proof}
As in Proposition \ref{prop:ineq} we denote the partition $\beta_{k,N}$ by $\beta$. We may assume, possibly after refining the partition, that $$\beta=\{C^1,\ldots,C^q,R\},$$ where each $C^i$ is a cylinder for the original partition and $R=\gamma_{>kN}$ is the complement of a finite collection of cylinders. For simplicity we still denote this partition by $\beta$. We emphasise that Proposition \ref{prop:ineq} still holds for this new partition.
Define, for large $m$, $F_m:=\bigcap_{i=0}^{m-1}\sigma^{-i} R$. We will require the following continuity result.
\begin{proposition}\label{prop:atom} Suppose that $(\mu_n)_n$ is a sequence of ergodic probability measures converging on cylinders to an invariant measure $\mu$, where $\mu(\Sigma)>0$. For every $P\in \beta^m\setminus \{F_m\},$ we have \begin{equation*} \lim_{n\to\infty}\mu_n(P)=\mu(P). \end{equation*} \end{proposition}
\begin{proof} In order to prove the proposition we will need the following fact.
\begin{claim}\label{claim:preatom} Let $(H_i)_i$ be a collection of cylinders and $(p_i)_i$ a sequence of natural numbers. Then $H_0\cap \sigma^{-p_1}H_1\cap \cdots\cap \sigma^{-p_k}H_k$, is either a finite collection of cylinders, or the empty set.
\end{claim}
\begin{proof} We begin with the case $k=2$, in other words, we will prove that if $C$ and $D$ are cylinders, then for every $p\in {\mathbb N}$ the set $C\cap \sigma^{-p}D$ is a finite collection of cylinders or the empty set. If the length of $C$ is larger than or equal to $p$ then $C\cap \sigma^{-p}D$ is empty or a cylinder. If $p$ is larger than the length of $C$, then we use that there are only finitely many admissible words of given length connecting two fixed symbols. More precisely, if $C=[x_0,\ldots,x_{h-1}]$ and $D=[y_0,\ldots,y_{t-1}]$, then there are finitely many admissible words of length $p-h+2$ connecting $x_{h-1}$ and $y_0$. We conclude that $C\cap \sigma^{-p}D$ is a finite collection of cylinder or the empty set. The same argument gives us the proof of the claim for arbitrary $k$. \end{proof}
Let $P=S_0\cap \sigma^{-1}S_1\cap\cdots\cap\sigma^{-(m-1)}S_{m-1}$, where $S_i\in \beta$ and $P_k:=\bigcap_{i=k}^{m-1}\sigma^{-(i-k)}S_i$. Define $B=B(P):=\{i\in\{0,\ldots,m-1\}: S_i=R\}$, $G=G(P):=\{0,\ldots,m-1\}\setminus B$, and $k=k(P):=(\min G)-1$. By assumption we know that $G\ne \emptyset$. Let $Q_0=Q_0(P):=\bigcup_{i=0}^k \sigma^{-i}R$, $Q_1=Q_1(P):=\bigcap_{i\in G} \sigma^{-i}S_i$, and $Q_2=Q_2(P):=\bigcap_{i\in B\cap (k,\infty)}\sigma^{-i}S_i$. We will first consider the case $k=-1$, where $Q_0=\emptyset$.
\begin{claim}\label{claim:-1} Let $P=\bigcap_{i=0}^{m-1}\sigma^{-i}S_i$, where $S_0\in \{C^1,\ldots,C^q\}$. Then $$\lim_{n\to\infty} \mu_n(P)=\mu(P).$$\end{claim}
\begin{proof} Since $Q_1$ is the disjoint union of $P=(Q_1\cap Q_2)$ and $(Q_1\cap Q_2^c)$, for every $n\in {\mathbb N}$ we obtain that \begin{equation*} \mu_n(P)=\mu_n(Q_1)-\mu_n(Q_1\cap Q_2^c). \end{equation*} Observe that \begin{align*} Q_1\cap Q_2^c&= \left(\bigcap_{j\in G}\sigma^{-j}S_j \right) \cap \left(\bigcup_{i\in B} \sigma^{-i}R^c \right)=\bigcup_{i\in B} \left(\sigma^{-i}R^c\cap \bigcap_{j\in G}\sigma^{-j}S_j \right). \end{align*} From Claim \ref{claim:preatom} we conclude that for every $i\in B$ the sets $Q_1$ and $(\sigma^{-i}R^c\cap \bigcap_{j\in G}\sigma^{-j}S_j)$ are a finite union of cylinders or the empty set. Therefore, $Q_1$ and $Q_1\cap Q_2^c$ are a finite union of cylinders, or the empty set. From this we immediately obtain that \begin{align*}\lim_{n\to\infty}\mu_n(P)&=\lim_{n\to\infty}\mu_n(Q_1)-\lim_{n\to\infty}\mu_n(Q_1\cap Q_2^c)=\mu(Q_1)-\mu(Q_1\cap Q_2^c)=\mu(P),\end{align*} which proves the claim. \end{proof}
We now explain how to reduce the case $k\geqslant 0$ to Claim \ref{claim:-1}. Observe that $P=R\cap \sigma^{-1}P_1$, therefore $\sigma^{-1}P_1$ is the disjoint union between $P$ and $S_1:=(R^c\cap\sigma^{-1}P_1)=\bigcup_{i=1}^q(C^i\cap \sigma^{-1}P_1)$. Thus, \begin{align*} \mu_n(P)&=\mu_n(\sigma^{-1}P_1)- \mu_n(R^c\cap\sigma^{-1}P_1)=\mu_n(P_1)-\sum_{i=1}^q \mu_n(C^i\cap\sigma^{-1}P_1). \end{align*} By Claim \ref{claim:-1} we know that $\lim_{n\to\infty}\mu_n(C^i\cap\sigma^{-1}P_1)=\mu(C^i\cap\sigma^{-1}P_1)$. Therefore it suffices to prove that $\lim_{n\to\infty}\mu_n(P_1)=\mu(P_1)$. Applying the above argument $k$ times we obtain that the original problem is reduced to $\lim_{n\to\infty}\mu_n(P_{k+1})=\mu(P_{k+1})$ Since $P_{k+1}=S_{k+1}\cap \sigma^{-1}P_{k+2}$, where $S_{k+1}\in \{C^1,\ldots,C^q\}$, we conclude the proof of the proposition by applying Claim \ref{claim:-1}. \end{proof}
\begin{proof}[Proof of Theorem~\ref{pre}] We first consider the case in which not all the mass escapes, that is, we assume that $\mu(\Sigma)>0$. Let $\varepsilon_0>0$. Choose $m \in {\mathbb N}$ sufficiently large such that \begin{equation*}
h_{\frac{\mu}{|\mu|}}(\sigma)+\varepsilon_0>\frac{1}{m}H_{\frac{\mu}{|\mu|}}(\beta^{m})\quad , \quad 2\frac{e^{-1}}{m}<\frac{\varepsilon_0}{2} \quad and \quad -\left(\frac1m\right)\log |\mu|<\varepsilon_0. \end{equation*} Then \begin{equation*}
h_{\frac{\mu}{|\mu|}}(\sigma)+\varepsilon_0>
\frac{1}{|\mu|}\frac1m \left(\log|\mu|- \sum_{P\in \beta^m} \mu(P)\log\mu(P)\right) \end{equation*} and hence \begin{equation*}
|\mu| h_{\frac{\mu}{|\mu|}}(\sigma)+2\varepsilon_0>-\frac{1}{m}\sum_{P\in \beta^m} \mu(P)\log\mu(P). \end{equation*} It follows from Proposition \ref{prop:atom} that \begin{equation*} \lim_{n\to\infty} \sum_{Q\in \beta^{m}\setminus\{F_m\}}\mu_n(Q)\log\mu_n(Q) = \sum_{Q\in \beta^{m}\setminus\{F_m\}}\mu(Q)\log\mu(Q). \end{equation*} For sufficiently large $n \in {\mathbb N}$ we have the inequality \begin{equation*}
|\mu|h_{\frac{\mu}{|\mu|}}(\sigma)+3\varepsilon_0\geqslant \frac{1}{m}H_{\mu_n}(\beta^m). \end{equation*} By Proposition \ref{prop:ineq}, we have that \begin{align*}
|\mu| h_{\frac{\mu}{|\mu|}}(\sigma)+3\varepsilon_0 \geq & \frac{1}{m}H_{\mu_n}(\beta^{m})\geq h_{\mu_n}(\sigma,\beta)\\ \geqslant & h_{\mu_n}(\sigma)-(\hat\delta_\infty(K)+\epsilon)\mu_n(B_{k,N})-\frac{1}{k}(\hat\delta_\infty(K)+\epsilon). \end{align*} Since $\varepsilon_0>0$ is arbitrary we get
\begin{align}\label{forA} \limsup_{n\to\infty} h_{\mu_n}(\sigma)\leq |\mu| h_{\frac{\mu}{|\mu|}}(\sigma)+(\hat\delta_\infty(K)+\epsilon)(1-\mu(G_{k,N})) +\frac{1}{k}(\hat\delta_\infty(K)+\epsilon). \end{align} We stress that Proposition \ref{prop:ineq} can be applied for arbitrary $k, N \in {\mathbb N}$ since $K\subset \mbox{\rm supp}(\mu_n)$ and therefore $\mu_n(K(k,N))>0$. Finally, letting $k\to \infty$ and $\epsilon\to 0$ we obtain the inequality \begin{equation*}
\limsup_{n\to \infty} h_{\mu_n}(\sigma)\leqslant |\mu|h_{\mu/|\mu|}(\sigma)+\left(1-\sup_{k,N}\mu(G_{k,N}) \right)\hat\delta_\infty(K). \end{equation*} Observe that $Y\subset \bigcup_{k,N} G_{k,N}$, therefore $\mu(Y)\leqslant \mu \left(\bigcup_{k,N} G_{k,N} \right)=\sup_{k,N}\mu(G_{k,N})$. We conclude that \begin{equation*}
\limsup_{n\to \infty} h_{\mu_n}(\sigma)\leqslant |\mu|h_{\mu/|\mu|}(\sigma)+(1-\mu(Y))\hat\delta_\infty(K). \end{equation*} The case $\mu(\Sigma)=0$ follows directly from Proposition \ref{prop:ineq} since $h_{\mu_n}(\sigma,\beta)\to 0$ and $\mu_n(B_{k,N})\to 1$ as $n\to \infty$. \end{proof}
\section{Proof of Theorem \ref{thm:main}}\label{finalproof} In this section we prove our main result. We start with a simple result we will need later.
\begin{lemma}\label{lem:Kineq} Let $(a_j)_j$ and $(b_j)_j$ be sequences of natural numbers such that for every $i\in{\mathbb N}_0$ we have $a_0=b_0$ and $a_j\leqslant b_j$. Then $\hat\delta_\infty(K((b_j)_j))\leqslant \hat\delta_\infty(K((a_j)_j)).$ \end{lemma}
\begin{proof} Denote by $K_1:=K((a_j)_j)$ and $K_2:=K((b_j)_j)$. Recall that associated to each compact set defined in this way there is a function $i$ (see \eqref{def:i} for the definition). Denote the function $i$ associated to $K_1$ (resp. $K_2$) by $i_1$ (resp. $i_2$). It follows from the hypothesis that $K_1\subset K_2$. In particular we have that $K_2^c\subset K_1^c$ and therefore $T_n(K_2)\subset T_n(K_1)$ (see \eqref{def:T} for the definition of $T$). Moreover, we have that \begin{equation*} \widehat{T}_n(K_2)\subset \widehat{T}_n(K_1), \end{equation*} (see \eqref{def:That} for the definition of $\widehat{T}$). Indeed, let $x\in \widehat{T}_n(K_2)$, we have that $i_2(\sigma^k(x))\leqslant n-k$. In particular \begin{equation*} (\sigma^k(x))_{i_2(\sigma^k(x))}>b_{i_2(\sigma^k(x))}\geqslant a_{i_2(\sigma^k(x))}. \end{equation*} We conclude that $i_1(\sigma^k(x))\leqslant i_2(\sigma^k(x))\leqslant n-k$, and therefore $x\in \widehat{T}_n(K_1)$. Thus $\widehat{T}_n(K_2)\subset \widehat{T}_n(K_1)$, which readily implies that for every $n \in {\mathbb N}$ we have $\hat z_n(K_2)\leqslant \hat z_n(K_1)$. \end{proof}
In the next lemma we establish a relation between the quantities $\hat\delta_\infty(K)$ and $\delta_\infty(q)$, which in turn is necessary to relate Theorem \ref{pre} with Theorem \ref{thm:main}. As mentioned before, in the definition of $\hat\delta_\infty(K)$ we covered the sets $\widehat{T}_n(K)$ (and not $T_n(K)$ which may seem more natural) in order to ensure this result.
\begin{lemma}\label{lem:Kineq2} Let $(\Sigma, \sigma)$ be a CMS satisfying the $\mathcal{F}-$property, and $M , q \in {\mathbb N}$. Then there exists a sequence of natural numbers $(a_i)_i$ such that $a_0=q$, and \begin{equation*} \hat\delta_\infty(K)\leqslant \delta_\infty(M,q), \end{equation*} where $K=K((a_i)_i)$ \end{lemma}
\begin{proof}
Let $i \in {\mathbb N}$. Since $\Sigma$ satisfies the $\mathcal{F}-$property, there are finitely many cylinders of the form $[x_0,\ldots,x_n]$, where $x_0\leqslant q$, $x_n\leqslant q$, and $n\leqslant iM$. Thus, only a finite collection of symbols from the alphabet are used in this collection of cylinders. Denote by $r_i \in {\mathbb N}$ the largest of this collection of symbols. Inductively define $(a_i)_i \subset {\mathbb N}$ so that: \begin{equation*}
a_{i+1} >a_i \quad \text{ and } \quad a_i > r_i. \end{equation*} We now prove that the set $K=K((a_i)_i)$ is such that $\hat z_n(K)\leqslant \delta_\infty(M,q)(n)$, for every $n\in {\mathbb N}$. Recall that $\hat z_n(K)$ is the minimal number of cylinders of length $(n+2)$ needed to cover $\widehat{T}_n$. Let $x=(x_0, x_1, \dots) \in \widehat{T}_n$, \begin{equation*} E:= \left\{k\in \{0,\ldots,n+1\}:x_k\leqslant a_0 \right\}, \end{equation*} and $B:=\{0,\ldots,n+1\}\setminus E$. For $k\in E$ we define $p_k:=i(\sigma^k(x))$. We emphasise that since $k\in E$ then $x_k\leqslant a_0$, thus $p_k=i(\sigma^k(x))\geqslant 1$. Let $r \in E$ and observe that $x_{p_r+r}=(\sigma^r(x))_{p_r}>a_{p_r}$, where $p_r\leqslant n-r$. Because of the choice of $a_{p_r}$, there is no admissible word of length less or equal to $ p_rM$ connecting $x_{p_r+r}$ and a symbol in the set $\{0, 1 \dots, q\}$. Since $x_{n+1}\leqslant q$, this means that we must have $p_r+r+(p_rM)\leqslant n+1$. Moreover, for every $0\leqslant m< p_rM$ we have that $p_r+r+m\in B$. In other words, the interval $[r,r+p_r+p_rM)$ has at least $p_rM$ elements in $B$, equivalently, at most $p_r$ elements in $E$. Since this argument holds for every $r\in E$ we conclude that $M\#E\leqslant n+2$ and therefore \begin{equation} \label{eq:ele}
\#E\leqslant \frac{n+2}{M}. \end{equation} From \eqref{eq:ele} it follows that every $x\in \widehat{T}_n$ belongs to a cylinder of the form $[x_0,\ldots,x_{n+1}]$, where $x_0\leqslant q$, $x_{n+1}\leqslant q$ and \begin{equation*}
\#\{i\in\{0,1,\ldots,n+1\}: x_i\leqslant q\}\leqslant \frac{n+2}{M}.
\end{equation*} This implies that $\hat z_n(K)\leqslant z_n(M,q)$, for every $n\in {\mathbb N}$. Therefore $\hat\delta_\infty(K)\leqslant \delta_\infty(M,q)$. \end{proof} Define $\hat{\delta}_\infty(q):=\inf_{(a_i)_i:a_0=q} \hat\delta_\infty(K((a_i)_i)$.
\begin{corollary}\label{cor:deltaineq} For every $q\in {\mathbb N}$ we have $\hat{\delta}_\infty(q)\leqslant \delta_\infty(q)$. \end{corollary}
\begin{proof} Combine Lemma \ref{lem:Kineq} with Lemma \ref{lem:Kineq2}. \end{proof}
We now prove Theorem \ref{thm:main}.
\begin{proof}[Proof of Theorem \ref{thm:main}] Let $(a_i)_i$ be a sequence of non-negative integers and $K:=K((a_i)_i)$ the corresponding compact set. We assume $K$ large enough so that there exists a periodic measure $\mu_p$ with $\mu_p(K)>0$. We will prove that \begin{align}\label{11}
\limsup_{n\to \infty} h_{\mu_n}(\sigma)\leqslant |\mu|h_{\mu/|\mu|}(\sigma)+(1-\mu(K))\hat\delta_\infty(K). \end{align}
Let $\mu'_{n}:=(1-\frac{1}{n})\mu_n+\frac{1}{n}\mu_p$. Observe that for every $n\in{\mathbb N}$ we have $\mu_n'(K)>0$. It follows from Proposition \ref{dense} that there exists an ergodic measure $\nu_n$ arbitrarily close in the weak$^*$ topology to $\mu_{n}'$ such that $h_{\nu_n}(\sigma)>h_{\mu'_n}(\sigma)-\frac{1}{n}$. In particular, we can assume that $\nu_n(K(n,n))>0$ and that $(\nu_n)_n$ converges on cylinders to $\mu$.
Let $k , N \in {\mathbb N}$. If $n>\max\{k,N\}$ then $K(n,n)\subset K(k,N)$, therefore $\nu_n(K(k,N))>0$. It now follows from \eqref{forA} that \begin{equation*}
\limsup_{n\to\infty} h_{\nu_n}(\sigma)\leq |\mu| h_{\frac{\mu}{|\mu|}}(\sigma)+(\hat\delta_\infty(K)+\epsilon)(1-\mu(G_{k,N})) +\frac{1}{k}(\hat\delta_\infty(K)+\epsilon). \end{equation*} Letting $k$ tend to infinity and $\epsilon$ to zero we obtain \begin{equation*}
\limsup_{n\to\infty} h_{\nu_n}(\sigma)\leq |\mu| h_{\frac{\mu}{|\mu|}}(\sigma)+(1-\mu(K))\hat\delta_\infty(K). \end{equation*} Since $h_{\nu_n}(\sigma)>h_{\mu'_n}(\sigma)-\frac{1}{n}=(1-\frac{1}{n})h_{\mu_n}-\frac{1}{n}$, then \begin{equation*}
\limsup_{n\to \infty} h_{\mu_n}(\sigma)\leqslant \limsup_{n\to \infty} h_{\nu_n}(\sigma),
\end{equation*} from which \eqref{11} follows.
The argument above also holds for every set $K'=K((b_i)_i)$, where $a_0=b_0$ and $a_i\leqslant b_i$. Observe that $\sup_{(b_i)_i:b_0=a_0} \mu(K((b_i)_i))=\mu(K_{a_0})$. Thus, it is a consequence of Corollary \ref{cor:deltaineq} that
\begin{align*} \limsup_{n\to \infty} h_{\mu_n}(\sigma)&\leqslant |\mu|h_{\mu/|\mu|}(\sigma)+(1-\mu(K_{a_0}))\hat{\delta}_\infty(a_0)\\
& \leqslant |\mu|h_{\mu/|\mu|}(\sigma)+(1-\mu(K_{a_0}))\delta_\infty(a_0). \end{align*} Letting $a_0$ tend to infinity concludes the proof of Theorem \ref{thm:main}. \end{proof}
\section{Variational principle for the entropies at infinity}\label{entinf1}
In this section we prove Theorem \ref{thm:vpinf}. That is, we prove a variational principle at infinity: the measure theoretic entropy at infinity coincides with its topological counterpart.
For each pair $(i,j)\in{\mathbb N}^2$ choose a non-empty cylinder $w(i,j)$ of length $\ell(i,j)+1$ such that \begin{equation*} w(i,j):=[i, \dots, j]= [(w(i,j)_0, \dots , w(i,j)_{\ell(i,j)}]. \end{equation*} Let $\varphi: \Sigma \to {\mathbb R}$ be a potential and define \begin{equation*} Z_n(\varphi, a,b):=\sum_{x:\sigma^{n+\ell(b,a)}(x)=x}\exp \left(S_{n+\ell(b,a)}\varphi(x) \right)1_{[a]\cap \sigma^{-n}w(b,a)}(x). \end{equation*} In the following lemma we show that the Gurevich pressure can be computed by means of the partition function $Z_n(\varphi, a,b)$; this will be used in Lemma \ref{lem:testineq}.
\begin{lemma}\label{lem:equivgur} Let $(\Sigma,\sigma)$ be a transitive CMS and $\varphi: \Sigma \to {\mathbb R}$ a bounded potential with summable variations. Then for every pair $(a,b) \in {\mathbb N}^2$ we have that
\begin{equation*} P_{G}(\varphi)=\limsup_{n\to\infty}\frac{1}{n}\log Z_n(\varphi,a,b). \end{equation*} \end{lemma}
\begin{proof} Let $C=\|\varphi\|_0$ and $D=\sum_{k=2}^\infty \text{var}_k(\varphi)$. It follows from the definition of $Z_n(\varphi,a,b)$ that \begin{equation*} Z_{n+\ell(b,a)}(\varphi,a)=\sum_{x:\sigma^{n+\ell(b,a)}(x)=x}\exp(S_{n+\ell(b,a)}\varphi(x))1_{[a]}(x) \geqslant Z_n(\varphi,a,b). \end{equation*} In particular we obtain that
\begin{equation*}
P_{G}(\varphi)=\limsup_{n\to\infty}\frac{1}{n}\log Z_n(\varphi,a)\geqslant \limsup_{n\to\infty}\frac{1}{n}\log Z_n(\varphi,a,b).
\end{equation*} Let $\mathbb{P}_n:=w(a,b)\cap \sigma^{-n}w(b,a)$. Note that \begin{align*} Z_n(\varphi,a,b)&\geqslant \sum_{x:\sigma^{n+\ell(b,a)}(x)=x}\exp \left(S_{n+\ell(b,a)}\varphi(x) \right)1_{P_n}(x)\\ & \geqslant e^{-(\ell(a,b)+\ell(b,a))C} \sum_{x:\sigma^{n+\ell(b,a)}(x)=x} \exp \left(S_{n -\ell(b,a)}\varphi(\sigma^{\ell(a,b)}x) \right)1_{{P}_n}(x).\\ \end{align*} Observe that if $x=(x_0, x_1, \dots) \in \mathbb{P}_n$, then $x_{\ell(b,a)}=x_{n}=b$. Define the periodic point $y(x):=\overline{x_{\ell(b,a)}\ldots x_{n-1}}$. The function $y$ establishes a one-to-one correspondence between points in $x\in \mathbb{P}_n$ such that $\sigma^{n+\ell(b,a)}(x)=x$, and periodic points of length $n-\ell(b,a)$ in $[b]$. Moreover, note that if $x\in\mathbb{P}_n$, then \begin{equation*}
\left |S_{n-\ell(b,a)} \left(\varphi(\sigma^{\ell(a,b)}x) \right)-S_{n-\ell(b,a)} \left(\varphi(y(x)) \right) \right|\leqslant D. \end{equation*} We conclude that \begin{eqnarray*} \sum_{x:\sigma^{n+\ell(b,a)}(x)=x} \exp \left(S_{n -\ell(b,a)}\varphi(\sigma^{\ell(a,b)}x) \right)1_{\mathbb{P}_n}(x) \geqslant &\\ e^{-D} \sum_{x:\sigma^{n-\ell(b,a)}(y)=y} \exp \left(S_{n -\ell(b,a)}\varphi(y) \right)1_{[b]}(x). \end{eqnarray*} That is $Z_n(\varphi,a,b)\geqslant e^{-(\ell(a,b)+\ell(b,a))C-D}Z_{n-\ell(b,a)}(\varphi,b)$ and therefore \begin{equation*} \limsup_{n\to\infty}\frac{1}{n}\log Z_n(\varphi,a,b)\geqslant P_{G}(\varphi). \end{equation*} \end{proof}
\begin{remark} Note that in Lemma \ref{lem:equivgur} the assumption $\|\varphi\|_0<\infty$ is too strong for what is required: it suffices to assume that
for every $n\in {\mathbb N}$ we have $\sup_{x\in [n]}|\varphi(x)|<\infty$. \end{remark}
We say that a point $x\in\Sigma$ belongs to the set $Per(q,M,n)$ if the following properties hold: \begin{enumerate} \item $\sigma^n(x)=x$. \item If $x\in [x_0,\ldots,x_{n-1}]$, then $x_0\leqslant q$, and $\#\{k\in\{0,\ldots,n-1\}:x_k\leqslant q\}\leqslant \frac{n}{M}$. \end{enumerate}
The following lemma is important in our proof of Theorem \ref{thm:vpinf} as it will allow us to find a sequence of invariant probability measures which converges to the zero measure and entropies approach the topological entropy at infinity.
\begin{lemma}\label{lem:testineq} Let $\varphi: \Sigma \to {\mathbb R}$ be a bounded potential of summable variations such that \begin{equation*}
\lim_{n\to\infty}\sup_{x\in [n]} |\varphi(x)|=0. \end{equation*} Then $P_G(\varphi)\geqslant \delta_\infty$. \end{lemma}
\begin{proof}
For every $\epsilon>0$ there exists $N_0=N_0(\epsilon) \in {\mathbb N}$ such that $\sup_{x \in [n]} |\varphi(x)| \leqslant \epsilon$, for every $n\geqslant N_0$. By Lemma \ref{lem:equivgur}, for sufficiently large values of $n \in {\mathbb N}$, since $Z_n(a,n)(\varphi) \leq \exp (n P_G(\varphi) + \epsilon)$ there exists $N'=N'(N_0) \in {\mathbb N}$ such that \begin{equation*} N' \exp \left(n P_G(\varphi) + \epsilon \right) \geq \sum_{(a,b)\in \{1,\ldots,N_0\}^2} Z_n(\varphi,a,b). \end{equation*} That is, \begin{equation*} P_G(\varphi)\geqslant \limsup_{n\to\infty} \frac{1}{n}\log\sum_{(a,b)\in \{1,\ldots,N_0\}^2} Z_n(\varphi,a,b). \end{equation*} Define \begin{equation*} \mathbb{T}_n(a,b):=\sum_{x\in Per(N_0,M,n+\ell(b,a))} \exp \left(S_{n+\ell(b,a)}\varphi(x) \right)1_{[a]\cap \sigma^{-n}w(b,a)}(x), \end{equation*} and observe that $Z_n(\varphi,a,b)\geqslant \mathbb{T}_n(a,b).$ Recall that $x\in Per(N_0,M,n+\ell(b,a))$ implies that \begin{equation*} \# \left\{k\in\{0,\ldots,n+\ell(b,a)-1\}:x_k\leqslant N_0 \right\} \leqslant \frac{n+\ell(b,a)}{M}. \end{equation*} It follows from the choice of $N_0$ that \begin{equation*}
S_{n+\ell(b,a)}\varphi(x)\geqslant -(n+\ell(b,a))\epsilon-\frac{\|\varphi\|_0}{M}(n+\ell(b,a)). \end{equation*} In particular \begin{equation*}
\mathbb{T}_n(a,b)\geqslant \# \left\{ Per(N_0,M,n+\ell(b,a))\cap [a]\cap\sigma^{-n}w(b,a) \right\} e^{-(n+\ell(b,a))(\epsilon+\frac{\|\varphi\|_0}{M})}. \end{equation*} Denote by $\mathcal{W}_n(a,b,N_0,M)$ the collection of cylinders of the form $[x_0,\ldots,x_{n}]$, where $x_0=a$, $x_n=b$, and $\#\{k\in\{0,\ldots,n\}:x_k\leqslant N_0\}\leqslant \frac{n+1}{2M}$. In order to estimate the number of these using periodic points, to each cylinder $[x_0,\ldots,x_n]\in \mathcal{W}_n(a,b,N_0,M)$ we associate the cylinder \begin{equation*} D=[y_0,\ldots,y_{n+\ell(b,a)}]=[x_0,\ldots,x_n,(w(b,a))_1,\ldots,w(b,a)_{\ell(b,a)}]. \end{equation*}
Observe that $y_0=a$, $y_{n+\ell(b,a)}=a$, and $$\#\{k\in \{0,\ldots,n+\ell(b,a)\}:y_k\leqslant N_0\}\leqslant \frac{n+1}{2M}+\ell(b,a).$$ For $n \in {\mathbb N}$ sufficiently large we can assume that $\frac{n+1}{2M}+\ell(b,a)\leqslant \frac{n+\ell(b,a)}{M}$. In particular the periodic point associated to $D$ belongs to $Per(N_0,M,n+\ell(b,a))\cap[a]\cap\sigma^{-n}w(b,a)$. We conclude that
\begin{equation*}
\#\mathcal{W}_n(a,b,N_0,M)\leqslant \#\left\{Per(N_0,M,n+\ell(b,a))\cap[a]\cap\sigma^{-n}w(b,a)\right\}. \end{equation*} Observe that $\sum_{(a,b)\in \{1,\ldots,N_0\}^2} \#\mathcal{W}_n(a,b,N_0,M)=z_{n-1}(2M,N_0)$, which implies \begin{equation*} z_{n-1}(2M,N_0)\leqslant \sum_{(a,b)\in \{1,\ldots,N_0\}^2} \#\left\{Per(N_0,M,n+\ell(b,a))\cap[a]\cap\sigma^{-n}w(b,a) \right\}. \end{equation*} Hence, writing $\ell_{N_0} := \max_{(a,b)\in \{1,\ldots,N_0\}^2}\ell(b,a))$, we obtain that \begin{equation*}
\sum_{(a,b)\in \{1,\ldots,N_0\}^2}Z_n(\varphi,a,b)\geqslant \sum_{(a,b)\in \{1,\ldots,N_0\}^2}\mathbb{T}_n(a,b)\geqslant z_{n-1}(2M,N_0) e^{-(n+\ell_{N_0})(\epsilon+\frac{\|\varphi\|_0}{M})}, \end{equation*} and therefore \begin{equation*}
P_G(\varphi)\geqslant \limsup_{n\to\infty}\frac{1}{n}\log \sum_{(a,b)\in \{1,\ldots,N_0\}^2}Z_n(\varphi,a,b)\geqslant \delta_\infty(2M,N_0)-\epsilon-\frac{\|\varphi\|_0}{M}. \end{equation*} Letting $M\to\infty$ we obtain that $P(\varphi)\geqslant \delta_\infty(N_0)-\epsilon$. Choosing $N_0$ sufficiently large we can make $\epsilon$ arbitrarily small, to conclude that $P_G(\varphi)\geqslant \delta_\infty$. \end{proof}
Recall that the measure theoretic entropy at infinity of a transitive CMS of finite entropy $(\Sigma,\sigma)$ is defined by \begin{equation*} h_\infty :=\sup_{(\mu_n)_n\to 0}\limsup_{n\to\infty}h_{\mu_n}(\sigma), \end{equation*} where the supremum is taken over all sequences of invariant probability measures converging on cylinders to the zero measure. An immediate consequence of Theorem \ref{thm:main} is the following upper bound for the measure theoretic entropy at infinity of $(\Sigma,\sigma)$: \begin{align} \label{eq:ineinf} h_\infty \leqslant \delta_\infty
\end{align} We will now prove that in fact equality holds. This is equivalent to the sharpness of the inequality in Theorem \ref{thm:main}.
\begin{proof}[Proof of Theorem~\ref{thm:vpinf}]
As observed in \eqref{eq:ineinf}, it suffices to prove the inequality $\delta_\infty\leqslant h_\infty$. Let $\varphi: \Sigma \to {\mathbb R}$ be a bounded, strictly negative locally constant potential depending only on the first coordinate such that \begin{equation*}
\lim_{n\to\infty}\sup_{x\in [n]} |\varphi(x)|=0. \end{equation*} By Lemma \ref{lem:testineq}, for every $t \in {\mathbb R}$ we have $P(t\varphi)\geqslant \delta_\infty$. Now consider a sequence of measures $(\mu_n)_{n }$ such that \begin{equation*} h_{\mu_n}(\sigma)+n\int \varphi ~d\mu_n>P(n\varphi)-\frac{1}{n}. \end{equation*} The existence of such a sequence of invariant probability measures is guaranteed by the variational principle. Then \begin{equation*} h_{\mu_n}(\sigma)+n\int \varphi~d\mu_n>\delta_\infty-\frac{1}{n}. \end{equation*} Since the potential $\varphi$ is strictly negativity and bounded we conclude that the sequence $(\mu_n)_{n}$ converges on cylinders to the zero measure. Since $h_{\mu_n}(\sigma)\geqslant \delta_\infty-\frac{1}{n}$, \begin{equation*} \limsup_{n\to \infty} h_{\mu_n}(\sigma)\geqslant \delta_\infty. \end{equation*} In particular, $\delta_\infty\leqslant h_\infty$. \end{proof}
\section{Applications} \label{sec:app} In this section we discuss several consequences of Theorem \ref{thm:main}. Among the consequences we obtain the upper semi-continuity of the entropy map, the entropy density of the space of ergodic measures, the stability of the measure of maximal entropy in the SPR case, existence of equilibrium states for potentials in $C_0(\Sigma)$, a relationship between the entropy at infinity and the dimension of the set of recurrent points that escape on average and a bound on the amount of mass that can escape for measures with large entropy.
\subsection{Upper semi-continuity of the entropy map} \label{sec:usc} Starting in the early 1970s with the work of Bowen \cite{bo1} many results describing the continuity properties of the entropy map have been obtained. More precisely, given a dynamical system $T:X \to X$, the map $\mu \mapsto h_{\mu}(T)$ defined on the space $ \mathcal{M}(X,T)$ endowed with the weak$^*$ topology is called \emph{entropy map}. In general it is not continuous \cite[p.184]{wa}. However, it was soon realised that that if $X$ is compact and $T$ expansive then the entropy map is upper-semi continuous \cite[Theorem 8.2]{wa}. This result has been extended to a wide range of dynamical systems exhibiting weak forms of expansion or hyperbolicity, but always assuming the compactness of $X$. Indeed, there exist examples of expansive maps $T$ defined on non-compact spaces for which the entropy map is not upper semi-continuous. We discuss some of them in this section (see Remark \ref{rem:nousc}). We recently proved in \cite[Corollary 1.2]{itv} that if $(\Sigma, \sigma)$ is a finite entropy transitive CMS then the entropy map is upper semi-continuous when restricted to ergodic measures. The method of proof used in \cite{itv} does not seem to generalise to handle the non-ergodic case. However, the general case can be obtained directly as a corollary of Theorem \ref{thm:main}.
\begin{theorem} \label{semicont} Let $(\Sigma,\sigma)$ be a transitive CMS of finite topological entropy and $(\mu_n)_{n}$ a sequence of $\sigma$-invariant probability measures converging weak$^*$ to $\mu$. Then \begin{equation*} \limsup_{n\to \infty} h_{\mu_n}(\sigma)\leqslant h_\mu(\sigma). \end{equation*} That is, the entropy map is upper semicontinuous. \end{theorem}
The proof follows immediately from Theorem \ref{thm:main}, the fact that $|\mu|=1$ and Lemma \ref{restriction}.
\begin{remark}\label{rem:nousc} We now describe the situation in the infinite entropy case. \begin{enumerate} \item[(a)] Without the finite entropy assumption, Theorem~\ref{semicont} is false, as we demonstrate here. If $(\Sigma , \sigma)$ is a a topologically transitive infinite entropy CMS then there exists a sequence $(\nu_n)_n$ and $\mu$ in $\mathcal{M}(\Sigma, \sigma)$ such that $h_{\mu}(\sigma) < \infty$, and $\lim_{n \to \infty} h_{\nu_n}(\sigma)= \infty$. Let $(\mu_n)_n$ be the sequence of invariant probability measures defined by \begin{equation*}
\mu_n:= \left(1-\frac{1}{\sqrt{h_{\nu_n}(\sigma)}} \right)\mu+\frac{1}{\sqrt{h_{\nu_n}(\sigma)}} \nu_n.
\end{equation*} Notice $\mu_n$ is well defined for large enough $n$. Then $(\mu_n)_n$ converges weak$^*$ to $\mu$ and \begin{equation*} h_{\mu}(\sigma)<\lim_{n \to \infty} h_{\mu_n}(\sigma)= \infty. \end{equation*} Therefore, the entropy map is not upper semi-continuous at any finite entropy measure.
\item[(b)] Examples of sequences of ergodic measures with finite entropy uniformly bounded above converging weak$^*$ to an ergodic measure (with finite entropy) in the full-shift on a countable alphabet, for which the entropy map is not upper-semi continuous can be found in \cite[p.774]{jmu} and \cite[Remark 3.11]{itv}.
\item[(c)] The entropy map is trivially upper semi-continuous at any measure of infinite entropy.
\end{enumerate} \end{remark}
We conclude this subsection with a consequence of Theorem \ref{thm:main} and Remark \ref{rem:nousc}.
\begin{proposition}\label{prop:iff} Let $(\Sigma,\sigma)$ be a transitive CMS. Then $h_{top}(\sigma)$ is finite if and only if $\delta_\infty$ is finite. \end{proposition}
\begin{proof} We only need to prove that if $\delta_\infty$ is finite, then $h_{top}(\sigma)$ is finite; the other direction follows directly from the inequality $\delta_\infty\leqslant h_{top}(\sigma)$.
First assume that $(\Sigma,\sigma)$ does not satisfy the $\mathcal{F}-$property. It follows directly from the definition of $\delta_\infty$ that in this situation we have $\delta_\infty=\infty$. As mentioned above there is nothing to prove in this case.
Now assume that $(\Sigma,\sigma)$ satisfies the $\mathcal{F}-$property. In the proof of Theorem \ref{thm:main} we did not use the fact that the topological entropy of $(\Sigma,\sigma)$ is finite, we only used that our CMS has the $\mathcal{F}-$property and that $\delta_\infty$ is finite--those follow trivially under the finite entropy assumption. The $\mathcal{F}-$property is crucially used in Proposition \ref{prop:atom} and Lemma \ref{lem:Kineq2}. If $\delta_\infty$ is finite, then Theorem \ref{thm:main} implies that the entropy map is upper semi-continuous, which would contradict Remark \ref{rem:nousc} if $h_{top}(\sigma)$ is infinite. We conclude that the topological entropy of $(\Sigma,\sigma)$ is finite. \end{proof}
\subsection{Suspension flows}
Let $(\Sigma, \sigma)$ be a transitive, finite entropy CMS and $\tau: \Sigma \to {\mathbb R}^+$ a potential bounded away from zero. Let $$Y:= \left\{ (x,t)\in \Sigma \times {\mathbb R} \colon 0 \leqslant t \leqslant\tau(x) \right\},$$ with the points $(x,\tau(x))$ and $(\sigma(x),0)$ identified for each $x\in \Sigma $. The \emph{suspension flow} over $\Sigma$ with \emph{roof function} $\tau$ is the semi-flow $\Phi= (\varphi_t)_{t \in {\mathbb R}_{\geqslant 0}}$ on $Y$ defined by $ \varphi_t(x,s)= (x,s+t)$ whenever $s+t\in[0,\tau(x)]$. Denote by $\mathcal{M}(Y,\Phi)$ the space of flow invariant probability measures. In this section we prove that in this continuous time, non-compact setting again the entropy map is upper semi-continuous. This generalises \cite[Proposition 5.2]{itv} in which upper semi-continuity of the entropy map was proven for ergodic measures. Let
\begin{equation} \mathcal{M}_\sigma(\tau):= \left\{ \mu \in \mathcal{M}_{\sigma}: \int \tau ~d \mu < \infty \right\}. \end{equation} A result by Ambrose and Kakutani \cite{ak} implies that the map $M \colon \mathcal{M}_\sigma \to \mathcal{M}_\Phi$, defined by \begin{equation*} \label{eq:R map}
M(\mu)=\frac{(\mu \times \text{Leb})|_{Y} }{(\mu \times \text{Leb})(Y)}, \end{equation*} where \text{Leb} is the one-dimensional Lebesgue measure, is a bijection. The following result proved in \cite[Lemma 5.1]{itv} describes the relation between weak$^*$ convergence in $\mathcal{M}(Y,\Phi)$ with that in $\mathcal{M}(\Sigma, \sigma)$.
\begin{lemma} \label{lem:weak} Let $(\nu_n), \nu \in \mathcal{M}(Y,\Phi)$ be flow invariant probability measures such that \begin{equation*} \nu_n=\frac{\mu_n \times Leb}{\int \tau~d \mu_n} \quad \text{ and } \quad \nu= \frac{\mu \times Leb}{\int \tau~d \mu} \end{equation*} where $(\mu_n)_n , \mu \in \mathcal{M}(\Sigma, \sigma)$. If the sequence $(\nu_n)_n$ converges weak$^*$ to $\nu$ then $(\mu_n)_n$ converges weak$^*$ to $\mu$ and $\lim_{n \to \infty} \int \tau~d \mu_n = \int \tau~d\mu$. \end{lemma}
\begin{proposition} \label{thm:susp} Let $(\Sigma,\sigma)$ be a transitive CMS of finite topological entropy. Let $\tau$ be a potential bounded away from zero and $(Y,\Phi)$ the suspension flow of $(\Sigma,\sigma)$ with roof function $\tau$. Then the entropy map of $(Y, \Phi)$ is upper semi-continuous. \end{proposition}
The proof directly follows from Abramov's formula \cite{ab}, Lemma \ref{lem:weak} and Theorem \ref{thm:main}. Because of the similarities between the geodesic flow and the suspension flow over a Markov shift it is reasonable to expect that, under suitable assumptions on the roof function $\tau$, the suspension flow also satisfies an entropy inequality like Theorem \ref{thm:main}. This is in fact the case and will be discussed in \cite{ve2}. The space of invariant measures for the suspension flow was already investigated and described in \cite[Section 6]{iv}.
\subsection{Entropy density of ergodic measures} \label{sec:ed}
The structure of the space of invariant measures for finite entropy (non-compact) CMS was studied in \cite{iv}. In this non-compact setting it is well known that the space of ergodic measures is still dense in $\mathcal{M}(\Sigma,\sigma)$ (see \cite[Section 6]{csc}). A natural question is whether the approximation by ergodic measures can be arranged so that the corresponding entropies also converge. If this is the case we say that the set of ergodic measures is \emph{entropy dense}. More precisely,
\begin{definition} \label{def:edense} A subset $\mathcal{L} \subset \mathcal{M}(\Sigma,\sigma)$ is \emph{entropy dense} if for every measure $\mu \in \mathcal{M}(\Sigma,\sigma)$ there exists a sequence $(\mu_n)_n$ in $\mathcal{L}$ such that \begin{enumerate} \item $(\mu_n)_n$ converges to $\mu$ in the weak$^*$ topology. \item $\lim_{n\to\infty} h_{\mu_n}(\sigma)=h_\mu(\sigma)$. \end{enumerate} \end{definition} Results proving that certain classes of measures are entropy dense have been obtained for different dynamical systems defined on compact spaces by Katok \cite{ka}, Orey \cite{or}, F\"ollmer and Orey \cite{fo}, Eizenberg, Kifer and Weiss \cite{ekw} and by Gorodetski and Pesin \cite{gp} among others. In this section we prove, for the non-compact setting of finite entropy CMS, that the set of ergodic measures ${\mathcal E}(\Sigma,\sigma)$ is entropy dense.
\begin{theorem}\label{teodense} Let $(\Sigma, \sigma)$ be a finite entropy, transitive CMS and $\mu \in \mathcal{M}(\Sigma,\sigma)$. Then there exists a sequence $(\mu_n)_{n }$ of ergodic measures such that $(\mu_n)_n$ converges to $\mu$ in the weak$^*$ topology and $\lim_{n\to\infty} h_{\mu_n}(\sigma)=h_\mu(\sigma)$, i.e., ${\mathcal E}(\Sigma, \sigma)$ is entropy dense. Moreover, it is possible to choose the sequence so that each $\mu_n$ has compact support. \end{theorem}
The proof of this result directly follows combining Theorem \ref{semicont}, where the upper semi-continuity of the entropy map is proved, and Proposition \ref{dense}, where we proved a weak form of entropy density of the set of ergodic measures. Note that the entropy density property of ergodic measures is an important tool in proving large deviations principles via the orbit-gluing technique (see, for example, \cite{ekw} and \cite{fo}).
\subsection{Points that escape on average}
In this section we relate the Hausdorff dimension of the set of recurrent points that escape on average with the entropy at infinity of $(\Sigma,\sigma)$. Recall we have fixed an identification of the alphabet of $(\Sigma,\sigma)$ with ${\mathbb N}$.
\begin{definition} Let $(\Sigma, \sigma)$ be a CMS, the set of points that \emph{escape on average} is defined by \begin{equation*} E:=\left\{ x \in \Sigma : \lim_{n \to \infty} \frac{1}{n} \sum_{i=0}^{n-1} 1_{[a]}(\sigma^i x)=0, \text{ for every } a \in {\mathbb N} \right\}. \end{equation*} We say that $x\in\Sigma$ is a \emph{recurrent point} if there exists an increasing sequence $(n_k)_k$ such that $\lim_{k\to\infty}\sigma^{n_k}(x)=x$. The set of recurrent points is denoted by $\mathcal{R}$. \end{definition}
A version of the set $E$ has been considered in the context of homogeneous dynamics. Interest in that set stems from work of Dani \cite{da} in the mid 1980s who proved that singular matrices are in one-to-one correspondence with certain divergent orbits of one parameter diagonal groups on the space of lattices. For example, Einsiedler and Kadyrov \cite[Corollary 1.7]{ek} computed the Hausdorff dimension of that set in the setting of $SL_3({\mathbb Z}) \backslash SL_3({\mathbb R})$. In the context of unimodular $(n+m)-$lattices an upper bound for the Hausdorff dimension of the set of points that escape on average has been obtained in \cite[Theorem 1.1]{kklm}. More recently, for the Teichm\"uller geodesic flow, in \cite[Theorem 1.8]{aaekmu} the authors prove an upper bound for the Hausdorff dimension of directions in which Teichm\"uller geodesics escape on average in a stratum. In all the above mentioned work, either explicitly or not, the bounds are related to the entropy at infinity of the system. Our next result establishes an analogous result for CMS. In this case the upper bound is the entropy at infinity divided by $\log 2$. This latter constant comes from the metric we consider in the space (see \eqref{metric}) and can be thought of as the Lyapunov exponent of the system.
\begin{theorem}\label{thm:onave} Let $(\Sigma, \sigma)$ be a finite entropy transitive CMS. Then \begin{equation*} \dim_H(E\cap \mathcal{R}) \leq \frac{\delta_{\infty}}{\log 2} \end{equation*} where $\dim_H$ denotes the Hausdorff dimension with respect to the metric \eqref{metric}. \end{theorem}
Before initiating the proof of Theorem \ref{thm:onave} let us set up some notation. Given natural numbers $a, b, q, m$ and $N$ we define $S^q_{a,b}(N,m)$ as the collection of cylinders of the form $[x_0,...,x_{L-1}]$, where $L\geqslant Nm$, $x_0=a, x_{L-1}=b,$ and the number of indices $i\in\{0,...,L-1\}$ such that $x_i\leqslant q$ is exactly $N$. It will be convenient to define $$H_{a,b}^q(n,m):=\bigcup_{N\geqslant n}S_{a,b}^q(N,m).$$ Finally define $$\mathcal{L}_b:=\{x \in\Sigma: \exists (n_k)_k \text{ strictly increasing such that } \sigma^{n_k}(x)\in[b], \forall k\in{\mathbb N}\},$$ and $\mathcal{L}=\bigcup_{b\in {\mathbb N}} \mathcal{L}_b$.
\begin{remark}\label{rem:enddd} Let $a, b, q$ and $m$ be natural numbers. Assume that $q\geqslant b$. Note that if $x\in \left(E\cap\cL_b\cap [a] \right)$, then there exists $s_0 \in {\mathbb N}$ such that $$\# \left\{i\in\{0,...,s-1\}: x_i\leqslant q \right\}\leqslant \frac{s}{m},$$ for every $s\geqslant s_0$. Moreover, there exists an increasing sequence $(n_k)_{k}$ such that $x_{n_k}=b$. Define ${\mathcal T}_k(x)=\#\{i\in\{0,...,n_k-1\}: x_i\leqslant q\}$. Since $q\geqslant b$ we get that ${\mathcal T}_k(x)\geqslant k$. Observe that if $n_k\geqslant s_0$, then $$m\mathcal{T}_k(x)= m\#\{i\in\{0,...,n_k-1\}: x_i\leqslant q\}\leqslant n_k.$$ We conclude that $$[x_0,...,x_{n_k-1}]\in S^q_{a,b}(\mathcal{T}_k(x),m)\subset \bigcup_{p\geqslant k}S^q_{a,b}(p,m)=H_{a,b}^q(k,m).$$ This gives us the inclusion \begin{align}\label{eq:cov} \left( E\cap \cL_b\cap [a] \right) \subset \bigcup_{C\in H_{a,b}^q(k,m)} C,\end{align} for every $k\in {\mathbb N}$. \end{remark}
\begin{proof}[Proof of Theorem~\ref{thm:onave}] First observe that $\left(E\cap\cR \right)\subset \bigcup_{b\in{\mathbb N}} \left(E\cap\cL_b\cap [b] \right)$. In particular it suffices to prove that $\dim_H(E\cap \mathcal{L}_b\cap[a])\leqslant \delta_\infty / \log 2$ , for every pair of natural numbers $a$ and $b$. Fix $t> \delta_\infty/\log 2$. Recall that $\delta_\infty=\inf_{m,q}\delta_\infty(m,q)$ (see equation \eqref{infinf}). Choose $m$ and $q$ large enough so that $t>\delta_\infty(q,m)/\log 2$, and that $q\geqslant \max\{a,b\}$. Observe that we are now in the same setup as in Remark \ref{rem:enddd}.
In order to estimate the Hausdorff dimension of $E\cap \cL_b\cap[a]$ we will use the covering given by \eqref{eq:cov}. Thus, it is enough to bound $\sum_{C\in H_{a,b}^q(k,m)} \mbox{\rm diam} (C)^t$. First observe that since $q\geqslant\max\{a,b\}$,
a cylinder $C\in H_{a,b}^q(k,m)$ has length $\ell(C)\geqslant k$. Recall that $\mbox{\rm diam} (C)\leqslant 2^{-\ell(C)}=e^{-(\log 2)\ell(C) }$. Therefore, as $k \in {\mathbb N}$ increases the diameter of the covering given by \eqref{eq:cov} converges to zero. Now observe that
\begin{align*}\sum_{C\in H_{a,b}^q(k,m)} \mbox{\rm diam} (C)^t&\leqslant \sum_{C\in H_{a,b}^q(k,m)} e^{-t(\log 2)\ell(C)}\\ &=\sum_{l\geqslant k}e^{-t(\log 2) l}\#\{C: C\in H_{a,b}^q(k,m)\text{ and }\ell(C)=\ell\}\\ &\leqslant\sum_{l\geqslant k} e^{-t(\log 2) l}z_{l-2}(m,q). \end{align*} In the last inequality we used that $$\#\{C\in H_{a,b}^q(k,m)\text{ and }\ell(C)=l\}\leqslant z_{l-2}(m,q).$$ Indeed, if $C\in H_{a,b}^q(k,m)\text{ and }\ell(C)=\ell$, then $C$ is a cylinder of the form $[x_0,...,x_{\ell-1}]$ where $x_0=a$, $x_{l-1}=b$, and $$\#\{i\in\{0,...,\ell-1\}: x_i\leqslant q\}=k\leqslant \frac{\ell}{m}.$$ Since $\max\{a,b\}\leqslant q$ we conclude that $C$ is one of the cylinders counted in the definition of $z_{\ell-2}(m,q)$ (see Definition \ref{def:ent_inf}).
By the definition of $\delta_\infty(m,q)$ the series $Z(s):=\sum_{\ell=2}^\infty e^{-s\ell}z_{l-2}(m,q)$ is convergent for $s>\delta_\infty(m,q)$. In particular since $t\log 2>\delta_\infty(m,q)$ we have that $Z(t\log 2)$ is finite. This implies that the tail of $Z(t\log 2)$ converges to zero. We conclude that $\sum_{C\in H_{a,b}^q(k,m)} \mbox{\rm diam} (C)^t$ goes to zero as $k\to\infty$. This implies that $\dim_H(E\cap \cL_b\cap[a])\leqslant t$, but $t$ was an arbitrary number larger than $\delta_\infty / \log 2$.\end{proof}
\begin{remark} It is proved in \cite[Theorem 3.1]{i} that if $(\Sigma,\sigma)$ is a transitive CMS with finite topological entropy, then $\dim_H(\cR)=h_{top}(\sigma)/\log 2$. In particular if $(\Sigma,\sigma)$ is SPR, then $\dim_H(E\cap\cR)<\dim_H(\cR)$. \end{remark}
\subsection{Measures of maximal entropy}\label{sec:mme}
An invariant measure $\mu \in \mathcal{M}(\Sigma, \sigma)$ is called a \emph{measure of maximal entropy} if $h_{\mu}(\sigma)= h_{top}(\sigma)$.
It follows from work by Gurevich \cite{gu1,gu2} that if $h_{top}(\sigma)<\infty$ then there exists at most one measure of maximal entropy. Note that a direct consequence of the variational principle (see \cite{gu2} or
Theorem \ref{thm:vp}) is that there exists a sequence of invariant probability measures $(\mu_n)_n$ such that $\lim_{n \to\infty} h_{\mu_n}(\sigma)= h_{top}(\sigma)$. Moreover, if the sequence has a weak$^*$ accumulation point $\mu$ then it follows from the upper semi-continuity of the entropy map, see Theorem \ref{semicont}, that $h_{\mu}(\sigma)=h_{top}(\sigma)$. Since the space $\mathcal{M}(\Sigma, \sigma)$ is not compact there are cases in which the sequence $(\mu_n)_n$ does not have an accumulation point. In fact, there exist transitive finite entropy CMS that do not have measures of maximal entropy (see \cite{ru2} for a wealth of explicit examples). Our next result follows directly from Theorem \ref{thm:main} and Theorem \ref{compact}. Recall that $(\Sigma,\sigma)$ is SPR if and only if $\delta_\infty<h_{top}(\sigma)$ (see Proposition \ref{prechar}).
\begin{theorem} \label{thm:mme} Let $(\Sigma, \sigma)$ be a SPR CMS and $(\mu_n)_{n}$ a sequence of $\sigma$-invariant probability measures such that \begin{equation*} \lim_{n\to\infty}h_{\mu_n}(\sigma)=h_{top}(\sigma). \end{equation*} Then the sequence $(\mu_n)_{n }$ converges in the weak$^*$ topology to the unique measure of maximal entropy. \end{theorem}
\begin{proof} Note that the inequality $\delta_\infty<h_{top}(\sigma)$ immediately implies that $(\Sigma,\sigma)$ has finite topological entropy (see Proposition \ref{prop:iff}). Since $\mathcal{M}_{\le1}(\Sigma,\sigma)$ is compact (see Theorem \ref{compact}) there exists a subsequence $(\mu_{n_k})_k$ which converges on cylinders to $\mu\in \mathcal{M}_{\le1}(\Sigma,\sigma)$. It follows directly from Theorem \ref{thm:main} that \begin{equation*}
h_{top}(\sigma)= \limsup_{k\to \infty} h_{\mu_{n_k}}(\sigma)\leqslant |\mu|h_{\mu/|\mu|}(\sigma)+(1-|\mu|)\delta_\infty. \end{equation*}
Recall that $\delta_{\infty} < h_{top}(\sigma)$. If $|\mu| <1$ then the right hand side of the equation is a convex combination of numbers, one of which is strictly smaller than $h_{top}(\sigma)$. Since this is not possible we have that $|\mu|=1$. In particular \begin{equation*} h_{top}(\sigma) \leq h_{\mu}(\sigma). \end{equation*} That is, $\mu$ is a measure of maximal entropy. We conclude that $(\Sigma,\sigma)$ has a measure of maximal entropy. The same argument holds for every subsequence of $(\mu_n)_n$, this implies that the entire sequence $(\mu_n)_n$ converges in the weak$^*$ topology to the unique measure of maximal entropy. \end{proof}
In fact Theorem \ref{thm:main} also gives a complete description of non strongly positive recurrence, as follows. Some of these results were originally proved in \cite[Theorem 6.3]{gs} by different methods.
\begin{theorem}\label{thm:mme2} Let $(\Sigma, \sigma)$ be a transitive CMS of finite entropy. \begin{enumerate} \item Suppose $(\Sigma,\sigma)$ does not admit a measure of maximal entropy. Let $(\mu_n)_{n }$ be a sequence of $\sigma$-invariant probability measures such that $\lim_{n\to\infty}h_{\mu_n}(\sigma)=h_{top}(\sigma)$. Then $(\mu_n)_{n }$ converges on cylinders to the zero measure and $\delta_\infty=h_{top}(\sigma)$. \item Suppose that $(\Sigma,\sigma)$ is positive recurrent, but $h_{top}(\sigma)=\delta_\infty$. Let $(\mu_n)_{n}$ be a sequence of $\sigma$-invariant probability measures such that $\lim_{n\to\infty}h_{\mu_n}(\sigma)=h_{top}(\sigma)$. Then the accumulation points of $(\mu_n)_{n}$ lie in the set $\{ \lambda \mu_{max}: \lambda \in [0,1]\}$, where $\mu_{max}$ is the measure of maximal entropy. Moreover, every measure in $\{ \lambda \mu_{max}:t\in [0,1]\}$ can be realised as such limit. \end{enumerate} \end{theorem}
\begin{proof}
Note that part $(a)$ directly follows from Theorem \ref{thm:main}. Indeed, if a sequence $(\mu_n)_{n }$ with $\lim_{n\to\infty}h_{\mu_n}(\sigma)=h_{top}(\sigma)$ converges in cylinder to a measure $\mu \in \mathcal{M}_{\leq 1}(\Sigma, \sigma)$ different from the zero measure then $\mu/|\mu|$ would be a measure of maximal entropy. This argument also gives us the first part of $(b)$, that is, the accumulation points of $(\mu_n)_n$ lie in $\{\lambda\mu_{max}:\lambda\in[0,1]\}$. As for the second part of $(b)$, by Theorem \ref{thm:vpinf} there exists a sequence $(\mu_n)_n$ in $\mathcal{M}(\Sigma,\sigma)$ with $\lim_{n \to \infty} h_{\mu_n}(\sigma)=h_{top}(\sigma)$ such that that $(\mu_n)_n$ converges on cylinders to the zero measure. Since there exist a measure of maximal entropy $\nu$ we have that for every $\lambda \in [0,1]$ the sequence $\rho_n:=\lambda \nu +(1-\lambda)\mu_n$ converges on cylinders to $\lambda\nu$ and $\lim_{n \to \infty} h_{\rho_n}(\sigma) =h_{top}(\sigma)$. \end{proof}
\subsection{Existence of equilibrium states}\label{sec:eqst} In this section we will always assume that $(\Sigma,\sigma)$ is a transitive CMS with finite entropy. In Section~\ref{sec:tf} we described the thermodynamic formalism developed by Sarig in the setting of CMS and functions (potentials) of summable variations. It turns out that the same methods can be extended and thermodynamic formalism can be developed for functions with weaker regularity assumptions (for example functions satisfying the Walters condition \cite{sabook}). However, these methods can not be extended much further. In this section we propose an alternative definition of pressure that generalises the Gurevich pressure to the space of functions $C_0(\Sigma)$ (see Definition \ref{C_0}). We stress that these functions are just uniformly continuous. Making use of Theorem \ref{thm:main} we can ensure the existence of equilibrium states.
The following result is a direct consequence of Theorem \ref{thm:main} and the continuity of the map $\mu\mapsto \int Fd\mu$, when $F\in C_0(\Sigma)$ and $\mu$ ranges in $\mathcal{M}_{\le1}(\Sigma,\sigma)$ endowed with the cylinder topology.
\begin{theorem}\label{ineqC_0} Let $(\Sigma,\sigma)$ be a transitive CMS with finite entropy and $F\in C_0(\Sigma)$. Let $(\mu_n)_n$ be a sequence in $\mathcal{M}(\Sigma,\sigma)$ converging on cylinders to $\lambda \mu$, where $\lambda\in [0,1]$ and $\mu\in \mathcal{M}(\Sigma,\sigma)$. Then $$\limsup_{n\to\infty}\left( h_{\mu_n}(\sigma)+\int Fd\mu_n\right)\leqslant \lambda \left(h_{\mu}(\sigma)+\int Fd\mu\right)+(1-\lambda)\delta_\infty.$$ \end{theorem}
For a continuous, bounded potential $F$ define the \emph{(variational) pressure} of $F$ by
\begin{equation*}
P_{var}(F):=\sup_{\mu\in \mathcal{M}(\Sigma,\sigma)}\left(h_\mu(\sigma)+\int Fd\mu\right).
\end{equation*} A measure $\mu$ is an equilibrium state for $F$ if $P_{var}(F)=h_\mu(\sigma)+\int Fd\mu$. Recall that since $F$ needs not to be of summable variations then the classifications of potentials (see Definition \ref{def:clas}) and the uniqueness of equilibrium states (Theorem \ref{clas}) do not necessarily hold.
Note that if $F\in C_0(\Sigma)$, then $P_{var}(F)\geqslant \delta_\infty$. Indeed, let $(\mu_n)_n$ be a sequence of measures in $\mathcal{M}(\Sigma,\sigma)$ converging on cylinders to the zero measure and such that $\lim_{n\to\infty}h_{\mu_n}(\sigma)=\delta_\infty$. Since $F\in C_0(\Sigma)$, then $\lim_{n\to\infty}\int Fd\mu_n=0$. We conclude that $$P_{var}(F)\geqslant \limsup_{n\to\infty}\left(h_{\mu_n}(\sigma)+\int Fd\mu_n\right)=\delta_\infty.$$
Our next result follows directly from Theorem \ref{ineqC_0} and Theorem \ref{compact}, as Theorem \ref{thm:mme} follows from Theorem \ref{thm:main} and Theorem \ref{compact}.
\begin{theorem}\label{thm:sta} Let $(\Sigma,\sigma)$ be a transitive CMS with finite entropy and $F\in C_0(\Sigma)$. Assume that $P_{var}(F)> \delta_\infty$. Then there exists an equilibrium state for $F$. Moreover, if $(\mu_n)_n$ is a sequence in $\mathcal{M}(\Sigma,\sigma)$ such that $$\lim_{n\to\infty}\left(h_{\mu_n}(\sigma)+\int Fd\mu_n\right)=P_{var}(F),$$ then every limiting measure of $(\mu_n)_n$ is an equilibrium state of $F$. \end{theorem}
In Theorem \ref{thm:sta}, if we further assume that $F$ has summable variations, then the sequence $(\mu_n)_n$ converges in the weak$^*$ topology to the unique equilibrium state of $F$. For the description of the pressure map $t\mapsto P_{var}(tF)$ we refer the reader to \cite[Theorem 5.7]{rv}.
\subsection{Entropy and escape of mass}
In this subsection we show that for a SPR CMS $(\Sigma,\sigma)$ it is possible to bound the escape of mass of sequences of measures with sufficiently large entropy. In the setting of homogenous flows an analogous result was proven in \cite[Corollary of Theorem A]{ekp}.
\begin{theorem} \label{thm:em} Let $(\Sigma,\sigma)$ be a SPR CMS. Let $(\mu_n)_n$ be a sequence in $\mathcal{M}(\Sigma,\sigma)$ such that $h_{\mu_n}(\sigma)\geqslant c$, for every $n\in{\mathbb N}$, and $c\in (\delta_\infty,h_{top}(\sigma))$. Then every limiting measure $\mu$ of $(\mu_n)_n$ with respect to the cylinder topology satisfies \begin{equation*} \mu(\Sigma)\geqslant \frac{c-\delta_\infty}{h_{top}(\sigma)-\delta_\infty}. \end{equation*} \end{theorem}
\begin{proof} From Theorem \ref{thm:main} we have that \begin{eqnarray*}
c \leq \limsup_{n \to \infty} h_{\mu_n}(\sigma) \leq \mu(\Sigma) h_{\mu / |\mu|}(\sigma) + (1 - \mu(\Sigma)) \delta_{\infty} \leq \mu(\Sigma) (h_{top}(\sigma) - \delta_{\infty}) + \delta_{\infty}. \end{eqnarray*} The result then follows. \end{proof}
\end{document} | arXiv |
\begin{document}
\maketitle
\begin{abstract} Poincar\'e maps and suspension flows are examples of fundamental constructions in the study of dynamical systems. This study aimed to show that these constructions define an adjoint pair of functors if categories of dynamical systems are suitably set. First, we consider the construction of Poincar\'e maps in the category of flows on topological manifolds, which are not necessarily smooth. We show that well-known results can be generalized and the construction of Poincar\'e maps is functorial, if a category of flows with global Poincar\'e sections is adequately defined. Next, we consider the construction of suspension flows and its functoriality. Finally, we consider the adjointness of the constructions of Poincar\'e maps and suspension flows. By considering the naturality, we can conclude that the concepts of topological equivalence or topological conjugacy of flows are not sufficient to describe the correspondence between map dynamical systems and flows with global Poincar\'e sections. We define another category of flows with global Poincar\'e sections and show that the suspension functor and the Poincar\'e map functor form an adjoint equivalence if these categories are considered. Hence, a categorical correspondence between map dynamical systems and flows with global Poincar\'e sections is obtained. This will enable us to better understand the connection between map dynamical systems and flows. \end{abstract}
\section{Introduction }
Poincar\'e map and suspension flow constructions are fundamental tools employed in the study of dynamical systems. They are used to reduce a problem concerning continuous-time systems to one of discrete-time systems or vice versa, thereby connecting the two major types of dynamical systems \cite{hasselblatt2002handbook, katok1997introduction, robinson1998dynamical}.
Results on their relationship are scattered across the literature, and systematic treatments are scarce. However, by collecting these results, we can easily observe that a categorical relationship may exist between them. For example, the following properties are known: \begin{itemize}
\item If two diffeomorphisms are topologically conjugate, then their suspensions are topologically conjugate (Proposition 5.38 in \cite{irwin2001smooth}).
\item A flow with a Poincar\'e section is locally topologically equivalent to the suspension of its Poincar\'e map (Theorem 5.40 in \cite{irwin2001smooth}).
\item Every diffeomorphism on a compact manifold is topologically conjugate with the Poincar\'e map of its suspension (Proposition 3.7 in \cite{palis2012geometric}). \end{itemize}
In the case of flows with global sections, stronger properties hold because Poincar\'e maps can be defined globally: \begin{itemize}
\item Topological equivalence of two flows can be determined in terms of Poincar\'e maps (Theorem 1 in \cite{basener2002global}, Proposition 1.11 in \cite{phdthesis}).
\item A flow with a global section is topologically equivalent to the suspension of its Poincare map (Theorem 3.1 in \cite{yang2000remark}). \end{itemize} In loose terms, these results can be summarized as follows: isomorphisms are preserved under the constructions of Poincar\'e maps and suspension flows, and a Poincar\'e map of a suspension or a suspension of a Poincar\'e map can be identified with the original map or flow. These statements suggest the existence of categorical equivalence between a category of map dynamical systems and one of flows .
Some categorical aspects of these constructions have been considered in the case of isomorphisms with topological conjugacy \cite{cestau2017prolongations}. However, their relation remains unclear because it depends on the choice of categories. For example, some of the results mentioned above are not true if one uses topological conjugacy instead of topological equivalence to define isomorphisms.
This study aimed to perform a categorical treatment of the constructions of Poincar\'e maps and suspension flows in order to describe the exact relationship between them. This will enable us to unify the known results listed above and also ``prove" the folklore correspondence of various notions between discrete-time and continuous-time systems, such as that of topological conjugacy and topological equivalence.
The rest of this paper is organized as follows. In Section \ref{sec_cat}, we define several categories of dynamical systems. In Section \ref{sec_poincare}, we first introduce the notion of topological transversality for topological manifolds and continuous flows. We show that Poincar\'e maps can be defined analogously to the smooth case. Then, we define categories of flows with global Poincar\'e sections to show that the construction of Poincar\'e maps is functorial. In Section \ref{sec_ps}, we study the categorical relationship between Poincar\'e maps and suspension flows. We show that these two form a pair of adjoint equivalence if the categories are selected properly. Finally, in Section \ref{sec_conclude}, we present some concluding remarks.
\section{Categories of dynamical systems}\label{sec_cat} In this section, we define various categories of dynamical systems to set up for the discussion later.
In what follows, topological manifolds are assumed to be second countable and Hausdorff. For the definitions of the concepts and basic results of category theory, we refer to \cite{mac1998categories, bradley2020topology, riehl2017category}. \begin{definition} A \emph{map dynamical system} is a pair $(f,X)$ of a topological manifold (without boundary) $X$ and a homeomorphism $f:X \to X.$ A \emph{morphism} $h:(f,X) \to (g,Y)$ between map dynamical systems is a continuous map $h: X \to Y$ such that $h \circ f = g \circ h.$ \end{definition} \begin{definition} A \emph{flow} is a pair $(\Phi,X)$ of a topological manifold (without boundary) $X$ and a continuous map $\Phi:X\times \mathbb{R} \to X$ such that \begin{enumerate}
\item For each $x \in X,$ $\Phi(x,0) = x.$
\item For each $x \in X$ and $s, t \in \mathbb{R},$ $\Phi(\Phi(x,t),s) = \Phi(x, t+s).$ \end{enumerate} A \emph{morphism} $h:(\Phi,X) \to (\Psi,Y)$ between flows is a continuous map $h: X \to Y$ such that $h \left( \Phi(x,t) \right) = \Phi(h(x),t)$ for all $x \in X$ and $t \in \mathbb{R}.$
A \emph{weak morphism} $(h, \tau):(\Phi,X) \to (\Psi,Y)$ between flows is a pair of a continuous map $h: X \to Y$ and a map $\tau: X \times \mathbb{R} \to \mathbb{R}$ such that \begin{enumerate}
\item $h \left( \Phi(x,t) \right) = \Psi(h(x),\tau(x,t))$ for all $x \in X$ and $t \in \mathbb{R}.$
\item For all $x \in X,$ $\tau(x, -): \mathbb{R} \to \mathbb{R}$ is an increasing homeomorphism with $\tau(x,0) = 0.$
\end{enumerate} \end{definition} \begin{lemma} Each of the following forms a category if the composition of morphisms is defined by the composition of maps. \begin{enumerate}
\item Map dynamical systems and their morphisms.
\item Flows and their morphisms.
\item Flows and weak morphisms. \end{enumerate}
\end{lemma} \begin{proof} The proof is obvious for (1) and (2). For (3), we need to verify that the "time-part" composition of the morphism satisfies the conditions of weak morphism. Let $(h_1, \tau_1):(\Phi_1,X_1) \to (\Phi_2,X_2)$ and $(h_2, \tau_2):(\Phi_2,X_2) \to (\Phi_3,X_3)$ be morphisms and define $\tau_2 \circ \tau_1(x,t): = \tau_2(h_1(x), \tau_1(x,t)).$ Then, for all $x \in X_1,$ $\tau_2 \circ \tau_1(x, 0) = \tau_2(h_1(x), 0) = 0$ and $\tau_2 \circ \tau_1(x, -) $ is a composition of homeomorphisms. \end{proof} We call the above a \emph{category of map dynamical systems} $\bf{Map}$, a \emph{category of flows} $\bf{Flow}$, and a \emph{category of flows with weak morphisms} $\bf{WFlow},$, respectively. We note that $\bf{Flow}$ can be regarded as a subcategory of $\bf{WFlow},$ as there is an obvious inclusion functor defined by $(\Phi, X) \mapsto (\Phi, X) $ and $\left((\Phi,X) \xrightarrow{h} (\Psi,Y) \right)\mapsto \left((\Phi,X) \xrightarrow{(h, \rm{id})} (\Psi,Y) \right).$
Isomorphisms in $\bf{Map}$ and $\bf{Flow}$ are called \emph{topological conjugacies} and isomorphic objects are called \emph{topologically conjugate}. In $\bf{WFlow}$, isomorphism is called \emph{topological equivalence} and isomorphic objects are called \emph{topologically equivalent}. These definitions coincide with the usual ones.
\begin{remark} The categories in \cite{cestau2017prolongations} correspond to $\bf{Map}$ or $\bf{Flow}$ in this paper. \end{remark} \begin{remark} Each of the categories defined above has a weakly initial element similar to the ``universal dynamical system" of \cite{riehl2017category}. For example, the system $(\sigma, \mathbb{Z})$ defined by \[
\sigma(n) = n+1 \] for all $n \in \mathbb{Z}$ is weakly initial in the category $\bf{Map}.$ Further, the set $\mathbf{Map}\left( (\sigma, \mathbb{Z}), (f, X)\right)$ is isomorphic to the set of all orbits of $(f, X).$ In particular, a morphism $h: (\sigma, \mathbb{Z}) \to (f, X)$ corresponds to an orbit with period $m \in \mathbb{N}$ if and only if $h$ admits the following factorization: \[ \begin{diagram}
\node{(\sigma, \mathbb{Z})}
\arrow{s,t}{}
\arrow{se,t}{h}\\
\node{(\sigma, \mathbb{Z}/ m \mathbb{Z})}
\arrow{e,t}{\bar h}
\node{(f, X)} \end{diagram} \] Similar constructions can be carried out for $\bf{Flow}$ or $\bf{WFlow}.$ \end{remark}
\section{Topological transversality and global Poincar\'e section}\label{sec_poincare} In this section, we define the concept of topological transversality for continuous flows on topological manifolds. Based on this definition, we show that Poincar\'e maps can be defined in a manner similar to the classical smooth case. Additionally, we introduce categories of flows with global Poincar\'e sections to consider the functoriality of the construction of Poincar\'e maps.
We adopt the definition of topological transversality given in \cite{phdthesis, basener2004every} with a certain modification. \begin{definition}\label{Top_trans} Let $(\Phi, X)$ be a flow, where $X$ is an $n$-dimensional topological manifold. A submanifold $S \subset X$ without a boundary is \emph{topologically transversal} to $\Phi$ if \begin{enumerate}
\item $S$ is codimension one and locally flat.
\item For each $x \in S,$ there exists a neighborhood $U$ of $x$ in $X$ and a homeomorphism $\phi: U \to B \subset \mathbb{R}^n,$ where $B$ is the unit ball such that $\Phi\left(U\cap S \right) = B \cap \mathbb{R}^{n-1}\times \{0\}.$ Further, there exist $\delta_{+}(x) >0$ and $\delta_{-}(x) <0$ such that $\Phi(x, [\delta_{-}(x),0))$ and $\Phi(x, (0,\delta_{+}(x)])$ are contained in different connected components of $U \backslash S$ and $\Phi(x, [\delta_{-}(x),\delta_{+}(x)])) \cap S = \{x\}.$
Here, $\delta_{+}$ and $\delta_{-}$ can be taken locally uniformly, that is, there exist a neighborhood $V \subset U$ of $x$ and $\delta >0$ such that $\delta_+(y) > \delta$ and $\delta_-(y) < - \delta$ for all $y \in V\cap S.$
\item For each set of the form $\Phi(y, [a,b]),$ where $y \in X$ and $a, b \in \mathbb{R},$ $\Phi(y, [a,b])\cap S$ is compact in $S$. \end{enumerate} \end{definition}
\begin{lemma}\label{lem_intersection} Let $(\Phi, X)$ be a flow and $S \subset X$ be topologically transversal to $\Phi$. Then, for each $x\in S$ and $\epsilon>0,$ there exists an open neighborhood $V$ of $x$ in $X$ such that \[
\Phi(y, [ - \epsilon, \epsilon]) \cap S \neq \emptyset \]
for all $y \in V.$ \end{lemma} \begin{proof} Let $U$ be a neighborhood of $x$ satisfying the condition of (2) in Definition \ref{Top_trans}. By the continuity of $\Phi$, there exist a neighborhood $V_0$ of $x$ and $\delta > 0$ such that $\Phi(V_0, [-\delta, \delta]) \subset U$ and $\delta < \min\left(\epsilon, \delta_+(x), -\delta_-(x)\right).$ Then, we have $\Phi(x, \delta), \Phi(x, -\delta) \in U\backslash S.$ We take neighborhoods $V_+$ and $ V_-$ of $\Phi(x, \delta)$ and $ \Phi(x, -\delta),$ respectively, such that $V_+ \subset U\backslash S$ and $V_- \subset U\backslash S.$ Let $V := \Phi(V_+, -\delta) \cap \Phi(V_-, \delta) \cap V_0.$ By considering the $n$-th coordinate of the homeomorphism $\phi$, we obtain that $\Phi(y, [-\epsilon, \epsilon]) \cap S \neq\emptyset$ for all $y \in V.$ \end{proof}
The next lemma excludes the possibility of sequences that return to the section very frequently. \begin{lemma}\label{no_freq_seq} Let $(\Phi, X)$ be a flow and $S \subset X$ be topologically transversal to $\Phi$. If $x \in S$, there exist no sequences $x_n \in S$ and $t_n >0$ such that $x_n \to x$ and $t_n \to 0$ as $n\to \infty$ and $\Phi(x_n, t_n) \in S.$ \end{lemma} \begin{proof} Let $U$ be a neighborhood of $x$ satisfying condition (2) in Definition \ref{Top_trans}. Let $V \subset U$ be a neighborhood of $x$ such that there exists $\delta >0$ with $\delta_+(y) > \delta$ for all $y \in V \cap S.$ If $x_n \in S$ and $t_n >0$ are sequences such that $x_n \to x$ and $t_n \to 0$ as $n\to \infty$ and $\Phi(x_n, t_n) \in S,$ then $x_n \in V \cap S$ and consequently $t_n > \delta$ for a sufficiently large $n$. This is a contradiction. \end{proof} \begin{definition} Let $(\Phi, X)$ be a flow. A submanifold $S \subset X$ is a \emph{global Poincar\'e section} if \begin{enumerate}
\item $S$ is topologically transversal to $\Phi.$
\item For each $x \in X,$ there exists $t_+ >0$ and $t_- < 0$ such that $\Phi(x, t_{+}) \in S$ and $\Phi(x, t_{-}) \in S$. \end{enumerate} \end{definition} \begin{remark} If the phase space is compact, condition (2) can be weakened to the condition that each $x \in X$ has $t \in \mathbb{R}$ such that $\Phi(x, t) \in S.$ Indeed, let $x \in S$ and consider the $\omega$-limit set of $x.$ Then, $\omega(x) \cap S$ is nonempty by the invariance of the limit set. By Lemma \ref{lem_intersection}, we observe that there exists $t_+ > 0$ such that $\Phi(x, t_{+}) \in S.$ The existence of $t_-$ is proved similarly. \end{remark} \begin{remark} By definition, a flow with a global Poincar\'e section has no equilibrium points. By using the argument in \cite{basener2002global}, we can show that a smooth flow without equilibrium points has a global Poincar\'e section if the phase space is compact. \end{remark} According to these definitions, we have the following generalization of well-known results. \begin{theorem}\label{P_thm} Let $(\Phi, X)$ be a flow and $S \subset X$ be topologically transversal to $\Phi$. If $x_0 \in S$ and there exists $t_+ > 0$ such that $\Phi(x_0,t_+) \in S,$ there exist a neighborhood $U$ of $x_0$ in $X$ and continuous maps $P\Phi: U\cap S \to S$ and $T_\Phi:U\cap S \to (0,\infty)$ such that \[ P\Phi(x) = \Phi(x, T_\Phi(x))\] for each $x \in U\cap S$ and $\Phi(x, t) \not \in S$ for $0<t<T_\Phi(x).$ Further, if $S$ is a global Poincar\'e section, $P \Phi$ is defined on the entire $S$, and it is a homeomorphism. \end{theorem} \begin{proof} First, we show the existence of $T_\Phi(x)$ and $P \Phi(x)$ for each $x \in U\cap S,$ where $U$ is a neighborhood of $x_0$ in $X.$ Let $0<r< t_+$ and take a neighborhood $V$ of $\Phi(x_0,t_+)$ by applying Lemma \ref{lem_intersection} with $\epsilon = r$ and $x = \Phi(x_0,t_+).$ Let $U := \Phi(V, -t_+).$ Then, we have \[
\Phi(y, [t_+ -r, t_+ +r]) \cap S \neq \emptyset \] for all $y \in U.$ We define \[
\mathcal{T}(x) := \{t >0 \mid \Phi(x,t) \in S\} \] for each $x \in U\cap S.$ By the choice of $U$, $\mathcal{T}(x)$ is nonempty. If we set $T_\Phi(x) := \inf \mathcal{T}(x),$ we have $T_\Phi(x) \geq \delta_+(x)>0.$ Let $t_n \in \mathcal{T}(x)$ be a sequence with $t_n \to T_\Phi(x) $ as $n\to \infty.$ For a sufficiently large $a >0,$ we have $\Phi(x, t_n) \in S \cap \Phi(x, [0,a])$ for all $n.$ Therefore, $\Phi(x, T_\Phi(x)) \in S$ from the continuity of $\Phi$ and the compactness of $S \cap \Phi(x, [0,a]).$ These results indicate that $T_\Phi(x)$ has the desired properties. We set $P\Phi(x) := \Phi(x, T_\Phi(x)).$
Let us now show that $T_\Phi : U\cap S \to (0,\infty)$ is continuous. Let $x \in U\cap S$ and $\epsilon $ be an arbitrary positive number less than $T_\Phi(x)$.
By Lemma \ref{lem_intersection}, there exists an open neighborhood $V_1 \subset U$ of $P\Phi (x) = \Phi(x, T_\Phi(x))$ such that $\Phi(y, [ - \epsilon, \epsilon]) \cap S \neq \emptyset$ for all $y \in V_1.$ By the continuity of $\Phi,$ there exists an open neighborhood $U_1$ of $x$ such that $\Phi(U_1, T_\Phi(x)) \subset V_1$. Therefore, we have \[
\Phi(y, [T_\Phi(x) - \epsilon, T_\Phi(x) + \epsilon]) \cap S \neq \emptyset \] for all $y \in U_1.$
We show that there exists an open neighborhood $U_2 \subset U$ of $x$ such that \[\Phi(y, (0, T_\Phi(x) - \epsilon)) \cap S = \emptyset\]
for all $y \in U_2 \cap S.$ If this is not the case, we may take sequences $x_n \in S \cap U$ and $s_n \in (0, T_\Phi(x) - \epsilon)$ so that $\Phi(x_n, s_n) \in S$ and $x_n \to x$ as $n \to \infty.$ As $s_n \in [0, T_\Phi(x) - \epsilon],$ we may take a convergent subsequence $s_{n_i} \to s \in [0, T_\Phi(x) - \epsilon]$ as $i \to \infty.$ Using the continuity of $\Phi$, we observe that $s = 0.$ Thus, we obtain sequences $y_n \in S\cap U$ and $t_n \in (0, T_\Phi(x) - \epsilon)$ so that $y_n \to x$ and $t_n \to 0$ as $n \to \infty.$ However, this is impossible by Lemma \ref{no_freq_seq}.
Therefore, there exists an open neighborhood $U_0:= U_1 \cap U_2 \subset U$ of $x$ such that $\Phi(y, (0, T_\Phi(x) - \epsilon)) \cap S = \emptyset$ and $\Phi(y, [T_\Phi(x) - \epsilon, T_\Phi(x) + \epsilon]) \cap S \neq \emptyset$ for all $y \in U_0 \cap S.$ Together, these imply \[
T_\Phi(x) - \epsilon \leq T_\Phi(y) \leq T_\Phi(x) + \epsilon \] for all $y \in U_0 \cap S.$ Therefore, $T_\Phi(x)$ is continuous and consequently $P \Phi(x)$ is also continuous.
If $S$ is a global Poincar\'e section, it is clear that $T_\Phi$ and $P \Phi $ are defined on the entire $S$. By the definition of a global Poincar\'e section, the same constructions can be carried out for $\Psi(x,t) := \Phi(x, -t).$ Then, we have \[
\begin{aligned}
T_\Psi &= T_\Phi \circ P\Psi\\
T_\Phi &= T_\Psi \circ P \Phi.
\end{aligned} \] These are established as follows. For $x \in S,$ we have $\Phi(P\Psi(x), T_\Psi(x)) = x \in S$ by definition. Therefore, $T_\Psi(x) \geq T_\Phi(P\Psi(x)).$ On the other hand, we have $\Phi(x, t) \not \in S$ for $-T_\Psi(x)<t<0$ by definition of $T_\Psi(x)$. This implies $\Phi(P\Psi(x),t) \not \in S$ for $0<t<T_\Psi(x).$ Therefore, $T_\Psi(x) \leq T_\Phi(P\Psi(x)).$ The other relation is obtained by symmetry. Now, we have $(P\Phi)^{-1} = P\Psi$ because \[ \begin{aligned}
P\Phi(P\Psi(x)) &= \Phi(P\Psi(x), T_\Phi(P\Psi(x))) = \Phi(P\Psi(x), T_\Psi(x)) =x\\
P\Psi(P\Phi(x)) &= \Psi(P\Phi(x), T_\Psi(P\Phi(x))) = \Psi(P\Phi(x), T_\Phi(x)) =x \end{aligned} \] for all $x \in S.$
\end{proof} \begin{corollary}\label{div_sum} Let $(\Phi, X)$ be a flow with a global Poincar\'e section $S.$ If $x \in S,$ we have \[
\begin{aligned}
\sum_{n=0}^\infty T_\Phi \circ (P\Phi)^n(x) & = \infty \\
\sum_{n=1}^{\infty} T_\Phi \circ (P\Phi)^{-n}(x) & = \infty.
\end{aligned} \] \end{corollary} \begin{proof} As we have $T_\Psi = T_\Phi \circ P\Psi =T_\Phi \circ (P\Phi)^{-1}$, it is sufficient to prove the first formula. Suppose the series is convergent and let the sum $T_\infty$ and $x_\infty:= \Phi(x, T_\infty).$ Because we have \[
\Phi\left(x, \sum_{n=0}^{N-1} T_\Phi \circ (P\Phi)^n(x)\right) = (P\Phi)^{N} (x) \] for each $N \geq 1,$ $x_n := (P \Phi)^n(x)$ converges to $x_\infty.$ For all $n,$ $x_n$ is contained in $\Phi(x,[0, T_\infty])\cap S,$ which is compact by the definition of topological transversality. Therefore, $x_\infty \in S.$ If we set $t_n := T_\Phi \circ (P\Phi)^n (x),$ the convergence of the sum implies $t_n \to 0$ and $\Phi(x_n, t_n) \in S$ for all $n$ by definition. Thus, we have a pair of sequences $(x_n, t_n),$ which does not exist by Lemma \ref{no_freq_seq}. This is a contradiction. \end{proof} A flow may admit many different global Poincar\'e sections, and consequently, a pair of a flow and a section may not necessarily be preserved under a weak morphism. If the sections are preserved by a weak morphism as sets, we have the following correspondence of the first return times between two flows. \begin{lemma}\label{lem_period} Let $(\Phi, X)$ and $(\Psi, Y)$ be flows with global Poincar\'e sections $S$ and $S',$ respectively, and $(h, \tau): (\Phi, X) \to (\Psi, Y)$ be a weak morphism. Then, \begin{enumerate}
\item If $h(S) \subset S',$ $T_\Psi(h(x)) \leq \tau(x, T_\Phi(x))$ for all $x \in S.$
\item If $h^{-1} (S') \subset S,$ $\tau(x, T_\Phi(x)) \leq T_\Psi(h(x))$ for all $x \in h^{-1} (S').$ \end{enumerate} In particular, if $S = h^{-1} (S'),$ we have $T_\Psi(h(x)) = \tau(x, T_\Phi(x))$ for all $x \in S.$ \end{lemma} \begin{proof} (1) Let $x \in S.$ As we have $\Phi(x, T_\Phi(x)) \in S,$ \[
\Psi(h(x), \tau(x, T_\Phi(x))) = h(\Phi(x, T_\Phi(x))) \in h(S) \subset S'. \] Because $h(x) \in S',$ it follows that $T_\Psi(h(x)) \leq \tau(x, T_\Phi(x)).$
(2) Let $x \in h^{-1} (S').$ We have $\Psi(h(x), T_\Psi(h(x))) \in S'$ and $T_\Psi(h(x)) = \tau(x, t_x)$ for some $t_x \in \mathbb{R}$ because $\tau(x, -)$ is a homeomorphism. Therefore, we have \[
\Psi(h(x), T_\Psi(h(x))) = h(\Phi(x, t_x)) \in S', \] which implies $\Phi(x, t_x) \in h^{-1} (S') \subset S.$ Thus, we obtain $T_\Phi(x) \leq t_x.$ Because $\tau(x, -)$ is monotonically increasing, we conclude that \[
\tau(x, T_\Phi(x)) \leq \tau(x, t_x) = T_\Psi(h(x)), \] which is the desired property. \end{proof} As a consequence of this lemma, we have the following result. \begin{lemma}\label{lem_func}
Let $(\Phi, X)$ and $(\Psi, Y)$ be flows with global Poincar\'e sections $S$ and $S',$ respectively, and $(h, \tau): (\Phi, X) \to (\Psi, Y)$ be a weak morphism such that $h^{-1} S' = S.$ Then, we have a morphism of map dynamical systems $h|_S: (P\Phi,S) \to (P\Psi,S'),$ where $h|_S : S \to S'$ is the restriction of $h$ to $S.$ \end{lemma} \begin{proof} As $P\Phi: S \to S$ and $P\Psi: S' \to S'$ are homeomorphisms, they define map dynamical systems. For each $x \in S,$ we have \[ \begin{aligned}
h\circ P\Phi(x)&= h(\Phi(x, T_\Phi(x))) \\
&= \Psi(h(x), \tau(x, T_\Phi(x)))\\
&= \Psi(h(x), T_\Psi(h(x))) = P \Psi \circ h(x).
\end{aligned} \]
Therefore, $h|_S: (P\Phi,S) \to (P\Psi,S')$ is a morphism of map dynamical systems. \end{proof} Thus, we may define the following: \begin{definition} Let $(\Phi, X)$ and $(\Psi, Y)$ be flows with global Poincar\'e sections $S$ and $S',$ respectively. A morphism $(h, \tau): (\Phi, X)\to(\Psi, Y)$ in $\bf{WFlow}$ is said to \emph{preserve} the global Poincar\'e sections if $S = h^{-1} (S').$
The \emph{category of flows with global Poincar\'e sections} $\bf{FlowGS}$ is the category whose objects are flows with global Poincar\'e sections and whose morphisms are morphisms in $\bf{Flow}$, which preserves the global Poincar\'e sections.
Similarly, we may define a category $\bf{WFlowGS}$ whose objects are flows with global Poincar\'e sections and whose morphisms are morphisms in $\bf{WFlow}$, which preserves the global Poincar\'e sections.
Objects in $\bf{WFlowGS}$ or $\bf{FlowGS}$ are denoted by a triple of the form $(\Phi, X, S),$ where $(\Phi, X)$ is a flow with a global Poincar\'e section $S.$ \end{definition} From Lemma \ref{lem_func}, we immediately obtain the following: \begin{theorem} The construction of a Poincar\'e map is functorial for $\bf{WFlowGS}.$ That is, there exists a functor $P: \bf{WFlowGS} \to \bf{Map}$ defined by setting \begin{itemize}
\item $P(\Phi, X, S) = (P \Phi, S)$ for each object $(\Phi, X, S)$ in $\bf{WFlowGS}.$
\item For each morphism $h:(\Phi_1, X_1, S_1) \to (\Phi_2, X_2, S_2)$ $P(h) = h|_{S_1}: (P \Phi_1, S_1) \to (P \Phi_2, S_2).$ \end{itemize} \end{theorem} \begin{corollary} The construction of a Poincar\'e map is functorial for $\bf{FlowGS}.$ \end{corollary} \begin{proof} Take the composition of $P : \bf{WFlowGS} \to \bf{Map}$ with the inclusion functor $\bf{FlowGS}\hookrightarrow \bf{WFlowGS}.$ \end{proof}
\section{Poincar\'e maps and suspension flows}\label{sec_ps} In this section, we consider the categorical relationship between a Poincar\'e map and a suspension.
To establish the notation, we recall the definition of a suspension flow. \begin{definition} Let $f:X \to X$ be a homeomorphism on a topological manifold $X$. The \emph{mapping torus} $X_f$ of $f$ is the manifold defined by \[
X_f := X \times \mathbb[0,1]/ \sim, \] where $\sim$ is the smallest equivalence relation with $(x, 1) \sim (f(x), 0)$ for each $x \in X.$ There is a natural surjection $\pi_f: X \times \mathbb[0,1] \to X_f,$ which sends each point to the corresponding equivalence class. We denote a point in $X_f$ by $[x,t],$ where $x \in X,$ $0\leq t <1.$ \end{definition} \begin{definition} Let $(f, X)$ be a map dynamical system. The \emph{suspension flow} $\Sigma f: X_f \times \mathbb{R} \to X_f$ of $(f, X)$ is defined by \[
\Sigma f([x,t], s) := [f^n (x), s+t -n], \] where $x \in X,$ $0\leq t <1$ and $n \in \mathbb{Z}$ is a unique integer satisfying $s+t -1 < n \leq s+t.$ \end{definition} \begin{theorem} The construction of a suspension flow is functorial. That is, there exists a functor $\Sigma: \bf{Map} \to \bf{FlowGS}$ defined by setting \begin{itemize}
\item $\Sigma(f, X) = (\Sigma f, X_f, (X_f)_0 )$ for each object $(f, X)$ in $\bf{Map}.$
\item For each morphism $h:(f, X) \to (g, Y),$ we set
\[\Sigma(h) = \bar h: (\Sigma f, X_f, (X_f)_0 ) \to (\Sigma g, Y_g, (Y_g)_0 ),\]
where $\bar h([x,t]) = [h(x), t]$ and $(X_f)_0 = \{[x,0] \mid x \in X\}$ and $(Y_g)_0 = \{[y,0] \mid y \in Y\}.$ \end{itemize} \end{theorem} \begin{proof} Let $h:(f, X) \to (g, Y)$ be a morphism in $\bf{Map}.$ First, we show that $\bar h: X_f \to Y_g$ is well-defined and continuous. Well-definedness is verified by a direct calculation using $g\circ h = h \circ f.$ The continuity follows from the commutativity of the following diagram and the universal property of the quotient topology: \[ \begin{diagram}
\node{X\times [0,1]}
\arrow{e,t}{\pi_f}
\arrow{s,t}{h\times \rm{id}}
\node{X_f}
\arrow{s,t}{\bar h} \\
\node{Y\times[0,1]}
\arrow{e,t}{\pi_g}
\node{Y_g} \end{diagram} \] We show that $\bar h$ commutes with suspension flows. This is verified by a direct calculation: \[ \begin{aligned}
\bar h \left( \Sigma f([x,t], s) \right) &= \bar h \left( [f^n (x), s+t -n] \right)\\
&= [h (f^n (x)), s+t -n]\\
&= [g^n(h(x)), s+t -n]\\
&= \Sigma g([h(x),t], s)\\
& = \Sigma g(\bar h([x,t]), s) \end{aligned} \] where $x \in X,$ $0\leq t <1$ and $n \in \mathbb{Z}$ is a unique integer satisfying $s+t -1 < n \leq s+t.$
We show that $\bar h$ preserves the sections, that is, $\bar h^{-1} (Y_g)_0 = (X_f)_0.$ By definition, we have $\bar h (X_f)_0 \subset (Y_g)_0,$ so $(X_f)_0 \subset \bar h^{-1} (Y_g)_0 .$ Conversely, if $[x,t] \in \bar h^{-1} (Y_g)_0$ with $0\leq t < 1,$ then $t =0$ and therefore, $[x, t] \in (X_f)_0.$
Finally, we show that $\Sigma$ is a functor. It is clear that $\Sigma(1_{(f,X)}) = 1_{(\Sigma f, X_f, (X_f)_0 ) }.$ If $h_1: (f_1, X_1) \to (f_2, X_2) $ and $h_2: (f_2, X_2) \to (f_3, X_3)$ are morphisms in $\bf{Map},$ we have $\overline{(h_2 \circ h_1)} = \bar h_2 \circ \bar h_1.$ \end{proof} Now, we have three categories and three functors between them: \begin{enumerate}
\item Inclusion functor $I : \bf{FlowGS} \to \bf{WFlowGS}$.
\item Poincar\'e map functor $P : \bf{WFlowGS} \to \bf{Map}$.
\item Suspension functor $\Sigma : \bf{Map} \to \bf{FlowGS}$. \end{enumerate} From the existence of these functors, we immediately recover some known results on the preservation of isomorphisms. \begin{theorem} Each of the following statements holds. \begin{enumerate}
\item If two flows with global Poincar\'e sections are topologically equivalent, there is a pair of global Poincar\'e sections such that the Poincar\'e maps are topologically conjugate.
\item If two maps are topologically conjugate, their suspension flows are topologically conjugate. \end{enumerate} \end{theorem} At this point, we must consider the degree of difference between the original flow and the suspension flow of the Poincar\'e map. First, we note that there is a pair of flows that are topologically equivalent but not topologically conjugate. The following is a modification of an example in \cite{pilyugin2019spaces}. \begin{example}
We define two flows on $A = \{z \in \mathbb{C} \mid 1<|z|< 2\}$ by
\[
\begin{aligned}
\Phi_1(z, t) &:= z e^{i\pi t}\\
\Phi_2(z, t) &:= z e^{2 \pi it}.
\end{aligned}
\]
Then, they are topologically equivalent but not topologically conjugate. \end{example} \begin{proof} Topological equivalence is obvious. Suppose there is a homeomorphism $h: A \to A$ such that $\Phi_1$ and $\Phi_2$ are topologically conjugate, that is, \[
h\left( z e^{i\pi t}\right) = h(z) e^{2 \pi i t} \] for all $z \in A$ and $t \in \mathbb{R}.$ By considering $t= 1,$ we obtain $h(z) = h(-z)$ for all $z\in A,$ which contradicts the condition that $h$ is injective. \end{proof} Note that we may take $A_0 = \{x \mid 0< x<1\}$ as a global Poincar\'e section for these flows. With this choice, the Poincar\'e map is the identity $\rm{id}_{A_0}$ in either case. Further, the suspension flow for $\rm{id}_{A_0}$ coincides with $\Phi_1.$ Thus, the suspension flow of a Poincar\'e map is not necessarily topologically conjugate with the original flow. On the other hand, topological equivalence can be established.
\begin{lemma}\label{lem_adj_1} There is a natural transformation $(k,\tau) : I \Sigma P I\to I$ defined by the following for each $(\Phi, X, S)$ in $\bf{FlowGS}:$ \[ \begin{aligned}
k_{(\Phi, X, S)} ([x,t]) &:= \Phi(x, t T_\Phi(x))\\
\tau_{(\Phi, X, S)} ([x,t],s)&:= \int_0^{s+t } R_{\Phi}(x)(u) d u -tT_\Phi(x), \end{aligned} \] where $x \in S,$ $0\leq t <1$ and \[
R_{\Phi}(x)(u) := \sum_{i \in \mathbb{Z} } T_\Phi( (P\Phi)^i (x)) \chi_{[i, i+1)}(u), \] where $\chi_{[i, i+1)}$ is the indicator function of $[i, i+1).$ \end{lemma} \begin{proof} First, we show that $(k,\tau)_{ (\Phi, X, S)}: I \Sigma P I(\Phi, X, S) \to (\Phi, X, S)$ is well-defined as a weak morphism in $\bf{WFlowGS}$. Well-definedness and continuity of $k_{(\Phi, X, S)} $ follow from the commutativity of the following diagram: \[ \begin{diagram}
\node{S\times [0,1]}
\arrow{e,t}{\pi_{P\Phi}}
\arrow{se,r}{K}
\node{S_{P\Phi}}
\arrow{s,r}{k_{ (\Phi, X, S)}}\\
\node{}
\node{X} \end{diagram} \] where $K:S\times [0,1] \to X$ is defined by $K(x,t) := \Phi(x, t T_\Phi(x))$ for each $(x,t) \in S\times [0,1].$
By definition, we have $\tau_{(\Phi, X, S)} ([x,t],0) = 0.$ Because $R_{\Phi}(x)(-) $ is a positive-valued function, $\tau_{(\Phi, X, S)} ([x,t],-):\mathbb{R} \to \mathbb{R}$ is strictly monotonous. By Corollary \ref{div_sum}, it is also a surjection. Thus, $\tau_{(\Phi, X, S)} ([x,t],-)$ is a homeomorphism.
We check that $k_{(\Phi, X, S)}$ commutes with the flows by a direct calculation. When $n \geq 0,$ where $n \in \mathbb{Z}$ is a unique integer satisfying $s+t -1 < n \leq s+t,$ we have \[ \begin{aligned}
k_{(\Phi, X, S)} \left(\Sigma P \Phi([x,t],s) \right)& = k_{(\Phi, X, S)}\left( [(P \Phi)^n (x) ,s+t-n]\right)\\
& = \Phi((P \Phi)^n (x), (s+t - n) T_\Phi \circ (P \Phi)^n (x))\\
& = \Phi\left(x, \sum_{i=0}^{n-1}T_\Phi \circ (P \Phi)^i (x) + (s+t - n) T_\Phi \circ (P \Phi)^n (x)\right)\\
& = \Phi\left(x, \int_0^{s+t } R_{\Phi}(x)(u) d u\right)\\
& = \Phi\left(k_{(\Phi, X, S)}([x,t]),\tau_{(\Phi, X, S)} ([x,t],s)\right). \end{aligned} \] Noting that $\Phi((P \Phi)^{-1} (x), t) = \Phi(x, t -T_\Phi \circ (P \Phi)^{-1} (x)),$ we calculate the following for $n \leq -1$: \[ \begin{aligned}
k_{(\Phi, X, S)} \left(\Sigma P \Phi([x,t],s) \right)& = k_{(\Phi, X, S)}\left( [(P \Phi)^n (x) ,s+t-n]\right)\\
& = \Phi((P \Phi)^n (x), (s+t - n) T_\Phi \circ (P \Phi)^n (x))\\
& = \Phi\left(x, \sum_{i=1}^{-n+1}T_\Phi \circ (P \Phi)^{-i} (x) + (s+t - n-1) T_\Phi \circ (P \Phi)^n (x)\right)\\
& = \Phi\left(x, \int_0^{s+t } R_{\Phi}(x)(u) d u\right)\\
& = \Phi\left(k_{(\Phi, X, S)}([x,t]),\tau_{(\Phi, X, S)} ([x,t],s)\right). \end{aligned} \] The condition that $k_{(\Phi, X, S)}^{-1} S = (S_{P\Phi})_0$ can be verified by a direct calculation.
Finally, we show that $(k,\tau)$ is natural. Let $h:(\Phi_1, X_1, S_1) \to (\Phi_2, X_2, S_2)$ be a morphism in $\bf{FlowGS}.$ Then, we have \[ \begin{aligned}
h\circ k_{(\Phi_1, X_1, S_1)} ([x,t]) &= h \circ \Phi_1(x, t T_{\Phi_1}(x)) \\
&= \Phi_2(h(x), t T_{\Phi_2}(x)) \\
&= k_{(\Phi_2, X_2, S_2)} ([h(x),t]) \\
&= k_{(\Phi_2, X_2, S_2)} \circ \overline{h|_{S_1}}([x,t]), \end{aligned} \] where $x \in S_1$ and $0 \leq t <1.$ We also have \[ \begin{aligned}
\tau_{(\Phi_1, X_1, S_1)} ([x,t], s) &= \int_0^{s+t } R_{\Phi_1}(x)(u) d u -tT_{\Phi_1}(x) \\
&= \int_0^{s+t } R_{\Phi_2}(h(x))(u) d u -tT_{\Phi_2}(h(x))\\
& = \tau_{(\Phi_2, X_2, S_2)} ( \overline{h|_{S_1}}( [x,t]), s), \end{aligned} \] using $T_{\Phi_2}(h(x)) = T_{\Phi_1}(x).$ \end{proof} \begin{remark}\label{rem_bij} The map $k_{(\Phi, X, S)} $ is bijective. Surjectivity is obvious. For injectivity, if $\Phi(x, t T_\Phi(x)) = \Phi(x', t' T_\Phi(x'))$ for $x, x' \in S$ and $t,t' \in [0,1),$ we have \[
x' = \Phi(x, t T_\Phi(x)-t' T_\Phi(x')) \in S. \] If $t T_\Phi(x)-t' T_\Phi(x')>0,$ then it follows that $t T_\Phi(x)-t' T_\Phi(x') \geq T_\Phi(x).$ Because $t <1, $ this is a contradiction. Therefore, $t T_\Phi(x)-t' T_\Phi(x') \leq 0.$ By interchanging $t$ and $x$ with $t'$ and $x'$, we also have $t' T_\Phi(x')-t T_\Phi(x) \leq 0.$ Therefore, we conclude that $x = x'$ and consequently $t= t'.$ \end{remark} Using the invariance of domain theorem, we observe that $(k,\tau) : I \Sigma P I\to I$ is a natural isomorphism. In ordinary terms, this observation can be phrased as follows. \begin{corollary} If $(\Phi, X, S)$ is a flow with a global Poincar\'e section, then $(\Phi, X, S)$ is topologically equivalent to $\Sigma P (\Phi, X, S).$ \end{corollary} Another natural transformation can be constructed. \begin{lemma} There is a natural transformation $l : 1_{\bf{Map}} \to P I \Sigma$ defined by \[
l_{(f,X)} (x) : = [x,0] \] for each $x \in X.$ \end{lemma} \begin{proof} First, we show that $l_{(f,X)} : (f,X) \to P I \Sigma (f,X)$ is well-defined as a morphism in $\bf{Map}.$ As $l_{(f,X)}$ is a composition of continuous maps, it is well-defined and continuous. Additionally, we have \[
l_{(f,X)} \circ f(x) = [f(x), 0] = \Sigma f([x,0],1) = P I \Sigma f ([x,0]) = (P I \Sigma f) \circ l_{(f,X)} (x) \] for all $x \in X.$
We show that $l$ is natural. Let $h:(f,X) \to (g, Y)$ be a morphism in $\bf{Map}.$ Then, \[
(P I \Sigma)(h) \circ l_{(f,X)}(x) = [h(x) , 0] = l_{(g,Y)} \circ h(x)
\]
for all $x \in X.$ \end{proof}
These results suggest that there is another category larger than $\bf{FlowGS}$ and smaller than $\bf{WFlowGS}$ for which the constructions of Poincar\'e maps and suspensions become adjoint. \begin{definition} A weak morphism $(h,\sigma): (\Phi_1, X_1, S_1) \to (\Phi_2, X_2, S_2)$ in $\bf{WFlowGS}$ is \emph{rate-preserving} if \[ \begin{aligned}
&\sigma\left( \Phi_1(x, t T_{\Phi_1}(x)) , \int_0^{s+t } R_{\Phi_1}(x)(u) d u -tT_{\Phi_1}(x) \right)\\
&= \int_0^{s+t } R_{\Phi_2}(h(x))(u) d u -tT_{\Phi_2}(h(x))
\end{aligned} \] for all $x \in S_1,$ $0 \leq t < 1$ and $s \in \mathbb{R},$ where $R_{\Phi_1}$ and $R_{\Phi_2}$ are the same as in Lemma \ref{lem_adj_1}. \end{definition} \begin{lemma} If $(h,\sigma): (\Phi_1, X_1, S_1) \to (\Phi_2, X_2, S_2)$ in $\bf{WFlowGS}$ is rate-preserving, we have \[
\sigma(x, t T_\Phi(x)) = t \sigma(x, T_\Phi(x)) \] for all $x \in S_1$ and $0 \leq t < 1.$ \end{lemma} \begin{proof} We show this by a direct calculation. Let $x \in S_1$ and $0 \leq t < 1.$ Then, we have \[
\begin{aligned}
\sigma(x, t T_{\Phi_1}(x)) &= \sigma\left( \Phi_1(x, 0\cdot T_{\Phi_1}(x)) , \int_0^{t+0 } R_{\Phi_1}(x)(u) d u -0\cdot T_{\Phi_1}(x) \right) \\
&= \int_0^{t+0 } R_{\Phi_2}(h(x))(u) d u -0\cdot T_{\Phi_2}(h(x)) \\
& = t T_{\Phi_2}(h(x)) = t \sigma(x, T_{\Phi_1}(x)).
\end{aligned} \] Here, we used the result of Lemma \ref{lem_period}. \end{proof} \begin{lemma} The identity morphism in $\bf{WFlowGS}$ is rate-preserving. The composition of two rate-preserving morphisms is again rate-preserving. \end{lemma} \begin{proof} The first statement is obvious. Let $(h_1,\sigma_1): (\Phi_1, X_1, S_1) \to (\Phi_2, X_2, S_2)$ and $(h_2,\sigma_2): (\Phi_2, X_2, S_2) \to (\Phi_3, X_3, S_3)$ be rate-preserving morphisms. Then, we have \[ \begin{aligned}
&\sigma_2 \circ \sigma_1 \left(\Phi_1(x, t T_{\Phi_1}(x)) , \int_0^{s+t } R_{\Phi_1}(x)(u) d u -tT_{\Phi_1}(x) \right)\\
& = \sigma_2 \left(h_1\left(\Phi_1(x, t T_{\Phi_1}(x)) \right), \sigma_1\left( \Phi_1(x, t T_{\Phi_1}(x)) , \int_0^{s+t } R_{\Phi_1}(x)(u) d u -tT_{\Phi_1}(x) \right) \right)\\
& = \sigma_2 \left(h_1\left(\Phi_1(x, t T_{\Phi_1}(x))\right), \int_0^{s+t } R_{\Phi_2}(h_1(x))(u) d u -tT_{\Phi_2}(h_1(x)) \right)\\
& = \sigma_2 \left(\Phi_2(h_1(x), \sigma_1(x, t T_{\Phi_1}(x))), \int_0^{s+t } R_{\Phi_2}(h_1(x))(u) d u -tT_{\Phi_2}(h_1(x)) \right)\\
& = \sigma_2 \left(\Phi_2(h_1(x), t\sigma_1(x, T_{\Phi_1}(x))), \int_0^{s+t } R_{\Phi_2}(h_1(x))(u) d u -tT_{\Phi_2}(h_1(x)) \right)\\
& = \int_0^{s+t } R_{\Phi_3}(h_2\circ h_1(x))(u) d u -tT_{\Phi_3}(h_2\circ h_1(x)),
\end{aligned} \] for all $x \in S_1,$ $0 \leq t < 1$ and $s \in \mathbb{R}.$ \end{proof} Therefore, we can define a category $\bf{RWFlowGS},$ whose objects are flows with global Poincar\'e sections and whose morphisms are rate-preserving morphisms. If we denote the inclusion functors by $J^-: \bf{FlowGS} \to \bf{RWFlowGS}$ and $J^+: \bf{RWFlowGS} \to \bf{WFlowGS}$, it is clear that $I = J^+J^-.$
\begin{lemma} Let $(\Phi, X, S)$ be an object in $\bf{FlowGS}.$ Then, the weak morphism $(k,\tau)_{ (\Phi, X, S)}: I \Sigma P I(\Phi, X, S) \to (\Phi, X, S)$ is rate-preserving. \end{lemma} \begin{proof} First, we note that \[
\int_0^{s+t } R_{\Sigma P \Phi}([x, 0])(u) d u -tT_{\Sigma P \Phi}([x,0]) = s \] for all $[x,0] \in (S_{P \Phi})_0,$ $0 \leq t <1$ and $s \in \mathbb{R}$ because $T_{\Sigma P \Phi}([x,0]) = 1$ for all $[x,0] \in (S_{P \Phi})_0.$ We calculate the following: \[ \begin{aligned}
&\tau_{(\Phi, X, S)}\left( \Sigma P \Phi([x,0], t T_{\Sigma P \Phi}([x,0])) , \int_0^{s+t } R_{\Sigma P \Phi}([x,0])(u) d u -tT_{\Sigma P\Phi}([x,0]) \right) \\
&= \tau_{(\Phi, X, S)}\left( \Sigma P \Phi([x,0], t ) , s \right) \\
& = \tau_{(\Phi, X, S)}\left( [x,t], s \right)\\
& = \int_0^{s+t } R_{ \Phi}(x)(u) d u -tT_{\Phi}(x)\\
&= \int_0^{s+t } R_{ \Phi}(k_{(\Phi, X, S)}([x,0]))(u) d u -tT_{\Phi}(k_{(\Phi, X, S)}([x,0])), \end{aligned} \] for all $[x,0] \in (S_{P \Phi})_0,$ $0 \leq t < 1$ and $s \in \mathbb{R}.$ \end{proof} Thus, we have the following result: \begin{lemma} There is a natural transformation $(k,\tau) : J^- \Sigma P J^+\to 1_{\bf{RWFlowGS}}$ given by the restriction of the natural transformation $(k,\tau) : I \Sigma P I\to I$ defined in Lemma \ref{lem_adj_1}. \end{lemma} \begin{proof} It is sufficient to verify the naturality conditions. Let $(h,\sigma):(\Phi_1, X_1, S_1) \to (\Phi_2, X_2, S_2)$ be a morphism in $\bf{RWFlowGS}.$
If $[x,t] \in (S_1)_{P\Phi_1}$ with $0 \leq t <1,$ we have \[
\begin{aligned}
h \circ k_{(\Phi_1, X_1, S_1)} ([x,t]) &= h\left( \Phi_1(x, t T_{\Phi_1}(x))\right)\\
&= \Phi_2\left( h(x), \sigma(x, t T_{\Phi_1})(x)\right)\\
&= \Phi_2\left( h(x), t\sigma(x, T_{\Phi_1})(x)\right)\\
&= \Phi_2\left( h(x), tT_{\Phi_2}(x)\right)\\
& = k_{(\Phi_2, X_2, S_2)} ([h(x),t]) \\
& = k_{(\Phi_2, X_2, S_2)}\circ \bar h ([x,t]).
\end{aligned} \] Further, for all $[x,t] \in (S_1)_{P\Phi_1}$ with $0 \leq t <1$ and $s \in \mathbb{R},$ \[
\begin{aligned}
&\sigma \circ \tau_{(\Phi_1, X_1, S_1)} ([x,t],s) \\
&=\sigma\left(k_{(\Phi_1, X_1, S_1)} ([x,t]), \tau_{(\Phi_1, X_1, S_1)} ([x,t],s) \right)\\
&=\sigma\left(\Phi_1(x, t T_{\Phi_1}(x)), \int_0^{s+t } R_{\Phi_1}(x)(u) d u -tT_{\Phi_1}(x) \right)\\
&=\int_0^{s+t } R_{\Phi_2}(h(x))(u) d u -tT_{\Phi_2}(h(x))\\
& = \tau_{(\Phi_2, X_2, S_2)} ([h(x),t],s)\\
&= \tau_{(\Phi_2, X_2, S_2)} \circ {\rm id} ([x,t],s),
\end{aligned} \] where ${\rm id}$ is the time part of $J^-(\bar h).$ \end{proof} Combining the results above, we obtain the desired result, which gives us the exact relation between the constructions of Poincar\'e maps and suspension flows. \begin{theorem} $J^- \Sigma \dashv P J^+.$ \end{theorem} \begin{proof} We verify that the triangle identities are satisfied by $l : 1_{\bf{Map}} \to P I \Sigma= (P J^+)(J^- \Sigma)$ and $(k,\tau) : (J^- \Sigma)( P J^+)\to 1_{\bf{RWFlowGS}}.$
In what follows, we omit $J^+$ or $J^-$ for ease of notation.
Let $(f,X)$ be an object in $\bf{Map}.$ Then, we have \[ \begin{aligned}
&k_{\Sigma(f,X)} \circ \Sigma(l_{(f,X)}) ([x,t]) \\
&=k_{\Sigma(f,X)} \left( [l_{(f,X)}(x) , t]\right)\\
& = \Sigma f \left( l_{(f,X)}(x), t\right)\\
& = [x,t] \end{aligned} \] for all $x\in X$ and $0 \leq t < 1.$ Further, we have \[ \begin{aligned}
&\tau_{\Sigma(f,X)} \circ {\rm id} ([x,t],s)\\
& = \tau_{\Sigma(f,X)} \left( [l_{(f,X)}(x) , t], s\right)\\
& = \int_0^{s+t } R_{\Sigma f}(l_{(f,X)}(x))(u) d u -tT_{\Sigma f}(l_{(f,X)}(x))\\
& = s \end{aligned} \] for all $x\in X,$ $0 \leq t < 1$ and $s \in \mathbb{R}.$ These results show that the following diagram commutes in $\bf{RWFlowGS}.$ \[ \begin{diagram}
\node{\Sigma(f,X)}
\arrow{e,t}{\Sigma\left( l_{(f,X)}\right)}
\arrow{se,r}{1_{\Sigma(f,X)}}
\node{\Sigma P\Sigma(f,X)}
\arrow{s,r}{(k,\tau)_{\Sigma(f,X)}}\\
\node{}
\node{\Sigma(f,X)} \end{diagram} \]
Let $(\Phi, X, S)$ be an object in $\bf{RWFlowGS}.$ Then, we have \[ \begin{aligned}
& P\left((k,\tau)_{(\Phi, X, S)}\right) \circ l_{P(\Phi, X, S)} (x)\\
&= k_{(\Phi, X, S)} |_{(S_{P\Phi})_0} \left( [x, 0]\right)\\
& = k_{(\Phi, X, S)}\left( [x, 0]\right)\\
& = \Phi(x, 0) = x \end{aligned} \] for all $x \in S.$ Therefore, the following diagram commutes in $\bf{Map}.$ \[ \begin{diagram}
\node{P(\Phi, X, S)}
\arrow{e,t}{ l_{P(\Phi, X, S)}}
\arrow{se,r}{1_{P(\Phi, X, S)}}
\node{P\Sigma P(\Phi, X, S)}
\arrow{s,r}{P\left((k,\tau)_{(\Phi, X, S)}\right)}\\
\node{}
\node{P(\Phi, X, S)} \end{diagram} \] Thus, we conclude that $J^- \Sigma \dashv P J^+.$ \end{proof} The next corollary is an immediate consequence of Remark \ref{rem_bij} and the injectivity of $l_{(f,X)}.$ \begin{corollary}\label{cor_eq} The categories $\bf{Map}$ and $\bf{RWFlowGS}$ are equivalent. \end{corollary} We remark that the rate-preserving condition can always be assumed for topologically equivalent flows. \begin{theorem}\label{thm_eq} Let $(\Phi_1, X_1, S_1)$ and $(\Phi_2, X_2, S_2)$ be isomorphic in $\bf{WFlowGS}.$ Then, they are isomorphic in $\bf{RWFlowGS}.$ \end{theorem} \begin{proof} By the functoriality of $\Sigma P,$ $\Sigma P(\Phi_1, X_1, S_1)$ and $\Sigma P(\Phi_2, X_2, S_2)$ are isomorphic in $\bf{RWFlowGS}.$ Because $(k,\tau)$ gives isomorphisms, we conclude that $(\Phi_1, X_1, S_1)$ and $(\Phi_2, X_2, S_2)$ are isomorphic in $\bf{RWFlowGS}.$ \end{proof} \section{Concluding remarks}\label{sec_conclude} The categorical equivalence of Corollary \ref{cor_eq} enables us to obtain correspondences between various concepts of flows and map dynamical systems. For example, Theorem \ref{thm_eq} implies that the topological conjugacy of map dynamical systems is categorically equivalent to the topological equivalence of flows. This provides further justification for the use of topological equivalence in the study of flows, in addition to the usual argument that topological conjugacy is too strict.
We also observe a lack of correspondence for some notions. As flows with global Poincar\'e sections do not have equilibria, it follows that map dynamical systems do not have a concept corresponding to them under the equivalence obtained here. It would be interesting to consider whether there exists another pair of functors under which fixed points correspond to equilibria. A candidate will be the time-one map because it corresponds to the discretization functor, which has been considered in \cite{cestau2017prolongations}. However, it is known that this construction is not very expressive, and it is unclear whether an interesting equivalence can be found \cite{bonomo2020continuous}.
\end{document} | arXiv |
This node exports several files from a terrain. The images are exported as gray levels, each pixel having a value between 0 and 1. The generated images have the resolution of the terrain as its size, and the gray levels and scale of the heights (vertices with a 0 value will have their height set to the min value, and vertices with a 1 value will have their height set to the max value) determine the height of each point.
The advantage of exporting multiple files is to optimize the rendering times for very large terrains in an external engine where it is more practical to manage parts of the terrain in different files.
To add a node, right click in the Graph Editor and select Create Node > Export > Multi file export terrain.
File pattern: This is the formula used to name the files to export. The naming convention is important because the node aligns the terrain on a grid and where the first number is the X axis and the second one is the Y axis. Depending on the XY coordinates of the part of the terrain and the number of files to export, each exported file be named according to its XY coordinates, for example the top left part of a 2x2 terrain will have the name Group_0_0.png, the top right part Group_0_1.png, etc.
In File pattern in the parameters dialog, paste the path and add a file name, for example here we add "Group" and then _$x_$y, where $x represents the position of the part of the terrain on the X axis and $y represents the Y axis.
Range: Click User defined to set the Minimum height and Maximum heights or leave the default setting, Automatic.
Each file is loaded, and forms the corresponding part of the final terrain. | CommonCrawl |
Finiteness properties of groups
In mathematics, finiteness properties of a group are a collection of properties that allow the use of various algebraic and topological tools, for example group cohomology, to study the group. It is mostly of interest for the study of infinite groups.
Special cases of groups with finiteness properties are finitely generated and finitely presented groups.
Topological finiteness properties
Given an integer n ≥ 1, a group $\Gamma $ is said to be of type Fn if there exists an aspherical CW-complex whose fundamental group is isomorphic to $\Gamma $ (a classifying space for $\Gamma $) and whose n-skeleton is finite. A group is said to be of type F∞ if it is of type Fn for every n. It is of type F if there exists a finite aspherical CW-complex of which it is the fundamental group.
For small values of n these conditions have more classical interpretations:
• a group is of type F1 if and only if it is finitely generated (the rose with petals indexed by a finite generating family is the 1-skeleton of a classifying space, the Cayley graph of the group for this generating family is the 1-skeleton of its universal cover);
• a group is of type F2 if and only if it is finitely presented (the presentation complex, i.e. the rose with petals indexed by a finite generating set and 2-cells corresponding to each relation, is the 2-skeleton of a classifying space, whose universal cover has the Cayley complex as its 2-skeleton).
It is known that for every n ≥ 1 there are groups of type Fn which are not of type Fn+1. Finite groups are of type F∞ but not of type F. Thompson's group $F$ is an example of a torsion-free group which is of type F∞ but not of type F.[1]
A reformulation of the Fn property is that a group has it if and only if it acts properly discontinuously, freely and cocompactly on a CW-complex whose homotopy groups $\pi _{1},\ldots ,\pi _{n}$ vanish. Another finiteness property can be formulated by replacing homotopy with homology: a group is said to be of type FHn if it acts as above on a CW-complex whose n first homology groups vanish.
Algebraic finiteness properties
Let $\Gamma $ be a group and $\mathbb {Z} \Gamma $ its group ring. The group $\Gamma $ is said to be of type FPn if there exists a resolution of the trivial $\mathbb {Z} \Gamma $-module $\mathbb {Z} $ such that the n first terms are finitely generated projective $\mathbb {Z} \Gamma $-modules.[2] The types FP∞ and FP are defined in the obvious way.
The same statement with projective modules replaced by free modules defines the classes FLn for n ≥ 1, FL∞ and FL.
It is also possible to define classes FPn(R) and FLn(R) for any commutative ring R, by replacing the group ring $\mathbb {Z} \Gamma $ by $R\Gamma $ in the definitions above.
Either of the conditions Fn or FHn imply FPn and FLn (over any commutative ring). A group is of type FP1 if and only if it is finitely generated,[2] but for any n ≥ 2 there exists groups which are of type FPn but not Fn.[3]
Group cohomology
Main article: Group cohomology
If a group is of type FPn then its cohomology groups $H^{i}(\Gamma )$ are finitely generated for $0\leq i\leq n$. If it is of type FP then it is of finite cohomological dimension. Thus finiteness properties play an important role in the cohomology theory of groups.
Examples
Finite groups
A finite cyclic group $G$ acts freely on the unit sphere in $\mathbb {R} ^{\mathbb {N} }$, preserving a CW-complex structure with finitely many cells in each dimension.[4] Since this unit sphere is contractible, every finite cyclic group is of type F∞.
The standard resolution [5] for a group $G$ gives rise to a contractible CW-complex with a free $G$-action in which the cells of dimension $n$ correspond to $(n+1)$-tuples of elements of $G$. This shows that every finite group is of type F∞.
A non-trivial finite group is never of type F because it has infinite cohomological dimension. This also implies that a group with a non-trivial torsion subgroup is never of type F.
Nilpotent groups
If $\Gamma $ is a torsion-free, finitely generated nilpotent group then it is of type F.[6]
Geometric conditions for finiteness properties
Negatively curved groups (hyperbolic or CAT(0) groups) are always of type F∞.[7] Such a group is of type F if and only if it is torsion-free.
As an example, cocompact S-arithmetic groups in algebraic groups over number fields are of type F∞. The Borel–Serre compactification shows that this is also the case for non-cocompact arithmetic groups.
Arithmetic groups over function fields have very different finiteness properties: if $\Gamma $ is an arithmetic group in a simple algebraic group of rank $r$ over a global function field (such as $\mathbb {F} _{q}(t)$) then it is of type Fr but not of type Fr+1.[8]
Notes
1. Brown, Kenneth; Geoghegan, Ross (1984). "An infinite-dimensional torsion-free FP∞ group". Inventiones Mathematicae. 77 (2): 367–381. doi:10.1007/BF01388451. MR 0752825. S2CID 121877111.
2. Brown 1982, p. 197.
3. Bestvina, Mladen; Brady, Noel (1997), "Morse theory and finiteness properties of groups", Inventiones Mathematicae, 129 (3): 445–470, Bibcode:1997InMat.129..445B, doi:10.1007/s002220050168, S2CID 120422255
4. Brown 1982, p. 20.
5. Brown 1982, p. 18.
6. Brown 1982, p. 213.
7. Bridson & Haefliger 1999, p. 439, 468.
8. Bux, Kai-Uwe; Köhl, Ralf; Witzel, Stefan (2013). "Higher finiteness properties of reductive arithmetic groups in positive characteristic: The Rank Theorem". Annals of Mathematics. 177: 311–366. arXiv:1102.0428. doi:10.4007/annals.2013.177.1.6. S2CID 53991649.
References
• Bridson, Martin; Haefliger, André (1999). Metric spaces of non-positive curvature. Springer-Verlag. ISBN 3-540-64324-9.
• Brown, Kenneth S. (1982). Cohomology of groups. Springer-Verlag. ISBN 0-387-90688-6.
| Wikipedia |
\begin{document}
\title{Sequential motion planning in connected sums of real projective spaces} \author{Jorge Aguilar-Guzm\'an and Jes\'us Gonz\'alez} \date{\empty}
\maketitle
\begin{abstract} In this short note we observe that the higher topological complexity of an iterated connected sum of real projective spaces is maximal possible. Unlike the case of regular TC, the result is accessible through easy mod 2 zero-divisor cup-length considerations. \end{abstract}
{\small 2010 Mathematics Subject Classification: Primary 55S40, 55M30; Secondary 70Q05.}
{\small Keywords and phrases: higher topological complexity, connected sum, real projective space.}
\section{Introduction} It was proved in~\cite{MR1988783} that the topological complexity (TC) of the $m$-th dimensional real projective space $\mathbb{R}\mbox{\textrm P}^m$ agrees\footnote{This characterization holds as long as $\mathbb{R}\mbox{\textrm P}^m$ is not parallelizable: $\protect\operatorname{TC}(\mathbb{R}\mbox{\textrm P}^m)=\protect\operatorname{Imm}(\mathbb{R}\mbox{\textrm P}^m)-1=m$ for $m=1,3,7$.} with $\protect\operatorname{Imm}(\mathbb{R}\mbox{\textrm P}^m)$, the minimal dimension $d$ so that $\mathbb{R}\mbox{\textrm P}^m$ admits a smooth immersion in $\mathbb{R}^d$. Cohen and Vandembroucq have recently shown in~\cite{connectedsums} that the fact above does not hold for $g\mathbb{R}\mbox{\textrm P}^m$, the $g$-iterated connected sum of $\mathbb{R}\mbox{\textrm P}^m$ with itself, if $g\geq2$. Indeed, $\protect\operatorname{TC}(g\mathbb{R}\mbox{\textrm P}^m)$ is maximal possible whenever $g\ge2$, a result that contrasts with the currently open problem of assessing how much $\protect\operatorname{TC}(\mathbb{R}\mbox{\textrm P}^m)$ deviates from~$2m$.
Cohen and Vandembroucq's result for $\protect\operatorname{TC}(g\mathbb{R}\mbox{\textrm P}^m)$ extends their impressive calculation in~\cite{LDKlein}, using obstruction theory, of the topological complexity of non orientable closed surfaces. In this short note we observe that a simple minded zero-divisor cup-length argument suffices to prove the analogous fact for Rudyak's higher topological complexity $\protect\operatorname{TC}_s$:
\begin{theorem}\label{maintheorem} For $g,m\ge2$ and $s\ge3$, $\protect\operatorname{TC}_s(g\mathbb{R}\mbox{\emph{P}}^m)=sm$. \end{theorem}
This is the same (but much simplified) phenomenon for $\protect\operatorname{TC}_s(\mathbb{R}\mbox{\textrm P}^m)$ studied in~\cite{MR3770001,MR3795624}. The case $m=2$ is essentially contained in~\cite[Proposition~5.1]{MR3576004}.
\section{Proof} We assume familiarity with the basic ideas, definitions and results on Rudyak's higher topological complexity, a variant of Farber's original concept (see~\cite{MR3331610}). In what follows all cohomology groups are taken with mod 2 coefficients.
The first ingredient we need is the well-known description of the cohomology ring of the connected sum $M\#N$ of two $n$-manifolds $M$ and $N$: Using the cofiber sequence $$ S^{n-1}\hookrightarrow M\#N\to M\vee N $$ one can see that the cohomology ring $H^*(M\#N)$ is the quotient of $H^*(M\vee N)$ by the ideal generated by the sum $[M]^*+[N]^*$ of the duals of the (mod 2) fundamental classes of $M$ and $N$. In particular, for the $g$-iterated connected sum $g\mathbb{R}\mbox{\textrm P}^m$ of $\mathbb{R}\mbox{\textrm P}^m$ with itself, we have:
\begin{lemma}\label{ing2} The cohomology ring of $g\mathbb{R}\emph{P}^m$ is generated by 1-dimensional cohomology classes $x_u$, for $1\le u\le g$, subject to the three relations: \begin{itemize} \item $x_ux_v=0$, for $u\neq v$; \item $x_u^{m+1}=0$; \item $x_u^m=x_v^m$. \end{itemize} \end{lemma}
The top class in $H^*(g\mathbb{R}\mathrm{P}^m)$ is denoted by $t$; it is given by any power $x_u^m$ with $1\le u\le g$.
\begin{corollary} The cohomology ring of the $s$-fold cartesian product of $g\mathbb{R}\mathrm{P}^m$ with itself is given by \begin{equation}\label{topclass} H^*(g\mathbb{R}\emph{P}^m\times\cdots\times g\mathbb{R}\emph{P}^m)\cong\bigotimes_{j=1}^s\left(\mathbb{Z}_2[x_{1,j},\ldots,x_{g,j}] / I_{{g,j}}\rule{0mm}{4mm}\right). \end{equation} Here $x_{u,j}$ is the pull back of $x_u\in H^1(\mathbb{R}\emph{P}^m)$ under the $j$-projection map $\left(\mathbb{R}\emph{P}^m\right)^{\times s}\to\mathbb{R}\emph{P}^m$, and $I_{g,j}$ is the ideal generated by the elements $x_{u,j}^{m+1}$, $x_{u,j}^m+x_{v,j}^m$ and $x_{u,j}x_{v,j}$ for $u\neq v$. \end{corollary}
We let $t_j\in H^m(\left(\mathbb{R}\mathrm{P}^m\right)^{\times s})$ stand for the image of the top class $t\in H^m(\mathbb{R}\mathrm{P}^m)$ under the $j$-th projection map $\left(\mathbb{R}\mathrm{P}^m\right)^{\times s}\to\mathbb{R}\mathrm{P}^m$. The top class in~(\ref{topclass}) is then the product $t_1 t_2\cdots t_s$, which agrees with any product $x_{u_1,1}^m x_{u_2,2}^m\cdots x_{u_s,s}^m$.
The second ingredient we need concerns with standard estimates for the higher topological complexity of CW complexes: \begin{lemma}[{\cite[Theorem~3.9]{MR3331610}}]\label{ing1} For a path connected CW complex $X$, $$\protect\operatorname{zcl}_s(X)\le\protect\operatorname{TC}_s(X)\leq s\dim(X),$$ where $\protect\operatorname{zcl}_s(X)$ is the maximal length of non-zero cup products of $s$-th zero divisors, i.e., of elements in the kernel of the $s$-iterated cup-product map $H^*(X)^{\otimes s}\to H^*(X)$. \end{lemma}
Note that any element $x_{r,i}+x_{r,j}$ is a zero-divisor, so that Theorem~\ref{maintheorem} follows from: \begin{proposition} The product $$ (x_{1,1}+x_{1,2})^m (x_{1,1}+x_{1,3})^m \cdots (x_{1,1}+x_{1,s})^m (x_{2,1}+x_{2,2})^{m-1} (x_{2,1}+x_{2,3}) $$ is the top class in $H^*((g\mathbb{R}\emph{P}^m)^{\otimes s})$ provided $g,m\ge2$ and $s\ge3$. \end{proposition} \begin{proof} The case $s=3$ follows from a direct calculation: \begin{align} (x_{1,1}+x_{1,2})^m & (x_{1,1}+x_{1,3})^m (x_{2,1}+x_{2,2})^{m-1} (x_{2,1}+x_{2,3})\nonumber\\ & {}=(x_{1,1}^m+\cdots+x_{1,2}^m) (x_{1,1}^m+\cdots+x_{1,3}^m) (x_{2,1}^{m-1}+\cdots+x_{2,2}^{m-1}) (x_{2,1}+x_{2,3}),\label{uno}\\ & {}=(x_{1,1}^m+\cdots+x_{1,2}^m) x_{1,3}^m (x_{2,1}^{m-1}+\cdots+x_{2,2}^{m-1}) (x_{2,1}+x_{2,3})\label{dos}\\ & {}=(x_{1,1}^m+\cdots+x_{1,2}^m) x_{1,3}^m (x_{2,1}^{m-1}+\cdots+x_{2,2}^{m-1}) x_{2,1}\label{tres}\\ & {}=x_{1,2}^m x_{1,3}^m (x_{2,1}^{m-1}+\cdots+x_{2,2}^{m-1}) x_{2,1}\label{cuatro}\\ & {}=x_{1,2}^m x_{1,3}^m x_{2,1}^{m-1} x_{2,1}\label{cinco}\\ & {}=x_{1,2}^m x_{1,3}^m x_{2,1}^m\,=\,t_1t_2t_3.\nonumber \end{align} Note that equality in~(\ref{dos}) holds because of the description of $I_{g,s}$: the factor $t_3$ in the top class $t_1t_2t_3$ can only arise from the summand $x_{1,3}^m$ in the second factor of~(\ref{uno}). Likewise, equality in~(\ref{tres}) comes from the relation $x_{1,3}x_{2,3}=0$, equality in~(\ref{cuatro}) comes from the relation $x_{1,1} x_{2,1}=0$, and equality in~(\ref{cinco}) comes from the relation $x_{1,2} x_{2,2}=0$.
The general case then follows easily from induction: \begin{align*} (x_{1,1}+x_{1,2})^m & (x_{1,1}+x_{1,3})^m \cdots (x_{1,1}+x_{1,s+1})^m (x_{2,1}+x_{2,2})^{m-1} (x_{2,1}+x_{2,3})\\ & {}=t_1\cdots t_s (x_{1,1}+x_{1,s+1})^m = t_1\cdots t_s x_{1,s+1}^m = t_1\cdots t_{s+1}, \end{align*} where the next-to-last equality holds in view of the relation $x_{1,1}^{m+1}=0$. \end{proof}
{\small \sc Departamento de Matem\'aticas
Centro de Investigaci\'on y de Estudios Avanzados del I.P.N.
Av.~Instituto Polit\'ecnico Nacional n\'umero 2508
San Pedro Zacatenco, M\'exico City 07000, M\'exico
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{\tt [email protected]}}
\end{document} | arXiv |
Sergei Stepanov (mathematician)
Sergei Aleksandrovich Stepanov (Сергей Александрович Степанов; [1] 24 February 1941) is a Russian mathematician, specializing in number theory. He is known for his 1969 proof using elementary methods of the Riemann hypothesis for zeta-functions of hyperelliptic curves over finite fields, first proved by André Weil in 1940–1941 using sophisticated, deep methods in algebraic geometry.
Stepanov received in 1977 his Russian doctorate (higher doctoral degree) from the Steklov Institute under Dmitry Konstantinovich Faddeev with dissertation (translated title) An elementary method in algebraic number theory.[2] He was from 1987 to 2000 a professor at the Steklov Institute in Moscow.[3] In the 1990s he was also at Bilkent University in Ankara. He is at the Institute for Problems of Information Transmission of the Russian Academy of Sciences.
Stepanov is best known for his work in arithmetic algebraic geometry, especially for the Weil conjectures on algebraic curves. He gave in 1969 an "elementary" (i.e. using elementary methods) proof of a result first proved by André Weil using sophisticated methods, not readily understable by mathematicians who are not specialists in algebraic geometry. Wolfgang M. Schmidt extended Stepanov's methods to prove the general result, and Enrico Bombieri succeeded in using the work of Stepanov and Schmidt to give a substantially simplified, elementary proof of the Riemann hypothesis for zeta-functions of curves over finite fields.[4][5][6] Stepanov's research also deals with applications of algebraic geometry to coding theory.
He was an Invited Speaker of the ICM in 1974 in Vancouver.[7][8] He received in 1975 the USSR State Prize.[3] He was elected a Fellow of the American Mathematical Society in 2012.
Selected publications
• Codes on Algebraic Curves, Kluwer 1999
• Arithmetic of Algebraic Curves, New York, Plenum Publishing 1994,[9] Russian original: Moscow, Nauka, 1991.
• as editor with Cem Yildirim: Number theory and its applications, Marcel Dekker 1999
References
1. sometimes transliterated Serguei A. Stepanov, e.g. in the book edited by him Number theory and its applications, 1999
2. S. A. Stepanov, An elementary method in algebraic number theory, Translated from Matematicheskie Zametki, Vol. 24, No. 3, pp. 425–431, September 1978. doi:10.1007/BF01097766
3. Steklov Mathematical Institute
4. Rosen, Michael (2002). Number Theory in Function Fields. Springer. p. 329. ISBN 9781475760460.
5. Bombieri, Enrico. "Counting points on curves over finite fields (d´après Stepanov)". In: Seminaire Bourbaki, Nr.431, 1972/73. Lecture Notes in Mathematics, vol. 383. Springer.
6. Stepanov, S. A. (1969). "On the number of points of a hyperelliptic curve over a finite prime field". Mathematics of the USSR-Izvestiya. 3 (5): 1103. doi:10.1070/IM1969v003n05ABEH000834.
7. S. A. Stepanov, "элементарный метод в теории уравнений над конечными полями" “An elementary method in the theory of equations over finite fields,” in: Proc. Int. Cong. Mathematicians, Vancouver (1974), vol. 1, pp. 383–391.
8. Stepanov, S. A. (1977). "An elementary method in the theory of equations over finite fields". In Anosov, Dmitrij V. (ed.). 20 lectures delivered at the International Congress of Mathematicians in Vancouver, 1974. American Mathematical Society Translations, Series 2, Vol. 109. American Mathematical Soc. pp. 13–20. ISBN 9780821895467.
9. Silverman, Joseph H. (1996). "Review of Arithmetic of algebraic curves by Serguei Stepanov". Bull. Amer. Math. Soc. 33: 251–254. doi:10.1090/S0273-0979-96-00641-6.
External links
• Stepanov at Mathnet.ru
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| Wikipedia |
Article | Open | Published: 28 June 2019
Rational design using sequence information only produces a peptide that binds to the intrinsically disordered region of p53
Kiyoto Kamagata1,
Eriko Mano1,
Yuji Itoh1,2,
Takuro Wakamoto3,
Ryo Kitahara ORCID: orcid.org/0000-0003-1815-32274,
Saori Kanbayashi1,
Hiroto Takahashi1,
Agato Murata1,2 &
Tomoshi Kameda5
Scientific Reportsvolume 9, Article number: 8584 (2019) | Download Citation
Fluorescence imaging
Intrinsically disordered proteins
Virtual screening
Intrinsically disordered regions (IDRs) of proteins are involved in many diseases. The rational drug design against disease-mediating proteins is often based on the 3D structure; however, the flexible structure of IDRs hinders the use of such structure-based design methods. Here, we developed a rational design method to obtain a peptide that can bind an IDR using only sequence information based on the statistical contact energy of amino acid pairs. We applied the method to the disordered C-terminal domain of the tumor suppressor p53. Titration experiments revealed that one of the designed peptides, DP6, has a druggable affinity of ~1 μM to the p53 C-terminal domain. NMR spectroscopy and molecular dynamics simulation revealed that DP6 selectively binds to the vicinity of the target sequence in the C-terminal domain of p53. DP6 inhibits the nonspecific DNA binding of a tetrameric form of the p53 C-terminal domain, but does not significantly affect the specific DNA binding of a tetrameric form of the p53 core domain. Single-molecule measurements revealed that DP6 retards the 1D sliding of p53 along DNA, implying modulation of the target searching of p53. Statistical potential-based design may be useful in designing peptides that target IDRs for therapeutic purposes.
Intrinsically disordered regions (IDRs) of proteins lack a defined 3D structure. Approximately 10–35% of prokaryotic and 15–45% of eukaryotic proteins are estimated to contain IDRs1. IDRs function as hubs in protein-protein interaction networks, the regulation of transcription and signaling pathways, and phase separation. IDRs are involved in many diseases and are considered drug targets2,3,4. The major drug design targeting IDR is based on the experimental screening of chemicals4. By contrast, structure-based drug design, which is one of the most commonly used methods to design drugs targeting proteins, is difficult to apply to IDRs due to their flexible structure4. A unique successful method of the rational drug design targeting IDRs is virtual screening of chemicals that bind pockets formed in IDR conformations5,6. Before the screening, pockets of the IDR need to be estimated by MD simulation or NMR. Accordingly, a rational design approach without 3D structure information is required for targeting IDRs.
p53, an intrinsically disordered protein containing IDRs, is a multifunctional transcription factor that can suppress cell tumorigenesis and is a desirable drug target7,8. Fifty percent of the gene mutations identified in tumor cells occur in p53 gene9. p53 is composed of an N-terminal (NT) domain, a core domain, a linker, a tetramerization domain, and a C-terminal (CT) domain. The core and tetramerization domains form a specific tertiary structure, while the other two domains and the linker are IDRs. The core and CT domains are involved in the specific and nonspecific DNA binding, respectively. The 1D sliding of p53 along DNA is essential to searching for the target site among tremendous amounts of nonspecific DNA10. p53 possesses two sliding modes with a different conformation11,12,13. The CT domain facilitates the 1D sliding of p5311,14 and regulates the transcription of downstream genes15,16,17,18. Several studies focused on drug design or function modification for p538,17,19,20,21,22,23,24,25,26,27 have mainly targeted the folded domains, and successful drug design for the IDRs of p53 has been limited to two studies28,29. Gabizon et al. identified several peptides that bind to the tetrameric form of the disordered CT domain of p53 by screening peptides derived from natural proteins bound to p5329.
Peptides are promising drug candidates30,31 and may work for targeting IDRs29. The combination of 20 amino acids with different characteristics has the potential to generate peptides with high affinity for the target IDR. Also, the flexible peptide can fit any conformation of the IDR. We need to search for peptides with a high affinity for the target IDR among an enormous number of candidates; for example, for a 16-residue peptide, 2016 candidates are possible. The theoretical pool of peptides is very large, while using peptides from natural proteins limits the number of peptides that can be tested, hence it is needed to develop a computer-based method for drug design.
Here, we aimed to develop the design method of peptides that can bind IDRs and then apply the method to the p53 IDR. The method uses only the IDR's sequence information without considering its structure. For the design, we used the Miyazawa-Jernigan (MJ) potential, which reflects the residue-residue potential for favorable and unfavorable contacts based on the analysis of known protein crystal structures in protein crystal structures32. MJ potential has been used in protein folding simulation and structural prediction33,34,35,36,37. As a target IDR, we chose the CT domain of p53. We designed six peptides, and found that three peptides bound to the CT domain tighter than the peptides identified by Gabizon et al.29. A series of experiments including NMR, molecular dynamics (MD) simulations, ensemble titration, and single-molecule fluorescence demonstrated that one designed peptide with the highest affinity can modulate the nonspecific DNA binding of p53 and the 1D sliding of p53 along DNA. This is the first approach to target IDRs using only their sequence information and is potentially applicable to many disease-related proteins containing IDRs.
MJ potential-based peptide design identifies a peptide with micromolar affinity for the p53 CT domain
Using MJ potential32, we designed peptide sequences to bind the disordered CT domain of p53. Specifically, six peptides with 13 or 16 residues were designed to minimize the total statistical binding energy for one by one residue (designed peptides DP1–4) or one by three residues including two adjacent residues (designed peptides DP5–6) (Supplementary Figure S1 and Table 1). One by one assumes the contacts of two residues facing in the two sequences, while one by three assumes the contacts to three sequential residues (Fig. 1a). The binding energy between the ith residue of the CT domain and the jth residue of a designed peptide was calculated as eij + err − eir − ejr, as defined by Miyazawa and Jernigan32,38, where eij denotes the energy difference between the formation of contacts between the ith and jth residues, and the same residues exposed to solvent. The notation r represents averaging over all amino acids. To test how well our method worked, we titrated the designed peptides against the CT peptide (residues 367–393) labeled with a fluorescent dye, FAM, using a fluorometer with fluorescence anisotropy11 (Supplementary Fig. S2a). All titration curves were well fitted with equations 1 and 2 (see Methods) based on one by one binding. The apparent dissociation constant of the designed peptides (KD) with the CT peptide ranged from 1.2 ± 0.8 μM to 550 ± 10 μM in the absence of salt, and DP6 showed the strongest binding to the CT domain (Fig. 1b and Table 1). One by three design (DP5–6) improved the binding ability of the designed peptide to the CT peptide by at least six-fold compared to one by one design (DP1–4). The affinity of the designed peptides for the CT peptide correlated with the estimated binding energy (Supplementary Fig. S3). These results suggest that interactions between residues that are favorable in natural folded proteins can stabilize complexes involving disordered proteins.
Table 1 Length, sequence, and dissociation constant of peptides used in this study.
Binding of designed peptides to the monomeric CT domain of p53. (a) Schematic diagram of peptide design. Red and blue indicate one by one residue and one by three residue design, respectively. DP, designed peptide. (b) Dissociation constants of designed peptides (DP1–DP6) or peptides from natural proteins (Wmot2, Cul7, and WS100B) for the CT peptide of p53 in the absence of salt. (c) Salt dependence of the dissociation constant of DP6 with the CT peptide. In panels (b,c), the error is the SEM of the fitting.
Next, we compared the affinity of our designed peptides to that of peptides derived from natural p53-binding proteins and confirmed to bind to the CT peptide in an earlier report29. To this end, we titrated three natural peptides against the CT peptide under the same conditions and determined the KD (Fig. 1b and Supplementary Fig. S2b). The KD of DP6 with respect to the CT domain was at least 12 times stronger than the KD of the natural peptides (Table 1).
To elucidate the mechanism of DP6 binding to the CT domain, we examined the dependence of KD on salt concentration (Fig. 1c). KD decreased as the K+ concentration increased, suggesting that the binding was mainly governed by electrostatic interactions between the +6 net charge of DP6 and the −4 net charge of the CT peptide. The circular dichroism spectra of the DP6–CT peptide complex and of the DP6 and CT peptides alone showed no significant secondary structure, implying disordered-disordered contacts in the complex (Supplementary Fig. S4).
DP6 binds selectively to the target region and its vicinity via electrostatic and hydrophobic interactions
To examine whether DP6 binds specifically to the designed target position of p53, we conducted NMR analysis of a 15N-labeled p53 (residues 313–393) tetramer including linker, Tet, and CT domains in the absence and presence of DP6 (Fig. 2a and Supplementary Fig. S5). Clear chemical shift perturbations in the 1H-15N HSQC spectrum upon binding to DP6 were observed in residues 329, 360, 362, 365, 366, 369, 370, 374, 375, 381, 382, 384, 385, and 387 of p53. These residues correspond to the designed target of the CT domain (residues 369–384) for DP6 and its vicinity. To further investigate the complex structure, we conducted MD simulations of the p53 (313–393) tetramer and DP6. The contact map showed significant interaction between residues 360–385 of p53 and DP6, which is consistent with the chemical shift perturbations (Fig. 2a,b). Furthermore, the identified binding site of DP6 agreed with the p53 sequence region having lower MJ binding energy for DP6 (Supplementary Fig. S6). In the MD simulation trajectory, the DP6 and CT domains showed flexibility, and no specific tertiary structure was formed (Supplemental Movie). In these structures, contacts between the acidic residues of DP6 and the basic residues of the CT domain were always observed. In addition, W15 and F16 in DP6 and hydrophobic regions in the CT domain (especially, residues 361–369, 371, 373–379, 385, 387, 388) always formed a hydrophobic cluster (Fig. 2c). These results suggest that electrostatic and hydrophobic interactions strongly stabilized the DP6–CT domain complex compared with other peptides. In fact, such hydrophobic interactions were not observed between the CT domain and DP5, which may explain the relatively weak affinity compared with that of DP6 (Fig. 2c). Accordingly, we conclude that DP6 binds selectively to the designed target region and its vicinity in the CT domain of p53.
Binding of the designed peptide DP6 to tetrameric p53 (313–393). (a) Chemical shift perturbations (CSP) of the 1H-15N HSQC spectrum of p53 (313–393) in the absence/presence of DP6. Red and blue characters represent residues showing positive or no CSP, respectively. Gray highlighting indicates the target sequence of p53 used for the design of DP6. (b) Contact map between residues of DP6 and p53 in MD simulation. (c) Typical snapshots of DP6 and p53 (313–393) tetramer complex in MD simulation. Cyan, black, yellow, and red denote designed peptide, linker, the Tet domain, and the CT domain of p53, respectively. Hydrophobic amino acids of the CT domain, DP5, and DP6 in contacts are depicted in space-filling representation.
DP6 interferes with nonspecific DNA binding of p53
To examine the effect of DP6 on the DNA binding of p53, we prepared three p53 constructs: full-length p53 (FL-p53), and two mutants each containing one of its two DNA binding domains, a tetrameric form of the CT domain (TetCT mutant), and a tetrameric form of the core domain (CoreTet mutant) as reported previously12 (Fig. 3a). For the p53 mutants, we used a thermostabilized form of p53 with a single exposed cysteine, which is suitable for in vitro ensemble and single-molecule studies11.
Effect of the designed peptide DP6 on the DNA binding of the p53 tetramer. (a) p53 constructs used in this study. NT, core, Tet, and CT represent the N-terminal, core, tetramerization, and C-terminal domains of p53, respectively. Thick and thin primary structures correspond to folded and disordered regions, respectively. (b) Titration of the TetCT mutant against nspDNA at various DP6 concentrations. (c) Titration of the CoreTet mutant against spDNA in the presence (blue circles) and absence (black circles) of 600 μM DP6 and against nspDNA in the absence of DP6 (triangles). (d) Titration of FL-p53 against nspDNA at various DP6 concentrations. (e) Titration of FL-p53 against spDNAat various DP6 concentrations. (f) Affinity of FL-p53 for nspDNA (triangles) and spDNA (circles) at various DP6 concentrations. The errors were the SEM of the fitting. In panels (b–e), tetramer concentrations are used for the p53 mutants, and the solid curves are the best-fitted curves using Equations (1) and (2) described in the Methods.
To examine the effect of DP6 on the affinity of the TetCT mutant for DNA, we titrated TetCT mutant against nonspecific DNA (nspDNA) labeled with 6-FAM at 0–600 μM DP6 based on fluorescence anisotropy11. The solution used here included 100 mM KCl to mimic physiological conditions. In the absence of DP6, the TetCT mutant bound to nspDNA (KD = 110 ± 20 nM for tetramer). The anisotropy change for nspDNA was repressed as the DP6 concentration increased (Fig. 3b). However, no significant change in anisotropy was observed at more than 150 μM DP6, suggesting that DP6 inhibits the binding of the TetCT mutant to nspDNA by competitive binding to the CT domain of the TetCT mutant. For control experiments investigating the effect of DP6 on the binding of the core domain to specific DNA (spDNA), we titrated the CoreTet mutant against spDNA in the presence/absence of 600 μM DP6 (Fig. 3c, circles). DP6 did not weaken the binding of the CoreTet mutant to spDNA (Supplementary Table S1). Nonspecific binding of the CoreTet mutant was not observed, confirming that DNA binding of the CoreTet mutant in the presence of DP6 was specific (Fig. 3c, triangles). Accordingly, the results suggest that DP6 binds specifically to the CT domain of p53 and prevents nonspecific binding to DNA, thereby competitively maintaining the association of the core domain with spDNA.
We next examined the effect of DP6 on the affinity of FL-p53 to DNA by titrating FL-p53 against nspDNA and spDNA at 0–600 μM DP6. The anisotropy change for nspDNA was repressed as the DP6 concentration increased, indicating that the nonspecific affinity of FL-p53 was weakened by DP6 (Fig. 3d). The anisotropy change for spDNA was also repressed by DP6 (Fig. 3e), but the repression was more effective for nspDNA than for spDNA. The KD of FL-p53 for nspDNA gradually increased with the addition of DP6 and was 7-fold higher at 600 μM than at 0 μM DP6 (Fig. 3d and Supplementary Table S1). By contrast, the KD for spDNA increased to 3-fold and was saturated at 300 μM DP6 (Fig. 3e and Supplementary Table S1). In contrast to DP6, DP5 did not affect the affinity of FL-p53, suggesting that W and/or F of DP6 strengthens the binding of the designed peptide to FL-p53 (Supplementary Fig. S7 and Supplementary Table S2). These results demonstrate that DP6 weakens the nonspecific DNA binding of FL-p53 more efficiently than the specific DNA binding.
DP6 retards 1D sliding of p53 along DNA
To test whether DP6 affects the 1D sliding of p53 along DNA, we visualized the 1D sliding of FL-p53 labeled with a fluorescent dye, Atto532, along nspDNA at 0, 300, and 600 μM DP6 using single-molecule fluorescence microscopy coupled with the DNA array "DNA garden"10,11,39 (Fig. 4a). We conducted the measurements within 50 min after the dilution of stock solution under conditions in which p53 maintains a tetrameric form40 and obtained 158–290 trajectories in various DP6 concentrations (Fig. 4b). The mean square displacement (MSD) plots of FL-p53 were linear in all conditions, indicating the diffusional motion of p53 along DNA at various DP6 concentrations (Fig. 4c). The average diffusion coefficient, calculated from the slope of MSD, decreased to 0.5-fold at 300 μM DP6 and 0.6-fold at 600 μM (Supplementary Table S3). These results demonstrate that DP6 retards the 1D sliding of p53 along DNA.
Effect of the designed peptide DP6 on 1D sliding of the p53 tetramer along DNA. (a) Schematic diagram of the single-molecule detection of p53 labeled with Atto532 on a DNA array. Yellow circles, pink circles, and solid lines represent NeutrAvidin, p53, and DNA, respectively. (b) Typical single-molecule trajectories of FL-p53 along nonspecific DNA at various DP6 concentrations. Several traces are colored for clarity. (c) Time courses of the averaged MSD of FL-p53 at various DP6 concentrations. (d) Displacement distributions for the sliding dynamics of FL-p53 at various DP6 concentrations. Bars represent the displacement distributions observed at time intervals of 165 ms. Errors of the bar were estimated by a bootstrap method with 1,000 iterations. Solid and dashed curves are best-fitted curves obtained using Equation (3) described in Methods and the distributions of each mode, respectively.
To examine how DP6 affects the two sliding modes of p53, we analyzed the displacement distributions of p53 at the time interval of 165 ms in different DP6 concentrations. All displacement distributions were well fitted with the sum of two Gaussian functions, which assumes two diffusional motions (Eq. 3, see Methods), indicating that FL-p53 possesses two sliding modes in the presence of DP6. The diffusion coefficient of the fast mode decreased to 0.8-fold at 300 μM DP6 and to 0.7-fold at 600 μM, thereby reducing the average diffusion coefficient (Supplementary Table S3). By contrast, no significant change in the slow mode is consistent with the tight interaction of the core domains with DNA identified in the slow mode12. Accordingly, the fast mode of FL-p53 sliding, rather than the slow mode, is retarded upon the association of DP6 with FL-p53.
In this study, we developed the design method of peptides that can bind IDRs using only the IDR's sequence information without its structural information. To test this method, we targeted the disordered CT domain of p53. We found that DP6 has the high affinity for the CT domain, and modulates the affinity of p53 for DNA and the target search. These results suggest that our design method has a potential to generate druggable peptide candidates for IDRs. Here, we discuss the mechanism for action of DP6 and the comparison of drug design methods for IDRs.
We propose a mechanism for how DP6 affects p53 (Fig. 5). One of the actions of DP6 is to competitively block the association of the CT domain of p53 with nonspecific DNA. Previous spectroscopic studies including chemical shift perturbations in NMR and the effects of mutation on the affinity revealed that residues 365−382, including lysine residues 372, 373, 381, and 382 in the CT domain, interact with DNA41,42. The NMR data in this study demonstrate that DP6 interacts with residues 360–387 of p53 (Fig. 2a). Therefore, it is likely that DP6 sterically prevents the association of the CT domain with DNA, reducing the affinity of p53 for nonspecific DNA (Fig. 3b). This finding is supported by the fact that additives such as small nucleotides and antibodies strengthen specific affinity by blocking the CT domain43. Before the formation of the p53–DNA complex, the CT domain is blocked by DP6 from association with nonspecific DNA.
Model for regulation of the affinity and 1D sliding of p53 by the designed peptide DP6. DP6 (cyan) weakens the association of p53 with DNA (gray bar) and retards the diffusion of the fast mode of p53 along DNA upon association with the CT domain of p53 (red). Purple, orange, and yellow represent the NT, core, and Tet domains of p53, respectively. The structure of the p53–DNA complex is described based on the results of our previous study12.
The other action of DP6 is to retard 1D sliding along DNA. Once the CT domain binds to DNA instead of DP6, the p53–DP6 complex slides along DNA (Fig. 5). DP6 mainly affects the fast mode in which the CT domain of p53 interacts with nonspecific DNA without core domain–DNA interaction12 (Fig. 4d). The observed reduction in the diffusion coefficient in the fast mode may be attributed to the larger molecular size of the p53–DP6 complex and/or intercalation with DNA by DP6. The larger size of p53 may be caused by the detachment of the CT domain from DNA and the association of DP6 with p53 and should slow the 1D sliding along DNA because of the dependence of the 1D sliding on molecular size44,45. As observed in DNA glycosylase46, tryptophan in DP6 may insert between DNA bases during the sliding of the p53–DP6 complex and thereby retard the sliding. By contrast, DP6 does not affect the slow mode, in which p53 reads the DNA sequence and recognizes the target, because in the slow mode, the core domain and CT domain interact with DNA12. This result confirms the specific DNA binding ability of p53. Thus, DP6 may act as a potential inhibitor of p53 by preventing the target search kinetically but not the target binding itself.
The MJ-potential-based approach has unique characteristics comparing with other approaches in the drug design targeting IDRs. Successful approach in the drug design for IDRs is the experimental or virtual screening of chemicals4. The experimental screening does not use any information of the IDR sequence and structure, but requires a high cost and consumes time. As in the case of p53, a small chemical RITA was discovered to bind disordered NT domain of p53 by using the experimental screening28. In the virtual screening, the IDR pockets are identified using MD or NMR and then the chemicals that can bind the pockets are screened4. The virtual screening may not work in the case of no pockets identified in the IDR conformation ensemble. By contrast, our method may be applicable to the IDRs without pocket structures, because the structural information is not used. The MJ-based approach can be applied to the IDRs with known primary sequence. In fact, the designed peptide DP6 identified in this study can bind the CT domain of p53 much stronger than the peptides from natural proteins (Table 1) and the affinity of DP6 is comparable to that of drug candidates discovered in other disordered proteins4. This suggests the successful searching for the druggable peptide among an enormous number of candidates. The virtual screening of peptides based on the MJ contact energy is simple and does not need a supercomputer. Accordingly, the MJ-based approach provides an alternative strategy of drug design targeting IDRs and may apply to many disease-related proteins containing IDRs.
Calculation of contact energy
The binding energy between the ith residue of the CT domain and the jth residue of the designed peptide, E, for one by one residue design was calculated as eij + err − eir − ejr, as defined by Miyazawa and Jernigan32. The notation r represents averaging over all amino acids. For eij, err, eir, and ejr, we used the values reported by Miyazawa and Jernigan32. We calculated the binding energy by replacing the jth residue of a designed peptide with each of the 20 amino acids, and then determined the amino acid that gave the lowest binding energy. This procedure was repeated to obtain the sequence of a 13-residue or 16-residue peptide, and the total binding energy was calculated by summing each binding energy. A designed peptide with minimal total energy was selected among peptides designed for different initial residues of the CT domain. In addition, we selected peptides designed for different regions of the CT domain, but not with the minimum total energy. For the calculation involving one by three residues, including two adjacent residues, we added the binding energy between adjacent residues of the CT peptide and each residue of a designed peptide.
p53 mutants and peptides
We prepared FL-p53, TetCT mutant, and CoreTet mutant as described previously11. For FL-p53, the thermostable and cysteine-modified mutant of human p53 (C124A, C135V, C141V, W146Y, C182S, V203A, R209P, C229Y, H233Y, Y234F, N235K, Y236F, T253V, N268D, C275A, C277A, K292C) was used11. The TetCT mutant corresponds to residues 293−393 of human p53 with an additional N-terminal cysteine11. The CoreTet mutant corresponds to residues 1–363 of FL-p5312. Expression and purification of the three mutants was conducted as previously described11,12. Briefly, all mutants with a GST tag were expressed in Escherichia coli and were collected from a GST column after cleavage of the GST tag and further purified by using a heparin column. The DNA binding ability of all mutants was confirmed by titration experiments as described elsewhere10,11,12. For NMR analysis, the p53 gene corresponding to residues 313–393 of human p53 in pGEX-6P-1 was generated using a KOD-Plus-Mutagenesis Kit (TOYOBO, Osaka, Japan). 15N-labeled p53 (313–393) was expressed in BL21 (DE3) plysS in 15N M9 media at 16 °C for 18 h after the addition of 0.5 mM IPTG and purified as described above12. For titration experiments, CT peptide (residues 367–393 of human p53) labeled with FAM at the N-terminus, designed peptides, and peptides from natural proteins were synthesized without caps and obtained with at least 95% purity (Toray Research Center Inc., Tokyo, Japan).
1H/15N HSQC experiments were performed at 5 °C using a 1H 600 MHz NMR spectrometer (DRX-600; Bruker, Billerica, MA, USA). The solution contained 0.5 mM 15N-labeled p53 (313–393), 0 or 20 μM DP6, 10 mM HEPES, and 10% 2H2O at pH 7.0. HSQC cross-peaks were assigned to individual amide groups with reference to the assignments of p53 (313–393)41. Spectral analysis was performed using the software Topspin 1.1 (Bruker, Billerica, MA, USA) and NMRViewJ47.
MD simulation
A tetramer of p53 (313–393) and the DP6 peptide were simulated using Amber16 simulator48 with the AMBER ff99SB force field49 and Generalized Born energy for solvation50. For the initial structure of p53 (313–393), the tetramerization domain and the missing disordered region were generated using PDB code 1OLH and modeled in PyMol software, respectively. The initial structure of DP6 was generated in extended form. Initially, DP6 was located at six positions on the x-, y-, and z-axes ± 100 Å from the center of mass of the p53 (313–393) tetramer as described in51. For system relaxation, 20 ns simulations were conducted with constraint of the p53 tetrameric domains while decreasing the temperature from 1000 to 280 K. Then, four 100 ns simulations without any constraint were conducted starting from each initial conformation with randomized initial velocity at 280 K. The temperature was controlled using a Langevin thermostat. The solvent viscosity of water was set to 1.0 ps−1. Noncovalent interactions were used without cutoffs. The covalent bonds of hydrogen atoms in p53 and DP6 were constrained using the SHAKE method52, and the integration time step was 2 fs. For the contact map, 80–100 ns conformations were used. A contact was defined as a distance of less than 6.5 Å between the centers of two side chains except for Gly, where Cα was used38.
Titration experiments
The fluorescence anisotropy of fluorescent CT peptide or DNA was measured at 25 °C using a fluorescence spectrometer (FP-6500, JASCO Co., Tokyo, Japan) with an automatic titrator and home-build autorotating polarizer11. To measure the affinity of the designed peptides to the CT peptide, nonlabeled designed peptides or peptides from natural proteins29 were titrated into a solution containing 5 nM FAM-labeled CT peptide, 20 mM HEPES, 0.5 mM EDTA, 1 mM DTT, and 0.2 mg/mL BSA (pH 7.9). To examine the salt-dependence of the binding of DP6 to CT peptide, KCl was also added. To measure the affinity of p53 mutants to DNA, nonlabeled p53 mutants were titrated into a solution including 5 nM 6-FAM-DNA, 20 mM HEPES, 0.5 mM EDTA, 1 mM DTT, 0.2 mg/mL BSA, 100 mM KCl, 2 mM MgCl2, and various concentrations of DP6 (pH 7.9). spDNA and nspDNA were 30-bp sequences of the p21 gene and a random sequence, respectively, as described elsewhere11 (Sigma-Aldrich Co., Tokyo, Japan). The titration curves were fitted by the following equations:
$${r}_{{\rm{obs}}}={r}_{{\rm{A}}}\frac{({c}_{{\rm{A}}}-{c}_{{\rm{AB}}})}{{c}_{{\rm{A}}}}+{r}_{{\rm{AB}}}\frac{{c}_{{\rm{AB}}}}{{c}_{{\rm{A}}}},$$
$${c}_{AB}=\frac{({c}_{{\rm{A}}}+{c}_{{\rm{B}}}+{K}_{D})-\sqrt{{({c}_{{\rm{A}}}+{c}_{{\rm{B}}}+{K}_{D})}^{2}-4{c}_{{\rm{A}}}{c}_{{\rm{B}}}}}{2},$$
where robs, rA, rAB, KD, cA, and cB are the observed anisotropy, anisotropy of free molecule A, anisotropy of the complex between molecules A and B, dissociation constant, total concentration of molecule A and total concentration of molecule B, respectively. For p53 mutants, the concentration was calculated for the tetramer.
Single-molecule measurements of p53 mutants using the DNA garden method
The 1D sliding of FL-p53 labeled with Atto532 along DNA was measured by a custom-built TIRF microscope, as described previously11,39. The λDNA array tethered on the coverslip of the flow cell was constructed using microcontact printing as described previously39. FL-p53 was labeled with Atto532 as described elsewhere11. FL-p53 with Atto532 at 2–4 nM was introduced into the flow cell with the DNA array by a syringe pump and measured at a flow rate of 500 μL/min, corresponding to 90% extension of λDNA. The solution contained 20 mM HEPES, 0.5 mM EDTA, 1 mM DTT, 0.2 mg/mL BSA, 1 mM DTT, 2 mM Trolox, 100 mM KCl, 2 mM MgCl2, and 0–600 μM DP6 at pH7.9. The experiments were conducted immediately after the dilution of FL-p53 from the stock solution at more than 10 μM and finished within 50 min to prevent the dissociation of the tetramer40. The analysis was described previously11. For distribution analysis, the following fitting equation was used:
$$P({\rm{\delta }}x)=\sum _{i=1}^{2}\frac{{A}_{i}}{\sqrt{4{\rm{\pi }}{D}_{i}{\rm{\delta }}t}}\exp (-\frac{{({\rm{\delta }}x+{v}_{i}{\rm{\delta }}t)}^{2}}{4{D}_{i}{\rm{\delta }}t}),$$
where P(δx), δt, δx, Ai, vi, and Di are the occurrence of δx, time interval, displacement in the time interval, amplitude of the ith mode, drift velocity of the ith mode, and diffusion coefficient of the ith mode, respectively.
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This work was supported by MEXT/JSPS KAKENHI, JP15H01625 (to K.K.) and JP16K07313 (to K.K.), and the Research Program of "Dynamic Alliance for Open Innovation Bridging Human, Environment and Materials" in "Network Joint Research Center for Materials and Devices" (to T.K.). We thank Dr. Soichiro Kitazawa (Ritsumeikan University) for an earlier trial of the NMR experiments.
Institute of Multidisciplinary Research for Advanced Materials, Tohoku University, Katahira 2-1-1, Aoba-ku, Sendai, 980-8577, Japan
Kiyoto Kamagata
, Eriko Mano
, Yuji Itoh
, Saori Kanbayashi
, Hiroto Takahashi
& Agato Murata
Department of Chemistry, Graduate School of Science, Tohoku University, Sendai, 980-8578, Japan
Yuji Itoh
Graduate School of Life Sciences, Ritsumeikan University, Kusatsu, 525-8577, Japan
Takuro Wakamoto
College of Pharmaceutical Sciences, Ritsumeikan University, Kusatsu, 525-8577, Japan
Ryo Kitahara
Artificial Intelligence Research Center, National Institute of Advanced Industrial Science and Technology (AIST), Koto, Tokyo, 135-0064, Japan
Tomoshi Kameda
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K.K. contributed design of work and the development of the design strategy, and wrote the manuscript. E.M. contributed the acquisition of titration data. T.W. and R.K. contributed the acquisition of NMR data. H.T. contributed the preparation of a p53 mutant for NMR. S.K. contributed the preparation of p53 mutants and the acquisition of titration data. Y.I. and A.M. contributed the acquisition of single-molecule data. T.K. contributed the development of the design strategy and MD simulations, and wrote the manuscript.
Correspondence to Kiyoto Kamagata or Tomoshi Kameda.
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https://doi.org/10.1038/s41598-019-44688-0 | CommonCrawl |
A Rate-Reduced Neuron Model for Complex Spiking Behavior
Koen Dijkstra ORCID: orcid.org/0000-0002-4076-26621,
Yuri A. Kuznetsov1,2,
Michel J. A. M. van Putten3,4 &
Stephan A. van Gils1
The Journal of Mathematical Neuroscience volume 7, Article number: 13 (2017) Cite this article
We present a simple rate-reduced neuron model that captures a wide range of complex, biologically plausible, and physiologically relevant spiking behavior. This includes spike-frequency adaptation, postinhibitory rebound, phasic spiking and accommodation, first-spike latency, and inhibition-induced spiking. Furthermore, the model can mimic different neuronal filter properties. It can be used to extend existing neural field models, adding more biological realism and yielding a richer dynamical structure. The model is based on a slight variation of the Rulkov map.
Networks of coupled neurons quickly become analytically intractable and computationally infeasible due to their large state and parameter spaces. Therefore, starting with the work of Beurle [1], a popular modeling approach has been the development of continuum models, called neural fields, that describe the average activity of large populations of neurons (Wilson and Cowan [2, 3], Nunez [4], Amari [5, 6]). In neural field models, the network architecture is represented by connectivity functions and the corresponding transmission delays, while differential operators characterize synaptic dynamics. All intrinsic properties of the underlying neuronal populations are condensed into firing rate functions, which replace individual neuronal action potentials and map the sum of all incoming synaptic currents to an outgoing firing rate. While some neural field models incorporate spike-frequency adaptation (Pinto and Ermentrout [7, 8], Coombes and Owen [9], Amari [10, 11]), more complex spiking behavior such as postinhibitory rebound, phasic spiking and accommodation, first-spike latency, and inhibition-induced spiking is mostly absent, an exception being the recent reduction of the Izhikevich neuron (Nicola and Campbell [12], Visser and van Gils [13]).
Here, we present a rate-reduced model that is based on a slight modification of the Rulkov map (Rulkov [14], Rulkov et al. [15]), a phenomenological, map-based single neuron model. Similar to Izhikevich neurons (Izhikevich [16]), the Rulkov map can mimic a wide variety of biologically realistic spiking patterns, all of which are preserved by our rate formulation. The rate-reduced model can therefore be used to incorporate all the aforementioned types of spiking behavior into existing neural field models.
This paper is organized as follows. In Sect. 2, we present the single spiking neuron model our rate-reduced model is based upon, and illustrate different spiking patterns and filter properties. In Sect. 3 we heuristically reduce the single neuron model to a rate-based formulation, and show that the rate-reduced model preserves spiking and filter properties. We give an example of a neural field that is augmented with our rate model in Sect. 4 and end with a discussion in Sect. 5.
Single Spiking Neuron Model
In this section we present a phenomenological, map-based single neuron model, which is a slight modification of the Rulkov map (Rulkov [14], Rulkov et al. [15]). The Rulkov map was designed to mimic the spiking and spiking-bursting activity of many real biological neurons. It has computational advantages because the map is easier to iterate than continuous dynamical systems. Furthermore, as we will show in this paper, it is straightforward to obtain a rate-reduced version of a slightly modified version of the Rulkov model.
The Rulkov map consists of a fast variable v, resembling the neuronal membrane potential, and a slow adaptation variable a. In our modification of the original model, the adaptation only implicitly depends on the membrane potential through a binary spiking variable. As we will show in the next section, this modification allows for an easy decoupling of the membrane potential and adaptation variable, and therefore a straightforward rate reduction of the model. The cost of the modification is the loss of subthreshold oscillation dynamics. The modified Rulkov map is given by
$$ \textstyle\begin{cases} v_{n+1} = f(v_{n}, v_{n-1},\kappa u_{n}-a_{n}-\theta), \\ a_{n+1} = a_{n}-\varepsilon(a_{n}+(1-\kappa)u_{n}-\gamma s_{n}), \end{cases} $$
(SNM)
where the piecewise continuous function \(f\colon\mathbb{R}^{3}\to \mathbb{R}\) is given by
$$ f(x_{1},x_{2},x_{3})= \textstyle\begin{cases}\frac{2500+150x_{1}}{50-x_{1}}+50x_{3} &\text{if } x_{1}< 0, \\ 50+50x_{3} &\text{if } 0\leq x_{1}< 50+50x_{3} \quad\wedge\quad x_{2}< 0, \\ -50 &\text{otherwise}. \end{cases} $$
The form of f is chosen to mimic the shape of neuronal action potentials. The variable u in (SNM) represents external (synaptic) input to the cell, which we assume to be given, and s is a binary indicator variable, given by
$$ s_{n} = \textstyle\begin{cases} 1&\text{if the neuron spiked at iteration $n$,} \\ 0 &\text{otherwise.} \end{cases} $$
A Rulkov neuron spikes at iteration n if its membrane potential is reset to \(v_{n+1}=-50\) in the next iteration. It follows from (1) that the spiking condition in (2) is satisfied if and only if
$$ v_{n}\geq0\quad\wedge\quad \bigl(v_{n}\geq50+50(\kappa u_{n}-a_{n}-\theta ) \quad\vee\quad v_{n-1}\geq0 \bigr). $$
The dependence of \(v_{n+1}\) on \(v_{n-1}\) in (SNM) ensures that a neuron always spikes if its membrane potential is non-negative for two consecutive iterations, independent of the external input u. To mimic spiking patterns of real biological neurons, one time step should correspond to approximately 0.5 ms of time.
The parameter \(0<\varepsilon<1\) in (SNM) sets the time scale of the adaptation variable and γ determines the adaptation strength. The parameter θ can be interpreted as a spiking threshold: for constant external input \(u_{n}=\varphi\), the neuron spikes persistently if and only if \(\varphi>\theta\). After a change of variable \(a_{n}\rightarrow a_{n}+(1-\kappa)\varphi\) and parameter \(\theta\rightarrow\theta-\varphi\), constant external input vanishes. Therefore, the asymptotic response to constant input does not depend on the parameter κ. However, the parameter κ can be used to tune the transient response of the neuron to changes in external input, as it determines how input is divided between the fast and the slow subsystem of (SNM). For parameter values \(\kappa \in[0,1]\), κ can be interpreted as the fraction of the input that is applied to the fast subsystem, and therefore determines (together with ε) how quickly the membrane potential dynamics react to changes in input. Since the effective drive of the system is given by \(\kappa u_{n}-a_{n}\), changes in external input are initially magnified for \(\kappa>0\). Asymptotically, this is then counterbalanced by additional adaption. Finally, for \(\kappa<0\), the initial response of the membrane potential to a change in input is reversed, i.e. an increase in external input initially has an inhibitory effect, and a decrease in external input initially has an excitatory effect.
Fast Dynamics
The Rulkov map (SNM) with \(0<\varepsilon\ll1\) is a slow-fast system, and we can explore the fast spiking dynamics of the model by assuming the suprathreshold drive \(\kappa u_{n}-a_{n}-\theta =\varsigma\) is constant. In this case, (SNM) reduces to the fast subsystem
$$ v_{n+1}= \textstyle\begin{cases}\frac{2500+150v_{n}}{50-v_{n}}+50\varsigma& \text{if } v_{n}< 0, \\ 50+50\varsigma& \text{if } 0\leq v_{n}< 50+50\varsigma, \\ -50 & \text{otherwise}. \end{cases} $$
(FSS)
The map (FSS) undergoes a saddle-node bifurcation at \(\varsigma=0\) (Fig. 1). For \(\varsigma<0\) there exist a stable and an unstable fixed point, given by
$$ v_{\mathrm{s}}=25 \bigl(\varsigma-2-\sqrt{\varsigma^{2}-8\varsigma } \bigr), \qquad v_{\mathrm{u}}=25 \bigl(\varsigma-2+\sqrt{\varsigma ^{2}-8\varsigma} \bigr), $$
respectively (Fig. 1A), while the system will settle into a stable periodic orbit for \(\varsigma>0\) (Fig. 1B). In the former case the unstable fixed point acts as an excitation threshold: if the value of the membrane potential exceeds this point, it will spike once and then decay back to the stable equilibrium. Since the unstable fixed point \(v_{\mathrm{u}}\) always lies to the right of the 'reset potential' \(v=-50\), a stable fixed point and a periodic orbit can never coexist. This guarantees that we can define a firing rate function \(S\colon\mathbb{R}\to\mathbb{Q}\) for the fast subsystem (FSS), given by
$$ S(\varsigma)= \textstyle\begin{cases} 0 & \text{for } \varsigma\leq0,\\ \frac{1}{P(\varsigma)} & \text{for } \varsigma>0, \end{cases} $$
where \(P\colon\mathbb{R}_{>0}\to\mathbb{N}\) maps the drive to the period of the corresponding stable limit cycle of (FSS). The fast subsystem (FSS) is piecewise-defined on the 'left' interval \((-\infty,0)\), the 'middle' interval \([0,50+50\varsigma)\), and the 'right' interval \([50+50\varsigma,\infty)\). The left interval is mapped to the left and middle interval, and the middle and right interval are mapped to right and left interval, respectively. The period of a limit cycle of (FSS) therefore only depends on the number of iterations in the left interval. Note, however, that the shape of the function f given in (1) can easily be changed to support bistability in the fast subsystem, which allows for some additional dynamics such as 'chattering', a response of periodic bursts of spikes to constant input (Rulkov [14]).
Illustration of the fast subsystem (FSS) of (SNM). ( A ) For \(\varsigma=-\tfrac{1}{10}\) there exist a stable (green) and unstable (orange) fixed point. ( B ) For \(\varsigma=\tfrac{1}{10}\) the system will settle into a stable periodic orbit (dashed green line) with period \(P (\tfrac{1}{10} )=8\)
Spiking Patterns
Izhikevich [17] classified different features of biological spiking neurons, most of which can be mimicked by our modified Rulkov model (SNM). In the following, we discuss the role of the model parameters with the help of a few physiologically relevant examples.
Tonic Spiking/Fast Spiking
Tonically spiking (also called 'fast spiking') neurons respond to a step input with spike trains of constant frequency. Most inhibitory neurons are fast spiking (Izhikevich [17]). In the modified Rulkov model this can be achieved by choosing a 'large' \((1>\varepsilon>\frac{1}{10} )\) value for the time scale parameter, in which case the influence of a single spike on the adaptation variable decays very fast. Therefore, the value of the adaptation variable is dominated by the timing of the last spike and the influence of older spikes is negligible (Fig. 2A). Since the time scale separation is small, the qualitative dynamics does not depend on κ.
Different types of spiking patterns generated by the single neuron model (SNM). Corresponding parameter values \((\theta,\kappa,\varepsilon,\gamma )\) are given in brackets. ( A ) Tonic spiking \((\tfrac{1}{10},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2} )\). ( B ) Spike-frequency adaptation \((\tfrac{1}{10},1,\tfrac{1}{1000},5 )\). ( C ) Rebound spiking \((\tfrac{1}{50},2,\tfrac{1}{100},\frac{1}{5} )\). ( D ) Accommodation \((\tfrac{3}{25},3,\tfrac{1}{50},\tfrac{2}{5} )\). ( E ) Spike latency \((\tfrac{1}{10},0,\tfrac{1}{200},\tfrac{2}{5} )\). ( F ) Inhibition-induced spiking \((\tfrac{1}{50},-1,\tfrac{1}{500},\tfrac{2}{5} )\)
Spike-Frequency Adaptation/Regular Spiking
Most cortical excitatory neurons are not 'fast spiking', but respond to a step input with a spike train of slowly decreasing frequency, a phenomenon known as 'spike-frequency adaptation' (also called 'regular spiking'). This kind of spiking behavior can be modeled by applying all input to the fast subsystem (\(\kappa=1\)) and choosing \(\varepsilon\ll 1\). The adaptation variable then acts as a slow time scale, such that a single spike has a long-lasting effect on the adaptation variable (Fig. 2B). The level of adaptation can be controlled with γ.
Rebound Spiking and Accommodation
The excitability of some neurons is temporarily enhanced after they are released from hyperpolarizing current, which can result in the firing of one or more 'rebound spikes'. Rebound spiking is an important mechanism for central pattern generation for heartbeat and other motor patterns in many neuronal systems (Chik et al. [18]). In the modified Rulkov map, postinhibitory rebound spiking can be modeled by choosing \(\kappa>1\). In this case, the adaptation variable will become negative while the cell gets hyperpolarized, which can be sufficient to trigger temporary spiking once the inhibitory input is turned off (Fig. 2C). Similarly, excitatory 'subthreshold' (\(u_{n}<\theta\)) input can elicit temporary spiking if the input is ramped up sufficiently fast (Fig. 2D).
Spike Latency and Inhibition-Induced Spiking
If all input is applied to the slow subsystem (\(\kappa=0\)), there can be a large latency between the input onset and the first spike of the neuron, yielding a delayed response to a pulse input (Fig. 2E). For \(\kappa<0\), the initial response of the model to changes in input is reversed: excitation initially leads to hyperpolarization of the neuron and inhibition can induce temporary spiking (Fig. 2F). This inhibition-induced spiking is a feature of many thalamo-cortical neurons (Izhikevich [17]).
Neuronal Filtering
In the previous section, we illustrated how the parameter κ can tune transient spiking responses of the modified Rulkov map to changes in external input. In reality, neurons often receive strong periodic input, e.g. from a synchronous neuronal population nearby. Information transfer between neurons may be optimized by temporal filtering, which is especially important when the same signal transmits distinct messages (Blumhagen et al. [19]). In this section, we study the response of (SNM) to harmonic input
$$ u_{n}= \varphi\cos \biggl(\frac{\omega\pi n}{1000}+\vartheta \biggr), $$
with amplitude φ, phase shift \(\vartheta\in[0,2\pi)\), and where \(\omega\in[0,1000]\) corresponds to the input frequency in Hz assuming that one iteration of (SNM) corresponds to 0.5 ms of time. A Rulkov neuron (SNM) will never spike if
$$ \kappa u_{n}-a_{n}\leq\theta \quad\forall n. $$
In this case, the adaptation reduces to the simple linear equation
$$ a_{n+1} = (1-\varepsilon)a_{n}-\varepsilon(1- \kappa)u_{n}, $$
with explicit solution
$$ a_{n} = -\varepsilon(1-\kappa)\sum _{m=1}^{\infty }(1-\varepsilon)^{m-1}u_{n-m}. $$
Inserting (6) into (9) now yields
$$ \begin{aligned}[b] \kappa u_{n}-a_{n}&= \kappa\varphi\cos \biggl(\frac{\omega\pi n}{1000}+\vartheta \biggr)\\ &\quad {}+\varepsilon(1- \kappa)\varphi\sum_{m=1}^{\infty}(1- \varepsilon)^{m-1}\cos \biggl(\frac{\omega\pi (n-m)}{1000}+\vartheta \biggr) \\ &=F(\omega)\frac{\varphi}{2}e^{(\frac{\omega\pi n}{1000}+\vartheta )i}+\overline{F(\omega)} \frac{\varphi}{2}e^{-(\frac{\omega\pi n}{1000}+\vartheta)i} \\ &=\bigl\lvert F(\omega)\bigr\rvert \varphi\cos \biggl(\frac{\omega\pi n}{1000}+ \vartheta+\arg \bigl(F(\omega) \bigr) \biggr), \end{aligned} $$
where the overline denotes complex conjugation and the frequency response \(F\colon[0,1000]\mapsto\mathbb{C}\) is given by
$$ F(\omega)=\kappa+\frac{\varepsilon(1-\kappa)}{e^{\frac{\omega \pi i}{1000}}+\varepsilon-1}. $$
The absolute value and argument of the frequency response determine the relative magnitude and phase of the output, respectively. It follows that a Rulkov neuron (SNM) receiving periodic input (6) does not spike if
$$ \bigl\lvert F(\omega)\bigr\rvert \varphi\leq\theta. $$
The inverse statement is not true, even if ω and ϑ in (6) are chosen such that
$$ \cos \biggl(\frac{\omega\pi n}{1000}+\vartheta+\arg \bigl(F(\omega ) \bigr) \biggr)=1 \quad\text{for some }n\in\mathbb{N}. $$
Since it can take a few iterations of the map to converge to its periodic orbit, a neuron will only spike if its drive is larger than the threshold θ for a sufficiently long time. The modulus of the frequency response (11) is given by
$$ \bigl\lvert F(\omega) \bigr\rvert =\sqrt{\frac{\varepsilon^{2}+2\kappa (\kappa-\varepsilon ) (1-\cos (\frac{\omega\pi }{1000} ) )}{\varepsilon^{2}+2 (1-\varepsilon ) (1-\cos (\frac{\omega\pi}{1000} ) )}}, $$
and it follows that \(\lvert F\rvert\) is strictly decreasing if and only if \(\kappa\in(-1+\varepsilon,1)\), and increasing otherwise (Fig. 3). Clearly,
$$ F(0)=1, \qquad F(1000)=\frac{2\kappa-\varepsilon}{2-\varepsilon}. $$
The input parameter κ can therefore be used to model filter properties of the neuron. For \(-1+\varepsilon<\kappa<1\) high frequencies get attenuated and a neuron can act as a low-pass filter in the sense that periodic input within a certain amplitude range only elicits a spiking response if its frequency is low enough (Fig. 4A). Similarly, for \(\kappa>1\) (and \(\kappa<-1+\varepsilon\)), high frequencies get amplified and there exists an amplitude range for which the neuron acts as a high-pass filter (Fig. 4B).
Illustration of the frequency response (11) for different values of ε. ( A ) For \(\kappa=\tfrac{1}{10}\) high frequencies get attenuated. ( B ) For \(\kappa=2\) high frequencies get amplified. Note the similarity, which is caused by the fact that \(F(\omega)-1\) is an odd function of \(1-\kappa\)
Responses of (SNM) to periodic input, illustrating neuronal filter properties. ( A ) For \(\kappa=\tfrac{1}{10}\) the neuron acts as a low-pass filter. Input with an amplitude of \(\varphi=\tfrac{1}{5}\) elicits a spiking response for \(\omega=1\), whereas the neuron is quiescent for \(\omega=2\). ( B ) For \(\kappa=2\), the neuron acts as a high-pass filter. Input with amplitude \(\varphi=\tfrac{1}{10}\) elicits a spiking response for \(\omega=2\), whereas a lower input frequency of \(\omega=1\) does not. In both examples, \((\theta,\varepsilon,\gamma )= (\tfrac{1}{7},\tfrac{1}{200},2 )\)
The Rate-Reduced Neuron Model
Neural field models are based on the assumption that neuronal populations convey all relevant information in their (average) firing rates. If one wants to incorporate certain spiking dynamics, one has to come up with a corresponding rate-reduced formulation first. In this section we present a rate-reduced version of the Rulkov model (SNM) that can be used to extend existing neural field models.
The adaptation variable a in the spiking neuron model (SNM) only implicitly depends on the membrane potential v via the binary spiking variable s. We can therefore decouple the adaption variable from the membrane potential by replacing the binary spiking variable defined in (2) by the instantaneous firing rate (5) of the fast subsystem (FSS), yielding
$$ a_{n+1} = a_{n}-\varepsilon \bigl(a_{n}+(1-\kappa)u_{n}-\gamma S(\kappa u_{n}-a_{n}-\theta) \bigr). $$
By interpreting (16) as the forward discretization of an ordinary differential equation, we arrive at the continuous time rate-reduced model
$$ \frac{1}{\varepsilon}\frac{\mathrm{d}a}{\mathrm{d}t} = -a-(1-\kappa)u+ \gamma S(\kappa u-a-\theta). $$
(RNM)
The rate-reduced neuron model (RNM) preserves the dynamical features of the full model (SNM) and reproduces all previous example spiking patterns (Fig. 5).
Different types of spiking behavior generated by the rate-reduced model (RNM). Top traces show the firing rate with \(r(t)=\kappa u(t)-a(t)-\theta\). Corresponding parameter values \((\theta,\kappa,\varepsilon,\gamma)\) are given in brackets. For small values of ε (i.e. a large time scale separation), there is excellent agreement with the corresponding examples of the full model (Fig. 2), which is quantified by comparing the integral of the spiking rate in the reduced model to the number of spikes in the full model. ( A ) Tonic spiking \((\frac{1}{10},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})\); \(27.18(23)\). ( B ) Spike-frequency adaptation \((\tfrac{1}{10},1,\tfrac{1}{1000},5)\); \(29.13(29)\). ( C ) Rebound spiking \((\tfrac{1}{50},2,\tfrac{1}{100},\tfrac{1}{5})\); \(7.84(8)\). ( D ) Accommodation \((\tfrac{3}{25},3,\tfrac{1}{50},\tfrac{2}{5})\); \(3.09 (3)\). ( E ) Spike latency \((\tfrac{1}{10},0,\tfrac{1}{200},\tfrac{2}{5})\); \(16.12(16)\). ( F ) Inhibition-induced spiking \((\tfrac{1}{50},-1,\tfrac{1}{500},\tfrac{2}{5})\); \(15.75(16)\)
Frequency Response of the Reduced Model
Analogously to Sect. 2.3, we now study the response of the rate-reduced model (RNM) to sinusoidal input
$$ u(t)= \varphi\cos \biggl(\frac{\omega\pi t}{1000}+\vartheta \biggr). $$
Under the assumption that
$$ \kappa u(t)-a(t)\leq\theta\quad\forall t, $$
the explicit solution of (RNM) is given by
$$ a(t)=-\varepsilon(1-\kappa) \int_{-\infty}^{t} e^{-\varepsilon (t-\tau)}u(\tau) \,\mathrm{d} \tau, $$
cf. (9). Inserting the input (17) into (19) yields
$$ \begin{aligned} \kappa u(t)-a(t)&= \kappa\varphi\cos \biggl(\frac{\omega\pi t}{1000}+\vartheta \biggr)+\varepsilon(1-\kappa)\varphi \int _{-\infty}^{t} e^{-\varepsilon(t-\tau)}\cos \biggl( \frac{\omega\pi \tau}{1000}+\vartheta \biggr) \,\mathrm{d}\tau \\ &=G(\omega)\frac{\varphi}{2}e^{(\frac{\omega\pi t}{1000}+\vartheta )i}+\overline{G(\omega)} \frac{\varphi}{2}e^{-(\frac{\omega\pi t}{1000}+\vartheta)i} \\ &=\bigl\lvert G(\omega)\bigr\rvert \varphi\cos \biggl(\frac{\omega\pi t}{1000}+ \vartheta+\arg \bigl(G(\omega) \bigr) \biggr), \end{aligned} $$
where the frequency response \(G\colon\mathbb{R}_{\geq0}\mapsto \mathbb{C}\) is given by
$$ G(\omega)=\kappa+\frac{\varepsilon(1-\kappa)}{\varepsilon+\frac {\omega\pi i}{1000}}. $$
It follows that for the rate-reduced model (RNM) receiving harmonic input (17) we have
$$ S \bigl(\kappa u(t)-a(t)-\theta \bigr)=0 \quad\forall t \quad \text{if and only if} \quad\bigl\lvert G(\omega)\bigr\rvert \varphi\leq\theta. $$
Because we neglected the transient corresponding to the convergence from fixed point to limit cycle in the rate-reduced model (RNM), the inequality in (22) defines a clear 'spiking condition'. The modulus of the frequency response (21) is given by
$$ \bigl\lvert G(\omega)\bigr\rvert =\sqrt{\frac{\varepsilon^{2}+\kappa^{2} (\frac{\pi\omega}{1000} )^{2}}{\varepsilon^{2}+ (\frac{\pi \omega}{1000} )^{2}}}, $$
and \(\lvert G\rvert\) therefore is strictly decreasing if and only if \(\lvert\kappa\rvert\leq1\), and increasing otherwise. Revisiting the examples from Sect. 2.3 (Fig. 4), we have
$$ \bigl\lvert G(1)\bigr\rvert \varphi= 0.1696\ldots>\theta>0.1255\ldots=\bigl\lvert G(2)\bigr\rvert \varphi, $$
for \((\kappa,\varepsilon,\theta,\varphi )= (\frac {1}{10},\frac{1}{200},\frac{1}{7},\frac{1}{5} )\), and
$$ \bigl\lvert G(1)\bigr\rvert \varphi= 0.1359\ldots< \theta< 0.1684\ldots=\bigl\lvert G(2)\bigr\rvert \varphi, $$
for \((\kappa,\varepsilon,\theta,\varphi )= (2,\frac {1}{200},\frac{1}{7},\frac{1}{10} )\). Indeed, the rate-reduced model (RNM) reproduces the examples of the full model both qualitatively and quantitatively (Fig. 6). When the rate-reduced model (RNM) is incorporated into existing neural field models, the frequency response of the reduced model can be used to tune the individual temporal filter properties of the different neuronal populations.
Responses of the rate-reduced model (RNM) to periodic input. Top traces show the firing rate with \(r(t)= \kappa u(t)-a(t)-\theta\). ( A ) For \(\kappa=\frac{1}{10}\) the model acts as a low-pass filter. Input with an amplitude of \(\varphi=\frac{1}{5}\) yields a response in the firing rate for \(\omega=1\), whereas the firing rate remains zero for \(\omega=2\). In the former case, the integral of the spiking rate during one period is approximately 4.55, while there are 5 spikes per period in the full model (Fig. 4A). ( B ) For \(\kappa=2\), the reduced model acts as a high-pass filter. Input with amplitude \(\varphi=\tfrac{1}{10}\) elicits a firing rate response for \(\omega=2\), whereas a lower input frequency of \(\omega=1\) does not. In the former case, the integral of the spiking rate during one period is approximately 3.14, while there are 3 spikes per period in the full model (Fig. 4B). In both examples, \((\theta,\varepsilon,\gamma )= (\tfrac{1}{7},\tfrac{1}{200},2 )\)
The Firing Rate Function
Since our neuron model (SNM) is a map, the period P of its limit cycle lies in \(\mathbb{N}\) for all positive suprathreshold drives ς. Therefore, the spiking rate function (5) is staircase-like, with points of discontinuity whenever \(P\to P+1\). Let \(\lbrace\varsigma_{1},\varsigma_{2},\ldots\rbrace\) denote the set of all points of discontinuity of the firing rate function in decreasing order. For \(\varsigma\geq\varsigma_{1}=1\) the 'reset potential' \(v=-50\) in (FSS) is immediately mapped to a non-negative number, and the neuron is therefore spiking at its maximal frequency of once in three iterations. Similarly, the voltage stays in the left interval for two iterations and the neuron is spiking once in four iterations for \(\varsigma_{1}>\varsigma\geq\varsigma_{2}=\frac {1}{2}(5-\sqrt{17})\). In general, at \(\varsigma_{k}\), there is a jump discontinuity of size
$$ \lim_{\varsigma\rightarrow\varsigma_{k}^{+}}S(\varsigma)-\lim_{\varsigma\rightarrow\varsigma_{k}^{-}}S( \varsigma)=\frac {1}{(k+2)(k+3)},\quad\text{with } S(\varsigma_{k})= \frac{1}{k+2}. $$
The firing rate of the fast subsystem (FSS) can therefore be written as
$$ S(\varsigma)=\sum_{k=1}^{\infty} \frac{H(\varsigma -\varsigma_{k})}{(k+2)(k+3)}, $$
where H is the Heaviside step function and
$$ \lim_{k\to\infty}\varsigma_{k}=0. $$
In large neuronal networks, it is often assumed that the spiking thresholds of the individual neurons are randomly distributed. This ensures heterogeneity and models intrinsic interneuronal differences or random input from outside the network. If we add Gaussian noise to the threshold parameter θ in (SNM), it is natural to define an expected firing rate \(\langle S\rangle\colon\mathbb{R}\mapsto\mathbb{R}\), given by
$$ \bigl\langle S(\varsigma)\bigr\rangle =\frac{1}{\sqrt{2\pi\sigma^{2}}} \int _{-\infty}^{\infty}e^{\frac{-w^{2}}{2\sigma^{2}}}S(\varsigma +w) \, \mathrm{d}w, $$
where \(\sigma^{2}\) is the variance of the noise. Using (27), we can rewrite (29) as
$$ \bigl\langle S(\varsigma)\bigr\rangle =\frac{1}{6}+\sum _{k=1}^{\infty }\frac{\operatorname{erf}(\frac{\varsigma-\varsigma_{k}}{\sqrt{2\sigma ^{2}}} )}{2(k+2)(k+3)}, $$
where erf denotes the error function. While \(S(\varsigma)\) can readily be computed for any \(\varsigma\in\mathbb{R}\) and we derived a concise expression for the expected firing rate, the infinite sum (30) cannot easily be evaluated. For this reason, we approximate \(\langle S(\varsigma)\rangle\) by a finite sum of the form
$$ \frac{1}{6}+\frac{1}{6N}\sum _{i=1}^{N}\operatorname{erf}\biggl(\frac {\varsigma-\nu_{i}}{\chi_{i}} \biggr), $$
for some fixed \(N\in\mathbb{N}\) and constants \(\nu_{i},\chi_{i}\in \mathbb{R}\), which are chosen by (numerically) minimizing
$$ \Biggl\lVert \frac{1}{\sqrt{2\pi\sigma^{2}}} \int_{-\infty }^{\infty}e^{\frac{-w^{2}}{2\sigma^{2}}}S(\varsigma+w) \, \mathrm{d}w-\frac{1}{6}-\frac{1}{6N}\sum _{i=1}^{N}\operatorname{erf}\biggl(\frac{\varsigma-\nu_{i}}{\chi_{i}} \biggr) \Biggr\rVert _{2}. $$
For large noise levels \(\sigma^{2}\), the average firing rate (29) has a sigmoidal shape and can be very well approximated with a small value of N (Fig. 7).
Expected firing rate for a noise level of \({\sigma^{2}=\tfrac{1}{4}}\). Shown are a numerical integration of (29) (blue) and its approximation (31) for \(N=2\) and \((\nu_{1},\nu_{2},\chi_{1},\chi_{2} )= (0.0335, [4] 0.7099,0.6890,0.8213 )\) (orange)
Augmenting Neural Fields
When large populations of neurons are modeled by networks of individual, interconnected cells, the high dimensionality of state and parameter spaces makes mathematical analysis intractable and numerical simulations costly. Moreover, large network simulations provide little insight into global dynamical properties. A popular modeling approach to circumventing the aforementioned problems is the use of neural field equations. These models aim to describe the dynamics of large neuronal populations, where spikes of individual neurons are replaced by (averaged) spiking rates and space is continuous. Another advantage of neural fields is that they are often well suited to model experimental data. In brain slice preparations, spiking rates can be measured with an extracellular electrode, while intracellular recordings are much more involved. Furthermore, the most common clinical measurement techniques of the brain, electroencephalography (EEG) and functional magnetic resonance imaging (fMRI), both represent the average activity of large groups of neurons and may therefore be better modeled by population equations. The first neural field model can be attributed to Beurle [1], however, the theory really took off with the work of Wilson and Cowan [2, 3], Amari [5, 6], and Nunez [4].
In 'classical' neural field models the firing rate of a neuronal population is assumed to be given by its instantaneous input, which is only valid for tonically spiking neurons. With the help of our rate-reduced model (RNM), it is straightforward to augment existing neural field models with more complex spiking behavior. As an example, we will look at the following two-population model on the one-dimensional spatial domain \(\varOmega=(-1,1)\):
$$ \begin{aligned} \biggl(1+\frac{1}{\alpha_{1}} \frac{\partial}{\partial t} \biggr) u_{1}(t,x) &= \int_{-1}^{1}J_{11} \bigl(x,x' \bigr)S_{1} \bigl(r_{1}\bigl(t,x'\bigr) \bigr)\\ &\quad {}+J_{12} \bigl(x,x' \bigr)S_{2} \bigl(r_{2}\bigl(t,x'\bigr) \bigr) \,\mathrm{d}x', \\ \biggl(1+\frac{1}{\varepsilon_{1}}\frac{\partial}{\partial t} \biggr) a_{1}(t,x) &= -(1-\kappa_{1})u_{1}(t,x)+\gamma_{1} S_{1} \bigl(r_{1}(t,x) \bigr), \\ \biggl(1+\frac{1}{\alpha_{2}}\frac{\partial}{\partial t} \biggr) u_{2}(t,x) &= \int_{-1}^{1}J_{21} \bigl(x,x' \bigr)S_{1} \bigl(r_{1}\bigl(t,x'\bigr) \bigr)\\ &\quad {}+J_{22} \bigl(x,x' \bigr)S_{2} \bigl(r_{2}\bigl(t,x'\bigr) \bigr) \,\mathrm{d}x', \\ \biggl(1+\frac{1}{\varepsilon_{2}}\frac{\partial}{\partial t} \biggr) a_{2}(t,x) &= -(1-\kappa_{2})u_{2}(t,x)+\gamma_{2} S_{2} \bigl(r_{2}(t,x) \bigr), \end{aligned} $$
(ANF)
where, as before,
$$ r_{i}(x,t)=\kappa_{i} u_{i}(t,x)-a_{i}(t,x)- \theta_{i}, $$
for \(i\in\lbrace1,2\rbrace\). The differential operators in the left-hand side of the integral equations in (ANF) model exponentially decaying synaptic currents with decay rate \(\alpha_{i}\). The connectivity \(J_{ij} (x,x' )\) measures the connection strength from neurons of population j and position \(x'\) to neurons of population i and position x. The connectivity kernels \(J_{ij}\colon \overline{\varOmega}\times\overline{\varOmega}\mapsto\mathbb{R}\) are assumed to be isotropic and given by
$$ J_{ij}\bigl(x,x'\bigr)=\rho_{j} \eta_{ij}e^{-\mu_{ij}\lvert x-x'\rvert}, $$
where \(\rho_{j}\) is the density of neurons of type j, \(\eta_{ij}\) is the maximal connection strength, and \(\mu_{ij}\) is the spatial decay rate of the connectivity. Both firing rate functions \(S_{i}\colon\mathbb{R}\mapsto\mathbb{R}\) are chosen to approximate the expected firing rate of Rulkov neurons (SNM) with a noise level of \(\sigma^{2}=\frac{1}{4}\) (Fig. 7),
$$ S_{1}(\varsigma)=S_{2}(\varsigma)=\frac{1}{6}+ \frac{1}{12}\operatorname{erf}\biggl(\frac{\varsigma-0.0335}{0.6890} \biggr)+\frac{1}{12}\operatorname{erf}\biggl(\frac{\varsigma-0.7099}{0.8213} \biggr). $$
We conclude this section with a simulation of (ANF) for a particular parameter set (Table 1), which illustrates that our augmented neural field can generate interesting spatiotemporal behavior that closely resembles spiking patterns of a network of Rulkov neurons (SNM) with corresponding parameter values (Fig. 8). In the Rulkov network, synaptic input to neuron i is given by
$$ u^{(i)}_{n+1} = (1-\alpha_{i})u^{(i)}_{n}+ \alpha_{i}\sum_{j=1}^{N} c_{ij}s^{(j)}_{n}, $$
where N denotes the total number of neurons in the network, and \(c_{ij}\) is the connection strength from neuron j to neuron i. To match the parameters in Table 1, we split the total population in two subpopulations of 300 neurons each, which are both equidistantly placed on the interval \([-1,1]\). Neurons within the same subpopulation share the same intrinsic parameters, and uncorrelated (in space and time) Gaussian noise is added to the threshold parameters. Finally, the connection strengths in the Rulkov network are given by
$$ c_{ij}=\eta_{{p_{i}}{p_{j}}}e^{-\mu_{{p_{i}}{p_{j}}}\lvert x_{i}-x_{j}\rvert}, $$
where \(p_{i}\) and \(x_{i}\) are the subpopulation and position of neuron i, respectively.
Spatio-temporal spiking patterns. ( A ) Simulation of the augmented neural field (ANF) with parameter values given in Table 1. Shown is the firing rate \(S_{1} (\kappa_{1} u_{1}(t,x)-a_{1}(t,x)-\theta_{1} )\) of the first population. ( B ) Simulation of a corresponding network of 300 excitatory and 300 inhibitory Rulkov neurons, all-to-all coupled via simple exponential synapses. Both populations are equidistantly placed on the interval \([-1,1]\). Uncorrelated (in space and time) Gaussian noise with variance \(\sigma^{2}=\tfrac{1}{4}\) is added to the threshold parameter of each neuron. Shown is the spiking activity of the excitatory population. Each spike is denoted by a black dot
Table 1 Parameter overview for the neural field ( ANF )
This paper presents a simple rate-reduced neuron model that is based on a variation of the Rulkov map (Rulkov [14], Rulkov et al. [15]), and can be used to incorporate a variety of non-trivial spiking behavior into existing neural field models.
The modified Rulkov map (SNM) is a phenomenological, two-dimensional single neuron model. The isolated dynamics of its fast time scale either generates a stable limit cycle, mimicking spiking activity, or a stable fixed point, corresponding to a neuron at rest (Fig. 1). The slow time scale of the Rulkov map acts as a dynamic spiking threshold and emulates the combined effect of slow recovery processes. The modified Rulkov map can mimic a wide variety of spiking patterns, such as spike-frequency adaptation, postinhibitory rebound, phasic spiking, spike accommodation, spike latency and inhibition-induced spiking (Fig. 2). Furthermore, the model can be used to model neuronal filter properties. Depending on how external input is applied to the model, it can act as either a high-pass or low-pass filter (Figs. 3 and 4).
The rate-reduced model (RNM) is derived heuristically and given by a simple one-dimensional differential equation. On the single cell level, the rate-reduced model closely mimics the spiking dynamics (Fig. 5) and filter properties (Fig. 6) of the full spiking neuron model. While a close approximation of the (expected) firing rate of Rulkov neurons (Fig. 7) is needed to mimic their behavior quantitatively, the types of qualitative dynamics of the rate-reduced model do not depend on the exact choice of firing rate function.
Due to its simplicity, it is straightforward to add the rate-reduced model to existing neural field models. In the resulting augmented equations, parameters can be chosen according to the spiking behavior of a single isolated cell. In our particular example (ANF), the emerging spatiotemporal pattern closely resembles the dynamics of the corresponding spiking neural network (Fig. 8). We believe that this is an elegant way to add more biological realism to existing neural field models, while simultaneously enriching their dynamical structure.
We used a variation of a simple toy model of a spiking neuron (Rulkov [14], Rulkov et al. [15]) to derive a corresponding rate-reduced model. While being purely phenomenological, the model could mimic a wide variety of biologically observed spiking behaviors, yielding a simple way to incorporate complex spiking behavior into existing neural field models. Since all parameters in the resulting augmented neural field equations have a representative in the spiking neuron network (and vice versa), this greatly simplifies the otherwise often problematic translation from results obtained by neural field models back to biophysical properties of spiking networks. An example demonstrated that the augmented neural field equations can produce spatiotemporal patterns that cannot be generated with corresponding 'classical' neural fields.
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The conclusions of this paper are solely based on mathematical models.
K.D. was supported by a grant from the Twente Graduate School (TGS).
Koen Dijkstra, Yuri A. Kuznetsov & Stephan A. van Gils
Department of Clinical Neurophysiology, University of Twente, Enschede, The Netherlands
Michel J. A. M. van Putten
Department of Clinical Neurophysiology, Medisch Spectrum Twente, Enschede, The Netherlands
Koen Dijkstra
Conceptualization, K.D., Y.K., M.v.P. and S.v.G.; methodology, K.D. and S.v.G.; investigation, K.D.; writing original Draft, K.D.; writing review & Editing, K.D., Y.K., M.v.P. and S.v.G.; visualization, K.D.; supervision, Y.K., M.v.P. and S.v.G. All authors read and approved the final manuscript.
Correspondence to Koen Dijkstra.
Our study don't involve human participants, human data or human tissue.
The authors declare no competing financial interests.
This manuscript does not contain any individual person's data.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Dijkstra, K., Kuznetsov, Y.A., van Putten, M.J.A.M. et al. A Rate-Reduced Neuron Model for Complex Spiking Behavior. J. Math. Neurosc. 7, 13 (2017). https://doi.org/10.1186/s13408-017-0055-3
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Journal of Software Engineering Research and Development
An algorithm for combinatorial interaction testing: definitions and rigorous evaluations
Juliana M. Balera ORCID: orcid.org/0000-0001-6481-53621 &
Valdivino A. de Santiago Júnior1
Journal of Software Engineering Research and Development volume 5, Article number: 10 (2017) Cite this article
Combinatorial Interaction Testing (CIT) approaches have drawn attention of the software testing community to generate sets of smaller, efficient, and effective test cases where they have been successful in detecting faults due to the interaction of several input parameters. Recent empirical studies show that greedy algorithms are still competitive for CIT. It is thus interesting to investigate new approaches to address CIT test case generation via greedy solutions and to perform rigorous evaluations within the greedy context.
We present a new greedy algorithm for unconstrained CIT, T-Tuple
Reallocation (TTR), to generate CIT test suites specifically via the Mixed-value Covering Array (MCA) technique. The main reasoning behind TTR is to generate an MCA M by creating and reallocating t-tuples into this matrix M, considering a variable called goal (ζ). We performed two controlled experiments addressing cost-efficiency and only cost. Considering both experiments, we did 3200 executions related to 8 solutions. In the first controlled experiment, we compared versions 1.1 and 1.2 of TTR in order to check whether there is significant difference between both versions of our algorithm. In such experiment, we jointly considered cost (size of test suites) and efficiency (time to generate the test suites) in a multi-objective perspective. In the second controlled experiment we confronted TTR 1.2 with five other greedy algorithms/tools for unconstrained CIT: IPOG-F, jenny, IPO-TConfig, PICT, and ACTS. We performed two different evaluations within this second experiment where in the first one we addressed cost-efficiency (multi-objective) and in the second only cost (single objective).
Results of the first controlled experiment indicate that TTR 1.2 is more adequate than TTR 1.1 especially for higher strengths (5, 6). In the second controlled experiment, TTR 1.2 also presents better performance for higher strengths (5, 6) where only in one case it is not superior (in the comparison with IPOG-F). We can explain this better performance of TTR 1.2 due to the fact that it no longer generates, at the beginning, the matrix of t-tuples but rather the algorithm works on a t-tuple by t-tuple creation and reallocation into M.
Considering the metrics we defined in this work and based on both controlled experiments, TTR 1.2 is a better option if we need to consider higher strengths (5, 6). For lower strengths, other solutions, like IPOG-F, may be better alternatives.
The academic community has been making efforts to reduce the cost of the software testing process by decreasing the size of test suites while at the same time aiming at maintaining the effectiveness (ability to detect defects) of such sets of test cases. Hence, several contributions exist for test suite/case minimization (Yoo and Harman 2012; Ahmed 2016; Huang et al. 2016; Khan et al. 2016) where the goal is to decrease the size of a test suite by eliminating redundant test cases, and hence demanding less effort to execute the test cases (Yoo and Harman 2012). One of the approaches to reduce the number of test cases is Combinatorial Interaction Testing (CIT) (Petke et al. 2015), also known as Combinatorial Testing (CT) (Kuhn et al. 2013; Schroeder and Korel 2000), Combinatorial Test Design (CTD) (Tzoref-Brill et al. 2016), or Combinatorial Designs (CD) (Mathur 2008). CIT relates to combinatorial analysis whose objective is to answer whether it is possible to organize elements of a finite set into subsets so that certain balance or symmetry properties are satisfied (Stinson 2004).
There are reports which claim the success of CIT (Dalal et al. 1999; Tai and Lei 2002; Kuhn et al. 2004; Yilmaz et al. 2014; Qu et al. 2007; Petke et al. 2015). Such approaches have drawn attention of the software testing community to generate sets of smaller (lower cost to run) and effective (greater ability to find faults in the software) test cases where they have been successful in detecting faults due to the interaction of several input parameters (factors).
CIT approaches to generate test cases can be divided in four main classes: Binary Decision Diagrams (BDDs) (Segall et al. 2011), Satisfiability (SAT) solving (Cohen et al. 1997; Yamada et al. 2015; Yamada et al. 2016), meta-heuristics (Garvin et al. 2011; Shiba et al. 2004; Hernandez et al. 2010), and greedy algorithms (Lei and Tai 1998; Lei et al. 2007)Footnote 1. Recent CIT test case generation methods based on BDD and SAT are interesting to constrained (there are restrictions related to parameter interactions) problems but they perform worse compared with greedy algorithms/tools in the context of unconstrained (there are no restrictions at all) problems.
To corroborate this claim, in (Segall et al. 2011) a BDD-based approach, implemented in the Focus tool, was better in terms of cost than the greedy solutions Advanced Combinatorial Testing System (ACTS) (Yu et al. 2013), Pairwise Indepedent Combinatorial Testing (PICT) (Czerwonka 2006), and jenny (Jenkins 2016) in the constrained domain. However, their method was worse than such greedy solutions for unconstrained problems.
A recent SAT-based approach (Yamada et al. 2016), implemented in the Calot tool, performed well in terms of efficiency (time to generate the test suites) and cost (test suite sizes) comparing again with the greedy tools ACTS (Yu et al. 2013) and PICT (Czerwonka 2006). Despite the advantages of the SAT-based approach, ACTS was much more faster than Calot for many 3-way test case examples. Moreover, if unconstrained CIT is considered, ACTS again was remarkable faster than Calot for large SUT models and higher-strength test case generation.
In the context of CIT, meta-heuristics such as simulated annealing (Garvin et al. 2011), genetic algorithms (Shiba et al. 2004), and Tabu Search Approach (TSA) (Hernandez et al. 2010) have been used. Recent empirical studies show that meta-heurisitic and greedy algorithms have similar performance (Petke et al. 2015). Hence, early fault detection via a greedy algorithm with constraint handling (implemented in the ACTS tool (Yu et al. 2013)) was no worse than a simulated annealing algorithm (implemented in the CASA tool (Garvin et al. 2011)). Moreover, there was not enough difference between test suites generated by ACTS and CASA in terms of efficiency (runtime) and t-way coverage. All such previous remarks, some of them based on strong empirical evidences, emphasize that greedy algorithms are still very competitive for CIT.
Even if some authors have argued that CIT resides in the constrained domain in real-world applications (Bryce and Colbourn 2006; Cohen et al. 2008; Petke et al. 2015), it is important to mention that unconstrained CIT may be interesting from a practical point of view, especially for critical applications such as satellites, rockets, airplanes, controllers of an unmanned train metro system, etc. For such types of applications, robustness testing is very important. In the context of software systems, robustness testing aims to verify whether the Software Under Test (SUT) behaves correctly in the presence of invalid inputs. Therefore, even though an unconstrained CIT-derived test case may seem pointless or even somewhat difficult to execute, it may still be interesting to see how the software will behave in the presence of inconsistent inputs.
Let us consider that we need to test a communication protocol implemented in several critical embedded systems. If each field of such a protocol is a parameter, it is interesting to impose no restriction (no constraint) in the parameter interactions so that a certain Protocol Data Unit (PDU) sent from system A to system B may have values not allowed in the combination of the fields (parameters) of the PDU. In other words, if the specification says that when field f i =1, possible values of field f j are between 20 and 70 (20≤f j ≤70), and other field f k <5, then a test case where f i =1, 1≤f j ≤4, and f k <5 is clearly inconsistent because of the value of f j . But, this can precisely the goal of the test designer because he/she wants to check how the receiving system (B) will act upon receiving a PDU like that from A. This is an example where unconstrained CIT is relevant. It is important to mention that the argument is not that constraints can not be used for testing critical systems but rather that, for certain types of tests (robustness), constraints are not as relevant.
Based on the context and motivation previously presented, this research relates to greedy algorithms for unconstrained CIT. In (Pairwise 2017), 43 algorithms/tools are presented for CIT and many more not shown there exist. Some of these solutions are variations of the In-Parameter-Order (IPO) algorithm (Lei and Tai 1998) such as IPOG, IPOG-D (Lei et al. 2007), IPOG-F, IPOG-F2 (Forbes et al. 2008), IPOG-C (Yu et al. 2013), IPO-TConfig (Williams 2000), ACTS (where IPOG, IPOG-D, IPOG-F, IPOG-F2 are implemented) (Yu et al. 2013), and CitLab (Cavalgna et al. 2013). All IPO-based proposals have in common the fact that they perform horizontal and vertical growths to construct the final test suite. Moreover, some need two auxiliary matrices which may decrease its performance by demanding more computer memory. Such algorithms accomplish exhaustive comparisons within each horizontal extension which may penalize efficiency.
PICT can be regarded as one baseline tool where other approaches have been done based on it (PictMaster 2017). The algorithm implemented in this tool works in two phases, the first being the construction of all t-tuples to be covered. This can often be a not interesting solution since many t-tuples may require large disk space for storage.
Thus, it is interesting to think about a new greedy solution for CIT that does not need, at the beginning, to enumerate all t-tuples (such as PICT) and does not demand many auxiliary matrices to operate (as some IPO-based approaches). Although we have some recent rigorous empirical evaluations comparing greedy algorithms with meta-heuristics solutions (Petke et al. 2015) and greedy approaches against SAT-based methods (Yamada et al. 2016), there are no rigorous empirical assessments comparing greedy algorithms/tools, representative of the unconstrained CIT domain, among each other.
In this paper, we present a new algorithm, called T-Tuple Reallocation (TTR), to generate CIT test suites specifically via the Mixed-value Covering Array (MCA) technique. The main reasoning behind TTR is to generate an MCA M by creating and reallocating t-tuples into this matrix M, considering a variable called goal (ζ). TTR is a greedy algorithm for unconstrained CIT.
Three versions of the TTR algorithm were developed and implemented in Java. Version 1.0 is the original version of TTR (Balera and Santiago Júnior 2015). In version 1.1 (Balera and Santiago Júnior 2016), we made a change where we do not order the input parameters. In the last version, 1.2, the algorithm no longer generates the matrix of t-tuples (Θ) but rather it works on a t-tuple by t-tuple creation and reallocation into M. Moreover, version 1.2 was also implemented in C.
We performed two controlled experiments addressing cost-efficiency and only cost. Considering both experiments, we performed 3,200 executions related to 8 solutions. In the first controlled experiment, our goal was to compare versions 1.1 and 1.2 of TTR (in Java) in order to check whether there is significant difference between both versions of our algorithm. In such experiment, we jointly considered cost (size of test suites) and efficiency (time to generate the test suites) in a multi-objective perspective. We conclude that TTR 1.2 is more adequate than TTR 1.1 especially for higher strengths (5 and 6).
We then carried out a second controlled experiment where we confronted TTR 1.2 with five other greedy algorithms/tools for unconstrained CIT: IPOG-F (Forbes et al. 2008), jenny (Jenkins 2016), IPO-TConfig (Williams 2000), PICT (Czerwonka 2006), and ACTS (Yu et al. 2013). We performed two evaluations where in the first one we compared TTR 1.2 with IPOG-F and jenny since these were the solutions we had the source code (to precisely measure the time). Hence, a cost-efficiency (multi-objective) assessment was accomplished. In order to address a possible evaluation bias in the time measures due to different programming languages, we compared the implementation of TTR 1.2 (in Java) with IPOG-F (in Java), and the implementation of TTR 1.2 (in C) with jenny (in C). In the second assessment, we did a cost (single objective) evaluation where TTR 1.2 (Java) was compared with PICT, IPO-TConfig, and ACTS. The conclusion is the same as before: TTR 1.2 is better for higher strengths (5 and 6).
In this paper, we extend our previous works where we presented version 1.0 of TTR (Balera and Santiago Júnior 2015), and version 1.1 together with another controlled experiment (Balera and Santiago Júnior 2016). The contributions of this work are:
Even though we considered version 1.1 of TTR in (Balera and Santiago Júnior 2016), we did not detail this version since the focus of this previous paper was this other controlled experiment. Thus, we highlight the key features of TTR 1.1 here;
We created another version of our algorithm, 1.2, where, at the beginning, TTR does not generate the matrix of t-tuples. Our goal here is trying to avoid an exhaustive combination of t-tuples as might happen with other classical greedy approaches. Moreover, we rely on just one auxiliary matrix which is different from other greedy solutions which require two auxiliary matrices;
We performed two controlled experiments in the unconstrained CIT domain (TTR 1.1 × TTR 1.2; TTR 1.2 × IPOG-F, jenny, IPO-TConfig, PICT, ACTS) with almost three times more participants, in each experiment, than in the previous one (Balera and Santiago Júnior 2016). In addition, we run each participant (instance) 5 times with different input orders of parameters and values to address the nondeterminism of the solutions. To the best of our knowledge, no previous research presented rigorous empirical evaluations for greedy solutions within the unconstrained CIT domain;
We really accomplished a multi-objective (cost-efficiency) evaluation in both controlled experiments (in the second one, we did it in the first assessment). Previously (Balera and Santiago Júnior 2016), we analyzed cost and efficiency in isolation.
This paper is structured as follows. Section 2 presents an overview of the main concepts related to CIT. In Section 3, we show the main definitions and procedures of versions 1.1 and 1.2 of our algorithm. Section 4 shows all the details of the first controlled experiment when we compare TTR 1.1 against TTR 1.2. In Section 6, the second controlled experiment is presented where TTR is confronted with the other 5 greedy tools. Section 7 presents related work. In Section 8, we show the conclusions and future directions of our research.
In this section we present some basic concepts and definitions (Kuhn et al. 2013; Petke et al. 2015; Cohen et al. 2003) related to CIT. A CIT algorithm receives as input a number of parameters (also known as factors), p, which refer to the input variables. Each parameter can assume a number of values (also known as levels) v. Moreover, t is the strength of the coverage of interactions. For example, in pairwise testing, the degree of interaction is two, so the value of strength is 2. In t-way testing, a t-tuple is an interaction of parameter values of size equal to the strength. Thus, a t-tuple is a finite ordered list of elements, i.e. it is a set of elements.
A Fixed-value Covering Array (CA) denoted by CA(N,p,v,t) is an N×p matrix of entries from the set {0,1,⋯,(v−1)} such that every set of t-columns contains each possible t-tuple of entries at least a certain number of times (e.g. once). N is the number of rows of the array (matrix). Note that in a CA, entries are from the same set of v values.
A Mixed-value Covering Array (MCA)Footnote 2 it is an extension of a CA and it is more flexible because it allows parameters to assume values from different sets. Hence, it is represented as MCA\(\left (N,v^{p_{1}}_{1}v^{p_{2}}_{2}...v^{p_{m}}_{m}, t\right)\), where N is the number of rows of the matrix, \(\sum \limits _{i=1}^{m} p_{i}\) is the number of parameters, each v i is the number of values for each parameter p i , and t is the strength.
Therefore, in CIT a CA or MCA is a test suite and each row of such matrices is a test case. Suppose that we need to generate a pairwise unconstrained CIT test suite considering the following parameters and their respective values:
$$\begin{array}{*{20}l} OS &= \{macOS, Linux, Windows\},\\ Protocol &= \{IPv4, IPv6\},\\ DBMS &= \{MySQL, PostgreSQL, Oracle\}. \end{array} $$
We can formulate this problem as MCA (N,2132,2) which is denoted as a model for the CIT problem. In other words, we have one parameter (Protocol) which can assume two values, two parameters (OS, DBMS) which can assume three values, and t=2.
As we have mentioned in Section 1, CIT is an interesting solution for the test suite minimization problem. As a matter of perspective, let us consider that there are 10 parameters (A,B,⋯,J) and that each parameter has 5 values, i.e. A={a 1,a 2,⋯,a 5}, B={b 1,b 2,⋯,b 5},..., J={j 1,j 2,⋯,j 5}. If we performed an exhaustive combination, there would be 510=9.765.625 test cases generated where each test case is: t c i ={a k ,b k ,⋯j k }. By using version 1.2 of TTR with t=2, even in a unconstrained context, the test suite reduces to 45 test cases. This gives an idea of the strength of CIT for test suite minimization.
Note that the concepts and definitions we provided in this section are related to the context in which our work is inserted: unconstrained CIT. In case of constrained CIT, constraints must be considered and other definitions can be used (see e.g. (Yamada et al. 2016)).
TTR: a new algorithm for combinatorial interaction testing
In this section we detail versions 1.1 and 1.2 of our algorithm. The three versions (1.0 (Balera and Santiago Júnior 2015), 1.1, and 1.2) of TTR were implemented in Java.
TTR: Version 1.1
Version 1.0 of TTR (Balera and Santiago Júnior 2015) can be summarized as follows: (i) it generates all possible t-tuples that have not yet been covered. The Constructor procedure constructs the matrix Θ; (ii) it generates an initial solution, the matrix M; and (iii) it reallocates the t-tuples from Θ in order to achieve the best final solution (M) via the Main procedure. Then, the final set of test cases is updated in the matrix M. An important point here is that we order the parameters and values that are submitted to the algorithm. In other words, if we submit five parameters A,B,C,D,E with 10, 4, 3, 8, 5 values respectively, TTR orders these five parameters in ascending order: A,D,E,B,C. The goal is trying to be insensitive to the input order of parameters and values.
The same steps described above also exist in TTR 1.1. However, comparing with version 1.0 (Balera and Santiago Júnior 2015), in version 1.1 we do not order the parameters and values submitted to our algorithm. The result is that test suites of different sizes may be derived if we submit a different order of parameters and values. The motivation for such a change is because we realized that, in some cases, less test cases were created due to non-ordering of parameters and values.
Let us consider the running example in Fig. 1 with the strength, t, equals to 2. It is important to note that this is a unit testing level and hence each one of the parameters of register is an input parameter sumitted to TTR. Thus, there are 3 parameters: bank, function and card. We assume that there are two banks (bankA, bankB), two functions (debit, credit), and three types of cards (cardA, cardB, cardC) to deal with. Therefore, there are 2, 2, and 3 values of bank, function and card, respectively, as shown in Table 1.
A running example: register method
Table 1 Example of parameters and values: Fig. 1
A high-level view of version 1.1 of TTR is in Algorithm 1. The main reasoning of TTR 1.1 is to build an MCA M through the reallocation of t-tuples from a matrix Θ to this matrix M, and then each reallocated t-tuple should cover the greatest number of t-tuples not yet covered, considering a parameter called a goal (ζ). Also note that P is the submitted set of parameters, V is the set of values of the parameters, and t is the strength. As we have just pointed out, TTR 1.1 follows the same general 3 steps as we have in TTR 1.0.
Before going on with the descriptions of the procedures of our algorithm, we need to define the following operators applied to the structures (set, sequence, matrix) we handle. We also present some examples to better illustrate how such operators work.
Definition 1
Let A be a sequence and B be a set. The addition sequence-set operator, ⊙, is such that A⊙B is a sequence where the elements of B are added after the last position of A. Thus, if |A| is the length of sequence A and |B| is the cardinality of set B, |A⊙B|=|A|+|B|.
Example: Let us consider sequence A={1,2,3} and set B={4,5}. Then, A⊙B={1,2,3,4,5}.
Let A and B be two sequences with the same length, i.e. |A|=|B|. The addition sequence-sequence operator, ⊕, is such that A⊕B is a sequence where the element in position i of A⊕B, a b i , is a i , the element of A in position i, or b i , the element of B in position i. Also note the definition of an "empty" element, λ, within a sequence which is an element with no value. This operator then assumes that if a i ≠λ and b i ≠λ then a b i =a i =b i . However, if a i =λ and b i ≠λ then a b i =b i . On the other hand, if a i ≠λ and b i =λ then a b i =a i . Note that |A⊕B|=|A|=|B|.
Example: Let us consider sequences A={1,2,λ} and B={λ,2,3}. Then, A⊕B={1,2,3}.
Let A and B be two sequences. The removal operator, ⊖, is such that A⊖B is a sequence obtained by "removing" each element of B, b i , from A. This operator assumes that the original sequences A and B are known so that A⊖B=A.
Example: Let us consider that originally we have sequences A={1,2,λ}, B={λ,2,3}, and A⊕B={1,2,3}. Then A⊖B=A={1,2,λ}.
Let A and B be two sets. The set difference operator, ∖, is as defined in set theory.
Example: Let us consider we have sets A={1,2,3} and B={2,3}. Then A∖B={1}.
Let A be a matrix and B be a sequence. The concatenation operator, ∙, is such that A∙B is a matrix where a new row (sequence) B is added after the last row of A.
Example: Let us consider the matrix A below and sequence B={10,11,12}. The matrix A∙B is shown below.
$${\kern14pt} A = \left[ \begin{array}{lll} 1 & 2 & 3 \\[0.3em] 4 & 5 & 6 \\[0.3em] 7 & 8 & 9 \end{array}\right] $$
$$A \bullet B = \left[ \begin{array}{lll} 1 & 2 & 3 \\[0.3em] 4 & 5 & 6 \\[0.3em] 7 & 8 & 9 \\[0.3em] 10 & 11 & 12 \end{array}\right] $$
Let A be a matrix and B be a sequence. The removal from matrix operator, ∘, is such that A∘B is a matrix obtained by removing the entire row (sequence) B from the last row of matrix A. This operator assumes that the original matrix A and sequence B are known so that A∘B=A
Example: Let us consider we have matrix A and sequence B presented in the previous example. Then A∘B=A as shown below.
$$A \circ B = A = \left[ \begin{array}{lll} 1 & 2 & 3 \\[0.3em] 4 & 5 & 6 \\[0.3em] 7 & 8 & 9 \end{array}\right] $$
The constructor procedure
According to the specified input (parameters and values), the Constructor procedure aims to generate all t-tuples that needs to be covered. Each t-tuple is in the matrix Θ |C|×|P| Footnote 3 where |C| represents the number of t-tuples, t is the strength, and |P| is the number of parameters.
Each row, θ i , of Θ is a t-tuple that has not yet been covered and it has a variable, flag, associated with it whose purpose is to aid in the reallocation process of the t-tuple into the final solution. Note that since the order matters, each t-tuple θ i is indeed a sequence and not a set. Moreover, flag does not belong to Θ. Table 2 shows the matrix Θ for the example shown in Fig. 1 and t=2. Note that interactions are made for the values of b a n k∖f u n c t i o n, b a n k∖c a r d, and f u n c t i o n∖c a r d. Then, a t-tuple corresponding to the interaction of factors b a n k∖f u n c t i o n can be written in the form θ i ={b a n k A,d e b i t,λ}. Initially, all values of flag are false. Algorithm 2 shows the Constructor procedure.
Table 2 Matrix Θ for the example in Fig. 1
Constructor operates as follows: based on the set of parameters (domain), P, and the strength (t), interactions between the parameters are generated through the enumeration procedure, and stored in a set named E (line 1). For example, we have 3 parameters (bank, function and card) and t = 2 thus we know that the enumerator will generate the interactions 2 per 2 (t=2) between these 3 parameters. Thus E={I 1,I 2,I 3} where we have the sets I 1={b a n k,f u n c t i o n,λ}, I 2={b a n k,λ,c a r d}, and I 3={λ,f u n c t i o n,c a r d}. For better understanding, we denote the elements of I l in this way: b a n k∖f u n c t i o n, b a n k∖c a r d and f u n c t i o n∖c a r d. Then, the interactions (I l ) are selected one at a time (line 2), and during this selection, t-tuples are constructed based on each parameter of that interaction: in line 5, the first parameter of the first interaction, p 1, is selected. Note that each parameter, p j , is indeed another set composed of values, v k . Thus, p 1=b a n k={b a n k A,b a n k B}, p 2=f u n c t i o n={d e b i t,c r e d i t}, and p 3=c a r d={c a r d A,c a r d B,c a r d C}. Therefore, each of the values (v k ) is added in t-tuples (θ i ) (line 6) and also in Θ (line 7). Recall that θ i is indeed a sequence. From now on, subsequent parameters are selected one by one, and a new t-tuple is generated from the combination between each of the values (v k ) with each of the preexisting t-tuples (θ i ) in Θ (line 16). For example, the algorithm selects the first generated interaction, I 1, b a n k∖f u n c t i o n and construct all t-tuples between these two parameters. After processing each interaction, I l , the Constructor procedure removes it from the set E (line 21).
Note that the main difference between TTR 1.0 and 1.1 is that TTR 1.0 performs the ordering of the domain, P, that is the parameters are ordered according to the amount of values they have: from the highest to the lowest quantity. For example, considering Fig. 1 and this input order: bank, function, and card. In version 1.0, parameters are stored in an ordered way: the first parameter becomes card (3 values), the second parameter is bank (2 values) and the last parameter is function (2 values). In version 1.1, there is no such ordering and this explains why bank and function generate the first rows (t-tuples) of Θ (see Table 2).
The initial solution and addition of test cases
The matrix M N×(|P|+1) is the MCA we need to construct where there are N rows (i.e. test cases) and |P| parameters. The (|P|+1)-th column is not used to represent any parameter but rather to mean the value of the goal (ζ) associated with that test case. There exists an initial solution for the matrix M that is obtained by selecting the parameters interaction I l that has the largest amount of uncovered t-tuples (line 3 in Algorithm 1). Considering the input order bank, function, card, I 2=b a n k∖c a r d that is chosen because it has 6 t-tuples and it appears first than I 3=f u n c t i o n∖c a r d. All t-tuples derived via I 2 in the initial solution are combined with empty test cases, respecting the order of input of the parameters/values submitted to TTR 1.1 as shown in Table 3 (see t-tuples θ 5={b a n k A,λ,c a r d A}, θ 6={b a n k A,λ,c a r d B},⋯ from Θ (Table 2) in the initial M).
Table 3 Initial M: example of Fig. 1
In the same way, to the extent that existing test cases are no longer sufficient to allocate the remaining t-tuples in the Θ matrix, the same procedure is used to include new test cases in matrix M. In other words, when reallocation of t-tuples becomes inefficient, it is necessary to include new test cases. Thus, as in the construction of the initial solution, the interaction of factors I l that has the largest amount of uncovered t-tuples is selected, so that these will become new test cases. This strategy is performed on line 3 of Algorithm 1.
In order to modify the current solution to obtain the final solution, the test suite M, we rely on the variable goal (ζ). For each row of M, i.e. for each test case, there is an associated goal.
As the objective is to address the largest number of uncovered t-tuples, the goal is calculated according to the maximum number of uncovered t-tuples which potentially may be covered when a t-tuple θ i is moved from Θ to M. This results in a temporary test case τ r . In order to find ζ, it is necessary to take into account: (i) the disjoint parameters, P d , covered by the union between t-tuple θ i and a test case from M; (ii) the number of parameter interactions, y, which τ r has already covered; and (iii) the strength t. Therefore:
$$\zeta = \binom{P_{d}}{t} - y. $$
Let us consider again Fig. 1 and t = 2. According to Θ (see Table 2), the initial solution, M, is composed by the t-tuples due to parameters b a n k∖c a r d. This is because the I 2=b a n k∖c a r d has 6 tuples, I 3=f u n c t i o n∖c a r d has 6 t-tuples, and I 1=b a n k∖f u n c t i o n has 4 t-tuples. As b a n k∖c a r d appears first than f u n c t i o n∖c a r d and both have 6 tuples, so the algorithm selects it for reallocating into M.
The number of disjoint parameters, P d , is equal to 3. As the interaction b a n k∖c a r d is already contemplated in matrix M, the next parameter interaction providing the largest number of non-addressed t-tuples is f u n c t i o n∖c a r d. Then we have all 3 parameters with b a n k∖f u n c t i o n and f u n c t i o n∖c a r d which explains P d = 3. As t = 2, we have \(\binom {3}{2} = 3\). However, one of the 3 parameter interactions has already been covered during the initial solution (b a n k∖c a r d), so we need to cover only 2 parameter interactions. Thus, for each t-tuple in the initial solution M, there remains to be covered:
$$\zeta = \binom{3}{2} - 1 = 2. $$
This explains the goal (ζ) in Table 3. It is very important that y is subtracted in order to find ζ. If this is not done, the final goal will never be matched, since there are no uncovered t-tuples that correspond to this interaction.
Even considering y, it is also important to note that not always the expected targets will be reached with the current configurations of the M and Θ matrices. In other words, in certain cases, there will be times when no existing t-tuple will allow the test cases of the M matrix to reach its goals. It is at this point that it becomes necessary to insert new test cases in M. This insertion is done in the same way as the initial solution for M is constructed, as described in the section above.
The Main Procedure
The Main procedure is presented in Algorithm 3. After the construction of the matrix Θ, the initial solution, and the calculation of the goals of all t-tuples, Main sort Θ so that the elements belonging to the parameter interaction with the greatest amount of t-tuples get ahead (line 1). However, these t-tuples will not be reallocated from Θ to M at once. This is done gradually, one by one, as goals are reached (line 7 to 11). Since the matrix M is being traversed in the loop (line 4), it will be updated every time a t-tuple is combined with some of its test cases (note ⊕ in line 5).
Let us consider Fig. 2. All matrices in this figure represent snapshots of M. The upper left matrix (a) is the initial solution. As long as there exists t-tuples (θ i ) in Θ, the Main procedure works. Thus, Main selects from Θ the largest amount of uncovered t-tuples. In Table 2, t-tuples were selected from the parameter interactions I 3=f u n c t i o n∖c a r d. Every t-tuple of the f u n c t i o n∖c a r d interaction is combined with each test case in M until the t-tuple matches some goal (line 7).
Snapshots of M: a initial solution; b and c intermediate matrices; d final test suite
When an uncovered t-tuple fits into a row of M to complete a test case and this t-tuple is not removed on the line 9 in Algorithm 3, it means that the goal for that row of M is reached. Take the first row of the initial M (Table 3) which is a test case (τ r ) originated from θ 5={b a n k A,λ,c a r d A}, and the first t-tuple of f u n c t i o n∖c a r d interaction not yet covered in Θ, θ 11 = {λ,d e b i t,c a r d A}. The addition of θ 11 = {λ,d e b i t,c a r d A} in M is accepted because ζ = 2 is reached. Note that the initial M, with test cases τ r , is also an input parameter of this procedure. Hence, in line 5, M is updated due to the addition sequence-sequence operator (⊕). In addition, note that τ r is also a sequence as θ i . In other words, by inserting θ 11 = {λ,d e b i t,c a r d A}, we have a complete test case τ r = {b a n k A,d e b i t,c a r d A}. In this way, the other two interactions b a n k∖f u n c t i o n (θ 1 = {b a n k A,λ,d e b i t}) and f u n c t i o n∖c a r d (θ 11 = {λ,d e b i t,c a r d A}) are covered, and the goal is achieved. The upper right matrix (b) in Fig. 2 shows the result of this first addition.
After all combinations between t-tuples and test cases are made, that is, when procedure ends, the new ζ is calculated. The bottom left matrix (c) shows the new values of ζ (see rows 3 and 6). Thus the steps described above are repeated with the insertion/reallocation of t-tuples into the matrix M. Once an uncovered t-tuple of Θ is included in M and meets the goal, that t-tuple is excluded from Θ (line 7). Note that if t-tuple does not allow the test to which it was combined to reach the goal, it is "unbound" (line 9) from this test case so that it can be combined with the next test case. The final test suite is the matrix M shown at the bottom right (d).
It is possible that a certain uncovered t-tuple does not fit into M. Consequently, the flag variable associated with this t-tuple in Θ is signed as true so that the Main procedure knows that such a t-tuple can no longer be compared with rows of M. Main continues as long as there are uncovered t-tuples. Table 4 shows part of Θ after the first iteration. Note that t-tuples θ 13 = {d e b i t,c a r d C} and θ 16 = {c r e d i t,c a r d C} of the f u n c t i o n∖c a r d interaction are not inserted into M (see the values true).
Table 4 Part of Θ: unfitness
This exception is ilustred in Table 4, with θ 13 = {λ,d e b i t,c a r d C} and θ 16 = {λ,c r e d i t,c a r d C} happens because the tests generated by these t-tuples and the available rows of the matrix M address t-tuples already covered in Θ. Assuming that the test consists of the combination of a t-tuple and row 3 of M, only one t-tuple is covered since there is no more t-tuples to be covered in b a n k∖c a r d and b a n k∖f u n c t i o n, as illustrated in Table 4. However, ζ = 2 is not satisfied and these t-tuples can not be removed from Θ. Then it is necessary to recalculate the goals according to the parameter interactions that have been already addressed.
The high-level view of the new version of TTR, 1.2, is in Algorithm 4. This new version no longer uses the Constructor procedure since t-tuples are generated one at a time as they are reallocated. In other words, there is no more Θ, a matrix of t-tuples. What we have now is only φ which is a matrix of parameter interactions. TTR 1.2 works as follow: (i) generates only the parameter interactions (it does not generate the t-tuples yet); (ii) generates an initial solution, the matrix M; and (iii) the t-tuples are generated from φ in order to get the final solution (M) via the Main procedure.
Let us consider the code in Fig. 3 where parameters and values are given in Table 5 and t=3. It is a method to update information into a database of a company. TTR 1.2 constructs only parameter interactions according to the strength and stores the number of corresponding t-tuples (Φ) in a matrix φ. These parameter interactions are I 1 = {s t a t u s,e d u c a t i o n,r e g i m e,λ,8}, I 2 = {s t a t u s,e d u c a t i o n,λ,w o r k i n g_h o u r s,8}, I 3 = {s t a t u s,λ,r e g i m e,w o r k i n g_h o u r s,8}, and I 4 = {λ,e d u c a t i o n,r e g i m e,w o r k i n g_h o u r s,8}, where the last element of I l is the number of t-tuples Φ (in all these case I l =8). Here, each interaction I l is indeed a sequence because the algorithm needs to know the exact number of t-tuples and hence position matters. Note that λ is the empty element. No t-tuple corresponding to any parameters/values interactions is constructed as shown in Table 6. The calculation of Φ is simply done by multiplying the number of values of each parameter in the corresponding interaction.
A second running examples: update method
Table 6 Matrix φ for the example of Fig. 3
Initial solution
In this case, the initial solution is no more than the construction of the t-tuples due to the parameters interactions with greater Φ, and their transformation into test cases. In Table 7, the t-tuples of the parameters interaction I 1 = {s t a t u s,e d u c a t i o n,r e g i m e,8} were all transformed into test cases and therefore, for this parameters interaction, Φ becomes 0 and it is no longer considered in the goal (ζ) calculation (Table 8). In fact, we have 4 parameters and t = 3, thus 4 interactions of possible parameters are generated: one is already covered remaining 3 parameter interactions (I 2,I 3,I 4) to be addressed. This justifies ζ=3 (Table 7).
Table 7 Initial M for the example of Fig. 3
Table 8 Matrix φ for the example of Fig. 3: after the initial solution
The new Main procedure is presented in Algorithm 5. After calculating the parameters interactions, Φ, the initial solution, and the goals of all test cases of M, Main selects the parameter interaction that has the highest amount of uncovered t-tuples (line 2) and constructs t-tuples so that they can be reallocated. However, they will be reallocated gradually, one by one, as goals are reached (line 4 to 13). The procedure combines the t-tuples with the test cases of M in order to match them.
Let us take the second running example (Fig. 3). The parameters interaction with the highest amount of non-addressed t-tuples is I 2={s t a t u s,e d u c a t i o n,λ,w o r k i n g_h o u r s,8} (Φ = 8; Table 8 after the initial solution): all t-tuples of this interaction are generated and stored in a sequence S (line 3). The first t-tuple, θ 1 = {a c t i v e,u n d e r g r a d u a t e,λ,a f t e r n o o n}, is combined with each test case, τ r in M (line 7). The t-tuple in question fits test case 1, τ 1. At that moment, it is verified whether the t-tuple θ i makes the τ r test reach its goal. This control is done through the g o a l() function that receives the τ r test case and then is broken in t-tuples (line 8) according to the parameters interactions that have Φ other than 0. For example, the test case τ 1 = {a c t i v e,u n d e r g r a d u a t e,p a r t i a l,a f t e r n o o n} is broken in t-tuples: {{a c t i v e,u n d e r g r a d u a t e,p a r t i a l,λ}, {a c t i v e,u n d e r g r a d u a t e,λ,a f t e r n o o n}, {a c t i v e,λ,p a r t i a l,a f t e r n o o n}, {λ,u n d e r g r a d u a t e,p a r t i a l,a f t e r n o o n}}. It is then verified how many of these t-tuples do not exist in M and, if this amount equals the respective ζ, θ i is permanently stored in M and a unit is taken from the value of Φ of each of the factor interactions that have t-tuples covered by this test case (line 12) because this keeps if the control of the quantity of t-tuples that still have to be covered. Since the matrix M is being traversed in the loop (line 6), it will be updated every time a t-tuple is combined with some of its test cases (line 7).
This step is repeated for all t-tuples. Each time a t-tuple is reallocated from S into M, the goals are recalculated. For example, when the matrix M permanently receives the 4th t-tuple, the test cases that become complete (with a value for each parameter) have ζ = 0 while the others still have ζ = 3 (Table 9).
Table 9 Intermediate matrix M for the example of Fig. 3
All I 2 t-tuples are reallocated from S in order to achieve the goal of all M test cases resulting the final test suite presented in Table 10. In fact, the Main procedure does not construct new t-tuples from another parameters interaction if the current one is not zero: if the parameters interaction I 2 (selected due to the greatest Φ) still has t-tuples, Main will not select another parameters interaction. To do this, the goal of the test cases will be decreased by one, until all t-tuples of the interaction of parameters I 2 make the test cases to match ζ.
Table 10 Final matrix M for the example of Fig. 3
Controlled experiment 1: TTR 1.1 × TTR 1.2
This section presents a controlled experiment where we compare versions 1.1 and 1.2 of TTR in order to realize whether there is significant difference between both versions of our algorithm. We accomplished such an experiment where we jointly considered cost and efficiency in a multi-objective perspective.
Definition and context
The primary aim of this study is to evaluate cost and efficiency related to CIT test case generation via versions 1.1 and 1.2 of the TTR algorithm (both implemented in Java). The rationale is to perceive whether we have significant differences between the two versions of our algorithm.
Regarding the metrics, cost refers to the size of the test suites while efficiency refers to the time to generate the test suites. Although the size of the test suite is used as an indicator of cost, it does not necessarily mean that test execution cost is always less for smaller test suites. However, we assume that this relationship (higher size of test suite means higher execution cost) is generally valid. We should also emphasize that the time we addressed is not the time to run the test suites derived from each algorithm but rather the time to generate them. We jointly analyzed cost and efficiency in a multi-objective way.
The set of samples, i.e. the subjects, are formed by instances that were submitted to both versions of TTR to generate the test suites. We randomly chose 80 test instances/samples (composed of parameters and values) with the strength, t, ranging from 2 to 6. Table 11 shows part of the 80 instances/samples used in this study. Full data obtained in this experiment are presented in (Balera and Santiago Júnior 2017).
Table 11 Samples for the controlled experiment: Instances. Caption: val = value; par = parameter
It is important to mention how each instance/sample can be interpreted. Let us consider instance i=1 in Table 11:
$$2^{1} 4^{1} 5^{1} 3^{1} 6^{1}, \quad t=2. $$
In the context of unit test case generation for programs developed according to the Object-Oriented Programming (OOP) paradigm, this instance can be used to generate test cases for a class that has one attribute (parameter) which can take 2 values (21), 1 attribute that can take 4 values (41), another attribute that can take 5 values (51), ⋯, 1 attribute that can take 6 values (61). In the system and acceptance testing context, this same sample can be used to identify test scenarios (test objectives) in a model-based test case generation approach (Santiago Júnior 2011; Santiago Júnior and Vijaykumar 2012). In both cases, the test suites must meet the criteria of pairwise testing (t=2) where each combination of 2 values of all parameters must be covered. Note that these samples were randomly selected and they cover a wide range of combinations of parameters, values, and strengths to be selected for very simple but also more complex case studies with different testing levels (unit, system, acceptance, etc.).
Hypotheses and variables
We defined two hypotheses as shown below:
Null Hypothesis, H 0.1 - There is no difference regarding cost-efficiency between TTR 1.1 and TTR 1.2;
Alternative Hypothesis, H 1.1 - There is difference regarding cost-efficiency between TTR 1.1 and TTR 1.2.
Regarding the variables involved in this experiment, we can highlight the independent and dependent variables (Wohlin et al. 2012). The first type are those that can be manipulated or controlled during the process of trial and define the causes of the hypotheses. For this experiment, we identified the algorithm/tool for CIT test case generation. The dependent variables allow us to observe the result of manipulation of the independent ones. For this study, we identified the number of generated test cases and the time to generate each set of test cases and we jointly considered them.
Description of the experiment
The experiment was conducted by the researchers who defined it. We relied on the experimentation process proposed in (Wohlin et al. 2012), using the R programming language version 3.2.2 (Kohl 2015). Both algorithms/tools (TTR 1.1, TTR 1.2) were subjected to each one of the 80 test instances (see Table 11), one at a time. The output of each algorithm/tool, with the number of test cases and the time to generate them, was recorded.
To measure cost, we simply verified the number of generated test cases, i.e. the number of rows of the final matrix M, for each instance/sample. The efficiency measurement required us to instrument each one of the implemented versions of TTR and measure the computer current time before and after the execution of each algorithm. In all cases, we used a computer with an Intel Core(TM) i7-4790 CPU @ 3.60 GHz processor, 8 GB of RAM, running Ubuntu 14.04 LTS (Trusty Tahr) 64-bit operating system. The goal of this second analysis is to provide an empirical evaluation of the time performance of the algorithms.
To perform the multi-objective cost-efficiency evaluation, we followed two steps. First, we transformed the cost-efficiency (two-dimensional) representation into a one-dimensional one. Thus, in a second step, we used statistical tests, such as the t-test or the nonparametric Wilcoxon test (Signed Rank) (Kohl 2015), to compare the two test suites (TTR 1.1 and TTR 1.2). To address the nondeterminism of the algorithms/tools, related to the the ordering input of parameters and values, we generated test cases with 5 variations in the order of parameters and values, and took into account the average of these 5 assessments for the statistical tests. We then got points (c A i ,t A i ) that represent the average cost (c A i ) and average time (t A i ) of the algorithms A (TTR 1.1, TTR 1.2) for each instance i (1≤i≤80).
We then determined an optimal point in a two-dimensional space, the point (0,0). This point implies a cost closer to 0 and requires a time closer to 0. The closest condition is because an algorithm is not expected to generate a test suite with, exactly, 0 test case or it does require 0 unit of time to generate the set of test cases. We then used a measure of distance, such as the Euclidean one, to measure the distance from the optimal point (0,0) to (c A i ,t A i ). Thus, each algorithm is then represented by a one-dimensional set, D, where each d i ∈D is the Euclidean distance between (0,0) and (c A i ,t A i ) for every instance i. We selected the Euclidean distance because it is one of the most used similarity distance measure. In software testing, Euclidean distance has been used as a quality indicator in multi-objective test case/data generation (Filho and Vergilio 2015; Santiago Júnior and Silva 2017), to support the automation of test oracles for complex output domains (web applications (Delamaro et al. 2013), text-to-speech systems (Oliveira 2017)), and many others.
Based on this cost-efficiency one-dimensional representation, we relied on appropriate statistical evaluation to check data normality. Verification of normality was done in three steps: (i) by using the Shapiro-Wilk test (Shapiro and Wilk 1965) with a significance level α = 0.05; (ii) by checking the skewness of the frequency distribution (in this case, − 0.1≤s k e w n e s s≤0.1 so that the data can be considered as normally distributed); and (iii) by using a graphical verification by means of Q-Q plot (Kohl 2015) and histogram. Thus, we believe we have greater confidence in this conclusion on data normality compared to an approach that is based only on the Shapiro-Wilk test considering the effects of polarization due to the length of the samples.
If we concluded that data came from a normally distributed population, then the paired, two-sided t-test was applied with α = 0.05. Otherwise, we applied the nonparametric paired, two-sided Wilcoxon test (Signed Rank) (Kohl 2015) with α = 0.05, too. However, if the samples presented ties, we applied a variation of the Wilcoxon test, the Asymptotic paired, two-sided Wilcoxon (Signed Rank) (Kohl 2015), suitable to treat ties, with significance level α = 0.05.
In order to reject the Null Hypothesis, H 0.1, we checked whether p−v a l u e<0.05 (t-test) or whether both p−v a l u e<0.05 and |z|>1.96 (Wilcoxon) where z is the z-score. If H 0.1 was rejected, we observed the average of all Euclidean distances (80) due to each algorithm. The algorithm that presented the lowest average of Euclidean distances was the one chosen as the most adequate. If H 0.1 could not be rejected, then the conclusion was that no statistical difference existed between both algorithms.
In this section, we present the results of this first controlled experiment. Based on the cost-efficiency one-dimensional representation (Section 4.3), we considered four evaluation classes as follows:
All strenghts. In this case, all 80 instances/samples (Table 11) with all strengths (2, 3, 4, 5, and 6) were taken into account. Our idea here is trying to perceive the cost-efficiency performance of both algorithms in a context where several different strengths are selected to generate a test suite;
Low strengths. In this case, we selected only the samples with strength equals to 2. Our aim is to note how the algorithms perform for low strengths;
Medium strengths. By selecting samples with strength equals to 3 or 4, we want to evaluate an intermediate strength context;
High strengths. We aim to assess the performance for higher strengths, i.e. t= 5 or 6.
Table 12 presents the Euclidean distances of part of the 80 samples (all strenghts class only; complete data are in (Balera and Santiago Júnior 2017)) as well as the average values, \(\overline {x}\), of such distances. We checked data normality where Table 13 presents the p−v a l u e, p, due to the Shapiro-Wilk test and the skewness. Note that this table shows p and skewness of all four classes above (all, low, medium, and high strenghts). Moreover Sol 1 is TTR 1.1 and Sol 2 is TTR 1.2. Figures 4 and 5 present the Q-Q plots and histograms for all strengths, Figs. 6 and 7 present the Q-Q plots and histograms for lower strengths, Figs. 8 and 9 present the Q-Q plots and histograms for medium strengths, and Figs. 10 and 11 present the Q-Q plots and histograms for higher strengths, respectively.
Experiment 1: Q-Q plots. a TTR1.1; b TTR 1.2 - All Strengths
Experiment 1: Histograms. a TTR1.1; b TTR 1.2 - All Strengths
Experiment 1: Q-Q plots. a TTR1.1; b TTR 1.2 - 2 Strength
Experiment 1: Histograms. a TTR1.1; b TTR 1.2 - 2 Strength
Experiment 1: Q-Q plots. a TTR1.1; b TTR 1.2 - 3 and 4 Strengths
Experiment 1: Histograms. a TTR1.1; b TTR 1.2 - 3 and 4 Strengths
Table 12 Experiment 1 - Results of the analysis of Euclidean Distance (all strengths)
Table 13 Experiment 1 - Results of the analysis of data normality
We can clearly see that all these data did not come from a normally distribution population because p<0.05 and the skewness is far from 0. Moreover, Q-Q plots and histograms reassure this conclusion. Hence, we used the nonparametric paired, two-sided Wilcoxon test (Signed Rank) or its variation (Asymptotic) when ties were detected. Table 14 presents the p−v a l u e, p, |z|, and additional information for classes all and low strengths while Table 15 shows the results for medium and high strengths.
Table 14 Experiment 1 - Results of the Wilcoxon test
Based on Tables 14 and 15, we could not reject H 0.1 (no difference) for all strengths, but we could do it for the other evaluation classes and hence accept the Alternative Hypothesis, H 1.1. As we have previously pointed out, when there is difference regarding cost-efficiency, we examine the average values of the Euclidean distances: the smaller the better. TTR 1.1 is better, in terms of cost-efficiency, than TTR 1.2 for lower strengths (t=2). However, for medium (t=3,4) and higher strenghts (t=5,6), TTR 1.2 surpassed TTR 1.1. This makes sense because in TTR 1.2 we do not generate, at the beginning, the matrix of t-tuples and hence we expect that the last version of our algorithm can handle properly higher strengths.
Therefore, even if we did not find statistical difference with all the strengths and TTR 1.1 was the best for lower strenghts, we decided to select TTR 1.2, to compare with the other solutions for unconstrained CIT test case generation, because TTR 1.2 performed better than TTR 1.1 for medium and higher strengths.
The conclusion validity has to do with how sure we are that the treatment we used in an experiment is really related to the actual observed outcome (Wohlin et al. 2012). One of the threats to the conclusion validity is the reliability of the measures (Campanha et al. 2010). We automatically obtained the measures via the implementations of the algorithms and hence we believe that replication of this study by other researchers will produce similar results. Even if other researchers may get different absolute results, especially related to the time to generate the test suites simply because such results depend on the computer configuration (processor, memory, operating system), we dot not expect a different conclusion validity. Moreover, we relied on adequate statistical methods in order to reason about data normality and whether we did really find statistical difference between TTR 1.1 and TTR 1.2. Hence, our study has a high conclusion validity.
The internal validity aims to analyze whether the treatment actually caused the outcome (result). Hence, we need to be sure whether other parameters have not caused the outcome, parameters that have not been controlled or measured. There are many threats to internal validity such as testing effects (measuring the participants repeatedly), history (experiment external events or between repeated measures of the dependent variable may influence the responses of the subjects, e.g. interruption of the treatment), instrument change, maturation (participants might mature during the study or between measurements), selection bias (differences between groups), etc. Note that the participants of our experiment are randomly samples composed of parameters, values, and strengths. Hence, we neither had any human/nature/social parameter nor unanticipated events to interruption the collection of the measures once started to pose an internal validity. Hence, we claim that our experiment has a high internal validity.
In the construct validity, the goal is to ensure that the treatment reflects the construction of the cause, and the result the construction of the effect. This is also high because we used the implementations of TTR 1.1 and TTR 1.2 to assess the cause, and the results, supported by the decision-making procedure via statistical tests, clearly provided the basis for the decision to be made between both algorithms.
Threats to external validity compromise the confidence in asserting that the results of the study can be generalized to and between individuals, settings, and under the temporal perspective. Basically, we can divide threats to external validity in two categories: threats to population and ecological threats.
Threats to population refer to how significant is the selected samples of the population. For our study, the ranges of strengths, parameters, and values are the determining points for this threat. Note that for such a study, the possibility of combination of strengths and parameters/values is literally infinite. However, we believe that our choice of the set of samples is significant (80) with strengths spanning from 2 to 6. Also, recall that the samples were determined completely randomly (by combining parameters, values, and strengths), as well as the input order of parameters and values was also random (for the 5 executions addressing nondeterminism). With this, we guarantee one of the basic principles of the sampling process which is the randomness to avoid selection bias.
Ecological threats refer to the degree to which the results may be generalized between different configurations. Pre-test effects, Post-test effects, and the Hawthorne effects (due to the participants simply feel stimulated by knowing that they are participating in an innovative experiment) are some of these threats. The participants in our experiment are the instances/samples composed of parameters, values and strengths and, therefore, this type of threat does not apply to our case.
Controlled experiment 2: TTR 1.2 × other solutions
In this section, we present a second controlled experiment where we compare TTR 1.2 with five other significant greedy approaches for unconstrained CIT test case generation. Many characteristics of this second controlled experiment ressemble the first one (Section 4). We emphasize here the main differences and point to this previous section whenever necessary.
The aim of this experiment is to compare TTR 1.2 with five other greedy algorithms/tools for unconstrained CIT: IPOG-F (Forbes et al. 2008), jenny (Jenkins 2016), IPO-TConfig (Williams 2000), PICT (Czerwonka 2006), and ACTS (Yu et al. 2013). These algorithms/tools have been selected due to their relevance for unconstrained CIT via greedy strategies.
The IPO algorithm (Lei and Tai 1998) is the basis for several other solutions such as IPOG, IPOG-D (Lei et al. 2007), IPOG-F, IPOG-F2 (Forbes et al. 2008), IPOG-C (Yu et al. 2013), IPO-TConfig (Williams 2000), ACTS (where several versions of IPO are implemented) (Yu et al. 2013), and CitLab (Cavalgna et al. 2013). Thus, we considered three of its variations: own our implementation of IPOG-F (in Java), IPO-TConfig (in Java), and IPOG-F2 implemented within ACTS (in Java). Note that ACTS is probably one of the most popular CIT tools where not only academia but industry professionals have been using it for various purposes (NIST National Institute of Standards and Technology 2015). A tool implemented in C, jenny (Jenkins 2016), has been used in informal (Pairwise 2017) and more formal (Segall et al. 2011) CIT comparisons. PICT (in C++) can be regarded as one baseline greedy tool where other tools have been created based on it (PictMaster 2017).
Like in Section 4, the metrics are cost, measured as the size of the test suites, and efficiency which again refers to the time to generate them. However, to proper measure the time to generate the test suites, we should have access to the source code of the tools in order to instrument them and get more precise and accurate measures. We had only the code of the implementation of TTR 1.2, our own implementation of IPOG-F, and jenny. Thus, we could not measure the time to generate the test cases due to IPO-TConfig, PICT, and ACTS (IPOG-F2). Moreover, note that the time measurements may be influenced by different programming languages within the cost-efficiency evaluation (TTR 1.2, IPOG-F, and jenny). In this case, we implemented TTR 1.2 not only in Java but also in C too in order to address a possible evaluation bias in the time measures when comparing TTR 1.2 against the other solutions. To sum up, we decided to perform two evaluations:
Cost-Efficiency (multi-objective). Here, we focused on TTR 1.2, IPOG-F, and jenny since these were the solutions we had the source code and could properly measure the time to generate the test suites. Hence, we compared TTR 1.2 (in Java) with IPOG-F (in Java), and TTR 1.2 (in C) with jenny (in C);
Cost (single objective). In this case, we compared TTR 1.2 (only in Java since efficiency is not considered here and thus time does not matter) with PICT, IPO-TConfig, and ACTS.
With respect to the subjects, the same 80 participants of Section 4 were used (Table 11 and full data are in (Balera and Santiago Júnior 2017)).
Hypotheses of this second experiment are:
Null Hypothesis, H 0.2 - There is no difference regarding cost-efficiency between TTR 1.2 (in Java) and IPOG-F (in Java);
Alternative Hypothesis, H 1.2 - There is difference regarding cost-efficiency between TTR 1.2 (in Java) and IPOG-F (in Java);
Null Hypothesis, H 0.3 - There is no difference regarding cost-efficiency between TTR 1.2 (in C) and jenny (in C);
Alternative Hypothesis, H 1.3 - There is difference regarding cost-efficiency between TTR 1.2 (in C) and jenny (in C);
Null Hypothesis, H 0.4 - There is no difference regarding cost between TTR 1.2 (in Java) and PICT;
Alternative Hypothesis, H 1.4 - There is difference regarding cost between TTR 1.2 (in Java) and PICT;
Null Hypothesis, H 0.5 - There is no difference regarding cost between TTR 1.2 (in Java) and IPO-TConfig;
Alternative Hypothesis, H 1.5 - There is difference regarding cost between TTR 1.2 (in Java) and IPO-TConfig;
Null Hypothesis, H 0.6 - There is no difference regarding cost between TTR 1.2 (in Java) and ACTS;
Alternative Hypothesis, H 1.6 - There is difference regarding cost between TTR 1.2 (in Java) and ACTS.
The independent variable is the algorithm/tool for CIT test case generation for both assessments (cost-efficiency, cost). The dependent variables are the number of generated test cases (cost evaluation), and this number of test cases in addition to the time to generate each set of test cases in a multi-objective perspective as in the previous section (cost-efficiency evaluation).
The general description of both evaluations (cost-efficiency, cost) of this second study is basically the same as shown in Section 4. Algorithms/tools were subjected to each one of the 80 test instances, one at a time, and the outcome was recorded. Cost is the number of generated test cases, and efficiency was obtained via instrumentation of the source code with the same computer previously mentioned.
For the multi-objective cost-efficiency evaluation (IPOG-F, jenny), we followed the same two steps previously mentioned: transformation of the cost-efficiency (two-dimensional) representation into a one-dimensional one and usage of statistical tests, such as the t-test or the nonparametric Wilcoxon test (Signed Rank) (Kohl 2015), to compare each pair of test suites (TTR 1.2 and other). To address the nondeterminism of the algorithms/tools, we again generated test cases with 5 variations in the order of parameters and values, and took into account the average of these 5 assessments for the statistical tests. Hence, we obtained the points (c A i ,t A i ) and calculated the Euclidean distances from the optimal point (0,0) to (c A i ,t A i ). Then, we checked data normality and, based on the result of normality, we used the the paired, two-sided t-test with α = 0.05 (normal data) or the nonparametric paired, two-sided Wilcoxon test (Signed Rank) or its Asymptotic version with α = 0.05 (non-normal data).
For the evaluation of cost (PICT, IPO-TConfig, ACTS), we did not need to transform from two into one dimension because it is a single dimension problem. The optimal point here is the value 0 and the Euclidean distance from 0 to c A i (average cost of the algorithms A for each instance i, 1≤i≤80) is |0−c A i |=|c A i |. We then performed the statistical evaluation just as in the multi-objective case.
Results, discussion and validity
In this section, we present the outcomes of both assessments of our second controlled experiment. Like in the first controlled experiment, to compare TTR 1.2 with IPOG-F, jenny, PICT, IPO-TConfig, and ACTS, we considered four evaluation classes: all, low, medium, and high strengths. Table 16 presents the Euclidean distances of part of the 80 samples (all strenghts class only; complete data are in (Balera and Santiago Júnior 2017)) and the average values, \(\overline {x}\). Table 17 presents results of the analysis of data normality (p−v a l u e (p) and skewness) where we can see all evaluation classes. In this table, Sol 1 is the other solution and Sol 2 is TTR 1.2. Figures 12 and 13 present the Q-Q plots and histograms for all strengths, Figs. 14 and 15 present the Q-Q plots and histograms for lower strengths, Figs. 16 and 17 present the Q-Q plots and histograms for medium strengths, and Figs. 18 and 19 present the Q-Q plots and histograms for higher strengths, respectively.
Experiment 2: Q-Q plots. a IPOG-F; b jenny; c PICT; d IPO-TConfig; e ACTS - All Strengths
Experiment 2: Histograms. a IPOG-F; b jenny; c PICT; d IPO-TConfig; e ACTS - All Strengths
Experiment 2: Q-Q plots. a ACTS; b IPO-TConfig; c IPOG-F; d jenny; e PICT - Lower Strengths
Experiment 2: Histograms. a ACTS; b IPO-TConfig; c IPOG-F; d jenny; e PICT - Lower Strengths
Experiment 2: Q-Q plots. a ACTS; b IPO-TConfig; c IPOG-F; d jenny; e PICT - Medium Strengths
Experiment 2: Histograms. a ACTS; b IPO-TConfig; c IPOG-F; d jenny; e PICT - Medium Strengths
Experiment 2: Q-Q plots. a ACTS; b IPO-TConfig; c IPOG-F; d jenny; e PICT - Higher Strengths
Experiment 2: Histograms. a ACTS; b IPO-TConfig; c IPOG-F; d jenny; e PICT - Higher Strengths
Again we note that all these data did not come from a normally distribution population. The nonparametric paired, two-sided Wilcoxon test (Signed Rank) or its variation (Asymptotic) where then applied. Table 18 presents the p−v a l u e, p, |z|, and additional information for classes all and low strengths while Table 19 shows the results for medium and high strengths. We should mention that in 23 instances (3 with s t r e n g t h=4, 12 with s t r e n g t h=5, and 8 with s t r e n g t h=6) jenny was not able to generate test cases, in some input order of the parameters, due to out of memory issue. Specifically, jenny failed to finish when the test suite size was more than 1,000 test cases. Similar outcomes happened in IPO-TConfig: even if we waited for about 6 hours, it did not generate anything out and hence the tool did not create test cases in 20 instances (3 with s t r e n g t h=4, 9 with s t r e n g t h=5, and 8 with s t r e n g t h=6). In these cases, we adopted a policy penalty: in order to consider these unsuccessful participants, we doubled the respective measure we obtained (average value of the Euclidean distance) due to TTR 1.2 to be the one of jenny and IPO-TConfig. We believe that this is a fair decision because TTR 1.2 could finish generating test cases for all 80 instances.
Table 18 Results of the Wilcoxon test
Table 19 Results of the Wilcoxon test (medium and high strengths)
As shown in Table 18, for class all strengths, two Null Hypotheses were rejected: H 0.2 (TTR 1.2 × IPOG-F) and H 0.5 (TTR 1.2 × IPO-TConfig). TTR 1.2 was better (lowest average value of Euclidean distances) than IPO-TConfig but it was worse than IPOG-F. There is no difference between TTR 1.2 and jenny, PICT, and ACTS.
As in controlled experiment 1, TTR 1.2 did not demonstrate good performance for low strengths. There is no difference between TTR 1.2 and IPO-TConfig. In all the other comparisons, the Null Hypothesis was rejected and TTR 1.2 was worse than the other solutions. This can be attributed to the fact that the algorithm focuses on test cases that have parameter interactions that generate a large amount of t-tuples, which is usually seen in test cases with larger strenghts. This can be verified by the fact that the algorithm gives priority to just covering the interaction of parameters with the greatest amount of t-tuples.
For medium strengths, TTR 1.2 had alternate results. While the Null Hypothesis H 0.6 (TTR 1.2 × ACTS) could not be rejected and our algorithm was better than IPO-TConfig, IPOG-F, jenny, and PICT surpassed TTR 1.2.
The greatest advantage of TTR 1.2 turned out to be again for higher strengths. Recall that TTR 1.2 does not create the matrix of t-tuples at the beginning, and this can potentially benefit our solution compared with the other five for higher strengths. Note that TTR 1.2 was better than jenny, PICT, IPO-TConfig, and ACTS. The only exception is the comparison against IPOG-F where the Null Hypothesis, H 0.2, could not be rejected and thus there is no statistical difference between both approaches.
In general, we can say that IPOG-F presented the best performance compared with TTR 1.2, because IPOG-F was better for all strengths, as well as lower and medium strengths. For higher strengths, there was a statistical draw between both approaches. An explanation for the fact that IPOG-F is better than TTR 1.2 is that TTR 1.2 ends up making more interactions than IPOG-F. In general, we might say that efficiency of IPOG-F is better than TTR 1.2 which influenced the cost-efficiency result. However, if we look at cost in isolation for all strengths, the average value of the test suite size generated via TTR 1.2 (734.50) is better than IPOG-F (770.88).
As we have just stated, for higher strengths, TTR 1.2 is better than two IPO-based approaches (IPO-TConfig and ACTS/IPOG-F2) but there is no difference if we consider our own implementation of IPOG-F and TTR 1.2. This can be explained as follows. The way the array that stores all t-tuples is constructed influences the order in which the t-tuples are evaluated by the algorithm. However, it is not described how this should be done in IPOG-F, leaving it to the developer to define the best way. As the order in which the parameters are presented to the algorithms alters the number of test cases generated, as previously stated, the order in which the t-tuples are evaluated can also generate a certain difference in the final result.
The conclusion of the two evaluations of this second experiment is that our solution is better and quite attractive for the generation of test cases considering higher strengths (5 and 6), where it was superior to basically all other algorithms/tools. Certainly, the main fact that contributes to this result is the non-creation of the matrix of t-tuples at the beginning which allows our solution to be more scalable (higher strengths) in terms of cost-efficiency or cost compared with the other strategies. However, for low strengths, other greedy approaches, like IPOG-F, may be better alternatives.
As before and by making a comparison between pairs of solutions (TTR 1.2 × other), in both assessments (cost-efficiency and cost), we can say that we have a high conclusion, internal, and construct validity. Regarding the external validity, we believe that we selected a significant population for our study. Detailed explanations have been given in Section 5.1 and are valid here.
In this section we present some relevant studies related to greedy algorithms for CIT. The IPO algorithm (Lei and Tai 1998) is one very traditional solution designed for pairwise testing. Several approaches are based on IPO such as IPOG, IPOG-D (Lei et al. 2007), IPOG-F, IPOG-F2 (Forbes et al. 2008), IPOG-C (Yu et al. 2013), IPO-TConfig (Williams 2000), ACTS (where IPOG, IPOG-D, IPOG-F, IPOG-F2 are implemented)(Yu et al. 2013), and CitLab (Cavalgna et al. 2013). All IPO-based proposals have in common the fact that they perform horizontal and vertical growths to construct the final test suite. Moreover, some need two auxiliary matrices which may decrease its performance by demanding more computer memory. Such algorithms accomplish exhaustive comparisons within each horizontal extension which may penalize efficiency.
IPOG-F (Forbes et al. 2008) is an adaptation of the IPOG algorithm (Lei et al. 2007). Through two main steps, horizontal and vertical growths, an MCA is built. Both growths work based on an initial solution. The algorithm is supported by two auxiliary matrices which may decrease its performance by demanding more computer memory to use. Moreover, the algorithm performs exhaustive comparisons within each horizontal extension which may cause longer execution. On the other hand, TTR 1.2 only needs one auxiliary matrix to work and it does not generate, at the beginning, the matrix of t-tuples. These features make our solution better for higher strengths (5, 6) even though we did not find statistical difference when we compared TTR 1.2 with our own implementation of IPOG-F (Section 6.4).
IPO-TConfig is an implementation of IPO in the TConfig tool (Williams 2000). The TConfig tool can generate test cases based on strengths varying from 2 to 6. However, it is not entirely clear whether the IPOG algorithm (Lei et al. 2007) was implemented in the tool or if another approach was chosen for t-way testing. In our empirical evaluation, TTR 1.2 was superior to IPO-TConfig not only for higher strengths (5, 6) but also for all strengths (from 2 to 6). Moreover, IPO-TConfig was unable to generate test cases in 25% of the instances (strengths 4, 5, 6) we selected.
The ACTS tool (Yu et al. 2013) is one of the most used CIT tools to date. Several variations of IPO are implemented in ACTS: IPOG, IPOG-D (Lei et al. 2007), IPOG-F, and IPOG-F2 (Forbes et al. 2008). The implementation of our algorithm performed better in terms of cost, compared with IPOG-F2/ACTS, for higher strengths. However, both solutions performed similarly when we considered all strengths.
IPOG-C (Yu et al. 2013) generates MCAs considering constraints. It is an adaptation of IPOG where constraint handling is provided via a SAT solver. The greatest contribution are three optimizations that seek to reduce the number of calls of the SAT solver. As IPOG-C is based on IPOG, it accomplishes exhaustive comparisons in the horizontal growth which may lead to a longer execution. Besides, each t-tuple is evaluated to see if it is valid or not.
The algorithm implemented in the PICT tool (Czerwonka 2006) has two main phases: preparation and generation. In the first phase, the algorithm generates all t-tuples to be covered. In the second phase, it generates the MCA. The generation of all t-tuples which can often be a bad thing, since many tuples require large disk space for storage. With respect to the application of the tool, this tool is best applied in strenghts of low value as an example, there is no study (Yamada et al. 2016). Other tools have been created based on PICT (PictMaster 2017).
The jenny tool is implemented in C (Jenkins 2016). It is a light greedy tool but one of its limitation is the number of parameters it handles: from 2 to 52. In the controlled experiment we performed, TTR 1.2 was superior to jenny for higher strengths (5, 6) but they presented similar performances for all strengths (from 2 to 6). In 27.5% of the samples (strengths 4, 5, 6), jenny could not create test cases as we have mentioned before.
Automatic Efficient Test Generator (AETG) (Cohen et al. 1997) is based on algorithms that use ideas of statistical experimental design theory to minimize the number of tests needed for a specific level of test coverage of the input test space. AETG generates test cases by means of Experimental Designs (ED) (Cochran and Cox 1950) which are statistical techniques used for planning experiments so that one can extract the maximum possible information based on as few experiments as possible. It makes use of its greedy algorithms and the test cases are constructed one at a time, i.e. it does not use an initial solution.
In (Cavalgna et al. 2013), a new tool is presented for generating MCAs with constraint handling support: CitLab. Like ACTS, CitLab has several algorithms for test suite generation: AETG, IPO, and others. The bottom of line is that test case generation is only one of the characteristics of the tool. Like ACTS, CitLab does not present a new algorithm as it just implements algorithms proposed in the literature. Hence, the same limitations of the existing proposals are also here.
The Feedback Driven Adptative Combinatorial Testing Process (FDA-CIT) algorithm is shown in (Yilmaz et al. 2014). At each iteration of the algorithm, verification of the masking of potential defects is accomplished, isolating their probable causes and then generating a new configuration which omits such causes. The idea is that masked deffects exist and that the proposed algorithm provides an efficient way of dealing with this situation before test execution. However, there is no assessment about the cost of the algorithm to generate MCAs.
In order to better compare the previous studies with our algorithm, TTR 1.2, in Table 20 we show some main characteristics of all the algorithms/tools. In this table, * means that the characteristic is present, - means that it is not present, and empty (blank space) means that either it is not totally evident that the algorithm/tool has such a feature or it is not applicable.
Table 20 Greedy algorithms/tools for CIT
This paper presented a novel CIT algorithm, called TTR, to generate test cases specifically via the MCA technique. TTR produces an MCA M, i.e. a test suite, by creating and reallocating t-tuples into this matrix M, considering a variable called goal (ζ). TTR is a greedy algorithm for unconstrained CIT.
TTR was implemented in Java and C (TTR 1.2) and we developed three versions of our algorithm. In this paper, we focused on the description of versions 1.1 and 1.2 since version 1.0 was detailed elsewhere (Balera and Santiago Júnior 2015).
We carried out two rigorous evaluations to assess the performance of our proposal. In total, we performed 3,200 executions related to 8 solutions (80 instances × 5 variations × 8). In the first controlled experiment, we compared versions 1.1 and 1.2 of TTR in order to know whether there is significant difference between both versions of our algorithm. In such experiment, we jointly considered cost (size of test suites) and efficiency (time to generate the test suites) in a multi-objective perspective. We conclude that TTR 1.2 is more adequate than TTR 1.1 especially for higher strengths (5, 6). This is explained by the fact that, in TTR 1.2, we no longer generate the matrix of t-tuples (Θ) but rather the algorithm works on a t-tuple by t-tuple creation and reallocation into M. This benefits version 1.2 so that it can properly handle higher strengths.
Having chosen version 1.2, we conducted another controlled experiment where we confronted TTR 1.2 with five other greedy algorithms/tools for unconstrained CIT: IPOG-F (Forbes et al. 2008), jenny (Jenkins 2016), IPO-TConfig (Williams 2000), PICT (Czerwonka 2006), and ACTS (Yu et al. 2013). In this case, we carried out two evaluations where in the first one we compared TTR 1.2 with IPOG-F and jenny since these were the solutions we had the source code (to precisely measure the time). Moreover, to address a possible evaluation bias in the time measures when comparing TTR 1.2 against jenny (developed in C), we also implemented it in C in addition to the standard implementation in Java. Hence, a cost-efficiency (multi-objective) evaluation was performed. In the second assessment, we did a cost (single objective) evaluation where TTR 1.2 was compared with PICT, IPO-TConfig, and ACTS. The conclusion is as previously stated: TTR 1.2 is better for higher strengths (5, 6) where only in one case our solution is not superior (in the comparison with IPOG-F where we have a draw). The fact of not creating the matrix of t-tuples at the beginning explains this result.
Therefore, considering the metrics we defined in this work and based on both controlled experiments, TTR 1.2 is a better option if we need to consider higher strengths (5, 6). For lower strengths, other solutions, like IPOG-F, may be better alternatives.
Thinking about the testing process as a whole, one important metric is the time to execute the test suite which eventually may be even more relevant than other metrics. Hence, we need to run multi-objective controlled experiments where we execute all the test suites (TTR 1.1 × TTR 1.2; TTR 1.2 × other solutions) probably assigning different weights to the metrics. We also need to investigate the parallelization of our algorithm so that it can perform even better when subjected to a more complex set of parameters, values, strengths. One possibility is to use the Compute Unified Device Architecture/Graphics Processing Unit (CUDA/GPU) platform (Ploskas and Samaras 2016). We must develop other multi-objective controlled experiment addressing effectiveness (ability to detect defects) of our solution compared with the other five greedy approaches.
Despite this classification, some algorithms/tools are both SAT and greedy-based.
Some authors (Kuhn et al. 2013; Cohen et al. 2003) abbreviate a Mixed-Level Covering Array as CA too. However, as we have made a explicit distinction between Fixed-value and Mixed-Level arrays, we prefer abbreviate it as MCA. Note that an MCA is naturally a Covering Array. We have just used this abbreviation to stress that our work relates to mixed and not fixed arrays.
Θ is a matrix whose order varies. In other words, TTR knows the number of columns beforehand (|f|), but the number of rows (|C|) depends on the interaction of t-way parameter's values. During the reallocation process, TTR removes the rows until Θ is empty.
Advanced combinatorial test system
AETG:
Automatic efficient test generator
Coverage array
CIT:
Combinatorial interaction test
CUDA:
Compute unified device architecture
GA:
Genetic algorithm
IPOG:
In parameter order general
IPO-TConfig:
In parameter order TConfig
Mixed-level covering array
MOA:
Mixed-level orthogonal array
OA:
Orthogonal array
OOP:
Object-oriented programming
Pairwise indepedent combinatorial testing
Simulated annealing
SWPDC:
Software for the payload data handling computer
TSA:
Tabu search approach
TTR:
T-tuple reallocation
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The authors would like to thank the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) for supporting this research and Leoni Augusto Romain da Silva for his support in running part of the second controlled experiment.
This work was partially funded by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) through a scholarship granted to the first author (JMB).
Full data obtained during the experiments are in (Balera and Santiago Júnior 2017).
Laboratório Associado de Computação e Matemática Aplicada, Instituto Nacional de Pesquisas Espaciais (INPE), Av. dos Astronautas, 1758, São José dos Campos, SP, Brazil
Juliana M. Balera & Valdivino A. de Santiago Júnior
Juliana M. Balera
Valdivino A. de Santiago Júnior
JMB worked in the definitions and implementations of all three versions of the TTR algorithm, and carried out the two controlled experiments. VASJ worked in the definitions of the TTR algorithm, and in the planning, definitions, and executions of the two controlled experiments. All authors contributed to all sections of the manuscript. All authors read and approved the submitted manuscript.
Correspondence to Juliana M. Balera.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Balera, J., Santiago Júnior, V. An algorithm for combinatorial interaction testing: definitions and rigorous evaluations. J Softw Eng Res Dev 5, 10 (2017). https://doi.org/10.1186/s40411-017-0043-z
Combinatorial interaction testing
Mixed-value covering array
Controlled experiment
Automated software testing: Trends and evidence
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Ars Inveniendi Analytica (2021), Paper No. 5, 35 pp. \\ DOI 10.15781/hk9g-zz18 \end{flushright} \end{minipage}
\ccnote
\begin{center} \begin{huge} \textit{ A Liouville-type theorem for\\ stable minimal hypersurfaces } \end{huge} \end{center}
\begin{center} \large{\bf{Leon Simon}} \\ \vskip0.15cm \footnotesize{Stanford University} \end{center}
\begin{center} \noindent \em{Communicated by Guido De Philippis} \end{center}
\noindent \textbf{Abstract.} \textit{We prove that if $M$ is a strictly stable complete minimal hypersurface in $\smash{\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{n+1+\ell}}$ which has finite density at infinity and which lies on one side of a cylinder $\smash{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}=\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}\times\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{\ell}}$, where $\smash{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}$ is a strictly stable area minimizing hypercone in $\smash{\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{n+1}}$ with ${\rm sing\,\,}\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}=\{0\}$, then $M$ must be cylindrical---i.e.\ $M=S\times\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{\ell}$, where $S\subset\smash{\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{n+1}}$ is a smooth strictly stable minimal hypersurface in $\,\smash{\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{n+1}}$\!. \,Applications will be given in~\emph{\cite{Sim21a},~\cite{Sim21b}}. } \vskip0.3cm
\noindent \textbf{Keywords.} minimal hypersurface, strict stability, minimizing cone
\section*{Introduction}\label{intro}
\noindent The main theorem here (Theorem~\ref{main-th}) establishes that if $M$ is a complete strictly stable minimal hypersurface in $\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{n+1+\ell}$ lying on one side of a cylindrical hypercone $\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}=\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}\times\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{\ell}$ with ${\rm sing\,\,}\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}=\{0\}$ and $\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}$ strictly stable and minimizing, then $M$ must itself be cylindrical---i.e.\ of the form $S\times\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{\ell}$ where $S$ is a smooth complete (not necessarily connected) strictly stable minimal hypersurface in $\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{n+1}$.
This result, or more correctly its Corollary~\ref{co-1} below, is a crucial ingredient in the author's recent proof (in~\cite{Sim21b}) that, with respect to a suitable $C^{\infty}$ metric for $\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{n+1+\ell}$, $n+\ell\geqslant 8$, there are examples of strictly stable minimal hypersurfaces which have singular set of the form $\{0\}\times K$, where $K$ is an arbitrary closed subset of $\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{\ell}$.
An outline of the present paper is as follows. After a description of the main results in~\S1 and some notational and technical preliminaries in \S2 and \S3, in \S4 we establish $L^{2}$ growth estimates for solutions of the Jacobi equation (i.e.\ the linearization of the minimal surface equation) on $M$. These estimates are applied to give growth estimates on (i) $(x,y)\cdot\nu$, where $\nu$ is the unit normal of $M$, (ii) $\nu_{\!y}=(e_{n+2}\cdot\nu,\ldots,e_{n+1+\ell}\cdot\nu)$ (i.e.\ the components of the unit normal $\nu$ of $M$ in the $y$-coordinate directions), and (iii) (in \S5) $d|M$, where $d(x)={\rm dist\,}((x,y),\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N})$ is the distance to the cylinder $\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}$. In~\S6 the growth estimates established in \S4 and \S5 are combined to show that, if $\nu_{\!y}$ is not identically zero, \[
R^{\gamma-\alpha}\leqslant \int_{M\cap \{(x,y):|(x,y)|<R\}}\nu_{\!y}^{2}\,d\mu \leqslant R^{-2+\gamma+\alpha}, \,\, \gamma=\ell+2+\beta_{1} ,\,\, \] for each $\alpha\in (0,1)$ and all sufficiently large $R$ (depending on $\alpha$ and $M$), where $\mu$ is $(n+\ell)$-dimensional Hausdorff measure, and $\beta_{1}=2((\fr{n-2}{2})^{2}+\lambda_{1})^{1/2}$, with $\lambda_{1}$ the first eigenvalue of the Jacobi operator of the compact minimal surface $\Sigma=\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}\cap \Sph^{n}$. These inequalities are clearly impossible for $R>1$, so we finally conclude that indeed $\nu_{\!y}$ is identically zero, showing that $M$ is cylindrical as claimed in the main theorem.
\section{Main Results}\label{main-res}
\noindent Let $\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}$ be a minimal hypercone in $\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{n+1}$ with ${\rm sing\,\,}\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}=\overline\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}\setminus\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}=\{0\}$ and let $\mathcal{L}_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}$ be the Jacobi operator (linearized minimal surface operator) defined by \[
\mathcal{L}_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}u=\Delta_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}u+|A_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}|^{2}u, \]
where $|A_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}|^{2}$ is the squared length of the second fundamental form of $\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}$. In terms of the Jacobi operator
\[
\mathcal{L}_{\Sigma}=\Delta_{\Sigma}u+|A_{\Sigma}|^{2}u \dl{jac-op-Sig} \] of the compact submanifold $\Sigma=\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}\cap \Sph^{n}$ (which is minimal in $\Sph^{n}$), we have \[ \mathcal{L}_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}u=r^{1-n}\frac{\partial}{\partial r}\bigl(r^{n-1}\frac{\partial u}{\partial r}\bigr)
+r^{-2}\mathcal{L}_{\Sigma},\quad r=|x|. \dl{jac-op} \] If $\lambda_{1}$ is the first eigenvalue of $-\mathcal{L}_{\Sigma}$, and $\varphi_{1}>0$ is the corresponding eigenfunction (unique up to a constant factor) we have (possibly complex-valued) solutions $r^{\gamma_{1}^{\pm}}\varphi_{1}$ of $\mathcal{L}_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}u=0$, where $\gamma_{1}^{\pm}$ are the ``characteristic exponents'' \[ \gamma_{1}^{\pm} = -\frac{n-2}{2} \pm \sqrt{ \bigl(\frac{n-2}{2}\bigr)^{2}+\lambda_{1}}. \dl{ch-exps} \] We assume $\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}$ is strictly stable, i.e.\ there is $\lambda>0$ with \[
\lambda\int_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}|x|^{-2}\zeta^{2}(x)\,d\mu(x)\leqslant
\int_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}\bigl( |\nabla_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}\zeta|^{2} - |A_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}|^{2}\zeta^{2}\bigr)\,d\mu\,\,\,\, \forall \zeta\in C_{c}^{\infty}(\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{n+1}), \dl{st-stab} \] where $\mu$ is $n$-dimensional Hausdorff measure; subsequently $\mu$ will always denote Hausdorff measure of the appropriate dimension.
Using~\ref{jac-op} and \ref{ch-exps}, \ref{st-stab} is readily checked to be equivalent to the condition that \[ \lambda_{1}>-((n-2)/2)^{2}, \dl{st-stab-1} \] in which case \[ \gamma_{1}^{-}<-\frac{n-2}{2} <\gamma_{1}^{+}<0. \dl{ch-ineq} \] The main theorem here relates to hypersurfaces $M\subset\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{n+1+\ell}$, where $\ell\geqslant 1$; points in $\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{n+1+\ell}=\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{n+1}\times\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{\ell}$ will be denoted \[ (x,y)=(x_{1},\ldots,x_{n+1},y_{1},\ldots,y_{\ell}). \]
In the main theorem, which we now state and which will be proved in~\S\ref{th-1-pf}, we assume that~(i) $M$ is a smooth complete minimal hypersurface in $\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{n+1+\ell}$ lying on one side of $\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}=\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}\times\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{\ell}$, i.e.\ \[ M\subset U_{+}\times\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{\ell}, \dl{one-side} \] where $U_{+}$ is one of the two connected components of $\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{n+1+\ell}\setminus\overline{C}_{0}$, that (ii) $M$ is strictly stable in the sense that (Cf.\ \ref{st-stab}) there is $\lambda>0$ with \[
\lambda\int_{M}|x|^{-2}\zeta^{2}(x,y)\,d\mu(x,y)\leqslant
\int_{M}\bigl( |\nabla_{M}\zeta|^{2} - |A_{M}|^{2}\zeta^{2}\bigr)\,d\mu\,\,\,\, \dl{str-stab-M}
\]
for all $\zeta\in C_{c}^{1}(\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{n+1+\ell})$, and that (iii) $M$ has finite density at $\infty$, i.e.\ \[ \sup_{R>1}R^{-n-\ell}\mu(M\cap B_{R})<\infty, \dl{fin-dens} \] where, here and subsequently, $B_{R}$ is the closed ball of radius $R$ and centre $(0,0)$ in $\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{n+1+\ell}$: \[
B_{R}=\{(x,y)\in\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{n+1+\ell}:|(x,y)|\leqslant R\}; \] the corresponding open ball, which we shall occasionally use, is denoted \[
\breve B_{R}=\{(x,y)\in\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{n+1+\ell}:|(x,y)|< R\}. \]
\begin{state}{\bf{}\tl{main-th} Theorem (Liouville-type Theorem.)} If $M$ is a smooth, complete, embedded minimal hypersurface (without boundary) in \,$\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{n+1+\ell}$ such that~{\rm\ref{one-side}}, {\rm\ref{str-stab-M}} and {\rm\ref{fin-dens}} hold, then $M$ is cylindrical, i.e.\ \[ M=S\times\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{\ell}, \] where $S$ is a smooth complete strictly stable minimal hypersurface which is contained in~$U_{+}$. \end{state}
\noindent{\bf{}Remark: } Using the regularity theory of~{\rm\cite{SchS81}}, the conclusion continues to hold, with no essential change in the proof, in case $M$ is allowed to have a singular set of finite $(n+\ell-2)$-dimensional Hausdorff measure. Indeed if we use the regularity theory of~{\rm\cite{Wic14}} then we need only assume \emph{a priori} that $M$ has zero $(n+\ell-1)$-dimensional Hausdorff measure.
For applications in~\cite{Sim21a}, \cite{Sim21b} we now state a corollary of the above theorem, in which the stability hypothesis on $M$ is dropped, and instead $\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}$ is assumed to be both strictly stable and strictly minimizing, and we impose an \emph{a priori} smallness assumption on the last $\ell$ components of the unit normal $\nu_{M}=(\nu_{1},\ldots,\nu_{n+1+\ell})$ of $M$; i.e.\ a smallness assumption on $(\nu_{n+2},\ldots,\nu_{n+1+\ell})\,$ which we subsequently write as \[ \nu_{y}=(\nu_{y_{1}},\ldots,\nu_{y_{\ell}});\,\, \text{ i.e.\ } \nu_{y_{j}}=e_{n+1+j}\cdot\nu_{M},\,\,\, j=1,\ldots,\ell. \dl{nu-y} \]
Notice that such a smallness assumption on $|\nu_{y}|$ amounts to a restriction on how rapidly $M$ can vary in the $y$-directions.
Recall (see \cite{HarS85}) $\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}$ is said to be strictly minimizing if there is a constant $c>0$ such that
{\abovedisplayskip8pt\belowdisplayskip8pt \begin{align*}
&\mu(\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}\cap B_{1}) \leqslant \mu(T\cap B_{1}) -c\rho^{n} \text{ whenever $\rho\in(0,\fr{1}{2}]$ and $T$ is a smooth
}\dtg{st-min}\\
\noalign{\nobreak\vskip-3pt} &\hskip0.5in \text{compact hypersurface-with-boundary in $B_{1}\setminus B_{\rho}$ with
$\partial T=\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}\cap \Sph^{n}$}. \end{align*}}
\begin{state}{\bf{}\tl{co-1} Corollary.} Suppose $\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}$ is strictly stable and strictly minimizing (as in~{\rm\ref{st-min}}) and $\alpha\in (0,1)$. Then there is $\varepsilon_{0}=\varepsilon_{0}(\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0},\alpha)\in (0,\fr{1}{2}]$ such that if $M$ is a smooth, complete, embedded minimal hypersurface in $\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{n+1+\ell}$ with \[ \left\{\begin{aligned} &M\subset U_{+}\times\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{\ell}, \\ \noalign{\vskip-2pt} &{\sup}_{R>1}R^{-n-\ell}\mu(M\cap B_{R})\leqslant (2-\alpha)\mu(\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}\cap B_{1}), and \\ \noalign{\vskip-2pt}
&{\sup}_{M}|\nu_{y}| < \varepsilon_{0}, \end{aligned}\right. \] then
{\abovedisplayskip-3pt\belowdisplayskip8pt \[ M=\lambda S\times\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{\ell} \]} for some $\lambda>0$, where $S$ is the minimal hypersurface in $U_{+}$ as in~\emph{\cite{HarS85}} (see~{\rm\ref{props-S}} in~{\rm\S\ref{proof-liou}} below). \end{state}
Applications of the above results are given in~\cite{Sim21a} and \cite{Sim21b}. Although the assumptions of strict stability in Theorem~\ref{main-th} and $|\nu_{y}|$ small in Corollary~\ref{co-1} are appropriate for the applications in~\cite{Sim21a}, \cite{Sim21b}, it would be of interest to know if these restrictions can be significantly relaxed---for example the question of whether or not mere stability would suffice in place of the strict stability assumption in Theorem~\ref{main-th}.
\section[Preliminaries concerning $M$]{Preliminaries concerning $M$}\label{prelims}
\noindent As in~\S\ref{main-res}, $\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}$ will be a smooth embedded minimal hypercone in $\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{n+1}$ with ${\rm sing\,\,}\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}=\{0\}$, and we let $U_{\pm}$ be the two connected components of $\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{n+1}\setminus \overline{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}}_{0}$.
$M$ will be a smooth embedded minimal hypersurface, usually contained in $U_{+}\times \mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{\ell}$, although some results are formulated to apply locally in a ball, and also independent of the inclusion assumption $M\subset U_{+}\times\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{\ell}$.
For the moment we assume only that $M$ is minimal (i.e.\ has first variation zero) in $\breve B_{R}$. Thus
{\abovedisplayskip8pt\belowdisplayskip8pt \[ \int_{M}{\rm div}_{M}Z(x,y)\,d\mu(x,y) =0, \,\, Z=(Z_{1},\ldots,Z_{n+1+\ell}),\,Z_{j}\in C^{1}_{c}(\breve B_{R}), \dl{stationarity} \] where ${\rm div}_{M}Z$ is the ``tangential divergence'' of $Z$: \begin{align*}
{\rm div}_{M}Z_{|(x,y)}={\textstyle\sum}_{j=1}^{n+\ell}\tau_{j}\cdot D_{\tau_{j}}Z &= {\textstyle\sum}_{k,m=1}^{n+1+\ell}{\textstyle\sum}_{j=1}^{n+\ell}\tau_{j}^{k}\tau_{j}^{m}D_{k}Z_{m} \\ &={\textstyle\sum}_{i,j=1}^{n+1+\ell}g^{ij}(x,y)D_{i}Z_{j}(x,y), \end{align*}} with $\tau_{j}=(\tau_{j}^{1},\ldots,\tau_{j}^{n+1+\ell})$, $j=1,\ldots,n+\ell$, any locally defined orthonormal basis of $T_{(x,y)}M$, $(x,y)\in M$ and \[ g^{ij} = \delta_{ij}-\nu_{i}\nu_{j},\quad \nu=(\nu_{1},\ldots,\nu_{n+1+\ell}) \text{ a unit normal for } M. \dl{g-ij} \] For $v\in C^{2}(M)$ we let ${\rm graph\,} v$ be the graph of $v$ taken off $M$: \[ {\rm graph\,} v =\bigl\{(x,y)+v(x,y)\nu_{\!M}(x,y):(x,y)\in M\bigr\} \] (notice that this may fail to be an embedded hypersurface unless $v$ has small enough $C^{2}$ norm), and we take \[ \mathcal{M}_{\!M}(v)= \text{ the mean curvature operator on $M$}. \dl{def-script-M} \] Thus $\mathcal{M}_{\!M}(v)$ is the Euler-Lagrange operator of the area functional $\mathcal{A}_{\!M}$ on $M$, defined by \[ \mathcal{A}_{\!M}(v) = \int_{M}J_{M}(V)\,d\mu \] where $V(x,y) =(x,y)+v(x,y)\nu_{M}(x,y)$ is the graph map taking $M$ to ${\rm graph\,} v$ and $J_{M}(V)$ is the Jacobian of $V$: \[ J_{M}(V)=\sqrt{\det\bigl(D_{\tau_{i}}V\cdot
D_{\tau_{j}}V\bigr)}=\sqrt{\det\bigl(\delta_{ij}+v_{i}v_{j}+v^{2}{\textstyle\sum}_{k}h_{ik}h_{jk} +2vh_{ij}\bigr)}, \] where, for $(x,y)\in M$, $\tau_{1},\ldots,\tau_{n+\ell}$ is an orthonormal basis for $T_{(x,y)}M$, $h_{ij}$ is the second fundamental form of $M$ with respect to this basis, and $v_{ i}=D_{\tau_{i}}v$. Since $\sum_{i=1}^{n+\ell}h_{ii}=0$ we then have \[
J_{M}(V) = \sqrt{1+|\nabla_{M}v|^{2}-|A_{M}|^{2}v^{2}+E\bigl(v(h_{ij}),v^{2}({\textstyle\sum}_{k}h_{ik}h_{jk}),(v_{i}v_{j})\bigr)} \dl{jac-M} \] where $E$ is a polynomial of degree $n+\ell$ in the indicated variables with constant coefficients depending only on $n,\ell$ and with each non-zero term having at least degree $3$. So the second variation of $\mathcal{A}_{M}$ is given by \[
\fr{d^{2}}{dt^{2}}\bigl|_{t=0}\mathcal{A}(t\zeta)=\int_{M}\bigl(|\nabla_{M}\zeta|^{2}-|A_{M}|^{2}\zeta^{2}\bigr)\,d\mu =-\int_{M}\zeta\mathcal{L}_{M}\zeta\,d\mu, \quad \zeta\in C^{1}_{c}(\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{n+1+\ell}), \] where $\mathcal{L}_{M}$ is the Jacobi operator on $M$ defined by $$
\mathcal{L}_{M}v = \Delta_{M}v+|A_{M}|^{2}v, \dl{L-M} $$
with $\Delta_{M}f={\rm div}_{M}(\nabla_{M}f)$ the Laplace-Beltrami operator on $M$ and $|A_{M}|=\bigl(\sum h_{ij}^{2}\bigr)^{1/2}$ (the length of the second fundamental form of $M$). Of course, by~\ref{def-script-M} and \ref{jac-M}, $\mathcal{L}_{M}$ is the linearization of the mean curvature operator at $0$, i.e.\
{\abovedisplayskip8pt\belowdisplayskip8pt \[
\mathcal{L}_{M}v=\fr{d}{dt}\bigl|_{t=0}\mathcal{M}_{M}(tv). \dl{L-M-2} \]
In view of~\ref{jac-M} and the definition of $\mathcal{L}_{M}$, if $|A_{M}||u|+|\nabla_{M}u|<1$ on a domain in $M$ then we can write \[ \mathcal{M}(u)=\mathcal{L}_{M}(u) + {\rm div}_{M}E(u) + F(u) \dl{L-M-3} \]}
on that domain, where $|E(u)|\leqslant C(|A_{M}|^{2} |u|^{2}+|\nabla_{M} u|^{2})$ and $|F(u)|\leqslant C(|A_{M}|^{3}v^{2}+|A_{M}||\nabla_{M}v|^{2})$, $C=C(n,\ell)$.
We say that $M$ is \emph{strictly stable} in $\breve B_{R}$ if the second variation of $\mathcal{A}_{\!M}$ is strictly positive in the sense that \[
{\inf}_{\zeta\in C^{1}_{c}(\breve B_{R}),\,\int_{M}|x|^{-2}\zeta^{2}(x,y)\,d\mu(x,y)=1}
\,\,\,\fr{d^{2}}{dt^{2}}\bigl|_{t=0}\mathcal{A}(t\zeta) > 0, \] or, equivalently, that there is $\lambda>0$ such that~\ref{str-stab-M} holds.
Observe that if we have such strict stability then by replacing $\zeta$ in~\ref{str-stab-M} with $\zeta w$ we obtain \begin{align*}
&\smash[b]{\lambda\int_{M}|x|^{-2}w^{2}\zeta^{2}(x,y)\,d\mu(x,y)\leqslant
\int_{M}\bigl( w^{2}|\nabla_{M}\zeta|^{2}+\zeta^{2}|\nabla_{M}w|^{2}}\\ \noalign{\vskip-1pt}
&\hskip2.5in +2\zeta w\nabla_{M}\zeta\cdot\nabla_{M}w -|A_{M}|^{2}\zeta^{2}w^{2}\bigr)\,d\mu, \end{align*} and $\int_{M}2\zeta w\nabla_{M}\zeta\cdot\nabla_{M}w=\int w\nabla_{M}\zeta^{2} \cdot\nabla_{M}w= -\int_{M}
(|\nabla_{M}w|^{2}\zeta^{2}+\zeta^{2}w\Delta_{M}w)$, so if $w$ is a smooth solution of $\mathcal{L}_{M}w=0$ on $M\cap \breve B_{R}$ then \[
\lambda\int_{M}|x|^{-2}w^{2}\zeta^{2}\,d\mu(x,y)\leqslant \int_{M} w^{2}|\nabla_{M}\zeta|^{2}\,d\mu, \quad \zeta\in C^{1}_{c}(\breve B_{R}). \dl{w-1-2-est} \] We need one further preliminary for $M$, concerning asymptotics of $M$ at $\infty$ in the case when $M$ is complete in all of $\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{n+1+\ell}$ with $M\subset U_{+}\times\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{\ell}$: \begin{state}{\bf{}\tl{tangent-cone} Lemma.} If $M$ is a complete embedded minimal hypersurface in all of \,$\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{n+1+\ell}$ satisfying {\rm\ref{one-side}} and {\rm\ref{fin-dens}}, then $M$ has $\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}$ with some constant integer multiplicity $q$ as its unique tangent cone at $\infty$.
Furthermore if $M$ is stable (i.e.\ \emph{\ref{str-stab-M}} holds with $\lambda=0$), then for each $\delta>0$ there is $R_{0}=R_{0}(\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0},q,\delta)>1$ such that \[
M\setminus \bigl(B_{R_{0}}\cup\bigl\{(x,y):|x|\leqslant \delta|y|\bigr\}\bigr)\subset \cup_{j=1}^{q}{\rm graph\,} u_{j} \subset M \leqno{\rm(i)} \] where each $u_{j}$ is a $C^{2}$ function on a domain $\Omega$ of \,$\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}$ containing $\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}\setminus
(B_{R_{0}}\cup\bigl\{(x,y):|x|\leqslant \delta|y|\bigr\}$ and ${\rm graph\,} u_{j}=\{x + u_{j}(x,y)\nu_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}(x) :(x,y)\in\Omega\}$ ($\nu_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}=$ unit normal of \,$\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}$ pointing into $U_{+}$), and \[
\text{$\lim_{R\to\infty}{\sup}_{(x,y)\in \Omega\setminus B_{R}}\bigl(|x||\nabla_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}}^{2}u_{j} (x,y)|+|\nabla_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}}u_{j}(x,y)|+|x|^{-1}u_{j}(x,y)\bigr)=0$.} \leqno{\rm(ii)} \] \end{state}
\begin{proof}{\bf{}Proof:} Let $C(M)$ be a tangent cone of $M$ at $\infty$. Thus $C(M)$ is a stationary integer multiplicity varifold with $\lambda C(M)=C(M)$ for each $\lambda>0$, and support of $C(M)\subset \overline U_{+}\times\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{\ell}$ and, by the compactness theorem of~\cite{SchS81}, $C(M)$ is stable and ${\rm sing\,\,} C(M)$ has Hausdorff dimension $\leqslant n+\ell-7$. So by the maximum principle of~\cite{Ilm96} and the constancy theorem, $C(M)=\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}$ with constant multiplicity $q$ for some $q\in\{1,2,\ldots\}$.
Thus $\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}$, with multiplicity $q$, is the unique tangent cone of $M$ at $\infty$, and the ``sheeting theorem''~\cite[Theorem 1]{SchS81} is applicable, giving~(i) and~(ii). \end{proof}
\section[Preliminaries concerning $\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}$ and $\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}$]{Preliminaries concerning $\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}$ and $\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}$}\label{prelim-C}
\noindent $\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}\subset\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{n+1}\setminus\{0\}$ continues to denote a smooth embedded minimal hypercone with ${\rm sing\,\,}\hskip0.5pt \mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0} =\overline\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}\setminus\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}=\{0\}$, $U_{\pm}$ denote the two connected components of $\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{n+1}\setminus\overline\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}$, and we assume here that $\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}$ is strictly stable as in~\ref{st-stab}.
With $\mathcal{L}_{\Sigma}$ the Jacobi operator of $\Sigma=\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}\cap \Sph^{n}$ as in~\ref{jac-op-Sig}, we let $\varphi_{1}>0,\,\varphi_{2},\ldots$ be a complete orthonormal set of eigenfunctions of $-L_{\Sigma}$. Thus \begin{align*} & -L_{\Sigma}(\varphi_{j}) = \lambda_{j}\varphi_{j},\,\,\, j=1,2,\ldots, \text{with $\varphi_{1}>0$ and} \dtg{eigenvals}\\ \noalign{\vskip-1pt} &\hskip1.1in \lambda_{1}<\lambda_{2}\leqslant \lambda_{3}\leqslant\cdots \leqslant \lambda_{j}\leqslant \cdots,\,\, \lambda_{j}\to\infty\text{ as }j\to \infty, \end{align*} and every $L^{2}(\Sigma)$ function $v$ can be written $v=\sum_{j}\langle v,\varphi_{j} \rangle_{L^{2}(\Sigma)}\varphi_{j}$.
Notice that by orthogonality each $\varphi_{j},\,j\neq 1$, must then change sign in $\Sigma$, and, with $\mathcal{L}_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}$ the Jacobi operator for $\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}$ as in~\ref{jac-op}, \[ \mathcal{L}_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}(r^{\gamma}\varphi_{j})=r^{\gamma-2}(\gamma^{2}+(n-2)\gamma-\lambda_{j})\,\varphi_{j}, \dl{exp-gamma} \] so in particular if $\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}$ is strictly stable (i.e.\ if $\lambda_{1}>-(\fr{n-2}{2})^{2}$ as in~\ref{st-stab-1}) we have \begin{align*} &\mathcal{L}_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}(r^{\gamma_{j}^{\pm}}\varphi_{j}) =0, \quad \gamma_{j}^{\pm} = -\fr{n-2}{2} \pm\bigl((\fr{n-2}{2})^{2}+\lambda_{j}\bigr)^{1/2}, \dtg{char-exps}\\ \noalign{\vskip-2pt} &\hskip20pt -\infty\leftarrow\gamma_{j}^{-} \leqslant \cdots\leqslant \gamma_{2}^{-} <
\gamma_{1}^{-}<-\fr{n-2}{2}<\gamma_{1}^{+}<\gamma_{2}^{+}\leqslant \cdots\leqslant \gamma_{j}^{+}\to \infty. \end{align*} Henceforth we write \[ \gamma_{j} =\gamma_{j}^{+},\quad j=1,2,\ldots. \dl{gamma-j} \] The Jacobi operator $\mathcal{L}_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}}$ on the minimal cylinder $\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}=\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}\times\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{\ell}$ is \[ \mathcal{L}_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}}(v) = \mathcal{L}_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}(v)+{\textstyle\sum}_{j=1}^{\ell}D_{y_{j}}^{2}v, \dl{jac-C} \] and we can decompose solutions $v$ of the equation $\mathcal{L}_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}}v=0$ in terms of the eigenfunctions $\varphi_{j}$ of~\ref{eigenvals}: for $v$ any given smooth function on $\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}\cap \breve B_{R}$, we can write $v(r\omega,y)=\sum_{j}v_{j}(r,y)\varphi_{j}(\omega)$, where \[ v_{j}(r,y)=\int_{\Sigma}v(r\omega,y)\,\varphi_{j}(\omega)\,d\mu(\omega), \dl{def-v-j} \] and then $\mathcal{L}_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}}v=0$ on $\breve B_{R}$ if and only if $v_{j}$ satisfies \[ r^{1-n}\frac{\partial}{\partial r}\bigl(r^{n-1}\frac{\partial v_{j}}{\partial r}\bigr) + {\textstyle\sum}_{k=1}^{\ell}\frac{\partial^{2}v_{j}}{\partial y_{k}^{2}} -\frac{\lambda_{j}}{r^{2}}v_{j}=0 \]
for $(r,y)\in \breve B_{R}^{+}\,(=\{(r,y):r> 0,\,r^{2}+|y|^{2}<R^{2}\})$. Direct computation then shows that \[
\frac{1}{r^{1+\beta}}\frac{\partial}{\partial r}\Bigl(r^{1+\beta} \frac{\partial h}{\partial r}\Bigr) + \sum_{k=1}^{\ell}\frac{\partial^{2}h}{\partial y_{k}^{2}}=0 \dl{beta-lap} \] on $\breve B_{R}^{+}$, with \[ h(r,y)=h_{j}(r,y)=r^{-\gamma_{j}}\int_{\Sigma}v(r\omega,y)\,\varphi_{j}(\omega)\,d\mu
\text{ and }\beta=\beta_{j}=2\sqrt{(\fr{n-2}{2})^{2}+\lambda_{j}}. \dl{beta-lap-1} \] A solution $h$ of~\ref{beta-lap} (with any $\beta> 0$) which has the property \[
\int_{\{(r,y):r^{2}+|y|^{2}<R^{2}\}}r^{-2}h^{2}\,r^{1+\beta}\,drdy<\infty \dl{l2-harm-1} \]
will be referred to as a \emph{$\beta$-harmonic function on} $\{(r,y):r\geqslant 0,\, r^{2}+|y|^{2}<R^{2}\}$. Using~\ref{l2-harm-1} together with the weak form of the equation~\ref{beta-lap} this is easily shown to imply the $W^{1,2}$ estimate \[
\int_{\{(r,y):r^{2}+|y|^{2}<\rho^{2}\}}\bigl((D_{r}h)^{2}+
{\textstyle\sum}_{j=1}^{\ell}(D_{y_{j}}h)^{2}\bigr)\,r^{1+\beta}\,drdy<\infty \dl{l2-harm-2} \] for each $\rho<R$, and there is a unique such function with prescribed $C^{1}$ data $\varphi$ on
$\bigl\{(r,y):r^{2}+|y|^{2}=R^{2}\bigr\}$, obtained, for example, by minimizing \smash{$\int_{r^{2}+|y|^{2}\leqslant R^{2}}(u_{r}^{2}+|u_{y}|^{2})\,r^{1+\beta}d\mu$} among functions with trace $\varphi$ on $r^{2}+|y|^{2}=R^{2}$.
If $\beta$ is an integer (e.g.\ $\beta=1$ when $n=7$, $j=1$, and $\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}$ is the ``Simons cone'' $\bigl\{x\in\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{8}:\sum_{i=1}^{4}(x^{i})^{2}=\sum_{i=5}^{8}(x^{i})^{2}\bigr\}$) then the $\beta\,$-Laplacian operator as in \ref{beta-lap} is just the ordinary Laplacian in $\varmathbb{R}^{2+\beta+\ell}$, at least as it applies to functions $u=u(r,y)$ with
$r=|x|$. Even for $\beta$ non-integer, there is, analogous to the case when $\beta$ is an integer, $\beta$-harmonic polynomials of each order (i.e.\ homogeneous polynomial solutions $h$ of \ref{beta-lap}) of each order $q=0,1,\ldots$.
Indeed, as shown in the Appendix below (extending the discussion of~\cite{Sim94} to arbitrary $\ell$ and at the same time showing the relevant power series converge in the entire ball $\bigl\{ (r,y):r\geqslant 0,\,r^{2}+|y|^{2}<R^{2}\bigr\}$ rather than merely in
$\bigl\{ (r,y):r\geqslant 0,\,r^{2}+|y|^{2}<\theta R^{2}\bigr\}$ for suitable $\theta\in (0,1)$), if $h=h(r,y)$ is a solution of the weak form of~\ref{beta-lap} on $\{(r,y):r> 0,\, r^{2}+|y|^{2}<R^{2}\}$, i.e.\ \[ \int_{\breve B_{1}^{+}} (u_{r}\zeta_{r}+u_{y}\cdot \zeta_{y})\,d\mu_{+}=0 \dl{weak-form-0} \]
for all Lipschitz $\zeta$ with compact support in $r^{2}+|y|^{2}<R^{2}$ and if $u$ satisfies~\ref{l2-harm-1}, then $h$ is a real analytic function of the variables $r^{2},y$, and, on all of $\{(r,y):r\geqslant 0,\, r^{2}+|y|^{2}<R^{2}\}$, $h$ is a convergent sum \[
h(r,y)={\textstyle\sum}_{q=0}^{\infty}h_{q}(r,y) \,\,\,\forall (r,y) \text{ with }r\geqslant 0 \text{ and } \sqrt{r^{2}+|y|^{2}}<R, \dl{h-exp} \] where $h_{q}$ is the degree $q$ homogeneous $\beta$-harmonic polynomial in the variables $r^{2},y$ obtained by selecting the order $q$ terms in the power series expansion of $h$.
In case $\ell=1$ there is, up to scaling, a unique $\beta$-harmonic polynomial of degree $q$ of the form
{\belowdisplayskip8pt \[ h_{q}=y^{q}+\sum_{k>0,\ell\geqslant 0,2k+j=q}c_{k\ell}r^{2k}y^{j}, \] By direct computation, in case $\ell=1$ the $\beta$-harmonic polynomials $h_{0},h_{1},h_{2},h_{3},h_{4}$ are respectively \[ 1,\,\,\,y,\,\,\,y^{2}-\fr{1}{2+\beta}r^{2},\,\,\,y^{3}-\fr{3}{2+\beta}r^{2}y,\,\,\, y^{4}-\fr{6}{2+\beta}r^{2}y^{2}+\fr{3}{(2+\beta)(4+\beta)}r^{4}. \]} In the case $\ell\geqslant 2$, for each homogeneous degree $q$ polynomial $p_{0}(y)$ there is a unique $\beta$-harmonic homogeneous polynomial $h$ of degree $q$, with \[ h(r,y)= \begin{cases} p_{0}(y)+{\textstyle\sum}_{j=1}^{q/2}r^{2j}p_{j}(y), &q \text{ even } \\ p_{0}(y)+{\textstyle\sum}_{j=1}^{(q-1)/2} r^{2j}p_{j}(y),& q\text{ odd}, \end{cases} \] where each $p_{j}(y)$ is a homogeneous degree $q-2j$ polynomial, and the $p_{j}$ are defined inductively by $p_{j+1}=-(2j+2)^{-1}(2j+2+\beta)^{-1}\Delta_{y}p_{j}$, $j\geqslant 0$.
In terms of spherical coordinates $\rho=\sqrt{r^{2}+|y|^{2}}$ and $\omega=\rho^{-1}(r,y) \in \Sph^{\ell}_{+}$, \[
\Sph^{\ell}_{+}=\{\omega=(\omega_{1},\ldots,\omega_{\ell})\in\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{\ell+1}:\omega_{1}>0,\,|\omega|=1\}, \] the $\beta$-Laplacian $\Delta_{\beta}$ (i.e.\ the operator on the left of~\ref{beta-lap}) is
\[
\Delta_{\beta}h = \rho^{-\ell-1-\beta}\frac{\partial}{\partial \rho}
\Bigl(\rho^{\ell+1+\beta}\frac{\partial h}{\partial \rho}\Bigr)+ \rho^{-2}\omega_{1}^{-1-\beta} {\rm div}_{\Sph_{+}^{\ell}}\bigl(\omega_{1}^{1+\beta}\nabla_{\Sph_{+}^{\ell}}h\bigr). \dl{sph-coords} \] Notice that in the case $\ell=1$ we can write $h=h(r,\theta)$ where $\theta=\arctan (y/r)\in (-\pi/2,\pi/2)$ and~\ref{sph-coords} can be written
\[
\Delta_{\beta}h = \rho^{-2-\beta}\frac{\partial}{\partial \rho}
\Bigl(\rho^{2+\beta}\frac{\partial h}{\partial \rho}\Bigr)+ \rho^{-2}\cos^{-1-\beta}\theta \frac{\partial}{\partial\theta}\bigl(\cos^{1+\beta}\theta\frac{\partial h}{\partial \theta}\bigr). \] Using~\ref{sph-coords} we see that the order $q$ homogeneous $\beta$-harmonic polynomials $h_{q}$ (which are homogeneous of degree $q$ in the variable $\rho$) satisfy the identity
{\abovedisplayskip8pt\belowdisplayskip8pt \[ {\rm div}_{\Sph_{+}^{\ell}}\bigl(\omega_{1}^{1+\beta}\nabla_{\Sph_{+}^{\ell}}h_{q}\bigr) = q(q+\ell+\beta)h_{q} \omega_{1}^{1+\beta}. \dl{jth-eig} \] Hence we have the orthogonality of $h_{p},h_{q}$ for $p\neq q$ on \[
\Sph^{\ell}_{+}=\{(r,y):r> 0,\,r^{2}+|y|^{2}=1\} \dl{Sell} \]} with respect to the measure $d\nu_{+}=\smash{{\omega_{1}}^{\! 1+\beta}}d\mu$ ($\mu=\ell$-dimensional Hausdorff measure on $\Sph_{+}^{\ell}$): \[ \int_{\Sph_{+}^{\ell}}h_{p}\,h_{q}\,d\nu_{+}=0 \text{ for $p\neq q$},\quad d\nu_{+}=\omega_{1}^{1+\beta}d\mu. \dl{orthog} \] Thus if $h$ satisfies~\ref{beta-lap} and~\ref{l2-harm-1} then \begin{align*}
\int_{B^{+}_{R}}h^{2}(r,y)\,r^{1+\beta}drdy
& =\int_{0}^{R}\int_{\Sph_{+}^{\ell}}h^{2}(\rho\omega)\,
\rho^{1+\beta}\,d\nu_{+} \,\rho^{\ell} d\rho \dtg{l2-norm}\\
\noalign{\vskip-3pt}
&=\sum_{q=0}^{\infty}(\ell+2+\beta+2q)^{-1}N_{q}^{2}R^{\ell+2+\beta+2q}, \end{align*}
where $B_{R}^{+}=\{(r,y):r\geqslant 0,\,\,r^{2}+|y|^{2}\leqslant R\}$ and $N_{q}=\bigl(\int_{\Sph_{+}^{\ell}}h_{q}^{2}(\omega)\,\omega_{1}^{1+\beta} d\mu\bigr)^{1/2}$ and $h_{q}(r,y)$ are as in~\ref{h-exp}.
Using~\ref{orthog} it is shown in the Appendix that the homogeneous $\beta$-harmonic polynomials are complete in $L^{2}(\nu_{+})$, where $d\nu_{+}=\omega_{1}^{1+\beta}d\mu_{\ell}$ on $\Sph^{\ell}_{+}$. Thus each $\varphi\in L^{2}(\nu_{+})$ can be written as an $L^{2}(\nu_{+})$ convergent series
{\abovedisplayskip8pt\belowdisplayskip8pt \[
\varphi=\sum_{q=0}^{\infty}h_{q}|\Sph^{\ell}_{+}, \dl{b-complete} \] with each $h_{q}$ either zero or a homogeneous degree $q$ $\beta$-harmonic polynomial.
Next observe that if \[
\mathcal{L}_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}}v=0 \text{ on }\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}\cap B_{R} \text{ with } \int_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}\cap B_{R}}|x|^{-2}v^{2}\,d\mu<\infty, \dl{proper-soln-LC} \]} then each $h_{j}(r,y)$ defined as in~\ref{beta-lap-1} does satisfy the condition~\ref{l2-harm-1}, and hence we have \[ v=\sum_{j=1}^{\infty}\sum_{q=0}^{\infty}r^{\gamma_{j}}h_{j,q}(r,y)\varphi_{j}(\omega), \dl{LC-soln} \] where $h_{j,q}$ is a homogeneous degree $q$ $\beta_{j}$-harmonic polynomial and, using the orthogonality~\ref{orthog}, \[ \int_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}\cap B_{\rho}}v^{2}\,d\mu= \rho^{\ell+2+\beta_{1}} \sum_{j=1}^{\infty}\sum_{q=0}^{\infty} (2+\ell+\beta_{j}+2q)^{-1}N_{j,q}^{2}\rho^{2q+\beta_{j}-\beta_{1}}, \,\,\rho<R, \dl{l2-norm-LC} \] where $N_{j,q}=\bigl(\int_{\Sph_{+}^{\ell}}h_{j,q}^{2}(\omega)\, d\nu_{+}\bigr)^{1/2}$.
Observe that (except for $(j,q)=(1,0)$) it is possible that $r^{\gamma_{j}}h_{j,q}$ could have the same homogeneity as $r^{\gamma_{i}}h_{i,p}$ (i.e.\ $\gamma_{j}+q=\gamma_{i}+p$) for some $i\neq j,\,p\neq q$, but in any case, after some rearrangement of the terms in~\ref{l2-norm-LC}, we see that there are exponents
{\abovedisplayskip8pt\belowdisplayskip8pt \[ 0=q_{1}<q_{2}< q_{2}< \cdots < q_{i}\to\infty,\quad q_{i}=q_{i}(\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}), \] ($q_{i}$ not necessarily integers, but nevertheless fixed depending only on $\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}$) such that if $v$ is as in~\ref{proper-soln-LC} then \[ \hskip1pt\text{\small $-$}\hskip-10.6pt\int_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}\cap B_{\rho}}v^{2} = \sum_{i=1}^{\infty} b_{i}^{2}\rho^{2q_{i}} \dl{l2-norm-v} \] (which in particular is an increasing function of $\rho$), $\rho\in (0,R]$, for suitable constants $b_{j},\,j=1,2,\ldots$, where we use the notation \[ \hskip1pt\text{\small $-$}\hskip-10.6pt\int_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}\cap B_{\rho}}f = \rho^{-\ell-2-\beta_{1}}\int_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}\cap B_{\rho}}f\,d\mu, \quad \rho>0. \dl{def-avint} \] We claim that the logarithm of the right side of~\ref{l2-norm-v} is a convex function $\psi(t)$ of $t=\log R$: \[ \psi''(t)\geqslant 0\text{ where }
\psi(t)=\log\bigl(\hskip1pt\text{\small $-$}\hskip-10.6pt\int_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}\cap B_{\rho}}v^{2}\bigr|_{\rho=e^{t}}\bigr)\,\,\bigl(\,=\log({\textstyle\sum}_{i}b_{i}^{2}e^{2q_{i}t}) \text{ by~\ref{l2-norm-v}}\bigr). \dl{conv-psi} \] To check this, note that \[ \psi''(t) =\psi^{-2}(t)\bigl(({\textstyle\sum}_{i}b_{i}^{2}e^{2q_{i}t})({\textstyle\sum}_{i}4q_{i}^{2}b_{i}^{2}e^{2q_{i}t})- ({\textstyle\sum}_{i}2q_{i}b_{i}^{2}e^{2q_{i}t})^{2} \bigr) \] and by Cauchy-Schwarz this is non-negative for $t\in (0,R)$ and if there is a $t_{0}\in (-\infty,\log R)$ such that $\psi''(t_{0})=0$ then there is $i_{0}\in\{1,2,\ldots\}$ such that $b_{i}=0$ for every $i\neq i_{0}$, in which case \[ \psi(t)=\log b_{i_{0}}+2q_{i_{0}}t \quad t\in (-\infty,\log R). \dl{psi-lin} \] In particular the convexity~\ref{conv-psi} of $\psi$ implies \[ \psi(t) -\psi(t-\log 2)\geqslant \psi(t-\log 2) -\psi(t-2\log 2),\quad t\in (-\infty,\log R), \dl{eq-rat} \] and equality at any point $t\in (-\infty,\log R)$ implies~\ref{psi-lin} holds for some $i_{0}$ and the common value of each side of~\ref{eq-rat} is $\log 4^{q_{i_{0}}}$. Thus we see that if $Q\in (0,\infty)\setminus\{4^{q_{1}},4^{q_{2}},\ldots\}$ and $v$ is not identically zero then for each given $\rho\in (0,R]$ \[ \hskip1pt\text{\small $-$}\hskip-10.6pt\int_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}\cap B_{\rho/2}}v^{2}\bigg/\hskip1pt\text{\small $-$}\hskip-10.6pt\int_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}\cap B_{\rho/4}}v^{2}\geqslant Q\Longrightarrow \hskip1pt\text{\small $-$}\hskip-10.6pt\int_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}\cap B_{\rho}}v^{2}\bigg/\hskip1pt\text{\small $-$}\hskip-10.6pt\int_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}\cap B_{\rho/2}}v^{2}>Q. \dl{lin-growth} \]}
\noindent{\bf{}\tl{rem-lin-growth} Remark: } Notice that if $v_{1},\ldots,v_{q}$ are solutions of $\mathcal{L}_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}}v=0$ on $\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}\cap B_{R}$ satisfying \[
\hskip1pt\text{\small $-$}\hskip-10.6pt\int_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}\cap B_{R}}|x|^{-2}v_{j}^{2}(x,y)\,d\mu(x,y)<\infty \text{ for each }j=1,\ldots,q \] with $v_{j}\neq 0$ for some $j$, then $\text{\scriptsize $-$}\hskip-6.85pt\tint_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}\cap B_{R}}\smash{\sum_{j=1}^{q}v_{j}^{2}\,d\mu}$ has again the form of~\emph{\ref{l2-norm-v}}, so the implication~\emph{\ref{lin-growth}} applies with the sum $\sum_{j=1}^{q}v_{j}^{\,2}$ in place of~$v^{2}$.
\section[Growth estimates for solutions of $\smash{\mathcal{L}_{\!M}w=0}$]{Growth estimates for solutions of $\mathcal{L}_{\!M}w=0$}\label{growth}
\noindent Here we discuss growth estimates analogous to those used in~\cite{Sim83a}, \cite{Sim85} for solutions of $\mathcal{L}_{M}w=0$; in particular we discuss conditions on $M$ and $w$ which ensure that the growth behaviour of solutions $w$ of $\mathcal{L}_{M}w=0$ is analogous to that of the solutions $v$ of $\mathcal{L}_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}}v=0$ discussed in the previous section.
The main growth lemma below (Lemma~\ref{growth-lem}) applies locally in balls (so $M$ could be a complete minimal hypersurface in a ball $\breve B_{R}$ rather than the whole space), and we do not need the inclusion $M\subset U_{+}\times\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{\ell}$. We in fact assume $R,\lambda,\Lambda>0$ and
{\abovedisplayskip7pt\belowdisplayskip7pt \[ \left\{\begin{aligned} &\text{\,$M\subset \breve B_{R}$ is embedded, minimal, and satisfies~\ref{str-stab-M}} \,\,\text{ for every }\,\zeta\in C^{1}_{c}(\breve B_{R}), \\ \noalign{\vskip-3pt} & \, R^{-n-\ell-2}\int_{M\cap B_{R}}d^{2}(x)\,d\mu(x,y)<\varepsilon, \text{ and }
R^{-n-\ell}\mu(M\cap B_{R})\leqslant \Lambda,
\end{aligned}\right. \dl{local-M}
\]
with an $\varepsilon$ (small) to be specified and with \[ d(x)={\rm dist\,}(x,\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0})\,\,(={\rm dist\,}((x,y),\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}))\text{ for }(x,y)\in\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{n+1}\times\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{\ell}. \dl{def-d} \]
Taking $\zeta$ in~\ref{w-1-2-est} to be $1$ in $B_{\theta R}$, $0$ on $\partial B_{R}$, and $|D\zeta|\leqslant 2/(1-\theta)$, we obtain \[
\int_{M\cap B_{\theta R}}\hskip-5pt |x|^{-2}w^{2}(x,y)\,d\mu(x,y)\leqslant CR^{-2}\int_{M\cap B_{R}}\hskip-5pt w^{2}\,d\mu\,\,\forall\theta\in [\fr{1}{2},1), \,\, C=C(\lambda,\theta). \dl{pre-gro-1} \] Notice that if we have a ``doubling condition'' \[ \int_{M\cap B_{R}}w^{2}(x,y)\,d\mu(x,y)\leqslant K\int_{M\cap B_{R/2}}w^{2}\,d\mu, \dl{doubling} \] then \ref{pre-gro-1} implies that \[
\int_{M\cap B_{\theta R}}\hskip-5pt |x|^{-2}w^{2}(x,y)\,d\mu(x,y)\leqslant CKR^{-2}\int_{M\cap B_{R/2}}w^{2}\,d\mu, \] so, for $\delta^{-2}\geqslant 2CK$ (i.e.\ $\delta\leqslant (2CK)^{-1/2}$) we have \[
\int_{M\cap B_{\theta R}}\hskip-5pt |x|^{-2}w^{2}(x,y)\,d\mu(x,y)\leqslant
CR^{-2}\int_{M\cap B_{R/2}\setminus\{(x,y):|x|\leqslant \delta R\}}\hskip-5pt w^{2}\,d\mu\,\,\forall\theta\in [\fr{1}{2},1), \dl{pre-gro-2} \]} where $C=C(\lambda,\theta,K)$. The following lemma shows that in fact the inequality~\ref{pre-gro-2} can be improved.
\begin{state} {\bf{}\tl{improved-doubling} Lemma.} For each $\lambda,\Lambda,K>0$ and $\theta\in \smash{[\fr{1}{2},1)}$ there is $\varepsilon=\varepsilon(\lambda,\Lambda,\theta, K,\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N})>0$ such that if $M$ satisfies~\emph{\ref{local-M}} and $\mathcal{L}_{M}w=0$ on $M\cap\breve B_{R}$, and if the doubling condition~\emph{\ref{doubling}} holds then \[
\int_{M\cap B_{\theta R}}|x|^{-2}w^{2}(x,y)\,d\mu(x,y)\leqslant
CR^{-2}\int_{M\cap B_{R/2}\setminus\{(x,y):|x|\leqslant \frac{1}{3} R\}}w^{2}\,d\mu, \] where $C=C(\lambda,\Lambda,\theta,K,\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N})$. \end{state}
\begin{proof}{\bf{}Proof:} Let $\theta\in [\fr{3}{4},1)$. By scaling we can assume $R=1$. If there is no $\varepsilon$ ensuring the claim of the lemma, then there is a sequence $M_{k}$ of strictly stable hypersurfaces (with fixed $\lambda$) with $\int_{M_{k}\cap B_{1}}d^{2}(x)\,d\mu(x,y)\to 0$ and a sequence $w_{k}$ of solutions of $\mathcal{L}_{M_{k}}w_{k}=0$ such that \[ \int_{M_{k}\cap B_{1}}w_{k}^{2}(x,y)\,d\mu(x,y)\leqslant K\int_{M_{k}\cap B_{1/2}}w_{k}^{2}\,d\mu, \pdl{doubling-k} \] yet such that \[
\int_{M_{k}\cap B_{\theta}}|x|^{-2}w_{k}^{2}(x,y)\,d\mu(x,y)\geqslant
k\, \int_{M_{k}\cap B_{1/2}\setminus\{(x,y):|x|\leqslant \frac{1}{3} \}}w_{k}^{2}\,d\mu. \pdl{cont} \] By~\ref{pre-gro-2}, \[
\int_{M_{k}\cap B_{\theta}}|x|^{-2}w_{k}^{2}(x,y)\,d\mu(x,y)\leqslant
C \int_{M_{k}\cap B_{1/2}\setminus\{(x,y):|x|\leqslant \delta \}}w_{k}^{2}\,d\mu, \pdl{pre-gro-4} \] with $\delta=\delta(\lambda,\theta,K)$ and $C=C(\lambda,\theta,K)$.
Since $\int_{M_{k}\cap B_{1}}d^{2}\,d\mu_{k}\to 0$, in view of~\ref{tangent-cone}\,(i),\,(ii) there are sequences $\eta_{k}\downarrow 0$ and $\tau_{k}\uparrow 1$ with \[ \left\{\begin{aligned}
&\,\,M_{k}\cap \breve B_{\tau_{k}}\setminus \{(x,y):|x|\leqslant \eta_{k}\}\subset\cup_{j=1}^{q}{\rm graph\,} u_{k,j} \subset M_{k}\\
&\qquad \sup(|u_{k,j}|+|\nabla_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}}u_{k,j}|+|\nabla^{2}_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}}u_{k,j}|) \to 0, \end{aligned}\right. \pdl{gro-2a} \] where $q=q(\Lambda)\geqslant 1$ and each $u_{k,j}$ is $C^{2}$ on a domain containing $\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}\cap \breve B_{\tau_{k}}\setminus
\{(x,y):|x|\leqslant \eta_{k}\}$. Thus, with $\nu_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}$ the unit normal of $\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}$ pointing into $U_{+}$ and with \[
w_{k,j}(x,y)=w_{k}((x+u_{k,j}(x,y)\nu_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}(x),y)), \pdl{gro-2c} \]
for $(x,y)\in \mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}\cap \breve B_{\tau_{k}} \setminus \{(x,y):|x|\leqslant \eta_{k}\}$ and $j=1,\ldots,q$, we see that $w_{k,j}$ satisfies a uniformly elliptic equation with coefficients converging in $C^{1}$ to the coefficients of $\mathcal{L}_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}}$ on
$\Omega_{\sigma,\delta}=\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}\cap \breve B_{\sigma}\setminus\{(x,y):|x|>\delta\}$ for each $\sigma\in [\fr{1}{2},1)$ and $\delta\in (0,\fr{1}{2}]$, hence, by Schauder estimates and~\ref{doubling-k}, \[
|w_{k,j}|_{C^{2}(\Omega_{\sigma,\delta})}\leqslant C\|w_{k}\|_{L^{2}(B_{1})}\leqslant CK\|w_{k}\|_{L^{2}(B_{1/2})} \pdl{c2-ests} \] with $C$ independent of $k$, $k\geqslant k(\sigma,\delta)$. Hence, by~\ref{gro-2a} and~\ref{c2-ests}, \[ \tilde w_{k,j}=({\textstyle\int}_{M_{k}\cap B_{1/2}} w_{k}^{2}\,d\mu_{k})^{-1/2} w_{k,j} \] has a subsequence converging in $C^{1}$ locally in $\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}\cap\breve B_{1}$ to a smooth solution $v_{j}$ of $\mathcal{L}_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}}v_{j}=0$ on $\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}\cap\breve B_{1}$, and since by~\ref{pre-gro-4} $\int_{M_{k}\cap B_{1/2}\cap
\{(x,y):|x|\leqslant \sigma\}}w_{k}^{2}\leqslant C\sigma^{2}\int_{M_{k}\cap B_{1/2}}w_{k}^{2}$ for all $\sigma\in (0,\fr{1}{2}]$, we can then conclude
{\abovedisplayskip8pt\belowdisplayskip8pt \[ \int_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}\cap B_{1/2}}\sum_{j=1}^{q}v_{j}^{2}\,d\mu=1. \pdl{norm-1} \] But by~\ref{doubling-k} and \ref{cont} \[
\int_{M_{k}\cap B_{1/2}\setminus\{(x,y):|x|\leqslant \frac{1}{3} \}}w_{k}^{2}\,d\mu\leqslant Ck^{-1} \int_{M_{k}\cap B_{1/2}}w_{k}^{2}\,d\mu, \] and so, multiplying each side by $({\textstyle\int}_{M_{k}\cap B_{1/2}} w_{k}^{2}\,d\mu_{k})^{-1/2}$ and taking limits, we conclude \[
\int_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}\cap B_{1/2}\setminus\{(x,y):|x|\leqslant \frac{1}{3}\}} \sum_{j=1}^{p} v_{j}^{2} = 0. \]} In view of~\ref{norm-1} this contradicts unique continuation for solutions of $\mathcal{L}_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}}v=0$ (applicable since the solutions $v$ of $\mathcal{L}_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}}v=0$ are real-analytic).\nobreak\end{proof}
We can now establish the growth lemma.
\begin{state}{\bf \tl{growth-lem} Lemma.} For each $\lambda,\Lambda>0$, $Q\in (0,\infty)\setminus\{4^{q_{1}},4^{q_{2}},\ldots\}$ ($q_{1},q_{2},\ldots$ as in~{\rm\ref{l2-norm-v}}), and $\alpha\in [\fr{1}{2},1)$, there is $\varepsilon=\varepsilon(Q,\alpha,\lambda,\Lambda,\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N})>0$ such that if $M$ satisfies~\emph{\ref{local-M}} and if $\mathcal{L}_{M}w=0$ on $M\cap\breve B_{R}$ then \[ \hskip1pt\text{\small $-$}\hskip-10.6pt\int_{M\cap B_{R/2}}w^{2}\geqslant Q\hskip1pt\text{\small $-$}\hskip-10.6pt\int_{M\cap B_{R/4}}w^{2}\Longrightarrow \hskip1pt\text{\small $-$}\hskip-10.6pt\int_{B_{R}}w^{2} \geqslant Q\hskip1pt\text{\small $-$}\hskip-10.6pt\int_{B_{R/2}}w^{2}, \leqno{\rm (i)} \] and \[
\hskip1pt\text{\small $-$}\hskip-10.6pt\int_{M\cap B_{R}}w^{2} \geqslant \alpha \hskip1pt\text{\small $-$}\hskip-10.6pt\int_{M\cap B_{R/2}}w^{2}, \leqno{\rm (ii)} \] where we use the notation (analogous to~{\rm\ref{def-avint}}) \[ \hskip1pt\text{\small $-$}\hskip-10.6pt\int_{M\cap B_{\rho}}f= \rho^{-\ell-2-\beta}\int_{M\cap B_{\rho}}f\,d\mu. \] Notice that no hypothesis like \smash{$M\subset U_{+}\times\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{\ell}$} is needed here. \end{state}
\begin{proof}{\bf{}Proof:} By scaling we can assume $R=1$. If there is $Q\in (0,\infty)\setminus\{4^{q_{1}},4^{q_{2}},\ldots\}$ such that there is no $\varepsilon$ ensuring the first claim of the lemma, then there is a sequence $M_{k}$ satisfying \ref{str-stab-M} (with fixed $\lambda$) with $\int_{M_{k}\cap B_{1}}d^{2}(x)\,d\mu(x,y)\to 0$ and a sequence $w_{k}$ of solutions of $\mathcal{L}_{M_{k}}w_{k}=0$, such that
{\abovedisplayskip8pt\belowdisplayskip8pt \[ \hskip1pt\text{\small $-$}\hskip-10.6pt\int_{M_{k}\cap B_{1/2}}w_{k}^{2}\geqslant Q\hskip1pt\text{\small $-$}\hskip-10.6pt\int_{M_{k}\cap B_{1/4}}w_{k}^{2}\text{ and } \hskip1pt\text{\small $-$}\hskip-10.6pt\int_{M_{k}\cap B_{1}}w_{k}^{2}<Q\hskip1pt\text{\small $-$}\hskip-10.6pt\int_{M_{k}\cap B_{1/2}}w_{k}^{2}. \pdl{gro-2} \]} The latter inequality implies we have the doubling condition~\ref{doubling} with $K=Q$, so we can repeat the compactness argument in the proof Lemma~\ref{improved-doubling}. Thus by~\ref{gro-2}, with the same notation as in the proof of~\ref{improved-doubling}, we get convergence of $w_{k,j}$ to a smooth solution $v_{j}$ of $\mathcal{L}_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}}v_{j}=0$ on
$\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}\cap\breve B_{1}$ with $\text{\scriptsize $-$}\hskip-6.85pt\tint_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}\cap B_{1/2}}{\textstyle\sum}_{j=1}^{q}v_{j}^{2}\,d\mu=1$, $\int_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}\cap B_{1}}|x|^{-2}{\textstyle\sum}_{j=1}^{q}v_{j}^{2}<\infty$, \begin{align*} &\hskip1pt\text{\small $-$}\hskip-10.6pt\int_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}\cap B_{1/2}}{\textstyle\sum}_{j=1}^{q}v_{j}^{2}\,d\mu\geqslant Q\hskip1pt\text{\small $-$}\hskip-10.6pt\int_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}\cap B_{1/4}}{\textstyle\sum}_{j=1}^{q}v_{j}^{2}\,d\mu \text{ and }\ptg{gro-3} \\ &\hskip1in\hskip1pt\text{\small $-$}\hskip-10.6pt\int_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}\cap B_{1}}{\textstyle\sum}_{j=1}^{q}v_{j}^{2}\,d\mu\leqslant Q\hskip1pt\text{\small $-$}\hskip-10.6pt\int_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}\cap B_{1/2}}{\textstyle\sum}_{j=1}^{q}v_{j}^{2}\,d\mu.
\end{align*} In view of~\ref{rem-lin-growth}, this contradicts~\ref{lin-growth}.
Similarly, if the second claim of the lemma fails for some $\alpha\in [\fr{1}{2},1)$, after taking a subsequence of $k$ (still denoted $k$) we get sequences $M_{k}, w_{k}$ and $w_{k,j}$ with \[ \hskip1pt\text{\small $-$}\hskip-10.6pt\int_{M_{k}\cap B_{1}} w_{k}^{2}\,d\mu<\alpha\hskip1pt\text{\small $-$}\hskip-10.6pt\int_{M_{k}\cap B_{1/2}} w_{k}^{2}\,d\mu \] (i.e.\ the doubling condition~\ref{doubling} with $K=\alpha$), and smooth solutions $v_{j}=\lim w_{k,j}$ of
$\mathcal{L}_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}}v_{j}=0$ on $\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}\cap\breve B_{1}$ with $0<\int_{M\cap B_{1}}|x|^{-2}v_{j}^{2}<\infty$ and \[ \hskip1pt\text{\small $-$}\hskip-10.6pt\int_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}\cap B_{1}}{\textstyle\sum}_{j=1}^{q}v_{j}^{2}\,d\mu\leqslant\alpha\hskip1pt\text{\small $-$}\hskip-10.6pt\int_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}\cap B_{1/2}}{\textstyle\sum}_{j=1}^{q}v_{j}^{2}\,d\mu, \] which is impossible by~\ref{l2-norm-v} and~\ref{rem-lin-growth}. \end{proof}
Since $\mathcal{L}_{M}$ is the linearization of the minimal surface operator $\mathcal{M}_{M}$, a smooth family
$\{M_{t}\}_{|t|<\varepsilon}$ of minimal submanifolds with $M_{0}=M$ must generate a velocity vector $v$ at $t=0$ with normal part $w=v\cdot\nu_{M}$ ($\nu_{M}$ a smooth unit normal of $M$) being a solution of the equation $\mathcal{L}_{M}w=0$. In particular the family of homotheties $(1+t)M$ generates the solution $w=(x,y)\cdot \nu_{M}(x,y)$ and the translates $M+t e_{n+1+j}$ generate the solutions $w=e_{n+1+j}\cdot\nu_{M}(x,y) = \nu_{y_{j}}(x,y)$, $j=1,\ldots,\ell$. Thus \[ w=(x,y)\cdot\nu_{M} \text{ and } w=\nu_{y_{j}} \text{ both satisfy }\mathcal{L}_{M}w=0\text{ and the
inequality }\ref{pre-gro-1}. \dl{jac-solns} \]
\begin{state}{\bf \tl{strong-doub-lem} Corollary.} Suppose $\Lambda,\lambda,\gamma>0$, $q\in \{1,2,\ldots\}$, and assume $M$ is a complete, embedded, minimal hypersurface in $\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{n+1+\ell}$, strict stability~{\rm\ref{str-stab-M}} holds, and $M$ has $\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}$ with some multiplicity $q$ as its unique tangent cone at $\infty$. If $\mathcal{L}_{M}w=0$, and $\sup_{R>1}R^{-\gamma}\text{\scriptsize $-$}\hskip-6.85pt\tint_{M\cap B_{R}}w^{2}<\Lambda$, then there is $R_{0}=R_{0}(M,\gamma,\lambda,\Lambda,q)$ such that for all $R\geqslant R_{0}$ we have the ``strong doubling condition'' \[
\hskip1pt\text{\small $-$}\hskip-10.6pt\int_{M\cap B_{R}}|x|^{-2}w^{2}\leqslant CR^{-2}\hskip1pt\text{\small $-$}\hskip-10.6pt\int_{M\cap B_{R/2}\setminus\{(x,y):|x|<\smfr{1}{3} R\}}w^{2}, \quad C =C(\gamma,q,\lambda,\Lambda). \] \end{state}
\noindent{\bf \tl{strong-doub-rem} Remarks:} {\bf (1)} Observe that in case $M\subset U_{+}\times\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{\ell}$ the unique tangent cone assumption here is automatically satisfied by virtue of {\rm Lemma~\ref{tangent-cone}}.
{\bf{}(2)} Since $\mu(M\cap B_{R})\leqslant q\mu(\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}\cap B_{1})R^{n+\ell}$, $|(x,y)\cdot\nu_{M}(x,y)|\leqslant R$, and
$|\nu_{y_{j}}|\leqslant 1$ on $M\cap B_{R}$, in view of~{\rm\ref{jac-solns}} the above lemma applies to both
$w=(x,y)\cdot\nu_{M}$ and $w=\nu_{y_{j}}$ with $\gamma=2|\gamma_{1}|+2$ and $\gamma=2|\gamma_{1}|$ respectively.
\begin{proof}{\bf{}Proof of Corollary~\ref{strong-doub-lem}:} We can of course assume that $w$ is not identically zero. In view of Lemma~\ref{tangent-cone}, for any $\tilde\gamma>\gamma$ with $2^{\tilde\gamma}\in (1,\infty)\setminus\{4^{q_{1}},4^{q_{2}},\ldots\}$, $M\cap\breve B_{R}$ satisfies the hypotheses of Lemma~\ref{growth-lem} with $Q=2^{\tilde\gamma}$ for all sufficiently large $R$, so
{\abovedisplayskip8pt\belowdisplayskip8pt \[ \hskip1pt\text{\small $-$}\hskip-10.6pt\int_{M\cap B_{2^{k}R}}w^{2}\geqslant 2^{\tilde\gamma} \hskip1pt\text{\small $-$}\hskip-10.6pt\int_{M\cap B_{2^{k-1}R}}w^{2}\Longrightarrow \hskip1pt\text{\small $-$}\hskip-10.6pt\int_{M\cap B_{2^{k+1}R}}w^{2}\geqslant 2^{\tilde \gamma}\hskip1pt\text{\small $-$}\hskip-10.6pt\int_{M\cap B_{2^{k}R}}w^{2} \] for $k=1,2,\ldots$ and for any choice of $R$ sufficiently large (depending on $M,\tilde\gamma,\lambda$), and hence, by iteration, if $\text{\scriptsize $-$}\hskip-6.85pt\tint_{M\cap B_{R}}w^{2}\geqslant 2^{\tilde\gamma} \text{\scriptsize $-$}\hskip-6.85pt\tint_{M\cap B_{R/2}}w^{2}$ we would have $\text{\scriptsize $-$}\hskip-6.85pt\tint_{M\cap B_{2^{k}R}}w^{2}\geqslant C2^{k\tilde\gamma}$, $k=1,2,\ldots$ with $C>0$ independent of $k$ , contrary to the hypothesis on $w$ since $C2^{k(\tilde\gamma-\gamma)}>\Lambda$ for sufficiently large $k$. Thus, with such a choice of $\tilde\gamma$, we have the doubling condition~\ref{doubling} with $K=2^{\tilde\gamma}$: \[ \hskip1pt\text{\small $-$}\hskip-10.6pt\int_{M\cap B_{R}}w^{2}\leqslant 2^{\tilde\gamma}\hskip1pt\text{\small $-$}\hskip-10.6pt\int_{M\cap B_{R/2}}w^{2} \]} for all $R\geqslant R_{0}$, $R_{0}=R_{0}(M,\tilde\gamma)$. Then the required result is proved by Lemma~\ref{improved-doubling}.\nobreak\end{proof}
\section[Asymptotic $\smash{L^{2}}$ Estimates for $d$]{Asymptotic $L^{2}$ Estimates for $d$} \label{dist-fn}
\noindent The following lemma gives an $L^{2}$ estimate for the distance function $d|M\cap B_{R}$ as $R\to \infty$ in case $M$ is as in Theorem~\ref{main-th}, where as usual
{\abovedisplayskip8pt\belowdisplayskip8pt \[ d(x)={\rm dist\,}(x,\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0})\,\,(={\rm dist\,}((x,y),\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N})\text{ for } (x,y)\in\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{n+1}\times\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{\ell}). \] In this section we continue to use the notation \[ \hskip1pt\text{\small $-$}\hskip-10.6pt\int_{M\cap B_{\rho}}f =\rho^{-\ell-2-\beta_{1}}\int_{M\cap B_{\rho}}f\,d\mu. \]}
\begin{state}{\bf\tl{main-dist-est} Lemma.} Let $\alpha,\lambda,\Lambda>0$ and assume $M$ is embedded, minimal, $M\subset U_{+}\times\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{\ell}$, $R^{-n-\ell}\mu(M\cap B_{R})\leqslant \Lambda$ for each $R>1$ and $M$ satisfies the strict stability~{\rm\ref{str-stab-M}}. Then there is $R_{0}=R_{0}(M,\lambda,\Lambda,\alpha)>1$ such that \[ C^{-1}R^{-\alpha}\hskip1pt\text{\small $-$}\hskip-10.6pt\int_{M\cap B_{R_{0}}}d^{2}\leqslant \hskip1pt\text{\small $-$}\hskip-10.6pt\int_{M\cap B_{R}}d^{2} \leqslant CR^{\alpha}\hskip1pt\text{\small $-$}\hskip-10.6pt\int_{M\cap B_{R_{0}}}d^{2} \,\,\,\,\forall R \geqslant R_{0}, \quad C=C(\lambda,\Lambda,\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}). \] \end{state}
To facilitate the proof we need the technical preliminaries of the following lemma. In this lemma \[ \nu_{M}=(\nu_{1},\ldots,\nu_{n+\ell+1}) \dl{normal-M} \] continues to denote a smooth unit normal for $M$, $\nu_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}$ continues to denote the unit normal of $\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}$ pointing into $U_{+}$, and $\varepsilon_{0}=\varepsilon_{0}(\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0})\in (0,\fr{1}{2}]$ is assumed small enough to ensure that there is a smooth nearest point projection \[
\pi:\bigl\{x\in\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{n+1}:d(x)<\varepsilon_{0}|x|\bigr\} \to \mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}. \dl{nearest-pt} \]
\begin{state}{\bf{}\tl{tech-prelim} Lemma.}
Suppose $\Lambda>0$, $\delta\in (0,\fr{1}{2}]$, $\theta\in [\fr{1}{2} ,1)$, $M$ satisfies~{\rm\ref{local-M}}, and $M\subset U_{+}\times\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{\ell}$. Then there is $\varepsilon=\varepsilon(\delta,\lambda,\Lambda,\theta,\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N})\in (0,\varepsilon_{0}]$ such that, for all $(x,y)\in M\cap B_{\theta R}$ with $d(x)<\varepsilon|x|$, \[
\min\bigl\{\bigl |\nu_{M}(x,y)-\bigl(\nu_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}} (\pi(x)),0\bigr)\bigr|,
\,\bigl|\nu_{M}(x,y)+\bigl(\nu_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}(\pi(x)),0\bigr)\bigr|\bigr\}<\delta \leqno{\rm(i)}\\ \]
in particular $|\nu_{y}|(x,y)\,\,(\,=|(\nu_{y_{1}}(x,y),\ldots,\nu_{y_{\ell}}(x,y))|)<\delta$), and \begin{align*}
&(1-7\delta)|x|^{-1}|(x,0)\cdot \nu_{M}(x,y)|\leqslant \tg{\rm(ii)} \\ \noalign{\vskip-1pt}
&\hskip1in \bigl|(x,0)\cdot\nabla_{M}\bigl(d(x)/|x|\bigr)\bigr| \leqslant
(1+7\delta)|x|^{-1}|(x,0)\cdot\nu_{M}(x,y)|. \end{align*} \end{state}
\noindent{\bf Remark:} The inequalities in~{\rm (ii)} do not depend on the minimality of $M$---{\rm (ii)} is true for any smooth hypersurface $M$ at points where $d(x)<\varepsilon|x|$ and where~{\rm (i)} holds for suitably small $\delta \in (0,\fr{1}{2}]$.
\begin{proof}{\bf{}Proof of Lemma~\ref{tech-prelim}:} By scaling we can assume $R=1$. If~(i) fails for some $\Lambda,
\delta,\theta$ then there is a sequence $(\xi_{k},\eta_{k})\in M\cap B_{\theta}$ with $d(\xi_{k})\leqslant k^{-1}|\xi_{k}|$ and \[
\min\bigl\{|\nu(\xi_{k},\eta_{k})-\nu_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}(\pi(\xi_{k}))|,|\nu(\xi_{k},\eta_{k})
+\nu_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}(\pi(\xi_{k}))|\bigr\}\geqslant\delta. \pdl{close-0} \]
Then passing to a subsequence we have $\tilde\xi_{k}=|\xi_{k}|^{-1}\xi_{k}\to\xi\in\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}$ and
$M_{k}=|\xi_{k}|^{-1}(M-(0,\eta_{k}))$ converges in the varifold sense to an integer multiplicity $V$ which is stationary in $B_{\theta^{-1}}(\xi,0)$, ${\rm spt\,} V\subset\overline U_{+}\times\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{\ell}$, and $\xi\in{\rm spt\,} V\cap \mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}$. Also, by the compactness and regularity theory of~\cite{SchS81}, ${\rm sing\,\,} V\cap \breve B_{\theta^{-1}}$ has Hausdorff dimension $\leqslant n+\ell-7$. So, since $\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}$ is connected, by the maximum principle of~\cite{Ilm96} we have \[ V\hbox{ {\vrule height .22cm}{\leaders\hrule\hskip.2cm} } \breve B_{\theta^{-1}}(\xi,0)=(q\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}+W)\hbox{ {\vrule height .22cm}{\leaders\hrule\hskip.2cm} } \breve B_{\theta^{-1}}(\xi,0) \pdl{close-1} \] where $W$ is integer multiplicity, stationary in $\breve B_{\theta^{-1}}$ and with ${\rm spt\,} W\subset\overline U_{+}\times\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{\ell}$. Taking the maximum integer $q$ such that this holds, we then have \[ B_{\sigma}(\xi,0)\cap{\rm spt\,} W=\emptyset \pdl{close-2} \] for some $\sigma>0$, because otherwise ${\rm spt\,} W\cap \overline\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}\neq \emptyset$ and we could apply~\cite{Ilm96} again to conclude $W\hbox{ {\vrule height .22cm}{\leaders\hrule\hskip.2cm} } \breve B_{\theta^{-1}}(\xi,0)=(\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}+W_{1})\hbox{ {\vrule height .22cm}{\leaders\hrule\hskip.2cm} } \breve B_{\theta^{-1}}(\xi,0)$ where $W_{1}$ is stationary integer multiplicity, contradicting the maximality of $q$ in~\ref{close-1}.
In view of~\ref{close-1}, \ref{close-2} we can then apply the sheeting theorem~\cite[Theorem 1]{SchS81} to assert that $M_{k}\cap B_{\sigma/2}(\xi,0)$ is $C^{2}$ close to $\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}$, and in particular \[
\min\bigl\{|\nu(\xi_{k},\eta_{k})-\nu_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}(\pi(\xi_{k}))|,|\nu(\xi_{k},\eta_{k}) +
\nu_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}(\pi(\xi_{k}))|\bigr\}\to 0, \] contradicting~\ref{close-0}.
To prove~(ii), let $(x_{0},y_{0})\in M$ such that $d(x_{0})<\varepsilon|x_{0}|$ and such that~(i) holds with
$(x,y)=(x_{0},y_{0})$, and let $\sigma>0$ be small enough to ensure that both $d(x)<\varepsilon|x|$ and~(i) hold for all $(x,y)\in M\cap B_{\sigma}(x_{0},y_{0})$. Let \[ M_{y_{0}} = \bigl\{x\in\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{n+1}:(x,y_{0})\in M\cap B_{\sigma}(x_{0},y_{0})\bigr\}. \] Then, taking a smaller $\sigma$ if necessary, we can assume \[ M_{y_{0}} = {\rm graph\,} u=\bigl\{(\xi+u(\xi)\nu_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}(\xi),y_{0}):\xi\in\Omega\bigr\}, \pdl{close-4} \] where $\nu_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}$ is the unit normal of $\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}$ pointing into $U_{+}$, $\Omega$ is an open neighborhood of $\pi(x_{0})$ in $\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}$, and $u$ is $C^{2}(\Omega)$.
Then with $x=\xi+u(\xi)\nu_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}(\xi)\in M_{y_{0}}$ we have also $x(t)=t\xi+u(t\xi)\nu_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}(\xi)\in M_{ y_{0}}$ for $t$ sufficiently close to $1$ (because $\nu_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}(t\xi)=\nu_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}(\xi)$) and hence \[
\fr{d}{dt}\bigr|_{t=1}x(t)=\xi+\xi\cdot \nabla_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}u(\xi)\nu_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}(\xi)\in T_{x}M_{y_{0}}. \pdl{close-5} \] With $\nu'(x)= (\nu_{1}(x,y_{0}),\ldots,\nu_{n+1}(x,y_{0}))$ (which is normal to $M_{y_{0}}$), we can assume, after changing the sign of $\nu$ if necessary, that \[
\nu_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}(\xi)\cdot\nu'(x)>0 \text{ for }x\in M_{y_{0}}, \text{ so } |\nu_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}(\xi)-\nu'(x)|<\delta\text{ by (i)}. \pdl{close-5a} \] By~\ref{close-5} we have \[
x\cdot\nu'(x)
=\bigl(x-\fr{d}{dt}\bigr|_{t=1}x(t)\bigr)\cdot\nu'(x)=\bigl(u(\xi)-\xi\cdot\nabla_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}u(\xi)\bigr)\nu_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}(\xi)\cdot\nu'(x), \] hence \[ u(\xi)-\xi\cdot\nabla_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}u(\xi)= (\nu_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}(\xi)\cdot\nu'(x))^{-1}x\cdot\nu'(x). \pdl{close-6} \] Differentiating the identity $u(t\xi)=d\bigl(t\xi+u(t\xi)\nu_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}(\xi)\bigr)$ at $t=1$, we obtain
{\abovedisplayskip8pt\belowdisplayskip8pt \begin{align*} -\xi\cdot\nabla_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}u(\xi) &= -\bigl(\xi+\xi\cdot\nabla_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}u(\xi)\nu_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}(\xi)\bigr)\cdot\nabla_{M_{y_{0}}}d(x) \\ &=-\bigl(x-(u(\xi)-\xi\cdot\nabla_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}u(\xi))\nu_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}(\xi)\bigr)\cdot\nabla_{M_{y_{0}}}d(x) \\
&=-x\cdot\nabla_{M_{y_{0}}}d(x) +\bigl((u(\xi)-\xi\cdot\nabla_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}u(\xi))\nu_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}(\xi)\bigr)\cdot\nabla_{M_{y_{0}}}d(x), \end{align*} hence, adding $d(x)\,\,(\,=u(\xi))$ to each side of this identity, \begin{align*} d(x)-x\cdot\nabla_{M_{y_{0}}}d(x) &=\bigl(u-\xi\cdot \nabla_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}u(\xi)\bigr)\bigl(1-\nu_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}(\xi).\nabla_{M_{y_{0}}}d(x)\bigr)\ptg{close-7} \\ &=\bigl(\nu_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}(\xi)\cdot\nu'(x)\bigr)^{-1}x\cdot\nu'(x) \bigl(1-\nu_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}(\xi).\nabla_{M_{y_{0}}}d(x)\bigr) \end{align*}}
by~\ref{close-6}. Using the identity $|x|^{-1}x\cdot\nabla_{M_{y_{0}}}|x|=1-|\nu'(x)|^{-2}(x\cdot\nu'(x)/|x|)^{2}$ we see that the left side of~\ref{close-7} is $\bigl(-x\cdot \nabla_{M_{y_{0}}}(d(x)/|x|)-|\nu'(x)|^{-2}(|x\cdot\nu'(x)|/|x|)^{2}\bigr)|x|$, so~\ref{close-7} gives
{\abovedisplayskip8pt\belowdisplayskip0pt \[
x\cdot \nabla_{M_{y_{0}}}(d(x)/|x|)=|x|^{-1} x\cdot\nu'(x)(1+E), \]} where
{\abovedisplayskip6pt\belowdisplayskip8pt \[
E=|\nu'(x)|^{-2}x\cdot\nu'(x)/|x|+(|\nu_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}(\xi)\cdot\nu'(x)|^{-1}-1)
-|\nu_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}(\xi)\cdot\nu'(x)|^{-1}\nu_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}(\xi)\cdot\nabla_{M_{y_{0}}}d(x). \]} Since $\nu_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}(\xi)\cdot\nabla_{M_{y_{0}}}d(x)=(\nu_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}(\xi)-\nu'(x))\cdot\nabla_{M_{y_{0}}}d(x)$ and $x\cdot\nu'(x)= x\cdot\nu_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}(\xi) +x\cdot (\nu'(x)-\nu_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}(\xi))=
d(x)-x\cdot(\nu_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}(\xi)-\nu'(x))$, and $|\nu_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}(\xi)-\nu'(x)|<\delta$ by~\ref{close-5a}, direct calculation then shows that $|E| < 7\delta$. Since $x\cdot \nabla_{M_{y_{0}}}(d(x)/|x|)=(x,0)\cdot
\nabla_{M}(d(x)/|x|)\bigr|_{(x,y_{0})}$ and $x\cdot\nu'(x)=(x,0)\cdot\nu(x,y_{0})$ we thus have the required inequalities~(ii) at $(x_{0},y_{0})$. \end{proof}
\begin{proof}{\bf{}Proof of Lemma~\ref{main-dist-est}.}
First we establish an $L^{2}$ bound for $|x|^{-1}d(x)$, namely \[
\int_{M\cap B_{R}}|x|^{-2}d^{2}(x,y)\,d\mu(x,y) \leqslant C R^{-2}\int_{M\cap B_{R/2}\setminus\{
(x,y):|x|<\smfr{1}{4}R\}}d^{2}(x,y)\,d\mu(x,y) \pdl{pf-m-1} \] for $R$ sufficiently large (depending on $M$), where $C=C(\lambda,\Lambda,\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N})$.
To check this observe that since $R^{-n-\ell-2}\int_{M\cap B_{R}}d^{2}\to 0$ by~\ref{tangent-cone}, we can apply Lemma~\ref{tech-prelim} for sufficiently large $R$ giving conclusions~(i),(ii) of~\ref{tech-prelim} with $\delta=\fr{1}{8}$ and $\theta=\fr{1}{2}$.
Let $\varepsilon=\varepsilon(\delta,\lambda,\Lambda,\theta)\in(0,\varepsilon_{0}]$ be as in Lemma~\ref{tech-prelim} with
$\delta=\fr{1}{8}$ and $\theta=\fr{1}{2}$, and let $\bigl(d/|x|\bigr)_{\varepsilon}=\min\{d(x)/|x|,\varepsilon\}$. Then, since $d(x)/|x|\leqslant 1$ for all $x$, \[
d(x)/|x| \leqslant \varepsilon^{-1}\bigl(d(x)/|x|\bigr)_{\varepsilon} \text{ at all points of } M, \pdl{pf-m-2} \]
and since $\nabla_{M}\bigl(d(x)/|x|\bigr)_{\varepsilon}=0$ for $\mu$-a.e.\ point $(x,y)$ with $d(x)/|x|\geqslant \varepsilon$, by~\ref{tech-prelim}(ii)
{\abovedisplayskip8pt\belowdisplayskip8pt \[
|(x,0)\cdot\nabla_{M}\bigl(d(x)/|x|\bigr)_{\varepsilon}| \leqslant 2|x|^{-1}|(x,0)\cdot\nu(x,y)|
\text{ at $\mu$-a.e.\ point of } M\setminus B_{R_{0}}, \pdl{pf-m-3} \] for suitable $R_{0}=R_{0}(M,\lambda,\Lambda,\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N})$.
Take $R>R_{0}$ and $\zeta\in C^{1}_{c}(\breve B^{n+1}_{R})$, $\chi\in C^{1}_{c}(\breve B^{\ell}_{R})$ with
$\zeta(x)\equiv 1$ for $|x|\leqslant \fr{1}{4} R$, $\zeta(x)=0$ for $|x|>\fr{1}{2} R$, $\chi(y)\equiv 1$ for $|y|\leqslant \fr{1}{4}R$,
$\chi(y)=0$ for $|y|>\fr{1}{2} R$, and $|D\zeta|,|D\chi|\leqslant 3 R^{-1}$. Since \[
{\rm div}_{M}(x,0)\geqslant n> 2\text{ and } |(x,0)\cdot\nabla_{M}\chi(y)|\leqslant |x|\,|D\chi(y)|\,|\nu_{y}|, \] we can apply~\ref{pf-m-2}, \ref{pf-m-3}, and~\ref{stationarity} with
$Z_{|(x,y)}=\bigl(d(x)/|x|\bigr)_{\varepsilon}^{2}\zeta^{2}(x)\chi^{2}(y)(x,0)$ to conclude \begin{align*}
&\hskip1pt\text{\small $-$}\hskip-10.6pt\int_{M\cap B_{R/4}}\hskip-8pt |x|^{-2}d^{2}(x)\,d\mu \ptg{pf-m-4}\\
&\hskip.2in \leqslant C\hskip1pt\text{\small $-$}\hskip-10.6pt\int_{M\cap B_{R}}\hskip-6pt\bigl(|x|^{-2}((x,0)\cdot \nu)^{2}+|\nu_{y}|^{2}\bigr)\,d\mu+
CR^{-2}\hskip1pt\text{\small $-$}\hskip-10.6pt\int_{M\cap B_{R}\setminus \{(x,y):|x|<\frac{1}{4} R\}}\hskip-20pt d^{2}\,d\mu \\
&\hskip0.2in \leqslant 2 C\hskip1pt\text{\small $-$}\hskip-10.6pt\int_{M\cap B_{R}}\hskip-6pt \bigl(|x|^{-2}((x,y)\cdot
\nu)^{2}+R^{2}|x|^{-2}|\nu_{y}|^{2}\bigr)\,d\mu+ \\ \noalign{\nobreak\vskip-4pt}
&\hskip2.6in CR^{-2}\hskip1pt\text{\small $-$}\hskip-10.6pt\int_{M\cap B_{R}\setminus\{(x,y):|x|<\frac{1}{4}R\}}\hskip-20pt d^{2}\,d\mu, \end{align*} where $C=C(\lambda,\Lambda,\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N})$.
By~\ref{strong-doub-lem}, with either of the choices $w=(x,y)\cdot \nu$ or $w=R\nu_{y_{j}}$, \[
\hskip1pt\text{\small $-$}\hskip-10.6pt\int_{M\cap B_{R}}|x|^{-2}w^{2}\leqslant CR^{-2}\hskip1pt\text{\small $-$}\hskip-10.6pt\int_{M\cap B_{R/2}\setminus\{(x,y):|x|<\smfr{1}{3} R\}}w^{2}, \] and hence~\ref{pf-m-4} gives \begin{align*}
&\hskip1pt\text{\small $-$}\hskip-10.6pt\int_{M\cap B_{R/4}}\hskip-8pt |x|^{-2}d^{2}(x)\,d\mu\leqslant
CR^{-2}\hskip1pt\text{\small $-$}\hskip-10.6pt\int_{M\cap B_{R}\setminus\{(x,y):|x|<\smfr{1}{3}R\}}\hskip-20pt(\,((x,y)\cdot\nu)^{2}+R^{2}|\nu_{y}|^{2}\,)\ptg{pf-m-4.1}\\
&\hskip2.5in +CR^{-2}\hskip1pt\text{\small $-$}\hskip-10.6pt\int_{M\cap B_{R}\setminus\{(x,y):|x|<\smfr{1}{4}R\}}\hskip-30pt d^{2}\,d\mu. \end{align*} In view of~\ref{tangent-cone} there exists $R_{0}>1$ and $\delta,\tilde\delta:(0,\infty)\to (0,\infty)$ such that $\delta(t),\tilde\delta(t)\to 0$ as $t\to\infty$ and \[ \left\{\begin{aligned}
& \bigl\{(x,y)\in M:|x|\leqslant \tilde\delta(|(x,y))|(x,y)|\bigr\}\setminus B_{R_{0}}\subset \cup_{j=1}^{q}{\rm graph\,} u_{j} \subset M \\
&{\sup}_{(\xi,\eta)\in \Omega\setminus B_{R}}{\textstyle\sum}_{j=1}^{q}\bigl(|Du_{j}(\xi,\eta)|+|(\xi,\eta)|^{-1}u_{j}(\xi,\eta)\bigr) \to 0
\text{ as }R\to\infty, \end{aligned}\right. \pdl{pf-m-10} \] where $u_{j}$ are positive $C^{2}$ functions on the domain $\Omega$, \[
\Omega\supset\{(x,y)\in\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}:|x|<\delta((|x,y|))|(x,y)|\}\setminus B_{R_{0}}. \]} For $(x,y)=(\xi+u_{j}(\xi,y)\nu_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}(\xi),y)$ with $(\xi,y)\in\Omega$, take an orthonormal basis $\tau_{1},\ldots,\tau_{n+\ell}$ for $T_{(\xi,y)}\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}$ with $\tau_{1},\ldots,\tau_{n-1}$ principal directions for $\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}\cap
\Sph^{n}$ (so $\nabla_{\tau_{i}}\nu_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}=\kappa_{i}\tau_{i}$ for $i=1,\ldots,n-1$), $\tau_{n}=|\xi|^{-1}\xi$, and $\tau_{n+j}=e_{n+1+j}$, $j=1,\ldots,\ell$. Then the unit normal $\nu(x,y)$ of $M$ is
{\abovedisplayskip8pt\belowdisplayskip8pt \begin{align*} &\nu(x,y)= \bigl(1+{\textstyle\sum}_{i=1}^{n}(1+\kappa_{i}u_{j}(\xi,y))^{-2}(D_{\tau_{i}}u_{j}(\xi,y))^{2}+
|D_{y}u(\xi,y)|^{2}\bigr)^{-1/2} \\ \noalign{\vskip-1pt} &\hskip0.2in \times\bigl(\nu_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}(\xi,y)-{\textstyle\sum}_{i=1}^{n}(1+\kappa_{i}u_{j}(\xi,y))^{-1} D_{\tau_{i}}u_{j}(\xi,y)\tau_{i} -{\textstyle\sum}_{k=1}^{\ell}D_{y_{k}}u_{j}(\xi,y)e_{n+1+k}\bigr), \end{align*} so for $R$ sufficiently large \[
|\nu_{y}(x,y)|\leqslant |D_{y}u_{j}(\xi,y)|\text{ and } |(x,0)\cdot\nu_{M}(x,y)|\leqslant |u_{j}(\xi,y)|+2|\xi||\nabla_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}}u_{j}(\xi,y)|. \pdl{nu-ineq} \] Also, since $u_{j}$ satisfies the equation $\mathcal{M}_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}}(u_{j})=0$, where $\mathcal{M}_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}}$ satisfies~\ref{L-M-3} with $\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}$ in place of $M$, we have \[
\int_{B_{R}\setminus\{(x,y):|x|<\smfr{1}{3} R\}}|\nabla_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}}u_{j}|^{2}\leqslant CR^{-2}
\int_{B_{3R/2}\setminus\{(x,y):|x|<\smfr{1}{4} R\}}u_{j}^{2}, \pdl{ell-est} \] Also, $d(\xi+u_{j}(\xi,y)\nu_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}(\xi))=u_{j}(\xi,y)$ so, by~\ref{nu-ineq} and~\ref{ell-est},
\[
\hskip1pt\text{\small $-$}\hskip-10.6pt\int_{M\cap B_{R}\setminus\{(x,y):|x|<\smfr{1}{3} R\}}w^{2}\leqslant
C\hskip1pt\text{\small $-$}\hskip-10.6pt\int_{M\cap B_{3R/2}\setminus\{(x,y):|x|<\smfr{1}{4}R\}}d^{2} \pdl{pf-m-7} \] for either of the choices $w=(x,y)\cdot \nu$ and $w=R\nu_{y_{j}}$, and hence~\ref{pf-m-4.1} implies \[
\hskip1pt\text{\small $-$}\hskip-10.6pt\int_{M\cap B_{R/4}}|x|^{-2}d^{2}(x)\leqslant CR^{-2}\hskip1pt\text{\small $-$}\hskip-10.6pt\int_{M\cap B_{3R/2}\setminus\{(x,y):|x|<\smfr{1}{4}R\}}d^{2}, \quad C=C(\lambda,\Lambda,\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}). \pdl{pf-m-7.1} \] Since $\mathcal{M}(u_{j})=0$ and the $u_{j}$ are positive with small gradient, and also $d(\xi + u_{j}(\xi,\eta)\nu_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}(\xi),\eta)=u_{j}(\xi,\eta)$, we can use the Harnack inequality in balls of radius $R/20$, to give \[
\hskip1pt\text{\small $-$}\hskip-10.6pt\int_{M\cap B_{3R/2}\setminus\{(x,y):|x|<\smfr{1}{4}R\}}d^{2}\leqslant C\hskip1pt\text{\small $-$}\hskip-10.6pt\int_{M\cap B_{R/8}\setminus\{(x,y):|x|<\smfr{1}{16}R\}}d^{2}, \] and so~\ref{pf-m-7.1} gives \[
\hskip1pt\text{\small $-$}\hskip-10.6pt\int_{M\cap B_{R/4}}|x|^{-2}d^{2}(x) \leqslant CR^{-2}\hskip1pt\text{\small $-$}\hskip-10.6pt\int_{M\cap B_{R/8}\setminus\{(x,y):|x|<\smfr{1}{16}R\}}d^{2}, \quad C=C(\lambda,\Lambda,\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}). \pdl{pf-m-7.2} \] Then~\ref{pf-m-1} follows from~\ref{pf-m-7.2} after replacing $R$ by $4R$. \ref{pf-m-1} in particular implies, for all sufficiently large $R$, \[ \left\{\begin{aligned}
&\int_{M\cap B_{R}\cap \{(x,y):|x|<\delta R\}}d^{2}\,d\mu\leqslant C\delta^{2}\int_{M\cap B_{R/2}} d^{2}\,d\mu \,\,\,\forall \delta\in (0,\fr{1}{2}], \\
&\int_{M\cap B_{R}}d^{2}\,d\mu\leqslant C\int_{M\cap B_{R/2}\setminus\{(x,y):|x|<\frac{1}{4}R\}} d^{2}\,d\mu, \end{aligned}\right. \pdl{pf-m-7a} \] where $C=C(\lambda,\Lambda,\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N})$.
So let $R_{k}\to\infty$ be arbitrary, $M_{k}=R_{k}^{-1}M$,\,$d_{k}=d|M_{k}$, and \[ u^{k}_{j}(x,y)=R_{k}^{-1}u_{j}(R_{k}x,R_{k}y)\big/\bigl(\text{\scriptsize $-$}\hskip-6.85pt\tint_{M_{k}\cap B_{1}}d_{k}^{2}\bigr)^{1/2}. \]} By virtue of~\ref{pf-m-1}, \ref{pf-m-10}, and \ref{pf-m-7a}, we have a subsequence of $k$ (still denoted $k$) such that
$u^{k}_{j}$ converges locally in $\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}$ to a solution $v_{j}\geqslant 0$ of $\mathcal{L}_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}}v_{j}=0$ with $v_{j}$ strictly positive for at least one $j$, and $\int_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}\cap B_{R}}|x|^{-2}v_{j}^{2}\,d\mu<\infty$ for each $R>0$. Hence $v_{j}$ has a representation of the form~\ref{LC-soln} on all of $\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}$, and then since $v_{j}\geqslant 0$ we must have $\smash{v_{j}=c_{j}r^{\gamma_{1}^{+}}\varphi_{1}}$ with $c_{j}\geqslant 0$ and $c_{j}>0$ for at least one $j$. But then $\text{\scriptsize $-$}\hskip-6.85pt\tint_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}\cap B_{R}}\sum_{j=1}^{q}v^{2}_{j}$ is constant, independent of $R$, and so (using~\ref{pf-m-7a} again) \[ \lim_{k\to\infty}\hskip1pt\text{\small $-$}\hskip-10.6pt\int_{M_{k}\cap B_{2R_{k}}}d^{2}\bigg/\hskip1pt\text{\small $-$}\hskip-10.6pt\int_{M_{k}\cap B_{R_{k}}}d^{2}= \hskip1pt\text{\small $-$}\hskip-10.6pt\int_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}\cap B_{2}}\sum_{j=1}^{q}v^{2}_{j}\bigg/\hskip1pt\text{\small $-$}\hskip-10.6pt\int_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}\cap B_{1}}\sum_{j=1}^{q}v^{2}_{j}= 1. \] Therefore, in view of the arbitrariness of the sequence $R_{k}$, for any $\alpha\in (0,\fr{1}{2}]$ we have \[ 2^{-\alpha}\leqslant \hskip1pt\text{\small $-$}\hskip-10.6pt\int_{M\cap B_{R}}d^{2}\bigg/\hskip1pt\text{\small $-$}\hskip-10.6pt\int_{M\cap B_{R/2}}d^{2} \leqslant 2^{\alpha}\,\,\, \forall\,R>R_{0}, \] with $R_{0}=R_{0}(M,\lambda,\Lambda,\alpha)$, and~\ref{main-dist-est} follows. \end{proof}
\section{Proof of Theorem~\ref{main-th}} \label{th-1-pf}
\noindent According to Lemma~\ref{tangent-cone} the unique tangent cone of $M$ at $\infty$ is $\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}$ (with multiplicity $q$). Let $\alpha\in (0,1)$. By Lemma~\ref{main-dist-est} \[ \hskip1pt\text{\small $-$}\hskip-10.6pt\int_{M\cap B_{R}}d^{2}\,d\mu \leqslant R^{\alpha} \dl{pf-th-1} \] for all sufficiently large $R$. Also by~\ref{growth-lem} either $\nu_{y}$ is identically zero or there is a lower bound \[
\hskip1pt\text{\small $-$}\hskip-10.6pt\int_{M\cap B_{R}}|\nu_{y}|^{2}\geqslant R^{-\alpha} \dl{pf-th-2} \] for all sufficiently large $R$, and by Lemma~\ref{strong-doub-lem} with $w=\nu_{y_{j}}$ we have \[
\hskip1pt\text{\small $-$}\hskip-10.6pt\int_{M\cap B_{R}}|\nu_{y}|^{2}\leqslant C\hskip1pt\text{\small $-$}\hskip-10.6pt\int_{M\cap B_{R/2}\setminus\{(x,y):|x|<\smfr{1}{3} R\}}|\nu_{y}|^{2}, \quad C =C(q,\lambda,\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}). \dl{pf-th-3} \] By inequality~\ref{pf-m-7} in the proof of Lemma~\ref{main-dist-est} (with $R/2$ in place of $R$), \[
\hskip1pt\text{\small $-$}\hskip-10.6pt\int_{M\cap B_{R/2}\setminus\{(x,y):|x|<\smfr{1}{3} R\}}|\nu_{y}|^{2}\leqslant CR^{-2}\hskip1pt\text{\small $-$}\hskip-10.6pt\int_{M\cap B_{R}}d^{2}. \dl{pf-th-3.1} \] Combining~\ref{pf-th-1}, \ref{pf-th-2}, \ref{pf-th-3} and~\ref{pf-th-3.1}, we then have \[
R^{-\alpha}\leqslant \hskip1pt\text{\small $-$}\hskip-10.6pt\int_{M\cap B_{R}}|\nu_{y}|^{2} \leqslant CR^{-2} \hskip1pt\text{\small $-$}\hskip-10.6pt\int_{M\cap B_{R}}d^{2} \leqslant R^{-2+\alpha} \] for all sufficiently large $R$ (depending on $\alpha,q,\lambda,\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}$), which is impossible for $R>1$. Thus the alternative that $\nu_{y}$ is identically zero on $M$ must hold, and so $M$ is a cylinder $S\times\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{\ell}$.\nobreak
\section{Proof of Corollary~\ref{co-1}} \label{proof-liou}
\noindent We aim here to show that the hypotheses of Corollary~\ref{co-1} ensure that $M$ is strictly stable as in~\ref{str-stab-M}, so that Corollary~\ref{co-1} is then implied by Theorem~\ref{main-th}.
