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November 2013 , Volume 33 , Issue 11&12
Jerry Goldstein on the occasion of his 70th birthday
Gisèle Ruiz Goldstein and Alain Miranville
2013, 33(11&12): i-ii doi: 10.3934/dcds.2013.33.11i +[Abstract](1201) +[PDF](83.8KB)
Jerome Arthur Goldstein (Jerry) was born on August 5, 1941 in Pittsburgh, PA. He attended Carnegie Mellon University, then called Carnegie Institute of Technology, where he earned his Bachelors of Science degree (1963), Masters of Science degree (1964) and Ph.D. (1967) in Mathematics. Jerry took a postdoctoral position at the Institute for Advanced Study in Princeton, followed by an Assistant Professorship at Tulane University. He became a Full Professor at Tulane University in 1975. After twenty-four years there, Jerry moved ``upriver" to join the faculty (and his wife, Gisèle) at Louisiana State University, where he was Professor of Mathematics from 1991 to 1996. In 1996 Jerry and Gisèle moved to the University of Memphis.
Gis\u00E8le Ruiz Goldstein, Alain Miranville. Preface. Discrete & Continuous Dynamical Systems - A, 2013, 33(11&12): i-ii. doi: 10.3934/dcds.2013.33.11i.
Floquet representations and asymptotic behavior of periodic evolution families
Fatih Bayazit, Ulrich Groh and Rainer Nagel
2013, 33(11&12): 4795-4810 doi: 10.3934/dcds.2013.33.4795 +[Abstract](1502) +[PDF](375.9KB)
We use semigroup techniques to describe the asymptotic behavior of contractive, periodic evolution families on Hilbert spaces. In particular, we show that such evolution families converge almost weakly to a Floquet representation with discrete spectrum.
Fatih Bayazit, Ulrich Groh, Rainer Nagel. Floquet representations and asymptotic behavior of periodic evolution families. Discrete & Continuous Dynamical Systems - A, 2013, 33(11&12): 4795-4810. doi: 10.3934/dcds.2013.33.4795.
Singularity formation and blowup of complex-valued solutions of the modified KdV equation
Jerry L. Bona, Stéphane Vento and Fred B. Weissler
The dynamics of the poles of the two--soliton solutions of the modified Korteweg--de Vries equation $$ u_t + 6u^2u_x + u_{xxx} = 0 $$ are investigated. A consequence of this study is the existence of classes of smooth, complex--valued solutions of this equation, defined for $-\infty < x < \infty$, exponentially decreasing to zero as $|x| \to \infty$, that blow up in finite time.
Jerry L. Bona, St\u00E9phane Vento, Fred B. Weissler. Singularity formation and blowup of complex-valued solutions of the modified KdV equation. Discrete & Continuous Dynamical Systems - A, 2013, 33(11&12): 4811-4840. doi: 10.3934/dcds.2013.33.4811.
Periodic traveling--wave solutions of nonlinear dispersive evolution equations
Hongqiu Chen and Jerry L. Bona
For a general class of nonlinear, dispersive wave equations, existence of periodic, traveling-wave solutions is studied. These traveling waveforms are the analog of the classical cnoidal-wave solutions of the Korteweg-de Vries equation. They are determined to be stable to perturbation of the same period. Their large wavelength limit is shown to be solitary waves.
Hongqiu Chen, Jerry L. Bona. Periodic traveling--wave solutions of nonlinear dispersive evolution equations. Discrete & Continuous Dynamical Systems - A, 2013, 33(11&12): 4841-4873. doi: 10.3934/dcds.2013.33.4841.
On the asymptotic behavior of variational inequalities set in cylinders
Michel Chipot and Karen Yeressian
We study the asymptotic behavior of solutions to variational inequalities with pointwise constraint on the value and gradient of the functions as the domain becomes unbounded. First, as a model problem, we consider the case when the constraint is only on the value of the functions. Then we consider the more general case of constraint also on the gradient. At the end we consider the case when there is no force term which corresponds to Saint-Venant principle for linear problems.
Michel Chipot, Karen Yeressian. On the asymptotic behavior of variational inequalities set in cylinders. Discrete & Continuous Dynamical Systems - A, 2013, 33(11&12): 4875-4890. doi: 10.3934/dcds.2013.33.4875.
Hopf bifurcation for a size-structured model with resting phase
Jixun Chu and Pierre Magal
2013, 33(11&12): 4891-4921 doi: 10.3934/dcds.2013.33.4891 +[Abstract](2044) +[PDF](1659.1KB)
This article investigates Hopf bifurcation for a size-structured population dynamic model that is designed to describe size dispersion among individuals in a given population. This model has a nonlinear and nonlocal boundary condition. We reformulate the problem as an abstract non-densely defined Cauchy problem, and study it in the frame work of integrated semigroup theory. We prove a Hopf bifurcation theorem and we present some numerical simulations to support our analysis.
Jixun Chu, Pierre Magal. Hopf bifurcation for a size-structured model with resting phase. Discrete & Continuous Dynamical Systems - A, 2013, 33(11&12): 4891-4921. doi: 10.3934/dcds.2013.33.4891.
On a Dirichlet problem in bounded domains with singular nonlinearity
Giuseppe Maria Coclite and Mario Michele Coclite
In this paper we prove the existence and regularity of positive solutions of the homogeneous Dirichlet problem \begin{equation*} -Δ u=g(x,u) in \Omega, u=0 on ∂ \Omega, \end{equation*} where $g(x,u)$ can be singular as $u\rightarrow0^+$ and $0\le g(x,u)\le\frac{\varphi_0(x)}{u^p}$ or $0\le$ $ g(x,u)$ $\le$ $\varphi_0(x)(1+\frac{1}{u^p})$, with $\varphi_0 \in L^m(\Omega), 1 ≤ m.$ There are no assumptions on the monotonicity of $g(x,\cdot)$ and the existence of super- or sub-solutions.
Giuseppe Maria Coclite, Mario Michele Coclite. On a Dirichlet problem in bounded domains with singular nonlinearity. Discrete & Continuous Dynamical Systems - A, 2013, 33(11&12): 4923-4944. doi: 10.3934/dcds.2013.33.4923.
Ultraparabolic equations with nonlocal delayed boundary conditions
Gabriella Di Blasio
A class of ultraparabolic equations with delay, arising from age--structured population diffusion, is analyzed. For such equations well--posedness as well as regularity results with respect to the space variables are proved.
Gabriella Di Blasio. Ultraparabolic equations with nonlocal delayed boundary conditions. Discrete & Continuous Dynamical Systems - A, 2013, 33(11&12): 4945-4965. doi: 10.3934/dcds.2013.33.4945.
Boundary value problem for elliptic differential equations in non-commutative cases
Angelo Favini, Rabah Labbas, Stéphane Maingot and Maëlis Meisner
This paper is devoted to abstract second order complete elliptic differential equations set on $\left[ 0,1\right] $ in non-commutative cases. Existence, uniqueness and maximal regularity of the strict solution are proved. The study is performed in $C^{\theta }\left( \left[ 0,1\right] ;X\right) $.
Angelo Favini, Rabah Labbas, St\u00E9phane Maingot, Ma\u00EBlis Meisner. Boundary value problem for elliptic differential equations in non-commutative cases. Discrete & Continuous Dynamical Systems - A, 2013, 33(11&12): 4967-4990. doi: 10.3934/dcds.2013.33.4967.
Resolution and optimal regularity for a biharmonic equation with impedance boundary conditions and some generalizations
Angelo Favini, Rabah Labbas, Keddour Lemrabet, Stéphane Maingot and Hassan D. Sidibé
In this work, a biharmonic equation with an impedance (non standard) boundary condition and more general equations are considered. The study is performed in the space $L^{p}(-1,0$ $;$ $X)$, $1 < p < \infty $, where $X$ is a UMD Banach space. The problem is obtained through a formal limiting process on a family of boundary and transmission problems $(P^{\delta})_{\delta > 0}$ set in a domain having a thin layer. The limiting problem models, for instance, the bending of a thin plate with a stiffness on a part of its boundary (see Favini et al. [13]).
We build an explicit representation of the solution, then we study its regularity and give a meaning to the non standard boundary condition.
Angelo Favini, Rabah Labbas, Keddour Lemrabet, St\u00E9phane Maingot, Hassan D. Sidib\u00E9. Resolution and optimal regularity for a biharmonic equation withimpedance boundary conditions and some generalizations. Discrete & Continuous Dynamical Systems - A, 2013, 33(11&12): 4991-5014. doi: 10.3934/dcds.2013.33.4991.
Local Hadamard well--posedness and blow--up for reaction--diffusion equations with non--linear dynamical boundary conditions
Alessio Fiscella and Enzo Vitillaro
The paper deals with local well--posedness, global existence and blow--up results for reaction--diffusion equations coupled with nonlinear dynamical boundary conditions. The typical problem studied is \[\begin{cases} u_{t}-\Delta u=|u|^{p-2} u in (0,\infty)\times\Omega,\\ u=0 on [0,\infty) \times \Gamma_{0},\\ \frac{\partial u}{\partial\nu} = -|u_{t}|^{m-2}u_{t} on [0,\infty)\times\Gamma_{1},\\ u(0,x)=u_{0}(x) in \Omega \end{cases}\] where $\Omega$ is a bounded open regular domain of $\mathbb{R}^{n}$ ($n\geq 1$), $\partial\Omega=\Gamma_0\cup\Gamma_1$, $2\le p\le 1+2^*/2$, $m>1$ and $u_0\in H^1(\Omega)$, ${u_0}_{|\Gamma_0}=0$. After showing local well--posedness in the Hadamard sense we give global existence and blow--up results when $\Gamma_0$ has positive surface measure. Moreover we discuss the generalization of the above mentioned results to more general problems where the terms $|u|^{p-2}u$ and $|u_{t}|^{m-2}u_{t}$ are respectively replaced by $f\left(x,u\right)$ and $Q(t,x,u_t)$ under suitable assumptions on them.
Alessio Fiscella, Enzo Vitillaro. Local Hadamard well--posedness and blow--up for reaction--diffusionequations with non--linear dynamical boundary conditions. Discrete & Continuous Dynamical Systems - A, 2013, 33(11&12): 5015-5047. doi: 10.3934/dcds.2013.33.5015.
On the Ornstein Uhlenbeck operator perturbed by singular potentials in $L^p$--spaces
Simona Fornaro and Abdelaziz Rhandi
In this paper we give sufficient conditions ensuring that the space of test functions $C_c^{\infty}(R^N)$ is a core for the operator $$L_0u=\Delta u-Mx\cdot \nabla u+\frac{\alpha}{|x|^2}u=:Lu+\frac{\alpha}{|x|^2}u,$$ and $L_0$ with domain $W_\mu^{2,p}(R^N)$ generates a quasi-contractive and positivity preserving $C_0$-semigroup in $L^p_\mu(R^N),\,1 < p < \infty$. Here $M$ is a positive definite $N\times N$ hermitian matrix and $\mu$ is the unique invariant measure for the Ornstein-Uhlenbeck operator $L$. The proofs are based on an $L^p$-weighted Hardy's inequality and perturbation techniques.