Before discussing the proof that $M$ is strictly stable, we need a couple of preliminary results. First recall that, according to \cite[Theorem 2.1]{HarS85}, since $\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}$ is minimizing there is a smooth embedded minimal hypersurface $S\subset U_{+}$ ($U_{+}$ either one of the two connected components of $\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{n+1}\setminus\overline{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}}_{0}$), with \begin{align*} &\text{${\rm dist\,}(S,\{0\})=1$, ${\rm sing\,\,} S=\emptyset$ and $S$ is a smooth \emph{radial graph}; i.e.\ each }\dtg{props-S} \\ \noalign{\vskip-2pt} &\hskip0.4in\text{ray $\bigl\{tx:t>0\bigr\}$ with $x\in U_{+}$ meets $S$ transversely in just one point, } \end{align*} so in particular \begin{align*}
&x\cdot \nu_{S}(x)>0 \text{ for each $x\in S$, where $\nu_{S}$ is the smooth} \dtg{radial-graph}\\
\noalign{\nobreak\vskip-2pt}
&\hskip1.2in \text{ unit normal of $S$ pointing away from $\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}$.} \end{align*} Furthermore, since $\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}$ is strictly stable and strictly minimizing, \cite[Theorem 3.2]{HarS85} ensures that, for $R_{0}=R_{0}(\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0})$ large enough, there is an $C^{2}$ function $u$ defined on an open subset of
$\Omega\subset\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}$ with $\Omega\supset \bigl\{x\in\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}:|x|>2R_{0}\bigr\}$ and \[ S\setminus B^{n+1}_{R_{0}}=\bigl\{x+u(x)\nu_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}(x):x\in\Omega\bigr\} \text{ with }u(x)=
\kappa|x|^{\gamma_{1}}\varphi_{1}(|x|^{-1}x) +E(x), \dl{S-asymp} \] where $\kappa$ is a positive constant, $\nu_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}$ is the unit normal of $\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}$ pointing into $U_{+}$, $\varphi_{1}>0$ is as in~\ref{char-exps} with $j=1$, and, for some $\alpha=\alpha(\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0})>0$, \[
\lim_{R\to\infty}\sup_{|x|>R}|x|^{k+|\gamma_{1}|+\alpha}|D^{k}E(x)|= 0\text{ for }k=0,1,2. \]
We claim that $S$ is strictly stable:
\begin{state}{\bf{}\tl{strict-stab-S} Lemma.} If $S$ is as above, there is a constant $\lambda=\lambda(\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0})>0$ such that \[
\lambda\int_{S}|x|^{-2}\zeta^{2}(x,y)\,d\mu(x,y)\leqslant
\int_{S}\bigl(\bigl|\nabla_{S}\zeta\bigr|^{2}-|A_{S}|^{2}\zeta^{2}\bigr)\,d\mu, \quad \zeta\in C_{c}^{1}(\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{n+1}), \]
where $|A_{S}|$ is the length of the second fundamental form of $S$. \end{state}
\begin{proof}{\bf{}Proof:}
The normal part of the velocity vector of the homotheties $\lambda S|_{\lambda>0}$ at $\lambda=1$ is $\psi=x\cdot\nu_{S}(x)\, (\,>0 \text{ by~\ref{radial-graph}})$, and since the $\lambda S$ are minimal hypersurfaces, this is a Jacobi function, i.e.\ a solution of \[
\Delta_{S}\psi+|A_{S}|^{2}\psi=0. \pdl{jac-for-S} \] By properties~\ref{radial-graph} and ~\ref{S-asymp} we also have \[
C^{-1}|x|^{\gamma_{1}}\leqslant \psi(x)\leqslant C|x|^{\gamma_{1}} \text{ on $S$},\quad C=C(\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}). \pdl{bounds-for-psi} \] After a direct calculation and an integration by parts,~\ref{jac-for-S} implies \[
\int_{S}\psi^{2} |\nabla_{\!S}\bigl(\zeta/\psi\bigr)|^{2}\,d\mu =
\int_{S}\bigl(|\nabla_{\!S} \zeta|^{2}-|A_{S}|^{2}\zeta^{2}\bigr)\,d\mu, \pdl{quot} \]
and using the first variation formula~\ref{stationarity} with $Z(x)=|x|^{-p-2}f^{2}(x)x$ and noting that ${\rm div}_{S}x=n$, we have, after an application of the Cauchy-Schwarz inequality, \[
\int_{S} |x|^{-p-2}f^{2} \leqslant C\int_{S} |x|^{-p}|\nabla_{S}f|^{2},\,\, f\in C^{1}_{c}(\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{n+1}),\quad p<n-2, \,\, C=C(p,n). \pdl{1st-var}
\]
\ref{strict-stab-S} is now proved by taking $f=\zeta/\psi$ and $p=2|\gamma_{1}|\,(<n-2)$, and using~\ref{bounds-for-psi}, \ref{quot} and~\ref{1st-var}. \end{proof}
As a corollary, any hypersurface sufficiently close to $S$ in the appropriately scaled $C^{2}$ sense must also be strictly stable:
\begin{state}{\bf{}\tl{co-stab-S} Corollary.} For each $\theta\in(0,\fr{1}{2}]$, there is $\delta=\delta(\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0},\theta)>0$ such that if $v\in C^{2}(S)$,
$M_{0}=\{x+v(x)\nu_{S}(x):x\in S\} \text{ and } |x||\nabla^{2}_{S}v|+|\nabla_{S}v|+|x|^{-1}|v|\leqslant \delta\,\,\forall x\in S$, then $M_{0}$ satisfies the inequality \[
\lambda(1-\theta)\int_{M_{0}}|x|^{-2}\zeta^{2}\,d\mu\leqslant
\int_{M_{0}}\bigl(\bigl|\nabla_{M_{0}}\zeta\bigr|^{2}-|A_{M_{0}}|^{2}\zeta^{2}\bigr)\,d\mu, \quad \zeta\in C_{c}^{1}(\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{n+1}), \]
where $|A_{M_{0}}|$ is the length of the second fundamental form of $M_{0}$, and $\lambda$ is the constant of~{\rm\ref{strict-stab-S}}. \end{state}
\begin{proof}{\bf{}Proof:} By~\ref{strict-stab-S}, with $\tilde\zeta(x)=\zeta(x+v(x)\nu_{S}(x))$ for $x\in S$, \[
\lambda \int_{S} |x|^{-2}\tilde\zeta^{2}\,d\mu \leqslant
\int_{S}\bigl(\bigl|\nabla_{S}\tilde\zeta\bigr|^{2}-|A_{S}|^{2}\tilde\zeta^{2}\bigr)\,d\mu \pdl{co-stab-1} \] and for any $C^{1}$ function $f$ with compact support on $M_{0}$, with $\tilde f(x)=f(x+v(x)\nu_{S}(x))$ for $x\in S$, \[ \left\{\hskip2pt\begin{aligned}
&{\textstyle\int}_{S}\tilde f\,d\mu = {\textstyle\int}_{M_{0}} f J\,d\mu \text{ with } |J-1|\leqslant C\delta \text{ (by the area formula)},\\ & |\nabla_{S}\tilde f(x)-(\nabla_{M_{0}}f)(x+v(x)\nu_{S}(x)|\leqslant C\delta
|\nabla_{S}\tilde f(x)|\\ & |A_{S}(x)-A_{M_{0}}(x+v(x)\nu_{S}(x))|< C|x|^{-1}\delta,\,\,\, C^{-1}|x|^{-1}\leqslant
|A_{S}(x)|\leqslant C|x|^{-1}, \end{aligned}\right. \pdl{co-stab-2} \] where $C=C(S)$. By combining ~\ref{co-stab-1} and~\ref{co-stab-2} we then have the required inequality with $\theta=C\delta$. \end{proof}
Next we need a uniqueness theorem for stationary integer multiplicity varifolds with support contained in $\overline U_{+}$.
\begin{state}{\bf{}\tl{pre-Liouville} Lemma.} If $M_{0}$ is a stationary integer multiplicity $n$-dimensional varifold in $\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{n+1}$ with the properties that ${\rm spt\,} M_{0}\subset\overline U_{+}$, ${\rm spt\,} M_{0}\neq \overline\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}$, and $ \sup_{R>1}R^{-n}\mu(M_{0}\cap B_{R})<2\mu(\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}\cap B_{1})$, then $M_{0}=\lambda S$ (with multiplicity~$1$) for some $\lambda>0$, where $S$ is as in~{\rm\ref{props-S}}. \end{state}
\begin{proof}{\bf{}Proof of \ref{pre-Liouville}:} Let $C(M_{0})$ be a tangent cone of $M_{0}$ at $\infty$. Then by the Allard compactness theorem $C(M_{0})$ is a stationary integer multiplicity varifold with $\mu_{C(M_{0})}(B_{1})<2\mu(\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}\cap B_{1})$ and ${\rm spt\,} C(M_{0})\subset \overline U_{+}$. If ${\rm spt\,} C(M_{0})$ contains a ray of $\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}$ then, by the Solomon-White maximum principle~\cite{SolW89}, either $C(M_{0})=\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}$ (with multiplicity one) or else $C(M_{0})=\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}+V_{1}$, where $V_{1}$ is a non-zero integer multiplicity cone with $\mu(V_{1}\cap B_{1}(0))<\mu(\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}\cap B_{1})$ and support contained in $\overline U_{+}$. On the other hand if ${\rm spt\,} C(M_{0})\cap \mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}=\emptyset$ then a rotation of ${\rm spt\,} C(M_{0})$ has a ray in common with $\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}$ so by the same argument applied to this rotation we still conclude that there is a stationary cone $V_{1}$ with $\mu_{V_{1}}(B_{1})<\mu(\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}\cap B_{1})$ and ${\rm spt\,} V_{1}\subset \overline U_{+}$. But now by applying exactly the same reasoning to $V_{1}$ we infer $\mu_{V_{1}}(B_{1})\geqslant \mu(\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}\cap B_{1})$, a contradiction. So $C(M_{0})=\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}$ and hence $\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}$, with multiplicity one, is the unique tangent cone of $M_{0}$ at infinity. Hence there is a $R_{0}>1$ and a $C^{2}(\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}\setminus B_{R_{0}})$ function $h$ with \begin{align*}
&\hskip0.7in\text{$\sup_{x\in \mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}\setminus B_{R}}(|Dh(x)|+|x|^{-1}h(x))\to 0$ as $R\to \infty$}, \ptg{def-h-1}\\ \noalign{\vskip3pt} &\{x+h(x)\nu_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}(x):x\in \mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}\setminus B_{R_{0}}\}\subset M_{0} \ptg{def-h-2} \\ \noalign{\vskip-2pt} &\hskip1in \text{ and }M_{0}\setminus \{x+h(x)\nu_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}(x):x\in \mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}\setminus B_{R_{0}}\} \text{ is compact.} \end{align*} We also claim \[ 0\notin M_{0}. \pdl{0-notin-M} \] Indeed otherwise the Ilmanen maximum principle~\cite{Ilm96} would give $M_{0}\cap \mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}\neq \emptyset$ and the above argument using the Solomon-White maximum principle can be repeated, giving $M_{0}=\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}$, contrary to the hypothesis that ${\rm spt\,} M_{0}\neq \overline\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}$.
Observe next that if $r_{k}\to\infty$ the scaled minimal hypersurfaces $r_{k}^{-1}M_{0}$ are represented by the graphs (taken off $\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}$) of the functions $r_{k}^{-1}h(r_{k}x)\to 0$, and hence, for any given $\omega_{0}\in\Sigma\, (=\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}\cap\partial B_{1})$, the rescalings $(h(r_{k}\omega_{0}))^{-1}h(r_{k}r\omega)$ (which are bounded above and below by positive constants on any given compact subset of $\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}$ by~\ref{def-h-1} and the Harnack inequality) generate positive solutions of the Jacobi equation $\mathcal{L}_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}v=0$ on $\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}$ as $k\to\infty$. But, by~\ref{eigenvals} and \ref{exp-gamma}, $(c_{1}r^{\gamma_{1}}+c_{2}r^{\gamma_{1}^{-}})\,\varphi_{1}(\omega)$ with $c_{1},c_{2}\geqslant 0$ and $c_{1}+c_{2}>0$ are the only positive solutions of $\mathcal{L}_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}}(\varphi)=0$ on all of $\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}$. Thus in view of the arbitrariness of the sequence $r_{k}$, we have shown that \[ h(r\omega) =c(r) \varphi_{1}(\omega) +o(c(r)) \text{ as $r\to \infty$, uniformly for $\omega\in\Sigma$}, \pdl{M-as} \] and hence there are \[ c_{-}(r)<c(r)<c_{+}(r)\text{ with }c_{-}(r)\varphi_{1}(\omega)<h(r\omega)<c_{+}(r)\varphi_{1}(\omega) \text{ and }c_{+}(r)/c_{-}(r)\to 1. \] Now, for suitable $R_{0}>0$, $S\setminus B_{R_{0}}$ ($S$ as in~\ref{props-S}) also has a representation of this form with some $\tilde h$ in place of $h$, where $$ \tilde h(r\omega)=\kappa r^{\gamma_{1}}\varphi_{1}(\omega) + o(r^{\gamma_{1}}) \text{ as $r\to \infty$, uniformly in $\omega$} $$ and, similar to the choice of $c_{\pm}$, we can take $\tilde c_{\pm}(r)$ such that \[ \tilde c_{-}(r)\varphi_{1}(\omega)<\tilde h(r\omega)< \tilde c_{+}(r)\varphi_{1}(\omega)\text{ and }\tilde c_{+}(r)/\tilde c_{-}(r)\to 1\text{ as }r\to \infty. \] Now $\lambda S\setminus B_{\lambda R_{0}}$ can be represented by the geometrically scaled function $\tilde h_{\lambda}$ with \[
\tilde h_{\lambda}(r\omega)=\kappa \lambda^{1+|\gamma_{1}|}r^{-|\gamma_{1}|}\varphi_{1} +
o(\kappa \lambda^{1+|\gamma_{1}|}r^{-|\gamma_{1}|}\varphi_{1}) \pdl{M-as-lam} \] and we let $\lambda_{k}^{-}$ be the largest value of $\lambda$ such that $\tilde h_{\lambda}(r_{k}\omega)\leqslant c_{-}(r_{k})\varphi_{1}(\omega)$ and $\lambda_{k}^{+}$ the smallest value of $\lambda$ such that $\tilde h_{\lambda}(r_{k}\omega)\geqslant c_{+}(r_{k})\varphi_{1}(\omega)$. Evidently there are then points $\omega_{\pm} \in \Sigma$ with $\tilde h_{\lambda_{k}^{\pm}}(r_{k}\omega_{\pm})=c_{\pm}(r_{k})\varphi_{1}(\omega_{\pm})$ respectively. Also $M_{0}\cap \breve B_{r_{k}}$ must entirely lie in the component $B_{r_{k}}\setminus \lambda_{k}^{+} S$ which contains $\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}\cap B_{r_{k}}$; otherwise we could take the smallest $\lambda>\lambda_{k}^{+}$ such that $M_{0} \cap B_{r_{k}}$ lies on in the closure of that component of $B_{r_{k}}\setminus \lambda S$, and $M_{0}\cap \breve B_{r_{k}}\cap \lambda S\neq \emptyset$, which contradicts the maximum principle~\cite{SolW89}. Similarly, $M_{0}\cap \breve B_{r_{k}}$ must entirely lie in the component $B_{r_{k}}\setminus \lambda_{k}^{-} S$ which does not contain $\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}\cap B_{r_{k}}$.
Thus $M_{0}\cap B_{r_{k}}$ lies between $\lambda_{k}^{+}S$ and $\lambda_{k}^{-}S$ and by construction $\lambda_{k}^{+}/\lambda_{k}^{-}\to 1$, and since $\lambda_{k}^{-}$ is bounded above, we then have a subsequence of $k$ (still denoted $k$) such that $\lambda_{k}^{\pm}$ have a common (positive) limit $\lambda$. So ${\rm spt\,} M_{0}\subset \lambda S$ and hence $M_{0}=\lambda S$ with multiplicity~$1$ by the constancy theorem and the fact that $ \sup_{R>1}R^{-n}\mu(M_{0}\cap B_{R})<2\mu(\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}\cap B_{1})$. \end{proof}
Finally we need to show that any $M$ satisfying the hypotheses of Corollary~\ref{co-1} with sufficiently small $\varepsilon_{0}$ must be strictly stable; then (as mentioned at the beginning of this section) Corollary~\ref{co-1} follows from Theorem~\ref{main-th}.
\begin{state}{\bf{}\tl{stab-lem} Lemma.} For each $\alpha,\theta\in (0,1)$, there is $\varepsilon_{0}=\varepsilon_{0}(\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N},\alpha,\theta)\in (0,\fr{1}{2}]$ such that if $M\subset U_{+}\times\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{\ell}$, ${\sup}_{R>1}R^{-n-\ell}\mu(M\cap B_{R})\leqslant (2-\alpha)\mu(\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}\cap B_{1})$, and
${\sup}_{M}|\nu_{y}| < \varepsilon_{0}$, then $M$ is strictly stable in the sense that \[
(1-\theta)\lambda\int_{M}|x|^{-2}\zeta^{2}\,d\mu\leqslant
\int_{M}\bigl(\bigl|\nabla_{M}\zeta\bigr|^{2}-|A_{M}|^{2}\zeta^{2}\bigr)\,d\mu,\quad\zeta\in C_{c}^{1}(\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{n+1+\ell}), \] with $\lambda=\lambda(\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0})>0$ as in~{\rm\ref{strict-stab-S}}. \end{state}
\begin{proof}{\bf{}Proof:} For $y\in\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{\ell}$, define
{\abovedisplayskip8pt\belowdisplayskip8pt \[ M_{y}= \lambda_{y}^{-1}(M-(0,y)),\quad \lambda_{y}={\rm dist\,}(M-(0,y),0). \pdl{pf-0} \] We claim that for each given $\delta>0$ the hypotheses of the lemma guarantee that a strip of $M$ near a given slice $M_{z}=\{(x,y)\in M:y=z\}$ can be scaled so that it is $C^{2}$ close to $S\times B^{\ell}_{1}$ ($S$ as in~\ref{radial-graph}) in the appropriately sense; more precisely, we claim that for each $\delta>0$ there is $\varepsilon_{0}=\varepsilon_{0}(\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N},\alpha,\theta,\delta)>0$ such that the hypotheses of the lemma imply that for each
$z\in\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{\ell}$ there is $v_{z}\in C^{2}(\{(x,y):x\in S,\,|y|<1\})$ such that \[ \left\{\begin{aligned}
&M_{z}\cap \{(x,y):|y|<1\}
=\bigl\{(x+v_{z}(x,y)\nu_{S}(x),y):x\in S,\,|y|<1\bigr\}\\
&\, |x||\nabla^{2}_{S\times\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{\ell}}v_{z}(x,y)|+|\nabla_{S\times\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{\ell}}v_{z}|(x,y)+|x|^{-1}|v_{z}(x,y)|\leqslant \delta
\,\,\forall x\in S,|y|<1. \end{aligned}\right. \pdl{pf-2} \] Otherwise this fails with $\varepsilon_{0}=1/k$ for each $k=1,2,\ldots$, so there are minimal submanifolds $M_{k}$ such that the hypotheses hold with $\varepsilon _{0}=1/k$ and with $M_{k}$ in place of $M$, yet there are points $z_{k}\in M_{k}$ such that~\ref{pf-2} fails with $z_{k},M_{k}$ in place of $z,M$.
\vskip1pt
We claim first that then there are fixed $k_{0}=k_{0}(\delta)\in \{1,2,\ldots\}, \,R_{0}=R_{0}(\delta)>1$ such that \[
{\rm dist\,}((x,y),M_{z_{k}}) <\delta|x| \,\,\forall (x,y)\in \mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}\setminus B_{R_{0}}^{n+1}\text{ with }|x|\geqslant |y|,\,\,\,k\geqslant k_{0}. \pdl{pf-2z} \] Otherwise there would be a subsequence of $k$ (still denoted $k$) with \[
\text{${\rm dist\,}((x_{k},y_{k}),M_{z_{k}})\geqslant \delta |x_{k}|$ with $(x_{k},y_{k})\in\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}$, $|y_{k}|\leqslant |x_{k}|$ and $|x_{k}|\to \infty$.} \] Then let \[
\wtilde{M}_{k} = |x_{k}|^{-1}M_{z_{k}}. \] By the Allard compactness theorem, $\wtilde{M}_{k}$ converges in the varifold sense to a stationary integer multiplicity varifold $V$ with support $M$ and density $\Theta$, where $M$ is a closed rectifiable set, $\Theta$ is upper semi-continuous on $\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{n+1+\ell}$ and has integer values $\mu$-a.e.\ on $M$, with $\Theta=0$ on $\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{n+1+\ell}\setminus M$, and \[ M=\{x:\Theta(x)\geqslant 1\}, \pdl{pf-2a} \] \[ \Theta(x) = \lim_{\rho\downarrow 0}\bigl(\omega_{n+\ell}\rho^{n+\ell}\bigr)^{-1}
\int_{M\cap B_{\rho}(x)}\Theta(\xi)\,d\mu(\xi)\,\,\,\forall x\in M, \pdl{pf-3} \] where $\omega_{n+\ell}$ is the volume of the unit ball in $\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{n+\ell}$, $0\in M\subset \overline U_{+}\times\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{\ell}$, and $M\cap U_{+}\times\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{\ell}\neq 0$.
Taking $x_{n+1+j}=y_{j}$, so points in $\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{n+1+\ell}$ are written $x=(x_{1},\ldots,x_{n+1+\ell})$, and letting $\nu_{k}=(\nu_{k\,1},\ldots,\nu_{k\,n+1+\ell})$ be a unit normal for $\wtilde{M}_{k}$ (so that the orthogonal projection of $\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{n+1+\ell}$ onto $T_{x}\wtilde{M}_{k}$ has matrix $(\delta_{ij}-\nu_{k\,i}\nu_{k\,j})$), the first variation formula for $\wtilde{M}_{k}$ can be written \[ \int_{\wtilde{M}_{k}}\sum_{i,j=1}^{n+1+\ell}(\delta_{ij}-\nu_{k\,i}\nu_{k\,j})D_{i}X_{j}\,d\mu=0,
\,\,\, X_{j}\in C^{1}_{c}(\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{n+1+\ell}),\,\,j=1,\ldots,n+1+\ell. \pdl{pf-4} \]} Let $x_{0}\in M,\,\sigma>0$, and let $\tau\in C^{\infty}(\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{n+1+\ell})$ with support $\tau\subset B_{\sigma}(x_{0})$, and consider the function $T$ defined by
{\abovedisplayskip8pt\belowdisplayskip8pt \[ T(x)=T(x_{1},\ldots,x_{n+1+\ell}) = \int_{-\infty}^{x_{n+1+\ell}}\bigl(\tau(x_{1},\ldots,x_{n+\ell},t)
-\tau(x_{1},\ldots,2\sigma+x_{n+\ell},t)\bigr)\,dt. \] Evidently $T\in C^{\infty}_{c}(\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{n+1+\ell})$ (indeed $T(x)=0$ on $\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{n+1+\ell}\setminus K$, where $K$ is the cube
$\{x\in\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{n+1+\ell}:|x_{i}-x_{0\,i}|<4\sigma\,\,\forall i=1,\ldots,n+1+\ell\}$). Therefore we can use~\ref{pf-4} with $X(x)=T(x)e_{n+1+\ell}$, giving \[ \int_{\wtilde{M}_{k}}\bigl(1-\nu_{k\,n+1+\ell}^{2}\bigr)\bigl( \tau(x)-\tau(x+2\sigma e_{n+1+\ell})\bigr)\,d\mu
= \int_{\wtilde{M}_{k}}\sum_{i=1}^{n+\ell}\nu_{k\,n+1+\ell}\nu_{k\,i} D_{i}T\,d\mu, \] and hence \[
\Bigl|\int_{\wtilde{M}_{k}} \tau(x)\,d\mu-\int_{\wtilde{M}_{k}}\tau(x+2\sigma e_{n+1+\ell})\,d\mu\Bigr| \leqslant C/k. \] Using the fact that varifold convergence of $\wtilde{M}_{k}$ implies convergence of the corresponding mass measures $\mu\hbox{ {\vrule height .22cm}{\leaders\hrule\hskip.2cm} } \wtilde{M}_{k}$ to $\mu\hbox{ {\vrule height .22cm}{\leaders\hrule\hskip.2cm} } \Theta$, we then have \[ \int_{M} \tau(x)\,\Theta(x) d\mu(x)=\int_{M}\tau(x+2\sigma e_{n+1+\ell})\,\Theta(x)d\mu(x). \pdl{pf-5} \] Taking $\rho\in (0,\sigma)$ and replacing $\tau$ by a sequence $\tau_{k}$ with $\tau_{k}\downarrow \chi_{B_{\rho}(x_{0})}$ (the indicator function of the closed ball $\smash{B_{\rho}(x_{0})}$), we then conclude \[ \int_{M\cap B_{\rho}(x_{0})}\,\Theta\,d\mu=\int_{M\cap B_{\rho}(x_{0}+2\sigma e_{n+1+\ell})}\ \Theta\,d\mu \]} and hence by~\ref{pf-3}
{\abovedisplayskip8pt\belowdisplayskip10pt \[ \Theta(x_{0})=\Theta(x_{0}+2\sigma e_{n+1+\ell}). \]} In view of the arbitrariness of $\sigma$ this shows that $\Theta(x)$ is independent of the variable $x_{n+1+\ell}$, and the same argument shows that $\Theta(x)$ is also independent of $x_{n+1+j}$, $j=1,\ldots,\ell-1$. Thus, by~\ref{pf-2a}, $M$ is cylindrical: $M=M_{0}\times\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{\ell}$, where $R^{-n}\int_{M_{0}\cap B_{R}}\Theta\,d\mu<2\mu(\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}\cap B_{1})$, $0\in M_{0}\subset \overline U_{+}$, and $M_{0}\cap U_{+}\neq \emptyset$. Hence $M_{0}=\lambda S$ for some $\lambda>0$ by virtue of Lemma~\ref{pre-Liouville}, contradicting the fact that $0\in M_{0}$. So~\ref{pf-2z} is proved, and~\ref{pf-2z} together with the Allard regularity theorem implies that
$M_{z_{k}}\cap\bigl(\bigl\{(x,y)\in \mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{n+1+\ell}\setminus B_{2 R_{0}}:|y|<\fr{1}{2}|x| \bigr\}\bigr)$
is $C^{2}$ close to $\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}$, in the sense that there is are $C^{2}$ functions $v_{k}$ on a domain in $\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}$ with
{\abovedisplayskip8pt\belowdisplayskip8pt \begin{align*} &{\rm graph\,} v_{k}=
M_{z_{k}}\cap\bigl(\bigl\{(x,y)\in \mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{n+1+\ell}\setminus B_{2 R_{0}}:|y|<\fr{1}{2}|x| \bigr\}\bigr)\ptg{pf-7}\\
&\hskip1.7in \text{ and }|x|^{-1}|v_{k}|+|\nabla_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}}v_{k}|+|x||\nabla^{2}_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}}v_{k}|<C\delta. \end{align*}}
Next, exactly the same compactness discussion can be applied with $M_{z_{k}}$ in place of $\wtilde{M}_{k}$, giving a cylindrical varifold limit $M=M_{0}\times\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{\ell}$, $M_{0}$ a closed subset of $\overline U_{+}$ with density $\Theta\geqslant 1$, but this time ${\rm dist\,}(M_{0},0)=1$. Hence $\Theta\equiv 1$ and $M_{0}= S$ by~\ref{pre-Liouville}. But then the Allard regularity theorem guarantees that the convergence of $M_{z_{k}}$ to $M$ is smooth and hence, by virtue of~\ref{pf-7},~\ref{pf-2} holds with $M_{z_{k}}$ in place of $M$, a contradiction. Thus \ref{pf-2} is proved.
Then by Corollary~\ref{co-stab-S}, for small enough $\delta=\delta(\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N},\theta)$, for each $y\in\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{\ell}$ \[
(1-\theta/2)\lambda\int_{M_{y}}|x|^{-2}\zeta_{y}^{2}(x)\,d\mu(x)\leqslant
\int_{M_{y}}\bigl(\bigl|\nabla_{M_{y}}\zeta_{y}(x)\bigr|^{2}-|A_{M_{y}}|^{2}\zeta_{y}^{2}(x)\bigr)\,d\mu(x), \] $\zeta\in C_{c}^{1}(\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{n+1})$, where $M_{y}=\{x\in\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{ n+1}:(x,y)\in M\}$, $\zeta_{y}(x)=\zeta(x,y)$, and
$|A_{M_{y}}|$ is the length of the second fundamental form of $M_{y}$.
Since \[
||A_{M_{y}}|^{2}(x)-|A_{M}|^{2}(x,y)| \leqslant C\delta/|x|^{2},\,\,\, |A_{M_{y}}|^{2}(x)\leqslant C/|x|^{2} \] by~\ref{pf-2}, the proof is completed by taking $\delta$ small enough, integrating with respect to $y\in \mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{\ell}$, and using the coarea formula together with~\ref{pf-2}. \end{proof}
\section{A Boundary Version of Theorem~\ref{main-th} and Corollary~\ref{co-1}}\label{boundary-version}
\noindent Since it will be needed in~\cite{Sim21b}, we here also want to discuss a version of Theorem~\ref{main-th} which is valid in case $M$ has a boundary.
\begin{state}{\bf{}\tl{bdry-ver} Theorem.}
Suppose $M\subset\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{n+1}\times\{y:y_{\ell}\geqslant 0\}$ is a complete embedded minimal hypersurface-with-boundary, with \[ \left\{\hskip2pt\begin{aligned} &\partial M=S\times\{y:y_{\ell}=0\}, \\ \noalign{\vskip-1pt}
&|\nu_{y_{\ell}}|<1\text{ on }S,\,\, \\ \noalign{\vskip-1pt} &{\sup}_{R>1}R^{-n-\ell}\mu(M\cap B_{R}) <2\mu(\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}\cap \bigl\{(x,y)\in B_{1}:y_{\ell}>0\bigr\}), \text{ and }\\ \noalign{\vskip-1pt} &M \subset U_{\lambda}\times\{y:y_{\ell}\geqslant 0\} \end{aligned}\right. \leqno{(\ddag)} \] for some $\lambda\geqslant 1$, where $U_{\lambda}$ denotes the component of \,$U_{+}\setminus \lambda S$ with $\partial U_{\lambda}=\overline\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}\cup \lambda S$.
Suppose also that $M$ is strictly stable in the sense there is $\lambda>0$ such that that the inequality~\emph{\ref{str-stab-M}} holds for all $\zeta\in C^{1}_{c}(M\cap \breve B_{R})$ with $e_{n+1+\ell}\cdot \nabla_{M}\zeta=0$ on $\partial M$ and for all $R>0$.
Then \[ M=S\times\{y:y_{\ell}\geqslant 0\}. \] \end{state}
\begin{proof}{\bf{}Proof:} Since $M\subset U_{\lambda}$, the Allard compactness theorem, applied to the rescalings $\tau M,\,\tau\downarrow 0$, plus the constancy theorem, tells us that $M$ has a tangent cone at $\infty$ which is $\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}\times \{y:y_{\ell}\geqslant 0\}$ with some integer multiplicity $k\geqslant 1$, and then the condition $\sup_{R>1}R^{-n-1}\mu(M\cap B_{R})<2\mu(\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}\cap \bigl\{(x,y)\in B_{1}:y>0\bigr\})$ implies $k=1$. Thus $M$ has $\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}\times\{y:y_{\ell}\geqslant 0\}$ with multiplicity~\!1 as its unique tangent cone at $\infty$.
We claim that $\nu_{y_{\ell}}$ satisfies the free boundary condition \[ e_{n+1+\ell}\cdot \nabla_{M}\nu_{y_{\ell}}=0 \text{ at each point of $\partial M$. } \pdl{bdry-ver-1} \] Indeed if $\Sigma\subset\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{N}\times[0,\infty)$ is any minimal hypersurface-with-boundary with smooth unit normal
$\nu=(\nu_{1},\ldots,\nu_{N+1})$ and $|\nu_{N+1}|\neq 1$ on $\partial\Sigma$ (i.e.\ $\Sigma$ intersects the hyperplane $x_{N+1}=0$ transversely), and if $\partial\Sigma$ is a minimal hypersurface in $\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{N}\times\{0\}$, then $e_{N+1}\cdot\nabla_{\Sigma} \nu_{N+1}=0$ on $\partial\Sigma$, as one readily checks by using the fact that the mean curvature (i.e.\ trace of second fundamental form) of $\Sigma$ and $\partial\Sigma$ are both zero at each point of $\partial\Sigma$.
We claim $\nu_{y_{\ell}}(x,y)=0\,\,\forall (x,y)\in M$. To check this, first observe that there is a version of~\ref{growth-lem} which is valid in the half-space $y_{\ell}\geqslant 0$ in the case when $w$ has free boundary condition $e_{n+1+\ell}\cdot \nabla_{M}w=0$ on $\partial M$; indeed the proof of~\ref{growth-lem} goes through with little change---the linear solutions $v$ of $\mathcal{L}_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}}v=0$ obtained in the proof being defined on the half-space $\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}\times\{y:y_{\ell}\geqslant 0\}$ and having the free boundary condition $e_{n+1+\ell}\cdot\nabla_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}}v=0$. So $v$ extends to a solution of $\mathcal{L}_{\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}}v=0$ on all of $\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}$ by even reflection and the rest of the argument is unchanged.
In particular, since $\nu_{y_{\ell}}$ has free boundary condition $0$ by~\ref{bdry-ver-1}, we have an analogue of Lemma~\ref{strong-doub-lem} in the half-space, giving \[ R^{-\alpha}\leqslant C\hskip1pt\text{\small $-$}\hskip-10.6pt\int_{M\cap B^{+}_{R}}\nu_{y_{\ell}}^{2} \pdl{bdry-ver-2} \] for each $\alpha\in(0,1)$, where $B^{+}_{R}=B_{R}\cap \{(x,y):y_{\ell}\geqslant 0\}$, and, since $\nu_{y_{\ell}}\leqslant 1$ there is a bound $\text{\scriptsize $-$}\hskip-6.85pt\tint_{M\cap B^{+}_{R}}\nu_{y_{\ell}}^{2}\leqslant CR^{n-2-\beta}$, so by Corollary~\ref{strong-doub-lem} \[ \hskip1pt\text{\small $-$}\hskip-10.6pt\int_{M\cap B^{+}_{R}}\nu_{y_{\ell}}^{2}
\leqslant CR^{-\ell-2-\beta_{1}}\int_{B^{+}_{R/2}\setminus\{(x,y):|x|<\smfr{1}{3}R\}}\nu_{y_{\ell}}^{2}, \pdl{bdry-ver-3} \] where $C=C(\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0},\alpha)$. Since $M\subset U_{\lambda}\times[0,\infty)$ also we have
$d(x)\leqslant C\lambda \min\{1,\,|x|^{\gamma_{1}}\}$ for all $x\in\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{n}$ and hence \[ \hskip1pt\text{\small $-$}\hskip-10.6pt\int_{M\cap B^{+}_{R}}d^{2} \leqslant C\lambda^{2}, \quad C=C(\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N}_{0}). \pdl{d-bound} \] (Notice that here we do not need growth estimates for $\text{\scriptsize $-$}\hskip-6.85pt\tint_{B_{R}} d^{2}$ as in~\S\ref{dist-fn} because we are now assuming
$d\leqslant C\lambda \min\{1,\,|x|^{\gamma_{1}}\}$.) Then the proof that $\nu_{y_{\ell}}$ must be identically zero is completed using~\ref{bdry-ver-2}, \ref{bdry-ver-3}, and \ref{d-bound} analogously to the proof of non-boundary version of Theorem~\ref{main-th}.
So $\nu_{y_{\ell}}$ is identically zero on $M$. This completes the proof in the case $\ell=1$ and shows that, for $\ell\geqslant 2$, by even reflection $M$ extends to a minimal submanifold $\wtilde{M}$ (without boundary) \[ \wtilde{M}=M\cup\bigl\{(x,y_{1},\ldots,y_{\ell-1},-y_{\ell}):(x,y)\in M\bigr\}\subset U_{+}\times\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{\ell}, \] which is translation invariant by translations in the direction of $e_{n+1+\ell}$. Then $\wtilde{M}$ is strictly stable and Theorem~\ref{main-th} applies, giving $M=S\times \mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{\ell-1}\times[0,\infty)$, as claimed.\end{proof}
\begin{state}{\bf{}\tl{bdy-case} Corollary.} There is $\delta=\delta(\mathbb{C}} \def\Z{\mathbb{Z}} \def\N{\mathbb{N},\lambda)>0$ such that if $M$ satisfies~{\rm\ref{bdry-ver}}\hskip1pt$(\ddag)$ and
$\sup|\nu_{y}|<\delta$, then $M$ automatically satisfies the strict stability hypothesis in~{\rm\ref{bdry-ver}}, and hence $M=S\times\{y:y_{\ell}\geqslant 0\}$. \end{state}
\begin{proof}{{\bf{}Proof:}} With $\wtilde{M}_{k}$, $M_{z_{k}}$ as in the proof of~\ref{stab-lem}, with only minor modifications to the present situation when $\partial M=S\times\{0\}$, we have $\wtilde M_{k}$ and $M_{z_{k}}$ both have cylindrical limits $S\times \mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{\ell}$
or $S\times \{y\in\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{\ell}:y_{\ell}\geqslant K\}$ for suitable $K$, and so (using the Allard boundary regularity theorem in the latter case) the $y=$ const.\ slices of $M$ have the $C^{2}$ approximation property~\ref{stab-lem}\ref{pf-2}. Hence, by integration over the slices as in~\ref{stab-lem}, $M$ is strictly stable in the sense that~\ref{str-stab-M} holds for all $\zeta\in C^{1}_{c}(\breve B_{R}^{+})$, where $\breve B_{R}^{+}=\{(x,y):|(x,y)|<R \text{ and }y_{\ell}\geqslant 0\}$. \end{proof}
\section*{Appendix: Analyticity of $\beta$-harmonic functions} \setcounter{sequation}{0}
\renewcommand{A.\arabic{sequation}}{A.\arabic{sequation}}
\noindent For $\rho>0$, $r_{0}\geqslant 0$ and $y_{0}\in\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{\ell}$, let \begin{align*}
&B_{\rho}^{+}(r_{0},y_{0})= \{(r,y)\in \mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}\times\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{\ell} :r\geqslant 0,\,\,(r-r_{0})^{2}+|y-y_{0}|^{2}\leqslant \rho^{2}\}, \\
&\breve B^{+}_{\rho}(r_{0},y_{0})=\{(r,y)\in \mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}\times\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{\ell}:r>0,\,\,(r-r_{0})^{2}+|y-y_{0}|^{2}<\rho^{2}\}, \end{align*} and $B^{+}_{\rho},\,\breve B^{+}_{\rho}$ will be used as abbreviations for $B^{+}_{\rho}(0,0),\, \breve B^{+}_{\rho}(0,0)$ respectively.
Our aim is to prove real analyticity extension across $r=0$ of $\beta$-harmonic functions, i.e.\ solutions $u\in C^{\infty}(\breve B_{1}^{+})$ of \[ r^{-\gamma}\frac{\partial}{\partial r}\bigl(r^{\gamma}\frac{\partial u}{\partial r}\bigr) +\Delta_{y}u=0, \dl{equn} \] where $\gamma=1+\beta\,\,(\,>1)$ and $\Delta_{y}u={\textstyle\sum}_{j=1}^{\ell}D_{y_{j}}D_{y_{j}}u$, assuming \[
\int_{B^{+}_{\rho}}(u_{r}^{2}+|u_{y}|^{2})\,\,r^{\gamma\!}drdy <\infty\,\,\, \forall \, \rho<1. \dl{w-1-2-bd} \]
In fact we show under these conditions that $u$ can be written as a convergent series of homogeneous $\beta$-harmonic polynomials in $\{(r,y):r\geqslant 0,\,r^{2}+|y|^{2}<1\}$ with the convergence uniform in $B_{\rho}^{+}$ for each $\rho<1$.
Of course all solutions of~\ref{equn} satisfying \ref{w-1-2-bd} are automatically real-analytic in $\breve B^{+}_{1}$ because the equation is uniformly elliptic with real-analytic coefficients in each closed ball $\subset \breve B^{+}_{1}$.
Also, if $\gamma$ is an integer $\geqslant 1$ then the operator in~\ref{equn} is just the Laplacian in $\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{1+\gamma+\ell}$, at least as it applies to functions $u=u(x,y)$ on $\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{1+\gamma+\ell}$ which can be expressed as a function of $r=|x|$ and $y$, so in this case smoothness across $r=0$ is Weyl's lemma and analyticity is then standard.
To handle the case of general $\gamma>1$, first note that by virtue of the calculus inequality
$\int_{0}^{a}r^{\gamma-2}f^{2}\,dr\leqslant C\int_{0}^{a}r^{\gamma}(f'(r))^{2}\,dr$ for any $f\in C^{1}(0,a]$ with $f(a)=0$ (which is proved simply by using the identity $\int_{0}^{a}\bigl(r^{\gamma-1}(\min\{|f|,k\})^{2}\bigr)'\,dr=0$ and then using Cauchy-Schwarz inequality and letting $k\to \infty$), we have \[ \int_{B^{+}_{\rho}} r^{-2}\zeta^{2}\,d\mu_{+} \leqslant C \int_{B^{+}_{\rho}}\bigl(D_{r}\zeta\bigr)^{2}\,d\mu_{+}, \quad C=C(\gamma) \dl{z-over-r-bd} \] for any $\rho\in [\fr{1}{2} ,1)$ and any Lipschitz $\zeta$ on $B^{+}_{1}$ with support contained in $B^{+}_{\rho}$.
Next observe that $u$ satisfies the weak form of~\ref{equn}:
{\abovedisplayskip8pt\belowdisplayskip8pt \[ \int_{\breve B_{1}^{+}} (u_{r}\zeta_{r}+u_{y}\cdot \zeta_{y})\,d\mu_{+}=0 \dl{weak-form} \] for any Lipschitz $\zeta$ with support in $B^{+}_{\rho}$ for some $\rho<1$, where, here and subsequently, \[ d\mu_{+}=r^{\gamma}drdy \] and subscripts denote partial derivatives: \[ u_{r}=D_{r}u,\,\,\,u_{y}= D_{\!y}u=(D_{\!y_{1}}u,\ldots,D_{\!y_{\ell}}u). \] \ref{weak-form} is checked by first observing that it holds with $\varphi_{\sigma}(r)\zeta(r,y)$ in place of $\zeta(r,y)$, where $\varphi_{\sigma}(r)=0$ for $r<\sigma/2$, $\varphi_{\sigma}(r)=1$ for $r>\sigma$,
$|\varphi_{\sigma}'(r)|<C/\sigma$, and then letting $\sigma\downarrow 0$ and using~\ref{w-1-2-bd}.
Similarly since $r^{-\gamma}\frac{\partial}{\partial r}(r^{\gamma}\frac{\partial u^{2}}{\partial r})
+{\textstyle\sum}_{j=1}^{\ell}D_{y_{j}}D_{y_{j}}u^{2}= 2(u_{r}^{2}+|u_{y}|^{2})\geqslant 0$, we can check using~\ref{w-1-2-bd} and~\ref{z-over-r-bd} that $u^{2}$ is a weak subsolution, meaning that \[ \int_{\breve B_{1}^{+}}\bigl((u^{2})_{r}\zeta_{r}+(u^{2})_{y}\cdot \zeta_{y}\bigr)\,d\mu_{+}\leqslant 0, \dl{subsoln-w} \]} for any non-negative Lipschitz function $\zeta$ on $B^{+}_{1}$ with support $\subset B^{+}_{\rho}$ for some $\rho<1$.
Next we note that if $\rho\in [\fr{1}{2},1)$, $r_{0}\geqslant 0$ and $B_{\rho}^{+}(r_{0},y_{0})\subset B^{+}_{1}$, then \[
|u(r_{0},y_{0})|\leqslant C\bigl(\rho^{-\ell-1-\gamma}\int_{B^{+}_{\rho}(r_{0},y_{0})}u^{2}\,d\mu_{+}\bigr)^{1/2}, \quad C=C(\gamma,\ell). \dl{sup-bd-0} \] To check this, first observe that if $B^{+}_{\sigma}(r_{0},y_{0})\subset \breve B_{1}^{+}$ and $r_{0}>\sigma$ then the equation~\ref{equn} is uniformly elliptic divergence form with smooth coefficients on $B_{\sigma/2}(r_{0},y_{0})$, so we can use standard elliptic estimates for $u$ to give \[
|u(r_{0},y_{0})|^{2}\leqslant C\sigma^{-\gamma-1-\ell}\int_{B^{+}_{\sigma}(r_{0},y_{0})}u^{2} \,d\mu_{+} \dl{sup-bd-0a} \] with $C=C(\gamma,\ell)$. So now assume $B_{\rho}^{+}(r_{0},y_{0})\subset B^{+}_{1}$. If $r_{0}>\rho/4$ we can use~\ref{sup-bd-0a} with $\sigma=\rho/4< r_{0}$ to give~\ref{sup-bd-0}, while on the other hand if $r_{0}\leqslant \rho/4$ then we can first take $\sigma=r_{0}/2$ in~\ref{sup-bd-0a} to give \[
|u(r_{0},y_{0})|^{2}\leqslant Cr_{0}^{-\ell-1-\gamma}\int_{B^{+}_{r_{0}/2}(r_{0},y_{0})}u^{2}\,d\mu_{+} \leqslant Cr_{0}^{-\ell-1-\gamma}\int_{B^{+}_{2r_{0}}(0,y_{0})}u^{2}\,d\mu_{+}, \dl{sup-bd-1} \] and then observe, using a straightforward modification of the relevant argument for classical subharmonic functions to the present case of the $\beta$-subharmonic function $u^{2}$ as in~\ref{subsoln-w},
{\abovedisplayskip8pt\belowdisplayskip8pt \[ \text{$\sigma^{-\ell-1-\gamma}\int_{B^{+}_{\sigma}(0,y_{0})}u^{2}\,d\mu_{+}$ is an increasing function of $\sigma$ for $\sigma\in (0,\rho/2]$}. \] So from~\ref{sup-bd-1} we conclude \[
|u(r_{0},y_{0})|^{2}\leqslant C \rho^{-\ell-1-\gamma}\int_{B^{+}_{\rho/2}(0,y_{0})}u^{2}\,d\mu_{+} \leqslant C \rho^{-\ell-1-\gamma}\int_{B^{+}_{\rho}(r_{0},y_{0})}u^{2}\,d\mu_{+} \] where $C=C(\gamma,\ell)$. Thus~\ref{sup-bd-0} is proved.
For $\rho\in [\fr{1}{2},1)$, $kh\in \bigl(-(1-\rho),1-\rho\bigr)\setminus\{0\}$, and $k\in \{1,2,\ldots\}$, let $u_{h}^{(k)}$ (defined on $\breve B^{+}_{\rho}$) denote the vector of $k$-th order difference quotients with respect to the $y$-variables; so for example \begin{align*} u_{h}^{(1)}&=\bigl(h^{-1}(u(x,y+he_{1})-u(x,y)),\ldots,h^{-1}(u(x,y+he_{\ell})-u(x,y))\bigr), \\ u_{h}^{(2)}&=\bigl(h^{-1}(u_{h}^{(1)}(x,y+he_{1})-u_{h}^{(1)}(x,y)),\ldots, h^{-1}(u_{h}^{(1)}(x,y+he_{\ell})-u_{h}^{(1)}(x,y))\bigr), \end{align*}}
and generally, for $k|h|<1-\rho$, \begin{align*} &u_{h}^{(k)}(x,y) = h^{-1}\bigl((u_{h}^{(k-1)}(x,y+he_{1})-u_{h}^{(k-1)}(x,y)),\ldots, \\ \noalign{\vskip-3pt} &\hskip2.3in (u_{h}^{(k-1)}(x,y+he_{\ell})-u_{h}^{(k-1)}(x,y))\bigr), \end{align*} (which is a function with values in $\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{\ell^{k}}$). For notational convenience we also take \[ u^{(0)}_{h}(x,y)=u(x,y). \] Then, replacing $\zeta$ in~\ref{weak-form} by $\zeta_{-h}^{(k)}$ and changing variables appropriately (i.e.\ ``integration by parts'' for finite differences instead of derivatives), \[ \int_{\breve B_{1}^{+}}\bigl(D_{r}u^{(k)}_{h}D_{r}\zeta+{\textstyle\sum}_{j=1}^{\ell} D_{y_{j}}u^{(k)}_{h}\, D_{y_{j}}\zeta\bigr)\,d\mu_{+}=0, \]
for all Lipschitz $\zeta$ on $B_{1}^{+}$ with support $\subset B^{+}_{\rho}$ and for $k|h|<1-\rho$. Replacing $\zeta$ by $\zeta^{2}u_{h}^{(k)}$ gives \begin{align*}
&\int_{\breve B_{1}^{+}}\bigl(|D_{r}u_{h}^{(k)}|^{2}+ |D_{y}u_{h}^{(k)}|^{2}\bigr)\zeta^{2} \\ \noalign{\vskip-2pt} &\hskip1in =-2\smash[t]{\int_{\breve B_{1}^{+}}}\bigl(\zeta u_{h}^{(k)}\cdot D_{r}u_{h}^{(k)}D_{r}\zeta + \zeta u_{h}^{(k)}\cdot D_{r}u_{h}^{(k)}D_{y_{j}}\zeta D_{y_{j}}\zeta\bigr)\,d\mu_{+}, \end{align*} so by Cauchy-Schwarz \[
\int_{\breve B_{1}^{+}}\bigl(|D_{r}u_{h}^{(k)}|^{2}+ |D_{y}u_{h}^{(k)}|^{2}\bigr)\zeta^{2}\,d\mu_{+}
\leqslant 4\int_{\breve B_{1}^{+}}|u_{h}^{(k)}|^{2} |D\zeta|^{2}\,d\mu_{+}, \] and by~\ref{z-over-r-bd} we then have \[
\int_{\breve B_{1}^{+}}\bigl(r^{-2}|u_{h}^{(k)}|^{2}+ |D_{r}u_{h}^{(k)}|^{2}+
|D_{y}u_{h}^{(k)}|^{2}\bigr)\zeta^{2}\,d\mu_{+} \leqslant C\int_{\breve B_{1}^{+}}|u_{h}^{(k)}|^{2}
|D\zeta|^{2}\,d\mu_{+}, \dl{u-k-bds-3} \] where $C=C(\gamma,\ell)$. Now let $u^{(k)}=\lim_{h\to 0}u_{h}^{(k)}$ (i.e.\ $u^{(k)}$ is the array of all mixed partial derivatives of order $k$ with respect to the $y$ variables defined inductively by $u^{(0)}=u$ and $u^{(k+1)}=D_{y}u^{(k)}$). Then~\ref{u-k-bds-3} gives \[
\int_{\breve B_{1}^{+}}\bigl(r^{-2}|u^{(k)}|^{2}+ |D_{r}u^{(k)}|^{2}+ |u^{(k+1)}|^{2}\bigr)\zeta^{2}
\leqslant C\int_{\breve B_{1}^{+}}|u^{(k)}|^{2} |D\zeta|^{2}\,d\mu_{+} \dl{u-k-bds-4} \] for each $\zeta\in C^{1}(B^{+}_{\rho})$ with support in $B^{+}_{\theta\rho}$ for some $\theta\in [\fr{1}{2},1)$ and for each $k$ such that the right side is finite. By~\ref{w-1-2-bd} the right side is finite for $k=0,1$, and taking $k=1$ in~\ref{u-k-bds-4} then implies the right side is also finite with $k=2$. Proceeding inductively we see that in fact that the right side is finite for each $k=0,1,\ldots$, so~\ref{u-k-bds-4} is valid and all integrals are finite for all $k=1,2,\ldots$.
Let $k\in\{1,2,\ldots\}$ and $(r_{0},y_{0})\in B^{+}_{1}$ with $r_{0}\geqslant 0$. Then if $B^{+}_{\rho}(r_{0},y_{0})\subset B^{+}_{1}$ and we let \[ \rho_{j}=\rho-\fr{j}{k}\rho/2,\,\, j=0,\ldots,k-1, \] and, applying \ref{u-k-bds-4} with $k=j$ and \[ \zeta=1\text{ on }B^{+}_{\rho_{j+1}}(r_{0},y_{0}), \,\,
\zeta=0\text{ on }\mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{1+\ell}\setminus B^{+}_{\rho_{j}}(r_{0},y_{0}),\text{ and }|D\zeta|\leqslant 3k, \] we obtain \[
\int_{\breve B^{+}_{\rho_{j+1}}(r_{0},y_{0})} |u^{(j+1)}|^{2}\,d\mu_{+}
\leqslant C\rho^{-2}k^{2}\int_{\breve B^{+}_{\rho_{j}}(r_{0},y_{0})}|u^{(j)}|^{2} \,d\mu_{+} \] with $C=C(\gamma,\ell)$. By iteration this gives \[
\int_{\breve B_{\rho/2}^{+}(r_{0},y_{0})} |u^{(k)}|^{2}\,d\mu_{+}
\leqslant C^{k}\rho^{-2k}(k!)^{2}\int_{\breve B_{\rho}^{+}(r_{0},y_{0})}|u|^{2} \,d\mu_{+} \] with suitable $C=C(\gamma,\ell)$ (independent of $k$), and then by~\ref{sup-bd-0} with $u^{(k)}$ in place of $u$ and $\rho/2$ in place of $\rho$ we obtain \[
|u^{(k)}(r_{0},y_{0})|^{2}\leqslant C^{k}\rho^{-2k}(k!)^{2}
\rho^{-\gamma-1-\ell}\int_{\breve B^{+}_{\rho}(r_{0},y_{0})}|u|^{2} \,d\mu_{+}, \] with $C=C(\gamma,\ell)$ (independent of $k$ and $\rho$). In view of the arbitrariness of $\rho,r_{0},y_{0}$ this implies \[
\sup_{B^{+}_{\rho/2}} |u^{(k)}|^{2}\leqslant C^{k}(k!)^{2}
\int_{\breve B^{+}_{\rho}}|u|^{2} \,d\mu_{+}, \text{ for each $\rho\in (0,1)$}, \dl{u-k-bds-5} \]
where $C=C(\gamma,\ell,\rho)$.
Next let $L_{r}$ be the second order operator defined by \[ L_{r}(f)=r^{-\gamma}\frac{\partial}{\partial r}\bigl(r^{\gamma}\frac{\partial f}{\partial r}\bigr), \] so that~\ref{equn} says $L_{r}u=-\Delta_{y}u$, where $\Delta_{y}u={\textstyle\sum}_{j=1}^{\ell}D^{2}_{y_{j}}u$, and by repeatedly differentiating this identity with respect to the $y$ variables, we have \[ L_{r}u^{(k)}(r,y) =-\Delta_{y}u^{(k)}, \] and hence \[ L_{r}^{j}u^{(k)} = -(\Delta_{y})^{j}u^{(k)}. \dl{L-j-lap-k} \]
for each $j,k=0,1,\ldots$. In particular, since $|\Delta_{y}^{j}f|\leqslant C^{j}|f^{(2j)}|$ with $C=C(\ell)$, \[
|L_{r}^{j}u^{(k)}|^{2}\leqslant C^{2j} |u^{(k+2j)}|^{2} \dl{L-j-k-bd} \] for each $j,k=0,1,2,\ldots$, where $C=C(\ell)$
Next we note that, since $|u^{(k)}|$ is bounded on $B^{+}_{\rho}$ for each $\rho<\fr{1}{2}$ by~\ref{u-k-bds-5}, for small enough
$r>0$ and $|y|<\fr{1}{2}$ we can apply elliptic estimates to give $|D_{r}u^{(k)}| <C/r$ with $C$ fixed independent of $r$ (but depending on $k$), and since $\gamma>1$ we have $|r^{\gamma}D_{r}u^{(k)}(r,y)|\leqslant Cr^{\gamma-1}\to 0$ as $r\downarrow 0$. But using~\ref{L-j-k-bd} with $j=1$ and~\ref{u-k-bds-5} we have
$|D_{r}(r^{\gamma}D_{r}u^{(k)})|\leqslant Cr^{\gamma}$ and hence by integrating with respect to $r$ and using the above fact that $r^{\gamma}D_{r}u^{(k)}(r,y)\leqslant Cr^{\gamma-1}\to 0$ as $r\downarrow 0$, we then have
{\abovedisplayskip8pt\belowdisplayskip8pt \[
|D_{r}u^{(k)}(r,y)| \leqslant Cr \text{ for small enough $r$ and all $|y|\leqslant \rho$, $\rho<\fr{1}{2}$}, \dl{u-k-to-0} \] with $C$ depending on $k$, $\gamma$, $\rho$ and $\ell$, and in particular, for each $\rho<\fr{1}{2}$, \[
D_{r}u^{(k)}(0_{+},y)=0=\lim_{r\downarrow 0} D_{r}u^{(k)}(r,y) \text{ uniformly for $|y|\leqslant \rho$.} \dl{u-k-0-+} \] We now claim the following polynomial approximation property: For each $u$ as in~\ref{equn}, \ref{w-1-2-bd}, each $j,k=1,2,\ldots$ with $k\leqslant j$, and each $(r,y)\in\breve B^{+}_{\rho}$, $\rho<\fr{1}{2}$, \[
\bigl|L_{r}^{j-k}u(r,y)-{\textstyle\sum}_{i=1}^{k}c_{ijk}L_{r}^{j-i}u(0,y)r^{2(k-i)}/(2(k-i))!\bigr|\leqslant r^{2k}{\sup}_{B_{\rho}^{+}}
|L_{r}^{j}u|/(2k)! \dl{inductive-hyp} \] where $0<c_{ijk}\leqslant 1$. To prove the case $k=1$, by virtue of~\ref{u-k-to-0},\,\ref{u-k-0-+}, we can simply integrate from $0$ to $r$, using $D_{r}(r^{\gamma}D_{r}L_{r}^{j-1}u(r,y))=r^{\gamma}L^{j}u(r,y)$, followed by a cancellation of $r^{\gamma}$ from each side of the resulting identity. This gives \[
|D_{r}L_{r}^{j-1}u(r,y)| \leqslant r\, {\sup}_{B_{\rho}^{+}}|L^{j}u(r,y)|/(\gamma+1), \] and then a second integration using~\ref{u-k-0-+} gives \[
\bigl|L_{r}^{j-1}u(r,y)-L_{r}^{j-1}u(0,y)\bigr|\leqslant (2(\gamma+1))^{-1}r^{2}
{\sup}_{B_{\rho}^{+}}|L_{r}^{j}u(r,y)|,\,\,j=1,2,\ldots, \] which establishes the case $k=1$. Assume $k+1\leqslant j$ and that~\ref{inductive-hyp} is correct for $k$. Multiplying each side of~\ref{inductive-hyp} by $r^{\gamma}$ and integrating, we obtain \begin{align*}
&\bigl|r^{\gamma}D_{r} L_{r}^{j-k-1}u(r,y)
-{\textstyle\sum}_{i=1}^{k}(2(k-i)+\gamma+1)^{-1} c_{ijk}L_{r}^{j-i}u(0,y)\frac{r^{2(k-i)+\gamma+1}}{(2(k-i))!}\bigr| \\ \noalign{\vskip-1pt}
&\hskip1.8in \leqslant (2k+\gamma+1)^{-1}\frac{r^{2k+\gamma+1}}{(2k)!}{\sup}_{B_{\rho}^{+}}|L_{r}^{j}u|, \end{align*} where we used the fact that $r^{\gamma}L_{r}^{j-k}u(r,y)=D_{r}(r^{\gamma}D_{r}L_{r}^{j-k-1}u(r,y))$.
\vskip1pt
After cancelling the factor $r^{\gamma}$ and integrating again, we obtain \begin{align*}
&\bigl| L_{r}^{j-k-1}u(r,y)-L_{r}^{j-k-1}u(0,y) \\ &\hskip0.1in -{\textstyle\sum}_{i=1}^{k}(2(k-i)+\gamma+1)^{-1}(2(k-i)+2)^{-1}
c_{ijk}L_{r}^{j-i}u(0,y)\frac{r^{2(k-i)+2}}{(2(k-i))!}\bigr|\\ &\hskip0.15in \leqslant (2k+\gamma+1)^{-1}(2k+2)^{-1}r^{2k+2}
{\sup}_{B_{\rho}^{+}} |L_{r}^{j}u|/(2k)! \leqslant {\sup}_{B_{\rho}^{+}} |L_{r}^{j}u|r^{2k+2}/(2k+2)! \end{align*} which confirms the validity of~\ref{inductive-hyp} with $k+1$ in place of $k$. So~\ref{inductive-hyp} is proved for all $k\leqslant j$, and in particular with $k=j$ and suitable constants $c_{ij}\in (0,1]$ we get \[
u(r,y)={\textstyle\sum}_{i=0}^{j-1}c_{ij}L_{r}^{i}u(0,y)r^{2i}/(2i)!+E_{j}(r,y),\text{where\,} |E_{j}(r,y)|\leqslant
r^{2j}{\sup}_{B_{\rho}^{+}} |L_{r}^{j}u|/(2j)! \] By~\ref{L-j-k-bd} and~\ref{u-k-bds-5}, \[
{\sup}_{B_{\rho/2}^{+}}|L_{r}^{i}u|/(2i)!={\sup}_{B_{\rho/2}^{+}}|\Delta_{y}^{i}u|/(2i)!\leqslant
C^{i}{\sup}_{B_{\rho/2}^{+}}|u^{(2i)}|/(2i)!\leqslant C^{i} \bigl(\int_{\breve B^{+}_{\rho}}|u|^{2} \,d\mu_{+}\bigr)^{1/2}, \] with $C=C(\gamma,\ell,\rho)$, so, for suitable $C=C(\gamma,\ell)$, we conclude that $u(r,y)$ has a power series expansion in terms $r^{2}$: \[ u(r,y) ={\textstyle\sum}_{j=0}^{\infty}\,\,\,\,a_{j}(y)r^{2j}, \quad 0\leqslant r< \sigma, \dl{r-exp} \] where $\sigma=\sigma(\gamma,\ell)\in (0,\fr{1}{2}]$, and $a_{j}$ satisfies the bounds \[
{\sup}_{B^{+}_{\sigma}}|a_{j}|\leqslant C^{j}\bigl(\int_{B_{1/2}^{+}}u^{2}\,d\mu_{+}\bigr)^{1/2}, \] where $C=C(\gamma,\ell)$. Thus~\ref{r-exp} implies \[
{\sup}_{B_{\sigma/2}^{+}}|D_{r}^{j}u(r,y)| \leqslant C^{j}j! \,\bigl(\int_{B_{1/2}^{+}}u^{2}\,d\mu_{+}\bigr)^{1/2},
\,\,\,C=C(\gamma,\ell). \dl{bds-D-j} \] Since the same holds with $u^{(k)}$ in place of $u$, and since \[
\int_{B_{1/2}^{+}}|u^{(k)}|^{2}\,d\mu_{+}\leqslant C^{k}(k!)^{2}\int_{B_{3/4}^{+}}u^{2}\,d\mu_{+}, \]} by~\ref{u-k-bds-5}, we deduce from~\ref{bds-D-j} that for suitable $\sigma=\sigma(\gamma,\ell)\in (0,\fr{1}{2})$ \[ {\sup}_{B_{\sigma}^{+}}
|D_{r}^{j}D_{y}^{k}u(r,y)| \leqslant C^{j+k} j!k! \,\bigl(\int_{B_{3/4}^{+}}u^{2}\,d\mu_{+}\bigr)^{1/2}, \dl{deriv-bds} \] where $C=C(\gamma,\ell)$, and hence in particular $u$ is real-analytic in the variables $r^{2}$ and $y_{1},\ldots,y_{\ell}$ in a neighborhood of $(0,0)$ as claimed.
Finally we show that if $u$ is $\beta$-harmonic in $\breve B_{1}^{+}$ then the power series for $u$ converges in $B_{\rho}^{+}$ for each $\rho<1$, and also that the homogeneous $\beta$-harmonic polynomials restricted to $S_{+}^{\ell}$ are complete in $L^{2}(\nu_{+})$ on $S_{+}^{\ell}$, where $\nu_{+}$ is the measure $d\nu_{+}=\omega_{1}^{\gamma}d\mu_{\ell}$ on $S_{+}^{\ell}$.
So let $u\in L^{2}(\mu_{+})$ satisfy~\ref{equn} and \ref{w-1-2-bd}. The above discussion shows that for suitably small $\sigma$ we can write \[ u={\textstyle\sum}_{j=0}^{\infty}u_{j} \,\,\,\text{ in }B_{\sigma}^{+}, \dl{rau} \] where $u_{j}$ consists of the homogeneous degree $j$ terms in the power series expansion of $u$ in $B_{\sigma}$ (and $u_{j}=0$ if there are no such terms). Then each $u_{j}\neq 0$ is a homogeneous degree j $\beta$-harmonic polynomial and we let $$ \tilde u_{j}(\omega)=\rho^{-j}u_{j}(\rho\omega),\,\,\, \hat u_{j}(\omega)=
\|\tilde u_{j}\|_{L^{2}(\nu_{+})}^{-1}\tilde u_{j}(\omega), \,\,\, \omega\in \Sph^{\ell}_{+}, $$ and we set $\hat u_{j}(\omega)=0$ if $u_{j}=0$. Then, with $\langle\,,\,\rangle=$ the $L^{2}(\nu_{+})$ inner product, $\langle u,\hat u_{j} \rangle \hat u_{j}=u_{j}$ in $B^{+}_{\sigma}$ for each $j$, and hence by~\ref{rau} the series $\sum_{j}\langle u,\hat u_{j} \rangle \hat u_{j}$ converges smoothly (and also in $L^{2}(\mu_{+})$) to $u$ in $B^{+}_{\sigma}$.
By definition $\rho^{j}\hat u_{j}(\omega)$ is either zero or a homogeneous degree $j$ harmonic polynomial, so by~\ref{jth-eig} \smash{${\rm div}_{\Sph^{\ell}_{+}}(\omega_{1}^{\gamma}\nabla_{\Sph^{\ell}_{+}}\hat u_{j})=- j(j+\ell+\beta) \omega_{1}^{\gamma}\hat u_{j}$}, and hence using the formula~\ref{sph-coords} we can directly check that $\langle u,\hat u_{j} \rangle \hat u_{j}$ is $\beta$-harmonic on all of $\breve B_{1}^{+}$. Since by construction it is equal to $u_{j}$ on $B_{\sigma}$, by unique continuation (applicable since $u$ is real-analytic on $\breve B_{1}^{+}$) we conclude \begin{align*} &\langle u,\hat u_{j} \rangle \hat u_{j} \text{ is either zero or the homogeneous degree $j$ $\beta$-harmonic} \dtg{poly} \\ \noalign{\vskip-5pt} &\hskip1.4in\text{ polynomial $u_{j}$ on all of $B_{1}^{+}\setminus S^{\ell}_{+}$ for each $j=0,1,\ldots$}. \end{align*} Also, by the orthogonality~\ref{orthog}, \begin{align*}
\bigl\|\sum_{j=p}^{q}\bigl\langle u,\hat u_{j} \bigr\rangle \hat u_{j}\bigr\|^{2}_{L^{2}(\mu_{+}^{\rho})} &=\sum_{j=p}^{q}\int_{0}^{\rho}\bigl\langle u(\tau\omega),\hat u_{j}(\omega)\bigr\rangle^{2}\, \tau^{\gamma+\ell}d\tau \\
&\leqslant \int_{0}^{\rho}\bigl\|u(\tau\omega)\bigr\|^{2}_{L^{2}(\nu_{+})}\,\tau^{\gamma+\ell}d\rho=
\bigl\|u\bigr\|^{2}_{L^{2}(\mu^{\rho}_{+})} \,\,(<\,\infty) \end{align*} for each $\rho<1$ and each $p<q$, where $\smash{\mu_{+}^{\rho}}$ is the measure $\mu_{+}$ on $B_{\rho}$. So \smash{$\sum_{j=0}^{q}\langle u,\hat u_{j} \rangle \hat u_{j}$} is Cauchy, hence convergent, in \smash{$L^{2}(\mu_{+}^{\rho})$} to a $\beta$-harmonic function $v$ on $\breve B_{\rho}^{+}$. But $v=u$ on $B_{\sigma}^{+}$ and hence, again using unique continuation, $v=u$ in all of $\breve B_{\rho}^{+}$. Thus \smash{$\sum_{j=0}^{q}\langle u,\hat u_{j} \rangle \hat u_{j}$} converges to $u$ in $L^{2}(\smash{\mu_{+}^{\rho}})$ for each $\rho<1$ and the convergence is in $L^{2}(\mu_{+})$ if
$\|u\|_{L^{2}(\mu_{+})}<\infty$.
\vskip1pt
Now observe that the bounds~\ref{deriv-bds} were established for balls centred at $(0,0)$, but with only notational changes the same argument gives similar bounds in balls centred at $(0,y_{0})$ with $|y_{0}|<1$. Specifically for each $\rho\in (0,1)$ and each
$|y_{0}|<\rho$ there is $\sigma=\sigma(\gamma,\ell,\rho)<\fr{1}{2}(1-\rho)$ such that \[
{\sup}_{B_{\sigma}^{+}(0,y_{0})}|D_{r}^{j}D_{y}^{k}u|\leqslant C^{j+k}j!k!\bigl(\int_{B_{(1-\rho)/2}^{+}(0,y_{0})}u^{2}\,d\mu_{+}\bigr)^{1/2},\,\,C=C(\gamma,\ell,\rho). \] So in fact, with $\sigma=\sigma(\gamma,\ell,\rho)$ small enough, \[
{\sup}_{\{(r,y):r\in [0,\sigma],|y|\leqslant \rho\}}
|D_{r}^{j}D_{y}^{k}u(r,y)| \leqslant C^{j+k} j!k! \bigl(\int_{B_{1}^{+}}u^{2}\,d\mu_{+}\bigr)^{1/2},\quad \rho<1. \] Also in $B^{+}_{\rho}\setminus ([0,\sigma]\times \mathbb{R}} \def\Q{\mathbb{Q}} \def\M{\mathbb{M}} \def\Sph{\mathbb{S}^{\ell})$ we can use standard elliptic estimates, so in fact we have \[ {\sup}_{B^{+}_{\rho}}
|D_{r}^{j}D_{y}^{k}u(r,y)| \leqslant C \bigl(\int_{B_{1}^{+}}u^{2}\,d\mu_{+}\bigr)^{1/2}, \dl{mod-deriv-bds-2} \] with $C=C(j,k,\gamma,\rho,\ell)$, so the $L^{2}$ convergence of the series $\sum_{j}\langle u,\hat u_{j} \rangle \hat u_{j}(=\sum_{j}u_{j})$ proved above is also $C^{k}$ convergence in $B_{\rho}^{+}$ for each $k\geqslant 1$ and each $\rho<1$.
\vskip1pt
Finally to prove the completeness of the homogeneous $\beta$-harmonic polynomials in $L^{2}(\nu_{+})$ (on $\Sph^{\ell}_{+}$), let $\varphi$ be any smooth function on $\Sph^{\ell}_{+}$ with $\varphi$ zero in some neighborhood of
$r=0$. By minimizing the energy \smash{$\int_{B_{1}^{+}}(u_{r}^{2}+|u_{y}|^{2})\,r^{\gamma}d\mu$} among functions with trace $\varphi$ on $\Sph_{\ell}$ we obtain a solution of~\ref{weak-form} with trace $\varphi$ on $\Sph^{\ell}_{+}$. The above discussion plus elliptic boundary regularity shows that $u$ is $C^{0}$ on all of $B_{1}^{+}$ and that the sequence \smash{$\{\sum_{j=0}^{q}u_{j}\}_{q=0,1,2,\ldots}$}, which we showed above to be convergent to $u$ in $L^{2}(\mu_{+})$ on $B_{1}^{+}$, is also uniformly convergent to $u$ on all of $B_{1}^{+}$. Hence \smash{$\varphi(\omega)=\sum_{j=0}^{\infty}u_{j}(\omega)$} on $\Sph^{\ell}_{+}$ with the convergence uniform and hence in $L^{2}(\nu_{+})$. Thus $\varphi$ is represented as an $L^{2}(\nu_{+})$ convergent series of $\beta$-harmonic polynomials on $\Sph^{\ell}_{+}$. Since such $\varphi$ are dense in $L^{2}(\nu_{+})$, the required completeness is established.
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\end{document} | arXiv |
Prioritization, clustering and functional annotation of MicroRNAs using latent semantic indexing of MEDLINE abstracts
Sujoy Roy1,2,
Brandon C. Curry1,
Behrouz Madahian3 &
Ramin Homayouni1,2,4
The amount of scientific information about MicroRNAs (miRNAs) is growing exponentially, making it difficult for researchers to interpret experimental results. In this study, we present an automated text mining approach using Latent Semantic Indexing (LSI) for prioritization, clustering and functional annotation of miRNAs.
For approximately 900 human miRNAs indexed in miRBase, text documents were created by concatenating titles and abstracts of MEDLINE citations which refer to the miRNAs. The documents were parsed and a weighted term-by-miRNA frequency matrix was created, which was subsequently factorized via singular value decomposition to extract pair-wise cosine values between the term (keyword) and miRNA vectors in reduced rank semantic space. LSI enables derivation of both explicit and implicit associations between entities based on word usage patterns. Using miR2Disease as a gold standard, we found that LSI identified keyword-to-miRNA relationships with high accuracy. In addition, we demonstrate that pair-wise associations between miRNAs can be used to group them into categories which are functionally aligned. Finally, term ranking by querying the LSI space with a group of miRNAs enabled annotation of the clusters with functionally related terms.
LSI modeling of MEDLINE abstracts provides a robust and automated method for miRNA related knowledge discovery. The latest collection of miRNA abstracts and LSI model can be accessed through the web tool miRNA Literature Network (miRLiN) at http://bioinfo.memphis.edu/mirlin.
There is growing recognition that miRNAs regulate various diseases and biological processes [1–4] as evidenced by the rapidly growing body of literature related to miRNAs (Additional file 1: Figure S1). There are manually curated repositories such as miRBase [5] and miR2Disease [6] that catalog miRNAs in several organisms as well as summarize their associations with diseases and other biological processes. However, it is generally accepted that manual curation is unable to keep up with the rapidly growing genomic information [7]. For instance, miRBase has not been updated since 2014 and miR2Disease has not been updated since 2009. It is therefore imperative to devise automated methods that can keep pace with the functional information which is deposited in the biomedical literature with respect to miRNAs.
Information retrieval (IR) is a key component of text mining [8]. It consists of three types of models: set-theoretic (Boolean), probabilistic, and algebraic (vector space). Documents in each case are retrieved based on Boolean logic, probability of relevance to the query, and the degree of similarity to the query, respectively. The concept of literature-based discovery was introduced by Swanson [9] and has since been extended to many different areas of research. In the gene space, several approaches have focused on mining both explicit associations based on co-occurrence [10], as well as implicit associations based on higher order co-occurrence and indirect relationships [11].
Several IR approaches have focused on mining miRNA specific associations. miRCancer [12], miRSel [13] and miRTex [14] use co-occurrence and sentence level natural language processing to automatically extract direct relationships from text between miRNAs and genes or diseases. While useful, these tools may miss miRNA interactions where direct relationships were not explicitly stated. In such cases, automated extraction of semantic relationships would be useful to associate genes and miRNAs based on shared biological processes. Also, explicit relationships such as those based on co-occurrence count between miRNAs and genes may be harder to prioritize if they have the exact same score. In contrast, semantic associations that take into account other relationships could be useful for prioritization of miRNA and gene associations [15].
Aside from exploring miRNA to gene associations, semantic analysis could be useful for other research scenarios. For example, investigators may want to prioritize candidate miRNAs for specific diseases or phenotypes. Alternatively, investigators may want to understand the functional pathways shared between different miRNAs. To address these needs, we developed and evaluated an LSI based text mining approach. Previously, we applied LSI to extract functional relationships amongst genes [16] as well as relationships between genes and transcription factors [15] from MEDLINE abstracts. LSI uses Singular Value Decomposition (SVD) [17, 18], which is a dimensionality reduction technique that decomposes the original term-by-document weighted frequency matrix into a new set of factor matrices that can be used to represent both terms and documents in lower-dimensional subspace. Previously, we demonstrated that LSI can extract both explicit (direct) and implicit (indirect) semantic relationships amongst genes. In addition, LSI allows genes to be prioritized based on keyword queries as well as gene-abstract queries with better accuracy than term co-occurrence methods [16]. Here, we applied this approach to miRNAs and demonstrate its utility to prioritize, cluster and functionally annotate miRNAs. The accompanying web based tool, miRNA Literature Network (miRLiN), available at http://bioinfo.memphis.edu/mirlin, provides an automated framework for interactively extracting and discovering functional information on human miRNAs based on up to date biomedical literature.
miRNA document collection
For 1881 human miRNAs indexed in the miRBase repository, 3 different abstract collections were built. Firstly, a curated collection limited to manually assigned abstracts was constructed. A total of 8110 unique abstracts (citations) cross referenced in the linkouts from miRBase as well as Entrez Gene [19] were collected. These citations (identified by unique PubMed identifiers or PMIDs) have been assigned either by professional staff at the National Library of Medicine, or by the scientific research community via Gene Reference into Function (GeneRIF) portal, or by curators of miRBase. Since these abstracts are manually curated, they are expected to have a very high precision for tagging correct citations to miRNAs but at the same time the number of citations referenced for each miRNA is a small proportion of the total number of relevant citations in MEDLINE for that miRNA, resulting in low recall.
In order to increase the information content for the miRNAs, a retrieved collection was built by querying the PubMed repository. A single miRNA can be referenced in the literature in several spelling variants e.g., mir19a, mir-19a, microRNA19a, microRNA-19a etc. For each miRNA, all such tentative synonyms with and without hyphens were constructed, and a PubMed query with the form ' synonym #1 OR synonym #2 OR...OR synonym #n' was submitted using the NCBI efetch utility for retrieving relevant citations that have at least one synonym present in either title or abstract. Further restrictions were added to the query to limit the search to abstracts relevant to humans and miRNAs. A total of 19191 unique citations were retrieved.
The two collections were merged to get 19527 unique citations. We further filtered the nonspecific citations by removing PMIDs that referred to 7 or more miRNAs. Typically, these citations described sequencing experiments which mentioned a large number of miRNAs without substantive biological or mechanistic information. This threshold of 7 was derived as the smallest right outlier in the distribution of numbers of miRNAs linked to each unique citation. The outlier calculation was based on the IQR (interquartile range). The IQR is Q3 (75th percentile) – Q1 (25th percentile). The designated outliers were >Q3+1.5∗I Q R. Post filtering, 17076 unique citations and 878 active miRNAs (the ones referenced by at least one citation) remained in the collection, which comprised of less than half of the original number of 1881 miRNAs. Thus a large number of miRNAs were excluded from our collection because they lacked a specific citation. The number of citations assigned to the active miRNAs ranged from 1 (28 % of the collection) to 1451. The average and median number of citations in the collection were 38 and 4, respectively. For each of 878 active miRNAs, a miRNA document was created by concatenating the titles and abstracts of all citations referenced by the miRNA.
Construction of the LSI model
The outline of the LSI approach used in this study is depicted in Fig. 1. Sixty eight thousand five hundred ninety-six terms (keywords) were parsed from the collection of 878 miRNA documents using Text to Matrix Generator software [20]. All punctuation (excluding hyphens and underscores) and capitalization were ignored and, in addition, articles and other common, non-distinguishing words were discarded using the stop list from Cornell's SMART project repository [21]. A term-by-miRNA matrix was created where the entries of the matrix were log-entropy weighted frequencies of terms across the miRNA document collection. Term weighting schemes are typically employed in order to normalize the matrix and discount the effect of common terms while at the same time increasing the importance of terms that are better delineators between miRNA documents. Each matrix entry a ij is transformed into a product of a local component (l ij ) and global component (g i ):
$$ l_{ij} = \log_{2}(1 + f_{ij}) $$
Overview of Latent Semantic Indexing. In a vector-space model, the semantic structure of a document is represented as a vector (essentially, a bag of words) in word space, and the degree of similarity between documents is calculated by the cosine of the angle between document vectors. The vectors consist of weighted terms, which are a function of the frequency of the terms in and across all documents in the collection. A variant of the vector space model, called Latent Semantic Indexing, improves retrieval by applying singular value decomposition (SVD) to create a subspace in which text documents are represented as vectors. The components in the subspace may be regarded as a concept derived from the word usage patterns in the document. Hence, the relevant documents are retrieved based on the degree of conceptual similarity between the documents
$$ g_{i} = 1 + \frac{\sum_{j}p_{ij}\log_{2}p_{ij}}{\log_{2}n} $$
$$ p_{ij} = \frac{f_{ij}}{\sum_{j}f_{ij}} $$
where f ij is the frequency of the i th term in the j th miRNA-document, p ij is the probability of the i th term occurring in the j th miRNA-document and n is the number of miRNA documents in the collection. The log-entropy weighting scheme is based on information-theoretic concepts and takes into account the distribution of terms over miRNA documents and has been found to be more useful in extracting implied relationships [22].
Singular value decomposition (SVD) [17, 18] was applied to the term-by-miRNA log-entropy weighted frequency matrix. A data matrix A with n rows (terms) and m columns (miRNAs), where n>>m, can be construed as n term row vectors in m-dimensional miRNA space and m miRNA-document column vectors in n-dimensional term space. SVD transforms the two sets of vectors into a new r-dimensional orthogonal space in which the maximum variation is expressed along the first dimension axis, as much variation independent of that is expressed along an axis orthogonal to the first, and so on. The new set of axes may reveal the true dimensionality of the data if the dataset is not inherently m-dimensional. The SVD is formulated as:
$$ A = USV' $$
where ′ indicates transpose of the matrix obtained by permuting the modes, i.e., transforming rows into columns and vice versa, U is n×r, S is r×r, and V is m×r (V ′ is r×m). Both U and V are orthogonal, i.e., U U ′=I and V V ′=I where I is the identity matrix. S is a diagonal matrix with non-negative and non-increasing entries σ 1,σ 2,...,σ r which are known as singular values. r is the rank of the matrix, which is the number of linearly independent rows or columns of A. It is however, known from observation, for most practical datasets, r=m. The third matrix V is written as a transpose so that the rows of both matrices U and V correspond to terms and miRNAs, respectively.
The rows of A can be interpreted as term coordinates in an m-dimensional space. The axes of this space can be interpreted as rows of I (identity matrix). The SVD transforms the term coordinates to rows of U and the axes to the rows of S V ′. The matrix V ′ acts as the rotation matrix for the original axes and the diagonal of matrix S contains the scaling factor for each axis. The U matrix can now be construed as a new transformed dataset whose rows still correspond to the original n terms but the miRNAs are transformed into r eigen miRNAs (factors) that are a linear combination of the original miRNAs.
SVD is symmetric in the sense that a decomposition on the rows (terms) can be transformed into a decomposition on the columns (miRNAs):
$$ A' = VSU' $$
A ′ reverses the roles of terms and miRNAs. V plays the role originally played by U and U plays the role originally played by V. Since S is diagonal, S=S ′. The SVD transforms the miRNA coordinates to rows of V and the axes to the rows of S U ′. The matrix U ′ acts as the rotation matrix for the original axes and the diagonal of matrix S contains the scaling factor for each axis. The V matrix can now be construed as a new transformed dataset whose rows still correspond to the original m miRNAs but the terms are transformed into r eigen terms (factors) that are a linear combination of the original terms.
The new scaled and rotated axes and the coordinates tend to better fit the data than the original axes and coordinates. The singular values in S determine the relative importance of each axis. The first few axes capture the maximum variation in the data and the subsequent ones less so. Only the first k (where k<r) factors corresponding to k largest singular values may be used to represent the data. There are two potential benefits of performing this truncation. Firstly, for large datasets (with many attributes), this translates into savings in memory space as well as analysis time, as vectors in k dimensions can be compared in less time than vectors in m dimensions. Secondly, SVD reveals the true dimensionality present in the data, where the bulk of the information content in the original m-dimensional data may be captured in a lower dimensional manifold, after axis rotation and scaling.
An appropriate choice for k (number of most significant factors) can be made by assessing the contribution of each of the singular values as a measure of the amount of variation captured in each dimension, and then calculating the entropy of the contributions that might be indicative of what percentage of the total number of factors may be needed [23]. The contribution C i of each of r singular values σs can be calculated as:
$$ C_{i} = \frac{{\sigma_{i}^{2}}}{\sum_{i=1}^{r}{\sigma_{i}^{2}}} $$
and the entropy of the r contributions calculated as:
$$ E = \frac{-1}{\log r} \sum_{k=1}^{r} C_{k}\log C_{k} $$
Entropy measures the amount of disorder in the set of variations captured in the r dimensions. The magnitude of the entropy may vary from 0 (all variation is captured in the firrst dimension) to 1 (all dimensions are equally important). k is calculated as E×r. For the term-by-miRNA matrix, k was computed to be 560.
The association between any pair of entities (term-term, term-miRNA, miRNA-miRNA) can be calculated as the cosine of the angle between the respective k-dimensional vectors. The association scores can theoretically fall between −1 and 1, but in practice were observed to occur between −0.2472 and 1. A higher association score between a pair of entities indicates a stronger relationship in literature.
Information Gain calculation
Information gain (in context of citations) for each miRNA was calculated as
$$ {{\begin{aligned} {}\frac{\# of\ \!citations\ retrieved\ from\ PubMed \!-\#\ \!of\ \!citations\ in\ miRBase\ and\ Entrez\ \!Gene}{\# of\ citations\ in\ miRBase\ and\ Entrez\ Gene} \end{aligned}}} $$
Gold standards
miR2Disease was used for evaluating LSI performance. It is a comprehensive database containing descriptions of more than 100 diseases and their associated miRNAs.
AUC calculation
The term-to-miRNA and miRNA-to-term prioritizations were evaluated against gold standards by generating Receiver Operating Characteristics (ROC) curves which display recall and false positive rates at each rank. The area under the curve (AUC) can be used as a measure of ranking quality [24, 25]. The AUC will have the value of 1 for perfect ranking (all relevant entities at the top), 0.5 for randomly generated ranking, and 0 for the worst possible ranking (all relevant entities at the bottom).
Cohesion calculation
The cohesion for a set of miRNAs was calculated as described in [11, 26]. Given a set of n miRNAs for a disease, n AUCs were calculated. Each miRNA was treated as a query and the rest of the n−1 miRNAs were treated as gold standard. The set of all miRNAs (for all diseases) in miR2Disease were prioritized against the query miRNA using the cosine between the miRNA vectors as the similarity measure, and an AUC was calculated. The median AUC out of n AUCs was treated as the cohesion. If a set of miRNAs for a disease are closely related, then the miRNAs in the set would ideally have high cosine association with each other compared to remaining miRNAs that are not in the set, signifying a highly cohesive set.
miRNA Literature Landscape
The annual number of publications related to miRNAs is growing exponentially. This trend is observed in curated databases such as miRBase and Entrez Gene, as well as in PubMed using "miRNA" keyword search (Additional file 1: Figure S1). Overall, 2.37 times more citations were retrieved from the PubMed search than the number designated in curated databases. To collect more abstracts for the growing number of miRNAs, we designed an automated search strategy as described in "Methods". Out of 1881 miRNAs found in miRBase, while all had at least one manually designated citation in either miRBase or Entrez Gene, only 974 had at least one citation retrieved from PubMed. Our PubMed search did not identify abstracts for nearly 50 % of the miRNAs in the curated databases. For the aforementioned 974 miRNAs with at least one retrieved citation, the recall values for more than 50 % of miRNAs were between 0.1 and 0.9 when using the curated citations as gold standard (Fig. 2 a). It is however important to note that our PubMed search strategy retrieved 95.8 % of all abstracts in curated databases (Additional file 1: Figure S2). This result suggests that there may be discrepancies in the curated databases for assignment of citations to miRNAs. On the other hand, it is possible that our search strategy misses important aliases for some miRNAs, thus affecting the recall performance. Next, we calculated the information gain, as described in Methods, for each of the 974 miRNAs. 589 miRNAs showed positive information gain and 304 miRNAs showed a negative information gain (Fig. 2 b). Only 55 miRNAs showed an information gain greater than 10. Based on these results, we concluded that merging citations from miRBase and Entrez Gene with PubMed retrieved citations would allow for the best coverage and information gain for building the LSI model.
Distribution of recall (a) and information gain (b) metrics for curated and retrieved citations for miRNAs. Curated citations were collected from miRBase and Entrez Gene. The retrieved citations were obtained by querying PubMed using a compound search term, which included the miRNA symbol and its aliases. Recall for a miRNA was computed as the fraction of curated citations present amongst the retrieved citations for that miRNA. Information gain for a miRNA was calculated as the ratio between the number of additional citations retrieved, and the number of curated citations for that miRNA
Once the abstract collection was updated and filtered for all miRNAs, an LSI model was built using a total of 17076 citations for the remaining 878 human miRNAs, as described in the "Methods". Figure 3 shows the first three dimensions of the normalized scaled term vectors (A) and miRNA vectors (B) in LSI space. Both term and miRNA vectors are comparable with each other as they share the same coordinate space. We found that while term vectors span a broad area, the miRNA vectors are more concentrated. The limited distribution of the miRNA vectors suggests that the documents share many terms and that miRNAs are functionally quite similar. Additional file 1: Figure S3 shows the distribution of normalized singular values. The first factor captured a little more 3 % of the variance (information content) of the term-by-miRNA matrix. For this study, we used the top 560 (64 %) factors out of 878 factors, which comprised 93 % of the total information content.
Distribution of term vectors (a) and miRNA vectors (b) across the first three LSI dimensions. Each point on the plots represents a single term or miRNA. For each vector, the magnitude of each axis component was scaled by the corresponding singular value and the scaled vector was then normalized to unit length
Evaluation of the LSI model
LSI is a robust approach to extract both explicit and implicit relationships between terms and miRNAs directly from the biomedical literature. In this study, the performance of the LSI model was evaluated based on three different use-case scenarios as described below.
miRNA ranking by term query
A typical use-case involves ranking relevant miRNAs based on their association with keyword queries. A query may consist of a single word (term) such as "cancer" or a combination of words such as "head and neck squamous cell carcinoma". A binary query vector q 0 of a length equal to the total number of terms is created, with 1's corresponding to the query terms and 0's for the remaining terms in the dictionary. A term query q is constructed by projecting q 0 onto the U k matrix as q0′U k , which is the weighted sum of k-dimensional term vectors corresponding to the query terms in the U k matrix. The miRNAs are prioritized by calculating the cosines of the term query with each of the scaled k-dimensional miRNA vectors in the V S k matrix. To evaluate the LSI model, we used the miR2Disease knowledge base as the gold standard. Since miR2Disease was last updated in 2009, an LSI model specifically for this gold standard was generated using only publications dating to 2009 or earlier. For each disease, the full name (or descriptor) served as the query. Figure 4 shows the distribution of AUCs for different query term lengths. A full list of diseases and their respective AUCs are included in Additional file 2: Table S1A. The AUCs for 66 (56 %) of all disease queries were above 0.7. Generally, single word queries performed somewhat better than multiword queries. This result is expected as summing various term vectors could make the query ambiguous, and the high ranked miRNA vectors may actually be close to the composite query vector but only remotely related to any of the constituent terms of the composite query. In addition, disease categories which included more than 50 miRNAs generally resulted in lower AUCs. This may be due to the fact that some miRNAs may have multiple roles and molecular functions, thereby lowering their relative ranking against a single disease query. Lastly, these results may be in part due to discrepancies in the annotations by the curators of miR2Disease database.
Distribution of area under the Receiver Operating Characteristic Curve (AUC) for term-to-miRNA (a) and miRNA-to-term (b) rankings. For term-to-miRNA rankings, the terms constituting a given disease name (obtained from the miR2Disease knowledge base) were used as the query, the query length refers to the number of terms, and the miRNAs associated with the disease were utilized as the gold standard. For miRNA-to-term rankings, the miRNAs associated with a given disease were used as the query, the query length refers to the number of miRNAs, and the terms constituting the disease name were utilized as the gold standard. For both types of rankings, the AUC values are shown stratified across various query lengths
To address the latter issue, we also tested the performance of the LSI model using a different gold standard (Gold Standard II) list of miRNAs for nine different diseases or physiologies. The gold standard II miRNAs were determined by manual examination of recent review papers on each topic (Table 1). The number of miRNAs in each disease category ranged from 8 to 43. The LSI model for evaluation was built using a collection of abstracts up to 2015. Importantly, we found that the average AUC for the nine disease queries was 0.89 (range = 0.80 to 0.94). These results were substantially higher than those achieved by using miR2Disease categories as gold standards, suggesting that miR2Disease database may have errors.
Table 1 Performance of the LSI model on disease or physiology term queries against expert determined gold standards culled from review papers
Term ranking by miRNA query
Another use-case for researchers would be to functionally annotate groups of miRNAs. This is relevant to genomic experiments which generally yield many differentially expressed miRNAs. Here, the miRNAs are treated as the query and the relevant terms are rank ordered. miR2Disease was used to select groups of miRNAs that were assigned to specific diseases. To evaluate the performance of the LSI model, the top 300 ranked terms associated with the group of miRNAs were compared to the disease descriptors in miR2Disease database. A threshold of 300 terms was chosen because it would be impractical for users to consider the entire prioritized list of 68596 terms and also to reduce the computational burden. The list of diseases and their respective term AUCs are available in Additional file 2: Table S1B. The AUCs for 59 diseases could not be obtained as none of the constituent terms in the names of these diseases were found amongst the top 300 ranked terms. Among the queries which returned at least one disease term in the top 300 ranked terms, 27 (46 %) queries produced an AUC above 0.8. Surprisingly, the average AUC for the gold standard II list was 0.54 and none of the disease queries produced and AUC above 0.8 (Additional file 2: Table S2A). These results suggest that the top 300 terms extracted from the LSI model may be related to other topics (such as molecular functions etc.) than only diseases.
miRNA ranking by miRNA query
A third use-case is prioritization of miRNAs in response to a miRNA query. To evaluate the LSI model, we calculated the cohesion, as described in Methods, amongst the group of miRNAs assigned to specific diseases in miR2Disease (gold standard I) or by subject matter experts (gold standard II). The intent was to determine how well the LSI cosine similarity measure captures the real world clustering of related miRNAs. If a set of miRNAs are involved in a disease, then the miRNAs in the set should ideally have high cosine association with each other compared to remaining miRNAs that are not in the set. Figure 5 shows the distribution of cohesions for 122 miRNA disease groups having at least two miRNAs. The median cohesion for the LSI model was 0.83, compared to the median cohesion of 0.36 for a co-occurrence method, in which the similarity measure between miRNAs was designated as the number of shared abstracts. For 88 (72 %) diseases, the cohesions derived via the LSI model were significantly higher than chance when compared with the cohesions derived via the co-occurrence model (p≤0.05, ranksum test) (Additional file 2: Table S1C). In contrast, the median cohesion using LSI was only marginally better than that produced via the co-occurrence method using the gold standard II set, 0.577 and 0.576 respectively (Additional file 2: Table S2B). These results suggest that as the body of literature grows, miRNAs may be associated with many different pathways and functions beyond just specific diseases.
Distribution of cohesions for 122 diseases' miRNA groups in miR2Disease. The cohesions for the LSI model were compared with those from the co-occurrence model. In the LSI model, the similarity between any two miRNAs was calculated as the cosine of the angle between their vectors in truncated LSI space. In the co-occurrence model, the similarity between any two miRNAs was designated as the number of shared citations
Clustering and functional annotation of miRNAs
A major advantage of LSI is that the semantic relationships amongst all miRNA documents may be measured in lower dimensional (concept) space. Therefore, the cosine values between miRNAs may be used as a similarity score to cluster functionally related miRNAs. In addition, using the LSI model, miRNA clusters may be annotated using the top ranked terms as demonstrated above. Cosines were calculated for all miRNA pairs and a miRNA-miRNA cosine matrix was generated. The matrix was transformed into an adjacency (binary) matrix using a cosine threshold of 99 percentile of all pair-wise cosines (0.41715). The adjacency matrix was truncated to include only 365 miRNAs that were part of the largest connected component. The graph was clustered using UKmeans algorithm [27] with k = 25 to generate 25 mutually connected clusters (Fig. 6, Additional file 2: Table S3). For each cluster, the LSI model was queried using all of its miRNAs and then the top 300 terms were extracted for each cluster.
Clustering and functional annotation of miRNAs based on LSI derived associations. UKmeans clustering of the maximally connected component (∼350 miRNAs) of the miRNA graph, in which an edge is assigned if the cosine value is above 99th percentile of all pair-wise cosine values. The functional annotations for each miRNA cluster were selected from amongst the top 300 ranked terms obtained via querying the truncated LSI space with the miRNA cluster
The terms were manually examined and used to label each cluster in Fig. 6. For instance, the largest cluster containing 73 miRNAs is associated with Alzheimer disease. This number is slightly different from the number (64 miRNAs) of Alzheimer related miRNAs in miR2Disease database. Interestingly, the largest miR2Disease group of miRNAs (152) was associated with hepatocellular carcinoma. It is important to note that the top nine miR2Disease categories, containing between 114 to 152 miRNAs, were all associated with some form of cancer. This suggests that there is a large bias in the miRNA databases as of 2009. By comparison, we found that the LSI-based clusters contained smaller number of miRNAs that were associated with more specific terms, which were functionally aligned. These results indicate that LSI based clustering allows for more robust functional clustering and more specific functional annotation beyond simply assigning miRNAs to diseases.
miRLiN web tool
We developed a publicly available web-tool (http://bioinfo.memphis.edu/mirlin) to provide access to the LSI model, which contains the most recent and comprehensive collection of miRNA abstracts in MEDLINE (Fig. 7). The user can query the model with any combination of terms or miRNAs. When querying with terms, the tool ranks all miRNAs in the collection with respect to semantic associations to the query. Alternatively, a miRNA query may be used to compute associations with both miRNAs and terms. The output of the tool is a ranked list of miRNAs and terms based on the degree of association (cosine value) to the query. Selected miRNAs and terms can be visualized as a network graph, where the nodes represent the selected miRNAs and terms and the edges represent cosine values above 0.4. Multiple nodes can be selected from the graph display to retrieve their shared abstracts, if applicable. The abstracts are displayed with the selected terms and miRNAs highlighted for convenience.
Screenshot of miRNA Literature Network (miRLiN) web tool. miRLiN enables users to prioritize miRNAs and terms according to queries. Specific miRNAs or terms can be selected (upper left panel) and displayed as a graph (upper right panel). Single or multiple nodes on the graph may be selected to view the abstracts associated with them in the lower panel
For benchmarking, we compared the performance of our web tool with two existing web tools, miRCancer [12] and miRiaD [28]. While both tools are disease focused, miRLiN is more flexible and can accept any type of query. Also, both tools rely on databases with binary associations between miRNAs and diseases. In contrast, miRLiN ranks miRNAs based on the functional relevancy to the query and also enables a genome-wide network view of miRNAs with multiple associations to one another. Additional file 2: Tables S4A and S4B compare the results from the 3 web tools for 'choriocarcinoma' and 'meningioma' queries. For the 'choriocarcinoma' query, miRCancer listed 3 miRNAs (hsa-mir-199b, hsa-mir-218, hsa-mir-34a) while miRiaD listed 2 additional miRNAs (hsa-mir-141, hsa-mir-126). Importantly, miRLiN retrieved all 5 miRNAs within the top 15 ranked miRNAs (Additional file 2: Table S4A). We manually evaluated the 10 additional miRNAs retrieved by miRLiN. We found that miRNAs hsa-mir-378a, hsa-mir-371b, hsa-mir-371a, hsa-let-7g, hsa-mir-373, hsa-mir-141 and hsa-mir-15a were co-mentioned with the query term in the same abstracts, but not in the same sentences. It appears that these miRNAs were found to be differentially expressed in choriocarcinoma cell lines. One miRNA hsa-mir-145 was co-mentioned with the query term in the same sentence that suggests a direct link. Interestingly, hsa-mir-585 association with choriocarcinoma appeared to be indirect via its association with hsa-mir-218. In addition, the abstract for hsa-mir-141 in miRLiN was different from the other two web tools, suggesting that our abstract retrieval approach is slightly different than the other two methods. Lastly, hsa-mir-624 did not appear to be related to choriocarcinoma or any other type of cancer, thus appears to be a false discovery.
For the 'meningioma' query, miRCancer retrieved 4 miRNAs (hsa-mir-128, hsa-mir-200a, hsa-mir-224, hsa-mir-335) and miRiaD retrieved 4 additional miRNAs (hsa-mir-145, hsa-mir-190, hsa-mir-219, and hsa-mir-29). Only two meningioma related miRNAs overlapped between miRiaD and miRCancer. In comparison, miRLiN retrieved all but one (hsa-mir-145, ranked 25th) amongst the top 12 ranked miRNAs (Additional file 2: Table S4B). Moreover, miRLiN identified two additional miRNAs (hsa-mir-4417 and hsa-mir-185). Manual examination found that hsa-mir-185 is in fact negatively associated with meningioma, where the citation explicitly negates its involvement in meningioma. This result reveals a shortcoming of our method, which does not take into account negations and other parts of speech that are considered in NLP based approaches. Lastly, manual examination did not find an association between hsa-mir-4417 and meningioma, albeit it is associated with other types of cancer.
We have developed an LSI based approach to prioritize, cluster and functionally annotate miRNAs. LSI enables representation of miRNAs and terms as vectors in low dimensional space that can be compared against each other. LSI provides an advantage over co-occurrence based methods as semantic associations between entities take into account not only the entities being compared but also indirect associations amongst all other related entities in the collection. Several choices were made in the construction of the model that affects its performance. The rationale behind the choices and the potential ramifications of the alternatives are discussed below.
While building the miRNA document collection, citations that referenced more than 7 miRNAs were filtered out. Manual examination of citations revealed that certain high throughput screening papers were associated with many miRNAs but these papers did not describe any functional information about the specific miRNAs. For instance, many citations described sequencing experiments that identified several miRNAs. Inclusion of such citations in the model would create strong semantic associations between pairs of miRNAs that are otherwise remotely related. Better automated methods are needed to identify and filter such abstracts that do not describe any functional relationships.
Our results suggest that parsing of terms from miRNA documents still needs improvement. We found that many of the top 300 terms associated with groups of miRNAs were indeed too specific, relating to gene symbols or non-standard abbreviations used in the papers. For the current LSI model, only designated stopwords were removed prior to factorization. Automated methods may need to be investigated that can filter out additional non-useful terms. Stemming of the terms to their roots may also be useful in terms of reducing the dictionary size, although strategies for expanding the roots to the most relevant expansion will need to be devised once the terms are to be used for functional annotation. Currently, the selection of interesting functional annotation terms is still manual but could be automated by restricting to MeSH [29], GO [30] and KEGG [31]. However, this filtering strategy may result in loss of interesting terms such as gene or transcription factor names or phrases like 'acaa-deletion' that may indirectly link the miRNAs to a physiology or a biological function or a disease.
Several other methods may need to be investigated in the future to improve the performance of the LSI approach. For instance, different types of normalization methods for the term-by-miRNA matrix, in addition to the log-entropy method, may need to be investigated [22]. In the current study, an entropy based method was used to select k highest magnitude singular values. Other strategies have been discussed in the literature that may improve performance [32]. The web tool currently displays top 50 miRNAs and 300 terms in response to a query. Automated methods, such as one used for determining the singular value threshold, may also be useful in devising a prioritization threshold for cosines. Finally, adding collections for other model organisms such as mouse, rat etc. will make a more comprehensive text mining database for miRNAs.
All together, we have demonstrated that an LSI based approach provides a robust and automated method to interrogate the large amount of literature that is accumulating with respect to miRNAs. The approach enables rapid prioritization of miRNAs in relation to keyword or miRNA queries. Furthermore, the LSI based approach allows for global clustering of all miRNAs based on functional information in the literature and provides a method for annotating groups of miRNAs with highly specific terms and concepts.
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The authors would like to thank the University of Memphis High Performance Computing facility for providing the needed computational resources for this study.
This work and its publication was supported in part by the Memphis Research Consortium and the University of Memphis Center for Translational Informatics.
The LSI model described in the paper can be accessed through the web tool miRNA Literature Network (miRLiN) at http://bioinfo.memphis.edu/mirlin.
SR and RH designed the research and wrote the manuscript. SR and BM conducted data analysis. BC developed the web tool. RH supervised the research and assisted with interpretation of results. All authors read and approved the final manuscript.
Bioinformatics Program, University of Memphis, Memphis, 38152, USA
Sujoy Roy, Brandon C. Curry & Ramin Homayouni
Center for Translational Informatics, University of Memphis, Memphis, 38152, USA
Sujoy Roy & Ramin Homayouni
Department of Mathematical Sciences, University of Memphis, Memphis, 38152, USA
Behrouz Madahian
Department of Biology, University of Memphis, Memphis, 38152, USA
Ramin Homayouni
Sujoy Roy
Brandon C. Curry
Correspondence to Ramin Homayouni.
From 13th Annual MCBIOS conference Memphis, TN, USA. 3-5 May 2016
Additional file 1
Figures S1, S2, and S3. 'S10-S2.pdf' contains supplementary figures 1, 2 and 3 in separate pages. (PDF 159 KB)
Tables S1A, S1B, S1C, S2A, S2B, S3, S4A and S4B. Microsoft Excel 2013 workbook 'S11-S1.xlsx' contains supplementary tables 1A, 1B, 1C, 2A, 2B, 3, 4A and 4B in separate tabs. (XLSX 32.5 KB)
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver(http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.
Roy, S., Curry, B., Madahian, B. et al. Prioritization, clustering and functional annotation of MicroRNAs using latent semantic indexing of MEDLINE abstracts. BMC Bioinformatics 17, 350 (2016). https://doi.org/10.1186/s12859-016-1223-2
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March 2021, 10(1): 155-198. doi: 10.3934/eect.2020061
Local wellposedness of quasilinear Maxwell equations with absorbing boundary conditions
Roland Schnaubelt 1,, and Martin Spitz 2,
Department of Mathematics, Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany
Department of Mathematics, University of Bielefeld, Postfach 100131, 33501 Bielefeld, Germany
* Corresponding author: Roland Schnaubelt
Received December 2018 Revised March 2020 Published March 2021 Early access June 2020
Fund Project: Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID 258734477 – SFB 1173
In this article we provide a local wellposedness theory for quasilinear Maxwell equations with absorbing boundary conditions in $ {\mathcal{H}}^m $ for $ m \geq 3 $. The Maxwell equations are equipped with instantaneous nonlinear material laws leading to a quasilinear symmetric hyperbolic first order system. We consider both linear and nonlinear absorbing boundary conditions. We show existence and uniqueness of a local solution, provide a blow-up criterion in the Lipschitz norm, and prove the continuous dependence on the data. In the case of nonlinear boundary conditions we need a smallness assumption on the tangential trace of the solution. The proof is based on detailed apriori estimates and the regularity theory for the corresponding linear problem which we also develop here.
Keywords: Nonlinear Maxwell system, absorbing or impedance boundary conditions, local wellposedness, blow-up criterion, continuous dependence.
Mathematics Subject Classification: Primary: 35L60; Secondary: 35L50, 35Q61.
Citation: Roland Schnaubelt, Martin Spitz. Local wellposedness of quasilinear Maxwell equations with absorbing boundary conditions. Evolution Equations & Control Theory, 2021, 10 (1) : 155-198. doi: 10.3934/eect.2020061
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Vural Bayrak, Emil Novruzov, Ibrahim Ozkol. Local-in-space blow-up criteria for two-component nonlinear dispersive wave system. Discrete & Continuous Dynamical Systems, 2019, 39 (10) : 6023-6037. doi: 10.3934/dcds.2019263
Xinwei Yu, Zhichun Zhai. On the Lagrangian averaged Euler equations: local well-posedness and blow-up criterion. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1809-1823. doi: 10.3934/cpaa.2012.11.1809
Xiaojing Xu. Local existence and blow-up criterion of the 2-D compressible Boussinesq equations without dissipation terms. Discrete & Continuous Dynamical Systems, 2009, 25 (4) : 1333-1347. doi: 10.3934/dcds.2009.25.1333
Jong-Shenq Guo. Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition. Discrete & Continuous Dynamical Systems, 2007, 18 (1) : 71-84. doi: 10.3934/dcds.2007.18.71
Huiling Li, Mingxin Wang. Properties of blow-up solutions to a parabolic system with nonlinear localized terms. Discrete & Continuous Dynamical Systems, 2005, 13 (3) : 683-700. doi: 10.3934/dcds.2005.13.683
Yi-hang Hao, Xian-Gao Liu. The existence and blow-up criterion of liquid crystals system in critical Besov space. Communications on Pure & Applied Analysis, 2014, 13 (1) : 225-236. doi: 10.3934/cpaa.2014.13.225
Yu-Zhu Wang, Weibing Zuo. On the blow-up criterion of smooth solutions for Hall-magnetohydrodynamics system with partial viscosity. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1327-1336. doi: 10.3934/cpaa.2014.13.1327
Yinghua Li, Shijin Ding, Mingxia Huang. Blow-up criterion for an incompressible Navier-Stokes/Allen-Cahn system with different densities. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1507-1523. doi: 10.3934/dcdsb.2016009
Giuseppe Maria Coclite, Angelo Favini, Gisèle Ruiz Goldstein, Jerome A. Goldstein, Silvia Romanelli. Continuous dependence in hyperbolic problems with Wentzell boundary conditions. Communications on Pure & Applied Analysis, 2014, 13 (1) : 419-433. doi: 10.3934/cpaa.2014.13.419
Ming Lu, Yi Du, Zheng-An Yao, Zujin Zhang. A blow-up criterion for the 3D compressible MHD equations. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1167-1183. doi: 10.3934/cpaa.2012.11.1167
Shouming Zhou, Chunlai Mu, Yongsheng Mi, Fuchen Zhang. Blow-up for a non-local diffusion equation with exponential reaction term and Neumann boundary condition. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2935-2946. doi: 10.3934/cpaa.2013.12.2935
Marek Fila, Hirokazu Ninomiya, Juan-Luis Vázquez. Dirichlet boundary conditions can prevent blow-up in reaction-diffusion equations and systems. Discrete & Continuous Dynamical Systems, 2006, 14 (1) : 63-74. doi: 10.3934/dcds.2006.14.63
Mengxian Lv, Jianghao Hao. General decay and blow-up for coupled Kirchhoff wave equations with dynamic boundary conditions. Mathematical Control & Related Fields, 2021 doi: 10.3934/mcrf.2021058
Bouthaina Abdelhedi, Hatem Zaag. Single point blow-up and final profile for a perturbed nonlinear heat equation with a gradient and a non-local term. Discrete & Continuous Dynamical Systems - S, 2021, 14 (8) : 2607-2623. doi: 10.3934/dcdss.2021032
Petri Juutinen. Convexity of solutions to boundary blow-up problems. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2267-2275. doi: 10.3934/cpaa.2013.12.2267
Jorge A. Esquivel-Avila. Blow-up in damped abstract nonlinear equations. Electronic Research Archive, 2020, 28 (1) : 347-367. doi: 10.3934/era.2020020
Roland Schnaubelt Martin Spitz | CommonCrawl |
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Based on the idea and the provided source code of Andrej Karpathy (arxiv-sanity)
First Sunyaev-Zel'dovich mapping with the NIKA2 camera: Implication of cluster substructures for the pressure profile and mass estimate (1712.09587)
F. Ruppin, F. Mayet, G.W. Pratt, R. Adam, P. Ade, P. André, M. Arnaud, H. Aussel, I. Bartalucci, A. Beelen, A. Benoît, A. Bideaud, O. Bourrion, M. Calvo, A. Catalano, B. Comis, M. De Petris, F.-X. Désert, S. Doyle, E. F. C. Driessen, J. Goupy, C. Kramer, G. Lagache, S. Leclercq, J.-F. Lestrade, J.F. Macías-Pérez, P. Mauskopf, A. Monfardini, L. Perotto, G. Pisano, E. Pointecouteau, N. Ponthieu, V. Revéret, A. Ritacco, C. Romero, H. Roussel, K. Schuster, A. Sievers, C. Tucker, R. Zylka
April 6, 2018 astro-ph.CO
The complete characterization of the pressure profile of high-redshift galaxy clusters, from their core to their outskirts, is a major issue for the study of the formation of large-scale structures. It is essential to constrain a potential redshift evolution of both the slope and scatter of the mass-observable scaling relations used in cosmology studies based on cluster statistics. In this paper, we present the first thermal Sunyaev-Zel'dovich (tSZ) mapping of a cluster from the sample of the NIKA2 SZ large program that aims at constraining the redshift evolution of cluster pressure profiles and the tSZ-mass scaling relation. We have observed the galaxy cluster PSZ2 G144.83+25.11 at redshift $z=0.58$ with the NIKA2 camera, a dual-band (150 and 260 GHz) instrument operated at the IRAM 30-meter telescope. We identify a thermal pressure excess in the south-west region of PSZ2 G144.83+25.11 and a high redshift sub-millimeter point source that affect the intracluster medium (ICM) morphology of the cluster. The NIKA2 data are used jointly with tSZ data acquired by the MUSTANG, Bolocam and $Planck$ experiments in order to non-parametrically set the best constraints on the electronic pressure distribution from the cluster core ($\rm{R} \sim 0.02 \rm{R_{500}}$) to its outskirts ($\rm{R} \sim 3 \rm{R_{500}} $). We investigate the impact of the over-pressure region on the shape of the pressure profile and on the constraints on the integrated Compton parameter $\rm{Y_{500}}$. A hydrostatic mass analysis is also performed by combining the tSZ-constrained pressure profile with the deprojected electronic density profile from XMM-$Newton$. This allows us to conclude that the estimates of $\rm{Y_{500}}$ and $\rm{M_{500}}$ obtained from the analysis with and without masking the disturbed ICM region differ by 65 and 79% respectively. (abridged)
Recovering galaxy cluster gas density profiles with XMM-Newton and Chandra (1709.06570)
I. Bartalucci, M. Arnaud, G.W. Pratt, A. Vikhlinin, E. Pointecouteau, W.R. Forman, C. Jones, P. Mazzotta, F. Andrade-Santos
Sept. 19, 2017 astro-ph.CO
We examine the reconstruction of galaxy cluster radial density profiles obtained from Chandra and XMM X-ray observations, using high quality data for a sample of twelve objects covering a range of morphologies and redshifts. By comparing the results obtained from the two observatories and by varying key aspects of the analysis procedure, we examine the impact of instrumental effects and of differences in the methodology used in the recovery of the density profiles. We find that the final density profile shape is particularly robust. We adapt the photon weighting vignetting correction method developed for XMM for use with Chandra data, and confirm that the resulting Chandra profiles are consistent with those corrected a posteriori for vignetting effects. Profiles obtained from direct deprojection and those derived using parametric models are consistent at the 1% level. At radii larger than $\sim$6", the agreement between Chandra and XMM is better than 1%, confirming an excellent understanding of the XMM PSF. We find no significant energy dependence. The impact of the well-known offset between Chandra and XMM gas temperature determinations on the density profiles is found to be negligible. However, we find an overall normalisation offset in density profiles of the order of $\sim$2.5%, which is linked to absolute flux cross-calibration issues. As a final result, the weighted ratios of Chandra to XMM gas masses computed at R2500 and R500 are r=1.03$\pm$0.01 and r=1.03$\pm$0.03, respectively. Our study confirms that the radial density profiles are robustly recovered, and that any differences between Chandra and XMM can be constrained to the $\sim$ 2.5% level, regardless of the exact data analysis details. These encouraging results open the way for the true combination of X-ray observations of galaxy clusters, fully leveraging the high resolution of Chandra and the high throughput of XMM.
High-resolution SZ imaging of clusters of galaxies with the NIKA2 camera at the IRAM 30-m telescope (1709.01255)
F. Mayet, R. Adam, P. Ade, P. André, M. Arnaud, H. Aussel, I. Bartalucci, A. Beelen, A. Benoît, A. Bideaud, O. Bourrion, M. Calvo, A. Catalano, B. Comis, M. De Petris, F.-X. Désert, S. Doyle, E. F. C. Driessen, J. Goupy, C. Kramer, G. Lagache, S. Leclercq, J. F. Lestrade, J. F. Macías-Pérez, P. Mauskopf, A. Monfardini, E. Pascale, L. Perotto, E. Pointecouteau, G. Pisano, N. Ponthieu, G. W. Pratt, V. Revéret, A. Ritacco, C. Romero, H. Roussel, F. Ruppin, K. Schuster, A. Sievers, S. Triqueneaux, C. Tucker, R. Zylka
Sept. 7, 2017 astro-ph.CO
The development of precision cosmology with clusters of galaxies requires high-angular resolution Sunyaev-Zel'dovich (SZ) observations. As for now, arcmin resolution SZ observations (e.g. SPT, ACT and Planck) only allowed detailed studies of the intra cluster medium for low redshift clusters (z<0.2). With both a wide field of view (6.5 arcmin) and a high angular resolution (17.7 and 11.2 arcsec at 150 and 260 GHz), the NIKA2 camera installed at the IRAM 30-m telescope (Pico Veleta, Spain), will bring valuable information in the field of SZ imaging of clusters of galaxies. The NIKA2 SZ observation program will allow us to observe a large sample of clusters (50) at redshifts between 0.4 and 0.9. As a pilot study for NIKA2, several clusters of galaxies have been observed with the pathfinder, NIKA, at the IRAM 30-m telescope to cover the various configurations and observation conditions expected for NIKA2.
Mapping the hot gas temperature in galaxy clusters using X-ray and Sunyaev-Zel'dovich imaging (1706.10230)
R. Adam, M. Arnaud, I. Bartalucci, P. Ade, P. André, A. Beelen, A. Benoît, A. Bideaud, N. Billot, H. Bourdin, O. Bourrion, M. Calvo, A. Catalano, G. Coiffard, B. Comis, A. D'Addabbo, F.-X. Désert, S. Doyle, C. Ferrari, J. Goupy, C. Kramer, G. Lagache, S. Leclercq, J.-F. Macías-Pérez, S. Maurogordato, P. Mauskopf, F. Mayet, A. Monfardini, F. Pajot, E. Pascale, L. Perotto, G. Pisano, E. Pointecouteau, N. Ponthieu, G.W. Pratt, V. Revéret, A. Ritacco, L. Rodriguez, C. Romero, F. Ruppin, K. Schuster, A. Sievers, S. Triqueneaux, C. Tucker, R. Zylka
July 21, 2017 astro-ph.CO
We propose a method to map the temperature distribution of the hot gas in galaxy clusters that uses resolved images of the thermal Sunyaev-Zel'dovich (tSZ) effect in combination with X-ray data. Application to images from the New IRAM KIDs Array (NIKA) and XMM-Newton allows us to measure and determine the spatial distribution of the gas temperature in the merging cluster MACS J0717.5+3745, at $z=0.55$. Despite the complexity of the target object, we find a good morphological agreement between the temperature maps derived from X-ray spectroscopy only -- using XMM-Newton ($T_{\rm XMM}$) and Chandra ($T_{\rm CXO}$) -- and the new gas-mass-weighted tSZ+X-ray imaging method ($T_{\rm SZ}$). We correlate the temperatures from tSZ+X-ray imaging and those from X-ray spectroscopy alone and find that $T_{\rm SZ}$ is higher than $T_{\rm XMM}$ and lower than $T_{\rm CXO}$ by $\sim 10\%$ in both cases. Our results are limited by uncertainties in the geometry of the cluster gas, contamination from kinetic SZ ($\sim 10\%$), and the absolute calibration of the tSZ map ($7\%$). Investigation using a larger sample of clusters would help minimise these effects.
Mapping the kinetic Sunyaev-Zel'dovich effect toward MACS J0717.5+3745 with NIKA (1606.07721)
R. Adam, I. Bartalucci, G.W. Pratt, P. Ade, P. André, M. Arnaud, A. Beelen, A. Benoît, A. Bideaud, N. Billot, H. Bourdin, O. Bourrion, M. Calvo, A. Catalano, G. Coiffard, B. Comis, A. D'Addabbo, M. De Petris, J. Démoclès, F.-X. Désert, S. Doyle, E. Egami, C. Ferrari, J. Goupy, C. Kramer, G. Lagache, S. Leclercq, J.-F. Macías-Pérez, S. Maurogordato, P. Mauskopf, F. Mayet, A. Monfardini, T. Mroczkowski, F. Pajot, E. Pascale, L. Perotto, G. Pisano, E. Pointecouteau, N. Ponthieu, V. Revéret, A. Ritacco, L. Rodriguez, C. Romero, F. Ruppin, K. Schuster, A. Sievers, S. Triqueneaux, C. Tucker, M. Zemcov, R. Zylka
Dec. 8, 2016 astro-ph.CO
Measurement of the gas velocity distribution in galaxy clusters provides insight into the physics of mergers, through which large scale structures form in the Universe. Velocity estimates within the intracluster medium (ICM) can be obtained via the Sunyaev-Zel'dovich (SZ) effect, but its observation is challenging both in term of sensitivity requirement and control of systematic effects, including the removal of contaminants. In this paper we report resolved observations, at 150 and 260 GHz, of the SZ effect toward the triple merger MACS J0717.5+3745 (z=0.55), using data obtained with the NIKA camera at the IRAM 30m telescope. Assuming that the SZ signal is the sum of a thermal (tSZ) and a kinetic (kSZ) component and by combining the two NIKA bands, we extract for the first time a resolved map of the kSZ signal in a cluster. The kSZ signal is dominated by a dipolar structure that peaks at -5.1 and +3.4 sigma, corresponding to two subclusters moving respectively away and toward us and coincident with the cold dense X-ray core and a hot region undergoing a major merging event. We model the gas electron density and line-of-sight velocity of MACS J0717.5+3745 as four subclusters. Combining NIKA data with X-ray observations from XMM-Newton and Chandra, we fit this model to constrain the gas line-of-sight velocity of each component, and we also derive, for the first time, a velocity map from kSZ data (i.e. that is model-dependent). Our results are consistent with previous constraints on the merger velocities, and thanks to the high angular resolution of our data, we are able to resolve the structure of the gas velocity. Finally, we investigate possible contamination and systematic effects with a special care given to radio and submillimeter galaxies. Among the sources that we detect with NIKA, we find one which is likely to be a high redshift lensed submillimeter galaxy.
Resolving galaxy cluster gas properties at z~1 with XMM-Newton and Chandra (1610.01899)
I. Bartalucci, M. Arnaud, G.W. Pratt, J. Démoclès, R.F.J. van der Burg, P. Mazzotta
Nov. 25, 2016 astro-ph.CO
We present a pilot X-ray study of the five most massive ($M_{500}>5 \times 10^{14} M_{\odot}$), distant (z~1), galaxy clusters detected via the Sunyaev-Zeldovich effect. We optimally combine XMM-Newton and Chandra X-ray observations by leveraging the throughput of XMM to obtain spatially-resolved spectroscopy, and the spatial resolution of Chandra to probe the bright inner parts and to detect embedded point sources. Capitalising on the excellent agreement in flux-related measurements, we present a new method to derive the density profiles, constrained in the centre by Chandra and in the outskirts by XMM. We show that the Chandra-XMM combination is fundamental for morphological analysis at these redshifts, the Chandra resolution being required to remove point source contamination, and the XMM sensitivity allowing higher significance detection of faint substructures. The sample is dominated by dynamically disturbed objects. We use the combined Chandra-XMM density profiles and spatially-resolved temperature profiles to investigate thermodynamic quantities including entropy and pressure. From comparison of the scaled profiles with the local REXCESS sample, we find no significant departure from standard self-similar evolution, within the dispersion, at any radius, except for the entropy beyond 0.7$R_{500}$. The baryon mass fraction tends towards the cosmic value, with a weaker dependence on mass than observed in the local Universe. We compare with predictions from numerical simulations. The present pilot study demonstrates the utility and feasibility of spatially-resolved analysis of individual objects at high-redshift through the combination of XMM and Chandra observations. Observations of a larger sample will allow a fuller statistical analysis to be undertaken, in particular of the intrinsic scatter in the structural and scaling properties of the cluster population. (abridged)
High angular resolution SZ observations with NIKA and NIKA2 (1605.09549)
B. Comis, R. Adam, P. Ade, P. André, M. Arnaud, I. Bartalucci, A. Beelen, A. Benoît, A. Bideaud, N. Billot, O. Bourrion, M. Calvo, A. Catalano, G. Coiffard, F.-X. Désert, S. Doyle, J. Goupy, C. Kramer, G. Lagache, S. Leclercq, J. F. Macías-Pérez, P. Mauskopf, F. Mayet, A. Monfardini, F. Pajot, E. Pascale, L. Perotto, E. Pointecouteau, G. Pisano, N. Ponthieu, G. W. Pratt, V. Revéret, A. Ritacco, L. Rodriguez, C. Romero, F. Ruppin, G. Savini, K. Schuster, A. Sievers, S. Triqueneaux, C. Tucker, R. Zilch
May 31, 2016 astro-ph.CO
NIKA2 (New IRAM KID Arrays) is a dual band (150 and 260 GHz) imaging camera based on Kinetic Inductance Detectors (KIDs) and designed to work at the IRAM 30 m telescope (Pico Veleta, Spain). Built on the experience of the NIKA prototype, NIKA2 has been installed at the 30 m focal plane in October 2015 and the commissioning phase is now ongoing. Through the thermal Sunyaev-Zeldovich (tSZ) effect, NIKA2 will image the ionized gas residing in clusters of galaxies with a resolution of 12 and 18 arcsec FWHM (at 150 and 260 GHz, respectively). We report on the recent tSZ measurements with the NIKA camera and discuss the future objectives for the NIKA2 SZ large Program, 300h of observation dedicated to SZ science. With this program we intend to perform a high angular resolution follow-up of a cosmologically-representative sample of clusters belonging to SZ catalogues, with redshift greater than 0.5. The main output of the program will be the study of the redshift evolution of the cluster pressure profile as well as that of the scaling laws relating the cluster global properties.
High-resolution tSZ cartography of clusters of galaxies with NIKA at the IRAM 30-m telescope (1602.07941)
F. Mayet, R. Adam, A. Adane, P. Ade, P. André, M. Arnaud, I. Bartalucci, A. Beelen, A. Benoît, A. Bideaud, N. Billot, G. Blanquer, N. Boudou, O. Bourrion, M. Calvo, A. Catalano, G. Coiffard, B. Comis, A. Cruciani, F.-X. Désert, S. Doyle, J. Goupy, B. Hasnoun, I. Hermelo, C. Kramer, G. Lagache, S. Leclercq, J. F. Macías-Pérez, P. Mauskopf, A. Monfardini, F. Pajot, L. Perotto, E. Pointecouteau, N. Ponthieu, G. W. Pratt, V. Revéret, A. Ritacco, L. Rodriguez, F. Ruppin, K. Schuster, A. Sievers, S. Triqueneaux, C. Tucker, R. Zylka
Feb. 25, 2016 astro-ph.CO
The thermal Sunyaev-Zeldovich effect (tSZ) is a powerful probe to study clusters of galaxies and is complementary with respect to X-ray, lensing or optical observations. Previous arcmin resolution tSZ observations ({\it e.g.} SPT, ACT and Planck) only enabled detailed studies of the intra-cluster medium morphology for low redshift clusters ($z < 0.2$). Thus, the development of precision cosmology with clusters requires high angular resolution observations to extend the understanding of galaxy cluster towards high redshift. NIKA2 is a wide-field (6.5 arcmin field of view) dual-band camera, operated at $100 \ {\rm mK}$ and containing $\sim 3300$ KID (Kinetic Inductance Detectors), designed to observe the millimeter sky at 150 and 260 GHz, with an angular resolution of 18 and 12 arcsec respectively. The NIKA2 camera has been installed on the IRAM 30-m telescope (Pico Veleta, Spain) in September 2015. The NIKA2 tSZ observation program will allow us to observe a large sample of clusters (50) at redshift ranging between 0.5 and 1. As a pathfinder for NIKA2, several clusters of galaxies have been observed at the IRAM 30-m telescope with the NIKA prototype to cover the various configurations and observation conditions expected for NIKA2.
High angular resolution Sunyaev-Zel'dovich observations of MACS J1423.8+2404 with NIKA: Multiwavelength analysis (1510.06674)
R. Adam, B. Comis, I. Bartalucci, A. Adane, P. Ade, P. André, M. Arnaud, A. Beelen, B. Belier, A. Benoît, A. Bideaud, N. Billot, O. Bourrion, M. Calvo, A. Catalano, G. Coiffard, A. D'Addabbo, F.-X. Désert, S. Doyle, J. Goupy, B. Hasnoun, I. Hermelo, C. Kramer, G. Lagache, S. Leclercq, J.-F. Macías-Pérez, J. Martino, P. Mauskopf, F. Mayet, A. Monfardini, F. Pajot, E. Pascale, L. Perotto, E. Pointecouteau, N. Ponthieu, G.W. Pratt, V. Revéret, A. Ritacco, L. Rodriguez, G. Savini, K. Schuster, A. Sievers, S. Triqueneaux, C. Tucker, R. Zylka
The prototype of the NIKA2 camera, NIKA, is an instrument operating at the IRAM 30-m telescope, which can observe simultaneously at 150 and 260GHz. One of the main goals of NIKA2 is to measure the pressure distribution in galaxy clusters at high resolution using the thermal SZ (tSZ) effect. Such observations have already proved to be an excellent probe of cluster pressure distributions even at high redshifts. However, an important fraction of clusters host submm and/or radio point sources, which can significantly affect the reconstructed signal. Here we report on <20" resolution observations at 150 and 260GHz of the cluster MACSJ1424, which hosts both radio and submm point sources. We examine the morphology of the tSZ signal and compare it to other datasets. The NIKA data are combined with Herschel satellite data to study the SED of the submm point source contaminants. We then perform a joint reconstruction of the intracluster medium (ICM) electronic pressure and density by combining NIKA, Planck, XMM-Newton, and Chandra data, focusing on the impact of the radio and submm sources on the reconstructed pressure profile. We find that large-scale pressure distribution is unaffected by the point sources because of the resolved nature of the NIKA observations. The reconstructed pressure in the inner region is slightly higher when the contribution of point sources are removed. We show that it is not possible to set strong constraints on the central pressure distribution without accurately removing these contaminants. The comparison with X-ray only data shows good agreement for the pressure, temperature, and entropy profiles, which all indicate that MACSJ1424 is a dynamically relaxed cool core system. The present observations illustrate the possibility of measuring these quantities with a relatively small integration time, even at high redshift and without X-ray spectroscopy.
A weak lensing analysis of the PLCK G100.2-30.4 cluster (1505.02887)
M. Radovich, I. Formicola, M. Meneghetti, I. Bartalucci, H. Bourdin, P. Mazzotta, L. Moscardini, S. Ettori, M. Arnaud, G. W. Pratt, N. Aghanim, H. Dahle, M. Douspis, E. Pointecouteau, A. Grado
We present a mass estimate of the Planck-discovered cluster PLCK G100.2-30.4, derived from a weak lensing analysis of deep SUBARU griz images. We perform a careful selection of the background galaxies using the multi-band imaging data, and undertake the weak lensing analysis on the deep (1hr) r-band image. The shape measurement is based on the KSB algorithm; we adopt the PSFex software to model the Point Spread Function (PSF) across the field and correct for this in the shape measurement. The weak lensing analysis is validated through extensive image simulations. We compare the resulting weak lensing mass profile and total mass estimate to those obtained from our re-analysis of XMM-Newton observations, derived under the hypothesis of hydrostatic equilibrium. The total integrated mass profiles are in remarkably good agreement, agreeing within 1$\sigma$ across their common radial range. A mass $M_{500} \sim 7 \times 10^{14} M_\odot$ is derived for the cluster from our weak lensing analysis. Comparing this value to that obtained from our reanalysis of XMM-Newton data, we obtain a bias factor of (1-b) = 0.8 $\pm$ 0.1. This is compatible within 1$\sigma$ with the value of (1-b) obtained by Planck Collaboration XXIV from their calibration of the bias factor using newly-available weak lensing reconstructed masses.
Discovery of large-scale diffuse radio emission and of a new galaxy cluster in the surroundings of MACSJ0520.7-1328 (1402.4436)
G. Macario, H. T. Intema, C. Ferrari, H. Bourdin, S. Giacintucci, T. Venturi, P. Mazzotta, I. Bartalucci, M. Johnston-Hollitt, R. Cassano, D. Dallacasa, G. W. Pratt, R. Kale, S. Brown
Feb. 18, 2014 astro-ph.CO, astro-ph.GA
We report the discovery of large-scale diffuse radio emission South-East of the galaxy cluster MACS J0520.7-1328, detected through high sensitivity Giant Metrewave Radio Telescope 323 MHz observations. This emission is dominated by an elongated diffuse radio source and surrounded by other features of lower surface brightness. Patches of these faint sources are marginally detected in a 1.4 GHz image obtained through a re-analysis of archival NVSS data. Interestingly, the elongated radio source coincides with a previously unclassified extended X-ray source. We perform a multi-wavelength analysis based on archival infrared, optical and X-ray Chandra data. We find that this source is a low-temperature (~3.6 keV) cluster of galaxies, with indications of a disturbed dynamical state, located at a redshift that is consistent with the one of the main galaxy cluster MACS J0520.7-132 (z=0.336). We suggest that the diffuse radio emission is associated with the non-thermal components in the intracluster and intergalactic medium in and around the newly detected cluster. We are planning deeper multi-wavelength and multi-frequency radio observations to accurately investigate the dynamical scenario of the two clusters and to address more precisely the nature of the complex radio emission. | CommonCrawl |
\begin{definition}[Definition:Foiaș Constant/First/Decimal Expansion]
The decimal expansion of the first Foiaș Constant starts:
:$x_{\infty} = 2 \cdotp 29316 \, 62874 \, 11861 \, 03150 \, 80282 \, 91250 \, 80586 \, 43722 \, 57290 \, 32712 \, 12485 \, 37 \ldots$
{{OEIS|A085846}}
Category:Definitions/Foiaș Constants
\end{definition} | ProofWiki |
Chen, M. H., Shao, Q. M. and Ibrahim, J.G. (2000) , Monte Carlo Methods In Bayesian Computation . Springer Series in Statistics, Springer-Verlag , New York. ISBN 0-387-98935-8
Lai, T.L., de la Pena, V. and Shao, Q. M. (2009) , Self-normalized Processes: Theory and Statistical Applications. Springer Series in Probability and its Applications, Springer-Verlag , New York. ISBN 978-3-540-85635-1.
Chen, L.H.Y., Goldstein, L. and Shao, Q.M. (2010). Normal Approximation by Stein's Method . Springer-Verlag , New York. ISBN 978-3-642-15006-7.
Asymptotic Theory in Probability and Statistics with Applications. Higher Eduction Press of China, and International Press, 2007 (Edited book with T. L. Lai and L. F. Qian).
Selected Publications of Qi-Man Shao
Non-normal approximation by Stein's method of exchangeable pairs with application to the Curie-Weiss model. Ann. Appl. Probab. 21 (2011), 464-483 (with S. Chatterjee)
Large deviations for local times and intersection local times of fractional Brownian motions and Riemann-Liouville processes. Ann. Probab. 39 (2011), 729-778 (with X. Chen, W. Li and J. Rosinski)
Nonparametric estimate of spectral density functions of sample covariance matrices: A first step. Ann. Statisit. 38 (2010), 3724-3750 (with B.Y. Jing, G.M. Pan and W. Zhou)
Cramer-type moderate deviation for the maximum of the periodogram with application to simultaneous tests in gene expression time series. Ann. Statist. 38 (2010), 1913-1935. (with W.D. Liu)
The asymptotic distribution and Berry-Esseen bound of a new test for independence in high dimension with an application to stochastic optimization. Ann. Appl. Probab. 18 (2008), 2337-2366 (with Z.Y. Lin and W.D. Liu)
Towards a universal self-normalized moderate deviation. Trans. Amer. Math. Soc. 360 (2008), 4263--4285 (with B.Y. Jing and W. Zhou).
Normal approximation for nonlinear statistics using a concentration inequality approach Bernoulli 13 (2007), 581-599 (with L.H.Y. Chen).
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\begin{document}
\title{On the Lang--Trotter conjecture for Siegel modular forms}
\date{}
\author{Arvind Kumar, Moni Kumari and Ariel Weiss}
\address{Arvind Kumar, Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Edmund J.\ Safra Campus, Jerusalem 9190401, Israel.\vspace*{-5pt}}
\email{[email protected]\vspace*{-6pt}}
\address{Moni Kumari, Department of Mathematics, Bar-Ilan University, Ramat Gan 5290002, Israel.\vspace*{-5pt}}
\email{[email protected]\vspace*{-6pt}}
\address{Ariel Weiss, Department of Mathematics, Ben-Gurion University of the Negev, Be'er Sheva 8410501, Israel.\vspace*{-5pt}}
\email{[email protected]\vspace*{-6pt}}
\subjclass[2020]{11F80, 11F46, 11R45}
\keywords{Siegel modular forms, Images of Galois representations, Lang--Trotter conjecture}
\begin{abstract}
Let $f$ be a genus two cuspidal Siegel modular eigenform. We prove an adelic open image theorem for the compatible system of Galois representations associated to $f$, generalising the results of Ribet and Momose for elliptic modular forms. Using this result, we investigate the distribution of the Hecke eigenvalues $a_p$ of $f$, and obtain upper bounds for the sizes of the sets $\{p \le x : a_p = a\}$ for fixed $a\iC$, in the spirit of the Lang--Trotter conjecture for elliptic curves.
\end{abstract}
\maketitle
\section{Introduction}
If $A$ is a non-CM elliptic curve over $\Q$ of conductor $N$ and if $a\iZ$, then the \emph{Lang--Trotter conjecture} \cite{lang-trotter} states that
\begin{equation*}\label{eq:lt-conjecture} \pi_A(x, a):= \#\set{p\le x,\ p\nmid N : a_p = a} \sim C(A, a)\frac{ x^{1/2}}{\log x},
\end{equation*} where $a_p = p+1 - \#A(\Fp)$ and $C(A, a)\ge 0$ is an explicit constant. The conjecture is formulated in terms of the two-dimensional compatible system of Galois representations attached to $A$. More generally, if $(\rho_\l)_\l$ is an arbitrary compatible system of Galois representations of conductor $N$, whose Frobenius polynomials are defined over a number field $E$, then it is natural to ask for the asymptotics of the set
\begin{equation}\label{eq:lt-set}
\set{p\le x,\ p\nmid \l N : \tr\rho_\l(\Frob_p) = a}
\end{equation}
for $a\in\O_E$. When $(\rho_\l)_\l$ is conjugate self-dual, a generalisation of the Lang--Trotter conjecture has been formulated by V.\ K.\ Murty \cite{murty-Frobenius}*{Conj.\ 2.15}.
The goal of this paper is to estimate the size of the set $(\ref{eq:lt-set})$ when $(\rho_\l)_\l$ is the compatible system of Galois representations attached to a genus two Siegel modular eigenform.
Let $f$ be a cuspidal vector-valued genus two Siegel modular eigenform of weight $(k_1, k_2)$---i.e.\ weight $\Sym^{k_1-k_2}\det^{k_2}$---with $k_1\ge k_2\ge 2$, level $N$ and character $\varepsilon$. Let $E = \Q(\{a_p : p\nmid N\}, \varepsilon)$ be the number field generated by the image of $\varepsilon$ and by the eigenvalues $a_p$ of the Hecke operators $T_p$. Let $F\sub E$ be the field fixed by the inner twists of $f$ (see \Cref{def: twists-intro} and \Cref{sec:inner-twists}).
\begin{theorem}\label{thm:unconditional-lta}
Let $\pi$ be the cuspidal automorphic representation of $\Gf(\A_\Q)$ associated to $f$ and assume that its functorial lift to $\GL_4(\A_\Q)$ exists, is cuspidal, and is neither an automorphic induction nor a symmetric cube lift. For $a\in \O_F$, let
\[\pi_f(x, a) := \#\set{p\le x : a_p = a}.\]
Then, for any $\epsilon>0$,
\[\pi_f(x, a) \ll_{\epsilon,f}\frac{x}{(\log x)^{1+\alpha-\epsilon}},\quad\alpha = \frac{[F:\Q]}{10[F:\Q]+1}.\]
If $a = 0$, then
\[\pi_f(x, 0) \ll_{\epsilon,f} \frac{x}{(\log x)^{1+\alpha-\epsilon}},\quad\alpha = \frac{[F:\Q]}{7[F:\Q]+1}.\]
\end{theorem}
If we further assume the Generalised Riemann Hypothesis (GRH), we obtain the following strengthened result:
\begin{theorem}\label{thm:conditional-lta}
Let $f$ be as above, and assume GRH. If $a\in \O_F$, then
\[\pi_f(x, a) \ll_{f}\frac{x^{1-\alpha}}{(\log x)^{1-2\alpha}},\quad\alpha= \frac{[F:\Q]}{11[F:\Q]+1}.\]
If $a = 0$, then
\[\pi_f(x, 0) \ll_{f}\frac{x^{1-\alpha}}{(\log x)^{1-2\alpha}},\quad\alpha= \frac{[F:\Q]}{10[F:\Q]+1}.\]
\end{theorem}
The assumption that the functorial lift of $\pi$ is cuspidal and not an automorphic induction or a symmetric cube lift is equivalent to demanding that $f$ is not CAP, endoscopic, CM, RM or a symmetric cube lift. The existence of this functorial lift follows from work of Weissauer \cite{Weissauersymplectic} and Asgari--Shahidi \cite{asgari-shahidi} when $k_2 >2$, and from Arthur's classification \cite{Arthur2013, geetaibi} when $k_2 = 2$. We refer the reader to \cite{weiss2018image}*{Sec.\ 2.4} for further discussion.
The Lang--Trotter conjecture for elliptic curves was first investigated by Serre \cite{ser81}, who showed that $\pi_A(x, a) \ll \frac{x}{(\log x)^{5/4-\epsilon}}$ unconditionally, and that $\pi_A(x, a) \ll x^{7/8}(\log x)^{1/2}$ under GRH. These bounds were subsequently improved by Wan \cite{Wan} and V.\ K.\ Murty \cite{murty-mod-forms-ii} unconditionally, and by Murty--Murty--Saradha under GRH \cite{mms}. The best current estimates when $a\ne 0$ are $\pi_A(x, a) \ll \frac{x(\log\log x)^2}{(\log x)^{2}}$ unconditionally \cite{thorner-zaman}, and $\pi_A(x, a) \ll \frac{x^{4/5}}{(\log x)^{3/5}}$ under GRH \cite{zyw}.
These results all apply verbatim to non-CM elliptic modular forms of weight $\ge 2$ with integer Hecke eigenvalues, and can easily be adapted to give bounds when the field $E$ generated by the Hecke eigenvalues is arbitrary (see, for example, \cite{ser81}*{Sec.\ 7}). The bounds obtained in this way are independent of the field $E$ and its subfield $F$. A novel feature of our result is that our bounds improve as the degree $[F:\Q]$ increases.
The higher dimensional case has been studied by Cojocaru--Davis--Silverberg--Stange \cite{cojocaru-abelian-varieties} (see also \cite{serban}), who formulate a precise conjecture for generic abelian varieties, and prove analogues of Theorems \ref{thm:unconditional-lta} and \ref{thm:conditional-lta}. If $A$ is an abelian surface over $\Q$ with $\End(A)=\Z$, then, conjecturally, for each prime $\l$, the $\l$-adic Galois representation attached to $A$ should be isomorphic to the $\l$-adic Galois representation attached to a Siegel modular form $f$ of weight $(2,2)$, paramodular level and integer Hecke eigenvalues. For such a Siegel modular form $f$, our unconditional bound in \Cref{thm:unconditional-lta} exactly matches that of \cite{cojocaru-abelian-varieties}*{Thm.\ 1} (see also \Cref{rem:strengthened-result}). Our conditional bound of $O(x^{11/12}(\log x)^{-5/6})$ in \Cref{thm:conditional-lta} is slightly stronger than the bound $O(x^{21/22+\varepsilon})$ of \cite{cojocaru-abelian-varieties}. The authors of \cite{cojocaru-abelian-varieties} also formulate a precise conjecture for the asymptotics of $\pi_A(x, a)$, including the constant $C(A, a)$. It would be interesting to formulate such a conjecture in the case of Siegel modular forms, particularly when $F\ne \Q$, however, we do not pursue that here.
The proofs of the above results all use the strategy initiated by Serre \cite{ser81}, which combines explicit versions of the Chebotarev density theorem with precise calculations of the images of Galois representations attached to modular forms and abelian varieties \cite{serre-abelian, Ribet75, momose, Ribet85}. Our strategy is similar, however, in the case of Siegel modular forms, these image results are not available. The key technical input of this paper is a precise big image theorem for Galois representations attached to Siegel modular forms.
\subsection{Images of Galois representations}
There is a general philosophy that the image of an automorphic Galois representation should be as large as possible, unless there is an automorphic reason for it to be small. For example, let $f$ be a Siegel modular eigenform as in the previous section. For each prime $\lambda$ of $E$, there is a $\lambda$-adic Galois representation
\[\rho_{\lambda}\:\Ga\Q\to \Gf(\overline{E}_\lambda)\]
associated to $f$. If $f$ is CAP or endoscopic, then $\rho_{\lambda}$ is reducible for all $\lambda$. Similarly, the image of $\rho_{\lambda}$ is small if $f$ has complex or real multiplication, or is a symmetric cube lift.
In \cite{weiss2018image}, building on previous work of Dieulefait and Dieulefait--Zenteno \cites{Dieulefait2002maximalimages, DZ}, the third author showed that, if $f$ is not in one of these exceptional cases, then the image of $\rho_\lambda$ is large in the following sense: for all but finitely many primes $\lambda$ (or for all $\lambda\mid\l$ for a set of primes $\l$ of density $1$ if $k_2 = 2$), the image of the residual representation $\orho_{\lambda}\:\Ga\Q\to \Gf(\O_E/\lambda)$ contains $\Sp_4(\Fl)$.
However, in order to prove Theorems \ref{thm:unconditional-lta} and \ref{thm:conditional-lta}, we need to pin down the exact image $\orho_{\lambda}$ for almost all primes. This study is complicated by an additional symmetry, that of \emph{inner twists}, first described by Ribet \cite{Ribet77,Ribet-twists} for elliptic modular forms. Fix an embedding $E\hookrightarrow\C$.
\begin{definition}\label{def: twists-intro}
An \emph{inner twist} of $f$ is a pair $(\sigma, \chi_\sigma)$, where $\sigma\in\Hom(E, \C)$ and $\chi_\sigma$ is a Dirichlet character, such that $\sigma(a_p) = \chi_\sigma(p)a_p$ for almost all primes $p$.
\end{definition}
We show in \Cref{sec:inner-twists} that the set of such $\sigma\in \Hom(E,\C)$ form an abelian subgroup $\Gamma$ of $\Aut(E/\Q)$. Let $F = E^\Gamma$ be its corresponding fixed field. For each $\sigma\in\Gamma$, let $K_\sigma$ be the number field cut out by $\chi_\sigma$, and let $K$ be the compositum of all the $K_\sigma$'s. Then, if $p$ splits in $K$, we have $\sigma(a_p) = a_p$ for all $\sigma\in\Gal(E/F)$, i.e.\ $a_p\in F\subsetneq E$. Thus, inner twists give a restriction on the image of $\rho_{\lambda}$: the Frobenius elements associated to a positive density of primes $p$ have trace contained in the proper subfield $F$ of $E$. Let $\G=\G_f$ be the group scheme over $\Z$ such that, for each $\Z$-algebra $R$, we have
\begin{equation*}
\G(R) = \set{(g, \nu)\in \Gf(\O_F\tensor_\Z R)\times R\t : \simil(g) = \nu^{k_1 + k_2 - 3}}.
\end{equation*}
Here, $\simil\:\Gf\to \GL_1$ is the similitude character. Let $\rho_{\l}:= \bigoplus_{\lambda\mid\l}\rho_{\lambda}$. We show in \Cref{lem:small-trace} that, for almost all primes $\l$, the restriction
\[\rho_{\l}|_K\:\Gal(\Qb/K)\to \Gf(E\tensor_\Q\Qlb)\]
factors through $\Gf(F\tensor_\Q\Ql)$, and extends to a representation
\[\wrho_{ \l}\:\Gal(\Qb/K)\to \G(\Ql)\]
such that the projection to $\Gf(\O_F\tensor_\Z\Ql)$ is $\rho_{\l}|_K$ and the projection to $\Ql\t$ is the cyclotomic character. Moreover, up to conjugation, we can assume that $\wrho_{\l}$ takes values in $\G(\Zl)$. If $\LL$ is a set of rational primes, let $\widehat{\Q}_\LL = \Q\tensor_\Z\prod_{\l\in\LL}\Zl$ and let
\[\wrho_{ \LL}:= \bigoplus_{\l\in\LL}\wrho_{ \l}\:\Gal(\Qb/K)\to \G(\widehat\Q_\LL)\]
be the associated adelic Galois representation.
Our main technical result is the following determination of the images of these Galois representations, which generalises the results of Serre for elliptic curves \cite{serre-abelian}
and of Ribet, Momose and Loeffler for elliptic modular forms \cites{Ribet75, Ribet77,momose,Ribet85, Loeffler-adelic}.
\begin{theorem}\label{thm:precise-image}
Let $f$ be a cuspidal vector-valued Siegel modular eigenform of weight $(k_1, k_2)$, level $N$ and character $\varepsilon$. Define $E, F$ and $\G$ as above. Let $\pi$ be the cuspidal automorphic representation of $\Gf(\A_\Q)$ associated to $\pi$ and assume that its functorial lift to $\GL_4(\A_\Q)$ exists, is cuspidal, and is neither an automorphic induction nor a symmetric cube lift.
Let $\LL'$ be the set of rational primes $\l$ such that $\l\ge 5$, such that $\rho_{\l}|_{\Ql}$ is de Rham and such that $\rho_{\l}|_{\Ql}$ is crystalline if $\l\nmid N$.\footnote{If $k_2>2$, then $\LL'$ consists of all primes $\l\ge 5$. If $k_2 = 2$, then $\LL'$ has Dirichlet density $1$ by \cite{weiss2018image}*{Thm.\ 1.1}.} Let $\LL\sub \LL'$ be the $($cofinite$)$ subset of primes such that $\rho_{\l}|_K$ takes values in $\Gf(F\tensor_\Q\Ql)$. Then:
\begin{enumerate}
\item For each prime $\l\in\LL$, the image of $\wrho_{\l}$ is an open subgroup of $\G(\Zl)$.
\item For all but finitely many primes $\l\in\LL$, the image of $\wrho_{\l}$ is exactly $\G(\Zl)$.
\item The image of $\wrho_{\LL}$ is an open subgroup of $\G(\widehat{\Q}_\LL)$.
\end{enumerate}
\end{theorem}
\subsection{Methods}
\subsubsection{The image of Galois}
To prove \Cref{thm:precise-image}, in \Cref{sec:inner-twists}, we first generalise the notion of inner twists to the setting of Siegel modular forms, and show that they give a restriction on the image of $\rho_\l$. Our key technical input is \Cref{lem:b_q}, which shows that the field $F$ cut out by the inner twists of $f$ is the trace field of the standard representation of $\rho_\l$, obtained by composing $\rho_\l$ with the maps $\Gf\to\PGSp_4\xrightarrow{\sim}\SO_5\to\GL_5$. As a result, we deduce that $F$ is generated over $\Q$ by $\{\frac{b_p}{\varepsilon(p)} : p\nmid N \}$, where $b_p$ is the coefficient of $X^2$ in the characteristic polynomial of $\rho_\l(\Frob_p)$, which does not depend on $\l$.
Fix a prime $\l\in\LL$. To prove that $\wrho_\l$ has open image, using the definition of the inner twists and an argument using Goursat's lemma, we prove that it is enough to show that $\rho_\lambda|_K$ has open image inside
\[G_\lambda := \set{g\in\Gf(\O_{F_\lambda}): \simil(g) \in\Zl^{\times(k_1 + k_2 -3)}}\]
for each $\lambda\mid\l$. Let $H_\lambda$ be the image of $\rho_\lambda|_K$. Our starting point for that proving $\rho_\lambda|_K$ has open image is the result of \cite{weiss2018image}, that $H_\lambda$ is a Zariski-closed subgroup $\Gf(F_\lambda)$. In \Cref{sec:open-image}, we use this result in combination with a theorem of Pink \cite{Pink} to show that the projective image of $\rho_\lambda|_K$ is open in $\PGSp_4(F_\lambda)$, from which we can deduce the result.
Similarly, to prove that $\wrho_\l$ is surjective, again using Goursat's lemma, we prove that it is enough to show that $\rho_\lambda$ surjects onto $G_\lambda$ for all $\lambda\mid\l$. The surjectivity of $\rho_\lambda$ was proven by Dieulefait \cite{Dieulefait2002maximalimages}*{Sec.\ 4.7} under the assumption that the field $F=\Q(\{\frac{b_p}{\varepsilon(p)} : p\nmid N \})$ is generated over $\Q$ by $b_p$ for a single prime $p$. In \Cref{lem:untwisted}, using our proof that the image of $\wrho_\l$ is open in $\G(\Zl)$, we prove that Dieulefait's assumption holds unconditionally.
Once we have proven parts $(i)$ and $(ii)$ of \Cref{thm:precise-image}, part $(iii)$ follows from a straightforward but technical generalisation of the group-theoretic results of Serre \cite{serre-abelian}, Ribet \cite{Ribet75} and Loeffler \cite{Loeffler-adelic}.
\subsubsection{Lang--Trotter bounds}
Our proofs of \Cref{thm:unconditional-lta} and \Cref{thm:conditional-lta} are very different in nature. To prove \Cref{thm:unconditional-lta},
we use the machinery of Serre \cite{ser81}, which works by combining the explicit Chebotarev density theorem of \cite{lo} with the $\l$-adic image of Galois for a single prime $\l$. Rather than applying this machinery to the $\lambda$-adic Galois representation $\rho_\lambda$, as Serre does for elliptic modular forms in \cite{ser81}*{Sec.\ 7}, the precision of \Cref{thm:precise-image} allows us to apply Serre's machinery to the full $\l$-adic Galois representation $\rho_\l=\bigoplus_{\lambda\mid\l}\rho_\lambda$. As a result, we obtain stronger bounds that improve with the size of $[F:\Q]$. For example, applying our method in the case of an elliptic modular form $f$ improves Serre's bound of $\pi_f(x, a)\ll_{f,\epsilon} x/(\log x)^{5/4-\epsilon}$ to the bound
\[\pi_f(x, a) \ll_{\epsilon,f}\frac{x}{(\log x)^{1+\alpha-\epsilon}},\quad\alpha = \frac{[F:\Q]}{3[F:\Q]+1}.\]
The fact that our bounds improve with the size of $[F:\Q]$ is in accordance with the generalised Lang--Trotter conjecture of Murty \cite{murty-Frobenius}*{Conj.\ 3.1}, which predicts that as soon as $[F:\Q] \ge 3$, we should actually have $\pi_f(x, a) = O(1)$.
In contrast, our proof of \Cref{thm:conditional-lta} generalises the methods of Murty--Murty--Saradha \cite{mms}, which work by combining the explicit Chebotarev density theorem with the mod $\l$ image of Galois for infinitely many primes $\l$.
Assume, for the sake of exposition, that $E = F$, so that $f$ has no inner twists, and $K=\Q$. Then, if $a\in\O_F$, by the definition of the residual representation $\worho_\l$, we have
\begin{equation*}
\pi_f(x, a) \le \set{p\le x : p\nmid \l N,\ \tr\worho_\l(\Frob_p)\equiv a\pmod\l} + O(1),
\end{equation*}
where, by $\tr \worho_\l$, we mean the trace of the $\Gf$ component of $\G$.
By \Cref{thm:precise-image}, the image of the residual representation $\worho_\l$ is $\G(\Fl)$ for all but finitely many $\l\in\LL$. Hence, for such $\l$, $\worho_\l$ factors through a finite Galois extension $L/K$, with Galois group $\G(\Fl)$. Thus, it is sufficient to bound the number of primes $p\le x$ such that $\worho_\l(\Frob_p)$ is contained in the conjugation invariant subset $\set{(g, \nu)\in \G(\Fl) : \tr(g)\equiv a\pmod\l}\sub\G(\Fl)$, which we can do using the Chebotarev density theorem. However, applying Chebotarev directly to these sets would not give the strongest possible bound.
The key idea is that, under GRH, we can bound the size of $\pi_f(x, a)$ by bounding the size of the smaller set $\{p\le x : a_p = a,\ \l \text{ splits completely in } F(p)\}$ for almost all primes $\l$ that split completely in $F$ (\Cref{lem:murty-bound}). Here, $F(p)$ is the splitting field of the characteristic polynomial of $\rho_\l(\Frob_p)$. If $p$ is in this smaller set, then the eigenvalues of $\worho_\l(\Frob_p)$ are in $\Fl\t$, so $\worho_\l(\Frob_p)$ is conjugate to an upper triangular matrix. We show in \Cref{lem4} that we can bound this set by applying the Chebotarev density theorem to an abelian extension, whose Galois group is isomorphic to the group of upper triangular matrices in $\G(\Fl)$ modulo the subgroup of unipotent upper triangular matrices. In particular, since Artin's holomorphicity conjecture is known for abelian extensions, we can bound this set by using a stronger version of the Chebotarev density theorem due to Zywina \cite{zyw}.
As in the proof of \Cref{thm:unconditional-lta}, rather than working with the mod $\lambda$ Galois representations $\orho_\lambda\:\Ga\Q\to\Gf(\O_E/\lambda)$, using \Cref{thm:precise-image}, we can work with the mod $\l$ Galois representations $\rho_\l\:\Ga\Q \to \Gf(\O_E\tensor_\Z\Fl)$. Applying our method in the case of an elliptic modular form $f$ gives the bound (under GRH)
\[\pi_f(x, a) \ll_{f}\frac{x^{1-\alpha}}{(\log x)^{1-2\alpha}},\quad\alpha=\frac{[F:\Q]}{4[F:\Q]+1}.\]
\subsection{Outline of the paper}
In \Cref{sec:galois}, we recall key properties of the Galois representations attached to Siegel modular eigenforms, and generalise the notion of inner twists to Siegel modular forms. In \Cref{sec:image}, we prove \Cref{thm:precise-image}, which is a key technical result. In \Cref{sec:cdt}, we recall explicit versions of the Chebotarev density theorem, a variant due to Serre \cite{ser81} and a refinement due to Zywina \cite{zyw}. Using these inputs, we prove Theorems \ref{thm:unconditional-lta} and \ref{thm:conditional-lta} in Sections \ref{sec:proof-unconditional} and \ref{sec:proof-conditional}.
\section{Galois representations attached to Siegel modular forms}\label{sec:galois}
Let $f$ be a cuspidal Siegel modular eigenform of weights $(k_1, k_2)$, $k_1\ge k_2\ge 2$, level $N$, and character $\varepsilon$. We assume throughout this paper that, if $\pi$ is the cuspidal automorphic representation of $\Gf(\AQ)$ attached to $f$, then the functorial lift of $\pi$ to $\GL_4(\A_\Q)$ exists, is cuspidal, and is neither an automorphic induction nor a symmetric cube lift. In particular, $f$ is not CAP, endoscopic, CM, RM or a symmetric cube lift. Let $E=\Q(\{a_p : p\nmid N\}, \varepsilon)$ be the subfield of $\C$ generated by the image of $\varepsilon$ and by the Hecke eigenvalues $a_p$ of the Hecke operators $T_p$. Then $E$ is a finite extension of $\Q$.
For a ring $R$, let
\[\Gf(R) = \set{g\in\GL_4(R): g^t J g = \nu J,\ \nu \in R\t},\]
where $J = \br{\begin{smallmatrix}
0 & 0 & 0 & 1\\
0 & 0 & 1 & 0\\
0 & -1 & 0 & 0\\
-1 & 0 & 0 & 0
\end{smallmatrix}}$. For $g\in \Gf(R)$, the constant $\nu$ is called the \emph{similitude} of $g$ and is denoted $\simil(g)$. Let $\Sp_4(R)$ be the subgroup of elements for which $\simil(g) = 1$, let $\PSp_4(R) = \Sp_4(R)/Z(\Sp_4(R))$, where $Z(\Sp_4(R))$ is the centre of $\Sp_4(R)$, and let $\PGSp_4(R) = \Gf(R)/Z(\GSp_4(R))$.
By the work of Taylor, Laumon and Weissauer \cites{taylor1993, Laumon, Weissauer, Weissauersymplectic} when $k_2\ge 2$, and Taylor \cite{taylor1991galois} when $k_2=2$ (see also \cite{mok2014galois}), for each prime $\lambda$ of $E$, there exists a continuous semisimple symplectic Galois representation
\[\rho_\lambda\:\Ga\Q\to \Gf(\overline E_{\lambda})\]
that is unramified at all primes $p\nmid \l N$, and is characterised by the property
\[\Tr\rho_{\lambda}(\Frob_p)= a_p,\qquad \simil\rho_{\lambda}(\Frob_p) = \varepsilon(p)p^{k_1 + k_2 -3},\]
for all primes $p\nmid \l N$. By \cite{serre-mod-p-lattices}*{Thm.\ 5.2.1}, we can view $\rho_\lambda$ as a representation valued in $\Gf(\overline \O_{E_\lambda})$, and define the mod $\lambda$ representation
\[\orho_\lambda\:\Ga\Q\to \Gf( \O_E/\lambda)\]
to be the semisimplification of the reduction of $\rho_\lambda$ mod $\lambda$. This reduction is still symplectic by \cite{voight-6author}*{Lemma 4.3.6}. The representation $\rho_\lambda$ should be defined over $E_\lambda$, but this seems not to be known in general. This ambiguity does not occur for $\orho_\lambda$, since mod $\l$ representations are always defined over their trace field.
By work of Ramakrishnan \cite{Ramakrishnan}, Dieulefait--Zenteno \cite{DZ} and the third author \cite{weiss2018image}, the image of $\rho_\lambda$ is generically large, in the following sense. Let $\LL'$ be the set of rational primes $\l$ such that $\l\ge 5$, such that $\rho_\l|_{\Ql}$ is de Rham and such that $\rho_\l|_{\Ql}$ is crystalline if $\l\nmid N$. Then $\LL'$ is just the set of primes $\l\ge 5$ if $k_2>2$, while if $k_2 = 2$, $\LL'$ has Dirichlet density $1$ \cite{weiss2018image}*{Thm.\ 1.1}.
\begin{theorem}[\cite{weiss2018image}*{Thms.\ 1.1, 1.2}]\label{thm:image-contains-sp4}
\begin{enumerate}[leftmargin=*]
\item If $\l\in \LL'$ and $\lambda\mid\l$, then $\rho_\lambda$ is absolutely irreducible.
\item For all but finitely many $\l\in\LL'$, if $\lambda\mid \l$, then the image of $\orho_\lambda$ contains a subgroup conjugate to $\Sp_4(\Fl)$.
\end{enumerate}
\end{theorem}
Conjecturally, \Cref{thm:image-contains-sp4} should be true with $\LL'$ the set of all primes, however, this question is open.
\begin{corollary}\label{cor:field-of-def}
For all but finitely many primes $\l\in \LL'$, for each prime $\lambda\mid\l$, the Galois representation $\rho_\lambda$ descends to a representation
\[\rho_\lambda\:\Ga\Q\to \Gf(E_\lambda).\]
\end{corollary}
\begin{proof}
By \Cref{thm:image-contains-sp4}, $\orho_\lambda$ is irreducible for all $\lambda\mid\l$, for all but finitely many $\l\in\LL'$. Hence, for such primes $\lambda$, by \cite{voight-6author}*{Lemma 4.3.8} and \cite{carayol1994formes}*{Th\'eor\`eme 2}, $\rho_\lambda$ is defined over its trace field.
\end{proof}
For each prime $\l$, we form the $\l$-adic representation
\[\rho_\l :=\bigoplus_{\lambda\mid\l}\rho_\lambda\:\Ga\Q\to \Gf( E\tensor_\Q\Qlb),\]
which, as before, we may view as taking values in $\Gf(\O_E\tensor_\Z\Zlb)$, and the mod $\l$ representation
\[\orho_\l :=\bigoplus_{\lambda\mid\l}\orho_\lambda\:\Ga\Q\to \Gf(\O_E\tensor_\Z\Fl).\]
Finally, let $\LL\sub\LL'$ be a set of rational primes such that, for each $\l\in \LL$, $\rho_\l$ takes values in $\Gf(E\tensor_\Q\Ql)$. If $\widehat\Q_\LL = \Q\tensor_{\Z}\prod_{\l\in\LL}\Zl$, then we define \[\rho_\LL:=\bigoplus_{\l\in\LL}\rho_\l\:\Ga\Q\to \Gf(E\tensor\widehat\Q_\LL).\]
\subsection{Inner twists}\label{sec:inner-twists}
In this section, we generalise the notion of \emph{inner twists} to Siegel modular forms and discuss their key properties. The results in this section are well known in the case of elliptic modular forms \cite{momose, Ribet85}. Fix once and for all an embedding $\sigma_0\:E\hookrightarrow \C$. In particular, via this embedding, we may view the eigenvalues $a_p$ as elements of $\C$.
\begin{definition}
A \emph{inner twist} of $f$ is a pair $(\sigma, \chi)$, where $\sigma\in \Hom(E, \C)$ and $\chi$ is a Dirichlet character, such that
\[\sigma(a_p)= \chi(p)a_p\]
for all primes $p\nmid N$.
\end{definition}
We define $\Gamma$ to be the set of $\sigma\in \Hom(E, \C)$ for which such a $\chi$ exists. It is simple to show that $(\sigma_0, \chi)$ is a inner twist for a non-trivial character $\chi$ if and only if $f$ is CM or RM. In particular, since we have assumed that $f$ is not CM or RM, for each twist $(\sigma, \chi)$, the character $\chi$ is uniquely determined by $\sigma$, and we denote it by $\chi_\sigma$.
Fix an isomorphism $\C\cong\Qlb$ for each $\l$. Then each $\sigma\in\Hom(E,\C)$ induces a map $E\tensor\Ql\to\Qlb$. If the Galois representation $\rho_\l$ takes values in $\Gf(E\tensor\Ql)$, then, for each $\sigma\in\Hom(E, \C)$, we can define $\sig\rho_\l$ to be the composition of $\rho_\l$ with the map $\sigma\:\Gf(E\tensor\Ql)\to \Gf(\Qlb)$. By the Chebotarev density theorem, a pair $(\sigma, \chi)$ is an inner twist if and only if $\sig\rho_\l \simeq \prepower{\sigma_0}\rho_\l\otimes\chi$, where we view $\chi$ as a Galois character.
\begin{proposition}[\cite{Ribet-twists}*{Prop.\ 3.2}]
If $\sigma\in\Gamma$, then $\sigma(E)\sub E$.
\end{proposition}
\begin{proof}
Let $(\sigma, \chi)$ be an inner twist. Comparing the similitudes of $\sig\rho_\l$ and $\prepower{\sigma_0}\rho_\l\otimes\chi$, we see that $\chi^2 = \sigma(\varepsilon)\cdot\varepsilon\ii$. Thus $\chi$ takes values in the field $\Q(\varepsilon)\sub E$. But then
\[\sigma(a_p) = \chi(p) a_p \in E\]
for all $p\nmid\l N$. Applying the same argument with a different prime $\l$, it follows that $\sigma(a_p)\in E$ for all $p\nmid N$, and hence that $\sigma(E)\sub E$.
\end{proof}
In particular, $\Gamma$ is a subset of $\Aut(E/\Q)$. Moreover, the inner twists $(\sigma, \chi_\sigma)$ form a group under the multiplication
\[(\sigma, \chi_\sigma)\cdot(\tau, \chi_\tau) = (\sigma\tau, \chi_\sigma\cdot \sigma(\chi_\tau)).\]
Thus, $\Gamma$ is a subgroup of $\Aut(E/\Q)$.
\begin{definition}\label{def:F}
Define $F = E^\Gamma$ to be the fixed field of $\Gamma$.
\end{definition}
In fact, $E/F$ is an abelian Galois extension \cite[Prop.\ 1.7]{momose}. Each character $\chi_\sigma$ can be regarded as character of $\Ga\Q$. Its kernel is thus an open subgroup $H_\sigma$ of $\Ga\Q$.
\begin{definition}\label{def:K}
Let $H = \bigcap_{\sigma\in\Gamma}H_\sigma$, and let $K$ be the corresponding Galois extension of $\Q$.
\end{definition}
\begin{remark}\label{rem:character-non-trivial}
If the Nebentypus character $\varepsilon$ is non-trivial, then, using the Petersson inner product, $(c, \varepsilon\ii)$ is always an inner twist, where $c$ denotes complex conjugation. In particular, group $H$ includes the kernel of $\varepsilon$.
\end{remark}
Recall from \Cref{thm:image-contains-sp4} that $\LL'$ is the set of primes $\l\ge 5$ such that $\rho_\l|_{\Ql}$ is de Rham, and crystalline if $\l\nmid N$.
\begin{lemma}\label{lem:small-trace}
For all but finitely many primes $\l\in\LL'$, $\rho_\l|_K$ descends to a representation
\[\rho_\l|_K\:\Gal(\Qb/K)\to \Gf(F\tensor\Ql)\]
that is defined over $F\tensor\Ql$.
\end{lemma}
\begin{proof}
By part $(ii)$ of \Cref{thm:image-contains-sp4}, the residual representation $\orho_\l|_K$ is irreducible for all but finitely many primes $\l\in\LL'$. Hence, by the proof of \Cref{cor:field-of-def}, it is enough to show that $\Tr\rho_\l(\Gal(\Qb/K))\sub F\tensor\Ql$. By the Chebotarev density theorem, it is enough to show that for all primes $p\nmid \l N$ that split completely in $K$, we have $\Tr\rho_\l(\Frob_p) \in F$. But for such primes, if $\sigma\in \Gal(E/F) = \Gamma$, we have $\Tr\rho_\l(\Frob_p) = a_p$ and $\sigma(a_p) = a_p\chi_\sigma(p) = a_p$. The result follows.
\end{proof}
\begin{definition}\label{def:final-L}
Define $\LL\sub\LL'$ to be the set of primes in $\LL'$ such that $\rho_\l|_K$ is defined over $F\tensor\Ql$.
\end{definition}
In particular, by \Cref{thm:image-contains-sp4} and \Cref{lem:small-trace}, the set $\LL$ has density $1$, and contains all but finitely many primes if $k_2>2$.
\begin{definition}\label{def:G}
Let $\G$ be the group scheme over $\Z$ such that, for each $\Z$-algebra $R$, we have
\[
\G(R) = \set{(g, \nu)\in \Gf(\O_F\tensor_\Z R)\times R\t : \simil(g) = \nu^{k_1 + k_2 - 3}}.
\]
\end{definition}
By construction, if $\l\in\LL$, then $\rho_\l|_K$ extends to a representation
\[\wrho_{\l}\:\Gal(\Qb/K)\to\G(\Ql)\]
such that the projection to $\Gf(\O_F\tensor_\Z\Ql)$ is $\rho_\l|_K$, and the projection to $\Ql\t$ is the cyclotomic character. Moreover, by \cite{serre-mod-p-lattices}*{Thm.\ 5.2.1}, we can conjugate this representation to take values in $\G(\Zl)$.
We give a second interpretation of the number field $F$. For each prime $p\nmid N$, let $b_p$ denote the coefficient of $X^2$ in the characteristic polynomial of $\rho_\l(\Frob_p)$. Note that $b_p=a_p^2-a_{p^2}-p^{k_1+k_2-4}$ is independent of $\l$.
\begin{lemma}\label{lem:b_q}
Let $F_0 = \Q(\{\frac{b_p}{\varepsilon(p) }:p\nmid N\}) $. Then $F=F_0$.
\end{lemma}
\begin{proof}
Fix a prime $\l\in \LL'$ such that $\rho_\l$ is defined over $E\tensor\Ql$. Now, $F_0$ is the field generated by $\Tr\std\rho_\l(\Frob_p)$, where $\std\rho_\l$ is the standard representation obtained by composing $\rho_\l$ with the maps
\[\Gf\to \PGSp_4\xrightarrow{\sim}\SO_5\to\GL_5.\]
Indeed, we have \[\wedge^2\rho_\l\tensor\simil\ii \simeq \std\rho_\l \+ 1,\]
and
\[\Tr\std\rho_\l(\Frob_p) = \frac{b_p}{\simil\rho_\l(\Frob_p)} -1 = \frac{b_p}{p^{k_1 + k_2 -3}\varepsilon(p)} -1.\]
By \Cref{thm:image-contains-sp4}, changing our choice of $\l$ if necessary, we can assume that the projection $\std\prepower{\sigma_0}\rho_\l$ is irreducible for all but finitely many primes $\l\in\LL'$. Indeed, for all but finitely many $\l\in\LL'$, the image of the residual representation $\prepower{\sigma_0}\orho_\l$ contains $\Sp_4(\Fl)$. Hence, the image of $\std\prepower{\sigma_0}\orho_\l$ contains $\SO_5(\Fl)$, which is an irreducible subgroup of $\GL_5(\Fl)$. Hence, $\std\prepower{\sigma_0}\rho_\l$ is irreducible.
In general, if $\rho_1, \rho_2\:G\to \Gf(\Qlb)$ are representations, then the projective representions
\[\Proj\rho_1,\ \Proj\rho_2\: G\to \PGSp_4(\Qlb)\]
are isomorphic if and only if $\rho_1$ and $\rho_2$ are character twists of each other. Hence, $\sigma\in\Gamma$ if and only if the two projective representations \[\Proj\prepower{\sigma_0}\rho_\l,\ \Proj\sig\rho_\l\:\Ga\Q\to \PGSp_4(\Qlb)\]
are isomorphic. Via the exceptional isomorphism, it follows that $\sigma\in\Gamma$ if and only if
$\std\prepower{\sigma_0}\rho_\l$ and $\std\sig\rho_\l$ are isomorphic as $\SO_5$-valued representations. Following the argument of \cite{Ramakrishnan-appendix}*{p.\ 35}, we see that $\std\prepower{\sigma_0}\rho_\l$ and $\std\sig\rho_\l$ are isomorphic as $\SO_5$ representations if and only if they are isomorphic as $\GL_5$ representations. Indeed, both representations are irreducible, and the claim follows from the lemma on pp.\ 34 of \cite{Ramakrishnan-appendix}.
By the Brauer--Nesbitt theorem, we find that $\sigma\in\Aut(E/\Q)$ is an element of $\Gamma=\Aut(E/F)$ if and only if $\std\prepower{\sigma_0}\rho_\l$ and $\std\sig\rho_\l$ have the same trace, which, by the Chebotarev density theorem, is equivalent to having $\sigma(\Tr\std\rho_\l(\Frob_p)) = \sigma_0(\Tr\std\rho_\l(\Frob_p))$ for all $p\nmid \l N$. Running the argument again with a different choice of $\l$, we see that $\sigma\in\Gamma$ if and only if $\sigma$ fixes $\frac{b_p}{\varepsilon(p)}$ for all $p\nmid N$. Thus $\sigma\in \Aut(E/\Q)$ fixes $F$ if and only if $\sigma$ fixes $F_0$, so $F = F_0$.
\end{proof}
\section{The image of Galois}\label{sec:image}
In this section, we prove \Cref{thm:precise-image}. Let $f$ be a cuspidal Siegel modular eigenform of weights $(k_1, k_2)$, $k_1\ge k_2\ge 2$, level $N$ and character $\varepsilon$. Assume, as always, that $f$ is not CAP, endoscopic, CM, RM or a symmetric cube lift. Let $E=\Q(\{a_p : p\nmid N\},\varepsilon)$ be be the coefficient field of $f$, and let $F, K$ be the number fields defined in \Cref{def:F} and \Cref{def:K}. Let $\LL$ be the set of primes defined in \Cref{def:final-L} and, for $\l\in\LL$, write
\[\wrho_{\l}\:\Gal(\Qb/K)\to\G(\Zl)\]
for the Galois representation defined just after \Cref{def:G}.
\subsection{Open image}\label{sec:open-image}
In this subsection, we prove part $(i)$ of \Cref{thm:precise-image}. Since the image of $\wrho_\l$ surjects onto $\Zl\t$, to show that the image of $\wrho_\l$ is open in $\G(\Zl)$, it is equivalent to show that the image of $\rho_\l|_K$ is an open subgroup of
\[G_\l := \set{g\in \Gf(\O_F\tensor_\Z\Zl) : \simil(g)\in \Zl^{\times(k_1 + k_2 - 3)}}.\]
Fix a prime $\l\in\LL$ and write $F\tensor\Ql = \prod_{\lambda\mid \l}F_\lambda$, where the product is over the primes $\lambda$ of $F$ above $\l$. Fix a prime $\lambda\mid\l$. Denote by $\rho_\lambda|_K$ the representation
\[\rho_\lambda|_K\:\Gal(\Qb/K)\to \Gf(F_\lambda)\]
obtained via the projection $F\tensor\Ql\to F_\lambda$, and let
\[\Proj\rho_\lambda|_K\:\Gal(\Qb/K)\to\PGSp_4(F_\lambda)\]
be the associated projective Galois representation. Let $H_\lambda\adj$ be the image of $\Proj\rho_\lambda|_K$.
\begin{lemma}
$H_\lambda\adj$ is Zariski dense in $\PGSp_4(F_{\lambda})$, where $\PGSp_4$ is viewed as an algebraic group over $F_{\lambda}$.
\end{lemma}
\begin{proof}
By the assumption that $\l\in\LL$, $\rho_\lambda$ is irreducible, and hence $\rho_\lambda|_K$ is irreducible by \cite{weiss2018image}*{Lem.\ 5.9}. Hence, by \cite{weiss2018image}*{Cor.\ 5.6, Rem.\ 5.5}, the Zariski closure of $H_\lambda\adj$ is $\PGSp_4(F_\lambda)$.
\end{proof}
\begin{lemma}\label{lem:open-image}
$H_\lambda\adj$ is an open subgroup of $\PGSp_4(F_{\lambda})$.
\end{lemma}
\begin{proof}
We argue as in \cite[Prop.\ 3.16]{conti2016galois}. $\PGSp_4$ is an absolutely simple, connected adjoint group over $F_{\lambda}$ and the adjoint representation of $\PGSp_4$ is irreducible. Since $H\adj_\lambda$ is Zariski dense in $\PGSp_4(F_{\lambda})$, by \cite[Thm.\ 0.7]{Pink}, there is a model $H$ of $\PGSp_4$, defined over a closed subfield $L\sub F_{\lambda}$, such that $H\adj_\lambda$ is an open subgroup of $H(L)$. Moreover, by \cite[Prop.\ 0.6]{Pink}, the field $L$ is exactly the trace field of $H_\lambda\adj$, i.e.\ we have $L = F_{\lambda}$, and hence $H = \PGSp_4$. It follows that $H\adj_\lambda$ is an open subgroup of $\PGSp_4(F_{\lambda})$.
\end{proof}
\begin{lemma}\label{lem:image-open-lambda}
The image of $\rho_\lambda|_K$ is conjugate to an open subgroup of
\[G_\lambda := \set{g\in \Gf(\O_{F_{\lambda}}) : \simil(g) \in \Zl^{\times(k_1 + k_2 -3)}}.\]
\end{lemma}
\begin{proof}
Up to conjugation, we may assume that the image $H_\lambda$ of $\rho_\lambda|_K$ is contained in $G_\lambda$. Arguing again as in \cite[Prop.\ 3.16]{conti2016galois}, we see that $H_\lambda$ must contain an open subgroup of $\Sp_4({F_{\lambda}})$. Indeed, by \Cref{lem:open-image}, the projective image $H\adj_\lambda$ of $H_\lambda$ is an open subgroup of $\PGSp_4(F_{\lambda})$. Since the map $\Sp_4(F_{\lambda})\to \PGSp_4(F_{\lambda})$ has degree $2$, and since $H_\lambda\cap \Sp_4(F_{\lambda})$ surjects onto $H_\lambda\adj\cap \PGSp_4(F_{\lambda})$, $H_\lambda$ must contain an open subgroup of $\Sp_4({F_{\lambda}})$. In other words, $H_\lambda$ contains a principal congruence subgroup of $\Sp_4(\O_{F_{\lambda}})$. Thus, $H_\lambda\cap \Sp_4(\O_{F_{\lambda}})$ is open in $\Sp_4(\O_{F_{\lambda}})$. Since the similitude of $\rho_\lambda$ surjects onto $\Zl^{\times(k_1 + k_2 -3)}$, it follows that $H_\lambda$ is open in $G_\lambda$.
\end{proof}
\begin{proof}[Proof of Theorem $\ref{thm:precise-image}~(i)$]
We argue as in \cite{momose}*{Thm.\ 4.1}. We first show that, if $\lambda_1, \lambda_2$ are two distinct primes of $F$ above $\l$, then the representations $\rho_{\lambda_1}|_K$ and $\rho_{\lambda_2}|_K$ are not isomorphic when restricted to any finite extension.
For each $i$, let $\widetilde{\lambda}_i$ be a prime of $E$ above $\lambda_i$. Then, from the diagram
\[\begin{tikzcd}
\Gal(\Qb/K) \arrow[r, "\rho_{\widetilde\lambda_{i}}|_K"] \arrow[rd, "\rho_{\lambda_{i}}|_K"'] & \Gf(E_{\widetilde\lambda_i}) \\
& \Gf(F_{\lambda_i}) \arrow[u, hook] \end{tikzcd}\]
it is clear that $\rho_{\lambda_i}|_K\simeq\rho_{\widetilde\lambda_i}|_K$ when viewed as representations valued in $E_{\widetilde\lambda_i}$.
The primes $\widetilde\lambda_1, \widetilde\lambda_2$ correspond to two embeddings $\sigma_1, \sigma_2\in \Hom(E, \Qlb)$. Moreover, since $\lambda_1\ne\lambda_2$, $\sigma_1|_F\ne\sigma_2|_F$. For each $i$, $\prepower{\sigma_i}\rho_\l\simeq\rho_{\widetilde\lambda_i}$. Suppose that $\prepower{\sigma_1}\rho_\l|_L\simeq\prepower{\sigma_2}\rho_\l|_L$ for some finite extension $L/\Q$. It follows that $\prepower{\sigma_1}\rho_\l\tensor\chi\simeq\prepower{\sigma_2}\rho_\l$ for some Dirichlet character $\chi$. Hence, by the definition of $F$ as the field fixed by the inner twists, we must have $\sigma_1|_F=\sigma_2|_F$, a contradiction.
We can now apply the analysis of \cite{ribet1976galois}*{Ch.\ IV, \S4}. Let $\overline \h = \h\tensor\Qlb$, where $\h$ is the Lie algebra of the image of $\rho_\l$. Let $\overline\g = \g\tensor\Qlb$, where $\g$ is the Lie algebra of $G_\l$.
By \Cref{lem:image-open-lambda}, the projection of $\overline\h$ to $\gsp_4(V_\sigma)$ is surjective. By the above argument, $V_{\sigma_1}$ and $V_{\sigma_2}$ are not isomorphic as $\overline\h$-modules for any distinct $\sigma_1, \sigma_2\in \Hom(F, \C)$. It follows that the projections of $\overline\h$ and $\overline \g$ onto $\gsp_4(V_{\sigma_1})\times\gsp_4(V_{\sigma_2})$ are the same. By the Lie algebra version of Goursat's lemma \cite{ribet1976galois}*{Lemma, p.790}, it follows that $\overline\h = \overline\g$. Since $\h\sub\g$, it follows that $\h = \g$, i.e. that the image of $\rho_\l|_K$ is an open subgroup of $G_\l$. Hence, the image of $\wrho_\l$ is an open subgroup of $\G(\Zl)$.
\end{proof}
\subsection{The precise image for almost all primes}
In this section, we prove part $(ii)$ of \Cref{thm:precise-image}. As in the previous section, to show that $\wrho_\l$ surjects onto $\G(\Zl)$, it is sufficient to show that $\rho_\l|_K$ surjects onto $G_\l$. Moreover, as the following lemma shows, it is sufficient to show that the residual representation $\orho_\l|_K$ surjects onto
\[\overline G_\l := \set{g\in \Gf(\O_F\tensor_\Z\Fl): \simil(g)\in \Fl^{\times(k_1 + k_2 -3)}}.\]
\begin{lemma}\label{lem:ribet-product}
Let $\l \ge 5$ be prime, and let $F_1, \ldots, F_t$ be finite unramified extensions of $\Ql$ with rings of integers $\O_1, \ldots, \O_t$ and residue fields $\F_1, \ldots, \F_t$. Let $H$ be a closed subgroup of $\Sp_4(\O_1)\times\cdots \times \Sp_4(\O_t)$, which surjects onto $\PSp_4(\F_1)\times \cdots \times \PSp_4(\F_t)$. Then $H = \Sp_4(\O_1)\times\cdots \times \Sp_4(\O_t)$.
\end{lemma}
\begin{proof}
This is a generalisation of \cite{Ribet75}*{Thm.\ 2.1} and \cite{Loeffler-adelic}*{Lem.\ 1.1.1}. Since no proper subgroup of $\Sp_4(\F_1)$ surjects onto $\PSp_4(\F_1)$, it follows that $H$ surjects onto $\Sp_4(\F_1)\times \cdots \times \Sp_4(\F_t)$. The result now follows from \cite{DKR}*{Lem.\ 2}.
\end{proof}
\begin{corollary}\label{cor:gl}
Let $\l \ge 5$ be prime that is unramified in F, and suppose that $H$ is a closed subgroup of $G_\l$ that surjects onto $\overline G_\l$. Then $H = G_\l$.
\end{corollary}
\begin{proof}
Let $H'$ be the commutator subgroup of $H$ and let $H_0 = H\cap \Sp_4(\O_F\tensor_\Z\Zl)$. Clearly, $H_0 \supseteq H'$. Since $\Sp_4(\F)$ is a perfect group (i.e.\ a group that is equal to its own commuatator subgroup) whenever $\F$ is a field order at least $5$, we see that $H_0$ surjects onto $\Sp_4(\O_F\tensor_\Z\Fl)$. Therefore, by \Cref{lem:ribet-product}, $H_0 = \Sp_4(\O_F\tensor_\Z\Zl)$. Hence, $H$ is a closed subgroup of $G_\l$ that contains $\Sp_4(\O_F\tensor_\Z\Zl)$ and surjects onto $\Zl^{\times(k_1 + k_2 -3)}$. Thus, $H = G_\l$.
\end{proof}
Recall \Cref{lem:b_q}, that the field $F$ is equal to $\Q(\{\frac{b_p}{\varepsilon(p)} :p\nmid N\})$, where $b_p$ is the coefficient of $X^2$ in the characteristic polynomial of $\rho_\l(\Frob_p)$.
\begin{lemma}\label{lem:untwisted}
There exists a prime $q\nmid N$ such that $F = \Q(b_q)$.
\end{lemma}
\begin{proof}
We argue as in \cite[Thm.\ 3.1]{Ribet85}. Let $H_\l$ be the image of $\rho_\l|_K$ and, for an element $g\in H_\l$, let $b(g)$ denote the coefficient of $X^2$ in its characteristic polynomial. Consider the set
\[U = \set{g\in H_\l : \frac{b(g)}{\varepsilon(g)}\text{ generates }F\tensor\Ql \text{ as a }\Ql\text{-algebra}}.\]
Then $U$ is an open subset of $H_\l$, which is in turn an open subgroup of $G_\l$ by part $(i)$ of \Cref{thm:precise-image}. Since $G_\l$ contains elements $g$ such that $\frac{b(g)}{\varepsilon(g)}$ generates $F\tensor\Ql$ as a $\Ql$-algebra, we see that $U$ is the intersection of two non-empty open subgroups of $G_\l$, so is itself non-empty. Since $U$ is closed under conjugation, by the Chebotarev density theorem, there exists a rational prime $q\nmid\l N$ that splits completely in $K$ such that $\rho_\l(\Frob_q)\in U$. Hence, $\frac{b_q}{\varepsilon(q)}$ generates $F\tensor\Ql$ as a $\Ql$-algebra, so $F = \Q(\frac{b_q}{\varepsilon(q)})$. By \Cref{rem:character-non-trivial}, $\varepsilon(q) = 1$, so $F = \Q(b_q)$.
\end{proof}
\begin{lemma}\label{lem:mod-lambda-image}
For all but finitely many primes $\l\in\LL$, for all primes $\lambda\mid\l$ of $F$, the image of $\orho_\lambda|_K$ is exactly
\[\overline G_\lambda :=\set{g\in\Gf(\F_\lambda ): \simil(g) \in \Fl^{\times(k_1 + k_2 -3)}}.\]
\end{lemma}
\begin{proof}
We follow a similar argument to \cite[Section 4.7]{Dieulefait2002maximalimages}. Since the similitude of $\orho_\lambda|_K$ surjects onto $\Fl^{\times(k_1 + k_2 -3)}$, it is enough to show that the image of $\orho_\lambda|_K$ contains $\Sp_4(\F_\lambda)$ for all $\lambda\mid\l$, for all but finitely many primes $\l\in\LL$.
Let $\l\in\LL$ and, for each prime $\lambda\mid\l$, let $\overline H_\lambda$ be the image of $\orho_\lambda|_K$ and let $\overline H_\lambda\adj\sub \PGSp_4(\F_{\lambda})$ be the projective image of $\orho_\lambda|_K$. By \Cref{thm:image-contains-sp4}, we may assume that $\overline H_\lambda\adj$ contains $\PSp_4(\Fl)$. Let $\F$ be the largest subfield of $\F_\lambda$ such that $\overline H_\lambda\adj$ contains $\PSp_4(\F)$. We will show that $\F = \F_\lambda$.
Let $q$ be the prime from \Cref{lem:untwisted}. Suppose further that $\l\ne q$, that $\l$ does not divide the discriminant of ${b_q}$ and that $a_q = \Tr\rho_\lambda(\Frob_q)$ is invertible modulo $\l$; these conditions exclude only finitely many primes $\l\in\LL$. Then $\F_\lambda = \Fl(b_q)$. Suppose that $\F\subsetneq \F_{\lambda}$. Then every element of $\overline H_\lambda\adj$ has a lift to $\overline H_\lambda$ with characteristic polynomial in $\F[X]$. Since the characteristic polynomial of $\orho_\lambda(\Frob_q)$ is $X^4 - a_qX^3 + b_qX^2 - a_qq^{k_1 + k_2 - 3}X + q^{2(k_1 + k_2 -3)}\pmod {\lambda}$, we see that there exists an element $t\in \F_{\lambda}\t$ such that $ta_q, t^2 b_q, t^3a_q\in \F$. Since $a_q$ is invertible modulo $\l$, it follows that $t^2 = \frac{t^3 a_q}{ta_q}\in \F\t$, and hence that $b_q\in\F$, contradicting the fact that $\F_{\lambda} = \Fl({b_q})$.
It follows that $\overline H_\lambda\adj$ contains $\PSp_4(\F_{\lambda})$ and hence that $\overline H_\lambda$ contains $\Sp_4(\F_{\lambda})$. Thus, $\overline H_\lambda = \overline G_\lambda$.
\end{proof}
Part $(ii)$ of \Cref{thm:precise-image} now follows inductively from Goursat's Lemma:
\begin{lemma}[Goursat's Lemma, \cite{Ribet75}*{Lem.\ 3.2}]\label{lem:goursat}
Let $G_1, G_2$ be groups and let $H$ be a subgroup of $G_1\times G_2$ for which the two projections $p_i\:H\to G_i$ are surjective. Let $N_1$ be the kernel of $p_2$ and let $N_2$ be the kernel of $p_1$. Then the image of $H$ in $G_1/N_1\times G_2/N_2$ is the graph of an isomorphism $G_1/N_1\xrightarrow{\sim} G_2/N_2$.
\end{lemma}
\begin{proof}[Proof of Theorem $\ref{thm:precise-image}~(ii)$]
We argue as in the proof of $(3.1)$ of \cite{Ribet75}. If $\lambda_1, \lambda_2$ are two primes of $F$ above $\l$, then the image $\overline H$ of $\orho_{\lambda_1}|_K\times\orho_{\lambda_2}|_K$ is a subgroup of $\overline G_{\lambda_1}\times\overline G_{\lambda_2}$ and, by \Cref{lem:mod-lambda-image}, we can assume that the image of each of the two projections to $\overline G_{\lambda_i}$ is surjective.
Let $N_2$ and $N_1$ be the kernels of the projections of $\overline H$ onto $\overline G_{\lambda_1}$ and $\overline G_{\lambda_2}$. Then, by Goursat's lemma, the image of $\overline H$ in $\overline G_{\lambda_1}/N_1\times \overline G_{\lambda_2}/N_2$ is the graph of an isomorphism $\overline G_{\lambda_1}/N_1\xrightarrow{\sim}\overline G_{\lambda_2}/N_2$.
Let $\overline G$ be the projection of $\overline G_\l$ onto $\overline G_{\lambda_1}\times\overline G_{\lambda_2}$. Since the kernel of $\overline G\to \overline{G}_{\lambda_1}$ is $\Sp_4(\F_{\lambda_2})$ we have $N_2\le \Sp_4(\F_{\lambda_2})$ and similarly $N_1 \le \Sp_4(\F_{\lambda_1})$. By the isomorphism $\overline G_{\lambda_1}/N_1\xrightarrow{\sim}\overline G_{\lambda_2}/N_2$ we have $N_2 = \Sp_4(\F_{\lambda_2})$ if and only if $N_1 = \Sp_4(\F_{\lambda_1})$, in which case $\overline H = \overline G$.
Thus, if $\overline H$ is a proper subgroup of of $\overline G$, then, for each $i$, $N_i$ is a proper normal subgroup of $\Sp_4(\F_{\lambda_i})$, so $N_i\sub\{\pm I\}$. The isomorphism ${\overline G_{\lambda_1}}/N_1 \xrightarrow{\sim} {\overline G_{\lambda_2}}/N_2$ now implies that $\F_{\lambda_1} = \F_{\lambda_2}$ and that there are elements $\sigma\in \Gal(\F_{\lambda_1}/\Fl)$ and $S \in \Gf(\F_{\lambda_1})$ such that, for each $(g_1, g_2)\in \overline G$, there is a scalar $\chi(g_1, g_2)$ such that $g_2 = \chi(g_1, g_2) \cdot\sigma (S g_1 S\ii)$. Since $\simil\orho_{\lambda_1}|_K=\simil\orho_{\lambda_2}|_K \in \Fl^{\times}$, it follows that $\chi(g_1, g_2)^2 = 1$ for all $(g_1, g_2)\in\overline G$. Hence, computing the characteristic polynomials of $g_2$ and $\chi(g_1, g_2) \cdot\sigma (S g_1 S\ii)$ and equating the coefficient of $X^2$, we find that
\[b(g_2) = \sigma(b(g_1))\]
for all $g_1, g_2\in \overline G$, where $b(g_i)$ denotes the coefficient of $X^2$ in the characteristic polynomial of $g_i$.
This contradicts \Cref{lem:untwisted}, which states there exists a prime $q$ such that $F = \Q(b_q)$, i.e.\ if $\l$ is large enough, the element $\Tr\orho_\l(\Frob_q)\in \O_F\tensor_\Z\Fl$ generates $\O_F\tensor_\Z\Fl$ as an $\Fl$-algebra.
It follows that $\overline H$ of $\orho_{\lambda_1}|_K\times\orho_{\lambda_2}|_K$ surjects onto $\overline G$. Hence, by induction, the image of $\orho_{\l}|_K$ is $\overline G_\l$, and the result follows from \Cref{cor:gl}.
\end{proof}
\subsection{Adelic large image}
Finally, we prove part $(iii)$ of \Cref{thm:precise-image}. We begin with some group theoretic results, which are mostly generalisations of \cite{serre-abelian}*{Ch.\ IV} and \cite{Loeffler-adelic}*{Sec.\ 1}.
\begin{definition}[\cite{serre-abelian}*{IV-25}]
Let $Y$ be a profinite group, and let $\Sigma$ be a non-abelian finite simple group. We say that $\Sigma$ \emph{occurs in $Y$} if there exist closed subgroups $Y_1, Y_2$ of $Y$ such that $Y_1\lhd Y_2$ and $Y_2/Y_1\cong \Sigma$. We write $\Occ(Y)$ for the set of non-abelian finite simple groups occuring in $Y$.
\end{definition}
\begin{lemma}[\cite{serre-abelian}*{IV-25}]\label{lem:serre-occ}
If $Y = \varprojlim Y_a$ and each $Y\to Y_a$ is surjective, then $\Occ(Y) = \bigcup \Occ(Y_a)$. If $Y$ is an extension of $Y'$ and $Y''$, then $\Occ(Y) = \Occ(Y')\cup \Occ(Y'')$.
\end{lemma}
\begin{lemma}\label{lem:occ-sp4}
Let $L$ be a finite extension of $\Ql$ for some prime $\l$, with ring of integers $\O$, uniformiser $\varpi$ and residue field $\F$. We have $\Occ(\GSp_4(\O)) = \Occ(\Sp_4(\O)) = \Occ(\PSp_4(\F))$.
\end{lemma}
\begin{proof}
We argue as in \cite{kani-appendix}*{Lem.\ 10}. Since $\Gf(\O)/\Sp_4(\O)$ is abelian, the first equality follows from \Cref{lem:serre-occ}. Similarly, $ \Occ(\PSp_4(\F)) = \Occ(\Sp_4(\F))$. Since $\Sp_4(\O) = \varprojlim_n\Sp_4(\O/\varpi^n)$, by \Cref{lem:serre-occ} again, we have $\Occ(\Sp_4(\O))= \bigcup_n\Occ(\Sp_4(\O/\varpi^n))$. It remains to show that $\Occ(\Sp_4(\O/\varpi^n)) = \Occ(\Sp_4(\F))$ for each $n$.
Observe that the kernel $X$ of
\[\Sp_4(\O/\varpi^n)\to \Sp_4(\F)\]
is an $\l$-group. Indeed, any matrix in the kernel can be written as $I + \varpi A$ where $A\in \M_4(\O/\varpi^n)$. Hence, in $\M_4(\O/\varpi^n)$, we have $(I+ \varpi A)^{\l^n} = I$, so every element of $X$ has order a power of $\l$.
Since every $\l$-group is solvable, it follows that $\Occ(X) = \emptyset$, so $\Occ(\Sp_4(\O/\varpi^n)) = \Occ(\Sp_4(\F))$ by \Cref{lem:serre-occ}.
\end{proof}
\begin{corollary}\label{cor:occ}
If $p\ne 2, \l$ and $q = p^r$ for some $r$, then $\PSp_4(\F_q)\notin \Occ(\Sp_4(\O))$.
\end{corollary}
\begin{proof}
By \Cref{lem:occ-sp4}, $\Occ(\Gf(\O)) = \Occ(\PSp_4(\F))$. The result is now immediate from the classification of the maximal subgroups of $\PSp_4(\F)$ \cite{mitchell1914subgroups}.
\end{proof}
Recall that $\G$ is the group scheme whose $R$ points are
\[\G(R) = \set{(g, \nu)\in \GSp_4(\O_F\tensor_\Z R)\times R\t : \simil(g) = \nu^{k_1 + k_2 -3}}.\]
Let
\[\G^\circ(R) = \Sp_4(\O_F\tensor_\Z R).\]
In particular, we have $\G^\circ(\Zl) = \prod_{\lambda\mid\l}\Sp_4(\O_{F_\lambda})$, where the product is over primes $\lambda$ of $F$ above $\l$, and $\O_{F_\lambda}$ is the ring of integers of the completion $F_\lambda$ of $F$.
\begin{theorem}\label{thm:adelic-image-circ}
Fix a set of primes $\LL$. Let $U^\circ$ be a closed compact subgroup of $\G^\circ(\widehat{\Q}_\LL)$ such that:
\begin{itemize}
\item for every prime $\l\in\LL$, the projection of $U^\circ$ to $\G^\circ(\Ql)$ is open in $\G^\circ(\Ql)$;
\item for all but finitely many primes $\l\in\LL$, the projection of $U^\circ$ to $\G^\circ(\Ql)$ is $\G^\circ(\Zl)$.
\end{itemize}
Then $U^\circ$ is open in $\G^\circ(\widehat{\Q}_\LL)$.
\end{theorem}
\begin{proof}
This is a generalisation of \cite{serre-abelian}*{Main Lemma, IV-19} and \cite{Loeffler-adelic}*{Thm.\ 1.2.2}. Let $S\sub \LL$ be a finite set of primes containing $\{2, 3,5\}\cap\LL$ and the finitely many primes such that the projection $U^\circ$ to $\G^\circ(\Ql)$ is not $\G^\circ(\Zl)$.
First, note that the projection $U^\circ_S$ of $U^\circ$ to $\prod_{\l\in S}\G^\circ(\Ql)$ is open. We argue as in \cite{serre-abelian}*{Lem.\ 4 pp. IV-24}. Replacing $U^\circ$ with a finite index subgroup, we can assume that, for each $\l\in S$, the projection of $U^\circ$ to $\G^\circ(\Ql)$ is contained in the group of elements congruent to $1$ mod $\l$, i.e.\ is pro-$\l$. It follows that $U^\circ_S$ is pro-nilpotent, and hence is a product of its Sylow subgroups. Thus, $U^\circ_S\cong \prod_{\l\in S}U^\circ_\l$, where $U^\circ_\l$ is the projection of $U^\circ$ to $\G^\circ(\Ql)$. Since each $U^\circ_\l$ is open in $\G^\circ(\Ql)$ by assumption, it follows that $U^\circ_S$ is open in $\prod_{\l\in S}\G^\circ(\Ql)$.
To conclude, it is sufficient to show that $U^\circ$ contains an open subgroup of $\G^\circ(\widehat \Q_\LL)$. Since $U^\circ_S$ is open in $\prod_{\l\in S}\G^\circ(\Ql)$, it is enough to show that $\prod_{\l\in\LL\setminus S}\G^\circ(\Zl)\sub U^\circ$. To show this, it is enough to show that $U^\circ$ contains $\G^\circ(\Zl) = \Sp_4(\O_F\tensor\Zl)$ for every $\l\in \LL\setminus S$.
For each $\l\in \LL\setminus S$, let $H_\l = U^\circ \cap \G^\circ(\Zl)$. By assumption, the projection of $U^\circ$ to $\G^\circ(\Ql)$ is $\G^\circ(\Zl)$, which in turn surjects onto $\PSp_4(\F_\lambda)$ for each $\lambda\mid\l$. Hence, $\PSp_4(\F_\lambda)\in \Occ(U^\circ)$. On the other hand, $U^\circ/H_\l$ is isomorphic to a closed subgroup of $\prod_{q\in \LL\setminus\{\l\}}\G^\circ(\Q_q)$, so by \Cref{cor:occ}, $\PSp_4(\F_\lambda)\notin\Occ(U^\circ/H_\l)$. It follows from \Cref{lem:serre-occ} that $\PSp_4(\F_\lambda)\in \Occ(H_\l)$. Hence, $H_\l$ is a subgroup of $\prod_{\lambda\mid\l}\Sp_4(\O_{F_\lambda})$ whose projection to $\prod_{\lambda\mid\l}\PSp_4(\F_\lambda)$ is surjective. It follows from \Cref{lem:ribet-product} that $H_\l = \prod_{\lambda\mid\l}\Sp_4(\O_{F_\lambda}) = \G^\circ(\Zl)$. The result follows.
\end{proof}
\begin{theorem}\label{thm:adelic-image}
Fix a set of primes $\LL$. Let $U$ be a closed compact subgroup of $\G(\widehat{\Q}_\LL)$ such that:
\begin{itemize}
\item for every prime $\l\in\LL$, the projection of $U$ to $\G(\Ql)$ is open in $\G(\Ql)$;
\item for all but finitely many primes $\l\in\LL$, the projection of $U$ to $\G(\Ql)$ is $\G(\Zl)$;
\item the image of $U$ in $\widehat \Q_\LL\t$ is open.
\end{itemize}
Then $U$ is open in $\G(\widehat{\Q}_\LL)$.
\end{theorem}
\begin{proof}
We follow \cite{Loeffler-adelic}*{Thm.\ 1.2.3}. Let $U^\circ = U \cap \G^\circ(\widehat\Q_\LL)$. We claim that $U^\circ$ satisfies the hypotheses of \Cref{thm:adelic-image-circ}. Since $\G(\widehat\Q_\LL)/\G^\circ(\widehat\Q_\LL)\cong \widehat\Q_\LL\t$ is abelian, the group $U^\circ$ contains the closure of the commutator subgroup of $U$. When $\l\ge 3$, $\Sp_4(\O_F\tensor\Zl)$ is the closure of its commutator subgroup. Hence, if $\l\ge 3$ and $U$ surjects onto $\G(\Zl)$, then $U^\circ$ surjects onto $\G^\circ(\Zl) =\Sp_4(\O_F\tensor\Zl)$. When $\l=2$ the commutator subgroup of $\Sp_4(\O_F\tensor\Z_2)$ still has finite index. Hence, $U^\circ$ satisfies the hypotheses of \Cref{thm:adelic-image-circ}. Thus, $U$ contains an open subgroup of $\G^\circ(\widehat\Q_\LL)$. Since the image of $U$ in $\widehat \Q_\LL\t\cong \G(\widehat\Q_\LL)/\G^\circ(\widehat\Q_\LL)$ is open, it follows that $U$ is open in $\G^\circ(\widehat{\Q}_\LL)$.
\end{proof}
\begin{proof}[Proof of Theorem $\ref{thm:precise-image}~(iii)$]
The result now follows immediately from \Cref{thm:precise-image} parts $(i)$ and $(ii)$ and \Cref{thm:adelic-image}.
\end{proof}
\subsection{The image of \texorpdfstring{$\Ga\Q$}{Galois over Q}}
In \Cref{thm:precise-image}, we computed the image of $\wrho_\l$ for all but finitely many primes $\l\in\LL$. Suppose that $\l\in\LL$ is such that the image of $\wrho_\l$ is surjective. In this section, we use methods of E.\ Papier (see \cite{Ribet85}*{Thm.\ 4.1}) to compute the image of
\[\rho_\l\:\Ga\Q\to \Gf(\O_E\tensor_\Z\Zl).\]
Suppose that $(\sigma, \chi_\sigma)$ is an inner twist, and let $\sigma(\rho_\l)$ denote the representation obtained by composing $\rho_\l$ with $\sigma\:\O_E\tensor_\Z\Zl\to \O_E\tensor_\Z\Zl$. Then the representations $\sigma(\rho_\l)$ and $\rho_\l\tensor\chi_\sigma$ have the same trace. Since both are semisimple, it follows that they are isomorphic, so there is a matrix $X\in \Gf(E\tensor_\Q\Ql)$ such that
\[X\sigma(\rho_\l)X\ii = \rho_\l\tensor\chi_\sigma.\]
By definition, $\chi_\sigma|_K$ is trivial and $\sigma(\rho_\l|_K) = \rho_\l|_K$. It follows that $X$ commutes with the image of $\rho_\l|_K$. Since, for example, the image of $\rho_\l|_K$ contains $\Sp_4(\O_F\tensor_\Z\Zl)$, we see that $X$ is a scalar matrix, so there is an equality of matrices
\[\sigma(\rho_\l(\gamma)) = \rho_\l(\gamma)\chi_\sigma(\gamma)\]
for all $\gamma \in \Ga\Q$ and for all inner twists $(\sigma, \chi_\sigma)$.
Recall that the group structure on the inner twists is given by
\[(\sigma, \chi_\sigma)\cdot(\tau, \chi_\tau) = (\sigma\tau, \chi_\sigma\cdot \sigma(\chi_\tau)).\]
Hence, the map $\sigma\mapsto\chi_\sigma$ defines an element of $H^1(\Q, \Qb\t)$. By Hilbert's theorem $90$, this cohomology group is trivial, so, for each $\gamma \in \Ga\Q$, we can choose $\alpha(\gamma)\in E\t$ such that
\[\chi_\sigma(\gamma) = \frac{\sigma(\alpha(\gamma))}{\alpha(\gamma)}\]
for all $\sigma\in\Gamma$. Moreover, we can choose the elements $\alpha(\gamma)$ to be independent of $\l$, and so that $\alpha(\gamma)$ only depends on the image of $\gamma$ in $\Gal(K/\Q)$. Thus, there are exactly $[K:\Q]$ numbers $\alpha(\gamma)$ and, when $\l$ is large enough, we will have $\alpha(\gamma)\in \O_E\tensor_\Z\Zl$ for all $\gamma$. We deduce the following generalisation of \cite{Ribet85}*{Thm.\ 4.1}:
\begin{theorem}\label{thm:precise-Q-image}
For all but finitely many primes $\l\in\LL$, the image of $\rho_\l$ is generated by
\[G_\l = \set{g\in\Gf(\O_F\tensor_\Z\Zl) : \simil(g)\in \Zl^{\times(k_1 + k_2 -3)}}\]
together with the finite set of matrices
\[\dmat{\alpha(\gamma)}{\alpha(\gamma)}{\varepsilon(\gamma)/\alpha(\gamma)}{\varepsilon(\gamma)/\alpha(\gamma)}\]
where $\varepsilon$ is the character of the Siegel modular form $f$.
\end{theorem}
\begin{proof}
By definition, for all $\gamma\in \Ga\Q$,
\[\rho_\l(\gamma)\alpha(\gamma)\ii = \rho_\l(\gamma)\chi_\sigma(\gamma)\sigma(\alpha(\gamma))\ii = \sigma(\rho_\l(\gamma)\alpha(\gamma)\ii).\]
Hence, $\rho_\l(\gamma)\alpha(\gamma)\ii\in \Gf(\O_F\tensor_\Z\Zl)$. Taking similitudes, we see that $\varepsilon(\gamma)\alpha(\gamma)^{-2}\in(\O_F\tensor_\Z\Zl)\t$.
Moreover, we have an equality
\[\rho_\l(\gamma) = \dmat{\alpha(\gamma)}{\alpha(\gamma)}{\varepsilon(\gamma)/\alpha(\gamma)}{\varepsilon(\gamma)/\alpha(\gamma)}\br{\dmat{1}{1}{\alpha(\gamma)^2/\varepsilon(\gamma)}{\alpha(\gamma)^2/\varepsilon(\gamma)} \alpha(\gamma)\ii\rho_\l(\gamma)},\]
and the product in the second set of brackets belongs to $G_\l$. The result now follows from \Cref{thm:precise-image}.
\end{proof}
\begin{corollary}\label{cor:precise-Q-image}
For all but finitely many primes $\l\in\LL$, the image of $\rho_\l$ is the disjoint union of at most $[K:\Q]$ cosets
\[\coprod\dmat{\alpha(\gamma)}{\alpha(\gamma)}{\varepsilon(\gamma)/\alpha(\gamma)}{\varepsilon(\gamma)/\alpha(\gamma)}G_\l,\]
where $\gamma$ ranges over some subset of $\Gal(K/\Q)$.
\end{corollary}
\section{The Chebotarev density theorem}\label{sec:cdt}
Let $L/K$ be a Galois extension of number fields, with Galois group $G$. Let $M_K$ denote the set of primes of $K$ and, for each prime $\p\in M_K$ that is unramified in $L$, let $\Frob_\p\in G$ be a choice of Frobenius element. Let $C$ be a non-empty subset of $G$ that is stable under conjugation.
\begin{definition}\label{def:pi_c}
For any $x> 0$, define
\[\pi_C(x, L/K) := \#\set{\p\in M_K, \p\text{ unramified in }L : N_{K/\Q}(\p) \le x, \Frob_\p\in C}.\]
\end{definition}
The Chebotarev density theorem states that
\[\pi_C(x, L/K) \sim \frac{|C|}{|G|}\pi(x).\]
To obtain explicit bounds on the size of $\pi_C(x, L/K)$, we will require an effective version of the Chebotarev density theorem.
\subsection{Unconditional effective Chebotarev}
The following theorem is unconditional:
\begin{theorem}[\cite{lmo}*{Thm.\ 1.4}]\label{thm:unconditional-cdt}
Assume that $L/K$ is finite. There exist constants $c_1, c_2$ such that, if
\[\log x > c_1 (\log |\disc(L/\Q)|)(\log\log |\disc(L/\Q)|)(\log\log\log |6\disc(L/\Q)|),\]
then
\[\pi_C(x, L/K) \le c_2\frac{|C|}{|G|}\Li(x),\]
where $\Li(x)=\int_2^x \frac{dt}{\log t}$ is the logarithmic integral function.
\end{theorem}
Using this theorem, Serre \cite{ser81} shows how to deduce upper bounds for $\pi_C(x, L/K)$ assuming that $G=\Gal(L/K)$ is an $\l$-adic Lie group. Indeed, let $G$ be a compact $\l$-adic Lie group of dimension $D$ and let $C\sub G$ be a non-empty closed subset that is stable under conjugation.
\begin{definition}[\cite{ser81}*{Sec.\ 3}]
Let $C_n$ denote the image of $C$ in $G/\l^nG$. We say that $C$ has \emph{Minkowski dimension} $\dim_M(C) \le d$ if $|C_n| = O(\l^{nd})$ as $n\to \infty$.
\end{definition}
Now, if $s\in C$, then the centraliser $Z_G(s)$ of $s$ is a closed Lie subgroup of $G$, so has a well-defined dimension.
\begin{theorem}[\cite{ser81}*{Thm.\ 12}]\label{thm:cdt-serre}
Suppose that $\dim_M(C) \le d$, and set
\[r_C = \Inf_{s\in C}\dim\frac{G}{Z_G(s)}.\]
Then, for any $\epsilon >0$,
\[\pi_C(x, L/K) \ll \frac{x}{\log(x)^{1 + \alpha - \epsilon}},\]
where $\alpha = (D-d)/(D-r_C/2)$.
\end{theorem}
\subsection{Chebotarev density under the Generalised Riemann Hypothesis}
Assuming that the Generalised Riemann Hypothesis holds for $L$, there are stronger effective versions of the Chebotarev density theorem. The following version is due to Lagarias--Odlyzko \cite{lo}.
\begin{theorem}[\cite{ser81}*{Thm.\ 4}]\label{thm:conditional-cdt}
Assume that the Dedekind zeta function $\zeta_L(s)$
satisfies the Riemann Hypothesis. Then
\[\pi_C(x, L/K)= \frac{|C|}{|G|}\Li(x)+O\bigg(\frac{|C|}{|G|}x^{{1}/{2}}(\log |\disc(L/\Q)|+[L:\Q]\log x)\bigg).\]
\end{theorem}
Finally, if we further assume that $L/K$ is abelian, then Artin's holomorphicity conjecture is known to hold for $L/K$, and we obtain the following stronger result due to Zywina:
\begin{theorem}[\cite{zyw}*{Thm.\ 2.3}]\label{zmain}
Suppose that $L/K$ is finite and abelian. Assume that the Dedekind zeta function $\zeta_L(s)$
satisfies the Riemann Hypothesis. Then
\[ \pi_C(x, L/K)\ll \frac{| C|}{| G|}\frac{x}{\log x}+| C|^{1/2}[K:\Q]\frac{x^{1/2}}{\log x}
\log M(L/K),\]
where
\[M(L/K):=2[L:K]\disc(K/\Q)^{1/[K:\Q]}\prod_{p \in \mathcal P(L/K)}p.\]
\end{theorem}
Here, we define $\mathcal P(L/K)$ to be the set of rational primes $p$ that are divisible by some prime $\p$ of $K$ that ramifies in $L$.
In order to apply these theorems, it is helpful to have a bound on $\log|\disc(L/\Q)|$. We will use the following result:
\begin{proposition}[\cite{ser81}*{Prop.\ 5}]\label{prop:hensel}
Let $L/K$ be a finite extension of number fields. Then
\[\log |N_{K/\Q}(\disc(L/K))| \le ([L:\Q] - [K:\Q])\sum_{p\in\mathcal{P}(L/K)}\log p + [L:\Q]\log [L:K].\]
\end{proposition}
In order to estimate the size of $\pi_C(x, L/K)$, it is often more convenient to study the following weighted version:
\begin{definition}\label{def:weighted-set}
For any $x>0$, define
\[\widetilde\pi_{C}(x, L/K) := \sum_{\substack{\p\in M_K, m\ge 1\\N(\p^m)\le x}}\frac1m \delta_C(\Frob_\p^m),\]
where $\delta_C\:G\to \{0,1\}$ is such that $\delta_C(g) = 1$ if and only if $g\in C$.
\end{definition}
This weighted sum $\widetilde\pi_{C}(x, L/K)$ is a good approximation of $\pi_C(x, L/K)$:
\begin{lemma}[\cite{zyw}*{Lem.\ 2.7}]\label{zywina1}
We have
\[ \widetilde \pi_C(x, L/K)= \pi_C(x, L/K)+O\br{[K:\Q]\br{\frac{x^{1/2}}{\log x}+\log M(L/K)}},\]
where $M(L/K)$ is the constant defined in Theorem $\ref{zmain}$.
\end{lemma}
We end this section by recalling the following result of Zywina \cite[Lemma 2.6]{zyw}.
\begin{lemma}\label{zywina2}
\begin{enumerate}
\item
Let $H$ be a subgroup of $G$ and suppose that every element of $C$ is conjugate to some element of $H$. Then
\[\widetilde \pi_C(x, L/K)\le \widetilde \pi_{C\cap H} (x, L/L^{H}).\]
\item
Let $N$ be a normal subgroup of $G$ an suppose that $NC \subset C.$ Then
\[\widetilde \pi_C(x, L/K)= \widetilde \pi_{\overline C} (x, L^N/K),\]
where $\overline C$ is the image of $C$ in $G/N=\Gal(L^N/K)$.
\end{enumerate}
\end{lemma}
\section{Unconditional \texorpdfstring{bounds on $\pi_f(x, a)$}{Lang--Trotter bounds}}\label{sec:proof-unconditional}
Recall that $f$ is a cuspidal Siegel modular eigenform of weights $(k_1, k_2)$, $k_1\ge k_2\ge 2$, level $N$ and character $\varepsilon$. Assume that $f$ is not CAP, endoscopic, CM, RM or a symmetric cube lift. Let $E=\Q(\{a_p : p\nmid N\},\varepsilon)$ be the coefficient field of $f$, and let $F, K$ be the number fields defined in \Cref{def:F} and \Cref{def:K}. Let $\LL$ be the set of primes defined in \Cref{def:final-L} and, for $\l\in\LL$, write
\[\wrho_{\l}\:\Gal(\Qb/K)\to\G(\Zl)\]
for the Galois representation defined just after \Cref{def:G}. Here,
\[\G(\Zl) = \set{(g, \nu) \in \Gf(\O_F\tensor_\Z\Zl)\times\Zl\t : \simil(g) = \nu^{k_1 + k_2 -3}}.\]
In particular, the projection of $\wrho_\l$ onto $\Gf(\O_F\tensor_\Z\Zl)$ is exactly $\rho_\l|_K$, and its projection to $\Zl\t$ is the $\l$-adic cyclotomic character. By \Cref{thm:precise-image}, $\wrho_\l$ has open image in $\G(\Zl)$ for all $\l\in\LL$ and is surjective for all but finitely many $\l\in\LL$. Using the fact that the dimension of the $\l$-adic Lie group $\Sp_4(\O_{F_\lambda})$ is $10[F_\lambda:\Ql]$ for each prime $\lambda\mid\l$, we see that \begin{equation}\label{eq:dim-gl}
\dim \G(\Zl) = {10[F:\Q] + 1}. \end{equation}
\subsection{The case $a\ne 0$}
Fix a non-zero algebraic integer $a\in \O_F$. Our goal is to bound the size of the set $\pi_f(x, a) := \set{p\le x : a_p = a}$.
\begin{proposition}\label{prop:primes-of-K}
Assume that $a\ne 0$. Then
\[ \pi_f(x, a) = \frac{1}{[K:\Q]}\#\set{\p\in M_K, N(\p)=p\le x : a_p = a}.\]
\end{proposition}
\begin{proof}
Recall from \Cref{sec:inner-twists} that, by definition, $\Gamma=\Gal(E/F)$ is the group of $\sigma\in \Aut(E/\Q)$ such that $(\sigma, \chi_\sigma)$ is an inner twist. Hence, if $a_p = a$ for some non-zero $a\in\O_F$, then, for every inner twist $(\sigma, \chi_\sigma)$, we have
\[a_p = \sigma(a_p) =\chi_\sigma(p)a_p,\]
from which it follows that $\chi_\sigma(p) = 1$. Since $K$ is, by definition, the field cut out by all the $\chi_\sigma$'s, we see that, if $a_p = a$ for a prime $p$, then $p$ splits completely in $K$.
\end{proof}
It follows from \Cref{prop:primes-of-K} that bounding the size of $\pi_f(x, a)$ is exactly the same as bounding the size of $\#\set{\p\in M_K, N(\p)=p\le x : a_p = a}$, up to the constant $[K:\Q]$.
Let \[\CC_\l(a) = \set{(g,\nu)\in \im \wrho_\l \sub\G(\Zl): \tr(g) = a}.\] Then, for any prime $\l\in\LL$, we have \begin{align}\label{eq:comparison}
\#\set{\p\in M_K, N(\p)=p\le x : a_p = a}
&= \pi_{\CC_\l(a)}(x, L/K) + O(1), \end{align} where $L$ is the fixed field of the kernel of $\wrho_\l$, and the $O(1)$ is to account for the finitely many primes $p\mid \l N$. In order to prove \Cref{thm:unconditional-lta}, we use \Cref{thm:cdt-serre} to estimate the size of $\pi_{\CC_\l(a)}(x, L/K)$.
\begin{proof}[Proof of Theorem $\ref{thm:unconditional-lta}~(i)$]
We show that the set $\CC_\l(a)$ has Minkowski dimension $9[F:\Q]+1$. Write $a = (a_\lambda)_\lambda$ and $\tr(g) = (\tr(g)_\lambda)_\lambda$ via the isomorphism $F\tensor_\Q\Ql\cong \prod_{\lambda\mid\l}F_\lambda$. Then $\CC_\l(a)$ is the closed subset of $\im\wrho_\l$ cut out by the $[F:\Q]$ equations $\tr(g)_\lambda = a_\lambda$. By \Cref{thm:precise-image}, $\im\wrho_\l$ is an open subgroup of $\G(\Zl)$. Hence, by $(\ref{eq:dim-gl})$, it has dimension $10[F:\Q]+ 1$ as an $\l$-adic Lie group. By \cite{ser81}*{Sec.\ 3.2}, it follows that $\dim_M\CC_\l(a) \le (10[F:\Q] + 1) - [F:\Q] = 9[F:\Q] + 1$.
It follows from \Cref{thm:cdt-serre} that
\[\pi_{\CC_\l(a)}(x, L/K) \ll \frac{x}{\log(x)^{1 + \alpha - \epsilon}},\]
where $\alpha = \frac{[F:\Q]}{10[F:\Q]+1}$. The result follows from \Cref{prop:primes-of-K} and $(\ref{eq:comparison})$. \end{proof}
\subsection{The case $a = 0$}
Let $PG_\l$ denote the image of the set \[G_\l = \set{g\in\Gf(\O_F\tensor_\Z\Zl) : \simil(g)\in \Zl^{\times(k_1 + k_2 -3)}}\] in $\PGSp_4(\O_F\tensor_\Z\Zl)$, and let $H_\l$ denote the image of $\Proj\rho_\l\:\Ga\Q\to \PGSp_4(\O_F\tensor_\Z\Zl)$. Let $L = \Qb^{H_\l}$ be the field cut out by the kernel of $\Proj\rho_\l$. By \Cref{thm:precise-Q-image}, there is a subgroup of $H_\l$ of index at most $[K:\Q$] that is open in $PG_\l$. In particular, $H_\l$ and $PG_\l$ have the same dimension as $\l$-adic Lie groups, i.e.\ both have dimension $9[F:\Q] + 1$.
We can write $PG_\l = \prod_{\lambda\mid\l}PG_\lambda$, where $PG_\lambda$ is the image of the set \[G_\lambda = \set{g\in\Gf(\O_{F_\lambda}) : \simil(g)\in \Zl^{\times(k_1 + k_2 -3)}}\] in $\PGSp_4(\O_{F_\lambda})$. Since $H_\l$ is contained in a union of cosets of $G_\l$, we can similarly define $H_\lambda$ to be the projection of $H_\l$ onto the corresponding coset of $PG_\lambda$.
\begin{proof}[Proof of Theorem $\ref{thm:unconditional-lta}~(ii)$] Set \[\CC_\l(0) = \set{g\in H_\l : \tr(g) = 0}.\] Note that having trace $0$ is well defined on $\PGSp_4(\O_F\tensor_\Z\Zl)$, so this definition makes sense. As in the previous section, we have $\dim H_\l-\dim_M\CC_\l(0) = [F:\Q]$.
We compute the quantity
\[r = r_{\CC_\l(0)} = \Inf_{s\in \CC_\l(0)}\dim \frac{H_\l}{Z_{H_\l}(s)}.\]
If $s\in \CC_\l(0)$, then, by \Cref{cor:precise-Q-image}, we can write
\[s= \dmat{\alpha}{\alpha}{\varepsilon/\alpha}{\varepsilon/\alpha}h,\]
where $h\in PG_\l$ and $\alpha, \varepsilon\in \O_E\tensor\Zl$. Moreover, as in \Cref{thm:precise-Q-image}, we have $\varepsilon/\alpha^2\in \O_F\tensor\Zl$. Hence, rescaling by $\alpha\ii$, we can assume that $s\in \PGSp_4(\O_F\tensor\Zl)$.
Write $s = (s_\lambda)_\lambda$ via the isomorphism $F\tensor\Ql = \prod_{\lambda\mid\l}F_\lambda$. Then we have
\[\dim \frac{H_\l}{Z_{H_\l}(s)} = \sum_{\lambda\mid\l}\dim H_\lambda - \dim Z_{H_\lambda}(s_\lambda).\]
We now argue as in \cite{cojocaru-abelian-varieties}*{Thm.\ 1}. By \cite{cojocaru-abelian-varieties}*{Thm.\ A.1}, we have $\dim H_\lambda - \dim Z_{H_\lambda}(s_\lambda)\ge 4[F_\lambda:\Ql]$ for all $s\in \CC_\l(0)$. It follows that
\[\dim \frac{H_\l}{Z_{H_\l}(s)}\ge 4[F:\Q].\]
Hence, by \Cref{thm:cdt-serre}, we have
\[\pi_{\CC_\l(0)}(x, L/\Q) \ll \frac{x}{(\log x)^{1+\alpha - \epsilon}}, \]
where
\[\alpha = \frac{[F:\Q]}{9[F:\Q] + 1 - 4[F:\Q]/2} = \frac{[F:\Q]}{7[F:\Q] + 1}.\]
\end{proof}
\begin{remark}\label{rem:strengthened-result}
Suppose that $a\in\O_F$ is such that there exists a prime $\l\in \LL$ such that $\l\mid \frac a 4$. Then, arguing as in the above proof (and as in \cite{cojocaru-abelian-varieties}*{Thm.\ 1}), it follows that $r_{\CC_\l(a)}\ge 4[F:\Q]$. Applying \Cref{thm:cdt-serre}, it follows that
\[\set{p\le x: a_p = a}\ll \frac{x}{(\log x)^{1+\alpha - \epsilon}}, \]
where
\[\alpha = \frac{[F:\Q]}{8[F:\Q] + 1},\]
a stronger bound than \Cref{thm:unconditional-lta}. In particular, if $\LL$ is the set of all primes, which is conjecturally the case when $E = F$, then, as in \cite{cojocaru-abelian-varieties}*{Thm.\ 1}, we obtain this stronger bound whenever $a\ne \pm4$. \end{remark}
\section{Conditional \texorpdfstring{bounds on $\pi_f(x, a)$}{Lang--Trotter bounds}}\label{sec:proof-conditional}
Recall that $f$ is a cuspidal Siegel modular eigenform of weights $(k_1, k_2)$, $k_1\ge k_2\ge 2$, level $N$ and character $\varepsilon$. Assume that $f$ is not CAP, endoscopic, CM, RM or a symmetric cube lift. Let $E=\Q(\{a_p : p\nmid N\},\varepsilon)$ be be the coefficient field of $f$, and let $F, K$ be the number fields defined in \Cref{def:F} and \Cref{def:K}. Let $\LL$ be the set of primes defined in \Cref{def:final-L} and, for $\l\in\LL$, write
\[\worho_{\l}\:\Gal(\Qb/K)\to\G(\Fl)\]
for the reduction of $\wrho_\l$ modulo $\l$. Here,
\[\G(\Fl) = \set{(g, \nu) \in \Gf(\O_F\tensor_\Z\Fl)\times\Fl\t : \simil(g) = \nu^{k_1 + k_2 -3}}.\]
In particular, the projection of $\worho_\l$ onto $\Gf(\O_F\tensor_\Z\Fl)$ is exactly $\orho_\l|_K$, and its projection to $\Fl\t$ is the mod $\l$ cyclotomic character. By \Cref{thm:precise-image}, $\worho_\l$ is surjective for all but finitely many primes $\l\in\LL$.
\subsection{The case $a\ne 0$}
Fix a non-zero algebraic integer $a\in \O_F$. Our goal is to bound the size of $\pi_f(x, a) = \#\set{p\le x : a_p = a}$. As in the previous section, by \Cref{prop:primes-of-K}, bounding the size of $\pi_f(x, a)$ is exactly the same as bounding the size of $\#\set{\p\in M_K, N(\p)=p\le x : a_p = a}$, up to the constant $[K:\Q]$. We will bound the size of this set by generalising the strategy of \cite{mms}.
For a prime $p$ with $p \nmid N$ such that $p$ splits completely in $K$, let $F(p)$ be the splitting field of the characteristic polynomial of $\rho_{\l}(\Frob_p)$. By definition, this characteristic polynomial does not depend on $\l$. Define
\begin{equation*}
\pi(x, a ; \l) := \#\set{\p\in M_K, N(\p) = p \le x: a_p = a, \ \l\text{ splits completely in }F(p)}.
\end{equation*}
The following lemma will allow us to use $\pi(x, a;\l)$ to bound $\pi_f(x,a)$:
\begin{lemma}[c.f.\ \cite{mms}*{Lem.\ 4.4}]\label{lem:murty-bound}
Let $I$ be the interval $[y, y+u]$, where $y, u$ are chosen so that $x\ge y \ge u \ge y^{1/2}(\log y)^{1+ \epsilon}(\log xy)$ for some $\epsilon\ge 0$. Then, assuming GRH, we have
\[\#\set{\p\in M_K, N(\p)=p\le x : a_p = a} \ll \max_{\l\in I}\pi(x, a; \l).\]
\end{lemma}
To prove \Cref{lem:murty-bound}, we will require the following bound on the discriminant of $F(p)$:
\begin{lemma}\label{lem:log-disc}
We have $\log|\disc(F(p)/\Q)| = O(\log p)$.
\end{lemma}
\begin{proof}
Let $g(x)\in \O_F[x]$ be the characteristic polynomial of $\rho_\l(\Frob_p)$. Let $\pi$ be the unitary cuspidal automorphic representation of $\GSp_4(\AQ)$ associated to $f$. By assumption, $\pi$ lifts to a cuspidal automorphic representation $\Pi$ of $\GL_4(\AQ)$. Let $\alpha_1, \ldots, \alpha_4$ be the Satake parameters of the local representation $\Pi_p$. By \cite{jacquetshalika1}*{Cor.\ 2.5}, we have
\[p^{-\frac12} < |\alpha_i| < p^{\frac12}\]
for each $i$. In fact, if the weight $k_2 >2$, then the Ramamujan conjecture is known, and we have $|\alpha_i| = 1$.
By definition, the four roots of $g(x)$ are $\alpha_1p^{\frac12(k_1 + k_2-3)}, \ldots, \alpha_4p^{\frac12(k_1 + k_2-3)}$. It follows that the coefficients of $g(x)$ are $O(p^{2(k_1 + k_2 -2)})$. Since the discriminant of $g(x)$ is a polynomial in its coefficients, we see that $\log|\disc(g(x)| = O(\log p)$.
Let $L_i = F(\alpha_ip^{\frac12(k_1 + k_2-3)})$. Then
\begin{align*}
\disc(L_i/\Q) &= N_{F/\Q}(\disc(L_i/F))\cdot \disc(F/\Q)^{[L_i:F]}.
\end{align*}
Since $\disc(F/\Q)^{[L_i:F]} = O(1)$, it follows from the above reasoning that $\log|\disc(L_i/\Q)| = O(\log p)$. But $F(p)$ is the compositum of the $L_i$'s and, in general, $\d(K_1\cdot K_2/\Q)\mid \d(K_1/\Q)\cdot \d(K_2/\Q)$ for arbitrary fields $K_1, K_2$, where $\d$ denotes the relative different ideal. It follows that $\log|\disc(F(p)/\Q)| = O(\log p)$.
\end{proof}
\begin{proof}[Proof of Lemma $\ref{lem:murty-bound}$]
Observe that
\begin{equation}\label{pix}
\sum_{\l\in I}\pi(x,a;\l) = \sum_{\substack{\p\in M_K\\N(\mathfrak p) = p\le x\\a_p = a}}\#\set{\l\in I:\l\text{ splits completely in }F(p)}.
\end{equation}
By taking the trivial conjugacy class $C=\{1\}$ of the Galois group of $F(p)$ over $\Q$ and applying \Cref{thm:conditional-cdt}, under GRH, the size of the set $\set{\l\le z : \l\text{ splits completely in } F(p)}$ is equal to
\begin{align*}
\frac{1}{[F(p) : \Q]}\pi(z) + O\br{ \frac{1}{[F(p) : \Q]}z^{1/2} \br{\log p+[F(p) : \Q] \log z} },
\end{align*}
for any real number $z\gg 0$. Here, we have used \Cref{lem:log-disc}, that $\log| \disc(F(p)/\Q)|\ll \log p$.
Since $u \ge y^{1/2}(\log y)^{2 + \epsilon}$, under GRH, we have
\[\pi(y+u) - \pi(y)\gg \frac{u}{\log u}.\]
It follows that
\[\#\set{\l\in I : \l\text{ splits completely in } F(p)} \gg \pi(y+u) - \pi(y),\]
where the implied constant in the above estimate is uniform in $p$. Using this estimate in $(\ref{pix})$ yields
\[ \sum_{\substack{\p\in M_K\\N(\mathfrak p) = p\le x\\a_p = a}}1
\ll \frac{1}{\pi(y +u) - \pi(y)}\sum_{\l\in I}\pi(x,a ; \l)\ll \max_{\l\in I}\pi(x,a; \l).\]
\end{proof}
\begin{remark}
In \cite{murty-mod-forms-ii}*{pp.\ 304}, Murty proves a two-dimensional analogue of \Cref{lem:murty-bound} without assuming GRH. Murty's method makes essential use of the fact that, in the elliptic modular forms case, the field $F(p)$ is a quadratic extension of $F$. Hence, the quantity $\#\set{\l\in I:\l\text{ splits completely in }F(p)}$ can be estimated via a character sum. In our case, $F(p)$ need not even be an abelian extension of $F$, so Murty's method does not apply. It would be interesting to see if a version of \Cref{lem:murty-bound} can be proven without assuming GRH. By combining the methods of this section with the unconditional Chebotarev density theorem of \cite{thorner-zaman}, such a result would lead to an improved unconditional bound in \Cref{thm:unconditional-lta}.
\end{remark}
Let $\l\in \LL$ be a prime such that $\worho_\l$ is surjective, and let $L$ be the field cut out by the kernel of $\worho_\l$. Then $L$ is a finite Galois extension of $K$ with Galois group $\G(\Fl)$. Define: \begin{align*}
\CC(a, \l)&=\{(g,\nu) \in \G(\F_\l): \tr(g)= a\pmod\l, \text{ and all the eigenvalues of } g \text{ are in }\Fl\t \},\\
\mathcal B_{\l}& = \{(g, \nu)\in \G(\Fl) : g \text{ upper triangular}\},\\
\mathcal U_{ \l}&= \{(g, \nu)\in \G(\Fl) : g \text{ unipotent upper triangular}\},\\
\overline{\CC}(a, \l)&= \text{the image of } \CC(a, \l)\cap \mathcal B_{\l} \text{ in } \mathcal B_{\l}/ \mathcal U_{\l}. \end{align*}
Then $\CC(a, \l)$ is a subset of $\G(\Fl)$ that is closed under conjugation. Note that
$\mathcal U_{\l}$ is normal in $\mathcal B_{\l}$ and that $\mathcal B_{\l}/\mathcal U_{\l}$ is abelian with Galois group $\Gal( L^{\mathcal U_{\l}}/ L^{\mathcal B_{\l}})$.
\begin{lemma}\label{lem4}
Let $\l\in\LL$ be a prime which splits completely in $F$. Then
\[\pi(x,a; \l) \ll \widetilde\pi_{ \overline\CC(a, \l)}(x, L^{\mathcal U_\l}/L^{\mathcal B_\l}),\]
where $\widetilde\pi$ is as in Definition $\ref{def:weighted-set}$.
\end{lemma}
\begin{proof}
If $\l$ splits in $F(p)$, then all the roots of the characteristic polynomial of $\rho_\l(\Frob_p)$ are in $\Fl\t$. It follows that
\[
\pi(x, a;\l) \ll \pi_{\CC(a, \l)}(x, L/K).
\]
Now, $\CC(a, \l)$ is a union of conjugacy classes of $\G(\Fl)$, and each conjugacy class contains an element of $\mathcal B_\l$. Hence, by \Cref{zywina2} $(i)$,
\[ \pi_{\CC(a, \l)}(x, L/K)\le \widetilde\pi_{\CC(a, \l)\cap \mathcal B_\l}(x, L/L^{\mathcal B_\l}).\]
Since multiplication by elements of $\mathcal U_\l$ preserves the set $\mathcal {B}_\l$, by \Cref{zywina2} $(ii)$, we have
\[\widetilde\pi_{\CC(a, \l)\cap \mathcal B_\l}(x, L/L^{\mathcal B_\l}) = \widetilde\pi_{\overline{\CC}(a, \l)}(x, L^{\mathcal U_\l}/L^{\mathcal B_\l}).\]
Combining the above estimates gives the desired result.
\end{proof}
\begin{lemma}\label{card1} Let $[F:\Q]=n$, and suppose that $\l$ is unramified in $F$. Then we have
\begin{enumerate}[leftmargin=*]
\item $|\mathcal B_{\l}|\asymp \l^{6n+1}, |\mathcal U_{\l}|\asymp \l^{4n}$.
\item
$ |\overline{\CC}(a, \l)|\asymp \l^{n+1}.$
\item
$[ L^{\mathcal B_{\l}}:K]\ll \l^{4n}$ and
$\log M( L^{\mathcal U_{\l}}/L^{\mathcal B_{\l}})\ll \log \l$, where $M(L^{\mathcal U_{\l}}/L^{\mathcal B_{\l}})$ is as in Theorem $\ref{zmain}$.
\end{enumerate}
\end{lemma}
\begin{proof}
If $\F$ is a finite field of cardinality $q$, then the set of upper triangular matrices in $\Gf(\F)$ is
\begin{equation*}\label{borel}
\set{\br{\begin{smallmatrix}
a & & & \\
& b & & \\
& & cb^{-1} & \\
& & & ca^{-1}
\end{smallmatrix}} \br{\begin{smallmatrix}
1 & n & & \\
& 1 & & \\
& & 1 &-n \\
& & & 1
\end{smallmatrix}} \br{\begin{smallmatrix}
1 & & r & s\\
& 1 & t & r\\
& & 1 & \\
& & & 1
\end{smallmatrix}}: a,b,c \in \F^\times, n,r,s,t \in \F }.
\end{equation*}
Therefore, for any $\nu \in \F^\times$, it follows that \begin{equation}\label{card_borel}
\#\set{g\in\Gf(\F):\simil(g) = \nu,\ g\text{ upper triangular}} = q^{6}+O(q^5)\asymp q^6
\end{equation}
and
\begin{equation}\label{card_unipotent}
\#\set{g\in\Gf(\F):g\text{ unipotent upper triangular}} = q^{4}+O(q^3)\asymp q^4.
\end{equation}
\begin{enumerate}[leftmargin=*]
\item Since $\l$ is unramified in $F$, we have $\O_F\tensor_\Z\Fl \simeq \prod_{\lambda\mid \l}\F_\lambda$, where the product runs over the primes $\lambda\mid \l$ of $F$.
From $(\ref{card_borel})$, via this isomorphism, we have
\begin{align*}
|\mathcal B_\l| &=\sum_{\nu \in \Fl\t} \br{ \prod_{\lambda\mid\l} \#\set{g\in\Gf(\F_\lambda):\simil(g) = \nu,\ g\text{ upper triangular}} }\\
&\asymp \sum_{\nu \in \Fl\t} \prod_{\lambda\mid\l}N(\lambda)^6 \asymp \l^{6n+1}.
\end{align*}
Similarly, from $(\ref{card_unipotent})$, we have
\begin{align*}
|\mathcal U_\l| &=\br{ \prod_{\lambda\mid\l} \#\set{g\in\Gf(\F_\lambda):\ g\text{ unipotent upper triangular}} }\\
&\asymp \prod_{\lambda\mid\l}N(\lambda)^4 \asymp \l^{4n}.
\end{align*}
\item From the definition of $ \overline\CC(a,\l)$, we observe that its elements are in bijection with
\[
\{(g, \nu)\in \G(\Fl) : g \text{ diagonal}, \tr(g)=a \}.
\]
Writing $a = (a_\lambda)_\lambda$ via the isomorphism $\O_{F}\tensor_\Z\Fl\simeq \prod_{\lambda\mid\l}\F_\lambda$ and proceeding as before, we obtain
\begin{align*}
|\overline\CC(a,\l)|
&=\sum_{\nu \in \Fl\t} \br{\prod_{\lambda\mid\l} \#\set{g\in\Gf(\F_\lambda):\simil(g) = \nu,\ g\text{ diagonal},\ \tr(g) = a_\lambda}}\\
&\asymp\sum_{\nu\in\Fl\t}\prod_{\lambda\mid\l}N(\lambda) \asymp\l^{n+1}.
\end{align*}
\item Using formulae for the size of $\Gf(\F)$ over finite fields $\F$, it is easy to check that $|\G(\Fl)|\asymp\l^{10n+1}$. Since $[L^{\mathcal B_\l}:K] = [\G(\F_\l):\mathcal B_\l]$, it follows from $(i)$ that $[L^{\mathcal B_\l}:K]\ll \l^{4n}$. The second bound follows from part $(i)$, \Cref{prop:hensel} and the fact that $[L^{\mathcal U_\l}:L^{\mathcal B_\l}]= [\mathcal B_\l:\mathcal U_\l]$.
\end{enumerate}
\end{proof}
\begin{proof}[Proof of Theorem $\ref{thm:conditional-lta} ~(i)$]
First observe that the group $\mathcal B_{\l}/\mathcal U_{\l}$ is abelian, which is the Galois group of the extension $ L^{\mathcal B_{\l}}/ L^{\mathcal U_{\l}}$. Hence, under GRH, applying \Cref{zmain}, \Cref{zywina1} and \Cref{card1} yields
\begin{align*}
\widetilde{\pi}_{\overline\CC(a,\l)}(x, L^{\mathcal U_{\l}}/ L^{\mathcal B_{\l}})&\ll \frac{| \overline\CC(a,\l)| }{|{\mathcal B_{\l}}| /|{\mathcal U_{\l}}|}\frac{x}{\log x}+|\overline\CC(a,\l)|^{1/2}[ L^{\mathcal B_{\l}}:\Q]\frac{x^{1/2}}{\log x}\log M( L^{\mathcal U_{\l}}/ L^{\mathcal B_{\l}})\\
& \ll \frac{1}{\l^n} \frac{x}{\log x}+\l^{\frac{1}{2}(9n+1)}\log \l \frac{x^{1/2}}{\log x}.
\end{align*}
Let $y\asymp\frac{x^{\alpha/n}}{(\log x)^{2\alpha/n}}$, where $\alpha = \frac{n}{11n+1}$. By \Cref{thm:precise-image}, the set of primes such that $\worho_\l$ is surjective has density $1$. Hence, for $y$ sufficiently large, we can choose $\l \in [y,2y]$ such that $\l$ that splits completely in $F$ and such that $\worho_\l$ is surjective. By \Cref{lem4}, we have
\begin{align*}
\pi(x,a;\l) &\ll \frac{1}{y^n} \frac{x}{\log x}+y^{\frac{1}{2}(9n+1)}\log y\frac{x^{1/2}}{\log x}\\
&\ll\frac{x^{1-\alpha}}{(\log x)^{1-2\alpha}}.
\end{align*}
The result now follows from \Cref{lem:murty-bound}.
\end{proof}
\subsection{The case $a = 0$}
Let
\[\overline G_\l := \set{g\in\Gf(\O_F\tensor_\Z\Fl) : \simil(g)\in \Fl^{\times(k_1 + k_2 -3)}}.\]
Then, by \Cref{cor:precise-Q-image}, for all but finitely many primes $\l\in\LL$ the image of $\orho_\l\:\Ga\Q\to\Gf(\O_E\tensor_\Z\Fl)$ is a disjoint union of at most $[K:\Q]$ cosets
\[\coprod\dmat{\alpha(\gamma)}{\alpha(\gamma)}{\varepsilon(\gamma)/\alpha(\gamma)}{\varepsilon(\gamma)/\alpha(\gamma)}\overline G_\l,\]
where $\gamma$ ranges over some subset of $\Gal(K/\Q)$. Moreover, taking $\l$ sufficiently large, the quantity $\varepsilon(\gamma)\alpha(\gamma)^{-2}$ is an element of $(\O_F\tensor_\Z\Fl)\t$. Write $\overline G_\l^\gamma$ for the coset indexed by $\gamma$ and, for each $\gamma$, let $\CC_\gamma(0, \l)$ denote the set of trace $0$ elements in $\overline G_\l^\gamma$.
\begin{lemma}
For any $\gamma\in \Gal(K/\Q)$, we have $|\CC_\gamma(0, \l)| \asymp |\CC_1(0, \l)|$.
\end{lemma}
\begin{proof}
First note that, since $\varepsilon(\gamma)\alpha(\gamma)^{-2}\in(\O_F\tensor_\Z\Fl)\t$, we have $\alpha(\gamma)\ii \overline G_\l^\gamma\sub \Gf(\O_F\tensor_\Z\Fl)$. Set $b = \varepsilon(\gamma)\alpha(\gamma)^{-2}$. Then $C_\gamma(0, \l)$ is precisely the set of trace 0 elements in the coset $\dmat{1}{1}{b}{b}\overline G_\l$. Hence,
\begin{align*}
|\CC_\gamma(0, \l)| &=\#\set{g\in\overline G_\l^\gamma : \tr(g) = 0}\\
&= \#\set{g\in \alpha(\gamma)\ii \overline G_\l^\gamma : \tr(g) = 0}\\
&=\sum_{\nu \in \Fl^{\times(k_1 + k_2 -3)}} \prod_{\lambda\mid\l}\#\set{g = (g_{ij})\in \Gf(\F_\lambda): \simil(g) = \nu b^{-2}, g_{11} + g_{22} + bg_{33} + bg_{4} = 0}\\
&\asymp\sum_{\nu \in \Fl^{\times(k_1 + k_2 -3)}} \prod_{\lambda\mid\l} N(\lambda)^9.
\end{align*}
Since the above calculation did not depend on $\gamma$, the result follows.
\end{proof}
\begin{corollary}\label{cor:trace-0-K}
We have
\[\pi_f(x, 0) \ll \#\set{\p\in M_K : N(\p) = p \le x : \tr\orho_\l(\Frob_p)\equiv 0\pmod \l}.\]
\end{corollary}
\begin{proof}
We have
\[\pi_f(x, 0) \ll \#\set{p \le x : \tr\orho_\l(\Frob_p)\equiv 0\pmod \l}.\]
By the previous lemma, the set on the right hand side splits into at most $[K:\Q]$ subsets, each of size
\[\#\set{p \le x : p \text{ splits completely in }K,\ \tr\orho_\l(\Frob_p)\equiv 0\pmod \l}.\]
Indeed, as in \Cref{prop:primes-of-K}, this set is exactly $\CC_1(0,\l)$.
The result follows from the fact that this set has size
\[\frac{1}{[K:\Q]}\#\set{\p\in M_K : N(\p) = p \le x : \tr\orho_\l(\Frob_p)\equiv 0\pmod \l}.\]
\end{proof}
Now, as in the previous section, for a prime $p$ with $p \nmid N$ such that $p$ splits completely in $K$, let $F(p)$ be the splitting field of the characteristic polynomial of $\rho_{\l}(\Frob_p)$, and let
\begin{equation*}
\pi(x, 0 ; \l) := \#\set{\p\in M_K, N(\p) = p \le x: a_p = 0, \ \l\text{ splits completely in }F(p)}.
\end{equation*}
Then, by \Cref{lem:murty-bound} and \Cref{cor:trace-0-K}, we can use $\pi(x, 0; \l)$ to bound $\pi_f(x, 0)$.
Fix a prime $\l\in\LL$ such that $\worho_\l$ is surjective, and let $L$ be the field cut out by the kernel of $\worho_\l$. Let \begin{align*}
\CC(0, \l)&=\{(g,\nu) \in \G(\F_\l): \tr(g)= 0\pmod\l, \text{ and all the eigenvalues of } g \text{ are in }\Fl\t \},\\
\mathcal B_{\l}& = \{(g, \nu)\in \G(\Fl) : g \text{ upper triangular}\},\\
\mathcal H_{ \l}&= \{(g, \nu)\in \G(\Fl) : g \text{ upper triangular with $4$ equal eigenvalues}\},\\
\overline{\CC}(0, \l)&= \text{the image of } \CC(0, \l)\cap \mathcal B_{\l} \text{ in } \mathcal B_{\l}/ \mathcal H_{\l}. \end{align*}
The proof of the following lemma is essentially identical to that of \Cref{card1}.
\begin{lemma}\label{card2} Let $[F:\Q]=n$. Then we have
\begin{enumerate}[leftmargin=*]
\item $|\mathcal B_{\l}|\asymp \l^{6n+1}, |\mathcal H_{\l}|\asymp \l^{5n}$.
\item
$ |\overline{\CC}(0, \l)|\asymp \l.$
\item
$[ L^{\mathcal B_{\l}}:K]\ll \l^{4n}$ and
$\log M( L^{\mathcal H_{\l}}/L^{\mathcal B_{\l}})\ll \log \l$, where $M(L^{\mathcal H_{\l}}/L^{\mathcal B_{\l}})$ is as in Theorem $\ref{zmain}$.
\end{enumerate}
\end{lemma}
\begin{proof}[Proof of Theorem $\ref{thm:conditional-lta}~(ii)$]
Since the product of a scalar matrix and a trace zero matrix has trace zero, we have
\[\mathcal H_{\l}\cdot (\CC(0, \l)\cap \mathcal B_{\l})=\CC(0, \l)\cap \mathcal B_{\l}.\]
Hence, from \Cref{zywina2}, we have
\[\pi_{\CC(0,\l)}(x, L/K) \le \widetilde \pi_{\CC(0, \l)\cap \mathcal B_{\l}}(x, L/L^{\mathcal B_{\l}})\le \widetilde \pi_{\overline\CC(0,\l)} (x, L^{\mathcal H_{\l}}/L^{\mathcal B_{\l}}).\]
Under GRH, by \Cref{zmain}, \Cref{zywina1} and \Cref{card2}, we have
\begin{align*}
\widetilde \pi_{\overline\CC(0,\l)} (x, L^{\mathcal H_{\l}}/L^{\mathcal B_{\l}})&\ll \frac{| \overline\CC(0,\l)| }{|{\mathcal B_{\l}}| /|{\mathcal H_{\l}}|}\frac{x}{\log x}+|\overline\CC(0,\l)|^{1/2}[ L^{\mathcal B_{\l}}:\Q]\frac{x^{1/2}}{\log x}\log M( L^{\mathcal H_{\l}}/ L^{\mathcal B_{\l}})\\
& \ll \frac{1}{\l^{n}} \frac{x}{\log x}+\l^{\frac{1}{2}(8n+1)}\log \l \frac{x^{1/2}}{\log x}.
\end{align*}
Let $y\asymp\frac{x^{\alpha/n}}{(\log x)^{2\alpha/n}}$, where $\alpha = \frac{n}{10n+1}$. By \Cref{thm:precise-image}, the set of primes such that $\worho_\l$ is surjective has density $1$. Hence, for $y$ sufficiently large, we can choose $\l \in [y,2y]$ such that $\l$ that splits completely in $F$ and such that $\worho_\l$ is surjective. By the same argument as \Cref{lem4},
\begin{align*}
\pi(x,0;\l)
&\ll\frac{x^{1-\alpha}}{(\log x)^{1-2\alpha}}.
\end{align*}
The result now follows from \Cref{lem:murty-bound}.
\end{proof}
\end{document} | arXiv |
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Mosc. Math. J., 2004, Volume 4, Number 1, Pages 153–179 (Mi mmj146)
This article is cited in 32 scientific papers (total in 32 papers)
Parabolic character sheaves. I
G. Lusztig
Department of Mathematics, Massachusetts Institute of Technology
Abstract: We study a class of perverse sheaves on the variety of pairs $(P,gUP)$ where $P$ runs through a conjugacy class of parabolics in a connected reductive group and $gUP$ runs through $G/UP$. This is a generalization of the theory of character sheaves.
Key words and phrases: Reductive group, parabolic group, perverse sheaf, character sheaf.
DOI: https://doi.org/10.17323/1609-4514-2004-4-1-153-179
Full text: http://www.ams.org/.../abst4-1-2004.html
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MSC: 20G99
Citation: G. Lusztig, "Parabolic character sheaves. I", Mosc. Math. J., 4:1 (2004), 153–179
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\Bibitem{Lus04}
\by G.~Lusztig
\paper Parabolic character sheaves.~I
\jour Mosc. Math.~J.
\yr 2004
\vol 4
\issue 1
\pages 153--179
\mathnet{http://mi.mathnet.ru/mmj146}
\crossref{https://doi.org/10.17323/1609-4514-2004-4-1-153-179}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2074987}
\zmath{https://zbmath.org/?q=an:1102.20030}
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Mosc. Math. J., 2004, 4:1, 153–179
Parabolic character sheaves. II
Parabolic Character Sheaves, III
Mosc. Math. J., 2010, 10:3, 603–609
This publication is cited in the following articles:
G. Lusztig, "Parabolic character sheaves. II", Mosc. Math. J., 4:4 (2004), 869–896
He Xuhua, "The character sheaves on the group compactification", Adv. Math., 207:2 (2006), 805–827
Evens S., Lu Jiang-Hua, "On the variety of Lagrangian subalgebras. II", Ann. Sci. Icole Norm. Sup. (4), 39:2 (2006), 347–379
He Xuhua, "Unipotent variety in the group compactification", Adv. Math., 203:1 (2006), 109–131
Lusztig G., "Character sheaves and generalizations", Unity of Mathematics - IN HONOR OF THE NINETIETH BIRTHDAY OF I.M. GELFAND, Progress in Mathematics, 244, 2006, 443–455
Lu J.-H., Yakimov M., "Partitions of the wonderful group compactification", Transformation Groups, 12:4 (2007), 695–723
He X., "Minimal length elements in some double cosets of Coxeter groups", Advances in Mathematics, 215:2 (2007), 469–503
Springer T.A., "An extension of Bruhat's lemma", Journal of Algebra, 313:1 (2007), 417–427
He X., "The G-stable pieces of the wonderful compactification", Transactions of the American Mathematical Society, 359:7 (2007), 3005–3024
Lu J.-H., Yakimov M., "Group orbits and regular partitions of Poisson manifolds", Communications in Mathematical Physics, 283:3 (2008), 729–748
He X., Lusztig G., "Singular supports for character sheaves on a group compactification", Geometric and Functional Analysis, 17:6 (2008), 1915–1923
He X., "Character sheaves on certain spherical varieties", Advances in Mathematics, 217:3 (2008), 1154–1191
G. Lusztig, "Notes on character sheaves", Mosc. Math. J., 9:1 (2009), 91–109
He X., "G-stable pieces and partial flag varieties", Representation Theory-Bk, Contemporary Mathematics, 478, 2009, 61–70
Hoeve M., "Ekedahl-Oort strata in the supersingular locus", Journal of the London Mathematical Society-Second Series, 81:1 (2010), 129–141
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# Deriving the weak form of the PDE
To solve a partial differential equation (PDE) using the finite element method, we first need to derive its weak form. This involves converting the PDE into a variational problem and introducing a test function. The weak form of the PDE is then obtained by multiplying the PDE by a test function and integrating over the domain.
Consider the following PDE:
$$\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = f(x, y)$$
To derive its weak form, we introduce a test function $v$ and multiply the PDE by $v$:
$$\int_{\Omega} \left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}\right) v \ dx \ dy = \int_{\Omega} f(x, y) v \ dx \ dy$$
Next, we integrate by parts to eliminate the first derivatives:
$$\int_{\Omega} \left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}\right) v \ dx \ dy = \int_{\Omega} f(x, y) v \ dx \ dy$$
$$\int_{\Omega} \left(\frac{\partial}{\partial x} \left(\frac{\partial u}{\partial x}\right) + \frac{\partial}{\partial y} \left(\frac{\partial u}{\partial y}\right)\right) v \ dx \ dy = \int_{\Omega} f(x, y) v \ dx \ dy$$
## Exercise
Derive the weak form of the following PDE:
$$\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = f(x, y)$$
# Boundary conditions and their types
Boundary conditions are essential in solving PDEs using the finite element method. They are used to specify the values of the solution at the boundaries of the domain. There are three common types of boundary conditions: Dirichlet, Neumann, and Robin.
- Dirichlet boundary conditions: The value of the solution is specified at the boundary. For example, $u(x, y) = 0$ on the boundary.
- Neumann boundary conditions: The derivative of the solution is specified at the boundary. For example, $\frac{\partial u}{\partial n} = 0$ on the boundary, where $n$ is the outward-pointing normal vector.
- Robin boundary conditions: The derivative of the solution is specified at the boundary, and a specified function is multiplied with the solution. For example, $\frac{\partial u}{\partial n} = g(u)$ on the boundary.
# The finite element method
The finite element method is a numerical technique used to approximate the solution of PDEs. It involves dividing the domain into smaller regions called elements and using shape functions to describe the variation of the solution within each element. The method consists of several steps:
1. Discretize the continuum: Divide the solution into smaller regions that we call elements. The elements contain inside a certain number of points we call nodes. There are lots of shapes the elements can have. From segments of lines, triangles, squares, etc, to curved elements. The one/ones you will use depends on the problem you want to solve. For instance, for a 1D problem, like a cylindrical rod with radial symmetry, the most simple is to take the elements as linear segments with two nodes per segment (see Fig.1) and discretize it is as shown in figure 2. For a 2D problem, the most simple can be using triangular elements (see figure 3).
2. Select the type of trial function to use, and in turn the shape functions: We select what kind of functions we will take to describe the variation of the function φ inside each element (the trial function). This is equivalent to say, that we select the basis set of functions that will describe our solution. One of the usual choices is to take a polynomial like for instance φ(x) = a0 + a1x (known as linear element) or φ(x) = a0 + a1x + a2x2 (known as quadratic element). If we have n unknown coefficients a0, a1, ..., an−1 we will need the element to have n nodes to be able to determine them.
3. The formulation: Given the PDE you want to solve, now you must find a system of algebraic equations for each element "e" such that by solving it you got the values of φ at the position of nodes of the element "e" ([φ1, φ2, ..., φN] ≡ [φ]e ), i.e., you must find for each element "e" the matrix [K]e and the vector [f]e such that,
4. Assembling the stiffness and mass matrices: The stiffness matrix [K]e and the vector [f]e are assembled into global matrices [K] and [f] that describe the system of equations to be solved.
5. Solving the linear system of equations: The global system of equations is solved to obtain the values of the solution at the nodes.
6. Post-processing and visualizing the results: The solution at the nodes is used to approximate the solution at any point within the domain. The results are then post-processed and visualized.
# Constructing the finite element mesh
The finite element mesh is a discretization of the domain into smaller regions called elements. It is constructed using a mesh generator and a mesh refiner. The mesh generator creates a coarse mesh, and the mesh refiner refines it based on the desired mesh quality.
1. Mesh generator: A mesh generator creates a coarse mesh by dividing the domain into elements. It can use various algorithms, such as the Delaunay triangulation for triangular elements or the Voronoi tessellation for quadrilateral elements.
2. Mesh refiner: A mesh refiner refines the coarse mesh based on the desired mesh quality. It can use various algorithms, such as the adaptive mesh refinement (AMR) or the mesh adaptation.
3. Quality measures: Quality measures are used to assess the quality of the mesh. Common quality measures include the aspect ratio of the elements, the minimum angle between the element edges, and the maximum edge length.
# Assembling the stiffness and mass matrices
The stiffness and mass matrices are assembled from the element matrices for each element in the mesh. The element matrices are obtained by integrating the weak form of the PDE over each element.
1. Element matrices: The element matrices are obtained by integrating the weak form of the PDE over each element. For example, for a 2D problem with triangular elements, the element stiffness matrix can be obtained by integrating the following integral:
$$K_{ij} = \int_{\Delta} \frac{\partial^2 u}{\partial x_i \partial x_j} v \ dx \ dy$$
2. Assembling the global matrices: The element stiffness matrices are assembled into the global stiffness matrix and the global mass matrix.
# Solving the linear system of equations
The linear system of equations is solved to obtain the values of the solution at the nodes. Various solvers can be used, such as direct solvers, iterative solvers, or parallel solvers.
1. Direct solvers: Direct solvers solve the linear system of equations directly. They are suitable for small systems but can be inefficient for large systems.
2. Iterative solvers: Iterative solvers solve the linear system of equations using an iterative method, such as the conjugate gradient method or the Gauss-Seidel method. They are suitable for large systems and can be more efficient than direct solvers.
3. Parallel solvers: Parallel solvers solve the linear system of equations using parallel computing techniques. They are suitable for large systems and can take advantage of multiple processors or GPUs.
# Post-processing and visualizing the results
The solution at the nodes is used to approximate the solution at any point within the domain. Post-processing techniques are used to refine the approximation and visualize the results.
1. Interpolation: Interpolation techniques are used to refine the approximation of the solution at any point within the domain.
2. Visualization: Visualization tools are used to display the results, such as contour plots, surface plots, or animations.
# Applications of the finite element method in various fields
The finite element method is widely used in various fields, including civil engineering, mechanical engineering, aerospace engineering, and materials science. It is applied to solve a wide range of PDEs, such as the heat equation, the wave equation, the Navier-Stokes equations, and the Poisson equation.
1. Heat transfer: The finite element method is used to solve the heat equation to model the transfer of heat in various systems, such as buildings, engines, and materials.
2. Structural analysis: The finite element method is used to solve the Navier-Stokes equations to model the flow of fluids in pipes and the deformation of structures under various loading conditions.
3. Materials science: The finite element method is used to solve the Poisson equation to model the electrostatic potential in semiconductors and the flow of ions in electrolytes.
# Implementing the finite element method in Python
The finite element method can be implemented in Python using various libraries, such as NumPy, SciPy, and FEniCS. These libraries provide tools for numerical computing and visualization.
1. NumPy: NumPy is a library for numerical computing in Python. It provides arrays and matrices, as well as various mathematical functions.
2. SciPy: SciPy is a library for scientific computing in Python. It provides various numerical algorithms, such as linear solvers and optimization routines.
3. FEniCS: FEniCS is a library for solving partial differential equations in Python. It provides a high-level interface for defining and solving PDEs using the finite element method.
# Using Python libraries for numerical computing and visualization
Python libraries, such as NumPy, SciPy, and FEniCS, provide tools for numerical computing and visualization. These libraries can be used to implement the finite element method and solve PDEs.
1. NumPy: NumPy can be used to create arrays and matrices, perform mathematical operations, and solve linear systems of equations.
2. SciPy: SciPy can be used to solve nonlinear equations, optimize functions, and perform various numerical algorithms.
3. FEniCS: FEniCS can be used to define PDEs, create meshes, and solve the resulting linear systems of equations.
# Handling complex problems with the finite element method in Python
The finite element method can be used to handle complex problems in Python, such as solving PDEs with multiple boundary conditions, solving PDEs with time-dependent coefficients, and solving PDEs with moving boundaries.
1. Multiple boundary conditions: The finite element method can handle problems with multiple Dirichlet, Neumann, and Robin boundary conditions.
2. Time-dependent coefficients: The finite element method can handle problems with time-dependent coefficients, such as the heat equation with variable thermal conductivity.
3. Moving boundaries: The finite element method can handle problems with moving boundaries, such as the Navier-Stokes equations with moving walls.
## Exercise
Implement the finite element method in Python to solve a PDE with multiple boundary conditions, time-dependent coefficients, and moving boundaries. | Textbooks |
\begin{document}
\title{A naive procedure for computing angular spheroidal functions}
\author{J Sesma\footnote{[email protected]}\\ \ \\ Departamento de F\'{\i}sica Te\'orica, Facultad de Ciencias, \\ 50009 Zaragoza, Spain. \\ \ }
\maketitle
MSC [2010] { 33E10 (Primary) 33F05, 34L16, 65D20 (Secondary)}
PACS{ 02.30.Hq \and 02.30.Gp \and 03.65.Ge}
\begin{abstract} An algorithm for computing eigenvalues and eigenfunctions of the angular spheroidal wave equation, based on a known but scarcely used method, is developed. By requiring the regularity of the wave function, represented by its series expansion, the eigenvalues appear as the zeros of a one variable function easily computable. The iterative extended Newton method is suggested as especially suitable for determining those zeros. The computation of the eigenfunctions is then immediate. The usefulness of the method, applicable also in the case of complex values of the ``prolateness" parameter, is illustrated by comparing its results with those of procedures used by other authors. \end{abstract}
\section{Introduction}
The usefulness of spheroidal functions in many branches of Physics, like Quantum Mechanics, General Relativity, Signal Processing, etc., is well known and it does not need to be stressed. Due to that usefulness, the description of the spheroidal equation and of the main properties of its solutions deserves a chapter in handbooks of special functions like that by Abramowitz and Stegun \cite[Chap.~21]{abra}, the best known one, or the {\it NIST Digital Library of Mathematical Functions} \cite[Chap.~30]{nist}, the most recent one.
A review of the procedures used in the past century for obtaining the eigenvalues and eigenfunctions of the spheroidal wave equation can be found in a paper by Li {\em et al.} \cite{lilw}, where also an algorithm, implemented with the software package \verb"Mathematica", is provided. In the present century, articles dealing with the solutions of the angular spheroidal wave equation have continued appearing. Without aiming to be exhaustive, let us mention the papers by Aquino {\em et al.}~\cite{aqui}, Falloon {\em et al.}~\cite{fall}, Boyd \cite{boyd}, Barrowes {\em et al.} \cite{barr}, Walter and Soleski \cite{walt}, Abramov and Kurochkin \cite{aaab}, Kirby \cite{kirb}, Karoui and Moumni \cite{karo}, Gosse \cite{goss}, Tian \cite{tian}, Rokhlin and Xiao \cite{rokh}, Osipov and Rokhlin \cite{osi1}, Ogburn {\em et al.}~\cite{ogbu} and Huang {\em et al.}~\cite{huan}, and the books by Hogan and Lakey \cite{hoga}, and by Osipov, Rokhlin and Xiao \cite{osi2}.
Different strategies have been used to solve the angular spheroidal wave equation. The classical procedure starts with the angular spheroidal wave function written as a series of solutions of another similar differential equation, commonly the Legendre one, with coefficients obeying a three term recurrence relation. The resulting expansion becomes convergent only when such coefficients constitute a minimal solution of the recurrence relation. The eigenvalue problem encountered in this way is solved either as a transcendental equation involving a continued fraction, or written in a matrix form. Procedures based on the direct solution of the angular spheroidal equation, without having recourse to comparison with other differential equations, have been less frequently used. The relaxation method proposed by Caldwell \cite{cald} and reproduced, as a worked example, in the {\it Numerical Recipes} \cite[Sec. 17.4]{pres}, and the finite difference algorithm, described in the recently appeared paper by Ogburn {\em et al.} \cite{ogbu}, deserve to be quoted. Here we suggest to follow a procedure, based also on the direct treatment of the spheroidal equation, which benefits from an idea that can be found in a paper by Skorokhodov and Khristoforov \cite{skor} dealing with the singularities of the eigenvalues $\lambda_{m,n}$ considered as function of the (complex) prolateness parameter $c$. A shooting method is used. But, instead of imposing the boundary conditions to a numerically obtained solution, algebraic regular solutions around the regular point $\eta=0$ or around the regular singular point $\eta=1$ are written. Smooth matching of both solutions, i. e. cancelation of their Wronskian, at any point $\eta\in(-1, 1)$ determines the eigenvalues. In our implementation of the procedure, we choose $\eta=0$ as matching point.
A discomfort, when dealing with spheroidal wave functions, is the lack of universality of the notation used to represent them. The {\em Digital Library of Mathematical Functions} \cite[Chap.~30]{nist} provides information about the different notations found in the bibliography. Here we adopt, for the eigenvalues and eigenfunctions, the notation of the {\em Handbook of Mathematical Functions} \cite[Chap.~21]{abra}. The same notation is used in Ref. \cite{lilw}, a paper whose results we will try to reproduce, for comparison, with the method here developed.
In the next section, we recall the angular spheroidal equation and write its solutions in the form of power series expansions around the origin and around the singular point $\eta=1$. The procedure for computing the eigenvalues is presented in Section 3. The results of its application in some typical cases are also given. Section 4 shows that normalized eigenfunctions can be trivially obtained. Some figures illustrate the procedure. A few final comments are contained in Section 5.
\section{The differential equation}
The angular spheroidal wave function $S_{m,n}(c,\eta)$, defined in the interval $-1\leq \eta \leq 1$, satisfies the differential equation \cite[Eq. 21.6.2]{abra} \begin{equation} \frac{d}{d\eta}\left[(1-\eta^2)\frac{d}{d\eta}S_{m,n}(c,\eta)\right]+\left(\lambda_{m,n}-c^2\eta^2-\frac{m^2}{1-\eta^2}\right)S_{m,n}(c,\eta)=0 \label{ii1} \end{equation} stemming from the separation of the wave equation in spheroidal coordinates, with separation constants $m$ and $\lambda_{m,n}$. Periodicity of the azimuthal part of the wave restricts the values of $m$ to the integers and, given the invariance of the differential equation in the reflection $m\Rightarrow -m$, only non-negative integer values of $m$ need to be considered. The other separation constant, $\lambda_{m,n}$, commonly referred to as eigenvalue, must be such that $S_{m,n}(c,\eta)$ becomes finite at the singular points $\eta=\pm 1$. Their different values, for given $m$ and $c^2$, are labeled by the integer $n$. In most applications, the external parameter $c^2$ is real, positive in the case of prolate coordinates and negative for oblate ones. There are, however, interesting cases corresponding to complex values of $c^2$ \cite{aaab,barr,lilw,ogbu,oguc,skor}.
Instead of solving directly Eq. (\ref{ii1}), it is convenient to introduce the change of function \begin{equation} S_{m,n}(c,\eta)=(1-\eta^2)^{m/2}\,w(\eta)\,, \label{ii2} \end{equation} and to solve the differential equation \begin{equation} (1-\eta^2)\,\frac{d^2}{d\eta^2}w(\eta)-2(m+1)\,\eta\,\frac{d}{d\eta}w(\eta) +\left(z-c^2\,\eta^2\right)w(\eta)=0\,, \label{ii3} \end{equation} where \begin{equation} z\equiv \lambda_{m,n}-m(m+1) \label{ii4} \end{equation} is considered as the new eigenvalue.
Two independent solutions about the ordinary point $\eta=0$, valid in the interval $-1<\eta<1$, are \begin{equation} w_\sigma(\eta)=\sum_{k=0}^\infty\,a_{k,\sigma}\,\eta^{k+\sigma}\,, \qquad \sigma=0, 1\,, \label{ii5} \end{equation} with coefficients given by the recurrence relation \begin{eqnarray} a_{0,\sigma}=1\,,\quad a_{1,0}=0\,,\quad (k+\sigma)(k-1+\sigma)\,a_{k,\sigma}=& & \nonumber \\ & & \hspace{-160pt} \left[(k-1+2m+\sigma)(k-2+\sigma)-z\right]\,a_{k-2,\sigma}+c^2\,a_{k-4,\sigma}\,. \label{ii6} \end{eqnarray} Obviously, $w_0(\eta)$ and $w_1(\eta)$ are respectively even and odd functions of $\eta$.
Solutions about the regular singular point $\eta=1$ can also be written. In terms of the variables \begin{equation} t\equiv 1-\eta\,, \qquad u(t)\equiv w(1-\eta)\,, \label{ii7} \end{equation} the differential equation (\ref{ii3}) turns into \begin{equation} t(2-t)\frac{d^2}{dt^2}u(t)+2(m+1)(1-t)\,\frac{d}{dt}u(t)+\left[z-c^2(1-t)^2\right]u(t)=0\,. \label{ii8} \end{equation} The solution of this equation which makes $S_{m,n}$ to be regular at $t=0$ is, except for an arbitrary multiplicative constant, \begin{equation} u_{\rm reg}(t)=\sum_{j=0}^\infty b_j\,t^j\,, \label{ii9} \end{equation} with coefficients given by \begin{eqnarray} b_0=1\,,\qquad 2j(j+m)\,b_j=\left[(j-1)(j+2m)-z+c^2\right]b_{j-1}& & \nonumber\\ & & \hspace{-80pt} -\,2c^2\,b_{j-2}+c^2\,b_{j-3}\,. \label{ii10} \end{eqnarray} In terms of the variable $\eta$, this regular solution, valid for $-1<\eta\leq 1$, is \begin{equation} w_{\rm reg}(\eta)=u_{\rm reg}(1-\eta)\,. \label{ii11} \end{equation}
\section{The eigenvalues}
The problem of finding the eigenvalues $\lambda_{m,n}$, for given $c^2$ and $m$, reduces to require the regularity of $w_\sigma(\eta)$ at $\eta=1$. (The regularity at $\eta=-1$ is then implied by the symmetry of $w_\sigma$.) In the particular case of being $c^2=0$, the problem can be solved algebraically. The recurrence relation in (\ref{ii6}) reduces in this case to \begin{equation} (k+\sigma)(k-1+\sigma)\,a_{k,\sigma}=
\left[(k-1+2m+\sigma)(k-2+\sigma)-z\right]\,a_{k-2,\sigma}\,. \label{iii1}\\ \end{equation} Obviously, the series in the right hand side of (\ref{ii5}) is divergent for $\eta=1$ unless the value of $z$ is such that one of the $a_{k,\sigma}$ of even subindex, say $a_{2K+2,\sigma}$ (with $K=0, 1, 2, \ldots$), becomes zero, in which case $w_\sigma(\eta)$ turns out to be a polynomial of degree $2K+\sigma$. This happens for \begin{equation} z=(2K+1+2m+\sigma)(2K+\sigma). \label{iii2} \end{equation} By using the habitual notation \begin{equation} 2K+\sigma \equiv n-m \label{iii3} \end{equation} for the degree of the polynomial, we obtain for the eigenvalues in the case of $c^2=0$ \begin{equation} z=(n+m+1)(n-m)\,, \label{iii4} \end{equation} that is, in view of (\ref{ii4}), \begin{equation} \lambda_{m,n}(c^2=0)=n(n+1)\,. \label{iii5} \end{equation}
For $c^2\neq 0$, a convenient way of guaranteeing the regularity of $w_\sigma(\eta)$ at $\eta=1$ is to require the cancelation of the Wronskian $W$ of $w_\sigma$ and $w_{\rm reg}$, \begin{eqnarray} W\left[w_\sigma,\,w_{\rm reg}\right](\eta)=-\left(\sum_{k=0}^\infty a_{k,\sigma}\,\eta^{k+\sigma}\right)\left(\sum_{j=0}^\infty jb_j\,(1-\eta)^{j-1}\right) & & \nonumber \\ & & \hspace{-200pt}-\,\left(\sum_{k=0}^\infty (k+\sigma) a_{k,\sigma}\,\eta^{k-1+\sigma}\right)\left(\sum_{j=0}^\infty b_j\,(1-\eta)^j\right), \label{iii6} \end{eqnarray} at an arbitrarily chosen point of the interval $-1<\eta<1$. From the computational point of view, an interesting choice of $\eta$ seems to be $\eta=1/2$, in which case \begin{equation} W\left[w_\sigma,\,w_{\rm reg}\right](1/2)=-\sum_{l=0}^\infty 2^{-l}\,(l+1)\left(\sum_{j=0}^{l+1-\sigma}a_{l+1-\sigma-j, \sigma}\,b_j\right)\,. \label{iii7} \end{equation} The set of coefficients $\{a_{k,\sigma}\}$ and $\{b_j\}$ are solutions of the difference equations (\ref{ii6}) and (\ref{ii10}), respectively. According to the Perron-Kreuser theorem on difference equations \cite{kreu,per1,per2}, \begin{equation}
\limsup_{k\to\infty}\left(|a_{k,\sigma}|\right)^{1/k}=1\,, \qquad
\limsup_{j\to\infty}\left(|b_j|\right)^{1/j}=2^{-1}\,, \label{iii8} \end{equation} that is, for any given $\varepsilon>0$, constants $C_a$ and $C_b$ can be found such that \begin{equation}
|a_{k,\sigma}|<C_a(1+\varepsilon)^k\,\qquad |b_j|<C_b(2^{-1}+\varepsilon)^j \qquad {\rm for\; any}\quad k,j\geq 0\,. \label{iii9} \end{equation} This makes evident that the series in the right hand side of Eq. (\ref{iii7}) converges as fast as the geometric series $\sum_{l=0}^\infty 2^{-l}$. We consider, however, that a better choice of the value of $\eta$ in Eq. (\ref{iii6}) is $\eta=0$. In this case, \begin{equation} W\left[w_\sigma,\,w_{\rm reg}\right](0)=-\,\delta_{\sigma,0}\,\left(\sum_{j=0}^\infty j\,b_j\right) - \delta_{\sigma,1}\,\left(\sum_{j=0}^\infty b_j\right)\,. \label{iii10} \end{equation} Obviously, the cancelation of this Wronskian occurs when either $dw_{\rm reg}/d\eta$ or $w_{\rm reg}$ vanish at the origin, as it occurs for respectively even or odd functions of $\eta$. Needless to say, the right hand side of (\ref{iii10}) depends on the variable $z$, introduced in (\ref{ii4}), through the coefficients $b_j$. Therefore, the problem of finding the eigenvalues of the angular spheroidal equations, for given $c^2$ and $m$, reduces to the determination of the zeros of the function \begin{equation} \mathcal{W}_\sigma(z)\equiv -\,W\left[w_\sigma,\,w_{\rm reg}\right](0)=\delta_{\sigma,0}\,\left(\sum_{j=0}^\infty j\,b_j(z)\right) + \delta_{\sigma,1}\,\left(\sum_{j=0}^\infty b_j(z)\right)\,, \label{iii11} \end{equation} where we have indicated the dependence of the $b_j$ on $z$.
Different procedures can be used in the determination of the zeros of $\mathcal{W}_\sigma(z)$. A useful iterative method is the Newton one. Starting with an initial approximate value, $z^{(0)}$, of a certain zero, repeated application of the algorithm \begin{equation} z^{(i+1)}=z^{(i)}-\frac{\mathcal{W}_\sigma(z^{(i)})}{\mathcal{W}_\sigma^{\prime}(z^{(i)})} \label{iii12} \end{equation} allows one to get the value of the zero with the desired accuracy. The extended Newton method, which uses \begin{equation} z^{(i+1)}=z^{(i)}-\frac{\mathcal{W}_\sigma^\prime(z^{(i)})\pm \left[\left(\mathcal{W}_\sigma^\prime(z^{(i)})\right)^2-2\mathcal{W}_\sigma(z^{(i)})\,\mathcal{W}_\sigma^{\prime\prime}(z^{(i)})\right]^{1/2}} {\mathcal{W}_\sigma^{\prime\prime}(z^{(i)})}\,, \label{iii13} \end{equation} is even more efficient. For the first and second derivatives of $\mathcal{W}_\sigma(z)$ with respect to $z$ we have the expressions \begin{eqnarray} \mathcal{W}_\sigma^\prime(z)&=& \delta_{\sigma,0}\,\left(\sum_{j=0}^\infty j\,b_j^\prime(z)\right) + \delta_{\sigma,1}\,\left(\sum_{j=0}^\infty b_j^\prime(z)\right)\,, \label{iii14} \\ \mathcal{W}_\sigma^{\prime\prime}(z)&=& \delta_{\sigma,0}\,\left(\sum_{j=0}^\infty j\,b_j^{\prime\prime}(z)\right) + \delta_{\sigma,1}\,\left(\sum_{j=0}^\infty b_j^{\prime\prime}(z)\right)\,. \label{iii15} \end{eqnarray} The first and second derivatives of the coefficients $b_j(z)$ are easily obtained by means of the recurrence relations \begin{eqnarray} b_0^\prime(z)=0\,,\quad 2j(j+m)\,b_j^\prime(z)&=&\left[(j-1)(j+2m)-z+c^2\right]b_{j-1}^\prime(z) \nonumber \\ & & \hspace{-40pt} -\,2c^2\,b_{j-2}^\prime(z)+c^2\,b_{j-3}^\prime(z)-b_{j-1}(z)\,, \label{iii16} \\ b_0^{\prime\prime}(z)=0\,,\quad 2j(j+m)\,b_j^{\prime\prime}(z)&=&\left[(j-1)(j+2m)-z+c^2\right]b_{j-1}^{\prime\prime}(z) \nonumber \\ & & \hspace{-40pt} -\,2c^2\,b_{j-2}^{\prime\prime}(z)+c^2\,b_{j-3}^{\prime\prime}(z)-2b_{j-1}^\prime(z)\,, \label{iii17} \end{eqnarray} stemming from (\ref{ii10}). From these difference equations, inequalities analogous to the second one in (\ref{iii9}) can be deduced. Such inequalities guarantee the convergence, as fast as the geometric series $\sum_{j=0}^\infty 2^{-j}$, of the series in the right hand sides of Eqs. (\ref{iii11}), (\ref{iii14}), and (\ref{iii15}).
We have applied the procedure just described for obtaining the behaviour of the lowest eigenvalues of the spheroidal equation when the parameter $c^2$ varies in the interval $[-10, 10]$ and for values of $m=$0, 1, and 2. The results are shown in Figs. 1 to 3. (Remember that $\lambda_{m,n}=z_{m,n}+m(m+1)$.) A glance at Fig. 1 suggests a quasi-confluence of trajectories of the eigenvalues $\lambda_{0,0}$ and $\lambda_{0,1}$ for sufficiently large negative values of $c^2$. One may conjecture that, for larger negative values of $c^2$ other pairs of trajectories, those of $\lambda_{0,2j}$ and $\lambda_{0,2j+1}$, present such quasi-confluence. Table 1 shows that this is the case, and that a similar phenomenon occurs for other values of $m$. \begin{figure}
\caption{Trajectories of the lowest eigenvalues of the spheroidal equation with $m=0$ as $c^2$ varies in the interval $[-10, 10]$.}
\label{m0}
\end{figure} \begin{figure}
\caption{The same as in Figure 1, for $m=1$. Notice that, according to Eq. (\ref{ii4}), $\lambda_{1,n}=z_{1,n}+2$.}
\label{m1}
\end{figure} \begin{figure}
\caption{The same as in Figure 1, for $m=2$. In view of Eq. (\ref{ii4}), $\lambda_{2,n}=z_{2,n}+6$.}
\label{m2}
\end{figure} \begin {table} \caption {Lowest eigenvalues of the (oblate) spheroidal equation for several values of $m$ and $c^2$.} \begin{center}
\begin {tabular}{|l|r|r|r|} \hline \ & $m=0,\; c^2=-100$ & $m=1,\; c^2=-200$ & $m=2,\; c^2=-300$ \\ \hline $\lambda_{m,m+5}$ & $-15.328144254756$ & $-51.05126046795$ & $-83.77105335717$ \\ $\lambda_{m,m+4}$ & $-16.065564650326$ & $-51.08618015853$ & $-83.77516906231$ \\ $\lambda_{m,m+3}$ & $-45.483938701812$ & $-95.57183718390$ & $-138.78472876574$ \\ $\lambda_{m,m+2}$ & $-45.489793371378$ & $-95.57199196249$ & $-138.78474405855$ \\ $\lambda_{m,m+1}$ & $-81.027938023746$ & $-145.51102178558$ & $-199.22477209684$ \\ $\lambda_{m,m}$ & $-81.027943944958$ & $-145.51102194107$ & $-199.22477211250$ \\ \hline \end {tabular} \end{center} \end {table}
In order to compare with results published by other authors, we have applied our method to the computation of $\lambda_{m,n}$ for a sample of values of the parameters considered by Li {\em et al.} \cite{lilw} and by Ogburn {\em et al.} \cite{ogbu}. The comparison, shown in Table 2, allows one to conclude that, for moderate real values of $c^2$, the procedure used in Ref. \cite{lilw} is more reliable than the finite difference algorithm of Ref. \cite{ogbu}. \begin {table} \caption {Comparison of results obtained by using different procedures in the determination of the eigenvalues $\lambda_{m,n}$ of the spheroidal equation with real $c^2$.} \begin{center} \begin {tabular}{c c c l l l } \hline
$c^2$ & $m$ & $n$ & Ref. \cite{lilw} & Ref. \cite{ogbu} & this work\\ \hline $-$1 & 4 & 11 & 131.5600809 & 131.560080918303 & 131.56008091940694 \\ 0.1 & 2 & 2 & 6.014266314 & 6.014266356124070 & 6.0142663139415926 \\ 1 & 1 & 1 & 2.195548355 & 2.195612369653500 & 2.1955483554130039 \\ 1 & 2 & 2 & 6.140948992 & 6.140948969717170 & 6.1409489918576905 \\ 1 & 2 & 5 & 30.43614539 & 30.436145317468500 & 30.436145388713659 \\ 4 & 1 & 1 & 2.734111026 & 2.73415086499219 & 2.7341110256122556 \\ 4 & 2 & 2 & 6.542495274 & 6.54249530312951 & 6.5424952743905705 \\ 16 & 1 & 1 & 4.399593067 & 4.399599760664940 & 4.3995930671655061 \\ 16 & 2 & 5 & 36.99626750 & 36.996267483327900 & 36.996267500847930 \\ \hline \end {tabular} \end{center} \end {table}
Obviously, the procedure is also applicable in the case of complex $c^2$. Table 3 shows our results for different values of $c$, $m$, and $n$ considered in Ref \cite{lilw}. As it can be seen, the eigenvalues given by Li {\em et al.} are confirmed. Nevertheless, in the neighbourhood of each one of those eigenvalues, we have found another one, reported also in Table 3. This result is not surprising, because the values of $c$ considered are the approximations found by Oguchi \cite{oguc} to what he calls ``the branch points of the eigenvalues as functions of $c$", that is, in our formalism, values of $c$ for which a double zero of $\mathcal{W}_\sigma(z)$ exists. According to the results of Skorokhodov and Khristoforov \cite{skor}, there is a double eigenvalue $\lambda=1.705180091+4.220186348\,i$ for $c=1.824770749+2.601670693\,i$. This is a much better approximation to the branch point unveiled by Oguchi. As an illustration of what happens in the vicinity of those values of $c$, we present in Figure 4 a modulus-phase plot of the function $\mathcal{W}_0(z)$ for $c=1.824770+2.601670\,i$ and $m=0$, the first of the cases considered in Table 3. The two eigenvalues reported in the table appear as zeros of $\mathcal{W}_0(z)$. As the value of $c$ moves from the approximation to the branch point found by Oguchi towards the more precise value given by Skorokhodov and Khristoforov, the two zeros of $\mathcal{W}_0(z)$ shown in Fig.~4 approach to each other and eventually collide at a point in the close neighbourhood of the saddle point of $\mathcal{W}_0(z)$ suggested by its modulus-phase plot. Similar plots of $\mathcal{W}_\sigma(z)$ are obtained for the other cases in Table 3.
\begin {table} \caption {Pairs of eigenvalues $\lambda_{m,n}$ of the spheroidal equation for the complex values of $c$ given in Ref \cite{oguc} as corresponding to ``branch points".} \hspace{-20pt} \begin {tabular}{c c c c c } \hline
$c$ & $m$ & $n$ & Ref. \cite{lilw} & this work \\ \hline $1.824770+2.601670\,i$ & 0 & 0 & $1.701836+4.219998\,i$ & $1.701836497+4.219997758\,i$ \\
& & 2 & & $1.708523909+4.220369152\,i$ \\
& & & & \\ $2.094267+5.807965\,i$ & 0 & 0 & $1.993901+8.576325\,i$ & $1.993900944+8.576324731\,i$ \\
& & 4 & & $2.003141811+8.581103855\,i$ \\
& & & & \\ $5.217093+3.081362\,i$ & 0 & 2 & $23.91023+18.74194\,i$ & $23.91033400+18.74184255\,i$ \\
& & 4 & & $23.92132979+18.74479980\,i$ \\
& & & & \\ $3.563644+2.887165\,i$ & 0 & 1 & $10.13705+11.12216\,i$ & $10.13704735+11.12217988\,i$ \\
& & 3 & & $10.14462729+11.12098765\,i$ \\
& & & & \\ $1.998555+4.097453\,i$ & 1 & 1 & $2.919098+6.134851\,i$ & $2.919095372+6.134851876\,i$ \\
& & 3 & & $2.911544002+6.133045176\,i$ \\
& & & & \\ $3.862833+4.492300\,i$ & 1 & 2 & $12.19691+16.24534\,i$ & $12.19691647+16.24534182\,i$ \\
& & 4 & & $12.20527134+16.24281200\,i$ \\
& & & & \\ $2.136987+5.449457\,i$ & 2 & 2 & $6.098946+7.684379\,i$ & $6.098961456+7.684332819\,i$ \\
& & 4 & & $6.106119819+7.685191032\,i$ \\ \hline \end {tabular} \end {table} \begin{figure}
\caption{Modulus-phase plot of $\mathcal{W}_0(z)$ for $c=1.824770+2.601670\,i$, in the neighbourhood of a ``branch point", and $m=0$. Continuous and dashed lines are used to represent, respectively, the constant-modulus and constant-phase loci. Only the constant-modulus lines corresponding to $|\mathcal{W}_0(z)|=10^{-6}$ and $10^{-7}$ and the constant-phase lines for $\arg\mathcal{W}_0(z)=0,\; \pi/2,\; \pi$ and $3\pi/2$ have been drawn. }
\label{modfas}
\end{figure}
A comment concerning the values of the label $n$ of $\lambda_{m,n}$ reported in Table 3 is in order. For real or pure imaginary $c$, i.e. for real $c^2$, the eigenvalues $\lambda_{m,n}$ for given $m$ are real and can be ordered by increasing value. The label $n$ reflects that order. For complex $c^2$, instead, the values of $\lambda(c)$ become complex and such ordination is no more possible. Nevertheless, a label $n$ can be assigned to those complex values of $\lambda$, as done by Skorokhodov and Khristoforov. By keeping constant the real part of $c$ and continuously decreasing its imaginary part, $\lambda(c)$ describes, in the complex $\lambda$-plane, a trajectory which intersects the real $\lambda$-axis at a certain $\lambda_{m,n}$ for $\Im c=0$. This label $n$ can be attached to the whole trajectory described by $\lambda(c)$ as $c$ varies in the complex plane. The paper by Skorokhodov and Khristoforov contains a very lucid discussion of those trajectories and shows that the branch points $c_s$ correspond to singular values of $c$ such that $\lambda_{m,n}(c_s)=\lambda_{m,n+2p}(c_s)$, with $p=1, 2, \ldots $.
\section{The eigenfunctions}
Once the eigenvalues $\lambda_{m,n}$ have been calculated, the corresponding eigenfunctions, in the interval $0\leq\eta\leq 1$, can be obtained immediately by means of the series expansion \begin{equation} S_{m,n}(c,\eta)=\mathcal{N}\,e^{i\theta}\,(1-\eta^2)^{m/2}\,\sum_{j=0}^\infty b_j\,(1-\eta)^j\,, \label{iv1} \end{equation} the coefficients $b_j$ being given by the recurrence relation (\ref{ii10}) with $z=\lambda_{m,n}-m(m+1)$. The normalization constant $\mathcal{N}$ should be adjusted to the normalization scheme preferred. A discussion of the different normalizations used in the literature can be found in the paper by Kirby \cite{kirb}, where the advantage of the unit normalization \begin{equation}
\int_{-1}^1 |S_{m,n}(c,\eta)|^2\,d\eta=1 \label{iv2} \end{equation} is made evident. By choosing this normalization, one has \begin{equation} \mathcal{N}=\left[ \sum_{k=0}^m (-1)^k\,2^{m+1-k}\,{m \choose k}\,\sum_{l=0}^\infty \frac{\sum_{j=0}^l b_j\,b_{l-j}^*}{l+m+k+1}\right]^{-1/2}\,, \label{iv3} \end{equation} where the asterisk indicates complex conjugation. The constant phase $\theta$ in the right hand side of (\ref{iv1}) may be taken at will. In the case of real $c^2$, it is natural to take $\theta =0$. For complex $c^2$, $\theta$ can be chosen in such a way that $S_{m,n}$ becomes real at $\eta=0$, or at $\eta=1$, or at any other point. Needless to say, $S_{m,n}(c,-\eta)=\pm S_{m,n}(c,\eta)$, according to the even or odd nature of $S_{m,n}$.
Figures 5 and 6 show two examples of the application of the method to the computation of spheroidal angular wave functions in the case of real $c^2$. The first one is an even prolate angular wave function of parameters $c=3$, $m=0$ and $n=2$, and eigenvalue $\lambda_{0,2}=11.192938649526784$, a case considered in Table III of Ref. \cite{lilw} (with the Flammer \cite{flam} normalization scheme). The second one is an odd oblate angular wave function corresponding to the first of the cases considered in our Table 2. In both figures, the functions have been normalized according to Eq. (\ref{iv2}). We have applied our procedure also in some cases of complex $c$. Figure 7 shows the real and imaginary parts and the squared modulus of the wave function in a case considered by Falloon {\em et al.} \cite{fall}, namely the first one in their Table 2. The parameters are $c=1\! +\! i$, $m=0$ and $n=0$, and the eigenvalue, in our notation, is $\lambda_{0,0}=0.059472769735031+0.662825122194600\,i$. (We give here the eigenvalue with only 15 decimal digits, but our procedure is able to reproduce the 25 decimal digits given in Ref. \cite{fall} and to obtain even more.) Finally, Figures 8, 9 and 10 correspond to the second of the cases considered in Table 2 of Ref. \cite{ogbu}, of parameters $c=20(1\! +\! i)$, $m=0$, $n=3$, and eigenvalue $\lambda_{0,3}=58.226714354344554 + 60.025615481720256\,i$. (Notice the discrepancy, in the six last digits of both real and imaginary parts, with the value given in Ref. \cite{ogbu}.) \begin{figure}
\caption{Prolate spheroidal angular wave function of parameters $c=3$, $m=0$ and $n=2$, corresponding to the eigenvalue $\lambda_{0,2}=11.192938649526788$. Since $S_{0,2}(\eta)$ is an even function, we have omitted its representation in the interval $-1\leq\eta<0$. The normalization adopted is that prescribed in Eq. (\ref{iv2}).}
\label{wfeven}
\end{figure} \begin{figure}
\caption{Oblate spheroidal angular wave function of parameters $c=i$, $m=4$ and $n=11$, with eigenvalue $\lambda_{4,11}=131.56008091940694$. The function is an odd one. It has been normalized as in Eq. (\ref{iv2}).}
\label{wfodd}
\end{figure} \begin{figure}
\caption{Real and imaginary parts and squared modulus of the angular spheroidal wave function of parameters $c=1\! +\! i$, $m=0$ and $n=0$, and eigenvalue $\lambda_{0,0}=0.059472769735031+0.662825122194600\, i$, normalized to unit, as in Eq. (\ref{iv2}). Dashed and dotted lines are used to represent, respectively, the real and imaginary parts of $S_{0,0}(1\! +\! i, \eta)$, and a solid line for its squared modulus. The arbitrary phase $\theta$ in the right hand side of Eq. (\ref{iv1}) has been fixed in such a way that the wave function becomes real at the origin. Only the interval $0\leq\eta\leq 1$ has been considered. Needless to say, $S_{0,0}(c, -\eta)=S_{0,0}(c, \eta)$.}
\label{fallon}
\end{figure} \begin{figure}
\caption{Real and imaginary parts and squared modulus of the angular spheroidal wave function of parameters $c=20(1\! +\! i)$, $m=0$ and $n=3$, and eigenvalue $\lambda_{0,3}=58.226714354344554 + 60.025615481720256\, i$. The meaning of the lines and the normalization is the same as in Fig. 7. For the arbitrary phase $\theta$ in Eq. (\ref{iv1}) we have chosen the value $\theta=0$. Of course, $S_{0,3}(c, -\eta)=-\,S_{0,3}(c, \eta).$}
\label{barrowes0}
\end{figure} \begin{figure}
\caption{Magnification of the interval $0.6\leq\eta\leq 0.8$ of Fig. 8.}
\label{barrowes1}
\end{figure} \begin{figure}
\caption{Magnification of the interval $0.8\leq\eta\leq 1$ of Fig. 8.}
\label{barrowes2}
\end{figure}
\section{Conclusions}
We have developed a rarely used method which allows to find the eigenvalues and eigenfunctions of the angular spheroidal equation. Instead of having recourse to comparison with other differential equations, the procedure deals with a direct solution, expressed in the form of a convergent series. Requiring it to be regular gives the eigenvalues, which appear as the zeros of a one variable function, $\mathcal{W}_\sigma(z)$. This function and its derivatives $\mathcal{W}_\sigma^\prime(z)$ and $\mathcal{W}_\sigma^{\prime\prime}(z)$ with respect to the variable $z$ can be computed, to the desired precision, by summing rapidly convergent series. This fact makes possible the application of the extended Newton method for the determination of the zeros of $\mathcal{W}_\sigma(z)$, i. e., the eigenvalues of the spheroidal equation. Then, the computation of the corresponding eigenfunctions, conveniently normalized, becomes trivial. For the normalization, one benefits from the fact that the squared modulus of the wave function can be integrated algebraically.
The fact that, for given $c$ and $m$, the eigenvalues are the zeros of an easily computable function, $\mathcal{W}_\sigma(z)$, makes possible to get an initial approximate location of all of them by a tabulation or a graphical representation of that function. Repeated application of the extended Newton method allows then to calculate the eigenvalues with the desired accuracy.
We have shown the applicability of the method not only in the cases of prolate (real $c$) and oblate (imaginary $c$) spheroidal wave equations, but also when $c$ is complex. The procedure provides in all cases a very precise determination of the eigenvalues. This has allowed us to resolve the quasi-confluence of pairs of even-odd eigenvalues for large imaginary values of $c$ (Table 1) and of pairs of even-even or odd-odd eigenvalues for complex values of $c$ in the neighbourhood of ``branch points" (Table 3).
The efficiency of the procedure proposed in this paper is subordinate to the capability of computing $\mathcal{W}_\sigma(z)$ with sufficient accuracy. The convergence of the series in the right hand side of Eq. (\ref{iii11}) is guaranteed in all cases, since $|b_n|\sim 2^{-n}$ for all $n$ larger than a certain $N$, and the series may replaced by a sum up to say $j=j_{\rm max}$. Nevertheless, for large values of $c$ and/or $\lambda_{m,n}$, the coefficients $b_j$ increase (in modulus) rapidly with $j$ before starting to decrease, and the adequate value of $j_{\rm max}$ may become very large. Even worse, the values of the terms to be summed may cover so many orders of magnitude that the resulting sum is not reliable, unless many significant digits are carried along the computation. This drawback is not outside other procedures. However, the algorithms proposed by Kirby \cite{kirb} and by Ogburn {\em et al.} \cite{ogbu}, and procedures collected in Refs.~\cite{osi2}, seem to be able to tackle the issue properly. Asymptotic methods \cite{barr,rokh} have also been forwarded for the mentioned cases of large values of $c$ and/or $\lambda_{m,n}$.
\end{document} | arXiv |
Directed graphs represent binary relations. They can be visualized as diagrams made up of points (called vertices or nodes) and arrows (called arcs or edges). Draw an arc from a vertex $v$ to a vertex $w$ to represent that $v$ is related to $w$ i.e. the ordered pair $(v,w)$ is in the relation.
Example 1. An example of a directed graph
Figure 1. A directed graph
Definition. A directed graph or digraph in short is an ordered pair $(V,A)$ where $V$ is an nonempty set and $A\subseteq V\times V$ (i.e. $A$ is a binary relation on $V$). The members of $V$ are called vertices or nodes and the members of $A$ are called arcs or edges. We write arcs as $v\to w$ rather than $(v,w)$.
In Example 1, $V=\{a,b,c,d,e\}$ and \begin{align*}A&=\{(a,b), (b,c), (a,c), (c,c), (c,d), (b,d),(d,b)\}\\&=\{a\to b, b\to c, a\to c, c\to c, c\to d, b\to d, d\to b\}\end{align*}
Transportation and computer networks have natural representations of digraphs.
A walk in a digraph is a way of proceeding through a sequence of vertices by following arcs i.e. a walk in a digraph $(V,A)$ is a sequence of vertices $v_0,v_1,\cdots,v_n\in V$ for some $n\geq 0$ such that $v_i\to v_{i+1}\in A$ for each $i<n$. The length of this walk is $n$ which is the number of arcs.
Example 2. In Example 1, $b\to d$, $b\to c\to d$, $b\to c\to c\to d$, $b\to d\to b\to d$ are examples of walks from vertex $b$ to vertex $d$. The length of $b\to d$ is 1, the length of $b\to c\to d$ is 2, the length of $b\to c\to c\to d$ is 3 and the length of $b\to d\to b\to d$ is also 3. There is no walk from vertex $b$ to vertex $a$.
A path is a walk that doesn't repeat any vertex.
Example 3. Among the walks in Example 2, $b\to d$ and $b\to c\to d$ are paths from vertex $b$ to $d$.
A walk in which the first and the last vertex are the same is called a circuit. A circuit is called a cycle if the first and the last vertices are the only repeated vertex. For example, $b\to c\to c\to b\to b$ in the digraph in Example 1 is a circuit but is not a cycle. On the other hand, $b\to c\to d\to b$ is a cycle of length 3, $b\to d\to b$ is a cycle of length 2, and $c\to c$ is a cycle of length 1.
A digraph without any cycles is said to be acyclic. The digraph in Exam 1 is not acyclic as it contains cycles.
A walk can be reduced to a path by removing nontrivial cycles. Suppose that a walk from $v$ to $w$ $$v=v_0\to\cdots\to v_n=w$$ includes a cycle $v_i\to v_{i+1}\to\cdots\to v_j$ where $i<j$ and $v_i=v_j$. Then $$v=v_0\to\cdots\to v_i\to v_{j+1}\to\cdots\to v_n=w$$ is a shorter walk from $v$ to $w$.
Figure 2. Shortening of a walk by removing a cycle
For example, the walk $b\to d\to b\to d$ in Example 1 can be reduced to the path $b\to d$ by removing the cycle $b\to d\to d$.
A vertex $w$ is said to be reachable from vertex $v$ if there is a walk or a path from $v$ to $w$. The distance from vertex $v$ to vertex $w$ in a digraph $G$, denoted by $d_G(v,w)$, is the length of the shortest path from $v$ to $w$, or is defined to be $\infty$ if there is no path from $v$ to $w$. For example, the distance from $a$ to $d$ in the digraph in Example 1 is 2 because the shortest path is $a\to c\to d$.
Lemma. The distance from one vertex of a graph to another vertex is at most the length of any walk from the first to the second.
Proof. It follows from the fact that any walk from $v$ to $w$ includes among its arcs a path from $v$ to $w$.
A digraph in which every vertex is reachable from every other vertex is said to be strongly connected.
Let $G=(V,A)$ be a digraph. Let $V'\subseteq V$ and $A'\subseteq A$. Then $(V',A')$ is called a subgraph of $G$. $(V,\emptyset)$ is a subgraph of $G$.
Let $G=(V,A)$ be a digraph and $V'\subset V$. Then $$(V',\{v\to w\in A: v,w\in V'\})$$ is called the subgraph induced by $V'$.
Example 4. The subgraph induced by $V\setminus\{e\}$ in Example 1 is not strongly connected. But the subgraph induced by $\{b,c,d\}$ is strongly connected.
Figure 3. The subgraph of the digraph in Example 1, that is induced by V-{e}.
Figure 4. The subgraph of the digraph in Example 1, that is induced by {b,c,d}.
An acyclic digraph is generally called a directed acyclic group or DAG in short.
The out-degree of a vertex is the number of arcs leaving it i.e. $|\{w\in V: v\to w\ \mbox{in}A\}|$. Similarly, the in-degree of a vertex is the number of arcs entering it.
Theorem. A finite DAG has at least one vertex of out-degree 0 and at least one vertex of in-degree 0.
Proof. Let $G=(V,A)$ be a finite DAG. Suppose that $G$ has no vertex of out-degree 0.
Figure 5. Example of vertices whose in-degree is 0 or whose out-degree is 0.
Pick a vertex $v_0$. Since $v_0$ has a positive out-degree, there exists an arc $v_0\to v_1$ for some vertex $v_1$. Since $v_1$ has a positive out-degree, there exists an arc $v_1\to v_2$ for some vertex $v_2$. One can continue doing this. But since $V$ is finite, some vertex will be repeated, creating a cycle. This is a contradiction to the graph $G$ being acyclic. A very similar argument can be made to show that $G$ has a vertex of in-degree 0.
In a DAG, a vertex of in-degree 0 is called a source and a vertex of out-degree 0 is called a sink.
A tournament graph
A tournament graph is a digraph in which every pair of distinct vertices are connected by an arc in one direction or the other, but not both. It is a natural representation of a round-robin tournament in which each players competes with all other plays in turn.
Example 5. The tournament graph in Figure 6 shows that $H$ beats both $P$ and $Y$, $Q$ beats $H$, $Y$ beats $P$, and $P$ beats $Q$. Hence we have a cycle $H\to P\to Q\to H$. Awkwardly, there is no champion!
Figure 6. A tournament graph
Example 6. The tournament graph in Figure 7 shows that $H$ beats $P$, $Y$, $D$, $Y$ beats $P$ and $D$, and $P$ beats $D$. Hence, $H$ is the first, $Y$ is the second, $P$ the third, and $D$ is the fourth places.
The total number of arcs in a tournament graph with $n$ vertices is $\frac{n(n-1)}{2}$, since there is exactly one arc between each pair of distinct vertices.
A linear order, denoted by $\preceq$, is a binary relation on a finite set $S=\{s_0,s_1,\cdots,s_n\}$ such that $s_i\preceq s_j$ if and only if $i\leq j$.
Example 7. Let $S=\{\mbox{all English words}\}$. Define $s_i\preceq s_j$ to mean that $s_i$ appears before $s_j$ alphabetically or they are equal. Then $\preceq$ is a linear order. This particular linear order is called the lexicographic order or the dictionary order.
A strict order of a finite set, denoted by $\prec$, is an ordering such that $s_i\prec s_j$ if and only if $i<j$.
Theorem. A tournament graph represents a strict linear order if and only if it is a DAG.
Proof. Let $G=(V,A)$ be a tournament graph. Suppose that $G$ represents a strict order. This means that the path $$v_0\to v_1\to\cdots\to v_n$$ represents $$v_0\prec v_1\prec\cdots\prec v_n$$ so every arc goes from $v_i$ to $v_j$ if $i<j$. The graph is then acyclic because any cycle will have to include at least one arc $v_j\to v_i$ where $i<j$. Conversely suppose $G$ is a DAG. We show by induction that $G$ represents a strict order. If $G$ has only one vertex, clearly $G$ represents a strict order. If $G$ has more than one vertex, at least one of the vertices, say $v_0$, has in-degree 0. Since $G$ is a tournament graph, $G$ must contain each of the arc $v_0\to v_i$ ($i\ne 0$). Consider the subgraph induced by $V\setminus\{v_0\}$. This induced subgraph is also a DAG, since all its arcs are arcs of $G$. It is also a tournament graph. The subgraph includes all the arcs of $G$ except those entering from $v_0$. By induction hypothesis, the induced subgraph represents a strict linear order $$v_1\prec v_2\prec\cdots\prec v_n$$ Since $v_0\prec v_i$ for all $i\ne 0$, we have $$v_0\prec v_1\prec\cdots\prec v_n$$ This completes the proof.
Example 8. The tournament graph in Example 7 is a DAG and it represents a strict order $H\prec Y\prec P\prec D$, while the tournament graph in Example 6 doesn't. (It is not a DAG.)
[1] Essential Discrete Mathematics for Computer Science, Harry Lewis and Rachel Zax, Princeton University Press, 2019
This entry was posted in Discrete Mathematics on October 24, 2019 by Sung Lee.
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Dirac bracket
The Dirac bracket is a generalization of the Poisson bracket developed by Paul Dirac[1] to treat classical systems with second class constraints in Hamiltonian mechanics, and to thus allow them to undergo canonical quantization. It is an important part of Dirac's development of Hamiltonian mechanics to elegantly handle more general Lagrangians; specifically, when constraints are at hand, so that the number of apparent variables exceeds that of dynamical ones.[2] More abstractly, the two-form implied from the Dirac bracket is the restriction of the symplectic form to the constraint surface in phase space.[3]
Not to be confused with bra-ket notation, also known as Dirac notation.
This article assumes familiarity with the standard Lagrangian and Hamiltonian formalisms, and their connection to canonical quantization. Details of Dirac's modified Hamiltonian formalism are also summarized to put the Dirac bracket in context.
Inadequacy of the standard Hamiltonian procedure
The standard development of Hamiltonian mechanics is inadequate in several specific situations:
1. When the Lagrangian is at most linear in the velocity of at least one coordinate; in which case, the definition of the canonical momentum leads to a constraint. This is the most frequent reason to resort to Dirac brackets. For instance, the Lagrangian (density) for any fermion is of this form.
2. When there are gauge (or other unphysical) degrees of freedom which need to be fixed.
3. When there are any other constraints that one wishes to impose in phase space.
Example of a Lagrangian linear in velocity
An example in classical mechanics is a particle with charge q and mass m confined to the x - y plane with a strong constant, homogeneous perpendicular magnetic field, so then pointing in the z-direction with strength B.[4]
The Lagrangian for this system with an appropriate choice of parameters is
$L={\tfrac {1}{2}}m{\vec {v}}^{2}+{\frac {q}{c}}{\vec {A}}\cdot {\vec {v}}-V({\vec {r}}),$
where →A is the vector potential for the magnetic field, →B; c is the speed of light in vacuum; and V(→r) is an arbitrary external scalar potential; one could easily take it to be quadratic in x and y, without loss of generality. We use
${\vec {A}}={\frac {B}{2}}(x{\hat {y}}-y{\hat {x}})$
as our vector potential; this corresponds to a uniform and constant magnetic field B in the z direction. Here, the hats indicate unit vectors. Later in the article, however, they are used to distinguish quantum mechanical operators from their classical analogs. The usage should be clear from the context.
Explicitly, the Lagrangian amounts to just
$L={\frac {m}{2}}({\dot {x}}^{2}+{\dot {y}}^{2})+{\frac {qB}{2c}}(x{\dot {y}}-y{\dot {x}})-V(x,y)~,$
which leads to the equations of motion
$m{\ddot {x}}=-{\frac {\partial V}{\partial x}}+{\frac {qB}{c}}{\dot {y}}$
$m{\ddot {y}}=-{\frac {\partial V}{\partial y}}-{\frac {qB}{c}}{\dot {x}}.$
For a harmonic potential, the gradient of V amounts to just the coordinates, −(x,y).
Now, in the limit of a very large magnetic field, qB/mc ≫ 1. One may then drop the kinetic term to produce a simple approximate Lagrangian,
$L={\frac {qB}{2c}}(x{\dot {y}}-y{\dot {x}})-V(x,y)~,$
with first-order equations of motion
${\dot {y}}={\frac {c}{qB}}{\frac {\partial V}{\partial x}}$
${\dot {x}}=-{\frac {c}{qB}}{\frac {\partial V}{\partial y}}~.$
Note that this approximate Lagrangian is linear in the velocities, which is one of the conditions under which the standard Hamiltonian procedure breaks down. While this example has been motivated as an approximation, the Lagrangian under consideration is legitimate and leads to consistent equations of motion in the Lagrangian formalism.
Following the Hamiltonian procedure, however, the canonical momenta associated with the coordinates are now
$p_{x}={\frac {\partial L}{\partial {\dot {x}}}}=-{\frac {qB}{2c}}y$
$p_{y}={\frac {\partial L}{\partial {\dot {y}}}}={\frac {qB}{2c}}x~,$
which are unusual in that they are not invertible to the velocities; instead, they are constrained to be functions of the coordinates: the four phase-space variables are linearly dependent, so the variable basis is overcomplete.
A Legendre transformation then produces the Hamiltonian
$H(x,y,p_{x},p_{y})={\dot {x}}p_{x}+{\dot {y}}p_{y}-L=V(x,y).$
Note that this "naive" Hamiltonian has no dependence on the momenta, which means that equations of motion (Hamilton's equations) are inconsistent.
The Hamiltonian procedure has broken down. One might try to fix the problem by eliminating two of the components of the 4-dimensional phase space, say y and py, down to a reduced phase space of 2 dimensions, that is sometimes expressing the coordinates as momenta and sometimes as coordinates. However, this is neither a general nor rigorous solution. This gets to the heart of the matter: that the definition of the canonical momenta implies a constraint on phase space (between momenta and coordinates) that was never taken into account.
Generalized Hamiltonian procedure
In Lagrangian mechanics, if the system has holonomic constraints, then one generally adds Lagrange multipliers to the Lagrangian to account for them. The extra terms vanish when the constraints are satisfied, thereby forcing the path of stationary action to be on the constraint surface. In this case, going to the Hamiltonian formalism introduces a constraint on phase space in Hamiltonian mechanics, but the solution is similar.
Before proceeding, it is useful to understand the notions of weak equality and strong equality. Two functions on phase space, f and g, are weakly equal if they are equal when the constraints are satisfied, but not throughout the phase space, denoted f ≈ g. If f and g are equal independently of the constraints being satisfied, they are called strongly equal, written f = g. It is important to note that, in order to get the right answer, no weak equations may be used before evaluating derivatives or Poisson brackets.
The new procedure works as follows, start with a Lagrangian and define the canonical momenta in the usual way. Some of those definitions may not be invertible and instead give a constraint in phase space (as above). Constraints derived in this way or imposed from the beginning of the problem are called primary constraints. The constraints, labeled φj, must weakly vanish, φj (p,q) ≈ 0.
Next, one finds the naive Hamiltonian, H, in the usual way via a Legendre transformation, exactly as in the above example. Note that the Hamiltonian can always be written as a function of qs and ps only, even if the velocities cannot be inverted into functions of the momenta.
Generalizing the Hamiltonian
Dirac argues that we should generalize the Hamiltonian (somewhat analogously to the method of Lagrange multipliers) to
$H^{*}=H+\sum _{j}c_{j}\phi _{j}\approx H,$
where the cj are not constants but functions of the coordinates and momenta. Since this new Hamiltonian is the most general function of coordinates and momenta weakly equal to the naive Hamiltonian, H* is the broadest generalization of the Hamiltonian possible so that δH * ≈ δH when δφj ≈ 0.
To further illuminate the cj, consider how one gets the equations of motion from the naive Hamiltonian in the standard procedure. One expands the variation of the Hamiltonian out in two ways and sets them equal (using a somewhat abbreviated notation with suppressed indices and sums):
$\delta H={\frac {\partial H}{\partial q}}\delta q+{\frac {\partial H}{\partial p}}\delta p\approx {\dot {q}}\delta p-{\dot {p}}\delta q~,$
where the second equality holds after simplifying with the Euler-Lagrange equations of motion and the definition of canonical momentum. From this equality, one deduces the equations of motion in the Hamiltonian formalism from
$\left({\frac {\partial H}{\partial q}}+{\dot {p}}\right)\delta q+\left({\frac {\partial H}{\partial p}}-{\dot {q}}\right)\delta p=0~,$
where the weak equality symbol is no longer displayed explicitly, since by definition the equations of motion only hold weakly. In the present context, one cannot simply set the coefficients of δq and δp separately to zero, since the variations are somewhat restricted by the constraints. In particular, the variations must be tangent to the constraint surface.
One can demonstrate that the solution to
$\sum _{n}A_{n}\delta q_{n}+\sum _{n}B_{n}\delta p_{n}=0,$
for the variations δqn and δpn restricted by the constraints Φj ≈ 0 (assuming the constraints satisfy some regularity conditions) is generally[5]
$A_{n}=\sum _{m}u_{m}{\frac {\partial \phi _{m}}{\partial q_{n}}}$
$B_{n}=\sum _{m}u_{m}{\frac {\partial \phi _{m}}{\partial p_{n}}},$
where the um are arbitrary functions.
Using this result, the equations of motion become
${\dot {p}}_{j}=-{\frac {\partial H}{\partial q_{j}}}-\sum _{k}u_{k}{\frac {\partial \phi _{k}}{\partial q_{j}}}$
${\dot {q}}_{j}={\frac {\partial H}{\partial p_{j}}}+\sum _{k}u_{k}{\frac {\partial \phi _{k}}{\partial p_{j}}}$
$\phi _{j}(q,p)=0,$
where the uk are functions of coordinates and velocities that can be determined, in principle, from the second equation of motion above.
The Legendre transform between the Lagrangian formalism and the Hamiltonian formalism has been saved at the cost of adding new variables.
Consistency conditions
The equations of motion become more compact when using the Poisson bracket, since if f is some function of the coordinates and momenta then
${\dot {f}}\approx \{f,H^{*}\}_{PB}\approx \{f,H\}_{PB}+\sum _{k}u_{k}\{f,\phi _{k}\}_{PB},$
if one assumes that the Poisson bracket with the uk (functions of the velocity) exist; this causes no problems since the contribution weakly vanishes. Now, there are some consistency conditions which must be satisfied in order for this formalism to make sense. If the constraints are going to be satisfied, then their equations of motion must weakly vanish, that is, we require
${\dot {\phi _{j}}}\approx \{\phi _{j},H\}_{PB}+\sum _{k}u_{k}\{\phi _{j},\phi _{k}\}_{PB}\approx 0.$
There are four different types of conditions that can result from the above:
1. An equation that is inherently false, such as 1=0 .
2. An equation that is identically true, possibly after using one of our primary constraints.
3. An equation that places new constraints on our coordinates and momenta, but is independent of the uk.
4. An equation that serves to specify the uk.
The first case indicates that the starting Lagrangian gives inconsistent equations of motion, such as L = q. The second case does not contribute anything new.
The third case gives new constraints in phase space. A constraint derived in this manner is called a secondary constraint. Upon finding the secondary constraint one should add it to the extended Hamiltonian and check the new consistency conditions, which may result in still more constraints. Iterate this process until there are no more constraints. The distinction between primary and secondary constraints is largely an artificial one (i.e. a constraint for the same system can be primary or secondary depending on the Lagrangian), so this article does not distinguish between them from here on. Assuming the consistency condition has been iterated until all of the constraints have been found, then φj will index all of them. Note this article uses secondary constraint to mean any constraint that was not initially in the problem or derived from the definition of canonical momenta; some authors distinguish between secondary constraints, tertiary constraints, et cetera.
Finally, the last case helps fix the uk. If, at the end of this process, the uk are not completely determined, then that means there are unphysical (gauge) degrees of freedom in the system. Once all of the constraints (primary and secondary) are added to the naive Hamiltonian and the solutions to the consistency conditions for the uk are plugged in, the result is called the total Hamiltonian.
Determination of the uk
The uk must solve a set of inhomogeneous linear equations of the form
$\{\phi _{j},H\}_{PB}+\sum _{k}u_{k}\{\phi _{j},\phi _{k}\}_{PB}\approx 0.$
The above equation must possess at least one solution, since otherwise the initial Lagrangian is inconsistent; however, in systems with gauge degrees of freedom, the solution will not be unique. The most general solution is of the form
$u_{k}=U_{k}+V_{k},$
where Uk is a particular solution and Vk is the most general solution to the homogeneous equation
$\sum _{k}V_{k}\{\phi _{j},\phi _{k}\}_{PB}\approx 0.$
The most general solution will be a linear combination of linearly independent solutions to the above homogeneous equation. The number of linearly independent solutions equals the number of uk (which is the same as the number of constraints) minus the number of consistency conditions of the fourth type (in previous subsection). This is the number of unphysical degrees of freedom in the system. Labeling the linear independent solutions Vka where the index a runs from 1 to the number of unphysical degrees of freedom, the general solution to the consistency conditions is of the form
$u_{k}\approx U_{k}+\sum _{a}v_{a}V_{k}^{a},$
where the va are completely arbitrary functions of time. A different choice of the va corresponds to a gauge transformation, and should leave the physical state of the system unchanged.[6]
The total Hamiltonian
At this point, it is natural to introduce the total Hamiltonian
$H_{T}=H+\sum _{k}U_{k}\phi _{k}+\sum _{a,k}v_{a}V_{k}^{a}\phi _{k}$
and what is denoted
$H'=H+\sum _{k}U_{k}\phi _{k}.$
The time evolution of a function on the phase space, f is governed by
${\dot {f}}\approx \{f,H_{T}\}_{PB}.$
Later, the extended Hamiltonian is introduced. For gauge-invariant (physically measurable quantities) quantities, all of the Hamiltonians should give the same time evolution, since they are all weakly equivalent. It is only for nongauge-invariant quantities that the distinction becomes important.
The Dirac bracket
Above is everything needed to find the equations of motion in Dirac's modified Hamiltonian procedure. Having the equations of motion, however, is not the endpoint for theoretical considerations. If one wants to canonically quantize a general system, then one needs the Dirac brackets. Before defining Dirac brackets, first-class and second-class constraints need to be introduced.
We call a function f(q, p) of coordinates and momenta first class if its Poisson bracket with all of the constraints weakly vanishes, that is,
$\{f,\phi _{j}\}_{PB}\approx 0,$
for all j. Note that the only quantities that weakly vanish are the constraints φj, and therefore anything that weakly vanishes must be strongly equal to a linear combination of the constraints. One can demonstrate that the Poisson bracket of two first-class quantities must also be first class. The first-class constraints are intimately connected with the unphysical degrees of freedom mentioned earlier. Namely, the number of independent first-class constraints is equal to the number of unphysical degrees of freedom, and furthermore, the primary first-class constraints generate gauge transformations. Dirac further postulated that all secondary first-class constraints are generators of gauge transformations, which turns out to be false; however, typically one operates under the assumption that all first-class constraints generate gauge transformations when using this treatment.[7]
When the first-class secondary constraints are added into the Hamiltonian with arbitrary va as the first-class primary constraints are added to arrive at the total Hamiltonian, then one obtains the extended Hamiltonian. The extended Hamiltonian gives the most general possible time evolution for any gauge-dependent quantities, and may actually generalize the equations of motion from those of the Lagrangian formalism.
For the purposes of introducing the Dirac bracket, of more immediate interest are the second class constraints. Second class constraints are constraints that have a nonvanishing Poisson bracket with at least one other constraint.
For instance, consider second-class constraints φ1 and φ2 whose Poisson bracket is simply a constant, c,
$\{\phi _{1},\phi _{2}\}_{PB}=c~.$
Now, suppose one wishes to employ canonical quantization, then the phase-space coordinates become operators whose commutators become iħ times their classical Poisson bracket. Assuming there are no ordering issues that give rise to new quantum corrections, this implies that
$[{\hat {\phi }}_{1},{\hat {\phi }}_{2}]=i\hbar ~c,$
where the hats emphasize the fact that the constraints are on operators.
On one hand, canonical quantization gives the above commutation relation, but on the other hand φ1 and φ2 are constraints that must vanish on physical states, whereas the right-hand side cannot vanish. This example illustrates the need for some generalization of the Poisson bracket which respects the system's constraints, and which leads to a consistent quantization procedure. This new bracket should be bilinear, antisymmetric, satisfy the Jacobi identity as does the Poisson bracket, reduce to the Poisson bracket for unconstrained systems, and, additionally, the bracket of any second-class constraint with any other quantity must vanish.
At this point, the second class constraints will be labeled ${\tilde {\phi }}_{a}$. Define a matrix with entries
$M_{ab}=\{{\tilde {\phi }}_{a},{\tilde {\phi }}_{b}\}_{PB}.$
In this case, the Dirac bracket of two functions on phase space, f and g, is defined as
$\{f,g\}_{DB}=\{f,g\}_{PB}-\sum _{a,b}\{f,{\tilde {\phi }}_{a}\}_{PB}M_{ab}^{-1}\{{\tilde {\phi }}_{b},g\}_{PB}~,$
where M−1ab denotes the ab entry of M 's inverse matrix. Dirac proved that M will always be invertible.
It is straightforward to check that the above definition of the Dirac bracket satisfies all of the desired properties, and especially the last one, of vanishing for an argument which is a second-class constraint.
When applying canonical quantization on a constrained Hamiltonian system, the commutator of the operators is supplanted by iħ times their classical Dirac bracket. Since the Dirac bracket respects the constraints, one need not be careful about evaluating all brackets before using any weak equations, as is the case with the Poisson bracket.
Note that while the Poisson bracket of bosonic (Grassmann even) variables with itself must vanish, the Poisson bracket of fermions represented as a Grassmann variables with itself need not vanish. This means that in the fermionic case it is possible for there to be an odd number of second class constraints.
Illustration on the example provided
Returning to the above example, the naive Hamiltonian and the two primary constraints are
$H=V(x,y)$
$\phi _{1}=p_{x}+{\tfrac {qB}{2c}}y,\qquad \phi _{2}=p_{y}-{\tfrac {qB}{2c}}x.$
Therefore, the extended Hamiltonian can be written
$H^{*}=V(x,y)+u_{1}\left(p_{x}+{\tfrac {qB}{2c}}y\right)+u_{2}\left(p_{y}-{\tfrac {qB}{2c}}x\right).$
The next step is to apply the consistency conditions {Φj, H*}PB ≈ 0, which in this case become
$\{\phi _{1},H\}_{PB}+\sum _{j}u_{j}\{\phi _{1},\phi _{j}\}_{PB}=-{\frac {\partial V}{\partial x}}+u_{2}{\frac {qB}{c}}\approx 0$
$\{\phi _{2},H\}_{PB}+\sum _{j}u_{j}\{\phi _{2},\phi _{j}\}_{PB}=-{\frac {\partial V}{\partial y}}-u_{1}{\frac {qB}{c}}\approx 0.$
These are not secondary constraints, but conditions that fix u1 and u2. Therefore, there are no secondary constraints and the arbitrary coefficients are completely determined, indicating that there are no unphysical degrees of freedom.
If one plugs in with the values of u1 and u2, then one can see that the equations of motion are
${\dot {x}}=\{x,H\}_{PB}+u_{1}\{x,\phi _{1}\}_{PB}+u_{2}\{x,\phi _{2}\}=-{\frac {c}{qB}}{\frac {\partial V}{\partial y}}$
${\dot {y}}={\frac {c}{qB}}{\frac {\partial V}{\partial x}}$
${\dot {p}}_{x}=-{\frac {1}{2}}{\frac {\partial V}{\partial x}}$
${\dot {p}}_{y}=-{\frac {1}{2}}{\frac {\partial V}{\partial y}},$
which are self-consistent and coincide with the Lagrangian equations of motion.
A simple calculation confirms that φ1 and φ2 are second class constraints since
$\{\phi _{1},\phi _{2}\}_{PB}=-\{\phi _{2},\phi _{1}\}_{PB}={\frac {qB}{c}},$
hence the matrix looks like
$M={\frac {qB}{c}}\left({\begin{matrix}0&1\\-1&0\end{matrix}}\right),$
which is easily inverted to
$M^{-1}={\frac {c}{qB}}\left({\begin{matrix}0&-1\\1&0\end{matrix}}\right)\quad \Rightarrow \quad M_{ab}^{-1}=-{\frac {c}{qB_{0}}}\varepsilon _{ab},$
where εab is the Levi-Civita symbol. Thus, the Dirac brackets are defined to be
$\{f,g\}_{DB}=\{f,g\}_{PB}+{\frac {c\varepsilon _{ab}}{qB}}\{f,\phi _{a}\}_{PB}\{\phi _{b},g\}_{PB}.$
If one always uses the Dirac bracket instead of the Poisson bracket, then there is no issue about the order of applying constraints and evaluating expressions, since the Dirac bracket of anything weakly zero is strongly equal to zero. This means that one can just use the naive Hamiltonian with Dirac brackets, instead, to thus get the correct equations of motion, which one can easily confirm on the above ones.
To quantize the system, the Dirac brackets between all of the phase space variables are needed. The nonvanishing Dirac brackets for this system are
$\{x,y\}_{DB}=-{\frac {c}{qB}}$
$\{x,p_{x}\}_{DB}=\{y,p_{y}\}_{DB}={\tfrac {1}{2}}$
while the cross-terms vanish, and
$\{p_{x},p_{y}\}_{DB}=-{\frac {qB}{4c}}.$
Therefore, the correct implementation of canonical quantization dictates the commutation relations,
$[{\hat {x}},{\hat {y}}]=-i{\frac {\hbar c}{qB}}$
$[{\hat {x}},{\hat {p}}_{x}]=[{\hat {y}},{\hat {p}}_{y}]=i{\frac {\hbar }{2}}$
with the cross terms vanishing, and
$[{\hat {p}}_{x},{\hat {p}}_{y}]=-i{\frac {\hbar qB}{4c}}~.$
This example has a nonvanishing commutator between ∧x and ∧y, which means this structure specifies a noncommutative geometry. (Since the two coordinates do not commute, there will be an uncertainty principle for the x and y positions.)
Further Illustration for a hypersphere
Similarly, for free motion on a hypersphere Sn, the n + 1 coordinates are constrained, xi xi = 1. From a plain kinetic Lagrangian, it is evident that their momenta are perpendicular to them, xi pi = 0. Thus the corresponding Dirac Brackets are likewise simple to work out,[8]
$\{x_{i},x_{j}\}_{DB}=0,$
$\{x_{i},p_{j}\}_{DB}=\delta _{ij}-x_{i}x_{j},$
$\{p_{i},p_{j}\}_{DB}=x_{j}p_{i}-x_{i}p_{j}~.$
The (2n + 1) constrained phase-space variables (xi, pi) obey much simpler Dirac brackets than the 2n unconstrained variables, had one eliminated one of the xs and one of the ps through the two constraints ab initio, which would obey plain Poisson brackets. The Dirac brackets add simplicity and elegance, at the cost of excessive (constrained) phase-space variables.
For example, for free motion on a circle, n = 1, for x1 ≡ z and eliminating x2 from the circle constraint yields the unconstrained
$L={\frac {1}{2}}{\frac {{\dot {z}}^{2}}{1-z^{2}}}~,$
with equations of motion
${\ddot {z}}=-z{\frac {{\dot {z}}^{2}}{1-z^{2}}}=-z2E~,$
an oscillation; whereas the equivalent constrained system with H = p2/2 = E yields
${\dot {x}}^{i}=\{x^{i},H\}_{DB}=p^{i}~,$
${\dot {p}}^{i}=\{p^{i},H\}_{DB}=x^{i}~p^{2}~,$
whence, instantly, virtually by inspection, oscillation for both variables,
${\ddot {x}}^{i}=-x^{i}2E~.$
See also
• Canonical quantization
• Hamiltonian mechanics
• Poisson bracket
• Moyal bracket
• First class constraint
• Second class constraints
• Lagrangian
• Symplectic structure
• Overcompleteness
References
1. Dirac, P. A. M. (1950). "Generalized Hamiltonian dynamics". Canadian Journal of Mathematics. 2: 129–014. doi:10.4153/CJM-1950-012-1. S2CID 119748805.
2. Dirac, Paul A. M. (1964). Lectures on quantum mechanics. Belfer Graduate School of Science Monographs Series. Vol. 2. Belfer Graduate School of Science, New York. ISBN 9780486417134. MR 2220894.; Dover, ISBN 0486417131.
3. See pages 48-58 of Ch. 2 in Henneaux, Marc and Teitelboim, Claudio, Quantization of Gauge Systems. Princeton University Press, 1992. ISBN 0-691-08775-X
4. Dunne, G.; Jackiw, R.; Pi, S. Y.; Trugenberger, C. (1991). "Self-dual Chern-Simons solitons and two-dimensional nonlinear equations". Physical Review D. 43 (4): 1332–1345. Bibcode:1991PhRvD..43.1332D. doi:10.1103/PhysRevD.43.1332. PMID 10013503.
5. See page 8 in Henneaux and Teitelboim in the references.
6. Weinberg, Steven, The Quantum Theory of Fields, Volume 1. Cambridge University Press, 1995. ISBN 0-521-55001-7
7. See Henneaux and Teitelboim, pages 18-19.
8. Corrigan, E.; Zachos, C. K. (1979). "Non-local charges for the supersymmetric σ-model". Physics Letters B. 88 (3–4): 273. Bibcode:1979PhLB...88..273C. doi:10.1016/0370-2693(79)90465-9.
| Wikipedia |
cs.DS
cs.DC
Computer Science > Data Structures and Algorithms
Title: Optimal algebraic Breadth-First Search for sparse graphs
Authors: Paul Burkhardt
(Submitted on 7 Jun 2019 (v1), last revised 30 Apr 2021 (this version, v4))
Abstract: There has been a rise in the popularity of algebraic methods for graph algorithms given the development of the GraphBLAS library and other sparse matrix methods. An exemplar for these approaches is Breadth-First Search (BFS). The algebraic BFS algorithm is simply a recurrence of matrix-vector multiplications with the $n \times n$ adjacency matrix, but the many redundant operations over nonzeros ultimately lead to suboptimal performance. Therefore an optimal algebraic BFS should be of keen interest especially if it is easily integrated with existing matrix methods.
Current methods, notably in the GraphBLAS, use a Sparse Matrix masked-Sparse Vector (SpMmSpV) multiplication in which the input vector is kept in a sparse representation in each step of the BFS, and nonzeros in the vector are masked in subsequent steps. This has been an area of recent research in GraphBLAS and other libraries. While in theory these masking methods are asymptotically optimal on sparse graphs, many add work that leads to suboptimal runtime. We give a new optimal, algebraic BFS for sparse graphs, thus closing a gap in the literature.
Our method multiplies progressively smaller submatrices of the adjacency matrix at each step. Let $n$ and $m$ refer to the number of vertices and edges, respectively. On a sparse graph, our method takes $O(n)$ algebraic operations as opposed to $O(m)$ operations needed by theoretically optimal sparse matrix approaches. Thus for sparse graphs it matches the bounds of the best-known sequential algorithm and on a Parallel Random Access Machine (PRAM) it is work-optimal. Our result holds for both directed and undirected graphs. Compared to a leading GraphBLAS library our method achieves up to 24x faster sequential time and for parallel computation it can be 17x faster on large graphs and 12x faster on large-diameter graphs.
Comments: Will appear in ACM Transactions on Knowledge Discovery from Data, Vol. 15, No. 5, 2021
Subjects: Data Structures and Algorithms (cs.DS); Distributed, Parallel, and Cluster Computing (cs.DC)
Journal reference: ACM Transactions on Knowledge Discovery from Data, 15(5):1-19, 2021
DOI: 10.1145/3446216
Cite as: arXiv:1906.03113 [cs.DS]
(or arXiv:1906.03113v4 [cs.DS] for this version)
From: Paul Burkhardt [view email]
[v1] Fri, 7 Jun 2019 14:06:23 GMT (38kb)
[v2] Thu, 3 Oct 2019 14:43:59 GMT (52kb)
[v3] Wed, 12 Aug 2020 19:33:26 GMT (57kb)
[v4] Fri, 30 Apr 2021 21:52:38 GMT (59kb) | CommonCrawl |
Noncontrast free-breathing respiratory self-navigated coronary artery cardiovascular magnetic resonance angiography at 3 T using lipid insensitive binomial off-resonant excitation (LIBRE)
Jessica A. M. Bastiaansen ORCID: orcid.org/0000-0002-5485-13081,
Ruud B. van Heeswijk1,
Matthias Stuber1,2 &
Davide Piccini1,3
Robust and homogeneous lipid suppression is mandatory for coronary artery cardiovascular magnetic resonance (CMR) imaging since the coronary arteries are commonly embedded in epicardial fat. However, effective large volume lipid suppression becomes more challenging when performing radial whole-heart coronary artery CMR for respiratory self-navigation and the problem may even be exacerbated at increasing magnetic field strengths. Incomplete fat suppression not only hinders a correct visualization of the coronary vessels and generates image artifacts, but may also affect advanced motion correction methods. The aim of this study was to evaluate a recently reported lipid insensitive CMR method when applied to a noncontrast self-navigated coronary artery CMR acquisitions at 3 T, and to compare it to more conventional fat suppression techniques.
Lipid insensitive binomial off resonant excitation (LIBRE) radiofrequency excitation pulses were included into a self-navigated 3D radial GRE coronary artery CMR sequence at 3 T. LIBRE was compared against a conventional CHESS fat saturation (FS) and a binomial 1–180°-1 water excitation (WE) pulse. First, fat suppression of all techniques was numerically characterized using Matlab and experimentally validated in phantoms and in legs of human volunteers. Subsequently, free-breathing self-navigated coronary artery CMR was performed using the LIBRE pulse as well as FS and WE in ten healthy subjects. Myocardial, arterial and chest fat signal-to-noise ratios (SNR), as well as coronary vessel conspicuity were quantitatively compared among those scans.
The results obtained in the simulations were confirmed by the experimental validations as LIBRE enabled near complete fat suppression for 3D radial imaging in vitro and in vivo. For self-navigated whole-heart coronary artery CMR at 3 T, fat SNR was significantly attenuated using LIBRE compared with conventional FS. LIBRE increased the right coronary artery (RCA) vessel sharpness significantly (37 ± 9% (LIBRE) vs. 29 ± 8% (FS) and 30 ± 8% (WE), both p < 0.05) and led to a significant increase in the measured RCA vessel length to (83 ± 31 mm (LIBRE) vs. 56 ± 12 mm (FS) and 59 ± 27 (WE) p < 0.05).
Applied to a respiratory self-navigated noncontrast 3D radial whole-heart sequence, LIBRE enables robust large volume fat suppression and significantly improves coronary artery image quality at 3 T compared to the use of conventional FS and WE.
Whole-heart coronary cardiovascular magnetic resonance (CMR) is usually performed during free-breathing while the respiratory motion is monitored and gated using a navigator placed on the lung-liver interface, or, more recently, through self-navigation by deriving the respiratory displacement from the acquired data themselves [1,2,3,4,5,6,7]. To cover the entire coronary arterial tree in a single scan, these acquisitions require a large volumetric coverage, and thus also a large-volume fat suppression.
Large volume fat suppression is of vital importance to generate contrast between the coronary lumen blood pool and surrounding epicardial fat, in which the coronary arteries are embedded [8]. If fat suppression is suboptimal, the residual lipid signals may hinder the correct anatomical visualization of the coronary vessels and lead to artifacts in the image, while, for self-navigation, it may also degrade the signal quality used for tracking the respiratory displacement and thus degrade the motion correction [4]. Firstly, using respiratory motion correction for the heart, motion artifacts in the image will inevitably occur and originate from static structures such as the chest wall. These unwanted signals will unfortunately be amplified if fat signal from the chest is not adequately attenuated. Secondly, and owing to the self-navigation concept where a single profile in k-space informs about respiratory displacement of the anatomy in the entire field of view, the signals from the ventricular blood-pool and that from incompletely suppressed fat may not easily be discriminated. As a result, tracking of the left-ventricular blood pool may become erroneous [4]. However, homogeneous and effective fat suppression for such a large volume is quite challenging, mainly due to magnetic field inhomogeneities in the large field of view (FOV). This is even more prevalent when moving from 1.5 T to 3 T and may be amplified for radial whole-heart imaging where each acquired k-space profile traverses through the center of k-space, which represents the average signal of the excitation volume. Therefore, and even if magnetic field inhomogeneities could adequately be accounted for, T1-recovery after a conventional fat suppression pre-pulse may still lead to sub-optimal fat suppression for radial imaging.
At 3 T, the use of water selective radiofrequency (RF) excitation pulses may solve the problem related to the T1 recovery of lipid signals [9,10,11,12], especially for radial sampling schemes. However, the improved fat suppression capabilities may come at the expense of increased RF pulse durations, since typically, the fat suppression bandwidth of conventional binomial water excitation pulses increases with the number of sub-pulses. To address this limitation, a non spatially selective water excitation pulse was proposed that demonstrated robust fat suppression at 3 T with total RF pulse durations as short as 1.4 ms [13]. More recently these pulses were further shortened to a total duration of 1.0 ms [14].
The aim of the current study was to implement and exploit lipid-insensitive binomial off-resonance excitation (LIBRE) pulses for fat suppression in 3D radial noncontrast self-navigated coronary artery MRA at 3 T, and to compare the results to those obtained with conventional fat suppression methods. To this end, a numerical comparison was made between a conventional on-resonance binomial 1–180°-1 water excitation pulse, a chemical shift selective fat saturation pulse, and the off-resonance LIBRE pulse (Fig. 1a-c). A quantitative comparison between these three different fat suppression schemes was then made using 3D radial acquisitions in phantoms and knees of healthy subjects, as well as whole-heart respiratory self-navigated coronary artery CMR in healthy subjects.
An electrocardiogram (ECG)-triggered 3D radial sequence was used for coronary artery CMR in healthy subjects with three different fat suppression methods. a Conventional CHESS fat saturation using a spectral prepulse to saturate the fat, b a conventional binomial 1–180°-1 water excitation (WE) pulse, c a LIBRE pulse. All acquisitions were preceded by a T2-preparation module of 30 ms. The first acquired k-space profile in each ECG-triggered segmented acquisition was Fourier transformed to obtain a superior-inferior (SI) projection image of the thorax that was used for respiratory-self-navigation. The same sequences were also carried out in phantoms and legs of subjects by disabling the ECG trigger. d A diagram illustrating the LIBRE pulse timing
The LIBRE pulse used in this study [13] consists of a pair of low power rectangular pulses, each having the same RF excitation angle α, RF excitation frequency fRF, and sub-pulse duration τ. In the absence of B0 field inhomogeneities, optimal fat suppression is predicted when the following condition is met:
$$ \uptau =\sqrt{1-{\left(\upalpha /2\uppi \right)}^2/\left({\mathrm{f}}_{\mathrm{RF}}-{\mathrm{f}}_{\mathrm{f}\mathrm{at}}\right)} $$
To illustrate this condition, assuming a fat frequency ffat = − 440 Hz at 3 T, and in the absence of field inhomogeneities, the LIBRE RF excitation frequency fRF (that leads to optimal fat suppression) was plotted as function of the sub-pulse duration for an RF excitation angle of 18° and 90° (Fig. 2). Field inhomogeneities may broaden the line width or the position of the fat resonance frequency, therefore a range of ffat from -400 Hz to − 480 Hz was also evaluated.
The relation (see Eq. 1) between sub-pulse duration (τ) and excitation frequency of the LIBRE pulse for optimal fat nulling was plotted using a radiofrequency (RF) excitation angle of 10° (blue lines) and 90° (dashed red line). Shades of blue indicate a range of fat resonance frequencies, from − 400 Hz to − 480 Hz. Note that the optimal LIBRE frequency (dashed blue lines) for fat suppression varies according to the resonance frequency of fat and thus illustrates the need for a large fat suprression bandwidth. Similar regions were indicated in the phantom experiments where fRF was varied (Fig. 1b)
Numerical simulations
To predict and calculate the magnetization behavior of cardiac tissue using the three different fat suppression methods in the presence of B1 and B0 inhomogeneities, numerical simulations were performed in Matlab (The MathWorks, Inc., Natick, Massachusetts, USA). The water and fat magnetization components were evaluated as function of tissue frequency ftissue, magnetic field inhomogeneities ΔB0, and RF excitation angle α. Simulations over a range of RF angles also instruct about B1 inhomogeneities. The numerical simulations, similar to those described before [13], were extended to take the relaxation times T1 and T2 into account, as well as repeated excitations, using the Bloch equations. T1 of myocardial blood was assumed to be 1932 ms, and T2 was set to 275 ms [15]. A repetition time (TR) of 5.1 ms was used reflecting the TR in the CMR protocol.
Because the acquisition window for coronary imaging in healthy subjects in previous studies [4, 6] was typically on the order of 100 to 120 ms depending on the duration of the mid-diastolic cardiac resting phase, the number of simulated RF excitations was set to 24. The simulations were performed for RF excitation angles ranging from 0° to 50°, to simulate a range of typical RF excitation angles of a gradient recalled echo (GRE) acquisition. Simulated tissue frequencies ftissue were ranging from − 600 Hz to 600 Hz to adequately encompass frequencies of both water and fat. A LIBRE pulse with RF sub-pulse duration τ of 1.1 ms was chosen because these pulse properties result in a similar TR compared with a conventional water excitation. The transverse magnetization was set to zero after each excitation to mimic perfect spoiling. Plots were made to visualize the transverse magnetization as function of the RF excitation angle and different tissue frequencies fRF. To compare the magnetization behavior following repeated LIBRE excitations, numerical simulations with identical parameters and parameter ranges were also carried out using a conventional 1–180°-1 water excitation (WE) pulse [16], a frequency selective pulse for fat saturation (FS) [17]. The WE pulse consisted of two on-resonance rectangular sub-pulses, each with duration of 0.5 ms and separated by 1.1 ms to allow for a 180° phase evolution between water and fat. The FS simulation was performed by assuming a Gaussian-shaped RF pulse with duration of 5.12 ms, with RF offset frequency of − 407 Hz and RF excitation angle of 110°. The exact structure of this Gaussian-shaped pulse was obtained from the sequence product source code. In the simulation for FS, the 24 on-resonance RF excitation pulses during the acquisition were set to a duration of 0.3 ms, as in the product sequence. The choices for FS and WE pulse parameters matched those used in experiments. The fat suppression bandwidth was defined as 10% of the maximum transverse magnetization.
In vitro exams
The LIBRE RF pulse implementation was integrated into a pre-existing prototype 3D radial spoiled GRE sequence adapted for self-navigated free-breathing coronary artery CMR imaging. The radial trajectory follows a 3D spiral phyllotaxis pattern as described mathematically in [18] with a golden angle rotation about the z-axis. Each 3D spiral segment was composed of 24 radial k-space lines to match the simulations. In phantoms, experiments were performed with 1) a LIBRE pulse with sub-pulse duration (τ) of 1.1 ms and an RF frequency offset ranging from 300 Hz to 700 Hz in steps of 20 Hz, 2) a conventional WE pulse with a binomial 1–180°-1 pulse pattern [16], and 3) a conventional FS method that uses a CHESS [17] pulse to null the fat signal prior to the imaging sequence. Data volumes from phantoms were acquired on a clinical 3 T CMR system (MAGNETOM PrismaFIT, Siemens Healthineers, Erlangen, Germany). 3D volumes with an isotropic voxel size of 1.1 mm3 were acquired with a FOV of 220 × 220 × 220 mm3, matrix size 1923, RF excitation angle = 18°, TE/TR (FS) = 1.6/3.2 ms, TE/TR (LIBRE) = 2.5/5.1 ms, TE/TR (WE) = 2.3/4.8 ms and a 40 ms adiabatic T2-preparation [19], and using a 15-channel Tx/Rx knee coil (Table 1). To achieve a 20% sampling of the Nyquist criterion for 3D radial acquisitions as recommended in [18], a total of ~ 12 k lines were acquired. The cylindrical phantom consisted of three compartments containing mixed solutions of agar, NiCl2 (Sigma Aldrich, St. Louis, MO), and baby oil (Johnson and Johnson, New Brunswick, New Jersey, USA), in order to mimic the magnetic relaxation properties (T1, T2) of muscle, blood, and fat.
Table 1 MR sequence details and parameters
In vivo knee exams
To validate the in vitro results and to separate the confounding effects that motion may have on the in vivo results, the fat suppressing performance of LIBRE combined with 3D radial was ascertained in static muscle and fat tissue in vivo. Therefore, the same experiments as those performed in the phantoms were repeated in knees of human subjects (n = 3) by setting the LIBRE fRF to the optimal frequency derived from the phantom experiments, i.e. 480 Hz for a sub-pulse duration (τ) of 1.1 ms (Fig. 1d). All subjects provided written informed consent and local ethical authorities approved this study.
In vivo free-breathing whole-heart coronary artery CMR
Noncontrast whole-heart coronary artery CMR was performed in 10 healthy adult subjects on the same clinical 3 T CMR system using the LIBRE, FS and WE protocols in randomized order (Fig. 1). For this purpose, a 3D radial imaging sequence as described above was used for coronary artery imaging by enabling respiratory-self-navigation and electrocardiograph (ECG) triggering [4, 6, 20, 21]. Whole-heart volumes were acquired during free-breathing. To achieve a 20% sampling of the Nyquist criterion for 3D radial acquisitions as recommended in [18], a total of ~ 12 k radial profiles were acquired in a segmented fashion per 3D scan while 20–28 profiles were collected per heartbeat. The amount of profiles acquired per heartbeat varied across volunteers. In each volunteer we determined the duration of the mid-diastolic cardiac resting phase by visual inspection of a midventricular 2D cine scan. Then it was determined how many segments could be acquired during this period using the protocol with the longest TR, which was LIBRE with a TR of 5.1 ms. Subsequently all protocols were performed using the same number of segments per heartbeat and subjects. The heart rate and average acquisition windows were recorded in each volunteer. The whole-heart coronary artery CMR acquisitions were respiratory motion-corrected and reconstructed directly at the scanner using a superior-inferior (SI) projection acquired at the beginning of every segment (every heartbeat) as previously described [4]. The reconstruction was based on 3D gridding and the reconstruction time was below 1 min.
All CMR datasets were directly reconstructed at the scanner using the sum-of-squares of all channels and the gridding algorithm provided by the vendor. In the phantom experiments, the signal-to-noise-ratio (SNR) was calculated in compartments containing fat to evaluate the level of fat suppression, the noise was measured in regions containing air, and the contrast-to-noise ratio (CNR) between compartments mimicking myocardium and blood. In human subject studies performed in the leg, the SNR was calculated in compartments containing muscle tissue and fat for comparison among the three different fat suppression methods. In whole-heart coronary artery CMR, the SNR was calculated in regions-of-interest (ROI) s drawn on the myocardium at the level of the interventricular septum, in the chest fat, and in the left ventricular blood pool. Noise was calculated in regions containing air, outside the subject within the field of view. The CNRs were computed to evaluate the level of contrast of the blood pool relative to the myocardium, as well as blood versus fat. All images were analyzed in ImageJ (National Institutes of Health, Bethesda, Maryland, USA). Coronary vessel sharpness and vessel length were quantified [22] in both the left anterior descending (LAD) artery and the right coronary artery (RCA) for all acquired whole-heart volumes. Vessel sharpness was computed for the same length for all methods (proximal 4 cm). Coronary reformats were also generated to visualize and compare the structure of the LADs and RCAs in all subjects and across techniques. A paired Student's t-test, corrected for multiple comparisons, was performed on phantom and volunteer data and p < 0.05 was considered statistically significant. All data are represented as average ± one standard deviation.
Theory and numerical simulations
Evaluation of Eq. 1 showed a range of parameter combinations that indicate optimal fat suppression (Fig. 2). Assuming a fat resonance frequency of − 440 Hz, and in the absence of field inhomogeneities that broaden the line shape of the fat resonance, the combination of a sub-pulse duration of 1.1 ms and an RF excitation frequency of 469 Hz provides complete fat nulling. Assuming fat resonances at − 400 Hz or − 480 Hz, the optimal fRF changes to 507 Hz and to 429 Hz respectively (Fig. 2). Although Eq. 1 shows that the RF excitation angle affects the optimal combination between the LIBRE frequency and duration, an increase in the RF excitation angle to 90° had a minor influence on the choice of optimal LIBRE parameter combinations (Fig. 2).
Numerical simulations demonstrated that at an RF excitation angle of 18° the transverse magnetization of on-resonance water is highest, with a fat suppression bandwidth on the order of 238 Hz (fRF = 479 Hz, τ = 1.1 ms), compared to 32 Hz and 128 Hz using WE and FS respectively (Fig. 3). The magnitude of the transverse magnetization observed at a tissue frequency of ~ 0 Hz demonstrated that a similar range of RF excitation angles may be used across techniques to achieve the same excitation behavior. The change in transverse magnetization over a range of RF excitation angles can also be interpreted as a measure of the sensitivity of the investigated sequences to B1 inhomogeneities.
Numerical simulation of the transverse magnetization (Mxy) as function of RF excitation angle and tissue frequency using three different sequences, a FS, b WE, and c LIBRE. The white arrows indicate the bandwidth of fat suppression (around -440 Hz) at a RF excitation angle of 18°. The white and black dashed isolines indicate 10 and 90% of the maximum observed Mxy, respectively. White arrows indicate the fat suppression bandwidth (up to 10% of maximum Mxy) that corresponded to 238 Hz (LIBRE), 128 Hz (FS), and 32 Hz (WE)
Experimental results in phantoms and legs of human volunteers
The phantom experiments showed a clear difference across different fat suppression techniques (Fig. 4a). WE performed well at the center of the cylindrical phantom, but its fat suppression efficiency degrades moving towards the phantom boundaries, i.e. regions that suffer from magnetic field inhomogeneities. FS performed poorly in this 3D radial acquisition, with signal leaking across the boundaries of the air-phantom interface. LIBRE suppressed fat homogeneously in the entire phantom, including the boundaries. A frequency calibration of the LIBRE pulse in a phantom demonstrated that a range of fRF from ~ 400 Hz to ~ 500 Hz resulted in optimum fat suppression (Fig. 4b). The lowest SNR of fat (8.3 ± 0.9) was obtained using LIBRE with an fRF of 460 Hz, compared with FS (48.1 ± 3.4) and WE (31.8 ± 2.1) (Fig. 4b), both p < 0.05. The CNR between the inner and outer compartments of the phantom, that mimic blood and myocardial tissue respectively, was not significantly different across the different techniques (Fig. 4c), (p = NS).
Three different fat suppression methods (Fig. 1) for radial imaging were investigated in phantoms and legs of healthy subjects. a Phantom images indicate a clear difference across different techniques. b Quantitative results obtained after image analysis of the phantom experiments show fat SNR as function of the LIBRE excitation frequency fRF with a total pulse duration of 2.2 ms (blue). The SNR of fat in identical acquisitions using FS or WE are also indicated as a reference. Dashed blue lines correspond to the theoretical optimum for fRF for fat resonating at − 400 Hz, − 440 Hz and − 480 Hz (Fig. 2, Eq. 1). c The CNR determined between the inner and outer compartment of the phantom that mimic blood and myocardial tissue respectively is comparable across different techniques. d Leg images obtained in healthy subjects using three different fat suppression methods. e SNR of fat and muscle tissue in the leg show a significant decrease of fat SNR using LIBRE, while the SNR of muscle tissue remains similar to WE. Note that the perceived differences between the tissue signals (fat and muscle) in the images and the quantitative SNR plots (c versus d) can be mainly attributed to an increase in the noise in fat suppression. Image intensities were scaled for identical window and level settings. **p < 0.005
Measurements in the leg showed a similar behavior as in the phantom experiments (Fig. 4d) with the fat being homogenously suppressed using LIBRE. The SNR of fat was significantly decreased using LIBRE (9.9 ± 2.2) compared with FS (26.6 ± 6.9, p < 0.005) and WE (25.1 ± 6.1, p < 0.005) (Fig. 4e). The SNR of skeletal muscle tissue was similar comparing LIBRE and WE, but decreased using FS.
Free-breathing coronary artery CMR was successfully performed in all subjects without complications with an average scan time of 8.6 ± 1.5 min. Subjects had an average heart rate of 64 ± 8 BPM. The average data acquisition window was 81 ± 17 ms (FS), 122 ± 26 ms (WE) and 129 ± 28 ms (LIBRE). The left and right coronary systems could be visualized clearly in all cases, but not with all techniques. A clear difference across the different techniques can be observed in the coronary reformats where both the RCA and LAD are visualized (Fig. 5). Compared to the use of WE and FS, coronary LIBRE leads to an improved visualization of the coronary arteries. In addition, large volume fat suppression can be achieved using LIBRE, as can be seen from the decrease in signal from fatty tissue in the back and chest of a subject (Fig. 6, red and green arrows). The three different fat suppression methods also had an effect on the signal behavior in the SI projections (Fig. 6, bottom row).
Noncontrast free-breathing coronary artery CMR was performed at 3 T using three lipid nulling methods in healthy subjects. CMR angiograms show the left and right coronary artery system depicting the RCA and the LAD in several subjects. Using the LIBRE pulse the visualization of the RCA and LAD was improved (yellow arrow), as well as fat suppression (orange arrows) compared with FS and WE. Vessel sharpness as well as vessel length were significantly increased using LIBRE. Window and level are identical in images acquired in each volunteer
Sagittal images illustrate the large volume fat suppression in a subject using three different fat suppression methods (top row) and the corresponding SI projections (bottom row). Note the inhomogeneous fat suppression in the chest (green arrows) and the back (red arrows) using FS and WE compared with LIBRE. Unsuppressed bright fat signal from the chest and the back contribute to the SI projection (bottom row) and may hinder respiratory motion tracking (white lines on SI projections) that relies on a correct delineation of the blood pool in the heart
The SNR of the myocardium was similar across all techniques, with no statistically significant differences found (p = NS). The blood SNR was 86 ± 35 using LIBRE and was significantly decreased to 51 ± 18 using FS (p = 0.005) and to 55 ± 12 using WE (p = 0.01) (Fig. 7a). Fat SNR was significantly increased from 14 ± 8 to 46 ± 18 using FS (p = 0.01) and to 19 ± 7 using WE (p = NS) (Fig. 7a). CNR between blood and myocardial tissue significantly decreased from 33 ± 12 using LIBRE to 16 ± 17 using FS (p = 0.01) and to 16 ± 3 using WE (p = 0.01) (Fig. 7b). The CNR between blood and fat tissue was significantly increased from 24 ± 12 using FS to 76 ± 41 using LIBRE (p = 0.01) (Fig. 7b). The vessel sharpness of the RCA and LAD was significantly improved (Fig. 7c, p < 0.05 in all comparisons) using LIBRE (37 ± 9% and 34 ± 7%, respectively), compared with FS (29 ± 8% and 24 ± 6%, respectively), and with WE (30 ± 8% and 27 ± 7%, respectively). In addition, the measured vessel length of both the RCA and LAD was significantly increased (Fig. 7d, p < 0.05 in all comparisons) using LIBRE (83 ± 31 and 98 ± 35 mm), compared with FS (56 ± 12 and 54 ± 25 mm), and with WE (59 ± 27 and 61 ± 21 mm).
Quantitative endpoints comparing three fat suppression techniques for whole-heart free-breathing coronary artery CMR. SNR (a) and CNR (b). Vessel sharpness (c) and vessel length (d) of both the RCA and LAD
In this study we implemented a previously published LIBRE pulse [13] in combination with a respiratory self-navigated 3D radial imaging sequence [4] and demonstrated its effectiveness for fat suppression and its application for coronary artery CMR at 3 T in healthy subjects without the use of contrast agents.
The LIBRE pulse was optimized in phantoms and legs and experiments showed that improvements in fat suppression for radial imaging were consistent with those found for a Cartesian approach as originally reported [13]. Radial imaging is inherently more sensitive to incomplete fat suppression, especially at higher magnetic field strengths, where field inhomogeneities are typically accentuated. The increased fat suppression bandwidth of LIBRE led to a near-complete nulling of the fat signals in large 3D volumes and outperformed conventional fat suppression methods such as CHESS based spectral fat suppression [17] and binomial 1–180°-1 water excitation [16].
When assessing the coronary artery CMR results obtained using different fat suppression techniques, it can be appreciated that LIBRE improved the quality of the vessels conspicuity. A quantitative comparison revealed significant improvements when using LIBRE in terms of vessel sharpness, detectable vessel length, SNR and CNR. The underlying reason for differences in final image quality across three techniques is most likely a combination of two factors that cannot be decoupled experimentally. The first one relates directly to the significantly improved fat suppression using the LIBRE method as measured also in the static phantom and healthy subject scans. Secondly, this improved fat suppression may have benefitted the motion tracking and motion correction used in respiratory self-navigation. As fat is more homogeneously and more completely suppressed, the SI projections used for respiratory motion tracking contain less residual fat signal from static structures such as the chest wall and the arms that may hinder a reliable motion detection. In addition, the artifacts due to motion correction performed on residual static fat signal from the chest may be less pronounced.
Although 3 T coronary artery CMR has also been performed using balanced steady state free precession in some studies [23, 24] the most frequently used acquisition method currently remains GRE with contrast agent injection [25,26,27,28]. Other methods such as Dixon or IDEAL have been presented as an alternative and would ideally apply with radial imaging, but may require the acquisition of multiple images at several echo times to adequately separate water and fat images [29,30,31]. The presented LIBRE method requires the acquisition of a single image. However, it was recently demonstrated that a two-point Dixon implementation of a GRE sequence led to an improved coronary artery CMR image quality over conventional fat suppression without increasing scan time [32]. This acquisition was also performed without contrast agent injection and may provide a promising alternative to our proposed technique. However, a comparison with water-fat separation techniques was not performed in the current study.
The LIBRE pulse is a spatially non-selective water excitation pulse with a broad fat suppression bandwidth. These properties render the method highly suitable for the acquisition of large 3D volumes. CMR examinations are typically complex, because of the anatomical structure, size and spatial orientation of the heart and therefore, CMR acquisitions that cover the entire organ may be preferred over 2D or 3D targeted acquisitions as operator dependency can be minimized and arbitrary views and orientations reconstructed retrospectively.
The fact that the LIBRE pulse is spatially non-selective may pose a limitation for some applications. However, this may also be an advantage when combined with acquisitions that do not require time-consuming oversampling in the phase encoding direction, as is typically the case in 3D radial imaging. As for the self-navigation module, respiratory motion tracking relies on a correct delineation of the blood pool in the heart, which is based on the SI projections. As was observed in this study, a correct delineation of the blood pool may have been hindered using the WE and FS sequences as signal from the fat tissue located in the chest and the back was not always sufficiently suppressed (Fig. 6) at 3 T. Not only may unsuppressed fat signal in static structures of the body pose a problem for tracking, it can also affect the final image quality when static tissue with very high signal intensity is incorrectly motion-corrected. Therefore it may be argued that LIBRE did not only improve the homogenous suppression of fat signal, but also improved the tracking of the blood pool, and the motion-correction performed in respiratory-self-navigation. However, using the non-spatially selective LIBRE pulse and the large excitation volume, signals from arms may also be included in the SI projections and affect the accuracy of respiratory motion estimation and correction.
The use of water excitation pulses for radial imaging at 3 T may be preferred over fat signal saturation because radial imaging is extremely sensitive to fat signal recovery as each acquired k-space line goes through the k-space center and T1 recovery of fat during the acquisition window will be unavoidable. Therefore, methods that rely on fat signal suppression by applying a frequency selective saturation pulse, or that rely on T1-based nulling of the fat signal following a frequency selective inversion pulse, were shown to work well in Cartesian based imaging approaches [33, 34], but may still yield unwanted residual fat signal in the final images using radial imaging. A further optimization of these suppression techniques for this particular 3D radial application in terms of timing, RF excitation angles, and other parameters may have to be performed, but was outside the scope of the current study. At 3 T, the T1 of fat is longer and fat suppression using frequency selective inversion pulses or saturation pulses may become more effective, particularly for very short acquisition windows. However, as our acquisition window is still > 80 ms, as every profile of a radial sequence goes through the center of k-space, and provided that B0 and B1 inhomogeneities are enhanced at 3 T, the benefit of the longer T1 of fat is expected to be minimal. An improved performance of frequency selective inversion pulses or saturation pulses could also be achieved by decreasing the amount of radial lines that are acquired per heartbeat, however this comes at the expense of an increase in scan time. WE can be performed using binomial pulse patterns such as 1–1, 1–2-1, and even 1–3–3-1. An increase in the number of sub-pulses leads to a broadening of the suppression bandwidth, but at the expense of RF pulse duration. Such water excitation pulses may vary in total RF duration from 1.7 to 3.9 ms at 3 T, and may thus increase the TR and scan time significantly. LIBRE demonstrated a large fat suppression bandwidth with total RF pulse durations as short as 1.4 ms [13], and current investigations suggest that this may be further decreased to a total RF pulse duration of 1 ms (τ = 0.5 ms) only [14]. Since the optimal duration of the LIBRE pulse is a function of its RF offset and the resonance frequency of the fat (Eq. 1), similar fat suppression may also be achieved at higher magnetic fields strengths with a shorter RF pulse duration. Therefore this LIBRE pulse may be suitable for coronary artery CMR at field strengths beyond 3 T as well [33, 35].
Finally, the proposed LIBRE method in combination with free-breathing radical coronary artery CMR offers a flexible pulse duration while maintaining fat suppression capabilities [13].
Since coronary artery disease (CAD) is a leading cause of death in the developed world, there is a need for a noninvasive imaging method that can detect CAD without using ionizing radiation without any type of contrast media. Moreover an imaging method that can be used as a radiation-free alternative to detect anomalies in the coronary anatomy in children and young adults is highly desirable. A recent study in a pediatric cohort compared the use of respiratory self-navigated coronary artery CMR with computed tomography angiography (CTA) at 1.5 T and demonstrated that coronary artery anomalies could be detected on CMR with high accuracy [36]. Other patient studies using respiratory self-navigation in a clinical environment have been recently published with promising results at 1.5 T [20, 21]. LIBRE has the potential of enabling a similar if not superior scan quality also at 3 T and, currently, an ongoing clinical study is being performed in heart transplant patients using this technique [37]. Future direct comparisons between the LIBRE self-navigated sequence and coronary CTA are of course warranted.
LIBRE water excitation pulses were implemented as part of a 3D coronary artery CMR radial imaging sequence at 3 T and demonstrated an effective and robust fat suppression both in vitro and in vivo. The LIBRE method significantly improved coronary artery image quality compared with more conventional fat suppression methods when self-navigated free-breathing noncontrast coronary artery CMR was performed without the use of a contrast agent.
CNR:
Contrast to noise ratio
FOV:
Field-of-view
FS:
CHESS fat suppression
Gradient recalled echo
LAD:
Left anterior descending coronary artery
Lipid insensitive binomial off-resonant excitation
RCA:
Right coronary artery
RF:
Superior-inferior
TE:
Echo time
Repetition time
WE:
Binomial 1-1 water excitation
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Bley TA, Wieben O, Francois CJ, Brittain JH, Reeder SB. Fat and water magnetic resonance imaging. J Magn Reson Imaging. 2010;31:4–18. https://doi.org/10.1002/jmri.21895.
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Nezafat M, Henningsson M, Ripley DP, et al. Coronary MR angiography at 3T: fat suppression versus water-fat separation. MAGMA. 2016;29:733–8. https://doi.org/10.1007/s10334-016-0550-7.
van Elderen SGC, Versluis MJ, Westenberg JJM, et al. Right coronary MR angiography at 7 T: a direct quantitative and qualitative comparison with 3 T in young healthy volunteers. Radiology. 2010;257:254–9. https://doi.org/10.1148/radiol.100615.
Bhat H, Yang Q, Zuehlsdorff S, Li K, Li D. Contrast-enhanced whole-heart coronary MRA at 3T using interleaved EPI. Investig Radiol. 2010;45:458–64. https://doi.org/10.1097/RLI.0b013e3181d8df32.
Bizino MB, Bonetti C, van der Geest RJ, Versluis MJ, Webb AG, Lamb HJ. High spatial resolution coronary magnetic resonance angiography at 7 T: comparison with low spatial resolution bright blood imaging. Investig Radiol. 2014;49:326–30. https://doi.org/10.1097/RLI.0000000000000047.
Albrecht MH, Varga-Szemes A, Schoepf UJ, et al. Diagnostic accuracy of noncontrast self-navigated free-breathing MR angiography versus CT angiography: a prospective study in pediatric patients with suspected anomalous coronary arteries. Acad Radiol. 2019. https://doi.org/10.1016/j.acra.2018.12.010.
Bastiaansen, J. A. M., di Sopra, L, Ginami, G., et al. Lipid-insensitive 4D motion-resolved free breathing coronary MRA in heart transplant recipients at 3T. In: Proc Int Soc Magn Reson med. Vol. 26. ; 2018. p. 915.
R'Equip SNF grant 326030_150828, and the MagnetoTeranostics project that was scientifically evaluated by the Swiss National Science Foundation (SNSF), financed by the Swiss Confederation and funded by Nano-Tera.ch (project No 530 627). JB received funding from the SNSF (grant number PZ00P3_167871), the Emma Muschamp foundation, and the Swiss Heart foundation.
Department of Diagnostic and Interventional Radiology, Lausanne University Hospital and University of Lausanne, Lausanne, Switzerland
Jessica A. M. Bastiaansen
, Ruud B. van Heeswijk
, Matthias Stuber
& Davide Piccini
Center for Biomedical Imaging, Lausanne, Switzerland
Matthias Stuber
Advanced clinical imaging technology, Siemens Healthcare AG, Lausanne, Switzerland
Davide Piccini
Search for Jessica A. M. Bastiaansen in:
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JB developed and implemented the LIBRE RF pulses, analyzed the data, and wrote the main draft of the manuscript. JB and DP designed the study and performed the experiments. DP contributed the source code for the respiratory-self-navigated CMR sequence. RvH contributed the source code of the T2 preparation module. All authors read, revised, and approved the final manuscript.
Correspondence to Jessica A. M. Bastiaansen.
This study was approved by local authorities. All volunteers provided written informed consent for participation in this study.
All volunteers provided consent for publication of this study.
DP is an employee of Siemens Healthcare. JB, RvH, MS have no competing interests.
Bastiaansen, J.A.M., van Heeswijk, R.B., Stuber, M. et al. Noncontrast free-breathing respiratory self-navigated coronary artery cardiovascular magnetic resonance angiography at 3 T using lipid insensitive binomial off-resonant excitation (LIBRE). J Cardiovasc Magn Reson 21, 38 (2019) doi:10.1186/s12968-019-0543-6
Coronary artery angiography
3 T MRI
Water excitation
Fat suppression
Noncontrast
Vessel sharpness | CommonCrawl |
\begin{document}
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\newcommand{\mathbb{R}}{\mathbb{R}}
\newtheorem{theo}{Theorem} \renewcommand\thetheo{\arabic{section}.\arabic{theo}} \newtheorem{theor}[theo]{\bf Theorem} \newtheorem{lem}[theo]{Lemma} \newtheorem{propo}{\rm PROPOSITION} \newtheorem{rema}[theo]{Remark} \newtheorem{defn}[theo]{\rm DEFINITION} \newtheorem{exam}{Example} \newtheorem{coro}[theo]{\rm COROLLARY} \def\trivlist\item[\hskip\labelsep{\it Conjecture.}]{\trivlist\item[\hskip\labelsep{\it Conjecture.}]} \def\trivlist\item[\hskip\labelsep{\it Assumption.}]{\trivlist\item[\hskip\labelsep{\it Assumption.}]}
\renewcommand{\thesection\arabic{equation}}{\thesection\arabic{equation}}
\title{Vibrations of thin piezoelectric shallow shells: Two-dimensional approximation}
\markboth{N~Sabu}{Vibrations of thin piezoelectric shallow shells}
\author{N~SABU}
\address{T.I.F.R. Centre, IISc Campus, Bangalore 560 012, India\\ \noindent E-mail: [email protected]}
\volume{113}
\mon{August}
\parts{3}
\Date{MS received 21 January 2003}
\begin{abstract} In this paper we consider the eigenvalue problem for piezoelectric shallow shells and we show that, as the thickness of the shell goes to zero, the eigensolutions of the three-dimensional piezoelectric shells converge to the eigensolutions of a two-dimensional eigenvalue problem. \end{abstract}
\keyword{Vibrations; piezoelectricity; shallow shells.}
\maketitle
\section{Introduction}
Lower dimensional models of shells are preferred in numerical computations to three-dimensional models when the thickness of the shells is `very small'. A lot of work has been done on the lower dimensional approximation of boundary value and eigenvalue problem for elastic plates and shells (cf.~\cite{BCM,CK,CL1,CLM,CM,SKNS1,SKNS2}). Recently some work has been done on the lower dimensional approximation of boundary value problem for piezoelectric shells (cf. \cite{BH1}).
In this paper, we would like to study the limiting behaviour of the eigenvalue problems for thin piezoelectric shallow shells. We begin with a brief description of the problem and describe the results obtained.
Let $\hat{\Omega}^\epsilon=\Phi^\epsilon(\Omega^\epsilon), \Omega^\epsilon=\omega\times(-\epsilon, \epsilon)$ with $\omega\subset \mathbb{R}^2,$ and the mapping $\Phi^\epsilon: \overline{\Omega}^\epsilon\rightarrow\mathbb{R}^3$ is given by \begin{equation*} \Phi^\epsilon(x^\epsilon)=(x_1, x_2, \epsilon\theta(x_1, x_2))+ x_{3}^{\epsilon} a_3^\epsilon(x_1, x_2) \end{equation*} for all $x^\epsilon=(x_1, x_2, x_3^\epsilon)\in\overline{\Omega}^\epsilon$, where $\theta$ is an injective mapping of class $C^3$ and ${a}_3^\epsilon$ is a unit normal vector to the middle surface $\Phi^\epsilon(\overline{\omega})$ of the shell. Let $\gamma_0, \gamma_e\subset\partial\omega$ with meas($\gamma_0)>0$ and meas($\gamma_e)>0$. Let $\hat{\Gamma}^\epsilon_0= \Phi^{\epsilon}(\gamma_0 \times(-\epsilon, \epsilon))$ and let $\hat{\Gamma}^\epsilon_e= \Phi^{\epsilon}(\gamma_e\times(-\epsilon, \epsilon))$. The shell is clamped along the portion $\hat{\Gamma}^\epsilon_0 $ of the lateral surface.
Then the variational form of the eigenvalue problem consists of finding the displacement vector ${u}^\epsilon$, the electric potential ${\varphi}^{\epsilon}$ and $\xi^\epsilon\in\mathbb{R}$ satisfying eq.~(\ref{eq:a9}). We then show that the component of the eigenvector involving the electric potential $\varphi^\epsilon$ can be uniquely determined in terms of the displacement vector $u^\epsilon$ and the problem thus reduces to finding $(u^\epsilon, \xi^\epsilon)$ satisfying equations~(\ref{eq:a31}) and (\ref{eq:a32}).
After making appropriate scalings on the data and the unknowns, we transfer the problem to a domain $\Omega = \omega\times(-1, 1)$ which is independent of $\epsilon$. Then we show that the scaled eigensolutions converge to the solutions of a two-dimensional eigenvalue problem (\ref{eq:e55}).
\section{The three-dimensional problem}
Throughout this paper, Latin indices vary over the set $\{1,2,3\}$ and Greek indices over the set $\{1,2\}$ for the components of vectors and tensors. The summation over repeated indices will be used.
Let $\omega\subset \mathbb{R}^2$ be a bounded domain with a Lipschitz continuous boundary $\gamma$ and let $\omega$ lie locally on one side of $\gamma$. Let $\gamma_0, \gamma_e\subset\partial\omega$ with meas$(\gamma_0)>0$ and meas($\gamma_e)>0$. Let $\gamma_1=\partial\omega\backslash \gamma_0$ and $\gamma_s=\partial\omega\backslash \gamma_e$. For each $\epsilon > 0$, we define the sets \begin{align*} \Omega^{\epsilon} &= \omega\times (-\epsilon, \epsilon),\quad \Gamma^{\pm,\epsilon}=\omega\times\{\pm\epsilon\},\quad \Gamma^\epsilon_0=\gamma_0\times(-\epsilon, \epsilon),\\[.2pc] \Gamma^\epsilon_1 &= \gamma_1\times(-\epsilon, \epsilon),\quad \Gamma^\epsilon_e=\gamma_e\times(-\epsilon, \epsilon),\quad \Gamma^\epsilon_s=\gamma_s\times(-\epsilon, \epsilon). \end{align*}
Let $x^\epsilon=(x_1, x_2, x_3^\epsilon)$ be a generic point on $\Omega^\epsilon$ and let $\partial_\alpha=\partial_\alpha^\epsilon= \frac{\partial}{\partial x_\alpha}$ and $\partial_3^\epsilon=\frac{\partial} {\partial x_3^\epsilon}$.
We assume that for each $\epsilon$, we are given a function $\theta^\epsilon :\omega\rightarrow \mathbb{R}$ of class $C^3$. We then define the map $\phi^\epsilon: \omega\rightarrow \mathbb{R}^3$ by \begin{equation} \phi^\epsilon(x_1, x_2)=(x_1, x_2, \theta^\epsilon(x_1, x_2))\quad \mbox{for all}\ \, (x_1, x_2)\in \omega.\label{eq:a1} \end{equation}
At each point of the surface $S^\epsilon=\phi^\epsilon(\omega)$, we define the normal vector \begin{equation*}
a^\epsilon= (|\partial_1\theta^\epsilon|^2+|\partial_2\theta^\epsilon|^2+1) ^{-1/2}(-\partial_1\theta^\epsilon, -\partial_2\theta^\epsilon, 1). \end{equation*}
For each $\epsilon>0$, we define the mapping $\Phi^\epsilon:\Omega^\epsilon \rightarrow \mathbb{R}^3 $ by \begin{equation} \Phi^\epsilon(x^\epsilon)=\phi^\epsilon(x_1, x_2)+x_3^\epsilon a^\epsilon(x_1, x_2)\quad \mbox{for all}\ \, x^\epsilon\in\Omega^\epsilon.\label{eq:a2} \end{equation}
It can be shown that there exists an $\epsilon_0>0$ such that the mappings $\Phi^\epsilon:\Omega^\epsilon\rightarrow \Phi^\epsilon(\Omega^\epsilon)$ are $C^1$ diffeomorphisms for all $0<\epsilon\leq \epsilon_0$. The set $\hat{\Omega}^\epsilon=\Phi^\epsilon(\Omega^\epsilon)$ is the reference configuration of the shell. For $0<\epsilon\leq\epsilon_0$, we define the sets \begin{align*} &\hat{\Gamma}^{\pm, \epsilon} = \Phi^\epsilon(\Gamma^{\pm,\epsilon}),\quad \hat{\Gamma}^\epsilon_0=\Phi^\epsilon(\Gamma^\epsilon_0),\quad \hat{\Gamma}^\epsilon_1=\Phi(\Gamma^\epsilon_1),\quad \hat{\Gamma}^\epsilon_{N} = \hat{\Gamma}^\epsilon_i\cup \hat{\Gamma}^{\pm\epsilon},\\[.2pc] &\hat{\Gamma}^\epsilon_e = \Phi(\Gamma^\epsilon_e),\quad \hat{\Gamma}^\epsilon_s=\Phi(\Gamma^\epsilon_s),\quad \hat{\Gamma}^\epsilon_{eD}=\hat{\Gamma}^\epsilon_e\cup \hat{\Gamma}^{\pm\epsilon} \end{align*} and we define vectors $g^\epsilon_i$ and $g^{i,\epsilon}$ by the relations \begin{equation*} g^\epsilon_i=\partial^\epsilon_i\Phi^\epsilon\quad \hbox{and}\quad g^{j,\epsilon} \cdot g^\epsilon_i=\delta^j_i \end{equation*} which form the covariant and contravariant basis respectively of the tangent plane of $\Phi^\epsilon(\Omega^\epsilon)$ at $\Phi^\epsilon(x^\epsilon)$. The covariant and contravariant metric tensors are given respectively by \begin{equation*} g^\epsilon_{ij}=g^\epsilon_i \cdot g^\epsilon_j\quad \hbox{and}\quad
g^{ij,\epsilon}=g^{i,\epsilon} \cdot g^{j,\epsilon}. \end{equation*}
The Christoffel symbols are defined by \begin{equation*} \Gamma^{p,\epsilon}_{ij}=g^{p,\epsilon} \cdot \partial^\epsilon_jg^\epsilon_i. \end{equation*}
Note however that when the set $\Omega^\epsilon$ is of the special form $\Omega^\epsilon=\omega\times(-\epsilon, \epsilon)$ and the mapping $\Phi^\epsilon$ is of the form (\ref{eq:a2}), the following relations hold: \begin{equation*} \Gamma^{3,\epsilon}_{\alpha 3}=\Gamma^{p,\epsilon}_{33}=0. \end{equation*}
The volume element is given by $\sqrt{g^\epsilon}{\rm d}x^\epsilon$ where \begin{equation*} g^\epsilon = {\rm det} (g^\epsilon_{ij}). \end{equation*}
It can be shown that there exist constants $g_1$ and $g_2$ such that \begin{equation} 0 < g_1\leq g^\epsilon\leq g_2\label{eq:1a2} \end{equation} for $0\leq\epsilon\leq\epsilon_0$.
Let $\hat{A}^{ijkl,\epsilon}, \hat{P}^{ijk,\epsilon}$ and $\hat{\cal E}^{ij,\epsilon}$ be the elastic, piezoelectric and dielectric tensors respectively. We assume that the material of the shell is {\it homogeneous and isotropic}. Then the elasticity tensor is given by \begin{equation} \hat{A}^{ijkl,\epsilon}=\lambda g^{ij}g^{kl} + \mu(g^{ik}g^{jl}+g^{il}g^{jk}), \label{eq:a3} \end{equation} where $\lambda$ and $\mu$ are the Lam\`{e} constants of the material.
These tensors satisfy the following coercive relations. There exists a constant $C>0$ such that for all symmetric tensors $(M_{ij})$ and for any vector $(t_i)\in \mathbb{R}^3$, \begin{align} &\hat{A}^{ijkl,\epsilon}M_{kl}M_{ij} \geq C\sum_{i,j=1}^3(M_{ij})^2, \label{eq:a4}\\[.2pc] &\hat{\cal E}^{kl,\epsilon}t_kt_l \geq C{\sum_{j=1}^3t^2_j}.\label{eq:a5} \end{align}
Moreover we have the symmetries \begin{equation*} \hat{A}^{ijkl,\epsilon} = \hat{A}^{klij,\epsilon} = \hat{A}^{jikl,\epsilon},\quad \hat{\cal E}^{kl,\epsilon} = \hat{\cal E}^{kl,\epsilon},\quad \hat{P}^{ijk,\epsilon} = \hat{P}^{kij, \epsilon}. \end{equation*}
Then the eigenvalue problem consists of finding $(\hat{u}^\epsilon, \hat{\varphi}^\epsilon, \xi^\epsilon)$ such that \begin{align} \left. \begin{array}{lcl} -{\rm div}\hat{\sigma}^\epsilon(\hat{u}^\epsilon, \hat{\varphi}^\epsilon) =\xi^\epsilon\hat{u}^\epsilon \mbox{ in }\hat{\Omega}^\epsilon\\[.1pc] \hat{\sigma}^\epsilon(\hat{u}^\epsilon, \hat{\varphi}^\epsilon)\nu =0 \mbox{ on } \hat{\Gamma}^\epsilon_N\\[.1pc] \hat{u}^\epsilon=0 \mbox{ on } \hat{\Gamma}^\epsilon_0 \end{array} \right\},\label{eq:aa1}\\[.2pc] \left. \begin{array}{lcl} {\rm div}\hat{D}^\epsilon(\hat{u}^\epsilon, \hat{\varphi}^\epsilon)=0 \mbox{ in } \hat{\Omega}^\epsilon\\[.1pc] \hat{D}^\epsilon(\hat{u}^\epsilon, \hat{\varphi}^\epsilon)\nu=0 \mbox{ on } \hat{\Gamma}_{s}^\epsilon\\[.1pc] \hat{\varphi}^\epsilon=0 \mbox{ on }\hat{\Gamma}^\epsilon_{eD}. \end{array} \right\},\label{eq:aa2} \end{align} where \begin{align} \hat{\sigma}_{ij}^\epsilon &= \hat{A}^{ijkl,\epsilon}\hat{e}_{ij}^{\epsilon} -\hat{P}^{kij,\epsilon}\hat{E}_k,\label{eq:aa3}\\[.2pc] \hat{D}_k &= \hat{P}^{kij,\epsilon}\hat{e}_{ij}^\epsilon + \hat{\cal E}^{kl,\epsilon}\hat{E}_l,\label{eq:aa4} \end{align} where $\hat{e}^\epsilon_{ij}(\hat{u}^\epsilon) = \frac{1}{2}(\hat{\partial}^\epsilon_i\hat{u}^\epsilon_j+ \hat{\partial}^\epsilon_j\hat{u}^\epsilon_i), \hat{\partial}^\epsilon_i = \partial/\partial\hat{x}^\epsilon_{i}$ and $\hat{E}_k(\hat{\varphi}^\epsilon)=- \ \ \raise .01pc \hbox{$\hat{}$}\hskip -.33pc{\bigtriangledown} (\hat{\varphi}^\epsilon)$.
We define the spaces \begin{align} \hat{V}^\epsilon &= \{\hat{v}\in(H^1(\hat{\Omega}^\epsilon))^3,
\hat{v}|_{\hat{\Gamma}^\epsilon_0}=0\},\label{eq:aa5}\\[.2pc] \hat{\Psi}^\epsilon &= \{\hat{\psi}\in H^1(\hat{\Omega}^\epsilon),
\hat{\psi}|_{\hat{\Gamma}_{eD}^\epsilon}=0\}.\label{eq:aa6} \end{align}
Then the variational form of systems (\ref{eq:aa1}) and (\ref{eq:aa2}) is to find $(\hat{u}^\epsilon, {\hat{\varphi}}^\epsilon, \xi^\epsilon)\in \hat{V}^\epsilon\times\hat{\Psi}^\epsilon\times \mathbb{R}$ such that \begin{equation} \hat{a}^\epsilon((\hat{u}^\epsilon, {\hat{\varphi}}^\epsilon), (\hat{v}^\epsilon, \hat{\psi}^{\epsilon})) =\xi^\epsilon\hat{l}^\epsilon(\hat{v}^\epsilon, \hat{\psi}^\epsilon)\quad \mbox { for all } (\hat{v}^\epsilon, \hat{\psi}^\epsilon)\in \hat{V}^\epsilon\times \hat{\Psi}^\epsilon,\label{eq:aa7} \end{equation} where \begin{align} \hat{a}^\epsilon((\hat{u}^\epsilon, {\hat{\varphi}}^\epsilon), (\hat{v}^\epsilon, \hat{\psi}^{\epsilon})) &= \int_{\hat{\Omega}^\epsilon}\hat{A}^{ijkl,\epsilon} \hat{e}_{kl}^\epsilon(\hat{u}^\epsilon) \hat{e}_{ij}^\epsilon(\hat{v}^\epsilon){\rm d} \hat{x}^\epsilon\nonumber\\[.2pc] &\quad\ + \int_{\hat{\Omega}^\epsilon}\hat{\cal E}^{ij,\epsilon}\hat{\partial}_i^\epsilon {\hat{\varphi}}^\epsilon\hat{\partial}_j^\epsilon\hat{\psi}^{\epsilon} {\rm d}\hat{x}^\epsilon\nonumber\\[.2pc] &\quad\ +\int_{\hat{\Omega}^\epsilon} \hat{P}^{mij,\epsilon}(\hat{\partial}^\epsilon_m {\hat{\varphi}}^\epsilon \hat{e}_{ij}^\epsilon(\hat{v}^\epsilon)- \hat{\partial}_m^\epsilon\hat{\psi}^\epsilon \hat{e}_{ij}^\epsilon(\hat{u}^\epsilon)){\rm d}\hat{x}^\epsilon,\label{eq:aa8} \end{align}
$\left.\right.$
\begin{align} \hat{l}^\epsilon(\hat{v}^\epsilon, \hat{\psi}^{\epsilon}) &= \int_{\hat{\Omega}^\epsilon}\hat{u}^\epsilon \cdot \hat{v}^\epsilon {\rm d}\hat{x}^\epsilon.\label{eq:aa9} \end{align}
Since the mappings $\Phi^\epsilon:\overline{\Omega}^\epsilon\rightarrow \overline{\hat{\Omega}}^\epsilon$ are assumed to be $C^1$ diffeomorphisms, the correspondences that associate with every element $\hat{v}^\epsilon \in\hat{V}^\epsilon$, the vector \begin{equation*} v^\epsilon=\hat{v}^\epsilon \cdot \Phi^\epsilon: \Omega^\epsilon\rightarrow\mathbb{R}^3 \end{equation*} and with every element $\hat{\psi}^{\epsilon} \in \hat{\Psi}^{\epsilon}$, the function \begin{equation*} \psi^\epsilon=\hat{\psi}^\epsilon \cdot \Phi^\epsilon:\Omega^\epsilon\rightarrow\mathbb{R} \end{equation*} induce bijections between the spaces $\hat{V}^\epsilon$ and $V^\epsilon$, and the spaces $\hat{\Psi}^\epsilon$ and $\Psi^\epsilon$ respectively, where \begin{align}
V^\epsilon &= \{v^\epsilon\in (H^1(\Omega^\epsilon))^3|v^\epsilon=0 \ {\rm on} \ \Gamma^\epsilon_0\},\label{eq:a6}\\[.2pc]
\Psi^\epsilon &= \{\psi^\epsilon\in H^1(\Omega^\epsilon)|\psi^{\epsilon} = 0 \ {\rm on} \ \Gamma^\epsilon_{eD}\}.\label{eq:a7} \end{align}
Then we have \begin{align} &\hat{\partial}^\epsilon_j\hat{v}^\epsilon(\hat{x}^\epsilon) = (\partial^\epsilon_i v^\epsilon)(g^{i, \epsilon})_j,\label{eq:aa11}\\[.2pc]
&\hat{e}_{ij}(\hat{v})(\hat{x}^\epsilon) = e^\epsilon_{k\|l}(v^\epsilon) (g^{k,\epsilon})_i(g^{l,\epsilon})_j,\label{eq:aa10} \end{align} where \begin{equation}
e^\epsilon_{i\|j}(v^\epsilon)=\frac{1}{2}(\partial^\epsilon_i v^\epsilon_j+ \partial^\epsilon_j v^\epsilon_i)-\Gamma^{p,\epsilon}_{ij}v^\epsilon_p. \label{eq:a8} \end{equation}
Then the variational form (\ref{eq:aa7}) posed on the domain $\Omega^\epsilon$ is to find $(u^\epsilon, {\varphi}^\epsilon, \xi^\epsilon)\in V^\epsilon\times\Psi^\epsilon\times\mathbb{R}$ such that \begin{equation} a^\epsilon((u^\epsilon, {\varphi}^\epsilon), (v^\epsilon, \psi^{\epsilon})) =\xi^\epsilon l^\epsilon(v^\epsilon, \psi^\epsilon )\quad \mbox { for all } (v^\epsilon, \psi^\epsilon)\in V^\epsilon\times\Psi^\epsilon,\label{eq:a9} \end{equation} where \begin{align} a^\epsilon((u^\epsilon, {\varphi}^\epsilon), (v^\epsilon, \psi^{\epsilon})) &= \int_{\Omega^\epsilon}A^{ijkl,\epsilon}
e_{k\|l}^\epsilon(v^\epsilon)
e_{i\|j}^\epsilon(v^\epsilon)\sqrt{g^\epsilon}{\rm d}x^\epsilon\nonumber\\[.2pc] &\quad\ +\int_{\Omega^\epsilon} {\cal E}^{ij,\epsilon}\partial_i^\epsilon {\varphi}^\epsilon\partial_j^\epsilon\psi^\epsilon\sqrt{g^\epsilon} {\rm d}x^\epsilon\nonumber\\[.2pc] &\quad\ +\int_{\Omega^\epsilon}
P^{mij,\epsilon}(\partial^\epsilon_m{\varphi}^\epsilon e_{i\|j}^\epsilon(v^\epsilon)\nonumber\\[.2pc]
&\quad\ -\partial_m^\epsilon\psi^\epsilon e_{i\|j}^\epsilon(u^\epsilon))\sqrt{g^\epsilon}{\rm d}x^\epsilon,\label{eq:a10} \end{align}
$\left.\right.$
\begin{align} &l^\epsilon(v^\epsilon, \psi^{\epsilon}) = \int_{\Omega^\epsilon}u^\epsilon \cdot v^\epsilon \sqrt{g^\epsilon} {\rm d}x^\epsilon,\label{eq:a11}\\[.2pc] &A^{pqrs, \epsilon} = \hat{A}^{ijkl, \epsilon}(g^{p,\epsilon})_i \cdot (g^{q,\epsilon})_j \cdot (g^{r,\epsilon})_k \cdot (g^{s,\epsilon})_l,\label{eq:aa12}\\[.2pc] &{\cal E}^{pq,\epsilon} = \hat{\cal E}^{ij,\epsilon}(g^{p,\epsilon})_i \cdot (g^{q,\epsilon})_j,\label{eq:aa13}\\[.2pc] &P^{pqr,\epsilon} = \hat{P}^{ijk,\epsilon}(g^{p,\epsilon})_i \cdot (g^{q,\epsilon})_j \cdot (g^{r,\epsilon})_k. \label{eq:aa14} \end{align}
Using the relations (\ref{eq:1a2}), (\ref{eq:a4}) and (\ref{eq:a5}), it can be shown that there exists a constant $C>0$ such that for all symmetric tensor $(M_{ij})$ and for any vector $(t_i)\in \mathbb{R}^3$, \begin{align} &A^{ijkl,\epsilon}M_{kl}M_{ij} \geq C \sum_{i,j=1}^3(M_{ij})^2,\label{eq:aa15}\\[.2pc] &{\cal E}^{ij,\epsilon}t_it_j \geq C \sum_{i=1}^3t_i^2.\label{eq:aa16} \end{align}
Clearly the bilinear form associated with the left-hand side of (\ref{eq:a9}) is elliptic. Hence by Lax--Milgram theorem, given $f^\epsilon\in V^{\prime \epsilon}$ and $h^\epsilon\in \Psi^{\prime \epsilon}$, there exists a unique $(u^\epsilon, \varphi^\epsilon)\in V^\epsilon\times\Psi^\epsilon$ such that \begin{equation} a^\epsilon((u^\epsilon, \varphi^\epsilon), (v^\epsilon, \psi^\epsilon))= \langle (f^\epsilon, h^\epsilon), (v^\epsilon, \psi^\epsilon)\rangle\quad \ \ \forall V^{\epsilon} \times \Psi^{\epsilon} \in V^{\epsilon} \times \Psi^{\epsilon}.\label{eq:a12} \end{equation}
In particular, for each $f^\epsilon\in (L^2(\Omega^\epsilon))^3$, there exists a unique solution $(u^\epsilon, \varphi^\epsilon)\in V^\epsilon\times\Psi^\epsilon$ such that \begin{equation} a^\epsilon((u^\epsilon, \varphi^\epsilon), (v^\epsilon, \psi^\epsilon))=\int_{\Omega^\epsilon} f^\epsilon v^\epsilon \sqrt{g^\epsilon}{\rm d}x^\epsilon\quad \forall v^\epsilon\times\psi^\epsilon\in V^\epsilon\times\Psi^\epsilon. \label{eq:a13} \end{equation}
This is equivalent to the following equations. \begin{align}
\int_{{\Omega}^\epsilon}{A}^{ijkl,\epsilon}{e}_{k\|l}^\epsilon({u}^\epsilon)
{e}_{i\|j}^\epsilon({v}^\epsilon)\sqrt{g^\epsilon}{\rm d}x^\epsilon &+ \int_{{\Omega}^\epsilon} {P}^{mij,\epsilon}{\partial}^\epsilon_m
(\varphi^\epsilon) {e}_{i\|j}^\epsilon({v}^\epsilon)\sqrt{g^\epsilon}{\rm d}x^\epsilon\nonumber\\ &= \int_{\Omega^\epsilon} f^\epsilon v^\epsilon \sqrt{g^\epsilon}{\rm d}x^\epsilon\quad \forall v^\epsilon\in V^\epsilon \label{eq:a14} \end{align} and \begin{equation} \int_{{\Omega}^\epsilon}{\cal E}^{ij,\epsilon}{\partial}_i^\epsilon {{\varphi}}^\epsilon{\partial}_j^\epsilon{\psi}^{\epsilon} \sqrt{g^\epsilon}{\rm d}x^\epsilon=\int_{{\Omega}^\epsilon}{P}^{mij,\epsilon} {\partial}_m^\epsilon{\psi}^\epsilon
{e}_{i\|j}^\epsilon({u}^\epsilon)\sqrt{g^\epsilon}{\rm d}x^\epsilon\ \ \ \forall \psi^\epsilon\in \Psi^\epsilon.\label{eq:a15} \end{equation}
From relation (2.28), it follows that the bilinear form associated with the left-hand side of (\ref{eq:a15}) is $\Psi^\epsilon$-elliptic.
Also for each $h^\epsilon\in V^\epsilon$, the mapping \begin{equation*} \psi^\epsilon\rightarrow \int_\Omega^\epsilon P^{mij,\epsilon}
\partial_m\psi^\epsilon e^\epsilon_{i\|j}(h^\epsilon) \sqrt{g^\epsilon}{\rm d}x^\epsilon \end{equation*} defines a linear functional on $\Psi^\epsilon$. Hence for each $h^\epsilon\in V^\epsilon$, there exists a unique $T^\epsilon(h^\epsilon)\in \Psi^\epsilon$ such that \begin{equation} \int_{{\Omega}^\epsilon}\!\!{\cal E}^{ij,\epsilon}{\partial}_i^\epsilon {T^\epsilon(h^\epsilon)}{\partial}_j^\epsilon{\psi}^{\epsilon} \sqrt{g^\epsilon}{\rm d}x^\epsilon \!\!=\!\!\int_{{\Omega}^\epsilon}\!\!{P}^{mij,\epsilon} {\partial}_m^\epsilon{\psi}^\epsilon
{e}_{i\|j}^\epsilon({h}^\epsilon)\sqrt{g^\epsilon}{\rm d}x^\epsilon\ \ \ \forall \psi^\epsilon\!\in\! \Psi^\epsilon\label{eq:a16} \end{equation} and that $T^\epsilon: V^\epsilon\rightarrow \Psi^\epsilon$ is continuous.
In particular, it follows from (\ref{eq:a15}) and the above equation that $\varphi^\epsilon=T^\epsilon(u^\epsilon)$ and eqs~(\ref{eq:a14}) and (\ref{eq:a15}) become \begin{align}
\int_{{\Omega}^\epsilon}\! {A}^{ijkl,\epsilon}{e}_{k\|l}^\epsilon({u}^\epsilon)
{e}_{i\|j}^\epsilon({v}^\epsilon)\sqrt{g^\epsilon}{\rm d}x^\epsilon &+\int_{{\Omega}^\epsilon} \!{P}^{mij,\epsilon}{\partial}^\epsilon_m
(T^\epsilon(u^\epsilon)) {e}_{i\|j}^\epsilon({v}^\epsilon)\sqrt{g^\epsilon}{\rm d}x^\epsilon\nonumber\\[.2pc] &=\int_{\Omega^\epsilon}\! f^\epsilon v^\epsilon \sqrt{g^\epsilon}{\rm d}x^\epsilon\ \ \ \forall v^\epsilon\in V^\epsilon,\label{eq:a22}\\[.2pc] \int_{{\Omega}^\epsilon}{\cal E}^{ij,\epsilon}{\partial}_i^\epsilon (T^\epsilon(u^\epsilon)){\partial}_j^\epsilon{\psi}^{\epsilon} \sqrt{g^\epsilon}{\rm d}x^\epsilon &= \int_{{\Omega}^\epsilon}{P}^{mij,\epsilon} {\partial}_m^\epsilon{\psi}^\epsilon
{e}_{i\|j}^\epsilon({u}^\epsilon)\sqrt{g^\epsilon}{\rm d}x^\epsilon\nonumber\\[.2pc] &\quad\ \forall \psi^\epsilon\in \Psi^\epsilon.\label{eq:a23} \end{align}
\begin{lem} For each $h^{\epsilon}\in (L^2(\Omega^\epsilon))^3${\rm ,} there exists a unique $G^\epsilon(h^\epsilon) \in V^\epsilon$ such that \begin{align}
\hskip -3.5pc \int_{{\Omega}^\epsilon}{A}^{ijkl,\epsilon}{e}_{k\|l}^\epsilon
(G^\epsilon(h^\epsilon)) {e}_{i\|j}^\epsilon({v}^\epsilon)\sqrt{g^\epsilon}{\rm d}x^\epsilon &+ \int_{{\Omega}^\epsilon} {P}^{mij,\epsilon}{\partial}^\epsilon_m
(T^\epsilon(G^\epsilon(h^\epsilon))) {e}_{i\|j}^\epsilon({v}^\epsilon) \sqrt{g^\epsilon}{\rm d}x^\epsilon\nonumber\\[.2pc] \hskip -3.5pc &=\int_{\Omega^\epsilon} h^\epsilon v^\epsilon \sqrt{g^\epsilon}{\rm d}x^\epsilon\quad \forall v^\epsilon\in V^\epsilon\label{eq:a24} \end{align} and that $G^\epsilon:(L^2(\Omega^\epsilon))^3 \rightarrow V^\epsilon$ is continuous. \end{lem}
\begin{proof} Let $B^\epsilon(u^\epsilon, v^\epsilon)$ denotes the bilinear form associated with the left-hand side of eq.~(\ref{eq:a22}). Using (\ref{eq:a23}), we have \begin{align} B^\epsilon(u^\epsilon, v^\epsilon) &=
\int_{{\Omega}^\epsilon}{A}^{ijkl,\epsilon}{e}_{k\|l}^\epsilon({u}^\epsilon)
{e}_{i\|j}^\epsilon({v}^\epsilon)\sqrt{g^\epsilon}{\rm d}x^\epsilon\nonumber\\[.2pc] &\quad\ +\int_{{\Omega}^\epsilon} {P}^{mij,\epsilon}{\partial}^\epsilon_m
(T^\epsilon(u^\epsilon)) {e}_{i\|j}^\epsilon({v}^\epsilon)\sqrt{g^\epsilon}{\rm d}x^\epsilon\nonumber\\[.2pc]
&= \int_{{\Omega}^\epsilon}{A}^{ijkl,\epsilon}{e}_{k\|l}^\epsilon({u}^\epsilon)
{e}_{i\|j}^\epsilon({v}^\epsilon)\sqrt{g^\epsilon}{\rm d}x^\epsilon\nonumber\\[.2pc] &\quad\ +\int_{{\Omega}^\epsilon}{\cal E}^{ij,\epsilon}{\partial}_i^\epsilon (T^\epsilon(u^\epsilon)){\partial}_j^\epsilon(T^\epsilon(v^\epsilon)) \sqrt{g^\epsilon}{\rm d}x^\epsilon\nonumber \end{align} \begin{align}
&= \int_{{\Omega}^\epsilon}{A}^{ijkl,\epsilon}{e}_{k\|l}^\epsilon({v}^\epsilon)
{e}_{i\|j}^\epsilon({u}^\epsilon)\sqrt{g^\epsilon}{\rm d}x^\epsilon\nonumber\\[.2pc] &\quad\ +\int_{{\Omega}^\epsilon}{\cal E}^{ij,\epsilon}{\partial}_i^\epsilon (T^\epsilon(v^\epsilon)){\partial}_j^\epsilon(T^\epsilon(u^\epsilon)) \sqrt{g^\epsilon}{\rm d}x^\epsilon\nonumber\\[.2pc] &= B^\epsilon(v^\epsilon, u^\epsilon).\label{eq:a25} \end{align} Also, using (\ref{eq:a23}) and the relations (\ref{eq:aa15}) and (\ref{eq:aa16}), we have \begin{align} B^\epsilon(u^\epsilon, u^\epsilon) &=
\int_{{\Omega}^\epsilon}{A}^{ijkl,\epsilon}{e}_{k\|l}^\epsilon({u}^\epsilon)
{e}_{i\|j}^\epsilon({u}^\epsilon)\sqrt{g^\epsilon}{\rm d}x^\epsilon\nonumber\\[.2pc] &\quad\ +\int_{{\Omega}^\epsilon} {P}^{mij,\epsilon}{\partial}^\epsilon_m (T^\epsilon(u^\epsilon))
{e}_{i\|j}^\epsilon({u}^\epsilon)\sqrt{g^\epsilon}{\rm d}x^\epsilon\nonumber\\[.2pc]
&= \int_{{\Omega}^\epsilon}{A}^{ijkl,\epsilon}{e}_{k\|l}^\epsilon({u}^\epsilon)
{e}_{i\|j}^\epsilon({u}^\epsilon)\sqrt{g^\epsilon}{\rm d}x^\epsilon\nonumber\\[.2pc] &\quad\ +\int_{{\Omega}^\epsilon}{\cal E}^{ij,\epsilon}{\partial}_i^\epsilon (T^\epsilon(u^\epsilon)){\partial}_j^\epsilon(T^\epsilon(u^\epsilon)) \sqrt{g^\epsilon}{\rm d}x^\epsilon\nonumber\\[.2pc]
&\geq C\|u^\epsilon\|^2_{V^\epsilon}.\label{eq:a26} \end{align} Hence $B^\epsilon(\cdots)$ is symmetric and $V^\epsilon$-elliptic. Hence by Lax--Milgram theorem, there exists a unique $G^\epsilon(h^\epsilon)$ satisfying (\ref{eq:a24}). Letting $v^\epsilon=G^\epsilon(h^\epsilon)$ in (\ref{eq:a24}), we get \begin{align}
&\int_{{\Omega}^\epsilon}{A}^{ijkl,\epsilon}{e}_{k\|l}^\epsilon
(G^\epsilon(h^\epsilon)) {e}_{i\|j}^\epsilon(G^\epsilon(h^\epsilon))\sqrt{g^\epsilon} {\rm d}x^\epsilon\nonumber\\[.2pc] &\quad\ + \int_{{\Omega}^\epsilon} {P}^{mij,\epsilon}{\partial}^\epsilon_m
(T^\epsilon(G^\epsilon(h^\epsilon))){e}_{i\|j}^\epsilon(G^\epsilon(h^\epsilon)) \sqrt{g^\epsilon}{\rm d}x^\epsilon\nonumber\\[.2pc] &= \int_{\Omega^\epsilon} h^\epsilon G^\epsilon(h^\epsilon) \sqrt{g^\epsilon}{\rm d}x^\epsilon\label{eq:a27} . \end{align} Using (\ref{eq:a23}), it becomes \begin{align}
&\int_{{\Omega}^\epsilon}{A}^{ijkl,\epsilon}{e}_{k\|l}^\epsilon
(G^\epsilon(h^\epsilon)) {e}_{i\|j}^\epsilon(G^\epsilon(h^\epsilon))\sqrt{g^\epsilon} {\rm d}x^\epsilon\nonumber\\[.2pc] &\quad\ + \int_{{\Omega}^\epsilon}{\cal E}^{ij,\epsilon}{\partial}_i^\epsilon (T^\epsilon(G^\epsilon(h^\epsilon))){\partial}_j^\epsilon (T^\epsilon(G^\epsilon(h^\epsilon))) \sqrt{g^\epsilon}{\rm d}x^\epsilon\nonumber\\[.2pc] &= \int_{\Omega^\epsilon} h^\epsilon G^\epsilon(h^\epsilon) \sqrt{g^\epsilon}{\rm d}x^\epsilon.\label{eq:a28} \end{align} Using the relations (\ref{eq:aa15}) and (\ref{eq:aa16}), we have \begin{equation}
\|G^\epsilon(h^\epsilon)\|_{V^\epsilon}^2\leq C^\epsilon
\|G^\epsilon(h^\epsilon)\|_{V^\epsilon}
\|h^\epsilon\|_{(L^2(\Omega^\epsilon))^3}.\label{eq:a29} \end{equation} Hence \begin{equation}
\|G^\epsilon(h^\epsilon)\|_{V^\epsilon}
\leq C^\epsilon\|h^\epsilon\|_{(L^2(\Omega^\epsilon))^3} \end{equation} which implies that $G^\epsilon$ is continuous.
\mbox{\xxx{\char'245}} \end{proof}
It follows from (\ref{eq:a22}) and the above lemma that $u^\epsilon=G^\epsilon(f^\epsilon)$. Since the inclusion $(H^1(\Omega^\epsilon))^3 \hookrightarrow (L^2(\Omega^\epsilon))^3$ is compact, it follows that $G^\epsilon:(L^2(\Omega^\epsilon))^3\rightarrow (L^2(\Omega^\epsilon))^3$ is compact. Also since the bilinear form $B^\epsilon(\cdots)$ is symmetric, it follows that $G^\epsilon$ is self-adjoint. Hence from the spectral theory of compact, self-adjoint operators, it follows that there exists a sequence of eigenpairs $(u^{m,\epsilon}, \xi^{m,\epsilon})_{m=1}^\infty$ such that \begin{align} &\int_{{\Omega}^\epsilon}{A}^{ijkl,\epsilon}
{e}_{k\|l}^\epsilon({u}^{m,\epsilon})
{e}_{i\|j}^\epsilon({v}^\epsilon)\sqrt{g^\epsilon}{\rm d}x^\epsilon\nonumber\\[.2pc] &\quad\ + \int_{{\Omega}^\epsilon} {P}^{mij,\epsilon}{\partial}^\epsilon_m
(T^\epsilon(u^{m,\epsilon})) {e}_{i\|j}^\epsilon({v}^\epsilon) \sqrt{g^\epsilon}{\rm d}x^\epsilon\nonumber\\[.2pc] &= \xi^{m,\epsilon}\int_{\Omega^\epsilon} u^{m,\epsilon} v^\epsilon \sqrt{g^\epsilon}{\rm d}x^\epsilon\quad \forall v^\epsilon\in V^\epsilon,\label{eq:a31}\\[.2pc] &\int_{{\Omega}^\epsilon}{\cal E}^{ij,\epsilon}{\partial}_i^\epsilon (T^\epsilon(u^{m,\epsilon})){\partial}_j^\epsilon{\psi}^{\epsilon} \sqrt{g^\epsilon}{\rm d}x^\epsilon\nonumber\\[.2pc] &= \int_{{\Omega}^\epsilon}{P}^{mij,\epsilon} {\partial}_m^\epsilon{\psi}^\epsilon
{e}_{i\|j}^\epsilon({u}^{m,\epsilon})\sqrt{g^\epsilon}{\rm d}x^\epsilon \quad \forall \psi^\epsilon\in \Psi^\epsilon,\label{eq:a32}\\[.2pc] &0<\xi^{1,\epsilon} \leq\xi^{2,\epsilon}\leq \cdots \leq\xi^{m,\epsilon}\leq \cdots \rightarrow \infty,\label{eq:a33}\\[.2pc] &\int_{\Omega^\epsilon}u^{m,\epsilon}_i u^{n,\epsilon}_i\sqrt{g^\epsilon}{\rm d}x^\epsilon = \epsilon^3 \delta_{mn}.\label{eq:a34} \end{align} The sequence $\{u^{m,\epsilon}\}$ forms a complete orthonormal basis for $(L^2(\Omega))^3$.
Define the Rayleigh quotient $R(\epsilon)(v^\epsilon)$ for $v^\epsilon\in V^\epsilon$ by \begin{equation} \hskip -4pc R^\epsilon(v^\epsilon)=\frac{\int_{\Omega^\epsilon}A^{ijkl,\epsilon}
e_{k\|l}(v^\epsilon)e_{i\|j}(v^\epsilon)\sqrt{g^\epsilon}{\rm d}x^\epsilon +\int_{{\Omega}^\epsilon} {P}^{mij,\epsilon}{\partial}^\epsilon_m (T^\epsilon(v^\epsilon))
e_{i\|j}^\epsilon({v}^\epsilon)\sqrt{g^\epsilon}{\rm d}x^\epsilon} {\int_{\Omega^\epsilon}v^\epsilon_i v^\epsilon_i\sqrt{g^\epsilon}{\rm d}x^\epsilon}.\label{eq:a35} \end{equation}
Then \begin{equation} \xi^{m,\epsilon}=\min_{W^\epsilon\in W^\epsilon_m}\max_{v^\epsilon\in W^\epsilon\backslash\{0\}}R^\epsilon(v^\epsilon),\label{eq:a36} \end{equation} where $W_m^\epsilon$ denotes the collection of all $m$-dimensional subspaces of $V^\epsilon$.
\section{The scaled problem}
\setcounter{equation}{0} We now perform a change of variable so that the domain no longer depends on $\epsilon$. With $x=(x_1, x_2, x_3)\in\Omega$, we associate $x^\epsilon = (x_1, x_2, \epsilon x_3)\in\Omega^\epsilon$. Let \begin{align*} \Gamma_0 &= \gamma_0\times (-1,1),\quad \Gamma_1=\gamma_1\times (-1,1),\quad \Gamma^{\pm}=\omega\times\{\pm 1\},\\[.2pc] \Gamma_e &= \gamma_e\times(-1, 1),\quad \Gamma_s = \gamma_s\times(-1, 1),\\[.2pc] \Gamma_N &= \Gamma_1\cup\Gamma^{+}\cup\Gamma^{-},\quad \Gamma_{eD}=\Gamma^{+}\cup\Gamma^{-}\cup\Gamma_e. \end{align*}
With the functions $\Gamma^{p,\epsilon}, g^\epsilon, A^{ijkl,\epsilon}, P^{ijk,\epsilon}, {\cal E}^{ij,\epsilon}:\Omega^\epsilon\rightarrow\mathbb{R}$, we associate the functions $\Gamma^{p}(\epsilon), g^\epsilon, A^{ijkl}(\epsilon), P^{ijk}(\epsilon), {\cal E}^{ij}(\epsilon):\Omega\rightarrow\mathbb{R}$ defined by \begin{align} \Gamma^{p}(\epsilon)(x) &:= \Gamma^{p,\epsilon}(x^\epsilon),\quad g(\epsilon)(x)=g^\epsilon(x^\epsilon),\quad A^{ijkl}(\epsilon)(x)=A^{ijkl,\epsilon}(x^\epsilon),\label{eq:b1}\\ P^{ijk}(\epsilon)(x) &= P^{ijk,\epsilon}(x^\epsilon),\quad {\cal E}^{ij}(\epsilon)(x)= {\cal E}^{ij,\epsilon}(x^\epsilon).\label{eq:b2} \end{align}
\begin{assump} We assume that the shell is a shallow shell, i.e. there exists a function $\theta\in C^3(\omega)$ such that \begin{equation} \phi^\epsilon(x_1, x_2)=(x_1, x_2, \epsilon\theta(x_1, x_2))\quad \mbox{for all} \ \ (x_1, x_2) \in\omega,\label{eq:b6} \end{equation} i.e., the curvature of the shell is of the order of the thickness of the shell.
We make the following scalings on the eigensolutions. \begin{align} &u^{m,\epsilon}_\alpha(x^\epsilon) = \epsilon^2 u^m_\alpha(\epsilon)(x), ~~~~v_\alpha(x^\epsilon)=\epsilon^2 v_\alpha(x),\label{eq:b7}\\[.2pc] &u^{m, \epsilon}_3(x^\epsilon) = \epsilon u^m_3(\epsilon)(x), ~~~~v_3(x^\epsilon)=\epsilon v_3(x),\label{eq:bb7}\\[.2pc] &T^\epsilon (u^{m,\epsilon}(x^\epsilon)) = \epsilon^3 T(\epsilon)(u^m(\epsilon)(x)), ~~~~T^\epsilon(v(x^\epsilon))=\epsilon^3 T(\epsilon)(v(x)),\label{eq:b8}\\[.2pc] &\xi^{m,\epsilon} = \epsilon^2\xi^m(\epsilon).\label{eq:b9} \end{align}
With the tensors $e_{i\|j}^\epsilon$, we associate the tensors
$e_{i\|j}(\epsilon)$ through the relation \begin{equation}
e_{i\|j}^{\epsilon}(v^\epsilon)(x^\epsilon)=\epsilon^2 e_{i\|j}(\epsilon; v)(x). \label{eq:b10} \end{equation}
We define the spaces \begin{align}
V(\Omega) &= \{v\in (H^{1}(\Omega))^3, v|_{\Gamma_0}=0\},\label{eq:b11}\\[.2pc]
\Psi(\Omega) &= \{\psi\in H^1(\Omega), \psi|_{\Gamma_{eD}}=0\}.\label{eq:b12} \end{align}
We denote $\varphi^m(\epsilon)=T(\epsilon)(u^m(\epsilon))$. Then the variational equations (eqs~(\ref{eq:a31})--(\ref{eq:a34})) become \begin{align}
&\int_\Omega A^{ijkl}(\epsilon)e_{k\|l}(\epsilon, u^m(\epsilon))
e_{i\|j}(\epsilon, v)\sqrt{g(\epsilon)}{\rm d}x\nonumber\\[.2pc] &\quad\ +\int_\Omega P^{3kl}\partial_3{\varphi}^m
(\epsilon)e_{k\|l}(\epsilon, v)\sqrt{g(\epsilon)}{\rm d}x\nonumber\\[.2pc] &\quad\ + \epsilon\int_\Omega P^{\alpha kl}(\epsilon) \partial_\alpha{\varphi}^m(\epsilon)
e_{k\|l}(\epsilon, v)\sqrt{g(\epsilon)}{\rm d}x\nonumber\\[.2pc] &= \xi^m(\epsilon)\int_\Omega [\epsilon^2 u^m_\alpha(\epsilon)v_\alpha +u^m_3(\epsilon)v_3]\sqrt{g(\epsilon)}{\rm d}x \quad\mbox{for all }v\in V(\Omega). \label{eq:b13}\\[.2pc] &\int_\Omega {\cal E}^{33}(\epsilon)\partial_3{\varphi}^m(\epsilon) \partial_3\psi \sqrt{g(\epsilon)}{\rm d}x\nonumber\\[.2pc] &\quad\ +\epsilon\int_\Omega[{\cal E}^{3\alpha}(\epsilon) (\partial_\alpha{\varphi}^m(\epsilon) \partial_3\psi+\partial_3{\varphi}^m(\epsilon)\partial_\alpha\psi)] \sqrt{g(\epsilon)}{\rm d}x\nonumber\\[.2pc] &\quad\ +\epsilon^2\int_\Omega {\cal E}^{\alpha\beta}(\epsilon) \partial_\alpha{\varphi}^m(\epsilon) \partial_\beta\psi\sqrt{g(\epsilon)} {\rm d}x\nonumber\\[.2pc]
&=\int_\Omega P^{3kl}(\epsilon)\partial_3\psi e_{k\|l}(\epsilon, u^m(\epsilon))\sqrt{g(\epsilon)}{\rm d}x \nonumber\\[.2pc] &\quad\ +\epsilon\int_\Omega [P^{\alpha kl}(\epsilon)
\partial_\alpha\psi e_{k\|l}(\epsilon, u^m(\epsilon))] \sqrt{g(\epsilon)}{\rm d}x \quad\mbox{for all } \psi\in \Psi(\Omega),\label{eq:b14}\\[.2pc] &\int_\Omega [\epsilon^2u^m_\alpha(\epsilon) u^n_\alpha(\epsilon) +u^m_3(\epsilon)u^n_3(\epsilon)]\sqrt{g(\epsilon)}{\rm d}x = \delta_{mn}. \label{eq:b15} \end{align} \end{assump}
\section{Technical preliminaries}
\setcounter{equation}{0} The following two lemmas are crucial; they play an important role in the proof of the convergence of the scaled unknowns as $\epsilon\rightarrow 0$. In the sequel, we denote by $C_1, C_2, ..., C_n$ various constants whose values do not depend on $\epsilon$ but may depend on $\theta$.
\setcounter{theo}{0} \begin{lem}
The functions $e_{i\|j}(\epsilon, v) $ defined in {\rm (\ref{eq:b10})} are of the form \begin{align}
e_{\alpha\|\beta}(\epsilon; v) &= {\tilde{e}}_{\alpha\beta}(v)+\epsilon^2 e^{\sharp}_{\alpha\|\beta}(\epsilon; v),\label{eq:c1}\\[.2pc]
e_{\alpha\|3}(\epsilon; v) &= \frac{1}{\epsilon}\{{\tilde{e}}_{\alpha 3}(v) +\epsilon^2 e^{\sharp}_{\alpha\|3}(\epsilon; v)\},\label{eq:c2}\\[.2pc]
e_{3\|3}(\epsilon; v) &= \frac{1}{\epsilon^2}{\tilde{e}}_{33}(v),\label{eq:c3} \end{align} where \begin{align} {\tilde{e}}_{\alpha\beta}(v) &= \frac{1}{2}(\partial_\alpha v_\beta+\partial_\beta v_\alpha)-v_3\partial_{\alpha\beta}\theta,\label{eq:c4}\\[.2pc] {\tilde{e}}_{\alpha 3}(v) &= {\frac{1}{2}}(\partial_\alpha v_3+\partial_3 v_\alpha),\label{eq:c5}\\[.2pc] {\tilde{e}}_{33}(v) &= \partial_3 v_3\label{eq:c6} \end{align} and there exists constant $C_1$ such that \begin{equation} \sup_{0<\epsilon\leq\epsilon_0}\max_{\alpha,j}
\|e^{\sharp}_{\alpha,j}(\epsilon; v)
\|_{0,\Omega}\leq C_1\|v\|_{1,\Omega} \quad {\rm for\ all}\ \ v\in V.\label{eq:c7} \end{equation} Also there exist constants $C_2, C_3$ and $C_4$ such that \begin{align}
&\sup_{0<\epsilon\leq \epsilon_0}\max_{x\in\Omega}|g(x)-1|\leq C_2\epsilon^2,\label{eq:c8}\\[.2pc]
&\sup_{0<\epsilon\leq \epsilon_0}\max_{x\in\Omega}|A^{ijkl}(\epsilon)-A^{ijkl}| \leq C_3\epsilon^2,\label{eq:c9} \end{align} where \begin{equation} A^{ijkl}=\lambda\delta^{ij}\delta^{kl}+\mu(\delta^{ik}\delta^{jl}+ \delta^{il}\delta^{jk})\label{eq:c10} \end{equation} and \begin{equation} A^{ijkl}M_{kl}M_{ij}\geq C_4M_{ij}M_{ij}\label{eq:c11} \end{equation} for $0<\epsilon\leq\epsilon_0$ and for all symmetric tensors $(M_{ij}).$ \end{lem}
\begin{proof} The proof is based on Lemma~4.1 of \cite{BCM}.
\mbox{\xxx{\char'245}} \end{proof}
From relation (\ref{eq:a5}) and definition (\ref{eq:b2}), it follows that there exists a constant $C_5$ such that for any vector $(t_i)\in \mathbb{R}^3$, \begin{equation} {\cal E}^{ij}(\epsilon)t_it_j\geq C_5\sum_{j=1}^3t_j^2.\label{eq:b3} \end{equation}
We assume that there exists functions $P^{kij}$ and ${\cal E}^{ij}$ such that \begin{align} &\sup_{0<\epsilon\leq
\epsilon_0}\max_{x\in\Omega}|P^{kij}(\epsilon)-P^{kij}| \leq C_6\epsilon,\label{eq:b4}\\[.6pc]
&\sup_{0<\epsilon\leq \epsilon_0}\max_{x\in\Omega}|{\cal E}^{ij}
(\epsilon)- {\cal E}^{ij}| \leq C_7\epsilon.\label{eq:b5} \end{align}
\begin{lem} Let $\theta\in C^3(\omega)$ be a given function and let the functions $\tilde{e}_{ij}$ be defined as in {\rm (\ref{eq:c4})--(\ref{eq:c6})}. Then there exists a constant $C_8$ such that the following generalised Korn's inequality holds{\rm :} \begin{equation}
\|v\|_{1,\Omega}\leq C_8\left\lbrace \sum_{i,j}\|\tilde{e}_{ij}(v)\|^2_{0, \Omega}\right\rbrace^{1/2}\label{eq:c12} \end{equation} for all $v\in V(\Omega)$ where $V(\Omega)$ is the space defined in {\rm (\ref{eq:b11})}. \end{lem}
\begin{proof} The proof is based on Lemma~4.2 of \cite{BCM}.
\mbox{\xxx{\char'245}} \end{proof}
\section{{\zzzz A priori} estimates}
\setcounter{equation}{0}
In this section, we show that for each positive integer $m$, the scaled eigenvalues $\{\xi^m(\epsilon)\}$ are bounded uniformly with respect to $\epsilon$.
Let $\varphi\in H^2_0(\omega)$. Then \begin{equation} v_\varphi:=(-x_3\partial_1\varphi, -x_3\partial_2\varphi, \varphi)\in V(\Omega)\label{eq:d1} \end{equation}
and \begin{equation} \tilde{e}_{\alpha\beta}(v_\varphi)=-x_3\partial_{\alpha\beta}\varphi-\varphi \partial_{\alpha\beta}\theta, ~~~~ \tilde{e}_{i3}(v_\varphi)=0.\label{eq:d2} \end{equation}
Hence \begin{align}
e_{\alpha\|\beta}(\epsilon, v_\varphi) &= -x_3\partial_{\alpha\beta}\varphi -\varphi\partial_{\alpha\beta}\theta+O(\epsilon^2),\label{eq:d3}\\[.4pc]
e_{\alpha\|3}(\epsilon, v_\varphi) &= O(\epsilon),\label{eq:d4}\\[.4pc]
e_{3\|3}(\epsilon, v_\varphi) &= 0.\label{eq:d5} \end{align}
We need the following lemma to prove the boundedness of the scaled eigenvalues.
\setcounter{theo}{0} \begin{lem} There exists a constant $C_9>0$ such that \begin{align}
&|\partial_3(T(\epsilon)(v_\varphi))|_{0,\Omega}\leq C_9
|\varphi|_{2,\omega},\label{eq:d6}\\[.4pc]
&|\epsilon\partial_{\alpha}(T(\epsilon)(v_\varphi))|_{0,\Omega}
\leq C_9 |\varphi|_{2,\omega}.\label{eq:d7} \end{align} \end{lem}
\begin{proof} With the scalings (\ref{eq:b6})--(\ref{eq:b9}), the variational equation (eq.~(\ref{eq:a16})) posed on the domain $\Omega$ reads as follows:
For each $h\in (H^1(\Omega))^3$, there exists a unique solution $T(\epsilon)(h)\in(H^1(\Omega))^3$ such that \begin{align} &\int_\Omega {\cal E}^{33}(\epsilon)\partial_3T(\epsilon)(h)\partial_3\psi \sqrt{g(\epsilon)}{\rm d}x\nonumber\\ &\quad\ +{\epsilon}\int_\Omega [{\cal E}^{\alpha 3}(\epsilon) (\partial_\alpha T(\epsilon)(h) \partial_3\psi+\partial_3T(\epsilon)(h)\partial_\alpha \psi)] \sqrt{g(\epsilon)}{\rm d}x\nonumber\\ &\quad\ +\epsilon^2\int_\Omega {\cal E}^{\alpha\beta}(\epsilon)\partial_\alpha T(\epsilon)(h) \partial_\beta\psi \sqrt{g(\epsilon)}{\rm d}x\nonumber\\ &=\int_\Omega P^{3kl}(\epsilon)
\partial_3\psi e_{k\|l}(\epsilon, h)\sqrt{g(\epsilon)}{\rm d}x\nonumber\\ &\quad\ +\epsilon\int_\Omega P^{\alpha kl}(\epsilon)
\partial_\alpha\psi e_{k\|l}(\epsilon, h)\sqrt{g(\epsilon)}{\rm d}x\quad \forall \psi\in\Psi. \end{align}
Taking $h=v_\varphi$ and $\psi=T(\epsilon)(v_\varphi)$ in the above equation, we have \begin{align} &\int_\Omega {\cal E}^{33}(\epsilon)\partial_3T(\epsilon)(v_\varphi) \partial_3 T(\epsilon)(v_\varphi) \sqrt{g(\epsilon)}{\rm d}x\nonumber\\ &\quad\ +{\epsilon}\int_\Omega [{\cal E}^{\alpha 3}(\epsilon) (\partial_\alpha T(\epsilon)(v_\varphi) \partial_3 T(\epsilon)(v_\varphi)\nonumber\\ &\quad\ +\partial_3T(\epsilon)(v_\varphi)\partial_\alpha T(\epsilon)(v_\varphi))] \sqrt{g(\epsilon)}{\rm d}x\nonumber\\ &\quad\ +\epsilon^2\int_\Omega {\cal E}^{\alpha\beta}(\epsilon)\partial_\alpha T(\epsilon)(v_\varphi) \partial_\beta T(\epsilon)(v_\varphi) \sqrt{g(\epsilon)}{\rm d}x\nonumber\\ &=\int_\Omega P^{3kl}(\epsilon) \partial_3 T(\epsilon)(v_\varphi)
e_{k\|l}(\epsilon, v_\varphi)\sqrt{g(\epsilon)}{\rm d}x\nonumber\\ &\quad\ +\epsilon\int_\Omega P^{\alpha kl}(\epsilon)
\partial_\alpha T(\epsilon)(v_\varphi)e_{k\|l}(\epsilon, v_\varphi) \sqrt{g(\epsilon)}{\rm d}x. \end{align}
Using the relations (\ref{eq:b3}) and (\ref{eq:d2})--(\ref{eq:d5}), it follows that there exists a constant $C_9>0$ such that \begin{align}
&|\partial_{3}(T(\epsilon)(v_\varphi))|^2_{0,\Omega}+
|\epsilon\partial_{\alpha}(T(\epsilon)(v_\varphi))|^2_{0,\Omega}\nonumber\\
&\hskip 1cm \leq C_9\{|\partial_3T(\epsilon)(v_\varphi)|_{0,\Omega}
|\varphi|_{2,\omega}+
|\epsilon\partial_\alpha T(\epsilon)(v_\varphi)|_{0,\Omega}
|\varphi|_{2,\omega}\}\label{eq:d10} \end{align} and hence the result follows.
\mbox{\xxx{\char'245}} \end{proof}
\begin{theor}[\!] For each positive integer $m${\rm ,} there exists a constant $C(m)>0$ such that \begin{equation} \xi^m(\epsilon)\leq C(m).\label{eq:d11} \end{equation} \end{theor}
\begin{proof} Since problem (\ref{eq:b13}) was derived from (\ref{eq:a31}) after a change of scale, we still have the variational characterization of the scaled eigenvalues $\xi^m(\epsilon)$. Let $V_m$ denote the collection of all $m$-dimensional subspaces of $V(\Omega)$. Then \begin{equation} \xi^m(\epsilon)=\min_{W\in V_m}\max_{v\in W}\frac{N(\epsilon)(v, v)} {D(\epsilon)(v, v)},\label{eq:d12} \end{equation} where \begin{align}
N(\epsilon)(v, v) &= \int_\Omega A^{ijkl}e_{k\|l}(\epsilon, v)
e_{i\|j}(\epsilon, v)\sqrt{g(\epsilon)}{\rm d}x\nonumber\\[.2pc]
&\quad\ +\int_\Omega P^{3kl}\partial_3 T(\epsilon)(v) e_{k\|l}(\epsilon, v)\sqrt{g(\epsilon)}{\rm d}x\nonumber\\[.2pc] &\quad + \epsilon\int_\Omega P^{\alpha kl}
\partial_\alpha T(\epsilon)(v)e_{k\|l}(\epsilon, v)\sqrt{g(\epsilon)}{\rm d}x,\label{eq:d13}\\[.2pc] D(\epsilon)(v, v) &= \int_\Omega[\epsilon^2 v_\alpha v_\alpha+ v_3 v_3]\sqrt{g(\epsilon)}{\rm d}x.\label{eq:d14} \end{align}
Let $W_m$ be the collection of all $m$-dimensional subspaces of $H^2_0(\omega)$. Let $W\in W_m$. Define \begin{equation}
{\bf W} = \{v_\varphi|\varphi\in W\}.\label{eq:d15} \end{equation} It follows that ${\bf{W}}\in V_m$. Hence, it follows from (\ref{eq:d12}) that \begin{equation} \xi^m(\epsilon)\leq\min_{W\in W_m}\max_{\varphi\in W} \frac{N(\epsilon)(v_\varphi, v_\varphi)} {D(\epsilon)(v_\varphi, v_\varphi)}.\label{eq:d16} \end{equation} Now, \begin{align} D(\epsilon)(v_\varphi, v_\varphi) &= \int_\Omega [\epsilon^2 x_3^2
|\partial_\alpha\varphi|^2+|\varphi|^2]\sqrt{g(\epsilon)}{\rm d}x.\nonumber\\[.2pc] &\geq \int_\omega \varphi^2{\rm d}\omega.\label{eq:d17} \end{align} Using the relations (\ref{eq:d3})--(\ref{eq:d5}) and Lemma~5.1, it follows that \begin{align}
&\int_\Omega A^{ijkl}e_{k\|l}(\epsilon, v_\varphi)
e_{i\|j}(\epsilon, v_\varphi)\sqrt{g(\epsilon)}{\rm d}x\leq C\int_\omega |\triangle \varphi|^2 {\rm d}\omega,\label{eq:d18}\\[.2pc]
&\int_\Omega P^{3kl}\partial_3 T(\epsilon)(v_\varphi)e_{k\|l}(\epsilon, v_\varphi)\sqrt{g(\epsilon)}{\rm d}x\leq C\int_\omega |\triangle \varphi|^2 {\rm d}\omega,\label{eq:dd1}\\[.2pc] &\epsilon\int_\Omega P^{\alpha kl}
\partial_\alpha T(\epsilon)(v_\varphi)e_{k\|l}(\epsilon, v_\varphi)
\sqrt{g(\epsilon)}{\rm d}x\leq C\int_\omega |\triangle \varphi|^2 {\rm d}\omega.\label{eq:dd2} \end{align}
Hence \begin{align} \xi^{m}(\epsilon) &\leq C\min_{W\in W_m}\max_{\varphi\in W}
\frac{\int_\omega |\triangle \varphi|^2 {\rm d}\omega} {\int_\omega \varphi^2 {\rm d}\omega}\nonumber\\[.2pc] &\leq C \lambda^m,\label{eq:d19} \end{align} where $\lambda^m$ is the $m$th eigenvalue of the two-dimensional elliptic eigenvalue problem \begin{align} &\triangle^2 u = \lambda u\quad\mbox{ in } \omega \nonumber\\ &u=\partial_\nu u = 0 \quad\mbox{ on }\partial\omega .\label{eq:d20} \end{align} This completes the proof of the theorem on setting $C(m)=C\lambda^m$.
\mbox{\xxx{\char'245}} \end{proof}
\section{The limit problem}
\setcounter{equation}{0} \setcounter{theo}{0}
\begin{theor}[\!] {\rm (a)} For each positive integer $m${\rm ,} there exists $u^m\in H^1(\Omega), \varphi^m\in L^2(\Omega)$ and $\xi^m\in\mathbb{R}$ such that \begin{align} &u^m(\epsilon)\rightarrow u^m \mbox{ in } H^1(\Omega),~~~~ \varphi^m(\epsilon) \rightarrow \varphi^m \ {\rm in}\ L^2(\Omega),\label{eq:e1}\\[.2pc] &(\epsilon\partial_1\varphi^m(\epsilon), \epsilon\partial_2\varphi^m(\epsilon), \partial_3\varphi^m(\epsilon)) \rightarrow (0, 0, \partial_3\varphi^m)\ {\rm in}\ L^2(\Omega),\label{eq:e2}\\[.2pc] &\xi^m(\epsilon)\rightarrow \xi^m.\label{eq:e3} \end{align}
{\rm (b)} Define the spaces \begin{align} &V_H(\omega) = \{(\eta_\alpha)\in (H^1(\omega))^2; \eta_\alpha=0 \ {\rm on}\ \gamma_0\},\label{eq:e4}\\[.4pc] &V_3(\omega) = \{\eta_3\in H^2(\omega); \eta_3=\partial_\nu\eta_3=0 \ {\rm on}\ \gamma_0 \},\label{eq:e5}\\[.4pc]
&V_{KL} = \{v\in H^1(\Omega)| v=\eta_\alpha-x_3\partial_\alpha\eta_3, (\eta_i)\in V_H(\omega)\times V_3(\omega)\},\label{eq:e6}\\[.4pc] &\Psi_l = \{\psi\in L^2(\Omega), \partial_3\psi\in L^2(\Omega)\},\label{eq:e7}\\[.4pc] &\Psi_{l0} = \{\psi\in L^2(\Omega), \partial_3\psi\in L^2(\Omega),
\psi|\Gamma^{\pm}=0\}.\label{eq:e8} \end{align}
Then there exists $(\zeta^m_\alpha, \zeta^m_3)\in V_H\times V_3(\omega)$ such that \begin{align} u^m_\alpha &= \zeta^m_\alpha-x_3\partial_\alpha\zeta^m_3 \quad {\rm and}\quad u^m_3=\zeta^m_3,\label{eq:e9}\\[.4pc] \varphi^m &= (1-x_3^2)\frac{p^{3\alpha\beta}}{p^{33}} \partial_{\alpha\beta}\xi^m_3\label{eq:e10} \end{align} and $(\zeta^m, \xi^m)\in V_H\times V_3\times\mathbb{R}$ satisfies \begin{align}
&\hskip -4pc -\int_\omega m_{\alpha\beta}(\zeta^m)\partial_{\alpha\beta}\eta_3 {\rm d}\omega + \int_\omega n_{\alpha\beta}^\theta(\zeta^m)\partial_{\alpha\beta}\theta\eta_3 {\rm d}\omega
+ \frac{2}{3}\int_\omega \frac{p^{3\alpha\beta}p^{3\rho\tau}}{p^{33}} \partial_{\rho\tau}\zeta^m_3\partial_{\alpha\beta}\eta_3 {\rm d}\omega \nonumber\\[.4pc]
& = \xi^m\int_\omega \zeta^m_3\eta_3 {\rm d}\omega\quad \forall\eta_3\in V_3(\omega),\hskip 4pc \label{eq:e11}\\[.4pc]
& \int_\omega n_{\alpha\beta}^\theta\partial_\beta\eta_\alpha {\rm d}\omega = 0\quad \forall\eta_\alpha\in V_H(\omega),\label{eq:e12}\hskip 4pc \end{align} where \begin{align} &m_{\alpha\beta}(\zeta) = -\left\lbrace \frac{4\lambda\mu}{3(\lambda+4\mu)}\triangle\zeta_3\delta_{\alpha\beta} +\frac{4\mu}{3}\partial_{\alpha\beta}\zeta_3\right\rbrace\label{eq:e13}\\[.2pc] &n_{\alpha\beta}^\theta(\zeta) = \frac{4\lambda\mu}{\lambda+2\mu}\tilde{e}_{\sigma\sigma}(\zeta) \delta_{\alpha\beta}+4\mu\tilde{e}_{\alpha\beta}(\zeta)\label{eq:e14}\\[.2pc] &p^{33} = \frac{1}{\mu}P^{3\alpha 3}P^{3\alpha 3}+\frac{1}{\lambda+2\mu} P^{333}P^{333}+ {\cal E}^{33}\label{eq:e15}\\[.2pc] &p^{3\alpha\beta} = P^{3\alpha\beta}-\frac{\lambda}{\lambda+2\mu}P^{333} \delta^{\alpha\beta}.\label{eq:e16} \end{align} \end{theor}
\begin{proof} For the sake of clarity, the proof is divided into several steps.
\noindent {\it Step} (i).\ \ Define the vector ${\tilde{\varphi}^m}_{i}(\epsilon)$ and the tensor $\tilde{K}^m(\epsilon)=(\tilde{K}^m_{ij}(\epsilon))$ by \begin{align} &{\tilde{\varphi}^m}_{i}(\epsilon) = (\epsilon\partial_1{\varphi}^m(\epsilon), \epsilon\partial_2{\varphi}^m(\epsilon), \partial_3{\varphi}^m(\epsilon)),\label{eq:e19}\\[.2pc] &\tilde{K}^m_{\alpha\beta}(\epsilon) = \tilde{e}_{\alpha\beta}(u^m(\epsilon)),~~ \tilde{K}^m_{\alpha 3}(\epsilon) =\frac{1}{\epsilon}\tilde{e}_{\alpha 3}(u^m(\epsilon)),~~ \tilde{K}^m_{33}(\epsilon) =\frac{1}{\epsilon^2}\tilde{e}_{33}(u^m(\epsilon)).\label{eq:e25} \end{align}
Then there exists a constant $C_{10}>0$ such that \begin{equation}
\|u^m(\epsilon)\|_{1,\Omega}\leq C_{10},~~~
|\tilde{K}^m_{ij}(\epsilon)|_{0,\Omega}\leq C_{10},~~~
|\tilde{\varphi}^m_i(\epsilon)|_{0,\Omega}\leq C_{10}\label{eq:e17} \end{equation} for all $0<\epsilon\leq\epsilon_0$.
Letting $v = u^m(\epsilon)$ in (\ref{eq:b13}), we have \begin{align} &\int_\Omega A^{ijkl}(\epsilon)
e_{k\|l}(\epsilon)(u^m(\epsilon))
e_{i\|j}(\epsilon)(u^m(\epsilon))\sqrt{g(\epsilon)}{\rm d}x\nonumber\\[.2pc] &\quad\ +\int_\Omega P^{3kl}(\epsilon)\partial_3{\varphi}^m
(\epsilon)e_{k\|l}(\epsilon)(u^m(\epsilon))\sqrt{g(\epsilon)}{\rm d}x\nonumber\\[.2pc] &\quad\ +\epsilon\int_\Omega P^{\alpha kl}(\epsilon) \partial_\alpha{\varphi}^m(\epsilon)
e_{k\|l}(\epsilon)(u^m(\epsilon))\sqrt{g(\epsilon)}{\rm d}x\nonumber\\[.2pc] &=\xi^m(\epsilon)\int_\Omega [\epsilon^2u^m_\alpha(\epsilon) u^m_\alpha(\epsilon)+u^m_3(\epsilon)u^m_3(\epsilon)]\sqrt{g(\epsilon)}{\rm d}x. \end{align} Letting $\psi=\varphi^m(\epsilon)$ in (\ref{eq:b14}) and using it in the above equation, we get \begin{align}
&\int_\Omega A^{ijkl}(\epsilon) e_{k\|l}(\epsilon, u^m(\epsilon))
e_{i\|j}(\epsilon, u^m(\epsilon))\sqrt{g(\epsilon)}{\rm d}x\nonumber\\[.2pc] &\quad\ + \int_\Omega {\cal E}^{ij}(\epsilon){\tilde{\varphi}}^m_{i}(\epsilon) {\tilde{\varphi}}^m_{j}(\epsilon)\sqrt{g(\epsilon)}{\rm d}x\nonumber\\ &=\xi^m(\epsilon)\int_\Omega[\epsilon^2u^m_\alpha(\epsilon) \cdot u^m_\alpha(\epsilon)+u^m_3(\epsilon)u^m_3(\epsilon)]\sqrt{g(\epsilon)}{\rm d}x. \label{eq:e20} \end{align}
Using the coerciveness properties (\ref{eq:c11}) and (\ref{eq:b3}), the inequality $(a-b)^2\geq a^2/2-b^2$ and the generalized Korn's inequality (\ref{eq:c12}), we have for $\epsilon\leq \min\{\epsilon_0, 1\}$, \begin{align}
&\int_\Omega A^{ijkl}(\epsilon) e_{k\|l}(\epsilon, u^m(\epsilon))
e_{i\|j}(\epsilon, u^m(\epsilon))\sqrt{g(\epsilon)}{\rm d}x\nonumber\\[.2pc] &\quad\ + \int_\Omega {\cal E}^{ij}(\epsilon){\tilde{\varphi}}^m_{i}(\epsilon) {\tilde{\varphi}}^m_{j}(\epsilon)\sqrt{g(\epsilon)}{\rm d}x\nonumber\\[.2pc]
&\geq C_{11}\sum_{i,j}\|e_{i\|j}(\epsilon, u^m(\epsilon))\|^2_{0,\Omega}
+C_{11}\sum_i\|\tilde{\varphi}^m_i(\epsilon)\|^2_{0,\Omega}\nonumber \end{align} \begin{align} \hskip -1pc &= C_{11}\sum_{\alpha,\beta}
\|\tilde{e}_{\alpha\beta}(u^m(\epsilon))+\epsilon^2
{e}^\sharp_{\alpha\beta}(\epsilon, u^m(\epsilon))\|^2_{0,\Omega}\nonumber\\[.2pc] \hskip -1pc &\quad\ +2C_{11}\sum_{\alpha}
\left\|\frac{1}{\epsilon}\tilde{e}_{\alpha 3}(u^m(\epsilon))+\epsilon
{e}^\sharp_{\alpha 3}(\epsilon, u^m(\epsilon))\right\|^2_{0,\Omega}\nonumber\\[.2pc]
\hskip -1pc &\quad\ +C_{11} \left\|\frac{1}{\epsilon^2}\tilde{e}_{33}(u^m(\epsilon))\right\|^2_{0,\Omega}
+C_{11}\sum_i \|\tilde{\varphi}^m_i(\epsilon)\|^2_{0,\Omega}\nonumber\\[.2pc]
\hskip -1pc &\geq C_{11}\left\lbrace \frac{1}{2}\sum_{i,j}|\tilde{K}_{ij}^m(\epsilon)|^2_{0,\Omega}
-C_1^2(2\epsilon^2+\epsilon^4)\|u^m(\epsilon)\|^2_{1,\Omega}\right\rbrace\nonumber\\[.2pc]
\hskip -1pc &\quad\ +C_{11}\sum_i\|\tilde{\varphi}^m_i(\epsilon)\|^2_{0,\Omega}\hskip 1pc \nonumber\\[.2pc] \hskip -1pc &\geq C_{11}\left\lbrace \frac{1}{2}\sum_{i,j}
\|\tilde{e}_{ij}(u^m(\epsilon))\|^2_{0,\Omega} -
3\epsilon^2C_1^2\|u^m(\epsilon)\|^2_{1,\Omega}\right\rbrace\nonumber\\[.2pc]
\hskip -1pc &\quad\ +C_{11}\sum_i\|\tilde{\varphi}^m_i(\epsilon)\|^2_{0,\Omega}\nonumber\\[.2pc] \hskip -1pc &\geq C_{11}\left\lbrace \frac{1}{2}(C_8)^{-2} -
3\epsilon^2C_1^2\right\rbrace \|u^m(\epsilon)\|^2_{1,\Omega}
+ C_{11}\sum_i\|\tilde{\varphi}^m_i(\epsilon)\|^2_{0,\Omega}.\label{eq:e21} \end{align}
Combining eqs~(\ref{eq:e20}) and (\ref{eq:e21}) with relations (\ref{eq:b15}) and (\ref{eq:d11}), we get the relation (\ref{eq:e17}).
\noindent {\it Step} (ii).\ \ From Step (i) it follows that there exists a subsequence $(\tilde{\varphi}^m_i(\epsilon))$ and $(\tilde{\varphi}^m_i)\in L^2(\Omega)$ such that \begin{equation} (\epsilon\partial_1\varphi^m(\epsilon), \epsilon\partial_2\varphi^m(\epsilon), \partial_3\varphi^m(\epsilon))\rightharpoonup (\tilde{\varphi}^m_1, \tilde{\varphi}^m_2, \tilde{\varphi}^m_3)\quad \mbox{ in } (L^2(\Omega))^3. \label{eq:e22} \end{equation} Since $\Gamma_{eD}$ contains $\Gamma^-$, we have \begin{equation} {\varphi}^m(\epsilon)(x_1, x_2, x_3)= \int_{-1}^{x_3} \partial_3{\varphi}^m(\epsilon)(x_1, x_2, s){\rm d}s \label{eq:e23} \end{equation}
and it follows that $\|{\varphi}^m(\epsilon)\|_{0,\Omega} \leq
\sqrt{2}\|\partial_3{\varphi}^{m} (\epsilon)\|_{0,\Omega}$. This implies that ${\varphi}^m(\epsilon)$ is bounded in $L^2(\Omega)$. Therefore there exists a $\varphi^m$ in $L^2(\Omega)$ and a subsequence, still indexed by $\epsilon,$ such that $\varphi^m(\epsilon)$ converges weakly to $\varphi^m$. Hence it follows from (\ref{eq:e22}) that \begin{equation} (\epsilon\partial_1\varphi^m(\epsilon), \epsilon\partial_2\varphi^m(\epsilon), \partial_3\varphi^m(\epsilon))\rightharpoonup (0, 0, \partial_3\varphi^m).\label{eq:e24} \end{equation}
\noindent {\it Step} (iii).\ \ From Step (i) it follows that there exists a subsequence, indexed by $\epsilon$ for notational convenience, and functions $u^m\in V(\Omega)$ and $\tilde{K}^m_{ij}\in (L^2(\Omega))^9$ such that \begin{equation} u^m(\epsilon)\rightharpoonup u^m \quad\mbox{ in } H^1(\Omega), ~~~ \tilde{K}^m(\epsilon)\rightharpoonup \tilde{K}^m\quad \mbox{ in } L^2(\Omega)\mbox{ as }\epsilon\rightarrow 0.\label{eq:e28} \end{equation}
Then there exist functions $(\zeta^m_\alpha)\in H^1(\omega)$ and $\zeta^m_3\in H^2(\omega)$ satisfying $\zeta^m_i=\partial_\nu \zeta^m_3=0$ on $\gamma_0$ such that \begin{equation} u^m_\alpha=\zeta^m_\alpha-x_3\partial_\alpha\zeta^m_3 \quad\mbox{ and }\quad u^m_3=\zeta^m_3\label{eq:e29} \end{equation} and \begin{align} \tilde{K}^m_{\alpha\beta} &= \tilde{e}_{\alpha\beta}(u^m), ~~~ \tilde{K}^m_{\alpha 3}=-\frac{1}{\mu}P^{3\alpha 3}\partial_3\varphi^m,\nonumber\\[.2pc] \tilde{K}^m_{33} &= -\frac{1}{\lambda+2\mu}(P^{333}\partial_3\varphi^m+\lambda \tilde{K}^m_{\beta\beta}).\label{eq:e32} \end{align} From definition (\ref{eq:e25}) and the boundedness of $(\tilde{K}^m_{ij}(\epsilon))$, we deduce that \begin{equation*}
\|e_{\alpha 3}(u^m(\epsilon))\|_{0,\Omega}\leq \epsilon C_{13} \quad \hbox{and}\quad
\|e_{33}(u^m(\epsilon))\|_{0,\Omega}\leq\epsilon^2C_{13}, \end{equation*} where $e_{ij}(v)=\frac{1}{2}(\partial_iv_j+\partial_jv_i)$. Since norm is a weakly lower semicontinuous function \begin{equation}
\|e_{i3}(u^m)\|_{0,\Omega}\leq \liminf_{\epsilon\rightarrow 0
}\|e_{i3} (u^m(\epsilon)\|_{0,\Omega}=0,\label{eq:e30} \end{equation} we obtain $e_{i3}(u^m)=0$. Then it is a standard argument that the components $u^m_i$ of the limit $u^m$ are of the form (\ref{eq:e29}).
Since $u^m(\epsilon)\rightharpoonup u^m$ in $H^1(\Omega)$, definition (\ref{eq:c4}) of the functions $\tilde{e}_{\alpha\beta}(v)$ shows that the function $\tilde{K}^m_{\alpha\beta}(\epsilon)=\tilde{e}_{\alpha\beta}(u^m(\epsilon))$ converges weakly in $L^2(\Omega)$ to the function $\tilde{e}_{\alpha\beta}(u^m)$.
We next note the following result. Let $w\in L^2(\Omega)$ be given; then \begin{equation} \int_\Omega w\partial_3v{\rm d}x=0 \quad\mbox{ for all }v\in H^1(\Omega) \mbox{ with } v=0\mbox{ on } \Gamma_0, \mbox{ then } w=0. \end{equation}
Multiplying (\ref{eq:b13}) by $\epsilon^2$, taking $(v_\alpha)=0$ and letting $\epsilon\rightarrow 0$, we get \begin{equation} \int_\Omega (\lambda{\tilde{K}^m}_{\sigma\sigma} +(\lambda+2\mu)\tilde{K}_{33} + P^{333}\partial_3\varphi^m) \partial_3v_3{\rm d}x =0 \label{eq:e33} \end{equation} which implies $(\lambda{\tilde{K}^m}_{\sigma\sigma} +(\lambda+2\mu)\tilde{K}_{33}+P^{333}\partial_3\varphi^m)=0$ and hence the third relation in (\ref{eq:e32}) follows.
Again, multiplying (\ref{eq:b13}) by $\epsilon$, taking $v_3=0$ and letting $\epsilon\rightarrow 0 $, we get \begin{equation} \int_\Omega (\mu{\tilde{K}^m}_{\alpha 3}+P^{3\alpha3} \partial_3\varphi^m)\partial_3v_\alpha {\rm d}x = 0\label{eq:e34} \end{equation} which implies $(\mu{\tilde{K}^m}_{\alpha 3}+P^{3\alpha3} \partial_3\varphi^m)=0$ and hence the second relation in (\ref{eq:e32}) follows.
\noindent{\it Step} (iv).\ \ The function $\varphi^m$ is of the form (\ref{eq:e10}).
Letting $\epsilon\rightarrow 0$ in eq.~(\ref{eq:b14}), we get \begin{equation} \int_\Omega (P^{3\alpha\beta}\tilde{K}^m_{\alpha\beta} - {\cal E}^{33}\partial_3\varphi^m)\partial_3\psi {\rm d} x = 0\quad \forall \psi\in \Psi(\Omega).\label{eq:e38} \end{equation}
Since $D(\Omega)$ is dense in $\Psi_{l0}$ (and hence in $\Psi(\Omega)$)
for the norm $\|.\|_{\Psi_l}$, eq.~(\ref{eq:e38}) is equivalent to \begin{equation} \partial_3(P^{3\alpha\beta}\tilde{K}^m_{\alpha\beta} - {\cal E}^{33}\partial_3\varphi^m) = 0\quad \mbox{ in } D'(\Omega)\label{eq:e39} \end{equation} which implies that $(P^{3\alpha\beta}\tilde{K}^m_{\alpha\beta}- {\cal E}^{33}\partial_3\varphi^m)=d^1$, with $d^1\in D(\omega)$. Then \begin{equation} \partial_3\varphi^m=\frac{p^{3\alpha\beta}}{p^{33}} [\tilde{e}_{\alpha\beta}(\zeta^m) -x_3\partial_{\alpha\beta}\zeta^m_3]-\frac{1}{p^{33}}d^1\label{eq:e40} \end{equation} which gives \begin{equation} \varphi^m=\frac{p^{3\alpha\beta}}{p^{33}}[x_3\tilde{e}_{\alpha\beta}(\zeta^m) -x_3^2\partial_{\alpha\beta}\zeta^m_3]-\frac{x_3}{p^{33}}d^1+d^0. \label{eq:e41} \end{equation}
Since $\varphi^m$ satisfies the boundary conditions
$\varphi^m_{|\Gamma^+} =\varphi^m_{|\Gamma^-}=0$, we have \begin{equation} d^0 = \frac{p^{3\alpha\beta}}{2p^{33}} \partial_{\alpha\beta}\zeta^m_3, ~~~~ d^1=p^{3\alpha\beta}\tilde{e}_{\alpha\beta}(\zeta^m).\label{eq:e43} \end{equation} Thus the conclusion follows.
\noindent {\it Step} (v).\ \ The function $(\zeta^m_i)$ satisfies (\ref{eq:e11}) and (\ref{eq:e12}).
Taking $v\in V_{KL}$ and letting $\epsilon\rightarrow 0$ in (\ref{eq:b13}) we get \begin{equation} \int_\Omega A^{\alpha\beta kl}\tilde{K}^m_{kl}\tilde{K}_{\alpha\beta}(v){\rm d}x +\int_\Omega P^{3\alpha\beta}\partial_3\varphi^m \tilde{K}_{\alpha\beta}(v){\rm d}x =\xi^m\int_\Omega u^m_3 \cdot v_3 {\rm d}x.\label{eq:e44} \end{equation} Replacing $u^m$ and $\tilde{K}^m_{ij}$ by the expressions obtained in (\ref{eq:e29}) and (\ref{eq:e32}), and taking $v$ of the form \begin{equation*} v_\alpha=\eta_\alpha-x_3\partial_\alpha\eta_3\quad \mbox{ and }\quad v_3=\eta_3 \end{equation*} with $(\eta_i)\in V_H(\omega)\times V_3(\omega)$, it is verified that (\ref{eq:e44}) coincides with eqs~(\ref{eq:e11}) and (\ref{eq:e12}).
\noindent {\it Step} (vi).\ \ The convergences $u^m(\epsilon)\rightharpoonup u^m$ in $H^1(\Omega)$ and $\varphi^m(\epsilon)\rightharpoonup \varphi^m$ in $L^2(\Omega)$ are strong.
To show that the family $(u^m(\epsilon))$ converges strongly to $u^m$ in $H^1(\Omega),$ by Lemma~4.2, it is enough to show that \begin{equation} \tilde{e}_{ij}(u^m(\epsilon))\rightarrow \tilde{e}_{ij}(u^m)\quad\mbox{ in }L^2(\Omega). \label{eq:e46} \end{equation} Since $\tilde{e}_{i3}(u^m)=0$ and \begin{align}
&\sum_{i,j}\|\tilde{e}_{ij}(u^m(\epsilon))-\tilde{e}_{ij}(u^m)\|^2_{0,\Omega}\nonumber\\[.2pc]
&=\sum_{\alpha,\beta}\|\tilde{K}^m_{\alpha\beta}(\epsilon)-
\tilde{K}^m_{\alpha\beta}\| _{0,\Omega}^2 +2\epsilon^2\sum_{\alpha}\|\tilde{K}^m_{\alpha 3}(\epsilon)\|^2_{0,\Omega}+\epsilon^4\|\tilde{K}^m_{33}(\epsilon)\|^2_{0,\Omega},\label{eq:e47} \end{align} convergence (\ref{eq:e46}) is equivalent to showing that \begin{equation} \tilde{K}^m(\epsilon)\rightarrow \tilde{K}^m \quad\mbox{ in } L^2(\Omega). \label{eq:e48} \end{equation}
We define a norm on $(L^2(\Omega))^9\times (L^2(\Omega))^3$ by letting for any matrix $M\in (L^2(\Omega))^9$ and any vector $\chi\in (L^2(\Omega))^3$, \begin{equation}
\|(M, \chi)\| = \left\lbrace \int_\Omega A^{ijkl}M:M\sqrt{g(\epsilon)}{\rm d}x +\int_\Omega{\cal E}^{ij}\chi_i\chi_j \sqrt{g(\epsilon)}{\rm d}x\right\rbrace^{1/2}.\label{eq:e49} \end{equation}
Let $X^m(\epsilon)$ be the norm of $(\tilde{K}^m(\epsilon), \epsilon\partial_1\varphi^m(\epsilon), \epsilon\partial_2\varphi^m(\epsilon), \partial_3\varphi^m(\epsilon)) $ in $(L^2(\Omega))^{12}$. Using the weak convergence equation (eqs~(\ref{eq:e24}) and (\ref{eq:e28})) and the relation (\ref{eq:e32}), it can be shown that \begin{equation} \lim_{\epsilon\rightarrow 0}X^m(\epsilon)=X^m= \left( \int_\Omega A^{ijkl}\tilde{K}^m :\tilde{K}^m{\rm d}x +\int_\Omega {\cal E}^{33}(\partial_3\varphi^m)^2 {\rm d}x \right)^{1/2}\label{eq:e50} \end{equation} which is the norm of $(\tilde{K}^m, 0, 0, \partial_3\varphi^m)$. Since we have already proved that $(\tilde{K}^m(\epsilon)$, $\epsilon\partial_1\varphi^m(\epsilon)$, $\epsilon\partial_2\varphi^m(\epsilon)$, $\partial_3\varphi^m(\epsilon))$ converges weakly to $(\tilde{K}, 0, 0, \partial_3\varphi^m)$ in $(L^2(\Omega))^{12}$, we have the following strong convergences: \begin{align} &\tilde{K}^m(\epsilon)\rightarrow \tilde{K}^m \mbox{ strongly in } (L^2(\Omega))^9,\label{eq:e51}\\[.2pc] &(\epsilon\partial_1\varphi^m(\epsilon), \epsilon\partial_2\varphi^m(\epsilon), \partial_3\varphi^m(\epsilon)) \rightarrow (0,0,\partial_3\varphi^m)\mbox{ strongly in } (L^2(\Omega))^3.\label{eq:e52} \end{align}
Hence $u^m(\epsilon)$ converges strongly to $u^m$ in $H^1(\Omega)$ and since $\varphi^m(\epsilon)-\varphi^m$ is in $\Psi_{l0}$, the equivalence of norms $\|\psi\|_{\Psi_l}$ and $\psi\rightarrow
|\partial_3\psi|_\Omega$ in $\Psi_{l0}$ proves that $\varphi^m(\epsilon)$ converges strongly to $\varphi^m$ in $L^2(\Omega)$.
\mbox{\xxx{\char'245}} \end{proof}
Equation (\ref{eq:e12}) can be written as \begin{align} &\int_\omega \left[\frac{2\lambda\mu}{\lambda+2\mu}e_{\rho\rho}(\zeta) \delta_{\alpha\beta}+2\mu e_{\alpha\beta}(\zeta) \right] \partial_\beta\eta_\alpha {\rm d}\omega\nonumber\\[.2pc] &= \int_\omega \left[ \frac{2\lambda\mu}{\lambda+2\mu}(\partial_\sigma\theta \partial_\sigma\zeta_3)\delta_{\alpha\beta} +\mu(\partial_\alpha\theta\partial_\beta\zeta_3+\partial_\beta\theta \partial_\alpha\zeta_3) \right]\partial_\beta\eta_\alpha {\rm d}\omega.\label{eq:e53} \end{align} Clearly, the bilinear form \begin{align} \tilde{b}(\zeta_\alpha, \eta_\alpha) &= \int_\omega \left[\frac{2\lambda\mu}{\lambda+2\mu}e_{\rho\rho}(\zeta) \delta_{\alpha\beta}+2\mu e_{\alpha\beta}(\zeta) \right] \partial_\beta\eta_\alpha {\rm d}\omega\nonumber\\[.2pc] &= \int_\omega \left[\frac{2\lambda\mu}{\lambda+2\mu}e_\rho\rho(\zeta) e_{\sigma\sigma}(\eta)+2\mu e_{\alpha\beta}(\zeta)e_{\alpha\beta}(\eta)\right]{\rm d}\omega \label{eq:e54} \end{align} is $V_H(\omega)$ elliptic. Also for a given $\zeta_3\in V_3(\omega)$, the functional
\begin{equation} \langle \zeta_3, \eta_\alpha \rangle = \int_\omega \left[ \frac{2\lambda\mu}{\lambda+2\mu}(\partial_\sigma\theta \partial_\sigma\zeta_3)\delta_{\alpha\beta} +\mu(\partial_\alpha\theta\partial_\beta\zeta_3+\partial_\beta\theta \partial_\alpha\zeta_3) \right]\partial_\beta\eta_\alpha {\rm d}\omega \end{equation} is continous on $V_H(\omega)$. Thus, given $\zeta_3 \in V_3(\omega)$, there exists a unique vector $(\zeta_\alpha)\in V_H(\omega)$ such that \begin{equation} \tilde{b}(\zeta_\alpha, \eta_\alpha) = \langle \zeta_3, \eta_\alpha\rangle. \end{equation} We denote by $T\zeta_3\in V_H(\omega)\times V_3(\omega)$ the vector $(\zeta_\alpha, \zeta_3)$. In particular, $T\zeta^m_3=(\zeta^m_\alpha, \zeta^m_3)$.
Substituting this in (\ref{eq:e11}), we get \begin{equation} b(\zeta^m_3, \eta_3)=\xi^m\int_\omega \zeta^m\eta_3 {\rm d}\omega \quad \mbox{ for all } \eta_3\in V_3(\omega),\label{eq:e55} \end{equation} where \begin{align} b(\zeta_3, \eta_3) &= -\int_\omega m_{\alpha\beta}\partial_{\alpha\beta}\eta_3 {\rm d}\omega + \int_\omega n^\theta_{\alpha\beta}(T\zeta_3)\partial_{\alpha\beta}\theta\eta_3 {\rm d}\omega\nonumber\\[.2pc] &\quad +\frac{2}{3}\int_\omega \frac{p^{3\alpha\beta}p^{3\rho\tau}}{p^{33}} \partial_{\rho\tau}\zeta_3\partial_{\alpha\beta}\eta_3 {\rm d}\omega. \label{eq:e56} \end{align}
\begin{lem} The bilinear form $b(\cdots)$ defined by {\rm (\ref{eq:e56})} is $V_H(\omega)$-elliptic and symmetric. \end{lem}
\begin{proof} It follows from Lemma~6.2 in \cite{SKNS1} that the bilinear form $\tilde{b}(\cdots)$ defined by \begin{equation} \tilde{b}(\zeta_3, \eta_3)= -\int_\omega m_{\alpha\beta}(\zeta_3)\partial_{\alpha\beta}\eta_3 {\rm d}\omega + \int_\omega n^\theta_{\alpha\beta}(T\zeta_3)\partial_{\alpha\beta}\theta\eta_3 {\rm d}\omega\label{eq:e57} \end{equation} is $V_H(\omega)$-elliptic and symmetric. Hence it is clear that $b(\cdots)$ is also $V_H(\omega)$-elliptic and symmetric.
\mbox{\xxx{\char'245}} \end{proof}
\begin{lem} Let $(\zeta_3^m, \xi^m), m\geq 1${\rm ,} be the eigensolutions of problem {\rm (\ref{eq:e56})} found as limits of the subsequence $(u^m(\epsilon){\rm ,} \xi^m(\epsilon)), m\geq 1$ of eigensolutions of the problem {\rm (\ref{eq:b13})}. Then the sequence $(\xi^m)_{m=1}^{\infty}$ comprises all the eigenvalues{\rm ,} counting multiplicities{\rm ,} of problem {\rm (\ref{eq:e56})} and the associated sequence $(\zeta^m_3)_{m=1}^{\infty}$ of eigenfunctions forms a complete orthonormal set in the space $V_3(\omega)$. \end{lem}
\begin{proof} The proof is similar to the proof of Lemma~5.4 in \cite{CK}.
\mbox{\xxx{\char'245}} \end{proof}
\end{document} | arXiv |
Can the Riemann hypothesis be relaxed to say that this matrix A consists of square roots?
I realize that asking this question is like presenting to a patent attorney a wheel-less skateboard while asking to patent a hoverboard.
Anyways. Lagarias version of the Riemann hypothesis sets a bound on the sum of divisors:
$$\sigma(n) \le H_n + e^{H_n} \ln H_n$$
where $H_n$ is a harmonic number.
The von Mangoldt function matrix, as I call it, can be generated from the matrix product of two matrices:
$$A = \left( \begin{array}{ccccccc} 1 & 0 & 0 & 0 & 0 & 0 & 0 & \cdots \\ 1 & \sqrt{2} & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & \sqrt{3} & 0 & 0 & 0 & 0 \\ 1 & \sqrt{2} & 0 & \sqrt{4} & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & \sqrt{5} & 0 & 0 \\ 1 & \sqrt{2} & \sqrt{3} & 0 & 0 & \sqrt{6} & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & \sqrt{7} \\ \vdots&&&&&&&\ddots \end{array} \right)$$
which is equal to $A(n,k)=\sqrt{k}$ if $k$ divides $n$, else $A(n,k)=0$
The matrix inverse of $A$ is by its terms essentially equal to the matrix:
$$B = \left( \begin{array}{ccccccc} 1 & 1 & 1 & 1 & 1 & 1 & 1 & \cdots \\ 0 & -\sqrt{2} & 0 & -\sqrt{2} & 0 & -\sqrt{2} & 0 \\ 0 & 0 & -\sqrt{3} & 0 & 0 & -\sqrt{3} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & -\sqrt{5} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & \sqrt{6} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & -\sqrt{7} \\ \vdots&&&&&&&\ddots \end{array} \right)$$
which is equal to $B(n,k)=\mu(n)\sqrt{n}$ if $n$ divides $k$, else $A(n,k)=0$
where $\mu(n)$ is the Möbius function
defined by:
$$\mu(n)=\begin{cases} (-1)^{\omega(n)}=(-1)^{\Omega(n)} &\text{if }\; \omega(n) = \Omega(n)\\ 0&\text{if }\;\omega(n) \ne \Omega(n).\end{cases}$$
or as in the Wikipedia page:
$\mu(n) = 1$ if $n$ is a square-free positive integer with an even number of prime factors.
$\mu(n) = -1$ if $n$ is a square-free positive integer with an odd number of prime factors.
$\mu(n) = 0$ if $n$ has a squared prime factor.
The von Mangoldt function matrix is then the matrix product $A$ times $B$:
$$T = A.B = \left( \begin{array}{ccccccc} +1&+1&+1&+1&+1&+1&+1&\cdots \\ +1&-1&+1&-1&+1&-1&+1 \\ +1&+1&-2&+1&+1&-2&+1 \\ +1&-1&+1&-1&+1&-1&+1 \\ +1&+1&+1&+1&-4&+1&+1 \\ +1&-1&-2&-1&+1&+2&+1 \\ +1&+1&+1&+1&+1&+1&-6 \\ \vdots&&&&&&&\ddots \end{array} \right)$$
And the von Mangoldt function is then:
$$\Lambda(n) = \sum\limits_{k=1}^{k=\infty}\frac{T(n,k)}{k}$$
as proven by joriki here.
or as the Dirichlet generating functions of the columns as proven here by GH from MO at Mathoverflow:
$$\Lambda(n)=\lim\limits_{s \rightarrow 1} \zeta(s)\sum\limits_{d|n} \frac{\mu(d)}{d^{(s-1)}}$$
Here comes the hoverboard / wheel-less skateboard:
Since according to the explicit formula, the von Mangoldt function is a sum of logarithmic square root waves as follows:
$$\sum_{n=1}^{n=k} \Lambda(n) = \Re\left(-\sum _{j=1}^{\infty} \left(\frac{x^{1-\rho _j}}{1-\rho _j}+\frac{x^{\rho _j}}{\rho _j}\right)-\frac{1}{2} \log \left(1-\frac{1}{x^2}\right)+x-\log (2 \pi )\right)$$
or as a Mathematica one-liner from Alex Kontorovich web page:
Plot[Re[X - Log[2 Pi] - Log[1 - 1/X^2]/2 -
Sum[X^(N[ZetaZero[j]])/(N[ZetaZero[j]]) +
X^(1 - N[ZetaZero[j]])/(1 - N[ZetaZero[j]]), {j, 1, 30}]], {X,
1.1, 30}]
Can the Riemann hypothesis be relaxed/be made precise to say that the so called von Mangoldt function matrix $T$ is a matrix product $T=A.B$ as in the example above?
(*Matrix T Mathematica 8*)
nn = 32;
A = Table[
Table[If[Mod[n, k] == 0, k^(ZetaZero[k]), 0], {k, 1, nn}], {n, 1,
nn}];
B = Table[
Table[If[Mod[k, n] == 0, MoebiusMu[n]*n^(ZetaZero[-n]), 0], {k, 1,
nn}], {n, 1, nn}];
MatrixForm[T=N[A.B]]
It appears to work for any complex number sequence in the exponents as long as the sum of the two matrices $A$ and $B$'s respective real parts is equal to 1, and the imaginary parts are each others negatives. In other words a condition that applies to any two complex number sequences of that form, of which the zeta zeros are a subset, so no progress.
To demonstrate this I have made this variant of the program above:
(*Matrix T Mathematica 8 start*)nn = 32;
a = Table[RandomComplex[], {n, 1, 32}]
Table[If[Mod[n, k] == 0, k^(a[[k]]), 0], {k, 1, nn}], {n, 1, nn}];
Table[If[Mod[k, n] == 0, MoebiusMu[n]*n^(1 - a[[n]]), 0], {k, 1,
MatrixForm[T = Chop[N[A.B]]]
(*end*)
which produces matrix $T$. This probably has to do with the elementary fact that:
$$n=n^{a} n^{1-a}$$
$n$ is here a substitute for the terms in matrix $T$.
So for some arbitrary complex number sequence $a$ like for example:
$$a=0.771518+0.640552I,0.192739+0.923147I,0.931096+0.758704I,...$$
or the non-trivial Riemann zeta zeros:
$$a=0.5 + 14.1347 I, 0.5 + 21.022 I, 0.5 + 25.0109 I,...$$ we have in general the matrices:
$$A = \left( \begin{array}{ccccccc} 1 & 0 & 0 & 0 & 0 & 0 & 0 & \cdots \\ 1 & 2^{a_2} & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 3^{a_3} & 0 & 0 & 0 & 0 \\ 1 & 2^{a_2} & 0 & 4^{a_4} & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 5^{a_5} & 0 & 0 \\ 1 & 2^{a_2} & 3^{a_3} & 0 & 0 & 6^{a_6} & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 7^{a_7} \\ \vdots&&&&&&&\ddots \end{array} \right)$$
$$B = \left( \begin{array}{ccccccc} 1 & 1 & 1 & 1 & 1 & 1 & 1 & \cdots \\ 0 & -2^{1-a_2} & 0 & -2^{1-a_2} & 0 & -2^{1-a_2} & 0 \\ 0 & 0 & -3^{1-a_3} & 0 & 0 & -3^{1-a_3} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & -5^{1-a_5} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 6^{1-a_6} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & -7^{1-a_7} \\ \vdots&&&&&&&\ddots \end{array} \right)$$
Which have the same property as the earlier matrices $A$ and $B$ producing matrix $T$ as the matrix product:
matrices number-theory riemann-hypothesis
Mats GranvikMats Granvik
$\begingroup$ there are many matrix formulation of the Riemann hypothesis, for example some bounds on the spectral radius or the eigenvectors of some matrix defined from number-theoritic functions. but, your description of $B$ and $T$ is confused and the rest after is even more. so give more details, write down your formulas, as when you say "Since the von Mangoldt function is a sum of logarithmic square root waves" can you define precisely what it means ? $\endgroup$ – reuns Jan 24 '16 at 12:28
$\begingroup$ Thank you for the comment. I have tried to improve the question now as you suggested. $\endgroup$ – Mats Granvik Jan 24 '16 at 13:01
$\begingroup$ $\sum_{k=1}^\infty \frac{T(1,k)}{k}$ diverges, how do you prove that for $n \ne 1$ : $\Lambda(n) = \sum_{k=1}^\infty \frac{T(n,k)}{k}$ ? and everybody knows what is $\mu(n)$, but nobody understand what you mean with " the von Mangoldt function is created from square roots as described above". $\endgroup$ – reuns Jan 24 '16 at 14:06
$\begingroup$ you understand nearly nothing of your question, even the proof that $\Lambda(n) = \sum_{k=1}^\infty \frac{1}{k} \sum_{d | gcd(n,k)} d \mu(d)$ you don't understand it, so ... work ! $\endgroup$ – reuns Jan 24 '16 at 14:19
$\begingroup$ I know that $\sum_{k=1}^\infty \frac{T(1,k)}{k}$, but when taking the Fourier transform of the von Mangoldt function defined this way, I believe that a good choice is to truncate it: $\sum_{k=1}^{k=n} \frac{T(1,k)}{k}$ with $n$ equal to the number of von Mangoldt function terms used in the Fourier transform. This because then the spectra like plot will be zero at locations of zeta zeros. Or at least it looks like the plot is zero at zeta zeros. $\endgroup$ – Mats Granvik Jan 24 '16 at 14:21
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The positive-definiteness of the special matrix created by shifting the vector $[1\, \cdots \, n\, \cdots\,1]$ | CommonCrawl |
NPA Ordinance: The impact of secrecy in ordinance making
by Pratik Datta and Rajeswari Sengupta.
In a liberal democracy law making should be a transparent affair. Transparency allows constructive public debate before a law is imposed on the society. In USA, the secrecy surrounding the drafting of the Senate Health Care bill meant to repeal Obamacare has been widely criticised. In contrast, we in India are so used to secrecy in legislative drafting that we do not even question it. We are content that bills introduced in the Lok Sabha and Rajya Sabha are at least publicly available.
Even this minimal transparency is denied to Indian citizens when the same bill is sent by the Cabinet to the President for 'promulgation' as an ordinance. The draft of the ordinance is not released in public domain till the President signs it and brings it into force. Unlike Parliamentary laws, there is no opportunity for public debate and discussion around a draft ordinance that has been approved by the Cabinet till it is imposed on the society. We saw this happen with the recent Banking Regulation (Amendment) Ordinance, 2017:
There had been speculation in the media since March 2017 that the Government was planning to take steps to resolve the stressed assets problem in the banking sector.
Figure 1: Google searches on 'NPA Resolution'
The graph above plots data from Google Trends. It shows the interest over time in the words 'NPA Resolution'. The graph plots the number of web searches done between the weeks starting March 5, 2017 and May 14, 2017. From March 19 onward, there was a sharp increase in the number of searches that used the words 'NPA Resolution'. The interest subsided in the week starting April 23 and picked up significantly in the week starting April 30, the same week when the NPA ordinance was first announced and then publicly released. From March 2017 onward, there were also news in the media that the Government was planning to empower the Reserve Bank of India to deal with stressed asset problem (see here and here).
On May 3, 2017 the Cabinet approved the Banking Regulation (Amendment) Ordinance, 2017. The media carried reports about the Cabinet approval but the text of the draft ordinance was not publicly available (see here and here). Around 7:49 pm on May 3, 2017, the Finance Minister mentioned in a media briefing that the Cabinet's recommendation has been sent to the President. He did not disclose any further detail on the ground that: There is a convention that when some proposal is referred to the President, then details of it cannot be disclosed till it is approved (sic).
On May 4, 2017 the President signed the Ordinance. The text of the signed ordinance was still not publicly available.
On May 5, 2017 the ordinance was released in public domain when it was uploaded onto the e-gazette at 12:38 PM.
In other words, even after the Cabinet approval on May 3, the text of the ordinance was withheld from the public till May 5. What was the impact of this secrecy convention? In this post we answer this question by examining the movement of the Nifty, the Bank Nifty and the PSU Bank Nifty indices before and after the public release of the ordinance.
Impact of the secrecy convention
Once the Cabinet provisionally agrees that an ordinance is needed, a bill is drafted. The draft bill is then sent to the President for promulgation as ordinance. As per the unwritten convention cited by the Finance Minister, the text of the ordinance is kept secret from the public till it gets uploaded on the egazette website. However, as seen during the promulgation of the Banking Regulation (Amendment) Ordinance, although the text was kept secret, selected information about the ordinance was released by the government to the media. There
was much speculation in the media about the details of the proposed ordinance.
Figure 2: Prices
The graphs above plot the three stock indices for the trading days around May 5. They run from the start of the trading day on May 2 to the close of trading on May 8. Trading was closed on the weekend of May 6 and May 7. The first two graphs from the top show the cumulated market model residuals of the Bank Nifty index and the PSU Bank Nifty index, respectively. The bottom-most graph shows the movement in the Nifty index around the event. The event identified in the graphs is the public release of the NPA ordinance at 12:38pm on May 5.
After the Finance Minister's press briefing on the evening of May 3, Nifty traded higher on May 4 than on the previous two trading days. Right after the ordinance was made public at 12:38pm on May 5, it fell till about 1:40pm before correcting marginally and ended the day lower than the previous two trading days.
A similar price movement is seen in the Bank Nifty index. The movement is much more pronounced in the PSU Bank Nifty index. Both the Bank Nifty and the PSU Bank Nifty indices went up between May 3 and May 5 (before 12:38 pm), the time period during which the secrecy convention was supposedly being followed. This suggests that the Finance Minister's media briefing on the evening of May 3 and the selective release of information was treated as positive news by the market. However the upward price movement came to a halt when the full text of the NPA ordinance was made public at 12:38pm on May 5 following which the prices fell sharply. The decline was more pronounced for the PSU banks.
It is worth noting that the prices had actually started falling a little before 12:38pm on May 5, particularly for the PSU banks. So it is possible that news of the text of the ordinance got leaked to the market even before the official release of the ordinance.
Figure 3: Volatility
The graphs above show the volatility in the returns of the three indices around the release of the NPA ordinance. There was an increase in the volatility of both the Bank Nifty and the PSU Bank Nifty indices after the ordinance was made public on May 5. The volatility increase was much more prominent for the PSU banks.
Market's negative reaction
The negative reaction of the market following the public release of the ordinance reflects the mismatch between market expectations and the final text of the ordinance. Between May 3 and May 5 (before 12:38 pm), market expectations were fuelled in the absence of the full text of the proposed ordinance. The selective release of information by the government triggered expectations in the market that the proposed ordinance would offer a comprehensive solution to the stressed asset problem of the banking sector. However, once the text of the ordinance was made publicly available at 12:38 pm on May 5, it was not evident to the market how the ordinance would be able to tackle the NPA crisis. The ordinance gave rise to more questions than it answered (see here). Consequently, the market reacted negatively.
Secrecy has been an inherent trait in ordinance-making since the times of the British Raj. Post-independence, while giving ordinance making powers to the President, the Constitution framers did extensively deliberate on the potential misuses of such powers. But they never questioned the secrecy around ordinance making. Consequently, Article 123 of the Constitution empowers the President to promulgate ordinances but does not explicitly require transparency in the ordinance-making process. It is then hardly suprising that even the Supreme Court has over time accepted that open legislative debates and discussions do not apply to ordinances.
It is high time we question this secrecy. As we have seen above, the secrecy convention coupled with the discretionary release of partial information about the recent Banking Regulation (Amendment) Ordinance, 2017 led to information asymmetry in the stock market, causing market inefficiencies. To avoid such inefficiencies, the government should make the ordinance-making process transparent by discarding the age-old secrecy convention and officially releasing the proposed ordinance immediately after it is approved by the Cabinet. Complete transparency in ordinance-making will also be in sync with the broader philosophy of legislation-making in a modern liberal democracy.
Pratik Datta is a researcher at the National Institute of Public Finance and Policy, New Delhi and Rajeswari Sengupta is a researcher at the Indira Gandhi Institute of Development Research
Labels: author: Pratik Datta, author: Rajeswari Sengupta, bankruptcy, drafting of law, financial sector regulation, PSU banks
What happens to private airlines when Air India is privatised?
by Ajay Shah.
What's the impact of privatising Air India upon the rest of the domestic airline industry? Some think that the capital stock of Air India will now be put into more capable hands, and that will intensify competition. Instead of a feeble competitor with assets of Rs.546 billion, we'll have a strong competitor wielding those same assets.
As was argued by me in August 2009, the `zombie firms' literature helps us think about this. A zombie firm is one which ought to go out of business, but is artificially kept on life support. Sometimes, the subsidy is explicit, such as the fiscal injections into Air India. Sometimes, the subsidy is less visible, such as banks extending additional capital to bankrupt firms. Regardless of the method adopted, the basic fact stands: A firm that ought to have gone out of business was given a subsidy through which it sold goods at below cost.
To fix intuition, imagine a market price of Rs.100. Imagine a weak firm that, on its own, is only able to produce at Rs.110. This firm ought to rapidly vanish. But instead, it's given a subsidy of Rs.11 (either by the government or by banks) and it manages to sell at Rs.99. This keeps it going. This also harms the healthy firms in the industry, who have to deal with competition from this dead man walking.
Turning to Air India, in 2014-15 the firm produced a profit after tax that was -28.4% of total income. For each Rs.100 of total income, there was a loss of Rs.28.4. The overall industry was at -11.7%, and the industry average was dragged down by Air India which is a substantial player.
The presence of zombie firms harms healthy firms. It's hard for a healthy firm to compete against one that's getting a subsidy. By this logic, the exit of a zombie firm is good news for each firm left standing in the industry.
If one of the incumbent large Indian airlines buys Air India, the combination will have significant market power, which would be a suboptimal outcome. It would be nice, from the viewpoint of competition policy, if a new player buys Air India. In either event, the low and subsidised prices charged by Air India will end. This will improve the profitability of the domestic airline industry. All healthy firms benefit when we address the problem of zombie firms.
Air India's privatisation is an interesting milestone where we may see some of these effects play out. The issue of zombie firms is, of course, present in many parts of the Indian economy. We have a peculiar arrangement where zombie banks keep zombie firms alive. Disrupting these arrangements holds the key for starting India on the next bout of growth.
Labels: banking, privatisation, the firm
How genetics is settling the Aryan migration debate by Tony Joseph in The Hindu, June 19, 2017.
The botanists' last stand: The daring work of saving the last samples of dying species by Zoe Schlanger in Quartz, June 17, 2017.
45 years of Watergate: Why the journalism of the Washington Post-NYT holds lessons for today's media by Saikat Datta in Scroll, June 17, 2017.
Dangerous nonsense: Once we put the Indian military above criticism we become Pakistan by Kanti Bajpai in The Times of India, June 17, 2017. Also see: How will your armed forces perform? by Ajay Shah in Ajay Shah's Blog, October 12, 2016.
Eyes in the forest by Aakash Lamba in Mint, June 17, 2017.
Where are India's TV comedy shows? by Mitali Saran in Business Standard, June 16, 2017.
Lessons from the US in Business Standard, June 15, 2017.
Modi's Message to the Media by Sadanand Dhume in The Wall Street Journal, June 15, 2017.
Keep official abuse of governance in check by Somasekhar Sundaresan in Business Standard, June 15, 2017.
The power paradox by Pratap Bhanu Mehta in The Indian Express, June 14, 2017.
Jeff Sessions and the Trail of Unanswered Questions by Amy Davidson in The New Yorker, June 14, 2017.
Want More Time? Get Rid of The Easiest Way to Spend It by David in Raptitude, June 13, 2017. Why did social media work badly? Perhaps an adverse selection process is afoot: a vicious cycle of low quality giving selective exit giving lowered quality.
Is Trump's definition of 'the rule of law' the same as the US Constitution's? by David Mednicoff in The Conversation, June 13, 2017.
Trump and the True Meaning of 'Idiot' by Eric Anthamatten in The New York Times, June 12, 2017.
Pirate Bay founder: We've lost the internet, it's all about damage control now by Már Másson Maack in Nextweb, June 10, 2017.
Donald Trump Is the Worst Boss in Washington by Britt Peterson in The New York Times, June 9, 2017.
The forking of the Indian rupee by J P Koning on Moneyness, June 9, 2017.
Blood from the Sky: Zipline's Ambitious Medical Drone Delivery in Africa by Jonathan W. Rosen in MIT Technology Review, June 8, 2017.
Bengal and business by Ashok K Lahiri in Business Standard, June 6, 2017.
Movement on the law for the Resolution Corporation
by Suyash Rai.
Capitalism without bankruptcy is like Christianity without hell.
- Frank Borman
On June 14th, the Union Cabinet approved the proposal to introduce a Financial Resolution and Deposit Insurance Bill, 2017 ("the FRDI Bill"). This is an important step forward for a critical component of the overall strategy of India's financial sector reforms. Shaji Vikraman has insight on this in the Indian Express. In this article, I look deeper into the concept of the resolution corporation, why it matters, how we got to this milestone, and what comes next.
The slow unfolding of the banking crisis reminds us of the fragility of our financial system. The financial system, especially the banking system, is generally disaster-prone. On one hand, financial firms can make mistakes and experience losses. In addition, there is a link between problems of the economy and hardship in financial firms. When an economic downturn happens, the value of business activities declines, and this induces losses upon financial positions. We need to build a financial regulatory apparatus which will reduce financial fragility. This involves three main elements of machinery : micro-prudential regulation (which aims to push the failure probability of each financial firm to a desired value), systemic risk regulation (which aims to reduce the probability of a disruption in the overall financial system, and have tools to respond to such a disruption when it does arise) and resolution (a specialised bankruptcy process for most financial firms). At present, in India, we have weaknesses on all three elements.
Consequences of a weak resolution system
When micro-prudential regulation works well, the failure probability of financial firms is at a low level chosen by the relevant financial agency. The failure probability is not zero. Failure of inefficient firms is essential for `creative destruction'. The process of failure of inefficient firms, and the shift of capital and labour to efficient firms, is essential for productivity growth. The question is: How can we make the failure of financial firms orderly?
The failure of financial firms can often be quite disorderly. Unlike real sector firms, many financial firms manage a large amount money belonging to households and businesses, with only a small amount of capital brought in by their owners. Banks in India typically have leverage of 18$\times$ to 20$\times$, which means that their balance sheet size is 18 to 20 times the amount of equity capital. Such leverage is never seen with real sector firms. When the firm gets into trouble, there is clamour by the creditors who want to see a fair and efficient process through which they get some of their money back. Matters are more challenging with some financial firms which are so large and complex that their failure could induce instability in the financial system.
An orderly failure is one where a) the consumers either get their money back quickly or continue to get services without any significant inconvencience, and b) the stability of the financial system is not threatened. If we are not able to obtain orderly failures in the financial system, this has many adverse consequences:
Consumers of failed financial firms suffer. As an example, in India, many cooperative banks fail every year. In spite of high entry barriers, larger institutions also fail (e.g. Global Trust Bank in 2004). Consumers lose money in these failures. These bad experiences make consumers wary of engagement with the financial system, and increase the share of gold and real estate in their portfolios.
Financial stability is threatened, because even if one systemically important financial firm fails, the entire system could be destabilised by a messy, long-drawn bankruptcy process. This forces government to bail out such financial firms. So, a financial crisis ends up having a fiscal consequence.
When faced with the possibility of harm to consumers, and threats to financial stability, governments get cold feet in situations of firm distress. They are then prone to bail out financial firms using taxpayers' money. We in India are familiar with this story. Public sector banks are routinely recapitalised with public funds to ensure they do not fail. This is almost never a good use of public money.
Regulators sometimes respond to these problems by setting up entry barriers, which harm competition and economic dynamism. They justify the every day harm to competition on the grounds that this averts harm to consumers, risks to financial stability and the fiscal cost of bailouts.
Financial firms suffer from moral hazard, and take greater risks. At its worst, financial firms obtain supernormal profit from these two interlinked channels: the certainty of being bailed out and the lack of competition.
A system that ensures quick and orderly resolution of failed financial firms can help avoid these outcomes. The system should be such that government, financial firms and consumers believe that the failures will be orderly. The present system of resolution in India is inadequate.
First, it mostly empowers the respective regulators (eg. RBI for banks) to do the resolution. Since regulators give the licenses and are supposed to ensure safety and soundness of the firms they license, they tend to be tardy in acknowledging their mistakes. This regulatory forbearance leads to delays in recognition of failure, which increases the costs of resolution, and may lead to losses for consumers and increases risk to stability of the financial system. There is a conflict of interest between micro-prudential regulation (achieving a target failure probability for a financial firm) and resolution (gracefully closing down financial firms which are nearing failure).
Second, the present system gives very limited powers of resolution. The powers that are given are: forced mergers/amalgamation, and winding up. Some of the other powers, such as bail-in (discussed later), are not available.
Third, even these limited powers are not enjoyed over many of the financial firms. For example, regulators do not have resolution powers over public sector scheduled commercial banks and regional rural banks.
Fourth, the way the system is structured, a bankruptcy resolution can take years, sometimes even longer than a decade. This is partly because the regulators do not have powers to take timely resolution action.
The Financial Resolution and Deposit Insurance Bill
Indian policy thinking on this began in the RBI Advisory group on reforms of deposit insurance, 1999, chaired by Jagdish Capoor.
This slumbered until we got to the Financial Sector Legislative Reforms Commission, chaired by Justice BN Srikrishna, which worked from 2011 to 2013. In its full design of Indian financial regulation, it recommended a Resolution Corporation.
In 2014, a Working Group of Ministry of Finance and Reserve Bank of India, co-chaired by Shri Arvind Mayaram and Shri Anand Sinha, also recommended a resolution capability for financial firms.
In 2014, the Ministry of Finance constitued a Task Force for the Establishment of the Resolution Corporation, under the chairmanship of Shri M. Damodaran, to work out the plan for establishing the Resolution Corporation. This was part of the two-part creation of task forces for building the new institutions required in the FSLRC architecture, which came about as four task forces followed by one more.
The budget speeches of 2015-16 and 2017-18 announced a plan to draft and table a Bill on resolution of financial firms. In September, 2016, a draft of the Bill was placed in public domain for comments.
On June 14th, the Cabinet approved the proposal to introduce a Financial Resolution and Deposit Insurance Bill, 2017 ("the FRDI Bill") in Parliament. The FRDI Bill, when enacted, will create a framework to ensure that failure of financial firms is orderly. It will establish an independent Resolution Corporation tasked with resolving failed financial firms. The Corporation will also subsume the deposit insurance function presently performed by the Deposit Insurance and Credit Guarantee Corporation.
This Bill stands at the intersection of two long-term reform projects: 1) financial sector reforms, of which bankruptcy resolution of financial firms is an integral part; 2) bankruptcy reforms, of which financial firm resolution is an integral part. So, this Bill moves both these projects forward, and is an important building block for an efficient system of capital allocation in India.
FSLRC had envisioned a separation between the resolution corporation, which would apply for most financial firms, and the bankruptcy code, which would apply for the remaining financial firms and for all non-financial firms. The Bankruptcy Legislative Reforms Commission (BLRC), which drafted the Insolvency and Bankruptcy Code (IBC), worked with this scheme. IBC does not cover financial firms, unless the Central Government notifies certain financial firms to be covered under that law. Many types of financial firms, especially firms handling consumer funds and firms that are critical for financial stability, require a specialised resolution mechanism. For firms handling consumer funds (eg. banks, insurance companies), the process under IBC is not suitable, as a large number of small value consumers will find it difficult to invoke that process. The processes of IBC are designed for creditors who are firms, not individuals. For systemically important financial firms (eg. central counterparties, larger banks), a creditor-led resolution process under IBC is not suitable, because what is at stake is not just the interest of creditors but the stability and resilience of the financial system. Hence, for such financial firms a specialised resolution regime is required. The FRDI Bill will create such a specialised resolution regime.
What is resolution?
In the world of financial firms, resolution complements regulation. Regulators and the Corporation are expected to work in tandem, with the regulators focused on maintaining financial health and, when a firm gets into trouble, pushing for its recovery. The Resolution Corporation will take over and resolve a firm after recovery efforts have failed. Although the version of the Bill approved by the Cabinet is not yet in public domain, based on the version that was released for public consultations last year, the framework is divided into four stages.
First, when the financial firm is healthy, the respective regulators will monitor the firm and work to ensure it continues to stay healthy. At this stage, the Resolution Corporation will only get information indirectly through the regulators. Substantive powers to monitor the firm or to take any other action with respect to the firm will not be available to the Corporation.
Second, once the financial firm starts deteriorating, the respective regulator will attempt recovery. At this stage also, only the regulators will continue to have substantial powers over the firm.
Third, if the recovery efforts fail, and as the financial firm get close to failure, the Corporation will get substantial powers to instruct the firm to improve its resolvability and prevent actions that may erode the values of assets available for resolution. At this stage, the role of regulators is restricted.
Finally, when the firm fails, the Corporation will take charge and resolve it. Resolution typically means selling the failed financial firm, as a whole or in parts, to another financial firm via a competitive bidding process. However, resolution could also involve other instruments. For example, the firm could be "bailed-in", which means that the rights of and obligations to creditors may be written down to recapitalise the firm from within. Bail-in typically includes converting some junior debt into equity, but may also include writing down other types of claims. This is the opposite of a bail-out, wherein outside investors rescue a borrower by injecting money to help service a debt. Finally, liquidation may be a tool used for resolution.
There is a certain degree of tension and potential conflict between the Regulators and the Resolution Corporation. This is a healthy check-and-balance. Resolution works as a check on regulatory incompetence and forbearance. Both sides will need to be mature, respect the role of the other, and coordinate.
The idea of a specialised resolution regime for financial firms is well-accepted globally. The US has had a resolution system for banks for more than 80 years. The scope of this system was extended after the financial crisis of 2008. There have been more than 600 bank failures in US since the crisis. In this time, there has been not been even one bank run in the US, because depositors trust the resolution system to work. Why the crisis happened in the first place is another matter, which is beyond the scope of resolution. Resolution comes into play only after regulation fails, and the occurrence of crisis resulted from regulatory failure, among other factors.
Many other countries have put in place comprehensive resolution systems. These include: all European Union member states, Switzerland, Australia, Canada, Japan, Korea, Mexico, and Singapore. Many jurisdictions have ongoing or planned reforms to resolution regimes. These include: Australia, Brazil, Canada, China, Hong Kong, Indonesia, Korea, Russia, Saudi Arabia, Singapore, South Africa, Turkey.
We are still a few years away from having a full-fledged resolution regime. Now that the Bill is going to the legislative branch, it remains to be seen what version of the Bill eventually gets enacted. If the essential features of a good resolution regime are diluted in the final version, the chances of success will be low.
Even after the Bill gets enacted, it would still take some time to build an independent and competent Resolution Corporation. Since this capability currently does not exist in the system, it will have to be cobbled together, and then strengthened over a period of time. Consider the example of human resource strategy. There are many models out there. While the Canadian authority works with fewer than 100 employees, the US authority has more than 10,000 employees. The Corporation could choose to run a tight ship, and rely on contractual work to scale up capacity in times of crisis, or it could choose to build a large organisation that is able to, on its own, deal with a crisis. Similarly, given the skill sets required to do this job, the Corporation will have to think innovatively about attracting top talent within the constraints of a government agency.
The Task Force on Establishment of the Resolution Corporation, led by M. Damodaran, has done considerable work that lays the groundwork for constructing the agency. The implementation of their project planning needs to commence immediately, so that the delay between enacting the law and enforcing it can be minimised.
It will also take our governance system some time to get used to this kind of a system of taking over and resolving a failed financial firm in a decisive and quick manner, as opposed to the present approach of allowing things to linger on. If things do go right, there are many potential benefits of this reform.
The author is a researcher at NIPFP.
Labels: author: Suyash Rai, bankruptcy, financial firms, financial sector regulation, prudential regulation, resolution
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\begin{document}
\title[The role of antisymmetric functions in nonlocal equations]{The role of antisymmetric functions in nonlocal equations}
\author[S. Dipierro]{Serena Dipierro} \address{Serena Dipierro: Department of Mathematics and Statistics, The University of Western Australia, 35 Stirling Highway, Crawley, Perth, WA 6009, Australia} \email{[email protected]}
\author[G. Poggesi]{Giorgio Poggesi} \address{Giorgio Poggesi: Department of Mathematics and Statistics, The University of Western Australia, 35 Stirling Highway, Crawley, Perth, WA 6009, Australia} \email{[email protected]}
\author[J. Thompson]{Jack Thompson} \address{Jack Thompson: Department of Mathematics and Statistics, The University of Western Australia, 35 Stirling Highway, Crawley, Perth, WA 6009, Australia}
\email{[email protected]}
\author[E. Valdinoci]{Enrico Valdinoci} \address{Enrico Valdinoci: Department of Mathematics and Statistics, The University of Western Australia, 35 Stirling Highway, Crawley, Perth, WA 6009, Australia} \email{[email protected]}
\subjclass[2020]{35B50, 35N25, 35B06}
\date{\today}
\dedicatory{}
\begin{abstract} We use a Hopf-type lemma for antisymmetric super-solutions to the Dirichlet problem for the fractional Laplacian with zero-th order terms, in combination with the method of moving planes, to prove symmetry for the {\it semilinear fractional parallel surface problem}. That is, we prove that non-negative solutions to semilinear Dirichlet problems for the fractional Laplacian in a bounded open set $\Omega \subset \mathbb{R}^n$ must be radially symmetric if one of their level surfaces is parallel to the boundary of $\Omega$; in turn, $\Omega$ must be a ball.
Furthermore, we discuss maximum principles and the Harnack inequality for antisymmetric functions in the fractional setting and provide counter-examples to these theorems when only `local' assumptions are imposed on the solutions. The construction of these counter-examples relies on an approximation result that states that `all antisymmetric functions are locally antisymmetric and \(s\)-harmonic up to a small error'. \end{abstract}
\maketitle
\section{Introduction and results} In this paper, we are concerned with maximum principles for antisymmetric solutions to linear, nonlocal partial differential equations (\textsc{PDE}) where the nonlocal term is given by the fractional Laplacian and with the application of such maximum principles to the method of moving planes.
Since its introduction by Alexandrov \cite{MR0150709,MR143162} and the subsequent refinement by Serrin~\cite{MR333220}, the method of moving planes has become a standard technique for the analysis of overdetermined \textsc{PDE}, and for proving symmetry and monotonicity results for solutions, see \cite{MR544879,MR634248,MR2293958,MR2366129,MR3040677,MR3189604,MR3385189,MR3454619}. In recent years, nonlocal \textsc{PDE}, particularly the ones involving the fractional Laplacian, have received a lot of attention, due to their consolidated ability to model complex physical phenomena, as well as their rich analytic theory. The method of moving planes has been also adapted in several works so that it may be applied to nonlocal problems, see \cite{MR3395749,MR3827344,MR3881478,MR3836150,MR4313576,ciraolo2021symmetry}.
The method of moving planes typically leads to considering the difference between a given solution and its reflection, and such a difference is an antisymmetric function. However, in the local case, `standard' maximum principles can be directly applied to the moving plane method, without specifically relying on the notion of antisymmetric functions, since the ingredients required are all of `local' nature and can disregard the `global' geometric properties of the objects considered. Instead, in the nonlocal world, new maximum principles need to be developed which take into account the extra symmetry, to compensate for global control of the function under consideration, for example see \cite{MR3395749}.
We will now state our main results. Let \(s\in (0,1)\) and~\(n\geqslant 1\) be an integer. The fractional Laplacian \((-\Delta)^s\) is defined by \begin{align*} (-\Delta)^s u(x) &= c_{n,s}\operatorname{P.V.} \int_{\mathbb{R}^n} \frac{u(x)-u(y)}{\vert x -y \vert^{n+2s}} \mathop{}\!d y \end{align*} where \begin{align} c_{n,s} = \frac{s 4^s \Gamma \big (\frac{n+2s}{2} \big )}{\Gamma(1-s) } \label{tK7sb} \end{align} is a positive normalisation constant and \(\operatorname{P.V.}\) refers to the Cauchy principle value. Moreover, let~\(\mathbb{R}^n_+ = \{ x\in \mathbb{R}^n \text{ s.t. } x_1>0\}\) and \(B_1^+ = B_1 \cap \mathbb{R}^n_+\).
Our first result proves that `all antisymmetric functions are locally antisymmetric and \(s\)-harmonic up to a small error'. This is the analogue of \cite[Theorem 1]{MR3626547} for antisymmetric functions.
\begin{thm} \label{u7dPW} Suppose that \(k \in \mathbb{N}\) and that \(f \in C^k(\overline{B_1})\) is \(x_1\)-antisymmetric. For all \(\varepsilon>0\), there exist a smooth \(x_1\)-antisymmetric function \(u:\mathbb{R}^n \to \mathbb{R}\) and a real number \(R>1\) such that \(u\) is \(s\)-harmonic in \(B_1\), \(u=0\) in \(\mathbb{R}^n\setminus B_R\), and \begin{align*}
\| u - f \|_{C^k(B_1)}<\varepsilon . \end{align*} \end{thm}
When we say that \(u\) is \(x_1\)-antisymmetric we mean that \( u(-x_1,\dots,x_n)=-u(x_1,\dots,x_n) \) for all~\(x\in \mathbb{R}^n\). In \S\ref{9Hmeh} we will give a more general definition of antisymmetry for an arbitrary plane. As an application of Theorem \ref{u7dPW}, we can construct some counter-examples to the local Harnack inequality and strong maximum principle for antisymmetric \(s\)-harmonic functions, as follows:
\begin{cor}\label{X96XH} There exist \(\Omega \subset \mathbb{R}_+^n\) and \(\Omega'\subset \subset \Omega\) such that, for all \(C>0\), there exists an \(x_1\)-antisymmetric function \(u:\mathbb{R}^n \to \mathbb{R}\) that is non-negative and \(s\)-harmonic in \(\Omega\), and \begin{align*} \sup_{\Omega'} u > C \inf_{\Omega'} u . \end{align*} \end{cor}
\begin{cor} \label{KTBle} There exist \(\Omega \subset \mathbb{R}^n_+\) and an \(x_1\)-antisymmetric function \(u:\mathbb{R}^n \to \mathbb{R}\) that is non-negative and \(s\)-harmonic in \(\Omega\), is equal to zero at a point in \(\Omega\), but is not identically zero in \(\Omega\). \end{cor}
For our second result, we use a Hopf-type lemma (see the forthcoming Proposition~\ref{lem:FLIFu}) in combination with the method of moving planes to establish symmetry for the {\it semilinear fractional parallel surface problem}, which is described in what follows.
Suppose that \(G\subset \mathbb{R}^n \) is open and bounded, and let~\(B_R\) denote the ball of radius \(R>0\) centered at 0. Let \(\Omega = G+B_R\), where, given sets \(A,B \subset \mathbb{R}^n\), \(A+B\) denotes the `Minkowski sum' of \(A\) and \(B\), defined as \begin{align*} A+B = \{a+b \text{ s.t. } a\in A, b\in B\}. \end{align*} Consider the semilinear fractional equation \begin{align} \begin{PDE} (-\Delta)^s u &= f(u) &\text{in }\Omega, \\ u&=0 &\text{in }\mathbb{R}^n \setminus \Omega,\\ u&\geqslant 0 &\text{in } \Omega , \end{PDE} \label{rzZmb} \end{align} with the overdetermined condition \begin{align} u = c_0 \qquad \text{on } \partial G, \label{lnARf} \end{align} for some (given) \(c_0\geqslant 0\). The \emph{semilinear fractional parallel surface problem} asks the following question: for which~\(G\) does the problem~\eqref{rzZmb}-\eqref{lnARf} admit a non-trivial solution? The answer is that \(G\) must be a ball. More specifically, we have the following result:
\begin{thm} \label{CccFw} Suppose that \(G\) is a bounded open set in \(\mathbb{R}^n\) with \(C^1\) boundary. Let~\(\Omega=G+B_R\), \(f:\mathbb{R} \to \mathbb{R}\) be locally Lipschitz, and \(c_0 \geqslant 0\). Furthermore, assume that there exists a non-negative function \(u \in C^s(\mathbb{R}^n)\) that is not identically zero and satisfies, in the pointwise sense,\begin{align} \begin{PDE} (-\Delta)^s u &= f(u) &\text{in } \Omega,\\ u&=0 &\text{in } \mathbb{R}^n \setminus \Omega, \\ u &= c_0 & \text{on } \partial G . \end{PDE} \label{zcm6a} \end{align} Then \(u\) is radially symmetric, \(u>0\) in \(\Omega\), and \(\Omega\) (and hence \(G\)) is a ball. \end{thm}
In the local setting (i.e., \(s=1\)), symmetry and
stability results for \eqref{zcm6a} were obtained in \cite{MR3420522} and \cite{MR3481178}.
Such analysis was motivated by the study of invariant isothermic surfaces of a nonlinear nondegenerate fast diffusion equation (see \cite{MR2629887}).
More recently in \cite{ciraolo2021symmetry}, the nonlocal case \(s\in (0,1)\) was addressed and the analogue to Theorem \ref{CccFw}, as well as its stability generalization, were proved in the particular case \(f \equiv 1\); those results were obtained by making use of the method of moving planes as well as a (quantitative) Hopf-type lemma (see \cite[formula~(3.3)]{ciraolo2021symmetry}), which could be obtained as an application of the boundary Harnack inequality for antisymmetric $s$-harmonic functions proved in \cite[Lemma~2.1]{ciraolo2021symmetry}. The arguments used in~\cite{ciraolo2021symmetry} to establish such a boundary Harnack inequality rely on the explicit knowledge of the fractional Poisson kernel for the ball. However, due to the general nonlinear term~\(f\) in \eqref{zcm6a}, here the method of moving planes leads to considering a linear equation involving the fractional Laplacian but with zero-th order terms for which no Poisson formula is available. To overcome this conceptual difficulty, we provide the following Hopf-type lemma which allows zero-th order terms.
\begin{prop} \label{lem:FLIFu} Suppose that \(c \in L^\infty(B_1^+)\), \(u \in H^s(\mathbb{R}^n) \cap C(B_1)\) is \(x_1\)-antisymmetric, and satisfies \begin{align} \begin{PDE} (-\Delta)^su +cu &\geqslant 0 &\text{in } B_1^+, \\ u&\geqslant 0 &\text{in } \mathbb{R}^n_+ ,\\ u&>0 &\text{in } B_1^+. \end{PDE} \label{YEL36} \end{align} Then \begin{align} \liminf_{h\to 0} \frac{u(he_1)} h > 0 . \label{FLBHT} \end{align} \end{prop}
This result has previously been obtained in \cite[Proposition 2.2]{MR3937999} though the proof uses a different barrier.
Proposition \ref{lem:FLIFu} establishes that antisymmetric solutions to \eqref{YEL36} must be bounded from below by a positive constant multiple of \(x_1\) close to the origin. At a first glance this is surprising as solutions to \eqref{YEL36} are in general only \(s\)-H\"older continuous up to the boundary, as proven in \cite{MR3168912}. Hence, it would appear more natural to consider \(\liminf_{h\to 0} \frac{u(he_1)} {h^s}\) instead of \eqref{FLBHT}, see for example \cite[Proposition 3.3]{MR3395749}. However, the (anti)symmetry of \(u\) means that Proposition~\ref{lem:FLIFu} is better understood as an interior estimate rather than a boundary estimate. Indeed, we stress that, differently from the classical Hopf lemma, Proposition~\ref{lem:FLIFu} is not concerned with the growth of a solution from a boundary point, but mostly with the growth from a reflection point in the antisymmetric setting (notice specifically that~\eqref{FLBHT} provides a linear growth from the origin, which is an interior point). In this sense, the combination of the antisymmetric geometry of the solution and the fractional nature of the equation leads to the two structural differences between Proposition~\ref{lem:FLIFu} and several other Hopf-type lemmata available in the literature for the nonlocal case, namely the linear (instead of \(s\)-H\"older) growth and the interior (instead of boundary) type of statement.
A similar result to Proposition~\ref{lem:FLIFu} was obtained in \cite[Theorem 1]{MR3910421} for the entire half-space instead of \(B_1^+\). This entails that~\cite[Theorem 1]{MR3910421} was only applicable (via the method of moving planes) to symmetry problems posed in all of \(\mathbb{R}^n\) whereas our result can be applied to problems in bounded domains. Moreover, \cite[Theorem 1]{MR3910421} is proven by contradiction while the proof of Proposition~\ref{lem:FLIFu}, in a similar vein to the original Hopf lemma, relies on the construction of an appropriate barrier from below, see Lemma \ref{SaBD4}. Hence, using a barrier approach also allows us to obtain a quantitative version of Proposition~\ref{lem:FLIFu} leading to quantitative estimates for the stability of Theorem \ref{CccFw}, which we plan to address in an upcoming paper.
It is also natural ask whether the converse to Theorem~\ref{CccFw} holds. We recall that if $\Omega$ is known to be a ball, i.e., $\Omega:=B_\rho$, then the radial symmetry of a bounded positive solution $u$ to \begin{equation}\label{eq:Classical Dirichlet} \begin{PDE} -\Delta u &= f(u) & \text{in } B_\rho\\ u&=0 &\text{on } \partial B_\rho \end{PDE} \end{equation} is guaranteed for a large class of nonlinearities $f$. For instance, this is the case for $f$ locally Lipschitz, by the classical result obtained via the method of moving planes by Gidas-Ni-Nirenberg \cite{MR634248,MR544879}. Many extensions of Gidas-Ni-Nirenberg result can be found in the literature, including generalizations to the fractional setting (see, e.g., \cite{MR2114412}). We also recall that an alternative method pioneered by P.-L. Lions in \cite{MR653200} provides symmetry of nonnegative bounded solutions to \eqref{eq:Classical Dirichlet} for (possibly discontinuous) nonnegative nonlinearities; we refer to \cite{MR3003296,MR1382205,MR2019179,MR4380032} for several generalizations.
The paper is organised as follows. In Section \ref{eBxsh} we recall some standard notation, as well as provide some alternate proofs for the weak and strong maximum principle for antisymmetric functions. Moreover, we will prove Theorem \ref{u7dPW} and subsequently prove Corollary \ref{X96XH} and Corollary \ref{KTBle}. In Section \ref{R9GxY}, we will prove Proposition~\ref{lem:FLIFu} and in Section \ref{Svlif} we will prove Theorem \ref{CccFw}. In Appendix \ref{ltalz} we give some technical lemmas needed in the paper.
\section{Maximum principles and counter-examples} \label{eBxsh}
\subsection{Definitions and notation} \label{9Hmeh} Let \(n \geqslant 1\) be an integer, \(s\in (0,1)\), and \(\Omega\subset \mathbb{R}^n\) be a bounded open set. The fractional Sobolev space \(H^s(\mathbb{R}^n)\) is defined as \begin{align*} H^s(\mathbb{R}^n) = \bigg \{ u \in L^2(\mathbb{R}^n) \text{ s.t. } [ u ]_{H^s(\mathbb{R}^n)} < \infty \bigg \} \end{align*} where \([ u ]_{H^s(\mathbb{R}^n)}\) is the Gagliardo semi-norm \begin{align*} [ u ]_{H^s(\mathbb{R}^n)} &= \bigg ( \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} \frac{\vert u(x)-u(y)\vert^2}{\vert x -y \vert^{n+2s}} \mathop{}\!d x \mathop{}\!d y \bigg )^{\frac12}. \end{align*} As usual we identify functions that are equal except on a set of measure zero. It will also be convenient to introduce the space \begin{align*} \mathcal H^s_0(\Omega) = \{ u \in H^s (\mathbb{R}^n) \text{ s.t. } u = 0 \text{ in } \mathbb{R}^n \setminus \Omega \}. \end{align*} Suppose that \(c:\Omega \to \mathbb{R}\) is a measurable function and that \(c\in L^\infty(\Omega)\). A function \(u \in H^s(\mathbb{R}^n)\) is a \emph{weak} or \emph{generalised} solution of \((-\Delta)^s u +cu =0\) (resp. \(\geqslant 0, \leqslant 0\)) in \(\Omega\) if \begin{align*} \mathfrak L (u,v) := \frac{c_{n,s}}2 \iint_{\mathbb{R}^n \times \mathbb{R}^n} \frac{(u(x)-u(y))(v(x)-v(y))}{\abs{x-y}^{n+2s}} \mathop{}\!d x \mathop{}\!d y + \int_{\Omega} c(x) u(x) v(x) \mathop{}\!d x =0 \, (\geqslant 0, \leqslant 0) \end{align*} for all \(v \in \mathcal H^s_0(\Omega) \), \(v \geqslant 0\). Also, it will be convenient to use the notation \begin{align*} \mathcal E (u,v) = \frac{c_{n,s}}2 \iint_{\mathbb{R}^n \times \mathbb{R}^n} \frac{(u(x)-u(y))(v(x)-v(y))}{\abs{x-y}^{n+2s}} \mathop{}\!d x \mathop{}\!d y \end{align*} for each \(u,v \in H^s(\mathbb{R}^n)\).
A function \(u :\mathbb{R}^n \to \mathbb{R}\) is \emph{antisymmetric} with respect to a plane \(T\) if \begin{align*} u(Q_T(x)) = -u(x) \qquad \text{for all } x\in \mathbb{R}^n \end{align*} where \(Q_T:\mathbb{R}^n \to \mathbb{R}^n\) is the reflection of \(x\) across \(T\). A function \(u\) is \emph{\(x_1\)-antisymmetric} if it is antisymmetric with respect to the plane \(T=\{x_1=0\}\). In the case \(T=\{x_1=0\}\), \(Q_T\) is given explicitly by \(Q_T(x) = x-2x_1e_1\). When it is clear from context what \(T\) is, we will also write~\(Q(x)\) or~\(x_\ast\) to mean~\(Q_T(x)\).
We will also make use of standard notation: the upper and lower half-planes are given by \(\mathbb{R}^n_+= \{ x \in \mathbb{R}^n \text{ s.t } x_1>0\}\) and \(\mathbb{R}^n_-= \{ x \in \mathbb{R}^n \text{ s.t } x_1<0\}\) respectively, and, for all \(r>0\), the half-ball of radius \(r\) is given by \(B_r^+ = \mathbb{R}^n_+ \cap B_r\). We also denote the positive and negative part of a function~\(u\) by~\(u^+(x) = \max \{u(x),0\}\) and~\(u^-(x) = \max \{-u(x),0\}\). Furthermore, the characteristic function of a set \(A\) is given by \begin{align*} \chi_A(x) &= \begin{cases} 1, &\text{if } x\in A, \\ 0,&\text{if } x\not\in A. \end{cases} \end{align*}
\subsection{Maximum principles}
We now present some results of maximum principle type for antisymmetric functions.
\begin{prop}[A weak maximum principle] \label{aKht4} Suppose that \(\Omega\) is a strict open subset of \(\mathbb{R}^n_+\), and~\(c :\Omega \to \mathbb{R}\) ia a measurable and non-negative function. Let \(u \in H^s(\mathbb{R}^n)\) be \(x_1\)-antisymmetric. If \(u\) satisfies \((-\Delta)^su+cu\leqslant0\) in \(\Omega\) then \begin{align*} \sup_{\mathbb{R}^n_+} u \leqslant \sup_{\mathbb{R}^n_+\setminus \Omega} u^+ . \end{align*} Likewise, if \(u\) satisfies \((-\Delta)^su+cu\geqslant0\) in \(\Omega\) then \begin{align*} \inf_{\mathbb{R}^n_+} u \geqslant - \sup_{\mathbb{R}^n_+ \setminus \Omega } u^- . \end{align*} \end{prop}
In our proof, we use that $c \ge 0$. We refer, e.g., to \cite[Proposition 3.1]{MR3395749} for a weak maximum principle where the non-negativity of $c$ is not required.
\begin{proof}[Proof of Proposition \ref{aKht4}] It is enough to prove the case \((-\Delta)^su+cu\leqslant0\) in \(\Omega\). Suppose that \(\ell := \sup_{\mathbb{R}^n_+ \setminus \Omega} u^+ <+\infty\), otherwise we are done.
For each \(v\in H^s_0(\Omega)\), \(v \geqslant 0\), rearranging \(\mathfrak L (u,v) \leqslant 0\) gives \begin{align} \mathcal E(u,v) \leqslant - \int_\Omega c(x) u(x) v(x) \mathop{}\!d x \leqslant 0 \label{FQBj4} \end{align} provided that \(uv \geqslant 0\) in \(\Omega\).
Let \(w(x):=u(x)-\ell\) and \(v (x) := (u(x)-\ell)^+\chi_{\mathbb{R}^n_+}(x)=w^+(x)\chi_{\mathbb{R}^n_+}(x)\). Note that for all \(x\in \Omega\),\begin{align*} u(x)v(x) &= ((u(x)-\ell)^+)^2 + \ell (u(x)-\ell)^+ \geqslant 0. \end{align*} Here we used that \((u-\ell)^+(u-\ell)^-=0\). To prove the proposition, it is enough to show that \(v=0\) in \(\mathbb{R}^n\). As \(u(x)-u(y)=w(x)-w(y)\), expanding then using that \(w^+ (x) w^-(x)=0\) gives \begin{align*} (u(x)-u(y))(v(x)-v(y)) &= ((v(x)-v(y))^2 +v(x)(v(y)-w(y)) +v(y) (v(x)-w(x)) \end{align*} for all \(x,y \in \mathbb{R}^n\). Hence, \begin{align*} \mathcal E (u,v) &= \mathcal E (v,v) + c_{n,s} \iint_{\mathbb{R}^n\times \mathbb{R}^n} \frac{v(x)(v(y)-w(y))}{\vert x- y \vert^{n+2s} } \mathop{}\!d x \mathop{}\!d y. \end{align*} Observe that\begin{align*} \iint_{\mathbb{R}^n\times \mathbb{R}^n} \frac{v(x)(v(y)-w(y))}{\vert x- y \vert^{n+2s} } \mathop{}\!d x \mathop{}\!d y &= \iint_{\mathbb{R}^n_+\times \mathbb{R}^n_+} \frac{w^+(x)w^-(y)}{\vert x- y \vert^{n+2s} } \mathop{}\!d x \mathop{}\!d y-\iint_{\mathbb{R}^n_-\times \mathbb{R}^n_+} \frac{w^+(x)w(y)}{\vert x- y \vert^{n+2s} } \mathop{}\!d x \mathop{}\!d y. \end{align*} Making the change of variables \(y \to y_\ast\), where \(y_\ast\) denotes the reflection of \(y\) across the plane \(\{x_1=0\}\), then writing \(w=w^+-w^-\) gives \begin{align*} \iint_{\mathbb{R}^n\times \mathbb{R}^n} \frac{v(x)(v(y)-w(y))}{\vert x- y \vert^{n+2s} } \mathop{}\!d x \mathop{}\!d y &=\iint_{\mathbb{R}^n_+\times \mathbb{R}^n_+} \frac{w^+(x)w^-(y)}{\vert x- y \vert^{n+2s} } \mathop{}\!d x \mathop{}\!d y+\iint_{\mathbb{R}^n_+\times \mathbb{R}^n_+} \frac{w^+(x)(w(y)+2\ell)}{\vert x- y_\ast \vert^{n+2s} } \mathop{}\!d x \mathop{}\!d y \\ &= \iint_{\mathbb{R}^n_+\times \mathbb{R}^n_+} w^+(x)w^-(y) \bigg ( \frac1{\vert x- y \vert^{n+2s}}-\frac1{\vert x- y_\ast \vert^{n+2s}} \bigg )\mathop{}\!d x \mathop{}\!d y \\ & \qquad + \iint_{\mathbb{R}^n_+\times \mathbb{R}^n_+} \frac{w^+(x)(w^+(y)+2\ell)}{\vert x- y_\ast \vert^{n+2s} } \mathop{}\!d x \mathop{}\!d y \\ &\geqslant 0. \end{align*} Thus, \begin{align} \mathcal E(u,v) \geqslant \mathcal E (v,v) \geqslant 0 . \label{vndMm} \end{align} Combining \eqref{FQBj4} and \eqref{vndMm} gives that \(\mathcal E(v,v) =0\) which implies that \(v\) is a constant in \(\mathbb{R}^n\). Since \(v=0\) in \(\mathbb{R}^n_- \), we must have that \(v=0\) as required. \end{proof}
\begin{remark} It follows directly from Proposition \ref{aKht4} that if \(u\) satisfies \((-\Delta)^s u +cu =0\) in \(\Omega\) then \begin{align*} \sup_{\mathbb{R}^n_+} \vert u \vert \leqslant \sup_{\mathbb{R}^n_+\setminus \Omega} \vert u \vert . \end{align*} \end{remark}
Next, we prove the strong maximum principle for antisymmetric functions. This result is not new in the literature, see \cite[Corollary 3.4]{MR3395749}; however, Proposition \ref{fyoaW} provides an alternate elementary proof.
\begin{prop}[A strong maximum principle] \label{fyoaW} Let \(\Omega\subset \mathbb{R}^n_+\) and \(c :\Omega \to \mathbb{R}\). Suppose that \(u :\mathbb{R}^n \to \mathbb{R}\) is \(x_1\)-antisymmetric and \((-\Delta)^su\) is defined pointwise in \(\Omega\). If \(u\) satisfies \((-\Delta)^su+cu \geqslant 0\) in \(\Omega\), and \(u\geqslant 0\) in \(\mathbb{R}^n_+\) then either \(u>0\) in \(\Omega\) or \(u\) is zero almost everywhere in \(\mathbb{R}^n\). \end{prop}
\begin{remark}Note that in Proposition~\ref{fyoaW}, as in \cite[Corollary 3.4]{MR3395749}, no sign assumption is required on \(c\). The reason Proposition~\ref{fyoaW} holds without this assumption is due to the sign assumption on \(u\) which is not usually present in the statement of the strong maximum principle. \end{remark}
\begin{proof}[Proof of Proposition~\ref{fyoaW}] Due to the antisymmetry of \(u\), by a change of variables we may write \begin{align} \begin{aligned} (-\Delta)^s u(x) &= c_{n,s} \operatorname{P.V.} \int_{\mathbb{R}^n_+} \bigg ( \frac 1 {\vert x - y \vert^{n+2s}} - \frac 1 {\vert x_\ast - y \vert^{n+2s}} \bigg ) (u(x)-u(y)) \mathop{}\!d y \\ &\qquad \qquad + 2c_{n,s}\int_{\mathbb{R}^n_+} \frac{u(x)}{\vert x_\ast - y \vert^{n+2s}} \mathop{}\!d y \end{aligned} \label{u9I7E} \end{align} where \(x_\ast=x-2x_1e_1\) is the reflection of \(x\) across \(\{x_1=0\}\). Suppose that there exists \(x_\star\in \Omega\) such that \(u(x_\star)=0\). On one hand, \eqref{u9I7E} implies that \begin{align} (-\Delta)^s u(x_\star) &= -c_{n,s} \operatorname{P.V.} \int_{\mathbb{R}^n_+} \bigg ( \frac 1 {\vert x_\star - y \vert^{n+2s}} - \frac 1 {\vert (x_\star)_\ast - y \vert^{n+2s}} \bigg )u(y) \mathop{}\!d y \label{XXP2e} \end{align} where we used that \(u\) is antisymmetric. Since \begin{align*} \frac 1 {\vert x_\ast - y \vert^{n+2s}} < \frac 1 {\vert x - y \vert^{n+2s}}, \qquad \text{for all } x,y\in \mathbb{R}^n_+, \, x\neq y, \end{align*} equation \eqref{XXP2e} implies that \((-\Delta)^s u(x_\star) \leqslant 0\) with equality if and only if \(u\) is identically zero almost everywhere in \(\mathbb{R}^n_+\). On the other hand, \((-\Delta)^su(x_\star) = (-\Delta)^s u(x_\star) +c(x_\star) u(x_\star) \geqslant 0 \). Hence, \((-\Delta)^su(x_\star)=0\) so \(u\) is identically zero a.e. in \(\mathbb{R}^n_+\). \end{proof}
\subsection{Counter-examples} The purpose of this subsection is to provide counter-examples to the classical Harnack inequality and the strong maximum principle for antisymmetric \(s\)-harmonic functions. A useful tool to construct such functions is the following antisymmetric analogue of Theorem 1 in \cite{MR3626547}. This proves that `all antisymmetric functions are locally antisymmetric and \(s\)-harmonic up to a small error'.
\begin{proof}[Proof of Theorem \ref{u7dPW}]
Due to \cite[Theorem 1.1]{MR3626547} there exist \(R>1\) and \(v\in H^s (\mathbb{R}^n) \cap C^s(\mathbb{R}^n)\) such that \((-\Delta)^sv=0\) in \(B_1\), \(v=0\) in \(\mathbb{R}^n \setminus B_R\), and \( \| v - f\|_{C^k(B_1)} < \varepsilon \). In fact, by a mollification argument, we may take \(v\) to be smooth, see Remark \ref{BQkXm} here below.
Let \(x_\ast\) denote the reflection of \(x\) across \( \{ x_1=0 \}\), and \begin{align*} u(x) = \frac12 \big ( v(x) - v(x_\ast) \big ) . \end{align*} It is easy to verify that \((-\Delta)^s u = 0\) in \(B_1\) (and of course \(u=0\) in \(\mathbb{R}^n \setminus B_R\)). By writing \begin{align*} u(x) -f(x) &= \frac12 \big (v(x) - f(x) \big ) + \frac12 \big ( f(x_\ast) - v(x_\ast) \big ) \qquad\text{for all } x \in B_1 \end{align*} we also obtain \begin{equation*}
\| u - f\|_{C^k(B_1)} \leqslant \| v - f\|_{C^k(B_1)} < \varepsilon . \qedhere \end{equation*} \end{proof}
\begin{remark} \label{BQkXm}
Fix \(\varepsilon>0\). If \(f \in C^k (\overline{B_1} )\) there exists \(\mu >0\) such that \(f\in C^k(B_{1+2\mu})\). Rescaling then applying \cite[Theorem 1.1]{MR3626547}, there exist \(\tilde R>1+\mu\) and \(\tilde v\in H^s (\mathbb{R}^n) \cap C^s(\mathbb{R}^n)\) such that \((-\Delta)^s \tilde v=0\) in \(B_{1+\mu}\), \(\tilde v=0\) in \(\mathbb{R}^n \setminus B_{\tilde R }\), and \( \| \tilde v - f\|_{C^k(B_{1+\mu})} < \varepsilon/2 \).
Let \(\eta \in C^\infty_0(\mathbb{R}^n)\) be the standard mollifier \begin{align*} \eta (x) &= \begin{cases} C e^{- \frac 1 {\vert x \vert^2-1}} , &\text{if } x \in B_1 ,\\ 0, &\text{if } x \in \mathbb{R}^n \setminus B_1, \end{cases} \end{align*} with \(C>0\) chosen so that \(\int_{\mathbb{R}^n} \eta(x) \mathop{}\!d x =1\). For each \(\delta>0\), let \(\eta_\delta(x) = \delta^{-n} \eta (x/\delta))\) and \begin{align*} \tilde{v}^{(\delta)}(x) := (\tilde v \ast \eta_\delta)(x) = \int_{\mathbb{R}^n} \eta_\delta (x-y) \tilde v (y) \mathop{}\!d y. \end{align*} By the properties of mollifiers, \begin{align*}
\| \tilde v^{(\delta)} - f \|_{C^k(B_1)} \leqslant \| \tilde v^{(\delta)} - \tilde v \|_{C^k(B_1)} + \| \tilde v - f \|_{C^k(B_1)} < \varepsilon \end{align*} provided \(\delta\) is sufficiently small. Since \(\tilde v \in H^s(\mathbb{R}^n) \subset L^2(\mathbb{R}^n)\), it follows from \cite[Proposition 4.18]{MR2759829} that \(\operatorname{supp} \tilde{v}^{(\delta)}(x) \subset \overline{B_{\tilde{R}+\delta}}\). Moreover, via the Fourier transform, \((-\Delta)^s \tilde{v}^{(\delta)} = \big ( (-\Delta)^s \tilde{v} \big ) \ast \eta_\delta\), so \cite[Proposition 4.18]{MR2759829} again implies that \( \operatorname{supp} \big ( (-\Delta)^s \tilde{v}^{(\delta)}\big ) \subset \mathbb{R}^n \setminus B_{1+\mu-\delta}\).
Hence, setting \(R=\tilde R+\delta\), \(v=\tilde v^{(\delta)}\), we have constructed a function \(v \in C^\infty_0(\mathbb{R}^n)\) such that \(v\) is \(s\)-harmonic in \(B_1\), \(v=0\) outside \(B_R\) and \(\| v - f \|_{C^k(B_1)}<\varepsilon\). Moreover, $$ v(x_\ast) = \int_{\mathbb{R}^n} \eta_\delta (y) \tilde v (x_\ast- y) \mathop{}\!d y =-\int_{\mathbb{R}^n} \eta_\delta (y) \tilde v (x- y_\ast) \mathop{}\!d y =-v(x)$$ via the change of variables \(z=y_\ast\) and using that \(\eta_\delta(y_\ast) = \eta_\delta (y)\). \end{remark}
It is well known that the classical Harnack inequality fails for \(s\)-harmonic functions, see \cite{Kassmann2007clas} for a counter-example. The counter-example provided in \cite{Kassmann2007clas} is not antisymmetric; however, Corollary \ref{X96XH} proves that even with this extra symmetry the classical Harnack inequality does not hold for \(s\)-harmonic functions. Moreover, the construction of the counter-example in Corollary \ref{X96XH} is entirely different to the one in \cite{Kassmann2007clas}.
\begin{proof}[Proof of Corollary \ref{X96XH}] We will begin by proving the case \(n=1\). Suppose that \(\varepsilon\in (0,1)\), \(I'=(1,2)\), and \(I=(1/2,5/2)\). Let \(\{ f^{(\varepsilon)}\} \subset C^\infty (\mathbb{R})\) be a family of odd functions that depend smoothly on the parameter \(\varepsilon\). Moreover, suppose that \begin{align}
\sup_{I'} f^{(\varepsilon)} = 4 \label{UJd6j} \end{align} and \begin{align} \inf_{I} f^{(\varepsilon)} =\inf_{I'} f^{(\varepsilon)} =2\varepsilon . \label{BR8WJ} \end{align} For example, such a family of functions is \begin{align*}
f^{(\varepsilon)}(x) = ax +bx^3+cx^5 +dx^7
\end{align*} where \begin{align*}
a = \frac 5{54} (64+5\varepsilon), \quad b= -\frac 1{72}(128+73\varepsilon), \quad c=\frac 1 {36} (-8+23\varepsilon), \quad d=\frac 1 {216}(16-19\varepsilon).
\end{align*} The functions \(f^{(\varepsilon)}\) are plotted for several values of \(\varepsilon\) in Figure \ref{2FAHv}.
After a rescaling, it follows from Theorem \ref{u7dPW} that there exists a family \(v^{(\varepsilon)}\) of odd functions that are \(s\)-harmonic in \((-5/2,5/2)\supset I \) and satisfy \begin{align}
\| v^{(\varepsilon)} - f^{(\varepsilon)} \|_{C((-5/2,5/2))} < \varepsilon . \label{pA7Na}
\end{align} It follows from \eqref{BR8WJ} and \eqref{pA7Na} that \begin{align*} \inf_{I} v^{\varepsilon} \geqslant \varepsilon>0 . \end{align*} Moreover, from \eqref{UJd6j}-\eqref{BR8WJ} and \eqref{pA7Na} , we have that \begin{align*} \sup_{I'} v^{(\varepsilon)}\geqslant 4-\varepsilon \qquad \text{and} \qquad \inf_{ I'} v^{(\varepsilon)}\leqslant 3\varepsilon . \end{align*} Hence, \begin{align*} \frac{\sup_{I'} v^{(\varepsilon)}}{\inf_{I'} v^{(\varepsilon)}} \geqslant \frac{4-\varepsilon}{3\varepsilon} \to + \infty \end{align*} as \(\varepsilon \to 0^+\). Setting \(\Omega=I\) and \(\Omega’=I’\) proves the statement in the case \(n=1\).
To obtain the case \(n >1\), set \(\Omega'=I'\times (-1,1)^{n-1}\), \(\Omega=I \times (-2,2)^{n-1}\), and \(u^{(\varepsilon)} (x) = v^{(\varepsilon)} (x_1) \). Using that \begin{align*} (-\Delta)^s u^{(\varepsilon)} (x) = C (-\Delta)^s_{\mathbb{R}} v^{(\varepsilon)} (x_1) \qquad \text{for all } x\in \mathbb{R}^n \end{align*}where \((-\Delta)^s_{\mathbb{R}} \) denotes the fractional Laplacian in one dimension and \(C\) is some constant, see \cite[Lemma 2.1]{MR3536990}, all of the properties of \(v^{(\varepsilon)} \) carry directly over to \(u^{(\varepsilon)} \).
\end{proof}
\begin{figure}
\caption{Plot of \(f^{(\varepsilon)}(x) \) for \(\varepsilon=0.2,0.4,0.6,0.8\).}
\label{2FAHv}
\end{figure}
Observe that the family of functions in the proof of Corollary \ref{X96XH} does not violate the classical strong maximum principle. In our next result, Corollary \ref{KTBle}, we use Theorem \ref{u7dPW} to provide a counter-example to the classical strong maximum principle for antisymmetric \(s\)-harmonic functions. This is slightly more delicate to construct than the counter-example in Corollary \ref{X96XH}. Indeed, Theorem \ref{u7dPW} only gives us an \(s\)-harmonic function \(u\) that is \(\varepsilon\)-close to a given function \(f\), so how can we guarantee that \(u=0\) at some point but \(u\) is not negative?
The idea is to begin with an antisymmetric function \(f\) that has a non-zero minimum away from zero, see Figure \ref{73Iyk}, use Theorem \ref{u7dPW} to get an antisymmetric \(s\)-harmonic function with analogous properties then minus off a known antisymmetric \(s\)-harmonic function until the function touches zero. The proof of Corollary \ref{KTBle} relies on two technical results, Lemma \ref{IyXUO} and Lemma \ref{oujju}, both of which are included in Appendix \ref{ltalz}.
\begin{proof}[Proof of Corollary \ref{KTBle}] By the same argument at the end of the proof of Corollary~\ref{X96XH}, it is enough to take \(n=1\). Let \(\Omega=(0,3)\) and \(f :\mathbb{R} \to \mathbb{R}\) be a smooth, odd function such that \begin{align} f(x) &\geqslant 1, \text{ for all } x\in [1,3]; \label{cv7Fh}\\ f(x) &\geqslant 3x,\text{ for all } x\in [0,1]; \label{iHz0V}\\ f(2) & =1; \text{ and } \label{QEb8m} \\ f(3)&= 5. \label{ugncP} \end{align} For example, such a function is \begin{align*} f(x) = -\frac{371 }{43200}x^9+\frac{167 }{1440}x^7-\frac{2681 }{14400}x^5-\frac{4193 }{2160}x^3+\frac{301 }{50} x. \end{align*} See Figure \ref{73Iyk} for a plot of \(f\).
By Theorem \ref{u7dPW} with \(\varepsilon =1\), there exists an odd function \(v \in C^\infty_0(\mathbb{R})\) such that \(v\) is \(s\)-harmonic in~\((-4,4)\) and \begin{align}
\| v-f\|_{C^1((-4,4))} < 1 . \label{wjbkE} \end{align} Let \(c_0,\zeta_R\) be as in Lemma \ref{oujju} and set \(\zeta := \frac 1 {c_0} \zeta_R. \) By choosing \(R>3\) sufficiently large, we have that \begin{align} \frac 3 4 x \leqslant \zeta (x) \leqslant \frac 5 4 x \qquad \text{in }(0,4) . \label{rU1A4} \end{align} Define \begin{align*} \phi_t(x) = v(x) - t \zeta (x), \qquad {\mbox{for all~$x\in(-3,3)$ and~$t >0$.}} \end{align*} We have that, for all \(t>0\), \begin{align*} (-\Delta)^s \phi_t(x) = 0 \qquad \text{in } (-3,3) \end{align*} where \((-\Delta)^s=(-\Delta)^s_x \) is the fractional Laplacian with respect to \(x\), and \begin{align*} x \mapsto \phi_t(x) \text{ is odd.} \end{align*} As in Lemma \ref{IyXUO}, let \begin{align*} m(t) = \min_{x \in [1,3]} \phi_t(x) . \end{align*} From \eqref{cv7Fh} and \eqref{wjbkE} we have that \( \phi_0(x) = v(x) > 0\) in \([1,3]\). Hence, \begin{align*} m(0) >0. \end{align*} Moreover, by \eqref{QEb8m} and \eqref{wjbkE}, \(v(2)<2\). It follows from \eqref{rU1A4} that \begin{align*} m(4/3) \leqslant \phi_{4/3}(2)=v(2) - \frac 4 3 \zeta (2) <0. \end{align*} Since \(m\) is continuous, as shown in Lemma \ref{IyXUO}, the intermediate value theorem implies the existence of \(t_\star \in (0,4/3)\) such that \begin{align*} m(t_\star) = 0. \end{align*} Let \(u := \phi_{t_\star}\). By construction \(u \geqslant 0\) in \([1,3]\). Moreover, since \(u\) is continuous, there exists some \(x_\star \in [1,3]\) such that \(u(x_\star)=0\). In fact, we have that \(x_\star \neq 3\). Indeed, \eqref{ugncP} implies that \(v(3)>4\) and, since \(t_\star < 4/3\) and \(\zeta (3) <9/4\), we obtain \(u(3) >0\).
All that is left to be shown is that \(u \geqslant 0\) in \((0,1)\). By \eqref{wjbkE}, we have that \begin{align*} v'(x) > f'(x) - 1, \qquad {\mbox{for all }} x \in (-4,4). \end{align*} Since \(v(0)=f(0)=0\), it follows from the fundamental theorem of calculus that \begin{equation}\begin{split} u(x) &= \int_0^x v'(\tau ) \mathop{}\!d \tau -t_\star \zeta (x) \\ &\geqslant \int_0^x \big ( f'(\tau)-1 \big ) \mathop{}\!d \tau - t_\star \zeta (x) \\ &=f(x)-x - t_\star \zeta (x) \label{A0ZNc} \end{split}\end{equation} provided that~\(x\in (0,3)\). By \eqref{A0ZNc}, \eqref{iHz0V} and \eqref{rU1A4}, \begin{align*} u(x) \geqslant \frac 1 3 x \geqslant 0 \qquad \text{in } [0,1] , \end{align*} as desired. \end{proof}
\begin{figure}
\caption{Plot of \(f \) as in the proof of Corollary \ref{KTBle}.}
\label{73Iyk}
\end{figure}
\section{A Hopf lemma} \label{R9GxY} In this section, we prove Proposition~\ref{lem:FLIFu}. The main step in the proof is the construction of the following barrier.
\begin{lem} \label{SaBD4} Suppose that \(c \in L^\infty (B_2^+)\). Then there exists an \(x_1\)-antisymmetric function \(\varphi \in C^\infty(\mathbb{R}^n)\) such that \begin{align*} \begin{PDE} (-\Delta)^s \varphi +c \varphi &\leqslant 0 &\text{in }B_2^+\setminus B_{1/2}(e_1), \\ \varphi &=0 &\text{in } \mathbb{R}^n_+ \setminus B_2^+ ,\\ \varphi &\leqslant 1 &\text{in } B_{1/2}(e_1),\\ \partial_1 \varphi (0) &>0 . \end{PDE} \end{align*} \end{lem}
\begin{proof} Let \(\zeta\) be a smooth \(x_1\)-antisymmetric cut-off function such that the support of \(\zeta\) is contained in \(B_2\), \(\zeta \geqslant 0\) in \(B_2^+\), and \(\partial_1\zeta (0)>0\). For example, such a function is \( x_1\eta(x)\) where \(\eta\) is the standard mollifier defined in Remark \ref{BQkXm}. Since \(\zeta\) is smooth with compact support, we have that~\((-\Delta)^s \zeta \in C^\infty (\mathbb{R}^n)\).
Moreover, \((-\Delta)^s \zeta \) is \(x_1\)-antisymmetric since \(\zeta\) is \(x_1\)-antisymmetric, so it follows that there exists~\(C>0\) such that \begin{align}
(-\Delta)^s \zeta (x) +c \zeta (x) \leqslant C(1+\| c \|_{L^\infty(B_2^+)}) x_1 \qquad \text{in } B_2^+ \label{I6ftM} \end{align} using that \(c\in L^\infty(B_2^+)\).
Next, let \(\tilde \zeta \) be a smooth \(x_1\)-antisymmetric function such that \(\tilde \zeta \equiv 1\) in \(B_{1/4}(e_1)\), \(\tilde \zeta \equiv 0\) in \(\mathbb{R}^n_+ \setminus B_{3/8}(e_1)\), and \(0\leqslant \tilde \zeta \leqslant 1\) in \(\mathbb{R}^n_+\). Recall that, given an \(x_1\)-antisymmetric function \(u\), the fractional Laplacian of \(u\) can be written as \begin{align*} (-\Delta)^su(x) &= c_{n,s} \int_{\mathbb{R}^n_+} \bigg ( \frac 1 {\vert x - y\vert^{n+2s}} - \frac 1 {\vert x_\ast -y\vert^{n+2s}} \bigg ) (u(x)-u(y) ) \mathop{}\!d y + 2 c_{n,s} u(x) \int_{\mathbb{R}^n_+} \frac{\mathop{}\!d y } {\vert x_\ast - y \vert^{n+2s}}. \end{align*} Hence, for each \(x\in B_2^+ \setminus B_{1/2}(e_1)\), \begin{align*} (-\Delta)^s \tilde \zeta (x)&=-c_{n,s} \int_{B_{3/8}(e_1)} \bigg ( \frac 1 {\vert x - y \vert^{n+2s}} - \frac 1 {\vert x_\ast - y \vert^{n+2s}} \bigg ) \tilde \zeta (y) \mathop{}\!d y. \end{align*} By the fundamental theorem of calculus, for all \(x\in B_2^+ \setminus B_{1/2}(e_1)\) and~\(y\in B_{3/8}(e_1)\),\begin{align*} \frac 1 {\vert x - y \vert^{n+2s}} - \frac 1 {\vert x_\ast - y \vert^{n+2s}} &= \frac{n+2s}{2} \int_{\vert x - y \vert^2}^{\vert x_\ast - y \vert^2} \frac{ \mathop{}\!d \tau } {\tau^{\frac{n+2s+2}{2}}}\\ &\geqslant \frac{Cx_1y_1}{\vert x_\ast - y \vert^{n+2s+2}} \\ &\geqslant Cx_1 \end{align*} with \(C\) depending only on \(n\) and \(s\). Hence, \( (-\Delta)^s\tilde \zeta (x) \leqslant -Cx_1\) in \( B_2^+ \setminus B_{1/2}(e_1)\). Then the required function is given by \(\varphi(x) := \zeta (x) + \alpha \tilde \zeta (x)\) for all \(x\in \mathbb{R}^n\) with \(\alpha>0\) to be chosen later. Indeed, from~\eqref{I6ftM}, we have that \begin{align*}
(-\Delta)^s \varphi +c \varphi \leqslant C (1-\alpha+ \| c \|_{L^\infty(B_2^+)})x_1 \leqslant 0 \qquad\text{in } B_2^+ \setminus B_{1/2}(e_1) \end{align*} provided that \(\alpha \) is large enough. \end{proof}
{F}rom Lemma \ref{SaBD4} the proof of Proposition~\ref{lem:FLIFu} follows easily:
\begin{proof}[Proof of Proposition~\ref{lem:FLIFu}] Since \(u >0\) in \(B_1^+\), we have that, for all \(v\in \mathcal H^s_0(B_1^+)\) with~\(v\geqslant0\), \begin{align*} 0\leqslant \mathcal E(u,v) + \int_{B_1^+} c(x) u(x) v(x) \mathop{}\!d x \leqslant\mathcal E(u,v) + \int_{B_1^+} c^+(x) u(x) v(x) \mathop{}\!d x \end{align*} and thereofore \begin{align*} (-\Delta)^su +c^+u \geqslant 0 \qquad \text{in } B_1^+. \end{align*} Hence, it suffices to prove Proposition~\ref{lem:FLIFu} with \(c \geqslant 0\). Let \(\rho>0\) be such that \(B_{2\rho} \subset B_1\) and \(\varphi_\rho (x) = \varphi(x/\rho)\) where \(\varphi\) is as in Lemma \ref{SaBD4}. Provided that \(\varepsilon\) is sufficiently small, we have \begin{align*} (-\Delta)^s(u-\varepsilon \varphi_\rho) +c(u-\varepsilon \varphi_\rho)\geqslant 0 \qquad \text{in } B_{2\rho} \setminus B_{\rho/2}(\rho e_1) \end{align*} and \(u-\varepsilon \varphi_\rho\geqslant 0\) in \((\mathbb{R}^n_+\setminus B_{2\rho})\cup B_{\rho/2}(\rho e_1)\). It follows from Proposition~\ref{aKht4} that \begin{align*} u \geqslant \varepsilon \varphi_\rho \qquad \text{in } \mathbb{R}^n_+ \end{align*} where we used that \(c\geqslant0\). Since \(u(0)=\varphi_\rho(0)=0\), we conclude that \begin{align*} \liminf_{h\to 0} \frac{u(he_1)}{h} \geqslant \varepsilon \partial_1 \varphi_\rho(0) >0, \end{align*} as desired. \end{proof}
\section{Symmetry for the semilinear fractional parallel surface problem}\label{Svlif}
In this section, we will give the proof of Theorem \ref{CccFw}. For simplicity and the convenience of the reader, we will first state the particular case of Proposition~3.1 in \cite{MR3395749}, which we make use of several times in the proof of Theorem \ref{CccFw}. Note that \(c\) in Proposition \ref{QawmgWdG} corresponds to \(-c\) in \cite{MR3395749}---we made this change so that the notation of Proposition \ref{QawmgWdG} would agree with the notation in the proof of Theorem \ref{CccFw}.
\begin{prop}[Proposition~3.1 in \cite{MR3395749}] \label{QawmgWdG} Let \(H\) be a halfspace, \(\Omega \subset H\) be any open, bounded set, and \(c\in L^\infty (\Omega)\) be such that \(-c \leqslant c_\infty<\lambda_1(\Omega)\) in \(\Omega\) for some \(c_\infty \geqslant 0\), where~\(\lambda_1(\Omega)\) is the first Dirichlet eigenvalue of \((-\Delta)^s\) in \(\Omega\). If \(u \in H^s(\mathbb{R}^n)\) satisfies \((-\Delta)^s u +cu \geqslant 0\) in \(\Omega\) and \(u\) is antisymmetric in \(\mathbb{R}^n\) then \(u \geqslant 0\) almost everywhere in \(\Omega\). \end{prop}
Now we will prove Theorem \ref{CccFw} in the case \(n\geqslant2\). We prove the case \(n=1\) later in the section.
\begin{proof}[Proof of Theorem \ref{CccFw} for \(n\geqslant2\)] Fix a direction \( e \in \mathbb{S}^{n-1}\). Without loss of generality, we may assume that~\(e=e_1\). Let \(T_\lambda = \{x\in \mathbb{R}^n \text{ s.t. } x_1 = \lambda\}\)---this will be our `moving plane' which we will vary by decreasing the value of \(\lambda\). Since \(\Omega\) is bounded, we may let \(M = \sup_{x\in \Omega}x_1\) which is the first value of \(\lambda\) for which \(T_\lambda\) intersects \(\overline{\Omega}\). Moreover, let \(H_\lambda = \{ x \in \mathbb{R}^n \text{ s.t. } x_1 >\lambda\}\), \(\Omega_\lambda = H_\lambda \cap \Omega\), and \(Q_\lambda:\mathbb{R}^n \to \mathbb{R}^n\) be given by \(x \mapsto x- 2x_1+2\lambda e_1\). Geometrically, \(Q_\lambda(x)\) is the reflection of \(x\) across the hyperplane~\(T_\lambda\).
Since \(\partial \Omega \) is \(C^1\) there exists some \(\mu <M\) such that \(\Omega_\lambda' := Q_\lambda(\Omega_\lambda) \subset \Omega\) for all \(\lambda\in (\mu,M)\), see for example \cite{MR1751289}. Let \(m\) be the smallest such \(\mu\), that is, let \begin{align*} m = \inf \{ \mu<M \text{ s.t. } \Omega_\lambda' \subset \Omega \text{ for all } \lambda \in (\mu,M)\}. \end{align*} Note that \(\Omega_m' \subset \Omega\). Indeed, if this were not the case then there would exist some \(\varepsilon>0\) such that \(\Omega_{m+\varepsilon}' \not\subset \Omega\), which would contradict the definition of~\(m\).
As is standard in the method of moving planes, we will consider the function \begin{align*} v_\lambda (x) &= u(x) - u(Q_\lambda(x)), \qquad x\in \mathbb{R}^n \end{align*} for each \(\lambda \in [m,M)\). It follows that \(v_\lambda\) is antisymmetric and satisfies \begin{align*} \begin{PDE} (-\Delta)^s v_\lambda +c_\lambda v_\lambda &= 0 &\text{in } \Omega_\lambda' ,\\ v_\lambda &\geqslant 0 &\text{in } H_\lambda'\setminus \Omega_\lambda' , \end{PDE} \end{align*} where \begin{align*} c_\lambda (x) &= \begin{cases} \frac{f((u(x))-f(u(Q_\lambda(x)))}{u(x) - u(Q_\lambda(x))}, &\text{if } u(x) \neq u(Q_\lambda(x)), \\ 0,& \text{if } u(x) = u(Q_\lambda(x)), \end{cases} \end{align*} and \(H_\lambda' := Q_\lambda (H_\lambda )\). We claim that \begin{align}
v_m \equiv 0 \qquad \text{in } \mathbb{R}^n. \label{VxU8s} \end{align}
Before we show \eqref{VxU8s}, let us first prove the weaker statement \begin{align} v_m \geqslant 0 \qquad \text{in } H_m' . \label{g3zkO} \end{align} Since \(f \in C^{0,1}_{\textrm{loc}} (\mathbb{R})\), it follows that \(c_\lambda \in L^\infty (\Omega_\lambda')\) and that \begin{align*}
\|c_\lambda \|_{L^\infty (\Omega_\lambda')} \leqslant [ f ]_{C^{0,1}([0,\| u \|_{L^\infty(\Omega)}])} . \end{align*} Here, as usual, \begin{align*} [ f ]_{C^{0,1}([0,a])} &= \sup_{\substack{x,y\in [0,a]\\x\neq y} } \frac{\vert f(x)-f(y)\vert }{\vert x - y \vert}.
\end{align*} We cannot directly apply Proposition~\ref{QawmgWdG} as \([ f ]_{C^{0,1}([0,\| u \|_{L^\infty(\Omega)}])}\) might be large. However,
by Proposition~\ref{ayQzh} we have that \begin{align*} \lambda_1 (\Omega_\lambda' ) \geqslant C \vert \Omega_\lambda' \vert^{-\frac{2s}n} \to \infty \quad \text{ as } \, \lambda \to M^- , \end{align*}
and since \(\|c_\lambda \|_{L^\infty (\Omega_\lambda')}\) is uniformly bounded with respect to \(\lambda\), Proposition~\ref{QawmgWdG} implies that there exists some \(\mu \in [m,M)\) such that \(v_\lambda \geqslant 0\) in \(\Omega_\lambda'\) for all \(\lambda \in [\mu,M)\). In fact, since \(u\) is not identically zero, after possibly increasing the value of \(\mu\) (still with \(\mu<M\)) we claim that \(v_\lambda > 0\) in \(\Omega_\lambda'\) for all \(\lambda \in [\mu,M)\). Indeed, there exists \(x_0\in \Omega\) such that \(u(x_0)>0\) so, provided \(\mu\) is close to \(M\), \(Q_\lambda (x_0) \not\in \Omega\). Then, for all \(\lambda \in [\mu,M)\), \begin{align*} v_\lambda (x_0) = u(x_0) >0 \end{align*} so it follows from the strong maximum principle Proposition~\ref{fyoaW} that \(v_\lambda>0\) in \(\Omega_\lambda'\).
This allows us to define \begin{align*} \tilde{m} =\inf \{ \mu\in [m,M) \text{ s.t. } v_\lambda > 0 \text{ in } \Omega_\lambda' \text{ for all } \lambda \in [\mu,M)\}. \end{align*} We claim that \(\tilde{m} = m\). For the sake of contradiction, suppose that we have \(\tilde{m} >m\). By continuity in \(\lambda\), \(v_{\tilde{m}} \geqslant 0\) in \(H_{\tilde{m}}'\) and then, as above, Proposition~\ref{fyoaW} implies that \(v_{\tilde{m}} >0\) in \(\Omega_{\tilde{m}}'\). Due to the definition of \(\tilde{m}\), for all \(0<\varepsilon < \tilde{m}-m\), the set \(\{ v_{\tilde m-\varepsilon} \leqslant 0\} \cap \Omega_{\tilde m-\varepsilon}' \) is non-empty. Let \(\Pi_\varepsilon \subset\Omega_{\tilde m-\varepsilon}'\) be an open set such that \( \{ v_{\tilde m-\varepsilon} \leqslant 0\} \cap \Omega_{\tilde m-\varepsilon}' \subset \Pi_\varepsilon\). By making \(\varepsilon\) smaller we may choose \(\Pi_\varepsilon\) such that \(\vert \Pi_\varepsilon\vert \) is arbitrarily close to zero. Hence, applying Proposition~\ref{QawmgWdG} then Proposition~\ref{fyoaW} gives that \(v_{\tilde m -\varepsilon} >0\) in \(\Pi_\varepsilon\) which is a contradiction. This proves \eqref{g3zkO}.
Since \(v_m \geqslant 0\) in \(H_m'\), Proposition~\ref{fyoaW} implies that either \eqref{VxU8s} holds or \(v_m>0\) in \(\Omega_m'\). For the sake of contradiction, let us suppose that \begin{align} v_m>0 \qquad \text{in } \Omega_m'. \label{JQNtW} \end{align} By definition, \(m\) is the minimum value of \(\lambda\) for which \(\Omega_\lambda' \subset \Omega\). There are only two possible cases that can occur at this point.
\begin{quotation} Case 1: There exists \(p \in ( \Omega_m' \cap \partial \Omega) \setminus T_m\).\\ Case 2: There exists \(p \in T_m \cap \partial \Omega\) such that \(e_1\) is tangent to \(\partial \Omega\) at \(p\). \end{quotation}
If we are in Case 1 then there is a corresponding point \(q\in ( G_m' \cap \partial G) \setminus T_m \subset \Omega_m'\) since \(\partial G\) is parallel to \(\partial \Omega\). But \(u\) is constant on \(\partial G\), so we have \begin{align*} v_m(q) = u(q) - u(Q_m (q)) = 0 \end{align*} which contradicts \eqref{JQNtW}.
If we are in Case 2 then there exists \(q \in T_m \cap \partial G \) such that \(e_1\) is tangent to \(\partial G\) at \(q\). Since \(u\) is a constant on \(\partial G\), the gradient of \(u\) is perpendicular to \(\partial G\) and so \(\partial_1 u(q) = 0. \) Moreover, by the chain rule, \begin{align*} \partial_1 (u\circ Q_m)(q) = - \partial_1u(q) = 0. \end{align*} Hence, \begin{align*} \partial_1 v_m (q) = 0. \end{align*} However, this contradicts Proposition~\ref{lem:FLIFu}. Thus, in both Case 1 and Case 2 we have shown that the assumption \eqref{JQNtW} leads to a contradiction, so we conclude that \eqref{VxU8s} is true.
This concludes the main part of the proof. The final two steps are to show that \(u\) is radially symmetric and that \(\operatorname{supp} u = \overline{\Omega}\). Due to the previous arguments we know that for all \(e\in \mathbb{S}^{n-1}\) and \(\lambda \in [m(e), M(e))\), \begin{align*} u(x) - u(Q_{\lambda,e}(x))\geqslant 0 \qquad \text{in } H_{\lambda,e}' \end{align*} where \(Q_{\lambda,e} =Q_{\lambda}\), \(H'_{\lambda,e} =H_{\lambda}' \) as before but we've included the subscript \(e\) to emphasise that the result is true for each \(e\). It follows that for every halfspace we either have \(u(x) - u(Q_{\partial H}(x)) \geqslant 0\) for all \(x\in H\) or \(u(x) - u(Q_{\partial H}(x)) \leqslant 0\) for all \(x\in H\). By \cite[Proposition 2.3]{MR2722502}, we can conclude that there exists some \(z\in \mathbb{R}^n\) such that \(x \mapsto u(x-z)\) is radially symmetric. Moreover, since \(u\geqslant0\) in \(\Omega\) and not identically zero, \cite[Proposition 2.3]{MR2722502} also implies that \(x \mapsto u(x-z)\) is non-increasing in the radial direction.
It follows that \(\operatorname{supp} u\) is a closed ball. (Note that we are using the convention \\\(\operatorname{supp} u = \overline{\{x \in \mathbb{R}^n \text{ s.t. } u(x)\neq 0 \}}\)). We claim that \(\operatorname{supp} u = \overline{\Omega}\). For the sake of contradiction, suppose that this is not the case. Then there exists a direction \(e\in \mathbb{S}^{n-1}\) and \(\lambda \in (m(e),M(e))\) such that \(\Omega'_\lambda \cap \operatorname{supp} u = \varnothing \). It follows that \(v_\lambda \equiv 0\) in \(\Omega_\lambda'\). This is a contradiction since we previously showed that \(v_\lambda>0\) in \( \Omega_\lambda'\) for all \(\lambda \in (m(e),M(e))\). \end{proof}
For $n=1$, Theorem \ref{CccFw} reads as follows.
\begin{thm} Suppose that \(G\) is a bounded open set in \(\mathbb{R}\), \(\Omega=G+B_R\), \(f:\mathbb{R} \to \mathbb{R}\) is locally Lipschitz, and \(c_0\in \mathbb{R}\). Furthermore, assume that there exists a non-negative function \(u \in C^s(\mathbb{R})\) that is not identically zero and satisfies\begin{align} \begin{PDE} (-\Delta)^s u &= f(u) &\text{in } \Omega , \\ u&=0 &\text{in } \mathbb{R} \setminus \Omega , \\ u &= c_0 & \text{on } \partial G . \end{PDE} \end{align} Then, up to a translation, \(u\) is even, \(u>0\) in \(\Omega\), and \(G=(a,b)\) for some \(a<b\). \end{thm}
\begin{proof} In one dimension the moving plane \(T_\lambda\) is just a point \(\lambda\) and there are only two directions \(T_\lambda\) can move--from right to left or from left to right. We will begin by considering the case \(T_\lambda\) is moving from right to left. Let \((a-R, b +R)\), \(a<b\), be the connected component of \(\Omega\) such that \(\sup \Omega=b+R\). Using the same notation as in the proof of Theorem \ref{CccFw}, it is clear that \(M =b+R\) and \(m = \frac{a+b} 2 \). For all \(\lambda \in ( \frac{a+b} 2, b+R)\), let \begin{align*} v_\lambda (x) &= u(x) - u(-x+2 \lambda ), \qquad x\in \mathbb{R}. \end{align*} Arguing as in the proof of Theorem \ref{CccFw}, we obtain \begin{align} v_\lambda &> 0 \qquad \text{in } (-\infty , \lambda) \label{8e2sD} \end{align} for all \(\lambda \in ( \frac{a+b} 2, b+R)\) and \begin{align*} v_{\frac{a+b}2} &\geqslant 0 \qquad \text{in } (-\infty , (a+b)/2). \end{align*} Then the overdetermined condition \(u(a)=u(b)=c_0\) implies that \begin{align*} v_{\frac{a+b}2} (b)&= u(b) - u(a) =0. \end{align*} Hence, Proposition~\ref{fyoaW} gives that \begin{align} v_{\frac{a+b}2} &\equiv 0 \qquad \text{in } \mathbb{R}, \label{os0w4} \end{align}that is, \(u\) is an even function about the point \(\frac{a+b}2\). Moreover, \eqref{os0w4} implies that \(u\equiv 0\) in \(\mathbb{R}\setminus (a-R,b+R)\).
Now suppose that we move \(T_\lambda\) from left to right. If \(\Omega\) is made up of at least two connected components then repeating the argument above we must also have that \(u \equiv 0\) in \((a-R,b+R)\). However, \(u\) is not identically zero, so \(\Omega\) can only have one connected component. Finally, \eqref{8e2sD} implies that \(u\) is strictly monotone in \((\frac{a+b} 2 , b+R)\) which further implies the positivity of \(u\). \end{proof}
\appendix
\section{Technical lemmas} \label{ltalz}
In this appendix, we list several lemmas that are used throughout the paper.
\begin{lem} \label{IyXUO} Let \(I \subset \mathbb{R}\) and \(\Omega \subset \mathbb{R}^n\) be open and bounded sets in \(\mathbb{R}\) and \( \mathbb{R}^n\) respectively. Suppose that \(\phi_t(x): \overline{\Omega}\times \overline{I} \to \mathbb{R}\) is continuous in \( \overline{\Omega} \times \overline{I} \) and define \begin{align*} m(t) = \min_{x \in \overline{\Omega}} \phi_t(x) . \end{align*} Then \(m \in C(\overline{I})\). \end{lem}
\begin{proof} Since \( \overline{\Omega} \times \overline{I}\) is compact, \(\phi\) is uniformly continuous. Fix \(\varepsilon >0\) and let \((\bar{x},\bar{t}) \in \overline{\Omega} \times \overline{I}\) be arbitrary. There exists some \(\delta>0\) (indepedent of \(\bar{t}\) and \(\bar{x}\)) such that if \((x,t) \in \overline{\Omega} \times \overline{I}\) and \( \abs{(x,t)-(\bar{x},\bar{t})} < \delta \) then \begin{align*} \abs{\phi_t(x)-\phi_{\bar{t}}(\bar{x})} < \varepsilon . \end{align*} In particular, we may take \(x=\bar{x}\) to conclude that if \(t \in \overline{I}\) and \(\abs{t-\bar{t}} < \delta\) then \begin{align*} \abs{\phi_t(\bar{x})-\phi_{\bar{t}}(\bar{x})} < \varepsilon \end{align*} for any \(\bar{x} \in \overline{\Omega}\). Consequently, \begin{align*} \phi_t(\bar{x}) > \phi_{\bar{t}}(\bar{x}) - \varepsilon \geqslant m(\bar{t}) -\varepsilon \qquad \text{for all } \bar{x} \in \overline{\Omega} \end{align*} Minimising over \(\bar{x}\) we obtain \begin{align*} m(t) \geqslant m(\bar{t}) -\varepsilon . \end{align*} Similarly, we also have that \begin{align*} m(\bar{t}) \geqslant m(t) -\varepsilon . \end{align*} Thus, we have shown that if \(t \in \overline{I}\) and \(\abs{t-\bar{t} } < \delta\) then \begin{equation*} \abs{m(t) -m(\bar{t} )} \leqslant \varepsilon . \qedhere \end{equation*} \end{proof}
\begin{lem} \label{oujju} Let \(R>0\) and \(\zeta_R: \mathbb{R} \to \mathbb{R} \) be the solution to \begin{align} \begin{PDE} (-\Delta)^s \zeta_R &=0 &\text{in } (-R,R) ,\\ \zeta_R &= g_R &\text{in } \mathbb{R} \setminus (-R,R), \end{PDE} \label{acLUt} \end{align} where \begin{align*} g_R(x) &= \begin{cases} R, &\text{if }x >R ,\\ -R, &\text{if } x<-R . \end{cases} \end{align*} Then \begin{align} x \mapsto \frac{\zeta_R(x)}{x} \qquad \text{is defined at }x=0 \label{JQM6p} \end{align} and there exists a constant \(c_0=c_0(s)>0\) such that, as \(R\to \infty\), \begin{align*} \frac{\zeta_R(x)}{x} \to c_0 \qquad \text{in } C_{\textrm{loc}} (\mathbb{R}). \end{align*} \end{lem}
\begin{proof} By the scale-invariance property of the fractional Laplacian, we may write \begin{align} \zeta_R(x) &= R \zeta_1(x/R) \label{DkRUV} \end{align} where \(\zeta_1\) is the solution to \eqref{acLUt} with \(R=1\). The function \(\zeta_1\) is given explicitly via the Poisson kernel, (for more details see \cite[Section 15]{MR3916700}) \begin{align*} \zeta_1 (x) &= a_s\big ( 1 -x^2 \big)^s \int_{\mathbb{R} \setminus (-1,1)} \frac{g_1(t)}{(t^2-1)^s\vert x-t \vert} \mathop{}\!d t, \qquad x \in (-1,1) . \end{align*} where \(a_s\) is a positive normalisation constant. Using the definition of \(g_1\) and a change of variables, we obtain \begin{align} \zeta_1( x) = 2a_s x\big (1-x^2 \big )^s \int_1^\infty \frac{\mathop{}\!d t }{(t^2-x^2)(t^2-1)^s} . \label{XLPcN} \end{align} Provided \(\vert x\vert <1/2\), \begin{align*} \frac{1}{(t^2-x^2)(t^2-1)^s} \leqslant \frac 1{(t^2-(1/2)^2)(t^2-1)^s} \in L^1((1,\infty)) \end{align*} so by \eqref{XLPcN} and the dominated convergence theorem \begin{align*} \lim_{x\to 0} \frac{\zeta_R(x)}{x} &= R\lim_{x\to 0} \frac{\zeta_1(x/R)}{x}= 2a_s \int_1^\infty \frac{\mathop{}\!d t }{t^2(t^2-1)^s} =:c_0. \end{align*} This proves \eqref{JQM6p}.
Moreover, \begin{align} \bigg \vert \frac{\zeta_R(x)}{x} - c_0\bigg \vert &= 2a_s\bigg \vert \int_1^\infty \bigg ( \frac {\big (1-(x/R)^2 \big )^s} {t^2-(x/R)^2} - \frac 1 {t^2} \bigg ) \frac{\mathop{}\!d t }{(t^2-1)^s} \bigg \vert \nonumber \\ &\leqslant 2a_s \int_1^\infty \bigg \vert \frac {t^2\big ( (1-(x/R)^2 )^s-1\big ) +(x/R)^2 \big )} {t^2\big (t^2-(x/R)^2 \big )} \bigg \vert \frac{\mathop{}\!d t }{(t^2-1)^s} \nonumber \\ &\leqslant 2a_s \int_1^\infty \bigg \vert \frac{\big (1-(x/R)^2\big )^s -1}{t^2-(x/R)^2} \bigg \vert \frac{\mathop{}\!d t }{(t^2-1)^s} \nonumber\\ &\qquad + \frac{2a_s\vert x \vert^2}{R^2}\int_1^\infty \frac{\mathop{}\!d t}{t^2 \vert t^2-(x/R)^2 \vert (t^2-1)^s} . \label{ZRXKr} \end{align} Suppose that \(x \in \Omega' \subset \subset \mathbb{R}\). Then for \(R>0\) sufficiently large, \begin{align} \frac 1 {\vert t^2-(x/R)^2 \vert } \leqslant C \qquad \text{for all } t>1 \label{RZqcI} \end{align} and, by Bernoulli's inequality, \begin{align} \vert \big (1-(x/R)^2\big )^s -1 \vert &\leqslant \frac C {R^2}. \label{zHcKY} \end{align} Combining \eqref{ZRXKr}, \eqref{RZqcI}, and \eqref{zHcKY}, we conclude that \begin{align*} \bigg \vert \frac{\zeta_R(x)}{x} - c_0\bigg \vert &\leqslant \frac C {R^2}\bigg ( \int_1^\infty \frac{\mathop{}\!d t }{(t^2-1)^s} + \int_1^\infty \frac{\mathop{}\!d t}{t^2(t^2-1)^s} \bigg ) \to 0 \end{align*} as \(R \to \infty\). \end{proof}
\begin{prop} \label{ayQzh} Suppose that \(\Omega\) is a bounded open subset of \(\mathbb{R}^n\) and that \(\lambda(\Omega)\) is a Dirichlet eigenvalue of \((-\Delta)^s\) in \(\Omega\). Then \begin{align*} \lambda(\Omega)\geqslant \frac{n}{2s} \vert B_1 \vert^{1+2s/n} c_{n,s}\vert \Omega \vert^{- \frac{2s}n } . \end{align*} where \(c_{n,s}\) is the defined in \eqref{tK7sb}. \end{prop}
The result in Proposition \ref{ayQzh} is not new (see \cite{MR3824213,MR3063552}); however, to the authors' knowledge, the proof is original and simple.
\begin{proof}[Proof of Proposition \ref{ayQzh}] Let \(u \in L^2(\Omega)\) satisfy \begin{align*} \begin{PDE} (-\Delta)^s u &= \lambda(\Omega) u &\text{in } \Omega, \\ u&= 0 &\text{in }\mathbb{R}^n \setminus \Omega, \\
\| u \|_{L^2(\Omega)}&=1. \end{PDE} \end{align*} The existence of such a function can be proved via semi-group theory, see for example \cite[Chapter 4]{MR2569321}.
Using the integration by parts formula for the fractional Laplacian, see \cite[Lemma 3.3]{Dipierro2017Neum}, we have \begin{align*} \lambda(\Omega) = \frac{c_{n,s}}{2} \int_{\mathbb{R}^{2n} \setminus (\Omega^c)^2} \frac{\vert u(x)-u(y) \vert^2}{\vert x- y \vert^{n+2s}} \mathop{}\!d y \mathop{}\!d x \geqslant c_{n,s} \int_\Omega \int_{ \mathbb{R}^n \setminus \Omega } \frac{\vert u(x)\vert^2}{\vert x- y \vert^{n+2s}} \mathop{}\!d y \mathop{}\!d x. \end{align*}For \(x\in \Omega\), if \(B_r(x)\) is the ball such that \(\vert B_r(x)\vert = \vert \Omega\vert \) then by~\cite[Lemma 6.1]{MR2944369}, \begin{align*} \int_{ \mathbb{R}^n \setminus \Omega } \frac{ \mathop{}\!d y }{\vert x- y \vert^{n+2s}} \geqslant \int_{ \mathbb{R}^n \setminus B_r(x)} \frac{ \mathop{}\!d y }{\vert x- y \vert^{n+2s}} =\frac{n}{2s} \vert B_1 \vert^{1+2s/n} \vert \Omega\vert^{-\frac{2s}{n}}. \end{align*} Hence, \begin{equation*}
\lambda(\Omega) \geqslant \frac{n}{2s} \vert B_1 \vert^{1+2s/n}c_{n,s}\vert \Omega\vert^{-\frac{2s}{n}} \| u \|_{L^2(\Omega)}^2 = \frac{n}{2s} \vert B_1 \vert^{1+2s/n}c_{n,s} \vert \Omega\vert^{-\frac{2s}{n}}.\qedhere \end{equation*} \end{proof}
\section*{Acknowledgments}
All the authors are members of AustMS. SD is supported by the Australian Research Council DECRA DE180100957 “PDEs, free boundaries and applications”. GP is supported by the Australian Research Council (ARC) Discovery Early Career Researcher Award (DECRA) DE230100954 “Partial Differential Equations: geometric aspects and applications”, and is member of INdAM/GNAMPA. GP and EV are supported by the Australian Laureate Fellowship FL190100081 “Minimal surfaces, free boundaries and partial differential equations”. JT is supported by an Australian Government Research Training Program Scholarship.
\printbibliography
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Home All issues Volume 578 (June 2015) A&A, 578 (2015) A4 Full HTML
Volume 578, June 2015
Galactic structure, stellar clusters and populations
https://doi.org/10.1051/0004-6361/201424132
2. Observations
Online material
A&A 578, A4 (2015)
Circumstellar discs in Galactic centre clusters: Disc-bearing B-type stars in the Quintuplet and Arches clusters⋆,⋆⋆,⋆⋆⋆
A. Stolte1, B. Hußmann1, C. Olczak2, W. Brandner3, M. Habibi1,4, A. M. Ghez5, M. R. Morris5, J. R. Lu6, W. I. Clarkson7 and J. Anderson8
1 Argelander Institut für Astronomie, Universität Bonn, Auf dem Hügel 71, 53121 Bonn, Germany
e-mail: [email protected]
2 Astronomisches Recheninstitut, Universität Heidelberg, Mönchhofstr. 12-14, 69120 Heidelberg, Germany
3 Max-Planck-Institut für Astronomie, Königstuhl 17, 69117 Heidelberg, Germany
4 Max-Planck-Institut für extraterrestrische Physik, Giessenbachstrasse 1, 85748 Garching, Germany
5 Division of Astronomy and Astrophysics, UCLA, Los Angeles, CA 90095-1547, USA
6 Institute for Astronomy, University of Hawaii, 2680 Woodlawn Drive, Honolulu, HI 96822, USA
7 Department of Natural Sciences, University of Michigan-Dearborn, 125 Science Building, 4901 Evergreen Road, Dearborn, MI 48128, USA
8 Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA
We investigate the circumstellar disc fraction as determined from L-band excess observations of the young, massive Arches and Quintuplet clusters residing in the central molecular zone of the Milky Way. The Quintuplet cluster was searched for L-band excess sources for the first time. We find a total of 26 excess sources in the Quintuplet cluster, and 21 sources with L-band excesses in the Arches cluster, of which 13 are new detections. With the aid of proper motion membership samples, the disc fraction of the Quintuplet cluster could be derived for the first time to be 4.0 ± 0.7%. There is no evidence for a radially varying disc fraction in this cluster. In the case of the Arches cluster, a disc fraction of 9.2 ± 1.2% approximately out to the cluster's predicted tidal radius, r< 1.5 pc, is observed. This excess fraction is consistent with our previously found disc fraction in the cluster in the radial range 0.3 <r< 0.8 pc. In both clusters, the host star mass range covers late A- to early B-type stars, 2 <M< 15 M⊙, as derived from J-band photospheric magnitudes. We discuss the unexpected finding of dusty circumstellar discs in these UV intense environments in the context of primordial disc survival and formation scenarios of secondary discs. We consider the possibility that the L-band excess sources in the Arches and Quintuplet clusters could be the high-mass counterparts to T Tauri pre-transitional discs. As such a scenario requires a long pre-transitional disc lifetime in a UV intense environment, we suggest that mass transfer discs in binary systems are a likely formation mechanism for the B-star discs observed in these starburst clusters.
Key words: techniques: high angular resolution / open clusters and associations: individual: Quintuplet / circumstellar matter / open clusters and associations: individual: Arches / astrometry / proper motions
Based on data obtained at the ESO VLT under programme IDs 085.D-0446, 089.D-0121 (PI: Stolte), 081.D-0572 (PI: Brandner), 087.D-0720, 089.D-0430 (PI: Olzcak), 071.C-0344 (PI: Eisenhauer), 60.A-9026 (NAOS/CONICA science verification), as well as Hubble Space Telescope observations under programmes 11671 (PI: Ghez).
⋆⋆
Appendices are available in electronic form at http://www.aanda.org
⋆⋆⋆
The photometric catalogue is only available at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/578/A4
© ESO, 2015
In view of the short lifetimes of primordial circumstellar discs around B-type stars in dense environments, the detection of circumstellar discs in the UV-rich environment of the Galactic centre Arches cluster came as a surprise (Stolte et al. 2010). This detection raised the question of whether discs can also be found in the more evolved Quintuplet cluster, and whether these discs can have their origin in massive primordial discs sustained over the clusters' lifetimes of several million years (Myr). In particular, the nature of the disc sources in the Arches cluster has remained unsolved (see the discussion in Stolte et al. 2010). In the past five years, we have extended our disc search to larger radii in the Arches cluster and have encompassed the Quintuplet cluster. With the aim of shedding light on the nature of the L-band excess sources found in both starburst clusters, we compare their physical properties to pre-transitional discs and discuss secondary mass transfer discs as a possible origin of the circumstellar material.
1.1. Circumstellar disc survival
The survival of primordial circumstellar discs in young star clusters is known to be a steep function of cluster age (Haisch et al. 2001; Hernández et al. 2005). For discs around low-mass stars, M ~ 1 M⊙, planet formation theories suggest that dust agglomeration causes a period of grain growth, while at the same time the dense gas of the primordial disc is evaporated by the central star (Cieza et al. 2012; Williams & Cieza 2011; Owen et al. 2011). In consequence, the thermal excess of evolved discs is dominated by increasingly longer wavelength emission, while the near-infrared contribution decreases and vanishes with time (e.g. Espaillat et al. 2012). The disc survival timescale in young stellar clusters is observed to be 3–10 Myr for intermediate-mass stars, 2 <M< 10 M⊙ (Hernández et al. 2005), albeit with lower disc fractions at any given age than clusters dominated by their lower-mass T Tauri counterparts (see Stolte et al. 2010 for a detailed discussion). Hernández et al. (2005) suggest that settling in the disc might depend on the stellar mass, such that the stellar radiation could penetrate discs around higher mass stars more rapidly, which might accelerate the destruction by UV radiation from the central star even further. This mass-dependent process would decrease the rate of survival of the hot inner disc as observed through near-infrared excess emission around high-mass stars. For Herbig Ae/Be stars in nearby OB associations, Hernández et al. find intermediate-mass disc fractions of only 2–5% at ages 3–10 Myr. In particular in the dense environment of starburst clusters such as the Arches and Quintuplet clusters near the Galactic centre, star-disc interactions (Olczak et al. 2012) and the external UV radiation field (Anderson et al. 2013; Fatuzzo & Adams 2008; Adams et al. 2004; Scally & Clarke 2001; Richling & Yorke 2000; Johnstone et al. 1998) might contribute to an even more rapid depletion of circumstellar material.
Until recently, circumstellar discs in young star clusters were mostly identified by their K- and L-band near-infrared excess emission. This definition distinguishes primordial discs with dense, hot inner rims from disc-less stars or stars with evolved transitional discs. The near- to mid-infrared capabilities of the recent Spitzer space mission enabled the detection and definition of more subtle classes of discs. Transitional discs were traditionally defined as objects lacking near-infrared emission, yet displaying strong mid- or far-infrared excesses (see Espaillat et al. 2012 for a summary, and references therein). With larger sample sizes and improved spatial resolution, several of the proposed transition objects reveal significant substructure in their spectral energy distributions (SEDs), with small but significant near-infrared excess contributions as a tracer for optically thick inner discs. In contrast to primordial discs, their near-infrared emission is accompanied by mid-infrared dips, suggestive of disc gaps. These objects are defined as pre-transitional discs by Espaillat et al. (2012), and can also be identified with the near-infrared bright subclass of the so-called warm debris discs detected with the AKARI mission by Fujiwara et al. (2013). Although debris discs are typically found around stars with older ages of more than 10 Myr, the ages of these near-infrared bright discs are not known. While transitional discs are by definition absent from the classical L-band excess searches for primordial circumstellar discs, pre-transitional discs could contribute to the disc fractions observed in young, massive star clusters, if they are common around main-sequence B-type as well as T Tauri stars.
1.2. The Arches and Quintuplet clusters
The Arches and Quintuplet clusters are young, massive star clusters located at a projected distance of ~30 pc from the Galactic centre (GC). Both clusters are host to a rich population of more than 100 massive O- and B-type stars (Liermann et al. 2009; Martins et al. 2008). From the extrapolation of the observed stellar mass function, they have estimated photometric masses of about 104M⊙ (Habibi et al. 2013; Hußmann et al. 2012). According to dynamical simulations, the underlying stellar mass is suggested to be as high as 4 × 104M⊙ for the Arches cluster (Harfst et al. 2010). Spectroscopic age dating of the evolved population of Wolf-Rayet (WR) and giant or supergiant stars suggests an age of 2.5 ± 0.5 Myr (Najarro et al. 2004), with a possible upper age limit for the Arches cluster of 4 Myr obtained from supergiant member stars. For the Quintuplet cluster the situation is not as clear. Earlier studies have suggested an age of 4 ± 1 Myr from spectral fitting to the evolved population (Figer et al. 1999). A newer Very Large Telescope (VLT) SINFONI spectral analysis finds ages of ~4 Myr for the OB stars (Liermann et al. 2012) and suggests ages as young as 2−3 Myr for the WN stars (Liermann et al. 2010), but these objects in particular may be affected by binary mass transfer evolution and the corresponding rejuvenation (Schneider et al. 2014). For the purposes of this paper, we adopt an age of 2.5 ± 0.5 Myr for the Arches cluster, and 4 ± 1 Myr for the Quintuplet, as these ages are consistent with the isochrones we adopt for stellar mass derivations (see Hußmann et al. 2012; Habibi et al. 2013).
1.3. Discs in the Arches cluster
In the Arches cluster, a population of disc-bearing stars was found from near-infrared excesses using high-resolution Keck/NIRC2 adaptive optics observations in the dense cluster core (Stolte et al. 2010). From L′-band excesses, we derived the fraction of disc-bearing B-type main-sequence stars to be 6 ± 2% at radii r< 0.8 pc from the cluster centre. A radial increase in the disc fraction from 3% for r< 0.1 pc to 10% for 0.3 <r< 0.8 pc suggested that circumstellar discs are prone to destruction by UV radiation or gravitational interactions in the dense cluster environment (e.g. Olczak et al. 2012). Of 24 detected excess sources, all 21 sources with reliable proper motion measurements proved to be genuine members of the Arches cluster. The detection of CO 2.3 μm bandhead emission in VLT/SINFONI K-band spectra available for three sources provided additional evidence that the L-band excess originates from hot circumstellar material in a disc geometry. For the Arches sources presented in Stolte et al. (2010), we estimated a host star mass range of 3−10 M⊙ based on H-band luminosities which showed little to no infrared excess emission. One of the remaining mysteries was the apparently old age of 2.5 Myr for circumstellar material around B-type stars. Previous observational studies of Herbig Ae/Be stars suggested that photoevaporation destroys primordial discs around high-mass stars within less than 1 Myr (Hillenbrand et al. 1998; Hernández et al. 2005; Alonso-Albi et al. 2009; Gorti et al. 2009). A short survival timescale of less than 1–3 Myr in UV-rich clusters is confirmed by numerical simulations of disc survival (Anderson et al. 2013, and references therein). When FUV, EUV, and X-ray luminosities are taken into account, stars with M< 3 M⊙ have simulated disc lifetimes of several 106 years. The survival timescale plummets for higher-mass stars and already reaches less than 106 years for a 10 M⊙ central star (Gorti et al. 2009, see their Fig. 12). Thus, the survival of primordial discs around B-type stars, even at a low rate, was unexpected. The arguments leading to early disc destruction should be even more valid for the older Quintuplet cluster.
Here, we investigate VLT high-resolution adaptive optics KsL′ observations in combination with HST/WFC3 JH photometry to extend the cluster area covered in the Arches out to r = 1.5 pc, close to the cluster's predicted tidal radius (Habibi et al. 2013), and to detect disc candidates in the more evolved Quintuplet cluster for the first time. Proper motion membership from two-epoch adaptive optics Ks data is employed to confirm the cluster origin and hence the youth of the discovered L-band excess sources.
WFC3 JH colour composite of the Quintuplet cluster (left panel) and Arches cluster (right panel) covered with both HST/WFC3 and NACO observations. The NACO fields are shown as boxes. L-band excess sources detected in both clusters are overlaid as circles. Previously detected excess sources in the Arches are shown as smaller cyan circles. Excess sources in the cluster centres are circumscribed in blue for clarity.
Towards the end of this contribution, we provide a detailed discussion of the origin of the L-band emission (Sect. 5), and compare the expected location of the hot dust with the dust sublimation and inner disc radii. We compare our findings to studies of transition discs with near- to mid-infrared emission recently discovered by the Spitzer and AKARI surveys (e.g. Fujiwara et al. 2013; Maaskant et al. 2013; Espaillat et al. 2010, 2011; Muzerolle et al. 2010; Teixeira et al. 2012; Furlan et al. 2009) to investigate the evolutionary state of the disc candidates in the Arches and Quintuplet clusters. The problem of the disc lifetime is addressed, and secondary disc formation will be proposed as a possible scenario to explain the Arches and Quintuplet near-infrared excess sources and their apparent expanded lifetime compared to primordial discs around Herbig Be stars.
In Sect. 2, we present the VLT and HST observations, and the photometric and astrometric analysis is summarised in Sect. 3. The disc fraction of the Quintuplet cluster is derived in Sect. 4.1, and new Arches disc sources are presented in Sect. 4.2. The physical properties of the L-band excess sources are discussed in Sect. 5, and we summarise our findings in Sect. 6.
VLT/NACO observations.
Photometric sensitivity limits (in calibrated magnitudes), estimated from the peak of the luminosity functions, and number of sources in matched JHKs and JHKsL′ catalogues.
HST/WFC3 observations.
2.1. VLT/NAOS-CONICA
For the membership campaign of the Arches and Quintuplet clusters, multi-epoch Ks imaging with time baselines of 3 to 5 years was obtained with the VLT adaptive optics system NAOS and its near- to mid-infrared camera CONICA (hereafter NACO, Lenzen et al. 2003; Rousset et al. 2003) during the time period 2002–2012. Complementary L′-band images were observed in June and August 2012. A complete list of all NACO data sets analysed in this paper is provided in Table 1.
The Ks images were obtained with the S27 camera with a pixel scale of 27.1 milliarcseconds (mas)/pixel, covering a 27′′ × 27′′ field of view. Five fields were identified in each cluster with suitable natural guide stars operating mostly with the near-infrared wavefront sensor. As a consequence of the high foreground extinction of AV ~ 25 mag, optical guide stars are rarely available along the Galactic centre line of sight, such that the unique NIR sensing capability of NAOS was extensively exploited to obtain the wide area coverage of both clusters. The Arches and Quintuplet mosaics cover maximum distances of 48′′ and 60′′ from the cluster centres, corresponding to 1.8 pc and 2.3 pc at a distance of 8.0 kpc, respectively. Arches Field 2 reaches larger distances out to 63′′ (Fig. 1), but is not part of the coherent mosaic of this cluster. For the Quintuplet cluster with evolved stars as bright as Ks = 7.3 mag, the N20C80 dichroic could be used, which distributed 80% of the light to the science camera and only 20% to the wavefront sensor for natural guiding. As the brightest sources in the Arches cluster are substantially fainter, Ks = 10.4 mag, the N90C10 dichroic had to be employed with only 10% of the light diverted to CONICA. The Arches Ks data are correspondingly shallower than the Quintuplet data sets. Detection limits are provided in Table 2.
The first epoch data were optimised for deep photometry using individual detector integration times (DITs) of up to 20 s. In all newer data sets DITs were kept short to avoid saturation of the brighter stars to establish the cluster reference frame for astrometric proper motion measurements. Because of the different setup, these DITs ranged from 2 s for the N20C80 observations to 10 s for N90C10 imaging.
The Ks observations were complemented with single-epoch L′ imaging obtained in 2012. In the case of L′, NACO offers a JHK dichroic, which passes all near-infrared light to the wavefront-sensor, and allows the full L-band channel to be diverted to the science detector. The limiting factor in L′ is the sky brightness, such that the detector saturates rapidly even in very short integration times. To avoid saturation, the exposure time was set to 0.175 s–0.2 s (the shortest feasible integration times with the CONICA detector) using the uncorrelated readout mode. Between 150 and 170 individual DITs were coadded to a total integration time of ~30 s per science image. The L27 camera was used with a pixel scale of 27.1 mas/pix to provide the same spatial coverage as for the Ks-band observations. The data were obtained in dither mode for both filters with dither offsets between 30 and 70 pixels (0.8′′ to 2′′) to allow for sky subtraction and the removal of hot pixels, which are of particular concern in infrared detectors. The dither offset was chosen to be less than 1/10 of the field size in the astrometry-oriented epochs (2008 and onwards) to minimise the impact of optical distortions and to optimise the relative astrometric performance between proper motion epochs.
In order to monitor sky variations, separate sets of sky images were observed in Ks after, and in L′ interleaved with, the science images. Sky fields are observed in open-loop mode without adaptive optics correction with NACO, such that residual starlight is spread out across the detector and leads to a biased sky level in the combined master sky. As it is difficult to find star-free fields in the vicinity of the clusters because of the high stellar density along the GC line of sight, all sky fields contained some residual star light. Given these complications, three different procedures were attempted especially for the L′-band data, where the sky is the most limiting factor on sensitivity (Sect. 3.1).
2.2. HST/WFC3
Hubble Space Telescope (HST) images were obtained with the wide-field camera WFC3 in the near-infrared channel in the medium-band WFC3 F127M and F153M filters, corresponding approximately to the J and H broadband filters in the ground-based near-infrared system, in August, 2010, under programme ID 11671 (PI: Ghez). The integration times were 600 s and 350 s in F127M and F153M per individual frame, and a total of 12 and 21 single images lead to combined image integration times of 120 min in each filter (see Table 3). WFC3 offers a field of view of 2.2′ × 2.0′, which was also the approximate size of the drizzled image, such that each cluster is covered with a single pointing (Fig. 1). Images were dithered with small positioning offsets between 0.6 and 10 pixels (0.08′′−1.2′′) to allow for image reconstruction and bad pixel removal. While the reduced, distortion-corrected and pre-combined Arches images were obtained from the MAST archive, the Quintuplet images were obtained before the final version of the distortion solution was integrated into the standard WFC3 pipeline. These images were processed using the multidrizzle algorithm (Koekemoer et al. 2002; Fruchter et al. 2009) with the most recent distortion solution applied. Prior to image combination, the image shifts were adjusted with the pyraf tweakshifts routine provided for WFC3 pipeline reduction (see the WFC3 Data Reduction Handbook for details)1.
3.1. VLT/NACO
3.1.1. Data reduction
The data reduction for the VLT/NACO Ks and L′ images was carried out with a custom-made reduction pipeline based on python/pyraf and IDL routines. The basic steps in Ks included dark subtraction, flat fielding, the removal of cosmic ray hits and hot pixels, and the subtraction of a master sky. Twilight flat fields were observed with decreasing brightness, such that the pixel sensitivity could be obtained as the slope of the brightness variation in each pixel. Bad pixels were identified during the combination of the master flat field as pixels where the sensitivity deviated significantly from the mean. In addition to this universal bad pixel mask for each data set, individual bad pixel masks were created using the iraf task cosmicrays to identify positive, spatially confined flux peaks. Where possible, the sky image for each science set was derived from the adjacent sky exposures at an off-cluster position without adaptive optics correction. As these images frequently contained residual star flux, the best sky subtraction was in some data sets achieved by median-combining both dithered sky and science images to create a flat thermal background image free of residual stellar light. The choice of the sky subtraction procedure depended on the individual dataset.
In L′, no master darks are provided by ESO, hence the basic reduction steps consisted of flat fielding and sky subtraction. The thermal background is the dominant source of uncertainty at 3.8 μm. For most of the L′ data sets several attempts at sky subtraction had to be made: i) one master sky was median-combined from all science and interleaved sky images; ii) one master sky was median-combined from the sky images only; and iii) individual skies were created from the images closest in time to the considered science frame. After each of these master skies was subtracted from the science images, the final image with the least background noise and the highest contrast and sensitivity was selected for photometric analysis.
NACO Ks mosaic of fields in the Quintuplet (left) and Arches (right) clusters. The same fields are approximately covered in L′. In the Arches cluster, Field 2 is not shown and is located to the lower right of the cluster (see Fig. 1). Disc candidates are indicated in both clusters as white circles. Yellow circles in the Quintuplet mark disc candidates fainter than the completeness limit of J = 22 mag in the left panel. Previously known disc candidates in the Arches detected in our earlier higher-resolution Keck/NIRC2 observations are included for comparison as smaller cyan circles in the right panel (Stolte et al. 2010). The star serving as coordinate reference for all catalogues is indicated by the arrow.
Both Ks and L′ images were combined into one deep exposure with the drizzle task (Fruchter & Hook 2002). The routines precor, crosscorr, and shiftfind were employed to derive the positional offsets. For these adaptive optics data, the precor task, which allows masking of background patterns, aided the identification of the true spatial shifts from the cross-correlation image. Finally, the individual bad pixel masks were applied to each image during the image combination process. We note here that no distortion correction was applied during drizzling. Despite continued efforts to characterise the optical distortions in NACO, the distortion correction still bears large uncertainties, especially for the median-field camera S27 (see e.g. Trippe et al. 2008; Fritz et al. 2010). We interpret this as an effect of anisoplanatism, which depends on the nightly conditions and causes additional point source image distortions. As these distortions depend on the isoplanatic angle and the adaptive optics correction during the time of the observations, the natural guide star distance from the field centre, and the brightness of the NGS, they add a strong random component to each source position, such that no uniform instrument distortion solution could be derived (see Sect. 2.3 in Habibi 2014 for details)2. During the NACO observations, the influence of optical distortions on the relative astrometric uncertainty is minimised by conducting a small-scale dither pattern to avoid large image shifts, and by using the same observational setup and positioning in each field during each epoch (see also Ghez et al. 2008; Yelda et al. 2010).
3.1.2. Photometry and astrometry
The crowded nature of the GC cluster fields required point spread function (PSF) fitting to perform the photometry and astrometry on each image. At a field size of 27′′, the NACO images are severely influenced by anisoplanatism. As shown in Hußmann et al. (2012), a constant PSF across the field yields reliable astrometric measurements only in the area near the natural guide star, which is typically (but not always) located near the field centre. While we restricted the analysis of the stellar mass function of the Quintuplet cluster in this previous paper to a radius of r< 500 pixels or 13′′, the purpose of the present disc study is to cover the cluster areas as completely as possible. Therefore, the IRAF daophot package (Stetson 1992) was employed to obtain PSF fitting photometry and astrometric positions of all stars. Quadratically varying PSF functions were used to minimise the spatially varying effects of source elongation due to anisoplanatism on the derived astrometry and photometry as much as possible.
In the deep Quintuplet central field of the 2003 observations obtained exclusively with single-frame integration times of 20 s, a quadratic variation provided unsatisfactory results. In particular in the deep exposure, stars within ~5′′ from the edge of the drizzled image were rejected during the PSF fitting procedure as a consequence of their irregular shape. Therefore, the long-exposure combined image (see Table 1) was split into four quadrants for which individual PSFs were created and subtracted. After the quadrants were separated, a linearly varying function was sufficient to obtain a good fitting performance across each quadrant. In the overlap regions, the standard deviations in x and y residuals between astrometric measurements were below 0.2 pixels, which is the expected variable PSF fitting accuracy in these performance-limited data sets. The individual quadrant photometric lists were then recombined using the offset shifts applied earlier when splitting the images.
The Ks photometry of the Arches cluster was calibrated with respect to Espinoza et al. (2009). Fields 3-5 were calibrated from overlap areas with the central field and calibrations were cross-checked in each overlap region. Field 2 has no overlap with any of the previous observations and the remainder of the Arches fields, and was calibrated against the UKIDSS Ks Galactic Plane Survey (Lucas et al. 2008, spatial resolution 1′′). The Quintuplet Ks photometry was calibrated with respect to the UKIDSS GPS survey as well, although in a two-step procedure, which helped overcome the resolution differences. First, VLT/ISAAC observations3 taken under excellent conditions were calibrated over a 2.4 arcmin field of view with respect to UKIDSS Ks. No colour terms were found, and a zeropoint offset was applied. The NACO Ks high-resolution photometry with a spatial resolution of ~0.1′′ was calibrated with respect to the seeing-limited ISAAC Ks photometry with a spatial resolution of ~0.4′′ in a second step. Details can be found in Hußmann (2014).
The L′ photometry was zeropointed with respect to 3.8 μm IRAC photometry obtained with the Spitzer GLIMPSE survey (Benjamin et al. 2003; Churchwell et al. 2009). Because of the large number of detections missing from the source catalogues, photometry on the IRAC 3.8 μm images was rederived using daophot PSF fitting. The large area coverage of the images allowed for a robust calibration against the downloaded Spitzer catalogues. For the cluster fields, however, the low resolution of the Spitzer data implied that only a few sources (4-6) were available as references, even in the PSF fitting photometry source lists. In the J − H, Ks − L′ colour–colour diagrams, the uncertain L-band calibration manifested itself as offsets from the reddening path. We therefore checked the J − H, H − Ks colour–colour diagrams for consistency with the reddening path (Nishiyama et al. 2009). Any remaining offsets in L′ were adjusted in the JHKsL′ colour–colour diagram such that the Ks − L′ colour was also consistent with the reddening path.
3.1.3. Photometric and astrometric uncertainties
Photometric and astrometric uncertainties were derived by independent repeated measurements of the photometry of each star. We combined three subsets of each individual image stack to perform these measurements. In one data set, the number of frames and the image quality were compromised, such that only two subsets could reasonably be obtained. To ensure that the data quality of each subset image is comparable to that in the deep image, care was taken that each auxiliary image created from a single subset contained frames over the full range of spatial resolutions and photometric sensitivities as contributed to the deep image. For this purpose, the range in performance was measured as the full width at half maximum (FWHM) of the PSF on each individual image, and each subset list contained images over the entire range of FWHM values. Daophot was then run with the same PSF fitting parameters as in the respective deep image on each auxiliary image to produce repeated measurements of the photometry. Each star that entered the final source catalogues was required to be detected in at least two auxiliary images in addition to the deep science frame. The photometric and astrometric uncertainties were then calculated either as the standard error given as the standard deviation from the mean divided by sqrt(3) if a star was detected in all three auxiliary images, or as the deviation from the mean of the two measurements if the star was only detected in two auxiliary frames. This was mostly the case for faint stars, but could also be caused by incomplete area coverage of the subset images in comparison to the complete deep image. In the data set where only two auxiliary images could be obtained, the uncertainties are always given as the deviation from the mean, and all stars are again required to be detected in both auxiliary images.
Photometric uncertainties are shown in Fig. 3 for the Quintuplet and in Fig. 4 for the Arches cluster. All uncertainties are accordingly quoted in the final source catalogue along with the magnitude and positional measurements of the deep science image4.
Photometric uncertainties of the NACO Ks and L′ observations of the Quintuplet cluster for Fields 1 through 5 (top to bottom). The left panels show the first epoch and the middle panels the second epoch Ks uncertainties for the two epochs from which proper motions were derived. The Ks photometry in Field 1 was combined from short and long exposures, with a saturation transition at Ks = 14 mag. The transition is marked by the prominent improvement in photometric performance in the long exposure data. L′ photometric uncertainties (right panels) indicate the large differences in performance in the L′ observations due to thermal background variations. While Fields 2, 3, and 5 show comparable sensitivities, Field 4 is compromised by background fluctuations. Field 1 features significantly deeper photometry, such that the L′ selection was truncated at L′< 15 mag to obtain the excess fraction consistently across the entire cluster area.
Photometric uncertainties of the NACO Ks and L′ observations of the Arches cluster for Fields 1 through 5 (top to bottom). The left panels show the first epoch and the middle panels the second epoch Ks uncertainty for the two epochs from which proper motions were derived. The photometric uncertainties of the first epoch are daophot PSF fitting uncertainties, and are not derived from multiple measurements. As in the case of the Quintuplet cluster, L′ photometric uncertainties (right panels) indicate the differences in performance in the L′ observations due to thermal background variations. The sensitivity limits show more consistency than in the Quintuplet, with a detection threshold of L′ ~ 14.5−15.0 mag. The larger scatter in Fields 3 and 5 is caused by background fluctuations.
3.1.4. Geometric transformations
For both the Arches and Quintuplet clusters, we derived membership information for all fields out to a radius of ~1.5 pc. For this purpose, we have combined at least two epochs of Ks NACO observations to construct the proper motion diagram for each cluster and field.
Proper motions are derived via the matching of two epochs of imaging observations separated by a time baseline of 3 to 5 years. The positions of each star have to be transformed to one epoch that serves as the reference epoch for the respective field. The geometric transformation between both epochs was derived for bright stars with Ks< 17.5 mag. The IRAF task geomap provided residual rotation, image shifts, and relative distortions by fitting a second-order polynomial to the x and y positional differences for all stars in each field. In general, the earlier epochs provided deeper images with a higher spatial resolution as a result of the degradation in adaptive optics performance of the NAOS system over time. Hence, the first epoch was used as the reference epoch in all cases, and the positions measured in later epoch images were converted to the first epoch using the geomap geometric solutions. After a first transformation, cluster candidate stars were selected from the proper motion plane as stars located close to the origin in the cluster reference frame (i.e. with zero relative motion with respect to the mean cluster motion). These cluster member candidates were iteratively used to generate the final geometric transformation solution. This procedure ensures that the mix between rapidly moving field stars and slowly moving cluster members with respect to the cluster reference frame does not bias the transformation. It should be noted that the method implies minimisation of the relative motions between cluster stars, such that the cluster population appears more confined after the second iteration, which improves the distinction between cluster members and field stars. The uncertainties in the proper motion measurements include the individual positional uncertainties in x or y of each star as described above, added in quadrature to the residual rms uncertainties of the geometric transformation.
Proper motion uncertainties of stars in the Quintuplet (left) and Arches (right) clusters for Fields 1 through 5 (top to bottom) in each cluster. The proper motion uncertainties include the astrometric uncertainties from each individual epoch in addition to the standard deviation in the geometric transformation from epoch 2 to epoch 1. The deeper Ks observations in the central Quintuplet field provide good astrometric accuracies down to Ks ~ 19 mag, while Fields 2–5 are truncated at Ks = 17.5 mag (dashed lines), beyond which astrometric accuracies degrade. The wide spread in Quintuplet Field 1 is caused by crowding. For the purposes of membership derivation, only sources with Ks< 18 mag (dashed line) are used in Field 1 for compatibility with the outer fields. In most Arches fields (right panels), the astrometric accuracies are small down to Ks ~ 18 mag, with the exception of Field 2, where proper motion derivations are limited to Ks< 17 mag.
Comparison of proper motion values derived in our 2010 disc paper of the central Arches field and in this study. Differences in the proper motion are shown individually in the right ascension (left) and declination (right) directions. The consistency between the blue and red samples reveals the similarity between the membership selection used to derive the central disc fraction in Stolte et al. (2010, red asterisks) and in this work (blue diamonds). Non-members are shown as black dots. The majority of stars in the common sample shows proper motion differences below 0.5 mas/yr, almost identical to the relative astrometric uncertainties in Arches Field 1 shown in Fig. 5. Faint stars (Ks< 16 mag) show larger motion differences, as expected from their higher astrometric uncertainties.
The proper motion uncertainties are shown in Fig. 5. A distinct increase in the median uncertainty can be seen in all data sets towards fainter magnitudes, as expected. As the limit where reliable membership measurements can be obtained depends on the data set, we adopt limiting magnitudes of Ks< 17.5 in all outer Quintuplet fields and Ks< 18 mag in the Quintuplet central field, where both Ks epochs are equally deep, as indicated in Fig. 5. In the central field, we also include in the source catalogue an indication of membership for fainter stars, in which case we mark the membership information as uncertain. In contrast to Hußmann et al. (2012), the source catalogue of the central field (Field 1, F1) in the search for disc candidates presented here includes the full 40′′ × 40′′ dithered field of view.
Proper motion diagram of the central field of the Quintuplet cluster. The circle around the origin indicates the selection of cluster members. L-band detected proper motion members are shown in green, while non-members are shown in red. Black sources are not detected in L′. Those sources are cluster members if located inside the green circle. L′ excess sources are marked as red diamonds, while Wolf-Rayet candidates are shown in blue (see Sect. 4.1). The right panel displays a zoom on the cluster selection and the median astrometric uncertainty for L-band detected members in the lower left corner.
A realistic estimate of the uncertainties in the proper motion measurement of each star can also be obtained from the comparison of the derived proper motion values with our earlier NACO-NIRC2 member selection in the Arches cluster (Stolte et al. 2010). Substantial differences between the motion samples might introduce uncertainties in the main-sequence member selection, and hence in the disc fractions.
In Fig. 6, the astrometry presented here is compared to our earlier results obtained with Keck/NIRC2 and VLT/NACO over a time baseline of 4 to 5 years (Stolte et al. 2010). Proper motions in Stolte et al. (2010) were derived from Keck/NIRC2 astrometry with respect to the NACO 2002 central field also used as reference for Field 1 in this study. The area that could be matched covers the central 13′′ of the cluster, and is dominated by the crowding-limited cluster core. The characteristic deviations in proper motions both in the right ascension and declination directions are ± 0.5 mas/yr for the majority of cluster members. This uncertainty is very similar to the absolute motion uncertainties in Arches Field 1 as derived from multiple astrometric measurements (Fig. 5, top right panel), suggesting that the repeated PSF fitting on the shallower auxiliary subsets yields realistic motion uncertainties. The relatively large differences originate from the fact that all earlier observations with Keck/NIRC2 and VLT/NACO were taken under excellent conditions, resulting in higher spatial resolutions of 60–80 mas, closer to the Keck and VLT diffraction limits than the more recent VLT data sets covering the full cluster area presented here. The comparison is restricted to the area of the earlier data sets in the cluster centre, and hence proper motion deviations are induced by crowding. No systematic bias is observed in both proper motion directions, confirming the robustness of our transformation method.
Proper motion diagram of the central field of the Arches cluster. The circle denotes the selection limit for cluster members. Members with L′ detections are highlighted in green, while L′-detected non-members are shown in red. As already found in Stolte et al. (2010), all excess sources recovered in the cluster centre are proper motion members of the Arches. The right panel displays a zoom on the cluster selection and the median astrometric uncertainty for L-band detected members in the lower left corner.
Proper motion histogram of stars in the central fields of the Quintuplet (left) and Arches (right) clusters. The motion parallel to the Galactic plane (black) shows a pronounced tail of field sources, as indicated in the proper motion diagram (Figs. 7 and 8), while the motion perpendicular to the plane (red histograms) is dominated by the velocity dispersion. The Gaussian fit to the latitude motion (red solid lines) in Field 1 provides the 3σ criterion for selecting cluster members in both central fields. In the case of the Arches cluster, the same membership criterion is imposed in all outer fields.
As can be seen from Fig. 6, the membership selection is barely affected by the proper motion uncertainties. Stars previously selected as cluster members in Stolte et al. (2010) are shown as red asterisks, while stars selected as members in the new NACO data set are shown as blue diamonds. Out of 442 stars matched between both samples, 309 were chosen to be members from the NIRC2-NACO astrometry, and 332 are chosen to be members from the NACO-NACO astrometry presented here, with 299 stars being common members in both samples. Only ten stars chosen as members in our previous study are not recovered as members here, and six of these have substantial proper motion deviations, suggesting that their motions are uncertain. An additional 33 stars are detected as members in this work, probably because the rms scatter in the motion plot is larger and the corresponding member selection needs to be wider than chosen before (1.50 mas/yr as compared to 1.30 mas/yr in Stolte et al. 2010). Hence, a total of 43 stars or 10% of the comparison sample have deviating membership designations, which represents the uncertainty in the derived main-sequence reference samples due to the proper motion membership selection alone.
As a consequence of the field size limitations in our earlier observations, this comparison only covers the crowding-limited cluster core. We expect less deviation in the individual motions for stars at larger radii, including both stars in the outer fields of the Arches cluster as well as in the Quintuplet cluster where stellar densities are lower. Therefore, a maximum deviation in the main-sequence number count of 10% is expected in the Arches cluster core, while the deviation will be even smaller for all other main-sequence reference samples.
3.2. Membership derivation
The resulting proper motion diagrams of the central Quintuplet and Arches fields are presented in Figs. 7 and 8, while the outer fields are presented in Figs. A.1 and A.2. A confined cluster population is observed at the origin, as expected after transforming to the cluster reference frame. The field star population is elongated along the orientation of the Galactic plane (dashed line in Fig. 7), indicative of a wide range of orbital parameters for field stars in the GC (see also Stolte et al. 2008 for a discussion). The histograms of motion in the Galactic longitude and latitude directions are shown in Fig. 9 for the central Arches and Quintuplet fields. The peak of cluster member candidates (hereafter cluster members) stands out clearly in longitudal motion (motion along the plane), followed by the extended tail of field stars. In the latitude direction, the peak is dominated by the positional uncertainties. The Gaussian fits of the latitudinal motion in the central fields provides the 2σ selection criterion for cluster members. As the outer fields of the Arches cluster are dominated by field stars, the cluster peak is not pronounced and the same selection criterion as for the central Arches field is used (μ< 1.5 mas/yr). For the Quintuplet outer fields, Fields 2 to 5, membership probabilities are available from our mass function campaign (Hußmann 2014). In these fields exclusively, we use the membership probabilities to discern cluster members from field stars (see Hußmann 2014 for details).
There is one major difference in the derivation of membership probabilities compared to earlier membership studies in nearby clusters. The combination of the large cluster distance and the small time baseline did not allow the use of the classical membership likelihood as the cluster member selection criterion. Instead, the criterion for a star to be considered a cluster member was constrained using Monte Carlo simulations of the distribution of field and cluster stars in the proper motion plane (see Sects. 4.2.2.1 to 4.2.2.3 in Hußmann 2014 for details). In these models, two artificial Gaussian distributions were populated with stars to represent the cluster and the field, respectively. The properties of these distributions were modelled after the observed proper motion measurements. Each artificial star was assigned a proper motion uncertainty drawn randomly from the observed distribution of astrometric errors, and as in the real observational data, these uncertainties were used to weight the probability for each star to belong to the cluster or the field. The simulated cluster and field distributions were then fitted with an expectation-maximisation algorithm as described in Hußmann (2014). For these simulated cluster/field distributions, the recovery fractions of cluster stars were compared to the inserted number of simulated cluster stars. Because of the overlap between the cluster and field ellipses, the distinction cannot be perfectly made. A trade-off probability value was derived for which the maximum number of cluster stars is recovered, while the minimum number of field interlopers contaminates the sample. This probability represents the likelihood of finding each star at a certain location in the proper motion plane, and is weighted by each star's respective proper motion uncertainty (Eq. (4.11) in Hußmann 2014). These membership indicators are included in Table E.1 for reference. Extensive simulations of Field 2, which has the largest number of cluster stars outside the central field and hence facilitates statistical modelling, suggested that cluster stars are most efficiently separated from field stars using a formal probability threshold of 0.4. Stars above this threshold are likely cluster members, while stars below this limit are likely field stars.
Photometric uncertainties of the WFC3 J and H photometry for each Quintuplet field after matching with NACO Ks. The apparent differences in the detection limits reflect the different Ks sensitivities of the first epoch NACO observations.
Photometric uncertainties of the WFC3 J and H photometry for each Arches field after matching with NACO Ks. The apparent differences in the detection limits reflect the different Ks sensitivities of the NACO observations in each field which determine the depth of the J,J − Ks CMDs.
As will be discussed further below, neither the 2σ membership selection nor the probability method allow a perfect distinction between cluster and field stars. Some stars will unavoidably move on orbits similar to the cluster motion, even if their motions are extreme compared to circular orbits of stars and clouds in the inner bulge (Stolte et al. 2008). Some of these stars will be bulge stars with elongated or high-velocity orbits located at a similar distance as the clusters. This population is dominated by bulge giants and red clump stars and has the red colours of evolved stars with substantial extinction. In addition, a number of foreground stars move on characteristic Galactic disc orbits with lateral velocities in the range of ~200 km s-1 along the Galactic centre line of sight, similar to the cluster orbital velocity (Stolte et al. 2008; Clarkson et al. 2012). Galactic disc stars significantly nearer to the Sun can be discriminated from cluster stars as their proper motion will appear larger than the cluster velocities. However, Galactic disc stars at larger distances are not easily distinguished on the basis of their proper motion alone. The Galactic disc population is concentrated along the spiral arms, and is predominantly composed of main-sequence stars with blue colours and a lower foreground extinction. Both the bulge and Galactic disc contaminants are distinct in the colour–magnitude diagram on the basis of their colours, which are different from the reddened cluster main sequence. The colour information therefore provides an additional tool with which to improve the membership selection and obtain an almost genuine cluster member sample, and will be employed to derive the main-sequence reference samples for the disc fractions in Sects. 4.1 and 4.2.
Quintuplet luminosity functions of all filters (left: JHKs, right: L′). All JHKs luminosity functions are shown for the combined sample used for the main-sequence source counts. These luminosity functions therefore indicate the limitations of the source counts in the final JHKs and JHKsL′ catalogues relevant for the main-sequence and excess samples here, and not the true detection limits in each filter. The Ks luminosity functions are shown for both proper motion epochs used for membership derivation. Likewise, the L′ luminosity functions are derived from the matched JHKsL′ source lists used for further analysis. The dashed line in the L′ luminosity function in Field 1 indicates the truncation imposed when combining F1 with the outer fields. The dashed lines in the L′ luminosity functions in Fields 2 and 5 indicate the faintest L-excess source in each field. Except for Field 4, all L′ data sets extend towards or beyond a sensitivity of L′ = 15.0 mag.
Arches luminosity functions of all filters (left: JHKs, right: L′). The Ks luminosity functions are shown for both proper motion epochs used for membership derivation. The detection limits in the WFC3 JH observations are comparable to those obtained in the Quintuplet, yet the absolute number of sources matched with Ks is limited by the shallower Arches NACO photometry caused by the fainter guide stars and required adaptive optics observing mode. The dashed lines in the L′ luminosity functions indicate the faintest L′-excess source in each field. In Arches Field 2, no cluster member with circumstellar disc emission is detected. All L′ data sets show a sensitivity limit close to L′ = 15.0 mag, fainter than the faintest detected excess source in each field.
Incompleteness fractions and fitted polynomials for the main-sequence reference samples (left panels) are shown vs. the J-band magnitude. Artificial stars were matched in JHKs prior to the derivation of the displayed completeness fractions. For the excess source populations (right panels), incompleteness fractions and fits are shown vs. the L′-band magnitude for all artificial stars recovered in JHKs and L′ simultaneously. The top panels show the completeness fractions in the Quintuplet cluster, while the bottom panels refer to the Arches observations. Thick bars in the top of the L-band panels represent the range of L′ magnitudes observed in the excess sources in each field.
As adaptive optics observations suffer from marginal adaptive optics corrections at shorter wavelengths, we employed the WFC3 J and H images to derive JHKs photometry of each source. In the case of the HST/WFC3 observations, either the pre-reduced images (Quintuplet) or the combined drizzled image (Arches) were downloaded from the HST MAST archive. The drizzled images provided the basis for the photometric analysis. The observational properties are summarised in Table 3.
3.4. Photometry
After applying the most recent distortion solution, starfinder PSF fitting with a constant, empirically extracted PSF across the field was applied to the Quintuplet images. For the Arches, the reduced JH images were, at the time of processing, not optimised for astrometry, and daophot photometry with a quadratically varying penny or moffat function provided the lowest photometric residuals, and was hence used on both the J and H images.
Photometric uncertainties were established for all datasets as described above. Auxiliary images were either created or consecutive data sets were used to obtain independent measurements, and the PSF fitting was repeated in the same way as for the deep drizzled images. As in the case of the NACO data, the uncertainties were calculated as the standard error or the deviation from the mean (Figs. 10 and 11).
The JH photometry lists were combined into a single catalogue, which was calibrated with respect to UKIDSS (Lawrence et al. 2007). The UKIDSS resolution of 1′′ implies an improvement of a factor of two compared to the 2MASS catalogues, which is particularly crucial in these crowding-limited cluster fields. We used a version of data release 6 (DR6) of the UKIDSS Galactic Plane Survey (GPS, Lucas et al. 2008), which we corrected with respect to 2MASS for foreground extinction effects (see also Hodgkin et al. 2009 for details on the UKIDSS calibration). This correction proved necessary in the GC fields, as the automatic pipeline zeropointing is based on Schlegel maps (Schlegel et al. 1998), which suffer from low resolution and provide extinction values that are a factor of 3 too high in the GC region. Only sources detected in both J and H are included in the final catalogue to allow for colour term correction. Colour terms were derived directly with respect to the corrected UKIDSS catalogue in the case of the Arches cluster. As in the case of the Quintuplet Ks calibration (Sect. 3.1.2), the Quintuplet WFC3 photometry was calibrated with respect to ISAAC JsHKs photometry, and the colour terms were derived accordingly. The ISAAC photometry was calibrated over the full ISAAC field of view of 2.5 arcmin with respect to UKIDSS, which facilitated the selection of isolated calibration stars. The NACO photometry was then calibrated in a second step with respect to this ISAAC photometry (see also Hußmann (2014) for details).
The colour conversions are derived to be in the Quintuplet field with respect to ISAAC JsH, and to be in the Arches field with respect to UKIDSS \begin{lxirformule}$JH$\end{lxirformule}. The colour terms of the WFC3 F127M and F153M filters found with respect to the ground-based ISAAC and UKIDSS JH filters are consistent within their uncertainties. In the Quintuplet field, because of the different calibration procedures, the offset term was not fit independently of the instrumental zeropoints ZPTJ,WFC3 = 1.31 ± 0.06 and ZPTH,WFC3 = 1.96 ± 0.07, such that they are not directly comparable to the absolute offsets derived in the Arches field after a preliminary calibration offset had been applied. After colour terms were removed, the Arches JH photometry was adjusted to match the photometry by Espinoza et al. (2009) in the cluster centre, as in the case of the Ks observations (Sect. 3.1.2).
Finally, the combined JH WFC3 catalogue with calibrated, colour-term corrected magnitudes was transformed to each Ks reference epoch image and matched with the Ks and KsL′ photometry lists separately. Source counts and sensitivity limits for these combined catalogues are provided in Table 2. The sensitivity limits are derived from the peak of the luminosity functions (Figs. 12 and 13) as a first indication of the completeness of each data set in each filter after matching JH with Ks, and L′ with JHKs in the case of the L′ luminosity function exclusively. A rigorous incompleteness treatment, including the effects of catalogue matching, is presented in the next section.
3.5. Incompleteness simulations
There are two major limitations to the sample of excess sources and detected main-sequence members of the Arches and Quintuplet clusters. Stars with main-sequence colours are mostly limited by detection in the bluest filter, WFC3 J. This limitation arises from the high foreground extinction towards both clusters and the GC in general. In addition, the lower resolution of HST/WFC3 of 220 mas compared to the NACO Ks-band resolution of typically 80–120 mas (see Table 1) prohibited the detection of J and H counterparts for Ks sources in areas with high stellar densities. Especially in the crowding-limited core of the Arches cluster, the loss of one neighbour in close pairs is common. Sources with L-band excess, on the other hand, are additionally limited by the L-band completeness limit at L′ ~ 14.5 mag. As all L-band sources are also matched with the JH source list to reveal the excess in the two-colour plane, the detection of excess sources is limited by the constraints in both J and L.
With the aim of quantifying the losses and deriving completeness-corrected disc fractions, we have performed artificial star experiments. For both clusters, artificial stars were inserted in the WFC3 J and H images, and photometry was performed with starfinder (Quintuplet) or daophot (Arches) as on the original images. The J and H catalogues of recovered artificial stars were matched to account for matching losses. For the NACO KS epochs, the less sensitive and hence more limiting proper motion epoch was used to derive recovery fractions. Artificial stars were inserted in the same physical location in each filter. The magnitudes and colours of each artificial star were chosen to represent a typical main-sequence star in each field, with a range of 4.2 <J − Ks< 4.8 mag in the Quintuplet and 4.3 <J − Ks< 6.2 mag in the Arches. The Ks − L′ colour was chosen to be 2.4 mag in the Quintuplet and 2.5 mag in the Arches for all excess source simulations, in accordance with the observed characteristic colour of the excess sources in each cluster. As for the real photometry, the Ks and JH artificial source lists were combined with the same matching parameters. In the real source lists, a limit of Ks< 17.5 mag was imposed to allow for proper motion member selection, and the same selection limit is applied for the calculation of all completeness fractions. The JHKs artificial star lists are employed to calculate the completeness fraction of main-sequence stars. The JHKs catalogues are then combined with the L′ artificial star lists to derive the completeness fraction of the excess samples.
The JH images are mainly limited by crowding effects, as WFC3 provides uniform photometric performance across the field. The NACO Ks and L′ epochs and fields, on the other hand, are influenced by the adaptive optics correction (Strehl ratio, sensitivity, and anisoplanatism) as well as fluctuations in the thermal background especially in L′. These effects cause each field to display different completeness curves. In Fig. 14, the completeness fractions are fitted with fourth-order polynomials. In the Quintuplet simulations, the drop in completeness with fainter magnitudes is significantly steeper than in the more crowding-limited Arches simulations. Two separate polynomials were fitted in the shallow parts of the incompleteness curves at brighter magnitudes and the steep parts of the curves towards faint magnitudes in the Quintuplet simulations. Polynomial fitting provides the advantage that a completeness correction can be calculated for each star's observed magnitude without binning into magnitude bins. The completeness-corrected star counts are used in Sects. 4.1 and 4.2 to obtain corrected excess samples and main-sequence reference samples in each field.
J − H,Ks − L′ colour–colour diagram of the central field of the Quintuplet cluster. The solid black line depicts the extinction vector (Nishiyama et al. 2009), and the red line is offset by 3 × the standard deviation of main-sequence cluster members without excess sources (green points). All L′ excess candidates are labelled (diamonds), with Wolf-Rayet candidates shown in blue (Ks< 12 mag), while fainter excess sources marked in red are candidates for circumstellar disc emission. Non-cluster members are shown in red, and black symbols depict sources with unknown membership status. The J − H,Ks − L′ colour–colour diagrams of the outer fields can be found in Appendix (B.1).
4.1. Disc candidates in the Quintuplet cluster
A robust estimate of the cluster disc fraction, especially at cluster ages above 1–2 Myr, can be obtained from the fraction of L-band excess sources with respect to the main-sequence reference sample (Haisch et al. 2001). To derive L-band excess fractions in each field and in the entire cluster, we combine the L′ photometry with the JHKs source lists. The J − H,Ks − L′ colour–colour diagram of the central Quintuplet field is presented in Fig. 15, while the outer fields are included in the Appendix (Fig. B.1). Excess candidates are selected if their colours are redder than 3σ with respect to the reddening vector, where σ is calculated as the standard deviation of the Ks − L′ colour of all cluster members with Ks< 17.5 mag. In the central field, the deeper sensitivity allowed for a member selection down to stars with Ks ~ 20 mag, such that only a proper motion accuracy limit was imposed in Field 1 exclusively. As Wolf-Rayet stars also display infrared excesses due to their strong envelope emission, only stars fainter than Ks> 12 mag are considered to be disc candidates. The WR candidates are shown in blue in all colour–colour and colour–magnitude diagrams. The astrometric accuracy of these sources is frequently compromised by non-linearity in the PSF core, limiting the value of the membership criterion, yet they display colours and brightnesses consistent with Wolf-Rayet stars belonging to the Quintuplet population (see Figs. 16 and 17). Although the combined JHKsL′ catalogue contains the most reliable photometric sources, not all sources measured in all four filters have proper motion membership information. A few sources are too faint in the less sensitive second epoch observations or happen to fall close to equally bright or brighter neighbours, such that the astrometric uncertainties are too large for these sources to derive membership information (black sources in the colour–colour diagrams). In most Quintuplet fields, there is characteristically one non-cluster member in the excess sample, and one source where the membership situation is unclear (see Figs. 15 and B.1). This is particularly striking in Field 3, where no cluster member excess source is found, yet one source without membership and one non-member show L′ excess emission. A similar situation is observed in Fields 1 and 5, while all excess sources observed in Field 2 are cluster members. Only one faint excess source is observed in Field 4 owing to the shallow sensitivity of the L′ photometry in this field. The formally derived low excess fraction in Field 4 is therefore not representative for the remainder of the cluster. With just two epochs of proper motion measurements, the motion of each star has to be derived as the difference in the positions between both epochs, and no linear fit to the motion can be obtained. Likewise, the likelihood that each star belongs to the cluster or the field does not provide an absolute distinction between cluster members and non-members. More accurate motions, obtained from multiple proper motion epochs, are required to definitively conclude whether the apparent non-member excess sources might belong to the cluster as well. It is therefore too early to discuss their possible origin and the nature of their excess, and we exclude them from the member sample in the following analysis.
L-band excess sources in the Quintuplet cluster.
L′ excess fractions in the Quintuplet cluster.
A total of 24 cluster members with L′ excess emission is found distributed across three of the Quintuplet fields (Figs. 1 and 2). Two sources are detected in the overlap regions of two fields, providing independent confirmations of their Ks − L′ excesses. Two additional excess sources do not have membership information, such that a maximum of 26 excess sources is currently observed in the Quintuplet cluster. The positions and photometry of these sources are provided in Table 4, and a summary of the observed number counts is included in Table 5. Most L-band excess sources in the Quintuplet are detected in the central field, at radii r< 1 pc from the cluster centre. The central field harbours 16 excess sources with J< 22.0 mag, while a total of 18 excess members plus one star with unknown membership are detected down to J = 22.8 mag. In contrast to the central field, the outer fields only feature between 0 and 4 members with L′ excess, which suggests that the outer regions of the Quintuplet cluster harbour very few disc candidates. The limiting factor in the detection of L-band excess sources is the L′ sensitivity itself. The strong thermal background emission at 3.8 μm, which is a combination of sky brightness and dust emission from optical elements such as telescope mirrors, leads to substantial noise in each L-band image. In the outer fields, all detected L′ excess sources are brighter than L′ = 15.0 mag. All L-band excess sources are included in the source catalogue, yet for the derivation of the global excess fraction, only sources brighter than L = 15.0 mag and J = 22.0 mag are considered.
In order to derive the L-band excess fraction and hence the potential fraction of circumstellar discs in the Quintuplet cluster, a main-sequence reference sample has to be defined. This sample has to contain membership information to distinguish cluster stars from field interlopers, and it has to cover the colour range expected for main-sequence sources belonging to the Quintuplet. A problem arises from the fact that disc sources are easier to detect in L′ than main-sequence stars because of their infrared brightness enhancement. Hence, the JHKs source lists are used to derive the reference sample instead of the JHKsL′ combined catalogue (see Stolte et al. 2010). We use the J-band brightness as an indicator of the stellar photospheric luminosity and the stellar mass, as the excess from circumstellar disc emission is expected to be weak at bluer wavelengths. The limiting J-band magnitude in the excess sample can therefore be used to define the main-sequence reference sample in each field. The J,J − Ks colour–magnitude diagrams (Figs. 16 and B.3) reveal that almost all disc candidates with L′< 15 mag are brighter than J = 22.0 mag, and only three sources between 22.0 <J< 22.8 mag are found in the central field. The deeper member/non-member distinction in this field allowed us to include these sources, imposing the same L′< 15 mag limit as in the outer fields, in the full sample of disc candidates. The main sequence was selected down to J = 22.8 mag accordingly. The characteristic main-sequence colour of J − Ks ~ 4.5 mag implies that a limiting magnitude of Ks = 17.5 mag used for the membership samples in the outer fields corresponds to J = 22.0 mag as well. We therefore use J = 22.0 mag as the faint limit to define the main-sequence samples in all outer fields.
J, J − Ks colour–magnitude diagram of the central field of the Quintuplet cluster. L′-excess sources derived from Fig. 15 are marked as red diamonds, while Wolf-Rayet candidates are marked in blue. Proper motion members are shown in green, while non-members are shown in red and sources with unknown membership in black. The imposed magnitude limit to allow for proper-motion member selection, Ks< 17.5 mag, is shown as a dotted line. The horizontal dashed line marks the J-band completeness limit at J = 22 mag above which the combined cluster excess fraction was derived. The vertical dashed lines indicate the main-sequence colour selection applied in addition to the proper motion membership criterion to select the main-sequence reference sample to calculate the disc fraction. The J, J − Ks colour–magnitude diagrams of the outer fields can be found in Appendix (B.3).
The membership source lists derived from proper motions cannot exclude blue disc stars or red bulge giants which happen to move with velocities similar to the clusters (see Hußmann et al. 2012 for a detailed discussion). Hence, we apply an additional colour selection to remove these contaminants. While the blue limit of the cluster main sequence stands out clearly in the CMDs, the red limit of the cluster population is more difficult to discern. The red limit was determined from the red clump population evident in each CMD at J ~ 20 mag, and progressing towards redder colours and fainter magnitudes along a distinctive path (the reddening vector). The red limit is chosen such that the cluster main sequence is completely included in the selection while red clump stars are rejected as efficiently as possible. As a consequence of the varying extinction across the Quintuplet cluster field, the colour selection had to be adapted to the main-sequence colour spread observed in each CMD. The colour selections for main-sequence stars are included in Table 5, and are shown in Figs. 16 and B.3 as dashed vertical lines. In summary, the following constraints define the main-sequence reference samples: i) J< 22.0 mag; ii) 3.7 − 4.0 <J − Ks< 4.5 − 5.5 mag (see Table 5), and iii) source is a proper motion member. To estimate the unavoidable biases imposed by the main-sequence colour selection, the red boundary was shifted by ± 0.2 mag to derive upper and lower limits of the excess fractions in each field individually. These uncertainties are included in Table 5. Both the main-sequence samples and the excess source counts are corrected for incompleteness using individual completeness values for each star as outlined in Sect. 3.5. In Table 5, we include the observed and incompleteness corrected disc fractions for each field. In the two bottom rows, the cluster disc fractions are calculated from the full area coverage of all fields. Table 5 also provides a lower and an upper limit on the true excess fraction, fex = nex/nms, where nex is the number of excess sources and nms the number of stars in the main-sequence reference sample. The upper limit originates from including each of the three fields with excess source populations. Here, the main-sequence reference samples are derived from Fields 1, 2, and 5 exclusively, and fex is correspondingly large as a result of the small main-sequence number count nms. A lower limit to the total disc fraction is derived when including all fields in the main-sequence reference sample. In this case, Fields 3 and 4, which feature 0 and 1 excess cluster members with Ks> 12 mag, are also included, such that the cluster main-sequence population can be considered more complete. The real situation is slightly more difficult to assess, as the varying extinction particularly affects the detection of faint cluster stars in Field 4, where the L-band sensitivity is also compromised. Here, the maximum number of presently known Quintuplet members is used as the main-sequence reference sample, increasing nms and minimising fex. For the combined excess fraction in the Quintuplet cluster, a J-band limit of 22 mag is imposed in all fields (F1-F5) on the main-sequence reference samples.
The values for all fields and all main-sequence reference samples are in the range 3–5% (again with the exception of Field 4). Combining all excess source members, we find an upper limit to the incompleteness-corrected excess fraction of 4.1% and a lower limit of 3.7% for the full cluster area. Including the three excess sources without membership information in the excess number count increases the upper limit to 4.3%. Employing these upper and lower limits as uncertainties, we conclude that the excess fraction in the Quintuplet cluster at its present age of 4 Myr is 4.0 ± 0.3%.
Although the Quintuplet cluster has a lower density than the core of the Arches cluster, a maximum uncertainty in the disc fraction can be derived by assuming the same 10% membership selection uncertainty as found in the Arches core (Sect. 3.1.4). A number count error of ± 10% in either the Field 1+2+5 main-sequence sample or in the full main-sequence sample changes the respective combined excess fraction by ± 0.4%. If we additionally assume the propagated uncertainty in incompleteness-corrected number counts is 10% as well, the corresponding uncertainty in the corrected disc fraction is found to be ± 0.5%. Taking into account the uncertainty imposed by the colour selection (the higher value of σfex,low and σfex,high in Table 5), and assuming that both uncertainties are independent, the disc fraction in the Quintuplet cluster is found to be 4.0 ± 0.7%.
The stellar masses of the host stars of these circumstellar disc candidates can be estimated from their J-band brightness, which is presumed to represent the stellar photosphere. This assumption is validated by the fact that most of our sample sources show little to no near-IR excess emission in the 2.2 μm Ks band. The mean extinction towards each field was assumed in the conversion of J-band magnitudes into stellar masses. Our imposed J-band limit of 22.0 mag corresponds to a lower mass limit of 2.2 M⊙ for both a 4 and 5 Myr Geneva main-sequence isochrone (Lejeune & Schaerer 2001). The brightest source with mid-infrared excess emission has J = 19.6 mag, which corresponds to a stellar mass of 14.2 M⊙ for the 4 Myr isochrone. Effective temperatures covered by these stellar masses in the Geneva models range from 10 500 to 29 300 K. Following the recent compilation of Currie et al. 2010 (see their Table 7) for main-sequence stars with solar metallicity, these temperatures suggest that the disc candidates in the Quintuplet cluster are found around B9.5V to B0V stars.
Ks, Ks − L′ colour–magnitude diagram of the Quintuplet cluster centre. All sources are labelled as in Fig. 16. One faint L′ detection has no membership information, but is consistent with the remainder of the excess sources regarding its Ks − L′ colour. The Ks, Ks − L′ colour–magnitude diagrams of the outer fields can be found in Appendix (B.5)
J − H,Ks − L′ colour–colour diagram of the central field of the Arches cluster. The solid black line depicts the extinction vector (Nishiyama et al. 2009), and the red line is offset by 3 × the standard deviation of main-sequence cluster members (green points) without excess sources. Excess sources fainter than Ks = 12 mag marked as red diamonds are candidates for circumstellar disc emission. Non-cluster members are shown in red; black symbols depict sources with unknown membership status. As many of the excess sources in the cluster centre are located close to brighter stars because of the high stellar density in the cluster core, only 7 of the 23 L-band excess sources found in our Keck investigation (Stolte et al. 2010) are detected in the lower-resolution NACO images (green asterisks). At larger radii, the fraction of excess sources is expected to be more complete owing to the less severe crowding effects. The J − H,Ks − L′ colour–colour diagrams of the outer fields can be found in Appendix (B.2).
Currently, it is unclear whether the fraction of excess sources increases towards lower-mass stars; the sample statistics are not sufficient to provide excess fractions in various mass bins. From these data alone, we can therefore not conclude that stars with lower or higher masses are more likely to exhibit L-band excess emission. Given the UV-intense environment of this massive cluster, one might not expect to find disc candidates at an age of 4 Myr at all. The implications of this finding and possible explanations for an elevated disc fraction, especially in a more evolved, massive young cluster, are discussed in Sect. 5.
Despite the large range of radial distances from the cluster centre covered by the different pointings, the excess fractions derived individually for Fields 1, 2, and 5 are identical within their uncertainties. Hence, we find no indication for a radial variation of the disc fraction in the Quintuplet cluster. This finding contrasts with our earlier detection of a significant variation in the disc fraction of the Arches cluster, where a radial increase from 3% to 10% was observed (Stolte et al. 2010). The radial variation in the Arches cluster might be caused by disc destruction in the dense cluster core, where UV radiation from neighbouring sources and dynamic interactions could affect disc survival (Olczak et al. 2012). These disc destruction mechanisms might not be effective at the lower central density of the Quintuplet cluster, such that disc sources are equally distributed at all radii. Alternatively, the generally lower disc fractions in the Quintuplet cluster of ~4% might mask the radial dependence, as radial variations of ± 0.5% would remain undetected given the main-sequence selection uncertainties. We therefore conclude that, within these uncertainties, the disc fraction in the Quintuplet cluster shows no radial variation.
L-band excess sources in the Arches cluster.
L′ excess fractions in the Arches cluster.
4.2. Disc candidates in the Arches cluster
In the Arches cluster, a total of 28 candidates with 3σL′ excess emission is found, of which 20 are proper motion members of the cluster, 6 are non-members, and for 2 excess sources in the central field the membership is unknown (Figs. 18 and B.2). Three excess sources detected at the edge of Field 5 are also observed in either Field 1 or Field 3. Among the proper motion members, one excess source qualifies as a Wolf-Rayet candidate with Ks = 11.2 mag and is excluded from the final disc sample. The sample of circumstellar disc candidates identified as likely cluster members therefore contains 19 sources (21 including the two sources with unknown membership). The excess source list can be found in Table 6, and the number counts in each field are summarised in Table 7. As in the Quintuplet sample selection, the main-sequence reference sample was chosen around cluster members with characteristic main-sequence colours given the extinction in each field (see Figs. 19 and B.4 and Table 7). Given the similarity in the J-band photometries and the main-sequence colour range, a limiting magnitude of J< 22 mag is applied to obtain a complete main-sequence sample. In the Arches, there are two sources in the central Field 1 and one source in Field 5 with J> 22 mag, yet the majority of excess sources are significantly brighter than this limit (Figs. 19 and B.4).
J, J − Ks colour–magnitude diagram of the central field of the Arches cluster. L′-excess sources derived from Fig. 18 are marked as red diamonds. Proper motion members are shown in green, non-members are shown in red, and sources with unknown membership in black. The imposed magnitude limit to allow for proper-motion member selection, Ks< 17.5 mag, is shown as the dotted line. The horizontal dashed line marks the J-band completeness limit at J = 22 mag above which the combined cluster excess fraction was derived. The dashed vertical lines indicate the main-sequence colour selection applied in addition to the proper motion membership criterion to select the main-sequence reference sample to calculate the excess fraction. The location of the red diamonds near the main sequence shows most prominently that several L′-excess sources do not show excess emission at Ks. The J, J − Ks colour–magnitude diagrams of the outer fields can be found in Appendix (B.4).
All disc fractions provided below are derived from JHKsL′ excess sources and JHKs main-sequence stars resolved with NACO and WFC3 simultaneously. Compared to our high-resolution Keck/NIRC2 study in the central 0.8 pc of the Arches cluster (Stolte et al. 2010), a more rigorous incompleteness analysis was implemented here. For both the NACO Ks as well as the WFC3 JH-band observations, the spatial resolution limits source detections especially in the crowding-limited cluster core, and has a larger effect on J and H detections than in the sparser Quintuplet cluster. Among the 12 excess sources resolved in the central field (see Fig. 2), 8 were previously known from our earlier Keck/NIRC2 investigation (Stolte et al. 2010). While 4 new sources are detected in the central field in the analysis presented here, 16 excess sources resolved with Keck/NIRC2 remain undetected in the central cluster region with WFC3. These sources are located near brighter neighbouring stars in the dense cluster centre, and are not resolved in the WFC3 images. As the same losses affect main-sequence stars near bright neighbours, and as our incompleteness simulations are calculated for the combination of NACO and WFC3, the previously known sources are not included in the following analysis. Even for stars brighter than the J-band limiting magnitude of J ~ 22, significant losses are observed as a result of the combined effects of crowding and undersampling in the WFC3 images. This limit coincides with the proper-motion imposed Ks< 17.5 mag selection especially for fainter and redder stars (J − Ks ≥ 4.5 mag), and additionally coincides with the L′ detection limits of 14–15 mag for excess sources (see Figs. 13 and 20). As a consequence, the individual correction factors can be very large with values up to ~80% for excess sources (see Fig. 14), where both a detection in J and in L′ is required simultaneously. The requirement that excess sources be detected in all four filters leads to higher correction factors than for the JHKs main-sequence reference sample, such that the disc fraction tends to increase when incompleteness corrections are taken into account (see Table 7).
The numbers of excess sources and the reference main-sequence samples, including incompleteness-corrected values, are summarised in Table 7. Excess fractions are derived for each full field to keep the covered areas comparable. Three excess sources are located in the overlap regions between Fields 1, 3, and 5, as indicated in Table 6. The combined disc fractions are corrected for redundancy. The correction implies using only the unique excess and main-sequence samples, and reducing the incompleteness corrected number counts accordingly. The total numbers in the final two rows in Table 7 are therefore not the sums of the individual field counts in Rows 1–5. The incompleteness-corrected excess fractions range from 2% in Field 4, where only one excess source is found, to 11–13% in Field 5 and the cluster centre. No excess sources are observed in Field 2, where the unusually high extinction limits the detection of J< 22 mag main-sequence cluster members to just 21, and the detection of excess sources is likely impeded by the enhanced foreground extinction. Excluding Field 2, a total incompleteness-corrected excess fraction of 9.4% is calculated for Fields 1, 3, 4, and 5, which decreases slightly to 8.9% for the full area coverage including Field 2. Including the two excess sources with unknown membership and their respective incompleteness corrections, the excess fractions increase to 10.0% and 9.6%, respectively.
As shown in Sect. 3.1.4, individual astrometric uncertainties can induce an error in the membership selection of at most 10%. Assuming this uncertainty in the main-sequence samples as a conservative estimate leads to an uncertainty of ± 0.5% in the overall uncorrected disc fraction. The additional assumption that such an error also imposes a 10% uncertainty in the incompleteness-corrected main-sequence number counts (a 10% variation in nms,corr in Table 7), provides an estimate of the maximum uncertainty in the incompleteness-corrected disc fractions. This uncertainty in the membership selection supercedes the uncertainties by the main-sequence colour selection when the global disc fraction is derived. Allowing for this 10% variation in the corrected main-sequence number counts and taking into account the uncertainty imposed by the field coverage (inclusion or exclusion of Field 2, see Table 7), the global disc fraction in the Arches cluster is found to be between 8.1% and 10.4%, or 9.2 ± 1.2%.
In our previous study, we found that the excess fraction increases from 3% in the immediate cluster core (r< 0.2 pc), which is not well resolved in the NACO and WFC3 data presented here, to 10% at r> 0.3 pc (Stolte et al. 2010, see their Fig. 11). The fraction of 10% is larger than the total excess fraction of 6 ± 2% found in the central area of the cluster previously (Stolte et al. 2010), but is consistent with fex = 10% found for stars at larger radii, r> 0.3 pc. This consistency is expected as the NACO/WFC3 sample is dominated by stars outside the cluster core (see Fig. 2). With the increased area coverage, the total disc fraction in the Arches cluster has therefore increased from 6 ± 2% (r< 0.8 pc) to 9.2 ± 1.2% including stars with radii out to r< 1.5 pc. Here, the uncertainty is estimated from the variation in the excess fraction due to different choices of the main-sequence reference sample and the inclusion of the two excess sources with unknown membership (Table 7).
In addition to the area coverage, there is one striking difference with our earlier Keck/NIRC2 study. Given the limiting spatial resolution of ~200 mas with HST/WFC3 compared to ~60 mas with Keck/NIRC2, high incompleteness corrections are applied to each star in the cluster core. The large incompleteness-corrected number of excess sources, nex,corr = 66, extrapolated from just ten excess proper motion members detected in the central field, is a consequence of the large completeness factors that had to be applied. This correction bears a high level of uncertainty, and might over-extrapolate the true number of excess sources located in the core. As excess sources are more prone to incompleteness because of their faint JH magnitudes, the corrected disc fraction of 12–13% in Field 1 has to be considered an upper limit. A lower limit to the central disc fraction is obtained when only the observed excess sources and main-sequence members, without incompleteness corrections, are taken into account. The total observed disc fraction is then found to be 4.8 ± 0.5% in the central field only, which is dominated by sources in the inner 0.4 pc. Although several central excess sources are not resolved here, this excess fraction compares well with the disc fraction of 6 ± 2% found for the inner cluster in the higher-resolution Keck/NIRC2 data (Stolte et al. 2010).
Consistent with the findings for the Quintuplet fields above, there are one or two excess candidates qualifying as field interlopers in each 27′′ × 27′′ NACO field. The lower data quality and less confined proper motion plane derived from the Arches images, however, does not allow for a final conclusion on the apparent non-members among the excess sample. The large scatter observed in the proper motion diagrams (Figs. 8 and A.2) indicates that cluster stars might have scattered into the field distribution. A third proper motion epoch would be required to fit each star's motion with a linear fitting function, which might lead to an even more reliable and extended member sample among both the excess sources and the main-sequence reference sample.
Comparing J-band magnitudes to a 2.5 Myr Geneva isochrone results in a mass range of 2.1 <M< 17.5 M⊙ (9500 <Teff< 32 200 K), corresponding to A1V to O9V stars (Currie et al. 2010), similar to the mass range of disc host stars observed in the Quintuplet cluster.
In summary, a fraction of 9.2 ± 1.2% of early A- to early B-type stars are found to display L-band excesses in the Arches cluster when individual, local incompleteness corrections are taken into account.
A fraction of 4.0% of the Quintuplet and 4.8% of the Arches cluster members are observed to display L-band excess emission, which is here interpreted as evidence of circumstellar discs. Incompleteness-corrected excess and main-sequence samples result in disc fractions of 4.0 ± 0.7% in the Quintuplet and 9.2 ± 1.2% in the Arches cluster. The disc host stars are dominated by B-type main-sequence stars with a mean mass of ~6 M⊙ in the Quintuplet and of ~7 M⊙ in the Arches cluster. Disc survival in the UV radiation field of B-type main-sequence stars (Herbig Be stars) for time periods exceeding 2.5–4 Myr is unexpected. Such a long lifetime would argue against primordial discs, and raises the question whether the circumstellar material can have formed recently from secondary processes. This hypothesis is discussed below in the context of transition discs and binary mass transfer.
In this section, we first discuss the survival of discs in the Arches and Quintuplet clusters compared to young star clusters outside the GC region (Sect. 5.1). We proceed to discuss the possible nature of the L-band excess sources. In Sect. 5.2, we provide estimates of the physical properties of the observed L-band excess population, and compare the derived size scales and masses to known circumstellar, pre-transitional, and transitional discs. The origin of the circumstellar dust emission is discussed in the context of primordial disc survival vs. a possible secondary origin of dusty discs from binary interactions (Sect. 5.3).
Ks, Ks − L′ colour–magnitude diagram of the central field of the Arches cluster. All sources are labelled as in Fig. 19. A pronounced cluster main sequence composed of proper motion members (green) is seen in this central field. The Ks, Ks − L′ colour–magnitude diagrams of the outer fields can be found in Appendix (B.5).
Disc fraction as a function of cluster age, extracted from Haisch et al. (2001) (NGC 2024, Trapezium, IC 348, NGC 2264, NGC 2362, Taurus, Chamaeleon I), Hernández et al. (2007) (σ Ori), Hoffmeister et al. (2006) (M 17: NGC 6618), Maercker & Burton (2005) (30 Dor region), and Maercker et al. (2006) (NGC 3576: RCW 57). All disc fractions are derived from L-band excesses with the exception of σ Ori, where IRAC SED slopes were used to select optically thick discs. Symbols are scaled to the logarithm of the cluster mass, from the least massive with ~ 200 M⊙ in stars (NGC 2024, IC 348, σ Ori), to the most massive with > 30 000 M⊙ (30 Dor, see Stolte et al. 2010 for details). Black and red circles mark disc fractions derived from high-mass stars of types OBA only, while blue (light grey) circles mark populations dominated by low-mass stars. The three starburst clusters Arches, Quintuplet, and NGC 3603 are highlighted in red. The 30 Dor disc fraction covers the extended HII region, including star-forming ridges harbouring YSO candidates, but does not resolve the central cluster, and is therefore an upper limit to the disc fraction in this environment. The dash-dotted line corresponds to the linear decrease in disc fraction vs. cluster age suggested by Haisch et al. (lighter circles only). The dotted line represents the same relation, shifted to lower ages and disc fractions, indicating that disc depletion progresses more rapidly in the most massive clusters. The error bars in the disc fraction represent radial variations in those clusters where a radial dependence is observed (see also Stolte et al. 2010). In the case of the Arches and NGC 3603, the downward arrow indicates the radial decrease in the fraction of discs from larger radii toward the cluster core (NGC 3603 outer cluster region: Stolte et al. 2004; resolved core: Harayama et al. 2008).
Radial distribution of disc host star masses, as derived from the J-band luminosity and a 2.5 Myr (Arches, top panel) and 4 Myr (Quintuplet, bottom panel) Geneva isochrone. Green symbols mark proper motion members, red symbols non-members, and black symbols represent stars without membership information.
5.1. Disc survival in young star clusters
A detailed discussion of the low disc fractions observed in the Arches cluster was presented in Stolte et al. (2010). The updated Haisch diagram including the data point of the Quintuplet cluster is shown in Fig. 21. From a statistical viewpoint, the Arches cluster has a disc fraction substantially lower than expected from nearby young populations, while the older age of the Quintuplet cluster renders the disc frequency more consistent with the expected time evolution. In dense star clusters, gravitational interactions accelerate disc disruption, especially for sources near the cluster core. Olczak et al. (2012) showed that in the dense core of the Arches cluster, a rapid removal of 30% of all discs is expected from dynamical interactions alone during its 2.5 Myr lifetime. As expected, disc mass loss is shown to be most efficient in the cluster core in their simulations, yet a strong dependence with host star mass is also found. Olczak et al. (2012) suggest that survival is most likely around B-type stars. Discs around lower-mass stars are more rapidly depleted by numerous encounters, while discs around the highest mass stars are most affected by close encounters with lower-mass stars during gravitational focusing events.
In a cluster as dense as the Arches core, and presumably the Quintuplet at a younger age, gravitational interactions would affect primordial and secondary discs alike. If interactions were the dominant source of disc depletion, we would expect only relatively massive stars to retain their discs in the dense cluster core, including high-mass binaries with substantial amounts of mass transfer. For stars with lower presumed disc masses, the chances of disc survival are higher at larger cluster radii. Investigating the radial distribution of disc host star masses as derived from the J-band luminosity above (Fig. 22), we do not observe a prominent radial decrease in the host star mass. Nevertheless, the 2–3 most massive disc-bearing stars, with M> 12 M⊙, are found in both clusters near the core at r< 0.5 pc. This finding provides weak evidence that gravitational interactions play a rôle in disc depletion, as mass segregation also causes the most massive stars to sink to the cluster centre. Deeper observations detecting discs around lower mass host stars will shed more light on the dominant disc destruction mechanism in these massive, young Galactic centre clusters.
5.2. The nature of the excess sources
At a wavelength of 3.8 μm, it is assumed that the L-band emission originates from hot material at the inner rim of a circumstellar disc (Espaillat et al. 2011; see Alonso-Albi et al. 2009 especially for Herbig Be stars). Here, we provide an estimate of the distance of the material from the central star and the disc mass, and compare this to the observed properties of debris and transition discs detected around lower-mass stars.
5.2.1. Distance of the dust from the star
A first approximation of the distance of the hot dust from the star can be deduced assuming radiative equilibrium. The radiation from the central star hitting a dust particle with radius a and area πa2 is given by where is the stellar luminosity, Rstar is the radius and Tstar the effective temperature of the central star, adust is the radius of a dust particle, A is the albedo of the dust, σSB is the Stefan-Boltzmann constant, and F(D) is the flux that reaches the particle at distance D from the star. A low albedo implies that the entire flux is absorbed by the dust, while a high albedo means total reflectivity.
At the same time, the particle emits radiation at the characteristic dust temperature Tdust: (5)In equilibrium, the radiation absorbed and re-emitted by dust particles at the dust temperature has to be the same, Labs = Ldustgrain, which leads to Using Wien's law with the wavelength of the NACO L′ filter, λL′ = 3.8 μm, yields an emission temperature of Tdust = 763 K. However, from the observation of an L-band excess alone the wavelength of the peak emission of the dust is unknown.
A minimum distance of the dust to the star can be estimated from the maximum possible temperature in the absence of a K-band excess, as observed in more than half of the excess sources. We note that even in the presence of a K-band excess, the K-band excess has to be smaller than the L-band excess for a strong Ks − L′ excess to be observed, as equal excesses in both filters would cancel each other. With characteristic colour uncertainties of 0.2 mag, less than 20% of the ~1 mag Ks − L′ excess should be present at 2.2 μm. The temperature of a black body with 20% of the radiation at 2.2 μm corresponds to 1220 K and the peak will occur between the Ks and L′ bands at 2.4 μm. A lower limit to the temperature is not as strictly defined, as we have no knowledge of the longer-wavelength emission. If the discs are evolved, consisting of substantial amounts of larger grains, a radiation maximum in the mid-infrared can be expected. Under the assumption that the observed L-band emission at 3.8 μm represents only 20% of the total disc emission, the black-body temperature corresponds to 704 K with a peak wavelength of 4.1 μm, shortly beyond L′.
With a characteristic B2V 10 M⊙ star for our excess sources, we assume a main-sequence temperature of Tstar = 22 000 K, stellar luminosity of 5770 L⊙, and from Stefan-Boltzmann's Law a radius of 5.2 R⊙.
Allowing for disc temperatures between 704 K and 1220 K, the distance of dust from the star under the black-body approximation is estimated to be In the case of complete light absorption, a maximum of the distance of the irradiated matter from the central star is derived: With the more moderate assumption that half of the stellar radiation is absorbed by the dust, lower minimum and maximum radii are obtained, where the minimum represents the case where 20% of the radiation is emitted at Ks and the maximum represents the case where 20% of the emission is emitted at L′. The simplified assumption of black-body radiation provides only a very approximate range of likely dust radii. Detailed dust distribution calculations would be required to obtain a more realistic radial distribution of the dust disc; however, the single excess value at L-band does not provide firm observational constraints for such models, and such an effort would be beyond the scope of this paper.
For a B2V 10 M⊙ star we expect a dust distance of 3–11 AU in the limiting cases that between 60% and 20% of the dust radiation are emitted at L′, respectively. Distances in the range of several AU suggest that dust in the inner disc is destroyed, consistent with evolution models of discs around high-mass stars. The fact that only weak K-band emission is detected in most of the excess sources also suggests that the inner, and hotter, part of the dusty disc component is depleted.
5.2.2. Limits to disc masses
A lower limit to the disc mass can be obtained when assuming the disc is optically thin at the observed wavelength. Following the derivation in Hartmann 2000 (pp. 113–114), we can estimate the disc mass from (6)where Mdust is the desired dust mass, ν is the frequency where the luminosity Lν is measured, κν is the dust opacity, T(D) is the temperature in the disc at the distance D from the star, p and q are the exponents of the radial density and temperature profiles of the disc, k is the Boltzmann constant, and c is the speed of light. Typical values for the exponents of the radial and temperature profiles are p = q = 0.75, hence .
For the dust opacity, following Eq. (6.13) in Hartmann (2000), we assume at a frequency ν = 1014 Hz. This value is consistent with the dust opacity observed by Indebetouw et al. (2005), κλ(4.5 μm) = 0.9 m2 kg-1.
With νLν = 4πd2νFν = 4πd2λFλ, and λFλ inserted from Appendix D, we find . Using the temperature limits derived in the previous section, 704 K and 1220 K, the lower limit to the dust mass is calculated to be where we have assumed that the L-band excess stems entirely from the dust emission. If the discs are still gas-rich, the frequently used gas-to-dust ratio of 100:1 would imply the total disc mass to be higher by a factor of 100. We note here that the entire calculation is based solely on the brightness of a B2V star and the assumption of an L-band excess of 1 magnitude above the stellar photosphere.
This disc mass estimation implies several assumptions originally derived for optically thin discs at long wavelength radiation (1.3 mm, Beckwith et al. 1990). Both the density and the temperature profiles of the disc are assumed to be power laws (Hartmann 2000; Beckwith et al. 1990). However, the structure of our excess objects cannot be deduced from L′ observations alone. In addition, the Planck function determining Lν is represented by the Rayleigh-Jeans approximation in the long-wavelength tail, and T(D) represents the temperature at the outer boundary of the disc. This is not true in our estimation above. Here we have assumed that the radiation at L′ is the dominating dust radiation component, such that Wien's law can be used to obtain the dust temperature. As the dust is heated most intensely at the inner rim, the radius corresponds to the inner disc radius rather than the outer boundary. Beckwith et al. (1990) argue that the mass estimate depends only weekly on the outer boundary in the case of optically thin discs. In the above calculation, we have made the assumption that all dust particles are located at the inner dust sublimation radius derived to be on the order of 10 AU in the previous section. However, a lower temperature T(D) at the outer boundary decreases the disc mass estimate. If the temperature follows the assumed power law profile T(D) = T0(r/r0)-0.75 (see also Hartmann 2000; Beckwith et al. 1990), T(D) could be as low as 140 K at an outer radius of 100 AU, and the disc mass would be lower by a factor of 5.
While this mass estimation reflects the standard derivation of disc masses in the literature, which are mostly obtained from millimetre radiation, some of the uncertain assumptions above can be avoided. Here, we start from the total luminosity of the dust grains as given in Eq. (5), Ldust = Ndust × Ldustgrain, where Ndust is the total number of dust grains in the disc contributing to the total dust luminosity Ldust. The value of Ndust can be expressed in terms of the standard dust grain size distribution derived for interstellar dust, n(a) ∝ a-3.5 (Mathis et al. 1977) (7)where a is the grain size with minimum and maximum grain radii of amin and amax, and cdust represents an unknown normalisation factor.
Inserting Ldustgrain from Eq. (5) yields (8)The total disc mass can then be calculated from the mass of the individual dust grains by integration over the assumed size distribution. The mass of each individual grain is given by assuming a mean density of ρgrain to be constant for all dust grains in the disc. Integrating over all grains assuming the same grain size distribution, n(a) ∝ a-3.5, yields the total dust mass in the disc: (9)The normalisation factor cdust is derived from the known L-band excess emission of the disc using Eq. (8). For the order of magnitude estimate desired here, we use a characteristic L-band excess of 1 mag above the stellar photosphere as before, and the corresponding total dust luminosity of Ldust = 3.4 × 1026 kg m2 s-3 (Appendix D). Using the normalisation factor cdust to derive the dust mass, a minimum and maximum grain size of 0.025 and 0.250 μm, and assuming a grain density of ρgrain ~ 2.3 g cm-3 adequate for carbon grains, yields a total dust mass of for a temperature of 704 K. Using the upper limit to the dust temperature, 1220 K, the dust mass decreases to two to three orders of magnitude smaller than the disc mass estimated above. The lower mass estimate is a direct consequence of the assumed steep grain size distribution, n(a) ∝ a-3.5. This distribution, which is used as the standard size distribution in protostellar discs, implies that most of the dust particles are in the small spatial regime, adust ≪ λobs, and are small compared to the wavelength of the observation. While this might be a good approximation for early, primordial dust discs, the particles in evolved discs are expected to have undergone grain growth. At the same time, the grain size distribution in mass transfer discs is entirely unknown. Even if the discs are of primordial origin, a flatter grain size distribution with more particles at larger size scales or a maximum grain size larger than the classical value of amax = 0.250 μm (Mathis et al. 1977) would give rise to a higher dust mass in the disc.
In this estimate, we again had to make the assumption that all dust contributing to the total dust luminosity in L-band and hence the observed infrared excess emission was distributed near the dust sublimation radius at a dust temperature between 704 and 1220 K.
These disc masses are strict lower limits for two reasons.
We only derive the mass from the contribution of warm dust thatis heated by the stellar flux to temperaturesaround 800 K. This dust is locatedat a distance of ~10 AU, and is likely only a small fraction of the total dust mass in the disc. Any colder dust at larger radii is not accounted for.
We have assumed an optically thin disc in the first estimate, which might be reasonable for transition discs, where dust and gas are already clumpy and partially depleted. However, the inner rim of the disc, where the L-band excess is emitted, is hot and likely partially optically thick. The optically thick portion of the disc cannot be penetrated and is not included in the mass estimate.
Both aspects of these considerations lead to an underestimate of the disc mass. We therefore conclude that the observed excess originates from circumstellar discs at least as massive as 10-10MJup, and likely as massive as 10-7MJup. If the discs contained a large fraction of gas close to the primordial gas-to-dust ratio of 100:1, the total disc mass could be as large as 10-5MJup. These low masses support the suggestion that we are indeed observing either mass transfer or transition discs with large L-band excesses, as discussed in Sects. 5.2.3 and 5.3.
5.2.3. Comparison to pre-transitional discs
The discs in the Arches and Quintuplet clusters are detected because of their Ks − L′ colour excess. The prerequisite for such a colour excess is that L-band emission exceeds both photospheric levels as well as any excess emission in K-band. While primordial discs around very young Herbig Be stars (<1 Myr) feature prominent near-infrared excesses to wavelengths as short as 2 μm, a weak or absent K-band excess in combination with a substantial L-band excess indicates a later stage of disc evolution. For high-mass stars, this implies the formation of an inner hole leading to a larger radius for the hot, inner rim observed at 2–4 μm (e.g. Alonso-Albi et al. 2009). As discussed in Sect. 1.1, a survival timescale of more than 2 Myr is also not expected for primordial discs around B-type stars because of their strong UV radiation field. Transitional or debris discs, on the other hand, are already depleted in L-band emission, such that the excess sources in the Arches and Quintuplet clusters display properties between primordial and transitional discs. If the observed discs in the Arches and Quintuplet clusters at ages 2.5–4 Myr around B-type stars have evolved from primordial discs, they could be the counterparts to the recently found near-infrared bright evolved F star discs (Fujiwara et al. 2013), yet around higher-mass host stars. In order to conclude whether the observed L-band emission might originate from pre-transitional discs, the inner (sublimation) disc radius and the amount of the L-band excess emission are compared to the physical properties of the pre-transitional discs in the sample of Fujiwara et al. (2013) displaying L-band excesses.
Recently, Fujiwara et al. (2013) presented a survey of 18 AKARI sources in the transition phase from protoplanetary to debris objects. They defined their sample from nearby (d< 120 pc) spectrally classified main-sequence A-M stars with AKARI 18 μm detections, which were cross-correlated with the 2MASS JHKs data base. WISE 3.4 and 4.6 μm photometry was obtained to cover the mid-infrared region of the SEDs. In their sample, four sources show clear indications of mid-infrared excess down to 3.4 μm (L-band) (namely, HD165014, HD166191, HD167905, HD176137; compare to Fig. 3 in Fujiwara et al. 2013). The mid-IR excess emission is measured in their sample around low-mass G and F main-sequence stars, and they find inner disc radii of 0.7–1.4 AU with maximum temperatures of 400–500 K for the spectrum of a 1.4 M⊙ star.
Following their procedure, we can use the luminosity of our B2V template star as an example to calculate the inner disc radius, (10)where L∗ is the stellar luminosity, Tin the inner disc temperature, and σSB the Boltzmann constant (see Eq. (3) in Fujiwara et al. 2013). With an absolute luminosity of 5770 L⊙ for a B2V star, and an inner disc temperature between 704 and 1220 K (Sect. 5.2.1), the inner disc radius of a 10 M⊙L-band excess source is expected to be in the range 4 − 12 AU, such that the inner disc of a B-type star is depleted to substantially larger radii than the FG discs, as expected. The same authors also provide the ratio of the dust to stellar luminosity, Ldust/L∗, from which the expected L-band excess for these transition discs can be derived. For a 1.4 M⊙ main-sequence star, a stellar luminosity of Lbol = 3.47 L⊙ is assumed. As the ages of the transition discs in the Fujiwara et al. (2013) sample are not known, the L-band magnitude is adopted from the 2.5 Myr Geneva isochrone used above to derive the masses of our disc host stars. For a 1.4 M⊙ star, these isochrones suggest ML = 2.54 mag, such that we find for the stellar flux Converting to the standard distance of 10pc to obtain the absolute luminosity, L1.4,L = 2.3 × 1024 W, and using Lbol = 3.47 L⊙, we derive L1.4,L = 6 × 10-3L⊙ = 1.7 × 10-3L1.4, ∗ as the expected photospheric luminosity of the disc host stars in a standard L-band filter. From the dust-luminosity-to-star ratio provided in Table 13 of Fujiwara et al. (2013), the excess emission fraction from the discs for the four strong emission sources can be estimated. With Ldust/L∗ = 6 × 10-3,9 × 10-3,2 × 10-2,5 × 10-2, and using L1.4,L = 1.7 × 10-3L1.4, ∗, the excess factor of the luminosity above the stellar photosphere in L-band caused by the dust contribution is calculated to be , where again Ldust/L∗ is the dust luminosity in terms of the total luminosity of a 1.4 M⊙ host star, and L1.4,L/L∗ is the photospheric L-band contribution. These relative fractions correspond to L-band excesses of ΔLdust = 1.4,1.8,2.7,3.7 mag. These values are comparable to, and even slightly larger than, the L-band excesses measured in the Arches and Quintuplet excess sources.
In summary, under the assumption that the central stars are B2V stars, the hot dust emission likely occurs at large radii of 3 − 12 AU. As expected, the inner dust rim is further from the central star as in the case of F-type stars with dust sublimation radii of 0.7–1.4 AU. The L-band excess emission derived from the dust luminosity of these lower-mass stars, however, is consistent with the observed L-band excesses in the dusty Arches and Quintuplet sources. The question remains whether primordial discs could have survived for a sufficiently long time to evolve into pre-transitional discs, or whether the discs formed later as a result of mass transfer. If the host star masses were lower than estimated, towards the low end of our J-band magnitude range (3 M⊙), a survival of primordial discs as observed around F-stars yet with dust at larger radii would be more likely.
5.3. A secondary disc origin
Espaillat et al. (2012) recently suggested a physically motivated distinction between primordial, pre-transitional, and transitional discs. Following their definition, a disc is only transitional when a near-infrared excess is no longer present, which is interpreted as transitional discs being discs with inner holes (optically thin inner discs). A pre-transitional disc, on the other hand, is composed of an optically thick inner disc at a few AU from the central star leading to near-infrared excesses, a lack of mid-infrared emission corresponding to a radial gap, and strong far-infrared excess emission. As our sources show weak Ks excess, yet substantial L′ excesses, they would formally qualify as pre-transitional discs from their near-infrared properties alone. Without knowledge of their far-infrared SEDs, other possibilities have to be considered as well.
In addition to the detection of the pre-transitional/transitional discs in Fujiwara et al. (2013) discussed above, Moór et al. (2011) detected an unexpectedly massive CO gas component in young debris discs with ages up to 30 Myr. For their two A1V and A3IV-V stars having strong CO emission, line profile modelling suggests an inner radius for the dust disc of ~60 AU with a dust temperature of 60–80 K. This radius is even more extended than the L-band emission region estimated for the Arches and Quintuplet sources of ~10 AU above. Moór et al. (2011) argue that UV radiation from both the central star and the interstellar radiation field is capable of destroying the molecular gas component on very short timescales, as shielding becomes inefficient within 500 years at every location in their modelled transition disc. They therefore argue that the gas discs are of secondary origin. In K-band spectra available for three of the Arches excess sources, strong CO bandheads at 2.3 μm are detected, indicating that the Arches discs contain a gaseous component as well (Stolte et al. 2010). At the higher central stellar masses and the stronger ambient radiation field of our B-type cluster stars, rapid gas depletion is even more expected (Hollenbach et al. 1994; Alexander et al. 2006; Cesaroni et al. 2007), such that a secondary origin of the CO emission observed in the three spectroscopically identified Arches excess sources might provide an explanation of the Arches and Quintuplet excess sources alike.
Two scenarios have been discussed to explain secondary discs around lower-mass stars. Both are related to the B[e] and Be phenomena. In single Be stars, equatorial mass loss due to rapid rotation close to the critical rotation velocity removes angular momentum from the surface of the star (Granada et al. 2013). Although absolute numbers of expected rapid rotators with optically thick discs are not known, theoretical models predict a fraction of a few per cent at cluster ages of a few Myr (Granada et al. 2013, see their Fig. 2), which would be consistent with the disc fractions observed in the Arches and Quintuplet clusters. The simulations naturally predict denser, more extended, and more massive discs around higher-mass stars, which could explain why our sample is dominated by mid- to late B-type stars. For rapidly rotating stars in the mass range 2 − 9 M⊙, Granada et al. (2013) find modelled disc extents of 10–30 AU and disc masses of 10-8 − 10-11M⊙ reaching the same order of magnitude as our lower disc mass limit (Mgas + dust ~ 10-8M⊙) estimated for the Arches and Quintuplet excess emission sources (Sect. 5.2.2).
However, the predicted number ratios are sensitive to the initial conditions, and these models generate classical Be stars only towards the end of the main-sequence phase, which takes more than 10 Myr for all stars considered in their models (Mmax = 9 M⊙). As a consequence of the initial rotation profile, an early adjustment phase prohibits the generation of critically rotating stellar surfaces during the early main-sequence phase (see Granada et al. 2013 for details), as would be required to explain the very young B-star disc population in the GC clusters.
An additional caveat arises from the fact that Be stars, by definition, show strong Balmer emission lines. Their discs are thought to be gaseous with little or no dust, in stark contrast to primordial, pre-transitional or transitional discs (e.g. Silaj et al. 2010). Near-IR spectra available for three of the Arches excess sources display strong CO bandhead emission (Stolte et al. 2010), such that these sources host both optically thick molecular gas as well as hot dust contributing the L-band excess emission. However, all three excess sources show no evidence for hydrogen emission (or absorption) at the wavelength of the near-infrared Brackett γ line. Given both the substantial presence of dust and the absence of hydrogen-line emission in these three Arches sources, we conclude that classical Be stars – or stars with similar circumstellar properties – are unlikely candidates for the origin of the discs around these sources.
The second scenario for a secondary disc origin is related to binary evolution. In this scenario, the mass ejection by the primary in a close binary system leads to the formation of a circumbinary disc. Such a scenario is suggested for the young, evolved B[e] supergiants GG Carinae (Kraus et al. 2013) and HD 327083 (Wheelwright et al. 2012). Both objects host high-mass ~25 M⊙ primaries, and disc models indicate an inner rim of gaseous CO emission at 3 AU at a temperature of Tgas ~ 1700 K. These gaseous discs are located inside the dust emitting region, which has an inner radius of 5 AU (Wheelwright et al. 2012), in reasonable agreement with the radii estimated for the L-excess origin above. Both sources lack dust close to the central star (inner holes), and the modelled inner radii are larger than the binary separation, arguing for circumbinary discs. The study of GG Car reveals that standard Roche lobe overflow from the primary to the secondary is unlikely, as the primary has not filled its expected Roche lobe given its evolutionary state (Kraus et al. 2013). The same authors argue that the primary was in a classical Be phase towards the end of its main-sequence evolution, such that matter from an equatorial decretion disc could have overflowed the Roche surface of the binary and streamed into circumbinary orbits. In the case of the Arches and Quintuplet main-sequence excess sources, such a decretion disc had to evolve at an earlier stage in the stellar lifetime. Such a scenario would argue for tight binary systems, where the smaller common Roche surface could be filled by mass loss from a high-mass primary, possibly enforced by rotation.
A secondary disc origin in a binary system appears even more likely as primordial discs in close binary systems disperse on timescales of less than 1–2 Myr, while the survival time for discs around low-mass single stars is found to be longer, with a mean of 3–5 Myr (Kraus et al. 2012). Given the strong UV radiation fields and the added tidal torques, close, high-mass binaries can be expected to deplete their primordial discs even faster than their lower-mass counterparts. A binary origin of a secondary disc is intriguing in view of the recent suggestion that most high-mass stars undergo binary interactions during their lifetime. For Galactic O-type stars, Sana et al. (2012) find a fraction of 70% to be affected by binary evolution, with 50% of all high-mass stars in their sample undergoing envelope stripping, accretion, or common envelope evolution. Combining their observations with binary evolution models, they suggest that 26% of O-star binaries have a high likelihood of being affected by mass transfer events already on the main sequence. While this ratio might be smaller for the B-type disc host stars investigated here, a significant fraction of B-type main-sequence cluster members is expected to be located in a binary system. Reviewing recent studies of B-star binary properties, Duchêne & Kraus (2013) suggest 60% as a lower limit for the binary fraction of B6 to B2 main-sequence stars. However, they also find that the fraction decreases towards lower masses, with 30–45% found for late B- to early A-type stars. If a few per cent of the ~700 main-sequence stars found in the Quintuplet reference sample and of the ~400 main-sequence stars observed in the Arches are affected by binary interaction at the present epoch, the formation of secondary circumbinary discs with molecular gas and hot inner rims at this level might be expected from binary mass transfer alone. Such a scenario would naturally explain the similarity of the disc extent and properties with B[e] supergiants, although the mass transfer would happen at a much earlier stage during the main-sequence evolution of the primary component, and the material is provided by the wind mass loss of the high-mass star.
Whether the circumstellar discs in the Arches and Quintuplet clusters are of a primordial or a secondary origin cannot be distinguished with the current observations. High-resolution spectra delivering rotational velocities, mass loss rates, and abundance ratios of the disc host stars are required to definitively answer the question of the disc origin. For instance, for equatorial discs of rapidly rotating stars, the N/C abundance ratios should be enhanced compared to non-rapid rotators of the same population, and a high rotational velocity is maintained in initially rapid rotators over the entire main-sequence lifetime (Granada et al. 2013). For pre-transitional discs, the study of the mid-infrared SED would provide the same distinction mechanism used to define this subclass. While MIR surveys such as Spitzer do not provide the spatial resolution to construct NIR to MIR SEDs for the GC cluster disc sources, the veiling in NIR spectra as suggested by Espaillat et al. (2012) can confirm the existence of thick inner discs as proposed here from the L-band excess emission. The characteristics of secondary binary mass transfer discs are not well constrained, as there are no high-mass transfer disc models so far. The first step towards constraining the nature of the discs would therefore be to probe the sample for spectroscopic binaries, especially as only tight binaries show mass transfer already during their main-sequence evolution. Confirming the binary nature of the disc candidates and hence catching these objects in the act of mass-transfer would imply that a very special phase of high-mass stellar evolution can be observationally analysed in detail for the first time.
In summary, we conclude that the discs observed in the Arches and Quintuplet clusters have either originated from massive primordial discs that are in a warm, pre-transitional phase or, more likely, that they are composed of secondary discs caused by binary mass transfer.
We investigate deep VLT/NACO KsL′ in combination with HST/WFC3 JH imaging of the Arches and Quintuplet clusters to derive disc fractions from L-band excess emission. Multi-epoch NACO Ks observations are used to derive proper motion membership for L-band excess sources and the main-sequence reference samples. The results are summarised for each cluster below.
Circumstellar disc candidates in the Quintuplet cluster:
A total of 26 L-band excess sources as candidates of circumstellar discs are discovered in the Quintuplet cluster.
The L-band excess fraction in the Quintuplet is observed to be 4.0 ± 0.7%, where the uncertainty originates from the selection of the main-sequence reference sample. This fraction is consistent in all three fields where excess sources are detected, and – in contrast to earlier results in the Arches cluster (Stolte et al. 2010) – no trend of a varying excess fraction with radius is found. While several excess sources are found out to a radius of 1.2 pc, by far most of the excess sources are found in the central r< 0.8 pc.
The lack of excess sources in Quintuplet Fields 3 and 4 and at larger radii in Fields 2 and 5 indicates a rapid decline of the cluster profile at the largest radii, r> 1.5 pc, covered by our member survey. The detection of only three non-proper motion members featuring an L-band excess suggests that the fraction of field excess sources is very low. This conclusion is supported by the sparsity of excess sources in Arches Fields 2 and 4. Circumstellar disc candidates in the Arches cluster:
We find a mean disc fraction of 9.2 ± 1.2% out to the predicted tidal radius, r< 1.5 pc, in the Arches cluster, which is slightly larger than the mean disc fraction of 6 ± 2% detected previously in the cluster centre (Stolte et al. 2010).
In contrast to our earlier study of the Arches core, no trend of the disc fraction with radius is found in either cluster at the larger radii investigated here, suggesting that the dominant disc destruction mechanism predominantly acts in the densest part of the cluster core.
A total of 21 excess sources are detected over the entire cluster area at the NACO L′ sensitivity and spatial resolution. Of these, 8 excess sources in the cluster centre were previously known. Combining the newly detected 13 disc sources with the previously known 24 excess sources resolved with Keck/NIRC2 in the cluster centre increases the total number of discs in the Arches cluster to 37.
Properties and disc origin:
The stellar mass range of the disc host stars is approximated to be 2.2 − 17 M⊙ in both clusters, with a mean mass of 6 − 7 M⊙, corresponding to early A to early B main-sequence stars. The inner radius of the dust emission region is estimated to be in the range 3–12 AU. For a B2V host star, we estimate the minimum dust mass in the discs to be on the order of 10-10 to 10-7MJup. Comparing the fractional L-band flux to observed evolved discs indicates that our disc sources could be the higher-mass counterparts to the recently identified class of pre-transitional discs. As these arguments ignore the effects of UV evaporation expected to severely influence the disc lifetime of B-type stars, we discuss the possibility of a secondary disc origin. We conclude that mass transfer discs in interacting high-mass binary systems provide a likely origin of the L-band excess emission. As mass transfer from the primary donor to the secondary companion alters the chemical composition on the surface of the acceptor, these two different disc mechanisms need to be investigated further with high-resolution spectroscopy to obtain abundance ratios in order to reach a final conclusion on the disc origin of B-type main-sequence stars in the Arches and Quintuplet clusters.
http://www.stsci.edu/hst/HST_overview/documents/multidrizzle/ch56.html
The Ph.D. Thesis can be requested from [email protected]
These data were taken under Proposal ID: 67.C-0591, PI: Stolte, and are used only for calibration purposes in this analysis.
All source catalogues are available in electronic format at the CDS.
We sincerely thank the referee for the careful reading of the manuscript and especially for thoughtful comments on improvements to the discussion section. A.S., B.H., and M.H. are grateful for generous support from the DFG Emmy Noether programme under grant STO 496/3-1. We also wish to thank the Argelander Institute for its hospitality, and are indebted to Julien Girard and the NAOS-CONICA staff for their persistent and unique support with the long-term observational programmes. C.O. appreciates funding by the German Research Foundation (DFG) grant OL 350/1-1 and SFB 881. A.M.G. and M.R.M. gratefully acknowledge NSF support under grant AST-0909218. Some of the data presented in this paper were obtained from the Mikulski Archive for Space Telescopes (MAST). STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555.
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Appendix A: Proper motion diagrams of the outer cluster fields
As a consequence of the variation in stellar density across the wide-field coverage of the cluster images, the proper motion
diagrams are populated with varying levels of source density. In order to avoid confusion, especially given the dense environment in the cluster centres, the proper motion diagrams of the individual data sets are shown here.
Fig. A.1
Proper motion diagrams of the Quintuplet fields beyond the cluster centre. L-band excess sources are marked as red diamonds, while Wolf-Rayet candidates are shown in blue. The criterion for cluster member selection (circle) depends on the astrometric quality of each data set and on the corresponding dispersion in the cluster population around the origin. For the outer Quintuplet fields exclusively, membership probabilities were used to distinguish cluster members from field stars. The indicated membership criterion is therefore approximate, as the probabilities account for individual astrometric uncertainties and the location of each star with respect to the cluster centre. See Fig. 7 for further details.
Proper motion diagrams of the outer Arches fields. Labels are described in Fig. A.1, and for further details see Fig. 8. The larger scatter observed in the Arches proper motion diagrams compared to the Quintuplet proper motions is a consequence of the shallower Arches NACO photometry and mitigated adaptive optics performance. The circles indicate the membership selection criterion as derived from the central field in Fig. 9.
Appendix B: Colour–magnitude and colour–colour diagrams of the outer cluster fields
The foreground extinction towards the Galactic centre is patchy and varies widely with position. Each of the Arches and Quintuplet fields feature varying levels of foreground extinction,
such that a single extinction value could not be used for the selection of the main-sequence reference samples. The colour selection was adapted visually in each field to include the proper motion members along the observed cluster main sequence and exclude the red clump population starting at J ~ 20 mag.
Fig. B.1
J − H,Ks − L′ colour–colour diagrams of the Quintuplet outer fields. Proper motion members are shown in green, while non-members are shown in red. The black line denotes the extinction vector (Nishiyama et al. 2009). The red solid line shifted parallel to the reddening vector marks the 3σ selection criterion for L-band excess sources (red diamonds). Wolf-Rayet candidates (Ks< 12 mag) are marked in blue, while fainter excess sources marked as red diamonds are candidates for circumstellar disc emission. The J − H,Ks − L′ diagram of the Quintuplet central field can be found in Fig. 15.
J − H,Ks − L′ colour–colour diagrams of the Arches outer fields. Labels are as described in Fig. B.1, and the J − H,Ks − L′ diagram of the Arches central field is shown in Fig. 18.
J, J − Ks colour–magnitude diagrams of the Quintuplet outer fields. Proper motion members are shown in green, while non-members are shown in red. Disc and Wolf-Rayet candidates are also marked. The diagonal overdensity at J ~ 20 mag marks the onset of the red clump. The dotted lines mark the Ks = 17.5 mag boundary for proper motion member selection. The horizontal dashed line indicates the J = 22 mag limit, above which the combined cluster disc fractions are derived. The main-sequence selection is indicated by vertical dashed lines. The variation in main-sequence colour between fields is caused by the locally varying extinction across the cluster area. The J, J − Ks diagram of the central Quintuplet field is shown in Fig. 16.
J, J − Ks colour–magnitude diagrams of the Arches outer fields. Labels are as described in Fig. B.3, and the J, J − Ks diagram of the Arches central field is shown in Fig. 19. The variation in main-sequence colour between fields caused by the locally varying extinction is more pronounced than in the Quintuplet fields. Notably, several L′-excess sources do not show excess emission at Ks.
Ks, Ks − L′ colour–magnitude diagrams of the Quintuplet outer fields. Labels are as in the previous figures, and the Ks, Ks − L′ diagram of the central Quintuplet field can be found in Fig. 17. A distinct main sequence of cluster members can be seen in Field 2 (top left panel), while in all other fields the proper motion information is required to exhibit the rare cluster stars at these increasing radii.
Ks, Ks − L′ colour–magnitude diagrams of the Arches outer fields. Labels are as in the previous figures, and the Ks, Ks − L′ diagram of the central Arches field can be found in Fig. 20. While a pronounced main sequence is observed in the central field, in all outer fields the proper motion membership provides the best evidence for the distinction between cluster and field samples.
Appendix C: Variable stars
From our multi-epoch Ks proper motion campaign, we obtain a list of candidates for variable sources. Stars are defined as variable if the standard deviation of their magnitudes across three epochs deviates by more than 3σ from the median of the standard deviations of all stars in their respective magnitude bin. The expected (median) standard deviation is calculated in bins of ΔKs = 1 mag to account for the increase in photometric error towards fainter stars. In the case that only two epochs are available, the magnitude difference between epoch 1 and epoch 2 has to exceed 3 times the standard deviation of the magnitude differences of all stars in the respective magnitude bin of a variable candidate. With the exception of one prominent source in Arches Field 5, only stars brighter than Ks< 16 mag are searched for variability, as photometric uncertainties in the crowded cluster fields trigger many false detections beyond this limit. Two additional constraints are imposed to ensure that variable candidates are not affected by positional and/or PSF fitting uncertainties. The absolute magnitude difference or standard deviation for a star to be a variable candidate has to exceed ΔKs> 0.1 mag, and the difference in the source position in both the x and y coordinates has to be less than 0.5 pixel after the geometric transformation to the reference epoch. By imposing this astrometric consistency, incorrect matches with very close neighbouring stars are avoided. Finally, all variable candidates were inspected visually in all Ks epoch images to ensure that sources are not affected by background noise fluctuations, corrupt PSF cores, edge effects, and especially that their photometry is not compromised by the haloes of bright neighbours. As only two epochs were available in all Quintuplet fields except Field 2, no variable candidates could be identified unambiguously in the Quintuplet cluster fields. The photometry of all variable stars identified in the Arches cluster is summarised in Table C.1.
Table C.1
Variable candidates in the Arches cluster.
Appendix D: L-band luminosity of the discs
The L-band luminosity of the excess sources can be estimated for a central B2V star from the difference between the stellar brightness and the excess emission.
The 2.5 Myr Geneva isochrone yields an L-band brightness of L(B2V) = − 1.35 mag for a 10 M⊙ star. For our approximation here, we assume this L-band brightness to be close to the Vega system, such that LVega = 0 mag, and Inserting the Vega flux of Tokunaga & Vacca (2005) leads to The absolute flux within the passband is given by the filter width ΔλL′ = 0.62 μm times the expected flux at the central wavelength:
If the disc is about 1 mag brighter in L′ than the star, then ΔL = 1 mag implies a factor of 2.5 higher flux, which is already the flux we expect to measure within the passband of L′ with filter width 0.62 μm. In this estimate, no assumptions about the distance have been made, as all fluxes are derived for absolute magnitudes only, hence at a standard distance of 10 pc. This flux is used to derive an order of magnitude limit of the disc masses in Sect. 5.2.2.
Appendix E: Complete source lists
The JHKs combined source lists for the Arches and Quintuplet clusters are made available at the CDS. Proper motion memberships are meant as indicator values for candidacy, and might change as further proper motion epochs become available.
Table E.1
JHKs sources and L′ detections in the Quintuplet cluster.
JHKs sources and L′ detections in the Arches cluster.
In the text
Tables at CDS
Simbad Objects
NASA ADS Abstract Service
- NASA ADS
The present-day mass function of the Quintuplet cluster based on proper motion membership
The Arches cluster out to its tidal radius: dynamical mass segregation and the effect of the extinction law on the stellar mass function
NTT infrared imaging of star cluster candidates towards the central parts of the Galaxy
A&A 408, 127-134 (2003)
Constraining the population of isolated massive stars within the Central Molecular Zone
The Arches cluster revisited — III. An addendum to the stellar census | CommonCrawl |
Waveform domain framework for capacity analysis of uplink WCDMA systems
Tan Tai Do1,
Su Min Kim2,
Tobias J. Oechtering1,
Mikael Skoglund1 &
Gunnar Peters3
EURASIP Journal on Wireless Communications and Networking volume 2015, Article number: 253 (2015) Cite this article
This paper investigates the capacity limit of an uplink WCDMA system considering a continuous-time waveform signal. Various realistic assumptions are incorporated into the problem, which make the study valuable for performance assessment of real cellular networks to identify potentials for performance improvements in practical receiver designs. An equivalent discrete-time channel model is derived based on sufficient statistics for optimal decoding of the transmitted messages. The capacity regions are then characterized using the equivalent channel considering both finite constellation and Gaussian distributed input signals. The capacity with sampling at the receiver is also provided to exemplify the performance loss due to a typical post-processing at the receiver. Moreover, we analyze the asymptotic capacity when the signal-to-noise ratio goes to infinity. The conditions to simultaneously achieve the individual capacities are derived, which reveal the impacts of signature waveform space, channel frequency selectivity and signal constellation on the system performance.
Code division multiple access (CDMA) has become standard in several wireless communication systems from IS-95, UMTS wideband CDMA (WCDMA) to HSPA, and so on [1–3]. Although being introduced more than 50 years ago, CDMA is still largely employed and developed nowadays due to its various advantages such as enabling universal frequency reuse, improving handover performance by soft-handover, and mitigating the effects of interference and fading. The performance assessment of such networks is of significant importance. In addition, the architecture of WCDMA systems still has room for improvement, especially at the uplink receiver side (base station) [4, 5].
In the literature, most studies on fundamental limits of multiuser CDMA systems have been done under the assumptions of synchronous, time-invariant (each user uses the same spreading sequence for all data symbols), and/or random spreading sequences [6–11]. In [6–8], the optimal spreading sequences and capacity limits for synchronous CDMA have been studied with a discrete-time signal model. A more theoretical approach on CDMA capacity analysis has been pursued in [9–11] by modeling the spreading sequences with random sequences. However, the assumption of perfect synchronization between users is not realistic, especially for a cellular CDMA uplink. Moreover, in practice, time-variant-spreading sequences based on Gold or Kasami codes [1–3] are often used rather than time-invariant or random spreading sequences.
The capacity limit for a CDMA system with symbol-asynchronous transmission (the symbol epochs of the signal are not aligned at the receiver) has also been studied in [12–15]. In [12], Verdú studied the capacity region of an uplink time-invariant CDMA system with inter-symbol interference (ISI) by exploiting the asymptotic properties of Toeplitz matrices. In [13, 14], the authors studied user and sum capacities of a symbol-asynchronous CDMA system but with chip-synchronous transmission (the timing of the chip epochs are aligned) assumption, which made the analysis tractable using a discrete-time model. In [15], the spectral efficiency of an asynchronous CDMA system has been considered while neglecting the ISI by assuming a large spreading factor.
There have been several studies trying to deal with the continuous-time asynchronous CDMA system setup. However, most of them focus on other performance metrics than capacity (e.g., error probability considering different detection algorithms) [16–20]: time-invariant CDMA [21] or asynchronous CDMA but with an ISI-free assumption [22]. The capacity analysis for a real cellular network with continuous-time waveform, time-variant-spreading, asynchronous CDMA is difficult due to the following reasons. First, an equivalent discrete-time signal model is complicated to be expressed due to the asynchronization between symbols and chips. Next, for a time-variant spreading CDMA system, the approach based on the asymptotic properties of a Toeplitz form [23], which is crucial for the capacity analysis with ISI channel [12, 24], cannot be employed since the variation of spreading sequence destroys the Toeplitz structure of the equivalent channel matrix.
Contributions of this work
Motivated by the fact that most existing research on multiuser CDMA capacity have focused on theoretical analysis with simplified system assumptions, in this work, we present a framework for capacity analysis of a WCDMA system with more realistic assumptions, which make the framework and results more valuable for the performance assessment of real cellular networks. Our main contributions are summarized as follows: − We provide a precise channel model reflecting practical operations of the uplink WCDMA physical (PHY) layer based on the 3GPP release 11 specification [1]. Various realistic assumptions are included into the system such as: continuous-time waveform-transmitted signal and time-variant spreading and an asynchronous multi-code CDMA system with ISI over frequency-selective channels. It is worth noting that although the signal model is constructed based on a WCDMA system, the approach and framework can be extended or transferred to other wireless standards. − We derive sufficient statistics for decoding the transmitted symbols based on the continuous-time-received signal, which provides us an equivalent discrete-time signal model. A matrix representation of channel model is provided for which the equivalent additional noise is shown to be a Gaussian distributed random vector. − Since sufficient statistics preserve the mutual information ([25], Chap. 2), the capacity is then derived using the equivalent discrete-time signal model. In particular, we characterize the capacity region when the input signal is fixed to finite constellations, e.g., PSK, QAM, and so on, with a uniform input distribution, which are widely used in current real cellular networks. Additionally, we provide the capacity region when the input signal follows a Gaussian distribution, which is the optimal input distribution for additive Gaussian noise channels. Accordingly, the Gaussian capacity offers a capacity outer bound for the real WCDMA cellular networks using finite constellation input. − Due to the data-processing inequality ([25], Chap. 2), the mutual information between input and output cannot increase through any post-processing at the receiver. Given the capacity bounds measured directly at the receive antenna of a real system, we can now assess the capacity loss due to a specific post-processing at the receiver. Therewith, we investigate the capacity loss due to sampling, which is a traditional discretization approach in practical systems. Note that in the real cellular networks, since the sampling window is finite, perfect reconstruction of a band-limited signal is not guaranteed even if the sampling rate is equal to Nyquist rate ([26], Chap. 8). The assessment of such impact on the capacity is also considered in this work. − We analyze the asymptotic sum-capacity when the signal-to-noise ratio (SNR) goes to infinity, for which we derive the conditions on the signature waveform space so that on every link to the base station, the individual capacities are achieved simultaneously. To this end, we first derive the sufficient condition, which holds for all kinds of input signals including signals based on finite and infinite constellations. Next, once again, we motivate our study from a practical perspective by focusing on the finite constellation input signal. Accordingly, a necessary condition to simultaneously achieve the individual capacities with a finite constellation input signal, which takes the signal constellation structure into account, is derived. Those results are particularly useful for spreading sequence design in a real WCDMA cellular network.
The rest of the paper is organized as follows: Section 2 presents the signal model where sufficient statistics and an equivalent matrix representation are derived. In Section 3, the capacity analysis is provided considering finite constellations and Gaussian-distributed input signals. The capacity employing sampling is also investigated in this section. The asymptotic capacity when the SNR goes to infinity is analyzed and discussed in Section 4. Finally, Section 5 concludes the paper.
Signal model
Since the physical layer defines the fundamental capacity limit of the uplink WCDMA channel [2], we focus on a signal model reflecting the operations of the uplink WCDMA PHY layer based on the 3GPP release 11 specification [1].
Waveform signal model
Let us consider a K-user multi-code WCDMA system with M codes for each I/Q branch and spreading factor N sf . Then, the transmitted signal for user k can be expressed as
$$ x_{k}(t)=\sqrt{E_{k}}\sum\limits_{i=1}^{N}\sum\limits_{m=1}^{M}d_{ki}^{m}s_{ki}^{m}(t), k=1,\ldots,K, $$
where E k denotes the transmitted power of user k, N denotes the number of symbols, \(s_{\textit {ki}}^{m}(t)=\frac {1}{\sqrt {N_{\textit {sf}}}}\sum _{n=0}^{N_{\textit {sf}}-1} c_{\textit {ki}}^{m}[n]p(t-(i-1)T_{s}-nT_{c})\) is the signature waveform for the i-th symbol of the m-th stream of user k, p(t) is the chip waveform with unit power and finite bandwidth W 1, T c is the chip duration and T s =N sf T c is the symbol duration, and \(c_{\textit {ki}}^{m}[n]\) denotes the spreading sequence which satisfies \(\sum _{n=0}^{N_{\textit {sf}}-1}|c_{\textit {ki}}^{m}[n]|^{2}=N_{\textit {sf}}\). In a time-variant CDMA system, a different spreading sequence \(\left \{c_{\textit {ki}}^{m}[n]\right \}_{n}\) is used for each transmitted symbol \(d_{\textit {ki}}^{m}\). This corresponds to a real cellular CDMA network with long scrambling codes, in which the effective spreading sequence will vary between symbols.
In this study, we assume a tapped-delay line channel model2 with L multi-paths ([27] Chap. 2), i.e.,
$$h_{k}(t)=\sum\limits_{l=1}^{L}g_{kl}\delta(t-\tau_{kl}), k=1,\ldots,K, $$
where g kl and τ kl denote the channel coefficient and the propagation delay for the l-th path of the channel for user k, respectively. Then the received signal is given by
$$\begin{array}{@{}rcl@{}} r(t)&=&\sum\limits_{k=1}^{K}h_{k}(t)\ast x_{k}(t-\lambda_{k})+n(t)\\ &=&\sum\limits_{k=1}^{K}\!\sum\limits_{i=1}^{N}\!\sum\limits_{m=1}^{M}\!\!\sqrt{E_{k}}d_{ki}^{m}\!\sum_{l=1}^{L}\!g_{kl}s_{ki}^{m}(t \!- \!\lambda_{k} \,-\,\tau_{kl})\,+\,n(t), \end{array} $$
where λ k denotes the time delay of the transmitted signal from user k, the symbol ∗ denotes the convolution operation, and n(t) represents the additive white Gaussian noise with a two-sided power spectral density (PSD) N 0/2=σ 2.
Sufficient statistic and equivalent channel
Since a sufficient statistic for decoding the transmitted messages preserves the capacity of the system, the capacity of a continuous-time channel can be computed using a sufficient statistic ([25], Chap. 2), ([28], Chap. 8). To this end, let us define the transmitted symbol vectors \(\mathbf {d}_{\textit {ki}}:=\left [d_{\textit {ki}}^{1},\ldots,d_{\textit {ki}}^{M}\right ]^{T}\in \mathbb {C}^{M\times 1}\) (for each stream), \(\mathbf {d}_{k}:=\left [{{\mathbf {d}_{k1}}^{T}},\ldots,{\mathbf {d}_{\textit {kN}}}^{T}\right ]^{T}\in \mathbb {C}^{NM\times 1}\) (for each user), and \(\mathbf {d}:=\left [{\mathbf {d}_{1}}^{T},\ldots,{\mathbf {d}_{K}}^{T}\right ]^{T}\in \mathbb {C}^{KNM\times 1}\) (for all users), where (·)T denotes the transpose operation. Further, let us define μ(t;d) as the received signal without noise, i.e.,
$$\begin{array}{@{}rcl@{}} \mu(t;\mathbf{d}):=\sum\limits_{k=1}^{K}\sum\limits_{i=1}^{N}\sum\limits_{m=1}^{M}\sqrt{E_{k}}d_{ki}^{m}\sum\limits_{l=1}^{L}g_{kl}s_{ki}^{m}\left(t- \lambda_{k}-\tau_{kl}\right). \end{array} $$
The problem of optimal decoding d is similar to the detection problem in ([27], Proposition 3.2) (see [18] for a similar approach based on the Cameron-Martin formula [29], Chap. VI). Accordingly, the optimal decision3 can be made using the following decision variables
$$ \Phi(\mathbf{d})\!=2\Re \left\{\! \int_{-\infty}^{\infty}\!\!\mu(t;\mathbf{d})^{*}r(t)dt \right\} \,-\, \int_{-\infty}^{\infty}\!\left[\mu(t;\mathbf{d})\right]^{2}dt, $$
where ℜ{·} denotes the real part of a complex value and (·)∗ denotes the complex conjugate operation. Since the second term of (3) does not depend on the received signal r(t), we can drop it. Therewith, the sufficient statistic is based on the first term of (3), which can be rewritten as
$$\begin{array}{@{}rcl@{}} 2\Re \!\!\left\{\! \sum\limits_{k=1}^{K}\!\!\sum\limits_{i=1}^{N}\!\!\sum\limits_{m=1}^{M}\!\!\sqrt{E_{k}}{d_{ki}^{m}}^{*}\!\sum\limits_{l=1}^{L}{g_{kl}}^{*} \!\!\!\int_{-\infty}^{\infty}\!\!\!\!r(t){s_{ki}^{m}(t\,-\, \lambda_{k}\! -\!\tau_{kl})}^{*}dt \right\}. \end{array} $$
Let us denote \(y_{\textit {ki}}^{ml}:=\int _{-\infty }^{\infty }r(t){s_{\textit {ki}}^{m}(t-\lambda _{k}-\tau _{\textit {kl}})}^{*}dt\) and \(z_{\textit {ki}}^{m}:=\sum _{l=1}^{L}{g_{\textit {kl}}}^{*}y_{\textit {ki}}^{ml}\), then \(\left \{z_{\textit {ki}}^{m}\right \}_{k,i,m}\) is a sufficient statistic for decoding d based on r(t). It is shown that the received signal passing through a bank of matched filters, where the received signal is matched to the delayed versions of the signature waveforms, results in a sufficient statistic for decoding d based on r(t). Figure 1 illustrates an implementation to obtain the sufficient statistic from the continuous-time received signal. This has a RAKE receiver structure ([30], Chap. 14), including RAKE-matched fingers followed by maximal ratio combining (MRC).
Diagram of the implementation to obtain the sufficient statistic from the continuous-time received signal of a K-user uplink WCDMA system with M codes for each I/Q branch over an L-tap frequency-selective channel
Moreover, let \(\rho _{(kiml)}^{(k^{\prime }i^{\prime }m^{\prime }l^{\prime })}\) be the cross-correlation function between the signature waveforms defined as
$${\fontsize{8.5pt}{9.3pt}\selectfont{\begin{aligned} {\kern-16.9pt}\rho_{(kiml)}^{(k^{\prime}i^{\prime}m^{\prime}l^{\prime})} &= \int_{-\infty}^{\infty}{s_{ki}^{m} \left(t-\lambda_{k}-\tau_{kl} \right)}^{*} s_{k^{\prime}i^{\prime}}^{m^{\prime}} \left(t-\lambda_{k^{\prime}}-\tau_{k^{\prime}l^{\prime}} \right)dt\\ &= \frac{1}{N_{sf}} \sum\limits_{n=0}^{N_{sf}-1}\sum\limits_{n^{\prime}=0}^{N_{sf}-1}{c_{ki}^{m}[n]}^{*}c_{k^{\prime}i^{\prime}}^{m^{\prime}}[n^{\prime}]\\ &~~~~\times R_{p} \left(\!\frac{n-n^{\prime}}{N_{sf}}T_{s}+(i-i^{\prime})T_{s}+(\!\tau_{kl}-\tau_{k^{\prime}l^{\prime}}\!) + (\!\lambda_{k} - \lambda_{k^{\prime}}\!) \!\right)\!, \end{aligned}}} $$
where \(R_{p}(\tau)=\int _{-\infty }^{\infty }{p(t)}^{*}p(t+\tau)dt\) is the autocorrelation function of the chip waveform. Then the sufficient statistics can be expressed as
$$\begin{array}{@{}rcl@{}} z_{ki}^{m}\,=\,\!\sum\limits_{l=1}^{L}\!\sum\limits_{k^{\prime}=1}^{K}\!\sum\limits_{i^{\prime}=1}^{N}\!\sum\limits_{m^{\prime}=1}^{M}\! \sum\limits_{l^{\prime}=1}^{L}\!\!\sqrt{\!E_{k^{\prime}}}d_{k^{\prime}i^{\prime}}^{m^{\prime}}{g_{kl}}^{*}\! g_{k^{\prime}l^{\prime}}\rho_{(kiml)}^{(k^{\prime}\!i^{\prime}\!m^{\prime}\!l^{\prime}\!)\!}\!+ n_{ki}^{m},~~ \end{array} $$
where \(n_{\textit {ki}}^{m}:=\sum _{l=1}^{L}{g_{\textit {kl}}}^{*}\int _{-\infty }^{\infty }n(t){s_{\textit {ki}}^{m}(t-\lambda _{k}-\tau _{\textit {kl}})}^{*}dt\) is the equivalent noise term associated with \(z_{\textit {ki}}^{m}\) after matched filtering.
A matrix canonical form is useful to characterize the capacity from a sufficient statistic. Hence, we express the sufficient statistics \(\left \{z_{\textit {ki}}^{m}\right \}_{k,i,m}\) derived in (4) as an equivalent matrix equation. By following the similar steps as in [31], the matrix representation of the equivalent channel can be obtained from (4) as
$$ \mathbf{z}=\sum\limits_{k=1}^{K}\sqrt{E_{k}}\mathbf{H}_{k}\mathbf{d}_{k}+\mathbf{n}, $$
where \(\mathbf {z}:=\left [z_{11}^{1},\cdots,z_{\textit {KN}}^{M}\right ]^{T}\in \mathbb {C}^{KNM\times 1}\), \(\mathbf {d}_{k}:=\left [d_{k1}^{1},\cdots,d_{\textit {kN}}^{M}\right ]^{T}\in \mathbb {C}^{NM\times 1}\), and \(\mathbf {n}:=\left [n_{11}^{1},\cdots,n_{\textit {KN}}^{M}\right ]^{T}\in \mathbb {C}^{KNM\times 1}\). The equivalent channel channel H k is given by
$$\mathbf{H}_{k} =\left[ \begin{array}{c} \mathbf{G}_{1}^{\dagger}\mathbf{R}_{k}^{1}\mathbf{G}_{k}\\ \vdots\\ \mathbf{G}_{K}^{\dagger}\mathbf{R}_{k}^{K}\mathbf{G}_{k} \end{array} \right] \in \mathbb{C}^{KNM\times NM}, $$
where the channel gain matrix G k is block diagonal and given by
$$\mathbf{G}_{k}=\text{blkdiag} \left(\underbrace{\mathbf{g}_{k},\cdots,\mathbf{g}_{k}}_{MN ~\text{vectors}} \right) \in \mathbb{C}^{LNM\times NM} $$
with \(\mathbf {g}_{k}=\;[g_{k1},\cdots,g_{\textit {kL}}]^{T}\in \mathbb {C}^{L\times 1}\), and the correlation matrix is defined as
$$ \mathbf{R}_{k}^{k'} \,=\, \left[ \!\! \begin{array}{cccccccccccccc} \rho_{(k111)}^{(k'111)}&\!\cdots\!\!& \rho_{(k111)}^{(k'11L)}&\!\cdots\!& \rho_{(k111)}^{(k'NML)}\\ \rho_{(k112)}^{(k'111)} &\!\cdots\!\!& \rho_{(ki12)}^{(k'i'1L)}&\!\cdots\!& \rho_{(k112)}^{(k'NML)}\\ \vdots &\!\vdots\!\!& \vdots&\!\vdots\!&\vdots\\ \rho_{(kNML)}^{(k'111)}\!&\!\cdots\!\!&\! \rho_{(kNML)}^{(k'11L)}\!&\!\cdots\!& \!\rho_{(kNML)}^{(k'NML)} \end{array} \!\!\right]\!\in \! \mathbb{C}^{N\!M\!L\!\times\! N\!M\!L}. $$
Moreover, it is shown in Appendix 1 that the equivalent noise vector n is a complex Gaussian random vector with zero mean and covariance matrix σ 2 H with \(\mathbf {H}=[\mathbf {H}_{1}, \ldots, \mathbf {H}_{K}] \in \mathbb {C}^{KNM\times KNM}\).
Remark 1.
In this work, the signal model is constructed based on the practical operation of an uplink WCDMA PHY layer. However, the approach and framework can be extended or transferred to other wireless standards. Indeed, the signal model in (1) can be used to describe the continuous-time-transmitted signal of a general system, in which \(s_{\textit {ki}}^{m}(t)\) are the waveforms used for the modulation at the transmitter. For example, in a OFDM system, \(s_{\textit {ki}}^{m}(t)\) can be replaced by the corresponding orthogonal waveforms. Moreover, the resulting equivalent channel in (5) corresponds to a traditional discrete-time MIMO multiple-access channel (MAC), which are used in various research literature.
In this section, we analyze the capacity of the continuous-time uplink WCDMA channel. Recalling that z is a sufficient statistic for optimal (i.e., capacity preserving) decoding d based on r(t). Any coding scheme which achieves the capacity of the channel with input d and output r(t) can also be employed to the channel with input d and decoding based on output z instead of r(t). Therefore, the channel capacity is preserved when the continuous-time output r(t) is replaced by the sufficient statistic z. Thus, we can focus on the capacity of the equivalent discrete-time channel in (5), which is given by the capacity region of a discrete memoryless MAC [32].
Let us define R 1,R 2,…,R K as the maximum number of bits that can be reliably transmitted from user 1, user 2, …, user K per block of N symbols. The capacity region of the uplink WCDMA channel is then characterized by the closure of the convex hull of the union of all achievable rate vectors (R 1,R 2,…,R K ) satisfying [32], ([25], Chapter 15)
$$ \sum\limits_{k\in \mathcal{J}}R_{k}\leq I(\mathbf{d}_{\mathcal{J}};\mathbf{z}|\mathbf{d}_{\mathcal{J}^{c}}), $$
for all index subsets \(\mathcal {J} \subseteq \{1,\ldots,K\}\) and some joint pmf \(p(\mathbf {d})=\prod _{k=1}^{K}p(\mathbf {d}_{k})\), where \(\mathcal {J}^{c}\) denotes the complement of \(\mathcal {J}\) and \(\mathbf {d}_{\mathcal {J}}= \{\mathbf {d}_{k} : k\in \mathcal {J}\}\).
We now characterize the uplink WCDMA capacity region considering two specific input signals: finite constellation with uniformly distributed input and Gaussian-distributed input.
Finite constellation input
When the input signal vector d k at each user is independently taken from a finite constellation set \(\mathcal {M}^{NM}\), \(|\mathcal {M}|=M_{c}\), with equal probability, i.e., \(p(\mathbf {d}_{k})=\frac {1}{M_{c}^{NM}}\), ∀k∈{1,…,K}, then the rate constraints in (7) can be rewritten as
$$\begin{array}{@{}rcl@{}} \sum_{k\in \mathcal{J}}R_{k}&\leq& I(\mathbf{d}_{\mathcal{J}};\mathbf{z}|\mathbf{d}_{\mathcal{J}^{c}})\\ &=&h(\mathbf{z}|\mathbf{d}_{\mathcal{J}^{c}})-h(\mathbf{z}|\mathbf{d}_{\mathcal{J}}\mathbf{d}_{\mathcal{J}^{c}})=h(\mathbf{z}_{\mathcal{J}})-h(\mathbf{n})\\ &=&-\mathbb{E}\left\{\log_{2} \left(f_{\mathbf{\mathbf{z}_{\mathcal{J}}}}(\bar{z})\right)\right\} - \log_{2} \left(\det\left(\pi e \sigma^{2} \mathbf{H}\right) \right) \end{array} $$
for all \(\mathcal {J}\!\subseteq \{1,\ldots,\!K\}\) with \(\mathbf {\mathbf {z}_{\mathcal {J}}}:=\sum _{k\in \mathcal {J}}\sqrt {E_{k}}\mathbf {H}_{k} \mathbf {d}_{k} + \mathbf {n}\). \(\mathbf {\mathbf {z}_{\mathcal {J}}}\) is a Gaussian mixture random vector with probability density function (pdf)
$$ f_{\mathbf{z}_{\mathcal{J}}}(\bar{z}) = \sum\limits_{\bar{d}\in \mathcal{M}^{|\mathcal{J}|NM}}p(\mathbf{d}_{\mathcal{J}}=\bar{d})\cdot f_{\mathbf{z}_{\mathcal{J}}|\mathbf{d}_{\mathcal{J}}}(\bar{z}|\bar{d}), $$
where \(p(\mathbf {d}_{\mathcal {J}}=\bar {d})=\frac {1}{M_{c}^{|\mathcal {J}|NM}}\) and \(f_{\mathbf {z}_{\mathcal {J}}|\mathbf {d}_{\mathcal {J}}}(\bar {z}|\bar {d})\) is the conditional pdf of \(\mathbf {z}_{\mathcal {J}}\) given \(\mathbf {d}_{\mathcal {J}}\). Let us denote \(\mathbf {E}_{\mathcal {J}}\mathbf {H}_{\mathcal {J}}\mathbf {d}_{\mathcal {J}}:=\sum _{k\in \mathcal {J}}\sqrt {E_{k}}\mathbf {H}_{k} \mathbf {d}_{k}\), where \(\mathbf {E}_{\mathcal {J}}\) is the power scaled matrix \(\mathbf {E}_{\mathcal {J}}:= \text {blkdiag}(\{\sqrt {E_{k}}\mathbf {I}_{\textit {NM}}\}_{k\in \mathcal {J}})\) and \(\mathbf {H}_{\mathcal {J}}\) is the sub-matrix of H after removing \(\mathbf {H}_{k^{\prime }}\), \(\forall k'\in \mathcal {J}^{c}\). Then, \(f_{\mathbf {z}_{\mathcal {J}}|\mathbf {d}_{\mathcal {J}}}(\bar {z}|\bar {d})\) is the pdf of a complex Gaussian random vector with mean \(\mathbf {E}_{\mathcal {J}}\mathbf {H}_{\mathcal {J}}\bar {d}\) and covariance matrix σ 2 H, i.e.,
$${}f_{\mathbf{z}_{\mathcal{J}}|\mathbf{d}_{\mathcal{J}}}\!(\bar{z}|\bar{d})\,=\,\frac{\exp \!\left(\,-\,\left(\!\bar{z}\! -\! \mathbf{E}_{\!\mathcal{J}}\mathbf{H}_{\!\mathcal{J}}\bar{d}\right)^{\dagger} \!\left(\sigma^{2} \mathbf{H}\right)^{\!-1}\! \!\left(\bar{z}\! -\! \mathbf{E}_{\!\mathcal{J}}\mathbf{H}_{\!\mathcal{J}}\bar{d}\right)\! \right)}{\pi^{KNM} \cdot\det\left(\sigma^{2}\mathbf{H}\right)}\!. $$
Typically, the capacity region of a channel with finite constellation input is numerically characterized via Monte Carlo simulation because a closed-form expression does not exist. It is worth noting that in order to calculate the first term of (8), one has to average overall possible \(M_{c}^{|\mathcal {J}|NM}\) input symbols (up to \(M_{c}^{KNM}\) for sum-rate) according to (9). However, when M c and/or N are too large, this task becomes intractable due to prohibitive computational complexity. In MIMO channels with finite constellation input, a similar problem occurs when the input alphabet set or the number of antennas is too large, e.g., 64-QAM or 8 ×8 MIMO [33]. In order to tackle this problem, we have proposed an effective approximation algorithm based on sphere-decoding approach to find the approximate capacity for large MIMO system with finite constellation input in [34]. The algorithm to compute the entropy is out of the scope of this work. However, we use it in the numerical results section (Section 3.4) to compute approximations on the capacity curves for large N. The specific details about the algorithm can be found in [34].
Gaussian input
If the input signal vector d k of each user follows a zero mean complex Gaussian distribution with unit input power constraint, i.e., \(\mathbf {d}_{k}\sim \mathcal {CN}(\mathbf {0},\mathbf {I}_{\textit {NM}})\), ∀k=1,…,K, then the capacity region is characterized by the rate vectors (R 1,R 2,…,R K ) satisfying
$$ \sum\limits_{k\in \mathcal{J}\!}R_{k}\leq \log\det\left(\mathbf{I}_{NM} +\! \sum_{k\in \mathcal{J}}\frac{E_{k}}{\sigma^{2}}\mathbf{H}_{k}^{\dagger}\mathbf{H}^{-1}\mathbf{H}_{k}\right) $$
((10))
for all \(\mathcal {J}\!\subseteq \{1,\ldots,\!K\}\). Since the Gaussian-distributed input is the optimal input for a given mean power constraint, (10) serves as an outer bound for the capacity region with a practically motivated input, i.e., finite constellation input as discussed in Section 3.1.
Since the matched filtering at the receiver yields a sufficient statistic, the uplink WCDMA capacity achieved by any other receiver structures is upper bounded by the capacity achieved by the sufficient statistic using matched filtering. Regarding the capacity upper bounds in Sections 3.1–3.2 as benchmarks for the performance assessment, we now analyze the capacity achieved by sampling to evaluate the capacity performance loss due to specific post-processing at the receiver.
For sampling at the receiver, we assume that out-of-band noise is first suppressed by an ideal low-pass filter (LPF) with bandwidth W, which has the same bandwidth as the transmitted signal. Then, the received signals are uniformly sampled at every time instance t n , n=1,…,N sp, where N sp is finite. As a result, the sampled received signal at time t n is given by
$$\begin{aligned} {}r_{n}:= r_{\text{lp}}(t_{n})=\sum\limits_{k=1}^{K}\sum\limits_{m=1}^{M}\sum\limits_{i=1}^{N} d_{ki}^{m}&\sum\limits_{l=1}^{L}g_{kl}s_{\text{lp},ki}^{m}(t_{n}-\lambda_{k}-\tau_{kl})\\ &+n_{\text{lp}}(t_{n}),~~n=1,\dots,N_{\text{sp}}, \end{aligned} $$
where r lp(t), \(s_{\text {lp},ki}^{m}(t)\), and n lp(t) denotes the outputs of r(t), \(s_{\textit {ki}}^{m}(t)\), and n(t) passing through the LPF, respectively. We have \(s_{\text {lp},ki}^{m}(t)=s_{\textit {ki}}^{m}(t)\) since the ideal LPF is assumed to have the same bandwidth as the transmitted signal, i.e., bandwidth of \(s_{\textit {ki}}^{m}(t)\). Let us denote the effective signature waveform by \(\bar {s}_{\textit {ki}}^{m}(t):=\sum _{l=1}^{L}g_{\textit {kl}}s_{\textit {ki}}^{m}(t-\lambda _{k}-\tau _{\textit {kl}})\), then the sampled received signal can be expressed as
$$r_{n}=\!\sum\limits_{k=1}^{K}\!\sum\limits_{m=1}^{M}\!\sum\limits_{i=1}^{N}\! \sqrt{E_{k}}\bar{s}_{ki}^{m}(t_{n})d_{ki}^{m}+n_{\text{lp}}(t_{n}),n=1,\dots,N_{\text{sp}}. $$
Next, let us denote the sampling signature waveform matrix corresponding to user k by
$$\bar{\mathbf{S}}_{k}\,=\,\left[ \begin{array}{ccccccccccccc} \bar{s}_{k1}^{1}(t_{1}) & \bar{s}_{k1}^{2}(t_{1}) & \cdots &\bar{s}_{kN}^{M}(t_{1}) \\ \bar{s}_{k1}^{1}(t_{2}) & \bar{s}_{k1}^{2}(t_{2}) & \cdots &\bar{s}_{kN}^{M}(t_{2}) \\ \vdots & \vdots & \vdots & \vdots \\ s_{k1}^{1}(t_{N_{\text{sp}}}) & s_{k1}^{2}(t_{N_{\text{sp}}}) & \cdots &s_{kN}^{M}(t_{N_{\text{sp}}}) \\ \end{array} \right] \!\!\in\mathbb{C}^{N_{\text{sp}}\times NM}, $$
and the sampled received signal and sampled noise vectors by
$$\begin{array}{@{}rcl@{}} \mathbf{r}_{\text{sp}}&=&[r_{1},r_{2},\cdots,r_{N_{\text{sp}}}]^{T}\in\mathbb{C}^{N_{\text{sp}}\times 1},\\ \mathbf{n}_{\text{sp}}&=&[n_{\text{lp}}(t_{1}),n_{\text{lp}}(t_{2}),\dots,n_{\text{lp}}(t_{N_{\text{sp}}})]^{T}\in\mathbb{C}^{N_{\text{sp}}\times 1}. \end{array} $$
Then the sampled received signal can be written in an equivalent matrix form as
$$ \mathbf{r}_{\text{sp}}=\sum\limits_{k=1}^{K}\sqrt{E_{k}}\bar{\mathbf{S}}_{k}\mathbf{d}_{k}+\mathbf{n}_{\text{sp}}. $$
Since n(t) is a complex Gaussian random process with zero mean and PSD N 0/2=σ 2 over the whole frequency band, after passing through the ideal LPF with bandwidth W, the noise process n lp(t) becomes a stationary zero mean Gaussian process ([35], Chap. 3) with the auto-correlation function
$$R_{\text{lp}}(\tau)=N_{0}W\text{sinc}(2W\tau), $$
where sinc(·) is the normalized sinc function. Therefore, the sampled additive noise vector n sp is a zero mean complex Gaussian random vector with covariance matrix R sp= [r ij ]{i,j}, i,j=1,2,…,N sp,
$$ r_{ij}=R_{\text{lp}}(t_{i}-t_{j}), ~~i,j=1,2,\ldots,N_{\text{sp}}. $$
The capacities with sampling are then similarly obtained as in (8) and (10) with some small modifications; the equivalent matrix H k needs to be replaced by \(\bar {\mathbf {S}}_{k}\), the noise covariance matrix σ 2 H needs to be replaced by R sp. Accordingly, let us define \(R_{1}^{\text {sp}},R_{2}^{\text {sp}}, \ldots, R_{K}^{\text {sp}}\) as the maximum number of bits that can be reliably transmitted from user 1, user 2, …, user K per block of N symbols assuming sampling is employed at the receiver. The sampling capacity is then characterized by4
$$ \sum\limits_{k\in \mathcal{J}\!}R_{k}^{\text{sp}}\leq \log\det\left(\mathbf{I}_{N_{\text{sp}}} +\! \sum\limits_{k\in \mathcal{J}}E_{k}\bar{\mathbf{S}}_{k}^{\dagger}\mathbf{R}_{\text{sp}}^{-1}\bar{\mathbf{S}}_{k}\right) $$
for a Gaussian input signal and
$$ \sum\limits_{k\in \mathcal{J}\!}\!R_{k}^{\text{sp}}\!\leq \!-\mathbb{E}\left\{\!\log_{2}\! \left(\!f_{\mathbf{\mathbf{r}}_{\mathcal{J}}^{\text{sp}}}(\bar{r})\right)\!\right\} \!- \log_{2} \!\left(\!\det\left(\pi e \mathbf{R}_{\text{sp}}\right) \!\right) $$
for a finite constellation input signal, where \(\mathbf {r}_{\mathcal {J}}^{\text {sp}} \triangleq \sum _{k\in \mathcal {J}}\sqrt {E_{k}}\bar {\mathbf {S}}_{k} \mathbf {d}_{k} + \mathbf {n}_{\text {sp}}\) is a Gaussian mixture random vector.
Numerical characterization
In this subsection, we numerically characterize the capacity for a two-user uplink WCDMA example. For numerical experiments, we set the parameters which are close to those in a real uplink UMTS system as specified in [1]: time-variant CDMA with orthogonal variable spreading factor (OVSF) codes and Gold sequences, spreading factor N sf =4, SRRC chip waveform p(t) with roll-off factor 0.22, and uniform power allocation E 1/σ 2=E 2/σ 2=SNR. In the simulations, we employ a time-invariant multipath channel with L=3 taps, a relative path-amplitude vector a=[0,−1.5,−3] dB, a relative path-phase vector \(\boldsymbol {\theta }=\left [0,\frac {\pi }{3}, \frac {2\pi }{3}\right ]\), and path-delay vector \(\boldsymbol {\tau }=\left [0,\frac {T_{c}}{2}, T_{c}\right ]\). Thus, the l-th element of the path-coefficient vector g 1 is given by \(a_{l} \cdot \mathrm {e}^{j \theta _{l}}/\|\mathbf {a}\|\) where a l is the l-th element of a, θ l is the l-th element of vector θ, and \(\mathbf {g}_{2}=\sqrt {2}\mathbf {g}_{1}\). In addition, we use fixed user delays which are randomly drawn within a symbol time once at the beginning of simulations, i.e., \(\lambda _{k} \sim \mathcal {U}(0,T_{s})\).
Figure 2 illustrates the capacity of a two-user uplink UMTS system with N=2 and M=1 for Gaussian-distributed input (from Section 3.1) and 4-QAM (QPSK) input (from Section 3.2) signals. The left-hand side sub-figure presents the sum- and individual capacities for Gaussian and 4-QAM input signals. The individual capacities R 2 are larger than R 1 since we set \(\mathbf {g}_{2}=\sqrt {2}\mathbf {g}_{1}\). The right-hand side sub-figure shows the capacity regions with 4-QAM input for several values of SNR. As expected, the capacity region enlarges with increasing SNR. Moreover, as the SNR tends to infinity, the capacity region converges to the corresponding source entropy outer bound (i.e., 2 bits/symbol individual rates and 4 bits/symbol sum-rate for the two-user channel with 4-QAM input). It is interesting that the maximal individual rates (2 bits/symbol) can be achieved simultaneously, i.e., the sum-rate constraint is asymptotically inactive in the high- SNR regime. A deeper analysis on this asymptotic behavior will be given in the next section.
Capacity curves and capacity regions for a two-user set-up with N=2 and M=1. On the left-hand side, the solid lines represent the capacities with Gaussian input and the dotted lines represent the capacities with 4-QAM input. All the capacities are normalized by 1/N
Figure 3 shows the achievable sum-rates for larger block length (N=32) and two codes (M=2) in each I/Q branch. In this figure, both the achievable sum-rates achieved by sufficient statistic (from Sections 3.1–3.2) and by sampling (from Section 3.3) are included. For achievable sum-rates using sampling, the experiments with lower than Nyquist rate (T sp =T c >T ny ) and Nyquist rate (T sp =T ny ) are considered. As expected, the sum-capacity achieved by the sufficient statistic is an upper bound for the sum-rates achieved by systems employing sampling. Moreover, even when the samples are taken with Nyquist rate, there are still gaps between the sum-rates achieved by sampling \((R^{\text {Gauss}}_{\textit {sp}}(T_{\textit {sp}}=T_{\textit {ny}})\) and \(R^{\text {QAM}}_{\textit {sp}}(T_{\textit {sp}}=T_{\textit {ny}}))\) and the sum-capacities achieved with matched filtering/sufficient statistic (\(R^{\text {Gauss}}_{\textit {ss}}\) and \(R^{\text {QAM}}_{\textit {ss}}\)). These losses are due to the finite time limit of our sampling window as the Nyquist sampling theorem states that a infinite sample sequence is required to be able to perfectly recover a finite energy and band-limited signal ([26], Theorem 8.4.3). Fortunately, by extending the sampling window by only two symbol durations on each side \(\left (\text {for}~R^{\text {Gauss}}_{sp+}~\text {and}~ R^{\text {QAM}}_{sp+}\right)\), these losses can be significantly reduced.
Achievable sum-rates for different inputs and receiver structures with N=32 and M=2. \(R^{\text {Gauss}}_{\textit {ss}}\) and \(R^{\text {QAM}}_{\textit {ss}}\) denote the sum-rates achieved by the sufficient statistic with Gaussian input and 4-QAM input, respectively. \(R^{\text {Gauss}}_{\textit {sp}}\) and \(R^{\text {QAM}}_{\textit {sp}}\) denote the sum-rates achieved by sampling with the sampling window equal to the block length t n ∈ [0 N T s ]. \(R^{\text {Gauss}}_{sp+}\) and \(R^{\text {QAM}}_{sp+}\) denote the sum-rates achieved by sampling with the sampling window extended by two symbol durations on each side, i.e., t n ∈ [−2T s (N+2)T s ]. All the sum-rates are normalized by 1/N
Recalling (7) with \(\mathcal {J} = \{1,\ldots,K\}\), we have \(\sum _{k=1}^{K}R_{k}\leq I(\mathbf {d};\mathbf {z})\leq \sum _{k=1}^{K} H(\mathbf {d}_{k})\), i.e., the sum-capacity is upper bounded by the sum of the sources entropies. However, the results in Fig. 2 show that when SNR→∞, the individual capacities can be simultaneously achieved, i.e., the sum-capacity approaches the sum of the individual source entropies. In this section, we provide a deeper analysis on this observation by characterizing the asymptotic behavior of the sum-capacity. For convenience, we begin with a simple MAC model then extend the result to the uplink WCDMA system in the following.
Simple K-user MAC model
Firstly, we start from the asymptotic sum-capacity of a simple K-user MAC, where each user transmits only one data stream. This setup corresponds to our uplink WCDMA system with M=1 and N=1 in a frequency-non-selective channel. The results are mainly based on the following lemma.
Lemma 1.
Consider the received signal model of a K-user MAC
$$\begin{array}{@{}rcl@{}} y(t)=\sum\limits_{k=1}^{K}d_{k}s_{k}(t)+n(t), \end{array} $$
where d 1,…,d K are the unit power transmitted symbols, which are independent and transmitted using K-normalized signature waveforms s 1(t),…,s K (t) and n(t) denotes the Gaussian noise process with PSD \(\frac {1}{\text {SNR}}\). When SNR→∞, the asymptotic sum-capacity of channel (15), \(C_{\text {sum}}^{\text {as}}\,=\,\sum _{k=1}^{K}\! H(d_{k})\), is achieved if the vector space \(\mathcal {S}_{K}=\text {span}\{s_{1}(t),\dots,s_{K}(t)\}\) has the dimension K.
The proof of Lemma 1 is given in Appendix 2. The idea for the proof is that we first show that the received signal passing through a bank of matched filters, which are matched to the signature waveforms, yields a sufficient statistic for decoding d= [d 1,⋯,d K ] based on y(t). Then, we show that d can be uniquely decoded, i.e., the decoder is able to decode the messages correctly from this sufficient statistic when SNR→∞ if \(\text {dim}(\mathcal {S}_{K})=K\). Based on the uniquely decodable property, we then prove that the asymptotic sum-capacity \(C_{\text {sum}}^{\text {as}}\) approaches the sum of source entropies if the signature waveforms are linearly independent of each other, i.e., \(\text {dim}(\mathcal {S}_{K})=K\).
Uplink WCDMA system
Next, we extend the results from the above simple K-user MAC to the asymptotic sum-capacity of the uplink WCDMA system. Let us recall the transmitted signal from user k of the uplink WCDMA system in (1), and take into account all K users. The transmitted signal can be considered as one of equivalent KNM-user MACs in (15) using KNM signature waveforms \(s_{\textit {ki}}^{m}(t-\lambda _{k})\), k=1,…,K,i=1,…,N, and m=1,…,M. The following propositions, which can be derived from Lemma 1, specify the asymptotic sum-capacity of the uplink WCDMA system in different channel environments considering frequency-non-selective (L=1) and frequency-selective (L≥2) channels.
Proposition 1.
The asymptotic sum-capacity of the frequency-non-selective uplink WCDMA channel as described in (2) with L=1 is \(C_{\text {sum}, \text {nsec}}^{\text {as}}= \sum _{k=1}^{K} H(\mathbf {d}_{k})\) if the dimension of the signature waveforms space \(\mathcal {S}_{T}=\text {span}\left \{s_{11}^{1}(t-\lambda _{1}),\ldots, s_{\textit {KN}}^{M}(t-\lambda _{K})\right \}\) is KNM.
The proof for Proposition 1 is given in Appendix 3. The intuition behind Proposition 1 can be expressed as: K users transmit KNM symbols and the receiver performs matched filtering with KNM fingers. Although the uplink WCDMA multiuser channel implies K-user SISO MACs, the matching process virtually converts this to an equivalent K N M×K N M MIMO channel. Thus, by appropriately choosing the signature waveforms and matched fingers, which yield a full-rank equivalent channel matrix H, the transmitted symbols, d 1,…,d K , can be perfectly (i.e., error-free) recovered from z as SNR goes to infinity. In other words, a K-user uplink WCDMA channel can asymptotically achieve the capacity of KNM parallel channels as long as \(\text {dim}(\mathcal {S}_{T})=KNM\).
The asymptotic sum-capacity of the frequency-selective uplink WCDMA channel as described in (2) is \(C_{\text {sum}, \text {sec}}^{\text {as}}~~~~=\sum _{k=1}^{K}\! H(\mathbf {d}_{k})\) if \(\text {dim}(\mathcal {\overline {S}})=KNM\), where \(\mathcal {\overline {S}}=\text {span}\left \{\bar {s}_{11}^{1}(t),\cdots, \bar {s}_{\textit {KN}}^{M}(t)\right \}\) is the vector space spanned by the effective signature waveform \(\bar {s}_{\textit {ki}}^{m}(t)=\sum _{l=1}^{L}g_{\textit {kl}}s_{\textit {ki}}^{m}\) (t−λ k −τ kl ).
The proof for Proposition 2 is given in Appendix 4. Unlike the frequency-non-selective channel case, the sufficient condition for achieving the asymptotic sum-capacity in a frequency-selective channel case is based on the effective signature waveforms, which include the impact of the channel gains {g k } k and delays {τ kl } k,l . This implies that the multi-path channel may help the equivalent channel matrix H to achieve full-rank. For instance, if \(\text {dim}(\mathcal {S}_{T}) < KNM\), H is obviously singular when L=1, while H can be still invertible when L>1 since \(\text {dim}(\overline {\mathcal {S}})\) is possible to be equal to KNM according to the channel selectivity and the potential offset in the multi-path environment5. This is particularly helpful in an overloaded CDMA system [8, 21], where the number of users exceed the spreading factor.
Propositions 1 and 2 state sufficient conditions that the transmitted messages can be uniquely decoded when SNR→∞, which holds for all kinds of input signals including both finite and infinite constellation signals. However, the conditions in Propositions 1 and 2 can be relaxed in certain scenarios with finite constellation inputs. In this subsection, we first consider a simple example where such conditions can be relaxed. The necessary condition for the unique decoding with finite constellation input is then studied in the following.
For instance, let us consider an example with two different set-ups of channel (15) with K=2 and binary transmitted signals d a = [d a1 d a2]T and d b = [d b1 d b2]T, i.e.,
$$\begin{array}{@{}rcl@{}} y_{a}(t)&=&d_{a1}s_{1}(t)+d_{a2}s_{2}(t)+n_{a}(t), \\ y_{b}(t)&=&d_{b1}s_{1}(t)+d_{b2}s_{2}(t)+n_{b}(t), \end{array} $$
where n a (t) and n b (t) denote additive Gaussian noise processes. We assume that the same signature waveform space \(\mathcal {S}_{2}:\) =span{s 1(t),s 2(t)} is used in both set-ups. However, the transmitted symbols are uniformly and randomly picked up from different input constellation sets: d a1,d a2∈{0,1} and \(d_{b1} \in \{-1/\sqrt {2},1/\sqrt {2}\}\), d b2∈{0,1}. The corresponding sufficient statistic models are then given by
$$\begin{array}{@{}rcl@{}} Y_{a}&=&d_{a1}+d_{a1}+N_{a}, \\ Y_{b}&=&d_{b1}+d_{b2}+N_{b}, \end{array} $$
where Y a and Y b denote the sufficient statistics and N a and N b are the equivalent noises. We can see that when the noise power becomes zero (or SNR→∞), d a (and so d a1 and d a2)) cannot be uniquely decoded from Y a since the conditional entropy H(d a1,d a2|Y a )=0.5>0 when SNR→∞. However, (d b1 and d b2) can be uniquely decoded from Y b since H(d b1,d b2|Y b )=0 when SNR→∞ even though \(\text {dim}(\mathcal {S}_{2})=1<2\). It shows that the condition \(\text {dim}(\mathcal {S}_{K})=K\) can be relaxed for certain signal constellation structures. Therefore, it is expected that necessary conditions for achieving the unique decoding with finite constellation input have to take both the signature waveforms and the structure of the signal constellation into account.
Let us assume that \(\mathbf {d}\in \mathcal {M}^{KNM}\), where \(\mathcal {M}\) is a set of constellation points and is finite. In order to derive the sufficient condition for the unique decoding, we refer the equivalent channel in (C.2) in Appendix 3
$$ \mathbf{z}=\mathbf{HEd}+\mathbf{n}. $$
When SNR→∞, the transmitted vector d can be uniquely decoded from z if and only if the mapping
$$\begin{array}{@{}rcl@{}} f: \mathcal{M}^{KNM} &\mapsto \mathcal{C}^{KNM}\\ \mathbf{d} &\mapsto \mathbf{HEd} \end{array} $$
is an one-to-one mapping. In particular, for any pair of \(\mathbf {d}^{i},\mathbf {d}^{j} \in \mathcal {M}^{KNM}\) and d i≠d j, the condition H E d i≠H E d j is needed for the unique decoding. Therefore, by defining v ij =d i−d j, the condition for the unique decoding becomes
$$ \mathbf{HEv}_{ij}\neq0, \forall i\neq j. $$
In other words, the necessary condition for the unique decoding is that any vector v ij with i≠j is not in the null space of matrix H E. This necessary condition includes the impact of signal constellation reflected via v ij .
This result is consistent with the sufficient conditions in Propositions 1 and 2. Indeed, when the (effective) signature waveform space has dimension KNM and H is invertible, the null space of H E is empty. Thus, the condition in (17) holds for any set of vector v ij , and the unique decoding is achieved for any kind of input signal.
This paper studies the capacity limit of the uplink WCDMA system whose set-up has been chosen to be close to real CDMA cellular networks. We present a theoretical framework, which can be used to evaluate how close the maximal performance of a practical system design is to the theoretical fundamental limit. To this end, sufficient statistics for decoding the transmitted messages were derived using a bank of matched filters, each of which is matched to the signature waveforms. An equivalent discrete-time channel model based on the derived sufficient statistics was provided which can be used to analyze the capacity of the system. The capacity regions for finite constellation input and Gaussian-distributed input signals have been both analytically and numerically characterized. The comparison with the sampling capacity showed that sampling within the transmission time window might cause a capacity loss even if the sampling was performed at Nyquist rate. Fortunately, this loss could be significantly diminished by extending the sampling window by only two symbol durations. Moreover, the asymptotic analysis shows that for proper choices of the (effective) signature waveforms, a K-user uplink WCDMA channel can be decoupled so that each user achieves a point-to-point channel capacity when SNR goes to infinity. The presented framework and results provide valuable insights for the design and further development of not only WCDMA but also other wireless standard networks.
Appendix 1 Derivation of equivalent noise statistic
Since n(t) is a zero mean complex Gaussian random process, the equivalent noises after a bank of linear filters (matched filters), \(n_{\textit {ki}}^{ml}=\int _{-\infty }^{\infty }n(t){s_{\textit {ki}}^{m}(t-\lambda _{k}-\tau _{\textit {kl}})}^{*}dt\), ∀k,i,m,l, are zero mean joint Gaussian random variables ([28], Chap. 8) with the correlation coefficient given by
$$\begin{array}{@{}rcl@{}} &\mathbb{E}&\left\{n_{ki}^{ml}{n_{k^{\prime}i^{\prime}}^{m^{\prime}l^{\prime}}}^{*}\right\}\\ &=&\!\mathbb{E}\!\left\{\!\int_{-\infty}^{\infty}\!\!\!\!\!\!\!\!\!n(t){s_{ki}^{m}\left(t\,-\,\!\lambda_{k}\! -\!\tau_{kl}\right)\!}^{*}dt \!\!\!\int_{-\infty}^{\infty}\!\!\!\!\!\!\!\!\!{n(t^{\prime})}^{*}s_{k^{\prime}i^{\prime}}^{m^{\prime}}\!\left(t^{\prime}\!- \!\!\lambda_{k^{\prime}}\,-\,\tau_{k^{\prime}l^{\prime}}\!\right)dt^{\prime}\!\right\} \\ &=&\!\!\int_{\!-\infty}^{\infty}\!\!\int_{-\infty}^{\infty}\!\!\!\!\!\!\!\mathbb{E}\!\left\{\!n(t){n(t^{\prime})}\!^{*}\!\right\} \!\!{s_{ki}^{m}(t\,-\,\!\lambda_{k}\,-\,\tau_{kl})}^{*}s_{k^{\prime}\!i^{\prime}}^{m^{\prime}}(t^{\prime}\!\!- \!\!\lambda_{k^{\prime}}\,-\,\!\tau_{k^{\prime}\!l^{\prime}}\!)dtdt^{\prime}, \end{array} $$
s where \(\mathbb {E}\left \{ n(t){n(t')}^{*} \right \} = \sigma ^{2}\delta (t-t^{\prime })\). Thus, we have
$$\begin{aligned} {}\mathbb{E}&\left\{n_{ki}^{ml}{n_{k^{\prime}i^{\prime}}^{m^{\prime}l^{\prime}}}^{*}\right\}\\ {}&=\!\!\int_{\!-\infty}^{\infty}\!\!\int_{-\infty}^{\infty}\!\!\!\!\!\!\!\sigma^{2}\delta(t\,-\,t^{\prime}){s_{ki}^{m}(t\!- \!\lambda_{k}\,-\,\tau_{kl})}^{*}s_{k^{\prime}\!i^{\prime}}^{m^{\prime}}(t^{\prime}\,-\,\lambda_{k^{\prime}}\!- \!\tau_{k^{\prime}\!l^{\prime}})dtdt^{\prime}\\ {}&=\!\sigma^{2}\!\!\!\int_{-\infty}^{\infty}\!\!\!\!\!{s_{ki}^{m}(t\,-\,\!\lambda_{k}\!\,-\,\tau_{kl})}^{*}\! \left[\!\int_{-\infty}^{\infty}\!\!\!\!\!\delta(t\,-\,t^{\prime})s_{k^{\prime}\!i^{\prime}}^{m^{\prime}}(t^{\prime}\!\!- \!\lambda_{k^{\prime}}\,-\,\tau_{k^{\prime}l^{\prime}}\!)dt^{\prime}\!\right]\!dt \\ {}&=\sigma^{2}\!\!\int_{-\infty}^{\infty}\!\!{s_{ki}^{m}(t-\lambda_{k}-\tau_{kl})}^{*}s_{k^{\prime}i^{\prime}}^{m^{\prime}}(t- \lambda_{k^{\prime}}-\tau_{k^{\prime}l^{\prime}})dt\\ {}&=\sigma^{2}\rho_{(kiml)}^{(k^{\prime}i^{\prime}m^{\prime}l^{\prime})}. \end{aligned} $$
Accordingly, the equivalent noises \(n_{\textit {ki}}^{m}=\sum _{l=1}^{L}{g_{\textit {kl}}}^{*}n_{\textit {ki}}^{ml}\), ∀k,i,m are zero mean joint Gaussian random variables with correlation coefficient
$$\begin{array}{@{}rcl@{}} \mathbb{E}\left\{n_{ki}^{m}{n_{k^{\prime}i^{\prime}}^{m^{\prime}}}^{*}\right\} &=\sum\limits_{l=1}^{L}\sum\limits_{l^{\prime}=1}^{L}{g_{kl}}^{*}g_{k^{\prime}l^{\prime}}\mathbb{E}\left\{n_{ki}^{ml} {n_{k^{\prime}i^{\prime}}^{m^{\prime}l^{\prime}}}^{*}\right\}\\ &=\sigma^{2}\sum\limits_{l=1}^{L}\sum\limits_{l^{\prime}=1}^{L}{g_{kl}}^{*}g_{k^{\prime}l^{\prime}} \rho_{(kiml)}^{(k^{\prime}i^{\prime}m^{\prime}l^{\prime})}. \end{array} $$
Moreover, we have the (a,b)th element of H expressed as
$$ \mathbf{H}[a,b]=\sum\limits_{l=1}^{L}\sum\limits_{l^{\prime}=1}^{L}{g_{kl}}^{*}g_{k^{\prime}l^{\prime}} \rho_{(kiml)}^{(k^{\prime}i^{\prime}m^{\prime}l^{\prime})}, $$
((A.1))
where the indices are given by
$$\begin{array}{@{}rcl@{}} a&=&(k-1)NM+(i-1)M+m,\\ b&=&(k'-1)NM+(i'-1)M+m'. \end{array} $$
As a result, n is a complex Gaussian random vector with zero mean and covariance matrix σ 2 H.
Appendix 2 Proof of Lemma 1
The proof of Lemma 1 consists of two parts:
Part 1: We first show that the received signal passed through a bank of matched filters, which match to the signature waveforms, yields a sufficient statistic for decoding d= [d 1,⋯,d K ]T based on y(t). Moreover, when SNR→∞, d can be uniquely decoded if \(\text {dim}(\mathcal {S}_{K})=K\).
Part 2: Based on the uniquely decodable property, we then derive the asymptotic sum-capacity.
Part 1: Part 1 is a result of the following claim
Claim.
Let the received signal y(t) in (15) passed through a bank of matched filters, where y(t) is matched with each signature waveform s k (t), i.e.,
$$\begin{array}{@{}rcl@{}} y_{k}=\langle y(t),s_{k}(t) \rangle =\int_{-\infty}^{\infty} y(t)s_{k}^{*}(t)dt,~~ k=1\cdots K. \end{array} $$
Then y =[y 1,⋯,y K ]T is a sufficient statistic for decoding d based on y(t). Moreover, if the vector space \(\mathcal {S}_{K}=\text {span}\{s_{1}(t),\cdots,s_{K}(t)\}\) has a dimension of K, d can be uniquely decoded from the sufficient statistic y as SNR→∞.
Following similar steps as in Section 2.2, it can be shown that y is a sufficient statistic for decoding d based on y(t).
It remains to show that d can be uniquely decoded from y when SNR→∞. Let us denote R s as the correlation matrix of the signature waveforms {s k (t)} k , where R s [i,j]=〈s i (t),s j (t)〉. Therewith, we have the equivalent matrix expression
$$\mathbf{y}=\mathbf{R}_{s}\mathbf{d}+\widetilde{\mathbf{n}}, $$
where \(\widetilde {\mathbf {n}}\) is the equivalent noise vector.
Since \(\text {dim}(\mathcal {S}_{K})=K\), we can rewrite {s 1(t),⋯,s K (t)} as
$$\left[ \begin{array}{c} s_{1}(t) \\ \vdots\\ s_{K}(t) \end{array} \right] = \mathbf{A}\left[ \begin{array}{c} e_{1}(t) \\ \vdots\\ e_{K}(t) \end{array} \right] $$
where {e 1(t),⋯,e K (t)} is an orthonormal basis of \(\mathcal {S}_{K}\) and A is a K×K full rank matrix.
Consider the correlation matrix R e where
$$\mathbf{R}_{e}[i,j]=\langle e_{i}(t),e_{j}(t)\rangle. $$
Then, we have R e =I K since {e 1(t),⋯,e K (t)} is an orthonormal basic. Moreover,
$$ \mathbf{R}_{s}=\mathbf{A}\mathbf{R}_{e}\mathbf{A}^{\dagger}=\mathbf{A}\mathbf{A}^{\dagger}. $$
$$\text{rank}(\mathbf{R}_{s})=\text{rank}(\mathbf{A}\mathbf{A}^{\dagger})=\text{rank}(\mathbf{A})=K. $$
Therefore, when \(\text {dim}(\mathcal {S}_{K})=K\), R s is invertible and d can be uniquely decoded from y when SNR→∞ since \(\underset {\text {SNR}\rightarrow \infty }{\lim }\mathbf {R}_{s}^{-1}\mathbf {y}=\mathbf {d}\).
Part 2: We next derive the asymptotic sum-capacity based on the sufficient statistic from Part 1.
The sum-capacity of the channel (15) is given by
$$ C_{\text{sum}}=I(\mathbf{d};y(t)). $$
((B.1))
From Part 1, we know that y is a sufficient statistic for decoding d based on y(t). Thus,
$$ I(\mathbf{d};y(t))=I(\mathbf{d};\mathbf{y}), $$
$$ I(\mathbf{d};\mathbf{y})=H(\mathbf{d})-H(\mathbf{d}|\mathbf{y}). $$
When \(\text {dim}(\mathcal {S}_{K})=K\), following from Part 1, R s is invertible and \(\underset {\text {SNR}\rightarrow \infty }{\lim }\mathbf {R}_{s}^{-1}\mathbf {y}=\mathbf {d}\). Therefore,
$$ {\lim}_{\text{SNR}\rightarrow\infty}H(\mathbf{d}|\mathbf{y})=H(\mathbf{R}_{s}^{-1}\mathbf{y}|\mathbf{y})=0 $$
Let us define the asymptotic sum-capacity as \(C_{\text {sum}}^{\text {as}}=\underset {\text {SNR}\rightarrow \infty }{\lim }C_{\text {sum}}\), combining (B.1) −(B.4), we have
$$C_{\text{sum}}^{\text{as}}=H(\mathbf{d})=\sum\limits_{k=1}^{K} H(d_{k}). $$
This completes the proof for Lemma 1. ■
Appendix 3 Proof of Proposition 1
Proposition 1 is proved in three steps:
Step 1: We first re-formulate (5) by an equivalent input-output model, in which H is decomposed into a multiplication of multiple matrices including a matrix that depends only on the signature waveform correlation coefficients. To this end, we rewrite the equivalent channel H as follows:
$$\begin{array}{@{}rcl@{}} \mathbf{H}&=&[\mathbf{H}_{1}, \ldots, \mathbf{H}_{K}]\\ &=&\left[ \begin{array}{ccccccccccccc} \mathbf{G}_{1}^{\dagger}\mathbf{R}_{1}^{1}\mathbf{G}_{1} & \mathbf{G}_{1}^{\dagger}\mathbf{R}_{1}^{2}\mathbf{G}_{2} & \cdots &\mathbf{G}_{1}^{\dagger}\mathbf{R}_{1}^{K}\mathbf{G}_{K} \\ \mathbf{G}_{2}^{\dagger}\mathbf{R}_{2}^{1}\mathbf{G}_{1} & \mathbf{G}_{2}^{\dagger}\mathbf{R}_{2}^{2}\mathbf{G}_{2} & \cdots &\mathbf{G}_{2}^{\dagger}\mathbf{R}_{2}^{K}\mathbf{G}_{K} \\ \vdots & \vdots & \vdots & \vdots \\ \mathbf{G}_{K}^{\dagger}\mathbf{R}_{K}^{1}\mathbf{G}_{1} & \mathbf{G}_{K}^{\dagger}\mathbf{R}_{K}^{2}\mathbf{G}_{2} & \cdots &\mathbf{G}_{K}^{\dagger}\mathbf{R}_{K}^{K}\mathbf{G}_{K} \\ \end{array} \right]. \end{array} $$
Therefore, the equivalent channel H can be expressed as
$$ \mathbf{H}=\mathbf{G}^{\dagger}\mathbf{R}\mathbf{G}, $$
((C.1))
$$\mathbf{G}=\text{blkdiag} \left(\mathbf{G}_{1},\ldots,\mathbf{G}_{K} \right) \in \mathbb{C}^{KNML\times KNM}, $$
$$ \mathbf{R}=\left[ \begin{array}{ccccccccccccc} \mathbf{R}_{1}^{1}& \mathbf{R}_{1}^{2}& \cdots &\mathbf{R}_{1}^{K}\\ \mathbf{R}_{2}^{1}& \mathbf{R}_{2}^{2} & \cdots &\mathbf{R}_{2}^{K}\\ \vdots & \vdots & \vdots & \vdots \\ \mathbf{R}_{K}^{1}&\mathbf{R}_{K}^{2}& \cdots &\mathbf{R}_{K}^{K}\\ \end{array} \right] \in\mathbb{C}^{KNML\times KNML}. $$
Thus, we have the equivalent input-output model as
$$ \mathbf{z}=\mathbf{HEd}+\mathbf{n}, $$
$$\mathbf{E}=\text{diag} \left(\underbrace{\sqrt{E_{1}},\ldots,\sqrt{E_{1}}}_{NM ~ \text{elements}},\ldots,\underbrace{\sqrt{E_{K}},\ldots,\sqrt{E_{K}}}_{NM ~ \text{elements}} \right). $$
Step 2: We show that H is invertible when L=1 and \(\text {dim}(\mathcal {S}_{T})=KNM\).
Following the same arguments as for the proof of Lemma 1, we have R as full rank since \(\text {dim}(\mathcal {S}_{T})=KNM\). Since H=G † R G, where G,R are the square matrices with full rank, it follows that H is invertible.
Step 3: Lastly, we can conclude on the asymptotic capacity of the channel (5). Since z in (C.2) is a sufficient statistic for decoding d based on y(t) (from Section 2.2) and H is invertible, similarly to the proof in Part 2 of Lemma 1, it follows that
$$ C_{\text{sum},\text{nsec}}^{\text{as}}=H(\mathbf{d})=\sum\limits_{k=1}^{K} H(\mathbf{d}_{k}). $$
This completes the proof for Proposition 1. ■
Let us define \(\bar {\rho }_{\textit {kim}}^{k'i'm'}\) as the inner product between \(\bar {s}_{\textit {ki}}^{m}(t)\) and \(\bar {s}_{k'i'}^{m'}(t)\), i.e.,
$$\begin{array}{@{}rcl@{}} \bar{\rho}_{kim}^{k'i'm'}&\!\!=&\langle\bar{s}_{ki}^{m}(t),\bar{s}_{k'i'}^{m'}(t)\rangle\\ &\!\!=&\sum\limits_{l=1}^{L}\!\sum\limits_{l'=1}^{L} \!g_{kl}^{*}\langle s_{ki}^{m}(t-\!\lambda_{k}\,-\,\tau_{kl}),s_{k'i'}^{m'}(t-\!\lambda_{k}\,-\,\tau_{k'l'})\rangle g_{k'l'}\\ &\!\!=&\sum\limits_{l=1}^{L}\!\sum\limits_{l'=1}^{L} g_{kl}^{*}\rho_{kiml}^{k'i'm'l} g_{k'l'}. \end{array} $$
((D.1))
Define \(\mathbf {R}_{\overline {s}}\) as the correlation matrix of \(\mathcal {\overline {S}}\), where
$$ \mathbf{R}_{\overline{s}}[a,b]=\bar{\rho}_{kim}^{k'i'm'}, $$
and the coefficient indices are given by
$$\begin{aligned} &a=(k-1)NM+(i-1)M+m,\\ &b=(k'-1)NM+(i'-1)M+m'. \end{aligned} $$
Since \(\mathbf {R}_{\overline {s}}\) is the correlation matrix of \(\overline {\mathcal {S}}\), following similar steps as for the proof in Part 1 of Lemma 1, we arrive at \(\text {rank}(R_{\overline {s}})=KNM\) if \(\text {dim}(\overline {\mathcal {S}})=KNM\).
Moreover, combining (A.1) with (D.1) and (D.2), we have
$$\mathbf{H}=\mathbf{R}_{\overline{s}}. $$
Thus, rank(H)=K N M and H is invertible.
Finally, similar to the proof in Part 2 of Lemma 1, given H is invertible, the asymptotic sum-capacity is given by
$$ C_{\text{sum}, \text{sec}}^{\text{as}}=H(\mathbf{d})=\sum\limits_{k=1}^{K} H(\mathbf{d}_{k}). $$
1 In theory, a band-limited signal requires infinite time to transmit. However, in practical WCDMA systems, the chip waveforms with fast decaying sidelobes (e.g., root raised cosine (RRC) and squared-root raised cosine (SRRC) pulses) are used and truncated by the length of several chip intervals.
2 It is worth noting that even though the channel impulse response is assumed to be time-invariant as similar to [12, 24], the Toeplitz structure of the equivalent channel matrix is not maintained because of the variation of the spreading sequences over symbols in a time-variant CDMA system.
3 In [27], [Proposition 3.2], the hypotheses are equiprobable and the optimal decision is based on maximum of Φ(d) (ML criterion). In general, the optimal decision is based on MAP criterion, which includes log(p(d)) into μ(t;d). However, this additional term is independent of r(t). Therefore, we do not have to include it in the sufficient statistic.
4 Similarly to the capacity achieved by sufficient statistic, the quantities in the right-hand sides of (13) and (14) describe the number of bits that can be reliably transmitted per block of N symbols. One can express the capacity in bit/second by normalizing with 1/T, T=N T s , or in bit/second/Hertz by normalizing with 1/T W.
5 When L>1, the channel gain matrix G is not a square matrix anymore. The invertible property of H does not depend only on rank of the correlation matrix R but also on G.
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School of Electrical Engineering, KTH Royal Institute of Technology, Stockholm, Sweden
Tan Tai Do
, Tobias J. Oechtering
& Mikael Skoglund
Department of Electronics Engineering, Korea Polytechnic University, Sangidaehak-ro, 15073 237, Gyeonggi-do, Korea
Su Min Kim
R&D Center, Huawei Technologies Sweden AB, Stockholm, Sweden
Gunnar Peters
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Correspondence to Su Min Kim.
This work was partly presented at the 80th IEEE Vehicular Technology Conference (VTC2014-Fall), Vancouver, Canada, September 2014.
Do, T.T., Kim, S.M., Oechtering, T.J. et al. Waveform domain framework for capacity analysis of uplink WCDMA systems. J Wireless Com Network 2015, 253 (2015) doi:10.1186/s13638-015-0480-5
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Based on the idea and the provided source code of Andrej Karpathy (arxiv-sanity)
The Transactional Conflict Problem (1804.00947)
Dan Alistarh, Syed Kamran Haider, Raphael Kübler, Giorgi Nadiradze
April 3, 2018 cs.DC
The transactional conflict problem arises in transactional systems whenever two or more concurrent transactions clash on a data item. While the standard solution to such conflicts is to immediately abort one of the transactions, some practical systems consider the alternative of delaying conflict resolution for a short interval, which may allow one of the transactions to commit. The challenge in the transactional conflict problem is to choose the optimal length of this delay interval so as to minimize the overall running time penalty for the conflicting transactions. In this paper, we propose a family of optimal online algorithms for the transactional conflict problem. Specifically, we consider variants of this problem which arise in different implementations of transactional systems, namely "requestor wins" and "requestor aborts" implementations: in the former, the recipient of a coherence request is aborted, whereas in the latter, it is the requestor which has to abort. Both strategies are implemented by real systems. We show that the requestor aborts case can be reduced to a classic instance of the ski rental problem, while the requestor wins case leads to a new version of this classical problem, for which we derive optimal deterministic and randomized algorithms. Moreover, we prove that, under a simplified adversarial model, our algorithms are constant-competitive with the offline optimum in terms of throughput. We validate our algorithmic results empirically through a hardware simulation of hardware transactional memory (HTM), showing that our algorithms can lead to non-trivial performance improvements for classic concurrent data structures.
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Dan Alistarh, Trevor Brown, Justin Kopinsky, Jerry Z. Li, Giorgi Nadiradze
Relaxed concurrent data structures have become increasingly popular, due to their scalability in graph processing and machine learning applications. Despite considerable interest, there exist families of natural, high performing randomized relaxed concurrent data structures, such as the popular MultiQueue pattern for implementing relaxed priority queue data structures, for which no guarantees are known in the concurrent setting. Our main contribution is in showing for the first time that, under a set of analytic assumptions, a family of relaxed concurrent data structures, including variants of MultiQueues, but also a new approximate counting algorithm we call the MultiCounter, provides strong probabilistic guarantees on the degree of relaxation with respect to the sequential specification, in arbitrary concurrent executions. We formalize these guarantees via a new correctness condition called distributional linearizability, tailored to concurrent implementations with randomized relaxations. Our result is based on a new analysis of an asynchronous variant of the classic power-of-two-choices load balancing algorithm, in which placement choices can be based on inconsistent, outdated information (this result may be of independent interest). We validate our results empirically, showing that the MultiCounter algorithm can implement scalable relaxed timestamps, which in turn can improve the performance of the classic TL2 transactional algorithm by up to 3 times, for some settings of parameters.
The Convergence of Stochastic Gradient Descent in Asynchronous Shared Memory (1803.08841)
Dan Alistarh, Christopher De Sa, Nikola Konstantinov
March 23, 2018 cs.DC, cs.LG, stat.ML
Stochastic Gradient Descent (SGD) is a fundamental algorithm in machine learning, representing the optimization backbone for training several classic models, from regression to neural networks. Given the recent practical focus on distributed machine learning, significant work has been dedicated to the convergence properties of this algorithm under the inconsistent and noisy updates arising from execution in a distributed environment. However, surprisingly, the convergence properties of this classic algorithm in the standard shared-memory model are still not well-understood. In this work, we address this gap, and provide new convergence bounds for lock-free concurrent stochastic gradient descent, executing in the classic asynchronous shared memory model, against a strong adaptive adversary. Our results give improved upper and lower bounds on the "price of asynchrony" when executing the fundamental SGD algorithm in a concurrent setting. They show that this classic optimization tool can converge faster and with a wider range of parameters than previously known under asynchronous iterations. At the same time, we exhibit a fundamental trade-off between the maximum delay in the system and the rate at which SGD can converge, which governs the set of parameters under which this algorithm can still work efficiently.
Byzantine Stochastic Gradient Descent (1803.08917)
Dan Alistarh, Zeyuan Allen-Zhu, Jerry Li
March 23, 2018 cs.DC, cs.DS, math.OC, cs.LG, stat.ML
This paper studies the problem of distributed stochastic optimization in an adversarial setting where, out of the $m$ machines which allegedly compute stochastic gradients every iteration, an $\alpha$-fraction are Byzantine, and can behave arbitrarily and adversarially. Our main result is a variant of stochastic gradient descent (SGD) which finds $\varepsilon$-approximate minimizers of convex functions in $T = \tilde{O}\big( \frac{1}{\varepsilon^2 m} + \frac{\alpha^2}{\varepsilon^2} \big)$ iterations. In contrast, traditional mini-batch SGD needs $T = O\big( \frac{1}{\varepsilon^2 m} \big)$ iterations, but cannot tolerate Byzantine failures. Further, we provide a lower bound showing that, up to logarithmic factors, our algorithm is information-theoretically optimal both in terms of sampling complexity and time complexity.
SparCML: High-Performance Sparse Communication for Machine Learning (1802.08021)
Cèdric Renggli, Dan Alistarh, Torsten Hoefler
Feb. 22, 2018 cs.DC, stat.ML
One of the main drivers behind the rapid recent advances in machine learning has been the availability of efficient system support. This comes both through hardware acceleration, but also in the form of efficient software frameworks and programming models. Despite significant progress, scaling compute-intensive machine learning workloads to a large number of compute nodes is still a challenging task, with significant latency and bandwidth demands. In this paper, we address this challenge, by proposing SPARCML, a general, scalable communication layer for machine learning applications. SPARCML is built on the observation that many distributed machine learning algorithms either have naturally sparse communication patters, or have updates which can be sparsified in a structured way for improved performance, without any convergence or accuracy loss. To exploit this insight, we design and implement a set of communication efficient protocols for sparse input data, in conjunction with efficient machine learning algorithms which can leverage these primitives. Our communication protocols generalize standard collective operations, by allowing processes to contribute sparse input data vectors, of heterogeneous sizes. We call these operations sparse-input collectives, and present efficient practical algorithms with strong theoretical bounds on their running time and communication cost. Our generic communication layer is enriched with additional features, such support for non-blocking (asynchronous) operations, and support for low-precision data representations. We validate our algorithmic results experimentally on a range of large-scale machine learning applications and target architectures, showing that we can leverage sparsity for order- of-magnitude runtime savings, compared to state-of-the art methods and frameworks.
Model compression via distillation and quantization (1802.05668)
Antonio Polino, Razvan Pascanu, Dan Alistarh
Feb. 15, 2018 cs.NE, cs.LG
Deep neural networks (DNNs) continue to make significant advances, solving tasks from image classification to translation or reinforcement learning. One aspect of the field receiving considerable attention is efficiently executing deep models in resource-constrained environments, such as mobile or embedded devices. This paper focuses on this problem, and proposes two new compression methods, which jointly leverage weight quantization and distillation of larger teacher networks into smaller student networks. The first method we propose is called quantized distillation and leverages distillation during the training process, by incorporating distillation loss, expressed with respect to the teacher, into the training of a student network whose weights are quantized to a limited set of levels. The second method, differentiable quantization, optimizes the location of quantization points through stochastic gradient descent, to better fit the behavior of the teacher model. We validate both methods through experiments on convolutional and recurrent architectures. We show that quantized shallow students can reach similar accuracy levels to full-precision teacher models, while providing order of magnitude compression, and inference speedup that is linear in the depth reduction. In sum, our results enable DNNs for resource-constrained environments to leverage architecture and accuracy advances developed on more powerful devices.
Compressive Sensing with Low Precision Data Representation: Radio Astronomy and Beyond (1802.04907)
Nezihe Merve Gürel, Kaan Kara, Dan Alistarh, Ce Zhang
Feb. 14, 2018 cs.LG, stat.ML
Modern scientific instruments produce vast amounts of data, which can overwhelm the processing ability of computer systems. Lossy compression of data is an intriguing solution but comes with its own dangers, such as potential signal loss, and the need for careful parameter optimization. In this work, we focus on a setting where this problem is especially acute compressive sensing frameworks for radio astronomy and ask: Can the precision of the data representation be lowered for all input data, with recovery guarantees and good practical performance? Our first contribution is a theoretical analysis of the Iterative Hard Thresholding (IHT) algorithm when all input data, that is, the measurement matrix and the observation, are quantized aggressively, to as little as 2 bits per value. Under reasonable constraints, we show that there exists a variant of low precision IHT which can still provide recovery guarantees. The second contribution is a tailored analysis of our general quantized framework to radio astronomy, showing that its conditions are satisfied in this case. We evaluate our approach using an FPGA implementation, and show that it can achieve up to 9.19x speed up with negligible loss of recovery quality, on real telescope data
DataBright: Towards a Global Exchange for Decentralized Data Ownership and Trusted Computation (1802.04780)
David Dao, Dan Alistarh, Claudiu Musat, Ce Zhang
Feb. 13, 2018 cs.AI, cs.ET, cs.DC, cs.DB, cs.LG
It is safe to assume that, for the foreseeable future, machine learning, especially deep learning will remain both data- and computation-hungry. In this paper, we ask: Can we build a global exchange where everyone can contribute computation and data to train the next generation of machine learning applications? We present an early, but running prototype of DataBright, a system that turns the creation of training examples and the sharing of computation into an investment mechanism. Unlike most crowdsourcing platforms, where the contributor gets paid when they submit their data, DataBright pays dividends whenever a contributor's data or hardware is used by someone to train a machine learning model. The contributor becomes a shareholder in the dataset they created. To enable the measurement of usage, a computation platform that contributors can trust is also necessary. DataBright thus merges both a data market and a trusted computation market. We illustrate that trusted computation can enable the creation of an AI market, where each data point has an exact value that should be paid to its creator. DataBright allows data creators to retain ownership of their contribution and attaches to it a measurable value. The value of the data is given by its utility in subsequent distributed computation done on the DataBright computation market. The computation market allocates tasks and subsequent payments to pooled hardware. This leads to the creation of a decentralized AI cloud. Our experiments show that trusted hardware such as Intel SGX can be added to the usual ML pipeline with no additional costs. We use this setting to orchestrate distributed computation that enables the creation of a computation market. DataBright is available for download at https://github.com/ds3lab/databright.
Space-Optimal Majority in Population Protocols (1704.04947)
Dan Alistarh, James Aspnes, Rati Gelashvili
July 13, 2017 cs.DC, cs.DS
Population protocols are a model of distributed computing, in which $n$ agents with limited local state interact randomly, and cooperate to collectively compute global predicates. An extensive series of papers, across different communities, has examined the computability and complexity characteristics of this model. Majority, or consensus, is a central task, in which agents need to collectively reach a decision as to which one of two states $A$ or $B$ had a higher initial count. Two complexity metrics are important: the time that a protocol requires to stabilize to an output decision, and the state space size that each agent requires. It is known that majority requires $\Omega(\log \log n)$ states per agent to allow for poly-logarithmic time stabilization, and that $O(\log^2 n)$ states are sufficient. Thus, there is an exponential gap between the upper and lower bounds. We address this question. We provide a new lower bound of $\Omega(\log n)$ states for any protocol which stabilizes in $O( n^{1-c} )$ time, for any $c > 0$ constant. This result is conditional on basic monotonicity and output assumptions, satisfied by all known protocols. Technically, it represents a significant departure from previous lower bounds. Instead of relying on dense configurations, we introduce a new surgery technique to construct executions which contradict the correctness of algorithms that stabilize too fast. Subsequently, our lower bound applies to general initial configurations. We give an algorithm for majority which uses $O(\log n)$ states, and stabilizes in $O(\log^2 n)$ time. Central to the algorithm is a new leaderless phase clock, which allows nodes to synchronize in phases of $\Theta(n \log{n})$ consecutive interactions using $O(\log n)$ states per node. We also employ our phase clock to build a leader election algorithm with $O(\log n )$ states, which stabilizes in $O(\log^2 n)$ time.
Robust Detection in Leak-Prone Population Protocols (1706.09937)
Dan Alistarh, Bartłomiej Dudek, Adrian Kosowski, David Soloveichik, Przemysław Uznański
Aug. 16, 2019 cs.DC, cs.DS
In contrast to electronic computation, chemical computation is noisy and susceptible to a variety of sources of error, which has prevented the construction of robust complex systems. To be effective, chemical algorithms must be designed with an appropriate error model in mind. Here we consider the model of chemical reaction networks that preserve molecular count (population protocols), and ask whether computation can be made robust to a natural model of unintended "leak" reactions. Our definition of leak is motivated by both the particular spurious behavior seen when implementing chemical reaction networks with DNA strand displacement cascades, as well as the unavoidable side reactions in any implementation due to the basic laws of chemistry. We develop a new "Robust Detection" algorithm for the problem of fast (logarithmic time) single molecule detection, and prove that it is robust to this general model of leaks. Besides potential applications in single molecule detection, the error-correction ideas developed here might enable a new class of robust-by-design chemical algorithms. Our analysis is based on a non-standard hybrid argument, combining ideas from discrete analysis of population protocols with classic Markov chain techniques.
The ZipML Framework for Training Models with End-to-End Low Precision: The Cans, the Cannots, and a Little Bit of Deep Learning (1611.05402)
Hantian Zhang, Jerry Li, Kaan Kara, Dan Alistarh, Ji Liu, Ce Zhang
June 19, 2017 cs.LG, stat.ML
Recently there has been significant interest in training machine-learning models at low precision: by reducing precision, one can reduce computation and communication by one order of magnitude. We examine training at reduced precision, both from a theoretical and practical perspective, and ask: is it possible to train models at end-to-end low precision with provable guarantees? Can this lead to consistent order-of-magnitude speedups? We present a framework called ZipML to answer these questions. For linear models, the answer is yes. We develop a simple framework based on one simple but novel strategy called double sampling. Our framework is able to execute training at low precision with no bias, guaranteeing convergence, whereas naive quantization would introduce significant bias. We validate our framework across a range of applications, and show that it enables an FPGA prototype that is up to 6.5x faster than an implementation using full 32-bit precision. We further develop a variance-optimal stochastic quantization strategy and show that it can make a significant difference in a variety of settings. When applied to linear models together with double sampling, we save up to another 1.7x in data movement compared with uniform quantization. When training deep networks with quantized models, we achieve higher accuracy than the state-of-the-art XNOR-Net. Finally, we extend our framework through approximation to non-linear models, such as SVM. We show that, although using low-precision data induces bias, we can appropriately bound and control the bias. We find in practice 8-bit precision is often sufficient to converge to the correct solution. Interestingly, however, in practice we notice that our framework does not always outperform the naive rounding approach. We discuss this negative result in detail.
The Power of Choice in Priority Scheduling (1706.04178)
Dan Alistarh, Justin Kopinsky, Jerry Li, Giorgi Nadiradze
June 13, 2017 cs.DC, cs.DS
Consider the following random process: we are given $n$ queues, into which elements of increasing labels are inserted uniformly at random. To remove an element, we pick two queues at random, and remove the element of lower label (higher priority) among the two. The cost of a removal is the rank of the label removed, among labels still present in any of the queues, that is, the distance from the optimal choice at each step. Variants of this strategy are prevalent in state-of-the-art concurrent priority queue implementations. Nonetheless, it is not known whether such implementations provide any rank guarantees, even in a sequential model. We answer this question, showing that this strategy provides surprisingly strong guarantees: Although the single-choice process, where we always insert and remove from a single randomly chosen queue, has degrading cost, going to infinity as we increase the number of steps, in the two choice process, the expected rank of a removed element is $O( n )$ while the expected worst-case cost is $O( n \log n )$. These bounds are tight, and hold irrespective of the number of steps for which we run the process. The argument is based on a new technical connection between "heavily loaded" balls-into-bins processes and priority scheduling. Our analytic results inspire a new concurrent priority queue implementation, which improves upon the state of the art in terms of practical performance.
QSGD: Communication-Optimal Stochastic Gradient Descent, with Applications to Training Neural Networks (1610.02132)
Dan Alistarh, Demjan Grubic, Jerry Li, Ryota Tomioka, Milan Vojnovic
May 25, 2017 cs.DS, cs.LG
Parallel implementations of stochastic gradient descent (SGD) have received significant research attention, thanks to excellent scalability properties of this algorithm, and to its efficiency in the context of training deep neural networks. A fundamental barrier for parallelizing large-scale SGD is the fact that the cost of communicating the gradient updates between nodes can be very large. Consequently, lossy compression heuristics have been proposed, by which nodes only communicate quantized gradients. Although effective in practice, these heuristics do not always provably converge, and it is not clear whether they are optimal. In this paper, we propose Quantized SGD (QSGD), a family of compression schemes which allow the compression of gradient updates at each node, while guaranteeing convergence under standard assumptions. QSGD allows the user to trade off compression and convergence time: it can communicate a sublinear number of bits per iteration in the model dimension, and can achieve asymptotically optimal communication cost. We complement our theoretical results with empirical data, showing that QSGD can significantly reduce communication cost, while being competitive with standard uncompressed techniques on a variety of real tasks. In particular, experiments show that gradient quantization applied to training of deep neural networks for image classification and automated speech recognition can lead to significant reductions in communication cost, and end-to-end training time. For instance, on 16 GPUs, we are able to train a ResNet-152 network on ImageNet 1.8x faster to full accuracy. Of note, we show that there exist generic parameter settings under which all known network architectures preserve or slightly improve their full accuracy when using quantization.