Simona Fornaro, Abdelaziz Rhandi. On the Ornstein Uhlenbeck operator perturbed by singular potentials in $L^p$--spaces. Discrete & Continuous Dynamical Systems - A, 2013, 33(11&12): 5049-5058. doi: 10.3934/dcds.2013.33.5049.
Prey-predator models with infected prey and predators
J. Gani and R. J. Swift
Some deterministic models for prey and predators are considered, when both may become infected, the infection of the prey being either of the SIS or SIR type. We also study a simplified model for surviving predators.
J. Gani, R. J. Swift. Prey-predator models with infected prey and predators. Discrete & Continuous Dynamical Systems - A, 2013, 33(11&12): 5059-5066. doi: 10.3934/dcds.2013.33.5059.
On a class of model Hilbert spaces
Fritz Gesztesy, Rudi Weikard and Maxim Zinchenko
A detailed description of the model Hilbert space $L^2(\mathbb{R}; d\Sigma; K)$, where $K$ represents a complex, separable Hilbert space, and $\Sigma$ denotes a bounded operator-valued measure, is provided. In particular, we show that several alternative approaches to such a construction in the literature are equivalent.
These spaces are of fundamental importance in the context of perturbation theory of self-adjoint extensions of symmetric operators, and the spectral theory of ordinary differential operators with operator-valued coefficients.
Fritz Gesztesy, Rudi Weikard, Maxim Zinchenko. On a class of model Hilbert spaces. Discrete & Continuous Dynamical Systems - A, 2013, 33(11&12): 5067-5088. doi: 10.3934/dcds.2013.33.5067.
Nonlocal phase-field systems with general potentials
Maurizio Grasselli and Giulio Schimperna
We consider a phase-field model of Caginalp type where the free energy depends on the order parameter $\chi$ in a nonlocal way. Therefore, the resulting system consists of the energy balance equation coupled with a nonlinear and nonlocal ODE for $\chi$. Such system has been analyzed by several authors, in particular when the configuration potential is a smooth double-well function. More recently, the first author has established the existence of a finite-dimensional global attractor in the case of a potential defined on $(-1,1)$ and singular at the endpoints. Here we examine both the case of regular potentials as well as the case of physically more relevant singular potentials (e.g., logarithmic). We prove well-posedness results and the eventual global boundedness of solutions uniformly with respect to the initial data. In addition, we show that the separation property holds in the case of singular potentials. Thanks to these results, we are able to demonstrate the existence of a finite-dimensional global attractor in the present cases as well.
Maurizio Grasselli, Giulio Schimperna. Nonlocal phase-field systems with general potentials. Discrete & Continuous Dynamical Systems - A, 2013, 33(11&12): 5089-5106. doi: 10.3934/dcds.2013.33.5089.
Partial reconstruction of the source term in a linear parabolic initial problem with Dirichlet boundary conditions
Davide Guidetti
We consider the problem of the reconstruction of the source term in a parabolic Cauchy-Dirichlet system in a cylindrical domain. The supplementary information, necessary to determine the unknown part of the source term together with the solution, is given by the knowledge of an integral of the solution with respect to some of the space variables.
Davide Guidetti. Partial reconstruction of the source term in a linear parabolic initial problem with Dirichlet boundary conditions. Discrete & Continuous Dynamical Systems - A, 2013, 33(11&12): 5107-5141. doi: 10.3934/dcds.2013.33.5107.
Well-posedness results for the Navier-Stokes equations in the rotational framework
Matthias Hieber and Sylvie Monniaux
Consider the Navier-Stokes equations in the rotational framework either on $\mathbb{R}^3$ or on open sets $\Omega \subset \mathbb{R}^3$ subject to Dirichlet boundary conditions. This paper discusses recent well-posedness and ill-posedness results for both situations.
Matthias Hieber, Sylvie Monniaux. Well-posedness results for the Navier-Stokes equations in the rotational framework. Discrete & Continuous Dynamical Systems - A, 2013, 33(11&12): 5143-5151. doi: 10.3934/dcds.2013.33.5143.
Multiplicity results for classes of singular problems on an exterior domain
Eunkyoung Ko, Eun Kyoung Lee and R. Shivaji
We study radial positive solutions to the singular boundary value problem \begin{equation*} \begin{cases} -\Delta_p u = \lambda K(|x|)\frac{f(u)}{u^\beta} \quad \mbox{in}~ \Omega,\\ ~~~u(x) = 0 \qquad \qquad \qquad \qquad~~\mbox{if}~|x|=r_0,\\ ~~~u(x) \rightarrow 0 \qquad\qquad \qquad \mbox{if}~|x|\rightarrow \infty, \end{cases} \end{equation*} where $\Delta_p u =$ div $(|\nabla u|^{p-2}\nabla u)$, $1 < p < N, N >2, \lambda > 0, 0 \leq \beta <1 ,\Omega= \{ x \in \mathbb{R}^{N} : |x| > r_0 \}$ and $ r_0 >0$. Here $f:[0, \infty)\rightarrow (0, \infty)$ is a continuous nondecreasing function such that $\lim_{u\rightarrow \infty} \frac{f(u)}{u^{\beta+p-1}}= 0$ and $ K \in C( (r_0, \infty),(0, \infty) ) $ is such that $\int_{r_0}^{\infty} r^\mu K(r) dr < \infty, $ for some $\mu > p-1$. We establish the existence of multiple positive solutions for certain range of $\lambda$ when $f$ satisfies certain additional assumptions. A simple model that will satisfy our hypotheses is $f(u)=e^{\frac{\alpha u}{\alpha+u}}$ for $ \alpha \gg 1.$ We also extend our results to classes of systems when the nonlinearities satisfy a combined sublinear condition at infinity. We prove our results by the method of sub-super solutions.
Eunkyoung Ko, Eun Kyoung Lee, R. Shivaji. Multiplicity results for classes of singular problems on an exterior domain. Discrete & Continuous Dynamical Systems - A, 2013, 33(11&12): 5153-5166. doi: 10.3934/dcds.2013.33.5153.
On the nonexistence of positive solutions to doubly nonlinear equations for Baouendi-Grushin operators
Ismail Kombe
The purpose of this paper is to study the nonexistence of positive solutions of the doubly nonlinear equation \[\begin{cases} \frac{\partial u}{\partial t}=\nabla_{\gamma}\cdot (u^{m-1}|\nabla_{\gamma} u|^{p-2}\nabla_{\gamma} u) +Vu^{m+p-2} & \text{in}\quad \Omega \times (0, T ) ,\\ u(x,0)=u_{0}(x)\geq 0 & \text{in} \quad\Omega, \\ u(x,t)=0 & \text{on}\quad \partial\Omega\times (0, T),\end{cases}\] where $\nabla_{\gamma}=(\nabla_x, |x|^{2\gamma}\nabla_y)$, $x\in \mathbb{R}^d, y\in \mathbb{R}^k$, $\gamma>0$, $\Omega$ is a metric ball in $\mathbb{R}^{N}$, $V\in L_{\text{loc}}^1(\Omega)$, $m\in \mathbb{R}$, $1 < p < d+k$ and $m + p - 2 > 0$. The exponents $q^{*}$ are found and the nonexistence results are proved for $q^{*} ≤ m+p < 3$.
Ismail Kombe. On the nonexistence of positive solutions to doubly nonlinear equations for Baouendi-Grushin operators. Discrete & Continuous Dynamical Systems - A, 2013, 33(11&12): 5167-5176. doi: 10.3934/dcds.2013.33.5167.
Coagulation and fragmentation processes with evolving size and shape profiles: A semigroup approach
Wilson Lamb, Adam McBride and Louise Smith
We investigate a class of bivariate coagulation-fragmentation equations. These equations describe the evolution of a system of particles that are characterised not only by a discrete size variable but also by a shape variable which can be either discrete or continuous. Existence and uniqueness of strong solutions to the associated abstract Cauchy problems are established by using the theory of substochastic semigroups of operators.
Wilson Lamb, Adam McBride, Louise Smith. Coagulation and fragmentation processes with evolving size and shape profiles: A semigroup approach. Discrete & Continuous Dynamical Systems - A, 2013, 33(11&12): 5177-5187. doi: 10.3934/dcds.2013.33.5177.
Maximal regularity and global existence of solutions to a quasilinear thermoelastic plate system
Irena Lasiecka and Mathias Wilke
We consider a quasilinear PDE system which models nonlinear vibrations of a thermoelastic plate defined on a bounded domain in $\mathbb{R}^n$. Global Well-posedness of solutions is shown by applying the theory of maximal parabolic regularity of type $L_p$. In addition, we prove exponential decay rates for strong solutions and their derivatives.
Irena Lasiecka, Mathias Wilke. Maximal regularity and global existence of solutions to a quasilinear thermoelastic plate system. Discrete & Continuous Dynamical Systems - A, 2013, 33(11&12): 5189-5202. doi: 10.3934/dcds.2013.33.5189.
Stability estimates for semigroups on Banach spaces
Yuri Latushkin and Valerian Yurov
For a strongly continuous operator semigroup on a Banach space, we revisit a quantitative version of Datko's Theorem and the estimates for the constant $M$ satisfying the inequality $||T(t)|| ≤ M e^{\omega t}$, for all $t\ge0$, in terms of the norm of the convolution and other operators involved in Datko's Theorem. We use techniques recently developed by B. Helffer and J. Sjöstrand for the Hilbert space case to estimate $M$ in terms of the norm of the resolvent of the generator of the semigroup in the right half-plane.
Yuri Latushkin, Valerian Yurov. Stability estimates for semigroups on Banach spaces. Discrete & Continuous Dynamical Systems - A, 2013, 33(11&12): 5203-5216. doi: 10.3934/dcds.2013.33.5203.