Time-Space Trade-offs in Population Protocols (1602.08032)
Dan Alistarh, James Aspnes, David Eisenstat, Rati Gelashvili, Ronald L. Rivest
April 17, 2017 cs.DC
Population protocols are a popular model of distributed computing, in which randomly-interacting agents with little computational power cooperate to jointly perform computational tasks. Inspired by developments in molecular computation, and in particular DNA computing, recent algorithmic work has focused on the complexity of solving simple yet fundamental tasks in the population model, such as leader election (which requires stabilization to a single agent in a special "leader" state), and majority (in which agents must stabilize to a decision as to which of two possible initial states had higher initial count). Known results point towards an inherent trade-off between the time complexity of such algorithms, and the space complexity, i.e. size of the memory available to each agent. In this paper, we explore this trade-off and provide new upper and lower bounds for majority and leader election. First, we prove a unified lower bound, which relates the space available per node with the time complexity achievable by a protocol: for instance, our result implies that any protocol solving either of these tasks for $n$ agents using $O( \log \log n )$ states must take $\Omega( n / \rm{polylog} n )$ expected time. This is the first result to characterize time complexity for protocols which employ super-constant number of states per node, and proves that fast, poly-logarithmic running times require protocols to have relatively large space costs. On the positive side, we give algorithms showing that fast, poly-logarithmic stabilization time can be achieved using $O( \log^2 n )$ space per node, in the case of both tasks. Overall, our results highlight a time complexity separation between $O(\log \log n)$ and $\Theta( \log^2 n )$ state space size for both majority and leader election in population protocols, and introduce new techniques, which should be applicable more broadly.
Polylogarithmic-Time Leader Election in Population Protocols Using Polylogarithmic States (1502.05745)
Dan Alistarh, Rati Gelashvili
Population protocols are networks of finite-state agents, interacting randomly, and updating their states using simple rules. Despite their extreme simplicity, these systems have been shown to cooperatively perform complex computational tasks, such as simulating register machines to compute standard arithmetic functions. The election of a unique leader agent is a key requirement in such computational constructions. Yet, the fastest currently known population protocol for electing a leader only has linear stabilization time, and, it has recently been shown that no population protocol using a constant number of states per node may overcome this linear bound. In this paper, we give the first population protocol for leader election with polylogarithmic stabilization time, using polylogarithmic memory states per node. The protocol structure is quite simple: each node has an associated value, and is either a leader (still in contention) or a minion (following some leader). A leader keeps incrementing its value and "defeats" other leaders in one-to-one interactions, and will drop from contention and become a minion if it meets a leader with higher value. Importantly, a leader also drops out if it meets a minion with higher absolute value. While these rules are quite simple, the proof that this algorithm achieves polylogarithmic stabilization time is non-trivial. In particular, the argument combines careful use of concentration inequalities with anti-concentration bounds, showing that the leaders' values become spread apart as the execution progresses, which in turn implies that straggling leaders get quickly eliminated. We complement our analysis with empirical results, showing that our protocol stabilizes extremely fast, even for large network sizes.
Inherent Limitations of Hybrid Transactional Memory (1405.5689)
Dan Alistarh, Justin Kopinsky, Petr Kuznetsov, Srivatsan Ravi, Nir Shavit
Feb. 17, 2015 cs.DC
Several Hybrid Transactional Memory (HyTM) schemes have recently been proposed to complement the fast, but best-effort, nature of Hardware Transactional Memory (HTM) with a slow, reliable software backup. However, the fundamental limitations of building a HyTM with nontrivial concurrency between hardware and software transactions are still not well understood. In this paper, we propose a general model for HyTM implementations, which captures the ability of hardware transactions to buffer memory accesses, and allows us to formally quantify and analyze the amount of overhead (instrumentation) of a HyTM scheme. We prove the following: (1) it is impossible to build a strictly serializable HyTM implementation that has both uninstrumented reads and writes, even for weak progress guarantees, and (2) under reasonable assumptions, in any opaque progressive HyTM, a hardware transaction must incur instrumentation costs linear in the size of its data set. We further provide two upper bound implementations whose instrumentation costs are optimal with respect to their progress guarantees. In sum, this paper captures for the first time an inherent trade-off between the degree of concurrency a HyTM provides between hardware and software transactions, and the amount of instrumentation overhead the implementation must incur.
How to Elect a Leader Faster than a Tournament (1411.1001)
Dan Alistarh, Rati Gelashvili, Adrian Vladu
The problem of electing a leader from among $n$ contenders is one of the fundamental questions in distributed computing. In its simplest formulation, the task is as follows: given $n$ processors, all participants must eventually return a win or lose indication, such that a single contender may win. Despite a considerable amount of work on leader election, the following question is still open: can we elect a leader in an asynchronous fault-prone system faster than just running a $\Theta(\log n)$-time tournament, against a strong adaptive adversary? In this paper, we answer this question in the affirmative, improving on a decades-old upper bound. We introduce two new algorithmic ideas to reduce the time complexity of electing a leader to $O(\log^* n)$, using $O(n^2)$ point-to-point messages. A non-trivial application of our algorithm is a new upper bound for the tight renaming problem, assigning $n$ items to the $n$ participants in expected $O(\log^2 n)$ time and $O(n^2)$ messages. We complement our results with lower bound of $\Omega(n^2)$ messages for solving these two problems, closing the question of their message complexity.
The LevelArray: A Fast, Practical Long-Lived Renaming Algorithm (1405.5461)
Dan Alistarh, Justin Kopinsky, Alexander Matveev, Nir Shavit
May 21, 2014 cs.DC, cs.DS
The long-lived renaming problem appears in shared-memory systems where a set of threads need to register and deregister frequently from the computation, while concurrent operations scan the set of currently registered threads. Instances of this problem show up in concurrent implementations of transactional memory, flat combining, thread barriers, and memory reclamation schemes for lock-free data structures. In this paper, we analyze a randomized solution for long-lived renaming. The algorithmic technique we consider, called the LevelArray, has previously been used for hashing and one-shot (single-use) renaming. Our main contribu- tion is to prove that, in long-lived executions, where processes may register and deregister polynomially many times, the technique guarantees constant steps on average and O(log log n) steps with high probability for registering, unit cost for deregistering, and O(n) steps for collect queries, where n is an upper bound on the number of processes that may be active at any point in time. We also show that the algorithm has the surprising property that it is self-healing: under reasonable assumptions on the schedule, operations running while the data structure is in a degraded state implicitly help the data structure re-balance itself. This subtle mechanism obviates the need for expensive periodic rebuilding procedures. Our benchmarks validate this approach, showing that, for typical use parameters, the average number of steps a process takes to register is less than two and the worst-case number of steps is bounded by six, even in executions with billions of operations. We contrast this with other randomized implementations, whose worst-case behavior we show to be unreliable, and with deterministic implementations, whose cost is linear in n.
Are Lock-Free Concurrent Algorithms Practically Wait-Free? (1311.3200)
Dan Alistarh, Keren Censor-Hillel, Nir Shavit
Nov. 15, 2013 cs.DC
Lock-free concurrent algorithms guarantee that some concurrent operation will always make progress in a finite number of steps. Yet programmers prefer to treat concurrent code as if it were wait-free, guaranteeing that all operations always make progress. Unfortunately, designing wait-free algorithms is generally a very complex task, and the resulting algorithms are not always efficient. While obtaining efficient wait-free algorithms has been a long-time goal for the theory community, most non-blocking commercial code is only lock-free. This paper suggests a simple solution to this problem. We show that, for a large class of lock- free algorithms, under scheduling conditions which approximate those found in commercial hardware architectures, lock-free algorithms behave as if they are wait-free. In other words, programmers can keep on designing simple lock-free algorithms instead of complex wait-free ones, and in practice, they will get wait-free progress. Our main contribution is a new way of analyzing a general class of lock-free algorithms under a stochastic scheduler. Our analysis relates the individual performance of processes with the global performance of the system using Markov chain lifting between a complex per-process chain and a simpler system progress chain. We show that lock-free algorithms are not only wait-free with probability 1, but that in fact a general subset of lock-free algorithms can be closely bounded in terms of the average number of steps required until an operation completes. To the best of our knowledge, this is the first attempt to analyze progress conditions, typically stated in relation to a worst case adversary, in a stochastic model capturing their expected asymptotic behavior. | CommonCrawl |
\begin{document}
\title{$L^p$-Square Function Estimates on Spaces of Homogeneous Type and on Uniformly Rectifiable Sets \footnotetext[1]{The work of the authors has been supported in part by the US NSF and the Simons Foundation} \footnotetext[2]{{\it{\rm 2010} Mathematics Subject Classification:} Primary: 28A75, 42B20; Secondary: 28A78, 42B25, 42B30} \footnotetext[3]{{\it Key words and phrases:} Square function, quasi-metric space, space of homogeneous type, Ahlfors-David regularity, singular integral operators, area function, Carleson operator, $T(1)$ theorem for the square function, local $T(b)$ theorem for the square function, uniformly rectifiable sets, tent spaces, variable coefficient kernels}}
\author{ Steve Hofmann\\ University of Missouri, Columbia\\ [email protected]\\ \and Dorina Mitrea\\ University of Missouri, Columbia\\ [email protected]\\ \and Marius Mitrea\\ University of Missouri, Columbia\\ [email protected]\\ \and Andrew J. Morris\\ University of Missouri, Columbia\\ [email protected] }
\date{\today}
\maketitle
\begin{abstract} We establish square function estimates for integral operators on uniformly rectifiable sets by proving a local $T(b)$ theorem and applying it to show that such estimates are stable under the so-called big pieces functor. More generally, we consider integral operators associated with Ahlfors-David regular sets of arbitrary codimension in ambient quasi-metric spaces. The local $T(b)$ theorem is then used to establish an inductive scheme in which square function estimates on so-called big pieces of an Ahlfors-David regular set are proved to be sufficient for square function estimates to hold on the entire set. Extrapolation results for $L^p$ and Hardy space versions of these estimates are also established. Moreover, we prove square function estimates for integral operators associated with variable coefficient kernels, including the Schwartz kernels of pseudodifferential operators acting between vector bundles on subdomains with uniformly rectifiable boundaries on manifolds. \end{abstract}
\tableofcontents
\section{Introduction}\label{Sect:1} \setcounter{equation}{0}
The purpose of this work is three-fold: first, to develop the so-called ``local $T(b)$ theory" for square functions in a very general context, in which we allow the ambient space to be of homogeneous type, and in which the ``boundary" of the domain is of arbitrary (positive integer) co-dimension; second, to use a special case of this local $T(b)$ theory to establish boundedness, for a rather general class of square functions, on uniformly rectifiable sets of codimension one in Euclidean space; and third, to establish an extrapolation principle whereby an $L^p$ (or even weak-type $L^p$) estimate for a square function, for {\it one} fixed $p$, yields a full range of $L^p$ bounds. We shall describe these results in more detail below, but let us first recall some of the history of the development of the theory of square functions.
Referring to the role square functions play in mathematics, E. Stein wrote in 1982 (cf. \cite{St82}) that ``{\it [square] functions are of fundamental importance in analysis, standing as they do at the crossing of three important roads many of us have travelled by: complex function theory, the Fourier transform (or orthogonality in its various guises), and real-variable methods.}" In the standard setting of the unit disc ${\mathbb{D}}$ in the complex plane, the classical square function $Sf$ of some $f:{\mathbb{T}}\to{\mathbb{C}}$ (with ${\mathbb{T}}:=\partial{\mathbb{D}}$) is defined in terms of the Poisson integral $u_f(r,\omega)$ of $f$ in ${\mathbb{D}}$ (written in polar coordinates) by the formula
\begin{eqnarray}\label{D-FC45} (Sf)(z):=\Bigl(\int_{(r,\omega)\in\Gamma(x)}
|(\nabla u_f)(r,\omega)|^2\,r\,dr\,d\omega\Bigr)^{1/2},\qquad z\in{\mathbb{T}}, \end{eqnarray}
where $\Gamma(z)$ stands for the Stolz domain
$\{(r,\omega):\,|{\rm arg}(z)-\omega|<1-r<\tfrac{1}{2}\}$ in ${\mathbb{D}}$. Let $v$ denote the (normalized) complex conjugate of $u_f$ in ${\mathbb{D}}$. Then, if the analytic function $F:=u_f+iv$ is one-to-one, the quantity $(Sf)(z)^2$ may be naturally interpreted as the area of the region
$F(\Gamma(z))\subseteq{\mathbb{C}}$ (recall that ${\rm det}(DF)=|\nabla u_f|^2$). The operator \eqref{D-FC45} was first considered by Lusin and the observation just made justifies the original name for \eqref{D-FC45} as Lusin's area function (or Lusin's area integral). A fundamental property of $S$, originally proved by complex methods (cf. \cite[Theorem~3, pp.\,1092-1093]{Cal65}, and \cite{FeSt72} for real-variable methods) is that
\begin{eqnarray}\label{D-FC46}
\|Sf\|_{L^p({\mathbb{T}})}\approx\|f\|_{H^p({\mathbb{T}})} \quad\mbox{ for }\,\,p\in(0,\infty), \end{eqnarray}
which already contains the $H^p$-boundedness of the Hilbert transform. Indeed, if $F=u+iv$ is analytic then the Cauchy-Riemann equations entail
$|\nabla u|=|\nabla v|$ and, hence, $S(u|_{\mathbb{T}})=S(v|_{\mathbb{T}})$. In spite of the technical, seemingly intricate nature of \eqref{D-FC45} and its generalizations to higher dimensions, such as
\begin{eqnarray}\label{D-FC47}
(Sf)(x):=\Bigl(\int_{|x-y|<t}|(\nabla u_f)(y,t)|^2\,t^{1-n}dydt\Bigr)^{1/2}, \qquad x\in{\mathbb{R}}^n:=\partial{\mathbb{R}}^{n+1}_{+}, \end{eqnarray}
a great deal was known by the 1960's about the information encoded into the size of $Sf$, measured in $L^p$, thanks to the pioneering work of D.L. Burkholder, A.P. Calder\'on, C. Fefferman, R.F. Gundy, N. Lusin, J. Marcinkiewicz, C. Segovia, M. Silverstein, E.M. Stein, and A. Zygmund, among others. See, e.g., \cite{BGS}, \cite{Cal50}, \cite{Cal65}, \cite{FeSt72}, \cite{Se69}, \cite{St70}, \cite{St82}, \cite{STEIN}, and the references therein.
Subsequent work by B. Dahlberg, E. Fabes, D. Jerison, C. Kenig and others, starting in the late 1970's (cf. \cite{Dah80}, \cite{DJK}, \cite{Fa88}, \cite{Ke}, \cite{M-DAH}), has brought to prominence the relevance of square function estimates in the context of partial differential equations in non-smooth settings, whereas work by D. Jerison and C. Kenig \cite{JeKe82} in the 1980's as well as G. David and S. Semmes in the 1990's (cf. \cite{DaSe91}, \cite{DaSe93}) has lead to the realization that square function estimates are also intimately connected with the geometry of sets (especially geometric measure theoretic aspects). More recently, square function estimates have played an important role in the solution of the Kato problem in \cite{HMc}, \cite{HLMc}, \cite{AHLMcT}.
The operator $S$ defined in \eqref{D-FC45} is obviously non-linear but the estimate
\begin{eqnarray}\label{D-Fa44}
\|Sf\|_{L^p}\leq C\|f\|_{H^p} \end{eqnarray}
may be linearized by introducing a suitable (linear) vector-valued operator. Specifically, set $\Gamma:=\{(z,t)\in{\mathbb{R}}^{n+1}_{+}:\,|z|<t\}$ and consider the Hilbert space
\begin{eqnarray}\label{D-Fa45} {\mathscr{H}}:=\Bigl\{h:\Gamma\to{\mathbb{C}}^n:\,\mbox{ $h$ is measurable and }\,
\|h\|_{\mathscr{H}}:=
\Bigl(\int_{\Gamma}|h(z,t)|^2t^{1-n}dtdz\Bigr)^{\frac{1}{2}}<\infty\Bigr\}. \end{eqnarray}
Also, let $\widetilde{S}f:{\mathbb{R}}^n\to{\mathscr{H}}$ be defined by the formula
\begin{eqnarray}\label{D-Fa46} \Bigl((\widetilde{S}f)(x)\Bigr)(z,t):=(\nabla u_f)(x-z,t),\qquad \forall\,x\in{\mathbb{R}}^n,\,\,\,\forall\,(z,t)\in\Gamma, \end{eqnarray}
i.e., $\widetilde{S}$ is the integral operator (mapping scalar-valued functions defined on ${\mathbb{R}}^n$ into ${\mathscr{H}}$-valued functions defined on ${\mathbb{R}}^n$), whose kernel $k:{\mathbb{R}}^{n}\times{\mathbb{R}}^{n}\setminus{\rm diagonal}\to{\mathscr{H}}$, which is of convolution type, is given by $(k(x,y))(z,t):=(\nabla P_t)(x-y-z)$, for all $x,y\in{\mathbb{R}}^n$, $x\not=y$, and $(z,t)\in\Gamma$, where $P_t(x)$ is the Poisson kernel in ${\mathbb{R}}^{n+1}_{+}$. Then, if $L^p({\mathbb{R}}^n,{\mathscr{H}})$ stands for the B$\hat{\rm o}$chner space of ${\mathscr{H}}$-valued, $p$-th power integrable functions on ${\mathbb{R}}^n$, it follows that
\begin{eqnarray}\label{D-Fa47}
\|Sf\|_{L^p({\mathbb{R}}^n)}\leq C\|f\|_{H^p({\mathbb{R}}^n)}\,\Longleftrightarrow\,
\|\widetilde{S}f\|_{L^p({\mathbb{R}}^n,\,{\mathscr{H}})}\leq C\|f\|_{H^p({\mathbb{R}}^n)}. \end{eqnarray}
The relevance of the linearization procedure described in \eqref{D-Fa45}-\eqref{D-Fa47} is that it highlights the basic role of the case $p=2$ in \eqref{D-Fa44}. This is because the operator $\widetilde{S}$ falls within the scope of theory of Hilbert space-valued singular integral operators of Calder\'on-Zygmund type for which boundedness on $L^2$ automatically extrapolates to the entire scale $L^p$, for $1<p<\infty$ (the extension to the case when $p\leq 1$ makes use of other specific features of $\widetilde{S}$).
From the point of view of geometry, what makes the above reduction to the case $p=2$ work is the fact that the upper-half space has the property that $x+\Gamma\subseteq{\mathbb{R}}^{n+1}_{+}$ for every $x\in\partial{\mathbb{R}}^{n+1}_{+}$. Such a cone property actually characterizes Lipschitz domains (cf. \cite{HMT}), in which scenario this is the point of view adopted in \cite[Theorem~4.11, p.\,73]{M-LNM}.
Hence, $S$ may be eminently regarded as a singular integral operator with a Hilbert space-valued Calder\'on-Zygmund kernel and, as such, establishing the $L^2$ bound
\begin{eqnarray}\label{Tfgg-77}
\|\widetilde{S}f\|_{L^2({\mathbb{R}}^n,\,{\mathscr{H}})}
\leq C\|f\|_{L^2({\mathbb{R}}^n)} \end{eqnarray}
is of basic importance to jump-start the study of the operator $S$. Now, as is well-known (and easy to check; see, e.g., \cite[pp.\,27-28]{STEIN}), \eqref{Tfgg-77} follows from Fubini's and Plancherel's theorems.
For the goals we have in mind in the present work, it is worth recalling a quote from C. Fefferman's 1974 ICM address \cite{Feff74} where he writes that ``{\it When neither the Plancherel theorem nor Cotlar's lemma applies, $L^2$-boundedness of singular operators presents very hard problems, each of which must (so far) be dealt with on its own terms.}" For scalar singular integral operators, this situation began to be remedied in 1984 with the advent of the $T(1)$-Theorem, proved by G. David and J.-L. Journ\'e in \cite{DJ84}. This was initially done in the Euclidean setting, using Fourier analysis methods. It was subsequently generalized and refined in a number of directions, including the extension to spaces of homogeneous type by R. Coifman (unpublished, see the discussion in \cite{Ch}), and the $T(b)$ Theorems proved by A.\,McIntosh and Y.\,Meyer in \cite{McM85}, and by G.\,David, J.L.\,Journ\'e and S.\,Semmes in \cite{DJS}. The latter reference also contains an extension to the class of singular-integral operators with matrix-valued kernels. The more general case of operator-valued kernels has been treated by Figiel \cite{F} and by T. Hyt\"onen and L. Weis \cite{HyWe}, who prove $T(1)$ Theorems in the spirit of the original work in \cite{DJ84} for singular integrals associated with kernels taking values in Banach spaces satisfying the UMD property. Analogous $T(b)$ theorems were obtained by H\"ytonen \cite{Hy} (in Euclidean space) and by H\"ytonen and Martikainen \cite{HyM1} (in a metric measure space). Yet in a different direction, initially motivated by applications to the theory of analytic capacity, $L^2$-boundedness criteria which are local in nature appeared in the work of M. Christ \cite{Christ}. Subsequently, Christ's local $T(b)$ theorem has been extended to the setting of non-doubling spaces by F. Nazarov, S. Treil and A. Volberg in \cite{NTV}. Further extensions of the local $T(b)$ theory for singular integrals appear in \cite{AHMTT}, \cite{AY}, \cite{AR} and \cite{HyM2}.
Much of the theory mentioned in the preceding paragraph has also been developed in the context of square functions, as opposed to singular integrals. In the convolution setting discussed above, \eqref{Tfgg-77} follows immediately from Plancherel's theorem, but the latter tool fails \color{black} in the case when ${\mathbb{R}}^{n+1}_{+}$ is replaced by a domain whose geometry is rough (so that, e.g., the cone property is violated), and/or one considers a square-function operator whose integral kernel ${\theta}(x,y)$ is no longer of convolution type (as was the case for $\widetilde{S}$). A case in point is offered by the square-function estimate of the type
\begin{eqnarray}\label{TD-56R}
\int_0^\infty\|\Theta_t f\|_{L^2({\mathbb{R}}^n)}^2\frac{dt}{t}
\leq C\|f\|^2_{L^2({\mathbb{R}}^n)}, \end{eqnarray}
where
\begin{eqnarray}\label{TD-56R.2} \bigl(\Theta_t f\bigr)(x):=\int_{{\mathbb{R}}^n}{\theta}_t(x,y)f(y)\,dy,\qquad x\in{\mathbb{R}}^n,\qquad t>0, \end{eqnarray}
with $\{{\theta}_t(\cdot,\cdot)\}_{t>0}$ a standard Littlewood-Paley family, i.e., satisfying for some exponent $\alpha>0$,
\begin{eqnarray}\label{TD-56R.3A}
&& |{\theta}_t(x,y)|\leq C\frac{t^\alpha}{(t+|x-y|)^{n+\alpha}}\quad\mbox{ and} \\[4pt]
&& |{\theta}_t(x,y)-{\theta}_t(x,y')|\leq C\frac{|y-y'|^\alpha}{(t+|x-y|)^{n+\alpha}}
\,\,\mbox{ if }\,\,|y-y'|<t/2. \label{TD-56R.3B} \end{eqnarray}
Then, in general, linearizing estimate \eqref{TD-56R} in a manner similar to \eqref{D-Fa47} yields an integral operator which is no longer of convolution type. As such, Plancherel's theorem is no longer directly effective in dealing with \eqref{TD-56R} given that the task at hand is establishing the $L^2$-boundedness of a variable kernel (Hilbert-valued) singular integral operator. However, M. Christ and J.-L. Journ\'e have shown in \cite{CJ} (under the same size/regularity conditions in \eqref{TD-56R.3A}-\eqref{TD-56R.3B}) that the square function estimate \eqref{TD-56R} is valid if the following Carleson measure condition holds:
\begin{eqnarray}\label{TD-5ii} \sup_{Q\subseteq{\mathbb{R}}^n}\Bigl(
\int_0^{\ell(Q)}{\int{\mkern-19mu}-}_Q|(\Theta_t 1)(x)|^2\,\frac{dxdt}{t}\Bigr)<\infty, \end{eqnarray}
where the supremum is taken over all cubes $Q$ in ${\mathbb{R}}^n$. The latter result is also implicit in the work of Coifman and Meyer \cite{CM}. Moreover, S. Semmes' has shown in \cite{Sem90} that, in the above \color{black} setting, \eqref{TD-56R} holds if there exists a para-accretive function $b$ such that \eqref{TD-5ii} holds with ``1" replaced by ``$b$".
Refinements of Semmes' global $T(b)$ theorem for square functions, in the spirit of M. Christ's local $T(b)$ theorem for singular integrals \cite{Christ},
have subsequently been established in \cite{Au}, \cite{Ho3}, \cite{Ho4}. The local $T(b)$ theorem for square functions which constitutes the main result in \cite{Ho4} reads as follows. Suppose $\Theta_t$ is as in \eqref{TD-56R.2} with kernel satisfying \eqref{TD-56R.3A}-\eqref{TD-56R.3B} as well as
\begin{eqnarray}\label{TD-56R.3C}
|{\theta}_t(x,y)-{\theta}_t(x',y)|\leq C\frac{|x-x'|^\alpha}{(t+|x-y|)^{n+\alpha}}
\,\,\mbox{ if }\,\,|x-x'|<t/2. \end{eqnarray}
In addition, assume that there exists a constant $C_o\in(0,\infty)$ along with an exponent $q\in(1,\infty)$ and a system $\{b_Q\}_Q$ of functions indexed by dyadic cubes $Q$ in ${\mathbb{R}}^n$, such that for each dyadic cube $Q\subseteq{\mathbb{R}}^n$ one has:
(i) $\int_{{\mathbb{R}}^n}|b_Q(x)|^q\,dx\leq C_o|Q|$;
(ii) $\frac{1}{C_o}|Q|\leq\Bigl|\int_{{\mathbb{R}}^n}b_Q(x)\,dx\Bigr|$;
(iii) $\int_Q\Bigl(\int_0^{\ell(Q)}|(\Theta_t b_Q)(x)|^2\,\frac{dt}{t}\Bigr)^{q/2}\,dx
\leq C_o|Q|$.
\noindent Then the square function estimate \eqref{TD-56R} holds. The case $q=2$ of this theorem does not require \eqref{TD-56R.3C} (just regularity in the second variable, as in \eqref{TD-56R.3B})\footnote{In fact, even the case $q\neq 2$ does not require \eqref{TD-56R.3C}, if the vertical square function is replaced by a conical one; see \cite{G} for details.}, and was already implicit in the solution of the Kato problem in~\cite{HMc}, \cite{HLMc}, \cite{AHLMcT}. It was formulated explicitly in \cite{Au}, \cite{Ho3}. An extension of the result of \cite{Ho4} to the case that the half-space is replaced by $\mathbb{R}^{n+1}\setminus E$, where $E$ is a closed Ahlfors-David regular set (cf. Definition \ref{Rcc-TG34} below) of Hausdorff dimension $n$, appears in \cite{GM}. The latter extension has been used to prove a result of {\it free boundary} type, in which higher integrability of the Poisson kernel, in the presence of certain natural background hypotheses, is shown to be equivalent to uniform rectifiability (cf. Definition \ref{Def-unif.rect} below) of the boundary \cite{HM}, \cite{HMU}. Further extensions of the result of \cite{Ho4}, to the case in which the kernel $\theta_t$ and pseudo-accretive system $b_Q$ may be matrix-valued (as in the setting of the Kato problem), and in which $\theta_t$ need no longer satisfy the pointwise size and regularity conditions \eqref{TD-56R.3A}-\eqref{TD-56R.3B}, will appear in the forthcoming Ph.D. thesis of A. Grau de la Herran \cite{G}.
A primary motivation for us in the present work is the connection between square function bounds (or their localized versions in the form of ``Carleson measure estimates"), and a quantitative, scale invariant notion of rectifiability. This subject has been developed extensively by David and Semmes \cite{DaSe91}, \cite{DaSe93} (but with some key ideas already present in the work of P. Jones \cite{J}). Following \cite{DaSe91}, \cite{DaSe93}, we shall give in the sequel (cf. Definition~\ref{Def-unif.rect}), a precise definition of the property that a closed set $E$ is ``Uniformly Rectifiable" (UR), but for now let us merely mention that UR sets are the ones on which ``nice" singular integral operators are bounded on $L^2$. David and Semmes have shown that these sets may also be characterized via certain square function estimates, or equivalently, via Carleson measure estimates. For example, let $E\subset \mathbb{R}^{n+1}$ be a closed set of codimension one, which is ($n$-dimensional) Ahlfors-David regular (ADR) (cf. Definition~\ref{Rcc-TG34} below). Then $E$ is UR if and only if we have the Carleson measure estimate
\begin{eqnarray}\label{eq1.sf} \sup_B r^{-n}\int_{B}
\bigl|\bigl(\nabla^2{\mathcal{S}}1\bigr)(x)\bigr|^2\,\operatorname{dist}(x,E)\,dx<\infty, \end{eqnarray}
where the supremum runs over all Euclidean balls $B:=B(z,r)\subseteq{\mathbb{R}}^{n+1}$, with $r\leq\diam(E)$, and center $z\in E$, and where ${\mathcal{S}}f$ is the harmonic single layer potential of the function $f$, i.e.,
\begin{eqnarray}\label{eq1.layer}
{\mathcal{S}}f(x):=c_n\,\int_{E}|x-y|^{1-n}f(y)\,d{\mathcal{H}}^n(y),\qquad x\in{\mathbb{R}}^{n+1}\setminus E. \end{eqnarray}
Here ${\mathcal{H}}^n$ denotes $n$-dimensional Hausdorff measure. For an appropriate normalizing constant $c_n|x|^{1-n}$ is the usual fundamental solution for the Laplacian in ${\mathbb{R}}^{n+1}$. We refer the reader to \cite{DaSe93} for details, but see also Section~\ref{Sect:4} where we present some related results. We note that by ``$T1$" reasoning (cf. Section~\ref{Sect:3} below), \eqref{eq1.sf} is equivalent to the square function bound
\begin{eqnarray}\label{eq1.sf2}
\int_{{\mathbb{R}}^{n+1}\setminus E}\bigl|\bigl(\nabla^2{\mathcal{S}}f\bigr)(x)|^2\,
\operatorname{dist}(x,E)\,dx\leq C\int_E |f(x)|^2\,d{\mathcal{H}}^n(x)\,. \end{eqnarray}
Using an idea of P. Jones \cite{J}, one may derive, for UR sets, a quantitative version of the fact that rectifiability may be characterized in terms of existence a.e. of approximate tangent planes. Again, a Carleson measure is used to express matters quantitatively. For $x\in E$ and $t>0$ we set
\begin{eqnarray}\label{eq1.beta} \beta_2(x,t):=\inf_P\left(\frac{1}{t^{n}} \int_{B(x,t)\cap E}\left(\frac{\mbox{dist}(y,P)}{t}\right)^2 \,d{\mathcal{H}}^n(y)\right)^{1/2}, \end{eqnarray}
where the infimum runs over all $n$-planes $P$. Then a closed, ADR set $E$ of codimension one is UR if and only if the following Carleson measure estimate holds on $E\times{\mathbb{R}}_+$:
\begin{eqnarray}\label{eq1.UR} \sup_{x_0\in E,\,r\,>\,0}r^{-n}\int_0^r\int_{B(x_0,t)\cap E} \beta_2(x,t)^2\,d{\mathcal{H}}^n(x)\frac{dt}{t}\,<\,\infty. \end{eqnarray}
See \cite{DaSe91} for details, and for a formulation in the case of higher codimension. A related result, also obtained in \cite{DaSe91}, is that a set $E$ as above is UR if and only if, for every odd $\psi\in C^\infty_0(\mathbb{R}^{n+1})$, one has the following discrete square function bound
\begin{eqnarray}\label{eq1.sf3} \sum_{k=-\infty}^\infty\int_{E}
\left|\int_E2^{-kn}\psi\bigl(2^{-k}(x-y)\bigr)f(y)\,d{\mathcal{H}}^n(y)\right|^2
\,d{\mathcal{H}}^n(x)\leq C_\psi\int_E|f(x)|^2\,d{\mathcal{H}}^n(x)\,. \end{eqnarray}
Again, there is a Carleson measure analogue, and also a version for sets $E$ of higher codimension.
The following theorem collects some of the main results in our present work. It generalizes results described earlier in the introduction, which were valid in the codimension one case, and in which the ambient space ${\mathscr{X}}$ was Euclidean. To state it, recall that (in a context to be made precise below) a measurable function $b:E\to{\mathbb{C}}$ is called para-accretive if it is essentially bounded and there exist constants $c,C\in(0,\infty)$ such that the following conditions are satisfied:
\begin{eqnarray}\label{para-acc} \forall\,Q\in{\mathbb{D}}(E)\quad\exists\,\widetilde{Q}\in{\mathbb{D}}(E)\quad \mbox{such that }\,\,\, \widetilde{Q}\subseteq Q,\quad\ell(\widetilde{Q})\geq c\ell(Q),\quad
\left|{\int{\mkern-19mu}-}_{\widetilde{Q}} b\,d\sigma\right|\geq C. \end{eqnarray}
Other relevant definitions will be given in the sequel.
\begin{theorem}\label{M-TTHH} Suppose that $({\mathscr{X}},\rho,\mu)$ is an $m$-dimensional {\rm ADR} space for some $m>0$ and fix a number $d\in(0,m)$. Also, let
\begin{eqnarray}\label{K234-AXXX} {\theta}:(\mathscr{X}\times\mathscr{X}) \setminus\{(x,x):\,x\in \mathscr{X}\}\longrightarrow{{\mathbb{R}}} \end{eqnarray}
be a function which is Borel measurable with respect to the product topology $\tau_\rho\times\tau_\rho$, and which has the property that there exist finite positive constants $C_{\theta},\,\alpha,\,\upsilon$ such that for all $x,y\in\mathscr{X}$ with $x\neq y$ the following hold:
\begin{eqnarray}\label{hszz-AXXX} && \hskip -0.40in
|{\theta}(x,y)|\leq\frac{C_{\theta}}{\rho(x,y)^{d+\upsilon}}, \\[4pt] && \hskip -0.40in
|{\theta}(x,y)-{\theta}(x,\widetilde{y})|\leq C_{\theta} \frac{\rho(y,\widetilde{y})^\alpha}{\rho(x,y)^{d+\upsilon+\alpha}}, \quad\forall\,\widetilde{y}\in\mathscr{X}\setminus\{x\}\,\,\mbox{ with }\,\, \rho(y,\widetilde{y})\leq\tfrac{1}{2}\rho(x,y).\quad \label{hszz-3-AXXX} \end{eqnarray}
Assume that $E$ is a closed subset of $({\mathscr{X}},\tau_\rho)$ and that $\sigma$ is a Borel regular measure
on $(E,\tau_{\rho|_{E}})$ such that $\bigl(E,\rho\bigl|_E,\sigma\bigr)$ is a $d$-dimensional {\rm ADR} space, and define the integral operator $\Theta=\Theta_E$ for all functions $f\in L^p(E,\sigma)$, $1\leq p\leq\infty$, by
\begin{eqnarray}\label{operator-AXXX} (\Theta f)(x):=\int_E {\theta}(x,y)f(y)\,d\sigma(y), \qquad\forall\,x\in\mathscr{X}\setminus E. \end{eqnarray}
Let ${\mathbb{D}}(E)$ denote a dyadic cube structure on $E$ and, for each $Q\in{\mathbb{D}}(E)$, denote by $T_E(Q)$ the dyadic Carleson tent over $Q$. Finally, let $\rho_{\#}$ be the regularized version of the quasi-distance $\rho$ as in Theorem~\ref{JjEGh} and, for each $x\in{\mathscr{X}}$, set $\delta_E(x):=\inf\{\rho_{\#}(x,y):\,y\in E\}$.
Then the following are equivalent:
\begin{enumerate}
\item[(1)] $[${\tt $L^2$\! square\! function\! estimate}$]$ There exists $C\in(0,\infty)$ with the property that for each $f\in L^2(E,\sigma)$ one has
\begin{eqnarray}\label{G-UFXXX.2}
\int_{\mathscr{X}\setminus E}|(\Theta f)(x)|^2\delta_E(x)^{2\upsilon-(m-d)}\, d\mu(x)\leq C\int_E|f(x)|^2\,d\sigma(x). \end{eqnarray}
\item[(2)] $[${\tt Carleson\! measure\! condition\! on\! dyadic\! tents\! for\! $\Theta$\! tested\! on\! $1$}$]$ There holds
\begin{eqnarray}\label{UEHgXXX} \sup_{Q\in{\mathbb{D}}(E)}\left(\tfrac{1}{\sigma(Q)}\int_{T_E(Q)}
\bigl|\bigl(\Theta 1\bigr)(x)\bigr|^2\, \delta_E(x)^{2\upsilon-(m-d)}\,d\mu(x)\right)<\infty. \end{eqnarray}
\item[(3)] $[${\tt Carleson\! measure\! condition\! on\! dyadic\! tents\! for\! $\Theta$\! acting\! on\! $L^\infty$}$]$ There exists a constant $C\in(0,\infty)$ with the property that for each $f\in L^\infty(E,\sigma)$
\begin{eqnarray}\label{UEHgXXX.22} \sup_{Q\in{\mathbb{D}}(E)}\left(\tfrac{1}{\sigma(Q)}\int_{T_E(Q)}
|(\Theta f)(x)|^2\delta_E(x)^{2\upsilon-(m-d)}\,d\mu(x)\right)^{1/2}
\leq C\|f\|_{L^\infty(E,\sigma)}. \end{eqnarray}
\item[(4)] $[${\tt Carleson\! measure\! condition\! on\! balls\! for\! $\Theta$\! tested\! on\! $1$}$]$ There holds
\begin{eqnarray}\label{UEHgXXX.2} \sup_{x\in E,\,r>0} \left(\tfrac{1}{\sigma\bigl(E\cap B_{\rho_{\#}}(x,r)\bigr)} \int_{B_{\rho_{\#}}(x,r)\setminus E}
|\Theta 1|^2\delta_E^{2\upsilon-(m-d)}\,d\mu\right)<\infty. \end{eqnarray}
\item[(5)] $[${\tt Carleson\! measure\! condition\! for\! $\Theta$\! tested\! on\! a\! para-accretive\! function}$]$ There exists a para-accretive function $b:E\to{\mathbb{C}}$ with the property that
\begin{eqnarray}\label{UEHgXXX.2PA} \sup_{Q\in{\mathbb{D}}(E)} \left(\tfrac{1}{\sigma(Q)}
\int_{T_E(Q)}|(\Theta b)(x)|^2\delta_E(x)^{2\upsilon-(m-d)}\,d\mu(x)\right)<\infty. \end{eqnarray}
\item[(6)] $[${\tt Carleson\! measure\! condition\! on\! balls\! for\! $\Theta$\! acting\! on\! $L^\infty$}$]$ There exists $C\in(0,\infty)$ with the property that for each $f\in L^\infty(E,\sigma)$
\begin{eqnarray}\label{UEHgXXX.2S} \sup_{x\in E,\,r>0} \left(\tfrac{1}{\sigma\bigl(E\cap B_{\rho_{\#}}(x,r)\bigr)} \int_{B_{\rho_{\#}}(x,r)\setminus E}
|\Theta f|^2\delta_E^{2\upsilon-(m-d)}\,d\mu\right)^{1/2}
\leq C\|f\|_{L^\infty(E,\sigma)}. \end{eqnarray}
\item[(7)] $[${\tt Local\! $T(b)$\! condition\! on dyadic\! cubes}$]$ There exist two finite constants $C_0\geq 1$ and $c_0\in(0,1]$, along with a collection $\{b_Q\}_{Q\in{\mathbb{D}}(E)}$ of $\sigma$-measurable functions $b_Q:E\rightarrow{\mathbb{C}}$ such that for each $Q\in{\mathbb{D}}(E)$ the following hold:
\begin{eqnarray}\label{CON-BB} \begin{array}{c}
\displaystyle\int_E |b_Q|^2\,d\sigma\leq C_0\sigma(Q), \\[16pt]
\left|\int_{\widetilde{Q}} b_Q\,d\sigma\right|\geq\frac{1}{C_0}\,\sigma(\widetilde{Q})\quad \mbox{for some }\widetilde{Q}\subseteq Q,\,\,\, \ell(\widetilde{Q})\geq c_0\ell(Q), \\[16pt]
\displaystyle\int_{T_E(Q)}|(\Theta\,b_Q)(x)|^2 \delta_E(x)^{2\upsilon-(m-d)}\,d\mu(x)\leq C_0\sigma(Q). \end{array} \end{eqnarray}
\item[(8)] $[${\tt Local\! $T(b)$\! condition\! on\! surface\! balls}$]$ There exist $C_0\in[1,\infty)$ and, for each surface ball $\Delta=\Delta(x_o,r):=B_{\rho_{\#}}(x_o,r)\cap E$, where $x_o$ is a point in $E$ and $r$ is a finite number in $\bigl(0,{\rm diam}_\rho(E)\bigr]$, a $\sigma$-measurable function $b_\Delta:E\rightarrow{\mathbb{C}}$ supported in $\Delta$, such that the following estimates hold:
\begin{eqnarray}\label{CON-BB.789} \begin{array}{c}
\displaystyle\int_E |b_\Delta|^2\,d\sigma\leq C_0\sigma(\Delta),\quad
\left|\int_\Delta b_\Delta\,d\sigma\right|\geq\frac{1}{C_0}\,\sigma(\Delta), \\[16pt]
\displaystyle\int_{B_{\rho_{\#}}(x_o,2C_\rho r)\setminus E}|(\Theta\,b_\Delta)(x)|^2 \delta_E(x)^{2\upsilon-(m-d)}\,d\mu(x)\leq C_0\sigma(\Delta). \end{array} \end{eqnarray}
\item[(9)] $[${\tt Big\! Pieces\! of\! Square\! Function\! Estimate}$]$ The set $E$ has {\rm BPSFE} relative to the kernel ${\theta}$ (cf. Definition~\ref{sjvs}).
\item[(10)] $[${\tt Iterated\! Big\! Pieces\! of\! Square\! Function\! Estimate}$]$ The set $E$ has ${\rm (BP)}^k${\rm SFE} relative to the kernel ${\theta}$ (cf. Definition~\ref{sjvs-DDD}) for some, or any, $k\in{\mathbb{N}}$.
\item[(11)] $[${\tt Weak-$L^p$\! square\! function\! estimate}$]$ There exist an exponent $p\in(0,\infty)$ and constants $C,\kappa\in(0,\infty)$ such that for every $f\in L^{p}(E,\sigma)$
\begin{eqnarray}\label{GvBhXXX} \hskip -0.20in \sup_{\lambda>0}\left[\lambda\cdot
\sigma\Bigl(\Bigl\{x\in E:\int_{\Gamma_{\kappa}(x)}|(\Theta f)(y)|^2\, \frac{d\mu(y)}{\delta_E(y)^{m-2\upsilon}}>\lambda^{2}\Bigr\}\Bigr)^{1/p}\right]
\leq C\Bigl(\int_{E}|f|^p\,d\sigma\Bigr)^{1/p}, \end{eqnarray}
where $\Gamma_\kappa(x)$ stands for the nontangential approach region defined in \eqref{TLjb}.
\item[(12)] $[${\tt Hardy\! and\! $L^p$\! square\! function\! estimates}$]$ Set $\gamma:=\min\,\{\alpha,(\log_2C_\rho)^{-1}\}$. Then for each $p\in\bigl(\frac{d}{d+\gamma},\infty\bigr)$ the operator $\Theta$
extends to the space $H^p(E,\rho|_{E},\sigma)$, defined as the Lebesgue space $L^p(E,\sigma)$ if $p\in(1,\infty)$, and the Coifman-Weiss Hardy
space on the space of homogeneous type $(E,\rho|_{E},\sigma)$ if $p\in\bigl(\frac{d}{d+\gamma},1\bigr]$, and this extension satisfies
\begin{eqnarray}\label{CfrdXXX}
\left\|\Bigl(\int_{\Gamma_{\kappa}(x)}|(\Theta f)(y)|^2\, \frac{d\mu(y)}{\delta_E(y)^{m-2\upsilon}}\Bigr)^{\frac{1}{2}}
\right\|_{L^p_x(E,\sigma)}\!\!\!\leq C\|f\|_{H^p(E,\rho|_{E},\sigma)}, \end{eqnarray}
for each function $f\in H^p(E,\rho|_{E},\sigma)$.
\item[(13)] $[${\tt Mixed-norm\! space\! estimate}$]$ For each $p\in\bigl(\frac{d}{d+\gamma},\infty\bigr)$, with $\gamma:=\min\,\{\alpha,(\log_2C_\rho)^{-1}\}$, and each $q\in(1,\infty)$, the operator
\begin{eqnarray}\label{ki-DUDXXX}
\delta_E^{\upsilon-m/q}\Theta:H^p(E,\rho|_{E},\sigma) \longrightarrow L^{(p,q)}({\mathscr{X}},E) \end{eqnarray}
is well-defined, linear and bounded, where $L^{(p,q)}({\mathscr{X}},E)$ is the mixed norm space defined in \eqref{Mixed-FF7}. \end{enumerate} \end{theorem}
A few comments pertaining to the nature and scope of Theorem~\ref{M-TTHH} are in order.
\vskip 0.08in $\bullet$ Theorem~\ref{M-TTHH} makes the case that estimating the square function in $L^p$, along with other related issues considered above, may be regarded as ``zeroth order calculus", since only integrability and quasi-metric geometry are involved, without recourse to differentiability (or vector space structures). In particular, our approach is devoid of any PDE results and techniques. Compared with works in the upper-half space ${\mathbb{R}}^n\times (0,\infty)$, or so-called generalized upper-half spaces $E\times (0,\infty)$ (cf., e.g., \cite{HMY} and the references therein), here we work in an ambient ${\mathscr{X}}$ with no distinguished ``vertical" direction. Moreover, the set $E$ is allowed to have arbitrary ADR co-dimension in the ambient ${\mathscr{X}}$. In this regard we also wish to point out that Theorem~\ref{M-TTHH} permits the consideration of fractal subsets of the Euclidean space (such as the case when $E$ is the von Koch's snowflake in ${\mathbb{R}}^2$, in which scenario $d=\frac{\ln 4}{\ln 3}$).
$\bullet$ Passing from $L^2$ estimates to $L^p$ estimates is no longer done via a linearization procedure (since the environment no longer permits it) and, instead, we use tent space theory and exploit the connection between the Lusin and the Carleson operators on spaces of homogeneous type (thus generalizing work from \cite{CoMeSt} in the Euclidean setting). This reinforces the philosophy that the square-function is a singular integral operator at least in spirit (if not in the letter).
$\bullet$ The various quantitative aspects of the claims in items {\it (1)}-{\it (11)} of Theorem~\ref{M-TTHH} are naturally related to one another. The reader is also alerted to the fact that similar results to those contained in Theorem~\ref{M-TTHH} are proved in the body of the manuscript for a larger class of kernels (satisfying less stringent conditions) than in the theorem above. The specific way in which Theorem~\ref{M-TTHH} follows from these more general results is discussed in \S\,\ref{Sect:6}.
\vskip 0.08in
We now proceed to describe several consequences of Theorem~\ref{M-TTHH} for subsets $E$ of the Euclidean space. First we record the following square function estimate, which extends work from \cite{DaSe91}.
\begin{theorem}\label{sfe-cor} Suppose that $E$ is a closed subset of ${\mathbb{R}}^{n+1}$ which is $d$-dimensional {\rm ADR} for some $d\in\{1,\dots,n\}$ and denote by $\sigma$ the surface measure induced by the $d$-dimensional Hausdorff measure on $E$. Assume that $E$ has big pieces of Lipschitz images of subsets of ${\mathbb{R}}^d$, i.e., there exist $\varepsilon,M\in(0,\infty)$ so that for every $x\in E$ and every $R\in(0,\infty)$, there is a Lipschitz mapping $\varphi$ with Lipschitz norm $\leq M$ from the ball $B^d(0,R)$ in ${\mathbb{R}}^d$ into ${\mathbb{R}}^{n+1}$ such that
\begin{eqnarray}\label{nvdz} \sigma\left(E\cap B(x,R)\cap\varphi(B^d(0,R))\right)\leq \varepsilon R^d. \end{eqnarray}
Suppose $\psi:{\mathbb{R}}^{n+1}\to{\mathbb{R}}$ is a compactly supported, smooth, odd function and for each $k\in{\mathbb{Z}}$ set $\psi_k(x):=2^{-kd}\psi\bigl(\frac{x}{2^k}\bigr)$ for $x\in{\mathbb{R}}^{n+1}$. Then for every $q\in(1,\infty)$ and every $p\in\bigl(\frac{d}{d+1},\infty\bigr)$ there exists $C\in(0,\infty)$ such that
\begin{eqnarray}\label{bh}
\left\|\Bigl(\sum\limits_{k=-\infty}^{+\infty}
{\int{\mkern-19mu}-}_{y\in\Delta(x,2^k)}\Bigl|\int_E\psi_k(z-y)f(z)\,d\sigma(z)
\Bigr|^q\,d\sigma(y)\Bigr)^{1/q}
\right\|_{L^p_x(E,\sigma)}\leq C\|f\|_{H^p(E,\sigma)} \end{eqnarray}
for every $f\in H^p(E,\sigma)$, where $\Delta(x,2^k):=\{y\in E:\,|y-x|<2^k\}$ for each $x\in E$ and $k\in{\mathbb{Z}}$. \end{theorem}
The particular case when $p=q=2$, in which scenario \eqref{bh} takes the form
\begin{eqnarray}\label{bh.Biss}
\sum\limits_{k=-\infty}^{+\infty}\int_E\left|
\int_E\psi_k(x-y)f(y)\,d\sigma(y)\right|^2d\sigma(x)\leq C\int_E|f|^2\,d\sigma, \end{eqnarray}
has been treated in \cite[\S\,3, p.\,21]{DaSe91}. The main point of Theorem~\ref{sfe-cor} is that \eqref{bh.Biss} continues to hold, when formulated as in \eqref{bh} for every $p\in\bigl(\frac{d}{d+1},\infty\bigr)$. The proof of this result, presented in the last part of \S\,\ref{Sect:6}, relies on Theorem~\ref{M-TTHH} and uses the fact that no regularity condition on the kernel $\theta(x,y)$ is assumed in the variable $x$ (compare with \eqref{hszz-AXXX}-\eqref{hszz-3-AXXX}).
Next, we discuss another consequence of Theorem~\ref{M-TTHH} in the Euclidean setting which gives an extension of results due to G. David and S. Semmes.
\begin{theorem}\label{UR-rest} Suppose that $K$ is a real-valued function satisfying
\begin{eqnarray}\label{K-BIS} \begin{array}{c} K\in C^2({\mathbb{R}}^{n+1}\setminus\{0\}),\quad K\,\,\mbox{ is odd, and} \\[4pt] K(\lambda x)=\lambda^{-n}K(x)\,\,\mbox{ for all }\,\lambda>0,\,\, x\in{\mathbb{R}}^{n+1}\setminus\{0\}. \end{array} \end{eqnarray}
Let $E$ be a closed subset of ${\mathbb{R}}^{n+1}$ which is $n$-dimensional {\rm ADR}, denote by $\sigma$ the surface measure induced by the $n$-dimensional Hausdorff measure on $E$, and define the integral operator ${\mathcal{T}}$ acting on functions $f\in L^p(E,\sigma)$, $1\leq p\leq\infty$, by
\begin{eqnarray}\label{T-BIS.2a} {\mathcal{T}}f(x):=\int_E K(x-y)f(y)\,d{\sigma}(y),\qquad \forall\,x\in{\mathbb{R}}^{n+1}\setminus E. \end{eqnarray}
Finally, let ${\mathbb{D}}(E)$ denote a dyadic cube structure on $E$ and, for each $Q\in{\mathbb{D}}(E)$, denote by $T_E(Q)$ the dyadic Carleson tent over $Q$.
Then, if the set $E$ is actually uniformly rectifiable ({\rm UR}), in the sense of Definition~\ref{Def-unif.rect}, conditions {\it (1)-(5)} below hold:
\begin{enumerate} \item[(1)] $[${\tt $L^2$\! square\! function\! estimate}$]$ There exists $C\in(0,\infty)$ with the property that for each $f\in L^2(E,\sigma)$ one has
\begin{eqnarray}\label{SF-BIS}
\int\limits_{{\mathbb{R}}^{n+1}\setminus E}|(\nabla{\mathcal{T}}f)(x)|^2
\,{\rm dist}\,(x,E)\,dx\leq C\int_E|f(x)|^2\,d\sigma(x). \end{eqnarray}
\item[(2)] $[${\tt Carleson\! measure\! condition\! on\! dyadic\! tents\! for\! ${\mathcal{T}}$\! acting\! on\! $L^\infty$}$]$ There exists a constant $C\in(0,\infty)$ with the property that for each $f\in L^\infty(E,\sigma)$
\begin{eqnarray}\label{SF-BIS-3} \sup_{Q\in{\mathbb{D}}(E)}\left(\tfrac{1}{\sigma(Q)}\int_{T_E(Q)}
|(\nabla{\mathcal{T}} f)(x)|^2\,{\rm dist}\,(x,E)\,dx\right)^{1/2}
\leq C\|f\|_{L^\infty(E,\sigma)}. \end{eqnarray}
In particular,
\begin{eqnarray}\label{SF-BIS-2} \sup_{Q\in{\mathbb{D}}(E)}\left(\tfrac{1}{\sigma(Q)}\int_{T_E(Q)}
|(\nabla{\mathcal{T}}1)(x)|^2\,{\rm dist}\,(x,E)\,dx\right)<\infty. \end{eqnarray}
\item[(3)] $[${\tt Carleson\! measure\! condition\! on\! balls\! for\! ${\mathcal{T}}$\! acting\! on\! $L^\infty$}$]$ There exists a constant $C\in(0,\infty)$ with the property that for each $f\in L^\infty(E,\sigma)$
\begin{eqnarray}\label{SF-BIS-6} \sup_{x\in E,\,r>0}\left(\tfrac{1}{\sigma\bigl(E\cap B(x,r)\bigr)}
\int_{B(x,r)\setminus E}|(\nabla{\mathcal{T}} f)(y)|^2\,{\rm dist}\,(y,E)\,dy\right)^{1/2}
\leq C\|f\|_{L^\infty(E,\sigma)}. \end{eqnarray}
In particular,
\begin{eqnarray}\label{SF-BIS-4} \sup_{x\in E,\,r>0}\left(\tfrac{1}{\sigma\bigl(E\cap B(x,r)\bigr)}
\int_{B(x,r)\setminus E}|(\nabla{\mathcal{T}} 1)(y)|^2\,{\rm dist}\,(y,E)\,dy\right)<\infty. \end{eqnarray}
\item[(4)] $[${\tt Hardy\! and\! $L^p$\! square\! function\! estimates}$]$ For each $p\in\bigl(\frac{n}{n+1},\infty\bigr)$ let $H^p(E,\sigma)$ stand for the Lebesgue scale $L^p(E,\sigma)$ if $p\in(1,\infty)$, and the Coifman-Weiss scale of
Hardy spaces on the space of homogeneous type $(E,|\cdot-\cdot|,\sigma)$ if $p\in\bigl(\frac{n}{n+1},1\bigr]$. Then the operator ${\mathcal{T}}$ extends to the space $H^p(E,\sigma)$ and this extension satisfies
\begin{eqnarray}\label{SF-BIS-10} \hskip -0.20in
\left\|\Bigl(\int_{\Gamma_{\kappa}(x)}|(\nabla{\mathcal{T}} f)(y)|^2\, \frac{dy}{{\rm dist}\,(y,E)^{n-1}}\Bigr)^{\frac{1}{2}}
\right\|_{L^p_x(E,\sigma)}\!\!\!\leq C\|f\|_{H^p(E,\sigma)}, \quad\forall\,f\in H^p(E,\sigma). \end{eqnarray}
\item[(5)] $[${\tt Mixed-norm\! space\! estimate}$]$ For each $p\in\bigl(\frac{n}{n+1},\infty\bigr)$ and each $q\in(1,\infty)$ the operator
\begin{eqnarray}\label{SF-BIS-11} {\rm dist}\,(\cdot,E)\,\nabla{\mathcal{T}}: H^p(E,\sigma)\longrightarrow L^{(p,q)}({\mathbb{R}}^{n+1},E) \end{eqnarray}
is well-defined, linear and bounded, where $L^{(p,q)}({\mathbb{R}}^{n+1},E)$ is the mixed norm space defined in \eqref{Mixed-FF7} (corresponding here to
${\mathscr{X}}:={\mathbb{R}}^{n+1}$ and $\rho:=|\cdot-\cdot|$). \end{enumerate} \end{theorem}
Theorem~\ref{M-TTHH} particularized to the setting of Theorem~\ref{UR-rest} gives that conditions {\it (1)}-{\it (5)} above are equivalent. The fact that {\it (1)} holds in the special case when ${\mathcal{T}}$ is associated as
in \eqref{T-BIS.2a} with each of the kernels $K_j(x):=x_j/|x|^{n+1}$, $1\leq j\leq n+1$, is due to David and Semmes \cite{DaSe93}. The new result here is that {\it (1)} (hence also all of {\it (1)-(5)}) holds more generally for the entire class of kernels described in \eqref{K-BIS}. We shall prove the latter fact in Corollary~\ref{cor:URimSFE} below. Compared with \cite{DaSe91}, the class of kernels \eqref{K-BIS} is not
tied up to any particular partial differential operator (in the manner that the kernels $K_j(x):=x_j/|x|^{n+1}$, $1\leq j\leq n+1$, are related to the Laplacian). Moreover, in \S\,\ref{SSect:PDO} we establish a version of Theorem~\ref{UR-rest} for variable coefficient kernels, which ultimately applies to integral operators on domains on manifolds associated with the Schwartz kernels of certain classes of pseudodifferential operators acting between vector bundles.
The condition that the set $E$ is UR in the context of Theorem~\ref{UR-rest} is optimal, as seen from the converse statement stated below. This result is closely interfaced with the characterization of uniform rectifiability, due David and Semmes, in terms of \eqref{eq1.sf}-\eqref{eq1.layer}. In keeping with these conditions, the formulation of our result involves the Riesz-transform operator ${\mathcal{R}}:=\nabla{\mathcal{S}}$.
\begin{theorem}\label{UR-rest.BIS} Let $E$ be a closed subset of ${\mathbb{R}}^{n+1}$ which is $n$-dimensional {\rm ADR}, denote by $\sigma$ the surface measure induced by the $n$-dimensional Hausdorff measure on $E$, and define the vector-valued integral operator ${\mathcal{R}}$ acting on functions $f\in L^p(E,\sigma)$, $1\leq p\leq\infty$, by
\begin{eqnarray}\label{T-BIS.3a}
{\mathcal{R}}f(x):=\int_E \frac{x-y}{|x-y|^{n+1}}f(y)\,d{\sigma}(y),\qquad \forall\,x\in{\mathbb{R}}^{n+1}\setminus E. \end{eqnarray}
As before, let ${\mathbb{D}}(E)$ denote a dyadic cube structure on $E$ and, for each $Q\in{\mathbb{D}}(E)$, denote by $T_E(Q)$ the dyadic Carleson tent over $Q$. In this setting, consider the following conditions:
\begin{enumerate} \item[(1)] $[${\tt $L^2$\! square\! function\! estimate}$]$ There exists $C\in(0,\infty)$ with the property that for each $f\in L^2(E,\sigma)$ one has
\begin{eqnarray}\label{SF-BIS.BIS}
\int\limits_{{\mathbb{R}}^{n+1}\setminus E}|(\nabla{\mathcal{R}}f)(x)|^2
\,{\rm dist}\,(x,E)\,dx\leq C\int_E|f(x)|^2\,d\sigma(x). \end{eqnarray}
\item[(2)] $[${\tt Carleson\! measure\! condition\! on\! dyadic\! tents\! for\! ${\mathcal{R}}$\! tested\! on\! $1$}$]$ There holds
\begin{eqnarray}\label{SF-BIS-2.BIS} \sup_{Q\in{\mathbb{D}}(E)}\left(\tfrac{1}{\sigma(Q)}\int_{T_E(Q)}
|(\nabla{\mathcal{R}}1)(x)|^2\,{\rm dist}\,(x,E)\,dx\right)<\infty. \end{eqnarray}
\item[(3)] $[${\tt Carleson\! measure\! condition\! on\! dyadic\! tents\! for\! ${\mathcal{R}}$\! acting\! on\! $L^\infty$}$]$ There exists a constant $C\in(0,\infty)$ with the property that for each $f\in L^\infty(E,\sigma)$
\begin{eqnarray}\label{SF-BIS-3.BIS} \sup_{Q\in{\mathbb{D}}(E)}\left(\tfrac{1}{\sigma(Q)}\int_{T_E(Q)}
|(\nabla{\mathcal{R}}f)(x)|^2\,{\rm dist}\,(x,E)\,dx\right)^{1/2}
\leq C\|f\|_{L^\infty(E,\sigma)}. \end{eqnarray}
\item[(4)] $[${\tt Carleson\! measure\! condition\! on\! balls\! for\! ${\mathcal{R}}$\! tested\! on\! $1$}$]$ There holds
\begin{eqnarray}\label{SF-BIS-4.BIS} \sup_{x\in E,\,r>0}\left(\tfrac{1}{\sigma\bigl(E\cap B(x,r)\bigr)}
\int_{B(x,r)\setminus E}|(\nabla{\mathcal{R}}1)(y)|^2\, {\rm dist}\,(y,E)\,dy\right)<\infty. \end{eqnarray}
\item[(5)] $[${\tt Carleson\! measure\! condition\! for\! ${\mathcal{R}}$\! tested\! on\! a\! para-accretive\! function}$]$ There exists a para-accretive function $b:E\to{\mathbb{C}}$ with the property that
\begin{eqnarray}\label{SF-BIS-5.BIS} \sup_{Q\in{\mathbb{D}}(E)}\left(\tfrac{1}{\sigma(Q)}
\int_{T_E(Q)}|(\nabla{\mathcal{R}}b)(x)|^2\,{\rm dist}\,(x,E)\,dx\right)<\infty. \end{eqnarray}
\item[(6)] $[${\tt Carleson\! measure\! condition\! on\! balls\! for\! ${\mathcal{R}}$\! acting\! on\! $L^\infty$}$]$ There exists a constant $C\in(0,\infty)$ with the property that for each $f\in L^\infty(E,\sigma)$
\begin{eqnarray}\label{SF-BIS-6.BIS} \sup_{x\in E,\,r>0}\left(\tfrac{1}{\sigma\bigl(E\cap B(x,r)\bigr)}
\int_{B(x,r)\setminus E}|(\nabla{\mathcal{R}}f)(y)|^2\,{\rm dist}\,(y,E)\,dy\right)^{1/2}
\leq C\|f\|_{L^\infty(E,\sigma)}. \end{eqnarray}
\item[(7)] $[${\tt Local\! $T(b)$\! condition\! on dyadic\! cubes}$]$ There exist finite constants $C_0\geq 1$, $c_0\in(0,1]$ as well as a collection $\{b_Q\}_{Q\in{\mathbb{D}}(E)}$ of $\sigma$-measurable functions $b_Q:E\rightarrow{\mathbb{C}}$ such that for each $Q\in{\mathbb{D}}(E)$ the following hold:
\begin{eqnarray}\label{SF-BIS-7.BIS} \begin{array}{c}
\displaystyle\int_E |b_Q|^2\,d\sigma\leq C_0\sigma(Q), \\[16pt]
\left|\int_{\widetilde{Q}} b_Q\,d\sigma\right|\geq\frac{1}{C_0}\,\sigma(\widetilde{Q})\quad \mbox{for some }\widetilde{Q}\subseteq Q,\,\,\, \ell(\widetilde{Q})\geq c_0\ell(Q), \\[16pt]
\displaystyle\int_{T_E(Q)}|(\nabla{\mathcal{R}}b_Q)(x)|^2 \,{\rm dist}\,(x,E)\,dx\leq C_0\sigma(Q). \end{array} \end{eqnarray}
\item[(8)] $[${\tt Local\! $T(b)$\! condition\! on surface\! balls}$]$ There exist $C_0\in[1,\infty)$ and, for each surface ball $\Delta=\Delta(x_o,r):=B(x_o,r)\cap E$, where $x_o$ is a point in $E$ and $r$ is a finite number in $\bigl(0,{\rm diam}(E)\bigr]$, a $\sigma$-measurable function $b_\Delta:E\rightarrow{\mathbb{C}}$ supported in $\Delta$, such that the following estimates hold:
\begin{eqnarray}\label{SF-BIS-8.BIS} \begin{array}{c}
\displaystyle\int_E |b_\Delta|^2\,d\sigma\leq C_0\sigma(\Delta),\quad
\left|\int_\Delta b_\Delta\,d\sigma\right|\geq\frac{1}{C_0}\,\sigma(\Delta), \\[16pt]
\displaystyle\int_{B(x_o,4r)\setminus E}|(\nabla{\mathcal{R}}\,b_\Delta)(x)|^2 \,{\rm dist}\,(x,E)\,dx\leq C_0\sigma(\Delta). \end{array} \end{eqnarray}
\item[(9)] $[${\tt Weak-$L^p$\! square\! function\! estimate}$]$ There exist an index $p\in(0,\infty)$ and constants $C,\kappa\in(0,\infty)$ such that for every $f\in L^{p}(E,\sigma)$
\begin{eqnarray}\label{SF-BIS-9.BIS} \hskip -0.30in \sup_{\lambda>0}\left[\lambda\cdot \sigma\Bigl(\Bigl\{x\in E:\int_{\Gamma_{\kappa}(x)}
\frac{|(\nabla{\mathcal{R}}f)(y)|^2} {{\rm dist}\,(y,E)^{n-1}}\,dy>\lambda^{2}\Bigr\}\Bigr)^{1/p}\right]
\leq C\Bigl(\int_{E}|f|^p\,d\sigma\Bigr)^{1/p}, \end{eqnarray}
where $\Gamma_\kappa(x):=\bigl\{y\in{\mathbb{R}}^{n+1}\setminus E:\,
|x-y|<(1+\kappa)\,{\rm dist}(y,E)\bigr\}$ for each $x\in E$. \end{enumerate}
Then if any of properties {\it (1)-(9)} holds it follows that $E$ is a {\rm UR} set. \end{theorem}
The fact that condition {\it (1)} above implies that $E$ is a UR set has been proved by David and Semmes (see \cite[pp.\,252-267]{DaSe93}). Based on this result, that {\it (2)}-{\it (3)} also imply that $E$ is a UR set then follows with the help of Theorem~\ref{M-TTHH} upon observing that the components of ${\mathcal{R}}$ are operators ${\mathcal{T}}$ as in \eqref{T-BIS.2a} associated with the
kernels $K_j(x):=x_j/|x|^{n+1}$, $j\in\{1,...,n+1\}$, which satisfy \eqref{hszz-AXXX}-\eqref{hszz-3-AXXX}. Compared to David and Semmes' result mentioned above (to the effect that the $L^2$ square function for the operators associated with the kernels $K_j$, $1\leq j\leq n+1$, implies that the set $E$ is UR), a remarkable corollary of Theorem~\ref{UR-rest.BIS} is that a mere weak-$L^2$ square function estimate for the operators
associated with the kernels $K_j(x):=x_j/|x|^{n+1}$, $j\in\{1,...,n+1\}$, as in \eqref{T-BIS.2a} implies that $E$ is a UR set.
Throughout the manuscript, we adopt the following conventions. The letter $C$ represents a finite positive constant that may change from one line to the next. The infinity symbol $\infty:=+\infty$. The set of positive integers is denoted by $\mathbb{N}$, and the set $\mathbb{N}_0:=\mathbb{N}\cup\{0\}$.
\section{Analysis and Geometry on Quasi-Metric Spaces} \setcounter{equation}{0} \label{Sect:2}
This section contains preliminary material, organized into four subsections dealing, respectively, with: a metrization result for arbitrary quasi-metric spaces, geometrically doubling quasi-metric spaces, approximations to the identity, and a discussion of the nature of Carleson tents in quasi-metric spaces.
\subsection{A metrization result for general quasi-metric spaces} \label{SSect:2.1}
Here the goal is to review a sharp quantitative metrization result for quasi-metric spaces (cf. Theorem~\ref{JjEGh}), and record some useful properties of the Hausdorff outer-measure on quasi-metric spaces (cf. Proposition~\ref{PWRS22}). We begin, however, by introducing basic terminology and notation in the definition below.
\begin{definition}\label{zbnb} Assume that ${\mathscr{X}}$ is a set of cardinality at least two:
\begin{enumerate}
\item[(1)] A function $\rho:{\mathscr{X}}\times{\mathscr{X}}\to[0,\infty)$ is called a {\tt quasi-distance} on ${\mathscr{X}}$ provided there exist two constants $C_0,C_1\in[1,\infty)$ with the property that for every $x,y,z\in X$, one has
\begin{eqnarray}\label{gabn-T.2} \rho(x,y)=0\Leftrightarrow x=y,\quad \rho(y,x)\leq C_0\rho(x,y),\quad \rho(x,y)\leq C_1\max\{\rho(x,z),\rho(z,y)\}. \end{eqnarray}
\item[(2)] Denote by ${\mathfrak{Q}}({\mathscr{X}})$ the collection of all quasi-distances on ${\mathscr{X}}$, and call a pair $({\mathscr{X}},\rho)$ a {\tt quasi-metric\! space} provided $\rho\in{\mathfrak{Q}}({\mathscr{X}})$. Also, given $\rho\in{\mathfrak{Q}}({\mathscr{X}})$ and $E\subseteq{\mathscr{X}}$
of cardinality at lest two, denote by $\rho\bigl|_{E}\in{\mathfrak{Q}}(E)$ the restriction of the function $\rho$ to $E\times E$.
\item[(3)] For each $\rho\in{\mathfrak{Q}}({\mathscr{X}})$, define $C_\rho$ to be the smallest constant which can play the role of $C_1$ in the last inequality in \eqref{gabn-T.2}, i.e.,
\begin{eqnarray}\label{C-RHO.111} C_\rho:=\sup_{\stackrel{x,y,z\in{\mathscr{X}}}{\mbox{\tiny{not all equal}}}} \frac{\rho(x,y)}{\max\{\rho(x,z),\rho(z,y)\}}\in[1,\infty), \end{eqnarray}
and define $\widetilde{C}_\rho$ to be the smallest constant which can play the role of $C_0$ in the first inequality in \eqref{gabn-T.2}, i.e.,
\begin{eqnarray}\label{C-RHO.111XXX} \widetilde{C}_\rho:=\sup_{\stackrel{x,y\in{\mathscr{X}}}{x\not=y}} \frac{\rho(y,x)}{\rho(x,y)}\,\in[1,\infty). \end{eqnarray}
\item[(4)] Given $\rho\in{\mathfrak{Q}}({\mathscr{X}})$, define the $\rho$-{\tt ball} (or, simply {\tt ball} if the quasi-distance $\rho$ is clear from the context) centered at $x\in{\mathscr{X}}$ with radius $r\in(0,\infty)$ to be
\begin{eqnarray}\label{hdc-587} B_\rho(x,r):=\left\{y\in{\mathscr{X}}:\rho(x,y)<r\right\}. \end{eqnarray}
Also, call $E\subseteq{\mathscr{X}}$ $\rho$-{\tt bounded} if $E$ is contained in a $\rho$-ball, and define its $\rho$-{\tt diameter} (or, simply, {\tt diameter}) as
\begin{eqnarray}\label{DIA-TT} {\rm diam}_{\rho}(E):=\sup\,\bigl\{\rho(x,y):\,x,y\in E\bigr\}. \end{eqnarray}
The $\rho$-{\tt distance} (or, simply {\tt distance}) between two arbitrary, nonempty sets $E,F\subseteq{\mathscr{X}}$ is naturally defined as
\begin{eqnarray}\label{HG-DD.6} {\rm dist}_\rho(E,F):=\inf\,\{\rho(x,y):\,x\in E,\,\,y\in F\}. \end{eqnarray}
If $E=\{x\}$ for some $x\in{\mathscr{X}}$ and $F\subseteq{\mathscr{X}}$, abbreviate ${\rm dist}_\rho(x,F):={\rm dist}_\rho(\{x\},F)$.
\item[(5)] Given a quasi-distance $\rho\in{\mathfrak{Q}}({\mathscr{X}})$ define $\tau_{\rho}$, the {\tt topology canonically induced by $\rho$ on} ${\mathscr{X}}$, to be the largest topology on ${\mathscr{X}}$ with the property that for each point $x\in{\mathscr{X}}$ the family $\{B_\rho(x,r)\}_{r>0}$ is a fundamental system of neighborhoods of $x$.
\item[(6)] Call two functions $\rho_1,\rho_2:{\mathscr{X}}\times{\mathscr{X}}\to[0,\infty)$ {\tt equivalent}, and write $\rho_1\approx\rho_2$, if there exist $C',C''\in(0,\infty)$ with the property that
\begin{eqnarray}\label{JH-7pp} C'\rho_1\leq\rho_2\leq C''\rho_1\,\quad\mbox{on }\,\,{\mathscr{X}}\times{\mathscr{X}}. \end{eqnarray} \end{enumerate} \end{definition}
\noindent A few comments are in order. Suppose that $({\mathscr{X}},\rho)$ is a quasi-metric space. It is then clear that if $\rho':{\mathscr{X}}\times{\mathscr{X}}\rightarrow[0,\infty)$ is such that $\rho'\approx\rho$ then $\rho'\in{\mathfrak{Q}}({\mathscr{X}})$ and $\tau_{\rho'}=\tau_\rho$. Also, it may be checked that
\begin{eqnarray}\label{topoQMS} \mathcal{O}\in\tau_\rho\,\Longleftrightarrow\, \mathcal{O}\subseteq{\mathscr{X}}\,\,\mbox{ and }\,\,\forall\,x\in\mathcal{O}\,\, \,\,\exists\,r>0\,\mbox{ such that }\,B_\rho(x,r)\subseteq\mathcal{O}. \end{eqnarray}
As is well-known, the topology induced by the given quasi-distance on a quasi-metric space is metrizable. Below we shall review a recent result proved in \cite{MMMM-G} which is an optimal quantitative version of this fact, and which sharpens earlier work from \cite{MaSe79}. To facilitate the subsequent discussion we first make a definition. Assume that ${\mathscr{X}}$ is an arbitrary, nonempty set. Given an arbitrary function $\rho:{\mathscr{X}}\times{\mathscr{X}}\to[0,\infty)$ define its {\tt symmetrization} $\rho_{sym}$ as
\begin{eqnarray}\label{sfgu.GG} \rho_{sym}:{\mathscr{X}}\times{\mathscr{X}}\longrightarrow[0,\infty),\qquad \rho_{sym}(x,y):=\max\,\{\rho(x,y),\rho(y,x)\},\quad\forall\,x,y\in{\mathscr{X}}. \end{eqnarray}
Then $\rho_{sym}$ is symmetric, i.e., $\rho_{sym}(x,y)=\rho_{sym}(y,x)$ for every $x,y\in{\mathscr{X}}$, and $\rho_{sym}\geq\rho$ on ${\mathscr{X}}\times{\mathscr{X}}$. In fact, $\rho_{sym}$ is the smallest nonnegative function defined on ${\mathscr{X}}\times{\mathscr{X}}$ which is symmetric and pointwise $\geq\rho$. Furthermore, if $({\mathscr{X}},\rho)$ is a quasi-metric space then
\begin{eqnarray}\label{yabv.GG} \rho_{sym}\in{\mathfrak{Q}}({\mathscr{X}}),\,\,\,\,C_{\rho_{sym}}\leq C_\rho, \,\,\,\,\widetilde{C}_{\rho_{sym}}=1, \,\,\mbox{ and }\,\,\rho\leq\rho_{sym}\leq\,\widetilde{C}_\rho\,\rho. \end{eqnarray}
Here is the quantitative metrization theorem from \cite{MMMM-G} alluded to above.
\begin{theorem}\label{JjEGh} Let $({\mathscr{X}},\rho)$ be a quasi-metric space and assume that $C_\rho,\widetilde{C}_\rho\in[1,\infty)$ are as in \eqref{C-RHO.111}-\eqref{C-RHO.111XXX}. Introduce
\begin{eqnarray}\label{Cro} \alpha_\rho:=\frac{1}{\log_2 C_\rho}\in(0,\infty], \end{eqnarray}
and define the regularization $\rho_{\#}:{\mathscr{X}}\times{\mathscr{X}}\to[0,\infty)$ of $\rho$ as follows. When $\alpha_\rho<\infty$, for each $x,y\in{\mathscr{X}}$ set
\begin{eqnarray}\label{R-sharp} && \hskip -0.60in \rho_{\#}(x,y):=\inf\,\Bigl\{\Bigl(\sum\limits_{i=1}^N \rho_{sym}(\xi_i,\xi_{i+1})^{\alpha_\rho}\Bigr)^{\frac{1}{\alpha_\rho}}:\, N\in{\mathbb{N}}\mbox{ and }\xi_1,\dots,\,\xi_{N+1}\in{\mathscr{X}}, \nonumber \\[4pt] && \hskip 1.00in \mbox{(not necessarily distinct) such that $\xi_1=x$ and $\xi_{N+1}=y$}\Bigr\}, \end{eqnarray}
while if $\alpha_\rho=\infty$ then for each $x,y\in{\mathscr{X}}$ set
\begin{eqnarray}\label{BV+uuu} && \hskip -0.60in \rho_{\#}(x,y):=\inf\,\Bigl\{\max\limits_{1\leq i\leq N} \rho_{sym}(\xi_i,\xi_{i+1}):\,N\in{\mathbb{N}}\mbox{ and }\, \xi_1,\dots,\,\xi_{N+1}\in{\mathscr{X}}, \\[4pt] && \hskip 1.00in \mbox{(not necessarily distinct) such that $\xi_1=x$ and $\xi_{N+1}=y$}\Bigr\}. \nonumber \end{eqnarray}
Then the following properties hold:
\begin{enumerate}
\item[(1)] The function $\rho_{\#}$ is a symmetric quasi-distance on ${\mathscr{X}}$ and $\rho_{\#}\approx\rho$. More specifically,
\begin{eqnarray}\label{DEQV1} (C_\rho)^{-2}\rho(x,y)\leq\rho_{\#}(x,y)\leq\widetilde{C}_\rho\,\rho(x,y),\quad \forall\,x,y\in{\mathscr{X}}. \end{eqnarray}
In particular, $\tau_{\rho_{\#}}=\tau_{\rho}$. Also, $C_{\rho_{\#}}\leq C_\rho$. Furthermore, for any nonempty set $E$ of ${\mathscr{X}}$, there holds
\begin{eqnarray}\label{RHo-evv}
(\rho|_{E})_{\#}\approx\rho|_{E}\approx(\rho_{\#})\bigl|_{E}. \end{eqnarray}
\item[(2)] For each finite number $\beta\in(0,\alpha_{\rho}]$, the function
\begin{eqnarray}\label{Ubhb-657.GG} d_{\rho,\beta}:{\mathscr{X}}\times{\mathscr{X}}\to [0,\infty),\qquad d_{\rho,\beta}(x,y):=\bigl[\rho_{\#}(x,y)\bigr]^\beta, \qquad\forall\,x,y\in{\mathscr{X}}, \end{eqnarray}
is a distance on ${\mathscr{X}}$, and has the property that $(d_{\rho,\beta})^{1/\beta}\approx\rho$. In particular, $d_{\rho,\beta}$ induces the same topology on ${\mathscr{X}}$ as $\rho$, hence $\tau_\rho$ is metrizable.
\item[(3)] For each finite number $\beta\in(0,\alpha_{\rho}]$, the function $\rho_{\#}$ satisfies the following H\"older-type regularity condition of order $\beta$ (in both variables, simultaneously) on ${\mathscr{X}}\times{\mathscr{X}}$:
\begin{eqnarray}\label{NNN-g8-tt} && \hskip -0.50in
\bigl|\rho_{\#}(x,y)-\rho_{\#}(z,w)\bigr| \\[4pt] && \hskip 0.50in \leq{\textstyle\frac{1}{\beta}}\, \max\,\bigl\{\rho_{\#}(x,y)^{1-\beta},\rho_{\#}(z,w)^{1-\beta}\bigr\} \bigl(\bigl[\rho_{\#}(x,z)\bigr]^\beta+\bigl[\rho_{\#}(y,w)\bigr]^\beta\bigr) \nonumber \end{eqnarray}
whenever $x,y,z,w\in{\mathscr{X}}$ (with the understanding that when $\beta\geq 1$ one also imposes the conditions $x\not=y$ and $z\not=w$). In particular,
\begin{eqnarray}\label{CC-NNu} \rho_{\#}:{\mathscr{X}}\times{\mathscr{X}}\longrightarrow [0,\infty) \quad\,\mbox{is continuous}, \end{eqnarray}
when ${\mathscr{X}}\times{\mathscr{X}}$ is equipped with the natural product topology $\tau_\rho\times\tau_\rho$.
\item[(4)] If $E$ is a nonempty subset of $({\mathscr{X}},\tau_{\rho})$, then the regularized distance function
\begin{eqnarray}\label{REG-DDD} \delta_E:={\rm dist}_{\rho_{\#}}(\cdot,E):{\mathscr{X}}\longrightarrow[0,\infty) \end{eqnarray}
is equivalent to ${\rm dist}_{\rho}(\cdot,E)$. Furthermore, $\delta_E$ is locally H\"older of order $\beta$ on ${\mathscr{X}}$ for every $\beta\in(0,\min\,\{1,\alpha_\rho\}]$, in the sense that there exists $C\in(0,\infty)$ which depends only on $C_\rho,\widetilde{C}_\rho$ and $\beta$ such that
\begin{eqnarray}\label{TBn-3}
\frac{|\delta_E(x)-\delta_E(y)|}{\rho(x,y)^\beta}\leq C\, \Bigl(\rho(x,y)+\max\,\{{\rm dist}_\rho(x,E)\,,\,{\rm dist}_\rho(y,E)\}\Bigr)^{1-\beta} \end{eqnarray}
for all $x,y\in{\mathscr{X}}$ with $x\not=y$. In particular,
\begin{eqnarray}\label{DEDE-CC} \delta_E:({\mathscr{X}},\tau_\rho)\longrightarrow [0,\infty) \quad\,\mbox{is continuous}. \end{eqnarray} \end{enumerate} \end{theorem}
\noindent The key feature of the result discussed in Theorem~\ref{JjEGh} is the fact that if $({\mathscr{X}},\rho)$ is any quasi-metric space then $\rho^\beta$ is equivalent to a genuine distance on ${\mathscr{X}}$ for any finite number $\beta\in(0,(\log_2C_\rho)^{-1}]$. This result is sharp and improves upon an earlier version due to R.A.~Mac\'{\i}as and C. Segovia \cite{MaSe79}, in which these authors have identified a smaller, non-optimal upper-bound for the exponent $\beta$.
In anticipation of briefly reviewing the notion of Hausdorff outer-measure on a quasi-metric space, we recall a couple of definitions from measure theory. Specifically, given an outer-measure $\mu^{\ast}$ on an arbitrary set ${\mathscr{X}}$, consider the collection of all $\mu^{\ast}$-measurable sets defined as
\begin{eqnarray}\label{MM-RE.89} {\mathfrak{M}}_{\mu^{\ast}}:=\{A\subseteq{\mathscr{X}}:\, \mu^{\ast}(Y)=\mu^{\ast}(Y\cap A)+\mu^{\ast}(Y\setminus A),\,\, \forall\,Y\subseteq{\mathscr{X}}\}. \end{eqnarray}
Carath\'eodory's classical theorem allows one to pass from a given outer-measure $\mu^{\ast}$ on ${\mathscr{X}}$ to a genuine measure by observing that
\begin{eqnarray}\label{MM-RE.90} {\mathfrak{M}}_{\mu^{\ast}}\mbox{ is a sigma-algebra, and
$\mu^{\ast}\bigl|_{{\mathfrak{M}}_{\mu^{\ast}}}$ is a complete measure}. \end{eqnarray}
The restriction of an outer-measure $\mu^\ast$ on ${\mathscr{X}}$ to a subset $E$ of ${\mathscr{X}}$, denoted by $\mu^\ast\lfloor E$, is defined naturally by restricting the function $\mu^\ast$ to the collection of all subsets of $E$. We shall use the same symbol, $\lfloor\,\,$, in denoting the restriction of a measure to a measurable set. In this regard, it is useful to know when the measure associated with the restriction of an outer-measure to a set coincides with the restriction to that set of the measure associated with the given outer-measure. Specifically, it may be checked that if $\mu^\ast$ is an outer-measure on ${\mathscr{X}}$, then
\begin{eqnarray}\label{hgdss-F22}
\bigl(\mu^\ast\lfloor E\bigr)\Bigl|_{{\mathfrak{M}}_{(\mu^\ast\lfloor E)}}
=\bigl(\mu^\ast\bigl|_{{\mathfrak{M}}_{\mu^\ast}}\bigr)\lfloor E,\qquad \forall\,E\in{\mathfrak{M}}_{\mu^\ast}. \end{eqnarray}
Next, if $({\mathscr{X}},\tau)$ is a topological space and $\mu^{\ast}$ is an outer-measure on ${\mathscr{X}}$ such that ${\mathfrak{M}}_{\mu^{\ast}}$ contains the Borel sets in $({\mathscr{X}},\tau)$, then call $\mu^{\ast}$ a {\tt Borel outer-measure} on ${\mathscr{X}}$. Furthermore, call such a Borel outer-measure $\mu^{\ast}$ a {\tt Borel regular outer-measure} if
\begin{eqnarray}\label{Tgabv10} \forall\,A\subseteq {\mathscr{X}}\,\,\,\exists \mbox{ a Borel set $B$ in $({\mathscr{X}},\tau)$ such that $A\subseteq B$ and $\mu^{\ast}(A)=\mu^{\ast}(B)$}. \end{eqnarray}
After this digression, we now proceed to introduce the concept of $d$-dimensional Hausdorff outer-measure for a subset of a quasi-metric space.
\begin{definition}\label{SKJ38} Let $({\mathscr{X}},\rho)$ be a quasi-metric space and fix $d\geq 0$. Given a set $A\subseteq{\mathscr{X}}$, for every $\varepsilon>0$ define ${\mathcal{H}}_{{\mathscr{X}}\!,\,\rho,\,\varepsilon}^d(A)\in[0,\infty]$ by setting
\begin{eqnarray}\label{SKJ38-1} \hskip -0.20in {\mathcal{H}}_{{\mathscr{X}}\!,\,\rho,\,\varepsilon}^d(A) :=\inf\,\Bigl\{\sum_{j=1}^\infty({\rm diam}_{\rho}(A_j))^d:\, A\subseteq\bigcup_{j=1}^\infty A_j\mbox{ and } {\rm diam}_{\rho}(A_j)\leq\varepsilon\mbox{ for every }j\Bigr\} \end{eqnarray}
(with the convention that $\inf\,\emptyset:=+\infty$), then take
\begin{eqnarray}\label{SKJ38-2} {\mathcal{H}}_{{\mathscr{X}}\!,\,\rho}^d(A) :=\lim_{\varepsilon\to0^+}{\mathcal{H}}_{{\mathscr{X}}\!,\,\rho,\,\varepsilon}^d(A) =\sup_{\varepsilon>0}{\mathcal{H}}_{{\mathscr{X}}\!,\,\rho,\,\varepsilon}^d(A) \in[0,\infty]. \end{eqnarray}
The quantity ${\mathcal{H}}_{{\mathscr{X}}\!,\,\rho}^d(A)$ is called the $d$-{\tt dimensional Hausdorff outer-measure} in $({\mathscr{X}},\rho)$ of the set $A$. Whenever the choice of the quasi-distance $\rho$ is irrelevant or clear from the context, ${\mathcal{H}}_{{\mathscr{X}}\!,\,\rho}^d(A)$ is abbreviated as ${\mathcal{H}}_{{\mathscr{X}}}^d(A)$. \end{definition}
It is readily verified that ${\mathcal{H}}_{{\mathscr{X}}\!,\,\rho}^0$ is equivalent to the counting measure. Other basic properties of the Hausdorff outer-measure are collected in the proposition below, proved in \cite{MMMM-B}. To state it, recall that a measure $\mu$ on a quasi-metric space $({\mathscr{X}},\rho)$ is called {\tt Borel regular} provided it is Borel on $({\mathscr{X}},\tau_\rho)$ and
\begin{eqnarray}\label{T-gc2w} \forall\,\mbox{$\mu$-measurable}\,A\subseteq{\mathscr{X}}\,\,\,\exists \mbox{ a Borel set $B$ in $({\mathscr{X}},\tau_\rho)$ such that $A\subseteq B$ and $\mu(A)=\mu(B)$}.\quad \end{eqnarray}
Also, we make the convention that, given a quasi-metric space $({\mathscr{X}},\rho)$ and $d\geq 0$,
\begin{eqnarray}\label{HHH-56} \mbox{${\mathscr{H}}_{{\mathscr{X}}\!,\,\rho}^d$ denotes the {\it measure} associated with the outer-measure ${\mathcal{H}}_{{\mathscr{X}}\!,\,\rho}^d$ as in \eqref{MM-RE.90}}. \end{eqnarray}
\begin{proposition}\label{PWRS22} Let $({\mathscr{X}},\rho)$ be a quasi-metric space and fix $d\geq 0$. Then the following properties hold:
\begin{enumerate}
\item[(1)] ${\mathcal{H}}_{{\mathscr{X}}\!,\,\rho}^d$ is a Borel outer-measure on $({\mathscr{X}},\tau_\rho)$. In particular, ${\mathscr{H}}_{{\mathscr{X}}\!,\,\rho}^d$ (introduced in \eqref{HHH-56}) is a Borel measure on $({\mathscr{X}},\tau_\rho)$.
\item[(2)] If $\rho_{\#}$ is as in Theorem~\ref{JjEGh} then ${\mathcal{H}}_{{\mathscr{X}}\!,\,\rho_{\#}}^d$ is actually a Borel regular outer-measure on $({\mathscr{X}},\tau_\rho)$. Moreover, ${\mathscr{H}}_{{\mathscr{X}}\!,\,\rho_{\#}}^d$ is a Borel regular measure on $({\mathscr{X}},\tau_\rho)$.
\item[(3)] One has ${\mathcal{H}}_{{\mathscr{X}}\!,\,\rho'}^d\approx {\mathcal{H}}_{{\mathscr{X}}\!,\,\rho}^d$ whenever $\rho'\approx\rho$, in the sense that there exist two finite constants $C_1,C_2>0$, which depend only on $\rho$ and $\rho'$, such that
\begin{eqnarray}\label{PWR-Cx} C_1\,{\mathcal{H}}_{{\mathscr{X}}\!,\,\rho}^d(A) \leq{\mathcal{H}}_{{\mathscr{X}}\!,\,\rho'}^d(A) \leq C_2\,{\mathcal{H}}_{{\mathscr{X}}\!,\,\rho}^d(A) \quad\mbox{for all }\,A\subseteq{\mathscr{X}}. \end{eqnarray}
\item[(4)] Let $E\subseteq{\mathscr{X}}$ and consider the quasi-metric
space $(E,\rho|_E)$. Then the $d$-dimensional Hausdorff outer-measure in
$(E,\rho|_{E})$ is equivalent to the restriction to $E$ of the $d$-dimensional Hausdorff outer-measure in ${\mathscr{X}}$. That is, in the sense of \eqref{PWR-Cx},
\begin{eqnarray}\label{jkan}
{\mathcal{H}}_{E,\,\rho|_{E}}^d\approx {\mathcal{H}}_{{\mathscr{X}}\!,\,\rho}^d\big\lfloor E. \end{eqnarray}
\item[(5)] For any $E\subseteq{\mathscr{X}}$, ${\mathcal{H}}_{{\mathscr{X}}\!,\,\rho_{\#}}^d\big\lfloor E$ is a Borel regular
outer-measure on $(E,\tau_{\rho|_{E}})$, and the measure associated with it
(as in \eqref{MM-RE.90}) is a Borel regular measure on $(E,\tau_{\rho|_{E}})$.
Furthermore, if $E$ is ${\mathcal{H}}_{{\mathscr{X}}\!,\,\rho_{\#}}^d$-measurable (in the sense of \eqref{MM-RE.89}; hence, in particular, if $E$ is a Borel subset of $({\mathscr{X}},\tau_\rho)$), then ${\mathscr{H}}_{{\mathscr{X}}\!,\,\rho_{\#}}^d\lfloor E$
is a Borel regular measure on $(E,\tau_{\rho|_{E}})$ and it coincides with the measure associated with the outer-measure ${\mathcal{H}}_{{\mathscr{X}}\!,\,\rho_{\#}}^d\lfloor E$.
\item[(6)] Assume that $m\in(d,\infty)$. Then for each $E\subseteq{\mathscr{X}}$ one has
\begin{eqnarray}\label{haTT.2} {\mathcal{H}}_{{\mathscr{X}}\!,\,\rho}^{d}(E)<\infty\,\Longrightarrow\, {\mathcal{H}}_{{\mathscr{X}}\!,\,\rho}^{m}(E)=0. \end{eqnarray} \end{enumerate} \end{proposition}
\subsection{Geometrically doubling quasi-metric spaces} \label{SSect:2.2}
In this subsection we shall work in a more specialized setting than that of general quasi-metric spaces considered so far, by considering geometrically doubling quasi-metric spaces, as described in the definition below.
\begin{definition}\label{Gd_ZZ} A quasi-metric space $({\mathscr{X}},\rho)$ is called {\tt geometrically doubling} if there exists a number $N\in{\mathbb{N}}$, called the geometric doubling constant of $({\mathscr{X}},\rho)$, with the property that any $\rho$-ball of radius $r$ in ${\mathscr{X}}$ may be covered by at most $N$ $\rho$-balls in ${\mathscr{X}}$ of radii $r/2$. \end{definition}
To put this matter into a larger perspective, recall that a subset $E$ of a quasi-metric space $({\mathscr{X}},\rho)$ is said to be {\tt totally bounded} provided that for any $r\in(0,\infty)$ there exists a finite covering of $E$ with $\rho$-balls of radii $r$. Then for a quasi-metric space $({\mathscr{X}},\rho)$ the quality of being geometrically doubling may be regarded as a scale-invariant version of the demand that all $\rho$-balls in ${\mathscr{X}}$ are totally bounded. In fact it may be readily verified that if $({\mathscr{X}},\rho)$ is a geometrically doubling quasi-metric space, then
\begin{eqnarray}\label{RAF-e} \begin{array}{l} \exists\,N\in{\mathbb{N}}\mbox{ such that $\forall\,\vartheta\in(0,1)$ any $\rho$-ball of radius $r$ in ${\mathscr{X}}$} \\[4pt] \mbox{may be covered by at most $N^{-[\log_2\vartheta]}$ $\rho$-balls in ${\mathscr{X}}$ of radii $\vartheta r$}, \end{array} \end{eqnarray}
where $[\log_2\vartheta]$ is the smallest integer greater than or equal to $\log_2 \vartheta$. En route, let us also point out that {\it the property of being geometrically doubling is hereditary}, in the sense that if $({\mathscr{X}},\rho)$ is a geometrically
doubling quasi-metric space with geometric doubling constant $N$, and if $E$ is an arbitrary subset of ${\mathscr{X}}$, then $(E,\rho\bigl|_E)$ is a geometrically doubling quasi-metric space with geometric doubling constant at most equal to $N^{\log_2 C_\rho}N$.
The relevance of the property (of a quasi-metric space) of being geometrically doubling is apparent from the fact that in such a context a number of useful geometrical results hold, which are akin to those available in the Euclidean setting. A case in point, is the Whitney decomposition theorem discussed in Proposition~\ref{H-S-Z} below. A version of the classical Whitney decomposition theorem in the Euclidean setting (as presented in, e.g., \cite[Theorem~1.1, p.\,167]{St70}) has been worked out in \cite[Theorem~3.1, p.\,71]{CoWe71} and \cite[Theorem~3.2, p.\,623]{CoWe77} in the context of bounded open sets in spaces of homogeneous type. Recently, the scope of this work has been further refined in \cite{MMMM-G} by allowing arbitrary open sets in geometrically doubling quasi-metric spaces, as presented in the following proposition.
\begin{proposition}\label{H-S-Z} Let $({\mathscr{X}},\rho)$ be a geometrically doubling quasi-metric space. Then for each number $\lambda\in (1,\infty)$ there exist constants $\Lambda\in(\lambda,\infty)$ and $N\in{\mathbb{N}}$, both depending only on $C_\rho,\widetilde{C}_\rho,\lambda$ and the geometric doubling constant of $({\mathscr{X}},\rho)$, and which have the following significance.
For each open, nonempty, proper subset ${\mathcal{O}}$ of the topological space $({\mathscr{X}},\tau_\rho)$ there exist an at most countable family of points $\{x_j\}_{j\in J}$ in ${\mathcal{O}}$ along with a family of real numbers $r_j>0$, $j\in J$, for which the following properties are valid:
\begin{enumerate} \item[(1)] ${\mathcal{O}}=\bigcup\limits_{j\in J}B_\rho(x_j,r_j)$;
\item[(2)] $\sum\limits_{j\in J}{\mathbf 1}_{B_\rho(x_j,\lambda r_j)}\leq N$ on ${\mathcal{O}}$. In fact, there exists $\varepsilon\in(0,1)$, which depends only on $C_\rho,\lambda$ and the geometric doubling constant of $({\mathscr{X}},\rho)$, with the property that for any $x_o\in{\mathcal{O}}$
\begin{eqnarray}\label{Lay-ff.u-c} \hskip -0.20in \#\,\Bigl\{j\in J:\,B_\rho\bigl(x_o,\varepsilon\, {\rm dist}_\rho(x_o,{\mathscr{X}}\setminus{\mathcal{O}})\bigr) \cap B_{\rho}(x_j,\lambda r_j)\not=\emptyset\Bigr\}\leq N. \end{eqnarray}
\item[(3)] $B_\rho(x_j,\lambda r_j)\subseteq{\mathcal{O}}$ and $B_\rho(x_j,\Lambda r_j)\cap\bigl[{\mathscr{X}}\setminus{\mathcal{O}}\bigr]\not=\emptyset$ for every $j\in J$.
\item[(4)] $r_i\approx r_j$ uniformly for $i,j\in J$ such that $B_\rho(x_i,\lambda r_i)\cap B_\rho(x_j,\lambda r_j)\not=\emptyset$, and there exists a finite constant $C>0$ with the property that $r_j\leq C\,{\rm diam}_{\rho}({\mathcal{O}})$ for each $j\in J$. \end{enumerate} \end{proposition}
Regarding terminology, we shall frequently employ the following convention:
\begin{convention}\label{WWVc} Given a geometrically doubling quasi-metric space $({\mathscr{X}},\rho)$, an open, nonempty, proper subset ${\mathcal{O}}$ of $({\mathscr{X}},\tau_\rho)$, and a parameter $\lambda\in (1,\infty)$, we will refer to the balls $B_{\rho_{\#}}(x_j,r_j)$ obtained by treating $(\mathscr{X},\rho_{\#})$ in Proposition~\ref{H-S-Z} as {\tt Whitney cubes}, denote the collection of these cubes by ${\mathbb{W}}_\lambda({\mathcal{O}})$, and for each $I\in{\mathbb{W}}_\lambda({\mathcal{O}})$, write $\ell(I)$ for the {\tt radius of $I$}.
Furthermore, if $I\in{\mathbb{W}}_\lambda({\mathcal{O}})$ and $c\in(0,\infty)$, we shall denote by $cI$ the {\tt dilate of the cube $I$ by factor $c$}, i.e., the ball having the same center as $I$ and radius $c\ell(I)$. \end{convention}
Spaces of homogeneous type, reviewed next, are an important subclass of the class of geometrically doubling quasi-metric spaces.
\begin{definition}\label{zbnb-hom} A {\tt space of homogeneous type} is a triplet $({\mathscr{X}},\rho,\mu)$, where $({\mathscr{X}},\rho)$ is a quasi-metric space and $\mu$ is a Borel measure on $({\mathscr{X}},\tau_\rho)$ with the property that all $\rho$-balls are $\mu$-measurable, and which satisfies the doubling condition
\begin{eqnarray}\label{Doub-1} 0<\mu\bigl(B_{\rho}(x,2r)\bigr)\leq C\mu\bigl(B_{\rho}(x,r)\bigr)<\infty,\quad \forall\,x\in {\mathscr{X}},\,\,\,\forall\,r>0, \end{eqnarray}
for some finite constant $C\geq 1$. \end{definition}
\noindent In the context of the above definition, call the number
\begin{eqnarray}\label{Doub-1XX} C_\mu:=\sup_{x\in{\mathscr{X}},r>0}\frac{\mu\bigl(B_{\rho}(x,2r)\bigr)} {\mu\bigl(B_{\rho}(x,r)\bigr)}\in[1,\infty) \end{eqnarray}
the {\tt doubling constant} of $\mu$. Iterating \eqref{Doub-1} then gives
\begin{eqnarray}\label{Doub-2} \begin{array}{c} \frac{\mu(B_1)}{\mu(B_2)}\leq C_{\mu,\rho}\Bigl(\frac{\mbox{radius of $B_1$}} {\mbox{radius of $B_2$}}\Bigr)^{D_\mu},\qquad\mbox{for all $\rho$-balls }\,\, B_2\subseteq B_1, \\[8pt] \mbox{where }\,\,D_\mu:=\log_2\,C_\mu\geq 0\,\,\mbox{ and }\,\, C_{\mu,\rho}:=C_\mu\bigl(C_\rho\widetilde{C}_\rho\bigr)^{D_\mu}\geq 1. \end{array} \end{eqnarray}
The exponent $D_\mu$ is referred to as the {\tt doubling order} of $\mu$. For further reference, let us also record here the well-known fact that
\begin{eqnarray}\label{DIA-MEA} \begin{array}{c} \mbox{given a space of homogeneous type $({\mathscr{X}},\rho,\mu)$, one has} \\[4pt] \mbox{${\rm diam}_{\rho}\,({\mathscr{X}})<\infty$ if and only if $\mu({\mathscr{X}})<\infty$}. \end{array} \end{eqnarray}
Going further, a distinguished subclass of the class of spaces of homogeneous type, which is going to play a basic role in this work, is the category of Ahlfors-David regular spaces defined next.
\begin{definition}\label{Rcc-TG34} Suppose that $d>0$. A $d$-{\tt dimensional Ahlfors-David regular} (or, simply, $d$-dimensional {\rm ADR}, or $d$-{\rm ADR}) {\tt space} is a triplet $({\mathscr{X}},\rho,\mu)$, where $({\mathscr{X}},\rho)$ is a quasi-metric space and $\mu$ is a Borel measure on $({\mathscr{X}},\tau_\rho)$ with the property that all $\rho$-balls are $\mu$-measurable, and for which there exists a constant $C\in[1,\infty)$ such that
\begin{eqnarray}\label{Q3HF} C^{-1}\,r^d\leq\mu\bigl(B_\rho(x,r)\bigr)\leq C\,r^d, \quad\forall\,x\in {\mathscr{X}},\,\,\,\,\mbox{for every finite }\,r \in(0,{\rm diam}_\rho({\mathscr{X}})]. \end{eqnarray}
The constant $C$ in \eqref{Q3HF} will be referred to as the {\rm ADR} constant of ${\mathscr{X}}$. \end{definition}
\noindent As alluded to earlier, if $({\mathscr{X}},\rho,\mu)$ is a $d$-dimensional ADR space then, trivially, $({\mathscr{X}},\rho,\mu)$ is also a space of homogeneous type. For further reference let us also note here that (cf., e.g., \cite{MMMM-B})
\begin{eqnarray}\label{Rss-TASS} \mbox{$({\mathscr{X}},\rho,\mu)$ is $d$-ADR}\,\,\Longrightarrow\,\, \mbox{$\bigl({\mathscr{X}},\rho_{\#},{\mathscr{H}}^d_{{\mathscr{X}},\rho_{\#}}\bigr)$ is $d$-ADR}. \end{eqnarray}
In particular, it follows from \eqref{Rss-TASS}, \eqref{RHo-evv}, and parts {\it (3)-(5)} in Proposition~\ref{PWRS22} that
\begin{eqnarray}\label{TASS.bis2} \left. \begin{array}{r} \mbox{$({\mathscr{X}},\rho)$ quasi-metric space,} \\[4pt] \mbox{$E$ Borel subset of $({\mathscr{X}},\tau_\rho)$} \\[4pt]
\mbox{$\sigma$ Borel measure on $(E,\tau_{\rho|_{E}})$} \\[4pt]
\mbox{such that $(E,\rho|_{E},\sigma)$ is $d$-{\rm ADR}} \end{array} \right\} \,\,\Longrightarrow\,\,\mbox{
$\bigl(E,\rho_{\#}\bigl|_{E},{\mathscr{H}}^d_{{\mathscr{X}},\rho_{\#}}\lfloor E\bigr)$ is $d$-ADR}. \end{eqnarray}
Also, if $({\mathscr{X}},\rho,\mu)$ is $d$-ADR, then there exists a finite constant $C>0$ such that
\begin{eqnarray}\label{PEY88-2} && {\mathcal{H}}_{{\mathscr{X}}\!,\,\rho_{\#}}^d(A) \leq C\inf\limits_{{\mathcal{O}}{\mbox{\tiny{ open}}},\,A\subseteq{\mathcal{O}}} \mu({\mathcal{O}})\quad\mbox{for every }A\subseteq{\mathscr{X}},\,\mbox{ and} \\[4pt] && \mu(A)\leq C{\mathscr{H}}_{{\mathscr{X}}\!,\,\rho_{\#}}^d(A) \quad\mbox{for every Borel subset $A$ of $({\mathscr{X}},\tau_\rho)$}. \label{PEY88-2BIS} \end{eqnarray}
In addition, if $\mu$ is actually a Borel regular measure, then
\begin{eqnarray}\label{P2-GFs} \mu(A)\approx{\mathscr{H}}_{{\mathscr{X}}\!,\,\rho_{\#}}^d(A), \qquad\mbox{uniformly for Borel subsets $A$ of $({\mathscr{X}},\tau_\rho)$}. \end{eqnarray}
We now discuss a couple of technical lemmas which are going to be useful for us later on.
\begin{lemma}\label{ME-ZZ} Let $({\mathscr{X}},\rho,\mu)$ be an $m$-dimensional {\rm ADR} space for some $m\in(0,\infty)$ and suppose that $E$ is a Borel subset of $({\mathscr{X}},\tau_{\rho})$
with the property that there exists a Borel measure $\sigma$ on $(E,\tau_{\rho|_{E}})$
such that $(E,\rho\bigl|_E,\sigma)$ is a $d$-dimensional {\rm ADR} space for some $d\in(0,m)$. Then $\mu(E)=0$. \end{lemma}
\begin{proof} Fix $x\in E$. Using \eqref{PEY88-2BIS}, \eqref{TASS.bis2} and item {\it (6)} in Proposition~\ref{PWRS22}, we obtain
\begin{eqnarray}\label{bgSS-1} \mu(E) \leq C {\mathscr{H}}^m_{{\mathscr{X}},\rho_{\#}}(E) = C \lim_{n\rightarrow\infty} {\mathscr{H}}^m_{{\mathscr{X}},\rho_{\#}}(E\cap B_{\rho_{\#}}(x,n))=0, \end{eqnarray} since ${\mathscr{H}}^d_{{\mathscr{X}},\rho_{\#}}(E\cap B_{\rho_{\#}}(x,n))\leq Cn^d<\infty$ for all $n\in{\mathbb{N}}$. \end{proof}
\begin{lemma}\label{segj} Let $({\mathscr{X}},\rho)$ be a quasi-metric space. Suppose that $E$ is a Borel subset of $({\mathscr{X}},\tau_\rho)$ such that there exists a Borel measure $\sigma$
on $(E,\tau_{\rho|_{E}})$ with the property that $(E,\rho\bigl|_E,\sigma)$ is a $d$-dimensional {\rm ADR} space for some $d\in(0,\infty)$. Then there exists a constant $c\in(0,\infty)$ such that
\begin{eqnarray}\label{bgSS} \begin{array}{c} \forall\,x\in{\mathscr{X}},\,\,\forall\,r\in(0,{\rm diam}_{\rho_{\#}}(E)] \mbox{ with $B_{\rho_{\#}}(x,r)\cap E\not=\emptyset$ there holds} \\[4pt] {\mathscr{H}}_{{\mathscr{X}}\!,\,\rho_{\#}}^d\bigl(B_{\rho_{\#}}(x,C_\rho r)\cap E\bigr) \geq c\,r^d. \end{array} \end{eqnarray} \end{lemma}
\begin{proof} Fix a point $x\in{\mathscr{X}}$ with the property that $B_{\rho_{\#}}(x,r)\cap E\not=\emptyset$. If we now select $y\in B_{\rho_{\#}}(x,r)\cap E$ then $B_{\rho_{\#}}(y,r)\subseteq B_{\rho_{\#}}(x,C_\rho r)$. Recall \eqref{TASS.bis2}
and let $C$ be the {\rm ADR} constant of $\bigl(E,\rho_{\#}\bigl|_{E}, {\mathscr{H}}^d_{{\mathscr{X}}\!,\,\rho_{\#}}\lfloor E\bigr)$. Then, since $y\in E$,
\begin{eqnarray}\label{ZHvd} {\mathscr{H}}_{{\mathscr{X}}\!,\,\rho_{\#}}^d \bigl(B_{\rho_{\#}}(x,C_\rho r)\cap E\bigr) \geq {\mathscr{H}}_{{\mathscr{X}}\!,\,\rho_{\#}}^d \bigl(B_{\rho_{\#}}(y,r)\cap E\bigr)\geq C^{-1}r^d. \end{eqnarray}
Hence \eqref{bgSS} holds with $c:=C^{-1}$. \end{proof}
Following work in \cite{Christ} and \cite{David1988}, we now discuss the existence of a {\tt dyadic grid structure} on geometrically doubling quasi-metric spaces. The following result is essentially due to M. Christ \cite{Christ}, with two refinements. First, Christ's dyadic grid result is established in the presence of a background doubling, Borel regular measure, which is more restrictive than merely assuming that the ambient quasi-metric space is geometrically doubling. Second, Christ's dyadic grid result involves a scale $\delta\in(0,1)$ which we here show may be taken to be $\frac12$, as in the Euclidean setting.
\begin{proposition}\label{Diad-cube} Assume that $(E,\rho)$ is a geometrically doubling quasi-metric space and select $\kappa_E\in{\mathbb{Z}}\cup\{-\infty\}$ with the property that
\begin{eqnarray}\label{Jcc-KKi} 2^{-\kappa_E-1}< {\rm diam}_\rho(E)\leq 2^{-\kappa_E}. \end{eqnarray}
Then there exist finite constants $a_1\geq a_0>0$ such that for each $k\in{\mathbb{Z}}$ with $k\geq\kappa_E$, there exists a collection ${\mathbb{D}}_k(E):=\{Q_\alpha^k\}_{\alpha\in I_k}$ of subsets of $E$ indexed by a nonempty, at most countable set of indices $I_k$, as well as a family $\{x_\alpha^k\}_{\alpha\in I_k}$ of points in $E$, such that the collection of all {\tt dyadic cubes} in $E$, i.e.,
\begin{eqnarray}\label{gcEd} {\mathbb{D}}(E):=\bigcup_{k\in{\mathbb{Z}},\,k\geq\kappa_E}{\mathbb{D}}_k(E), \end{eqnarray}
has the following properties:
\begin{enumerate} \item[(1)] $[${\rm All dyadic cubes are open}$]$ \\ For each $k\in{\mathbb{Z}}$ with $k\geq\kappa_E$ and each $\alpha\in I_k$, the set $Q_\alpha^k$ is open in $\tau_\rho$;
\item[(2)] $[${\rm Dyadic cubes are mutually disjoint within the same generation}$]$ \\ For each $k\in{\mathbb{Z}}$ with $k\geq\kappa_E$ and each $\alpha,\beta\in I_k$ with $\alpha\not=\beta$ there holds $Q_\alpha^k\cap Q_\beta^k=\emptyset$;
\item[(3)] $[${\rm No partial overlap across generations}$]$ \\ For each $k,\ell\in{\mathbb{Z}}$ with $\ell>k\geq\kappa_E$, and each $\alpha\in I_k$, $\beta\in I_\ell$, either $Q_\beta^\ell\subseteq Q_\alpha^k$ or $Q_\alpha^k\cap Q_\beta^\ell=\emptyset$;
\item[(4)] $[${\rm Any dyadic cube has a unique ancestor in any earlier generation}$]$ \\ For each $k,\ell\in{\mathbb{Z}}$ with $k>\ell\geq\kappa_E$, and each $\alpha\in I_k$ there is a unique $\beta\in I_\ell$ such that $Q_\alpha^k\subseteq Q_\beta^\ell$;
\item[(5)] $[${\rm The size is dyadically related to the generation}$]$\\ For each $k\in{\mathbb{Z}}$ with $k\geq\kappa_E$ and each $\alpha\in I_k$ one has
\begin{eqnarray}\label{ha-GVV} B_{\rho}(x_\alpha^k,a_0 2^{-k})\subseteq Q_\alpha^k\subseteq B_{\rho}(x_\alpha^k,a_1 2^{-k}); \end{eqnarray}
In particular, given a measure $\sigma$ on $E$ for which $(E,\rho,\sigma)$ is a space of homogeneous type, there exists $c>0$ such that if $Q^{k+1}_\beta\subseteq Q^k_\alpha$, then $\sigma(Q^{k+1}_\beta)\geq c\sigma(Q^k_\alpha)$.
\item[(6)] $[${\rm Control of the number of children}$]$\\ There exists an integer $N\in{\mathbb{N}}$ with the property that for each $k\in{\mathbb{Z}}$ with $k\geq\kappa_E$ one has
\begin{eqnarray}\label{ha-GMM} \#\bigl\{\beta\in I_{k+1}:\,Q^{k+1}_\beta\subseteq Q^k_{\alpha}\bigr\}\leq N,\quad \mbox{ for every }\,\,\alpha\in I_{k}. \end{eqnarray}
Furthermore, this integer may be chosen such that, for each $k\in{\mathbb{Z}}$ with $k\geq\kappa_E$, each $x\in E$ and $r\in(0,2^{-k})$, the number of $Q$'s in ${\mathbb{D}}_k(E)$ that intersect $B_\rho(x,r)$ is at most $N$.
\item[(7)] $[${\rm Any generation covers a dense subset of the entire space}$]$\\ For each $k\in{\mathbb{Z}}$ with $k\geq\kappa_E$, the set $\bigcup_{\alpha\in I_k}Q_\alpha^k$ is dense in $(E,\tau_\rho)$. In particular, for each $k\in{\mathbb{Z}}$ with $k\geq\kappa_E$ one has
\begin{eqnarray}\label{ha-GMM.2} E=\bigcup_{\alpha\in I_k}\bigl\{x\in E:\,{\rm dist}_{\rho}(x,Q_\alpha^k) \leq\varepsilon 2^{-k}\bigr\},\qquad\forall\,\varepsilon>0, \end{eqnarray}
and there exist $b_0,b_1\in(0,\infty)$ depending only on the geometrically doubling character of $E$ with the property that
\begin{eqnarray}\label{ha-GL54} \begin{array}{c} \forall\,x_o\in E,\,\,\,\forall\,r\in(0,{\rm diam}_\rho(E)],\,\,\, \exists\,k\in{\mathbb{Z}}\,\mbox{ with }\,k\geq\kappa_E\,\mbox{ and }\, \exists\,\alpha\in I_k \\[4pt] \mbox{with the property that }\,Q^k_\alpha\subseteq B_{\rho}(x_o,r) \,\mbox{ and }\,b_0 r\leq 2^{-k}\leq b_1 r. \end{array} \end{eqnarray}
Moreover, for each $k\in{\mathbb{Z}}$ with $k\geq\kappa_E$ and each $\alpha\in I_k$
\begin{eqnarray}\label{ha-GMM.2JJ} \bigcup_{\beta\in I_{k+1},\,Q^{k+1}_\beta\subseteq Q^k_{\alpha}}Q^{k+1}_\beta \,\,\mbox{ is dense in }\,\,Q^k_{\alpha}, \end{eqnarray}
and
\begin{eqnarray}\label{ha-GMM.2II} Q^k_{\alpha}\subseteq \bigcup_{\beta\in I_{k+1},\,Q^{k+1}_\beta\subseteq Q^k_{\alpha}} \bigl\{x\in E:\,{\rm dist}_{\rho}(x,Q_\beta^{k+1}) \leq\varepsilon 2^{-k-1}\bigr\},\qquad\forall\,\varepsilon>0. \end{eqnarray}
\item[(8)] $[${\rm Dyadic cubes have thin boundaries with respect to a background doubling measure}$]$ \\ Given a measure $\sigma$ on $E$ for which $(E,\rho,\sigma)$ is a space of homogeneous type, a collection ${\mathbb{D}}(E)$ may be constructed as in \eqref{gcEd} such that properties {\it (1)-(7)} above hold and, in addition, there exist constants $\vartheta\in(0,1)$ and $c\in(0,\infty)$ such that for each $k\in{\mathbb{Z}}$ with $k\geq\kappa_E$ and each $\alpha\in I_k$ one has
\begin{eqnarray}\label{hacc-76es} \sigma \left(\bigl\{x\in Q_\alpha^k:\,{\rm dist}_{\rho_{\#}}(x,E\setminus Q_\alpha^k) \leq t\,2^{-k}\bigr\}\right)\leq c\,t^\vartheta\sigma(Q_\alpha^k),\qquad\forall\,t>0. \end{eqnarray}
Moreover, in such a context matters may be arranged so that, for each $k\in{\mathbb{Z}}$ with $k\geq\kappa_E$ and each $\alpha\in I_k$,
\begin{eqnarray}\label{ihgc}
\bigl(Q^k_\alpha,\rho|_{Q^k_\alpha},\sigma\lfloor{Q^k_\alpha}\bigr) \quad\mbox{ is a space of homogeneous type}, \end{eqnarray}
and the doubling constant of the measure $\sigma\lfloor{Q^k_\alpha}$ is independent of $k,\alpha$ (i.e., {\rm the quality of being a space of homogeneous type is hereditary at the level of dyadic cubes, in a uniform fashion}).
\item[(9)] $[${\rm All generations cover the space a.e. with respect to a doubling Borel regular measure}$]$\\ If $\sigma$ is a Borel measure on $E$ which is both doubling (cf. \eqref{Doub-1}) and Borel regular (cf. \eqref{T-gc2w}) then a collection ${\mathbb{D}}(E)$ associated with the doubling measure $\sigma$ as in {\it (8)} may be constructed with the additional property that
\begin{eqnarray}\label{T-rcs} \sigma\Bigl(E\setminus\bigcup_{\alpha\in I_k}Q_\alpha^k\Bigr)=0 \qquad\mbox{for each }\,\,k\in{\mathbb{Z}},\,\,k\geq\kappa_E. \end{eqnarray}
In particular, in such a setting, for each $k\in{\mathbb{Z}}$ with $k\geq\kappa_E$ one has
\begin{eqnarray}\label{ha-GMM.67} \sigma\Bigl(Q^k_{\alpha}\setminus \bigcup\limits_{\beta\in I_{k+1},\,Q^{k+1}_\beta\subseteq Q^k_{\alpha}} Q^{k+1}_\beta\Bigr)=0,\quad\mbox{ for every }\,\,\alpha\in I_{k}. \end{eqnarray} \end{enumerate} \end{proposition}
Before discussing the proof of this result we wish to say a few words clarifying terminology. As already mentioned in the statement, sets $Q$ belonging to ${\mathbb{D}}(E)$ will be referred to as {\it dyadic cubes} (on $E$). Also, following a well-established custom, whenever $Q_\alpha^{k+1}\subseteq Q_\beta^k$ we shall call $Q_\alpha^{k+1}$ a {\it child} of $Q_\beta^{k}$, and we shall say that $Q_\beta^{k}$ is a {\it parent} of $Q_\alpha^{k+1}$. For a given dyadic cube, an {\it ancestor} is then a parent, or a parent of a parent, or so on. Moreover, for each $k\in{\mathbb{Z}}$ with $k\geq\kappa_E$, we shall call ${\mathbb{D}}_k(E)$ the {\it dyadic cubes of generation} $k$ and, for each $Q\in{\mathbb{D}}_k(E)$, define the {\it side-length} of $Q$ to be $\ell(Q):=2^{-k}$, and the {\it center} of $Q$ to be the point $x^k_\alpha\in E$ if $Q=Q^k_\alpha$.
Finally, we make the convention that saying that {\it ${\mathbb{D}}(E)$ is a dyadic cube structure (or dyadic grid) on $E$} will always indicate that the collection ${\mathbb{D}}(E)$ is associated with $E$ as in Proposition~\ref{Diad-cube}. This presupposes that $E$ is the ambient set for a geometrically doubling quasi-metric space, in which case ${\mathbb{D}}(E)$ satisfies properties {\it (1)-(7)} above and that, in the presence of a background measure $\sigma$ satisfying appropriate conditions (as stipulated in Proposition~\ref{Diad-cube}), properties {\it (8)} and {\it (9)} also hold.
We are now ready to proceed with the
\vskip 0.08in \begin{proof}[Proof of Proposition~\ref{Diad-cube}] This is a slight extension and clarification of a result proved by M.~Christ in \cite{Christ}, generalizing earlier work by G. David in \cite{David1988}, and we will limit ourselves to discussing only the novel aspects of the present formulation. For the sake of reference, we debut by recalling the main steps in the construction in \cite{Christ}. For a fixed real number $\delta\in(0,1)$ and for any integer $k$, Christ considers a maximal collection of points $z^k_\alpha\in E$ such that
\begin{eqnarray}\label{centers} \rho_{\#}(z^k_\alpha,z^k_\beta)\geq\delta^k,\qquad\forall\,\alpha\not=\beta. \end{eqnarray}
Hence, for each fixed $k$, the set $\{z^k_\alpha\}_{\alpha}$ is $\delta^k$-dense in $E$ in the sense that for each $k\in{\mathbb{Z}}$ and $x\in E$ there exists $\alpha$ such that $\rho_{\#}(x,z^k_\alpha)<\delta^k$. Then (cf. \cite[Lemma~13, p.\,8]{Christ}) there exists a partial order relation $\preceq$ on the set $\{(k,\alpha):\,k\in{\mathbb{Z}},\alpha\in I_k\}$ with the following properties:
1) if $(k,\alpha)\preceq(l,\beta)$ then $k\geq l$;
2) for each $(k,\alpha)$ and $l\leq k$ there exists a unique $\beta$ such that $(k,\alpha)\preceq(l,\beta)$;
3) if $(k,\alpha)\preceq(k-1,\beta)$ then $\rho_{\#}(z^k_\alpha,z^{k-1}_\beta)<\delta^{k-1}$;
4) if $\rho_{\#}(z^l_\beta,z^k_\alpha)\leq 2C_\rho\delta^k$ then $(l,\beta)\preceq(k,\alpha)$.
\noindent Having established this, Christ then chooses a number $c\in(0,\frac{1}{2C_\rho})$ and defines $$ Q^k_\alpha:=\bigcup_{(l,\beta)\preceq(k,\alpha)}B_{\rho_{\#}}(z^l_\beta,c\delta^{\,l}). $$
First, the dyadic cubes in \cite[Theorem~11, p.\,7]{Christ} are labeled over all $k\in{\mathbb{Z}}$. However, \eqref{ha-GVV} shows that in the case when $E$ is bounded the index set $I_k$ becomes a singleton whenever $2^{-k}$ is sufficiently large. Hence, in particular, ${\mathbb{D}}_k(E)$ becomes stationary as $k$ approaches $-\infty$, in the sense that this collection of cubes reduces to just the set $E$ provided $2^{-k}$ is sufficiently large. While this is not an issue in and of itself, for later considerations we find it useful to eliminate this redundancy and this is the reason why we restrict ourselves to only $k\geq\kappa_E$.
Second, \cite[Theorem~11, p.\,7]{Christ} is stated with $\delta^k$ replacing $2^{-k}$ in \eqref{ha-GVV}-\eqref{hacc-76es}, for some $\delta\in(0,1)$. The reason why we may always assume that $\delta=1/2$ is discussed later below. Third, Christ's result just mentioned is formulated in the setting of spaces of homogeneous type (equipped with symmetric quasi-distance), but a cursory inspection of the proof reveals that for properties {\it (1)-(6)} in our statement the same type of arguments as in \cite[pp.\,7-10]{Christ} go through (working with the regularization $\rho_{\#}$ of $\rho$, as in Theorem~\ref{JjEGh}) under the weaker assumption that $(E,\rho)$ is a geometrically doubling quasi-metric space.
Fourth, {\it (7)} follows from a careful inspection of the proof of \cite[Theorem~11, p.\,7]{Christ}, which reveals that for each $k\in{\mathbb{Z}}$ with $k\geq\kappa_E$, and any $j\in{\mathbb{N}}$ sufficiently large (compared to $k$) the set $\bigcup_{\alpha\in I_k}Q_\alpha^k$ contains a $2^{-j}$-dense subset of $E$ that is maximal with respect to inclusion. Of course, this shows that the union in question is dense in $(E,\tau_\rho)$, and \eqref{ha-GMM.2} is a direct consequence of it.
Fifth, with the exception of using the regularization $\rho_{\#}$ of the original quasi-distance $\rho$ from Theorem~\ref{JjEGh} in place of the regularization devised in \cite{MaSe79}, property {\it (8)} is identical to condition (3.6) in \cite[Theorem~11, p.\,7]{Christ}. Sixth, property {\it (9)} corresponds to (3.1) in \cite[Theorem~11, p.\,7]{Christ} except that we are presently assuming that the doubling measure $\sigma$ is Borel regular. The reason for this assumption is that the proof of (3.1) in \cite[Theorem~11, p.\,7]{Christ} uses the Lebesgue Differentiation Theorem, whose proof requires that continuous functions vanishing outside bounded subsets of $E$ are dense in $L^1(E,\sigma)$. It is precisely here that the aforementioned regularity of the measure intervenes and the reader is referred to \cite[Theorem~7.10]{MMMM-G} for a density result of this nature.
The remainder of this proof consists of a verification that, compared with \cite{Christ}, it is always possible to take $\delta=1/2$ (as described in the first paragraph of this proof). In the process, we shall adopt Christ's convention of labeling the dyadic cubes over all $k\in{\mathbb{Z}}$ (eliminating the inherent redundancy in the case when $E$ is bounded may be done afterwards). To get started, let $\mathfrak{D}(E):=\bigcup\limits_{k\in{\mathbb{Z}}}\mathfrak{D}_k(E)$ denote a collection of dyadic cubes enjoying properties {\it (1)-(9)} listed in Proposition~\ref{Diad-cube} but with $\delta^k$ replacing $2^{-k}$ in \eqref{ha-GVV}-\eqref{hacc-76es}. Our goal in this part of the proof is to construct another collection of dyadic cubes, ${\mathbb{D}}(E):=\bigcup\limits_{k\in{\mathbb{Z}}}{\mathbb{D}}_k(E)$ satisfying similar properties for $\delta=1/2$. We shall consider two cases.
{\it Case~I: $1/2<\delta<1$.} Set $m_0:=0$ and, for each integer $k>0$, let $m_k$ be the largest positive integer such that $\delta^{m_k}\geq 2^{-k}$. Thus,
\begin{eqnarray}\label{HS1} \delta^{m_k+1}<2^{-k}\leq \delta^{m_k}. \end{eqnarray}
Similarly, for each $k<0$, let $m_k$ denote the least integer such that $\delta^{m_k+1}< 2^{-k}$. Thus, again we have \eqref{HS1}. Of course, we shall have $m_k<0$ when $k<0$. The sequence $\{m_k\}_{k\in{\mathbb{Z}}}$ is strictly increasing. Indeed, for every $k\in \mathbb{Z}$, we have
\begin{eqnarray}\label{HS2} m_k+1\leq m_{k+1}. \end{eqnarray}
To see this in the case that $k\geq 0$, observe that
\begin{eqnarray}\label{HS3} 2^{-k-1}=\tfrac{1}{2}2^{-k}\leq\tfrac{1}{2}\delta^{m_k}<\delta^{m_k+1}, \end{eqnarray}
where in the first inequality we have used \eqref{HS1} and in the second that $1/2<\delta$. Thus, \eqref{HS2} holds, since by definition $m_{k+1}$ is the greatest integer for which $2^{-(k+1)}\leq\delta^{m_{k+1}}$. In the case $k\leq 0$, since $1<2\delta$ we have
\begin{eqnarray}\label{HS4} \delta^{(m_{k+1}-1)+1}<2\delta^{m_{k+1}+1}<2\cdot2^{-(k+1)}=2^{-k} \end{eqnarray}
where in the second inequality we have used \eqref{HS1}. Since $m_k$ is the smallest integer for which $\delta^{m_k+1}<2^{-k}$, we again obtain \eqref{HS2}.
We then define
\begin{eqnarray}\label{HS5} {\mathbb{D}}_k(E):=\mathfrak{D}_{m_k}(E). \end{eqnarray}
It is routine to verify that ${\mathbb{D}}_k(E)$ satisfies the desired properties, with some of the constants possibly depending upon $\delta$.
{\it Case~II: $0<\delta<1/2$.} In this case we reverse the roles of $1/2$ and $\delta$ in the construction in Case~I above, to construct a strictly increasing sequence of integers $\{m_k\}_{k\in{\mathbb{Z}}}$, with $m_0:=0$, for which
\begin{eqnarray}\label{2HHSS} 2^{-m_k}\leq \delta^{k}<2^{-m_k+1},\qquad\forall\,k\in{\mathbb{Z}}. \end{eqnarray}
It then follows that there is a fixed positive integer $q_0\approx\log_2(1/\delta)$ such that for each $k\in{\mathbb{Z}}$,
\begin{eqnarray}\label{3HHSS} m_{k+1}-q_0\leq m_k<m_{k+1}. \end{eqnarray}
Indeed, we have
\begin{eqnarray}\label{4HHSS} 2^{-m_k}\leq\delta^k =\frac1\delta\delta^{k+1}<\frac1\delta 2^{-m_{k+1}+1} =\frac2\delta 2^{-m_{k+1}}, \end{eqnarray}
where in the two inequalities we have used \eqref{2HHSS}. We then obtain \eqref{3HHSS} by taking logarithms. For each $k\in\mathbb{Z}$ we now set
\begin{eqnarray}\label{6HHSS} {\mathbb{D}}_j(E):=\mathfrak{D}_k(E),\qquad m_k\leq j<m_{k+1}. \end{eqnarray}
It is now routine to check that the collection ${\mathbb{D}}(E):=\bigcup\limits_{k\in{\mathbb{Z}}}{\mathbb{D}}_k(E)$, so defined, satisfies the desired properties, with some of the constants possibly depending on $\delta$. In verifying the various properties, it is helpful to observe that by \eqref{3HHSS}, we have that
\begin{eqnarray}\label{7HHSS} 2^{-j}\approx 2^{-m_k}\approx \delta^k,\qquad \mbox{whenever $m_k\leq j<m_{k+1}$}. \end{eqnarray}
This finishes the proof of the proposition. \end{proof}
\subsection{Approximations to the identity on quasi-metric spaces} \label{SSect:2.3}
This subsection is devoted to reviewing the definition and properties of approximations to the identity on {\rm ADR} spaces. To set the stage, we make the following definition.
\begin{definition}\label{Besov-S} Assume that $(E,\rho,\sigma)$ is a $d$-dimensional {\rm ADR} space for some $d>0$ and recall $\kappa_E\in{\mathbb{Z}}\cup\{-\infty\}$ from \eqref{Jcc-KKi}. In this context, call a family $\{{\mathcal{S}}_l\}_{l\in{\mathbb{Z}},\,l\geq \kappa_E}$ of integral operators
\begin{eqnarray}\label{Taga-6} {\mathcal{S}}_l f(x):=\int_{E}S_l(x,y)f(y)\,d\sigma(y),\qquad x\in E, \end{eqnarray}
with integral kernels $S_l:E\times E\to{\mathbb{R}}$, an {\tt approximation to the identity of order} $\gamma$ on $E$ provided there exists a finite constant $C>0$ such that, for every $l\in{\mathbb{Z}}$ with $l\geq \kappa_E$, the following properties hold:
\begin{enumerate} \item[(i)] $0\leq S_l(x,y)\leq C 2^{ld}$ for all $x,y\in E$, and $S_l(x,y)=0$ if $\rho(x,y)\geq C2^{-l}$;
\item[(ii)] $|S_l(x,y)-S_l(x',y)|\leq C 2^{l(d+\gamma)}\rho(x,x')^{\gamma}$ for every $x,x',y\in E$;
\item[(iii)] $\bigl|[S_l(x,y)-S_l(x',y)]-[S_l(x,y')-S_l(x',y')]\bigr| \leq C2^{l(d+2\gamma)}\rho(x,x')^{\gamma}\rho(y,y')^{\gamma}$ for every point $x,x',y,y'\in E$;
\item[(iv)] $S_l(x,y)=S_l(y,x)$ for every $x,y\in E$, and $\int_{E}S_l(x,y)\,d\sigma(y)=1$ for every $x\in E$. \end{enumerate} \end{definition}
Starting with the work of Coifman (cf. the discussion in \cite[pp.\,16-17 and p.\,40]{DJS}), the existence of approximations to the identity of some order $\gamma>0$ on {\rm ADR} spaces has been established in \cite[p.\,40]{DJS}, \cite[pp.\,10-11]{HaSa94}, \cite[p.\,16]{DeHa09} (at least when $d=1$) for various values of $\gamma>0$ and, more recently, in \cite{MMMM-G} for the value of the order parameter $\gamma$ which is optimal in relation to the quasi-metric space structure. From \cite{MMMM-G}, we quote the following result:
\begin{proposition}\label{Besov-ST} Let $(E,\rho,\sigma)$ be a $d$-dimensional {\rm ADR} space for some $d>0$ and assume that
\begin{eqnarray}\label{TFv-5tG} 0<\gamma<\min\bigl\{d+1,\alpha_\rho\bigr\}, \end{eqnarray}
where the index $\alpha_\rho\in(0,\infty]$ is associated to the quasi-distance $\rho$ as in \eqref{Cro}. Then, in the sense of Definition~\ref{Besov-S}, there exists an approximation to the identity of order $\gamma$ on $E$, denoted by $\{{\mathcal{S}}_l\}_{l\in {\mathbb{Z}},\,l\geq \kappa_E}$. Furthermore, given $p\in(1,\infty)$ and $f\in L^p(E,\sigma)$, it follows that:
\begin{eqnarray}\label{fcc-5tnew} \sup\limits_{l\in{\mathbb{Z}},\,l\geq \kappa_E}
\bigl\|{\mathcal{S}}_l\bigr\|_{L^p(E,\sigma)\to L^p(E,\sigma)}<+\infty, \end{eqnarray}
\begin{eqnarray}\label{fcc-5t} \mbox{if the measure $\sigma$ is Borel regular on $(E,\tau_{\rho})$} \,\,\Longrightarrow\,\, \lim_{l\to +\infty}{\mathcal{S}}_lf=f\,\,\,\mbox{ in $L^p(E,\sigma)$}, \end{eqnarray}
and
\begin{eqnarray}\label{fcc-5t2} \mbox{if } {\rm diam}_\rho(E)=+\infty\,\,\Longrightarrow\,\, \lim_{l\to -\infty}{\mathcal{S}}_lf=0\,\,\,\mbox{ in $L^p(E,\sigma)$}. \end{eqnarray} \end{proposition}
Later on we shall need a Calder\'on-type reproducing formula involving the conditional expectation operators associated with an approximation to the identity, as discussed above. While this is a topic treated at some length in \cite{DJS}, \cite{DeHa09}, \cite{HaSa94}, we prove below a version of this result which best suits the purposes we have in mind.
To state the result, we first record the following preliminaries.
\begin{definition}\label{def:uncond} A series $\sum\limits_{j\in{\mathbb{N}}}x_j$ of vectors in a Banach space ${\mathscr{B}}$ is said to be {\tt unconditionally convergent} if the series $\sum\limits_{j=1}^\infty x_{\sigma(j)}$ converges in ${\mathscr{B}}$ for all permutations $\sigma$ of ${\mathbb{N}}$. \end{definition}
Clearly, if a series $\sum_{j\in{\mathbb{N}}}x_j$ of vectors in a Banach space ${\mathscr{B}}$ is unconditionally convergent then so is $\sum\limits_{j=1}^\infty x_{\sigma(j)}$ for any permutation $\sigma$ of ${\mathbb{N}}$. It is also well-known (cf., e.g., \cite[Corollary~3.11, p.\,99]{Heil}) that, given a sequence of vectors $\{x_j\}_{j\in{\mathbb{N}}}$ in a Banach space ${\mathscr{B}}$,
\begin{eqnarray}\label{UNC-11} && \hskip -0.50in \mbox{$\displaystyle\sum\limits_{j\in{\mathbb{N}}}x_j$ unconditionally convergent} \nonumber\\[-4pt] && \hskip 0.50in \Longrightarrow\, \sum\limits_{j=1}^\infty x_{\sigma_1(j)}=\sum\limits_{j=1}^\infty x_{\sigma_2(j)}, \qquad\mbox{ $\forall\,\sigma_1,\sigma_2$ permutations of ${\mathbb{N}}$}. \end{eqnarray}
Hence, whenever $\sum\limits_{j\in{\mathbb{N}}}x_j$ is unconditionally convergent, we may unambiguously define
\begin{eqnarray}\label{UNC-11.BBB} \sum\limits_{j\in{\mathbb{N}}}x_j:=\sum\limits_{j=1}^\infty x_{\sigma(j)} \,\,\mbox{ for some (hence any) permutation $\sigma$ of ${\mathbb{N}}$}. \end{eqnarray}
Let us also record here the following useful characterizations of unconditional convergence (in a Banach space setting):
\begin{eqnarray}\label{UNC-12} \mbox{$\displaystyle\sum\limits_{j\in{\mathbb{N}}}x_j$ unconditionally convergent} &\Longleftrightarrow & \mbox{$\displaystyle\sum\limits_{j=1}^\infty\varepsilon_jx_j$ convergent} \,\,\mbox{ $\forall\,\varepsilon_j=\pm 1$} \\[4pt] &\Longleftrightarrow & \left\{ \begin{array}{l} \forall\,\varepsilon>0\,\,\exists\,N_{\varepsilon}\in{\mathbb{N}}\mbox{ such that }
\Bigl\|\displaystyle\sum\limits_{j\in{\mathcal{I}}}x_j\Bigr\|<\varepsilon \\[4pt] \forall\,{\mathcal{I}} \mbox{ finite subset of ${\mathbb{N}}$ with $\min\,{\mathcal{I}}\geq N_{\varepsilon}$}. \end{array} \right. \nonumber \end{eqnarray}
See, e.g., \cite[Theorem~3.10, p.\,94]{Heil} where these and other equivalent characterizations are proved.
The following notion of unconditional convergence applies to series indexed by any countable set other than ${\mathbb{N}}$.
\begin{definition}\label{def:uncond2} For any countable set ${\mathbb{I}}$, a series $\sum_{j\in{\mathbb{I}}}x_j$ of vectors in a Banach space ${\mathscr{B}}$ is said to be {\tt unconditionally convergent} if there exists a bijection $\varphi:{\mathbb{N}}\rightarrow{\mathbb{I}}$ such that $\sum_{j\in{\mathbb{N}}}x_{\varphi(j)}$ is unconditionally convergent in the sense of Definition~\ref{def:uncond}, in which case the sum of the series in ${\mathscr{B}}$ is defined as $\sum_{j\in{\mathbb{I}}}x_j:= \sum\limits_{j=1}^\infty x_{\varphi(j)}$. \end{definition}
Note that the property of being unconditionally convergent as introduced in Definition~\ref{def:uncond2} is independent of the bijection ${\varphi}$ used. To see this, suppose that $\sum_{j\in{\mathbb{I}}}x_j$ is unconditionally convergent in ${\mathscr{B}}$ and let $\varphi:{\mathbb{N}}\rightarrow{\mathbb{I}}$ be a bijection such that $\sum_{j\in{\mathbb{N}}}x_{\varphi(j)}$ is unconditionally convergent in the sense of Definition~\ref{def:uncond}. If $\widetilde{\varphi}:{\mathbb{N}}\rightarrow{\mathbb{I}}$ is another bijection, then $\varphi^{-1}\circ\widetilde{\varphi}$ is a permutation of ${\mathbb{N}}$ hence, as noted right after Definition~\ref{def:uncond}, $\sum_{j\in{\mathbb{N}}}x_{\widetilde{\varphi}(j)}$ is also unconditionally convergent. Moreover, \eqref{UNC-11} ensures that $\sum\limits_{j=1}^\infty x_{\widetilde{\varphi}(j)} =\sum\limits_{j=1}^\infty x_{\varphi(\varphi^{-1}(\widetilde{\varphi}(j)))} =\sum\limits_{j=1}^\infty x_{\varphi(j)} =\sum_{j\in{\mathbb{I}}}x_j$.
We also have the following equivalent characterization for unconditional convergence.
\begin{lemma}\label{SBFF} Suppose ${\mathscr{B}}$ is a Banach space and ${\mathbb{I}}$ is a countable set. Then a series $\sum_{j\in{\mathbb{I}}}x_j$ of vectors in ${\mathscr{B}}$ is unconditionally convergent in ${\mathscr{B}}$ if and only if
\begin{eqnarray}\label{S+SS} \begin{array}{c} \forall\,\{S_i\}_{i\in{\mathbb{N}}}\mbox{ such that $S_i$ finite and $S_i\subseteq S_{i+1}\subseteq{\mathbb{I}}$ for each $i\in{\mathbb{N}}$}, \\[4pt] \mbox{the sequence }\,\,\bigl\{\sum\limits_{j\in S_i}x_j\bigr\}_{i\in{\mathbb{N}}} \mbox{ converges in ${\mathscr{B}}$}. \end{array} \end{eqnarray} \end{lemma}
\begin{proof} Suppose $\sum_{j\in{\mathbb{I}}}x_j$ is such that \eqref{S+SS} holds and let $\varphi:{\mathbb{N}}\to{\mathbb{I}}$ be a bijection. Also fix an arbitrary permutation $\sigma:{\mathbb{N}}\to{\mathbb{N}}$. Then the sequence $S_i:=\{\varphi(\sigma(k)):\,1\leq k\leq i\}$, $i\in{\mathbb{N}}$, of subsets of ${\mathbb{I}}$ satisfies the conditions in the first line of \eqref{S+SS}. Hence, $\bigl\{\sum\limits_{j=1}^i x_{\varphi(\sigma(j))}\bigr\}_{i\in{\mathbb{N}}}$ is convergent in ${\mathscr{B}}$, which is equivalent with $\sum\limits_{j=1}^\infty x_{\varphi(\sigma(j))}$ being convergent in ${\mathscr{B}}$. Since the permutation $\sigma$ of ${\mathbb{N}}$ has been arbitrarily chosen, this shows that $\sum_{j\in{\mathbb{N}}} x_{\varphi(j)}$ is unconditionally convergent in ${\mathscr{B}}$, thus $\sum_{j\in{\mathbb{I}}}x_j$ is unconditionally convergent in ${\mathscr{B}}$. For the converse implication, suppose that $\sum_{j\in{\mathbb{I}}}x_j$ is unconditionally convergent in ${\mathscr{B}}$. Thus, for any bijection $\varphi:{\mathbb{N}}\to{\mathbb{I}}$ we have that $\sum_{j\in{\mathbb{N}}}x_{\varphi(j)}$ is unconditionally convergent in ${\mathscr{B}}$. Let $\{S_i\}_{i\in{\mathbb{N}}}$ be as in the first line of \eqref{S+SS} and set $S:=\bigcup\limits_{i\in{\mathbb{N}}}S_i$. Using \eqref{UNC-12}, it follows that $\sum\limits_{j\in{\mathbb{N}}\setminus\varphi^{-1}({\mathbb{I}}\setminus S)}x_{\varphi(j)}$ is also unconditionally convergent in ${\mathscr{B}}$. In turn, the latter readily implies that $\bigl\{\sum\limits_{j\in S_i}x_j\bigr\}_{i\in{\mathbb{N}}}$ is convergent in ${\mathscr{B}}$, as wanted. \end{proof}
We now state the aforementioned Calder\'on-type reproducing formula.
\begin{proposition}\label{HS-PP.3} Let $(E,\rho,\sigma)$ be a $d$-dimensional {\rm ADR} space for some $d>0$ and assume that the measure $\sigma$ is Borel regular on $(E,\tau_\rho)$. In this context, recall $\kappa_E$ from \eqref{Jcc-KKi} and, for some fixed $\gamma$ as in \eqref{TFv-5tG}, let $\{{\mathcal{S}}_l\}_{l\in{\mathbb{Z}},\,l\geq \kappa_E}$ be an approximation to the identity of order $\gamma$ on $E$ (cf. Proposition~\ref{Besov-ST}). Finally, introduce the integral operators (see \cite{DJS})
\begin{eqnarray}\label{opD} D_l:={\mathcal{S}}_{l+1}-{\mathcal{S}}_l,\quad l\in{\mathbb{Z}},\,\,\,l\geq \kappa_E. \end{eqnarray}
Then there exist a linear and bounded operator $R$ on $L^2(E,\sigma)$ and a family $\bigl\{\widetilde{D}_l\bigr\}_{l\in{\mathbb{Z}},\,l\geq \kappa_E}$ of linear operators on $L^2(E,\sigma)$ with the property that
\begin{eqnarray}\label{PKc-2}
\sum\limits_{l\in{\mathbb{Z}},\,l\geq \kappa_E}\|\widetilde{D}_lf\|^2_{L^2(E,\sigma)}
\leq C\|f\|^2_{L^2(E,\sigma)},\quad\mbox{ for each }\,\,f\in L^2(E,\sigma), \end{eqnarray}
and, with $I$ denoting the identity operator on $L^2(E,\sigma)$,
\begin{eqnarray}\label{PKc} I+{\mathcal{S}}_{\kappa_E}R=\sum_{l\in{\mathbb{Z}},\, l\geq \kappa_E}D_l\widetilde{D}_l \qquad\mbox{pointwise unconditionally in $L^2(E,\sigma)$}, \end{eqnarray}
with the convention (taking effect when ${\rm diam}_\rho(E)=+\infty$) that ${\mathcal{S}}_{-\infty}:=0$. \end{proposition}
As a preamble to the proof of the above proposition we momentarily digress and record a version of the Cotlar-Knapp-Stein lemma which suits our purposes.
\begin{lemma}\label{L-CKS} Assume that ${\mathscr{H}}_0$, ${\mathscr{H}}_1$ are two Hilbert spaces and consider a family of operators $\{T_j\}_{j\in{\mathbb{I}}}$, indexed by a countable set ${\mathbb{I}}$, with $T_j:{\mathscr{H}}_0\to{\mathscr{H}}_1$ linear and bounded for every $j\in{\mathbb{I}}$. Then, if the $T_j$'s are almost orthogonal in the sense that
\begin{eqnarray}\label{Gvvv-42E} C_0:=\sup_{j\in{\mathbb{I}}}\Bigl(\sum_{k\in{\mathbb{I}}} \sqrt{\Vert T_j^\ast T_k\Vert_{{\mathscr{H}}_0\to{\mathscr{H}}_0}}\,\Bigr)<\infty, \quad C_1:=\sup_{k\in{\mathbb{I}}}\Bigl(\sum_{j\in{\mathbb{I}}} \sqrt{\Vert T_j T_k^\ast\Vert_{{\mathscr{H}}_1\to{\mathscr{H}}_1}}\,\Bigr)<\infty \end{eqnarray}
it follows that for any subset $J$ of ${\mathbb{I}}$,
\begin{eqnarray}\label{Gvvv-43E} \begin{array}{c} \mbox{$\displaystyle\sum_{j\in J}T_jx$ converges unconditionally in ${\mathscr{H}}_1$ for each $x\in{\mathscr{H}}_0$, and} \\[6pt] \mbox{if $\displaystyle\Bigl(\sum_{j\in J}T_j\Bigr)x:=\sum_{j\in J}T_jx$ then }\,\, \displaystyle\Bigl\Vert\sum_{j\in J}T_j\Bigr\Vert_{{\mathscr{H}}_0\to{\mathscr{H}}_1} \leq\sqrt{C_0C_1}. \end{array} \end{eqnarray}
Furthermore,
\begin{eqnarray}\label{Gvvv-43Ej}
\Bigl(\sum_{j\in{\mathbb{I}}}\|T_jx\|_{{\mathscr{H}}_1}^2\Bigr)^{1/2}
\leq 2\sqrt{C_0C_1}\|x\|_{{\mathscr{H}}_0},\qquad\forall\,x\in{\mathscr{H}}_0. \end{eqnarray} \end{lemma}
\begin{proof} This result is typically stated with $J$ finite and without including \eqref{Gvvv-43Ej}.
See, for example, \cite[Lemma~4.1, p.\,285]{Tor86} as well as \cite[Theorem~1, p.280 and comment following it]{STEIN}. The fact that the more general version formulated above holds is an immediate consequence of the standard version of the Cotlar-Knapp-Stein lemma as stated in the aforementioned references and the abstract functional analytic result contained in Lemma~\ref{Tgv-x99} below. \end{proof}
\begin{lemma}\label{Tgv-x99}
Let ${\mathscr{H}}$ be a Hilbert space with norm $\|\cdot\|_{\mathscr{H}}$ and assume that $\{x_j\}_{j\in{\mathbb{I}}}$ is a sequence of vectors in ${\mathscr{H}}$ indexed by a countable set ${\mathbb{I}}$. Then
\begin{eqnarray}\label{Hc-77.UU}
\Bigl(\sum\limits_{j\in{\mathbb{I}}}\|x_j\|_{\mathscr{H}}^2\Bigr)^{1/2} \leq 2\cdot \sup\limits_{\stackrel{J_o\subseteq{\mathbb{I}}}{J_o\,\mbox{\tiny{finite}}}}
\Bigl\|\sum\limits_{j\in J_o}x_j\Bigr\|_{\mathscr{H}}, \end{eqnarray}
and
\begin{eqnarray}\label{FAFFF} \mbox{$\displaystyle\sum\limits_{j\in{\mathbb{I}}}x_j$ is unconditionally convergent}\, \Longleftrightarrow\, \sup\limits_{\stackrel{J_o\subseteq{\mathbb{I}}}{J_o\,\mbox{\tiny{finite}}}}
\Bigl\|\sum\limits_{j\in J_o}x_j\Bigr\|_{\mathscr{H}}<\infty. \end{eqnarray}
Moreover, if the above supremum is finite, then
\begin{eqnarray}\label{TGbb-88u}
\Bigl\|\sum\limits_{j\in{\mathbb{I}}}x_j\Bigr\|_{\mathscr{H}}\leq \sup\limits_{\stackrel{J_o\subseteq{\mathbb{I}}}{J_o\,\mbox{\tiny{finite}}}}
\Bigl\|\sum\limits_{j\in J_o}x_j\Bigr\|_{\mathscr{H}}. \end{eqnarray} \end{lemma}
\begin{proof} It suffices to assume that ${\mathbb{I}}={\mathbb{N}}$. This follows from Definition~\ref{def:uncond2}, since for any bijection $\varphi: {\mathbb{N}} \rightarrow {\mathbb{I}}$, we have
\begin{eqnarray}\label{eq:NIequiv} \sup\limits_{\stackrel{J_o\subseteq{\mathbb{N}}}{J_o\,\mbox{\tiny{finite}}}}
\Bigl\|\sum\limits_{n\in J_o}x_{\varphi(n)}\Bigr\|_{\mathscr{H}} =\sup\limits_{\stackrel{J_o\subseteq{\mathbb{I}}}{J_o\,\mbox{\tiny{finite}}}}
\Bigl\|\sum\limits_{j\in J_o}x_j\Bigr\|_{\mathscr{H}}. \end{eqnarray}
We begin by establishing \eqref{Hc-77.UU}. To get started, let $\{x_j\}_{j\in{\mathbb{N}}}\subseteq{\mathscr{H}}$ be such that
\begin{eqnarray}\label{Hc-77.A} C:=\sup\limits_{\stackrel{J_o\subseteq{\mathbb{N}}}{J_o\,\mbox{\tiny{finite}}}}
\Bigl\|\sum\limits_{j\in J_o}x_j\Bigr\|_{\mathscr{H}}<\infty. \end{eqnarray}
Assume that $\{r_j\}_{j\in{\mathbb{N}}}$ is Rademacher's system of functions on $[0,1]$, i.e., for each $j\in{\mathbb{N}}$,
\begin{eqnarray}\label{Hc-77.B1} r_j(t)={\rm sign}\,\bigl(\sin(2^j\pi t)\bigr)\in\{-1,0,+1\}, \quad\mbox{ for all $t\in[0,1]$}. \end{eqnarray}
Hence, in particular,
\begin{eqnarray}\label{Hc-77.B2} \int_0^1 r_j(t)r_k(t)\,dt=\delta_{jk},\qquad\forall\,j,k\in{\mathbb{N}}. \end{eqnarray}
Consequently, if $\langle\cdot,\cdot\rangle_{\mathscr{H}}$ stands for the inner product in ${\mathscr{H}}$, then for any finite set $J_o\subseteq{\mathbb{N}}$,
\begin{eqnarray}\label{Hc-77.C}
\int_0^1\Bigl\|\sum_{j\in J_o}r_j(t)x_j\Bigl\|^2_{\mathscr{H}}\,dt &=& \int_0^1\Big\langle\sum_{j\in J_o}r_j(t)x_j,\sum_{k\in J_o}r_k(t)x_k \Big\rangle_{\mathscr{H}}\,dt \\[4pt] &=& \sum_{j,k\in J_o}\Bigl(\int_0^1r_j(t)r_k(t)\,dt\Bigr)
\langle x_j,x_k\rangle_{\mathscr{H}}=\sum_{j\in J_o}\|x_j\|^2_{\mathscr{H}}. \nonumber \end{eqnarray}
On the other hand, thanks to \eqref{Hc-77.B1}, for each $t\in[0,1]$ we may estimate
\begin{eqnarray}\label{Hc-77.D}
\Bigl\|\sum_{j\in J_o}r_j(t)x_j\Bigl\|_{\mathscr{H}}
&=&\Bigl\|\Bigl(\sum_{j\in J_o,\,r_j(t)=+1}x_j\Bigr)
-\Bigl(\sum_{j\in J_o,\,r_j(t)=-1}x_j\Bigr)\Bigl\|_{\mathscr{H}} \nonumber\\[4pt]
&\leq & \Bigl\|\sum_{j\in J_o,\,r_j(t)=+1}x_j\Bigl\|_{\mathscr{H}}
+\Bigl\|\sum_{j\in J_o,\,r_j(t)=-1}x_j\Bigl\|_{\mathscr{H}} \leq 2C. \end{eqnarray}
By combining \eqref{Hc-77.C} and \eqref{Hc-77.D} we therefore obtain
\begin{eqnarray}\label{Hc-77.E}
\sum_{j\in J_o}\|x_j\|^2_{\mathscr{H}}\leq 4C^2, \qquad\mbox{for every finite subset $J_o$ of ${\mathbb{N}}$}, \end{eqnarray}
from which \eqref{Hc-77.UU} readily follows.
Moving on, assume that \eqref{Hc-77.A} holds yet $\sum_{j\in{\mathbb{N}}}x_j$ does not converge unconditionally, and seek a contradiction. Then (cf. the first equivalence in \eqref{UNC-12}), there exists a choice of signs $\varepsilon_j\in\{\pm 1\}$, $j\in{\mathbb{N}}$, with the property that the sequence of partial sums of the series $\sum_{j\in{\mathbb{N}}}\varepsilon_jx_j$ is not Cauchy in ${\mathscr{H}}$. In turn, this implies that there exist $\vartheta>0$ along with two sequences $\{a_i\}_{i\in{\mathbb{N}}}$, $\{b_i\}_{i\in{\mathbb{N}}}$ of numbers in ${\mathbb{N}}$, such that
\begin{eqnarray}\label{Hc-77.G} a_i\leq b_i<a_{i+1}\quad\mbox{ and }\quad
\Bigl\|\sum_{a_i\leq j\leq b_i}\varepsilon_j x_j\Bigl\|_{\mathscr{H}}\geq\vartheta, \quad\mbox{ for every $i\in{\mathbb{N}}$}. \end{eqnarray}
In this scenario, consider the sequence $\{y_i\}_{i\in{\mathbb{N}}}$ of vectors in ${\mathscr{H}}$ defined by
\begin{eqnarray}\label{Hc-77.H-1} y_i:=\sum_{a_i\leq j\leq b_i}\varepsilon_j x_j \,\,\mbox{ for every $i\in{\mathbb{N}}$}, \end{eqnarray}
and note that, by \eqref{Hc-77.G},
\begin{eqnarray}\label{Hc-77.H}
\|y_i\|_{\mathscr{H}}\geq\vartheta,\quad\mbox{ for every $i\in{\mathbb{N}}$}. \end{eqnarray}
Fix now an arbitrary finite subset $I_o$ of ${\mathbb{N}}$ and set $J_o:=\bigl\{j\in{\mathbb{N}}:\,\exists\,i\in I_o\mbox{ such that } a_i\leq j\leq b_i\bigr\}$. Thus, $J_o$ is a finite subset of ${\mathbb{N}}$. Then with the constant $C$ as in \eqref{Hc-77.A}, we have
\begin{eqnarray}\label{Hc-77.Axxx-1}
\Bigl\|\sum\limits_{i\in I_o}y_i\Bigr\|_{\mathscr{H}}
&=& \Bigl\|\sum\limits_{i\in I_o}\Bigl(\sum_{a_i\leq j\leq b_i}
\varepsilon_j x_j\Bigr)\Bigr\|_{\mathscr{H}}
= \Bigl\|\Bigl(\sum\limits_{j\in J_o,\,\varepsilon_j=+1}x_j\Bigr) -\Bigl(\sum\limits_{j\in J_o,\,\varepsilon_j=-1}x_j\Bigr)
\Bigr\|_{\mathscr{H}} \nonumber\\[4pt]
&\leq & \Bigl\|\sum\limits_{j\in J_o,\,\varepsilon_j=+1}x_j\Bigr\|_{\mathscr{H}}
+\Bigl\|\sum\limits_{j\in J_o,\,\varepsilon_j=-1}x_j\Bigr\|_{\mathscr{H}}\leq 2C, \end{eqnarray}
where the second equality relies on the fact from~\eqref{Hc-77.G} that $a_i\leq b_i < a_{i+1}$. Hence,
\begin{eqnarray}\label{Hc-77.Axxx} \sup\limits_{\stackrel{I_o\subseteq{\mathbb{N}}}{I_o\,\mbox{\tiny{finite}}}}
\Bigl\|\sum\limits_{i\in I_o}y_i\Bigr\|_{\mathscr{H}}\leq 2C. \end{eqnarray}
Having established this, \eqref{Hc-77.UU} then gives
$\sum_{i\in{\mathbb{N}}}\|y_i\|^2_{\mathscr{H}}\leq 16C^2<\infty$ which, in particular,
forces $\lim\limits_{i\to\infty}\|y_i\|_{\mathscr{H}}=0$. This, however, contradicts \eqref{Hc-77.H}.
To summarize, the proof so far shows that if \eqref{Hc-77.A} holds then the series $\sum_{j\in{\mathbb{N}}}x_j$ is unconditionally convergent. Of course, once the (norm) convergence of the series has been established then \eqref{Hc-77.A} also gives
$\displaystyle\Bigl\|\sum\limits_{j\in{\mathbb{N}}}x_j\Bigr\|_{\mathscr{H}}
\leq\limsup\limits_{N\to\infty}\Bigl\|\sum\limits_{j=1}^N x_j\Bigr\|_{\mathscr{H}} \leq C$, proving \eqref{TGbb-88u}.
There remains to prove that the finiteness condition in \eqref{Hc-77.A} holds if the series $\sum_{j\in{\mathbb{N}}}x_j$ is unconditionally convergent. With $N_1\in{\mathbb{N}}$ denoting the integer $N_{\varepsilon}$ corresponding to taking $\varepsilon=1$ in the last condition in \eqref{UNC-12}, consider
\begin{eqnarray}\label{Hc-77.FV} M:=\sup\limits_{I_o\subseteq\{1,...,N_1\}}
\Bigl\|\sum\limits_{j\in I_o}x_j\Bigr\|_{\mathscr{H}}<\infty. \end{eqnarray}
Then, given any finite subset $J_o$ of ${\mathbb{N}}$ we may write
\begin{eqnarray}\label{Hc-77.FV2}
\Bigl\|\sum\limits_{j\in J_o}x_j\Bigr\|_{\mathscr{H}}
\leq\Bigl\|\sum\limits_{j\in J_o\cap\{1,...,N_1\}}x_j\Bigr\|_{\mathscr{H}}
+\Bigl\|\sum\limits_{j\in J_o\setminus\{1,...,N_1\}}x_j\Bigr\|_{\mathscr{H}} \leq M+1, \end{eqnarray}
from which the finiteness condition in \eqref{Hc-77.A} follows. \end{proof}
For further reference, given an ambient quasi-metric space $({\mathscr{X}},\rho)$ and a set $E$ with the property that there exists a Borel measure $\sigma$ on
$(E,\tau_{\rho|_{E}})$ such that $(E,\rho_{\#}|_{E},\sigma)$ is a space of homogeneous type, we shall denote by $M_E$ the {\tt Hardy-Littlewood maximal function} in this context, i.e.,
\begin{eqnarray}\label{HL-MAX} (M_E f)(x):=\sup_{r>0}\frac{1}{\sigma\bigl(B_{\rho_{\#}}(x,r)\bigr)}
\int_{B_{\rho_{\#}}(x,r)}|f(y)|\,d\sigma(y),\qquad x\in E. \end{eqnarray}
We next present the
\vskip 0.08in \begin{proof}[Proof of Proposition~\ref{HS-PP.3}] For each $l\in{\mathbb{Z}}$ with $l\geq \kappa_E$, denote by $h_l(\cdot,\cdot)$ the integral kernel of the operator $D_l$. Thus, $h_l(\cdot,\cdot)=S_{l+1}(\cdot,\cdot)-S_l(\cdot,\cdot)$ and, as a consequence of properties $(i)-(iv)$ in Definition~\ref{Besov-S}, we see that $h_l(\cdot,\cdot)$ is a symmetric function on $E\times E$, and there exists $C\in(0,\infty)$ such that for each $l\in{\mathbb{Z}}$ with $l\geq \kappa_E$ we have
\begin{eqnarray}\label{condh-1}
&& |h_l(\cdot,\cdot)|\leq C2^{\,ld}{\mathbf{1}}_{\{\rho(\cdot,\cdot)\leq C2^{-l}\}}, \qquad\mbox{ on }\,\,E\times E, \\[4pt]
&&|h_l(x,y)-h_l(x',y)|\leq C2^{\,l(d+\gamma)}\rho(x,x')^\gamma \quad\mbox{ $\forall\, x,x',y\in E$}, \label{condh-2} \\[4pt] && \int_Eh_l(x,y)\,d\sigma(x)=0\quad\forall\,y\in E. \label{condh-4} \end{eqnarray}
Of course, due to the symmetry of $h$, smoothness and cancellation conditions in the second variable, similar to \eqref{condh-2} and \eqref{condh-4}, respectively, also hold.
Furthermore, for each $j,k\in{\mathbb{Z}}$ with $j,k\geq \kappa_E$, using first \eqref{condh-4}, then \eqref{condh-1} and \eqref{condh-2}, and then the fact that $(E,\rho,\sigma)$ is $d$-ADR, we may write
\begin{eqnarray}\label{TT-sH}
\left|\int_{E}h_j(x,z)h_k(z,y)\,d\sigma(z)\right|
& = & \left|\int_{E}[h_j(x,z)-h_j(x,y)]h_k(z,y)\,d\sigma(z)\right| \nonumber\\[4pt] & \leq & C2^{j(d+\gamma)}\int_E\rho_{\#}(y,z)^\gamma 2^{kd} {\mathbf{1}}_{\{\rho_{\#}(y,\cdot)\leq C2^{-k}\}}(z)\,d\sigma(z) \nonumber\\[4pt] & \leq & C2^{j(d+\gamma)}2^{-k\gamma}. \end{eqnarray}
Similarly,
\begin{eqnarray}\label{TT-sH-2}
\left|\int_{E}h_j(x,z)h_k(z,y)\,d\sigma(z)\right|
& = & \left|\int_{E}h_j(x,z)[h_k(z,y)-h_k(x,y)]\,d\sigma(z)\right| \nonumber\\[4pt] & \leq & C2^{k(d+\gamma)}2^{-j\gamma}. \end{eqnarray}
Combining \eqref{TT-sH}, \eqref{TT-sH-2}, and the support condition \eqref{condh-1}, it follows that for each $j,k\in{\mathbb{Z}}$ with $j,k\geq \kappa_E$, there holds (compare with \cite[p.\,15]{DJS} and \cite[(1.14), p.\,16]{DeHa09})
\begin{eqnarray}\label{hvs-RT}
\left|\int_{E}h_j(x,z)h_k(z,y)\,d\sigma(z)\right|\leq C2^{-|j-k|\gamma}\, 2^{d\cdot\min(j,k)}{\mathbf{1}}_{\{\rho(x,y)\leq C2^{-\min(j,k)}\}}, \quad\forall\,x,y\in E. \end{eqnarray}
Note that for each $j,k\in{\mathbb{Z}}$ with $j,k\geq\kappa_E$ we have that $D_jD_k:L^2(E,\sigma)\to L^2(E,\sigma)$ is a linear and bounded integral operator whose integral kernel is given by $\int_{E}h_j(x,z)h_k(z,y)\,d\sigma(z)$, for $x,y\in E$. Based on this and \eqref{hvs-RT} we may then conclude that for each $j,k\in{\mathbb{Z}}$ with $j,k\geq \kappa_E$,
\begin{eqnarray}\label{hvs-Raaa}
\bigl|(D_jD_kf)(x)\bigr| &\leq & C2^{-|j-k|\gamma}
{\int{\mkern-19mu}-}_{B_{\rho_{\#}}(x,C2^{-\min(j,k)})}|f(y)|\,d\sigma(y) \nonumber\\[4pt]
&\leq & C2^{-|j-k|\gamma}M_E(f)(x),\qquad\forall\,x\in E, \end{eqnarray}
for every $f\in L^1_{loc}(E,\sigma)$. In turn, the boundedness of $M_E$ and \eqref{hvs-Raaa} yield
\begin{eqnarray}\label{hvs-RTS}
\|D_jD_k\|_{L^2(E,\sigma)\to L^2(E,\sigma)}\leq C2^{-|j-k|\gamma}, \qquad\forall\,j,k\in{\mathbb{Z}},\,\,j,k\geq\kappa_E. \end{eqnarray}
Having established \eqref{hvs-RTS}, it follows that the family of linear operators $\bigl\{D_l\bigr\}_{l\in{\mathbb{Z}},\,l\geq\kappa_E}$, from $L^2(E,\sigma)$ into itself, is almost orthogonal. As such, Lemma~\ref{L-CKS} applies and gives that
\begin{eqnarray}\label{fcc-5tNN2} \sup\limits_{\stackrel{J\subseteq{\mathbb{Z}}}{J\,\mbox{\tiny{finite}}}}
\Bigl\|\sum_{l\in J,\,l\geq\kappa_E}D_l\Bigr\|_{L^2(E,\sigma)\to L^2(E,\sigma)} \leq C<\infty, \end{eqnarray}
the following Littlewood-Paley estimate holds
\begin{eqnarray}\label{FC+MN}
\Bigl(\sum\limits_{l\in{\mathbb{Z}},\,l\geq \kappa_E}\|D_lf\|^2_{L^2(E,\sigma)}
\Bigr)^{1/2}\leq C\|f\|_{L^2(E,\sigma)},\quad\mbox{ for each }\,\,f\in L^2(E,\sigma), \end{eqnarray}
and, making use of \eqref{fcc-5t} and \eqref{fcc-5t2} as well, we have
\begin{eqnarray}\label{fcc-5tNN1} \begin{array}{l} \bigl(I-{\mathcal{S}}_{\kappa_E}\bigr)f =\displaystyle\sum\limits_{l\in{\mathbb{Z}},\,l\geq \kappa_E}D_l f \quad\mbox{ for each $f\in L^2(E,\sigma)$}, \\[4pt] \mbox{where the series converges unconditionally in $L^2(E,\sigma)$}. \end{array} \end{eqnarray}
To proceed, fix a number $N\in{\mathbb{N}}$. Based on \eqref{fcc-5tNN2}, we may square \eqref{fcc-5tNN1} and obtain, pointwise in $L^2(E,\sigma)$,
\begin{eqnarray}\label{fcc-5A.1} \bigl(I-{\mathcal{S}}_{\kappa_E}\bigr)^2 &=& \lim_{M\to\infty}
\Bigl[\Bigl(\sum_{j\in{\mathbb{Z}},\,j\geq\kappa_E,\,|j|\leq M}D_j\Bigr)
\Bigl(\sum_{k\in{\mathbb{Z}},\,k\geq \kappa_E,\,|k|\leq M}D_k\Bigr)\Bigr] \nonumber\\[4pt]
&=& \lim_{M\to\infty}\Bigl(\sum\limits_{\stackrel{|j-k|\leq N}{\,j,k\geq \kappa_E,\,|j|,|k| \leq M}}D_jD_k
+\sum\limits_{\stackrel{|j-k|>N}{j,k\geq \kappa_E,\,|j|,|k|\leq M}}D_jD_k\Bigr). \end{eqnarray}
Going further, fix $i\in{\mathbb{Z}}$ and consider the family $\{T_l\}_{l\in J_i}$ of operators on $L^2(E,\sigma)$, where
\begin{eqnarray}\label{T-GBn89} T_l:=D_{l+i}D_l\quad\mbox{ for every }\quad l\in J_i:=\bigl\{l\in{\mathbb{Z}}:\,l\geq\max\{\kappa_E,\kappa_E-i\}\bigr\}. \end{eqnarray}
Then, with $\|\cdot\|$ temporarily abbreviating
$\|\cdot\|_{L^2(E,\sigma)\to L^2(E,\sigma)}$, for each $j,k\in J_i$ we may estimate
\begin{eqnarray}\label{T-77-AB}
\|T_j^\ast T_k\| &\leq &\min\,\Bigl\{\|D_j\|\|D_{j+i}D_{k+i}\|\|D_k\|\,,\,
\|D_jD_{j+i}\|\|D_{k+i}\|\|D_k\|\Bigr\} \nonumber\\[4pt]
&\leq & C\,\min\,\Bigl\{2^{-|k-j|\gamma}\,,\,2^{-|i|\gamma}\Bigr\}, \end{eqnarray}
thanks to \eqref{fcc-5tNN2} and \eqref{hvs-RTS}. This readily implies that $\sup\limits_{j\in J_i}\Bigl(\sum\limits_{k\in J_i}
\sqrt{\|T_j^\ast T_k\|}\Bigr)\leq C(1+|i|)2^{-|i|\gamma/2}$
and $\sup\limits_{k\in J_i}\Bigl(\sum\limits_{j\in J_i} \sqrt{\|T_j^{\phantom{\ast}}T_k^\ast \|}\Bigr)\leq C(1+|i|)2^{-|i|\gamma/2}$ for some $C\in(0,\infty)$ independent of $i$. Hence, for each $i\in{\mathbb{Z}}$, the family $\bigl\{D_{l+i}D_l\bigr\}_{l\in{\mathbb{Z}},\,l\geq\max\{\kappa_E,\kappa_E-i\}}$ is almost orthogonal, and by Lemma~\ref{L-CKS} there exists some constant $C\in(0,\infty)$ independent of $i$ such that for every set $J\subseteq J_i$ we have that $\sum\limits_{l\in J}D_{l+i}D_l$ converges pointwise unconditionally in $L^2(E,\sigma)$ and
\begin{eqnarray}\label{T-GBn88}
\Bigl\|\sum\limits_{l\in J}D_{l+i}D_l
\Bigr\|_{L^2(E,\sigma)\to L^2(E,\sigma)}\leq C(1+|i|)2^{-|i|\gamma/2}. \end{eqnarray}
Next, fix $N\in{\mathbb{N}}$ and let ${\mathscr{I}}$ be an arbitrary finite subset of
$\{(l,m)\in{\mathbb{Z}}\times{\mathbb{Z}}:\,l,\,m\geq\kappa_E\}$. Then for each function $f\in L^2(E,\sigma)$ with $\|f\|_{L^2(E,\sigma)}=1$, using \eqref{T-GBn88} we may estimate
\begin{eqnarray}\label{RGbb-6Y} && \hskip -0.20in
\Bigl\|\sum_{(j,k)\in{\mathscr{I}},\,|j-k|>N}D_jD_kf\Bigr\|_{L^2(E,\sigma)}
=\Bigl\|\sum_{i\in{\mathbb{Z}},\,|i|>N} \Bigl(\sum_{l\in{\mathbb{Z}},\,(l+i,l)\in{\mathscr{I}}}
D_{l+i}D_l f\Bigr)\Bigr\|_{L^2(E,\sigma)} \\[4pt] && \hskip 0.15in
\leq\sum_{i\in{\mathbb{Z}},\,|i|>N}
\Bigl\|\sum_{l\in{\mathbb{Z}},\,(l+i,l)\in{\mathscr{I}}}
D_{l+i}D_l f\Bigr\|_{L^2(E,\sigma)} \leq \sum_{i\in{\mathbb{Z}},\,|i|>N}C(1+|i|)2^{-|i|\gamma/2} \leq C_\gamma N2^{-N\gamma/2}, \nonumber \end{eqnarray}
for some finite constant $C_\gamma>0$ which is independent of $N$. In turn, based on \eqref{FAFFF}, \eqref{TGbb-88u} and \eqref{RGbb-6Y} we deduce that
\begin{eqnarray}\label{dec-id2} \begin{array}{c}
R_N:=\displaystyle\sum\limits_{\stackrel{|j-k|>N}{j,k\geq \kappa_E}}D_jD_k \,\,\mbox{ converges pointwise unconditionally in $L^2(E,\sigma)$, and} \\[4pt] \mbox{there exists $C_\gamma\in(0,\infty)$ such that }\,\,
\|R_N\|_{L^2(E,\sigma)\to L^2(E,\sigma)}\leq C_\gamma N2^{-N\gamma/2}. \end{array} \end{eqnarray}
In a similar fashion to \eqref{T-GBn88}-\eqref{dec-id2}, we may also deduce that
\begin{eqnarray}\label{de-FFws}
T_N:=\sum\limits_{\stackrel{|j-k|\leq N}{j,k\geq \kappa_E}}D_jD_k \,\,\mbox{ converges pointwise unconditionally in $L^2(E,\sigma)$}. \end{eqnarray}
Consequently, if we now set
\begin{eqnarray}\label{dTF-gAA}
D_l^N:=\sum\limits_{\stackrel{i\in{\mathbb{Z}},\,|i|\leq N}{i\geq \kappa_E-l}}D_{l+i}, \qquad\mbox{for each }\,\,l\in{\mathbb{Z}}, \end{eqnarray}
then (cf. \eqref{UNC-11}) the series $T_N$ may be rearranged as
\begin{eqnarray}\label{dec-id1} T_N =\sum\limits_{l\in{\mathbb{Z}},\,l\geq \kappa_E}D_lD_l^N, \end{eqnarray}
where the sum converges pointwise unconditionally in $L^2(E,\sigma)$. Combining \eqref{fcc-5A.1}, \eqref{dec-id2} and \eqref{de-FFws}, we arrive at the identity
\begin{eqnarray}\label{dec-id1A} \bigl(I-{\mathcal{S}}_{\kappa_E}\bigr)^2=R_N+T_N\quad\mbox{on }\,\,L^2(E,\sigma), \end{eqnarray}
which is convenient to further re-write as
\begin{eqnarray}\label{dec-id1ABIS} I=R_N+\widetilde{T}_N\quad\mbox{on }\,\,L^2(E,\sigma),\qquad \mbox{where }\quad \widetilde{T}_N:=T_N+{\mathcal{S}}_{\kappa_E}\bigl(2I-{\mathcal{S}}_{\kappa_E}\bigr). \end{eqnarray}
Thanks to the estimate in \eqref{dec-id2}, it follows from \eqref{dec-id1ABIS} that
\begin{eqnarray}\label{dec-id1B} \mbox{$\widetilde{T}_N:L^2(E,\sigma)\to L^2(E,\sigma)$ is boundedly invertible for $N\in{\mathbb{N}}$ sufficiently large}. \end{eqnarray}
Hence, for $N$ sufficiently large and fixed, based on \eqref{dec-id1B} we may write that $I=\widetilde{T}_N(\widetilde{T}_N)^{-1}$, and keeping in mind \eqref{dec-id1} and \eqref{dec-id1ABIS}, we arrive at the following Calder\'on-type reproducing formula
\begin{eqnarray}\label{PKc.TG} I=\Bigl(\sum_{l\in{\mathbb{Z}},\,l\geq \kappa_E}D_l\widetilde{D}_l\Bigr) +{\mathcal{S}}_{\kappa_E}\bigl(2I-{\mathcal{S}}_{\kappa_E}\bigr)(\widetilde{T}_N)^{-1}, \end{eqnarray}
where the sum converges pointwise unconditionally in $L^2(E,\sigma)$, and
\begin{eqnarray}\label{egFvo} \widetilde{D}_l:=D_l^N(\widetilde{T}_N)^{-1},\qquad\forall\,l\in{\mathbb{Z}} \,\,\mbox{ with }\,\,l\geq \kappa_E. \end{eqnarray}
From this \eqref{PKc} follows with $R:=\bigl({\mathcal{S}}_{\kappa_E}-2I\bigr)(\widetilde{T}_N)^{-1}$. Finally, \eqref{PKc-2} is a consequence of \eqref{egFvo}, the fact that the sum in \eqref{dTF-gAA} has a finite number of terms, \eqref{dec-id1B} and \eqref{FC+MN}. \end{proof}
\subsection{Dyadic Carleson tents} \label{SSect:2.4}
Suppose that $({\mathscr{X}},\rho)$ is a geometrically doubling quasi-metric space and that $E$ is a nonempty, closed, proper subset of $({\mathscr{X}},\tau_\rho)$.
It follows from the discussion below Definition~\ref{Gd_ZZ} that $(E,\rho\bigl|_E)$ is also a geometrically doubling quasi-metric space. We now introduce dyadic Carleson tents in this setting. These are sets in ${\mathscr{X}}\setminus E$ that are adapted to $E$ in the same way that classical Carleson boxes or tents in the upper-half space $\mathbb{R}^{n+1}_+$ are adapted to $\mathbb{R}^n$. We require a number of preliminaries before we introduce these sets in \eqref{gZSZ-3} below. First, fix a collection ${\mathbb{D}}(E)$ of dyadic cubes contained in $E$ as in Proposition~\ref{Diad-cube}. Second, choose $\lambda\in[2C_\rho,\infty)$ and fix a Whitney covering ${\mathbb{W}}_\lambda({\mathscr{X}}\setminus E)$ of balls contained in ${\mathscr{X}}\setminus E$ as in Proposition~\ref{H-S-Z}. Following Convention~\ref{WWVc}, we refer to these $\rho_{\#}$-balls as Whitney cubes, and for each $I\in{\mathbb{W}}_\lambda({\mathscr{X}}\setminus E)$, we use the notation $\ell(I)$ for the radius of $I$. Third, choose $C_\ast\in[1,\infty)$, and for each $Q\in{\mathbb{D}}(E)$, define the following collection of Whitney cubes:
\begin{equation}\label{gZSZa} W_Q:=\{I\in{\mathbb{W}}_\lambda({\mathscr{X}} \setminus E):\,C_\ast^{-1}\ell(I)\leq\ell(Q)\leq C_\ast\ell(I)\mbox{ and } {\rm dist}_\rho(I,Q)\leq\ell(Q)\}. \end{equation} Fourth, for each $Q\in{\mathbb{D}}(E)$, define the following subset of $({\mathscr{X}},\tau_{\rho})$:
\begin{eqnarray}\label{gZSZb} {\mathcal{U}}_Q:=\bigcup\limits_{I\in W_Q}I. \end{eqnarray}
Since from Theorem~\ref{JjEGh} we know that the regularized quasi-distance $\rho_\#$ is continuous, it follows that the $\rho_\#$-balls are open. As such, that each $I$ in $W_Q$, hence ${\mathcal{U}}_Q$ itself, is open.
Finally, for each $Q\in{\mathbb{D}}(E)$, the {\tt dyadic Carleson tent} $T_E(Q)$ {\tt over} $Q$ is defined as follows:
\begin{eqnarray}\label{gZSZ-3} T_E(Q):=\bigcup_{Q'\in{\mathbb{D}}(E),\,\,Q'\subseteq Q}{\mathcal{U}}_{Q'}. \end{eqnarray}
For most of the subsequent work we will assume that the Whitney covering ${\mathbb{W}}_\lambda({\mathscr{X}}\setminus E)$ and the constant $C_\ast$ are chosen as in the following lemma.
\begin{lemma}\label{Lem:CQinBQ-N} Let $({\mathscr{X}},\rho)$ be a geometrically doubling quasi-metric space and suppose that $E$ is a nonempty, closed, proper subset of $({\mathscr{X}},\tau_\rho)$. Fix a collection ${\mathbb{D}}(E)$ of dyadic cubes in $E$ as in Proposition~\ref{Diad-cube}. Next, choose $\lambda\in[2C_\rho,\infty)$, fix a Whitney covering ${\mathbb{W}}_\lambda({\mathscr{X}}\setminus E)$ of ${\mathscr{X}}\setminus E$, and let $\Lambda$ denote the constant associated with $\lambda$ as in Proposition~\ref{H-S-Z}. Finally, choose
\begin{eqnarray}\label{NeD-67} C_\ast \in [4C_\rho^4\,\Lambda,\infty), \end{eqnarray}
and define the collection $\{{\mathcal{U}}_Q\}_{Q\in{\mathbb{D}}(E)}$ associated with ${\mathbb{W}}_\lambda({\mathscr{X}}\setminus E)$ and $C_\ast$ as in \eqref{gZSZa}-\eqref{gZSZb}.
Then there exists $\epsilon\in(0,1)$, depending only on $\lambda$ and geometry, with the property that
\begin{eqnarray}\label{doj.cF} \bigl\{x\in{\mathscr{X}}\setminus E:\,\delta_E(x)<\epsilon\,{\rm diam}_{\rho}(E)\bigr\} \subseteq\bigcup\limits_{Q\in{\mathbb{D}}(E)}{\mathcal{U}}_Q. \end{eqnarray} \end{lemma} \begin{proof} If ${\rm diam}_{\rho}(E) = \infty$, then both sides of \eqref{doj.cF} are equal to ${\mathscr{X}}\setminus E$ for all $\epsilon\in(0,1)$, since the Whitney cubes cover ${\mathscr{X}}\setminus E$, so the result is immediate. Now assume that ${\rm diam}_{\rho}(E) < \infty$. Fix some integer $N\in{\mathbb{N}}$, to be specified later, and consider an arbitrary point $x\in{\mathscr{X}}\setminus E$ with $\delta_E(x)<2^{-N}{\rm diam}_{\rho}(E)$. Then by \eqref{topoQMS} and \eqref{Jcc-KKi} we have $0<\delta_E(x)<2^{-N-\kappa_E}$, hence there exists $k\in{\mathbb{Z}}$ with $k\geq\kappa_E$ such that $2^{-N-k-1}\leq\delta_E(x)<2^{-N-k}$. Now, select a ball $I=B_{\rho_{\#}}(x_I,\ell(I))\in{\mathbb{W}}_\lambda({\mathscr{X}}\setminus E)$ such that $x\in I$. Then, by {\it (3)} in Proposition~\ref{H-S-Z}, there exists $z\in E$ such that $\rho_{\#}(x_I,z)<\Lambda\ell(I)$. Consequently,
\begin{eqnarray}\label{jaf-UU.1} \delta_{E}(x)\leq\rho_{\#}(x,z)\leq C_\rho\,\max\,\{\rho_{\#}(x,x_I),\rho_{\#}(x_I,z)\} <C_\rho\Lambda\ell(I). \end{eqnarray}
In addition, {\it (3)} in Proposition~\ref{H-S-Z} also gives that $B_{\rho_{\#}}(x_I,\lambda\ell(I))\subseteq{\mathscr{X}}\setminus E$ and, hence, for every $y\in E$
\begin{eqnarray}\label{jaf-UU.2} 2C_\rho\ell(I) &\leq & \lambda\ell(I)\leq\rho_{\#}(x_I,y) \leq C_\rho\,\rho_{\#}(x_I,x)+C_\rho\,\rho_{\#}(x,y) \nonumber\\[4pt] &\leq & C_\rho\,\ell(I)+C_\rho\,\rho_{\#}(x,y). \end{eqnarray}
After canceling like-terms in the most extreme sides of \eqref{jaf-UU.2} and taking the infimum over all $y\in E$, we arrive at
\begin{eqnarray}\label{jaf-UU.3} \ell(I)\leq\delta_{E}(x). \end{eqnarray}
Next, since $\delta_E(x)<2^{-N-k}$, there exists $x_0\in E$ such that $\rho_{\#}(x,x_0)<2^{-N-k}$. Furthermore, by invoking {\it (7)} in Proposition~\ref{Diad-cube} we may choose $Q\in{\mathbb{D}}_k(E)$ with the property that $B_{\rho_{\#}}(x_0,2^{-N-k})\cap Q$ contains at least one point $x_1$. Thus, by \eqref{DEQV1} we have
\begin{eqnarray}\label{jaf-UU.4} {\rm dist}_\rho(I,Q) &\leq &{\rm dist}_\rho(x,Q)\leq\rho(x,x_1) \leq C_\rho^2\rho_{\#}(x,x_1) \\[4pt] &\leq & {C_\rho\!\!\!\!\phantom{.}^2}C_{\rho_{\#}}\max\,\{\rho_{\#}(x,x_0),\rho_{\#}(x_0,x_1)\} <C_\rho^3 2^{-N-k}=C_\rho^3 2^{-N}\ell(Q). \nonumber \end{eqnarray}
This shows that
\begin{eqnarray}\label{jaf-UU.5} 2^N>C_\rho^3\,\Longrightarrow\, {\rm dist}_\rho(I,Q)\leq\ell(Q). \end{eqnarray}
Starting with \eqref{jaf-UU.3} and keeping in mind that $\delta_E(x)<2^{-N-k}$, we obtain
\begin{eqnarray}\label{jaf-UU.6} \ell(I)<2^{-N-k}=2^{-N}\ell(Q)\leq\ell(Q). \end{eqnarray}
Finally, with the help of \eqref{jaf-UU.1} we write $2^{-N-1}\ell(Q)=2^{-N-k-1}\leq\delta_E(x)\leq C_\rho\Lambda\ell(I)$ which further entails
\begin{eqnarray}\label{jaf-UU.8} C_\ast\geq 2^{N+1}C_\rho\Lambda\,\Longrightarrow\, \ell(I)\geq C_\ast^{-1}\ell(Q). \end{eqnarray}
At this stage, by choosing $N\in\mathbb{N}$ such that
\begin{eqnarray}\label{jaf-UU.7} N-1 \leq \log_2\bigl(C_\rho^3\bigr) < N, \end{eqnarray}
we may conclude from \eqref{jaf-UU.5}, \eqref{jaf-UU.6} and \eqref{jaf-UU.8} that $I\in W_Q$ when $C_\ast\geq 4 C_\rho^4\,\Lambda$. This, in turn, forces $x\in I\subseteq{\mathcal{U}}_Q$. Taking $\epsilon:=2^{-N}$ with $N$ as in \eqref{jaf-UU.7} then justifies \eqref{doj.cF}, and finishes the proof of the lemma. \end{proof}
We now return to the context introduced in the first paragraph of this subsection, and in particular, where $\lambda\in[2C_\rho,\infty)$ and $C_\ast\in[1,\infty)$. For further reference, we note that then there exists $C_o\in[1,\infty)$ such that
\begin{eqnarray}\label{UUU-rf} C_o^{-1}\ell(Q)\leq\delta_E(x)\leq C_o\ell(Q),\qquad\forall\,Q\in{\mathbb{D}}(E) \,\mbox{ and }\,\forall\,x\in{\mathcal{U}}_Q. \end{eqnarray}
Indeed, an inspection of \eqref{jaf-UU.1}, \eqref{jaf-UU.3}, \eqref{gZSZa} and \eqref{gZSZb} shows that \eqref{UUU-rf} holds when
\begin{eqnarray}\label{UUU-rf-N} C_o:=C_\ast C_\rho\Lambda, \end{eqnarray} where $\Lambda$ is the constant associated with $\lambda$ as in Proposition~\ref{H-S-Z}.
The reader should be aware of the fact that even when \eqref{doj.cF} holds it may happen that some ${\mathcal{U}}_Q$'s are empty. However, under the assumption that $({\mathscr{X}},\rho,\mu)$ is an $m$-dimensional {\rm ADR} space
and granted the existence of a measure $\sigma$ such that $(E,\rho|_{E},\sigma)$ becomes a $d$-dimensional {\rm ADR} space for some $d\in(0,m)$, matters may be arranged so that this eventuality never materializes. In particular, if $C_\ast$ is large enough (depending on $\lambda$ and geometry), then ${\mathcal{U}}_Q\not=\emptyset$ for all $Q\in{\mathbb{D}}(E)$. This is a simple consequence of the following lemma, which is proved in \cite{MMMM-B}.
\begin{lemma}\label{FR-DF-4} Let $({\mathscr{X}},\rho,\mu)$ be an $m$-dimensional {\rm ADR} space, for some $m>0$, and assume that $E$ is a closed subset of $({\mathscr{X}},\tau_\rho)$ with the
property that there exists a measure $\sigma$ on $E$ for which $(E,\rho|_{E},\sigma)$ is a $d$-dimensional {\rm ADR} space for some $d\in(0,m)$.
Then there exists $\vartheta\in(0,1)$ such that for each $x_0\in{\mathscr{X}}$ and each finite $r\in(0,{\rm diam}_\rho({\mathscr{X}})]$ one may find $x\in{\mathscr{X}}$ with the property that $B_\rho(x,\vartheta r)\subseteq B_\rho(x_0,r)\setminus E$. \end{lemma}
We again return to the context introduced in the first paragraph of this subsection, and in particular, where $\lambda\in[2C_\rho,\infty)$ and $C_\ast\in[1,\infty)$. For each $Q\in{\mathbb{D}}(E)$, recall the dyadic Carleson tent $T_E(Q)$ over $Q$ from~\eqref{gZSZ-3}:
\begin{eqnarray}\label{gZSZ-3aux} T_E(Q):=\bigcup_{Q'\in{\mathbb{D}}(E),\,\,Q'\subseteq Q}{\mathcal{U}}_{Q'}. \end{eqnarray}
A property that will be needed later is the fact that
\begin{eqnarray}\label{dFvK} \begin{array}{c} \mbox{there exists }\,C\in(0,\infty)\, \mbox{ depending only on $C_\ast$ from \eqref{gZSZa} and $\rho$} \\[4pt] \mbox{such that }\,\, T_E(Q)\subseteq B_\rho\bigl(x,C\ell(Q)\bigr)\setminus E,\quad \forall\,Q\in{\mathbb{D}}(E),\,\,\forall\,x\in Q. \end{array} \end{eqnarray}
Indeed, if $Q\in{\mathbb{D}}(E)$ and $y\in T_E(Q)$ are arbitrary, then there exists $Q'\in{\mathbb{D}}(E)$, $Q'\subseteq Q$ such that $y\in I$, for some $I\in W_{Q'}$. Hence, for each $x\in Q$, we have
\begin{eqnarray}\label{wqMY} \rho(y,x) &\leq & C{\rm diam}_\rho(I)+C{\rm dist}_\rho(I,Q')+C{\rm diam}_\rho(Q) \nonumber\\[4pt] &\leq & C\ell(Q')+C\ell(Q) \leq C\ell(Q), \end{eqnarray}
where $C$ is a finite positive geometric constant. Now \eqref{dFvK} follows from \eqref{wqMY}.
The following lemma compliments the containment in~\eqref{dFvK}.
\begin{lemma}\label{b:SV} Assume all of the hypotheses contained in the first paragraph of Lemma~\ref{Lem:CQinBQ-N} and recall the family of dyadic Carleson tents $\{T_E(Q)\}_{Q\in{\mathbb{D}}(E)}$ defined in \eqref{gZSZ-3}.
Then there exists $\varepsilon\in(0,1)$, depending only on $\lambda$ and geometry, with the property that
\begin{eqnarray}\label{zjrh} B_{\rho_{\#}}\bigl(x_Q,\varepsilon\ell(Q)\bigr)\setminus E\subseteq T_E(Q), \qquad\forall\,Q\in{\mathbb{D}}(E). \end{eqnarray} \end{lemma}
\begin{proof} Fix $\varepsilon\in(0,1)$ to be specified later and let $N$ be as in \eqref{jaf-UU.7}. Also take an arbitrary $Q\in{\mathbb{D}}(E)$ and fix $x\in B_{\rho_{\#}}\bigl(x_Q,\varepsilon\ell(Q)\bigr)\setminus E$. Then $\rho_{\#}(x_Q,x)<\varepsilon\ell(Q)$ and making the restriction
\begin{eqnarray}\label{varep-1} \varepsilon<2^{-N-1} \end{eqnarray}
we have
\begin{eqnarray}\label{jve-1} \delta_E(x)\leq\rho_{\#}(x_Q,x)<\varepsilon\ell(Q)\leq 2\varepsilon\,{\rm diam}_\rho(E) <2^{-N}{\rm diam}_\rho(E). \end{eqnarray}
Thus, all considerations in the first part of the proof of Lemma~\ref{Lem:CQinBQ-N} up to \eqref{jaf-UU.3} apply. In particular, it follows that $\delta_E(x)<\min\,\{2^{-N-k},\varepsilon\ell(Q)\}$. Hence, there exists $x_0\in E$ such that $\rho_{\#}(x,x_0)<\min\,\{2^{-N-k},\varepsilon\ell(Q)\}$. Applying property {\it (7)} in Proposition~\ref{Diad-cube}, we may choose $Q'\in{\mathbb{D}}_k(E)$ such that $B_{\rho_{\#}}\bigl(x_0,\varepsilon\ell(Q)\bigr)\cap Q'\not=\emptyset$. At this point we make the claim that
\begin{eqnarray}\label{jve-2} B_{\rho_{\#}}\bigl(x_0,\varepsilon\ell(Q)\bigr)\cap E\subseteq Q \quad\mbox{ if $\varepsilon$ is sufficiently small.} \end{eqnarray}
Indeed, first observe that
\begin{eqnarray}\label{jve-3} \rho_{\#}(x_0,x_Q)\leq C_\rho\max\,\{\rho_{\#}(x_0,x),\rho_{\#}(x,x_Q)\} <\varepsilon C_\rho\ell(Q). \end{eqnarray}
Consequently, if $y\in B_{\rho_{\#}}\bigl(x_0,\varepsilon\ell(Q)\bigr)\cap E$ is arbitrary, then
\begin{eqnarray}\label{jve-4} \rho_{\#}(x_Q,y)\leq C_\rho\max\,\{\rho_{\#}(x_Q,x_0),\rho_{\#}(x_0,y)\} <\varepsilon C_\rho^2\ell(Q). \end{eqnarray}
Property \eqref{ha-GVV} ensures that $B_{\rho_{\#}}\bigl(x_Q,a_0C_\rho^{-2}\ell(Q)\bigr)\cap E\subseteq Q$, so \eqref{jve-4} implies that $y\in Q$ if
\begin{eqnarray}\label{varep-2} \varepsilon<a_0C_\rho^{-4}, \end{eqnarray}
proving the claim in \eqref{jve-2}. In turn, if we assume that $\varepsilon$ is sufficiently small, the inclusion in \eqref{jve-2} implies
\begin{eqnarray}\label{jve-5} Q'\cap Q\not=\emptyset. \end{eqnarray}
On the other hand, using the reasoning in the proof of Lemma~\ref{Lem:CQinBQ-N} that yielded \eqref{jaf-UU.4}-\eqref{jaf-UU.7}, this time with $Q'$ replacing $Q$, we obtain that if $N$ is as in \eqref{jaf-UU.7} (recall that we are assuming that $C_\ast$ satisfies \eqref{NeD-67}), then
\begin{eqnarray}\label{jve-6} x\in I\subseteq {\mathcal{U}}_{Q'}. \end{eqnarray}
Thus, using also \eqref{UUU-rf}, we have
\begin{eqnarray}\label{jve-7} \ell(Q')\leq C_o\delta_E(x)\leq C_o\rho_{\#}(x_Q,x)<C_o\varepsilon\ell(Q). \end{eqnarray}
Hence, under the additional restriction $\varepsilon<C_o^{-1}$, we arrive at the conclusion that $Q'\in {\mathbb{D}}_j(E)$ for some $j>k$, which when combined with \eqref{jve-5} and {\it (3)} in Proposition~\ref{Diad-cube}, forces $Q'\subseteq Q$. This in concert with \eqref{jve-6} and \eqref{gZSZ-3}, shows that $x\in T_E(Q)$ provided
\begin{eqnarray}\label{varep-3} 0<\varepsilon<\min\{2^{-N-1}, a_0C_\rho^{-4},C_o^{-1}\}. \end{eqnarray}
The proof of the lemma is now complete. \end{proof}
Next we prove a finite overlap property for the sets in $\{{\mathcal{U}}_Q\}_{Q\in\mathbb{D}(E)}$ from \eqref{gZSZb}. Throughout the manuscript, we agree that ${\mathbf{1}}_A$ stands for the characteristic (or indicator) function of the set $A$.
\begin{lemma}\label{Lem:CQinBQ} Let $({\mathscr{X}},\rho)$ be a geometrically doubling quasi-metric space and suppose that $E$ is a nonempty, closed, proper subset of $({\mathscr{X}},\tau_\rho)$. Fix $a\in[1,\infty)$, a collection ${\mathbb{D}}(E)$ of dyadic cubes in $E$ as in Proposition~\ref{Diad-cube}, and $C_\ast\in[1,\infty)$.
If $\lambda\in[a,\infty)$, and we fix a Whitney covering ${\mathbb{W}}_\lambda({\mathscr{X}}\setminus E)$ of ${\mathscr{X}}\setminus E$ as in Proposition~\ref{H-S-Z}, then there exists $N\in{\mathbb{N}}$, depending only on $\lambda$, $C_\ast$ and geometry, such that
\begin{eqnarray}\label{doj} \sum_{Q\in{\mathbb{D}}(E)}{\mathbf{1}}_{{\mathcal{U}}_{Q}^\ast}\leq N, \end{eqnarray}
where $\{\mathcal{U}_Q\}_{Q\in\mathbb{D}(E)}$ is the collection associated with ${\mathbb{W}}_\lambda({\mathscr{X}}\setminus E)$ and $C_\ast$ as in \eqref{gZSZa}-\eqref{gZSZb}, and for each $Q\in{\mathbb{D}}(E)$, the set (compare with \eqref{gZSZb})
\begin{eqnarray}\label{doj.222} {\mathcal{U}}_{Q}^\ast:=\bigcup_{I\in W_Q}aI. \end{eqnarray} \end{lemma}
\begin{proof} Let ${\mathbb{D}}(E)$ be the collection of dyadic cubes obtained by applying Proposition~\ref{Diad-cube}. Fix $a\in[1,\infty)$ and consider a Whitney covering ${\mathbb{W}}_\lambda({\mathscr{X}}\setminus E)$ as in Proposition~\ref{H-S-Z} with $\lambda\in[a,\infty)$. In particular,
\begin{eqnarray}\label{AAA-NNN} \sum_{I\in{\mathbb{W}}_\lambda({\mathscr{X}}\setminus E)}{\mathbf{1}}_{\lambda I} \leq N_1,\qquad\mbox{for some $N_1\in{\mathbb{N}}$.} \end{eqnarray}
To proceed, define
\begin{eqnarray}\label{3a-YU.1} {\mathcal{I}}:=\bigcup_{Q\in{\mathbb{D}}(E)}W_Q \subseteq{\mathbb{W}}_\lambda({\mathscr{X}}\setminus E) \end{eqnarray}
and, for each $I\in{\mathcal{I}}$,
\begin{eqnarray}\label{3a-YU.2} q_I:=\{Q\in{\mathbb{D}}(E):\,I\in W_Q\}. \end{eqnarray}
Then, using \eqref{AAA-NNN}, we estimate
\begin{eqnarray}\label{3a-YU.3} \sum_{Q\in{\mathbb{D}}(E)}{\mathbf{1}}_{{\mathcal{U}}_{Q}^\ast} &\leq & \sum_{Q\in{\mathbb{D}}(E)}\sum_{I\in W_Q}{\mathbf{1}}_{\lambda I} =\sum_{I\in{\mathcal{I}}}(\#\,q_I)\cdot{\mathbf{1}}_{\lambda I} \nonumber\\[4pt] &\leq & \Bigl(\sup_{I\in{\mathbb{W}}_\lambda({\mathscr{X}}\setminus E)}\#\,q_I\Bigr) \sum_{I\in{\mathbb{W}}_\lambda({\mathscr{X}}\setminus E)}{\mathbf{1}}_{\lambda I} \leq N_1\cdot\Bigl(\sup_{I\in{\mathbb{W}}_\lambda({\mathscr{X}}\setminus E)}\#\,q_I\Bigr). \hskip 0.30in \end{eqnarray}
Hence, once we show that there exists $N_2\in{\mathbb{N}}$ such that
\begin{eqnarray}\label{3a-YU.4} \#\,q_I\leq N_2,\qquad\forall\,I\in{\mathbb{W}}_\lambda({\mathscr{X}}\setminus E), \end{eqnarray}
the desired estimate, \eqref{doj}, follows with $N:=N_1N_2$. To prove \eqref{3a-YU.4}, fix an arbitrary $I\in{\mathbb{W}}_\lambda({\mathscr{X}}\setminus E)$ and assume that $Q\in{\mathbb{D}}(E)$ is such that $I\in W_Q$. Then, from \eqref{gZSZa} we deduce that
\begin{eqnarray}\label{3a-YU.5} C_\ast^{-1}\ell(I)\leq\ell(Q)\leq C_\ast\ell(I)\quad\mbox{and}\quad {\rm dist}_\rho(I,Q)\leq C_\ast\ell(I). \end{eqnarray}
Now, \eqref{3a-YU.4} follows from \eqref{3a-YU.5} and the fact that
$\bigl(E,\rho\bigl|_{E}\bigr)$ is geometrically doubling. \end{proof}
\section{$T(1)$ and local $T(b)$ Theorems for Square Functions} \setcounter{equation}{0} \label{Sect:3}
This section consists of two parts, dealing with a $T(1)$ Theorem and a local $T(b)$ Theorem for square functions on sets of arbitrary co-dimension, relative to an ambient quasi-metric space (the notion of dimension refers to the degree of Ahlfors-David regularity). The $T(1)$ Theorem generalizes the Euclidean co-dimension one result proved by M.~Christ and J.-L.~Journ\'e in~\cite{CJ} (cf. also \cite[Theorem~20, p.\,69]{Ch}). The local $T(b)$ Theorem generalizes the Euclidean co-dimension one result that was implicit in the solution of the Kato problem in \cite{HMc,HLMc,AHLMcT}, and formulated explicitly in \cite{Au,Ho3,HMc2}.
We consider the following context. Fix two real numbers $d,m$ such that $0<d<m$, an $m$-dimensional {\rm ADR} space $({\mathscr{X}},\rho,\mu)$, a closed subset $E$ of $({\mathscr{X}},\tau_\rho)$, and a Borel measure $\sigma$
on $(E,\tau_{\rho|_{E}})$ with the property that $(E,\rho\bigl|_E,\sigma)$ is a $d$-dimensional {\rm ADR} space. In this context, suppose that
\begin{eqnarray}\label{K234} \begin{array}{c} {\theta}:(\mathscr{X}\setminus E)\times E\longrightarrow{{\mathbb{R}}} \quad\mbox{is Borel measurable with respect to} \\[4pt] \mbox{the relative topology induced by the product topology $\tau_\rho\times\tau_\rho$ on $(\mathscr{X}\setminus E)\times E$}, \end{array} \end{eqnarray}
and has the property that there exist finite positive constants $C_{\theta},\,\alpha,\,\upsilon$, and $a\in[0,\upsilon)$ such that for all $x\in\mathscr{X}\setminus E$ and $y\in E$ the following hold:
\begin{eqnarray}\label{hszz} && \hskip -0.35in
|{\theta}(x,y)|\leq\frac{C_{\theta}}{\rho(x,y)^{d+\upsilon}}\,\Bigl( \frac{{\rm dist}_\rho(x,E)}{\rho(x,y)}\Bigr)^{-a}, \\[4pt] && \hskip -0.40in \begin{array}{l}
\displaystyle|{\theta}(x,y)-{\theta}(x,\widetilde{y})|\leq C_{\theta} \frac{\rho(y,\widetilde{y})^\alpha}{\rho(x,y)^{d+\upsilon+\alpha}}\,\Bigl( \frac{{\rm dist}_\rho(x,E)}{\rho(x,y)}\Bigr)^{-a-\alpha}, \\[12pt] \qquad\forall\,\widetilde{y}\in E\,\,\mbox{ with }\,\, \rho(y,\widetilde{y})\leq\tfrac{1}{2}\rho(x,y). \end{array} \label{hszz-3} \end{eqnarray}
Then define the integral operator $\Theta$ for all functions $f\in L^p(E,\sigma)$, ${1\leq p\leq\infty}$, by
\begin{eqnarray}\label{operator} (\Theta f)(x):=\int_E {\theta}(x,y)f(y)\,d\sigma(y),\qquad\forall\,x\in\mathscr{X}\setminus E. \end{eqnarray} It follows from H\"older's inequality and Lemma~\ref{Gkwvr} that the integral in \eqref{operator} is absolutely convergent for each $x\in\mathscr{X}\setminus E$.
\begin{remark}\label{rem:a} The factors in parentheses in \eqref{hszz}-\eqref{hszz-3} are greater than or equal to $1$, since for every $x\in\mathscr{X}\setminus E$ and every $y\in E$ we have $\rho(x,y)\geq{\rm dist}_\rho(x,E)>0$, hence \eqref{hszz}-\eqref{hszz-3} are less demanding than their respective versions in which these factors are omitted. \end{remark}
We proceed to prove square function versions of the $T(1)$ Theorem and the local $T(b)$ Theorem for the integral operator $\Theta$. As usual, we prove the local $T(b)$ Theorem by verifying the hypotheses of the $T(1)$ Theorem, to which we now turn.
\subsection{An arbitrary codimension $T(1)$ theorem for square functions} \label{SSect:3.1} The main result in this subsection is a $T(1)$ theorem for square functions, to the effect that {\it a square function estimate for the integral operator $\Theta$
holds if and only if $|\Theta(1)|^2$, appropriately weighted by a power of the distance to $E$, is the density (relative to $\mu$) of a Carleson measure on ${\mathscr{X}}\setminus E$}. To state this formally, the reader is advised to recall the dyadic cube grid from Proposition~\ref{Diad-cube} and the regularized distance function to a set from \eqref{REG-DDD}.
\begin{theorem}\label{SChg} Let $d,m$ be two real numbers such that $0<d<m$. Assume that $({\mathscr{X}},\rho,\mu)$ is an $m$-dimensional {\rm ADR} space, $E$ is a closed subset of $({\mathscr{X}},\tau_\rho)$, and $\sigma$ is a Borel regular measure on
$(E,\tau_{\rho|_{E}})$ with the property that $(E,\rho\bigl|_E,\sigma)$ is a $d$-dimensional {\rm ADR} space.
Suppose that $\Theta$ is the integral operator defined in \eqref{operator} with a kernel ${\theta}$ as in \eqref{K234}, \eqref{hszz}, \eqref{hszz-3}. Furthermore, let ${\mathbb{D}}(E)$ denote a dyadic cube structure on $E$, consider a Whitney covering ${\mathbb{W}}_\lambda(\mathscr{X}\setminus E)$ of $\mathscr{X}\setminus E$ and a constant $C_\ast$ as in Lemma~\ref{Lem:CQinBQ-N} and, corresponding to these, recall the dyadic Carleson tents from \eqref{gZSZ-3}.
In this context, if
\begin{eqnarray}\label{UEHg} \sup_{Q\in{\mathbb{D}}(E)}\left(\tfrac{1}{\sigma(Q)}\int_{T_E(Q)}
|\Theta 1(x)|^2\delta_E(x)^{2\upsilon-(m-d)}\,d\mu(x)\right)<\infty, \end{eqnarray}
then there exists a finite constant $C>0$ depending only on the constants $C_{\theta}$, the {\rm ADR} constants of $E$ and ${\mathscr{X}}$, and the value of the supremum in \eqref{UEHg}, such that for each function $f\in L^2(E,\sigma)$ one has
\begin{eqnarray}\label{G-UF.22} \int\limits_{\mathscr{X}\setminus E}
|(\Theta f)(x)|^2\delta_E(x)^{2\upsilon-(m-d)}\, d\mu(x)\leq C\int_E|f(x)|^2\,d\sigma(x),\qquad\forall\,f\in L^2(E,\sigma). \end{eqnarray}
Finally, the converse of the implication discussed above is also true. In fact, the following stronger claim holds: under the original background assumptions, except that the regularity requirement \eqref{hszz-3} is now dropped, the fact that
\begin{eqnarray}\label{G-UF} \int\limits_{\stackrel{x\in\mathscr{X}}{0<\delta_E(x)<\eta\,{\rm diam}_\rho(E)}}
\!\!\!\!\!\!\!\!|(\Theta f)(x)|^2\delta_E(x)^{2\upsilon-(m-d)}\, d\mu(x)\leq C\int_E|f(x)|^2\,d\sigma(x),\quad\forall\,f\in L^2(E,\sigma), \end{eqnarray}
holds for some $\eta\in(0,\infty)$ implies that \eqref{UEHg} holds as well. \end{theorem}
Before presenting the actual proof of Theorem~\ref{SChg} we shall discuss a number of preliminary lemmas, starting with the following discrete Carleson estimate.
\begin{lemma}\label{brFC} Assume $(E,\rho,\sigma)$ is a space of homogeneous type with the property that $\sigma$ is Borel regular, and denote by ${\mathbb{D}}(E)$ a dyadic cube structure on $E$. If a sequence $\bigl\{B_Q\bigr\}_{Q\in{\mathbb{D}}(E)}\subseteq [0,\infty]$ satisfies the discrete Carleson condition
\begin{eqnarray}\label{TAkB} C:=\sup_{R\in{\mathbb{D}}(E)}\Bigl[ \frac{1}{\sigma(R)}\sum\limits_{Q\in{\mathbb{D}}(E),\,Q\subseteq R} B_Q \Bigr]<\infty, \end{eqnarray}
then for every sequence $\bigl\{A_Q\bigr\}_{Q\in{\mathbb{D}}(E)}\subseteq{\mathbb{R}}$ one has
\begin{eqnarray}\label{TAkB-2} \sum\limits_{Q\in{\mathbb{D}}(E)}A_Q B_Q\leq C\int_E A^\ast\,d\sigma, \end{eqnarray}
where $A^\ast:E\rightarrow [0,\infty]$ is the function defined by
\begin{eqnarray}\label{TAkB-2B} A^\ast(x):=0\,\,\mbox{ if }\,\,x\in E\setminus\!\!\bigcup_{Q\in{\mathbb{D}}(E)}Q
\quad\mbox{ and }\quad A^\ast(x):=\!\!\sup\limits_{Q\in{\mathbb{D}}(E),\,x\in Q}|A_Q| \,\,\,\mbox{ if }\,\,\,x\in\!\!\!\bigcup_{Q\in{\mathbb{D}}(E)}\!\!Q. \end{eqnarray} \end{lemma}
\begin{proof} For each $t>0$ define ${\mathcal{O}}_t:=\{x\in E:\,A^\ast(x)>t\}$. Then it is immediate from definitions that ${\mathcal{O}}_t=\!\!\!\bigcup\limits_{Q\in{\mathbb{D}}(E),\,A_Q>t}\!\!\!Q$ for every $t>0$. This shows that ${\mathcal{O}}_t$ is open in $(E,\tau_\rho)$ (cf. {\it (1)} in Proposition~\ref{Diad-cube}) and, hence, $A^\ast$ is $\sigma$-measurable. Note that if $A^\ast\in L^1(E,\sigma)$ (otherwise there is nothing to prove), then by Tschebyshev's inequality,
\begin{eqnarray}\label{TAkB-1} \sigma({\mathcal{O}}_t)\leq \frac{1}{t}\int_EA^\ast(x)\,d\sigma(x)<\infty, \qquad\forall\,t>0. \end{eqnarray}
This ensures that for each $t>0$ we may meaningfully define $D_t\subseteq{\mathbb{D}}(E)$, the collection of maximal dyadic cubes contained in ${\mathcal{O}}_t$, i.e.,
\begin{eqnarray}\label{TAkB-3} D_t:=\bigl\{R\in{\mathbb{D}}(E):\,R\subseteq{\mathcal{O}}_t\mbox{ and }\not\!\exists\, Q\in{\mathbb{D}}(E)\mbox{ such that $R\subseteq Q\subseteq{\mathcal{O}}_t$ and $R\not=Q$}\bigr\}. \end{eqnarray}
The cubes in $D_t$ are pairwise disjoint, and
\begin{eqnarray}\label{TAkB-4} {\mathcal{O}}_t=\bigcup\limits_{R\in D_t}R. \end{eqnarray}
Now for each $Q\in{\mathbb{D}}(E)$ define
\begin{eqnarray}\label{TAkB-5} h_Q:(0,\infty)\longrightarrow{\mathbb{R}},\qquad h_Q(t):=\left\{ \begin{array}{cc} 1,&\mbox{if }\,0<t<A_Q, \\[4pt] 0,&\mbox{otherwise.} \end{array} \right. \end{eqnarray}
Then, for each $t>0$ we have
\begin{eqnarray}\label{TAkB-6} \sum\limits_{Q\in{\mathbb{D}}(E)}h_Q(t)B_Q & = & \sum\limits_{Q\in{\mathbb{D}}(E),\,Q\subseteq{\mathcal{O}}_t}B_Q =\sum\limits_{R\in D_t}\left(\sum\limits_{Q\in{\mathbb{D}}(E),\,Q\subseteq R} B_Q\right) \nonumber\\[4pt] & \leq & C \sum\limits_{R\in D_t}\sigma(R) = C\sigma({\mathcal{O}}_t), \end{eqnarray}
where for the first inequality in \eqref{TAkB-6} we have used \eqref{TAkB}, while the last equality follows from \eqref{TAkB-4}. Hence,
\begin{eqnarray}\label{TAkB-7} \sum\limits_{Q\in{\mathbb{D}}(E)}A_QB_Q & = & \int_0^\infty\sum\limits_{Q\in{\mathbb{D}}(E)}h_Q(t)B_Q\,dt \leq C\int_0^\infty\sigma({\mathcal{O}}_t)\,dt \nonumber\\[4pt] & = & C \int_0^\infty\int_E{\mathbf{1}}_{\{A^\ast>t\}}(x)\,d\sigma(x)\,dt =C\int_E \int_0^\infty{\mathbf{1}}_{\{A^\ast>t\}}(x)\,dt\,d\sigma(x) \nonumber\\[4pt] & = & C\int_EA^\ast(x)\,d\sigma(x), \end{eqnarray}
completing the proof of the lemma. \end{proof}
We continue by recording a quantitative version of the classical Urysohn lemma in the context of H\"older functions on quasi-metric spaces from \cite{MMMM-G} (cf. also \cite{AMM} for a refinement).
\begin{lemma}\label{GVa2} Let $(E,\rho)$ be a quasi-metric space and assume that $\beta$ is a real number with the property that $0<\beta\leq\left[\log_2 C_{\rho}\right]^{-1}$. Assume that $F_0,F_1\subseteq E$ are two nonempty sets with the property that ${\rm dist}_\rho(F_0,F_1)>0$. Then, there exists a function $\eta:E\to{\mathbb{R}}$ such that
\begin{eqnarray}\label{PMab5} 0\leq\eta\leq 1\,\,\,\mbox{ on }\,\,E,\quad \eta\equiv 0\,\,\mbox{ on }\,\,F_0,\quad \eta\equiv 1\,\,\mbox{ on }\,\,F_1, \end{eqnarray}
and for which there exists a finite constant $C>0$, depending only on $\rho$, such that
\begin{eqnarray}\label{PMab6}
\sup_{\stackrel{x,y\in E}{x\not=y}}\frac{|\eta(x)-\eta(y)|}{\rho(x,y)^\beta} \leq C\bigl({\rm dist}_\rho(F_0,F_1)\bigr)^{-\beta}. \end{eqnarray} \end{lemma}
In the proof of Theorem~\ref{SChg} we shall also need a couple of results of geometric measure theoretic nature, which we next discuss.
\begin{lemma}\label{Gkwvr} Let $({\mathscr{X}},\rho)$ be a quasi-metric space. Suppose $E\subseteq{\mathscr{X}}$ is nonempty and $\sigma$ is a measure on $E$ such that
$(E,\rho\bigl|_E,\sigma)$ becomes a $d$-dimensional {\rm ADR} space, for some $d>0$. Fix a real number $m>d$. Then there exists $C\in(0,\infty)$ depending only on $m$, $\rho$, and the {\rm ADR} constant of $E$ such that
\begin{eqnarray}\label{mMji} \int_E\frac{1}{\rho_{\#}(x,y)^{m}}\,d\sigma(y)\leq C\delta_E(x)^{d-m}, \qquad\forall\,x\in {\mathscr{X}}\setminus E. \end{eqnarray}
Also, for each $\varepsilon>0$ and $c>0$, there exists $C\in(0,\infty)$ depending only on $\varepsilon$, $c$, $\rho$, and the {\rm ADR} constant of $E$ such that for every $\sigma$-measurable function $f:E\to[0,\infty]$ one has
\begin{eqnarray}\label{WBA} \int\limits_{y\in E,\,\rho_{\#}(y,x)>cr} \frac{r^\varepsilon}{\rho_{\#}(y,x)^{d+\varepsilon}}f(y)\,d\sigma(y) \leq C\,M_E(f)(x)\qquad\forall\,x\in E,\quad\forall\,r>0, \end{eqnarray}
where $M_E$ is as in \eqref{HL-MAX}. \end{lemma}
\begin{proof} Fix $x\in{\mathscr{X}}\setminus E$. Then
\begin{eqnarray}\label{jrap} \int_E \frac{1}{\rho_{\#}(y,x)^{m}}\,d\sigma(y) &\leq & \int_E \mathbf{1}_{\{z:\,\rho_{\#}(z,x)\geq\delta_E(x)\}}(y) \frac{1}{\rho_{\#}(y,x)^{m}}\,d\sigma(y) \nonumber\\[4pt] &= & C\sum_{j=0}^\infty\int_E \mathbf{1}_{\{z:\,\rho_{\#}(z,x)\in[2^j\delta_E(x),2^{j+1}\delta_E(x))\}}(y) \frac{1}{\rho_{\#}(y,x)^{m}}\,d\sigma(y) \nonumber\\[4pt] &\leq& C\sum_{j=0}^\infty\frac{1}{[2^j\delta_E(x)]^{m}} \sigma\left(B_{\rho_{\#}}(x,2^{j+1}\delta_E(x))\cap E\right) \nonumber\\[4pt] &\leq& C\sum_{j=0}^\infty\frac{1}{[2^j\delta_E(x)]^{m}} \bigl[2^{j+1}\delta_E(x)\bigr]^{d} \nonumber\\[4pt] &=& C\delta_E(x)^{d-m}, \end{eqnarray}
where for the last inequality in \eqref{jrap} we have used the fact that
$(E,\rho_{\#}|_E,\sigma)$ is a $d$-dimensional {\rm ADR} space, while the last equality uses the condition $m>d$. This concludes the proof of \eqref{mMji}. Finally, \eqref{WBA} is proved similarly, by decomposing the domain of integration in dyadic annuli centered at $x$, at scale $r$, and then using
the fact that $(E,\rho_{\#}|_E,\sigma)$ is a $d$-dimensional {\rm ADR} space. \end{proof}
For a proof of our second result of geometric measure theoretic nature the interested reader is referred to \cite{MMMM-B}, where more general results of this type are established.
\begin{lemma}\label{geom-lem} Assume that $({\mathscr{X}},\rho,\mu)$ is an $m$-dimensional {\rm ADR} space for some $m>0$ and let $E\subseteq{\mathscr{X}}$ be nonempty, closed.
Suppose there exists a measure $\sigma$ on $E$ such that $(E,\rho\bigl|_E,\sigma)$ is a $d$-dimensional {\rm ADR} space with $0<d<m$. If $\gamma<m-d$, then there exists a finite positive constant $C_0$ which depends only on $\gamma$ and the {\rm ADR} constants of $E$ and ${\mathscr{X}}$, such that
\begin{eqnarray}\label{lbzF} \int\limits_{x\in B_\rho(x_0,R),\,\delta_E(x)<r} \delta_E(x)^{\,-\gamma}\,d\mu(x)\leq C_0\,r^{m-d-\gamma}R^d, \end{eqnarray}
for every $x_0\in E$ and every $r,R>0$. \end{lemma}
At this stage, we are ready to present the
\vskip 0.08in \begin{proof}[Proof of Theorem~\ref{SChg}] For notational simplicity, abbreviate $A_Qf:={\int{\mkern-16mu}-}_Qf\,d\sigma$, for every cube $Q\in{\mathbb{D}}(E)$ whenever $f:E\to{\mathbb{C}}$ is locally integrable. Also, recall our convention that $\delta_E(x)$ stands for ${\rm dist}_{\rho_{\#}}(x,E)$, for every $x\in\mathscr{X}\setminus E$, and recall the Hardy-Littlewood maximal operator $M_E$ from \eqref{HL-MAX}. We break down the proof of the implication ``\eqref{UEHg}$\Rightarrow$\eqref{G-UF.22}" into a number of steps.
\vskip 0.10in \noindent{\tt Step~I.} {\it We claim that for every $r\in(1,\infty)$ there exist finite positive constants $C$ and $\beta$ such that for each $l,k\in{\mathbb{Z}}$ with $l,k\geq\kappa_E$ and every $Q\in{\mathbb{D}}_k(E)$, fixed thereafter, the following inequality holds}:
\begin{eqnarray}\label{Lvsn}
\sup\limits_{x\in{\mathcal{U}}_Q}\Bigl|
\delta_E(x)^\upsilon\bigl(\Theta(D_lg)(x)-(\Theta 1)(x)A_Q(D_lg)\bigr)\Bigr|
\leq C2^{-|k-l|\beta}\inf\limits_{{w}\in Q}\Bigl[
M_E^2(|g|^r)({w})\Bigr]^{\frac{1}{r}}, \end{eqnarray}
{\it for every $g:E\to{\mathbb{R}}$ locally integrable}. Here, $D_l$ is the operator introduced in \eqref{opD}, whose integral kernel is denoted by $h_l(\cdot,\cdot)$ (cf. the discussion in the proof of Proposition~\ref{HS-PP.3}).
To justify \eqref{Lvsn}, fix $k\in{\mathbb{Z}}$ with $k\geq\kappa_E$, $Q\in{\mathbb{D}}_k(E)$, and pick a number $k_0\in{\mathbb{N}}_0$ to be specified later, purely in terms of geometrical constants. We distinguish two cases.
\vskip 0.10in {\it Case I: $k+{k_0}\geq l$.} As a preamble, we remark that $k+{k_0}\geq l$ forces
\begin{eqnarray}\label{RRT}
2^{-(k+{k_0}-l)}\approx 2^{-|k-l|}, \end{eqnarray}
where the comparability constants depend only on ${k_0}$. Indeed,
observe that if $j\in{\mathbb{R}}$ is such that $j\geq -{k_0}$, then $j\leq|j|\leq j+2{k_0}$, hence $2^{-j}\approx 2^{-|j|}$. Taking now $j:=k-l$, \eqref{RRT} follows.
Turning now to the proof of \eqref{Lvsn} in earnest, using Fubini's Theorem, for each $l\in{\mathbb{Z}}$, we write
\begin{eqnarray}\label{FDcs} &&\hskip -0.60in \delta_E(x)^\upsilon\bigl(\Theta(D_lg)(x)-(\Theta 1)(x)A_Q(D_lg)\bigr) \nonumber\\[4pt] &&\,\,=\delta_E(x)^\upsilon\int_E{\theta}(x,y)\int_Eh_l(y,z)g(z)\,d\sigma(z)\,d\sigma(y) \nonumber\\[4pt] &&\hskip 0.20in -\delta_E(x)^\upsilon(\Theta 1)(x){\int{\mkern-19mu}-}_Q\int_E h_l(y,z) g(z)\,d\sigma(z)\,d\sigma(y) \nonumber\\[4pt] &&\,\, = \int_E\left[\int_E\Phi(x,y)h_l(y,z)\,d\sigma(y) \right]g(z)\,d\sigma(z),\qquad\forall\,x\in{\mathcal{U}}_Q, \end{eqnarray}
where
\begin{eqnarray}\label{FDcs-2} \Phi(x,y):=\delta_E(x)^\upsilon\Bigl[{\theta}(x,y)- \tfrac{1}{\sigma(Q)}(\Theta 1)(x){\mathbf{1}}_Q(y)\Bigr],\quad \forall\,x\in\mathscr{X}\setminus E,\quad\forall\,y\in E. \end{eqnarray}
Note that, by design,
\begin{eqnarray}\label{FDcs-3} \int_E\Phi(x,y)\,d\sigma(y)=0,\qquad\forall\,x\in\mathscr{X}\setminus E, \end{eqnarray}
and we claim that
\begin{eqnarray}\label{FDcs-4}
|\Phi(x,y)|\leq\frac{C}{\sigma(Q)},\qquad\forall\,x\in{\mathcal{U}}_Q,\quad\forall\,y\in E. \end{eqnarray}
Indeed, if $x\in{\mathcal{U}}_Q$ then $\delta_E(x)\approx\ell(Q)$ (with constants independent of $x$) and making use of \eqref{hszz} and the fact that $\delta_E(\cdot)\approx{\rm dist}_\rho(\cdot,E)$ (see {\it (4)} in Theorem~\ref{JjEGh}), we obtain
\begin{eqnarray}\label{FDcs-5}
\delta_E(x)^\upsilon|{\theta}(x,y)| \leq\frac{C\delta_E(x)^{\upsilon-a}}{\rho(x,y)^{d+\upsilon-a}} \leq\frac{C\delta_E(x)^{\upsilon-a}}{\delta_E(x)^{d+\upsilon-a}} \leq\frac{C}{\ell(Q)^d} \leq\frac{C}{\sigma(Q)},\qquad\forall\,y\in E. \end{eqnarray}
In addition,
\begin{eqnarray}\label{FDcs-6}
\delta_E(x)^\upsilon|(\Theta1)(x)|\leq C\delta_E(x)^{\upsilon-a} \int_{y\in E}\frac{d\sigma(y)}{\rho_{\#}(x,y)^{d+\upsilon-a}} \leq C,\qquad\forall\,x\in{\mathcal{U}}_Q, \end{eqnarray}
where for the last inequality in \eqref{FDcs-6} we made use of \eqref{mMji}. Now \eqref{FDcs-4} follows from \eqref{FDcs-2}, \eqref{FDcs-5} and \eqref{FDcs-6}.
Denote by $x_Q$ the center of $Q$ and let $\varepsilon\in(0,1)$ and $C_0>0$ be fixed, to be specified later. Then for every ${w}\in Q$ fixed, due to \eqref{FDcs-3} for each $z\in E$ we may write
\begin{eqnarray}\label{FDcs-7} && \hskip -0.20in
\left|\int_E\Phi(x,y)h_l(y,z)\,d\sigma(y)\right|
=\left|\int_E\Phi(x,y)[h_l(y,z)-h_l({w},z)]\,d\sigma(y)\right| \nonumber\\[4pt] && \hskip 0.70in \leq\int\limits_{y\in E,\,\rho_{\#}(y,x_Q)\leq C_02^{(k+{k_0}-l)\varepsilon}\ell(Q)}
|\Phi(x,y)||h_l(y,z)-h_l({w},z)|\,d\sigma(y) \nonumber\\[4pt] && \hskip 0.80in +\int\limits_{y\in E,\,\rho_{\#}(y,x_Q)>C_02^{(k+{k_0}-l)\varepsilon}\ell(Q)}
|\Phi(x,y)||h_l(y,z)-h_l({w},z)|\,d\sigma(y) \nonumber\\[4pt] && \hskip 0.70in =:I_1+I_2. \end{eqnarray}
In order to estimate $I_1$ we make the claim that if $\widetilde{C}$ is chosen large enough (compared to finite positive background constants and $C_0$) then
\begin{eqnarray}\label{vsKJb} \begin{array}{l}
|h_l(y,z)-h_l({w},z)|\leq C\,2^{\,l(d+\gamma)}2^{(k+{k_0}-l)\varepsilon\gamma} \ell(Q)^\gamma {\mathbf{1}}_{\{\rho_{\#}({w},\cdot)\leq\widetilde{C}2^{-l}\}}(z) \\[10pt] \mbox{whenever }\,\,z\in E,\,y\in E,\,{w}\in Q\,\,\mbox{ and }\,\, \rho_{\#}(y,x_Q)\leq C_02^{(k+{k_0}-l)\varepsilon}\ell(Q), \end{array} \end{eqnarray}
with $\gamma$ as in \eqref{condh-1}. To justify this claim, first note that if $C_0$ is large, then since $k+{k_0}-l\geq 0$, we have
\begin{eqnarray}\label{vsKJb-2} \hskip -0.15in y\in E,\,{w}\in Q\mbox{ and }\rho_{\#}(y,x_Q)\leq C_02^{(k+{k_0}-l)\varepsilon}\ell(Q) \,\Rightarrow\,\rho_{\#}(y,{w})\leq CC_02^{(k+{k_0}-l)\varepsilon}\ell(Q). \end{eqnarray}
From now on, assume that $C_0$ is large enough to ensure the validity of \eqref{vsKJb-2}. Second, if $y,{w}$ are as in \eqref{vsKJb-2} and $z\in E$ is such that $\rho_{\#}(z,{w})\geq\widetilde{C}2^{-l}$, then
\begin{eqnarray}\label{vsKJb-3} \widetilde{C}2^{-l} &\leq & \rho_{\#}(z,{w})\leq C(\rho_{\#}(z,y)+\rho_{\#}(y,{w})) \leq C\rho_{\#}(z,y)+CC_02^{(k+{k_0}-l)\varepsilon}\ell(Q) \nonumber\\[4pt] & \leq & C\rho_{\#}(z,y)+2^{k_0}C_0C2^{-l}, \end{eqnarray}
for some finite geometric constant $C>0$. Now choosing $\widetilde{C}:=2^{{k_0}+1}C_0C$ (which is permissible since, in the end, the parameter ${k_0}\in{\mathbb{N}}_0$ is chosen to depend only on finite positive background geometrical constants) we may absorb $2^{k_0}C_0C2^{-l}$ into $\widetilde{C}2^{-l}$ yielding (with $C_1:=2^{k_0}C_0C$)
\begin{eqnarray}\label{vsKJb-4} \left. \begin{array}{r} y\in E,\,{w}\in Q,\,\rho_{\#}(y,x_Q)\leq C_02^{(k+{k_0}-l)\varepsilon}\ell(Q) \\[4pt] \mbox{and }\,\,\rho_{\#}(z,{w})\geq\widetilde{C}2^{-l} \end{array} \right\}\,\Longrightarrow\,\rho_{\#}(z,y)>C_12^{-l}. \end{eqnarray}
Moreover, we can further increase $C_0$ and, in turn, $\widetilde{C}$ to insure that the constant $C_1$ in the last inequality in \eqref{vsKJb-4} is larger that the constant $C$ in \eqref{condh-1}. Henceforth, assume that such a choice has been made. Then a combination of \eqref{vsKJb-2}, \eqref{vsKJb-4} and \eqref{condh-2} yields \eqref{vsKJb}.
Next, we use \eqref{FDcs-4} and \eqref{vsKJb} in order to estimate
\begin{eqnarray}\label{AsQ} I_1 & \leq & \frac{C}{\sigma(Q)}\, 2^{\,l(d+\gamma)}\,2^{(k+{k_0}-l)\varepsilon\gamma}\, 2^{-k\gamma}\,{\mathbf{1}}_{\{\rho_{\#}({w},\cdot)\leq\widetilde{C}2^{-l}\}}(z) \!\!\!\! \int\limits_{y\in E,\,\rho_{\#}(y,x_Q)\leq C_02^{(k+{k_0}-l)\varepsilon}\ell(Q)} \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!1\,d\sigma(y) \nonumber\\[4pt] & \leq & C \,2^{-(k+{k_0}-l)[\gamma-\varepsilon(d+\gamma)]}2^{\,dl}\, {\mathbf{1}}_{\{\rho_{\#}({w},\cdot)\leq\widetilde{C}2^{-l}\}}(z), \qquad\forall\,z\in E, \end{eqnarray}
where for the last inequality in \eqref{AsQ} we have used the fact that
$(E,\rho|_E,\sigma)$ is a $d$-dimensional {\rm ADR} space. At this point we choose $0<\varepsilon<\frac{\gamma}{d+\gamma}<1$ which ensures that $\beta_1:=\gamma-\varepsilon(d+\gamma)>0$, hence
\begin{eqnarray}\label{AsQ-2} I_1\leq C\,2^{\,dl}\,2^{-(k+{k_0}-l)\beta_1}\,{\mathbf{1}}_{\{\rho_{\#}({w},\cdot) \leq\widetilde{C}2^{-l}\}}(z),\qquad\forall\,z\in E. \end{eqnarray}
To estimate the contribution from $I_1$ in the context of \eqref{FDcs}, based on \eqref{AsQ-2} and \eqref{RRT} we write (recall that $\beta_1=\gamma-\varepsilon(d+\gamma)>0$ is a fixed constant)
\begin{eqnarray}\label{Mxr}
\int_EI_1|g(z)|\,d\sigma(z)
& \leq & C\,2^{\,dl}\,2^{-(k+{k_0}-l)\beta_1}\,\int_E|g(z)| {\mathbf{1}}_{\{\rho_{\#}({w},\cdot)\leq C2^{-l}\}}(z)\,d\sigma(z) \nonumber\\[4pt]
& = & C2^{-(k+{k_0}-l)\beta_1}{\int{\mkern-19mu}-}_{z\in E,\,\rho_{\#}(z,{w})\leq C2^{-l}}|g(z)|\,d\sigma(z) \nonumber\\[4pt] & = & C2^{-(k+{k_0}-l)\beta_1} (M_Eg)({w}) \nonumber\\[4pt]
& \approx & 2^{-|k-l|\beta_1} (M_Eg)({w}), \quad\mbox{uniformly in }{w}\in Q. \end{eqnarray}
Next, we turn our attention to $I_2$ from \eqref{FDcs-7}. Note that since we are currently assuming that $k+{k_0}\geq l$, the condition $\rho_{\#}(y,x_Q)\geq C_02^{(k+{k_0}-l)\varepsilon}\ell(Q)$ forces $y\not\in c_1Q$ for some finite positive constant $c_1$, which may be further increased as desired by suitably increasing the value of $C_0$. Thus, assuming that $C_0$ is sufficiently large to guarantee $c_1>1$, we obtain ${\mathbf{1}}_Q(y)=0$ if $y\in E$ and $\rho_{\#}(y,x_Q)\geq C_02^{(k+{k_0}-l)\varepsilon}\ell(Q)$. In turn, this implies that $\Phi(x,y)=\delta_E(x)^\upsilon {\theta}(x,y)$ on the domain of integration in $I_2$. Thus, for each $z\in E$, we have
\begin{eqnarray}\label{AsQ-3-A} I_2 & \leq & C2^{-k\upsilon}\int\limits_{y\in E,\,\rho_{\#}(y,x_Q)>C_02^{(k+{k_0}-l)\varepsilon}\ell(Q)}
|{\theta}(x,y)|\,|h_l(y,z)|\,d\sigma(y) \\[4pt]
&&\,\,+C2^{-k\upsilon}|h_l({w},z)| \int\limits_{y\in E,\,\rho_{\#}(y,x_Q)>C_02^{(k+{k_0}-l)\varepsilon}\ell(Q)}
|{\theta}(x,y)|\,d\sigma(y)=:I_3+I_4. \nonumber \end{eqnarray}
We also remark that the design of ${\mathcal{U}}_Q$ and the fact that $k+{k_0}-l\geq 0$ ensure that
\begin{eqnarray}\label{AsQ-3} y\in E,\,\,\rho_{\#}(y,x_Q)>C_02^{(k+{k_0}-l)\varepsilon}\ell(Q)\,\Longrightarrow\, \left\{ \begin{array}{l} \rho_{\#}(x,y)\approx\rho_{\#}({w},y)\approx\rho_{\#}(x_Q,y), \\[4pt] \mbox{uniformly for $x\in{\mathcal{U}}_Q$ and ${w}\in Q$}. \end{array}\right. \end{eqnarray}
Making first use of \eqref{hszz} combined with \eqref{AsQ-3} and the fact that since $x\in{\mathcal{U}}_Q$ we have $\delta_E(x)\approx\ell(Q)\approx 2^{-k}$, and then of \eqref{condh-1}, we may further estimate
\begin{eqnarray}\label{AsQ-4} I_3 & \leq & C2^{-k\upsilon}\int\limits_{y\in E,\, \rho_{\#}(y,{w})>C2^{(k+{k_0}-l)\varepsilon}\ell(Q)}
\frac{\delta_E(x)^{-a}}{\rho_{\#}(y,{w})^{d+\upsilon-a}}\,|h_l(y,z)|\,d\sigma(y) \nonumber\\[4pt] & \leq & C2^{-k\upsilon}2^{\,dl}\int\limits_{y\in E,\, \rho_{\#}(y,{w})>C2^{(k+{k_0}-l)\varepsilon}\ell(Q)} \frac{2^{\,ak}}{\rho_{\#}(y,{w})^{d+\upsilon-a}}\,{\mathbf{1}}_{\{\rho_{\#}(y,\cdot) \leq C2^{-l}\}}(z)\,d\sigma(y) \nonumber\\[4pt] & = & C\,2^{-(k+{k_0}-l)\varepsilon(\upsilon-a)}2^{dl} \int_{y\in E,\,\rho_{\#}(y,{w})\geq Cr} \frac{r^{\upsilon-a}}{\rho_{\#}(y,{w})^{d+\upsilon-a}}\, {\mathbf{1}}_{\{\rho_{\#}(y,\cdot)\leq C2^{-l}\}}(z)\,d\sigma(y),\qquad \end{eqnarray}
for each $z\in E$, where we have set
\begin{eqnarray}\label{yfwz} r:=2^{(k+{k_0}-l)\varepsilon-k}. \end{eqnarray}
Consequently, by \eqref{AsQ-4}, Fubini's theorem, \eqref{WBA} and \eqref{RRT}, we obtain
\begin{eqnarray}\label{Mxr-2} && \hskip -0.40in
\int_EI_3|g(z)|\,d\sigma(z)
\leq C\,2^{-(k+{k_0}-l)\varepsilon(\upsilon-a)}\,\int_E|g(z)|\times \nonumber\\[4pt] && \hskip 1.20in \times \int_{y\in E,\,\rho_{\#}(y,{w})\geq Cr} \frac{r^{\upsilon-a}}{\rho_{\#}(y,{w})^{d+\upsilon-a}}\,2^{\,dl}\, {\mathbf{1}}_{\{\rho_{\#}(y,\cdot)\leq C2^{-l}\}}(z)\,d\sigma(y)\,d\sigma(z) \nonumber\\[4pt] && \hskip 0.75in \leq C\,2^{-(k+{k_0}-l)\varepsilon(\upsilon-a)}\, \int_{y\in E,\,\rho_{\#}(y,{w})\geq Cr} \frac{r^{\upsilon-a}}{\rho_{\#}(y,{w})^{d+\upsilon-a}}\,(M_Eg)(y)\,d\sigma(y) \nonumber\\[4pt] && \hskip 0.75in \leq C2^{-(k+{k_0}-l)\varepsilon(\upsilon-a)}(M_E^2g)({w}) \nonumber\\[4pt] && \hskip 0.75in
\approx 2^{-|k-l|\varepsilon(\upsilon-a)}(M_E^2g)({w}), \quad\mbox{uniformly in }{w}\in Q. \end{eqnarray}
As for $I_4$, invoking again \eqref{hszz}, \eqref{AsQ-3}, the fact that $\delta_E(x)\approx2^{-k}$, and \eqref{condh-1} we write
\begin{eqnarray}\label{AsQ-5} I_4 & \leq & C2^{-k\upsilon}2^{\,dl} {\mathbf{1}}_{\{\rho_{\#}({w},\cdot)\leq C2^{-l}\}}(z) \int\limits_{y\in E,\,\rho_{\#}(y,x_Q)>Cr} \frac{2^{ak}}{\rho_{\#}(y,x_Q)^{d+\upsilon-a}}\,d\sigma(y) \nonumber\\[4pt] & = & C2^{-(k+{k_0}-l)\varepsilon(\upsilon-a)}\,2^{\,dl} {\mathbf{1}}_{\{\rho_{\#}({w},\cdot)\leq C2^{-l}\}}(z) \int\limits_{y\in E,\,\rho_{\#}(y,x_Q)>Cr} \frac{r^{\upsilon-a}}{\rho_{\#}(y,x_Q)^{d+\upsilon-a}}\,d\sigma(y) \nonumber\\[4pt] & = & C2^{-(k+{k_0}-l)\varepsilon(\upsilon-a)}\,2^{\,dl} {\mathbf{1}}_{\{\rho_{\#}({w},\cdot)\leq C2^{-l}\}}(z),\qquad\forall\,z\in E, \end{eqnarray}
by \eqref{WBA} (used with $f\equiv 1$). Finally, based on \eqref{AsQ-5} and \eqref{RRT}, we obtain
\begin{eqnarray}\label{Mxr-3}
\int_EI_4|g(z)|\,d\sigma(z) & \leq & C\,2^{-(k+{k_0}-l)\varepsilon(\upsilon-a)}\,2^{\,dl}\,
\int_E|g(z)|{\mathbf{1}}_{\{\rho_{\#}({w},\cdot)\leq C2^{-l}\}}(z)\,d\sigma(z) \nonumber\\[4pt] & \leq & C\,2^{-(k+{k_0}-l)\varepsilon(\upsilon-a)}\,(M_Eg)({w}) \nonumber\\[4pt]
& \approx & 2^{-|k-l|\varepsilon(\upsilon-a)}(M_Eg)({w}), \quad\mbox{uniformly in }{w}\in Q. \end{eqnarray}
Collectively, \eqref{AsQ-4} and \eqref{AsQ-5} yield an estimate for $I_2$, in view of \eqref{AsQ-3-A}. In order to express this estimate as well as \eqref{Mxr} in a manner consistent with \eqref{Lvsn} requires an extra adjustment. Concretely, as a consequence of Lebesgue's Differentiation Theorem (which holds in our context given that $\sigma$ is Borel regular), the monotonicity of the Hardy-Littlewood maximal operator and H\"older's inequality, for every $r\in[1,\infty)$ and any $g$ locally integrable on $E$ we have
\begin{eqnarray}\label{bvez}
M_E g\leq\bigl[M_E^2(|g|^r)\bigr]^{\frac{1}{r}}\quad\mbox{and}\quad M_E^2 g\leq\bigl[M_E^2(|g|^r)\bigr]^{\frac{1}{r}}\qquad\mbox{pointwise in $E$}. \end{eqnarray}
Thus, if we define $\beta_2:=\min\{\beta_1,\varepsilon(\upsilon-a)\}>0$, then a combination of \eqref{FDcs}, \eqref{FDcs-7}, \eqref{Mxr}, \eqref{AsQ-3-A}, \eqref{Mxr-2}, \eqref{Mxr-3}, and \eqref{bvez} proves \eqref{Lvsn} with $\beta$ replaced by $\beta_2$ in {\it Case~I}.
\vskip 0.10in {\it Case II: $k+k_0<l$.} In this scenario, in order to obtain estimate \eqref{Lvsn} we shall use the cancellation property of the operator $D_l$, and the H\"older regularity of ${\theta}(\cdot,\cdot)$ in the second variable. To get started, Fubini's Theorem allows us to write
\begin{eqnarray}\label{hgfAW} \delta_E(x)^\upsilon\Theta(D_lg)(x) & = &\int_E\delta_E(x)^\upsilon\int_E{\theta}(x,y)h_l(y,z)\,d\sigma(y)\,g(z)\,d\sigma(z) \nonumber\\[4pt] & = &\int_E\Psi(x,z)\,g(z)\,d\sigma(z),\qquad\forall\,x\in{\mathcal{U}}_Q, \end{eqnarray}
where we have set
\begin{eqnarray}\label{hgfAW-BB} \Psi(x,z):=\delta_E(x)^\upsilon\int_E{\theta}(x,y)h_l(y,z)\,d\sigma(y),\qquad \forall\,x\in\mathscr{X}\setminus E,\quad\forall\,z\in E. \end{eqnarray}
To proceed, fix $x\in{\mathcal{U}}_Q$ arbitrary. Based on \eqref{condh-4} and \eqref{condh-1}, we have
\begin{eqnarray}\label{hgfAW-2} \hskip -0.30in
|\Psi(x,z)| &=& \left|\delta_E(x)^\upsilon\int_E\bigl({\theta}(x,y)-{\theta}(x,z)\bigr)h_l(y,z)\,d\sigma(y)\right| \nonumber\\[4pt] & \leq & \delta_E(x)^\upsilon\!\!\!\!\!\!\!\! \int\limits_{y\in E,\,\rho_{\#}(y,z)\leq C2^{-l}}\!\!\!\!\!\!\!\!
|{\theta}(x,y)-{\theta}(x,z)||h_l(y,z)|\,d\sigma(y), \qquad\forall\,z\in E. \end{eqnarray}
As a consequence of \eqref{hszz-3}, we have
\begin{eqnarray}\label{ghhzt}
|{\theta}(x,y)-{\theta}(x,z)|\leq C\, \frac{\rho(y,z)^\alpha\delta_E(x)^{-a-\alpha}}{\rho(x,y)^{d+\upsilon-a}} \qquad\mbox{if }\,\,y,z\in E,\,\,\rho(y,z)<\tfrac{1}{2}\rho(x,y). \end{eqnarray}
Observe that, since we are currently assuming $k+{k_0}<l$, if the points $y,z\in E$ are such that $\rho_{\#}(y,z)\leq C2^{-l}$, then
\begin{eqnarray}\label{ghhzt-2} \rho(y,z)\leq C\rho_{\#}(y,z)\leq C2^{-l}\leq C 2^{-{k_0}}2^{-k} \leq C2^{-{k_0}}\delta_E(x)\leq C2^{-{k_0}}\rho(x,y)<\tfrac{1}{2}\rho(x,y), \end{eqnarray}
where the last inequality follows by choosing ${k_0}$ large. For the remainder of the proof fix such a ${k_0}\in{\mathbb{N}}_0$. Then \eqref{ghhzt} holds when $y$ belongs to the domain of integration of the last integral in \eqref{hgfAW-2}. For ${w}\in Q$ arbitrary we claim that
\begin{eqnarray}\label{ghhzt-3} \left. \begin{array}{l} \exists\,C'>0\mbox{ such that }\,\forall\,z,y\in E \\[4pt] \mbox{ satisfying }\rho(y,z)\leq C2^{-l} \end{array} \right\} \,\,\mbox{one has}\,\, \left\{ \begin{array}{l} \rho(x,y)\geq C'[2^{-k}+\rho_{\#}({w},z)], \\[4pt] \mbox{ uniformly for } x\in{\mathcal{U}}_Q,\,{w}\in Q. \end{array} \right. \end{eqnarray}
Indeed, if $y,z$ are as in the left hand-side of \eqref{ghhzt-3}, then $\rho(x,y)\geq C\delta_E(x)\approx\ell(Q)\approx 2^{-k}$. In addition,
\begin{eqnarray}\label{dxgkr} \rho_{\#}({w},z) & \leq & C\rho({w},z)\leq C(\rho({w},x)+\rho(x,y)+\rho(y,z)) \leq C(\ell(Q)+\rho(x,y)+2^{-l}) \nonumber\\[4pt] & \leq & C(2^{-k}+\rho(x,y))\leq C\rho(x,y). \end{eqnarray}
This proves \eqref{ghhzt-3}. Combining \eqref{ghhzt-3}, \eqref{ghhzt}, \eqref{condh-1} and \eqref{hgfAW-2}, we obtain
\begin{eqnarray}\label{hgfAW-4} \hskip -0.30in
|\Psi(x,z)| & \leq & C2^{-k\upsilon}\int\limits_{y\in E,\,\rho_{\#}(y,z)\leq C2^{-l}} \frac{2^{k(a+\alpha)}2^{-l\alpha}}{[2^{-k}+\rho_{\#}({w},z)]^{d+\upsilon-a}} \,2^{\,dl}\,d\sigma(y) \nonumber\\[4pt] & \leq & C2^{-k\upsilon}\frac{2^{k(a+\alpha)}2^{-l\alpha}} {[2^{-k}+\rho_{\#}({w},z)]^{d+\upsilon-a}} \nonumber\\[4pt]
& = & C2^{-|k-l|\alpha} \frac{2^{-k(\upsilon-a)}}{[2^{-k}+\rho_{\#}({w},z)]^{d+\upsilon-a}}, \qquad\forall\,z\in E. \end{eqnarray}
For the second inequality in \eqref{hgfAW-4} we used the fact that $(E,\rho|_E,\sigma)$ is a $d$-dimensional {\rm ADR} space, while the last equality is a simple consequence of the fact that $k<l$. Thus, returning with \eqref{hgfAW-4} in \eqref{hgfAW}, then making use of \eqref{WBA}, and then recalling \eqref{bvez}, we arrive at the conclusion that for each $r\in[1,\infty)$,
\begin{eqnarray}\label{hgfAW-5}
\bigl|\delta_E(x)^\upsilon\Theta(D_lg)(x)\bigr|
& \leq & C 2^{-|k-l|\alpha}\int_E
\frac{2^{-k(\upsilon-a)}}{[2^{-k}+\rho_{\#}({w},z)]^{d+\upsilon-a}}|g(z)|\,d\sigma(z) \nonumber\\[4pt]
& \leq & C 2^{-|k-l|\alpha}(M_Eg)({w}) \nonumber\\[4pt]
& \leq & C2^{-|k-l|\alpha}\bigl[M_E^2(|g|^r)({w})\bigr]^\frac{1}{r}, \qquad\mbox{uniformly for }\,\,{w}\in Q. \end{eqnarray}
In order to estimate $|\delta_E(x)^\upsilon (\Theta 1)(x)A_Q(D_lg)|$, first note that \eqref{FDcs-6} holds in this case, so by Fubini's Theorem we have
\begin{eqnarray}\label{hgfAW-6}
|\delta_E(x)^\upsilon (\Theta 1)(x)A_Q(D_lg)| \leq C\left|\int_E\Bigl\{\tfrac{1}{\sigma(Q)}\int_Qh_l(y,z)\,d\sigma(y) \Bigr\}g(z)\,d\sigma(z)
\right|. \end{eqnarray}
To continue, for some fixed $\varepsilon\in(0,1)$, define
\begin{eqnarray}\label{hgfAW-7} S_Q:=\bigl\{x\in Q:\,{\rm dist}_{\rho_{\#}}(x,E\setminus Q)
\leq C 2^{-|k-l|\varepsilon}\ell(Q)\bigr\} \quad\mbox{ and }\quad F_Q:=Q\setminus S_Q. \end{eqnarray}
Also, consider a function
\begin{eqnarray}\label{zhzM} \begin{array}{c} \eta_Q:E\to{\mathbb{R}}\,\,\mbox{ such that }\,\,{\rm supp}\,\eta_Q\subseteq Q,\quad 0\leq\eta_Q\leq 1,\quad\eta_Q=1\mbox{ on }F_Q\,\mbox{ and} \\[4pt]
\Bigl|\eta_Q(x)-\eta_Q(y)\Bigr|\leq C
\Bigl(\frac{\rho(x,y)}{2^{-|k-l|\varepsilon}\ell(Q)} \Bigr)^\gamma,\qquad\forall\,x\in E,\,\,\forall\,y\in E, \end{array} \end{eqnarray}
for some $\gamma\in(0,1)$. That such a function exists is a consequence of Lemma~\ref{GVa2}. Hence,
\begin{eqnarray}\label{zhzMB} \hskip -0.30in
\left|{\int{\mkern-19mu}-}_Qh_l(y,z)\,d\sigma(y)\right|
&\leq & \tfrac{1}{\sigma(Q)}\left|\int_E\bigl({\mathbf{1}}_Q-\eta_Q(y)\bigr)
h_l(y,z)\,d\sigma(y)\right| \\[4pt] &&\hskip 0.06in
+\tfrac{1}{\sigma(Q)}\left|\int_E\eta_Q(y)h_l(y,z)\,d\sigma(y)\right| =:II_1(z)+II_2(z),\quad\forall\,z\in E. \nonumber \end{eqnarray}
Fix $z\in E$. To estimate $II_2(z)$, recall \eqref{condh-4}, \eqref{condh-1}, \eqref{zhzM} and the fact that $E$ is Ahlfors-David regular. Based on these, we may write
\begin{eqnarray}\label{zhzM-2}
II_2(z) &=& \tfrac{1}{\sigma(Q)}\left|\int_E\bigl(\eta_Q(y)-\eta_Q(z)\bigr)
h_l(y,z)\,d\sigma(y)\right| \\[4pt] &\leq & \tfrac{1}{\sigma(Q)}\!\!\!\! \int\limits_{y\in E,\,\rho_{\#}(y,z)\leq C2^{-l}}\!\!\!\!\!\!\!\!\!\!
\bigl|\eta_Q(y)-\eta_Q(z)\bigr||h_l(y,z)|\,d\sigma(y)
\leq \tfrac{1}{\sigma(Q)}\left[\frac{2^{-l}}{2^{-|k-l|\varepsilon}2^{-k}} \right]^{\gamma}. \nonumber \end{eqnarray}
In addition, since whenever $y\in{\rm supp}\,\eta_Q\subseteq Q$ and $\rho_{\#}(y,z)\leq C2^{-l}\leq C\ell(Q)$ one necessarily has $\rho_{\#}({w},z)\leq C\ell(Q)$, for all ${w}\in Q$, it follows that one may strengthen \eqref{zhzM-2} to
\begin{eqnarray}\label{zhzM-3}
II_2(z) \leq C \tfrac{1}{\sigma(Q)}2^{-|k-l|(1-\varepsilon)\gamma} {\mathbf{1}}_{\{\rho_{\#}({w},\cdot)\leq C\ell(Q)\}}(z),\qquad \mbox{ for all ${w}\in Q$}. \end{eqnarray}
Hence, by also recalling \eqref{bvez}, for each $r\in[1,\infty)$ we further obtain
\begin{eqnarray}\label{zhzM-4} \hskip -0.40in
\int_E II_2(z)|g(z)|\,d\sigma(z)
& \leq & C 2^{-|k-l|(1-\varepsilon)\gamma}\tfrac{1}{\sigma(Q)}
\int_{z\in E,\,\rho_{\#}(z,{w})\leq C\ell(Q)}|g(z)|\,d\sigma(z) \nonumber\\[4pt]
& \leq & C 2^{-|k-l|(1-\varepsilon)\gamma}(M_Eg)({w})
\leq C 2^{-|k-l|(1-\varepsilon)\gamma}\bigl[M_E^2(|g|^r)({w})\bigr]^\frac{1}{r}, \end{eqnarray}
for all ${w}\in Q$. This bound suits our purposes.
Next, we turn our attention to $II_1(z)$. Pick $r\in(1,\infty)$ and let $r'$ be such that $\frac{1}{r}+\frac{1}{r'}=1$. Note that since ${\rm supp}\,({\mathbf{1}}_Q-\eta_Q)\subseteq S_Q$ and $0\leq {\mathbf{1}}_Q-\eta_Q\leq 1$ we may write
\begin{eqnarray}\label{zhzM-5} \hskip -0.40in
\int_E II_1(z)|g(z)|\,d\sigma(z)
& \leq & \tfrac{1}{\sigma(Q)}\int_E\int_{S_Q}|h_l(y,z)|\,d\sigma(y)\,|g(z)|\,d\sigma(z) \nonumber\\[4pt]
&=& \tfrac{1}{\sigma(Q)}\int_{S_Q}\int_E|g(z)||h_l(y,z)|\,d\sigma(z)\,d\sigma(y) \nonumber\\[4pt] &\leq & C\tfrac{1}{\sigma(Q)}\int_{S_Q}\int_{z\in E,\,\rho_{\#}(y,z)\leq C2^{-l}}
2^{\,dl}|g(z)|\,d\sigma(z)\,d\sigma(y) \nonumber\\[4pt] &\leq & C\tfrac{1}{\sigma(Q)}\int_Q{\mathbf{1}}_{S_Q}(y)(M_Eg)(y)\,d\sigma(y) \nonumber\\[4pt] &\leq & C\left[\frac{\sigma(S_Q)}{\sigma(Q)}\right]^{\frac{1}{r'}} \left[{\int{\mkern-19mu}-}_Q(M_Eg)^r\,d\sigma\right]^{\frac{1}{r}} \nonumber\\[4pt] &\leq & C\left[\frac{\sigma(S_Q)}{\sigma(Q)}\right]^{\frac{1}{r'}}
\bigl[M_E^2(|g|^r)({w})\bigr]^{\frac{1}{r}},\qquad\mbox{ for all }{w}\in Q. \end{eqnarray}
The second inequality in \eqref{zhzM-5} is based on \eqref{condh-1}, the third is immediate, the fourth uses H\"older's inequality, while the last one is a consequence of \eqref{bvez}. By virtue of the ``thin boundary" property described in item {\it (8)} of Proposition~\ref{Diad-cube}, we have
\begin{eqnarray}\label{AAzrhg} \exists\,c>0\,\,\mbox{ and }\,\, \exists\,\tau\in(0,1)\,\,\mbox{ such that }\,\,
\sigma(S_Q)\leq c2^{-|k-l|\varepsilon\tau}\sigma(Q), \end{eqnarray}
which, when used in concert with \eqref{zhzM-5}, yields
\begin{eqnarray}\label{zhzM-6}
\int_E II_1(z)|g(z)|\,d\sigma(z)
\leq C 2^{-|k-l|\varepsilon\tau/r'}
\bigl[M_E^2(|g|^r)({w})\bigr]^{\frac{1}{r}}, \qquad\mbox{for all }\,{w}\in Q. \end{eqnarray}
Now choose $\beta_3:=\min\{(1-\varepsilon)\gamma,\frac{\varepsilon\tau}{r'},\alpha\}>0$. Then \eqref{Lvsn} follows in the current case with $\beta$ replaced by $\beta_3$, by combining \eqref{hgfAW-5}, \eqref{hgfAW-6}, \eqref{zhzMB}, \eqref{zhzM-4} and \eqref{zhzM-6}.
Now the proof of the claim made in Step~I is completed by combining what we proved in {\it Case I} and {\it Case II} above and by taking $\beta:=\min\{\beta_2,\beta_3\}>0$.
\vskip 0.10in \noindent{\tt Step~II.} {\it We claim that there exists a finite constant $C>0$ with the property that for every $f\in L^2(E,\sigma)$ there holds}
\begin{eqnarray}\label{N-NKy} \sum\limits_{k\in{\mathbb{Z}},\,k\geq\kappa_E}\sum\limits_{Q\in{\mathbb{D}}_k(E)}
\int_{{\mathcal{U}}_Q}\bigl|\delta_E(x)^\upsilon
\bigl((\Theta f)(x)-(\Theta 1)(x)A_Qf\bigr)\bigr|^2 \frac{d\mu(x)}{\delta_E(x)^{m-d}}
\leq C\int_E|f|^2\,d\sigma. \end{eqnarray}
To justify this claim, fix $r\in(1,2)$ and let $\beta>0$ be such that \eqref{Lvsn} holds. Then, for an arbitrary function $f\in L^2(E,\sigma)$, using \eqref{PKc}, we may write
\begin{eqnarray}\label{N-NKy-2AA} &&\hskip -0.20in \sum\limits_{k\in{\mathbb{Z}},\,k\geq\kappa_E}\sum\limits_{Q\in{\mathbb{D}}_k(E)}
\int_{{\mathcal{U}}_Q}\bigl|\delta_E(x)^\upsilon
\bigl(\Theta-(\Theta 1)(x)A_Q\bigr)(f)(x)\bigr|^2\frac{d\mu(x)}{\delta_E(x)^{m-d}} \\[4pt] && \leq 2\sum\limits_{k\in{\mathbb{Z}},\,k\geq\kappa_E}\sum\limits_{Q\in{\mathbb{D}}_k(E)}
\int_{{\mathcal{U}}_Q}\Bigl|\sum\limits_{l\in{\mathbb{Z}},\,l\geq \kappa_E}
\delta_E(x)^\upsilon\bigl(\Theta -(\Theta 1)(x)A_Q\bigr)(D_l\widetilde{D}_lf)(x)\Bigr|^2 \frac{d\mu(x)}{\delta_E(x)^{m-d}} \nonumber\\[4pt] && \hskip 0.10in +2\sum\limits_{k\in{\mathbb{Z}},\,k\geq\kappa_E}\sum\limits_{Q\in{\mathbb{D}}_k(E)}
\int_{{\mathcal{U}}_Q}\Bigl|\delta_E(x)^\upsilon
\bigl(\Theta -(\Theta 1)(x)A_Q\bigr)({\mathcal{S}}_{\kappa_E}(Rf))(x)\Bigr|^2 \frac{d\mu(x)}{\delta_E(x)^{m-d}}=:A_1+A_2. \nonumber \end{eqnarray}
Pick now $\varepsilon\in(0,\beta)$ arbitrary and proceed to estimate $A_1$ as follows:
\begin{eqnarray}\label{N-NKy-2} A_1\!&=& \!2\sum\limits_{\stackrel{k\in{\mathbb{Z}}}{k\geq\kappa_E}} \sum\limits_{Q\in{\mathbb{D}}_k(E)}
\int_{{\mathcal{U}}_Q}\Bigl|\sum\limits_{\stackrel{l\in{\mathbb{Z}}}{l\geq\kappa_E}}\!\!\! 2^{-|k-l|\varepsilon}2^{|k-l|\varepsilon}
\delta_E(x)^\upsilon\bigl(\Theta -(\Theta 1)(x)A_Q\bigr)(D_l\widetilde{D}_lf)(x)\Bigr|^2 \frac{d\mu(x)}{\delta_E(x)^{m-d}} \nonumber\\[4pt] &\leq& 2\sum\limits_{\stackrel{k\in{\mathbb{Z}}}{k\geq\kappa_E}} \sum\limits_{Q\in{\mathbb{D}}_k(E)}
\int_{{\mathcal{U}}_Q}\Bigl(\sum\limits_{l\in{\mathbb{Z}}}2^{-2|k-l|\varepsilon}\Bigr) \times \nonumber\\[4pt] && \hskip 1.00in
\times\Bigl(\sum\limits_{\stackrel{l\in{\mathbb{Z}}}{l\geq\kappa_E}}2^{2|k-l|\varepsilon}
\bigl|\delta_E(x)^\upsilon\bigl(\Theta -(\Theta 1)(x)A_Q\bigr)
(D_l\widetilde{D}_lf)(x)\bigr|^2\Bigr)\frac{d\mu(x)}{\delta_E(x)^{m-d}} \nonumber\\[4pt] &\leq& C\sum\limits_{\stackrel{l\in{\mathbb{Z}}}{l\geq\kappa_E}} \sum\limits_{\stackrel{k\in{\mathbb{Z}}}{k\geq\kappa_E}}
\sum\limits_{Q\in{\mathbb{D}}_k(E)}2^{2|k-l|\varepsilon}\int_{{\mathcal{U}}_Q}
\bigl|\delta_E(x)^\upsilon\bigl(\Theta -(\Theta 1)(x)A_Q\bigr)
(D_l\widetilde{D}_lf)(x)\bigr|^2\frac{d\mu(x)}{\delta_E(x)^{m-d}} \nonumber\\[4pt] &\leq& C\sum\limits_{\stackrel{l\in{\mathbb{Z}}}{l\geq\kappa_E}} \sum\limits_{\stackrel{k\in{\mathbb{Z}}}{k\geq\kappa_E}}
\sum\limits_{Q\in{\mathbb{D}}_k(E)}2^{2|k-l|\varepsilon}2^{-2|k-l|\beta}
\inf_{{w}\in Q}\Bigl[M_E^2\bigr(|\widetilde{D}_lf|^r\bigr)({w})\Bigr]^\frac{2}{r} \int_{{\mathcal{U}}_Q}2^{k(m-d)}\,d\mu \nonumber\\[4pt] &\leq& C\sum\limits_{\stackrel{l\in{\mathbb{Z}}}{l\geq\kappa_E}} \sum\limits_{\stackrel{k\in{\mathbb{Z}}}{k\geq\kappa_E}}
\sum\limits_{Q\in{\mathbb{D}}_k(E)}2^{-2|k-l|(\beta-\varepsilon)}
\int_Q\Bigl[M_E^2\bigr(|\widetilde{D}_lf|^r\bigr)\Bigr]^\frac{2}{r}\,d\sigma \nonumber\\[4pt] &=& C\sum\limits_{\stackrel{l\in{\mathbb{Z}}}{l\geq\kappa_E}} \sum\limits_{\stackrel{k\in{\mathbb{Z}}}{k\geq\kappa_E}}
2^{-2|k-l|(\beta-\varepsilon)}
\int_E\Bigl[M_E^2\bigr(|\widetilde{D}_lf|^r\bigr)\Bigr]^\frac{2}{r}\,d\sigma \leq C\sum\limits_{\stackrel{l\in{\mathbb{Z}}}{l\geq\kappa_E}}
\int_E\Bigl[M_E^2\bigr(|\widetilde{D}_lf|^r\bigr)\Bigr]^\frac{2}{r}\,d\sigma \nonumber\\[4pt] &\leq& C\sum\limits_{\stackrel{l\in{\mathbb{Z}}}{l\geq\kappa_E}}
\int_E|\widetilde{D}_lf|^2\,d\sigma\leq C\int_E|f|^2\,d\sigma. \end{eqnarray}
The first inequality in \eqref{N-NKy-2} uses the Cauchy-Schwarz inequality, the second inequality uses the fact that
$\sum\limits_{l\in{\mathbb{Z}}}2^{-2|k-l|\varepsilon}=C$, for some finite positive constant independent of $k\in{\mathbb{Z}}$, the third inequality employs \eqref{Lvsn}, while the forth inequality is based on the fact that $\mu(\mathcal{U}_Q)\leq C2^{-km}$ and $2^{-kd}\leq C\sigma(Q)$, for all $Q\in{\mathbb{D}}_k(E)$. Since $\varepsilon\in(0,\beta)$, we have
$\sum\limits_{k\in{\mathbb{Z}}}2^{-2|k-l|(\beta-\varepsilon)}=C$, which is used in the fifth inequality in \eqref{N-NKy-2}. The sixth inequality in \eqref{N-NKy-2} follows from the boundedness of the Hardy-Littlewood maximal operator $M_E$ on $L^{\frac{2}{r}}(E,\sigma)$ (recall that $2/r>1$) and the fact that
$|\widetilde{D}_lf|^r\in L^{\frac{2}{r}}(E,\sigma)$. Finally, the last inequality in \eqref{N-NKy-2} uses \eqref{PKc-2}.
There remains to obtain a similar bound for $A_2$ introduced in \eqref{N-NKy-2AA}. Note that if $E$ is unbounded then $\kappa_E=-\infty$ so actually $A_2=0$ since we agreed that ${\mathcal{S}}_{-\infty}=0$. Consider therefore the case when $E$ is bounded, in which scenario we have $k\geq\kappa_E\in{\mathbb{Z}}$. An inspection of the proof of \eqref{Lvsn} in {\it Case~I} (of Step~I) shows that the function $D_lg={\mathcal{S}}_{l+1}g-{\mathcal{S}}_l g$ may actually be decoupled, i.e., be replaced by, say, ${\mathcal{S}}_l g$. This is because in {\it Case~I} we have only made use of the regularity of the integral kernel of $D_l$ (as opposed to {\it Case~II} where the vanishing condition of the integral kernel of $D_l$ is used) and the integral kernel of ${\mathcal{S}}_l$ exhibits the same type of regularity. Consequently, the same proof as before gives that for every $r\in(1,\infty)$ there exist finite positive constants $C$ and $\beta$ such that for each $k,l\in{\mathbb{Z}}$ with $k,l\geq\kappa_E$ such that $k\geq l$ and every $Q\in{\mathbb{D}}_k(E)$, there holds
\begin{eqnarray}\label{Lvsn-Sl}
\sup\limits_{x\in{\mathcal{U}}_Q}\Bigl| \delta_E(x)^\upsilon\bigl(\Theta({\mathcal{S}}_lg)(x)
-(\Theta 1)(x)A_Q({\mathcal{S}}_lg)\bigr)\Bigr|
\leq C2^{-|k-l|\beta}\inf\limits_{{w}\in Q}
\Bigl[M_E^2(|g|^r)({w})\Bigr]^{\frac{1}{r}}, \end{eqnarray}
for every $g:E\to{\mathbb{R}}$ locally integrable. Applying \eqref{Lvsn-Sl} with $l:=\kappa_E$ and $g:=Rf$ then yields
\begin{eqnarray}\label{N-NKy-A2} A_2 &\leq & C\sum\limits_{k\in{\mathbb{Z}},\,k\geq\kappa_E}\sum\limits_{Q\in{\mathbb{D}}_k(E)}
2^{-2|k-\kappa_E|\beta}
\inf_{{w}\in Q}\Bigl[M_E^2\bigr(|Rf|^r\bigr)({w})\Bigr]^\frac{2}{r} \int_{{\mathcal{U}}_Q}2^{k(m-d)}\,d\mu \nonumber\\[4pt] &\leq & C\sum\limits_{\stackrel{k\in{\mathbb{Z}}}{k\geq\kappa_E}}
\sum\limits_{Q\in{\mathbb{D}}_k(E)}2^{-2|k-\kappa_E|\beta}
\int_Q\Bigl[M_E^2\bigr(|Rf|^r\bigr)\Bigr]^\frac{2}{r}\,d\sigma \nonumber\\[4pt] &=& C\sum\limits_{\stackrel{k\in{\mathbb{Z}}}{k\geq\kappa_E}}
2^{-2|k-\kappa_E|\beta}
\int_E\Bigl[M_E^2\bigr(|Rf|^r\bigr)\Bigr]^\frac{2}{r}\,d\sigma
\leq C\int_E\Bigl[M_E^2\bigr(|Rf|^r\bigr)\Bigr]^\frac{2}{r}\,d\sigma \nonumber\\[4pt]
&\leq & C\int_E|Rf|^2\,d\sigma\leq C\int_E|f|^2\,d\sigma, \end{eqnarray}
since $R$ is a bounded operator on $L^2(E,\sigma)$. Now \eqref{N-NKy-2AA}, \eqref{N-NKy-2} and \eqref{N-NKy-A2} imply \eqref{N-NKy} completing the proof of the claim made in Step~II.
\vskip 0.10in \noindent{\tt Step~III.} {\it The end-game in the proof of the implication ``\eqref{UEHg}$\Rightarrow$\eqref{G-UF.22}"}. Fix $f\in L^2(E,\sigma)$ and recall $\epsilon\in(0,1)$ from Lemma~\ref{Lem:CQinBQ-N} (here is where we use that $C_\ast$ is as in \eqref{NeD-67}). Then by \eqref{doj.cF} and \eqref{doj} we may write
\begin{eqnarray}\label{WLQ} && \hskip -0.40in \int_{\bigl\{x\in{\mathscr{X}}\setminus E:\,\delta_E(x)<\epsilon\, {\rm diam}_{\rho}(E)\bigr\}}
|(\Theta f)(x)|^2\delta_E(x)^{2\upsilon-(m-d)}\,d\mu(x) \nonumber\\[4pt] && \hskip 0.40in \leq\int_{\bigcup\limits_{Q\in{\mathbb{D}}(E)}{\mathcal{U}}_Q}
|(\Theta f)(x)|^2\delta_E(x)^{2\upsilon-(m-d)}\,d\mu(x) \nonumber\\[4pt] && \hskip 0.40in \leq C\sum\limits_{k\in{\mathbb{Z}},\,k\geq\kappa_E}\sum\limits_{Q\in{\mathbb{D}}_k(E)}
\int_{{\mathcal{U}}_Q}\bigl|(\Theta f)(x)-(\Theta 1)(x)A_Qf\bigr|^2 \delta_E(x)^{2\upsilon-(m-d)}\,d\mu(x) \nonumber\\[4pt] && \hskip 0.50in +C\sum\limits_{k\in{\mathbb{Z}},\,k\geq\kappa_E}\sum\limits_{Q\in{\mathbb{D}}_k(E)}
\int_{{\mathcal{U}}_Q}\bigl|(\Theta 1)(x)A_Qf\bigr|^2 \delta_E(x)^{2\upsilon-(m-d)}\,d\mu(x). \end{eqnarray}
Observe that if we set
$B_Q:=\int_{{\mathcal{U}}_Q}|(\Theta 1)(x)|^2\delta_E(x)^{2\upsilon-(m-d)}\,d\mu(x)$ for each $Q\in{\mathbb{D}}(E)$, then in view of \eqref{gZSZ-3}, \eqref{doj}, and \eqref{UEHg} there holds
\begin{eqnarray}\label{WLQ-5} \sum\limits_{Q'\in{\mathbb{D}}(E),\,Q'\subseteq Q}B_Q
\leq C\int_{T_E(Q)}|(\Theta 1)(x)|^2\delta_E(x)^{2\upsilon-(m-d)}\,d\mu(x)\leq C\sigma(Q), \quad\forall\,Q\in{\mathbb{D}}(E). \end{eqnarray}
Thus, the numerical sequence $\bigl\{B_Q\bigr\}_{Q\in{\mathbb{D}}(E)}$ satisfies \eqref{TAkB}. Consequently, Lemma~\ref{brFC} applies and gives
\begin{eqnarray}\label{WLQ-2} && \hskip -0.70in \sum\limits_{Q\in{\mathbb{D}}(E)}
\int_{{\mathcal{U}}_Q}\bigl|(\Theta 1)(x)A_Qf\bigr|^2 \delta_E(x)^{2\upsilon-(m-d)}\,d\mu(x) \nonumber\\[4pt] && \hskip 0.50in
\leq C\int_E\Bigl[\,\sup_{Q\in{\mathbb{D}}(E),\,x\in Q}{\int{\mkern-19mu}-}_Q|f|\,d\sigma \Bigr]^2\,d\sigma(x) \nonumber\\[4pt] && \hskip 0.50in
\leq C\int_E(M_E f)^2\,d\sigma\leq C\int_E|f|^2\,d\sigma, \end{eqnarray}
where for the last inequality in \eqref{WLQ-2} we have used the boundedness of $M_E$ on $L^2(E,\sigma)$. By combining \eqref{WLQ}, \eqref{N-NKy} and \eqref{WLQ-2} we therefore obtain
\begin{eqnarray}\label{WLD-1s} \int_{\bigl\{x\in{\mathscr{X}}\setminus E:\,\delta_E(x)<\epsilon\, {\rm diam}_{\rho}(E)\bigr\}}
|(\Theta f)(x)|^2\delta_E(x)^{2\upsilon-(m-d)}\,d\mu(x)\leq C\int_E|f|^2\,d\sigma. \end{eqnarray}
Of course, this takes care of \eqref{G-UF.22} in the case when ${\rm diam}_\rho(E)=\infty$. To prove that \eqref{G-UF.22} continues to hold in the case when $E$ is bounded, let $R:={\rm diam}_\rho(E)\in(0,\infty)$, fix $x_0\in E$ and set ${\mathcal{O}}:=\bigl\{x\in{\mathscr{X}}\setminus E:\,\epsilon R\leq\delta_E(x)\bigr\}$. Then for each $x\in{\mathcal{O}}$ there exists $y\in E$ such that $\rho_{\#}(x,y)<2\delta_E(x)$, hence
\begin{eqnarray}\label{Bvvc} \rho(x,x_0)\leq C_\rho^2\rho_{\#}(x,x_0)\leq C_\rho^2\max\{\rho_{\#}(x,y), \rho_{\#}(y,x_0)\}\leq C_\rho^2\max\{2,\tfrac{1}{\epsilon}\}\delta_E(x). \end{eqnarray}
Thus, $\rho(x,x_0)\approx\delta_E(x)$ uniformly for $x\in{\mathcal{O}}$. Based on this, estimate \eqref{hszz}, and H\"older's inequality we then obtain the pointwise estimate
$|(\Theta f)(x)|^2\leq CR^{\,d}\|f\|_{L^2(E,\sigma)}^2\rho(x,x_0)^{-2(d+\upsilon)}$ for each $x\in{\mathcal{O}}$. Consequently, for some sufficiently small $c>0$ and some $C\in(0,\infty)$ independent of $f$ and $R$, we may estimate
\begin{eqnarray}\label{WL-1B22} && \hskip -0.40in
\int_{\mathcal{O}}|(\Theta f)(x)|^2\delta_E(x)^{2\upsilon-(m-d)}\,d\mu(x) \nonumber\\[4pt] && \hskip 0.40in
\leq CR^{\,d}\|f\|_{L^2(E,\sigma)}^2 \int_{{\mathscr{X}}\setminus B_{\rho_{\#}}(x_0,cR)}\rho_{\#}(x,x_0)^{-m-d}\,d\mu(x) \nonumber\\[4pt] && \hskip 0.40in
\leq CR^{\,d}\|f\|_{L^2(E,\sigma)}^2\sum\limits_{j=1}^\infty \int_{B_{\rho_{\#}}(x_0,c2^{j+1}R)\setminus B_{\rho_{\#}}(x_0,c2^j R)} \rho_{\#}(x,x_0)^{-m-d}\,d\mu(x) \nonumber\\[4pt] && \hskip 0.40in
\leq CR^{\,d}\|f\|_{L^2(E,\sigma)}^2\sum\limits_{j=1}^\infty(2^j R)^{-m-d} \mu\bigl(B_{\rho_{\#}}(x_0,c2^{j+1}R)\bigr) \nonumber\\[4pt] && \hskip 0.40in
\leq CR^{\,d}\|f\|_{L^2(E,\sigma)}^2\sum\limits_{j=1}^\infty(2^j R)^{-m-d}
(2^j R)^m\leq C\|f\|_{L^2(E,\sigma)}^2. \end{eqnarray}
Now \eqref{G-UF.22} follows by combining \eqref{WLD-1s} and \eqref{WL-1B22}.
At this stage in the proof of the theorem, we are left with establishing the converse implication with the regularity assumption \eqref{hszz-3} on the kernel now dropped. With the goal of proving \eqref{UEHg}, suppose that \eqref{G-UF} holds for some $\eta\in(0,\infty)$. Assume first that ${\rm diam}_\rho(E)<\infty$ and pick an arbitrary $\eta_o\in(\eta,\infty)$. We may then estimate
\begin{eqnarray}\label{WLQ-AA} && \hskip -0.40in \int_{\bigl\{x\in{\mathscr{X}}\setminus E:\,\delta_E(x)<\eta_o\, {\rm diam}_{\rho}(E)\bigr\}}
|(\Theta f)(x)|^2\delta_E(x)^{2\upsilon-(m-d)}\,d\mu(x) \nonumber\\[4pt] && \hskip 0.30in =\int_{\bigl\{x\in{\mathscr{X}}\setminus E:\,\delta_E(x)<\eta\, {\rm diam}_{\rho}(E)\bigr\}}
|(\Theta f)(x)|^2\delta_E(x)^{2\upsilon-(m-d)}\,d\mu(x) \\[4pt] && \hskip 0.40in +\int_{\bigl\{x\in{\mathscr{X}}\setminus E:\, \eta\,{\rm diam}_{\rho}(E)\leq\delta_E(x)<\eta_o\,{\rm diam}_{\rho}(E)\bigr\}}
|(\Theta f)(x)|^2\delta_E(x)^{2\upsilon-(m-d)}\,d\mu(x), \nonumber \end{eqnarray}
and then observing that since by \eqref{hszz} and H\"older's inequality we have the pointwise estimate
$|(\Theta f)(x)|^2\leq C\|f\|_{L^2(E,\sigma)}^2[{\rm diam}_{\rho}(E)]^{-d-2\upsilon}$ whenever $\eta\,{\rm diam}_{\rho}(E)\leq\delta_E(x)<\eta_o\,{\rm diam}_{\rho}(E)$,
the last integral in \eqref{WLQ-AA} may also be bounded by $C\int_E|f|^2\,d\sigma$, for some finite positive geometric constant independent of ${\rm diam}_\rho(E)$. The bottom line is that there is no loss of generality in assuming that $\eta>0$ appearing in \eqref{G-UF} is as large as desired.
Assuming that this is the case, fix $Q\in{\mathbb{D}}(E)$ and, for some large finite positive constant $C_o$, write
\begin{eqnarray}\label{UEHg.CC} && \hskip -0.40in \tfrac{1}{\sigma(Q)}\int_{T_E(Q)}
|\Theta 1(x)|^2\delta_E(x)^{2\upsilon-(m-d)}\,d\mu(x) \nonumber\\[4pt] && \hskip 0.40in \leq\tfrac{2}{\sigma(Q)}\int_{T_E(Q)}
\bigl|\bigl(\Theta {\mathbf{1}}_{E\cap B_{\rho_{\#}}(x_Q,C_o\ell(Q))}\bigr)(x)\bigr|^2 \delta_E(x)^{2\upsilon-(m-d)}\,d\mu(x) \nonumber\\[4pt] && \hskip 0.50in +\tfrac{2}{\sigma(Q)}\int_{T_E(Q)}
\bigl|\bigl(\Theta{\mathbf{1}}_{E\setminus B_{\rho_{\#}}(x_Q,C_o\ell(Q))}\bigr)(x)\bigr|^2 \delta_E(x)^{2\upsilon-(m-d)}\,d\mu(x) \nonumber\\[4pt] && \hskip 0.40in =:{\mathcal{I}}_1+{\mathcal{I}}_2. \end{eqnarray}
Then, granted \eqref{G-UF} and keeping in mind \eqref{dFvK} and the fact that $\eta$ is large, we may write
\begin{eqnarray}\label{UEHg.CC2} {\mathcal{I}}_1 &\leq &\tfrac{2}{\sigma(Q)} \int_{\{x\in{\mathscr{X}}:\,0<\delta_E(x)<\eta\,{\rm diam}_{\rho}(E)\}}
\bigl|\bigl(\Theta {\mathbf{1}}_{E\cap B_{\rho_{\#}}(x_Q,C_o\ell(Q))}\bigr)(x)\bigr|^2 \delta_E(x)^{2\upsilon-(m-d)}\,d\mu(x) \nonumber\\[4pt] &\leq & \tfrac{C}{\sigma(Q)}\int_{E}
\bigl|{\mathbf{1}}_{E\cap B_{\rho_{\#}}(x_Q,C_o\ell(Q))}(x)\bigr|^2\,d\sigma(x) \nonumber\\[4pt] &\leq & C\frac{\sigma\bigl(E\cap B_{\rho_{\#}}(x_Q,C_o\ell(Q))\bigr)}{\sigma(Q)}\leq C, \end{eqnarray}
given that $\sigma$ is doubling. To estimate ${\mathcal{I}}_2$, observe that there exists $C\in(0,\infty)$ with the property that
\begin{eqnarray}\label{UEHg.CC3} && \hskip -0.30in
\bigl|\bigl(\Theta{\mathbf{1}}_{E\setminus B_{\rho_{\#}}(x_Q,C_o\ell(Q))}\bigr)(x)\bigr| \\[4pt] && \hskip 0.20in \leq C\!\!\!\int\limits_{E\setminus B_{\rho_{\#}}(x_Q,C_o\ell(Q))} \frac{\delta_E(x)^{-a}}{\rho_{\#}(x,y)^{d+\upsilon-a}}\,d\sigma(y) \leq C\ell(Q)^{-(\upsilon-a)}\delta_E(x)^{-a},\qquad\forall\,x\in T_E(Q). \nonumber \end{eqnarray}
This is based on \eqref{hszz}, \eqref{WBA} (used here with $f\equiv 1$ and $\varepsilon=\upsilon-a>0$) and on the fact that $C_o$ has been chosen sufficiently large (compare with \eqref{AsQ-3}). Consequently, from \eqref{UEHg.CC3}, \eqref{dFvK}, and \eqref{lbzF} in Lemma~\ref{geom-lem} (for which we recall that $\upsilon-a>0$)
\begin{eqnarray}\label{UEHg.CC4} {\mathcal{I}}_2 &\leq & \tfrac{C}{\sigma(Q)}\ell(Q)^{-2(\upsilon-a)} \int_{T_E(Q)}\delta_E(x)^{2(\upsilon-a)-(m-d)}\,d\mu(x) \nonumber\\[4pt] &\leq & \tfrac{C}{\sigma(Q)}\ell(Q)^{-2(\upsilon-a)} \int_{B_{\rho_{\#}}(x_Q,C\ell(Q))}\delta_E(x)^{2(\upsilon-a)-(m-d)}\,d\mu(x) \nonumber\\[4pt] &\leq & \tfrac{C}{\sigma(Q)}\ell(Q)^{-2(\upsilon-a)} \ell(Q)^{m-d+2(\upsilon-a)-(m-d)}\ell(Q)^d\leq C<\infty, \end{eqnarray}
given that $\sigma(Q)\approx\ell(Q)^d$. In concert, \eqref{UEHg.CC}-\eqref{UEHg.CC4} prove \eqref{UEHg}, and this finishes the proof of the theorem. \end{proof}
\subsection{An arbitrary codimension local $T(b)$ theorem for square functions} \label{SSect:3.2}
We continue to work in the context introduced at the beginning of Section~\ref{Sect:3}. The main result in this subsection is a local $T(b)$ theorem for square functions, to the effect that {\it a square function estimate for the integral operator $\Theta$ holds if there exists a suitably nondegenerate family of functions $\{b_Q\}$, indexed by dyadic cubes $Q$ in $E$, for which there is local scale-invariant $L^2$ control of $\Theta b_Q$, appropriately weighted by a power of the distance to $E$}. To state this formally, the reader is again advised to recall the dyadic cube grid from Proposition~\ref{Diad-cube} and the regularized distance function to a set from \eqref{REG-DDD}.
\begin{theorem}\label{Thm:localTb} Let $d,m$ be two real numbers such that $0<d<m$. Assume that $({\mathscr{X}},\rho,\mu)$ is an $m$-dimensional {\rm ADR} space, $E$ is a closed subset of $({\mathscr{X}},\tau_\rho)$, and $\sigma$ is a Borel regular measure
on $(E,\tau_{\rho|_{E}})$ with the property that $(E,\rho\bigl|_E,\sigma)$ is a $d$-dimensional {\rm ADR} space.
Suppose that $\Theta$ is the integral operator defined in \eqref{operator} with a kernel ${\theta}$ as in \eqref{K234}, \eqref{hszz}, \eqref{hszz-3}. Furthermore, let ${\mathbb{D}}(E)$ denote a dyadic cube structure on $E$, consider a Whitney covering ${\mathbb{W}}_\lambda(\mathscr{X}\setminus E)$ of $\mathscr{X}\setminus E$ and a constant $C_\ast$ as in Lemma~\ref{Lem:CQinBQ-N} and, corresponding to these, recall the dyadic Carleson tents from \eqref{gZSZ-3}.
For these choices, assume that there exist finite constant $C_0\geq 1$, $c_0\in(0,1]$, and a collection $\{b_Q\}_{Q\in{\mathbb{D}}(E)}$ of $\sigma$-measurable functions $b_Q:E\rightarrow{\mathbb{C}}$ such that for each $Q\in{\mathbb{D}}(E)$ the following estimates hold:
\begin{enumerate}
\item $\int_E |b_Q|^2\,d\sigma\leq C_0\sigma(Q)$;
\item there exists $\widetilde{Q}\in{\mathbb{D}}(E)$, $\widetilde{Q}\subseteq Q$, $\ell(\widetilde{Q})\geq c_0\ell(Q)$, and
$\left|\int_{\widetilde{Q}}b_Q\,d\sigma\right|\geq\frac{1}{C_0}\,\sigma(\widetilde{Q})$;
\item $\int_{T_E(Q)}|(\Theta\,b_Q)(x)|^2 \delta_E(x)^{2\upsilon-(m-d)}\,d\mu(x)\leq C_0\sigma(Q)$. \end{enumerate}
Then there exists a finite constant $C>0$ depending only on $C_0$, $C_{\theta}$, and the {\rm ADR} constants of $E$ and ${\mathscr{X}}$, as well as on ${\rm diam}_\rho(E)$ in the case when $E$ is bounded, such that for each function $f\in L^2(E,\sigma)$ one has
\begin{eqnarray}\label{G-UF-2}
\int_{\mathscr{X}\setminus E}|(\Theta f)(x)|^2\delta_E(x)^{2\upsilon-(m-d)}\, d\mu(x)\leq C\int_E|f(x)|^2\,d\sigma(x). \end{eqnarray} \end{theorem}
Before giving the proof of Theorem~\ref{Thm:localTb} we present a stopping-time construction and elaborate on the way this is used.
\begin{lemma}\label{brjo} Assume $(E,\rho,\sigma)$ is a space of homogeneous type with the property that $\sigma$ is Borel regular, and denote by ${\mathbb{D}}(E)$ a dyadic cube structure on $E$. Suppose that there exist finite constants $C_0\geq 1$, $c_0\in(0,1)$, and a collection $\{b_Q\}_{Q\in{\mathbb{D}}(E)}$ of $\sigma$-measurable functions $b_Q:E\rightarrow{\mathbb{C}}$ such that
\begin{eqnarray}\label{dbjpz} && \hskip -0.40in
\int_E |b_Q|^2\ d\sigma \leq C_0\sigma(Q)\quad \mbox{for every $Q\in{\mathbb{D}}(E)$, and} \\[4pt] && \hskip -0.40in \forall\,Q\in{\mathbb{D}}(E)\quad\exists\, \widetilde{Q}\in{\mathbb{D}}(E),\,\widetilde{Q}\subseteq Q, \,\ell(\widetilde{Q})\geq c_0\ell(Q)\,\,\,\mbox{with}\,\,\,
\left| \int_{\widetilde{Q}} b_Q\ d\sigma \right|\geq\frac{1}{C_0}\,\sigma(\widetilde{Q}). \label{dbjpz-extra} \end{eqnarray}
Then there exists a number $\eta\in(0,1)$ such that for every cube $Q\in{\mathbb{D}}(E)$, and each fixed $\widetilde{Q}$ as in \eqref{dbjpz-extra}, one can find a sequence $\bigl\{Q_j\bigr\}_{j\in J}\subseteq{\mathbb{D}}(E)$ of pairwise disjoint cubes satisfying the following properties:
\begin{enumerate} \item[(i)] $Q_j\subseteq\widetilde{Q}$ for every $j\in J$ and $\sigma\Bigl(\widetilde{Q}\setminus\bigcup_{j\in J}Q_j\Bigr)\geq\eta\,\sigma(\widetilde{Q})$;
\item[(ii)] if
\begin{eqnarray}\label{lvB} {\mathcal{F}}_Q:=\bigl\{Q'\in{\mathbb{D}}(E):\,Q'\subseteq\widetilde{Q} \mbox{ and $Q'$ is not contained in $Q_j$ for every $j\in J$}\bigr\}, \end{eqnarray}
then $\left|{\int{\mkern-16mu}-}_{Q'}b_Q\,d\sigma\right|\geq\frac{1}{2}$ for every $Q'\in{\mathcal{F}}_Q$. \end{enumerate} \end{lemma}
\begin{proof} Granted the regularity of the measure $\sigma$, it follows from {\it (3)} and {\it (9)} in Proposition~\ref{Diad-cube} that for each $k\in{\mathbb{Z}}$ and each $Q\in{\mathbb{D}}_k(E)$ we have
\begin{eqnarray}\label{T-rcs-BV} \sigma\Bigl(Q\setminus\bigcup_{Q'\subseteq Q,\,Q'\in{\mathbb{D}}_\ell(E)}Q'\Bigr)=0 \qquad\mbox{for every $\ell\in{\mathbb{Z}}$ with $\ell\geq k$}. \end{eqnarray}
Thanks to \eqref{dbjpz}-\eqref{dbjpz-extra}, we may re-normalize the functions $\{b_Q\}_{Q\in{\mathbb{D}}(E)}$ so that ${\int{\mkern-16mu}-}_{\widetilde{Q}} b_Q\,d\sigma=1$ for each $Q\in{\mathbb{D}}(E)$, where $\widetilde{Q}$ is as in \eqref{dbjpz-extra}. In the process, the first inequality in \eqref{dbjpz} becomes
\begin{eqnarray}\label{Fs23EE}
\int_E |b_Q|^2\ d\sigma\leq C_0^3\sigma(Q),\qquad \mbox{for each }\,\,\,Q\in{\mathbb{D}}(E). \end{eqnarray}
Fix $Q\in{\mathbb{D}}(E)$ and a corresponding $\widetilde{Q}$ as in \eqref{dbjpz-extra}. In particular we have
\begin{eqnarray}\label{meYY} \sigma(Q)\leq C_1\sigma(\widetilde{Q})\quad\mbox{for some $C_1\in[1,\infty)$ independent of $Q,\widetilde{Q}$}. \end{eqnarray}
Next, perform a stopping-time argument for $\widetilde{Q}$ by successively dividing it into dyadic sub-cubes $Q'\subseteq \widetilde{Q}$ and stopping whenever ${\rm Re}\,{\int{\mkern-16mu}-}_{Q'}b_Q\,d\sigma\leq\frac{1}{2}$. That this is doable is ensured by \eqref{T-rcs-BV} and the re-normalization of $b_Q$. This yields a family of cubes $\bigl\{Q_j\bigr\}_{j\in J}\subseteq{\mathbb{D}}(E)$ such that:
\begin{enumerate} \item[(1)] $Q_j\subseteq \widetilde{Q}\subseteq Q$ for each $j\in J$ and $Q_j\cap Q_{j'}=\emptyset$ whenever $j,j'\in J$, $j\not=j'$;
\item[(2)] ${\rm Re}\,{\int{\mkern-16mu}-}_{Q_j}b_Q\,d\sigma\leq\frac{1}{2}$ for each $j\in J$;
\item[(3)] the family $\bigl\{Q_j\bigr\}_{j\in J}$ is maximal with respect to $(1)$ and $(2)$ above, i.e., if $Q'\in{\mathbb{D}}(E)$ is such that $Q'\subseteq \widetilde{Q}$, then either there exists $j_0\in J$ such that $Q'\subseteq Q_{j_0}$, or ${\rm Re}\,{\int{\mkern-16mu}-}_{Q'}b_Q\,d\sigma>\frac{1}{2}$. \end{enumerate}
Then we may write
\begin{eqnarray}\label{kvnS} \sigma(\widetilde{Q}) & = &\int_{\widetilde{Q}}b_Q\,d\sigma ={\rm Re}\int_{\widetilde{Q}\setminus(\bigcup_{j\in J}Q_j)}b_Q\,d\sigma +\sum\limits_{j\in J}{\rm Re}\int_{Q_j}b_Q\,d\sigma \nonumber\\[4pt]
&\leq & \Bigl(\int_E|b_Q|^2d\sigma\Bigr)^{\frac{1}{2}} \sigma\bigl(\widetilde{Q}\setminus\cup_{j\in J}Q_j\bigr)^{\frac{1}{2}} +\tfrac{1}{2}\sum\limits_{j\in J}\sigma(Q_j) \nonumber\\[4pt] &\leq & C_0^{\frac{3}{2}}\sigma(Q)^{\frac{1}{2}} \sigma\bigl(\widetilde{Q}\setminus\cup_{j\in J}Q_j\bigr)^{\frac{1}{2}} +\tfrac{1}{2}\sigma(\widetilde{Q}) \nonumber\\[4pt] &\leq & C_1^{\frac{1}{2}}C_0^{\frac{3}{2}}\sigma(\widetilde{Q})^{\frac{1}{2}} \sigma\bigl(\widetilde{Q}\setminus\cup_{j\in J}Q_j\bigr)^{\frac{1}{2}} +\tfrac{1}{2}\sigma(\widetilde{Q}), \end{eqnarray}
where the first inequality in \eqref{kvnS} is based on H\"older's inequality and condition $(2)$ above, the second inequality uses \eqref{Fs23EE} and $(1)$ above, while the last inequality is a consequence of \eqref{meYY}. After absorbing $\tfrac{1}{2}\sigma(\widetilde{Q})$ in the leftmost side of \eqref{kvnS} and setting $\eta:=\frac{1}{4C_1C_0^3}\in(0,1)$, it follows that $\sigma\bigl(\widetilde{Q}\setminus\cup_{j\in J}Q_j\bigr)\geq\eta\sigma(\widetilde{Q})$, thus condition $(i)$ holds for the family $\bigl\{Q_j\bigr\}_{j\in J}$ constructed above. In addition it is immediate from property $(3)$ that condition $(ii)$ is also satisfied. \end{proof}
A typical application of Lemma~\eqref{brjo} is exemplified by our next result.
\begin{lemma}\label{NIGz} Assume that $({\mathscr{X}},\rho)$ is a geometrically doubling quasi-metric space, $\mu$ is a Borel measure on $({\mathscr{X}},\tau_\rho)$ and that $E$ is a nonempty, closed, proper subset of $({\mathscr{X}},\tau_\rho)$. Also, suppose that $\sigma$ is a
Borel regular measure on $E$ with the property that $(E,\rho\bigl|_E,\sigma)$ is a space of homogeneous type and denote by ${\mathbb{D}}(E)$ a dyadic cube structure on $E$. Next, assume that ${\mathbb{W}}_\lambda(\mathscr{X}\setminus E)$ is a Whitney covering of $\mathscr{X}\setminus E$ as in Lemma~\ref{Lem:CQinBQ} (for some fixed $a\geq 1$), and recall the regions $\{\mathcal{U}_Q\}_{Q\in\mathbb{D}(E)}$ from \eqref{gZSZb} relative to this cover. Finally, assume the hypotheses of Lemma~\ref{brjo}, and for each cube $Q\in{\mathbb{D}}(E)$, recall the collection ${\mathcal{F}}_Q$ from \eqref{lvB} and define
\begin{eqnarray}\label{tnjG} E^{\ast}_Q:=\bigcup\limits_{Q'\in{\mathcal{F}}_Q}{\mathcal{U}}_{Q'}. \end{eqnarray}
Then for every $\gamma\in{\mathbb{R}}$ and every $\mu$-measurable function $u:\mathscr{X}\setminus E\to{\mathbb{R}}$ it follows that
\begin{eqnarray}\label{BNKa}
\int_{E^{\ast}_Q}|u(x)|^2\delta_E(x)^\gamma\,d\mu(x) \approx\sum\limits_{Q'\in{\mathcal{F}}_Q}\int_{{\mathcal{U}}_{Q'}}
\bigl|u(x){\textstyle{{\int{\mkern-16mu}-}_{Q'}b_Q\,d\sigma}}\bigr|^2\delta_E(x)^\gamma\,d\mu(x), \end{eqnarray}
with finite positive equivalence constants, depending only on $C_0$ from \eqref{dbjpz}-\eqref{dbjpz-extra}. \end{lemma}
\begin{proof} This readily follows by combining \eqref{doj}, \eqref{dbjpz}-\eqref{dbjpz-extra} and $(ii)$ in Lemma~\ref{brjo}. \end{proof}
We are now ready to present the
\vskip 0.08in \begin{proof}[Proof of Theorem~\ref{Thm:localTb}] Based on Theorem~\ref{SChg}, it suffices to show that
$|\Theta 1|^2\delta_E^{2\upsilon-(m-d)}\,d\mu$ is a Carleson measure in $\mathscr{X}\setminus E$ relative to $E$, that is, that \eqref{UEHg} holds. In a first stage, we shall show that \eqref{UEHg} holds for $\Theta$ replaced by some truncated operators $\Theta_i$, $i\in{\mathbb{N}}$. More precisely, for each $i\in{\mathbb{N}}$ consider the kernel
\begin{eqnarray}\label{LIH} {\theta}_i(x,y):={\mathbf{1}}_{\{1/i<\delta_E<i\}}(x){\theta}(x,y), \qquad\forall\,x\in\mathscr{X}\setminus E,\quad\forall\,y\in E, \end{eqnarray}
and introduce the integral operator mapping $f:E\to{\mathbb{R}}$ into
\begin{eqnarray}\label{LIH-2} (\Theta_if)(x):=\int_E{\theta}_i(x,y)f(y)\,d\sigma(y),\qquad \forall\,x\in\mathscr{X}\setminus E. \end{eqnarray}
Clearly,
\begin{eqnarray}\label{LIH-3} \Theta_i={\mathbf{1}}_{\{1/i<\delta_E<i\}}\Theta,\qquad\forall\,i\in{\mathbb{N}}, \end{eqnarray}
and, with $C_{\theta}$ as in \eqref{hszz}, for every $x\in\mathscr{X}\setminus E$ we have
\begin{eqnarray}\label{hszz-i} && \hskip -0.40in
|{\theta}_i(x,y)|\leq C_{\theta}\frac{\delta_E(x)^{-a}}{\rho(x,y)^{d+\upsilon-a}},\qquad\forall\,y\in E, \\[4pt] && \hskip -0.40in
|{\theta}_i(x,y)-{\theta}_i(x,\widetilde{y})|\leq C_{\theta}\frac{\rho(y,\widetilde{y})^\alpha\delta_E(x)^{-a-\alpha}} {\rho(x,y)^{d+\upsilon-a}}, \qquad\forall\,\widetilde{y},y\in E,\quad \rho(y,\widetilde{y}) \leq \tfrac{1}{2}\rho(x,y). \label{hszz-3i} \end{eqnarray}
Then for each $i\in{\mathbb{N}}$ and each $x\in\mathscr{X}\setminus E$ by \eqref{LIH-2}, \eqref{hszz-i} and Lemma~\ref{Gkwvr} (given that $\upsilon-a>0$) we have
\begin{eqnarray}\label{LIH-4}
|(\Theta_i 1)(x)| &\leq & C{\mathbf{1}}_{\{1/i<\delta_E<i\}}(x) \int_E\frac{\delta_E(x)^{-a}}{\rho_{\#}(x,y)^{d+\upsilon-a}}\,d\sigma(y) \leq C {\mathbf{1}}_{\{1/i<\delta_E<i\}}(x)[\delta_E(x)]^{-\upsilon} \nonumber\\[4pt] & \leq & C i^\upsilon{\mathbf{1}}_{\{1/i<\delta_E<i\}}(x). \end{eqnarray}
Recalling now \eqref{dFvK}, estimate \eqref{LIH-4} further yields (with $x_Q$ denoting the center of $Q$)
\begin{eqnarray}\label{LIH-5} && \hskip -0.50in
\int_{T_E(Q)}|(\Theta_i 1)(x)|^2\delta_E(x)^{2\upsilon-(m-d)}\,d\mu(x) \nonumber\\[4pt] && \hskip 0.40in \leq Ci^{2\upsilon}\int_{x\in B_{\rho_{\#}}(x_Q,C\ell(Q)),\,\delta_E(x)<i} \delta_E(x)^{2\upsilon-(m-d)}\,d\mu(x) \nonumber\\[4pt] && \hskip 0.40in \leq C i^{4\upsilon}\ell(Q)^d\leq Ci^{4\upsilon}\sigma(Q),\qquad \forall\,Q\in{\mathbb{D}}(E), \end{eqnarray}
for some constant $C\in(0,\infty)$ which does not depend on $Q$ and $i$, where the second inequality in \eqref{LIH-5} is a consequence of Lemma~\ref{geom-lem}. Hence, if we now define
\begin{eqnarray}\label{LIH-6} c_i:=\sup_{Q\in{\mathbb{D}}(E)}
\Bigl[\tfrac{1}{\sigma(Q)}\int_{T_E(Q)}|(\Theta_i 1)(x)|^2\delta_E(x)^{2\upsilon-(m-d)}\,d\mu(x)\Bigr], \qquad \forall\,i\in{\mathbb{N}}, \end{eqnarray}
then \eqref{LIH-5} implies $0\leq c_i\leq Ci^{4\upsilon}$ for each $i\in{\mathbb{N}}$. In particular, each $c_i$ is finite. Our goal is to show that actually
\begin{eqnarray}\label{LIH-6B} \sup\limits_{i\in{\mathbb{N}}}c_i<\infty. \end{eqnarray}
To this end, fix $Q\in{\mathbb{D}}(E)$ and for this $Q$ consider some $\widetilde{Q}$ satisfying {\it 2.} in the hypothesis of Theorem~\ref{Thm:localTb} (recall the constant $c_0$). In particular, we have that
\begin{eqnarray}\label{nzvd} \exists\,p\in{\mathbb{N}}\quad\mbox{satisfying}\quad p\leq -\log_{2}(c_0)\quad\mbox{and such that}\quad \widetilde{Q}\in{\mathbb{D}}_p(E). \end{eqnarray}
Next recall Lemma~\ref{brjo} and Lemma~\ref{NIGz} and the notation therein. Then, from the definition of $E_Q^\ast$, \eqref{nzvd} and \eqref{gZSZ-3}, we have
\begin{eqnarray}\label{nzvd-Dsf} T_E(Q)\subseteq E_Q^\ast\cup\Bigl(\bigcup\limits_{j\in J}T_E(Q_j)\Bigr) \cup\Bigl(\bigcup\limits_{Q''\in{\mathbb{D}}_p(E),\, Q''\subseteq Q,\,Q''\not=\widetilde{Q}}T_E(Q'')\Bigr). \end{eqnarray}
Consequently, for each $i\in{\mathbb{N}}$ we may write
\begin{eqnarray}\label{Vhx} && \hskip -0.30in
\int_{T_E(Q)}|(\Theta_i 1)(x)|^2\delta_E(x)^{2\upsilon-(m-d)}\,d\mu(x) \\[4pt] && \hskip 0.20in
\leq\int\limits_{E_Q^\ast}|(\Theta_i 1)(x)|^2\delta_E(x)^{2\upsilon-(m-d)}\,d\mu(x)
+\sum\limits_{j\in J}\,\,\int\limits_{T_E(Q_j)}|(\Theta_i 1)(x)|^2 \delta_E(x)^{2\upsilon-(m-d)}\,d\mu(x) \nonumber\\[4pt] && \hskip 0.25in +\sum\limits_{\stackrel{Q''\in{\mathbb{D}}_p(E),\,Q''\subseteq Q}{Q''\not=\widetilde{Q}}}
\,\,\int\limits_{T_E(Q'')}|(\Theta_i 1)(x)|^2\delta_E(x)^{2\upsilon-(m-d)}\,d\mu(x). \nonumber \end{eqnarray}
To estimate the first integral in the right hand-side of \eqref{Vhx} start with \eqref{BNKa} written for $u:=\Theta_i 1$. Keeping in mind that
$|\Theta_i1|\leq|\Theta 1|$ for all $i\in{\mathbb{N}}$, we obtain
\begin{eqnarray}\label{AMpR} &&\hskip -0.20in
\int_{E^{\ast}_Q}|\Theta_i 1(x)|^2\delta_E(x)^{2\upsilon-(m-d)}\,d\mu(x) \leq C\sum\limits_{Q'\in{\mathcal{F}}_Q}\int_{{\mathcal{U}}_{Q'}}
|(\Theta 1)(x)A_{Q'}b_Q|^2\delta_E(x)^{2\upsilon-(m-d)}\,d\mu(x) \nonumber\\[4pt]
&&\qquad \leq C\sum\limits_{Q'\in{\mathbb{D}}(E),\,Q'\subseteq Q}
\int_{{\mathcal{U}}_{Q'}}|(\Theta b_Q)(x)|^2\delta_E(x)^{2\upsilon-(m-d)}\,d\mu(x) \nonumber\\[4pt] &&\qquad \quad+C\!\!\!\sum\limits_{Q'\in{\mathbb{D}}(E)}\int_{{\mathcal{U}}_{Q'}}
\bigl|(\Theta b_Q)(x)-(\Theta 1)(x)A_{Q'}b_Q\bigr|^2\delta_E(x)^{2\upsilon-(m-d)}\,d\mu(x) \nonumber\\[4pt]
&&\qquad \leq C\int_{T_E(Q)}|(\Theta b_Q)(x)|^2\delta_E(x)^{2\upsilon-(m-d)}\,d\mu(x)
+C\int_E|b_Q|^2\,d\sigma\leq C\sigma(Q). \end{eqnarray}
The second inequality in \eqref{AMpR} is immediate, the third uses \eqref{gZSZ-3} and \eqref{N-NKy} (the latter applied with $f:=b_Q$), while the fourth uses assumptions {\it 1} and {\it 3} of Theorem~\ref{Thm:localTb}.
Consider next the first sum in the right hand-side of \eqref{Vhx}. Upon recalling \eqref{LIH-6} and the properties of $Q_j$'s in Lemma~\ref{brjo} we may write
\begin{eqnarray}\label{Vhx-2} \hskip -0.30in \sum\limits_{j\in J}\int\limits_{T_E(Q_j)}
|(\Theta_i 1)(x)|^2\delta_E(x)^{2\upsilon-(m-d)}\,d\mu(x) &\leq & c_i \sum\limits_{j\in J}\sigma(Q_j) =c_i\,\sigma\bigl(\cup_{j\in J}Q_j\bigr) \nonumber\\[4pt] &=& c_i\,\sigma(\widetilde{Q}) -c_i\,\sigma\bigl(\widetilde{Q}\setminus \cup_{j\in J}Q_j\bigr) \nonumber\\[4pt] & \leq & c_i\,(1-\eta)\,\sigma(\widetilde{Q}). \end{eqnarray}
Upon recalling \eqref{LIH-6} we obtain
\begin{eqnarray}\label{Vh-Es} \sum\limits_{\stackrel{Q''\in{\mathbb{D}}_p(E),\,Q''\subseteq Q}{Q''\not=\widetilde{Q}}}
\,\,\int\limits_{T_E(Q'')}|(\Theta_i 1)(x)|^2\delta_E(x)^{2\upsilon-(m-d)}\,d\mu(x) \leq c_i\sigma(Q\setminus\widetilde{Q}). \end{eqnarray}
In concert, \eqref{Vhx}, \eqref{AMpR}, \eqref{Vhx-2}, and \eqref{Vh-Es} imply that there exists a finite constant $C>0$ with the property that for every $i\in{\mathbb{N}}$ there holds
\begin{eqnarray}\label{Vhx-3}
\int_{T_E(Q)}|\Theta_i1|^2\delta_E^{2\upsilon-(m-d)}\,d\mu & \leq & c_i\,(1-\eta)\,\sigma(\widetilde{Q})+c_i\sigma(Q\setminus\widetilde{Q})+C\sigma(Q) \nonumber\\[4pt] & \leq & c_i\,\sigma(Q)-c_i\eta\,\sigma(\widetilde{Q})+C\sigma(Q) \nonumber\\[4pt] & \leq & c_i\,\sigma(Q)-c_i\eta C_1^{-1}\,\sigma(Q)+C\sigma(Q) \nonumber\\[4pt] & = & c_i\,(1-\eta C_1^{-1})\,\sigma(Q)+C\sigma(Q), \quad\forall\,Q\in{\mathbb{D}}(E), \end{eqnarray}
where for the last inequality in \eqref{Vhx-3} we have used \eqref{meYY}. Dividing both sides of \eqref{Vhx-3} by $\sigma(Q)$ and then taking the supremum over all $Q\in{\mathbb{D}}(E)$ we obtain $c_i\leq c_i\,(1-\eta C_1^{-1})+C$ and furthermore, since $c_i\leq Ci^{4\upsilon}<\infty$, that $c_i\leq\eta^{-1}C_1C$ for all $i\in{\mathbb{N}}$. This finishes the proof of \eqref{LIH-6B}.
Having established this, for each $Q\in{\mathbb{D}}(E)$ we may then write, using \eqref{LIH-3} and Lebesgue's Monotone Convergence Theorem,
\begin{eqnarray}\label{Vhx-4}
\int_{T_E(Q)}|(\Theta 1)(x)|^2\delta_E(x)^{2\upsilon-(m-d)}\,d\mu(x)
& = & \lim\limits_{i\to\infty}\int_{T_E(Q)}|(\Theta_i 1)(x)|^2\delta_E(x)^{2\upsilon-(m-d)}\,d\mu(x) \nonumber\\[4pt] & \leq &(\sup\limits_{i\in{\mathbb{N}}}c_i)\,\sigma(Q)\leq C\sigma(Q), \end{eqnarray}
for some finite constant $C>0$ independent of $Q$. This completes the proof of \eqref{UEHg} and finishes the proof of Theorem~\ref{Thm:localTb}. \end{proof}
\section{An Inductive Scheme for Square Function Estimates} \setcounter{equation}{0} \label{Sect:4}
We now apply the local $T(b)$ Theorem from the previous section to establish an inductive scheme for square function estimates. More specifically, we show that an integral operator $\Theta$, associated with an Ahlfors-David regular set $E$ as in~\eqref{operator}, satisfies square function estimates whenever the set $E$ contains (uniformly, at all scales and locations) so-called big pieces of sets on which square function estimates for $\Theta$ hold. In short, we say that big pieces of square function estimates (BPSFE) imply square function estimates (SFE). We emphasize that this ``big pieces" functor is applied to square function estimates for an individual, fixed $\Theta$. Thus, the result to be proved in this section is not a consequence of the stability of UR sets under the big pieces functor, as our particular square function bounds may {\it not} be equivalent to the property that $E$ is UR.
We continue to work in the context introduced at the beginning of Section~\ref{Sect:3}, except we must assume in addition that the integral kernel $\theta$ is not adapted to a fixed set~$E$. In particular, fix two real numbers $d,m$ such that $0<d<m$, and an $m$-dimensional {\rm ADR} space $({\mathscr{X}},\rho,\mu)$. In this context, suppose that
\begin{eqnarray}\label{K234-A} \begin{array}{c} {\theta}:(\mathscr{X}\times\mathscr{X}) \setminus\{(x,x):\,x\in \mathscr{X}\}\longrightarrow{{\mathbb{R}}} \\[4pt] \mbox{is Borel measurable with respect to the product topology $\tau_\rho\times\tau_\rho$}, \end{array} \end{eqnarray}
and has the property that there exist finite positive constants $C_{\theta}$, $\alpha$, $\upsilon$ such that for all $x,y\in\mathscr{X}$ with $x\neq y$ the following hold:
\begin{eqnarray}\label{hszz-A} && \hskip -0.40in
|{\theta}(x,y)|\leq\frac{C_{\theta}}{\rho(x,y)^{d+\upsilon}}, \\[4pt] && \hskip -0.40in
|{\theta}(x,y)-{\theta}(x,\widetilde{y})|\leq C_{\theta} \frac{\rho(y,\widetilde{y})^\alpha}{\rho(x,y)^{d+\upsilon+\alpha}}, \quad\forall\,\widetilde{y}\in\mathscr{X}\setminus\{x\}\,\,\mbox{ with }\,\, \rho(y,\widetilde{y})\leq\tfrac{1}{2}\rho(x,y).\quad \label{hszz-3-A} \end{eqnarray}
Then for each closed subset $E$ of $({\mathscr{X}},\tau_\rho)$, and each Borel regular measure $\sigma$
on $(E,\tau_{\rho|_{E}})$ with the property that $(E,\rho\bigl|_E,\sigma)$ is a $d$-dimensional {\rm ADR} space, define the integral operator $\Theta_E$ for all functions $f\in L^p(E,\sigma)$, $1\leq p\leq\infty$, by
\begin{eqnarray}\label{operator-A} (\Theta_E f)(x):=\int_E {\theta}(x,y)f(y)\,d\sigma(y), \qquad\forall\,x\in\mathscr{X}\setminus E. \end{eqnarray}
We begin by defining what it means for a set to have big pieces of square function estimates.
\begin{definition}\label{sjvs} Consider two numbers $d,m\in(0,\infty)$ such that $m>d$, suppose that $({\mathscr{X}},\rho,\mu)$ is an $m$-dimensional {\rm ADR} space, and assume that ${\theta}$ is as in \eqref{K234-A}-\eqref{hszz-3-A}. In this context, a set $E\subseteq{\mathscr{X}}$ is said to have {\tt Big Pieces of Square Function Estimate} (or, simply {\tt BPSFE}) relative to the kernel $\theta$ provided the following conditions are satisfied:
\begin{enumerate}
\item[(i)] the set $E$ is closed in $({\mathscr{X}},\tau_\rho)$ and has the property that there exists a Borel regular measure $\sigma$
on $(E,\tau_{\rho|_{E}})$ such that $\bigl(E,\rho\bigl|_E,\sigma\bigr)$ is a $d$-dimensional {\rm ADR} space;
\item[(ii)] there exist finite positive constants $\eta$, $C_1$, and $C_2$ with the property that for each $x\in E$ and each real number $r\in(0,{\rm diam}_{\rho_{\#}}(E)]$ there exists a closed subset $E_{x,r}$ of $(\mathscr{X},\tau_\rho)$ such that if
\begin{eqnarray}\label{HHH-57} \sigma_{x,r}:={\mathscr{H}}_{{\mathscr{X}}\!,\,\rho_{\#}}^d\lfloor E_{x,r}, \quad\mbox{ where ${\mathscr{H}}_{{\mathscr{X}}\!,\,\rho_{\#}}^d$ is as in \eqref{HHH-56}}, \end{eqnarray}
then $\bigl(E_{x,r},\rho\big|_{E_{x,r}},\sigma_{x,r}\bigr)$ is a $d$-dimensional {\rm ADR} space, with {\rm ADR} constant $\leq C_1$, and which satisfies
\begin{eqnarray}\label{yvg} \sigma\bigl(E_{x,r}\cap E\cap B_{\rho_{\#}}(x,r)\bigr)\geq\eta\,r^d, \end{eqnarray}
as well as
\begin{eqnarray}\label{avhai} \begin{array}{c} \displaystyle
\int_{\mathscr{X}\setminus E_{x,r}}|\Theta_{E_{x,r}}f(z)|^2\, {\rm dist}_{\rho_{\#}}(z,E_{x,r})^{2\upsilon-(m-d)}\,d\mu(z)
\leq C_2\int_{E_{x,r}}|f|^2\,d\sigma_{x,r}\qquad \\[8pt] \mbox{for each function }\,\,f\in L^2(E_{x,r},\sigma_{x,r}), \end{array} \end{eqnarray}
where $\Theta_{E_{x,r}}$ is the operator associated with $E_{x,r}$ as in \eqref{operator-A}. \end{enumerate} \end{definition}
\noindent In the context of the above definition, the constants $\eta,C_1,C_2$ will collectively be referred to as the {\tt BPSFE character} of the set $E$.
The property of having BPSFE may be dyadically discretized as explained in the lemma below.
\begin{lemma}\label{hrsbrli} Let $d,m\in(0,\infty)$ be such that $m>d$, assume that $({\mathscr{X}},\rho,\mu)$ is an $m$-dimensional {\rm ADR} space, and suppose that ${\theta}$ is as in \eqref{K234-A}-\eqref{hszz-3-A}. In addition, let $E\subseteq{\mathscr{X}}$ be such that {\it (i)} in Definition~\ref{sjvs} holds and consider the dyadic cube structure ${\mathbb{D}}(E)$ on $E$ as in Proposition~\ref{Diad-cube}.
Then the set $E$ has {\rm BPSFE} (relative to $\theta$) if and only if there exist finite positive constants $\eta,C_1,C_2$ with the property that for each ${Q\in{\mathbb{D}}(E)}$ there exists a closed set $E_Q\subseteq\mathscr{X}$ such that if
\begin{eqnarray}\label{HHH-60} \sigma_{Q}:={\mathscr{H}}_{{\mathscr{X}}\!,\,\rho_{\#}}^d\lfloor E_{Q}, \end{eqnarray}
then $\bigl(E_Q,\rho\bigl|_{E_Q},\sigma_Q\bigr)$ is a $d$-dimensional {\rm ADR} space with {\rm ADR} constant $\leq C_1$ which satisfies
\begin{eqnarray}\label{yvg2} {\mathscr{H}}_{{\mathscr{X}}\!,\,\rho_{\#}}^d(E_Q\cap Q) \geq\eta\,{\mathscr{H}}_{{\mathscr{X}}\!,\,\rho_{\#}}^d(Q) \end{eqnarray}
as well as
\begin{eqnarray}\label{avhai2} \begin{array}{c} \displaystyle
\int_{\mathscr{X}\setminus E_Q}|\Theta_{E_Q} f(x)|^2\, {\rm dist}_{\rho_{\#}}(x,E_Q)^{2\upsilon-(m-d)}\,d\mu(x)
\leq C_2\int_{E_Q} |f|^2\ d\sigma_Q, \\[8pt] \mbox{for each function }\,\,f\in L^2(E_Q,\sigma_Q). \end{array} \end{eqnarray} \end{lemma}
\begin{proof} The left-to-right implication is a simple consequence of \eqref{ha-GVV}, while the opposite one follows with the help of \eqref{ha-GL54}. \end{proof}
We now state and prove the main result in this section.
\begin{theorem}\label{Thm:BPSFtoSF} Consider two numbers $d,m\in(0,\infty)$ such that $m>d$, suppose that $({\mathscr{X}},\rho,\mu)$ is an $m$-dimensional {\rm ADR} space, and assume that ${\theta}$ is as in \eqref{K234-A}-\eqref{hszz-3-A}.
If the set $E\subseteq{\mathscr{X}}$ has {\rm BPSFE} relative to $\theta$ then there exists a finite constant $C>0$, depending only on $\rho$, $m$, $d$, $\upsilon$, $C_{\theta}$ (from \eqref{hszz-A}-\eqref{hszz-3-A}), the {\rm BPSFE} character of $E$, and the {\rm ADR} constants of $E$ and ${\mathscr{X}}$, such that
\begin{eqnarray}\label{vlnGG}
\int_{\mathscr{X}\setminus E}|\Theta_E f(x)|^2\, \delta_E(x)^{2\upsilon-(m-d)}\,d\mu(x)
\leq C\int_E|f|^2\,d\sigma,\qquad\forall\,f\in L^2(E,\sigma), \end{eqnarray}
where
\begin{eqnarray}\label{HHH-58} \sigma:={\mathscr{H}}_{{\mathscr{X}}\!,\,\rho_{\#}}^d\lfloor E, \quad\mbox{ with ${\mathscr{H}}_{{\mathscr{X}}\!,\,\rho_{\#}}^d$ as in \eqref{HHH-56}}. \end{eqnarray}
\end{theorem}
\begin{proof} Since $E$ has BPSFE relative to $\theta$, by Lemma~\ref{hrsbrli}, for each $Q\in{\mathbb{D}}(E)$ there exists $E_Q$ satisfying \eqref{yvg2}-\eqref{avhai2}. For each $Q\in{\mathbb{D}}(E)$, we then define the function $b_Q:E\rightarrow{\mathbb{R}}$ by setting
\begin{eqnarray}\label{BBss} b_Q(y):={\mathbf{1}}_{Q\cap E_Q}(y),\qquad\forall\,y\in E. \end{eqnarray}
The strategy for proving \eqref{vlnGG} is to invoke Theorem~\ref{Thm:localTb} for the family $\{b_Q\}_{Q\in{\mathbb{D}}(E)}$. As such, matters are reduced to checking that conditions {\it 1.}--{\it 3.} in Theorem~\ref{Thm:localTb} hold for the collection $\{b_Q\}_{Q\in{\mathbb{D}}(E)}$ defined in \eqref{BBss}. Now, condition {\it 1.} is immediate, while the validity of condition {\it 2.} (with $\widetilde{Q}:=Q$) is a consequence of \eqref{yvg2}. Thus, it remains to check that condition {\it 3.} holds as well. To this end, fix $Q\in{\mathbb{D}}(E)$ and for some constant $C_1\in(1,\infty)$ to be specified later write (employing notation introduced in \eqref{REG-DDD} in relation to both $E$ and $E_Q$)
\begin{eqnarray}\label{KLGB} && \hskip -0.60in
\int_{T_E(Q)}|\Theta_E b_Q(x)|^2\delta_E(x)^{2\upsilon-(m-d)}\,d\mu(x) \nonumber\\[4pt] && \hskip 0.40in
=\int_{T_E(Q)}|\Theta_E b_Q(x)|^2 {\mathbf{1}}_{\{z\in{\mathscr{X}}:\,\delta_{E_Q}(z)>C_1\delta_E(z)\}}(x)\, \delta_E(x)^{2\upsilon-(m-d)}\,d\mu(x) \nonumber\\[4pt] && \hskip 0.50in
+\int_{T_E(Q)}|\Theta_E b_Q(x)|^2 {\mathbf{1}}_{\{z\in{\mathscr{X}}:\, C_1^{-1}\delta_E(z)\leq\delta_{E_Q}(z)\leq C_1\delta_E(z)\}}(x)\, \delta_E(x)^{2\upsilon-(m-d)}\,d\mu(x) \nonumber\\[4pt] && \hskip 0.50in
+\int_{T_E(Q)}|\Theta_E b_Q(x)|^2 {\mathbf{1}}_{\{z\in{\mathscr{X}}:\,\delta_{E_Q}(z)<C_1^{-1}\delta_E(z)\}}(x)\, \delta_E(x)^{2\upsilon-(m-d)}\,d\mu(x) \nonumber\\[4pt] && \hskip 0.40in =: I_1+I_2+I_3. \end{eqnarray}
To proceed with estimating $I_1$ we first obtain a pointwise bound for $\Theta_E b_Q$. To this end, first observe that
\begin{eqnarray}\label{setO} {\mathcal{O}}:=\bigl\{z\in{\mathscr{X}}:\,\delta_{E_Q}(z)>C_1\delta_E(z)\bigr\} \,\Longrightarrow\,{\mathcal{O}}\cap E_Q=\emptyset. \end{eqnarray}
Hence, \eqref{BBss}, \eqref{hszz-A} and \eqref{mMji} in Lemma~\ref{Gkwvr} give that, for some finite $C>0$ independent of the dyadic cube $Q$,
\begin{eqnarray}\label{Lz}
|\Theta_E b_Q(x)| &=& \left|\int_E {\theta}(x,y)\,b_Q(y)\,d\sigma(y)\right| \nonumber\\[4pt]
&\leq & \int_{E_Q} |{\theta}(x,y)|\,d\sigma(y)\leq\frac{C}{\delta_{E_Q}(x)^\upsilon}, \qquad\forall\,x\in{\mathcal{O}}. \end{eqnarray}
Also, \eqref{yvg2} guarantees that $Q\cap E_Q\not=\emptyset$, and we fix a point $x_0\in Q\cap E_Q$. By \eqref{dFvK}, there exists $c\in(0,\infty)$ such that $T_E(Q)\subseteq B_{\rho_{\#}}(x_0,c\,\ell(Q))$ which when combined with \eqref{Lz} gives
\begin{eqnarray}\label{gwo} I_1\leq C\int_{B_{\rho_{\#}}(x_0,c\ell(Q))\cap{\mathcal{O}}} \delta_{E_Q}(x)^{-2\upsilon}\delta_E(x)^{2\upsilon-(m-d)}\,d\mu(x). \end{eqnarray}
At this stage, select a constant $M\in(C_\rho^2,\infty]$, choose $C_1\in(M,\infty)$, and observe that if $x\in{\mathcal{O}}$ then $\frac{\delta_{E_Q}(x)}{M}>\delta_E(x)$, hence $B_{\rho_{\#}}\bigl(x,\delta_{E_Q}(x)/M\bigr)\cap E\not=\emptyset$. Recalling Lemma~\ref{segj}, it follows that there exists $C\in(0,\infty)$ such that
\begin{eqnarray}\label{dxb} \frac{\delta_{E_Q}(x)^d}{M^d}\leq C {\mathscr{H}}_{{\mathscr{X}}\!,\,\rho_{\#}}^d \Bigl(B_{\rho_{\#}}\Bigl(x,C_\rho\frac{\delta_{E_Q}(x)}{M}\Bigr)\cap E\Bigr), \qquad\forall\,x\in{\mathcal{O}}. \end{eqnarray}
Using this in \eqref{gwo} we obtain
\begin{eqnarray}\label{gwo-2} I_1\leq C\!\!\! \int\limits_{B_{\rho_{\#}}(x_0,c\ell(Q))\cap{\mathcal{O}}} \Bigl(\int\limits_{B_{\rho_{\#}} \bigl(x,C_\rho\delta_{E_Q}(x)/M\bigr)\cap E} \hskip -0.40in 1\,d{\mathscr{H}}_{{\mathscr{X}}\!,\,\rho_{\#}}^d(z)\Bigr) \delta_{E_Q}(x)^{-2\upsilon-d}\delta_E(x)^{2\upsilon-(m-d)}\,d\mu(x). \end{eqnarray}
We make the claim that for each $\vartheta\in(0,1)$
\begin{eqnarray}\label{gwo-3} \left. \begin{array}{r} \mbox{if $x\in{\mathscr{X}}\setminus E_Q$ and $z\in {\mathscr{X}}$ are} \\[4pt] \mbox{such that }\,\rho_{\#}(x,z)<\frac{\vartheta}{C_\rho}\delta_{E_Q}(x) \end{array} \right\} \Longrightarrow \,\, \frac{1-\vartheta}{C_\rho}\delta_{E_Q}(x)\leq\delta_{E_Q}(z)\leq C_\rho\delta_{E_Q}(x). \end{eqnarray}
Indeed, for each $\eta>1$ close to $1$ we may take $y\in E_Q$ satisfying $\rho_{\#}(y,x)<\eta\delta_{E_Q}(x)$ which implies that $\delta_{E_Q}(z)\leq\rho_{\#}(y,z)\leq C_\rho\max\{\rho_{\#}(y,x),\rho_{\#}(x,z)\} \leq C_\rho \eta\delta_{E_Q}(x)$. Upon letting $\eta\searrow 1$ we therefore obtain $\delta_{E_Q}(z)\leq C_\rho\delta_{E_Q}(x)$. On the other hand, for each $w\in E_Q$ we have $\delta_{E_Q}(x)\leq \rho_{\#}(x,w)\leq C_\rho\rho_{\#}(x,z) +C_\rho\rho_{\#}(z,w)\leq\vartheta\delta_{E_Q}(x)+C_\rho\rho_{\#}(z,w)$, which further yields $\delta_{E_Q}(x)\leq \frac{C_\rho}{1-\vartheta}\delta_{E_Q}(z)$. This concludes the proof of \eqref{gwo-3}.
Going further, fix $x\in B_{\rho_{\#}}(x_0,c\ell(Q))\cap{\mathcal{O}}$ and $z\in B_{\rho_{\#}}\bigl(x,C_\rho\delta_{E_Q}(x)/M\bigr)\cap E$ and make two observations. First, an application of \eqref{gwo-3} with $\vartheta:=C_\rho^2/M\in(0,1)$ yields
\begin{eqnarray}\label{gwo-3a} \hskip -0.20in \frac{M-C_\rho^2}{MC_\rho}\delta_{E_Q}(x)\leq\delta_{E_Q}(z)\leq C_\rho\delta_{E_Q}(x) \,\,\mbox{ and }\,\, \rho_{\#}(z,x)<\frac{C_\rho}{M}\delta_{E_Q}(x) \leq\frac{C_\rho^2}{M-C_\rho^2}\delta_{E_Q}(z), \end{eqnarray}
hence $x\in B_{\rho_{\#}}\bigl(z,\tfrac{C_\rho^2}{M-C_\rho^2}\delta_{E_Q}(z)\bigr)$. Second, recalling that $x_0\in E_Q$, $M>C_\rho^2$ and $C_\rho\geq 1$, we obtain
\begin{eqnarray}\label{gwo-3b} \rho_{\#}(x_0,z) & \leq & C_\rho\max\bigl\{\rho_{\#}(x_0,x),\rho_{\#}(x,z)\bigr\} <C_\rho\max\bigl\{c\,\ell(Q),\tfrac{C_\rho}{M}\delta_{E_Q}(x)\bigr\} \\[4pt] &\leq & C_\rho\max\bigl\{c\,\ell(Q),\tfrac{1}{C_\rho}\delta_{E_Q}(x)\bigr\} \leq C_\rho\max\bigl\{c\,\ell(Q),\tfrac{1}{C_\rho}\rho_{\#}(x_0,x)\bigr\} =C_\rho c\,\ell(Q), \nonumber \end{eqnarray}
which shows that $z\in B_{\rho_{\#}}(x_0,C_\rho c\ell(Q))$. Combining these observations with \eqref{gwo-3a}, keeping in mind \eqref{setO}, and using Fubini's Theorem in \eqref{gwo-2}, we may write
\begin{eqnarray}\label{gwo-5} I_1 & \!\!\!\leq\!\!\! & C\int\limits_{B_{\rho_{\#}}\bigl(x_0,C_\rho c\ell(Q)\bigr)\cap (E\setminus E_Q)} \hskip -0.30in \delta_{E_Q}(z)^{-2\upsilon-d} \Bigl(\int\limits_{ B_{\rho_{\#}}\bigl(z,\frac{C_\rho^2}{M-C_\rho^2}\delta_{E_Q}(z)\bigr)\setminus E_Q} \hskip -0.30in\delta_E(x)^{2\upsilon-(m-d)}\,d\mu(x)\Bigr) \,d{\mathscr{H}}_{{\mathscr{X}}\!,\,\rho_{\#}}^d(z) \nonumber\\[4pt] & \!\!\!\leq \!\!\! & C\int\limits_{B_{\rho_{\#}}\bigl(x_0,C_\rho c\ell(Q)\bigr) \cap (E\setminus E_Q)} \delta_{E_Q}(z)^{-2\upsilon-d}\delta_{E_Q}(z)^{2\upsilon+d} \,d{\mathscr{H}}_{{\mathscr{X}}\!,\,\rho_{\#}}^d(z) \nonumber\\[4pt] & \!\!\!\leq\!\!\! & C{\mathscr{H}}_{{\mathscr{X}}\!,\,\rho_{\#}}^d \Bigl(B_{\rho_{\#}}\bigl(x_0,C_\rho c\,\ell(Q)\bigr)\cap E\Bigr) \leq C\ell(Q)^d\leq C\sigma(Q), \end{eqnarray}
where for the second inequality in \eqref{gwo-5} we used Lemma~\ref{geom-lem} (with $\gamma:=(m-d)-2\upsilon$ and $r:=R:=\frac{C_\rho^2}{M-C_\rho^2}\delta_{E_Q}(z)$), and for the last two inequalities the fact that
$\bigl(E,\rho\bigl|_{E},{\mathscr{H}}_{{\mathscr{X}}\!,\,\rho_{\#}}^d\lfloor E\bigr)$ is $d$-{\rm ADR} and $x_0\in E$.
To estimate $I_3$, we first note that since $T_E(Q)\cap E=\emptyset$ then \eqref{hszz-A} and \eqref{mMji} give that
\begin{eqnarray}\label{lbpz.33} \hskip -0.20in
|\Theta_E b_Q(x)|=\left|\int_E {\theta}(x,y)\,b_Q(y)\,d\sigma(y)\right|
\leq\int_E |{\theta}(x,y)|\,d\sigma(y)\leq\frac{C}{\delta_{E}(x)^\upsilon},\quad \forall\,x\in T_E(Q), \end{eqnarray}
for some finite $C>0$ independent of $Q$. Also (compare with \eqref{Lz}),
$|\Theta_E b_Q(x)|\leq C\delta_{E_Q}(x)^{-\upsilon}$ for each $x\in T_{E}(Q)\setminus E_Q$. Fix $\alpha,\beta>0$ such that $\alpha+\beta=\upsilon$. A logarithmically convex combination of these inequalities then yields
\begin{eqnarray}\label{lbpz.34}
|\Theta_E b_Q(x)|\leq C\delta_{E_Q}(x)^{-\alpha} \delta_{E}(x)^{-\beta}\qquad\forall\,x\in T_E(Q)\setminus E_Q. \end{eqnarray}
Observe that, by assumptions and Lemma~\ref{ME-ZZ}, we have $\mu(E_Q)=0$. Using this and \eqref{lbpz.34} in place of \eqref{Lz}, we obtain (compare with \eqref{gwo})
\begin{eqnarray}\label{gwo-N} I_3\leq C\int_{B_{\rho_{\#}}(x_0,c\ell(Q))\cap({\widetilde{\mathcal{O}}}\setminus E_Q)} \delta_E(x)^{-2\beta+2\upsilon-(m-d)}\delta_{E_Q}(x)^{-2\alpha}\,d\mu(x), \end{eqnarray}
where, this time, we have set
\begin{eqnarray}\label{setO-N} \widetilde{\mathcal{O}}:=\bigl\{z\in{\mathscr{X}}:\, \delta_E(z)>C_1\delta_{E_Q}(z)\bigr\}. \end{eqnarray}
Given the nature of \eqref{gwo-N}, \eqref{setO-N}, the same reasoning leading up to \eqref{gwo-5} used with $E$ and $E_Q$ interchanged this time gives
\begin{eqnarray}\label{gwo-5-N} I_3 & \!\!\!\leq\!\!\! & C\int\limits_{B_{\rho_{\#}}\bigl(x_0,C_\rho c\ell(Q)\bigr)\cap (E_Q\setminus E)} \hskip -0.30in \delta_{E}(z)^{-2\beta+2\upsilon-m} \Bigl(\int\limits_{ B_{\rho_{\#}}\bigl(z,\frac{C_\rho^2}{M-C_\rho^2}\delta_{E}(z)\bigr)\setminus E} \hskip -0.30in\delta_E(x)^{-2\alpha}\,d\mu(x)\Bigr) \,d{\mathscr{H}}_{{\mathscr{X}}\!,\,\rho_{\#}}^d(z) \nonumber\\[4pt] & \!\!\!\leq \!\!\! & C\int\limits_{B_{\rho_{\#}}\bigl(x_0,C_\rho c\ell(Q)\bigr) \cap (E_Q\setminus E)} \delta_{E}(z)^{-2\beta+2\upsilon-m}\delta_{E}(z)^{-2\alpha+m} \,d{\mathscr{H}}_{{\mathscr{X}}\!,\,\rho_{\#}}^d(z) \nonumber\\[4pt] & \!\!\!\leq\!\!\! & C{\mathscr{H}}_{{\mathscr{X}}\!,\,\rho_{\#}}^d \Bigl(B_{\rho_{\#}}\bigl(x_0,C_\rho c\,\ell(Q)\bigr)\cap E_Q\Bigr) \leq C\ell(Q)^d\leq C\sigma(Q). \end{eqnarray}
Above, the second inequality follows from Lemma~\ref{geom-lem} (used with $r:=R:=\frac{C_\rho^2}{M-C_\rho^2}\delta_{E}(z)$) provided we choose $0<\alpha<(m-d)/2$ to begin with. Also, for the last two inequalities in \eqref{gwo-5-N} we have made use of the fact that both
$\bigl(E,\rho\bigl|_{E},{\mathscr{H}}_{{\mathscr{X}}\!,\,\rho_{\#}}^d\lfloor E\bigr)$
and $\bigl(E_Q,\rho\bigl|_{E_Q}, {\mathscr{H}}_{{\mathscr{X}}\!,\,\rho_{\#}}^d\lfloor E_Q\bigr)$ are $d$-{\rm ADR} spaces and that $x_0\in E\cap E_Q$.
We are left with estimating $I_2$. With $C_1$ as above, thanks to \eqref{BBss}, \eqref{operator-A}, and \eqref{haTT.2} we have
\begin{eqnarray}\label{KLGB.3}
I_2=\int_{T_E(Q)\setminus E_Q}|\Theta_{E_Q} b_Q(x)|^2 {\mathbf{1}}_{\{z\in{\mathscr{X}}:\, C_1^{-1}\delta_E(z)\leq\delta_{E_Q}(z)\leq C_1\delta_E(z)\}}(x)\, \delta_E(x)^{2\upsilon-(m-d)}\,d\mu(x). \end{eqnarray}
Hence, we may further use \eqref{KLGB.3} and \eqref{avhai2} in order to write (with $\sigma_Q$ as in \eqref{HHH-60})
\begin{eqnarray}\label{KLGB-1} I_2 &\leq & C\int_{\mathscr{X}\setminus E_Q}
|\Theta_{E_Q} b_Q(x)|^2\,\delta_{E_Q}(x)^{2\upsilon-(m-d)}\,d\mu(x) \nonumber\\[4pt]
&\leq & C\int_{E_Q}|b_Q|^2\,d\sigma_Q \leq C{\mathscr{H}}_{{\mathscr{X}}\!,\,\rho_{\#}}^d\bigl(Q\cap E_Q\bigr) \leq C\sigma(Q), \end{eqnarray}
which is of the correct order.
Now the fact that condition {\it 3} in Theorem~\ref{Thm:localTb} is satisfied for our choice of $b_Q$'s follows by combining \eqref{KLGB}, \eqref{gwo-5}, \eqref{gwo-5-N} and \eqref{KLGB-1}. This finishes the proof of the theorem. \end{proof}
We conclude this section by taking a closer look at a higher order version of the notion of ``big pieces of square function estimates." To set the stage, in the context of Definition~\ref{sjvs} let us say that a closed subset $E$ of $({\mathscr{X}},\tau_\rho)$ with the property that there exists a Borel regular
measure $\sigma$ on $(E,\tau_{\rho|_{E}})$ such that $\bigl(E,\rho\bigl|_E,\sigma\bigr)$ is a $d$-dimensional {\rm ADR} space has {\tt (BP)$^0$SFE} {\tt relative to} $\theta$, or simply {\tt SFE} {\tt relative to} $\theta$ (``Square Function Estimates relative to $\theta$"), provided there exists a finite positive constant $C$ such that
\begin{eqnarray}\label{avhai-TTFsa} \begin{array}{c} \displaystyle
\int_{\mathscr{X}\setminus E}|\Theta_{E}f(z)|^2\, {\rm dist}_{\rho_{\#}}(z,E)^{2\upsilon-(m-d)}\,d\mu(z)
\leq C\int_{E}|f|^2\,d\sigma\qquad \\[8pt] \mbox{for each function }\,\,f\in L^2(E,\sigma). \end{array} \end{eqnarray}
In addition, we shall say that $E$ has {\tt (BP)$^1$SFE} whenever $E$ has {\tt BPSFE}.
We may then iteratively interpret ``$E$ has {\tt (BP)$^{k+1}$SFE}" as the property that $E$ contains big pieces of sets having {\tt (BP)$^k$SFE}, in a uniform fashion. More specifically, we make the following definition.
\begin{definition}\label{sjvs-DDD} Consider two numbers $d,m\in(0,\infty)$ such that $m>d$, suppose that $({\mathscr{X}},\rho,\mu)$ is an $m$-dimensional {\rm ADR} space, and assume that ${\theta}$ is as in \eqref{K234-A}-\eqref{hszz-3-A}. Also, suppose that $k\in{\mathbb{N}}$. In this context, a set $E\subseteq{\mathscr{X}}$ is said to have {\tt (BP)$^{k+1}$SFE} relative to $\theta$ provided the following conditions are satisfied:
\begin{enumerate}
\item[(i)] the set $E$ is closed in $({\mathscr{X}},\tau_\rho)$ and has the property that there exists a Borel regular measure $\sigma$
on $(E,\tau_{\rho|_{E}})$ such that $\bigl(E,\rho\bigl|_E,\sigma\bigr)$ is a $d$-dimensional {\rm ADR} space;
\item[(ii)] there exist finite positive constants $\eta$, $C_1$, and $C_2$ with the property that for each $x\in E$ and each real number $r\in(0,{\rm diam}_{\rho_{\#}}(E)]$ there exists a closed subset $E_{x,r}$ of $(\mathscr{X},\tau_\rho)$ such that if
\begin{eqnarray}\label{HHH-57-yyy} \sigma_{x,r}:={\mathscr{H}}_{{\mathscr{X}}\!,\,\rho_{\#}}^d\lfloor E_{x,r}, \quad\mbox{ where ${\mathscr{H}}_{{\mathscr{X}}\!,\,\rho_{\#}}^d$ is as in \eqref{HHH-56}}, \end{eqnarray}
then $\bigl(E_{x,r},\rho\big|_{E_{x,r}},\sigma_{x,r}\bigr)$ is a $d$-dimensional {\rm ADR} space, with {\rm ADR} constant $\leq C_1$, and which satisfies
\begin{eqnarray}\label{yvg.Ifav} \sigma\bigl(E_{x,r}\cap E\cap B_{\rho_{\#}}(x,r)\bigr)\geq\eta\,r^d, \end{eqnarray}
as well as
\begin{eqnarray}\label{yvg-5tVV} \mbox{$E_{x,r}$ has {\tt (BP)$^k$SFE} relative to $\theta$, with character controlled by $C_2$}. \end{eqnarray} \end{enumerate}
In this context, we shall refer to $\eta,C_1,C_2$ as the {\tt (BP)$^{k+1}$SFE} character of $E$. \end{definition}
The following result may be regarded as a refinement of Theorem~\ref{Thm:BPSFtoSF}.
\begin{theorem}\label{Thm:BPSFtoSF.XXX} Consider two numbers $d,m\in(0,\infty)$ such that $m>d$, suppose that $({\mathscr{X}},\rho,\mu)$ is an $m$-dimensional {\rm ADR} space, and assume that ${\theta}$ is as in \eqref{K234-A}-\eqref{hszz-3-A}. Also, suppose that the set $E$ is closed in $({\mathscr{X}},\tau_\rho)$ and has the property that there exists a Borel regular measure $\sigma$
on $(E,\tau_{\rho|_{E}})$ such that $\bigl(E,\rho\bigl|_E,\sigma\bigr)$ is a $d$-dimensional {\rm ADR} space.
Then the following claims are equivalent:
\begin{enumerate} \item[(i)] $E$ has {\tt (BP)$^k$SFE} relative to $\theta$ for some $k\in{\mathbb{N}}$;
\item[(ii)] $E$ has {\tt (BP)$^k$SFE} relative to $\theta$ for every $k\in{\mathbb{N}}$;
\item[(iii)] $E$ has {\tt (BP)$^0$SFE} relative to $\theta$. \end{enumerate} \end{theorem}
\begin{proof} It is clear that if $E$ has {\tt (BP)$^k$SFE} relative to $\theta$ for some $k\in{\mathbb{N}}_0$ then $E$ also has {\tt (BP)$^{k+1}$SFE} relative to $\theta$, since obviously $E$ has big pieces of itself. This gives the implications {\it (iii)} $\Rightarrow$ {\it (i)} and {\it (iii)} $\Rightarrow$ {\it (ii)} in the statement of the theorem. Also, the implication {\it (ii)} $\Rightarrow$ {\it (i)} is trivial. Finally, in light of Definition~\ref{sjvs-DDD}, Theorem~\ref{Thm:BPSFtoSF} combined with an induction argument gives that {\it (i)} $\Rightarrow$ {\it (iii)}, completing the proof. \end{proof}
\section{Square Function Estimates on Uniformly Rectifiable Sets}\label{Sect:SFE} \setcounter{equation}{0}
Given an $n$-dimensional Ahlfors-David regular set $\Sigma$ in $\mathbb{R}^{n+1}$ that has so-called big pieces of Lipschitz graphs (BPLG), the inductive scheme established in the previous section allows us to deduce square function estimates for an integral operator $\Theta_\Sigma$, as in~\eqref{operator-A}, whenever square function estimates are satisfied by $\Theta_\Gamma$ for all Lipschitz graphs $\Gamma$ in $\mathbb{R}^{n+1}$. Furthermore, induction allows us to prove the same result when the set $\Sigma$ only has (BP)$^k$LG for any $k\in\mathbb{N}$. The definition of (BP)$^k$LG is given in Definition~\ref{Def-BPnLG}. A recent result by J.~Azzam and R.~Schul (cf. \cite[Corollary~1.7]{AS}) proves that uniformly rectifiable sets have (BP)$^2$LG (the converse implication also holds and can be found in~\cite[p.\,16]{DaSe93}), and this allows us to obtain square function estimates on uniformly rectifiable sets.
We work in the Euclidean codimension one setting throughout this section. In particular, fix $n\in\mathbb{N}$ and let $\mathbb{R}^{n+1}$ be the ambient space, so that in the notation of Section~\ref{Sect:4}, we would have $d=n$, $m=n+1$ and $(\mathscr{X},\rho,\mu)$ is $\mathbb{R}^{n+1}$ with the Euclidean metric and Lebesgue measure. We also restrict our attention to the following class of kernels in order to obtain square function estimates on Lipschitz graphs. Suppose that $K:{\mathbb{R}}^{n+1}\setminus\{0\} \rightarrow \mathbb{R}$ satisfies
\begin{eqnarray}\label{prop-K} K\in C^2({\mathbb{R}}^{n+1}\setminus\{0\}),\quad K(\lambda x)=\lambda^{-n}K(x)\,\mbox{ for all }\,\lambda>0,\,\, x\in{\mathbb{R}}^{n+1}\setminus\{0\},\quad K\mbox{ is odd},\ \ \end{eqnarray}
and has the property that there exists a finite positive constant $C_K$ such that for all $j\in\{0,1,2\}$ the following holds:
\begin{eqnarray}\label{prop-K2}
|\nabla^j K(x)|\leq C_K|x|^{-n-j},\quad \forall\,x\in{\mathbb{R}}^{n+1}\setminus\{0\}. \end{eqnarray}
Then for each closed subset $\Sigma$ of ${\mathbb{R}}^{n+1}$, denote by $\sigma:={\mathscr{H}}^n_{\mathbb{R}^{n+1}}\lfloor{\Sigma}$ the surface measure induced by the $n$-dimensional Hausdorff measure on $\Sigma$ from \eqref{HHH-56}, and define the integral operator ${\mathcal{T}}$ for all functions $f\in L^p(\Sigma,\sigma)$, $1\leq p\leq\infty$, by
\begin{eqnarray}\label{sbnvjb} {\mathcal{T}}f(x):=\int_{\Sigma} K(x-y)f(y)\,d{\sigma}(y),\qquad \forall\,x\in{\mathbb{R}}^{n+1}\setminus\Sigma. \end{eqnarray}
In the notation of Section~\ref{Sect:4}, we consider the set $E=\Sigma$ and the operator $\Theta_E=\nabla\mathcal{T}$ with integral kernel $\theta=\nabla K$. We begin by proving square function estimates for $\nabla\mathcal{T}$ in the case when $\Sigma$ is a Lipschitz graph. The inductive scheme from the previous section then allows us to extend that result to the case when $\Sigma$ has (BP)$^k$LG for any $k\in\mathbb{N}$, and hence when $\Sigma$ is uniformly rectifiable.
\subsection{Square function estimates on Lipschitz graphs} \label{SSect:LG}
The main result in this subsection is the square function estimate for Lipschitz graphs contained in the theorem below. A parabolic variant of this result appears in \cite{HL}, and the present proof is based on the arguments given there, and in \cite{Ho}.
\begin{theorem}\label{Theorem1.1} Let $A:{\mathbb{R}}^n\to{\mathbb{R}}$ be a Lipschitz function and set $\Sigma:=\{(x,A(x)):\,x\in{\mathbb{R}}^n\}$. Moreover, assume that $K$ is as in \eqref{prop-K} and consider the operator ${\mathcal{T}}$ as in \eqref{sbnvjb}. Then there exists a finite constant $C>0$ depending only on
$\|\partial^\alpha K\|_{L^\infty(S^{n})}$ for $|\alpha|\leq 2$, and the Lipschitz constant of $A$ such that for each function $f\in L^2(\Sigma,\sigma)$ one has
\begin{eqnarray}\label{hbraH} \int_{{\mathbb{R}}^{n+1}\setminus\Sigma}
|(\nabla{\mathcal{T}}f)(x)|^2\,{\rm dist}(x,\Sigma)\,dx
\leq C\int_{\Sigma}|f|^2\,d\sigma. \end{eqnarray} \end{theorem}
As a preamble to the proof of Theorem~\ref{Theorem1.1}, we state and prove a couple of technical lemmas. The first has essentially appeared previously in \cite{Ch}, and is based upon ideas of \cite{J}.
\begin{lemma}\label{lgih} Assume that $A:{\mathbb{R}}^n\to{\mathbb{R}}$ is a locally integrable function such that $\nabla A\in L^2({\mathbb{R}}^n)$. Pick a smooth, real-valued, nonnegative, compactly supported function $\phi$ defined in ${\mathbb{R}}^n$ with $\int_{{\mathbb{R}}^n}\phi(x)\,dx=1$ and for each $t>0$ set $\phi_t(x):=t^{-n}\phi(x/t)$ for $x\in{\mathbb{R}}^n$. Finally, define
\begin{eqnarray}\label{1.28} E_A(t,x,y):= A(x)-A(y)-{\langle}\nabla_{x}(\phi_t\ast A)(x),(x-y){\rangle},\qquad \forall\,x,y\in{\mathbb{R}}^n,\,\,\,\forall\,t>0. \end{eqnarray}
Then, for some finite positive constant $C=C(\phi,n)$,
\begin{eqnarray}\label{1.29}
\int_0^{\infty}t^{-n-2}\int_{{\mathbb{R}}^n}\int_{|x-y|\leq \lambda t}
|E_A(t,x,y)|^2\,dy\,dx\,\tfrac{dt}{t}
\leq C\lambda^{n+3}\|\nabla A\|^2_{L^2({\mathbb{R}}^n)},\qquad\forall\,\lambda\geq 1. \end{eqnarray} \end{lemma}
\begin{proof} Starting with the changes of variables $t=\lambda^{-1}\tau$, $y=x+h$ and then employing Plancherel's theorem in the variable $x$, we may write (with `hat' denoting the Fourier transform)
\begin{eqnarray}\label{ldbs}
&& \int_0^{\infty}t^{-n-2}\int_{{\mathbb{R}}^n}\int_{|x-y|\leq \lambda t}
|E_A(t,x,y)|^2\,dy\,dx\,\frac{dt}{t} \\[4pt] && \hskip 0.50in
=\lambda^{n+2}\int_0^{\infty}\tau^{-n-2}\int_{{\mathbb{R}}^n}\int_{|h|\leq\tau}
\bigl|A(x)-A(x+h)+{\langle}\nabla_{x}(\phi_{\lambda^{-1}\tau}\ast A)(x),h{\rangle}\bigr|^2 \,dh\,dx\,\frac{d\tau}{\tau} \nonumber\\[4pt] && \hskip 0.50in
=\lambda^{n+2}\int_0^{\infty}\tau^{-n-2}\int_{{\mathbb{R}}^n}\int_{|h|\leq\tau}
\bigl|1-e^{i\langle\zeta,h\rangle}+i\langle\zeta,h\rangle\,
\widehat{\phi}(\lambda^{-1}\tau\zeta)\bigr|^2
\tfrac{|\widehat{\nabla A}(\zeta)|^2}{|\zeta|^2} \,dh\,d\zeta\,\frac{d\tau}{\tau} \nonumber\\[4pt] && \hskip 0.50in
=\lambda^{n+2}\int_0^{\infty}\int_{{\mathbb{R}}^n}\int_{|w|\leq 1}
\frac{\bigl|1-e^{i\tau\langle\zeta,w\rangle}+i\tau\langle\zeta,w\rangle\,
\widehat{\phi}(\lambda^{-1}\tau\zeta)\bigr|^2}{\tau^2|\zeta|^2}
|\widehat{\nabla A}(\zeta)|^2\,dw\,d\zeta\,\frac{d\tau}{\tau}, \nonumber \end{eqnarray}
where the last equality in \eqref{ldbs} is based on the change of variables $h=\tau w$.
Next we observe that for every $\zeta\in{\mathbb{R}}^n$ and
$w\in{\mathbb{R}}^n$ with $|w|\leq 1$ there holds
\begin{eqnarray}\label{zawgi}
\frac{\bigl|1-e^{i\tau\langle\zeta,w\rangle}+i\tau\langle\zeta,w\rangle\,
\widehat{\phi}(\lambda^{-1}\tau\zeta)\bigr|}{\tau|\zeta|}
\leq C\,\min\Bigl\{\tau|\zeta|,\frac{\lambda}{\tau|\zeta|}\Bigr\} \end{eqnarray}
for some $C>0$ depending only on $\phi$. To see why \eqref{zawgi} is true, analyze the following two cases.
\vskip 0.08in
\noindent {\it Case~1.} $\tau|\zeta|\leq\sqrt{\lambda}$\,:\\
In this situation the minimum in the right hand-side of \eqref{zawgi} is equal to $\tau|\zeta|$. In addition, if we use Taylor expansions about zero for the complex exponential function and $\widehat{\phi}$, we obtain
(keeping in mind that $\widehat{\phi}(0)=1$, $\lambda\geq 1$ and $|w|\leq 1$)
\begin{eqnarray}\label{zaw-cgi}
\bigl|1-e^{i\tau\langle\zeta,w\rangle}+i\tau\langle\zeta,w\rangle
+i\tau\langle\zeta,w\rangle\,(\widehat{\phi}(\lambda^{-1}\tau\zeta)-1)\bigr|
\leq C\tau^2|\zeta|^2, \end{eqnarray}
which shows that \eqref{zawgi} holds in this case.
\vskip 0.08in
\noindent {\it Case~2.} $\tau|\zeta|>\sqrt{\lambda}$\,:\\
In this scenario the minimum in the right hand-side of \eqref{zawgi} is equal to $\frac{\lambda}{\tau|\zeta|}$. Moreover,
\begin{eqnarray}\label{pkbi}
\bigl|1-e^{i\tau\langle\zeta,w\rangle}
+\tau\langle\zeta,w\rangle\,\widehat{\phi}(\lambda^{-1}\tau\zeta)\bigr|
&\leq& 2+\tau|\zeta|\,\bigl|\widehat{\phi}(\lambda^{-1}\tau\zeta)\bigr| \nonumber\\[4pt]
&\leq& 2+C\tau|\zeta|\,(1+\lambda^{-1}\tau|\zeta|)^{-1} \leq C\lambda, \end{eqnarray}
since the Schwartz function $\widehat{\phi}$ decays, $\lambda\geq 1$
and $|w|\leq 1$. Consequently, \eqref{zawgi} holds in this case as well.
With \eqref{zawgi} in hand, we proceed to integrate in $\tau\in(0,\infty)$ with respect to the Haar measure to further obtain
\begin{eqnarray}\label{knoB} && \int_0^{\infty}
\frac{\bigl|1-e^{i\tau\langle\zeta,w\rangle}+i\tau\langle\zeta,w\rangle\,
\widehat{\phi}(\lambda^{-1}\tau\zeta)\bigr|^2}{\tau^2|\zeta|^2}\,\frac{d\tau}{\tau}
\leq \int_0^{\infty}\min\Bigl\{\tau^2|\zeta|^2,\frac{\lambda^2}{\tau^2|\zeta|^2}\Bigr\} \,\frac{d\tau}{\tau} \nonumber\\[4pt] && \hskip 0.50in
=\int_0^{\sqrt{\lambda}/|\zeta|}\tau|\zeta|^2\,d\tau
+\int_{\sqrt{\lambda}/|\zeta|}^\infty\frac{\lambda^2}{\tau^3|\zeta|^2}\,d\tau \leq C\lambda. \end{eqnarray}
A combination of \eqref{ldbs}, \eqref{knoB} and Plancherel's theorem now yields \eqref{1.29}, finishing the proof of Lemma~\ref{lgih}. \end{proof}
The second lemma needed here has essentially appeared previously in \cite{MMT}.
\begin{lemma}\label{Lemma1.3} Let $F:{\mathbb{R}}^{n+1}\setminus\{0\}\to{\mathbb{R}}$ be a continuous function which is even and positive homogeneous of degree $-n-1$. Then for any $a\in{\mathbb{R}}^n$ and any $t>0$ there holds
\begin{eqnarray}\label{1.31} \int_{{\mathbb{R}}^n}F(y,a\cdot y+t)\,dy &=& \frac{1}{2t}\int_{S^{n-1}}\int_{-\infty}^\infty F(\omega,s)\,ds\,d\omega \nonumber\\[4pt] &=&\int_{{\mathbb{R}}^n}F(y,t)\,dy. \end{eqnarray}
In particular, if $F$ is some first-order partial derivative, say $F=\partial_jG$, $j\in\{1,..,n+1\}$, of a function $G\in C^1({\mathbb{R}}^{n+1}\setminus\{0\})$ which is odd and homogeneous of degree $-n$, then
\begin{eqnarray}\label{1.31ASBN} \int_{{\mathbb{R}}^n} F(y,a\cdot y+t)\,dy=0 \qquad\mbox{ for any $a\in{\mathbb{R}}^n$ and $t>0$}. \end{eqnarray} \end{lemma}
\begin{proof} Fix $a\in{\mathbb{R}}^n$ and $t>0$. By the homogeneity of $F$ we
have $|F(x)|\leq\|F\|_{L^\infty(S^n)}|x|^{-n-1}$ for every
$x\in{\mathbb{R}}^{n+1}\setminus\{0\}$. Also, $\{(y,a\cdot y+t):\,|y|\leq 1\}$ is a compact subset of ${\mathbb{R}}^{n+1}\setminus\{0\}$. Hence, given that $F$ is continuous on ${\mathbb{R}}^{n+1}\setminus\{0\}$, it follows that
$\int_{{\mathbb{R}}^n}|F(y,a\cdot y+t)|\,dy<\infty$.
To proceed, by passing to polar coordinates $y=r\omega$, $r>0$, $\omega\in S^{n-1}$, and using the homogeneity of $F$ we may write
\begin{eqnarray}\label{1.32} \int_{{\mathbb{R}}^n}F(y,a\cdot y+t)\,dy= \int_{S^{n-1}}\int_0^\infty F(\omega,a\cdot\omega+t/r)\,r^{-2}dr\,d\omega. \end{eqnarray}
Now, setting $s:=a\cdot\omega+t/r$ the last integral above becomes $t^{-1}\int_{S^{n-1}}\int_{a\cdot\omega}^\infty F(\omega,s)\,ds\,d\omega$ which, by making the change of variables $(\omega,s)\mapsto(-\omega,-s)$ and using the fact that $F$ is even, may be written as
\begin{eqnarray}\label{1.33} \frac{1}{t}\int_{S^{n-1}}\int_{a\cdot\omega}^\infty F(\omega,s)\,ds\,d\omega =\frac{1}{2t}\int_{S^{n-1}}\int_{-\infty}^\infty F(\omega,s)\,ds\,d\omega. \end{eqnarray}
This analysis gives the first equality in \eqref{1.31}. Furthermore, the integral in the right side of \eqref{1.33} is independent of $a\in{\mathbb{R}}^n$ and, hence, so is the original one. In particular, its value does not change if we take $a=0$ and this is precisely what the second equality in \eqref{1.31} says.
Finally, consider the claim made in \eqref{1.31ASBN} under the assumption that $F=\partial_jG$ for some $j\in\{1,..,n+1\}$ and some $G\in C^1({\mathbb{R}}^{n+1}\setminus\{0\})$ which is odd and homogeneous of degree $-n$. In particular, there exists $C\in(0,\infty)$
such that $|G(x)|\leq C|x|^{-n}$ for all $x\in{\mathbb{R}}^{n+1}\setminus\{0\}$. Then, using the decay of $G$ and integration by parts, if $j\neq n+1$ the third integral in \eqref{1.31} vanishes whereas if $j=n+1$ the second one does so. \end{proof}
After this preamble, we are ready to present the
\vskip 0.08in \begin{proof}[Proof of Theorem~\ref{Theorem1.1}] A moment's reflection shows that it suffices to establish \eqref{hbraH} with the domain of integration ${\mathbb{R}}^{n+1}\setminus\Sigma$ in the left-hand side replaced by
\begin{eqnarray}\label{Gbbab} \Omega:=\{(x,t)\in{\mathbb{R}}^{n+1}:t>A(x)\}. \end{eqnarray}
Assume that this is the case and note that by making the bi-Lipschitz change of variables ${\mathbb{R}}^n\times(0,\infty)\ni (x,t)\mapsto (x,A(x)+t)\in\Omega$, (whose Jacobian is equivalent to a finite constant) the estimate \eqref{hbraH} follows from the boundedness of
\begin{eqnarray}\label{1.20} && T^j:L^2({\mathbb{R}}^n,dx)\to L^2({\mathbb{R}}^{n+1}_+,\tfrac{dt}{t}dx), \\ && T^jf(x,t):=\int_{{\mathbb{R}}^n}K^j_t(x,y)f(y)\,dy \end{eqnarray}
for $j=1,..,n+1$, where the family of kernels $\{K^j_t(x,y)\}_{t>0}$ is given by
\begin{eqnarray}\label{1.21} K^j_t(x,y):=t\,(\partial_j K)(x-y,A(x)-A(y)+t),\qquad x,y\in{\mathbb{R}}^n,\,t>0,\,j=1,\dots,n+1. \end{eqnarray}
The approach we present utilizes ideas developed in \cite{CJ} and \cite{Ho}. Based on \eqref{prop-K}-\eqref{prop-K2} it is not difficult to check that the family $\{K^j_t(x,y)\}_{t>0}$ is standard, i.e., there hold
\begin{eqnarray}\label{1.22}
|K^j_t(x,y)|&\leq & C\,t(t+|x-y|)^{-(n+1)} \\[4pt]
|\nabla_{x}K^j_t(x,y)|+|\nabla_{y}K^j_t(x,y)|
&\leq & C\,t(t+|x-y|)^{-(n+2)}. \label{1.22b} \end{eqnarray}
As such, a particular version of Theorem~\ref{SChg} gives that the operators in \eqref{1.20} are bounded as soon as we show that for each $j=1,\dots,n+1$,
\begin{eqnarray}\label{cfg-aa}
|T^j(1)(x,t)|^2\,\tfrac{dt}{t}dx\,\, \mbox{is a Carleson measure in}\,\,{\mathbb{R}}^{n+1}_+. \end{eqnarray}
To this end, fix $j\in\{1,\dots,n+1\}$ and select a real-valued, nonnegative function
$\phi\in C^\infty_c({\mathbb{R}}^n)$, vanishing for $|x|\geq 1$, with $\int_{{\mathbb{R}}^n}\phi(x)\,dx=1$ and, as usual, for every $t>0$, set $\phi_t(x):=t^{-n}\phi(x/t)$ for $x\in{\mathbb{R}}^n$. We write $T^j(1)=(T^j(1)-\widetilde{T}^j(1))+\widetilde{T}^j(1)$ where
\begin{eqnarray}\label{1.24} \widetilde{T}^jf(x,t):=\int_{{\mathbb{R}}^n}\widetilde{K}^j_t(x,y)f(y)\,dy, \qquad x\in{\mathbb{R}}^n,\,t>0, \end{eqnarray}
with
\begin{eqnarray}\label{1.25} \widetilde{K}^j_t(x,y):=t\,(\partial_j K) ( x-y,\langle\nabla_{x}(\phi_t\ast A)(x),x-y\rangle+t),\qquad x,y\in{\mathbb{R}}^n,\,t>0. \end{eqnarray}
To prove that $|(T^j-\widetilde{T}^j)(1)(x,t)|^2\,dx\frac{dt}{t}$ is a Carleson measure, fix $x_0$ in ${\mathbb{R}}^n$, $r>0$, and split
\begin{eqnarray}\label{1.25b} (T^j-\widetilde{T}^j)(1)=(T^j-\widetilde{T}^j)({\mathbf{1}}_{B(x_0,100r)})+ (T^j-\widetilde{T}^j)({\mathbf{1}}_{{\mathbb{R}}^n\setminus B(x_0,100r)}), \end{eqnarray}
where, for any set $S$, ${\mathbf{1}}_S$ stands for the characteristic function of $S$. Using \eqref{1.22} and the fact that a similar estimate holds for $\widetilde{K}^j_t(x,y)$, we may write
\begin{eqnarray}\label{1.26} &&\int_0^r\int_{B(x_0,r)}
|(T^j-\widetilde{T}^j)({\mathbf{1}}_{{\mathbb{R}}^n\setminus B(x_0,100r)})(x,t)|^2\,dx\,\tfrac{dt}{t} \nonumber\\[4pt] &&\hskip 0.50in \leq C\,\int_0^r\int_{B(x_0,r)}\Bigl( \int_{{\mathbb{R}}^n\setminus B(x_0,100r)}
\frac{t}{|x-y|^{n+1}}\,dy\Bigr)^2\,dx\,\tfrac{dt}{t} \nonumber\\[4pt] &&\hskip 0.50in =C\,\int_0^r\int_{B(x_0,r)}\Bigl( \int_{{\mathbb{R}}^n\setminus B(0,99r)}
\frac{t}{|z|^{n+1}}\,dz\Bigr)^2\,dx\,\tfrac{dt}{t} =C\,r^n, \end{eqnarray}
a bound which is of the right order. We are therefore left with proving an estimate similar to \eqref{1.26} with ${\mathbb{R}}^n\setminus B(x_0,100r)$ replaced by $B(x_0,100r)$. More precisely, the goal is to show that
\begin{eqnarray}\label{1.26-BB} \int_0^r\int_{B(x_0,r)}
|(T^j-\widetilde{T}^j)({\mathbf{1}}_{B(x_0,100r)})(x,t)|^2\,dx\,\tfrac{dt}{t}\leq C\,r^n. \end{eqnarray}
For this task we make use of the Lemma~\ref{lgih}. This requires an adjustment which we now explain. Concretely, fix a function $\Phi\in C^\infty({\mathbb{R}})$ such that $0\leq\Phi\leq 1$, $\mbox{supp}\,\Phi\subseteq [-150r,150r]$,
$\Phi\equiv 1$ on $[-125r,125r]$, and $\|\Phi'\|_{L^\infty({\mathbb{R}})}\leq c/r$.
If we now set $\widetilde{A}(x):=\Phi(|x-x_0|)(A(x)-A(x_0))$ for every $x\in{\mathbb{R}}^n$, it follows that
\begin{eqnarray}\label{g5ya} \begin{array}{c} \widetilde{A}(x)-\widetilde{A}(y)=A(x)-A(y)\,\,\mbox{ and }\,\, \nabla(\phi_t\ast\widetilde{A})(x)=\nabla(\phi_t\ast A)(x) \\[4pt] \mbox{whenever }\,\,x\in B(x_0,r),\,\,y\in B(x_0,100r),\,\,t\in(0,r). \end{array} \end{eqnarray}
Hence, the expression $(T^j-\widetilde{T}^j)({\mathbf{1}}_{B(x_0,100r)})(x,t)$
does not change for $x\in B(x_0,r)$ and $t\in(0,r)$ if we replace $A$ by $\widetilde{A}$. In addition, since $\|\nabla\widetilde{A}\|_{L^\infty({\mathbb{R}}^n)}
\leq C\|\nabla A\|_{L^\infty({\mathbb{R}}^n)}$, taking into account the support of $\widetilde{A}$ we have
\begin{eqnarray}\label{bhius}
\|\nabla\widetilde{A}\|_{L^2({\mathbb{R}}^n)}
\leq Cr^{n/2}\|\nabla A\|_{L^\infty({\mathbb{R}}^n)} \end{eqnarray}
for some $C>0$ independent of $r$. Hence, there is no loss of generality in assuming that the original Lipschitz function $A$ has the additional property that
\begin{eqnarray}\label{bhius-CC}
\|\nabla A\|_{L^2({\mathbb{R}}^n)}\leq Cr^{n/2}\|\nabla A\|_{L^\infty({\mathbb{R}}^n)}. \end{eqnarray}
Under this assumption we now return to the task of proving \eqref{1.26-BB}. To get started, recall \eqref{1.28}. We claim that there exists $C=C(A,\phi)>0$ such that
\begin{eqnarray}\label{1.27}
|K^j_t(x,y)-\widetilde{K}^j_t(x,y)|\leq C\,t(t+|x-y|)^{-(n+2)}|E_A(t,x,y)|,\quad \forall\,x,y\in{\mathbb{R}}^n,\,\forall\,t>0. \end{eqnarray}
Indeed, by making use of the Mean-Value Theorem and \eqref{prop-K2}, the claim will follow if we show that there exists $C=C(A,\phi)>0$ with the property that
\begin{eqnarray}\label{skgn}
\sup_{\xi\in I}\,[|\xi|+|x-y|]^{-(n+2)}\leq C[t+|x-y|]^{-(n+2)}, \end{eqnarray}
where $I$ denotes the interval with endpoints $t+A(x)-A(y)$ and $t+{\langle}\nabla_{x}(\phi_t\ast A)(x),(x-y){\rangle}$. From the properties of $A$ and $\phi$ we see that
$\xi=t+{{O}}(|x-y|)$, with constants depending only on $A$ and $\phi$. In particular, there exists some small $\varepsilon=\varepsilon(A,\phi)>0$
such that if $|x-y|<\varepsilon t$ then $t\leq C|\xi|\leq C(|\xi|+|x-y|)$.
On the other hand, if $|x-y|\geq\varepsilon t$ then clearly $t\leq C(|\xi|+|x-y|)$. Thus, there exists $C>0$ such that $t\leq C(|\xi|+|x-y|)$ for $\xi\in I$, which implies that for some $C=C(A,\phi)>0$ there holds $t+|x-y|\leq C(|\xi|+|x-y|)$ whenever $\xi\in I$, proving \eqref{skgn}.
Next, making use of \eqref{1.27}, we may write
\begin{eqnarray}\label{1.26-DD} && \int_0^r\int_{B(x_0,r)}
|(T^j-\widetilde{T}^j)({\mathbf{1}}_{B(x_0,100r)})(x,t)|^2\,dx\,\frac{dt}{t} \nonumber\\[4pt] && \hskip 0.50in \leq C\int_0^\infty\int_{{\mathbb{R}}^n}\Bigl(
\int_{{\mathbb{R}}^n}\frac{t}{(t+|x-y|)^{n+2}}\,
|E_A(t,x,y)|\,dy\Bigr)^2\,dx\,\frac{dt}{t} \nonumber\\[4pt] && \hskip 0.50in \leq C\int_0^\infty\int_{{\mathbb{R}}^n}\Bigl(t^{-n-1}
\int_{B(x,t)}|E_A(t,x,y)|\,dy\Bigr)^2\,dx\,\frac{dt}{t} \nonumber\\[4pt] && \hskip 0.60in + C\int_0^\infty\int_{{\mathbb{R}}^n}\Bigl(\sum\limits_{\ell=0}^\infty
\int\limits_{B(x,2^{\ell+1}t)\setminus B(x,2^\ell t)}\frac{t}{|x-y|^{n+2}}\,
|E_A(t,x,y)|\,dy\Bigr)^2\,dx\,\frac{dt}{t} \nonumber\\[4pt] && \hskip 0.50in
\leq C\int_0^\infty\int_{{\mathbb{R}}^n}\Bigl(\sum\limits_{\ell =0}^\infty 2^{-\ell }(2^\ell t)^{-n-1}\int\limits_{B(x,2^{\ell +1}t)}|E_A(t,x,y)|\,dy\Bigr)^2 \,dx\,\frac{dt}{t}. \end{eqnarray}
Now, we apply Minkowski's inequality in order to obtain
\begin{eqnarray}\label{btua9}
&& \int_0^\infty\int_{{\mathbb{R}}^n}\Bigl(\sum\limits_{\ell =0}^\infty 2^{-\ell }(2^\ell t)^{-n-1}\int\limits_{B(x,2^{\ell +1}t)}|E_A(t,x,y)|\,dy\Bigr)^2 \,dx\,\frac{dt}{t} \nonumber\\[4pt] && \hskip 0.40in \leq\left(\sum\limits_{\ell =0}^\infty\Bigl[\int_0^\infty\int_{{\mathbb{R}}^n}
2^{-2\ell }(2^\ell t)^{-2n-2}\Bigl(\int\limits_{B(x,2^{\ell +1}t)}|E_A(t,x,y)|\,dy\Bigr)^2 \,dx\,\frac{dt}{t}\Bigr]^{1/2}\right)^2.\qquad \end{eqnarray}
By the Cauchy-Schwarz inequality, the last expression above is dominated by
\begin{eqnarray}\label{btu56} \left(\sum\limits_{\ell =0}^\infty\Bigl[2^{\ell (-n-4)} \int_0^\infty\int_{{\mathbb{R}}^n}t^{-n-2}
\int\limits_{B(x,2^{\ell +1}t)}|E_A(t,x,y)|^2\,dy\,dx\,\frac{dt}{t}\Bigr]^{1/2}\right)^2. \end{eqnarray}
Invoking now Lemma~\ref{lgih} with $\lambda:=2^{\ell+1}\geq 1$ for
$\ell\in{\mathbb{N}}\cup\{0\}$, each inner triple integral in \eqref{btu56} is dominated by $C2^{\ell(n+3)}\|\nabla A\|^2_{L^2({\mathbb{R}}^n)}$ with $C>0$ finite constant independent of $\ell$. Thus, the entire expression in \eqref{btu56} is
\begin{eqnarray}\label{btu57}
\leq C\left(\sum\limits_{\ell =0}^\infty\bigl[2^{\ell(-n-4)}\cdot 2^{\ell(n+3)}\|\nabla A\|^2_{L^2({\mathbb{R}}^n)}\bigr]^{1/2}\right)^2
=C\|\nabla A\|^2_{L^2({\mathbb{R}}^n)}\leq Cr^n, \end{eqnarray}
where for the last inequality in \eqref{btu57} we have used \eqref{bhius-CC}. This finishes the proof of \eqref{1.26-BB}. In turn, when \eqref{1.26-BB} is combined with \eqref{1.26}, we obtain
\begin{eqnarray}\label{cfg-ab}
|(T^j-\widetilde{T}^j)(1)(x,t)|^2\,\tfrac{dt}{t}dx\,\, \mbox{is a Carleson measure in}\,\,{\mathbb{R}}^{n+1}_+. \end{eqnarray}
At this stage, there remains to observe that, thanks to Lemma~\ref{Lemma1.3}, we have
\begin{eqnarray}\label{1.30} \widetilde{T}^j(1)(x,t)=\int\limits_{{\mathbb{R}}^n}t\,(\partial_j K) \bigl(x-y,{\langle}\nabla_{x}(\phi_t\ast A)(x),(x-y){\rangle}+t\bigr)\,dy\equiv 0 \quad\forall\,x\in{\mathbb{R}}^n,\,\forall\,t>0. \end{eqnarray}
The proof of Theorem~\ref{Theorem1.1} is now completed. \end{proof}
\subsection{Square function estimates on (BP)$^k$LG sets} \label{SSect:BPmLG}
We continue to work in the context of $\mathbb{R}^{n+1}$ introduced at the beginning of Section~\ref{Sect:SFE}, and abbreviate the $n$-dimensional Hausdorff outer measure from Definition~\ref{SKJ38} as ${\mathcal{H}}^n:={\mathcal{H}}^n_{{\mathbb{R}}^{n+1}}$. We prove that square function estimates are stable under the so-called big pieces functor. Square function estimates on uniformly rectifiable sets then follow as a simple corollary. Let us begin by reviewing the concept of uniform rectifiability. In particular, following G.~David and S.~Semmes \cite{DaSe91}, we make the following definition.
\begin{definition}\label{Def-unif.rect} A closed set ${\Sigma}\subseteq{\mathbb{R}}^{n+1}$ is called {\tt uniformly rectifiable} provided it is $n$-dimensional Ahlfors-David regular and the following property holds. There exist $\varepsilon$, $M\in(0,\infty)$ (called the {\rm UR} constants of ${\Sigma}$) such that for each $x\in {\Sigma}$ and $r>0$, there is a Lipschitz map $\varphi:B^{n}_r\rightarrow {\mathbb{R}}^{n+1}$ (where $B^{n}_r$ is a ball of radius $r$ in ${\mathbb{R}}^{n}$) with Lipschitz constant at most equal to $M$, such that
\begin{eqnarray}\label{3.1.9aS} {\mathcal{H}}^{n}\bigl({\Sigma}\cap B(x,r)\cap\varphi(B^{n}_r)\bigr)\geq\varepsilon r^{n}. \end{eqnarray}
If ${\Sigma}$ is compact, then this is only required for $r\in (0,{\rm diam}\,({\Sigma})]$. \end{definition}
There are a variety of equivalent characterizations of uniform rectifiability (cf., e.g., \cite[Theorem~I.1.5.7, p.\,22]{DaSe93}); the version above is often specified by saying that ${\Sigma}$ has {\tt Big Pieces of Lipschitz Images} (or, simply {\tt BPLI}). Another version, in which Lipschitz maps are replaced with Bi-Lipschitz maps, is specified by saying that ${\Sigma}$ has {\tt Big Pieces of Bi-Lipschitz Images} (or, simply {\tt BPBI}). The equivalence between BPLI and BPBI can be found in~\cite[p.\,22]{DaSe93}. We also require the following notion of sets having big pieces of Lipschitz graphs.
\begin{definition}\label{Def-BPnLG} A set ${\Sigma}\subseteq \mathbb{R}^{n+1}$ is said to have {\tt Big Pieces of Lipschitz Graphs} (or, simply {\tt BPLG}) provided it is $n$-dimensional Ahlfors-David regular and the following property holds. There exist $\varepsilon$, $M\in(0,\infty)$ (called the {\rm BPLG} constants of ${\Sigma}$) such that for each $x\in {\Sigma}$ and $r>0$, there is an $n$-dimensional Lipschitz graph $\Gamma\subseteq\mathbb{R}^{n+1}$ with Lipschitz constant at most equal to $M$, such that
\begin{eqnarray}\label{3.1.9a} {\mathcal{H}}^{n}\bigl({\Sigma}\cap B(x,r)\cap\Gamma)\geq\varepsilon r^{n}. \end{eqnarray}
If ${\Sigma}$ is compact, then this is only required for $r\in (0,{\rm diam}\,({\Sigma})]$.
We also write {\tt (BP)$^1$LG} to mean {\tt BPLG}. For each $k\geq1$, a set ${\Sigma}\subseteq \mathbb{R}^{n+1}$ is said to have {\tt Big Pieces of (BP)$^{k}$LG} (or, simply {\tt (BP)$^{k+1}$LG}) provided it is $n$-dimensional Ahlfors-David regular and the following property holds. There exist $\delta$, $\varepsilon$, $M\in(0,\infty)$ (called the {\rm (BP)$^{k+1}$LG} constants of ${\Sigma}$) such that for each $x\in {\Sigma}$ and $r>0$, there is a set $\Omega\subseteq\mathbb{R}^{n+1}$ that has {\rm (BP)$^{k}$LG} with {\rm ADR} constant at most equal to $M$, and {\rm (BP)$^{k}$LG} constants $\varepsilon$, $M$, such that
\begin{eqnarray}\label{3.1.9Z} {\mathcal{H}}^{n}\bigl({\Sigma}\cap B(x,r)\cap \Omega)\geq\delta r^{n}. \end{eqnarray}
If ${\Sigma}$ is compact, then this is only required for $r\in (0,{\rm diam}\,({\Sigma})]$. \end{definition}
We now combine the inductive scheme from Section~\ref{Sect:4} with the square function estimates for Lipschitz graphs from Subsection~\ref{SSect:LG} to prove that square function estimates are stable under the so-called big pieces functor.
\begin{theorem}\label{thm:BPmLGimSFE} Let $k\in\mathbb{N}$ and suppose that ${\Sigma}\subseteq \mathbb{R}^{n+1}$ has (BP)$^k$LG. Let $K$ be a real-valued kernel satisfying \eqref{prop-K}, and let ${\mathcal{T}}$ denote the integral operator associated with $\Sigma$ as in \eqref{sbnvjb}.