Recovering damping and potential coefficients for an inverse non-homogeneous second-order hyperbolic problem via a localized Neumann boundary trace
Shitao Liu and Roberto Triggiani
We consider a second-order hyperbolic equation defined on an open bounded domain $\Omega$ in $\mathbb{R}^n$ for $n \geq 2$, with $C^2$-boundary $\Gamma = \partial \Omega = \overline{\Gamma_0 \cup \Gamma_1}$, $\Gamma_0 \cap \Gamma_1 = \emptyset$, subject to non-homogeneous Dirichlet boundary conditions for the entire boundary $\Gamma$. We then study the inverse problem of determining both the damping and the potential (source) coefficients simultaneously, in one shot, by means of an additional measurement of the Neumann boundary trace of the solution, in a suitable, explicit sub-portion $\Gamma_1$ of the boundary $\Gamma$, and over a computable time interval $T > 0$. Under sharp conditions on the complementary part $\Gamma_0 = \Gamma \backslash \Gamma_1$, $T > 0$, and under sharp regularity requirements on the data, we establish the two canonical results in inverse problems: (i) uniqueness and (ii) Lipschitz-stability. The latter (ii) is the main result of the paper. Our proof relies on a few main ingredients: (a) sharp Carleman estimates at the $H^1(\Omega) \times L^2(\Omega)$-level for second-order hyperbolic equations [23], originally introduced for control theory issues; (b) a correspondingly implied continuous observability inequality at the same energy level [23]; (c) sharp interior and boundary regularity theory for second-order hyperbolic equations with Dirichlet boundary data [14,15,16]. The proof of the linear uniqueness result (Section 3) also takes advantage of a convenient tactical route ``post-Carleman estimates" proposed by V. Isakov in [8, Thm. 8.2.2, p. 231]. Expressing the final results for the nonlinear inverse problem directly in terms of the data offers an additional challenge.
Shitao Liu, Roberto Triggiani. Recovering damping and potential coefficients for an inverse non-homogeneous second-order hyperbolic problem via a localized Neumann boundary trace. Discrete & Continuous Dynamical Systems - A, 2013, 33(11&12): 5217-5252. doi: 10.3934/dcds.2013.33.5217.
An identification problem for a nonlinear one-dimensional wave equation
Alfredo Lorenzi and Eugenio Sinestrari
We prove the existence of a spatial coefficient in front of a nonlinear term in a one-dimensional wave equation when, in addition to classical initial and boundary condition, an integral mean involving the displacement is prescribed.
Alfredo Lorenzi, Eugenio Sinestrari. An identification problem for a nonlinear one-dimensional wave equation. Discrete & Continuous Dynamical Systems - A, 2013, 33(11&12): 5253-5271. doi: 10.3934/dcds.2013.33.5253.
A thermo piezoelectric model: Exponential decay of the total energy
Gustavo Alberto Perla Menzala and Julian Moises Sejje Suárez
We consider a linear evolution model describing a piezoelectric phenomenon under thermal effects as suggested by R. Mindlin [13] and W. Nowacki [16]. We prove the equivalence between exponential decay of the total energy and an observability inequality for an anisotropic elastic wave system. Our strategy is to use a decoupling method to reduce the problem to an equivalent observability inequality for an anisotropic elastic wave system and assume a condition which guarantees that the corresponding elliptic operator has no eigenfunctions with null divergence.
Gustavo Alberto Perla Menzala, Julian Moises Sejje Su\u00E1rez. A thermo piezoelectric model: Exponential decay of the total energy. Discrete & Continuous Dynamical Systems - A, 2013, 33(11&12): 5273-5292. doi: 10.3934/dcds.2013.33.5273.
How to distinguish a local semigroup from a global semigroup
J. W. Neuberger
For a given autonomous time-dependent system that generates either a global, in time, semigroup or else only a local, in time, semigroup, a test involving a linear eigenvalue problem is given which determines which of `global' or `local' holds. Numerical examples are given. A linear transformation $A$ is defined so that one has `global' or `local' depending on whether $A$ does not or does have a positive eigenvalue. There is a possible application to Navier-Stokes problems..
J. W. Neuberger. How to distinguish a local semigroup from a global semigroup. Discrete & Continuous Dynamical Systems - A, 2013, 33(11&12): 5293-5303. doi: 10.3934/dcds.2013.33.5293.
Rational approximations of semigroups without scaling and squaring
Frank Neubrander, Koray Özer and Teresa Sandmaier
We show that for all $q\ge 1$ and $1\le i \le q$ there exist pairwise conjugate complex numbers $b_{q,i}$ and $\lambda_{q,i}$ with $\mbox{Re} (\lambda_{q,i}) > 0$ such that for any generator $(A, D(A))$ of a bounded, strongly continuous semigroup $T(t)$ on Banach space $X$ with resolvent $R(\lambda,A) := (\lambda I-A)^{-1}$ the expression $\frac{b_{q,1}}{t}R(\frac{\lambda_{q,1}}{t},A) + \frac{b_{q,2}}{t}R(\frac{\lambda_{q,2}}{t},A) + \cdots + \frac{b_{q,q}}{t}R(\frac{\lambda_{q,q}}{t},A)$ provides an excellent approximation of the semigroup $T(t)$ on $D(A^{2q-1})$. Precise error estimates as well as applications to the numerical inversion of the Laplace transform are given.
Frank Neubrander, Koray \u00D6zer, Teresa Sandmaier. Rational approximations of semigroups without scaling and squaring. Discrete & Continuous Dynamical Systems - A, 2013, 33(11&12): 5305-5317. doi: 10.3934/dcds.2013.33.5305.
Understanding Thomas-Fermi-Like approximations: Averaging over oscillating occupied orbitals
John P. Perdew and Adrienn Ruzsinszky
The Thomas-Fermi equation arises from the earliest density functional approximation for the ground-state energy of a many-electron system. Its solutions have been carefully studied by mathematicians, including J.A. Goldstein. Here we will review the approximation and its validity conditions from a physics perspective, explaining why the theory correctly describes the core electrons of an atom but fails to bind atoms to form molecules and solids. The valence electrons are poorly described in Thomas-Fermi theory, for two reasons: (1) This theory neglects the exchange-correlation energy, ``nature's glue". (2) It also makes a local density approximation for the kinetic energy, which neglects important shell-structure effects in the exact kinetic energy that are responsible for the structure of the periodic table of the elements. Finally, we present a tentative explanation for the fact that the shell-structure effects are relatively unimportant for the exact exchange energy, which can thus be more usefully described by a local density or semilocal approximation (as in the popular Kohn-Sham theory): The exact exchange energy from the occupied Kohn-Sham orbitals has an extra sum over orbital labels and an extra integration over space, in comparison to the kinetic energy, and thus averages out more of the atomic individuality of the orbital oscillations.
John P. Perdew, Adrienn Ruzsinszky. Understanding Thomas-Fermi-Like approximations: Averaging over oscillating occupied orbitals. Discrete & Continuous Dynamical Systems - A, 2013, 33(11&12): 5319-5325. doi: 10.3934/dcds.2013.33.5319.
An interface problem: The two-layer shallow water equations
Madalina Petcu and Roger Temam
The aim of this article is to study a model of two superposed layers of fluid governed by the shallow water equations in space dimension one. Under some suitable hypotheses the governing equations are hyperbolic. We introduce suitable boundary conditions and establish a result of existence and uniqueness of smooth solutions for a limited time for this model.
Madalina Petcu, Roger Temam. An interface problem: The two-layer shallow water equations. Discrete & Continuous Dynamical Systems - A, 2013, 33(11&12): 5327-5345. doi: 10.3934/dcds.2013.33.5327.
Global existence via a multivalued operator for an Allen-Cahn-Gurtin equation
Michel Pierre and Morgan Pierre
The main goal of this paper is to prove existence of global solutions in time for an Allen-Cahn-Gurtin model of pseudo-parabolic type. Local solutions were known to ``blow up" in some sense in finite time. It is proved that the equation is actually governed by a monotone-like operator. It turns out to be multivalued and measure-valued. The measures are singular with respect to the Lebesgue measure. This operator allows to extend the local solutions globally in time and to fully solve the evolution problem. The asymptotic behavior is also analyzed.
Michel Pierre, Morgan Pierre. Global existence via a multivalued operator for an Allen-Cahn-Gurtin equation. Discrete & Continuous Dynamical Systems - A, 2013, 33(11&12): 5347-5377. doi: 10.3934/dcds.2013.33.5347.
Singular limits for the two-phase Stefan problem
Jan Prüss, Jürgen Saal and Gieri Simonett
We prove strong convergence to singular limits for a linearized fully inhomogeneous Stefan problem subject to surface tension and kinetic undercooling effects. Different combinations of $\sigma \to \sigma_0$ and $\delta\to\delta_0$, where $\sigma,\sigma_0\ge 0$ and $\delta,\delta_0\ge 0$ denote surface tension and kinetic undercooling coefficients respectively, altogether lead to five different types of singular limits. Their strong convergence is based on uniform maximal regularity estimates.
Jan Pr\u00FCss, J\u00FCrgen Saal, Gieri Simonett. Singular limits for the two-phase Stefan problem. Discrete & Continuous Dynamical Systems - A, 2013, 33(11&12): 5379-5405. doi: 10.3934/dcds.2013.33.5379.
On the manifold of closed hypersurfaces in $\mathbb{R}^n$
Jan Prüss and Gieri Simonett
Several results from differential geometry of hypersurfaces in $\mathbb{R}^n$ are derived to form a tool box for the direct mapping method. The latter technique has been widely employed to solve problems with moving interfaces, and to study the asymptotics of the induced semiflows.
Jan Pr\u00FCss, Gieri Simonett. On the manifold of closed hypersurfaces in $\\mathbb{R}^n$. Discrete & Continuous Dynamical Systems - A, 2013, 33(11&12): 5407-5428. doi: 10.3934/dcds.2013.33.5407.
Integration with vector valued measures
M. M. Rao
Of the many variations of vector measures, the Fréchet variation is finite valued but only subadditive. Finding a `controlling' finite measure for these in several cases, it is possible to develop a useful integration of the Bartle-Dunford-Schwartz type for many linear metric spaces. These include the generalized Orlicz spaces, $L^{\varphi}(\mu)$, where $\varphi$ is a concave $\varphi$-function with applications to stochastic measures $Z(\cdot)$ into various Fréchet spaces useful in prediction theory. In particular, certain $p$-stable random measures and a (sub) class of these leading to positive infinitely divisible ones are detailed.
M. M. Rao. Integration with vector valued measures. Discrete & Continuous Dynamical Systems - A, 2013, 33(11&12): 5429-5440. doi: 10.3934/dcds.2013.33.5429.
Estimating eigenvalues of an anisotropic thermal tensor from transient thermal probe measurements
Steve Rosencrans, Xuefeng Wang and Shan Zhao
We propose a new method for estimating the eigenvalues of the thermal tensor of an anisotropically heat-conducting material, from transient thermal probe measurements of a heated thin cylinder.
We assume the principal axes of the thermal tensor to have been identified, and that the cylinder is oriented parallel to one of these axes (but we outline what is needed to overcome this limitation). The method involves estimating the first two Dirichlet eigenvalues (exponential decay rates) from transient thermal probe data. These implicitly determine the thermal diffusion coefficients (thermal tensor eigenvalues) in the directions of the other two axes. The process is repeated two more times with cylinders parallel to each of the remaining axes.
The method is tested by simulating a temperature probe time-series (obtained by solving the anisotropic heat equation numerically) and comparing the computed thermal tensor eigenvalues with their true values. The results are generally accurate to less than $1\%$ error.
Steve Rosencrans, Xuefeng Wang, Shan Zhao. Estimating eigenvalues of an anisotropic thermal tensor from transient thermal probe measurements. Discrete & Continuous Dynamical Systems - A, 2013, 33(11&12): 5441-5455. doi: 10.3934/dcds.2013.33.5441.
Hardy type inequalities and hidden energies
Juan Luis Vázquez and Nikolaos B. Zographopoulos
We obtain new insights into Hardy type Inequalities and the evolution problems associated to them. Surprisingly, the connection of the energy with the Hardy functionals is nontrivial, due to the presence of a Hardy singularity energy. This corresponds to a loss for the total energy. These problems are defined on bounded domains or the whole space.
We also consider equivalent problems with inverse square potential on exterior domains or the whole space. The extra energy term is then present as an effect that comes from infinity, a kind of hidden energy. In this case, in an unexpected way, this term is additive to the total energy, and it may even constitute the main part of it.
Juan Luis V\u00E1zquez, Nikolaos B. Zographopoulos. Hardy type inequalities and hidden energies. Discrete & Continuous Dynamical Systems - A, 2013, 33(11&12): 5457-5491. doi: 10.3934/dcds.2013.33.5457.
Semi linear parabolic equations with nonlinear general Wentzell boundary conditions
Mahamadi Warma
Let $A$ be a uniformly elliptic operator in divergence form with bounded coefficients. We show that on a bounded domain $Ω ⊂ \mathbb{R}^N$ with Lipschitz continuous boundary $∂Ω$, a realization of $Au-\beta_1(x,u)$ in $C(\bar{Ω})$ with the nonlinear general Wentzell boundary conditions $[Au-\beta_1(x,u)]|_{∂Ω}-\Delta_\Gamma u+\partial_\nu^au+\beta_2(x,u)= 0$ on $∂Ω$ generates a strongly continuous nonlinear semigroup on $C(\bar{Ω})$. Here, $\partial_\nu^au$ is the conormal derivative of $u$, and $\beta_1(x,\cdot)$ ($x \in Ω$), $\beta_2(x,\cdot)$ ($x \in ∂Ω$) are continuous on $\mathbb{R}$ satisfying a certain growth condition.
Mahamadi Warma. Semi linear parabolic equations with nonlinear general Wentzell boundary conditions. Discrete & Continuous Dynamical Systems - A, 2013, 33(11&12): 5493-5506. doi: 10.3934/dcds.2013.33.5493.
Positive solutions of nonlinear equations via comparison with linear operators
Jeffrey R. L. Webb
We discuss positive solutions of problems that arise from nonlinear boundary value problems in the particular situation where the nonlinear term $f(t,u)$ depends explicitly on $t$ and this dependence is crucial. We give new fixed point index results using comparisons with linear operators. These prove new results on existence of positive solutions under some conditions which can be sharp.
Jeffrey R. L. Webb. Positive solutions of nonlinear equations via comparison with linear operators. Discrete & Continuous Dynamical Systems - A, 2013, 33(11&12): 5507-5519. doi: 10.3934/dcds.2013.33.5507.
An example of large global smooth solution of 3 dimensional Navier-Stokes equations without pressure
Qi S. Zhang
We construct a global smooth solution of 3 dimensional Navier-Stokes equations in the torus, which also solves the heat equation. The solution is three dimensional and it can be arbitrarily large.
Qi S. Zhang. An example of large global smooth solution of 3 dimensional Navier-Stokes equations without pressure. Discrete & Continuous Dynamical Systems - A, 2013, 33(11&12): 5521-5523. doi: 10.3934/dcds.2013.33.5521.
Weighted pseudo almost automorphic mild solutions to semilinear integral equations with $S^{p}$-weighted pseudo almost automorphic coefficients
Rui Zhang, Yong-Kui Chang and G. M. N'Guérékata
In this paper, we consider the existence of weighted pseudo almost automorphic solutions of the semilinear integral equation $x(t)= \int_{-\infty}^{t}a(t-s)[Ax(s) + f(s,x(s))]ds, \ t\in\mathbb{R}$ in a Banach space $\mathbb{X}$, where $a\in L^{1}(\mathbb{R}_{+})$, $A$ is the generator of an integral resolvent family of linear bounded operators defined on the Banach space $\mathbb{X}$, and $f : \mathbb{R}\times\mathbb{X} \rightarrow \mathbb{X}$ is a weighted pseudo almost automorphic function. The main results are proved by using integral resolvent families, suitable composition theorems combined with the theory of fixed points.
Rui Zhang, Yong-Kui Chang, G. M. N\'Gu\u00E9r\u00E9kata. Weighted pseudo almost automorphic mild solutions to semilinear integral equations with $S^{p}$-weighted pseudo almost automorphic coefficients. Discrete & Continuous Dynamical Systems - A, 2013, 33(11&12): 5525-5537. doi: 10.3934/dcds.2013.33.5525. | CommonCrawl |
\begin{document}
\numberwithin{theorem}{section}
\begin{abstract}
Recently Bowden, Hensel and Webb defined the {\em fine curve graph} for
surfaces, extending the notion of curve graphs for the study of
homeomorphism or diffeomorphism groups of surfaces.
Later Long, Margalit, Pham, Verberne and Yao proved that for a closed
surface of genus~$g\geqslant 2$,
the automorphism group of the fine graph is naturally isomorphic
to the homeomorphism group of the surface.
We extend this result to the torus case $g=1$; in fact our method works for
more general surfaces, compact or not, orientable or not.
We also discuss the case of a smooth version of the fine graph.
\end{abstract}
\title{Automorphisms of some variants of fine graphs}
\sloppy
\section{Introduction}
\subsection{Context and results}
For a connected, compact surface $\Sigma_g$ of genus $g\geqslant 1$, Bowden, Hensel and Webb~\cite{BHW} recently introduced the {\em fine curve graph} $\mathcal{C}^\dagger(\Sigma)$, as the graph whose vertices are all the essential closed curves on $\Sigma$, with an edge between two vertices $a$ and $b$ whenever $a\cap b=\emptyset$, if $g\geqslant 2$, and whenever
$|a\cap b|\leqslant 1$ if $g=1$. They proved that for every $g\geqslant 1$, the graph $\mathcal{C}^\dagger(\Sigma)$ is hyperbolic, and derived a construction of an infinite dimensional family of quasi-morphisms on $\Homeo_0(\Sigma)$, thereby answering long standing questions of Burago, Ivanov and Polterovich.
The ancestor of the fine graph is the usual curve complex of a surface $\Sigma$, \textsl{i.e.}, the complex whose vertices are the isotopy classes of essential curves, with an edge (or a simplex, more generally) between some vertices if and only if they have disjoint representatives. Since its introduction by Harvey~\cite{Harvey}, the curve complex of a surface has been an extremely useful tool for the study of the mapping class group $\mathrm{Mod}(\Sigma)$ of that surface, as it acts on it naturally. In particular, the fact that this complex is hyperbolic, discovered by Masur and Minsky, has greatly improved the understanding of the mapping class groups (see~\cite{MM1,MM2}). The result of Bowden, Hensel and Webb, promoting the hyperbolicity of the curve complex to that of the fine curve graph, opens the door both to the study of what classical properties of usual curve complexes have counterparts in the fine curve graph, and to the use of this graph to derive properties of homeomorphism groups. A first step in this direction was taken by Bowden, Hensel, Mann, Militon and Webb~\cite{BHMMW}, who explored the metric properties of the action of $\Homeo(\Sigma)$ on this hyperbolic graph.
A classical theorem by Ivanov~\cite{Ivanov} states that, when $\Sigma$ is a closed surface of genus $g\geqslant 2$, the natural map $\mathrm{Mod}(\Sigma)\to\mathrm{Aut}(\mathcal{C}(\Sigma))$ is an isomorphism. Recently Long, Margalit, Pham, Verberne and Yao~\cite{LMPVY} proved the following natural counterpart of Ivanov's theorem for fine graphs: provided $\Sigma$ is a compact orientable surface of genus $g\geqslant 2$, the natural map \[ \Homeo(\Sigma)\longrightarrow\mathrm{Aut}(\mathcal{C}^\dagger(\Sigma)) \] is an isomorphism. They also suggested that this map (with the appropriate version of $\mathcal{C}^\dagger$) may also be an isomorphism when $g=1$, and conjectured that the automorphism group of the fine curve graph of smooth curves, should be nothing more than $\mathrm{Diff}(\Sigma)$.
In this article, we address both these questions. Our motivation originates from the case of the torus: excited by~\cite{BHMMW},
we wanted to understand more closely the relation between the rotation set of homeomorphisms isotopic to the identity and the metric properties of their actions on the fine graph. This subject will be treated in another article, joint with Passeggi and Sambarino~\cite{Fantome}. The methods developed in the present article are valid not only for the torus but for a large class of surfaces.
We work on nonspherical surfaces (\textsl{i.e.}, surfaces not embeddable in the $2$-sphere, or equivalently, containing at least one nonseparating simple closed curve), orientable or not, compact or not. We consider the graph $\NCfin{\pitchfork}(\Sigma)$, whose vertices are the nonseparating simple closed curves, and with an edge between two vertices $a$ and $b$ whenever they are either disjoint, or have exactly one, topologically transverse intersection point (see the beginning of Section~\ref{sec:Lemmes1-2-3} for more detail). Our first result answers a problem raised in~\cite{LMPVY}. \begin{theorem}\label{thm:AutC1}
Let $\Sigma$ be a connected, nonspherical surface, without boundary.
Then the natural map $\Homeo(\Sigma)\to\mathrm{Aut}(\NCfin{\pitchfork}(\Sigma))$
is an isomorphism. \end{theorem} Our second result concerns the smooth version of fine graphs. We consider the graph $\mathcal{NC}_\pitchfork^{\dagger\infty}(\Sigma)$ whose vertices are the smooth nonseparating curves in $\Sigma$, with an edge between $a$ and $b$ if they are disjoint or have one, transverse intersection point, in the differentiable sense (in particular, $\mathcal{NC}_\pitchfork^{\dagger\infty}(\Sigma)$ is not the subgraph of $\NCfin{\pitchfork}(\Sigma)$ induced by the vertices corresponding to smooth curves: it has fewer edges). The following result partially confirms the
conjecture of~\cite{LMPVY}; here we restrict to the case of orientable surfaces for simplicity. \begin{theorem}\label{thm:AutCFinLisse}
Let $\Sigma$ be a connected, orientable, nonspherical surface, without boundary.