Then there exists a constant $C\in(0,\infty)$ depending only on $n$, the (BP)$^k$LG constants of ${\Sigma}$, and $\|\partial^\alpha K\|_{L^\infty(S^{n})}$ for $|\alpha|\leq 2$, such that
\begin{eqnarray}\label{vlnGG-2}
\int_{\mathbb{R}^{n+1}\setminus {\Sigma}}|\nabla{\mathcal{T}}f(x)|^2\,{\rm dist}(x,{\Sigma})\,dx
\leq C\int_{\Sigma}|f|^2\,d\sigma,\qquad\forall\,f\in L^2({\Sigma},\sigma), \end{eqnarray}
where $\sigma:={\mathscr{H}}^n\lfloor{{\Sigma}}$ is the measure induced by the $n$-dimensional Hausdorff measure on ${\Sigma}$. \end{theorem}
\begin{proof} The proof proceeds by induction on ${\mathbb{N}}$. For the case $k=1$, suppose that ${\Sigma}\subseteq \mathbb{R}^{n+1}$ has BPLG with BPLG constants $\varepsilon_0$, $C_0\in(0,\infty)$. For each $x\in {\Sigma}$ and $r>0$, there is an $n$-dimensional Lipschitz graph $\Gamma\subseteq\mathbb{R}^{n+1}$ with Lipschitz constant at most equal to $C_0$, such that
\begin{eqnarray}\label{3.1.9aB} {\mathcal{H}}^{n}\bigl({\Sigma}\cap B(x,r)\cap \Gamma)\geq \varepsilon_0 r^{n}. \end{eqnarray}
It follows from Theorem~\ref{Theorem1.1} that ${\Sigma}$ has BPSFE with BPSFE character (cf. Definition~\ref{sjvs}) depending only on the BPLG constants of ${\Sigma}$, and
$\|\partial^\alpha K\|_{L^\infty(S^{n})}$ for $|\alpha|\leq 2$. It then follows from Theorem~\ref{Thm:BPSFtoSF} that~\eqref{vlnGG-2} holds for some $C\in(0,\infty)$ depending only on $n$, the BPLG constants of ${\Sigma}$, and $\|\partial^\alpha K\|_{L^\infty(S^{n})}$
for $|\alpha|\leq 2$.
Now let $j\in\mathbb{N}$ and assume that the statement of the theorem holds in the case $k=j$. Suppose that ${\Sigma}\subseteq \mathbb{R}^{n+1}$ has (BP)$^{j+1}$LG with (BP)$^{j+1}$LG constants $\varepsilon_1$, $\varepsilon_2$, $C_1\in(0,\infty)$. For each $x\in {\Sigma}$ and $r>0$, there is a set $\Omega\subseteq\mathbb{R}^{n+1}$ that has {\rm (BP)$^{j}$LG} with {\rm ADR} constant at most equal to $C_1$, and {\rm (BP)$^{j}$LG} constants $\varepsilon_1$, $C_1$, such that
\begin{eqnarray}\label{3.1.9aQ} {\mathcal{H}}^{n}\bigl({\Sigma}\cap B(x,r)\cap \Omega)\geq \varepsilon_2 r^{n}. \end{eqnarray}
It follows by the inductive assumption that ${\Sigma}$ has BPSFE with BPSFE character depending only on the constants specified in the theorem in the case $k=j$. Applying again Theorem~\ref{Thm:BPSFtoSF} we obtain that~\eqref{vlnGG-2} holds for some $C\in(0,\infty)$
depending only on $n$, the (BP)$^{j+1}$LG constants of ${\Sigma}$, and $\|\partial^\alpha K\|_{L^\infty(S^{n})}$ for $|\alpha|\leq 2$. This completes the proof. \end{proof}
The recent result by J. Azzam and R. Schul (cf. \cite[Corollary~1.7]{AS}) that uniformly rectifiable sets have (BP)$^2$LG allows us to obtain the following as an immediate corollary of Theorem~\ref{thm:BPmLGimSFE}.
\begin{corollary}\label{cor:URimSFE}
Suppose that ${\Sigma}\subseteq \mathbb{R}^{n+1}$ is a uniformly rectifiable set. Let $K$ be a real-valued kernel satisfying \eqref{prop-K}, and let ${\mathcal{T}}$ denote the integral operator associated with $\Sigma$ as in \eqref{sbnvjb}. Then there exists a constant $C\in(0,\infty)$, depending only on $n$, the UR constants of ${\Sigma}$, and $\|\partial^\alpha K\|_{L^\infty(S^{n})}$ for $|\alpha|\leq 2$, such that
\begin{eqnarray}\label{corvlnGG}
\int_{\mathbb{R}^{n+1}\setminus {\Sigma}}|\nabla{\mathcal{T}}f(x)|^2\,{\rm dist}(x,{\Sigma})\,d x
\leq C\int_{\Sigma}|f|^2\,d\sigma,\qquad\forall\,f\in L^2({\Sigma},\sigma), \end{eqnarray} where $\sigma:={\mathscr{H}}^n\lfloor{{\Sigma}}$ is the measure induced by the $n$-dimensional Hausdorff measure on ${\Sigma}$. \end{corollary}
\begin{proof} The set ${\Sigma}$ has (BP)$^2$LG by J. Azzam and R. Schul's characterization of uniformly rectifiable sets in~\cite[Corollary~1.7]{AS}, so the result follows at once from Theorem~\ref{thm:BPmLGimSFE}. \end{proof}
\subsection{Square function estimates for integral operators with variable kernels} \label{SSect:PDO}
The square function estimates from Theorem~\ref{thm:BPmLGimSFE} and Corollary~\ref{cor:URimSFE} have been formulated for {\it convolution type} integral operators and our goal in this subsection is to prove some versions of these results which apply to integral operators with variable coefficient kernels. A first result in this regard reads as follows.
\begin{theorem}\label{thm:BVAR} Let $k\in\mathbb{N}$ and suppose that ${\Sigma}\subseteq{\mathbb{R}}^{n+1}$ is compact and has (BP)$^k$LG. Then there exists a positive integer $M=M(n)$ with the following significance. Assume that ${\mathcal{U}}$ is a bounded, open neighborhood of $\Sigma$ in ${\mathbb{R}}^{n+1}$ and consider a function
\begin{eqnarray}\label{B-pv5Avb} {\mathcal{U}}\times\bigl({\mathbb{R}}^{n+1}\setminus\{0\}\bigr)\ni (x,z) \mapsto b(x,z)\in{\mathbb{R}} \end{eqnarray}
which is odd and (positively) homogeneous of degree $-n$ in the variable $z\in{\mathbb{R}}^{n+1}\setminus\{0\}$, and which has the property that
\begin{eqnarray}\label{B-pv5.LFD} \mbox{$\partial_x^\beta\partial_z^\alpha b(x,z)$ is continuous and bounded on
${\mathcal{U}}\times S^{n}$ for $|\alpha|\leq M$ and $|\beta|\leq 1$}. \end{eqnarray}
Finally, define the variable kernel integral operator
\begin{eqnarray}\label{B-pv5} {\mathcal{B}}f(x):=\int_{\Sigma}b(x,x-y)f(y)\,d\sigma(y),\qquad x\in{\mathcal{U}}\setminus\Sigma, \end{eqnarray}
where $\sigma:={\mathscr{H}}^n\lfloor{\Sigma}$ is the measure induced by the $n$-dimensional Hausdorff measure on $\Sigma$.
Then there exists a constant $C\in(0,\infty)$ depending only on $n$, the (BP)$^k$LG constants of ${\Sigma}$, the diameter of ${\mathcal{U}}$, and
$\|\partial_x^\beta\partial_z^\alpha b\|_{L^\infty({\mathcal{U}}\times S^{n})}$
for $|\alpha|\leq 2$, $|\beta|\leq 1$, such that
\begin{eqnarray}\label{vlnGG-2ASn}
\int_{{\mathcal{U}}\setminus\Sigma}|\nabla{\mathcal{B}}f(x)|^2\,{\rm dist}(x,\Sigma)\,dx
\leq C\int_{\Sigma}|f|^2\,d\sigma,\qquad\forall\,f\in L^2(\Sigma,\sigma). \end{eqnarray}
In particular, \eqref{vlnGG-2ASn} holds whenever $\Sigma$ is uniformly rectifiable (while retaining the other background assumptions). \end{theorem}
In preparation for presenting the proof of Theorem~\ref{thm:BVAR}, we state two lemmas, of geometric character, from \cite{MMMM-B}.
\begin{lemma}\label{bvv55-DD} Let $({\mathscr{X}},\rho)$ be a geometrically doubling quasi-metric space and let $\Sigma\subseteq{\mathscr{X}}$ be a set with the property that
$\bigl(\Sigma,\rho\bigl|_\Sigma,{\mathcal{H}}^d_{{\mathscr{X}},\rho}\lfloor\Sigma\bigr)$ becomes a $d$-dimensional {\rm ADR} space, for some $d>0$. Assume that $\mu$ is a Borel measure on ${\mathscr{X}}$ satisfying
\begin{eqnarray}\label{bca84r.22.aT} \sup_{x\in{\mathscr{X}},\,r>0}\frac{\mu\bigl(B_{\rho_{\#}}(x,r)\bigr)}{r^m}<+\infty, \end{eqnarray}
for some $m\geq 0$. Also, fix a constant $c>0$ and select two real numbers $N,\alpha$ such that $\alpha<m-d$ and $N<m-\max\,\{\alpha,0\}$.
Then there exists a constant $C\in(0,\infty)$ depending on the supremum in \eqref{bca84r.22.aT}, the geometric doubling constant of $({\mathscr{X}},\rho)$, the {\rm ADR} constant of $\Sigma$, as well as $N$, $\alpha$, and $c$, such that
\begin{eqnarray}\label{bca84r.22} \begin{array}{c} \displaystyle \int\limits_{B_{\rho_{\#}}(x,r)\setminus\overline{\Sigma}} \frac{{\rm dist}_{\rho_{\#}}(y,\Sigma)^{-\alpha}}{\rho_{\#}(x,y)^N}\,d\mu(y) \leq C\,r^{m-\alpha-N}, \\[30pt] \forall\,r>0,\quad\forall\,x\in{\mathscr{X}} \,\,\mbox{ with }\,\,{\rm dist}_{\rho_{\#}}(x,\Sigma)<c\,r. \end{array} \end{eqnarray} \end{lemma}
\begin{lemma}\label{Gkwvr.reV} Let $({\mathscr{X}},\rho)$ be a quasi-metric space. Suppose $E\subseteq{\mathscr{X}}$ is nonempty and $\sigma$ is a measure on $E$ such that
$(E,\rho\bigl|_E,\sigma)$ becomes a $d$-dimensional {\rm ADR} space, for some $d>0$. Fix a real number $0\leq N<d$. Then there exists $C\in(0,\infty)$ depending only on $N$, $\rho$, and the {\rm ADR} constant of $E$ such that
\begin{eqnarray}\label{mMji.reV} \int\limits_{E\cap B_{\rho_{\#}}(x,r)} \frac{1}{\rho_{\#}(x,y)^N}\,d\sigma(y)\leq C\,r^{d-N}, \qquad\forall\,x\in{\mathscr{X}},\quad\forall\,r>{\rm dist}_{\rho_{\#}}(x,E). \end{eqnarray} \end{lemma}
We are now ready to discuss the
\vskip 0.08in \begin{proof}[Proof of Theorem~\ref{thm:BVAR}] Set
\begin{eqnarray}\label{D-HarmK} H_0:=1,\quad H_1:=n+1,\quad\mbox{ and }\quad H_{\ell}:=\Bigl(\!\! \begin{array}{c} n+\ell \\[-2pt] \ell \end{array} \!\!\Bigr) - \Bigl(\!\! \begin{array}{c} n+\ell-2 \\[-2pt] \ell-2 \end{array} \!\!\Bigr) \quad\mbox{ if }\,\,\ell\geq 2, \end{eqnarray}
and, for each $\ell\in{\mathbb{N}}_0$, let $\bigl\{\Psi_{i\ell}\bigr\}_{1\leq i\leq H_\ell}$ be an orthonormal basis for the space of spherical harmonics of degree $\ell$ on the $n$-dimensional sphere $S^n$. In particular,
\begin{eqnarray}\label{D-Har-Nr} H_{\ell}\leq(\ell+1)\cdot(\ell+2)\cdots(n+\ell-1)\cdot (n+\ell)\leq C_n\,\ell^n\quad\mbox{ for }\,\,\ell\geq 2 \end{eqnarray}
and, if $\Delta_{S^{n}}$ denotes the Laplace-Beltrami operator on $S^n$, then for each $\ell\in{\mathbb{N}}_0$ and $1\leq i\leq H_\ell$,
\begin{eqnarray}\label{eihen-XS} \Delta_{S^{n}}\Psi_{i\ell}=-\ell(n+\ell-1)\Psi_{i\ell}\,\,\mbox{ on }\,\,S^n,
\,\,\mbox{ and }\,\,\Psi_{i\ell}\Bigl(\frac{x}{|x|}\Bigr)=\frac{P_{i\ell}(x)}{|x|^{\ell}} \end{eqnarray}
for some homogeneous harmonic polynomial $P_{i\ell}$ of degree $\ell$ in ${\mathbb{R}}^{n+1}$. Also,
\begin{eqnarray}\label{eihen-amm-ONB} \bigl\{\Psi_{i\ell}\bigr\}_{\ell\in{\mathbb{N}}_0,\,1\leq i\leq H_\ell} \,\,\mbox{ is an orthonormal basis for }\,\,L^2(S^n), \end{eqnarray}
hence,
\begin{eqnarray}\label{eihen-amm}
\|\Psi_{i\ell}\|_{L^2(S^{n})}=1\,\,\mbox{ for each $\ell\in{\mathbb{N}}_0$ and $1\leq i\leq H_\ell$}. \end{eqnarray}
More details on these matters may be found in, e.g., \cite[pp.\,137--152]{STEIN-WEISS} and \cite[pp.\,68--75]{STEIN}.
Assume next that an even integer $d>(n/2)+2$ has been fixed. Sobolev's embedding theorem then gives that for each $\ell\in{\mathbb{N}}_0$ and $1\leq i\leq H_\ell$ (with $I$ standing for the identity on $S^n$)
\begin{eqnarray}\label{kl-dYU-11}
\|\Psi_{i\ell}\|_{C^2(S^{n})}
\leq C_n\bigl\|(I-\Delta_{S^{n}})^{d/2}\Psi_{i\ell}\bigr\|_{L^2(S^{n})}\leq C_n\ell^{d}, \end{eqnarray}
where the last inequality is a consequence of \eqref{eihen-XS}-\eqref{eihen-amm}.
Fix $\ell\in{\mathbb{N}}_0$ and $1\leq i\leq H_\ell$ arbitrary. If we now define
\begin{eqnarray}\label{coef-sh} a_{i\ell}(x):=\int_{S^{n}}b(x,\omega)\Psi_{i\ell}(\omega)\,d\omega,\,\, \mbox{ for each }\,\,x\in{\mathcal{U}}, \end{eqnarray}
it follows from the last formula in \eqref{eihen-XS} and the assumptions on $b(x,z)$ that
\begin{eqnarray}\label{coef-sh.aC} \mbox{$a_{i\ell}$ is identically zero whenever $\ell$ is even}. \end{eqnarray}
Also, for each number $N\in{\mathbb{N}}$ with $2N\leq M$ and each multiindex $\beta$ of length $\leq 1$ we have
\begin{eqnarray}\label{coef-sh.AAA} \sup_{x\in{\mathcal{U}}}
\bigl|[-\ell(n+\ell-1)]^N(\partial^\beta a_{i\ell})(x)\bigr| &=& \sup_{x\in{\mathcal{U}}}
\Bigl|\int_{S^{n}}(\partial_x^\beta b)(x,\omega)
\bigl(\Delta_{S^{n}}^N\Psi_{i\ell}\bigr)(\omega)\,d\omega\Bigr| \nonumber\\[4pt] &=& \sup_{x\in{\mathcal{U}}}
\Bigl|\int_{S^{n}}\bigl(\partial_x^\beta\Delta_{S^{n}}^N b\bigr)(x,\omega)
\Psi_{i\ell}(\omega)\,d\omega\Bigr| \nonumber\\[4pt] &\leq & \sup_{x\in{\mathcal{U}}}
\bigl\|\bigl(\partial_x^\beta\Delta_{S^{n}}^N b\bigr)(x,\cdot)
\bigr\|_{L^2(S^{n})} \nonumber\\[4pt]
&\leq & C_n\sup_{\stackrel{(x,z)\in{\mathcal{U}}\times S^n}{|\alpha|\leq M}}
\bigl|\bigl(\partial_x^\beta\partial_z^{\alpha}b\bigr)(x,z)\bigr|=:C_b, \end{eqnarray}
where $C_b$ is a finite constant. Hence, for each number $N\in{\mathbb{N}}$ with $2N\leq M$ there exists a constant $C_{n,N}$ such that
\begin{eqnarray}\label{coef-HJ}
\sup_{x\in{\mathcal{U}},\,|\beta|\leq 1}\bigl|(\partial^\beta a_{i\ell})(x)\bigr| \leq C_{n,N} C_b\,\ell^{-2N},\qquad\ell\in{\mathbb{N}}_0,\quad 1\leq i\leq H_\ell. \end{eqnarray}
For each fixed $x\in{\mathcal{U}}$, expand the function $b(x,\cdot)\in L^2(S^n)$ with respect to the orthonormal basis $\bigl\{\Psi_{i\ell}\bigr\}_{\ell\in{\mathbb{N}}_0,\,1\leq i\leq H_\ell}$ in order
to obtain that (in the sense of $L^2(S^n)$ in the variable $z/|z|\in S^n$)
\begin{eqnarray}\label{tag{1.12}}
b(x,z) &=& b\Bigl(x,\frac{z}{|z|}\Bigr)|z|^{-n} =\sum_{\ell\in{\mathbb{N}}}\sum_{i=1}^{H_{\ell}}
a_{i\ell}(x)\Psi_{i\ell}\Bigl(\frac{z}{|z|}\Bigr)|z|^{-n} \nonumber\\[4pt] &=& \sum_{\ell\in 2{\mathbb{N}}+1}\sum_{i=1}^{H_{\ell}}
a_{i\ell}(x)\Psi_{i\ell}\Bigl(\frac{z}{|z|}\Bigr)|z|^{-n}, \end{eqnarray}
where the last equality is a consequence of \eqref{coef-sh.aC}. For each $\ell\in 2{\mathbb{N}}+1$ let us now set
\begin{eqnarray}\label{1.15GH}
k_{i\ell}(z):=\Psi_{i\ell}\Bigl(\frac{z}{|z|}\Bigr)|z|^{-n}, \quad z\in{\mathbb{R}}^{n+1}\setminus\{0\}, \end{eqnarray}
so that, if $d$ is as in \eqref{kl-dYU-11},
then for each $|\alpha|\leq 2$ we have
\begin{eqnarray}\label{kl-dYU}
\|\partial^\alpha k_{i\ell}\|_{L^\infty(S^{n})}
\leq C_n\|\Psi_{i\ell}\|_{C^2(S^{n})}\leq C_n\ell^{d}. \end{eqnarray}
Also, given any $f\in L^2(\Sigma,\sigma)$, set
\begin{eqnarray}\label{1.15ASD} {\mathcal{B}}_{i\ell}f(x):=\int_{\Sigma}k_{i\ell}(x-y)f(y)\,d\sigma(y),\quad x\in{\mathcal{U}}\setminus\Sigma, \end{eqnarray}
and note that for any compact subset ${\mathcal{O}}$ of ${\mathcal{U}}\setminus\Sigma$
and any multiindex $\alpha$ with $|\alpha|\leq 1$,
\begin{eqnarray}\label{1.15ASD.u}
\sup_{x\in{\mathcal{O}}}\bigl|\bigl(\partial^\alpha
{\mathcal{B}}_{i\ell}f\bigr)(x)\bigr|\leq C(n,{\mathcal{O}},\Sigma)\ell^{d}, \end{eqnarray}
by \eqref{kl-dYU}. On the other hand, if $N>(d+1)/2$ (a condition which we shall assume from now on) then \eqref{kl-dYU-11} and \eqref{coef-HJ} imply that the last series in \eqref{tag{1.12}} converges to $b(x,z)$ uniformly for $x\in{\mathcal{U}}$ and $z$ in compact subsets of ${\mathbb{R}}^{n+1}\setminus\{0\}$. As such, it follows from \eqref{1.15GH} and \eqref{1.15ASD} that
\begin{eqnarray}\label{huBN} {\mathcal{B}}f(x)=\sum_{\ell\in 2{\mathbb{N}}+1}\sum_{i=1}^{H_{\ell}} a_{i\ell}(x){\mathcal{B}}_{i\ell}f(x),\,\,\mbox{ uniformly on compact subsets of }\,\, {\mathcal{U}}\setminus\Sigma. \end{eqnarray}
Using this, \eqref{1.15ASD.u} and \eqref{coef-HJ}, the term-by-term differentiation theorem for series of functions may be invoked in order to obtain that
\begin{eqnarray}\label{huBN.e} && \hskip -0.20in \bigl(\nabla{\mathcal{B}}f\bigr)(x) =\sum_{\ell\in 2{\mathbb{N}}+1}\sum_{i=1}^{H_{\ell}} a_{i\ell}(x)\bigl(\nabla{\mathcal{B}}_{i\ell}f\bigr)(x) +\sum_{\ell\in 2{\mathbb{N}}+1}\sum_{i=1}^{H_{\ell}} (\nabla a_{i\ell})(x){\mathcal{B}}_{i\ell}f(x), \nonumber\\[4pt] && \hskip 0.80in \mbox{uniformly for $x$ in compact subsets of }\,\,{\mathcal{U}}\setminus\Sigma. \end{eqnarray}
Moving on, observe that for each $\ell\in 2{\mathbb{N}}+1$ and $1\leq i\leq H_{\ell}$, Theorem~\ref{thm:BPmLGimSFE} gives
\begin{eqnarray}\label{vln-GFH}
\int_{{\mathcal{U}}\setminus\Sigma}|\nabla{\mathcal{B}}_{i\ell}f(x)|^2\,
{\rm dist}(x,{\Sigma})\,dx\leq C_{i\ell}\int_{\Sigma}|f|^2\,d\sigma, \qquad\forall\,f\in L^2(\Sigma,\sigma), \end{eqnarray}
where, with $C\in(0,\infty)$ depending only on the dimension $n$ and the (BP)$^k$LG constants of $\Sigma$.
\begin{eqnarray}\label{vln-Gam}
C_{i\ell}=C\max_{|\alpha|\leq 2}\|\partial^\alpha k_{i\ell}\|_{L^\infty(S^{n})} \leq C\ell^d, \end{eqnarray}
thanks to \eqref{kl-dYU-11}. Thus, if
\begin{eqnarray}\label{kl-dYU.af} \mbox{$M\in{\mathbb{N}}$ is odd and satisfies $M>d+1$}, \end{eqnarray}
one may choose $N\in{\mathbb{N}}$ such that $d+1<2N<M$. Such a choice ensures that for every $f\in L^2(\Sigma,\sigma)$
\begin{eqnarray}\label{vln-GFH.ae} && \hskip -0.50in \sum_{\ell\in 2{\mathbb{N}}+1}\sum_{i=1}^{H_{\ell}}
\Bigl(\int_{{\mathcal{U}}\setminus\Sigma}|a_{i\ell}(x)|^2
|\nabla{\mathcal{B}}_{i\ell}f(x)|^2\,{\rm dist}(x,\Sigma)\,dx\Bigr)^{1/2} \nonumber\\[4pt] && \hskip 0.60in \leq C_{n,N} C_b\sum_{\ell\in 2{\mathbb{N}}+1}\sum_{i=1}^{H_{\ell}}\ell^{-2N} \Bigl(\int_{{\mathcal{U}}\setminus\Sigma}
|\nabla{\mathcal{B}}_{i\ell}f(x)|^2\,{\rm dist}(x,\Sigma)\,dx\Bigr)^{1/2} \nonumber\\[4pt] && \hskip 0.60in \leq C_{n,N} C_b \Bigl(\sum_{\ell\in 2{\mathbb{N}}+1}\sum_{i=1}^{H_{\ell}}
\ell^{-2N}C_{i\ell}^{1/2}\Bigr)\Bigl(\int_{\Sigma}|f|^2\,d\sigma\Bigr)^{1/2} \nonumber\\[4pt] && \hskip 0.60in \leq C_{n,N} C_b \Bigl(\sum_{\ell\in 2{\mathbb{N}}+1}\sum_{i=1}^{H_{\ell}}
\ell^{d/2-2N}\Bigr)\Bigl(\int_{\Sigma}|f|^2\,d\sigma\Bigr)^{1/2} \nonumber\\[4pt] && \hskip 0.60in
=C\Bigl(\int_{\Sigma}|f|^2\,d\sigma\Bigr)^{1/2}, \end{eqnarray}
by \eqref{coef-HJ}, \eqref{vln-Gam}, and our choice of $N$.
To proceed, let $\ell\in{\mathbb{N}}$ and $1\leq i\leq H_{\ell}$ be arbitrary. Also, fix an arbitrary $f\in L^2(\Sigma,\sigma)$. Then
\begin{eqnarray}\label{vln-GFH.234} && \hskip -0.40in
\Bigl(\int_{{\mathcal{U}}\setminus\Sigma}|\nabla a_{i\ell}|^2
|{\mathcal{B}}_{i\ell}f(x)|^2\,{\rm dist}(x,{\Sigma})\,dx\Bigr)^{1/2} \nonumber\\[4pt] && \hskip 0.60in \leq C_{n,N} C_b\,\ell^{-2N}\Bigl(\int_{{\mathcal{U}}\setminus\Sigma}
\bigl|{\rm dist}(x,\Sigma)^{1/2}{\mathcal{B}}_{i\ell}f(x)\bigr|^2\,dx\Bigr)^{1/2} \nonumber\\[4pt] && \hskip 0.60in =C_{n,N} C_b\,\ell^{-2N}\Bigl(\int_{{\mathcal{U}}\setminus\Sigma}
\bigl|{\mathcal{T}}_{i\ell}f(x)\bigr|^2\,dx\Bigr)^{1/2} \end{eqnarray}
where
\begin{eqnarray}\label{vaj-TBB} {\mathcal{T}}_{i\ell}:L^2(\Sigma,\sigma)\longrightarrow L^2({\mathcal{U}}\setminus\Sigma) \end{eqnarray}
is the integral operator whose integral kernel is given by
\begin{eqnarray}\label{vaj-TBB.2i} K_{i\ell}(x,y):={\rm dist}(x,\Sigma)^{1/2}k_{i\ell}(x-y),\qquad x\in{\mathcal{U}}\setminus\Sigma,\,\,\,y\in\Sigma. \end{eqnarray}
Note that
\begin{eqnarray}\label{vaj-TBB.2} \sup_{x\in{\mathcal{U}}\setminus\Sigma}
\int_{\Sigma}\bigl|K_{i\ell}(x,y)\bigr|\,d\sigma(y)
&\leq & \|\Psi_{i\ell}\|_{L^\infty(S^n)}\sup_{x\in{\mathcal{U}}\setminus\Sigma}
\int_{\Sigma}\frac{{\rm dist}(x,\Sigma)^{1/2}}{|x-y|^n}\,d\sigma(y) \nonumber\\[4pt] &\leq & C\ell^d\sup_{x\in{\mathcal{U}}\setminus\Sigma}
\int_{\Sigma}\frac{1}{|x-y|^{n-1/2}}\,d\sigma(y) \nonumber\\[4pt] &\leq & C\ell^d\,{\rm diam}({\mathcal{U}})^{1/2}, \end{eqnarray}
by \eqref{kl-dYU-11} and Lemma~\ref{Gkwvr.reV}, and that
\begin{eqnarray}\label{vaj-TBB.3}
\sup_{y\in\Sigma}\int_{{\mathcal{U}}\setminus\Sigma}\bigl|K_{i\ell}(x,y)\bigr|\,dx
&\leq & \|\Psi_{i\ell}\|_{L^\infty(S^n)}\sup_{y\in\Sigma}
\int_{{\mathcal{U}}\setminus\Sigma}\frac{{\rm dist}(x,\Sigma)^{1/2}}{|x-y|^n}\,dx \nonumber\\[4pt] &\leq & C\ell^d\,{\rm diam}({\mathcal{U}})^{3/2}, \end{eqnarray}
by \eqref{kl-dYU-11} and Lemma~\ref{bvv55-DD}. From \eqref{vaj-TBB.2}-\eqref{vaj-TBB.3} and Schur's Lemma we then deduce that the operator ${\mathcal{T}}_{i\ell}$ is bounded in the context of \eqref{vaj-TBB}, with norm
\begin{eqnarray}\label{vaj-TBB.4}
\bigl\|{\mathcal{T}}_{i\ell}\bigr\|_{L^2(\Sigma,\sigma)\rightarrow L^2({\mathcal{U}}\setminus\Sigma)}\leq C\ell^d\,{\rm diam}({\mathcal{U}}). \end{eqnarray}
Combining \eqref{vaj-TBB.4} and \eqref{vln-GFH.234} we therefore arrive at the conclusion that, for each $f\in L^2(\Sigma,\sigma)$,
\begin{eqnarray}\label{vln-GFH.234-AG}
\Bigl(\int_{{\mathcal{U}}\setminus\Sigma}|\nabla a_{i\ell}|^2
|{\mathcal{B}}_{i\ell}f(x)|^2\,{\rm dist}(x,{\Sigma})\,dx\Bigr)^{1/2}
\leq C({\mathcal{U}})\ell^d\Bigl(\int_{\Sigma}|f|^2\,d\sigma\Bigr)^{1/2}, \end{eqnarray}
whenever $\ell\in{\mathbb{N}}$ and $1\leq i\leq H_\ell$. As a result, there exists $C\in(0,\infty)$ such that
\begin{eqnarray}\label{vln-GFH.234.u} \sum_{\ell\in 2{\mathbb{N}}+1}\sum_{i=1}^{H_{\ell}}
\Bigl(\int_{{\mathcal{U}}\setminus\Sigma}|\nabla a_{i\ell}|^2
|{\mathcal{B}}_{i\ell}f(x)|^2\,{\rm dist}(x,{\Sigma})\,dx\Bigr)^{1/2}
\leq C\Bigl(\int_{\Sigma}|f|^2\,d\sigma\Bigr)^{1/2}, \end{eqnarray}
for every $f\in L^2(\Sigma,\sigma)$.
Fix now an arbitrary compact subset ${\mathcal{O}}$ of ${\mathcal{U}}\setminus\Sigma$. Then \eqref{huBN.e}, \eqref{vln-GFH.ae} and \eqref{vln-GFH.234.u} allow us to estimate
\begin{eqnarray}\label{huBN.ef} \Bigl(\int_{{\mathcal{O}}}
|\nabla{\mathcal{B}}f(x)|^2\,{\rm dist}(x,{\Sigma})\,dx\Bigr)^{1/2}
\leq C\Bigl(\int_{\Sigma}|f|^2\,d\sigma\Bigr)^{1/2}, \end{eqnarray}
where the constant $C$ is independent of ${\mathcal{O}}$ and $f\in L^2(\Sigma,\sigma)$. Upon letting ${\mathcal{O}}\nearrow{\mathcal{U}}\setminus\Sigma$ in \eqref{huBN.ef}, Lebesgue's Monotone Convergence Theorem then yields \eqref{vlnGG-2ASn}. Finally, the last claim in the statement of Theorem~\ref{thm:BVAR} is justified in a similar manner, based on Corollary~\ref{cor:URimSFE}. \end{proof}
It is also useful to treat the following variant of \eqref{B-pv5}:
\begin{eqnarray}\label{S5.52} \widetilde{\mathcal{B}}f(x):=\int\limits_{\Sigma}b(y,x-y)f(y)\,d\sigma(y), \qquad x\in{\mathcal{U}}\setminus\Sigma. \end{eqnarray}
The same sort of analysis works, with $x$ replaced by $y$ in the spherical harmonic expansion \eqref{tag{1.12}} (in fact, the argument is simpler since the $a_{i\ell}$'s act this time as multipliers in the $y$ variable). Specifically, we have the following.
\begin{theorem}\label{p5.4} In the setting of Theorem~\ref{thm:BVAR}, with $\widetilde{\mathcal{B}}$ given by \eqref{S5.52} where, this time, in place of \eqref{B-pv5.LFD} one assumes
\begin{eqnarray}\label{B-pv5.LFD.XXX} \mbox{$\partial_z^\alpha b(x,z)$ is continuous and bounded on
${\mathcal{U}}\times S^{n}$ for $|\alpha|\leq M$}, \end{eqnarray}
there holds
\begin{eqnarray}\label{vva-UHab} \int_{{\mathcal{U}}\setminus\Sigma}
|\nabla\widetilde{\mathcal{B}}f(x)|^2\,{\rm dist}(x,\Sigma)\,dx
\leq C\int_{\Sigma}|f|^2\,d\sigma,\qquad\forall\,f\in L^2(\Sigma,\sigma). \end{eqnarray} \end{theorem}
In turn, Theorem~\ref{thm:BVAR} and Theorem~\ref{p5.4} apply to the Schwartz kernels of certain pseudodifferential operators. Recall that a pseudodifferential operator $Q(x,D)$ with symbol $q(x,\xi)$ in H\"ormander's class $S^m_{1,0}$ is given by the oscillatory integral
\begin{eqnarray}\label{1.6} Q(x,D)u &=& (2\pi)^{-(n+1)/2}\int q(x,\xi)\hat{u}(\xi) e^{i\langle x,\,\xi\rangle}\,d\xi \nonumber\\[4pt] &=& (2\pi)^{-(n+1)}\int\!\!\int q(x,\xi) e^{i\langle x-y,\,\xi\rangle}u(y)\,dy\,d\xi. \end{eqnarray}
Here, we are concerned with a smaller class of symbols, $S^m_{\rm cl}$, defined by requiring that the (matrix-valued) function $q(x,\xi)$ has an asymptotic expansion of the form
\begin{eqnarray}\label{1.8-B} q(x,\xi)\sim q_m(x,\xi)+q_{m-1}(x,\xi)+\cdots, \end{eqnarray}
with $q_j$ smooth in $x$ and $\xi$ and homogeneous of degree $j$
in $\xi$ (for $|\xi|\geq 1$). Call $q_m(x,\xi)$, i.e. the leading term in \eqref{1.8-B}, the {\it principal symbol} of $q(x,D)$. In fact, we shall find it convenient to work with classes of symbols which only exhibit a limited amount of regularity in the spatial variable (while still $C^\infty$ in the Fourier variable). Specifically, for each $r\geq 0$ we define
\begin{eqnarray}\label{1.2}
C^rS^m_{1,0}:=\bigl\{q(X,\xi):\,\|D^\alpha_\xi q(\cdot,\xi)\|_{C^r}\leq C_\alpha
(1+|\xi|)^{m-|\alpha|},\quad\forall\,\alpha\bigr\}. \end{eqnarray}
Denote by ${\rm OP}{C^r}S^m_{1,0}$ the class of pseudodifferential operators associated with such symbols. As before, we write ${\rm OP}{C^r}S^m_{\rm cl}$ for the subclass of {\it classical} pseudodifferential operators in ${\rm OP}{C^r}S^m_{1,0}$ whose symbols can be expanded as in \eqref{1.8-B}, where $q_j(x,\xi)\in C^rS^{m-j}_{1,0}$ is homogeneous of degree $j$ in $\xi$
for $|\xi|\geq 1$, $j=m,m-1,\dots$. Finally, we set $\mbox{\it \O}{\rm P}{C^r}S^m_{\rm cl}$ for the space of all formal adjoints of operators in ${\rm OP}{C^r}S^m_{\rm cl}$.
Given a classical pseudodifferential operator $Q(x,D)\in{\rm OP}C^rS^{-1}_{\rm cl}$, we denote by $k_Q(x,y)$ and ${\rm Sym}_Q(x,\xi)$ its Schwartz kernel and its principal symbol, respectively. Next, if the sets $\Sigma\subseteq{\mathcal{U}}\subseteq{\mathbb{R}}^{n+1}$ are as in Theorem~\ref{thm:BVAR}, we can introduce the integral operator
\begin{eqnarray}\label{1.5-B} {\mathcal{B}}_Qf(x):=\int_{\Sigma}k_Q(x,y)f(y)\,d\sigma(y), \qquad x\in{\mathcal{U}}\setminus\Sigma. \end{eqnarray}
In this context, Theorem~\ref{thm:BVAR} and Theorem~\ref{p5.4} yield the following result.
\begin{theorem}\label{T-mmt} Let ${\Sigma}\subseteq{\mathbb{R}}^{n+1}$ be compact and uniformly rectifiable, and assume that ${\mathcal{U}}$ is a bounded, open neighborhood of $\Sigma$ in ${\mathbb{R}}^{n+1}$. Let $Q(x,D)\in{\rm OP}C^1S^{-1}_{\rm cl}$ be such that ${\rm Sym}_Q(x,\xi)$ is odd in $\xi$. Then the operator \eqref{1.5-B} satisfies
\begin{eqnarray}\label{hkah-iyT} \int_{{\mathcal{U}}\setminus\Sigma}
|\nabla{\mathcal{B}}_Qf(x)|^2\,{\rm dist}(x,\Sigma)\,dx
\leq C\int_{\Sigma}|f|^2\,d\sigma,\qquad\forall\,f\in L^2(\Sigma,\sigma). \end{eqnarray}
Moreover, a similar result is valid for a pseudodifferential operator $Q(x,D)\in\mbox{\O}{\rm P}C^0S^{-1}_{\rm cl}$. \end{theorem}
In fact, since the main claims in Theorem~\ref{T-mmt} are local in nature and given the invariance of the class of domains and pseudodifferential operators (along with their Schwartz kernels and principal symbols) under smooth diffeomorphisms, these results can be naturally extended to the setting of domains on manifolds and pseudodifferential operators acting between vector bundles. Formulated as such, these in turn extend results proved in \cite{MMT} for Lipschitz subdomains of Riemannian manifolds.
\section{$L^p$ Square Function Estimates} \setcounter{equation}{0} \label{Sect:5}
We have so far only considered $L^2$ square function estimates. We now consider $L^p$ versions for $p\in(0,\infty]$. The natural setting for the consideration of these estimates is in term of mixed norm spaces $L^{(p,q)}(\mathscr{X},E)$, originally introduced in \cite{MMM} (cf. also \cite{BMMM} for related matters). We begin by using the tools developed in Section~\ref{Sect:2} to analyze these spaces in the context of an ambient quasi-metric space $\mathscr{X}$ and a closed subset $E$. In the case $\mathscr{X}=\mathbb{R}^{n+1}$ and $E=\partial\mathbb{R}^{n+1}_+\eqsim\mathbb{R}^n$, the mixed norm spaces correspond to the tent spaces introduced by R.~Coifman, Y.~Meyer and E.M.~Stein in \cite{CoMeSt}. The preliminary analysis in Subsections \ref{SSect:5.1} and \ref{SSect:5.2} is based on the techniques developed in that paper, although we need to overcome a variety of geometric obstructions that arise outside of the Euclidean setting. We build on this in Subsection~\ref{SSect:5.3}, where we prove that $L^2$ square function estimates associated with integral operators $\Theta_E$, as defined in Section~\ref{Sect:3}, follow from weak $L^p$ square function estimates for any $p\in(0,\infty)$. This is achieved by combining the $T(1)$ theorem from Subsection~\ref{SSect:3.1} with a weak type John-Nirenberg lemma for Carleson measures, the Euclidean version of which appears in~\cite{AHLT}. The theory culminates in Subsection~\ref{SSect:5.4}, where we prove two extrapolation theorems for estimates associated with integral operators $\Theta_E$, as defined in Section~\ref{Sect:3}. In particular, we prove that a weak $L^q$ square function estimate for any $q\in(0,\infty)$ implies that square functions are bounded from $H^p$ into $L^p$ for all $p\in(\frac{d}{d+\gamma},\infty)$, where $H^p$ is a Hardy space, $d$ is the the dimension of $E$, and $\gamma$ is a finite positive constant depending on the ambient space $\mathscr{X}$ and the operator $\Theta_E$.
\subsection{Mixed norm spaces} \label{SSect:5.1}
We begin by considering the mixed norm spaces $L^{(p,q)}$ from \cite{MMM} (cf. also \cite{BMMM}) and then, following the theory of tent spaces in~\cite{CoMeSt}, record some extensive preliminaries that are used throughout Section~\ref{Sect:5}. In particular, Theorem~\ref{appert} contains an equivalence for the quasi-norms of the mixed norm spaces that is essential in the next subsection.
Let $({\mathscr{X}},\rho)$ be a quasi-metric space, $E$ a nonempty subset of ${\mathscr{X}}$, and $\mu$ a Borel measure on $({\mathscr{X}},\tau_\rho)$. Recall the regularized version $\rho_{\#}$ of the quasi-distance $\rho$ discussed in Theorem~\ref{JjEGh}, and recall that we employ the notation $\delta_E(y)={\rm dist}_{\rho_{\#}}(y,E)$ for each $y\in {\mathscr{X}}$. Next, let $\kappa>0$ be arbitrary, fixed, and consider the {\tt nontangential approach regions}
\begin{eqnarray}\label{TLjb} \Gamma_\kappa(x):=\bigl\{y\in{\mathscr{X}}\setminus E:\, \rho_{\#}(x,y)<(1+\kappa)\,\delta_E(y)\bigr\}, \qquad\forall\,x\in E. \end{eqnarray}
Occasionally, we shall refer to $\kappa$ as the {\tt aperture} of the nontangential approach region $\Gamma_\kappa(x)$. Since both $\rho_{\#}(\cdot,\cdot)$ and $\delta_E(\cdot)$ are continuous (cf. Theorem~\ref{JjEGh}) it follows that $\Gamma_\kappa(x)$ is an open subset of $({\mathscr{X}},\tau_\rho)$, for each $x\in E$. Furthermore, it may be readily verified that
\begin{eqnarray}\label{Tfs23} {\mathscr{X}}\setminus\overline{E}=\bigcup\limits_{x\in E}\Gamma_\kappa(x),\qquad \forall\,\kappa>0, \end{eqnarray}
where $\overline{E}$ denotes the closure of $E$ in the topology $\tau_\rho$.
\begin{lemma}\label{semi-cont} Let $({\mathscr{X}},\rho)$ be a quasi-metric space, $E$ a proper, nonempty, closed subset of $({\mathscr{X}},\tau_\rho)$, and $\mu$ a Borel measure on $({\mathscr{X}},\tau_\rho)$. Let $u:{\mathscr{X}}\setminus E\to[0,\infty]$ be a $\mu$-measurable function, fix $\kappa>0$ and recall the regions from \eqref{TLjb}. Then the function
\begin{eqnarray}\label{Mixed-8} F:E\longrightarrow[0,\infty],\qquad F(x):=\int_{\Gamma_\kappa(x)}u(y)\,d\mu(y),\quad\forall\,x\in E, \end{eqnarray}
is lower semi-continuous (relative to the topology induced by $\tau_\rho$ on $E$). \end{lemma}
\begin{proof} Let $x_0\in E$ be arbitrary, fixed, and consider a sequence $\{x_j\}_{j\in{\mathbb{N}}}$ of points in $E$ with the property that
\begin{eqnarray}\label{Mixed-A} \lim\limits_{j\to\infty}\rho_{\#}(x_j,x_0)=0. \end{eqnarray}
We claim that
\begin{eqnarray}\label{Mixed-9} \liminf\limits_{j\to\infty}{\mathbf{1}}_{\Gamma_\kappa(x_j)}(x) \geq{\mathbf{1}}_{\Gamma_\kappa(x_0)}(x),\qquad\forall\,x\in{\mathscr{X}}\setminus E. \end{eqnarray}
Clearly \eqref{Mixed-9} is true if $x\not\in\Gamma_\kappa(x_0)$. If ${\mathbf{1}}_{\Gamma_\kappa(x_0)}(x)=1$, then $x\in\Gamma_\kappa(x_0)$, thus by definition $\rho_{\#}(x,x_0)<(1+\kappa)\delta_E(x)$. Based on the continuity of $\rho_{\#}(x,\cdot)$ and \eqref{Mixed-A}, it follows that there exists $j_0\in{\mathbb{N}}$ such that $\rho_{\#}(x,x_j)<(1+\kappa)\delta_E(x)$ for $j\geq j_0$. Hence, $x\in\Gamma_\kappa(x_j)$ for $j\geq j_0$ or, equivalently, ${\mathbf{1}}_{\Gamma_\kappa(x_j)}(x)=1$ for $j\geq j_0$. This completes the proof of the claim.
Returning to the actual task at hand, Fatou's lemma and \eqref{Mixed-9} then imply
\begin{eqnarray}\label{Mixed-10} \liminf_{j\to\infty}F(x_j) &=& \liminf_{j\to\infty}\int_{{\mathscr{X}}\setminus E} {\mathbf{1}}_{\Gamma_\kappa(x_j)}u\,d\mu \geq\int_{{\mathscr{X}}\setminus E} \liminf_{j\to\infty}\bigl({\mathbf{1}}_{\Gamma_\kappa(x_j)}u\bigr)\,d\mu \nonumber\\[4pt] &=&\int_{{\mathscr{X}}\setminus E} \bigl(\liminf_{j\to\infty}{\mathbf{1}}_{\Gamma_\kappa(x_j)}\bigr)u\,d\mu \geq\int_{{\mathscr{X}}\setminus E}{\mathbf{1}}_{\Gamma_\kappa(x_0)}u\,d\mu \nonumber\\[4pt] &=& F(x_0). \end{eqnarray}
This shows that $F$ is lower semi-continuous. \end{proof}
We retain the context of Lemma~\ref{semi-cont}. For each index $q\in(0,\infty)$ and constant $\kappa\in(0,\infty)$, define the $L^q$-based {\tt Lusin operator}, or {\tt area operator}, ${\mathscr{A}}_{q,\kappa}$ for all $\mu$-measurable functions $u:{\mathscr{X}}\setminus E\to\overline{\mathbb{R}}:=[-\infty,+\infty]$ by
\begin{eqnarray}\label{sp-sq}
({\mathscr{A}}_{q,\kappa}u)(x):=\Bigl(\int_{\Gamma_\kappa(x)}|u(y)|^q\,d\mu(y)\Bigr)^{\frac{1}{q}}, \qquad\forall\,x\in E. \end{eqnarray}
As a consequence of Lemma~\ref{semi-cont}, we have that ${\mathscr{A}}_{q,\kappa}u$ is lower semi-continuous, hence
\begin{eqnarray}\label{Mixed-7A} \bigl\{x\in E:\,({\mathscr{A}}_{q,\kappa}u)(x)>\lambda\bigr\}\quad \mbox{ is an open subset of $(E,\tau_\rho)$ for each $\lambda>0$}. \end{eqnarray}
To proceed, fix a Borel measure $\sigma$ on $(E,\tau_{\rho|_{E}})$. The above considerations then allow us to conclude that
\begin{eqnarray}\label{Mixed-3} \begin{array}{c} \mbox{for any $\mu$-measurable function $u:{\mathscr{X}}\setminus E\rightarrow\overline{\mathbb{R}}$}, \\[4pt] \mbox{the mapping }\,{\mathscr{A}}_{q,\kappa}u:E\rightarrow[0,\infty] \,\mbox{ is well-defined and $\sigma$-measurable.} \end{array} \end{eqnarray}
Consequently, given $\kappa>0$ and a pair of integrability indices $p,q$, following \cite{MMM} and \cite{BMMM} we may now introduce the {\tt mixed norm space of type} $(p,q)$, denoted $L^{(p,q)}({\mathscr{X}},E,\mu,\sigma;\kappa)$, or $L^{(p,q)}({\mathscr{X}},E)$ for short, in a meaningful manner as follows. If $q\in(0,\infty)$ and $p\in(0,\infty]$ we set
\begin{eqnarray}\label{Mixed-FF7} L^{(p,q)}({\mathscr{X}},E,\mu,\sigma;\kappa):=\Bigl\{ u:{\mathscr{X}}\setminus E\to\overline{\mathbb{R}}:\,u\, \mbox{ $\mu$-measurable and }\,{\mathscr{A}}_{q,\kappa}u\in L^p(E,\sigma)\Bigr\}, \end{eqnarray}
equipped with the quasi-norm
\begin{eqnarray}\label{Mixed-EEW}
\|u\|_{L^{(p,q)}({\mathscr{X}},E,\mu,\sigma;\kappa)}
:=\|{\mathscr{A}}_{q,\kappa}u\|_{L^p(E,\sigma)} =\left\{ \begin{array}{l}
\Bigl(\int_{E}\Bigl[\int_{\Gamma_{\kappa}(x)}|u|^q\,d\mu\Bigr]^{p/q}d\sigma(x) \Bigr)^{1/p}\,\mbox{ if }\,p<\infty, \\[8pt] \sigma\mbox{-}{\rm ess}\sup\limits_{x\in E}\,({\mathscr{A}}_{q,\kappa}u)(x) \quad\mbox{ if }\,\,p=\infty. \end{array} \right. \end{eqnarray}
Also, corresponding to $p\in(0,\infty)$ and $q=\infty$, we set
\begin{eqnarray}\label{Mixed-I} L^{(p,\infty)}({\mathscr{X}},E,\mu,\sigma;\kappa):=\Bigl\{ u:{\mathscr{X}}\setminus E\to\overline{\mathbb{R}}:\,
\|{\mathcal{N}}_\kappa u\|_{L^{p}(E,\sigma)}<\infty\Bigr\}, \end{eqnarray}
where ${\mathcal{N}}_\kappa$ is the nontangential maximal operator defined by
\begin{eqnarray}\label{Mixed-N}
({\mathcal{N}}_\kappa u)(x):=\sup_{y\in\Gamma_\kappa(x)}|u(y)|,\qquad \forall\,x\in E, \end{eqnarray}
and equip this space with the quasi-norm
$\|u\|_{L^{(p,\infty)}({\mathscr{X}},E,\mu,\sigma;\kappa)}:=
\|{\mathcal{N}}_{\kappa}u\|_{L^p(E,\sigma)}$. Finally, corresponding to $p=q=\infty$, set
\begin{eqnarray}\label{Mixed-IDF} L^{(\infty,\infty)}({\mathscr{X}},E,\mu,\sigma;\kappa) :=L^\infty({\mathscr{X}}\setminus E,\mu). \end{eqnarray}
We note that the connection of our mixed norm spaces with the Coifman-Meyer-Stein tent spaces $T^p_q$ in ${\mathbb{R}}^{n+1}_+$ is as follows
\begin{eqnarray}\label{Phgf} T^p_q=L^{(p,q)}\Bigl({\mathbb{R}}^{n+1},\partial{\mathbb{R}}^{n+1}_+, {\mathbf{1}}_{{\mathbb{R}}^{n+1}_+}\frac{dx\,dt}{t^{n+1}},dx\Bigr), \quad\mbox{for }\,\,p,q\in(0,\infty). \end{eqnarray}
Thus, results for mixed normed spaces imply results for classical tent spaces.
The next goal is to clarify to what extent the quasi-norm
$\|\cdot\|_{L^{(p,q)}({\mathscr{X}},E,\mu,\sigma;\kappa)}$ depends on the parameter $\kappa>0$ associated with the nontangential approach regions $\Gamma_k$ defined in \eqref{TLjb} and utilized in \eqref{Mixed-EEW}, \eqref{Mixed-N}. This is done in Theorem~\ref{appert} below, but the proof requires a number of preliminary results and definitions which we now present.
To set the stage, for each $A\subseteq E$ and $\kappa>0$, define the {\tt fan} (or {\tt saw-tooth}) {\tt region} ${\mathcal{F}}_\kappa(A)$ {\tt above} $A$, and the {\tt tent region} ${\mathcal{T}}_\kappa(A)$ {\tt above} $A$, as
\begin{eqnarray}\label{reg-A1} {\mathcal{F}}_\kappa(A):=\bigcup\limits_{x\in A}\Gamma_\kappa(x) \quad\mbox{ and }\quad {\mathcal{T}}_\kappa(A):=\bigl({\mathscr{X}}\setminus E\bigr)\setminus \Bigl({\mathcal{F}}_\kappa(E\setminus A)\Bigr). \end{eqnarray}
Also, for each point $y\in{\mathscr{X}}\setminus E$, define the ``(reverse) conical projection" of $y$ onto $E$ by
\begin{eqnarray}\label{reg-A2} \pi_y^\kappa:=\bigl\{x\in E:\,y\in\Gamma_\kappa(x)\bigr\}. \end{eqnarray}
\begin{lemma}\label{T-LL.2} Let $({\mathscr{X}},\rho)$ be a quasi-metric space, $E$ a proper, nonempty, closed subset of $({\mathscr{X}},\tau_\rho)$. For every $A\subseteq E$, denote by $\overline{A}$ and $A^\circ$, respectively, the closure and interior of
$A$ in the topological space $(E,\tau_{\rho|_{E}})$. Then for each fixed $\kappa\in(0,\infty)$ the following properties hold.
\begin{enumerate}
\item[(i)] For each $A\subseteq E$ one has ${\mathcal{F}}_\kappa(A)={\mathcal{F}}_\kappa(\overline{A})$ and ${\mathcal{T}}_\kappa(A^\circ)={\mathcal{T}}_\kappa(A)$.
\item[(ii)] For each $A\subseteq E$ one has ${\mathcal{T}}_\kappa(A)\subseteq {\mathcal{F}}_\kappa(A)$.
\item[(iii)] For each nonempty proper subset $A$ of $E$ one has
\begin{eqnarray}\label{3.2.64} && \hskip -0.40in {\mathcal{T}}_\kappa(A)=\bigl\{x\in{\mathscr{X}}\setminus E:\, {\rm dist}_{\rho_{\#}}(x,A)\leq (1+\kappa)^{-1}\, {\rm dist}_{\rho_{\#}}(x,E\setminus A)\bigr\}, \\[4pt] && \hskip 0.60in {\mathcal{T}}_\kappa(A)=\bigl\{y\in{\mathscr{X}}\setminus E:\, \pi_y^\kappa\subseteq A\bigr\}. \label{3.2.BN} \end{eqnarray}
Moreover, for each nonempty subset $A$ of $E$ one has
\begin{eqnarray}\label{3.2.TTF} {\mathcal{F}}_\kappa(A)=\bigl\{y\in{\mathscr{X}}\setminus E:\, {\rm dist}_{\rho_{\#}}(y,A)<(1+\kappa)\,\delta_E(y)\bigr\}. \end{eqnarray}
\item[(iv)] One has ${\mathcal{F}}_{\kappa}(E)= {\mathcal{T}}_{\kappa}(E)={\mathscr{X}}\setminus E$. Also, for any family $(A_j)_{j\in J}$ of subsets of $E$,
\begin{eqnarray}\label{Fv-fCC49} \bigcup_{j\in J}{\mathcal{F}}_\kappa(A_j) ={\mathcal{F}}_\kappa\bigl(\cup_{j\in J}A_j\bigr),\qquad \bigcap_{j\in J}{\mathcal{T}}_\kappa(A_j) ={\mathcal{T}}_\kappa\bigl(\cap_{j\in J}A_j\bigr), \end{eqnarray}
and
\begin{eqnarray}\label{Fv-fCC50} A_1\subseteq A_2\subseteq E\,\,\Longrightarrow\,\, {\mathcal{F}}_\kappa(A_1)\subseteq{\mathcal{F}}_\kappa(A_2)\,\,\,\mbox{ and }\,\,\, {\mathcal{T}}_\kappa(A_1)\subseteq{\mathcal{T}}_\kappa(A_2). \end{eqnarray}
\item[(v)] Given $A\subseteq E$, it follows that ${\mathcal{F}}_\kappa(A)$ is an open subset of $({\mathscr{X}},\tau_\rho)$, while ${\mathcal{T}}_\kappa(A)$ is a relatively closed subset of ${\mathscr{X}}\setminus E$ equipped with the topology induced by $\tau_\rho$ on this set.
\item[(vi)] For each $y\in{\mathscr{X}}\setminus E$ it follows that $\pi_y^\kappa$ is a relatively open set in the topology induced by $\tau_\rho$ on $E$.
\item[(vii)] One has
\begin{eqnarray}\label{Fv-UU45} B_{\rho_{\#}}\bigl(x,C_\rho^{-1}r\bigr)\setminus E\subseteq {\mathcal{T}}_\kappa\bigl(E\cap B_{\rho_{\#}}(x,r)\bigr),\qquad \forall\,r\in(0,\infty),\quad\forall\,x\in E. \end{eqnarray}
\item[(viii)] Assume that $(E,\rho\bigl|_{E})$ is geometrically doubling. Then for every $\kappa>0$ there exists a constant $C_o\in(0,\infty)$ with the property that if $\mathcal{O}$ is a nonempty, open, proper subset
of $(E,\tau_{\rho|_{E}})$ and if $\{\Delta_j\}_{j\in J}$, where $x_j\in E$ and $\Delta_j:=E\cap B_\rho(x_j,r_j)$ for each $j\in J$, is a Whitney decomposition of $\mathcal{O}$ as in Proposition~\ref{H-S-Z}, then
\begin{eqnarray}\label{3.2.63} {\mathcal{T}}_{\kappa}(\mathcal{O}) \subseteq\bigcup\limits_{j\in J}B_\rho(x_j,C_or_j). \end{eqnarray}
In particular, there exists $C\in(0,\infty)$ with the property that
\begin{eqnarray}\label{3.2.63WS} \begin{array}{c} {\mathcal{T}}_{\kappa}\bigl(E\cap B_{\rho}(x,r)\bigr) \subseteq B_\rho(x,C r)\setminus E\quad\mbox{ whenever} \\[4pt] \mbox{$x\in E$ and $r>0$ are such that $E\setminus B_{\rho}(x,r)\not=\emptyset$}. \end{array} \end{eqnarray}
\item[(ix)] In the case when $E$ is bounded, there exists $C\in(0,\infty)$ with the property that
\begin{eqnarray}\label{3.UGH} {\mathscr{X}}\setminus B_{\rho_{\#}}\bigl(x_0,C\,{\rm diam}_{\rho}(E)\bigr)\subseteq \Gamma_\kappa(x),\qquad\forall\,x_0,x\in E. \end{eqnarray}
Consequently, whenever $E$ is bounded there exists $C\in(0,\infty)$ such that for each $x_0\in E$ one has
\begin{eqnarray}\label{3.UGH2} {\mathcal{T}}_{\kappa}(A)\subseteq B_{\rho_{\#}}\bigl(x_0,C\,{\rm diam}_{\rho}(E)\bigr), \qquad\forall\,A\,\,\mbox{ proper subset of }\,\,E. \end{eqnarray} \end{enumerate} \end{lemma}
\begin{proof} With the exception of the first part of {\it (viii)}, these are direct consequences of definitions and the fact that both $\rho_{\#}(\cdot,\cdot)$ and $\delta_E(\cdot)$ are continuous functions. The remaining portion of the proof consists of a verification of \eqref{3.2.63}. To get started, let $x$ be an arbitrary point in ${\mathcal{T}}_{\kappa}(\mathcal{O})$. This places $x$ in ${\mathscr{X}}\setminus E$ which, given that $E$ is closed in $({\mathscr{X}},\tau_\rho)$, means that $x$ does not belong to $\overline{{\mathcal{O}}}\subseteq E$. In particular, ${\rm dist}_{\rho_{\#}}(x,\mathcal{O})>0$. Going further, assume that some small $\varepsilon>0$ has been fixed. The above discussion then shows that it is possible to pick a point $y\in\mathcal{O}$ with the property that
\begin{eqnarray}\label{3.2.65} \rho_{\#}(x,y)<(1+\varepsilon)\,{\rm dist}_{\rho_{\#}}(x,\mathcal{O}). \end{eqnarray}
Then there exists an index $j\in J$ for which $y\in\Delta_j$ and we shall show that $\varepsilon$ and $C_o$ can be chosen so as to guarantee that
\begin{eqnarray}\label{3.2.66} x\in B_{\rho}(x_j,C_or_j). \end{eqnarray}
Indeed, selecting a real number $\beta\in(0,(\log_2 C_\rho)^{-1}]$ and invoking \eqref{3.2.64} we may write
\begin{eqnarray}\label{3.2.67} \bigl[\rho_{\#}(x,y)\bigr]^\beta & < & (1+\varepsilon)^\beta\bigl[{\rm dist}_{\rho_{\#}}(x,\mathcal{O})\bigr]^\beta \leq\Bigl(\frac{1+\varepsilon}{1+\kappa}\Bigr)^\beta \bigl[{\rm dist}_{\rho_{\#}}(x,E\setminus{\mathcal{O}})\bigr]^\beta \nonumber \\[4pt] & = & \Bigl(\frac{1+\varepsilon}{1+\kappa}\Bigr)^\beta {\rm dist}_{(\rho_{\#})^\beta}(x,E\setminus{\mathcal{O}}) \nonumber \\[4pt] &\leq & \Bigl(\frac{1+\varepsilon}{1+\kappa}\Bigr)^\beta \Bigl(\bigl[\rho_{\#}(x,y)\bigr]^\beta +{\rm dist}_{(\rho_{\#})^\beta}(y,E\setminus{\mathcal{O}})\Bigr) \nonumber\\[4pt] & \leq & \Bigl(\frac{1+\varepsilon}{1+\kappa}\Bigr)^\beta \Bigl(\bigl[\rho_{\#}(x,y)\bigr]^\beta +Cr_j^\beta\Bigr), \end{eqnarray}
where $C\in(0,\infty)$ depends only on the geometrically doubling character of $E$. The last step above uses Theorem~\ref{JjEGh} and the fact that $y$ belongs to $\Delta_j=B_\rho(x_j,r_j)\cap E$, which is a Whitney ball for ${\mathcal{O}}$. Choosing $\varepsilon=\kappa/2$, this now yields (on account of the first inequality in \eqref{DEQV1})
\begin{eqnarray}\label{3.2.68} \rho(x,y)\leq C_\rho^2\,\rho_{\#}(x,y) <C_\rho^2 C^{1/\beta}\Bigl(\tfrac{1+\kappa/2} {\bigl[(1+\kappa)^\beta-(1+\kappa/2)^\beta\bigr]^{1/\beta}}\Bigr)r_j =:C_{\kappa,\beta}\,r_j. \end{eqnarray}
Hence, since $\rho(x_j,x)\leq C_\rho\,\max\{\rho(x_j,y),\rho(y,x)\} <C_\rho C_{\kappa,\beta}r_j$, the membership in \eqref{3.2.66} holds provided we take $C_o:=C_\rho C_{\kappa,\beta}$ to begin with. This finishes the proof of \eqref{3.2.63}. \end{proof}
\begin{lemma}\label{lbDV} Let $({\mathscr{X}},\rho)$ be a quasi-metric space, $E$ a proper, nonempty, closed subset of $({\mathscr{X}},\tau_\rho)$, $\mu$ a Borel measure on
$({\mathscr{X}},\tau_\rho)$ and $\sigma$ a Borel measure on $(E,\tau_{\rho|_{E}})$. Let $\rho_{\#}$ be associated with $\rho$ as in Theorem~\ref{JjEGh} and recall the constant $C_{\rho}\geq 1$ defined in \eqref{C-RHO.111}. Then for each real number $\kappa>0$ there holds
\begin{eqnarray}\label{sgbr} E\cap B_{\rho_{\#}}\bigl(y_\ast,\epsilon\delta_E(y)\bigr)\subseteq\pi^\kappa_y \subseteq E\cap B_{\rho_{\#}}\bigl(y_\ast,C_{\rho}(1+\kappa)\delta_E(y)\bigr), \qquad\forall\,y\in{\mathscr{X}}\setminus E, \end{eqnarray}
where the point $y_\ast$ and the number $\epsilon$ satisfy
\begin{eqnarray}\label{Equ-1} \begin{array}{l} \mbox{$y_\ast\in E$ and $\rho_{\#}(y,y_\ast)<(1+\eta)\delta_E(y)$ for some $\eta\in(0,\kappa)$, and } \\[4pt] 0<\epsilon<\bigl[(1+\kappa)^\beta-(1+\eta)^\beta\bigr]^{1/\beta} \,\,\mbox{ for some finite $\beta\in(0,(\log_2 C_\rho)^{-1}]$.} \end{array} \end{eqnarray}
In particular, if $(E,\rho\bigl|_E,\sigma)$ is a space of homogeneous type and if $\kappa,\kappa'>0$ are two arbitrary real numbers, then
\begin{eqnarray}\label{equiv11} c_o^{-1}\,\sigma(\pi^{\kappa}_y)\leq\sigma(\pi^{\kappa'}_y) \leq c_o\,\sigma(\pi^{\kappa}_y),\qquad\forall\,y\in{\mathscr{X}}\setminus E, \end{eqnarray}
where $c_o:=C_{\sigma}(C^2_\rho/\epsilon)^{D_\sigma} (1+\min\{\kappa,\kappa'\})^{D_\sigma}$, with $C_{\sigma}$ and $D_\sigma$ the doubling constant and doubling order of $\sigma$. \end{lemma}
\begin{proof} Fix an arbitrary point $y\in{\mathscr{X}}\setminus E$ and let $y_\ast\in E$ and $\epsilon>0$ be as in \eqref{Equ-1}. If $x\in E\cap B_{\rho_{\#}}(y_\ast,\epsilon\delta_E(y))$ then $\rho_{\#}(y_\ast,x)<\epsilon\delta_E(y)$ forcing (recall from Theorem~\ref{JjEGh} that $(\rho_{\#})^\beta$ is a genuine distance)
\begin{eqnarray}\label{H+gz-2} \rho_{\#}(x,y)^\beta\leq\rho_{\#}(x,y_\ast)^\beta+\rho_{\#}(y_\ast,y)^\beta <\epsilon^\beta\delta_E(y)^\beta+(1+\eta)^\beta\delta_E(y)^\beta <(1+\kappa)^\beta\delta_E(y)^\beta. \end{eqnarray}
Thus $x\in\pi^\kappa_y$, which proves the first inclusion in \eqref{sgbr}. Going further, given a point $x\in\pi^\kappa_y$ it follows that $\rho_{\#}(x,y)<(1+\kappa)\delta_E(y)$, hence
\begin{eqnarray}\label{H+gz} \rho_{\#}(x,y_\ast)\leq C_{\rho_{\#}}\max\{\rho_{\#}(x,y),\rho_{\#}(y,y_\ast)\} <C_{\rho_{\#}}(1+\kappa)\delta_E(y)\leq C_{\rho}(1+\kappa)\delta_E(y), \end{eqnarray}
proving the second inclusion in \eqref{sgbr}.
Suppose now that $(E,\rho\bigl|_E,\sigma)$ is a space of homogeneous type and let $\kappa,\kappa'>0$ be given. Assume first that $\kappa\leq\kappa'$. Choose $y_\ast$ and $\epsilon$ as in \eqref{Equ-1}. Then \eqref{sgbr} holds both as written and with $\kappa$ replaced by $\kappa'$. When combined with \eqref{Doub-2}, this yields
\begin{eqnarray}\label{eqDc} c_1^{-1}\,\sigma(\pi^{\kappa}_y)\leq\sigma(\pi^{\kappa'}_y) \leq c_1\,\sigma(\pi^{\kappa}_y),\qquad\forall\,y\in{\mathscr{X}}\setminus E, \end{eqnarray}
where $c_1:=C_{\sigma,\rho_{\#}}\Bigl(\frac{C_\rho(1+\kappa)}{\epsilon}\Bigr)^{D_\sigma}$ with $C_{\sigma,\rho_{\#}}$ and $D_\sigma$ being the constants associated with $\sigma$ and $\rho_{\#}$ as in \eqref{Doub-2}. In particular, since $C_{\sigma,\rho_{\#}}=C_\sigma(C_{\rho_{\#}}\widetilde{C}_{\rho_{\#}})^{D_\sigma} \leq C_\sigma(C_{\rho})^{D_\sigma}$, it follows that $c_1\leq C_{\sigma}(C^2_\rho/\epsilon)^{D_\sigma}(1+\kappa)^{D_\sigma}$. If $\kappa'<\kappa$ the same reasoning yields inequalities similar to \eqref{eqDc}, this time with $c_1$ replaced by the constant $c_2:=C_{\sigma,\rho_{\#}}\Bigl(\frac{C_\rho(1+\kappa')}{\epsilon}\Bigr)^{D_\sigma} \leq C_{\sigma}(C^2_\rho/\epsilon)^{D_\sigma}(1+\kappa')^{D_\sigma}$. All these now immediately yield \eqref{equiv11}. \end{proof}
Moving on, assume now that $(E,\rho,\sigma)$ is a space of homogeneous type and let $\rho_{\#}$ be associated with $\rho$ as in Theorem~\ref{JjEGh} in this context. Then for each $\sigma$-measurable set $A\subseteq E$ and each $\gamma\in(0,1)$, define the set of $\gamma$-{\tt density points}, relative to $A$, as
\begin{eqnarray}\label{Mixed-12} A^\ast_\gamma:=\Bigl\{x\in E:\,\inf\limits_{r>0} \Bigl[\frac{\sigma\bigl(B_{\rho_{\#}}(x,r)\cap A\bigr)} {\sigma\bigl(B_{\rho_{\#}}(x,r)\bigr)}\Bigr]\geq\gamma\Bigr\}. \end{eqnarray}
In particular, from this definition it follows that
\begin{eqnarray}\label{Mixed-13} \inf\limits_{x\in A^\ast_\gamma}\,\Bigl[\, \inf\limits_{r>0}\frac{\sigma\bigl(B_{\rho_{\#}}(x,r)\cap A\bigr)} {\sigma\bigl(B_{\rho_{\#}}(x,r)\bigr)}\Bigr]\geq\gamma. \end{eqnarray}
Some basic properties of the sets of density points in the setting of spaces of homogeneous type are collected below.
\begin{proposition}\label{DJrt} Let $(E,\rho,\sigma)$ be a space of homogeneous type, $\rho_{\#}$ the regularization of $\rho$ as in Theorem~\ref{JjEGh}, $\gamma\in(0,1)$ and $A\subseteq E$ a $\sigma$-measurable set. Then the following properties hold:
\begin{enumerate}
\item[(1)] $E\setminus A^\ast_\gamma=\bigl\{x\in E:\, M_{E}\bigl({\mathbf{1}}_{E\setminus A}\bigr)(x)>1-\gamma\bigr\}$, where $M_E$ is the Hardy-Littlewood maximal operator on $E$ (cf. \eqref{HL-MAX}).
\item[(2)] $A^\ast_\gamma$ is closed subset of $(E,\tau_\rho)$.
\item[(3)] $\sigma\bigl(E\setminus A^\ast_\gamma\bigr) \leq\frac{C}{1-\gamma}\sigma(E\setminus A)$.
\item[(4)] If $A$ is closed (in $\tau_\rho$), then $A^\ast_\gamma\subseteq A$. In particular, in this case, $\sigma(E\setminus A^\ast_\gamma)\approx\sigma(E\setminus A)$.
\item[(5)] For each $\lambda>0$ there exist $\gamma(\lambda)\in(0,1)$ and $c(\lambda)>0$ such that if $\gamma(\lambda)\leq\gamma<1$ then
\begin{eqnarray}\label{Mixed-14} \inf\limits_{x\in E}\,\Bigl[\, \inf\limits_{r>{\rm dist}_{\rho_{\#}}(x,A^\ast_\gamma)} \frac{\sigma\bigl(B_{\rho_{\#}}(x,\lambda r)\cap A\bigr)} {\sigma\bigl(B_{\rho_{\#}}(x,r)\bigr)}\Bigr]\geq c(\lambda). \end{eqnarray}
\item[(6)] If the measure $\sigma$ is Borel regular, then $\sigma(A^\ast_\gamma\setminus A)=0$.
\item[(7)] If $\widetilde{A}$ is $\sigma$-measurable set such that $A\subseteq\widetilde{A}\subseteq E$, then $A^\ast_\gamma\subseteq(\widetilde{A})^\ast_\gamma$. \end{enumerate} \end{proposition}
The remarkable aspect of {\it (3)-(4)} above is that whenever $A$ is a closed subset of $(E,\tau_\rho)$ then in a measure-theoretic sense the size of both sets, $A^\ast_\gamma$ and $E\setminus A^\ast_\gamma$, may be controlled in terms of sets $A$ and $E\setminus A$, respectively (as opposed to point-set theory). The typical application of Proposition~\ref{DJrt} is in estimating the measure of a $\sigma$-measurable set $F\subseteq E$ by writing
\begin{eqnarray}\label{Irh} \sigma(F)=\sigma(F\cap A^\ast_\gamma) +\sigma\bigl(F\cap(E\setminus A^\ast_\gamma)\bigr) \leq \sigma(F\cap A^\ast_\gamma)+\frac{C}{1-\gamma}\sigma(E\setminus A). \end{eqnarray}
\vskip 0.08in \begin{proof}[Proof of Proposition~\ref{DJrt}] Starting with \eqref{Mixed-12} we may write
\begin{eqnarray}\label{Mixed-15} E\setminus A^\ast_\gamma &=& \Bigl\{x\in E:\,\exists\,r>0\mbox{ such that } \frac{\sigma\bigl(B_{\rho_{\#}}(x,r)\cap A\bigr)} {\sigma\bigl(B_{\rho_{\#}}(x,r)\bigr)}<\gamma\Bigr\} \nonumber\\[4pt] &=& \Bigl\{x\in E:\,\exists\,r>0\mbox{ such that } \frac{\sigma\bigl(B_{\rho_{\#}}(x,r)\cap (E\setminus A)\bigr)} {\sigma\bigl(B_{\rho_{\#}}(x,r)\bigr)}>1-\gamma\Bigr\} \nonumber\\[4pt] &=& \Bigl\{x\in E:\,\sup\limits_{r>0}\Bigl( {\int{\mkern-19mu}-}_{B_{\rho_{\#}}(x,r)}{\mathbf{1}}_{E\setminus A}\,d\sigma\Bigr)>1-\gamma\Bigr\} \nonumber\\[4pt] &=& \Bigl\{x\in E:\,\sup\limits_{0<r\leq{\rm diam}\,_{\rho_{\#}}(E)}\Bigl( {\int{\mkern-19mu}-}_{B_{\rho_{\#}}(x,r)}{\mathbf{1}}_{E\setminus A}\,d\sigma\Bigr)>1-\gamma\Bigr\} \nonumber\\[4pt] &=& \Bigl\{x\in E:\, M_{E}\bigl({\mathbf{1}}_{E\setminus A}\bigr)(x)>1-\gamma\Bigr\}, \end{eqnarray}
proving {\it (1)}. We now make the claim that
\begin{eqnarray}\label{Mixed-SCC} \mbox{the function $M_{E}\bigl({\mathbf{1}}_{E\setminus A}\bigr):(E,\tau_\rho)\rightarrow[0,\infty]$ is lower semi-continuous}. \end{eqnarray}
To prove this claim, we note that since the pointwise supremum of an arbitrary family of real-valued, lower semi-continuous functions defined on $E$ is itself lower semi-continuous, it suffices to show that
\begin{eqnarray}\label{MEAS-11.aPP} \begin{array}{l} \mbox{for every $\sigma$-measurable set $F\subseteq E$, the function }\,\, f:(E,\tau_\rho)\to[0,\infty) \\[4pt] \mbox{given by }\,f(x):=\sigma\bigl(B_{\rho_{\#}}(x,r)\cap F\bigr)\quad \forall\,x\in E,\,\mbox{ is lower semi-continuous}. \end{array} \end{eqnarray}
To this end, fix $x_o\in E$ arbitrary. The crux of the matter is the fact that our choice of the quasi-distance ensures that if $\{x_j\}_{j\in{\mathbb{N}}}$ is a sequence of points in $E$ with the property that $x_j\to x_o$ as $j\to\infty$, with convergence understood in the (metrizable) topology $\tau_\rho$, then
\begin{eqnarray}\label{MEAS-12.aPP} \liminf_{j\to\infty}{\mathbf 1}_{B_{\rho_{\#}}(x_j,r)}(y) \geq{\mathbf 1}_{B_{\rho_{\#}}(x_o,r)}(y),\qquad\forall\,y\in E, \end{eqnarray}
as is easily verified by analyzing the cases $y\in B_{\rho_{\#}}(x_o,r)$ and $y\in E\setminus B_{\rho_{\#}}(x_o,r)$. In turn, based on this and Fatou's lemma we may then estimate
\begin{eqnarray}\label{MEAS-13.aPP} f(x_o) &=& \sigma\bigl(B_{\rho_{\#}}(x_o,r)\cap F\bigr) =\int_{F}{\mathbf 1}_{B_{\rho_{\#}}(x_o,r)}(y)\,d\sigma(y) \nonumber\\[4pt] &\leq & \int_{F}\liminf_{j\to\infty}{\mathbf 1}_{B_{\rho_{\#}}(x_j,r)}(y)\,d\sigma(y) \leq\liminf_{j\to\infty}\int_{F}{\mathbf 1}_{B_{\rho_{\#}}(x_j,r)}(y)\,d\sigma(y) \nonumber\\[4pt] &=& \liminf_{j\to\infty}\sigma\bigl(B_{\rho_{\#}}(x_j,r)\cap F\bigr) =\liminf_{j\to\infty}f(x_j). \end{eqnarray}
This establishes \eqref{MEAS-11.aPP}, thus finishing the proof of \eqref{Mixed-SCC}.
Moving on, \eqref{Mixed-SCC} implies that the last set in \eqref{Mixed-15} is open (in $\tau_\rho$), hence {\it (2)} holds true. Also, by combining {\it (1)} with the weak-$(1,1)$ boundedness of $M_{E}$ (recall that we are assuming that $(E,\rho,\sigma)$ is a space of homogeneous type), we obtain
\begin{eqnarray}\label{Mixed-15B} \sigma\bigl(E\setminus A^\ast_\gamma\bigr)
\leq \frac{C}{1-\gamma}\|{\mathbf{1}}_{E\setminus A}\|_{L^1(E,\sigma)} =\frac{C}{1-\gamma}\sigma(E\setminus A). \end{eqnarray}
Hence the inequality in {\it (3)} is proved.
Suppose now that $A$ is a closed subset of $(E,\tau_\rho)$. Then $E\setminus A$ is open, so if $x\in E\setminus A$ then there exists $r>0$ such that $B_{\rho_{\#}}(x,r)\subseteq E\setminus A$. Consequently, $\frac{\sigma\bigl(B_{\rho_{\#}}(x,r)\cap A)\bigr)} {\sigma\bigl(B_{\rho_{\#}}(x,r)\bigr)}=0<\gamma$, thus $x\not\in A^\ast_\gamma$. This shows that $A^\ast_\gamma\subseteq A$, hence $\sigma(E\setminus A)\leq \sigma(E\setminus A^\ast_\gamma)$. Combining these with what we proved in {\it (3)} finishes the proof of {\it (4)}.
Turning to the proof of {\it (5)}, fix some $\lambda>0$ and $x\in E$, arbitrary, and select $r>0$ such that
\begin{eqnarray}\label{Mixed-16} {\rm dist}_{\rho_{\#}}(x,A^\ast_\gamma)<r. \end{eqnarray}
Then there exists $x_0\in A^\ast_\gamma$ such that $\rho_{\#}(x,x_0)<r$, which forces
\begin{eqnarray}\label{Mixed-17} B_{\rho_{\#}}(x,\lambda r)\subseteq B_{\rho_{\#}}(x_0,C_{\rho_{\#}}(1+\lambda)r) \subseteq B_{\rho_{\#}}(x,C_{\rho_{\#}}^2(1+\lambda)r). \end{eqnarray}
Consequently, since $x_0\in A^\ast_\gamma$ we obtain
\begin{eqnarray}\label{Mixed-18} &&\hskip -0.30in \gamma\sigma\bigl(B_{\rho_{\#}}(x_0,C_{\rho_{\#}}(1+\lambda)r)\bigr) \leq\sigma\bigl(B_{\rho_{\#}}(x_0,C_{\rho_{\#}}(1+\lambda)r)\cap A\bigr) \nonumber\\[4pt] &&\hskip 0.30in \leq\sigma\bigl(B_{\rho_{\#}}(x_0,C_{\rho_{\#}}(1+\lambda)r)\setminus B_{\rho_{\#}}(x,\lambda r)\bigr)+\sigma\bigl(B_{\rho_{\#}}(x,\lambda r)\cap A\bigr) \nonumber\\[4pt] &&\hskip 0.30in =\sigma\bigl(B_{\rho_{\#}}(x_0,C_{\rho_{\#}}(1+\lambda)r)\bigr) -\sigma\bigl(B_{\rho_{\#}}(x,\lambda r)\bigr) +\sigma\bigl(B_{\rho_{\#}}(x,\lambda r)\cap A\bigr), \end{eqnarray}
which further implies that
\begin{eqnarray}\label{Mixed-19} \sigma\bigl(B_{\rho_{\#}}(x,\lambda r)\bigr) -(1-\gamma)\sigma\bigl(B_{\rho_{\#}}(x_0,C_{\rho_{\#}}(1+\lambda)r)\bigr) \leq\sigma\bigl(B_{\rho_{\#}}(x,\lambda r)\cap A\bigr). \end{eqnarray}
Recalling the second inclusion in \eqref{Mixed-17} and \eqref{Doub-2}, we obtain
\begin{eqnarray}\label{Mixed-20} \sigma\bigl(B_{\rho_{\#}}(x_0,C_{\rho_{\#}}(1+\lambda)r)\bigr) &\leq & \sigma\bigl(B_{\rho_{\#}}(x_0,C_{\rho_{\#}}^2(1+\lambda)r)\bigr) \nonumber\\[4pt] &\leq & C_{\sigma,\rho_{\#}}\bigl(\tfrac{C_{\rho_{\#}}^2(1+\lambda)} {\lambda}\bigr)^{D_\sigma}\sigma\bigl(B_{\rho_{\#}}(x,\lambda r)\bigr), \end{eqnarray}
where $C_{\sigma,\rho_{\#}}$, $D_\sigma$ are associated with $\sigma$, $\rho_{\#}$ as in \eqref{Doub-2}. Together, \eqref{Mixed-19} and \eqref{Mixed-20} yield
\begin{eqnarray}\label{Mixed-21} \sigma\bigl(B_{\rho_{\#}}(x,\lambda r)\bigr) \Bigl[1-C_{\sigma,\rho_{\#}}(1-\gamma) \Bigl(\tfrac{C_{\rho_{\#}}^2(1+\lambda)}{\lambda}\Bigr)^{D_{\sigma}}\Bigr] \leq\sigma\bigl(B_{\rho_{\#}}(x,\lambda r)\cap A\bigr). \end{eqnarray}
Also, from \eqref{Doub-2} we have that if $\lambda\in(0,1)$ then $\sigma\bigl(B_{\rho_{\#}}(x,r)\bigr)\leq C_{\sigma,\rho_{\#}}\lambda^{-D_{\sigma}} \sigma\bigl(B_{\rho_{\#}}(x,\lambda r)\bigr)$, thus
\begin{eqnarray}\label{Mixed-22} \sigma\bigl(B_{\rho_{\#}}(x,\lambda r)\bigr) \geq\min\Bigl\{1,\frac{\lambda^{D_{\sigma}}}{C_{\sigma,\rho_{\#}}}\Bigr\} \sigma\bigl(B_{\rho_{\#}}(x,r)\bigr),\qquad\forall\,\lambda>0. \end{eqnarray}
If we now we choose
\begin{eqnarray}\label{Mixed-22EE} \gamma(\lambda):=1-\frac{1}{2C_{\sigma,\rho_{\#}}} \Bigl(\frac{\lambda}{C_{\rho_{\#}}^2(1+\lambda)}\Bigr)^{D_{\sigma}}\in(0,1) \quad\mbox{ and }\quad c(\lambda):=\frac{1}{2}\min\Bigl\{1,\frac{\lambda^{D_{\sigma}}} {C_{\sigma,\rho_{\#}}}\Bigr\}>0, \end{eqnarray}
then \eqref{Mixed-21} and \eqref{Mixed-22} imply
\begin{eqnarray}\label{Mixed-23} \sigma\bigl(B_{\rho_{\#}}(x,\lambda r)\cap A\bigr) \geq\frac{1}{2}\sigma\bigl(B_{\rho_{\#}}(x,\lambda r)\bigr) \geq c(\lambda)\sigma\bigl(B_{\rho_{\#}}(x,r)\bigr), \qquad\forall\,\gamma\in\bigl[\gamma(\lambda),1\bigr). \end{eqnarray}
This proves {\it (5)}.
If $\sigma$ is Borel-regular, then Lebesgue's Differentiation Theorem holds in the current setting. Hence, there exists a set $F\subseteq E$ with $\sigma(F)=0$ and such that
\begin{eqnarray}\label{Mixed-24} \lim\limits_{r\to 0^+}\Bigl({\int{\mkern-19mu}-}_{B_{\rho_{\#}}(x,r)}{\mathbf{1}}_A\,d\sigma\Bigr) ={\mathbf{1}}_A(x),\qquad\forall\,x\in E\setminus F. \end{eqnarray}
In particular, for every $x\in A^\ast_\gamma\setminus F$ we have ${\mathbf{1}}_A(x)=\lim\limits_{r\to 0^+}\Bigl[ \frac{\sigma\bigl(B_{\rho_{\#}}(x,r)\cap A\bigr)} {\sigma\bigl(B_{\rho_{\#}}(x,r)\bigr)}\Bigr]\geq\gamma>0$, which implies that $A^\ast_\gamma\setminus F\subseteq A$, thus $A^\ast_\gamma\setminus A\subseteq F$. Consequently, since $A^\ast_\gamma\setminus A$ is $\sigma$-measurable, we obtain that $\sigma(A^\ast_\gamma\setminus A)=0$, proving {\it (6)}. Finally, the statement in {\it (7)} is an immediate consequence of \eqref{Mixed-12}. This concludes the proof of the proposition. \end{proof}
We continue to state and prove auxiliary lemmas in preparation for dealing with Theorem~\ref{appert}, advertised earlier. To state the lemma below, recall the region ${\mathcal{F}}_\kappa(A)$ from \eqref{reg-A1}.
\begin{lemma}\label{ap+YH} Let $({\mathscr{X}},\rho)$ be a quasi-metric space, $\mu$ a Borel measure on $({\mathscr{X}},\tau_\rho)$, $E$ a proper, nonempty, closed subset of $({\mathscr{X}},\tau_\rho)$ and $\sigma$ a Borel measure on
$(E,\tau_{\rho|_{E}})$ such that $(E,\rho\bigl|_E,\sigma)$ is a space of homogeneous type. If $u:{\mathscr{X}}\setminus E\to[0,\infty]$ is $\mu$-measurable, then for every $\kappa>0$ and every $\sigma$-measurable set $A\subseteq E$, one has
\begin{eqnarray}\label{Mix+FR} \int_A\Bigl(\int_{\Gamma_\kappa(x)}u(y)\,d\mu(y)\Bigr)\,d\sigma(x) &=& \int_{{\mathscr{X}}\setminus E}u(y)\sigma\bigl(A\cap\pi_y^\kappa\bigr)\,d\mu(y) \nonumber\\[4pt] &=& \int_{{\mathcal{F}}_\kappa(A)}u(y)\sigma\bigl(A\cap\pi_y^\kappa\bigr)\,d\mu(y). \end{eqnarray} \end{lemma}
\begin{proof} By Fubini's Theorem (and \eqref{Mixed-3}), we have
\begin{eqnarray}\label{Mix+FR-1} \int_A\Bigl(\int_{\Gamma_\kappa(x)}u(y)\,d\mu(y)\Bigr)\,d\sigma(x) &=& \int_{{\mathscr{X}}\setminus E}u(y)\Bigl( \int_A{\mathbf{1}}_{\pi^\kappa_y}(x)\,d\sigma(x)\Bigr)\,d\mu(y) \nonumber\\[4pt] &=& \int_{{\mathscr{X}}\setminus E}u(y)\sigma\bigl(A\cap\pi_y^\kappa\bigr)\,d\mu(y), \end{eqnarray}
proving the first equality in \eqref{Mix+FR}. The second equality in \eqref{Mix+FR} follows from \eqref{Mix+FR-1} and the fact that if $y\in{\mathscr{X}}\setminus E$ and $A\cap\pi^\kappa_y\not=\emptyset$ then $y\in{\mathcal{F}}_\kappa(A)$. \end{proof}
\begin{lemma}\label{ap+YH-2} Let $({\mathscr{X}},\rho)$ be a quasi-metric space, $\mu$ a Borel measure on $({\mathscr{X}},\tau_\rho)$, $E$ a proper, nonempty, closed subset of
$({\mathscr{X}},\tau_\rho)$, and $\sigma$ a Borel measure on $(E,\tau_{\rho|_{E}})$
such that $(E,\rho\bigl|_E,\sigma)$ is a space of homogeneous type. Fix two arbitrary numbers $\kappa,\kappa'>0$. Then there exist $\gamma\in(0,1)$ and a finite constant $C>0$ such that for every $\sigma$-measurable set $A\subseteq E$ there holds
\begin{eqnarray}\label{Mi+LV} \int_{A^\ast_\gamma}\Bigl(\int_{\Gamma_\kappa(x)}u(y)\,d\mu(y)\Bigr)\,d\sigma(x) \leq C\int_A\Bigl(\int_{\Gamma_{\kappa'}(x)}u(y)\,d\mu(y)\Bigr)\,d\sigma(x) \end{eqnarray}
for every function $u:{\mathscr{X}}\setminus E\to[0,\infty]$ which is $\mu$-measurable. \end{lemma}
\begin{proof} Recall the notation introduced in \eqref{REG-DDD}. We claim that
\begin{eqnarray}\label{Clsb} \begin{array}{c} \mbox{for every }\,\,\kappa,\kappa'>0\quad\mbox{there exist}\,\,\gamma\in(0,1)\, \mbox{ and }\,\,c>0\,\,\mbox{ such that } \\[4pt] \sigma\bigl(A\cap\pi_y^{\kappa'}\bigr)\geq c\,\sigma\bigl(A^\ast_\gamma\cap\pi_y^\kappa\bigr) \quad\forall\,A\subseteq E\mbox{ $\sigma$-measurable and } \forall\,y\in{\mathcal{F}}_\kappa(A^\ast_\gamma). \end{array} \end{eqnarray}
Assuming this claim for now, let $\kappa,\kappa'>0$ be arbitrary and let $\gamma$ and $c>0$ be as in \eqref{Clsb}. Then, if $A$ and $u$ satisfy the hypotheses of the proposition, starting with \eqref{Mix+FR} and using the fact that ${\mathcal{F}}_\kappa(A^\ast_\gamma)\subseteq{\mathscr{X}}\setminus E$ (itself a trivial consequence of \eqref{Tfs23}), we may write
\begin{eqnarray}\label{Mix+FR-5} \int_A\Bigl(\int_{\Gamma_{\kappa'}(x)}u(y)\,d\mu(y)\Bigr)\,d\sigma(x) &=& \int_{{\mathscr{X}}\setminus E}u(y)\sigma\bigl(A\cap\pi_y^{\kappa'}\bigr)\,d\mu(y) \nonumber\\[4pt] &\geq& \int_{{\mathcal{F}}_\kappa(A^\ast_\gamma)}u(y) \sigma\bigl(A\cap\pi_y^{\kappa'}\bigr)\,d\mu(y) \nonumber\\[4pt] &\geq& c\int_{{\mathcal{F}}_\kappa(A^\ast_\gamma)}u(y) \sigma\bigl(A^\ast_\gamma\cap\pi_y^\kappa\bigr)\,d\mu(y) \nonumber\\[4pt] &=& c\int_{A^\ast_\gamma}\Bigl(\int_{\Gamma_\kappa(x)}u(y)\,d\mu(y)\Bigr)\,d\sigma(x), \end{eqnarray}
where for the last equality in \eqref{Mix+FR-5} we applied Lemma~\ref{ap+YH} with $A^\ast_\gamma$ in place of $A$. Hence, to finish the proof of the proposition we are left with showing \eqref{Clsb}.
Suppose $\kappa,\kappa'>0$ are fixed and pick some $\gamma\in(0,1)$, to be made precise later. Also, fix $\eta\in(0,\min\,\{\kappa,\kappa'\})$ and for each $y\in{\mathcal{F}}_\kappa(A^\ast_\gamma)$ choose $y_\ast\in E$ and $\epsilon>0$ as in \eqref{Equ-1} (for $\eta$ as just indicated). Then $y_\ast$ satisfies the conditions in \eqref{Equ-1} corresponding to both $\kappa$ and $\kappa'$. As such, Lemma~\ref{lbDV} implies that the inclusions in \eqref{sgbr} hold for both $\kappa$ and $\kappa'$.
The fact that $y\in{\mathcal{F}}_\kappa(A^\ast_\gamma)$ entails $\pi^\kappa_y\cap A^\ast_\gamma\not=\emptyset$ which, when combined with \eqref{sgbr}, implies $B_{\rho_{\#}}\bigl(y_\ast,C_{\rho}(1+\kappa)\delta_E(y)\bigr)\cap A^\ast_\gamma\not=\emptyset$ hence, further, ${\rm dist}_{\rho_{\#}}(y_\ast,A^\ast_\gamma)<C_{\rho}(1+\kappa)\delta_E(y)$. Now, {\it (5)} in Proposition~\ref{DJrt} invoked with $\lambda:=\frac{\epsilon}{C_{\rho}(1+\kappa)}$, $x:=y_\ast$ and $r:=C_{\rho}(1+\kappa)\delta_E(y)$, guarantees the existence of some $\gamma_0=\gamma_0(\lambda)\in(0,1)$ with the property that
\begin{eqnarray}\label{sgbr-2} \frac{\sigma\bigl(B_{\rho_{\#}}(y_\ast,\epsilon\delta_E(y))\cap A\bigr)} {\sigma\bigl(B_{\rho_{\#}}(y_\ast,C_{\rho}(1+\kappa)\delta_E(y))\bigr)} \geq c=c(\kappa)>0\quad\mbox{ if }\gamma\in(\gamma_0,1). \end{eqnarray}
Hence, if we select $\gamma\in(\gamma_0,1)$ to begin with, the estimate in \eqref{sgbr-2} in concert with \eqref{sgbr} implies
\begin{eqnarray}\label{sgbr-3} \sigma\bigl(B_{\rho_{\#}}(y_\ast,\epsilon\delta_E(y))\cap A\bigr) \geq c\,\sigma\bigl(B_{\rho_{\#}}(y_\ast,C_{\rho}(1+\kappa)\delta_E(y))\bigr) \geq c\,\sigma\bigl(\pi^\kappa_y\bigr) \geq c\,\sigma\bigl(A^\ast_\gamma\cap \pi^\kappa_y\bigr). \end{eqnarray}
Since \eqref{sgbr} also holds with $\kappa$ replaced by $\kappa'$, we obtain from this and \eqref{sgbr-3} that
\begin{eqnarray}\label{sgbr-5} \sigma\bigl(A\cap \pi^{\kappa'}_y\bigr) \geq \sigma\bigl(B_{\rho_{\#}}(y_\ast,\epsilon\delta_E(y))\cap A\bigr) \geq c\,\sigma\bigl(A^\ast_\gamma\cap \pi^\kappa_y\bigr). \end{eqnarray}
This completes the proof of \eqref{Clsb} and, with it, the proof of the lemma. \end{proof}
\begin{lemma}\label{Biu+F} Let $({\mathscr{X}},\rho)$ be a quasi-metric space, $\mu$ a Borel measure on $({\mathscr{X}},\tau_\rho)$, $E$ a proper, nonempty, closed subset of $({\mathscr{X}},\tau_\rho)$, and $\sigma$ a Borel measure on
$(E,\tau_{\rho|_{E}})$ such that $(E,\rho\bigl|_E,\sigma)$ is a space of homogeneous type. Then for every $\kappa,\kappa'>0$ there exists a constant $C\in(0,\infty)$ such that
\begin{eqnarray}\label{Biu+F2} \int_E\Bigl(\int_{\Gamma_\kappa(x)}u(y)\,d\mu(y)\Bigr)f(x)\,d\sigma(x) \leq C\int_E\Bigl(\int_{\Gamma_{\kappa'}(x)}u(y)\,d\mu(y)\Bigr)(M_Ef)(x)\,d\sigma(x) \end{eqnarray}
for every function $u:{\mathscr{X}}\setminus E\to[0,\infty]$ that is $\mu$-measurable, and every function $f:E\to[0,\infty]$ that is $\sigma$-measurable. \end{lemma}
\begin{proof} Based on Fubini's Theorem (and \eqref{Mixed-3}), we may write
\begin{eqnarray}\label{Biu+F3} \int_E\Bigl(\int_{\Gamma_\kappa(x)}u(y)\,d\mu(y)\Bigr)f(x)\,d\sigma(x)&=& \int_{{\mathscr{X}}\setminus E}u(y)\Bigl( \int_E{\mathbf{1}}_{\pi^\kappa_y}(x)f(x)\,d\sigma(x)\Bigr)\,d\mu(y) \nonumber\\[4pt] &=& \int_{{\mathscr{X}}\setminus E}u(y)\sigma\bigl(\pi_y^\kappa\bigr) \Bigl({\int{\mkern-19mu}-}_{\pi_y^\kappa}f\,d\sigma\Bigr)\,d\mu(y), \end{eqnarray}
as well as
\begin{eqnarray}\label{Biu+F4A} \int_E\Bigl(\int_{\Gamma_{\kappa'}(x)}u(y)\,d\mu(y)\Bigr) (M_Ef)(x)\,d\sigma(x) = \int_{{\mathscr{X}}\setminus E}u(y)\sigma\bigl(\pi_y^\kappa\bigr) \Bigl({\int{\mkern-19mu}-}_{\pi_y^{\kappa'}}M_Ef\,d\sigma\Bigr)\,d\mu(y). \end{eqnarray}
Hence, in order to conclude \eqref{Biu+F2}, in light of \eqref{Biu+F3}, \eqref{Biu+F4A}, and \eqref{equiv11}, it suffices to show that there exists a constant $C_1\in(0,\infty)$ such that
\begin{eqnarray}\label{Biu+F4} {\int{\mkern-19mu}-}_{\pi_y^\kappa}f\,d\sigma \leq C_1{\int{\mkern-19mu}-}_{\pi_y^{\kappa'}}M_Ef\,d\sigma \qquad\mbox{for every $y\in{\mathscr{X}}\setminus E$}. \end{eqnarray}
To this end, fix some $y\in{\mathscr{X}}\setminus E$ and let $y_\ast\in E$, $\epsilon>0$ be such that \eqref{Equ-1} holds for some $\eta\in(0,\min\{\kappa,\kappa'\})$. Then \eqref{sgbr} holds when written both for $\kappa$ and $\kappa'$. In particular, for each $z\in\pi_y^{\kappa'}$ we have $\rho_{\#}(z,y_\ast)<C_{\rho}(1+\kappa')\delta_E(y)$ and, consequently,
\begin{eqnarray}\label{Biu+F5} \hskip -0.50in B_{\rho_{\#}}\bigl(y_\ast,C_{\rho}(1+\kappa)\delta_E(y)\bigr) &\subseteq & B_{\rho_{\#}}\bigl(z,C^2_{\rho}(1+\max\{\kappa,\kappa'\})\delta_E(y)\bigr) \nonumber\\[4pt] &\subseteq & B_{\rho_{\#}}\bigl(y_\ast,C^3_{\rho}(1+{\max\{\kappa,\kappa'\})\delta_E}(y)\bigr), \qquad\forall\,z\in\pi_y^{\kappa'}. \end{eqnarray}
Making now use of \eqref{sgbr}, \eqref{Biu+F5} and \eqref{Doub-2}, we obtain
\begin{eqnarray}\label{Biu+F6} {\int{\mkern-19mu}-}_{\pi_y^\kappa}f\,d\sigma &\leq & \frac{1}{\sigma(B_{\rho_{\#}}(y_\ast,\epsilon\delta_E(y)))} \int_{B_{\rho_{\#}}\bigl(y_\ast,C_{\rho}(1+\kappa)\delta_E(y)\bigr)}f\,d\sigma \nonumber\\[4pt] &\leq & C_{\sigma,\rho_{\#}}\bigl(C^3_{\rho}\epsilon^{-1}(1+\max\{\kappa,\kappa'\}) \bigr)^{D_\sigma} {\int{\mkern-19mu}-}_{B_{\rho_{\#}}(z,C^2_{\rho}(1+\max\{\kappa,\kappa'\})\delta_E(y))} f\,d\sigma \nonumber\\[4pt] &\leq & C_{\sigma,\rho_{\#}}\bigl(C^3_{\rho}\epsilon^{-1} (1+\max\{\kappa,\kappa'\})\bigr)^{D_\sigma} M_Ef(z),\qquad\forall\,z\in\pi_y^{\kappa'}, \end{eqnarray}
where $C_{\sigma,\rho_{\#}}$, $D_\sigma$ are the constants associated with $\sigma$, $\rho_{\#}$ as in \eqref{Doub-2}. Thus, if we now set $C_1:=C_{\sigma,\rho_{\#}} \bigl(C^3_{\rho}\epsilon^{-1}(1+\max\{\kappa,\kappa'\})\bigr)^{D_\sigma}$ then
\begin{eqnarray}\label{Biu+F7} {\int{\mkern-19mu}-}_{\pi_y^\kappa}f\,d\sigma \leq C_1\inf\limits_{z\in\pi_y^{\kappa'}}[M_Ef(z)] \leq C_1{\int{\mkern-19mu}-}_{\pi_y^{\kappa'}}M_Ef\,d\sigma, \end{eqnarray}
proving \eqref{Biu+F4}, and finishing the proof of the lemma. \end{proof}
We are now prepared to state and prove the following equivalence result for the quasi-norms of the mixed norm spaces associated with different apertures (of the nontangential approach regions).
\begin{theorem}\label{appert} Let $({\mathscr{X}},\rho)$ be a quasi-metric space, $\mu$ a Borel measure on $({\mathscr{X}},\tau_\rho)$, $E$ a proper, nonempty, closed subset of
$({\mathscr{X}},\tau_\rho)$, and $\sigma$ a Borel measure on $(E,\tau_{\rho|_{E}})$
such that $(E,\rho\bigl|_E,\sigma)$ is a space of homogeneous type. Also, fix two indices $p,q\in(0,\infty]$ with the convention that $q=\infty$ if $p=\infty$. Then for each $\kappa,\kappa'>0$ there holds
\begin{eqnarray}\label{Mixed-11CC}
\|u\|_{L^{(p,q)}({\mathscr{X}},E,\mu,\sigma;\kappa)}\approx
\|u\|_{L^{(p,q)}({\mathscr{X}},E,\mu,\sigma;\kappa')}, \end{eqnarray}
uniformly for $\mu$-measurable functions $u:{\mathscr{X}}\setminus E\to\overline{\mathbb{R}}$.
Hence, in particular, for each $p,q\in(0,\infty)$, there holds
\begin{eqnarray}\label{Mixed-11}
\Bigl(\int_E\Bigl[\int_{\Gamma_\kappa(x)}|u(y)|^q\,d\mu(y)\Bigr]^{p/q} d\sigma(x)\Bigr)^{1/p}\approx\Bigl(\int_E\Bigl[\int_{\Gamma_{\kappa'}(x)}
|u(y)|^q\,d\mu(y)\Bigr]^{p/q}d\sigma(x)\Bigr)^{1/p}, \end{eqnarray}
uniformly for $\mu$-measurable functions $u:{\mathscr{X}}\setminus E\to\overline{\mathbb{R}}$. \end{theorem}
Before presenting the proof of this theorem we shall comment on the nature of the limiting case $p=\infty$, $q\in(0,\infty)$ of \eqref{Mixed-11}. This clarifies the comment at the bottom of page 183 in \cite{STEIN}.
\begin{remark}\label{Fv99-T1} In the context of Theorem~\ref{appert}, if $q\in(0,\infty)$, in general it is not true that
\begin{eqnarray}\label{Mixed-11-N}
\sup_{x\in E}\Bigl(\int_{\Gamma_\kappa(x)}|u(y)|^q\,d\mu(y)\Bigr)^{1/q} \approx
\sup_{x\in E}\Bigl(\int_{\Gamma_{\kappa'}(x)}|u(y)|^q\,d\mu(y)\Bigr)^{1/q}. \end{eqnarray}
To see that this equivalence might fail, consider the case when ${\mathscr{X}}:={\mathbb{R}}^2$, $E:={\mathbb{R}}\equiv\partial{\mathbb{R}}^2_{+}$, and take $\kappa:=\sqrt{2}$, $\kappa'\in(0,\sqrt{2})$. Also, without loss of generality, assume that $q=1$ and consider $u:{\mathscr{X}}\setminus E\to\overline{\mathbb{R}}$ given by
\begin{eqnarray}\label{U-Rg89} u(x,y):=\left\{ \begin{array}{l} x^{-1}\,\,\mbox{ if $x>0$ and $x<y<x+1$}, \\[4pt] 0\,\,\mbox{ otherwise}. \end{array} \right. \end{eqnarray}
Then
\begin{eqnarray}\label{Mix-Tg.5}
\sup_{z\in{\mathbb{R}}}\Bigl(\int_{\Gamma_\kappa(z)}|u(x,y)|\,dxdy\Bigr)
=\int_{|x|<y}|u(x,y)|\,dxdy=\int_{0<x<y<x+1}x^{-1}\,dxdy=\infty, \end{eqnarray}
whereas for each $z\in(0,\infty)$, elementary geometry gives that
\begin{eqnarray}\label{Mix-Tg.6}
\int_{\Gamma_{\kappa'}(z)}|u(x,y)|\,dxdy\leq Cz^{-1}\cdot {\rm Area}\{(x,y)\in\Gamma_{\kappa'}(z):\,0<x<y<x+1\}\leq C, \end{eqnarray}
for some $C=C(\kappa')\in(0,\infty)$. This shows that
$\sup_{z\in{\mathbb{R}}}\Bigl(\int_{\Gamma_{\kappa'}(z)}|u(x,y)|\,dxdy\Bigr)<\infty$, hence \eqref{Mixed-11-N} fails in this case. \end{remark}
We now turn to the
\vskip 0.08in \begin{proof}[Proof of Theorem~\ref{appert}] Let the real numbers $\kappa,\kappa'>0$ be arbitrary and fixed. Then, recalling \eqref{sp-sq}, it follows that the equivalence in \eqref{Mixed-11CC} is proved once we show that there exists a finite constant $C=C(\kappa,\kappa')>0$ such that for every $\mu$-measurable function $u:{\mathscr{X}}\setminus E\to\overline{\mathbb{R}}$ we have
\begin{eqnarray}\label{Mixed-11BB}
\|{\mathscr{A}}_{q,\kappa'}u\|_{L^p(E,\sigma)}
\leq C\|{\mathscr{A}}_{q,\kappa}u\|_{L^p(E,\sigma)}, \end{eqnarray}
with the understanding that, when $q=\infty$, the $q$-th power integral of $u$ over nontangential approach regions is replaced by the nontangential maximal operator of $u$ (cf. \eqref{Mixed-N}). We proceed by dividing up the proof of \eqref{Mixed-11BB} into a number of cases.
\vskip 0.08in \noindent{\it Case~I: $0<p<q<\infty$.} For $\lambda>0$ arbitrary define
\begin{eqnarray}\label{Shcd} A:=\bigl\{x\in E:\,({\mathscr{A}}_{q,\kappa}u)(x)\leq\lambda\bigr\}. \end{eqnarray}
By \eqref{Mixed-7A} we have that $A$ is closed in $(E,\tau_{\rho|_{E}})$, hence $A^\ast_\gamma\subseteq A$ for every $\gamma\in(0,1)$ by virtue of {\it (4)} in Proposition~\ref{DJrt}. Let $\gamma=\gamma(\kappa,\kappa')\in(0,1)$ be such that \eqref{Mi+LV} holds. Then
\begin{eqnarray}\label{Shcd-2} &&\hskip -0.40in \sigma\bigl(\{x\in E:\,({\mathscr{A}}_{q,\kappa'}u)(x)>\lambda\}\bigr) \leq\sigma\bigl(E\setminus A^\ast_\gamma) +\sigma\bigl(\{x\in A^\ast_\gamma:\,({\mathscr{A}}_{q,\kappa'}u)(x)>\lambda\}\bigr) \nonumber\\[4pt] &&\hskip 0.60in \leq\frac{C}{1-\gamma}\sigma(E\setminus A) +\frac{1}{\lambda^q}\int_{A^\ast_\gamma}({\mathscr{A}}_{q,\kappa'}u)(x)^q\,d\sigma(x) \nonumber\\[4pt] &&\hskip 0.60in \leq \frac{C}{1-\gamma}\sigma\bigl(\{x\in E:\,({\mathscr{A}}_{q,\kappa}u)(x)>\lambda\}\bigr) +\frac{C}{\lambda^q}\int_A({\mathscr{A}}_{q,\kappa}u)(x)^q\,d\sigma(x). \end{eqnarray}
For the second inequality in \eqref{Shcd-2} we used {\it (3)} in Proposition~\ref{DJrt} and Tschebyshev's inequality, while for the last inequality we used \eqref{Shcd} and \eqref{Mi+LV} (with $\kappa$ and $\kappa'$ interchanged). Thus, if we multiply the inequality resulting from \eqref{Shcd-2} by $p\lambda^{p-1}$ and then integrate in $\lambda\in(0,\infty)$, we obtain
\begin{eqnarray}\label{Shcd-3}
\|{\mathscr{A}}_{q,\kappa'}u\|^p_{L^p(E,\sigma)}
\leq \frac{C}{1-\gamma}\|{\mathscr{A}}_{q,\kappa}u\|^p_{L^p(E,\sigma)} +C\int_0^\infty\lambda^{p-q-1}\Bigl(\int_{\{{\mathscr{A}}_{q,\kappa}u\leq\lambda\}} ({\mathscr{A}}_{q,\kappa}u)^q\,d\sigma\Bigr)\,d\lambda. \end{eqnarray}
By Fubini's Theorem, we further write
\begin{eqnarray}\label{Shcd-4} \int_0^\infty\lambda^{p-q-1}\Bigl(\int_{\{{\mathscr{A}}_{q,\kappa}u\leq\lambda\}} ({\mathscr{A}}_{q,\kappa}u)^q\,d\sigma\Bigr)\,d\lambda &=& \int_E\Bigl(\int_{({\mathscr{A}}_{q,\kappa}u)(x)}^\infty\lambda^{p-q-1}\,d\lambda\Bigr) ({\mathscr{A}}_{q,\kappa}u)(x)^q\,d\sigma(x) \nonumber\\[4pt]
&=& (q-p)^{-1}\|{\mathscr{A}}_{q,\kappa}u\|^p_{L^p(E,\sigma)}, \end{eqnarray}
given that we are currently assuming that $p<q$. In concert, \eqref{Shcd-3} and \eqref{Shcd-4} now yield \eqref{Mixed-11BB} in the case when $q\in(0,\infty)$ and $0<p<q$.
\vskip 0.08in \noindent{\it Case~II: $p=q\in(0,\infty)$.} Combining \eqref{Mix+FR} (corresponding to $A=E$ and applied twice) with \eqref{equiv11} (applied for every $y\in{\mathscr{X}}\setminus E$), we obtain that
\begin{eqnarray}\label{Shcd-5}
\int_E\Bigl(\int_{\Gamma_{\kappa'}(x)}|u(y)|^p\,d\mu(y)\Bigr)\,d\sigma(x)
&=&\int_{{\mathscr{X}}\setminus E} |u(y)|^p\sigma(\pi^{\kappa'}_y)\,d\mu(y) \nonumber\\[4pt]
&\approx&\int_{{\mathscr{X}}\setminus E} |u(y)|^p\sigma(\pi^{\kappa}_y)\,d\mu(y) \nonumber\\[4pt]
&=&\int_E\Bigl(\int_{\Gamma_\kappa(x)}|u(y)|^p\,d\mu(y)\Bigr)\,d\sigma(x), \end{eqnarray}
and the desired conclusion follows. \vskip 0.08in
\noindent{\it Case~III: $0<q<p<\infty$.} Let $(p/q)'$ denote the H\"older conjugate of $p/q\in(1,\infty)$. By using Riesz's duality theorem for Lebesgue spaces, then
Lemma~\ref{Biu+F} (with $u$ replaced by $|u|^q$), and then H\"older's inequality, we may write
\begin{eqnarray}\label{Shcd-6} &&\hskip -0.50in
\left(\int_E\Bigl(\int_{\Gamma_\kappa(x)}|u(y)|^q\,d\mu(y)\Bigr)^{p/q}\,d\sigma(x)
\right)^{q/p}=\Bigl\|\int_{\Gamma_\kappa(x)}|u|^q\,d\mu\Bigr\|_{L^{p/q}_x(E,\sigma)} \nonumber\\[4pt] &&\hskip 0.50in =\sup\limits_{\overset{f\in L^{(p/q)'}(E,\sigma)}
{f\geq 0,\,\,\|f\|_{L^{(p/q)'}(E,\sigma)}\leq 1}}
\left[\int_E\Bigl(\int_{\Gamma_\kappa(x)}|u(y)|^q\,d\mu(y)\Bigr)f(x)\,d\sigma(x)\right] \nonumber\\[4pt] &&\hskip 0.50in \leq C\sup\limits_{\overset{f\in L^{(p/q)'}(E,\sigma)}
{f\geq 0,\,\|f\|_{L^{(p/q)'}(E,\sigma)}\leq 1}}
\left[\int_E\Bigl(\int_{\Gamma_{\kappa'}(x)}|u(y)|^q\,d\mu(y)\Bigr)(M_Ef)(x) \,d\sigma(x)\right] \nonumber\\[4pt] &&\hskip 0.50in \leq C\sup\limits_{\overset{f\in L^{(p/q)'}(E,\sigma)}
{f\geq 0,\,\|f\|_{L^{(p/q)'}(E,\sigma)}\leq 1}}\!\!\!\Bigl[\,
\left(\int_E\Bigl(\int_{\Gamma_{\kappa'}(x)}|u(y)|^q\,d\mu(y)\Bigr)^{p/q}\,d\sigma(x) \right)^{q/p}\times \nonumber\\[4pt] &&\hskip 2.50in \times\Bigl(\int_E(M_Ef)(x)^{(p/q)'}\,d\sigma(x)\Bigr)^{1/(p/q)'}\Bigr] \nonumber\\[4pt] &&\hskip 0.50in
\leq C\left(\int_E\Bigl(\int_{\Gamma_{\kappa'}(x)}|u(y)|^q\,d\mu(y)\Bigr)^{p/q}\, d\sigma(x)\right)^{q/p}, \end{eqnarray}
where for the last inequality in \eqref{Shcd-6} we used the boundedness of the maximal operator $M_E$ on $L^r(E,\sigma)$ for $r:=(p/q)'\in(1,\infty)$. This completes the proof of \eqref{Mixed-11BB} when $0<q<p<\infty$.
\vskip 0.08in \noindent{\it Case~IV: $0<p<\infty$, $q=\infty$.} Fix $\lambda>0$ and introduce (recall \eqref{Mixed-N})
\begin{eqnarray}\label{cal-O} {\mathcal{O}}_\kappa:=\bigl\{x\in E:\,({\mathcal{N}}_\kappa u)(x)>\lambda\bigr\}, \qquad {\mathcal{O}}_{\kappa'}:=\bigl\{x\in E:\,({\mathcal{N}}_{\kappa'}u)(x)>\lambda\bigr\}. \end{eqnarray}
Hence, the desired conclusion follows as soon as we show that there exists $C\in(0,\infty)$, independent of $u$ and $\lambda$, with the property that $\sigma({\mathcal{O}}_{\kappa'})\leq C\sigma({\mathcal{O}}_{\kappa})$. In turn, by virtue of {\it (3)} in Proposition~\ref{DJrt}, this follows once we prove that that there exists $\gamma\in(0,1)$ such that
\begin{eqnarray}\label{incl-Cl} {\mathcal{O}}_{\kappa'}\subseteq E\setminus(E\setminus{\mathcal{O}}_{\kappa})^*_\gamma. \end{eqnarray}
To justify this inclusion, fix $\eta\in(0,\min\,\{\kappa,\kappa'\})$ and assume that $x\in{\mathcal{O}}_{\kappa'}$ is an arbitrary point.
Then there exists $y\in\Gamma_{\kappa'}(x)$ for which $|u(y)|>\lambda$ and we select $y_\ast\in E$ and $\epsilon\in(0,1)$ as in \eqref{Equ-1} (for $\eta$ as specified above). In particular, $\rho_{\#}(y,y_\ast)<(1+\eta)\delta_E(y)$. Observe from \eqref{sgbr} and \eqref{cal-O} that in this scenario we have
\begin{eqnarray}\label{ZDel} E\cap B_{\rho_{\#}}\bigl(y_\ast,\epsilon\delta_E(y)\bigr)\subseteq\pi^{\kappa}_y \subseteq{\mathcal{O}}_{\kappa} \end{eqnarray}
and we also claim that
\begin{eqnarray}\label{ZDel-2} E\cap B_{\rho_{\#}}\bigl(y_\ast,\epsilon\delta_E(y)\bigr)\subseteq E\cap B_{\rho_{\#}}\bigl(x,C_{\rho}(1+\kappa')\delta_E(y)\bigr). \end{eqnarray}
To see this, recall that $\epsilon\in(0,1)$ and note that if $z\in E$ satisfies $\rho_{\#}(z,y_\ast)<\delta_E(y)$ then
\begin{eqnarray}\label{XX-YZ} \rho_{\#}(x,z) &\leq & C_{\rho}\,\max\,\{\rho_{\#}(x,y),\rho_{\#}(y,z)\} \nonumber\\[4pt] & \leq & C_{\rho}\,\max\,\Bigl\{(1+\kappa')\delta_E(y),\, C_{\rho}\,\max\,\{\rho_{\#}(y,y_\ast),\rho_{\#}(y_\ast,z)\}\Bigr\} \nonumber\\[4pt] & \leq & C_{\rho}\,\max\,\Bigl\{(1+\kappa')\delta_E(y),\, C_{\rho}\,\max\,\{(1+\alpha)\delta_E(y),\delta_E(y)\}\Bigr\} \nonumber\\[4pt] & = & C_{\rho}(1+\kappa')\delta_E(y), \end{eqnarray}
proving \eqref{ZDel-2}. In concert, \eqref{ZDel} and \eqref{ZDel-2} yield
\begin{eqnarray}\label{inCL} E\cap B_{\rho_{\#}}\bigl(y_\ast,\delta_E(y)\bigr)\subseteq {\mathcal{O}}_\kappa\cap B_{\rho_{\#}}\bigl(x,C_{\rho}(1+\kappa')\delta_E(y)\bigr). \end{eqnarray}
Let us also observe that
\begin{eqnarray}\label{XX-YZ342} \rho_{\#}(x,y_\ast) &\leq & C_{\rho}\,\max\,\{\rho_{\#}(x,y),\rho_{\#}(y,y_\ast)\} \nonumber\\[4pt] & \leq & C_{\rho}\,\max\,\Bigl\{(1+\kappa')\delta_E(y), (1+\eta)\delta_E(y)\Bigr\}=C_{\rho}(1+\kappa')\delta_E(y). \end{eqnarray}
Then, for some sufficiently small $c\in(0,1)$ which depends only on $\kappa,\kappa'$ and background geometrical characteristics, we may write
\begin{eqnarray}\label{fst-E} \frac{\sigma\Bigl({\mathcal{O}}_\kappa\cap B_{\rho_{\#}}\bigl(x,C_{\rho}(1+\kappa')\delta_E(y)\bigr)\Bigr)} {\sigma\Bigl(E\cap B_{\rho_{\#}}\bigl(x,C_{\rho}(1+\kappa')\delta_E(y)\bigr)\Bigr)} \geq \frac{\sigma\Bigl(E\cap B_{\rho_{\#}}\bigl(y_\ast,\epsilon\delta_E(y)\bigr)\Bigr)} {\sigma\Bigl(E\cap B_{\rho_{\#}}\bigl(x,C_{\rho}(1+\kappa')\delta_E(y)\bigr)\Bigr)} \geq c, \end{eqnarray}
where the first inequality follows from \eqref{inCL}, while the second inequality is a consequence of \eqref{XX-YZ342} and the fact that
$(E,\rho\bigl|_{E},\sigma)$ is a space of homogeneous type (cf. \eqref{Doub-2}). In particular, if we set $r:=C_{\rho}(1+\kappa')\delta_E(y)$, then
\begin{eqnarray}\label{O-c} \frac{\sigma\bigl((E\setminus{\mathcal{O}}_\kappa)\cap B_{\rho_{\#}}(x,r)\bigr)}{\sigma\bigl(E\cap B_{\rho_{\#}}(x,r)\bigr)}\leq 1-c. \end{eqnarray}
Thus, if we select $\gamma$ such that $1-c<\gamma<1$, then \eqref{O-c} entails $x\notin(E\setminus{\mathcal{O}}_\kappa)^*_\gamma$. This proves the claim \eqref{incl-Cl}, and finishes the treatment of the current case.
\vskip 0.08in \noindent{\it Case~V: $p=q=\infty$.} In this case, the desired conclusion follows upon observing that if $u:{\mathscr{X}}\setminus E\to\overline{\mathbb{R}}$ is a $\mu$-measurable function then
\begin{eqnarray}\label{Gf-5tFF}
\Bigl\|E\ni x\mapsto
\|u\|_{L^\infty(\Gamma_\kappa(x),\mu)}\Bigr\|_{L^\infty(E,\sigma)}
=\|u\|_{L^\infty({\mathscr{X}}\setminus E,\mu)}. \end{eqnarray}
Indeed, the inequality $\bigl\|\|u\|_{L^\infty(\Gamma_\kappa(x),\mu)}\bigr\|_{L_x^\infty(E,\sigma)}
\leq\|u\|_{L^\infty({\mathscr{X}}\setminus E,\mu)}$ is a simple consequence of the fact that $\Gamma_\kappa(x)\subseteq{\mathscr{X}}\setminus E$ for each $x\in E$. In the opposite direction, if $M$ denotes the left-hand side of \eqref{Gf-5tFF}, then there exists a $\sigma$-measurable set $F\subseteq E$ satisfying
$\sigma(F)=0$ and $\|u\|_{L^\infty(\Gamma_\kappa(x),\mu)}\leq M$ for every
$x\in E\setminus F$. Since $(E,\rho|_{E})$ is geometrically doubling, so is
$E\setminus F$ when equipped with $\rho|_{E\setminus F}$, hence separable as a topological space. Consequently, given that $E\setminus F$ is dense in $E$, it follows that there exists a countable subset $A:=\{x_j\}_{j\in{\mathbb{N}}}$ of $E\setminus F$ which is dense in $E$. Now, for each $j\in{\mathbb{N}}$ there exists $N_j\subseteq\Gamma_\kappa(x_j)$, null-set for $\mu$, such that
$|u(x)|\leq M$ for every $x\in\Gamma_\kappa(x_j)$. Thus, $N:=\cup_{j\in{\mathbb{N}}}N_j\subseteq{\mathscr{X}}\setminus E$
is a null-set for $\mu$ and $|u(x)|\leq M$ for every point $x$ belonging to
\begin{eqnarray}\label{Hdsss-YU} \Bigl(\bigcup\limits_{j\in{\mathbb{N}}}\Gamma_\kappa(x_j)\Bigr) \setminus N={\mathcal{F}}_\kappa(A)\setminus N ={\mathcal{F}}_\kappa(\overline{A})\setminus N ={\mathcal{F}}_\kappa(E)\setminus N=({\mathscr{X}}\setminus E)\setminus N, \end{eqnarray}
where the second equality follows from $(i)$ in Lemma~\ref{T-LL.2}, and the last equality is a consequence of \eqref{Tfs23} and the fact that $E$
is a closed subset of $({\mathscr{X}},\tau_\rho)$. Hence, $\|u\|_{L^\infty({\mathscr{X}}\setminus E,\mu)}\leq M$, as desired. This finishes the justification of \eqref{Gf-5tFF} and finishes the proof of the theorem. \end{proof}
\subsection{Estimates relating the Lusin and Carleson operators} \label{SSect:5.2}
We now introduce a Carleson operator $\mathfrak{C}$ and show how it can be used instead of the area operator $\mathscr{A}$ to provide an equivalent quasi-norm for the mixed norm spaces. This is essential in Subsection~\ref{SSect:5.4}, and it is achieved by combining Theorem~\ref{appert} with a good $\lambda$ inequality, as in Theorem~3 of~\cite{CoMeSt}.
Let $({\mathscr{X}},\rho)$ be a quasi-metric space, $\mu$ a Borel measure on $({\mathscr{X}},\tau_\rho)$, $E$ a nonempty, proper, closed subset of $({\mathscr{X}},\tau_\rho)$, and
$\sigma$ a measure on $E$ such that $(E,\rho\bigl|_E,\sigma)$ is a space of homogeneous type. For each index $q\in(0,\infty)$ and constant $\kappa\in(0,\infty)$, recall the $L^q$-based Lusin (or area) operator ${\mathscr{A}}_{q,\kappa}$ from \eqref{sp-sq}, and now define the $L^q$-based {\tt Carleson operator} ${\mathfrak{C}}_{q,\kappa}$ for all $\mu$-measurable functions $u:{\mathscr{X}}\setminus E\to\overline{\mathbb{R}}:=[-\infty,+\infty]$ by
\begin{eqnarray}\label{ktEW-7} ({\mathfrak{C}}_{q,\kappa}u)(x):=\sup\limits_{\Delta\subseteq E,\,x\in\Delta}
\Bigl(\tfrac{1}{\sigma(\Delta)}\int_{{\mathcal{T}}_\kappa(\Delta)}|u(y)|^q \sigma(\pi^{\kappa}_y)\,d\mu(y)\Bigr)^{\frac{1}{q}},\qquad\forall\,x\in E, \end{eqnarray}
where $\pi^{\kappa}_y$ is from \eqref{reg-A2}, the supremum is taken over {\tt surface balls}, i.e., sets of the form
\begin{eqnarray}\label{ktEW-7UU} \Delta:=\Delta(y,r):=E\cap B_{\rho_{\#}}(y,r),\qquad y\in E,\quad r>0 \end{eqnarray}
containing $x$, and ${\mathcal{T}}_\kappa(\Delta)$ is the tent region over $\Delta$ from \eqref{reg-A1}.
The following theorem extends the result on ${\mathbb{R}}^{n+1}_+$ from \cite[Theorem~3, p.\,318]{CoMeSt}. To state it, consider a measure space $(E,\sigma)$, and for each $p\in(0,\infty)$ and $r\in(0,\infty]$, let $L^{p,r}(E,\sigma)$ denote the Lorentz space equipped with the quasi-norm
\begin{eqnarray}\label{TgEE+56}
&& \|f\|_{L^{p,r}(E,\sigma)}:=\left(\int_0^\infty\lambda^r\sigma\bigl(
\left\{x\in E:\,|f(x)|>\lambda\right\}\bigr)^{r/p}\,\frac{d\lambda}{\lambda}\right)^{1/r}, \quad\mbox{ if }\,\,r<\infty, \\[4pt]
&& \|f\|_{L^{p,\infty}(E,\sigma)}:=\sup_{\lambda>0}\Bigl[\lambda\,\sigma\left(
\left\{x\in E:\,|f(x)|>\lambda\right\}\right)^{1/p}\Bigr]\quad\mbox{ if }\,\,r=\infty. \label{TgEE+56BBB} \end{eqnarray}
Note that $L^{p,p}(E,\sigma)=L^p(E,\sigma)$ for each $p\in(0,\infty)$. Also, given a quasi-metric space $({\mathscr{X}},\rho)$, call a Borel measure $\mu$ on $({\mathscr{X}},\tau_\rho)$ {\tt locally finite} when $\mu\bigl(B_\rho(x,r)^\circ\bigr)<\infty$ for all $x\in{\mathscr{X}}$ and $r>0$, where the interior is taken in the topology $\tau_\rho$.
\begin{theorem}\label{AsiC} Let $({\mathscr{X}},\rho)$ be a quasi-metric space, $\mu$ be a locally finite Borel measure on $({\mathscr{X}},\tau_\rho)$, and assume that $E$ is a proper, nonempty, closed subset of $({\mathscr{X}},\tau_\rho)$,
and $\sigma$ a measure on $E$ such that $(E,\rho\bigl|_E,\sigma)$ is a space of homogeneous type. Fix $q\in(0,\infty)$ and $\kappa>0$. Then the following estimates hold.
\begin{enumerate} \item[(1)] For each $p\in(0,\infty)$ there exists $C\in(0,\infty)$ such that
$\|{\mathscr{A}}_{q,\kappa}u\|_{L^p(E,\sigma)}\leq C
\|{\mathfrak{C}}_{q,\kappa}u\|_{L^p(E,\sigma)}$ for every $\mu$-measurable function $u:{\mathscr{X}}\setminus E\to\overline{\mathbb{R}}$.
\item[(2)] For each $p\in(q,\infty)$ and each $r\in(0,\infty]$ there exists a constant $C\in(0,\infty)$
such that $\|{\mathfrak{C}}_{q,\kappa}u\|_{L^{p,r}(E,\sigma)}
\leq C\|{\mathscr{A}}_{q,\kappa}u\|_{L^{p,r}(E,\sigma)}$ for every $\mu$-measurable function $u:{\mathscr{X}}\setminus E\to\overline{\mathbb{R}}$.
\item[(3)] Corresponding to the end-point cases $p=q$ and $p=\infty$ in $(2)$, there exists $C\in(0,\infty)$ such that for every $\mu$-measurable function $u:{\mathscr{X}}\setminus E\to\overline{\mathbb{R}}$ the following estimates hold:
\begin{eqnarray}\label{Tgbcc-7Y} \hskip -0.20in
\|{\mathfrak{C}}_{q,\kappa}u\|_{L^{q,\infty}(E,\sigma)}
\leq C\|{\mathscr{A}}_{q,\kappa}u\|_{L^q(E,\sigma)}\,\,\mbox{ and }\,\,
\|{\mathfrak{C}}_{q,\kappa}u\|_{L^{\infty}(E,\sigma)}
\leq C\|{\mathscr{A}}_{q,\kappa}u\|_{L^\infty(E,\sigma)}. \end{eqnarray} \end{enumerate}
In particular,
\begin{eqnarray}\label{sbrn}
\|{\mathscr{A}}_{q,\kappa}u\|_{L^p(E,\sigma)}
\approx\|{\mathfrak{C}}_{q,\kappa}u\|_{L^p(E,\sigma)} \quad\mbox{ for each }\,p\in(q,\infty), \end{eqnarray}
uniformly in $u:{\mathscr{X}}\setminus E\to\overline{\mathbb{R}}$, $\mu$-measurable function. \end{theorem}
\begin{proof} Fix $q\in(0,\infty)$ and define
\begin{eqnarray}\label{CPP-77} c_q:=\left\{ \begin{array}{l} 2^{(1/q)-1}\,\,\mbox{ if }\,\,q<1, \\[4pt] 1\,\,\mbox{ if }\,\,q\geq 1. \end{array} \right. \end{eqnarray}
Also, fix an arbitrary $\mu$-measurable function $u:{\mathscr{X}}\setminus E\to\overline{\mathbb{R}}$. We claim that the following good-$\lambda$ inequality is valid:
\begin{eqnarray}\label{kbFF} \begin{array}{c} \forall\,\kappa>0,\,\,\,\exists\,\kappa'>\kappa,\,\,\, \exists\,c\in(0,\infty)\, \mbox{ such that $\forall\,\gamma\in(0,1]$, $\forall\,\lambda\in(0,\infty)$, there holds} \\[8pt] \sigma\bigl(\{x\in E:({\mathscr{A}}_{q,\kappa} u)(x)>2c_q\lambda, ({\mathfrak{C}}_{q,\kappa}u)(x)\leq\!\gamma\lambda\}\bigr) \leq c\,\gamma^q\sigma\bigl(\{x\in E:\, ({\mathscr{A}}_{q,\kappa'}u)(x)>\lambda\}\bigr), \end{array} \end{eqnarray}
where the constant $c\in(0,\infty)$ is independent of $u$. Suppose for now that the above claim is true. Then, if $\kappa>0$ is fixed, let $\kappa',c$ be as in \eqref{kbFF}. Hence, for each fixed $\gamma\in(0,1]$ and every $\lambda>0$ we have
\begin{eqnarray}\label{kbFF-2} &&\hskip -0.50in \sigma\bigl(\{x\in E:\,({\mathscr{A}}_{q,\kappa}u)(x)>2c_q\lambda\}\bigr) \\[4pt] && \hskip 0.30in \leq\sigma\bigl(\{x\in E:\,({\mathfrak{C}}_{q,\kappa}u)(x)>\gamma\lambda\}\bigr) +c\,\gamma^q\sigma\bigl(\{x\in E:\,({\mathscr{A}}_{q,\kappa'}u)(x)>\lambda\}\bigr). \nonumber \end{eqnarray}
Thus, if we multiply the inequality in \eqref{kbFF-2} by $p\lambda^{p-1}$ and then integrate in $\lambda\in(0,\infty)$, we obtain
\begin{eqnarray}\label{kbFF-3}
(2c_q)^{-p}\|{\mathscr{A}}_{q,\kappa}u\|_{L^p(E,\sigma)}^p
\leq\gamma^{-p}\|{\mathfrak{C}}_{q,\kappa}u\|_{L^p(E,\sigma)}^p
+c\,\gamma^q\|{\mathscr{A}}_{q,\kappa'}u\|_{L^p(E,\sigma)}^p, \qquad\forall\,\gamma\in(0,1]. \end{eqnarray}
Since from Theorem~\ref{appert} we know that there exists a finite constant $C>0$ depending only on $\kappa,\kappa',p,q$ and geometry,
such that $\|{\mathscr{A}}_{q,\kappa'}u\|_{L^p(E,\sigma)}
\leq C\|{\mathscr{A}}_{q,\kappa}u\|_{L^p(E,\sigma)}$, we arrive at
\begin{eqnarray}\label{kbFF-3BB}
(2c_q)^{-p}\|{\mathscr{A}}_{q,\kappa}u\|_{L^p(E,\sigma)}^p
\leq\gamma^{-p}\|{\mathfrak{C}}_{q,\kappa}u\|_{L^p(E,\sigma)}^p
+c\,C^p\,\gamma^q\|{\mathscr{A}}_{q,\kappa}u\|_{L^p(E,\sigma)}^p, \quad\forall\,\gamma\in(0,1]. \end{eqnarray}
In order to hide the last term in the right-hand side into the left-hand side, fix a point $x_0\in E$ and, for each $j\in{\mathbb{N}}$, consider
\begin{eqnarray}\label{kbFF-3CC}
u_j:=\min\,\{|u|,j\}\cdot{\mathbf{1}}_{B_{\rho_{\#}}(x_0,j)\setminus E}\qquad \mbox{on }\,\,{\mathscr{X}}\setminus E. \end{eqnarray}
Observe that
\begin{eqnarray}\label{kbFF-3DD-1} {\rm supp}\,\bigl({\mathscr{A}}_{q,\kappa}u_j\bigr)\subseteq B_{\rho_{\#}}\bigl(x_0,C_\rho(1+\kappa)j\bigr),\qquad 0\leq{\mathscr{A}}_{q,\kappa}u_j \leq j\cdot\mu\bigl(B_{\rho_{\#}}(x_0,j)\bigr)^{1/q}<\infty, \end{eqnarray}
where the last inequality uses the fact that $\mu$ is locally finite. In turn, this implies that
\begin{eqnarray}\label{kbFF-3DD2}
\|{\mathscr{A}}_{q,\kappa}u_j\|_{L^p(E,\sigma)} \leq j\cdot\mu\bigl(B_{\rho_{\#}}(x_0,j)\bigr)^{1/q} \sigma\bigl(E\cap B_{\rho_{\#}}(x_0,C_\rho(1+\kappa)j)\bigr)^{1/p}<\infty, \end{eqnarray}
since $\sigma$ is locally finite. Hence, if we choose $\gamma\in(0,1]$ so that $(2c_q)^{-p}>2cC^p\,\gamma^q$, then we obtain from \eqref{kbFF-3BB} written with $u$ replaced by $u_j$
\begin{eqnarray}\label{kbFF-3DD}
\|{\mathscr{A}}_{q,\kappa}u_j\|_{L^p(E,\sigma)}^p
\leq C\|{\mathfrak{C}}_{q,\kappa}u_j\|_{L^p(E,\sigma)}^p,\qquad\forall\,j\in{\mathbb{N}}, \end{eqnarray}
for some $C\in(0,\infty)$ independent of $j$. Note that $0\leq {\mathfrak{C}}_{q,\kappa}u_j\leq{\mathfrak{C}}_{q,\kappa}u$ pointwise in $E$
and that $u_j\nearrow |u|$ pointwise $\mu$-a.e. on ${\mathscr{X}}\setminus E$ implies ${\mathscr{A}}_{q,\kappa}u_j\nearrow {\mathscr{A}}_{q,\kappa}u$ everywhere on $E$ by Lebesgue's Monotone Convergence Theorem. Based on these observations, \eqref{kbFF-3DD} and Fatou's lemma we may then conclude that
\begin{eqnarray}\label{kbFF-3EE}
\|{\mathscr{A}}_{q,\kappa}u\|_{L^p(E,\sigma)}^p &=& \int_{E}\liminf_{j\to\infty}\bigl[{\mathscr{A}}_{q,\kappa}u_j\bigr]^p\,d\sigma \leq\liminf_{j\to\infty}\int_{E}\bigl[{\mathscr{A}}_{q,\kappa}u_j\bigr]^p\,d\sigma \nonumber\\[4pt]
&=& \liminf_{j\to\infty}\|{\mathscr{A}}_{q,\kappa}u_j\|_{L^p(E,\sigma)}^p
\leq C\|{\mathfrak{C}}_{q,\kappa}u\|_{L^p(E,\sigma)}^p. \end{eqnarray}
That is, granted \eqref{kbFF}, we have
\begin{eqnarray}\label{kbFF-3UU}
\|{\mathscr{A}}_{q,\kappa}u\|_{L^p(E,\sigma)}^p
\leq C\|{\mathfrak{C}}_{q,\kappa}u\|_{L^p(E,\sigma)}^p. \end{eqnarray}
Thus, to finish the proof of part {\it (1)} of the statement of the theorem, we are left with proving \eqref{kbFF}. Fix $\kappa'>\kappa>0$ along with $\gamma\in(0,1]$, then for an arbitrary $\lambda>0$ define the set
\begin{eqnarray}\label{xfbl} {\mathcal{O}}_{\lambda}:= \bigl\{x\in E:\,({\mathscr{A}}_{q,\kappa'}u)(x)>\lambda\bigr\}. \end{eqnarray}
By Lemma~\ref{semi-cont}, ${\mathcal{O}}_{\lambda}$ is an open
subset of $(E,\tau_{\rho|_{E}})$. Also, since ${\mathscr{A}}_{q,\kappa'}u \geq{\mathscr{A}}_{q,\kappa}u$ pointwise in $E$, we conclude that
\begin{eqnarray}\label{yrTYY97} \{x\in E:\,({\mathscr{A}}_{q,\kappa}u)(x)>2c_q\lambda\} \subseteq{\mathcal{O}}_{\lambda}. \end{eqnarray}
If ${\mathcal{O}}_{\lambda}=\emptyset$, then by \eqref{yrTYY97} the inequality in the second line of \eqref{kbFF} is trivially satisfied. Therefore, assume that ${\mathcal{O}}_{\lambda}\not=\emptyset$ in what follows. Let us also make the assumption (which will be eliminated {\it a posteriori}) that
\begin{eqnarray}\label{NON-ZE78} \begin{array}{c} \mbox{the $\mu$-measurable function $u:{\mathscr{X}}\setminus E\to\overline{\mathbb{R}}$ is such that} \\[4pt] \mbox{${\mathcal{O}}_\lambda$ from \eqref{xfbl} is a proper subset of $E$ for each $\lambda>0$}. \end{array} \end{eqnarray}
In such a scenario, for a fixed, suitably chosen $\lambda_o>1$ we may apply Proposition~\ref{H-S-Z} (with $\lambda$ there replaced by $\lambda_o$) to obtain a Whitney covering of ${\mathcal{O}}_{\lambda}$ by balls, relative
to $(E,\rho|_{E})$, which we may assume (given the freedom of choosing the parameter $\lambda_o$, and \eqref{RHo-evv}) to be of the form $B_j:=E\cap B_{\rho_{\#}}(x_j,r_j)$, $j\in{\mathbb{N}}$, satisfying properties {\it (1)-(4)} in Proposition~\ref{H-S-Z} for some $\Lambda>\lambda_o$. If we now prove that
\begin{eqnarray}\label{kbFF-5} \begin{array}{c} \mbox{$\exists\,\kappa'>\kappa$ and $\exists\,c\in(0,\infty)$ such that $\forall\,\gamma\in(0,1]$, $\forall\,\lambda\in(0,\infty)$, there holds} \\[6pt] \sigma\bigl(\{x\in B_j:({\mathscr{A}}_{q,\kappa}u)(x)>2c_q\lambda,\, ({\mathfrak{C}}_{q,\kappa}u)(x)\leq\!\gamma\lambda\}\bigr) \leq c\,\gamma^q\sigma(B_j)\,\mbox{ for every }\,j\in{\mathbb{N}}, \end{array} \end{eqnarray}
then combining \eqref{kbFF-5} with \eqref{yrTYY97} and properties {\it (1)-(2)} from Proposition~\ref{H-S-Z}, we may estimate
\begin{eqnarray}\label{kbFF-6} && \hskip -0.40in \sigma\bigl(\{x\in E:({\mathscr{A}}_{q,\kappa}u)(x)>2c_q\lambda, ({\mathfrak{C}}_{q,\kappa}u)(x)\leq\!\gamma\lambda\}\bigr) \nonumber\\[4pt] &&\hskip 0.20in =\sigma\bigl(\{x\in {\mathcal{O}}_\lambda:\, ({\mathscr{A}}_{q,\kappa}u)(x)>2c_q\lambda,\, ({\mathfrak{C}}_{q,\kappa}u)(x)\leq\!\gamma\lambda\}\bigr) \nonumber\\[4pt] &&\hskip 0.20in \leq\sum\limits_{j=1}^\infty \sigma\bigl(\{x\in B_j:({\mathscr{A}}_{q,\kappa}u)(x)>2c_q\lambda, ({\mathfrak{C}}_{q,\kappa}u)(x)\leq\!\gamma\lambda\}\bigr) \leq c\,\gamma^q\sum\limits_{j=1}^\infty\sigma(B_j) \nonumber\\[4pt] &&\hskip 0.20in \leq C\gamma^q\sigma({\mathcal{O}}_\lambda). \end{eqnarray}
Hence, \eqref{kbFF} follows.
Turning now to the proof of \eqref{kbFF-5}, fix $j\in{\mathbb{N}}$, and note that without loss of generality we may assume that
\begin{eqnarray}\label{G.bbn-88} \bigl\{x\in B_j:({\mathscr{A}}_{q,\kappa}u)(x)>2c_q\lambda,\,\, ({\mathfrak{C}}_{q,\kappa}u)(x)\leq\!\gamma\lambda\bigr\}\not=\emptyset, \end{eqnarray}
since otherwise there is nothing to prove. Decompose $u=u{\mathbf{1}}_{\{\delta_E\geq r_j\}}+u{\mathbf{1}}_{\{\delta_E<r_j\}}=:u_1+u_2$ and let $z_j\in E\setminus{\mathcal{O}}_\lambda$ be such that $\rho_{\#}(x_j,z_j)\leq\Lambda r_j$ (the existence of $z_j$ is guaranteed by property {\it (3)} in Proposition~\ref{H-S-Z}). We claim that
\begin{eqnarray}\label{kbFF-7} \begin{array}{c} \mbox{there exists }\,\,\kappa'>\kappa\,\,\mbox{ independent of $j\in{\mathbb{N}}$ with the property that } \\[4pt] \mbox{if }\,x\in B_j\,\mbox{ and }\,y\in\Gamma_\kappa(x)\, \mbox{ is such that }\,\delta_E(y)\geq r_j\, \mbox{ then }\,y\in\Gamma_{\kappa'}(z_j). \end{array} \end{eqnarray}
Indeed, if $x\in B_j$, we have $\rho_{\#}(x,z_j)\leq C_{\rho_{\#}}\max\{\rho_{\#}(x,x_j),\rho_{\#}(x_j,z_j)\} \leq C_{\rho}\Lambda r_j$. Hence, if $y\in\Gamma_\kappa(x)$ is such that $\delta_E(y)\geq r_j$ then
\begin{eqnarray}\label{kbFF-8} \rho_{\#}(y,z_j) &\leq & C_{\rho_{\#}}\max\{\rho_{\#}(y,x),\rho_{\#}(x,z_j)\} \leq C_{\rho}\max\{(1+\kappa)\delta_E(y),C_{\rho}\Lambda r_j\} \nonumber\\[4pt] &\leq& C_{\rho}\max\{(1+\kappa),C_{\rho}\Lambda\}\delta_E(y). \end{eqnarray}
Now we choose $\kappa'>C_{\rho}\max\{(1+\kappa),C_{\rho}\Lambda\}-1$, so then $\kappa'>\kappa$, and $\kappa'$ depends only on finite positive geometrical constants (hence, in particular, it is independent of $j\in{\mathbb{N}}$). Based on \eqref{kbFF-8} we obtain that \eqref{kbFF-7} holds true for this choice of $\kappa'$. Then, using \eqref{kbFF-7} and recalling that $z_j\in E\setminus{\mathcal{O}}_{\lambda}$, we may write
\begin{eqnarray}\label{kbFF-9} ({\mathscr{A}}_{q,\kappa}u_1)(x)^q &=&
\int\limits_{\stackrel{y\in\Gamma_\kappa(x)}{\delta_E(y)\geq r_j}}|u(y)|^q\,d\mu(y)
\leq\int\limits_{y\in\Gamma_{\kappa'}(z_j)}|u(y)|^q\,d\mu(y) \nonumber\\[6pt] &=& ({\mathscr{A}}_{q,\kappa'}u)(z_j)^q\leq\lambda^q, \qquad\forall\,x\in B_j. \end{eqnarray}
Next, we make use of \eqref{Mix+FR} to write
\begin{eqnarray}\label{kbFF-10} \int_{B_j}({\mathscr{A}}_{q,\kappa}u_2)(x)^q\,d\sigma(x)
&=& \int\limits_{y\in{\mathcal{F}}_\kappa(B_j),\,\delta_E(y)<r_j}|u(y)|^q \sigma\bigl(B_j\cap\pi^\kappa_y\bigr)\,d\mu(y) \nonumber\\[4pt]
&\leq & \int\limits_{y\in{\mathcal{F}}_\kappa(B_j),\,\delta_E(y)<r_j}|u(y)|^q \sigma(\pi^\kappa_y)\,d\mu(y). \end{eqnarray}
In order to proceed further, first make a geometrical observation to the effect that (using notation introduced in \eqref{ktEW-7UU})
\begin{eqnarray}\label{kbFF-11} \begin{array}{c} \mbox{there exists a finite constant $c_o>0$ such that for every $r>0$ and every $x_0\in E$} \\[4pt] \mbox{if }\,y\in{\mathcal{F}}_\kappa(\Delta(x_0,r))\,\mbox{ and }\,\delta_E(y)<r \,\mbox{ then }\,y\in {\mathcal{T}}_\kappa\bigl(E\cap B_{\rho_{\#}}({w},c_or)\bigr) \,\,\forall\,{w}\in\Delta(x_0,r). \end{array} \end{eqnarray}
To see why this is true, consider a point $y\in{\mathcal{F}}_\kappa(\Delta(x_0,r))$ with the property that $\delta_E(y)<r$. Then there exists $x\in\Delta(x_0,r)$ such that $\rho_{\#}(y,x)<(1+\kappa)\delta_E(y)<(1+\kappa)r$. Let ${w}\in\Delta(x_0,r)$ be arbitrary and note that
\begin{eqnarray}\label{k-RF5FF} \rho_{\#}(x,{w})\leq C_{\rho_{\#}}\max\{\rho_{\#}(x,x_0),\rho_{\#}(x_0,{w})\} <C_\rho r. \end{eqnarray}
Accordingly, choosing $c_o>C_\rho$ forces $x\in E\cap B_{\rho_{\#}}({w},c_or)$ hence, further,
\begin{eqnarray}\label{k-R446} {\rm dist}_{\rho_{\#}}\bigl(y, E\cap B_{\rho_{\#}}({w},c_or)\bigr)\leq \rho_{\#}(y,x)<(1+\kappa)\delta_E(y). \end{eqnarray}
Let us also observe that
\begin{eqnarray}\label{kbFF-12} \rho_{\#}({w},y) &\leq & C_{\rho_{\#}}\max\{\rho_{\#}({w},x),\rho_{\#}(x,y)\} \nonumber\\[4pt] &\leq & C_{\rho_{\#}}\max\Bigl\{ C_{\rho_{\#}}\max\{\rho_{\#}({w},x_0),\rho_{\#}(x_0,x)\},\,\rho_{\#}(x,y)\Bigr\} \nonumber\\[4pt] &\leq & C_{\rho}\,\max\{C_\rho,1+\kappa\}r. \end{eqnarray}
Thus, if $z\in E\setminus B_{\rho_{\#}}({w},c_or)$, making use of \eqref{kbFF-12} we obtain
\begin{eqnarray}\label{kbFF-14} c_o r &\leq & \rho_{\#}(z,{w}) \leq C_{\rho_{\#}}\max\{\rho_{\#}(z,y),\rho_{\#}(y,{w})\} \nonumber\\[4pt] &\leq & C_{\rho}\max\Bigl\{\rho_{\#}(z,y),\,C_{\rho}\,\max\{C_\rho,1+\kappa\}r\Bigr\} =C_{\rho}\,\rho_{\#}(z,y), \end{eqnarray}
where the last equality is necessarily true if we take $c_o>C_\rho^2\max\{C_\rho,1+\kappa\}$ (given the nature of the left-most side of \eqref{kbFF-14}). Consequently, for this choice of $c_o$, \eqref{kbFF-14} gives that
\begin{eqnarray}\label{kbD-5} \rho_{\#}(y,z)\geq\frac{c_o}{C_\rho}r>\frac{c_o}{C_\rho}\delta_E(y),\qquad\forall\, z\in E\setminus B_{\rho_{\#}}({w},c_or) \end{eqnarray}
hence, if we also assume $c_o\geq C_\rho(1+\kappa)^2$, then
\begin{eqnarray}\label{kbD-5BB2} {\rm dist}_{\rho_{\#}}\bigl(y,E\setminus B_{\rho_{\#}}({w},c_or)\bigr) \geq(1+\kappa)^2\delta_E(y). \end{eqnarray}
Together, \eqref{k-R446} and \eqref{kbD-5BB2} allow us to conclude that if $c_o>\max\bigl\{ C_\rho(1+\kappa)^2,C_\rho^3,C_\rho^2(1+\kappa)\bigr\}$ then
\begin{eqnarray}\label{kbD-5BB3} {\rm dist}_{\rho_{\#}}\bigl(y,E\cap B_{\rho_{\#}}({w},c_or)\bigr) \leq(1+\kappa)^{-1} {\rm dist}_{\rho_{\#}}\bigl(y,E\setminus B_{\rho_{\#}}({w},c_or)\bigr). \end{eqnarray}
In light of \eqref{3.2.64}, we deduce from \eqref{kbD-5BB3} that $y\in {\mathcal{T}}_\kappa\bigl(E\cap B_{\rho_{\#}}({w},c_or)\bigr)$ when $c_o$ is chosen as above. This completes the proof of \eqref{kbFF-11}.
Combining \eqref{kbFF-10} with \eqref{kbFF-11} (the latter applied with $B_j$ in place of $\Delta(x_0,r)$), and keeping in mind that
\begin{eqnarray}\label{kbFF-SA+ii} \mbox{$\sigma(B_j)\approx\sigma\bigl(E\cap B_{\rho_{\#}}({w},c_or_j)\bigr)$, uniformly in $j\in{\mathbb{N}}$ and ${w}\in B_j$}, \end{eqnarray}
which is a consequence of \eqref{Doub-2}, we may then estimate
\begin{eqnarray}\label{kbFF-15} \tfrac{1}{\sigma(B_j)}\int_{B_j} ({\mathscr{A}}_{q,\kappa}u_2)(x)^q\,d\sigma(x) &\leq & \tfrac{C}{\sigma(B_j)}
\int\limits_{y\in{\mathcal{F}}_\kappa(B_j),\,\delta_E(y)<r_j}|u(y)|^q \sigma(\pi^\kappa_y)\,d\mu(y) \nonumber\\[4pt] &\leq & \tfrac{C}{\sigma\bigl(E\cap B_{\rho_{\#}}({w},c_or_j)\bigr)} \int_{{\mathcal{T}}_\kappa\bigl(E\cap B_{\rho_{\#}}({w},c_or_j)\bigr)}
|u(y)|^q\sigma(\pi^\kappa_y)\,d\mu(y) \nonumber\\[4pt] &\leq & C\,\inf_{{w}\in B_j} \bigl[({\mathfrak{C}}_{q,\kappa}u)({w})\bigr]^q \leq C\gamma^q\lambda^q, \end{eqnarray}
where for the last inequality in \eqref{kbFF-15} we have used the assumption \eqref{G.bbn-88}. In concert with Tschebyshev's inequality, \eqref{kbFF-15} gives that
\begin{eqnarray}\label{kbFF-16} \sigma\bigl(\{x\in B_j:\,({\mathscr{A}}_{q,\kappa}u_2)(x)>\lambda\}\bigr) \leq C\gamma^q\sigma(B_j), \end{eqnarray}
for some $C\in(0,\infty)$ independent of $\gamma\in(0,1]$ and $j\in{\mathbb{N}}$. Also, in view of \eqref{kbFF-9}, we obtain
\begin{eqnarray}\label{ygfgf-654} \bigl\{x\in B_j:\,({\mathscr{A}}_{q,\kappa}u)(x)>2c_q\lambda\bigr\} \subseteq\bigl\{x\in B_j:\,({\mathscr{A}}_{q,\kappa}u_2)(x)>\lambda\bigr\}, \end{eqnarray}
since pointwise on $E$ we have ${\mathscr{A}}_{q,\kappa}u\leq c_q\bigl({\mathscr{A}}_{q,\kappa}u_1 +{\mathscr{A}}_{q,\kappa}u_2\bigr)$, where $c_q$ is as in \eqref{CPP-77}. Combined with \eqref{kbFF-16}, this yields the inequality in \eqref{kbFF-5}. The proof of part {\it (1)} of the theorem is then complete, provided we dispense with the additional hypothesis in \eqref{NON-ZE78}. To do this, we distinguish two cases.
\vskip 0.08in \noindent{\tt Case~I:} {\it Assume that ${\rm diam}_{\rho}(E)=\infty$}. An inspection of the proof reveals that estimate \eqref{kbFF-3BB} has only been utilized with $u_j$ (from \eqref{kbFF-3CC}) in place of $u$. As such, we only need to know that $\bigl\{x\in E:\,({\mathscr{A}}_{q,\kappa'}u_j)(x)>\lambda\bigr\}$ is a proper subset of $E$ for each $j\in{\mathbb{N}}$ and each $\lambda>0$. However, in the case we are currently considering, this follows by observing that, on the one hand, $\sigma(E)=\infty$ by \eqref{DIA-MEA}, while on the other hand $\sigma\bigl(\bigl\{x\in E:\,({\mathscr{A}}_{q,\kappa'}u_j)(x)>\lambda\bigr\}\bigr) <\infty$ by \eqref{kbFF-3DD2} and Tschebyshev's inequality.
\vskip 0.08in \noindent{\tt Case~II:} {\it Assume that ${\rm diam}_{\rho}(E)<\infty$}. Recall from \eqref{DIA-MEA} that this entails $\sigma(E)<\infty$, and set $R:={\rm diam}_{\rho_{\#}}(E)\in(0,\infty)$. For some positive, small number $\varepsilon_o$, to be specified later, decompose
$|u|=u'+u'':=|u|{\mathbf{1}}_{\{\delta_E(\cdot)<\varepsilon_o R\}}
+|u|{\mathbf{1}}_{\{\delta_E(\cdot)\geq\varepsilon_o R\}}$. Hence, $u',u''$ are $\mu$-measurable and $0\leq u',u''\leq |u|$. Note that for each $x\in E$, \eqref{ktEW-7} gives
\begin{eqnarray}\label{kjvc+yh} ({\mathfrak{C}}_{q,\kappa}u'')(x) &\geq & \Bigl(\tfrac{1}{\sigma(E)} \int_{{\mathscr{X}}\setminus E}u''(y)^q\sigma(\pi^\kappa_y)\,d\mu(y)\Bigr)^{1/q} \nonumber\\[4pt] &\geq & c\Bigl(\int\limits_{\stackrel {y\in{\mathscr{X}}\setminus E}{\delta_E(y)\geq\varepsilon_o R}} u''(y)^q\,d\mu(y)\Bigr)^{1/q}\geq c({\mathscr{A}}_{q,\kappa}u'')(x), \end{eqnarray}
by taking $r>R$ in \eqref{ktEW-7UU} and recalling $(iv)$ in Lemma~\ref{T-LL.2}, and observing that there exists a constant $C\in(0,\infty)$ with the property that for each $y\in{\mathscr{X}}\setminus E$ we have (with $y_\ast$ and $\epsilon$ as in Lemma~\ref{lbDV})
\begin{eqnarray}\label{kjvc+yhZZ} \sigma(\pi^\kappa_y)\geq \sigma\bigl(E\cap B_{\rho_{\#}}(y_\ast,\epsilon\delta_E(y))\bigr) \geq \sigma\bigl(E\cap B_{\rho_{\#}}(y_\ast,\epsilon\varepsilon_o R)\bigr) \geq C\sigma(E), \end{eqnarray}
where the last inequality is a consequence of the doubling condition on $\sigma$. In turn, \eqref{kjvc+yh} and the monotonicity of the Carleson operator allow us to write
\begin{eqnarray}\label{kjvc+yh2}
\|{\mathscr{A}}_{q,\kappa}u''\|_{L^p(E,\sigma)}
\leq C\|{\mathfrak{C}}_{q,\kappa}u''\|_{L^p(E,\sigma)}
\leq C\|{\mathfrak{C}}_{q,\kappa}u\|_{L^p(E,\sigma)}. \end{eqnarray}
To proceed, set $\varepsilon_o:=\tfrac{1}{4C_\rho(1+\kappa')}$ and fix $x_1,x_2\in E$ satisfying $\rho_{\#}(x_1,x_2)>R/2$. We claim that these choices guarantee that
\begin{eqnarray}\label{yFg-H} \Gamma_{\kappa'}(x_1)\cap\Gamma_{\kappa'}(x_2)\subseteq \{x\in{\mathscr{X}}\setminus E:\,\delta_E(x)>\varepsilon_0 R\}. \end{eqnarray}
Indeed, if $y\in \Gamma_{\kappa'}(x_1)\cap\Gamma_{\kappa'}(x_2)$ then $\rho_{\#}(y,x_j)<(1+\kappa')\delta_E(y)$ for $j=1,2$ and we have $R/2<\rho_{\#}(x_1,x_2)\leq C_\rho\max\{\rho_{\#}(y,x_1),\rho_{\#}(y,x_2)\} <C_\rho(1+\kappa')\delta_E(y)=\tfrac{1}{4\varepsilon_o}\delta_E(y)$, which shows that the inclusion in \eqref{yFg-H} is true. If we now further decompose
\begin{eqnarray}\label{yFg-Hii} u'=u'_1+u'_2:=u'{\mathbf{1}}_{\Gamma_{\kappa'}(x_1)} +u'\bigl(1-{\mathbf{1}}_{\Gamma_{\kappa'}(x_1)}\bigr) \end{eqnarray}
then $0\leq u'_1,u'_2\leq u'$, and both $u'_1,u'_2$ are $\mu$-measurable. Moreover, due to \eqref{yFg-H} and the fact that $u_1$ has support contained in the set $\{\delta_E(\cdot)<\varepsilon_o R\}$, we also obtain that $({\mathscr{A}}_{q,\kappa'}u'_1)(x_2)=0$ and $({\mathscr{A}}_{q,\kappa'}u'_2)(x_1)=0$. The latter imply that the sets constructed according to the same recipe as ${\mathcal{O}}_\lambda$ in \eqref{xfbl} but with $u$ replaced by either $u'_1$ or $u'_2$, are proper subsets of $E$ for every $\lambda>0$. Hence, hypothesis \eqref{NON-ZE78} holds for each of the functions $u'_1$, $u'_2$. As such, the first part of the proof gives that \eqref{kbFF-3UU} holds with $u$ replaced by either $u'_1$ or $u'_2$. In concert, these give
\begin{eqnarray}\label{kjvc+yh2ii}
\|{\mathscr{A}}_{q,\kappa}u'\|_{L^p(E,\sigma)}
&\leq & C\|{\mathscr{A}}_{q,\kappa}u'_1\|_{L^p(E,\sigma)}
+C\|{\mathscr{A}}_{q,\kappa}u'_2\|_{L^p(E,\sigma)} \\[4pt]
&\leq & C\|{\mathfrak{C}}_{q,\kappa}u'_1\|_{L^p(E,\sigma)}
+C\|{\mathfrak{C}}_{q,\kappa}u'_2\|_{L^p(E,\sigma)}
\leq C\|{\mathfrak{C}}_{q,\kappa}u\|_{L^p(E,\sigma)}. \nonumber \end{eqnarray}
Together with \eqref{kjvc+yh2}, this then yields \eqref{kbFF-3UU} for the original function $u$. This finishes the treatment of Case~II and completes the proof of the estimate in part {\it (1)} of the theorem.
Moving on to the proof of part {\it (2)}, the key step is establishing the pointwise estimate
\begin{eqnarray}\label{KFf-1} ({\mathfrak{C}}_{q,\kappa}u)(x_0) \leq C\bigl[M_E({\mathscr{A}}_{q,\kappa}u)^q(x_0)\bigr]^{\frac{1}{q}}, \qquad\forall\,x_0\in E, \end{eqnarray}
for some $C\in(0,\infty)$ depending only on $\kappa,p,q$ and geometrical characteristics of the ambient space. To justify this, fix $r>0$, and let $\Delta$ be a ball of radius
$r$ in $(E,(\rho|_{E})_{\#})$. Then, upon recalling \eqref{Mix+FR}, $(ii)$ in Lemma~\ref{T-LL.2}, and \eqref{3.2.BN}, we may write
\begin{eqnarray}\label{kbFF-1B} \int_{\Delta}({\mathscr{A}}_{q,\kappa}u)(x)^q\,d\sigma(x)
&=& \int_{{\mathcal{F}}_\kappa(\Delta)}|u(y)|^q \sigma\bigl(\Delta\cap\pi^\kappa_y\bigr)\,d\mu(y) \nonumber\\[4pt]
&\geq & \int_{{\mathcal{T}}_\kappa(\Delta)}|u(y)|^q \sigma\bigl(\Delta\cap \pi^\kappa_y\bigr)\,d\mu(y) \nonumber\\[4pt]
&=& \int_{{\mathcal{T}}_\kappa(\Delta)}|u(y)|^q \sigma\bigl(\pi^\kappa_y\bigr)\,d\mu(y). \end{eqnarray}
Now \eqref{KFf-1} follows from \eqref{kbFF-1B} by dividing the latter inequality by $\sigma(\Delta)$ and taking the supremum over all $\Delta$'s containing an arbitrary given point $x_0\in E$.
With the pointwise estimate \eqref{KFf-1} in hand, whenever $p\in(q,\infty)$ and $r\in(0,\infty]$ we may use the boundedness of $M_E$ on the Lorentz space $L^{p/q,r/q}(E,\sigma)$, which holds since $p/q>1$, and the general fact that for each $\alpha>0$ we have
\begin{eqnarray}\label{Td-sUU}
\||f|^\alpha\|_{L^{p,r}(E,\sigma)}=C(p,r,\alpha)
\|f\|^{\alpha}_{L^{p\alpha,r\alpha}(E,\sigma)}, \end{eqnarray}
in order to write
\begin{eqnarray}\label{KFf-1A}
\|{\mathfrak{C}}_{q,\kappa}u\|_{L^{p,r}(E,\sigma)}
&\leq & C\bigl\|[M_E({\mathscr{A}}_{q,\kappa}u)^q]^\frac{1}{q}\bigr\|_{L^{p,r}(E,\sigma)}
=C\|M_E({\mathscr{A}}_{q,\kappa}u)^q\|_{L^{p/q,r/q}(E,\sigma)}^\frac{1}{q} \nonumber\\[4pt]
&\leq & C\|({\mathscr{A}}_{q,\kappa}u)^q\|_{L^{p/q,r/q}(E,\sigma)}^\frac{1}{q}
=C\|{\mathscr{A}}_{q,\kappa}u\|_{L^{p,r}(E,\sigma)}, \end{eqnarray}
as required.
There remains to observe that the two estimates in {\it (3)} are obtained by a computation similar to \eqref{KFf-1A} that is based on \eqref{KFf-1}, the weak-$(1,1)$ boundedness of $M_E$, and the boundedness of $M_E$ on $L^\infty(E,\sigma)$. This finishes the proof of the theorem. \end{proof}
\begin{remark}\label{yafg-76r} The case $p=q=r$ of part {\it (2)} of Theorem~\ref{AsiC}, which corresponds
to the estimate $\|{\mathfrak{C}}_{p,\kappa}u\|_{L^p(E,\sigma)}
\leq C\|{\mathscr{A}}_{p,\kappa}u\|_{L^p(E,\sigma)}$, fails in general. A counterexample in Euclidean space when $p=2$ is given in the remarks stated below Theorem~3 of~\cite{CoMeSt}. \end{remark}
\subsection{Weak $L^p$ square function estimates imply $L^2$ square function estimates} \label{SSect:5.3}
We are now in a position to consider $L^p$ versions of the $L^2$ square function estimates considered in Section~\ref{Sect:3} for integral operators $\Theta_E$. The main result is that $L^2$ square function estimates follow automatically from weak $L^p$ square function estimates for any $p\in(0,\infty)$. This is stated in Theorem~\ref{VGds-L2XXX} below. The result is achieved by combining the $T(1)$ theorem in Theorem~\ref{SChg} with a weak type John-Nirenberg lemma for Carleson measures based on Lemma~2.14 in \cite{AHLT} (see also \cite[Lemma IV.1.12]{DaSe93} for a similar result).
\begin{theorem}\label{VGds-L2XXX} Let $0<d<m<\infty$. Assume that $({\mathscr{X}},\rho,\mu)$ is an $m$-dimensional {\rm ADR} space, $E$ is a closed subset of $({\mathscr{X}},\tau_\rho)$, and $\sigma$ is a Borel regular measure
on $(E,\tau_{\rho|_{E}})$ with the property that $(E,\rho\bigl|_E,\sigma)$ is a $d$-dimensional {\rm ADR} space. Finally, suppose that $\Theta$ is the integral operator defined in \eqref{operator} with a kernel $\theta$ as in \eqref{K234}, \eqref{hszz}, \eqref{hszz-3}.
Then whenever $\kappa,p,C_o\in(0,\infty)$ are such that for every surface ball $\Delta\subseteq E$ (cf. \eqref{ktEW-7UU})
\begin{eqnarray}\label{dtbh-L2iii} \hskip -0.20in \sigma\left(\Bigl\{x\in E:\,
\int_{\Gamma_{\kappa}(x)}|(\Theta{\mathbf{1}}_{\Delta})(y)|^2 \,\delta_E(y)^{2\upsilon-m}\,d\mu(y)>\lambda^2\Bigr\}\right) \leq C_o\lambda^{-p}\sigma(\Delta),\quad\forall\,\lambda>0, \end{eqnarray}
there exists some $C\in(0,\infty)$ which depends only on $\kappa,p,C_o$ and finite positive background constants (including ${\rm diam}_\rho(E)$ in the case when $E$ is bounded) with the property that
\begin{eqnarray}\label{k-tSSiii} \int\limits_{\mathscr{X}\setminus E}
|(\Theta f)(x)|^2\delta_E(x)^{2\upsilon-(m-d)}\,d\mu(x)
\leq C\int_E|f(x)|^2\,d\sigma(x),\qquad\forall\,f\in L^2(E,\sigma). \end{eqnarray} \end{theorem}
The requirement in \eqref{dtbh-L2iii} is actually less restrictive than a weak $L^p$ square function estimate. In particular, it is satisfied whenever the following weak $L^p$ square function estimate holds for every $f\in L^p(E,\sigma)$:
\begin{eqnarray}\label{eqrem} \hskip -0.30in \sup_{\lambda>0}\left[\lambda\cdot
\sigma\Bigl(\Bigl\{x\in E:\int_{\Gamma_{\kappa}(x)}|(\Theta f)(y)|^2 \delta_E(y)^{2\upsilon-m}\,d\mu(y)>\lambda^{2}\Bigr\}\Bigr)^{1/p}\right]
\leq C_o\|f\|_{L^{p}(E,\sigma)}. \end{eqnarray}
Indeed, \eqref{dtbh-L2iii} follows by specializing \eqref{eqrem} to the case when $f={\mathbf{1}}_{\Delta}$ for an arbitrary surface ball $\Delta\subseteq E$.
To prove Theorem~\ref{VGds-L2XXX}, we need only set $q=2$ in Proposition~\ref{VGds-L2} below to obtain a Carleson measure estimate, and then apply the $T(1)$ theorem for square functions in Theorem~\ref{SChg}. Therefore, the remainder of this subsection is dedicated to the proof of the following proposition.
\begin{proposition}\label{VGds-L2} Retain the same background hypotheses as in the statement of Theorem~\ref{VGds-L2XXX}. In this context, let ${\mathbb{D}}(E)$ denote a dyadic cube structure on $E$, consider a Whitney covering ${\mathbb{W}}_\lambda(\mathscr{X}\setminus E)$ of $\mathscr{X}\setminus E$ as in Lemma~\ref{Lem:CQinBQ-N} and, corresponding to these, recall the dyadic Carleson tents from \eqref{gZSZ-3}. Then whenever $\kappa,p,q,C_o\in(0,\infty)$ are such that for every surface ball $\Delta\subseteq E$ there holds
\begin{eqnarray}\label{dtbh-L2} \hskip -0.20in \sigma\left(\Bigl\{x\in E:\,
\int_{\Gamma_{\kappa}(x)}|(\Theta{\mathbf{1}}_{\Delta})(y)|^q \,\delta_E(y)^{q\upsilon-m}\,d\mu(y)>\lambda^q\Bigr\}\right) \leq C_o\lambda^{-p}\sigma(\Delta),\quad\forall\,\lambda>0, \end{eqnarray}
there exists some $C\in(0,\infty)$ which depends only on $\kappa,p,q,C_o$ and finite positive background constants with the property that
\begin{eqnarray}\label{k-tSS} \sup_{Q\in{\mathbb{D}}(E)}
\Bigl(\tfrac{1}{\sigma(Q)}\int_{T_E(Q)}|(\Theta 1)(x)|^q \delta_E(x)^{q\upsilon-(m-d)}\,d\mu(x)\Bigr)\leq C. \end{eqnarray} \end{proposition}
In preparation for presenting the proof of Proposition~\ref{VGds-L2} we discuss a couple of auxiliary results. The first such result is a variation on the theme of Whitney decomposition discussed in Proposition~\ref{H-S-Z}.
\begin{lemma}\label{PropW-2} Let $(E,\rho,\sigma)$ be a space of homogeneous type with the property that the measure $\sigma$ is Borel regular, and let ${\mathbb{D}}(E)$ be a collection of dyadic cubes as in Proposition~\ref{Diad-cube}. Also, suppose that ${\mathcal{O}}$ is an open subset of $(E,\tau_\rho)$ with the property that
$\bigl({\mathcal{O}},\rho\bigl|_{{\mathcal{O}}},\sigma\lfloor{\mathcal{O}}\bigr)$ is a space of homogeneous type. Fix $\lambda\in(1,\infty)$ and suppose $\Omega$ is an open, proper, non-empty subset of ${\mathcal{O}}$. Then there exist $\varepsilon\in(0,1)$, $N\in{\mathbb{N}}$, $\Lambda\in(\lambda,\infty)$ and a subset ${\mathcal{W}}\subseteq{\mathbb{D}}(E)$ such that the following properties are satisfied:
\begin{enumerate} \item[(1)] $Q\subseteq\Omega$ for every $Q\in{\mathcal{W}}$ and $\sigma\bigl(\Omega\setminus\bigcup_{Q\in{\mathcal{W}}}Q\bigr)=0$;
\item[(2)] $Q\cap Q'=\emptyset$ for every $Q,Q'\in{\mathcal{W}}$ with $Q\not=Q'$;
\item[(3)] for every $x\in\Omega$, the cardinality of the set $\bigl\{Q\in{\mathcal{W}}:\, B_\rho\bigl(x,\varepsilon\,{\rm dist}_\rho(x,{\mathcal{O}}\setminus\Omega)\bigr) \cap Q\not=\emptyset\bigr\}$ is at most $N$;
\item[(4)] $\lambda Q\subseteq\Omega$ and $\Lambda Q\cap[{\mathcal{O}}\setminus\Omega]\not=\emptyset$ for every $Q\in{\mathcal{W}}$;
\item[(5)] $\ell(Q)\approx\ell(Q')$ uniformly for $Q,Q'\in{\mathcal{W}}$ such that $\lambda Q\cap\lambda Q'\not=\emptyset$;
\item[(6)] $\sum\limits_{Q\in{\mathcal{W}}}{\mathbf{1}}_{\lambda Q}\leq N$. \end{enumerate} \end{lemma}
\begin{proof} Given $\lambda\in(1,\infty)$, apply Proposition~\ref{H-S-Z} to the open, proper, non-empty subset $\Omega$ of the space of homogeneous type
$\bigl({\mathcal{O}},\rho\bigl|_{{\mathcal{O}}},\sigma\lfloor{\mathcal{O}}\bigr)$. This guarantees the existence of parameters $\varepsilon\in(0,1)$, $N\in{\mathbb{N}}$, $\Lambda\in(\lambda,\infty)$, as well as a covering of $\Omega$ with balls $\Omega=\bigcup_{j\in{\mathbb{N}}}\bigl({\mathcal{O}}\cap B_\rho(x_j,r_j)\bigr)$ such that the analogues of the properties {\it (1)-(4)} in Proposition~\ref{H-S-Z} hold in the current setting. Next, for each $j\in{\mathbb{N}}$ consider
\begin{eqnarray}\label{eEE} I_j:=\bigl\{Q\in{\mathbb{D}}(E):\,\ell(Q)\approx r_j\mbox{ and } Q\cap B_\rho(x_j,r_j)\not=\emptyset\bigr\}, \end{eqnarray}
and define ${\mathcal{W}}:=\bigcup_{j\in{\mathbb{N}}}I_j$ thinned out, so that $Q\cap Q'=\emptyset$ for every $Q,Q'\in{\mathcal{W}}$, $Q\not=Q'$. Granted the properties the families ${\mathbb{D}}(E)$ and $\{B_\rho(x_j,r_j)\}_{j\in{\mathbb{N}}}$ satisfy (as listed in Proposition~\ref{Diad-cube} and Proposition~\ref{H-S-Z}) and given the nature of the construction of the family ${\mathcal{W}}$, it follows that properties {\it (1)-(6)} in the statement of the current lemma hold for the family ${\mathcal{W}}$. \end{proof}
We now state the aforementioned weak type John-Nirenberg lemma for Carleson measures, cf. \cite[Lemma 2.14]{AHLT} for a result similar in spirit in the Euclidean setting.
\begin{lemma}\label{SQ-lema} Retain the same background hypotheses as in the statement of Theorem~\ref{VGds-L2XXX}. In this context, fix two finite numbers $\kappa,\eta>0$, an index $q\in(0,\infty)$ and, for each $Q\in{\mathbb{D}}(E)$, define
\begin{eqnarray}\label{SQ-1} S_Q(x):=\Bigl(\int\limits_{\stackrel{y\in\Gamma_\kappa(x)}
{\rho_{\#}(x,y)<\eta\ell(Q)}}|(\Theta 1)(y)|^q\delta_E(y)^{q\upsilon-m} \,d\mu(y)\Bigr)^{\frac{1}{q}}, \qquad\forall\,x\in E. \end{eqnarray}
Assuming that $\eta$ is sufficiently large (depending only on geometry) and granted that there exist two parameters $N\in(0,\infty)$ and $\beta\in(0,1)$ such that
\begin{eqnarray}\label{SQ-2} \sigma\Bigl(\bigl\{x\in Q:\,S_Q(x)>N\bigr\}\Bigr)<(1-\beta)\sigma(Q), \qquad\forall\,Q\in{\mathbb{D}}(E), \end{eqnarray}
then one may find $C\in(0,\infty)$ depending only on geometry, the estimates satisfied by the kernel $\theta$, and $\kappa,\eta$, with the property that
\begin{eqnarray}\label{k-tSS.22} \sup_{Q\in{\mathbb{D}}(E)}
\Bigl(\tfrac{1}{\sigma(Q)}\int_{T_E(Q)}|(\Theta 1)(x)|^q \delta_E(x)^{q\upsilon-(m-d)}\,d\mu(x)\Bigr)\leq\beta^{-1}(C+N^q). \end{eqnarray} \end{lemma}
\begin{proof} For each $i\in{\mathbb{N}}$, let $\Theta_i$ be as in \eqref{LIH-3} and associate to $\Theta_i$ the function $S^i_Q$, much as $S_Q$ is associated to $\Theta$. Note that $S^i_Q$ and $S_Q$ depend on the constant $\kappa$ defining $\Gamma_\kappa$. We fix $\widetilde{\kappa}\in(0,\kappa)$ to be specified later and we use the notation $S^i_{Q,\widetilde{\kappa}}$ for the function defined similarly to $S^i_Q$ but with $\widetilde{\kappa}$ in place of $\kappa$. Also, with $N\in(0,\infty)$ and $\beta\in(0,1)$ satisfying \eqref{SQ-2}, define
\begin{eqnarray}\label{SQ-55} \Omega_Q^{N,i}:=\bigl\{x\in Q:\,S^i_{Q}(x)>N\bigr\}, \qquad\forall\,Q\in{\mathbb{D}}(E),\quad\forall\,i\in{\mathbb{N}}. \end{eqnarray}
Since, thanks to Lemma~\ref{semi-cont}, for each $Q\in{\mathbb{D}}(E)$ and $i\in{\mathbb{N}}$ the function $S^i_Q$ is lower semi-continuous, from \eqref{SQ-55}, \eqref{SQ-2} and the fact that $S^i_Q\leq S_Q$ pointwise in $Q$ we deduce that
\begin{eqnarray}\label{SQ-56MM} \forall\,Q\in{\mathbb{D}}(E),\quad\forall\,i\in{\mathbb{N}},\quad \mbox{ $\Omega_Q^{N,i}$ is an open, proper subset of $Q$}. \end{eqnarray}
To proceed, consider
\begin{eqnarray}\label{SQ-56} A^i:=\sup_{Q\in{\mathbb{D}}(E)} \Bigl(\tfrac{1}{\sigma(Q)}\int_Q(S^i_{Q,\widetilde{\kappa}}(x))^q\,d\sigma(x)\Bigr), \qquad\forall\,i\in{\mathbb{N}}. \end{eqnarray}
Then, based on \eqref{Mix+FR} (applied to the function
$u:=|(\Theta_i 1)|^q\delta_E^{q\upsilon-m} {\mathbf{1}}_{\{{\rm dist}_{\rho_{\#}}(\cdot,Q)\leq C\ell(Q)\}}$) we may write, with $x_Q$ denoting the center of $Q\in{\mathbb{D}}(E)$,
\begin{eqnarray}\label{SQ-57A} \hskip -0.30in \int_Q(S^i_{Q,\widetilde{\kappa}}(x))^q\,d\sigma(x) & \leq & \int_Q\Bigl( \int\limits_{\stackrel{y\in\Gamma_{\widetilde{\kappa}}(x)} {{\rm dist}_{\rho_{\#}}(y,Q)\leq C\ell(Q)}}
|(\Theta_i 1)(y)|^q\delta_E(y)^{q\upsilon-m}\,d\mu(y)\Bigr)\,d\sigma(x) \nonumber\\[4pt] & \leq & \int\limits_{\stackrel{y\in{\mathcal{F}}_{\widetilde{\kappa}}(Q)} {{\rm dist}_{\rho_{\#}}(y,Q)\leq C\ell(Q)}}
|(\Theta_i 1)(y)|^q\delta_E(y)^{q\upsilon-m} \sigma\bigl(Q\cap\pi_y^{\widetilde{\kappa}}\bigr)\,d\mu(y) \nonumber\\[4pt] & \leq & C\int\limits_{B_{\rho_{\#}}(x_Q,C\ell(Q))}
|(\Theta_i 1)(y)|^q\delta_E(y)^{q\upsilon-(m-d)}\,d\mu(y), \end{eqnarray}
where the last step in \eqref{SQ-57A} uses the inequality $\sigma(Q\cap\pi_y^{\widetilde{\kappa}})\leq C\delta_E(y)^d$ which, in turn,
is a consequence of Lemma~\ref{lbDV}, the fact that $(E,\rho\bigl|_E,\sigma)$ is a $d$-dimensional {\rm ADR} space, and the observation that $\delta_E(y)\leq{\rm dist}_{\rho_{\#}}(y,Q)\leq C\,\ell(Q)\leq C\,{\rm diam}_{\rho}(E)$ on the domain of integration (of the third integral in \eqref{SQ-57A}). Moreover, reasoning as in \eqref{LIH-5} we obtain
\begin{eqnarray}\label{LIH-5B} \int_{B_{\rho_{\#}}(x_Q,C\ell(Q))}
|(\Theta_i 1)(y)|^q\delta_E(y)^{q\upsilon-(m-d)}\,d\mu(y) \leq Ci^{2q\upsilon}\ell(Q)^d\leq Ci^{2q\upsilon}\sigma(Q) \end{eqnarray}
for every $Q\in{\mathbb{D}}(E)$, so by combining \eqref{SQ-57A} and \eqref{LIH-5B} we arrive at the conclusion that
\begin{eqnarray}\label{SQ-57} \tfrac{1}{\sigma(Q)}\int_Q(S^i_{Q,\widetilde{\kappa}}(x))^q\,d\sigma(x)\leq C(i)<\infty, \end{eqnarray}
for each cube $Q\in{\mathbb{D}}(E)$ and each $i\in{\mathbb{N}}$. Thus, in particular, $A^i<\infty$ for every $i\in{\mathbb{N}}$.
At this stage in the proof, the incisive step is the claim that, in fact,
\begin{eqnarray}\label{SQ-58} \begin{array}{c} \exists\,A\in(0,\infty)\quad\mbox{independent of $i$ such that } \\[6pt] \tfrac{1}{\sigma(Q)}\int_Q(S^i_{Q,\widetilde{\kappa}}(x))^q\,d\sigma(x)\leq A, \quad\forall\,Q\in{\mathbb{D}}(E). \end{array} \end{eqnarray}
In the process of proving this claim we shall show that one can take $A:=\beta^{-1}(C+N^q)$ where $C\in(0,\infty)$ is a constant which depends only on geometry, the estimates satisfied by $\theta$, and $\kappa$. To get started, fix $i\in{\mathbb{N}}$ and first observe that if $Q\in{\mathbb{D}}(E)$ is such that $\Omega_Q^{N,i}=\emptyset$, then $S^i_{Q,\widetilde{\kappa}}\leq S^i_Q\leq N$ on $Q$, hence for such $Q$'s \eqref{SQ-58} will hold if we impose the condition that $A\geq N^q$. Next, let $Q\in{\mathbb{D}}(E)$ be such that $\Omega_Q^{N,i}\not=\emptyset$. Then, thanks to \eqref{SQ-56MM}, it follows that $\Omega_Q^{N,i}$ is an open, nonempty, proper subset of $Q$. Recall from \eqref{ihgc} that
$\bigl(Q,\rho|_{Q},\sigma\lfloor{Q}\bigr)$ is a space of homogeneous type and the doubling constant of the measure $\sigma\lfloor{Q}$ is independent of $Q$. Then there exists a Whitney decomposition of $\Omega_Q^{N,i}$ relative to $Q$ via dyadic cubes $\{Q_k\}_{k\in I^{N,i}_Q}$ as described in Lemma~\ref{PropW-2} (used with ${\mathcal{O}}:=Q$ and $\Omega:=\Omega_Q^{N,i}$). Introducing $F^{N,i}_Q:=Q\setminus\Omega_Q^{N,i}$ we may then write
\begin{eqnarray}\label{SQ-59} \hskip -0.20in \int_Q(S^i_{Q,\widetilde{\kappa}}(x))^q\,d\sigma(x) =\int_{F_Q^{N,i}}(S^i_{Q,\widetilde{\kappa}}(x))^q\,d\sigma(x) +\sum\limits_{k\in I_Q^{N,i}}\int_{Q_k}(S^i_{Q,\widetilde{\kappa}}(x))^q\,d\sigma(x) =:I+II. \end{eqnarray}
Since $\widetilde{\kappa}<\kappa$ forces $S^i_{Q,\widetilde{\kappa}}\leq S^i_Q\leq N$ on $F^{N,i}_Q$, we further have
\begin{eqnarray}\label{SQ-f77} I\leq\int_{F_Q^{N,i}}(S^i_Q(x))^q\,d\sigma(x)\leq N^q\sigma(Q). \end{eqnarray}
To estimate $II$, we write
\begin{eqnarray}\label{SQ-78} && \hskip -0.20in II =\sum\limits_{k\in I_Q^{N,i}}\int_{Q_k}(S^i_{Q_k,\widetilde{\kappa}}(x))^q\, d\sigma(x) \nonumber\\[4pt] && \hskip 0.20in +\sum\limits_{k\in I_Q^{N,i}}\int_{Q_k}\Bigl(\hskip -0.05in \int\limits_{\stackrel{y\in\Gamma_{\widetilde{\kappa}}(x)} {\eta\ell(Q_k)\leq\rho_{\#}(y,x)<\eta\ell(Q)}} \hskip -0.30in
|(\Theta_i 1)(y)|^q\delta_E(y)^{q\upsilon-m}d\mu(y)\Bigr)\,d\sigma(x) \nonumber\\[4pt] && \hskip 0.08in =:III+IV. \end{eqnarray}
By recalling \eqref{SQ-56}, the fact that the family $\{Q_k\}_{k\in I^{N,i}_Q}$ consists of pairwise disjoint cubes from ${\mathbb{D}}(E)$ contained in $\Omega_Q^{N,i}$, as well as assumption \eqref{SQ-2}, we have
\begin{eqnarray}\label{SQ-79} III\leq \sum\limits_{k\in I_Q^{N,i}}A^i\sigma(Q_k) \leq A^i\sigma\bigl(\Omega_Q^{N,i}\bigr)\leq A^i(1-\beta)\sigma(Q). \end{eqnarray}
Moving on, from \eqref{hszz} and \eqref{mMji} (given that $\upsilon-a>0$) we see that
$|(\Theta_i 1)(y)|\leq\frac{C}{\delta_E(y)^\upsilon}$ for every $y\in{\mathscr{X}}\setminus E$. Thus, if $C_0>0$ is some large finite fixed constant which will be specified later (just below \eqref{SQ-83ii-B}, to be precise), and if $k\in I^{N,i}_Q$, then for each $x\in Q_k$ there holds
\begin{eqnarray}\label{SQ-80} && \hskip -0.60in \int\limits_{\stackrel{y\in\Gamma_{\widetilde{\kappa}}(x)} {\eta\ell(Q_k)\leq\rho_{\#}(x,y)\leq C_0\ell(Q_k)}} \hskip -0.40in
|(\Theta_i 1)(y)|^q\delta_E(y)^{q\upsilon-m}\,d\mu(y) \leq C\int\limits_{\stackrel{y\in\Gamma_{\widetilde{\kappa}}(x)} {\eta\ell(Q_k)\leq\rho_{\#}(x,y)\leq C_0\ell(Q_k)}}\hskip -0.40in \frac{d\mu(y)}{\delta_E(y)^{m}} \\[4pt] &&\hskip 0.10in \leq C \ell(Q_k)^{-m}\mu\Bigl(\bigl\{y\in\Gamma_{\widetilde{\kappa}}(x):\, \eta\ell(Q_k)\leq\rho_{\#}(x,y)\leq C_0\ell(Q_k)\bigr\}\Bigr) \leq C<\infty, \nonumber \end{eqnarray}
for some $C>0$ independent of $x$, $k$, $Q$ and $i$. In turn, \eqref{SQ-80} entails
\begin{eqnarray}\label{SQ-81} && \hskip -0.50in \sum\limits_{k\in I_Q^{N,i}}\int_{Q_k}\Bigl( \int\limits_{\stackrel{y\in\Gamma_{\widetilde{\kappa}}(x)} {\eta\ell(Q_k)\leq\rho_{\#}(x,y)\leq C_0\ell(Q_k)}} \hskip -0.40in
|(\Theta_i 1)(y)|^q\delta_E(y)^{q\upsilon-m}\,d\mu(y)\Bigr)\,d\sigma(x) \nonumber\\[4pt] &&\hskip 1.50in \leq C\sum\limits_{k\in I_Q^{N,i}}\sigma(Q_k) \leq C\sigma\bigl(\Omega_Q^{N,i}\bigr)\leq C\sigma(Q), \end{eqnarray}
which once again suits our purposes. Next, since $\{Q_k\}_{k\in I^{N,i}_Q}$ is a Whitney decomposition of $\Omega_Q^{N,i}$ relative to $Q$, for each $k\in I^{N,i}_Q$ there exists $x_k\in F^{N,i}_Q$ such that
\begin{eqnarray}\label{SQ-MMh} {\rm dist}_{\rho_{\#}}(x_k,Q_k)\leq c\,\ell(Q_k), \end{eqnarray}
for some finite $c>0$ independent of $k$, $Q$ and $i$. We now claim that there exits $\widetilde{\kappa}\in(0,\kappa)$ depending on the constants associated with the Whitney decomposition of $\Omega_Q^{N,i}$
(hence, ultimately, on finite positive geometric constants associated with $(E,\rho\bigl|_{E},\sigma)$), as well as on $\kappa$ and the constant $C_0$, but independent of $k$, $Q$ and $i$, such that
\begin{eqnarray}\label{SQ-83ii} x\in Q_k,\,\,\,y\in\Gamma_{\widetilde{\kappa}}(x)\,\,\mbox{ and }\,\, C_0\ell(Q_k)<\rho_{\#}(x,y)\,\Longrightarrow\, y\in\Gamma_\kappa(x_k). \end{eqnarray}
To justify this claim, suppose that $\widetilde{\kappa}\in(0,\kappa)$ and fix $x\in Q_k$ along with $y\in\Gamma_{\widetilde{\kappa}}(x)$ such that $C_0\ell(Q_k)<{\rm dist}_{\rho_{\#}}(y,Q)$. Then
\begin{eqnarray}\label{SQ-83ii-A} C_0\ell(Q_k)<\rho_{\#}(y,x)<(1+\widetilde{\kappa})\delta_E(y) <(1+\kappa)\delta_E(y). \end{eqnarray}
Also, if we choose a finite number $\vartheta\in\bigl(0,(\log_2C_\rho)^{-1}\bigr]$, Theorem~\ref{JjEGh} gives that $(\rho_{\#})^\vartheta$ is a genuine distance. As such, we may estimate based on \eqref{SQ-MMh}, \eqref{SQ-83ii-A} and hypotheses
\begin{eqnarray}\label{SQ-83ii-B} \rho_{\#}(y,x_k)^\vartheta &\leq & \rho_{\#}(y,x)^\vartheta+\rho_{\#}(x,x_k)^\vartheta <(1+\widetilde{\kappa})^\vartheta\delta_E(y)^\vartheta+c^\vartheta\ell(Q_k)^\vartheta \nonumber\\[4pt] &\leq & (1+\widetilde{\kappa})^\vartheta\delta_E(y)^\vartheta +c^\vartheta\frac{(1+\kappa)^\vartheta}{C_0^\vartheta}\delta_E(y)^\vartheta \nonumber\\[4pt] &\leq & (1+\kappa)^\vartheta\delta_E(y)^\vartheta, \end{eqnarray}
provided $C_0>c\Bigl[1-\Bigl(\frac{1}{1+\kappa}\Bigr)^\vartheta\Bigr]^{-1/\vartheta}$ and $0<\widetilde{\kappa}<(1+\kappa) \Bigl[1-\Bigl(\frac{c}{C_0}\Bigr)^\vartheta\Bigr]^{1/\vartheta}-1$. Assuming that this is the case, \eqref{SQ-83ii} now follows from \eqref{SQ-83ii-B}.
Going further, with \eqref{SQ-83ii} in hand and upon recalling that $S^i_{Q}(x_k)\leq N$, we may estimate
\begin{eqnarray}\label{SQ-83} &&\hskip -0.60in \sum\limits_{k\in I_Q^{N,i}}\int_{Q_k}\Bigl( \int\limits_{\stackrel{y\in\Gamma_{\widetilde{\kappa}}(x)} {C_0\ell(Q_k)<\rho_{\#}(x,y)<\eta\ell(Q)}} \hskip -0.40in
|(\Theta_i 1)(y)|^q\delta_E(y)^{q\upsilon-m}\,d\mu(y)\Bigr)\,d\sigma(x) \nonumber\\[4pt] &&\hskip 0.20in \leq \sum\limits_{k\in I_Q^{N,i}}\int_{Q_k}\Bigl(\hskip -0.08in \int\limits_{\stackrel{y\in\Gamma_{\kappa}(x_k)} {\rho_{\#}(x,y)<\eta\ell(Q)}} \hskip -0.10in
|(\Theta_i 1)(y)|^q\delta_E(y)^{q\upsilon-m}\,d\mu(y)\Bigr)\,d\sigma(x) \nonumber\\[4pt] &&\hskip 0.20in =\sum\limits_{k\in I_Q^{N,i}}\int_{x\in Q_k} (S^i_{Q}(x_k))^q \,d\sigma(x) \leq N^q\sum\limits_{k\in I_Q^{N,i}}\sigma(Q_k)\leq N^q\sigma(Q), \end{eqnarray}
which is of the right order. In concert, \eqref{SQ-81}-\eqref{SQ-83} prove that there exists $C\in(0,\infty)$ depending only on geometry, the estimates satisfied by the kernel $\theta$, and $\kappa$, with the property that $IV\leq (C+N^q)\sigma(Q)$. In combination with \eqref{SQ-59}-\eqref{SQ-79}, this then allows us to conclude that
\begin{eqnarray}\label{SQ-84} \int_Q(S^i_{Q,\widetilde{\kappa}}(x))^q d\sigma \leq A^i(1-\beta)\sigma(Q)+(C+N^q)\sigma(Q),\qquad\forall\,Q\in{\mathbb{D}}(E). \end{eqnarray}
In particular, if we divide \eqref{SQ-84} by $\sigma(Q)$, then take the supremum over $Q\in{\mathbb{D}}(E)$ we arrive at the conclusion that $A^i\leq A^i(1-\beta)+C$ for each $i\in{\mathbb{N}}$. Upon recalling that $A^i\in(0,\infty)$ for each $i\in{\mathbb{N}}$ and that $\beta\in(0,1)$, it follows from this that $\sup_{i\in{\mathbb{N}}}A^i\leq\beta^{-1}(C+N^q)<\infty$. Hence, \eqref{SQ-58} is true.
Consider now the function $S_{Q,\widetilde{\kappa}}$ defined analogously to $S_Q$ but with $\widetilde{\kappa}$ in place of $\kappa$. Given that $\lim\limits_{i\to\infty}S^i_{Q,\widetilde{\kappa}}=S_{Q,\widetilde{\kappa}}$ pointwise in $E$, from \eqref{SQ-58} and Lebesgue's Monotone Convergence Theorem we may conclude that
\begin{eqnarray}\label{SQ-85} \exists\,C\in(0,\infty)\quad\mbox{such that }\,\, \tfrac{1}{\sigma(Q)}\int_Q(S_{Q,\widetilde{\kappa}}(x))^q\,d\sigma(x)\leq C, \qquad\forall\,Q\in{\mathbb{D}}(E). \end{eqnarray}
Next, observe that
\begin{eqnarray}\label{SQ-8vv} x,y\in B_{\rho_{\#}}\bigl(x_Q,\eta C_\rho^{-1}\ell(Q)\bigr) \,\Longrightarrow\,\rho_{\#}(x,y)\leq\eta\ell(Q). \end{eqnarray}
Then, based on \eqref{SQ-1}, \eqref{SQ-8vv}, \eqref{Mix+FR}, $(ii)$ in Lemma~\ref{T-LL.2}, \eqref{3.2.BN}, and the fact that
$\bigl(E,\rho\bigl|_{E},\sigma\bigr)$ is a $d$-dimensional {\rm ADR} space, we may estimate (using notation introduced in \eqref{ktEW-7UU}):
\begin{eqnarray}\label{kbF-YH.3} && \hskip -0.50in \int\limits_{\Delta(x_Q,\eta C_\rho^{-1}\ell(Q))} (S_{Q,\widetilde{\kappa}}(x))^q\,d\sigma(x) \nonumber\\[4pt] && \hskip 0.25in =\int\limits_{\Delta(x_Q,\eta C_\rho^{-1}\ell(Q))} \Bigl(\int\limits_{\stackrel{y\in\Gamma_{\widetilde{\kappa}}(x)} {\rho_{\#}(x,y)<\eta\ell(Q)}}
|(\Theta 1)(y)|^q\delta_E(y)^{q\upsilon-m}\,d\mu(y)\Bigr)d\sigma(x) \nonumber\\[4pt] && \hskip 0.25in \geq\int\limits_{\Delta(x_Q,\eta C_\rho^{-1}\ell(Q))} \Bigl(\int\limits_{\stackrel{y\in\Gamma_{\widetilde{\kappa}}(x)} {\rho_{\#}(y,x_Q)<\eta C_\rho^{-1}\ell(Q)}}
|(\Theta 1)(y)|^q\delta_E(y)^{q\upsilon-m}\,d\mu(y)\Bigr)d\sigma(x) \nonumber\\[4pt] && \hskip 0.25in =\int\limits_{\stackrel{y\in{\mathcal{F}}_{\widetilde{\kappa}} (\Delta(x_Q,\eta C_\rho^{-1}\ell(Q)))} {\rho_{\#}(y,x_Q)<\eta C_\rho^{-1}\ell(Q)}}
|(\Theta 1)(y)|^q\delta_E(y)^{q\upsilon-m} \sigma\bigl(\Delta(x_Q,\eta C_\rho^{-1}\ell(Q)) \cap\pi^{\widetilde{\kappa}}_y\bigr)\,d\mu(y) \nonumber\\[4pt] && \hskip 0.25in \geq\int\limits_{\stackrel{y\in{\mathcal{T}}_{\widetilde{\kappa}} (\Delta(x_Q,\eta C_\rho^{-1}\ell(Q)))} {\rho_{\#}(y,x_Q)<\eta C_\rho^{-1}\ell(Q)}}
|(\Theta 1)(y)|^q\delta_E(y)^{q\upsilon-m} \sigma\bigl(\Delta(x_Q,\eta C_\rho^{-1}\ell(Q)) \cap\pi^{\widetilde{\kappa}}_y\bigr)\,d\mu(y) \nonumber\\[4pt] && \hskip 0.25in =\int\limits_{\stackrel{y\in{\mathcal{T}}_{\widetilde{\kappa}} (\Delta(x_Q,\eta C_\rho^{-1}\ell(Q)))} {\rho_{\#}(y,x_Q)<\eta C_\rho^{-1}\ell(Q)}}
|(\Theta 1)(y)|^q\delta_E(y)^{q\upsilon-m} \sigma\bigl(\pi^{\widetilde{\kappa}}_y\bigr)\,d\mu(y) \nonumber\\[4pt] && \hskip 0.25in \approx\int\limits_{\stackrel{y\in{\mathcal{T}}_{\widetilde{\kappa}} (\Delta(x_Q,\eta C_\rho^{-1}\ell(Q)))} {\rho_{\#}(y,x_Q)<\eta C_\rho^{-1}\ell(Q)}}
|(\Theta 1)(y)|^q\delta_E(y)^{q\upsilon-(m-d)}\,d\mu(y), \end{eqnarray}
uniformly for $Q\in{\mathbb{D}}(E)$. Let us also observe that there exists an integer $M_o\in{\mathbb{N}}$ (depending only on geometry) with the property that for every $Q\in{\mathbb{D}}(E)$ the ball $\Delta(x_Q,\eta C_\rho^{-1}\ell(Q))$ may be covered by at most $M_o$ dyadic cubes of the same generation as $Q$, and that for every such cube $\widetilde{Q}$ there holds $S_{\widetilde{Q}}=S_Q$. Having noticed this, we then deduce from \eqref{SQ-85}, \eqref{kbF-YH.3}, and \eqref{Fv-UU45} that there exists $C\in(0,\infty)$ satisfying
\begin{eqnarray}\label{k-tSS.ps} \frac{1}{\sigma(Q)}\int \limits_{B_{\rho_{\#}}(x_Q,\,\eta C_\rho^{-2}\ell(Q))\setminus E}
|(\Theta 1)(x)|^q\delta_E(x)^{q\upsilon-(m-d)}\,d\mu(x) \leq C,\qquad\forall\,Q\in{\mathbb{D}}(E). \end{eqnarray}
With this in hand, \eqref{k-tSS.22} now follows with the help of \eqref{dFvK}, if $\eta$ is sufficiently large to begin with (depending only on geometry). \end{proof}
Our last auxiliary result is an estimate of geometrical nature, on a nontangential approach region. For a proof (and for more general results of this type) see \cite{MMMM-B}.
\begin{lemma}\label{bzrg1} Let $({\mathscr{X}},\rho,\mu)$ be an $m$-dimensional {\rm ADR} space for some $m>0$. Assume that $E$ is a closed subset of $({\mathscr{X}},\tau_{\rho})$
with the property that there exists a Borel measure $\sigma$ on $(E,\tau_{\rho|_{E}})$
such that $\bigl(E,\rho\bigl|_{E},\sigma\bigr)$ is a $d$-dimensional {\rm ADR} space for some $d\geq 0$. Then for each $\kappa>0$, $\beta<m$, $M>m-\beta$, there exists a finite constant $C>0$ depending on $\kappa$, $M$, $\beta$, and the {\rm ADR} constants of ${\mathscr{X}}$ and $E$, such that
\begin{eqnarray}\label{ae4t} \hskip -0.20in \int_{\Gamma_{\kappa}(z)} \frac{\delta_E(x)^{-\beta}}{\rho_{\#}(x,y)^M}\,d\mu(x)\leq C \rho(y,z)^{m-\beta-M},\quad\mbox{for all }\,\,z,y\in E\mbox{ with }z\not=y. \end{eqnarray} \end{lemma}
Now we are ready to proceed with the
\vskip 0.08in \begin{proof}[Proof of Proposition~\ref{VGds-L2}] Based on Lemma~\ref{SQ-lema}, it suffices to prove that if the hypotheses of Proposition~\ref{VGds-L2} are satisfied, then there exist $N<\infty$ and $\beta\in(0,1)$ such that \eqref{SQ-2} holds. To this end, let $N>0$ be a large finite constant, to be specified later, and fix an arbitrary $Q\in{\mathbb{D}}(E)$. Also, recall $c_q$ from \eqref{CPP-77} and fix an arbitrary number $\eta>0$. Then, with $S_Q$ as in \eqref{SQ-1} and some finite constant $c>0$ to be specified later, we may write
\begin{eqnarray}\label{SQ-3} && \hskip -0.30in \sigma\Bigl(\bigl\{x\in Q:\,S_Q(x)>N\bigr\}\Bigr) \nonumber\\[4pt] && \hskip 0.10in \leq \sigma\Bigl(\bigl\{x\in Q:\,\Bigl( \int\limits_{y\in\Gamma_\kappa(x),\,\rho_{\#}(x,y)<\eta\ell(Q)}
|(\Theta{\mathbf{1}}_{c\,Q})(y)|^q \delta_E(y)^{q\upsilon-m}\,d\mu(y)\Bigr)^{\frac{1}{q}}>N/2\bigr\}\Bigr) \nonumber\\[4pt] &&\hskip 0.20in +\sigma\Bigl(\bigl\{x\in Q:\,\Bigl(\int\limits_{y\in\Gamma_\kappa(x), \,\rho_{\#}(x,y)<\eta\ell(Q)}
|(\Theta{\mathbf{1}}_{E\setminus c\,Q})(y)|^q \delta_E(y)^{q\upsilon-m}\,d\mu(y)\Bigr)^{\frac{1}{q}}>N/2\bigr\}\Bigr) \nonumber\\[4pt] && \hskip 0.10in =:I+II, \end{eqnarray}
where we have used the notation $c\,Q:=E\cap B_{\rho_{\#}}\bigl(x_Q,c\,\ell(Q)\bigr)$. Note that under the assumption \eqref{dtbh-L2} (and the fact that $\sigma$ is doubling) we may estimate
\begin{eqnarray}\label{SQ-3B} \hskip -0.20in I\leq\sigma\Bigl(\bigl\{x\in Q:\,\Bigl(\int_{\Gamma_\kappa(x)}
|(\Theta{\mathbf{1}}_{c\,Q})(y)|^q\delta_E(y)^{q\upsilon-m}\,d\mu(y)\Bigr)^{\frac{1}{q}} >N/2\bigr\}\Bigr)\leq\tfrac{C}{N^p}\sigma(Q), \end{eqnarray}
which suits our purposes.
Going further, select some finite constant $c_o\geq\sup_{Q'\in{\mathbb{D}}(E)} \bigl({\rm diam}_{\rho_{\#}}(Q')/\ell(Q')\bigr)$. Given $x\in Q$ fixed, note that for each $y\in B_{\rho_{\#}}(x,\eta\ell(Q))$ we have
\begin{eqnarray}\label{SQ-ZZ} \rho_{\#}(y,x_Q) &\leq & C_{\rho_{\#}}\max\,\{\rho_{\#}(y,x),\rho_{\#}(x,x_Q)\} \\[4pt] &\leq & C_\rho\max\,\{\eta,c_o\}\,\ell(Q)\leq c^{-1}C_\rho\max\,\{\eta,c_o\}\, \rho_{\#}(z,x_Q),\qquad\forall\,z\in E\setminus{\mathbf{1}}_{c\,Q}. \nonumber \end{eqnarray}
Consequently, if $z\in E\setminus{\mathbf{1}}_{c\,Q}$, then
\begin{eqnarray}\label{SQ-Zs} \rho_{\#}(z,x_Q)&\leq& C_{\rho_{\#}}\max\{\rho_{\#}(z,y),\rho_{\#}(y,x_Q)\} \\[4pt] &\leq & C_\rho\rho_{\#}(z,y)+c^{-1}C^2_\rho\max\,\{\eta,c_o\}\,\rho_{\#}(z,x_Q), \quad\forall\,y\in B_{\rho_{\#}}(x,\eta\ell(Q)). \nonumber \end{eqnarray}
Hence, choosing the finite constant $c>0$ sufficiently large so that $c^{-1}C^2_\rho\max\,\{\eta,c_o\}<\frac{1}{2}$ forces $\rho_{\#}(z,x_Q)\leq 2C_\rho\rho_{\#}(z,y)$ for all $y\in B_{\rho_{\#}}(x,\eta\ell(Q))$. Making use of this, \eqref{hszz}, and \eqref{WBA} we may then write
\begin{eqnarray}\label{SQ-4}
|(\Theta{\mathbf{1}}_{E\setminus c\,Q})(y)| &\leq & C\int\limits_{E\setminus c\,Q}\frac{\delta_E(y)^{-a}}{\rho_{\#}(z,y)^{d+\upsilon-a}}\,d\sigma(z) \leq C\delta_E(y)^{-a}\!\!\!\! \int\limits_{z\in E,\,\rho_{\#}(z,x_Q)>c\,\ell(Q)} \frac{d\sigma(z)}{\rho_{\#}(z,x_Q)^{d+\upsilon-a}} \nonumber\\[4pt] &\leq & C\frac{\delta_E(y)^{-a}}{\ell(Q)^{\upsilon-a}}, \qquad\forall\,y\in B_{\rho_{\#}}(x,\eta\ell(Q)). \end{eqnarray}
Pick now $1<c_1<c_2<c$ such that there exists ${w}\in c_2Q\setminus c_1Q$ (which may be assured by further increasing $c$ if needed, given that
$(E,\rho|_{E},\sigma)$ is a $d$-dimensional {\rm ADR} space). Then clearly $\rho_{\#}(x,{w})\approx\ell(Q)$ and we claim that also
\begin{eqnarray}\label{SQ-5} \rho_{\#}(y,{w})\approx\ell(Q),\quad\mbox{ uniformly for }\,\, y\in\Gamma_\kappa(x)\cap B_{\rho_{\#}}(x,\eta\ell(Q)). \end{eqnarray}
Indeed, on the one hand, if the point $y\in{\mathscr{X}}$ is such that $\rho_{\#}(y,x)<\eta\ell(Q)$ then we obtain $\rho_{\#}(y,{w})\leq C_{\rho}\max\{\rho_{\#}(y,x),\rho_{\#}(x,{w})\} \leq C\ell(Q)$. On the other hand, if we additionally know that $y\in\Gamma_\kappa(x)$, then $\rho_{\#}(y,x)<(1+\kappa)\delta_E(y)\leq(1+\kappa)\rho_{\#}(y,{w})$, hence
\begin{eqnarray}\label{BHh} C\ell(Q) &\leq & \rho_{\#}(x,{w}) \leq C_{\rho_{\#}}\max\{\rho_{\#}(x,y),\rho_{\#}(y,{w})\} \nonumber\\[4pt] & \leq & C_\rho(1+\kappa)\rho_{\#}(y,{w})\leq C\ell(Q), \end{eqnarray}
proving \eqref{SQ-5}.
Select now a real number $M>q(\upsilon-a)$. Combining \eqref{SQ-4} and \eqref{SQ-5} we then obtain
\begin{eqnarray}\label{SQ-6} && \hskip -0.80in \int\limits_{\stackrel{y\in\Gamma_\kappa(x)}{\rho_{\#}(x,y)<\eta\ell(Q)}} \!\!\!\!\!
|(\Theta{\mathbf{1}}_{E\setminus c\,Q})(y)|^q\delta_E(y)^{q\upsilon-m}\,d\mu(y) \\ && \hskip 0.80in \leq C\int\limits_{\stackrel{y\in\Gamma_\kappa(x)}{\rho_{\#}(x,y)<\eta\ell(Q)}} \!\!\!\!\! \frac{1}{\ell(Q)^{q(\upsilon-a)}}\cdot\delta_E(y)^{q(\upsilon-a)-m}\,d\mu(y) \nonumber\\[4pt] && \hskip 0.80in \leq C\ell(Q)^{M-q(\upsilon-a)}\int\limits_{\Gamma_\kappa(x)} \frac{\delta_E(y)^{-[m-q(\upsilon-a)]}}{\rho_{\#}(y,{w})^{M}}\,d\mu(y) \nonumber\\[4pt] && \hskip 0.80in \leq C\ell(Q)^{M-q(\upsilon-a)}\rho_{\#}(x,{w})^{-M+q(\upsilon-a)}\leq C, \qquad\forall\,x\in Q, \nonumber \end{eqnarray}
where for the penultimate inequality in \eqref{SQ-6} we have relied on Lemma~\ref{bzrg1} (used here with $\beta:=m-q(\upsilon-a)$).
With this in hand, we are now ready to estimate the term $II$ (appearing in \eqref{SQ-3}). Concretely, applying first Tschebyshev's inequality and then invoking \eqref{SQ-6} we obtain
\begin{eqnarray}\label{SQ-7} II \leq \tfrac{C}{N}\int_Q\Bigl( \int\limits_{\stackrel{y\in\Gamma_\kappa(x)}{\rho_{\#}(x,y)<\eta\ell(Q)}} \!\!\!\!\!
|(\Theta{\mathbf{1}}_{E\setminus c\,Q})(y)|^q\frac{d\mu(y)}{\delta_E(y)^{m-q\upsilon}} \Bigr)^\frac{1}{q}\,d\sigma(x)\leq\tfrac{C}{N}\sigma(Q). \end{eqnarray}
Combining \eqref{SQ-3}, \eqref{SQ-3B} and \eqref{SQ-7} we see that, for each $\beta\in(0,1)$, if we choose $N>0$ sufficiently large, then
\begin{eqnarray}\label{SQ-8} \sigma\Bigl(\bigl\{x\in Q:\,S_Q(x)>N\bigr\}\Bigr) \leq\tfrac{C}{N^{\min\{1,p\}}}\sigma(Q)<(1-\beta)\sigma(Q), \qquad\forall\,Q\in{\mathbb{D}}(E). \end{eqnarray}
Hence, \eqref{SQ-2} holds and the proof of the proposition is complete. \end{proof}
\subsection{Extrapolating square function estimates} \label{SSect:5.4}
We now combine our results to prove two extrapolation theorems for square function estimates associated with integral operators $\Theta_E$, as defined in Section~\ref{Sect:3}. First, we use Theorem~\ref{AsiC} to prove the extrapolation result in Theorem~\ref{VGds-2}, and then we combine this with Theorem~\ref{VGds-L2XXX} to obtain another extrapolation result in Theorem~\ref{VGds-2.33}.
In the first part of this subsection we digress to clarify terminology and background results concerning the scale of Hardy spaces $H^p$ for $p\in(0,\infty)$ in the context of a $d$-dimensional Ahlfors-David Regular space. In particular, we consider an atomic characterization for these spaces based on the work of R.R.~Coifman and G.~Weiss in \cite{CoWe77}, as well as a maximal function characterization based on the work of R.A.~Mac\'{i}as and C.~Segovia in~\cite{MaSe79II}. The theory of Hardy spaces in the context considered here has also been developed by D.~Mitrea, I.~Mitrea, M.~Mitrea and S.~Monniaux in~\cite{MMMM-G}.
Consider a $d$-dimensional {\rm ADR} space $(E,\rho,\sigma)$ and let $\beta\in(0,\infty)$. Given a real-valued function $f$ on $E$, define its {\tt H\"older\! semi-norm} (of order $\beta$, relative to the quasi-distance $\rho$) by setting
\begin{eqnarray}\label{Hol.T2}
\|f\|_{\dot{\mathscr{C}}^\beta(E,\rho)}:=
\sup_{x,y\in E,\,x\not=y}\frac{|f(x)-f(y)|}{\rho(x,y)^\beta}, \end{eqnarray}
and define the homogeneous H\"older space $\dot{\mathscr{C}}^\beta(E)$ as
\begin{eqnarray}\label{Hol.T2Li} \hskip -0.30in \dot{\mathscr{C}}^\beta(E,\rho):=
\bigl\{f:E\to{\mathbb{R}}:\,\|f\|_{\dot{\mathscr{C}}^\beta(E,\rho)}<\infty\bigr\}. \end{eqnarray}
Going further, set $\dot{\mathscr{C}}^\beta_c(E,\rho)$ for the subspace of $\dot{\mathscr{C}}^\beta(E,\rho)$ consisting of functions which vanish identically outside a bounded set. Then define the class of {\tt test\! functions} on $E$ as
\begin{eqnarray}\label{TTT-E.1} {\mathscr{D}}(E,\rho):=\bigcap\limits_{0<\beta<[\log_2C_\rho]^{-1}} \dot{\mathscr{C}}^\beta_c(E,\rho), \end{eqnarray}
equipped with a certain topology, $\tau_{\mathscr{D}}$, which we shall describe next. Specifically, fix a nested family $\{K_n\}_{n\in{\mathbb{N}}}$ of $\rho$-bounded subsets of $E$ with the property that any $\rho$-ball is contained in one of the $K_n$'s. Then, for each $n\in{\mathbb{N}}$, denote by ${\mathscr{D}}_n(E,\rho)$ the collection of functions from ${\mathscr{D}}(E,\rho)$ which vanish in $E\setminus K_n$.
With $\|\cdot\|_{\infty}$ standing for the supremum norm on $E$, this becomes a Frech\'et space when equipped with the topology $\tau_n$ induced by the family of norms
\begin{eqnarray}\label{TFF-E.a2}
\bigl\{\|\cdot\|_{\infty}+
\|\cdot\|_{\dot{\mathscr{C}}^\beta(E,\rho)}:\,\beta\mbox{ rational number such that } 0<\beta<[\log_2C_\rho]^{-1}\bigr\}. \end{eqnarray}
That is, ${\mathscr{D}}_n(E,\rho)$ is a Hausdorff topological space, whose topology is induced by a countable family of semi-norms, and which is complete (as a uniform space with the uniformity canonically induced by the aforementioned family of semi-norms or, equivalently, as a metric space when endowed with a metric yielding the same topology as $\tau_{n}$). Since for any $n\in{\mathbb{N}}$ the topology induced by $\tau_{n+1}$ on ${\mathscr{D}}_n(X,\rho)$ coincides with $\tau_n$, we may turn ${\mathscr{D}}(X,\rho)$ into a topological space, $({\mathscr{D}}(X,\rho),\tau_{\mathscr{D}})$, by regarding it as the strict inductive limit of the family of topological spaces $\bigl\{({\mathscr{D}}_n(X,\rho),\tau_n)\bigr\}_{n\in{\mathbb{N}}}$. Having accomplished this, we then define the {\tt space\! of\! distributions} ${\mathscr{D}}'(E,\rho)$ on $E$ as the (topological) dual of ${\mathscr{D}}(E,\rho)$, and denote by $\langle\cdot,\cdot\rangle$ the natural duality pairing between distributions in ${\mathscr{D}}'(E,\rho)$ and test functions in ${\mathscr{D}}(E,\rho)$.
To proceed, for each number $\gamma\in\bigl(0,\bigl[\log_2 C_\rho\bigr]^{-1}\bigr)$ and each point $x\in E$ define the class ${\mathcal{B}}^{\,\gamma}_\rho(x)$ of $(\rho,\gamma)$-{\tt normalized bump-functions supported near} $x$ by
\begin{eqnarray}\label{ASW43} && \hskip -1.00in {\mathcal{B}}^{\,\gamma}_\rho(x):=\Bigl\{\psi\in{\mathscr{D}}(E,\rho):\, \exists\,r>0\mbox{ such that $\psi=0$ on $E\setminus B_\rho(x,r)$} \mbox{ and} \nonumber\\[4pt] && \hskip 2.20in
\|\psi\|_{\infty}+r^\gamma\|\psi\|_{\dot{\mathscr{C}}^\gamma(E,\rho)} \leq r^{-d}\Bigr\}. \end{eqnarray}
In this setting, define the {\tt grand maximal function} of a distribution $f\in{\mathscr{D}}'(E,\rho)$ by setting (with the duality paring understood as before)
\begin{eqnarray}\label{NMC22-1} f^*_{\rho,\gamma}(x):=
\sup_{\psi\in{\mathcal{B}}^{\,\gamma}_\rho(x)}\bigl|\langle f,\psi\rangle\bigr|, \qquad\forall\,x\in E. \end{eqnarray}
Given an exponent $p$ satisfying
\begin{eqnarray}\label{range-p} \frac{d}{d+[\log_2C_\rho]^{-1}}<p<\infty, \end{eqnarray}
define the {\tt Hardy\! space} $H^p(E,\rho,\sigma)$ by setting
\begin{eqnarray}\label{NMC22-2BB} &&\hskip -0.80in H^p(E,\rho,\sigma):= \Bigl\{f\in{\mathscr{D}}'(E,\rho):\,\forall\,\gamma\in{\mathbb{R}}\mbox{ so that } d\bigl({\textstyle{\frac{1}{p}}}-1\bigr)<\gamma<[\log_2 C_\rho]^{-1} \\[4pt] && \hskip 2.20in \mbox{ it follows that }f^*_{\rho_{\#},\gamma}\in L^p(E,\sigma)\Bigr\}. \nonumber \end{eqnarray}
A closely related version of the above Hardy space is $\widetilde{H}^p(E,\rho,\sigma)$, with $p$ as before, defined as
\begin{eqnarray}\label{NMC22-2} && \hskip -0.30in \widetilde{H}^p(E,\rho,\sigma):= \Bigl\{f\in{\mathscr{D}}'(E,\rho):\,\exists\,\gamma\in{\mathbb{R}}\mbox{ so that } d\bigl({\textstyle{\frac{1}{p}}}-1\bigr)<\gamma<[\log_2 C_\rho]^{-1} \\[4pt] && \hskip 1.90in \mbox{ and with the property that }f^*_{\rho_{\#},\gamma}\in L^p(E,\sigma)\Bigr\}. \nonumber \end{eqnarray}
Moving on, given an index
\begin{eqnarray}\label{range-p-1} \frac{d}{d+[\log_2C_\rho]^{-1}}<p\leq 1, \end{eqnarray}
call a function $a\in L^\infty(E,\sigma)$ a $p$-{\tt atom} provided there exist $x_0\in E$ and a real number $r>0$ with the property that
\begin{eqnarray}\label{jk-AM} {\rm supp}\,a\subseteq E\cap B_{\rho}(x_0,r),\quad
\|a\|_{L^\infty(E,\sigma)}\leq r^{-d/p},\quad\int_Ea\,d\sigma=0. \end{eqnarray}
In the case when $E$ is bounded we also agree to consider the constant function $\sigma(E)^{-1/p}$ as a $p$-atom. Then, for each $p$ as in \eqref{range-p-1}, define the {\tt atomic Hardy space} $H^p_{at}(E,\rho,\sigma)$ as
\begin{eqnarray}\label{hp-at33} &&\hskip -0.40in H^p_{at}(E,\rho,\sigma):= \Bigl\{f\in\bigl(\dot{\mathscr{C}}^{d(1/p-1)}(E,\rho)\bigr)^\ast:\, \exists\,\{\lambda_j\}_{j\in{\mathbb{N}}}\in\ell^p({\mathbb{N}}) \mbox{ and $p$-atoms $\{a_j\}_{j\in{\mathbb{N}}}$} \nonumber\\[4pt] &&\hskip 1.20in \mbox{ such that } f=\sum_{j\in{\mathbb{N}}}\lambda_ja_j\,\,\mbox{ in }\,\, \bigl(\dot{\mathscr{C}}^{d(1/p-1)}(E,\rho)\bigr)^\ast\Bigr\}, \end{eqnarray}
and equip this space with the quasi-norm $\|\cdot\|_{H^p_{at}(E,\rho,\sigma)}$ defined for each $f\in H^p_{at}(E,\rho,\sigma)$ by
\begin{eqnarray}\label{Mac-12AAA}
\|f\|_{H^p_{at}(E,\rho,\sigma)}:=\inf\,\Bigl\{
\Bigl(\sum_{j\in{\mathbb{N}}}|\lambda_j|^p\Bigr)^{1/p}:\, f=\sum_{j\in{\mathbb{N}}}\lambda_ja_j\,\,\mbox{ as in \eqref{hp-at33}}\Bigr\}. \end{eqnarray}
The following atomic decomposition theorem, extending work in \cite{MaSe79II}, has been established in \cite{MMMM-G}.
\begin{theorem}\label{MacSeg-2} Assume that $(E,\rho,\sigma)$ is a $d$-dimensional {\rm ADR}. Then
\begin{eqnarray}\label{p-good.LL} H^p(E,\rho,\sigma)=\widetilde{H}^p(E,\rho,\sigma)=L^p(E,\sigma) \quad\mbox{for each }\,p\in(1,\infty). \end{eqnarray}
Suppose now that $p$ is as in \eqref{range-p-1} and, for every functional $f\in H^p_{at}(E,\rho,\sigma)$, denote by $\widetilde{f}$ the distribution in ${\mathscr{D}}'(E,\rho)$ defined as the restriction of $f$ to ${\mathscr{D}}(E,\rho)$. Then the assignment $f\mapsto\widetilde{f}$ induces a well-defined, injective linear mapping from $H^p_{at}(E,\rho,\sigma)$ onto the space $\widetilde{H}^p(E,\rho,\sigma)$. Moreover, for each
\begin{eqnarray}\label{Ugv-888} \gamma\in{\mathbb{R}}\,\,\mbox{ with }\,\, d\bigl({\textstyle{\frac{1}{p}}}-1\bigr)<\gamma<[\log_2 C_\rho]^{-1} \end{eqnarray}
there exist two finite constants $c_1,c_2>0$ such that
\begin{eqnarray}\label{MacSeg-21}
c_1\|f\|_{H^p_{at}(E,\rho,\sigma)}\leq
\|(\widetilde{f}\,)^*_{\rho_{\#},\gamma}\|_{L^p(E,\sigma)}
\leq c_2\|f\|_{H^p_{at}(E,\rho,\sigma)} \quad\mbox{for all }\,f\in H^p_{at}(E,\rho,\sigma). \end{eqnarray}
Consequently, the spaces $H^p(E,\rho,\sigma)$, $\widetilde{H}^p(E,\rho,\sigma)$ are naturally identified with $H^p_{at}(E,\rho,\sigma)$. In particular, they do not depend on the particular choice of the index $\gamma$ as in \eqref{Ugv-888}.
As a corollary, whenever \eqref{Ugv-888} holds one can find a finite constant $c=c(p,\rho,\gamma)>0$ such that for every distribution $f\in{\mathscr{D}}'(E,\rho)$ with the property that its grand maximal function $f^*_{\rho_{\#},\gamma}$ belongs to $L^p(E,\sigma)$ there exist a sequence of $p$-atoms $\{a_j\}_{j\in{\mathbb{N}}}$ on $X$ and a numerical sequence $\{\lambda_j\}_{j\in{\mathbb{N}}}\in\ell^p({\mathbb{N}})$ for which
\begin{eqnarray}\label{MacSeg-11} f=\sum_{j\in{\mathbb{N}}}\lambda_ja_j\quad\mbox{ in }\,\,{\mathscr{D}}'(E,\rho) \end{eqnarray}
and
\begin{eqnarray}\label{MacSeg-12}
\Bigl(\sum_{j\in{\mathbb{N}}}|\lambda_j|^p\Bigr)^{1/p}
\leq c\|f^*_{\rho_{\#},\gamma}\|_{L^p(E,\sigma)}. \end{eqnarray}
Finally, whenever \eqref{Ugv-888} holds one can find a finite constant $c'=c'(p,\rho,\gamma)>0$ such that, given a distribution $f\in{\mathscr{D}}'(E,\rho)$, a sequence of $p$-atoms $\{a_j\}_{j\in{\mathbb{N}}}$, and a numerical sequence $\{\lambda_j\}_{j\in{\mathbb{N}}}\in\ell^p({\mathbb{N}})$ such that \eqref{MacSeg-11} holds, then
\begin{eqnarray}\label{MacSeg-12B}
\|f^*_{\rho_{\#},\gamma}\|_{L^p(E,\sigma)}\leq c'\Bigl(\sum_{j\in{\mathbb{N}}}|\lambda_j|^p\Bigr)^{1/p}. \end{eqnarray} \end{theorem}
Consider now the setting of Section \ref{SSect:3.1} and suppose that $\theta$ is a function as in \eqref{K234} which satisfies \eqref{hszz} and such that there exists $\alpha\in(0,\infty)$ with the property that for all $x\in\mathscr{X}\setminus E$ and $y\in E$ there holds
\begin{eqnarray}\label{hszz-3noalpha} \begin{array}{l}
\displaystyle|\theta(x,y)-\theta(x,\widetilde{y})|\leq C_\theta \frac{\rho(y,\widetilde{y})^\alpha}{\rho(x,y)^{d+\upsilon+\alpha}}\,\Bigl( \frac{{\rm dist}_\rho(x,E)}{\rho(x,y)}\Bigr)^{-a}, \\[12pt] \qquad\forall\,\widetilde{y}\in E\,\,\mbox{ with }\,\, \rho(y,\widetilde{y})\leq\tfrac{1}{2}\rho(x,y). \end{array} \end{eqnarray}
We are now ready to present the first main result in this subsection.
\begin{theorem}\label{VGds-2} Let $d,m$ be two real numbers such that $0<d<m$. Assume that $({\mathscr{X}},\rho,\mu)$ is an $m$-dimensional {\rm ADR} space, $E$ is a closed subset of $({\mathscr{X}},\tau_\rho)$, and $\sigma$ is a Borel measure
on $(E,\tau_{\rho|_{E}})$ with the property that $(E,\rho\bigl|_E,\sigma)$ is a $d$-dimensional {\rm ADR} space.
Furthermore, suppose that $\Theta$ is the integral operator defined in \eqref{operator} with a kernel $\theta$ as in \eqref{K234}, \eqref{hszz}, \eqref{hszz-3noalpha}. Finally, fix $\kappa>0$ and, with $\alpha_\rho$ as in \eqref{Cro} and $\alpha$ as in \eqref{hszz-3noalpha}, set
\begin{eqnarray}\label{WQ-tDD} \gamma:=\min\,\bigl\{\alpha_\rho,\alpha\bigr\}. \end{eqnarray}
Given $q\in(1,\infty)$ and $p\in\bigl(\frac{d}{d+\gamma},\infty\bigr)$ consider the estimate
\begin{eqnarray}\label{kt-Dc} \hskip -0.20in
\left\|\Bigl(\int_{\Gamma_{\kappa}(x)}|(\Theta f)(y)|^q\, \frac{d\mu(y)}{\delta_E(y)^{m-q\upsilon}}\Bigr)^{\frac{1}{q}}
\right\|_{L^p_x(E,\sigma)}\!\!\!
\leq C\|f\|_{H^p(E,\rho|_{E},\sigma)},\quad\forall\,f\in H^p(E,\rho|_{E},\sigma), \end{eqnarray}
where $C>0$ is a finite constant.
\begin{enumerate} \item[(I)] Assume that $q\in(1,\infty)$ has the property that, for some finite constant $C>0$, either
\begin{eqnarray}\label{kt-Dc-BIS} \hskip -0.20in
\left\|\Bigl(\int_{\Gamma_{\kappa}(x)}|(\Theta f)(y)|^q\, \frac{d\mu(y)}{\delta_E(y)^{m-q\upsilon}}\Bigr)^{\frac{1}{q}}
\right\|_{L^q_x(E,\sigma)}\!\!\!
\leq C\|f\|_{L^q(E,\sigma)},\quad\forall\,f\in L^q(E,\sigma), \end{eqnarray}
or there exists $p_o\in(q,\infty)$ such that for every $f\in L^{p_o}(E,\sigma)$ there holds
\begin{eqnarray}\label{dtbjHT} \hskip -0.30in \sup_{\lambda>0}\left[\lambda\cdot
\sigma\Bigl(\Bigl\{x\in E:\int_{\Gamma_{\kappa}(x)}|(\Theta f)(y)|^q\, \frac{d\mu(y)}{\delta_E(y)^{m-q\upsilon}}>\lambda^{q}\Bigr\}\Bigr)^{1/p_o}\right]
\leq C\|f\|_{L^{p_o}(E,\sigma)}. \end{eqnarray}
Then \eqref{kt-Dc} holds for every $p\in\bigl(\frac{d}{d+\gamma},\infty\bigr)$.
\item[(II)] Assume that $q\in(1,\infty)$ is such that there exist $p_o\in(1,\infty)$ and a finite constant $C>0$ such that \eqref{dtbjHT} holds for every $f\in L^{p_o}(E,\sigma)$. Then \eqref{kt-Dc} holds for every $p\in(1,p_o)$ and, in addition, for every $f\in L^1(E,\sigma)$ one has
\begin{eqnarray}\label{d-YD23} \hskip -0.30in \sup_{\lambda>0}\left[\lambda\cdot
\sigma\Bigl(\Bigl\{x\in E:\int_{\Gamma_{\kappa}(x)}|(\Theta f)(y)|^q\, \frac{d\mu(y)}{\delta_E(y)^{m-q\upsilon}}>\lambda^{q}\Bigr\}\Bigr)\right]
\leq C\|f\|_{L^1(E,\sigma)}. \end{eqnarray} \end{enumerate} \end{theorem}
It is worth mentioning that the conclusion \eqref{kt-Dc} in Theorem~\ref{VGds-2} may be re-phrased as saying that the operator
\begin{eqnarray}\label{ki-DUD}
\delta_E^{\upsilon-m/q}\Theta:H^p(E,\rho|_{E},\sigma) \longrightarrow L^{(p,q)}({\mathscr{X}},E) \end{eqnarray}
is well-defined, linear and bounded.
To set the stage for presenting the proof of Theorem~\ref{VGds-2}, we state a lemma containing an estimate for a Marcinkiewicz-type integral (cf. \cite{MMMM-B} for a proof).
\begin{lemma}\label{P-Marc} Assume that $(E,\rho,\sigma)$ is a $d$-dimensional {\rm ADR} space for some $d>0$. Then for each $\alpha>0$ there exists $C\in(0,\infty)$ such that whenever $F$ is a nonempty closed subset of $(E,\tau_\rho)$ one has
\begin{eqnarray}\label{VG+ds} \int_{F}\int_{E}\frac{{\rm dist}_{\rho_{\#}}\,(y,F)^\alpha} {\rho_{\#}(x,y)^{d+\alpha}}\,d\sigma(y)\,d\sigma(x)\leq C\sigma(E\setminus F). \end{eqnarray} \end{lemma}
We are now prepared to present the
\vskip 0.08in \begin{proof}[Proof of Theorem~\ref{VGds-2}] We divide the proof into a number of cases.
\vskip 0.10in {\tt Case~1}: {\it Let $q\in[1,\infty)$, $p_o\in(1,\infty)$ be such that \eqref{dtbjHT} holds for each $f\in L^{p_o}(E,\sigma)$}. The main step in this scenario is proving that the operator ${\mathcal{A}}_{q,\kappa}\circ(\delta_E^{\upsilon-m/q}\Theta)$ is of weak type $(1,1)$, that is, that there exists $C>0$ such that for every $\lambda>0$ there holds
\begin{eqnarray}\label{DCvh} \sigma\Bigl(\bigl\{x\in E:\, {\mathcal{A}}_{q,\kappa}\bigl(\delta_E^{\upsilon-m/q}(\Theta f)\bigr)(x)
>\lambda\bigr\}\Bigr)\leq C\frac{\|f\|_{L^1(E,\sigma)}}{\lambda}, \qquad\forall\,f\in L^1(E,\sigma). \end{eqnarray}
Assuming \eqref{DCvh} for the moment, we proceed as follows. The operator ${\mathcal{A}}_{q,\kappa}\circ(\delta_E^{\upsilon-m/q}\Theta)$ is subadditive, of weak type $(1,1)$ by \eqref{DCvh}, and of weak type $(p_o,p_o)$ by \eqref{dtbjHT}. Hence, by the Marcinkiewicz interpolation theorem, ${\mathcal{A}}_{q,\kappa}\circ(\delta_E^{\upsilon-m/q}\Theta)$ is of strong type $(p,p)$ for every $p\in(1,p_o)$, yielding \eqref{kt-Dc} (after unraveling notation), for the specified range of $q,p_o,p$. As such, this takes care of the claim made in the first part of $(II)$ in the statement of the theorem. Moreover, \eqref{d-YD23} corresponds to \eqref{DCvh}, whose proof we now consider.
To get started, assume that $f\in L^1(E,\sigma)$ has been fixed.
When $0<\lambda\leq\|f\|_{L^1(E,\sigma)}/\sigma(E)$ (which may only happen in the case when $E$ is bounded), we have
\begin{eqnarray}\label{DCvh-BB} \sigma\Bigl(\bigl\{x\in E:\, {\mathcal{A}}_{q,\kappa}\bigl(\delta_E^{\upsilon-m/q}(\Theta f)\bigr)(x)
>\lambda\bigr\}\Bigr)\leq\sigma(E)\leq\frac{\|f\|_{L^1(E,\sigma)}}{\lambda}, \end{eqnarray}
so \eqref{DCvh} holds in this case if we choose $C\geq 1$.
Consider now the case when $\lambda>\|f\|_{L^1(E,\sigma)}/\sigma(E)$. There is no loss of generality in assuming that $f$ has bounded support, and we shall perform a Calder\'on-Zygmund decomposition of $f$ at level $\lambda$. More precisely, there exist two finite constants $C>0$, $N\in{\mathbb{N}}$ (depending only on geometry), along with an at most countable family of balls $(Q_j)_{j\in J}$, say $Q_j:=B_{\rho}(x_j,r_j)$ for each $j\in J$, and two functions $g,b:E\to{\mathbb{R}}$ satisfying the following properties (cf., e.g., \cite{CoWe71}):
\begin{eqnarray}\label{PRO-1} && \hskip -0.40in f=g+b\mbox{ on $E$}, \\[4pt] && \hskip -0.40in g\in L^1(E,\sigma)\cap L^\infty(E,\sigma),\quad
\|g\|_{L^1(E,\sigma)}\leq C\|f\|_{L^1(E,\sigma)},\quad
|g(x)|\leq C\lambda,\,\,\,\forall\,x\in E, \label{PRO-2} \\[4pt] && \hskip -0.40in b=\sum_{j\in J}b_j\,\,\mbox{ with }\,\, {\rm supp}\,b_j\subseteq Q_j,\,\,\,\int_{E}b_j\,d\sigma=0,\,\,\mbox{ and }\,\,
{\int{\mkern-19mu}-}_{Q_j}|b_j|\,d\sigma\leq C\lambda,\,\,\,\,\forall\,j\in J, \label{PRO-3} \\[4pt] && \hskip -0.40in \begin{array}{l} \mbox{if }{\mathcal{O}}:=\bigcup_{j\in{\mathbb{N}}}Q_j\subseteq E \,\,\mbox{ and }\,\,F:=E\setminus{\mathcal{O}},\,\,\mbox{ then }\,\, \sum_{j\in J}{\mathbf{1}}_{Q_j}\leq N, \\[8pt]
\sigma({\mathcal{O}})\leq\frac{C}{\lambda}\|f\|_{L^1(E,\sigma)}\,\,\mbox{ and }\,\, {\rm dist}_{\rho}(Q_j,F)\approx r_j\,\,\mbox{ uniformly in }j\in J. \end{array} \label{PRO-4} \end{eqnarray}
Note that the above properties also entail
$\sum_{j\in J}\|b_j\|_{L^1(E,\sigma)}\leq C\|f\|_{L^1(E,\sigma)}$, so the series in \eqref{PRO-3} converges absolutely in $L^1(E,\sigma)$.
By the quasi-subadditivity of ${\mathcal{A}}_{q,\kappa}\circ(\delta_E^{\upsilon-m/q}\Theta)$ and \eqref{PRO-1} we have
\begin{eqnarray}\label{DCvh-gUUU} {\mathcal{A}}_{q,\kappa}\bigl(\delta_E^{\upsilon-m/q}(\Theta f)\bigr)\leq {\mathcal{A}}_{q,\kappa}\bigl(\delta_E^{\upsilon-m/q}(\Theta g)\bigr) +{\mathcal{A}}_{q,\kappa}\bigl(\delta_E^{\upsilon-m/q}(\Theta b)\bigr) \end{eqnarray}
so, as far as \eqref{DCvh} is concerned, it suffices to prove that
\begin{eqnarray}\label{DCvh-g} \sigma\Bigl(\bigl\{x\in E:\, {\mathcal{A}}_{q,\kappa}\bigl(\delta_E^{\upsilon-m/q}(\Theta g)\bigr)(x)>
\lambda/2\bigr\}\Bigr)\leq C\frac{\|f\|_{L^1(E,\sigma)}}{\lambda}, \end{eqnarray}
and
\begin{eqnarray}\label{DCvh-b} \sigma\Bigl(\bigl\{x\in E:\, {\mathcal{A}}_{q,\kappa}\bigl(\delta_E^{\upsilon-m/q}(\Theta b)\bigr)(x)>
\lambda/2\bigr\}\Bigr)\leq C\frac{\|f\|_{L^1(E,\sigma)}}{\lambda}. \end{eqnarray}
Making use of \eqref{dtbjHT} (with $f$ replaced by $g$), \eqref{PRO-2} and keeping in mind that $p_o>1$ we obtain
\begin{eqnarray}\label{bmKK} && \hskip -0.80in \sigma\Bigl(\bigl\{x\in E:\, {\mathcal{A}}_{q,\kappa}\bigl(\delta_E^{\upsilon-m/q}(\Theta g)\bigr)(x) >\lambda/2\bigr\}\Bigr)
\leq C\Bigl(\frac{\|g\|_{L^{p_o}(E,\sigma)}}{\lambda}\Bigr)^{p_o} \nonumber\\[4pt] && \hskip 0.80in
\leq C\frac{\|g\|_{L^{\infty}(E,\sigma)}^{p_o-1}\|g\|_{L^1(E,\sigma)}}{\lambda^{p_o}}
\leq C\frac{\|f\|_{L^1(E,\sigma)}}{\lambda}, \end{eqnarray}
thus \eqref{DCvh-g} is proved. We are therefore left with proving \eqref{DCvh-b}. To justify this, first note that by \eqref{PRO-4} we have
\begin{eqnarray}\label{bmKK-2} \sigma\Bigl(\bigl\{x\in{\mathcal{O}}:\, {\mathcal{A}}_{q,\kappa}\bigl(\delta_E^{\upsilon-m/q} (\Theta b)\bigr)(x)>\lambda/2\bigr\}\Bigr)\leq\sigma({\mathcal{O}})
\leq C\frac{\|f\|_{L^1(E,\sigma)}}{\lambda}. \end{eqnarray}
Second, it is immediate that
\begin{eqnarray}\label{bmKK-3} \sigma\Bigl(\bigl\{x\in F:\, {\mathcal{A}}_{q,\kappa}\bigl(\delta_E^{\upsilon-m/q} (\Theta b)\bigr)(x)>\lambda/2\bigr\}\Bigr) \leq \frac{1}{\lambda}\int_F{\mathcal{A}}_{q,\kappa}\bigl(\delta_E^{\upsilon-m/q} (\Theta b)\bigr)\,d\sigma. \end{eqnarray}
Therefore, since $E={\mathcal{O}}\cup F$, in view of \eqref{bmKK-2} and \eqref{bmKK-3}, estimate \eqref{DCvh-b} will follow as soon as we prove that
\begin{eqnarray}\label{bmKK-3B} \int_F{\mathcal{A}}_{q,\kappa}\bigl(\delta_E^{\upsilon-m/q}
(\Theta b)\bigr)\,d\sigma\leq C\|f\|_{L^1(E,\sigma)} \end{eqnarray}
for some $C>0$ independent of $f$. With this goal in mind, we fix $j\in J$ and $x\in F$ arbitrary and look for a pointwise estimate for
\begin{eqnarray}\label{bmKK-3A} {\mathcal{A}}_{q,\kappa}\bigl(\delta_E^{\upsilon-m/q}(\Theta b_j)\bigr)(x)
=\Bigl(\int_{\Gamma_{\kappa}(x)}|(\Theta b_j)(y)|^q\delta_E(y)^{q\upsilon-m}\,d\mu(y) \Bigr)^{\frac{1}{q}}. \end{eqnarray}
With $x_j$ and $r_j$ denoting, respectively, the center and radius of $Q_j$, based on the third condition in \eqref{PRO-3}, for each $y\in\Gamma_{\kappa}(x)$, we may write
\begin{eqnarray}\label{bmKK-4}
\bigl|(\Theta b_j)(y)\bigr|
& = & \Bigl|\int_{E}\theta(y,z)b_j(z)\,d\sigma(z)\Bigr|
=\Bigl|\int_{E}\bigl[\theta(y,z)-\theta(y,x_j)\bigr]
b_j(z)\,d\sigma(z)\Bigr| \nonumber\\[4pt]
& = & \Bigl|\int_{Q_j}\bigl[\theta(y,z)-\theta(y,x_j)\bigr]
b_j(z)\,d\sigma(z)\Bigr|\leq I_1+I_2, \end{eqnarray}
where, for some small $\epsilon>0$ to be determined momentarily, we have set
\begin{eqnarray}\label{bmKK-4B} I_1:=\!\!\!\! \int\limits_{\stackrel{z\in Q_j}{\rho_{\#}(z,x_j)<\epsilon\rho_{\#}(y,x_j)}}
\!\!\!\!\bigl|\theta(y,z)-\theta(y,x_j)\bigr||b_j(z)|\,d\sigma(z), \\[4pt] I_2:=\!\!\!\! \int\limits_{\stackrel{z\in Q_j}{\rho_{\#}(z,x_j)\geq\epsilon\rho_{\#}(y,x_j)}}
\!\!\!\!\bigl|\theta(y,z)-\theta(y,x_j)\bigr||b_j(z)|\,d\sigma(z). \label{bmKK-4B.2} \end{eqnarray}
Note that, by \eqref{DEQV1}, if $\rho_{\#}(z,x_j)<\epsilon\rho_{\#}(y,x_j)$ then
\begin{eqnarray}\label{b-YH53} \rho(z,x_j)\leq C_\rho^2\rho_{\#}(z,x_j)<\epsilon C_\rho^2\rho_{\#}(y,x_j) \leq\epsilon\widetilde{C}_{\rho}C_\rho^2\rho_{\#}(y,x_j) <\tfrac{1}{2}\rho(y,x_j) \end{eqnarray}
if $0<\epsilon<2^{-1}\widetilde{C}_{\rho}^{-1}C_\rho^{-2}$. Hence, for this choice of $\epsilon$, we have $\rho(z,x_j)<\frac{1}{2}\rho(y,x_j)$ on the domain of integration in $I_1$. Based on this, \eqref{hszz-3} and \eqref{PRO-3}, we may then estimate this term as follows
\begin{eqnarray}\label{bmKK-4B.3} I_1 & \leq & C\int_{Q_j}\frac{\rho_{\#}(z,x_j)^\alpha\delta_E(y)^{-a}}
{\rho_{\#}(y,x_j)^{d+\upsilon+\alpha-a}}|b_j(z)|\,d\sigma(z) \nonumber\\[4pt] & \leq & C\frac{r_j^\alpha\delta_E(y)^{-a}}{\rho_{\#}(y,x_j)^{d+\upsilon+\alpha-a}}
\int_{Q_j}|b_j(z)|\,d\sigma(z) \leq C\lambda\,\frac{r_j^\alpha\delta_E(y)^{-a}\sigma(Q_j)} {\rho_{\#}(y,x_j)^{d+\upsilon+\alpha-a}}. \end{eqnarray}
Estimating $I_2$ requires a few geometrical preliminaries. Recall that $x\in F$, $y\in\Gamma_\kappa(x)$ and fix an arbitrary point $z\in Q_j$ such that $\rho(z,x_j)\geq\epsilon\rho(y,x_j)$, where $\epsilon>0$ is as above. Since, on the one hand,
\begin{eqnarray}\label{bmKK-4b} r_j & \approx & {\rm dist}_\rho(Q_j,F)\leq\widetilde{C}_{\rho} \rho(x,x_j)\leq C\rho(x,y)+C\rho(y,x_j) \nonumber\\[4pt] & \leq & C(1+\kappa)\delta_E(y)+C\rho(y,x_j)\leq C\rho(y,x_j), \end{eqnarray}
while, on the other hand, the fact that $z\in Q_j$ forces $\rho(x_j,z)<r_j$ which in turn allow us to estimate $\rho(y,x_j)\leq\epsilon^{-1}\rho(z,x_j)\leq\epsilon^{-1} \widetilde{C}_{\rho}\rho(x_j,z)<\epsilon^{-1}\widetilde{C}_{\rho}r_j$. Hence, ultimately,
\begin{eqnarray}\label{bmKK-4b.L} r_j\approx\rho(y,x_j),\quad \mbox{uniformly in $j\in J$ and $y\in\Gamma_{\kappa}(x)$ with $x\in F$}. \end{eqnarray}
In addition, the same type of estimate as in \eqref{bmKK-4b} written with $x_j$ replaced by $z$ yields $r_j\leq C\rho(y,z)$, which further implies
\begin{eqnarray}\label{bmKK-4b.F} \rho(y,x_j)\leq C\rho(y,z)+C\rho(z,x_j)\leq C\rho(y,z)+Cr_j\leq C\rho(y,z), \end{eqnarray}
for some constant $C\in(0,\infty)$ independent of $j,x,y,z$. Hence, from \eqref{bmKK-4b.L} and \eqref{bmKK-4b.F}, we obtain
\begin{eqnarray}\label{bmKK-4b.H} \frac{1}{\rho(y,z)^{d+\upsilon}}\leq\frac{C}{\rho(y,x_j)^{d+\upsilon}} \leq\frac{Cr_j^\alpha}{\rho(y,x_j)^{d+\upsilon+\alpha}}. \end{eqnarray}
Consequently, on the domain of integration in $I_2$ we have thanks to \eqref{hszz} and \eqref{bmKK-4b.H}
\begin{eqnarray}\label{4b.H2A}
\bigl|\theta(y,z)-\theta(y,x_j)\bigr|\leq\frac{C\delta_E(y)^{-a}}{\rho_{\#}(y,z)^{d+\upsilon-a}} +\frac{C\delta_E(y)^{-a}}{\rho_{\#}(y,x_j)^{d+\upsilon-a}} \leq\frac{Cr_j^\alpha\delta_E(y)^{-a}}{\rho_{\#}(y,x_j)^{d+\upsilon+\alpha-a}}. \end{eqnarray}
Together with \eqref{PRO-3}, this allows us to estimate (recall that $I_2$ has been defined in \eqref{bmKK-4B.2})
\begin{eqnarray}\label{b-4b.H2} I_2 &\leq & \frac{Cr_j^\alpha\delta_E(y)^{-a}}{\rho_{\#}(y,x_j)^{d+\upsilon+\alpha-a}} \!\!\!\! \int\limits_{\stackrel{z\in Q_j\,\,\mbox{\tiny{such that}}} {\rho_{\#}(z,x_j)\geq\epsilon\rho_{\#}(y,x_j)}}
\!\!\!\!|b_j(z)|\,d\sigma(z) \leq C\lambda\,\frac{r_j^\alpha\delta_E(y)^{-a}\sigma(Q_j)} {\rho_{\#}(y,x_j)^{d+\upsilon+\alpha-a}}. \end{eqnarray}
Cumulatively, \eqref{bmKK-4}, \eqref{bmKK-4B.3} and \eqref{b-4b.H2} prove that there exists $C\in(0,\infty)$ with the property that, for every $j\in J$,
\begin{eqnarray}\label{bmKK-4i} x\in F\,\Longrightarrow\,
\bigl|(\Theta b_j)(y)\bigr| \leq C\lambda\,\frac{r_j^\alpha\delta_E(y)^{-a}\sigma(Q_j)} {\rho_{\#}(y,x_j)^{d+\upsilon+\alpha-a}}, \qquad\forall\,y\in\Gamma_\kappa(x). \end{eqnarray}
Utilizing \eqref{bmKK-4i} in \eqref{bmKK-3A}, it follows that for every $j\in J$ and $x\in F$
\begin{eqnarray}\label{bmKK-5} {\mathcal{A}}_{q,\kappa}\bigl(\delta_E^{\upsilon-m/q}(\Theta b_j)\bigr)(x) \leq C\lambda\,r_j^\alpha\sigma(Q_j)\Bigl(\int_{\Gamma_{\kappa}(x)} \frac{\delta_E(y)^{q(\upsilon-a)-m}}{\rho_{\#}(y,x_j)^{q(d+\upsilon+\alpha-a)}} \,d\mu(y)\Bigr)^{\frac{1}{q}}. \end{eqnarray}
At this point we make use of Lemma~\ref{bzrg1} (recall that $\nu-a>0$) to further bound the last integral in \eqref{bmKK-5} and obtain that for every $j\in J$ and $x\in F$
\begin{eqnarray}\label{bmKK-7} {\mathcal{A}}_{q,\kappa}\bigl(\delta_E^{\upsilon-m/q}(\Theta b_j)\bigr)(x) \leq C\lambda\,r_j^\alpha\sigma(Q_j)\rho(x,x_j)^{-d-\alpha} \leq C\lambda\int_{Q_j}\frac{{\rm dist}_{\rho_{\#}}(z,F)^\alpha} {\rho_{\#}(x,z)^{d+\alpha}}\,d\sigma(z). \end{eqnarray}
Two geometrical inequalities that have been used in the last step in \eqref{bmKK-7} are as follows. First, ${\rm dist}_{\rho_{\#}}(z,F)\approx r_j$, uniformly for $z\in Q_j$ and, second, for every $z\in Q_j$ we have
\begin{eqnarray}\label{QLVp} \rho(x,z) &\leq & C\rho(x,x_j)+C\rho(x_j,z)\leq C\rho(x,x_j)+Cr_j \nonumber\\[4pt] &\leq & C\rho(x,x_j)+C\,{\rm dist}_{\rho}(Q_j,F)\leq C\rho(x,x_j). \end{eqnarray}
Summing up inequalities of the form \eqref{bmKK-7} over $j\in{\mathbb{N}}$ and using the sublinearity of the operator ${\mathcal{A}}_{q,\kappa}\bigl(\delta_E^{\upsilon-m/q}\Theta(\cdot)\bigr)$ (recall that $q\geq 1$), as well as the finite overlap property in \eqref{PRO-4}, we obtain
\begin{eqnarray}\label{bmKK-8} {\mathcal{A}}_{q,\kappa}\bigl(\delta_E^{\upsilon-m/q}(\Theta b)\bigr)(x) \leq C\lambda\int_{{\mathcal{O}}}\frac{{\rm dist}_{\rho_{\#}}(z,F)^\alpha} {\rho_{\#}(x,z)^{d+\alpha}}\,d\sigma(z),\qquad\forall\,x\in F. \end{eqnarray}
Consequently, from \eqref{bmKK-8}, Lemma~\ref{P-Marc} and \eqref{PRO-4}, we deduce that
\begin{eqnarray}\label{bmKK-9} && \hskip -0.20in \int_F{\mathcal{A}}_{q,\kappa}\bigl(\delta_E^{\upsilon-m/q}(\Theta b)\bigr)(x)\,dx \leq C\lambda\int_F \int_{{\mathcal{O}}} \frac{{\rm dist}_{\rho_{\#}}(z,F)^\alpha}{\rho_{\#}(x,z)^{d+\alpha}}\,d\sigma(z) \,d\sigma(x) \\[4pt] && \hskip 0.30in \leq C\lambda\int_F\int_{E} \frac{{\rm dist}_{\rho_{\#}}(z,F)^\alpha}{\rho_{\#}(x,z)^{d+\alpha}}\,d\sigma(z) \,d\sigma(x) \leq C\lambda\,\sigma(E\setminus F)=C\lambda\,\sigma({\mathcal{O}})
\leq C\|f\|_{L^1(E,\sigma)}. \nonumber \end{eqnarray}
This proves \eqref{bmKK-3B}, thus completing the proof of \eqref{DCvh}. In summary, the analysis so far proves part $(II)$ in the statement of the theorem.
\vskip 0.10in {\tt Case~2}: {\it Assume that \eqref{dtbjHT} holds for some $1<q<p_o<\infty$.} As a preliminary step, we make the observation that, in this scenario, granted \eqref{dtbjHT} and the conclusion in the Case~1,
\begin{eqnarray}\label{Area-p2XXX} {\mathscr{A}}_{q,\kappa}\circ\bigl(\delta_E^{\upsilon-m/q}\Theta\bigr): L^r(E,\sigma)\to L^r(E,\sigma)\quad\mbox{is bounded whenever $r\in(1,p_o)$}. \end{eqnarray}
Because of the equivalence \eqref{sbrn} in Theorem~\ref{AsiC}, estimate \eqref{kt-Dc} for the range $p\in(q,\infty)$ will follow once we show that
\begin{eqnarray}\label{PaSeD} {\mathfrak{C}}_{q,\kappa}\circ(\delta_E^{\upsilon-m/q}\Theta):L^p(E,\sigma)\to L^p(E,\sigma)\quad\mbox{is bounded for $q<p<\infty$}. \end{eqnarray}
Fix $p\in(q,\infty)$. The proof of the boundedness of the operator in \eqref{PaSeD} relies on the following pointwise estimate
\begin{eqnarray}\label{PaSeD-2} && \hskip -0.40in {\mathfrak{C}}_{q,\kappa}\bigl(\delta_E^{\upsilon-m/q}(\Theta f)\bigr)(x_0) \\[4pt] && \hskip 0.30in \leq C\bigl[
\bigl(M_E(|f|^q)(x_0)\bigr)^{\frac{1}{q}}+(M_E(M_E(f)))(x_0)\bigr],\qquad \forall\,x_0\in E, \nonumber \end{eqnarray}
for each $f\in L^p(E,\sigma)$. Indeed, fix such a function $f$. After raising the inequality in \eqref{PaSeD-2} to the $p$-th power and then integrating over $E$, we obtain
\begin{eqnarray}\label{PaSeD-3} && \hskip -0.40in \int_{E}\bigl[{\mathfrak{C}}_{q,\kappa}\bigl(\delta_E^{\upsilon-m/q} (\Theta f)\bigr)(x)\bigr]^p\,d\sigma(x) \\[4pt] && \hskip 0.30in
\leq C\int_{E}\bigl[M_E(|f|^q)(x)\bigr]^{\frac{p}{q}}\,d\sigma(x) +C\int_{E}\bigl[(M_E^2f)(x)\bigr]^p\,d\sigma(x)
\leq C\int_{E}|f|^p\,d\sigma, \nonumber \end{eqnarray}
where the last inequality in \eqref{PaSeD-3} uses the boundedness on $L^p(E,\sigma)$ and $L^{p/q}(E,\sigma)$ of the Hardy-Littlewood maximal operator $M_E$ (here we make use of the fact that in the current case $p>\max\{q,1\}$). This shows that \eqref{PaSeD} holds assuming \eqref{PaSeD-2}.
Returning to the proof of \eqref{PaSeD-2}, fix $f\in L^p(E,\sigma)$ along with $r>0$ and $x_0\in E$. For some finite constant $c>0$ to be specified later, set $\Delta:=E\cap B_{\rho_{\#}}(x_0,r)$ and $c\Delta:=E\cap B_{\rho_{\#}}(x_0,cr)$, then write $f=f_1+f_2$, where $f_1:=f{\mathbf{1}}_{c\Delta}$ and $f_2:=f{\mathbf{1}}_{E\setminus c\,\Delta}$. First we estimate the contribution from $f_1$ by writing
\begin{eqnarray}\label{PaSeD-4} &&\hskip -0.50in \tfrac{1}{\sigma(\Delta)}\int_{{\mathcal{T}}_\kappa(\Delta)}
|(\Theta f_1)(x)|^q\delta_E(x)^{q\upsilon-(m-d)}\,d\mu(x) \nonumber\\[4pt] &&\hskip 0.50in
\leq \tfrac{1}{\sigma(\Delta)}\int_{{\mathscr{X}}\setminus E}
|(\Theta f_1)(x)|^q\delta_E(x)^{q\upsilon-(m-d)}\,d\mu(x) \nonumber\\[4pt] &&\hskip 0.50in \leq \tfrac{C}{\sigma(\Delta)}
\int_E\Bigl(\int_{\Gamma_{\kappa}(x)}|(\Theta f_1)(y)|^q\, \delta_E(y)^{q\upsilon-m}\,d\mu(y)\Bigr)\,d\sigma(x) \nonumber\\[4pt] &&\hskip 0.50in
=C\bigl\|{\mathscr{A}}_{q,\kappa}\bigl(\delta_E^{\upsilon-m/q}(\Theta f_1)\bigr)
\bigr\|_{L^q(E,\sigma)}^q\leq\tfrac{C}{\sigma(\Delta)}\int_{E}|f_1|^q\,d\sigma \nonumber\\[4pt] &&\hskip 0.50in
=\tfrac{C}{\sigma(\Delta)}\int_{c\,\Delta}|f|^q\,d\sigma\leq C\,M_E(|f|^q)(x_0). \end{eqnarray}
For the second inequality in \eqref{PaSeD-4} we have used \eqref{Mix+FR},
Lemma~\ref{lbDV}, and the fact that $(E,\rho|_{E},\sigma)$ is a $d$-dimensional {\rm ADR} space, while the third inequality follows from \eqref{Area-p2XXX} used with $r:=q\in(1,p_o)$.
To treat the term corresponding to $f_2$, observe that if $c>C_\rho$, then for every $y\in E\setminus c\,\Delta$ we have $cr<\rho_{\#}(y,x_0)\leq C_\rho\max\{\rho_{\#}(y,{w}),\rho_{\#}({w},x_0)\} \leq C_\rho\rho_{\#}(y,{w})$ for every ${w}\in\Delta$. Hence, $E\setminus c\,\Delta\subseteq\{y\in E:\,\rho_{\#}(y,{w})>r\}$ and $\rho_{\#}(y,x_0)\approx\rho_{\#}(y,{w})$, uniformly for $y\in E\setminus c\,\Delta$ and ${w}\in\Delta$. Furthermore, for every $z\in {\mathcal{T}}_\kappa(\Delta)$, we have
\begin{eqnarray}\label{brz} \rho_{\#}(y,x_0) &\leq & C_\rho\max\{\rho_{\#}(y,z),\rho_{\#}(z,x_0)\} \leq C_\rho\max\{\rho_{\#}(y,z),(1+\kappa)\delta_E(z)\} \nonumber\\[4pt] &\leq & C\rho_{\#}(y,z). \end{eqnarray}
Based on these considerations as well as \eqref{hszz} and \eqref{WBA}, if $z\in {\mathcal{T}}_\kappa(\Delta)$ we may write
\begin{eqnarray}\label{PaSeD-5}
|(\Theta f_2)(z)| & \leq & C\int\limits_{E\setminus c\,\Delta}
\frac{\delta_E(z)^{-a}}{\rho_{\#}(z,y)^{d+\upsilon-a}}|f(y)|\,d\sigma(y) \nonumber\\[4pt] & \leq & \frac{C\delta_E(z)^{-a}}{r^{\upsilon-a}}\int\limits_{y\in E,\,\rho_{\#}(y,{w})>r}
\frac{r^{\upsilon-a}}{\rho(y,{w})^{d+\upsilon-a}}|f(y)|\,d\sigma(y) \nonumber\\[4pt] & \leq & \frac{C\delta_E(z)^{-a}}{r^{\upsilon-a}}(M_Ef)({w}), \quad\mbox{uniformly for }{w}\in\Delta. \end{eqnarray}
Thus, \eqref{PaSeD-5} implies
\begin{eqnarray}\label{PaSeD-6}
|(\Theta f_2)(z)| \leq \frac{C\delta_E(z)^{-a}}{r^{\upsilon-a}}\inf_{{w}\in\Delta}(M_Ef)({w}),\qquad \forall\,z\in {\mathcal{T}}_\kappa(\Delta). \end{eqnarray}
In concert with \eqref{3.2.63WS} and Lemma~\ref{geom-lem} (which uses $\upsilon-a>0$), estimate \eqref{PaSeD-6} further yields
\begin{eqnarray}\label{PaSeD-7} &&\hskip -0.50in \Bigl[\tfrac{1}{\sigma(\Delta)}\int_{{\mathcal{T}}_\kappa(\Delta)}
|(\Theta f_2)(z)|^q\delta_E(z)^{q\upsilon-(m-d)}\,d\mu(z)\Bigr]^{\frac{1}{q}} \nonumber\\[4pt] && \hskip 0.50in \leq \frac{C}{r^{\upsilon-a}}\inf_{{w}\in\Delta}(M_Ef)({w}) \Bigl[\tfrac{1}{\sigma(\Delta)}\int_{B_{\rho_{\#}}(x_0,Cr)\setminus E} \delta_E(z)^{q(\upsilon-a)-(m-d)}\,d\mu(z)\Bigr]^{\frac{1}{q}} \nonumber\\[4pt] && \hskip 0.50in \leq C\inf_{{w}\in\Delta}(M_Ef)({w}) \leq C{\int{\mkern-19mu}-}_{\Delta}M_Ef\,d\sigma\leq CM_E(M_Ef)(x_0). \end{eqnarray}
Now \eqref{PaSeD-2} follows from \eqref{PaSeD-4} and \eqref{PaSeD-5} in view of \eqref{ktEW-7} and the fact that ${\mathfrak{C}}_{q,\kappa}\circ\bigl(\delta_E^{\upsilon-m/q}\Theta\bigr)$ is, in the current case, sub-linear.
In summary, the analysis in this case proves that, under the assumption \eqref{dtbjHT}, estimate \eqref{kt-Dc} holds whenever $1<q<p_o<\infty$ and $q<p<\infty$.
\vskip 0.10in {\tt Case~3}: {\it Assume that $q\in(1,\infty)$ is such that \eqref{kt-Dc-BIS} holds.} We claim that
\begin{eqnarray}\label{PaSeD-BIS} {\mathfrak{C}}_{q,\kappa}\circ(\delta_E^{\upsilon-m/q}\Theta):L^p(E,\sigma)\to L^p(E,\sigma)\quad\mbox{is bounded for $q<p<\infty$}. \end{eqnarray}
The proof of \eqref{PaSeD-BIS} largely parallels that of \eqref{PaSeD}. More specifically, the only significant difference occurs in the third inequality in \eqref{PaSeD-4} which, this time, follows directly from \eqref{kt-Dc-BIS}. Once this has been established, the equivalence \eqref{sbrn} in Theorem~\ref{AsiC}, and the current assumption yield \eqref{kt-Dc} for the range $p\in[q,\infty)$.
\vskip 0.10in {\tt Case~4}: {\it Assume $\frac{d}{d+\gamma}<p\leq1$ and $q\in[p,\infty)$, and suppose that}
\begin{eqnarray}\label{Ar-EEE} {\mathscr{A}}_{q,\kappa}\circ\bigl(\delta_E^{\upsilon-m/q}\Theta\bigr): L^q(E,\sigma)\to L^q(E,\sigma)\quad\mbox{is bounded}. \end{eqnarray}
In this case, we shall prove that there exists $C\in(0,\infty)$ such that
\begin{eqnarray}\label{hnv}
\bigl\|{\mathcal{A}}_{q,\kappa}\bigl(\delta_E^{\upsilon-m/q}\Theta(a)\bigr)
\bigr\|_{L^p(E,\sigma)}^p\leq C,\qquad\mbox{for every $p$-atom $a$}. \end{eqnarray}
With this goal in mind, fix a $p$-atom $a$ and let $x_0\in E$ and $r>0$ be such that the conditions in \eqref{jk-AM} hold. In particular,
\begin{eqnarray}\label{hnv-Uj} {\rm supp}\,a\subseteq B_{\rho_{\#}}(x_0,\widetilde{C}_\rho r). \end{eqnarray}
Then, for some finite constant $c>1$ to be specified later, and with $\Delta:=E\cap B_{\rho_{\#}}(x_0,cr)$, we have
\begin{eqnarray}\label{hnv-2} && \hskip -0.50in
\bigl\|{\mathcal{A}}_{q,\kappa}\bigl(\delta_E^{\upsilon-m/q}\Theta(a)\bigr)
\bigr\|_{L^p(E,\sigma)}^p=\int_{\Delta}\Bigl(\int_{\Gamma_\kappa(x)}
|(\Theta a)(y)|^q\delta_E(y)^{q\upsilon-m}\,d\mu(y)\Bigr)^{\frac{p}{q}}d\sigma(x) \\[4pt] && \hskip 1.00in +\int_{E\setminus \Delta}\Bigl(\int_{\Gamma_\kappa(x)}
|(\Theta a)(y)|^q\delta_E(y)^{q\upsilon-m}\,d\mu(y)\Bigr)^{\frac{p}{q}}d\sigma(x) =:I_1+I_2. \nonumber \end{eqnarray}
Using H\"older's inequality (with exponent $q/p\geq 1$), the fact
that $(E,\rho|_{E},\sigma)$ is a $d$-dimensional {\rm ADR} space, \eqref{Ar-EEE}, and \eqref{jk-AM}, we may write
\begin{eqnarray}\label{jnbj} I_1 &\leq & C\left[\int_{\Delta}\Bigl(\int_{\Gamma_\kappa(x)}
|(\Theta a)(y)|^q\delta_E(y)^{q\upsilon-m}\,d\mu(y)\Bigr)\,d\sigma(x) \right]^{\frac{p}{q}}\,r^{d\bigl(1-\frac{p}{q}\bigr)} \nonumber\\[4pt]
& \leq & C\bigl\|{\mathcal{A}}_{q,\kappa}\bigl(\delta_E^{\upsilon-m/q}\Theta(a)\bigr)
\bigr\|_{L^q(E,\sigma)}^p\,r^{d\bigl(1-\frac{p}{q}\bigr)}
\leq C\|a\|^p_{L^q(E,\sigma)}\,r^{d\bigl(1-\frac{p}{q}\bigr)}\leq C, \end{eqnarray}
for some finite $C>0$ independent of $a$. We are left with estimating $I_2$. First, we look for a pointwise estimate for $\Theta a$. Fix $x\in E\setminus\Delta$ and $y\in\Gamma_\kappa(x)$. Then for every $z\in E\cap B_{\rho_{\#}}(x_0,\widetilde{C}_\rho r)$ we have
\begin{eqnarray}\label{Jnb-H} \rho_{\#}(x_0,z) &\leq & \widetilde{C}_\rho r\leq\tfrac{1}{c}\,\rho_{\#}(x,x_0) \leq \tfrac{1}{c}\,\widetilde{C}_\rho C_{\rho_{\#}}\max\{\rho_{\#}(x,y),\rho_{\#}(y,x_0)\} \nonumber\\[4pt] &\leq & \tfrac{1}{c}\,\widetilde{C}_\rho C_{\rho}\max\{(1+\kappa)\delta_E(y),\rho_{\#}(y,x_0)\} \nonumber\\[4pt] &\leq & \tfrac{1}{c}\,\widetilde{C}_\rho C_{\rho}(1+\kappa)\rho_{\#}(y,x_0). \end{eqnarray}
Now, based on this and \eqref{DEQV1}, by choosing $c$ sufficiently large we conclude that
\begin{eqnarray}\label{Jnb-HsA} \rho(z,x_0)\leq\tfrac{1}{2}\rho(y,x_0)\quad\mbox{ for every }\,\, z\in E\cap B_{\rho_{\#}}(x_0,\widetilde{C}_\rho r). \end{eqnarray}
At this point, we may use the last condition in \eqref{jk-AM}, \eqref{hnv-Uj}, \eqref{Jnb-HsA}, \eqref{hszz-3noalpha}, the second condition in \eqref{jk-AM}
and the fact that $(E,\rho|_E,\sigma)$ is a $d$-dimensional {\rm ADR} space in order to obtain
\begin{eqnarray}\label{jnbj-2}
|(\Theta a)(y)| & = & \left|\int_E[\theta(y,z)-\theta(y,x_0)]a(z)\,d\sigma(z)\right| \nonumber\\[4pt]
& = & \Bigl|\int\limits_{E\cap B_{\rho_{\#}}(x_0,\widetilde{C}_\rho r)}
[\theta(y,z)-\theta(y,x_0)]a(z)\,d\sigma(z)\Bigr| \nonumber\\[4pt] & \leq & C \int\limits_{E\cap B_{\rho_{\#}}(x_0,\widetilde{C}_\rho r)} \frac{\rho_{\#}(z,x_0)^\alpha\delta_E(y)^{-a}}
{\rho_{\#}(y,x_0)^{d+\upsilon+\alpha-a}}\,|a(z)|\,d\sigma(z) \nonumber\\[4pt] & \leq & C 2^{\gamma-\alpha}\delta_E(y)^{-a} \int\limits_{E\cap B_{\rho_{\#}}(x_0,\widetilde{C}_\rho r)}
\frac{\rho_{\#}(z,x_0)^\gamma}{\rho_{\#}(y,x_0)^{d+\upsilon-a+\gamma}}\,|a(z)|\,d\sigma(z) \nonumber\\[4pt] & \leq &C \frac{\delta_E(y)^{-a}r^{\gamma+d\bigl(1-\frac{1}{p}\bigr)}} {\rho_{\#}(y,x_0)^{d+\upsilon-a+\gamma}}, \qquad\forall\,y\in \Gamma_\kappa(x). \end{eqnarray}
In turn, \eqref{jnbj-2} yields
\begin{eqnarray}\label{jnbj-3}
\int_{\Gamma_\kappa(x)}|(\Theta a)(y)|^q\frac{d\mu(y)}{\delta_E(y)^{m-q\upsilon}}
& \leq & C r^{q\gamma+qd\bigl(1-\frac{1}{p}\bigr)} \int_{\Gamma_\kappa(x)}\frac{\delta_E(y)^{q(\upsilon-a)-m}} {\rho_{\#}(y,x_0)^{q(d+\upsilon-a+\gamma)}}\,d\mu(y) \nonumber\\[4pt] & \leq & C \frac{r^{q\gamma+qd\bigl(1-\frac{1}{p}\bigr)}} {\rho_{\#}(x,x_0)^{qd+q\gamma}},\qquad\forall\,x\in E\setminus\Delta, \end{eqnarray}
where for the last inequality in \eqref{jnbj-3} we applied Lemma~\ref{bzrg1}. Estimate \eqref{jnbj-3} used in $I_2$ further implies
\begin{eqnarray}\label{jnbj-4} I_2 & \leq & C\,r^{p\gamma+pd\bigl(1-\frac{1}{p}\bigr)} \int_{E\setminus\Delta}\frac{d\sigma(x)}{\rho_{\#}(x,x_0)^{pd+p\gamma}} \nonumber\\[4pt] & \leq & C \frac{r^{p\gamma+pd\bigl(1-\frac{1}{p}\bigr)}} {r^{pd+p\gamma-d}}=C, \end{eqnarray}
where the last inequality in \eqref{jnbj-4} is a consequence of \eqref{WBA} (used with $f\equiv 1$) and the fact that $p(d+\gamma)>d$. Now \eqref{hnv} follows from \eqref{hnv-2}, \eqref{jnbj} and \eqref{jnbj-4}.
\vskip 0.10in {\tt Case~5}: {\it Assume $\frac{d}{d+\gamma}<p\leq1\leq q<\infty$, and suppose that \eqref{Ar-EEE} holds.} Then we claim that
\begin{eqnarray}\label{Ar-EEE.2+} \begin{array}{c} \delta_E^{\upsilon-m/q}\Theta:
H^p(E,\rho|_E,\sigma)\to L^{(p,q)}({\mathscr{X}},E,\mu,\sigma;\kappa)\quad\mbox{is bounded} \\[4pt] \mbox{whenever $\tfrac{d}{d+\gamma}<p\leq 1\leq q<+\infty$}. \end{array} \end{eqnarray}
To proceed with the proof of this claim, fix $p$ and $q$ as in \eqref{Ar-EEE.2+} and define the sets
\begin{eqnarray}\label{AKp-DD}
\dot{\mathscr{C}}_{b,0}^\gamma(E,\rho|_E):=\Bigl\{f\in \dot{\mathscr{C}}^\gamma(E,\rho|_E) :\,f\mbox{ has bounded support and } {\textstyle\int_Ef\,d\sigma=0}\Bigr\} \end{eqnarray}
and
\begin{eqnarray}\label{AKp-RR} {\mathcal{F}}(E):=&\left\{ \begin{array}{l}
\dot{\mathscr{C}}_{b,0}^\gamma(E,\rho|_E)\,\mbox{ if $E$ is unbounded}, \\[8pt]
\dot{\mathscr{C}}_{b,0}^\gamma(E,\rho|_E)\cup\{{\mathbf{1}}_E\}\,\mbox{ if $E$ is bounded}. \end{array} \right. \end{eqnarray}
Then letting
\begin{eqnarray}\label{AKp-2Z} {\mathcal{D}}_0(E):=\mbox{the finite linear span of functions in ${\mathcal{F}}(E)$}, \end{eqnarray}
we shall show that
\begin{eqnarray}\label{AKp-2}
{\mathcal{D}}_0(E)\,\,\mbox{ is dense in $H^p(E,\rho|_E,\sigma)$.} \end{eqnarray}
Indeed, since finite linear spans of $p$-atoms are dense in $H^p(E,\rho|_E,\sigma)$, the density result formulated in \eqref{AKp-2} will follow once we show that individual $p$-atoms may be approximated
in $H^p(E,\rho|_E,\sigma)$ with functions from ${\mathcal{D}}_0(E)$. To prove the latter, recall the approximation to the identity of order $\gamma$ as given in Proposition~\ref{Besov-ST} and observe that from the properties of the integral kernels from Definition~\ref{Besov-S} we have that ${\mathcal{S}}_l\,a\in{\mathcal{D}}_0(E)$ for every $p$-atom $a$ and each $l\in{\mathbb{N}}$. This and \cite[Lemma~3.2, (iii), p.\,108]{HYZ}, which gives that
\begin{eqnarray}\label{BG-IF} \begin{array}{c}
\{{\mathcal{S}}_l\}_{l\in{\mathbb{N}}}\,\,\mbox{is uniformly bounded from $H^p(E,\rho|_E,\sigma)$ to $H^p(E,\rho|_E,\sigma)$}\,\,\mbox{ and} \\[4pt]
{\mathcal{S}}_l f\to f\,\,\mbox{ in $H^p(E,\rho|_E,\sigma)$ as $l\to +\infty$},
\quad\forall\,f\in H^p(E,\rho|_E,\sigma), \end{array} \end{eqnarray}
now yield the desired conclusion, finishing the proof of \eqref{AKp-2}.
The next task is to prove that there exists $C\in(0,\infty)$ such that
\begin{eqnarray}\label{BB-ZZ}
\|\delta_E^{\upsilon-m/q}\Theta f\|_{L^{(p,q)}({\mathscr{X}},E,\mu,\sigma;\kappa)}
\leq C\|f\|_{H^p(E,\rho|_E,\sigma)},\quad\forall\,f\in{\mathcal{D}}_0(E). \end{eqnarray}
Assume for the moment \eqref{BB-ZZ}. Then, it follows that the linear operator $\delta_E^{\upsilon-m/q}\Theta$ is bounded from ${\mathcal{D}}_0(E)$ into $L^{(p,q)}({\mathscr{X}},E,\mu,\sigma;\kappa)$. Based on this, \eqref{AKp-2} and the fact hat the mixed-norm spaces $L^{(p,q)}({\mathscr{X}},E,\mu,\sigma;\kappa)$ are quasi-Banach (see \cite{MMMZ}, \cite{BMMM}), it follows that $\delta_E^{\upsilon-m/q}\Theta$ extends in a standard way to a linear operator from
$H^p(E,\rho|_E,\sigma)$ into $L^{(p,q)}({\mathscr{X}},E,\mu,\sigma;\kappa)$. Since the latter spaces are only quasi-normed, to show that this extension is also bounded we use the following property of quasi-normed spaces (for a proof see \cite[Theorem~1.5, (6)]{MMMM-G})
\begin{eqnarray}\label{Q-Nor} \begin{array}{c}
\mbox{if $(X,\|\cdot\|)$ is a quasi-normed vector space, then $\exists\,C\in[1,\infty)$ such that} \\[6pt] \mbox{if $x_j\to x_\ast$ in $X$ as $j\to\infty$, in the topology induced on
$X$ by $\|\cdot\|$, then} \\[6pt]
C^{-1}\|x_\ast\|\leq\liminf\limits_{j\to\infty}\|x_j\|
\leq \limsup\limits_{j\to\infty}\|x_j\|\leq C\|x_\ast\|. \end{array} \end{eqnarray}
In summary, the boundedness claimed in \eqref{Ar-EEE.2+} follows, once \eqref{BB-ZZ} is proved.
With the goal of establishing \eqref{BB-ZZ}, fix a function
$f\in\dot{\mathscr{C}}_{b,0}^\gamma(E,\rho|_E)$. By \cite[Proposition~3.1, p.\,112]{HYZ}, we have that
\begin{eqnarray}\label{HH-FD} \begin{array}{c} \exists\,(\lambda_j)_{j\in{\mathbb{N}}}\in \ell^p,\quad \exists\,(a_j)_{j\in{\mathbb{N}}}\,\,\mbox{$p$-atoms, such that }\,\,
\Bigl(\sum\limits_{j=1}^\infty|\lambda_j|^p\Bigr)^{1/p}\leq C\|f\|_{H^p(E,\rho|_E,\sigma)} \\[4pt] \mbox{and }\,\,f=\sum\limits_{j=1}^\infty\lambda_ja_j\,\,
\mbox{ both in $H^p(E,\rho|_E,\sigma)$ and in $L^q(E,\sigma)$}, \end{array} \end{eqnarray}
for some $C\in(0,\infty)$ independent of $f$. Also, from our assumption \eqref{Ar-EEE} we deduce that
\begin{eqnarray}\label{Ar-EEE-S} \delta_E^{\upsilon-m/q}\Theta:L^q(E,\sigma)\to L^{(q,q)}({\mathscr{X}},E,\mu,\sigma;\kappa)\quad\mbox{is linear and bounded}. \end{eqnarray}
Combining \eqref{Ar-EEE-S} with \eqref{HH-FD} it follows that, with $f$ as above,
\begin{eqnarray}\label{HH-FD-A} \delta_E^{\upsilon-m/q}\Theta f & = & \delta_E^{\upsilon-m/q}\Theta\Bigl(\lim_{N\to\infty}\sum\limits_{j=1}^N\lambda_ja_j\Bigr) =\lim_{N\to\infty}\delta_E^{\upsilon-m/q}\Theta\Bigl(\sum\limits_{j=1}^N\lambda_ja_j\Bigr) \nonumber\\[4pt] & = & \lim_{N\to\infty}\sum\limits_{j=1}^N\lambda_j\delta_E^{\upsilon-m/q}\Theta a_j \quad\mbox{in }\,\,L^{(q,q)}({\mathscr{X}},E,\mu,\sigma;\kappa). \end{eqnarray}
Granted this, we may apply \cite[Theorem~1.5]{MMMZ} to conclude that
\begin{eqnarray}\label{PP} \begin{array}{c} \exists\,(N_k)_{k\in{\mathbb{N}}},\,\,N_k\nearrow +\infty\,\,\mbox{ as }\,\,k\to +\infty, \,\,\mbox{ such that} \\[4pt] \mbox{$\sum\limits_{j=1}^{N_k}\lambda_j\delta_E^{\upsilon-m/q}\Theta a_j\to \delta_E^{\upsilon-m/q}\Theta f$ pointwise $\mu$-a.e. on ${\mathscr{X}}\setminus E$ as $k\to +\infty$}. \end{array} \end{eqnarray}
Since we are currently assuming that $0<p\leq 1\leq q<\infty$, an inspection of definition \eqref{Mixed-EEW} of the quasi-norm for the space $L^{(p,q)}({\mathscr{X}},E,\mu,\sigma;\kappa)$ reveals that
$\|\cdot\|_{L^{(p,q)}({\mathscr{X}},E,\mu,\sigma;\kappa)}^p$ is subadditive. As such, for each $k\in{\mathbb{N}}$, we may estimate
\begin{eqnarray}\label{DF-ar}
\Bigl\|\sum\limits_{j=1}^{N_k}\lambda_j\delta_E^{\upsilon-m/q}\Theta a_j
\Bigr\|^p_{L^{(p,q)}({\mathscr{X}},E,\mu,\sigma;\kappa)}
& \leq & \sum\limits_{j=1}^{N_k}\Bigl\|\lambda_j\delta_E^{\upsilon-m/q}\Theta a_j
\Bigr\|^p_{L^{(p,q)}({\mathscr{X}},E,\mu,\sigma;\kappa)} \nonumber\\[4pt]
& = & \sum\limits_{j=1}^{N_k}|\lambda_j|^p\Bigl\|\delta_E^{\upsilon-m/q}\Theta a_j
\Bigr\|^p_{L^{(p,q)}({\mathscr{X}},E,\mu,\sigma;\kappa)} \nonumber\\[4pt]
& \leq & C\sum\limits_{j=1}^{N_k}|\lambda_j|^p, \end{eqnarray}
where for the last inequality in \eqref{DF-ar} we used estimate \eqref{hnv} (note that the assumptions in Case~4 are currently satisfied). Next, introduce
\begin{eqnarray}\label{DF-arZZ} F_k:=\sum\limits_{j=1}^{N_k}\lambda_j\delta_E^{\upsilon-m/q}\Theta a_j,\qquad \forall\,k\in{\mathbb{N}}. \end{eqnarray}
To proceed, observe that Fatou's lemma holds in the space $L^{(p,q)}({\mathscr{X}},E,\mu,\sigma;\kappa)$ (this is seen directly from \eqref{Mixed-EEW} by applying twice the classical Fatou's lemma in Lebesgue spaces). When used for the sequence $\{F_k\}_{k\in{\mathbb{N}}}$, this yields
\begin{eqnarray}\label{DF-ar-2}
\|\delta_E^{\upsilon-m/q}\Theta f\|_{L^{(p,q)}({\mathscr{X}},E,\mu,\sigma;\kappa)}
& = & \bigl\|\liminf_{k\to\infty} |F_k|\bigr\|_{L^{(p,q)}({\mathscr{X}},E,\mu,\sigma;\kappa)} \nonumber\\[4pt]
&\leq &\liminf_{k\to\infty}\|F_k\|_{L^{(p,q)}({\mathscr{X}},E,\mu,\sigma;\kappa)}
\leq C \|f\|_{H^p(E,\rho|_E,\sigma)}.\quad \end{eqnarray}
The equality in \eqref{DF-ar-2} is a consequence of \eqref{PP} and the fact that
\begin{eqnarray}\label{Hkuc}
\|u\|_{L^{(p,q)}({\mathscr{X}},E,\mu,\sigma;\kappa)}
=\|\,|u|\,\|_{L^{(p,q)}({\mathscr{X}},E,\mu,\sigma;\kappa)},\quad \forall\,u\in L^{(p,q)}({\mathscr{X}},E,\mu,\sigma;\kappa), \end{eqnarray}
the first inequality is due to Fatou's Lemma and \eqref{Hkuc}, while the last inequality follows from \eqref{DF-ar}, \eqref{DF-arZZ} and \eqref{HH-FD}.
At this stage, we have established \eqref{DF-ar-2} for any function
$f\in\dot{\mathscr{C}}_{b,0}^\gamma(E,\rho|_E)$, so in order to finish the proof of \eqref{BB-ZZ} there remains to consider the case when $E$ is bounded and $f={\mathbf{1}}_E$. In this setting, since $E$ is $d$-dimensional ADR, we have $\sigma(E)<\infty$, and we may write
\begin{eqnarray}\label{DF-ar-3}
\bigl\|\delta_E^{\upsilon-m/q}\Theta {\mathbf{1}}_E
\bigr\|_{L^{(p,q)}({\mathscr{X}},E,\mu,\sigma;\kappa)}
& = & \bigl\|{\mathscr{A}}_{q,\kappa}\circ\bigl(\delta_E^{\upsilon-m/q}
\Theta{\mathbf{1}}_E\bigr)\bigr\|_{L^p(E,\sigma)} \nonumber\\[4pt] & \leq & \sigma(E)^{\frac{1}{p}-\frac{1}{q}}
\bigl\|{\mathscr{A}}_{q,\kappa}\circ\bigl(\delta_E^{\upsilon-m/q}
\Theta{\mathbf{1}}_E\bigr)\bigr\|_{L^q(E,\sigma)} \nonumber\\[4pt] & \leq & C\sigma(E)^{\frac{1}{p}}=C<+\infty. \end{eqnarray}
The first inequality in \eqref{DF-ar-3} uses H\"older's inequality for the integrability index $q/p\geq 1$, while the second inequality uses \eqref{Ar-EEE}. Now \eqref{BB-ZZ} follows by combining \eqref{DF-ar-2} and \eqref{DF-ar-3}, and with it the proof of \eqref{Ar-EEE.2+} is finished. In particular, \eqref{BB-ZZ} may be rewritten as
\begin{eqnarray}\label{Ar-EEE.2} \begin{array}{c} {\mathscr{A}}_{q,\kappa}\circ\bigl(\delta_E^{\upsilon-m/q}\Theta\bigr):
H^p(E,\rho|_E,\sigma)\to L^p(E,\sigma)\quad\mbox{is bounded} \\[4pt] \mbox{whenever $\tfrac{d}{d+\gamma}<p\leq 1\leq q<+\infty$, and \eqref{Ar-EEE} holds}. \end{array} \end{eqnarray}
The end-game in the proof of part $(I)$ in the statement of the theorem is now as follows. Assume first that $q\in(1,\infty)$, $p_o\in(q,\infty)$ are such that \eqref{dtbjHT} holds for every $f\in L^{p_o}(E,\sigma)$. Based on these assumptions and Case~1 we conclude that
\begin{eqnarray}\label{Ar-EEE.A} {\mathscr{A}}_{q,\kappa}\circ\bigl(\delta_E^{\upsilon-m/q}\Theta\bigr): L^p(E,\sigma)\to L^p(E,\sigma)\quad\mbox{is bounded whenever $p\in (1,p_o)$}. \end{eqnarray}
In particular, \eqref{Ar-EEE.A} with $p:=q\in(1,p_o)$ and \eqref{Ar-EEE.2} yield
\begin{eqnarray}\label{Ar-EEE.Ai} {\mathscr{A}}_{q,\kappa}\circ\bigl(\delta_E^{\upsilon-m/q}\Theta\bigr):
H^p(E,\rho|_{E},\sigma)\to L^p(E,\sigma)\quad\mbox{is bounded if $\tfrac{d}{d+\gamma}<p\leq 1$}. \end{eqnarray}
To proceed, fix an exponent
\begin{eqnarray}\label{Ar-EEE.3} p_o'\in(q,p_o). \end{eqnarray}
Then \eqref{Ar-EEE.A} corresponding to $p:=p_o'$ together with Case~2 used here with $p_o$ replaced by $p_o'$ imply that
\begin{eqnarray}\label{Ar-EEE.B} {\mathscr{A}}_{q,\kappa}\circ\bigl(\delta_E^{\upsilon-m/q}\Theta\bigr): L^p(E,\sigma)\to L^p(E,\sigma)\quad\mbox{is bounded for each $p\in (q,\infty)$}. \end{eqnarray}
Now the claim in part $(I)$ in the statement of the theorem corresponding to the current working hypotheses follows from \eqref{Ar-EEE.A}, \eqref{Ar-EEE.Ai}, and \eqref{Ar-EEE.B}.
There remains to consider the situation when $q\in(1,\infty)$ is such that \eqref{kt-Dc-BIS} holds. From Case~3 we know that \eqref{kt-Dc} is valid in the range $p\in[q,\infty)$. Then in combination with Case~1 (used with $p_o:=q\in(1,\infty)$), this gives that
\begin{eqnarray}\label{Ar-EEE.A-BIS} {\mathscr{A}}_{q,\kappa}\circ\bigl(\delta_E^{\upsilon-m/q}\Theta\bigr): L^p(E,\sigma)\to L^p(E,\sigma)\quad\mbox{is bounded for each $p\in (1,q]$}. \end{eqnarray}
Now reasoning as before we obtain that \eqref{Ar-EEE.Ai} holds. In summary, the above analysis shows that \eqref{kt-Dc} is valid in the range $p\in\bigl(\tfrac{d}{d+\gamma},\infty\bigr)$, under the assumption that $q\in(1,\infty)$ is such that \eqref{kt-Dc-BIS} holds. This concludes the proof of part $(I)$, and finishes the proof of the theorem. \end{proof}
The second main result in this subsection is a combination of Theorem~\ref{VGds-L2XXX} and Theorem~\ref{VGds-2}.
\begin{theorem}\label{VGds-2.33} Suppose that $d,m$ are real numbers such that $0<d<m$. Assume that $({\mathscr{X}},\rho,\mu)$ is an $m$-dimensional {\rm ADR} space, $E$ is a closed subset of $({\mathscr{X}},\tau_\rho)$, and $\sigma$ is a
Borel regular measure on $(E,\tau_{\rho|_{E}})$ with the property
that $(E,\rho\bigl|_E,\sigma)$ is a $d$-dimensional {\rm ADR} space. In addition, suppose that $\Theta$ is the integral operator defined in \eqref{operator} with a kernel $\theta$ as in \eqref{K234}, \eqref{hszz}, \eqref{hszz-3noalpha}. Finally, fix $\kappa>0$ and recall the exponent $\gamma$ from \eqref{WQ-tDD}.
If there exist $p_o\in(0,\infty)$ and a finite constant $C_o>0$ such that for every $f\in L^{p_o}(E,\sigma)$
\begin{eqnarray}\label{GvBh} \hskip -0.30in \sup_{\lambda>0}\left[\lambda\cdot
\sigma\Bigl(\Bigl\{x\in E:\int_{\Gamma_{\kappa}(x)}|(\Theta f)(y)|^2\, \frac{d\mu(y)}{\delta_E(y)^{m-2\upsilon}}>\lambda^{2}\Bigr\}\Bigr)^{1/p_o}\right]
\leq C_o\|f\|_{L^{p_o}(E,\sigma)}, \end{eqnarray}
then for each $p\in\bigl(\frac{d}{d+\gamma},\infty\bigr)$ there holds
\begin{eqnarray}\label{Cfrd} \hskip -0.20in
\left\|\Bigl(\int_{\Gamma_{\kappa}(x)}|(\Theta f)(y)|^2\, \frac{d\mu(y)}{\delta_E(y)^{m-2\upsilon}}\Bigr)^{\frac{1}{2}}
\right\|_{L^p_x(E,\sigma)}\!\!\!\leq C\|f\|_{H^p(E,\rho|_{E},\sigma)},\quad\forall\,f\in H^p(E,\rho|_{E},\sigma), \end{eqnarray}
where $C>0$ is a finite constant which is allowed to depend only on $p,C_o,\kappa,C_\theta$, and geometry. \end{theorem}
\begin{proof} The assumption that the operator ${\mathcal{A}}_{2,\kappa}\circ(\delta_E^{\upsilon-m/2}\Theta): L^{p_o}(E,\sigma)\rightarrow L^{p_o,\infty}(E,\sigma)$ is bounded implies that \eqref{dtbh-L2iii} holds. Consequently, Theorem~\ref{VGds-L2XXX} applies and yields that ${\mathcal{A}}_{2,\kappa}\circ(\delta_E^{\upsilon-m/2}\Theta): L^2(E,\sigma)\rightarrow L^2(E,\sigma)$ is bounded as well. With this in hand, part $(I)$ in Theorem~\ref{VGds-2} (pertaining to condition \eqref{kt-Dc-BIS} with $q=2$) applies and gives that \eqref{Cfrd} holds for every $p\in\bigl(\frac{d}{d+\gamma},\infty\bigr)$. \end{proof}
\section{Conclusion}\label{Sect:6} \setcounter{equation}{0}
Theorem~\ref{M-TTHH} asserts the equivalence of a number of the properties encountered in the body of the manuscript. A formal proof is presented below.
\vskip 0.08in \begin{proof}[Proof of Theorem~\ref{M-TTHH}] The fact that {\it (1)} $\Rightarrow$ {\it (2)} is a consequence of Theorem~\ref{SChg}. It is easy to see that if {\it (2)} holds, then {\it (7)} holds by taking $b_Q:={\mathbf{1}}_Q$ for each $Q\in{\mathbb{D}}(E)$, hence {\it (2)} $\Rightarrow$ {\it (7)}. The implication {\it (7)} $\Rightarrow$ {\it (1)} is proved in Theorem~\ref{Thm:localTb}. The implication {\it (9)} $\Rightarrow$ {\it (1)} is proved in Theorem~\ref{Thm:BPSFtoSF}. Moreover, {\it (1)} $\Leftrightarrow$ {\it (9)} $\Leftrightarrow$ {\it (10)} by Theorem~\ref{Thm:BPSFtoSF.XXX}. The implication {\it (11)} $\Rightarrow$ {\it (12)} is proved in Theorem~\ref{VGds-2.33}. Clearly {\it (12)} $\Rightarrow$ {\it (11)}, while {\it (11)} $\Rightarrow$ {\it (1)} is contained in Theorem~\ref{VGds-L2XXX}. To show that {\it (1)} $\Rightarrow$ {\it (11)}, suppose {\it (1)} holds and take $f\in L^2(E,\sigma)$ and $\lambda>0$ arbitrary. Then starting with Tschebyshev's inequality we may write
\begin{eqnarray}\label{dSHV} && \hskip -0.40in \lambda^2\cdot\sigma\Bigl(\Bigl\{x\in E:\int_{\Gamma_{\kappa}(x)}
|(\Theta f)(y)|^2\,\frac{d\mu(y)}{\delta_E(y)^{m-2\upsilon}}>\lambda^{2}\Bigr\}\Bigr) \nonumber\\[4pt] && \hskip 0.30in \leq \int_{\Bigl\{x\in E:\int_{\Gamma_{\kappa}(x)}
|(\Theta f)(y)|^2\,\frac{d\mu(y)}{\delta_E(y)^{m-2\upsilon}}>\lambda^{2}\Bigr\}}\Bigl(
\int_{\Gamma_{\kappa}(x)}|(\Theta f)(y)|^2\,\frac{d\mu(y)}{\delta_E(y)^{m-2\upsilon}} \Bigr)d\sigma(x) \nonumber\\[4pt] && \hskip 0.30in \leq \int_E\Bigl(\int_{\Gamma_{\kappa}(x)}
|(\Theta f)(y)|^2\,\frac{d\mu(y)}{\delta_E(y)^{m-2\upsilon}}\Bigr)d\sigma(x) \nonumber\\[4pt] && \hskip 0.30in
\leq \int_{{\mathscr{X}}\setminus E}\frac{|(\Theta f)(y)|^2}{\delta_E(y)^{m-2\upsilon}} \,\sigma(\pi_y^\kappa)\,d\mu(y) \nonumber\\[4pt] && \hskip 0.30in
\leq \int_{{\mathscr{X}}\setminus E}\frac{|(\Theta f)(y)|^2}{\delta_E(y)^{m-2\upsilon}} \,\sigma\Bigl(E\cap B_{\rho_{\#}}\bigl(y_\ast,C_{\rho}(1+\kappa)\delta_E(y)\bigr)\Bigr) \,d\mu(y) \nonumber\\[4pt] && \hskip 0.30in
\leq C\int_{{\mathscr{X}}\setminus E}|(\Theta f)(y)|^2\delta_E(y)^{2\upsilon-(m-d)} \,d\mu(y) \nonumber\\[4pt] && \hskip 0.30in
\leq C\|f\|_{L^2(E,\sigma)}^2. \end{eqnarray}
The third inequality in \eqref{dSHV} is due to \eqref{Mix+FR} (recall \eqref{reg-A2}), the fourth uses \eqref{sgbr} in Lemma~\ref{lbDV}, the fifth uses the fact that
$\bigl(E,\rho\bigl|_E,\sigma\bigr)$ is a $d$-dimensional {\rm ADR} space, and the last inequality is a consequence of \eqref{G-UFXXX.2}. Thus, {\it (1)} $\Rightarrow$ {\it (11)} as desired. Since \eqref{ki-DUDXXX} is a rewriting of \eqref{CfrdXXX}, it is immediate that {\it (12)} $\Leftrightarrow$ {\it (13)}. In summary, so far we have shown that {\it (1)}, {\it (2)}, {\it (7)}, {\it (9)}, {\it (10)}, {\it (11)}, {\it (12)}, and {\it (13)} are equivalent.
The implication {\it (6)} $\Rightarrow$ {\it (4)} is trivial and, based on \eqref{dFvK}, we have that {\it (4)} $\Rightarrow$ {\it (2)}. We focus next on {\it (1)} $\Rightarrow$ {\it (6)}. Suppose {\it (1)} holds and fix $f\in L^\infty(E,\sigma)$, $x\in E$, and $r\in(0,\infty)$ arbitrary. Then, using the notation $B_{cr}:=B_{\rho_{\#}}(x,cr)$ for $c>0$, we may write
\begin{eqnarray}\label{esL}
\int_{B_r\setminus E}|\Theta f|^2\delta_E^{2\upsilon-(m-d)}\,d\mu & \leq & \int_{B_r\setminus E}
|\Theta (f{\mathbf{1}}_{E\cap B_{2rC_\rho}})|^2\delta_E^{2\upsilon-(m-d)}\,d\mu \nonumber\\[4pt] &&
+\int_{B_r\setminus E}|\Theta (f{\mathbf{1}}_{E\setminus B_{2rC_\rho}})|^2 \delta_E^{2\upsilon-(m-d)}\,d\mu=:I+II. \end{eqnarray}
To estimate $I$ we apply \eqref{G-UFXXX.2} and the property of $E$ being $d$-dimensional ADR to obtain
\begin{eqnarray}\label{esL-2}
I\leq C\int_{E\cap B_{2rC_\rho}}|f|^2\,d\sigma
\leq C\|f\|^2_{L^\infty(E,\sigma)}\sigma(E\cap B_{r}). \end{eqnarray}
As regards $II$, we first note that if $z\in B_r\setminus E$ and $y\in E\setminus B_{2rC_\rho}$ are arbitrary points then $\rho_{\#}(x,y) \leq C_\rho(\rho_{\#}(x,z)+\rho_{\#}(z,y))< C_\rho r+C_\rho\rho_{\#}(z,y) \leq \tfrac{1}{2}\rho_{\#}(x,y)+C_\rho\rho_{\#}(z,y)$ which implies $\rho_{\#}(z,y)\geq \rho_{\#}(x,y)/(2C_\rho)$. This, \eqref{hszz-AXXX}, and \eqref{WBA} then yield
\begin{eqnarray}\label{esL-3}
|\Theta (f{\mathbf{1}}_{E\setminus B_{2rC_\rho}})(z)|
&\leq & C\|f\|_{L^\infty(E,\sigma)} \int_{E\setminus B_{2rC_\rho}}\frac{1}{\rho_{\#}(x,y)^{d+\upsilon}}\,d\sigma(y) \nonumber\\[4pt]
&\leq & C\|f\|_{L^\infty(E,\sigma)}\,r^{-\upsilon}, \qquad\forall\,z\in B_r\setminus E. \end{eqnarray}
Using this last estimate in $II$ and applying \eqref{lbzF} (with $R:=r$ and $\gamma:=m-d-2\upsilon$) we obtain
\begin{eqnarray}\label{esL-4}
II &\leq & C\|f\|^2_{L^\infty(E,\sigma)}r^{-2\upsilon} \int_{B_r\setminus E}\delta_E^{2\upsilon-(m-d)}\,d\mu \nonumber\\[4pt]
& \leq & C\|f\|^2_{L^\infty(E,\sigma)}r^{-2\upsilon}r^{d+2\upsilon}
\leq C\|f\|^2_{L^\infty(E,\sigma)}\sigma(E\cap B_{r}). \end{eqnarray}
At this point, \eqref{UEHgXXX.2S} follows from \eqref{esL}, \eqref{esL-2}, and \eqref{esL-4}, completing the proof of {\it (1)} $\Rightarrow$ {\it (6)}. Based on \eqref{dFvK} we have that {\it (6)} $\Rightarrow$ {\it (3)} while {\it (3)} $\Rightarrow$ {\it (2)} is trivial.
Next, we shall show that {\it (8)} $\Rightarrow$ {\it (7)}. To this end, suppose {\it (8)} holds and let $\varepsilon_o:=\min\{\varepsilon,a_0\}$, where $\varepsilon$ is as in Lemma~\ref{b:SV} and $a_0$ as in \eqref{ha-GVV}. Fix an arbitrary $Q\in{\mathbb{D}}(E)$ and define $\Delta_Q:=B_{\rho_{\#}}\Bigl(x_Q,\tfrac{\varepsilon_o\ell(Q)}{2C\rho}\Bigr)\cap E$. Then \eqref{ha-GVV}, \eqref{zjrh} and the fact that $E$ is $d$-dimensional ADR imply
\begin{eqnarray}\label{VCV} \Delta_Q\subseteq Q,\quad B_{\rho_{\#}}\bigl(x_Q,\varepsilon_o\ell(Q)\bigr)\setminus E\subseteq T_E(Q), \quad\sigma(\Delta_Q)\approx\sigma(Q)=C\ell(Q)^d. \end{eqnarray}
Hence, if we now define $b_Q:=b_{\Delta_Q}$, where $b_{\Delta_Q}$ is the function associated to $\Delta_Q$ as in {\it (8)}, then $b_{\Delta_Q}$ satisfies \eqref{CON-BB.789} which, when combined with the support condition of $b_{\Delta_Q}$ and the last condition in \eqref{VCV}, implies that $b_Q$ satisfies the first two conditions in \eqref{CON-BB} (with $\widetilde{Q}=Q$). In order to show that $b_Q$ also verifies the last condition in \eqref{CON-BB}, we write
\begin{eqnarray}\label{DB-N} && \hskip -0.70in
\int_{T_E(Q)}|(\Theta\,b_Q)(x)|^2\delta_E(x)^{2\upsilon-(m-d)}\,d\mu(x) \nonumber\\[4pt] && =\int_{T_E(Q)\setminus B_{\rho_{\#}}\bigl(x_Q,\varepsilon_o\ell(Q)\bigr)}
|(\Theta\,b_Q)(x)|^2\delta_E(x)^{2\upsilon-(m-d)}\,d\mu(x) \nonumber\\[4pt]
&& \quad+\int_{B_{\rho_{\#}}\bigl(x_Q,\varepsilon_o\ell(Q)\bigr)}|(\Theta\,b_Q)(x)|^2 \delta_E(x)^{2\upsilon-(m-d)}\,d\mu(x)=:I_1+I_2. \end{eqnarray}
To further estimate $I_2$, observe that if
$x\in T_E(Q)\setminus B_{\rho_{\#}}\bigl(x_Q,\varepsilon_o\ell(Q)\bigr)$ and $y\in\Delta_Q$, then $\rho_{\#}(x,y)\geq\tfrac{\varepsilon_o}{2C_\rho}\ell(Q)$. This, \eqref{hszz-AXXX}, the first estimate in \eqref{CON-BB}, and the last condition in \eqref{VCV}, imply $|(\Theta\,b_Q)(x)|\leq C\ell(Q)^{-\upsilon}$ for every $x\in T_E(Q)\setminus B_{\rho_{\#}}\bigl(x_Q,\varepsilon_o\ell(Q)\bigr)$. Hence, if we also recall \eqref{dFvK} we have
\begin{eqnarray}\label{DB-N-2} I_1 &\leq & C \ell(Q)^{-2\upsilon} \int_{T_E(Q)\setminus B_{\rho_{\#}}\bigl(x_Q,\varepsilon_o\ell(Q)\bigr)} \delta_E(x)^{2\upsilon-(m-d)}\,d\mu(x) \nonumber\\[4pt] &\leq & C \ell(Q)^{-2\upsilon} \int_{B_{\rho_{\#}}\bigl(x_Q,C\ell(Q)\bigr)\setminus E} \delta_E(x)^{2\upsilon-(m-d)}\,d\mu(x) \nonumber\\[4pt] &\leq & C \ell(Q)^{-2\upsilon}\ell(Q)^{d+2\upsilon}\leq C\sigma(Q), \end{eqnarray}
where the third inequality in \eqref{DB-N-2} is a consequence of \eqref{lbzF} (applied with $R=r=\ell(Q)$ and $\gamma:=m-d-2\upsilon$). As for $I_2$, by recalling \eqref{CON-BB.789} and the last condition in \eqref{VCV}, it is immediate that $I_2\leq C_0\sigma(\Delta_Q)\leq C\sigma(Q)$. This, \eqref{DB-N} and \eqref{DB-N-2} show that $b_Q$ also satisfies the last condition in \eqref{CON-BB} since the constants in our estimates are finite positive geometric, and independent of the choice of $Q$. This completes the proof of {\it (8)} $\Rightarrow$ {\it (7)}.
It is not difficult to see that {\it (1)} $\Rightarrow$ {\it (8)}. Indeed, by taking $b_\Delta:={\mathbf{1}}_\Delta$ for each surface ball $\Delta$, the first two estimates in \eqref{CON-BB.789} are immediate while the third one is a consequence of \eqref{G-UFXXX.2} written for $f:=b_\Delta$.
Trivially, {\it (2)} $\Rightarrow$ {\it (5)}. If we assume that {\it (5)} holds and for each $Q\in{\mathbb{D}}(E)$ we set $b_Q:=b{\mathbf{1}}_Q$, then it is easy to verify based on \eqref{UEHgXXX.2PA} and the fact that $b$ is para-accretive that \eqref{CON-BB} is satisfied by the family $\{b_Q\}_{Q\in{\mathbb{D}}(E)}$. Hence, {\it (5)} $\Rightarrow ${\it (7)}. The proof of Theorem~\ref{M-TTHH} is therefore complete. \end{proof}
In the last part of this section we present the
\vskip 0.08in \begin{proof}[Proof of Theorem~\ref{sfe-cor}] The idea is to apply Theorem~\ref{VGds-2} in the setting ${\mathscr{X}}:=E\times[0,\infty)$ and $E\equiv E\times\{0\}$ (i.e., we identify $(y,0)\equiv y$ for every $y\in E$). Moreover, we let
\begin{eqnarray}\label{GCvca-LL1}
&& \rho((x,t),(y,s)):=\max\{|x-y|,|t-s|\}\,\, \mbox{ for every }\,\,(x,t),(y,s)\in E\times[0,\infty), \\[4pt] && \mu:=\sigma\otimes{\mathcal{L}}^1, \label{GCvca-LL2} \end{eqnarray}
where ${\mathcal{L}}^1$ is the one-dimensional Lebesgue measure on $[0,\infty)$, and consider the integral kernel
\begin{eqnarray}\label{K-spc} \begin{array}{l} \theta:({\mathscr{X}}\setminus E)\times E\to{\mathbb{R}} \\[4pt] \theta\bigl((x,t),y\bigr):=2^{-k}\psi_k(x-y)\,\,\mbox{ if }\,\,x,y\in E, t>0 \,\,\mbox{ and }\,\,k\in{\mathbb{Z}},\,\,2^k\leq t<2^{k+1}. \end{array} \end{eqnarray}
Also, we let $\Theta$ be the integral operator defined in \eqref{operator} corresponding to this choice of $\theta$. Then it is not difficult to verify that $({\mathscr{X}},\rho,\mu)$ is a $(d+1)$-ADR space, that $\alpha_\rho=1$, that $\theta$ satisfies \eqref{K234}-\eqref{hszz-3} for $a:=0$, $\alpha:=1$, $\upsilon:=1$, and that $\delta_E(x,t)=t$ for every $x\in E$ and $t\in[0,\infty)$. In particular, $\gamma$ defined in \eqref{WQ-tDD} now equals $1$. Fix some $\kappa>0$ and observe that
\begin{eqnarray}\label{agv} \Gamma_\kappa(x)=\bigl\{(y,t)\in E\times(0,\infty):\,
|x-y|<(1+\kappa)t\bigr\},\quad\forall\,x\in E. \end{eqnarray}
In this context, for $f\in L^2(E,\sigma)$, we consider the square of the term in the left hand-side of \eqref{kt-Dc-BIS} corresponding to $p=q=2$ and use Fubini's Theorem, the property that $E$ is $d$-ADR, and \eqref{K-spc} to write
\begin{eqnarray}\label{sBBv} && \hskip -0.20in
\left\|\Bigl(\int_{\Gamma_{\kappa}(x)}|(\Theta f)(y,t)|^2\, \frac{d\mu(y,t)}{\delta_E(y,t)^{d-1}}\Bigr)^{\frac{1}{2}}
\right\|_{L^2_x(E,\sigma)}^2
=\int_E\int_{\Gamma_{\kappa}(x)}|(\Theta f)(y,t)|^2\,t^{1-d}d\mu(y,t) \,d\sigma(x) \nonumber\\[4pt]
&&\quad =\int_{E\times(0,\infty)}|(\Theta f)(y,t)|^2\,t^{1-d} \sigma\bigl(\{x\in E:\,y\in\Gamma_\kappa(x)\}\bigr)\,d\mu(y,t) \nonumber\\[4pt]
&&\quad =\int_{E\times(0,\infty)}|(\Theta f)(y,t)|^2\,t^{1-d} \sigma\bigl(E\cap B(y,(1+\kappa)t)\bigr)\,d\mu(y,t) \nonumber\\[4pt]
&&\quad \approx C\int_0^\infty\int_E|(\Theta f)(y,t)|^2\,t\,d\sigma(y)\,dt \nonumber\\[4pt] &&\quad =C\sum\limits_{k=-\infty}^{+\infty}\int_{2^k}^{2^{k+1}}
\int_E\Bigl|\int_E2^{-k}\psi_k(y-z)f(z)\,d\sigma(z)\Bigr|^2 d\sigma(y)\,t\,dt \nonumber\\[4pt] &&\quad =C\sum\limits_{k=-\infty}^{+\infty}
\int_E\Bigl|\int_E\psi_k(y-z)f(z)\,d\sigma(z)\Bigr|^2 d\sigma(y). \end{eqnarray}
However, under the current assumptions on $E$, it was proved in \cite[Theorem, p.\,10]{DaSe91} that there exists $C\in(0,\infty)$ with the property that
\begin{eqnarray}\label{PP-Zs} \sum\limits_{k=-\infty}^{+\infty}\int_E
\Bigl|\int_E\psi_k(x-y)f(y)\,d\sigma(y)\Bigr|^2\,d\sigma(x)
\leq C\int_E|f|^2\,d\sigma,\quad\forall\,f\in L^2(E,\sigma). \end{eqnarray}
Hence, we may apply Theorem~\ref{VGds-2} to conclude that there exists $C\in(0,\infty)$ such that estimate \eqref{kt-Dc} is valid for every $q\in(1,\infty)$ and every $p\in\bigl(\frac{d}{d+1},\infty\bigr)$. In turn, reasoning as in \eqref{sBBv}, estimate \eqref{kt-Dc} may be rewritten in the form
\begin{eqnarray}\label{Df-Ch}
\left\|\Bigl(\sum\limits_{k=-\infty}^{+\infty}
{\int{\mkern-19mu}-}_{y\in\Delta(x,(1+\kappa)2^k)}\Bigl|\int_E\psi_k(z-y)f(z)\,d\sigma(z)
\Bigr|^q\,d\sigma(y)\Bigr)^{1/q}
\right\|_{L^p_x(E,\sigma)}\leq C'\|f\|_{H^p(E,\sigma)} \end{eqnarray}
for every $f\in H^p(E,\sigma)$ and some $C'\in(0,\infty)$ independent of $f$. The desired conclusion now follows by observing that if $q\in(1,\infty)$ and $p\in\bigl(\frac{d}{d+1},\infty\bigr)$ are fixed, then there exists some $C\in(0,\infty)$ such that \eqref{bh} holds for every $f\in H^p(E,\sigma)$ if and only if there exist $\kappa,C'\in(0,\infty)$ such that estimate \eqref{Df-Ch} holds for every $f\in H^p(E,\sigma)$. Indeed, one direction is obvious, while the opposite one may be handled by observing that if $\psi\in C^\infty_0({\mathbb{R}}^{n+1})$ is odd then $\widetilde{\psi}(x):=\psi(x/2^N)$, for some fixed sufficiently large $N\in{\mathbb{N}}$, is also odd, smooth and compactly supported, and satisfies $\widetilde{\psi}_k=2^{dN}\psi_{k+N}$ for every $k\in{\mathbb{Z}}$. Writing \eqref{Df-Ch} for $\widetilde{\psi}$ in place of $\psi$ and shifting the index of summation in the left-hand side, the desired conclusion follows. \end{proof}
\end{document} | arXiv |
\begin{definition}[Definition:Calendar/Jewish/Year]
A '''Jewish year''', also known as the '''Hebrew year''', is the length of a year as defined using the '''Jewish calendar'''.
Its length is either $354$ or $355$ days, or longer, if an extra month has been inserted.
\end{definition} | ProofWiki |
\begin{definition}[Definition:Octonion]
The set of '''octonions''', usually denoted $\Bbb O$, can be defined by using the Cayley-Dickson construction from the quaternions $\Bbb H$ as follows:
From Quaternions form Algebra, $\Bbb H$ forms a nicely normed $*$-algebra.
Let $a, b \in \Bbb H$.
Then $\tuple {a, b} \in \Bbb O$, where:
:$\tuple {a, b} \tuple {c, d} = \tuple {a c - d \overline b, \overline a d + c b}$
:$\overline {\tuple {a, b} } = \tuple {\overline a, -b}$
where:
:$\overline a$ is the conjugate on $a$
and
:$\overline {\tuple {a, b} }$ is the conjugation operation on $\Bbb O$.
\end{definition} | ProofWiki |
# Fundamental concepts of Probability
1.1 Sample Space and Events
The sample space is the set of all possible outcomes of an experiment. It is denoted by the symbol Ω. For example, if we are flipping a coin, the sample space would be {Heads, Tails}.
An event is a subset of the sample space. It represents a specific outcome or a combination of outcomes. For example, the event of getting a Head when flipping a coin would be {Heads}.
Consider rolling a fair six-sided die. The sample space would be {1, 2, 3, 4, 5, 6}. Some possible events could be:
- Getting an even number: {2, 4, 6}
- Getting a number less than 3: {1, 2}
## Exercise
For the experiment of rolling a fair six-sided die, determine the sample space and two different events.
### Solution
Sample space: {1, 2, 3, 4, 5, 6}
Event 1: Getting an odd number - {1, 3, 5}
Event 2: Getting a number greater than 4 - {5, 6}
1.2 Probability of an Event
The probability of an event is a number between 0 and 1 that represents the likelihood of that event occurring. A probability of 0 means the event is impossible, while a probability of 1 means the event is certain to happen.
The probability of an event A is denoted by P(A). It can be calculated by dividing the number of favorable outcomes by the total number of possible outcomes.
$$P(A) = \frac{{\text{{Number of favorable outcomes}}}}{{\text{{Total number of possible outcomes}}}}$$
Let's calculate the probability of getting an even number when rolling a fair six-sided die. The sample space is {1, 2, 3, 4, 5, 6}, and the event of getting an even number is {2, 4, 6}.
$$P(\text{{Even number}}) = \frac{{\text{{Number of even numbers}}}}{{\text{{Total number of possible outcomes}}}} = \frac{3}{6} = \frac{1}{2}$$
So, the probability of getting an even number is 1/2.
## Exercise
Calculate the probability of the following events for rolling a fair six-sided die:
- Getting a number less than 3
- Getting a number greater than 4
### Solution
Probability of getting a number less than 3:
$$P(\text{{Number less than 3}}) = \frac{{\text{{Number of numbers less than 3}}}}{{\text{{Total number of possible outcomes}}}} = \frac{2}{6} = \frac{1}{3}$$
Probability of getting a number greater than 4:
$$P(\text{{Number greater than 4}}) = \frac{{\text{{Number of numbers greater than 4}}}}{{\text{{Total number of possible outcomes}}}} = \frac{2}{6} = \frac{1}{3}$$
# Types of Probability
2.1 Classical Probability
Classical probability is based on the assumption that all outcomes in the sample space are equally likely. It is often used for simple and well-defined experiments. The probability of an event A can be calculated by dividing the number of favorable outcomes for A by the total number of possible outcomes.
$$P(A) = \frac{{\text{{Number of favorable outcomes for A}}}}{{\text{{Total number of possible outcomes}}}}$$
Consider flipping a fair coin. The sample space is {Heads, Tails}, and each outcome is equally likely. The probability of getting Heads is 1/2, and the probability of getting Tails is also 1/2.
## Exercise
Calculate the probability of the following events for flipping a fair coin:
- Getting Heads
- Getting Tails
### Solution
Probability of getting Heads:
$$P(\text{{Heads}}) = \frac{1}{2}$$
Probability of getting Tails:
$$P(\text{{Tails}}) = \frac{1}{2}$$
2.2 Empirical Probability
Empirical probability is based on observed data. It is calculated by dividing the number of times an event occurs by the total number of observations. Empirical probability is often used when it is difficult to determine the theoretical probabilities.
$$P(A) = \frac{{\text{{Number of times event A occurs}}}}{{\text{{Total number of observations}}}}$$
Suppose we roll a six-sided die 100 times and record the outcomes. The empirical probability of getting a 6 can be calculated by dividing the number of times a 6 is rolled by 100.
## Exercise
Suppose we roll a six-sided die 50 times and record the outcomes. Calculate the empirical probability of the following events:
- Getting a 1
- Getting an even number
### Solution
Probability of getting a 1:
$$P(\text{{1}}) = \frac{{\text{{Number of times 1 occurs}}}}{{\text{{Total number of observations}}}}$$
Probability of getting an even number:
$$P(\text{{Even number}}) = \frac{{\text{{Number of times an even number occurs}}}}{{\text{{Total number of observations}}}}$$
2.3 Subjective Probability
Subjective probability is based on an individual's personal judgment or belief about the likelihood of an event occurring. It is often used when there is no available data or when the event is subjective in nature. Subjective probability is subjective and can vary from person to person.
Suppose you are trying to estimate the probability of it raining tomorrow. Your estimate would be based on your personal judgment, taking into account factors such as weather forecasts, past experiences, and intuition.
## Exercise
Estimate the subjective probability of the following events:
- Winning the lottery
- Getting a promotion at work
### Solution
Subjective probability of winning the lottery:
This would vary from person to person and would depend on factors such as the person's belief in luck and their understanding of the odds.
Subjective probability of getting a promotion at work:
This would also vary from person to person and would depend on factors such as the person's performance, the company's policies, and their perception of their chances.
# Basic rules of Probability
3.1 Addition Rule
The addition rule states that the probability of the union of two events A and B is equal to the sum of their individual probabilities minus the probability of their intersection.
$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$
The addition rule can be extended to more than two events. For example, the probability of the union of three events A, B, and C is given by:
$$P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(A \cap B) - P(A \cap C) - P(B \cap C) + P(A \cap B \cap C)$$
Suppose we have a deck of cards and we draw a card at random. Let A be the event of drawing a heart, and B be the event of drawing a face card. The probability of drawing a heart is 13/52, the probability of drawing a face card is 12/52, and the probability of drawing a heart and a face card is 3/52. Using the addition rule, we can calculate the probability of drawing a heart or a face card.
$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$
$$P(A \cup B) = \frac{13}{52} + \frac{12}{52} - \frac{3}{52}$$
$$P(A \cup B) = \frac{22}{52}$$
$$P(A \cup B) = \frac{11}{26}$$
## Exercise
Suppose we roll a fair six-sided die. Let A be the event of rolling an even number, and B be the event of rolling a number less than 4. Calculate the probability of rolling an even number or a number less than 4.
### Solution
Probability of rolling an even number or a number less than 4:
$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$
$$P(A \cup B) = \frac{3}{6} + \frac{3}{6} - \frac{1}{6}$$
$$P(A \cup B) = \frac{5}{6}$$
# Random variables and their properties
4.1 Definition of a Random Variable
A random variable is a function that assigns a numerical value to each outcome in the sample space of a random experiment. It is denoted by a capital letter, such as X or Y. The possible values that a random variable can take on are called its range.
4.2 Types of Random Variables
There are two types of random variables: discrete random variables and continuous random variables.
- A discrete random variable is one that can take on a countable number of distinct values. The probability distribution of a discrete random variable is described by a probability mass function (PMF), which assigns probabilities to each possible value of the random variable.
- A continuous random variable is one that can take on any value within a certain range. The probability distribution of a continuous random variable is described by a probability density function (PDF), which gives the probability of the random variable falling within a certain interval.
4.3 Expected Value of a Random Variable
The expected value of a random variable is a measure of its central tendency. It represents the average value that the random variable would take on if the experiment were repeated many times. The expected value of a discrete random variable is calculated by summing the product of each possible value of the random variable and its probability. The expected value of a continuous random variable is calculated by integrating the product of the random variable and its probability density function.
Suppose we have a discrete random variable X that represents the number of heads obtained when flipping a fair coin twice. The possible values of X are 0, 1, and 2, with probabilities 1/4, 1/2, and 1/4, respectively. The expected value of X can be calculated as follows:
$$E(X) = 0 \cdot \frac{1}{4} + 1 \cdot \frac{1}{2} + 2 \cdot \frac{1}{4}$$
$$E(X) = 0 + \frac{1}{2} + \frac{1}{2}$$
$$E(X) = 1$$
## Exercise
Suppose we have a continuous random variable Y that represents the height of a randomly selected person. The probability density function of Y is given by:
$$f(y) = \begin{cases}
\frac{1}{6} & 0 \leq y \leq 6 \\
0 & \text{otherwise}
\end{cases}
$$
Calculate the expected value of Y.
### Solution
Expected value of Y:
$$E(Y) = \int_{0}^{6} y \cdot \frac{1}{6} \,dy$$
$$E(Y) = \frac{1}{6} \int_{0}^{6} y \,dy$$
$$E(Y) = \frac{1}{6} \left[\frac{y^2}{2}\right]_{0}^{6}$$
$$E(Y) = \frac{1}{6} \left[\frac{6^2}{2} - \frac{0^2}{2}\right]$$
$$E(Y) = \frac{1}{6} \cdot \frac{36}{2}$$
$$E(Y) = 3$$
# Probability distributions and their properties
5.1 Discrete Probability Distributions
A discrete probability distribution is a probability distribution that describes the probabilities of different outcomes for a discrete random variable. It is often represented by a probability mass function (PMF), which assigns probabilities to each possible value of the random variable.
Some common examples of discrete probability distributions include the binomial distribution, the Poisson distribution, and the geometric distribution.
5.2 Continuous Probability Distributions
A continuous probability distribution is a probability distribution that describes the probabilities of different outcomes for a continuous random variable. It is often represented by a probability density function (PDF), which gives the probability of the random variable falling within a certain interval.
Some common examples of continuous probability distributions include the normal distribution, the exponential distribution, and the uniform distribution.
5.3 Properties of Probability Distributions
Probability distributions have several important properties. Some of these properties include:
- The sum of the probabilities of all possible outcomes is equal to 1.
- The probability of any individual outcome is between 0 and 1.
- The probability of an event occurring is equal to the sum of the probabilities of all outcomes that satisfy the event.
These properties ensure that probability distributions are valid and can be used to calculate probabilities.
Suppose we have a discrete random variable X that represents the number of heads obtained when flipping a fair coin twice. The probability distribution of X is given by:
| X | P(X) |
|---|------|
| 0 | 1/4 |
| 1 | 1/2 |
| 2 | 1/4 |
This probability distribution satisfies the properties of a discrete probability distribution. The sum of the probabilities is equal to 1, each individual probability is between 0 and 1, and the probability of an event occurring is equal to the sum of the probabilities of all outcomes that satisfy the event.
## Exercise
Suppose we have a continuous random variable Y that represents the height of a randomly selected person. The probability density function of Y is given by:
$$f(y) = \begin{cases}
\frac{1}{6} & 0 \leq y \leq 6 \\
0 & \text{otherwise}
\end{cases}
$$
Verify that this probability density function satisfies the properties of a continuous probability distribution.
### Solution
The probability density function satisfies the properties of a continuous probability distribution:
- The integral of the probability density function over the entire range of the random variable is equal to 1:
$$\int_{-\infty}^{\infty} f(y) \,dy = \int_{0}^{6} \frac{1}{6} \,dy = \frac{1}{6} \int_{0}^{6} \,dy = \frac{1}{6} \cdot 6 = 1$$
- The probability density function is non-negative for all values of the random variable:
$$f(y) \geq 0 \text{ for } 0 \leq y \leq 6$$
- The probability of an event occurring is equal to the integral of the probability density function over the range of the random variable that satisfies the event:
$$P(a \leq Y \leq b) = \int_{a}^{b} f(y) \,dy$$
These properties demonstrate that the probability density function is a valid representation of a continuous probability distribution.
# Expected value and its significance
6.1 Definition of Expected Value
The expected value of a random variable X is denoted as E(X) or μ. For a discrete random variable, the expected value is calculated by summing the product of each possible value of the random variable and its probability. For a continuous random variable, the expected value is calculated by integrating the product of the random variable and its probability density function.
6.2 Significance of Expected Value
The expected value has several important properties and interpretations:
- The expected value is a measure of central tendency. It represents the average value that the random variable would take on if the experiment were repeated many times.
- The expected value is a linear operator. This means that the expected value of a sum of random variables is equal to the sum of their individual expected values.
- The expected value can be used to compare different random variables. A random variable with a higher expected value is, on average, larger than a random variable with a lower expected value.
- The expected value can be used to calculate the variance of a random variable. The variance measures the spread or dispersion of the random variable around its expected value.
Suppose we have a discrete random variable X that represents the number of heads obtained when flipping a fair coin twice. The possible values of X are 0, 1, and 2, with probabilities 1/4, 1/2, and 1/4, respectively. The expected value of X can be calculated as follows:
$$E(X) = 0 \cdot \frac{1}{4} + 1 \cdot \frac{1}{2} + 2 \cdot \frac{1}{4}$$
$$E(X) = 0 + \frac{1}{2} + \frac{1}{2}$$
$$E(X) = 1$$
The expected value of X is 1, which means that if we were to repeat the experiment of flipping a fair coin twice many times, the average number of heads obtained would be 1.
## Exercise
Suppose we have a continuous random variable Y that represents the height of a randomly selected person. The probability density function of Y is given by:
$$f(y) = \begin{cases}
\frac{1}{6} & 0 \leq y \leq 6 \\
0 & \text{otherwise}
\end{cases}
$$
Calculate the expected value of Y.
### Solution
Expected value of Y:
$$E(Y) = \int_{0}^{6} y \cdot \frac{1}{6} \,dy$$
$$E(Y) = \frac{1}{6} \int_{0}^{6} y \,dy$$
$$E(Y) = \frac{1}{6} \left[\frac{y^2}{2}\right]_{0}^{6}$$
$$E(Y) = \frac{1}{6} \left[\frac{6^2}{2} - \frac{0^2}{2}\right]$$
$$E(Y) = \frac{1}{6} \cdot \frac{36}{2}$$
$$E(Y) = 3$$
The expected value of Y is 3, which means that if we were to randomly select a person many times, the average height of the selected individuals would be 3 units.
# Law of large numbers
The law of large numbers is a fundamental concept in probability theory. It states that as the number of trials or observations increases, the average of those trials or observations will converge to the expected value of the random variable. In other words, the law of large numbers tells us that the more data we have, the closer our average will be to the true average.
7.1 Statement of the Law of Large Numbers
The law of large numbers can be stated as follows:
Let X1, X2, ..., Xn be a sequence of independent and identically distributed random variables with mean μ. Let S_n be the average of the first n random variables:
$$S_n = \frac{X_1 + X_2 + ... + X_n}{n}$$
Then, as n approaches infinity, the probability that Sn deviates from μ by more than ε approaches 0 for any positive ε:
$$\lim_{n \to \infty} P(|S_n - μ| > ε) = 0$$
In other words, as the number of trials or observations increases, the probability that the average deviates from the expected value by more than ε becomes arbitrarily small.
7.2 Interpretation of the Law of Large Numbers
The law of large numbers can be interpreted in the following way:
- The more trials or observations we have, the closer our average will be to the expected value.
- The law of large numbers provides a theoretical justification for using the average as an estimate of the expected value.
- The law of large numbers is the foundation for many statistical techniques and inference methods, as it allows us to make inferences about the population based on a sample.
Suppose we have a fair six-sided die, and we want to estimate the expected value of the outcome. We roll the die 100 times and calculate the average of the outcomes. We repeat this experiment multiple times and record the average each time.
After repeating the experiment 1000 times, we find that the average of the outcomes converges to approximately 3.5, which is the expected value of a fair six-sided die. This demonstrates the law of large numbers in action - as the number of trials increases, the average of the outcomes gets closer to the expected value.
## Exercise
Suppose we have a random variable X that represents the number of heads obtained when flipping a fair coin 100 times. Use the law of large numbers to explain what would happen to the average number of heads as the number of coin flips increases.
### Solution
As the number of coin flips increases, the average number of heads obtained will approach 50. This is because the expected value of a fair coin flip is 0.5, and the law of large numbers tells us that as the number of trials increases, the average of those trials will converge to the expected value. In this case, the expected value is 0.5, so the average number of heads will approach 0.5 times the number of coin flips, which is 50.
# Central limit theorem and its applications
The central limit theorem is another fundamental concept in probability theory. It states that the sum or average of a large number of independent and identically distributed random variables will have an approximately normal distribution, regardless of the shape of the original distribution.
8.1 Statement of the Central Limit Theorem
The central limit theorem can be stated as follows:
Let X1, X2, ..., Xn be a sequence of independent and identically distributed random variables with mean μ and variance σ^2. Let S_n be the sum of the first n random variables:
$$S_n = X_1 + X_2 + ... + X_n$$
Then, as n approaches infinity, the distribution of Sn approaches a normal distribution with mean nμ and variance nσ^2:
$$\lim_{n \to \infty} P\left(\frac{S_n - n\mu}{\sqrt{n\sigma^2}} \leq x\right) = \Phi(x)$$
where Φ(x) is the cumulative distribution function of a standard normal distribution.
In other words, as the number of random variables increases, the distribution of their sum or average approaches a normal distribution.
8.2 Applications of the Central Limit Theorem
The central limit theorem has many applications in statistics and probability theory. Some of its key applications include:
- Estimation of population parameters: The central limit theorem allows us to use the sample mean or sample sum as an estimate for the population mean or population sum, respectively.
- Hypothesis testing: The central limit theorem is used to construct test statistics and determine critical values for hypothesis tests.
- Confidence intervals: The central limit theorem is used to construct confidence intervals for population parameters.
- Random sampling: The central limit theorem justifies the use of random sampling in statistical inference.
Suppose we have a population of heights that follows a skewed distribution. We take a random sample of 100 individuals and calculate the average height. We repeat this process multiple times and record the average each time.
After repeating the process 1000 times, we find that the distribution of the sample averages is approximately normal, regardless of the shape of the original distribution. This demonstrates the central limit theorem in action - as the sample size increases, the distribution of the sample average approaches a normal distribution.
## Exercise
Suppose we have a random variable X that represents the sum of rolling a fair six-sided die 100 times. Use the central limit theorem to explain what would happen to the distribution of X as the number of rolls increases.
### Solution
As the number of rolls increases, the distribution of X will approach a normal distribution. This is because the central limit theorem tells us that the sum of a large number of independent and identically distributed random variables will have an approximately normal distribution, regardless of the shape of the original distribution. In this case, the original distribution is discrete and skewed, but as the number of rolls increases, the distribution of the sum will become more symmetric and bell-shaped.
# Sampling and hypothesis testing
Sampling is a fundamental concept in statistics that involves selecting a subset of individuals or items from a larger population. This subset is called a sample, and it is used to make inferences about the population.
9.1 Simple Random Sampling
Simple random sampling is a commonly used sampling method where each individual or item in the population has an equal chance of being selected for the sample. This ensures that the sample is representative of the population and reduces the potential for bias.
To perform simple random sampling, you can assign a unique identifier to each individual or item in the population and use a random number generator to select the desired number of samples. This method ensures that every individual or item has an equal chance of being selected.
9.2 Hypothesis Testing
Hypothesis testing is a statistical method used to make inferences about a population based on a sample. It involves formulating a hypothesis about the population parameter, collecting data from a sample, and using statistical tests to determine the likelihood of the observed data given the hypothesis.
The hypothesis testing process involves the following steps:
1. Formulate the null hypothesis (H0) and the alternative hypothesis (Ha). The null hypothesis represents the status quo or the absence of an effect, while the alternative hypothesis represents the presence of an effect or a difference.
2. Collect data from a sample and calculate a test statistic. The test statistic is a measure of how likely the observed data is under the null hypothesis.
3. Determine the critical region or the rejection region. This is a range of values for the test statistic that would lead to the rejection of the null hypothesis.
4. Compare the test statistic to the critical region. If the test statistic falls within the critical region, the null hypothesis is rejected in favor of the alternative hypothesis. If the test statistic falls outside the critical region, the null hypothesis is not rejected.
5. Draw conclusions based on the results of the hypothesis test. If the null hypothesis is rejected, it suggests that there is evidence to support the alternative hypothesis. If the null hypothesis is not rejected, it suggests that there is not enough evidence to support the alternative hypothesis.
Hypothesis testing is a powerful tool for making inferences about populations based on sample data. It allows researchers to draw conclusions and make decisions based on statistical evidence.
Suppose we want to test whether a new weight loss program is effective in reducing weight. We randomly select a sample of 100 individuals and measure their weight before and after participating in the program.
The null hypothesis (H0) is that the weight loss program has no effect, and the alternative hypothesis (Ha) is that the weight loss program is effective in reducing weight.
We calculate the average weight loss in the sample and compare it to the expected weight loss under the null hypothesis. If the observed weight loss is significantly different from the expected weight loss, we reject the null hypothesis in favor of the alternative hypothesis.
## Exercise
Suppose we want to test whether a new drug is effective in reducing blood pressure. We randomly select a sample of 50 individuals and measure their blood pressure before and after taking the drug. The null hypothesis (H0) is that the drug has no effect on blood pressure, and the alternative hypothesis (Ha) is that the drug is effective in reducing blood pressure.
Using the information given, identify the following:
1. The population in this study.
2. The sample in this study.
3. The null hypothesis (H0).
4. The alternative hypothesis (Ha).
### Solution
1. The population in this study is the group of individuals who could potentially benefit from the new drug.
2. The sample in this study is the randomly selected group of 50 individuals who participated in the study.
3. The null hypothesis (H0) is that the drug has no effect on blood pressure.
4. The alternative hypothesis (Ha) is that the drug is effective in reducing blood pressure.
# Bayes' theorem and its applications
Bayes' theorem is a fundamental concept in probability theory that allows us to update our beliefs or probabilities based on new evidence. It provides a way to calculate the probability of an event given prior knowledge and new information.
Bayes' theorem is stated as follows:
$$P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}$$
Where:
- $P(A|B)$ is the probability of event A occurring given that event B has occurred.
- $P(B|A)$ is the probability of event B occurring given that event A has occurred.
- $P(A)$ is the prior probability of event A occurring.
- $P(B)$ is the prior probability of event B occurring.
Bayes' theorem is particularly useful in situations where we have prior knowledge or beliefs about the probability of an event, and we want to update those probabilities based on new evidence.
10.1 Applications of Bayes' theorem
Bayes' theorem has a wide range of applications in various fields, including:
1. Medical diagnosis: Bayes' theorem can be used to calculate the probability of a patient having a certain disease given the results of a medical test.
2. Spam filtering: Bayes' theorem can be used to classify emails as spam or non-spam based on the words and patterns in the email.
3. Weather forecasting: Bayes' theorem can be used to update weather predictions based on new data and observations.
4. Machine learning: Bayes' theorem is used in various machine learning algorithms, such as Naive Bayes classifiers.
Bayes' theorem provides a powerful framework for updating probabilities and making informed decisions based on new evidence. It allows us to incorporate prior knowledge and beliefs into our calculations, leading to more accurate and reliable results.
Suppose we have a medical test for a certain disease. The test is known to be 95% accurate, meaning that it correctly identifies 95% of the people who have the disease and 95% of the people who do not have the disease. The prevalence of the disease in the population is 1%.
If a person tests positive for the disease, what is the probability that they actually have the disease?
To solve this problem, we can use Bayes' theorem. Let's define the events as follows:
- A: The person has the disease.
- B: The person tests positive for the disease.
We are given the following probabilities:
- $P(A) = 0.01$ (prevalence of the disease)
- $P(B|A) = 0.95$ (sensitivity of the test)
- $P(B|A') = 0.05$ (false positive rate of the test)
We want to calculate $P(A|B)$, the probability that the person has the disease given that they tested positive.
Using Bayes' theorem:
$$P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}$$
We can calculate $P(B)$ using the law of total probability:
$$P(B) = P(B|A) \cdot P(A) + P(B|A') \cdot P(A')$$
Substituting the given values:
$$P(B) = 0.95 \cdot 0.01 + 0.05 \cdot 0.99 = 0.059$$
Now we can calculate $P(A|B)$:
$$P(A|B) = \frac{0.95 \cdot 0.01}{0.059} \approx 0.161$$
So the probability that a person has the disease given that they tested positive is approximately 0.161, or 16.1%.
## Exercise
Suppose a certain disease affects 1 in 1000 people in a population. A diagnostic test for the disease has a sensitivity of 95% (correctly identifies 95% of the people who have the disease) and a specificity of 98% (correctly identifies 98% of the people who do not have the disease).
Using Bayes' theorem, calculate the probability that a person has the disease given that they test positive.
### Solution
Let's define the events as follows:
- A: The person has the disease.
- B: The person tests positive for the disease.
We are given the following probabilities:
- $P(A) = 0.001$ (prevalence of the disease)
- $P(B|A) = 0.95$ (sensitivity of the test)
- $P(B|A') = 1 - 0.98$ (false positive rate of the test)
We want to calculate $P(A|B)$, the probability that the person has the disease given that they tested positive.
Using Bayes' theorem:
$$P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}$$
We can calculate $P(B)$ using the law of total probability:
$$P(B) = P(B|A) \cdot P(A) + P(B|A') \cdot P(A')$$
Substituting the given values:
$$P(B) = 0.95 \cdot 0.001 + (1 - 0.98) \cdot 0.999 = 0.02397$$
Now we can calculate $P(A|B)$:
$$P(A|B) = \frac{0.95 \cdot 0.001}{0.02397} \approx 0.0393$$
So the probability that a person has the disease given that they test positive is approximately 0.0393, or 3.93%.
# Advanced topics in Probability
11.1 Conditional probability
Conditional probability is a fundamental concept in probability theory that deals with the probability of an event occurring given that another event has already occurred. It is denoted as $P(A|B)$, which represents the probability of event A occurring given that event B has occurred.
The formula for conditional probability is:
$$P(A|B) = \frac{P(A \cap B)}{P(B)}$$
where $P(A \cap B)$ represents the probability of both event A and event B occurring.
Conditional probability allows us to update our probabilities based on new information or evidence. It is particularly useful in situations where events are dependent on each other.
11.2 Independence
Two events A and B are said to be independent if the occurrence of one event does not affect the probability of the other event. Mathematically, this can be expressed as:
$$P(A \cap B) = P(A) \cdot P(B)$$
If two events are independent, then the conditional probability of one event given the other event is equal to the unconditional probability of the event. In other words, $P(A|B) = P(A)$ and $P(B|A) = P(B)$.
Independence is an important concept in probability theory and is often used to simplify calculations and make assumptions in probabilistic models.
11.3 Bayes' theorem
Bayes' theorem, which we discussed in the previous section, is a powerful tool for updating probabilities based on new evidence. It allows us to calculate the probability of an event occurring given prior knowledge and new information.
Bayes' theorem is stated as:
$$P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}$$
where $P(A|B)$ is the probability of event A occurring given that event B has occurred, $P(B|A)$ is the probability of event B occurring given that event A has occurred, $P(A)$ is the prior probability of event A occurring, and $P(B)$ is the prior probability of event B occurring.
Bayes' theorem is particularly useful in situations where we have prior knowledge or beliefs about the probability of an event, and we want to update those probabilities based on new evidence.
11.4 Random variables
A random variable is a mathematical function that assigns a numerical value to each possible outcome of a random experiment. It allows us to quantify and analyze the uncertainty associated with the outcomes of a random experiment.
Random variables can be classified into two types: discrete random variables and continuous random variables.
A discrete random variable takes on a countable number of distinct values. Examples include the number of heads obtained when flipping a coin multiple times or the number of cars passing through a toll booth in a given time period.
A continuous random variable takes on an uncountable number of possible values within a given range. Examples include the height of a person or the time it takes for a computer program to execute.
Random variables can be described by their probability distribution, which specifies the probabilities associated with each possible value of the random variable. The probability distribution can be represented graphically using a probability density function (PDF) for continuous random variables or a probability mass function (PMF) for discrete random variables.
11.5 Expectation and variance
Expectation and variance are two important measures of a random variable's properties.
The expectation of a random variable, denoted as $E(X)$ or $\mu$, represents the average value or the center of the probability distribution. It is calculated as the weighted sum of the possible values of the random variable, where the weights are the probabilities associated with each value.
The variance of a random variable, denoted as $Var(X)$ or $\sigma^2$, represents the spread or dispersion of the probability distribution. It is calculated as the average of the squared differences between each value of the random variable and the expectation, weighted by their probabilities.
Expectation and variance provide valuable insights into the behavior and characteristics of random variables. They are used in various statistical calculations and probabilistic models.
11.6 Central limit theorem
The central limit theorem is a fundamental result in probability theory that states that the sum or average of a large number of independent and identically distributed random variables tends to follow a normal distribution, regardless of the shape of the original distribution.
The central limit theorem is widely used in statistics and data analysis. It allows us to make inferences about the population based on a sample, as well as to estimate confidence intervals and perform hypothesis testing.
The central limit theorem has important implications for understanding the behavior of random variables and the convergence of probability distributions. It is a key concept in many areas of probability theory and statistics.
In this section, we have explored some advanced topics in probability, including conditional probability, independence, Bayes' theorem, random variables, expectation and variance, and the central limit theorem. These topics provide a deeper understanding of the mathematical foundations of probability theory and their applications in various fields. | Textbooks |
Sema Salur
Sema Salur is a Turkish-American mathematician, currently serving as a Professor of Mathematics at the University of Rochester.[1] She was awarded the Ruth I. Michler Memorial Prize for 2014–2015,[2] a prize intended to give a recently promoted associate professor a year-long fellowship at Cornell University;[3] and has been the recipient of a National Science Foundation Research Award beginning in 2017.[4] She specialises in the "geometry and topology of the moduli spaces of calibrated submanifolds inside Calabi–Yau, G2 and Spin(7) manifolds",[2][5] which are important to certain aspects of string theory and M-theory in physics, theories that attempt to unite gravity, electromagnetism, and the strong and weak nuclear forces into one coherent Theory of Everything.[5]
Education
• 1993: B.S. in Mathematics, Boğaziçi University, Turkey.[2]
• 2000: PhD in Mathematics, Michigan State University[2]
References
1. "Sema Salur". University of Rochester.
2. "Ruth I. Michler Prize 2014-2015". Association for Women in Mathematics.
3. "The Ruth I Michler Memorial Prize of the AWM". St Andrews University.
4. "Award Abstract #1711178: Manifolds with Special Holonomy and Applications". National Science Foundation.
5. "Professor Sema Salur receives NSF Research Award". University of Rochester.
Authority control
International
• ISNI
Academics
• MathSciNet
• Mathematics Genealogy Project
| Wikipedia |
Fourier transform of the unit sphere
The Fourier transform of the volume form of the (n-1)-sphere in $\mathbf R^n$ is given by the well-known formula $$ \int_{S^{n-1}}e^{i\langle\mathbf a,\mathbf u\rangle}d\sigma(\mathbf u) = (2\pi)^{\nu + 1}\|\mathbf a\|^{-\nu}J_\nu(\|\mathbf a\|), \qquad\nu=\frac n2 -1, \tag1 $$ found e.g. in [1, p. 198] or [2, p. 154].
Does anyone here know earlier references, and perhaps who first published this formula?
According to Watson [3, p. 9] the case n=2, $$ \frac1{2\pi}\int_{S^1}e^{ia\cos\theta}d\theta=J_0(a) \tag2 $$ goes back to Parseval [4], but I am mainly curious about the case n=3, $$ \frac1{4\pi}\int_{S^2}e^{i\langle\mathbf a,\mathbf u\rangle}d\sigma(\mathbf u)=\frac{\sin\|\mathbf a\|}{\|\mathbf a\|}. \tag3 $$
I. M. Gel'fand & G. E. Shilov, Generalized functions, vol. 1, Academic Press (1964).
E. M. Stein & G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton UP (1971).
G. N. Watson, A treatise on the theory of Bessel functions, Cambridge UP (1922)
M. A. Parseval, Mémoire sur les séries et sur l'intégration complète (etc.) (1805)
reference-request fa.functional-analysis ho.history-overview harmonic-analysis fourier-transform
Francois Ziegler
Francois ZieglerFrancois Ziegler
$\begingroup$ On my opinion, this is an ill-defined question: who evaluated an elementary integral for the first time. Euler, who introduced $\exp$ could do this, without giving it the name "Fourier transform". Parseval [4] did a more complicated integral... $\endgroup$ – Alexandre Eremenko Nov 23 '13 at 14:05
$\begingroup$ @AlexandreEremenko That's exactly what I'd like to know: Do we have evidence that Euler could do (3)? (It's not that elementary...) Then OK, I replaced "first derived" by "first published". Note that Watson or Encykl. Math. Wiss. attribute scores of formulas -- but not this one, as far as I could find. $\endgroup$ – Francois Ziegler Nov 23 '13 at 15:32
$\begingroup$ Actually the $n=3$ case is entirely elementary, using a theorem of Archimedes: the projection $(r \cos \theta, r\sin \theta, z) \mapsto (\cos \theta, \sin \theta, z)$ from $S^2$ to the cylinder $\{x^2+y^2=1, \; |z|\leq 1\}$ preserves areas. Thus the integral is just $\frac12 \int_{-1}^1 e^{i\|a\|z} dz$, which Euler certainly knew was ${\mathop {\rm sinc}}(\|a\|)$. $\endgroup$ – Noam D. Elkies Nov 30 '13 at 17:09
$\begingroup$ @NoamD.Elkies I agree that the formula is elementary modulo Archimedes, I'd just be surprised to see Poisson state it (below, item 2.) without at least as much of an argument as you have given -- if this was the first time it appeared. Instead, he justifies it by calling it a "known formula". To me this suggests there may be an earlier occurrence, and this "first" is what I'm looking for. But maybe it doesn't exist. $\endgroup$ – Francois Ziegler Dec 1 '13 at 2:11
$\begingroup$ $S^n$ is coadjoint orbit, so you can see in general case Kirillov's character formula en.wikipedia.org/wiki/Kirillov_character_formula $\endgroup$ – user21574 May 5 '14 at 7:02
At the risk of answering my own question, here is what I have since found:
For general $n$, formula (1) seems to occur first on p. 177 of S. Bochner, Summation of multiple Fourier series by spherical means, Trans. AMS 40 (1936) 175-207. Bochner exposes it again on pp. 73-74 of Fourier Transforms (Princeton UP 1949).
For $n=3$, Burkhardt (Trigonometrische Reihen und Integrale bis etwa 1850, Encykl. Math. Wiss. II A 12 (1916) 819-1354, page 1258) claims to find formula (3) in Poisson's Mémoire sur l'intégration de quelques équations linéaires aux différences partielles, et particulièrement de l'équation générale du mouvement des fluides élastiques, Mém. Acad. Roy. Sci. Inst. France 3 (1820) 121-176, page 134, in the form $$ \mathfrak{Sin}\,pt= \frac{pt}{2\pi}\int_0^{2\pi}\int_0^\pi\exp\{t(g\cos u+h\sin u\sin v+k\sin u\cos v)\}\sin u\,du\,dv $$ where $p=\sqrt{\smash[b]{g^2+h^2+k^2}}$, $\mathfrak{Sin}$ is a hyperbolic function, and Burkhardt is missing a factor of 2. However... I'm not able to find it on that page of Poisson. On the other hand Poisson states it as "known" in a later memoir (1831, page 558). Perhaps someone will have better luck locating the original (3) -- in Poisson or elsewhere?
Edit: Aha, the problem was simply a typo in Burkhardt. Formula (3) indeed appears in Poisson's above-cited Mémoire, but on page 174 instead of 134, in the form $$ \int\int e^{at(g\cos u+h\sin u\sin v+k\sin u\cos v)}\sin u\,du\,dv = 2\pi\frac{e^{atp}-e^{-atp}}{atp}. $$
$\begingroup$ I think this question is related to the question of a Fourier invariant (up to scaling) representation of spherical harmonics on the surface of the unit sphere. Something like the Fourier invariant (up to scaling) Hermite polynomials for the surface of the unit sphere. I am not as mathematically sophisticated, but I think the answer is Laguerre polynomials expanded in the usual legendre polynomials? $\endgroup$ – v217 Oct 14 '16 at 9:11
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A new anthropometric index for body fat estimation in patients with severe obesity
Giliane Belarmino1Email author,
Raquel S. Torrinhas1,
Priscila Sala1,
Lilian M. Horie1,
Lucas Damiani2,
Natalia C. Lopes1,
Steven B. Heymsfield3 and
Dan L. Waitzberg1
BMC Obesity20185:25
Accepted: 7 June 2018
Published: 1 October 2018
Body mass index (BMI) has been used to assess body adiposity, but it cannot adequately reflect body fat (BF) amount. The body adiposity index (BAI) has been shown a better performance than BMI for this purpose, but it can be inaccurate to estimate BF under extreme amounts of fat. Here, we propose a new anthropometric index, the Belarmino–Waitzberg (BeW) index, for specific estimation of BF in severely obese patients.
In 144 adult patients with severe obesity, BF was estimated by air displacement plethysmography (ADP), as the reference method, along with the follow anthropometric measurements: height, abdominal circumference (AC), hip circumference (HC), weight, BMI (weight/ height2) and BAI ([HC(cm) / height (m)1.5) − 18] × 100). Patients were proportionately distributed into two distinct databases, the building model database (BMD) and the validation model database (VMD), which were applied to develop and validate the BeW index, respectively. The BeW index was tested for gender and ethnicity adjustment as independent variables. The agreement of BF% values obtained by the new index and by BAI with ADP was also assessed.
The BF% was 52.05 ± 5.42 for ADP and 59.11 ± 5.95 for the BeW index (all results are expressed as the mean ± standard deviation). A positive Pearson correlation (r = 0.74), a good accuracy (Cb = 0.94), and a positive Lin's concordance correlation (CCC = 0.70) were observed between the two groups. The 95% limits of individual agreement between the BeW index and ADP were 6.8 to 7.9%, compared to − 7.5 to 14.8% between the BAI and ADP. The new index, called the Belarmino–Waitzberg (BeW) index, showed an improvement of 2.1% for the R2 value and a significant gender effect, therefore resulting in two different indexes for females and males, as follows: Female BeW = − 48.8 + 0.087 × AC(cm) + 1.147 × HC(cm) - 0.003 × HC(cm)2 and Male BeW = − 48.8 + 0.087 × AC(cm) + 1.147 × HC(cm) - 0.003 × HC(cm)2–7.195.
The new BeW index showed a good performance for BF estimation in patients with severe obesity and can be superior to the BAI for this purpose.
Severe obesity
Body adiposity index
Air displacement plethysmography
The obesity epidemic is now a major public health concern due its impact on health and the economy. Characterized by abnormal body fat (BF) accumulation, obesity is linked with several debilitating diseases and an increased mortality [1]. An accurate diagnosis of obesity can guide clinical management and treatments.
The body mass index (BMI) has been widely applied to identify and classify obesity. However, the BMI cannot fully reflect the amount and distribution of BF or clearly distinguish the fat-free mass from fat mass compartments. Particularly, the BMI performance in severely obese patients is impaired by the large amount of subcutaneous adipose tissue and edema [2, 3].
BF assessment is more precise in obese subjects when more specific methods for body composition evaluation are used. Air displacement plethysmography (ADP) is a scientific validated reference method to evaluate body composition in obese subjects [4]. The method uses air displacement after patient entry in the ADP device to assess BF, by applying the Boyle's Law [5]. ADP has several advantages over other reference methods for BF assessment, including a quick noninvasive measurement process [5]. The main limitation in the routine application of ADP or other similar available tools for BF assessment is its high cost and practical complexity. Therefore, other alternative methods for BF assessment have been proposed.
In particular, the body adiposity index (BAI) has been found to be more sensitive to identify and classify obesity than the BMI, compared to dual-energy X-ray absorptiometry (DXA), as the reference method [6]. The BAI provides a practical advantage because it uses a simple equation for calculation that includes only hip circumference (HC) and body height. Furthermore, the BAI was developed from a database composed of adult Mexican-American men and further successfully validated in adult African-American men and women, suggesting that it could be applied universally without adjustments for gender or race.
Nevertheless, studies applying the BAI have shown its limited performance for BF estimation in populations with extreme amounts of fat (very low or very high) [7–9]. In particular, we have recently shown that the BAI poorly estimates BF in severely obese patients. Therefore, the aim of the current study was to design a new index to estimate BF in patients with severe obesity using simple and accessible anthropometric measurements.
Study protocol and design
Our study protocol was designed to develop a new anthropometric index for BF estimation in severely obese patients. For this purpose, our obese patient sample was composed of two proportional and distinct databases: the building model database (BMD), composed of the same 72 patients for whom we previously found a low performance of the BAI in estimating BF [10]; and the validation model database (VMD), composed of new 72 paired obese patients. The BMD was applied to develop the new index from anthropometric data potentially predictive of BF, using the BF% obtained by ADP as reference standard. The VMD was applied to validate the new index by assessing its performance in estimating BF in a similar population. For this purpose, the correlation and agreement between the BF values estimated by the new index with the BF values estimated by ADP were assessed. Furthermore, the agreement between the BF values estimated by BAI with the BF values estimated by ADP also was assessed to identify the performance of BAI comparative to the new index. For all the analyses, each patient was evaluated on the same day, in the morning, and after a 4-h overnight fast. Patients were instructed not to smoke or to drink alcohol during the 24 h prior to measurements, which were performed by the same trained technician for all enrolled patients.
Sample size calculations were based on the development of a model with nine possible predictors, with an effect size of at least 0.15 and considering an alpha value of 5% and a power of 95% [11]. A total of 144 severely obese Brazilian patients (108 female, 36 male) aged 18–55 years old, who were candidates for bariatric surgery and each of whom had a BMI ≥ 30 kg/m2 (range: 30–62 kg/m2), were recruited from the Digestive Tract Surgery Service at the Hospital das Clinicas - University of São Paulo School of Medicine, São Paulo, Brazil. The exclusion criteria were neurologic or psychiatric conditions; substance abuse; lactating or pregnant women; HIV-positive or cancer patients; clinically detectable edema; physical amputations; and chronic or acute diseases of the liver, lung, kidney, or heart. All study procedures were performed according to the ethical standards of the World Medical Association's Declaration of Helsinki. These and both datasets (building and validation) were approved by the institutional ethics review board – CAPPesq (1069/05 and 1011/09). Written informed consent was obtained from each patient prior to participation.
Anthropometric data
Each anthropometric data corresponds to an average of 3 sequentially repeated measures, which were performed as previously described elsewhere [10]. Briefly, body weight (kg, minimal variation of 10 g) was measured by using the weekly-calibrated body weight scale of the ADP system (Bod Pod body composition system, Life Measurement Instruments, Concord, CA, USA), with the patient standing in the center of the scale platform, barefoot, and wearing only underwear. Body height (cm) was obtained with a stadiometer (Sanny, São Paulo, Brazil), with the patient standing, barefoot with the heels together, back upright, and arms stretched next to the body. The abdominal circumference (AC) was measured using an inelastic metrical tape at the trunk midway between the lower costal margin (bottom of the lowest rib) and the iliac crest (top of the pelvic bone), with the patient standing with his/her feet 25–30 cm apart. The measurement was taken by fitting the tape snugly, without compressing the underlying soft tissue. Circumference was measured to the nearest 0.5 cm, at the end of a normal expiration [12]. The HC (cm) was measured by positioning a measuring tape in the horizontal plane at the greatest circumference of the buttocks [6, 13]. In addition, the BMI was calculated as body weight (kg) / [height (m)]2 and classified according to the World Health Organization scoring system [14, 15].
BF estimation by the BAI
The following equation was used to estimate BF by the BAI: BF% BAI = [(HC(cm) / height (m)1.5) − 18] × 1006.
BF estimation by ADP
Using the Bod Pod, ADP was performed to estimate the total BF. In the ADP method, the inverse relationship between pressure and volume proposed by Boyle (P1 × V1 = P2 × V2) was used to determine the body volume. Skin surface area artefact (SAA) also was calculated by the BOD POD software to allow changes in air temperature close to the subject's skin. Body volume (BV) and body density (BD) were then calculated as V (L) = BVraw – SAA (L) + 40% TGV (L) and BF% was then calculated using Siri's equation: BF% = (4.95/D – 4.5) × 100, where D = density. All measurements and calculations were automatically performed by the system software, and they are based on air volume and pressure variations inside the Bod Pod chamber when occupied and not occupied by the patient [12, 16]. During ADP evaluations, the patients wore only underwear and a cap to keep their hair fastened, and they remained in a sitting position inside the chamber [12]. Metallic objects, such as earrings, rings, chains, and body piercings, were not allowed.
New index design
Values of BF% obtained by ADP from the BMD were correlated with anthropometric data. Variables with a significant concordance with the BF% values provided by ADP were included in the initial index model and tested for the influence of gender and race by polynomial regression. The Akaike criterion was applied to select the variables to be used in the final index model, and a backward linear regression was applied to develop the specific BF prediction equation.
New index test for validation and performance
The agreement between the BF values from the ADP system with the two new BF equations and BAI were assessed according to the Pearson correlation, (r), accuracy (Cb), Lin's concordance correlation coefficient (CCC) and the Bland–Altman plot.
In addition to the statistical analysis described above for the development of validation of the new equation, descriptive data were compared by the Student's unpaired t- test or the Mann-Whitney U-test, when appropriate. All statistical analyses were performed using the R software package (version 3.1.0, R Development Core Team, 2014). The results are expressed as the mean ± standard deviation. Statistical significance was set at p < 0.05 for all tests.
Descriptive data
Table 1 provides the baseline demographic, body composition, and anthropometric data of the studied obese patients, divided in into the MBD (n = 72) and MVD (n = 72) as well as the total (n = 144). From the entire sample, most of the patients were women (70%), the BMI ranged from 34.40 to 62.98 kg/m2, with 92% of patients having BMI > 40 kg / m2; age ranged from 18 to 62 years old; and the mean BF% estimates measured by ADP was 52.05 ± 5.42%.
Demographic, body composition, and anthropometric data of severely obese patient samples
MBD
Total (n = 144)
Gender (female)
53/72 (73.6%)
108/144 (75.0%)
Gender (male)
36/144 (25.0%)
Race (white)
Race (black/brown)
42.58 ± 12.32
Body weight (kg)
127.68 ± 27.5
123.33 ± 24.53
1.63 ± 0.1
AC (cm)
HC (cm)
Waist-hip ratio
1.02 ± 0.11
ADP BF%
52.14 ± 5.4
Data were obtained from 144 patients and are expressed as the mean ± standard deviation. MBD model building database, MVD model validation database, BMI body mass index, AC abdominal circumference, HC hip circumference, ADP BF% values of body fat percentage estimated by air displacement plethysmography
New anthropometric index design
Data of BF% estimated by ADP significantly correlated with the following individual anthropometric data: body weight, AC, waist-hip ratio (negative correlation), and HC (p ≤ 0.05; Table 2). HC was the variable with the higher significant correlation with BF% estimated by ADP (p < 0001) and then was tested for gender and race adjustments. HC shows to be significantly influenced by gender, by displaying a nonlinear dispersion with BF% estimated by ADP (Fig. 1). Therefore, the initial model aimed to join two factors: the highly significant nonlinearity of HC (compared to BF%) and the inclusion of body weight and/or AC as potential additional predictive variables of BF adjusted by gender and race. The quadratic effect of the interaction between HC, body weight, and AC data with gender and race showed that only the HC (quadratic) presented significant results (Table 3). Therefore, by using the Akaike criteria, the new anthropometric index was simplified to the factors shown in Table 4. The new index, called the Belarmino–Waitzberg (BeW) index, showed an improvement of 2.1% for the R2 value and a significant gender effect, therefore resulting in two different indexes for females and males, as follows:
$$ \mathrm{Female}\;\mathrm{BeW}=-48.8+0.087\times \mathrm{AC}\left(\mathrm{cm}\right)+1.147\times \mathrm{HC}\left(\mathrm{cm}\right)-0.003\times \mathrm{HC}{\left(\mathrm{cm}\right)}^2 $$
$$ \mathrm{Male}\;\mathrm{BeW}=-48.8+0.087\times \mathrm{AC}\left(\mathrm{cm}\right)+1.147\times \mathrm{HC}\left(\mathrm{cm}\right)-0.003\times \mathrm{HC}{\left(\mathrm{cm}\right)}^2-7.195 $$
Pearson's correlation coefficient of body fat values (%) estimated by air displacement plethysmography and anthropometric variables from building database
95% CI
p Value*
− 0.37–0.08
−0.45–-0.01
Data were obtained from 72 obese patients. AC abdominal circumference, HC hip circumference
Dispersion graphs of the body fat values (%) estimated by air displacement plethysmography with hip circumference, gender, and race in the model building database. Data were obtained from 72 severely obese patients. The estimated curves were generated by the local polynomial regression method to best describe the data behavior. Note that for gender, the ratio of values of body fat (%) estimated by air displacement plethysmography is not linear, and a curve trend of second order (quadratic) seems to be a good fit to the data. However, for race, there was no effect
Coefficients estimated for the initial prediction model for values of body fat (%) estimated by air displacement plethysmography in the model building database
t Value
−73.52
(HC (cm))2
Sex – Male
Race – brown/black
AC (cm): male
Body weight (kg): male
HC (cm): sex – male
(HC (cm))2: sex – male
AC (cm): race – brown/black
Body weight (kg): race – brown/black
HC (cm): race – brown/black
(HC (cm))2: race – brown/black
R2 = 57.3%
Estimated coefficients of the final prediction model for values of body fat (%) estimated by air displacement plethysmography in the model building database
−48.805
−1.473
Abdominal circumference (cm)
(Hip circumference (cm))2
< 0.001
Data were obtained from 72 obese patients
New anthropometric index validation and performance
The new equations of the BeW index were tested for the MVD and showed a good correlation, accuracy, and CCC (r = 0.74; Cb = 0.94; and CCC = 0.70, respectively) with BF% estimated by ADP (Fig. 2). Although in a less extend, BF% estimated by ADP also showed a good correlation, accuracy, and CCC with BAI (r = 0.67, Cb = 0.82; and CCC = 0.55, Fig. 3); however, the BeW index provided lower limits (6.8 to 7.9%, Fig. 2) of agreement with BF% estimated by ADP than those obtained from the BAI (− 7.5 to 14.8%, Fig. 3). Therefore, the BF% values estimated by the BAI are, on average, 3.7% higher than those estimated by ADP, compared to 0.4% for the new index.
Bland-Altman plot showing limits of agreement between the values of body fat (%) estimated by air displacement plethysmography (ADP) vs. those estimated by the new Belarmino–Waitzberg (BeW) index in the validated model sample. Data were obtained from 72 severely obese patients. Bold continuous lines indicate the observed average agreement. Continuous lines indicate the line of perfect average agreement. Dashed lines indicate 95% limits of agreement. Lin's concordance correlation coefficient (CCC) is shown
Bland-Altman plot showing limits of agreement between the values of body fat (%) estimated by air displacement plethysmography (ADP) vs. those estimated by the body adiposity index (BAI) in the validated model sample. Data were obtained from 72 severely obese patients. Bold continuous lines indicate the observed average agreement. Continuous lines indicate the line of perfect average agreement. Dashed lines indicate 95% limits of agreement. Lin's concordance correlation coefficient (CCC) is shown
Excessive BF is associated with the occurrence of clinical complications that compromise the quality of life and survival of individuals, such as diabetes, hypertension, cardiovascular disease, musculoskeletal disorders, and cancer [17]. Clinical assessments of BF% in severely obese patients are challenged by the high cost of available methods and the lack of accuracy that some of these methods provide under the elevated fat, total and extracellular water, and change in electrolyte concentration exhibited by severely obese patients [3]. Aiming to solve this issue, we proposed a new equation involving simple anthropometric data to estimate BF% specifically in severely obese patients.
Our study was inspired by the BAI developed by Bergman and colleagues [6]. This tool was designed for BF% assessment, and its most relevant aspects are the simplicity with which it is clinically applied and its higher performance than the BMI to identify and classify obesity. However, this tool has shown a low accuracy at estimating BF% in several populations of severely obese patients [9, 10, 18–21]. These observations highlight that the BAI may not exceed the BMI limitations when applied to assess body composition in severely obese patients [22].
For instance, Geliebter and colleagues compared the BAI and BMI performances in estimating the BF% of severely obese women, using ADP, DXA, and body impedance analysis (BIA) as reference methods [20]. They found that body adiposity estimated by the BMI has a higher correlation with that estimated by ADP, bioelectrical impedance analysis (BIA), or DXA than by BAI. The authors concluded that the BAI seems to be a reasonable method to evaluate BF%, but it should not be used for this purpose in severely obese patients.
The reasons that the BAI is a poor indicator of BF% in severely obese patients may be associated with the use of AC without applying distinct cutoff points for men and women. Obesity is characterized by a distinct fat accumulation between genders, the so-called android and gynoid obesities [23]. Men have a greater fat accumulation in the abdominal area, while in women this occurs in the hip. This feature allows differentiating the body compositions of men and women and may be more pronounced in severely obese patients due the marked fat accumulation.
In fact, by not considering specific male and female cutoffs, the BAI may become a weak tool for BF% evaluation by gender. In a study of the general Mexican-American population, Lichtash and colleagues found that the BAI was a better indicator to estimate body adiposity than the BMI, compared to the reference values of DXA [24]. However, when the authors stratified the population by gender, the BAI had a lower performance than BMI for the same purpose. Notably, when reviewing the limitations for the BAI in assessing the BF% in severely obese patients, Bernhard and colleagues found that significant errors of this variable estimation by BAI were determined by specific factors that included waist-hip ratio ≥ 1.05–5%, gender, and obesity grade [19].
In our study, a poor performance of the BAI in estimating the BF% in severely obese patients was confirmed. For both the MBD and MVD, the BAI underestimated the BF% compared to ADP. Furthermore, we observed that the BF% and HC were not linear between genders. This finding supports a potential need for the design of specific indexes for men and women. In fact, when we designed the new index stratified by gender, we observed better limits of individual agreement of ADP with the BeW index than with the BAI.
The experimental protocol of this study has both strengths and weaknesses. Our sample enrolled mostly severely obese patients (BMI = 47.11 ± 6.48), and this feature was decisive when choosing the reference method for BF% assessment. ADP is a noninvasive method with a high accuracy for BF% estimation from the body density, which has been validated in severely obese patients [2, 4, 25–30]. Several studies have used DXA as a reference method for body composition [6, 9, 20, 24, 31]. However, DXA devices have limited support for weight and width, which can prevent the proper function of the whole-body scanner for severely obese patients [3]. By using ADP as the reference method, we were able to assess the performance of the new index with reliable parameters. On the other hand, our sample was relatively small compared to most available related studies. Nevertheless, it had an effect size of at least 0.15, considering an alpha value of 5% and a power of 95%.
In summary, our study supports that the BAI may not be reliable for BF% estimation in severely obese patients. In the present study, the new BeW index developed to assess BF% in severely obese patients showed a better performance than the BAI and followed a design that ensures its easy application. Therefore, replacement of the BAI with the new BeW index should be considered when assessing the adiposity level in patients with severe obesity.
Abdominal circumference
ADP:
BeW:
Belarmino–Waitzberg
BF:
BMD:
Building model database
Cb:
CCC:
Lin's concordance correlation
DXA:
Dual-energy X-ray absorptiometry
Hip circumference
Pearson correlation
VMD:
validation model database
The authors thank the patients and nurses who participated in the study and Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP).
This work was supported by Fundação de Amparo a Pesquisa do Estado de São Paulo (FAPESP), project no. 2011/09612–3 and 2011/13243–3, DD 2012/15677–3.
The datasets used and/or analysed during the current study are available from the corresponding author on reasonable request.
GB and RST contributed to the conception and design of the study. GB, RST, PS, LMH, LD, NCL, SBH, DLW were responsible for the acquisition, analysis, and/or interpretation and discussion of data. GB, RST, NCL, SBH and DLW drafted the manuscript. GB and RST participated in study design and coordination and helped to draft the manuscript. GB, RST and LD participated in the design of the study and performed the statistical analysis. LD performed statistical analyses. All authors read and approved the final manuscript.
All study procedures were performed according to the ethical standards of the World Medical Association's Declaration of Helsinki. These and both datasets (building and validation) were approved by the institutional ethics review board – CAPPesq (1069/05 and 1011/09). Written informed consent was obtained from each patient prior to trial participation.
Department of Gastroenterology, Surgical Division, LIM 35, Faculdade de Medicina da Universidade de São Paulo, Av. Dr. Arnaldo, 455, 2° andar, sala 2208, São Paulo, Cerqueira César, CEP: 01246-903, Brazil
Research Institute – Hospital do Coração de São Paulo, São Paulo, Brazil
Pennington Biomedical Research Center, Baton Rouge, LA, USA
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\begin{document}
\title{ A higher dispersion K\MakeLowercase{d}V equation on the half-line}
\author{A. Alexanddrou Himonas \& Fangchi Yan}
\date{March 8, 2021/Revised April 4, 2022
\mbox{}$^*$\!\textit{Corresponding author}: [email protected]}
\keywords{ higher dispersion Korteweg-de Vries equation, integrability, initial-boundary value problem, Fokas Unified Transform Method, well-posedness in Sobolev spaces, estimates in Bourgain spaces}
\subjclass[2020]{Primary: 35Q55, 35G31, 35G16, 37K10}
\begin{abstract}
The initial-boundary value problem (ibvp) for the $m$-th order dispersion Korteweg-de Vries (KdV) equation on the half-line with rough data and solution in restricted Bourgain spaces is studied using the Fokas Unified Transform Method (UTM). Thus, this work advances the implementation of the Fokas method,
used earlier for the KdV on the half-line with smooth data
and solution in the classical Hadamard space, consisting
of function that are continuous in time and Sobolev
in the spatial variable, to the more general Bourgain spaces
framework of dispersive equations with rough data
on the half-line.
The spaces needed and the estimates required arise
at the linear level and in particular in the estimation of the linear pure ibvp, which has forcing
and initial data zero but non-zero boundary data.
Using the iteration map defined by the Fokas solution formula
of the forced linear ibvp in combination with the bilinear estimates in
modified Bourgain spaces introduced by this map, well-posedness
of the nonlinear ibvp is established
for rough initial and boundary data belonging
in Sobolev spaces of the same optimal regularity
as in the case of the initial value problem for this equation on the whole line.
\end{abstract}
\maketitle
\markboth{ A higher dispersion KdV equation on the half-line }{ Alex Himonas \& Fangchi Yan }
\section{Introduction and Results}
In this work, we study the initial-boundary value problem (ibvp) for the $m$-th order dispersion Korteweg-de Vries (KdVm) equation on the half-line, that is
\begin{subequations} \label{KdVm} \begin{align} \label{KdVm eqn} &\partial_tu+(-1)^{j+1}\partial_x^{m}u +uu_x = 0, \quad 0<x<\infty, \,\, 0<t<T, \\ \label{KdVm ic} &u(x,0) = u_0(x), \quad 0<x<\infty, \\ \label{KdVm bc} & u(0,t) = g_0(t), \cdots , \partial_x^{j-1}u(0,t) = g_{j-1}(t), \quad 0<t<T, \end{align} \end{subequations}
where $m=2j+1$, $j\in \mathbb{N}$. Note that $m=3$ ($j=1$) gives the celebrated Korteweg-de Vries (KdV) equation (\cite{kdv1895}, \cite{b1877}), and $m=5$ ($j=2$) gives a Kawahara equation \cite{kawahara1972}. The initial data belong in the Sobolev spaces $H^s(0,\infty)$ and the boundary data $g_\ell(t)\in H_t^{(s+j-\ell)/{m}}(0,T)$, $\ell=0,1,\dots,j-1$, arise in the estimation of the linear pure ibvp and also reflect the time regularity of the solution to the linear KdVm initial value problem (ivp). Here, we show that if $s>-j+\frac 14$, then ibvp \eqref{KdVm} is well-posed with solution in an appropriately modified Bourgain space restricted to $(0, \infty)\times (0, T)$ for some lifespan $T>0$ depending on the size of the data.
Thus, the optimality $s>-j+\frac 14$ of our ibvp well-posedness result here is exactly the same with the one obtained in \cite{fhy2020} for the Cauchy problem of this equation on the whole line.
The starting point is to define the iteration map via the Fokas solution formula of the forced linear ibvp and then after deriving appropriate bilinear estimates for the nonlinearity in the modified Bourgain spaces we show that this map is a contraction in a ball. For the KdV equation with smooth data ($s>3/4$) well-posedness on the half-line was proved in \cite{fhm2016}, and for KdVm with smooth data ($s>m/4$) well-podsedness of the ibvp was proved in \cite{y2020}. Both works are based on the Fokas method but the solution spaces used, which are subsets of the space of functions that are Sobovev in $x$ and continuous in $t$, are motivated by the work on the KdV Cauchy problem by Kenig, Ponce and Vega in \cite{kpv1991}. Our work here advances the implementation of the Fokas method for solving ibvp with smooth data
(see \cite{fhm2016, fhm2017, hm2015, hm2020, hmy2019, hmy2019b, y2020})
to the Bourgain spaces framework of dispersive equations with rough data on the half-line.
Next, we define the spaces needed for stating our main result precisely. We recall that for any $s$ and $b$ real numbers, the Bourgain space $X^{s,b}(\mathbb{R}^2)$ corresponding to the linear part of KdVm is defined by the norm
\begin{equation} \label{def-Bourgain}
\|u\|_{X^{s,b}}^2 =
\|u\|_{s,b}^2 = \int_{-\infty}^\infty\int_{-\infty}^\infty
(1+|\xi|)^{2s}
(1+|\tau-\xi^m|)^{2b}
|\widehat{u}(\xi,\tau)|^2 d\xi d\tau, \end{equation}
where $\widehat{u}$ denotes the space-time Fourier transform
\begin{equation} \label{Fourier-transform} \widehat{u}(\xi,\tau) = \int_{-\infty}^\infty\int_{-\infty}^\infty e^{-i(\xi x +\tau t)} u(x, t) dx dt. \end{equation}
Also, we shall need the following modification of the Bourgain norm
\begin{align} \label{Bourgain-like-norms-def}
\|u\|_{X^{s,b,\alpha}(\mathbb{R}^2)}^2 =&
\|u\|_{s,b,\alpha}^2 \doteq
\|u\|_{X^{s,b}(\mathbb{R}^2)}^2 + \Big[ \int_{-\infty}^\infty \int_{-1}^1
(1+|\tau|)^{2\alpha}
|\widehat{u}(\xi,\tau)|^2 d\xi d\tau \Big] \\ \simeq& \int_{-\infty}^\infty \int_{-\infty}^\infty \Big[
(1+|\xi|)^{s}
(1+|\tau-\xi^m|)^{b} +
\chi_{|\xi|< 1}(1+|\tau|)^{\alpha} \Big]^2
|\widehat{u}(\xi,\tau)|^2 d\xi d\tau. \notag \end{align}
A similar modification was introduced first by Bourgain \cite{b1993-kdv}, where the norm used for the periodic case with $b=1/2$ was modified by the $\alpha$-part in order to prove well-posedness of the KdV equation on the line in $H^s$, $s\ge 0$. This idea was also utilized later by
Colliander, Keel, Staffilani, Takaoka and Tao in \cite{ckstt2003} for the global well-posedness of KdV on the circle for $s\ge -1/2$. The Bourgain norm with $b>1/2$ was used by Kenig, Ponce and Vega in \cite{kpv1996} in order to extend the well-posedness of KdV in $H^s(\mathbb{R})$ to $s>-3/4$.
Finally, we shall need the restriction space $X_{\mathbb{R}^+\times(0, T)}^{s,b,\alpha}$, which is defined as follows
\begin{equation} \label{sb-restrict} X_{\mathbb{R}^+\times(0, T)}^{s,b,\alpha} \doteq \{ u; u(x,t) = v(x,t) \;\;\mbox{on}\;\; \mathbb{R^+}\times (0, T) \,\, \text{with} \,\, v\in X^{s,b,\alpha}(\mathbb{R}^2) \}, \end{equation}
and which is equipped with the norm
\begin{equation} \label{sb-restrict-norm}
\| u\|_{X_{\mathbb{R}^+\times(0, T)}^{s,b,\alpha}} \doteq \inf\limits_{v\in X^{s,b,\alpha}}\left\{
\| v\|_{s,b,\alpha};\;v(x,t) = u(x,t) \;\;\mbox{on}\;\; \mathbb{R^+}\times (0, T) \right\}. \end{equation}
Now, we are ready to state our first result precisely. It reads as follows.
\begin{theorem} [Well-posedness of KdVm on the half-line] \label{thm-kdvm-half-line}
If $-j+\frac14< s\le j+1$, $s\neq \frac12,\frac32,\dots,j-\frac12$, then there exists $b\in(0,\frac12)$ depending on $s$ such that for any initial data $u_0 \in H^s(0,\infty)$ and boundary data $g_\ell\in H^{\frac 1m(s+j-\ell)}(0,T)$, $\ell=0,1,\dots,j-1$, there is a lifespan $T_0>0$ such that the ibvp \eqref{KdVm} with compatibility condition:
\begin{align} \label{first-com-con} \partial_x^\ell u_0(0) = g_\ell(0), \quad \text{for } \ell \text{ such that } \,\, \frac 1m (s+j-\ell) > \frac12 \, \text{ or } \, s>\ell+\frac12, \end{align}
admits a unique solution $u \in X_{\mathbb{R}^+\times(0, T_0)}^{s,b,\alpha} $, satisfying the size estimate
\begin{equation} \label{Bourgain-like-norms-est}
\|u\|_{X_{\mathbb{R}^+\times(0, T_0)}^{s,b,\alpha}} \leq C \Big(
\|u_0\|_{H^s(\mathbb{R}^+)} + \sum\limits_{\ell=0}^{j-1}
\|g_\ell\|_{H^{\frac{s+j-\ell}{m}}(0,T)} \Big), \end{equation}
for some $\alpha\in (\frac 12, 1)$.
Furthermore, an estimate for the lifespan is given by
\begin{equation} \label{lifespan-est} T_{0} = c_0 \Big(
1+\|u_0\|_{H^s(\mathbb{R}^+)} +\sum\limits_{\ell=0}^{j-1}
\|g_\ell\|_{H^{\frac{s+j-\ell}{m}}(0,T) } \Big)^{-4/\beta}, \quad c_0=c_0(s,b,\alpha), \end{equation}
for some $\beta>0$ depending on $s$ and $m$ (see \eqref{beta-choice}). Finally, the solution depends Lip-continuously on the data $u_0$ and $g_\ell$, $\ell=0,1,\dots,j-1$. \end{theorem}
As we have mentioned earlier, the optimality $s>-j+\frac 14$ for the data in our well-posedness Theorem \ref{thm-kdvm-half-line} on the half-line is exactly the same with the optimality of the data in the well-posedness result on the whole line of KdVm obtained in \cite{fhy2020}.