Then all the automorphisms of $\mathcal{NC}_\pitchfork^{\dagger\infty}(\Sigma)$ are realized
by homeomorphisms of~$\Sigma$. \end{theorem}
In other words, if we denote by $\Homeo_{\infty \pitchfork}(\Sigma)$ the subgroup of $\Homeo(\Sigma)$ preserving the collection of smooth curves and preserving transversality, then the natural map \[ \Homeo_{\infty \pitchfork}(\Sigma) \longrightarrow \mathrm{Aut}(\mathcal{NC}_\pitchfork^{\dagger\infty}(\Sigma)) \] is an isomorphism. We were surprised to realize however that $\Homeo_{\infty \pitchfork}(\Sigma)$ is strictly larger than $\mathrm{Diff}(\Sigma)$.
\begin{proposition}\label{prop:PasDiff}
Every surface $\Sigma$ admits a homeomorphism $f$
such that
$f$ and $f^{-1}$ preserve the set of smooth curves, and preserve
transversality, but such that neither $f$ nor $f^{-1}$ is differentiable.
In particular, the natural map
$$\mathrm{Diff}(\Sigma)\longrightarrow \mathrm{Aut}(\mathcal{NC}_\pitchfork^{\dagger\infty}(\Sigma))$$
is not surjective. \end{proposition}
\subsection{Idea of the proof of Theorem~\ref{thm:AutC1}}
The main step in this proof is the following. \begin{proposition}\label{prop:Types}
If $\{a,b\}$, or $\{a,b,c\}$ is a $2$-clique or
a $3$-clique of $\NCfin{\pitchfork}(\Sigma)$ then, from the graph
structure of $\NCfin{\pitchfork}(\Sigma)$, we can tell the type of the clique. \end{proposition} If $\{a_1,\ldots, a_n\}$ is an $n$-clique in the graph $\NCfin{\pitchfork}(\Sigma)$, the homeomorphism type of the subset $\cup_{j=1}^n a_j$ of $\Sigma$ will be called the {\em type} of the $n$-clique. We will explore this only for $2$ and $3$-cliques. A $2$-clique $\{a,b\}$, \textsl{i.e.}, an edge of the graph $\NCfin{\pitchfork}(\Sigma)$, may have two distinct types: the intersection $a\cap b$ may be empty or not. For a $3$-clique $\{a,b,c\}$, up to permuting the curves $a$, $b$ and $c$, the cardinals of the intersections $a\cap b$, $a\cap c$ and $b\cap c$, respectively, may be $(1,1,1)$, or $(1,1,0)$, or $(1,0,0)$, or $(0,0,0)$. This determines the type of the $3$-clique, except in the case $(1,1,1)$, where the intersection points $a\cap b$, $a\cap c$ and $b\cap c$ may be pairwise distinct, in which case we will speak of a $3$-clique of type {\em necklace}, or these intersection points may be equal, in which case we will speak of a $3$-clique of type {\em bouquet}, see Figure~\ref{fig:Bouquet-Collier}. \begin{figure}
\caption{A bouquet (left) and a necklace (right) of three circles.}
\label{fig:Bouquet-Collier}
\end{figure}
The main bulk of the proof of Proposition~\ref{prop:Types} consists in distinguishing the $3$-cliques of type necklace from any other $3$-clique of $\NCfin{\pitchfork}(\Sigma)$. Here, the key is that among all the $3$-cliques, the cliques $\{a,b,c\}$ of type necklace are exactly those such that the union $a\cup b\cup c$ contains nonseparating simple closed curves other than $a$, $b$ and $c$. In terms of the graph structure, this leads to the following property,
denoted by $N(a,b,c)$, which turns out to characterize these cliques:
{\em There exists a finite set $F$ of at most $8$ vertices of $\NCfin{\pitchfork}(\Sigma)$, all distinct from $a$, $b$ and $c$, such that every vertex $d$ connected to $a$, $b$ and $c$ in this graph, is connected to at least one element of $F$.}
From this,
we will easily characterize all the configurations of $2$-cliques and $3$-cliques
in terms of similar statements in the first order logic of
the graph $\NCfin{\pitchfork}(\Sigma)$.
Now, let $T(\NCfin{\pitchfork}(\Sigma))$ denote the set of edges $\{a,b\}$ of
$\NCfin{\pitchfork}(\Sigma)$ satisfying $|a\cap b|=1$. Then we have a map \[ \mathrm{Point}\colon \ T(\NCfin{\pitchfork}(\Sigma))\to\Sigma, \] which to each edge $\{a,b\}$ of $\NCfin{\pitchfork}(\Sigma)$, associates the intersection point $a\cap b$. The next step in the the proof of Theorem~\ref{thm:AutC1} now consists in characterizing the equality $\mathrm{Point}(a,b)=\mathrm{Point}(c,d)$ in terms of the structure of the graph. This characterization shows that every automorphism of $\NCfin{\pitchfork}(\Sigma)$ is realized by some bijection of $\Sigma$; then we prove that such a bijection is necessarily a homeomorphism (see Proposition~\ref{prop:BijHomeo}).
In order to characterize the equality $\mathrm{Point}(a,b)=\mathrm{Point}(c,d)$, we introduce on $T(\NCfin{\pitchfork}(\Sigma))$ the relation $\diamondvert$ generated, essentially (see section~\ref{sec:Adjacency} for details), by $(a,b)\diamondvert(b,c)$ if $(a,b,c)$ is a $3$-clique of type bouquet. Obviously, if $(a,b)\diamondvert(c,d)$ then $\mathrm{Point}(a,b)=\mathrm{Point}(c,d)$. Interestingly, the converse is false, but we can still use this idea in order to characterize the points of $\Sigma$ in terms of the graph structure of $\NCfin{\pitchfork}(\Sigma)$.
This subtlety between the relation $\diamondvert$ and the equality of points is related to the non smoothness of the curves involved, and more precisely, to the fact that a curve may spiral infinitely with respect to another curve in a neighborhood of a common point. We think that this phenomenon is of independent interest and we investigate it in Section~\ref{sec:LocalSubgraphs}. In particular, we can easily state, in terms of the graph structure of $\NCfin{\pitchfork}(\Sigma)$, an obstruction for a homeomorphism to be conjugate to a $C^1$-diffeomorphism, see Section~\ref{ssec:Tourbillons}.
\subsection{Ideas of the proof of Theorem~\ref{thm:AutCFinLisse}}
In the smooth case, the adaptation of our proof of Theorem~\ref{thm:AutC1} fails from the start: indeed, the closed curves contained in the union $a\cup b\cup c$ of a necklace, and distinct from $a$, $b$ and $c$, are not smooth. This suggests the idea to use sequences of curves (at the expense of losing the characterizations of configurations in terms of first order logic).
This time it is easiest to first characterize disjointness of curves (see Lemma~\ref{lem:DisjointLisse}), and then recover the different types of $3$-cliques. Then the strategy follows the $C^0$ case.
Once we start to work with sequences, it is natural to say that a sequence $(f_n)$ of curves not escaping to infinity converges to $a$ in some weak sense, if for every vertex $d$ such that $\{a,d\}$ is an edge of the graph, $\{f_n,d\}$ is also an edge for all $n$ large enough. As it turns out, this property implies convergence in $C^0$-sense to $a$, and is implied by convergence in $C^1$-sense. But it is not equivalent to the convergence in $C^1$-sense, and it is precisely this default of $C^1$-convergence that enables us to distinguish between disjoint or transverse pairs of curves.
Interestingly, this simple criterion for disjointness has no counterpart in the $C^0$-setting. Indeed, in that setting, no sequence of curves converges in this weak sense: given a curve $a$, and a sequence $(f_n)$ of curves with, say, some accumulation point in $a$, we can build a curve $d$ intersecting $a$ once transversally (topologically), but oscillating so much that it itersects every $f_n$ several times. From this perspective, none of our approaches in the $C^0$-setting and in the $C^\infty$-setting are directly adaptable to the other.
\subsection{Further comments}
We can imagine many variants of fine graphs. For example, in the arXiv version of~\cite{BHW}, for the case of the torus they worked with the graph $\NCfin{\pitchfork}(\Sigma)$ on which we are working here, whereas in the published version, they changed to a fine graph in which two curves $a$ and $b$ are still related by an edge when they have one intersection, not necessarily transverse.
More generally, in the spirit of Ivanov's metaconjecture, we expect that the group of automorphisms should not change from any reasonable variant to another. And indeed, using the ideas of \cite[Section~2]{LMPVY} and those presented here, we can navigate between various versions of fine graphs, and recover, from elementary properties of one version, the configurations defining the edges in another version, thus proving that their automorphism groups are naturally isomorphic. From this perspective, it seems satisfying to recover the group of homeomorphisms of the surface as the automorphism group of any reasonable variant of the fine graph. In this vein, we should mention that the results of~\cite[Section~2]{LMPVY} directly yield a natural map $\mathrm{Aut}(\Cfin{}(\Sigma))\to\mathrm{Aut}(\NCfin{\pitchfork}(\Sigma))$, and from there, our proof of Theorem~\ref{thm:AutC1} may be used as an alternative proof of their main result.
All reasonable variants of the fine graphs should be quasi-isometric, and a unifying theorem (yet out of reach today, as it seems to us) would certainly be a counterpart of the theorem by Rafi and Schleimer~\cite{RafiSchleimer}, which states that every quasi-isometry of the usual curve graph is bounded distance from an isometry.
\subsection{Organization of the article} Section~\ref{sec:Lemmes1-2-3} is devoted to the recognition of the $3$-cliques in the $C^0$-setting, and of some other configurations regarding the nonorientable case. We encourage the reader to skip, at first reading, everything that concerns the nonorientable case: these points shoud be easily identified, and this halves the length of the proof. In Section~\ref{sec:AutHomeo} we prove Theorem~\ref{thm:AutC1}. In Section~\ref{sec:LocalSubgraphs} we characterize, from the topological viewpoint, the relation $\diamondvert$ introduced above in terms of the graph structure, and deduce our obstruction to differentiability. Finally in Section~\ref{sec:AutCFinLisse} we prove Theorem~\ref{thm:AutCFinLisse} and Proposition~\ref{prop:PasDiff}.
\subsection*{Acknowledgments}
We thank Kathryn Mann for encouraging discussions, and Dan Margalit for his extensive feedback on a preliminary version of this manuscript.
\section{Recognizing configurations of curves}\label{sec:Lemmes1-2-3}
\subsection{Standard facts and notation} We will use, often without mention, the following easy or standard facts for curves on surfaces.
The first is the classification of connected, topological surfaces with boundary (not necessarily compact). In particular, every topological surface admits a smooth structure. Given a closed curve $a$ in a surface $\Sigma$, we can apply this classification to $\Sigma\smallsetminus a$ and understand all possible configurations of simple curves; this is the so-called {\em change of coordinates principle} in the vocabulary of the book of Farb and Margalit~\cite{FarbMargalit}.
In particular, every closed curve $a$ has a neighborhood homeomorphic to an annulus or a M\"obius strip in which $a$ is the ``central curve''. Very often in this article, we will consider the curves $a'$ obtained by deforming $a$ in such a neighborhood, so that $a'$ is disjoint from $a$ in the first case, or intersects it once, transversely, in the second, as in Figure~\ref{fig:FigureFacts}. We will say that $a'$ is obtained by {\em pushing $a$ aside}.
The change of coordinates principle also applies to
finite graphs embedded in $\Sigma$: there is a homeomorphism of $\Sigma$ that sends any given graph to a smooth graph, such that all edges connected to a given vertex leave it in distinct directions. In the simple case when the graph is the union of two or three simple closed curves that pairwise intersect at most once, this observation justifies the description of the possible configurations of cliques in the introduction. This also enables, provided two curves $a$ and $b$ intersect at a single point (or more generally at a finite number of points), to speak of a {\em transverse} (also called {\em essential}), or to the contrary {\em inessential}, intersection point, as we did in the introduction.
Here are two other useful facts. \begin{fact}\label{fact:NonSep}
A simple closed curve $a$ in a surface, is nonseparating if and only
if there exists a closed curve $b$, such that $a\cap b$ is a
single point and this intersection is transverse. \end{fact} \begin{fact}\label{fact:TroisArcsSep}
Let $p,q$ be two distinct points, and $x,x',x''$ three simple arcs,
each with end-points $p$ and $q$, such that
\[ x \cap x' = x' \cap x'' = x \cap x'' = \{p,q\}. \]
If two of the three curves $x \cup x', x' \cup x'', x'' \cup x$ are
separating, then the third one is also separating. \end{fact} \begin{proof}
Denote $y=x\smallsetminus\{p,q\}$, the arc $x$ without its ends, and
similarly, define $y'$ and $y''$. Suppose $x\cup x'$ and $x\cup x''$ are
separating.
Denote by $\Sigma_1, \Sigma_2$, resp. $\Sigma_3, \Sigma_4$,
the components of
$\Sigma\smallsetminus(x\cup x')$, resp. $\Sigma\smallsetminus(x\cup x'')$,
where $\Sigma_2$ contains $y''$ and $\Sigma_4$ contains $y'$.
By looking at neighborhoods of $p$ and $q$
(see Figure~\ref{fig:FigureFacts}, left), we see that
$\Sigma'=\Sigma_2\cap\Sigma_4$ is non empty, and that the arc $y$
bounds $\Sigma_1$ on one side, and $\Sigma_3$ on the other, so
$\Sigma''=\Sigma_1\cup y\cup \Sigma_3$ is a surface.
Now, $\Sigma\smallsetminus(x'\cup x'')=\Sigma'\cup\Sigma''$, and
$\Sigma'$ and $\Sigma''$ are disjoint by construction. \end{proof} \begin{figure}
\caption{Left: a neighborhood of $p$. Center: pushing a two-sided curve $a$. Right: pushing a one-sided curve $a$.}
\label{fig:FigureFacts}
\end{figure}
\subsection{Properties characterizing geometric configurations} Now we list the properties, in terms of the graph $\NCfin{\pitchfork}(\Sigma)$, that will be used as characterizations of certain configurations of curves. This allows us to specify the statement of Proposition~\ref{prop:Types}, which will be proved in the next paragraph, and define the relation $\diamondvert$ in terms of the graph $\NCfin{\pitchfork}(\Sigma)$.
In the following, the letters $N, D, T, B$ respectively stand for necklace, disjoint, transverse, and bouquet. If $a,b,c$ are vertices of this graph, we will denote by: \begin{itemize} \item[$\bullet$] $N(a,b,c)$ the property that $\{a,b,c\}$ is a $3$-clique
of $\NCfin{\pitchfork}(\Sigma)$ and there exists a finite set $F$ of at most $8$
vertices of $\NCfin{\pitchfork}(\Sigma)$, all distinct from $a$, $b$ and $c$,
such that for every vertex $d$ such that $\{a,b,c,d\}$ is a $4$-clique,
there is an edge from $d$ to at least one element of $F$, \item[$\bullet$] $D(a,b)$ the property that $\{a,b\}$ is an edge of
$\NCfin{\pitchfork}(\Sigma)$ and there does not
exist a vertex $d$ such that $N(a,b,d)$ holds, \item[$\bullet$] $T(a,b)$ the property that $\{a,b\}$ is an edge of
$\NCfin{\pitchfork}(\Sigma)$ and $D(a,b)$ does not hold, \item[$\bullet$] $B(a,b,c)$ the property that $T(a,b)$, $T(a,c)$, $T(b,c)$
all hold but $N(a,b,c)$ does not. \end{itemize}
The following proposition is the main part of Proposition~\ref{prop:Types}. \begin{proposition}\label{prop:TypesPrecis}
Let $a$, $b$ and $c$ be vertices of the graph $\NCfin{\pitchfork}(\Sigma)$.
Property $N(a,b,c)$ holds if and only if $\{a,b,c\}$
is a $3$-clique of type necklace of $\NCfin{\pitchfork}(\Sigma)$.
\end{proposition}
The proof of Proposition~\ref{prop:TypesPrecis} will occupy the next paragraph. The following corollary complements Proposition~\ref{prop:TypesPrecis} and provides a precise version of Proposition~\ref{prop:Types}. \begin{corollary}\label{coro:TypesPrecis}~
\begin{itemize}
\item Property $D(a,b)$ holds if and only if the curves $a$ and
$b$ are disjoint.
\item Property $T(a,b)$ holds if and only if $a$ and $b$ have
a unique intersection point and the intersection is transverse.
\item Property $B(a,b,c)$ holds if and only if $\{a, b, c\}$
is a $3$-clique of type bouquet.
\end{itemize} \end{corollary}
\begin{proof}
Let $a$ and $b$ be neighbors in the graph $\NCfin{\pitchfork}(\Sigma)$.
Of course, if $a$ and $b$ are disjoint, then $D(a,b)$ holds: there does
not exist a curve $d$ such that $N(a,b,d)$, since this would mean
that $\{a,b,d\}$ is of type necklace and then, by definition, $a$ and $b$
would intersect.
Conversely, suppose that $a$ and $b$ are not disjoint, and let us prove
that $D(a,b)$ does not hold, \textsl{i.e.}, let us find a curve $c$
such that $\{a,b,c\}$ is a $3$-clique of type necklace.
In the case when one of $a$ or $b$ is one-sided, up to exchanging the two,
suppose $a$ is one-sided. Then we may push $a$ in order to find a curve $c$
which makes a $3$-clique of type necklace with $a$ and $b$, see
Figure~\ref{fig:OnComplete} (left). In the case when both $a$ and $b$ are
two-sided, then by the change of coordinates principle,
a regular neighborhood of $a\cup b$ is homeomorphic to a
one-holed torus, embedded in $\Sigma$, with a
choice of meridian and longitude
coming from $a$ and $b$.
In this torus, a curve $c$ with
slope $1$ will form a $3$-clique of type necklace with $a$ and $b$,
see Figure~\ref{fig:OnComplete}, right.
\begin{figure}
\caption{Completing $(a,b)$ to a $3$-clique of type necklace}
\label{fig:OnComplete}
\end{figure}
This proves the first point.
The second point is a straightforward
consequence of the first, and the third simply follows from the second
point together with Proposition~\ref{prop:TypesPrecis}.
\end{proof}
\subsection{Proof of Proposition~\ref{prop:TypesPrecis}}
The following lemma is a key step in the proof of the direct implication in Proposition~\ref{prop:TypesPrecis}.
\begin{lemma}\label{lem:drenccc}
Let $\{a,b,c\}$ be a $3$-clique of $\NCfin{\pitchfork}(\Sigma)$ which is not
of type necklace. Then there exists a vertex $d$ of
$\NCfin{\pitchfork}(\Sigma)$ such that $\{a,b,c,d\}$ is a $4$-clique of
$\NCfin{\pitchfork}(\Sigma)$, and such that $d$ meets every connected
component of $\Sigma\smallsetminus(a\cup b\cup c)$. \end{lemma} Before entering the proof, we note that we cannot remove the hypothesis that $\{a, b, c\}$ is not of type necklace. Indeed, in the flat torus $\Sigma = \mathbb{R}^2/\mathbb{Z}^2$, consider three closed geodesics $a,b,c$ respectively directed by $(1,0)$, $(0,1)$, and $(1, 1)$. By pushing $c$ aside if necessary, we obtain a $3$-clique of type necklace. The complement of $a \cup b \cup c$ in $\Sigma$ has three connected components, and there is no curve $d$ satisfying the conclusion of the lemma. \begin{proof}
In all the proof, we will denote
$\Sigma'=\Sigma\smallsetminus(a\cup b\cup c)$.
Up to permuting the curves $a$, $b$ and $c$, we may suppose that the triple
of cardinals of intersections, $(|a\cap b|, |a\cap c|, |b\cap c|)$, equals
$(1,1,1)$, or $(1,1,0)$, or $(1,0,0)$, or~$(0,0,0)$. We will deal with
these cases separately.
Let us begin with the case $(0, 0, 0)$. If $\Sigma'$ is connected,
then any curve $d$ making a $4$-clique with $(a,b,c)$ satisfies the Lemma.
Such a curve can be found, for example, by pushing $a$ aside.
If $\Sigma'$ has two connected components, denote them by $\Sigma_1$ and
$\Sigma_2$. Since $a$, $b$ and $c$ are each nonseparating, at least two of
the curves $a, b, c$ (say, $a$ and $b$) correspond to boundary components of
both $\Sigma_1'$ and $\Sigma_2'$.
Choose one point $x_a$ in $a$ and one point $x_b$ in $b$. For $i=1,2$, there
is an arc $\gamma_i$ connecting $x_a$ to $x_b$ in $\Sigma_i$, and disjoint
from the boundary of $\Sigma_i$ except at its ends. Then the
curve $d=\gamma_1\cup\gamma_2$ satisfies the lemma
(see Figure~\ref{fig:(0,0,0)}, left).
It may also happen that $\Sigma'$ has three connected components, in which
case we find a curve $d$ exactly in the same way, see
Figure~\ref{fig:(0,0,0)}, right.
\begin{figure}
\caption{Finding $d$ in the case $(0,0,0)$}
\label{fig:(0,0,0)}
\end{figure}
Next we deal with the case of intersections $(1, 0, 0)$. In this case,
$a$ and $b$ intersect transversely, once, and $c$ is disjoint from $a\cup b$.
By hypothesis, the curve $c$ is (globally) nonseparating.
Consider the union $a\cup b$. If $a$ or $b$ is two-sided, then
$a\cup b$ does not disconnect its regular neighborhoods.
This is seen by travelling
along a small band on one side of $a\cup b$,
(see Figure~\ref{fig:DeuxCourbes}, left).
In this case, $\Sigma'$ cannot have more connected components than
$\Sigma\smallsetminus c$, hence $\Sigma'$ is connected, and any
curve $d$ obtained by pushing $c$, as in the preceding case, satisfies
the lemma. If both $a$ and $b$ are one-sided, then $a\cup b$ is locally
disconnecting, so $\Sigma'$ may have up to two connected components.