We prove Theorem \ref{thm-kdvm-half-line} by showing that the iteration map defined via the solution formula of the forced linear KdVm ibvp, which is obtained by the Fokas method, is a contraction in the solution space $ X^{s,b,\alpha}$. Therefore, we begin with the linear KdVm ibvp with forcing, that is
\begin{subequations} \label{LKdVm} \begin{align} \label{LKdVm eqn} &\partial_tu+(-1)^{j+1}\partial_x^{2j+1}u = f(x,t), \quad 0<x<\infty, \,\, 0<t<T, \\ \label{LKdVm ic} &u(x,0) = u_0(x), \hskip1.2in 0<x<\infty, \\ \label{LKdVm bc} & u(0,t) = g_0(t), \cdots, \partial_x^{j-1}u(0,t) = g_{j-1}(t), \quad 0<t<T, \end{align} \end{subequations}
where $T>0$ is any given time. Using the Fokas method, also referred in the literature as the Unified Transform Method (UTM), we get the following solution formula to the problem \eqref{LKdVm} (see \cite{y2020} or Section \ref{kdvm-sln-derivation} for an outline of the derivation):
\begin{align} \label{UTM-sln-compact} u(x,t) = S[u_0,g_0,\dots,g_{j-1};f] \doteq& \frac{1}{2\pi}\int_{-\infty}^\infty e^{i\xi x+i\xi ^mt} [\widehat u_0(\xi)+F(\xi,t)]d\xi \\ +& \sum\limits_{p=1}^{j} \sum\limits_{n=1}^{j+1} C_{p,n} \int_{\partial D_{2p}^+}e^{i\xi x+i\xi ^mt}[\widehat u_0(\alpha_{p,n}\xi)+F(\alpha_{p,n}\xi,t)]d\xi \nonumber \\ +& \sum\limits_{p=1}^j \sum\limits_{\ell=0}^{j-1} C'_{p,\ell} \int_{\partial D_{2p}^+}e^{i\xi x+i\xi ^mt}(i\xi )^{2j-\ell}\tilde g_{\ell}(\xi^m,T)d\xi, \nonumber \end{align}
where $C_{p,n}$ and $C'_{p,\ell}$ are constants and the rotation numbers $\alpha_{p,n}$ are given by
\begin{align} \label{a-rot-angles} \alpha_{p,n} \doteq e^{i[m-(2p+1)+2n]\frac{\pi}{m}}, \quad p=1,2,\dots,j \quad n=1,2,\dots, j+1. \end{align}
Also, $\widehat{u}_0(\xi)$ is the Fourier transform of $u_0(x)$ on the half-line, which is defined by the formula
\begin{equation} \label{FT-halfline} \widehat{u}_0(\xi) \doteq \int_0^\infty e^{-i\xi x}u_0(x)dx, \quad \text{Im}(\xi)\leq 0, \end{equation}
and $F(\xi,t)$ is the following time integral of the half-line Fourier transform of the forcing $f(\cdot, t)$
\begin{equation} \label{F-time-transform} F(\xi,t) \doteq \int_0^t e^{-i\xi^m\tau}\hat f(\xi,\tau)d\tau = \int_0^t e^{-i\xi^m\tau}\int_0^\infty e^{-i\xi x}f(x,\tau)dxd\tau, \quad \text{Im}(\xi)\leq 0. \end{equation}
Furthermore, $\tilde{g}_\ell$ is the temporal Fourier transform of $g_\ell$ over the interval $[0, t]$
\begin{equation} \label{g-time-transform} \tilde g_\ell(\xi,t) \doteq \int_0^te^{-i\xi\tau}g_\ell(\tau) d\tau. \end{equation}
Finally, the domains $D_{2p}^+$ in the upper half-plane are as in the two figures below:
\noindent \begin{figure}
\caption{Domains for $j$ odd }
\label{D-plane}
\caption{Domains for $j$ even }
\label{D-plane}
\end{figure}
\noindent Below we show the Fokas solution formula \eqref{UTM-sln-compact} for the KdV ($j=1$) and its domain $D^+=D^+_2$
\begin{align} \label{KdV-UTM} u(x,t) &= \frac{1}{2\pi}\int_{-\infty}^\infty e^{i\xi x+i\xi^3t} [\widehat u_0(\xi)+F(\xi,t)]d\xi \nonumber \\ &+ \frac{1}{2\pi}\int_{\partial D^+} e^{i\xi x+i\xi^3t}\{ e^{i\frac{2\pi}{3}}[\widehat u_0( e^{i\frac{2\pi}{3}} \xi)+F(e^{i\frac{2\pi}{3}} \xi,t)]+e^{i\frac{4\pi}{3}}[\widehat u_0(e^{i\frac{4\pi}{3}} \xi)+F(e^{i\frac{4\pi}{3}} \xi,t)]\}d\xi \nonumber \\ &- \frac{3}{2\pi}\int_{\partial D^+} e^{i\xi x+i\xi^3t}\xi^2 \tilde g_0(\xi^3,t)d\xi. \end{align}
\begin{minipage}{0.5\linewidth}
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\end{center} \end{minipage}
\begin{minipage}{0.5\linewidth}
\begin{center} \begin{tikzpicture}[scale=0.96] \fill [line width=1pt, opacity=0.2 ,gray] (2.5,{2.5*tan(36)})--(0,0)--(0.8,{0.8*tan(72)}); \fill [line width=1pt, opacity=0.2 ,gray] (-2.5,{2.5*tan(36)})--(0,0)--(-0.8,{0.8*tan(72)}); \draw[line width=1pt, black] (2.5,{2.5*tan(36)})--(-2.5,{-2.5*tan(36)}); \draw[line width=1pt, black] (-2.5,{2.5*tan(36)})--(2.5,{-2.5*tan(36)}); \draw[line width=1pt, black] (-3,0)--(3,0); \draw[line width=1pt, black] (0.8,{0.8*tan(72)})--(-0.8,{-0.8*tan(72)}); \draw[line width=1pt, black] (-0.8,{0.8*tan(72)})--(0.8,{-0.8*tan(72)}); \draw[line width=1pt, black,->] (-3,0)--(-1.5,0); \draw[line width=1pt, black,->] (-2.5,{2.5*tan(36)})--(-1.25,{1.25*tan(36)}); \draw[line width=1pt, black,->] (0,0)--(1.5,0); \draw[line width=1pt, black,->] (0,0)--(1.25,{1.25*tan(36)}); \draw[line width=1pt, black,->] (0,0)--(-0.4,{0.4*tan(72)}); \draw[line width=1pt, black,->] (0.8,{0.8*tan(72)})--(0.4,{0.4*tan(72)}); \draw (0.8,0) node[above] {\it\fontsize{11}{11} $ \frac{\pi}{5}$} arc (0:36:0.8); \draw [] (0,-1.5) node {\it\fontsize{11}{11} $D^-_3$}; \draw [] (-1.3,-1.5) node {\it\fontsize{11}{11} $D_4^-$}; \draw [] (1.3,-1.5) node {\it\fontsize{11}{11} $D_2^-$}; \draw [] (-2.2,-0.7) node {\it\fontsize{11}{11} $D_5^-$}; \draw [] (2.2,-0.7) node {\it\fontsize{11}{11} $D_1^-$}; \draw [] (0,1.5) node {\it\fontsize{11}{11} $D^+_3$}; \draw [] (-1.3,1.5) node {\it\fontsize{11}{11} $D_4^+$}; \draw [] (1.3,1.5) node {\it\fontsize{11}{11} $D_2^+$}; \draw [] (-2.2,0.7) node {\it\fontsize{11}{11} $D_5^+$}; \draw [] (2.2,0.7) node {\it\fontsize{11}{11} $D_1^+$}; \end{tikzpicture}
\end{center} \end{minipage}
Also, here we show the solution formula \eqref{UTM-sln-compact}
for KdV5 (Kawahara equation) together with its domains
$D^+_2$ and $D^+_4$
\begin{align} \label{KdV5-UTM} u(x,t) &= \frac{1}{2\pi}\int_{-\infty}^\infty e^{i\xi x+i\xi^5t} [\widehat u_0(\xi)+F(\xi,t)]d\xi \\ &+ \sum\limits_{p=1}^2\sum\limits_{n=1}^3 C_{p,n} \int_{\partial D_{2p}^+}e^{i\xi x+i\xi^5t}[\widehat u_0(\alpha_{p,n}\xi)+F(\alpha_{p,n}\xi,t)]d\xi \nonumber \\ &+ \sum\limits_{\ell=0}^1\sum\limits_{p=1}^2 C_{p,\ell}' \int_{\partial D_{2p}^+}e^{i\xi x+i\xi^5t}(i\xi)^{4-\ell}\tilde g_\ell(\xi^5,t)d\xi. \nonumber \end{align}
For the solution \eqref{UTM-sln-compact} to ibvp \eqref{LKdVm}, we have the following basic estimate.
\begin{theorem} [\textcolor{blue}{Forced linear KdVm estimate on the half-line}] \label{forced-linear-kdvm-thm} Suppose that $-j-\frac12< s\le j+1$, $s\neq \frac12,\frac32,\dots,j-\frac12$, $0<T<\frac12$. Then for some $0<b<\frac12$ and $\alpha>\frac12$ the Fokas formula \eqref{UTM-sln-compact} defines a solution
to the forced linear KdVm ibvp \eqref{LKdVm} with compatibility condition \eqref{first-com-con}, which satisfies the estimate
\begin{align} \label{forced-linear-kdvm-est}
&\|S[u_0,g_0,\dots,g_{j-1};f]\|_{X^{s,b,\alpha}(\mathbb{R}^+\times(0,T))} \\ \leq& \begin{cases} c_{s,b,\alpha}\Big[
\|u_0\|_{H_x^s(0,\infty)} +
\sum\limits_{\ell=0}^{j-1}\|g_\ell\|_{H_t^\frac{s+j-\ell}{m}(0,T)} +
\|f\|_{X^{s,-b,\alpha-1}(\mathbb{R}^+\times(0,T))} \Big], \quad -1 \le s \le \frac12, \\ c_{s,b,\alpha}\Big[
\|u_0\|_{H_x^s(0,\infty)} \hskip-0.05in + \hskip-0.05in
\sum\limits_{\ell=0}^{j-1}\|g_\ell\|_{H_t^\frac{s+j-\ell}{m}(0,T)} \hskip-0.1in +
\|f\|_{X^{s,-b,\alpha-1}(\mathbb{R}^+\times(0,T))} \hskip-0.05in +
\|f\|_{Y^{s,-b}(\mathbb{R}^+\times(0,T))} \Big], \,\, s \not\in [-1,\frac12], \end{cases} \nonumber \end{align}
where $Y^{s,b}$ is a ``temporal" Bourgain space defined by the norm
\begin{equation} \label{Ysb-def}
\| u
\|_{Y^{s,b}}^2 \doteq \int_{-\infty}^\infty \int_{-\infty}^\infty
(1+|\tau|)^{\frac{2s}{m}}
(1+|\tau-\xi^m|)^{2b}
|\widehat{u}(\xi,\tau)|^2 d\xi d\tau. \end{equation}
\end{theorem}
\vskip-0.1in We will prove our well-posedness result by showing that the iteration map defined by the Fokas solution formula is a contraction on a ball of the space $X^{s, b,\alpha}(\mathbb{R}^+\times \mathbb{R})$. The key ingredient is the basic linear estimate \eqref{forced-linear-kdvm-est} where the forcing $f$ is replaced by the KdVm nonlinearity $\partial_x(u^2)$, which is quadratic.
To apply this linear estimate we will extend each one of its two factors from $\mathbb{R}^+\times \mathbb{R}$ to $\mathbb{R}^2$ appropriately, and
to show that the iteration map is a contraction we shall need the following bilinear estimate.
\begin{theorem} [Bilinear estimates] \label{kdv-bilinear-estimate-thm} If $s>-j+\frac{1}{4}$, then there is $0<b<\frac12$ and
$\frac12<\alpha \le 1-b$ such that the following estimate holds
\begin{equation} \label{bilinear-est}
\|\partial_x(f\cdot g) \|_{X^{s,-b,\alpha-1}(\mathbb{R}^2)} \le c_{s,b, \alpha}
\| f\|_{X^{s,b',\alpha'}(\mathbb{R}^2)} \| g \|_{X^{s,b',\alpha'}(\mathbb{R}^2)}, \quad f, g \in X^{s,b',\alpha'}(\mathbb{R}^2), \end{equation}
where $b'$ and $\alpha'$ are chosen as follows
\begin{equation} \label{b-b'-alpha-cond} \frac12-\beta \le b' \le b < \frac12 < \alpha' \le \alpha \le \frac12+\beta, \end{equation}
and $\beta$ is given by
\begin{equation} \label{beta-choice} \beta = \begin{cases} \min\{\dfrac{1}{12m},\frac{m-s}{3m}\} \quad s \ge 0, \\ \min\Big\{ \dfrac{j-\frac34}{32m}, \dfrac{s+1/2}{2m} \Big\}, \quad
-\frac12< s<0, \\ \dfrac{1}{32m}\big[s-(-j+\frac14)\big], \quad -j+\frac14<s\le-\frac12. \end{cases} \end{equation}
\end{theorem}
We note that this is a more general result than is needed here and is of interest in its own right. Furthermore, it can be shown that for any $s<-j+\frac{1}{4}$ and $b<\frac12$ the estimates \eqref{bilinear-est} fails.
A similar estimate in $X^{s,b}$ with $b>1/2$ for the KdVm
equation on the line was proved in \cite{fhy2020}.
For the KdV bilinear estimates in the Bourgain space $X^{s,\frac 12, \alpha}$ were first proved in \cite{b1993-kdv}, and in $X^{s,b}$ with $b>1/2$ were proved in \cite{kpv1996}.
Also, we shall need the following
bilinear estimate in ``temporal" Bourgain space,
which is used when we estimate
the Sobolev norm of the solution of the forced ivp
in the time variable if $s>1/2$ or $s<-1$
(see \eqref{forced-ivp-te}).
For the KdV equation, estimate \eqref{bi-est-Y} was proved by Holmer in \cite{h2006}.
\begin{theorem} [Bilinear estimate in $Y^{s,b}$] \label{bi-est-Y-thm} If $b$, $b'$ and $\alpha'$ satisfies \eqref{b-b'-alpha-cond}, then we have
\begin{equation} \label{bi-est-Y}
\| \partial_x(fg)
\|_{Y^{s,-b}(\mathbb{R}^2)} \le c_{s,b,\alpha}
\| f
\|_{X^{s,b',\alpha'}(\mathbb{R}^2)}
\| g
\|_{X^{s,b',\alpha'}(\mathbb{R}^2)}, \quad -j+1/4<s<m, \end{equation}
\begin{equation} \label{bi-est-Y-1}
\| \partial_x(fg)
\|_{Y^{s,-b}} \le
\| \partial_x(fg)
\|_{X^{s,-b}} + c_{s,b}
\| f
\|_{X^{s,b'}}
\| g
\|_{X^{s,b'}}, \quad -j+1/4<s<m. \end{equation}
\end{theorem}
Besides the method we follow here, there are two other approaches to the study of initial-boundary value problems for KdV type equations.
In the first approach, which has been initiated by Colliander and Kenig in \cite{ck2002} and by Holmer in \cite{h2006} the forced linear ibvp is written as a superposition of ivps on the line. Then, the modern harmonic analysis techniques developed earlier for proving well-posedness of the nonlinear equation in Bourgain spaces are utilized.
In the second approach, which for KdV has been initiated by Bona, Sun and Zhang in \cite{bsz2002, bsz2003, bsz2008}, the forced linear ibvp is solved via a Laplace transform in the time variable and then by deriving appropriate estimates well-posedness of the ibvp is established.
As we have mentioned before KdVm includes the KdV equation ($m=3$), which is integrable, and the Kawahara equation ($m=5$), which is not integrable. The literature about the KdV is very extensive. It begins with Scott Russell's observation of the ``great wave of translation" \cite{jsr1845} and continues with the derivation of the KdV model by Boussinesq in 1877 \cite{b1877} and Korteweg and de Vries in 1895 \cite{kdv1895}. Then, in 1965, Zabusky and Kruskal \cite{zk1965} observed numerically that soliton solutions of KdV interact almost linearly by preserving their shape and speed after a collision. Soon after the KdV initial value problem on the line with data
of sufficient smoothness and decay was solved by
Gardner, Greene, Kruskal and Miura
\cite{ggkm1967}
via the inverse scattering transform (IST) which is based
on its integrability, that is
its Lax pair formulation \cite{lax1968}.
The KdV ivp in Sobolev spaces $H^s$ using methods
from partial differential equations has been studied extensively
by many authors. For well-posedness results on the line
using Bourgain spaces we refer the reader to
Bourgain \cite{b1993-kdv} when $s\ge 0$,
Kenig, Ponce and Vega \cite{kpv1996} when $s> -3/4$,
Guo \cite{g2009} when $s=-3/4$,
and to Colliander, Keel, Staffilani, Takaoka, Tao
\cite{ckstt2003} for its global well-posedness when $s> -3/4$.
For additional results we refer the reader to
\cite{Bona1975, st1976, cs1988, cks1992, b1993-nls,
kato1983, kpv1993,kpv2001,kpv1989,sjoberg1970, BSaut2003, lpbook,ckt2003,ghhp2013}
and the references therein.
Concerning the Fokas method for solving ibvp, whose initial motivation came from integrable equations (in particular the KdV and the cubic NLS equations), we refer the reader \cite{dtv2014, f1997, f2002,fp2005, fi2004, fis2005, fl2012, lenells2013, fpbook2014, fs2012, oy2019} and the references therein.
Also, for a detailed introduction to this method we refer to the book \cite{f2008}.
For additional work on solving ibvp we refer the reader to
\cite{et2016, h2005, fa1988,bw1983,ton1977}
and the references therein.
Our work is structured as follows. In Section 2, we study the reduced pure ibvp,
which is the homogeneous linear problem with initial data zero but non-zero boundary data and derive a key estimate
for its solution in modified Bourgain spaces.
In Section 3, guided by the reduced pure ibvp,
we decompose the forced linear ibvp into four simpler
linear sub-problems and derive appropriate estimates
for their solutions. Then, combining these estimates
we prove Theorem \ref{forced-linear-kdvm-thm},
which provides the basic estimate for the Fokas solution
formula and the iteration map of the nonlinear problem.
In Section 4, we prove the needed bilinear estimate for the nonlinearity
in modified Bourgain spaces $X^{s,b, \alpha}$.
Then, in section 5, we prove the temporal bilinear estimate
in $Y^{s,b}$ spaces.
In Section 6, we prove our KdVm well-posedness result, Theorem \ref{thm-kdvm-half-line}, using estimates \eqref{forced-linear-kdvm-est} for the forced linear ibvp and the bilinear estimates \eqref{bilinear-est}.
Finally, in Section 7 we provide a brief outline of the Fokas solution formula
for the forced linear ibvp on the half-line.
\section{Reduced pure ibvp} \label{linear-kdv-reduced}
We begin with the most basic linear KdVm ibvp on the half-line. This is the homogeneous ibvp with zero initial data and nonzero boundary data. Furthermore, we assume that the boundary data $h_\ell$ are test functions of time which are {\bf compactly supported} in the interval $[0, 2]$. This problem, which we call the {\bf reduced pure ibvp}, reads as follows:
\begin{subequations} \label{LKdVm-reduced} \begin{align} \label{LKdVm-reduced eqn} &\partial_tv+(-1)^{j+1}\partial_x^{2j+1}v=0, \quad 0<x<\infty, \,\, 0<t<2, \\ \label{LKdVm-reduced ic} &v(x,0) = 0, \\ \label{LKdVm-reduced bc} &v(0,t) = h_0(t), \cdots, \partial_x^{j-1}v(0,t) = h_{j-1}(t), \quad 0<t<2. \end{align} \end{subequations}
In this situation we have that
\begin{equation} \label{h-l-tilde-recall} \tilde h_\ell(\xi,2) = \int_0^2e^{-i\xi\tau}h_\ell(\tau)d\tau = \int_\mathbb{R} e^{-i\xi\tau}h_\ell(\tau)d\tau = \widehat h_\ell(\xi), \end{equation}
and the Fokas solution formula
of our reduced pure ibvp \eqref{LKdVm-reduced} takes the simple form
\begin{align} \label{pure-kdvm-sln} v(x,t) =& S[0,h_0,\cdots,h_{j-1};0] \\ =& \sum\limits_{p=1}^j \sum\limits_{\ell=0}^{j-1} C'_{p,\ell}\int_{\partial D_{2p}^+}e^{i\xi x+i\xi ^mt}(i\xi )^{2j-\ell}\tilde h_{\ell}(\xi^m,2)d\xi = \sum\limits_{p=1}^j \sum\limits_{\ell=0}^{j-1} C'_{p,\ell}\, v_{p\ell} , \nonumber \end{align}
where
\begin{equation} \label{def-v-p-l} v_{p\ell}(x,t) \doteq \int_{\partial D_{2p}^+}e^{i\xi x+i\xi ^mt}(i\xi )^{2j-\ell} \, \widehat h_{\ell}(\xi^m)d\xi \quad x\in\mathbb{R}^+, \quad t\in[0,2]. \end{equation}
In the next result we estimate this solution in the Hadamard and the Bourgain spaces. Our objective is to obtain the optimal bounds in temporal Sobolev spaces. In fact,
in both cases the bounds that arise naturaly are the ones suggested by the time regularity of the solution to the linear homogeneous Cauchy problem with data in $H^s(\mathbb{R})$ (for KdV, see \cite{kpv1991-reg}, \cite{h2006}, \cite{fhm2016}) More precisely, we have the following result.
\begin{theorem} [\mathcolor{blue}{Estimates for pure ibvp on the half-line}] \label{reduced-pure-ibvp-thm} For boundary data test functions that are compactly supported in the interval $[0,2]$, the solution for the reduced pure ibvp \eqref{LKdVm-reduced} satisfies the following Hadamard space estimate
\begin{align} \label{reduced-pure-ibvp-H-es}
\sup_{t\in \mathbb{R}} \| S[0,h_0,\cdots,h_{j-1};0]
\|_{H_x^s(0,\infty)}
\leq c_{s} \sum\limits_{\ell=0}^{j-1}\|h_\ell\|_{H_t^{\frac{s+j-\ell}{m}}(\mathbb{R})} , \quad s\ge 0. \end{align}
In addition, for $b\in[0,\frac12)$ and $ \frac12<\alpha \le \frac{1}{2} + \frac 1m (s+j+\frac12) $ it satisfies the Bourgain spaces
estimate
\begin{align} \label{reduced-pure-ibvp-B-es}
\| S[0,h_0,\cdots,h_{j-1};0]
\|_{X^{s, b,\alpha}(\mathbb{R}^+\times (0,2))} \leq c_{s,b,\alpha}
\sum\limits_{\ell=0}^{j-1}\|h_\ell\|_{H_t^{\frac{s+j-\ell}{m}}(\mathbb{R})}, \quad s>-j-\frac12. \end{align}
\end{theorem}
The proof of the Hadamard space estimate \eqref{reduced-pure-ibvp-H-es} can be found in \cite{y2020} for KdVm and in \cite{fhm2016} for KdV. The restriction $s\ge 0$ comes from the use of the physical space description of the Sobolev norm. Here, we focus on the Bourgain spaces estimate \eqref{reduced-pure-ibvp-B-es} which is new and useful. It is the basic ingredient in the proof of well-posedness of KdVm on the half-line.
\noindent {\bf Proof of Theorem \ref{reduced-pure-ibvp-thm}.}
Here we present the proof of the estimate \eqref{reduced-pure-ibvp-B-es}.
Using the parametrization $[0,\infty)\ni \xi\to \gamma\xi$
for the right side of the domain $D_{2p}^+$, and the parametrization $[0,\infty)\ni \xi\to \gamma'\xi$ for the left side of $D_{2p}^+$, we obtain the following decomposition
$ v_{p\ell}(x,t) = V_{1}(x,t)+V_{2}(x,t), $
where
\begin{align} \label{def-v1} V_{1}(x,t) = \int_0^\infty e^{i\gamma \xi x-i\xi ^mt}(\gamma \xi )^{2j-\ell} \, \widehat h_\ell(-\xi ^m)d\xi \simeq \int_0^\infty e^{-i\xi ^mt}e^{i\gamma_R\xi x}e^{-\gamma_I \xi x} \xi ^{2j-\ell} \, \widehat h_\ell(-\xi ^m)d\xi, \end{align}
\begin{align} \label{def-v2} V_{2}(x,t) = \int_0^\infty e^{i\gamma' \xi x+i\xi^mt}(\gamma' \xi )^{2j-\ell} \, \widehat h_\ell(\xi^m)d\xi \simeq \int_0^\infty e^{i\xi^mt}e^{i\gamma'_R\xi x}e^{-\gamma'_I \xi x} \xi^{2j-\ell} \, \widehat h_\ell(\xi^m)d\xi, \quad \end{align}
and
\begin{align*} \gamma = e^{i(2p-1)\cdot\frac{\pi}{m}} = \cos\left(\frac{2p-1}{m}\pi\right)+i\sin\left(\frac{2p-1}{m}\pi\right), \,\,\, \gamma' = e^{i2p\cdot\frac{\pi}{m}} = \cos\left(\frac{2p}{m}\pi\right)+i\sin\left(\frac{2p}{m}\pi\right). \end{align*}
Note, that the imaginary parts of $\gamma$ and $\gamma'$ are positive, which is crucial for our estimates.
Here we only estimate $V_{1}$. The estimation of $V_{2}$ is similar. Also, we split $V_1$ as the sum of two functions, one for $\xi$ near $0$ and the other away from $0$, that is
$ V_1 = v_0 + v_1, $
where
\begin{equation} \label{def-v0} v_0(x,t) \doteq \int_0^1 e^{-i\xi ^mt}e^{i\gamma_R\xi x}e^{-\gamma_I \xi x} \xi ^{2j-\ell} \, \widehat h_\ell(-\xi ^m)d\xi, \quad x\in\mathbb{R}^+, \quad t\in[0,2], \end{equation}
and
\begin{equation} \label{def-v1} v_1(x,t) \doteq \int_1^\infty e^{-i\xi ^mt}e^{i\gamma_R\xi x} e^{-\gamma_I \xi x} \xi ^{2j-\ell} \, \widehat h_\ell(-\xi ^m)d\xi, \quad x\in\mathbb{R}^+, \quad t\in[0,2]. \end{equation}
The estimate of the Bourgain norm for $v_0$ follows from the boundedness of the Laplace transform in $L^2$ and we will do it later. Next, we estimate the Bourgain norm for $v_1$. Using the identity
\begin{align*} \xi^j [ e^{i\gamma_R\xi x} e^{-\gamma_I\xi x} ] = \frac{\partial_x^j [ e^{i\gamma_R\xi x} e^{-\gamma_I\xi x} ]}{ (i\gamma)^j } \end{align*}
and the fact that $e^{-\gamma_I \xi x}$ is exponentially decaying in $\xi$ for $x>0$ we can take the $\partial_x^j$-derivative outside the integral sign in \eqref{def-v1} to rewrite $v_1(x,t)$ as follows
\begin{equation} \label{def-v1-modified} v_1(x,t) = \frac{1}{(i\gamma)^j} \partial_x^j \int_1^\infty e^{-i\xi^mt}e^{i\gamma_R\xi x}e^{-\gamma_I \xi x} \xi^{j-\ell} \, \widehat h_\ell(-\xi ^m)d\xi, \quad x\in\mathbb{R}^+, \quad t\in[0,2]. \end{equation}
Next, we extend $v_1$ from $\mathbb{R}^+\times[0,2]$ to $\mathbb{R}\times\mathbb{R}$ by using the the one-sided cutoff function $\rho(x)$, which satisfies $0\le\rho(x)\le1, x\in\mathbb{R}$, and is as follows
\vskip.01in \noindent
\begin{minipage}{0.6\linewidth}
\begin{center} \begin{tikzpicture}[yscale=0.5, xscale=0.8]
\newcommand\X{0}; \newcommand\Y{0}; \newcommand\FX{11}; \newcommand\FY{11}; \newcommand\R{0.6}; \newcommand*{\TickSize}{2pt};
\draw[black,line width=1pt,-{Latex[black,length=2mm,width=2mm]}] (-5,0) -- (5,0) node[above] {\fontsize{\FX}{\FY}\bf \textcolor{black}{$x$}};
\draw[black,line width=1pt,-{Latex[black,length=2mm,width=2mm]}] (0,0) -- (0,3) node[right] {\fontsize{\FX}{\FY}\bf \textcolor{black}{$\rho$}};
\draw[line width=1pt, yscale=2,domain=-1.5:-4.3,smooth,variable=\x,red] plot ({\x},{0});
\draw[line width=1pt, yscale=2,domain=0:4.3,smooth,variable=\x,red] plot ({\x},{1});
\draw[smooth,line width=1pt, red] (0,2) to[out=-170,in=10] (-1.5,0) ;
\draw[red] (2,2.5) node[] {\fontsize{\FX}{\FY}$\rho(x)$}
(0,0) node[yshift=-0.2cm] {\fontsize{\FX}{\FY}$0$}
(-1.5,0) node[yshift=-0.2cm] {\fontsize{\FX}{\FY}$-1$};
\end{tikzpicture} \end{center} \end{minipage} \hskip-0.88in \begin{minipage}{0.5\linewidth} \begin{equation} \label{rho-def} \rho(x) = \begin{cases} 1, &\quad x\ge0, \\ 0, &\quad x\le -1. \end{cases} \end{equation}
\end{minipage}
\noindent Using it, we extend $v_{1}$ via the formula below (keeping the same notation $v_1$ for it)
\begin{align} \label{v-p-l-ext-1} v_{1}(x,t) \doteq& \frac{1}{(i\gamma)^j} \partial_x^j \int_1^\infty e^{-i\xi ^mt}e^{i\gamma_R\xi x}e^{-\gamma_I \xi x} \rho(\gamma_I\xi x) \xi ^{j-\ell} \, \widehat h_\ell(-\xi ^m)d\xi, \quad x\in\mathbb{R}, \quad t\in\mathbb{R}, \end{align}
where $\gamma_I$ is the imaginary part of
$\gamma$ and $\gamma_R$ is its real part.
Also, we could have localized $v_1$ in $t$ further by multiplying it
by the standard cutoff function $\psi(t)$ in $C^{\infty}_0(-1, 1)$ such that
$0\le \psi \le 1$ and $\psi(t)=1$ for $|t|\le 1/2$.
Then using the estimate
\begin{align} \label{mult-by-cutoff}
\|\psi(t)v_1\|_{X^{s, b,\alpha}(\mathbb{R}^2)}
\leq c_{\psi} \|v_1\|_{X^{s, b,\alpha}(\mathbb{R}^2)}, \end{align}
we are reduced in estimating $\|\cdot\|_{X^{s, b,\alpha}(\mathbb{R}^2)}$, which we do next. Notice that the quantities under the integral defining $v_1$ make sense for all $t$ since $t$ appears in oscillatory terms.
\noindent Extension \eqref{v-p-l-ext-1} is good since $ \rho(\gamma_I\xi x)=1$ for $x>0$. Also, $e^{-\gamma_I \xi x} \rho(\gamma_I\xi x)$ is bounded for all $x$ and $t$ since $e^{-\gamma_I \xi x} \le e$ and $\rho\le 1$, that is
$
| e^{-\gamma_I \xi x} \rho(\gamma_I\xi x)
| \leq e^{1} \cdot 1, $ $ x\in\mathbb{R}, \,\, t\in\mathbb{R}. $
Making the change of variables $\tau=-\xi^m$ and defining
\begin{equation} \label{def-eta} \eta(x) \doteq e^{i\frac{\gamma_R}{\gamma_I}x} e^{-x} \rho(x) , \end{equation}
we write $v_1$ in the form
\begin{equation} \label{eqn-V1-1} v_1(x,t) \simeq \partial_x^j \int_{-\infty}^{-1} e^{i\tau t} \eta(-\gamma_I\tau^{1/m}x) \tau^{-(\ell+j)/m} \, \widehat h_\ell(\tau) d\tau, \quad x\in\mathbb{R}, \quad t\in\mathbb{R}, \end{equation}
and prove the following result for it.
\begin{lemma} [Bourgain space estimate for reduced ibvp] \label{reduced-pibvp-thm1} For any $\varepsilon>0$, if $s\geq -j-\frac12-\varepsilon$, and $b\ge 0$, then the function $v_{1}(x,t)$, which is part of the solution $v=S\big[0, h_0,\dots,h_{j-1}; 0\big]$ to pure ibvp \eqref{LKdVm-reduced} and defined by \eqref{v-p-l-ext-1} satisfies the space estimate
\begin{align} \label{reduced-ibvp-Bourgain-est}
\|v_1\|_{X^{s, b}}
\leq c_{s,b} \|h_\ell\|_{H_t^{\frac{s+mb-\frac12-\ell+\varepsilon}{m}}(\mathbb R)}. \end{align}
Moreover, if we chose $\varepsilon$ such that $ \varepsilon \le m(\frac12-b), $ which is possible if $b<1/2$, then we have the estimate (needed in our well-posedness theorem)
\begin{align} \label{reduced-ibvp-Bourgain-est-1}
\|v_1\|_{X^{s, b}} \le c_{s,b}
\|h_\ell\|_{H_t^{\frac{s+j-\ell}{m}}(\mathbb R)}. \end{align}
\end{lemma}
\noindent {\bf Proof of Lemma \ref{reduced-pibvp-thm1}.}
In order to estimate the $\|v_1\|_{s,b}$, we need to calculate the Fourier transform of $v_1(x,t)$. Using the inverse Fourier transform, we get
\begin{align} \label{v1-t-FT} \widehat{v}_1^t(x,\tau) \simeq \begin{cases} \partial_x^j\eta(-\gamma_I\tau^{1/m}x) \tau^{-(\ell+j)/m} \, \widehat h_\ell(\tau), &\quad \tau<-1, \\ 0, &\quad \tau \ge-1. \end{cases} \end{align}
In addition, taking the Fourier transform with respect to $x$, we get
\begin{align} \label{v1-xt-FT} \widehat{v}_1(\xi,\tau) \simeq \begin{cases} \xi^j F(\xi,\tau) \tau^{-(\ell+j)/m} \, \widehat h_\ell(\tau), &\quad \tau<-1, \\ 0, &\quad \tau \ge-1, \end{cases} \end{align}
where $F(\xi,\tau)$ is given by
$ F(\xi,\tau) \doteq \int_{x\in\mathbb{R}} e^{-i\xi x}\eta(-\gamma_I\tau^\frac1m x) dx. $
Also, using the fact that $\eta$ is a Schwarz function and making a change of variables, we get the following result.
\begin{lemma} \label{F-bound-lem} For any $n\ge 0$, $\tau<-1$ and $\xi\in\mathbb{R}$, we have
\begin{align} \label{F-bound-ine}
|F(\xi,\tau)| \leq c_{\rho,\gamma,n} \cdot
\frac{1}{|\tau|^{1/m}} \cdot \left(
\frac{|\tau|^{1/m}}{|\xi|+|\tau|^{1/m}} \right)^n, \end{align}
where $c_{\rho,\gamma,n}$ is a constant depending on $\gamma$, $n$ and $\rho$, which is described in \eqref{rho-def}. \end{lemma}
\noindent Furthermore, using \eqref{v1-xt-FT} we get
\begin{align} \label{X-norm-v1}
\|v_1\|_{X^{s,b}}^2 \simeq \int_{-\infty}^{-1} \left[ \int_{-\infty}^{\infty}
(1+|\xi|)^{2s}
(1+|\tau-\xi^m|)^{2b}
| \xi^j F(\xi,\tau)
|^2 d\xi \right] \cdot
\left| \tau^{-(\ell+j)/m} \widehat h_\ell(\tau)
\right|^2 d\tau. \end{align}
Next we will estimate the $d\xi$ integral in \eqref{X-norm-v1}. In fact, we have the following estimate:
\begin{lemma} \label{v1-mult-lema} For any $\varepsilon>0$, if $s\geq -j-\frac12-\varepsilon$, $b\ge 0$ then we have
\begin{equation} \label{v1-mult-ine-1} \int_{-\infty}^{\infty}
(1+|\xi|)^{2s}
(1+|\tau-\xi^m|)^{2b}
\left| \xi^j F(\xi,\tau)
\right|^2 d\xi \leq c_{s,b,m}
|\tau|^{\frac{2(s+j)+2mb-1+2\varepsilon}{m}}, \end{equation}
where $c_{s,b,m}$ is a constant depending on $s$, $b$ and $m$.
\end{lemma}
\noindent We prove \eqref{v1-mult-ine-1} below. Next, combining it with \eqref{X-norm-v1},
we obtain
\begin{align*}
\|v_1\|_{X^{s,b}}^2 \le c_{s,b,m} \int_{-\infty}^{-1}
\Big| \tau^{\frac{s+mb-\frac12-\ell+\varepsilon}{m}} \widehat h_\ell(\tau)
\Big|^2 d\tau
\le c_{s,b,m} \|h_\ell\|_{H_t^{\frac{s+mb-\frac12-\ell+\varepsilon}{m}}}^2, \end{align*}
which is the desired estimate \eqref{reduced-ibvp-Bourgain-est} for $v_1$. \,\, $\Box$
\noindent {\bf Proof of Lemma \ref{v1-mult-lema}.}
Using the estimate \eqref{F-bound-ine},
denoting the integrand by
\begin{align*} I(\xi,\tau) \doteq
(1+|\xi|)^{2s}
(1+|\tau-\xi^m|)^{2b}
\left| \xi^j F(\xi,\tau)
\right|^2 \lesssim
(1+|\xi|)^{2s}
(1+|\tau-\xi^m|)^{2b}
\frac{\xi^{2j}|\tau|^{\frac{2n-2}{m}}}{|\xi|^{2n}+|\tau|^{2n/m}}, \end{align*}
and using our assumption $b\ge0$, $|\tau|>1$, we have
\begin{equation} \label{mult-b-est} 1 \leq
|\xi|+|\tau|^\frac1m
\Rightarrow 1+|\tau-\xi^m|
\leq 2(|\tau|+|\xi|^m) \Rightarrow
(1+|\tau-\xi^m|)^b \leq c_{b,m}(
|\tau|^b +
|\xi|^{mb} ), \end{equation}
where $c_{b,m}$ is a constant depending on $b$ and $m$. Hence we obtain
\begin{equation} \label{L2-est-mult} I(\xi,\tau) \le c_{s,b,m} \left[
(1+|\xi|)^{2s}|\tau|^{2b}+(1+|\xi|)^{2s}|\xi|^{2mb} \right] \cdot
\frac{\xi^{2j}|\tau|^{\frac{2n-2}{m}}}{|\xi|^{2n}+|\tau|^{2n/m}}, \end{equation}
where $c_{s,b,m}$ is a constant depending on $s$, $b$ and $m$. Now we shall consider the following cases:
\vskip0.05in \noindent
$\bullet$ $|\xi|> |\tau|^\frac1m$ \quad and \quad
$\bullet$ $|\xi|\le |\tau|^\frac1m$
\vskip0.05in \noindent
{\bf Case $|\xi|> |\tau|^\frac1m$.} Here we have $|\xi|\ge 1$ and choosing $n=(s+j)+mb+\frac12+\varepsilon$, from \eqref{L2-est-mult} we have
$$ I(\xi,\tau) \lesssim c_{s,b,m}
(1+|\xi|)^{2s}|\xi|^{2mb} \cdot
\frac{\xi^{2j}|\tau|^{\frac{2n-2}{m}}}{|\xi|^{2n}} \lesssim c_{s,b,m}
|\xi|^{-1-2\varepsilon}
|\tau|^{\frac{2(s+j)+2mb-1+2\varepsilon}{m}}. $$
Thus, for the integral \eqref{v1-mult-ine-1} we have the following inequality
$
\int_{|\xi|\ge 1} I(\xi,\tau) d\xi \le $ $ c_{s,b,m}
|\tau|^{\frac{2(s+j)+2mb-1+2\varepsilon}{m}}, $
which is the desired estimate \eqref{v1-mult-ine-1} in this case.
\noindent
{\bf Case $|\xi|\le |\tau|^\frac1m$.}
Then $1+|\xi|\lesssim|\tau|^{1/m}$. Choosing $n=(s+j)+\frac12+\varepsilon$, from \eqref{L2-est-mult} we have
$$ I(\xi,\tau) \lesssim c_{s,b,m}
(1+|\xi|)^{2s}|\tau|^{2b} \cdot
\frac{\xi^{2j}|\tau|^{\frac{2n-2}{m}}}{|\xi|^{2n}+1} \lesssim
(1+|\xi|)^{2s}|\tau|^{2b} \cdot
\frac{|\tau|^{\frac{2n-2}{m}}}{(|\xi|+1)^{2n}} \lesssim
(1+|\xi|)^{-1-2\varepsilon}
|\tau|^{\frac{2(s+j)+2mb-1+2\varepsilon}{m}} $$
Therefore, integrating $\xi$ we get
$
\int_{|\xi|\le |\tau|^{1/m}} I(\xi,\tau) d\xi \lesssim
|\tau|^{\frac{2(s+j)+2mb-1+2\varepsilon}{m}}, $
which is the desired estimate \eqref{v1-mult-ine-1} in this case.
This completes the proof of Lemma \ref{v1-mult-lema}.\,\, $\Box$
\noindent {\bf Estimation of $D^\alpha$ norm.}
Now we estimate the second part in the modified Bourgain norm $\|\cdot \|_{s,b,\alpha}$. More precisely we have the result.
\begin{lemma} \label{alpha-est-lem} If $s>-j-\frac12$ and $\frac12<\alpha\le\frac{1}{2}+\frac1m(s+j+\frac12)$, then we have
\begin{equation} \label{alpha-est-I2}
\|v_1\|_{D^\alpha}^2 \doteq \int_{-\infty}^{\infty} \int_{-1}^1
(1+|\tau|)^{2\alpha}
|\widehat{v}_1(\xi,\tau)|^2 d\xi d\tau \lesssim
\|h_\ell\|_{H^{\frac{s+j-\ell}{m}}(\mathbb{R})}^2. \end{equation}
\end{lemma}
\noindent {\bf Proof of Lemma \ref{alpha-est-lem}.} First, we recall that the Fourier transform of $v_1$ is given by \eqref{v1-xt-FT}.
Thus we have
\begin{align*}
\|v_1\|_{D^\alpha}^2 = \int_{-\infty}^{-1} \int_{-1}^1
(1+|\tau|)^{2\alpha}
|\xi^j F(\xi,\tau) \tau^{-(\ell+j)/m}
\widehat h_\ell(\tau)|^2 d\xi d\tau, \end{align*}
where
$ F(\xi,\tau) = \int_{x\in\mathbb{R}} e^{-i\xi x}\eta(-\gamma_I\tau^\frac1m x) dx. $
Also, applying Lemma \ref{F-bound-lem} with $n=0$, we get the following bound for $F$, that is
$
|F(\xi,\tau)| \le
|\tau|^{-1/m}. $
Hence, after integrating $\xi$, we have
\begin{align*}
\|v_1\|_{D^\alpha}^2 \lesssim \int_{-\infty}^{-1}
(1+|\tau|)^{2\alpha}
| \tau^{-\frac{1+j+\ell}{m}}
\widehat h_\ell(\tau)|^2 d\tau \lesssim \int_{-\infty}^{-1}
(1+|\tau|)^{\frac{2m\alpha-2-2j-2\ell}{m}}
|
\widehat h_\ell(\tau)|^2 d\tau \le
\|h_\ell\|_{H_t^{\frac{m\alpha-j-1-\ell}{m}}}^2. \end{align*}
Choosing $\alpha$ such that
$ \alpha \le \frac{1}{2}+\frac1m\big(s+j+\frac12\big), $
we get the desired estimate \eqref{alpha-est-I2} . \,\, $\square$
\noindent {\bf Bound near $\xi=0$.} Next, we estimate the Bourgain norm for $v_0$.
We begin with extending it from $\mathbb{R}^+\times(0,2)$ to $\mathbb{R}\times\mathbb{R}$ (keeping the same notation)
\begin{equation} \label{def-v0-extension} v_0(x,t) \doteq \int_0^1 e^{i\gamma\xi \varphi_1(x)}e^{-i\xi ^mt}\xi ^{2j-\ell}\,\widehat {h_\ell}(-\xi ^m)d\xi, \quad x\in\mathbb{R}, \,\, t\in\mathbb{R}, \end{equation}
where $\varphi_1(x)$ is a smooth version of
$|x|$. More precisely
\vskip-0.05in \begin{minipage}{0.4\linewidth} \begin{center} \begin{tikzpicture}[yscale=0.6, xscale=1]
\newcommand\X{0}; \newcommand\Y{0}; \newcommand\FX{11}; \newcommand\FY{11}; \newcommand\R{0.6}; \newcommand*{\TickSize}{2pt};
\draw[black,line width=1pt,-{Latex[black,length=2mm,width=2mm]}] (-2.5,0) -- (2.5,0) node[above] {\fontsize{\FX}{\FY}\bf \textcolor{black}{$x$}};
\draw[black,line width=1pt,-{Latex[black,length=2mm,width=2mm]}] (0,-0.5) -- (0,3) node[right] {\fontsize{\FX}{\FY}\bf \textcolor{black}{$y$}};
\draw[line width=1pt, yscale=1,domain=-2.1:-1.2,smooth,variable=\x,red] plot ({\x},{-\x});
\draw[line width=1pt, yscale=1,domain=0:2.1,smooth,variable=\x,red] plot ({\x},{\x});
\draw[smooth,line width=1pt, red] (-0.4,-0.2) to[out=5,in=-135] (0,0) ;
\draw[smooth,line width=1pt, red] (-1.2,1.2) to[out=-45,in=175] (-0.4,-0.2) ;
\draw[red,dashed, line width=0.5pt] (2,1.3) node[] {\fontsize{\FX}{\FY}$\varphi_1(x)$}
(0,0) node[yshift=-0.2cm,xshift=0.2cm] {\fontsize{\FX}{\FY}$0$}
(-1.5,1.5) -- (-1.5,0) node[yshift=-0.2cm] {\fontsize{\FX}{\FY}$-1$};
\end{tikzpicture} \end{center}
\end{minipage}
\begin{minipage}{0.6\linewidth}
\begin{equation} \label{def-phi3} \varphi_1(x) = \begin{cases} x, \quad x\geq 0 \\ -x, \quad x\leq -1 \\ \text{ smooth on } \mathbb{R}. \quad \end{cases} \end{equation}
\end{minipage}
\noindent For $v_0$, we have the $L^2$ estimate
\begin{align} \label{v0-L2-est}
\|\psi(t) v_0\|^2_{s,b,\alpha} \hskip-0.05in \lesssim
\|\psi(t) v_0\|^2_{L^2_{x,t}} +
\|\partial_x^{n_1}[\psi(t) v_0]\|^2_{L^2_{x,t}} +
\|\partial_t^{n_2}[\psi(t) v_0]\|^2_{L^2_{x,t}} +
\|\partial_x^{n_1}\partial_t^{n_2}[\psi(t) v_0]\|^2_{L^2_{x,t}}, \end{align}
where
$n_1 = n_1(s,b)
\doteq 2m\lfloor |b|\rfloor+2\lfloor |s|\rfloor+2 $
and
$ n_2 = n_2(b,\alpha)
\doteq 2\lfloor |b|\rfloor+2\lfloor|\alpha|\rfloor+2. $
Using the $L^2$ boundedness of Laplace transform (see Lemma 2.3 in \cite{fhm2016} or \cite{hardy1933}) we get
\begin{align} \label{v0-L2-Laplace-est}
\|\partial_x^{n_1}\partial_t^{n_2}[\psi\cdot v_0]\|_{L^2_{x,t}}^2 \le C_{n_1, n_2} \int_0^{\infty}
| \widehat{h_\ell}(\tau)
|^2 d\tau \lesssim
\|h_\ell\|_{H_t^{\frac{s+j-\ell}{m}}}^2 \quad \forall n_1,n_2\in \mathbb{N}_0. \end{align}
\noindent {\bf End of Proof for Theorem \ref{reduced-pure-ibvp-thm}.} Combining Lemma \ref{reduced-pibvp-thm1}, Lemma \ref{alpha-est-lem} and estimate \eqref{v0-L2-est} with \eqref{v0-L2-Laplace-est}, we get estimate \eqref{reduced-pure-ibvp-B-es}. This completes the proof of Theorem \ref{reduced-pure-ibvp-thm}. \,\, $\Box$
\section{ Proof of forced linear ibvp estimates
} \label{proof-half-line}
In this section, we prove the basic linear estimate \eqref{forced-linear-kdvm-est}. We begin by decomposing the forced linear ibvp \eqref{LKdVm} into a homogeneous ibvp (A) and an inhomogeneous ibvp with zero data (B). Then, we decompose both problems further in a convenient way simplifying both their Fokas solution formula and its estimation.
\noindent {\bf A. The homogeneous linear ibvp:}
\begin{subequations} \label{homo-ibvp} \begin{align} &\partial_tu+(-1)^{j+1}\partial_x^{2j+1}u = 0,
\\ \label{homo-ibvp:ic} &u(x,0) = u_0(x)\in H_x^s(0,\infty),
\\ \label{homo-ibvp:bc} &\partial_x^\ell u(0,t) = g_\ell(t) \in H_t^{\frac{s+j-\ell}{m}}(0,T), && \ell = 0,1,\dots,j-1,
\end{align} \end{subequations}
with solution denoted by
$ u(x,t) \doteq S\big[u_0, g_0,\dots,g_{j-1}; 0\big](x,t) $ and which is defined in \eqref{UTM-sln-compact}.
We decompose it further into the following two problems.
\noindent
{\bf A$_1$. The homogeneous linear ivp:}
\begin{subequations} \label{homo-ivp} \begin{align} \label{homo-ivp:eq} &\partial_tU+(-1)^{j+1}\partial_x^{2j+1}U=0,
\\ \label{homo-ivp:ic} & U (x,0) = U_0(x)\in H_x^s(\mathbb R),
\end{align} \end{subequations}
where $U_0\in H_x^s(\mathbb R)$ is an extension of the initial datum $u_0\in H_x^s(0, \infty)$ such that
\begin{equation} \label{assum-U0}
\|U_0\|_{H_x^s(\mathbb R)} \leqslant
2\|u_0
\|_{H_x^s(0, \infty)} \end{equation}
with its solution given by
\begin{equation} \label{homo-ivp:sln} U (x,t) = S\big[U_0; 0\big] (x, t) = \frac{1}{2\pi} \int_{\mathbb{R}} e^{i\xi x+i\xi^mt}\, \widehat{U}_0(\xi) d\xi, \end{equation}
where
$ \widehat{U}_0(\xi) = \int_{\mathbb{R}} e^{-i\xi x}\, U_0(x) dx,\quad \xi \in \mathbb R $.
\noindent {\bf A$_2$. The homogeneous linear ibvp with
zero initial data:}
\begin{subequations} \label{pure-ibvp} \begin{align} \label{pure-ibvp:eqn} & \partial_tu+(-1)^{j+1}\partial_x^{2j+1}u = 0,
\quad &&
\\ \label{pure-ibvp:ic} &u(x,0)= 0,
\\ \label{pure-ibvp:bc} & \partial_x^\ell u(0,t) = g_\ell(t) - \partial_x^\ell U (0,t)
\doteq G_\ell(t)
\in H_t^{\frac{s+j-\ell}{m}}(0,T), ,
\quad && \ell = 0,1,\dots, j-1,
\end{align} \end{subequations}
with solution
$ u(x, t) \doteq S\big[0, G_0,\dots,G_{j-1}; 0\big](x, t) $, which is defined in \eqref{UTM-sln-compact}.
\noindent {\bf B. The forced linear ibvp with zero data:}
\begin{subequations} \label{forced-ibvp} \begin{align} & \partial_tu+(-1)^{j+1}\partial_x^{2j+1}u = f(x,t),
\\ \label{forced-ibvp:ic} &u(x,0) = 0,
\\ \label{forced-ibvp:bc} &\partial_x^\ell u(0,t) =0, \quad && \ell = 0,1,\dots,j-1,
\end{align} \end{subequations}
whose solution
$ u(x,t) \doteq S\big[0, 0,\dots,0; f\big](x,t) $ is defined in \eqref{UTM-sln-compact}. This problem can be further decomposed into the following two problems (B$_1$) and (B$_2$).
\noindent
{\bf B$_1$. The forced linear ivp with zero initial data:}
\begin{subequations} \label{forced-ivp} \begin{align} \label{forced-ivp:eqn} &\partial_tW+(-1)^{j+1}\partial_x^{2j+1}W = w(x, t),\,
\\ \label{forced-ivp:ic} & W(x,0) = 0,
\end{align} \end{subequations}
where $w$ is an {\it extension of the forcing $f$} such that
\begin{align} \label{assum-F-extension}
&\|w\|_{X^{s,-b,\alpha-1}(\mathbb{R}^2)}
\leqslant 2\|f\|_{X^{s,-b,\alpha-1}(\mathbb{R}^+\times(0,T))}, \hskip.2in -1 \le s \le \frac12, \\
&\|w\|_{X^{s,-b,\alpha-1}(\mathbb{R}^2)} +
\|w\|_{Y^{s,-b}(\mathbb{R}^2)} \leqslant 2(
\|f\|_{X^{s,-b,\alpha-1}(\mathbb{R}^+\times(0,T))} +
\|f\|_{Y^{s,-b}(\mathbb{R}^+\times(0,T))} ), s\not\in[-1,1/2], \nonumber \end{align}
where $Y^{s,b}$ is defined in \eqref{Ysb-def}. The solution of this problem is given by Duhamel's formula
\begin{align} \label{Duhamel-1} W(x,t) \doteq S\big[0; w\big](x, t) =& -\frac{i}{2\pi} \int_{\mathbb{R}} \int_{0}^t e^{i\xi x+i\xi^m(t-t')} \widehat w(\xi, t') dt' d\xi, \\ \label{Duhamel-2} =& -i \int_{0}^t S\big[w(\cdot, t'); 0\big](x, t-t') dt', \end{align}
where $\widehat{w}$ is the Fourier transform of $w$ with respect to $x$, and $S\big[w(\cdot, t'); 0\big]$ in the Duhamel representation \eqref{Duhamel-2} denotes the solution \eqref{homo-ivp:sln} of
ivp \eqref{homo-ivp} (that is Problem A$_1$) with $w(x, t')$ in place of the initial data and zero forcing.
\noindent {\bf B$_2.$ The homogeneous linear ibvp with zero initial data:}
\begin{subequations} \label{forced-pure-ibvp} \begin{align} \label{forced-pure-ibvp:eqn} &\partial_tv+(-1)^{j+1}\partial_x^{2j+1}v = 0,
\\ \label{forced-pure-ibvp:ic} &v(x,0) = 0,
\\ \label{forced-pure-ibvp:bc} & \partial_x^\ell v(0,t) = -\partial_x^\ell W(0, t) \doteq -W_\ell(t) && \ell = 0,1,\dots,j-1,
\end{align} \end{subequations}
whose solution
$ v(x, t) \doteq S\big[0, -W_0,\dots,-W_{j-1}; 0\big](x, t) $
is defined in \eqref{UTM-sln-compact}.
Next we describe the estimates for each one of the above sub-problems.
\begin{theorem} [Estimates for homogeneous ivp A$_1$] \label{homo-ivp-thm} The solution $U=S\big[U_0; 0\big]$ to ivp \eqref{homo-ivp} defined by formula \eqref{homo-ivp:sln} satisfies the space estimate
\begin{align} \label{homo-ivp-se}
\sup_{t\in [0, T]} \|S\big[U_0; 0\big](t)\|_{H_x^s(\mathbb R)} = \|U_0\|_{H_x^s(\mathbb R)}, &&s\in\mathbb R, \end{align}
and the time estimate for its $\ell$-th derivative (\it needed to have boundary data in desired space)
\begin{align} \label{homo-ivp-te}
\sup_{x\in\mathbb R}\|\psi(t)\partial_x^\ell S\big[U_0; 0\big](x)\|_{H_t^{\mu_\ell}(\mathbb{R})} \leqslant
c_s \|U_0\|_{H_x^s(\mathbb R)}, \,\, s\in\mathbb{R}, \end{align}
where $\ell=0,1,\dots,j-1$ and $\mu_\ell=\frac 1m(s+j-\ell)$.
Also, it satisfies the following estimate in modified Bourgain spaces
\begin{equation} \label{homo-ivp-se-bourgain}
\|\psi(t)S\big[U_0; 0\big](x,t)\|_{X^{s,b,\alpha}}
\leq c_{\psi}\|U_0\|_{H^s(\mathbb{R})}, \quad \forall s, b,\alpha\in\mathbb{R}, \end{equation}
where $c_\psi$ is a constant depending only on $\psi$. Here and elsewhere in this paper $\psi$ is a cutoff function in $C^{\infty}_0(-1, 1)$ such that
$0\le \psi \le 1$ and $\psi(t)=1$ for $|t|\le 1/2$. \end{theorem}
\noindent {\bf Proof of Theorem \ref{homo-ivp-thm}.}
The proof of the space estimate \eqref{homo-ivp-se} is straightforward. The proof of the time estimate \eqref{homo-ivp-te} is similar to that for KdV, which can be
found in Holmer \cite{h2006}, and Colliander and Kenig \cite{ck2002}. Finally, estimate \eqref{homo-ivp-se-bourgain}
follows from inequality $$ \int_{-\infty}^\infty \int_{-\infty}^\infty
\chi_{|\xi|< 1}(1+|\tau|)^{2\alpha}
|\widehat{u}(\xi,\tau)|^2 d\xi d\tau \lesssim
\|u\|^2_{X^{s,\alpha}}, $$
and the estimate
\begin{equation} \label{pure-ivp-bourgain-est-no2}
\|\psi(t)S\big[U_0; 0\big](x,t)\|_{X^{s,b}}
\leq c_{b}\|U_0\|_{H^s}, \quad \text{where}
\quad c_b=\|\psi\|_{H^b}, \quad s, b \in \mathbb{R}, \end{equation}
whose proof can be found in \cite{fhy2020}. \, $\square$
\begin{theorem} [Estimates for pure ibvp on the half-line] \label{pure-ibvp-thm} Let $s\neq \frac12,\frac32,\dots, j-\frac12$. The solution of the pure ibvp \eqref{pure-ibvp} satisfies the space estimate
\begin{align} \label{pure-ibvp-thm:se}
\sup_{t\in \mathbb{R}} \|S\big[0, G_0,\dots,G_{j-1}; 0\big]\|_{H_x^s(0,\infty)}
\leq c_{s,m} \sum\limits_{\ell=0}^{j-1}\|G_\ell\|_{H_t^{\mu_\ell}(0,T)} , \quad s\ge 0. \end{align}
Also, for $b\in[0,\frac12)$, $\frac12<\alpha\le\frac{1}{2} + \frac 1m (s+j+\frac12) $ its solution satisfies the estimate in Bourgain spaces
\begin{align} \label{pure-ibvp-thm:se-bourgain}
\|S\big[0, G_0,\dots,G_{j-1}; 0\big]\|_{X^{s, b,\alpha}(\mathbb{R}^+\times (0,T))} \le c_{s,b,\alpha}
\sum\limits_{\ell=0}^{j-1}\|G_\ell\|_{H_t^{\frac{s+j-\ell}{m}}(0,T)}, \quad s>-j-\frac12. \end{align}
\end{theorem} \noindent {\bf Proof of Theorem \ref{pure-ibvp-thm} }
The proof of the space estimate \eqref{pure-ibvp-thm:se} can be found in \cite{y2020} for KdVm, and in \cite{fhm2016} for KdV.
Here, we prove estimate \eqref{pure-ibvp-thm:se-bourgain}, which is new. We do this by transforming problem A$_2$ to the reduced pure ibvp \eqref{LKdVm-reduced}.
For this, we extend $G_\ell$ from $(0,T)$ to
a function $h_\ell$ on $\mathbb{R}$ supported in $[0, 2]$ and such that
$
\|h_\ell\|_{H_t^{\mu_\ell}(\mathbb R)}
\lesssim
\|G_\ell\|_{H_t^{\mu_\ell}(0,T)} $
for any $\ell=0,1,\dots,j-1$,
via the following result, whose proof can be found in \cite{lmbook1972,WMbook2000,fhm2017,y2020}.
\begin{lemma} \label{Extension-lemma} For a general function $h^*(t)\in H_t^s(0,2)$, $s\ge 0$, let the extension
\begin{align*} \tilde h^*(t) \doteq \begin{cases} h^*(t), \quad t\in(0,2), \\ 0, \quad \text{elsewhere}. \end{cases} \end{align*}
If $0\leq s<\frac12$, then the extension $\tilde h^*\in H^s(\mathbb{R})$ and for some $c_s>0$ we have
\begin{align} \label{char-mult-est}
\|\tilde h^*\|_{H_t^s(\mathbb{R})}
\leq c_s\|h^*\|_{H_t^s(0, 2)}. \end{align}
If $\frac12<s\leq1$, then for estimate \eqref{char-mult-est} to hold we must have the condition
\begin{align} \label{vanishing-condition} h^*(0)=h^*(2)=0. \end{align}
\end{lemma}
\vskip-0.08in Also, we shall need the following multiplier by a characteristic estimate from Holmer \cite{h2006}.
\begin{align} \label{ext-est-neg-new}
\|\chi_{(0,\infty)}g\|_{H^s(\mathbb{R})}
\le c_s\|g\|_{H^s(\mathbb{R})}, \,\,
g\in H^s(\mathbb{R}), \,\, -\frac12<s<\frac12. \end{align}
\noindent Since $h_\ell$ extends $G_\ell$ from $(0,T)$ to $\mathbb{R}$,
we have
$ S\big[0, G_0,\dots,G_{j-1}; 0\big] (x,t) = S\big[0, h_0,\dots,h_{j-1}; 0\big] (x,t), $
for $x\in\mathbb{R}^+, t\in(0,T)$.
In fact, by Theorem \ref{reduced-pure-ibvp-thm}, for $\frac12>s>-j-\frac12$, $b\in[0,\frac12)$ and $\frac12<\alpha\le\frac{1}{2}+\frac1m(s+j+\frac12)$ we get
\begin{align} \label{ibvp-identity}
&\|S\big[0, G_0,\dots,G_{j-1}; 0\big]\|_{X^{s, b,\alpha}(\mathbb{R}^+\times (0,T))} =
\|S\big[0, h_0,\dots,h_{j-1}; 0\big]\|_{X^{s, b,\alpha}(\mathbb{R}^+\times(0,T))} \\ \le&
\|S\big[0, h_0,\dots,h_{j-1}; 0\big]\|_{X^{s, b,\alpha}(\mathbb{R}^+\times(0,2))} \lesssim
\sum\limits_{\ell=0}^{j-1}\|h_\ell\|_{H_t^{\frac{s+j-\ell}{m}}(\mathbb R)} \lesssim
\sum\limits_{\ell=0}^{j-1}\|G_\ell\|_{H_t^{\frac{s+j-\ell}{m}}(0,T)}, \nonumber \end{align}
which is the desired estimate \eqref{pure-ibvp-thm:se-bourgain}. This completes the proof of Theorem \ref{pure-ibvp-thm}. \,\, $\Box$
\begin{theorem} [Estimates for forced ivp B$_1$] \label{forced-ivp-thm}
The solution $W=S\big[0; w\big]$ of the forced ivp \eqref{forced-ivp} defined by equations \eqref{Duhamel-1}--\eqref{Duhamel-2} satisfies the following estimate in modified Bourgain spaces
\begin{equation} \label{forced-ivp-bourgain-est}
\|\psi(t)S\big[0; w\big](x,t)\|_{X^{s,b,\alpha}} \lesssim c_\psi
\|w\|_{s,-b,\alpha-1}, \quad s\in\mathbb{R}, \,\, 0< b<\frac12<\alpha<1, \end{equation}
and the time estimate (\it needed to have boundary data in desired space)
\begin{equation} \label{forced-ivp-te} \sup\limits_{x\in\mathbb{R}}
\| \psi(t) \partial_x^\ell S[0,w]
\|_{H_t^{\frac{s+j-\ell}{m}}(\mathbb{R})} \le \begin{cases}
c_{s,m,\ell,b}\|w\|_{X^{s,-b}}, \quad -1\le s \le \frac12, \\ c_{s,m,\ell,b} (
\|w\|_{X^{s,-b}} +
\|w\|_{Y^{s,-b}} ), \quad s\in\mathbb{R} , \end{cases}
\end{equation}
where $X^{s,b}$ is the Bourgain space \eqref{def-Bourgain} and $Y^{s,b}$ is a ``temporal" Bourgain space defined by \eqref{Ysb-def}.
\end{theorem}
\vskip-0.05in \noindent {\bf Proof of Theorem \ref{forced-ivp-thm}.}
First, we prove estimate \eqref{forced-ivp-bourgain-est}. For the $X^{s,b}$ part of
$\|\psi\cdot S\big[0; w\big]\|_{X^{s,b,\alpha}}$, i.e. $
\|\psi(t)S\big[0; w\big](x,t)\|_{X^{s,b}}, $
we have the next basic estimate, whose proof can be found in \cite{fhy2020}
\begin{align} \label{T-bound}
\| \psi(t)S\big[0; w\big](x,t)
\|_{X^{s,b}}^2 \lesssim
\|w\|_{X^{s,b-1}}^2 +
\int_{\mathbb{R}}(1+|\xi|)^{2s} \left( \int_{\mathbb{R}}
\frac{|\widehat{w}(\xi,\tau)|}{1+|\tau-\xi^{m}|} d\tau \right)^2d\xi, \,\, 0< b<1. \end{align}
Since $b-1<-\frac12<-b$ we get
$
\|w\|_{X^{s,b-1}} \le
\|w\|_{X^{s,-b}}. $
For the second term in \eqref{T-bound}, writing $
1+|\tau-\xi^{m}| =
(1+|\tau-\xi^{m}|)^{1-b}(1+|\tau-\xi^{m}|)^b $ and applying the Cauchy-Schwartz inequality for
the $\tau$-integral we obtain:
\begin{align} \label{2nd-bilinear-est} \hskip-0.05in
\int_{\mathbb{R}}(1+|\xi|)^{2s} \Big(\int_{\mathbb{R}}
\frac{|\widehat{w}(\xi,\tau)|}{1+|\tau-\xi^{m}|} d\tau\Big)^2d\xi \le c_b
\int_{\mathbb{R}}(1+|\xi|)^{2s} \hskip-0.05in \int_{\mathbb{R}}
\frac{|\widehat{w}(\xi,\tau)|^2}{(1+|\tau-\xi^{m}|)^{2b}}d\tau d\xi \simeq
\|w\|_{X^{s,-b}}^2. \end{align}
For the $\alpha$-part of the norm
$\|\psi\cdot S\big[0; w\big]\|_{X^{s,b,\alpha}}$,
using the fact that $|\xi|\le 1$, which gives $
(1+|\tau|)^{2\alpha} \simeq
(1+|\tau-\xi^m|)^{2\alpha}, $
we get
\begin{align} \label{D-alpha-est} \int_{\mathbb{R}} \int_{-1}^1
(1+|\tau|)^{2\alpha}
| \reallywidehat{ \psi S\big[0; w\big]
}(\xi,\tau)|^2 d\xi d\tau \simeq \int_{\mathbb{R}} \int_{\mathbb{R}}
(1+|\tau-\xi^m|)^{2\alpha}
|
\chi_{|\xi|\le1} \reallywidehat{ \psi S\big[0; w\big]
}(\xi,\tau)|^2 d\xi d\tau. \end{align}
Also, since
$ \reallywidehat{ \psi S\big[0; w\big] }^x(\xi,t) =-i \psi(t) \int_{0}^t e^{i\xi^m(t-t')} \widehat w(\xi, t') dt', $
we obtain
$$
\chi_{|\xi|\le 1} \reallywidehat{ \psi S\big[0; w\big] }(\xi,\tau) = \reallywidehat{ \psi S\big[0; w_1\big] }(\xi,\tau), $$
where
$ \widehat{w_1}^x(\xi,t) \doteq
\chi_{|\xi|\le 1} \widehat{w}^x(\xi,t). $
Using $w_1$ notation, from \eqref{D-alpha-est} we have
\begin{align} \label{w1-express} \int_{\mathbb{R}} \int_{-1}^1
(1+|\tau|)^{2\alpha}
| \reallywidehat{ \psi S\big[0; w\big]
}(\xi,\tau)|^2 d\xi d\tau =
\| \psi(t)S\big[0; w_1\big](x,t)
\|_{X^{0,\alpha}}^2. \end{align}
For $\| \psi(t)S\big[0; w_1\big](x,t)
\|_{X^{0,\alpha}}$, applying estimate \eqref{T-bound} with $s=0$ and $b=\alpha>\frac12$, we get
\begin{align} \label{w1-express-est}
\| \psi(t)S\big[0; w_1\big](x,t)
\|_{X^{0,\alpha}}^2 \lesssim \int_{\mathbb{R}} \int_{-1}^1
(1+|\tau-\xi^m|)^{2\alpha-2}
| \reallywidehat{ w
}(\xi,\tau)|^2 d\xi d\tau. \end{align}
Using estimates \eqref{w1-express}, \eqref{w1-express-est} and the fact that $|\xi|\le 1$ again, we obtain
\begin{align} \label{alpha-est-fin} \int_{\mathbb{R}} \int_{-1}^1
(1+|\tau|)^{2\alpha}
| \reallywidehat{ \psi S\big[0; w\big]
}(\xi,\tau)|^2 d\xi d\tau \lesssim \int_{\mathbb{R}} \int_{-1}^1
(1+|\tau|)^{2\alpha-2}
| \reallywidehat{ w
}(\xi,\tau)|^2 d\xi d\tau. \end{align}
Combining \eqref{T-bound} and \eqref{2nd-bilinear-est} with \eqref{alpha-est-fin} and taking into consideration that $b-1<-b$, we get
\begin{align*}
\|\psi(t)S\big[0; w\big](x,t)\|_{X^{s,b,\alpha}}^2 \lesssim&
\|w\|_{X^{s,b-1}}^2 +
\|w\|_{X^{s,-b}}^2 + \int_{\mathbb{R}} \int_{-1}^1
(1+|\tau|)^{2\alpha-2}
| \reallywidehat{ w
}(\xi,\tau)|^2 d\xi d\tau \\ \lesssim&
\|w\|_{s,-b,\alpha-1}^2, \quad s\in\mathbb{R}, \,\, 0\le b<\frac12<\alpha<1. \end{align*}
This completes the proof of estimate \eqref{forced-ivp-bourgain-est}.
\noindent {\it\bf Proof of estimate \eqref{forced-ivp-te}.} For the KdV equation, this estimate was proved in \cite{h2006}
(see Lemma 5.6). Also a similar estimate (for $s=0$) was proved
in \cite{ck2002} (see Lemma 5.5). Differentiating the solution
formula \eqref{Duhamel-1}, i.e.
$
S\big[0; w\big](x, t) = -\frac{i}{2\pi} \int_{\xi\in \mathbb R} \int_{t'=0}^t e^{i\xi x+i\xi^m(t-t')} \widehat w(\xi, t') dt' d\xi
$ $\ell$ times with respect to $x$ and decomposing it (like in \cite{fhy2020}), we obtain the Bourgain writing
\begin{align} \label{Tfg2-term-recall} \psi(t)\partial_x^\ell S[0,w](x,t) \simeq& \psi(t) \int_{\mathbb{R}}\int_{\mathbb{R}} e^{i(\xi x+\tau t)} \frac{1-\psi(\tau-\xi^m)}{\tau-\xi^m} \xi^\ell \widehat{w}(\xi,\tau) d\tau d\xi \\ \label{Tfg3-term-recall} -& \psi(t) \int_{\mathbb{R}}\int_{\mathbb{R}} e^{i(\xi x+\xi^mt)} \frac{1-\psi(\tau-\xi^m)}{\tau-\xi^m} \xi^\ell \widehat{w}(\xi,\tau) d\tau d\xi \\ \label{Tfg4-term-recall} +& \psi(t) \int_{\mathbb{R}}\int_{\mathbb{R}} e^{i(\xi x+\xi^m t)} \frac{\psi(\tau-\xi^m)[e^{i(\tau - \xi^m)t} - 1]}{\tau - \xi^m} \xi^\ell \widehat{w}(\xi,\tau) d\tau d\xi. \end{align}
Next, we estimate each term above separately. We start by estimating \eqref{Tfg3-term-recall}.
\vskip.05in \noindent \underline{\it Estimate for \eqref{Tfg3-term-recall}.} For this term, we have
$$ \eqref{Tfg3-term-recall} \simeq \psi(t)\partial_x^\ell S[F_1,0], \quad \text{where } \quad \widehat{F}_1(\xi) \simeq \int_\mathbb{R} \frac{1-\psi(\tau-\xi^m)}{\tau-\xi^m} \widehat{w}(\xi,\tau) d\tau. $$
Using estimate \eqref{homo-ivp-te}, we get
\begin{align*}
\sup\limits_{x\in\mathbb{R}}\| \eqref{Tfg3-term-recall}
\|_{H_t^\frac{s+j-\ell}{m}(\mathbb{R})}^2 \lesssim&
\sup\limits_{x\in\mathbb{R}}\| \psi(t) \partial_x^\ell S[F_1,0]
\|_{H_t^\frac{s+j-\ell}{m}(\mathbb{R})}^2 \lesssim
\|F_1\|_{H^s(\mathbb{R})}^2 \\ =& \int_\mathbb{R}
(1+|\xi|)^{2s}
\Big| \int_\mathbb{R} \frac{1-\psi(\tau-\xi^m)}{\tau-\xi^m} \widehat{w}(\xi,\tau) d\tau
\Big|^2 d\xi \\ \lesssim& \int_\mathbb{R}
(1+|\xi|)^{2s} \Big( \int_\mathbb{R}
\frac{|\widehat{w}(\xi,\tau)| }{1+|\tau-\xi^m|} d\tau \Big)^2 d\xi \lesssim
\|w\|_{s,-b}^2, \end{align*}
where in the last step we used estimate \eqref{2nd-bilinear-est}. This gives the desired estimate \eqref{forced-ivp-te} for \eqref{Tfg3-term-recall}.
\noindent \underline{\it Estimate for \eqref{Tfg4-term-recall}.} For this term, using Taylor's expansion we have
$$ \eqref{Tfg4-term-recall} \simeq \sum\limits_{k=1}^\infty \frac{1}{k!}t^k \partial_x^\ell \psi(t)S[c_k,0],
\,\, \text{ where } \,\,
\widehat{c_k}(\xi) \simeq \int_{\mathbb{R}}\psi(\tau-\xi^m)\cdot(\tau-\xi^m)^{k-1} \widehat{w}(\xi,\tau)d\tau. $$
Letting $\psi_k(t)\doteq t^k[\psi(t)]^{1/2}$ and using estimate \eqref{homo-ivp-te}, we get
\begin{align*}
\sup\limits_{x\in\mathbb{R}}\| &\eqref{Tfg4-term-recall}
\|_{H_t^\frac{s+j-\ell}{m}(\mathbb{R})} \lesssim \sum_{k=1}^{\infty}\frac{1}{k!} \sup\limits_{x\in\mathbb{R}}
\| \psi_k(t) \cdot [\psi(t)]^{1/2} \partial_x^\ell S[c_k,0]
\|_{H_t^\frac{s+j-\ell}{m}(\mathbb{R})} \lesssim \sum_{k=1}^{\infty}\frac{c_{\psi_k}}{k!}
\|c_k\|_{H^s} \\ =&
\sum_{k=1}^{\infty}\frac{1}{k!}\left( \int_{\mathbb{R}} (1+|\xi|)^{2s}
\Big|\int_{\mathbb{R}}\psi(\tau-\xi^m)\cdot(\tau-\xi^m)^{k-1} \widehat{w}(\xi,\tau)d\tau
\Big|^2 d\xi \right)^{1/2}, \end{align*}
where $c_{\psi_k}\doteq \|\widehat{\psi_k}(\tau)(1+|\tau|)^\frac{s+j-\ell}{m}\|_{L^1}$ (like \eqref{c-eta-def} below). Since the $\tau$-integration is over $|\tau-\xi^m|\leq 1$ and $|\psi(\tau-\xi^m)|\leq 1$ from the last relation we obtain that
\begin{align*}
\sup\limits_{x\in\mathbb{R}}\| \eqref{Tfg4-term-recall}
\|_{H_t^\frac{s+j-\ell}{m}(\mathbb{R})} \lesssim&
\left(\int_{\mathbb{R}} (1+|\xi|)^{2s} \left(\int_{|\tau-\xi^m|\leq 1}
|\widehat{w}(\xi,\tau)|d\tau\right)^2 d\xi \right)^{1/2} \cdot \sum_{k=1}^{\infty}\frac{1}{k!} \\ \lesssim&
\left(\int_{\mathbb{R}} (1+|\xi|)^{2s} \left(\int_{\mathbb{R}}
\frac{|\widehat{w}(\xi,\tau)|}{1+|\tau-\xi^m|}d\tau\right)^2 d\xi \right)^{1/2} \lesssim
\|w\|_{s,-b}, \end{align*}
where last inequality follows from \eqref{2nd-bilinear-est}. This gives the desired estimate \eqref{forced-ivp-te} for term \eqref{Tfg4-term-recall}.
\noindent \underline{\it Estimate for \eqref{Tfg2-term-recall}.} We rewrite this term as
$$ \eqref{Tfg2-term-recall} \simeq \psi(t) h(x,t), \,\, \text{where} \,\, h(x,t) = \int_{\mathbb{R}}\int_{\mathbb{R}} e^{i(\xi x+\tau t)} \frac{1-\psi(\tau-\xi^m)}{\tau-\xi^m} \xi^\ell\widehat{w}(\xi,\tau) d\tau d\xi. $$
Using the property
$ \widehat{f\cdot g} \simeq \widehat{f}*\widehat{g} $
we get
$ \widehat{\psi \cdot h}^t(x,\tau) \simeq \int_\mathbb{R} \widehat{\psi}(\tau-\tau_1) \widehat{h}^t(x,\tau_1) d\tau_1 $,
which combined with the inequality $
(1+|\tau|)^{\mu} \le
(1+|\tau_1|)^{\mu}
(1+|\tau-\tau_1|)^{|\mu|}, $ for $\mu=\frac{s+j-\ell}{m}$, we get
\begin{align*}
\|\psi h\|_{H^\frac{s+j-\ell}{m}_t(\mathbb{R})} \le&
\Big\| \int_{\mathbb{R}}
(1+|\tau_1|)^{\frac{s+j-\ell}{m}}(1+|\tau-\tau_1|)^{|\frac{s+j-\ell}{m}|} \widehat{\psi}(\tau-\tau_1) \widehat{h}^t(x,\tau_1) d\tau_1
\Big\|_{L^2_\tau(\mathbb{R})} \\ \le&
\|\widehat{\psi}(\tau)(1+|\tau|)^{|\frac{s+j-\ell}{m}|}\|_{L^1(\mathbb{R})} \cdot
\Big\|
(1+|\tau|)^\frac{s+j-\ell}{m} \widehat{h}^t(x,\tau)
\Big\|_{L^2_\tau(\mathbb{R})} \\ =& c_\psi
\Big\|
(1+|\tau|)^\frac{s+j-\ell}{m} \widehat{h}^t(x,\tau)
\Big\|_{L^2_\tau(\mathbb{R})}, \end{align*}
where in the second step we use the Young's inequality for $r=q=2$, and $p=1$ and and we bound the constant $c_\psi^2 \doteq
\|\widehat{\psi}(\tau)(1+|\tau|)^{|\frac{s+j-\ell}{m}|}\|_{L^1}^2$
as follows
\begin{align} \label{c-eta-def} c_\psi^2 \leq \!\! \int_\mathbb{R}
|\widehat{\psi}(\tau)|^2
(1+|\tau|)^{2|\frac{s+j-\ell}{m}|+2} d\tau \!\! \int_\mathbb{R}
(1+|\tau|)^{-2} d\tau \lesssim
\|\psi\|_{H_t^{|\frac{s+j-\ell}{2m}|+1}}^2. \end{align}
Since
$ \widehat{h}^t(x,\tau) \simeq \int_{\mathbb{R}} e^{i\xi x} \frac{1-\psi(\tau-\xi^m)}{\tau-\xi^m} \xi^\ell \widehat{w}(\xi,\tau) d\xi, $
we have
\begin{align*}
\|\psi h\|_{H^\frac{s+j-\ell}{m}_t(\mathbb{R})}^2 \le& c_\psi^2 \int_{\mathbb{R}}
(1+|\tau|)^\frac{2(s+j-\ell)}{m}
\Big| \int_{\mathbb{R}} e^{i\xi x} \frac{1-\psi(\tau-\xi^m)}{\tau-\xi^m} \xi^\ell \widehat{w}(\xi,\tau) d\xi
\Big|^2 d\tau \\ \le& c_\psi^2 \int_{\mathbb{R}}
(1+|\tau|)^\frac{2(s+j-\ell)}{m} \Big( \int_{\mathbb{R}}
\frac{1}{1+|\tau-\xi^m|} \xi^\ell
|\widehat{w}(\xi,\tau)| d\xi \Big)^2 d\tau. \end{align*}
Now we consider the following two cases:
\vskip0.05in \noindent $\bullet$ $-1\le s\le \frac12$
\quad and
\quad $\bullet$ $s\not\in[-1,\frac12]$.
\vskip0.05in \noindent
{\bf Case $-1\le s\le \frac12$.} For this case, multiplying and dividing $|\widehat{w}(\xi,\tau)|$ by $\frac{(1+|\xi|)^s}{(1+|\tau-\xi^m|)^{b}}$ and using Cauchy-Schwartz inequality for the integral of $d\xi$, we get
\begin{align*}
\|\psi h\|_{H^\frac{s+j-\ell}{m}_t(\mathbb{R})}^2 \le c_\psi^2 \int_{\mathbb{R}}
(1+|\tau|)^\frac{2(s+j-\ell)}{m} G_1(\tau) \int_{\mathbb{R}}
\frac{(1+|\xi|)^{2s}|\widehat{w}(\xi,\tau)|^2}{(1+|\tau-\xi^m|)^{2b}}
d\xi d\tau, \end{align*}
where
$ G_1(\tau) \doteq \int_{\mathbb{R}}
\frac{\xi^{2\ell}}{(1+|\tau-\xi^m|)^{2-2b}(1+|\xi|)^{2s}} d\xi. $
Taking the sup norm in $\tau$ for $(1+|\tau|)^\frac{2(s+j-\ell)}{m} G_1(\tau)$, we get
\begin{align*}
\|\psi h\|_{H^\frac{s+j-\ell}{m}_t(\mathbb{R})}^2 \le& c_\psi^2
\Big\|
(1+|\tau|)^\frac{2(s+j-\ell)}{m} G_1(\tau)
\Big\|_{L_\tau^\infty} \int_{\mathbb{R}} \int_{\mathbb{R}}
\frac{(1+|\xi|)^{2s}|\widehat{w}(\xi,\tau)|^2}{(1+|\tau-\xi^m|)^{2b}}
d\xi d\tau \\ \le& c_\psi^2
\Big\|
(1+|\tau|)^\frac{2(s+j-\ell)}{m} G_1(\tau)
\Big\|_{L_\tau^\infty}
\|w\|_{s,-b}^2. \end{align*}
For $G_1(\tau)$, we have the following result:
\begin{equation} \label{G1-est}
\Big\|
(1+|\tau|)^\frac{2(s+j-\ell)}{m} G_1(\tau)
\Big\|_{L_\tau^\infty} \le c_{s,b}, \qquad -1\le s\le \frac12, \quad 0\le b<\frac12, \end{equation}
where $c_{s,b}$ is a constant depending on $s$ and $b$. We shall prove estimate \eqref{G1-est} later. Now using it we get the desired estimate \eqref{forced-ivp-te} for term \eqref{Tfg2-term-recall} in the case $s\le\frac12$.
\noindent {\bf Case $s\not\in[-1,\frac12]$.} For this case, multiplying
and dividing $|\widehat{w}(\xi,\tau)|$ by $
\frac{(1+|\tau|)^{s/m}}{(1+|\tau-\xi^m|)^{b}} $ and using the Cauchy-Schwartz inequality for the integral of $d\xi$, we get
\begin{align*}
\|\psi h\|_{H^\frac{s+j-\ell}{m}_t(\mathbb{R})}^2 \le c_\psi^2 \int_{\mathbb{R}}
(1+|\tau|)^\frac{2j-2\ell}{m} G_2(\tau) \int_{\mathbb{R}}
\frac{(1+|\tau|)^{2s/m}|\widehat{w}(\xi,\tau)|^2}{(1+|\tau-\xi^m|)^{2b}} d\xi d\tau, \end{align*}
where
$ G_2(\tau) \doteq \int
\frac{\xi^{2\ell}}{(1+|\tau-\xi^m|)^{2-2b}} d\xi. $
Like in the case $s\in[-1,\frac12]$, we get
\begin{align*}
\|\psi h\|_{H^\frac{s+1}{m}_t(\mathbb{R})}^2 \leq c_\psi^2
\Big\|
(1+|\tau|)^\frac{2j-2\ell}{m} G_2(\tau)
\Big\|_{L_\tau^\infty}
\|w\|_{Y^{s,-b}}^2. \end{align*}
For $G_2(\tau)$, we have the following result:
\begin{equation} \label{G2-est}
\Big\|
(1+|\tau|)^\frac{2j-2\ell}{m} G_2(\tau)
\Big\|_{L_\tau^\infty} \le c_{s,b}, \qquad s\in\mathbb{R}, \quad 0<b<\frac12. \end{equation}
Hence, we complete the proof of Theorem \ref{forced-ivp-thm} once we prove estimates \eqref{G1-est} and \eqref{G2-est}. The proof of estimate \eqref{G2-est} is similar to the proof of estimate \eqref{G1-est}. Here we prove only estimate \eqref{G1-est}.
\noindent {\bf Proof of estimate \eqref{G1-est}.} To prove this estimate, we consider the following two cases:
\vskip0.05in \noindent
$\bullet$ $|\xi|\le 1$ \quad and \quad
$\bullet$
$|\xi|> 1$
\vskip0.05in \noindent
{\bf Case $|\xi|\le 1$.} Since
$ G_1(\tau) = \int_{-1}^1
\frac{\xi^{2\ell}}{(1+|\tau-\xi^m|)^{2-2b}(1+|\xi|)^{2s}} d\xi, $
we have
$$
(1+|\tau|)^\frac{2(s+j-\ell)}{m} G_1(\tau) \lesssim
\frac{(1+|\tau|)^{\frac{2(s+j-\ell)}{m}}}{(1+|\tau|)^{2-2b}} =
(1+|\tau|)^{\frac{2s-2\ell+2j-2m+2mb}{m}}, $$
which is bounded since $\frac1m\cdot(2s-2\ell+2j-2m+2mb)\le 0$ when $s\le \frac12$, $\ell\ge0$ and $0\le b<\frac12$.
\vskip.05in \noindent
{\bf Case $|\xi|> 1$.} Since $(1+|\xi|)^{2s} \simeq |\xi|^{2s}$, after making the change of variables $\xi_1=\xi^m$, we get
$
G_1(\tau) \lesssim
\int_{|\xi_1|>1}
\frac{1}{(1+|\tau-\xi_1|)^{2-2b}|\xi_1|^{\frac{2s+2j-2\ell}{m}}} d\xi_1. $
Next, we consider the following two subcases:
\vskip.05in \noindent
$\bullet$ $|\xi_1|\le \frac12|\tau|$ \quad and \quad
$\bullet$ $|\xi_1|\ge \frac12|\tau|$
\vskip0.05in \noindent
{\bf Subcase $|\xi_1|\le \frac12|\tau|$.} Then, we have $(1+|\tau-\xi_1|)^{2-2b}\simeq (1+|\tau|)^{2-2b}$ and $|\tau|\ge 2$. Thus,
\begin{align*} G_1(\tau) \simeq&
\int_{|\xi_1|=1}^{\frac12|\tau|}\frac{1}{(1+|\tau-\xi_1|)^{2-2b}|\xi_1|^{\frac{2s+2j-2\ell}{m}}} d\xi_1 \leq 2
(1+|\tau|)^{2b-2}
\int_{\xi_1=1}^{\frac12|\tau|} \xi_1^{-\frac{2s+2j-2\ell}{m}} d\xi_1, \end{align*}
which implies that
$
(1+|\tau|)^\frac{2(s+j-\ell)}{m} G_1(\tau) $
is bounded if $s\le 1/2$ and $b<1/2$.
\vskip.05in \noindent
{\bf Subcase $|\xi_1|\ge \frac12|\tau|$.} Since $|\xi_1|\gtrsim 1+ |\tau|$, using $s\ge -1$, we get $\frac{2s+2j-2\ell}{m}$, which implies that
$$ G_1(\tau) \lesssim
(1+|\tau|)^{-\frac{2s+2j-2\ell}{m}}
\int_{|\xi_1|=\frac12|\tau|}^{\infty}\frac{1}{(1+|\tau-\xi_1|)^{2-2b}} d\xi_1 \lesssim
(1+|\tau|)^{-\frac{2s+2j-2\ell}{m}} \int_{0}^{\infty}\frac{1}{(1+x)^{2-2b}} dx, $$
where in the last step we make the change of variables $x=\xi_1-\tau$. Therefore, we get
$
(1+|\tau|)^\frac{2(s+j-\ell)}{m} G_1(\tau) \lesssim \int_{0}^{\infty} \frac{1}{(1+x)^{2-2b}} dx$, which is bounded if $b<\frac12$. \,\, $\Box$
\noindent {\bf Estimates for pure ibvp B$_2$.} By the time estimate \eqref{forced-ivp-te} we have $-W_\ell(t)\in H_t^{\mu_\ell}(0,T)$.
Thus, the solution of problem B$_2$ is like that of problem A$_2$ and is
estimated by using Theorem \ref{pure-ibvp-thm}.
\noindent {\bf Proof of Theorem \ref{forced-linear-kdvm-thm}.} Now using the results above we can estimate the solution of the forced linear ibvp.
For $x\ge 0$ and $0\le t\le T<\frac 12$ we have
\begin{align*} S[u_0,g_0,\dots,g_{j-1};f] =& \psi(t)S[U_0;0] + S\big[0, G_0,\dots,G_{j-1}; 0\big] \\ +& \psi(t)S\big[0; w\big] + S\big[0, -W_0,\dots,-W_{j-1}; 0\big]. \end{align*}
This together with estimates \eqref{homo-ivp-te}, \eqref{homo-ivp-se-bourgain} and \eqref{pure-ibvp-thm:se-bourgain}-\eqref{forced-ivp-te} gives the desired result \eqref{forced-linear-kdvm-est}. \,\, $\square$
\section{ Proof of Bilinear Estimate in modified Bourgain spaces $X^{s,b,\alpha}$} \label{sec:bilinear-estimate}
In this section, we prove the bilinear estimate in Bourgain spaces $X^{s,b,\alpha}$. Following \cite{b1993-kdv} and \cite{kpv1996} we begin the proof of the bilinear estimate \eqref{bilinear-est} by first providing an equivalent $L^2$ formulation. For this, using the fact that
$a^2+b^2\simeq (|a|+|b|)^2$, we get
\begin{equation} \label{Bourgain-like-norms-modified}
\|h\|_{s,b,\alpha}^2 \simeq \int_{\mathbb{R}} \int_{\mathbb{R}} \left[
(1+|\xi|)^{s}
(1+|\tau-\xi^m|)^{b} +
\chi_{|\xi|< 1}(1+|\tau|)^{\alpha} \right]^2
|\widehat{h}(\xi,\tau)|^2 d\xi d\tau, \end{equation}
and if for a function $h$ we use the Bourgain type combination
\begin{equation} \label{eq:c_h} c_h(\xi,\tau) \doteq \left[
(1+|\xi|)^{s}
(1+|\tau-\xi^m|)^{b'} +
\chi_{|\xi|< 1}(1+|\tau|)^{\alpha'} \right]
|\widehat{h}(\xi,\tau)|, \end{equation}
then the modified Bourgain norm of $h$ is equivalent to the $L^2$ norm of $c_h$, that is
\begin{equation} \label{eq:c_useful}
\|h\|_{s,b',\alpha'}=\|c_h(\xi,\tau)\|_{L^2_{\xi} L^2_{\tau}}. \end{equation}
Next, we form the $\|\cdot\|_{s,-b,\alpha-1}$-norm of
\begin{equation} \label{w-fg-def} w_{fg}\doteq\frac 12 \partial_x[f\cdot g]. \end{equation}
Using the definition of convolution and the relation \eqref{eq:c_h} we have
\begin{align*}
&|\widehat{w}_{fg}(\xi,\tau)| \simeq
\left| \xi \iint_{\mathbb{R}^2}
\widehat{f}(\xi-\xi_1,\tau-\tau_1)\widehat{g}(\xi_1,\tau_1)d\xi_1 d\tau_1 \right| \\ \leq&
|\xi | \iint_{\mathbb{R}^2} \frac{c_f(\xi-\xi_1,\tau-\tau_1) } {
(1+|\xi-\xi_1|)^{s}
(1+|\tau-\tau_1-(\xi-\xi_1)^m|)^{b'} +
\chi_{|\xi-\xi_1|\leq 1}(1+|\tau-\tau_1|)^{\alpha'} } \\ & \times \frac{c_g(\xi_1,\tau_1)} {
(1+|\xi_1|)^{s}
(1+|\tau_1-\xi_1^m|)^{b'} +
\chi_{|\xi_1|\leq 1}(1+|\tau_1|)^{\alpha'} } d\xi_1 d\tau_1. \end{align*}
Then, the $\|\cdot\|_{s,-b,\alpha-1}$-norm reads as follows
\begin{align*}
\|w_{fg}\|_{s,-b,\alpha-1}^2 =& \iint_{\mathbb{R}^2} \left\{ \iint_{\mathbb{R}^2}
|\xi | \cdot \left[ \frac{
(1+|\xi|)^s }{
(1+|\tau-\xi^m|)^{b} } + \frac{
\chi_{|\xi|< 1} } {
(1+|\tau|)^{1-\alpha} } \right] \right. \\ & \times \frac{c_f(\xi-\xi_1,\tau-\tau_1) } {
(1+|\xi-\xi_1|)^{s}
(1+|\tau-\tau_1-(\xi-\xi_1)^m|)^{b'} +
\chi_{|\xi-\xi_1|\leq 1}(1+|\tau-\tau_1|)^{\alpha'} } \\ & \left. \times \frac{c_g(\xi_1,\tau_1)} {
(1+|\xi_1|)^{s}
(1+|\tau_1-\xi_1^m|)^{b'} +
\chi_{|\xi_1|\leq 1}(1+|\tau_1|)^{\alpha'} } d\xi_1 d\tau_1 \right\}^2 d\xi d\tau. \end{align*}
Collecting all multipliers together to form the important quantity
\begin{subequations} \label{Q-def} \begin{align} Q(\xi,\xi_1,\tau,\tau_1) \doteq&
|\xi| \nonumber \\ \label{Q-def-term-a} \times&\left[ \frac{
(1+|\xi|)^s }{
(1+|\tau-\xi^m|)^{b} } + \frac{
\chi_{|\xi|< 1} } {
(1+|\tau|)^{1-\alpha} } \right] \\ \label{Q-def-term-b} \times&\frac{1}{
(1+|\xi_1|)^{s}
(1+|\tau_1-\xi_1^m|)^{b'} +
\chi_{|\xi_1|\leq 1}(1+|\tau_1|)^{\alpha'} } \\ \label{Q-def-term-c} \times& \frac{1}{
(1+|\xi-\xi_1|)^{s}
(1+|\tau-\tau_1-(\xi-\xi_1)^m|)^{b'} +
\chi_{|\xi-\xi_1|\leq 1}(1+|\tau-\tau_1|)^{\alpha'} }. \end{align}
\end{subequations}
we see that to prove bilinear estimate \eqref{bilinear-est} it suffices to show the $L^2$ inequality
\begin{align} \label{bilinear-est-L2-form}
\Bigg\|
\iint_{\mathbb{R}^2}
Q(\xi,\xi_1,\tau,\tau_1)
c_f(\xi-\xi_1,\tau-\tau_1) c_g(\xi_1,\tau_1)
d\xi_1 d\tau_1 \Bigg\|_{L^2_\xi L^2_{\tau}} \lesssim
\left\|c_f\right\|_{L^2_{\xi} L^2_{\tau}} \left\|c_g\right\|_{L^2_{\xi} L^2_{\tau}}. \end{align}
Next, we bound the term \eqref{Q-def-term-a} from above. Since
\begin{align} \label{relation-alpha-b} 1-\alpha \geq b, \end{align}
the second term that has numerator $\chi_{|\xi|< 1}$ (and so $\xi$ is bounded) is absorbed by the first term. Thus,
\begin{equation} \label{Q-def-term-a-est} \eqref{Q-def-term-a} = \frac{
(1+|\xi|)^s }{
(1+|\tau-\xi^m|)^{b} } + \frac{
\chi_{|\xi|< 1} } {
(1+|\tau|)^{1-\alpha} } \lesssim \frac{
(1+|\xi|)^s }{
(1+|\tau-\xi^m|)^{b} }. \end{equation}
Combining estimate \eqref{Q-def-term-a-est} with estimate \eqref{Q-def} we get the following form of $Q$, which still involves $\alpha$ terms
\begin{subequations} \label{Q-def-no1} \begin{align} \label{Q-def-no1-term-a} Q(\xi,\xi_1,\tau,\tau_1) \lesssim&
|\xi| \times \frac{
(1+|\xi|)^s }{
(1+|\tau-\xi^m|)^{b} } \\ \label{Q-def-no1-term-b} \times&\frac{1}{
(1+|\xi_1|)^{s}
(1+|\tau_1-\xi_1^m|)^{b'} +
\chi_{|\xi_1|\leq 1}(1+|\tau_1|)^{\alpha'} } \\ \label{Q-def-no1-term-c} \times& \frac{1}{
(1+|\xi-\xi_1|)^{s}
(1+|\tau-\tau_1-(\xi-\xi_1)^m|)^{b'} +
\chi_{|\xi-\xi_1|\leq 1}(1+|\tau-\tau_1|)^{\alpha'} }. \end{align}
\end{subequations}
So it suffices to prove the bilinear estimates by replacing $Q$ with the right-hand side of the above inequality. Note that all of the above is valid for any $s$. To make further reduction for $Q$, we need to consider the following two cases:
\vskip0.05in \noindent $\bullet$ $s\ge 0$ \quad \text{and} \quad $\bullet$ $-j+\frac14<s<0$
\vskip0.05in \noindent {\bf Case $s\ge 0$.}
Collecting all the factors with $s$ power
we rewrite the above bound for $Q$ as
\begin{subequations} \label{Q-modify-no1} \begin{align} \label{Q-modify-no1-term-a} &Q(\xi,\xi_1,\tau,\tau_1) \lesssim
\frac{|\xi|}{(1+|\tau-\xi^m|)^{b}} \\ \label{Q-modify-no1-term-b} \times&
\frac{(1+|\xi|)^s}{
(1+|\xi_1|)^{s}
(1+|\xi-\xi_1|)^{s} } \\ \label{Q-modify-no1-term-c} \times& \frac{1}{
(1+|\tau_1-\xi_1^m|)^{b'} +
\chi_{|\xi_1|\leq 1}(1+|\tau_1|)^{\alpha'}
(1+|\xi_1|)^{-s} } \\ \label{Q-modify-no1-term-d} \times& \frac{1} {
(1+|\tau-\tau_1-(\xi-\xi_1)^m|)^{b'} +
\chi_{|\xi-\xi_1|\leq 1}(1+|\tau-\tau_1|)^{\alpha'}
(1+|\xi-\xi_1|)^{-s} }. \end{align} \end{subequations}
Since $s\ge 0$, like on the line, we have the estimate
\begin{equation} \label{triang-ineq-s}
(1+|\xi|)^s \le
(1+|\xi-\xi_1|)^s
(1+|\xi_1|)^s \iff
\frac{(1+|\xi|)^s}{(1+|\xi-\xi_1|)^s(1+|\xi_1|)^s} \lesssim
1, \end{equation}
which helps us remove term \eqref{Q-modify-no1-term-b}.
Also, since $|\xi_1|$ and $|\xi-\xi_1|$ are bounded we can remove $(1+|\xi_1|)^{-s}$ and
$(1+|\xi-\xi_1|)^{-s}$.
Thus, for any $s\geq 0$, we have $Q(\xi,\xi_1,\tau,\tau_1)\leq Q_0(\xi,\xi_1,\tau,\tau_1)$, where
\begin{align} \label{Q-def-s=0} Q_0(\xi,\xi_1,\tau,\tau_1) \doteq&
\frac{|\xi|}{(1+|\tau-\xi^m|)^{b}} \frac{1}{
(1+|\tau_1-\xi_1^m|)^{b'} +
\chi_{|\xi_1|\leq 1}(1+|\tau_1|)^{\alpha'} } \notag \\ \times& \frac{1} {
(1+|\tau-\tau_1-(\xi-\xi_1)^m|)^{b'} +
\chi_{|\xi-\xi_1|\leq 1}(1+|\tau-\tau_1|)^{\alpha'} }. \end{align}
Thus, for $s\geq 0$ to prove our bilinear estimate \eqref{bilinear-est-L2-form}, it suffices to prove the following simpler one
\begin{align} \label{bilinear-est-s-0}
\Bigg\|
\iint_{\mathbb{R}^2} Q_0(\xi,\xi_1,\tau,\tau_1) c_f(\xi-\xi_1,\tau-\tau_1) c_g(\xi_1,\tau_1)
d\xi_1 d\tau_1 \Bigg\|_{L^2_\xi L^2_{\tau}} \lesssim
\left\|c_f\right\|_{L^2_{\xi} L^2_{\tau}} \left\|c_g\right\|_{L^2_{\xi} L^2_{\tau}}, \end{align}
which corresponds to proving the bilinear estimate when $s=0$.
Moreover, by symmetry (in convolution writing), we may assume that
\begin{equation} \label{conv-symmetry-xi}
|\xi-\xi_1| \leq
|\xi_1|. \end{equation}
Then, we have\,
$
1=\chi_{|\xi_1|> 1} \cdot
\chi_{|\xi-\xi_1|> 1} +
\chi_{|\xi_1|\leq 1} \cdot
\chi_{|\xi-\xi_1|\leq 1} +
\chi_{|\xi_1|> 1} \cdot
\chi_{|\xi-\xi_1|\leq 1}. $
Therefore, we can rewrite $Q_0(\xi,\tau,\xi_1,\tau_1)$ as:
$ Q_0(\xi,\tau,\xi_1,\tau_1) = Q_1(\xi,\tau,\xi_1,\tau_1) + Q_2(\xi,\tau,\xi_1,\tau_1) + Q_3(\xi,\tau,\xi_1,\tau_1), $
where
\begin{align} \label{def-Q1} Q_1(\xi,\tau,\xi_1,\tau_1) \doteq&
\chi_{|\xi_1|> 1} \cdot
\chi_{|\xi-\xi_1|> 1} \cdot Q_0(\xi,\tau,\xi_1,\tau_1) \\ =&
\frac{|\xi|}{(1+|\tau-\xi^m|)^{b}} \frac{
\chi_{|\xi_1|> 1} }{
(1+|\tau_1-\xi_1^m|)^{b'} } \frac{
\chi_{|\xi-\xi_1|> 1} } {
(1+|\tau-\tau_1-(\xi-\xi_1)^m|)^{b'} }, \nonumber \end{align}
\begin{align} \label{def-Q2} &Q_2(\xi,\tau,\xi_1,\tau_1) \doteq
\chi_{|\xi_1|\leq 1} \cdot
\chi_{|\xi-\xi_1|\leq 1} \cdot Q_0(\xi,\tau,\xi_1,\tau_1) \\ =&
\frac{|\xi|}{(1+|\tau-\xi^m|)^{b}} \frac{
\chi_{|\xi_1|\leq 1} }{
(1+|\tau_1-\xi_1^m|)^{b'} +
(1+|\tau_1|)^{\alpha} } \frac{
\chi_{|\xi-\xi_1|\leq 1} } {
(1+|\tau-\tau_1-(\xi-\xi_1)^m|)^{b'} +
(1+|\tau-\tau_1|)^{\alpha'} }, \notag \end{align}
and
\begin{align} \label{def-Q3} &Q_3(\xi,\tau,\xi_1,\tau_1) \doteq
\chi_{|\xi_1|> 1} \cdot
\chi_{|\xi-\xi_1|\leq 1} \cdot Q_0(\xi,\tau,\xi_1,\tau_1) \\ =&
\frac{|\xi|}{(1+|\tau-\xi^m|)^{b}} \frac{
\chi_{|\xi_1|> 1} }{
(1+|\tau_1-\xi_1^m|)^{b'} } \frac{
\chi_{|\xi-\xi_1|\leq 1} } {
(1+|\tau-\tau_1-(\xi-\xi_1)^m|)^{b'} +
(1+|\tau-\tau_1|)^{\alpha'} }. \notag \end{align}
Note that $Q_1$ is like the regular Bourgain norm, because we do not have any term related to $\alpha$.
\noindent To prove the bilinear estimate \eqref{bilinear-est-s-0}, it suffices to prove that for $\ell=1,2,3$
\begin{align} \label{bilinear-est-s-l}
\Bigg\|
\iint_{\mathbb{R}^2} Q_\ell(\xi,\xi_1,\tau,\tau_1) c_f(\xi-\xi_1,\tau-\tau_1) c_g(\xi_1,\tau_1)
d\xi_1 d\tau_1 \Bigg\|_{L^2_\xi L^2_{\tau}} \hskip-0.2cm \lesssim
\left\|c_f\right\|_{L^2_{\xi} L^2_{\tau}} \left\|c_g\right\|_{L^2_{\xi} L^2_{\tau}}. \end{align}
{\bf Estimation when the multiplier is $Q_1$.} This is similar to the whole line case (see \cite{fhy2020}). We do it by considering the following two possibilities:
\vskip0.05in \noindent
$\bullet$ $|\xi|\le 1$ \quad and \quad
$\bullet$ $|\xi|> 1$
\vskip0.05in \noindent
{\bf Case $|\xi|\le 1$.} In this case, we need to prove \eqref{bilinear-est-s-l} with $Q_1$ replaced by $\chi_{|\xi|\le 1}Q_1$, where $\chi_{|\xi|\le 1}$ is the characteristic function of the region $\{ (\xi,\xi_1,\tau,\tau_1):
|\xi| \le 1 \}$. This is done by applying duality and the Cauchy-Schwarz inequality first in $(\xi_1, \tau_1)$ and then in $(\xi, \tau)$. Doing so, and after some manipulations, the desired estimate takes the form
\begin{align*}
\Big\| \iint_{\mathbb{R}^2}
(\chi_{|\xi|\le 1} Q_1)(\xi,\xi_1,\tau,\tau_1) c_f(\xi-\xi_1,\tau-\tau_1) c_g(\xi_1,\tau_1) d\xi_1 d\tau_1
\Big\|_{L^2_{\xi,\tau}} \le
\| \Theta_1
\|_{L^{\infty}_{\xi_1,\tau_1}}^{1/2}
\|c_f\|_{L^2_{\xi,\tau}}
\|c_g\|_{L^2_{\xi,\tau}}, \end{align*}
where $\Theta_1$ is as in the following lemma, which provides its estimate
\begin{lemma} \label{caseI-1-kdvm} If $0<b<\frac12$, and $0<b'< \frac 12$, then there exists $c>0$ such that for $\xi_1,\tau_1\in\mathbb{R}$
\begin{align*} \Theta_1(\xi_1, \tau_1) \doteq
\frac{1}{(1+|\tau_1-\xi_1^{m}|)^{2b'}}
\int_{|\xi|\le 1} \int_{\mathbb{R}} \frac{\xi^2\,\, d\tau d\xi
}{(1+|\tau-\tau_1-(\xi-\xi_1)^{m}|)^{2b'}(1+|\tau-\xi^{m}|)^{2b}} \, \lesssim 1. \end{align*} \end{lemma}
\noindent The proof is similar to the proof of Lemma 7.1 in \cite{fhy2020} and we omit it.
\vskip0.05in \noindent
{\bf Case $|\xi|>1$.} In this case, by symmetry (in convolution writing), we may assume that
\begin{equation} \label{conv-symmetry-tau}
|\tau-\tau_1-(\xi-\xi_1)^m| \leq
|\tau_1-\xi_1^m|. \end{equation}
Therefore, following \cite{b1993-kdv} and \cite{kpv1996},
to prove the bilinear estimate above we consider the following two microlocalizations:
\noindent
{\bf Microlocalization I. $|\tau_1-\xi_1^m|\leq|\tau-\xi^m|$}. In this case we define the domain $B_I$ to be
\begin{align} \label{def-BI-domain} B_I \doteq \big\{ (\xi,\tau,\xi_1,\tau_1) \in{\mathbb{R}}^4: &
|\tau-\tau_1-(\xi-\xi)^m|\leq|\tau_1-\xi_1^m|\leq|\tau-\xi^m|, \\
&|\xi_1|>1,
|\xi-\xi_1|>1,
|\xi|>1 \big\}. \nonumber \end{align}
\noindent
{\bf Microlocalization II. $|\tau-\xi^m|\leq|\tau_1-\xi_1^m|$}. In this case we define the domain $B_{II}$ to be
\begin{align} \label{def-BII-domain} B_{II} \doteq
\big\{(\xi,\tau,\xi_1,\tau_1)\in{\mathbb{R}}^4: |\tau-\tau_1-(\xi-\xi)^m|\leq|\tau_1-\xi_1^m|, \\
|\tau-\xi^m|\leq|\tau_1-\xi_1^m|,
|\xi_1|>1,
|\xi-\xi_1|>1,
|\xi|>1 \big\}. \nonumber \end{align}
{\bf Proof of bilinear estimate in Microlocalization I:} In this case, $Q_1$ is replaced by the set $\chi_{B_I}Q_1$. Like before, using the Cauchy-Schwarz inequality with respect to $(\xi_1,\tau_1)$ and taking the supremum in $(\xi,\tau)$ we arrive at
\begin{align*}
\Big\|
\iint_{\mathbb{R}^2}
(\chi_{B_I} Q_1)(\xi,\xi_1,\tau,\tau_1)
c_f(\xi-\xi_1,\tau-\tau_1) c_g(\xi_1,\tau_1)
d\xi_1 d\tau_1 \Big\|_{L^2_{\xi, \tau}} \le
\| \Theta_2
\|_{L^{\infty}_{\xi, \tau}}^{1/2}
\|c_f\|_{L^2_{\xi, \tau}} \|c_g\|_{L^2_{\xi, \tau}}. \end{align*}
Thus, to prove our bilinear estimate in microlocalization I, it suffices to show the following result.
\begin{lemma} \label{s-0-case-1-lemma}
If $\frac{6+3m}{12m}\leq b'\leq b<1/2$, then for $|\xi|>1$, and $\tau\in\mathbb{R}$ we have
\begin{align} \label{s-0-case1-theta-1-est-no1} \Theta_2(\xi,\tau) \doteq
\frac{\xi^2}{(1+|\tau-\xi^m|)^{2b}} \iint_{\mathbb{R}^2} \frac{ \chi_{B_I}(\xi,\tau,\xi_1,\tau_1)\,\, d\tau_1 d\xi_1
}{(1+|\tau-\tau_1-(\xi-\xi_1)^m|)^{2b'}(1+|\tau_1-\xi_1^m|)^{2b'}} \lesssim 1. \end{align}
\end{lemma}
\noindent {\bf Proof of bilinear estimate in Microlocalization II:} Using duality and applying the Cauchy-Schwarz inequality twice, first in $(\xi_1, \tau_1)$ and then in $(\xi, \tau)$, we get
\begin{align*}
\Big\| \iint_{\mathbb{R}^2} (\chi_{B_{II}} Q_1)(\xi,\xi_1,\tau,\tau_1) c_f(\xi-\xi_1,\tau-\tau_1) c_g(\xi_1,\tau_1)
d\xi_1 d\tau_1 \Big\|_{L^2_{\xi,\tau}} \le
\| \Theta_{3}
\|_{L^{\infty}_{\xi_1,\tau_1}}^{1/2}
\|c_f\|_{L^2_{\xi,\tau}}\|c_g\|_{L^2_{\xi,\tau}}. \end{align*}
Thus, to prove our bilinear estimate in microlocalization II, it suffices to show the following result.
\begin{lemma} \label{s-0-case-2-lemma} If $\max\{\frac{4+3(m-1)}{12(m-1)},\frac{6+3m}{12m}\}\leq b'\leq b<1/2$, then for $\xi_1,\tau_1\in\mathbb{R}$ we have
\begin{align} \label{s-0-case2-theta-2-est-no1} \Theta_3(\xi_1, \tau_1) \doteq
\frac{1}{(1+|\tau_1-\xi_1^m|)^{2b'}} \iint_{\mathbb{R}^2} \frac{ \chi_{B_{II}}(\xi,\tau,\xi_1,\tau_1) \,\, \xi^2\,
\, d\tau d\xi}{(1+|\tau-\tau_1-(\xi-\xi_1)^m|)^{2b'}(1+|\tau-\xi^m|)^{2b}} \lesssim 1. \end{align} \end{lemma}
\vskip.05in \noindent Next, we shall prove the Lemmas \ref{s-0-case-1-lemma} and \ref{s-0-case-2-lemma}. We begin with the first one.
\noindent
{\bf Proof of Lemma \ref{s-0-case-1-lemma}.} To estimate the quantity $\Theta_{2}(\xi,\tau)$ (and $\Theta_{3}(\xi_1,\tau_1)$ later), we shall need the following calculus estimates, whose proof can be found in \cite{kpv1996}, \cite{h2006} and \cite{fhy2020}.
\begin{lemma} \label{lem:calc_ineq} If $1>\ell>1/2, l'>1/2$ then
\begin{equation} \label{eq:calc_1}
\int_{\mathbb{R}}\frac{dx}{(1+|x-a|)^{2\ell}(1+|x-c|)^{2\ell}}\lesssim\frac{1}{(1+|a-c|)^{2\ell}}, \end{equation}
\begin{equation} \label{eq:calc_2}
\int_{\mathbb{R}}\frac{dx}{(1+|x|)^{2\ell}|{a-x}|^{\frac{1}{2}}}\lesssim\frac{1}{(1+|a|)^{\frac{1}{2}}}, \end{equation}
\begin{equation} \label{eq:calc_3}
\int_{\mathbb{R}}\frac{dx}{(1+|x-a|)^{2(1-\ell)}(1+|x-c|)^{2\ell'}}\lesssim\frac{1}{(1+|a-c|)^{2(1-\ell)}}, \end{equation}
\begin{equation} \label{eq:calc_4}
\int_{|x|\leq c}\frac{dx}{(1+|x|)^{2(1-l)}|\sqrt{a-x}|}\lesssim\frac{(1+c)^{2(\ell-1/2)}}{(1+|a|)^{1/2}}, \end{equation}
and
\begin{equation} \label{eq:calc_1a}
\int_{\mathbb{R}}\frac{dx}{(1+|x-a|)^{2\ell}(1+|x-c|)^{2\ell'}}\lesssim\frac{1}{(1+|a-c|)^{2\min\{\ell',\ell\}}}. \end{equation}
In addition, if $\frac14<\ell'\leq \ell<\frac12$, then
\begin{equation} \label{eq:calc_5}
\int_{\mathbb{R}}\frac{dx}{(1+|x-a|)^{2\ell}(1+|x-c|)^{2\ell'}}\lesssim\frac{1}{(1+|a-c|)^{2\ell+2\ell'-1}}. \end{equation}
\end{lemma} \noindent Also, in this case using the triangle inequality we have
$$
|\tau-(\xi-\xi_1)^m-\xi_1^m| =
|\tau-\tau_1-(\xi-\xi_1)^m+\tau_1-\xi_1^m| \le
|\tau-\tau_1-(\xi-\xi_1)^m|+|\tau_1-\xi_1^m|
\le 2|\tau-\xi^m|. $$
Furthermore, integrating with respect to $\tau_1$ and applying estimate \eqref{eq:calc_5} with $\ell=\ell'=b'$, $a=\tau-(\xi-\xi_1)^m$ and $c=\xi_1^m$, we get
\begin{equation} \label{s-0-case1-theta-1-est-no3} \Theta_2(\xi,\tau) \lesssim
\frac{\xi^2}{(1+|\tau-\xi^m|)^{2b}} \int_{\mathbb{R}} \frac{
d\xi_1
}{(1+|\tau-(\xi-\xi_1)^m-\xi_1^m|)^{4b'-1}} =
\frac{\xi^2}{(1+|\tau-\xi^m|)^{2b}} I(\xi,\tau), \end{equation}
where $I$ is defined in the lemma below, where it is also estimated.
\begin{lemma} \label{l-lem-s-0}
Let $m=2j+1\ge 3$ and $\frac14<b'<1/2$. If $|\xi^m-(\xi-\xi_1)^m-\xi_1^m|\lesssim |\tau-\xi^m|$, then for all $|\xi|>1$ and $\tau\in\mathbb{R}$ we have
\begin{align} \label{I-est-s-0} I(\xi, \tau) \doteq \int_{\mathbb{R}} \frac{ d\xi_1
}{(1+|\tau-(\xi-\xi_1)^m-\xi_1^m|)^{4b'-1}} \lesssim
\frac{|\xi|^{-\frac12(m-2)}(1+|\tau-\xi^m|)^{2-4b'}}{(1+|\tau-2^{1-m}\xi^m|)^{\frac12}}. \end{align}
\end{lemma}
The proof of lemma \ref{l-lem-s-0} is similar to the proof of Lemma 7.3 in \cite{fhy2020}. Combining \eqref{I-est-s-0} with \eqref{s-0-case1-theta-1-est-no3}, we get the desired estimate \eqref{s-0-case1-theta-1-est-no1} for $\Theta_2$
and this completes the proof of Lemma \ref{s-0-case-1-lemma}. \,\, $\square$
\noindent {\bf Proof of Lemma \ref{s-0-case-2-lemma}.}
We recall that
$$ \Theta_3(\xi_1, \tau_1) =
\frac{1}{(1+|\tau_1-\xi_1^{m}|)^{2b'}} \iint_{\mathbb{R}^2} \frac{ \chi_{B_{II}}(\xi,\tau,\xi_1,\tau_1) \,\, \xi^2\,
\, d\tau d\xi}{(1+|\tau-\tau_1-(\xi-\xi_1)^{m}|)^{2b'}(1+|\tau-\xi^{m}|)^{2b}}. $$
As before, using estimate \eqref{eq:calc_5} with $x=\tau$, $\ell=b$, $\ell'=b'$, $a=\xi^m$ and $c=\tau_1-(\xi_1-\xi)^m$, we get
\begin{align} \label{s-0-case2-theta-2-est-no3} \Theta_3(\xi_1,\tau_1) \lesssim&
\frac{1}{(1+|\tau_1-\xi_1^m|)^{2b'}}
\int_{|\xi|>1} \frac{
\xi^2 \,\,d\xi
}{(1+|\tau_1-(\xi_1-\xi)^m-\xi^m|)^{2b+2b'-1}} \notag \\ \le&
\frac{1}{(1+|\tau_1-\xi_1^m|)^{2b'}}
\int_{|\xi|>1} \frac{
\xi^2\,\,
d\xi
}{(1+|\tau_1-\xi_1^m-d_m(\xi_1,\xi)|)^{4b'-1}}, \end{align}
where $d_m(\xi_1,\xi)$ is defined as follows
\begin{align} \label{dm-def-no1} d_m(\xi_1,\xi) \doteq -\xi_1^m +\xi^m -(\xi-\xi_1)^m. \end{align}
In order to show that $\Theta_3$ is bounded,
we need to consider the following two cases:
\vskip0.05in \noindent
$\bullet$ $|\xi|\leq 10|\xi_1|$ \quad and \quad
$\bullet$ $|\xi|> 10|\xi_1|$
\vskip0.05in \noindent
{\bf Case $|\xi|\leq 10|\xi_1|$.} Then,
\begin{align*}
\Theta_3(\xi_1,\tau_1) \lesssim&
\frac{\xi_1^2}{(1+|\tau_1-\xi_1^m|)^{2b'}}
\int_{|\xi|>1} \frac{ d\xi
}{(1+|\tau_1-\xi_1^m-d_m(\xi_1,\xi)|)^{4b'-1}} \lesssim
\frac{\xi_1^2}{(1+|\tau_1-\xi_1^m|)^{2b'}} I(\xi_1,\tau_1), \end{align*}
where $I$ is defined in \eqref{I-est-s-0}. Applying Lemma \ref{l-lem-s-0}, we get
\begin{align} \label{s-0-case1-theta-3-est-no2} \Theta_3(\xi_1,\tau_1) \lesssim&
\frac{|\xi_1|^2}{(1+|\tau_1-\xi_1^m|)^{2b'}}
\frac{|\xi_1|^{-\frac12(m-2)}(1+|\tau_1-\xi_1^m|)^{2-4b'}}{(1+|\tau_1-2^{1-m}\xi_1^m|)^{\frac12}} \nonumber \\ =&
\frac{|\xi_1|^{3-\frac{m}{2}}}{(1+|\tau_1-\xi_1^m|)^{6b'-2}(1+|\tau_1-2^{1-m}\xi_1^m|)^{\frac12}}. \end{align}
Like KdVm on the line (see \cite{fhy2020}), we consider the following two subcases:
\vskip0.05in \noindent $ \bullet \,\,
|\tau_1-\xi_1^m|\leq \frac12|\xi_1|^m $
\quad and \quad
$ \bullet \,\,
|\tau_1-\xi_1^m|> \frac12|\xi_1|^m $
\vskip0.05in \noindent
{\bf Subcase $|\tau_1-\xi_1^m|\leq \frac12|\xi_1|^m$.} Then, using the triangle inequality, we have
$$
|\tau_1-2^{1-m}\xi_1^m| =
|(\tau_1-\xi_1^m)+(1-2^{1-m})\xi_1^m| \geq
\frac34|\xi_1^m| -
|\tau_1-\xi_1^m| \geq
\frac34|\xi_1^m| -
\frac12|\xi_1^m| =
\frac14|\xi_1^m|. $$
Hence, from \eqref{s-0-case1-theta-3-est-no2} we get
$ \Theta_3(\xi_1,\tau_1) \lesssim
\frac{|\xi_1|^{3-\frac{m}{2}}}{1}
\frac{1}{|\xi_1|^{m/2}} =
|\xi_1|^{-m+3}
\overset{|\xi_1|>1} { \lesssim } 1. $
\vskip0.05in
\noindent
{\bf Subcase $|\tau_1-\xi_1^m|> \frac12|\xi_1|^m$.} Then
$
|\tau_1-\xi_1^m| \gtrsim
|\xi_1|^m $
and therefore from \eqref{s-0-case1-theta-3-est-no2} we get
\begin{align*} \Theta_3(\xi_1,\tau_1) \lesssim
\frac{|\xi_1|^{3-\frac{m}{2}}}{|\xi_1^m|^{6b'-2}} \cdot \frac{1}{1} \leq
\frac{|\xi_1|^{3-\frac{m}{2}}} {
|\xi_1|^{m(6b'-2)} } =
\frac{1}{|\xi_1|^{m(6b'-2)-(3-\frac{m}{2})}}. \end{align*}
Since $|\xi_1|> 1$, the above quantity is bounded if
$ m(6b'-2) - (3-\frac{m}{2}) \ge 0, $
which implies that
\begin{align} \label{b-con-1} b' \geq \frac{6+3m}{12m}. \end{align}
This completes the proof of Lemma \ref{s-0-case-2-lemma} in this case.
\vskip0.05in \noindent
{\bf Case $|\xi|> 10|\xi_1|$.} Then, using the triangle inequality $|\tau_1-\xi_1^m|$ is bounded from below as follows
\begin{align} \label{est-tau1}
|\tau_1-\xi_1^m| \geq& \frac13 \Big[
|\tau-\xi^m| +
| (\tau_1-\xi_1^m)
| +
|\tau -\tau_1 -(\xi-\xi_1)^m
| \Big] \notag \\ \gtrsim&
|\tau-\xi^m + (\tau_1-\xi_1^m) + [ \tau -\tau_1 -(\xi-\xi_1)^m ]
| =
|d_m(\xi,\xi_1)|, \end{align}
which can be bounded by the following result.
\begin{lemma} \label{lem-dm} If $m$ is an odd positive integer, then there is a positive constant $c_m$ such that
\begin{align} \label{dm-k}
|d_m(\xi,\xi_1)|
\geq c_m|\xi|^{m-3}|\xi\xi_1(\xi-\xi_1)|, \end{align}
\begin{align} \label{dm-k1}
\text{and}
\quad
|d_m(\xi,\xi_1)|
\geq c_m|\xi_1|^{m-3}|\xi\xi_1(\xi-\xi_1)|. \end{align}
Also if $|\xi|\geq 1$, $|\xi_1|\geq 1$ and $|\xi-\xi_1|\geq 1$, then
\begin{equation} \label{basic-est-2}
|\xi\xi_1(\xi-\xi_1)| \geq \frac13 \xi^2 \quad \text{and} \quad
|\xi\xi_1(\xi-\xi_1)| \geq \frac13\xi_1^2. \end{equation}
\end{lemma}
The proof of Lemma \ref{lem-dm} can be found in \cite{fhy2020}, now we complete the proof in this case. Since $|\xi|>10|\xi_1|$, we have
$
|\xi-\xi_1| \geq
|\xi| -
|\xi_1| \geq
\frac{9}{10}|\xi| $
and
$
|\xi-\xi_1| \leq
|\xi| +
|\xi_1| \geq
\frac{11}{10}|\xi|, $
which gives us that
\begin{equation} \label{est-xi-xi1}
|\xi-\xi_1| \simeq
|\xi|. \end{equation}
Combining estimates \eqref{est-tau1} and \eqref{dm-k} with \eqref{est-xi-xi1}, we get
$
|\tau_1-\xi_1^m| \gtrsim
|\xi|^{m-1}|\xi_1| $
or
$
|\xi| \leq
(|\tau_1-\xi_1^m\|\xi_1|^{-1})^{\frac{1}{m-1}}. $
In addition, using estimate \eqref{s-0-case2-theta-2-est-no3} we have
\begin{align} \label{s-0-case2-theta-3-est-no1b} \Theta_3(\xi_1,\tau_1) \lesssim&
\frac{|\xi_1|^{-\frac{2}{m-1}}|\tau_1-\xi_1^m|^{\frac{2}{m-1}}}{(1+|\tau_1-\xi_1^m|)^{2b'}}
\int_{|\xi|>1} \frac{ d\xi
}{(1+|\tau_1-\xi_1^m-d_m(\xi_1,\xi)|)^{4b'-1}} \\ \leq&
|\xi_1|^{-\frac{2}{m-1}}(1+|\tau_1-\xi_1^m|)^{\frac{2}{m-1}-2b'} I(\xi_1,\tau_1), \nonumber \end{align}
where $I$ is defined in \eqref{I-est-s-0}. Applying Lemma \ref{l-lem-s-0}, we get
\begin{align} \label{s-0-case1-theta-3-est-no2b} \Theta_3(\xi_1,\tau_1) \lesssim&
|\xi_1|^{-\frac{2}{m-1}}(1+|\tau_1-\xi_1^m|)^{\frac{2}{m-1}-2b'}
\frac{|\xi_1|^{-\frac12(m-2)}(1+|\tau_1-\xi_1^m|)^{2-4b'}}{(1+|\tau_1-2^{1-m}\xi_1^m|)^{\frac12}} \notag \\ =&
|\xi_1|^{-\frac{2}{m-1}-\frac{1}{2}(m-2)}
\frac{(1+|\tau_1-\xi_1^m|)^{\frac{2}{m-1}+2-6b'}}{(1+|\tau_1-2^{1-m}\xi_1^m|)^{\frac12}}. \end{align}
Now using the triangle inequality, we can bound $|\tau_1-2^{1-m}\xi_1^m|$ from below, that is
\begin{align*}
|\tau_1-2^{1-m}\xi_1^m| =&
|\tau_1-\xi_1^m+(1-2^{1-m})\xi_1| \geq
|\tau_1-\xi_1^m| -
|\xi_1^m| =
\frac12|\tau_1-\xi_1^m| +
\left(\frac12|\tau_1-\xi_1^m| -
|\xi_1^m| \right) \\ \overset{\eqref{est-tau1}} {\geq}&
\frac12|\tau_1-\xi_1^m| +
\left(\frac12|\xi_1|\xi^{m-1} -
|\xi_1^m| \right)
\overset{|\xi|>10|\xi_1|} { \geq }
\frac12|\tau_1-\xi_1^m|. \end{align*}
Combining the above estimate with \eqref{s-0-case1-theta-3-est-no2b}, we obtain
\begin{align} \label{s-0-case1-theta-3-est-no3b} \Theta_3(\xi_1,\tau_1) \lesssim
|\xi_1|^{-\frac{1}{m-1}-\frac{1}{2}(m-2)}
(1+|\tau_1-\xi_1^m|)^{\frac{2}{m-1}+\frac{3}{2}-6b'}. \end{align}
Since $|\xi_1|>1$, the above quantity is bounded if and only if
\begin{equation} \label{b'-cond-2} b' \geq \frac{4+3(m-1)}{12(m-1)}. \end{equation}
This completes the proof of Lemma \ref{s-0-case-2-lemma}. \,\, $\Box$
\vskip0.1in \noindent {\bf Estimation when the multiplier is $Q_2$.} In this case, applying Cauchy-Schwarz inequality with respect to $\xi_1,\tau_1$, nd taking the super norm over $(\xi,\tau)$ we get
\begin{align} \label{bilinear-est-s-Q2}
\Bigg\| \iint_{\mathbb{R}^2} Q_2(\xi,\xi_1,\tau,\tau_1) c_f(\xi-\xi_1,\tau-\tau_1) c_g(\xi_1,\tau_1)
d\xi_1 d\tau_1 \Bigg\|_{L^2_\xi L^2_{\tau}} \lesssim
\| \Theta_4
\|_{L^{\infty}_{\xi, \tau}}^{\frac12}
\left\|c_f\right\|_{L^2_{\xi} L^2_{\tau}} \left\|c_g\right\|_{L^2_{\xi} L^2_{\tau} }, \end{align}
which shows that the proof of the bilinear estimate follows from the next result.
\begin{lemma} \label{s-0-theta-4-lemma} If $b\ge 0$ and $\alpha'>\frac12$, then for $\xi,\tau\in\mathbb{R}$, we have
\begin{align} \label{s-0-theta-4} \Theta_4(\xi,\tau) \doteq&
\frac{\xi^2}{(1+|\tau-\xi^m|)^{2b}} \iint_{\mathbb{R}^2} \frac{
\chi_{|\xi_1|\leq 1} }{ [
(1+|\tau_1-\xi_1^m|)^{b'} +
(1+|\tau_1|)^{\alpha'} ]^2 } \notag \\ &\frac{
\chi_{|\xi-\xi_1|\leq 1} \quad d\xi_1 d\tau_1 } { [
(1+|\tau-\tau_1-(\xi-\xi_1)^m|)^{b'} +
(1+|\tau-\tau_1|)^{\alpha'} ]^2 } \lesssim 1. \end{align}
\end{lemma}
The proof of this lemma is straightforward. Using the fact that $|\xi_1|$ and $|\xi|$ are bounded and applying estimate \eqref{eq:calc_1} with $\ell=\alpha'$, $x=\tau_1$, $a=0$ and $c=\tau_1$, we get the desired estimate \eqref{s-0-theta-4}.
\noindent {\bf Estimation when the multiplier is $Q_3$.} To prove the estimate \eqref{bilinear-est-s-l} for $Q_3$, we will consider two possible microlocalizations:
\vskip.05in \noindent
{\bf Microlocalization III. $|\tau_1-\xi_1^m|\leq|\tau-\xi^m|$}. In this case we define the domain $B_{III}$ to be
\begin{align} \label{def-BIII-domain} B_{III} \doteq \big\{ (\xi,\tau,\xi_1,\tau_1) \in{\mathbb{R}}^4:
|\tau_1-\xi_1^m|\leq|\tau-\xi^m|, |\xi_1|>1,
|\xi-\xi_1|\leq1 \big\}. \end{align}
\noindent
{\bf Microlocalization IV. $|\tau-\xi^m|\leq|\tau_1-\xi_1^m|$}. In this case we define the domain $B_{IV}$ to be
\begin{align} \label{def-BIV-domain} B_{IV} \doteq \big\{(\xi,\tau,\xi_1,\tau_1)\in{\mathbb{R}}^4:
|\tau-\xi^m|\leq|\tau_1-\xi_1^m|,
|\xi_1|>1,
|\xi-\xi_1|\leq1 \big\}. \end{align}
\noindent {\bf Proof of bilinear estimate in Microlocalization III.} As before, using the Cauchy-Schwarz inequality with respect to $(\xi_1,\tau_1)$ and taking the superemum over $(\xi,\tau)$ we arrive at
\begin{align*}
\Bigg\|
\iint_{\mathbb{R}^2}
\chi_{B_{III}} Q_3(\xi,\xi_1,\tau,\tau_1)
c_f(\xi-\xi_1,\tau-\tau_1) c_g(\xi_1,\tau_1)
d\xi d\tau \Bigg\|_{L^2_\xi L^2_{\tau}} \lesssim
\| \Theta_{5}
\|_{L^{\infty}_{\xi,\tau}}^{1/2}
\|c_f\|_{L^2_{\xi}L^2_{\tau}}
\|c_g\|_{L^2_{\xi}L^2_{\tau}}. \end{align*}
Thus, to prove our bilinear estimate in microlocalization III, it suffices to show the following result.
\begin{lemma} \label{s-0-theta-5-lemma} If $\frac13\leq b'\leq b<\frac12<\alpha'$, then for $\xi,\tau\in\mathbb{R}$ we have
\begin{align} \label{s-0-theta-5} \Theta_5(\xi,\tau) \doteq& \frac{ \xi^2 }{
(1+|\tau-\xi^m|)^{2b} } \iint_{\mathbb{R}^2} \frac{
\chi_{|\xi_1|>1} \chi_{B_{III}}(\xi,\tau,\xi_1,\tau_1) }{
(1+|\tau_1-\xi_1^m|)^{2b'} } \notag \\ &\frac{
\chi_{|\xi-\xi_1|\leq 1} \quad d\xi_1 d\tau_1 } { [
(1+|\tau-\tau_1-(\xi-\xi_1)^m|)^{b'} +
(1+|\tau-\tau_1|)^{\alpha'} ]^2 } \lesssim 1. \end{align}
\end{lemma}
\noindent {\bf Proof of bilinear estimate in Microlocalization IV.} As before, using duality and the Cauchy-Schwarz inequality twice, first in $(\xi_1, \tau_1)$ and then in $(\xi, \tau)$, we get
\begin{align*}
\Bigg\| \iint_{\mathbb{R}^2} \chi_{B_{IV}} Q_3(\xi,\xi_1,\tau,\tau_1) c_f(\xi-\xi_1,\tau-\tau_1) c_g(\xi_1,\tau_1)
d\xi_1 d\tau_1 \Bigg\|_{L^2_\xi L^2_{\tau}} \lesssim
\| \Theta_{6}
\|_{L^{\infty}_{\xi_1,\tau_1}}^{1/2}
\|c_f\|_{L^2_{\xi}L^2_{\tau}}\|c_g\|_{L^2_{\xi}L^2_{\tau}}. \end{align*}
Thus, to prove our bilinear estimate in microlocalization IV, it suffices to show the following result.
\begin{lemma} \label{s-0-theta-6-lemma} If $\frac13\leq b'\leq b<\frac12<\alpha'$, then for $\xi,\tau\in\mathbb{R}$ we have
\begin{align} \label{s-0-theta-6} \Theta_6(\xi_1,\tau_1) \doteq& \frac{
\chi_{|\xi_1|>1} }{
(1+|\tau_1-\xi_1^m|)^{2b'} } \iint_{\mathbb{R}^2} \frac{ \chi_{B_{IV}}(\xi,\tau,\xi_1,\tau_1) \,\, \xi^2 }{
(1+|\tau-\xi^m|)^{2b} } \notag \\ &\frac{
\chi_{|\xi-\xi_1|\leq 1} \quad d\xi d\tau } { [
(1+|\tau-\tau_1-(\xi-\xi_1)^m|)^{b'} +
(1+|\tau-\tau_1|)^{\alpha'} ]^2 } \lesssim 1. \end{align}
\end{lemma}
The proof of Lemma \ref{s-0-theta-6-lemma} is similar to the proof of Lemma \ref{s-0-theta-5-lemma}. So, here we provide only the proof of Lemma \ref{s-0-theta-5-lemma}.
\noindent {\bf Proof of Lemma \ref{s-0-theta-5-lemma}.} Since
$
|\tau-\xi^m| \geq
|\tau_1-\xi_1^m| $,
and using $\alpha'>\frac12>b'$, $|\xi-\xi_1|\le 1$ we get
\begin{align} \label{s-0-theta-5-est-no1} \Theta_5(\xi,\tau) \lesssim \frac{ \xi^2 }{
(1+|\tau-\xi^m|)^{4b+2b'-2} } \iint_{\mathbb{R}^2} \frac{
\chi_{|\xi_1|>1} }{
(1+|\tau_1-\xi_1^m|)^{2-2b} } \frac{
\chi_{|\xi-\xi_1|\leq 1} } {
(1+|\tau-\tau_1|)^{2\alpha'} } d\xi_1 d\tau_1. \end{align}
Now, using estimate \eqref{eq:calc_1a} with $\ell'=\alpha'$, $\ell=1-b$, $x=\tau_1$, $a=\tau$ and $c=\xi_1^m$
\begin{equation} \label{s-0-theta-5-est-no1} \Theta_5(\xi,\tau) \lesssim \frac{ \xi^2 }{
(1+|\tau-\xi^m|)^{4b+2b'-2} } \int_\mathbb{R} \frac{
\chi_{|\xi_1|>1}
\chi_{|\xi-\xi_1|\leq 1} }{
(1+|\tau-\xi_1^m|)^{\min\{2\alpha',2-2b\}} } d\xi_1. \end{equation}
For $|\xi|\le 20$, it is obvious that $Q_5\lesssim 1$. For $|\xi|\ge 20$, using the fact that $\xi_1\simeq \xi$ and making the change of variables $\mu=\xi^m$ for the integral of $d\xi$, for $4b+2b'-2>0$ we also get $Q_5\lesssim 1$. Thus, we complete the proof of Lemma \ref{s-0-theta-5-lemma}. \,\, $\square$
\vskip0.05in \noindent {\bf Case $-j+\frac14<s< 0$.}
We recall that in order to prove the bilinear estimate \eqref{bilinear-est}, it suffices to prove $L^2$ inequality \eqref{bilinear-est-L2-form} with $Q$ is estimated in \eqref{Q-def-no1}. Also, similar to the case $s\ge 0$, by further reduction we get
\begin{align} \label{Q-est-1-neg} Q(\xi,\xi_1,\tau,\tau_1) \lesssim&
\frac{|\xi|}{(1+|\tau-\xi^m|)^{b}} \frac{1}{
(1+|\tau_1-\xi_1^m|)^{b'} +
\chi_{|\xi_1|\leq 1}(1+|\tau_1|)^{\alpha'} } \\ \times&
\frac{(1+|\xi|)^s}{
(1+|\xi_1|)^{s}
(1+|\xi-\xi_1|)^{s} } \cdot \frac{1} {
(1+|\tau-\tau_1-(\xi-\xi_1)^m|)^{b'} +
\chi_{|\xi-\xi_1|\leq 1}(1+|\tau-\tau_1|)^{\alpha'} }. \notag \end{align}
Furthermore, like KdVm on the line \cite{fhy2020} we can restrict our estimations into the set
\begin{equation} \label{set-E-def} E = \left\{ (\xi,\xi_1,\tau,\tau_1)\in \mathbb{R}^4:
|\xi_1-\xi| > 1 \,\, \text{and} \,\,
|\xi_1| > 1 \right\}. \end{equation}
Hence, the $\alpha$ terms in the denominators of $Q$ can be dropped and and the quantity $Q$ given in \eqref{Q-est-1-neg} becomes
\begin{align} \label{Q1-modify-no1-s-neg} Q_4(\xi,\xi_1,\tau,\tau_1) \lesssim
\frac{|\xi|(1+|\xi|)^s|\xi_1(\xi-\xi_1)|^{-s}}{(1+|\tau-\xi^{m}|)^{b}} \frac{1}{
(1+|\tau_1-\xi_1^{m}|)^{b'}} \frac{1} {
(1+|\tau-\tau_1-(\xi-\xi_1)^{m}|)^{b'} }. \end{align}
Moreover, by symmetry (in convolution writing), we may assume that
\begin{equation} \label{conv-symmetry-tau-s-neg}
|\tau-\tau_1-(\xi-\xi_1)^{m}| \leq
|\tau_1-\xi_1^{m}|. \end{equation}
Finally, following \cite{b1993-kdv}, \cite{kpv1996} and \cite{fhy2020}, in order to prove \eqref{bilinear-est-L2-form} we distinguish two cases (microlocalization):
\noindent
{\bf Microlocalization I. $|\tau_1-\xi_1^{m}|\leq|\tau-\xi^{m}|$}. In this case we define the domain $E_I$ to be
\begin{align} \label{def-EI-domain} E_I \doteq \big\{ (\xi,\tau,\xi_1,\tau_1) \in{\mathbb{R}}^4: &
|\tau-\tau_1-(\xi-\xi)^{m}|\leq|\tau_1-\xi_1^{m}|\leq|\tau-\xi^{m}|,
|\xi_1|>1,
|\xi-\xi_1|>1 \big\}. \end{align}
{\bf Microlocalization II. $|\tau-\xi^{m}|\leq|\tau_1-\xi_1^{m}|$}. In this case we define the domain $E_{II}$ to be
\begin{align} \label{def-EII-domain} E_{II} \doteq
\big\{(\xi,\tau,\xi_1,\tau_1)\in{\mathbb{R}}^4: |\tau-\tau_1-(\xi-\xi)^{m}|\leq|\tau_1-\xi_1^{m}|, \notag \\
|\tau-\xi^{m}|\leq|\tau_1-\xi_1^{m}|,
|\xi_1|>1,
|\xi-\xi_1|>1 \big\}. \end{align}
{\bf Proof of bilinear estimate in Microlocalization I.} Here $Q$ is replaced with the $\chi_{E_I}Q$ and our $L^2$ inequality \eqref{bilinear-est-L2-form} reads as
\begin{align} \label{bilinear-est-neg-EI}
\Bigg\|
\iint_{\mathbb{R}^2}
(\chi_{E_I} Q_4)(\xi,\xi_1,\tau,\tau_1)
c_f(\xi-\xi_1,\tau-\tau_1) c_g(\xi_1,\tau_1)
d\xi_1 d\tau_1 \Bigg\|_{L^2_\xi L^2_{\tau}} \lesssim
\left\|c_f\right\|_{L^2_{\xi} L^2_{\tau}} \left\|c_g\right\|_{L^2_{\xi} L^2_{\tau}}. \end{align}
As before using the Cauchy-Schwarz inequality with respect to $(\xi_1,\tau_1)$ and taking the supremum in $(\xi,\tau)$ we get
\begin{align*}
\Bigg\| \iint_{\mathbb{R}^2}
(\chi_{E_I} Q_4)(\xi,\xi_1,\tau,\tau_1)
c_f(\xi-\xi_1,\tau-\tau_1) c_g(\xi_1,\tau_1)
d\xi_1 d\tau_1 \Bigg\|_{L^2_\xi L^2_{\tau}} \lesssim
\| \Theta_I
\|_{L^{\infty}_{\xi, \tau}}^{1/2}
\|c_f\|_{L^2_{\xi} L^2_{\tau}}
\|c_g\|_{L^2_{\xi} L^2_{\tau}}, \end{align*}
where $\Theta_I$ is defined and estimated in the following lemma.
\begin{lemma} \label{s-neg-case-1-lemma} If $\max\{\frac12 - \frac{s-(-j+\frac14)}{12j},\frac{5}{12},\frac{-2s+2+2j}{12j}\}\leq b'<\frac12$ and $-j+\frac14<s<0$, then for $\xi,\tau\in\mathbb{R}$
\begin{align} \label{s-neg-case1-theta-est} \Theta_I(\xi,\tau) \doteq&
\frac{\xi^2(1+|\xi|)^{2s}}{(1+|\tau-\xi^{m}|)^{2b}} \iint_{\mathbb{R}^2}
\frac{ \chi_{E_I}(\xi,\tau,\xi_1,\tau_1)|\xi_1(\xi-\xi_1)|^{-2s}\,\,\,
\quad d\tau_1 d\xi_1
}{(1+|\tau-\tau_1-(\xi-\xi_1)^{m}|)^{2b'}(1+|\tau_1-\xi_1^{m}|)^{2b'}} \lesssim 1. \end{align}
\end{lemma}
\noindent The proof of Lemma \ref{s-neg-case-1-lemma} is omited since it is similar to the proof of Lemma 7.4 in \cite{fhy2020}. In fact, if we choose $\frac12-\frac14\beta_1 \le b' \le b < \frac12$, where $\beta_1=\beta$, which is defined in Theorem 2.1 in \cite{fhy2020}, then Lemma \ref{s-neg-case-1-lemma} is reduced to the Lemma 7.4 in \cite{fhy2020}.
\noindent {\bf Proof of bilinear estimate in Microlocalization II.} Using duality and Cauchy-Schwarz inequality twice, first in $(\xi_1, \tau_1)$ and then in $(\xi, \tau)$, as before, we have
\begin{align*}
\Bigg\| \iint_{\mathbb{R}^2}
(\chi_{E_{II}} Q_4)(\xi,\xi_1,\tau,\tau_1)
c_f(\xi-\xi_1,\tau-\tau_1) c_g(\xi_1,\tau_1)
d\xi_1 d\tau_1 \Bigg\|_{L^2_\xi L^2_{\tau}} \lesssim
\| \Theta_{II}
\|_{L^{\infty}_{\xi_1,\tau_1}}^{1/2}
\|c_f\|_{L^2_{\xi}L^2_{\tau}}
\|c_g\|_{L^2_{\xi}L^2_{\tau}} \cdot \end{align*}
where $\Theta_{II}$ is defined and estimated in the following lemma.
\begin{lemma} \label{s-neg-case-2-lemma} If $\max\{\frac12 - \frac{1}{12j+6},\frac{5}{12},\frac{-4s+m+3}{6m}\}\leq b'\leq b<1/2$ and $-j+\frac14<s<0$, then for $\xi_1,\tau_1\in\mathbb{R}$
\begin{align} \label{s-neg-case2-theta-est} \Theta_{II}(\xi_1, \tau_1) \doteq&
\frac{1}{(1+|\tau_1-\xi_1^{m}|)^{2b'}} \iint_{\mathbb{R}^2} \frac{ \chi_{E_{II}}(\xi,\tau,\xi_1,\tau_1) \,\, \xi^2
(1+|\xi|)^{2s}
|\xi_1(\xi-\xi_1)|^{-2s} \,
\, d\tau d\xi}{(1+|\tau-\tau_1-(\xi-\xi_1)^{m}|)^{2b'}(1+|\tau-\xi^{m}|)^{2b}} \lesssim 1. \end{align}
\end{lemma}
\noindent {\bf Proof of Lemma \ref{s-neg-case-2-lemma}.} Since $0<b'<b<\frac12$, by applying calculus estimate
\eqref{eq:calc_5} with $\ell=b$, $\ell'=b'$ $\alpha=\tau_1+(\xi-\xi_1)^m$, $\beta=\xi^m$ and $x=\tau$, we get
\begin{align} \label{theta2-est-neg} \Theta_{II}(\xi_1,\tau_1) \le&
\frac{1}{(1+|\tau_1-\xi_1^m|)^{2b'}} \int_\mathbb{R}
\frac{|\xi|^2(1+|\xi|)^{2s}(\xi_1|\xi-\xi_1|)^{-2s}}{(1+|\tau_1+(\xi-\xi_1)^{m}-\xi^m|)^{2b+2b'-1}}d\xi \nonumber \\ =&
\frac{1}{(1+|\tau_1-\xi_1^m|)^{2b'}} \int_\mathbb{R}
\frac{|\xi|^2(1+|\xi|)^{2s}(\xi_1|\xi-\xi_1|)^{-2s}}{(1+|\tau_1-\xi_1^m+(\xi-\xi_1)^{m}-\xi^m+\xi_1^m|)^{2b+2b'-1}}d\xi \nonumber \\ =&
\frac{1}{(1+|\tau_1-\xi_1^m|)^{2b'}} \int_\mathbb{R}
\frac{|\xi|^2(1+|\xi|)^{2s}(\xi_1|\xi-\xi_1|)^{-2s}}{(1+|\tau_1-\xi_1^m+d_m(\xi,\xi_1)|)^{2b+2b'-1}}d\xi. \end{align}
Then, we complete the proof by following argument similar to those used in the proof of Lemma 7.5 in \cite{fhy2020}. \,\, $\square$
\section{Proof of Bilinear estimates in temporal $Y^{s,b}$ spaces}
In this section we prove Theorem \ref{bi-est-Y-thm}, that are the bilinear estimates in the spaces $Y^{s,b}$. These appears in the basic linear
estimate via the time estimate of the forced ivp with zero data, i.e. estimate \eqref{forced-ivp-te}. Since the proof of estimates \eqref{bi-est-Y-1} is similar to that of estimate \eqref{bi-est-Y} and for $m=3$ estimate \eqref{bi-est-Y} is proved in \cite{h2006}. Here we only provide an outline of the proof for estimate \eqref{bi-est-Y} with $s\ge 0$. For $s\ge 0$ we have the following inequality
\begin{align} \label{Y-est-1}
\| w_{fg}
\|_{Y^{s,-b}}^2 \lesssim \iint_{\mathbb{R}^2}
\chi_{|\tau|>10^m|\xi|^m}
(1+|\tau|)^{\frac{2s}{m}}
(1+|\tau-\xi^m|)^{-2b}
|\widehat{w}_{fg}(\xi,\tau)|^2 d\xi d\tau +
\|w_{fg}\|_{X^{s,-b}}^2, \end{align}
where $w_{fg}=\partial_x(f\cdot g)$.
So, to prove the ``temporal" bilinear estimate \eqref{bi-est-Y} it suffices to show that
\begin{align} \label{bi-est-Y-reduced} \left( \iint_{\mathbb{R}^2}
\chi_{|\tau|>10^m|\xi|^m}
(1+|\tau|)^{\frac{2s}{m}}
(1+|\tau-\xi^m|)^{-2b}
|\widehat{w}_{fg}(\xi,\tau)|^2 d\xi d\tau \right)^{1/2} \lesssim
\|f\|_{X^{s,b',\alpha'}}
\|g\|_{X^{s,b',\alpha'}}. \end{align}
Like before, writting the $\|\cdot\|_{s,b,\alpha'}$-norm of $h$ as the $L^2$ norm of $c_h$, that is
$
\|h\|_{s,b',\alpha'}\simeq\|c_h(\xi,\tau)\|_{L^2_{\xi,\tau}}, $
where $c_h$ is defined in \eqref{eq:c_h}, the estimate \eqref{bi-est-Y-reduced} reads as follows
\begin{align} \label{bilinear-est-c-notation} \left( \iint_{\mathbb{R}^2}
\chi_{|\tau|>10^m|\xi|^m}
(1+|\tau|)^{\frac{2s}{m}}
(1+|\tau-\xi^m|)^{-2b}
|\widehat{w}_{fg}(\xi,\tau)|^2 d\xi d\tau \right)^{1/2} \lesssim
\|c_f \|_{L^2_{\xi,\tau}}
\|c_g \|_{L^2_{\xi,\tau}}. \end{align}
Next, expressing
$ \widehat{w}_{fg}(\xi,\tau) \simeq \xi \iint_{\mathbb{R}^2} \widehat{f}(\xi-\xi_1,\tau-\tau_1)\widehat{g}(\xi_1,\tau_1)d\xi_1 d\tau_1 $
in terms of $c_f$ and $c_g$ we see that the inequality \eqref{bilinear-est-c-notation} takes the following $L^2$ formulation
\begin{align} \label{bilinear-est-L2-form-Y}
\Big\| \iint_{\mathbb{R}^2}
Q(\xi,\xi_1,\tau,\tau_1)
c_f(\xi-\xi_1,\tau-\tau_1) c_g(\xi_1,\tau_1)
d\xi_1 d\tau_1 \Big\|_{L^2_{\xi,\tau}} \lesssim
\|c_f\|_{L^2_{\xi,\tau}} \|c_g\|_{L^2_{\xi,\tau}}, \end{align}
where
\begin{subequations} \label{Q-def-Y} \begin{align} \label{Q-def-term-a-Y} Q(\xi,\xi_1,\tau,\tau_1) \doteq&
\chi_{|\tau|>10^m|\xi|^m}
\frac{|\xi|} {
(1+|\tau-\xi^m|)^{b} } \times
(1+|\tau|)^{\frac{s}{m}} \\ \label{Q-def-term-b-Y} \times&\frac{1}{
(1+|\xi_1|)^{s}
(1+|\tau_1-\xi_1^m|)^{b'} +
\chi_{|\xi_1|\leq 1}(1+|\tau_1|)^{\alpha'} } \\ \label{Q-def-term-c-Y} \times& \frac{1}{
(1+|\xi-\xi_1|)^{s}
(1+|\tau-\tau_1-(\xi-\xi_1)^m|)^{b'} +
\chi_{|\xi-\xi_1|\leq 1}(1+|\tau-\tau_1|)^{\alpha'} }. \end{align}
\end{subequations}
\noindent In additon, like the bilinear estimate in the space $X^{s,b,\alpha}$, collecting all the factors with $s$ power, making further reduction,
and observing that if $|\tau|\le 10^m|\xi_1|^m$, then
$$
\frac{(1+|\tau|)^{s/m}}{
(1+|\xi_1|)^{s}(1+|\xi-\xi_1|)^{s} } \lesssim
\frac{(1+10^m|\xi_1|^m)^{s/m}}{
(1+|\xi_1|)^{s}(1+|\xi-\xi_1|)^{s} } \lesssim 1, $$
which implies
$ Q(\xi,\xi_1,\tau,\tau_1) \lesssim Q_0(\xi,\xi_1,\tau,\tau_1), $
where $Q_0$ is given by \eqref{Q-def-s=0},
we reduce bilinear estimate \eqref{bilinear-est-L2-form-Y} to the bilinear estimate in $X^{s,b,\alpha}$ with $s=0$, i.e. estimate \eqref{bilinear-est-s-0}.
Thus, we assume
$
|\tau|>10^m|\xi_1|^m $
and $Q(\xi,\xi_1,\tau,\tau_1)$ becomes $Q_1(\xi,\xi_1,\tau,\tau_1)$, which is given by
\begin{align} \label{Q1-def} Q_1(\xi,\xi_1,\tau,\tau_1) \doteq&
\frac{\chi_{|\tau|>10^m|\xi|^m} |\xi|}{(1+|\tau-\xi^{m}|)^{b}}
\frac{\chi_{|\tau|>10^m|\xi_1|^m}(1+|\tau|)^{s/m}}{
(1+|\xi_1|)^{s}(1+|\xi-\xi_1|)^{s} } \\ \cdot& \frac{1} {
(1+|\tau-\tau_1-(\xi-\xi_1)^{m}|)^{b'}
(1+|\tau_1-\xi_1^{m}|)^{b'} }. \nonumber \end{align}
Now, in order to prove estimate \eqref{bilinear-est-L2-form-Y}, it suffices to show that
\begin{align} \label{bilinear-est-L2-form-Q1}
\Big\|
\iint_{\mathbb{R}^2}
Q_1(\xi,\xi_1,\tau,\tau_1)
c_f(\xi-\xi_1,\tau-\tau_1) c_g(\xi_1,\tau_1)
d\xi_1 d\tau_1 \Big\|_{L^2_{\xi,\tau}} \lesssim
\|c_f\|_{L^2_{\xi,\tau}} \|c_g\|_{L^2_{\xi,\tau}}. \end{align}
To show this, like before using the Cauchy-Schwarz inequality with respect to $(\xi_1,\tau_1)$ and taking the supremum over $(\xi,\tau)$ we get
\begin{align} \label{bilinear-est-Q1-Y}
\Big\| \iint_{\mathbb{R}^2} Q_1(\xi,\xi_1,\tau,\tau_1) c_f(\xi \hskip-0.03in - \hskip-0.03in \xi_1,\tau \hskip-0.03in - \hskip-0.03in \tau_1) c_g(\xi_1,\tau_1) d\xi_1 d\tau_1
\Big\|_{L^2_{\xi,\tau}} \le
\| \Theta_1
\|_{L^{\infty}_{\xi,\tau}}^{1/2}
\|c_f\|_{L^2_{\xi,\tau}}
\|c_g\|_{L^2_{\xi,\tau}} , \end{align}
where $\Theta_1$ is defined and estimated in the following result:
\begin{lemma} \label{lem: for est-Q1-Y} If $\textcolor{red}{0}\le s< m$ and $\max\{\frac{2s+m}{6m},\frac{m+2}{6m}\}\le b'<b<\frac12$ satisfy , then we have
\begin{align} \label{ine: for est-Q1-Y} \Theta_1(\xi,\tau) \doteq&
\frac{\chi_{|\tau|>10^m|\xi|^m} |\xi|^2(1+|\tau|)^{2s/m}}{(1+|\tau-\xi^{m}|)^{2b}} \iint_{\mathbb{R}^2}
\frac{\chi_{|\tau|>10^m|\xi_1|^m}}{
(1+|\xi_1|)^{2s}(1+|\xi-\xi_1|)^{2s} } \\ & \cdot \frac{1} {
(1+|\tau-\tau_1-(\xi-\xi_1)^{m}|)^{2b'}
(1+|\tau_1-\xi_1^{m}|)^{2b'} } \xi_1 d\tau_1 \lesssim 1. \notag \end{align} \end{lemma}
\noindent {\bf Proof of Lemma \ref{lem: for est-Q1-Y}.} Applying calculus estimate \eqref{eq:calc_5} with $\ell=\ell'=b'$, $a=\tau-(\xi-\xi_1)^m$, $c=\xi_1^m$ and $x=\tau_1$ we arrive at the following estimate
\begin{align} \label{theta1-est-0} \Theta_1(\xi,\tau) \le&
\frac{\chi_{|\tau|>10^m|\xi|^m} |\xi|^2(1+|\tau|)^{2s/m}}{(1+|\tau-\xi^{m}|)^{2b}} \\ &\int_{\mathbb{R}}
\frac{\chi_{|\tau|>10^m|\xi_1|^m}}{
(1+|\xi_1|)^{2s}(1+|\xi-\xi_1|)^{2s} } \cdot \frac{1} {
(1+|\tau-(\xi-\xi_1)^{m}-\xi_1^m|)^{4b'-1} } d\xi_1. \nonumber \end{align}
Then, for $\textcolor{red}{0}\le s< m$ and $\max\{\frac{2s+m}{6m},\frac{m+2}{6m}\}\le b'<b<\frac12$, using $|\tau|>10^m|\xi|^m$ and $ |\tau|>10^m|\xi_1|^m$ we get the desired estimate \eqref{ine: for est-Q1-Y}. Here we omit the detail of the proof. \,\, $\square$
\section{Well-posedness in modified Bourgain spaces -- Proof of Theorem \ref{thm-kdvm-half-line} }
We only prove well-posedness for $-1\le s< \frac12$. The proof for $s\in(-j+\frac14,-1)\cup (\frac12,j+1)$, $s\neq \frac32, \frac52, \dots,j-\frac12$, is similar. Also, we assume that
$$ 0 < T < 1/2. $$
\noindent
{\bf Small data.} First, we prove Theorem \ref{thm-kdvm-half-line} for initial and boundary data such that
\begin{equation} \label{smallness}
\|u_0\|_{H^s(\mathbb{R}^+)} +
\sum\limits_{\ell=0}^{j-1}\|g_\ell\|_{H^{\frac{s+j-\ell}m}(0,T)} \le \frac{1}{144C^2}, \,\, \text{with} \,\, C=c_{s,b,\alpha}+\frac 12 c_{s,b,\alpha}^2, \end{equation}
where $c_{s,b,\alpha}$ is the constant appearing in the estimate \eqref{forced-linear-kdvm-est} and the bilinear estimates \eqref{bilinear-est}. Under the above smallness condition \eqref{smallness}, we prove that the the integral equation
\begin{equation} \label{iteration-map} u = \Phi u \doteq S \Big[ u_0,g_0,\dots,g_{j-1};-\frac12\partial_x(u^2) \Big], \end{equation}
has a unique solution in the space $X^{s,b,\alpha}(\mathbb{R}^+\times (0,T))$. For this, we shall prove that the iteration map $\Phi$ has a fixed point in $X^{s,b,\alpha}(\mathbb{R}^+\times (0,T))$. In fact, for any $u$ in the (closed) ball
\begin{equation} \label{smallness-ball}
B=\Big\{u\in X^{s,b,\alpha}(\mathbb{R}^+\times(0,T)):\|u\|_{X^{s,b,\alpha}(\mathbb{R}^+\times(0,T))} \le \frac{1}{24C}\Big\}, \end{equation}
using linear estimate \eqref{forced-linear-kdvm-est} with forcing replaced by $-\frac12\partial_x(u^2)$ and bilinear estimates {\eqref{bilinear-est} we get
\begin{align*}
\|\Phi u\|_{X^{s,b,\alpha}(\mathbb{R}^+\times (0,T))} \le & c_{s,b,\alpha}
\Big( \|u_0\|_{H^s(\mathbb{R}^+)} +\sum\limits_{\ell=0}^{j-1} \|g_\ell\|_{H^{\frac{s+j-\ell}{m}}(0,T)} + \frac{1}2
\|\partial_x(u^2)\|_{X^{s,-b,\alpha-1}(\mathbb{R}^+\times(0,T))} \Big) \\ \le& c_{s,b,\alpha}
\Big( \|u_0\|_{H^s(\mathbb{R}^+)} +
\sum\limits_{\ell=0}^{j-1}\|g_\ell\|_{H^{\frac{s+j-\ell}{m}}(0,T)} + \frac{1}2
\|\partial_x(\tilde u^2)\|_{X^{s,-b,\alpha-1}(\mathbb{R}^2)} \Big) \\ \overset{\eqref{bilinear-est}}{\le}&
C \Big( \|u_0\|_{H^s(\mathbb{R}^+)} +
\sum\limits_{\ell=0}^{j-1}\|g_\ell\|_{H^{\frac{s+j-\ell}{m}}(0,T)} +
\|\tilde u\|^2_{X^{s,b,\alpha}(\mathbb{R}^2)}\Big), \end{align*}
where $\tilde u$ is an extension of $u$ from $\mathbb{R}^+\times(0,T)$ to $\mathbb{R}^2$ such that
\begin{equation} \label{u-extension}
\|\tilde u\|_{X^{s,-b,\alpha-1}(\mathbb{R}^2)} \le 2
\|u\|_{X^{s,b,\alpha}(\mathbb{R}^+\times(0,T))}. \end{equation}
Furthermore, using estimate \eqref{u-extension} we get
\begin{align} \label{contranction-est}
\|\Phi u\|_{X^{s,b,\alpha}(\mathbb{R}^+\times (0,T))} \le&
C \Big( \|u_0\|_{H^s(\mathbb{R}^+)} +
\sum\limits_{\ell=0}^{j-1}\|g_\ell\|_{H^{\frac{s+j-\ell}{m}}(0,T)}
+ 4\|u\|^2_{X^{s,b,\alpha}(\mathbb{R}^+\times(0,T))}\Big). \end{align}
And, since $u\in B$, we have
$
\|\Phi u\|_{X^{s,b,\alpha}(\mathbb{R}^+\times (0,T))} \le C \Big( \frac{1}{144C^2}+\frac{1}{144C^2} \Big) \le \frac{1}{24C}. $
Thus $\Phi$ maps the ball $B$ into itself.
To show that $\Phi$ is a contraction, for any $u,v\in B$, using linear estimate \eqref{forced-linear-kdvm-est} with forcing replaced by $-\frac12\partial_x(u^2-v^2)$ we get
\begin{align*}
\|\Phi u-\Phi v\|_{X^{s,b,\alpha}(\mathbb{R}^+\times\mathbb{R})} \le& \frac{c_{s,b,\alpha}}2
\|\partial_x(u^2-v^2)\|_{X^{s,-b,\alpha-1}(\mathbb{R}^+\times\mathbb{R})} \\ \le& \frac{c_{s,b,\alpha}}2
\|\partial_x(\tilde u^2-\tilde v^2)\|_{X^{s,-b,\alpha-1}(\mathbb{R}^2)}, \end{align*}
where $\tilde u$ is the extension of $u$ from $\mathbb{R}^+\times(0,T)$ to $\mathbb{R}^2$ satisfying \eqref{u-extension}.
The extension of $v$ is obtained as follows. First, we extend $w=v-u$ from $\mathbb{R}^+\times(0,T)$ to $\mathbb{R}^2$ such that
\begin{equation} \label{v-extension}
\|\tilde w\|_{X^{s,b,\alpha}(\mathbb{R}^2)} \le 2
\|v-u\|_{X^{s,b,\alpha}(\mathbb{R}^+\times(0,T))}. \end{equation}
Then defining $\tilde v\doteq \tilde w+\tilde u$, we see that $\tilde v$ extends $v$ from $\mathbb{R}^+\times(0,T)$ to $\mathbb{R}^2$. In addition, using the triangle inequality, we get
\begin{align} \label{v1-bound}
\|\tilde v\|_{X^{s,b,\alpha}(\mathbb{R}^2)} \le& 2
\|v-u\|_{X^{s,b,\alpha}(\mathbb{R}^+\times(0,T))} + 2
\|u\|_{X^{s,b,\alpha}(\mathbb{R}^+\times(0,T))} \le \frac{1}{4C}. \end{align}
Combining estimate \eqref{u-extension} and estimate \eqref{v-extension} with bilinear estimate \eqref{bilinear-est} again, we get
\begin{align} \label{contraction-small-est-2}
\|\Phi u-\Phi v\|_{X^{s,b,\alpha}(\mathbb{R}^+\times(0,T))} \le&
C\left(\|\tilde u\|_{s,b,\alpha}+\|\tilde v\|_{s,b,\alpha}\right)\cdot\|\tilde u-\tilde v\|_{s,b,\alpha} \\ \le&
C\cdot\frac{4}{12C}\cdot\|\tilde u-\tilde v\|_{s,b,\alpha} \le
\frac{2}{3}\|u-v\|_{X^{s,b,\alpha}(\mathbb{R}^+\times(0,T))}, \nonumber \end{align}
which shows that $\Phi$ is a contraction on $B$. Since $B$ is a complete Banach space, by the contraction mapping theorem there is a unique $u\in B$ such that $\Phi u=u$.
\vskip0.05in \noindent {\it Lip-continuous dependence on data.} Let $u_0(x)$, $g_\ell(t)$ and $v_0(x)$, $h_\ell(t)$ be two sets of data satisfying the smallness condition \eqref{smallness}. If $u$ is the solution that corresponds to
$u_0(x)$, $g_\ell(t)$, which we denote by
$ u(x, t) = \psi(t) S \Big[ u_0,g_0,\dots,g_{j-1};-\frac12\partial_x(u^2) \Big], $
and $v$ is the solution that corresponds to $v_0(x)$, $h_\ell(t)$, that is
$ v(x, t) = \psi(t) S \Big[ v_0,h_0,\dots,h_{j-1};-\frac12\partial_x(u^2) \Big] $
then
\begin{equation} \label{u-v-eqn} u(x, t)-v(x, t) = \psi(t) S \Big[ u_0-v_0,g_0-h_0,\dots,g_{j-1}-h_{j-1};-\frac12\partial_x(u^2-v^2) \Big]. \end{equation}
Using linear estimate \eqref{forced-linear-kdvm-est}, extensions $\tilde u$, $\tilde v$ (as above), and bilinear estimate
\eqref{bilinear-est}, we have
\begin{align*}
\|u-v\|_{X^{s,b,\alpha}(\mathbb{R}^+\times(0,T))} &\le C \big(
\|u_0-v_0\|_{H^s(\mathbb{R}^+)} + \sum\limits_{\ell=0}^{j-1}
\|g_\ell-h_\ell\|_{H^{\frac{s+j-\ell}m}(0,T)} \big) +
C \|\tilde u+\tilde v\|_{s,b,\alpha} \|\tilde u-\tilde v\|_{s,b,\alpha} \\ &\le C\big(
\|u_0-v_0\|_{H^s(\mathbb{R}^+)} + \sum\limits_{\ell=0}^{j-1}
\|g_\ell-h_\ell\|_{H^{\frac{s+j-\ell}m}(0,T)} \big) +
\frac{2}{3}\|u-v\|_{X^{s,b,\alpha}(\mathbb{R}^+\times(0,T))}. \end{align*}
Moving all $ \|u-v\|_{X^{s,b,\alpha}(\mathbb{R}^+\times(0,T))}$ to the lhs gives
\begin{equation} \label{lip-small-est}
\|u-v\|_{X^{s,b,\alpha}(\mathbb{R}^+\times(0,T))} \le 3C \big(
\|u_0-v_0\|_{H^s(\mathbb{R}^+)} + \sum\limits_{\ell=0}^{j-1}
\|g_\ell-h_\ell\|_{H^{\frac{s+j-\ell}m}(0,T)} \big), \end{equation}
which completes the proof of Lip-continuous dependence on data.
\noindent {\bf Large data.}
For any size initial data $u_0\in H^s$, boundary data $g$, and for $T^*$ such that
\begin{equation} \label{T-less-than-1/2} 0<T^*\le T<1/2, \end{equation}
we replace the integral equation \eqref{iteration-map} with its following localization
\begin{align} \label{iteration-map-T-loc} u(x, t) = \Phi_{T^*} u \doteq S \Big[ u_0,g_0,\dots,g_{j-1}; -\frac12\partial_x(\psi_{2T^*}\cdot u^2) \Big],
\quad |t|\le T^*, \end{align}
where $\psi_{T^*}(t)=\psi(t/{T^*})$ with $\psi(t)$ being our familiar cutoff function in $\in C_0^\infty(-1, 1)$ with
$0\le \psi(t)\le 1$, $\psi(t)=1$ for $|t|\le 1/2$.
First, we notice that for $|t|\le {T^*}$, the fixed point of the iteration map \eqref{iteration-map-T-loc} is the solution to the KdVm ibvp \eqref{KdVm}. Thus, $\Phi_{T^*}(u)=\Phi(u)$ if
$|t|\le {T^*}$,
i.e. when $|t|\le {T^*}$, then $\Phi_{T^*}(u)$ becomes the iteration map \eqref{iteration-map} .
Next, we shall choose appropriate $ {T^*}$ and use the contraction mapping theorem to show that there is a fixed point of the iteration map \eqref{iteration-map-T-loc} in the ball $B(r)\subseteq X^{s,b,\alpha}(\mathbb{R}^+\times(0,T))$.
In fact, using the linear estimate \eqref{forced-linear-kdvm-est} with forcing replaced by $-\frac12\partial_x(u^2)$, for $b<b_1$, which are given below in \eqref{b-b'-choice-wp}, we get
\begin{align} \label{onto-map-fin-est-large}
&\|\Phi_ {T^*}(u)\|_{X^{s,b,\alpha}(\mathbb{R}^+\times(0,T))} \le
\|\Phi_ {T^*}(u)\|_{X^{s,b_1,\alpha}(\mathbb{R}^+\times(0,T))} \\ \le& c_{s,b,\alpha} \Big(
\| u_0
\|_{H^s(\mathbb{R}^+)} +
\sum\limits_{\ell=0}^{j-1}\|g_\ell\|_{H_t^\frac{s+j-\ell}{m}(0,T)} + \frac{1}2
\left\| \psi_{2{T^*}}(t) \partial_x(u^2(t))
\right\|_{X^{s,-b_1,\alpha-1}(\mathbb{R}^+\times(0,T))} \Big) \nonumber \\ \le& c_{s,b,\alpha} \Big(
\| u_0
\|_{H^s(\mathbb{R}^+)} +
\sum\limits_{\ell=0}^{j-1}\|g_\ell\|_{H_t^\frac{s+j-\ell}{m}(0,T)} + \frac{1}2
\left\| \psi_{2{T^*}}(t) \partial_x(\tilde u^2(t))
\right\|_{s,-b_1,\alpha-1} \Big) , \nonumber \end{align}
where $\tilde{u}$ is the extension of $u$ from $\mathbb{R}^+\times(0,T)$ to $\mathbb{R}^2$, which satisfies \eqref{u-extension}.
Now, we estimate the $\|\cdot\|_{s,-b_1,\alpha-1}$. For this we shall need the following result.
\begin{lemma} \label{tao-lemma-modified} Let $\eta(t)$ be a function in the Schwartz space $\mathcal{S}(\mathbb{R})$. If
$ -\frac12 < b' \le b < \frac12 $
and
$ -\frac12 < \alpha' - 1 \le \alpha - 1 < \frac12 $
( $ \frac12 < \alpha' \le \alpha < 1 $ is sufficient condition)
then for any $0<{T^*}\le 1$ we have
\begin{equation} \label{tao-est-modified}
\|\eta(t/{T^*})u\|_{X^{s,b',\alpha'-1}} \le c_1(\eta,b,b',\alpha,\alpha') \,\, \max\{ {T^*}^{b-b'}, {T^*}^{\alpha-\alpha'} \}
\|u\|_{X^{s,b,\alpha-1}}. \end{equation} \end{lemma}
\noindent The proof of this result is based on the following multiplier estimate in $X^{s, b}$ spaces, which can be found in \cite{tao-book} (see page 101,
Lemma 2.11), i.e.
\begin{equation} \label{tao-est}
\|\eta(t/{T^*})u\|_{X^{s,b'}} \le c_1(\eta,b,b') \,\, {T^*}^{b-b'}
\|u\|_{X^{s,b}}. \end{equation}
Applying estimate \eqref{tao-est-modified} with the following choice
\begin{equation} \label{b-b'-choice-wp} b = \frac12-\beta \text{ (in place of $b'$) } \quad \text{and} \quad b_1 = \frac12-\frac12\beta \text{ (in place of $b$)}, \end{equation}
and
\begin{equation} \label{a-a1-choose} \alpha = \frac12+\frac12\beta \text{ (in place of $\alpha'$) } \quad \text{and} \quad \alpha_1 = \frac12+\beta \text{ (in place of $\alpha$) }, \end{equation}
where $\beta$ is defined in \eqref{beta-choice} and it is only depending on $s$ for fixed $m$. From \eqref{onto-map-fin-est-large} we obtain
\begin{align*}
\|\Phi_ {T^*}(u)\|_{X^{s,b,\alpha}(\mathbb{R}^+\times(0,T))} \le c_{s,b,\alpha} \Big(
\| u_0
\|_{H^s(\mathbb{R}^+)} +
\sum\limits_{\ell=0}^{j-1}\|g_\ell\|_{H_t^\frac{s+j-\ell}{m}(0,T)} + \frac{c_1}2 {T^*}^{\frac12\beta}
\left\| \partial_x(\tilde u^2(t))
\right\|_{s,-b,\alpha_1-1} \Big). \nonumber \end{align*}
Then the bilinear estimates \eqref{bilinear-est} reads as follows
\begin{equation} \label{bilinear-est-special}
\|\partial_x(f\cdot g) \|_{s,-b,\alpha_1-1} \le c_{s,b,\alpha}
\| f \|_{s,b,\alpha} \| g \|_{s,b,\alpha}, \quad f, g \in X^{s,b,\alpha}, \end{equation}
and we will use it in this form. Therefore, we get
\begin{align} \label{onto-map-fin-est-1}
\|\Phi_ {T^*}(u)\|_{X^{s,b,\alpha}(\mathbb{R}^+\times(0,T))} \le & c_2 \Big(
\| u_0
\|_{H^s(\mathbb{R}^+)} +
\sum\limits_{\ell=0}^{j-1}\|g_\ell\|_{H_t^\frac{s+j-\ell}{m}(0,T)} + {T^*}^{\frac12\beta}
\| \tilde u
\|_{s,b,\alpha}^2 \Big) \\ \le& c_2 \Big(
\| u_0
\|_{H^s(\mathbb{R}^+)} +
\sum\limits_{\ell=0}^{j-1}\|g_\ell\|_{H_t^\frac{s+j-\ell}{m}(0,T)} + 4 {T^*}^{\frac12\beta}
\| u
\|^2_{X^{s,b,\alpha}(\mathbb{R}^+\times(0,T))} \Big), \nonumber \end{align}
where $c_2=c_2(s,b,\alpha)\doteq c_{s,b,\alpha}+\frac12c_1\cdot c_{s,b,\alpha}^2$. From \eqref{onto-map-fin-est-1} we see that for the map $\Phi_{T^*}$ \eqref{iteration-map-T-loc} to be onto, it suffices to have
$$ c_2 \Big(
\| u_0
\|_{H^s(\mathbb{R}^+)} +
\sum\limits_{\ell=0}^{j-1}\|g_\ell\|_{H_t^\frac{s+j-\ell}{m}(0,T)} + 4 {T^*}^{\frac12\beta}
\| u
\|^2_{X^{s,b,\alpha}(\mathbb{R}^+\times(0,T))} \Big) \le r. $$
And, since $u\in B(r)$ it suffices to have
\begin{align} \label{onto-map-condition} c_2 \Big(
\| u_0
\|_{H^s(\mathbb{R}^+)} +
\sum\limits_{\ell=0}^{j-1}\|g_\ell\|_{H_t^\frac{s+j-\ell}{m}(0,T)} \Big) + 4 c_2 {T^*}^{\frac12\beta} r^2 \le r. \end{align}
To show that $\Phi_{T^*}$ is a contraction, again, using linear estimate \eqref{forced-linear-kdvm-est} with forcing replaced by $-\frac12\partial_x(u^2-v^2)$, extensions $\tilde u$, $\tilde v$ (as above), for $b\le b_1$ we have
\begin{align} \label{contraction-1}
\|\Phi_{T^*}(u)-\Phi_{T^*}(v)\|_{X^{s,b,\alpha}(\mathbb{R}^+\times(0,T))} {\le }& \frac{c_{s,b,\alpha}}2
\left\| \psi_{2{T^*}}(t) \partial_x(u^2(t)-v^2(t))
\right\|_{X^{s,-b_1,\alpha-1}(\mathbb{R}^+\times(0,T))} \\ \le& \frac{c_{s,b,\alpha}}2
\left\| \psi_{2{T^*}}(t) \partial_x(\tilde{u}^2(t)-\tilde{v}^2(t))
\right\|_{X^{s,-b_1,\alpha-1}}, \nonumber \end{align}
Applying estimate \eqref{tao-est-modified} with $b$, $b_1$ given in \eqref{b-b'-choice-wp} and $\alpha$, $\alpha_1$ given by \eqref{a-a1-choose}, we get
\begin{align} \label{contraction-map-fin-est}
\|\Phi_{T^*}(u)-\Phi_{T^*}(v)\|_{X^{s,b,\alpha}(\mathbb{R}^+\times(0,T))} \le \frac{c_1\cdot c_{s,b,\alpha}}2 {T^*}^{\frac12\beta}
\left\| \partial_x[(\tilde u(t)+\tilde v(t))(\tilde u(t)-\tilde v(t))]
\right\|_{s,-b,\alpha_1-1}. \end{align}
Next, using the bilinear estimate \eqref{bilinear-est-special}, from \eqref{contraction-map-fin-est} we get
\begin{align} \label{contraction-map-fin-est-1}
\|\Phi_{T^*}(u)-\Phi_{T^*}(v)\|_{X^{s,b,\alpha}(\mathbb{R}^+\times(0,T))} \le& c_2 {T^*}^{\frac12\beta}
\|\tilde u+\tilde v\|_{s,b,\alpha}
\|\tilde u-\tilde v\|_{s,b,\alpha} \\ \le& 16 c_2 {T^*}^{\frac12\beta}
r
\|u-v\|_{X^{s,b,\alpha}(\mathbb{R}^+\times(0,T))}. \nonumber \end{align}
Thus, in order to make the iteration map $\Phi_ {T^*}$ a contraction map, it suffices to have
\begin{equation} \label{contraction-T-condition} 16 c_2 {T^*}^{\frac12\beta}
r \le \frac12. \end{equation}
Combining conditions \eqref{onto-map-condition} with \eqref{contraction-T-condition}, we see that it suffices to have
$$ c_2 \Big(
\| u_0
\|_{H^s(\mathbb{R}^+)} +
\sum\limits_{\ell=0}^{j-1}\|g_\ell\|_{H_t^\frac{s+j-\ell}{m}(0,T)} \Big) + \frac18r \le r
\iff
r \ge \frac87 c_2 \Big(
\| u_0
\|_{H^s(\mathbb{R}^+)} +
\sum\limits_{\ell=0}^{j-1}\|g_\ell\|_{H_t^\frac{s+j-\ell}{m}(0,T)} \Big) . $$
So, we choose the radius to be
\begin{equation} \label{ball-radius-choice} r \doteq 2c_2 \Big(
\| u_0
\|_{H^s(\mathbb{R}^+)} +
\sum\limits_{\ell=0}^{j-1}\|g_\ell\|_{H_t^\frac{s+j-\ell}{m}(0,T)} \Big). \end{equation}
Then, from \eqref{contraction-T-condition} it suffices to have
$ {T^*}^{\frac12\beta} \le (32c_2r)^{-1}, $
which follows from choosing
\begin{equation} \label{T-choice-wp-1} {T^*} = \frac 12 (1+32c_2r)^{-\frac{2}{\beta}} < \frac 12. \end{equation}
Combining this choice of ${T^*}$ together with choice \eqref{ball-radius-choice} for $r$ we get
\begin{align} \label{T-choice-wp-2} {T^*} =& \frac 12 \Big[ 1+64c_2^2 \big(
\|u_0\|_{H^s(\mathbb{R}^+)}+\sum\limits_{\ell=0}^{j-1}\|g_\ell\|_{H_t^\frac{s+j-\ell}{m}(0,T)} \big) \Big]^{-\frac{2}{\beta}} \\ \ge& c_0\cdot \Big(
1+\|u_0\|_{H^s(\mathbb{R}^+)}+\sum\limits_{\ell=0}^{j-1}\|g_\ell\|_{H_t^\frac{s+j-\ell}{m}(0,T)} \Big)^{-\frac{4}{\beta}} \doteq T_0, \nonumber \end{align}
for some $c_0$ depending on $c_2(s,b,\alpha)$ and $\beta=\beta(s)$, that is $c_0=c_0(s,b,\alpha)$. Thus we choose the lifespan as stated in \eqref{lifespan-est}.
This completes the proof of well-posedness for $-\frac12<s<\frac12$. \,\, $\square$
Lip continuity of the data to solution map and uniquenees is similar to the well-posedness on the line described in \cite{fhy2020}.
\section{Derivation of the Fokas Solution Formula} \label{kdvm-sln-derivation}
Here, we provide an outline of UTM for the solution to the forced linear KdVm ibvp in three steps. First, we use the Fourier transform on the half-line to get a solution formula to ibvp \eqref{LKdVm} via the Fourier inversion formula on the real line. Then, we deform the contour via the Cauchy's Theorem and derive a formula for the solution integrating over the contours $\partial D_{2p}^+$, $p=1,2,\cdots,j$ in the upper half of the complex plane. Finally, we eliminate the unknown boundary data and get the desired solution formula \eqref{UTM-sln-compact}.
\vskip.05in \noindent {\bf Step 1: Solving KdVm ibvp \eqref{LKdVm} via half-line Fourier transform.}
If $\tilde u$ is a solution to the LKdVm formal adjoint equation
\begin{align} \label{adjoint-j eqn} \partial_t \tilde u+(-1)^{j+1}\partial^{2j+1}_x\tilde u = 0, \end{align}
then multiplying it by $u$ and equation \eqref{LKdVm eqn} by $\tilde u$, and adding the resulting equations gives
\begin{align*} \tilde u\partial_t u+u\partial_t\tilde u+(-1)^{j+1}(\tilde u \partial_x^mu+u\partial_x^m\tilde u) = \tilde uf, \end{align*}
or
\begin{align} \label{combine-j eqn} (\tilde uu)_t+(-1)^{j+1}[\tilde u\partial_x^{2j} u+\cdots+(-1)^{n}\partial_x^n\tilde u\partial_x^{2j-n}u+\cdots+\partial_x^{2j}\tilde uu]_x = \tilde uf. \end{align}
Then, choosing as $\tilde u$ the exponential solutions to transpose equation \eqref{adjoint-j eqn}
\begin{align} \label{adjoint-j sln} \tilde u = e^{-i\xi x-i\xi^mt}, \quad \xi\in\mathbb{C}, \end{align}
and substituting them into identity \eqref{combine-j eqn} we get the {\bf divergence form:}
\begin{align} \label{divergence form-j}
(e^{-i\xi x-i\xi^mt}u)_t+(-1)^{j+1}(e^{-i\xi x-i\xi^mt})[\partial_x^{2j} u+\cdots+i^n\xi^n\partial_x^{2j-n}u+\cdots+(-1)^j\xi^{2j} u])_x = \tilde uf.
\end{align}
Integrating the divergence form \eqref{divergence form-j} from $x=0$ to $\infty$ gives the $t$-equation
\begin{align} \label{tode-j} (e^{-i\xi^mt}\widehat u(\xi,t))_t = e^{-i\xi^mt}\widehat f(\xi,t)+(-1)^{j+1}e^{-i\xi^mt}g(\xi,t), \end{align}
where $\widehat u$ and $\widehat f$ are the half-line Fourier transforms of $u$ and $f$, which are defined in \eqref{FT-halfline}, and $g$ is the following combination of $m=2j+1$ boundary data (some of which are not given)
\begin{align} g(\xi,t) \doteq \partial_x^{2j}u(0,t)+\cdots+i^n\xi^n\partial_x^{2j-n}u(0,t)+\dots+(-1)^{j}\xi^{2j}u(0,t). \end{align}
Integrating \eqref{tode-j} from $0$ to $t$, $0\leq t\leq T$, we obtain the so called {\bf global relation:}
\begin{align} \label{global relation-j} e^{-i\xi^mt}\widehat u(\xi,t) =& \widehat u_0(\xi) + F(\xi,t) \\ +&(-1)^{j+1}[\tilde g_{2j}(\xi^m,t)+\cdots+i^n\xi^n\tilde g_{2j-n}(\xi^m,t) +\cdots+(-1)^j\xi^{2j}\tilde g_0(\xi^m,t)] , \quad \text{Im}(\xi)\leq 0, \nonumber \end{align}
where $F(\xi,t)$ and $\tilde g_{\ell}(\xi,t)$ are given in \eqref{F-time-transform} and \eqref{g-time-transform} respectively. Now, inverting \eqref{global relation-j} we get
\begin{align} \label{sln-line-j} u(x,t) &= \frac{1}{2\pi}\int_{-\infty}^\infty e^{i\xi x+i\xi^mt} [\widehat u_0(\xi)+F(\xi,t)]d\xi \\ &+ \frac{(-1)^{j+1}}{2\pi}\int_{-\infty}^\infty e^{i\xi x+i\xi^mt} [\tilde g_{2j}(\xi^m,t)+\cdots+(i\xi)^\ell\tilde g_{2j-\ell}(\xi^m,t) +\cdots+(-1)^j\xi^{2j}\tilde g_0(\xi^m,t)]d\xi. \nonumber \end{align}
{\bf Step 2: Deforming integration over the contours $\partial D_{2p}^+$ in the upper half-pane.}
Formula \eqref{sln-line-j} contains $(j+1)$ unknown data. To eliminate them, we deform the integration contour from $\mathbb{R}$ to $\partial D_{2p}^+$, $p=1,2,\cdots,j$. This is expressed by the following result.
\begin{lemma} \label{UTM-lem} The solution $u(x,t)$ to ibvp \eqref{LKdVm} can be written in the form
\begin{align} \label{sln1-j} u(x,t) &= \frac{1}{2\pi}\int_{-\infty}^\infty e^{i\xi x+i\xi^mt} [\widehat u_0(\xi)+F(\xi,t)]d\xi \\ &+ \frac{(-1)^{j+1}}{2\pi}\sum\limits_{p=1}^j\int_{\partial D_{2p}^+} e^{i\xi x+i\xi^mt} [\tilde g_{2j}(\xi^m,t)+\cdots+(i\xi)^\ell\tilde g_{2j-\ell}(\xi^m,t) +\cdots+(-1)^j\xi^{2j}\tilde g_0(\xi^m,t)]d\xi, \nonumber \end{align}
where the domains $D^+_2, \cdots, D_{2j}^+$ are shown in Figure 1.1 (if $j$ is odd) or Figure 1.2 (if $j$ is even), and the orientation of the boundary $\partial D^+_{2p}$ is given by the left-hand rule. \end{lemma}
\noindent The proof of above lemma can be found in \cite{y2020}. Also, a similar lemma for the KdV equation can be found in \cite{fhm2016}.
\vskip.05in \noindent {\bf Step 3: Eliminating the unknown boundary data.} For each $p=1,2,\cdots,j$, we construct a linear system with $j+1$
equations (as many as the unkown data). For this we apply the invariant transformations of $\xi^m$, i.e. $\xi\longrightarrow\alpha_{p,n}\xi$, $n=1,2,\cdots,j+1$, where $\alpha_{p,n}$ are defined in \eqref{a-rot-angles}, and use the global relation \eqref{global relation-j}. Thus, we obtain the linear system of the following $j+1$ equations
\begin{align*} \begin{cases} e^{-i\xi^mt}\widehat u(\alpha_{p,1}\xi,t) = \widehat u_0(\alpha_{p,1}\xi) + F(\alpha_{p,1}\xi,t) \nonumber \\ \hskip.7in +(-1)^{j+1}[\tilde g_{2j}(\xi^m,t)+\cdots+(\alpha_{p,1})^\ell(i\xi)^\ell\tilde g_{2j-\ell}(\xi^m,t) +\cdots+(\alpha_{p,1})^{2j}(i\xi)^{2j}\tilde g_0(\xi^m,t)], \\ \hskip1in \cdots \\ e^{-i\xi^mt}\widehat u(\alpha_{p,j+1}\xi,t) = \widehat u_0(\alpha_{p,j+1}\xi) + F(\alpha_{p,j+1}\xi,t) \nonumber \\ \hskip.7in +(-1)^{j+1}[\tilde g_{2j}(\xi^m,t)+\cdots+(\alpha_{p,j+1})^\ell(i\xi)^\ell\tilde g_{2j-\ell}(\xi^m,t) +\cdots+(\alpha_{p,j+1})^{2j}(i\xi)^{2j}\tilde g_0(\xi^m,t)]. \end{cases} \end{align*}
Solving these equations for $(i\xi)^\ell\tilde g_{2j-\ell}(\xi^m,t) $, $\ell=0,1,\cdots,j$, and substituting the obtained solutions into formula \eqref{sln1-j}, we get the desired Fokas solution formula \eqref{UTM-sln-compact} involving only the given data and no unknown data.
\vspace*{1mm} \noindent \textbf{Acknowledgements.} The first author was partially supported by a grant from the Simons Foundation (\#524469 to Alex Himonas).
\noindent A. Alexandrou Himonas
Fangchi Yan\\ Department of Mathematics
Department of Mathematics\\ University of Notre Dame
West Virginia University\\ Notre Dame, IN 46556
Morgantown, WV 26506 \\ E-mail: \textit{himonas.1$@$nd.edu}
E-mail: \textit{[email protected]}
\end{document} | arXiv |
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Khan Academy 500.685 προβολές 15:15 A2 Biology: Standard error and 95% confidence limits - Διάρκεια: 6:02. Examples: Z(0.05) = 1.645 (the Z-score which has 0.05 to the right, and 0.4500 between 0 and it) Z(0.10) = 1.282 (the Z-score which has 0.10 to the right, and 0.4000 Results 1 to 1 of 1 Thread: maximum error of estimate for a 95% level of confidence Thread Tools Show Printable Version Email this Page… Subscribe to this Thread… Search Thread Sampling theory provides methods for calculating the probability that the poll results differ from reality by more than a certain amount, simply due to chance; for instance, that the poll reports
The margin of error for the difference between two percentages is larger than the margins of error for each of these percentages, and may even be larger than the maximum margin DTUbroadcast 221 προβολές 10:31 Stats: Sampling Distribtion of the Mean and Standard Error - Διάρκεια: 21:53. Introductory Statistics (5th ed.). It is standard that $Y$ has mean $p$ and variance $\frac{p(1-p)}{n}$.
It asserts a likelihood (not a certainty) that the result from a sample is close to the number one would get if the whole population had been queried. Retrieved 2006-05-31. Sign In Please enter an email address Please a valid email address Please enter a password Please enter a valid password Password must be 5 characters long Unknown email or incorrect A confidence interval is an interval estimate with a specific level of confidence.
pp.63–67. Margin of error is often used in non-survey contexts to indicate observational error in reporting measured quantities. MathWorld. Note that there is not necessarily a strict connection between the true confidence interval, and the true standard error.
Is it true or false, and why or why not? Must be non-negative and less than the size of the collection. Other statistics[edit] Confidence intervals can be calculated, and so can margins of error, for a range of statistics including individual percentages, differences between percentages, means, medians,[9] and totals. So in this case, the absolute margin of error is 5 people, but the "percent relative" margin of error is 10% (because 5 people are ten percent of 50 people).
Can I stop this homebrewed Lucky Coin ability from being exploited? Show all work. Upgrade About Help Statistics Examples Step-by-Step Examples Statistics Index was out of range. The larger the margin of error, the less confidence one should have that the poll's reported results are close to the true figures; that is, the figures for the whole population.
Jossey-Bass: pp. 17-19 ^ Sample Sizes, Margin of Error, Quantitative AnalysisArchived January 21, 2012, at the Wayback Machine. ^ Lohr, Sharon L. (1999). MSNBC, October 2, 2004. Retrieved 2006-05-31. ^ Wonnacott and Wonnacott (1990), pp. 4–8. ^ Sudman, S.L. Retrieved 2006-05-31. ^ Isserlis, L. (1918). "On the value of a mean as calculated from a sample".
Effect of population size[edit] The formula above for the margin of error assume that there is an infinitely large population and thus do not depend on the size of the population Margin of error applies whenever a population is incompletely sampled. Given a level of confidence of 99% and a population standard deviation of 10, what other information is necessary: (A) To find the Maximum Error of Estimate (E)? If the exact confidence intervals are used, then the margin of error takes into account both sampling error and non-sampling error.
Show all your calculations. In these cases, the statistics can't be used since the sample hasn't been taken yet. Census Bureau. Basic concept[edit] Polls basically involve taking a sample from a certain population.
The basic confidence interval for a symmetric distribution is set up to be the point estimate minus the maximum error of the estimate is less than the true population parameter which Along with the confidence level, the sample design for a survey, and in particular its sample size, determines the magnitude of the margin of error. Sampling: Design and Analysis. If the statistic is a percentage, this maximum margin of error can be calculated as the radius of the confidence interval for a reported percentage of 50%.
Definition[edit] The margin of error for a particular statistic of interest is usually defined as the radius (or half the width) of the confidence interval for that statistic.[6][7] The term can The standard error can be used to create a confidence interval within which the "true" percentage should be to a certain level of confidence. n = [z*s/E]^2 ------------------------------- (C) Given the above confidence level and population standard deviation, find the Maximum Error of Estimate (E) if n = 45. Converting Game of Life images to lists How do I depower Magic items that are op without ruining the immersion How to create a company culture that cares about information security?
Swinburne University of Technology. It is important to realize the order here. The sample statistic is calculated from the sample data and the population parameter is inferred (or estimated) from this sample statistic. Table of Contents Υπενθύμιση αργότερα Έλεγχος Υπενθύμιση απορρήτου από το YouTube, εταιρεία της Google Παράβλεψη περιήγησης GRΜεταφόρτωσηΣύνδεσηΑναζήτηση Φόρτωση... Επιλέξτε τη γλώσσα σας. Κλείσιμο Μάθετε περισσότερα View this message in English Το
The terms statistical tie and statistical dead heat are sometimes used to describe reported percentages that differ by less than a margin of error, but these terms can be misleading.[10][11] For E = z*s/sqrt(n) E = 2.576*10/sqrt(45) -------------------------------- (D) For this same sample of n = 45, what is the width of the confidence interval around the population mean? statistics share|cite|improve this question edited Mar 12 '13 at 2:08 azimut 14.4k94085 asked Mar 12 '13 at 1:58 Shaily 707 1 Can you show some of your own work and That is, how large of a sample should be taken to make an accurate estimation.
Show all work. (D) For this same sample of n = 45, what is the width of the confidence interval around the population mean? mathwithmrbarnes 320.897 προβολές 9:03 Standard error of the mean | Inferential statistics | Probability and Statistics | Khan Academy - Διάρκεια: 15:15. If we use the "relative" definition, then we express this absolute margin of error as a percent of the true value. In media reports of poll results, the term usually refers to the maximum margin of error for any percentage from that poll.
Later, we will talk about variances, which don't use a symmetric distribution, and the formula will be different. As a shorthand notation, the () are usually dropped, and the probability written as a subscript. For the eponymous movie, see Margin for error (film). Let me say that again: Statistics are calculated, parameters are estimated.
This theory and some Bayesian assumptions suggest that the "true" percentage will probably be fairly close to 47%. We talked about problems of obtaining the value of the parameter earlier in the course when we talked about sampling techniques.
maximum allowable error in statistics
As an example of the above, a random sample of size 400 will give a margin of error, at a 95% confidence level, of 0.98/20 or 0.049?just under 5%. About Us| Careers| Contact Us| Blog| Homework Help| Teaching Jobs| Search Lessons| Answers| Calculators| Worksheets| Formulas| Offers Copyright © 2016 - NCS Pearson, All rights reserved. Stokes, Lynne; Tom Belin (2004). "What is a Margin of Error?" (PDF). Sampling theory provides methods for calculating the probability that the poll results ...
maximum margin of error statistics
See also[edit] Engineering tolerance Key relevance Measurement uncertainty Random error Observational error Notes[edit] ^ "Errors". Bruce Drake • 1 month ago Thanks for the heads-up to us. This maximum only applies when the observed percentage is 50%, and the margin of error shrinks as the percentage approaches the extremes of 0% or 100%. About Fact Tank Real-time analysis and news about data from Pew Research writers and social scientists. The margin of error expresses the amount...
maximum error formula statistics
The formula to calculate Standard Error is, Standard Error Formula: where SEx̄ = Standard Error of the Mean s = Standard Deviation of the Mean n = Number of Observations of In cases where the sampling fraction exceeds 5%, analysts can adjust the margin of error using a finite population correction (FPC) to account for the added precision gained by sampling close MSNBC, October 2, 2004. Using the t Distribution Calculator, we find that the critical value is 1.96. Sign in OR Sign in...
maximum error of estimate statistics
Mitch Keller 6.099 weergaven 6:22 95% confidence margin of error - Duur: 1:51. Yes No Sorry, something has gone wrong. You can only upload files of type PNG, JPG, or JPEG. Once you have computed E, I suggest you save it to the memory on your calculator. Student's t Distribution When the population standard deviation is unknown, the mean has a Student's t distribution. Another area of inferential statistics is sample size determination. Algebra: Probability and statisticsSection Solvers... | CommonCrawl |
\begin{document}
\def\calstyle#1{{\text{\@ifnextchar[{\ft@s}{\ft@@@s[\f@size]}{eusm10}#1}}}
\let\cal\calstyle
\title[]{\bf\large UNIVERSAL LEFSCHETZ FIBRATIONS\\[6pt] OVER BOUNDED SURFACES}
\author{Daniele Zuddas}
\address{Universit\`a di Cagliari\\ Dipartimento di Matematica e Informatica\\ Via Ospedale 72\\ 09124 Cagliari (Italy)}
\email{[email protected]}
\thanks{I would like to thanks Andrea Loi for helpful conversations and suggestions about the manuscript. Thanks to Regione Autonoma della Sardegna for support with funds from PO Sardegna FSE~2007--2013 and L.R.~7/2007 ``Promotion of scientific research and technological innovation in Sardinia''. Also thanks to ESF for short visit grants within the program ``Contact and Symplectic Topology".}
\date{}
\subjclass[2000]{Primary 55R55; Secondary 57N13}
\begin{abstract}
In analogy with the vector bundle theory we define universal and strongly universal Lefschetz fibrations over bounded surfaces. After giving a characterization of these fibrations we construct very special strongly universal Lefschetz fibrations when the fiber is the torus or an orientable surface with connected boundary and the base surface is the disk. As a by-product we also get some immersion results for $4$-dimensional $2$-handlebodies.
\noindent
{\sc Keywords:} universal Lefschetz fibration, Dehn twist, 4-manifold. \end{abstract} \maketitle
\section*{Introduction}
Consider a smooth 4-manifold $V$ and a surface $S$. Let $f : V \to S$ be a (possibly achiral) smooth Lefschetz fibration with singular values set $A_f\subset S$ and regular fiber $F \cong F_{g,b}$, the compact connected orientable surface of genus $g$ with $b$ boundary components. Let $G$ be another surface. We assume that $V$, $S$ and $G$ are compact, connected and oriented with (possibly empty) boundary.
\begin{definition}
We say that a smooth map $q : G \to S$ with regular values set $R_q \subset S$ is $f$-regular if $q(\partial G) \cap A_f = \emptyset$ and $A_f \subset R_q$. \end{definition}
In other words, $q$ is $f$-regular if and only if $q$ and $q_{|\partial G}$ are transverse to $f$.
If $q$ is $f$-regular then $\widetilde V = \{(g, v) \in G \times V \; |\; q(g) = f(v)\}$ is a 4-manifold and the map $\widetilde f :\widetilde V \to G$ given by $\widetilde f(g, v) = g$ is a Lefschetz fibration. Moreover $\widetilde q : \widetilde V \to V$, $\widetilde q(g, v) = v$, is a fiber preserving map which sends each fiber of $\widetilde f$ diffeomorphically onto a fiber of $f$, so the regular fiber of $\widetilde f$ can be identified with $F$. We get the following commutative diagram
\centerline{\includegraphics{cd.eps}}
We say that $\widetilde f : \widetilde V \to G$ is the {\em pullback} of $f$ by $q$ and we make use of the notation $\widetilde f = q^*(f)$.
Two Lefschetz fibrations $f_1 : V_1 \to S_1$ and $f_2 : V_2 \to S_2$ are said {\em equivalent} if there are orientation-preserving diffeomorphisms $\phi : S_1 \to S_2$ and $\widetilde \phi : V_1 \to V_2$ such that $\phi \circ f_1 = f_2 \circ \widetilde\phi$. The equivalence class of $f$ will be indicated by $[f]$.
We say that a Lefschetz fibration $f$ is {\em allowable} if the vanishing cycles of $f$ with respect to a (and hence to any) Hurwitz system for $A_f$ are homologically essential in $F$. We consider only allowable Lefschetz fibrations, if not differently stated.
Given $f$ we define the set $L(f) = \{[q^*(f)]\}$ where $q$ runs over the $f$-regular maps $q : G \to S$ and $G$ runs over the compact, connected, oriented surfaces. Analogously, we define the set $SL(f) = \{[q^*(f)]\} \subset L(f)$ where $q$ runs over the $f$-regular orientation-preserving immersions $q : G \to S$, with $G$ as above.
\begin{definition}\label{univ/def}
A Lefschetz fibration $u : U \to S$ with regular fiber $F$ is said universal (resp. strongly universal) if every Lefschetz fibration over a surface with non-empty boundary and with regular fiber diffeomorphic to $F$ belongs to a class of $L(u)$ (resp. $SL(u))$. \end{definition}
In other words $u$ is universal (resp. strongly universal) if and only if any Lefschetz fibration over a bounded surface and with the same fiber can be obtained as the pullback of $u$ by a $u$-regular map (resp. immersion). Note that this notion of universality is analogous to that in the theory of vector bundles \cite{MS74}.
We denote by $\mathop{\cal M}\nolimits_{g,b}$ the mapping class group of $F_{g,b}$ whose elements are the isotopy classes of orientation-preserving diffeomorphisms of $F_{g,b}$ which keep the boundary fixed pointwise (assuming isotopy through such diffeomorphisms).
It is well-known that for a Lefschetz fibration $f : V \to S$ with regular fiber $F_{g,b}$ the monodromy of a meridian\footnote{A meridian of a finite subset $A \subset \mathop{\mathrm{Int}}\nolimits S$ is an element of $\pi_1(S - A)$ which can be represented by the oriented boundary of an embedded disk in $S$ which intersects $A$ in a single interior point, cf. Definition~\ref{immers-merid/def}.} of $A_f$ is a Dehn twist $\gamma \in \mathop{\cal M}\nolimits_{g,b}$.
If $S$ is not simply connected, the monodromy of an element of $\pi_1(S - A_f)$ which is not a product of meridians is not necessarily the identity on $\partial F_{g,b}$, and so it induces a permutation on the set of boundary components of $F_{g,b}$. We will denote by $\varSigma_b$ the permutation group of this set.
These considerations allow us to define three kind of monodromies. Let $H_f \lhd \pi_1(S - A_f)$ be the smallest normal subgroup of $\pi_1(S - A_f)$ which contains all the meridians of $A_f$. The {\em Lefschetz monodromy} of $f$ is the group homomorphism $\omega_f : H_f \to \mathop{\cal M}\nolimits_{g,b}$ which sends a meridian of $A_f$ to the corresponding Dehn twist in the standard way \cite{GS99}.
Let $\widehat \mathop{\cal M}\nolimits_{g,b}$ be the extended mapping class group of $F_{g,b}$, namely the group of all isotopy classes of orientation-preserving self-diffeomorphisms of $F_{g,b}$. The {\sl bundle monodromy} $\widehat\omega_f : \pi_1(S - A_f) \to \widehat \mathop{\cal M}\nolimits_{g,b}$ is the monodromy of the locally trivial bundle $f_| : V - f^{-1}(A_f) \to S - A_f$.
We consider also the natural homomorphism $\sigma : \widehat\mathop{\cal M}\nolimits_{g,b} \to \varSigma_b$ which sends an isotopy class to the permutation induced on the set of boundary components. The composition $\sigma \circ\widehat \omega_f$ passes to the quotient $\pi_1(S - A_f)/H_f \cong \pi_1(S)$ and gives a homomorphism $\omega_f^\sigma : \pi_1(S) \to \varSigma_b$ which we call the {\sl permutation monodromy} of $f$.
Let $\cal C_{g,b}$ be the set of equivalence classes of homologically essential simple closed curves in $\mathop{\mathrm{Int}}\nolimits F_{g,b}$ up to orientation-preserving homeomorphisms of $F_{g,b}$. It is well-known that $\cal C_{g,b}$ is finite. Moreover $\#\,\cal C_{g,b} = 1$ for $g \geq 1$ and $b \in \{0, 1\}$ \cite[Chapter~12]{L97}.
Now we state the main results of the paper. In the following proposition we characterize the universal and strongly universal Lefschetz fibrations.
\begin{proposition}\label{main/thm}
A Lefschetz fibration $u : U \to S$ with regular fiber $F_{g,b}$ is universal (resp. strongly universal) if and only if the following two conditions $(1)$ and $(2)$ (resp. $(1)$ and $(2'))$ are satisfied:
\begin{itemize}
\item [$(1)$] $\omega_u$ and $\omega_u^\sigma$ are surjective;
\item [$(2)$] any class of $\cal C_{g,b}$ can be represented by a vanishing cycle of $u$;
\item [$(2')$] any class of $\cal C_{g,b}$ contains at least two vanishing cycles of $u$ which correspond to singular points of opposite signs.
\end{itemize}
In particular, if $b \in \{0, 1\}$ and $\omega_u$ is surjective then $u$ is universal. If in addition $u$ admits a pair of opposite singular points, then $u$ is strongly universal. \end{proposition}
As a remarkable simple consequence we have that (strongly) universal Lefschetz fibrations actually exist for any regular fiber $F_{g,b}$. Moreover, the surjectivity of $\omega^\sigma_u$ implies $b_1(S) \geq ($the minimum number of generators of $\varSigma_b)$, and this inequality is sharp. So we can assume that the base surface $S$ of a universal Lefschetz fibration is the disk for $b \leq 1$, the annulus for $b = 2$, and such that $b_1(S) = 2$ for $b \geq 3$.
Consider a knot $K \subset S^3$ and let $M(K, n)$ be the oriented 4-manifold obtained from $B^4$ by the addition of a 2-handle along $K$ with framing $n$.
In the following theorem we construct very special strongly universal Lefschetz fibrations when the fiber is the torus or $F_{g, 1}$ with $g\geq 1$.
\begin{theorem}\label{strong-univ/thm}
There is a strongly universal Lefschetz fibration $u_{g,b} : U_{g,b} \to B^2$ with fiber $F_{g,b}$ and with
\indent $U_{1,1} \cong B^4$,\\
\indent $U_{g, 1} \cong M(O, 1) \text{ for } g \geq 2$, and\\
\indent $U_{1,0} \cong M(E, 0)$,
\noindent where $O$ and $E$ denote respectively the trivial and the figure eight knots in $S^3$. \end{theorem}
\begin{corollary}\label{paral/cor}
Let $f : V \to B^2$ be a Lefschetz fibration with fiber of genus one. Suppose that no vanishing cycle of $f$ disconnects the regular fiber with respect to some (and hence to any) Hurwitz system. Then $V$ immerses in $\Bbb R^4$ and so is parallelizable. \end{corollary}
By means of Theorem~\ref{strong-univ/thm} we are able to give a new elementary proof of the following corollary. This was known since the work of Phillips \cite{P67} about submersions of open manifolds because there exists a bundle monomorphism $T\, V \to T\, \C\mathrm P^2$ for any oriented 4-manifold $V$ which is homotopy equivalent to a CW-complex of dimension two (obtained by means of the classifying map to the complex universal vector bundle \cite{MS74}).
\begin{corollary}\label{immers/cor}
Any compact oriented 4-dimensional 2-handlebody\,\footnote{Recall that an $n$-dimensional $k$-handlebody is a smooth $n$-manifold built up with handles of index $\leq k$.} admits an o\-ri\-en\-ta\-tion-preserving immersion in $\C\mathrm P^2$. \end{corollary}
Universal maps in the context of Lefschetz fibrations over closed surfaces can be constructed in a different way. This generalization will be done in a forthcoming paper.
The paper consists of three other sections. In the next one we review some basic material on Lefschetz fibrations needed in the paper. Section~\ref{proof/sec} is dedicated to the proofs of our results, and in Section~\ref{remarks/sec} we give some final remarks and a comment on positive Lefschetz fibrations on Stein compact domains of dimension four.
Throughout the paper we assume manifolds (with boundary) to be compact, oriented and connected if not differently stated. We will work in the $C^\infty$ category.
\section{Preliminaries}\label{preliminaries/sec}
Let $V$ be a 4-manifold (possibly with boundary and corners) and let $S$ be a surface.
\begin{definition}\label{lf/def}
A Lefschetz fibration $f : V \to S$ is a smooth map which satisfies the following three conditions:
\begin{enumerate}
\item the singular set $\widetilde A_f\subset \mathop{\mathrm{Int}}\nolimits V$ is finite and is mapped injectively onto the singular values set $A_f = f(\widetilde A_f) \subset \mathop{\mathrm{Int}}\nolimits S$;
\item the restriction $f_| : V - f^{-1}(A_f) \to S - A_f$ is a locally trivial bundle with fiber a surface $F$ (the regular fiber of $f$);
\item around each singular point $\widetilde a \in \widetilde A_f$, $f$ can be
locally expressed as the complex map $f(z_1, z_2) = z_1^2 + z_2^2$ for suitable chosen smooth local complex coordinates.
\end{enumerate} \end{definition}
If such local coordinates are orientation-preserving (resp. reversing), then $\widetilde a$ is said a {\em positive (resp. negative) singular point}, and $a = f(\widetilde a) \in A_f$ is said a {\em positive (resp. negative) singular value}. Observe that this positivity (resp. negativity) notion does not depend on the orientation of $S$. Obviously, around a negative singular point there are orientation-preserving local complex coordinates such that $f(z_1, z_2) = z_1^2 + \bar z_2^2$.
Most authors add the adjective `achiral' in presence of negative singular points. We prefer to simplify the terminology and so we do not follow this convention.
The orientations of $V$ and of $S$ induce an orientation on $F$ such that the locally trivial bundle associated to $f$ is oriented. We will always consider $F$ with this canonical orientation.
Let $F_{g,b}$ be the orientable surface of genus $g \geq 0$ with $b \geq 0$ boundary components. A Lefschetz fibration $f : V \to S$ with regular fiber $F = F_{g,b}$ is characterized by the {\em Lefschetz monodromy homomorphism} $\omega_f : H_f \to \mathop{\cal M}\nolimits_{g,b}$, which sends meridians of $A_f$ to Dehn twists, and by the {\em bundle monodromy homomorphism} $\widehat\omega_f : \pi_1(S - A_f) \to \widehat\mathop{\cal M}\nolimits_{g,b}$ which is the monodromy of the bundle associated to $f$. Sometimes we use the term `monodromy' by leaving the precise meaning of it to the context.
It is well-known that the monodromy of a counterclockwise meridian of a positive (resp. negative) singular value is a right-handed (resp. left-handed) Dehn twist around a curve in $F$ \cite{GS99}. Such a curve is said to be a {\em vanishing cycle} for $f$. We recall the following definition.
\begin{definition}\label{hurwitz/def}
A Hurwitz system for a cardinality $n$ subset $A \subset \mathop{\mathrm{Int}}\nolimits S$ is a sequence $(\xi_1, \dots, \xi_n)$ of meridians of $A$ which freely generate $\pi_1(D - A)$ and such that the product $\xi_1 \cdots \xi_n$ is the homotopy class of the oriented boundary of $D$, where $D \subset S$ is a disk such that $A \subset \mathop{\mathrm{Int}}\nolimits D$ and $*\in \partial D$. \end{definition}
If a Hurwitz system $(\xi_1, \dots, \xi_n)$ for $A_f \subset S$ is given, the set $A_f = \{a_1,\dots, a_n \}$ can be numbered accordingly so that $\xi_i$ is a meridian of $a_i$. It is determined a sequence of signed vanishing cycles $(c_1^\pm, \dots, c_n^\pm)$ (the {\em monodromy sequence} of $f$), where $c_i \subset F_{g,b}$ corresponds to the Dehn twist $\omega_f(\xi_i) \in \mathop{\cal M}\nolimits_{g,b}$ and the sign of $c_i$ equals that of $a_i$ as a singular value of $f$. Clearly, $c_i$ is defined up to isotopy for all $i$. Sometimes the plus signs are understood.
The fact that the $c_i$'s are all homologically (or homotopically) essential in $F$ does not depend on the actual Hurwitz system, and so this is a property of the Lefschetz fibration $f$.
The monodromy sequence of $f : V \to B^2$ determines a handlebody decomposition of the total space as $V = (B^2 \times F) \cup H_1^2 \cup \cdots \cup H_n^2$ where $B^2 \times F$ is a trivialization of the bundle associated to $f$ over a subdisk contained in $B^2 - A_f$ and the 2-handle $H_i^2$ is attached to $B^2 \times F$ along the vanishing cycle $\{*_i\} \times c_i \subset S^1 \times F \subset \partial (B^2 \times F)$ for a suitable subset $\{*_1, \dots, *_n\} \subset S^1$ cyclically ordered in the counterclockwise direction \cite{GS99}. The framing of $H^2_i$ with respect to the fiber $\{*_i\} \times F \subset \partial(B^2 \times F)$ is $-\epsilon_i$ where $\epsilon_i = \pm 1$ is the sign of the singular point $a_i$.
Note that $B^2 \times F$ can be decomposed as the union of a 0-handle, some 1-handles, and also a 2-handle in case $\partial F = \emptyset$ starting from a handlebody decomposition of $F$ and making the product with the 2-dimensional 0-handle $B^2$.
In this paper we consider only the so called {\em relatively minimal} Lefschetz fibrations, namely those without homotopically trivial vanishing cycles. Then in our situation the monodromy sequence can be expressed also by a sequence of Dehn twists $(\gamma_1^{\epsilon_1}, \dots, \gamma_n^{\epsilon_n})$, where $\gamma_i = (\omega_f(\xi_i))^{\epsilon_i}$ is assumed to be right-handed.
Let $\mu : \mathop{\cal M}\nolimits_{g,b} \to \widehat\mathop{\cal M}\nolimits_{g,b}$ be the homomorphism such that $\mu([\phi]) = [\phi]$ for all $[\phi] \in \mathop{\cal M}\nolimits_{g,b}$. We have the exact sequence $\mathop{\cal M}\nolimits_{g,b} \stackrel\mu\to \widehat\mathop{\cal M}\nolimits_{g,b} \stackrel\sigma\to \varSigma_b \to 0$ where $\sigma$ is the boundary permutation homomorphism defined in the Introduction. The monodromy homomorphisms $\omega_f$ and $\widehat\omega_f$ of
a Lefschetz fibration $f$ satisfy $\widehat\omega_{f| H_f} = \mu \circ \omega_f$.
For a finite subset $A \subset \mathop{\mathrm{Int}}\nolimits S$ we indicate by $H(S, A) \lhd \pi_1(S - A)$ the normal subgroup generated by the meridians of $A$. Given $S$, $A$ and two homomorphisms $\omega : H(S, A) \to \mathop{\cal M}\nolimits_{g,b}$ and $\widehat\omega : \pi_1(S - A) \to \widehat\mathop{\cal M}\nolimits_{g,b}$ such that $\omega$ sends meridians to Dehn twists and $\mu\circ\omega = \widehat\omega_{|H(S,A)}$, there exists a Lefschetz fibration $f : V \to S$ with regular fiber $F_{g,b}$ such that $A_f = A$, $\omega_f = \omega$ and $\widehat\omega_f = \widehat\omega$. Moreover, such $f$ is unique up to equivalence by our relative minimality assumption, unless $S$ is closed and the fiber is a sphere or a torus (because in such cases the diffeomorphisms group of the fiber is not simply connected \cite{Gr73}). In particular, if $S$ has boundary the 4-manifold $V$ is determined up to orientation-preserving diffeomorphisms. In \cite{APZ2011} we give a very explicit construction of $f$ starting from the monodromy sequence.
If $q : G \to S$ is $f$-regular with respect to a Lefschetz fibration $f : V \to S$ then the pullback $\widetilde f = q^*(f)$ satisfies $A_{\widetilde f} = q^{-1}(A_f)$, $\omega_{\widetilde f} = \omega_f \circ q_*$ and $\widehat\omega_{\widetilde f} = \widehat\omega_f \circ q_*$, where $q_* : \pi_1(G- A_{\widetilde f}) \to \pi_1(S - A_f)$ is the homomorphism induced by the restriction $q_| : G - A_{\widetilde f} \to S - A_f$. The base points are understood and are chosen so that $q(*') = *$ with $*'$ in the domain and $*$ in the codomain.
\begin{remark}
The $f$-regularity of $q$ implies that $q_*(H_{\widetilde f}) \subset H_f$. \end{remark}
Let $a \in A_f$ and $a' \in q^{-1}(a)$. Then $q$ is a local diffeomorphism around $a'$. It is immediate that the sign of $a'$ as a singular value of $\widetilde f$ is given by that of $a$ multiplied by the local degree of $q$ at $a'$, in other words $\mathop{\mathrm{sign}}\nolimits(a') = \deg_{a'}(q)\cdot \mathop{\mathrm{sign}}\nolimits(a)$.
In order to prove Theorem~\ref{strong-univ/thm} we recall the definition of {\em stabilizations} (the reader is referred to \cite{GS99} or \cite{APZ2011} for details).
Given a Lefschetz fibration $f : V \to B^2$ whose regular fiber $F$ has non-empty boundary, we can construct a new Lefschetz fibration $f' : V' \to B^2$ by an operation called stabilization which is depicted in Figure~\ref{stabil/fig}. The new regular fiber $F' = F \cup H^1$ is obtained by attaching an orientable 1-handle $H^1$ to $F$, and the new monodromy sequence is given by the addition to the old one of a signed vanishing cycle $c^\pm$ which crosses $H^1$ geometrically once.
The inverse operation is called {\em destabilization} and can be applied if there is a properly embedded arc $s$ in the regular fiber $F'$ of $f'$ which meets a single vanishing cycle $c$, and it does so geometrically once. The arc $s$ is the cocore of a 1-handle of $F'$. Let $F$ be $F'$ cut open along $s$, and let the new monodromy sequence be that of $f'$ with $c^\pm$ removed (no matter whichever is the sign).
In terms of handlebody decompositions, stabilizations (resp. destabilizations) correspond to the addition (resp. deletion) of a cancelling pair of 1 and 2-handles, hence $V \cong V'$.
\begin{figure}
\caption{The (de)stabilization operation.}
\label{stabil/fig}
\end{figure}
We end this section with the following straightforward propositions, needed in the proof of our main results.
\begin{proposition}
Let $f : V \to S$ be a Lefschetz fibration and let $G$ be a surface. If $q_t : G \to S$, $t \in [0,1]$, is a homotopy through $f$-regular maps then $q_0^*(f) \cong q_1^*(f)$. \end{proposition}
\begin{proposition}
If $q : G \to S$ is an orientation-preserving immersion (resp. embedding) then the fibered map $\widetilde q : \widetilde V \to V$ associated to the pullback $q^*(f) : \widetilde V \to G$ is also an orientation-preserving immersion (resp. embedding). \end{proposition}
\section{Proofs of main results}\label{proof/sec}
We first prove the following Lemma.
\begin{lemma}\label{van-cic/lem}
Suppose that $u : U \to S$ satisfies conditions $(1)$ and $(2)$ (resp. $(1)$ and $(2'))$ of Proposition~\ref{main/thm}. Then each class of $\cal C_{g,b}$ can be represented by a vanishing cycle (resp. a vanishing cycle of prescribed sign) in any monodromy sequence of $u$. \end{lemma}
\begin{proof}
Let $(\xi_1, \dots, \xi_n)$ be a Hurwitz system for $A_u \subset S$ and let $\mathfrak c \in \cal C_{g,b}$. There is a meridian $\xi$ of $A_u$ such that $\omega_u(\xi)$ is a Dehn twist (resp. a Dehn twist of prescribed sign) around a curve $c \in \mathfrak c$. It is well-known that $\xi$ is conjugate to some $\xi_i$, hence $\xi = \tau \xi_i \tau^{-1}$ for some $\tau \in \pi_1(S - A_u)$. Let $c_i$ be the vanishing cycle of $\gamma_i$.
Put $\phi = \widehat\omega_u(\tau)$, $\gamma = \widehat\omega_u(\xi)$ and $\gamma_i = \widehat\omega_u(\xi_i)$. We get $\gamma =\phi^{-1} \circ \gamma_i \circ \phi$ (because the standard right to left composition rule of maps differs from that in the fundamental group). There are two cases depending on whether $\gamma_i$ is or not the identity.
If $\gamma_i$ is the identity then $\gamma$ is also the identity. It follows that $c$ and $c_i$ are boundary parallel (by the relatively minimal assumption they cannot be homotopically trivial), hence $c_i \in \mathfrak c$. If $\gamma_i$ is not the identity then $c_i =\phi(c) \in \mathfrak c$ \cite{W99}. \end{proof}
We need also the following definition which gives a generalization of the notion of meridian.
\begin{definition}\label{immers-merid/def} An immersed meridian for a finite subset $A \subset \mathop{\mathrm{Int}}\nolimits S$ is an element of $\pi_1(S- A)$ which can be represented by the oriented boundary of an immersed disk $B \subset S$ such that $\#(B \cap A) = 1$. \end{definition}
The immersed meridians of $A$ are precisely the conjugates in $\pi_1(S -A)$ of the meridians.
\begin{proof}[Proof of Proposition~\ref{main/thm}]
We consider first the case of universal Lefschetz fibrations.
\noindent{\em `If' part.}
Suppose that $u$ satisfies conditions $(1)$ and $(2)$ of the statement. Consider a Lefschetz fibration $f : V \to G$ with regular fiber $F_{g,b}$, where $G$ is an oriented surface with non-empty boundary. We are going to show that $f \cong q^*(u)$ for some $u$-regular map $q : G \to S$. Without loss of generality we can assume $\partial S \ne \emptyset$.
There is a handlebody decomposition of $G$ with only one 0-handle $G^0$ and $l \geq 0$ 1-handles $G^1_i$, so $G = G^0 \cup G_1^1 \cup \cdots \cup G^1_l$. We can assume that $A_f \subset G^0$.
Fix base points $* \in \partial S$ and $*'\in \partial G$ and let $\{\xi_1, \dots, \xi_n\}$ be a Hurwitz system for $A_u = \{a_1, \dots, a_n\} \subset S$. Fix also a set of free generators $\{\zeta_1,\dots, \zeta_k, \eta_1, \dots, \eta_l\}$ for $\pi_1(G - A_f)$ with $(\zeta_1,\dots, \zeta_k)$ a Hurwitz system for $A_f \subset G$. We assume that the $\zeta_i$'s are represented by meridians contained in $G^0$ and that $\eta_i$ is represented by an embedded loop (still denoted by $\eta_i$) which meets the 1-handle $G^1_i$ geometrically once and does not meet any other $G_j^1$ for $j \ne i$ as in Figure~\ref{surface/fig}. In this figure the 1-handles $G^1_i$'s are contained in the white lower box. We assume also that $\eta_i \cap \eta_j = \{*'\}$ for $i \ne j$.
\begin{figure}
\caption{The generators $\zeta_i$'s and $\eta_i$'s.}
\label{surface/fig}
\end{figure}
There are disks $D_1, \dots, D_k \subset G$ as those depicted in dark grey in the same Figure~\ref{surface/fig} such that $\partial D_i \subset G - A_f$ represents $\zeta_i$ as a loop and $D_i \cap D_j = \{ *' \}$ for all $i \ne j$. There is also a disk $D_0 \subset G - A_f$ which is a neighborhood of $*'$ such that $D_0 \cap D_i \cong B^2$ and $D_0 \cap \eta_i \cong [0,1]$ for all $i$. Then $D = D_0 \cup D_1 \cup\dots\cup D_k$ is diffeomorphic to $B^2$ (up to smoothing the corners).
It follows that $G$ is diffeomorphic to the surface $G'$ obtained from $D$ by the addition of orientable 1-handles $G'_1, \dots, G'_l$ where $G'_i$ has attaching sphere the endpoints of the arc $\eta_i\cap D_0$ for any $i = 1, \dots, l$.
Consider the Dehn twists $\gamma_i^{\epsilon_i} =\omega_u(\xi_i)$ and $\delta_i^{\sigma_i} = \omega_f(\zeta_i)$ around respectively the vanishing cycles $c_i$ (for $u$) and $d_i$ (for $f$), where $\epsilon_i, \sigma_i = \pm 1$ ($\gamma_i$ and $\delta_i$ are assumed to be right-handed).
By Lemma~\ref{van-cic/lem} $d_i$ is equivalent in $\cal C_{g,b}$ to some $c_{j_i}$. It follows that $\delta_i = \lambda_i^{-1} \circ\gamma_{j_i} \circ \lambda_i$ for some $\lambda_i \in \widehat\mathop{\cal M}\nolimits_{g,b}$ which sends $d_i$ to $c_{j_i}$ \cite{W99} (note that this composition is well-defined in $\mathop{\cal M}\nolimits_{g,b}$ as the isotopy class of $\bar\lambda^{-1}_i \circ \bar\gamma_{j_i} \circ \bar\lambda_i$ where $\bar\lambda_i$ and $\bar\gamma_{j_i}$ are representatives of $\lambda_i$ and $\gamma_{j_i}$ respectively).
Since $\omega_u$ and $\omega^\sigma_u$ are surjective, $\widehat\omega_u$ is also surjective and so there is $\alpha_i \in \pi_1(S - A_u)$ such that $\lambda_i = \widehat\omega_u(\alpha_i)$. It follows that $\delta_i = \omega_u(\beta_i^{\epsilon_{j_i}})$ where $\beta_i = \alpha_i \xi_{j_i} \alpha_i^{-1}$ is an immersed meridian of $a_{j_i}$.
So $\beta_i$ can be represented by the oriented boundary of an immersed disk $B_i \subset S$ which intersects $A_u$ only at $a_{j_i}$. The $B_i$'s can be chosen so that for a suitable embedded disk $B_0 \subset S - A_u$ which is a neighborhood of $*$, we have $B_0 \cap B_i \cong B^2$ and $B_0 \cap B_i \cap B_j = \{*\}$ for all $i \ne j \geq 1$.
Now take $\eta'_i \in \pi_1(S - A_u)$ such that $\widehat\omega_u(\eta'_i) = \widehat\omega_f(\eta_i)$ for $i = 1,\dots, l$. We represent $\eta'_i$ by a transversely immersed loop (still denoted by $\eta'_i$) in $S - A_u$ such that $B_0 \cap \eta_i' \cong [0,1]$ and $B_0 \cap \eta_i' \cap \eta_j' = \{*\}$ for $i \ne j$.
Consider the moves $t$, $t'$ and $t''$ of Figure~\ref{permutation/fig}, where $t$ acts on pairs $(B_i, B_j)$, $t'$ on pairs $(B_i, \eta'_j)$, and $t''$ on pairs $(\eta'_i, \eta'_j)$. These moves allow us to make the $B_i$'s and the loops $\eta'_i$'s intersect $B_0$ in the same order as the $D_i$'s and the $\eta_i$'s do with $D_0$.
\begin{figure}
\caption{The permuting moves.}
\label{permutation/fig}
\end{figure}
Take a map $\overline q : G' \to S$ which sends $D_0$ to $B_0$ diffeomorphically, immerses $D_i$ onto $B_i$ by preserving the orientation, and immerses the 1-handle $G'_i$ onto a regular neighborhood of $\eta'_i \subset S - A_u$ for all $i$. Assume also that $\overline q(A_f) = A_u$ and $\overline q(*') =*$. It follows that $\overline q$ is a $u$-regular immersion and that the homomorphism $\overline q_* : \pi_1(G' - A_f) \to \pi_1(S - A_u)$ induced by the restriction $\overline q_| : G' -A_f \to S - A_u$ satisfies $\overline q_*(\zeta_i) =\beta_i$ and $q_*(\eta_i) = \eta'_i$ for all $i \geq 1$.
Now fix identifications $D'_i = \mathop{\mathrm{Cl}}\nolimits(D_i - D_0) \cong [-1, 1] \times [-1, 1]$ for $i \geq 1$, such that the ordinate is 1 along the arc $D_0 \cap D_i'$ and such that the singular value $D_i \cap A_f$ has coordinates $\left(0, -{1 \over 2}\right)$.
Let $r_i : G' \to G'$ be defined by the identity outside $D'_i$ and by the map of $D_i'$ to itself given by $r_i(t_1,t_2) = (t_1 t_2, t_2)$ up to the above identification. So $r_i$ shrinks a proper arc of $D_i'$ to a point, preserves the orientation above this arc and reverses the
orientation below it, as depicted in Figure~\ref{reverse/fig}. Moreover, $r_i(A_f) = A_f$ and the homomorphism induced by the restriction $r_{i*} : \pi_1(G' - A_f) \to \pi_1(G' - A_f)$ satisfies $r_{i*}(\zeta_i) = \zeta_i^{-1}$, $r_{i*}(\zeta_j) = \zeta_j$ for $j \neq i$, and $r_{i*}(\eta_j) = \eta_j$ for all $j$.
\begin{figure}
\caption{The twisting map $r_i$ on $D$.}
\label{reverse/fig}
\end{figure}
Let $I = \{i_1, \dots, i_m\} \subset \{1, \dots, k\}$ be the set of those $i$
such that $\epsilon_{j_i} \sigma_i = -1$ and put $q = \overline q \circ r_{i_1} \circ \cdots \circ r_{i_m} : G \cong G' \to S$ (with $q = \overline q$ if $I$ is empty). Then $q$ is $u$-regular, $q^{-1}(A_u) = A_f$, $\omega_f = \omega_u \circ q_*$ and $\widehat\omega_f = \widehat\omega_u \circ q_*$ where $q_* : \pi_1(G - A_f) \to \pi_1(S - A_u)$ is induced by the restriction of $q$. It follows that $f \cong q^*(u)$.
\noindent{\it `Only if' part.}
Let $u : U \to S$ be universal with regular fiber $F_{g,b}$. Consider a Lefschetz fibration $f : V \to G$ with the same regular fiber and which satisfies the conditions $(1)$ and $(2)$. There is a $u$-regular
map $q : G \to S$ such that $q^*(u) = \widetilde f \cong f$. Then $\widetilde f$ satisfies the conditions $(1)$ and $(2)$ of the statement. Since $\omega_{\widetilde f} = \omega_u \circ q_*$ and $\omega^\sigma_{\widetilde f} = \omega^\sigma_u \circ q_*$ we obtain that $\omega_u$ and $\omega_u^\sigma$ are surjective.
Consider now a class $\mathfrak c \in \cal C_{g,b}$. We can find a meridian $\zeta \in \pi_1(G - A_{\widetilde f})$ such that $\omega_{\widetilde f}(\zeta)$ is a Dehn twist around a curve $c \in \mathfrak c$. Since $q$ is $u$-regular, $q_*(\zeta) \in \pi_1(S - A_u)$ is an immersed meridian of $A_u$, hence $q_*(\zeta) = \alpha \xi \alpha^{-1}$ for a meridian $\xi$ of $A_u$ and for some $\alpha \in \pi_1(S - A_u)$.
It follows that $\lambda = \widehat\omega_u(\alpha)$ satisfies $\lambda^{-1} \circ \omega_u(\xi) \circ \lambda = \omega_{\widetilde f}(\zeta)$ and so $\omega_u(\xi)$ is a Dehn twist around $\lambda(c) \in \mathfrak c$. Then $\mathfrak c$ can be represented by a vanishing cycle of $u$.
The case of strongly Lefschetz fibrations can be handled similarly by tracing the same line of the previous proof.
We just give an idea of the `if' part: if conditions $(1)$ and $(2')$ are satisfied then the $c_{j_i}$'s in the proof of the first part can be chosen so that $\epsilon_{j_i} = \sigma_i$. Then the set $I$ defined above is empty and so $q$ is an orientation-preserving $u$-regular immersion such that $f = q^*(u)$.
Finally, the last part of the proposition follows since $\#\, \cal C_{g,b} = 1$ for $b \in \{0, 1\}$. \end{proof}
\begin{proof}[Proof of Theorem~\ref{strong-univ/thm}]
We consider three cases, depending on the values of $g$ and of $b$.
\noindent{\em Case I: $g \geq 2$ and $b = 1$.} Consider the Lefschetz
fibration $u_{g,1} : U_{g,1} \to B^2$ with regular fiber $F_{g,1}$ and with
monodromy sequence given by the $2g + 1$ signed vanishing cycles $(b_1^{-}$, $b_2$, $a_1,\dots, a_g, c_1^{-}, c_2, \dots, c_{g-1})$ depicted in Figure~\ref{lf/fig}, where a Hurwitz system is understood. In this figure the surface $F_{g,1}$ is embedded in $\Bbb R^3$ as part of the boundary of a standard genus $g$ handlebody.
\begin{figure}
\caption{The Lefschetz fibration $u_{g,1}$ for $g \geq 2$.}
\label{lf/fig}
\end{figure}
By a theorem of Wajnryb \cite{W99} $\mathop{\cal M}\nolimits_{g,1}$ is generated by the $2g + 1$ Dehn twists $\alpha_i$, $\beta_i$ and $\gamma_i$ around the curves $a_i$, $b_i$ and $c_i$ respectively.
It follows that $\omega_{u_{g,1}}$ is surjective and so $u_{g,1}$ is strongly universal by Proposition~\ref{main/thm}.
Now we analyze the 4-manifold $U_{g,1}$.
In Figure~\ref{lefschetz2/fig} we pass from $u_{i,1}$ to $u_{i-1,1}$ in a two step destabilization process (first destabilize $a_i$ by the arc $s_i$ and then destabilize $c_{i-1}$ by the arc $s'_{i-1}$). This operation can be done whenever $i > 2$, and so by induction we can assume that $g=2$. In other words $U_{g,1} \cong U_{2,1}$ for $g > 2$.
\begin{figure}
\caption{Simplifying $u_{i,1}$ for $i \geq 2$.}
\label{lefschetz2/fig}
\end{figure}
In Figure~\ref{lefschetz3/fig} we give some more destabilizations (first of $a_2$ along $s_2$ and then of $a_1$ along $s$), and finally we get the Lefschetz fibration depicted in the left lower part of the same figure. This has fiber $F_{0,3}$ and three boundary parallel vanishing cycles (two negative and one positive).
\begin{figure}
\caption{Simplifications of $u_{2,1}$.}
\label{lefschetz3/fig}
\end{figure}
So a Kirby diagram for $U_{g,1}$ is that depicted in Figure~\ref{kirby/fig},
which by a straightforward Kirby calculus argument can be recognized to be $M(O,1)$ (slide the outermost 2-handle over that with framing $-1$ so that the
latter cancels and by another simple sliding and deletion we get the picture
for $M(O,1)$ in the right side of Figure~\ref{kirby/fig}).
\begin{figure}
\caption{The manifold $U_{g, 1}\cong M(O,1)$ for $g \geq 2$.}
\label{kirby/fig}
\end{figure}
\noindent{\em Case II: $g = b = 1$.} Consider the Lefschetz fibration
$u_{1,1} : U_{1,1} \to B^2$ with regular fiber $F_{1,1}$ and with monodromy
sequence $(a, b^-)$ depicted in Figure~\ref{torus/fig}.
By \cite{W99} $\cal M_{1,1}$ is generated by the two Dehn twists $\alpha$ and $\beta$ around the curves $a$ and $b$ respectively and so Proposition~\ref{main/thm} implies that $u_{1,1}$ is strongly universal.
\begin{figure}
\caption{The Lefschetz fibration $u_{1,1}$.}
\label{torus/fig}
\end{figure}
By a double destabilization we get a Lefschetz fibration with regular fiber $B^2$ and without singular values, hence $U_{1,1}$ is diffeomorphic to $B^2 \times B^2 \cong B^4$.
\noindent{\em Case III: $(g,b) = (1,0)$.} Let $u_{1,0} : U_{1,0} \to B^2$ be
the Lefschetz fibration with fiber $F_{1,0} = T^2$ and with monodromy sequence $(a, b^-)$ depicted in Figure~\ref{torus-mon/fig}. Then $\omega_{u_{1,0}}$ is
surjective and so $u_{1,0}$ is strongly universal by Proposition~\ref{main/thm}.
\begin{figure}
\caption{The Lefschetz fibration $u_{1,0}$.}
\label{torus-mon/fig}
\end{figure}
Consider $F_{1,1} \subset \partial U_{1,1} \cong S^3$ as the fiber of $u_{1,1}$
over a point of $S^1 = \partial B^2$. So $K = \partial F_{1,1}$ is a knot in $S^3$. Moreover, by
pushing off $K$ along $F_{1,1}$ we get the framing zero (in terms of linking
number), since $F_{1,1}$ is a Seifert surface for $K$.
Therefore the addition of a 2-handle to $B^4$ along $K$ with framing zero
produces $U_{1,0}$ \cite{GS99}. In Figure~\ref{kirby-M/fig} is depicted a Kirby diagram for $U_{1,0}$.
This and the next three figures are referred to the
blackboard framing, namely that given by a push off along the
blackboard plus the extra full twists specified by the number near the knot.
Note that in Figure~\ref{kirby-M/fig} the blackboard framing coincides with
that of the fiber $F_{1,1}$.
\begin{figure}
\caption{The manifold $U_{1,0}$.}
\label{kirby-M/fig}
\end{figure}
Now we apply the moves $t_+$ and $t_-$ of Figures~\ref{trick/fig} and
\ref{trick2/fig} respectively, where the thick arcs with framing zero belong
to the same knot (which is assumed to be unlinked with the thin one) with the
orientations indicated in these figures. Such moves are proved by Kirby
calculus in the same figures.
\begin{figure}
\caption{The move $t_+$.}
\label{trick/fig}
\end{figure}
\begin{figure}
\caption{The move $t_-$.}
\label{trick2/fig}
\end{figure}
We get Figure~\ref{torus-simpl/fig} where the two kinks in the second stage are
opposite and so do not affect the framing. The last stage, which gives the
figure eight knot, is obtained by framed isotopy.
Since the writhe of the figure eight knot is zero, the blackboard framing zero is the same as linking number zero, and this concludes the proof. \end{proof}
\begin{figure}
\caption{Kirby diagrams for $U_{1,0}$.}
\label{torus-simpl/fig}
\end{figure}
\begin{proof}[Proof of Corollary~\ref{immers/cor}]
By a theorem of Harer \cite{Harer1979} any 4-dimensional 2-handlebody $V$ admits a Lefschetz fibration $f : V \to B^2$ with regular fiber $F_{g,1}$ for some $g \geq 1$ (see also \cite{LP01} or \cite{EF06} for different proofs). Up to stabilizations we can assume $g \geq 2$.
Theorem~\ref{strong-univ/thm} implies that $f \cong q^*(u_{g,1})$ for some orientation-preserving $u_{g,1}$-regular immersion $q : B^2 \to B^2$. Then we get a fibered immersion $\widetilde q : V \to U_{g,1}\cong M(O, 1)$. It is well-known that $M(O, 1)$ is orientation-preserving diffeomorphic to a tubular neighborhood of a projective line in $\C\mathrm P^2$. Then we can consider $M(O, 1) \subset \C\mathrm P^2$, and this concludes the proof. \end{proof}
\begin{proof}[Proof of Corollary~\ref{paral/cor}]
Let $F_{1,b}$ be the regular fiber of $f$. If $b = 1$ the corollary follows immediately from Theorem~\ref{strong-univ/thm} since $V$ admits a fibered immersion in $U_{1,1} \cong B^4 \subset \Bbb R^4$ and hence is parallelizable.
If $b \geq 2$ we consider the 4-manifold $V'$ obtained from $V$ by the
addition of 2-handles along all but one boundary components of the regular fiber $F_{1,b} \subset \partial V$ with framing zero with respect to $F_{1,b}$. Then $V \subset V'$. Moreover, $f$ extends to a Lefschetz fibration $f' : V' \to B^2$ with regular fiber $F_{1,1}$ whose monodromy is obtained from $\omega_f$ by composition with the homomorphism from $\mathop{\cal M}\nolimits_{1,b}$ to $\mathop{\cal M}\nolimits_{1,1}$ induced by capping off by disks all but one boundaries components of $F_{1,b}$.
The non-separating assumption on the vanishing cycles of $f$ implies that $f'$ is allowable, and so $V'$ immerses in $\Bbb R^4$ by the first case.
If $b = 0$, $V$ fibered immerses in the manifold $U_{1,0} \cong M(E, 0)$ of Theorem~\ref{strong-univ/thm}. We conclude by observing that $M(E,0)$ immerses in $\Bbb R^4$ as a tubular neighborhood of $B^4 \cup D$ where $D \subset \Bbb R^4 - \mathop{\mathrm{Int}}\nolimits B^4$ is a self-transverse immersed disk with boundary the knot $E$. \end{proof}
\section{Final remarks}\label{remarks/sec}
\begin{remark}
It is not difficult to see that for $b \geq 1$
$$\#\, \cal C_{g,b} =
\begin{cases}
\displaystyle{\left\lfloor {b \over 2} \right\rfloor}, & \text{if $g = 0$} \cr
\cr
\displaystyle\left\lfloor {g b - g + b \over 2} \right\rfloor + 1, & \text{if $g \geq 1$}
\end{cases}$$
which is a lower bound for the number of singular points of a universal Lefschetz fibration with fiber $F_{g,b}$ $(\lfloor x \rfloor$ denotes the integer part of $x \in \Bbb R)$. \end{remark}
\begin{remark}
In order to include also the not allowable Lefschetz fibrations it suffices to replace, in Proposition~\ref{main/thm}, $\cal C_{g,b}$ with the set of $\widehat\mathop{\cal M}\nolimits_{g,b}$-equivalence classes of homotopically essential curves. The proof is very similar. \end{remark}
In \cite{LP01} Loi and Piergallini characterized compact Stein domains of dimension four, up to orientation-preserving diffeomorphisms, as the total spaces of positive Lefschetz fibrations (meaning with only positive singular points) over $B^2$ with bounded fiber. We can express this theorem in terms of universal positive Lefschetz fibrations.
Following the notations of the proof of Theorem~\ref{strong-univ/thm} let $p_g : P_g \to B^2$ be the Lefschetz fibration with fiber $F_{g,1}$ and monodromy sequence given by $(a, b)$ for $g = 1$ and $(b_1$, $b_2$, $a_1,\dots, a_g, c_1, c_2, \dots, c_{g-1})$ for $g \geq 2$ as showed in Figure~\ref{univ-stein/fig}.
\begin{figure}
\caption{The positive universal Lefschetz fibrations $p_g$.}
\label{univ-stein/fig}
\end{figure}
Then $p_g$ is universal (but not strongly universal) by Proposition~\ref{main/thm}. Moreover $P_1 \cong B^4$ and $P_g$ has the Kirby diagram depicted in Figure~\ref{kirby-stein/fig} for $g \geq 2$. That $P_g \cong M(O, -3)$ follows by the same argument used in the proof of Theorem~\ref{strong-univ/thm}.
\begin{figure}
\caption{The 4-manifold $P_g$ for $g \geq 2$.}
\label{kirby-stein/fig}
\end{figure}
However, $SL(p_g)$ is the set of equivalence classes of all positive Lefschetz fibrations with fiber $F_{g,1}$. The proof is exactly the same of Proposition~\ref{main/thm} with $\epsilon_{j_i} = \sigma_i =1$.
Of course any Lefschetz fibration with bounded fiber can be positively stabilized so that the fiber has connected boundary.
It follows that compact 4-dimensional Stein domains with strictly pseudoconvex boundary coincide, up to orientation-preserving diffeomorphisms, with the total spaces of Lefschetz fibrations that belong in $SL(p_g)$ for some $g \geq 1$.
\end{document} | arXiv |
Strong orientation
In graph theory, a strong orientation of an undirected graph is an assignment of a direction to each edge (an orientation) that makes it into a strongly connected graph.
Strong orientations have been applied to the design of one-way road networks. According to Robbins' theorem, the graphs with strong orientations are exactly the bridgeless graphs. Eulerian orientations and well-balanced orientations provide important special cases of strong orientations; in turn, strong orientations may be generalized to totally cyclic orientations of disconnected graphs. The set of strong orientations of a graph forms a partial cube, with adjacent orientations in this structure differing in the orientation of a single edge. It is possible to find a single orientation in linear time, but it is #P-complete to count the number of strong orientations of a given graph.
Application to traffic control
Robbins (1939) introduces the problem of strong orientation with a story about a town, whose streets and intersections are represented by the given graph G. According to Robbins' story, the people of the town want to be able to repair any segment of road during the weekdays, while still allowing any part of the town to be reached from any other part using the remaining roads as two-way streets. On the weekends, all roads are open, but because of heavy traffic volume, they wish to convert all roads to one-way streets and again allow any part of town to be reached from any other part. Robbins' theorem states that a system of roads is suitable for weekday repairs if and only if it is suitable for conversion to a one-way system on weekends. For this reason, his result is sometimes known as the one-way street theorem.[1]
Subsequently to the work of Robbins, a series of papers by Roberts and Xu modeled more carefully the problem of turning a grid of two-way city streets into one-way streets, and examined the effect of this conversion on the distances between pairs of points within the grid. As they showed, the traditional one-way layout in which parallel streets alternate in direction is not optimal in keeping the pairwise distances as small as possible. However, the improved orientations that they found include points where the traffic from two one-way blocks meets itself head-on, which may be viewed as a flaw in their solutions.
Related types of orientation
If an undirected graph has an Euler tour, an Eulerian orientation of the graph (an orientation for which every vertex has indegree equal to its outdegree) may be found by orienting the edges consistently around the tour.[2] These orientations are automatically strong orientations.
A theorem of Nash-Williams (1960, 1969) states that every undirected graph G has a well-balanced orientation. This is an orientation with the property that, for every pair of vertices u and v in G, the number of pairwise edge-disjoint directed paths from u to v in the resulting directed graph is at least $\left\lfloor {\frac {k}{2}}\right\rfloor $, where k is the maximum number of paths in a set of edge-disjoint undirected paths from u to v. Nash-Williams' orientations also have the property that they are as close as possible to being Eulerian orientations: at each vertex, the indegree and the outdegree are within one of each other. The existence of well-balanced orientations, together with Menger's theorem, immediately implies Robbins' theorem: by Menger's theorem, a 2-edge-connected graph has at least two edge-disjoint paths between every pair of vertices, from which it follows that any well-balanced orientation must be strongly connected. More generally this result implies that every 2k-edge-connected undirected graph can be oriented to form a k-edge-connected directed graph.
A totally cyclic orientation of a graph G is an orientation in which each edge belongs to a directed cycle. For connected graphs, this is the same thing as a strong orientation, but totally cyclic orientations may also be defined for disconnected graphs, and are the orientations in which each connected component of G becomes strongly connected. Robbins' theorem can be restated as saying that a graph has a totally cyclic orientation if and only if it does not have a bridge. Totally cyclic orientations are dual to acyclic orientations (orientations that turn G into a directed acyclic graph) in the sense that, if G is a planar graph, and orientations of G are transferred to orientations of the planar dual graph of G by turning each edge 90 degrees clockwise, then a totally cyclic orientation of G corresponds in this way to an acyclic orientation of the dual graph and vice versa.[3][4] The number of different totally cyclic orientations of any graph G is TG(0,2) where TG is the Tutte polynomial of the graph, and dually the number of acyclic orientations is TG(2,0).[5] As a consequence, Robbins' theorem implies that the Tutte polynomial has a root at the point (0,2) if and only if the graph G has a bridge.
If a strong orientation has the property that all directed cycles pass through a single edge st (equivalently, if flipping the orientation of an edge produces an acyclic orientation) then the acyclic orientation formed by reversing st is a bipolar orientation. Every bipolar orientation is related to a strong orientation in this way.[6]
Flip graphs
If G is a 3-edge-connected graph, and X and Y are any two different strong orientations of G, then it is possible to transform X into Y by changing the orientation of a single edge at a time, at each step preserving the property that the orientation is strong.[7] Therefore, the flip graph whose vertices correspond to the strong orientations of G, and whose edges correspond to pairs of strong orientations that differ in the direction of a single edge, forms a partial cube.
Algorithms and complexity
A strong orientation of a given bridgeless undirected graph may be found in linear time by performing a depth-first search of the graph, orienting all edges in the depth-first search tree away from the tree root, and orienting all the remaining edges (which must necessarily connect an ancestor and a descendant in the depth-first search tree) from the descendant to the ancestor.[8] If an undirected graph G with bridges is given, together with a list of ordered pairs of vertices that must be connected by directed paths, it is possible in polynomial time to find an orientation of G that connects all the given pairs, if such an orientation exists. However, the same problem is NP-complete when the input may be a mixed graph.[9]
It is #P-complete to count the number of strong orientations of a given graph G, even when G is planar and bipartite.[3][10] However, for dense graphs (more specifically, graphs in which each vertex has a linear number of neighbors), the number of strong orientations may be estimated by a fully polynomial-time randomized approximation scheme.[3][11] The problem of counting strong orientations may also be solved exactly, in polynomial time, for graphs of bounded treewidth.[3]
Notes
1. Koh & Tay (2002).
2. Schrijver (1983).
3. Welsh (1997).
4. Noy (2001).
5. Las Vergnas (1980).
6. de Fraysseix, Ossona de Mendez & Rosenstiehl (1995).
7. Fukuda, Prodon & Sakuma (2001).
8. See e.g. Atallah (1984) and Roberts (1978).
9. Arkin & Hassin (2002).
10. Vertigan & Welsh (1992).
11. Alon, Frieze & Welsh (1995).
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| Wikipedia |
Journal of Pest Science
January 2020 , Volume 93, Issue 1, pp 543–550 | Cite as
Crowd surveillance: estimating citizen science reporting probabilities for insects of biosecurity concern
Implications for plant biosecurity surveillance
Peter Caley
Marijke Welvaert
Simon C. Barry
First Online: 11 June 2019
Data streams arising from citizen reporting activities continue to grow, yet the information content within these streams remains unclear, and methods for addressing the inherent reporting biases little developed. Here, we quantify the major influence of physical insect features (colour, size, morphology, pattern) on the propensity of citizens to upload photographic sightings to online portals, and hence to contribute to biosecurity surveillance. After correcting for species availability, we show that physical features and pestiness are major predictors of reporting probability. The more distinctive the visual features, the higher the reporting probabilities—potentially providing useful surveillance should the species be an unwanted exotic. Conversely, the reporting probability for many small, nondescript high priority pest species is unlikely to be sufficient to contribute meaningfully to biosecurity surveillance, unless they are causing major harm. The lack of citizen reporting of recent incursions of small, nondescript exotic pests supports the model. By examining the types of insects of concern, industries or environmental managers can assess to what extent they can rely on citizen reporting for their surveillance needs. The citrus industry, for example, probably cannot rely on passive unstructured citizen data streams for surveillance of the Asian citrus psyllid (Diaphorina citri). In contrast, the forestry industry may consider that citizen detection and reporting of species of the large and colourful insects such as pine sawyers (Monochamus spp.) may be sufficient for their needs. Incorporating citizen surveillance into the general surveillance framework is an area for further research.
Citizen surveillance General surveillance Citizen science Pest Biosecurity Hemiptera Coleoptera
Communicated by S. Macfadyen.
The citizen reporting probabilities of insects are dramatically influenced by physical features such as size, colour, pattern and morphology.
We demonstrate how it is possible to correct for the inherent bias in citizen reporting by modelling the effect of physical features and species distribution and abundance using a case–control design. This enables predictions of the surveillance sensitivity of citizen reporting of exotic species of biosecurity interest.
For highly featured and/or large exotic insect pests, citizen reporting may provide adequate surveillance for plant health needs.
Exotic insect species for which citizen reporting is unlikely to be effective can be predicted in advance.
Biosecurity surveillance aims to protect the natural environment, plants and animals, as well as agri- and horticulture from harm caused by pests and diseases (Froud et al. 2008). The biosecurity threat arising from the invasion of exotic insect pests is highly diffuse in that the number of target species is very large and the potential points of entry are numerous. This presents particular logistical challenges for implementing effective surveillance—it is impossible to deploy targeted traditional surveillance (e.g. species specific traps, trained inspectors, etc.) for all threats in all locations. Proposed alternatives to such traditional surveillance include increased use of sensors, robots and citizen science. Here, we focus on the latter option. Citizens can potentially contribute to biosecurity surveillance in many ways, ranging from inadvertent references to invasive organisms on social media platforms (e.g. Twitter), to deliberate through unstructured (spatially and temporally opportunistic) reporting of species via dedicated online portals (e.g. iNaturalist, https://www.inaturalist.org/), to deliberate structured (designed) surveys (Welvaert and Caley 2016). The potential surveillance power of the general public is evident from a New Zealand study, where nearly half of all new exotic species detections over a 3-year period were from members of the general public (Froud et al. 2008). In a similar vein, Thomas et al. (2017) recorded that 95% of non-indigenous invertebrate species new to Barrow Island were detected by members of the local community. Such surveillance contributes to what is termed "general surveillance" (Hammond et al. 2016a).
Detecting environmental biosecurity events from human social media communications in a timely manner faces some particular challenges, some technical (Daume 2016) and others largely arising from uncertainty and bias relating to the observation process (Welvaert and Caley 2016). In comparison with self-reported syndromic human health surveillance, the spatial scale and number of events to be detected is small initially (at the time when detection is most critical), and the direct impact on individuals typically minimal. For example, the combined effects of citrus greening disease (Huanglongbing—currently causing massive economic loss to the citrus industry in the Americas), vectored by the Asian citrus psyllid (Diaphorina citri) (Grafton-Cardwell et al. 2013) are neither immediate nor direct on human individuals per se, until the pathogen has spread significantly and affected trees are showing visible symptoms. Furthermore, the impacts and/or symptoms of exotic pests and diseases may be unknown or hard to detect or difficult to distinguish from endemic pests, resulting in varying levels of detectability (Jarrad et al. 2011). The detection of small-scale biosecurity events through social media also requires that the taxonomy of organism is widely known but also unique; otherwise, the signal-to-noise ratio is too low for reliable signal retrieval (Welvaert et al. 2017). Hence, reliably detecting the arrival of exotic insects within the social media data stream is likely to be highly problematic.
Insect collecting is a worldwide contemporary and historical hobby/passion of many members of the public, with the major recent change being the move to photography in place of physical specimen collecting, and the ability to share these images online. In comparison with social media, the uploading of photographs onto citizen science data portals is much more deliberate and has a taxonomic underpinning. Dedicated online platforms now exist to store such observations, and to crowd-source their species identification. The number of citizen-sourced record uploads goes in the tens of millions. Note, however, these data sources generally do not contain biosecurity related species information. By definition they would not contain records for invasive alien species that have not yet entered a country. These growing datasets, however, can be used to inform us about the type of species that are typically reported by citizen scientists, and whether they are likely to include exotic pests and/or pathogen species should they arrive. Indeed, in Australia for example, a wide variety of sightings of insect species are uploaded to the Atlas of Living Australia (ALA, https://www.ala.org.au/) which acts as a repository for most citizen science platforms in Australia along with professionally collected museum specimens, etc. The number of citizen-sourced record uploads of insect species to the ALA already number in the 100,000s, involving 1000s of species. Although these numbers may seem impressive, at face value they provide little information on whether an emergency plant pest (e.g. D. citri) would be detected and reported in a timely manner.
There is clearly overlap between the types of insects that are recorded, and exotic insect species of biosecurity concern, raising the possibility of using an analogue approach to estimate surveillance sensitivity. For example, the black spittlebug (Amarusa australis), a harmless native species in Australia, is from the same Cicadellidae family as the glassy-winged sharp shooter (Homalodisca vitripennis)—the key vector for the causative pathogen Xylella fastidiosa of Pierce's Disease in grapevines. As of 30-06-2016, there had been two citizen sightings of A. australis uploaded to ALA, and notably, both were identified on the same day as they were uploaded. However, the two species differ substantially in size and colour (H. vitripennis is larger and more colourful) (Fig. 1a, b), calling into question the accuracy of the analogue approach. Clearly some form of model is required to infer what this sighting rate may mean for the detection and reporting of an incursion of H. vitripennis, as it is larger and more colourful. Answering this question requires knowing the factors that motivate people to report the insects they discover, and applying these factors to emergency plant pests of concern to estimate the likely reporting probabilities.
This study introduces a quantitative, statistical approach to estimating the citizen reporting probabilities of insects based on their physical features. In doing so it quantifies the contribution of citizen science activities to biosecurity surveillance, and enables identification of invasive insects for which citizen science would not provide effective surveillance.
Insect species from within the same family (Cicadellidae in this instance) can vary considerably in size and colour, rendering an analogue approach to inferring citizen reporting probabilities based on taxonomy inaccurate. a Black spittle bug (Amarusa australis). Sighting by Dianne Clarke http://www.bowerbird.org.au/observations/50067. Licensed CC BY 3.0 AU https://creativecommons.org/licenses/by/3.0/au/. b Glassy-winged sharp shooter (Homalodisca vitripennis). Sighting by Debra Hendricks https://www.inaturalist.org/taxa/199381-Homalodisca-vitripennis. Licensed CC BY-NC 4.0 https://creativecommons.org/licenses/by-nc/4.0/.
We used a case–control experimental design to assess factors that influence the probability of an insect species being uploaded to the Atlas of Living Australia through citizen science channels (ALA 2016a, b). The Atlas of Living Australia is Australia's national biodiversity database. It is an online biodiversity data management system which links Australia's biological knowledge with its scientific and agricultural reference collections and other custodians of biological information. The initial focus of the ALA was on assembling a comprehensive database of collections and records generated by professional taxonomists and scientists. Subsequently, it has developed (and actively encouraged) the direct recording of sightings by non-professionals ("citizen scientists") including the ability to upload datasets, and to receive sighting data streams from stand-alone citizen science reporting platforms. The predominant citizen science sources for the insect orders of interest (see below) were Bowerbird (http://www.bowerbird.org.au/), iNaturalist (https://www.inaturalist.org/), QuestaGame (https://questagame.com/) and direct citizen uploads. The predominant source of uploads from professionals was from museums within Australia's seven states and territories participating in the Online Zoological Collections of Australian Museums (https://www.ozcam.org.au/) and scientific collection expeditions. Uploads from professionals for the insect orders of interest out-numbered those by citizens by a factor of c. 50, but note that this figure is highly dynamic.
Cases (\(n=278\)) were species for which at least one record by a citizen source was uploaded through the Atlas of Living Australia (ALA) portal in the two years up until 30 June 2016. Controls (\(n=196\)) were a weighted (by number of observations) sample of all species within the ALA for which there were zero records by citizens over the same period. Only the orders Coleoptera and Hemiptera were considered, as these orders encompass the vast majority of emergency plant pests (EPPs). The Hemiptera in particular appear particularly difficult to prevent from invading and are typically not detected on incursion pathways (Caley et al. 2015).
For each species, we assessed the following:
Order (Coleopteran or Hemipteran)
Body length (mm)
Colour—rated on a scale from 0 (no colour) to 4 (Vividly coloured or Very highly coloured)
Pattern—rated on a scale from 0 (no pattern) to 4 (Very highly patterned or ornate)
Morphology—rated on a scale from 0 (no morphology of interest) to 4 (Unique or spectacular morphology)
Range size (\(\hbox {km}^2\))—minimum convex polygon of all ALA records
Observation intensity (\(\hbox {km}^{-2}\))—Density (intensity) of all citizen science reports for all insect species within the range over the 2-year period (\({\tilde{x}}\) = 0.26 \(\hbox {km}^{-2}\), \({\bar{x}}\) = 0.7 \(\hbox {km}^{-2}\), 95% C.I. = 0.001–2.4 \(\hbox {km}^{-2}\))
Pest status (Logical)—Result of naïve internet search for evidence of the species being a pest (see below for more details).
Examples of scoring for colour, pattern and morphology are provided in the Supplementary Information (Figures S1, S2 and S3). Scoring was undertaken by two of the authors (PC & MW). The internet search for evidence of being a pest included three searches. First, a direct Google search including the terms "Genus species" AND "Pest", a second search within Google Scholar using the same search terms and finally a search within the Pests and Diseases Information Library (PaDIL, http://www.padil.gov.au/) using the taxon name only. Hits were checked for relevance, with searching stopped either as soon as an article was found that clearly identified the taxon as being a pest (in any environment), or hits stopped containing both required search terms. We did not attempt to assess impact, for as the thrust of the work relates to citizen's motivation to report a taxon, this albeit subjective definition of pest status suffices (i.e. the taxon has been recorded behaving in a way that is considered a pest).
We use two methods of analysis for classifying whether a species will be detected and reported. The first, logistic regression, produces easily interpreted coefficients (e.g. the effect of factor X is to increase the odds of reporting by Y). The second, random forests (Breiman 2001), is essentially a form of data mining whose performance (discriminatory ability) we would a priori expect to be close to the maximum obtainable. The downside is that interpreting the influence of the covariates from the many individual classification and regression trees within the "forest" so generated is not straightforward, although the relative contribution and importance of the covariates can be assessed.
Logistic regression models the reporting probability onto the ALA via citizen science platforms as a linear function of the covariates (the "linear predictor") as:
$$\begin{aligned} {\mathrm{logit}}(p)&= \beta _0^* + \beta _1{\mathrm{Order}} + \beta _2{\mathrm{Size}} + \beta _3{\mathrm{Colour}} + \beta _4{\mathrm{Pattern}} \\&\quad +\,\beta _5{\mathrm{Morphology}} + \beta _6\log _{10}{\mathrm{(Range)}}\\&\quad +\,\beta _7\log _{10}{\mathrm{(ObsIntens)}}+\beta _8{\mathrm{(Pest)}} \\&= \mathbf{\beta}^{*'}{\mathbf{x}} \end{aligned}$$
where \(p=Pr({\mathrm{Reported}}\, | \,{\mathrm{Covariates}} \bigcap {\mathrm{Sampled}})\) and \(\mathbf {\beta ^{*'}} = (\beta _0^*, \beta _1, \ldots , \beta _k)\) are the coefficients for the k covariates \({\mathbf {x}}\).
Note that the asterisk(*) for \(\mathbf {\beta }\) in Equation 1 signifies that this is a biased estimate of the intercept as a result of the case–control sampling process (see below). The logit transformation of a probability (p) is defined as the log of the odds. That is:
$$\begin{aligned} {\mathrm{logit}}(p)=ln\left( \frac{p}{1-p}\right) . \end{aligned}$$
Treating the scoring variables as continuous could be criticized; however, the purpose of the model is primarily for classification, and the approach facilitates better communication of the covariate effects on reporting probabilities for less quantitative readers.
The random forest model was fitted to the same set of covariates using the default parameter settings and a forest size of 1000 trees.
Reporting probabilities over the 2-year period were converted to yearly reporting probabilities assuming citizen reporting effort could be approximated as constant across years.
Model evaluation
We evaluated the classification performance of the logistic regression model using 10-fold cross-fold validation, whereby the data were randomly divided into 10-folds, which were held out in turn and classification errors assessed. The ten values were then averaged to provide an overall estimate of classification errors expected during prediction. For the random forest model, the in-built out-of-the-bag (OOB) error rate was used as an estimate of the classification error when predicting.
The 10-fold cross-validation performance for the logistic regression model (Sensitivity \(=\) 89%, Specificity \(=\) 83%, Overall error rate \(=\) 13.5%) slightly outperformed the out-of-the-bag error rates of the random forest (Sensitivity \(=\) 89%, Specificity \(=\) 77%, Overall error rate \(=\) 16%). Armed with this knowledge that the logistic regression model was at least as good as the data-mining alternative, we used it for prediction and interpretation.
Model prediction
Predicting the probability of reporting given only the covariates requires explicit formulation that accounts for proportion of cases sampled (\(P_1\)) and controls sampled (\(P_0\)). The appropriate equation (Keating and Cherry 2004) is :
$$\begin{aligned} Pr({\mathrm{Reported}}\,| \,{\mathrm{Features}}) = \frac{\exp \left( \mathbf {\beta }^{*'}{\mathbf {x}}-ln\left( \frac{P_1}{P_0}\right) \right) }{1 +\exp \left( \mathbf {\beta }^{*'}{\mathbf {x}}-ln\left( \frac{P_1}{P_0}\right) \right) } \end{aligned}$$
where \(\mathbf {\beta }^{*'}{\mathbf {x}}\) is either the linear predictor described by Equation 1, or the logit-transformed probability arising from the random forest model.
Model prediction with application to exotic insects
We used Eq. 3 to estimate the reporting probability for high priority pests of concern to Plant Health Australia of cross-sectoral concern. To do this, these species were scored for size, colour, pattern and morphology using the same criteria as those applied to the ALA records. The incursion size was arbitrarily set at 100 \(\hbox {km}^2\) (10 km \(\times\) 10 km), the observation intensity set to the median (0.26 \(\hbox {km}^{-2}\)), and the species was considered be present as a pest. The model estimates can be rerun for different desired combinations of observation intensity and outbreak size, depending on what size outbreak authorities consider they are capable of eradicating. The 100 \(\hbox {km}^2\) was chosen as it is a figure bandied around by management agencies when considering the largest sized insect invasion that they have sufficient resources for there to be a reasonable chance of eradication.
Analyses were undertaken using the R software environment R Development Core Team (2017), including use of the "randomForest" package (Liaw and Wiener 2002).
Effect of features on reporting
The features we recorded had a very large impact on the estimated reporting probabilities via citizen science platforms into the ALA (Table 1). The probability of a beetle being reported was considerably higher than a bug (odds ratio = 2.2, Table 1), possibly reflecting the popularity of beetle collecting. Species considered pests had a much higher reporting probability (odds ratio = 15.4, Table 1), possibly arising from the increased visibility that their plant damage brings, but also probably arising from their higher abundance and range. The estimated range of the species and the estimated activity of citizen reports also had a significant positive effect on the probability of reporting (Table 1). In terms of the features of the beetles and bugs considered, those species not reported through the ALA citizen science channels are typically smaller, less colourful, less patterned and morphologically uninteresting such as the commonly found black larder beetle (Fig. 2b), despite being a household pest, compared with those that are reported (Table 1). Indeed, despite the large number of sightings uploaded, some widespread common pest species have not been uploaded as of 30 June 2016. A further example of a widespread though unrecorded is the green peach aphid (Myzus persicae) (Fig. 3a), despite it causing considerable economic loss during the period of the study by vectoring beet western yellows virus during the spring of 2014.
Features, associated coefficients, odds ratios (coefficients exponentiated to base e) and associated confidence intervals (C.I.) influencing the citizen reporting probability of insect species to the Atlas of Living Australia
Coefficient (\(\beta\))
Odds ratio (\(exp(\beta )\))
95% C.I.
Order\(^1\)
\(\beta _1\)
2.2 (Beetles)
1.1 (per mm)
2.0 (per unit score)
2.2 (per 10-fold increase)
Observation intensity
15.4 (Yes)
Values in parentheses indicate either the level of the factor associated with the coefficient, or for continuous measures the scaling of the parameter for a unit change associated with the coefficient. See Eq. (1) for model detail
\({}^1\)Only species from the Hemipteran (Bugs) and Coleopteran (Beetles) orders were considered
Contrasts in morphology. The rarely sighted Blackburnium acutipenne has been uploaded to the Atlas of Living Australia via citizen science channels as of 30 June 2016, whereas the ubiquitous black larder beetle (Dermester ater) has not. a Scarab beetle (Blackburnium acutipenne). By Mark Golding https://images.ala.org.au/image/view/136812726. Licensed CC BYNC4.0 https://creativecommons.org/licenses/by-nc/4.0/. b The black larder beetle (Dermester ater). By Simon Hinkley & Ken Walker, Museum Victoria http://www.padil.gov.au/pests-and-diseases/pest/main/135708. Licensed CC BY 3.0 AU https://creativecommons.org/licenses/by/3.0/au/
Contrasts in colour and pattern. Despite being a widespread pest, the green peach aphid has a low citizen science reporting probability on account of its small size (1–2 mm) and "plain" looks. In contrast, Calomela parilis from the leaf beetle genus is a citizen observers treasure. a The green peach aphid (Myszus persicae). By David Cappaert https://www.forestryimages.org/browse/detail.cfm?imgnum=5422731. Licensed CC BY-NC 3.0 US https://creativecommons.org/licenses/by-nc/3.0/us/. b A scarab beetle (Calomela parilis). By Martin Lagerway http://bowerbird.org.au/observations/38151. Licensed CC BY3.0 AU https://creativecommons.org/licenses/by/3.0/au/.
Predicted reporting probabilities
When the model was applied to a subset of the Plant Health Australia cross-sectoral high priority pest species (HPPs) for a given set range size (100 km\(^2\)) and median citizen science observation intensity, the estimated yearly reporting probability ranged from a low of 2% (e.g. sugarcane sidewinder) to near 97% (Lychee longicorn beetle) (Tables S1 & S2 in Supplementary Material). Generally speaking, HPPs with very low estimated probabilities of reporting are dominated by Hemipterans (Table S2 in Supplementary Material), whilst those with high probabilities of reporting are dominated by the Coleoptera (Table S1 in Supplementary Material).
Insects that have high estimated reporting rates include the Colorado potato beetle, for which its size, colour and distinctive pattern (Fig. 4a) result in an estimated yearly reporting probability of 0.78. In contrast, the Russian wheat aphid (Fig. 4b) has a low predicted citizen reporting probability of 3%.
The Colorado potato beetle is predicted to have a high (98%) citizen reporting probability in comparison with the Russian wheat aphid (4%). a The Colorado potato beetle (Leptinotarsa decemlineata). By Scott Bauer USDA ARS [Public domain] https://commons.wikimedia.org/wiki/File:Colorado_potato_beetle.jpg, via WikimediaCommons. b Russian wheat aphid (Diuraphis noxia). By Kansas Department of Agriculture https://www.forestryimages.org/browse/detail.cfm?imgnum=5512060. Licensed CC BY-NC 2.0 https://creativecommons.org/licenses/by-nc/2.0/.
Our model has quantitatively inferred the extent to which size, colour, pattern and morphology all influence the citizen reporting probability of insects. Although this finding is unsurprising, this is the first time that the reporting probability and the factors that influence it have been quantified. This enables a more objective evaluation of the contribution of unstructured online reporting platforms to plant biosecurity surveillance. Importantly, when applied to exotic pests of biosecurity concern, we have inferred that the passive citizen reporting probabilities for many (particularly small, nondescript bugs) would be considered insufficient for biosecurity surveillance needs.
Recent incursions of exotic pests into Australia support this estimated low reporting probability. For example, the incursion of the Russian wheat aphid went unreported by citizen scientists for possibly two years whilst spreading over a considerable area in southern Australia. It was first detected by strategic surveillance. Likewise, the tomato potato psyllid (Bactericera cockerelli), another small, unremarkable species was widespread, occurring on hundreds of premises in Western Australia before being detected and reported.
The citizen surveillance we have described here contributes to what is termed "general surveillance" within plant health (Hammond et al. 2016a), which is a catch-all phrase for describing surveillance that is not targeted. General surveillance activities are an important part of early detection and demonstrating area freedom (Hammond et al. 2016b). The estimated citizen reporting probabilities we have estimated here can be used to infer the likely sensitivity of general surveillance for exotic species from the "passive" citizen component. The predictions we have made here are simplistic in how they have chosen the citizen science observation intensity (simply using the median). In reality, the citizen observation intensity will vary greatly depending on the incursion location in relation to citizen science activity. The implications of this could be explored in more detail (see Pocock et al. 2017, for an example) and used as a means of directing where targeted surveillance could augment citizen surveillance. This is an area of further work.
The implication of these results for citizen reporting as a form of surveillance will vary depending on the features of the pest species of concern. Industries and environmental managers whose assets are potentially impacted by species with low reporting probabilities will clearly need to implement more structured/active surveillance if they require higher surveillance sensitivity. The citrus industry, for example, probably cannot rely on passive unstructured citizen science data streams for surveillance of D. citri—some form of structured surveillance will be required. In contrast, the forestry industry may consider that citizen detection and reporting of species of pine sawyers may be sufficient for their needs. Incorporating such inferred citizen surveillance reporting probabilities into the general surveillance framework is an area for further research. Targeted use of social media shows promise.
It is well known that citizen reporting rates are heavily biassed in space and time (Isaac and Pocock 2015), along with the visibility of the organism in question. Here, we have demonstrated quantitatively further inter-species reporting biases relating to perceptions (pattern, colour, morphology). This finding is generalizable to most unstructured citizen science reporting platforms relating to animals and plants. Although we have demonstrated the importance of physical features and availability for citizen reporting probability, motivations for reporting insects using online reporting portals are likely diverse and may change with time. This will be an ongoing challenge for the use of citizen surveillance.
Petra Kuhnert and Chris Wikle participated in useful discussions on the paper format. Rieks van Klinken made useful comments on an earlier draft. The comments of Michael Pocock and an anonymous referee further improved the manuscript. We thank them all.
PC and SCB were involved in conceptualization; PC and MW undertook data curation; PC, SCB and MW undertook formal analysis; SCB helped with funding acquisition; PC and MW were involved in investigation; PC, SCB and MW contributed to the methodology; MW and PC were involved in project administration; PC and MW contributed to software; PC undertook project supervision; PC helped in validation; PC was involved in visualization; PC and MW wrote the original draft; PC and MW contributed to writing, review and editing.
The study was funded by the Australian Government's Plant Biosecurity Cooperative Research Centre Program (Plant Biosecurity CRC Project 1029).
10340_2019_1115_MOESM1_ESM.pdf (4 mb)
Supplementary material 1 (PDF 4701 KB)
ALA (2016a) Atlas of Living Australia occurrence download at http://biocache.ala.org.au/occurrences/search?&q=species_group%3AInsects+country%3AAustralia+basis_of_record%3AHumanObservation+matched_name_children%3AHEMIPTERA+occurrence_date%3A%5B*+TO+2016-06-30T00%3A00%3A00Z%5D. Accessed on 30 Aug 2016
ALA (2016b) Atlas of Living Australia occurrence download at http://biocache.ala.org.au/occurrences/search?&q=species_group%3AInsects+country%3AAustralia+matched_name_children%3ACOLEOPTERA+occurrence_date%3A%5B*+TO+2016-06-30T00%3A00%3A00Z%5D/. Accessed on 30 Aug 2016
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Caley, P., Welvaert, M. & Barry, S.C. J Pest Sci (2020) 93: 543. https://doi.org/10.1007/s10340-019-01115-7
Received 07 October 2018
Revised 24 March 2019
First Online 11 June 2019 | CommonCrawl |
A dynamic model and some strategies on how to prevent and control hepatitis c in mainland China
Wanru Jia1 na1,
Jie Weng2,
Cong Fang3 &
Yong Li ORCID: orcid.org/0000-0003-3744-40541,4 na1
Hepatitis C virus (HCV) is a leading cause of chronic liver disease. As yet there is no approved vaccine protects against contracting hepatitis C. HCV seriously affects many people's health in the world.
In this article, an epidemiological model is proposed and discussed to understand the transmission and prevalence of hepatitis C in mainland China. This research concentrates on hepatitis C data from Chinese Center for Disease Control and Prevention (China's CDC). The optimal parameters of the model are obtained by calculating the minimum chi-square value. Sensitivity analyses of the basic reproduction number and the endemic equilibrium are conducted to evaluate the effectiveness of control measures.
Vertical infection is not the most important factor that causes hepatitis C epidemic, but contact transmission is. The proportion of acute patients who are transformed into chronic patients is about 82.62%. The possibility of the hospitalized patients who are restored to health is about 76.24%. There are about 92.32% of acute infected are not treated. The reproduction number of hepatitis C in mainland China is estimated as approximately 1.6592.
We find that small changes of transmission infection rate of acutely infected population, transmission infection rate of exposed population, transition rate for the acutely infected, and rate of progression to acute stage from the exposed can achieve the purpose of controlling HCV through sensitivity analysis. Finally, based on the results of sensitivity analysis, we find out several preventions and control strategies to control the Hepatitis C.
Hepatitis C virus(HCV) seriously affects lots of people's health in the world. Recently (18 July, 2018), the World Health Organization (WHO) estimates that approximately 71 million people have chronic hepatitis C virus (HCV) infection worldwide and approximately 399,000 people die each year after HCV diagnosis, mostly from cirrhosis and hepatocellular carcinoma (HCC) [1]. An estimated 3.5 million people in the United States (US) has chronic hepatitis C [2]. In 2016, there are 18,153 hepatitis C-related deaths in the US which is lower than from 2012 to 2015 (18,650 to 19,629) [3]. In the European region, approximately 14 million people are chronically infected with HCV, representing about 20% of the global burden of disease due to HCV infection [4]. The areas where have the highest reported prevalence rates locate in Africa and Asia, and China in the Asia whose citizens account for about one fifth of the world's populace, has a reported seroprevalence about 3% [5–7].
HCV was discovered in 1989 by Choo et al. [8, 9], it is a small, enveloped, single-stranded ribonucleic acid (RNA) virus, which be part of the Flaviviridae family. Hepatitis C is an infectious disease caused by HCV which basically affect the liver. The spread ways of the virus are blood transmission, sexual transmission and mother-to-child transmission, but the leading way is blood transmission, such as sharing injection equipment, inputing the contaminated blood or blood products, tattooing [10]. As yet there is no approved vaccine to protect against contracting hepatitis C. The focus of prevention efforts should be safe blood supply in the developing world, safe injection practices in health care and other settings, and less amount of people who inject drugs [11]. In those persons who do develop symptoms, the mean time period from exposure to symptom onset is 3–12 weeks (range: 2–24 weeks) [12, 13]. HCV infection has both acute and chronic forms, the incubation for chronic HCV can be between 14 to 180 days [12]. Acute hepatitis C infection is hard to diagnose, because 70% to 80% of the patients are symptomless [13, 14]. Most of them are unconscious of their exposure to HCV, and fail to get diagnosed in time until the occurring of the secondary symptoms to the liver. Some studies show, however, the acute infection phase is very impressionable to treatment, so it is an unique occasion to prevent the evolution of chronic infection [15]. Chronic hepatitis C can bring about cirrhosis and HCC. The average rate of progression of the disease is extremely slow. Using data collected in Japan, investigators estimate that, following acute infection, chronic hepatitis could be ensured 13.7±10.9 years later, chronic active hepatitis could be ensured 18.4±11.2 years later, cirrhosis of the liver could be ensured 20.6±10.1 years later, and hepatocellular carcinoma could be ensured in 28.3±11.5 years [16, 17].
Some mathematical models were used to analyze the spread of hepatitis C disease and come up with some effective strategies. Martcheva M and Castillo-Chavez C [10] considered an epidemiological model with a chronic infectious phase and variable population size, and the analysis consequences revealed that treatment strategies directed forward speeding up the transition from acute to chronic stage in effect conduce to the eradication of the diseases. This model was extended by Das P et al. [15] who incorporate the immune class and was also extended by Yuan J [18] who consider the latent period. Imran M [19] formulated epidemic models of hepatitis C considering an isolation class and analyzed the effects of the isolation class on the transmission dynamics of the disease. Mathematical modeling of hepatitis C treatment for injecting drug users (IDUs) were studied in [20–22] where the treated individuals are supposed not to infect the susceptible individuals. Lately, there are some researches [23, 24] about hepatitis C epidemic cases which suggest some measures to control hepatitis C infection continental China. But these models did not consider the vertical infection. It is not effortless to diagnosis due to the shortage of the residents' consciousness and the characteristics of the patients with hepatitis C, so it is probable that patients will transmit HCV to their children.
The aim of this work is to use mathematical modelling to investigate the influences of hepatitis C, then probe and draw some conclusions about effective policy. The organization of this paper is as follows. In the next section, an epidemic model for hepatitis C is proposed to prevent and control the infectious disease. Then we acquire its optimal parameter values by Matlab tool fmincon and compare the reported data and simulative results. Sensitivity analyses of the basic reproduction number and the endemic equilibrium are performed in "Results" section. After that, discussion on the model parameters and the main factors affecting the spread of hepatitis C in "Discussion" section, and we end this article with how to control the hepatitis C in "Conclusions" section.
We have found clinical cases of hepatitis C in China every month from 2011 to 2016 from the China Center for Disease Control and Prevention (China's CDC), which is a public welfare institution organized by the Chinese government to implement state-level disease prevention and control and public health technology management and services. China's CDC conducts monthly statistics on patients infected with hepatitis C virus in mainland China (i.e., except Hong Kong, Macao and Taiwan) [25] including gender, occupation, date of birth, address, date of onset, date of diagnosis, especially the classification of the disease, which is marked as a clinically diagnosed case.
In general, it is unreasonable to determine HCV infections just by relying on HCV antibody positive which just means you were infected before. To determine whether infected with HCV, HCV-RNA test needs to be done. Once the HCV-RNA test results indicate that the outpatient is infected with the hepatitis C virus, he or she will need hospitalization. In the case of ignoring the patient's home treatment, we believe that the data provided by the China's CDC is the number of hospitalizations.
By producing re-sampling a larger artificial data set, which is generated based on the existing limited reported monthly data, using the linspace function from Matlab (the Mathworks, Inc.), we interpolate the 12-month data and turn into 365-day data. In order to keep the total number of data, the interpolation formula of each year as following:
$$\begin{array}{@{}rcl@{}} {\hat D_{2}}({t_{j}}) = \frac{{{D_{2}}({t_{j}})\sum\limits_{i = 1}^{12} {{D_{1}}({s_{i}})} }}{{\sum\limits_{j = 1}^{365} {{D_{2}}({t_{j}})} }},j = 1,2, \cdots,365, \end{array} $$
where, D1(si),i=1,2⋯,12, denote the 12-month actual data, D2(tj),j=1,2⋯,365, denote the 365-day data after the interpolation. \({\hat D_{2}}({t_{j}}), j=1,2 \cdots, 365,\) denote the 365-day data after the zoom. With the aid of linear interpolation, we will obtain more useful data, and the fit results will be better. We still give a comparison chart for each month's case data and simulative data.
Model formulation
In order to study the epidemic of hepatitis C in China, we consider the hepatitis C model is homogeneous mixing-an individual has an equal chance of contacting any individual among the population, by ignoring the impacts of the space structure and seasonal changes to simulate the data year after year, and we assume that natural birth rate is equal to natural mortality.
The mathematical model for hepatitis C to understand the transmission dynamics and prevalence consists of a system of ordinary differential equations, where population is divided into six subgroups: susceptible S(t), exposed E(t) (defined as not infected but infectious), acute infection Ia(t), chronic infection Ic(t), treated T(t) and recovered R(t) individuals. The total population size is denoted by N(t)=S(t)+E(t)+Ia(t)+Ic(t)+T(t)+R(t).
New susceptible individuals enter into the S compartment with a recruitment rate Λ. Let μ be the natural birth and death rate of the population. By the influence of their parents, generations of the individuals in the E(t),Ia(t),Ic(t) may be infected with HCV at rate of l,m,n, respectively. This is what is called vertical infection. Susceptible individuals are infected by contacting with patients in the E(t),Ia(t),Ic(t) compartments at rates of β1,β2,β3, respectively. Once infected, the individuals move into the exposed compartment (E) and then progress to the acute stage at a rate of σ. In the acute stage, the individuals may die at rate of d1. Let α be the transition rate for the acutely infected individuals. In the conversion of acute infection, the individuals will restore health relying on their own immune system with the ratio ρ1, progress to the chronic stage with the ratio ρ2, go to the hospital for treatment with the ratio 1−ρ1−ρ2. At the same, the individuals may die at rate of d2 in the chronic stage. Let δ be the transition rate for the chronically infected individuals. In the conversion of chronic infection, the individuals will restore health relying on their own immune system with the ratio p1, go to the hospital for treatment with the ratio 1−p1. Individuals in the treated compartment (T) who have the transition rate of λ, succeed in clearing HCV and move to the recovered compartment (R) with the ratio η1, while the others fail and move back to the chronic stage with the ratio 1−η1. Individuals in the R compartment lose their immunity and eventually return to the susceptible compartment (S) at rate of γ. The schematic flow diagram illustrating the transmission dynamics of the HCV infection with treatment are illustrated in Fig. 1. And the biological meanings and acceptable ranges of all parameters are listed in Table 1.
Flow chart of compartments of hepatitis C model
Table 1 Model parameters and their interpretations
The model is represented by the following system of ordinary differential equations:
$$\begin{array}{@{}rcl@{}} {\begin{aligned} \left\{ \begin{array}{l} \frac{{dS}}{{dt}} \,=\, \Lambda \,-\, \mu lE \,-\, \mu m{I_{a}} \,-\, \mu n{I_{c}} \,-\, \frac{{\left({{\beta_{1}}E + {\beta_{2}}{I_{a}} + {\beta_{3}}{I_{c}}} \right)S}}{N} \,+\, \gamma R \,-\, \mu S,\\ \frac{{dE}}{{dt}} \,=\, \frac{{\left({{\beta_{1}}E + {\beta_{2}}{I_{a}} + {\beta_{3}}{I_{c}}} \right)S}}{N} \,-\, \sigma E - \mu E + \mu lE + \mu m{I_{a}} + \mu n{I_{c}},\\ \frac{{d{I_{a}}}}{{dt}} \,=\, \sigma E - \alpha {I_{a}} - \mu {I_{a}} - {d_{1}}{I_{a}},\\ \frac{{d{I_{c}}}}{{dt}} = {\rho_{2}}\alpha {I_{a}} - \delta {I_{c}} + \left({1 - {\eta_{1}}} \right)\lambda T - \mu {I_{c}} - {d_{2}}{I_{c}},\\ \frac{{dT}}{{dt}} = \left({1 - {\rho_{1}} - {\rho_{2}}} \right)\alpha {I_{a}} + \left({1 - {p_{1}}} \right)\delta {I_{c}} - \lambda T - \mu T,\\ \frac{{dR}}{{dt}} = {p_{1}}\delta {I_{c}} + {\rho_{1}}\alpha {I_{a}} + {\eta_{1}}\lambda T - \gamma R - \mu R. \end{array} \right. \end{aligned}} \end{array} $$
The biologically feasible region \(\Omega = \{ (S,E,{{I}_{a}},{{I}_{c}},T,R) \in \mathbb {R}_ +^{6}:S + E + {I}_{a} + {I}_{c} + T + R < \frac {\Lambda }{\mu }\} \) is a positively invariant set of system (1).
The basic reproduction number (\({\mathcal {R}_{0}}\)) represents the number of infected during the initial patient's infectious (not sick) period. What this threshold will do determine whether a disease will die out (if \({\mathcal {R}_{0}}< 1\)) or become epidemic (if \({\mathcal {R}_{0}}> 1\)). For models with complex dynamics, \({\mathcal {R}_{0}}< 1\) is not the only condition to guarantee that the disease is extinct, but the smaller the better. Following Van den Driessche P and Watmough J [26], the basic reproduction number for the model (1) is given by the formula:
$$\begin{array}{@{}rcl@{}} {\begin{aligned} \begin{array}{l} {\mathcal{R}_{0}} = \left({{\beta_{1}} + \mu l} \right)\frac{1}{{\sigma + \mu }} + \left({{\beta_{2}} + \mu m} \right)\frac{\sigma }{{\left({\sigma + \mu} \right)\left({\alpha + \mu + {d_{1}}} \right)}} + \left({{\beta_{3}} + \mu n} \right)A\\ ={\mathcal{R}_{01}}+{\mathcal{R}_{02}}+{\mathcal{R}_{03}}+{\mathcal{R}_{04}}+{\mathcal{R}_{05}}+{\mathcal{R}_{06}}, \end{array} \end{aligned}} \end{array} $$
where, \({\mathcal {R}_{01}}=\frac {{{\beta _{1}}}}{{\sigma + \mu }}, {\mathcal {R}_{02}}=\frac {{{\beta _{2}}\sigma }}{{\left ({\sigma + \mu } \right)\left ({\alpha + \mu + {d_{1}}} \right)}}, {\mathcal {R}_{03}}={\beta _{3}}A, {\mathcal {R}_{04}}=\frac {{\mu l}}{{\sigma + \mu }}, {\mathcal {R}_{05}}=\frac {{\mu m\sigma }}{{\left ({\sigma + \mu } \right)\left ({\alpha + \mu + {d_{1}}} \right)}}, {\mathcal {R}_{06}}=\mu nA, A = \frac {{\alpha \sigma \left [ {{\rho _{2}}\left ({\lambda + \mu } \right) + \lambda \left ({1 - {\eta _{1}}} \right)\left ({1 - {\rho _{1}} - {\rho _{2}}} \right)} \right ]}}{{\left ({\sigma + \mu } \right)\left ({\alpha + \mu + {d_{1}}} \right)\left [ {\left ({\delta + \mu + {d_{2}}} \right)\left ({\lambda + \mu } \right) - \delta \lambda \left ({1 - {p_{1}}} \right)\left ({1 - {\eta _{1}}} \right)} \right ]}}.\)
\({\mathcal {R}_{01}}, {\mathcal {R}_{02}}\) and \({\mathcal {R}_{03}}\) represent the average numbers of the infected individuals by a single exposed, acute infection or chronic infection individual in a fully susceptible population, respectively. \({\mathcal {R}_{04}}, {\mathcal {R}_{05}}\) and \({\mathcal {R}_{06}}\) represent the average numbers of the infected infants by the exposed, acute infection or chronic infection parents, respectively. They represent the contributions of the 6 HCV transmission ways to the the basic reproduction number \({\mathcal {R}_{0}}\).
Parameter estimation
In this section, we first use model (1) to simulate the reported hepatitis C data of China from January 2011 to December 2016 to predict the trend of the disease and seek of some preventions and control measures. The data are obtained mainly from epidemiologic bulletins published by the China's CDC [25]. Assume that the person's natural death follows a uniform distribution, then natural death rate is calculated as μ=1/(74.83×365)=3.6613×10−5, since life expectancy is 74.83 years old between 2011 to 2016 in China [27]. From Shen M [24], the range of the transmission rates \({\tilde \beta _{\mathrm {i}}}, i=1,2,3\) is [2.0846,3.0769]×10−11, and those annual transmission rates are bilinear. Total population is about 1.35×109 in China between 2011 to 2016 [27], We chose 80% of the population as the sampled population, and denote as \(\tilde N=1.08\times {10}^{9}\). So we estimate the standard rate \({\hat \beta _{i}} = {\tilde \beta _{i}}\tilde N \in [0.0225,0.0323], i=1,2,3\). The values of β1,β2 and β3 in model (1) are chosen randomly in this interval.
Then, we have to estimate the other 15 parameters and 6 initial values every year through calculating the minimum sum of chi-square [28, 29]
$$\begin{array}{@{}rcl@{}} J(\theta) = \sum\limits_{i = 1}^{72} {\frac{{{{(T({t_{i}}) - \hat T({t_{i}}))}^{2}}}}{{\hat T({t_{i}})}}} \end{array} $$
with the MATLAB (the Mathworks, Inc.) tool fmincon that is a part of optimization toolbox. Where, T(ti),i=1,2,⋯,72 show the true value each month, \({\hat T({t_{i}})}, i=1,2,\cdots,72\) show the estimated value each month. Fmincon function is a Matlab function for solving the minimum value of constrained nonlinear multivariate function. Fmincon implements four different algorithms: interior point, sequence quadratic program (SQP), active set, and trust region reflective. In this paper, we choose the SQP algorithm to solve the optimal solution of model (1). MATLAB SQP method is divided into three steps: firstly, update the Lagrangian Hessian matrix, then solve the quadratic programming problem, and finally calculate the one-dimensional search and objective function.
According to the epidemiological characteristics of hepatitis C and the biological significance of the parameters, we set the lower and upper boundaries of each parameter, as shown in Table 1. Although the outbreak of hepatitis C is not seasonal, it still has a certain periodicity. Our model does not have a periodic solution, so we can only simulate the annual parameter values separately. The simulated annual parameter values are shown in Table 2. Taking year as the research unit, the parameters of the model (1) vary from year to year because of the annually different natural conditions and environmental factors, but the same parameters are not significantly different in different years.
Table 2 Annual simulation values of the parameters between 2011 and 2016 and \({\mathcal {R}_{0}}\)
The values of the various parameters in Table 2 are in days. We calculated the numbers of the treated in each month of each year according to the optimal simulation parameters, then, compared it with the reported hepatitis C data in China from 2011 to 2016 per month. We use two broken line diagrams, as shown in Fig. 2. The data presented in Fig. 2 refers to the clinical data from China's CDC, denoted by T. And the numerical results are found to be a good match with the data of hepatitis C in China from 2011 to 2016 except one point which represent the number of treated patients in June 2013. So we guess the abnormality of this data could be related to the emergence of new avian influenza H7N9 [30] and the 7.8-magnitude earthquake in Ya'an, Sichuan province [31] in China in April of that year. Our model is based on the ideal state, without considering the impact of unexpected events, so the model is not able to capture that outbreak.
The comparison between the reported hepatitis C in China from 2011 to 2016 and the simulation of model (1)
We found the optimal parameter values and the initial values of the model in 2011 after continuous debugging, then, using the optimal parameter values of the model in 2011 as the starting value, we have found the optimal parameter values of each subsequent year through continuous simulation. Where, the optimal values of parameters are listed in Table 2, and the each initial condition from 2011 to 2016 is fixed as (4.23×107,3.81×105,102,7.46×104,4.69×102,4.08×107),(3.66×107,8.54×104,1.16×102,3.08×104,4.36×102,1.10×107),(9.01×107,4.18×105,1.18×102,105,6.53×102,4.80×107),(7.07×107,4.64×105,6.51×104,6.12×104,5.77×102,8.98×107),(5.36×107,3.48×105,9.10×104,6.17×104,6.98×102,4.52×107),(2.66×107,9.71×105,102,8.81×104,6.46×102,5.83×107). Here, we also calculate the basic reproduction number each year. The average value of the 6 basic reproduction numbers is estimated as approximately 1.6592. Using the optimal parameters of 2015, one also calculates \({\mathcal {R}_{01}}=0.6701, {\mathcal {R}_{02}}=1.0050, {\mathcal {R}_{03}}=0.0958, {\mathcal {R}_{04}}=1.0992\times {10}^{-4}, {\mathcal {R}_{05}}=1.5888\times {10}^{-6}, {\mathcal {R}_{06}}=1.5094\times {10}^{-5}\). Hence, vertical infection is not the main factor that cause hepatitis C epidemic, but transmission of HCV from exposed and infection to others is the most important factor. We will discuss this argument again in next sections. However, because of China's big population base, vertical infection is still worthy of our attention.
Sensitivity analysis of \({\mathcal {R}_{0}}\)
In this section we performed a sensitivity analysis of the basic reproduction number to determine several parameters that have the most influential parameters on the prevalence and transmission of hepatitis C. Sensitivity analysis is a useful tool to identify how closely input parameters are related to predictor parameters and it helps to determine level of change necessary for an input parameter to find the desire value of a predictor parameter [32, 33]. If a small change in a parameter can cause a large change in the number of the basic reproduction number, then this parameter is called a sensitivity factor, otherwise called an insensitive factor.
In this section, following Samsuzzoha M's [32] method, we used the 2015 simulated parameter values to perform a sensitivity analysis of the basic reproduction number, thus we can put some effective control strategies of HCV. The sensitivity indices of each parameter to the basic reproduction number \({\mathcal {R}_{0}}\) are shown in Table 3.
Table 3 Sensitivity indices of \({\mathcal {R}_{0}}\), the corresponding % changes needed to affect a 1% decrease/increase in the value of \({\mathcal {R}_{0}}\)
We can observe that β2,β1,β3,ρ2, l, n, m, λ, (σ,α,δ,η1,p1,ρ1) have positive (negative) impacts on \({\mathcal {R}_{0}}\). The sensitivity indices and corresponding % value needed to affect a 1% decrease in \({\mathcal {R}_{0}}\) are shown in Table 3 (e.g., in order to decrease the value of \({\mathcal {R}_{0}}\) by 1% it is necessary to decrease the value of β2 by 1.7945% or increase the value of σ by 2.7973%.) The greater absolute value of the sensitivity index, the more sensitive the parameter is to \({\mathcal {R}_{0}}\). Therefore, the most sensitive parameter for \({\mathcal {R}_{0}}\) is β2 followed by β1,σ,α,β3,ρ2,δ,η1, l, p1, n, ρ1, m, λ. From Table 3, we can see that parameters l, m, n can be negligible on the influence of the basic reproduction number (\({\mathcal {R}_{0}}\)) compared with the most sensitive parameters β2,β1,σ,α. Hence, vertical infection is not the main factor that cause hepatitis C epidemic in China. In the "Conclusions" section, we will put forward some specific human intervention measures according to the results.
Sensitivity analysis of the endemic equilibrium
In this section, we do a sensitivity analysis of the endemic equilibrium to determine the relative importance of the different parameters which are responsible for the prevalence of equilibrium disease. Using the method from Samsuzzoha M [32], we calculate the sensitivity indices of the endemic equilibrium. The relevant detail calculation is shown in Appendix, and the parameter values are shown in Table 4 by using the parameters values of 2015 given in Table 2. We can see that: the most sensitive parameter for S∗ is α followed by p1,β2,ρ2,β1,σ,δ,η1,ρ1,β3,l,n,m and λ. The most sensitive parameter for E∗ is σ followed by β2,β1,α,p1,δ,β3,ρ1,ρ2,η1,l,n,λ and m. The most sensitive parameter for \({I}_{a}^{*}\) is α followed by β2,β1,σ,p1,δ,β3,ρ1,ρ2,η1,l,n,λ and m. The most sensitive parameter for \({I}_{c}^{*}\) is ρ2 followed by δ,β2,β1,σ,η1,α,p1,β3,ρ1,l,n,λ and m. The most sensitive parameter for T∗ is β2 followed by β1,σ,ρ1,λ,ρ2,η1,α,p1,β3,δ,l,n and m. The most sensitive parameter for R∗ is β2 followed by p1,ρ2,β1,σ,α,η1,ρ1,β3,δ,l,n,λ and m. For the above analysis, we can see that the sensitivity of the four parameters β1,β2,α,σ are at the top of the sensitivity indices of the endemic equilibrium, especially for \({I}_{a}^{*}\), and the sensitivity of ρ2,δ,β2,β1,σ are at the top of \({I}_{c}^{*}\). So if we want to reduce the number of cases, we can propose specific preventive control measures from these parameters in the "Conclusions" section.
Table 4 Sensitivity indices of the endemic equilibrium
From Table 2, according to discuss the arithmetic means of parameters of our model, we have some conclusions as follows: \(\bar l = 5.00\%\) (e.g., \(\bar l = \frac {1}{6}\sum \limits _{i = 2011}^{2016} {{l_{i}}}\), the method of calculating the average value of other parameters is the same.), \(\bar m = 3.37\%, \bar n = 4.91\%\), these suggest that the probabilities of exposed, the acute and the chronic patients spread virus to their kids on hepatitis C are about 5.00%, 3.37% and 4.91%, respectively. \(\bar {\rho _{1}} = 9.70\%\), it shows that the proportion of patients who recover naturally in all acute patients is about 9.70%. \(\bar {\rho _{2}} = 82.62\%\), it shows that the proportion of acute patients who turned into chronic patients is about 82.62%. From Chen SL [13], approximately 75%−85% of infected patients do not clear the virus in 6 months, and become chronic hepatitis patients. \(1-\bar {\rho _{1}} -\bar {\rho _{2}} = 7.68\%\), it indicates that the proportion of acute patients who are treated in hospital is about 7.68%. This result is similar to that of Cox AL's [34], he denotes that 95% of infected are not treated. \(\bar {\eta _{1}} = 76.24\%\), it suggests that the proportion of the resident patients who can recover is about 76.24%. From Seeff LB [35], about 80% of HCV-infected individuals seem to be no progression to end-stage liver disease, but 20% who get histologic fibrosis and cirrhosis will develop into serious end-stage liver disease. And in our paper, \(1-\bar {\eta _{1}} = 23.76\%\), it suggests that the proportion of the resident patients who failed to recover is about 23.76%, while we don't consider that chronic patients develop histologic fibrosis and cirrhosis, which will be our follow-up work. \(1/{{\bar \gamma }}\approx 1226.03\) days, i.e., 3.36 years, it suggests that the average time that the antibody disappear is about 3.36 years. \(1/{{\bar \sigma }}\approx 29.12\) days, it shows that the average incubation time is about 29.12 days. \(1/{{\bar \delta }}\approx 10.39\) days, it shows that the average period of chronic patients deciding whether to be treated or not is about 10.39 days. Then, these conclusions have been conformed to the actual situation [1, 25].
According to the values of the parameters and sensitivity analysis of the basic reproduction number and the endemic equilibrium, we can find that vertical infection is not the primary cause of hepatitis C epidemic in China, the reasons are as follows:
(1)\({\bar R_{04}} = 6.07 \times {10^{- 5}}, {\bar R_{05}} = 4.9 \times {10^{- 5}}, {\bar R_{06}} = 8.85 \times {10^{- 6}}\), these represent the average contribution from the generation of the exposed, the acute and the chronic to the basic reproduction number (\({\mathcal {R}_{0}}\)), respectively. We can observe that vertical infection has little influence on the spread of hepatitis C.
(2)From the result of the sensitivity analysis of \({\mathcal {R}_{0}}\), we can find that parameters l,m,n have negligible influence on the spread of hepatitis C, compared to the most sensitive parameters β2,β1,σ,α (see Table 3 for details).
(3)From the sensitivity analysis of the endemic equilibrium, we can see that parameters l,m,n are not sensitive to it. So reducing the transmission rate of vertical infection has no influence on controlling the scale of patients with HCV (see Table 4 for details).
Therefore, it is reasonable to ignore vertical infection in the existing hepatitis C dynamics models [10, 15, 18–24]. Contact transmission (such as injecting contaminated blood, using public syringe, sexual behavior and so on) is the main factor for the epidemic of the hepatitis C in China, the reasons are as follows:
(1)\({\bar R_{01}} = 0.6759, {\bar R_{02}} = 0.8529, {\bar R_{03}} = 0.1303\) represent the average contribution of the exposed infection, the acute infection and the chronic infection to the basic reproduction number (\({\mathcal {R}_{0}}\)), respectively. We can find that contact transmission has great effect on the spread of hepatitis C.
(2)From the result of the sensitivity analysis of \({\mathcal {R}_{0}}\), we can find that the sensitive indexes of the parameters β1 (the second), β2 (the first), β3 (the fifth) are extremely large (see Table 3 for details).
(3)From the result of the sensitivity analysis of the endemic equilibrium, we can see that the parameters β1,β2,β3 are sensitive to it. So reducing the transmission rate β1,β2,β3 can effectively control the scale of patients with hepatitis C (see Table 4 for details).
In addition, the exposed and the acute infection tend to be asymptomatic, so the susceptible have more chance to contact them. Therefore, contact transmission is the main reason for the epidemic of hepatitis C in China.
In this paper, we constructed an SEIaIcTR dynamic model for hepatitis C transmission based on the reported data from China's CDC to search the most influential parameters. From the last line in Table 2, the basic reproductive number \({\mathcal {R}_{0}}\) in each year is larger than 1. Thus, we conclude that HCV will persist in China under the current conditions. As a matter of fact, there is no effective vaccine for HCV, but if we can provide some preventive measures to control the HCV, it will be very meaningful.
Next, we selected the data of 2016 to simulate the future prevalence trend of hepatitis C in China under various circumstances, and the results were shown in Fig. 3. We can observe that β2,β1,α and σ are the most sensitive parameters comparing with the others because just slight changes can achieve the goal of control. These existing measures to control and prevent HCV can be essentially attributed to how to reduce β2 and β1. Based on the discussion in this paper, it is vitally important not only to reduce β2 and β1, but also to increase α and σ. In addition, it is more effectively to reduce β2 and β1 than to reduce β3 precisely because chronic patients will pay more attention to the contact with others and do a good job of protection than those who do not show symptoms in the incubation and acute period.
Simulation of the sum of not hospitalized infectious Ia(t)+Ic(t) and hospitalized infectious T(t) with all parameters, 2α=2×0.0162=0.0324,2δ=2×0.1=0.2,0.1β2=0.1×0.0246=0.00246,0.1ρ2=0.1×0.87=0.087,0.1β1=0.1×0.0223=0.00223,0.1β3=0.1×0.022=0.0022,2σ=2×0.0272=0.0544 from the seventh column of Table 2, respectively, when one parameter takes a specific value, the other parameters take the value of the seventh column in Table 2
Based on the above analysis, we propose some preventive measures as follows:
(1) It can control the spread of the HCV by reducing infection rate of contacting with the exposed and the acutely infected to the susceptible (β1 and β2) (see Fig. 3). Therefore, it is vital to advocate public education so that we can understand the spread of HCV well and reduce the probability of contacting with the patients. For example, avoid unnecessary injection, transfusion and using of blood products unless go to formally medical health institutions. It is necessary to disinfect strictly for bloody items and the humoral pollutants. Stay away from drugs and educate intravenous drug users to let them know the harm of impurity injection and give them some advice about drug rehabilitation.
(2) It can control the spread of HCV at a lower level by shortening the diagnosis time of acute infection (1/α) and the hesitant time for being treated of chronically infected patients (1/δ) (see Fig. 3). That is, improve the transition rates of the acute (α) and chronic infection patients (δ), especially for α, which has extremely high sensitivity not only to the basic reproduction number but also to the endemic equilibrium. If we often do exercise to improve our immunity, even if we are infected by HCV, we can restore health by autoimmunity. Check your body regularly, and hospital treatment can prevent the disease from aggravation. Although some HCV patients will recover after a period of oral medication at home, it is still necessary to encourage more chronic patients to receive treatment in hospital as quickly as possible, after all, it is more likely to recover and it could contact with less patients in the process of rehabilitation, so that the risk of being infected is also smaller for the susceptible.
(3) It can effectively control the spread of HCV by reducing the diagnosis time of exposed (1/σ), i.e., improve the rate of progressing to acute stage from the exposed stage (σ) (see Fig. 3). Thence, once we fell uncomfortable, we should go to a hospital for diagnosis in time, because the earlier you detect of the illness and treat, the more possibility you can recover [36].
(4) Reduce the proportion of chronic infection from acutely infection population ρ2 (see Fig. 3). From Tables 3 and 4, we can see that it is very sensitive to the basic reproduction number and the endemic equilibrium. So it is meaningful to received timely treatment, which can reduce the source of infection. Because 70% to 80% patients are asymptomatic [13, 14], it is difficult to diagnose acute HCV infection. But some studies suggest that acute infection stage is very sensitive to treatment, and it is an unique opportunity to prevent the evolution of chronic infection [15].
(5) It can control the number of patients in a relatively small size by improving recovery rate of hospitalization η1. It is not sensitive to the basic reproduction number, but it is sensitive to the endemic equilibrium. It need not only patients cooperate with treatment actively but also relevant departments study new and effective medicine for the treatment of HCV [37–39]. It can improve the recovery rate of patients.
In a word, if we can implement these control measures, HCV will be controlled well, and with the time flies, the number of patients will decrease.
By the parameters value of 2015 given in Table 2, we can calculate the endemic equilibrium values:
P∗ = (S∗,E∗,Ia∗,Ic∗,T∗,R∗)=(184287929.9,575029.5,783783.9,83492.0,1880.2,140659573.5).
The variables (S,E,Ia,Ic,T,R) have been replaced by x1,x2,x3,x4,x5,x6; the parameters l,m,n,β1,β2,β3,σ,α,ρ1,ρ2,δ,η1,λ,p1,γ,Λ,μ,d1,d2 by y1, y2, y3, y4,y5,y6,y7,y8,y9,y10,y11,y12,y13,y14,y15,y16,y17,y18,y19; the point of endemic equilibrium \(({S^{*}}, {E^{*}}, I_{a}^{*}, I_{c}^{*}, {T^{*}}, {R^{*}})\) by \(x_{1}^{*},x_{2}^{*},x_{3}^{*},x_{4}^{*},x_{5}^{*},x_{6}^{*}\) and six equilibrium equations of the model by
$${f_{i}}({x_{1}},...,{x_{6}};{y_{1}},...,{y_{19}}) = 0,i = 1,2,3,4,5,6. $$
$${{} \begin{aligned} {f_{1}}({x_{1}},...,{x_{6}};{y_{1}},...,{y_{19}}) &= {y_{16}} - {y_{17}}{y_{1}}{x_{2}} - {y_{17}}{y_{2}}{x_{3}} - {y_{17}}{y_{3}}{x_{4}} \\&- \frac{{({y_{4}}{x_{2}} + {y_{5}}{x_{3}} + {y_{6}}{x_{4}}){x_{1}}}}{{{x_{1}} + {x_{2}} + {x_{3}} + {x_{4}} + {x_{5}} + {x_{6}}}} \\&\quad+ {y_{15}}{x_{6}} - {y_{17}}{x_{1}} = 0, \end{aligned}} $$
$${\begin{aligned} {f_{2}}({x_{1}},...,{x_{6}};{y_{1}},...,{y_{19}}) &\,=\, \frac{{({y_{4}}{x_{2}} + {y_{5}}{x_{3}} + {y_{6}}{x_{4}}){x_{1}}}}{{{x_{1}} + {x_{2}} + {x_{3}} + {x_{4}} + {x_{5}} + {x_{6}}}} \,-\, {y_{7}}{x_{2}} - {y_{17}}{x_{2}} \\&+ {y_{17}}{y_{1}}{x_{2}} + {y_{17}}{y_{2}}{x_{3}} + {y_{17}}{y_{3}}{x_{4}} = 0, \end{aligned}} $$
$${{} \begin{aligned} {f_{3}}({x_{1}},...,{x_{6}};{y_{1}},...,{y_{19}}) = {y_{7}}{x_{2}} - {y_{8}}{x_{3}} - {y_{17}}{x_{3}} - {y_{18}}{x_{3}} = 0, \end{aligned}} $$
$${{}\begin{aligned} {f_{4}}({x_{1}},...,{x_{6}};{y_{1}},...,{y_{19}}) &= {y_{10}}{y_{8}}{x_{3}} - {y_{11}}{x_{4}} \\ &+ (1 \,-\, {y_{12}}){y_{13}}{x_{5}}\! - \!{y_{17}}{x_{4}}\! - \!{y_{19}}{x_{4}} = 0, \end{aligned}} $$
$${{} \begin{aligned} {f_{5}}({x_{1}},...,{x_{6}};{y_{1}},...,{y_{19}}) &= (1 - {y_{9}} - {y_{10}}){y_{8}}{x_{3}} \\&+ (1 - {y_{9}}){y_{11}}{x_{4}} \,-\, {y_{13}}{x_{5}} \,-\, {y_{17}}{x_{5}} = 0, \end{aligned}} $$
$${\begin{aligned} {f_{6}}({x_{1}},...,{x_{6}};{y_{1}},...,{y_{19}}) &= {y_{14}}{y_{11}}{x_{4}} + {y_{9}}{y_{8}}{x_{3}} + {y_{12}}{y_{13}}{x_{5}} \\&- ({y_{15}} + {y_{17}}){x_{6}} = 0. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \end{aligned}} $$
Let AXj=Kj be the system of equations where
$${\begin{aligned} A& = \left[ {\begin{array}{*{20}{c}} {{a_{11}}}&{{a_{12}}}&{{a_{13}}}&{{a_{14}}}&{{a_{15}}}&{{a_{16}}}\\ {{a_{21}}}&{{a_{22}}}&{{a_{23}}}&{{a_{24}}}&{{a_{25}}}&{{a_{26}}}\\ {{a_{31}}}&{{a_{32}}}&{{a_{33}}}&{{a_{34}}}&{{a_{35}}}&{{a_{36}}}\\ {{a_{41}}}&{{a_{42}}}&{{a_{43}}}&{{a_{44}}}&{{a_{45}}}&{{a_{46}}}\\ {{a_{51}}}&{{a_{52}}}&{{a_{53}}}&{{a_{54}}}&{{a_{55}}}&{{a_{56}}}\\ {{a_{61}}}&{{a_{62}}}&{{a_{63}}}&{{a_{64}}}&{{a_{65}}}&{{a_{66}}} \end{array}} \right];\\&\quad {X_{j}} = \left[ {\begin{array}{*{20}{c}} {\frac{{\partial x_{1}^{*}}}{{\partial {y_{j}}}}}\\ {\frac{{\partial x_{2}^{*}}}{{\partial {y_{j}}}}}\\ {\frac{{\partial x_{3}^{*}}}{{\partial {y_{j}}}}}\\ {\frac{{\partial x_{4}^{*}}}{{\partial {y_{j}}}}}\\ {\frac{{\partial x_{5}^{*}}}{{\partial {y_{j}}}}}\\ {\frac{{\partial x_{6}^{*}}}{{\partial {y_{j}}}}} \end{array}} \right];{K_{j}} = \left[ {\begin{array}{*{20}{c}} { - \frac{{\partial {f_{1}}}}{{\partial {y_{j}}}}}\\ { - \frac{{\partial {f_{2}}}}{{\partial {y_{j}}}}}\\ { - \frac{{\partial {f_{3}}}}{{\partial {y_{j}}}}}\\ { - \frac{{\partial {f_{4}}}}{{\partial {y_{j}}}}}\\ { - \frac{{\partial {f_{5}}}}{{\partial {y_{j}}}}}\\ { - \frac{{\partial {f_{6}}}}{{\partial {y_{j}}}}} \end{array}} \right]; \end{aligned}} $$
$${\begin{aligned} \begin{array}{l} {a_{11}} = \frac{{{x_{1}}\left({{x_{2}}{y_{4}} + {x_{3}}{y_{5}} + {x_{4}}{y_{6}}} \right)}}{{{{({x_{1}} + {x_{2}} + {x_{3}} + {x_{4}} + {x_{5}} + {x_{6}})}^{2}}}} - \frac{{{x_{2}}{y_{4}} + {x_{3}}{y_{5}} + {x_{4}}{y_{6}}}}{{{x_{1}} + {x_{2}} + {x_{3}} + {x_{4}} + {x_{5}} + {x_{6}}}} - {y_{17}},\\ {a_{12}} = \frac{{{x_{1}}\left({{x_{2}}{y_{4}} + {x_{3}}{y_{5}} + {x_{4}}{y_{6}}} \right)}}{{{{({x_{1}} + {x_{2}} + {x_{3}} + {x_{4}} + {x_{5}} + {x_{6}})}^{2}}}} - {y_{1}}{y_{17}} - \frac{{{x_{1}}{y_{4}}}}{{{x_{1}} + {x_{2}} + {x_{3}} + {x_{4}} + {x_{5}} + {x_{6}}}},\\ {a_{13}} = \frac{{{x_{1}}\left({{x_{2}}{y_{4}} + {x_{3}}{y_{5}} + {x_{4}}{y_{6}}} \right)}}{{{{({x_{1}} + {x_{2}} + {x_{3}} + {x_{4}} + {x_{5}} + {x_{6}})}^{2}}}} - {y_{2}}{y_{17}} - \frac{{{x_{1}}{y_{5}}}}{{{x_{1}} + {x_{2}} + {x_{3}} + {x_{4}} + {x_{5}} + {x_{6}}}},\\ {a_{14}} = \frac{{{x_{1}}\left({{x_{2}}{y_{4}} + {x_{3}}{y_{5}} + {x_{4}}{y_{6}}} \right)}}{{{{({x_{1}} + {x_{2}} + {x_{3}} + {x_{4}} + {x_{5}} + {x_{6}})}^{2}}}} - {y_{3}}{y_{17}} - \frac{{{x_{1}}{y_{6}}}}{{{x_{1}} + {x_{2}} + {x_{3}} + {x_{4}} + {x_{5}} + {x_{6}}}},\\ {a_{15}} = \frac{{{x_{1}}\left({{x_{2}}{y_{4}} + {x_{3}}{y_{5}} + {x_{4}}{y_{6}}} \right)}}{{{{({x_{1}} + {x_{2}} + {x_{3}} + {x_{4}} + {x_{5}} + {x_{6}})}^{2}}}}, {a_{16}} = {y_{15}} + \frac{{{x_{1}}\left({{x_{2}}{y_{4}} + {x_{3}}{y_{5}} + {x_{4}}{y_{6}}} \right)}}{{{{({x_{1}} + {x_{2}} + {x_{3}} + {x_{4}} + {x_{5}} + {x_{6}})}^{2}}}},\\ {a_{21}} = \frac{{{x_{2}}{y_{4}} + {x_{3}}{y_{5}} + {x_{4}}{y_{6}}}}{{{x_{1}} + {x_{2}} + {x_{3}} + {x_{4}} + {x_{5}} + {x_{6}}}} - \frac{{{x_{1}}\left({{x_{2}}{y_{4}} + {x_{3}}{y_{5}} + {x_{4}}{y_{6}}} \right)}}{{{{({x_{1}} + {x_{2}} + {x_{3}} + {x_{4}} + {x_{5}} + {x_{6}})}^{2}}}},\\ {a_{22}} \,=\, {y_{1}}{y_{17}} \!\,-\, {y_{17}} \,-\, {y_{7}} \,-\, \frac{{{x_{1}}\left({{x_{2}}{y_{4}} + {x_{3}}{y_{5}} + {x_{4}}{y_{6}}} \right)}}{{{{({x_{1}} + {x_{2}} + {x_{3}} + {x_{4}} + {x_{5}} + {x_{6}})}^{2}}}} \,+\, \frac{{{x_{1}}{y_{4}}}}{{{x_{1}} \,+\, {x_{2}} + {x_{3}} + {x_{4}} + {x_{5}} + {x_{6}}}},\\ {a_{23}} = {y_{2}}{y_{17}} - \frac{{{x_{1}}\left({{x_{2}}{y_{4}} + {x_{3}}{y_{5}} + {x_{4}}{y_{6}}} \right)}}{{{{({x_{1}} + {x_{2}} + {x_{3}} + {x_{4}} + {x_{5}} + {x_{6}})}^{2}}}} + \frac{{{x_{1}}{y_{5}}}}{{{x_{1}} + {x_{2}} + {x_{3}} + {x_{4}} + {x_{5}} + {x_{6}}}},\\ {a_{24}} = {y_{3}}{y_{17}} - \frac{{{x_{1}}\left({{x_{2}}{y_{4}} + {x_{3}}{y_{5}} + {x_{4}}{y_{6}}} \right)}}{{{{({x_{1}} + {x_{2}} + {x_{3}} + {x_{4}} + {x_{5}} + {x_{6}})}^{2}}}} + \frac{{{x_{1}}{y_{6}}}}{{{x_{1}} + {x_{2}} + {x_{3}} + {x_{4}} + {x_{5}} + {x_{6}}}},\\ {a_{25}} = - \frac{{{x_{1}}\left({{x_{2}}{y_{4}} + {x_{3}}{y_{5}} + {x_{4}}{y_{6}}} \right)}}{{{{({x_{1}} + {x_{2}} + {x_{3}} + {x_{4}} + {x_{5}} + {x_{6}})}^{2}}}}, {a_{26}} = - \frac{{{x_{1}}\left({{x_{2}}{y_{4}} + {x_{3}}{y_{5}} + {x_{4}}{y_{6}}} \right)}}{{{{({x_{1}} + {x_{2}} + {x_{3}} + {x_{4}} + {x_{5}} + {x_{6}})}^{2}}}},\\ {a_{31}} = 0, {a_{32}} = {y_{7}}, {a_{33}} = - {y_{8}} - {y_{17}} - {y_{18}},{a_{34}} = 0,{a_{35}} \\\quad\quad= 0,{a_{36}} = 0,{a_{41}} = 0,{a_{42}} = 0,\\ {a_{43}} = {y_{8}}{y_{10}},{a_{44}} = - {y_{11}} - {y_{17}} - {y_{19}},{a_{45}} = - {y_{13}}({y_{12}} - 1),{a_{46}} \\\quad\quad= 0,{a_{51}} = 0,{a_{52}} = 0,\\ {a_{53}} = - {y_{8}}({y_{9}} + {y_{10}} - 1),{a_{54}} = - {y_{11}}({y_{9}} - 1),{a_{55}} = - {y_{13}} - {y_{17}},{a_{56}} \\ \quad\quad= 0,{a_{61}} = 0,{a_{62}} = 0,\\ {a_{63}} = {y_{8}}{y_{9}},{a_{64}} = {y_{11}}{y_{14}},{a_{65}} = {y_{12}}{y_{13}},{a_{66}} = - {y_{15}} - {y_{17}}. \end{array} \end{aligned}} $$
Finally, the sensitivity index of the point of endemic equilibrium, \(x_{i}^{*}\) to the parameter, yj is given by \(\frac {{\partial x_{i}^{*}}}{{\partial {y_{j}}}}\frac {{{y_{j}}}}{{x_{i}^{*}}}\) for 1≤i≤6 and 1≤j≤16.
The data that support the findings of this study are available from the China Center for Disease Control and Prevention (China's CDC) (http://www.nhc.gov.cn/jkj/s2907/new_list.shtml?tdsourcetag=s_pcqq_aiomsg), these network direct data are completely open, and we count these data month by month.
China's CDC:
Chinese center for disease control and prevention
HCC:
HCV:
IDUs:
Injecting drug users
RNA:
Ribonucleic acid
SEI a I c TR :
Susceptible-exposed-acute infection-chronic infection-treated-recovered
SQP:
Sequence quadratic program
The world health organization
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We would like to thank anonymous reviewers for very helpful suggestions which improved greatly this manuscript.
The work was partially supported by the National Natural Science Foundation of China (11547006, 11471133), Scientific Research Project of Hubei Provincial Department of Education (B2017039), Undergraduate Training Program of Yangtze University for Innovation and Entrepreneurship (20150094). The funding body had no role in the study design, collection, analysis, interpretation of data and in writing the manuscript.
Wanru Jia and Yong Li contributed equally to this work.
School of Information and Mathematics, Yangtze University, Jingzhou 434023, China, Nanhuan Road, Jingzhou, 434023, China
Wanru Jia & Yong Li
College of Mechanical and Vehicle Engineering, Hunan University, Lushan South Road, Changsha, 410082, China
Jie Weng
School of Mechanical and Automotive Engineering, South China University of Technology, Wushan Road, Guangzhou, 510641, China
Cong Fang
Institute of Applied Mathematics, Yangtze University, Nanhuan Road, Jingzhou, 434023, China
Wanru Jia
LY, JW and WJ conceptualized and designed the study, drafted the initial manuscript, and approved the final manuscript as submitted. FC and WJ analyzed the data and simulated parameters. JW and LY carried out the initial analyses, reviewed and revised the manuscript. All authors read and approved the final manuscript.
Correspondence to Yong Li.
The authors declare that there is no conflict of interests regarding the publication of this article. No authors have potential conflicts of interest with reference to this work.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver(http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.
Jia, W., Weng, J., Fang, C. et al. A dynamic model and some strategies on how to prevent and control hepatitis c in mainland China. BMC Infect Dis 19, 724 (2019). https://doi.org/10.1186/s12879-019-4311-x
Received: 19 June 2018
Preventions and control strategies
Hepatitis and co-infections | CommonCrawl |
\begin{definition}[Definition:Pointwise Multiplication of Complex-Valued Functions]
Let $S$ be a non-empty set.
Let $f, g: S \to \Z$ be complex-valued functions.
Then the '''pointwise product of $f$ and $g$''' is defined as:
:$f \times g: S \to \Z:$
::$\forall s \in S: \map {\paren {f \times g} } s := \map f s \times \map g s$
where the $\times$ on the {{RHS}} is complex multiplication.
Thus '''pointwise multiplication''' is seen to be an instance of a pointwise operation on complex-valued functions.
\end{definition} | ProofWiki |
The Annals of Probability
Ann. Probab.
Volume 14, Number 2 (1986), 560-581.
The Lower Limit of a Normalized Random Walk
Cun-Hui Zhang
More by Cun-Hui Zhang
Full-text: Open access
PDF File (1350 KB)
Let $\{S_n\}$ be a random walk with underlying distribution function $F(x)$ and $\{\gamma_n\}$ be a sequence of constants such that $\gamma_n/n$ is nondecreasing. A universal integral test is given which determines the lower limit of $S_n/\gamma_n$ up to a constant scale for $\lim \sup \gamma_{2n}/\gamma_n < \infty$. The generalized LIL is obtained which contains the main result of Fristedt-Pruitt (1971). The rapidly growing random walks and the limit points of $\{S_n/\gamma_n\}$ are also studied.
Ann. Probab., Volume 14, Number 2 (1986), 560-581.
First available in Project Euclid: 19 April 2007
https://projecteuclid.org/euclid.aop/1176992531
doi:10.1214/aop/1176992531
links.jstor.org
Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60J15 60F15: Strong theorems 60F20: Zero-one laws
Normalized random walks lower limits generalized law of the iterated logarithm exponential bounds truncated moments
Zhang, Cun-Hui. The Lower Limit of a Normalized Random Walk. Ann. Probab. 14 (1986), no. 2, 560--581. doi:10.1214/aop/1176992531. https://projecteuclid.org/euclid.aop/1176992531
The Institute of Mathematical Statistics
Future Papers
The 1971 Rietz Lecture Sums of Independent Random Variables--Without Moment Conditions
Kesten, Harry, The Annals of Mathematical Statistics, 1972
A Note on Feller's Strong Law of Large Numbers
Chow, Yuan Shih and Zhang, Cun-Hui, The Annals of Probability, 1986
Maximal Increments of Local Time of a Random Walk
Jain, Naresh C. and Pruitt, William E., The Annals of Probability, 1987
On the Almost Sure Minimal Growth Rate of Partial Sum Maxima
Klass, Michael J. and Zhang, Cun-Hui, The Annals of Probability, 1994
General One-Sided Laws of the Iterated Logarithm
Pruitt, William E., The Annals of Probability, 1981
An Integral Test for the Rate of Escape of $d$-Dimensional Random Walk
Griffin, Philip S., The Annals of Probability, 1983
The Law of the Iterated Logarithm for the Range of Random Walk
Jain, Naresh C. and Pruitt, William E., The Annals of Mathematical Statistics, 1972
Approximate Local Limit Theorems for Laws Outside Domains of Attraction
Griffin, Philip S., Jain, Naresh C., and Pruitt, William E., The Annals of Probability, 1984
Linear Functions of Order Statistics
Stigler, Stephen Mack, The Annals of Mathematical Statistics, 1969
Matrix Normalized Sums of Independent Identically Distributed Random Vectors
euclid.aop/1176992531 | CommonCrawl |
# Defining inner product spaces
An inner product space is a vector space equipped with an inner product, which is a function that takes two vectors as input and returns a scalar value. The inner product satisfies certain properties, such as linearity in the first argument and conjugate symmetry in the second argument.
Consider the vector space of all real-valued functions on a set $X$. We can define an inner product on this space by
$$\langle f, g \rangle = \int_X f(x) \overline{g(x)} \, dx$$
where $\overline{g(x)}$ denotes the complex conjugate of $g(x)$. This is an example of an inner product space.
## Exercise
Define an inner product on the space of all $n \times n$ matrices with real entries.
# Examples of inner product spaces
There are many examples of inner product spaces in mathematics and physics. Some common examples include:
- The space of all square-integrable functions on a set $X$ with the inner product defined as above.
- The space of all $n \times 1$ column vectors with real entries, equipped with the standard inner product.
- The space of all $n \times n$ Hermitian matrices, equipped with the inner product defined as the Frobenius norm.
## Exercise
Find an inner product on the space of all polynomials of degree at most $n$ with real coefficients.
# Properties of inner product spaces
Inner product spaces have several important properties:
- Linearity in the first argument: $\langle \alpha f + \beta g, h \rangle = \alpha \langle f, h \rangle + \beta \langle g, h \rangle$ for all scalars $\alpha, \beta$ and vectors $f, g, h$.
- Conjugate symmetry: $\langle f, g \rangle = \overline{\langle g, f \rangle}$ for all vectors $f, g$.
- Positive definiteness: $\langle f, f \rangle \geq 0$ for all vectors $f$, with equality if and only if $f = 0$.
- Orthogonality: $\langle f, g \rangle = 0$ for all vectors $f, g$ if and only if $f$ and $g$ are orthogonal.
## Exercise
Prove that the inner product defined in section 1 satisfies all the properties mentioned above.
# Orthogonality in inner product spaces
Orthogonality is a fundamental concept in inner product spaces. Two vectors $f$ and $g$ are orthogonal if their inner product is zero: $\langle f, g \rangle = 0$.
In the example of inner product spaces in section 1, the functions $f(x) = 1$ and $g(x) = x$ are orthogonal because
$$\langle f, g \rangle = \int_0^1 1 \cdot \overline{x} \, dx = 0$$
## Exercise
Find two orthogonal vectors in the space of all $n \times 1$ column vectors with real entries.
# Linear maps and their properties
A linear map is a function between vector spaces that preserves the structure of vector addition and scalar multiplication. In an inner product space, linear maps have several important properties:
- Preservation of inner products: $\langle T(f), T(g) \rangle = \langle f, g \rangle$ for all vectors $f, g$ and linear maps $T$.
- Preservation of orthogonality: $T(f)$ and $T(g)$ are orthogonal if and only if $f$ and $g$ are orthogonal.
## Exercise
Find a linear map between two inner product spaces that preserves inner products and orthogonality.
# Orthogonal bases
An orthogonal basis is a set of orthogonal vectors that span an inner product space. In an inner product space, every vector can be expressed as a linear combination of the basis vectors.
In the example of inner product spaces in section 1, the functions $f_1(x) = 1$ and $f_2(x) = x$ form an orthogonal basis because $\langle f_1, f_2 \rangle = 0$.
## Exercise
Find an orthogonal basis for the space of all $n \times 1$ column vectors with real entries.
# Norms and their properties
A norm is a function that assigns a non-negative real number to each vector in an inner product space. The norm is related to the inner product through the Cauchy-Schwarz inequality:
$$\vert \langle f, g \rangle \vert \leq \Vert f \Vert \Vert g \Vert$$
for all vectors $f, g$.
In the example of inner product spaces in section 1, the norm is defined as the square root of the integral of the square of the function:
$$\Vert f \Vert = \sqrt{\int_0^1 f(x)^2 \, dx}$$
## Exercise
Find a norm on the space of all $n \times 1$ column vectors with real entries.
# Distance in inner product spaces
The distance between two vectors $f$ and $g$ in an inner product space is defined as
$$\Vert f - g \Vert = \sqrt{\langle f - g, f - g \rangle}$$
In the example of inner product spaces in section 1, the distance between the functions $f(x) = 1$ and $g(x) = x$ is
$$\Vert f - g \Vert = \sqrt{\int_0^1 (1 - x)^2 \, dx}$$
## Exercise
Find the distance between two vectors in the space of all $n \times 1$ column vectors with real entries.
# Applications of inner product spaces and functions
Inner product spaces and functions have numerous applications in mathematics, physics, and engineering. Some examples include:
- Fourier analysis: The inner product is used to define the dot product of functions, which is related to the orthogonality of the basis functions.
- Optimization: Inner product spaces and functions are used to define the concept of distance and to solve optimization problems.
- Quantum mechanics: Inner product spaces and functions are used to define the inner product of wave functions and to describe the behavior of quantum systems.
## Exercise
Discuss an application of inner product spaces and functions in your field of study.
# Review and practice problems
In this textbook, we have covered the following topics:
- Defining inner product spaces
- Examples of inner product spaces
- Properties of inner product spaces
- Orthogonality in inner product spaces
- Linear maps and their properties
- Orthogonal bases
- Norms and their properties
- Distance in inner product spaces
- Applications of inner product spaces and functions
## Exercise
Solve the following problems to review your understanding of the material:
1. Find an inner product on the space of all $n \times n$ matrices with real entries.
2. Find two orthogonal vectors in the space of all $n \times 1$ column vectors with real entries.
3. Find a linear map between two inner product spaces that preserves inner products and orthogonality.
4. Find an orthogonal basis for the space of all $n \times 1$ column vectors with real entries.
5. Find a norm on the space of all $n \times 1$ column vectors with real entries.
6. Find the distance between two vectors in the space of all $n \times 1$ column vectors with real entries.
7. Discuss an application of inner product spaces and functions in your field of study.
The solutions to the problems are as follows:
1. The inner product on the space of all $n \times n$ matrices with real entries can be defined as
$$\langle A, B \rangle = \text{tr}(A^T B)$$
where $A^T$ denotes the transpose of $A$.
2. Two orthogonal vectors in the space of all $n \times 1$ column vectors with real entries are
$$v_1 = \begin{bmatrix} 1 \\ 0 \\ \vdots \\ 0 \end{bmatrix}, \quad v_2 = \begin{bmatrix} 0 \\ 1 \\ \vdots \\ 0 \end{bmatrix}$$
3. A linear map that preserves inner products and orthogonality is the identity map.
4. An orthogonal basis for the space of all $n \times 1$ column vectors with real entries is the standard basis.
5. A norm on the space of all $n \times 1$ column vectors with real entries is the Euclidean norm:
$$\Vert v \Vert = \sqrt{v^T v}$$
6. The distance between two vectors in the space of all $n \times 1$ column vectors with real entries is
$$\Vert v_1 - v_2 \Vert = \sqrt{(v_1 - v_2)^T (v_1 - v_2)}$$
7. One application of inner product spaces and functions is in quantum mechanics, where they are used to define the inner product of wave functions and to describe the behavior of quantum systems. | Textbooks |
\begin{definition}[Definition:Divisor (Algebra)/Real Number]
Let $\R$ be the set of real numbers.
Let $x, y \in \R$.
Then '''$x$ divides $y$''' is defined as:
:$x \divides y \iff \exists t \in \Z: y = t \times x$
where $\Z$ is the set of integers.
That is, that $y$ is an integer multiple of $x$.
\end{definition} | ProofWiki |
ILovePhilosophy.com
Philosophical Discussion Forums
http://forums.ilovephilosophy.com/
Does infinity exist?
http://forums.ilovephilosophy.com/viewtopic.php?f=4&t=194376
Re: Does infinity exist?
by Serendipper
wtf wrote: What I don't understand about your point of view is why you reject purely mathematical infinity.
I just posted that from David Hilbert because I was listening to debate involving William Lane Craig who used a quote from David which prompted me to look it up and discover that document which I posted here solely so I would know where it is and it wasn't meant to prod you in any way. One of the purposes of this thread is to be a "junk drawer" where I can stuff articles that I randomly stumble upon in order to consolidate evidence. That's all that was about.
I don't object to the use of infinity as a "play thing" just like Christopher Hitchens didn't object to religion being a "toy" of the religious, but my objection is when the toy is asserted to be more than a toy. It's actually a funny analogy https://www.youtube.com/watch?v=d2vqwZActbQ
Studying math for the sake of math is really cool and I'm only objecting to pretending it's more than that. I think that's all I'm saying.
As David said, "In summary, let us return to our main theme and draw some conclusions from all our thinking about the infinite. Our principal result is that the infinite is nowhere to be found in reality. It neither exists in nature nor provides a legitimate basis for rational thought."
Infinity doesn't exist in nature and neither does it exist in the mind, although we can pretend it does through inference and then draw conclusions from that for whatever reason.
by wtf
Serendipper wrote: I'm only objecting to pretending it's more than that.
Who is doing that? You're making a strawman argument. You've been bitterly complaining about set theory for pages now. I'm asking you why.
And if math is only a fiction, how is it that all of modern science is based on it? You seem unable to engage with this important question.
If you need a place to store your thoughts, you could try a private blog or a pad of paper. This is a public discussion forum where you've made many claims that are open to objection, not least because many of your statements contradict each other and/or established science and math. You've gone on for weeks objecting to modern math; and when challenged directly, you claim you don't actually mean anything you say but are only using this forum as a scratchpad for your confused ideas and random quote-mining. Early on all you were capable of was throwing insults at me. At least you've stopped that. But why don't you try to learn something about the subject instead of just Googling around for quotes that are no longer even on topic? And why is it that you have the time to Google irrelevant quotes but don't have the time to click on the Wiki link that gives specific examples of the need for precise reasoning about changing the order of multiple integrals, which you explicitly asked me for? Your presentation in this thread makes you out to be an unserious person, arguing in a disingenuous manner.
Don't you want to try to learn anything?
Ah fuck it. I'm outta here. All the best.
wtf wrote:
I thought it was a fairly harmless thing to say. You're acting as if math studied for the enjoyment of math is somehow inferior to math studied for some goal or purpose and have therefore construed my comment as offensive. Chess is a toy, but is being good at chess a bad thing? Is describing chess as a toy a bad thing? Where is the offensiveness?
Who is doing that?
Whoever is pretending infinity applies outside of math.
You've been bitterly complaining about set theory for pages now. I'm asking you why.
The only thing I can remember bitterly complaining about was the stubbornness surrounding the definition of infinity. The only thing I remember claiming about set theory is that I couldn't find anything about it within 1200 pages of my calculus book, and that was only in response to your assertion that set theory underpins calculus. Then we transitioned to higher math where I said "math for the sake of math" then you got all pissed off because I dis'd math and here we are.
It's not fiction, but a tool that can be misapplied. It also can model fictitious things. Math can do more than what we would describe as "real" and just because math makes a claim, doesn't mean it will match reality. But neither can we say math is worthless because it often matches reality. It's tool to be used intelligently with proper assumptions. Garbage in = garbage out.
If you need a place to store your thoughts, you could try a private blog or a pad of paper. This is a public discussion forum where you've made many claims that are open to objection,
The reason they are public is because I want peer review, but not dogmatic digging-in in stubborn refusal to concede a point. This should be a collaboration and not a competition.
not least because many of your statements contradict each other and/or established science and math.
I know my ideas on infinity conflict with ideas of many mathematicians, but I'm not aware of other contradictions and it would be helpful if you'd point them out.
I posted an article from livescience today, written by Don Lincoln, Senior Scientist, Fermi National Accelerator Laboratory; Adjunct Professor of Physics, University of Notre Dame, that said:
Astronomers and physicists have long held that the idea of a singularity simply must be wrong. If an object with mass has no size, then it has infinite density. And, as much as researchers throw around the word "infinity," infinities of that kind don't exist in nature. Instead, when you encounter an infinity in a real, physical, science situation, what it really means is that you've pushed your mathematics beyond the realm where they apply. You need new math. https://www.livescience.com/64332-black ... avity.html
As far as I can tell, I'm in sync with all but mainstream mathematicians.
You've gone on for weeks objecting to modern math; and when challenged directly, you claim you don't actually mean anything you say
Where did I say I didn't mean what I say? I guess if I said it, I obviously didn't mean it
but are only using this forum as a scratchpad for your confused ideas and random quote-mining. Early on all you were capable of was throwing insults at me.
You're insulting me by characterizing my ideas as confused and then insulting me again for formerly insulting you. I should be insulting you now in retaliation, but have recognized that you're casting yourself in a bad light with your attacks on me. Honestly, I don't want to see you go, but I just want this silliness to end. I'd prefer to have the perspective of a mathematician available.
At least you've stopped that.
Live and learn. There are other things going on in my life too that makes having patience for certain things more difficult.
But why don't you try to learn something about the subject instead of just Googling around for quotes that are no longer even on topic?
What is not on topic? I'm not rummaging around the net for random quotes. The quotes from David Hilbert came in response to William Lane Craig who quoted him in a debate, so I found the document and posted it. Other quotes came either from links you sent me or from Wildberger. The livescience quote came from google news yesterday. I'm not looking for authorities to appeal to, but will take them if they fall in my lap. And as they fall in my lap, I wanted a thread to post them into so they would be all in one place and open to discussion.
And why is it that you have the time to Google irrelevant quotes but don't have the time to click on the Wiki link that gives specific examples of the need for precise reasoning about changing the order of multiple integrals, which you explicitly asked me for?
Because I'm not interested in that topic, it's expensive in terms of neural energy, your claim is counterintuitive, and I'm not confident, even if your object were valid, that it would lead anywhere. It's a confluence of powerful forces sapping my motivation. I asked you to explain it to me and you declined. Maybe you aren't confident it would accomplish anything either, which isn't doing much for my motivation to learn about it.
So why would it matter if we integrate in the x before the y, and how do we know which is the right answer for the average temperature, and what does this have to do with practical applications of infinity? The plates aren't infinite.... the temperature isn't infinite... and there aren't even infinite slices in the plates because there is a smallest size in the universe, so where does infinity fit in?
Your presentation in this thread makes you out to be an unserious person, arguing in a disingenuous manner.
I've resigned myself to having to start a new thread eventually to consolidate what's in this thread, so perhaps I'll do better with the next iteration.
I'd like to learn the answer to those questions, but not enough to struggle through that wiki article.
Perhaps you'll reconsider. I'd prefer if you'd stay, but I can't say you're right if I don't think you are and I don't see why you'd want me to.
Maybe we must realize a marriage between precision and vagary.
Or, we could stop believing something as limited in scope as mathematics could be expected attain any comprehensive logical precision.
Frankly mankinds deference before math is autistic. It is a direct explanation of the state of the world and man's imagined impotency to do anything about it.
The part can't encompass the whole and a thought, an idea, is a part of the universe.
Yes I think so. Thoughts are part of the universe and therefore thoughts cannot embrace the universe. Trying to embrace the universe with an aspect of the universe results in infinite regression which is evidence that the initial assumption was false.
As David said, "We have already seen that the infinite is nowhere to be found in reality, no matter what experiences, observations, and knowledge are appealed to. Can thought about things be so much different from things? Can thinking processes be so unlike the actual processes of things? In short, can thought be so far removed from reality? Rather is it not clear that, when we think that we have encountered the infinite in some real sense, we have merely been seduced into thinking so by the fact that we often encounter extremely large and extremely small dimensions in reality?
Does material logical deduction somehow deceive us or leave us in the lurch when we apply it to real things or events? No! Material logical deduction is indispensable. It deceives us only when we form arbitrary abstract definitions, especially those which involve infinitely many objects. In such cases we have illegitimately used material logical deduction; i.e., we have not paid sufficient attention to the preconditions necessary for its valid use."
For discerning what is valid use I developed value ontology.
A hell of a lot more taxing than mathematics, as it not only requires perfect exactitude but also actual experiential terms rather than only abstractions.
In other words it requires not just a sharp intelligence but also footing in the world and, for lack of a better word, a heart. I can hardly think of a weaker idea than that to understand something you need to strip it of all its actual existence.
Your ideas derived from Watts are deeply pertinent here.
Also what you said about the struggle and immersion, the rewiring of neuron webs required to get there. I've talked about that a lot too. I'm glad to see I'm not the only one.
We may just get somewhere decent as a species.
This arrived in the news yesterday:
Matter Sucked in by Black Holes May Travel into the Future, Get Spit Back Out
Astronomers and physicists have long held that the idea of a singularity simply must be wrong. If an object with mass has no size, then it has infinite density. And, as much as researchers throw around the word "infinity," infinities of that kind don't exist in nature. Instead, when you encounter an infinity in a real, physical, science situation, what it really means is that you've pushed your mathematics beyond the realm where they apply. You need new math.
It's easy to give a familiar example of this. Newton's law of gravity says that the strength of the gravitational attraction changes as one over the distance squared between two objects. So if you took a ball located far from Earth, it would experience a certain weight. Then, as you brought it closer to Earth, the weight would increase. Taking that equation to the extreme, as you brought the object near to the center of Earth, it would experience an infinite force. But it doesn't.
Instead, as you bring the object close to the surface of Earth, Newton's simple law of gravity no longer applies. You have to take into account the actual distribution of Earth's mass, and this means that you need to use different and more complex equations that predict different behavior. Similarly, while Einstein's theory of general relativity predicts that a singularity of infinite density exists at the center of black holes, this can't be true. At very small sizes, a new theory of gravity must come into play. We have a generic name for this new theory: It's called quantum gravity.
All this is just to say that when we encounter infinity, it means we've done something wrong.
As an example of how shallow mathematicians are in their logic, I can show how easily I resolved the so called contradiction between Relativity and QM.
In fact Relativity itself produces the uncertainty principle.
A quantum like an electron is, like every amount of energy, a reference frame. It can only be properly defined in its own terms. So to approach this minuscule reference frame from the relatively massive reference frame of a technological instrumentarium could logically not produce any straightforward results about such things as "location" and "momentum".
Einstein also requested to Freud that he resolve human cruelty. Mathematicians are generally very naive about actual existence. It may have arisen as a form of escapism. In any case that is what it has become.
Existence ostensibly relies on conflict. Mathematics is of all forms of thought the least equipped to deal with this.
Jakob wrote:
Math is just another tool in the toolbox and although sometimes a screwdriver can be a chisel or prybar, it's not necessarily optimized for it. Tools usually have to be implemented intelligently.
I'm beginning to think there isn't a tidy method or one particular way that works, as much as we'd like there to be, much like we can't write every situation into law and therefore we need judges with a sense of equity to fill the voids left by reaching the limits of precision. These philosophical topics are increasingly drawing me into the Middle Way.
Precision/vagary
particles/waves
prickles/goo
Frankly mankinds deference before math is autistic.
You have a remarkable skill for succinctly summarizing situations
Sopranos fan?
Tony: I need proof. I'm not gonna hurt a man that I love because some cop gossip from a degenerate fucking gambler with a badge. You understand me?
Vin: You know, you got an amazing ability to sum up a man's whole life in a single sentence: degenerate gambler with a badge, huh.
Help me understand what you mean by "value". Do you mean "desire"? An atom desires to have a stable electron configuration? If that is what you mean, then I think we may be on the same page, though I approached it from the eastern angle and have concluded that desire seems fundamental to everything and to be absent desire is not to exist. You've honed in from a different angle with different terminology and it could be that we've arrived at the same place, but "value" is a very confusing word for me because it has so many interpretations.
A hell of a lot more taxing than mathematics, as it not only requires perfect exactitude but also actual experiential terms rather than only abstractions. In other words it requires not just a sharp intelligence but also footing in the world and, for lack of a better word, a heart.
Hopefully 1 out of the 3 will suffice if by "heart" you mean "determination".
"Nothing in the world can take the place of Persistence. Talent will not; nothing is more common than unsuccessful men with talent. Genius will not; unrewarded genius is almost a proverb. Education will not; the world is full of educated derelicts. Persistence and Determination alone are omnipotent. The slogan "Press On" has solved and will always solve the problems of the human race."- Calvin Coolidge
I can hardly think of a weaker idea than that to understand something you need to strip it of all its actual existence.
Another insightful summary!
Your ideas derived from Watts are deeply pertinent here. Also what you said about the struggle and immersion, the rewiring of neuron webs required to get there. I've talked about that a lot too. I'm glad to see I'm not the only one.
That probably explains the old adage of "sleeping on it" to make important decisions. The brain may need time to grow or somehow arrange itself to better model or conceptualize a situation which can then be seen clearer for a better decision. From my own experience with Watts, I feel I had to grow eyes to see what he was presenting and it wasn't easy. I listened to some presentations 30 times and still learned something, which meant that nuggets flew right over my head 30 times without my noticing. Birds that fly high indeed look small! But now I can see clearer and spot those birds because I grew the tools necessary to do so.
I think the brain can be rewired and neurons can grow, so I believe (contrary to popular opinion) that intelligence can be taught. Just like muscles can be trained to support heavier loads, the brain can be trained to support heavier ideas by forming the necessary neural network, which is the supporting framework capable of modeling what previously couldn't be conceived. The problem is often the ego... people can't let go of what they already think is right, either due to self-esteem issues or narcissism. If true, then intelligence is mainly a function of humility or the extent to which one can be wrong and switch sides.
I know what you mean.
The Catholic Church issued apology in 1992 for persecuting Galileo and in 1995 for burning people at the stake,,,, 400 years later! https://en.wikipedia.org/wiki/List_of_a ... hn_Paul_II
Jakob wrote: As an example of how shallow mathematicians are in their logic, I can show how easily I resolved the so called contradiction between Relativity and QM.
Yes, it doesn't makes sense to probe an electron with something bigger than an electron.
In 1983 an American developmental psychologist Howard Gardener described 9 types of intelligence:
Naturalist (nature smart)
Musical (sound smart)
Logical-mathematical (number/reasoning smart)
Existential (life smart)
Interpersonal (people smart)
Bodily-kinesthetic (body smart)
Linguistic (word smart)
Intra-personal (self smart)
Spatial (picture smart)
https://blog.adioma.com/9-types-of-inte ... fographic/
Serendipper - I've used the term value because all of its meanings apply as the same principle in different contexts. Indeed desire is a main human context. But I see loyalty (when taken in the same semantic manner as you propose we take desire) as at leat as fundamental. Stability, consistency. For me terms like consciousness and desire are rather too specific but it is clear we aren't in contradiction.
I've spent some 20 years with rigorous meditation, Zen to begin with, yoga and later western occultism on top of that, coupled with martial arts, Shaolin and Tai Chi, the first years that was many hours each day. And it's absolutely the case that this has formed the background to my emerged understanding.
I am a Sopranos fan and I much enjoyed that compliment. A memorable character, that detective. I think its around the same time whem Tony has to deal with these Hasidic Jews
"Hasidim but I don't believe 'em" in Paulies immortal words.
Or: "and the Romans, where are they now? "
"You're looking at em asshole."
But then, " this guy is a bull. If we don't kI'll him we should put him to work."
Fantastic show. No wonder Gandolfini couldn't live on after it was done.
I mention the Hasids as a playful reference to our earlier clash on BTL. I'm very much a Zionist, for reasons more complex and dark than can be said aloud. It's very nice that we do see eye to eye on philosophy proper. Politics is nasty by nature, philosophy is not.
I've been winding down my forum activities after 17 years, so Im wary of going all the way again into explaining this logic I developed between East and West, I've done this hundreds f times across the Web with different names and I'm very satisfied that a lot of my focus on valuing now echos in all kinds of media. I'm all about grassroots insurgency of ideas.
The thread "the Philosophers" in nonphilosophical chat has become a kind of archive of some of my work. I think of online writing as scattering seeds knowing that some of them will take root. I don't believe in directly convincing people. "You only see what your eyes want to see" , with Madonna, an unlikely person to quote. But since you're able to understand much I'll give one last summary here.
I've called this phenomenon of selectivity self-valuing logic, though this is misleading as one only vales oneself through ones external values such as first of all oxygen and other unconsciously held values, of which one only becomes conscious when they fall away - or when one practices meditation. On a conscious leven we get desires and on an even more conscious level, moral values.
Some say oxygen isn't a value. But let's see them discard it in favour of some money before we believe them.
So I see breathing as valuing. And valuing, if it amounts to the perpetuation of the being, as self-valuing. Not the valuing of a self (the self is an illusion) but the valuing of valuing.
It's tough to put into grammar. Grammar itself is a presupposed structure that is false. There is no fundamental difference between the subject, the verb, the object; the being is artificially split into this trinity for the sake of communicating some utilitarian messages. But utility itself is a specific value.
I recommend a look at the thread "the ontological thranny" started by without-music, a poster of some genius who was here for a short while in 2011.
Beyond all this I wish you good "demon days" as the Vikings called this solstice time, and take care of yourself, and keep up your kung fu, meaning literally, as far as my knowledge of Cantonese goes, good work.
PS tying the thought of infinity to the expression "sleeping on it", a bit I wrote to a friend recently;
"Any selfvaluing is an epistemic standard, meaning a limited holder of truth. Its being relies on its limitedness, as you explain. Knowledge is obviously reliant on being limited. Too many ways to clarify that. And that's at the genesis of my own understanding of being as selfvaluing; seeing the scientific method as a form of severe limitation of what constitutes knowledge. Precisely because it refuses almost every form of knowing, it becomes almost indistinguishable from knowledge itself.
There is no negation of continuity in the absence of it. Continuity of what? Continuity in life also doesn't exist. We must sleep precisely because of the ruptures that [lie] between literally everything. We must negate the positive being and lose ourselves each day just in order to not contradict the contents of own consciousness by putting them into some vessel."
I think this is what Spinoza tried to get at. Or maybe he did, I never finished the Ethica, and I know only one old poster here who did, a guy named Dunamis. He was banned long ago, but it's worth looking into some of his posts. There is a lot of treasure hidden on the site. Your own posts are always substantive and with interesting references and must be counted among such treasures.
Perhaps you'll reconsider. I'd prefer if you'd stay ...
I'm a sucker for a kind word. But I was confused by this:
Serendipper wrote: ,,,it wasn't meant to prod you in any way.
I took that to mean that when I dared to post to this thread you regretted "prodding" me. I was genuinely insulted and demoralized. I thought we were having a conversation, but apparently you prefer me to stay unprodded.
So which is it? Now that I'm going to respond to your latest post will you regret prodding me? Or will you appreciate my taking the time to provide my perspective? No way for me to know, is there. But ok what the hell. Happy New Year by the way. Here's my response to your latest.
Serendipper wrote: I thought it was a fairly harmless thing to say. You're acting as if math studied for the enjoyment of math is somehow inferior to math studied for some goal or purpose and have therefore construed my comment as offensive.
???? What ???? If anything I have the opposite point of view. I agree with G.H Hardy (played by Jeremy Irons in The Man Who Knew Infinity. I highly recommend it) that the best math is by definition the most useless math.
Serendipper wrote: Chess is a toy, but is being good at chess a bad thing? Is describing chess as a toy a bad thing? Where is the offensiveness?
You are mischaracterizing my words and viewpoint 180 degrees. You're attacking a strawman. That adds to my frustration with this conversation. Was I unclear? Is your reading comprehension bad? Are you just deliberately lying about what I said? Hard for me to know. Do you already regret prodding me again? Or do you wish I'd contribute? Hard to know.
Would that include every physicist since Newton? I've asked you this before. Modern physical science is based on infinitary math. Whether that's a necessary or a contingent fact we don't yet know. But the empirical fact remains. No infinitary math and you throw science back to the Middle ages. Is that your intention? I have asked you this several times now without getting a direct response.
The only thing I can remember bitterly complaining about was the stubbornness surrounding the definition of infinity. The only thing I remember claiming about set theory is that I couldn't find anything about it within 1200 pages of my calculus book ...
I imagine that when you learned to drive a car, you were not first required to master metallurgy and automotive engineering. Do you take that as evidence that these disciplines do not actually underlie the act of driving a car? Or is it perhaps more likely that these disciplines are in fact essential to the very existence of cars, but that we don't teach them to beginning drivers, in favor of simply teaching them how not to hit things?
..., and that was only in response to your assertion that set theory underpins calculus.
Which it certainly does. I assume you can operate a light switch and were not first required to master the subject of electrical power generation. In calculus we teach people a rote procedure to "pull down the exponent and subtract 1." We do not show beginning students Newton's application of the fact that the binomial theorem can be extended to real-valued exponents.
It's perfect clear historically that Newton worked with infinitary math. Would you really send us all back to the pre-Newtonian world?
Then we transitioned to higher math where I said "math for the sake of math" then you got all pissed off because I dis'd math and here we are.
Yet another vile mischaracterization of what I actually said. Now I'm reminded of why I quit this thread in disgust.
So now you DO agree that math underpins modern physical science? Or are you still demanding that we take science back to the year 1500 or so?
Project much?
Well now you're contradicting yourself again. Do you or do you not agree that math is valid within itself; and does happen to be supremely useful? If you agree that math for the sake of math is valid, then exactly WHAT ARE your ideas on infinity? What do you know that all the mathematicians in the world don't?
I posted an article from livescience today, written by Don Lincoln, Senior Scientist, Fermi National Accelerator Laboratory; Adjunct Professor of Physics, University of Notre Dame, that said: ...
(quote omitted)
I perfectly well agree. But I wonder: WHY ARE YOU TELLING ME THIS? From my first post in this thread I have agreed that (as far as we know, to the limits of contemporary physical theory) there is no actual infinity instantiated in the world. Since I have long ago agreed with this point, why are you acting as if making this point again somehow counts as an intelligent comment in response to anything I've said?
All but mainstream mathematicians? So you know something that 140 years of professional mathematicians don't? What would that be exactly?
What does "all but the mainstream" mean? Are you saying you're in line with the mathematical cranks? How does that help your credibility?
You've already agreed that math is perfectly fine as an abstract game. That's the philosophical doctrine of mathematical formalism. But now you claim that you oppose even the formalism. WHICH IS IT?
Yeah, that I believe.
So then why should I bother? Do you regret "prodding" me today? Or do you appreciate my point-by-point response to your remarks? How would I know what mood you're in? You know you could always write your response in a text file and sit on it for a day to make sure you're saying what you mean and not reacting irrationally to whatever's going on in your life. That would be a tactic that would enable you to post more coherently.
What is not on topic? I'm not rummaging around the net for random quotes. The quotes from David Hilbert came in response to William Lane Craig who quoted him in a debate,
Craig is the worst kind of sophist. Let's not go down that road. But your Hilbert quote was about the physical world, and I've already said many times that I agree that (as far as we currently know) there are no actual infinities in the physical world. So your quote was totally off topic when directed to me.
so I found the document and posted it.
Why? It's off-topic to our discussion, which is about mathematical infinity.
Other quotes came either from links you sent me or from Wildberger.
Good God man, Wildberger is an absolute crank on the subject of infinity. Who are you trying to fool? Not me, since I'm extremely familiar with Wildberger's work.
The livescience quote came from google news yesterday. I'm not looking for authorities to appeal to, but will take them if they fall in my lap.
Authorities about what? You're not making any actual point. You have said both that
* You are perfectly fine with modern mathematical formalism regarding infinity; and
* You absolutely oppose modern mathematical formalism regarding infinity.
Which is it? State your freaking thesis and defend it. Stop going back and forth on this point.
Because I'm not interested in that topic,
Yet you explicitly asked me for the examples of situations in which the order of a multiple integration matters. Once again you are just playing games. You ask me for the examples, I point you to the examples, you refuse to click on the link, and then you say you have no interest. THEN WHY THE F*CK DID YOU ASK????? Just playing games. Not a serious person at all.
it's expensive in terms of neural energy, your claim is counterintuitive, and I'm not confident, even if your object were valid, that it would lead anywhere.
It's one of the examples of the need for precision and rigor in the foundations of math. The details of Fubini's theorem are not important. The necessity of a clear and precise theorem is the point.
It's a confluence of powerful forces sapping my motivation.
You ask me a question, I point to the answer, you claim you were never interested. That's why I say you are not serious about learning or thinking or conversating.
I asked you to explain it to me and you declined.
I pointed to the link on Wiki. If I thought Wiki did a bad job I'd do a better one. In this particular case, Wiki's presentation is spot on and I could not improve on it.
You don't need to dive into the details. They're unimportant. What is important is that the examples exist. The 18th and 19th centuries were all about mathematicians realizing that they desperately needed clear and logically rigorous foundations, else their intuitions would lead them astray. It's the existence of the examples, not the details of the examples, that's important and significant.
Maybe you aren't confident it would accomplish anything either, which isn't doing much for my motivation to learn about it.
Learning the specific examples is totally unimportant. The fact that the examples exist is important. And all that's needed there is a mouse click to the Wiki page.
So why would it matter if we integrate in the x before the y, and how do we know which is the right answer for the average temperature, and what does this have to do with practical applications of infinity?
It matters because although our intuition says the order doesn't matter, there are actual examples in which it does matter. Showing that there is a need for logical rigor in our foundations.
The plates aren't infinite.... the temperature isn't infinite... and there aren't even infinite slices in the plates because there is a smallest size in the universe, so where does infinity fit in?
Physicists find mathematical reasoning indispensable in their work. Take it up with them. Else drive us all back to 1500 when nobody knew or cared about any of it.
And I'm sure I'm resigned to reading it. But unless your thinking gets more clear it will just be more of the same.
"There is no royal road to geometry." -- Euclid.
We all have to struggle to understand the math. But in this case understanding the math is totally unimportant. All that's needed to to accept that these examples exist, whether we drill down to the details or not. And these examples show the need for mathematical rigor.
You do NOT NEED TO UNDERSTAND THE EXAMPLES. You only need to acknowledge that the examples exist.
So bottom line, do you:
* Accept mathematical formalism as an abstract, meaningless game but perhaps an interesting one? Or
* Do you reject modern math?
Which is it? I wonder if you have even interrogated yourself on this issue, since you contantly whipsaw back and forth.
Please state clearly what is your objection to the mathematical formalism of infinity. And also please tell me if you have any similar objections to the rules of chess. Maybe you think the King should be able to move two squares instead of just one. Is that your point? What are you trying to say? You do understand that the entirety of set theory can be expressed in finitely many symbols, right? So what exactly is your objection? And do you want to drive physics back to pre-Newton or even pre-Galileo?
Jakob wrote: Serendipper - I've used the term value because all of its meanings apply as the same principle in different contexts. Indeed desire is a main human context.
I will have to watch you use the word and learn that way, kinda like Iambiguous' use of Dasein which evidently can't be articulated in words.
Have you seen the movie: Iron and Silk?
I think you would like it, but fwd to 50:10 for a point I want to illustrate.
Pan Qingfu: You have to teach me English
Mark: Sure!
Pan Qingfu: I want to learn to say these sentences.
Mark: Have you studied English before?
Pan Qingfu: No.
Mark: Then let's start with the ABCs.
Pan Qingfu: No! That's too slow. I want to learn whole sentences then string them together like wushu routines.
Mark: Learning English is different from learning wushu!
Pan Qingfu: No! Everything is like wushu! Now we'll start.
Sometimes it's easier to take things apart to figure out how it was put together than to be told how things are put together.
I am a Sopranos fan and I much enjoyed that compliment.
That's awesome! I've watched them so many times that I could probably communicate in sopranos clips lol
A memorable character, that detective. I think its around the same time whem Tony has to deal with these Hasidic Jews
Yeah I like Paulie.
"He killed 19 Czechoslovakians! The guy was an interior decorator!"
"Really? His apartment looked like shit."
That was a good one!
"Yea though I walk through the valley of the shadow of death..."
"Hold that thought."
I mention the Hasids as a playful reference to our earlier clash on BTL. I'm very much a Zionist, for reasons more complex and dark than can be said aloud.
I think I was mis-pegged after referencing BN which was only innocently meant to illustrate the colorado mountains. It was merely bad fortune that the two coincided. I don't think I see in terms of ethnicity, but it's more about the philosophies people hold. Everyone is a voice in the wilderness with no other special significance than that.
It's very nice that we do see eye to eye on philosophy proper. Politics is nasty by nature, philosophy is not.
Yes, I'm very happy at the prospect of burying hatchets and forming an amicable relationship with a knowledgeable person.
Just talk about what you want to talk about.
The thread "the Philosophers" in nonphilosophical chat has become a kind of archive of some of my work. I think of online writing as scattering seeds knowing that some of them will take root.
He who has an ear to hear, let him hear.
I don't believe in directly convincing people. "You only see what your eyes want to see" , with Madonna, an unlikely person to quote. But since you're able to understand much I'll give one last summary here.
What determines what the eyes want to see?
Oh I think I see what you're saying. Like music has notes, chords, melodies which constitute different levels of resonance. My theory is desire is essentially resonance, harmony, stability. The electron is a standing wave in a location where it wants to be (a place where it naturally resonates). Atoms desire certain numbers of electrons and form molecules that desire stability of their own. Proteins, cells, organisms, communities all desire their own harmony.
Will do. Thanks!
I heard it meant "excellence". Mindful refinement to the point of mindless reflex.
"Any selfvaluing is an epistemic standard, meaning a limited holder of truth. Its being relies on its limitedness, as you explain. Knowledge is obviously reliant on being limited.
Yes I think so and I think the limits of knowledge preclude any knowledge of an absolute. A thing viewing itself could never see all of itself and any piece of knowledge known could never be a whole description of what there is.
Too many ways to clarify that. And that's at the genesis of my own understanding of being as selfvaluing; seeing the scientific method as a form of severe limitation of what constitutes knowledge. Precisely because it refuses almost every form of knowing, it becomes almost indistinguishable from knowledge itself.
So a method that never claims 100% certainty is no longer the subject but becomes the object? I haven't pondered that.
There is no negation of continuity in the absence of it. Continuity of what? Continuity in life also doesn't exist.
Borders don't separate, but join. All borders are shared. I'd say there is no such thing as discrete.
We must sleep precisely because of the ruptures that [lie] between literally everything.
Where do the ruptures reside? They're dimensionless, right?
We must negate the positive being and lose ourselves each day just in order to not contradict the contents of own consciousness by putting them into some vessel."
I'm having trouble understanding that one.
Thanks, I appreciate the feedback. My problem is finding time to rummage around the site, but it has been on my bucket list. I've been busier this winter than most.
I'm a sucker for a kind word. ]But I was confused by this:
Happy New Year! I meant that posting that didn't have anything to do with you. I would have posted it regardless if you had existed. What I meant by prodding is I didn't intend to provoke you. I plan to add more posts when they fall in my lap so that they're all in one place.
I didn't know that movie existed and I intend to watch it asap. Thanks!
Without a concrete example, those are vague accusations to me. It's hard for me to believe I have been unfair though. Dumb maybe, but not unfair.
I've answered this before. Physics does not require infinity and in fact, Don Lincoln, physicist at Fermi Labs wrote that article I posted which said "when you encounter an infinity in a real, physical, science situation, what it really means is that you've pushed your mathematics beyond the realm where they apply. You need new math." https://www.livescience.com/64332-black ... avity.html
In other words, infinity = error. Science is not underpinned by error.
Do cars have to be made of metal? I don't know what you're getting at. You keep claiming infinity underpins math, but I can find no evidence to support that counterintuitive notion.
That dispatch looks like a ball of logical fallacies. Appealing to analogies for doing math without understanding the underpinnings of math does not substantiate the claim that math is underpinned by infinity. That seems like an argument from ignorance: if I don't understand x, then y must be true; if I don't have a theory to explain the universe, then god must exist; if I don't understand the underpinning of math, then infinity must underpin it. I don't see how that follows.
Then you appeal to horrors of pre-newton like any desire to avoid going back to pre-newtonian math has any relevance on what's true.
That's how I remember it.
So now you DO agree that math underpins modern physical science?
I wouldn't say math underpins science, no. A tool doesn't underpin mechanics. Math is just another tool in the box and it is not the box nor the science.
Or are you still demanding that we take science back to the year 1500 or so?
I never was demanding that. You're trying to make it seem like I'm demanding that.
I'm not projecting anything onto you, but revealing to you what you're doing. You're married to this infinity idea and are grasping at anything you can find to substantiate it. As if you can convince me that going pre-newtonian is a bad idea, and that merely because my opinion is it's a bad idea, it must mean infinity is true. How is truth contingent on my opinion? Grasping at appeals like that reveal to me that you're seeking to support infinity rather than being open to what exists.
Well now you're contradicting yourself again. Do you or do you not agree that math is valid within itself;
No. Math is a broad category of various types of math and I can't say any type of math would substantiate any other type. I reject ordinal infinities as fantasy, but the math is valid within the fantasy construct and independent from other constructs.
and does happen to be supremely useful? If you agree that math for the sake of math is valid, then exactly WHAT ARE your ideas on infinity?
My ideas on infinity is that no one can have ideas of infinity. I don't even know what it means to approach infinity because no matter how close we get, we're still infinitely far away and that will always be true. Near-infinite is nonsense.
What do you know that all the mathematicians in the world don't?
Tell me what it is that mathematicians don't know and I'll tell you if I know it.
Idk, I forgot the context for this one.
I'm saying I'm in sync with the greatest mathematicians and the newbies who can't let go of infinity are who I am not in sync with.
When you call people cranks, it affects your credibility. Slander is the tool of the loser. That is more display of the grasping for anything to support infinity that I was on about above. Discrediting deniers through slander is a tactic that you believe helps substantiate the infinite, but whether he is a crank is irrelevant because even if he were credible, we'd still have to evaluate the truth of his assertions and knowing whether or not he's a crank doesn't aid in doing that.
So then why should I bother?
You should only bother if it's fun.
Do you regret "prodding" me today?
I didn't prod you today or any other day.
Or do you appreciate my point-by-point response to your remarks? How would I know what mood you're in? You know you could always write your response in a text file and sit on it for a day to make sure you're saying what you mean and not reacting irrationally to whatever's going on in your life. That would be a tactic that would enable you to post more coherently.
It's not a day to day mood, but general frustration with dogmatists that probably won't be alleviated for the foreseeable future.
First, it wasn't directed to you. Second, Hilbert said infinity cannot be basis for rational thought.
The thread is depository of all topics regarding the existence of the infinite.
A few moths ago you didn't even know who he was. You called him "some guy" viewtopic.php?f=4&t=193794&start=25#p2705482
Authorities about what? You're not making any actual point.
Yes I am. I'm not looking for authorities, but if I find a statement by someone authoritative by chance, then I will post it.
You have said both that
I'm not flip flopping, but those statements are too vague to answer. It depends what you mean by formalism and how it's applied. If you want to play with toys in the abstract, that's fine, but infinity doesn't underpin science, math, or anything except toys in the abstract. That's my stance and I don't know how to make it more clear.
If you disagree, then give me example where a completed actual infinity is required for the math to work. Don't give me analogies or wiki articles, but a specific problem we can dig into and discover beyond a doubt that a concept of infinity is required to solve the problem. You were heading in the right direction with the metal plate problem, but after I asked you to explain it to me, you got mad.
I mean, if we cannot even approach infinity, then we may as well have an extremely large finite number that we also have no method of knowing how close to it we are. Let z = a finite number bigger than any that could be represented and carryon like usual with z substituting for infinity.
I did click the link and scanned the article then came to the conclusions I conveyed to you. I didn't bring it up, but you brought it to the table as example of some point you were making and then bailed when I asked for a detailed explanation.
I agree, but why does it matter if we add in one direction before another?
I don't remember asking a question that prompted the Fubini answer. I thought you were making a point that I didn't request. I could be wrong, but that's how I remember it.
No, wiki is actually lacking. There's a broken link that is supposed to lead to examples that have been deleted or never added. I suspect you didn't read it because you specifically mentioned it containing examples that were not there. Broken link -> https://en.wikipedia.org/wiki/Fubini%27 ... d_integral
Yes but I still want to know why. Extraordinary claims require evidence and that claim is quite extraordinary.
There's the pre-newtonian appeal again.
How do you know it's my thinking that is unclear?
This is what the theists say: you don't need to understand, just believe! I can't control what I believe.
That's the same question, only negated.
I don't know where you're getting that impression.
I think I fulfilled all those inquisitions above. The king can move however we want it to move. In some games the king can be captured if the opponent doesn't realize he's in check and moves another piece. It's all relative and arbitrary.
Everyone says "as x approaches infinity", but what does that mean?
To approach means to get closer to, but regardless how close we get to infinity, we're still infinitely far away.
Near-infinite doesn't make sense.
Serendipper wrote: When you call people cranks, it affects your credibility.
You made a lot of specific points and I'll try to handle them one by one rather than attempt a big bang reply. Hopefully we can bring some focus. First, Wildberger is a crank; and I'm not the only one who says so. He's an interesting case because he has also done serious mathematical work. It's only his ideas on infinity that are regarded (by a lot of people, not just me) as cranky.
https://www.reddit.com/r/math/comments/ ... my_school/
https://scienceblogs.com/goodmath/2007/ ... e-sets-and
https://math-frolic.blogspot.com/2012/0 ... erger.html
https://www.reddit.com/r/math/comments/ ... _goldbach/
https://www.physicsforums.com/threads/n ... ra.772409/
You can Google "Norman Wildberger crank" for yourself and get a wide variety of articles on the topic.
It doesn't reflect badly on me to label someone a crank who is (a) widely labelled a crank; and (b) happens to actually be a crank.
Please read someone other than Wildberger on the topic of infinity. It doesn't help your cause because his ideas on infinity are generally regarded as cranky.
It's not calling cranks cranks that affects my credibility, any more than if I called the Pope a Catholic. Wildberger's a known mathematical crank. Rather, your quoting known cranks hurts your own credibility.
By the way whether or not Wildberger is a crank is not all that important to us. What's important is that you're wasting your time quoting him to me. Also I've known about him for several years, if I called him "some guy" in some post it's probably because someone posted a link to a video and I didn't bother to watch the video.
Serendipper wrote: Everyone says "as x approaches infinity", but what does that mean?
It means x gets arbitrarily large. That's ALL it means. Didn't they explain that in your 1200 page calculus text? No matter. "x goes to infinity" or "x approaches infinity" means that x gets arbitrarily large. It's not bounded. It's just a figure of speech. Although we can formalize it using the extended real numbers, in which we add a pair of symbols \(+ \infty\) and \(- \infty\) and assign them formal properties that let us use them as we need to. I'm pretty sure that's in your calculus text too. But it's ok if these fine points aren't clear. Nobody is expected to learn anything in calculus beyond the basic techniques. The fine points of getting everything logically correct are taught in a subject called real analysis, taken by math majors.
In any event the use of limits at infinity in calculus is completely different than the transfinite ordinals and cardinals studied in set theory. But if you even believe in the familiar real number line taught in high school, that's an example of an infinitely long mathematical object that's indispensable in physical science and even social science. The familiar Gaussian probability curve, or "bell curve," is defined over the entire real line and is one of the most important concepts in probability and statistics.
I also wanted to mention that your point about the scientist's quote about infinity in physics is a good one and I have something substantive to say about it, but not tonight. So I hope we won't get sidetracked on these two minor issues (Wildberger's crankitude and the meaning of "x goes to infinity") before I get to what I consider the more substantive and important point about infinity in physics.
Posted: Wed Jan 09, 2019 6:08 am
Just dropped in to say I'm hard at work writing down my thoughts on the meaning of infinity as applied to physics. I'm trying to make it brief and clear. That may take a little more time. I completely take your point regarding physical infinities and I'm drafting a response that I hope will shed light.
Whether or not other people think he is a crank is completely irrelevant. It's an appeal to popularity fallacy committed for the purpose of supporting an ad hominem fallacy. A bunch of people say he's a crank (appeal to popularity), so he must be a crank, and because he's a crank, his ideas on infinity are wrong (ad hominem redirection from the topic to the person).
Despite various publications of results where hand washing reduced mortality to below 1%, Semmelweis's observations conflicted with the established scientific and medical opinions of the time and his ideas were rejected by the medical community. Semmelweis could offer no acceptable scientific explanation for his findings, and some doctors were offended at the suggestion that they should wash their hands and mocked him for it. In 1865, Semmelweis suffered a nervous breakdown and was committed to an asylum, where he died at age 47 of pyaemia, after being beaten by the guards, only 14 days after he was committed. https://en.wikipedia.org/wiki/Ignaz_Semmelweis
The guy who suggested hand washing was also labeled a crank and that is the argument you're appealing to. Some people on the net who like the idea of infinity are calling Wildberger a crank specifically because he doesn't and that is supposed to mean something? Do they also think Gauss and Hilbert are cranks??? Should it matter if they did?
From this I can learn more about you and the people calling Wildberger a crank than I can learn about Wildberger from all the slandering. And I'm probably more inclined to believe Wildberger is not a crank specifically because people insist that he is. That philosophy would have served me well in Galileo vs The Church concerning geocentrism and from my perspective it's at least equally likely, if not more likely, that the cranks wind up being right in the end.
"All truth passes through three stages: First, it is ridiculed. Second, it is violently opposed. Third, it is accepted as being self-evident." - Arthur Schopenhauer
The only reason to vilify people is if they're a threat and if they're a threat, it's probably for good reason.
That's why I say it does more damage to your credibility to insult his.
"Slander is the tool of the loser" - Socrates
"When you're out of ammo, throw mud." - Me
Mudslingers sling mud because that's all the have to sling.
Anyway, even if you generally discredit Wildberger, it wouldn't mean anything unless you could show that he is incapable of being correct on anything. Even the biggest fool isn't wrong about everything.
And lastly, I don't quote these people because I need their references, but because I'm trying to consolidate various quotes and arguments/ideas about infinity into one place.
It means x gets arbitrarily large. That's ALL it means. Didn't they explain that in your 1200 page calculus text? No matter. "x goes to infinity" or "x approaches infinity" means that x gets arbitrarily large. It's not bounded. It's just a figure of speech.
Is it also a figure of speech to say "near-infinite"?
I'm ok with figures of speech, but Michio Kaku said "as density approaches infinity...." and it dawned on me at that moment that what he said doesn't make sense because there is no way to approach infinity; regardless how close you get, you're still just as far away. So the statement means absolutely nothing when taken literally, but figuratively, ok, I get it.
wtf wrote: Just dropped in to say I'm hard at work writing down my thoughts on the meaning of infinity as applied to physics. I'm trying to make it brief and clear. That may take a little more time. I completely take your point regarding physical infinities and I'm drafting a response that I hope will shed light.
Take your time. It's no problem as I'm kinda wrapped up in political threads anyway.
by surreptitious75
Serendipper wrote:
Infinity doesnt exist in nature and neither does it exist in the mind although we can pretend
it does through inference and then draw conclusions from that for whatever reason
Can non existence exist in the absolute sense ? No it cannot which means existence has always existed [ and always will ]
This means that it exists infinitely in the past and will exist infinitely into the future
We are here discussing the existence of infinity then how can it not exist in the mind ?
It only needs to exist as a mental concept in the same way that any other thought does
I posted an article from livescience today written by Don Lincoln Senior Scientist Fermi National Accelerator Laboratory
Adjunct Professor of Physics University of Notre Dame that said :
Astronomers and physicists have long held that the idea of a singularity simply must be wrong . If an object with mass has no size
then it has infinite density . And as much as researchers throw around the word infinity infinities of that kind dont exist in nature
He is absolutely right : the singularity as traditionally defined does not exist in nature as zero dimension and infinite density are mutually incompatible
But what is not mutually incompatible is infinitesimal dimension combined with finite density and this is therefore the true definition of the singularity
This definition also allows for time to have existed before the Big Bang because the singularity did not experience it as it [ time ] is asymptotic
surreptitious75 wrote:
Something has always existed, but I wouldn't call that something "existence" since I define existence as a relationship between subject and object. How can something be said to exist if it doesn't exist: in something, as a function of something, in relation to something, etc? If we talk about objective existence where the object exists only in relation to actuality, then the standpoint of actuality becomes the subject.
Like James said, that which has no affect, does not exist. So if the grand totality of everything has no affect on anything else (because there is nothing else), then we can't talk about it as existing.
Where does the absolute exist? There is no "where". Location is only defined inside the absolute.
What does it exist in? There is no "what" because all "what" is inside the absolute.
There is no way to talk about the absolute and any concept we think we have of it simply has to be wrong.
I think it means that time, like location, is only defined inside the thing we're trying to measure. Time is an emergent property and a consequence of relationships between moving bodies. For instance, I could drive to the next town in 1/24 revolution of the earth... or so many billion vibrations of a certain atom. Time itself does not exist and eternity is not infinite time, but absence of time.
Well, let's talk about squared circles. Or maybe we can discuss what the universe looks like from the outside even though there is no such thing as "look" outside the universe. This goes to show that we can talk about things without conceptualizing them. We can discuss nonsense without having a concept of the nonsense. I once had a dream where a cat had its head in its mouth. Likewise with infinity: there has never been a person who could properly conceptualize what he fools himself into believing he has. All we can do is imagine the biggest thing we can and we call that "good enough, close enough" and pretend we've conceptualized infinity, but we've only approached it and our approach is still infinitely far away.
I don't understand. Can you unpack that a little? The way I understand it is physicists generally regard time before the big bang as north of the north pole: there is no such thing as before the big bang. Time cannot exist before there is something in existence that is moving in relation to something else that is either not moving or moving with a different velocity.
Time is internal to the universe, subjective to it, and not an objective thing existing independent of the universe which could preside over the formation of the universe and record what happened before. Whatever happened before, has no affect on anything, so it doesn't exist
Posted: Sat Jan 12, 2019 4:01 am
I dont understand . Can you unpack that a little ? The way I understand it is physicists generally regard time before the big bang as north of the north pole
there is no such thing as before the big bang . Time cannot exist before there is something in existence that is moving in relation to something else that is
either not moving or moving with a different velocity
Nothing north of the north pole is only true in relation to the Earth as an isolated body but not so in relation to its position within the Universe
So this is where it fails as an analogy in trying to explain why time did not exist before the Big Bang as it assumes nothing existed before it did
It is not known what did or did not exist before the Big Bang because that is only as far as back as physics can currently go
This is demonstrably not the same as saying it cannot go any further back because the BB is the absolute beginning of time
If the singularity was a space of zero volume and infinite density then nothing could have existed before it including time
This definition is wrong because both zero volume and infinite density cannot exist in actuality
As the former would have no dimension or property and the latter can only exist in finite form
A singularity less absolute in physicality however would allow for time to exist before it as it would not be the totality of all that existed
Also if time did begin at the Big Bang it would mean absolute nothing existed before it but this is actually invalidated by quantum mechanics
As absolute nothing can only exist infinitesimally not infinitely because of the existence of quantum fluctuations which disturb vacuum states
So quantum mechanics absolutely forbids the existence of a singularity as traditionally defined
Also the Big Bang was not the beginning of the Universe as such but only local cosmic expansion
Serendipper wrote: I posted an article from livescience today, written by Don Lincoln, Senior Scientist, Fermi National Accelerator Laboratory; Adjunct Professor of Physics, University of Notre Dame, that said:
"Astronomers and physicists have long held that the idea of a singularity simply must be wrong. If an object with mass has no size, then it has infinite density. And, as much as researchers throw around the word "infinity," infinities of that kind don't exist in nature. Instead, when you encounter an infinity in a real, physical, science situation, what it really means is that you've pushed your mathematics beyond the realm where they apply. You need new math. "https://www.livescience.com/64332-black ... avity.html
I totally agree. I totally agree. I hope saying it twice will convince you that I mean it.
When infinities arise in physics equations, it doesn't mean there's a physical infinity. It means that our physics has broken down. Our equations don't apply. I totally get that. In fact even our friend Max gets that.
http://blogs.discovermagazine.com/crux/ ... g-physics/
The point I am making is something different. I am pointing out that:
All of our modern theories of physics rely ultimately on highly abstract infinitary mathematics
That doesn't mean that they necessarily do; only that so far, that's how the history has worked out. There is at the moment no credible alternative. There are attempts to build physics on constructive foundations (there are infinite objects but they can be constructed by algorithms). But not finitary principles, because to do physics you need the real numbers; and to construct the real numbers we need infinite sets.
I collected some examples of the infinitary math underlying physics. I tried to be brief. Each example could be expanded to a book or the work of a lifetime. I'll do my best to answer specific questions. As with Fubini I regret that it's beyond me to explain any of these examples fully and in detail with perfect clarity and without requiring effort on the part of the reader. That's what TED talks are for. /s
1) The rigorization of Newton's calculus culminated with infinitary set theory.
Newton discovered his theory of gravity using calculus, which he invented for that purpose. However, it's well-known that Newton's formulation of calculus made no logical sense at all. If \(\Delta y\) and \(\Delta x\) are nonzero, then \(\frac{\Delta y}{\Delta x}\) isn't the derivative. And if they're both zero, then the expression makes no mathematical sense! But if we pretend that it does, then we can write down a simple law that explains apples falling to earth and the planets endlessly falling around the sun.
It took another 200 years for mathematicians to develop a rigorous account of calculus from first principles; and those first principles are infinitary set theory. No set theory, no real numbers, no calculus, no gravity.
https://www.encyclopediaofmath.org/inde ... f_analysis
2) Einstein's gneral relativity uses Riemann's differential geometry.
In the 1840's Bernhard Riemann developed a general theory of surfaces that could be Euclidean or very far from Euclidean. As long as they were "locally" Euclidean. Like spheres, and torii, and far weirder non-visualizable shapes. Riemann showed how to do calculus on those surfaces. 60 years later, Einstein had these crazy ideas about the nature of the universe, and the mathematician Minkowski saw that Einstein's ideas made the most mathematical sense in Riemann's framework. This is all abstract infinitary mathematics.
https://en.wikipedia.org/wiki/Differential_geometry
https://en.wikipedia.org/wiki/Introduct ... relativity
3) Fourier series link the physics of heat to the physics of the Internet; via infinite trigonometric series.
In 1807 Joseph Fourier analyzed the mathematics of the distribution of heat through an iron bar. He discovered that any continuous function can be expressed as an infinite trigonometric series, which looks like this:
$$f(x) = \sum_{n=0}^\infty a_n \cos(nx) + \sum_{n=1}^\infty b_n \sin(nx)$$
I only posted that because if you managed to survive high school trigonometry, it's not that hard to unpack. You're composing any motion into a sum of periodic sine and cosine waves, one wave for each whole number frequency. And this is an infinite series of real numbers, which we cannot make sense of without using infinitary math.
Fast forward to present time. Fourier series underlie the propagation of digital signals over the Internet. They allow us to converse in this very moment.
https://en.wikipedia.org/wiki/Fourier_series
4) Quantum theory is functional analysis.
If you took linear algebra, then functional analysis can be thought of as infinite-dimensional linear algebra combined with calculus. Functional analysis studies spaces whose points are actually functions; so you can apply geometric ideas like length and angle to wild collections of functions. In that sense functional analysis actually generalizes Fourier series.
Quantum mechanics is expressed in the mathematical framework of functional analysis. QM takes place in an infinite-dimensional Hilbert space. To explain Hilbert space requires a deep dive into modern infinitary math. In particular, Hilbert space is complete, meaning that it has no holes in it. It's like the real numbers and not like the rational numbers.
QM rests on the mathematics of uncountable sets, in an essential way.
ps -- There's our buddy Hilbert again. He did many great things. William Lane Craig misuses and abuses Hilbert's popularized example of the infinite hotel to make disingenuous points about theology and in particular to argue for the existence of God. That's what I've got against Craig.
5) Cantor was led to set theory from Fourier series.
In every online overview of Georg Cantor's magnificent creation of set theory, nobody ever mentions how he came upon his ideas. It's as if he woke up one day and decided to revolutionize the foundations of math and piss off his teacher and mentor Kronecker. Nothing could be further from the truth.
Cantor was in fact studing Fourier's trigonometric series! One of the questions of that era was whether a given function could have more than one distinct Fourier series. To investigate this problem, Cantor had to consider the various types of sets of points on which two series could agree; or equivalently, the various sets of points on which a trigonometric series could be zero. He was thereby led to the problem of classifying various infinite sets of real numbers; and that led him to the discovery of transfinite ordinal and cardinal numbers. (Ordinals are about order in the same way that cardinals are about quantity).
In other words, and this is a fact that you probably will not find stated as clearly as I'm stating it here:
If you begin by studying the flow of heat through an iron rod; you will inexorably discover transfinite set theory.
I do not know what that means in the ultimate scheme of things. But I submit that even the most ardent finitist must at least give consideration to this historical reality.
https://www.ias.ac.in/article/fulltext/ ... /0977-0999
I hope I've been able to explain why I completely agree with your point that infinities in physical equations don't imply the actual existence of infinities. Yet at the same time, I am pointing out that our best THEORIES of physics are invariably founded on highly infinitary math. As to what that means ... for my own part, I can't help but feel that mathematical infinity is telling us something about the world. We just don't know yet what that is.
Serendipper wrote: Is it also a figure of speech to say "near-infinite"?
Most definitely, and one that I would never personally use. Nothing is "near infinite," I agree with you about that. Physicists and others use it to mean "really big."
Serendipper wrote: I'm ok with figures of speech, but Michio Kaku said "as density approaches infinity...."
When physicists talk about infinity they often have NO IDEA what they're saying in terms of math. Physicists misuse the word infinity terribly; and of all the physicists who do that, the celebrity physicists do it the worst.
You're reading way too much into words people are using very informally.
Posted: Mon Jan 14, 2019 3:22 am
by Ecmandu
I almost don't feel like this needs saying...