In this case, a curve $d$ obtained by pushing
$a$ satisfies the lemma (See Figure~\ref{fig:DeuxCourbes}, right).
\begin{figure}
\caption{Finding $d$ in case $(1,0,0)$}
\label{fig:DeuxCourbes}
\end{figure}
Now assume we are in the case $(1, 1, 1)$ or $(1, 1, 0)$.
Since $\{a, b, c\}$ is not of type necklace, note that in any
case $b$ and $c$ do not meet outside $a$. We first treat the sub-case
when $a$ is two-sided. For this we consider any curve $d$ obtained by
pushing $a$ aside, and we claim that $d$ meets every connected component
of $\Sigma'$. Indeed, let $C$ be such a component.
Of course the closure of $C$ meets $a$ or $b$ or $c$.
Since both $b$ and $c$ meet $a$, it actually has to meet $a$, as we can see
by traveling along $b$ or $c$ in $C$. More precisely, by following $b$
or $c$ in both directions, we see that $C$ meets any neighborhood of $a$
from both sides. Thus it meets $d$.
It remains to treat the sub-case when $a$ is one-sided, first for the
$(1,1,1)$ case, and then for the $(1,1,0)$ case. In the $(1,1,1)$ case, the
curves $a$, $b$, $c$ play symmetric roles, and by the above argument it
just remains to consider the case when they are all one-sided.
Then the situation is depicted on Figure~\ref{fig:(1,1,1)Et(1,1,0)}, left:
$a \cup b \cup c$ disconnects its regular neighborhoods into three connected
components, and the figure shows a curve $d$, obtained by pushing $a$ aside,
which intersects all three components and such that
$\{a, b, c, d \}$ is a 4-clique.
In the remaining case the curves
$b$ and $c$ play symmetric roles,
and there are three different cases to
consider, regarding whether $b$ and $c$ are one or two-sided.
These three cases are pictured in Figure~\ref{fig:(1,1,1)Et(1,1,0)},
and in each case, we obtain $d$ by pushing $a$ aside.
\begin{figure}
\caption{Finding $d$ in cases $(1,1,1)$ and $(1,1,0)$}
\label{fig:(1,1,1)Et(1,1,0)}
\end{figure} \end{proof} We deduce the following. \begin{lemma}\label{lem:CestLeBouquet}
Let $\{a,b,c\}$ be a $3$-clique of $\NCfin{\pitchfork}(\Sigma)$, not of type
necklace. Let $(\alpha_1,\ldots,\alpha_n)$ be a finite family
of vertices of $\NCfin{\pitchfork}(\Sigma)$, all distinct from $a$,
$b$ and~$c$. Then there exists a vertex $d$ of $\NCfin{\pitchfork}(\Sigma)$, such that
\begin{itemize}
\item[$\bullet$] $\{a,b,c,d\}$ is a $4$-clique of $\NCfin{\pitchfork}(\Sigma)$,
\item[$\bullet$] for all $j\in\{1,\ldots,n\}$, the intersection
$d\cap \alpha_j$ is infinite; in particular, $\{d, \alpha_{j}\}$ is not an
edge of $\NCfin{\pitchfork}(\Sigma)$.
\end{itemize} \end{lemma} As a corollary, we get the direct implication in Proposition~\ref{prop:TypesPrecis}. \begin{corollary}
If $\{a,b,c\}$ is a $3$-clique not of type necklace, then $N(a,b,c)$ does not
hold. \end{corollary}
\begin{proof}[Proof of Lemma~\ref{lem:CestLeBouquet}]
The hypotheses that $\{a,b,c\}$ is not of type necklace, and
$\alpha_j\not\in\{a,b,c\}$, impose that for every $j$,
the curve $\alpha_j$ is not contained in the union $a\cup b\cup c$.
Hence, there exists a small subarc $\beta_j\subset\alpha_j$
lying in the complement of $a\cup b\cup c$, and we may
further suppose that these $n$ arcs are pairwise disjoint,
and choose a point $x_j$ in $\beta_j$ for each $j$.
Now, let $d_0$ be a vertex of $\NCfin{\leqslant 1}(\Sigma)$ as from
Lemma~\ref{lem:drenccc}. Since $d_0$ meets every component of
$\Sigma'= \Sigma\smallsetminus(a\cup b\cup c)$,
we may perform a surgery on $d_0$, far from $a\cup b\cup c$,
to obtain a new curve $d_1$ such that $\{a,b,c,d_1\}$ is still a
$4$-clique, and $d_1$ still meets every component of
$\Sigma'$, and $d_1$ passes through $x_1$. We may iterate this
process, to get a curve $d_n$ which passes through $x_j$
for every $j$, and such that $\{a,b,c,d_n\}$ is a $4$-clique.
Finally, we may perform a last surgery on $d_n$, in the
neighborhood of all $x_j$, in order to obtain a curve $d$
such that for each $j$, $|\beta_j\cap d|$ is
infinite. \end{proof}
It remains to prove the converse implication in Proposition~\ref{prop:TypesPrecis}, which we restate as Lemma~\ref{lem:CestLeCollier}. \begin{lemma}\label{lem:CestLeCollier}
Let $\{a,b,c\}$ be a $3$-clique of $\NCfin{\pitchfork}(\Sigma)$ of type
necklace. Then there exists a finite set $F$, of at most 8 vertices of
$\NCfin{\pitchfork}(\Sigma)$ all distinct from $a$, $b$ and $c$, and
such that every $d$ such that $\{a,b,c,d\}$ is a $4$-clique of
$\NCfin{\pitchfork}(\Sigma)$ is connected by an edge to some element
of~$F$. \end{lemma} In fact this set $F$ can be chosen explicitely, with cardinal at most~8, as follows. If $\{a,b,c\}$ is a $3$-clique of type necklace, then there exists a family of arcs $(x, X, y, Y, z, Z)$, all embedded in $\Sigma$, such that $a=x\cup X$, $b=y\cup Y$, $c=z\cup Z$, and such that these six arcs pairwise intersect at most at their ends. The union $a\cup b\cup c$ may be viewed as a graph embedded in $\Sigma$, and these six arcs are the edges of this embedded graph. We let $F$ be the set of nonseparating curves, among the~8 curves $X\cup Y\cup Z$, $X\cup Y\cup z$, $X\cup y\cup Z$ \textsl{etc.}, (there is one choice of upper/lower case for each letter). In the course of the proof of Lemma~\ref{lem:CestLeCollier}, we will see that $F$ is nonempty, and satisfies the Lemma. \begin{proof}[Proof of Lemma~\ref{lem:CestLeCollier}]
Let $\{a,b,c\}$ be a $3$-clique of type necklace.
Let $F$ be the set of nonseparating curves, as above, among all the~8
curves $x\cup y\cup z$, $X\cup y\cup z$, $X\cup Y\cup z$, etc.
Let $d$ be such that $\{a,b,c,d\}$ is a $4$-clique.
Up to permuting the curves $a$, $b$ and $c$, we may suppose that
$(|a\cap d|, |b\cap d|, |c\cap d|)$ equals
$(1,1,1)$, or $(1,1,0)$, or $(1,0,0)$ or $(0,0,0)$~; our proof proceeds
case by case.
The easiest case is $(1, 0, 0)$. In this case,
up to exchanging the arcs $X$ and $x$, we may suppose that
$d$ intersects $a$ at an interior point of $X$, and is disjoint from
all the other arcs. Consider $f=X\cup y\cup z$.
This curve intersects
$d$ at a unique point, transversely. It follows that $f$ is nonseparating.
Hence $f\in F$ and $f$ satisfies the conclusion of the lemma.
Now let us deal simultaneously with the cases $(1,1,1)$ and $(1,1,0)$.
Suppose first that the intersections of
$d$ with $a\cup b\cup c$ do not occur at the intersection points
$a\cap b$, $a\cap c$ or $b\cap c$. Up to exchanging $x$ with $X$,
$y$ with $Y$ and $z$ with $Z$, we may suppose that the intersections
occur in the interior of the arcs $X$, $Y$, and $Z$ in the case $(1,1,1)$,
and in the interior of the arcs $X$ and $Y$ in the case $(1,1,0)$.
Now the curve
$f=X\cup y\cup z$, for instance, satisfies the conclusion of the lemma.
Now suppose that $d$ contains one of the points $a\cap b$, $a\cap c$ or $b\cap c$.
In case $(1,1,1)$ we may suppose, up to permuting $a$, $b$ and $c$, that $d$
contains the point $a\cap b$, and in the case $(1,1,0)$, this is automatic,
as $d$ is disjoint from $c$.
Now in any case, $d$ cannot
contain $a\cap c$ nor $b\cap c$, because it intersects $a$ and $b$ only once.
Hence, up to exchanging $z$ with $Z$, we may suppose $d\cap z=\emptyset$.
In the neighborhood of the point $a\cap b$, up to homeomorphism, the configuration of
our curves is as depicted in
Figure~\ref{fig:dTransverse}, because all the intersections are supposed
to be transverse. Then, up to exchanging $X$ with $x$ or $Y$ with $y$,
we can suppose that the arc $X\cup Y$ has a transverse intersection with
$d$, and then the arc $f = X\cup Y \cup z$ satisfies the conclusion of the lemma.
\begin{figure}
\caption{A suitable choice of $X$ and $Y$}
\label{fig:dTransverse}
\end{figure}
We are left with the case $(0,0,0)$. In this case, any curve in $F$ will
satisfy the conclusion of the lemma, and hence all we have to do is to prove that $F$ is
nonempty. If $X\cup Y\cup Z$ and $X\cup Y\cup z$ were both separating,
then so would be $c=z\cup Z$, by Fact~\ref{fact:TroisArcsSep}.
Hence, among these two curves, at least one is nonseparating, and $F$ is
nonempty (in fact, it contains at least 4 elements). \end{proof}
\subsection{One or two-sided curves, and extra bouquets}\label{ssec:xB}
In this last paragraph of this section, we will see how to recognize, from the graph strucutre of $\NCfin{\pitchfork}(\Sigma)$, some additional configurations. We insist that the work in this paragraph is useful only in the case when $\Sigma$ is non orientable; it is needed in order to make our proof of Theorem~\ref{thm:AutC1} work in that case (see Remark~\ref{rmk:Klein} below).
We start with a simple characterization of one-sided and two-sided curves.
\begin{itemize} \item[$\bullet$] $\mathrm{Two}(a)$ the property that for all $b$ such that $T(a,b)$
holds, there exists $c$ such that $T(b,c)$ and $D(a,c)$ both hold, \item[$\bullet$] $\mathrm{One}(a)$ the negation of $\mathrm{Two}(a)$:
there exists $b$ such that $T(a,b)$ and such that there does not
exist $c$ satisfying $T(b,c)$ and $D(a,c)$. \end{itemize}
\begin{observation}\label{obs:1-sided}
Let $a$ be a vertex of $\NCfin{\pitchfork}(\Sigma)$. Then the curve $a$ is
one-sided, if and only if $\mathrm{One}(a)$ holds. \end{observation} \begin{proof}
If $a$ is two-sided, and $b$ satisfying $T(a,b)$, by pushing $a$
aside we find another curve $c$ as in the definition of $\mathrm{Two}(a)$.