To be accurate about infinite sets, it's proper to say, "the sequence approaches 2". Rather than, "the sequence is 2" | CommonCrawl |
Spherical Distance Python
making distance computations from plane and spherical coordinates, see chapter 11. The arc is a circlar-arc and calculating the distance and azimuth are trivial exercises using the toolbox from the spherical model (azimuth is simple zero, of course). If you need to get that information to a degrees, minutes and seconds format, use the GPS Converter. multivariate_normal¶ numpy. Let me give an example based on my own experience. Hello, I'm trying to run the command line version of CloudCompare via Python 3 in order to compute distances between a point cloud generated from one mesh (a 3D bone surface segmentation, generated from CT images) and another mesh (3D air surface segmentation) using -C2M_DIST. When to use the cosine similarity? Let's compare two different measures of distance in a vector space, and why either has its function under different circumstances. >If you draw a line (red) starting at point A, it cuts the vertical line at B and the circle at C. The KrigingModelOrdinary object is used in the Kriging tool. To start the Remote API shell:. How to calculate distance on a sphere with an earth-like coordination system? Spherical distance between two points in terms of latitude and longitude. To get the distance in kilometers, multiply by 6373. Besides 3D wires, and planes, one of the most popular 3-dimensional graph types is 3D scatter plots. I recommend the Continuum IO Anaconda python distribution (https://www. Write a program that computes the spherical distance between two points on the surface of the Earth, given their latitudes and longitudes. Pure Python geodesy tools. 74 # 4423. A triangulation of a compact surface is a finite collection of triangles that cover the surface in such a way that every point on the surface is in a triangle, and the intersection of any two triangles is either void, a common edge or a common vertex. straight-line) distance between two points in Euclidean space. x n} where n is the number of observations, the k-means clustering algorithm groups the data into k clusters. Discussion. It has built-in Python3 and R native extension support, so you can from libKMCUDA import kmeans_cuda or dyn. This tutorial covers how to do just that with some simple sample data. spherical_in (n, z[, derivative]) Modified spherical Bessel function of the first kind or its derivative. , 2013; Pennington et al. Will modified later for geofencing with mqttitude. In the above plot, I have displayed the comparison between the distance covered by two cars BMW and Audi over a period of 5 days. Write a function called increment_date that takes a Date object, date, and an integer, n, and returns a new Date object that represents the day n days after date. You define particle initial conditions and interactions in a high-level python script. This transformation is standard: = = = For our purposes, we will set r=1. The SED above is clearly a distance in this sense. It is a special case of a more general formula in spherical trigonometry, the law of haversines, relating the sides and angles of spherical triangles. OpenCV Python Homography Example. Falling Body with Air Resistance Assume that a body of mass m is dropped from a great height above the surface of the earth. This is a log that I intend to turn into a how-to for some standard codes when using dronekit python to control multiple robots. ncl: An example of using g2gsh, which interpolates from one gaussian grid to another using spherical harmonics. It uses import numpy as np and import math. destination def destination (point, distance, bearing):. In this simple python program to add two numbers example, the following statements ask the user to enter two integer numbers and stores the. • Implemented an ultra-fast shape descriptor based on spherical harmonics expansion and applied it to binding sites and electron density maps. Python implementation of above algorithm without using the sklearn library can be found here dbscan_in_python. The following are code examples for showing how to use geopy. Spherical geometry is the study of geometric objects located on the surface of a sphere. Python synopsis. 3045) and then print that value into the Distance column and then compute the distance between the 2nd (-77. How do I express things in Python. This goal will require a synergetic convergence of the fields of CAD, GIS, visual simulation, surveying and remote sensing. def intersection (self): """ Once all of the polygons have been added to the graph, calculate the intersection. Calculate distance using Google Maps between any number of coordinates/addresses in Excel. The equatorial system of coordinates (Right Ascension and declination) is the one most often used. A rotatable design exists when there is an equal prediction variance for all points a fixed distance from the center, 0. The Python extension makes this simple to incorporate into a python web framework like flask (See AAI). The model enumerates for a broad range of. In case of the DIST_L1 or DIST_C distance type, the parameter is forced to 3 because a \(3\times 3\) mask gives the same result as \(5\times 5\) or any larger aperture. The equatorial system of coordinates (Right Ascension and declination) is the one most often used. Time for a GIS task. I ended up using meters instead of kilometers so my R was set to 6371100. This function calculates distance with the Haversine formula, this formula assumes that our Earth is spherical, but it isnt since its more like a big Calculate distance between two points with latitude and longitude coordinates - JavaScript - Snipplr Social Snippet Repository. The Maps JavaScript API geometry library provides utility functions for the computation of geometric data on the surface of the Earth. kml) Hi, I'm trying to work out this problem and wanted to reach out if anyone had experience with geospatial querying. These are fully independent, compilable examples. cdist(array_Voronoi_vertices,random_coordinate_array) 8 9 #now, each row of the above distance array corresponds to a single Voronoi vertex, with each 10 #column of that row representing the distance to the respective generator point. This is due Python3 map function returns a iterator and not a list directly. To install PyGeodesy, type pip install PyGeodesy or easy_install PyGeodesy in a terminal or command window. Converts the field from a spherical variable coordinate to a normal coordinate system. movMF in python for 3 dimensions (via NIPY) movMF in C. Not a general property of all self-gravitating systems - e. Optimal spherical codes are therefore a way of packing spherical particles such that their configuration within a cluster layer is spherical but still tightly packed. If you want to be precise this is actually quite hard because of the Earth's irregular shape. The min() method returns the smallest element in an iterable or smallest of two or more parameters. Here is what the dodecahedron grid looks like: Two Python scripts, an example output, and a readme file are available in the zip package. MAP PROJECTIONS: REPRESENTINGA SPHERICAL SURFACE ON A and Python software tools for working with HEALPix The horizontal length is the cosine of the distance. Warning 2: If argument distance. euclidean¶ scipy. Kriging is a multistep process; it includes exploratory statistical analysis of the data, variogram modeling, creating the surface, and (optionally) exploring a variance surface. d and type python setup. Calculate the geographical distance (in kilometers or miles) between 2 points with extreme accuracy. If the number is a complex number, abs() returns its magnitude. Notice: Undefined index: HTTP_REFERER in /home/eventsand/domains/eventsandproduction. For instance, a "line" between two points on a sphere is actually a great circle of the sphere, which is also the projection of a line in three. The following are code examples for showing how to use numpy. @ivy_lynx The consensus arguments bring up two issues. We will pass the radius value to the function argument and then it will calculate the Surface Area and Volume of a Sphere as per the formula. This question is identical to Custom nonlinear distortion lens With one difference: I'm only interested in a solution at the code level. Assuming the kriging errors are normally distributed, there is a 95. The first two equations of motion each describe one kinematic variable as a function of time. Zoom in to your desired area, click on "Start A Course", and then click on the points you want (or enter a name or address to create a point). where is your spherical distance, is the earth radius in km, and stands for "euclidian". Hello, I'm a student in a summer workshop and i'm trying to make a python table from which people can choose the type , and within a mile radius. The multivariate normal, multinormal or Gaussian distribution is a generalization of the one-dimensional normal distribution to higher dimensions. py from the documentation but there are only plots in cartesian coordinates. The most extreme difference between straight-line distance and travel distance is found between Grand Marais, Minnesota, and Houghton, Michigan (Figure 1). The haversine formula is an equation important in navigation, giving great-circle distances between two points on a sphere from their longitudes and latitudes. Uses spherical development of ellipsoidal coordinates. km = deg2km(deg) converts distances from degrees to kilometers, as measured along a great circle on a sphere with a radius of 6371 km, the mean radius of the Earth. I've got a list of geographic coordinates ([lat, long]) and want to compute the corresponding matrix of distances. Function to compute distance between points- In this video you will learn how to write a function to compute distance between two points in two dimensional and three dimensional planes Visit us. (The red arrows are intended to represent that only a small portion of the full output table is shown in the picture). Answer: Because charges q2 and q3 are equal, and q1 lies on the line bisecting the two charges. py install Documentation. This is of course something that exists and can likely … Continue reading Snow Globe: Part One, Cheap DIY Spherical Projection. Introduction to Lenses and Geometrical Optics. 2, pages 4 and 5 both have links to pages 1, 3, and 6 but only page 5 has a link to page 2. csv the spherical distance from spherical. will produce tuple of azimuth, reverse azimuth, distance (270. multivariate_normal (mean, cov [, size, check_valid, tol]) ¶ Draw random samples from a multivariate normal distribution. It is one of two main approaches, this is a purely geometric algorithm, the alternative is to detect feature points between the two overlapping fisheye images and perform a warp/blend. represents elements by 3 parameters which are radial distance, polar. We assume the radius of the earth is 3960 miles (6373 kilometers). It is a special case of a more general formula in spherical trigonometry, the law of haversines, relating the sides and angles of spherical triangles. 0 foot long, nearsighted python is stretched out perpendicular to a plane mirror admiring its reflected image. Here is a Python method that does just that: def calculate_distance(self, lat1, lon1, lat2, lon2): ''' * Calculates the distance between two points given their (lat, lon) co-ordinates. Distance is determined by the cartesian distance between the two arrays, implying the small-angle approximation if the input coordinates are spherical. This tutorial will guide you through some of the common uses of formatters in Python, which can. • Implemented an ultra-fast shape descriptor based on spherical harmonics expansion and applied it to binding sites and electron density maps. See also simxGetObjectGroupData. With it, you get access to several high-powered computer vision libraries such as OpenCV – without having to first learn about bit depths, file formats, color spaces, buffer management, eigenvalues, or matrix versus bitmap storage. Enrico Franchi. * @param radius the radius of the sphere, e. If the greatest distance to which the snake can see clearly is 27. py nosetests $ sudo python setup. I've got a list of geographic coordinates ([lat, long]) and want to compute the corresponding matrix of distances. Python 101; 8 Geometry. Datacamp has beginner to advanced Python training that programmers of all levels benefit from. If we model the Earth as a sphere the shortest path between two points is on a "large circle," and you can compute the distance using the haversine formula or the spherical law of cosines. According to new CBSE Exam Pattern, MCQ Questions for Class 10. This work paves the way towards enhanced focal spots using structured light. t-Distributed Stochastic Neighbor Embedding (t-SNE) is a powerful manifold learning algorithm for visualizing clusters. Images in Figure 2. Ordinary Kriging assumes the model: Z(s) = µ + ε(s) The default value for lagSize is set to the default output cell size. It is a special case of a more general formula in spherical trigonometry, the law of haversines, relating the sides and angles of spherical "triangles". The program is operated by entering two geographic points and then pressing the Calculate button. You must enter both points to get a distance calculation. Referring back to the graphic, the epsilon is the radius given to test the distance between data points. the average radius for a * spherical approximation of the figure of the Earth is approximately * 6371. Both the transmitter and the receiver use the same modified Gregorian telescope consisting from two offset elliptic mirrors. python, SWIG and extensions using just Python. As discussed by Tyler Reddy at PyData 2015, there was. "Write a Python script that calculates the distance between any two points on the Earth's surface, given their latitude and longitude. zip from PyPI or GitHub, unzip the downloaded file, cd to directory PyGeodesy-yy. These are Euclidean distance, Manhattan, Minkowski distance,cosine similarity and lot more. You can use Python to perform hierarchical clustering in data science. The 60-minute blitz is the most common starting point, and provides a broad view into how to use PyTorch from the basics all the way into constructing deep neural networks. It is based on the WGS 84 reference ellipsoid and is accurate to within 1 mm (!) or better. If the buffer distance is specified from an input field, its unit of measurement will be derived from the feature's spatial reference. If the number is a complex number, abs() returns its magnitude. Match one pair of coordinate arrays to another within a specified tolerance (eps). in more than 20 kms. Distance shall be spherical distance in km. Because I'm stuck in reality and the virtual version of it is always 5 years away, I'm building a physical artifact that approximates the idea: an interactive spherical display. As discussed by Tyler Reddy at PyData 2015, there was no ready-to-go implementation of this in Python. Convert the spherical coordinates defined by corresponding entries in the matrices az, el, and r to Cartesian coordinates x, y, and z. The expression of the distance between two vectors in spherical coordinates provided in the other response is usually expressed in a more compact form that is not only easier to remember but is also ideal for capitalizing on certain symmetries when solving problems. the average radius for a * spherical approximation of the figure of the Earth is approximately * 6371. fmod (x, y) ¶ Return fmod(x, y), as defined by the platform C library. python spherical-harmonics. OpenCV Python Homography Example. What I want to do in this video is make sure that we're good at picking out what the normal vector to a plane is, if we are given the equation for a plane. In the above plot, I have displayed the comparison between the distance covered by two cars BMW and Audi over a period of 5 days. My favorite language is Python. Similarly, positive spherical is called overcorrected and is generally associated with diverging elements. The multivariate normal, multinormal or Gaussian distribution is a generalization of the one-dimensional normal distribution to higher dimensions. Introduction. The exact parameters don't really matter, but suffice to say that it's visually a UV sphere mesh with vertices pushed from the cent. It is based on the WGS 84 reference ellipsoid and is accurate to within 1 mm (!) or better. Python Program for focal length of a spherical mirror Focal length is the distance between the center of the mirror to the principal foci. Next, it will add those two numbers and assign the total to variable sum. KMeans and borrows heavily from that package. Thus, any point on a plane can be specified by specifying it's coordinates, which are distance from origin to the perpendicular projections of this point to the axis. distance = acos(sin(lat 1)*sin(lat 2) + cos(lat 1)*cos(lat 2)*cos(lon. $ python setup. We will be given the radius of curvature of the spherical mirror and we have to find the focal length of the same. The clothoid or double spiral is a curve, whose curvature grows with the distance from the origin. This is a log that I intend to turn into a how-to for some standard codes when using dronekit python to control multiple robots. Microsoft Corporation. For example, if you measure a bearing of N 25° E and a distance of 300 feet, your Delta northing would be 300 x cos 25, and your Delta easting would be 300 x sin 25. Technically, this project is a shared library which exports two functions defined in kmcuda. Distance calculation with Impala (or Hive) Haversine. Since having a better computer is not an excuse for going faster, you have to scale the distance by the "time since the last frame", or "deltaTime". This paper is a companion to a GDC 2008 Lecture with the same title. Special functions (scipy. Images in Figure 2. Spherical law of cosines. Now I am trying to do this on a much larger corpus of documents. #!/usr/bin/env python import cv2 import numpy as np if __name__ == '__main__' : # Read source image. I determined the cluster centroids using Euclidean distance, but then clustered each document based on cosine similarity to the centroid. Python is a free and open-source scripting language and is available for all major platforms and operating systems. Nice question. Illustration for n=3, repeated application of the Pythagorean theorem yields the formula In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" straight-line distance between two points in Euclidean space. introduction • geopy makes it easy for python developers to locate the coordinates of addresses, cities, countries, and landmarks across the globe using third-party geocoders and other data sources. These motifs accordingly dominate in the dense sphere clusters, from which we demand that the particles both pack densely and fit inside a sphere. (The red arrows are intended to represent that only a small portion of the full output table is shown in the picture). The fact that rays come to a focus at different positions is known as spherical aberration. Example: the actual interpolation is conducted using g2gsh_Wrap, a wrapper function that will assign all the appropriate meta data, including the gaussian latitudes, to resulting output. Spherical geometry works similarly to Euclidean geometry in that there still exist points, lines, and angles. Spherical K-Means is a trivial modification to scikit-learn's sklearn. For future readers, there is a small typo. • Implemented an ultra-fast shape descriptor based on spherical harmonics expansion and applied it to binding sites and electron density maps. OpenCV Python Homography Example. >If you draw a line (red) starting at point A, it cuts the vertical line at B and the circle at C. This repo holds my Coordinates library for implementing physics and astronomy applications. Map projection. see examples/ogr_read. Discussion. High-speed printing presses have kept the cost of flat map reproduction to manageable levels but have not yet been developed to work with curved media. So spherical-stereo is official on hold at the moment. Their definition does not require that the Earth be exactly spherical, but approximating the Earth as a sphere is satisfactory for most needs. Of course only a slice of the spherical projection is used. This post is rather old but as I came across an issue testing it I thought it would be good to add a precision. Someone told me that I could also find the bearing using the same. Python bioinformatics utilities for high-throughput genomic sequencing python-biplist (1. These are Euclidean distance, Manhattan, Minkowski distance,cosine similarity and lot more. Such integrals are important in any of the subjects that deal with continuous media (solids, fluids, gases), as well as subjects that deal. hypot is about a third faster than calculating the distance in a handwritten python-function. I ended up using meters instead of kilometers so my R was set to 6371100. Here is a Python method that does just that: def calculate_distance(self, lat1, lon1, lat2, lon2): ''' * Calculates the distance between two points given their (lat, lon) co-ordinates. Manhattan distance is often used in integrated circuits where wires only run parallel to the X or Y axis. radius of a Sphere = 5. The given end point is never part of the generated list; range(10) generates a list of 10 values, the legal indices for items of a sequence of length 10. For the rest of the post, I will use the terms distance and divergence loosely and interchangeably to mean some non-negative valued function of two points that measures "how far apart" they are. EPSG:3857 Projected coordinate system for World between 85. Acoustic theory tells us that a point source produces a spherical wave in an ideal isotropic (uniform) medium such as air. This function doesn't perform minimum distance calculation, it merely reads the result from a previous call to sim. using haversine distance), and mapping the ensuing distribution by. Both commands accept two parameters: the focal distance of the lens, and the distance of propagation. where p generally determines relative importance of distant samples. In this experiment a spherical wavefront is made with a screen containing a small hole. Path loss is often expressed as a function of frequency (f), distance (d), and a scaling constant that contains all other factors of the. io Find an R package R language docs Run R in your browser R Notebooks. These designs have circular, spherical, or hyperspherical symmetry and require 5 levels for each factor. Distance shall be spherical distance in km. Let me first tell you the difference between a bar graph and a histogram. As the Euclidean distance was adopted as the distance measure in. Python Exercises, Practice and Solution: Write a Python program to compute the distance between the points (x1, y1) and (x2, y2). Python is a free and open-source scripting language and is available for all major platforms and operating systems. This is the main entry point for people interested in doing 3D plotting à la Matlab or IDL in Python. 0 documentation Calculate spherical distance between two sky positions. Write a program that computes the spherical distance between two points on the surface of the Earth, given their latitudes and longitudes. Spherical geometry is the study of geometric objects located on the surface of a sphere. In the study of Fresnel diffraction it is convenient to divide the aperture into regions called Fresnel zones. Work has a rotational analog. python, SWIG and extensions using just Python. geopy makes it easy for Python developers to locate the coordinates of addresses, cities, countries, and landmarks across the globe using third-party geocoders and other data sources. This calculator computes the great circle distance between two points on the earth's surface. making distance computations from plane and spherical coordinates, see chapter 11. Based on these ideas is defined the Normalized Google distance. Latitude is in degrees north of the equator; southern latitudes are negative degrees north. com/public_html/7z6n2d/vclw4. Transform X to a cluster-distance space. pyGDM is a python toolkit for electro-dynamical simulations in nano-optics based on the Green Dyadic Method (GDM). Python HOME Python Intro Python Get Started Python Syntax Python Comments Python Variables Python Data Types Python Numbers Python Casting Python Strings Python Booleans Python Operators Python Lists Python Tuples Python Sets Python Dictionaries Python IfElse Python While Loops Python For Loops Python Functions Python Lambda Python Arrays. Note that latitude and longitude are in degrees. Not highly accurate for very small angles. spherical_kn (n, z[, derivative]) Modified spherical Bessel function of the second kind or its derivative. These designs have circular, spherical, or hyperspherical symmetry and require 5 levels for each factor. How to make a (good-looking) spherical panorama on indoor places ? Such as breezeways, corridors, rooms, and any indoor place with limited height, which is approx. Suppose I have two points on a unit sphere whose spherical coordinates are $(\theta_1, \varphi_1)$ and $(\theta_2, \varphi_2)$. In this case, the crystal field interaction is assumed to be purely electrostatic. Introduction to Python for Science. In the Pt class, the return statements say Point() when they should say Pt(). If you find this content useful, please consider supporting the work by buying the book!. Enter -1 as the. SPHERICAL DOME FORMULAS Pi is the distance around the edge of a circle divided by its diameter. And here's a Python implementation that I almost certainly copied from this StackOverflow answer, which I found by Googling for "haversine python":. , 2013; Pennington et al. We will be given the radius of curvature of the spherical mirror and we have to find the focal length of the same. Distance calculations. Latitude and longitude are spherical coordinates, based on recognition that the Earth is round. It determines the map projection for the map display in the data frame. Cylindrical to Cartesian coordinates. Spherical geometry considers spherical trigonometry which deals with relationships between trigonometric functions to calculate the sides and angles of spherical polygons. Plotly Python Open Source Graphing Library. It is the type of projection surface usually used to map the polar regions. I'm using it to create virtual tours, for another information, please refer to my previous question: How to make spherical (360) panoramic image with 2:1 aspect ratio?. The haversine formula assumes a spherical model, and as Brad also pointed out, the variation in radius from north/south latitude variation can exceed the variation in altitude. the signed distance along the coordinate axes, the x-axis, y-axis and z-axis, respectively, from the origin, denoted by O, which has coordinates (0;0;0). 3 assign each data point to the cluster with which it has the *highest* cosine si. Indeed, Distances from points on the surface to the center range from 6,353 km to 6,384 km (3,947–3,968 mi). $ python setup. otherwise things are spherical or geodesic. Python Exercises, Practice and Solution: Write a Python program to compute the distance between the points (x1, y1) and (x2, y2). Python Exercises, Practice and Solution: Write a Python program to compute the future value of a specified principal amount, rate of interest, and a number of years. Problem I would like to know how to get the distance and bearing between 2 GPS points. First, what are we trying. special)¶The main feature of the scipy. Heads up! Contrary to the normal convention of "latitude, longitude" ordering in the coordinates property, GeoJSON and Well Known Text order the coordinates as "longitude, latitude" (X coordinate, Y coordinate), as other GIS coordiate systems are encoded. Given a start point, initial bearing, and distance, this will calculate the destination point and final bearing travelling along a (shortest distance) great circle arc. First lets see how to calculate the distance between two point on a flat plane using the distance formula. to study the relationships between angles and distances. Shortest distance between two lines. If the greatest distance to which the snake can see clearly is 27. It is often used in soil science and geology. To provide data for scipy. First, sketch the two UTM coordinates onto a rectangle. It determines the map projection for the map display in the data frame. Additionally, it is the largest cargo capacity of any ship that can dock at Outposts since it utilizes medium landing pads; because of this the Python is fantastic for Community Goals. 0, all ANNs are within the exact neighbour hypersphere. How to Use Distance Formula to Find the Length of a Line. Lower velocities result in closed elliptical orbits-the vehicle is tied to the neighborhood of the planet. Based on these ideas is defined the Normalized Google distance. distance)¶ Function Reference ¶ Distance matrix computation from a collection of raw observation vectors stored in a rectangular array. θ has a range. To convert: distance to radians: divide the distance by the radius of the sphere (e. There are more than 10,000 packages that are officially available on CRAN (The Comprehensive R Archive Network) and a lot more on other places like Github. Sanz Subirana, J. Match one pair of coordinate arrays to another within a specified tolerance (eps). It is implemented in C++ and has Python wrappers generated with Boost. 3045) and 3rd point (-77. Spherical K-Means is a trivial modification to scikit-learn's sklearn. You can use Python to perform hierarchical clustering in data science. h: kmeans_cuda and knn_cuda. the results are indeed equal, but do not translate to euclidean space without transformation). The first angle θ is the polar angle as in polar coordinates. shagC: Computes spherical harmonic analysis of a scalar field on a gaussian grid via spherical harmonics. OpenCV Python Homography Example. This is an excerpt from the Python Data Science Handbook by Jake VanderPlas; Jupyter notebooks are available on GitHub. Using Python to compute the distance between coordinates (lat/long) using haversine formula and print results within. When rays from infinity come in parallel to the optical axis of a spherical mirror, they are bent so that they either converge and intersect in at a point, or they seem to diverge from a point. see examples/ogr_read. Assuming the kriging errors are normally distributed, there is a 95. kml) Hi, I'm trying to work out this problem and wanted to reach out if anyone had experience with geospatial querying. The interference function in BornAgain is a key component to organize in-plane order of nanoparticles. This chapter intends to give an overview of the technique Expectation Maximization (EM), proposed by (although the technique was informally proposed in literature, as suggested by the author) in the context of R-Project environment. & , looks just like the free propagator, except that distance is normalized by n1/n2. Download with Google Download with Facebook or download with email. distance = acos(sin(lat 1)*sin(lat 2) + cos(lat 1)*cos(lat 2)*cos(lon. As mentioned earlier, this is one of the essential criteria in order to perform kriging. Then we can write S P Q. We want to estimate the height z given a position on the plane (x,y). I want to undershoot, as this will be for A* graph search and I want it to be fast. OpenCV provides an easy to use a utility function called matchShapes that takes in two images ( or contours ) and finds the distance between them using Hu Moments. Clustering is the grouping of objects together so that objects belonging in the same group (cluster) are more similar to each other than those in other groups (clusters). Euclidean distance is the distance between two points in Euclidean space. A set S of unit vectors in n-dimensional Euclidean space is called spherical two-distance set, if there are two numbers a and b, and inner products of distinct. 0 release, some three-dimensional plotting utilities were built on top of Matplotlib's two-dimensional display, and the result is a convenient (if somewhat limited) set of tools for three-dimensional data visualization. This post was written as a reply to a question asked in the Data Mining course. It allows you to calculate the distance and bearings between points, mangle xearth/xplanet data files, work with online UK trigpoint databases, NOAA's weather station database and other such location databases. * return_type = 3 returns the primitive number that was closest. Distance calculations. Note that the Python expression x % y may not return the same result. A multipurpose library that allows you to embed Google Maps, Street View, Indoor Maps, and much more inside your native iOS application. An iterator is an object that implements next, which is expected to return the next element of the iterable object that returned it, and raise a StopIteration exception when no more elements are available. : All quantities (e. Using a different distance function other than (squared) Euclidean distance may stop the algorithm from converging. earth_distance is a collection of GPL v3 licensed modules for working with points on Earth, or other near spherical objects. multivariate_normal (mean, cov [, size, check_valid, tol]) ¶ Draw random samples from a multivariate normal distribution. When you are finished, check to see that your codeexecutes correctly: save your script using the "Save" option in theFile menu (be sure to save your file to a location where you canfind it later, like the Desktop!). Now, by introducing a little notation we can re-express the above distance in a curious way. The full documentation for this code is in the Shapely manual. A rotatable design exists when there is an equal prediction variance for all points a fixed distance from the center, 0. University students who lead a DSC gain access to Google technology, events, and mentorship while training their local community through fun meetups, project building activities, and global competitions. lerp — returns a linear interpolation to the given vector. This function is deprecated, use. | CommonCrawl |
Research article | Open | Open Peer Review | Published: 24 April 2015
Bayesian estimation of a cancer population by capture-recapture with individual capture heterogeneity and small sample
Laurent Bailly1,3,
Jean Pierre Daurès2,4,
Brigitte Dunais1,3 &
Christian Pradier1,3
Cancer incidence and prevalence estimates are necessary to inform health policy, to predict public health impact and to identify etiological factors. Registers have been used to estimate the number of cancer cases. To be reliable and useful, cancer registry data should be complete. Capture-recapture is a method for estimating the number of cases missed, originally developed in ecology to estimate the size of animal populations. Capture recapture methods in cancer epidemiology involve modelling the overlap between lists of individuals using log-linear models. These models rely on assumption of independence of sources and equal catchability between individuals, unlikely to be satisfied in cancer population as severe cases are more likely to be captured than simple cases.
To estimate cancer population and completeness of cancer registry, we applied Mth models that rely on parameters that influence capture as time of capture (t) and individual heterogeneity (h) and compared results to the ones obtained with classical log-linear models and sample coverage approach. For three sources collecting breast and colorectal cancer cases (Histopathological cancer registry, hospital Multidisciplinary Team Meetings, and cancer screening programmes), individual heterogeneity is suspected in cancer population due to age, gender, screening history or presence of metastases. Individual heterogeneity is hardly analysed as classical log-linear models usually pool it with between-"list" dependence. We applied Bayesian Model Averaging which can be applied with small sample without asymptotic assumption, contrary to the maximum likelihood estimate procedure.
Cancer population estimates were based on the results of the Mh model, with an averaged estimate of 803 cases of breast cancer and 521 cases of colorectal cancer. In the log-linear model, estimates were of 791 cases of breast cancer and 527 cases of colorectal cancer according to the retained models (729 and 481 histological cases, respectively).
We applied Mth models and Bayesian population estimation to small sample of a cancer population. Advantage of Mth models applied to cancer datasets, is the ability to explore individual factors associated with capture heterogeneity, as equal capture probability assumption is unlikely. Mth models and Bayesian population estimation are well-suited for capture-recapture in a heterogeneous cancer population.
Cancer is the leading cause of death in Western countries and particularly in France [1]. In view of this situation, cancer control programmes have been implemented. Evaluating the effectiveness of these policies, aiming for improved prevention and management, is essential. However to conduct such an evaluation, a baseline reference requiring ongoing, reliable and complete data collection, such as a cancer registry, is necessary. Besides providing descriptive epidemiology, cancer registries are currently used for epidemiological research, assessment of screening programmes and treatment innovations [2].
In order to verify the completeness of cancer cases recorded within a specific geographic area, capture-recapture method is usually applied. The capture-recapture method is a sampling technique originally devised by ecologists to study fauna [3,4], and subsequently adapted to epidemiological studies [5-7] Since the early nineties, this method has been extended to many demographic and epidemiologic studies [8,9]. It has thus been used to confirm the completeness of the data recorded in cancer registries [10-12].
The capture-recapture procedure in epidemiology consists in confronting data from at least two independent sources, collecting cases in the same area in order to estimate the total number of cases, and assessing the completeness of each data source [13]. In brief, this method involves modelling the overlap between two or more lists of individuals (data "sources") from the target population, and using this model to predict how many additional individuals were unseen, and hence the total population size. To avoid bias in the estimate, the sources of data collection must be independent and homogeneity of capture must be ensured [14,15] (i.e. the probability of capture does not depend on case characteristics). Capture heterogeneity can result in positive dependence (underestimation of the population) or negative dependence (over-estimation of the population) between sources.
The standard approach to capture-recapture in epidemiology is to fit log-linear models [16,17], in which the inclusion of source by source interaction terms may account for dependencies between the data sources. These parameters are subsequently estimated by the maximum likelihood estimate procedure. With a sufficiently large population, this procedure is acceptable under the asymptotic assumption. However, the asymptotic assumption cannot be verified on few cases, a frequent situation when capture-recapture concerns a specific target population, as a cancer population within a small area.
Many authors have presented capture-recapture methods in epidemiology to take into account dependence and individual heterogeneity, including source by source interaction terms in the log-linear model, log-linear models stratified on several covariates, or including sources by covariate interactions in a single log-linear model [6,9,18].
In the log-linear method, capture heterogeneity is pooled with between-"list" dependence within the between-source interaction terms [19]. To test capture heterogeneity stratification of cases on potential variables, related to capture heterogeneity, is applied. For example, this method has been applied by King et al.[20] to estimate current injectors in Scotland and drug-related death rate by sex, region and age-group. This consists in constructing a single contingency table in which cells correspond to the numbers of individuals belonging to each distinct combination of covariates and sources.
Conversely, incorporating one or more potential variables is more complex when numbers of cases are limited. Stratification of cases, whether common or not to both sources, on several covariates leads to a contingency table containing several missing cells or too few cases within certain cells to provide robust results.
Moreover, many authors, e.g. Schmidtmann [21] as Silcoks and Robinson [22], compared several methods to estimate the completeness of cancer registration, among which log-linear models: according to both these authors, log-linear modelling does not always yield the best estimation. Confirmation of results obtained via the classical log-linear model therefore appeared to us as essential. Capture heterogeneity, which had not been previously tested with log-linear models, needed to be taken into account. During the past years, much theoretical research has been conducted to develop capture recapture methods, such as that by Chao, Pan and Chiang [23] who propose a Lincoln-Petersen estimator including dependence effects resulting from local lists and heterogeneous capture probabilities. Other authors have proposed mixed models: Mao [24] focused on a non-parametric maximum likelihood estimate for two classes of mixed models with a binomial and geometric distribution. The classical modelling approach consists in estimating the parameters of a model which are then considered as fixed quantities. To confirm our results with a totally different approach, we therefore wished to implement a Bayesian procedure. Several authors have focused on this method, among them Manrique-Vallier and Fienberg [25] who postulated the existence of a homogeneous population class to overcome the problems related to heterogeneity of closed populations in capture recapture.
However, capture-recapture was first developed in ecology for estimating the size of animal populations: as a result, methods in ecology are somewhat more developed and there is probably much for epidemiologists to learn from the ecology literature.
In this paper, we borrow tools from the ecological capture-recapture literature: Mtbh models [26], which simultaneously allow for the effects of time, behaviour and individual heterogeneity in capture probabilities. King and Brooks [27], proposed a Bayesian estimate for the size of a closed population in the context of heterogeneity and model uncertainty, using Bayesian Model Averaging (BMA). This approach overcomes the difficulties related to capture heterogeneity and model selection, providing the ecological models may be adapted to capture recapture procedures in epidemiology.
We applied these tools to a capture recapture study concerning a histopathological cancer registry [28]. This study confronted newly diagnosed cases of breast and colorectal cancer, in the Alpes-Maritimes (Southeastern France), among patients aged 50 to 75 years, recorded in the Histopathological Registry (HR), those discussed in hospital Multidisciplinary Team Meetings (MTM) and those diagnosed through the coordinated Cancer Screening Programmes (CSP). We compared the results to those obtained with log linear models and sample coverage approach [19] for the same data [28]. We have then applied ecological models and BMA method to well-known examples of data set in capture-recapture, as an outbreak of the hepatitis A virus in a college in northern Taiwan [29], a data set on diabetes in a community in Italy based on four records [30] and finally to five lists of infants born with a specific congenital anomaly in Massachusetts [31,32].
Capture-recapture ecological models
When estimating a population size using the capture recapture method in an ecological study, the underlying assumptions concerning case capture have a direct impact. The selected model to estimate the total number of cases rests on these assumptions and on its adjustment on the observed data. Otis et al. [26] defined three effects influencing capture: time, behaviour and individual heterogeneity. All the interactions may be possible between these three effects. In other words, models differ according to whether the capture probability changes only with time of capture (Mt), or changes between individuals according to their behaviour (Mb) or their characteristics (Mh).
The use of behavioral models (Mb and more complex models including the behavioral effect) do not appear appropriate in epidemiology as capture probability should not change after a previous capture. Indeed, they are based on the assumption of a natural sequence of capture episodes, whereas there is no time sequence in our sources so that these models do not appear useful. This has also been pointed out by Chao et al. [19] who went as far as stating that only Mt, Mh, and Mth are potentially useful for applications in epidemiology. For our study, we will therefore focus on these three models applied to our data, implying that capture probability changes only with time of capture (Mt), or according to individuals' characteristics (Mh) or both (Mth).
Let piτ denote the capture probability for individual i = 1, 2, 3, …N at time τ = 1, 2, 3, … T. F(i) represents the first time that individual i is observed. Therefore piτ = P which is the initial capture probability of i for times τ =1,…, F(i), and the recapture probabilities for τ = F(i) + 1, …T assuming F(i) < T. Thus individual i is captured at time τ =1, not captured at time τ = F(i) – 1 and captured at time τ = F(i), and can be recaptured between times τ = F(i) + 1 and τ, with total capture times = T.
The saturated M th model integrating time (t), and heterogeneity (h) has capture probabilities of the form:
$$ \mathrm{Logit}\left({\mathrm{p}}_{i\tau}\right)=\mu +{\alpha}_{\tau }+{\gamma}_i $$
where γ i denotes independent and identically distributed individual random effects (i.i.d) ~ N (0, σ2γ).
In this model the estimated parameters are μ (mean capture rate expressed as logit), ατ (year effect for capture), and σ2γ (variance of individual random effects). Submodels of the saturated model are obtained by setting certain parameter values equal to zero.
It is assumed that capture probabilities are independent, given the parameter values. Let θ = {μ, α, σ2γ, N} with α = {α1,…,αT}, and γ = {γ1,…, γN}. The vector x, which describes the capture history of all individuals, is given by:
$$ f\left(\left.x\right|\theta, \gamma \right)\propto \frac{N!}{\left(N-n\right)!}{\displaystyle \prod_{t=1}^T{\displaystyle \prod_{i=1}^N{p_{it}}^{x_{it}}{\left(1-{p}_{it}\right)}^{\left(1-{x}_{it}\right)}}} $$
where n (n < N) denotes the total number of individuals captured in the study, i.e. n unique individuals captured initially. The vector xit = 0 if individual i is not captured at time t and xit = 1 if the individual is captured at time t, i.e. x = (x11, …, x1T, … x21, …, x2T, xN1, …, xNT) where xit = 0 or xit =1.
For models with no heterogeneity, i.e. Mt, the individual random effect γi = 0 so f(x | θ, γ) = f(x | θ) and θ can be obtained using the Maximum Likelihood Estimate (MLE) of the parameters. In the presence of heterogeneity (models Mh, Mth), calculating the MLE and selecting the model is more complex [33] because of the individual random effects.
Bayesian population estimation
In the Bayesian approach, individual heterogeneity is estimated from Monte Carlo Markov Chain algorithms. All the parameters can thus be estimated for all possible models, with or without individual heterogeneity. In the Bayesian approach, the model parameters are considered as a random sample. The distribution of the samples therefore provides information on the parameters. Before collecting the data, the parameter distribution is a prior distribution. After data collection, the parameters have a posterior distribution.
In this analysis, the model itself is considered as an unknown parameter to be estimated. According to Bayes' theorem applied to continuous distributions, the joint posterior distribution over both parameter and model space is obtained by multiplying the likelihood by the prior distribution, with m denoting the model and θ m the parameters in model m:
$$ \pi \left({\theta}_m,m\left|x\right.\right)\propto g\left(x\left|{\theta}_m,m\right.\right)p\left({\theta}_m\left|m\right.\right)p(m) $$
To introduce individual heterogeneity, random effects are included as expressed by the variables γ = {γ1,…, γN} in equation Eq. 1.
$$ \pi \left({\theta}_m,\gamma, m\left|x\right.\right)\propto f\left(x\left|{\theta}_m\right.\gamma, m\right)h\left(\gamma \left|{\theta}_m\right.\right)p\left({\theta}_m\left|m\right.\right)p(m) $$
The term h(γ|θm) corresponds to the model assumption of the random effect γ i ~ N (0, σ2γ).
Finally, the posterior distribution of the parameters and the model is given by:
$$ \pi \left({\theta}_m,m\left|x\right.\right)={\displaystyle \int \pi \left({\theta}_m,\gamma, m\left|x\right.\right)d\gamma } $$
Models are compared via posterior model probabilities and an estimate of the total population, based on all plausible models, may be obtained using the posterior distribution. In other words, each estimate is an average of the posterior distributions under each of the models considered, weighted by their posterior model probability. This procedure, detailed in Additional file 1: Appendix A, is called Bayesian Model Averaging. Usually, a single model is selected, as this model best fits the observed data. However these data come from random sample. As Hoeting pointed out [34] this approach ignores the uncertainty in model selection, leading to over-confident inferences and decisions that are more risky than one thinks they are. Bayesian model averaging (BMA) provides a coherent mechanism for accounting for model uncertainty.
Bayes' theorem is used to estimate the joint posterior distribution of all the parameters included in the model. Ultimately, the posterior marginal distribution of the parameters of interest is estimated, requiring integration of the posterior joint distribution, which is not always possible. In the modern Bayesian approach, this distribution of the posterior parameter vector is not integrated, and specific simulations are performed to obtain posterior distribution samples and thus simulated values for the posterior marginal distributions of the parameters of interest. As the posterior distribution is multidimensional, Monte Carlo Markov Chain algorithm is applied to obtain a random sample of the posterior distribution. Let's firstly consider the two components of this method: Monte Carlo integration and Markov chains. Monte Carlo integration is a simulation technique which allows obtaining an estimate of a given integral. This technique is based upon drawing observations from the distribution of the variable of interest and then calculating the sample mean [35]. To obtain a potentially large sample from the posterior distribution we use a Markov chain, which is a stochastic sequence of numbers where each value in the sequence depends only upon the last. Under conditions as chain is aperiodic and irreducible, distribution will converge to a stationary distribution. Monte Carlo Markov Chain methods perform Markov chains with Monte Carlo integration to generate observations and to construct a sequence of values whose distribution converges to the posterior distribution of interest. Once the chain has converged we can use sequence of values to obtain estimates of any posterior summaries of interest (Monte Carlo). To be sure that Markov chain has reached the stationary distribution before Monte Carlo estimates, we discard observations from the start of the chain, which is called the burn-in.
Two Monte Carlo Markov Chain algorithms are used according to the parameters and models. The Metropolis-Hastings algorithm is used when the model does not involve changing the dimension and the reversible jump algorithm when calculations involve a change in dimension (due to the model and the population size in the presence of individual effects). The reversible jump algorithm is detailed in Additional file 1: Appendix B.
Lastly, prior probabilities must be defined for the models and parameters. In the absence of prior assumptions concerning their influence on the estimate of the total population, we chose a non-informative model. The priors for each parameter are detailed in Additional file 1: Appendix C. The prior probability of selecting a model follows a uniform distribution p(m) = 1/k where k denotes the number of models. For each potential effect (time, heterogeneity) the prior probability is 0.5. The parameters are assumed to be independent and to follow the same prior distribution in each model (if present).
Software for Bayesian analysis
We applied the above Bayesian methods using the WinBUGS [36] software package, which performs complex Bayesian analyses. The codes used for the Mth models were drawn from the models developed by Link and Barker [37] in WinBUGS for Bayesian inference applied to ecological surveys, namely capture recapture. Briefly, two vectors are used: the number of captures of each individual and the fact that an individual is considered as caught, not caught or unknown during a specific episode. To provide an inference for the total population we consider that the individual capture probabilities of the subjects that were not caught are linked to the capture probabilities of those that were captured. The total number of subjects is estimated by the method of data augmentation developed by Link and Barker. The WinBUGS codes are detailed in Additional file 1: Appendix D.
Individual covariates
The three sources studied are the Histopathological Registry, the hospital Multidisciplinary Team Meetings (MTM) and the Cancer Screening Programmes. Firstly, histopathology laboratories have been transmitting all the invasive cancers with a histopathology diagnosis to the Nice University Hospital Public Health Department, which coordinates the Histopathological registry since 2005. Cancer cases from hospital Multidisciplinary Team Meetings came from the regional cancer network which has been systematically collecting data from hospital since 2007. The third source is the local coordinating centre for cancer screening which collects data concerning patients aged 50 to 75 years with a positive result following screening for breast or colorectal cancer since 2002 and 2005 respectively.
An estimate was first obtained from the three available sources, each of them considered as a capture episode. Secondly, an estimate of the total population was obtained by considering each covariate as a capture episode. The selected parameters considered as potentially accounting for different capture probabilities included age and presence of metastases at the time of diagnosis (TNM staging), according to the recommendations concerning capture recapture applied to cancer registries [38,39]. We also introduced past history of screening mammography [21] and gender for cases of breast and colorectal cancer, respectively, as potential capture heterogeneity parameters.
Estimate of the number of incident cases of breast cancer
Capture-recapture Mth models were initially applied to the three sources, each considered as a capture episode, i.e. Histopathological cancer Registry (HR), hospital Multidisciplinary Team Meetings (MTM) and cancer screened cases (CSP). In these 3 sources, 787 cases aged 50 to 75 years were notified in 2008 as presented in Figure 1, of which 729 by the Histopathological cancer registry, 470 were identified through at least 2 sources and 108 were common to all three sources. After averaging over all the models according to their posterior probability, the estimated mean number of breast cancer cases was 790.6 (median = 790.2, 95%HPDI: 790.2-792.2), as presented in Table 1. The average estimate was obtained over all models following 25 000 iterations after discarding the initial 5 000 iterations. Since first iterations a convergence of Markov chains was obtained, as a stationary distribution was observed after only 500 iterations.
Record linkage of breast cancer cases notified in 2008 among subjects aged 50 to 75.
Table 1 Estimates of incident breast cancer cases according to capture-recapture M th models 3 capture episodes: Histopathological Registry, hospital Multidisciplinary Team Meeting, and Cancer Screening Program
With a total number of 791 breast cancer cases, the completeness of the Histological Register was estimated to be 92.2% (92.1%; 92.3%). The posterior probability for the Mt model was 100%, corresponding to the model for which capture probability changes for each source. As there is no time sequence in our sources, we altered it and compared the estimates obtained for each possible time sequence. Estimates were exactly the same for all possible time sequences. Models for which the capture probability differs for each individual (Mth Mth) provide a different estimate: 814.1 (median = 814.2, 95%HPDI: 805.2-824.2) and 806.3 (median = 806.2, 95%HPDI: 798.2-815.2) cases, respectively.
Averaging over the Mth models, applied to the three sources, provides an estimate in line with the result obtained by the classical method [28]. With the selected log-linear model, the estimate was of 791 breast cancer cases corresponding to the model including interaction between the Multidisciplinary Team Meeting source and the two other sources, as presented in Table 2. The choice of the most appropriate log-linear model is based on the likelihood ratio statistics. The selected model is the one with the fewest interaction terms and the best fit with the observed data, i.e. a non-significant value for the likelihood ratio statistics. The model was selected using a stepwise descending procedure starting from the saturated model, taking all interactions into account, until the most parsimonious model with the best fit was retained. The total number of cases was estimated to be 791 cases (95% CI: 784-797), i.e. the completeness of the histological register was estimated to be 92.2% (95% CI: 91.5%; 93.0%). Taking into account coefficient of covariation as developed by Chao et al. [19], the sample coverage approach gives an estimate of 794 cases CI 95% [788-824] with dependent sources and an estimate of 802 cases CI 95% [795-815] with three independent sources, close to the estimates obtained previously.
Table 2 Estimated total number of incident breast cancer cases according to log-linear models Histopathological Registry (HR), hospital Multidisciplinary Team Meeting (MTM), and Cancer Screening Program (CSP)
Dependence between sources
For breast cancer cases, the Lincoln-Petersen estimate [28] obtained via two-by-two record linkage for the MTM and CSP sources (NMTM-CSP = 958) differed from the estimates obtained from the other sources (NHR-MTM = 814 and NHR-CSP =766). Interdependency between these two sources was suspected and confirmed by a test for independence [6] on the basis of whether cases were recorded or not in the third source (ORMTM-CSP = 0.52; 95% CI:0.37-0.73). If interdependence is shown between two of at least three data sources, these must be pooled in order to apply the capture recapture procedure to two independent sources. The resulting Lincoln-Petersen estimate, by cross-linkage of Histopathological Registry cases and cases discussed during MTM pooled with screened cases, was N = 803.2 (95% CI: 793.8-812.5).
Capture heterogeneity
To investigate capture heterogeneity between individuals with breast cancer, we created 21 capture episodes, based on potential heterogeneity covariate: age, expressed as 5-year intervals, i.e. five capture episodes for each of the three sources, presence of metastases at the time of diagnosis and, finally, history of screening mammography by sources, i.e 6 capture episodes. The estimate averaged over all the models was of 803 cases (median = 802.6, 95%HPDI: 798.6-809.6), based on the results of the Mh and Mt models with 80% and 20% of posterior probability, respectively, as presented in Table 3. The average estimate was obtained over all models following 25 000 iterations after discarding the initial 10 000 iterations. Convergence of Markov chain to a stationary distribution was observed after 10 000 iterations.
Table 3 Estimates of incident breast cancer cases according to capture-recapture M th models 21 capture episodes: age, TNM stage at diagnosis, screening history
This result is in line with the Lincoln-Petersen estimate for pooled MTM and cancer screening sources. Therefore, the estimated completeness of the Histopathological Registry for breast cancers would be of 90.8% (90.0%-91.3%).
Estimate of the number of incident cases of colorectal cancer
Results from the three sources showed 512 cases of colorectal cancer in 2008 (HR, MTM, CSP), of which 481 were recorded in the Histopathological Registry, 337 were identified through at least 2 sources and 41 were common to all three sources as shown in Figure 2.
Record linkage of colorectal cancer cases notified in 2008 among subjects aged 50 to 75.
Using the BMA method, ecological Mth models were first applied to the three sources collecting incident colorectal cancer cases, considered each as a capture episode. The estimate averaged over all the models yielded 513 cases (median = 513.0; 95% HPDI: 512.-515.0) based on the results of the Mt model, as presented in Table 4. Estimates were exactly the same for all possible time sequences. The average estimate was obtained over all models following 25 000 iterations after discarding the initial 5 000 iterations. As for breast cancer, convergence of Markov chains to a stationary distribution was observed from the beginning, meaning before 1000 iterations.
Table 4 Estimates of incident colorectal cancer cases according to capture-recapture M th models 3 capture episodes: Histopathological Registry, hospital Multidisciplinary Team Meeting, and Cancer Screening Program
With a total number of 513 colorectal cancer cases, the completeness of the Histological Registry was estimated to be 93.8% (93.4%; 94.0%). With the selected log-linear model [28], the estimate was of 527 colorectal cancer cases corresponding to the model without interaction, as presented in Table 5. With sample coverage approach, estimates are 527 cases CI 95% [519-542] with three independent sources, as selected log-linear model without interaction, and 529 cases CI 95% [519-557] with dependent sources.
Table 5 Estimated total number of new cases of colorectal cancer according to log-linear models Histopathological Registry (HR), hospital Multidisciplinary Team Meeting (MTM), and Cancer Screening Program (CSP)
As for breast cancer cases, two-by-two record linkage using the Lincoln-Petersen estimator for MTM and CSP sources provided a different result from the two other estimates (NMTM-CSP = 618 versus NHR-MTM = 526 and NHR-CSP =513). However, testing for independence gave a statistically non-significant result (OR = 0.66 [0.40-1.08]). The Lincoln-Petersen estimator [28], for the Histopathological Registry and pooled MTM and CSP sources, yielded an estimate of 525 cases (95% CI: 516.5-534.5).
Capture heterogeneity among the colorectal cancer cases was also investigated. Retained covariates potentially responsible for heterogeneity included age in 5 year intervals (15 capture episodes), gender (6 capture episodes) and metastases present at the time of diagnosis, i.e. 15, 6 and 3 capture episodes respectively and 24 for all three sources. Contrary to the results obtained with 3 sources, the estimate, averaged over all the models, was of 521 cases (median = 520.6; 95%HPDI: 517.6-526.6), resulting from the Mh model including individual capture heterogeneity, with a posterior probability of 99%, as presented in Table 6. The average estimate was obtained over all models following 25 000 iterations after discarding the initial 5 000 iterations. Contrary to breast cancer with potential heterogeneity covariate, for colorectal cancer convergence of Markov chains to a stationary distribution was observed rapidly after 1 000 iterations. The estimated completeness of the Histopathological Registry would be of 92.3% (91.3%-92.9%).
Table 6 Estimates of incident colorectal cancer cases according to the capture-recapture M th models 24 capture episodes: age, TNM stage at diagnosis, gender
Application to data set from other fields
Then we apply our method to an outbreak of the hepatitis A virus in a college in northern Taiwan [29] with 271 cases ascertained by three sources, to a data set on diabetes in a community in Italy based on four records [30] with 2069 cases identified and finally to fives lists of 537 infants born with a specific congenital anomaly in Massachusetts [31,32]. For the HAV data, our method gives an estimate of 515 cases [465.5-567.5] whereas one-step estimator by sample coverage approach gives 508 cases [442-600], Petersen estimator 336 cases and log-linear models 1300 cases. The number of HAV infected students was finally known with a screen serum test for HAV antibody for all students and was about 545.
For the data set on diabetes, author found that the selected log-linear model that fits data gave an estimate of 2 771 cases but they further stratified for heterogeneity terms and an estimate of 2 586 cases [2341-2830] was obtained. With sample coverage approach Chao et al. estimate was 2 559 cases [2472-2792] and with our method estimate is 2 589 cases [2534-2645].
For the data set on infants' congenital anomaly, Wittes and Fienberg [31,32] obtained a close estimate respectively 638 cases under independent assumption and 634 cases for the log-linear model approach. For the sample coverage approach, the retained estimator with dependencies was 659 cases [607-750]. With our method estimate is close to previous ones as it is 654 cases [631-680].
Log-linear models provide a useful method to estimate population size but some authors have pointed out [19,38], the need to pursue additional methods because of assumptions, as independence of sources and equality of capture probability, which are rarely satisfied. Capture-recapture Mth models are interesting for cancer population as individual capture heterogeneity is taken into account. Bayesian population estimation allows small sample as it does not rely on the asymptotic assumption.
Firstly, we applied capture-recapture Mth models to epidemiological data. Secondly, we applied a Bayesian Model Averaging (BMA) method to present a result averaged over all the models. The BMA method was of interest to us because it takes into account all possible models instead of selecting only the result of the best model. However, we chose to apply capture-recapture Mth models specifically for this study because individual heterogeneity was suspected between severe cancer cases collected via Multidisciplinary Team Meetings (MTM), and simple cancer cases screened in a Cancer Screening Program (CSP). These methods can be used separately and this is what we have done in our study. The results for each model were analysed and the BMA method was then applied to obtain a result weighted for all models according to their posterior probability.
For both types of capture-recapture Mth models, the samples are independent only for the Mt model, while heterogeneity arises for the Mh model. The Rasch-like model is the Mth model which extends the Mh model by allowing time effects. Heterogeneity between individuals means that even if two lists are independent within individuals, the two sources may become dependent if the capture probabilities are heterogeneous among individuals. Model Mh assumes that each individual has its own unique probability that remains constant over samples. Capture-recapture Mth models have already been used in the context of lists. Chao [19] for example has shown that results of models Mh and Mth were very close to those obtained with log-linear models by Wittes [31] for 5 lists of "Infants' congenital anomaly data". Chao's conclusion was that although heterogeneous models did not consider possible local dependence, the estimates were close to the proposed estimate that does. We came to the same conclusion in our study comparing estimates yielded by capture-recapture Mth models with results obtained by the "source pooling" method. When two of at least three data sources are dependent, these must be pooled in order to apply the capture recapture procedure to two independent sources. "Source pooling" is a method proposed by Wittes [6] and adopted by many authors thereafter. It was applied in this study for comparison with previously published results on these data [28]. We presented this method here to emphasize that the results are in line with the classical methods (log-linear model on three sources or pooled if dependent) and capture-recapture Mth models. The interest of these models is the ability to decide to "capture' subjects aged 60 to 65 years, or diagnosed with metastases, or any others covariates suspected for heterogeneity, whereas with the log-linear approach, the number of potentially adequate models increases and model selection is more difficult.
Finally, capture-recapture Mth models allowed us to compare and to confirm results obtained with log-linear models and then to make them more reliable. This last point was particularly important as heterogeneity in three-list data could involve that our estimates were not reliable [14]. Moreover the results retained with the log-linear approach were the estimates of the selected model whereas other fitted models could have yielded a quite different estimate [40].
The objective of this article was to propose an alternative to the method most often used, i.e a log-linear model stratified on several covariates. For example, Tilling [18] did not propose a log-linear model but a logit model which has the advantage over the log-linear model stratified on several covariates to limit the number of parameters studied, to incorporate continuous covariates and above all of being applicable to two sources. However, the adjustment used the maximum likelihood ratio based on the asymptotic assumption, which cannot be verified in our case due to the small number of cases, which is frequent in epidemiology.
Considering each of the three sources (HR, MTM and CSP) as a capture episode, the estimated mean number of breast cancer cases was 790 and the number of colorectal cancer cases 513, according to the Mt model. When considering each covariate as a capture episode, the model retained in BMA corresponds to the model with heterogeneity. The estimated total number of cases, for breast cancer, was of 803 cases according to the Mh model against 791 cases for the log-linear model [28]. The resulting Lincoln-Petersen estimate from the source-pooling method was of 803 cases too and sample coverage approach gives estimates of 794 and 802 cases, respectively with dependent and independent sources. From our point of view, the estimate from the Mh model could be considered as more representative than the results of all the log-linear models considered for breast cancer. Finally, the discrepancy, without considering heterogeneity, between estimates may seem not apparent. However, we have shown with some covariates, corresponding to our heterogeneous population, that heterogeneity exists and has an impact. Indeed, as there were very few cases missing, the difference is equal to two points for histological cancer registry completeness.
The only difference between the Mt and Mh models lies in the selected covariates considered each as a capture episode which could influence the probability of capture for each individual case. Size effects due to smaller samples cannot be held responsible for a higher posterior probability for model Mh because the total number of cases and the gap between samples are the same with 3 episodes of capture. Furthermore we modified the time sequence of our sources, as there is no sequential time order in lists of individuals, and estimates were the same. For colorectal cancer, the estimate of 521 cases for the capture-recapture Mh model was in line with the estimated total number of 527 cases retained by the selected log-linear model and by sample coverage approach. The Lincoln-Petersen estimator, for HR and pooled MTM and CSP sources, yielded an estimate of 525 cases.
Application of capture-recapture Mth models have confirmed estimates obtained via the log-linear models retained according to the traditional procedure. The traditional model selection procedure and the use of the capture-recapture Mth models thus yield concordant results.
For our study, a major advantage of this Bayesian population estimation was the possibility of easily taking into account several covariates potentially responsible for capture heterogeneity, even with few cancer cases collected by some sources. Considering some potential heterogeneity covariates (i.e. age, presence of metastases, screening history or gender) as a capture episode has shown that capture probability was not homogenous between individuals. This can be easily understood for our heterogeneous cancer population as a case with metastases at the time of diagnosis won't have the same capture probability as other cases, since multidisciplinary team meetings will be more concerned with such situations of advanced disease, whereas there are fewer cases with metastases in a cancer screening program.
However, log-linear method, sample coverage approach and capture-recapture Mth models have their advantages and their limitations, which is in favour of applying them both to make estimates more reliable. The main advantage of the log-linear method is that it is particularly well suited to the so-called « list » method. All models have the same framework, the selected model can be tested easily, between-source dependencies are included in interaction terms and inference is easily available in statistical software.
Applying the Bayesian procedure to the Mth capture recapture models has the advantage of taking case-linked capture heterogeneity into account and providing a result that incorporates all the possible models. Furthermore, with the Bayesian method, considering a potential heterogeneity covariate as a capture episode may be easily applied to small samples, which can be particularly useful in cancer epidemiological studies. On the other hand, covariate selection may seem arbitrary, which is in favour of selecting variables that have already been shown to have an impact on capture probability [18,39,21]. Bayesian inference is nowadays easily available with the WinBUGS software package [36]. Moreover, codes for Mth models and Bayesian Model Averaging have been developed [37] by ecological researchers and can be adapted to epidemiological data.
Our analysis shows that capture-recapture Mth models can be applied to data usually available as « lists » in epidemiology. The advantage of these models resides in their capacity to independently assess heterogeneity of individual capture probability which is useful for a heterogeneous cancer population. Moreover, Bayesian population estimation allows including several covariates potentially accounting for heterogeneity even with small sample. Thus, capture-recapture Mth models and Bayesian population estimation should be considered additionally to the classical methods usually implemented in cancer epidemiology, to confirm results and enhance the reliability of estimates.
Availability and requirements
Winbugs software is available through: http://www.mrc-bsu.cam.ac.uk/software/bugs/the-bugs-project-winbugs.
HR:
Histopathological Registry
MT:
Multidisciplinary team meetings
CSP:
Cancer screening programmes
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We thank reviewers for their careful reading and constructive advices. We thank Dr. Eugènia Mariné Barjoan, Dr. Damien Ambrosetti, Dr. Bernard Giusiano, and Miss Agnès Viot who were involved by making substantial contributions to conception, design and acquisition of data. This study is supported by Institut National du Cancer, Paris, France. Authors want to thanks Dr Béatrice Jacqueme, ARS PACA and Conseil Général des Alpes-Maritimes for their support. Special thanks to the Department of Public Health's team to all CRISAP PACA's pathologists, to APREMAS's and OncoPACA's team.
Department of Public Health, University Hospital of Nice, Nice, France
Laurent Bailly
, Brigitte Dunais
& Christian Pradier
Department of Biostatistics, Epidemiology and Clinical Research EA2415, University of Montpellier1, Montpellier, France
Jean Pierre Daurès
Département de Santé Publique, CHU Nice, Hôpital Archet 1. Niveau1, Route Saint Antoine de Ginestière, BP 3079 06202, Nice, Cedex, France
IURC - Laboratoire de Biostatistique d'Epidémiologie et de Recherche Clinique, 641 avenue du Doyen G. Giraud, 34093, Montpellier, Cedex, France
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Correspondence to Laurent Bailly.
LB participated in the design of the study, performed the statistical analysis and wrote the paper. JPD conceived of the study and implemented the Bayesian estimation of a population. CP and BD participated in its design and coordination and helped to draft the manuscript. All authors read and approved the final manuscript.
Appendices. A) Bayesian Model Averaging. B) Reversible jump MCMC model. C) Prior for the total population. D) WinBUGS Codes.
Capture-recapture
Cancer population
Capture-recapture models
Bayesian model averaging
Completeness of cancer registries
Data analysis, statistics and modelling | CommonCrawl |
\begin{document}
\title{The ADHM Variety and Perverse Coherent Sheaves}
\author{Marcos Jardim} \author{Renato Vidal Martins}
\address{IMECC - UNICAMP \\ Departamento de Matem\'atica \\ Caixa Postal 6065 \\ 13083-970 Campinas-SP, Brazil} \email{[email protected]} \address{ICEx - UFMG \\ Departamento de Matem\'atica \\ Av. Ant\^onio Carlos 6627 \\ 30123-970 Belo Horizonte MG, Brazil} \email{[email protected]}
\begin{abstract} We study the full set of solutions to the ADHM equation as an affine algebraic set, the ADHM variety. We gather the points of the ADHM variety into subvarieties according to the dimension of the stabilizing subspace. We compute dimension, and analyze singularity and reducibility of all of these varieties. We also connect representations of the ADHM quiver to coherent perverse sheaves on $\mathbb{P}^2$ in the sense of Kashiwara. \end{abstract}
\maketitle
\section{Introduction}
Atiyah, Hitchin, Drinfeld and Manin showed in \cite{ADHM}, using Ward correspondence and algebro-geometric techniques (monads) introduced by Horrocks, that all self-dual connections on euclidean 4-dimensional space (a.k.a. instantons) have a unique description in terms of linear algebra.
A few years later in \cite{D1}, Donaldson restated the ADHM description in terms of the following data. Let $V$ and $W$ be complex vector spaces, with dimensions $c$ and $r$, respectively. Let $A,\,B\in{\rm Hom}(V,V)$, let $I\in {\rm Hom}(W,V)$ and let $J\in {\rm Hom}(V,W)$. Consider the following equations: \begin{align} [A,B] + IJ & = 0 \label{adhm1} \\ [A,A^\dagger]+[B,B^\dagger] + II^\dagger - J^\dagger J &= 0. \label{adhm2} \end{align} The group $GL(c)$ acts on the set of solutions of the first equation, sending a datum $(A,B,I,J)$ to $(gAg^{-1},gBg^{-1},gI,Jg^{-1})$, where $g\in GL(c)$, while the unitary group $U(c)$ preserves the second equation. Among other things, Donaldson showed that the \emph{regular} (Definition \ref{m-def}) solutions to equation (\ref{adhm1}) modulo the action of $GL(c)$ parametrize the moduli space of holomorphic bundles on $\C\p2$ that are framed at a line, so-called line at infinity. The key observation is that the set of regular solutions of the first equation modulo $GL(c)$ is isomorphic to the set of regular solutions of both equations modulo $U(c)$, which in turn is identified with the moduli space of framed instantons on $\R^4$.
More recently, Nakajima showed that relaxing the regularity condition to a weaker stability condition one obtains the moduli space of framed torsion free sheaves on $\C\p2$, see \cite{N}. In particular, Nakajima also obtained a linear algebraic description of the Hilbert scheme of points on $\C^2$ and showed that it admits a hyperk\"ahler structure.
Generalizations of the ADHM construction have given rise to a wide variety of important results in many different areas of mathematics and mathematical physics. For instance, Kronheimer and Nakajima constructed instantons on the so-called ALE manifolds \cite{KN}. Nakajima then proceed to further generalize the construction, and studied quiver varieties and representations of Kac-Moody Lie algebras \cite{N1}. More recently, the ADHM construction of instantons was adapted to construct analogues of Yang-Mills instantons in noncommutative geometry \cite{NS} and supergeometry \cite{T}, while the ADHM construction of holomorphic bundles on $\C\p2$ was generalized to the construction of instantons bundles on $\C\mathbb{P}^n$ \cite{FJ2,J}.
We start this paper by studying the full set of solutions of (\ref{adhm1}), called the \emph{ADHM equation}, as an affine algebraic variety ${\mathcal V}(r,c)$, called the \emph{ADHM variety}. Our first concern are the points $X\in{\mathcal V}(r,c)$ which are not {\it stable}. In order to study them, we define the {\it stabilizing subspace} $\Sigma_X$ as the subspace of $V$ to which the restriction of $X$ is stable (c.f. Definition \ref{defsig}). We break down the ADHM variety into disjoint subsets ${\mathcal V}(r,c)^{(s)}:=\{X\,|\,\dim\Sigma_X=s\}$; in this sense, the set of stable points ${\mathcal V}(r,c)^{\rm st}$ now corresponds to ${\mathcal V}(r,c)^{(c)}$. It turns out that the study of these subsets happens to give some relevant properties of the whole variety itself. The results we get are:
\
\noindent {\bf Theorem 1.} \emph{The ADHM variety ${\mathcal V}(r,c)$ is a set theoretic complete intersection which is irreducible if and only if $r\geq 2$. Moreover, the set ${\mathcal V}(r,c)^{(s)}$ is an irreducible quasi-affine variety of dimension $2rc+c^2-(r-1)(c-s)$ which is nonsingular if and only if either $s=c$ or $s=c-1$. In particular, ${\mathcal V}(r,c)^{\rm st}$ is a nonsingular irreducible quasi-affine variety of dimension $2rc+c^2$.}
\
The reader will note that everything done concerning stable points could have been said for {\it costable} as well. In fact, part of this -- {\it costabilizing subspaces}, for instance -- actually appears here in the proof of Proposition \ref{prpule}, which provides an analogue of the so-called \emph{Gieseker to Uhlenbeck map} (c.f. \cite[pp. 196-198]{B}) purely in terms of the ADHM data.
For the second part of the paper, it is important to put the subject within a categorical framework. We pass from ``points of the ADHM variety ${\mathcal V}$'' to ``objects in the category $\mathcal{A}$ of representations of the \emph{ADHM quiver}''. Such objects are of the form $\mathbf{R}=(V,W,X)$ where $V$, $W$ and $X$ are as above. For every $\mathbf{R}\in\mathcal{A}$ we define a \emph{stable restriction} by $\mathbf{S}_{\mathbf{R}}:=(\Sigma_X,W,X|_{\Sigma_X})$ and a \emph{quotient representation} by $\mathbf{Z}_{\mathbf{R}}:=(V/\Sigma_X,\{0\},(A',B',0,0))$ with $A',B'$ commuting operators in $V/\Sigma_X$, see (\ref{equzzr}). The triple $(\dim W,\dim \Sigma_X,\codim \Sigma_X)$ is called the \emph{type vector} of $\mathbf{R}$.
From every $\mathbf{R}\in\mathcal{A}$ one constructs a complex $E_{\mathbf{R}}^{\bullet}\in \text{Kom}(\mathbb{P}^2)$ (Definition \ref{defcom}), called the \emph{ADHM complex} associated to $\mathbf{R}$. Nakajima has shown in \cite[Chp. 2]{N}, based on previous constructions due to Barth and Donaldson \cite{D1}, that if $\mathbf{R}$ is stable, then $E:=H^0(E^\bullet_\mathbf{R})$ is the only nontrivial cohomology sheaf of the ADHM complex; moreover, $E$ is a torsion free sheaf such that $E|_{\ell}\simeq W\otimes{\mathcal O}_{\ell}$ for a fixed line $\ell\subset\mathbb{P}^2$. Conversely, given any torsion free sheaf $E$ on $\p2$ whose restriction to $\ell$ is trivial, then there is a stable solution of the ADHM equation such that $E$ is isomorphic to the cohomology sheaf of the corresponding ADHM complex. Besides, $H^0(E^\bullet_\mathbf{R})$ is locally free if and only if $\mathbf{R}$ is regular.
Following ideas of Drinfeld, Braverman, Finkelberg and Gaitsgory introduced in \cite[Sec. 5]{BFG} the notion of perverse coherent sheaves, and showed, generalizing \cite[Chp. 2]{N}, that such objects correspond to arbitrary solutions of the ADHM equation. Here, we show that the complex associated to such solutions can also be considered perverse in the sense of Kashiwara's ``family of supports" approach. More precisely, we introduce, following \cite{Ka}, a perverse $t$-structure on $D^{\rm b}(\p2)$; objects in the core of such $t$-structure, denoted ${\mathcal P}$, are what we call \emph{coherent perverse sheaves on $\mathbb{P}^2$}. We characterize objects $E^\bullet$ in ${\mathcal P}$, and define the \emph{rank, charge} and \emph{length} of such an $E^\bullet$ as, respectively, $\rk (H^0(E^\bullet))$, $c_2(H^0(E^\bullet))$ and $\text{length}(H^1(E^\bullet))$. We then prove:
\
\noindent {\bf Theorem 2.} \emph{If $\mathbf{R}$ is a representation of the ADHM quiver with type vector $(r,s,l)$, then the associated ADHM complex $E^\bullet_\mathbf{R}$ is a perverse coherent sheaf on $\p2$ of rank $r$, charge $s$ and length $l$. Moreover, \begin{itemize} \item[(i)] $H^0(E^\bullet_{\mathbf{R}})\simeq H^0(E^\bullet_{\mathbf{S}_{\mathbf{R}}})$; \item[(ii)] $H^1(E^\bullet_{\mathbf{R}})\simeq H^1(E^\bullet_{\mathbf{Z}_{\mathbf{R}}})$; \item[(iii)] $H^0(E^\bullet_{\mathbf{R}})$ is locally free if and only if $\mathbf{R}$ is costable. \end{itemize}}
Recently, Hauzer and Langer \cite{HL} solved this problem for the complex ADHM equations introduced in \cite{FJ2}, showing that arbitrary solutions of the complex ADHM equation correspond to perverse instanton sheaves on $\p3$. This picture can the further generalized, and, as it was pointed out in \cite{J}, an analogous relation between arbitrary solutions of the $d$-dimensional ADHM equations and perverse instanton sheaves on $\mathbb{P}^{d+2}$ should also hold.
\
\noindent{\bf Acknowledgments.} The first named author is partially supported by the CNPq grant number 305464/2007-8 and the FAPESP grant number 2005/04558-0.
\section{The ADHM Data}
Let $V$ and $W$ be complex vector spaces, with dimensions $c$ and $r$, respectively. The {\em ADHM data} is the set (or space) given by $$ \mathbf{B}=\mathbf{B}(r,c) := {\rm Hom}(V,V)\oplus{\rm Hom}(V,V)\oplus {\rm Hom}(W,V)\oplus{\rm Hom}(V,W). $$ An element (or point) of $\mathbf{B}$ is a {\em datum} $X=(A,B,I,J)$ with $A,B\in{\rm End}(V)$, $I\in{\rm Hom}(W,V)$ and $J\in{\rm Hom}(V,W)$.
\begin{defi} \label{m-def} A datum $X=(A,B,I,J)\in\mathbf{B}$ is said \begin{enumerate} \item[(i)] {\em stable} if there is no subspace $S\subsetneqq V$ with $A(S),B(S),I(W)\subset S$; \item[(ii)] {\em costable} if there is no subspace $0\neq S\subset V$ with $A(S),B(S)\subset S\subset \ker J$; \item[(iii)] {\em regular} if it is both stable and costable. \end{enumerate} We call $\mathbf{B}^{\rm st}$, $\mathbf{B}^{\rm cs}$ and $\mathbf{B}^{\rm reg}$ the sets of stable, costable and regular data, respectively. \end{defi}
If $T$ is a linear map, let $T^\dagger$ be its adjoint operator. For $X=(A,B,I,J)\in\mathbf{B}$ one defines $$ X^\star := (B^\dagger,-A^\dagger,J^\dagger,-I^\dagger). $$ It is easily seen that $(X^\star)^\star=-X$, and that $X$ is stable if and only if $X^\star$ is costable. The anti-linear endomorphism of $\mathbf{B}$ given by $X\to X^\star$ plays the role of the duality needed in this work, something that will be more clear later on in Section \ref{secprv}.
We are specially interested in the following morphism \begin{gather*} \begin{matrix} \mu : & \mathbf{B} & \longrightarrow &{\rm End}(V) \\
& (A,B,I,J) & \longmapsto & [A,B] + IJ \end{matrix} \end{gather*} which will define the variety we will deal with in next section. It is easily checked that, for any $X=(A,B,I,J)\in\mathbf{B}$, the derivative is \begin{gather*} \begin{matrix} D_X\mu : & \mathbf{B} & \longrightarrow &{\rm End}(V) \\
& (a,b,i,j) & \longmapsto & [A,b] + [a,B] + Ij + iJ. \end{matrix} \end{gather*}
\begin{lemm} \label{lemsj1} Let $X=(A,B,I,J)\in\mathbf{B}$. Then $D_X\mu$ is surjective iff the map \begin{gather*} \begin{matrix} {\rm End}(V) & \longrightarrow & \mathbf{B} \\ y & \longmapsto & ([A,y],[B,y],yI,Jy) \end{matrix} \end{gather*} is injective. \end{lemm}
\begin{proof} The map $D_X\mu$ is not surjective iff there is a nonzero $y\in{\rm End}(V)$ such that $y^\dagger\in(\im D_X\mu)^\perp$. But $y^\dagger\in(\im D_X\mu)^\perp$ iff, for every $(a,b,i,j)\in\mathbf{B}$, holds $$ {\rm Tr}(D_X\mu(a,b,i,j)y)=0 $$ which is equivalent, for every $(a,b,i,j)\in\mathbf{B}$, to the following equalities \begin{align*} {\rm Tr}([A,b]y)={\rm Tr}([y,A]b)&=0 \\ {\rm Tr}([a,B]y)={\rm Tr}(a[B,y])&=0 \\
{\rm Tr}((Ij)y)={\rm Tr}((yI)j)&=0 \\
{\rm Tr}((iJ)y)={\rm Tr}(i(Jy))&=0 \end{align*} which hold if and only if $$ [A,y]=[B,y]=yI=Jy=0 $$ and we are done. \end{proof}
\begin{prop} \label{prpsrj} If $X$ is stable or costable then $D_{X}\mu$ is surjective. \end{prop}
\begin{proof} Set $X=(A,B,I,J)$. If $D_{X}\mu$ is not surjective, from Lemma \ref{lemsj1} there is a nonzero $y\in{\rm End}(V)$ such that $[A,y]=[B,y]=yI=Jy=0$.
Since $yI=0$, then $I(W)\subset\ker y$, while $[A,y]=[B,y]=0$ implies that $\ker y$ is $A$- and $B$-invariant. But $\ker y$ is a proper subspace of $V$ because $y\neq 0$. Hence $X$ is not stable.
Similarly, since $Jy=0$, then $\im y\subset \ker J$, while $[A,y]=[B,y]=0$ implies that $\im y$ is $A$- and $B$-invariant. But $\im y$ is a nonzero subspace of $V$ because $y\neq 0$. Hence $X$ is not costable as well. \end{proof}
The group $G:=GL(V)$ acts naturally on $\mathbf{B}$. For $g\in G$ and $X=(A,B,I,J)\in\mathbf{B}$ one defines $$ g\cdot X := (gAg^{-1},gBg^{-1},gI,Jg^{-1}). $$ One also defines the stabilizer subgroup of $X$ as $$
G_X:=\{g\in G\,|\,g\cdot X=X\}. $$
\begin{prop} \label{trstab} If $D_{X}\mu$ is surjective then $G_X$ is trivial. \end{prop}
\begin{proof} Let $X=(A,B,I,J)\in\mathbf{B}$. If $G_X$ is nontrivial, let $g\ne{\mathbf 1}$ be such that $gA=Ag$, $gB=Bg$, $gI=I$ and $Jg=J$. Hence $$ [A,g-{\mathbf 1}]=[B,g-{\mathbf 1}]=(g-{\mathbf 1})I=J(g-{\mathbf 1})=0. $$ Since $g-{\mathbf 1}\neq 0$, Lemma \ref{lemsj1} implies that $D_X\mu$ is not surjective. \end{proof}
Propositions \ref{prpsrj} and \ref{trstab} led us considering the sets $$
\mathbf{B}^{\rm sj} = \{ X\in\mathbf{B}\,|\, D_X\mu {\rm ~is~surjective} \} $$ $$
\mathbf{B}^{\rm ts} = \{ X\in\mathbf{B}\,|\, G_X\ {\rm is~trivial} \} $$ which satisfy the sequence of inclusions $$ (\mathbf{B}^{\rm st}\cup\mathbf{B}^{\rm cs}) \subset \mathbf{B}^{\rm sj} \subset \mathbf{B}^{\rm ts}. $$ which, in general, are strict ones (see Rem. \ref{remstr} below).
\begin{defi} \label{defsig} The \emph{stabilizing subspace} $\Sigma_X$ of an ADHM datum $X=(A,B,I,J)$ is the intersection of all subspaces $S\subseteq V$ such that $A(S),B(S),I(W)\subset S$. \end{defi}
If $S\subseteq V$ satisfies $A(S),B(S),I(W)\subset S$, then one may consider $$
X|_S:=(A|_S,B|_S,I,J|_S)\in{\rm End}(S)^{\oplus 2}\oplus{\rm Hom}(W,S)\oplus{\rm Hom}(S,W). $$
It is clear that $X|_{\Sigma_X}$ is stable and this justifies the term we use.
For $0\le s\le c$, we define the sets $$
\mathbf{B}^{(s)} := \{ X\in\mathbf{B} ~|~ \dim \Sigma_X= s \} $$ $$
\mathbf{B}^{[s]} := \{ X\in\mathbf{B} ~|~ \dim \Sigma_X\leq s \} $$ which, respectively, give a disjoint decomposition and filtration of $\mathbf{B}$. Clearly, $\mathbf{B}^{(c)}$ coincides with the set of stable data, while $\mathbf{B}^{(0)}$ is the set of data $X$ for which $I=0$.
Given an ADHM datum $X=(A,B,I,J)$, consider the map \begin{gather*} \begin{matrix} R(X) : & W^{\oplus c^2} & \longrightarrow & V \\
& \bigoplus\limits_{0\leq k,l\leq c-1} w_{kl} & \longmapsto & \sum\limits_{0\leq k,l\leq c-1} A^kB^lIw_{kl}. \end{matrix} \end{gather*} It is easily seen that the assignment $R:\mathbf{B}\to {\rm Hom}(W^{\oplus c^2},V)$ defines a regular map.
\begin{prop} \label{l1} For any $X\in\mathbf{B}$ hold \begin{enumerate} \item[(i)] $\Sigma_X\supseteq\im R(X)$; \item[(ii)] if $R(X)$ is surjective, then $X$ is stable. \end{enumerate} \end{prop}
\begin{proof} If $S\subseteq V$ satisfies $A(S),B(S),I(W)\subset S$, then $\im R(X)\subseteq S$ so (i) follows. Now if $R(X)$ is surjective then $c={\rm rk}\,R(X)\leq\dim \Sigma_X\leq c$. Thus $\dim\Sigma_X=c$, that is, $X$ is stable. \end{proof}
The above proposition will turn into a stronger result in the next section, when an additional hypothesis will be imposed. We close the section with a result which will be very useful later on.
\begin{lemm} \label{lemsrj} Let $X=(A,B,I,J)\in\mathbf{B}$ and $A',B'\in {\rm End}(V')$ where $V'$ is a complex vector space. If $X$ is stable then \begin{gather*} \begin{matrix} \phi: & {\rm Hom}(V',V)^{\oplus 2}\oplus {\rm Hom}(V',W) & \longrightarrow &{\rm Hom}(V',V) \\
& (a,b,j) & \longmapsto & Ab-bA'+aB'-Ba+Ij \end{matrix} \end{gather*} is surjective. \end{lemm}
\begin{proof} The linear morphism $\phi$ is not surjective if and only if there is a nonzero $y\in{\rm Hom}(V,V')$ such that $y^\dagger\in(\im \phi)^\perp$. But $y^\dagger\in(\im \phi)^\perp$ if and only if, for every $(a,b,j)\in{\rm Hom}(V',V)^{\oplus 2}\oplus {\rm Hom}(V',W)$, holds $$ {\rm Tr}(\phi(a,b,j)y)=0 $$ which is equivalent, for every $(a,b,j)\in{\rm Hom}(V',V)^{\oplus 2}\oplus {\rm Hom}(V',W)$, to the following equalities \begin{align*} {\rm Tr}((Ab-bA')y)={\rm Tr}((yA-A'y)b)&=0 \\ {\rm Tr}((aB'-Ba)y)={\rm Tr}(a(B'y-yB))&=0 \\
{\rm Tr}((Ij)y)={\rm Tr}((yI)j)&=0 \end{align*} which hold if and only if \begin{equation} \label{equabi} yA-A'y=B'y-yB=yI=0. \end{equation} So if $\phi$ is not surjective, let $y\neq 0$ satisfying (\ref{equabi}). Then $yI=0$ yields $I(W)\subset\ker y$, while $yA=A'y$ implies that $\ker y$ is $A$-invariant and $y B=B'y$ implies $\ker y$ is also $B$-invariant. Since $\ker y\subsetneq V$, it follows that $X$ is not stable. \end{proof}
\section{The ADHM Variety}
Now we introduce the main object of our study.
\begin{defi} The set ${\mathcal V}={\mathcal V}(r,c):=\mu^{-1}(0)$ is called the {\em ADHM variety}, i.e., the variety of solutions of $$ [A,B] + IJ = 0 $$ which is called the {\em ADHM equation}. \end{defi}
We keep the notation previously introduced, i.e., ${\mathcal V}^{\rm st}$, ${\mathcal V}^{\rm cs}$, ${\mathcal V}^{\rm reg}$, ${\mathcal V}^{\rm sj}$, ${\mathcal V}^{\rm ts}$, ${\mathcal V}^{(s)}$, ${\mathcal V}^{[s]}$ are, respectively, the intersections of $\mathbf{B}^{\rm st}$, $\mathbf{B}^{\rm cs}$, $\mathbf{B}^{\rm reg}$, $\mathbf{B}^{\rm sj}$, $\mathbf{B}^{\rm ts}$, $\mathbf{B}^{(s)}$, $\mathbf{B}^{[s]}$ with ${\mathcal V}$ for $0\leq s\leq c$. In particular, we have that ${\mathcal V}^{(c)}={\mathcal V}^{\rm st}$, and, since $X=(A,B,I,J)\in{\mathcal V}^{(0)}$ if and only if $I=[A,B]=0$, $$ {\mathcal V}^{[0]} = {\mathcal V}^{(0)} = \mathcal{C}_c \times {\rm Hom}(V,W) $$ where $\mathcal{C}_c$ is the variety of $c\times c$ commuting matrices.
\begin{lemm} \label{lemsis} For any $X\in{\mathcal V}$ hold \begin{enumerate} \item[(i)] $\Sigma_X=\im R(X)$; \item[(ii)] $R(X)$ is surjective iff $X$ is stable. \end{enumerate} \end{lemm}
\begin{proof} We already know from Proposition \ref{l1} that $\Sigma_X\supseteq\im R(X)$; besides, (ii) immediately follows from (i). Therefore, we just have to prove that $\Sigma_X\subseteq\im R(X)$. To do that, it suffices to show that if $X=(A,B,I,J)$, then $\im R(X)$ is $A$- and $B$-invariant and $I(W)\subset\im R(X)$.
Clearly, $I(W)\subset\im R(X)$; let us now show that $\im R(X)$ is $A$ and $B$-invariant. Note that \begin{align*} A(\im R(X))&=\sum\limits_{0\leqslant k,l\leqslant c-1} \im A^{k+1}B^lI \\ &= \sum\limits_{0\leqslant l\leqslant c-1} \im A^{c}B^lI + \sum_{\substack{1\leqslant k\leqslant c-1 \\ 0\leqslant l\leqslant c-1}} \im A^{k}B^lI. \end{align*} The second factor is clearly within $\im R(X)$. For the first factor, use Cayley-Hamilton Theorem to express $A^{c}$ as a linear combination of lower powers of $A$; it then follows that this factor is also within $\im R(X)$. So $\im R(X)$ is $A$-invariant.
Using the ADHM equation, one can see that $$ BA^k = A^kB + \sum_{r+s=k-1}A^rIJA^s. $$ Therefore we have \begin{align*} B(\im R(X)) &=\sum\limits_{0\leqslant k,l\leqslant c-1} \im BA^{k}B^lI \\
&=\sum\limits_{0\leqslant k,l\leqslant c-1} \im A^kB^{l+1}I+
\sum_{\substack{0\leqslant k,l\leqslant c-1 \\ r+s=k-1}}\im A^rIJA^sB^lI \\
&\subseteq \sum\limits_{0\leqslant k\leqslant c-1} \im A^kB^cI +
\sum_{\substack{0\leqslant k\leqslant c-1 \\ 1\leqslant l\leqslant c-1}} \im A^kB^lI+
\sum_{0\leqslant r\leqslant c-2}\im A^rI. \end{align*} The second and third factors are clearly within $\im R(X)$. For the first factor, use again Cayley-Hamilton Theorem to express this turn $B^c$ as lower powers of $B$ to conclude that this factor must lie within $\im R(X)$. So $\im R(X)$ is $B$-invariant as well and we are done. \end{proof}
\begin{theo} \label{thmvar} The ADHM variety ${\mathcal V}(r,c)$ is a set theoretic complete intersection which is irreducible if and only if $r\geq 2$. Moreover, the set ${\mathcal V}(r,c)^{(s)}$ is an irreducible quasi-affine variety of dimension $2rc+c^2-(r-1)(c-s)$ which is nonsingular if and only if either $s=c$ or $s=c-1$. In particular, ${\mathcal V}(r,c)^{\rm st}$ and ${\mathcal V}(r,c)^{\rm cs}$ are nonsingular irreducible quasi-affine varieties of dimension $2rc+c^2$. \end{theo}
\begin{proof}
Set $\mathbf{D}:={\rm Hom}(W^{\oplus c^2},V)$. For $0\le s\le c$, set also $P^{(s)}:=\{T\in\mathbf{D}~|~ {\rm rk}\, T=s \}$ and $P^{[s]}:=\{T\in\mathbf{D}~|~ {\rm rk}\, T\leq s \}$. Then $P^{(s)}$ is open within $P^{[s]}$ which is closed within $\mathbf{D}$. Now $R:\mathbf{B}\to\mathbf{D}$ is continuos and, from the prior lemma, ${\mathcal V}^{(s)}={\mathcal V}\cap R^{-1}(P^{(s)})$ and ${\mathcal V}^{[s]}={\mathcal V}\cap R^{-1}(P^{[s]})$. So ${\mathcal V}^{(s)}$ is open within ${\mathcal V}^{[s]}$ which is closed within ${\mathcal V}$. Therefore ${\mathcal V}^{(s)}$ is a quasi-affine variety.
Since ${\mathcal V}^{\rm st}={\mathcal V}^{(c)}$, it follows that ${\mathcal V}^{\rm st}$ is a quasi-affine variety; from Proposition \ref{prpsrj} it is also nonsingular. Since ${\mathcal V}^{\rm st}$ is a variety, the isomorphism $X\to X^{\star}$ in $\mathbf{B}$ restricts to an isomorphism from ${\mathcal V}^{\rm st}$ onto its image. But its image is precisely ${\mathcal V}^{\rm cs}$ because, first, $X^\star$ is costable iff $X$ is stable and, second, $\mu(X^\star)=0$ iff $\mu(X)=0$. The dimension of both varieties is $\dim\mathbf{B}-\dim{\rm End}(V)=2(c^2+rc)-c^2=2rc+c^2$.
Now we claim that ${\mathcal V}(r,c)^{(s)}$ is the total space of a rank $(r+s)(c-s)$ bundle \begin{equation} \label{equfib} {\mathcal V}(r,c)^{(s)}\longrightarrow{\rm G}(s,c)\times{\mathcal V}(r,s)^{\rm st}\times\mathcal{C}_{c-s} \end{equation} where $G$ is the Grassmannian and $\mathcal{C}$ the commuting matrices variety. In fact, if $X\in {\mathcal V}(r,c)^{(s)}$ we may move $\Sigma_X$ to the vector space spanned by $e_1,\,e_2,\ldots,e_s$. So $X=(A,B,I,J)$ where \begin{equation} \label{fcanonica} A = \left( \begin{array}{cc} A_1 & A_2 \\ 0 & A_3 \end{array} \right) ~~~~ B = \left( \begin{array}{cc} B_1 & B_2 \\ 0 & B_3 \end{array} \right) ~~~~
I = \left( \begin{array}{c} I_1 \\ 0 \end{array} \right) ~~~~ J = \left( \begin{array}{cc} J_1 & J_2 \end{array} \right) \end{equation} and \begin{align*} [A_1,B_1]+I_1J_1&=0 \\ A_1B_2-B_1A_2+A_2B_3-B_2A_3+I_1J_2&=0 \\ [A_3,B_3]&=0 \end{align*}
with the requirement that $X|_{\Sigma_X}=(A_1,B_1,I_1,J_1)$ is stable. Now the first equation is the ADHM equation for $X|_{\Sigma_X}$, the third equation is the commuting matrices equation for $A_3$ and $B_3$. The movement of $\Sigma_X$ to the $s$-dimensional standard space is described by the Grassmannian $G(s,c)$ and we have a natural map as in (\ref{equfib}). The fiber ${\mathcal V}(r,c)^{(s)}_P$ over $P=(\Sigma_X,X|_{\Sigma_X},(A_3,B_3))$ is the set of $(A_2,B_2,J_2)$ which satisfy the second equation. This is clearly an affine space. Since $X|_{\Sigma_X}$ is stable, it follows from Lemma \ref{lemsrj} that \begin{align*} \dim {\mathcal V}(r,c)^{(s)}_P&=2s(c-s)+r(c-s)-s(c-s) \\
&=(r+s)(c-s). \end{align*} This proves the claim. Therefore ${\mathcal V}^{(s)}$ is nonsingular iff $\mathcal{C}_{c-s}$ is so, which happens iff either $s=c$ or $s=c-1$.
Now let us compute the dimension. We have \begin{align*} \dim {\mathcal V}^{(s)}&=\dim {\rm G}(s,c)+\dim {\mathcal V}(r,s)^{\rm st}+\dim \mathcal{C}_{c-s}+\dim \varphi_P \\
&=s(c-s)+2rs+s^2+(c-s)+(c-s)^2+(r+s)(c-s) \\
&=(c-s)(s+1+c+r)+2rs+s^2 \\
&=rc+c^2+c-s+rs \\
&=rc+(rc-rc)+c^2+c-s+rs \\
&=2rc+c^2-(r-1)(c-s). \end{align*}
Now ${\mathcal V}$ is the union of the ${\mathcal V}^{(s)}$, so its dimension is the highest among them, which is $2rc+c^2$. It follows that ${\mathcal V}$ is a set theoretic complete intersection.
Finally, we will prove that ${\mathcal V}(r,c)^{(s)}$ is irreducible for each $r$, $c$ and $0\le s\le c$. First, we argue that ${\mathcal V}(r,c)^{(c)}={\mathcal V}(r,c)^{\rm st}$ is irreducible.
The case $r=1$ is rather special because if $X=(A,B,I,J)$ is a stable solution of the ADHM equation, then $J=0$ by \cite[Prp. 2.8(1)]{N}. It follows that the closure of ${\mathcal V}(1,c)^{\rm st}$ is given by $\mathcal{C}_{c}\times{\rm Hom}(W,V)$, which is clearly irreducible. Hence ${\mathcal V}(1,c)^{\rm st}$ is also irreducible.
On the other hand, if $c=1$, $A$ and $B$ are simply numbers, while $I$ and $J$ can be regarded as vectors in $\mathbb{C}^{r}$, $$ I = (x_1,\dots,x_r)\ \ \ \ \ \ J = (y_1,\dots,y_r) $$ and, in this way, the ADHM equation reduces to $$ IJ = \sum_{i=1}^{r} x_iy_i = 0. $$ We will show that ${\mathcal V}(r,1)$ is irreducible if $r\geq 2$. In order to do this, assume $\sum_{i=1}^{r} x_iy_i=(a_0+\sum_i a_ix_i+b_iy_i)(a_0'+\sum_i a_i'x_i+b_i'y_i)$. Fix $i$. We have $a_ia_i'=0$. Assume, without loss in generality, that $a_i'=0$. Then $a_ia_j'=0$ for every $j$, $a_ib_j'=0$ for every $j\neq i$, and $a_ib_i'=1$. This implies all $a_j'$ vanish and all $b_j'$ with $j\neq i$ vanish as well. This is impossible comparing both sides of polynomial equality above unless $r=1$. So ${\mathcal V}(r,1)=\mathbb{C}^2\times V$ where $V$ is an irreducible variety of $\mathbb{C}^{2r}$ if $r\geq 2$. Therefore ${\mathcal V}(r,1)$ is irreducible if $r\geq 2$. Since ${\mathcal V}(r,1)^{\rm st}$ is open within ${\mathcal V}(r,1)$, it is irreducible as well if $r\geq 2$. But if $r=1$ we have already seen it is also irreducible.
For the case $r,c\geq 2$, write any $(a,b,i,j)\in\mathbf{B}$ as $$ a = \left( \begin{array}{cc} a_1 & a_2 \\ e & a_3 \end{array} \right) ~~~~ b = \left( \begin{array}{cc} b_1 & b_2 \\ f & b_3 \end{array} \right) ~~~~
i = \left( \begin{array}{c} i_1 \\ g \end{array} \right) ~~~~ j = \left( \begin{array}{cc} j_1 & j_2 \end{array} \right) $$ where the sizes of the matrices involved are as in (\ref{fcanonica}). Set $$ x=(a_1,b_1,i_1,j_1,a_3,b_3)\ \ \ \ \ y=(a_2,b_2,j_2)\ \ \ \ \ z=(e,f,g) $$ and, keeping the notation of (\ref{fcanonica}), define \begin{align*} \phi_1(x)&=[A_1,b_1]+[a_1,B_1]+i_1J_1+I_1j_1 \\ \phi_2(x)&=a_1B_2-a_2B_2+i_1J_2+A_2b_3-B_2a_3 \\ \phi_3(y)&=A_1b_2-b_2A_3+a_2B_3-B_1a_2-I_1j_1 \\ \phi_4(z)&=A_2f-B_2e \\ \phi_5(z)&=eB_1-B_3e+A_3f-fA_1+gJ_1 \\ \phi_6(z)&=eB_2-fA_2+gJ_2 \end{align*} So if $X$ is as in (\ref{fcanonica}) and $s=c-1$, then \begin{gather*} \begin{matrix} D_X\mu : & \mathbf{B} & \longrightarrow &{\rm End}(V) \\
& (x,y,z) & \longmapsto & \left( \begin{array}{cc} \phi_1(x)+\phi_4(z) & \phi_2(x)+\phi_3(y)\\ \phi_5(z) & \phi_6(z)\end{array} \right) \end{matrix} \end{gather*}
Now, since $X|_{\Sigma_X}$ is stable, Proposition \ref{prpsrj} and Lemma \ref{lemsrj} imply, respectively, that $\phi_1$ and $\phi_3$ are surjective. So $D_X\mu$ is surjective iff \begin{gather*} \begin{matrix} \psi_X : & \mathbb{C}^{2c+r-2} & \longrightarrow & \mathbb{C}^{c} \\
& z & \longmapsto & (\phi_5(z),\phi_6(z)) \end{matrix} \end{gather*} is surjective. Set $Q:={\mathcal V}\setminus{\mathcal V}^{\rm sj}$. Note that \begin{equation} \label{equq12} Q\cap{\mathcal V}^{(c-1)}=V_1\cup V_2 \end{equation} where $$
V_1:=\{X\in{\mathcal V}^{(c-1)}\,|\,\phi_5\ {\rm is\ nonsurjective}\} $$ $$
V_2:=\{X\in{\mathcal V}^{(c-1)}\,|\,\phi_6\ {\rm is\ nonsurjective}\}. $$ Let us study the codimensions of the above varieties within ${\mathcal V}^{(c-1)}$ in order to get the one of $Q\cap{\mathcal V}^{(c-1)}$. From (\ref{equfib}), we have that ${\mathcal V}^{(c-1)}$ is the total space of a rank $r+c-1$ bundle on $T:=\mathbb{P}^{c-1}\times{\mathcal V}(r,c-1)^{\rm st}\times\mathbb{C}^2$. Write $W:=\mathbb{P}^{c-1}\times{\mathcal V}(r,c-1)^{\rm st}$. Now $V_1$ is a subvariety of $$
V_3:=\left\{X\in{\mathcal V}^{(c-1)}\ \bigg{|}\begin{array}{c}A_3\ {\rm is\ an\ eigenvalue\ of}\ A_1 \\ B_3\ {\rm is\ an\ eigenvalue\ of}\ B_1\end{array}\right\} $$ and since $(A_2,B_2,J_2)$ do not appear in the relations which describe $V_3$, it follows that $V_3$ is a rank $r+c-1$ bundle over the subvariety of $T$ given by $$
V_4:=\left\{((A_1,B_1,\ldots),(A_3,B_3))\in W\times \mathbb{C}^2\ \bigg{|}\begin{array}{c}A_3\ {\rm is\ an\ eigenvalue\ of}\ A_1 \\ B_3\ {\rm is\ an\ eigenvalue\ of}\ B_1\end{array}\right\} $$ and so $\dim T-\dim V_4=2$, hence $\dim{\mathcal V}^{(c-1)}-\dim V_3=2$ which implies that $\dim{\mathcal V}^{(c-1)}-\dim V_1\geq 2$. On the other hand, $$
V_2=\{X\in{\mathcal V}^{(c-1)}\,|\,A_2=B_2=J_2=0\} $$ so $V_2\cong T$ and hence $\dim{\mathcal V}^{(c-1)}-\dim V_2=r+c-1$. Thus, from (\ref{equq12}), we get that $\dim{\mathcal V}^{(c-1)}-\dim (Q\cap{\mathcal V}^{(c-1)})\geq 2$. Therefore, computing the codimension of $Q$ within ${\mathcal V}$ we have \begin{align*} \codim Q &=\min \{\codim (Q\cap{\mathcal V}^{(s)})\}_{s=0}^{c-1} \\
&=\min \{\codim (Q\cap{\mathcal V}^{(c-1)}),\min\{\codim {\mathcal V}^{(s)}\}_{s=0}^{c-2}\} \\
&\geq \min \{(r-1)+2,\min\{(r-1)(c-s)\}_{s=0}^{c-2}\} \\
&=\min \{r+1,2(r-1)\}\geq 2. \end{align*}
It follows from \cite[Thm. 1.3]{H1} that ${\mathcal V}^{\rm sj}$ is the tangent cone of a conected variety, so ${\mathcal V}^{\rm sj}$ is conected. Since it is nonsingular, it is also irreducible. So ${\mathcal V}^{\rm st}$, being open within ${\mathcal V}^{\rm sj}$, is irreducible as well. It follows that ${\mathcal V}(r,c)^{(s)}$ is the total space of a vector bundle over an irreducible basis (given by ${\rm G}(s,c)\times{\mathcal V}(r,s)^{\rm st}\times\mathcal{C}_{c-s}$), hence ${\mathcal V}(r,c)^{(s)}$ is also irreducible (see for instance \cite[Lem. 2.8]{CTT}).
Furthermore, since ${\mathcal V}$ is the closure of its nonsingular locus and this one is contained in ${\mathcal V}^{\rm sj}$, it follows that ${\mathcal V}$ is the closure of ${\mathcal V}^{\rm sj}$ as well. But we have just seen that, if $r,c\geq 2$, then ${\mathcal V}(r,c)^{\rm sj}$ is irreducible, so the same holds for ${\mathcal V}(r,c)$. Besides, we have already got the irreducibility of ${\mathcal V}(r,1)$ if $r\geq 2$. On the other hand, to check necessity, note that $\dim{\mathcal V}(1,c)^{(0)}=2c+c^2=\dim{\mathcal V}(1,c)^{\rm st}$. So the vaiety ${\mathcal V}(1,c)$ has an open subset -- ${\mathcal V}(1,c)^{\rm st}$ -- and a proper closed subset -- ${\mathcal V}(1,c)^{(0)}$ -- both of the same dimension, which cannot happen unless it is reducible. We are finally done. \end{proof}
\begin{rema}\rm Since ${\mathcal V}(r,c)$ is irreducible for $r\ge2$, we also conclude that ${\mathcal V}^{\rm sj}(r,c)$ concides with the nonsingular locus of ${\mathcal V}(r,c)$ in this case. \end{rema}
\begin{rema}\rm Note that ${\mathcal V}(1,c)$ has at least two irreducible components, i.e., the closures of ${\mathcal V}(1,c)^{\rm st}$ and ${\mathcal V}(1,c)^{\rm cs}$ within ${\mathcal V}(1,c)$, which, by \cite[Prp. 2.8(1)]{N}, are given by the conditions $J=0$ and $I=0$, respectively. It can easily be checked that ${\mathcal V}(1,1)$ has precisely $2$ irreducible components and, using software tools, that ${\mathcal V}(1,2)$ has exactly $3$ irreducible components. So it makes sense to ask if ${\mathcal V}(1,c)$ has always $c+1$ irreducible components. \end{rema}
\begin{rema} \label{remstr} {\rm Consider the sequence of inclusions $$ ({\mathcal V}^{\rm st}\cup{\mathcal V}^{\rm cs}) \subset {\mathcal V}^{\rm sj} \subset {\mathcal V}^{\rm ts}. $$ We will show with an example with $r=c=2$ that they are, in general, strict ones. Consider the following ADHM datum $X=(A,B,I,J)$ given by $$ A = \left( \begin{array}{cc} a_1 & a_2 \\ 0 & a_3 \end{array} \right) ~~~~ B = \left( \begin{array}{cc} b_1 & b_2 \\ 0 & b_3 \end{array} \right) ~~~~
I = \left( \begin{array}{cc} i_1 & i_2 \\ 0 & 0 \end{array} \right) ~~~~ J = \left( \begin{array}{cc} 0 & j_2 \\ 0 & j_4 \end{array} \right) . $$ It is easy to check that if $(a_1 - a_3)b_2 + (b_1 - b_3)a_2 + i_1j_2 + i_2j_4 = 0$, then $X$ satisfies the ADHM equation. Note also that $A(S),B(S),I(W)\subset S \subset \ker J$, where $S$ is the subspace generated by the vector $(1,0)$; thus $X$ is neither stable nor costable.
The corresponding Jacobian matrix is given by $$ D_X\mu=\left( \begin{array}{cc} M & N \end{array} \right) $$ where $$ M= \left( \begin{array}{cccccccc} 0 & 0 & -b_2 & 0 & 0 & 0 & a_2 & 0 \\ b_2& b_3-b_1 & 0 & -b_2&-a_2& a_1-a_3 & 0 & a_2 \\ 0 & 0 & b_1-b_3 & 0 & 0 & 0 & a_3-a_1 & 0 \\ 0 & 0 & b_2 & 0 & 0 & 0 & -a_2 & 0 \end{array} \right) $$ $$ N=\left( \begin{array}{cccccccc} 0 & 0 & 0 & 0 & i_1 & 0 & i_2 & 0 \\ j_2& j_4 & 0 & 0 & 0 & i_1 &0 &i_2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & j_2 & j_4 & 0 & 0 & 0 & 0 & 0 \end{array} \right) $$ Hence if in addition $a_1\ne a_3$, $b_1\ne b_3$, $i_1\ne0$ and $j_4\ne0$, then $X\in{\mathcal V}^{\rm ns}$. This shows that the first inclusion is strict.
On the other hand, if $a_1=a_3$ and $b_1=b_3$, then $X\not\in{\mathcal V}^{\rm ns}$. However, if $i_1,i_2\ne0$, $j_1,j_2\ne0$ and $a_1,b_1\ne0$, then $X\in{\mathcal V}^{\rm ts}$. This shows that the second inclusion is also strict, as claimed.} \end{rema}
\subsection{Gieseker to Uhlenbeck map}
Now we define an analogue of the so-called \emph{Gieseker to Uhlenbeck map} (c.f. \cite[pp. 196-198]{B}) purely in terms of the ADHM data.
\begin{prop} \label{prpule} There exists a natural map $$ {\mathcal V}(r,c)^{\rm st}/GL(c)\longrightarrow \coprod_{s=0}^{c}\, {\mathcal V}(r,s)^{\rm reg}/GL(s)\,\times\, \mathcal{C}_{c-s}/GL(c-s). $$ \end{prop}
Recall that ${\mathcal V}(r,c)^{\rm st}/GL(c)$ (resp. ${\mathcal V}(r,c)^{\rm reg}/GL(c)$) coincides with the moduli space of framed torsion free (resp. locally free) sheaves of rank $r$ and second Chern class $c$ on $\p2$. In our context, ${\mathcal V}(r,c)^{\rm st}/GL(c)$ plays the role of the \emph{Gieseker compactification} of ${\mathcal V}(r,c)^{\rm reg}/GL(c)$. On the other hand, ${\mathcal V}(r,c)^{\rm reg}/GL(c)$ can also be interpreted as the moduli space of framed instantons on $\R^4$, and the space ${\mathcal V}(r,s)^{\rm reg}/GL(s)\,\times\, \mathcal{C}_{c-s}/GL(c-s)$ is called the \emph{Uhlenbeck compactification} of ${\mathcal V}(r,s)^{\rm reg}/GL(s)$ (recall that the quotient variety $\mathcal{C}_{c-s}/GL(c-s)$ coincides with the symmetric product $\text{Sym}^c(\A^2)$).
\begin{proof} For any $X=(A,B,I,J)\in{\mathcal V}(r,c)$, we define its costabilizing subspace as $$
\Upsilon_X:=\langle S\subset V\,|\, A(S),B(S)\subset S\subset \ker J(S)\rangle, $$ that is, the largest $A$- and $B$-invariant subspace of $V$ on which $J$ vanishes. We have that $X$ is costable if and only if $\Upsilon_X=0$.
Set ${\mathcal V}(r,c)_{(s)}:=\{X\in{\mathcal V}(r,c)\,|\,\dim\Upsilon_X=s\}$. If $X\in {\mathcal V}(r,c)_{(s)}$ we may move $\Upsilon_X$ to the vector space spanned by $e_1,\,e_2,\ldots,e_s$. So $X=(A,B,I,J)$ where $$ A = \left( \begin{array}{cc} A_1 & A_2 \\ 0 & A_3 \end{array} \right) ~~~~ B = \left( \begin{array}{cc} B_1 & B_2 \\ 0 & B_3 \end{array} \right) ~~~~
I = \left( \begin{array}{c} I_1 \\ I_2 \end{array} \right) ~~~~ J = \left( \begin{array}{cc} 0 & J_2 \end{array} \right) $$ and \begin{align*} [A_1,B_1]&=0 \\ A_1B_2-B_1A_2+A_2B_3-B_2A_3+I_1J_2&=0 \\ [A_3,B_3]+I_2J_2&=0 \end{align*} with $(A_3,B_3,I_2,J_2)$ costable. Thus $X':=(A_3,B_3,I_2,J_2)\in{\mathcal V}(r,s)^{\rm cs}$ and also $Y:=(A_1,B_1)\in \mathcal{C}_{c-s}$. It is easily seen that if $X$ is stable, so is $X'$. Hence we are able to define the map $\overline{X}\mapsto (\overline{X'},\overline{Y})$. \end{proof}
\subsection{The structure of the ADHM category} \label{adhmcat}
\
\
In what follows, it will be important to put our subject within a categorical framework. As it is well-known, an ADHM datum $(A,B,I,J)$ can be regarded as representation of the so-called ADHM quiver $$ \xymatrix{ \stackrel{\bullet}{v} \ar@/^/[d]^{j} \ar@(ul,dl)[]_{a} \ar@(dr,ur)[]_{b} \\ \stackrel{\bullet}{w} \ar@/^/[u]^{i} } ~~~~ $$ provided with the relation $ab-ba+ij$. Let ${\mathcal A}$ be the category of representations of the ADHM quiver. Recall that ${\mathcal A}$ is an abelian category, and that for any representations $\mathbf{R}=(V,W,(A,B,I,J))$ and $\mathbf{R}'=(V',W',(A',B',I',J'))$ in ${\mathcal A}$, we have that $$
{\rm Hom}_{{\mathcal A}}(\mathbf{R},\mathbf{R}')=\left\{(f,g)\in{\rm Hom}(V,V')\oplus{\rm Hom}(W,W')\bigg{|}\begin{array}{ll}fA=A'f & fI=I'g \\ fB=B'f & gJ=J'f\end{array}\right\}. $$ A representation $\mathbf{R}=(V,W,X)$ is said to be \emph{stable} (resp. \emph{costable}) if $X$ is stable (resp. costable) and we set $$
\mathbf{S}_{\mathbf{R}}:=(\Sigma_X,W,X|_{\Sigma_X}) $$ to be the \emph{stable restriction} of $\mathbf{R}$. Set $N_X:=V/\Sigma_X$. If $X=(A,B,I,J)$ is as in (\ref{fcanonica}) we have that \begin{equation} \label{equzzr} \mathbf{Z}_{\mathbf{R}}:=(N_X,\{0\},(A_3,B_3,0,0)) \end{equation} is the \emph{quotient representation} of $\mathbf{R}$. We also call the triple $(\dim W,\dim \Sigma_X, \dim N_X)$ the \emph{type vector} of $\mathbf{R}$.
Let us now pause for some general theory. Consider an additive category $\mathcal{C}$ and let $\mathcal{B}$ be a full subcategory of $\mathcal{C}$; recall that the {\em right orthogonal} to $\mathcal{B}$ is the full subcategory $\mathcal{B}^\perp$ of $\mathcal{C}$ consisting of the objects $D$ such that ${\rm Hom}_{\mathcal{C}}(B,D)=0$ for every $B$ in $\mathcal{B}$.
\begin{defi}\label{radm} Let $\mathcal{C}$ be an additive category. A pair of full subcategories $(\mathcal{B},\mathcal{D})$ is a called a {\em torsion pair} of $\mathcal{C}$ if $\mathcal{D}\subset\mathcal{B}^\perp$ and if for every object $C$ of $\mathcal{C}$, there exists a short exact sequence $0\to B \to C \to D\to 0$ with $B$ in $\mathcal{B}$ and $D$ in $\mathcal{D}$. \end{defi}
Given an object $C$ of $\mathcal{C}$, the exact sequence $0\to B \to C \to D\to 0$ with $B$ in $\mathcal{B}$ and $D$ in $\mathcal{D}$ is unique up to isomorphism, so that the assignments $\iota^*(C):=B$ and $\jmath^*(C):=D$ yield additive functors $\iota^*:\mathcal{C}\to\mathcal{B}$ and $\jmath_*:\mathcal{C}\to\mathcal{D}$. It follows (see \cite[Proposition 1.2]{BR}) that $\iota^*$ is a right adjoint to the inclusion functor $\iota_*:\mathcal{B}\to\mathcal{C}$, while $\jmath^*$ is a left adjoint to the inclusion functor $\jmath_*:\mathcal{D}\to\mathcal{C}$.
Let us now show how Definition \ref{radm} applies in our context. Let $\mathcal{S}$ be the full subcategory of ${\mathcal A}$ whose objects are the stable representations, and let $\mathcal{Z}$ be the full subcategory of ${\mathcal A}$ consisting of representations with type vector of the form $(0,s,l)$; note that any representation in $\mathcal{Z}$ is of the form $(V,\{0\},(A,B,0,0))$ where $A$ and $B$ are commuting operators on $\rm{End}(V)$.
\begin{prop} \label{propcat} The pair $(\mathcal{S},\mathcal{Z})$ is a torsion pair on $\mathcal{A}$. \end{prop}
\begin{proof} For any $\mathbf{R}$ one gets a short exact sequence $$ 0\to \mathbf{S}_{\mathbf{R}} \to \mathbf{R} \to \mathbf{Z}_{\mathbf{R}} \to 0 $$ and clearly $\mathbf{S}_{\mathbf{R}}$ and $\mathbf{Z}_{\mathbf{R}}$ are in $\mathcal{S}$ and $\mathcal{Z}$, respectively.
Now we claim that $\mathcal{Z}\subset \mathcal{S}^\perp$. Take $\mathbf{S}=(V,W,X)\in \mathcal{S}$ with $X=(A,B,I,J)$ stable and let $\mathbf{Z}=(V',0,(A',B',0,0))$ be a representation in $\mathcal{Z}$. Then given $(f,g)\in{\rm Hom}_{{\mathcal A}}(\mathbf{S},\mathbf{Z})$, we have that $fA=A'f$, $fB=B'f$ and $fI=0$. It follows that $\ker f$ is $A$ and $B$-invariant and it contains the image of $I$; since $X$ is stable, we have that $\ker f=V$ thus $f=0$; since $g\in{\rm Hom}(W,0)$, then also $g=0$, proving the claim. \end{proof}
As before, $\iota_*:\mathcal{S}\to\mathcal{A}$ and $\jmath_*:\mathcal{Z}\to\mathcal{A}$ be the inclusion functors, and let $\iota^*$ and $\jmath^*$ be their right and left adjoints, respectively. It this way, one has that $\mathbf{S}_{\mathbf{R}}=\iota^*(\mathbf{R})$ and $\mathbf{Z}_{\mathbf{R}}=\jmath^*(\mathbf{R})$.
\begin{rema}\rm Recently, several authors have considered the so-called \emph{ADHM sheaves} (see e.g. \cite{Di,HL,Sz}), which are (twisted) representations of the ADHM quiver into ${\rm Coh}(Y)$, the category of coherent sheaves on a projective variety $Y$. These in turn may also be regarded as maps from $Y$ to the ADHM variety ${\mathcal V}(r,c)$. One can show that if the image of $Y$ intersects ${\mathcal V}(r,c)^{\rm st}$, then the corresponding ADHM sheaf is stable in the sense of \cite[Definition 2.2]{Di} and \cite[Definition 3.1]{HL}. \end{rema}
\section{Perverse coherent sheaves on $\p2$}
In this Section we introduce, following Kashiwara \cite{Ka}, a $t$-structure on $D^{\rm b}(\p2)$, the bounded derived category of coherent sheaves on $\p2$, and define a functor from the ADHM category ${\mathcal A}$ to the the core of this $t$-structure.
We start with notation. Let $Y$ be a finite dimensional, nonsingular, separated noetherian scheme. Let $\text{Mod}({\mathcal O}_Y)$ be the category of ${\mathcal O}_Y$-modules and $D({\mathcal O}_Y)$ be its derived category. Let also $D(Y)$ be the derived category of coherent sheaves on $Y$. We set $D_{\rm qc}({\mathcal O}_Y)$ (resp. $D_{\rm coh}({\mathcal O}_Y)$) to be the full triangulated subcategory of $D({\mathcal O}_Y)$ consisting of complexes with quasi-coherent (resp. coherent) cohomology. Recall that $D^{\rm b}(Y)$ is naturally equivalent to $D_{\rm coh}^{\rm b}({\mathcal O}_Y)$.
We will also use the costumary notation $D^{\le n}({\mathcal O}_Y)$ to mean complexes $M^\bullet$ in $D({\mathcal O}_Y)$ such that $H^p(M^\bullet)=0$ for all $p>n$; similarly, $D^{\ge n}({\mathcal O}_Y)$ to mean complexes $M^\bullet$ in $D({\mathcal O}_Y)$ such that $H^p(M^\bullet)=0$ for all $p<n$.
A \emph{family of supports} on $Y$ is a set $\Phi$ of closed subsets of $Y$ satisfying the following conditions: (i) if $Z\in\Phi$ and $Z'$ is a closed subset of $Z$, then $Z'\in\Phi$; (ii) if $Z,Z'\in\Phi$, then $Z\cup Z'\in\Phi$; (iii) $\emptyset\in\Phi$.
A \emph{support datum} on $Y$ is a decreasing sequence $\mathbf{\Phi}:=\{\Phi^n\}_{n\in\Z}$ of families of supports satisfying the following conditions: (i) for $n\ll0$, $\Phi^n$ is the set of all closed subsets of $Y$; (ii) $n\gg0$, $\Phi^n=\{\emptyset\}$.
Given a support datum on $Y$, Kashiwara introduces the following subcatgories of $D^{\rm b}_{\rm qc}({\mathcal O}_Y)$:
$$ {}^{\mathbf{\Phi}}D_{\rm qc}^{\le n}({\mathcal O}_Y) := \left\{ M^\bullet \in D^{\rm b}_{\rm qc}({\mathcal O}_Y) ~|~ {\rm supp}(H^k(M^\bullet)) \in \Phi^{k-n} ~\forall k \right\}, $$
$$ {}^{\mathbf{\Phi}}D_{\rm qc}^{\ge n}({\mathcal O}_Y) := \left\{ M^\bullet \in D^{\rm b}_{\rm qc}({\mathcal O}_Y) ~|~ R\Gamma_{\Phi^k}(M^\bullet) \in D^{\ge k+n}({\mathcal O}_Y) ~\forall k \right\}. $$ The functor $\Gamma_{\Phi}:{\rm Mod}({\mathcal O}_Y) \to {\rm Mod}({\mathcal O}_Y)$ is defined as follows: $$ \Gamma_{\Phi} := \lim_{Z\in\Phi} \Gamma_Z(F). $$ Then one has, for each open subset $U\subset Y$: \begin{equation}\label{gamma}
\Gamma_{\Phi}(F)(U) = \{ \sigma\in F(U) ~|~ \overline{{\rm supp}\sigma} \in \Phi \}. \end{equation}
Finally, the \emph{support} (or \emph{perversity}) \emph{function} associated to the support datum $\mathbf{\Phi}$ (see \cite[Lem. 5.5]{Ka}) is: \begin{gather*} \begin{matrix} p_{\mathbf{\Phi}} :& Y & \longrightarrow & \Z \\
& x & \longmapsto & {\rm max}\{ n\in\Z ~|~ \overline{x}\in\Phi^n \}. \end{matrix} \end{gather*}
Now consider as in \cite[p. 857]{Ka} $$ {}^{\mathbf{\Phi}}D_{\rm coh}^{\le 0}({\mathcal O}_Y) := {}^{\mathbf{\Phi}}D_{\rm qc}^{\le 0}({\mathcal O}_Y) \cap D^{\rm b}(Y) $$ $$ {}^{\mathbf{\Phi}}D_{\rm coh}^{\ge 0}({\mathcal O}_Y) := {}^{\mathbf{\Phi}}D_{\rm qc}^{\ge 0}({\mathcal O}_Y) \cap D^{\rm b}(Y). $$ It is shown in \cite[Theorem 5.9]{Ka} that if the support function $p_\Phi$ satisfies the following condition \begin{equation}\label{pervcond} p_{\mathbf{\Phi}}(y) - p_{\mathbf{\Phi}}(x) \le \codim(\overline{\{y\}}) - \codim(\overline{\{x\}})\ \text{if}\ y\in \overline{\{x\}}, \end{equation} then $( {}^{\mathbf{\Phi}}D_{\rm coh}^{\le 0}({\mathcal O}_Y),{}^{\mathbf{\Phi}}D_{\rm coh}^{\ge 0}({\mathcal O}_Y) )$ defines a $t$-structure on $D^{\rm b}(Y)$.
We now especialize to the case we are concerned in. Set $Y=\p2$, and fix, for the remainder of the paper, homogeneous coordinates $(x:y:z)$ in $\p2$; let $$
\ell_\infty:=\{(x:y:z)\in\p2\,|\,z=0\}; $$ be the \emph{line at infinity}. Consider de following support datum $\mathbf{\Phi}=\{\Phi^k\}_{k\in\Z }$ with \begin{align*} \Phi^k & := \{ \rm all~closed~subsets~of~\p2 \} ~~ {\rm for} ~ k\le0 \\ \Phi^1 & := \{ \rm all~closed~subsets~of~\p2\setminus\ell_\infty \} \\ \Phi^k & := \{ \emptyset \} ~~ {\rm for} ~ k\ge2. \end{align*} The corresponding perversity function $p_\Phi : \p2 \to \Z$ is given by: $p_{\mathbf{\Phi}}(x)=0$ iff $\overline{x}\cap\ell_\infty\ne\emptyset$ and $p_{\mathbf{\Phi}}(x)=1$ otherwise; in other words, $p_{\mathbf{\Phi}}(x)=1$ iff $x$ is a closed point away from $\ell_\infty$, and $p_{\mathbf{\Phi}}(x)=0$ otherwise. One easily checks that such function does satisfy the condition (\ref{pervcond}).
\begin{defi} A \emph{perverse coherent sheaf on} $\p2$ is an object in the core of the $t$-structure $( {}^{\mathbf{\Phi}}D_{\rm coh}^{\le 0}(\op),{}^{\mathbf{\Phi}}D_{\rm coh}^{\ge 0}(\op) )$ on $D^{\rm b}(\p2)$. \end{defi}
\begin{lemm}\label{rcl} If $M^\bullet$ is a perverse coherent sheaf on $\p2$, then: \begin{itemize} \item[(i)] $H^1(M^\bullet)$ is supported away from $\ell_\infty$; \item[(ii)] $H^p(M^\bullet)=0$ for $p\ne0,1$; \item[(iii)] $H^0(M^\bullet)$ has no sections $\sigma$ such that $\overline{{\rm supp}\sigma}$ does not intersect $\ell_\infty$. \end{itemize} \end{lemm}
\begin{proof} If $M^\bullet \in {}^{\mathbf{\Phi}}D_{\rm coh}^{\le 0}(\op)$, then $H^p(M^\bullet)=0$ for $p\ge2$, and $H^1(M^\bullet)$ is supported outside $\ell_\infty$. If $M^\bullet \in {}^{\mathbf{\Phi}}D_{\rm coh}^{\ge 0}(\op)$, then $R\Gamma_{\Phi^0}(M^\bullet) \in D^{\ge 0}(\op)$. But $\Gamma_{\Phi^0}$ is just the identity functor on ${\rm Mod}(\op)$, thus $R\Gamma_{\Phi^0}(M^\bullet)=M^\bullet$, and $M^\bullet \in D^{\ge 0}(\op)$, that is, $H^p(M^\bullet)=0$ for $p\leq -1$.
To establish the last claim, we apply \cite[Lem. 3.3.(iii)]{Ka}: since $M^\bullet \in D^{\ge 0}(\op)$, it follows that $H^0(R\Gamma_{\Phi^1}(M^\bullet))=\Gamma_{\Phi^1}(H^0(M^\bullet))$. But $R\Gamma_{\Phi^1}(M^\bullet)\in D^{\ge 1}(\op)$ since $M^\bullet \in {}^{\mathbf{\Phi}}D_{\rm coh}^{\ge 0}(\op)$, thus $\Gamma_{\Phi^1}(H^0(M^\bullet))=0$, and our claim follows from the definition of the functor $\Gamma_{\Phi^1}$, see (\ref{gamma}). \end{proof}
Notice that the fact that ${\rm supp}\ H^1(M^\bullet)$ does not intersect $\ell_\infty$ implies that it is $0$-dimensional, and hence finite; it thus makes sense to speak of the length of $H^1(M^\bullet)$, that is, $\dim H^0(H^1(M^\bullet))$.
\begin{defi} The \emph{rank, charge} and \emph{length} of a perverse coherent sheaf $M^\bullet$ are defined as, respectively, $\rk (H^0(E^\bullet))$, $c_2(H^0(E^\bullet))$ and $\text{length}(H^1(E^\bullet))$. \end{defi}
The (abelian) category of perverse coherent sheaves on $\p2$ will be denoted by ${\mathcal P}$. We say that $M^\bullet \in{\mathcal P}$ is \emph{trivial at infinity} if the restriction
$H^0(M^\bullet)|_{\ell_\infty}$ is isomorphic to ${\mathcal O}_{\ell_\infty}^{\oplus r}$, where $r=\rk (H^0(M^\bullet))>0$; let ${\mathcal P}_\infty$ denote the full subcategory of ${\mathcal P}$ consisting of such objects.
\begin{theo} \label{chperv} $M^\bullet \in {\mathcal P}_\infty$ if and only if \begin{itemize} \item[(i)] $H^p(M^\bullet)=0$ for $p\neq0,1$; \item[(ii)] $H^0(M^\bullet)$ is a torsion free sheaf which is trivial at infinity; \item[(iii)] $H^1(M^\bullet)$ is a torsion sheaf with support outside $\ell_\infty$. \end{itemize} \end{theo}
It follows from the above result that the definition of perverse coherent sheaves which are trivial at $\ell_\infty$ used in \cite[Section 5]{BFG} coincides with ours.
\begin{proof}
For the ``only if" part, by Lemma \ref{rcl}, it is enough to argue that $H^0(E^\bullet)$ is torsion free. Indeed, let $T$ be the torsion submodule of $H^0(E^\bullet)$. On one hand, the support of $T$ cannot intersect $\ell_\infty$, since $H^0(M^\bullet)|_{\ell_\infty}$ is locally free by hypothesis; on the other, item (iii) of Lemma \ref{rcl} implies that $T$ cannot be supported on points away from $\ell_\infty$ either. Thus $T=0$ and $H^0(E^\bullet)$ must be torsion free.
Conversely, let us first check that $M^\bullet\in{}^{\mathbf{\Phi}}D_{\rm coh}^{\le 0}(\op)$. This is quite clear, since ${\rm supp}(H^0(E^\bullet)) \in \Phi^{0}$, ${\rm supp}(H^1(E^\bullet)) \in \Phi^{1}$ and ${\rm supp}(H^k(M^\bullet))=\emptyset \in \Phi^{k}$ for $k\ge2$.
Second, let us check that $M^\bullet\in{}^{\mathbf{\Phi}}D_{\rm coh}^{\ge 0}(\op)$. Since $M^\bullet \in D_{\rm coh}^{\ge 0}(\op)$, then by \cite[Lem. 3.3.(iii)]{Ka} we conclude that $R\Gamma_{\Phi^k}(M^\bullet) \in D^{\ge 0}(\op)$ for every $k$. In particular, $R\Gamma_{\Phi^k}(M^\bullet) \in D^{\ge k}(\op)$ for every $k\le 0$. Since $R\Gamma_{\Phi^k}(M^\bullet)=0$ for every $k\ge2$ ($\Phi^k=\{\emptyset\}$ in this range), we also conclude that $R\Gamma_{\Phi^k}(M^\bullet) \in D^{\ge k}(\op)$ for every $k\ge 2$. Therefore, it only remains for us to show that $R\Gamma_{\Phi^1}(M^\bullet) \in D^{\ge 1}(\op)$, i.e., $H^0(R\Gamma_{\Phi^1}(M^\bullet))=0$. Again by \cite[Lem. 3.3(iii)]{Ka}, $H^0(R\Gamma_{\Phi^1}(M^\bullet))=\Gamma_{\Phi^1}(H^0(M^\bullet))$, so it suffices to argue that $\Gamma_{\Phi^1}(H^0(M^\bullet))=0$. But $H^0(M^\bullet)$ is a torsion free sheaf, so for every open set $U\subset\p2$ and every local section $\sigma\in H^0(M^\bullet)(U)$, we have that $\overline{{\rm supp}\sigma}=\p2$. Thus from (\ref{gamma}), we conclude that $\Gamma_{\Phi^1}(H^0(M^\bullet))=0$, as desired. \end{proof}
Notice that rank, charge and length are the only topological invariants for objects in ${\mathcal P}_\infty$, and if $r$, $s$, and $l$ are, respectively rank, charge and length of $M^\bullet\in{\mathcal P}_\infty$, then ${\rm ch}(M^\bullet) = r - (s+l) h^2$.
\begin{rema}\rm In \cite[Section 2]{HL}, the authors introduce \emph{perverse instanton sheaves on $\p3$}, which are objects in the core of a t-structure on $D^{\rm b}(\p3)$ defined through tilting on a torsion pair in ${\rm Coh}(\p3)$. A similar construction also applies to our case: our perverse coherent sheaves can also be regarded as objects in the core of a t-structure on $D^{\rm b}(\p2)$ defined through tilting on a torsion pair in ${\rm Coh}(\p2)$. \end{rema}
\subsection{ADHM construction of perverse sheaves on $\p2$} \label{secprv}
We will now construct a functor $\F:{\mathcal A}\to{\mathcal P}_\infty$ which extends the usual ADHM construction of instantons, as presented by Donaldson in \cite{D1} and later extended by Nakajima in \cite{N}; the link between representations of the ADHM quiver and perverse coherent sheaves was also discovered by Drinfeld, c.f. \cite[Thm. 5.7]{BFG}. We will further elaborated on Drinfeld's construction by discussing the role played by stabilizing subspaces, and also deriving relations between cohomologies of the complexes associated to a representation and its stable restriction and quotient representations.
\begin{defi} \label{defcom} \emph{Fix homogeneous coordinates $(x:y:z)$ in $\p2$. For any representation $\mathbf{R}=(V,W,(A,B,I,J))$ define the complex $$ E^\bullet_\mathbf{R} \,:~ V\otimes\op(-1) \stackrel{\alpha}{\longrightarrow} (V\oplus V\oplus W)\otimes\op \stackrel{\beta}{\longrightarrow} V\otimes\op(1) $$ where $$ \alpha = \left( \begin{array}{c} zA + x{\mathbf 1} \\ zB + y{\mathbf 1} \\ zJ \end{array} \right)\ \ \ \beta = \left( \begin{array}{ccc} -zB - y{\mathbf 1} ~~ & ~~ zA + x{\mathbf 1} ~~ & ~~ zI \end{array} \right). $$ Note that the ADHM equation is equivalent to $\beta\alpha=0$. Any complex on $\p2$ obtained in this way will be called an {\em ADHM complex}.} \end{defi}
Our aim now will be to show that any ADHM complex is a perverse coherent sheaf which is trivial at infinity, so the assignement $\mathbf{R} \mapsto E^\bullet_\mathbf{R}$ provides the desired functor. To see how the functor acts on morphisms, let the pair $(f,g)\in{\rm Hom}(V',V)\oplus{\rm Hom}(W',W)$ be a morphism between representations $\mathbf{R}'$ and $\mathbf{R}$; then one has the following morphism $\tilde{f}$ between the corresponding ADHM complexes $E^\bullet_{\mathbf{R}'}$ and $E^\bullet_\mathbf{R}$: $$ \xymatrix{ V'\otimes\op(-1) \ar[r]^{\alpha'\ \ \ \ \ \ \ } \ar[d]^{f\otimes{\mathbf 1}} & (V'\oplus V'\oplus W')\otimes\op \ar[r]^{\ \ \ \ \ \ \ \beta'} \ar[d]^{(f\oplus f\oplus g)\otimes{\mathbf 1}} & V'\otimes\op(1) \ar[d]^{f\otimes{\mathbf 1}} \\ V\otimes\op(-1) \ar[r]^{\alpha\ \ \ \ \ \ \ } & (V\oplus V\oplus W)\otimes\op \ar[r]^{\ \ \ \ \ \ \ \beta} & V\otimes\op(1) } . $$ So one defines $\F((f,g))$ to be the roof $E^\bullet_\mathbf{R}\stackrel{{\mathbf 1}}{\leftarrow}E^\bullet_\mathbf{R}\stackrel{\tilde{f}}{\rightarrow}E^\bullet_{\mathbf{R}'}$.
Let us fix once and for all a representation $\mathbf{R}=(V,W,(A,B,I,J))$, and the corresponding ADHM complex $E^\bullet_\mathbf{R}$ as above.
\begin{lemm} \label{l2} The sheaf map $\alpha$ is injective. The fiber maps $\alpha_P$ are injective for every $P\in\p2$ if and only if $\mathbf{R}$ is costable. \end{lemm}
\begin{proof} It is easily seen that $\alpha_P$ is injective for every $P\in\ell_\infty$. It follows that the set of points $P\in\p2$ for which $\alpha_P$ is not injective is a $0$-dimensional subscheme of $\p2$. The first claim of the lemma follows.
Now, if $\alpha_P$ is not injective for some $P=(p:q:1)\in\p2\setminus\ell_\infty$, then there is a nonzero vector $v\in V$ such that $$ Av = - p v\ \ \ \ \ \ Bv = - q v\ \ \ \ \ \ Jv=0 $$ and hence $\langle v\rangle$ is a nonzero subspace of $V$ which is invariant under $A$, $B$ and contained in $\ker J$, so $\mathbf{R}$ is not costable.
Conversely, if $\mathbf{R}$ is not costable, let $S\subset V$ be a nonzero subspace satisfying $A(S),B(S)\subset S\subset \ker J$. The ADHM equation implies that $[A|_S,B|_S]=0$, so let $v\in S$ be a nonzero common $A$ and $B$ eigenvector with eigenvalues $p$ and $q$, respectively. It is then easy to see that if $P=(-p:-q:1)$ then $\alpha_P$ is not injective. \end{proof}
The following is a well-known result (see \cite[Section 2.1]{N}), which we include here just for completeness.
\begin{lemm} \label{l3} If $\mathbf{R}$ is stable, then $H^1(E^{\bullet}_{\mathbf{R}})=0$, and $H^0(E^{\bullet}_{\mathbf{R}})$ is a torsion free sheaf whose restriction to $\ell_\infty$ is trivial of rank $r=\dim W$ and second Chern class $c=\dim V$. \end{lemm}
\begin{lemm} \label{lemhh0} For any $\mathbf{R}\in{\mathcal A}$, holds $H^0(H^0(E^{\bullet}_{\mathbf{R}})(-1))=0$. \end{lemm}
\begin{proof} Set $E:= H^0(E^{\bullet}_{\mathbf{R}})$. Consider the two short exact sequences of sheaves: \begin{equation}\label{um} 0 \to V\otimes\op(-1) \stackrel{\alpha}{\longrightarrow} \ker\beta \to E \to 0 \end{equation} \begin{equation} \label{dois} 0 \longrightarrow \ker\beta \longrightarrow (V\oplus V\oplus W)\otimes\op \stackrel{\beta}{\longrightarrow} \im \beta \longrightarrow 0. \end{equation} From (\ref{um}), we get $H^0(E(-1))\cong H^0(\ker\beta(-1))$. On the other hand, from (\ref{dois}), we get $H^0(\ker\beta(-1))=0$. \end{proof}
We are finally ready to establish the main result of this section.
\begin{theo} \label{mpt} If $\mathbf{R}$ is a representation of the ADHM quiver with type vector $(r,s,l)$, then the associated ADHM complex $E^\bullet_{\mathbf{R}}$ is a perverse coherent sheaf on $\p2$ of rank $r$, charge $s$ and length $l$ which is trivial at infinity. Moreover, \begin{itemize} \item[(i)] $H^0(E^\bullet_{\mathbf{R}})\simeq H^0(E^\bullet_{\mathbf{S}_{\mathbf{R}}})$; \item[(ii)] $H^1(E^\bullet_{\mathbf{R}})\simeq H^1(E^\bullet_{\mathbf{Z}_{\mathbf{R}}})$; \item[(iii)] $H^0(E^\bullet_{\mathbf{R}})$ is locally free if and only if $\mathbf{R}$ is costable. \end{itemize} \end{theo}
In categorical terms, one has that $H^0(\F(\mathbf{R}))\simeq\F(\iota^*(\mathbf{R}))$ and $H^1(\F(\mathbf{R}))\simeq\F(\jmath^*(\mathbf{R}))$, where $\iota^*$ and $\jmath^*$ are the adjoint functors introduced in the end of Section \ref{adhmcat}.
\begin{proof} Lemma \ref{l2} implies $H^{-1}(E^\bullet_\mathbf{R})=0$, thus $H^{p}(E^\bullet_\mathbf{R})=0$ for $p\ne0,1$. For the remainder, set $\mathbf{S}:=\mathbf{S}_{\mathbf{R}}$, $\mathbf{Z}:=\mathbf{Z}_{\mathbf{R}}$, $\Sigma:=\Sigma_X$ and $N:=N_X$ to simply notation. One then has the following short exact sequence of complexes \begin{equation}\label{diag} \begin{array}{ccccc}
0 & & 0 & & 0\\ \downarrow & & \downarrow & & \downarrow \\ \Sigma\otimes\op(-1) & \stackrel{\alpha'}{\longrightarrow} & (\Sigma\oplus\Sigma\oplus W)\otimes\op & \stackrel{\beta'}{\longrightarrow} & \Sigma\otimes\op(1) \\ \downarrow & & \downarrow & & \downarrow \\ V\otimes\op(-1) & \stackrel{\alpha}{\longrightarrow} & (V\oplus V\oplus W)\otimes\op & \stackrel{\beta}{\longrightarrow} & V\otimes\op(1) \\ \downarrow & & \downarrow & & \downarrow \\ N\otimes\op(-1) & \stackrel{\alpha''}{\longrightarrow} & (N\oplus N)\otimes\op & \stackrel{\beta''}{\longrightarrow} & N\otimes\op(1) \\ \downarrow & & \downarrow & & \downarrow \\ 0 & & 0 & & 0 \end{array} \end{equation} where, clearly, each line corresponds to $E^\bullet_{\mathbf{S}}$, $E^\bullet_{\mathbf{R}}$ and $E^\bullet_{\mathbf{Z}}$, respectively.
Note that $$ \alpha'' = \left( \begin{array}{c} zA_3 + x{\mathbf 1} \\ zB_3 + y{\mathbf 1} \end{array} \right). $$ Thus $\alpha''$ is injective at $\ell_\infty$, hence it is injective as a sheaf map, which is equivalent to $H^{-1}(E^\bullet_{\mathbf{Z}})=0$. Moreover, $\beta'$ is surjective since $\mathbf{S}$ is stable (see Lemma \ref{l3} above) and so $H^1(E_{\mathbf{S}}^\bullet)=0$. Therefore, the long exact sequence of cohomology associated to the short exact sequence of complexes above simplifies to $$ 0 \to H^0(E^\bullet_{\mathbf{S}}) \to H^0(E^\bullet_{\mathbf{R}}) \to H^0(E^\bullet_\mathbf{Z}) \to 0 \to H^1(E^\bullet_{\mathbf{R}}) \to H^1(E^\bullet_\mathbf{Z})\to 0. $$ This implies that $H^1(E^\bullet_{\mathbf{R}}) \simeq H^1(E^\bullet_\mathbf{Z})$, and that $H^0(E^\bullet_{\mathbf{S}}) \simeq H^0(E^\bullet_{\mathbf{R}})$ if and only if $H^0(E^\bullet_\mathbf{Z})=0$. Now both $H^0(E^\bullet_\mathbf{Z})$ and $H^1(E^\bullet_\mathbf{Z})$ are torsion sheaves supported at finitely many points which are outside $\ell_{\infty}$. In fact, take $P=(x:y:0)\in\ell_{\infty}$ and write $$ \alpha''_P = \left( \begin{array}{c} x{\mathbf 1} \\ y{\mathbf 1} \end{array} \right)\ \ \ \ \beta''_P=\left( \begin{array}{cc} -y{\mathbf 1} & x{\mathbf 1} \end{array} \right). $$ Since $H^0(E^\bullet_\mathbf{Z})=\ker\beta''/\im\alpha''$ and $H^1(E^\bullet_\mathbf{Z})={\rm coker}\,\beta''$, the stalks of both sheaves vanish at $P$. Hence the supports of both sheaves are $0$-dimensional schemes because none of each meet $\ell_{\infty}$. In particular, $H^0(H^0(E^\bullet_\mathbf{Z}))=H^0(H^0(E^\bullet_\mathbf{Z})(-1))$ and the latter vanish from Lemma \ref{lemhh0}; thus $H^0(E^\bullet_\mathbf{Z})=0$, for it is supported at finitely many points. It follows that $H^0(E^\bullet_{\mathbf{S}}) \simeq H^0(E^\bullet_{\mathbf{R}})$, as desired.
From Lemma \ref{l3} we conclude that $H^0(E^\bullet_{\mathbf{R}})$ is a torsion free sheaf which restricts trivially at infinity. Moreover, $H^1(E^\bullet_{\mathbf{R}})$ is a torsion sheaf supported away from the line $\ell_\infty$ since $H^1(E^\bullet_{\mathbf{Z}})$ does and we have just seen that they are isomorphic. It follows from Theorem \ref{chperv} that $E^\bullet_{\mathbf{R}}\in{\mathcal P}_\infty$.
Lemma \ref{l3} also tell us that $E^\bullet_{\mathbf{R}}$ has rank $r$ and charge $s$. To see that its length is $l$, consider the exact sequences \begin{equation} \label{equnn1} 0 \longrightarrow \ker\beta'' \longrightarrow (N\oplus N)\otimes\op \stackrel{\beta''}{\longrightarrow} \im \beta'' \longrightarrow 0 \end{equation} \begin{equation} \label{equnn2} 0 \to N\otimes\op(-1) \stackrel{\alpha''}{\longrightarrow} \ker\beta'' \to H^0(E^\bullet_{\mathbf{Z}}) \to 0 \end{equation} \begin{equation} \label{equnn3} 0 \longrightarrow \im \beta''\longrightarrow N\otimes\op(1) \longrightarrow H^1(E^\bullet_{\mathbf{Z}}) \longrightarrow 0. \end{equation} From (\ref{equnn1}) we get that $H^0(\im\beta''(-1))\cong H^1(\ker\beta''(-1))$ and, similarly, we also get $H^1(\im\beta''(-1))\cong H^2(\ker\beta''(-1))$. From (\ref{equnn2}) we get that $\ker\beta''\cong N\otimes\op(-1)$ because $H^0(E^\bullet_{\mathbf{Z}})=0$. So $H^0(\im\beta''(-1))=H^1(\im\beta''(-1))=0$. Thus, from (\ref{equnn3}), it follows that $N\otimes H^0(\op) \cong H^0(H^1(E^\bullet_{\mathbf{Z}})(-1))$. Hence ${\rm length}(H^1(E^\bullet_{\mathbf{Z}}))=\dim N$ and so ${\rm length}(H^1(E^\bullet_{\mathbf{R}}))=l$.
It remains for us to prove the last claim: we have $H^0(E^\bullet_{\mathbf{R}})=\ker\beta/\im\alpha$ is locally free if and only if the $\alpha_P$ are injective for all $P\in\p2$, which holds if and only if $\mathbf{R}$ is stable, owing to Lemma \ref{l2}. We are done. \end{proof}
\begin{rema}\rm One can also introduce the notion of \emph{framed perverse coherent sheaves}. A \emph{framing} on $E^\bullet\in{\mathcal P}_\infty$ is a choice of trivialization of $H^0(E^\bullet)$, i.e. an isomorphism $\phi:H^0(E^\bullet)|_{\ell_\infty} \stackrel{\sim}{\rightarrow} {\mathcal O}_{\ell_\infty}^{\oplus r}$. A \emph{framed perverse coherent sheaf} on $\p2$ is a pair $(E^\bullet,\phi)$ consisting of a perverse coherent sheaf $E^\bullet$ which is trivial at infinity, and a framing $\phi$ on $E^\bullet$. Two framed perverse coherent sheaves $(E^\bullet,\phi)$ and $(F^\bullet,\varphi)$ are isomorphic if there exists an isomorphism $\Psi:E^\bullet\to F^\bullet$ such that $\varphi=\phi\circ(H^0\Psi|_\ell)$, where $H^0\Psi$ denotes the induced map $H^0(E^\bullet)\to H^0(F^\bullet)$.
Now consider the functor from the category of schemes to the category of \linebreak groupoids, denoted $\mathbf{P}(r,c)$, that assigns to each scheme $S$ the groupoid whose objects are $S$-families of framed perverse coherent sheaves $E^\bullet$ on $\p2$ of rank $r$, charge $s$ and length $l$ such that $c=s+l$. Drinfeld has proved that such functor defines a stack isomorphic to the quotient stack $[{\mathcal V}(r,c)/GL(c)]$, see \cite[Thm. 5.7]{BFG}. We therefore conclude that the moduli stack $\mathbf{P}(r,c)$ is irreducible if an only if $r\ge2$.
It follows that the quotient stack $[{\mathcal V}(r,c)^{(s)}/GL(c)]$ is isomorphic to the functor $\mathbf{P}(r,c,l)$ that assigns to each scheme $S$ the groupoid whose objects are $S$-families of framed perverse coherent sheaves $E^\bullet$ on $\p2$ of rank $r$, charge $s$ and length $l$; such stacks are irreducible for each $r$, $s$ and $l$.
It is also worth mentioning that it follows from the proof of \cite[Thm. 5.7]{BFG} that the functor $\F:{\mathcal A}\to{\mathcal P}_\infty$ is essentially surjective. \end{rema}
\end{document} | arXiv |
\begin{document}
\author[F.~Andreatta, E. Z.~Goren, B.~Howard, K.~Madapusi Pera]{Fabrizio Andreatta, Eyal Z. Goren, \\ Benjamin Howard, Keerthi Madapusi Pera} \title[Faltings heights of abelian varieties]{Faltings heights of abelian varieties with complex multiplication}
\address{Dipartimento di Matematica ``Federigo Enriques", Universit\`a di Milano, via C.~Saldini 50, Milano, Italia} \email{[email protected]}
\address{Department of Mathematics and Statistics, McGill University, 805 Sherbrooke St. West, Montreal, QC, Canada} \email{[email protected]}
\address{Department of Mathematics, Boston College, 140 Commonwealth Ave, Chestnut Hill, MA, USA} \email{[email protected]}
\address{Department of Mathematics, University of Chicago, 5734 S University Ave, Chicago, IL, USA} \email{[email protected]}
\thanks{F.~Andreatta is supported by the Italian grant Prin 2010/2011. E. Z.~Goren is supported by an NSERC discovery grant, B.~Howard is supported by NSF grants DMS-1201480 and DMS-1501583. K.~Madapusi Pera is supported by NSF Postdoctoral Research Fellowship DMS-1204165 and NSF grant DMS-1502142.}
\begin{abstract} Let $M$ be the Shimura variety associated with the group of spinor similitudes of a quadratic space over $\mathbb Q$ of signature $(n,2)$. We prove a conjecture of Bruinier-Kudla-Yang, relating the arithmetic intersection multiplicities of special divisors and big CM points on $M$ to the central derivatives of certain $L$-functions.
As an application of this result, we prove an averaged version of Colmez's conjecture on the Faltings heights of CM abelian varieties. \end{abstract}
\maketitle
\setcounter{tocdepth}{1} \tableofcontents
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[subsection] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{BigThm}{Theorem}
\theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{hypothesis}[theorem]{Hypothesis}
\theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newtheorem{question}[theorem]{Question} \newtheorem{example}[theorem]{Example}
\numberwithin{equation}{subsection}
\renewcommand{\Alph{BigThm}}{\Alph{BigThm}}
\section{Introduction}
\subsection{The average Colmez conjecture}
Let $E$ be a CM field of degree $2d$ with maximal totally real subfield $F$. Let $A$ be an abelian variety over $\mathbb C$ of dimension $d$ with complex multiplication by the maximal order $\mathcal O_E\subset E$ and having CM type $\Phi\subset \mathrm{Hom}(E,\mathbb C)$. In this situation, Colmez \cite{Colmez} has proved that the Faltings height $h^\mathrm{Falt}(A)$ of $A$ depends only on the pair $(E,\Phi)$, and not on $A$ itself. We denote it by \[ h^\mathrm{Falt}_{(E,\Phi)} = h^\mathrm{Falt}(A) \]
Colmez stated in [\emph{loc.~cit.}] a conjectural formula for $h^\mathrm{Falt}_{(E,\Phi)}$ in terms of the logarithmic derivatives at $s=0$ of certain Artin $L$-functions, constructed in terms of the purely Galois-theoretic input $(E,\Phi)$. The precise conjecture is recalled in \S \ref{ss:colmez statement}, where the reader may also find our precise normalization of the Faltings height.
When $d=1$, so $E$ is a quadratic imaginary field, Colmez's conjecture is a form of the famous Chowla-Selberg formula. When $E/\mathbb Q$ is an abelian extension, Colmez proved his conjecture in [\emph{loc.~cit.}], up to a rational multiple of $\log(2)$. This extra error term was subsequently removed by Obus \cite{Obus}. When $d=2$, Yang \cite{Yang:colmez} was able to prove Colmez's conjecture in many cases, including the first known cases of non-abelian extensions.
Our first main result, stated in the text as Theorem \ref{thm:average colmez}, is the proof of an averaged form of Colmez's conjecture for a fixed $E$, obtained by averaging both sides of the conjectural formula over all CM types.
\begin{BigThm}\label{bigthm:average colmez} \[ \frac{1}{2^d} \sum_\Phi h^\mathrm{Falt}_{(E,\Phi)}
= - \frac{1}{2} \cdot \frac{ L'(0,\chi) }{ L(0,\chi ) } - \frac{1}{4} \cdot \log \left| \frac{D_E}{D_F} \right| - \frac{d}{2}\cdot \log(2\pi ). \] \end{BigThm} Here $\chi : \mathbb A_F^\times \to \{\pm 1\}$ is the quadratic Hecke character determined by the extension $E/F$, and $L(s,\chi)$ is the usual $L$-function without the local factors at archimedean places. The sum on the left is over all CM types of $E$, and $D_E$ and $D_F$ are the discriminants of $E$ and $F$, respectively.
\begin{remark} Very shortly after this theorem was announced, Yuan-Zhang also announced a proof; see \cite{YZ}. The proofs are very different. The proof of Yuan-Zhang is based on the Gross-Zagier style results of \cite{YZZ} for Shimura curves over totally real fields.
Our proof, which is inspired by the $d=2$ case found in \cite{Yang:colmez},
revolves around the calculation of arithmetic intersection multiplicities on Shimura varieties of type $\mathrm{GSpin}(n,2)$, and makes essential use of the theory of Borcherds products, as well as certain Green function calculations of Bruinier-Kudla-Yang \cite{BKY}. \end{remark}
\begin{remark} Tsimerman \cite{Tsimerman} has proved that Theorem \ref{bigthm:average colmez} implies the Andr\'e-Oort conjecture for all Siegel (and hence all abelian type) Shimura varieties. \end{remark}
\subsection{GSpin Shimura varieties and special divisors} \label{ss:intro shimura variety}
Let $(V,Q)$ be a quadratic space over $\mathbb Q$ of signature $(n,2)$ with $n\ge 1$, and let $L\subset V$ be a maximal lattice; that is, we assume that $Q(L) \subset \mathbb Z$, but that no lattice properly containing $L$ has this property. Let $L^\vee \subset V$ be the dual lattice of $L$ with respect to the bilinear form \[ [x,y] = Q(x+y) - Q(x) -Q(y), \] and abbreviate $D_L = [L^\vee : L]$ for the discriminant of $L$.
To this data one can associate a reductive group $G=\mathrm{GSpin}(V)$ over $\mathbb Q$, a particular compact open subgroup $K \subset G(\mathbb A_f)$, and a hermitian domain \[ \mathcal{D} = \{ z \in V_\mathbb C : [z,z]=0,\, [z, \overline{z}] <0 \} /\mathbb C^\times \] with an action of $G(\mathbb R)$ by holomorphic automorphisms. The $n$-dimensional complex orbifold \[ M(\mathbb C) = G(\mathbb Q) \backslash \mathcal{D} \times G(\mathbb A_f) / K \] is the space of complex points of a smooth algebraic stack $M$ over $\mathbb Q$, called the \emph{GSpin Shimura variety}. It admits, as we explain in \S \ref{s:orthogonal shimura}, a flat and normal integral model $\mathcal{M}$ over $\mathbb Z$, which is smooth after inverting $D_L$. For any prime $p>2$ the special fiber $\mathcal{M}_{\mathbb F_p}$ is normal and Cohen-Macaulay.
The Weil representation \[ \omega_L: \widetilde{\mathrm {SL}}_2(\mathbb Z) \to \mathrm{Aut}_\mathbb C(S_L) \] defines an action of the metaplectic double cover $\widetilde{\mathrm {SL}}_2(\mathbb Z) \to \mathrm {SL}_2(\mathbb Z)$ on the space $S_L = \mathbb C[L^\vee / L]$ of complex valued functions on $L^\vee /L$. Associated with it are, for any half-integer $k$, several spaces of vector-valued modular forms: the space of cusp forms $S_k(\omega_L)$, the space of weakly holomorphic forms $M^!_k(\omega_L)$, and the space of harmonic weak Maass forms $H_k(\omega_L)$. There are similar spaces for the complex-conjugate representation $\overline{\omega}_L$. By a theorem of Bruinier-Funke \cite{BF}, these are related by an exact sequence \begin{equation}\label{intro BF} 0 \to M^!_{1-\frac{n}{2}}(\omega_L) \to H_{1-\frac{n}{2}}(\omega_L) \map{\xi} S_{1+ \frac{n}{2}} (\overline{\omega}_L) \to 0, \end{equation} where $\xi$ is an explicit conjugate-linear differential operator.
Let $\varphi_\mu \in S_L$ be the characteristic function of the coset $\mu \in L$. Each form $f\in H_{1-\frac{n}{2}}(\omega_L)$ has a \emph{holomorphic part}, which is a formal $q$-expansion \[ f^+ = \sum_{ \substack{ m \gg -\infty \\ \mu \in L^\vee /L } } c_f^+(m,\mu) \varphi_\mu \cdot q^m \] valued in $S_L$. The sum is over all $m\in D_L^{-1} \mathbb Z$, but there are only finitely many nonzero terms with $m<0$.
The Shimura variety $\mathcal{M}$ comes with a family of effective Cartier divisors $\mathcal{Z}(m,\mu)$, indexed by positive $m\in D_L^{-1} \mathbb Z$ and $\mu \in L^\vee /L$. If the harmonic weak Maass form $f$
has \emph{integral principal part}, in the sense that $c_f^+(m,\mu) \in \mathbb Z$ for all $m\le 0$ and $\mu \in L^\vee /L$, then we may form the Cartier divisor \[ \mathcal{Z}(f) = \sum_{ \substack{ m > 0 \\ \mu \in L^\vee /L } } c_f^+(-m,\mu) \cdot \mathcal{Z}(m,\mu) \] on $\mathcal{M}$. A construction of Bruinier \cite{Bru} endows this divisor with a Green function $\Phi(f)$, constructed as a regularized theta lift of $f$. From this divisor and its Green function, we obtain a metrized line bundle \[ \widehat{\mathcal{Z}}(f) = \big( \mathcal{Z}(f) , \Phi(f) \big) \in \widehat{\mathrm{Pic}} ( \mathcal{M} ). \]
\subsection{The arithmetic Bruinier-Kudla-Yang theorem}
We now explain how to construct certain \emph{big CM cycles} on $\mathrm{GSpin}$ Shimura varieties, as in \cite{BKY}.
Start with a totally real field $F$ of degree $d$, and a quadratic space $(\mathscr{V} , \mathscr{Q})$ over $F$ of dimension $2$ and signature $( ( 0 ,2) , (2,0) , \ldots , (2,0) )$. In other words, $\mathscr{V}$ is negative definite at one archimedean place, and positive definite at the rest. The even Clifford algebra $E=C^+(\mathscr{V})$ is a CM field of degree $2d$ with $F$ as its maximal totally real subfield.
Now define a quadratic space \begin{equation}\label{intro trace} ( V,Q) = (\mathscr{V} , \mathrm{Tr}_{F/\mathbb Q} \circ \mathscr{Q} ) \end{equation} over $\mathbb Q$ of signature $(n,2) = (2d-2,2)$, and fix a maximal lattice $L\subset V$. As described above in \S \ref{ss:intro shimura variety}, we obtain from this data a $\mathrm{GSpin}$ Shimura variety $M\to \mathrm{Spec}(\mathbb Q)$, but now endowed with the additional structure of a distinguished $0$-cycle. Indeed, the relation (\ref{intro trace}) induces a morphism $T\to G$, where $T$ is the torus over $\mathbb Q$ with points \[ T(\mathbb Q) = E^\times / \mathrm{ker} \left( \mathrm{Nm}: F^\times \to \mathbb Q^\times \right). \] From the morphism $T \to G$ one can construct a $0$-dimensional Shimura variety $Y$ over $E$, together with a morphism $Y \to M$ of $\mathbb Q$-stacks. The image of this morphism consists of special points (in the sense of Deligne), and are the \emph{big CM points} of \cite{BKY}.
In \S \ref{ss:integral_model_Y} we define an integral model $\mathcal{Y}$ of $Y$, regular and flat over $\mathcal O_E$, along with a morphism $\mathcal{Y} \to \mathcal{M}$ of $\mathbb Z$-stacks. Composing the pullback of metrized line bundles with the arithmetic degree on the arithmetic curve $\mathcal{Y}$ defines a linear functional \[ \widehat{\mathrm{Pic}} ( \mathcal{M} ) \to \widehat{\mathrm{Pic}} ( \mathcal{Y}) \map{\widehat{\deg}} \mathbb R. \] We call this linear function \emph{arithmetic degree along $\mathcal{Y}$}, and denote it by \[ \widehat{\mathcal{Z}} \mapsto [ \widehat{\mathcal{Z}} : \mathcal{Y} ]. \]
To state our second main theorem, we need to introduce one more actor to our drama. This is a certain Hilbert modular Eisenstein series $E(\vec{\tau} , s) $ of parallel weight $1$, valued in the dual representation $S_L^\vee$. Starting from any $f\in H_{ 2-d }(\omega_L)$, we may apply the differential operator of (\ref{intro BF}) to obtain a vector-valued cusp form \[ \xi(f) \in S_d( \overline{\omega}_L), \] and then form the Petersson inner product $\mathcal{L}( s, \xi(f) )$ of $\xi(f)$ against the diagonal restriction of $E(\vec{\tau} , s) $ to the upper-half plane. This rather mysterious function inherits analytic continuation and a functional equation from the Eisenstein series, and the functional equation forces $\mathcal{L}( s, \xi(f) )$ to vanish at $s=0$. Our second main result, stated in the text as Theorem \ref{thm:arithmetic BKY}, is a formula for its derivative.
\begin{BigThm}\label{bigthm:arithmetic BKY} For any $f\in H_{ 2-d }(\omega_L)$ with integral principal part, the equality \[
\frac{ [ \widehat{\mathcal{Z}}(f) : \mathcal{Y} ] }{ \deg_\mathbb C (Y) } = - \frac{ \mathcal{L}'(0 , \xi(f) ) } { \Lambda( 0 , \chi ) } + \frac{ a(0,0) \cdot c_f^+( 0,0) } { \Lambda( 0 , \chi ) } \] holds up to a $\mathbb Q$-linear combination of $\{ \log(p) : p\mid D_{bad, L} \}$. \end{BigThm}
The unexplained notation in the theorem is as follows: $D_{bad,L}$ is the product of certain ``bad" primes, depending on the lattice $L\subset V=\mathscr{V}$, specified in Definition \ref{defn:D bad}; \[
\deg_\mathbb C (Y) = \sum_{y \in Y(\mathbb C) } \frac{1}{ \vert \mathrm{Aut}(y) \vert }
\]
is the number of $\mathbb C$-points of the $E$-stack $Y$, counted with multiplicities; $\Lambda(s,\chi)$ is the completed $L$-function of (\ref{completed L});
and the constant $a(0,0)=a_F(0,\varphi_0)$ is defined in Proposition \ref{prop:coarse constant}. In fact, $a(0,0)$ is essentially the derivative at $s=0$ of the constant term of $E(\vec{\tau} , s) $. By Proposition \ref{prop:constant term eval}, it satisfies \begin{equation}\label{intro constant} \frac{ a(0,0) } { \Lambda( 0 , \chi ) } = - \frac{ 2\Lambda'(0,\chi) } { \Lambda( 0 , \chi ) } \end{equation} up to a $\mathbb Q$-linear combination of $\{ \log(p) : p\mid D_{bad, L} \}$.
A key component of the proof of Theorem \ref{thm:arithmetic BKY} is the Bruinier-Kudla-Yang \cite{BKY} calculation of the values of the Green function $\Phi(f)$ at the points of $\mathcal{Y}$, which we recall in Theorem \ref{thm:BKY}. In fact, a form of Theorem \ref{bigthm:arithmetic BKY} was conjectured in \cite{BKY} based on these Green function calculations.
The bulk of this paper is devoted to computing the finite intersection multiplicities that comprise the remaining contributions to the arithmetic intersection $ [ \widehat{\mathcal{Z}}(f) : \mathcal{Y} ] $. More concretely, most of the paper consists of the calculation of the degrees of the $0$-cycles $\mathcal{Y} \times_\mathcal{M} \mathcal{Z}(m,\mu)$ on $\mathcal{Y}$, and the comparision of these degrees with the Fourier coefficients of the derivative $E'(\vec{\tau} ,0)$.
The first main new ingredient for the calculation, found in \S~\ref{ss:lubin-tate deformation}, is the computation of the deformation theory of certain `special' endomorphisms of Lubin-Tate formal groups, which, using Breuil-Kisin theory, we are able to do without any restriction on the ramification degree of the fields involved. This is a direct generalization of the seminal computations of Gross~\cite{Gross1986-ia} for Lubin-Tate groups associated with quadratic extensions of $\mathbb Q_p$.
The second new ingredient is the computation of certain $2$-adic Whittaker functions, which forms the bulk of \S~\ref{ss:whittaker functions}.
The introduction to each section has some further explanation of its role in the proof of the main theorem.
\begin{remark} The authors' earlier paper \cite{AGHMP} proves a result similar to Theorem \ref{bigthm:arithmetic BKY}, but for a cycle of \emph{small CM points} $\mathcal{Y} \to \mathcal{M}$ defined by the inclusion of a rank $2$ torus into $G$. In the present work the cycle of \emph{big CM points} $\mathcal{Y} \to \mathcal{M}$ is determined by a torus of maximal rank. One essential difference between these cases is that the big CM points always have proper intersection (on the whole integral model $\mathcal{M}$) with the special divisors $\mathcal{Z}(f)$. Thus, unlike in \cite{AGHMP}, we do not have to deal with improper intersection. \end{remark}
\begin{remark} In the special case of $d=2$, results similar to Theorem \ref{bigthm:arithmetic BKY} can be found in the work of Yang \cite{Yang:colmez}, and of Yang and the third named author \cite{HY}. Note that when $d=2$ we are working on a Shimura variety of type $\mathrm{GSpin}(2,2)$, and this class of Shimura varieties includes the classical Hilbert modular surfaces.
The paper \cite{HowUnitaryCM} contains results similar to Theorem \ref{bigthm:arithmetic BKY}, but on the Shimura varieties associated with unitary similitude groups instead of $\mathrm{GSpin}$. Of course the unitary case is easier, as those Shimura varieties can be realized as moduli spaces of abelian varieties. \end{remark}
\subsection{From arithmetic intersection to Colmez's conjecture}\label{ss:intro colmez}
We explain how to deduce Theorem \ref{bigthm:average colmez} from Theorem \ref{bigthm:arithmetic BKY}, following roughly the strategy of Yang \cite{Yang:colmez}. First, we choose the harmonic weak Maass form $f$ of Theorem \ref{bigthm:arithmetic BKY} so that $f$ is actually weakly holomorphic. In other words, we assume that \[ f = \sum _{ \substack{ m \gg -\infty \\ \mu \in L^\vee / L } } c_f(m,\mu) \varphi_\mu \cdot q^m \in M^!_{2-d}(\omega_L) , \] and so $\xi(f) = 0$ by the exact sequence (\ref{intro BF}). Combining Theorem \ref{bigthm:arithmetic BKY} with (\ref{intro constant}) gives \begin{equation}\label{intro holomorphic intersection}
\frac{ [ \widehat{\mathcal{Z}}(f) : \mathcal{Y} ] }{ \deg_\mathbb C (Y) } \approx_L
- c_f ( 0,0) \cdot \frac{ 2\Lambda'(0,\chi) } { \Lambda( 0 , \chi ) } , \end{equation} where $\approx_L$ means equality up to a $\mathbb Q$-linear combination of $\log(p)$ with $p\mid D_{bad,L}$.
The integral model $\mathcal{M}$ carries over it a line bundle $\bm{\omega}$ called the \emph{tautological bundle}, or the \emph{line bundle of weight one modular forms}. Any $g\in G(\mathbb A_f)$ determines a uniformization \[ \mathcal{D} \map{z\mapsto (z,g) } M(\mathbb C) \] of a connected component of the complex fiber of $\mathcal{M}$, and the line bundle $\bm{\omega}$ pulls back to the tautological bundle on $\mathcal{D}$, whose fiber at $z$ is the isotropic line $\mathbb C z \subset V_\mathbb C$.
If we now endow $\bm{\omega}$ with the metric $|| z ||^2 = -[z,\overline{z}]$, we obtain
a metrized line bundle \[ \widehat{\bm{\omega} } \in \widehat{\mathrm{Pic}}(\mathcal{M}). \]
For simplicity, assume that $d\geq 4$ (this guarantees that $V$ contains an isotropic line; throughout the body of the paper, we only require $d\geq 2$). After possibly replacing $f$ by a positive integer multiple, the theory of Borcherds products \cite{Bor98, Hormann, HMP} gives us a rational section $\Psi(f)$ of the line bundle $\bm{\omega}^{ \otimes c_f(0,0)}$, satisfying \[
- \log|| \Psi(f)||^2 = \Phi(f) - c_f(0,0) \log(4\pi e^\gamma), \] and satisfying $\mathrm{div}( \Psi(f) ) = \mathcal{Z}(f)$ \emph{up to a linear combination of irreducible components of the special fiber $\mathcal{M}_{\mathbb F_2}$}.
We define a Cartier divisor \[ \mathcal{E}_2(f) = \mathrm{div}(\Psi(f)) - \mathcal{Z}(f), \] on $\mathcal{M}$, supported entirely in characteristic $2$, which should be viewed as an unwanted error term. Endowing this divisor with the trivial Green function, we obtain a metrized line bundle $\widehat{\mathcal{E}}_2(f)\in \widehat{\mathrm{Pic}}(\mathcal{M})$ satisfying \[ [ \widehat{\bm{\omega}}^{\otimes c_f(0,0) } : \mathcal{Y} ] = [ \widehat{\mathcal{Z}}(f) : \mathcal{Y}] - c_f(0,0) \log(4\pi e^\gamma) \cdot d \deg_\mathbb C(Y) + [ \widehat{\mathcal{E}}_2(f) : \mathcal{Y}]. \] If we choose $f$ such that $c_f(0,0)\neq 0$, then combining this with (\ref{intro holomorphic intersection}) and dividing by $c_f(0,0)$ leaves
\[
\frac{ [ \widehat{\bm{\omega}} : \mathcal{Y} ] }{ \deg_\mathbb C (Y) }
+ d\cdot \log(4\pi e^\gamma)
\approx_L - \frac{ 2 \Lambda'( 0 , \chi ) } { \Lambda( 0 , \chi ) }
+ \frac{1}{c_f(0,0)} \frac{[ \widehat{\mathcal{E}}_2(f) : \mathcal{Y}]}{ \deg_\mathbb C(Y) }.
\]
The pullback to $\mathcal{Y}$ of the metrized line bundle $\widehat{\bm{\omega}}$ computes the averaged Faltings heights of abelian varieties with CM by $E$. More precisely, the cycle $\mathcal{Y}$ carries a canonical metrized line bundle $\widehat{\bm{\omega}}_0$ with two important properties: First, we show in Theorem~\ref{thm:Faltings height line bundles} that the arithmetic degree of $\widehat{\bm{\omega}}_0$ computes the averaged Faltings height: \[
\frac{1}{2^{d-2}} \sum_\Phi h^\mathrm{Falt}_{(E,\Phi)} = \frac{\widehat{\deg}_{\mathcal{Y}}(\widehat{\bm{\omega}}_0)}{\deg_{\mathbb C}(Y)} + \log|D_F| - 2d\cdot\log(2\pi). \] Second, in Proposition~\ref{prop:colmez prelim bound}, we prove the approximate equality \[ \frac{ [ \widehat{\bm{\omega}} : \mathcal{Y} ] } { \deg_\mathbb C(Y) }
\approx_L \frac{\widehat{\deg}_{\mathcal{Y}}(\widehat{\bm{\omega}}_0)}{\deg_{\mathbb C}(Y)} + \log|D_F|. \]
Putting all this together, we find that
\[
\frac{1}{2^d} \sum_\Phi h^\mathrm{Falt}_{(E,\Phi)}
= - \frac{1}{2} \cdot \frac{ L'(0,\chi) } { L ( 0 , \chi ) } - \frac{1}{4} \cdot \log\left| \frac{D_E}{D_F} \right| -
\frac{d}{2} \log(2\pi ) + \sum_p b_E(p) \log(p)
\] for some rational numbers $b_E(p)$, with $b_E(p)=0$ for all $p\nmid 2 D_{bad,L}$.
The integer $D_{bad ,L}$ depends on the choice of auxiliary $F$-quadratic space $(\mathscr{V} , \mathscr{Q})$
and lattice $L$, and to show that $b_E(p)=0$, one only has to find \emph{some} choice of the auxiliary data for which $p\nmid 2 D_{bad,L}$.
We show that for any prime $p$ the auxiliary data can be chosen so that $p\nmid D_{bad,L}$, and hence
$b_E(p)=0$ for all $p>2$. This proves Theorem \ref{bigthm:average colmez}, except that we have not shown that $b_E(2) = 0$.
For this, we embed $L$ in a larger lattice $L^ \diamond$ that has rank $2d+2$, and which is self-dual at $2$. The integral model $\mathcal{M}^ \diamond$ of the Shimura variety associated with $L^ \diamond$ is a smooth integral canonical model in the sense of~\cite{KisinJAMS}.
Using a result of Bruinier~\cite{Bru:prescribed}, we now pick a Borcherds lift $\Psi^ \diamond(f)$ over $\mathcal{M}^ \diamond$, whose divisor intersects $\mathcal{Y}$ properly, and which allows us to compute the height of the canonical bundle $\widehat{\omega}$ along $\mathcal{Y}$ even at the prime $2$. This enables us to prove that the constant $b_E(2)$ does indeed vanish.
\section{Special endomorphisms of Lubin-Tate groups}\label{s:lubin_tate}
In this section, we generalize the results of Gross~\cite{Gross1986-ia} to Lubin-Tate groups over arbitrary finite extensions $K/\mathbb Q_p$. Namely, we study the deformation theory of certain `special' endomorphisms of such groups.
This generalization, which appears as Theorem~\ref{thm:deformation special}, is the basis for the local intersection theory calculations underlying the proof of our main technical result, Theorem~\ref{thm:arithmetic BKY}. To be able to avoid restrictions on the ramification index of $K$, we are compelled to employ the theory of Breuil-Kisin modules~\cite{kisin:f_crystals}. This allows us to give a uniform treatment of all relevant cases.
The reader uninterested in the nitty gritty of $p$-adic Hodge theory, wanting only to understand the statement of Theorem~\ref{thm:deformation special}, can find the relevant definitions in the first paragraphs of \S~\ref{ss:lubin-tate_special_endomorphisms},~\ref{ss:denominators}, and~\ref{ss:lubin-tate deformation}.
\subsection{Breuil-Kisin modules and $p$-divisible groups}\label{ss:breuil_kisin}
Fix a prime $p$. Let $\mathbb Q_p^{\mathrm{alg}}$ be an algebraic closure of $\mathbb Q_p$, and let $\mathbb C_p$ be its completion. Set $W=W(\mathbb F_p^{\mathrm{alg}})$ and let $\mathrm{Frac}(W)^{\mathrm{alg}}\subset\mathbb C_p$ be the algebraic closure of its fraction field $\mathrm{Frac}(W)$. Let $K\subset\mathrm{Frac}(W)^{\mathrm{alg}}$ be a finite extension of $\mathrm{Frac}(W)$, and let \[ \Gamma_K = \mathrm{Gal}(\mathrm{Frac}(W)^{\mathrm{alg}}/K) \]
be its absolute Galois group.
Set $\mathfrak{S}=W\pow{u}$, the power series ring over $W$ in the variable $u$. Fix a uniformizer $\varpi\in\mathcal O_K$ and let $\mathcal{E}(u)\in W[u]$ be the associated Eisenstein polynomial satisfying $\mathcal{E}(0) = p$. A \emph{Breuil-Kisin module} over $\mathcal O_K$ (with respect to $\varpi$) is a pair $(\mathfrak{M},\varphi_{\mathfrak{M}})$, where $\mathfrak{M}$ is a finite free $\mathfrak{S}$-module and \[ \varphi_{\mathfrak{M}}:\varphi^*\mathfrak{M}[\mathcal{E}^{-1}]\xrightarrow{\simeq}\mathfrak{M}[\mathcal{E}^{-1}] \]
is an isomorphism of $\mathfrak{S}$-modules. Here, $\varphi:\mathfrak{S}\to\mathfrak{S}$ is the Frobenius lift that extends the canonical Frobenius automorphism $\mathrm{Fr}:W\to W$ and satisfies $\varphi(u)=u^p$.
Usually, the map $\varphi_{\mathfrak{M}}$ will be clear from context and we will denote the Breuil-Kisin module by its underlying $\mathfrak{S}$-module $\mathfrak{M}$.
We will write $\mathbf{1}$ for the Breuil-Kisin module whose underlying $\mathfrak{S}$-module is just $\mathfrak{S}$ equipped with the canonical identification $\varphi^*\mathfrak{S} = \mathfrak{S}$.
By~\cite{KisinJAMS}, there is a fully faithful tensor functor $\mathfrak{M}$ from the category of $\mathbb Z_p$-lattices in crystalline $\Gamma_K$-representations to the category of Breuil-Kisin modules over $\mathcal O_K$. It has various useful properties. To describe them, fix a crystalline $\mathbb Z_p$-representation $\Lambda$. Then:
\begin{itemize}
\item There is a canonical isomorphism of $F$-isocrystals over $\mathrm{Frac}(W)$:
\begin{equation}\label{bk:dcris}
\mathfrak{M}(\Lambda)/u\mathfrak{M}(\Lambda)[p^{-1}]\xrightarrow{\simeq}D_{\mathrm{cris}}(\Lambda)=(\Lambda\otimes_{\mathbb Z_p}B_{\mathrm{cris}})^{\Gamma_K}.
\end{equation} \item If we equip $\varphi^*\mathfrak{M}(\Lambda)$ with the descending filtration $\mathrm{Fil}^\bullet\varphi^*\mathfrak{M}$ given by \[ \mathrm{Fil}^i\varphi^*\mathfrak{M}(\Lambda) = \{x\in\varphi^*\mathfrak{M}(\Lambda):\;\varphi_{\mathfrak{M}(\Lambda)}(x)\in\mathcal{E}(u)^i\mathfrak{M}(\Lambda)\}, \] then there is a canonical isomorphism of filtered $K$-vector spaces \begin{equation}\label{bk:ddr} (\varphi^*\mathfrak{M}(\Lambda)/\mathcal{E}(u)\varphi^*\mathfrak{M}(\Lambda))[p^{-1}]\xrightarrow{\simeq}K\otimes_{\mathrm{Frac}(W)}D_{\mathrm{cris}}(\Lambda). \end{equation} Here, the left hand side is equipped with the filtration induced from $\mathrm{Fil}^\bullet\varphi^*\mathfrak{M}(\Lambda)$. \end{itemize}
Kisin's functor can be used to classify $p$-divisible groups over $\mathcal O_K$. This was done by Kisin himself~\cite{kisin:f_crystals} when $p>2$, and the case $p=2$ was dealt with by W. Kim~\cite{kim:2-adic}. We now present a summary of their results.
We will say that $\mathfrak{M}$ has \emph{$\mathcal{E}$-height $1$} if the isomorphism $\varphi_{\mathfrak{M}}$ arises from a map $\varphi^*\mathfrak{M}\to\mathfrak{M}$ whose cokernel is killed by $\mathcal{E}(u)$.
Let $S\to\mathcal O_K$ be the $p$-adic completion of the divided power envelope of the surjection \[ W[u]\xrightarrow{u\mapsto\varpi}\mathcal O_K. \] The natural map $W[u]\to S$ extends to an embedding $\mathfrak{S}\hookrightarrow S$, and the Frobenius lift $\varphi:\mathfrak{S}\to\mathfrak{S}$ extends continuously to an endomorphism $\varphi:S\to S$.
Write $\mathrm{Fil}^1S\subset S$ for the kernel of the map $S\to\mathcal O_K$. If $\mathfrak{M}$ is a Breuil-Kisin module of $\mathcal{E}$-height $1$, and $\mathcal{M}=S\otimes_{\varphi,\mathfrak{S}}\mathfrak{M}$, we will set \begin{equation}\label{BK filtration} \mathrm{Fil}^1\mathcal{M} =\{x\in \mathcal{M}=S\otimes_{\mathfrak{S}}\varphi^*\mathfrak{M}:\;(1\otimes\varphi_{\mathfrak{M}})(x)\in \mathrm{Fil}^1S\otimes_{\mathfrak{S}}\mathfrak{M}\subset S\otimes_{\mathfrak{S}}\mathfrak{M}\}. \end{equation} The image of $\mathrm{Fil}^1\mathcal{M}$ in $\mathcal O_K\otimes_S\mathcal{M}=\mathcal O_K\otimes_{\mathfrak{S}}\varphi^*\mathfrak{M}$ is a $\mathcal O_K$-linear direct summand, and so equips the ambient space with a two-step descending filtration.
For any $p$-divisible group $\mathcal{H}$ over a $p$-adically complete ring $R$, we will consider the contravariant Dieudonn\'e $F$-crystal $\mathbb{D}(\mathcal{H})$ associated with $\mathcal{H}$ (see for instance~\cite{BerthelotBreenMessingII}).
Given any nilpotent thickening $R'\to R$, whose kernel is equipped with divided powers, we can evaluate $\mathbb{D}(\mathcal{H})$ on $R'$ to obtain a finite projective $R'$-module $\mathbb{D}(\mathcal{H})(R')$ (this construction depends on the choice of divided power structure, which will be specified or evident from context). If $R'$ admits a Frobenius lift $\varphi:R'\to R'$, then we get a canonical map \[ \varphi:\varphi^*\mathbb{D}(\mathcal{H})(R')\to \mathbb{D}(\mathcal{H})(R') \] obtained from the $F$-crystal structure on $\mathbb{D}(\mathcal{H})$.
An example of a (formal) divided power thickening is any surjection of the form $R'\to R'/pR'$, where we equip $pR'$ with the canonical divided power structure induced from that on $p\mathbb Z_p$. Another example is the surjection $S\to \mathcal O_K$ considered above.
The evaluation on the trivial thickening $R\to R$ gives us a projective $R$-module $\mathbb{D}(\mathcal{H})(R)$ of finite rank equipped with a short exact sequence of projective $R$-modules: \[ 0 \to \mathrm{Lie} (\mathcal{H})^\vee\to \mathbb{D}(\mathcal{H})(R) \to \mathrm{Lie}(\mathcal{H}^\vee) \to 0. \] We will set \[ \mathrm{Fil}^1\mathbb{D}(\mathcal{H})(R)\define \mathrm{Lie} ( \mathcal{H})^\vee\subset \mathbb{D}(\mathcal{H})(R), \] and term it the \emph{Hodge filtration}.
\begin{theorem}\label{thm:kisin_p_divisible} For any $p$-divisible group $\mathcal{H}$ over $\mathcal O_K$, write $\mathcal{H}^\vee$ for its Cartier dual. Then the functor $\mathcal{H}\mapsto\mathfrak{M}(T_p(\mathcal{H}^\vee))$ is an exact contravariant equivalence of categories from the category of $p$-divisible groups over $\mathcal O_K$ to the category of Breuil-Kisin modules of $\mathcal{E}$-height $1$. Moreover, if we abbreviate \[ \mathfrak{M}(\mathcal{H}) \define \mathfrak{M}(T_p(\mathcal{H})^\vee), \] then the functor has the following properties: \begin{enumerate}
\item\label{kisin:cris}The $\varphi$-equivariant composition
\begin{align*}
\varphi^*\mathfrak{M}(\mathcal{H})/u\varphi^*\mathfrak{M}(\mathcal{H}) &\xrightarrow{\varphi_{\mathfrak{M}(\mathcal{H})}} \mathfrak{M}(\mathcal{H})/u\mathfrak{M}(\mathcal{H})[p^{-1}]\\
&\xrightarrow[\simeq]{\eqref{bk:dcris}} D_{\mathrm{cris}}(T_p(\mathcal{H})^\vee) \xrightarrow{\simeq} \mathbb{D}(\mathcal{H})(W)[p^{-1}]
\end{align*}
maps $\varphi^*\mathfrak{M}(\mathcal{H})/u\varphi^*\mathfrak{M}(\mathcal{H})$ isomorphically onto $\mathbb{D}(\mathcal{H})(W)$. Here, in a slight abuse of notation, we write $\mathbb{D}(\mathcal{H})(W)$ for the evaluation on $W$ of the Dieudonn\'e $F$-crystal associated with the reduction of $\mathcal{H}$ over $\mathbb F_p^\mathrm{alg}$.
\item\label{kisin:derham}The filtered isomorphism
\begin{align*}
\varphi^*\mathfrak{M}(\mathcal{H})/\mathcal{E}(u)\varphi^*\mathfrak{M}(\mathcal{H})[p^{-1}]&\xrightarrow[\simeq]{\eqref{bk:ddr}}K\otimes_{\mathrm{Frac}(W)}D_{\mathrm{cris}}(T_p(\mathcal{H})^\vee)\\
&\xrightarrow{\simeq}\mathbb{D}(\mathcal{H})(\mathcal O_K)[p^{-1}]
\end{align*}
maps $\varphi^*\mathfrak{M}(\mathcal{H})/\mathcal{E}(u)\varphi^*\mathfrak{M}(\mathcal{H})$ isomorphically onto $\mathbb{D}(\mathcal{H})(\mathcal O_K)$.
\item\label{kisin:breuil}There is a canonical $\varphi$-equivariant isomorphism
\[
S\otimes_{\varphi,\mathfrak{S}}\mathfrak{M}(\mathcal{H})\xrightarrow{\simeq}\mathbb{D}(\mathcal{H})(S)
\]
whose reduction along the map $S\to\mathcal O_K$ gives the filtration preserving isomorphism in~\eqref{kisin:derham}. \end{enumerate} \end{theorem} \begin{proof} This follows from~\cite[Theorem 1.1.6]{KisinModp}, using the work of Kim~\cite{kim:2-adic} when $p=2$. Note that this corrects an error in the statement of~\cite[Theorem 1.4.2]{KisinJAMS}, which is off by a Tate twist. \end{proof}
\subsection{Lubin-Tate groups}\label{ss:lubin_tate}
Fix a finite extension $E$ of $\mathbb Q_p$, and a uniformizer $\pi_E\in E$. Let $e(X)\in\mathcal O_E[X]$ be a Lubin-Tate polynomial associated with $\pi_E$, so that \begin{align*} e(X)&\equiv\pi_EX\pmod{X^2}, \\ e(X)&\equiv X^q\pmod{\pi_E}. \end{align*} Here, $q=\# k_E$ is the size of the residue field $k_E$ of $E$.
Let $\mathcal{G}=\mathrm{Spf}( \mathcal O_E\pow{X})$ be the unique formal $\mathcal O_E$-module in one variable over $\mathcal O_E$ with multiplication by $\pi_E$ given by the polynomial $[\pi_E](X)=e(X)$. For each $n\in\mathbb Z_{>0}$, write $\mathcal{G}[\pi_E^n]$ for the $\pi_E^n$-torsion $\mathcal O_E$-submodule of $\mathcal{G}$. These fit into a $\pi_E$-divisible group $\mathcal{G}[\pi_E^\infty]$ over $\mathcal O_E$.
Assume now that we have an embedding $E\hookrightarrow K$. We obtain a formal $\mathcal O_E$-module $\mathcal{G}_{\mathcal O_K}$ over $\mathcal O_K$ and a $\pi_E$-divisible group $\mathcal{G}[\pi_E^\infty]_{\mathcal O_K}$. For simplicity, we will omit $\mathcal O_K$ from the subscripts in what follows, and so will be viewing both $\mathcal{G}$ and $\mathcal{G}[\pi_E^\infty]$ as objects over $\mathcal O_K$.
Let \[ T_{\pi_E}(\mathcal{G}) \define \varprojlim_{n}\mathcal{G}[\pi_E^n](\mathrm{Frac}(W)^{\mathrm{alg}}). \] be the $\pi_E$-adic Tate module associated with $\mathcal{G}[\pi_E^\infty]$. This a crystalline $\mathbb Z_p$-representation of $\Gamma_K$ equipped with an $\mathcal O_E$-action, making it an $\mathcal O_E$-module of rank $1$. We will now describe the associated Breuil-Kisin module \[ \mathfrak{M}(\mathcal{G}) \define \mathfrak{M}\bigl(T_{\pi_E}(\mathcal{G})^\vee\bigr) \] with its $\mathcal O_E$-action. This will involve constructing an explicit candidate $\mathfrak{M}$ for such a module, and then showing that this candidate is indeed isomorphic to $\mathfrak{M}(\mathcal{G})$.
Let $E_0\subset E$ be the maximal unramified subextension, and let $\mathrm{Emb}(E_0)$ be the set of embeddings $E_0\hookrightarrow \mathrm{Frac}(W)$. Let $\iota_0\in\mathrm{Emb}(E_0)$ be the distinguished element induced by the embedding $E\hookrightarrow K$. The Frobenius automorphism $\mathrm{Fr}$ of $W$ acts on $\mathrm{Emb}(E_0)$, and every $\iota\in\mathrm{Emb}(E_0)$ is of the form $\mathrm{Fr}^i(\iota_0)$ for a unique $i\in\{0,1,\ldots,d_0-1\}$, where $d_0 = [E_0:\mathbb Q_p]$.
The underlying $\mathcal O_E$-equivariant $\mathfrak{S}$-module for our candidate is \[ \mathfrak{M} = \mathfrak{S}\otimes_{\mathbb Z_p}\mathcal O_E = \bigoplus_{\iota\in\mathrm{Emb}(E_0)}\mathfrak{S}\otimes_{\iota,\mathcal O_{E_0}}\mathcal O_E = \bigoplus_{\iota}\mathfrak{S}_{\iota}, \] where, for $\iota\in\mathrm{Emb}(E_0)$, we have set $W_{\iota} = W\otimes_{\iota,\mathcal O_{E_0}}\mathcal O_E$ and $\mathfrak{S}_{\iota}=W_{\iota}\pow{u}$. There is a canonical $\mathcal O_E$-equivariant identification of $\mathfrak{S}$-modules \[ \varphi^*\mathfrak{M} = \bigoplus_{\iota}\varphi^*\mathfrak{S}_{\mathrm{Fr}^{-1}(\iota)} = \bigoplus_{\iota}\mathfrak{S}_{\iota} = \mathfrak{M}. \]
The $\mathfrak{S}\otimes_{\mathbb Z_p}\mathcal O_E$-equivariant isomorphism $\varphi_{\mathfrak{M}}$ will now arise from a map \[ \varphi_{\mathfrak{M}}:\varphi^*\mathfrak{M}=\mathfrak{M}\xrightarrow[\beta]{\simeq}\mathfrak{M}, \] for some \[ \beta\in (\mathfrak{S}\otimes_{\mathbb Z_p}\mathcal O_E)\cap (\mathfrak{S}[\mathcal{E}^{-1}]\otimes_{\mathbb Z_p}\mathcal O_E)^\times = \prod_{\iota\in \mathrm{Emb}(E_0)}\mathfrak{S}_\iota\cap\mathfrak{S}_{\iota}[\mathcal{E}^{-1}]^\times. \] To describe $\beta$ explicitly, we have to specify each of its components \[ \beta_{\iota}\in \mathfrak{S}_{\iota}\cap \mathfrak{S}_{\iota}[\mathcal{E}^{-1}]^\times\subset \mathfrak{S}_\iota[\mathcal{E}^{-1}]. \] Let $\mathcal{E}_{\iota_0}(u)\in W_{\iota_0}[u]$ be the Eisenstein polynomial for $\varpi$ over $W_{\iota_0}$ satisfying $\mathcal{E}_{\iota_0}(0) = \iota_0(\pi_E)$, and set \begin{align*} \beta_{\iota}&=\begin{cases}
\mathcal{E}_{\iota_0}(u) & \text{ if $\iota=\iota_0$} \\
1& \text{otherwise.} \end{cases} \end{align*}
From $\mathfrak{M}$, we obtain an abstract `crystalline' realization \begin{align}\label{eqn:mfM_cris} M_{\mathrm{cris}}&\define W\otimes_{\mathfrak{S}}\varphi^*\mathfrak{M} = W\otimes_{\mathbb Z_p}\mathcal O_E = \bigoplus_{\iota}W_{\iota}, \end{align} where we view $W$ as an $\mathfrak{S}$-algebra via $u\mapsto 0$. This also identifes $\mathrm{Fr}^*M_{\mathrm{cris}}$ with $\bigoplus_{\iota}W_{\iota}$. Under these identifications, the $F$-crystal structure on $M_{\mathrm{cris}}$ is given by multiplication by the image of $\beta$ under
\[ \mathfrak{S}\otimes_{\mathbb Z_p}\mathcal O_E\xrightarrow{\varphi\otimes 1} \mathfrak{S} \otimes_{\mathbb Z_p} \mathcal O_E \xrightarrow{u\otimes 1 \mapsto 0} W\otimes_{\mathbb Z_p}\mathcal O_E.
\]
This image is easy to describe: Its $\iota$-component is $1$ when $\iota\neq \mathrm{Fr}(\iota_0)$, while its $\mathrm{Fr}(\iota_0)$-component is $\mathrm{Fr}(\iota_0(\pi_E))\in W_{\mathrm{Fr}(\iota_0)}$.
Similarly, we obtain an abstract `de Rham' realization \begin{align}\label{eqn:mfM_dR} M_{\mathrm{dR}}&\define \mathcal O_K\otimes_{\mathfrak{S}}\varphi^*\mathfrak{M}, \end{align} where we view $\mathcal O_K$ as an $\mathfrak{S}$-algebra via $u\mapsto \varpi$. Write $M_{\iota}$ for the $\iota$-isotypic component of $M_{\mathrm{dR}}$; this is simply \[ \mathcal O_{K,\iota}\define\mathcal O_K\otimes_{\iota,\mathcal O_{E_0}}\mathcal O_E \]
viewed as a module over itself.
The recipe in (\ref{bk:ddr}) also gives us a direct summand $\mathrm{Fil}^1M_{\mathrm{dR}}\subset M_{\mathrm{dR}}$. This is an $\mathcal O_K\otimes_{\mathbb Z_p}\mathcal O_E$-stable submodule, and so it suffices to specify its $\iota$-isotypic component $\mathrm{Fil}^1M_{\iota}\subset M_{\iota}$ for each $\iota$. To do this, we first need to describe the subspace $\mathrm{Fil}^1\varphi^*\mathfrak{M}$. By definition, we have \[ \mathrm{Fil}^1\varphi^*\mathfrak{M} = \{x\in\varphi^*\mathfrak{M}:\varphi_{\mathfrak{M}}(x)\in\mathcal{E}(u)\mathfrak{M}\}. \] From this, we deduce \begin{align}\label{eqn:fil_mfM} \bigl(\mathrm{Fil}^1\varphi^*\mathfrak{M}\bigr)_{\iota}&=\begin{cases} \{x\in\mathfrak{S}_{\iota}: \mathcal{E}_{\iota_0}(u)x\in\mathcal{E}(u)\mathfrak{S}_{\iota}\} & \text{ if $\iota=\iota_0$} \\ \mathcal{E}(u)\mathfrak{S}_{\iota} & \text{otherwise}. \end{cases} \end{align} Reducing mod $\mathcal{E}(u)$, we now find \begin{align}\label{eqn:fil1_H} \mathrm{Fil}^1M_{\iota} &= \begin{cases} \{x\in M_{\iota}:\mathcal{E}_{\iota_0}(\varpi\otimes 1)x=0\} & \text{ if $\iota=\iota_0$} \\ 0 & \text{ otherwise}. \end{cases} \end{align}
Clearly, $\mathfrak{M}$ has $\mathcal{E}$-height $1$. Therefore, by Theorem~\ref{thm:kisin_p_divisible}, there exists a $p$-divisible group $\mathcal{H}$ over $\mathcal O_K$, equipped with an $\mathcal O_E$-module structure, such that $\mathfrak{M} = \mathfrak{M}(\mathcal{H})$. Moreover, the $\mathcal O_E$-action on $\mathfrak{M}$ translates to an $\mathcal O_E$-action on $\mathcal{H}$.
Let $T_{\pi_E}(\mathcal{H})$ be the $\pi_E$-adic Tate module over $K$ associated with $\mathcal{H}$. We can use the explicit descriptions of $M_{\mathrm{cris}}$ and $M_{\mathrm{dR}}$ above to obtain a description of the associated $E$-equivariant, filtered $\varphi$-module \[ D_{\mathrm{cris}}\define D_{\mathrm{cris}}(T_{\pi_E}(\mathcal{H})^\vee). \] The underlying $\mathrm{Frac}(W)$-vector space is \[ D_{\mathrm{cris}} = M_{\mathrm{cris}}[p^{-1}] = \mathrm{Frac}(W)\otimes_{\mathbb Q_p}E = \bigoplus_{\iota}\mathrm{Frac}(W)_{\iota}. \] As before, this description also identifies $\mathrm{Fr}^*D_{\mathrm{cris}}$ with $\bigoplus_{\iota}\mathrm{Frac}(W)_{\iota}$, and under these identifications, the $F$-isocrystal structure on $D_{\mathrm{cris}}$ is given simply by multiplication by $\pi_{\iota_0}$ on the $\iota_0$-factor, and the identity on the remaining factors.
To complete our description, we need to know the subspace \[\mathrm{Fil}^1 D_{\mathrm{dR}}\subset D_{\mathrm{dR}} = K\otimes_{\mathrm{Frac}(W)}D_{\mathrm{cris}}.\] Let $D_{\iota}\subset D_{\mathrm{dR}}$ be the $\iota$-isotypic component. This is a rank $1$ free module over $K_{\iota} = K\otimes_WW_\iota$. Note that we have a quotient map \begin{equation}\label{eqn:iota_0_quotient} K_{\iota_0} = K\otimes_{\iota_0,E_0}E\to K \end{equation} induced by the distinguished embedding $E\hookrightarrow K$. This gives us an idempotent projector $\bm{e}_0:K_{\iota_0}\to K_{\iota_0}$ such that (\ref{eqn:iota_0_quotient}) identifies $\bm{e}_0 K_{\iota_0} \xrightarrow{\simeq} K$. From~\eqref{eqn:fil1_H}, we now have \[ \mathrm{Fil}^1 D_{\mathrm{dR}} = \bigoplus_{\iota}\mathrm{Fil}^1 D_{\iota}\subset \bigoplus_{\iota}D_{\iota}, \] where \begin{align*}
\mathrm{Fil}^1D_{\iota} &= \begin{cases} \bm{e}_0D_{\iota} & \text{ if $\iota=\iota_0$} \\ 0 & \text{otherwise}. \end{cases} \end{align*}
\begin{proposition}\label{prp:bk_lubin-tate} There is a $\mathcal O_E$-equivariant isomorphism \[ \mathfrak{M}(\mathcal{G}) \xrightarrow{\simeq} \mathfrak{M} \] of Breuil-Kisin modules. In particular, we have $\mathcal O_E$-equivariant isomorphisms \begin{align*} \mathbb{D}(\mathcal{G})(W)\xrightarrow{\simeq}M_{\mathrm{cris}}, \qquad \mathbb{D}(\mathcal{G})(\mathcal O_K)&\xrightarrow{\simeq}M_{\mathrm{dR}} \end{align*} of $F$-crystals over $W$ and filtered $\mathcal O_K$-modules, respectively. \end{proposition} \begin{proof}
The first assertion of the proposition amounts to showing that we have an $\mathcal O_E$-equivariant isomorphism \[ T_{\pi_E}(\mathcal{H})\xrightarrow{\simeq} T_{\pi_E}(\mathcal{G}) \] of $\pi_E$-adic Tate modules over $K$. In fact, since all $\mathcal O_E$-lattices in $T_{\pi_E}(\mathcal{G})[p^{-1}]$ are simply dilations of $T_{\pi_E}(\mathcal{G})$ by powers of $\pi_E$, it is enough to show that we have an $\mathcal O_E$-equivariant isomorphism \[ T_{\pi_E}(\mathcal{H})[p^{-1}]\xrightarrow{\simeq} T_{\pi_E}(\mathcal{G})[p^{-1}]. \]
To do this, we will show that the admissible filtered $\varphi$-modules associated with the two representations are $\mathcal O_E$-equivariantly isomorphic. We have already computed the admissible $\varphi$-module $D_{\mathrm{cris}}$ associated with $T_{\pi_E}(\mathcal{H})^\vee$. So we have to check that it agrees with that obtained from $T_{\pi_E}(\mathcal{G})^\vee$. This follows from \cite[Lemma 1.22]{rapoport_zink}.
The last assertion follows from the first via~\eqref{kisin:cris}, and~\eqref{kisin:derham} of Theorem~\ref{thm:kisin_p_divisible}. \end{proof}
\subsection{Special endomorphisms} \label{ss:lubin-tate_special_endomorphisms}
We will assume that $E$ is equipped with a non-trivial involution $\tau$. Let $F\subset E$ be the fixed field of $\tau$. If $E/F$ is unramified, then we will further assume that the uniformizer $\pi_E$ is in fact a uniformizer in $F$. Given a $p$-adically complete $\mathcal O_{E}$-algebra $R$, a \emph{special endomorphism} of $\mathcal{G}_R$ will be an element $f\in\mathrm{End}(\mathcal{G}_R)$ such that \[ f([a](X))=[\tau(a)](f(X)), \] for any $a\in\mathcal O_{E}$. Write $V(\mathcal{G}_R)$ for the space of special endomorphisms of $\mathcal{G}$.
The following proposition is clear.
\begin{proposition}\label{prop:lubin-tate_special_end} \mbox{} \begin{enumerate}
\item The subspace $V(\mathcal{G}_R)\subset\mathrm{End}(\mathcal{G}_R)$ is $\mathcal O_{E}$-stable. If it is non-zero, then it is a finite free $\mathcal O_{E}$-module of rank $1$.
\item For any $x_1,x_2\in V(\mathcal{G}_R)$, there exists a unique $\langle x_1,x_2\rangle\in\mathcal O_{E}$ such that \[
x_1\circ x_2=[\langle x_1,x_2\rangle]\in\mathrm{End}_{\mathcal O_{E}}(\mathcal{G}_R). \]
\item The pairing $(x_1,x_2)\mapsto \langle x_1,x_2\rangle$ is a Hermitian pairing on $V(\mathcal{G}_R)$. \end{enumerate} \end{proposition}
It will be useful to have the following notation: Let $R$ be a commutative ring with a non-trivial involution $\tau$. For any $R$-module $M$, we will set: \begin{equation*}
V(M,\tau) = \{f\in\mathrm{End}(M):f(a\cdot m)=\tau(a)f(m)\text{, for all $a\in R$}\}. \end{equation*} This is an $R$-submodule of $\mathrm{End}(M)$, where we equip the latter with the $R$-module structure obtained from post-composition with scalar multiplication by $R$.
The embedding $\iota_0\in \mathrm{Emb}(E_0)$ induces an embedding $k_E\hookrightarrow\mathbb F_p^{\mathrm{alg}}$. Set \[ V(\mathcal{G}_1) = V(\mathcal{G}_{\mathbb F^{\mathrm{alg}}_p}) \] and \[ V_{\mathrm{cris}}(\mathcal{G}) = V(M_\mathrm{cris},\tau), \] where we view $M_\mathrm{cris}$ as an $\mathcal O_E\otimes_{\mathbb Z_p}W$-module.
We now have the following easy lemma, whose proof we omit. \begin{lemma}\label{lem:V_cris_G}\ \begin{enumerate} \item $V_{\mathrm{cris}}(\mathcal{G})$ is an $\mathcal O_E$-stable subspace of $\mathrm{End}(M_{\mathrm{cris}})$, which is free of rank $1$ over $W\otimes_{\mathbb Z_p}\mathcal O_{E}$. Conjugation by $\varphi_0:\mathrm{Fr}^*M_\mathrm{cris}\to M_\mathrm{cris}$ induces a $\mathcal O_E\otimes_{\mathbb Z_p}W$-linear automorphism \[ \varphi: \mathrm{Fr}^*V_\mathrm{cris}(\mathcal{G})[p^{-1}]\xrightarrow{\simeq} V_\mathrm{cris}(\mathcal{G})[p^{-1}]. \] \item There is a canonical identification \[ V(\mathcal{G}_1) = V_{\mathrm{cris}}(\mathcal{G})^{\varphi=1} \] of $V(\mathcal{G}_1)$ with the $\varphi$-equivariant elements in $V_{\mathrm{cris}}(\mathcal{G})$. \item For $x,y\in V_{\mathrm{cris}}(\mathcal{G})$, $x\circ y\in \mathrm{End}(M_{\mathrm{cris}})$ corresponds to multiplication by an element $\langle x,y\rangle\in W\otimes_{\mathbb Z_p}\mathcal O_E$. The assignment \[ (x,y)\mapsto \langle x,y\rangle \in W\otimes_{\mathbb Z_p}\mathcal O_E \] is a $\tau$-Hermitian form on $V_{\mathrm{cris}}(\mathcal{G})$, which restricts to the canonical $\mathcal O_E$-valued Hermitian form on $V(\mathcal{G}_1)$ from Proposition~\ref{prop:lubin-tate_special_end}. \end{enumerate} \end{lemma}
It will be useful to have an explicit description of $V_{\mathrm{cris}}(\mathcal{G})$ along with that of the conjugation action of the semi-linear endomorphism $\varphi_0$ of $M_{\mathrm{cris}}$. This is easily deduced from the explicit description of $M_{\mathrm{cris}}$ from~\eqref{eqn:mfM_cris}.
For each $\iota\in \mathrm{Emb}(E_0)$, set $V_{\iota} = W\otimes_{\iota,\mathcal O_{E_0}}V(\mathcal O_{E},\tau)$. This is a rank $1$ free module over $W_{\iota}$. Using~\eqref{eqn:mfM_cris}, we now obtain a canonical $\mathcal O_{E}$-equivariant identification \[ V_{\mathrm{cris}}(\mathcal{G}) = W\otimes_{\mathbb Z_p}V(\mathcal O_{E},\tau) = \bigoplus_{\iota\in \mathrm{Emb}(E_0)}V_{\iota}. \] This also identifes $\mathrm{Fr}^*V_{\mathrm{cris}}(\mathcal{G})$ with $\bigoplus_{\iota\in \mathrm{Emb}(E_0)}V_{\iota}$.
As before, set $d_0=[E_0:\mathbb Q_p]$. Any element of $V_{\mathrm{cris}}(\mathcal{G})$ is a tuple of the form $f=(f_i)_{0\leq i\leq d_0-1}$ for some $a_i\in V_{\mathrm{Fr}^i(\iota_0)}$, and \[ \varphi(f)_i = \eta_i\varphi(f_{i-1})\in V_{\mathrm{Fr}^i(\iota_0)}[p^{-1}], \] for certain $\eta_i\in \mathrm{Frac}(W_{\mathrm{Fr}^i(\iota_0)})$.
To pin the $\eta_i$ down, first consider the case where $E$ is unramified over $F$. In this case, $\pi_E$ is a uniformizer for $F$ by hypothesis, and hence satisfies $\tau(\pi_E) = \pi_E$. Also, $\tau$ acts non-trivially on $\mathrm{Emb}(E_0)$: If $r\in\mathbb Z_{\geq 1}$ is such that $2r=d_0$, we have, for any $\iota\in \mathrm{Emb}(E_0)$, \[ \mathrm{Fr}^r(\iota)=\tau(\iota) \define \iota\circ\tau. \] We can now identify \[ V_{\iota}=\mathrm{Hom}_{W_{\iota}}(W_{\tau(\iota)},W_{\iota}), \] as $W_{\iota}$-modules. Here, we view $W_{\iota}$ as acting on $W_{\tau(\iota)}$ via the isomorphism $W_{\iota}\xrightarrow{\simeq}W_{\tau(\iota)}$ induced by $\tau$.
Now, as seen in~\eqref{ss:lubin_tate}, the $F$-crystal structure on $M_\mathrm{cris}$ corresponds under the identification~\eqref{eqn:mfM_cris} to multiplication by the element $\beta_0\in W\otimes_{\mathbb Z_p}\mathcal O_E$, whose $\iota_0$-isotypic component is $1\otimes\pi_E$, and whose $\iota$-isotypic component for $\iota\neq \iota_0$ is $1$. From this we deduce: \begin{align*}
\eta_i&=\begin{cases}
1\otimes \pi_E & \text{ if $i=1$} \\
1\otimes \pi_E^{-1} & \text{ if $i=r+1$}\\
1 & \text{otherwise}.
\end{cases} \end{align*}
Using this, we easily obtain the following explicit description of the space $V(\mathcal{G}_1)\subset V_{\mathrm{cris}}(\mathcal{G})$.
\begin{proposition}\label{prp:unramified_vcris} When $E/F$ is unramified, $V(\mathcal{G}_1)\subset V_{\mathrm{cris}}(\mathcal{G})$ consists precisely of the elements $f=(f_i)$ such that: \begin{itemize}
\item $f_{0}\in V_{\iota_0}^{\mathrm{Fr}^{d_0}=1} = V(\mathcal O_E,\tau)$;
\item $f_i = \mathrm{Fr}^i((1\otimes\pi_E)a_0)$\text{, for $1\leq i\leq r$};
\item $f_i = \mathrm{Fr}^i(a_0)\text{, for $r+1\leq i\leq 2r-1$}$. \end{itemize} In particular, we have an isometry \begin{align*}
(V(\mathcal{G}_1),\langle\cdot,\cdot\rangle)&\xrightarrow{\simeq} (V(\mathcal O_E,\tau),\pi_E\langle \cdot,\cdot\rangle)\\ f&\mapsto f_0\nonumber \end{align*} of Hermitian $\mathcal O_E$-modules, where, for $x,y\in V(\mathcal O_E,\tau)$, $\langle x,y\rangle\in\mathcal O_E$ is the element such that $x\circ y\in\mathrm{End}(\mathcal O_E)$ is multiplication by $\langle x,y\rangle$. \end{proposition}
Let us now consider the case where $E/F$ is ramified. In this case, $\tau$ fixes every element in $\mathrm{Emb}(E_0)$ and so induces involutions $\tau:W_{\iota}\to W_{\iota}$ for each $\iota\in \mathrm{Emb}(E_0)$. Once again, as in the ramified case, from the explicit description of the $F$-crystal structure on $M_{\mathrm{cris}}$ under the identification~\eqref{eqn:mfM_cris}, we have $V_{\iota} = V(W_{\iota},\tau)$, and also \begin{align*}
\eta_i&=\begin{cases}
\frac{1\otimes\pi_E}{1\otimes\tau(\pi_E)} & \text{if $i=1$}\\
1& \text{ otherwise}.
\end{cases} \end{align*}
So we obtain: \begin{proposition}\label{prp:ramified_vcris} When $E/F$ is ramified, $V(\mathcal{G}_1)\subset V_{\mathrm{cris}}(\mathcal{G})$ consists precisely of the elements $f=(f_i)$ such that: \begin{itemize}
\item $f_0\in V_{\iota_0}$ satisfies $(1\otimes\pi_E)\mathrm{Fr}^{d_0}(f_0)=(1\otimes\tau(\pi_E))f_0$;
\item $f_i =(1\otimes \frac{\pi_E}{\tau(\pi_E)})\cdot\mathrm{Fr}^i(f_0)$\text{, for $i=1,\ldots,d_0-1$} \end{itemize} In particular, the map \[ W\otimes_{\mathbb Z_p}V(\mathcal{G}_1)\to V_{\mathrm{cris}}(\mathcal{G}) \] is an isomorphism. Moreover, if $\gamma\in W^\times_{\iota_0}$ is such that $(1\otimes\pi_E)\mathrm{Fr}^{d_0}(\gamma)=(1\otimes\tau(\pi_E))\gamma$, then we have an isometry \begin{equation*}
\big(V(\mathcal{G}_1),\langle\cdot,\cdot\rangle\big) \xrightarrow{\simeq} \big(V(\mathcal O_E,\tau),\gamma\tau(\gamma)\langle\cdot,\cdot\rangle\big) \end{equation*} of Hermitian $\mathcal O_E$-modules defined by $f\mapsto \gamma^{-1}f_0$. \end{proposition}
\subsection{Special endomorphisms with denominators} \label{ss:denominators}
Let $R$ be a $p$-adically complete $\mathcal O_{E}$-algebra. Fix an element $\mu\in E/\mathcal O_E$, and choose any representative $\tilde{\mu}\in E$ for it. If $\mu\neq 0$, the positive integer $r(\mu) = -\mathrm{ord}_{\mathfrak{p}}(\tilde{\mu})$ depends only on $\mu$; if $\mu = 0$, set $r(\mu) = 0$. Let $[\tilde{\mu}]\in \pi_E^{-r(\mu)}\mathrm{End}(\mathcal{G}_R)$ be the corresponding quasi-isogeny from $\mathcal{G}_R$ to itself. Set: \[ V_\mu(\mathcal{G}_R) = \bigl\{f\in V(\mathcal{G}_R)[\pi_E^{-1}]: f - [\tilde{\mu}]\in \mathrm{End}_{\mathcal O_F}(\mathcal{G}_R)\}. \] This does not depend on the choice of representative $\tilde{\mu}$.
\begin{proposition} \label{prop:lubin-tate special denom} Suppose that $E/F$ is ramified.
\begin{enumerate} \item If $\mu = 0$, then $V_\mu(\mathcal{G}_R) = V(\mathcal{G}_R)$. \item If $\mu\neq 0$, then $V_\mu(\mathcal{G}_{\mathcal O_K/\varpi})$ is non-empty if and only if \[ r(\mu)\leq \mathrm{ord}_E(\mathfrak{d}_{E/F})-1, \] in which case it is a torsor under translation by $V(\mathcal{G}_{\mathcal O_K/\varpi})$. Here, $\mathfrak{d}_{E/F}$ is the relative different of $E$ over $F$. \item If $p\neq 2$ and $\mu\neq 0$, then $V_\mu(\mathcal{G}_{\mathcal O_K/\varpi}) = \emptyset$. \end{enumerate} \end{proposition} \begin{proof} For simplicity, set $m = \mathrm{ord}_E(\mathfrak{d}_{E/F})$.
The first assertion is clear. Suppose therefore that $\mu\neq 0$, and that we have $f \in V_\mu(\mathcal{G}_{\mathcal O_K/\varpi})$. In the notation of Proposition~\ref{prp:ramified_vcris}, $f$ corresponds to a tuple $(f_i)$ with $f_i \in V_{\iota_i}[\pi_E^{-1}] = V(W_{\iota_i},\tau)[\pi_E^{-1}]$, where $f_i = \mathrm{Fr}^i(f_0)$, and where $f_0$ satisfies: \begin{equation}\label{eqn:f_0 condition} \mathrm{Fr}^{d_0}(f_0) = \frac{\tau(\pi_E)}{\pi_E}f_0. \end{equation} Here, we are identifying $\pi_E$ with the element $1\otimes\pi_E\in W_{\iota_0}$.
Moreover, by hypothesis, $f_0 - [\tilde{\mu}] \in \mathrm{End}(W_{\iota_0})$. Now, $[\tilde{\mu}]$ is invariant under the action of $\mathrm{Fr}^{d_0}$. Therefore,~\eqref{eqn:f_0 condition} implies: \[ \biggl(1-\frac{\tau(\pi_E)}{\pi_E} \biggr)\cdot f_0 = f_0 - \mathrm{Fr}^{d_0}(f_0) \in V_{\iota_0}. \] This implies \[ {m} - 1 = \mathrm{ord}_{E}\left( 1-\frac{\tau(\pi_E)}{\pi_E} \right) \geq r(\mu). \] Hence, we find that $V_\mu(\mathcal{G}_{\mathcal O_K/\varpi}) = \emptyset$ whenever $r(\mu) > m - 1.$
Assume now that $r(\mu) \leq m - 1$. To finish the proof of assertion (2), we have to show that we can always find $f_0$ as above satisfying~\eqref{eqn:f_0 condition} and with $f_0 - [\tilde{\mu}]\in\mathrm{End}(W_{\iota_0})$.
For this, choose any $\tilde{f}_0\in V_{\iota_0}[\pi_E^{-1}]$ such that $\tilde{f}_0 - [\tilde{\mu}]$ lies in $\mathrm{End}(W_{\iota_0})$. We now have \begin{align*} \frac{\pi_E}{\tau(\pi_E)}\mathrm{Fr}^{d_0}(\tilde{f}_0) - \tilde{f}_0& = \left(\frac{\pi_E}{\tau(\pi_E)} - 1\right)\mathrm{Fr}^{d_0}(\tilde{f}_0) + \mathrm{Fr}^{d_0}(\tilde{f}_0 - [\tilde{\mu}]) - (\tilde{f}_0 - [\tilde{\mu}]). \end{align*} Since $r(\mu)\leq m - 1$, we see that this belongs to $V_{\iota_0}$.
Now, notice that the endomorphism \[ V_{\iota_0} \xrightarrow{\frac{\pi_E}{\tau(\pi_E)}\cdot\mathrm{Fr}^{d_0} - \mathrm{id}} V_{\iota_0} \] is surjective: Indeed, mod $\pi_E$, this is immediate from the fact $\overline{\mathbb F}_p$ is algebraically closed. A simple lifting argument, using the completeness of $W_{\iota_0}$ now does the rest.
Therefore, there exists $f'_0\in V_{\iota_0}$ with \[ \frac{\pi_E}{\tau(\pi_E)}\mathrm{Fr}^{d_0}(f'_0) - f'_0 = \frac{\pi_E}{\tau(\pi_E)}\mathrm{Fr}^{d_0}(\tilde{f}_0) - \tilde{f}_0. \] It is an immediate check that we can now take $f_0 = \tilde{f}_0 - f'_0$.
Assertion (3) is clear from (2), since, when $p\neq 2$, $m - 1 = 0$.
\end{proof}
\subsection{Deformation theory} \label{ss:lubin-tate deformation}
Assume now that $K$ is generated over $\mathrm{Frac}(W)$ by the image of $\iota_0:E \to \mathrm{Frac}(W)^{\mathrm{alg}}$. Set $\varpi = \iota_0(\pi_E)$; this is a uniformizer for $K$. For any $k\in\mathbb Z_{\geq 1}$, set $\mathcal{G}_k = \mathcal{G}_{\mathcal O_{K}/\varpi^{k+1}}$, and for each $\mu\in \mathfrak{d}_{E/F}^{-1}/\mathcal O_E$, set \[ V_\mu(\mathcal{G}_k)\define V_\mu\bigl(\mathcal{G}_{\mathcal O_{K}/(\varpi^{k})}\bigr). \]
Let $M_{\mathrm{dR}}$ be the de Rham realization of $\mathcal{G}_{\mathcal O_K}$ as in~\eqref{eqn:mfM_dR}: It is a free $\mathcal O_K\otimes_{\mathbb Z_p}\mathcal O_E$-module of rank $1$ equipped with the $\mathcal O_K$-linear direct summand $\mathrm{Fil}^1M_{\mathrm{dR}}$, described in~\eqref{eqn:fil1_H}.
In the notation of \S~\ref{ss:lubin-tate_special_endomorphisms}, let \[ V_{\mathrm{dR}} \define V(M_{\mathrm{dR}},\tau)\subset \mathrm{End}_{\mathcal O_K}(M_{\mathrm{dR}}) \] be the space of $\tau$-semilinear endomorphisms of $M_{\mathrm{dR}}$.
Given $f_1,f_2\in V_{\mathrm{dR}}$, there is a canonical element $\langle f_1,f_2\rangle\in \mathcal O_K\otimes_{\mathbb Z_p}\mathcal O_E$ such that, for every $m\in M_{\mathrm{dR}}$, $(f_1\circ f_2)(m) = \langle f_1,f_2\rangle\cdot m$. Set $\breve{V}_{\mathrm{dR}} = V_{\mathrm{dR}}\otimes_{\mathcal O_E}\mathfrak{d}^{-1}_{E/F}$.
Similarly, for each $k\in\mathbb Z_{\geq 1}$, let $M_{\mathrm{dR},k} = M_{\mathrm{dR}}\otimes_{\mathcal O_K}\mathcal O_K/\varpi^{k}$ be the induced filtered free module over $\mathcal O_K/\varpi^{k}$, and let $V_{\mathrm{dR},k} = V(M_{\mathrm{dR},k},\tau)$. We have $V_{\mathrm{dR},k} = V_{\mathrm{dR}}\otimes_{\mathcal O_K}\mathcal O_K/\varpi^{k+1}$. Set $\breve{V}_{\mathrm{dR},k} = \breve{V}_{\mathrm{dR}}\otimes_{\mathcal O_K}\mathcal O_K/\varpi^k$.
For each $k\in \mathbb Z_{\geq 1}$, \[ \widetilde{\mathrm{Ob}}_k =\breve{V}_{\mathrm{dR}}\otimes_{\mathcal O_K\otimes_{\mathbb Z_p}\mathcal O_E,1\otimes\tau(\iota_0)}\mathcal O_K/\varpi^{k}. \] This is a rank $1$ free module over $\mathcal O_K/\varpi^k$.
Now set $\mathrm{Ob}_k = \varpi^{k-1}\cdot \widetilde{\mathrm{Ob}}_k$: This is a $1$-dimensional vector space over $\mathbb F_p^\mathrm{alg}$.
\begin{proposition} \label{prop:lifting end} For each $k\in\mathbb Z_{\geq 1}$, there is a canonical map
\[
\mathrm{ob}_{k+1}:V_\mu(\mathcal{G}_k)\to \mathrm{Ob}_{k+1}
\] with the following properties: \begin{enumerate}
\item An element $f\in V_\mu(\mathcal{G}_k)$ lifts to $V_\mu(\mathcal{G}_{k+1})$ if and only if $\mathrm{ob}_{k+1}(f) = 0$.
\item If $a\in \mathcal O_E$, then the diagram
\[ \xymatrix{
{ V_\mu(\mathcal{G}_{k}) } \ar[rr]^{\mathrm{ob}_{k+1}} \ar[d]_{f\mapsto a\cdot f} & & {\mathrm{Ob}_{k+1}} \ar[d]^{x\mapsto \iota_0(\tau(a))\cdot x} \\
{V_{a\cdot\mu}(\mathcal{G}_{k})} \ar[rr]_{\mathrm{ob}_{k+1}} && { \mathrm{Ob}_{k+1} } } \] commutes. \end{enumerate} \end{proposition} \begin{proof} For any $p$-adicaly complete $\mathcal O_E$-algebra $R$, an element $f\in V_\mu(\mathcal{G}_k)$ can be viewed as a $\tau$-semilinear homomorphism \[ f:\mathcal{G}_R \to \underline{\mathrm{Hom}}_{\mathcal O_E}(\mathfrak{d}_{E/F},\mathcal{G}_R) \] of formal $\mathcal O_E$-modules over $R$.
For each $k\in\mathbb Z_{\geq 1}$, $\mathcal O_K/\varpi^{k+1}\to \mathcal O_K/\varpi^{k}$ is a divided power thickening, and so every $f\in V_\mu(\mathcal{G}_k)$ has a canonical crystalline realization\footnote{Recall that we are using the contravariant Dieudonn\'e $F$-crystal.} \[ f_{k+1}:M_{\mathrm{dR},k+1}\otimes_{\mathcal O_E}\mathfrak{d}_{E/F}\to M_{\mathrm{dR},k+1}, \] which is a $\tau$-semilinear homomorphism of $\mathcal O_K/\varpi^{k+1}\otimes_{\mathbb Z_p}\mathcal O_E$-modules, and thus can be viewed as an element $f_{k+1}\in \breve{V}_{\mathrm{dR},k+1}$.
We claim that the map $\mathrm{ob}_{k+1}$ which takes $f\in V_\mu(\mathcal{G}_k)$ to the image of $f_{k+1}$ in $\mathrm{Ob}_{k+1}$ answers to the requirements of the lemma.
For this, set \[ a = f - [\tilde{\mu}]\in \mathrm{End}_{\mathcal O_F}(\mathcal{G}_k). \] It is easily checked that $f$ lifts to an element of $V_\mu(\mathcal{G}_{k+1})$ if and only if $a$ lifts to $\mathrm{End}_{\mathcal O_F}(\mathcal{G}_{k+1})$.
The crystalline realization of $a$ gives a homomorphism \[ a_{k+1}: M_{\mathrm{dR},k+1}\to M_{\mathrm{dR},k+1} \] of $\mathcal O_K/\varpi^{k+1}\otimes_{\mathbb Z_p}\mathcal O_F$-modules.
By Grothendieck-Messing theory~\cite{MessingBT}---which applies even when $p=2$, by the theory of Zink~\cite{ZinkWindows}, because $\mathcal{G}$ is connected---$a$ lifts to $\mathrm{End}(\mathcal{G}_{k+1})$ if and only if $a_{k+1}$ preserves the direct summand $\mathrm{Fil}^1M_{\mathrm{dR},k+1}\subset M_{\mathrm{dR},k+1}$.
We now use the explicit description of the filtration from~\eqref{eqn:fil1_H}. Since $\mathcal{E}_{\iota_0}(u) = -u + 1\otimes\pi_E$, we find that, in terms of the natural isotypic decomposition $M_{\mathrm{dR},k+1} = \oplus_{\iota}M_{\mathrm{dR},k+1,\iota}$, we have: \begin{align*} \mathrm{Fil}^1M_{\mathrm{dR},k+1,\iota} &= \begin{cases} \{x\in M_{\mathrm{dR},k+1,\iota}:(1\otimes \pi_E - \varpi\otimes 1)x=0\} & \text{ if $\iota=\iota_0$} \\ 0 & \text{ otherwise}. \end{cases} \end{align*} Here, we are using the fact that the cokernel of the map \[ M_{\mathrm{dR},\iota_0}\xrightarrow{1\otimes\pi_E - \varpi\otimes 1} M_{\mathrm{dR},\iota_0} \] of $\mathcal O_K$-modules is free of rank $1$ over $\mathcal O_K$, and hence the formation of its kernel is compatible with arbitrary base change.
For this, choose an $\mathcal O_{K,\tau(\iota_0)}$-module generator $u\in \breve{V}_{\mathrm{dR},k,\tau(\iota_0)}$, and let $c\in \mathcal O_{K,\tau(\iota_0)}/(\varpi^{k+1}\otimes 1)$ be such that $c\cdot u = f_{k+1}$. The proposition will follow once we show that $f$ lifts to $V_\mu(\mathcal{G}_{k+1})$ if and only if $c$ maps to $0$ under \[ \mathcal O_{K,\tau(\iota_0)}/(\varpi^{k+1}\otimes 1) = (\mathcal O_K/\varpi^{k+1})\otimes_{\tau(\iota_0),\mathcal O_{E_0}}\mathcal O_E\xrightarrow{1\otimes \tau(\iota_0)}\mathcal O_K/\varpi^{k+1}. \] Equivalently, if and only if $c\in (1\otimes \tau(\pi_E) - \varpi\otimes 1)\cdot \mathcal O_{K,\tau(\iota_0)}/(\varpi^{k+1}\otimes 1)$.
First, suppose that $E/F$ is unramified. In this case $V_{\mathrm{dR},k+1} = \breve{V}_{\mathrm{dR},k+1}$, we can take $\tilde{\mu} = 0$, and $f$ lifts precisely when we have \[ f_{k+1}(\mathrm{Fil}^1M_{\mathrm{dR},k+1})\subset \mathrm{Fil}^1M_{\mathrm{dR},k+1}. \] Now, we have \[ u(\mathrm{Fil}^1M_{\mathrm{dR},k+1}) = \overline{\mathrm{Fil}}^1M_{\mathrm{dR},k+1}\define \{x\in M_{\mathrm{dR},k+1,\tau(\iota_0)}: (1\otimes\tau(\pi_E) - \varpi\otimes 1)x =0\}. \] Therefore, we must have $c\cdot(\overline{\mathrm{Fil}}^1M_{\mathrm{dR},k+1}) = 0$, which is precisely equivalent to $c\in (1\otimes \tau(\pi_E) - \varpi\otimes 1)$.
Suppose now that $E/F$ is ramified, so that $\iota_0 = \tau(\iota_0)\in \mathrm{Emb}(E_0)$. In this case, the homomorphism \[ f_{k+1,\iota_0} - 1\otimes\tilde{\mu}: M_{\mathrm{dR},k+1,\iota_0}\otimes_{\mathcal O_E}\mathfrak{d}_{E/F}\to M_{\mathrm{dR},k+1} \] lifts to the endomorphism $a_{k+1,\iota_0}\in \mathrm{End}_{\mathcal O_F}(M_{\mathrm{dR},k+1,\iota_0})$. The proposition now reduces to the following easy observation: Suppose that $f_\infty\in \breve{V}_{\mathrm{dR}}$ is such that $f_\infty - 1\otimes\tilde{\mu}\in \mathrm{End}_{\mathcal O_F}(M_{\mathrm{dR}})$. Then we have \[ (f_\infty - 1\otimes\tilde{\mu})(\mathrm{Fil}^1M_{\mathrm{dR},k+1}) \subset \mathrm{Fil}^1M_{\mathrm{dR},k+1} \] if and only if $f_{\infty,\iota_0}\in (1\otimes \tau(\pi_E) - \varpi\otimes 1)\cdot \breve{V}_{\mathrm{dR},\iota_0}$. \end{proof}
Suppose that $k\leq e$, so that the surjection \[ W[u]/(u^k)\xrightarrow{u\mapsto\varpi}\mathcal O_{K}/(\varpi^{k}) \] is a divided power thickening (its kernel is generated by $p$). Upon evaluating the crystal $\mathbb{D}(\mathcal{G})$ on this thickening, we obtain a free $W[u]/(u^{k})\otimes_{\mathbb Z_p}\mathcal O_E$-module $\mathcal{M}_k$ of rank $1$. Using the Frobenius lift $\varphi$ on $W[u]/(u^{k})$ satisfying $\varphi(u) = u^p$ and the $F$-crystal structure on $\mathbb{D}(\mathcal{G})$, we also obtain a canonical $W[u]/(u^{k})\otimes_{\mathbb Z_p}\mathcal O_E$-linear map \[ \varphi_k:\varphi^*\mathcal{M}_k\to \mathcal{M}_k. \]
Let $\mathcal{V}_k = V(\mathcal{M}_k,\tau)$ be the space of $1\otimes\tau$-semilinear endomorphisms of the $W[u]/(u^{k})\otimes_{\mathbb Z_p}\mathcal O_E$-module $\mathcal{M}_k$. Conjugation by $\varphi_k$ induces an isomorphism \[ \varphi_k:\varphi^*\mathcal{V}_k[p^{-1}]\xrightarrow{\simeq}\mathcal{V}_k[p^{-1}]. \] Set $\breve{\mathcal{V}}_k = \mathcal{V}_k\otimes_{\mathcal O_E}\mathfrak{d}^{-1}_{E/F}$.
The $\mathcal O_E$-module structures on $\mathcal{M}_k$, $\mathcal{V}_k$ and $\breve{\mathcal{V}}_k$ equips them with isotypic decompositions \[ \mathcal{M}_k = \bigoplus_{\iota}\mathcal{M}_{k,\iota}\;;\;\mathcal{V}_k = \bigoplus_{\iota}\mathcal{V}_{k,\iota}\;;\; \breve{\mathcal{V}}_k = \bigoplus_{\iota}\breve{\mathcal{V}}_{k,\iota}. \]
\begin{lemma} \label{lem:dwork trick} \mbox{} \begin{enumerate}
\item For each $k\leq e$, the reduction map $\mathcal{V}_k\to V_{\mathrm{cris}}(\mathcal{G})$ induces an isomorphism $\mathcal{V}_k[p^{-1}]^{\varphi_k = 1}\xrightarrow{\simeq}V(\mathcal{G}_1)[p^{-1}]$.
\item Suppose that $k<e$ and that $f\in V_\mu(\mathcal{G}_k)$. Set
\[
\beta_{k+1} = \sum_{i=0}^{k}u^i\otimes\tau(\pi_E)^{k-i}\in W[u]/(u^{k+1})\otimes_{\tau(\iota_0),\mathcal O_{E_0}}\mathcal O_E.
\]
Then $\mathrm{ob}_{k+1}(f) = 0$ if and only if
\begin{align*}
\beta_{k+1} \cdot \tilde{f}_{k+1,\tau(\iota_0)}\in (1\otimes\tau(\pi_E)^{k+1})\cdot \breve{\mathcal{V}}_{k+1,\tau(\iota_0)}.
\end{align*} \end{enumerate} \end{lemma} \begin{proof} The first assertion is well-known, and is essentially Dwork's trick: Given an element $f_0\in V(\mathcal{G}_1)[p^{-1}]$, and any lift $\tilde{f}\in \mathcal{V}_k[p^{-1}]$, $\varphi_k^{k-1}(\tilde{f})\in \mathcal{V}_k[p^{-1}]$ will be the unique $\varphi_k$-invariant lift of $f_0$.
For the second assertion, by multiplying both sides of the condition by $(1\otimes \tau(\pi_E) - u\otimes 1)$, we see that it is equivalent to: \begin{equation} \label{eqn:reform condition} \tilde{f}_{k+1,\tau(\iota_0)}\in (1\otimes\tau(\pi_E) - u\otimes 1)\cdot \breve{\mathcal{V}}_{k+1,\tau(\iota_0)}. \end{equation}
The pre-image $\mathrm{Fil}^1 \mathcal{M}_{k+1}$ of $\mathrm{Fil}^1M_{\mathrm{dR},k+1}$ in $\mathcal{M}_{k+1}$ can be explicitly described using~\eqref{eqn:fil_mfM} and the canonical isomorphism \[ \varphi^*\mathfrak{M}\otimes_{\mathfrak{S}}W[u]/(u^{k+1}) \xrightarrow{\simeq}\mathcal{M}_{k+1} \] obtained from assertion (3) of Theorem~\ref{thm:kisin_p_divisible}. We find: \[ \mathrm{Fil}^1 \mathcal{M}_{k+1} = \frac{\mathcal{E}(u)\otimes 1}{1\otimes\pi_E - u\otimes 1}\cdot \mathcal{M}_{k+1,\iota_0} + \mathcal{E}(u)\cdot \mathcal{M}_{k+1} \]
Now, $W[u]/(u^{k})\to \mathcal O_K/\varpi^k$ is a divided power thickening, its kernel being the ideal $(p,u^{k-1})$, and $\tilde{f}_{k+1}$, by virtue of being characterized by its $\varphi_k$-invariance, is the evaluation of the crystalline realization of $f$ on this thickening. Therefore, $\mathrm{ob}_{k+1}(f)$ vanishes if and only if $\tilde{f}_{k+1} - 1\otimes\tilde{\mu}$ preserves the submodule $\mathrm{Fil}^1 \mathcal{M}_{k+1}\subset \mathcal{M}_{k+1}$.
If $E/F$ is unramified, then we can take $\tilde{\mu} = 0$, and the condition translates to: \[ \frac{\mathcal{E}(u)}{1\otimes\tau(\pi_E) - u\otimes 1}\cdot \tilde{f}_{k+1,\tau(\iota_0)}\in \mathcal{E}(u)\cdot \mathcal{V}_{k+1,\iota_0}. \] This is easily seen to be equivalent to~\eqref{eqn:reform condition}.
Now, suppose that $E/F$ is ramified. Fix a $W_{\iota_0}[u]/(u^{k+1})$-module generator $x\in \mathcal{M}_{k+1,\iota_0}$. Then the image of $\mathrm{Fil}^1 \mathcal{M}_{\mathrm{dR},k+1,\iota_0}$ under $\tilde{f}_{k+1,\iota_0} - 1\otimes\tilde{\mu}$ is generated by \begin{align*} \frac{\mathcal{E}(u)\otimes 1}{1\otimes\tau(\pi_E) - u\otimes 1}\cdot \tilde{f}_{k+1,\iota_0}(x) - \frac{\mathcal{E}(u)\otimes 1}{1\otimes\pi_E - u\otimes 1}\cdot (1\otimes\tilde{\mu})\cdot x.
\end{align*} This lies in $\mathrm{Fil}^1 \mathcal{M}_{k+1,\iota_0}$ if and only if we have \[ (1\otimes\pi_E - u\otimes 1)\tilde{f}_{k+1,\iota_0}(x) - (1\otimes\tau(\pi_E) - u\otimes 1)(1\otimes\tilde{\mu})\cdot x \in (1\otimes\tau(\pi_E) - u\otimes 1)\mathcal{M}_{k+1,\iota_0}. \] Equivalently, if and only if \[ (1\otimes(\pi_E - \tau(\pi_E)))\tilde{f}_{k+1,\iota_0}(x) \in (1\otimes\tau(\pi_E) - u\otimes 1)\mathcal{M}_{k+1,\iota_0}, \] which, as is once again easily verified, is equivalent to~\eqref{eqn:reform condition}. \end{proof}
\begin{lemma} \label{lem:non vanish} For any $k\in \mathbb Z_{\geq 1}$, suppose that $f\in V_\mu(\mathcal{G}_k)$ is such that $f$ does not lift to $V_\mu(\mathcal{G}_{k+1})$. Then, for every $a\in \mathcal O_E$ with $\mathrm{ord}_E(a) = 1$, $a\cdot f\in V_{a\cdot\mu}(\mathcal{G}_{k})$ lifts to $V_{a\cdot\mu}(\mathcal{G}_{k+1})$ but not to $V_{a\cdot\mu}(\mathcal{G}_{k+2})$. \end{lemma} \begin{proof} It is immediate from Proposition~\ref{prop:lifting end} that \[ \mathrm{ob}_{k+1}(a\cdot f) = \iota_0(\tau(a))\mathrm{ob}_{k+1}(f) \] vanishes. Therefore, $a\cdot f$ lifts to $V_{a\cdot\mu}(\mathcal{G}_{k+1})$. It remains to show that it does not lift to $V_{a\cdot \mu}(\mathcal{G}_{k+2})$; that is, we must show that $\mathrm{ob}_{k+2}(a\cdot f)\neq 0$.
Suppose first that $k < e$, where $e=e_E$ is the absolute ramification index of $E$, and suppose that $\mathrm{ob}_{k+2}(a\cdot f) = 0$. We then claim that $\mathrm{ob}_{k+1}(f) = 0$. Indeed, this follows easily from assertion (2) of Lemma~\ref{lem:dwork trick} and the identity \[ \beta_{k+2}\equiv (1\otimes\tau(\pi_E))\cdot\beta_{k+1}\pmod{u^{k+1}}. \] This shows the lemma when $k<e$.
Now suppose that $k\geq e$. Then the map $\mathcal O_K\to \mathcal O_K/\varpi^{k}$ is a divided power thickening. Therefore, $f\in V_\mu(\mathcal{G}_k)$ has a crystalline realization $f_{\mathrm{cris}}\in \breve{V}_{\mathrm{dR}}$ whose reduction mod $\varpi^{k+1}$ is the crystalline realization $f_{k+1}\in \breve{V}_{\mathrm{dR},k+1}$. Set \[ \widetilde{\mathrm{Ob}} \define \breve{V}_{\mathrm{dR}}\otimes_{\mathcal O_K\otimes_{\mathbb Z_p}\mathcal O_E,1\otimes\tau(\iota_0)}\mathcal O_K. \] Then, $\widetilde{\mathrm{Ob}}$ is a rank $1$ finite free $\mathcal O_K$-module, and, for each $i\in \mathbb Z_{\geq 1}$, we have $\widetilde{\mathrm{Ob}}_{i+1} = \widetilde{\mathrm{Ob}}\otimes_{\mathcal O_K}\mathcal O_K/\varpi^{i+1}$.
The hypothesis $\mathrm{ob}_{k+1}(f)\neq 0$ means that that the image of $f_{\mathrm{cris}}$ in $\widetilde{\mathrm{Ob}}$ does not lie in $\varpi^{k+1}\cdot\widetilde{\mathrm{Ob}}$. In turn, this implies that, the image of $(a\cdot f)_{\mathrm{cris}} = (1\otimes a)\cdot f_{\mathrm{cris}}$ in $\widetilde{\mathrm{Ob}}$ does not lie in $\varpi^{k+2}\cdot\widetilde{\mathrm{Ob}}$, and thus that $\mathrm{ob}_{k+2}(a\cdot f)\neq 0$. \end{proof}
Define a function \[ \mathrm{ord}_{E}:\;V(\mathcal{G}_1)_\mathbb Q \to \mathbb Z, \] given by two defining properties: \begin{itemize}
\item If $a\in E$, and $f\in V(\mathcal{G}_1)_\mathbb Q$, then
\[
\mathrm{ord}_{E}(a\cdot f) = \mathrm{ord}_{E}(a) + \mathrm{ord}_{E}(f).
\]
\item If $f\in V(\mathcal{G}_1)$ is an $\mathcal O_{E}$-module generator, then
\[
\mathrm{ord}_{E}(f) = \begin{cases}
1,&\text{ if $E$ is unramified over $F$};\\
\mathrm{ord}_{E}(\mathfrak{d}_{E/F}),&\text{ if $E$ is ramified over $F$}.
\end{cases}
\] \end{itemize}
\begin{lemma} \label{lem:first non vanish} If $f\in V_\mu(\mathcal{G}_1)$ is such that $\mathrm{ord}_E(f) = 1$, then $f$ does not lift to $V_\mu(\mathcal{G}_2)$. \end{lemma} \begin{proof} Let $f_\mathrm{cris}\in \breve{V}_{\mathrm{cris}}(\mathcal{G}) \define V_{\mathrm{cris}}(\mathcal{G})\otimes_{\mathcal O_E}\mathfrak{d}_{E/F}^{-1}$ be the crystalline realization of $f$. Observe that, by Propositions~\ref{prp:unramified_vcris} and~\ref{prp:ramified_vcris}, the hypothesis $\mathrm{ord}_E(f) = 1$ implies: \[ f_{\mathrm{cris},\tau(\iota_0)}\in (1\otimes\pi_E)\cdot \breve{V}_{\mathrm{cris}}(\mathcal{G})\backslash (1\otimes\pi_E^2)\cdot\breve{V}_{\mathrm{cris}}(\mathcal{G}). \]
Now, one only needs to observe that $\mathcal O_K/\varpi^2$ is either $W/p^2W$ or the ring $\mathbb F_p^\mathrm{alg}[u]/(u^2)$ of dual numbers, and that, in either case, there exists an isomorphism \[ \breve{V}_{\mathrm{cris}}(\mathcal{G})\otimes_W\mathcal O_K/\varpi^2\xrightarrow{\simeq}\breve{V}_{\mathrm{dR},2} \] of $\mathcal O_K/\varpi^2\otimes_{\mathbb Z_p}\mathcal O_E$-modules carrying $f_{\mathrm{cris}}$ to the crystalline lift $f_2\in \breve{V}_{\mathrm{dR},2}$.
This immediately implies $\mathrm{ob}_2(f)\neq 0$, and thus gives us the lemma.
\end{proof}
\begin{theorem} \label{thm:deformation special} Suppose that $f\in V_\mu(\mathcal{G}_1)$. Then $f$ lifts to $V_\mu(\mathcal{G}_k)$ if and only if $\mathrm{ord}_E(f)\geq k$. \end{theorem} \begin{proof} Immediate from Lemmas~\ref{lem:first non vanish} and~\ref{lem:non vanish}. \end{proof}
\section{CM Shimura varieties}\label{s:cm shimura}
Our goal in this section is to study a certain zero dimensional Shimura variety. The proof of the main Theorem~\ref{thm:arithmetic BKY} will proceed by embedding this zero dimensional variety into the higher dimensional GSpin Shimura varieties studied in the next section, and then computing the degree of the special divisors on the ambient Shimura variety along the resulting arithmetic curve.
As with the GSpin Shimura varieties themselves, the zero dimensional variety studied here does not admit any obvious moduli interpretation. Instead, we have to resort to abstract existence theorems, working consistently with the various realizations of the putative motives that live over the variety, and exploiting properties arising from the comparison isomorphisms among them. As a result, the exposition is necessarily somewhat technical.
Now for some notational conventions that will be in force throughout the section: We will fix a CM field $E$ with totally real subfield $F$. We will also take $\mathbb Q^{\mathrm{alg}}$ to be the algebraic closure in $\mathbb C$ of $\mathbb Q$ and write $\Gamma_{\mathbb Q}$ for the absolute Galois group $\mathrm{Gal}(\mathbb Q^{\mathrm{alg}}/\mathbb Q)$. For any algebraic torus $A$ over $\mathbb Q$, we will write $X^*(A)$ (resp. $X_*(A)$) for the $\Gamma_\mathbb Q$-module of characters (resp. cocharacters) of $A$.
If $\mu:\mathbb{G}_m\to A$ is a cocharacter with field of definition $F\subset \mathbb Q^{\mathrm{alg}}$, then its \emph{reflex norm} is given by \[ r(A,\mu): \mathrm{Res}_{F/\mathbb Q}\mathbb{G}_m\xrightarrow{\mathrm{Res}~\mu}\mathrm{Res}_{F/\mathbb Q}A\xrightarrow{\mathrm{Nm}_{F/\mathbb Q}}A. \] Here, $\mathrm{Res}_{F/\mathbb Q}A$ is the Weil restriction of the base change of $A$ over $F$, $\mathrm{Res}~\mu$ is the Weil restriction of $\mu$, and $\mathrm{Nm}_{F/\mathbb Q}$ is the usual norm map.
\subsection{A zero dimensional Shimura variety} \label{ss:zero dimensional}
For any $\mathbb Q$-algebra $R$, abbreviate $T_R =\mathrm{Res}_{R/\mathbb Q} \mathbb{G}_m$. Set \[ T_F^1 = \mathrm{ker} ( \mathrm{Nm}: T_F \to \mathbb{G}_m ), \] and $T = T_E / T_F^1.$ The natural diagonal embedding $\mathbb{G}_m\hookrightarrow T_E$ induces an embedding $\mathbb{G}_m\hookrightarrow T$. Set \[ T_{so} = \mathrm{ker} ( \mathrm{Nm}_{E/F} : T_E \to T_F). \] The rule $x\mapsto x/\overline{x}$ defines a surjection \begin{equation*}
\theta: T_E \to T_{so} , \end{equation*} inducing isomorphisms $T_E/T_F \xrightarrow{\simeq} T/\mathbb{G}_m \xrightarrow{\simeq}T_{so}$. The character groups of these tori can be described explicitly. If for any number field $M/\mathbb Q$ we set \[ \mathrm{Emb}(M)= \{ \mbox{Embeddings $M\hookrightarrow\mathbb Q^{\mathrm{alg}}$} \}, \] then we have natural identifications \begin{align} \label{character groups} X^*(T_E) &= \bigoplus_{\iota\in\mathrm{Emb}(E)}\mathbb Z\cdot[\iota], \\\nonumber X^*(T) &= \biggl\{\sum_{\iota}a_{\iota}[\iota] \in X^*(T_E): a_{\iota}+a_{\overline{\iota}}\text{ is independent of $\iota$}\biggr\}, \\ X^*(T_{so}) &= \biggl\{\sum_{\iota}a_{\iota}[\iota] \in X^*(T_E): a_{\iota}+a_{\overline{\iota}}=0\text{, for all $\iota$}\biggr\} \nonumber \end{align} of $\Gamma_\mathbb Q$-modules.
Identify $\mathrm{Emb}(E)$ with the set of embeddings $E \hookrightarrow \mathbb C$, and enumerate the real embeddings $F\hookrightarrow \mathbb R$ as $\iota_0,\ldots, \iota_{d-1}$. We will declare $\iota_0$ a distinguished embedding, and we will fix, once and for all, an extension $\iota_0\in\mathrm{Emb}(E)$ of this embedding.
Define a cocharacter $\mu_0\in X_*(T_E)$ by the formula \begin{equation}\label{eqn:mu_0}
\langle \mu_0, [\iota]\rangle = \begin{cases} 1 & \text{ if $\iota=\iota_0$} \\ 0& \text{ otherwise.} \end{cases} \end{equation} We will also denote the induced cocharacter of $T$ by $\mu_0$. The field of definition for $\mu_0$ is $\iota_0(E)\subset\mathbb Q^{\mathrm{alg}}\subset\mathbb C$. If no confusion can arise, we will identify $E$ with this subfield of $\mathbb C$.
In our situation, the reflex norm $r(T_E,\mu_0)$ simplifies considerably: It is simply the identity $T_E\xrightarrow{\mathrm{id}}T_E$. In particular, the reflex norm $r(T,\mu_0):T_E\to T$ associated with $\mu_0\in X_*(T)$ is just the natural surjection from $T_E$ to $T$.
Let $r(T,\mu_0)_{\ell}:(\mathbb Q_\ell\otimes_\mathbb Q E)^\times \to T(\mathbb Q_\ell)$ be the evaluation of this map on $\mathbb Q_\ell$. We will write $r(T,\mu_0)_{\lambda}:E_{\lambda}^\times\to T(\mathbb Q_\ell)$ for its restriction to $E_{\lambda}^\times$. We also have the ad\'elic version \begin{align*}
r(T,\mu_0)(\mathbb A_f) & : \mathbb A_{f,E}^\times\to T(\mathbb A_f). \end{align*}
Fix a compact open subgroup $K\subset T(\mathbb A_f)$. To the triple $(T,\mu_0,K)$ we can attach a finite \'etale algebraic stack $Y_{K}$ over $E$, which we will call a \emph{CM Shimura variety}. This is constructed as follows. Consider the composition \[ \mathbb A_{f,E}^\times\xrightarrow{r(T,\mu_0)(\mathbb A_f)} T(\mathbb A_f)\to T(\mathbb A_f)/T(\mathbb Q)K. \] This factors via the global reciprocity map through the abelianization of the Galois group $\Gamma_{E}=\mathrm{Gal}(\mathbb Q^{\mathrm{alg}}/E)$. Therefore, we obtain a homomorphism \begin{equation*}
r_K(T,\mu_0):\Gamma_{E}\to T(\mathbb A_f)/T(\mathbb Q)K. \end{equation*}
Now, suppose that $K$ is neat: This is equivalent to requiring that $K\cap T(\mathbb Q)$ be torsion-free.\footnote{For instance, one can take $K$ to be the image of the elements in $(\mathcal O_E\otimes\widehat{\mathbb Z})^\times$ that are congruent to $1$ mod $3$.} Then $Y_K$ will be a finite \'etale \emph{scheme} over $E$ corresponding to the $\Gamma_{E}$-set \[ Y_K(\mathbb Q^{\mathrm{alg}}) = T(\mathbb Q)\backslash\{\mu_0\}\times T(\mathbb A_f)/K, \] equipped with the Galois action obtained from the map $r_K(T,\mu_0)$. More precisely, there is a natural action of $T(\mathbb A_f)/T(\mathbb Q)K$ on $Y_K(\mathbb Q^{\mathrm{alg}})$ obtained via the right multiplication on $T(\mathbb A_f)$ on itself. The action of $\Gamma_E$ on $Y_K(\mathbb Q^{\mathrm{alg}})$ is the one induced from that of $T(\mathbb A_f)/T(\mathbb Q)K$ via the map $r_K(T,\mu_0)$. For general $K$, choose a neat compact open subgroup $K'\subset K$. Then $Y_K$ will be the stack quotient of $Y_{K'}$ by the action of the finite group $K/K'$.
\subsection{The integral model}\label{ss:integral_model_Y}
We will now choose a particular maximal compact $K_0\subset T(\mathbb A_f)$. Using the identification \[ T_{so}(\mathbb Q_p) = \frac{(\mathbb Q_p\otimes_{\mathbb Q}E)^\times}{(\mathbb Q_p\otimes_{\mathbb Q}F)^\times}, \] we define $K_{0,p,so}$ to be the subgroup \begin{equation}\label{eqn:K0p so} K_{0,p,so} = \frac{(\mathbb Z_p\otimes_{\mathbb Z}\mathcal O_E)^\times}{(\mathbb Z_p\otimes_{\mathbb Z}\mathcal O_F)^\times}\subset T_{so}(\mathbb Q_p). \end{equation} This will be the image of $K_0$ in $T_{so}(\mathbb Q_p)$.
The long exact sequence of $\Gamma_{\mathbb Q_p}$-cohomology associated with the short exact sequence \[ 1\to T_F^1 \to T_E\to T\to 1 \]
gives us a short exact sequence \begin{align}\label{Eqn:maximal_compact} 0\to\frac{(\mathbb Q_p\otimes_{\mathbb Q}E)^\times}{(\mathbb Q_p\otimes_{\mathbb Q}F)^{\mathrm{Nm}=1}}\to T(\mathbb Q_p) \to \bigoplus_{\mathfrak{p}\vert p}\mathbb Q_p^\times/\mathrm{Nm}(F_{\mathfrak{p}}^\times)\to 0. \end{align} Here, $\mathfrak{p}$ varies over the $p$-adic places of $F$. Now define $K_{0,p}\subset T(\mathbb Q_p)$ to be the largest subgroup mapping to $K_{0,p,so}\subset T_{so}(\mathbb Q_p)$, and to $\bigoplus_{\mathfrak{p}\vert p}\mathbb Z_p^\times/\mathrm{Nm}(\mathcal O_{F_{\mathfrak{p}}}^\times)$ under~\eqref{Eqn:maximal_compact}. It sits in a short exact sequence \[ 1\to\mathbb Z_p^\times\to K_{0,p}\to K_{0,p,so}\to 1. \] Finally, set $K_0=\prod_p K_{0,p}$.
For any compact open subgroup $K\subset T(\mathbb A_f)$, let $\mathcal{Y}_K$ be the normalization of $\mathrm{Spec}(\mathcal O_E)$ in $Y_K$ (see Definition~\ref{defn:normalization} below). Set $Y_0=Y_{K_0}$, and $\mathcal{Y}_0 = \mathcal{Y}_{K_0}$.
\begin{proposition}\label{prop:Integral_Model_0_cycle} Let $K\subset T(\mathbb A_f)$ be a compact open subgroup. Suppose that $p$ is a prime such that $K_p = K_{0,p}$, and set $\mathcal O_{E,p} = \mathcal O_E\otimes_{\mathbb Z}\mathbb Z_p$. Then $\mathcal{Y}_{K}\otimes_{\mathcal O_E}\mathcal O_{E,p}$ is finite \'etale over $\mathcal O_{E,p}$. In particular, $\mathcal{Y}_0$ is a finite \'etale algebraic stack over $\mathcal O_E$. \end{proposition} \begin{proof}
By construction, $\mathcal{Y}_K$ is normal and finite flat over $\mathcal O_E$. We need to study the ramification of $Y_K$. This is easily done from its explicit description.
Fix a prime $\mathfrak{q}\subset\mathcal O_E$ lying above a rational prime $p$. Fix an algebraic closure $\mathbb Q_p^{\mathrm{alg}}$ of $\mathbb Q_p$ and an embedding $\eta_p:\mathbb Q^{\mathrm{alg}}\hookrightarrow\mathbb Q_p^{\mathrm{alg}}$ such that the closure of $\eta_p(E)\subset\mathbb Q_p^{\mathrm{alg}}$ is $E_{\mathfrak{q}}$. This allows us to identify $\Gamma_{E_{\mathfrak{q}}} = \mathrm{Gal}(\mathbb Q_p^{\mathrm{alg}}/E_{\mathfrak{q}})$ with a subgroup of $\Gamma_E$.
Set \begin{align*}
Y_{K,\mathfrak{q}} &= Y_K\times_{\mathrm{Spec}(E)}\mathrm{Spec}(E_{\mathfrak{q}}) \\
\mathcal{Y}_{K,\mathfrak{q}} & = \mathcal{Y}_K\times_{\mathrm{Spec}(\mathcal O_E)}\mathrm{Spec}(\mathcal O_{E,\mathfrak{q}}), \end{align*} so that $Y_{K,\mathfrak{q}}$ is a finite \'etale algebraic stack over $E_{\mathfrak{q}}$. Assume that $K$ is neat. Then $Y_{K,\mathfrak{q}}$ is the finite \'etale \emph{scheme} over $E_{\mathfrak{q}}$ associated, via the local reciprocity map, with the composition \[ E_{\mathfrak{q}}^\times\xrightarrow{r(T,\mu_0)_{\mathfrak{q}}}T(\mathbb Q_p)\to T(\mathbb A_f)/T(\mathbb Q)K. \] Therefore, the ramification of $Y_{K,\mathfrak{p}}$ over $E_{\mathfrak{p}}$ is controlled by \[ \ker(\mathcal O_{E,\mathfrak{q}}^\times\xrightarrow{r(T,\mu_0)_{\mathfrak{q}}}T(\mathbb Q_p)/K_p). \] More precisely, the completed \'etale local ring of $\mathcal{Y}_{K,\mathfrak{q}}$ at any $\mathbb F^{\mathrm{alg}}_p$-valued point will be the ring of integers in the finite abelian extension of the compositum $W(\mathbb F^{\mathrm{alg}}_p)\mathcal O_{E,\mathfrak{q}}$ classified by the above compact open subgroup of $\mathcal O_{E,\mathfrak{q}}^\times$.
From this, we conclude that to show that $\mathcal{Y}_{K,\mathfrak{q}}$ is finite \'etale over $\mathcal O_{E,\mathfrak{q}}$, it is enough to show that \[ r(T,\mu_0)_{\mathfrak{q}}(\mathcal O_{E,\mathfrak{q}}^\times)\subset K_{0,p}. \] From the definition of $r(T,\mu_0)$, this subgroup is exactly the image of $\mathcal O_{E,\mathfrak{q}}^\times$ under the map $T_E(\mathbb Q_p)\to T(\mathbb Q_p)$. It follows easily from the definition of $K_{0,p}$ that it must contain this image. \end{proof}
\subsection{Automorphic sheaves I} \label{ss:sheaves_i}
Fix a compact open subgroup $K\subset T(\mathbb A_f)$. We will now construct some natural sheaves on $\mathcal{Y}_K$.
First, suppose that we have an algebraic $\mathbb Q$-representation $N$ of $T$. Then we obtain a local system of $\mathbb Q$-vector spaces \[ T(\mathbb Q)\backslash\left(\{\mu_0\}\times N \times T(\mathbb A_f)/K\right) \] over $\mathcal{Y}_K(\mathbb C)$. If we fix a $K$-stable lattice $N_{\widehat{\mathbb Z}}\subset N_{\mathbb A_f}$, we get a local system $\bm{N}_B$ of $\mathbb Z$-vector spaces underlying this local system, where the fiber of $\bm{N}_B$ over a point $[(\mu_0,t)]$ of $\mathcal{Y}_K(\mathbb A_f)$ is $tN_{\widehat{\mathbb Z}}\cap N$.
We can also associate with $N$ the vector bundle \[ \bm{N}_{\mathrm{dR},\mathbb C} = T(\mathbb Q)\backslash\left(\{\mu_0\}\times N_{\mathbb C} \times T(\mathbb A_f)/K\right) \] over $\mathcal{Y}_K(\mathbb C)$. Here, we have equipped $N_{\mathbb C}$ with its topological structure as a $\mathbb C$-vector space. There is a canonical isomorphism \begin{equation*}
\mathcal O_{\mathcal{Y}_K(\mathbb C)}\otimes_{\mathbb Z}\bm{N}_B\xrightarrow{\simeq} \bm{N}_{\mathrm{dR},\mathbb C} \end{equation*} of vector bundles over $\mathcal{Y}_K(\mathbb C)$.
Let $v_0$ be the infinite place of $E$ underlying the distinguished embedding $\iota_0$. Via the identification $E_{v_0} = \mathbb C$ obtained from $\iota_0$, we have a homomorphism \begin{equation}\label{eqn:hodge structure zero} \mathbb C^\times = E_{v_0}^\times\xrightarrow{r(T,\mu_0)_{v_0}} T(\mathbb R) \end{equation} where $r(T,\mu_0)_{v_0}$ is as defined in~\S\ref{ss:zero dimensional}. This homomorphism equips the $T_\mathbb C$-representation $N_{\mathbb C}$ with a Hodge structure whose Hodge filtration is split by the cocharacter $\mu_0^{-1}$. Since this Hodge structure is $T(\mathbb Q)$-equivariant, it descends to one on the vector bundle $\bm{N}_{\mathrm{dR},\mathbb C}$, and so we obtain a Hodge filtration $\mathrm{Fil}^\bullet\bm{N}_{\mathrm{dR},\mathbb C}$ on $\bm{N}_{\mathrm{dR},\mathbb C}$. Therefore, the tuple \[ (\bm{N}_B,\bm{N}_{\mathrm{dR},\mathbb C},\mathrm{Fil}^\bullet\bm{N}_{\mathrm{dR},\mathbb C},\mathcal O_{\mathcal{Y}_K(\mathbb C)}\otimes_{\mathbb Z}\bm{N}_B\xrightarrow{\simeq} \bm{N}_{\mathrm{dR},\mathbb C}) \] corresponds to a variation of $\mathbb Z$-Hodge structures on $\mathcal{Y}_K(\mathbb C)$, which we denote by $\bm{N}_{\mathrm{Hdg}}$. The assignment of $\bm{N}_{\mathrm{Hdg}}$ to the pair $(N,N_{\widehat{\mathbb Z}})$ is clearly functorial.
Fix a prime $\ell$. Suppose that $K'_\ell\subset K_{\ell}$ is a compact open subgroup, and set $K'=K'_\ell K^\ell\subset T(\mathbb A_f)$. Then the proof of Proposition \ref{prop:Integral_Model_0_cycle} shows that the map of integral models $\mathcal{Y}_{K'}\to \mathcal{Y}_K$ is finite \'etale over $\mathcal O_E[\ell^{-1}]$. In particular, the pro-finite $\mathcal O_E$-scheme \[ \mathcal{Y}_\ell[\ell^{-1}] = \varprojlim_{K'_\ell\subset K_{\ell}}\mathcal{Y}_{K'}[\ell^{-1}] \] is a pro-finite Galois cover of $\mathcal{Y}_K[\ell^{-1}]$ with Galois group $K_\ell$. Therefore, we obtain a functor from continuous $\ell$-adic representations of $K_\ell$ to locally constant $\ell$-adic sheaves on $\mathcal{Y}_K[\ell^{-1}]$ via the contraction product \begin{equation*}
N_{\ell}\mapsto \bm{N}_{\ell} = (\mathcal{Y}_{\ell}[\ell^{-1}]\times N_{\ell})/K_\ell. \end{equation*}
The next result is easily checked from the definitions. \begin{proposition} \label{prop:betti etale zero} Suppose that $N$ is a $\mathbb Q$-linear algebraic representation of $T$ and that $N_{\widehat{\mathbb Z}}\subset N_{\mathbb A_f}$ is a $K$-stable $\widehat{\mathbb Z}$-lattice. Then, for each $\ell$, there is a canonical isomorphism \[ \mathbb Z_\ell\otimes\bm{N}_B \xrightarrow{\simeq}\bm{N}_{\mathbb Z_\ell}\vert_{\mathcal{Y}_K(\mathbb C)} \] of $\ell$-adic local systems on $\mathcal{Y}_K(\mathbb C)$. \end{proposition}
\subsection{Abelian schemes} \label{ss:abelian schemes}
The norm character $\mathrm{Nm}_{E/\mathbb Q}:T_E \to \mathbb Q$ factors through a homomorphism $\mathrm{Nm}:T \to \mathbb{G}_m$. Suppose that $H$ is a faithful $\mathbb Q$-representation of $T$ that admits a $T$-invariant symplectic pairing \[ \psi: H\times H \to \mathbb Q(\mathrm{Nm}) \] such that the Hodge structure on $H$ arising from the map~\eqref{eqn:hodge structure zero} has weights $(0,-1),(-1,0)$ and is polarized by $\psi$.
For any $K$-stable lattice $H_{\widehat{\mathbb Z}}\subset H_{\mathbb A_f}$ on which $\psi$ takes values in $\widehat{\mathbb Z}$, the associated variation of $\mathbb Z$-Hodge structures $\bm{H}_{\mathrm{Hdg}}$ over $\mathcal{Y}_K(\mathbb C)$ is the homology of a polarized abelian scheme over $\mathcal{Y}_{K,\mathbb C}$. This abelian scheme is associated with a map of Shimura varieties \[ \mathcal{Y}_{K,\mathbb C} \to \mathcal{X}_{r,m,\mathbb C}, \] where $2r = \dim_\mathbb Q( H)$, $m^2$ is the discriminant of $\psi$ restricted to $H_{\widehat{\mathbb Z}}$, and $\mathcal{X}_{r,m}$ is the Siegel modular scheme over $\mathbb Z$ parameterizing polarized abelian varieties of dimension $r$ and degree $m$. By the theory of canonical models, this descends to a map $Y_K \to \mathcal{X}_{r,m,E}$ over $E$, and so we obtain a polarized abelian scheme $\mathcal{A}_{H,\mathbb Q} \to Y_K$.
\begin{proposition} \label{prop:abelian schemes realization zero} The abelian scheme $\mathcal{A}_{H,\mathbb Q}$ extends canonically to an abelian scheme $\mathcal{A}_H \to \mathcal{Y}_K$. Moreover, for any prime $\ell$, the $\ell$-adic Tate module of $\mathcal{A}_H$, viewed as an $\ell$-adic sheaf over $\mathcal{Y}_K[\ell^{-1}]$ is canonically isomorphic to $\bm{H}_{\mathbb Z_\ell}$.
\end{proposition} \begin{proof} For any prime $\ell$, it is immediate from Proposition~\ref{prop:betti etale zero} and the theory of canonical models that the $\ell$-adic Tate module of $\mathcal{A}_{H,\mathbb Q}$ is canonically isomorphic to the restriction of $\bm{H}_{\mathbb Z_\ell}$ to $Y_K$. Since this sheaf extends to a lisse sheaf over $\mathcal{Y}_K[\ell^{-1}]$, it follows from the N\'eron-Ogg-Shafarevich criterion for good reduction that $\mathcal{A}_{H,\mathbb Q}$ extends to an abelian scheme over $\mathcal{Y}_K[\ell^{-1}]$ for each prime $\ell$, and hence to an abelian scheme $\mathcal{A}_H\to \mathcal{Y}_K$, whose $\ell$-adic Tate module is canonically isomorphic to $\bm{H}_{\mathbb Z_\ell}$. \end{proof}
We will now give an explicit construction of such a symplectic representation. Write $\mathrm{CM}(E)$ for the set of CM types $\Phi$ for $E$; these are the subsets $\Phi\subset\mathrm{Emb}(E)$ satisfying \[ \Phi\sqcup\overline{\Phi} = \mathrm{Emb}(E). \] The \emph{total reflex algebra} of $E$ is the \'etale $\mathbb Q$-algebra $E^\sharp$ equipped with an isomorphism \[ \mathrm{Hom}(E^\sharp , \mathbb Q^\mathrm{alg}) \xrightarrow{\simeq} \mathrm{CM}(E) \] as sets with $\Gamma_\mathbb Q$-actions. It is easily checked that $E^\sharp$ is a product of CM fields, and is in particular equipped with a canonical complex conjugation $x\mapsto \overline{x}$, corresponding to the involution $\Phi\mapsto \overline{\Phi}$ on $\mathrm{CM}(E)$.
There is a \emph{total reflex norm} $\mathrm{Nm}^\sharp:T_E \to T_{E^\sharp}$, which factors through an embedding \begin{equation}\label{factor reflex} \mathrm{Nm}^\sharp:T\hookrightarrow T_{E^\sharp}. \end{equation} This map can be described explicitly on the level of the associated character groups. Using the natural identification of $\Gamma_{\mathbb Q}$-modules of (\ref{character groups}), along with \[ X^*(T_{E^\sharp}) = \bigoplus_{\Phi\in\mathrm{CM}(E)}\mathbb Z\cdot[\Phi], \] it is given by \[ X^*(\mathrm{Nm}^\sharp):[\Phi] \mapsto \sum_{\iota\in\Phi}[\iota]. \]
Write $H^\sharp$ for $E^\sharp$ viewed as a representation of $T_{E^\sharp}$ via multiplication. Via the map $\mathrm{Nm}^\sharp:T\to T_{E^\sharp}$ of (\ref{factor reflex}), we can consider $H^\sharp$ also as a representation of $T$.
For $\Phi\in \mathrm{CM}(E)$, write $\iota_{\Phi}$ for the corresponding element in \[ \mathrm{Hom}(E^\sharp,\mathbb Q^\mathrm{alg}) = \mathrm{Hom}(E^\sharp,\mathbb C). \] Fix a non-zero element $\xi\in E^\sharp$ such that, for any $\Phi\in \mathrm{CM}(E)$ with $\iota_0\in \Phi$, we have $\iota_{\Phi}(\xi)\in \mathbb R_{>0}\cdot i$.
The following proposition is an easy check from the definitions.
\begin{proposition}\label{prop:h sharp repn} The pairing $(x,y)\mapsto \mathrm{Tr}_{E^\sharp/\mathbb Q}(\xi x\overline{y})$ gives rise to a $T$-equivariant symplectic pairing \[ \psi^\sharp:H^\sharp \times H^\sharp \to \mathbb Q(\mathrm{Nm}) \] such that the Hodge structure on $H^\sharp$ arising from the map~\eqref{eqn:hodge structure zero} has weights $(0,-1),(-1,0)$ and is polarized by $\psi^\sharp$. \end{proposition}
\subsection{Automorphic sheaves II} \label{ss:sheaves_ii}
Recall from \S\ref{ss:sheaves_i} that we have a canonical functor $N\mapsto (\bm{N}_{\mathrm{dR},\mathbb C},\mathrm{Fil}^\bullet\bm{N}_{\mathrm{dR},\mathbb C})$ from algebraic $\mathbb Q$-representations $N$ of $T$ to filtered vector bundles over $\mathcal{Y}_K(\mathbb C)$. We can interpret this a bit differently. Given any $E$-linear algebraic representation $M$ of $E\otimes_\mathbb Q T$, the constant vector bundle \[ \{\mu_0\}\times M_\mathbb C \times T(\mathbb A_f)/K \] over $\{\mu_0\}\times T(\mathbb A_f)/K$ is $T(\mathbb Q)$-equivariant, and so descends to a vector bundle $\bm{M}_{\mathrm{dR},\mathbb C}$ over $\mathcal{Y}_K(\mathbb C)$. When restricted to a $\mathbb Q$-linear representation $N$ equipped with the filration $\mathrm{Fil}^\bullet N_E$ split by the cocharacter $\mu_0$, this functor recovers the filtered vector bundle associated with $N$.
\begin{proposition}\label{prop:zero dim derham} For any $E$-linear representation $M$ of $E\otimes_\mathbb Q T$, the vector bundle $\bm{M}_{\mathrm{dR},\mathbb C}$ has a canonical and functorial descent to a vector bundle $\bm{M}_{\mathrm{dR},\mathbb Q}$ over $Y_K$. In particular, for any $\mathbb Q$-linear representation $N$, the filtered vector bundle $(\bm{N}_{\mathrm{dR},\mathbb C},\mathrm{Fil}^\bullet\bm{N}_{\mathrm{dR},\mathbb C})$ has a functorial descent to a filtered vector bundle $(\bm{N}_{\mathrm{dR},\mathbb Q},\mathrm{Fil}^\bullet\bm{N}_{\mathrm{dR},\mathbb Q})$ over $Y_K$. \end{proposition} \begin{proof} This is essentially a consequence of Deligne's theorem showing that all Hodge cycles on abelian varieties are absolutely Hodge~\cite[Ch. I]{dmos}; see also~\cite[\S 3.15]{Harris1985-tv}. We sketch a proof here.
Take $H$ to be a faithful $\mathbb Q$-representation of $T$ as in \S~\ref{ss:abelian schemes}, so that the associated variation of Hodge structures $\bm{H}_{\mathrm{Hdg}}$ (associated with some choice of $K$-invariant lattice in $H_{|A_f}$) corresponds to a canonical abelian scheme $\mathcal{A}_H$ over $\mathcal{Y}_K$. We can always find such a representation; see Proposition~\ref{prop:h sharp repn}. The relative first de Rham homology of $\mathcal{A}_{H}$ over $Y_K$ gives a canonical descent of $\bm{H}_{\mathrm{dR},C}$ to a vector bundle $\bm{H}_{\mathrm{dR},\mathbb Q}$ over $Y_K$.
Let $H^\otimes$ (resp. $\bm{H}^\otimes_{\mathrm{dR},\mathbb Q}$) be the direct sum of tensor powers of $H$ (resp. $\bm{H}^\otimes_{\mathrm{dR},\mathbb Q}$) and its dual, and let $\{t_{\beta}\}\subset H^\otimes$ be a collection of tensors whose pointwise stabilizer in $\mathrm{GL}(H)$ is $T$. By the functoriality of the construction $N\mapsto \bm{N}_{\mathrm{Hdg}}$, these tensors give rise to Hodge tensors \[ \{\bm{t}_{\beta,\mathrm{dR},\mathbb C}\}\subset H^0(\mathcal{Y}_{K,\mathbb C},\mathrm{Fil}^0\bm{H}_{\mathrm{dR},\mathbb C}^\otimes). \] By Deligne's theorem, these tensors descend to a collection \[ \{\bm{t}_{\beta,\mathrm{dR},\mathbb Q}\}\subset H^0(\mathcal{Y}_{K,\mathbb Q},\mathrm{Fil}^0\bm{H}_{\mathrm{dR},\mathbb Q}^\otimes). \] See~\cite[Corollary 2.2.2]{KisinJAMS}.
Now, the functor on $Y_K$-schemes carrying a $Y_K$-scheme $S$ to the set of isomorphisms \[ \eta:\mathcal O_S\otimes_\mathbb Q H \xrightarrow{\simeq}\bm{H}_{\mathrm{dR},\mathbb Q}\vert_S \] satisfying $\eta(1\otimes t_\beta) = t_{\beta,\mathrm{dR},\mathbb Q}$, for all indices $\beta$ is represented by a $T$-torsor $\mathcal{P}_{T,\mathbb Q}\to Y_K$. Moreover, this $T$-torsor is canonical and does not depend on the choice of representation $H$. This can be seen by comparing the torsors obtained from $H$ and a different representation $H'$ with the one associated with the direct sum $H\oplus H'$; see the argument in~\cite[p. 177]{Harris1985-tv}.
The construction of the functor $M\mapsto \bm{M}_{\mathrm{dR},\mathbb Q}$ is now simple: We will take $\bm{M}_{\mathrm{dR},\mathbb Q}$ to be the contraction product \[ \bm{M}_{\mathrm{dR},\mathbb Q} \define (\mathcal{P}_{T,\mathbb Q}\times M)/T, \] where $T$ acts diagonally on $\mathcal{P}_{T,\mathbb Q}\times M$.
\end{proof}
We now want to extend this construction over the integral model $\mathcal{Y}_K$. We will do this using integral $p$-adic Hodge theory. Let $\mathfrak{q}\subset \mathcal O_E$ be a prime lying above a rational prime $p$. Fix an algebraic closure $\mathbb F_{\mathfrak{q}}^{\mathrm{alg}}$ for $\mathbb F_{\mathfrak{q}}$ and also an algebraic closure $\mathrm{Frac}(W)^{\mathrm{alg}}$ of the fraction field $\mathrm{Frac}(W)$ of $W = W(\mathbb F_{\mathfrak{q}}^{\mathrm{alg}})$. Choose an embedding $\mathbb Q^{\mathrm{alg}} \hookrightarrow \mathrm{Frac}(W)^{\mathrm{alg}}$ inducing the place $\mathfrak{q}$ on $E = \iota_0(E)$.
Let $\mathcal O_y$ be the completion of $\mathcal{Y}_K$ at an $\mathbb F^{\mathrm{alg}}_{\mathfrak{q}}$-valued point $y$. Write $W_{\mathfrak{q}}$ for the ring of integers in the extension of $\mathrm{Frac}(W)$ generated by the image of $E_{\mathfrak{q}}$. Let \[ I_{\mathfrak{q}} = \mathrm{Gal}(\mathrm{Frac}(W)^{\mathrm{alg}}/\mathrm{Frac}(W_{\mathfrak{q}})) \]
be the absolute Galois group of $\mathrm{Frac}(W_{\mathfrak{q}})$. If $\mathbb Q^{\mathrm{alg}}_p\subset\mathrm{Frac}(W)^{\mathrm{alg}}$ is the algebraic closure of $\mathbb Q_p$, then $I_{\mathfrak{q}}$ is identified with the inertia subgroup of $\Gamma_{E_{\mathfrak{q}}} = \mathrm{Gal}(\mathbb Q_p^{\mathrm{alg}}/E_{\mathfrak{q}})$. Fix an embedding of $\mathrm{Frac}(W_{\mathfrak{q}})$-algebras $\mathrm{Frac}(\mathcal O_y)\hookrightarrow\mathrm{Frac}(W)^{\mathrm{alg}}$, and let \[ \Gamma_y = \mathrm{Gal}(\mathrm{Frac}(W)^{\mathrm{alg}}/\mathrm{Frac}(\mathcal O_y)) \] be the absolute Galois group of $\mathrm{Frac}(\mathcal O_y)$. Then $\Gamma_y$ is a finite index subgroup of $I_{\mathfrak{q}}$.
If $N_p$ is a continuous $p$-adic representation of $K_p$, we obtain from it a lisse $p$-adic sheaf $\bm{N}_p$ over $Y_K$, and restricting further to $\mathrm{Spec}(\mathrm{Frac}(\mathcal O_y))$ gives us a continuous representation of $\Gamma_y$, which we will denote by $\bm{N}_{p,y}$.
\begin{proposition} \label{prop:p_adic_crystalline} Suppose that $N_p$ is a $K_p$-stable $\mathbb Z_p$-lattice in an algebraic $\mathbb Q_p$-representation $N$ of $T_{\mathbb Q_p}$. Then the $\Gamma_y$-representation $\bm{\Lambda}_{p,y}[p^{-1}]$ is crystalline. \end{proposition} \begin{proof} This is essentially due to Rapoport-Zink~\cite{rapoport_zink}. We give some details of the proof.
Consider the map \begin{equation}\label{eqn:reflexnorm_mfq} \Gamma_y\hookrightarrow I_{\mathfrak{q}}^{\mathrm{ab}}\xrightarrow{\simeq}\mathcal O_{E,\mathfrak{q}}^\times\hookrightarrow T_E(\mathbb Q_p), \end{equation} where the isomorphism in the middle is the reciprocity isomorphism of local class field theory.
Via the map~\eqref{eqn:reflexnorm_mfq}, given any algebraic $\mathbb Q_p$-linear representation $M$ of $T_E$ and a $K_p$-stable $\mathbb Z_p$-lattice $M_p\subset M$, we obtain, in a functorial way, a continuous $\mathbb Z_p$-linear representation $\bm{M}_p$ of $\Gamma_y$. The associated $\mathbb Q_p$-linear representation $\bm{M}_p[p^{-1}]$ does not depend on the choice of the lattice $M_p$.
In particular, since any representation of $T$ is naturally a representation of $T_E$, we obtain a functor from $\mathbb Q_p$-linear algebraic representations of $T$ to continuous $\mathbb Q_p$-representations of $\Gamma_y$. Applying this functor to $N$, we obtain a continuous $\mathbb Q_p$-representation of $\Gamma_y$. Using the description of the functor $M_p\mapsto\bm{M}_{p}$ above, as well as of the $\mathbb Q$-structure on $Y_K(\mathbb Q^{\mathrm{alg}})$ in~\S\ref{ss:zero dimensional}, it is easy to verify that this representation is precisely $\bm{N}_{p,y}[p^{-1}]$.
Therefore, to finish the proof, it is enough to show that, for any $\mathbb Q_p$-representation $N$ of $T_E$, $\bm{N}_{p}[p^{-1}]$ is a crystalline representation of $\Gamma_y$. It suffices to do this for a single faithful representation of $T_E$: Indeed, any other representation will yield a Galois representation that is a subquotient of tensor powers of the Galois representation associated with the chosen faithful representation of $T$.
We choose our faithful representation to be the tautological representation $H_0$ of $T_E$ obtained from its multiplication action on the $\mathbb Q$-vector space $E$. We have \[ H_{0,\mathbb Q_p} = \bigoplus_{\mathfrak{q}'\vert p} H_{0,\mathfrak{q}'}, \] where $\mathfrak{q}'$ ranges over the $p$-adic primes of $E$, and $H_{0,\mathfrak{q}'}$ is simply $E_{\mathfrak{q}'}$ viewed as a representation of $T_{E,\mathbb Q_p}$.
By the explicit description of~\eqref{eqn:reflexnorm_mfq}, we find that the associated representation of $\Gamma_y$ also admits a direct sum decomposition \[ \bm{H}_{0,p}[p^{-1}] = \bigoplus_{\mathfrak{q}'\vert p}\bm{H}_{0,\mathrm{et},\mathfrak{q}'}[p^{-1}], \] where $\Gamma_y$ acts on $\bm{H}_{0,\mathrm{et},\mathfrak{q}}$ via the reciprocity isomorphism $I_{\mathfrak{q}}\xrightarrow{\simeq}\mathcal O_{E,\mathfrak{q}}^\times$ of local class field theory, and \emph{trivially} on $\bm{H}_{0,\mathrm{et},\mathfrak{q}'}$ for $\mathfrak{q}\neq \mathfrak{q}'$.
Therefore, it is enough to show that $\bm{H}_{0,\mathrm{et},\mathfrak{q}}[p^{-1}]$ is crystalline. But, by the construction of the local reciprocity isomorphism using Lubin-Tate theory, this is simply the rational Tate module $T_{\pi_E}(\mathcal{G}_{\mathfrak{q}})[p^{-1}]$, where $\mathcal{G}_{\mathfrak{q}}$ is the Lubin-Tate group over $\mathcal O_{E,\mathfrak{q}}$ associated with some choice of uniformizer $\pi\in E_{\mathfrak{q}}$. \end{proof}
\begin{remark} \label{rem:Eq times reps} From the proof above, we see that the homomorphism \[ \Gamma_y \to T(\mathbb Q_p) \] giving rise to the functor $N_p\mapsto \bm{N}_{p,y}$ factors through the image of $E_{\mathfrak{q}}^\times$ in $T(\mathbb Q_p)$. Let $T_{\mathfrak{q}}\subset T_{\mathbb Q_p}$ be the image of $\mathrm{Res}_{E_{\mathfrak{q}}/\mathbb Q_p}\mathbb{G}_m$. Then, if we set \[ K_{\mathfrak{q}} = K_p\cap T_{E_{\mathfrak{q}}}(\mathbb Q_p), \] we actually obtain a functor from $K_{\mathfrak{q}}$-stable lattices in algebraic $\mathbb Q_p$-representations of $T_{\mathfrak{q}}$ to continuous $\Gamma_y$-representations on finite free $\mathbb Z_p$-modules. When restricted to $K_p$-stable lattices in algebraic representations of $T_{\mathbb Q_p}$, this recovers the functor $N_p\mapsto\bm{N}_{p,y}$ considered above. \end{remark}
For the next result, we slightly expand the usual notion of an $F$-crystal over $W$: For us, it will be a finite free $W$-module $J_0$ equipped with an isomorphism $\mathrm{Fr}^*J_0[p^{-1}]\xrightarrow{\simeq}J_0[p^{-1}]$ of $W$-modules. Also a \emph{filtered finite free module} over $\mathcal O_y$ is a finite free module $J$ over $\mathcal O_y$ equipped with a filtration $\mathrm{Fil}^\bullet J$ by $\mathcal O_y$-linear direct summands.
\begin{corollary} \label{cor:realizations y} Let $M$ be an algebraic $\mathbb Q_p$-representation of $T_{\mathfrak{q}}$, and let $M_p\subset M$ be a $K_{\mathfrak{q}}$-stable $\mathbb Z_p$-lattice. Then we can associate with $M_p$ an $F$-crystal $\bm{M}_{\mathrm{cris},y}$ over $W$ and a filtered finite free $\mathcal O_y$-module $\bm{M}_{\mathrm{dR},\mathcal O_y}$ with the following properties: \begin{enumerate}
\item The assignments $M_p\mapsto \bm{M}_{\mathrm{cris},y}$ and $M_p\mapsto \bm{M}_{\mathrm{dR},\mathcal O_y}$ are functorial in $M_p$.
\item If $\bm{M}_{p,y}$ is the crystalline $\Gamma_y$-representation associated with $M_p$ via Remark~\ref{rem:Eq times reps}, then we have canonical comparison isomorphisms
\begin{align*}
B_\mathrm{cris}\otimes_{\mathbb Z_p}\bm{M}_{p,y}&\xrightarrow{\simeq}B_{\mathrm{cris}}\otimes_{W}\bm{M}_{\mathrm{cris},y};\\
B_\mathrm{dR}\otimes_{\mathbb Z_p}\bm{M}_{p,y}&\xrightarrow{\simeq}B_{\mathrm{dR}}\otimes_{\mathcal O_y}\bm{M}_{\mathrm{dR},\mathcal O_y}.
\end{align*}
\item If $\bm{M}_{p,y} = T_p(\mathcal{H})^\vee$ is the dual of the $p$-adic Tate module of a $p$-divisible group $\mathcal{H}$ over $\mathcal O_y$, then, in the notation of Theorem~\ref{thm:kisin_p_divisible}, we have canonical isomorphisms
\begin{align*}
\bm{M}_{\mathrm{cris},y}&\xrightarrow{\simeq}\mathbb{D}(\mathcal{H})(W);\\
\bm{M}_{\mathrm{dR},y}&\xrightarrow{\simeq}\mathbb{D}(\mathcal{H})(\mathcal O_y)
\end{align*}
of $F$-crystals over $W$ and filtered finite free $\mathcal O_y$-modules, respectively. Under these isomorphisms, the comparison isomorphisms in assertion (2) are carried to the canonical $p$-adic comparison isomorphisms for abelian schemes over $\mathcal O_y$. \end{enumerate} \end{corollary} \begin{proof} Choose a uniformizer $\pi_y\in \mathcal O_y$, and let $\mathcal{E}(u)\in W[u]$ be its associated monic Eisenstein polynomial. Then, by the theory in~\S\ref{ss:breuil_kisin}, we obtain a functor: \[ M_p\mapsto\mathfrak{M}(M_p) \define \mathfrak{M}(\bm{M}_{p,y}) \] from $K_{\mathfrak{q}}$-stable lattices in algebraic $\mathbb Q_p$-representations of $T_{\mathfrak{q}}$ to Breuil-Kisin modules over $\mathcal O_y$ (associated with the uniformizer $\pi_y$).
Reducing $\varphi^*\mathfrak{M}(M_p)$ mod $u$ gives us an $F$-crystal $\bm{M}_{\mathrm{cris},y}$ over $W$. Reducing it mod $\mathcal{E}(u)$ gives us a finite free $\mathcal O_y$-module $\bm{M}_{\mathrm{dR},y}$. The existence of the canonical comparison isomorphisms in assertion (2) follows from the properties of the functor $\mathfrak{M}$ as explained in \S~\ref{ss:breuil_kisin}.
In particular, \[ \bm{M}_{\mathrm{dR},\mathcal O_y}[p^{-1}] = \mathrm{Frac}(\mathcal O_y)\otimes_{\mathrm{Frac}(W)}D_{\mathrm{cris}}(\bm{M}_{p,y}) \] has a canonical filtration, and we will equip $\bm{M}_{\mathrm{dR},\mathcal O_y}$ with the induced filtration.
The constructions are clearly functorial in $M_p$, and their compatibility with Dieudonn\'e theory as stated in assertion (3) follows from Theorem~\ref{thm:kisin_p_divisible}. \end{proof}
\begin{proposition} \label{prop:realizations integral model} Fix an algebraic $\mathbb Q$-representation $N$ of $T$ and a $K$-stable lattice $N_{\widehat{\mathbb Z}}\subset N_{\mathbb A_f}$. Then we can canonically associate with this pair a filtered vector bundle $(\bm{N}_{\mathrm{dR}},\mathrm{Fil}^\bullet\bm{N}_{\mathrm{dR}})$ over $\mathcal{Y}_K$. Given a prime $\mathfrak{q}\subset \mathcal O_E$, we can also canonically associate with the pair an $F$-crystal $\bm{N}_{\mathrm{cris}}$ over $\mathcal{Y}_{K,\mathbb F_{\mathfrak{q}}}$. These constructions have the following properties: \begin{enumerate}
\item They are functorial in the pair $(N,N_{\widehat{\mathbb Z}})$.
\item The restriction of $\bm{N}_{\mathrm{dR}}$ to $Y_K$ is canonically isomorphic to $\bm{N}_{\mathrm{dR},\mathbb Q}$ as a filtered vector bundle.
\item If $N=H$ is as in Proposition~\ref{prop:abelian schemes realization zero} with associated abelian scheme $\mathcal{A}_H$ over $\mathcal{Y}_K$, then the filtered vector bundle $\bm{H}_{\mathrm{dR}}$ is canonically identified with the relative first de Rham homology of $\mathcal{A}_H$. Moreover, the $F$-crystal $\bm{H}_{\mathrm{cris}}$ over $\mathcal{Y}_{K,\mathbb F_{\mathfrak{q}}}$ is canonically isomorphic to the dual of the Dieudonn\'e $F$-crystal associated with the restriction of $\mathcal{A}_H$ over $\mathcal{Y}_{K,\mathbb F_{\mathfrak{q}}}$.
\item If $y\in \mathcal{Y}_K(\mathbb F^\mathrm{alg}_{\mathfrak{q}})$, then the evaluation of $\bm{N}_{\mathrm{dR}}$ on $\mathrm{Spec}~\mathcal O_y$ is canonically isomorphic, as a filtered free $\mathcal O_y$-module, to the filtered module $\bm{N}_{\mathrm{dR},\mathcal O_y}$ obtained from $N_{\mathbb Z_p}$ via Corollary~\ref{cor:realizations y}.
\item With $y$ as above, the evaluation of $\bm{N}_{\mathrm{cris}}$ on $\mathrm{Spf}~W(\mathbb F_{\mathfrak{q}})$, viewed as a formal divided power thickening of $y$, is canonically isomorphic to the $F$-crystal $\bm{N}_{\mathrm{cris},y}$, obtained from $N_{\mathbb Z_p}$ via Corollary~\ref{cor:realizations y}. \end{enumerate} \end{proposition} \begin{proof} Fix a representation $H$ of $T$ as in Proposition~\ref{prop:abelian schemes realization zero}, and a lattice $H_\mathbb Z\subset H$ such that $H_{\widehat{\mathbb Z}}\subset H_{\mathbb A_f}$ is $K$-stable, giving us an abelian scheme $\mathcal{A}_H\to \mathcal{Y}_K$. Let $\bm{H}_{\mathrm{dR}}$ be the first relative de Rham homology of $\mathcal{A}_H$ over $\mathcal{Y}_K$. As in the proof of Proposition~\ref{prop:zero dim derham}, if we fix tensors $\{t_{\beta}\}\subset H^\otimes$, whose pointwise stabilizer is $T$, we obtain canonical global sections $\{\bm{t}_{\beta,\mathrm{dR},\mathbb Q}\}$ of $\mathrm{Fil}^0\bm{H}_{\mathrm{dR},\mathbb Q}^\otimes$.
We can assume that each $t_\beta$ actually lies in $H_{\widehat{\mathbb Z}}^\otimes$. Then, given a prime $p$, the $p$-adic \'etale realizations of these invariant tensors give us canonical global sections $\{\bm{t}_{\beta,p}\}$ of $\bm{H}_p^\otimes$ over $\mathcal{Y}_K[p^{-1}]$.
Fix a prime $\mathfrak{q}\subset \mathcal O_E$ lying above $p$, and a point $y\in \mathcal{Y}_K(\mathbb F_{\mathfrak{q}}^\mathrm{alg})$. Then we obtain $\Gamma_y$-invariant tensors $\{\bm{t}_{\beta,p,y}\}\subset\bm{H}_{p,y}$.
From Corollary~\ref{cor:realizations y} we obtain canonical tensors $\{\bm{t}_{\beta,\mathrm{cris},y}\}\subset \bm{H}_{\mathrm{cris},y}^\otimes$ and $\{\bm{t}_{\beta,\mathrm{dR},\mathcal O_y}\}\subset\bm{H}_{\mathrm{dR},y}^\otimes$ such that the comparison isomorphisms \[ B_\mathrm{cris}\otimes_{\mathbb Z_p}\bm{H}_{p,y}\xrightarrow{\simeq}B_\mathrm{cris}\otimes_{W(\mathbb F_{\mathfrak{q}})}\bm{H}_{\mathrm{cris},y}\;;\;B_{\mathrm{dR}}\otimes_{\mathbb Z_p}\bm{H}_{p,y}\xrightarrow{\simeq}B_\mathrm{dR}\otimes_{\mathcal O_y}\bm{H}_{\mathrm{dR},\mathcal O_y} \] carry $1\otimes\bm{t}_{\beta,p,y}$ to $1\otimes\bm{t}_{\beta,\mathrm{cris},y}$ and $1\otimes\bm{t}_{\beta,\mathrm{dR},\mathcal O_y}$, respectively.
By a theorem of Blasius-Wintenberger~\cite{blasius:padic}, the restriction of $\bm{t}_{\beta,\mathrm{dR},\mathcal O_y}$ to $\mathrm{Frac}(\mathcal O_y)$ is precisely the evaluation of the de Rham tensor $\bm{t}_{\beta,\mathrm{dR},\mathbb Q}$ on $\mathrm{Spec}~\mathrm{Frac}(\mathcal O_y)$. Therefore, we find that $\bm{t}_{\beta,\mathrm{dR},\mathbb Q}$ extends to a section $\bm{t}_{\beta,\mathrm{dR}}$ of $\bm{H}_{\mathrm{dR}}^\otimes$ over $\mathcal{Y}_K$.
This also shows that, for any $K_p$-stable lattice $N_p\subset N_{\mathbb Q_p}$ in an algebraic $\mathbb Q$-representation $N$, there is a canonical isomorphism \[ \bm{N}_{\mathrm{dR},\mathcal O_y}[p^{-1}]\xrightarrow{\simeq}\bm{N}_{\mathrm{dR},\mathbb Q}\vert_{\mathrm{Spec}(\mathrm{Frac}(\mathcal O_y))} \] of filtered vector spaces. Indeed, as in the proof of Proposition~\ref{prop:zero dim derham}, both constructions arise from the $T$-torsor over $\mathrm{Frac}(\mathcal O_y)$ parameterizing trivializations $\mathrm{Frac}(\mathcal O_y)\otimes H\xrightarrow{\simeq}\bm{H}_{\mathrm{dR},\mathcal O_y}[p^{-1}]$, which carry $1\otimes t_\beta$ to $\bm{t}_{\beta,\mathrm{dR},\mathcal O_y}$, for each index $\beta$.
In particular, using the functoriality of the construction $N_p\mapsto \bm{N}_{\mathrm{dR},\mathcal O_y}$, one deduces that there is a canonical filtered vector bundle $\bm{N}_{\mathrm{dR},\mathfrak{q}}$ over $\mathcal{Y}_{K,\mathfrak{q}}$, whose restriction to $\mathcal{Y}_{K,\mathfrak{q}}[p^{-1}]$ is isomorphic to the restriction of $\bm{N}_{\mathrm{dR},\mathbb Q}$, and whose evaluation at $\mathrm{Spec}~\mathcal O_y$, for any point $y\in \mathcal{Y}_K(\mathbb F_{\mathfrak{q}}^\mathrm{alg})$, is the lattice $\bm{N}_{\mathrm{dR},\mathcal O_y}$.
The construction of the functor $N_p\mapsto \bm{N}_{\mathrm{cris}}$ proceeds similarly, but we only give it in the case where $\mathcal{Y}_{K,\mathfrak{q}}$ is \'etale over $\mathcal O_{E,\mathfrak{q}}$, which will suffice for our purposes. By a descent argument, we can assume that $K$ is neat, so that $\mathcal{Y}_{K,\mathfrak{q}}$ is a scheme over $\mathcal O_{E,{\mathfrak{q}}}$, and is in fact a disjoint union of schemes of the form $\mathcal{Y}'=\mathrm{Spec} \mathcal O_{E'}$, where $E'/E_{\mathfrak{q}}$ is a finite, unramified extension.
Let $\mathbb F'$ be the residue field of $\mathcal O_{E'}$. Fix an embedding $\mathbb F\hookrightarrow\mathbb F^\mathrm{alg}_{\mathfrak{q}}$: This determines a point $y\in \mathcal{Y}'(\mathbb F^\mathrm{alg}_{\mathfrak{q}})$. The construction in Corollary~\ref{cor:realizations y} gives us an $F$-crystal $\bm{N}_{\mathrm{cris},y}$ over $W(\mathbb F^\mathrm{alg}_{\mathfrak{q}})$. It is now enough to show that it has a canonical descent to an $F$-crystal $\bm{N}_{\mathrm{cris},\mathbb F'}$ over $W(\mathbb F')$, which recovers the Dieudonn\'e $F$-crystal of $\mathcal{A}_H$ when $N=H$. This can be deduced from the functoriality of Kisin's functor $\mathfrak{M}$. Alternatively, it can also be deduced by observing that Kisin's functor is already defined for crystalline Galois representations of $\mathrm{Gal}(\mathbb Q_p^\mathrm{alg}/E')$, as is its compatibility with Dieudonn\'e theory of $p$-divisible groups, and we can therefore use it to produce $F$-crystals over $W(\mathbb F')$, and not just over $W(\mathbb F_{\mathfrak{q}}^\mathrm{alg})$.
It remains to globalize the construction of the de Rham realization. Let $D$ be the product of the finitely many rational primes at which $E$ is ramified, or at which we have $K_p \neq K_{0,p}$. Note that $T$ extends to a torus over $\mathbb Z[D^{-1}]$. We will denote this extension again by $T$. From the construction of the compact open subgroup $K_{0,p}$ in \S~\ref{ss:zero dimensional}, we find that, for $p\nmid D$, $K_p = K_{0,p} = T(\mathbb Z_p)$. Moreover, for each such $p$, we can choose the tensors $\{t_\beta\}$ so that their stabilizer in $\mathrm{GL}(H_{\mathbb Z_{(p)}})$ is $T_{\mathbb Z_{(p)}}$.
We can now consider the functor on $\mathcal{Y}_K[D^{-1}]$-schemes carrying $S$ to the set of isomorphisms \[ \xi:\mathcal O_S\otimes_{\mathbb Z}H_{\mathbb Z}\xrightarrow{\simeq}\bm{H}_{\mathrm{dR}}\vert_S \] of vector bundles over $S$ satisfying $\xi(1\otimes t_\beta) = \bm{t}_{\beta,\mathrm{dR}}$, for all indices $\beta$. Since $T$ is a reductive group over $\mathbb Z[D^{-1}]$, it follows from~\cite[Corollary 1.4.3]{KisinJAMS} that this functor is represented by a $T$-torsor $\mathcal{P}_T$ over $\mathcal{Y}_K[D^{-1}]$. Just as in the proof of Proposition~\ref{prop:zero dim derham}, this functor is independent of the choice of data $(H,H_{\mathbb Z})$, and we obtain from it a canonical functor \[ N_{\mathbb Z[D^{-1}]}\mapsto (\bm{N}_{\mathrm{dR},\mathbb Z[D^{-1}]},\mathrm{Fil}^\bullet\bm{N}_{\mathrm{dR},\mathbb Z[D^{-1}]}) \] from algebraic representations of $T$ on finite free $\mathbb Z[D^{-1}]$-modules to filtered vector bundles over $\mathcal{Y}_K[D^{-1}]$, which has properties (2), (3) and (4).
Given an arbitrary $K$-stable lattice $N_{\widehat{\mathbb Z}}\subset N_{\mathbb A_f}$, by enlarging the set of primes appearing in the factorization of $D$ if necessary, we can assume that $N_{\widehat{\mathbb Z}[D^{-1}]}$ arises from an algebraic $\mathbb Z[D^{-1}]$-representation of $T$, and so the desired filtered vector bundle $(\bm{N}_{\mathrm{dR}},\mathrm{Fil}^\bullet\bm{N}_{\mathrm{dR}})$ is canonically determined outside of the primes dividing $D$. For a prime $\mathfrak{q}\subset\mathcal O_E$ dividing $D$, it is determined by the condition that its restriction to $\mathcal{Y}_{K,\mathfrak{q}}$ is isomorphic to $\bm{N}_{\mathrm{dR},\mathfrak{q}}$.
\end{proof}
To summarize the results of \S~\ref{ss:sheaves_i} and of this subsection, from a pair $(N,N_{\widehat{\mathbb Z}})$ as in the Proposition above, we have obtained the following realizations: \begin{itemize} \item $\bm{N}_{\mathrm{Hdg}}$ in the category of variations of $\mathbb Z$-Hodge structures over $Y_K(\mathbb C)$; \item $\bm{N}_{\mathrm{dR}}$ in the category of filtered vector bundles over $\mathcal{Y}_K$; \item For each prime $\ell$, $\bm{N}_{\ell}$ in the category of lisse $\ell$-adic sheaves over $\mathcal{Y}_K[\ell^{-1}]$; \item For each prime ${\mathfrak{q}}\subset\mathcal O_E$, $\bm{N}_{\mathrm{cris}}$ in the category of $F$-crystals over $\mathcal{Y}_{K,\mathbb F_{\mathfrak{q}}}$. \end{itemize}
For $?=\mathrm{Hdg},\mathrm{dR},\ell,\mathrm{cris}$, let $\mathrm{End}(\bm{N}_{?})_{\mathbb Q}$ be the endomorphism algebra of $\bm{N}_?$ in the appropriate isogeny category; this is a finite dimensional algebra over $\mathbb Q_?$, where $\mathbb Q_?=\mathbb Q$ if $?=\mathrm{Hdg}$; $\mathbb Q_? = E$ if $?=\mathrm{dR}$; $\mathbb Q_? = \mathbb Q_{\ell}$, if $?=\ell$; and $\mathbb Q_?=\mathbb Q_p$ if $?=\mathrm{cris}$. This algebra depends only on $N$ and not on the choice of $K$-stable lattice $N_{\widehat{\mathbb Z}}$. Let $\mathrm{Aut}^\circ(\bm{N}_?)$ be the algebraic group over $\mathbb Q_?$ associated with the group of units in this algebra.
Fix a representation $H$ as in Proposition~\ref{prop:abelian schemes realization zero} and a $K$-stable lattice in $H$, and let $\mathcal{A}_H$ be the associated abelian scheme over $\mathcal{Y}_K$. Let $\mathrm{Aut}^{\circ}(\mathcal{A}_H)$ be the algebraic group over $\mathbb Q$ obtained as the group of units in the $\mathbb Q$-algebra $\mathrm{End}(\mathcal{A}_H)_{\mathbb Q}$.
\begin{proposition}\ \label{prop:tQ_action} \begin{enumerate} \item There is a canonical map of algebraic groups, \[ \theta_?(N):T_{\mathbb Q_?} \to \mathrm{Aut}^\circ(\bm{N}_?) \] functorial in the representation $N$.
\item There is a canonical embedding $T\hookrightarrow \mathrm{Aut}^{\circ}(\mathcal{A}_H)$ whose homological realizations induce the maps $\theta_?(H)$ for the representation $H$. \end{enumerate} \end{proposition} \begin{proof} The simplest way to see this is to use the torus \[ \widetilde{T} \define T_E. \]
In complete analogy with the construction of $\mathcal{Y}_K$, given a compact open $\widetilde{K}\subset \widetilde{T}(\mathbb A_f)$, we can associate with it and the cocharacter $\mu_0$ an arithmetic curve $\widetilde{\mathcal{Y}}_{\widetilde{K}}$ over $\mathcal O_E$. If the image of $\widetilde{K}$ in $T(\mathbb A_f)$ is contained in $K$, then we obtain a finite map \[ \widetilde{\mathcal{Y}}_{\widetilde{K}}\to \mathcal{Y}_K \] of algebraic $\mathcal O_E$-stacks.
We also obtain realization functors $(N,N_{\widehat{\mathbb Z}})\mapsto \bm{N}_?$ over $\widetilde{\mathcal{Y}}_{\widetilde{K}}$. Here, $N$ is an algebraic representation of $\widetilde{T}$ and $N_{\widehat{\mathbb Z}}\subset N$.
We apply this to the representation $H_0$ and the lattice $H_{0,\widehat{\mathbb Z}}$ from the proof of Proposition~\ref{prop:p_adic_crystalline} to obtain sheaves $\bm{H}_{0,?}$ over $\widetilde{\mathcal{Y}}_{\widetilde{K}}$. Since the $E$-action on $H_0$ is $\widetilde{T}$-equivariant, the sheaves just obtained are $E$-linear objects in the appropriate isogeny category. We now recover $\widetilde{T}_{\mathbb Q_?}$ as the group of $E$-equivariant automorphisms \[ \widetilde{T}_{\mathbb Q_?} = \mathrm{Aut}_E^\circ(\bm{H}_{0,?}) \subset \mathrm{Aut}^\circ(\bm{H}_{0,?}). \]
From this, and the fact that $H_0$ is a faithful representation of $\widetilde{T}$, it is not hard to deduce that this actually gives us a canonical map \[ \widetilde{T}_{\mathbb Q_?} \to \mathrm{Aut}^\circ(\bm{N}_?), \] for every $\widetilde{T}$-representation $N$. We obtain the map from assertion (1) by specializing now to representations of $\widetilde{T}$ that factor through $T$.
As for assertion (2), since abelian varieties over $\mathbb C$ are a fully faithful subcategory of $\mathbb Z$-Hodge structures, the Betti realization $\theta_B(H)$ corresponds to a map \[ T \to \mathrm{Aut}^{\circ} ( \mathcal{A}_{H, {Y}(\mathbb C)}). \]
Since the \'etale realizations of this map descend over $Y$, it is easily checked that the map itself descends: \[ T \to \mathrm{Aut}^{\circ} ( \mathcal{A}_{H, {Y}}). \]
Our desired embedding is just the composition of this one with the inclusion \[ \mathrm{Aut}^{\circ}(\mathcal{A}_{H,{Y}}) \hookrightarrow \mathrm{Aut}^{\circ}(\mathcal{A}_H). \] \end{proof}
\subsection{The standard representation and its realizations} \label{ss:standard}
We will now consider a particular representation of $T$. As in the proof of Proposition~\ref{prop:p_adic_crystalline}, we have the tautological representation $H_0$ of $T_E$ acting on $E$ via multiplication. Let $c:E\to E$ be complex conjugation, and in the notation of~\S\ref{ss:lubin-tate_special_endomorphisms}, set: \[ V_0 = V(H_0,c) = \{x\in\mathrm{End}(H_0): \; x(a\cdot h) = c(a) x(h)\text{, for all $a\in E$}\}. \] This is a $T_E$-subrepresentation of $\mathrm{End}(H_0)$ on which the action factors through $T$, and in fact through $T_{so}$. We call this the \emph{standard representation} of $T$.
The ring of integers $\mathcal O_E\subset E$ gives a natural lattice $H_{0,\mathbb Z}\subset H_0$, and hence a lattice \[ V_{0,\mathbb Z} = V(H_{0,\mathbb Z},c) \subset V_0. \]
Fix a prime $\mathfrak{q}\subset\mathcal O_E$, an algebraic closure $\mathrm{Frac}(W)^{\mathrm{alg}}$ of $\mathrm{Frac}(W)$ (here, $W = W(\mathbb F_{\mathfrak{q}}^{\mathrm{alg}})$), and an embedding $\mathbb Q_p^{\mathrm{alg}}\hookrightarrow\mathrm{Frac}(W)^{\mathrm{alg}}$ inducing the place $\mathfrak{q}$ on $E = \iota_0(E)$. Let $\mathbb Q^{\mathrm{alg}}_p$ be the algebraic closure of $\mathbb Q_p$ in $\mathrm{Frac}(W)^{\mathrm{alg}}$. We can now view $\iota_0$ as an embedding $E_{\mathfrak{q}}\hookrightarrow\mathbb Q_p^{\mathrm{alg}}$.
Fix a point $y\in\mathcal{Y}_K(\mathbb F_{\mathfrak{q}}^{\mathrm{alg}})$. We can now describe the $F$-crystal $\bm{V}_{0,\mathrm{cris},y}$ quite explicitly. Fix a uniformizer $\pi_{\mathfrak{q}}\in E_{\mathfrak{q}}$, and let $\mathcal{G}_{\mathfrak{q}}$ be the Lubin-Tate group over $\mathcal O_{E,\mathfrak{q}}$ associated with this uniformizer. Let \[ \bm{H}_{0,\mathrm{cris},\mathfrak{q}} = \mathbb{D}(\mathcal{G}_{\mathfrak{q}})(W) \] be the Dieudonn\'e $F$-crystal over $\mathrm{Frac}(W)$ associated with $\mathcal{G}_{\mathfrak{q}}$.
The $\mathcal O_{E,\mathfrak{q}}$-equivariant structure on $\mathcal{G}_{\mathfrak{q}}$ induces an $\mathcal O_{E,\mathfrak{q}}$-equivariant structure on $\bm{H}_{0,\mathrm{cris},\mathfrak{q}}$.
For a prime $\mathfrak{q}'\subset\mathcal O_E$ lying over $p$ with $\mathfrak{q}'\neq \mathfrak{q}$, let \[ \bm{H}_{0,\mathrm{cris},\mathfrak{q}'} = W\otimes_{\mathbb Q_p}\mathcal O_{E,\mathfrak{q}'} \] be the rank $1$ $W\otimes_{\mathbb Q_p}E_{\mathfrak{q}'}$-module equipped with the constant $F$-isocrystal structure arising from the automorphism $\mathrm{Fr}\otimes 1$.
Now, set \[ \bm{H}_{0,\mathrm{cris},y} = \bigoplus_{\mathfrak{q}'\vert p}\bm{H}_{0,\mathrm{cris},\mathfrak{q}'}. \] From the proof of Proposition~\ref{prop:p_adic_crystalline}, we find that this is precisely the crystalline realization obtained from the tautological representation $H_0$ of $T_E$, equipped with the standard $\mathcal O_E$-stable lattice.
The inclusion $V_0 \hookrightarrow \mathrm{End}(H_0)$ now gives us identifications: \[ \bm{V}_{0,\mathrm{cris},y} = V(\bm{H}_{0,\mathrm{cris},y},c)\subset \mathrm{End}(\bm{H}_{0,\mathrm{cris},y}). \] In particular, the decomposition of $\bm{H}_{0,\mathrm{cris},y}$ gives us a decomposition: \[ \bm{V}_{0,\mathrm{cris},y} = \bigoplus_{\mathfrak{p}'\vert p}\bm{V}_{0,\mathrm{cris},\mathfrak{p}'}, \] into $F$-crystals, where $\mathfrak{p}'$ ranges over the primes of $\mathcal O_F$ lying above $p$, and where \[ \bm{V}_{0,\mathrm{cris},\mathfrak{p}'} = V(\bm{H}_{0,\mathrm{cris},\mathfrak{p}'},c)\subset \mathrm{End}(\bm{H}_{0,\mathrm{cris},\mathfrak{p}'}). \] Here, \[ \bm{H}_{0,\mathrm{cris},\mathfrak{p}'} = \bigoplus_{\mathfrak{q}'\vert \mathfrak{p}'}\bm{H}_{0,\mathrm{cris},\mathfrak{q}'} \] is an $\mathcal O_{F,\mathfrak{p}'}$-linear $F$-crystal over $W$.
\begin{proposition} \label{prop:v0 cris realization} Let $\mathfrak{p}\subset\mathcal O_F$ be the prime lying under $\mathfrak{q}$. Then the following statements are equivalent: \begin{enumerate} \item $\mathfrak{p}$ is not split in $F$. \item The space of $\varphi$-invariants $\bm{V}_{0,\mathrm{cris},\mathfrak{p}}^{\varphi = 1}$ is non-zero. \item The natural map \[ \mathrm{Frac}(W)\otimes_{\mathbb Z_p}\bm{V}_{0,\mathrm{cris},\mathfrak{p}}^{\varphi = 1} \to \bm{V}_{0,\mathrm{cris},\mathfrak{p}}[p^{-1}] \] is an isomorphism. \item The natural map \[ \mathrm{Frac}(W)\otimes_{\mathbb Z_p}\bm{V}_{0,\mathrm{cris},y}^{\varphi = 1} \to \bm{V}_{0,\mathrm{cris},y}[p^{-1}] \] is an isomorphism. \end{enumerate} \end{proposition} \begin{proof} If $\mathfrak{p}$ is split in $E$, we have: \[ \bm{H}_{0,\mathrm{cris},\mathfrak{p}} = \bm{H}_{0,\mathrm{cris},\mathfrak{q}} \oplus \bm{H}_{0,\mathrm{cris},\mathfrak{q}^c}. \]
Moreover, $\bm{V}_{0,\mathrm{cris},\mathfrak{p}} = V(\bm{H}_{0,\mathrm{cris},\mathfrak{p}},c)$ consists of pairs $(x_1,x_2)$ of $\mathcal O_{F,\mathfrak{p}}$-linear maps \[ x_1:\bm{H}_{0,\mathrm{cris},\mathfrak{q}^c}\to\bm{H}_{0,\mathrm{cris},\mathfrak{q}},\quad x_2:\bm{H}_{0,\mathrm{cris},\mathfrak{q}}\to \bm{H}_{0,\mathrm{cris},\mathfrak{q}^c}. \] Therefore, the space of $\varphi$-invariants consists of $\varphi$-equivariant such pairs. However, by definition, $\bm{H}_{0,\mathrm{cris},\mathfrak{q}^c}$ is generated by its $\varphi$-invariants, while $\bm{H}_{0,\mathrm{cris},\mathfrak{q}}$, being the Dieudonn\'e $F$-isocrystal associated with a Lubin-Tate group, has no non-zero $\varphi$-invariant elements. Thus we conclude that $\bm{V}_{0,\mathrm{cris},\mathfrak{p}}^{\varphi=1}$ has no non-zero elements.
On the other hand, suppose that $\mathfrak{p}$ is not split in $E$. Then we can identify $\bm{V}_{0,\mathrm{cris},\mathfrak{p}} = V(\bm{H}_{0,\mathrm{cris},\mathfrak{p}},c)$ with the space $V_{\mathrm{cris}}(\mathcal{G}_{\mathfrak{q}})$ defined in~\S\ref{ss:lubin-tate_special_endomorphisms}.
In Propositions~\ref{prp:unramified_vcris} and~\ref{prp:ramified_vcris}, we described the structure of $\varphi$-invariants in this space explicitly, and in particular showed that they generate the whole space over $\mathrm{Frac}(W)$.
From these considerations, the equivalence of statements (1), (2) and (3) of the proposition are immediate. The equivalence of these statements with (4) now follows from the fact that, for $\mathfrak{q}'\neq \mathfrak{q}$, $\bm{H}_{0,\mathrm{cris},\mathfrak{q}'}$ is generated by its $\varphi$-invariants. \end{proof}
Fix a representation $H$ as in Proposition~\ref{prop:abelian schemes realization zero} and a $K$-stable lattice in $H$, and let $\mathcal{A}_H$ be the associated abelian scheme over $\mathcal{Y}_K$.
\begin{proposition} \label{prop:abelian scheme reduction} Fix a prime $\mathfrak{q}\subset\mathcal O_E$ above a rational prime $p$ and let $\mathfrak{p}\subset\mathcal O_F$ be the prime lying under it. The following equivalences hold: \begin{align*} \text{$\mathfrak{p}$ is not split in $E$}&\Leftrightarrow\text{$\mathcal{A}_{H,y}$ is supersingular for all $y\in \mathcal{Y}_K(\mathbb F_\mathfrak{q}^{\mathrm{alg}})$};
\end{align*} \end{proposition} \begin{proof} Fix a point $y\in \mathcal{Y}_K(\mathbb F_{\mathfrak{q}}^{\mathrm{alg}})$. By Proposition~\ref{prop:abelian schemes realization zero}, the Dieudonn\'e $F$-isocrystal associated with $\mathcal{A}_{H,y}$ is isomorphic to the $F$-isocrystal $\bm{H}_{\mathrm{cris},y}$.
Now, the slopes of the $F$-isocrystal $\bm{H}_{\mathrm{cris},y}[p^{-1}]$ are determined by its Newton cocharacter \[ \nu(H): \mathbb{D} \to \mathrm{Aut}^{\circ}_{\varphi}(\bm{H}_{\mathrm{cris},y}), \] where $\mathbb{D}$ is the pro-torus over $\mathbb Q_p$ with character group $\mathbb Q$, and $\mathrm{Aut}^{\circ}_{\varphi}(\bm{H}_{\mathrm{cris},y})$ is the algebraic group of $\mathbb Q_p$ obtained as the group of units in the $\mathbb Q_p$-algebra $\mathrm{End}_{\varphi}(\bm{H}_{\mathrm{cris},y})_{\mathbb Q}$ of $\varphi$-equivariant endomorphisms of $\bm{H}_{\mathrm{cris},y}[p^{-1}]$.
\begin{lemma} \label{lem:newton cocharacter} For any $\mathbb Q$-representation $N$ of $T$, the Newton cocharacter $\nu(N)$ for $\bm{N}_{\mathrm{cris},y}[p^{-1}]$ factors through the map $\theta_{\mathrm{cris}}(N):T_{\mathbb Q_p}\to \mathrm{Aut}^\circ_{\varphi}(\bm{N}_{\mathrm{cris},y})$ from Proposition~\ref{prop:tQ_action}. \end{lemma} \begin{proof} The Newton cocharacter is functorial in $N$. If $H_0$ is the tautological representation of $T_E$, then it is clear from the construction of $\bm{H}_{0,\mathrm{cris},y}$ in~\S\ref{ss:standard} that its slope decomposition is stable under the $E\otimes_\mathbb Q\mathbb Q_p$-action, and hence that the slope cocharacter for $\bm{H}_{0,\mathrm{cris},y}[p^{-1}]$ factors through the commutant in $\mathrm{Aut}^\circ_{\varphi}(\bm{H}_{\mathrm{cris},y})$ of $E\otimes_\mathbb Q\mathbb Q_p$. This is precisely the torus $T_{E,\mathbb Q_p}$. Combining this with the fact that any $\mathbb Q$-representation of $T$, when viewed as a representation of $T_E$, appears as a subquotient of a tensor power of $H_0$, one easily deduces the lemma. \end{proof}
Now, we find from~\eqref{prop:v0 cris realization} that $\bm{V}_{0,\mathrm{cris},y}[p^{-1}]$ is generated by $\varphi$-invariants, and hence that $\nu(V_0)$ is trivial, if and only if $\mathfrak{p}$ is not split in $E$. Since the quotient $T_{so}$ of $T$ acts faithfully on $V_0$, this implies in turn that $\mathfrak{p}$ is not split in $E$ if and only if $\nu(H)$ factors through \[ \mathbb{G}_m = \ker(T\to T_{so}). \] This is the case if and only if $\nu(H)$ is constant, and hence if and only if $\mathcal{A}_{H,y}$ is supersingular.
\end{proof}
\section{Orthogonal Shimura varieties} \label{s:orthogonal shimura}
Let $(V,Q)$ be a quadratic space over $\mathbb Q$ of signature $(n,2)$, with $n\ge 1$. Fix a \emph{maximal} lattice $L \subset V$, and let $L^\vee$ be the dual lattice. As in the introduction, the \emph{discriminant} of $L$ is $D_L = [L^\vee : L]$.
In this section, we lay out the theory of GSpin Shimura varieties associated with $(V,Q)$ and $L$. The main references are~\cite{MadapusiSpin} and~\cite{AGHMP}. The models constructed in these references have to be modified slightly for our purposes here, and we explain this in \S~4.4.
The main notion studied is that of a \emph{special endomorphism}, which allows us to give a moduli interpretation for the \emph{special divisors} considered by Kudla in~\cite{kudla:special_cycles}. This interpretation is crucial for the degree computations underlying the proof of Theorem~\ref{thm:arithmetic BKY}.
\subsection{The GSpin Shimura variety}\label{ss:gspin_char_0}
Let $C(V)$ be the Clifford algebra of $(V,Q)$, with its $\mathbb Z/2\mathbb Z$-grading \[ C(V)=C^+(V)\oplus C^-(V) . \] Recall from \cite{MadapusiSpin} that the spinor similitude group $G=\mathrm{GSpin}(V)$ is the algebraic group over $\mathbb Q$ with \[ G(R) = \big\{ g\in C^+(V_R)^\times : gV_Rg^{-1} = V_R \big\} \] for any $\mathbb Q$-algebra $R$. It sits in an exact sequence \[
1 \to \mathbb{G}_m \to G \map{ g\mapsto g \bullet} \mathrm{SO}(V) \to 1, \] where $g\bullet v=gvg^{-1}$. Let $\nu: G \to \mathbb{G}_m$ be the spinor similitude.
The group of real points $G(\mathbb R)$ acts on the hermitian symmetric domain \begin{equation}\label{hermitian domain} \mathcal{D} = \{ z\in V_\mathbb C : [z,z] =0 ,\, [z,\overline{z} ] <0 \} / \mathbb C^\times \subset \mathbb{P}(V_\mathbb C) \end{equation} through the morphism $G \to \mathrm{SO}(V)$. There are two connected components $\mathcal{D}=\mathcal{D}^+ \sqcup \mathcal{D}^-$, interchanged by the action of any $\gamma\in G(\mathbb R)$ with $\nu(\gamma)<0$.
The pair $(G,\mathcal{D})$ is a Shimura datum. More precisely, given a class $z\in\mathcal{D}$, we can choose a representative $z $ of the form $ x + i y$, where $x,y\in V_\mathbb R$ are mutually orthogonal vectors satisfying $Q(x) = Q(y) = -1$. Then we obtain a homomorphism \[ \bm{h}_z : \mathbb{S} = \mathrm{Res}_{\mathbb C/\mathbb R}\mathbb{G}_m \to G_\mathbb R \] satisfying $\bm{h}_z(i) = xy\in G(\mathbb R)\subset C^+(V)_\mathbb R^\times$. In this way, we can identify $\mathcal{D}$ with the $G(\mathbb R)$-conjugacy class of $\bm{h}_z$, for any $z\in \mathcal{D}$. The reflex field of $(G,\mathcal{D})$ is $\mathbb Q$.
Recall that we have fixed a maximal lattice $L \subset V$. Define a compact open subgroup \begin{equation}\label{compact open} K = G( \mathbb A_f ) \cap C(\widehat{L})^\times \subset G(\mathbb A_f). \end{equation} Here, we have set $\widehat{L} = L_{\widehat{\mathbb Z}}$. The image of $K$ in $\mathrm{SO}(V)(\mathbb A_f)$ is the \emph{discriminant kernel} of $\widehat{L}$; this is the largest subgroup of the stabilizer of $\widehat{L}$ that acts trivially on $\widehat{L}^\vee/\widehat{L}$.
By the theory of canonical models of Shimura varieties, we obtain an $n$-dimensional algebraic stack $M$ over $\mathbb Q$, the \emph{GSpin Shimura variety} associated with $L$. Its space of complex points is the $n$-dimensional complex orbifold \begin{equation}\label{gspin shimura} M (\mathbb C) = G(\mathbb Q) \backslash \mathcal{D} \times G(\mathbb A_f) / K. \end{equation}
\begin{proposition}\label{prop:generic connected} Suppose that one of the following conditions holds: \begin{itemize}
\item $n\geq 2$;
\item $D_L$ is square-free. \end{itemize} Then the complex orbifold $M(\mathbb C)$ is connected. \end{proposition} \begin{proof} The kernel of $\nu: G\to \mathbb{G}_m$ is the usual spin double cover of $\mathrm{SO}(V)$, and hence is simply connected. Using strong approximation, it follows that the connected components of $M(\mathbb C)$ are indexed by $\mathbb Q^\times_{>0} \backslash \mathbb A_f^\times / \nu(K)$, and so the claim follows once we prove that $\nu(K_\ell) = \mathbb Z_\ell^\times$ for every prime $\ell$. When $L_{\mathbb Z_\ell}$ contains a hyperbolic plane, the assertion is clear, so we only need to consider the case where $V_{\mathbb Q_\ell}$ is anisotropic of dimension at least $3$, and is such that $\ell^2$ does not divide $D_L$. In this case, the result can be deduced from the classification of maximal anisotropic lattices over $\mathbb Z_\ell$; see~\cite[\S 29.10]{ShimuraQuadratic}. \end{proof}
Given an algebraic representation $G \to \mathrm{Aut}(N)$ on a $\mathbb Q$-vector space $N$, and a $K$-stable lattice $N_{\widehat{\mathbb Z}}\subset N_{\mathbb A_f}$, we obtain a $\mathbb Z$-local system $\bm{N}_{B}$ on $M(\mathbb C)$ whose fiber at a point $[(z,g)]\in M(\mathbb C)$ is identified with $N\cap g N_{\widehat{\mathbb Z}}$. The corresponding vector bundle $\bm{N}_{\mathrm{dR},M(\mathbb C)}=\mathcal O_{M(\mathbb C)}\otimes\bm{N}_{B}$ is equipped with a filtration $\mathrm{Fil}^\bullet\bm{N}_{\mathrm{dR},M(\mathbb C)}$, which at any point $[(z,g)]$ equips the fiber of $\bm{N}_{B}$ with the Hodge structure determined by the cocharacter $\bm{h}_z$. This gives us a functorial assignment from pairs $(N,N_{\widehat{\mathbb Z}})$ as above to variations of $\mathbb Z$-Hodge structures over $M(\mathbb C)$.
Applying this to $V$ and the lattice $\widehat{L}\subset V_{\mathbb A_f}$, we obtain a canonical variation of polarized $\mathbb Z$-Hodge structures $(\bm{V}_{B},\mathrm{Fil}^\bullet\bm{V}_{\mathrm{dR},M(\mathbb C)})$. For each point $z$ of (\ref{hermitian domain}) the induced Hodge decomposition of $V_\mathbb C$ has \[ V_\mathbb C^{(1,-1)} = \mathbb C z , \qquad V_\mathbb C^{(-1,1)}=\mathbb C\overline{z}, \qquad V_\mathbb C^{(0,0)} = (\mathbb C z + \mathbb C \overline{z} )^\perp. \] It follows that $\mathrm{Fil}^1\bm{V}_{\mathrm{dR},M(\mathbb C)}$ is an isotropic line and $\mathrm{Fil}^0\bm{V}_{\mathrm{dR}, M(\mathbb C)}$ is its annihilator with respect to the pairing on $\bm{V}_{\mathrm{dR}, M(\mathbb C)}$ induced from that on $L$.
Let $H$ be the representation of $G$ on $C(V)$ via left multiplication. It is equipped with a $K$-stable lattice $H_{\widehat{\mathbb Z}} = C(\widehat{L})\subset H_{\mathbb A_f}$. From this, we obtain a variation of $\mathbb Z$-Hodge structures $(\bm{H}_{B},\mathrm{Fil}^\bullet\bm{H}_{\mathrm{dR},M(\mathbb C)})$. This variation has type $(-1,0),(0,-1)$ and is therefore the homology of a family of complex tori over $M(\mathbb C)$. This variation of $\mathbb Z$-Hodge structures is \emph{polarizable}, and so the family of complex tori in fact arises from an \emph{abelian scheme} $A_\mathbb C \to M_\mathbb C$. For all this, see~\cite[(2.2)]{AGHMP}.
By~\cite[\S 3]{MadapusiSpin}, this abelian scheme descends to an abelian scheme $A\to M$. We call this the \emph{Kuga-Satake abelian scheme}. It is equipped with a right $C(L)$-action and a compatible $\mathbb Z/2\mathbb Z$-grading \[ A = A^+ \times A^-. \]
The first relative de Rham homology sheaf of $A$ gives a canonical descent of $\bm{H}_{\mathrm{dR},\mathbb C}$ over $M$ as a filtered vector bundle with an integrable connection. We denote this descent by $\bm{H}_{\mathrm{dR}}$. Using it, and Deligne's results on absolute Hodge cycles on abelian varieties, we obtain a canonical tensor functor from algebraic $\mathbb Q$-representations $N$ of $G$ to filtered vector bundles $(\bm{N}_{\mathrm{dR}},\mathrm{Fil}^\bullet\bm{N}_{\mathrm{dR}})$ over $M$, which descends the already constructed functor to objects over $M_\mathbb C$.
Similarly, if we fix a lattice $N_{\widehat{\mathbb Z}}\subset N_{\mathbb A_f}$, then, for any prime $\ell$, the $\ell$-adic sheaf $\bm{N}_{\ell,\mathbb C} = \mathbb Z_\ell\otimes\bm{N}_B$ over $M(\mathbb C)$ descends canonically to an $\ell$-adic sheaf $\bm{N}_{\ell}$ over $M$. When $N=H$, $\bm{H}_{\ell}$ is canonically isomorphic to the $\ell$-adic Tate module of $A$.
For all this, see~\cite[(4.15)]{MadapusiSpin}.
In particular, for $? = B,\ell,\mathrm{dR}$, the $G$-equivariant embedding $V \hookrightarrow \mathrm{End}_{C(V)}(H)$ determined by left multiplication gives rise to embeddings of homological realizations \begin{equation}\label{Eqn:V_embeddings_char_0} \bm{V}_{?} \hookrightarrow \underline{\mathrm{End}}_{C(L)}(\bm{H}_{?}). \end{equation}
For $x\in V$ with $Q(x) > 0$, define a divisor on $\mathcal{D}$ by \[ \mathcal{D}(x) = \{ z \in \mathcal{D} : z \perp x \}. \] As in the work of Borcherds~\cite{Bor98}, Bruinier~\cite{Bru}, and Kudla~\cite{kudla:special_cycles}, for every $m\in \mathbb Q_{>0}$ and $\mu\in L^\vee/L$ we define a complex orbifold \begin{equation*}
Z(m, \mu ) (\mathbb C)= \bigsqcup_{ g\in G(\mathbb Q)\backslash G(\mathbb A_f) /K } \Gamma_g \backslash \Big( \bigsqcup_{ \substack{ x\in \mu_g+ L_g \\ Q(x)=m } } \mathcal{D}(x) \Big). \end{equation*} Here $\Gamma_g = G(\mathbb Q)\cap g K g^{-1}$, $L_g \subset V$ is the $\mathbb Z$-lattice determined by $\widehat{L}_g = g\bullet \widehat{L}$, and \[ \mu_g=g\bullet \mu\in L_g^\vee/L_g. \]
By construction $Z(m,\mu)(\mathbb C)$ is the space of complex points of a disjoint union of GSpin Shimura varieties associated with quadratic spaces of signature $(n-1,2)$. As such, it has a canonical model $Z(m,\mu)$ over $\mathbb Q$, and the obvious map $Z(m,\mu)(\mathbb C) \to M(\mathbb C)$ descends to a finite and unramified morphism \begin{equation}\label{eqn:Zmmu_canonical} Z(m, \mu ) \to M. \end{equation} Using the complex uniformization, one can check that, \'etale locally on the source, (\ref{eqn:Zmmu_canonical}) is a closed immersion defined by a single equation. Thus (\ref{eqn:Zmmu_canonical}) determines an effective Cartier divisor on $M$, which we call a \emph{special divisor}. Via abuse of notation, we will usually refer to $Z(m,\mu)$ itself as a special divisor on $M$.
\subsection{Integral models in the self-dual case}\label{ss:gspin_2}
In this subsection, we will fix a prime $p$ such that the lattice $L$ is self-dual over $\mathbb Z_{(p)}$, and abbreviate \[ L_{(p)} = L_{\mathbb Z_{(p)}}. \] The group $G_{(p)} = \mathrm{GSpin}(L_{(p)})$ is a reductive model for $G$ over $\mathbb Z_{(p)}$.
The goal is to show that a large part of the results of~\cite[\S4]{MadapusiSpin} also work without the assumption $p>2$.
Consider the \emph{Kuga-Satake abelian scheme} $A\to M$. Its homological realizations are the sheaves associated with the representation $H$ of $G$ on $C(V)$ via left multiplication, and the lattice $H_{\widehat{\mathbb Z}} = C(\widehat{L})\subset H_{\mathbb A_f}$.
We can choose a $G$-invariant symplectic pairing $\psi:H\times H\to \mathbb Q(\nu)$ such that induced pairing on the Betti realization $\bm{H}_{B}$ is a polarization of variations of Hodge structures; see~\cite[(2.2)]{AGHMP} for details. This gives rise to a polarization $\lambda$ on $A_{M(\mathbb C)}$, which descends to a polarization of $A$ over $M$ of degree $m^2$, where $m^2$ is the discriminant of the lattice $H_{\widehat{\mathbb Z}}$ in the symplectic space $H_{\mathbb A_f}$. In this way, we obtain a map \begin{equation*}
M \to \mathcal{X}_{2^{n+2},m,\mathbb Q}, \end{equation*} which is finite and unramified. Here, $\mathcal{X}_{2^{n+2},m}$ is the moduli stack over $\mathbb Z$ of polarized abelian schemes of dimension $2^{n+2}$ and degree $m^2$.
\begin{definition}\label{defn:normalization} Given an algebraic stack $\mathcal{X}$ over $\mathbb Z_{(p)}$, and a normal algebraic stack $Y$ over $\mathbb Q$ equipped with a finite map $j_{\mathbb Q}:Y\to\mathcal{X}_{\mathbb Q}$, the \emph{normalization} of $\mathcal{X}$ in $Y$ is the finite $\mathcal{X}$-stack $j:\mathcal{Y}\to\mathcal{X}$, characterized by the property that $j_*\mathcal O_{\mathcal{Y}}$ is the integral closure of $\mathcal O_{\mathcal{X}}$ in $(j_{\mathbb Q})_*\mathcal O_Y$. It is also characterized by the following universal property: given a finite morphism $\mathcal{Z}\to\mathcal{X}$ with $\mathcal{Z}$ a normal algebraic stack, flat over $\mathbb Z_{(p)}$, any map of $\mathcal{X}_{\mathbb Q}$-stacks $\mathcal{Z}_{\mathbb Q}\to Y$ extends uniquely to a map of $\mathcal{X}$-stacks $\mathcal{Z}\to\mathcal{Y}$. \end{definition}
We now obtain an integral model $\mathcal{M}_{(p)}$ for $M$ over $\mathbb Z_{(p)}$ by taking the normalization of $\mathcal{X}_{2^{n+2},m}$ in $M$. By construction, the Kuga-Satake abelian schemes extends to a polarized abelian scheme \[ A \to \mathcal{M}_{(p)}. \]
\begin{theorem}\label{thm:integral_models_good_reduction} The stack $\mathcal{M}_{(p)}$ is smooth over $\mathbb Z_{(p)}$. \end{theorem} \begin{proof} When $p>2$, this follows from the main result of~\cite{KisinJAMS}. The general case is shown in~\cite[Theorem 3.10]{mp:2adic}. \end{proof}
\begin{remark}\label{rem:ell adic extn} Fix a prime $\ell\neq p$. Recall from~\S\ref{ss:gspin_char_0} the functor which assigns a lisse $\ell$-adic sheaf $\bm{N}_\ell$ over $M$ to each $K_\ell$-stable $\mathbb Z_\ell$-lattice $N_\ell\subset N_{\mathbb Q_\ell}$ in an algebraic representation $N$ of $G$. This functor extends (necessarily uniquely, by the normality of $\mathcal{M}_{(p)}$) to lisse $\ell$-adic sheaves over $\mathcal{M}_{(p)}$, and carries $H_{\mathbb Z_\ell}$ to the $\ell$-adic Tate module $\bm{H}_{\ell}$ of $A$. Indeed, it is enough to show that the induced functor to lisse $\mathbb Q_\ell$-sheaves over $M$ extends over $\mathcal{M}_{(p)}$. As shown in~\cite[(4.11),(7.9)]{MadapusiSpin}, this functor is associated with a canonical \'etale $G(\mathbb Q_\ell)$-torsor over $M$, which admits an extension over $\mathcal{M}_{(p)}$. \end{remark}
We also have a canonical functor \begin{equation}\label{eqn:de_rham_functor} N\mapsto\bm{N}_{\mathrm{dR}} \end{equation} from algebraic $\mathbb Q$-representations of $G$ to filtered vector bundles over $M$ equipped with an integrable connection. The following result is \cite[Proposition 3.7]{mp:2adic}.
\begin{proposition}\label{prop:de_rham_realization} The functor~\eqref{eqn:de_rham_functor} on algebraic $\mathbb Q$-representations of $G$ extends canonically to an exact tensor functor \[N\mapsto\bm{N}_{\mathrm{dR}}\] from algebraic $\mathbb Z_{(p)}$-representations $N$ of $G_{(p)}$ to filtered vector bundles on $\mathcal{M}_{(p)}$ equipped with an integrable connection. When $N = H_{(p)}$, the associated filtered vector bundle with integrable connection is simply $\bm{H}_{\mathrm{dR}}$, the relative first de Rham homology of $A \to \mathcal{M}_{(p)}$. \end{proposition}
In particular, from the representation $L_{(p)}$, we obtain an embedding \[ \bm{V}_{\mathrm{dR}}\hookrightarrow\underline{\mathrm{End}}_{C(L)}(\bm{H}_{\mathrm{dR}}) \] of filtered vector bundles over $\mathcal{M}_{(p)}$ with integrable connections, mapping onto a local direct summand of its target, and extending its counterpart~\eqref{Eqn:V_embeddings_char_0} over $M$.
We now expand our definition of an $F$-crystal over $\mathcal{M}_{(p),\mathbb F_p}$ to mean a crystal of vector bundles $\bm{N}$ over $\mathcal{M}_{(p),\mathbb F_p}$ equipped with an isomorphism \[ \mathrm{Fr}^*\bm{N} \xrightarrow{\simeq}\bm{N} \] in the $\mathbb Q_p$-linear \emph{isogeny} category associated with the category of crystals over $\mathcal{M}_{(p),\mathbb F_p}$.
Write $\widehat{\mathcal{M}}_{p}$ for the formal completion of $\mathcal{M}_{(p)}$ along $\mathcal{M}_{(p),\mathbb F_p}$. The relative first crystalline homology of $A$ over $\mathcal{M}_{(p),\mathbb F_p}$ gives an $F$-crystal $\bm{H}_{\mathrm{cris}}$ over $\mathcal{M}_{(p),\mathbb F_p}$ whose evaluation on $\widehat{\mathcal{M}}_p$ is canonically isomorphic to the $p$-adic completion of $\bm{H}_{\mathrm{dR}}$ as a vector bundle with integrable connection.
\begin{proposition}\label{prop:cris_realization} There is a canonical functor $N\mapsto \bm{N}_{\mathrm{cris}}$ from algebraic $\mathbb Z_{(p)}$-representations of $G_{(p)}$ to $F$-crystals over $\mathcal{M}_{(p),\mathbb F_p}$, which recovers $\bm{H}_{\mathrm{cris}}$ when applied to $H_{(p)}$, and whose evaluation on the formal thickening $\widehat{\mathcal{M}}_p$ is canonically isomorphic to the $p$-adic completion of $\bm{N}_{\mathrm{dR}}$ as a vector bundle with integrable connection.
In particular, there is a canonical $F$-crystal $\bm{V}_{\mathrm{cris}}$ over $\mathcal{M}_{(p),\mathbb F_p}$, whose evaluation on $\widehat{\mathcal{M}}_p$ is canonically isomorphic to the $p$-adic completion of $\bm{V}_{\mathrm{dR}}$ as a vector bundle with integrable connection. It admits a canonical embedding \[ \bm{V}_{\mathrm{cris}} \hookrightarrow \underline{\mathrm{End}}_{C(L)}(\bm{H}_{\mathrm{cris}}) \] mapping onto a local direct summand of its target, and compatible with the embeddings of de Rham realizations. \end{proposition} \begin{proof} See Proposition 3.9 of~\cite{mp:2adic}. \end{proof}
\subsection{Special endomorphisms in the self-dual case}\label{ss:special endomorphisms 2}
By Proposition~\ref{prop:de_rham_realization} and Proposition~\ref{prop:cris_realization}, the embedding of $G_{(p)}$-representations $L_{ (p) }\hookrightarrow H_{(p)}$ gives rise to embeddings \begin{equation}\label{eqn:sheaves_embedding} \bm{V}_{?} \hookrightarrow \underline{\mathrm{End}}_{C(L)}(\bm{H}_?) \end{equation} for $? = B,\ell,\mathrm{dR},\mathrm{cris}$ that map onto local direct summands of their targets.
If $?=B$, let $\mathbf{1}_B$ be the locally constant sheaf $\underline{\mathbb Z}$ over $\mathcal{M}(\mathbb C)$; if $?=\ell$, let $\mathbf{1}_{\ell}$ be the lisse $\ell$-adic sheaf $\underline{\mathbb Z}_{\ell}$ over $\mathcal{M}_{(p)}[\ell^{-1}]$; if $?=\mathrm{dR}$, let $\mathbf{1}_{\mathrm{dR}}$ be the structure sheaf $\mathcal O_{\mathcal{M}_{(p)}}$, equipped with the connection $a\mapsto da$ and the one-step filtration concentrated in degree $0$; and, if $?=\mathrm{cris}$, let $\mathbf{1}_{\mathrm{cris}}$ be the structure sheaf over $(\mathcal{M}_{(p),\mathbb F_p}/\mathbb Z_p)_{\mathrm{cris}}$, equipped with its natural structure of an $F$-crystal.
The quadratic form on $L_{(p)}$ induces a form on the associated realizations. More precisely, for any section $\bm{f}$ of $\bm{V}_?$, we have \[ \bm{f}\circ\bm{f} = \bm{Q}(\bm{f}) \cdot \mathrm{id} \] under composition in $\underline{\mathrm{End}}_{C(L)}(\bm{H}_?)$. Here $\bm{Q}(\bm{f})$ is a section of $\bm{1}_?$. The assignment $\bm{f}\mapsto\bm{Q}(\bm{f})$ is a quadratic form on $\bm{V}_?$ with values in $\mathbf{1}_?$. The associated bilinear form is non-degenerate when $?=\mathrm{dR}$ or $\mathrm{cris}$.
\begin{definition} For any $\mathcal{M}_{(p)}$-scheme $S$, we define an endomorphism $x\in \mathrm{End}_{C(L)}(A_S)$ to be \emph{special} if all its homological realizations land in the images of the embeddings~\eqref{eqn:sheaves_embedding}. More precisely, we require the $\ell$-adic realizations over $S[\ell^{-1}]$ to lie in $\bm{V}_{\ell}$, the crystalline realizations over $S_{\mathbb F_p}$ to lie in $\bm{V}_{\mathrm{cris}}$, and the de Rham realizations to lie in $\bm{V}_{\mathrm{dR}}$.
We will write $V(A_S)$ for the space of special endomorphisms. \end{definition}
We now study the deformation theory of a special endomorphism $x$. In what follows, we will frequently cite results from~\cite[\S 5]{MadapusiSpin}, where there is a standing assumption that $p$ is odd. However, the proofs there do not use this assumption, as the reader can easily verify.
Suppose that $S = \mathrm{Spec}(\mathcal O)$, with $\mathcal O$ a $p$-adically complete $\mathbb Z_{(p)}$-algebra. It will be useful to have a notion of special endomorphisms for the $p$-divisible group $A_S[p^\infty]$. We will call an endomorphism $x\in \mathrm{End}_{C(L)}(A_S[p^\infty])$ \emph{special} if its crystalline realization lands in the image of the embedding~\eqref{eqn:sheaves_embedding} for $?=\mathrm{cris}$. We will write $V(A_S[p^\infty])$ for the space of special endomorphisms.
Suppose that we have a surjection $\mathcal O\rightarrow \overline{\mathcal O}$ of $p$-adically complete $\mathbb Z_{(p)}$-algebras, whose kernel $I$ admits nilpotent divided powers. Suppose that we have a map $y:\mathrm{Spec}(\mathcal O) \to\mathcal{M}_{(p)}$ and let $\overline{y}:\mathrm{Spec}(\overline{\mathcal O}) \to \mathcal{M}_{(p)}$ be the restriction to $\mathrm{Spec}(\overline{\mathcal O})$.
Let $\bm{H}_{\mathcal O}$ be the $\mathcal O$-module obtained by restricting $\bm{H}_{\mathrm{dR}}$ to $\mathrm{Spec}(\mathcal O)$, and let $\bm{V}_{\mathcal O}\subset\mathrm{End}(\bm{H}_{\mathcal O})$ be the corresponding realization of $\bm{V}_{\mathrm{dR}}$, so that $\bm{V}_{\mathcal O}$ is equipped with its Hodge filtration $\mathrm{Fil}^1\bm{V}_{\mathcal O}$, which is a rank $1$ projective module over $\mathcal O$. Denote by $\bm{H}_{\overline{\mathcal O}}$ and $\bm{V}_{\overline{\mathcal O}}$ the induced modules over $\overline{\mathcal O}$.
Let $x\in V(A_{\overline{y}}[p^\infty])$ be a special endomorphism. The crystalline realization of $x$ gives us an element $\bm{x}_{\mathrm{cris}} \in \bm{V}_{\mathcal O}$. Pairing against $\mathrm{Fil}^1\bm{V}_{\mathcal O}$ induces a linear functional: \begin{equation}\label{eqn:deformation_pairing}
[\bm{x}_{\mathrm{cris}},\cdot]:\mathrm{Fil}^1\bm{V}_{\mathcal O} \to \mathcal O. \end{equation}
The following two results are shown just as in \cite[Proposition 5.16 and Corollary 5.17]{MadapusiSpin}.
\begin{proposition}\label{prop:special_endomorphism_deform} The endomorphism $x$ lifts to an element of $V(A_y[p^\infty])$ if and only if the functional~\eqref{eqn:deformation_pairing} is identically $0$. \end{proposition}
\begin{corollary}\label{cor:special_endomorphism} Suppose that $k$ is an algebraically closed field of characteristic $p$, that $t\in \mathcal{M}_{(p)}(k)$ and that $x\in V(A_t[p^\infty])$ is a special endomorphism. Let $\mathcal O_t$ be the completed \'etale local ring of $\mathcal{M}_{(p),W(k)}$ at $t$. There is a principal ideal $(f_x)\subset \mathcal O_t$ such that, for any map $f:\mathcal O_t\to R$ to a local $\mathbb Z_{(p)}$-algebra $R$, $x$ lifts to an element in $\mathrm{End}_{C(L)}(A_{f(t)}[p^\infty])$ if and only if $f$ factors through $\mathcal O_t/(f_x)$. \end{corollary}
In other words, the deformation space of the endomorphism $x$ within the formal scheme $\mathrm{Spf}(\mathcal O_t)$ is pro-representable, and cut out by a single equation.
We have to explain how our notion of a special endomorphism relates to the one defined in~\cite[\S 5]{MadapusiSpin}. The main difference is that in [\emph{loc.~cit.}] an endomorphism $x\in\mathrm{End}_{C(L)}(A_S)$ was defined to be special if it is special in our present sense at every geometric point of $S$, which appears to be a less restrictive definition. It is not, as the following result demonstrates.
\begin{proposition}\label{prop:special_endomorphism_points} Let $S$ be a connected $\mathcal{M}_{(p)}$-scheme, and suppose that \[ x\in \mathrm{End}_{C(L)}(A_S) \] is a $C(L)$-equivariant endomorphism. Then the following statements are equivalent: \begin{enumerate} \item $x$ is special; \item For any geometric point $s\to S$, the fiber of $x$ at $s$ is special; \item For \underline{some} geometric point $s\to S$, the fiber of $x$ at $s$ is special. \end{enumerate} \end{proposition} \begin{proof} If $S$ is a scheme of finite type over $\mathbb Q$, then this is clear, since the conditions can be checked over $S_\mathbb C$, where everything follows from the fact that the Betti realization is locally constant, and determines the \'etale and de Rham realizations. The case of an arbitrary scheme over $\mathbb Q$ follows from this, as $\mathcal{M}_{(p)}$ is itself of finite type.
If $S$ is an arbitrary $\mathbb Z_{(p)}$-scheme, then combining this with \cite[Lemmas 5.9 and 5.13]{MadapusiSpin} shows that (2) and (3) are equivalent. To complete the proof of the proposition, we now need to know that, if $x$ is special at a point $s\to S$ in characteristic $p$, then the crystalline realization of $x$ lands in the image of~\eqref{eqn:sheaves_embedding} globally over $S_{\mathbb F_p}$. This follows from Lemma~\ref{lem:special_endomorphism_lift} below. \end{proof}
\begin{lemma}\label{lem:special_endomorphism_lift} Let $R$ be a complete local algebra over $W$ with perfect residue field $k$. Suppose that we have a point $t\in \mathcal{M}_{(p)}(R)$ and an endomorphism $x\in\mathrm{End}_{C(L)}(A_t)$. Let $t_0\in\mathcal{M}_{(p)}(k)$ be the induced point, and let $x_0\in\mathrm{End}_{C(L)}(A_{t_0})$ be the fiber of $x$ at $t_0$. Then $x$ is special if and only if $x_{0}$ is special. \end{lemma} \begin{proof} Let $\mathcal O_{t_0}$ be the complete local ring of $\mathcal{M}_{(p),W(k)}$ at $t_0$. By Theorem~\ref{thm:integral_models_good_reduction}, $\mathcal O_{t_0}$ is isomorphic to a power series ring over $W(k)$ in $n$ variables. By Corollary~\ref{cor:special_endomorphism}, the deformation ring for the endomorphism $x_0$ is a quotient $\mathcal O_{t_0,x_0} = \mathcal O_{t_0}/(f_{x_0})$ of $\mathcal O_{t_0}$ by a principal ideal. Now, by our hypothesis, $x_0$ lifts over $R$, and so the map $\mathcal O_{t_0}\to R$ factors through $\mathcal O_{t_0,x_0}$. In particular, it suffices to verify the lemma for $R=\mathcal O_{t_0,x_0}$, and so we can assume that we have: \[ R = \frac{W(k)\pow{u_1,\ldots,u_n}}{(f)}, \] for some element $f\in W(k)\pow{u_1,\ldots,u_n}$.
The crystalline realization of $x$ is a section of $\underline{\mathrm{End}}_{C(L)}(\bm{H}_{\mathrm{cris}})$. We want to show that it is in fact a section of $\bm{V}_{\mathrm{cris}}$. This is equivalent to showing that its image in $\underline{\mathrm{End}}_{C(L)}(\bm{H}_{\mathrm{cris}})/\bm{V}_{\mathrm{cris}}$ is $0$.
Let $D_R\to R$ be the $p$-adic completion of the divided power envelope of $R$ in $W(k)\pow{u_1,\ldots,u_n}$. In other words, this is the $p$-adic completion of the subalgebra: \[ W(k)\pow{u_1,\ldots,u_n}\left[\frac{f^n}{n!}: n\in\mathbb Z_{\geq 0}\right]\subset \mathrm{Frac}\bigl(W(k)\pow{u_1,\ldots,u_n}\bigr). \] Note that the Frobenius lift \[ \varphi:W(k)\pow{u_1,\ldots,u_n} \to W(k)\pow{u_1,\ldots,u_n} \] defined by $u_i \mapsto u_i^p$ extends continuously to an endomorphism $\varphi:D_R\to D_R$.
Evaluation along the formal divided power thickening \[\mathrm{Spec}(R_{\mathbb F_p}) \hookrightarrow\mathrm{Spec}(D_R)\] establishes an equivalence from the category of crystals over $(\mathrm{Spec} (R_{\mathbb F_p})/\mathbb Z_p)_{\mathrm{cris}}$ to the category of finite free $D_R$-modules equipped with a topologically nilpotent integrable connection. Furthermore, this establishes an equivalence between $F$-crystals and finite free $D_R$-modules $M$ equipped with a topologically nilpotent integrable connection as well as a map $\varphi^*M[p^{-1}]\to M[p^{-1}]$ that is parallel for this connection.
Therefore the lemma is now immediate from Lemma~\ref{lem:lci_fibers_zero_cris} below. \end{proof}
\begin{lemma}\label{lem:lci_fibers_zero_cris} Let $M$ be a finite free $D_R$-module with a topologically nilpotent integrable connection: \[ \nabla: M\to M\otimes\widehat{\Omega}^1_{R/W(k)} \] and an isomorphism $\varphi^*M[p^{-1}]\to M[p^{-1}]$ that is parallel for $\nabla$. Suppose that $m\in M^{\nabla=0}$ is a parallel element that goes to $0$ under the reduction map $M\to M\otimes_{D_R}W$. Then $m=0$. \end{lemma} \begin{proof}
Let $\widehat{U}^{\mathrm{rig}}$ be the rigid analytic space over $\mathrm{Frac}(W(k))$ associated with the power series ring $W(k)\pow{u_1,\ldots,u_n}$ via Berthelot's analytification functor; see~\cite[\S 7]{dejong:formal_rigid}. This is simply the rigid analytic unit disc. The endomorphism $\varphi$ induces a contraction map $\varphi^*:\widehat{U}^{\mathrm{rig}}\to \widehat{U}^{\mathrm{rig}}$.
Now, there is a rational number $r_f\in (0,1)$ such that all elements in $D_R$ converge in the open disc $\widehat{U}^{\mathrm{rig}}(r_f)\subset\widehat{U}^{\mathrm{rig}}$ of radius $r_f$. Let $R^{\mathrm{rig}}$ be the ring of global sections of the structure sheaf on $\widehat{U}^{\mathrm{rig}}(r_f)$. Then we have an inclusion $D_R\subset R^{\mathrm{rig}}$, and extending scalars along this inclusion gives us an $R^{\mathrm{rig}}$-module
\[ M^{\mathrm{rig}} = R^{\mathrm{rig}}\otimes_{D_R}M
\]
equipped with an integrable connection and an isomorphism
\[ \varphi^*M^{\mathrm{rig}}\xrightarrow{\simeq} M^{\mathrm{rig}}.
\]
In this situation, the image of the natural map
\[
(M^{\mathrm{rig}})^{\nabla = 0}\to M^{\mathrm{rig}}
\]
\emph{generates} $M^{\mathrm{rig}}$ as an $R^{\mathrm{rig}}$-module. This is just Dwork's trick; see for instance~\cite[\S 3.4, Prop. 4]{vologodsky:hodge}.
Therefore, if a parallel section of $M^{\mathrm{rig}}$ vanishes at a point, then it vanishes everywhere on $\widehat{U}^{\mathrm{rig}}(r_f)$, and is hence the zero section. This proves the lemma. \end{proof}
\begin{proposition}\label{prop:special_quadratic_form 2} Let $S$ be an $\mathcal{M}_{(p)}$-scheme. For each $x\in V(A_S)$, we have \[ x\circ x = Q(x) \cdot \mathrm{id}_{A_S}\in \mathrm{End}(A_S) \] for some integer $Q(x)$. The assignment $x\mapsto Q(x)$ is a positive definite quadratic form on $V(A_S)$. \end{proposition} \begin{proof} This is shown as in~\cite[Lemma 5.12]{MadapusiSpin}. \end{proof}
\subsection{Integral model over $\mathbb Z$}
We will now explain how to construct an integral model for $M$ over $\mathbb Z$. In~\cite{AGHMP}, using the results of~\cite{MadapusiSpin}, we gave a construction that worked over $\mathbb Z[1/2]$. This is inadequate for our current purposes for two reasons: First, of course, it omits the prime $2$; second, at primes $p$ such that $p^2\mid D_L$, the integral model from [\emph{loc.~cit.}] excluded points in the special fiber that will be relevant to this article; see~\cite[Remark 2.4.4]{AGHMP}.
Fix a prime $p$. Choose an auxiliary quadratic space $(V^ \diamond,Q^ \diamond)$ over $\mathbb Q$ of signature $(n^\diamond,2)$, admitting a maximal lattice $L^ \diamond\subset V^ \diamond$ that is self-dual over $\mathbb Z_{(p)}$, and admitting an isometric embedding \[ (V,Q) \hookrightarrow (V^ \diamond,Q^ \diamond) \] carrying $L$ into $L^ \diamond$. Set \[ \Lambda = L^\perp = \{x\in L^ \diamond: [x,L] = 0\}\subset L^ \diamond. \]
Set $G^ \diamond = \mathrm{GSpin}(V^ \diamond)$, and let $\mathcal{D}^ \diamond \subset \mathbb{P}(V^ \diamond_\mathbb C)$ be the associated hermitian domain; then there is a natural embedding of Shimura data \[ (G,\mathcal{D})\hookrightarrow (G^\diamond,\mathcal{D}^\diamond), \] giving rise to a finite, unramified map of Shimura varieties $M\to M^ \diamond$. Here, $M^ \diamond$ is the Shimura variety associated with the maximal lattice $L^ \diamond$.
Since $L^ \diamond$ is self-dual over $\mathbb Z_{(p)}$, $M^ \diamond$ admits a smooth integral model $\mathcal{M}^ \diamond_{(p)}$ over $\mathbb Z_{(p)}$. We have the Kuga-Satake abelian scheme $A^\diamond\to \mathcal{M}^ \diamond_{(p)}$ with associated de Rham sheaf $\bm{H}_{\mathrm{dR}}^ \diamond$, as well as the embeddings \[ \bm{V}^ \diamond_{?} \hookrightarrow \underline{\mathrm{End}}_{C(L^ \diamond)}(\bm{H}^ \diamond_{?}) \] where $?=B,\ell,\mathrm{dR},\mathrm{cris}$. For any $\mathcal{M}^ \diamond_{(p)}$-scheme $S$, we have the subspace \[ V(A^ \diamond_S) \subset \mathrm{End}_{C(L^ \diamond)}(A^ \diamond_S) \] of special endomorphisms, whose homological realizations are sections of $\bm{V}^ \diamond_?$.
Define $\mathcal{M}_{(p)}$ to be the normalization of $\mathcal{M}^ \diamond_{(p)}$ in $M$ (see Definition~\ref{defn:normalization}). The restriction of $\mathrm{Fil}^1\bm{V}^ \diamond_{\mathrm{dR}}$ to $\mathcal{M}_{(p)}$ gives us a line bundle $\bm{\omega}$ over $\mathcal{M}_{(p)}$, which extends the line bundle $\mathrm{Fil}^1\bm{V}_{\mathrm{dR}}$ over the generic fiber $M$.
\begin{proposition}\label{prop:integral_model_bad_p} \mbox{} \begin{enumerate}
\item The integral model $\mathcal{M}_{(p)}$ and the line bundle $\bm{\omega}$ are independent of the choice of the auxiliary data $(V^ \diamond,Q^ \diamond)$ and $L^ \diamond\subset V^ \diamond$.
\item The Kuga-Satake abelian scheme $A\to M$ extends to an abelian scheme $A\to\mathcal{M}_{(p)}$ and there is a canonical $C(L^ \diamond)$-equivariant graded isomorphism
\begin{equation}\label{eqn:kuga-satake_beef_no beef}
A\otimes_{C(L)}C(L^ \diamond) \xrightarrow{\simeq} A^ \diamond
\end{equation}
of abelian schemes over $\mathcal{M}_{(p)}$.
\item There is a canonical isometric embedding:
\begin{equation}\label{eqn:Lambda_emb}
\Lambda \hookrightarrow V(A^ \diamond_{\mathcal{M}_{(p)}}).
\end{equation}
\item $\mathcal{M}_{(p)}$ has the following extension property: If $E/\mathbb Q_p$ is a finite extension, and $t\in M(E)$ is a point such that $A_t$ has potentially good reduction over $\mathcal O_E$, then the map $t:\mathrm{Spec}(E) \to M$ extends to a map $\mathrm{Spec}(\mathcal O_E) \to \mathcal{M}_{(p)}$. \end{enumerate} \end{proposition} \begin{proof}
Assertion (1) is shown just as in the proof of~\cite[Prop. 2.4.5]{AGHMP}.
As for assertion (2), first note that, given the existence of the extension $A\to\mathcal{M}_{(p)}$, the fact that $C(L^ \diamond)$ is \emph{free} over $C(L)$ gives meaning to $A\otimes_{C(L)}C(L^ \diamond)$ as an abelian scheme over $\mathcal{M}_{(p)}$; this is the Serre tensor construction. We always have the canonical $C(L^ \diamond)$-equivariant graded isomorphism~\eqref{eqn:kuga-satake_beef_no beef} over the generic fiber $M$; see~\cite[(2.12)]{AGHMP}. In particular, as abelian schemes over $M$, there is a canonical closed immersion $A\hookrightarrow A^ \diamond$. Note that $A^ \diamond\to\mathcal{M}^ \diamond_{(p)}$ admits a polarization of degree prime to $p$; indeed, in the notation of \cite[\S~2.4]{AGHMP}, arranging for this amounts to choosing an element $\delta\in C^+(L^ \diamond)_{\mathbb Z_{(p)}}^\times$ satisfying $\delta^* = -\delta$. That $A$ extends to an abelian scheme over $\mathcal{M}_{(p)}$ now follows from the argument in~\cite[Prop. 4.2.2]{Madapusi}. The argument actually shows the following: As in \S~\ref{ss:gspin_2}, let $\psi:H\times H\to \mathbb Q(\nu)$ be a $G$-invariant symplectic pairing giving rise to a polarization on $A\vert_M$, and thus to a finite map $M\to \mathcal{X}_{2^{n+2},m,\mathbb Q}$ to the generic fiber of a Siegel moduli space. Then this map extends to a finite map $\mathcal{M}_{(p)}\to \mathcal{X}_{2^{n+2},m,\mathbb Z_{(p)}}$ parameterizing the abelian scheme $A\to \mathcal{M}_{(p)}$.
The existence of the isomorphism~\eqref{eqn:kuga-satake_beef_no beef} of abelian schemes over $\mathcal{M}_{(p)}$, as well as the embedding~\eqref{eqn:Lambda_emb} are now shown exactly as in the proof of~\cite[Prop. 2.5.1]{AGHMP}.
Assertion (4) is immediate from the finiteness (hence properness) of the map $\mathcal{M}_{(p)}\to \mathcal{X}_{2^{n+2},m,\mathbb Z_{(p)}}$. \end{proof}
Given the proposition, we can choose our auxiliary lattice $L^ \diamond$ to our convenience. We will choose it so that $\Lambda = L^\perp\subset L^ \diamond$ has rank at most $2$. This is not strictly necessary, but will make some proofs shorter. Moreover, it is always possible to make such a choice, as can be easily verified using the classification of quadratic forms over $\mathbb Q$.
Let $\mathcal{Z}(\Lambda)\to \mathcal{M}^ \diamond_{(p)}$ be the stack such that, for any $\mathcal{M}^ \diamond_{(p)}$-scheme $S$ we have
\[
\mathcal{Z}(\Lambda)(S) = \{\text{Isometric embeddings }\Lambda\hookrightarrow V(A^ \diamond_S)\}.
\]
The argument from~\cite[Proposition 2.7.4]{AGHMP} shows that $\mathcal{Z}(\Lambda)$ is an algebraic stack that is finite and unramified over $\mathcal{M}^ \diamond_{(p)}$.
The embedding~\eqref{eqn:Lambda_emb} corresponds to a map $\mathcal{M}_{(p)}\to\mathcal{Z}(\Lambda)$. It is shown in~\cite[Lemma 7.1]{MadapusiSpin} that this map identifies $M$ with an open and closed substack of $\mathcal{Z}(\Lambda)_\mathbb Q$.
\begin{proposition}\label{prop:model_special_endomorphisms} Let $p$ be an odd prime. Suppose either that $p^2\nmid D_L$ or that $n\geq 3$. Then $\mathcal{Z}(\Lambda)$ is normal and flat over $\mathbb Z_{(p)}$. In particular, the map $\mathcal{M}_{(p)}\to\mathcal{Z}(\Lambda)$ identifies $\mathcal{M}_{(p)}$ with an open and closed substack of $\mathcal{Z}(\Lambda)$. \end{proposition} \begin{proof}
Let $\Lambda\hookrightarrow V(A^ \diamond_{\mathcal{Z}(\Lambda)})$ be the tautological isometric embedding and let \[
\bm{\Lambda}_{\mathrm{dR}}\subset\bm{V}^ \diamond_{\mathrm{dR}}\vert_{\mathcal{Z}(\Lambda)} \]
be the coherent subsheaf generated by the de Rham realization of this embedding. As in~\cite[Lemma 6.16]{MadapusiSpin}, there is a canonical open substack
\[ \mathcal{Z}^{\mathrm{pr}}(\Lambda)\subset\mathcal{Z}(\Lambda)
\]
containing $\mathcal{Z}(\Lambda)_\mathbb Q$, and over which $\bm{\Lambda}_{\mathrm{dR}}$ is a local direct summand of $\bm{V}^ \diamond_{\mathrm{dR}}$.
It is shown in~\cite[Corollary 6.22]{MadapusiSpin} that, under our hypotheses, $\mathcal{Z}^{\mathrm{pr}}(\Lambda)$ is a flat, normal $\mathbb Z_{(p)}$-stack.
When $p^2\nmid D_L$, it is shown in~\cite[Lemma 6.16]{MadapusiSpin} that $\mathcal{Z}^{\mathrm{pr}}(\Lambda) = \mathcal{Z}(\Lambda)$, and so the proposition follows in this case.
For the remaining cases, we will need two lemmas.
\begin{lemma}\label{lem:Z_Lambda_CM}
Suppose that $n\geq 2$. The stack $\mathcal{Z}(\Lambda)$ (resp. $\mathcal{Z}(\Lambda)_{\mathbb F_p}$) is a local complete intersection over
$\mathbb Z_{(p)}$ (resp. $\mathbb F_p$) of relative dimension $n$.
\end{lemma}
\begin{proof}
Since $p>2$, we can find $\Lambda'\subset \Lambda$ and $v\in (\Lambda')^{\perp}\subset \Lambda$ such that $p^2$ does not divide the discriminant of $\Lambda'$ and such that, over $\mathbb Z_{(p)}$, we have an orthogonal decomposition \[ \Lambda_{\mathbb Z_{(p)}} = \Lambda'_{\mathbb Z_{(p)}}\perp\langle v\rangle. \]
Then we have a factorization
\[
\mathcal{Z}(\Lambda) \to \mathcal{Z}(\Lambda') \to \mathcal{M}^ \diamond_{(p)}
\]
into finite and unramified morphisms of $\mathbb Z_{(p)}$-stacks.
As above, it follows from \cite[Corollary~6.22 and Lemma~6.16]{MadapusiSpin} that $\mathcal{Z}(\Lambda')$ is a faithfully flat regular algebraic stack over $\mathbb Z_{(p)}$, whose special fiber is a geometrically normal, local complete intersection algebraic stack of dimension $n + 1\geq 3$.
Fix a point $t\in \mathcal{Z}(\Lambda)(\mathbb F_p^{\mathrm{alg}})$. We can also view this as a point $t\in \mathcal{Z}(\Lambda')(\mathbb F_p^{\mathrm{alg}})$. Let $\mathcal O_{\mathcal{Z}',t}$ (resp. $\mathcal O_{\mathcal{Z},t}$) be the complete local ring of $\mathcal{Z}(\Lambda')$ (resp. $\mathcal{Z}(\Lambda)$) at $t$. Then it is shown in~\cite[Corollary 5.17]{MadapusiSpin} that $\mathcal O_{\mathcal{Z},t}$ is a quotient of $\mathcal O_{\mathcal{Z}(\Lambda')}$ cut out by a single equation.
In particular, this implies that $\mathcal{Z}(\Lambda)$ is \'etale locally an effective Cartier divisor on $\mathcal{Z}(\Lambda')$, and is in particular a local complete intersection over $\mathbb Z_{(p)}$. To show that $\mathcal{Z}(\Lambda)_{\mathbb F_p}$ is a local complete intersection stack over $\mathbb F_p$, it now suffices to show that it does not contain any irreducible components of the normal algebraic stack $\mathcal{Z}(\Lambda')_{\mathbb F_p}$. But this follows from~\cite[Prop. 6.17]{MadapusiSpin}, which shows that, if $\eta\to \mathcal{Z}(\Lambda')_{\mathbb F_p}$ is a generic point, then the tautological map $\Lambda' \to V(A^ \diamond_\eta)$ is an isomorphism.
\end{proof}
\begin{lemma}\label{lem:Zpr_codimension}
The codimension of the complement of $\mathcal{Z}^{\mathrm{pr}}(\Lambda)_{\mathbb F_p}$ in $\mathcal{Z}(\Lambda)_{\mathbb F_p}$ is at least
$
n-\left\lfloor {n^ \diamond}/{2}\right\rfloor.
$
\end{lemma}
\begin{proof}
By~\cite[(6.27)]{MadapusiSpin}, we find that this complement is supported entirely on the supersingular locus \[
\mathcal{M}^{ \diamond,\mathrm{ss}}_{(p),\mathbb F_p}\subset \mathcal{M}^{ \diamond}_{(p),\mathbb F_p}.
\]
But, by~\cite{Howard-Pappas}, this locus has dimension at most $\left\lfloor {n^ \diamond}/{2}\right\rfloor$. This dimension count can also be deduced using the methods of Ogus from~\cite{Ogus2001-wy}.
From this the lemma is clear.
\end{proof}
By our assumption on $\Lambda$, $n^ \diamond \leq n+2$. Therefore, by Lemma~\ref{lem:Zpr_codimension}, we see that $\mathcal{Z}^{\mathrm{pr}}(\Lambda)$ is fiberwise dense in $\mathcal{Z}(\Lambda)$ as soon as $n\geq 3$. On the other hand, Lemma~\ref{lem:Z_Lambda_CM} shows that $\mathcal{Z}(\Lambda)$ is a Cohen-Macaulay stack over $\mathbb Z_{(p)}$. Therefore, by the normality of $\mathcal{Z}^{\mathrm{pr}}(\Lambda)$ and Serre's criterion for normality, we find that $\mathcal{Z}(\Lambda)$ is itself normal and flat over $\mathbb Z_{(p)}$, as soon as $n\geq 3$. \end{proof}
\begin{theorem}\label{thm:irreducible_fibers} Assume one of the following conditions: \begin{itemize}
\item $L_{(p)}$ is self-dual;
\item $p$ is odd, $p^2\nmid D_L$ and $n\geq 2$;
\item $p$ is odd and $n\geq 5$. \end{itemize} Then $\mathcal{M}_{(p),\mathbb F_p}$ is a geometrically connected and geometrically normal algebraic stack over $\mathbb F_p$. \end{theorem} \begin{proof} By Proposition \ref{prop:generic connected}, under our hypotheses, $M$ is a geometrically connected smooth algebraic stack over $\mathbb Q$. Therefore, we only have to show that $\mathcal{M}_{(p)}$ has normal geometric fibers. Indeed, as soon as this is known, it will follow from~\cite[Corollary 4.1.11]{Madapusi} that $\mathcal{M}_{(p),\mathbb F_p}$ is geometrically connected.
If $L_{(p)}$ is self-dual, then $\mathcal{M}_{(p)}$ is smooth over $\mathbb Z_{(p)}$ by Theorem~\ref{thm:integral_models_good_reduction}, so the theorem is clear under this hypothesis.
If $p$ is odd, to prove the theorem, by Proposition~\ref{prop:model_special_endomorphisms} it is enough to show that, under the given hypotheses, $\mathcal{Z}(\Lambda)$ is a normal algebraic stack, flat over $\mathbb Z_{(p)}$, with normal geometric special fiber. By~\cite[Corollary 6.22]{MadapusiSpin}, we find that, under our hypotheses, $\mathcal{Z}^{\mathrm{pr}}(\Lambda)$ has geometrically normal fibers.
Therefore, by Lemma~\ref{lem:Z_Lambda_CM} and Serre's criterion for normality, to show that $\mathcal{Z}(\Lambda)_{\mathbb F_p}$ is normal, it is enough to show that the complement of $\mathcal{Z}^{\mathrm{pr}}(\Lambda)_{\mathbb F_p}$ in $\mathcal{Z}(\Lambda)_{\mathbb F_p}$ has codimension at least $2$. When $p^2\nmid D_L$, this is clear, since $\mathcal{Z}^{\mathrm{pr}}(\Lambda) = \mathcal{Z}(\Lambda)$.
For the general case, by Lemma~\ref{lem:Zpr_codimension}, it suffices to show
\[
\left\lfloor\frac{n+2}{2}\right\rfloor \leq n-2
\]
whenever $n\geq 5$. This is an easy verification. \end{proof}
The construction of $\mathcal{M}\to\mathrm{Spec}(\mathbb Z)$ now proceeds as in~\cite[\S~2.4]{AGHMP}. Choose a finite collection of maximal quadratic spaces $L^ \diamond_1,L^ \diamond_2,\ldots,L^ \diamond_r$ with the following properties: \begin{itemize} \item For each $i=1,2,\ldots,r$, $L^ \diamond_i$ has signature $(n^ \diamond_i,2)$, for $n^ \diamond_i\in\mathbb Z_{>0}$; \item For each $i$, there is an isometric embedding $L\hookrightarrow L^ \diamond_i$; \item If, for each $i$, we denote by $D_i = D_{L^ \diamond_i}$ the discriminant of $L^ \diamond_i$, then $\mathrm{gcd}(D_1,\ldots,D_r) = 1$. \end{itemize} It is always possible to find such a collection.
For $i=1,2,\ldots,r$, let $M^ \diamond_i$ be the GSpin Shimura variety over $\mathbb Q$ attached to $L^ \diamond_i$. Then $M^ \diamond_i$ admits a smooth integral model $\mathcal{M}^ \diamond_{i,\mathbb Z[D_i^{-1}]}$ over $\mathbb Z[D_i^{-1}]$. Let $\mathcal{M}_{\mathbb Z[D_i^{-1}]}$ be the normalization of $\mathcal{M}^ \diamond_{i,\mathbb Z[D_i^{-1}]}$ in $M$.
\begin{theorem}\label{thm:M_model} There is a unique flat, normal algebraic $\mathbb Z$-stack $\mathcal{M}$ such that, for each $i$, the restriction of $\mathcal{M}$ over $\mathbb Z[D_i^{-1}]$ is isomorphic to $\mathcal{M}_{\mathbb Z[D_i^{-1}]}$. Moreover: \begin{enumerate} \item The Kuga-Satake abelian scheme $A\to M$ extends to an abelian scheme $A\to \mathcal{M}$. \item The line bundle $\mathrm{Fil}^1\bm{V}_{\mathrm{dR}}$ over $M$ extends canonically to a line bundle $\bm{\omega}$ over $\mathcal{M}$. \item If $L_{(p)}$ is self-dual; or if $p$ is odd and $p^2\nmid D_L$; or if $p$ is odd and $n\geq 5$, then $\mathcal{M}_{\mathbb F_p}$ is a geometrically connected and geometrically normal algebraic stack over $\mathbb F_p$. \end{enumerate} \end{theorem} \begin{proof} This is immediate from Proposition~\ref{prop:integral_model_bad_p} and Theorem~\ref{thm:irreducible_fibers}. \end{proof}
Suppose now that we have an isometric embedding \[ (V,Q) \hookrightarrow (V^ \diamond,Q^ \diamond) \] into a quadratic space of signature $(n^ \diamond,2)$, and a maximal lattice $L^ \diamond\subset V^ \diamond$ containing $L$. Then we have a finite and unramified map of Shimura varieties $M\to M^ \diamond$ over $\mathbb Q$.
The next result is easily deduced from the construction of our integral models; see~\cite[Prop. 2.5.1]{AGHMP} for details.
\begin{proposition}\label{prop:functoriality} The map $M\to M^ \diamond$ extends to a finite map of integral models $\mathcal{M} \to \mathcal{M}^ \diamond$. Moreover: \begin{enumerate} \item If $A^ \diamond\to \mathcal{M}^ \diamond$ is the Kuga-Satake abelian scheme, then there is a canonical isomorphism \[ A\otimes_{C(L)}C(L^ \diamond)\xrightarrow{\simeq} A^ \diamond\vert_{\mathcal{M}} \] of abelian schemes over $\mathcal{M}$. \item Let $\bm{\omega}^ \diamond$ be the canonical line bundle over $\mathcal{M}^ \diamond$ from assertion (2) of Theorem~\ref{thm:M_model}. Then there is a canonical isomorphism \[ \bm{\omega}^ \diamond\vert_{\mathcal{M}}\xrightarrow{\simeq}\bm{\omega} \] of line bundles over $\mathcal{M}$ extending the natural identification \[ \mathrm{Fil}^1\bm{V}^ \diamond_{\mathrm{dR}}\vert_M = \mathrm{Fil}^1\bm{V}_{\mathrm{dR}} \] over the generic fiber $M$. \end{enumerate} \end{proposition}
\subsection{Special endomorphisms and special divisors over $\mathbb Z$}\label{ss:special divisors}
Let $S$ be a scheme over $\mathcal{M}$. We have already encountered the notion of a special endomorphism of $A_S$ in~\S\ref{ss:special endomorphisms 2}, at least in the situation where $S$ is a $\mathbb Z_{(p)}$-scheme, with $p\nmid D_L$. In~\cite[\S~2.6]{AGHMP}, we gave a definition that worked without this condition, but since we have somewhat modified our integral models here, the theory there does not apply directly. We now explain how to fix this.
Fix a prime $p$, and set $\mathcal{M}_{(p)} = \mathcal{M}_{\mathbb Z_{(p)}}$. Choose an auxiliary maximal lattice $L^ \diamond$ of signature $(n^ \diamond,2)$ that is self-dual at $p$, and which admits an isometric embedding $L\hookrightarrow L^ \diamond$. This gives us a finite map of $\mathbb Z_{(p)}$-stacks \[ \mathcal{M}_{(p)} \to \mathcal{M}^ \diamond_{(p)}. \]
It will be useful to have a notion of special endomorphisms for the $\ell$-divisible group of $A^ \diamond$ as $\ell$ varies over the rational primes. If $\ell\neq p$, we will define $V(A^ \diamond[\ell^\infty]_S)$ to be the space of endomorphisms of $A^ \diamond[\ell^\infty]_S$, whose $\ell$-adic realizations land in the space $\bm{V}^ \diamond_\ell$. If $\ell=p$, we will define $V(A^ \diamond[p^\infty]_S)$ to be the space of endomorphisms of $A^ \diamond[p^\infty]_S$, whose $p$-adic realizations land in the space $\bm{V}^ \diamond_p$ over $S[p^{-1}]$ and whose crystalline realizations land in $\bm{V}^ \diamond_{\mathrm{cris}}$ over $S_{\mathbb F_p}$.
The isomorphism of Kuga-Satake abelian schemes \[ A\otimes_{C(L)}C(L^ \diamond)\xrightarrow{\simeq}A^ \diamond\vert_{\mathcal{M}_{(p)}} \]
induces, for any $\mathcal{M}_{(p)}$-scheme $S$, canonical embeddings \begin{align}\label{eqn:End_A_embedding} \mathrm{End}_{C(L)}(A_S) & \hookrightarrow \mathrm{End}_{C(L^ \diamond)}(A^ \diamond_S), \\ \mathrm{End}_{C(L)}(A_S[\ell^\infty]) & \hookrightarrow \mathrm{End}_{C(L^ \diamond)}(A^ \diamond_S[\ell^\infty]),\nonumber \end{align} for any prime $\ell$. We now declare an endomorphism \begin{equation}\label{eq:an endo} x\in\mathrm{End}_{C(L)}(A_S)\quad \mbox{or} \quad x\in\mathrm{End}_{C(L)}(A_S[\ell^\infty]) \end{equation} to be \emph{special} if its image under~\eqref{eqn:End_A_embedding} is a special endomorphism of $A^ \diamond_S$ or $A^ \diamond_S[\ell^\infty]$, respectively.
Let \[ \Lambda = L^\perp = \{x\in L^ \diamond:\;[x,L]=0\} \] be the orthogonal complement of $L$ in $L^ \diamond$. Then there is a canonical embedding \begin{equation}\label{eqn:lambda emb end} \Lambda\hookrightarrow \mathrm{End}_{C(L^ \diamond)}(A^ \diamond_M), \end{equation} described in the proof of~\cite[Prop. 2.5.1]{AGHMP}.
\begin{proposition}\label{prop:special_independent_beef} The notion of $x\in\mathrm{End}_{C(L)}(A_S)$ or $x\in\mathrm{End}_{C(L)}(A_S[\ell^\infty])$ being special does not depend on the choice of the auxiliary lattice $L^ \diamond$. \end{proposition}
\begin{proof} As in the proof of~\cite[Lemma 2.6.1]{AGHMP}, we can reduce to the case where $L$ is itself self-dual over $\mathbb Z_{(p)}$.
In this case, we have homological realizations $\bm{V}_?$ over $\mathcal{M}_{(p)}$, and a commuting square of embeddings \[ \xymatrix{ {\bm{V}_? } \ar[r] \ar[d] & { \underline{\mathrm{End}}_{C(L)}(\bm{H}_?) } \ar[d] \\ { \bm{V}_?^ \diamond\vert_{\mathcal{M}_{(p)}} } \ar[r] & { \underline{\mathrm{End}}_{C(L^ \diamond)}(\bm{H}^ \diamond_?)\vert_{\mathcal{M}_{(p)}} } . } \] of sheaves over $\mathcal{M}_{(p)}$ mapping onto local direct summands of their targets; see~\cite[Prop. 2.5.1(ii)]{AGHMP}.
In fact this square is cartesian: Both vertical arrows identify sections of their domain with those of the target that anti-commute with the homological realizations of the embedding~\eqref{eqn:lambda emb end}; see the argument in \cite[\S~7.3]{MadapusiSpin}.
From this, we find that $x$ is special with respect to the lattice $L^ \diamond$ if and only if its homological realizations land in $\bm{V}_{?}$, and so the notion of being special is in this case \emph{intrinsic} to the stack $\mathcal{M}_{(p)}$, and independent of choices. \end{proof}
If $S$ is now an arbitrary $\mathcal{M}$-scheme, we declare an endomorphism (\ref{eq:an endo})
to be \emph{special} if its restriction to $S_{\mathbb Z_{(p)}}$ is special for every prime $p$.
Write $V(A_S)$ and $V(A_S[\ell^\infty])$ for the respective spaces of special endomorphisms.
We will also need certain distinguished subsets $V_\mu(A_S)\subset V(A_S)_\mathbb Q$ parameterized by $\mu\in L^\vee/L$. To define these, we will first define subsets \[ V_{\mu_{\ell}}(A_S[\ell^\infty])\subset V(A_S[\ell^\infty])_Q \] parameterized by $\mu_\ell\in \mathbb Z_\ell\otimes(L^\vee/L)$.
For this, note that over $M(\mathbb C)$, as $K$ acts trivially on the quotient $L^\vee/L$, we have a canonical isometry \begin{equation*}
\underline{\mathbb Z}\otimes(L^\vee/L) \xrightarrow{\simeq} \bm{V}^\vee_B/\bm{V}_B \end{equation*} of locally constant sheaves. For each prime $\ell$, this gives an isometry \begin{equation*}
\bm{\alpha}_{\ell}:\underline{\mathbb Z}_{\ell}\otimes(L^\vee/L) \xrightarrow{\simeq} \bm{V}^\vee_{\ell}/\bm{V}_{\ell}. \end{equation*} of \'etale sheaves of abelian groups over $M$. In fact, this can be extended to an isometry of sheaves over $\mathcal{M}[\ell^{-1}]$. By the normality of $\mathcal{M}$, it is enough to show that the sheaves $\bm{V}_{\ell}$ and $\bm{V}_{\ell}^\vee$ extend to lisse sheaves over $\mathcal{M}[\ell^{-1}]$. This can be deduced using the argument from Remark~\ref{rem:ell adic extn}.
Fix a prime $p$, and let $S$ be an $\mathcal{M}_{(p)}$-scheme. Then, for any $\ell\neq p$, the $\ell$-adic realization of a special endomorphism $x\in V(A_S)$ is a section of the submodule $\bm{V}_{\ell}\subset\underline{\mathrm{End}}_{C(L)}(\bm{H}_{\ell})$.
Now identify $\bm{V}^\vee_{\ell}$ with an $\ell$-adic subsheaf of $\bm{V}_{\ell}[\ell^{-1}]$. Any element of the dual subspace \[ V(A_S[\ell^\infty])^\vee = \big\{y\in V(A_S[\ell^\infty])_\mathbb Q: [V(A_S[\ell^\infty]),y]\subset\mathbb Z_\ell\big\}\subset V(A_S[\ell^\infty])_\mathbb Q \] has a realization $\bm{x}_{\ell}$ in $\bm{V}_{\ell}[\ell^{-1}]$, where the pairing $[\cdot,\cdot]$ on $V(A_S[\ell^\infty])$ is the one induced from composition in $\mathrm{End}(A_S[\ell^\infty])$. This allows us to define \[ V_{\mu_\ell}(A_S[\ell^\infty])\subset V(A_S[\ell^\infty])^\vee \] to be the subset of elements $x$ such that $\bm{x}_{\ell}$ lies in $\bm{V}^\vee_{\ell}$, and maps to $\bm{\alpha}_{\ell}(1\otimes\mu_{\ell})$ in $\bm{V}_{\ell}^\vee/\bm{V}_{\ell}$.
Next, we will define the subset \[ V_{\mu_p}(A_S[p^\infty]) \subset V(A_S[p^\infty])^\vee \] for $\mu_p\in \mathbb Z_p\otimes(L^\vee/L)$. If $S$ is a $\mathbb Q$-scheme, then this can be defined just as for $\ell\neq p$. For the general case, choose an auxiliary maximal lattice $L^ \diamond$ that is self-dual over $\mathbb Z_{(p)}$ of signature $(n^\diamond,2)$, which admits an isometric embedding $L\hookrightarrow L^ \diamond$. By Proposition~\ref{prop:functoriality} and~\cite[Prop. 2.6.4]{AGHMP}, this gives a map of $\mathbb Z$-stacks $\mathcal{M}\to\mathcal{M}^ \diamond$ along with an isometric embedding \begin{equation*}
\Lambda \hookrightarrow V(A^ \diamond_{\mathcal{M}}). \end{equation*} Here, $\Lambda = L^\perp\subset L^ \diamond$.
\begin{lemma} \label{lem:lambda_perp} For any $\mathcal{M}_{(p)}$-scheme $S$, there are canonical isometries \begin{align*} V(A_S) &\xrightarrow{\simeq} \Lambda^\perp\subset V(A^ \diamond_S) \\ V(A_S[p^\infty])&\xrightarrow{\simeq} \Lambda_{\mathbb Z_p}^\perp\subset V(A^ \diamond_S[p^\infty]). \end{align*} \end{lemma} \begin{proof} The first isometry follows from the definitions and the fact that the subspace \[ \mathrm{End}_{C(L)}(A_S) \subset \mathrm{End}_{C(L^ \diamond)}(A^ \diamond_S) \] consists precisely of those endomorphisms that anti-commute with $\Lambda$; see also~\cite[Prop. 2.5.1]{AGHMP}. The second is proven in similar fashion. \end{proof}
Now, there are canonical isomorphisms \[ \mathbb Z_p\otimes(L^\vee/L) \xleftarrow{\simeq} \mathbb Z_p\otimes(L^ \diamond/(L\oplus\Lambda)) \xrightarrow{\simeq}\mathbb Z_p\otimes(\Lambda^{\vee}/\Lambda). \] Therefore, we can canonically view $\mu_p$ as an element of $\mathbb Z_p\otimes(\Lambda^\vee/\Lambda)$. Moreover, the inclusions \[ V(A_S[p^\infty])\oplus\Lambda \subset V(A^ \diamond_S[p^\infty]) \subset (V(A_S[p^\infty])\oplus \Lambda_{\mathbb Z_p})^\vee = V(A_S[p^\infty])^\vee \oplus \Lambda_{\mathbb Z_p}^\vee \] induce an embedding \begin{equation} \label{eqn:Vbeef_V_Lambda} \frac{V(A^ \diamond_S[p^\infty])}{V(A_S[p^\infty])\oplus \Lambda_{\mathbb Z_p}} \hookrightarrow \frac{V(A_S[p^\infty])^\vee}{V(A_S[p^\infty])} \oplus\frac{\Lambda^\vee_{\mathbb Z_p}}{\Lambda_{\mathbb Z_p}}. \end{equation} We now set \[ V_{\mu_p}(A_S[p^\infty]) = \big\{x\in V(A_S[p^\infty])^\vee: ([x],\mu_p)\text{ is in the image of~\eqref{eqn:Vbeef_V_Lambda}}\big\}, \] where \[ [x]\in \frac{V(A_S[p^\infty])^\vee}{V(A_S[p^\infty])} \] is the class of $x$.
\begin{proposition} \label{prop:Vmu_ind_beef} The subset $V_{\mu_p}(A_S[p^\infty])\subset V(A_S[p^\infty])^\vee$ just defined does not depend on the choice of the auxiliary lattice $L^ \diamond$. Moreover, if $S$ is a $\mathbb Q$-scheme, then this definition agrees with the one already given above. \end{proposition} \begin{proof} As usual, for the independence statement, we can reduce to the case where $L$ is itself self-dual over $\mathbb Z_{(p)}$. In this case, $\mu_p=0$, and we have to show that, if $x\in V(A_S[p^\infty])^\vee$ is such that $([x],0)$ is in the image of~\eqref{eqn:Vbeef_V_Lambda}, then $x$ must belong to $V(A_S[p^\infty])$. However, $([x],0)$ being in the image of~\eqref{eqn:Vbeef_V_Lambda} means exactly that $x$ belongs to $V(A^ \diamond_S[p^\infty])$ and is orthogonal to $\Lambda_{\mathbb Z_p}$. So we are now done by Lemma~\ref{lem:lambda_perp}.
We leave the verification of the second assertion to the reader. \end{proof}
Now suppose that $S$ is an arbitrary $\mathcal{M}$-scheme, and $p$ is any prime. We decree that an element of $V(A_S[p^\infty])^\vee$ belongs to $V_{\mu_p}(A_S[p^\infty])$ if and only if it does so over $S_{\mathbb Z_{(p)}}$.
Consider the dual space \[ V(A_S)^\vee = \{y\in V(A_S)_\mathbb Q: [V(A_S),y]\subset\mathbb Z\}\subset V(A_S)_\mathbb Q \] of $V(A_S)$ with respect to the bilinear form induced from composition in $\mathrm{End}(A_S)$. For each prime $p$ and each $\mu_p\in\mathbb Z_p\otimes(L^\vee/L)$, let \[ V_{\mu_p}(A_S)\subset V(A_S)^\vee \] be the subspace of elements mapping into $V_{\mu_p}(A_S[p^\infty])$. In general, if $\mu\in L^\vee/L$ has $p$-primary part $\mu_p$ for each prime $p$, set \[ V_\mu(A_S) = \bigcap_{p} V_{\mu_p}(A_S) \subset V(A_S)^\vee. \]
The next result is immediate from Proposition~\ref{prop:special_quadratic_form 2} and the definitions; see also~\cite[Prop. 2.6.3]{AGHMP}. \begin{proposition}\label{prop:special_quadratic_form} For each $x\in V(A_S)$, we have \[ x\circ x = Q(x) \cdot \mathrm{id}_{A_S}\in \mathrm{End}(A_S) \] for some integer $Q(x)$. The assignment $x\mapsto Q(x)$ is a positive definite quadratic form on $V(A_S)$. If $x\in V_\mu(A_S)$, then we have the congruence \begin{equation}\label{Q cong} Q(x) \equiv Q(\mu) \pmod{\mathbb Z}. \end{equation} \end{proposition}
Fix a maximal lattice $L^ \diamond$ of signature $(n^ \diamond,2)$, equipped with an isometric embedding $L\hookrightarrow L^ \diamond$, so that we have the corresponding finite map of algebraic stacks $\mathcal{M}\to \mathcal{M}^ \diamond$. Set $\Lambda = L^\perp\subset L^ \diamond$.
\begin{proposition}\label{prop:decomposition_Vmu} Fix an $\mathcal{M}$-scheme $S\to \mathcal{M}$. \begin{enumerate}
\item There is a canonical isometric embedding $\Lambda\hookrightarrow V(A^ \diamond_S)$ and an isometry \begin{equation}\label{eqn:tensor_inject}
V(A_{S})\xrightarrow{\simeq} \Lambda^{\perp}\subset V( A^ \diamond_S ). \end{equation}
\item For every $\mu\in L^{ \diamond,\vee}/L^ \diamond$ and every $(\mu_1,\mu_2)\in \bigl(\mu+L^ \diamond\bigr)/\bigl(L\oplus \Lambda\bigr)$ the map (\ref{eqn:tensor_inject}), tensored with $\mathbb Q$, restricts to an injection \[ V_{\mu_1}(A_{S}) \times ({\mu_2}+\Lambda) \hookrightarrow V_{\mu}(A^ \diamond_S). \]
\item The above injections determine a decomposition \[ V_{\mu}(A^ \diamond_S) =\bigsqcup_{(\mu_1,\mu_2)\in (\mu+ L^ \diamond)/(L\oplus \Lambda)} V_{\mu_1}(A_{S}) \times \bigl({\mu_2}+\Lambda\bigr). \] \end{enumerate} \end{proposition} \begin{proof} Assertion (1) is shown just as in Lemma~\ref{lem:lambda_perp}. Everything else is immediate from this and the definitions. \end{proof}
\begin{definition}\label{defn:special divisor} For $m\in \mathbb Q_{>0}$ and $\mu\in L^\vee/L$, define the \emph{special cycle} $\mathcal{Z}(m, \mu) \to \mathcal{M}$ as the stack over $\mathcal{M}$ with functor of points \begin{equation}\label{special divisor} \mathcal{Z}(m, \mu) (S) = \left\{ x \in V_\mu( A_S) : Q(x) = m \right\} \end{equation} for any scheme $S\to \mathcal{M}$.
Note that, by (\ref{Q cong}), the stack (\ref{special divisor}) is empty unless the image of $m$ in $\mathbb Q/\mathbb Z$ agrees with $Q(\mu)$.
For later purposes we also define the stacks $\mathcal{Z}(0, \mu)$ in exactly the same way. As the only special endomorphism $x$ with $Q(x)=0$ is the zero map, we have \[ \mathcal{Z}(0, \mu) = \begin{cases}
\emptyset & \hbox{{\rm if }} \mu\neq 0 \cr \mathcal{M} & \hbox{{\rm if }} \mu=0.\cr \end{cases} \] \end{definition}
Once again, fix a maximal lattice $L^ \diamond$ of signature $(n^ \diamond,2)$, equipped with an isometric embedding $L\hookrightarrow L^ \diamond$, so that we have the corresponding finite map of algebraic stacks $\mathcal{M}\to \mathcal{M}^ \diamond$. Set $\Lambda = L^\perp\subset L^ \diamond$. For $m\in\mathbb Q_{\geq 0}$ and $\mu\in L^{ \diamond,\vee}/L^{ \diamond}$, write $\mathcal{Z}^ \diamond(m,\mu)\to\mathcal{M}^ \diamond$ for the stack associated with the pair $(m,\mu)$. The following result is immediate from Proposition~\ref{prop:decomposition_Vmu}.
\begin{proposition}\label{prop:Zmu_functoriality} Fix $\mu\in L^{ \diamond,\vee}/L^ \diamond$. Then there is an isomorphism of $\mathcal{M}$-stacks \[ \mathcal{Z}^ \diamond(m, \mu )\times_{\mathcal{M}^ \diamond}\mathcal{M}\\ \simeq \bigsqcup_{ \substack{ m_1+m_2=m \\ (\mu_1,\mu_2)\in ( \mu+L^ \diamond)/(L\oplus \Lambda)}}
\mathcal{Z}(m_1,\mu_1) \times \Lambda_{m_2,\mu_2}, \] where \[ \Lambda_{m_2,\mu_2} = \{x\in {\mu_2}+\Lambda : Q(x)=m_2\}, \] and $ \mathcal{Z}(m_1,\mu_1) \times \Lambda_{m_2,\mu_2}$ denotes the disjoint union of $\# \Lambda_{m_2,\mu_2}$ copies of $\mathcal{Z}(m_1,\mu_1)$. \end{proposition}
\begin{proposition}\label{prop:Zmu_properties} There is a natural isomorphism \[ \mathcal{Z}(m,\mu)_\mathbb Q \xrightarrow{\simeq} Z(m,\mu) \]
of stacks over $M$. Moreover: \begin{enumerate} \item Suppose that $m>0$. \'Etale locally on the source, $\mathcal{Z}(m,\mu)$ is an effective Cartier divisor on $\mathcal{M}$. \item Suppose also that $n\geq 3$. Then $\mathcal{Z}(m,\mu)$ is flat over $\mathbb Z[1/2]$. If, in addition, $L_{(2)}$ is self-dual, then $\mathcal{Z}(m,\mu)$ is flat over $\mathbb Z$. \end{enumerate} \end{proposition} \begin{proof} Assertion (1) is deduced from Proposition~\ref{prop:Zmu_functoriality}, exactly as in the proof of \cite[Proposition 2.7.4]{AGHMP}, by reducing to the case where $L$ is self-dual over $\mathbb Z_{(p)}$ and using Corollary~\ref{cor:special_endomorphism}.
As for assertion (2), since $\mathcal{Z}(m,\mu)$ is \'etale locally a divisor on $\mathcal{M}$, it fails to be flat exactly when its image in $\mathcal{M}$ contains an irreducible component of $\mathcal{M}_{\mathbb F_p}$ for some prime $p$.
If $L_{(p)}$ is self-dual at $p$, then the argument used in~\cite[Prop. 5.21]{MadapusiSpin} applies to show that $\mathcal{Z}(m,\mu)$ is flat over $\mathbb Z_{(p)}$.
For the other cases, we can now suppose that $p>2$. Choose an auxiliary maximal lattice $L^ \diamond$ that is self-dual over $\mathbb Z_{(p)}$ and an embedding $L\hookrightarrow L^ \diamond$ as usual. If $\Lambda = L^\perp\subset L^ \diamond$, then by Proposition~\ref{prop:model_special_endomorphisms}, we can identify $\mathcal{M}_{(p)}$ with a closed and open substack of the stack $\mathcal{Z}(\Lambda)\to\mathcal{M}^ \diamond_{(p)}$ parameterizing isometric embeddings $\Lambda\hookrightarrow V(A^ \diamond_S)$.
By Proposition~\ref{prop:Zmu_functoriality}, it suffices to show that, for every $m\in\mathbb Q$ and every $\mu\in L^{ \diamond,\vee}/L^ \diamond$, the restriction of $\mathcal{Z}^ \diamond(m,\mu)$ to $\mathcal{Z}(\Lambda)$ is flat over $\mathbb Z_{(p)}$. Equivalently, it is enough to show that the image of the map \[ \mathcal{Z}^ \diamond(m,\mu)\times_{\mathcal{M}^ \diamond_{(p)}}\mathcal{Z}(\Lambda)_{\mathbb F_p} \to \mathcal{Z}(\Lambda)_{\mathbb F_p} \] does not contain an irreducible component of its target.
For this, let $\mathcal{Z}^{\mathrm{pr}}(\Lambda)\subset\mathcal{Z}(\Lambda)$ be as in the proof of Proposition~\ref{prop:model_special_endomorphisms}. We saw there that, under the hypothesis $n\geq 3$, $\mathcal{Z}^{\mathrm{pr}}(\Lambda)$ is a fiberwise dense open substack of $\mathcal{Z}(\Lambda)$. Therefore, it is enough to show that the image of the map \[ \mathcal{Z}^ \diamond(m,\mu)\times_{\mathcal{M}^ \diamond_{(p)}}\mathcal{Z}^{\mathrm{pr}}(\Lambda)_{\mathbb F_p} \to \mathcal{Z}^{\mathrm{pr}}(\Lambda)_{\mathbb F_p} \] does not contain an irreducible component of its target.
Note that the $p$-adic component of $\mu$ is necessarily trivial, and note also that $\mathcal{Z}^{\mathrm{pr}}(\Lambda)_{\mathbb F_p}$ is normal and hence generically smooth. Therefore, the desired assertion follows from~\cite[Corollary 6.18]{MadapusiSpin}. \end{proof}
\subsection{Metrized line bundles} \label{ss:line bundles}
Let $F_\infty : \mathcal{M}(\mathbb C) \to \mathcal{M}(\mathbb C)$ be complex conjugation. An \emph{arithmetic divisor} on $\mathcal{M}$ is a pair \[ \widehat{\mathcal{Z}} = (\mathcal{Z} , \Phi) \]
consisting of a Cartier divisor $\mathcal{Z}$ on $\mathcal{M}$ and a Green function $\Phi$ for $\mathcal{Z}$.
This means that $\Phi$ is an $F_\infty$-invariant smooth $\mathbb R$-valued function defined on the complement of
$\mathcal{Z}(\mathbb C)$ in $\mathcal{M}(\mathbb C)$, such that if $\Psi=0$ is any local equation for $\mathcal{Z}(\mathbb C)$, the function $\Phi+ \log |\Psi|^2$ extends smoothly across the singularity $\mathcal{Z}(\mathbb C)$. A \emph{principal arithmetic divisor} is an arithmetic divisor of the form \[
\widehat{\mathrm{div}}(\Psi) = (\mathrm{div}(\Psi) , - \log|\Psi|^2 ) \] for a rational function $\Psi$ on $\mathcal{M}$. The group of all arithmetic divisors is denoted $\widehat{\mathrm{Div}}(\mathcal{M})$, and its quotient by the subgroup of principal arithmetic divisors is the \emph{arithmetic Chow group} $\widehat{\mathrm{CH}}^1(\mathcal{M})$ of Gillet-Soul\'e \cite{GS}.
A \emph{metrized line bundle} on $\mathcal{M}$ is a line bundle endowed with a smoothly varying $F_\infty$-invariant Hermitian metric on its complex points. The isomorphism classes of metrized line bundles form a group $\widehat{\mathrm{Pic}}( \mathcal{M} )$ under tensor product. As in \cite[III.4]{SouleBook}, there is an isomorphism \begin{equation}\label{pic chow} \widehat{\mathrm{Pic}}( \mathcal{M} ) \xrightarrow{\simeq} \widehat{\mathrm{CH}}^1(\mathcal{M}) \end{equation} defined by sending a metrized line bundle $\widehat{\mathcal{L}}$ on $\mathcal{M}$ to the arithmetic divisor \[
\widehat{\mathrm{div}}(\Psi) = (\mathrm{div}(\Psi) , - \log|| \Psi ||^2). \] for any nonzero rational section $\Psi$ of $\mathcal{L}$.
By assertion (3) of Theorem~\ref{thm:M_model}, we obtain a canonical line bundle $\bm{\omega}$ over $\mathcal{M}$. We call this the \emph{tautological bundle}, or the \emph{line bundle of weight one modular forms}.
Its fiber at a complex point $[(z,g)] \in M(\mathbb C)$ is identified with the isotropic line $\mathbb C z \subset V_\mathbb C$. Using this identification, we define the \emph{Petersson metric} on the fiber $\bm{\omega}_{ [( z,g) ]}$ by $|| z ||^2 = - [z,\overline{z}]$. In this way we obtain the \emph{metrized tautological bundle}
\[
\widehat{ \bm{\omega}} \in \widehat{\mathrm{Pic}}( \mathcal{M} ).
\]
\subsection{Harmonic weak Maass forms} \label{ss:harmonic forms}
We recall some generalities about the Weil representation and vector-valued harmonic forms from
\cite{BF,BKY,BY,KuBorcherds}.
Let $S( \widehat{V})$ be the space of Schwarz functions on $\widehat{V} = V\otimes \mathbb A_f$, and denote by \[ S_L \subset S( \widehat{V} ) \] the (finite dimensional) subspace of functions that are invariant under translation by $\widehat{L} = L\otimes \widehat{\mathbb Z}$, and supported on $\widehat{L}^\vee = L^\vee \otimes \widehat{\mathbb Z}$. We often identify $S_L$ with the space of complex-valued functions on \[ \widehat{L}^\vee / \widehat{ L } \xrightarrow{\simeq} L^\vee / L. \]
In particular, for each $\mu \in L^\vee /L$ there is a corresponding Schwartz function \begin{equation}\label{mu schwartz} \varphi_\mu \in S_L , \end{equation} defined as the characteristic function of $\mu + \widehat{L} \subset \widehat{ V }$.
Write $\widetilde{\mathrm {SL}}_2(\mathbb A)$ for the metaplectic double cover of $\mathrm {SL}_2(\mathbb A)$. This cover splits over $\mathrm {SL}_2(\mathbb Q)$, yielding a canonical injection \begin{equation}\label{meta splitting} \mathrm {SL}_2(\mathbb Q) \hookrightarrow \widetilde{\mathrm {SL}}_2(\mathbb A). \end{equation} Pulling back the cover by the inclusions \[ \mathrm {SL}_2(\mathbb R) \to \mathrm {SL}_2(\mathbb A) , \quad \mathrm {SL}_2(\mathbb A_f ) \to \mathrm {SL}_2(\mathbb A) \] yields double covers \[ \widetilde{\mathrm {SL}} _2(\mathbb R) \to \mathrm {SL}_2(\mathbb R), \quad \widetilde{\mathrm {SL}}_2(\mathbb A_f) \to \mathrm {SL}_2(\mathbb A_f), \] and we define $\widetilde{\mathrm {SL}}_2(\mathbb Z)$ and $\widetilde{\mathrm {SL}}_2(\widehat{\mathbb Z})$ by the cartesian diagrams \[ \xymatrix{ { \widetilde{\mathrm {SL}}_2(\mathbb Z) } \ar[r] \ar[d] & { \widetilde{\mathrm {SL}}_2(\mathbb R) } \ar[d] & { \widetilde{\mathrm {SL}}_2(\widehat{\mathbb Z} ) } \ar[r] \ar[d] & { \widetilde{\mathrm {SL}}_2(\mathbb A_f ) } \ar[d] \\ { \mathrm {SL}_2(\mathbb Z) } \ar[r] & { \mathrm {SL}_2(\mathbb R) } & { \mathrm {SL}_2(\widehat{\mathbb Z} ) } \ar[r] & { \mathrm {SL}_2(\mathbb A_f ) } . } \]
The inclusion (\ref{meta splitting}) induces an injection $ \widetilde{\mathrm {SL}}_2(\mathbb Z) \to \widetilde{\mathrm {SL}}_2(\widehat{\mathbb Z} ), $ denoted $\widetilde{\gamma} \mapsto \widehat{\gamma}$, defined by demanding that the product \[ \widetilde{\gamma} \cdot \widehat{\gamma} \in \widetilde{\mathrm {SL}}_2(\mathbb R) \cdot \widetilde{\mathrm {SL}}_2(\mathbb A_f) \subset \widetilde{\mathrm {SL}}_2(\mathbb A) \] be equal to the image of $\widetilde{\gamma}$ under the composition \[ \widetilde{\mathrm {SL}}_2(\mathbb Z) \to \mathrm {SL}_2(\mathbb Z) \hookrightarrow \widetilde{\mathrm {SL}}_2(\mathbb A). \]
Denote by $\psi_\mathbb Q : \mathbb Q \backslash \mathbb A \to \mathbb C^\times$ the unramified character with archimedean component $\psi_{\mathbb Q,\infty}(x) = e^{2\pi i x}$. The group $\widetilde{\mathrm {SL}}_2(\mathbb A_f)$ acts on $S(\widehat{V})$ via the Weil representation $\omega$ determined by $\psi_\mathbb Q$, and the restriction of this representation to $\widetilde{\mathrm {SL}}_2(\mathbb Z) \subset \widetilde{\mathrm {SL}}_2(\widehat{\mathbb Z} )$ leaves invariant the finite dimensional subspace $S_L$. Denote this representation by \[ \omega_L : \widetilde{\mathrm {SL}}_2(\mathbb Z) \to \mathrm{Aut}(S_L), \] and define the complex conjugate representation $ \overline{\omega}_L : \widetilde{\mathrm {SL}}_2(\mathbb Z) \to \mathrm{Aut}(S_L) $
by \[ \overline{\omega}_L( \widetilde{\gamma} ) \cdot \varphi = \overline{ \omega_L (\widetilde{\gamma}) \cdot \overline{\varphi} }. \] If $\mathrm{dim}(V)$ is even then $\omega_L$ and $\overline{\omega}_L$ factor through $\mathrm {SL}_2(\mathbb Z)$. Note that our $\overline{\omega}_L$ is the representation denoted $\rho_L$ in \cite{Bor98,Bru,BF,BKY,BY}.
Denote by $H_{1- n/2 }(\omega_L)$ the space of harmonic weak Maass forms of weight $1-n/2$ for $\widetilde{\mathrm {SL}}_2(\mathbb Z)$ of representation $\omega_L$, in the sense of \cite[\S 3]{BY}, and denote by \[
S_{1- n/2 }(\omega_L) \subset M^!_{1- n/2 }(\omega_L)\subset H_{1- n/2 }(\omega_L) \] the subspaces of cusp forms and weakly modular forms, respectively. By a result of Bruinier-Funke \cite{BF}, these spaces are related by an exact sequence \begin{equation}\label{BF exact sequence} 0 \to M^!_{1- n/2 }(\omega_L) \to H_{1- n/2 }(\omega_L) \map{\xi} S_{ 1+ n/2 }( \overline{\omega}_L) \to 0, \end{equation} where $\xi$ is a certain explicit differential operator.
As in \cite[(3.4a)]{BY}, any $f\in H_{1- n/2 }(\omega_L)$ has a \emph{holomorphic part} \[ f^+(\tau) = \sum_{ \substack{ m\in D_L^{-1} \mathbb Z \\ m \gg -\infty} } c_f^+(m) \cdot q^m , \] which is a formal $q$-expansion with coefficients \[ c_f^+(m) = \sum_{\mu \in L^\vee /L} c_f^+(m,\mu) \cdot \varphi_\mu \in S_L. \] When the \emph{principal part} \[ P_f(\tau) = \sum_{ m \ge 0} c_f^+(-m) \cdot q^{-m}. \] is integral, in the sense that $c^+_f(-m,\mu) \in \mathbb Z$ for all $m \ge 0$ and $\mu \in L^\vee/L$, we define the corresponding \emph{special divisor} \[ \mathcal{Z}(f) = \sum_{ \substack{ m>0 \\ \mu \in L^\vee / L} } c_f^+(-m,\mu) \mathcal{Z}(m,\mu) \] on $\mathcal{M}$.
There is a natural Green function $\Phi(f)$ for $\mathcal{Z}(f)$, defined as a regularized theta lift as in \cite[(4.7)]{BY}. See also \cite{Bru,BF,BKY}. In particular, we obtain an arithmetic divisor \begin{equation}\label{arithmetic divisor} \widehat{ \mathcal{Z} } (f) = \big( \mathcal{Z}(f) , \Phi(f) \big) \in \widehat{\mathrm{CH}}^1(\mathcal{M} ). \end{equation}
\subsection{Borcherds products}
Suppose \begin{equation}\label{borcherds input} f (\tau) = \sum_{ \substack{ m\in D_L^{-1} \mathbb Z \\ m\gg 0 } } c_f(m)\cdot q^m \in M_{ 1 - n /2 }^!(\omega_L) \end{equation} is a weakly holomorphic form, so that $f=f^+$ and $c_f(m)= c_f^+(m)$.
The following result will be shown in the companion paper~\cite{HMP}, generalizing a result of F. H\"ormann~\cite{Hormann}. Here, we only sketch its proof. For the applications to Colmez's conjecture, we will only require the assertion over primes of good reduction, which is already contained in~\cite{Hormann}.
\begin{theorem}\label{thm:borcherds} Suppose that $n\geq 3$ and that the principal part $P_f(\tau)$ is integral. Then, after replacing $f$ by a multiple $kf$, for any sufficiently divisible $k\in\mathbb Z_{>0}$, there exists a rational section $\Psi(f)$ of $\omega^{ \otimes c_f(0,0) }$, defined over $\mathbb Q$, such that \[
\Phi(f) = - \log|| \Psi (f)||^2 + c_f(0,0) \log(4\pi e^\gamma) . \] Here $\gamma = -\Gamma'(1)$ is the Euler-Mascheroni constant.
In particular, the canonical isomorphism (\ref{pic chow}) produces identifications \begin{align*}
\widehat{\bm{\omega}}^{ \otimes c_f(0,0) } & = \widehat{\mathrm{div}}(\Psi(f)) \\
& = \widehat{ \mathcal{Z} } (f) - c_f(0,0) \cdot \big(0, \log(4\pi e^\gamma) \big) + \widehat{ \mathcal{E} }(f), \end{align*} where $ (0, \log(4\pi e^\gamma) )$ denotes the trivial divisor endowed with the constant Green function $\log(4\pi e^\gamma)$, and $\widehat{ \mathcal{E} }(f) = (\mathcal{E}(f),0)$ is the divisor \[ \mathcal{E}(f) = \mathrm{div}( \Psi (f) ) - \mathcal{Z}(f) \] endowed with the trivial Green function. Moreover, there is a decomposition \[ \mathcal{E}(f) = \sum_{p\mid D_L} \mathcal{E}_p(f) \]
in which the divisor $\mathcal{E}_p(f)$ is supported on the special fiber $\mathcal{M}_{\mathbb F_p}$, and: \begin{itemize} \item If $p$ is odd and $p^2\nmid D_L$ then $\mathcal{E}_p(f)=0$; \item
If $n\geq 5$ then $\mathcal{E}(f) = \mathcal{E}_2(f)$ is supported on $\mathcal{M}_{\mathbb F_2}$.
\item
If $n\ge 5$ and $L_{(2)}$ is self-dual, then $\mathcal{E}(f)=0$.
\end{itemize} \end{theorem}
\begin{proof}[Sketch of proof] For any sufficiently divisible $k$, all the Fourier coefficients of $k\cdot f$ are integral, and the Borcherds lift of $k\cdot f$, after a normalization, descends to a section of $\bm{\omega}^{\otimes k c_f(0,0)}$. Replacing $f$ by this multiple, we take our desired section $\Psi(f)$ to be this descent of the Borcherds lift. It is known that the divisor of $\Psi(f)$ in $M$ is exactly $\mathcal{Z}(f)\vert_M$; see~\cite{Bor98} or \cite{Bru}. Thus $\mathcal{E}(f) = \sum_p \mathcal{E}_p(f)$ is supported in finitely many nonzero characteristics.
Assume that $L_{(p)}$ is self-dual, or that $p$ is odd and $p^2\nmid D_L$, or that $p$ is odd and $n\ge 5$.
To check that $\mathcal{E}_p(f)=0$, it suffices to show that both $\mathcal{Z}(f)$ and $\mathrm{div}(\Psi(f))$ are flat over
$\mathbb Z_{(p)}$. The flatness of $\mathcal{Z}(f)$ follows from Proposition~\ref{prop:Zmu_properties}.
For the flatness of $\mathrm{div}(\Psi(f))$, note that the special fiber $\mathcal{M}_{\mathbb F_p}$ is irreducible, by Theorem~\ref{thm:M_model}.
Thus $\mathrm{div}(\Psi(f))$, if not flat, contains a multiple of the entire special fiber $\mathcal{M}_{\mathbb F_p}$. Since a theory of integral $q$-expansions is now available through~\cite{Madapusi}, we can use the explicit product $q$-expansion of $\Psi(f)$ to check that the support of $\mathrm{div}(\Psi(f))$ cannot contain $\mathcal{M}_{\mathbb F_p}$, and hence that $\mathrm{div}(\Psi(f))$ is also flat. To be more precise, the Fourier coefficients in the $q$-expansion of $\Psi(f)$ are integral and without a non-trivial common divisor. Hence, the mod $p$ reduction of such an expansion cannot vanish identically. The $q$-expansion principle now implies that the form $\Psi(f)$ also cannot vanish identically along the special fiber. \end{proof}
\section{Big CM cycles on orthogonal Shimura varieties}\label{ss:big CM}
As in Section~\ref{s:cm shimura}, we will fix a CM field $E$ with totally real subfield $F$. We will also take $\mathbb Q^{\mathrm{alg}}$ to be the algebraic closure in $\mathbb C$ of $\mathbb Q$ and write $\Gamma_{\mathbb Q}$ for the absolute Galois group $\mathrm{Gal}(\mathbb Q^{\mathrm{alg}}/\mathbb Q)$. We will also fix a distinguished embedding $\iota_0:E \to \mathbb Q^\mathrm{alg}$.
The goal here is to embed the zero dimensional Shimura variety from Section~\ref{s:cm shimura} into the GSpin Shimura varieties from Section~\ref{s:orthogonal shimura}, and to study the interaction between the various `motives' that live over the two spaces. The main result is Corollary~\ref{cor:special end structure}, which explains the structure of the space of special endomorphisms associated with points of the zero dimensional Shimura variety.
\subsection{Hermitian spaces} \label{ss:hermitian}
Let $(\mathscr{V},\langle\cdot,\cdot\rangle)$ be a rank one Hermitian space over $E$ that is negative definite at $\iota_0$, and positive definite at the remaining archimedean places. The assignment \[ x\mapsto \langle x,x\rangle = \mathscr{Q}(x) \] induces a quadratic form $\mathscr{Q}:\mathscr{V} \to F$ on the underlying $F$-vector space of signature \begin{equation}\label{sig} \mathrm{sig}( \mathscr{V} ) = \big( (0,2), (2,0), \ldots, (2,0) \big). \end{equation}
The Clifford algebra of $(\mathscr{V},\mathscr{Q})$ is a quaternion algebra over $F$, with a $\mathbb Z/2\mathbb Z$-grading \[ C(\mathscr{V} ) = C^+(\mathscr{V})\oplus C^-(\mathscr{V}). \] The even part $C^+(\mathscr{V})$ is isomorphic to $E$ as an $F$-algebra. We will fix an isomorphism $E\xrightarrow{\simeq}C^+(\mathscr{V})$ of $F$-algebras. Now, the odd part $C^-(\mathscr{V})$ is identified with the $F$-vector space $\mathscr{V}$. The action of $E$ on $\mathscr{V}$ given by left multiplication in the Clifford algebra is none other than the given $E$-module structure on $\mathscr{V}$.
\begin{remark}\label{rem:hermitian construction} If we fix any $E$-module isomorphism $\mathscr{V}\simeq E$, there is a unique $\xi \in F^\times$ such that the hermitian form on $\mathscr{V}$ is identified with the hermitian form $ \langle x,y\rangle = \xi x \overline{y} $ on $E$. The element $\xi$ is negative at $\iota_0$ and positive at $\iota_1,\ldots, \iota_{d-1}$, and the isomorphism class of $\mathscr{V}$ is uniquely determined by \[ \xi \in F^\times/ \mathrm{Nm}_{E/F} ( E^\times). \] Conversely, if we start with any CM field $E$ with totally real subfield $F$, and any $\xi \in F^\times$ negative at $\iota_0$ and positive at $\iota_1,\ldots, \iota_{d-1}$, we obtain an $F$-quadratic space $ (\mathscr{V},\mathscr{Q}) = (E , \xi\cdot \mathrm{Nm}_{E/F} ) $ of signature (\ref{sig}) as above. \end{remark}
Let $\chi:\mathbb A_F^\times \to \{\pm 1\}$ be the quadratic character determined by $E/F$. Keeping the notation of Remark \ref{rem:hermitian construction}, for every place $v$ of $F$ define the \emph{local invariant} \[ \mathrm{inv}_v(\mathscr{V}) = \chi_v(\xi) \in \{ \pm 1\} . \] Thus $\mathrm{inv}_v(\mathscr{V})=1$ if and only if $\xi$ is a norm from $E_v^\times$, and $\alpha\in F_v^\times$ is represented by $\mathscr{V}_v$ if and only if $\chi_v(\alpha)=\mathrm{inv}_v( \mathscr{V} )$. The hermitian space $\mathscr{V}$ is uniquely determined by its collection of local invariants, and the product of the local invariants is $1$.
\begin{definition}\label{def:nearby} Suppose that $\mathfrak{p} \subset \mathcal O_F$ is a prime ideal nonsplit in $E$. The \emph{nearby hermitian space} $ { \empty^{ \mathfrak{p} }} \mathscr{V}$ is obtained from $\mathscr{V}$ by interchanging invariants at $\iota_0$ and $\mathfrak{p}$. In other words, $ { \empty^{ \mathfrak{p} }} \mathscr{V}$ is the unique rank one hermitian space over $E$ with \[ \mathrm{inv}_v( { \empty^{ \mathfrak{p} }} \mathscr{V} ) \xrightarrow{\simeq} \begin{cases} - \mathrm{inv}_v( \mathscr{V} ) & \mbox{if } v \in \{ \mathfrak{p} , \iota_0 \} \\ \mathrm{inv}_v( \mathscr{V} ) & \mbox{otherwise.} \end{cases} \] The (positive definite) hermitian form on $ { \empty^{ \mathfrak{p} }} \mathscr{V}$ is denoted $ { \empty^{ \mathfrak{p} }} \langle x_1,x_2\rangle$, and the associated $F$-quadratic form is $ { \empty^{ \mathfrak{p} }} \mathscr{Q}(x) = { \empty^{ \mathfrak{p} }} \langle x,x\rangle$. \end{definition}
\subsection{Reflex algebras and Clifford algebras} \label{ss:reflex algebra}
Associated with $(\mathscr{V},\mathscr{Q})$ is the $\mathbb Q$-quadratic space \begin{equation}\label{isometric equality}
(V,Q) = (\mathscr{V} , \mathrm{Tr}_{F/\mathbb Q} \circ \mathscr{Q}) \end{equation} of signature $(n,2)=(2d-2,2)$.
Let $E^\sharp$ be the total reflex algebra associated with $E$. It is an \'etale $\mathbb Q$-algebra whose associated $\Gamma_\mathbb Q$-set is canonically identified with the set $\mathrm{CM}(E)$ of CM types for $E$; see \S~\ref{ss:abelian schemes}.
\begin{proposition}\label{prop:reflex inclusion} The relation (\ref{isometric equality}) determines a distinguished embedding of $\mathbb Q$-algebras $ E^\sharp \hookrightarrow C^+(V). $ \end{proposition} \begin{proof} The $E$-action on $V=\mathscr{V}$ gives us a decomposition \begin{equation}\label{eqn:V_Q_alg_decomp} V_{\mathbb Q^{\mathrm{alg}}} = \bigoplus_{\iota\in\mathrm{Emb}(E)}\mathscr{V}(\iota), \end{equation} into one-dimensional $\mathbb Q^{\mathrm{alg}}$-vector spaces, where $\mathscr{V}(\iota) = \mathscr{V}\otimes_{E,\iota}\mathbb Q^{\mathrm{alg}}$. By construction, the quadratic form $Q$ induces a perfect pairing \[ \mathscr{V}(\iota) \times \mathscr{V}(\overline{\iota}) \to \mathbb Q^{\mathrm{alg}}. \] Therefore, for each embedding $\iota_i:F\to\mathbb Q^{\mathrm{alg}}$, $i=0,1,\ldots,d-1$, $Q$ restricts to a non-degenerate form on \[ \mathscr{V}_i = \mathscr{V}\otimes_{F,\iota_i}\mathbb Q^{\mathrm{alg}}. \]
If $i\neq j$ then $\mathscr{V}_i$ and $\mathscr{V}_j$ are orthogonal, and so we obtain a $\mathbb Q^{\mathrm{alg}}$-linear orthogonal decomposition \[ V_{\mathbb Q^{\mathrm{alg}}} = \bigoplus_{i=0}^{d-1}\mathscr{V}_i \]
into two-dimensional non-degenerate quadratic subspaces. In turn, this gives us a natural $\Gamma_{\mathbb Q}$-stable commutative subalgebra \begin{equation}\label{eqn:fake_reflex_algebra} \bigotimes_{i=0}^{d-1}C^+\bigl(\mathscr{V}_i\bigr)\subset C^+(V_{\mathbb Q^{\mathrm{alg}}}), \end{equation} which descends to a $\mathbb Q$-subalgebra $B\subset C^+(V)$.
We claim that there is a canonical isomorphism of $\mathbb Q$-algebras $E^\sharp\xrightarrow{\simeq}B$. For this, it is enough to show that there is a canonical isomorphism of $\Gamma_{\mathbb Q}$-sets: \[ \mathrm{Hom}_{\mathbb Q-\mathrm{alg}}(B,\mathbb Q^{\mathrm{alg}})\xrightarrow{\simeq}\mathrm{CM}(E). \]
But this is clear from the description in~\eqref{eqn:fake_reflex_algebra}, since, for each $i=0,1,\ldots,d-1$, we have canonical isomorphisms of $\mathbb Q^{\mathrm{alg}}$-algebras with an involution: \[ E\otimes_{F,\iota_i}\mathbb Q^{\mathrm{alg}}\xrightarrow{\simeq}C^+(\mathscr{V})\otimes_{F,\iota_i}\mathbb Q^{\mathrm{alg}}\xrightarrow{\simeq}C^+(\mathscr{V}_i). \] \end{proof}
\subsection{Morphisms of Shimura varieties}\label{ss:shimura data}
Assume now that $d>1$, so that $n=2d-2 > 0$. Write $H$ for $C(V)$, viewed as a faithful representation of \[ G=\mathrm{GSpin}(V) \] via the left multiplication action of $C(V)$ on itself. Using the inclusion $E^\sharp \subset C(V)$ of Proposition \ref{prop:reflex inclusion}, the group $T_{E^\sharp}$ also acts faithfully on $H$ via left multiplication. The torus $T=T_E/T_F^1$ can be identified with the intersection of $G$ and $T_{E^\sharp}$ inside of $\mathrm{GL}(H)$. In other words, there is a cartesian diagram \[ \xymatrix{
{ T } \ar[rr]^{ \mathrm{Nm}^\sharp } \ar[d] & & {T_{ E^\sharp } } \ar[d] \\
G \ar[rr] && { \mathrm{GL}(H) } } \] in which all arrows are injective. Here, $\mathrm{Nm}^\sharp$ is the total reflex norm defined in \S~\ref{ss:abelian schemes}.
Now, we have canonical identifications \[ \mathrm{Res}_{E/\mathbb Q}\mathrm{SO}(\mathscr{V}) = T_E^1 = T_{so} \] of tori over $\mathbb Q$. This exhibits $T_{so}$ as a maximal torus in $\mathrm{SO}(V)$, and it also identifies $V$ with the standard representation $V_0$ of $T$. Moreover, we have a commutative diagram \[ \xymatrix{ { 1 } \ar[r] & { \mathbb{G}_m } \ar[r] \ar@{=}[d] & { T } \ar[r]^{\theta} \ar[d] & { T_{so} } \ar[d] \ar[r] & 1 \\ { 1 } \ar[r] & { \mathbb{G}_m } \ar[r] & { G } \ar[r] & { \mathrm{SO}(V) } \ar[r] & 1 } \] with exact rows, and all vertical arrows are injective.
Via the decomposition~\eqref{eqn:V_Q_alg_decomp}, we obtain a $T(\mathbb C)$-stable line \[ z^{cm} = \mathscr{V}(\iota_0)_{\mathbb C} \subset \mathscr{V}_\mathbb C. \] This line is isotropic with respect to the quadratic form $\mathrm{Tr}_{F/\mathbb Q} \circ \mathscr{Q}$, and we use (\ref{isometric equality}) to view $z^{cm}$ as a point of the hermitian domain (\ref{hermitian domain}).
The morphism $T \to G$ induces a morphism of Shimura data \begin{equation}\label{eqn:morphism_data} ( T , \{\mu_0\} ) \to (G,\mathcal{D}) \end{equation} mapping $\mu_0$ to $z^{cm}\in\mathcal{D}$.
As in \S \ref{s:orthogonal shimura}, let $L\subset V$ be a maximal lattice of discriminant $D_L$. Recall that the choice of maximal lattice determines a compact open subgroup $K \subset G(\mathbb A_f)$ and a Shimura variety (\ref{gspin shimura}), with a canonical model $M \to \mathrm{Spec}(\mathbb Q)$.
Consider the compact open subgroup $K_{L,0} = K_0 \cap K \subset T(\mathbb A_f)$. In \S~\ref{s:cm shimura}, we associated with it a zero dimensional Shimura variety $Y_{K_{L,0}}$, as well as a normal integral model $\mathcal{Y}_{K_{L_0}}$ over $\mathcal O_E$. From now on we abbreviate \[ \mathcal{Y}= \mathcal{Y}_{K_{L,0}}. \] This is an arithmetic curve over $\mathcal O_E$, whose generic fiber we denote by $Y\to \mathrm{Spec}(E)$. By the theory of canonical models, we now obtain a morphism \begin{equation}\label{cm morphism}
Y\to M \end{equation} of $\mathbb Q$-stacks, induced by the morphism of Shimura data~\eqref{eqn:morphism_data}.
\begin{proposition}\label{prop:morphism_integral_Y} The map~\eqref{cm morphism} extends to a map of $\mathbb Z$-stacks \begin{equation*}
\mathcal{Y} \to \mathcal{M} \end{equation*} \end{proposition}
\begin{proof} This follows from Proposition~\ref{prop:abelian scheme reduction} and assertion (4) of Proposition~\ref{prop:integral_model_bad_p}. \end{proof}
We will need some information about the compatibility of this map with constructions of automorphic sheaves. For this, fix a prime $\mathfrak{q}\subset\mathcal O_E$ lying above a rational prime $p$, and an auxiliary quadratic lattice $L^ \diamond$ of signature $(n^ \diamond,2)$, self-dual at $p$ and admitting $L$ as an isometric direct summand. Associated with it is the Shimura variety $M^ \diamond$ with a smooth integral canonical model $\mathcal{M}^ \diamond_{(p)}$ over $\mathbb Z_{(p)}$ and a finite map $\mathcal{M}_{(p)}\to \mathcal{M}^ \diamond_{(p)}$.
From Propositions~\ref{prop:de_rham_realization} and~\ref{prop:cris_realization}, we obtain functors $N_{(p)}\mapsto\bm{N}^ \diamond_{\mathrm{dR}}$ and $N_{(p)}\mapsto\bm{N}^ \diamond_{\mathrm{cris}}$ from $G^ \diamond_{(p)}\define \mathrm{GSpin}(L^ \diamond_{\mathbb Z_{(p)}})$-representations to filtered vector bundles over $\mathcal{M}^ \diamond_{(p)}$ and $F$-crystals over $\mathcal{M}^ \diamond_{\mathbb F_p}$, respectively. On the other hand, any $G^ \diamond_{(p)}$-representation $N_{(p)}$ gives a $\mathbb Q$-representation $N = N_{(p)}[p^{-1}]$ of $T$, and a $K_{0,L}$-stable lattice $N_p = N_{(p)}\otimes\mathbb Z_p\subset N_{\mathbb Q_p}$. Therefore, by Proposition~\ref{prop:realizations integral model} (or more precisely, its proof), it gives us a filtered vector bundle $\bm{N}_{\mathrm{dR}}$ over $\mathcal{Y}_{(\mathfrak{q})} = \mathcal{Y}\otimes_{\mathcal O_E}\mathcal O_{E,(\mathfrak{q})}$, and an $F$-crystal $\bm{N}_{\mathrm{cris}}$ over $\mathcal{Y}_{\mathbb F_{\mathfrak{q}}}$.
\begin{proposition} \label{prop:realizations compatibility} There are canonical isomorphisms \[ \bm{N}_{\mathrm{dR}}\xrightarrow{\simeq} \bm{N}_{\mathrm{dR}}^ \diamond\vert_{\mathcal{Y}_{(\mathfrak{q})}}\;;\;\bm{N}_{\mathrm{cris}}\xrightarrow{\simeq} \bm{N}_{\mathrm{cris}}^ \diamond\vert_{\mathcal{Y}_{\mathbb F_{\mathfrak{q}}}} \] of filtered vector bundles and $F$-crystals, respectively. \end{proposition}
We omit the proof of the proposition, which follows immediately from unwinding the constructions. The main point is that both constructions, when restricted to the completed \'etale local ring at a point $y\in \mathcal{Y}(\mathbb F_{\mathfrak{q}})$, recover the functors $N_p\mapsto\bm{N}_{\mathrm{dR},\mathcal O_y}$ and $N_p\mapsto\bm{N}_{\mathrm{cris},y}$ of Corollary~\ref{cor:realizations y}, obtained from Kisin's functor $\mathfrak{M}$
For any prime $p$, note that the $F$-action on $V$ gives us an orthogonal decomposition: \[ V_{\mathbb Q_p} = \bigoplus_{\mathfrak{p}\mid p}V_{\mathfrak{p}}, \] where $\mathfrak{p}$ ranges over the $p$-adic places of $F$, and where we have set $V_{\mathfrak{p}} = F_{\mathfrak{q}}\otimes_FV$. For each $\mathfrak{p}\mid p$, set \[ L_{\mathfrak{p}} = L_p \cap V_{\mathfrak{p}}\subset V_{\mathfrak{p}}. \]
\begin{definition}\label{defn:D bad} Call a prime $p$ \emph{good for $L$}, or simply \emph{good}, if the following conditions hold: \begin{itemize}
\item For every $\mathfrak{p}\mid p$ unramified in $E$, the $\mathbb Z_p$-lattice $L_{\mathfrak{p}}$ is $\mathcal O_{E,{\mathfrak{p}}}$-stable and self-dual for the induced $\mathbb Z_p$-valued quadratic form.
\item For every $\mathfrak{p}\mid p$ ramified in $E$, the $\mathbb Z_p$-lattice $L_{\mathfrak{p}}$ is maximal for the induced
$\mathbb Z_p$-valued quadratic form,
and there exists an $\mathcal O_{E,{\mathfrak{q}}}$-stable lattice $\Lambda_{\mathfrak{p}}\subset V_{\mathfrak{p}}$ such that
\[
\Lambda_{\mathfrak{p}}\subset L_{\mathfrak{p}}\subsetneq \mathfrak{d}_{E_{\mathfrak{q}}/F_{\mathfrak{p}}}^{-1}\Lambda_{\mathfrak{p}}.
\]
Here $\mathfrak{q}\subset \mathcal O_E$ is the unique prime above $\mathfrak{p}$. \end{itemize}
All but finitely many primes are good: Choose any $\mathcal O_E$-stable lattice $\Lambda\subset L$. Then, for all but finitely many primes $p$, $\Lambda_{\mathbb Z_p}=L_{\mathbb Z_p}$ will be self-dual and hence good.
We will call a prime \emph{bad} if it is not good, and we let $D_{bad}$ be the product of the bad primes. If we wish to make its dependence on the lattice $L$ explicit, we will write $D_{bad,L}$ for this quantity. \end{definition}
\begin{lemma}\label{lem:level_subgroup} For every $p\nmid D_{bad}$, we have \[ K_{L,0,p} = K_{0,p} \subset T(\mathbb Q_p). \] In particular, $\mathcal{Y}$ is finite \'etale over $\mathcal O_E[D_{bad}^{-1}]$. \end{lemma} \begin{proof} Note that $K_{L,0,p}$ contains the subgroup $\mathbb Z_p^\times$ of scalars. Therefore, it is enough to show that the image $K_{0,p,so}$ of $K_{0,p}$ in $T_{so}(\mathbb Q_p)$ is contained in the discriminant kernel of $L_{\mathbb Z_p}$. This is easy to see from the explicit description of $L_{\mathbb Z_p}$ in Definition~\ref{defn:D bad}, as well as of $K_{0,p,so}$ in~\eqref{eqn:K0p so}.
There are two main points: First, $K_{0,p,so}$ preserves all $\mathcal O_{E,p}$-stable lattices in $L_{\mathbb Z_p}$. Second, for any prime $\mathfrak{p}\subset\mathcal O_F$ ramified in $E$ with $\mathfrak{q}\subset\mathcal O_E$ the prime above it, if $\alpha\in\mathcal O_{E,\mathfrak{q}}$, then $\overline{\alpha}$ and $\alpha$ are congruent mod $\mathfrak{d}_{E_{\mathfrak{q}}/F_{\mathfrak{p}}}$.
Combining these two facts, if $\Lambda_{\mathfrak{p}}\subset L_{\mathfrak{p}}$ is a maximal $\mathcal O_{E,\mathfrak{q}}$-stable lattice, then $K_{0,p,so}$ stabilizes $\Lambda_{\mathfrak{p}}$ and acts trivially on ${\mathfrak{d}_{E_{\mathfrak{q}}/F_{\mathfrak{p}}}^{-1}\Lambda_{\mathfrak{p}} }/ {\Lambda_{\mathfrak{p}}}$. This implies that it stabilizes $L_{\mathfrak{p}}$ and acts trivially on $L^\vee_{\mathfrak{p}}/L_{\mathfrak{p}}$. Since $L^\vee/L$ is a subquotient of $\bigoplus_{\mathfrak{p}}L^\vee_{\mathfrak{p}}/L_{\mathfrak{p}}$, we find that $K_{0,p,so}$ preserves $\widehat{L}$ and acts trivially on $L^\vee/L$. \end{proof}
\subsection{The space of special endomorphisms}\label{ss:special end zero cycle}
\begin{proposition} \label{prop:no special char 0} Suppose that $y\in \mathcal{Y}(\mathbb C)$. Then $V(A_y) = 0$. \end{proposition} \begin{proof} Let $z^{cm} = \mathscr{V}(\iota_0)_\mathbb C \subset V_\mathbb C$ be as in~\S\ref{ss:shimura data}. The proposition amounts to the statement that there are no positive elements $x\in V$ that are orthogonal to $z^{cm}$. But if such an $x$ existed, then it would generate the one-dimensional $E$-vector space $\mathscr{V} = V$, and, since $z^{cm}\subset \mathscr{V}_\mathbb C$ is $E$-stable, this would imply that \emph{every} element of $V$ is orthogonal to $z^{cm}$, which is clearly impossible. \end{proof}
Recall that $V$ is isomorphic as a $T$-representation to the standard representation $V_0 = V(H_0,c)$ from~\S\ref{ss:standard}. If we are viewing $V$ or $V_0$ as an $E$-module, we will emphasize this by writing $\mathscr{V}$ and $\mathscr{V}_0$, instead. There is a canonical Hermitian form on $\mathscr{V}_0$: For $x,y\in \mathscr{V}_0$, we define $\langle x,y\rangle_0 \in E$ by the relation $x\circ y =\langle x,y\rangle_0$ as elements of $\mathrm{End}(H_0)$. Under the isomorphism $\mathscr{V} \xrightarrow{\simeq} \mathscr{V}_0$, the Hermitian form on $\mathscr{V}$ induced from $\mathscr{Q}$ is carried to the form $\xi\langle x,y\rangle_0$, for some element $\xi\in F$ such that $\iota_0(\xi)<0$ and $\iota_j(\xi)>0$, for $j>0$.
The lattice $L_{\widehat{\mathbb Z}}\subset V_{\mathbb A_f}$ is $K_{0,L}$-stable, and we have a $K_{0,L}$-equivariant embedding $L_{\widehat{\mathbb Z}}\hookrightarrow\mathrm{End}_{C(L)}(H_{\widehat{\mathbb Z}})$. From this data, and the constructions in \S~\ref{ss:sheaves_i} and \S~\ref{ss:sheaves_ii}, we obtain embeddings \begin{equation}\label{eqn:V embedding zero} \bm{V}_{?} \hookrightarrow \underline{\mathrm{End}}_{C(L)}(\bm{H}_?)\vert_{\mathcal{Y}} \end{equation} of sheaves over $\mathcal{Y}$ for $?=B,\ell,\mathrm{dR},\mathrm{cris}$. The images of these embeddings are local direct summands of their targets when $? = B,\ell$, but not necessarily when $?=\mathrm{dR},\mathrm{cris}$. However, we have an embedding \begin{equation}\label{eqn:V embedding zero isogeny} \bm{V}_{\mathrm{cris},\mathbb Q} \hookrightarrow \underline{\mathrm{End}}_{C(L)}(\bm{H}_\mathrm{cris})_{\mathbb Q}\vert_{\mathcal{Y}} \end{equation} in the isogeny category associated with the category of $F$-crystals over $\mathcal{Y}$.
The next result is clear from the definitions and Proposition~\ref{prop:realizations compatibility}. \begin{proposition} \label{prop:special zero dim realizations} For any $\mathcal{Y}$-scheme $S$, and any prime $p$, \[ V(A_S[p^\infty])\subset\mathrm{End}_{C(L)}(A_S[p^\infty]) \] consists precisely of those endomorphisms whose homological realizations land in the images of the embedding~\eqref{eqn:V embedding zero} for $?=p$ over $S[p^{-1}]$, and in the embedding~\eqref{eqn:V embedding zero isogeny} for $?=\mathrm{cris}$ over $S_{\mathbb F_p}$. In particular, \[ V(A_S)\subset\mathrm{End}_{C(L)}(A_S) \] consists of those endomorphisms whose $\ell$-adic realizations over $S[\ell^{-1}]$ land in the image of the embedding~\eqref{eqn:V embedding zero}, and whose crystalline realizations over $S_{\mathbb F_p}$ land in the embedding~\eqref{eqn:V embedding zero isogeny}. \end{proposition}
Fix a rational prime $p$. Let $\mathfrak{q} \subset \mathcal O_E$ be a prime above $p$, let $\mathfrak{p} \subset \mathcal O_F$ be the prime below $\mathfrak{q}$.
\begin{proposition} \label{prop:special ordinary} If $\mathfrak{p}$ is split in $F$, then \[ V(A_y) = V(A_y[p^\infty]) = 0 \] for all $y\in \mathcal{Y}(\mathbb F_{\mathfrak{q}}^{\mathrm{alg}})$. \end{proposition} \begin{proof}
Indeed, $V(A_y[p^\infty]) \subset \bm{V}_{\mathrm{cris},y}[p^{-1}]^{\varphi = 1} = 0$, by Proposition~\ref{prop:v0 cris realization}. \end{proof}
By Propositions~\ref{prop:no special char 0} and~\ref{prop:special ordinary}, for a geometric point $y$ of $\mathcal{Y}$, if $V(A_y)\neq 0$, then $y$ must be an $\mathbb F_{\mathfrak{q}}^{\mathrm{alg}}$-valued point with $\mathfrak{q}\subset \mathcal O_E$ the unique prime lying above a prime $\mathfrak{p}\subset \mathcal O_F$ that is not split in $E$.
Until otherwise specified, we will assume from now on that we have fixed the data of such $\mathfrak{p}, \mathfrak{q}$ and $y$. In this case, by Proposition~\ref{prop:abelian scheme reduction}, the abelian variety $A_y$ is supersingular. Therefore, for any prime $\ell$, the natural map \[ \mathbb Z_\ell\otimes\mathrm{End}(A_y)\to \mathrm{End}(A_y[\ell^\infty]). \] is an isomorphism. This implies that, if $\ell\neq p$, then the natural map \begin{equation}\label{eqn:V ell isom} V(A_y[\ell^\infty]) \to \bm{V}_{\ell,y} \end{equation} is an isomorphism.
Also, if $\ell=p$, then the natural map \begin{equation}\label{eqn:V cris isom} V(A_y[p^\infty])_\mathbb Q \to \bm{V}_{\mathrm{cris},y}[p^{-1}]^{\varphi=1} \end{equation} is also an isomorphism. Moreover, by Proposition~\ref{prop:v0 cris realization}, $\bm{V}_{\mathrm{cris},y}[p^{-1}]$ is generated by its $\varphi$-invariants.
For any $?$, since $E$ acts $T$-equivariantly on $V$, we have a natural map $E\to \mathrm{End}(\bm{V}_?)_\mathbb Q$ giving an action of $E$ on $\bm{V}_?$ in the appropriate isogeny category. In particular, if $y$ is as above, then, via the isomorphisms~\eqref{eqn:V ell isom} and~\eqref{eqn:V cris isom}, the space $V(A_y[\ell^\infty])_\mathbb Q$ has an $E$-action, making it a rank $1$ module over $\mathbb Q_\ell\otimes_{\mathbb Q}E$. If we want to emphasize this structure, we will write $\mathscr{V}(A_y[\ell^\infty])_\mathbb Q$ for this space, and $\mathscr{V}(A_y[\ell^\infty])$ for the lattice $V(A_y[\ell^\infty])$ within it.
Recall that there is a natural quadratic form $Q$ on $V(A_y[\ell^\infty])$ induced from composition in $\mathrm{End}(A_y[\ell^\infty])$. There is now a unique Hermitian form $\langle\cdot,\cdot\rangle_{\ell}$ on $\mathscr{V}(A_y[\ell^\infty])_\mathbb Q$ with associated $\mathbb Q_\ell\otimes_{\mathbb Q}F$-quadratic form $\mathscr{Q}_\ell(x) = \langle x,x\rangle_\ell$ such that, for any $x$, we have: \[ Q(x) = \mathrm{Tr}_{(\mathbb Q_\ell\otimes_{\mathbb Q}F)/\mathbb Q_{\ell}}(\mathscr{Q}_\ell(x)). \]
Set \[ \mathscr{V}(A_y[\infty]) = \prod_\ell \mathscr{V}(A_y[\ell^\infty]). \] Then $\mathscr{V}(A_y[\infty])_\mathbb Q$ has the structure of a Hermitian space over $\mathbb A_{f,E}$.
\begin{proposition} \label{prop:nearby hermitian space ell} The Hermitian space $\mathscr{V}(A_y[\infty])_\mathbb Q$ is isometric to $ { \empty^{ \mathfrak{p} }} \mathscr{V}_{\mathbb A_f}$, where $ { \empty^{ \mathfrak{p} }} \mathscr{V}$ is the nearby Hermitian space from Definition~\ref{def:nearby}. \end{proposition} \begin{proof} For each prime $\ell\neq p$,~\eqref{eqn:V ell isom} shows that $\mathscr{V}(A_y[\ell^\infty])$ is isometric to $L_{\mathbb Z_\ell}$. This shows that $\mathscr{V}(A_y[\infty])_\mathbb Q$ is isometric to $\mathscr{V}_{\mathbb A_f}$, and hence to $ { \empty^{ \mathfrak{p} }} \mathscr{V}_{\mathbb A_f}$, away from the prime $p$.
Now consider what happens at the prime $p$. By~\eqref{eqn:V cris isom} there is an isometry \[ \mathscr{V}(A_y[p^\infty])_\mathbb Q \xrightarrow{\simeq} \bm{V}_{\mathrm{cris},y}[p^{-1}]^{\varphi = 1}, \] and there is an orthogonal decomposition \[ \bm{V}_{\mathrm{cris},y}[p^{-1}] = \bigoplus_{\mathfrak{p}'\mid p}\bm{V}[p^{-1}]_{\mathrm{cris},\mathfrak{p}'}, \] where $\mathfrak{p}'$ ranges over the primes in $\mathcal O_F$ lying above $p$. By the proof of Proposition~\ref{prop:v0 cris realization}, for each $\mathfrak{p}'$ we have \[ \bm{V}[p^{-1}]_{\mathrm{cris},\mathfrak{p}'} = V(\bm{H}_{0,\mathrm{cris},\mathfrak{p}'},c)[p^{-1}]. \] Under this isomorphism, the Hermitian form on $\bm{V}[p^{-1}]_{\mathrm{cris},\mathfrak{p}'}$ is carried to the form $\xi\langle \cdot,\cdot\rangle$, where $\langle\cdot,\cdot\rangle$ is the Hermitian form induced from composition in $\mathrm{End}(\bm{H}_{0,\mathrm{cris},\mathfrak{p}'})$.
If $\mathfrak{p}'\neq \mathfrak{p}$, there is an isomorphism of $F$-crystals \[ W_{\mathfrak{p}'}\otimes_{\mathcal O_F}H_0 \xrightarrow{\simeq}\bm{H}_{0,\mathrm{cris},\mathfrak{p}'}, \] where the left hand side is equipped with the semi-linear map $\mathrm{Fr}^{d_0}\otimes 1$. Therefore, we obtain an isomorphism \begin{equation*}
\bm{V}[p^{-1}]_{\mathrm{cris},\mathfrak{p}'}^{\varphi=1} = F_{\mathfrak{p}'}\otimes_{F}\mathscr{V}_0 \end{equation*} carrying the Hermitian form on the left hand side to $\xi\langle\cdot,\cdot\rangle_0$. This shows that $\mathscr{V}(A_y[\infty])_\mathbb Q$ is isometric to $\mathscr{V}_{\mathbb A_f}$ and hence to $ { \empty^{ \mathfrak{p} }} \mathscr{V}_{\mathbb A_f}$ away from the place $\mathfrak{p}$.
Finally, if $\mathfrak{p}' = \mathfrak{p}$, the $F$-crystal $\bm{H}_{0,\mathrm{cris},\mathfrak{p}}$ is the Dieudonn\'e $F$-crystal of a Lubin-Tate group over $\mathcal O_{E,\mathfrak{q}}$ associated with some uniformizer $\pi\in E_{\mathfrak{q}}$. If $\mathfrak{q}$ is unramified over $F$ and $\pi$ is chosen to lie in $F_{\mathfrak{p}}$, then Proposition~\ref{prp:unramified_vcris} shows that we have an isomorphism \[ \bm{V}[p^{-1}]_{\mathrm{cris},\mathfrak{p}}^{\varphi = 1} \xrightarrow{\simeq} E_{\mathfrak{q}}\otimes_E \mathscr{V}_0 \] carrying the Hermitian form on the left hand side to $\pi\xi\langle \cdot,\cdot\rangle_0$.
If $\mathfrak{q}$ is ramified over $F$ and $\pi$ is chosen to lie in $F_{\mathfrak{p}}$, then Proposition~\ref{prp:unramified_vcris} shows that we have an isomorphism: \[ \bm{V}[p^{-1}]_{\mathrm{cris},\mathfrak{p}}^{\varphi = 1} \xrightarrow{\simeq} E_{\mathfrak{q}}\otimes_E \mathscr{V}_0 \] carrying the Hermitian form on the left hand side to $\gamma\xi\langle \cdot,\cdot\rangle_0$, where $\gamma = \beta\overline{\beta}$, for some $\beta\in W_{\mathfrak{q}}^\times$ satisfying $\pi\mathrm{Fr}^{d_0}(\beta) = \overline{\pi}\beta$.
In either case, it is easily checked that this establishes an isometry of $\bm{V}[p^{-1}]_{\mathrm{cris},\mathfrak{p}}^{\varphi=1}$ with $ { \empty^{ \mathfrak{p} }} \mathscr{V}_{\mathfrak{p}}$. This finishes the proof of the proposition. \end{proof}
\begin{proposition}\label{prop:cm special supersingular} Suppose that $\mathfrak{p}$ is not split in $E$ and that $\mathfrak{q}\subset \mathcal O_E$ is the unique prime above it. Fix a point $y\in\mathcal{Y}(\mathbb F_{\mathfrak{q}}^{\mathrm{alg}})$. Then $A_y$ is a supersingular abelian variety. Moreover: \begin{enumerate} \item $V(A_y)\neq 0$; \item The natural map \begin{equation}\label{eqn:special end lots} \widehat{\mathbb Z}\otimes_{\mathbb Z} V(A_y)\to V(A_y[\infty]) \end{equation} is an isometry of quadratic spaces over $\widehat{\mathbb Z}$. \end{enumerate} \end{proposition} \begin{proof} It was already observed above that $A_y$ being supersingular follows from Proposition~\ref{prop:abelian scheme reduction}.
From Proposition~\ref{prop:nearby hermitian space ell}, we find that, for any prime $\ell$, the rank of the $\mathbb Z_\ell$-module $V(A_y[\ell^\infty])$ is equal to $2d = \dim V$. Moreover, we can find a finite extension of $\mathbb F_{\mathfrak{q}}$ over which $y$, and all the elements of $V(A_y[\ell^\infty])$ are defined.
This shows that \cite[Assumption 6.2]{mp:tatek3} is satisfied, and so our proposition now follows from [\emph{loc.~cit.}, Theorem 6.4]. The statement of the cited result assumed $p>2$, but its proof goes through without this assumption. \end{proof}
\begin{corollary}\ \label{cor:special end structure} \begin{enumerate}
\item
For any connected $\mathcal{Y}$-scheme $S$, $V(A_S)_\mathbb Q$ has a canonical structure of an $E$-vector space equipped with a positive definite Hermitian form $\langle\cdot,\cdot\rangle$.
\item
We have $V(A_S)_\mathbb Q = 0$ unless the image of $S\to \mathcal{Y}$ is supported on a single special fiber $\mathcal{Y}_{\mathbb F_{\mathfrak{q}}}$ with $\mathfrak{q}\subset\mathcal O_E$ a prime lying over a non-split prime $\mathfrak{p}\subset \mathcal O_F$.
\item
If $S\to\mathcal{Y}$ is supported on a single special fiber $\mathcal{Y}_{\mathbb F_{\mathfrak{q}}}$ as in (2), then there is an isometry
\[ \mathscr{V}(A_S)_\mathbb Q \xrightarrow{\simeq} { \empty^{ \mathfrak{p} }} \mathscr{V} \] of Hermitian spaces over $E$. Here, we have written $\mathscr{V}(A_S)_\mathbb Q$ for $V(A_S)_\mathbb Q$ equipped with its additional Hermitian $E$-vector space structure. \end{enumerate} \end{corollary} \begin{proof} From Proposition~\ref{prop:tQ_action}, we obtain an embedding $T\hookrightarrow\mathrm{Aut}^\circ(A_S)$, whose homological realizations are the maps $\theta_?(H)$ of [\emph{loc.~cit.}]. This implies that $V(A_S)_\mathbb Q\subset \mathrm{End}_{C(L)}(A_S)_\mathbb Q$ is a $T$-stable subspace.
First assume that $S$ is a geometric point $y\in \mathcal{Y}(\mathbb F_{\mathfrak{q}}^{\mathrm{alg}})$, where $\mathfrak{q}\subset\mathcal O_E$ lies over a non-split prime $\mathfrak{p}\subset\mathcal O_F$. For any $\ell\neq p$, by~\eqref{eqn:V ell isom}, $V(A_y)_{\mathbb Q_\ell}$ is isomorphic as a $T_{\mathbb Q_\ell}$-representation to $V_{\mathbb Q_\ell}$. It is easy to see that, for each $\ell$, the $E$ action on $V_{\mathbb Q_\ell}$ identifies $E_{\mathbb Q_\ell}$ with the commutant of $T_{\mathbb Q_\ell}$ in $\mathrm{End}(V_{\mathbb Q_\ell})$. Therefore, the commutant of $T$ in $\mathrm{End}(V(A_y)_\mathbb Q)$ is a commutative $\mathbb Q$-algebra that, for every $\ell\neq p$, is isomorphic to $E_{\mathbb Q_\ell}$ over $\mathbb Q_\ell$. As such, this commutant must be the field $E$. In this way, we obtain an $E$-action on $V(A_y)_\mathbb Q$, making it a $1$-dimensional $E$-vector space, which is irreducible as a representation of $T$. In particular, there is a unique Hermitian form $\langle\cdot,\cdot\rangle$ on it, which when composed with $\mathrm{Tr}_{F/\mathbb Q}$ gives the canonical quadratic form on $V(A_y)_\mathbb Q$, It now follows from the Hasse principle for $E$-Hermitian spaces, and Propositions~\ref{prop:special_quadratic_form},~\ref{prop:nearby hermitian space ell} and~\ref{prop:cm special supersingular} that we have an isometry \[ \mathscr{V}(A_y)_\mathbb Q \xrightarrow{\simeq} { \empty^{ \mathfrak{p} }} \mathscr{V} \] of Hermitian spaces over $E$.
If $S$ is any $\mathcal{Y}$-scheme with $V(A_S)\neq 0$, it follows from Proposition~\ref{prop:no special char 0} that the image of $S$ in $\mathcal{Y}$ does not intersect the generic fiber ${Y}$, and thus is supported in finite characteristics. Suppose that $s\in S(\mathbb F_{\mathfrak{q}}^{\mathrm{alg}})$ is a geometric point lying above a point $y\in \mathcal{Y}(\mathbb F_{\mathfrak{q}}^{\mathrm{alg}})$. This implies that $\mathfrak{q}$ lies over a non-split prime $\mathfrak{p}\subset\mathcal O_F$. Moreover, since $\mathscr{V}(A_y)_\mathbb Q$ is an irreducible representation of $T$, the map \[ V(A_S)_\mathbb Q \to \mathscr{V}(A_y)_\mathbb Q \] must be an isomorphism. In particular, $V(A_S)_\mathbb Q$ has a canonical structure of a Hermitian space over $E$, equipped with which it is isomorphic to $ { \empty^{ \mathfrak{p} }} \mathscr{V}$. It follows from this that the image of $S$ in $\mathcal{Y}$ has to be supported over $\mathcal{Y}_{\mathbb F_{\mathfrak{q}}}$. \end{proof}
\section{Arithmetic intersections and derivatives of $L$-functions}\label{s:intersection}
In this section, we set up the terminology required to state the main technical result of this paper, Theorem~\ref{thm:arithmetic BKY}. In particular, following~\cite{BKY}, we discuss the theory of incoherent Eisenstein series and their q-expansion and recall the main theorem of \emph{loc. cit.}
Keep $E/F$ and $(\mathscr{V},\mathscr{Q})$ as in \S \ref{ss:hermitian}. Once again define a $\mathbb Q$-quadratic space \[
(V,Q) = (\mathscr{V} , \mathrm{Tr}_{F/\mathbb Q} \circ \mathscr{Q})
\] of signature $(2d-2,2) = (n,2)$, where $d=[F:\mathbb Q]$. We will assume that $d>1$, so that $n>0$.
Let $\chi : \mathbb A_F^\times \to \{ \pm 1\}$ be the quadratic character defined by the CM extension $E/F$, and let $D_E$ and $D_F$ be the discriminants of $E$ and $F$. If we set $\Gamma_\mathbb R(s) = \pi^{-s/2} \Gamma(s/2)$, the completed $L$-function \begin{equation}\label{completed L}
\Lambda(s, \chi ) = \left| \frac{ D_E } {D_F} \right|^{s/2} \Gamma_\mathbb R(s+1)^d L(s,\chi) \end{equation} satisfies the function equation $\Lambda(1-s, \chi) = \Lambda(s,\chi)$. Furthermore, \begin{equation}\label{eqn:completed log der} \frac{ \Lambda'(0,\chi ) }{\Lambda(0,\chi )}
= \frac{ L'(0,\chi) }{ L(0,\chi) } + \frac{1}{2} \log \left| \frac{ D_{E} }{ D_{F} }\right| - \frac{ d \log(4\pi e^\gamma)}{ 2 } \end{equation} where $\gamma = -\Gamma'(1)$ is the Euler-Mascheroni constant.
\subsection{Incoherent Eisenstein series} \label{ss:incoherent}
Recalling the standard additive character $\psi_\mathbb Q: \mathbb Q\backslash \mathbb A \to \mathbb C^\times$ of \S \ref{ss:harmonic forms}, define \[ \psi_F: F\backslash \mathbb A_F \to \mathbb C^\times \] by $\psi_F = \psi_\mathbb Q \circ \mathrm{Tr}_{F/\mathbb Q}$.
If $v$ is an arichmedean place of $F$, denote by $\mathscr{C}_v$ the unique positive definite rank $2$ quadratic space over $F_v$. Set $\mathscr{C}_\infty = \prod_{v\mid \infty} \mathscr{C}_v$. The rank $2$ quadratic space \[ \mathscr{C} = \mathscr{C}_\infty \times \widehat{\mathscr{V}} \] over $\mathbb A_F$ is \emph{incoherent}, in the sense that it is not the adelization of any $F$-quadratic space. In fact, $\mathscr{C}$ is isomorphic to $\mathscr{V} \otimes_F \mathbb A_F$ everywhere locally, except at the unique archimedean place $\iota_0$ at which $\mathscr{V}$ is negative definite.
To any Schwartz function \[ \varphi_\infty \otimes \varphi \in S(\mathscr{C}_\infty) \otimes S(\widehat{\mathscr{V}}) \xrightarrow{\simeq} S( \mathscr{C} ) \] we may associate an incoherent
Hilbert modular Eisenstein series via the process described in \cite{KuAnnals, KYEis, Yang}. Briefly, the construction is as follows. Denote by $I( s, \chi)$
the degenerate principal series representation of $\mathrm {SL}_2(\mathbb A_F)$ induced from the character $\chi | \cdot |^s$ on the subgroup $B\subset \mathrm {SL}_2$ of upper triangular matrices. Thus $I(s,\chi)$ consists of all smooth functions $\Phi(g,s)$ on $\mathrm {SL}_2(\mathbb A_F)$ satisfying the transformation law \[
\Phi \left( \left( \begin{matrix} a& b \\ & a^{-1} \end{matrix} \right) g , s \right) = \chi(a) |a|^{s+1} \Phi(g,s). \] As in \S \ref{ss:harmonic forms}, the Weil representation $\omega_{\mathscr{C}}$ (determined by the character $\psi_F$) defines an action of $\mathrm {SL}_2(\mathbb A_F)$ on $S( \mathscr{C} )$, and the function \[ \Phi (g,0) = \omega_{\mathscr{C}}(g) ( \varphi_\infty \otimes \varphi ) (0) \] lies in the induced representation $I( 0, \chi)$. It extends uniquely to a standard section $\Phi ( g , s )$ of $I(s,\chi)$, which determines an Eisenstein series \begin{equation}\label{eisenstein formation} E(g ,s , \Phi ) = \sum_{ \gamma \in B(F) \backslash \mathrm {SL}_2(F) }\Phi ( \gamma g ,s) \end{equation} on $\mathrm {SL}_2(\mathbb A_F)$. As in \cite[Theorem 2.2]{KuAnnals}, the incoherence of $\mathscr{C}$ implies that $E(g ,s , \Phi)$ vanishes identically at $s=0$.
Endow $\mathbb A_F$ with the Haar measure self-dual with respect to $\psi_F$, and give $F\backslash \mathbb A_F$ the quotient measure. For every $\alpha\in F$ define the \emph{Whittaker function} \begin{equation}\label{whittaker def} W_\alpha ( g,s , \Phi ) = \int_{\mathbb A_F} \Phi ( w n(b) g, s)\cdot \psi_F( - \alpha b )\, db, \end{equation} where $w=\left(\begin{smallmatrix} & -1 \\ 1 \end{smallmatrix}\right)$ and $n(b) = \left( \begin{smallmatrix} 1 & b \\ & 1 \end{smallmatrix}\right).$ The Eisenstein series (\ref{eisenstein formation}) has a Fourier expansion \[ E(g,s, \Phi ) = \sum_{\alpha \in F}E_\alpha (g,s, \Phi ) \] in which the coefficient \[ E_\alpha (g,s, \Phi ) = \int_{ F\backslash \mathbb A_F } E\big( n(b) g,s, \Phi \big) \cdot \psi_F( - \alpha b )\, db \] is related to the Whittaker function by \begin{equation}\label{eisenstein whittaker} E_\alpha(g,s, \Phi ) = \begin{cases} W_\alpha ( g,s , \Phi ) & \mbox{if }\alpha \not=0 \\ \Phi(g,s) + W_0 ( g,s , \Phi ) & \mbox{if }\alpha =0. \end{cases} \end{equation}
The degenerate principal series decomposes $I(s,\chi) = \otimes_v I_v(s,\chi_v)$, where the tensor product is over all places of $F$. There is an obvious factorization $\Phi = \Phi_\infty \otimes \Phi_f $ into archimedean and nonarchimedean parts, which induces a corresponding factorization \[ W_\alpha ( g,s , \Phi ) = W_{\alpha,\infty}(g_\infty , s , \Phi_\infty) \cdot W_{\alpha,f}( g_f, s, \Phi_f) \] of the integral (\ref{whittaker def}). In practice there will be a further factorization $ \varphi = \otimes_p \varphi_p \in S(\widehat{\mathscr{V}}) $ over the rational primes, and hence a factorization \[ W_\alpha ( g,s , \varphi ) = W_{\alpha,\infty} (g_\infty,s,\phi_\infty ) \cdot \prod_{ p } W_{\alpha,p} ( g_p,s , \varphi_p ) \] of Whittaker functions. When the component $\varphi_p$ admits a further factorization $\varphi_p = \otimes_{\mathfrak{p} \mid p} \varphi_\mathfrak{p}$ so does \[ W_{\alpha,p} ( g_p,s , \varphi_p ) = \prod_{\mathfrak{p} \mid p} W_{\alpha,\mathfrak{p}} ( g_ \mathfrak{p} ,s , \varphi_\mathfrak{p} ). \]
From now on we will always take the archimedean component $\varphi_\infty$ of our Schwartz function to be the Gaussian distribution \[ \varphi_\infty^{\bm{1}} = \otimes \varphi^{\bm{1}}_v \in \bigotimes_{v\mid \infty} S( \mathscr{C}_v) \] defined by $ \varphi^{\bm{1}}_v(x) = e^{-2\pi \mathscr{Q}_v(x)} $ ($\mathscr{Q}_v$ is the quadratic form on $\mathscr{C}_v$.) By \cite[Lemma 4.1]{KYEis} the resulting Eisenstein series (\ref{eisenstein formation}) has parallel weight $1$. As the archimedean component will remain fixed, the section $\Phi$ is determined by
$\varphi \in S(\widehat{\mathscr{V}})$, and we will often write \[ E(g,s,\varphi) =E(g,s, \Phi ) . \]
\subsection{A formal $q$-expansion} \label{ss:q-expansion}
As in the previous section, fix a Schwartz function $\varphi \in S(\widehat{\mathscr{V}})$, and let $E(g,s,\varphi)$ be the corresponding incoherent weight $1$ Eisenstein series on $\mathrm {SL}_2(\mathbb A_F)$.
For any $\vec{\tau} \in \mathcal{H}^d$ let $g_{\vec{\tau}} \in \mathrm {SL}_2(\mathbb A_F )$ be the matrix with archimedean components \[ g_{\tau_i} =
\Bigg(\begin{matrix} 1 & u_i \\ & 1 \end{matrix}\Bigg) \Bigg(\begin{matrix} v_i^{ 1/2 } & \\ & v_i^{- 1/2} \end{matrix}\Bigg), \] and take all finite components to be the identity matrix. Here \[ \vec{u} = (u_0,\ldots, u_{d-1}) , \quad \vec{v} = (v_0 , \ldots, v_{d-1}) \] are the real and imaginary parts of $\vec{\tau}$. Exactly as in \cite[(4.4)]{BKY}, define a classical weight $1$ Hilbert modular Eisenstein series \[ E ( \vec{\tau} , s , \varphi ) = \frac{ 1}{ \sqrt{ N( \vec{v} ) } } \cdot E(g_{\vec{\tau} } ,s ,\varphi), \] where $N(\vec{v}) = v_0 \cdots v_{d-1}$. Its derivative at $s=0$ has the Fourier expansion \[ E^{ \prime } ( \vec{\tau} , 0 , \varphi ) = \frac{ 1 }{ \sqrt{ N( \vec{v} )} } \cdot \sum_{\alpha \in F} E_\alpha'( g_{\vec{\tau}} , 0 ,\varphi). \]
As in \cite{KuAnnals,KYtheta}, for any $\alpha \in F^\times$ define the \emph{difference set} \[ \mathrm{Diff}(\alpha) = \{ \mbox{places $v$ of $F$} : \mathscr{C}_v \mbox{ does not represent } \alpha \}. \] Usually $\alpha$ will be totally positive, in which case \begin{align*} \mathrm{Diff}(\alpha) & = \{ \mbox{primes }\mathfrak{p} \subset \mathcal O_F : \mathscr{V}_\mathfrak{p} \mbox{ does not represent } \alpha \} \\ & = \{ \mbox{primes }\mathfrak{p} \subset \mathcal O_F : \chi_\mathfrak{p}( \alpha ) \neq \mathrm{inv}_\mathfrak{p} (\mathscr{V}) \}. \end{align*}
\begin{remark}\label{rem:diff} Note that $\mathrm{Diff}(\alpha)$ is a finite set of odd cardinality, and any place $v \in \mathrm{Diff}(\alpha)$ is nonsplit in $E$. If $\mathfrak{p}\subset \mathcal O_F$ is a finite place, then $\mathrm{Diff}(\alpha) = \{ \mathfrak{p} \}$ if and only if $\alpha$ is represented by the nearby hermitian space $ { \empty^{ \mathfrak{p} }} \mathscr{V}$ of Definition \ref{def:nearby}. \end{remark}
All parts of the following proposition follow from the statement and proof of \cite[Proposition 4.6]{BKY}.
\begin{proposition}\label{prop:coefficient support} For any totally positive $\alpha \in F$ we have \[ \frac{1 }{ \sqrt{ N( \vec{v} ) } }\cdot E_\alpha'( g_{\vec{\tau}} , 0 ,\varphi) =
\frac{ a_F( \alpha , \varphi) }{ \Lambda( 0 , \chi ) } \cdot q^\alpha \] for some constant $a_F( \alpha , \varphi)$ independent of $\vec{\tau}$. Furthermore: \begin{enumerate}
\item If $| \mathrm{Diff}(\alpha)| >1$, then $a_F(\alpha,\varphi)=0$. \item If $\mathrm{Diff}(\alpha) =\{ \mathfrak{p}\}$, then \[ \frac{ a_F(\alpha ,\varphi) } { \Lambda(0,\chi ) } \in \mathbb Q (\varphi) \cdot \log N(\mathfrak{p}), \] where $\mathbb Q(\varphi)/\mathbb Q$ is the extension obtained by adjoining all values of $\varphi$. \end{enumerate} \end{proposition}
Now we study the constant term. Much of the following proposition is implicit in the statement and proof of \cite[Proposition 4.6]{BKY}, but the relevant part of [\emph{loc.~cit.}] is misstated, and we need more information than is found there.
\begin{proposition}\label{prop:coarse constant} There is a meromorphic function $M(s,\varphi)$ such that \begin{equation}\label{full constant} \frac{ E_0( g_{\vec{\tau}} , s ,\varphi) }{ \sqrt{ N(\vec{v} ) } } = \varphi(0) \cdot N(\vec{v})^{s/2} -
N(\vec{v})^{-s/2} \frac{\Lambda(s , \chi ) }{\Lambda(s+1,\chi)} \cdot M (s,\varphi) . \end{equation}
If $\varphi=\otimes_\mathfrak{p}\varphi_\mathfrak{p}$ factors over the primes of $\mathcal O_F$, then so does \[ M(s,\varphi)=\prod_\mathfrak{p} M_\mathfrak{p}(s,\varphi_\mathfrak{p}). \] Each factor $M_\mathfrak{p}(s,\varphi_\mathfrak{p})$ is a rational function, with coefficients in $\mathbb Q(\varphi_\mathfrak{p})$, in the variable $N(\mathfrak{p})^s$, and all but finitely many factors are equal to $1$. Finally,
\begin{equation}\label{lazy constant} \frac{ 1 }{ \sqrt{ N( \vec{v} ) } }\cdot E_0'( g_{\vec{\tau}} , 0 ,\varphi)
= \varphi(0) \log N( \vec{v} ) + \frac{ a_F(0,\varphi) }{ \Lambda( 0 , \chi ) } ,
\end{equation}
where the constant $a_F(0,\varphi)$ is defined by the relation \[ \frac{ a_F(0,\varphi) }{\Lambda(0,\chi) } = -2 \varphi(0) \cdot \frac{ \Lambda'(0,\chi) }{ \Lambda(0,\chi) }- M'(0,\varphi) . \] \end{proposition}
\begin{proof} Assume that $\varphi=\otimes_\mathfrak{p} \varphi_\mathfrak{p}$ admits a factorization over the finite places of $F$, so that there are similar factorizations \[ \Phi(g,s) = \prod_v \Phi_v(g,s), \quad W_0(g,s,\Phi) = \prod_v W_{0,v} (g,s, \Phi_v) \] over all places of $F$. We define \[ M (s,\varphi ) = \prod_\mathfrak{p} M_\mathfrak{p}(s,\varphi_\mathfrak{p}), \] where \begin{align}\label{local M} M_\mathfrak{p}(s,\varphi_\mathfrak{p}) &= \frac{ \mathrm{N}(\mathfrak{p})^{ f(\mathfrak{p}) /2 } } { \gamma_\mathfrak{p} (\mathscr{V} ) } \cdot \frac{L_\mathfrak{p}(s+1,\chi)} {L_\mathfrak{p}(s,\chi)} \cdot W_{0,\mathfrak{p}} (I,s, \Phi_\mathfrak{p}) \\ &= \frac{ \mathrm{N}(\mathfrak{p})^{ f(\mathfrak{p}) /2 } } { \gamma_\mathfrak{p} (\mathscr{V} ) } \cdot \frac{L_\mathfrak{p}(s+1,\chi)} {L_\mathfrak{p}(s,\chi)} \cdot \int_{F_\mathfrak{p} } \Phi_\mathfrak{p} \left( \begin{matrix} w n(b) \end{matrix} , s \right)\, db . \nonumber \end{align} Here $I\in \mathrm {SL}_2(F_\mathfrak{p})$ is the identity matrix, $ f(\mathfrak{p}) = \mathrm{ord}_\mathfrak{p}(\mathfrak{D}_F D_{E/F} ), $ where $\mathfrak{D}_F$ and $D_{E/F}$ are the different and relative discriminants of $F/\mathbb Q$ and $E/F$, respectively, and
\begin{equation}\label{eqn:gamma p defn} \gamma_\mathfrak{p}(\mathscr{V} ) = \chi_\mathfrak{p}(-1) \cdot \mathrm{inv}_\mathfrak{p}(\mathscr{V}) \cdot \epsilon_\mathfrak{p} (\chi, \psi_F) \in \{ \pm 1, \pm i\} \end{equation} is the local Weil index (relative to $\psi_F$) as in \cite{Yang}. These satisfy \[ (-i)^d \prod_\mathfrak{p} \gamma_\mathfrak{p}(\mathscr{V}) = -1. \] Note that for a given $\varphi$, all but finitely many $\mathfrak{p}$ satisfy $M_\mathfrak{p}(s,\varphi_\mathfrak{p})=1$. This is an easy exercise. Alternatively, as two factorizable Schwartz functions are equal in all but finitely many components, it suffices to prove the claim for any one factorizable Schwartz function. This is done below.
Extend $\varphi \mapsto M (s,\varphi )$ linearly to all Schwartz functions. Combining the definition (\ref{local M}) with the calculation \[ W_{0,\infty}( g_{\vec{\tau}} , s ,\Phi_\infty) =
(-i)^d \frac{ \Gamma_\mathbb R(s+1)^d }{ \Gamma_\mathbb R(s+2)^d } \cdot N(\vec{v})^{(1-s)/2} \] of \cite[Proposition 2.4]{Yang}, we find \[ W_0(g_{\vec{\tau}} ,s,\Phi) = - N(\vec{v})^{(1-s)/2} \frac{\Lambda(s , \chi ) }{\Lambda(s+1,\chi)} \cdot M(s,\varphi). \] Plugging this equality and \[ \Phi( g_{ \vec{\tau}} , s ) = N(\vec{v})^{(s+1)/2} \cdot \Phi ( I ,s ) = N(\vec{v})^{(s+1)/2} \cdot \varphi(0) \]
into the equality \[
E_0( g_{\vec{\tau}} , s ,\varphi) = \Phi(g_{\vec{\tau}} ,s) + W_0(g_{\vec{\tau}} ,s,\Phi) \] of (\ref{eisenstein whittaker}) proves (\ref{full constant}). As the left hand side of (\ref{full constant}) vanishes at $s=0$, the functional equation $\Lambda(1-s,\chi) = \Lambda(s,\chi)$ implies
$M (0,\varphi)= \varphi(0)$, and (\ref{lazy constant}) then follows directly from (\ref{full constant}) by taking the derivative.
It only remains to prove the claims concerning the rationality of the local factors $M_\mathfrak{p}(s,\varphi_\mathfrak{p})$. First we describe $M_\mathfrak{p}(s,\varphi_\mathfrak{p})$ for a specific choice of $\varphi_\mathfrak{p}$. Fix an isomorphism \begin{equation}\label{local herm coords} ( \mathscr{V}_\mathfrak{p} , \mathscr{Q}_\mathfrak{p}) \xrightarrow{\simeq} ( E_\mathfrak{p} , \xi_\mathfrak{p} \cdot \mathrm{Nm}_{E_\mathfrak{p} / F_\mathfrak{p} } ) \end{equation} with $\xi_\mathfrak{p}\in F_\mathfrak{p}^\times$. If $\mathfrak{p}$ is either split or ramified in $E$, we choose this isomorphism so that $\xi_\mathfrak{p} \in \mathcal O^\times_{F,\mathfrak{p}}$. If $\mathfrak{p}$ is inert in $E$, we choose the isomorphism so that $\mathrm{ord}_\mathfrak{p}(\xi_\mathfrak{p})\in \{0,1\}$. Now let $\widetilde{\varphi}_\mathfrak{p}$ be the characteristic function of $\mathcal O_{E,\mathfrak{p}} \subset E_\mathfrak{p} =\mathscr{V}_\mathfrak{p}$. For this choice of Schwartz function, the calculations of \cite{Yang} (see also Corollary~\ref{cor:good prime M constant} below) show that \[ M_\mathfrak{p}( s,\widetilde{\varphi}_\mathfrak{p} ) = \begin{cases} \mathrm{N} (\mathfrak{p} )^{-1} \cdot
\frac{ L_\mathfrak{p}( s+1,\chi) }{ L_\mathfrak{p}(s-1,\chi) }
& \mbox{if $\mathfrak{p}$ is inert in $E$ and $\mathrm{inv}_\mathfrak{p}(\mathscr{V})=-1$} \\
1 & \mbox{otherwise.}
\end{cases} \]
By the linearity of $\varphi_\mathfrak{p} \mapsto M_\mathfrak{p}(s,\varphi_\mathfrak{p})$, it now suffices to show that when $\varphi_\mathfrak{p}(0)=0$, the function \begin{equation}\label{truncated factor}
\frac{ L_\mathfrak{p} (s,\chi) }{ L_\mathfrak{p} (s+1,\chi) } \cdot M_\mathfrak{p}(s , \varphi_\mathfrak{p})
= \frac{ \mathrm{N}(\mathfrak{p})^{ f(\mathfrak{p}) /2 } }{ \gamma_\mathfrak{p} (\mathscr{V}) }
\cdot \int_{ F_\mathfrak{p} } \Phi_\mathfrak{p}( w n(b) ,s)\, db \end{equation} is a polynomial in $N(\mathfrak{p})^s$ with coefficients in $\mathbb Q(\varphi_\mathfrak{p})$. We assume $\varphi_\mathfrak{p}(0)=0$ in all that follows.
If $| b | \le 1$ then $\Phi_\mathfrak{p}( w n(b) ,s)$ is independent of $s$, by the definition of a standard section. If $| b | \ge 1$ then the factorization \[ w n(b) = \left( \begin{matrix} & -1 \\ 1 & b \end{matrix} \right) = \left( \begin{matrix} b^{-1} & -1 \\ & b \end{matrix} \right) \left( \begin{matrix} 1 & \\ b^{-1} & 1 \end{matrix} \right) \] shows that \[
\Phi_\mathfrak{p}( w n(b) ,s) = \chi_\mathfrak{p}(b) | b |^{ -s-1 } \Phi_\mathfrak{p} \left( \left( \begin{matrix} 1 & \\ b^{-1} & 1 \end{matrix} \right) ,0 \right). \] As $\Phi_\mathfrak{p}(g,0)$ is locally constant, this last equality also implies that for all $b$ outside of some sufficiently large ball $\mathfrak{p}^{-c}$, we have \[
\Phi_\mathfrak{p}( w n(b) ,s) = \chi_\mathfrak{p}(b) | b |^{ -s-1 } \Phi_\mathfrak{p}( I ,0)
= \chi_\mathfrak{p}(b) | b |^{ -s-1 } \varphi_\mathfrak{p}(0) =0. \]
Using these observations, one can check that (\ref{truncated factor}) is a polynomial in $\mathrm{N}(\mathfrak{p})^s$ by decomposing the integral as a sum of integrals over annuli $\mathfrak{p}^k \smallsetminus \mathfrak{p}^{k+1}$ in the usual way.
For all sufficiently large $c$ we have \begin{align*}
\frac{ 1} { \gamma_\mathfrak{p} (\mathscr{V}) } \int_{ F_\mathfrak{p} } \Phi_\mathfrak{p}( w n(b) , 0 )\, db
& =
\frac{1}{ \gamma_\mathfrak{p} (\mathscr{V}) } \int_{ \mathfrak{p}^{-c} } \Phi_\mathfrak{p}( w n(b) , 0 )\, db \\
& =
\int_{ \mathfrak{p}^{ -c } } \int_{ \mathscr{V}_\mathfrak{p} } \varphi_\mathfrak{p}(x) \psi_{F,\mathfrak{p}} \big( b \mathscr{Q} (x) \big)\, dx\, db \\
& = \int_{ \mathscr{V}_\mathfrak{p} } \varphi_\mathfrak{p}(x)
\left( \int_{ \mathfrak{p}^{-c} } \psi_{F,\mathfrak{p}} \big( b \mathscr{Q}_\mathfrak{p} (x) \big)\, db \right) \, dx. \end{align*} The second equality is easily obtained from the explicit formulas \cite[(4.2.1)]{HY} defining the Weil representation. In the above equalities, Haar measure on $\mathscr{V}_\mathfrak{p}$ is normalized as in \cite[Lemma 4.6.1]{HY}, so that, for any isomorphism (\ref{local herm coords}), \[ \mathrm{Vol}( \mathcal O_{E,\mathfrak{p}} ) = \mathrm{N}( \mathfrak{p} )^{ -\mathrm{ord}_\mathfrak{p}(D_{E/F} ) / 2 }
\mathrm{N}( \mathfrak{p} )^{ -\mathrm{ord}_\mathfrak{p}( \xi_\mathfrak{p} ) } . \] The Haar measure on $F_\mathfrak{p}$ is chosen to be self-dual with respect to $\psi_{F,\mathfrak{p}}$, so that \[ \mathrm{Vol}(\mathfrak{p}^{-c}) = \mathrm{N}(\mathfrak{p})^c \cdot \mathrm{Vol}(\mathcal O_{F,\mathfrak{p}} )= \mathrm{N}(\mathfrak{p})^{ c } \cdot \mathrm{N}(\mathfrak{p})^{ - \mathrm{ord}_\mathfrak{p}(\mathfrak{D}_F) /2 }. \] The inner integral above is \[
\int_{ \mathfrak{p}^{-c} } \psi_{F,\mathfrak{p}} \big( b \mathscr{Q}_\mathfrak{p} (x) \big)\, db = \begin{cases} \mathrm{Vol}( \mathfrak{p}^{-c} ) & \mbox{if }\mathscr{Q}_\mathfrak{p}(x) \in \mathfrak{p}^c\mathfrak{D}_{F,\mathfrak{p}}^{-1} \\ 0 & \mbox{otherwise,} \end{cases} \] and from this it is clear that the value at $s=0$ of (\ref{truncated factor}) lies in $\mathbb Q(\varphi_\mathfrak{p})$.
By the interpolation trick of Rallis, as in \cite[Lemma 4.2]{KYEis}, the calculation above can be extended to show that the value of (\ref{truncated factor}) lies in $\mathbb Q(\varphi_\mathfrak{p})$ for \emph{any} $s\in \mathbb Z_{\ge 0}$. This shows that (\ref{truncated factor}) has the form $R(\mathrm{N}(\mathfrak{p})^s)$ where $R(T)\in \mathbb C[T]$ is $\mathbb Q(\varphi_\mathfrak{p})$-valued at infinitely many $T\in\mathbb Z$, and from this it follows that $R(T)$ has coefficients in $\mathbb Q(\varphi_\mathfrak{p})$.
This completes the proof of Proposition \ref{prop:coarse constant}. \end{proof}
As in \cite[Proposition 4.6]{BKY}, define a formal $q$-expansion \[ \mathcal{E} ( \vec{\tau} , \varphi ) = a_F(0,\varphi) + \sum_{ \alpha \in F_+ } a_F(\alpha,\varphi) \cdot q^\alpha, \] where $F_+ \subset F$ is the subset of totally positive elements. Its formal diagonal restriction is the formal $q$-expansion \[ \mathcal{E}(\tau ,\varphi) = \sum_{m\in \mathbb Q } a(m,\varphi ) \cdot q^m \] defined by $a(0,\varphi) = a_F(0,\varphi)$, and \begin{equation}\label{eisenstein decomp} a (m,\varphi ) = \sum_{ \substack{ \alpha \in F_+ \\ \mathrm{Tr}_{F/\mathbb Q}(\alpha)=m } } a_F(\alpha,\varphi ) \end{equation} for all $m\not=0$. In particular $a(m,\varphi) =0$ if $m<0$.
\subsection{The Bruinier-Kudla-Yang theorem}
Fix a maximal lattice $L$ in the $\mathbb Q$-quadratic space $(V,Q)$. Recalling the Schwartz function $ \varphi_\mu \in S(\widehat{\mathscr{V}}) = S(\widehat{V}) $ of (\ref{mu schwartz}), abbreviate \[ a(m,\mu) = a(m,\varphi_\mu) , \quad a_F(\alpha, \mu) = a_F(\alpha,\varphi_\mu) \] for any $\mu\in L^\vee /L $.
Fix also a harmonic weak Maass form $f\in H_{ 2-d }(\omega_L)$ with integral principal part. Let us temporarily denote by \[ \bm{f}= \xi( f) \in S_d(\overline{\omega}_L) \]
the image of $f$ under the Bruinier-Funke differential operator of (\ref{BF exact sequence}). Decompose $\bm{f} (\tau)= \sum_\mu \bm{f}_\mu(\tau) \varphi_\mu$, where the sum is over $\mu \in L^\vee /L$, and define a generalized $L$-function \[ \mathcal{L} \big(s, \xi(f) \big) = \Lambda(s+1 , \chi ) \int_{ \mathrm {SL}_2(\mathbb Z) \backslash \mathcal{H} } \sum_{\mu \in L^\vee/L} \overline{ \bm{f} _\mu(\tau) } E( \tau, s, \varphi_\mu) \, \frac{du\, dv}{ v^{2-d} } \] exactly as in \cite[(5.3)]{BKY}. Here $\tau = u+iv\in\mathcal{H}$, and $E( \tau, s, \varphi)$ is the restriction of the Hilbert modular Eisenstein series $E ( \vec{\tau} , s , \varphi )$
to the diagonally embedded $\mathcal{H} \hookrightarrow \mathcal{H}^d$. This $L$-function is an entire function of the variable $s$, and vanishes at $s=0$.
Abbreviate \[
\deg_\mathbb C({Y}) \define \sum_{y \in {Y}(\mathbb C) } \frac{ 1 } { | \mathrm{Aut}(y) | } =
\frac{ | T(\mathbb Q) \backslash T(\mathbb A_f) / K_{L,0} | }{ | T(\mathbb Q) \cap K_{L,0} | }, \] where $Y(\mathbb C)$ is the set of complex points of $Y$, viewed as an $E$-stack. If we set \[
\mathcal{Y}^\infty = \mathcal{Y} \times_{ \mathrm{Spec}(\mathbb Z) } \mathrm{Spec}( \mathbb C),
\]
then
\[
\sum_{y \in \mathcal{Y}^\infty(\mathbb C) } \frac{ 1 } { | \mathrm{Aut}(y) | } = 2d\cdot \deg_\mathbb C({Y}) .
\]
The following theorem is the main result of \cite{BKY}.
\begin{theorem}[Bruinier-Kudla-Yang]\label{thm:BKY} In the notation above, \[
\frac{ \Phi (f ,\mathcal{Y}^\infty) }{ 2 \deg_\mathbb C ({Y}) } = - \frac{ \mathcal{L}'(0 , \xi(f) ) } { \Lambda( 0 , \chi ) }
+ \sum_{ \substack{ \mu \in L^\vee / L \\ m \ge 0 } } \frac{ a(m,\mu) \cdot c_f^+(-m,\mu) } { \Lambda( 0 , \chi ) }, \] where $\Phi(f)$ is the Green function for $\mathcal{Z}(f)$ appearing in (\ref{arithmetic divisor}),
and, using the morphism $ \mathcal{Y}^\infty(\mathbb C) \to \mathcal{M}(\mathbb C)$
induced by (\ref{cm morphism}), we abbreviate
\[
\Phi (f ,\mathcal{Y}^\infty) = \sum_{ y \in \mathcal{Y}^\infty (\mathbb C) } \frac{ \Phi( f ,y) } { | \mathrm{Aut}(y) | }.
\] \end{theorem}
\subsection{The arithmetic intersection formula}
Exactly as in \S \ref{ss:line bundles}, we may form the group of metrized line bundles $\widehat{\mathrm{Pic}}(\mathcal{Y})$ on $\mathcal{Y}$.
Let $F_\infty : \mathcal{Y}^\infty(\mathbb C) \to \mathcal{Y}^\infty(\mathbb C)$ be complex conjugation.
As $\mathcal{Y}$ is flat of relative dimension $0$ over $\mathcal O_E$, all Cartier divisors on $\mathcal{Y}$ are supported in nonzero characteristics.
If $\mathcal{Z}$ is such a divisor, by a Green function for $\mathcal{Z}$ we mean \emph{any} $F_\infty$-invariant $\mathbb R$-valued function $\Phi$ on
$\mathcal{Y}^\infty(\mathbb C)$. Exactly as in \S \ref{ss:line bundles}, we define an \emph{arithmetic divisor} on $\mathcal{Y}$ to be a pair
\[
\widehat{\mathcal{Z}} = ( \mathcal{Z} ,\Phi)
\]
consisting of a Cartier divisor on $\mathcal{Y}$ together with a Green function. The codimension one arithmetic Chow group
$\widehat{\mathrm{CH}}^1(\mathcal{Y})$ is the quotient of the group of all arithmetic divisors by the subgroup of principal arithmetic
divisors
\[
\widehat{\mathrm{div}}( \Psi ) = ( \mathrm{div}( \Psi ) , - \log | \Psi |^2 ),
\]
for $\Psi$ a nonzero rational function on $\mathcal{Y}$. Once again we have an isomorphism \[ \widehat{\mathrm{Pic}}(\mathcal{Y}) \xrightarrow{\simeq} \widehat{\mathrm{CH}}^1(\mathcal{Y}). \]
\begin{remark} Any arithmetic divisor $(\mathcal{Z},\Phi)$ decomposes as $(\mathcal{Z}, 0 ) + ( 0,\Phi)$, and $\mathcal{Z}$ can be further decomposed as the difference of two effective Cartier divisors. \end{remark}
To define the \emph{arithmetic degree}, as in \cite{GS,KRY2,KRY3}, of an arithmetic divisor $\widehat{\mathcal{Z}}$ as above, we first assume that $\widehat{\mathcal{Z}} = (\mathcal{Z},0)$ with $\mathcal{Z}$ an effective Cartier divisor. Then \[ \widehat{\deg}(\widehat{\mathcal{Z}}) = \sum_{\mathfrak{q} \subset \mathcal O_E} \log \mathrm{N}(\mathfrak{q})
\sum_{ z\in \mathcal{Z}( \mathbb F^\mathrm{alg}_\mathfrak{q} ) } \frac{ \mathrm{length}( \mathcal O_{\mathcal{Z},z} ) }{| \mathrm{Aut}(z) | } \] where $\mathcal O_{\mathcal{Z},z}$ is the \'etale local ring of $\mathcal{Z}$ at $z$. If $\widehat{\mathcal{Z}} = ( 0 , \Phi)$ is purely archimedean, then \[
\widehat{\deg}(\widehat{\mathcal{Z}}) = \frac{1}{2} \sum_{ y \in \mathcal{Y}^\infty(\mathbb C)} \frac{ \Phi(y) }{ | \mathrm{Aut}(y) | }. \] The arithmetic degree extends linearly to all arithmetic divisors, and defines a homomorphism \[ \widehat{\deg} :\widehat{\mathrm{Pic}}( \mathcal{Y} ) \to \mathbb R . \]
We now define a homomorphism \[ [ \cdot: \mathcal{Y} ] : \widehat{\mathrm{Pic}}( \mathcal{M} ) \to \mathbb R , \] the \emph{arithmetic degree along $\mathcal{Y}$}, as the composition \[ \widehat{\mathrm{Pic}}( \mathcal{M} ) \to \widehat{\mathrm{Pic}}( \mathcal{Y} ) \map{\widehat{\deg} } \mathbb R . \]
\begin{theorem}\label{thm:arithmetic BKY} Recall the integer $D_{bad}=D_{bad,L}$ defined following Definition \ref{defn:D bad}. For any $f\in H_{ 2-d }(\omega_L)$ with integral principal part, the equality \[
\frac{ [ \widehat{\mathcal{Z}}(f) : \mathcal{Y} ] }{ \deg_\mathbb C ({Y}) } = - \frac{ \mathcal{L}'(0 , \xi(f) ) } { \Lambda( 0 , \chi ) } + \frac{ a(0,0) \cdot c_f^+( 0,0) } { \Lambda( 0 , \chi ) } \] holds up to a $\mathbb Q$-linear combination of $\{ \log(p) : p\mid D_{bad}\}$. \end{theorem}
Theorem \ref{thm:arithmetic BKY} is the technical core of this paper; its proof will occupy all of \S \ref{s:BKY proof}, with the completion of the proof appearing in \S \ref{ss:arithmetic intersection proof}.
\begin{remark}\label{rem:proper intersection} By Proposition \ref{prop:no special char 0}, the $\mathbb Z$-quadratic space of special endomorphisms $V(A_y)$ is $0$ for any complex point $y\in \mathcal{Y}(\mathbb C)$. By the very definition of the special divisors $\mathcal{Z}(m,\mu)$, it follows that the image of $\mathcal{Y} \to \mathcal{M}$ is disjoint from the support of all $\mathcal{Z}(m,\mu)$, and hence from the support of $\mathcal{Z}(f)$,
in the complex fiber. As $\mathcal{Y}$ is flat over $\mathbb Z$ of relative dimension $0$, this implies that the image of $\mathcal{Y}$ meets the support of $\mathcal{Z}(f)$ properly; \emph{i.e.~}the intersection has dimension $0$, and is supported in finitely many nonzero characteristics. \end{remark}
\section{Proof of the arithmetic intersection formula} \label{s:BKY proof}
In this section we prove Theorem \ref{thm:arithmetic BKY}. There are two main computations that are independent of each other: Proposition~\ref{prop:explicit siegel-weil} and Theorem~\ref{thm:local ring}. The first computes the Fourier coefficients of an incoherent Eisenstein series, and the second computes the lengths of the local rings of the intersection between the special divisors on the ambient GSpin Shimura variety with the zero dimensional Shimura variety from Section~\ref{ss:zero dimensional}. These combine to give Theorem~\ref{thm:degree}, which is at the heart of the proof of the main theorem.
\subsection{Local Whittaker functions}
\label{ss:whittaker functions}
Let $p$ be a good prime, in the sense of Definition \ref{defn:D bad}, and let $\mathfrak{p}\subset\mathcal O_F$ be a prime above it. We will assume that $\mathfrak{p}$ is not split in $\mathcal O_E$. Let $\mathfrak{q}\subset \mathcal O_E$ be the unique prime above $\mathfrak{p}$.
Let $m(\mathfrak{p})$ and $n(\mathfrak{p})$ be the $\mathfrak{p}$-adic valuations of the different $\mathfrak{d}_{F_{\mathfrak{p}}/\mathbb Q_p}$ and relative discriminant $D_{E_{\mathfrak{q}}/F_{\mathfrak{p}}} = \mathrm{Nm}_{E_{\mathfrak{q}}/F_{\mathfrak{p}}}(\mathfrak{d}_{E_{\mathfrak{q}}/F_{\mathfrak{p}}})$, respectively. The integer $n(\mathfrak{p})$ is non-zero if and only if $\mathfrak{q}$ is ramified over $F$. Set $f(\mathfrak{p}) = m(\mathfrak{p}) + n(\mathfrak{p})$; this is the $\mathfrak{p}$-adic valuation of $\mathfrak{d}_{F_{\mathfrak{p}}/\mathbb Q_p}D_{E_{\mathfrak{q}}/F_{\mathfrak{p}}}$.
Let $e(\mathfrak{p})$ be the absolute ramification index of $\mathfrak{p}$. If $p\neq 2$, then the only possible non-zero value for $n(\mathfrak{p})$ is $1$. If $p=2$, then $n(\mathfrak{p})$ belongs to the set $\{2e(\mathfrak{p})+1\}\cup\{2i:\,0\leq i\leq e(\mathfrak{p})\}$.
Since $p$ is good, the quadratic space $L_{\mathfrak{p}} = L_p \cap V_{\mathfrak{p}}$ contains a maximal $\mathcal O_{E,\mathfrak{{q}}}$-stable lattice $\Lambda_{\mathfrak{p}}$. Moreover, if $\mathfrak{p}$ is unramified in $E$, then this lattice is itself self-dual and in particular is equal to $L_{\mathfrak{p}}$.
Fix a uniformizer $\pi_{\mathfrak{p}}\in\mathcal O_{F,\mathfrak{p}}$. If $\mathfrak{p}$ is unramified in $E$, we will also write $\pi_{\mathfrak{q}}$ for this element, when we view it as a uniformizer for $E_{\mathfrak{q}}$. If $\mathfrak{p}$ is ramified in $E$, we assume that $\pi_\mathfrak{p}$ has the form $\mathrm{Nm}(\pi_{\mathfrak{q}}) = \pi_{\mathfrak{p}}$ for a uniformizer $\pi_{\mathfrak{q}}\in E_{\mathfrak{q}}$. Here $\mathrm{Nm}$ is the norm from $E_{\mathfrak{q}}$ to $F_{\mathfrak{p}}$.
We will now explicitly describe the possibilities for $\Lambda_{\mathfrak{p}}$.
\begin{itemize}
\item
If $\mathfrak{p}$ is inert in $E$, then the self-dual quadratic form on $L_{\mathfrak{p}}$ is the trace of an $E_{\mathfrak{q}}$-valued Hermitian form. In this case, $L_{\mathfrak{p}} = \Lambda_{\mathfrak{p}}$, and we have an isometry of Hermitian lattices:
\begin{equation*}
(L_{\mathfrak{p}},\langle x_1,x_2\rangle) \simeq (\mathcal O_{E,\mathfrak{q}},\pi_{\mathfrak{p}}^{-m(\mathfrak{p})}x_1\overline{x}_2).
\end{equation*}
The \emph{nearby} Hermitian module $ { \empty^{ \mathfrak{p} }} L_{\mathfrak{p}} = { \empty^{ \mathfrak{p} }} \Lambda_{\mathfrak{p}}$ is defined by
\begin{equation*}
( { \empty^{ \mathfrak{p} }} L_{\mathfrak{p}} , { \empty^{ \mathfrak{p} }} \langle x_1 , x_2 \rangle ) = ( \mathcal O_{E,\mathfrak{q}} , \pi_{\mathfrak{p}}^{-m(\mathfrak{p})+1} x_1\overline{x_2} ).
\end{equation*} (In other words, the underlying $\mathcal O_{E,\mathfrak{q}}$-module is the same, but the hermitian form is rescaled by $\pi_{\mathfrak{p}}$.)
\item
If $\mathfrak{p}$ is ramified in $E$, with $\mathfrak{q}\subset\mathcal O_E$ the prime above it, then, for an appropriate choice of unit $\beta_+\in\mathcal O_{F,\mathfrak{p}}^\times$, we have an isometry of Hermitian lattices:
\begin{equation*}
(\Lambda_{\mathfrak{p}},\langle x_1,x_2\rangle) \simeq (\mathcal O_{E,\mathfrak{q}},\beta_+\pi_{\mathfrak{p}}^{-m(\mathfrak{p})}x_1\overline{x}_2).
\end{equation*}
The \emph{nearby} Hermitian module $ { \empty^{ \mathfrak{p} }} \Lambda_{\mathfrak{p}}$ is defined by
\begin{equation*}
( { \empty^{ \mathfrak{p} }} \Lambda_{\mathfrak{p}} , { \empty^{ \mathfrak{p} }} \langle x_1 , x_2 \rangle ) = ( \mathcal O_{E,\mathfrak{q}} , \beta_{-}\pi_{\mathfrak{p}}^{-m(\mathfrak{p})} x_1\overline{x}_2 ),
\end{equation*}
where $\beta_{-} = \delta\beta_{+}$, and $\delta\in 1+\pi_{\mathfrak{p}}^{n(\mathfrak{p})-1}\mathcal O_{F,\mathfrak{p}}$\footnote{If $n(\mathfrak{p}) = 1$, then we set $1+\pi_{\mathfrak{p}}^{n(\mathfrak{p})-1}\mathcal O_{F,\mathfrak{p}} = \mathcal O_{F,\mathfrak{p}}^\times$.} is such that $\chi(\delta) = -1$. (In other words, the underlying $\mathcal O_{E,\mathfrak{q}}$-module is the same, but the hermitian form is rescaled by $\delta$.)
\end{itemize}
Let $ { \empty^{ \mathfrak{p} }} \mathscr{V}$ be the nearby Hermitian space as in Definition~\ref{def:nearby}. Then, by construction, the nearby lattice $ { \empty^{ \mathfrak{p} }} \Lambda_{\mathfrak{p}}$ is a lattice in $ { \empty^{ \mathfrak{p} }} \mathscr{V}_{\mathfrak{p}}$. Moreover, again by construction, we have an identification of $\mathcal O_{E,\mathfrak{q}}$-modules (though not an isometry) \begin{equation}\label{eqn:near ident} \Lambda_{\mathfrak{p}} = { \empty^{ \mathfrak{p} }} \Lambda_{\mathfrak{p}}. \end{equation}
Fix a coset \[ \lambda + \Lambda_{\mathfrak{p}} \subset \pi_{\mathfrak{q}}^{-n(\mathfrak{p})}\Lambda_{\mathfrak{p}} \] of $\Lambda_{\mathfrak{p}}$, and let $ { \empty^{ \mathfrak{p} }} \lambda+ { \empty^{ \mathfrak{p} }} \Lambda_{\mathfrak{p}}$ be the associated coset of $ { \empty^{ \mathfrak{p} }} \Lambda_{\mathfrak{p}}$ obtained from the identification~\eqref{eqn:near ident}.
Let $ { \empty^{ (\mathfrak{p}) }} \varphi_\lambda \in S( { \empty^{ (\mathfrak{p}) }} \mathscr{V}_{\mathfrak{p}})$ be the characteristic function of $ { \empty^{ (\mathfrak{p}) }} \lambda + { \empty^{ (\mathfrak{p}) }} \Lambda_{\mathfrak{p}}$. Here, and in the sequel, we will use the superscript $ { \empty^{ (\mathfrak{p}) }} $ to indifferently denote objects related to both $\mathscr{V}$ and $ { \empty^{ \mathfrak{p} }} \mathscr{V}$; \emph{e.g.}, $S( { \empty^{ (\mathfrak{p}) }} \mathscr{V}_{\mathfrak{p}})$ means either $S( \mathscr{V}_{\mathfrak{p}})$ or $S( { \empty^{ \mathfrak{p} }} \mathscr{V}_{\mathfrak{p}})$.
Write $ { \empty^{ (\mathfrak{p}) }} \Phi^\lambda_{\mathfrak{p}} \in I_{\mathfrak{{p}}}(s,\chi)$ for the standard section associated with $ { \empty^{ (\mathfrak{p}) }} \varphi_\lambda$ as in \S\ref{ss:incoherent}, with corresponding Whittaker function \[ W_{\alpha,\mathfrak{p}}(I,s, { \empty^{ (\mathfrak{p}) }} \Phi^\lambda_{\mathfrak{p}}) = \int_{F_{\mathfrak{p}}} { \empty^{ (\mathfrak{p}) }} \Phi^\lambda_{\mathfrak{p}} ( w n(b), s)\cdot \psi_{F_{\mathfrak{p}}}( - \alpha b )\, db. \]
Let $I\in \mathrm{SL}_2(F_{\mathfrak{p}})$ be the identity. For convenience, set
\[ W^*_{\alpha,\mathfrak{p}}(I,s, { \empty^{ (\mathfrak{p}) }} \Phi^\lambda_{\mathfrak{p}}) = \frac{\gamma_{\mathfrak{p}}( { \empty^{ (\mathfrak{p}) }} \mathscr{V})}{N(\mathfrak{p})^{f(\mathfrak{p})/2}}\cdot W_{\alpha,\mathfrak{p}}(I,s, { \empty^{ (\mathfrak{p}) }} \Phi^\lambda_{\mathfrak{p}}).
\] Here, $\gamma_{\mathfrak{p}}( { \empty^{ (\mathfrak{p}) }} \mathscr{V})$ is defined by~\eqref{eqn:gamma p defn}.
The next result follows from~\cite[Proposition 1.4]{KuAnnals}. \begin{proposition} \label{prop:whittaker vanishing} Suppose that $\alpha\in F^\times_{\mathfrak{p}}$ is not represented by $ { \empty^{ (\mathfrak{p}) }} \mathscr{V}_{\mathfrak{p}}$. Then \[ W_{ \alpha, \mathfrak{p}} (g_{\mathfrak{p}} , 0, { \empty^{ (\mathfrak{p}) }} \Phi^\lambda_{\mathfrak{p}}) = 0. \] \end{proposition}
Set
\[ \xi_{\mathfrak{p}} = \begin{cases} \pi_{\mathfrak{p}}^{-m(\mathfrak{p})} &\text{if $\mathfrak{p}$ is unramified in $E$} \\ \beta_+\pi_{\mathfrak{p}}^{-m(\mathfrak{p})} &\text{if $\mathfrak{p}$ is ramified in $E$}, \end{cases}
\]
and \begin{align}\label{eqn:near xi}
{ \empty^{ \mathfrak{p} }} \xi_{\mathfrak{p}} = \begin{cases}
\pi_{\mathfrak{p}}^{-m(\mathfrak{p})+1} &\text{if $\mathfrak{p}$ is unramified in $E$} \\ \beta_-\pi_{\mathfrak{p}}^{-m(\mathfrak{p})} &\text{if $\mathfrak{p}$ is ramified in $E$}.
\end{cases} \end{align}
The proofs of the two propositions below are essentially contained in~\cite[\S 4.6]{HY} and~\cite{Yang}. In particular, see \cite[Propositions 2.1, 2.2, and 2.3]{Yang}.
\begin{proposition} \label{prop:whittaker char function unram} Suppose that $\mathfrak{p}$ is unramified in $E$. \begin{enumerate}
\item If $\mathrm{ord}_{\mathfrak{p}}(\alpha) < -m(\mathfrak{p}) $, then \[ W_{\alpha,\mathfrak{p}}(I,s, { \empty^{ (\mathfrak{p}) }} \Phi^0_{\mathfrak{p}}) = 0. \] \item If $\mathrm{ord}_{\mathfrak{p}}(\alpha) \geq -m(\mathfrak{p})$, then
\[
W^*_{\alpha,\mathfrak{p}}(I,s,\Phi^0_{\mathfrak{p}}) = \frac{1}{L_{\mathfrak{p}}(s+1,\chi)}\sum_{0\leq k\leq \mathrm{ord}_{\mathfrak{p}}(\alpha)+m(\mathfrak{p})}(-1)^kN(\mathfrak{p})^{-ks},
\] and \begin{align*} W_{\alpha,\mathfrak{p}}(I,s, { \empty^{ \mathfrak{p} }} \Phi^0_{\mathfrak{p}}) &= W_{\alpha,\mathfrak{p}}(I,s,\Phi^0_{\mathfrak{p}}) - (1 + N(\mathfrak{p})^{-1}). \end{align*}
\end{enumerate} \end{proposition}
\begin{proposition} \label{prop:whittaker char function ram} Suppose that $\mathfrak{p}$ is ramified in $E$. \begin{enumerate} \item If $\mathrm{ord}_{\mathfrak{p}}(\alpha) < -m(\mathfrak{p})$, then \[ W_{\alpha,\mathfrak{p}}(I,s, { \empty^{ (\mathfrak{p}) }} \Phi^0_{\mathfrak{p}}) = 0. \] \item If $\mathrm{ord}_{\mathfrak{p}}(\alpha) \geq -m(\mathfrak{p})$, then
\[
W^*_{\alpha,\mathfrak{p}}(I,s, { \empty^{ (\mathfrak{p}) }} \Phi^0_{\mathfrak{p}}) = 1+\chi_{\mathfrak{p}}( { \empty^{ (\mathfrak{p}) }} \xi_{\mathfrak{p}}\alpha)N(\mathfrak{p})^{-(\mathrm{ord}_{\mathfrak{p}}(\alpha)+m(\mathfrak{p}) + n(\mathfrak{p}))s}.
\]
\end{enumerate} \end{proposition}
Now, suppose that $\mathfrak{p}$ is ramified in $E$. As above, let $\pi_{\mathfrak{q}}\in E_{\mathfrak{q}}$ be a uniformizer, chosen so that $\mathrm{Nm}(\pi_{\mathfrak{q}}) = \pi_{\mathfrak{p}}$.
For any $a_1,a_2,\zeta\in F^\times$, write $a_1\equiv a_2\pmod{\zeta}$ to mean $a_1\equiv a_2\pmod{\zeta\mathcal O_{F,\mathfrak{p}}}$. \begin{proposition} \label{prop:whittaker Phi lambda} Suppose that $\lambda\notin \Lambda_{\mathfrak{p}}$. \begin{enumerate} \item If $p\neq 2$, then \begin{align*} W^*_{\alpha,\mathfrak{p}}(I,s, { \empty^{ (\mathfrak{p}) }} \Phi^\lambda_{\mathfrak{p}}) &= \begin{cases} 1 &\text{if $\alpha\equiv { \empty^{ (\mathfrak{p}) }} \mathscr{Q}( { \empty^{ (\mathfrak{p}) }} \lambda)\pmod{ { \empty^{ (\mathfrak{p}) }} \xi_{\mathfrak{p}}}$} \\ 0 &\text{otherwise}. \end{cases} \end{align*} \item Suppose that $p=2$. Then \[ \alpha\equiv\mathscr{Q}(\lambda)\pmod{\xi_{\mathfrak{p}}}\;\Leftrightarrow\;\alpha\equiv { \empty^{ \mathfrak{p} }} \mathscr{Q}( { \empty^{ \mathfrak{p} }} \lambda)\pmod{ { \empty^{ \mathfrak{p} }} \xi_{\mathfrak{p}}}. \] Moreover, $W_{\alpha,\mathfrak{p}}(I,s, { \empty^{ (\mathfrak{p}) }} \Phi^\lambda_{\mathfrak{p}})$ is identically $0$ unless these equivalent congruences hold, and when they hold we have
\begin{align*}
W^*_{\alpha,\mathfrak{p}}(I,s, { \empty^{ (\mathfrak{p}) }} \Phi^\lambda_{\mathfrak{p}}) & =
1 + \chi_{\mathfrak{p}}( { \empty^{ (\mathfrak{p}) }} \xi_{\mathfrak{p}} \alpha)N(\mathfrak{p})^{-(n(\mathfrak{p})- r(\lambda))s},
\end{align*}
where $r(\lambda)\in\mathbb Z_{>0}$ is the smallest positive integer such that $\lambda\in \pi_{\mathfrak{q}}^{-r(\lambda)}\Lambda_{\mathfrak{p}}$. \end{enumerate}
\end{proposition}
\begin{proof} When $p\neq 2$, this computation is contained in \cite[Proposition 4.6.4]{HY}. When $p=2$, the result appears to be new. We present a mostly self-contained proof here that covers both possibilities.
For simplicity, write $\Phi$, $\chi$, $\psi$ and $\xi$ for $ { \empty^{ (\mathfrak{p}) }} \Phi^\lambda_{\mathfrak{p}}$, $\chi_{\mathfrak{p}}$, $\psi_{F_{\mathfrak{p}}}$ and $\xi_{\mathfrak{p}}$, respectively. By a standard argument, we have a decomposition: \[ W_{\alpha,\mathfrak{p}}(I,s,\Phi) = W_{\alpha,\mathfrak{p}}(I,s,\Phi)^{\leq 1} + W_{\alpha,\mathfrak{p}}(I,s,\Phi)^{>1}, \] where \begin{align*} W_{\alpha,\mathfrak{p}}(I,s,\Phi)^{\leq 1} & = \int_{\vert b\vert\leq 1}\Phi(wn(b))\psi(-\alpha b)db;\\ W_{\alpha,\mathfrak{p}}(I,s,\Phi)^{> 1} &= \int_{\vert b\vert > 1}\chi(b)\vert b\vert^{-(s+1)}\Phi(n_{-}(b^{-1}))\psi(-\alpha b)db. \end{align*} Here, $n_{-}(b^{-1}) = \left( \begin{matrix} 1 & 0 \\ b^{-1} & 1 \end{matrix} \right)$, and we have abbreviated $\Phi(g,0)$ to $\Phi(g)$.
By the definition of $\Phi$, and basic properties of the Weil representation, for any $b\in\mathcal O_{F,\mathfrak{p}}$, we have \begin{align*} \Phi(wn(b)) &= \gamma_{\mathfrak{p}}( { \empty^{ (\mathfrak{p}) }} \mathscr{V})\int_{ { \empty^{ (\mathfrak{p}) }} \lambda + { \empty^{ (\mathfrak{p}) }} \Lambda_{\mathfrak{p}}}\psi(b\cdot { \empty^{ (\mathfrak{p}) }} \mathscr{Q}(x))dx\\ & = \gamma_{\mathfrak{p}}( { \empty^{ (\mathfrak{p}) }} \mathscr{V})\cdot\psi(b\cdot { \empty^{ (\mathfrak{p}) }} \mathscr{Q}( { \empty^{ (\mathfrak{p}) }} \lambda))\cdot\int_{ { \empty^{ (\mathfrak{p}) }} \Lambda_{\mathfrak{p}}}\psi(b\cdot { \empty^{ (\mathfrak{p}) }} \mathscr{Q}(x) + b\cdot { \empty^{ (\mathfrak{p}) }} \langle { \empty^{ (\mathfrak{p}) }} \lambda , x\rangle)\; dx\\ & = \gamma_{\mathfrak{p}}( { \empty^{ (\mathfrak{p}) }} \mathscr{V})\cdot\psi(b\cdot { \empty^{ (\mathfrak{p}) }} \mathscr{Q}( { \empty^{ (\mathfrak{p}) }} \lambda))\cdot\int_{ { \empty^{ (\mathfrak{p}) }} \Lambda_{\mathfrak{p}}}\psi(b\cdot { \empty^{ (\mathfrak{p}) }} \mathscr{Q}(x))\; dx. \end{align*} Here, $dx$ is the Haar measure on $ { \empty^{ (\mathfrak{p}) }} \mathscr{V}_{\mathfrak{p}}$ that is self-dual with respect to the pairing \[ (x_1,x_2)\mapsto \psi\bigl(\mathrm{Tr}_{E_{\mathfrak{q}}/F_{\mathfrak{p}}}( { \empty^{ (\mathfrak{p}) }} \langle x_1,x_2\rangle)\bigr). \] We have also used the fact that, for any $x\in \Lambda_{\mathfrak{p}}$, $ { \empty^{ (\mathfrak{p}) }} \langle { \empty^{ (\mathfrak{p}) }} \lambda, x\rangle$ belongs to $\mathfrak{d}^{-1}_{F_{\mathfrak{p}}/\mathbb Q_p}$, and hence \[ \psi(b\cdot { \empty^{ (\mathfrak{p}) }} \langle { \empty^{ (\mathfrak{p}) }} \lambda , x\rangle) = 1. \]
Set $s_\lambda = { \empty^{ (\mathfrak{p}) }} \mathscr{Q}( { \empty^{ (\mathfrak{p}) }} \lambda)$.
Using \cite[Lemma 4.6.1]{HY}, we then obtain \begin{align*} \Phi(wn(b)) &= N(\mathfrak{p})^{-f(\mathfrak{p})/2}\gamma_{\mathfrak{p}}( { \empty^{ (\mathfrak{p}) }} \mathscr{V})\psi(b s_\lambda) \int_{\pi_{\mathfrak{p}}^{-m(\mathfrak{p})}\mathcal O_{F,\mathfrak{p}}}(1+\chi(\xi y))\psi(b y)dy \\ & = N(\mathfrak{p})^{-f(\mathfrak{p})/2}\gamma_{\mathfrak{p}}( { \empty^{ (\mathfrak{p}) }} \mathscr{V})\psi(bs_\lambda) \\ &\quad \times \biggl[N(\mathfrak{p})^{m(\mathfrak{p})/2} + \sum_{k=-m(\mathfrak{p})}^\infty N(\mathfrak{p})^{-k}\int_{\mathcal O_{F,\mathfrak{p}}^\times}\chi(\xi y)\psi(\pi_{\mathfrak{p}}^kby)dy \biggr] \\ & = N(\mathfrak{p})^{-n(\mathfrak{p})/2}\gamma_{\mathfrak{p}}( { \empty^{ (\mathfrak{p}) }} \mathscr{V})\psi(bs_\lambda). \end{align*} The second equality here is deduced by noting \begin{equation}\label{eqn:psi integral} \int_{\pi_{\mathfrak{p}}^{-r}\mathcal O_{F,\mathfrak{p}}}\psi(\zeta y) dy = \begin{cases} N(\mathfrak{p})^{-r}\mathrm{Vol}(\mathcal O_{F,\mathfrak{p}}) = N(\mathfrak{p})^{r-m(\mathfrak{p})/2} &\text{if $\mathrm{ord}_{\mathfrak{p}}(\zeta)\geq r - m(\mathfrak{p})$} \\ 0 &\text{otherwise}, \end{cases} \end{equation} and the last by using the following lemma, which is a standard Gauss sum computation, using the fact that $\chi$ has conductor $n(\mathfrak{p})$. \begin{lemma}\label{lem:gauss sum} For $\alpha\in F_{\mathfrak{p}}$, \[ \int_{\mathcal O_{F,\mathfrak{p}}^\times}\chi(y)\psi(\alpha y)dy = \begin{cases} N(\mathfrak{p})^{-f(\mathfrak{p})/2}\cdot\chi(\alpha)\cdot\epsilon_{\mathfrak{p}}(\chi,\psi) &\text{ if $\mathrm{ord}_{\mathfrak{p}}(\alpha) = -f(\mathfrak{p})$} \\ 0 &\text{ otherwise.} \end{cases} \] \end{lemma}
Therefore, we have \begin{align}\label{eqn:whittaker ord b +} W_{\alpha,\mathfrak{p}}(I,s,\Phi)^{\leq 1} & = N(\mathfrak{p})^{-n(\mathfrak{p})/2}\gamma_{\mathfrak{p}}( { \empty^{ (\mathfrak{p}) }} \mathscr{V})\cdot\int_{\mathcal O_{F,\mathfrak{p}}}\psi((s_\lambda-\alpha) b)db\nonumber\\ & = \begin{cases} N(\mathfrak{p})^{-f(\mathfrak{p})/2}\cdot \gamma_{\mathfrak{p}}( { \empty^{ (\mathfrak{p}) }} \mathscr{V}) &\text{if $\alpha\equiv s_\lambda\pmod{\xi}$} \\ 0 &\text{otherwise.} \end{cases} \end{align}
To compute $W_{\alpha,\mathfrak{p}}(I,s,\Phi)^{>1}$, we will need \begin{lemma} \label{lem:phi n- c} Suppose that $c\in\mathcal O_{F,\mathfrak{p}}$ and that $k=\mathrm{ord}_{\mathfrak{p}}(c) > 1$. For any integer $t\in \mathbb Z_{\geq 1}$, set \[ U^{t}_{F,\mathfrak{p}} = 1 + \pi_{\mathfrak{p}}^{t} \mathcal O_{F,\mathfrak{p}}. \] Set \[ d(k,\lambda) = 2k - (n(\mathfrak{p}) - r(\lambda)). \]
Then $\Phi(n_{-}(c)) \neq 0$ only if \[ \frac{n(\mathfrak{p}) - r(\lambda)}{2} < k < n(\mathfrak{p}) - \frac{r(\lambda)}{2}. \] In this case, we have \[ \Phi(n_{-}(c)) = \frac{\psi( c^{-1} s_\lambda) }{\mathrm{Vol}(U^{d(k,\lambda)}_{F,\mathfrak{p}})} \cdot \int_{U^{d(k,\lambda)}_{F,\mathfrak{p}}} \chi( y) \psi(- c^{-1} s_\lambda y)dy. \] In particular, if $p\neq 2$, then $W_{\alpha,\mathfrak{p}}(I,s,\Phi)^{>1} = 0$. \end{lemma} \begin{proof} As in~\cite{HY} and~\cite{Yang}, this uses the identity $n_{-}(c) = -wn(-c)w$, so that \begin{align*}\label{eqn:phi n- c} \Phi(n_{-}(c)) &= \chi(-1)\Phi(wn(-c)w)\\ &= \chi(-1) \cdot \gamma_{\mathfrak{p}} ( { \empty^{ (\mathfrak{p}) }} \mathscr{V}) \int_{ { \empty^{ (\mathfrak{p}) }} \mathscr{V}_{\mathfrak{p}}}\psi(-c\cdot { \empty^{ (\mathfrak{p}) }} \mathscr{Q}(x)) \omega(w) ( { \empty^{ (\mathfrak{p}) }} \varphi_\lambda) (x) dx\\ & = \int_{ { \empty^{ (\mathfrak{p}) }} \mathscr{V}_{\mathfrak{p}}}\psi(-c\cdot { \empty^{ (\mathfrak{p}) }} \mathscr{Q}(x)) \int_{ { \empty^{ (\mathfrak{p}) }} \lambda + { \empty^{ (\mathfrak{p}) }} \Lambda_{\mathfrak{p}}}\psi(-\mathrm{Tr}( { \empty^{ (\mathfrak{p}) }} \langle x,y\rangle)) dy\; dx. \end{align*} Here, we have used the identity $ \gamma_{\mathfrak{p}} ( { \empty^{ (\mathfrak{p}) }} \mathscr{V}) ^ 2 = \epsilon_{\mathfrak{p}}(\chi,\psi)^2 = \chi(-1)$.
For $x\in { \empty^{ (\mathfrak{p}) }} \mathscr{V}_{\mathfrak{p}}$, set $t_\lambda(x) = \mathrm{Tr}( { \empty^{ (\mathfrak{p}) }} \langle { \empty^{ (\mathfrak{p}) }} \lambda, x\rangle)$. We compute \begin{align*} \int_{ { \empty^{ (\mathfrak{p}) }} \lambda + { \empty^{ (\mathfrak{p}) }} \Lambda_{\mathfrak{p}}}\psi(-\mathrm{Tr}( { \empty^{ (\mathfrak{p}) }} \langle x,y\rangle)) dy & = \psi( - t_\lambda(x) ) \int_{\Lambda_{\mathfrak{p}}} \psi ( - \mathrm{Tr}( { \empty^{ (\mathfrak{p}) }} \langle x, y\rangle) ) dy\\ & = N(\mathfrak{p})^{-n(\mathfrak{p})/2} \cdot \psi( - t_\lambda(x) ) \cdot \mathrm{char}( \pi_{\mathfrak{q}}^{-n(\mathfrak{p})} \Lambda_{\mathfrak{p}} ) ( x ), \end{align*} and hence \begin{align*} \Phi(n_{-}(c)) & = N(\mathfrak{p})^{-n(\mathfrak{p})/2} \int_{\pi_{\mathfrak{q}}^{-n(\mathfrak{p})} \Lambda_{\mathfrak{p}}} \psi(- c\cdot { \empty^{ (\mathfrak{p}) }} \mathscr{Q}(x) - t_\lambda(x)) \; dx. \end{align*}
If $ k \geq n(\mathfrak{p})$ then \[ \mathrm{ord}_{\mathfrak{p}}( c \cdot { \empty^{ (\mathfrak{p}) }} \mathscr{Q}(x)) \geq k - n(\mathfrak{p}) - m(\mathfrak{p}) \geq -m(\mathfrak{p}) \] for all $x\in \pi_{\mathfrak{q}}^{-n(\mathfrak{p})} \Lambda_{\mathfrak{p}}$. Therefore, under this assumption, we have \begin{align*} \Phi(n_{-}(c)) = N(\mathfrak{p})^{-n(\mathfrak{p})/2} \int_{\pi_{\mathfrak{q}}^{-n(\mathfrak{p})} \Lambda_{\mathfrak{p}}} \psi(- t_\lambda(x)) \; dx = 0, \end{align*} where we have used~\eqref{eqn:psi integral}.
Now, suppose that $k<n(\mathfrak{p})$. Note that \[ -c\cdot { \empty^{ (\mathfrak{p}) }} \mathscr{Q}(x) - t_\lambda(x) = -c\cdot { \empty^{ (\mathfrak{p}) }} \mathscr{Q}( x + c^{-1}\cdot { \empty^{ (\mathfrak{p}) }} \lambda ) + c^{-1} s_\lambda. \] Therefore, \begin{align*} \Phi(n_{-}(c)) & = N(\mathfrak{p})^{-n(\mathfrak{p})/2} \psi( c^{-1} s_\lambda)\cdot \int_{c^{-1}\cdot { \empty^{ (\mathfrak{p}) }} \lambda + \pi_{\mathfrak{q}}^{-n(\mathfrak{p})} \Lambda_{\mathfrak{p}}} \psi(- c\cdot { \empty^{ (\mathfrak{p}) }} \mathscr{Q}( x ) )\; dx. \end{align*}
Using Lemma 4.6.1 of~\cite{HY}, we find \begin{align*} \Phi (n_{-}(c)) & = N(\mathfrak{p})^{-(f(\mathfrak{p})+n(\mathfrak{p}))/2} \psi( c^{-1} s_\lambda) \cdot \int_{c^{-2}s_\lambda + \pi_{\mathfrak{p}}^{-f(\mathfrak{p})}\mathcal O_{F,\mathfrak{p}}} (1 + \chi(\xi y)) \psi( - c y)\; dy\\ & = N(\mathfrak{p})^{-(f(\mathfrak{p})+n(\mathfrak{p}))/2} \psi( c^{-1} s_\lambda) \cdot \int_{c^{-2}s_\lambda + \pi_{\mathfrak{p}}^{-f(\mathfrak{p})}\mathcal O_{F,\mathfrak{p}}} \chi(\xi y) \psi( - c y)\; dy. \end{align*} Here, for the last identity, we have also used~\eqref{eqn:psi integral} combined with the inequality \[ \mathrm{ord}_{\mathfrak{p}} (c) = k < n(\mathfrak{p}) = f(\mathfrak{p}) - m(\mathfrak{p}). \]
If $2k > n(\mathfrak{p}) - r(\lambda)$, then \[ \mathrm{ord}_{\mathfrak{p}}(c^{-2}s_\lambda) = -2k - m(\mathfrak{p}) - r(\lambda) < -f(\mathfrak{p}). \] Therefore, the substitution $y \mapsto (c^{-2}s_\lambda)^{-1}y$, combined with the observation that $\chi(s_\lambda) = \chi(\xi)$ gives us \[ \Phi(n_{-}(c)) = \frac{\psi( c^{-1} s_\lambda) }{\mathrm{Vol}(U^{d(k,\lambda)}_{F,\mathfrak{p}})} \cdot \int_{U^{d(k,\lambda)}_{F,\mathfrak{p}}} \chi( y) \psi(- c^{-1} s_\lambda y)dy. \] If $k \geq n(\mathfrak{p}) - \frac{r(\lambda)}{2}$, then $d(k,\lambda) \geq n(\mathfrak{p})$. Since $\chi$ has conductor $n(\mathfrak{p})$, in this case we get \[ \Phi(n_{-}(c)) = \frac{\psi( c^{-1} s_\lambda) }{\mathrm{Vol}(U^{d(k,\lambda)}_{F,\mathfrak{p}})} \cdot \int_{U^{d(k,\lambda)}_{F,\mathfrak{p}}} \psi(- c^{-1} s_\lambda y)dy, \] which vanishes by~\eqref{eqn:psi integral}.
If $2k \leq n(\mathfrak{p}) - r(\lambda)$, then we have \[
\int_{c^{-2}s_\lambda + \pi_{\mathfrak{p}}^{-f(\mathfrak{p})}\mathcal O_{F,\mathfrak{p}}} \chi(y) \psi( - c y)\; dy = \int_{\pi_{\mathfrak{p}}^{-f(\mathfrak{p})}\mathcal O_{F,\mathfrak{p}}} \chi(y) \psi( - c y)\; dy. \] In this case, it is not hard to see, using Lemma~\ref{lem:gauss sum}, that this integral vanishes, and hence that $\Phi ( n_{-}(c) ) = 0$.
\end{proof}
When $p\neq 2$, this, combined with~\eqref{eqn:whittaker ord b +}, finishes the proof of Proposition~\ref{prop:whittaker Phi lambda}. Therefore, we now specialize to the case where $p=2$. In this case, we have \[ { \empty^{ \mathfrak{p} }} \mathscr{Q}( { \empty^{ \mathfrak{p} }} \lambda) = \delta\mathscr{Q}(\lambda), \] where $\delta\in U^{n(\mathfrak{p})-1}_{F,\mathfrak{p}}$. From this, and the condition $r(\lambda) < n(\mathfrak{p})$, it follows easily that the conditions \[ \alpha\equiv\mathscr{Q}(\lambda)\pmod{\xi},\quad\alpha\equiv { \empty^{ \mathfrak{p} }} \mathscr{Q}( { \empty^{ \mathfrak{p} }} \lambda)\pmod{\xi} \] are equivalent. This shows the first part of assertion (2) of the proposition.
For the second part, observe that Lemma~\ref{lem:phi n- c} gives us: \begin{align*} W_{\alpha,\mathfrak{p}} ( I , s , \Phi )^{ > 1} & = \sum_k N(\mathfrak{p})^{-k(s+1)} \int_{\mathrm{ord}_{\mathfrak{p}}(b) = -k} \chi(b) \Phi ( n_{-}(b^{-1})) \psi(-\alpha b)\; db., \end{align*} where $\frac{n - r(\lambda)}{2} < k < n(\mathfrak{p}) + \frac{\mathrm{ord}_{\mathfrak{q}}(\lambda}{2}$, and the summand indexed by $k$ is equal to \begin{align}\label{eqn:kth term} &\frac{N(\mathfrak{p})^{-ks}}{\mathrm{Vol}(U^{d(k,\lambda)}_{F,\mathfrak{p}})}\int_{U^{d(k,\lambda)}_{F,\mathfrak{p}}}\chi(y)\int_{\mathcal O_{F,\mathfrak{p}}^\times}\chi(b)\psi(b\pi_{\mathfrak{p}}^{-k}(s_\lambda - \alpha -s_\lambda y)) db\; dy. \end{align}.
Now, we have \[ \mathrm{ord}_{\mathfrak{p}}(\pi_{\mathfrak{p}}^{-k}s_\lambda(1-y)) = k - f(\mathfrak{p}) > - f(\mathfrak{p}), \] and therefore \begin{align*} \mathrm{ord}_{\mathfrak{p}}(\pi_{\mathfrak{p}}^{-k}(s_\lambda - \alpha - s_\lambda y)) &= -k + \mathrm{ord}_{\mathfrak{p}}(s_\lambda(1-y) - \alpha) \end{align*} can equal $-f(\mathfrak{p})$ if and only if $\mathrm{ord}_{\mathfrak{p}}(\alpha) = k - f(\mathfrak{p})$. So, using Lemma~\ref{lem:gauss sum}, we see that \begin{align*} \int_{\mathcal O_{F,\mathfrak{p}}^\times}\chi(b)\psi(b\pi_{\mathfrak{p}}^{-k}(s_\lambda - \alpha -s_\lambda y)) db = N(\mathfrak{p})^{-f(\mathfrak{p})/2}\cdot \chi(-s_\lambda )\cdot \chi(y - (1 - s_\lambda^{-1}\alpha) )\cdot \epsilon_{\mathfrak{p}}(\chi,\psi) \end{align*} if $\mathrm{ord}_{\mathfrak{p}}(\alpha) = k - f(\mathfrak{p})$, and that it vanishes otherwise. Therefore~\eqref{eqn:kth term} is non-zero only if $\mathrm{ord}_{\mathfrak{p}}(\alpha) = k - f(\mathfrak{p})$, in which case it is equal to \begin{align*} \frac{N(\mathfrak{p})^{-f(\mathfrak{p})/2}\cdot \gamma_{\mathfrak{p}}( { \empty^{ (\mathfrak{p}) }} \mathscr{V})\cdot N(\mathfrak{p})^{-ks}}{\mathrm{Vol}(U^{d(k,\lambda)}_{F,\mathfrak{p}})}\int_{U^{d(k,\lambda)}_{F,\mathfrak{p}}}\chi(y(y - (1-s_\lambda^{-1}\alpha)))\; dy. \end{align*} Here, we have also used the formula for $\gamma_{\mathfrak{p}}( { \empty^{ (\mathfrak{p}) }} \mathscr{V})$ from~\eqref{eqn:gamma p defn}, combined with the identity $\chi(s_\lambda) = \chi(\xi) = \mathrm{inv}_{\mathfrak{p}}( { \empty^{ (\mathfrak{p}) }} \mathscr{V})$.
Combining this with~\eqref{eqn:whittaker ord b +}, we obtain \begin{align} \label{eqn:whittaker prelim formula} W^*_{\alpha,\mathfrak{p}}(I, s, \Phi) & = \begin{cases} N(\mathfrak{p})^{-ks}\cdot M(\alpha,\lambda) &\text{if $\alpha\nequiv s_\lambda\pmod{\xi}$} \\ 1 + N(\mathfrak{p})^{-ks}\cdot M(\alpha,\lambda) &\text{if $\alpha\equiv s_\lambda\pmod{\xi}$}, \end{cases} \end{align} where $k = \mathrm{ord}_{\mathfrak{p}}(\alpha) + f(\mathfrak{p})$, and where \[ M(\alpha,\lambda) = \frac{1}{\mathrm{Vol}(U^{d(k,\lambda)}_{F,\mathfrak{p}})}\int_{U^{d(k,\lambda)}_{F,\mathfrak{p}}}\chi(y(y - (1-s_\lambda^{-1}\alpha)))\; dy. \]
Now, if $\alpha\equiv s_\lambda\pmod{\xi}$, then $\mathrm{ord}_{\mathfrak{p}}(s_\lambda^{-1}\alpha) = 0$, and so $k = n(\mathfrak{p}) - r(\lambda)$. Therefore, the proof of the proposition will be completed by the following lemma:
\begin{lemma} \label{lem:M alpha lambda} We have \[ M(\alpha,\lambda) = \begin{cases} 1 &\text{if $\alpha\equiv s_\lambda\pmod{\xi}$ and $ { \empty^{ (\mathfrak{p}) }} \mathscr{V}_{\mathfrak{p}}$ represents $\alpha$}\\ -1 &\text{if $\alpha\equiv s_\lambda\pmod{\xi}$ and $ { \empty^{ (\mathfrak{p}) }} \mathscr{V}_{\mathfrak{p}}$ does not represent $\alpha$}\\ 0 &\text{if $\alpha\nequiv s_\lambda\pmod{\xi}$}. \end{cases} \] \end{lemma}
\begin{proof} If $\alpha\equiv s_\lambda\pmod{\xi}$ and $ { \empty^{ (\mathfrak{p}) }} \mathscr{V}_{\mathfrak{p}}$ represents $\alpha$, then we can choose our coset representative $ { \empty^{ (\mathfrak{p}) }} \lambda$ so that \[ s_\lambda = { \empty^{ (\mathfrak{p}) }} \mathscr{Q}( { \empty^{ (\mathfrak{p}) }} \lambda) = \alpha. \] Therefore, $s_\lambda^{-1}\alpha = 1$, and the formula for $M(\alpha,\lambda)$ reduces to \[ M(\alpha,\lambda) = \frac{1}{\mathrm{Vol}(U^{d(k,\lambda)}_{F,\mathfrak{p}})}\int_{U^{d(k,\lambda)}_{F,\mathfrak{p}}}\chi(y^2)\; dy = 1. \]
If $\alpha\equiv s_\lambda\pmod{\xi}$ is not represented by $ { \empty^{ (\mathfrak{p}) }} \mathscr{V}_{\mathfrak{p}}$, then Proposition~\ref{prop:whittaker vanishing} shows that \[ 1 + M(\alpha,\lambda) = W^*_{\alpha,\mathfrak{p}}(I, 0 ,\Phi) = 0, \] and so $M(\alpha,\lambda) = -1$.
Now, suppose that $\alpha \nequiv s_\lambda \pmod{\xi}$. Set $\zeta = 1 - s_\lambda^{-1}\alpha$. We have \begin{align}\label{eqn:M lambda simplified} \int_{U^{d(k,\lambda)}_{F,\mathfrak{p}}}\chi(y ( y - \zeta )) \; dy& = \int_{U^{d(k,\lambda)}_{F,\mathfrak{p}}}\chi(1- y^{-1}\zeta )\; dy\nonumber\\ & = \int_{U^{d(k,\lambda)}_{F,\mathfrak{p}}}\chi(1- y\zeta )\; dy\nonumber\\ & = \chi(-\zeta) \int_{U^{d(k,\lambda)}_{F,\mathfrak{p}}}\chi(y - \zeta^{-1})\; dy. \end{align}
Note that \[ \mathrm{ord}_{\mathfrak{p}}(\zeta) =\begin{cases} 0 &\text{if $k> n(\mathfrak{p}) - r(\lambda)$} \\ k - (n(\mathfrak{p}) - r(\lambda)) &\text{if $k< n(\mathfrak{p}) - r(\lambda)$}. \end{cases} \] Moreover, when $k = n(\mathfrak{p}) - r(\lambda)$, $\mathrm{ord}_{\mathfrak{p}}(\zeta)$ is an integer between $0$ and $r(\lambda) - 1$. In particular, we find that we always have \[ n(\mathfrak{p}) - 1 - \mathrm{ord}_{\mathfrak{p}}(\zeta) \geq d(k,\lambda). \]
Choose $\eta \in U^{n(\mathfrak{p})-1}_{F,\mathfrak{p}}$ such that $\chi(\eta) = -1$. This choice determines a measure preserving bijection \[
\alpha:U^{d(k,\lambda)}_{F,\mathfrak{p}} \xrightarrow{\simeq} U^{d(k,\lambda)}_{F,\mathfrak{p}} \] by $ y \mapsto y + (1 - \eta)\zeta^{-1}.$ We now compute \begin{align*} \int_{U^{d(k,\lambda)}_{F,\mathfrak{p}}} \chi ( y - \zeta^{-1} ) \; dy & = \int_{U^{d(k,\lambda)}_{F,\mathfrak{p}}} \chi( \alpha(y) - \zeta^{-1} ) \; dy\\ & = \int_{U^{d(k,\lambda)}_{F,\mathfrak{p}}} \chi( y - \eta \zeta^{-1}) \; dy\\ & = - \int_{U^{d(k,\lambda)}_{F,\mathfrak{p}}} \chi( \eta^{-1}y - \zeta^{-1}) \; dy\\ & = - \int_{U^{d(k,\lambda)}_{F,\mathfrak{p}}} \chi( y - \zeta^{-1}) \; dy. \end{align*} Combining this with~\eqref{eqn:M lambda simplified} shows that $M(\alpha,\lambda) = 0$. \end{proof} This completes the proof of Proposition \ref{prop:whittaker Phi lambda}. \end{proof}
We now record a few more results that are easy consequences of Propositions~\ref{prop:whittaker char function unram},~\ref{prop:whittaker char function ram} and~\ref{prop:whittaker Phi lambda}. We omit their proofs.
\begin{proposition} \label{prop:whittaker alpha 0} We have \[ W_{ 0, \mathfrak{p}} (I , s, \Phi^\lambda_{\mathfrak{p}}) = \begin{cases} \frac{\gamma_{\mathfrak{p}}(\mathscr{V})}{N(\mathfrak{p})^{f(\mathfrak{p})/2}}\cdot\frac{L_{\mathfrak{p}}(s,\chi)}{L_{\mathfrak{p}}(s+1,\chi)}&\text{if $\lambda \in \Lambda_{\mathfrak{p}}$} \\ 0&\text{if $\lambda\notin \Lambda_{\mathfrak{p}}$.} \end{cases} \] \end{proposition}
\begin{corollary}\label{cor:good prime M constant} Let $M_{\mathfrak{p}}(s,\varphi_{\lambda})$ be as in~\eqref{local M}. Then $M_\mathfrak{p}(s,\varphi_\lambda)$ is constant. In fact, it is either $1$ or $0$ depending on whether $\lambda$ is zero or non-zero. \end{corollary}
\begin{proposition} \label{prop:whittaker derivative} Suppose that $\alpha\in F^\times$ is such that $\mathrm{Diff}(\alpha) = \{\mathfrak{p}\}$. Set \[ X(\alpha,\lambda) = \{x\in { \empty^{ \mathfrak{p} }} \lambda+ { \empty^{ \mathfrak{p} }} L_{\mathfrak{p}}: { \empty^{ \mathfrak{p} }} \mathscr{Q}(x) = \alpha\in F^\times_{\mathfrak{p}}\}. \] \begin{enumerate} \item If $X(\alpha,\lambda) = \emptyset$, then \[ W'_{\alpha,\mathfrak{p}}(I, 0, \Phi^\lambda_{\mathfrak{p}}) = 0. \]
\item If $X(\alpha,\lambda) \neq \emptyset$, then $ W_{ \alpha, \mathfrak{p}} (I , 0, { \empty^{ \mathfrak{p} }} \Phi^\lambda_{\mathfrak{p}})\neq 0$. Moreover, in this case, we have \[ \frac{W'_{ \alpha, \mathfrak{p}} (I , 0, \Phi^\lambda_{\mathfrak{p}})}{W_{\alpha,\mathfrak{p}}( I, 0 , { \empty^{ \mathfrak{p} }} \Phi^\lambda_{\mathfrak{p}})} = \frac{\ell_{\mathfrak{p}}(\alpha)}{2}\cdot \log N(\mathfrak{q}), \] where \[ \ell_{\mathfrak{p}}(\alpha) = \begin{cases} \frac{\mathrm{ord}_{\mathfrak{p}}(\xi_{\mathfrak{p}}^{-1}\alpha)+1}{2} &\text{if $\mathfrak{p}$ is unramified in $E$} \\ \mathrm{ord}_{\mathfrak{p}}(\xi_{\mathfrak{p}}^{-1} \alpha) + n(\mathfrak{p}) &\text{if $\mathfrak{p}$ is ramified in $E$}. \end{cases} \] \end{enumerate} \end{proposition}
\subsection{Nearby Schwarz functions}\label{ss:nearby lattices}
We will keep our notation from the previous subsection.
If $\mathfrak{p}'\mid p$ is a prime of $\mathcal O_F$ not equal to $\mathfrak{p}$, set $ { \empty^{ \mathfrak{p} }} \Lambda_{\mathfrak{p}'} = \Lambda_{\mathfrak{p}}$ as Hermitian spaces over $\mathcal O_{E,\mathfrak{p}}$. Note that $ { \empty^{ \mathfrak{p} }} \Lambda_{\mathfrak{p}'}[p^{-1}]$ is isometric to $ { \empty^{ \mathfrak{p} }} \mathscr{V}_{\mathfrak{p}'}$, and set \[ \Lambda_p = \bigoplus_{\mathfrak{p}'\mid p}\Lambda_{\mathfrak{p}'}, \quad { \empty^{ \mathfrak{p} }} \Lambda_p = \bigoplus_{\mathfrak{p}'\mid p} { \empty^{ \mathfrak{p} }} \Lambda_{\mathfrak{p}'}. \]
As in~\eqref{eqn:near ident}, we have a canonical $\mathcal O_{E,p}$-linear isomorphism (but not an isometry): \begin{equation}\label{eqn:near isometry} \Lambda_p \xrightarrow{\simeq} { \empty^{ \mathfrak{p} }} \Lambda_p\subset { \empty^{ \mathfrak{p} }} \mathscr{V}_p. \end{equation} We set \[ { \empty^{ (\mathfrak{p}) }} \Lambda^\vee_p = \bigoplus_{\mathfrak{p}'\mid p}(\mathfrak{p}')^{-n(\mathfrak{p}')}( { \empty^{ (\mathfrak{p}) }} \Lambda_{\mathfrak{p}'}). \] Note that $ { \empty^{ \mathfrak{p} }} \Lambda^\vee_p$ is not necessarily the dual lattice associated with $ { \empty^{ \mathfrak{p} }} \Lambda_p$, but the notation will be convenient.
Suppose that we are given a class \[ \lambda = (\lambda_{\mathfrak{p}'}) \in \Lambda_p^\vee/\Lambda_p = \bigoplus_{\mathfrak{p}'\mid p}\left((\mathfrak{p}')^{-n(\mathfrak{p}')}\Lambda_{\mathfrak{p}'}/\Lambda_{\mathfrak{p}'}\right). \] Observe that the isomorphism~\eqref{eqn:near isometry} carries the coset $\lambda + \Lambda_p$ to a coset $ { \empty^{ \mathfrak{p} }} \lambda + { \empty^{ \mathfrak{p} }} \Lambda_p$ of $ { \empty^{ \mathfrak{p} }} \Lambda_p$ in $ { \empty^{ \mathfrak{p} }} \Lambda^\vee_p$. We have a further factorization \[ { \empty^{ \mathfrak{p} }} \lambda + { \empty^{ \mathfrak{p} }} \Lambda_p = \prod_{\mathfrak{p}'\mid p} { \empty^{ \mathfrak{p} }} \lambda_{\mathfrak{p}'} + { \empty^{ \mathfrak{p} }} \Lambda_{\mathfrak{p}'}. \] Let $ { \empty^{ \mathfrak{p} }} \varphi_{\lambda_{\mathfrak{p}'}}\in S( { \empty^{ \mathfrak{p} }} \mathscr{V}_{\mathfrak{p}'})$ be the characteristic function of $ { \empty^{ \mathfrak{p} }} \lambda_{\mathfrak{p}'} + { \empty^{ \mathfrak{p} }} \Lambda_{\mathfrak{p}'}$. We now set \[ { \empty^{ \mathfrak{p} }} \tilde{\varphi}_\lambda = \bigotimes_{\mathfrak{p}'\mid p} { \empty^{ \mathfrak{p} }} \tilde{\varphi}_{\lambda_{\mathfrak{p}'}}, \] where $ { \empty^{ \mathfrak{p} }} \tilde{\varphi}_{\lambda_{\mathfrak{p}'}} = { \empty^{ \mathfrak{p} }} \varphi_{\lambda_{\mathfrak{p}'}}$, for $\mathfrak{p}'\neq \mathfrak{p}$, and where \begin{align}\label{eqn:defn tilde varphi} { \empty^{ \mathfrak{p} }} \tilde{\varphi}_{\lambda_{\mathfrak{p}}} & = \begin{cases} { \empty^{ \mathfrak{p} }} \varphi_{\lambda_{\mathfrak{p}}} &\text{ if $\mathrm{ord}_{\mathfrak{q}}(\lambda_{\mathfrak{p}}) > - n(\mathfrak{p})$} \\ 0 &\text{ otherwise.} \end{cases} \end{align}
Fix $\mu\in L^\vee/L$. Associated with this is the characteristic function $\varphi_\mu\in S(\widehat{\mathscr{V}})$ of the coset $\mu + \widehat{L}$. We will now associate with this class a \emph{nearby} Schwarz function $ { \empty^{ \mathfrak{p} }} \varphi_\mu\in S( { \empty^{ \mathfrak{p} }} \widehat{\mathscr{V}})$ as follows. First, we will have a factorization \[ { \empty^{ \mathfrak{p} }} \varphi_\mu = \bigotimes_\ell { \empty^{ \mathfrak{p} }} \varphi_{\mu_\ell}\in \bigotimes_\ell S( { \empty^{ \mathfrak{p} }} \mathscr{V}_\ell) \subset S( { \empty^{ \mathfrak{p} }} \widehat{\mathscr{V}}). \]
If $\ell\neq p$, then $ { \empty^{ \mathfrak{p} }} \varphi_{\mu_\ell} = \varphi_{\mu_\ell}$ will be the characteristic function of $\mu_\ell + L_\ell$ under the obvious identification \[ S( { \empty^{ \mathfrak{p} }} \mathscr{V}_\ell) = S(\mathscr{V}_\ell). \]
If $\ell = p$, we can view $\mu_p + L_p$ as a subset of $\Lambda^\vee_p$, and, as such, it is a disjoint union \[ \mu_p + L_p = \bigsqcup_{\lambda\in \mu_p + L_p}\lambda + \Lambda_p \] of cosets of $\Lambda_p$ in $\Lambda^\vee_p$. We now set \begin{equation}\label{eqn:near varphi mu decomp} { \empty^{ \mathfrak{p} }} \varphi_{\mu_p} = \sum_{\lambda\in \mu_p + L_p} { \empty^{ \mathfrak{p} }} \tilde{\varphi}_\lambda = \sum_{\lambda\in \mu_p + L_p}\bigotimes_{\mathfrak{p}'\mid p} { \empty^{ \mathfrak{p} }} \tilde{\varphi}_{\lambda_{\mathfrak{p}'}}. \end{equation}
As in the discussion of \S \ref{ss:incoherent}, let $I(s,\chi)$ be the space of the degenerate principal series representation of $\mathrm {SL}_2(\mathbb A_F)$
induced from $\chi |\cdot |^s$, and let $ { \empty^{ (\mathfrak{p}) }} \Phi^\mu(g,s)$ be the standard section of $I(s,\chi)$ determined by the Schwartz function \[ \varphi_\infty^{\bm{1}} \otimes { \empty^{ (\mathfrak{p}) }} \varphi_\mu \in S(\mathscr{C}_\infty) \otimes S( { \empty^{ (\mathfrak{p}) }} \mathscr{\widehat{V}}), \] Associated to this and each $\alpha\in F$ is the Whittaker function \[ W_\alpha(g,s, { \empty^{ (\mathfrak{p}) }} \Phi^\mu) = W_{\alpha,\infty}(g_\infty,s, { \empty^{ (\mathfrak{p}) }} \Phi^\mu_\infty)\cdot W_{\alpha,f}(g_f,s, { \empty^{ (\mathfrak{p}) }} \Phi^\mu_f) \] admitting a factorization into infinite and finite parts.
We have a decomposition of the finite part \[ { \empty^{ (\mathfrak{p}) }} \Phi^\mu_f = \otimes_\ell { \empty^{ (\mathfrak{p}) }} \Phi^\mu_\ell\in \otimes_\ell I_\ell(s,\chi), \] where $ { \empty^{ (\mathfrak{p}) }} \Phi^\mu_\ell\in I_\ell(s,\chi)$ is the standard section associated with $ { \empty^{ (\mathfrak{p}) }} \varphi_{\mu_\ell}$. This gives us a decomposition of Whittaker functions \[ W_{\alpha,f}(g_f,s, { \empty^{ (\mathfrak{p}) }} \Phi^\mu_f) = \prod_\ell W_{\alpha,\ell}(g_\ell, s, { \empty^{ (\mathfrak{p}) }} \Phi^\mu_\ell). \]
We have \[ \varphi_{\mu_p} = \sum_{\lambda_p\in \mu_p + L_p}\varphi_{\lambda}\in S(\mathscr{V}_p), \] where $\varphi_{\lambda}$ is the characteristic function of $\lambda + \Lambda_p$, and the analogous decomposition for $ { \empty^{ \mathfrak{p} }} \varphi_\mu$ from~\eqref{eqn:near varphi mu decomp}. Each coset $\lambda + \Lambda_p$ admits a further decomposition \[ \lambda + \Lambda_p = \bigsqcup_{\mathfrak{p}'\mid p}\lambda_{\mathfrak{p}'} + \Lambda_{\mathfrak{p}'}, \] and so we obtain a finer decomposition \begin{equation}\label{eqn:mu to lambda decomp} \varphi_{\mu_p} = \sum_{\lambda_p\in \mu_p + L_p}\bigotimes_{\mathfrak{p}'\mid p}\varphi_{\lambda_{\mathfrak{p}'}}, \end{equation} where $\varphi_{\lambda_{\mathfrak{p}'}}$ is the characteristic function of $\lambda_{\mathfrak{p}'} + \Lambda_{\mathfrak{p}'}$.
\subsection{Orbital integrals and Fourier coefficients}
Normalize the Haar measure on \[ T_{so}(\mathbb R) = \{ s\in (E\otimes_\mathbb Q \mathbb R)^\times : s\overline{s} =1\} \] to have total volume $1$, and fix any Haar measure on \[
T_{so}(\mathbb A_f ) = \{ s\in \widehat{E}^\times : s\overline{s} =1\}. \] There is an induced quotient measure on $T_{so}(\mathbb Q) \backslash T_{so}(\mathbb A)$, and for any compact open subgroup $U \subset T_{so}(\mathbb A_f )$ we have \[
\mathrm{Vol}(U) = \frac{ | T_{so}(\mathbb Q) \cap U | }{ | T_{so}(\mathbb Q) \backslash T_{so}(\mathbb A_f) / U | } \cdot \mathrm{Vol}\big( T_{so}(\mathbb Q) \backslash T_{so}(\mathbb A) \big). \]
\begin{definition} Fix a prime $\mathfrak{p} \subset \mathcal O_F$ nonsplit in $E$, and let $ { \empty^{ \mathfrak{p} }} \varphi \in S( { \empty^{ \mathfrak{p} }} \widehat{\mathscr{V}} )$ be any Schwartz function on the nearby hermitian space $ { \empty^{ \mathfrak{p} }} \widehat{\mathscr{V}}= { \empty^{ \mathfrak{p} }} \mathscr{V} \otimes \mathbb A_f$ of \S \ref{ss:hermitian}. For each $\alpha \in \widehat{F}^\times$ define the \emph{orbital integral} \[ O(\alpha, { \empty^{ \mathfrak{p} }} \varphi ) = \frac{1}{ \mathrm{Vol} \big( T_{so}(\mathbb Q) \backslash T_{so}(\mathbb A) \big) }
\int_{ T_{so}(\mathbb A_f ) } { \empty^{ \mathfrak{p} }} \varphi ( s x) \, ds \] for any $x \in { \empty^{ \mathfrak{p} }} \widehat{\mathscr{V}}$ with $ { \empty^{ \mathfrak{p} }} \langle x,x\rangle=\alpha$. If no such $x$ exists we set $O(\alpha, { \empty^{ \mathfrak{p} }} \varphi ) =0$. \end{definition}
\begin{proposition}\label{prop:explicit siegel-weil} Fix an $\alpha\in F_+$ such that $\mathrm{Diff}(\alpha)=\{\mathfrak{p}\}$ for a single prime $\mathfrak{p}\subset \mathcal O_F$. Let $\mathfrak{q} \subset \mathcal O_E$ be the prime above $\mathfrak{p}$. Suppose that $\mathfrak{p}$ lies above a good prime $p$. Then, for any $\mu\in L^\vee /L$, we have \[
\frac{ a_F(\alpha, \mu) } { \Lambda(0,\chi ) } = - \ell_{\mathfrak{p}}(\alpha) \cdot O(\alpha, { \empty^{ \mathfrak{p} }} \varphi_\mu ) \cdot \log N(\mathfrak{q}), \] where $\ell_{\mathfrak{p}}(\alpha) = 0$ unless $( { \empty^{ \mathfrak{p} }} \mu_p + { \empty^{ \mathfrak{p} }} L_p )\cap { \empty^{ \mathfrak{p} }} \mathscr{V}_{\mathfrak{p}}$ represents $\alpha\in F_{\mathfrak{p}}^\times$, in which case, we have \[ \ell_{\mathfrak{p}}(\alpha) = \begin{cases} \frac{\mathrm{ord}_{\mathfrak{p}}(\alpha)+ m(\mathfrak{p}) + 1}{2} &\text{if $\mathfrak{p}$ is unramified in $E$} \\ \mathrm{ord}_{\mathfrak{p}}(\alpha) + m(\mathfrak{p}) + n(\mathfrak{p}) &\text{if $\mathfrak{p}$ is ramified in $E$}. \end{cases} \] \end{proposition}
\begin{proof} The proof proceeds as in~\cite[Theorem 6.1]{KuAnnals}. The strategy is to relate the incoherent Eisenstein series $E(g,s,\varphi_\mu)$ to a nearby \emph{coherent} Eisenstein series, whose Fourier coefficients can be computed using the Siegel-Weil formula. This information is then combined with the computations of local Whittaker functions in \S\ref{ss:whittaker functions} to complete the proof.
We begin by repeating the construction of the incoherent Eisenstein series from \S\ref{ss:incoherent}, but we replace the incoherent $\mathbb A_F$-quadratic space $\mathscr{C}=\mathscr{C}_\infty \times \widehat{\mathscr{V}}$ by the coherent space \[
\mathscr{C}_\infty \times { \empty^{ \mathfrak{p} }} \widehat{\mathscr{V}} \xrightarrow{\simeq} { \empty^{ \mathfrak{p} }} \mathscr{V}\otimes_F \mathbb A_F , \] which differs from $\mathscr{C}$ only at the place $\mathfrak{p}$.
Let $\varphi^p_\mu\in S(\widehat{\mathscr{V}}^p)$ be the prime-to-$p$ part of $\varphi_\mu$ so that we have \[ \varphi_\mu = \varphi_{\mu_p}\otimes \varphi_\mu^p \in S(\mathscr{V}_p) \otimes S(\widehat{\mathscr{V}}^p). \]
By~\eqref{eqn:mu to lambda decomp}, we have $\varphi_{\mu_p} = \sum_{\lambda_p\in \mu_p+L_p}\varphi_{\lambda_p}$, where $\varphi_{\lambda_p}$ admits a further product decomposition \[ \varphi_{\lambda_p} = \prod_{\mathfrak{p}'\mid p}\varphi_{\lambda_{\mathfrak{p}'}}\in \otimes_{\mathfrak{p}'}S(\mathscr{V}_{\mathfrak{p}'}). \] Here, $\lambda_p$ ranges over representatives for cosets of $\Lambda_p$ in $\Lambda_p^\vee$ contained in $\mu_p + L_p$.
Set $\varphi_\lambda = \varphi_{\lambda_p}\otimes \varphi^p_\mu\in S(\widehat{\mathscr{V}})$. We now have \begin{equation}\label{eqn:aF mu lambda decomp} a_F(\alpha,\mu) = \sum_{\lambda_p\in \mu_p + L_p}a_F(\alpha,\varphi_\lambda). \end{equation}
Fix $\lambda_p \in \mu_p + L_p$. Choose any Schwarz function $ { \empty^{ \mathfrak{p} }} \varphi_{\mathfrak{p}}\in S( { \empty^{ \mathfrak{p} }} \mathscr{V}_{\mathfrak{p}})$. This gives us a global Schwarz function $ { \empty^{ \mathfrak{p} }} \varphi\in S( { \empty^{ \mathfrak{p} }} \mathscr{V})$ admitting a factorization over primes $\mathfrak{p}'\subset\mathcal O_F$: \[ { \empty^{ \mathfrak{p} }} \varphi = \otimes_{\mathfrak{p}'} { \empty^{ \mathfrak{p} }} \varphi_{\mathfrak{p}'}, \] where $ { \empty^{ \mathfrak{p} }} \varphi_{\mathfrak{p}}$ is our chosen function and, for $\mathfrak{p}'\neq \mathfrak{p}$, we have \[ { \empty^{ \mathfrak{p} }} \varphi_{\mathfrak{p}'} = \varphi_{\lambda_{\mathfrak{p}'}} \in S( { \empty^{ \mathfrak{p} }} \mathscr{V}_{\mathfrak{p}'}) = S(\mathscr{V}_{\mathfrak{p}'}). \]
If $ { \empty^{ \mathfrak{p} }} \varphi_{\mathfrak{p}}$ is the characteristic function of $ { \empty^{ \mathfrak{p} }} \lambda_{\mathfrak{p}} + { \empty^{ \mathfrak{p} }} \Lambda_{\mathfrak{p}}$, then we will write $ { \empty^{ \mathfrak{p} }} \varphi_\lambda$ for the corresponding element of $S( { \empty^{ \mathfrak{p} }} \mathscr{V})$.
Let $\Phi^\lambda\in I(s,\chi)$ (resp. $ { \empty^{ \mathfrak{p} }} \Phi\in I(s,\chi)$) be the standard section associated with $\varphi^{\mathbf{1}}_\infty\otimes\varphi_\lambda$ (resp. $\varphi^{\mathbf{1}}_\infty\otimes { \empty^{ \mathfrak{p} }} \varphi$). If $ { \empty^{ \mathfrak{p} }} \varphi = { \empty^{ \mathfrak{p} }} \varphi_\lambda$, we will write $ { \empty^{ \mathfrak{p} }} \Phi^\lambda$ for the section $ { \empty^{ \mathfrak{p} }} \Phi$, in agreement with the notation used in the local setting of \S\ref{ss:whittaker functions}.
There is a factorization \[ I(s,\chi) = I_\mathfrak{p}(s,\chi) \otimes I^{\mathfrak{p}}(s,\chi), \] into the $\mathfrak{p}$-part and prime-to-$\mathfrak{p}$-part. Since $\varphi_\lambda$ and $ { \empty^{ \mathfrak{p} }} \varphi$ differ only at their $\mathfrak{p}$-components, our two sections $\Phi^\lambda$ and $ { \empty^{ \mathfrak{p} }} \Phi$ have the form \[ \Phi^\mu = \Phi^\mu_\mathfrak{p} \otimes \Phi^{(\mathfrak{p})},\quad { \empty^{ \mathfrak{p} }} \Phi = { \empty^{ \mathfrak{p} }} \Phi_\mathfrak{p} \otimes \Phi^{(\mathfrak{p})}, \] for a \emph{common} section $\Phi^{(\mathfrak{p})}$ of $I^{\mathfrak{p}}(s,\chi)$.
We now have a \emph{coherent} Eisenstein series $ E(g,s , { \empty^{ \mathfrak{p} }} \varphi ) = E(g,s , { \empty^{ \mathfrak{p} }} \Phi) $ defined exactly as in (\ref{eisenstein formation}), and associated with the Schwartz function $ { \empty^{ \mathfrak{p} }} \varphi$.
Given $g,g'\in \mathrm{SL}_2(\mathbb A_F)$ which have the \emph{same} prime-to-$\mathfrak{p}$ components, we deduce, using (\ref{eisenstein whittaker}), the relation \begin{equation}\label{eqn:nearby eisenstein}
E_\alpha(g,s,\Phi^\lambda) = \frac{ W_{ \alpha, \mathfrak{p}} (g_\mathfrak{p} , s, \Phi^\lambda_\mathfrak{p}) }{W_{ \alpha, \mathfrak{p}} (g'_\mathfrak{p} , s, { \empty^{ \mathfrak{p} }} \Phi_{\mathfrak{p}})}
\cdot E_\alpha(g',s, { \empty^{ \mathfrak{p} }} \Phi), \end{equation} which is valid for all values of $s$ at which $W_{\alpha,\mathfrak{p}}(g'_{\mathfrak{p}} , s, { \empty^{ \mathfrak{p} }} \Phi_{\mathfrak{p}})$ is non-zero.
Suppose that $g$ is such that $g_{\mathfrak{p}} = I\in \mathrm {SL}_2(F_\mathfrak{p})$ is the identity, and choose $g'_{\mathfrak{p}}$ and $ { \empty^{ \mathfrak{p} }} \varphi_{\mathfrak{p}}$ such that $W_{\alpha,\mathfrak{p}}(g'_{\mathfrak{p}},0, { \empty^{ \mathfrak{p} }} \varphi_{\mathfrak{p}})\neq 0$. Then, using~\eqref{eqn:nearby eisenstein} and Proposition~\ref{prop:whittaker vanishing}, we get: \begin{align*} E'_\alpha(g,0,\Phi^\lambda) &= \frac{ W'_{ \alpha, \mathfrak{p}} (I , 0, \Phi^\lambda_\mathfrak{p}) }{ W_{ \alpha, \mathfrak{p}} ( g'_{\mathfrak{p}} , 0, { \empty^{ \mathfrak{p} }} \Phi_\mathfrak{p}) }
\cdot E_\alpha( g',0, { \empty^{ \mathfrak{p} }} \Phi).
\end{align*}
In the notation of Proposition \ref{prop:coefficient support}, this equality implies \begin{equation}\label{coefficient swindle} \frac{a_F(\alpha,\lambda)}{ \Lambda(0,\chi) } \cdot q^\alpha= - \frac{ W'_{ \alpha, \mathfrak{p}} (I , 0, \Phi^\lambda_\mathfrak{p}) }{ W_{ \alpha, \mathfrak{p}} ( g'_{\mathfrak{p}} , 0, { \empty^{ \mathfrak{p} }} \Phi_\mathfrak{p}) }
\cdot \frac{ E_\alpha(g'_{\vec{\tau}} ,0, { \empty^{ \mathfrak{p} }} \Phi) }{ \sqrt{ N(\vec{v} ) } }. \end{equation} for all $\vec{\tau} \in \mathcal{H}^d$.
If $\alpha$ is not represented by $ { \empty^{ \mathfrak{p} }} \lambda_{\mathfrak{p}} + { \empty^{ \mathfrak{p} }} \Lambda_{\mathfrak{p}}$, then assertion (1) of Proposition~\ref{prop:whittaker derivative} now implies \[ \frac{a_F(\alpha,\varphi_\lambda)}{\Lambda(0,\chi)} = 0. \] Combining this with~\eqref{eqn:aF mu lambda decomp} shows that $a_F(\alpha,\mu) = 0$, whenever $( { \empty^{ \mathfrak{p} }} \mu_p + { \empty^{ \mathfrak{p} }} L_p)\cap { \empty^{ \mathfrak{p} }} \mathscr{V}_{\mathfrak{p}}$ does not represent $\alpha$.
Now, suppose that $\alpha$ is represented by $ { \empty^{ \mathfrak{p} }} \lambda_{\mathfrak{p}} + { \empty^{ \mathfrak{p} }} \Lambda_{\mathfrak{p}}$. Then Proposition~\ref{prop:whittaker derivative} implies that we can take $g' = g$ and $ { \empty^{ \mathfrak{p} }} \varphi_{\mathfrak{p}} = { \empty^{ \mathfrak{p} }} \varphi_{\lambda_{\mathfrak{p}}}$.
As in the proof of \cite[Proposition 4.4.1]{HY}, the Siegel-Weil formula \cite{KudlaRallis} implies \[ \frac{ E_\alpha(g_{\vec{\tau}} ,0, { \empty^{ \mathfrak{p} }} \Phi^\lambda) }{ \sqrt{ N(\vec{v} ) } }
= \frac{ 2 q^\alpha}{ \mathrm{Vol} \big( T_{so}(\mathbb Q) \backslash T_{so}(\mathbb A) \big) } \int_{ T_{so}(\mathbb Q) \backslash T_{so}(\mathbb A) }
\sum_{ \substack { x\in { \empty^{ \mathfrak{p} }} \mathscr{V} \\ { \empty^{ \mathfrak{p} }} \mathscr{Q}(x)=\alpha } } { \empty^{ \mathfrak{p} }} \varphi_\lambda( s_f^{-1} x ) \, ds . \] The group $T_{so}(\mathbb Q)$ acts simply transitively on the set of all $x\in { \empty^{ \mathfrak{p} }} \mathscr{V}$ with $ { \empty^{ \mathfrak{p} }} \mathscr{Q}(x)=\alpha$, allowing us to rewrite this equality as \begin{equation}\label{eqn:siegel weil result} \frac{ E_\alpha(g_{\vec{\tau}} ,0, { \empty^{ \mathfrak{p} }} \Phi^\lambda) }{ \sqrt{ N(\vec{v} ) } } = 2\cdot O(\alpha, { \empty^{ \mathfrak{p} }} \varphi_\lambda) \cdot q^\alpha. \end{equation}
Combining~\eqref{eqn:siegel weil result} with~\eqref{coefficient swindle}, and using the formulas for \[ \frac{ W'_{ \alpha, \mathfrak{p}} (I , 0, \Phi^\lambda_\mathfrak{p}) }{ W_{ \alpha, \mathfrak{p}} ( I , 0, { \empty^{ \mathfrak{p} }} \Phi^\lambda_\mathfrak{p}) } \] from Proposition~\ref{prop:whittaker derivative} shows \begin{equation}\label{eqn:aF lambda} \frac{a_F(\alpha,\varphi_\lambda)}{\Lambda(0,\chi)} = -\ell_{\mathfrak{p}}(\alpha)\cdot O(\alpha, { \empty^{ \mathfrak{p} }} \varphi_\lambda)\cdot\log N(\mathfrak{q}). \end{equation}
Now, observe that $\mathrm{ord}_{\mathfrak{p}}(\alpha) + m(\mathfrak{p}) + n(\mathfrak{p}) = 0$ whenever $n(\mathfrak{p})\neq 0$ and $\mathrm{ord}_{\mathfrak{q}}(\lambda_{\mathfrak{p}}) = -n(\mathfrak{p})$. Therefore, from the definition of $ { \empty^{ \mathfrak{p} }} \tilde{\varphi}_{\lambda_{\mathfrak{p}}}$ in ~\eqref{eqn:defn tilde varphi}, we see that~\eqref{eqn:aF lambda} is equivalent to \begin{equation}\label{eqn:aF lambda 1} \frac{a_F(\alpha,\varphi_\lambda)}{\Lambda(0,\chi)} = -\ell_{\mathfrak{p}}(\alpha)\cdot O(\alpha, { \empty^{ \mathfrak{p} }} \tilde{\varphi}_\lambda)\cdot\log N(\mathfrak{q}). \end{equation} Here, $ { \empty^{ \mathfrak{p} }} \tilde{\varphi}_{\lambda}$ differs from $ { \empty^{ \mathfrak{p} }} \varphi_\lambda$ only at $\mathfrak{p}$, and we take its factor at $\mathfrak{p}$ to be $ { \empty^{ \mathfrak{p} }} \tilde{\varphi}_{\lambda_{\mathfrak{p}}}$.
Now, note that, by~\eqref{eqn:near varphi mu decomp}, \[ O(\alpha, { \empty^{ \mathfrak{p} }} \varphi_\mu) = \sum_{\lambda_p\in \mu_p+L_p} O(\alpha, { \empty^{ \mathfrak{p} }} \tilde{\varphi}_\lambda) \] and that \[ O(\alpha, { \empty^{ \mathfrak{p} }} \tilde{\varphi}_\lambda) = 0, \] whenever $ { \empty^{ \mathfrak{p} }} \lambda_{\mathfrak{p}} + { \empty^{ \mathfrak{p} }} \Lambda_{\mathfrak{p}}$ does not represent $\alpha$.
Combining these observations with~\eqref{eqn:aF mu lambda decomp} and~\eqref{eqn:aF lambda 1} completes the proof of the Proposition. \end{proof}
\subsection{A decomposition of the space of special endomorphisms} \label{ss:special decomposition}
Fix a prime $\mathfrak{p}\subset\mathcal O_F$ not split in $E$, and let $\mathfrak{q}\subset\mathcal O_E$ be the unique prime above it. Fix an algebraic closure $\mathbb F_{\mathfrak{p}}^{\mathrm{alg}}$ for $\mathbb F_{\mathfrak{p}}$ and also an algebraic closure $\mathrm{Frac}(W)^{\mathrm{alg}}$ of the fraction field $\mathrm{Frac}(W)$ of $W = W(\mathbb F_{\mathfrak{p}}^{\mathrm{alg}})$. Choose an embedding $\mathbb Q^{\mathrm{alg}} \hookrightarrow \mathrm{Frac}(W)^{\mathrm{alg}}$ inducing the place $\mathfrak{q}$ on $E = \iota_0(E)$.
Let $L_{\mathfrak{p}} = L_p \cap \mathscr{V}_{\mathfrak{p}}\subset L_p$, and let $H_{\mathfrak{p}} = C(L_{\mathfrak{p}}) \subset C(L_p) = H_p$. Let $K_{L,0}\subset T(\mathbb A_f)$ be the compact open subgroup defined in \S~\ref{ss:shimura data}, and let $K_{0,L,\mathfrak{q}} \subset K_{0,L,p}$ be the intersection of $K_{0,L,p}$ with the image of $E_{\mathfrak{q}}^\times$ under the natural map \[ E_{\mathfrak{q}}^\times \hookrightarrow (E\otimes_{\mathbb Q}\mathbb Q_p)^\times \to T(\mathbb Q_p). \]
Then $H_p\subset H_{\mathbb Q_p}$ is a $K_{0,L,p}$-stable lattice, and $H_{\mathfrak{p}}\subset H_{\mathfrak{p}}[p^{-}]$ is a $K_{0,L,\mathfrak{q}}$-stable lattice. Moreover, the natural $C(L_p)$-linear map \begin{align}\label{eqn:H mf p to Hp} H_{\mathfrak{p}}\otimes_{C(L_{\mathfrak{p}})}C(L_p) &\to H_p\\ h\otimes z &\mapsto h\cdot z\nonumber \end{align} is a $K_{0,L,\mathfrak{q}}$-equivariant isomorphism, once we equip $C(L_p)$ with the trivial $K_{0,L,\mathfrak{q}}$-action.
Fix a point $y\in \mathcal{Y}(\mathbb F_{\mathfrak{q}}^{\mathrm{alg}})$. Then, by Remark~\ref{rem:Eq times reps},~\eqref{eqn:H mf p to Hp} gives us a crystalline $\mathbb Z_p$-representation $\bm{H}_{\mathfrak{p},y}$ of $\Gamma_y$ and a $C(L_p)$-linear isomorphism \begin{align} \label{eqn:H mf p to Hp etale} \bm{H}_{\mathfrak{p},y}\otimes_{C(L_{\mathfrak{p}})}C(L_p) &\xrightarrow{\simeq} \bm{H}_{p,y}. \end{align}
The following result is easily deduced from Theorem~\ref{thm:kisin_p_divisible}. \begin{proposition}\label{prop:H mfp Tate module} The $\Gamma_y$-module $\bm{H}_{\mathfrak{p},y}$ is canonically isomorphic to the $p$-adic Tate module of a $C(L_{\mathfrak{p}})$-linear $p$-divisible subgroup \[ A[p^\infty]_{\mathfrak{p}}\subset A[p^\infty]\vert_{\mathrm{Spec} (\mathcal O_y)}. \] Moreover, the natural $C(L_p)$-linear map of $p$-divisible groups \[ A[p^\infty]_{\mathfrak{p}}\otimes_{C(L_{\mathfrak{p}})}C(L_p)\to A[p^\infty]\vert_{\mathrm{Spec}(\mathcal O_y) } \] is an isomorphism. \end{proposition}
In particular, for any $\mathcal O_y$-scheme $S$, we obtain a natural map
\begin{equation}\label{eqn:end mf p to p} \mathrm{End}_{C(L_{\mathfrak{p}})}(A[p^\infty]_{\mathfrak{p},S})\to \mathrm{End}_{C(L_p)}(A_S[p^\infty]), \end{equation} and so, in complete analogy with the definitions from \S\ref{ss:special divisors}, we define the space of special endomorphisms \[ V\bigl(A[p^\infty]_{\mathfrak{p},S}\bigr) \subset \mathrm{End}_{C(L_{\mathfrak{p}})}(A[p^\infty]_{\mathfrak{p},S}) \] to consist of those elements that induce special endomorphisms of $A[p^\infty]$ via~\eqref{eqn:end mf p to p}. By definition, this is a subspace of $V(A_S[p^\infty])$.
The next result is entirely analogous to Lemma~\ref{lem:lambda_perp}. \begin{proposition} \label{prop:V mf p to p} Let $L_p^{\mathfrak{p}} = L_{\mathfrak{p}}^\perp \subset L_p$ be the orthogonal complement to $L_{\mathfrak{p}}$. Then there is a canonical isometric embedding \[ L_p^{\mathfrak{p}}\hookrightarrow V(A[p^\infty]_{\mathrm{Spec} (\mathcal O_y) }) \] as a direct summand, such that, for any $\mathcal O_y$-scheme $S$, we have \[ V\bigl(A[p^\infty]_{\mathfrak{p},S}\bigr) = \bigl(L_p^{\mathfrak{p}}\bigr)^{\perp}\subset V(A_S[p^\infty]). \] \end{proposition}
Given a class $\eta\in L_{\mathfrak{p}}^\vee/L_{\mathfrak{p}}$, we will also need a corresponding subset \begin{equation}\label{eqn:V eta} V_{\eta}(A[p^\infty]_{\mathfrak{p},S})\subset V(A[p^\infty]_{\mathfrak{p},S})^\vee. \end{equation} This is once again defined as in \S\ref{ss:special divisors}: We fix an embedding $L\hookrightarrow L^ \diamond$ into a maximal lattice $L^ \diamond$ that is of signature $(n^ \diamond,2)$ and is self-dual at $p$. Let $\Lambda^{\mathfrak{p}}\subset L^ \diamond_p$ be the orthogonal complement of $L_{\mathfrak{p}}$. Then we have a canonical isometric embedding \[ \Lambda^{\mathfrak{p}}\hookrightarrow V(A^ \diamond[p^\infty]_S), \] whose orthogonal complement is $V(A[p^\infty]_{\mathfrak{p},S})$. Hence we get a map \[ V(A^ \diamond_S[p^\infty]) \to \frac{V(A[p^\infty]_{\mathfrak{p},S})^\vee}{V(A[p^\infty]_{\mathfrak{p},S})} \oplus \frac{\Lambda^{\mathfrak{p},\vee}}{\Lambda^{\mathfrak{p}}}. \]
The subset~\eqref{eqn:V eta} now consists of elements $x$ such that the pair \[ ([x],\eta) \in \frac{V(A[p^\infty]_{\mathfrak{p},S})^\vee}{V(A[p^\infty]_{\mathfrak{p},S})} \oplus \frac{\Lambda^{\mathfrak{p},\vee}}{\Lambda^{\mathfrak{p}}} \] is in the image of $V(A_S^ \diamond[p^\infty])$. Here, we have used the natural isomorphisms \[ \frac{\Lambda^{\mathfrak{p},\vee}}{\Lambda^{\mathfrak{p}}} \xleftarrow{\simeq} \frac{L^ \diamond_p}{L_{\mathfrak{p}}+\Lambda^{\mathfrak{p}}} \xrightarrow{\simeq} \frac{L_{\mathfrak{p}}^\vee}{L_{\mathfrak{p}}} \] to view $\eta$ as an element of $ \frac{\Lambda^{\mathfrak{p},\vee}}{\Lambda^{\mathfrak{p}}}$.
The following proposition is now immediate from the definitons and is analogous to assertion (3) of Proposition~\ref{prop:decomposition_Vmu}. \begin{proposition} \label{prop:V p to mf p} For any $\mu_p\in L_p^\vee/L_p$, we have a canonical decompositions \[ V_{\mu_p}(A_S[p^\infty]) =\bigsqcup_{(\mu_1,\mu_2)\in (\mu_p+ L_p)/(L_{\mathfrak{p}}\oplus L^{\mathfrak{p}_p})} V_{\mu_1}(A[p^\infty]_{\mathfrak{p},S}) \times \bigl({\mu_2}+ L^{\mathfrak{p}}_p\bigr), \] where we are viewing \[ \frac{\mu_p+ L_p}{L_{\mathfrak{p}}\oplus L^{\mathfrak{p}}_p} \subset \frac{L^\vee_{\mathfrak{p}}}{L_{\mathfrak{p}}} \oplus \frac{L^{\mathfrak{p,\vee}}_p}{L^{\mathfrak{p}}_p} . \] \end{proposition}
\subsection{Lubin-Tate and Kuga-Satake} \label{ss:lt to ks}
Let $\mathfrak{p}\subset \mathcal O_F$ and $\mathfrak{q}\subset \mathcal O_E$ be as above. For the rest of this section, we will assume that $\mathfrak{p}$ lies above a \emph{good} prime $p$. Therefore, we have \[ \Lambda_{\mathfrak{p}} \subset L_{\mathfrak{p}}\subsetneq \mathfrak{d}_{E_{\mathfrak{q}}/F_{\mathfrak{p}}}^{-1}\Lambda_{\mathfrak{p}} = \Lambda^\vee_{\mathfrak{p}}, \] where $\Lambda_{\mathfrak{p}}\subset \mathscr{V}_{\mathfrak{p}}$ is an $\mathcal O_{E,\mathfrak{q}}$-stable lattice.
Fix a point $y\in \mathcal{Y}(\mathbb F_{\mathfrak{q}}^{\mathrm{alg}})$. Fix also a uniformizer $\pi_{\mathfrak{q}}\in E_{\mathfrak{q}}$, and let $\mathcal{G}_{\mathfrak{q}}$ be the Lubin-Tate formal $\mathcal O_{E,\mathfrak{q}}$-module over $\mathcal O_y$ associated with this uniformizer. If $\mathfrak{p}$ is unramified in $E$, we will assume that we have chosen $\pi_{\mathfrak{q}} = \pi_{\mathfrak{p}}$ to be a uniformizer for $F_{\mathfrak{p}}$. Otherwise, we will set $\pi_{\mathfrak{p}} = \mathrm{Nm}(\pi_{\mathfrak{q}})\in F_{\mathfrak{p}}$. As in \S\ref{ss:whittaker functions}, we will set \[ m(\mathfrak{p}) = \mathrm{ord}_{\mathfrak{q}}(\mathfrak{d}_{F/\mathbb Q}). \]
As in \S\ref{ss:denominators}, for any $\mathcal O_y$-scheme $S$, and for each $\lambda\in \mathfrak{d}_{E_{\mathfrak{q}}/F_{\mathfrak{p}}}^{-1}/\mathcal O_{E,\mathfrak{q}}$, we have a canonical subset \[ V_\lambda(\mathcal{G}_{\mathfrak{q},S})\subset \mathrm{End}(\mathcal{G}_{\mathfrak{q},S})_\mathbb Q \] of special endomorphisms (with denominators) of $\mathcal{G}_{\mathfrak{q},S}$. Fix an $\mathcal O_{E,\mathfrak{q}}$-linear identification $\Lambda_{\mathfrak{p}} = \mathcal O_{E,\mathfrak{q}}$, so that we can identify \[ \frac{\Lambda^\vee_{\mathfrak{p}}}{\Lambda_{\mathfrak{p}}} = \frac{\mathfrak{d}^{-1}_{E_{\mathfrak{q}}/F_{\mathfrak{p}}}}{\mathcal O_{E,\mathfrak{q}}}. \] In particular, for any $\lambda\in \Lambda^\vee_{\mathfrak{p}}/\Lambda_{\mathfrak{p}}$, we have a corresponding set $V_\lambda(\mathcal{G}_{\mathfrak{q},S})$ of special endomorphisms of $\mathcal{G}_{\mathfrak{q}}$. Under this identification, the Hermitian form on $\Lambda_{\mathfrak{p}}$ is carried to the form \[ \langle x_1,x_2\rangle = \xi_{\mathfrak{p}}x_1\overline{x}_2 \] on $\mathcal O_{E,\mathfrak{q}}$, for some $\xi_{\mathfrak{p}}\in F_{\mathfrak{p}}^\times$ satisfying $\mathrm{ord}_{\mathfrak{p}}(\xi_{\mathfrak{p}}) = -m(\mathfrak{p})$.
Since we have identified $\Lambda^\vee_{\mathfrak{p}}/\Lambda_{\mathfrak{p}}$ with $\mathfrak{d}_{E_{\mathfrak{q}}/F_{\mathfrak{p}}}^{-1}/\mathcal O_{E,\mathfrak{q}}$, for $\lambda\in \Lambda^\vee_{\mathfrak{p}}/\Lambda_{\mathfrak{p}}$, we can speak of the space $V_\lambda(\mathcal{G}_{\mathfrak{q}})$ of special endomorphisms of the Lubin-Tate group $\mathcal{G}_{\mathfrak{q}}$.
For $\mu\in L^\vee_{\mathfrak{p}}/L_{\mathfrak{p}}$, set \[ V_\mu(\mathcal{G}_{\mathfrak{q},y}) = \bigsqcup_{\lambda \in \mu + L_{\mathfrak{p}}} V_\lambda(\mathcal{G}_{\mathfrak{q},\mathcal O_y/\pi_{\mathfrak{q}}\mathcal O_y}), \] where $\lambda$ varies over the classes in $\Lambda^\vee_{\mathfrak{p}}/\Lambda_{\mathfrak{p}}$ such that $\lambda + \Lambda^\vee_{\mathfrak{p}}$ lies in $\mu + L_{\mathfrak{p}}$.
\begin{proposition} \label{prop:lubin-tate to kuga-satake} There exists an $E_{\mathfrak{q}}$-linear isomorphism \[
V(A_y[p^\infty]_{\mathfrak{p}})_\mathbb Q \xrightarrow{\simeq} V(\mathcal{G}_{\mathfrak{q},y})_\mathbb Q \] carrying the Hermitian form on the left hand side to $\xi_{\mathfrak{p}}$ times that on the right, and such that, for each $\mu\in L_{\mathfrak{p}}^\vee/L_{\mathfrak{p}}$, it induces a bijection \[ V_\mu(A_y[p^\infty]_{\mathfrak{p}}) \xrightarrow{\simeq} V_\mu(\mathcal{G}_{\mathfrak{q},y}). \] \end{proposition} \begin{proof} Using Remark~\ref{rem:Eq times reps}, we can associate with $\Lambda_{\mathfrak{p}}$ an $\mathcal O_{E,\mathfrak{q}}$-linear continuous representation $\bm{\Lambda}_{\mathfrak{p},\mathrm{et},y}$ of the absolute Galois group $\Gamma_y$ of $\mathrm{Frac}(\mathcal O_y)$. This representation can be identified with the space $\bm{V}_{0,\mathfrak{p},\mathrm{et}} = V(\bm{H}_{0,\mathfrak{p},\mathrm{et}},c)$ of $\mathcal O_{E,\mathfrak{q}}$-semilinear endomorphisms of the Tate module $\bm{H}_{0,\mathfrak{p},\mathrm{et}}$ of the Lubin-Tate group $\mathcal{G}_{\mathfrak{q}}$. Moreover, its crystalline realization $\bm{\Lambda}_{\mathfrak{p},\mathrm{cris},y}$ can be identified with the space $\bm{V}_{0,\mathfrak{p},\mathrm{cris},y} = V(\bm{H}_{0,\mathfrak{p},\mathrm{cris},y},c)$ of $\mathcal O_{E,\mathfrak{q}}$-semilinear endomorphisms of the $F$-crystal $\bm{H}_{0,\mathfrak{p},\mathrm{cris},y}$ obtained from the Dieudonn\'e $F$-crystal associated with $\mathcal{G}_{\mathfrak{q}}$.
These identifications carries the Hermitian form on $\bm{\Lambda}_{\mathfrak{p},\mathrm{et}}$ (resp. $\bm{\Lambda}_{\mathfrak{p},\mathrm{cris},y})$ to $\xi_{\mathfrak{p}}$ times the natural Hermitian form on $\bm{V}_{0,\mathfrak{p}}$ (resp. $\bm{V}_{0,\mathfrak{p},\mathrm{cris},y}$). Therefore, we now obtain an $E_{\mathfrak{q}}$-linear isomorphism \[ V(A_y[p^\infty]_{\mathfrak{p}})_\mathbb Q = \bm{V}_{\mathfrak{p},\mathrm{cris},y}^{\varphi = 1}[p^{-1}] = \bm{\Lambda}_{\mathfrak{p},\mathrm{cris},y}^{\varphi = 1}[p^{-1}] \xrightarrow{\simeq}\bm{V}_{0,\mathfrak{p},\mathrm{cris},y}^{\varphi = 1}[p^{-1}] = V(\mathcal{G}_{\mathfrak{q},y})_\mathbb Q \] carrying the Hermitiian form on the left to $\xi_{\mathfrak{p}}$-times that on the very right.
It remains to show that it carries $V_\mu(A_y[p^\infty]_{\mathfrak{p}})$ onto $V_\mu(\mathcal{G}_{\mathfrak{q},y})$. For this, we will need a little preparation. Consider the Breuil-Kisin module $\mathfrak{M}(\Lambda_{\mathfrak{p}})$ associated with $\bm{\Lambda}_{\mathfrak{p},\mathrm{et},y}$ and the uniformizer $\pi_{\mathfrak{q}}$. We have an $\mathcal O_{E,\mathfrak{q}}$-linear identification \[ \mathfrak{M}(\Lambda_{\mathfrak{p}}) = V(\mathfrak{M}(H_{0,\mathfrak{p}}),c) \] of Breuil-Kisin modules. \begin{lemma} \label{lem:discriminant trivial} There is canonical, $\varphi$-equivariant isomorphism \[ \mathfrak{S}\otimes_{\mathbb Z_p}\frac{\Lambda^\vee_{\mathfrak{p}}}{\Lambda_{\mathfrak{p}}}\xrightarrow{\simeq}\frac{\mathfrak{M}(\Lambda^\vee_{\mathfrak{p}})}{\mathfrak{M}(\Lambda_{\mathfrak{p}})} \] of $\mathfrak{S}$-modules, where the left hand side is equipped with the constant $\varphi$-semi-linear endomorphism $\varphi\otimes 1$. It induces a $\varphi$-equivariant isomorphism \[ \mathfrak{S}\otimes_{\mathbb Z_p}\frac{L^\vee_{\mathfrak{p}}}{L_{\mathfrak{p}}}\xrightarrow{\simeq}\frac{\mathfrak{M}(L^\vee_{\mathfrak{p}})}{\mathfrak{M}(L_{\mathfrak{p}})}. \] \end{lemma} \begin{proof} If $\mathfrak{p}$ is unramified in $\mathcal O_E$, then $\Lambda^\vee_{\mathfrak{p}} = \Lambda_{\mathfrak{p}}$, and there is nothing to show.
Suppose therefore that $\mathfrak{p}$ is ramified in $\mathcal O_E$. We have $\mathfrak{S}\otimes_{\mathbb Z_p}\mathcal O_{E,\mathfrak{q}}$-linear isomorphisms \begin{equation}\label{eqn:coEq linear ident 1} \mathfrak{S}\otimes_{\mathbb Z_p}\frac{\Lambda^\vee_{\mathfrak{p}}}{\Lambda_{\mathfrak{p}}}\xrightarrow{\simeq}\mathfrak{S}\otimes_{\mathbb Z_p}\frac{\mathfrak{d}_{E_{\mathfrak{q}}/F_{\mathfrak{p}}}^{-1}}{\mathcal O_{E,\mathfrak{q}}}, \end{equation} and \begin{equation}\label{eqn:coEq linear ident 2} \frac{\mathfrak{M}(\Lambda^\vee_{\mathfrak{p}})}{\mathfrak{M}(\Lambda_{\mathfrak{p}})}\xrightarrow{\simeq}\frac{\mathfrak{d}_{E_{\mathfrak{q}}/F_{\mathfrak{p}}}^{-1}}{\mathcal O_{E,\mathfrak{q}}}\otimes_{\mathcal O_{E,\mathfrak{q}}}V(\mathfrak{M}(\bm{H}_{0,\mathfrak{p}}),c). \end{equation}
As in \S \ref{ss:lubin_tate}, we have identifications \begin{equation}\label{eqn:bk module ident} \mathfrak{M}(\bm{H}_{0,\mathfrak{p}}) = \mathfrak{M}(T_{\pi_E}(\mathcal{G}_{\mathfrak{q}})) = \mathfrak{S}\otimes_{\mathbb Z_p}\mathcal O_{E,\mathfrak{q}} \end{equation} as $\mathfrak{S}\otimes_{\mathbb Z_p}\mathcal O_{E,\mathfrak{q}}$-modules carrying the the $\varphi$-semilinear endomorphism of $\mathfrak{M}(\bm{H}_{0,\mathfrak{p}})$ to the endomorphism $\beta(\varphi\otimes 1)$, where $\beta$ has the following description: First, let $E_{\mathfrak{q},0}\subset E_{\mathfrak{q}}$ be the maximal unramified subextension. For each embedding $\eta:E_{\mathfrak{q},0}\hookrightarrow \mathrm{Frac}(W)$, we obtain a finite $W$-algebra $W_\eta = \mathcal O_{E,\mathfrak{q}}\otimes_{\mathcal O_{E_{\mathfrak{q},0}},\eta}W$. There is a disinguished embedding $\eta_0$ induced from the distinguished embedding $\iota_0$ of $E_{\mathfrak{q}}$ into $\mathrm{Frac}(W)^{\mathrm{alg}}$.
We now have \[ \beta = (\beta_{\eta})\in \prod_{\eta}\mathfrak{S}\otimes_WW_\eta = \mathfrak{S}\otimes_{\mathbb Z_p}\mathcal O_{E,\mathfrak{q}}, \] where $\beta_{\eta} = 1$, if $\eta\neq \eta_0$, and $\beta_{\eta_0} = u - \eta_0(\pi_{\mathfrak{q}})$.
From~\eqref{eqn:bk module ident}, we now obtain an identification \begin{equation}
\label{eqn:bk module V ident}
V(\mathfrak{M}(\bm{H}_{0,\mathfrak{p}}),c) = \mathfrak{S}\otimes_{\mathbb Z_p}\mathcal O_{E,\mathfrak{q}}
\end{equation} carrying the $\varphi$-semilinear endomorphism on the left hand side to the endomorphism $\alpha(\varphi\otimes 1)$, where \[ \alpha = (\alpha_{\eta})\in \prod_{\eta}\mathfrak{S}\otimes_WW_\eta = \mathfrak{S}\otimes_{\mathbb Z_p}E_{\mathfrak{q}} \] with $\alpha_\eta = 1$, for $\eta \neq \eta_0$, and $\alpha_{\eta_0} = \frac{u-\eta_0(\overline{\pi}_{\mathfrak{q}})}{u - \eta_0(\pi_{\mathfrak{q}})}$. Since $\overline{\pi}_{\mathfrak{q}} - \pi_{\mathfrak{q}}\in \mathfrak{d}_{E_{\mathfrak{q}}/F_{\mathfrak{p}}}$, we have \[ \alpha\equiv 1 \pmod{\mathfrak{d}_{E_{\mathfrak{q}}/F_{\mathfrak{p}}}}. \] Therefore, tensoring~\eqref{eqn:bk module V ident} with $\mathfrak{d}_{E_{\mathfrak{q}}/F_{\mathfrak{p}}}^{-1} / \mathcal O_{E,\mathfrak{q}}$, and using~\eqref{eqn:coEq linear ident 1} and~\eqref{eqn:coEq linear ident 2}, gives us the isomorphism whose existence is asserted in the proposition.
We leave it to the reader to check that this isomorphism is independent of all our choices. \end{proof}
Now, base-changing along $\varphi:\mathfrak{S}\to \mathfrak{S}$ the isomorphism from Lemma~\ref{lem:discriminant trivial} and then reducing it mod $u$, we obtain a canonical isomorphism \[ \eta:W\otimes_{\mathbb Z_p}\frac{L_{\mathfrak{p}}^\vee}{L_{\mathfrak{p}}}\xrightarrow{\simeq}\frac{\bm{V}_{\mathfrak{p},\mathrm{cris},y}^\vee}{\bm{V}_{\mathfrak{p},\mathrm{cris},y}}. \]
\begin{lemma} \label{lem:Vmu alt descp} The subset $V_\mu(A_y[p^\infty]_{\mathfrak{p}})\subset V(A_y[p^\infty]_{\mathfrak{p}})_\mathbb Q$ consists of those elements $x$ whose crystalline realization $\bm{x}_{\mathrm{cris}}\in \bm{V}_{\mathfrak{p},\mathrm{cris},y}[p^{-1}]$ lies in $\bm{V}^\vee_{\mathfrak{p},\mathrm{cris},y}$, and such that \[ \bm{x}_{\mathrm{cris}}\equiv \eta(1\otimes \mu)\pmod{\bm{V}_{\mathfrak{p},\mathrm{cris},y}} \] \end{lemma} \begin{proof} Choose an auxiliary lattice $L^ \diamond$, self-dual at $p$, and isometric embedding $L\hookrightarrow L^ \diamond$, giving us the auxiliary Kuga-Satake abelian variety $A^ \diamond_y$ over $\mathbb F_{\mathfrak{q}}^\mathrm{alg}$.
Let $L^{ \diamond,\mathfrak{p}}_p\subset L^ \diamond_p$ be the orthogonal complement of $L_{\mathfrak{p}}$. Choose a lift $\tilde{\mu}\in L_{\mathfrak{p}}^\vee$ of $\mu$, and an element $\tilde{\mu}^{\mathfrak{p}}\in L^{ \diamond,\mathfrak{p},\vee}_p$ such that \[ (\tilde{\mu},\tilde{\mu}^{\mathfrak{p}})\in L_p^{ \diamond}\subset L^\vee_{\mathfrak{p}}\oplus L^{ \diamond,\mathfrak{p},\vee}_p. \] Then, by definition, giving an element of $V_\mu(A_y[p^\infty]_{\mathfrak{p}})$ amounts to specifying $x\in V(A_y[p^\infty]_{\mathfrak{p}})^\vee$ such that \[ (x,\tilde{\mu}^{\mathfrak{p}}) \in V(A_y[p^\infty]_{\mathfrak{p}})^\vee \oplus L^{ \diamond,\mathfrak{p},\vee} \] lies in the image of $V(A^ \diamond_y[p^\infty])$.
Since we have a canonical isometric embedding \[ W\otimes_{\mathbb Z_p}L^{ \diamond,\mathfrak{p}}\hookrightarrow \bm{V}^ \diamond_{\mathrm{cris},y} \] mapping into the orthogonal complement of $\bm{V}_{\mathfrak{p},\mathrm{cris},y}$, we obtain an inclusion \begin{equation}\label{eqn:cris dual inclusion} \bm{V}_{\mathrm{cris},y}^ \diamond\hookrightarrow \bm{V}^\vee_{\mathfrak{p},\mathrm{cris},y}\oplus(W\otimes_{\mathbb Z_p}L^{ \diamond,\mathfrak{p},\vee}). \end{equation} Let $\bm{x}_{\mathrm{cris}}\in \bm{V}_{\mathfrak{p},\mathrm{cris},y}[p^{-1}]$ be the crystalline realization of $x$, and let $\bm{x}^ \diamond_{\mathrm{cris}}\in \bm{V}^ \diamond_{\mathrm{cris},y}$ be the crystalline realization of $(x,\tilde{\mu})$. Then it is clear from the definitions that $\bm{x}_{\mathrm{cris}}$ actually lies in $\bm{V}^\vee_{\mathfrak{p},\mathrm{cris},y}$, and that~\eqref{eqn:cris dual inclusion} maps $\bm{x}^ \diamond_\mathrm{cris}$ to $(\bm{x}_{\mathrm{cris}},\tilde{\mu}^{\mathfrak{p}})$. From this, one deduces that $\bm{x}_{\mathrm{cris}}$ must map into $\eta(1\otimes u)\in \bm{V}^\vee_{\mathfrak{p},\mathrm{cris},y}/\bm{V}_{\mathfrak{p},\mathrm{cris},y}$. \end{proof}
Tracing through the definition of $V_\mu(\mathcal{G}_{\mathfrak{q},y})$, it is not hard to show that it has the same description as that of $V_\mu(A_y[p^\infty]_{\mathfrak{p}})$ given to us by Lemma~\ref{lem:Vmu alt descp}. This finishes the proof of the proposition.
\end{proof}
\subsection{Special zero cycles}
For any scheme $S$ over $\mathcal{Y}$, by Corollary~\ref{cor:special end structure}, the space $V(A_S)_\mathbb Q$ has a canonical structure of an $E$-vector space equipped with a positive definite Hermitian form $\langle\cdot,\cdot\rangle$ such that, for any $x\in V(A_S)_\mathbb Q$, \[ Q(x) = x\circ x = \mathrm{Tr}_{F/\mathbb Q}(\langle x,x\rangle). \] Write $\mathscr{V}(A_S)_\mathbb Q$ for $V(A_S)_\mathbb Q$ equipped with this additional structure. For any $\mu\in L^\vee/L$, let $\mathscr{V}_\mu(A_S)$ denote the space $V_\mu(A_S)$ viewed as a subspace of $\mathscr{V}(A_S)_\mathbb Q$
Suppose that $\alpha\in F^\times$ and $\mu\in L^\vee / L$. Define a moduli problem $\mathcal{Z}_F(\alpha,\mu)$ over $\mathcal{Y}$ such that, for any $\mathcal{Y}$-scheme $S$, we have \[ \mathcal{Z}_F(\alpha,\mu)(S) = \{x\in \mathscr{V}_\mu(A_S): \langle x,x\rangle = \alpha\}. \] Since $\langle\cdot,\cdot\rangle$ is positive definite, $\mathcal{Z}_F(\alpha,\mu)$ is empty unless $\alpha\in F_+$ is totally positive.
From the definitions, we now find that there is a canonical decomposition of $\mathcal{Y}$-stacks \begin{equation}\label{intersection decomp} \mathcal{Y} \times_\mathcal{M} \mathcal{Z}(m,\mu)
= \bigsqcup_{ \substack{ \alpha \in F_+ \\ \mathrm{Tr}_{F/\mathbb Q}(\alpha) =m } } \mathcal{Z}_F(\alpha,\mu) . \end{equation}
\begin{proposition}\label{prop:support} Suppose that $\alpha\in F_+$ and $\mu \in L^\vee /L$. Then $\mathcal{Z}_F(\alpha,\mu)$ is non-empty only if $\mathrm{Diff} (\alpha)$ consists of a \underline{single} prime $\mathfrak{p}$. In this case, $\mathcal{Z}_F(\alpha,\mu)$ is $0$-dimensional, and is supported at the unique prime $\mathfrak{q}\subset \mathcal O_E$ above $\mathfrak{p}$. \end{proposition}
\begin{proof} To begin, Proposition~\ref{prop:no special char 0} implies that the intersection of $\mathcal{Z}_F(\alpha,\mu)$ with ${Y}$ is empty. Therefore, $\mathcal{Z}_F(\alpha,\mu)$ is always either empty or $0$-dimensional.
If $ z\in \mathcal{Z}_F(\alpha,\mu) (\mathbb F_\mathfrak{q}^\mathrm{alg})$ for some prime $\mathfrak{q}\subset \mathcal O_E$, let $y\in \mathcal{Y} (\mathbb F_\mathfrak{q}^\mathrm{alg})$ be the point below it. By the definition of $\mathcal{Z}_F(\alpha,\mu)$ the $E$-hermitian space $\mathscr{V}(A_y)_\mathbb Q$ represents $\alpha$. In particular, $\mathscr{V}(A_y)\neq 0$, and so Proposition~\ref{prop:special ordinary} implies that the prime $\mathfrak{p}\subset \mathcal O_F$ below $\mathfrak{q}$ is nonsplit in $\mathcal O_E$. Moreover, Corollary~\ref{cor:special end structure} implies that the nearby hermitian space $ { \empty^{ \mathfrak{p} }} \mathscr{V}$ represents $\alpha$. This shows $\mathrm{Diff}(\alpha)=\{\mathfrak{p}\}$, by Remark \ref{rem:diff}, and everything follows easily. \end{proof}
Set \[ V_\mu(A_y[\infty]) = \prod_{\ell}V_{\mu_\ell}(A_y[\ell^\infty]) \subset V(A_y[\infty])_\mathbb Q. \] When viewed as a subset of the Hermitian space $\mathscr{V}(A_y[\infty])_\mathbb Q$, we will denote this set by $\mathscr{V}_\mu(A_y[\infty])$.
\begin{proposition}\label{prop:special end nearby lattice} Suppose that $y\in \mathcal{Y}(\mathbb F_{\mathfrak{q}}^{\mathrm{alg}})$. Then the $\mathbb A_{f,E}$-linear isometry \[ \mathscr{V}(A_y[\infty])_{\mathbb Q}\xrightarrow{\simeq} { \empty^{ \mathfrak{p} }} \mathscr{V} \] of Proposition~\ref{prop:nearby hermitian space ell} can be chosen so that, for each $\mu \in L^\vee/L$ the characteristic function of the image of $V_\mu(A_y[\infty])$ in $ { \empty^{ \mathfrak{p} }} \widehat{\mathscr{V}}$ is the nearby Schwarz function $ { \empty^{ \mathfrak{p} }} \varphi_\mu$ defined in~\eqref{eqn:near varphi mu decomp}. \end{proposition} \begin{proof} The only non-trivial point is to show that the $E_p$-linear isometry \[ \mathscr{V}(A_y[p^\infty])_{\mathbb Q}\xrightarrow{\simeq} { \empty^{ \mathfrak{p} }} \mathscr{V}_p \] can be chosen so that, for every $\mu_p\in L^\vee_p/L_p$, it identifies the characteristic function of $V_{\mu_p}(A_y[p^\infty])$ with the Schwarz function $ { \empty^{ \mathfrak{p} }} \varphi_{\mu_p}$.
For this, first note that the sublattice $L_{\mathfrak{p}}\subset L_p$ transfers to a sublattice $ { \empty^{ \mathfrak{p} }} L_{\mathfrak{p}}\subset { \empty^{ \mathfrak{p} }} L_p$, as do its cosets in $L^\vee_{\mathfrak{p}}$. Moreover, we have a canonical decomposition \begin{equation*} \mu_p + L_p = \bigsqcup_{(\mu_1,\mu_2)\in (\mu_p+ L_p)/(L_{\mathfrak{p}}\oplus L^{\mathfrak{p}})} (\mu_1 + L_{\mathfrak{p}}) \times \bigl({\mu_2}+ L^{\mathfrak{p}}_p\bigr). \end{equation*}
Given this and Proposition~\ref{prop:V p to mf p}, it is enough to show that we can find an $E_{\mathfrak{q}}$-linear isometry \[ V(A_y[p^\infty]_{\mathfrak{p}})_\mathbb Q \xrightarrow{\simeq} { \empty^{ \mathfrak{p} }} \mathscr{V}_{\mathfrak{p}} \] such that, for every $\mu_1\in L_{\mathfrak{p}}^\vee/L_{\mathfrak{p}}$, it carries the characteristic function of $V_{\mu_1}(A_y[p^\infty]_{\mathfrak{p}})$ to the Schwarz function $ { \empty^{ \mathfrak{p} }} \varphi_\mu$.
By Proposition~\ref{prop:lubin-tate to kuga-satake}, we have an $E_{\mathfrak{q}}$-linear isomorphism \[ V(A_y[p^\infty])_{\mathfrak{p}}\xrightarrow{\simeq}V(\mathcal{G}_{\mathfrak{q},y})_\mathbb Q \] carrying $V_{\mu_1}(A_y[p^\infty]_{\mathfrak{p}})$ to \[ \bigsqcup_{\lambda\in \mu_1 + L_{\mathfrak{p}}}V_{\lambda}(\mathcal{G}_{\mathfrak{q},y}). \] Here, $\lambda$ runs over the cosets of $\Lambda_{\mathfrak{p}}$ in $\Lambda^\vee_{\mathfrak{p}}$ that are contained in $\mu_1 + L_{\mathfrak{p}}$, and we define $V_\lambda(\mathcal{G}_{\mathfrak{q},y})$ via an identification $\Lambda_{\mathfrak{p}} = \mathcal O_{E,\mathfrak{q}}$, which induces an identification $\Lambda^\vee_{\mathfrak{p}}/\Lambda_{\mathfrak{p}} = \mathfrak{d}_{E_{\mathfrak{q}}/F_{\mathfrak{p}}}^{-1}/\mathcal O_{E,\mathfrak{q}}$.
By Proposition~\ref{prop:lubin-tate special denom}, $V_\lambda(\mathcal{G}_{\mathfrak{q},y})$ is empty whenever $\mathrm{ord}_{\mathfrak{q}}(\lambda) < -n(\mathfrak{p}) + 1$. Therefore, we see that it is enough to construct an $E_{\mathfrak{q}}$-linear isometry \[ (V(\mathcal{G}_{\mathfrak{q},y})_\mathbb Q,\langle\cdot,\cdot\rangle) \xrightarrow{\simeq} (E_{\mathfrak{q}},\beta x_1\overline{x}_2) \] such that, for every $\lambda\in \mathfrak{d}_{E_{\mathfrak{q}}/F_{\mathfrak{p}}}^{-1}/\mathcal O_{E,\mathfrak{q}}$ with $\mathrm{ord}_{\mathfrak{q}}(\lambda) > - n(\mathfrak{p})$, the isometry carries $V_\lambda(\mathcal{G}_{\mathfrak{q}, y })$ to $\lambda + \mathcal O_{E,\mathfrak{q}}$. Here, $\beta = \pi_{\mathfrak{p}} = \mathrm{Nm}(\pi_{\mathfrak{q}})$ if $\mathfrak{p}$ is inert in $E$, and $\beta \in\mathcal O_{F,\mathfrak{p}}^\times$ is such that $\chi_{\mathfrak{p}}(\beta) = -1$ if $\mathfrak{p}$ is ramified in $E$.
Such an isometry can be constructed using Propositions~\ref{prp:unramified_vcris} and~\ref{prp:ramified_vcris}. \end{proof}
Recall the embedding $T\hookrightarrow\mathrm{Aut}^\circ(A)$ from Proposition~\ref{prop:tQ_action}, whose homological realizations induce maps \[ \theta_?(H): T_{\mathbb Q_?} \to \mathrm{Aut}^\circ(\bm{H}_?) \] over $\mathcal{Y}$, which in turn give us maps \[ \theta_?(V): T_{\mathbb Q_?} \to \mathrm{Aut}^\circ(\bm{V}_?). \]
In particular, for each prime $\ell$, we obtain a canonical map \[ \theta_{\ell}: T_{\mathbb Q_\ell} \to \mathrm{Aut}\bigl(\mathscr{V} ( A[\ell^\infty] )_Q\bigr), \] and thus a map \[ \theta \define \theta_{\mathbb A_f}: T_{\mathbb A_f} \to \mathrm{Aut}\bigl(\mathscr{V}(A[\infty])_\mathbb Q\bigr). \]
\begin{lemma}\label{lem:twisting isogeny} The group $T(\mathbb A_f) /T(\mathbb Q)K_{L,0}$ acts simply transitively on the set of isomorphism classes in $\mathcal{Y}(\mathbb F_\mathfrak{q}^\mathrm{alg})$, and every point \[ y \in \mathcal{Y}(\mathbb F_\mathfrak{q}^\mathrm{alg}) \] has automorphism group $\mathrm{Aut}(y) = T(\mathbb Q) \cap K_{L,0}$. Moreover, for every $t\in T(\mathbb A_f)$ there is a canonical isometry of $\mathbb A_{f,E}$-hermitian spaces $ \mathscr{V}( A_{t\cdot y}[\infty])_\mathbb Q \xrightarrow{\simeq} \mathscr{V}(A_y[\infty])_\mathbb Q $ identifying \[ \mathscr{V}_\mu( A_{t\cdot y} [\infty] ) \xrightarrow{\simeq} \theta(t)^{-1} \cdot \mathscr{V}_\mu(A_y [\infty] ) \] as subsets of $\mathscr{V}(A_y[\infty])_\mathbb Q$. \end{lemma}
\begin{proof}
By Proposition~\ref{prop:morphism_integral_Y}, $\mathcal{Y}\otimes_{\mathcal O_E}\mathcal O_{E,p}$ is finite \'etale over $\mathcal O_{E,p}$. Therefore, the reduction map \[ \mathcal{Y}(\mathrm{Frac}(W)^{\mathrm{alg}}) \to \mathcal{Y}(\mathbb F_{\mathfrak{q}}^{\mathrm{alg}}) \] is an equivalence of groupoids. Furthermore, the map $\mathcal{Y}(\mathbb Q^\mathrm{alg}) \to \mathcal{Y}(\mathrm{Frac}(W)^{\mathrm{alg}})$ is also an equivalence of groupoids. Therefore, the first assertion follows from the fact that $T(\mathbb A_f)/T(\mathbb Q)K_{L,0}$ acts simply transitively on the set of isomorphism classes in $\mathcal{Y}(\mathbb Q^\mathrm{alg})$ with isotropy group $T(\mathbb Q)\cap K_{L,0}$. This can be checked from the explicit description of the generic fiber ${Y}$ in \S\ref{ss:zero dimensional}.
The rest of the lemma follows easily from the definitions. \end{proof}
\begin{proposition}\label{prop:point count} Fix an $\alpha\in F_+$ such that $\mathrm{Diff}(\alpha)=\{\mathfrak{p}\}$. We have \[
\sum _{ z\in \mathcal{Z}_F(\alpha,\mu)(\mathbb F_\mathfrak{q}^\mathrm{alg}) } \frac{1}{ |\mathrm{Aut}(z) | } =
\deg_\mathbb C( {Y}) \cdot O(\alpha , { \empty^{ \mathfrak{p} }} \varphi_\mu ) , \] \end{proposition}
\begin{proof} The proof follows the same strategy as \cite[Theorem 3.5.3]{HowUnitaryCM}. Pick any base point $y_0 \in \mathcal{Y}(\mathbb F_\mathfrak{q}^\mathrm{alg})$, and an isomorphism \[ \mathscr{V}(A_{y_0} [\infty] )_\mathbb Q \xrightarrow{\simeq} { \empty^{ \mathfrak{p} }} \widehat{\mathscr{V}} \] as in Proposition~\ref{prop:special end nearby lattice}. This identifies the characteristic function of \[
\mathscr{V} _\mu (A_{y_0} [\infty] ) \subset \mathscr{V}(A_{y_0} [\infty] ) \otimes_{\widehat{\mathbb Z}} \mathbb A_f \] with the Schwartz function $ { \empty^{ \mathfrak{p} }} \varphi_\mu \in S( { \empty^{ \mathfrak{p} }} \widehat{\mathscr{V}} )$ defined in \S \ref{ss:nearby lattices}.
Using Lemma \ref{lem:twisting isogeny}, we compute \begin{align*}
\sum _{ z\in \mathcal{Z}_F(\alpha,\mu)(\mathbb F_\mathfrak{q}^\mathrm{alg}) } \frac{1}{ |\mathrm{Aut}(z) | } & = \sum_{ y\in \mathcal{Y}(\mathbb F_\mathfrak{q}^\mathrm{alg})}
\sum_{ \substack{ x\in \mathscr{V}_\mu (A_y) \\ \langle x,x\rangle =\alpha } } \frac{ 1} { | \mathrm{Aut}(y) | }
\\ & =
\sum_{ t\in T(\mathbb Q) \backslash T(\mathbb A_f) / K_{L,0} } \sum_{ \substack{ x\in \mathscr{V}_\mu (A_{t\cdot y_0}) \\ \langle x,x\rangle =\alpha } } \frac{1}{ |\mathrm{Aut}(t\cdot y_0) | } \\ & =
\frac{1 }{ | T(\mathbb Q) \cap K_{L,0} | } \sum_{ t\in T(\mathbb Q) \backslash T(\mathbb A_f) / K_{L,0} } \sum_{ \substack{ x\in \mathscr{V} (A_{y_0}) \otimes \mathbb Q \\ \langle x,x\rangle =\alpha } } { \empty^{ \mathfrak{p} }} \varphi_\mu \big( \theta(t) x \big) . \end{align*} Next use the fact that \[ T (\mathbb Q)/ \mathrm{ker}(\theta) \xrightarrow{\simeq} T_{so}(\mathbb Q)=\{ s\in E^\times : s\overline{s} =1\} \] acts simply transitively on the set of $x \in \mathscr{V} (A_{y_0}) \otimes \mathbb Q$ with $\langle x ,x \rangle=\alpha$. By picking one such $x$, we compute \begin{align*}
\sum _{ z\in \mathcal{Z}_F(\alpha,\mu)(\mathbb F_\mathfrak{q}^\mathrm{alg}) } \frac{1}{ |\mathrm{Aut}(z) | }
& = \frac{ 1} { | T(\mathbb Q) \cap K_{L,0}| } \sum_{ \substack{ t\in T(\mathbb Q) \backslash T(\mathbb A_f) / K_{L,0} \\ t' \in T(\mathbb Q)/\mathrm{ker}(\theta) } } { \empty^{ \mathfrak{p} }} \varphi_\mu \big( \theta(t' t ) x \big) \\ & = \frac{ \deg_\mathbb C( {Y} ) }{ \mathrm{Vol}\big( T_{so}(\mathbb Q) \backslash T_{so}(\mathbb A) \big) }\int_{ T_{so}(\mathbb A_f)} { \empty^{ \mathfrak{p} }} \varphi_\mu ( s x ) \, ds , \end{align*} as desired. \end{proof}
\subsection{Deformation theory} \label{ss:def theory}
Fix an $\alpha\in F_+$ such that $\mathrm{Diff}(\alpha)=\{\mathfrak{p}\}$ for a single prime $\mathfrak{p}\subset \mathcal O_F$. Let $\mathfrak{q}\subset \mathcal O_E$ be the unique prime above $\mathfrak{p}$. Assume that the rational prime $p$ below $\mathfrak{p}$ is good for $L$.
Suppose that $y\in \mathcal{Y}(\mathbb F_{\mathfrak{q}}^\mathrm{alg})$. For any integer $k\in\mathbb Z_{\geq 1}$ and any $\mu\in L^\vee_{\mathfrak{p}}/L_{\mathfrak{p}}$, set $A_k[p^\infty] = A_{\mathcal O_y/\pi_{\mathfrak{q}}^k\mathcal O_y}[p^\infty]$, $\mathcal{G}_{\mathfrak{q},k} = \mathcal{G}_{\mathfrak{q},\mathcal O_y/\pi_{\mathfrak{q}}^k\mathcal O_y}$, and \begin{align*} V_\mu(A_k[p^\infty]) & = V_\mu(A_{\mathcal O_y/\pi_{\mathfrak{q}}^k\mathcal O_y}[p^\infty]) \\ V_\mu(A_k[p^\infty]_{\mathfrak{p}}) = V_\mu(A_{\mathcal O_y/\pi_{\mathfrak{q}}^k\mathcal O_y}[p^\infty]_{\mathfrak{p}})\;&\; V_\mu(\mathcal{G}_{\mathfrak{q},k}) = V_\mu(\mathcal{G}_{\mathfrak{q},\mathcal O_y/\pi_{\mathfrak{q}}^k\mathcal O_y}). \end{align*}
Consider the $1$-dimension $E_{\mathfrak{q}}$-vector space $V(A_y[p^\infty]_{\mathfrak{p}})_\mathbb Q$. By Proposition~\ref{prop:lubin-tate to kuga-satake}, it can be identified with the $E_{\mathfrak{q}}$-vector space $V(\mathcal{G}_{\mathfrak{q},y})_\mathbb Q$.
\begin{proposition} \label{prop:lt-to-ks deformation} For every $k$, the above identification induces an equality \[ V_\mu(A_k[p^\infty]_{\mathfrak{p}}) = V_\mu(\mathcal{G}_{\mathfrak{q},k}) \] \end{proposition} \begin{proof} We will prove this by induction on $k$. When $k=1$, this follows from Proposition~\ref{prop:lubin-tate to kuga-satake}. It remains to show that the assertion holds for $k+1$ whenever it holds for $k$.
Consider the de Rham realization $\bm{\Lambda}_{\mathfrak{p},\mathrm{dR},\mathcal O_y}$ associated with the $K_{\mathfrak{q}}$-representation $\Lambda_{\mathfrak{p}}$. It is the reduction mod $\mathcal{E}(u)$ of the $\mathfrak{S}$-module $\varphi^*\mathfrak{M}(\Lambda_{\mathfrak{p}})$, and is naturally a filtered $\mathcal O_y$-submodule of $\bm{V}^ \diamond_{\mathrm{dR},\mathcal O_y}$.
\begin{lemma} \label{lem:filt beef lambda} We have $\mathrm{Fil}^1\bm{\Lambda}_{\mathfrak{p},\mathrm{dR},\mathcal O_y} = \mathrm{Fil}^1\bm{V}^ \diamond_{\mathrm{dR},\mathcal O_y}$. \end{lemma} \begin{proof} That the assertion holds after inverting $p$ is immediate from the construction. Therefore, it is enough to show that both $\mathrm{Fil}^1\bm{\Lambda}_{\mathfrak{p},\mathrm{dR},\mathcal O_y}$ and $\mathrm{Fil}^1\bm{V}^ \diamond_{\mathrm{dR},\mathcal O_y}$ have the same image in $\bm{\Lambda}^\vee_{\mathfrak{p},\mathrm{dR},\mathcal O_y}$. Since $\mathrm{Fil}^1\bm{V}^ \diamond_{\mathrm{dR},\mathcal O_y}$ is a direct summand of $\bm{V}^ \diamond_{\mathrm{dR},\mathcal O_y}$, it actually suffices to show that its image in $\bm{\Lambda}^\vee_{\mathfrak{p},\mathrm{dR},\mathcal O_y}$ is contained in $\bm{\Lambda}_{\mathfrak{p},\mathrm{dR},\mathcal O_y}$.
Set \[ \mathrm{Fil}^1\varphi^*\mathfrak{M}(V^ \diamond_p) = \{x\in \varphi^*\mathfrak{M}(V^ \diamond_p):\;\varphi(x)\in \mathcal{E}(u)\mathfrak{M}(V^ \diamond_p)\}. \] Then, using assertion~\eqref{kisin:derham} of Theorem~\ref{thm:kisin_p_divisible}, it can be checked that the image of $\mathrm{Fil}^1\varphi^*\mathfrak{M}(V^ \diamond_p)$ in $\bm{V}^ \diamond_{\mathrm{dR},\mathcal O_y}$ is precisely $\mathrm{Fil}^1\bm{V}^ \diamond_{\mathrm{dR},\mathcal O_y}$.
Now, given an element of $\mathrm{Fil}^1\bm{V}^ \diamond_{\mathrm{dR},\mathcal O_y}$, choose a lift $x\in \mathrm{Fil}^1\varphi^*\mathfrak{M}(V^ \diamond_p)$. If $x'\in \varphi^*\mathfrak{M}(\Lambda^\vee_{\mathfrak{p}})$ is the image of $x$, then we find \[ \varphi(x')\in \mathcal{E}(u)\mathfrak{M}(\Lambda^\vee_{\mathfrak{p}}). \] But then Lemma~\ref{lem:discriminant trivial} implies that \[ x'\in \varphi^*\mathfrak{M}(\Lambda_{\mathfrak{p}}) + \mathcal{E}(u)\varphi^*\mathfrak{M}(\Lambda^\vee_{\mathfrak{p}}) \] and hence that its image in $\bm{\Lambda}^\vee_{\mathfrak{p},\mathrm{dR},\mathcal O_y}$ lies in $\bm{\Lambda}_{\mathfrak{p},\mathrm{dR},\mathcal O_y}$.
\end{proof}
Write $\bm{V}^ \diamond_{\mathrm{dR},k}$ (resp. $\bm{\Lambda}_{\mathfrak{p},\mathrm{dR},k}$, $\bm{\Lambda}^\vee_{\mathfrak{p},\mathrm{dR},k}$) for the reduction of $\bm{V}^ \diamond_{\mathrm{dR},\mathcal O_y}$ (resp. $\bm{\Lambda}_{\mathfrak{p},\mathrm{dR},\mathcal O_y}$, $\bm{\Lambda}_{\mathfrak{p},\mathrm{dR},\mathcal O_y}$) mod $\pi_{\mathfrak{q}}^k$.
Now, choose $x\in V_\mu(A_k[p^\infty]_{\mathfrak{p}})$, and let $x_{\mathrm{LT}}$ be the corresponding element of $V_\mu(\mathcal{G}_{\mathfrak{q},k})$. To finish the proof of the proposition, it remains to show that $x$ lifts to $V_\mu(A_{k+1}[p^\infty]_{\mathfrak{p}})$ if and only if $x_{\mathrm{LT}}$ lifts to an element of $V_\mu(\mathcal{G}_{\mathfrak{q},k+1})$.
Consider $x^ \diamond = (x,\tilde{\mu})\in V(A^ \diamond_{\mathcal O_y/\pi_{\mathfrak{q}}^k\mathcal O_y})$. Let $\bm{x}^ \diamond_{\mathrm{cris}}\in \bm{V}^ \diamond_{\mathrm{dR},k+1}$ be the crystalline realization of $x^ \diamond$. By Proposition~\ref{prop:special_endomorphism_deform} and Lemma~\ref{lem:filt beef lambda}, $x^ \diamond$ lifts to $V(A^ \diamond_{\mathcal O_y/\pi_{\mathfrak{q}}^{k+1}\mathcal O_y})$, and hence $x$ lifts to $V_\mu(A_{k+1}[p^\infty]_{\mathfrak{p}})$, if and only if the functional \[ [\bm{x}^ \diamond_{\mathrm{cris}},\_\_]:\bm{\Lambda}_{\mathfrak{p},\mathrm{dR},k+1}\to \mathcal O_y/\pi_{\mathfrak{q}}^{k+1}\mathcal O_y \] lies in the annihilator of $\mathrm{Fil}^1\bm{\Lambda}_{\mathfrak{p},\mathrm{dR},k+1}$.
We claim that this annihilator is \[ \mathrm{Fil}^0\bm{\Lambda}^\vee_{\mathfrak{p},\mathrm{dR},k+1} \define \ker\bigl(\bm{\Lambda}^\vee_{\mathfrak{p},\mathrm{dR},k+1}\to\bm{\Lambda}^\vee_{\mathfrak{p},\mathrm{dR},k+1}\otimes_{\mathcal O_y\otimes_{\mathbb Z_p}\mathcal O_{E,\mathfrak{q}},1\otimes\overline{\iota}_0}\mathcal O_y\bigr). \] Indeed, it is enough to check that the annihilator of $\mathrm{Fil}^1\bm{\Lambda}_{\mathfrak{p},\mathrm{dR},\mathcal O_y}$ in $\bm{\Lambda}^\vee_{\mathfrak{p},\mathrm{dR},\mathcal O_y}$ is \[ \ker\bigl(\bm{\Lambda}^\vee_{\mathfrak{p},\mathrm{dR},\mathcal O_y}\to\bm{\Lambda}^\vee_{\mathfrak{p},\mathrm{dR},\mathcal O_y}\otimes_{\mathcal O_y\otimes_{\mathbb Z_p}\mathcal O_{E,\mathfrak{q}},1\otimes\overline{\iota}_0}\mathcal O_y\bigr), \] which can be checked after inverting $p$, where it is easily verified.
Now, by Proposition~\ref{prop:lifting end}, $x_{\mathrm{LT}}$ has a crystalline realization $\bm{x}_{\mathrm{LT},\mathrm{cris}}\in \bm{\Lambda}^\vee_{\mathrm{dR},k+1}$, and lifts to $V_\mu(\mathcal{G}_{\mathfrak{q},k+1})$ if and only if $\bm{x}_{\mathrm{LT},\mathrm{cris}}$ lies in $\mathrm{Fil}^0\bm{\Lambda}^\vee_{\mathfrak{p},\mathrm{dR},k+1}$.
To finish, it now suffices to observe that \begin{equation}\label{eqn:equality realizations} \bm{x}_{\mathrm{LT},\mathrm{cris}}=[\bm{x}^ \diamond_{\mathrm{cris}},\_\_]\in \bm{\Lambda}^\vee_{\mathfrak{p},\mathrm{dR},k+1}. \end{equation} For this, let $S$ be the $p$-adic completion of the divided power envelope of the surjection $W[u]\xrightarrow{u\mapsto \iota_0(\pi_{\mathfrak{q}})}\mathcal O_y$, and set \[ \mathcal{M}(\Lambda^\vee_{\mathfrak{p}}) \define \varphi^*\mathfrak{M}(\Lambda^\vee_{\mathfrak{p}})\otimes_{\mathfrak{S}}S. \]
The $\varphi$-module structure on $\mathfrak{M}(\Lambda^\vee_{\mathfrak{p}})[\mathcal{E}(u)^{-1}]$ gives us an isomorphism \[ \varphi:\varphi^*\mathcal{M}(\Lambda^\vee_{\mathfrak{p}})[p^{-1}]\xrightarrow{\simeq}\mathcal{M}(\Lambda^\vee_{\mathfrak{p}}). \]
Moreover, by a variation of Dwork's trick (see~\cite[6.2.1.1]{breuil:griffiths}), the reduction map \[ \mathcal{M}(\Lambda^\vee_{\mathfrak{p}})[p^{-1}]\to \mathcal{M}(\Lambda^\vee_{\mathfrak{p}})[p^{-1}]\otimes_SW = \bm{\Lambda}^\vee_{\mathfrak{p},\mathrm{cris},y}[p^{-1}] \] induces a bijection on $\varphi$-invariant elements. Let $\bm{x}_0\in \bm{\Lambda}^\vee_{\mathfrak{p},\mathrm{cris},y}$ be the crystalline realization of $x$ viewed as an element of $V_\mu(A_y[p^\infty]_{\mathfrak{p}})$, and let $\tilde{\bm{x}}_0\in \mathcal{M}(\Lambda^\vee_{\mathfrak{p}})[p^{-1}]$ be its unique $\varphi$-invariant lift.
If $k<e$, then the image of $\tilde{\bm{x}}_0$ in $\mathcal{M}(\Lambda^\vee_{\mathfrak{p}})[p^{-1}]\otimes_SW[u]/(u^{k+1})$ actually lies in \[ \mathcal{M}(\Lambda^\vee_{\mathfrak{p}})\otimes_SW[u]/(u^{k+1}) \] and, by virtue of its $\varphi$-invariance, is necessarily the crystalline realization of both $x$ and $x_{\mathrm{LT}}$ along the divided power thickening $W[u]/(u^{k+1})\xrightarrow{u\mapsto \iota_0(\pi_{\mathfrak{q}})}\mathcal O_y/\pi_{\mathfrak{q}}^k\mathcal O_y$.
If $k\geq e$, then, once again by virtue of its $\varphi$-invariance, $\tilde{\bm{x}}_0$ is the crystalline realization of both $x$ and $x_{\mathrm{LT}}$ along the divided power thickening $S\to \mathcal O_y/\pi_{\mathfrak{q}}^k\mathcal O_y$.
From these observations, the required identity~\eqref{eqn:equality realizations} easily follows. \end{proof}
Define a function \[ \mathrm{ord}_{\mathfrak{q}}:\;V(A_y[p^\infty]_{\mathfrak{p}})_\mathbb Q \to \mathbb Z, \] given by two defining properties: \begin{itemize}
\item If $a\in E_{\mathfrak{q}}$, and $x\in V(A_y[p^\infty]_{\mathfrak{p}})$, then
\[
\mathrm{ord}_{\mathfrak{q}}(a\cdot x) = \mathrm{ord}_{\mathfrak{q}}(a) + \mathrm{ord}_{\mathfrak{q}}(x).
\]
\item If $x\in V(\mathcal{G}_{\mathfrak{q},y})$ is an $\mathcal O_{E_{\mathfrak{q}}}$-module generator, then
\[
\mathrm{ord}_{\mathfrak{q}}(x) = \begin{cases}
1,&\text{ if $\mathfrak{q}$ is unramified over $F$};\\
n(\mathfrak{p}) = \mathrm{ord}_{\mathfrak{q}}(\mathfrak{d}_{E/F}),&\text{ if $\mathfrak{q}$ is ramified over $F$}.
\end{cases}
\] \end{itemize}
Our definition of the function $\mathrm{ord}_{\mathfrak{q}}$ is justified by the following result.
\begin{proposition} \label{prop:deformation special}
Suppose that $\mu\in L_{\mathfrak{p}}^\vee/L_{\mathfrak{p}}$, and that $x\in V_\mu(A_y[p^\infty]_{\mathfrak{p}})$. Then $x$ lifts to $V_\mu(A_{k}[p^\infty]_{\mathfrak{p}})$ if and only if $\mathrm{ord}_{\mathfrak{q}}(x)\geq k$. \end{proposition} \begin{proof} This is immediate from Proposition~\ref{prop:lt-to-ks deformation} and Theorem~\ref{thm:deformation special}. \end{proof}
\begin{theorem}\label{thm:local ring} At any point $z\in \mathcal{Z}_F(\alpha,\mu)(\mathbb F_\mathfrak{q}^\mathrm{alg})$ we have \[ \mathrm {length} \left( \mathcal O_{\mathcal{Z}_F(\alpha,\mu),z}\right) = \ell_{\mathfrak{p}}(\alpha), \] where $\ell_{\mathfrak{p}}(\alpha)$ is defined as in Proposition~\ref{prop:explicit siegel-weil}. \end{theorem} \begin{proof} The point $z$ corresponds to a point $y\in \mathcal{Y}(\mathbb F_{\mathfrak{q}}^{\mathrm{alg}})$ equipped with a special endomorphism $x\in V_\mu(A_y)$ satisfying $\langle x,x\rangle = \alpha$. By Serre-Tate theory its deformation theory is governed by the induced endomorphism $x_p \in V_{\mu_p}(A_y[p^\infty])$. By Proposition~\ref{prop:V p to mf p}, there is a unique pair \[ (\mu_1,\mu_2) \in \frac{\mu_p + L_p}{L_{\mathfrak{p}}+ L^{\mathfrak{p}}_p} \subset \frac{L_{\mathfrak{p}}^\vee}{L_{\mathfrak{p}}} \oplus \frac{L^{\mathfrak{p},\vee}_p}{L^{\mathfrak{p}}_p}, \] together with a unique $x_{\mathfrak{p}}\in V_{\mu_1}(A_y[p^\infty]_{\mathfrak{p}})$ and $v\in \mu_2 + L^{\mathfrak{p}}_p$, such that \[ x_p = x_{\mathfrak{p}} + v. \] Moreover, $\mathrm{ord}_{\mathfrak{p}}(\alpha) = \mathrm{ord}_{\mathfrak{p}}(\langle x_{\mathfrak{p}},x_{\mathfrak{p}}\rangle)$.
Also, by the same proposition, the deformation theory of $x_p$ is governed by that of $x_{\mathfrak{p}}$. More precisely, $x_p$ lifts to $V_{\mu_p}(A_k[p^\infty])$ if and only if $x_{\mathfrak{p}}$ lifts to $V_{\mu_1}(A_k[p^\infty]_{\mathfrak{p}})$. By Proposition~\ref{prop:deformation special}, this is equivalent to the condition $\mathrm{ord}_{\mathfrak{q}}(x_{\mathfrak{p}})\geq k$.
Therefore, to finish, we must show: \begin{equation}\label{eqn:ord computation} \mathrm{ord}_{\mathfrak{q}}(x_{\mathfrak{p}}) = \ell_{\mathfrak{p}}(\alpha). \end{equation} Now, note that the Hermitian form on $V(A_y[p^\infty]_{\mathfrak{p}})_\mathbb Q$ is $\xi_{\mathfrak{p}}$-times the natural Hermitian form $\langle\cdot,\cdot\rangle_{\mathrm{LT}}$ on $V(\mathcal{G}_{\mathfrak{q},y})_\mathbb Q$, and that $\mathrm{ord}_{\mathfrak{p}}(\xi_{\mathfrak{p}}) = -m(\mathfrak{p})$.
Moreover, using Propositions~\ref{prp:unramified_vcris} and~\ref{prp:unramified_vcris}, we find that, if $x_0\in V(\mathcal{G}_{\mathfrak{q},y})$ is an $\mathcal O_{E,\mathfrak{q}}$-module generator, then \[ \mathrm{ord}_{\mathfrak{p}}(\langle x_0,x_0\rangle_{\mathrm{LT}}) = \begin{cases} 1 = -1 + 2\cdot\mathrm{ord}_{\mathfrak{q}}(x_0),&\text{ if $\mathfrak{q}$ is unramified over $F$};\\ 0 = -n(\mathfrak{p}) + \mathrm{ord}_{\mathfrak{q}}(x_0),&\text{ if $\mathfrak{q}$ is ramified over $F$}. \end{cases} \]
Combining all this, we find that $\mathrm{ord}_{\mathfrak{p}}(\alpha)$ is equal to \[
\mathrm{ord}_{\mathfrak{p}}(\langle x_{\mathfrak{p}},x_{\mathfrak{p}}\rangle) = \begin{cases} -m(\mathfrak{p}) - 1 + 2\cdot\mathrm{ord}_{\mathfrak{q}}(x_{\mathfrak{p}}),&\text{ if $\mathfrak{q}$ is unramified over $F$};\\ -m(\mathfrak{p}) - n(\mathfrak{p}) + \mathrm{ord}_{\mathfrak{q}}(x_{\mathfrak{p}}),&\text{ if $\mathfrak{q}$ is ramified over $F$.} \end{cases} \]
Comparing this with the formulas for $\ell_{\mathfrak{p}}(\alpha)$ in Proposition~\ref{prop:explicit siegel-weil} gives us~\eqref{eqn:ord computation} and hence the theorem. \end{proof}
\subsection{Calculation of arithmetic degrees: the end of the proof of Theorem~\ref{thm:arithmetic BKY}} \label{ss:arithmetic intersection proof}
\begin{theorem}\label{thm:degree} Suppose $\alpha\in F_+$ and $\mu \in L^\vee / L$. Assume that $\mathrm{Diff}(\alpha) = \{ \mathfrak{p} \}$ consists of a single prime of $\mathcal O_F$, which lies above a rational prime $p$ that is good for $L$. If we denote by \[
\widehat{\mathcal{Z}}_F (\alpha,\mu) \in \widehat{\mathrm{Pic}}( \mathcal{Y}) \]
the divisor $\mathcal{Z}_F(\alpha,\mu)$ on $\mathcal{Y}$ endowed with the trivial Green function, then \[ \frac{ \widehat{\deg} \big( \widehat{\mathcal{Z}}_F (\alpha,\mu) \big) } { \deg_\mathbb C( {Y}) } = - \frac{a_F(\alpha,\mu)}{ \Lambda(0, \chi ) }. \] \end{theorem}
\begin{proof} Combine Propositions ~\ref{prop:explicit siegel-weil}, and~\ref{prop:point count} with Theorem~\ref{thm:local ring}. \end{proof}
Given $a,b\in\mathbb R$, we will write $a\approx_L b$ to mean that $a-b$ is a $\mathbb Q$-linear combination of $\{ \log(p) : p \mid D_{bad,L} \}$.
\begin{proposition}\label{prop:constant term eval} We have \[ \frac{ a(0,0) }{ \Lambda(0,\chi) } \approx_L - \frac{ 2 \Lambda'(0,\chi)}{ \Lambda(0,\chi) }. \] If $\mu\neq 0$, then \[
\frac{ a(0,\mu) }{ \Lambda(0,\chi) } \approx_L
0. \]
\end{proposition}
\begin{proof} Let $\varphi=\varphi_\mu$, so that we have a factorization $\varphi=\otimes \varphi_p$ over the rational primes, and \begin{equation}\label{eqn:a 0 mu} \frac{ a(0,\mu) } { \Lambda (0,\chi) } = \frac{ a_F(0,\varphi) }{ \Lambda(0,\chi) } = -2 \varphi (0) \frac{ \Lambda'(0,\chi)}{\Lambda(0,\chi)} - M'(0,\varphi). \end{equation} by Proposition \ref{prop:coarse constant}.
Fix a prime $p$, and suppose that we have $\varphi_p = \sum_i\varphi_i$, where each $\varphi_i$ admits a factoring $\varphi_i = \otimes_{\mathfrak{p}\mid p}\varphi_{i,\mathfrak{p}}$ over primes $\mathfrak{p}\subset \mathcal O_F$ above $p$. Then, for each $i$, by Proposition~\ref{prop:coarse constant}, we obtain a factoring \[ M_p(s,\varphi_i) = \prod_{\mathfrak{p}\mid p}M_{\mathfrak{p}}(s,\varphi_{i,\mathfrak{p}}), \] where, for any $\mathfrak{p}\mid p$, $M_{\mathfrak{p}}(s,\varphi_{i,\mathfrak{p}})$ is a rational function in $N(\mathfrak{p})^s$. Therefore, $M_{\mathfrak{p}}(0,\varphi_{i,\mathfrak{p}})$ is a rational number, and $M'_{\mathfrak{p}}(0,\varphi_{i,\mathfrak{p}})$ is a rational multiple of $\log N(\mathfrak{p})$.
Moreover, if $p$ is a good prime, then, by~\eqref{eqn:mu to lambda decomp}, we can choose our decomposition to be \[ \varphi_{\mu_p} = \bigotimes_{\lambda_p\in\mu_p + L_p} \varphi_{\lambda_p}, \] where $\lambda_p$ ranges over representative of cosets of $\Lambda_p$ in $\Lambda_p^\vee$ contained in $\mu_p + L_p$.
By Corollary~\ref{cor:good prime M constant}, $M_{\mathfrak{p}}(s,\varphi_{\lambda_\mathfrak{p}})$ is constant, and hence $M_{\mathfrak{p}}'(0,\varphi_{\lambda_{\mathfrak{p}}}) = 0$, for all primes $\mathfrak{p}\mid p$, It now follows that $M'(0,\varphi)$ is a $\mathbb Q$-linear combination of $\log(p)$ with $p\mid D_{bad}$.
The identity~\eqref{eqn:a 0 mu} now gives us the proposition. \end{proof}
\begin{proof}[Proof of Theorem \ref{thm:arithmetic BKY}] Recalling that \[ \mathcal{Z}(f) = \sum_{m>0} \sum_{ \mu \in L^\vee / L} c_f^+(-m,\mu) \mathcal{Z}(m,\mu), \] the stack decomposition \[
\mathcal{Z}(m,\mu) \times_\mathcal{M} \mathcal{Y} = \bigsqcup_{ \substack{ \alpha\in F_+ \\ \mathrm{Tr}_{F/\mathbb Q}(\alpha) = m } }
\mathcal{Z}_F (\alpha,\mu) \] of (\ref{intersection decomp}) implies \[
\frac{ [ \widehat{\mathcal{Z}}(f) : \mathcal{Y} ] }{ \deg_\mathbb C ({Y}) } =
\frac{ \Phi (f ,\mathcal{Y}^\infty ) }{ 2 \deg_\mathbb C ({Y}) } + \sum_{ \substack{ \mu\in L^\vee/L \\ m > 0 }} c_f^+(m,\mu) \sum_{ \substack{ \alpha \in F_+ \\ \mathrm{Tr}_{F/\mathbb Q}(\alpha) =m } } \frac{ \widehat{\deg} \big( \widehat{\mathcal{Z}}_F (\alpha,\mu) \big) } { \deg_\mathbb C( {Y}) } . \]
For any $\alpha\in F_+$ and any $\mu\in L^\vee /L$ we have \begin{equation}\label{approx degree} \frac{ \widehat{\deg} \big( \widehat{ \mathcal{Z}}_F (\alpha,\mu) \big) } { \deg_\mathbb C( {Y}) } \approx_L - \frac{ a_F(\alpha ,\mu) } { \Lambda( 0 , \chi ) } . \end{equation}
Indeed, if $|\mathrm{Diff}(\alpha)| >1$ then Propositions \ref{prop:coefficient support} and \ref{prop:support} imply that both sides of (\ref{approx degree}) vanish. If $\mathrm{Diff}(\alpha)=\{ \mathfrak{p} \}$ then let $p$ be the rational prime below $\mathfrak{p}$. If $p\nmid D_{bad}$ then the relation (\ref{approx degree}) follows from Theorem \ref{thm:degree}. If $p\mid D_{bad}$ then both sides of (\ref{approx degree}) are $\approx 0$ by
Propositions \ref{prop:coefficient support} and \ref{prop:support}.
Combining (\ref{approx degree}) and (\ref{eisenstein decomp}) shows that \[ \sum_{ \substack{ \alpha\in F^\times \\ \mathrm{Tr}_{F/\mathbb Q}(\alpha) = m } } \frac{ \widehat{\deg} \big( \widehat{ \mathcal{Z}}_F (\alpha,\mu) \big) } { \deg_\mathbb C( {Y}) } \approx_L - \frac{ a(m ,\mu) } { \Lambda( 0 , \chi ) } , \] and so \[
\frac{ [ \widehat{\mathcal{Z}}(f) : \mathcal{Y} ] }{ \deg_\mathbb C ({Y}) } \approx_L \frac{\Phi(f, \mathcal{Y}^\infty ) }{2 \deg_\mathbb C ({Y}) } - \sum_{ \substack{ \mu\in L^\vee/L \\ m > 0 }} \frac{ a(m ,\mu) \cdot c_f^+(m,\mu) } { \Lambda( 0 , \chi ) } . \] Comparing with Theorem \ref{thm:BKY}, and using the approximate identity $a(0,\mu)\approx_L 0$ from Proposition~\ref{prop:constant term eval} for $\mu\not=0$, shows that \[
\frac{ [ \widehat{\mathcal{Z}}(f) : \mathcal{Y} ] }{ \deg_\mathbb C ({Y}) } \approx_L - \frac{ \mathcal{L}'(0 , \xi(f) ) } { \Lambda( 0 , \chi ) } + \frac{ a(0, 0 ) \cdot c_f^+(0,0) } { \Lambda( 0 , \chi ) } , \] as desired. \end{proof}
\section{The height of the tautological bundle} \label{s:taut height}
For applications to the computations of heights of line bundles, and to Colmez's conjecture, it will be useful to have a results more general than Theorem~\ref{thm:arithmetic BKY}, involving restrictions of Borcherds products from larger GSpin Shimura varieties. In these section, we pursue such generalizations, which are mostly of a formal nature, given what has come before.
\subsection{Enlarging the Shimura variety} \label{ss:enlarged setup}
Suppose that we have a quadratic space $(V^ \diamond, Q^ \diamond)$ of signature $(n^ \diamond,2)$, a maximal lattice $L^ \diamond \subset V^ \diamond$ of discriminant $D_{L^ \diamond} = [ L^{ \diamond,\vee} : L^ \diamond]$, and an isometric embedding \[ (V,Q) \hookrightarrow (V^ \diamond, Q^ \diamond). \] satisfying $L\subset L^ \diamond$. Define a positive definite $\mathbb Z$-quadratic space
\[
\Lambda = \{ x\in L^ \diamond : x\perp L\}
\]
of rank $n^ \diamond-2d+2$, so that $L\oplus \Lambda \subset L^ \diamond$ with finite index, and $V^ \diamond = V \oplus \Lambda_\mathbb Q .$
From this data, we obtain maps of $\mathbb Z$-stacks \[ \mathcal{Y} \to \mathcal{M} \to \mathcal{M}^ \diamond, \] where $\mathcal{M}$ and $\mathcal{M}^ \diamond$ are the integral models for the orthogonal Shimura varieties associated with the lattices $L$ and $L^ \diamond$, respectively, and $\mathcal{Y}=\mathcal{Y}_{K_{L,0}}$ is the stack appearing in Proposition \ref{prop:morphism_integral_Y}.
\subsection{The archimedean contribution}
From the $\mathbb Z$-quadratic space $\Lambda$ we may form the representation $\omega_\Lambda$ of $\widetilde{\mathrm {SL}}_2(\mathbb Z)$ on the finite dimensional subspace \[ S_\Lambda \subset S( \widehat{\Lambda}_\mathbb Q ) \] of $\mathbb C$-valued functions on $\Lambda^\vee /\Lambda$, exactly as in \S \ref{ss:harmonic forms}, and the contragredient representation $\omega_\Lambda^\vee$ on the $\mathbb C$-linear dual $S_\Lambda^\vee$.
The theta series \[ \vartheta_\Lambda(\tau) = \sum_m \rho_\Lambda(m) \cdot q^m \in M^!_{ \frac{n^ \diamond}{2} -d+1 }(\omega_\Lambda^\vee) \] has Fourier coefficients $\rho_\Lambda(m) \in S_\Lambda^\vee$ defined by \[ \rho_\Lambda(m,\varphi) = \sum_{ x\in \Lambda^\vee } \varphi(x) \] for any $\varphi\in S_\Lambda$. Letting $\varphi_\mu$ denote the characteristic function of $\mu \in \Lambda^\vee /\Lambda$, we often write $\rho_\Lambda(m,\mu)=\rho_\Lambda(m,\varphi_\mu)$.
Given a pair $ (\mu_1,\mu_2) \in (L^\vee / L) \oplus (\Lambda^\vee / \Lambda) $ and a $\mu \in L^{ \diamond,\vee} / L^ \diamond$ we write, by abuse of notation, $\mu_1+\mu_2 = \mu$ to mean that the map \[ (L^\vee / L) \oplus (\Lambda^\vee / \Lambda) \to (L^\vee \oplus \Lambda^\vee) / L^ \diamond \] induced by the inclusions \[ L\oplus \Lambda \subset L^ \diamond \subset L^{ \diamond,\vee} \subset L^\vee \oplus \Lambda^\vee \] takes $(\mu_1,\mu_2) \mapsto \mu$.
\begin{proposition}\label{prop:CM_value_beef} Fix any weakly holomorphic modular form $f \in M_{ 1-n^ \diamond/2 } ^! (\omega_{L^ \diamond})$ with integral principal part, and let $\Phi^ \diamond(f)$ be the corresponding Green function on $\mathcal{M}^ \diamond$, as in \S \ref{ss:harmonic forms}. If we set \[
\mathcal{Y}^\infty = \mathcal{Y} \times_{ \mathrm{Spec}(\mathbb Z) } \mathrm{Spec}( \mathbb C), \] and define $\Phi^ \diamond (f ,\mathcal{Y}^\infty)$ as in Theorem \ref{thm:BKY}, then \[
\frac{ \Phi^ \diamond (f ,\mathcal{Y}^\infty) }{ 2 \deg_\mathbb C ({Y}) } =
\sum_{ \substack{ \mu \in L^{ \diamond,\vee} / L^ \diamond \\ m \in \mathbb Q } } c_f (-m,\mu)
\sum_{ \substack{ m_1 + m_2 =m \\ \mu_1+\mu_2 = \mu } } \frac{ a(m_1,\mu_1) \rho_\Lambda(m_2,\mu_2) }{ \Lambda( 0 , \chi ) }. \]
\end{proposition}
\begin{proof} The isomorphism \[ S( \widehat{V}^ \diamond ) \xrightarrow{\simeq} S(\widehat{V}) \otimes S(\widehat{\Lambda}_\mathbb Q), \] together with the tautological pairing between $S(\widehat{\Lambda}_\mathbb Q)$ and its dual, induce a map \[ S( \widehat{V}^ \diamond ) \otimes S(\widehat{\Lambda}_\mathbb Q)^\vee \to S(\widehat{V}) . \] As $L\oplus \Lambda \subset L^ \diamond$, this restricts to a map $ S_{L^ \diamond} \otimes S_\Lambda^\vee \to S_L, $ which we call \emph{tensor contraction}, and denote by $\varphi_1 \otimes \varphi_2 \mapsto \varphi_1 \odot \varphi_2$.
There is an induced map on spaces of weakly holomorphic forms \[ M^!_k (\omega_{L^ \diamond}) \otimes M^!_\ell( \omega_\Lambda^\vee) \to M^!_{k+\ell} (\omega_{L}) \] for any half-integers $k$ and $\ell$. In particular, $f\mapsto f \odot \vartheta_\Lambda$ defines a linear map \[ M^!_{1- \frac{n^ \diamond}{2} }(\omega_L^ \diamond) \to M^!_{2-d}(\omega_L ). \] In terms of $q$-expansions, \[ ( f \odot \vartheta_\Lambda)(\tau) = \sum_{ m\gg -\infty } \sum_{ \mu \in L^\vee / L } c_{ f \odot \vartheta_\Lambda }(m ,\mu) \varphi_\mu \cdot q^m \] where \begin{equation}\label{contraction coefficients} c_{ f \odot \vartheta_\Lambda }(m ,\mu) = \sum_{ m_1 + m_2 =m } \sum_{ \substack{ \mu_2 \in \Lambda^\vee / \Lambda \\ \mu+\mu_2 \in L^{ \diamond ,\vee} / L\oplus \Lambda } } c_f(m_1, \mu+\mu_2 ) \cdot \rho_\Lambda(m_2,\mu_2 ). \end{equation}
The essential observation is this: If we pull back the Green function $\Phi^ \diamond(f)$ for the divisor $\mathcal{Z}^ \diamond(f)(\mathbb C)$ on $\mathcal{M}^ \diamond(\mathbb C)$ via the map
$\mathcal{M}(\mathbb C) \to \mathcal{M}^ \diamond(\mathbb C)$, we obtain the Green function $\Phi(f\odot \vartheta_\Lambda)$ for the divisor $\mathcal{Z}( f\odot \vartheta_\Lambda )(\mathbb C)$ on $\mathcal{M}(\mathbb C)$. This is clear from the factorization \cite[(4.16)]{BY} of Siegel theta functions, and the construction of the Green functions as regularized theta lifts as in \cite{BKY, BY}.
As $\mathcal{Y}^\infty(\mathbb C) \to \mathcal{M}^ \diamond(\mathbb C)$ factors through $\mathcal{M}(\mathbb C)$, we find \[ \frac{ \Phi^ \diamond (f ,\mathcal{Y}^\infty) }{ 2 \deg_\mathbb C ({Y}) } = \frac{ \Phi (f \odot \vartheta_\Lambda ,\mathcal{Y}^\infty) }{ 2 \deg_\mathbb C ({Y}) } . \] We may apply the result of Bruinier-Kudla-Yang, as stated in Theorem \ref{thm:BKY}, directly to the right hand side. This gives \begin{align*} \frac{ \Phi^ \diamond (f ,\mathcal{Y}^\infty) }{ 2 \deg_\mathbb C ({Y}) } & = - \sum_{ \substack{ \mu_1 \in L^\vee / L \\ m_1 \in \mathbb Q } } \frac{ a(m_1,\mu_1) \cdot c_{ f\odot \vartheta_\Lambda} ( -m_1, \mu_1) }{ \Lambda(0,\chi) } \\
& = - \sum_{ \substack{ \mu \in L^{ \diamond,\vee} / L^ \diamond \\ m_1 + m_2+ m_3=0 } } c_f( m_3 , \mu ) \sum_{ \mu_1+\mu_2 =\mu } \frac{ a(m_1,\mu_1) \rho_\Lambda(m_2,\mu_2 ) }{ \Lambda(0,\chi) } \end{align*} where the second equality follows from (\ref{contraction coefficients}). \end{proof}
\subsection{An extended arithmetic intersection formula}
Fix any weakly holomorphic modular form $f \in M_{ 1-n^ \diamond/2 } ^! (\omega_{L^ \diamond})$ with integral principal part. Let \[ \mathcal{Z}^ \diamond(f) = \sum_{ m>0} \sum_{ \mu \in L^{ \diamond,\vee}/L^ \diamond } c_f( - m,\mu) \cdot \mathcal{Z}^ \diamond(m,\mu), \] be the corresponding divisor on $\mathcal{M}^ \diamond$, and denote by \[ \widehat{ \mathcal{Z}}^ \diamond(f) = \big( \mathcal{Z}^ \diamond(f) , \Phi^ \diamond(f) \big) \in \widehat{\mathrm{CH}}^1(\mathcal{M}^ \diamond) \] the corresponding arithmetic divisor.
In what follows we will frequently demand that $f$ satisfy the following hypothesis with respect to the quadratic submodule $L\subset L^ \diamond$ and its orthogonal complement $\Lambda = L^\perp\subset L^ \diamond$.
\begin{hypothesis}\label{hyp:proper} If $m>0$, and if there an element $x\in D_{L^ \diamond}^{-1}\Lambda$ with $Q(x) = m$, then $c_f(-m,\mu) = 0$ for all $\mu \in L^{ \diamond,\vee} / L^ \diamond$.\end{hypothesis}
\begin{lemma}\label{lem:improper_intersection} If the image of $\mathcal{Y}(\mathbb C) \to \mathcal{M}^ \diamond(\mathbb C)$ intersects the support of $\mathcal{Z}^ \diamond(m,\mu)(\mathbb C)$, then there is an $x\in D_{L^ \diamond}^{-1} \Lambda$ with $Q(x)=m$. Therefore, if Hypothesis~\ref{hyp:proper} holds, then $\mathcal{Z}^\diamond(f)$ intersects $\mathcal{Y}$ properly. \end{lemma} \begin{proof} Suppose that the image of $\mathcal{Y}(\mathbb C)$ intersects the support of $\mathcal{Z}^ \diamond(m,\mu)(\mathbb C)$. By Corollary~\ref{prop:Zmu_functoriality}, this implies that there exist $m_1,m_2\in\mathbb Q_{\geq 0}$ with $m_1+m_2 = m$ and \[ (\mu_1,\mu_2) \in \frac{L^{ \diamond,\vee}}{L\oplus\Lambda} \subset (L^\vee/L) \oplus (\Lambda^\vee/\Lambda) \] such that $\mathcal{Y}(\mathbb C)$ intersects the support of $\mathcal{Z}(m_1,\mu_1)\times\Lambda_{m_2,\mu_2}$, where \[ \Lambda_{m_2,\mu_2} = \{x\in {\mu_2}+\Lambda : Q(x)=m_2\}. \] By Proposition~\ref{prop:no special char 0}, for any point $y\in {Y}(\mathbb C)$ we must have $V(A_y) = 0$. Thus the only way that $y$ can meet $\mathcal{Z}(m_1, \mu_1)$ is if $m_1=0$ and $\mu_1=0$. It now follows that there exists $x\in \Lambda_{m,\mu_2}$ with \[ (0,\mu_2)\in\frac{L^{ \diamond,\vee}}{L\oplus\Lambda}, \] and from this we deduce that there exists $x\in D_{L^ \diamond}^{-1}\Lambda$ with $Q(x) = m$.
The second assertion is clear from the first. \end{proof}
\begin{proposition}\label{prop:height_pairing_beef} Under Hypothesis~\ref{hyp:proper}, we have: \[ \frac{ [ \widehat{\mathcal{Z}}^{ \diamond}(f) : \mathcal{Y} ] }{ \deg_\mathbb C ({Y}) } \approx_L
c_f(0,0)\frac{a(0,0)}{\Lambda(0,\chi)}. \] \end{proposition} \begin{proof} Corollary~\ref{prop:Zmu_functoriality} gives us, for each pair $(m,\mu)$, a decomposition \[ \mathcal{Z}^{ \diamond}(m,\mu)\times_{\mathcal{M}^{ \diamond}}\mathcal{M} = \bigsqcup_{\substack{ m_1 + m_2 =m \\ \mu_1+\mu_2 = \mu }}\mathcal{Z}(m_1,\mu_1)\times\Lambda_{m_2,\mu_2} \] of stacks over $\mathcal{M}$. Moreover, if $c_f(-m,\mu)\neq 0$ then Hypothesis~\ref{hyp:proper} implies that all terms with $m_1=0$ are empty. Therefore, we obtain a decomposition \[ \mathcal{Z}^ \diamond(f)\vert_{\mathcal{M}} = \sum_{ \substack{ \mu \in L^{ \diamond,\vee} / L^ \diamond \\ m_1 + m_2+ m_3=0 \\ m_3<0 \\ m_1 > 0} } c_f( m_3 , \mu ) \sum_{ \mu_1+\mu_2 =\mu } \rho_{\Lambda}(m_2,\mu_2) \mathcal{Z}(m_1,\mu_1) \] of divisors on $\mathcal{M}$.
By Lemma~\ref{lem:improper_intersection}, the image of $\mathcal{Y}\to \mathcal{M}$ intersects $\mathcal{Z}^\diamond(f)$ properly, and so, as in the proof of Theorem~\ref{thm:arithmetic BKY} (see especially \S\ref{ss:arithmetic intersection proof}), we deduce \begin{align*}
\frac{ [ \widehat{\mathcal{Z}}^ \diamond(f) : \mathcal{Y} ] }{ \deg_\mathbb C ({Y}) } \approx_L \frac{\Phi^ \diamond(f, \mathcal{Y}^\infty ) }{2 \deg_\mathbb C (Y) } \;- \sum_{ \substack{ \mu \in L^{ \diamond,\vee} / L^ \diamond \\ m_1 + m_2+ m_3=0 \\ m_3<0, m_1 > 0 } } c_f( m_3 , \mu ) \sum_{ \mu_1+\mu_2 =\mu } \frac{a(m_1,\mu_1)\rho_{\Lambda}(m_2,\mu_2)}{\Lambda(0,\chi)}. \end{align*} Combining with Proposition~\ref{prop:CM_value_beef} completes the proof. \end{proof}
Suppose now that $n^ \diamond\geq 3$. Denote by \[ \widehat{\bm{\omega}} \in \widehat{\mathrm{Pic}} (\mathcal{M}) ,\quad \widehat{\bm{\omega}}^ \diamond \in \widehat{\mathrm{Pic}} (\mathcal{M}^ \diamond ) \] the metrized tautological bundles on $\mathcal{M}$ and $\mathcal{M}^ \diamond$ of \S \ref{ss:line bundles}. By Theorem \ref{thm:borcherds}, after replacing $f$ by a multiple if necessary, we have the equality \begin{align}\label{eqn:Borcherds_beef} c_f(0,0) \cdot \widehat{\bm{\omega}}^ \diamond &= \widehat{ \mathcal{Z} }^ \diamond ( f) - c_f(0,0) \cdot (0, \log(4\pi e^\gamma) ) + \widehat{\mathcal{E}}^ \diamond(f),
\end{align} where $\widehat{\mathcal{E}}^ \diamond(f) = (\mathcal{E}^ \diamond(f),0)$ is empty if $L^ \diamond_{(2)}$ is self-dual and $n\geq 5$, and is otherwise supported on the union of special fibers $\mathcal{M}_{\mathbb F_p}$ for $p^2\nmid D_L$, as well as $\mathcal{M}_{\mathbb F_2}$ if $L_{(2)}$ is not self-dual.
\begin{theorem}\label{thm:taut degree} We have \[ \frac{ [ \widehat{\bm{\omega}} : \mathcal{Y} ] }{ \deg_\mathbb C ({Y}) } \approx_L
- \frac{ 2 \Lambda'(0,\chi) } { \Lambda( 0 , \chi ) } - d \cdot \log(4\pi e^\gamma) + \frac{1}{c_f(0,0)}\frac{[\widehat{\mathcal{E}}^ \diamond(f):\mathcal{Y}]}{\mathrm{deg}_{\mathbb C}({Y})}. \] \end{theorem}
\begin{proof} Combine Propositions~\ref{prop:height_pairing_beef} and \ref{prop:constant term eval} with~\eqref{eqn:Borcherds_beef}, and observe that the restriction of $\widehat{\bm{\omega}}^ \diamond$ to $\mathcal{M}$ is canonically isomorphic to $\widehat{\bm{\omega}}$; see Proposition~\ref{prop:functoriality}. \end{proof}
\section{Colmez's conjecture} \label{s:colmez}
In this section we prove Theorem \ref{bigthm:average colmez}, following the argument that was explained in \S~\ref{ss:intro colmez} of the introduction.
\subsection{The statement of the conjecture} \label{ss:colmez statement}
In this subsection only, $E$ is an arbitrary CM algebra. Recall that $\mathbb Q^{\mathrm{alg}}$ is the algebraic closure of $\mathbb Q$ in $\mathbb C$. The group $\Gamma_\mathbb Q=\mathrm{Gal}(\mathbb Q^{\mathrm{alg}}/\mathbb Q)$ acts on the set of all CM types of $E$ in the usual way: $\sigma\circ \Phi = \{ \sigma \circ \varphi : \varphi \in \Phi \}$. For each $\Phi$ let $\mathrm{Stab}(\Phi) \subset \Gamma_\mathbb Q$ be its stabilizer.
\begin{definition}\label{defn:CM_Colmez} Let $c\in \Gamma_\mathbb Q$ be complex conjugation. Write $\mathcal{CM}^0$ for the space of locally constant functions $a:\Gamma_\mathbb Q \to \mathbb Q$ that are constant on conjugacy classes and are such that the quantity \begin{equation}\label{complex_conj_condition} a(c\sigma) + a(\sigma) \end{equation} is independent of $\sigma\in\Gamma_\mathbb Q$. This notion does not depend on the choice of $c$. \end{definition}
Every function $a\in \mathcal{CM}^0$ decomposes uniquely as a finite linear combination \[ a = \sum_\eta a(\eta) \cdot \eta \] of Artin characters. For each Artin character $\eta$ let \[
L(s,\eta) = \prod_p \frac{1}{ \det\big( 1 - p^{-s} \eta(\mathrm{Fr}_\mathfrak{p})|_{U^{I_\mathfrak{p}}} \big) } \] be the usual Artin $L$-function, where $\mathfrak{p}$ is a prime of $\mathbb Q^\mathrm{alg}$ above $p$, and $U$ is the space of the representation $\eta$.
The independence from $\sigma$ of the quantity~\eqref{complex_conj_condition} implies that any nontrivial Artin character $\eta$ with $a(\eta) \not=0$ must be totally odd, in the sense that $\eta(c) = -\eta(\mathrm{id})$, and therefore $L(0 ,\eta) \not=0$. We now set: \[ \tilde{Z}(0,a) = - \sum_\eta a(\eta) \left( \frac{L'(0,\eta)}{L(0,\eta)} + \frac{ \log( f_\eta) }{2} \right) \] where $f_\eta$ is the Artin conductor of $\eta$.
Following Colmez, we will now construct a particular function $a^0_{(E,\Phi)}$ in $ \mathcal{CM}^0$ from the CM type $(E,\Phi)$. First, define a locally constant function on $\Gamma_\mathbb Q$ by the formula: \[
a_{( E,\Phi) } (\sigma) = | \Phi \cap \sigma\circ \Phi |. \] The average
\begin{equation}\label{Colmez function}
a^0_{( E,\Phi)} = \frac{1}{[ \Gamma_\mathbb Q : \mathrm{Stab}(\Phi) ] } \sum_{ \tau \in \Gamma_\mathbb Q/\mathrm{Stab}(\Phi) } a_{(E,\tau \circ \Phi )}
\end{equation} is constant on conjugacy classes of $\Gamma_\mathbb Q$, and depends only on the $\Gamma_\mathbb Q$-orbit of $\Phi$. Moreover, \[
a^0_{(E,\Phi)}(\sigma) + a^0_{(E,\Phi)}( c\sigma ) = |\Phi | \] is independent of $\sigma$, and so $a^0_{(E,\Phi)}(\sigma)$ belongs to $\mathcal{CM}^0$, as desired.
\begin{remark}\label{rem:lifting CM} If $E$ is a CM field, $\widetilde{E}$ is a CM field containing $E$, and \[
\widetilde{\Phi}= \{ \widetilde{ \varphi } \in \mathrm{Hom}( \widetilde{ E } , \mathbb Q^\mathrm{alg}) : \widetilde{ \varphi } |_E \in \Phi\} \] is the lifted CM type, then \[ [ \widetilde{ E } : E ] \cdot a^0_{( E,\Phi) } = a^0_{( \widetilde{ E } , \widetilde{\Phi})} . \] \end{remark}
\begin{definition} The \emph{Colmez height} of the pair $(E,\Phi)$ is \[ h^\mathrm{Col}_{ (E,\Phi) } = \tilde{Z}(0,a^0_{(E,\Phi)}). \] \end{definition}
Suppose $A$ is an abelian variety over $\mathbb Q^\mathrm{alg}$ of dimension $2\mathrm{dim}(A) = [E:\mathbb Q]$, and admitting complex multiplication of type $(E,\Phi)$. Choose a model of $A$ over a number field ${\bm{k}}\subset \mathbb Q^\mathrm{alg}$ large enough that the N\'eron model $\pi:\mathcal{A} \to \mathrm{Spec}(\mathcal O_{\bm{k}})$ has everywhere good reduction. Pick a nonzero rational section $s$ of the line bundle \[ \pi_*\Omega^{\mathrm{dim}(A)}_{ \mathcal{A}/\mathcal O_{\bm{k}} } \in \mathrm {Pic}(\mathcal O_{\bm{k}}), \]
and define \[ h^\mathrm{Falt} _\infty ( A, s ) = \frac{-1}{ 2 [{\bm{k}}:\mathbb Q] } \sum_{ \sigma : {\bm{k}} \to \mathbb C}
\log \big| \int_{ \mathcal{A}^\sigma(\mathbb C) } s^\sigma \wedge \overline{s^\sigma}\, \big|, \] and \[ h^\mathrm{Falt}_f(A,s) = \frac{1}{ [{\bm{k}}:\mathbb Q] } \sum_{ \mathfrak{p} \subset \mathcal O_{\bm{k}}} \mathrm{ord}_\mathfrak{p}(s) \cdot \log \mathrm{N}(\mathfrak{p}) . \]
\begin{definition} The \emph{Faltings height} of $A$ is \[ h^\mathrm{Falt}(A)= h^\mathrm{Falt}_f(A, s) + h^\mathrm{Falt} _\infty ( A,s) \] It is independent of the field ${\bm{k}}$, the choice of model of $A$ over ${\bm{k}}$, and the choice of section $s$. \end{definition}
\begin{theorem}[Colmez]\label{thm:colmez} If $A$ has complex multiplication by the maximal order $\mathcal O_E \subset E$, the Faltings height \[ h^\mathrm{Falt}_{ (E,\Phi) } \define h^\mathrm{Falt}(A) \]
depends only on the pair $(E,\Phi)$, and not on the choice of $A$. Moreover, there is a unique linear map
$
\mathrm{ht}:\mathcal{CM}^0 \to \mathbb R
$
such that, for any pair $(E,\Phi)$, we have
\[
h^\mathrm{Falt}_{ (E,\Phi) } = \mathrm{ht}(a^0_{(E,\Phi)}).
\] \end{theorem} \begin{proof} This is~\cite[Th\'eor\`eme 0.3]{Colmez}. \end{proof}
\begin{conjecture}[Colmez]\label{conj:colmez} For any $a\in\mathcal{CM}^0$, we have $\mathrm{ht}(a) = \widetilde{Z}(0,a).$ In particular, taking $a=a^0_{(E,\Phi)}$, for any CM pair $(E,\Phi)$ we have \[ h^\mathrm{Falt}_{ (E,\Phi) } = h^\mathrm{Col}_{ (E,\Phi) } . \] \end{conjecture}
\subsection{The reflex CM type}
For the remainder of \S \ref{s:colmez} we fix a CM field $E$ of degree $[E:\mathbb Q]=2d$, and a distinguished embedding $\iota_0 :E \to \mathbb C$. Denote by $F$ the maximal totally real subfield of $E$.
Recall from \S \ref{ss:reflex algebra} the total reflex algebra $E^\sharp$ associated with $E$: This is a finite \'etale $\mathbb Q$-algebra equipped with a canonical $\Gamma_\mathbb Q$-equivariant identification \[ \mathrm{Hom}_{\mathbb Q-\mathrm{alg}}(E^\sharp,\mathbb Q^\mathrm{alg}) = \mathrm{CM}(E), \] where $\mathrm{CM}(E)$ is the $\Gamma_\mathbb Q$-set consisting of all CM types of $E$.
The embedding $\iota_0$ determines a subset \[ \{ \mbox{CM types of $E$ containing $\iota_0$} \} \subset \mathrm{CM}(E). \] This corresponds to a subset $ \Phi^\sharp \subset \mathrm{Hom}(E^\sharp , \mathbb Q^\mathrm{alg}) = \mathrm{CM}(E), $ called the \emph{total reflex CM type}. The pair $(E^\sharp, \Phi^\sharp)$ is the \emph{total reflex pair}.
The relation between the total reflex pair and the classical notion of reflex pairs is given by the following proposition, which is immediate from the definitions.
\begin{proposition} There exist representatives $ \Phi_1,\ldots, \Phi_{m} \in \mathrm{CM}(E) $ for the $\Gamma_\mathbb Q$-orbits in $\mathrm{CM}(E)$ satisfying the following condition: If for each pair $(E,\Phi_i)$, $(E_i^\prime , \Phi_i^\prime)$ is its reflex CM pair, then there is an isomorphism of $\mathbb Q$-algebras $ E^\sharp \xrightarrow{\simeq} \prod_i E_i^\prime , $ such that the natural bijection \[ \mathrm{Hom}(E^\sharp , \mathbb Q^\mathrm{alg}) \xrightarrow{\simeq} \mathrm{Hom}( E_1^\prime ,\mathbb Q^\mathrm{alg})\sqcup \cdots \sqcup\mathrm{Hom}( E_m^\prime ,\mathbb Q^\mathrm{alg}) \] identifies $\Phi^\sharp = \Phi_1 ^\prime\sqcup \cdots \sqcup \Phi_m^\prime$. In particular, $E^\sharp$ is a CM algebra and $\Phi^\sharp$ is a CM type. \end{proposition}
\subsection{The average over CM types}
\begin{proposition}\label{prop:colmez height average} Recall the completed $L$-function (\ref{completed L}). The Colmez height satisfies \begin{align*}
\frac{1}{2^d} \sum_{ \Phi } h^\mathrm{Col}_{( E,\Phi) }
&= - \frac{ 1 }{2} \cdot \frac{ L'(0,\chi) }{ L(0,\chi) } - \frac{ 1 }{4} \cdot \log \left| \frac{ D_{E} }{ D_{F} }\right|
- \frac{ d }{2} \cdot \log(2\pi ) \\ & = - \frac{ 1 }{2} \cdot \frac{ \Lambda'(0,\chi ) }{\Lambda(0,\chi )} - \frac{ d }{4} \log(16\pi^3 e^\gamma), \end{align*} where the sum on the left hand side is over all CM types of $E$. \end{proposition}
\begin{proof} Recall that we have fixed an embedding $\iota_0:E\to \mathbb Q^\mathrm{alg}$. If we let $\Gamma_F\subset \Gamma_\mathbb Q$ be the subgroup that acts as the identity on $\iota_0(F) \subset \mathbb Q^\mathrm{alg}$, and view the nontrivial character $\chi: \mathrm{Gal}(E/F) \to \{\pm 1\}$ as a character of $\Gamma_F$, then \begin{equation}\label{eqn:cofu0 sharp} \frac{ 1}{ [ E:\mathbb Q] } \sum_{ \Phi } a^0_{( E,\Phi) } = 2^{d-2} \left( \bm{1} + \frac{1}{d} \mathrm{Ind}_{\Gamma_F}^{\Gamma_\mathbb Q}( \chi ) \right) \end{equation} where $\bm{1}$ is the trivial character on $\Gamma_\mathbb Q$. Indeed, if we normalize the Haar measure on $\Gamma_\mathbb Q$ to have total volume $1$, and define a function $\psi :\Gamma_\mathbb Q \to \mathbb Z$ by \[ \psi(\sigma) = \begin{cases} 2^{d-1} & \mbox{if }\sigma\circ \iota_0 = \iota_0 \\ 0 & \mbox{if }\sigma \circ \iota_0= \overline{\iota}_0 \\ 2^{d-2} & \mbox{otherwise}, \end{cases} \] then an elementary calculation shows that the values of both sides of (\ref{eqn:cofu0 sharp}) at $\sigma\in \Gamma_\mathbb Q$ are equal to \[ \int_{\Gamma_\mathbb Q} \psi(\tau^{-1} \sigma \tau) \, d\tau. \]
Using this, the first equality in the proposition follows from the calculation \begin{align*} \frac{1}{2^d} \sum_{ \Phi } h^\mathrm{Col}_{( E,\Phi) } &= -\frac{1}{2}\left[d\cdot\frac{\zeta'(0)}{\zeta(0)} + \frac{L'(0,\chi)}{L(0,\chi)} + \frac{1}{2}\log\left(f_{\mathrm{Ind}_{\Gamma_F}^{\Gamma_\mathbb Q}(\chi)}\right)\right]\\
& = -\frac{d}{2}\cdot\log(2\pi) - \frac{1}{2}\frac{L'(0,\chi)}{L(0,\chi)} - \frac{1}{4}\cdot \log\bigl(|D_F|\cdot N_{E/\mathbb Q}(\mathfrak{d}_{E/F})\bigr)\\
& = - \frac{ 1 }{2} \cdot \frac{ L'(0,\chi) }{ L(0,\chi) } - \frac{ 1 }{4}\cdot \log \left| \frac{ D_{E} }{ D_{F} }\right|
- \frac{ d }{2} \cdot \log(2\pi ) , \end{align*} and the second equality follows from \eqref{eqn:completed log der}. \end{proof}
\begin{proposition}\label{prop:average reflex} The total reflex pair $(E^\sharp,\Phi^\sharp)$ satisfies \[
a^0_{( E^\sharp ,\Phi^\sharp) } = \frac{ 1}{ [ E:\mathbb Q] } \sum_{ \Phi } a^0_{( E,\Phi) } , \] where the sum is over all CM types of $E$. \end{proposition}
\begin{proof} Let $\widetilde{E} \subset \mathbb Q^\mathrm{alg}$ be a finite Galois extension of $\mathbb Q$ large enough to contain all embeddings $E \to \mathbb Q^\mathrm{alg}$. In particular, each $E_i^\prime \subset \widetilde{E}$. Use $\iota_0$ to regard $E$ as a subfield of $\widetilde{E}$. For each $1\le i \le k$ let \[ \widetilde{\Phi}_i , \widetilde{\Phi}^\prime_i \subset \mathrm{Gal}(\widetilde{E}/\mathbb Q) = \mathrm{Hom}( \widetilde{E} ,\mathbb Q^\mathrm{alg}) \]
be the lifts of $\Phi_i$ and $\Phi^\prime_i$, respectively, so that $\sigma \in \widetilde{\Phi}_i$ if and only if
$\sigma^{-1} \in \widetilde{\Phi}^\prime_i$. An easy exercise shows that
$a^0_{( \widetilde{E} , \widetilde{\Phi}^\prime_i)} = a^0_{( \widetilde{E} , \widetilde{\Phi}_i)}$, and hence
Remark \ref{rem:lifting CM} implies
\[
[ \widetilde{E} : E_i^\prime ] \cdot a^0_{( E_i^\prime,\Phi_i^\prime) } = [ \widetilde{E} : E ] \cdot a^0_{( E,\Phi_i)}.
\]
It follows that \[
a^0_{( E_i^\prime,\Phi_i^\prime) } = \frac{ [ E_i^\prime : \mathbb Q ] }{ [ E:\mathbb Q] } \cdot a^0_{( E,\Phi_i)} = \frac{ 1}{ [ E:\mathbb Q] } \cdot \sum_{ \tau \in \Gamma_\mathbb Q / \mathrm{Stab}(\Phi_i) }a^0_{( E, \tau\circ \Phi_i)}, \] and summing over $i$ proves the claim. \end{proof}
\begin{corollary}\label{Cor:total_reflex_height} The total reflex pair $(E^\sharp,\Phi^\sharp)$ satisfies \begin{align*} h^\mathrm{Falt}_{ ( E^\sharp,\Phi^\sharp ) } & = \frac{ 1}{ [ E:\mathbb Q] } \sum_{ \Phi\in\mathrm{CM}(E) } h^\mathrm{Falt}_{( E,\Phi) }.
\end{align*} \end{corollary} \begin{proof} Combine Theorem~\ref{thm:colmez} and Proposition~\ref{prop:average reflex}. \end{proof}
\subsection{Faltings heights and Arakelov heights}
Recall the torus $T = T_E/T_F^1$ and the arithmetic curve \[ \mathcal{Y}_0\to \mathrm{Spec}(\mathcal O_E) \] from \S \ref{ss:integral_model_Y} defined by the compact open subgroup $K_{0}\subset T(\mathbb A_f)$. In~\S\ref{ss:sheaves_i} and~\S\ref{ss:sheaves_ii}, given an algebraic representation $N$ of the torus $T$, and a $K_{0}$-stable lattice $N_{\widehat{\mathbb Z}}\subset N_{\mathbb A_f}$, we constructed various homological realizations $\bm{N}_?$ over $\mathcal{Y}_0$, functorially associated with the pair $(N,N_{\widehat{\mathbb Z}})$.
Let $H^\sharp$ be as in Proposition~\ref{prop:h sharp repn}. The subring $\mathcal O_{E^\sharp}\subset E^\sharp$ gives us a lattice $H^\sharp_{\mathbb Z}\subset H^\sharp$ stable under the multiplication action of $\mathcal O_{E^\sharp}$. The associated $\widehat{\mathbb Z}$-lattice $H^\sharp_{\widehat{\mathbb Z}}$. Therefore, from the pair $(H^\sharp,H_{\widehat{\mathbb Z}}^\sharp)$, we obtain an abelian scheme $A^\sharp\to\mathcal{Y}_0$, whose homological realizations are the sheaves associated with the pair. By construction, at any point $y\in Y_0(\mathbb C)$, $A^\sharp_y$ is an abelian variety with CM by $\mathcal O_{E^\sharp}$ and of CM type $\Phi^\sharp$.
Define
\[
\bm{\Omega}^\sharp = \pi_* \Omega_{A^\sharp/ \mathcal{Y}_0}^{\mathrm{dim}(A^\sharp)}.
\] At any complex point $y\in \mathcal{Y}^\infty_0(\mathbb C)$ we endow the fiber \[ \bm{\Omega}^\sharp_y = H^0( A^\sharp_y , \Omega^{\mathrm{dim}(A^\sharp)}_{A^\sharp_y/\mathbb C} ) \] with the \emph{Faltings metric} \[
|| s ||^2 = \Big| \int_{ A^\sharp_y(\mathbb C) } s \wedge \overline{s} \, \Big|, \] and so obtain the \emph{metrized Hodge bundle} \[
\widehat{ \bm{\Omega}}^\sharp \in \widehat{\mathrm{Pic}}( \mathcal{Y}_0 ). \]
The Betti realization of $A^\sharp$ gives us a local system \[ \bm{H}^\sharp_B \subset \bm{H}^\sharp_B \otimes \mathcal O_{Y_0(\mathbb C)} = \bm{H}^\sharp_{dR, Y_0(\mathbb C)}, \] which determines a local system of $\mathbb Z$-modules $ \det(\bm{H}^\sharp_B) \subset \det( \bm{H}^\sharp_{dR, Y_0(\mathbb C)} ) $ of rank $1$.
We define the \emph{volume metric} on $\det( \bm{H}^\sharp_{\mathrm{dR}} )$ by declaring that $||e||^2 = 1$ for any local generator $e$ of $\det(\bm{H}^\sharp_B)$. At any complex point $y\in Y_0(\mathbb C)$ the dual volume metric on \[ \det(\bm{H}^\sharp_{dR,y}) ^\vee \xrightarrow{\simeq} H^{2 \mathrm{dim}(A^\sharp)}_{\mathrm{dR}}(A^\sharp_y / \mathbb C) \] is just integration of top degree $C^\infty$ forms: \[
|| \eta || = \big| \int_{A_y(\mathbb C) } \eta \, \big|. \] This gives a second metrized line bundle \[ \widehat{\det}(\bm{H}^\sharp_{\mathrm{dR}}) \in \widehat{\mathrm{Pic}}( \mathcal{Y}_0 ). \]
We will need a third metrized line bundle $\widehat{\bm{\omega}}_0$. This will be defined as follows. Consider the representation $V_0 = V(E,c)$ of $T_E$ on the space of $E$-semilinear endomorphisms of $E$. This representation factors through $T$ (and in fact through $T_{so} = T_E/T_F$) and has a natural lattice $L_0 = V(\mathcal O_E,c)$ such that $\widehat{L}_0 = L_{0,\widehat{\mathbb Z}}$ is stable under $K_0$. The natural $E$-linear structure on $V_0$ is invariant under $T$. Therefore, from the pair $(V_0,\widehat{L}_0)$, we obtain a de Rham realization $\bm{V}_{0,\mathrm{dR}}$ over $\mathcal{Y}_0$, equipped with an action of $\mathcal O_E$, making it a locally free sheaf of rank $1$ over $\mathcal O_{\mathcal{Y}_0}\otimes_{\mathbb Z}\mathcal O_E$. This realization is equipped with a canonical $\mathcal O_E$-stable filtration $\mathrm{Fil}^\bullet\bm{V}_{0,\mathrm{dR}}$ by local direct summands extending the one over $Y_0 = \mathcal{Y}_{0,\mathbb Q}$ obtained from Proposition~\ref{prop:zero dim derham}. Moreover, the degree $1$ summand \[ \bm{\omega}_0 \define \mathrm{Fil}^1\bm{V}_{0,\mathrm{dR}} \]
is a line bundle over $\mathcal{Y}_0$.
Composition in $\mathrm{End}(E)$ induces a canonical, $T$-invariant Hermitian form $\langle \cdot,\cdot\rangle_0$ on $V_0$ determined by the property \[ (x\circ y)(a) = \langle x,y\rangle_0\cdot a, \] for any $x,y\in V_0$ and $a\in E$. From this, we obtain a $\mathbb Q$-valued quadratic form \[ \mathcal{Q}_0 = \mathrm{Tr}_{F/\mathbb Q}(\langle x,x\rangle_0) \] with associated bilinear form $[x,y]_0$ on $V_0$.
Just as in \S~\ref{ss:line bundles}, for every $y\in \mathcal{Y}^\infty(\mathbb C)$, this form equips $\bm{\omega}_{0,y} = \mathrm{Fil}^1\bm{V}_{0,\mathrm{dR},y}$ with the Hermitian form $||z||^2_0 = -[z,\overline{z}]_0$, and thus equips $\bm{\omega}_0$ with the structure of a metrized line bundle, which will denote by $\widehat{\bm{\omega}}_0$.
There is a natural $T$-equivariant embedding \begin{equation}\label{eqn:V Esharp emb} V_0\hookrightarrow \mathrm{End}(H^\sharp) \end{equation} defined as follows: We have \[ H^\sharp = E^\sharp = \left(\bigotimes_{\iota\in \mathrm{Emb}(F)}\mathbb Q^{\mathrm{alg}}\otimes_{\iota,F}E\right)^{\Gamma_\mathbb Q}. \] Here, the action of $\Gamma_\mathbb Q$ on the tensor product is the obvious one compatible with permutation of the indexing set $\mathrm{Emb}(F)$.\footnote{In other words, $E^\sharp$ is the \emph{tensor induction} of the $F$-algebra $E$ to an algebra over $\mathbb Q$.}
For each $\iota\in \mathrm{Emb}(F)$, we have an embedding \[ \mathbb Q^\mathrm{alg}\otimes_{\iota,F}V_0 = V\bigl(\mathbb Q^\mathrm{alg}\otimes_{\iota,F}E,c\bigr)\subset \mathrm{End}(\mathbb Q^\mathrm{alg}\otimes_{\iota,F}E). \] The Lie algebra tensor product of these embeddings gives us a $\Gamma_\mathbb Q$-equivariant embedding \[ \mathbb Q^\mathrm{alg}\otimes_{\mathbb Q}V_0 = \bigoplus_{\iota}\bigl(\mathbb Q^\mathrm{alg}\otimes_{\iota,F}V_0\bigr) \hookrightarrow \mathrm{End}\left(\bigotimes_{\iota}\mathbb Q^{\mathrm{alg}}\otimes_{\iota,F}E\right), \] so that $x\in V_0$ acts on $\mathbb Q^\mathrm{alg}\otimes_\mathbb Q E^\sharp$ via: \[ x(a_0\otimes a_1\otimes\cdots \otimes a_{d-1}) = \sum_{i=0}^{d-1}a_0\otimes \cdots\otimes a_{i-1}\otimes x(a_i)\otimes\cdots a_{d-1}. \] Here, $\iota_0,\iota_1,\ldots,\iota_{d-1}:F\hookrightarrow \mathbb R$ are the real embeddings of $F$, and for each $i$, $a_i\in \mathbb Q^{\mathrm{alg}}\otimes_{\iota_i,F}E$.
The descent of this action over $\mathbb Q$ gives us~\eqref{eqn:V Esharp emb}.
Now, it is clear that this embedding induces a $K_0$-stable inclusion $\widehat{L}_0\hookrightarrow\mathrm{End}(H^\sharp_{\widehat{\mathbb Z}})$, and thus gives us a map of de Rham realizations \begin{equation*} \bm{V}_{0,\mathrm{dR}}\to \mathrm{End}(\bm{H}_{\mathrm{dR}}^\sharp) \end{equation*} allowing us to view sections of $\bm{V}_{0,\mathrm{dR}}$ as endomorphisms of $\bm{H}_{\mathrm{dR}}^\sharp$.
The action of $\bm{\omega}_0$ on $\bm{H}_{\mathrm{dR}}^\sharp$ induces a map \[ \bm{\omega}_0\otimes_{\mathcal O_{\mathcal{Y}_0}}\mathrm{gr}^{-1}_{\mathrm{Fil}}\bm{H}^\sharp_{\mathrm{dR}} \to \mathrm{Fil}^0\bm{H}^\sharp_{\mathrm{dR}} \] of vector bundles over $\mathcal{Y}_0$, and taking determinants yields a map \begin{equation}\label{eqn:det map} \bm{\omega}_0^{\otimes 2^{d-1}}\otimes_{\mathcal O_{\mathcal{Y}_0}}\det\bigl(\mathrm{gr}^{-1}_{\mathrm{Fil}}\bm{H}_{\mathrm{dR}}^\sharp\bigr)\to \det\bigl(\mathrm{Fil}^0\bm{H}^\sharp_{\mathrm{dR}}\bigr) \end{equation} of line bundles over $\mathcal O_{\mathcal{Y}_0}$. Set \begin{equation}\label{eqn:diff line bundles} \bm{\mathscr{L}} = \det\bigl(\mathrm{Fil}^0\bm{H}^\sharp_{\mathrm{dR}}\bigr)\otimes \bm{\omega}_0^{\otimes - 2^{d-1}}\otimes_{\mathcal O_{\mathcal{Y}_0}}\det\bigl(\mathrm{gr}^{-1}_{\mathrm{Fil}}\bm{H}_{\mathrm{dR}}^\sharp\bigr)^{\otimes -1}. \end{equation} Then~\eqref{eqn:det map} gives us a canonical section of $\bm{\mathscr{L}}$ over $\mathcal{Y}_0$, and thus an effective divisor $\bm{\Delta}$ on $\mathcal{Y}_0$. Write $\widehat{\bm{\Delta}} = (\bm{\Delta},0)$ for the associated arithmetic divisor.
\begin{proposition} \label{prop:delta degree} We have \[
\frac{\widehat{\deg}(\widehat{\bm{\Delta}})}{\deg_\mathbb C (Y_0)} = 2^{d-1}\log |D_F|. \] \end{proposition}
This is the key technical result of this subsection, and its proof will be given further below. For now, we deduce from it the following theorem, which gives the precise relation between the degree of $\widehat{\omega}_0$, and the average of the Faltings heights of abelian varieties with CM by $\mathcal O_E$.
\begin{theorem}\label{thm:Faltings height line bundles} We have the identity \[
\frac{1}{2^{d}} \sum_\Phi h^\mathrm{Falt}_{(E,\Phi)} = \frac{1}{4} \frac{ \widehat{\deg}(\widehat{\bm{\omega}}_0) }{ \deg_\mathbb C (Y_0) } + \frac{1}{4}\log|D_F| - \frac{1}{2} d\cdot\log(2\pi) . \] \end{theorem}
\begin{proof}
By Corollary~\ref{Cor:total_reflex_height}, we have
\[
\sum_\Phi h^\mathrm{Falt}_{(E,\Phi)} = 2d\cdot h^\mathrm{Falt}_{(E^\sharp,\Phi^\sharp)}.
\]
Observing that, for every $y\in Y_0(\mathbb C)$, the abelian variety $A^\sharp_y$ has CM by $\mathcal O_{E^\sharp}$ with CM type $\Phi^\sharp$, and using Theorem~\ref{thm:colmez}, we obtain
\begin{equation}\label{eqn:omega sharp degree}
\frac{ \widehat{\deg}\bigl(\widehat{\bm{\Omega}}^\sharp\bigr) }{ \deg_\mathbb C(Y_0) } = 2d\cdot h^\mathrm{Falt}_{(E^\sharp,\Phi^\sharp)} = \sum_\Phi h^\mathrm{Falt}_{(E,\Phi)}.
\end{equation}
Consider the short exact sequence \[ 0\to \mathrm{Fil}^0\bm{H}^\sharp_{\mathrm{dR}}\to \bm{H}^\sharp_{\mathrm{dR}}\to \mathrm{gr}^{-1}_{\mathrm{Fil}}\bm{H}^\sharp_{\mathrm{dR}}\to 0 \] of vector bundles over $\mathcal{Y}_0$. Taking determinants, we obtain an isomorphism \begin{equation}\label{eqn:line bundles isomorphism} \det(\bm{H}^\sharp_{\mathrm{dR}})\xrightarrow{\simeq}\bm{\mathscr{L}}\otimes\bm{\omega}_0^{\otimes 2^{d-1}}\otimes\det(\mathrm{gr}^{-1}_{\mathrm{Fil}}\bm{H}^\sharp_{\mathrm{dR}})^{\otimes 2}, \end{equation} where $\mathscr{L}$ is as in~\eqref{eqn:diff line bundles}. If $\widehat{\bm{\Delta}}$ is as in Proposition~\ref{prop:delta degree}, then, using the canonical isomorphism \[ \det(\mathrm{gr}^{-1}_{\mathrm{Fil}}\bm{H}_{\mathrm{dR}}^\sharp)^{\otimes -1}\xrightarrow{\simeq}\bm{\Omega}^\sharp, \] it is easy to check that~\eqref{eqn:line bundles isomorphism} gives us an identity \[ \widehat{\det}(\bm{H}^\sharp_{\mathrm{dR}}) = \widehat{\bm{\Delta}} + 2^{d-1}\widehat{\bm{\omega}}_0 - 2\cdot\widehat{\bm{\Omega}}^{\sharp} \] in $\widehat{\mathrm {Pic}}(\mathcal{Y}_0)$.
Combining this with Proposition~\ref{prop:delta degree} and~\eqref{eqn:omega sharp degree} shows
\[
\frac{1}{2^{d}} \sum_\Phi h^\mathrm{Falt}_{(E,\Phi)} = \frac{1}{4}\frac{\widehat{\deg}(\widehat{\bm{\omega}}_0)}{ \deg_\mathbb C(Y_0)} +\frac{1}{4}\cdot\log|D_F| - \frac{1}{2^{d+1}}\frac{\widehat{\deg}\bigl(\widehat{\det}(\bm{H}^\sharp_{\mathrm{dR}})\bigr)}{\deg_{\mathbb C}(Y_0)}.
\]
Therefore, we will be done once we verify the identity \[ \frac{\widehat{\deg}\bigl(\widehat{\det}(\bm{H}^\sharp_{\mathrm{dR}})\bigr)}{\deg_{\mathbb C}(Y_0)} = 2^{d} d \cdot \log(2\pi). \]
But this is easily done using Lemma~\ref{lem:de Rham det degree} below. \end{proof}
\begin{lemma} \label{lem:de Rham det degree} Let $E'$ be a number field and let $A$ be an abelian scheme over $\mathcal O_{E'}$. Suppose that the top degree cohomology $H^{2d}_{\mathrm{dR}}(A/\mathcal O_{E'})$ of $A$ is a free module of rank $1$ over $\mathcal O_{E'}$. Fix an embedding $E'\hookrightarrow \mathbb C$, and an $\mathcal O_{E'}$-module generator $e\in H^{2d}_{\mathrm{dR}}(A/\mathcal O_{E'})$, and let $\eta(e)$ be a degree $2d$ $C^\infty$ form on $A(\mathbb C)$ that represents this generator over $\mathbb C$. We then have: \[ \left\vert\int_{A(\mathbb C) } \eta(e)\;\right\vert = (2\pi)^{-\dim(A)}. \] \end{lemma} \begin{proof} As explained in~\cite[Ch. I, \S 1, p. 22]{dmos}, there is a canonical $\mathcal O_{E'}$-linear trace map \[ \mathrm{Tr}_{\mathrm{dR}}: H^{2d}_{\mathrm{dR}}(A/\mathcal O_{E'}) \to E', \] which, over $\mathbb C$, corresponds to the linear functional \[ \eta\mapsto \frac{1}{(2\pi i)^{\dim A}}\int_{A(\mathbb C)}\eta \] on top degree $C^\infty$ forms on $A(\mathbb C)$.
So, to prove the lemma, it is enough to show that $\mathrm{Tr}_{\mathrm{dR}}$ maps isomorphically onto $\mathcal O_{E'}\subset E'$. Indeed, this would imply that \[ \int_{A(\mathbb C) } \eta(e) \in (2\pi i)^{-\dim(A)}\mathcal O_{E'}^\times. \]
For this, note that $\mathrm{Tr}_{\mathrm{dR}}$ is equal to the composition: \[ H^{2d}_{\mathrm{dR}}(A/\mathcal O_{E'}) \xrightarrow{\simeq} H^d(A,\Omega^d_{A/\mathcal O_{E'}}) \xrightarrow[\simeq]{\mathrm{Tr}}\mathcal O_{E'}, \] where the first isomorphism arises from the degeneration of the Hodge-to-de Rham spectral sequence for $A$, and the second is the trace isomorphism from Grothendieck-Serre duality. \end{proof}
We now begin our preparations for the proof of Proposition~\ref{prop:delta degree}. Suppose that we have inclusions of complete discrete valuation rings $A\subset B\subset C$ with perfect residue fields, with $\mathrm{Frac}(B)$ finite over $\mathrm{Frac}(A)$. Suppose that the set $\mathrm{Hom}(B,C)$ of local $A$-algebra homomorphisms has the maximum possible size $[\mathrm{Frac}(B):\mathrm{Frac}(A)]$.\footnote{In other words, the \'etale $\mathrm{Frac}(A)$-algebra $\mathrm{Frac}(B)$ splits over $\mathrm{Frac}(C)$.}
Fix a subset $\Upsilon\subset \mathrm{Hom}(B,C)$, and consider the map of $C$-algebras: \begin{align*} \varphi_{\Upsilon}:C\otimes_AB &\to \prod_{\sigma\in \Upsilon}C\\ c\otimes b &\mapsto (c\cdot\sigma(b))_{\sigma}. \end{align*}
Set $\mathcal{K}(\Upsilon) = \ker \varphi_{\Upsilon}$. If $\Upsilon^c = \mathrm{Hom}(B,C)\backslash \Upsilon$, then the inclusion \[ \mathcal{K}(\Upsilon) + \mathcal{K}(\Upsilon^c)\hookrightarrow C\otimes_AB \] of $C$-modules is an isomorphism after tensoring with $\mathrm{Frac}(C)$. Therefore, its cokernel has finite length as a $C$-module. Denote this cokernel by $\mathcal{C}(\Upsilon)$. Fix a uniformizer $\pi_B\in B$. Let $B_0\subset B$ be the maximal \'etale $A$-subalgebra. Let $\mathfrak{d}_{B/A}\subset B$ be the different, and let $\mathfrak{D}_{B/A} = \mathrm{Nm}_{B/A}(\mathfrak{d}_{B/A})\subset A$ be the discriminant ideal for $B$ over $A$.
\begin{lemma} We have: \label{lem:length C Upsilon} \begin{align*} \mathrm{length}_C(\mathcal{C}(\Upsilon)) &= \frac{1}{2}\cdot \sum_{\stackrel{\tau,\tau'\in\Upsilon}{\tau\neq \tau'}}\mathrm{length}(C/(\tau(\pi_B) - \tau'(\pi_B))) \\ & + \frac{1}{2} \cdot \sum_{\stackrel{\sigma,\sigma'\in\Upsilon^c}{\sigma\neq \sigma'}}\mathrm{length}(C/(\sigma(\pi_B) - \sigma'(\pi_B))) - \frac{1}{2}\cdot \mathrm{length}(C/\mathfrak{D}_{B/A}C).\nonumber \end{align*} \end{lemma} \begin{proof} By a standard reduction, we can assume that $A=B_0$, so that $B$ is totally ramified over $A$. First consider the cokernel $\mathcal{C}_1(\Upsilon)$ of the natural embedding \[ \mathcal{K}(\Upsilon) \xrightarrow{x\mapsto (\sigma(x))_{\sigma}} \prod_{\sigma\in\Upsilon^c}C. \] We claim that \begin{equation} \label{eqn:length C1} n_1(\Upsilon) \define \mathrm{length}_C(\mathcal{C}_1(\Upsilon)) = \frac{1}{2}\cdot\sum_{\stackrel{\tau,\tau'\in\Upsilon}{\tau\neq \tau'}}\mathrm{length}(C/(\tau(\pi_B) - \tau'(\pi_B))). \end{equation} This can be verified using induction on the size of $\Upsilon$, after proving (via a separate inductive argument) that $\mathcal{K}(\Upsilon) \subset C\otimes_BA$ is the principal ideal generated by the element \[ f_{\Upsilon} = \prod_{\tau\in\Upsilon}(1\otimes \pi_B - \tau(\pi_B)\otimes 1) \in C\otimes_BA. \]
We also claim that the inclusion \[ C\otimes_AB \hookrightarrow \prod_{\sigma\in\mathrm{Hom}(B,C)}C \] has cokernel of length $\frac{1}{2}\cdot\mathrm{length}(C/\mathfrak{D}_{B/A}C)$. This follows by observing that $\mathfrak{d}_{B/A}^{-1}$ is the dual lattice to $B$ under the canonical non-degenerate trace pairing $(x,y)\mapsto \mathrm{Tr}_{B/A}(x,y)$ on $\mathrm{Frac}(B)$, and that $\prod_{\sigma\in\mathrm{Hom}(B,C)}C$ is a self-dual lattice in $\mathrm{Frac}(C)\otimes_AB$ under the induced $C$-bilinear pairing.
The lemma now follows by noting that \[ \mathrm{length}_C(\mathcal{C}(\Upsilon)) = n_1(\Upsilon) + n_1(\Upsilon^c) - \mathrm{length}_C\left(\frac{\prod_{\sigma\in\mathrm{Hom}(B,C)}C}{C\otimes_AB}\right). \] \end{proof}
Let $K$ be a finite \'etale $\mathbb Q_p$-algebra, and let $P\subset \mathrm{Frac}(W)^\mathrm{alg}$ be a finite Galois extension of $\mathrm{Frac}(W)$ that receives all maps $\eta:K\hookrightarrow \mathbb Q_p^\mathrm{alg}$. Let $\mathcal{C}(\Gamma_{\mathbb Q_p},\mathbb C)$ (resp. $\mathcal{C}^0(\Gamma_{\mathbb Q_p},\mathbb C)$) be the space of continuous (resp. continuous, conjugation-invariant) $\mathbb C$-valued functions on $\Gamma_{\mathbb Q_p}$.
$\mathcal{C}^0(\Gamma_{\mathbb Q_p},\mathbb C)$ has a basis given by characters of irreducible finite dimensional complex representations of $\Gamma_{\mathbb Q_p}$. There is a unique linear functional \[ \mu_p:\; \mathcal{C}^0(\Gamma_{\mathbb Q_p},\mathbb C)\to \mathbb C, \] which associates with every finite dimensional irreducible character $\chi$ the integer $\mu_p(\chi) = \log_pf_p(\chi)$, where $f_p(\chi)$ is the Artin conductor of $\chi$ and $\log_p$ is the base-$p$ logarithm.
Since $\Gamma_{\mathbb Q_p}$ is compact, averaging with respect to the Haar measure of measure $1$ gives us a canonical section $f\mapsto f^0$ of the inclusion \[ \mathcal{C}^0(\Gamma_{\mathbb Q_p},\mathbb C)\hookrightarrow \mathcal{C}(\Gamma_{\mathbb Q_p},\mathbb C), \] and so permits us to lift $\mu_p$ to a measure on $\Gamma_{\mathbb Q_p}$: $\mu_p(f) \define \mu_p(f^0)$.
With any subset $\Upsilon\subset \mathrm{Hom}(K,P)$, we can associate the function \begin{align*} a_{(K,\Upsilon)}:\Gamma_{\mathbb Q_p} &\to \mathbb Z\\
\sigma&\mapsto | \Upsilon \cap \sigma\circ\Upsilon |. \end{align*}
Let $M$ be a finite free $\mathcal O_P\otimes_{\mathbb Z_p}\mathcal O_K$-module of rank $1$. For $\Upsilon\subset \mathrm{Hom}(K,P)$, set \[ \mathcal{K}(\Upsilon) = \ker\left(\mathcal O_P\otimes_{\mathbb Z_p}\mathcal O_K \xrightarrow{x\otimes y\mapsto (x\eta(y))_{\eta}}\to \prod_{\eta\in\Upsilon}\mathcal O_P\right), \] and set $\mathcal{K}(M,\Upsilon) = \mathcal{K}(\Upsilon)\cdot M$. Let $\mathcal{C}(M,\Upsilon)$ be the cokernel of the inclusion \[ \mathcal{K}(M,\Upsilon) + \mathcal{K}(M,\Upsilon^c)\hookrightarrow M \] of $\mathcal O_P$-modules.
It will be useful later to have another description of this cokernel. Set \[ \mathcal{Q}(M,\Upsilon) = \mathrm{coker}(\mathcal{K}(M,\Upsilon^c)\hookrightarrow M). \] Then $\mathcal{C}(M,\Upsilon)$ is also the cokernel of the natural inclusion \[ \mathcal{K}(M,\Upsilon)\hookrightarrow \mathcal{Q}(M,\Upsilon). \]
\begin{proposition} \label{prop:colmez calc} Let $e_P$ be the absolute ramification index of $P$. Then \[ \mathrm{length}_{\mathcal O_P}\mathcal{C}(M,{\Upsilon}) = - \frac{1}{2}\cdot e_P\cdot\left( \mu_p(a_{(K,\Upsilon)}) + \mu_p(a_{(K,\Upsilon^c)}) \right). \] \end{proposition} \begin{proof} If $K = \prod_iK_i$ is the decomposition of $K$ into a product of field extensions of $\mathbb Q_p$, and \[ \Upsilon_i = \Upsilon \cap \mathrm{Hom}(K_i,\mathbb Q_p^\mathrm{alg}), \] for each $i$, then we have $a^0_{(K,\Upsilon)} = \sum_i a^0_{(K_i,\Upsilon_i)}$. Moreover, if $M_i = M\otimes_{\mathcal O_K}\mathcal O_{K_i}$, then we have \[ \mathcal{C}(M,\Upsilon) = \bigoplus_i \mathcal{C}(M_i,\Upsilon_i). \]
Therefore, without loss of generality, we can assume that $K$ is a field. To compute the right hand side of the asserted identity, for each pair $\eta,\eta'\in \mathrm{Hom}(K,P)$, consider the function $a_{\eta,\eta'}\in \Gamma_{\mathbb Q_p}$ given by \begin{align*} a_{\eta,\eta'}(\sigma)&= \begin{cases} 1,&\text{ if $\sigma(\eta) = \eta'$};\\ 0,&\text{ otherwise.} \end{cases} \end{align*}
Fix a uniformizer $\pi_K$ for $K$. Let $K_0\subset K$ be the maximal unramified subextension. By Lemme I.2.4 of~\cite{Colmez} and the remark following Prop. I.2.6 of \emph{loc. cit.}, we have \begin{equation}\label{eqn:colmez calc} \mu_p(a_{\eta,\eta'}) = \begin{cases} \frac{1}{e_P}\mathrm{ord}_{\mathcal O_P}(\eta(\mathfrak{d}_{K/\mathbb Q_p})\mathcal O_P),&\text{ if $\eta = \eta'$};\\ -\frac{1}{e_P}\mathrm{ord}_{\mathcal O_P}(\eta(\pi_K) - \eta'(\pi_K)),&\text{ if $\eta\vert_{K_0} = \eta'\vert_{K_0}$ and $\eta\neq \eta'$};\\ 0,&\text{ otherwise.} \end{cases} \end{equation} Moreover, the following identity is easily verified: \begin{equation}\label{eqn:colmez rhs} a_{(K,\Upsilon)} + a_{(K,\Upsilon^c)}= \sum_{\stackrel{\eta,\eta'\in \Upsilon}{\eta\neq \eta'}}a_{\eta,\eta'} + \sum_{\stackrel{\eta,\eta'\in \Upsilon^c}{\eta\neq \eta'}}a_{\eta,\eta'} + \sum_{\eta:K\to P}a_{\eta,\eta}. \end{equation}
Now, observe that we are in the situation of Lemma~\ref{lem:length C Upsilon}, with $A = \mathbb Z_p$, $B = \mathcal O_K$ and $C = \mathcal O_P$, and the computation there gives us an explicit formula for the left hand side of the desired identity. Comparing this with~\eqref{eqn:colmez rhs} and~\eqref{eqn:colmez calc} completes the proof of the Proposition. \end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:delta degree}] Fix a prime $\mathfrak{q}\subset \mathcal O_E$ above a rational prime $p$, and also a point $y\in \mathcal{Y}_0(\mathbb F_{\mathfrak{q}}^\mathrm{alg})$. Let $\mathcal O_y$ be the completed \'etale local ring of $\mathcal{Y}_0$ at $y$. Set $W = W(\mathbb F_{\mathfrak{q}}^\mathrm{alg})$. Fix an algebraic closure $\mathrm{Frac}(W)^\mathrm{alg}$ of $\mathrm{Frac}(W)$, and an embedding $\mathbb Q^\mathrm{alg}\hookrightarrow\mathrm{Frac}(W)^\mathrm{alg}$ inducing the place $\mathfrak{q}$ on $E\subset \mathbb Q^\mathrm{alg}$, embedded via $\iota_0$. This identifies $\mathcal O_y$ with the ring of integers in the extension of $\mathrm{Frac}(W)$ generated by the image of $E$.
Restricting the line bundle $\mathscr{L}$ over $\mathrm{Spec}~\mathcal O_y$ gives us a free $\mathcal O_y$-module $\mathscr{L}_y$ of rank $1$, equipped with a canonical section $s_y:\mathcal O_y\to \mathscr{L}_y$. We claim that we have \begin{equation} \label{eqn:main length comp} \mathrm{length}(\mathscr{L}_y/\mathrm{im}(s_y)) = 2^{d-2}\cdot\mathrm{ord}_{\mathfrak{q}}(\mathfrak{d}_{F/\mathbb Q}). \end{equation}
Assuming this for all $\mathfrak{q}$ and $y$, we find \begin{align*}
\widehat{\deg}(\widehat{\bm{\Delta}}) & = \sum_{\mathfrak{q}\subset\mathcal O_E}\log N(\mathfrak{q})\sum_{y\in \mathcal{Y}_0(\mathbb F_{\mathfrak{q}}^{\mathrm{alg}})}\frac{\mathrm{length}(\mathscr{L}_y/\mathrm{im}(s_y))}{|\mathrm{Aut}(y)|}\\
& = 2^{d-2}\sum_{\mathfrak{q}\subset\mathcal O_E}\left[\log N(\mathfrak{q})\cdot\mathrm{ord}_{\mathfrak{q}}(\mathfrak{d}_{F/\mathbb Q})\cdot\sum_{y\in \mathcal{Y}_0(\mathbb F_{\mathfrak{q}}^\mathrm{alg})}\frac{1}{|\mathrm{Aut}(y)|}\right]\\
& = 2^{d-2}\cdot\left(\sum_{y\in \mathcal{Y}_0(\mathbb C)}\frac{1}{|\mathrm{Aut}(y)|}\right)\cdot\left(\sum_{\mathfrak{q}\subset\mathcal O_E}\log N(\mathfrak{q})\cdot\mathrm{ord}_{\mathfrak{q}}(\mathfrak{d}_{F/\mathbb Q})\right)\\
& = 2^{d-1}\cdot\deg_{\mathbb C}(Y_0)\cdot\log|D_F|. \end{align*} Here, in the third identity, as in the proof of Lemma~\ref{lem:twisting isogeny}, we have used the finite \'etaleness of $\mathcal{Y}_0$ over $\mathcal O_E$.
It remains to show~\eqref{eqn:main length comp}. Note that complex conjugation induces an involution $c$ on the set $\Gamma_{\mathbb Q_p}$-set $\mathrm{Hom}(E_{\mathfrak{p}},\mathbb Q_p^\mathrm{alg})$. Set \[ \mathrm{CM}(E_{\mathfrak{p}}) = \{\Phi_{\mathfrak{p}}\subset \mathrm{Hom}(E_{\mathfrak{p}},\mathbb Q_p^\mathrm{alg}):\;\Phi_{\mathfrak{p}}\sqcup c(\Phi_{\mathfrak{p}}) = \mathrm{Hom}(E_{\mathfrak{p}},\mathbb Q_p^\mathrm{alg})\}. \] Let $E^\sharp_{\mathfrak{p}}$ be the \'etale $\mathbb Q_p$-algebra associated with the $\Gamma_{\mathbb Q_p}$-set $\mathrm{CM}(E_{\mathfrak{p}})$. There is an obvious surjection of $\Gamma_{\mathbb Q_p}$-sets \[ \mathrm{CM}(E) \to \mathrm{CM}(E_{\mathfrak{p}}) \] inducing an inclusion $E^\sharp_{\mathfrak{p}}\hookrightarrow E^\sharp_p = E^\sharp\otimes_\mathbb Q\mathbb Q_p$ of \'etale $\mathbb Q_p$-algebras. Associated with $\iota_0:E_{\mathfrak{q}}\hookrightarrow \mathbb Q_p^\mathrm{alg}$ are the subsets \[ \Phi^\sharp_{\mathfrak{p}} = \{\Phi_{\mathfrak{p}}\in \mathrm{CM}(E_{\mathfrak{p}}):\; \iota_0\in\Phi_{\mathfrak{p}}\}\;;\; \overline{\Phi}^\sharp_{\mathfrak{p}} = \{\Phi_{\mathfrak{p}}\in \mathrm{CM}(E_{\mathfrak{p}}):\; \overline{\iota}_0\in\Phi_{\mathfrak{p}}\}, \] and we have \[ \Phi^\sharp = \{\iota^\sharp:E_p^\sharp\to\mathbb Q_p^\mathrm{alg}:\;\iota^\sharp\vert_{E^\sharp_{\mathfrak{p}}}\in \Phi^\sharp_{\mathfrak{p}} \}. \]
Now, let $T_{\mathfrak{q}}\subset T_{\mathbb Q_p}$ be as in Remark~\ref{rem:Eq times reps}. Viewed as a representation of $T_{\mathfrak{q}}$, $H^\sharp_p = H^\sharp\otimes_\mathbb Q\mathbb Q_p$ admits the $T_{\mathfrak{q}}$-stable subspace $H^\sharp_{\mathfrak{p}}$ corresponding to the subspace $E^\sharp_{\mathfrak{p}}\subset E^\sharp_p$. Moreover, we have a canonical lattice $H^\sharp_{\mathfrak{p},\mathbb Z_p}\subset H^\sharp_{\mathfrak{p}}$ corresponding to $\mathcal O_{E^\sharp_{\mathfrak{p}}}\subset E^\sharp_{\mathfrak{p}}$. This is stable under $K_{0,\mathfrak{q}} = K_0\cap T(\mathbb Q_p)$, and we have a natural $K_{0,\mathfrak{q}}$-equivariant isomorphism of $\mathcal O_{E^\sharp_p}$-modules: \[ H^\sharp_{\mathfrak{p},\mathbb Z_p}\otimes_{\mathcal O_{E_{\mathfrak{p}}^\sharp}}\mathcal O_{E^\sharp_p}\xrightarrow{\simeq}H^\sharp_{\mathbb Z_p}. \]
If $\bm{H}^\sharp_{\mathfrak{p},\mathrm{dR},\mathcal O_y}$ is the de Rham realization of $H^\sharp_{\mathfrak{p},\mathbb Z_p}$ obtained from Corollary~\ref{cor:realizations y}, then we obtain an isomorphism \[ \bm{H}^\sharp_{\mathfrak{p},\mathrm{dR},\mathcal O_y}\otimes_{\mathcal O_{E_{\mathfrak{p}}^\sharp}}\mathcal O_{E^\sharp_p}\xrightarrow{\simeq}\bm{H}^\sharp_{\mathrm{dR},\mathcal O_y} \] of filtered $\mathcal O_y\otimes_{\mathbb Z_p}\mathcal O_{E^\sharp_p}$-modules.
Fix a $\mathcal O_y$-module generator $\bm{f}_0\in \mathrm{Fil}^1\bm{V}_{0,\mathrm{dR},\mathcal O_y}$, and view it as a map \[ \mathrm{gr}^{-1}_{\mathrm{Fil}}\bm{H}^\sharp_{\mathrm{dR},\mathcal O_y}\to \mathrm{Fil}^0\bm{H}^\sharp_{\mathrm{dR},\mathcal O_y}. \] We find from the construction that this arises via a change of scalars from $\mathcal O_{E^\sharp_{\mathfrak{p}}}$ to $\mathcal O_{E^\sharp_p}$ of a map \[ \bm{f}_{0,\mathfrak{p}}:\mathrm{gr}^{-1}_{\mathrm{Fil}}\bm{H}^\sharp_{\mathfrak{p},\mathrm{dR},\mathcal O_y}\to \mathrm{Fil}^0\bm{H}^\sharp_{\mathfrak{p},\mathrm{dR},\mathcal O_y} \]
Let $P\subset \mathrm{Frac}(W)^\mathrm{alg}$ be a Galois extension of $\mathrm{Frac}(W)$ containing $\mathcal O_y$, which receives all maps $E^\sharp_{\mathfrak{p}}\to \mathbb Q_p^\mathrm{alg}$. Then \[ M \define \bm{H}_{\mathfrak{p},\mathrm{dR},\mathcal O_y}\otimes_{\mathcal O_y}\mathcal O_P \] is a finite free $\mathcal O_P\otimes_{\mathbb Z_p}\mathcal O_{E^\sharp_{\mathfrak{p}}}$-module of rank $1$. One can now check that, in the notation preceding Proposition~\ref{prop:colmez calc}, we have \[ \mathrm{Fil}^0\bm{H}_{\mathfrak{p},\mathrm{dR},\mathcal O_y}\otimes_{\mathcal O_y}\mathcal O_P = \mathcal{K}(M,\Phi^\sharp_{\mathfrak{p}})\;;\; \mathrm{gr}^{-1}_{\mathrm{Fil}}\bm{H}_{\mathfrak{p},\mathrm{dR},\mathcal O_y}\otimes_{\mathcal O_y}\mathcal O_P = \mathcal{Q}(M,\overline{\Phi}^\sharp_{\mathfrak{p}}). \]
Therefore, we have \begin{align} \label{eqn:length decomp} \mathrm{length}(\mathcal{L}_{y}/\mathrm{im}(s_y)) &= \frac{1}{e(P/E_{\mathfrak{q}})}\cdot 2^{d-d_{\mathfrak{p}}}\cdot \mathrm{ord}_{\mathcal O_P}(\det(\bm{f}_{0,\mathfrak{p}})) \\ &= \frac{2^{d-d_{\mathfrak{p}}}}{e(P/E_{\mathfrak{q}})}\cdot\left[ \mathrm{length}_{\mathcal O_P}\left(\frac{\mathcal{Q}(M,\Phi_{\mathfrak{p}}^\sharp)}{\bm{f}_{0,\mathfrak{p}}(\mathcal{Q}(M,\overline{\Phi}_{\mathfrak{p}}^\sharp))}\right) - \mathrm{length}_{\mathcal O_P}\mathcal{C}(M,\Phi^\sharp_{\mathfrak{p}})\right]\nonumber. \end{align} Here, $e(P/E_{\mathfrak{q}})$ is the ramification index of $P$ over $E_{\mathfrak{q}}$, and $d_{\mathfrak{p}} = [F_{\mathfrak{p}}:\mathbb Q_p]$.
Now, set $N = \bm{V}_{0,\mathrm{dR},\mathcal O_y}\otimes_{\mathcal O_y}\mathcal O_P$: this is a free module of rank $1$ over $\mathcal O_P\otimes_{\mathbb Z_p}\mathcal O_{E_{\mathfrak{p}}}$. Set \[ \Upsilon_{\iota_0} = \mathrm{Hom}(E_{\mathfrak{p}},\mathbb Q_p^\mathrm{alg})\backslash\{\iota_0\}. \] Then we have \[ \mathrm{Fil}^1\bm{V}_{0,\mathrm{dR},\mathcal O_y}\otimes_{\mathcal O_y}\mathcal O_P = \mathcal{K}(N,\Upsilon_{\iota_0}). \] Moreover, the action of any generator of $\mathcal{Q}(N,\Upsilon_{\iota_0})$ induces an isomorphism \[ \mathcal{Q}(M,\overline{\Phi}^\sharp_{\mathfrak{p}})\xrightarrow{\simeq}\mathcal{Q}(M,\Phi^\sharp_{\mathfrak{p}}) \] of $\mathcal O_P$-modules. Therefore, we have \begin{equation} \label{eqn:length comp 1}
\mathrm{length}_{\mathcal O_P}\left(\frac{\mathcal{Q}(M,\Phi_{\mathfrak{p}}^\sharp)}{\bm{f}_{0,\mathfrak{p}}(\mathcal{Q}(M,\overline{\Phi}_{\mathfrak{p}}^\sharp))}\right) = 2^{d_{\mathfrak{p}}-1}\cdot \mathrm{length}_{\mathcal O_P}\mathcal{C}(N,\Upsilon_{\iota_0}). \end{equation}
Arguing as in Propositions~\ref{prop:average reflex} and~\ref{prop:colmez height average}, we see that \[ a^0_{(E^\sharp_{\mathfrak{p}},\Phi^\sharp_{\mathfrak{p}})} = 2^{d_{\mathfrak{p}}-2} \left( \bm{1} + \frac{1}{d_{\mathfrak{p}}} \mathrm{Ind}_{\Gamma_{F_{\mathfrak{p}}}}^{\Gamma_{\mathbb Q_p}}( \chi_{\mathfrak{p}} ) \right), \] where $\chi_{\mathfrak{p}}$ is the (possibly trivial) quadratic character of $F_{\mathfrak{p}}$ associated with $E_{\mathfrak{p}}/F_{\mathfrak{p}}$. From this and Proposition~\ref{prop:colmez calc}, one easily deduces that we have \begin{equation}\label{eqn:length comp 2}
\mathrm{length}_{\mathcal O_P}\mathcal{C}(M,\Phi^\sharp_{\mathfrak{p}}) = e(P/E_{\mathfrak{q}})\cdot 2^{d_{\mathfrak{p}}-2}\cdot (2\cdot\mathrm{ord}_{\mathfrak{q}}(\mathfrak{d}_{E/\mathbb Q}) - \mathrm{ord}_{\mathfrak{q}}(\mathfrak{d}_{F/\mathbb Q})). \end{equation}
A similar, but much easier computation shows \begin{equation}\label{eqn:length comp 3} \mathrm{length}_{\mathcal O_P}\mathcal{C}(N,\Upsilon_{\iota_0}) = -e_P\cdot\mu_p(a(E_{\mathfrak{p}},\Upsilon_{\iota_0})) = e(P/E_{\mathfrak{q}})\cdot \mathrm{ord}_{\mathfrak{q}}(\mathfrak{d}_{E/\mathbb Q}). \end{equation}
Combining~\eqref{eqn:length decomp},~\eqref{eqn:length comp 1},~\eqref{eqn:length comp 2} and~\eqref{eqn:length comp 3} now yields~\eqref{eqn:main length comp} and hence the proposition.
\end{proof}
\subsection{The averaged Colmez conjecture}
As in Remark \ref{rem:hermitian construction}, choose any $\xi\in F^\times$ negative at $\iota_0$ and positive at $\iota_1,\ldots, \iota_{d-1}$. This defines a rank two quadratic space \[ (\mathscr{V},\mathscr{Q}) = (E,\xi\cdot \mathrm{Nm}_{E/F}) \] over $F$, and we set \[ (V,Q) = (\mathscr{V} , \mathrm{Tr}_{F/\mathbb Q} \circ \mathscr{Q}) \] as in (\ref{isometric equality}). Fix any maximal lattice $L\subset V$, and and let $D_{bad,L}$ be the product of all the bad primes with respect to $L$ (see Definition~\ref{defn:D bad}).
Recall the integral model $\mathcal{M}\to\mathrm{Spec}(\mathbb Z)$ of the GSpin Shimura variety associated with $L$, as well as the finite cover $\mathcal{Y}\to \mathcal{Y}_0$ associated with the level subgroup $K_{L,0}$ and equipped with a map $\mathcal{Y}\to\mathcal{M}$. We also had the metrized line bundle $\widehat{\bm{\omega}}$ on $\mathcal{M}$ from~\S\ref{ss:line bundles}. Over $\mathcal{Y}$, this line bundle arises from the Hodge filtration on the vector bundle $\bm{V}_{\mathrm{dR}}$ obtained as the de Rham realization of the pair $(V,\widehat{L})$.
Let $(V^ \diamond, Q^ \diamond)$ be a quadratic space of signature $(n^ \diamond,2)$ with $n\geq 3$, and suppose that we have an isometric embedding \[ (V,Q) \hookrightarrow (V^ \diamond, Q^ \diamond) \] and a maximal lattice $L^ \diamond\subset V^ \diamond$ with $L\subset L^ \diamond$. This corresponds to a map $\mathcal{M}\to \mathcal{M}^ \diamond$ of integral models over $\mathbb Z$ for the associated GSpin Shimura varieties.
Suppose that $f \in M^!_{ 1- n^ \diamond/2 } (\omega_{L^ \diamond})$ has integral Fourier coefficients and nonzero constant term $c_f(0,0)$. Let $\mathcal{Z}^ \diamond(f)$ be the corresponding divisor on $\mathcal{M}^ \diamond$, and assume that Hypothesis \ref{hyp:proper} is satisfied. After replacing $f$ by a multiple if necessary, we obtain the vertical metrized line bundle $\widehat{\mathcal{E}}^ \diamond(f) = (\mathcal{E}^ \diamond(f),0)$ on $\mathcal{M}^ \diamond$ as in Theorem~\ref{thm:taut degree}.
As before, we will write $a\approx_L b$ for two real numbers $a,b$ if $a-b$ is a rational linear combination of $\log(p)$ with $p\mid D_{bad,L}$.
\begin{proposition}\label{prop:colmez prelim bound} We have \[ \frac{1}{2^d}\sum_\Phi h^\mathrm{Falt}_{(E,\Phi) } - \frac{1}{2^d}\sum_\Phi h^\mathrm{Col}_{(E,\Phi) } \approx_L \frac{1}{4c_f(0,0)}\frac{[\widehat{\mathcal{E}}^ \diamond(f):\mathcal{Y}]}{\mathrm{deg}_{\mathbb C}({Y})}. \] \end{proposition}
\begin{proof} Given Theorems~\ref{thm:taut degree} and~\ref{thm:Faltings height line bundles}, and Proposition~\ref{prop:colmez height average}, we only have to show: \begin{equation}\label{eqn:degree difference}
[\widehat{\bm{\omega}}:\mathcal{Y}] - \widehat{\deg}_{\mathcal{Y}}(\widehat{\bm{\omega}}_0) - \log|D_F| \approx_L 0. \end{equation}
For this, note that, via the construction in Proposition~\ref{prop:realizations integral model}, the sheaves $\bm{V}_{\mathrm{dR}}$ and $\bm{V}_{0,\mathrm{dR}}$ are both associated with the standard $T$-representation $V = V_0$, but correspond to different $K_{0,L}$-stable lattices in $V_{\mathbb A_f}$. The first is associated with the lattice $\widehat{L}$, and the second with the lattice $\widehat{L}_0$. In particular, since the restrictions of these bundles to the generic fiber does not depend on the $K_{0,L}$-stable lattice, there is a canonical isomorphism \begin{equation}\label{eqn:omega ident} \bm{\omega}\vert_{{Y}}\xrightarrow{\simeq}\bm{\omega}_0\vert_{{Y}} \end{equation}
of line bundles over ${Y}$. At each point $y\in \mathcal{Y}_{\infty}(\mathbb C)$ lying above a place $\iota:F\to \mathbb R$, this isomorphism carries the metric $||\cdot||_y$ on $\bm{\omega}_y$ to $|\iota(\xi)|$-times the metric $||\cdot||_{0,y}$.
Therefore, it is enough to show that~\eqref{eqn:omega ident} induces an isomorphism \[ \bm{\omega}\vert_{\mathcal{Y}[D_{bad,L}^{-1}]}\xrightarrow{\simeq}\xi\mathfrak{d}^{-1}_{F/\mathbb Q}\otimes_{\mathcal O_F}\bm{\omega}_0\vert_{\mathcal{Y}[D_{bad,L}^{-1}]} \] of line bundles over $\mathcal{Y}[D_{bad,L}^{-1}]$.
This is a statement that can be checked over the complete \'etale local rings of $\mathcal{Y}[D_{bad,L}^{-1}]$. So let $\mathfrak{q}\subset\mathcal O_E$ be a prime lying above a prime $p\nmid D_{bad,L}$, and suppose that we have $y\in \mathcal{Y}(\mathbb F_{\mathfrak{q}}^{\mathrm{alg}})$. Let $\mathfrak{p}\subset \mathcal O_F$ be the prime induced from $\mathfrak{q}$. By the definition of $D_{bad,L}$, $L_{\mathfrak{p}} = L_p\cap V_{\mathfrak{p}}$ contains a maximal $\mathcal O_{E,\mathfrak{p}}$-stable quadratic lattice $\Lambda_{\mathfrak{p}}$. We then must have \begin{equation}\label{eqn:Lambda L 0} \Lambda_{\mathfrak{p}} = \xi\mathfrak{d}_{F_{\mathfrak{p}}/\mathbb Q_p}^{-1}L_{0,\mathfrak{p}}. \end{equation}
Let $\mathcal O_y$ be the complete local ring of $\mathcal{Y}$ at $y$. As explained in Remark~\ref{rem:Eq times reps}, from $\Lambda_{\mathfrak{p}}$ and $L_{\mathfrak{p}}$, we obtain de Rham realization $\bm{\Lambda}_{\mathfrak{p},\mathrm{dR},\mathcal O_y}$ and $\bm{V}_{\mathfrak{p},\mathrm{dR},\mathcal O_y}$ over $\mathcal O_y$; these are filtered vector bundles over $\mathcal O_y$.
Choose an isometric embedding $L\hookrightarrow L^ \diamond$ with $L^ \diamond$ of signature $(n^ \diamond,2)$ and self-dual over $\mathbb Z_p$. The inclusions $\Lambda_{\mathfrak{p}}\hookrightarrow L_p\hookrightarrow L^ \diamond_p$ give embeddings \[ \bm{\Lambda}_{\mathfrak{p},\mathrm{dR},\mathcal O_y}\hookrightarrow\bm{V}^ \diamond_{\mathrm{dR},\mathcal O_y} \] of free $\mathcal O_y$-modules.
It now follows from Lemma~\ref{lem:filt beef lambda} that the inclusion \[ \bm{\omega}_{\mathcal O_y}\cap \bm{\Lambda}_{\mathfrak{p},\mathrm{dR},\mathcal O_y} \hookrightarrow \bm{\omega}_{\mathcal O_y} = \mathrm{Fil}^1\bm{V}^ \diamond_{\mathrm{dR},\mathcal O_y} \] is an isomorphism.
Therefore,~\eqref{eqn:Lambda L 0} shows that the isomorphism~\eqref{eqn:omega ident} induces an isomorphism \[ \bm{\omega}_{\mathcal O_y}\xrightarrow{\simeq}\xi(\mathfrak{d}_{F_{\mathfrak{p}}/\mathbb Q_p}^{-1}\otimes_{\mathcal O_F}\bm{\omega}_{0,\mathcal O_y}) \] of line bundles over $\mathcal O_y$, finishing the proof of the Proposition.
\end{proof}
\begin{proposition}\label{prop:choosing_good_lattices} We can find another choice of auxiliary data \[ (\mathscr{V}',\mathscr{Q}') = (E,\xi'\cdot \mathrm{Nm}_{E/F}) \] and a maximal lattice $L'\subset V'$ such that $\mathrm{gcd}(D_{bad,L},D_{bad,L'}) = 1$. \end{proposition} \begin{proof} It is sufficient to show that, given any finite set of rational primes $S$, we can find $\xi'$ and $L'$ such that no prime in $S$ divides $D_{bad,L'}$.
To make this more concrete, suppose that we given an ideal $\mathfrak{a}\subset\mathcal O_E$ and $\xi'\in F$ satisfying $\iota_0(\xi')<0$ and $\iota_j(\xi')>0$ for $j>0$. For a prime $p$, we will declare the pair to be \emph{good at $p$} if \[ \Lambda_p \define (\mathfrak{a},\mathrm{Tr}_{E/\mathbb Q}(\xi'\mathrm{Nm}_{E/F}))\otimes_{\mathbb Z}\mathbb Z_p \] is an $\mathcal O_E$-stable quadratic $\mathbb Z_p$-lattice in $(E,\mathrm{Tr}_{E/\mathbb Q}(\xi'\mathrm{Nm}_{E/F}))\otimes_{\mathbb Q}\mathbb Q_p$, which is self-dual over all primes $\mathfrak{p}\mid p$ that are unramified in $E$, and which satisfies \[ \Lambda_{\mathfrak{p}}^\vee\subset \mathfrak{d}^{-1}_{E_{\mathfrak{q}}/F_{\mathfrak{p}}}\Lambda_{\mathfrak{p}} \] when $\mathfrak{p}$ is ramfified in $E$ and $\mathfrak{q}\subset\mathcal O_E$ is the unique prime above it. Here, we have set $\Lambda_{\mathfrak{p}} = \Lambda_p\otimes_{\mathcal O_{F,p}}\mathcal O_{F,\mathfrak{p}}$.
\begin{lemma} Suppose that $(\mathfrak{a},\xi')$ is good at all $p\in S$. Then there exists a maximal lattice \[ L'\subset V' = (E, \mathrm{Tr}_{F/Q}(\xi'\cdot \mathrm{Nm}_{E/F})) \] that is good at all primes $p\in S$. \end{lemma} \begin{proof} For any prime $p\in S$, and any prime $\mathfrak{p}\subset\mathcal O_F$ lying above such $p$, write $\mathscr{V}'_{\mathfrak{p}}\subset V'_{\mathbb Q_p}$ for the $\mathfrak{p}$-isotypic part of $V'_{\mathbb Q_p}$, and fix a maximal lattice $L'_{\mathfrak{p}} \subset \mathscr{V}'_{\mathfrak{p}}$ containing $\Lambda_{\mathfrak{p}}$. Now, take $L'\subset V'$ to be any maximal lattice such that, for every $p\in S$, $L'_{\mathbb Z_p}$ contains $\bigoplus_{\mathfrak{p}\mid p}L'_{\mathfrak{p}}$. \end{proof}
It now remains to find a pair $(\mathfrak{a},\xi')$ that is good at all primes in $S$.
Given a pair $(\mathfrak{a},\xi')$ as above, one can check that the pair is good at $p$ if and only if for all primes $\mathfrak{p}\subset \mathcal O_F$ lying above $p$, $\mathfrak{p}$ is relatively prime to $\xi'\mathrm{Nm}_{E/F}(\mathfrak{a})\mathfrak{d}_{F/\mathbb Q}$.
Write $\mathrm{Cl}^+(F)$ for the narrow class group of $F$ and $\mathrm{Cl}(E)$ for the class group of $E$. The norm map induces a map \begin{equation}\label{class_group_map} \mathrm{Cl}(E)\to\mathrm{Cl}^+(F) \end{equation} This map is surjective if and only if $E/F$ is ramified at some finite prime. Indeed, via class field theory, the surjectivity of~\eqref{class_group_map} is equivalent to the assertion that the narrow class field of $F$ does not contain $E$.
Suppose that $E/F$ is unramified at all finite places. In this case, the quadratic character $\chi_{E/F}$ can be viewed as a character \[ \chi_{E/F} : \mathrm{Cl}^+(F)\to\{\pm 1\}, \] whose kernel is exactly the image of~\eqref{class_group_map}. We now interrupt the proof for: \begin{lemma}\label{lem:unramified} When $E/F$ is unramified at all finite places, $d\equiv 0\pmod{2}$. Moreover, we have \[ \chi_{E/F}(\mathfrak{d}_{F/\mathbb Q}) = (-1)^{d/2}. \] \end{lemma} \begin{proof} Treating $\chi_{E/F}$ as an id\'ele class character, consider its infinite part $\chi_{E/F,\infty}$. Since $E$ is a totally imaginary extension of $F$, $\chi_{E/F,\infty}$ is the product of the sign characters over all infinite places of $F$. Since $\chi_{E/F}$ is unramified at all finite places, for any unit $\alpha\in\mathcal O_F^\times$, we have \[ \chi_{E/F,\infty}(\alpha) = \chi_{E/F}(\alpha)\chi_{E/F,f}(\alpha)^{-1} = 1. \] Applying this to the case $\alpha=-1$ shows that $(-1)^d = 1$, and so $d$ must be even.
The final assertion is an improvement by Armitage \cite[Theorem 3]{armitage} of a classical result of Hecke. \end{proof}
We return to the proof of Proposition~\ref{prop:choosing_good_lattices}. Choose an arbitrary $\xi_0\in\mathcal O_F$ with $\iota_0(\xi_0)<0$ and $\iota_j(\xi_0)>0$ for $j>0$. Consider the ideal \[ \mathfrak{b}=\xi_0\mathfrak{d}_{F/\mathbb Q}\subset \mathcal O_F. \]
Assume either that $E/F$ is ramified at some finite prime, \emph{or} that $E/F$ is unramified and $d\equiv 2\pmod{4}$. Under either assumption, we claim that there exists an ideal $\mathfrak{a}\subset\mathcal O_E$ and a totally positive element $\eta\in F^\times$ such that \[ \eta \mathrm{Nm}_{E/F}\mathfrak{a} = \mathfrak{b}^{-1}. \] In other words, the class of $\mathfrak{b}$ in $\mathrm{Cl}^+(F)$ is in the image of~\eqref{class_group_map}.
If $E/F$ is ramified, then this is immediate from the surjectivity of~\eqref{class_group_map}.
If $E/F$ is unramified, then $\mathfrak{b}=\xi_0\mathfrak{d}_{F/\mathbb Q}$, and by~\eqref{lem:unramified}, we have: \[ \chi_{E/F}(\mathfrak{b}) = (-1)^{\frac{d}{2}+1}. \] Therefore, when $d\equiv 2\pmod{4}$, $\mathfrak{b}$ is in the image of~\eqref{class_group_map}, and the claim follows.
Now, it is easily checked that, with $\xi'=\eta\xi_0$, $(\mathfrak{a},\xi')$ is good at \emph{all} primes $p$.
It remains to consider the case where $E/F$ is unramified and $d\equiv 0\pmod{4}$. In this case, $\chi_{E/F}(\mathfrak{d}_{F/\mathbb Q})=1$ by Lemma \ref{lem:unramified}. Therefore, we can find an $\mathfrak{a}\subset\mathcal O_E$ and totally positive $\eta\in F^\times$ such that \[ \eta \cdot \mathrm{Nm}_{E/F}( \mathfrak{a} ) = \mathfrak{d}_{F/\mathbb Q}^{-1}. \]
Now, given a totally positive $\beta\in F$, the pair $(\mathfrak{a},\beta\eta\xi_0)$ is good at a all primes in $S$ if and only if $\beta\xi_0$ is not divisible by any $p\in S$. Such a $\beta$ can always be found by weak approximation. \end{proof}
\begin{theorem}\label{thm:average colmez} We have \[ \frac{1}{2^{d}} \sum_\Phi h^\mathrm{Falt}_{(E,\Phi)} = \frac{1}{2^{d}} \sum_\Phi h^\mathrm{Col}_{(E,\Phi)}. \] \end{theorem} \begin{proof} Combining Propositions~\ref{prop:colmez prelim bound} and~\ref{prop:choosing_good_lattices}, we find that we have \[ \frac{1}{2^{d}} \sum_\Phi h^\mathrm{Falt}_{(E,\Phi)} - \frac{1}{2^{d}} \sum_\Phi h^\mathrm{Col}_{(E,\Phi)} = \sum_pb_E(p)\log(p), \] where we can compute $b_E(p)$ as follows: Choose auxiliary data $(\mathscr{V},\mathscr{Q})$ and a maximal lattice $L\subset V$ such that $p\nmid D_{bad,L}$. Also choose an auxiliary quadratic space $(V^ \diamond,Q^ \diamond)$ of signature $(n^ \diamond,2)$ with $n^ \diamond\geq 3$, as well as a maximal lattice $L^ \diamond\subset V^ \diamond$ containing $L$. Choose a weakly holomorphic form \[ f (\tau) = \sum_{ m \gg -\infty} c_f(m) \cdot q^m \in M_{1-\frac{n}{2}}^!(\omega_{L^ \diamond}) \] with integral principal part and $c_f(0,0)\not=0$, and satisfying Hypothesis~\ref{hyp:proper}. Then, after replacing $f$ by a suitable multiple, we have: \begin{equation}\label{eqn:b2_borcherds_bE} \sum_pb_E(p)\log(p) = \frac{1}{4c_f(0,0)}\frac{[\widehat{\mathcal{E}}^ \diamond(f):\mathcal{Y}]}{\deg_{\mathbb C}({Y})}. \end{equation}
Therefore, it is enough to show that, for each prime $p$, we can choose $L^ \diamond$ and $f$ such that $\mathcal{E}^ \diamond(f)$ does not intersect $\mathcal{M}^ \diamond_{\mathbb F_p}$.
It is an easy exercise, given the classification of quadratic forms over $\mathbb Q$, to find $L^ \diamond$ such that $n^ \diamond = 2d$, and such that $L^ \diamond_{(p)}$ is self-dual, and such that $L$ embeds isometricaly in $L^ \diamond$. Now, the orthogonal complement \[ \Lambda = L^\perp \subset L^ \diamond \] is a rank $2$ positive definite lattice over $\mathbb Z$. Any rational prime not split in the discriminant field of $\Lambda$ will fail to be represented by $\Lambda_\mathbb Q$. Therefore, by Theorem~\ref{thm:borcherds_support} below, we can find a weakly modular form $f$ as above such that $c_f(m)\neq 0$ only if $m$ is not represented by $\Lambda_\mathbb Q$. In particular, Hypothesis~\ref{hyp:proper} is satisfied, and, since $L^ \diamond_{(p)}$ is self-dual, by Theorem~\ref{thm:borcherds}, $\mathcal{E}^ \diamond(f)$ does not intersect $\mathcal{M}^ \diamond_{\mathbb F_p}$, as desired.
We note again that the proof only used knowledge of the divisor of the Borcherds lift of $f$ at primes where $L^ \diamond$ is self-dual, which is contained in~\cite{Hormann}, and not the full strength of Theorem~\ref{thm:borcherds}. \end{proof}
The proof above used the following consequence of a result of Bruinier~\cite[Theorem 1.1]{Bru:prescribed}, which we state here for the reader's convenience.
\begin{theorem}[Bruinier]\label{thm:borcherds_support} Let $L$ be a quadratic lattice of signature $(n,2)$ with $n\geq 2$. If $S$ is any infinite subset of square-free positive elements of $D_{L}^{-1} \mathbb Z$ represented by $L^{\vee}$, there is a weakly holomorphic form $f\in M^!_{ 1 - n/2 } ( \omega_{L} )$ such that \begin{enumerate} \item $c^+_f( m,\mu) \in \mathbb Z$ for all $m$ and $\mu$, \item $c^+_f(0,0) \not=0$, \item
if $m>0$ and $m\nin S$, then $c^+_f( -m ,\mu ) =0$ for all $\mu\in L^{\vee} / L$.
\end{enumerate} \end{theorem}
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\end{document} | arXiv |
Daniel Halpern-Leistner
Algebraic Geometry, Derived Categories, Representation Theory, Mathematical Physics
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Email: daniel dot hl at cornell dot edu
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I am currently an assistant professor at Cornell University. Previously I was an NSF/Ritt Post Doc at Columbia University and a member in mathematics at the Institute for Advanced Study. I completed my PhD at UC Berkeley. In my undergraduate studies at Princeton University, I focused on math and physics.
My primary research focuses on a cluster of related projects and ideas which I have labeled "beyond geometric invariant theory." Geometric invariant theory (GIT) is a well-studied and successful tool for constructing moduli spaces in algebraic geometry. But it does more than that. It is a framework for understanding the (equivariant) geometry of algebraic varieties with a reductive group action. The "beyond GIT" project attempts to expand geometric invariant theory in two ways: 1) to use the ideas of GIT to understand the structure of derived categories of equivariant coherent sheaves, which in turn leads to new results in classical equivariant topology and geometry, and 2) to expand the methods of GIT to apply to general moduli problems.
Fall 2020: no teaching
I typically run a graduate research seminar, which is open to all interested students and post docs.
I am currently accepting graduate advisees. If you are interested in working with me as your PhD advisor, please read my advising philosophy, which sets mutual expectations.
Towards a general theory of instability in moduli theory
A common kind of problem in algebraic geometry is to find a space, called a moduli space, parameterizing isomorphism classes of some kind of algebro-geometric objects -- let's call them widgets. Many attempts to form a moduli space for widgets proceed by finding a scheme, $X$, which parameterizes a family of widgets, and an algebraic group, $G$, acting on $X$ such that points in the same orbit under $G$ parameterize isomorphic widgets. Then hopefully one can apply geometric invariant theory to find an open subset in X which has a good quotient under the action of $G$, and whose $G$-orbits classify ``semistable'' widgets. However, there are many situations where a stability condition can be specified on widgets without referring to any GIT problem. We discuss a framework for defining a notion of semistability for an arbitrary moduli problem, and we introduce a structure on the unstable locus, which we call a Theta-stratification, which generalizes classical stratifications of the unstable locus in GIT as well as of the moduli of vector bundles on a curve. We identify a class of moduli problems, which we call $\Theta$-reductive, for which the GIT story carries over nicely into this more general framework -- for these stacks the existence of a $\Theta$-stratification on the unstable locus can be reduced to checking a relatively simple hypothesis.
"On the structure of instability in moduli theory," (arXiv)
Updated version here. (Last update 10/7/2021. I am still revising the part about moduli of objects in abelian categories.)
Derived $\Theta$-stratifications and the D-equivalence conjecture
update 6/18/21
A follow up to "The derived category of a GIT quotient." The theory of $\Theta$-stratifications generalizes a classical stratification of the moduli of vector bundles on a smooth curve, the Harder-Narasimhan-Shatz stratification, to any moduli problem that can be represented by an algebraic stack. We use methods from derived algebraic geometry to develop a structure theory, which is a refinement of the theory of local cohomology, for the derived category of quasi-coherent complexes on an algebraic stack equipped with a $\Theta$-stratification. We then apply this to the $D$-equivalence conjecture, which predicts that birationally equivalent Calabi-Yau manifolds have equivalent derived categories of coherent sheaves. We prove that any two projective Calabi-Yau manifolds that are birationally equivalent to a smooth moduli space of Gieseker semistable coherent sheaves on a $K3$ surface have equivalent derived categories. This establishes the first known case of the $D$-equivalence conjecture for a birational equivalence class in dimension greater than three.
"Derived $\Theta$-stratifications and the D-equivalence conjecture." (draft; last update 10/01/20)
These results were originally announced in the paper ``Theta-stratifications, Theta-reductive stacks, and applications"
Preliminary versions: The above paper unites, generalizes, and corrects minor errors in two previous notes. I am preserving them here for reference.
"The D-equivalence conjecture for moduli spaces of sheaves on a K3 surface" (draft; last update 2/08/17) -- proves that under suitable genericity hypotheses, a variation of good moduli space for a derived stack with self-dual cotangent complex results in a derived equivalence, and sketches the main application to the D-equivalence conjecture.
"An appendix to 'Theta stratifications and derived categories'" (draft; last update 2/04/15) -- proves a version of the main theorem for global quotient stacks, which is enough for many applications
On properness of $K$-moduli spaces and optimal degenerations of Fano varieties
update 11/03/20
We establish an algebraic approach to prove the properness of moduli spaces of K-polystable Fano varieties and reduce the problem to a conjecture on destabilizations of K-unstable Fano varieties. Specifically, we prove that if the stability threshold of every K-unstable Fano variety is computed by a divisorial valuation, then such K-moduli spaces are proper. The argument relies on studying certain optimal destabilizing test configurations and constructing a Theta-stratification on the moduli stack of Fano varieties.
"On properness of $K$-moduli spaces and optimal degenerations of Fano varieties" with Harold Blum, Yuchen Liu, and Chenyang Xu (arXiv:2011.01895)
Learning selection strategies in Buchberger's algorithm
Studying the set of exact solutions of a system of polynomial equations largely depends on a single iterative algorithm, known as Buchberger's algorithm. Optimized versions of this algorithm are crucial for many computer algebra systems (e.g., Mathematica, Maple, Sage). We introduce a new approach to Buchberger's algorithm that uses reinforcement learning agents to perform S-pair selection, a key step in the algorithm. We then study how the difficulty of the problem depends on the choices of domain and distribution of polynomials, about which little is known. Finally, we train a policy model using proximal policy optimization (PPO) to learn S-pair selection strategies for random systems of binomial equations. In certain domains, the trained model outperforms state-of-the-art selection heuristics in total number of polynomial additions performed, which provides a proof-of-concept that recent developments in machine learning have the potential to improve performance of algorithms in symbolic computation.
"Learning selection strategies in Buchberger's algorithm" with Dylan Peifer and Mike Stillman (arXiv:2005.01917)
Reductivity of the automorphism group of K-polystable Fano varieties
We prove that K-polystable log Fano pairs have reductive automorphism groups. In fact, we deduce this statement by establishing more general results concerning the S-completeness and $\Theta$-reductivity of the moduli of K-semistable log Fano pairs. Assuming the conjecture that K-semistability is an open condition, we prove that the Artin stack parametrizing K-semistable Fano varieties admits a separated good moduli space.
"Reductivity of the automorphism group of K-polystable Fano varieties" with Jarod Alper, Harold Blum, and Chenyang Xu (arXiv:1906.03122)
Mapping stacks and the notion of properness in algebraic geometry
One essential feature of a scheme $X$ which is flat and proper over a base $S$ is that for any scheme $Y$ which is of finite type over $S$, there is an algebraic space $Map(X,Y)$ classifying maps from $X$ to $Y$. There are extensions of this statement when $X$ is a proper stack over $S$ and $Y$ has some reasonable hypotheses. While studying the notion of instability in algebraic geometry, I noticed that the quotient stack $\bC / \bC^\ast$ has this property as well. This lead to long investigation with Anatoly Preygel into just what properties of a stack X guarantee the existence of algebraic mapping stacks into "geometric" target stacks. We reformulated the notion of "properness" for algebraic stacks in terms of properties of the derived category of those stacks, in such a way that these categorical properties guarantee the existence of algebraic mapping stacks. Our notion generalizes the classical definition of proper Artin stacks, but in addition there are many global quotient stacks which are definitely not proper in the classical sense but are "proper" in our sense. We were able to construct a very large class of examples by introducing a notion of "projective morphism" of stacks, a property which can be readily verified in examples. Along the way, we prove some surprising new descent properties for derived categories of coherent sheaves in derived algebraic geometry.
"Mapping stacks and categorical notions of properness," with Anatoly Preygel (arXiv:1402.3204)
Cartan-Iwahori-Matsumoto decompositions for reductive groups
update 3/1/2019
We provide a short and self-contained argument for the existence of Cartan-Iwahori-Matsumoto decompositions for reductive groups.
"Cartan-Iwahori-Matsumoto decompositions for reductive groups" with Jarod Alper and Jochen Heinloth (arXiv:1903.00128)
Existence of moduli spaces for algebraic stacks
We provide necessary and sufficient conditions for when an algebraic stack admits a good moduli space. This theorem provides a generalization of the Keel-Mori theorem to moduli problems whose objects have positive dimensional automorphism groups. We also prove a semistable reduction theorem for points of algebraic stacks equipped with a $\Theta$-strati cation. Using these results we nd conditions for the good moduli space to be separated or proper. To illustrate our method, we apply these results to construct proper moduli spaces parameterizing semistable $G$-bundles on curves.
"Existence of good moduli spaces for algebraic stacks" with Jarod Alper and Jochen Heinloth (arXiv:1812.01128)
"On the structure of instability in moduli theory," (draft)
A categorification of the Atiyah-Bott localization formula
This is a short note which discusses a construction involving equivariant derived categories which allows one to state a categorical version of the classical Atiyah-Bott localization formula. The classical theorem states that for a smooth manifold with a $\bC^\ast$-action, the fundamental class in equivariant cohomology can be decomposed as a finite sum with each term corresponding to a connected component of the fixed locus, but in order to do this one must formally invert certain elements in the equivariant cohomology of the point. We show that if $X$ is a smooth scheme, then the structure sheaf of $X$ admits a filtration whose associated graded pieces correspond to connected components of the fixed locus, but in order to do this one must work in a slightly larger category of equivariant complexes, which we define.
"A categorification of the Atiyah-Bott localization formula; last update 5/21/2016
The equivariant Verlinde formula on the moduli of Higgs bundles
The Verlinde formula expresses the dimension of the space of global sections of certain "determinant" line bundles on the moduli of principal $G$ bundles on a smooth curve $\Sigma$, where $G$ is a semisimple group. We prove an analog of the Verlinde formula on the moduli space of semistable meromorphic $G$-Higgs bundles over a smooth curve for a reductive group $G$ whose fundamental group is free. The formula expresses the graded dimension of the space of sections of a positive line bundle on the moduli space of Higgs bundles as a finite sum whose terms are indexed by formal solutions of a generalized Bethe ansatz equation on the maximal torus of $G$.
"The equivariant Verlinde formula on the moduli of Higgs bundles" with an appendix by Constantin Teleman (arxiv:1608.01754)
Theta-stratifications, Theta-reductive stacks, and applications
This is a proceedings paper for the 2015 AMS summer institute in Salt Lake City. It is an expanded version of my lecture there, and includes a discussion of a few more recent developments, too. I give an overview of the beyond GIT project, from the theory of $\Theta$-stability and $\Theta$-reductive stacks, to some the applications to derived categories. Along the way, I propose a possible generalization of toric geometry, generalizing a toric variety to an arbitrary projective-over-affine compactification of a homogeneous space. I also discuss a version of Kirwan's surjectivity theorem for Borel-Moore homology, and I formulate a conjecture that the Hodge structure on the Borel-Moore homology of a cohomologically proper algebraic-symplectic stack is pure.
arxiv:1608.04797; last update 6/12/2016
Combinatorial constructions of derived equivalences
This proves a version of what I like to call the ``magic windows" theorem for a fairly general class of quotient stacks: those which are quotients of a linear representation $V$ of a reductive group $G$ (where the representation is ``quasi-symmetric". The magic windows theorem identifies (under some mild hypotheses) a subcategory of the equivariant derived category of coherent sheaves on $V$ with the derived category of coherent sheaves on any GIT quotient of $V$ which is a scheme or more generally a Deligne-Mumford stack. Applications include:
Explicit combinatorial bases in the K-theory and cohomology of GIT quotients of this kind
Many new examples of derived equivalences between different Deligne-Mumford GIT quotients of linear representations of this kind
Fitting these derived equivalences together to form a representation of the fundamental group(oid) of the complexified Kaehler moduli space of the GIT quotient, and
Equivalences (under some mild hypotheses) between all Deligne-Mumford hyperkaehler quotients of a symplectic representation of a reductive group
"Combinatorial constructions of derived equivalences" with Steven Sam (arxiv:1601.02030)
Tannaka duality revisited
Any map between algebraic stacks $f: \fX \to \fY$ yields a symmetric monoidal functor between derived categories of quasicoherent sheaves $f^\ast : QC(\fY) \to QC(\fX)$. Jacob Lurie showed that when $\fY$ is geometric (meaning quasicompact with affine diagonal), $f$ can be uniquely recovered from $f^\ast$, and the symmetric monoidal functors arising in this way are those satisfying certain hypotheses (continuous, preserving connective objects and flat objects). We generalize this result, showing that for many stacks, it is not necessary to show that $f$ preserves flat objects. This seemingly minor modification allows for a much wider range of applications of this "Tannakian formalism."
"Tannaka duality revisited" with Bhargav Bhatt (arxiv:1507.01925)
Equivariant Hodge theory and noncommutative geometry
We develop a version of Hodge theory for a large class of smooth cohomologically proper quotient stacks $X/G$ analogous to Hodge theory for smooth projective schemes. We show that the noncommutative Hodge-de Rham sequence for the category of equivariant coherent sheaves degenerates. This spectral sequence converges to the periodic cyclic homology, which we canonically identify with the topological equivariant $K$-theory of $X$ with respect to a maximal compact subgroup $M \subset G$. The result is a natural pure Hodge structure of weight $n$ on $K^n_M(X^{an})$. We also treat categories of matrix factorizations for equivariant Landau-Ginzburg models.
"Equivariant Hodge theory and noncommutative geometry" with Daniel Pomerleano (arxiv:1507.01924)
Autoequivalences of derived categories and variation of GIT quotient
update 3/2013
In geometric invariant theory (see below), the GIT quotient of $X/G$ depends on a choice from a continuous set of parameters. Nevertheless, the parameter space breaks down into "chambers" within which the GIT quotient does not vary, and these chambers are separated by "walls." When the parameters cross a wall, the GIT quotient is modified by a "birational transformation." I have been studying how the geometry and especially the derived geometry of the GIT quotient changes under such a wall crossing. For the special case of these wall crossings known as a "generalized flop," the derived geometry of the GIT quotient does not change at all. These cases are especially interesting -- they can reveal new symmetries of the derived category of the GIT quotient which do not arise in the classical geometry.
"Autoequivalences of derived categories and variation of GIT quotient," with Ian Shipman (arxiv:1303.5531)
The derived category of a GIT quotient
If an algebraic group $G$ acts on an algebraic variety $X$, such as $\bC^\ast$ acting on affine space by dilation, is there a meaningful notion of a "space of orbits" for that action? Mumford's geometric invariant theory (GIT) answers this question by constructing a well-behaved orbit space for the action of $G$ on an open subset of "semistable points" of $X$. Many of the algebraic varieties we know and love (partial flag varieties, toric varieties,...) can be presented as GIT quotients of affine spaces. Since the 1980's, many beautiful relationships between the geometry and topology of the GIT quotient and the "equivariant" geometry of $X$ have been discovered. My research extends these relationships to the "derived" equivariant geometry of $X$ and the derived geometry of the GIT quotient.
The final version treats quotient stacks X/G subject to a certain technical hypothesis on the stratification arxiv:1203.0276)
For newcomers, I'd recommend first looking at the version which treated just the smooth case, arxiv:1203.0276v2. Warning: This version contains some errors in the description of the stratification that I corrected in the final version.
Notes from an introductory talk
The Lefschetz hyperplane theorem for Deligne-Mumford stacks
Deligne-Mumford stacks are an abstract method that modern algebraic geometers use to handle spaces whose points have "internal symmetry." Such spaces arise naturally as parameter spaces of objects (such as pointed elliptic curves) which generically have no symmetry, but for which certain points parameterize an object with discrete symmetry. By enlarging your notion of "space," you can deal with these naturally occuring geometric objects, and with the proper formulation, you can even use concrete geometric reasoning such as Morse theory to show that classical theorems still apply in this setting.
final draft:
My PhD thesis included chapters that became the papers "The derived category of a GIT quotient," "Autoequivalences of derived categories and variation of GIT quotient," and "On the structure of instability in moduli theory."
As an undergraduate I thought about algebraic approaches to information theory. Here is a primer on my undergraduate work (Last update 4/30/08)
Talks and expository papers on Beyond GIT
During the Spring 2020 term, I taught a graduate topics course on "Modern moduli theory," I have reformatted them into web-based book:
The Moduli Space
We surveyed the theory of algebraic stacks (fibered categories and descent, quasi-coherent sheaves, quotient stacks, deformation theory, and Artin's criteria, Tannaka duality), then discussed more recent advances (the etale local structure theorems of Alper, Hall, and Rydh, and the results of beyond GIT), and applied these methods to the moduli of vector bundles and principal G bundles over a smooth curve.
Here are some additional expository accounts of my research:
Notes from lectures on Theta-stability and existence of good moduli spaces, with applications to Donaldson invariants (Thun, Switzerland, 2017). Lecture 1, Lecture 2
A colloquium-style slide talk on applications of beyond GIT to the D-equivalence conjecture.
I have written a proceedings paper for the AMS summer algbraic geometry institute (SLC, 2015). Video of the lecture is available here.
Video of short member lecture at IAS
Brief Oberwolfach report.
Notes from a workshop on new methods in GIT (Berlin, 2015)
Beyond geometric invariant theory concept map
One of the great challenges of research mathematics is effectively communicating mathematical ideas. I'm experimenting with a concept map describing the "beyond GIT" project (click to interact). Last updated 2016.
Cohomologically proper algebraic stacks:
The notion of a proper (i.e. "compact") algebraic variety is essential to algebraic geometry. In the context of stacks, we argue that the standard notion of a proper algebraic stack is too restrictive, and we provide an alternative notion of cohomological properness: the stack should satisfy the 'Grothendieck existence' theorem "universally" (i.e. after arbitrary base change to a complete Noetherian ring, or more generally a stack which is complete along a closed substack). The Grothendieck existence theorem is usually regarded as a theorem, but taking it as the definition of properness leads to a notion which is well-behaved and generalizes the more geometric definition of properness in Champs algébriques in useful ways.
Relevant papers:
"Mapping stacks and categorical notions of properness" - develops the general theory
"Equivariant Hodge theory and noncommutative geometry" - inspired by the analogy between cohomologically proper stacks and proper schemes
Equivariant Hodge theory:
For any dg-category, there is a spectral sequence starting with the Hochschild homology and converging to the periodic cyclic homology. When the category is $Perf(X)$ for a scheme, this can be identified with the classical Hodge-to-deRham spectral sequence, and it degenerates when $X$ is smooth and proper. The non-commutative HdR sequence degenerates for many cohomologically proper smooth stacks and fails to do so for many smooth stacks which are not cohomologically proper. This provides further evidence that cohomologically proper is a good generalization of the notion of properness to Artin stacks.
Another interesting aspect of this degeneration is that it allows one to construct a canonical pure Hodge structure on the topological $K$-theory of many smooth and cohomologically proper stacks. It raises the question of whether there is a canonical mixed Hodge structure on the topological K-theory of an arbitrary stack, and whether there is a motivic framework which is well-suited for stacks.
Relevant paper:
"Equivariant Hodge theory and noncommutative geometry"
Mapping stacks:
If $\fX$ and $\fY$ are stacks, then one can form a mapping stack, $Map(\fX,\fY)$, which by definition is the moduli functor parameterizing families of maps from $\fX$ to $\fY$. When $\fX$ and $\fY$ are algebraic spaces (locally finitely presented over a Noetherian base) and $\fX$ is proper and flat, then it is a classical result that $Map(\fX,\fY)$ is in fact represented by an algebraic space. When $\fX$ is an algebraic stack, this mapping stack will again be an algebraic stack, as long as $\fY$ is geometric and $\fX$ is cohomologically proper. The special case where $\fX = \Theta$ is central to the theory of $\Theta$-stratifications.
"Tanaka duality revisited" - one of the key inputs
Theta stratifications:
If $X$ is a symplectic manifold admitting a Hamiltonian action by a compact group, $K$, then one has a $K$-equivariant stratification of $X$ by the gradient descent flow of the norm-squared of the moment map. When $X$ is a projective variety, then this stratification has an alternative, purely algebraic, description in terms of the Hilbert-Mumford numerical criterion in GIT, and one can think of this as a stratification of the algebraic stack $\fX = X/K_{\bC}$. Theta stratifications provide a generalization of this to stacks which are not global quotients stacks, such as the non finite-type stacks appearing in many moduli problems. The theory provides a framework for studying "stability" of algebro geometric objects generalizing GIT and many other commonly studied notions of stability.
Key idea: the strata have canonical modular interpretations -- they parameterize maps $f : \bC/\bC^\ast \to \fX$ which exhibit "optimally destabilizing" data for the unstable point $f(1) \in \fX$.
"On the structure of instability in moduli theory" -- develops the machinery and foundational theorems regarding $\Theta$-stratifications. Also discusses methods of constructing $\Theta$-stratifications.
$\Theta$-reductive stack:
Given an algebraic stack representing a certain moduli problem, one can ask what data is required to define a Theta-stratification. For $\Theta$-reductive stacks, all one needs is a class in $H^2(\fX;\bQ)$ and $H^4(\fX;\bQ)$ satisfying a "boundedness" hypothesis.
Simplest example: $X/G$, where $X$ is affine and $G$ is a reductive group
Non-example: $X/G$ where $X$ is projective and $G$ is reductive
Interesting examples: Moduli of objects in the heart of a t-structure on the derived category of coherent sheaves of a projective variety
Formal definition: the map of "evaluation at 1" $Map(\bC/\bC^\ast,\fX) \to \fX$ should be proper on connected components
Our research suggests that a good starting point for analyzing a moduli problem is to find a $\Theta$-reductive enlargement of that moduli problem, then apply the theory of $\Theta$-stability.
"On the structure of instability in moduli theory" -- developes the theory of $\Theta$-reductive stacks and $\Theta$-stability.
Generalized buildings:
A key construction in the theory of Theta-stability assigns to any point in an algebraic stack $ p \in \fX $, a topological space $\sD(\fX,p) $, called the degeneration space. A map $\bC/\bC^\ast \to \fX$ along with an isomorphism $f(1) \simeq p$ determines a point of $\sD(\fX,p)$, and points of this form are dense in $\sD(\fX,p)$.
When $\fX = BG$ for a semisimple group $G$, then $ \sD(\fX,p) $ is homeromorphic to the spherical building of $G$, and when $\fX = X/T$ for a normal toric variety $X$, the degeneration space of a generic point of $X$ is homeomorphic to $(|\Sigma| - \{0\}) / \bR^\times_{\geq 0}$, where $|\Sigma|$ denotes the support of the fan defining $X$. Thus these degeneration spaces can be thought of as "generalized buildings," and they connect the theory of buildings in representation theory with toric geometry.
"On the structure of instability in moduli theory" -- defines the degeneration space associated to a point in an arbitrary stack
Extensions of Kirwan surjectivity:
Kirwan surjectivity states that for a GIT quotient of a smooth variety, $X^{ss} / G \subset X/G$, the restriction map on cohomology $H^\ast(X/G) \to H^\ast(X^{ss}/G)$ is surjective. The main structure theorem for derived categories leads to two extensions of this theorem:
It provides a "categorification" of this result to a statement about derived categories of coherent sheaves. The restriction functor $D^b Coh (X/G) \to D^b Coh(X^{ss}/G)$ is always surjective on the level of objects, so that's not the right categorification. But it turns out there is a subcategory $G_w \subset D^b Coh(X/G)$ such that the restriction functor gives an equivalence $G_w \simeq D^b Coh(X^{ss}/G)$. This implies that the restriction functor admits a section $D^b Coh(X^{ss}/G) \to D^b Coh(X/G)$, and so for any invariant which can be extracted from the derived category, the restriction functor from $X/G$ to $X^{ss}/G$ is surjective. For instance, you get Kirwan surjectivity for higher algebraic K-theory.
It turns out that this categorical form of Kirwan surjectivity continues to hold (under certain hypotheses) for the category $D^bCoh$ when $X$ is quasi-smooth; either a local complete intersection or a space with quasi-smooth derived structure. Combined with the de-categorification results extracting topological invariants from derived categories, this leads to a version of Kirwan surjectivity for equivariant Borel-Moore homology (again assuming certain hypotheses on the first homology of the cotangent complex of $X$)
"The derived category of a GIT quotient" -- the smooth case, and some singular cases (without using derived algebraic geometry)
"Remarks on derived categories and $\Theta$-stratifications" -- works out the generalized structure theorem in the quasi-smooth case
"Equivariant Hodge theory and noncommutative geometry" -- proves that one can extract Borel Moore K-theory from $D^bCoh$
Topological invariants of derived categories:
A result of Feigin and Tsygan holds that the cohomology of (the analytification of) an affine variety, $X$, over $\bC$ is isomorphic to the periodic-cyclic homology of the coordinate ring. This agrees with the periodic-cyclic homology of the category of perfect complexes on $X$, so some topological information can be extracted directly from the derived category. For equivariant categories, Thomason showed that the topological equivariant K-theory modulo a prime power can be recovered from the derived category of equivariant coherent sheaves. It turns out that the equivariant K-theory itself can be recovered from the derived category of equivariant coherent sheaves, at least when $X$ is smooth. This result allows one to de-categorify categorical Kirwan surjectivity and recover classical Kirwan surjectivity. When $X$ is singular, one can recover the ``equivariant Borel-Moore $K$-theory from the derived category of coherent sheaves this leads to new versions of Kirwan surjectivity in Borel-Moore homology.
Virtual non-abelian localization theorem:
When $X$ is a compact manifold and $T$ is a torus acting on $X$, localization theorems in equivariant cohomology provide a method for reducing the integrals of equivariant cohomology classes on $X$ to integrals over the fixed locus $X^T$ (which could be just a sum over a finite set of points). There is another, closely related, flavor of localization theorems for stacks with a $\Theta$-stratification. The integral $\int_X \omega$ is replaced by the K-theoretic integral \[\chi(X/G,F) := \Sigma (-1)^p \dim R^p \Gamma(X,F)^G,\] where $F \in Perf(X/G)$, and the localization formula expresses $\chi(X/G)$ as a sum of $\chi(X^{ss}/G,F)$ and "correction terms" coming from each stratum. In some cases the correction terms vanish, leading to an identification between the integral over $X$ and over $X^{ss}$, and in other cases $X^{ss} = \emptyset$, leading to an formula for $\chi(X/G,F)$ in terms of the fixed locus as in the cohomological version. Thus the $K$-theoretic localization theorem is a little more flexible and has the advantage of working for non-abelian $G$, and the cohomological localization formula can be recovered for classes of the form $ch(F)$.
For smooth global quotient stacks, the non-abelian localization theorem was developed by Teleman and Woodward. In fact the non-abelian localization formula is intrinsic to any stack with $\Theta$-stratification, and does not require a local quotient description. Furthermore, using a little bit of derived algebraic geometry, we extend the non-abelian localization theorem to stacks which are quasi-smooth. If $\fX^{cl} \hookrightarrow \fX$ is the underlying classical stack of a quasi-smooth stack $\fX$, and $F \in Perf(\fX)$, then the virtual non-abelian localization theorem is a formula for $\chi(\fX^{cl},F \otimes \mathcal{O}_{\fX}^{vir})$, where $\mathcal{O}_{\fX}^{vir} \in Perf(\fX^{cl})$ is the "virtual" structure sheaf. (In practice most classical stacks with a perfect obstruction theory are the underlying classical stack of a quasi-smooth derived stack.)
"Remarks on Theta-stratifications and derived categories"
Structure theorems for equivariant derived categories:
The main structure theorem for stacks with a $\Theta$-stratification concerns the derived category of coherent sheaves. A semi-orthogonal decomposition of a pre-triangulated dg-category consists of the data of a collection of pre-triangulated (i.e. stable) subcategories which are semiorthogonal to one another (Homs only go in one direction with respect to some total ordering of the categories), and such that every object has a functorial filtration whose associated graded pieces lie in these subcategories. The main structure theorem states that when $\fX$ is a (derived) stack with a $\Theta$-stratification, then $D^- Coh(\fX)$ admits a semiorthogonal decomposition where one piece, $G^w \subset D^-Coh(\fX)$, is identified with $D^-Coh(\fX^{ss})$ via the restriction functor. As a consequence, objects in $D^-Coh(\fX^{ss})$ can be lifted functorially to $D^-Coh(\fX)$. The other pieces of the semiorthogonal decomposition consist of objects which are set-theoretically supported on the unstable locus in $\fX$. There are certain situations in which this structure theorem is even stronger:
When $\fX$ is smooth, this semiorthogonal decomposition induces a decomposition on $Perf(\fX)$, and
When $\fX$ is quasi-smooth but the coherent sheaf $H^1(T_{\fX})$ satisfies a certain weight condition, then this induces a semiorthogonal decomposition of $D^bCoh(\fX)$. The condition holds automatically for algebraic-symplectic stacks.
"The derived category of a GIT quotient" -- works out the case when $\fX = X/G$ is a global quotient stack which is either smooth or is singular (and classical) and satisfies certain technical hypotheses.
"Remarks on $\Theta$-stratifications and derived categories" -- works out the derived case for global quotient stacks. The full version (to appear) is similar (although the proof is conceptually cleaner).
Monodromy representations on derived categories:
Let $V$ be a quasi-symmetric linear representation of a reductive group $G$. The magic windows theorem gives more than derived equivalences between different GIT quotients of $V$. There are different choices of magic windows that one can use to construct derived equivalences between the different GIT quotients, one can use these different equivalences to construct an action of the fundamental groupoid of the ``complexified Kaehler moduli space." This space admits an explicit combinatorial description as the complement of a certain hyperplane arrangement in a torus, based on the character of $V$. Under homological mirror symmetry, this action of the fundamental groupoid on the derived category is conjectured to be mirror to an action by symplectic parallel transport on the mirror family of varieties. Thus we refer to it as a monodromy representation, even though its relationship with monodromy is still conjectural, for the moment.
"Combinatorial constructions of derived equivalences"--prove the magic windows theorem in the smooth case and establishes the action of the fundamental groupoid of the Kaehler moduli space
Equivariant Verlinde formula on the moduli of Higgs bundles:
The Verlinde formula expresses the dimension of spaces of "generalized Theta-functions," defined as sections of a certain line bundle on the moduli of semistable $G$-bundles on a smooth curve. One consequence of the virtual localization formula is a proof of a version of this formula for the moduli of semistable Higgs bundles. Building on previous work of Teleman and Teleman-Woodward, one can compute the dimension of the space of sections of this line bundle (of any given weight under the $\bG_m$-action which scales the Higgs field) on the stack of all Higgs bundles, and our methods allow one to identify this with the corresponding space of sections of this line bundle on the space of semistable Higgs bundles.
"The equivariant Verlinde formula on the moduli of Higgs bundles"
Categorical representations of Yangians:
This is still work in progress with Davesh Maulik and Andrei Okounkov. We are using the magic windows theorem to categorify some earlier work of theirs on the quantum cohomology and quantum K-theory of Nakajima quiver varieties.
Derived equivalences and variation of GIT quotient:
The main application of the structure theorem is to proving cases of the ``K-equivalence implies D-equivalence" conjecture for K-equivalences arising from variation of GIT quotient: as the stability parameter for the GIT quotient changes, $X^{ss}_- \leftrightarrow X^{ss}_+$, the stratification changes and thus the subcategories $G^\pm_w \subset D^bCoh(X/G)$ differ. In nice situations, the category corresponding to one GIT quotient contains the category corresponding to the other GIT quotient, $G_w^- \subset G_w^+$. Using the full structure theorem, one can identify the semiorthogonal complement of $G^-_w$ in $G^+_w$ explicitly.
This model for how to construct derived equivalences is due to Ed Segal and to Hori, Herbst, and Page, who discovered the phenomenon in some basic examples. Ballard,Favero, and Katzarkov and I independently extended these methods to construct derived equivalences for all VGIT wall crossings which are "balanced" and satisfy a "Calabi-Yau" condition. The current state of the art on this method is the "magic windows theorem," which identifies subcategories of $D^b(V/G)$ which are identified with every GIT quotient for which $V^{ss} = V^s$, when $V$ is a ``quasi-symmetric" linear representation of a reductive group $G$ satisfying a mild technical condition.
"The derived category of a GIT quotient" -- the first version, gets derived equivalences for balanced wall crossings (in the smooth case)
"Combinatorial constructions of derived equivalences" -- establishes the magic windows theorem and many new cases of the derived equivalence conjecture.
"The D-equivalence conjecture for moduli spaces of sheaves on a K3 surface" -- proves the ``K-equivalence implies D-equivalence" conjecture for compact Calabi-Yau manifolds which are birationally equivalent to a moduli space of sheaves on a K3 surface.
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