This proves the revers implication.
If $a$ is one-sided, let $b$ be a curve obtained by pushing $a$ aside.
We have $T(a,b)$, and $a$ and $b$ bound a disk. Any curve $c$
disjoint from $a$, and with $T(b,c)$, has to enter this disk: but then,
it has to get out, which is impossible without touching $a$ and
without intersecting $b$ another time. \end{proof}
Our next objective is to characterize when two one-sided curves $a$ and $b$ meet exactly once, non-transversely. We will do this in several steps. \begin{lemma}\label{lem:acapbnonconnexe}
Let $a$, $b$ be one-sided simple curves of $\Sigma$. Suppose the intersection
$a\cap b$ is not connected.
Then there exists a vertex $c$ of $\NCfin{\pitchfork}(\Sigma)$,
distinct from $a$ and $b$,
such that for every neighbor $d$ of both $a$ and $b$ in this graph,
and such that $D(a,d)$ or $D(b,d)$ (or both), the vertices $c$ and $d$ are
neighbors in this fine graph. \end{lemma} \begin{proof}
Let $x$ be a subarc of $b$, whose endpoints lie in $a$, and disjoint from $a$
otherwise. Since $a\cap b$ is disconnected, such an arc exists, and has two
distinct end points, $p$ and $q$. Let $x'$ and $x''$ be the two subarcs of $a$
whose ends are $p$ and $q$.
From Fact~\ref{fact:TroisArcsSep}, we know that $x\cup x'$ or $x\cup x''$
(or both) is a nonseparating curve; denote it by $c$. By construction,
we have $c\neq a$ and $c\neq b$.
Now let $d$ be a curve satisfying the hypothesis of the lemma. If $d$ is
disjoint from $a$ and $b$, then it is disjoint from $c$; otherwise $d$
intersects exactly one of $a$, $b$, far away from the other. So the intersection
between $d$ and $c$, if any, is still transverse, and $d$ is a neighbor of
$c$ in~$\NCfin{\pitchfork}(\Sigma)$. \end{proof} This contrasts with the situation we want to characterize, as we see now. \begin{lemma}\label{lem:I(a,b)-lemme1}
Let $a$ and $b$ be two one-sided curves, and suppose that $a\cap b$
consists in one, inessential intersection point. Then, for every
nonseparating curve $c\not\in\{a,b\}$, there exists $d$ such
that $T(a,d)$ and $D(b,d)$
hold but such that the intersection $c\cap d$ is infinite. \end{lemma} \begin{proof}
We first observe that $\Sigma'=\Sigma\smallsetminus(a\cup b)$ is connected.
This is seen by following the curves $a$ and $b$ in both directions: the
union $a\cup b$ does not disconnect its small neighborhoods.
Let $c$ be a curve as above. Then, we may consider a first curve $d_0$,
obtained by pushing $a$ aside, in such a way that $d_0$ is disjoint from
$b$ (this is possible since the intersection $a\cap b$ is inessential).
Since $c\not\in\{a,b\}$, the curve $c$ intersects $\Sigma'$.
Since $d_0$ meets every component of $\Sigma'$ (there is only one), we may
deform it into a curve $d$ which intersects $c$ infinitely many times,
exactly as in the proof of Lemma~\ref{lem:CestLeBouquet}. \end{proof} After these two lemmas, we have a simple sentence in terms of the graph $\NCfin{\pitchfork}(\Sigma)$, which holds when $a\cap b$ is a single inessential intersection point, and which guarantees that $a\cap b$ is connected. In order to upgrade this into a characterization of the first situation, we need to be able to exclude as well the cases when $a\cap b$ is a non degenerate arc. These cases fall into two subcases: the intersection arc $a\cap b$ can be essential or inessential, exactly as an intersection point. One way to formalize this, is to say that
the intersection $a\cap b$ is essential if $a$ cuts a regular neighborhood of $a\cap b$ into two regions both containing a subarc of $b$, and inessential otherwise.
\begin{lemma}\label{lem:ArcEssentiel}
Let $a$ and $b$ be one-sided curves.
Suppose that $a\cap b$ is a non degenerate arc, and suppose this intersection
is essential. Then there exist curves $\alpha,\alpha',\beta,\beta'$ obtained
by pushing $a$ aside, such that $B(a,\alpha,\beta)$, $B(b,\alpha,\beta)$,
$B(a,\alpha',\beta')$, $B(b,\alpha',\beta')$, and $N(a,\alpha,\beta')$. \end{lemma}
\begin{proof}[Proof of Lemma~\ref{lem:ArcEssentiel}]
The curves $\alpha$, $\beta$, $\alpha'$ and $\beta'$ may be taken in a
neighborhood of $a\cup b$, as pictured in Figure~\ref{fig:PropI-1}.
\begin{figure}
\caption{The curves $\alpha,\beta,\alpha',\beta'$}
\label{fig:PropI-1}
\end{figure} \end{proof} Finally, we deal with inessential arcs. \begin{lemma}\label{lem:IntNonEssentielle}
Let $a$ and $b$ be one-sided curves, such that $a\cap b$ is
an inessential arc or intersection point. Let $c$ be such that $T(a,c)$.
Then there exists $d$ such that $B(a,c,d)$ and $D(b,d)$, if and only
if the intersection point $a\cap c$ does not belong to~$b$. \end{lemma} \begin{proof}
If $a\cap c$ belongs to $b$, then obviously, any curve $d$ satisfying
$B(a,b,d)$ will meet $b$. Otherwise, we may push $a$ aside, in order to
obtain a curve $d$, as in Figure~\ref{fig:I-xB}, left.
\end{proof} Putting all together, this yields the following characterization of inessential intersection points between one-sided curves. \begin{corollary}\label{cor:I(a,b)}
Let $a$ and $b$ be one-sided curves. Then, $a\cap b$ consists of one,
inessential intersection point, if and only if the following conditions are
satisfied:
\begin{enumerate}
\item $a$ and $b$ are not neighbors in $\NCfin{\pitchfork}(\Sigma)$,
\item for every $c\not\in\{a,b\}$, there exists $d$ such that $d$ is a
neighbor of $a$ and $b$ in $\NCfin{\pitchfork}(\Sigma)$, and $D(a,d)$ or $D(b,d)$
(or both), but $d$ is not a neighbor of $c$ in that graph,
\item there do not exist $\alpha,\beta,\alpha',\beta'$ such that
$B(a,\alpha,\beta)$, $B(b,\alpha,\beta)$, $B(a,\alpha',\beta')$,
$B(b,\alpha',\beta')$ and $N(a,\alpha,\beta')$ all hold,
\item there do not exist $c_1,c_2$ with $D(c_1,c_2)$ and with the following
property:
for $i=1,2$ we have: $T(a,c_i)$ and for all $d$, $B(a,c_i,d)$ and $D(b,d)$
do not both hold.
\end{enumerate} \end{corollary} This enumeration of conditions expressed only in terms of the graph structure of $\NCfin{\pitchfork}(\Sigma)$, with the addition of the conditions $\mathrm{One}(a)$ and $\mathrm{One}(b)$, will be also denoted by $I(a,b)$, for {\em inessential intersection} (of one-sided curves). \begin{proof}
First, let us check that if $a$ and $b$ have one, inessential intersection
point then $I(a,b)$ holds. Condition~(1) holds by definition,
and~(2) follows from Lemma~\ref{lem:I(a,b)-lemme1}.
The negation of condition~(3) would imply that the cardinal of $a\cap b$ is at least~2.
Indeed, $B(a,\alpha,\beta)$ implies that $\alpha\cap\beta$ is a point lying
in $a$. Thus, the bouquet conditions imply that both $\alpha\cap\beta$ and
$\alpha'\cap\beta'$ lie in $a\cap b$. And the condition $N(a,\alpha,\beta')$
then implies that $a\cap\alpha$ and $a\cap\beta'$ are disjoint, hence
the two points $\alpha\cap\beta$ and $\alpha'\cap\beta'$ are distinct.
Finally, condition~(4) follows from Lemma~\ref{lem:IntNonEssentielle}.
Indeed, this lemma implies that the two curves $c_1$ and $c_2$ should
both contain a point of $a\cap b$, hence they cannot be disjoint.
Now, let $a$ and $b$ be any two nonseparating curves and suppose that $I(a,b)$.
By conditions~(1) and~(2), the intersection $a\cap b$ is non empty, and
connected. Along the lines of the proof of Lemma~\ref{lem:ArcEssentiel},
we can see that condition~(3) implies that $a\neq b$, so $a\cap b$
is an inessential intersection point, or an arc. Suppose for contradiction
that it is a nondegenerate arc. By Lemma~\ref{lem:ArcEssentiel} and
condition~(3), this intersection arc cannot be essential.
Now Figure~\ref{fig:I-xB}, right,
shows the desired contradiction with condition~(4).
\begin{figure}
\caption{Some configurations of curves for properties $I$ and $xB$}
\label{fig:I-xB}
\end{figure} \end{proof}
Finally, we deal with extra bouquets. We denote by $xB(a,b,c)$ the property that $T(a,b)$, $T(a,c)$, $I(b,c)$ all hold and moreover: for all $d$ such that $B(a,b,d)$ holds, $D(c,d)$ does not. \begin{lemma}
Let $a$, $b$ and $c$ be such
that $T(a,b)$, $T(a,c)$ and $I(b,c)$, with $b$ and $c$ one-sided.
Then $xB(a,b,c)$ holds if and only if the intersection points
$a\cap b$, $a\cap c$ and $b\cap c$ coincide. \end{lemma} \begin{proof}
Of course if these points coincide, then property $xB(a,b,c)$
holds: every curve $d$ such that $B(a,b,d)$ holds, must contain
this point and hence cannot be disjoint from $c$.
Now suppose that these points do not coincide, hence, are three
pairwise distinct points. Then, we may push $b$ aside, in order
to find a curve $d$ which does not intersect $c$ any more, as
the intersection $b\cap c$ is not essential. This curve $d$,
obtained by pushing $b$, can be made to satisfy $T(b,d)$,
while crossing $b$ precisely at the point $a\cap b$, and
this intersection can be made transverse;
the illustration
of this situation is similar to Figure~\ref{fig:I-xB}, left,
and this time we leave it to the reader.
This yields a curve $d$
such that $B(a,b,d)$ holds and $d$ disjoint from $c$.
\end{proof}
\section{Proof of Theorem~\ref{thm:AutC1}}\label{sec:AutHomeo}
Here as above, $\Sigma$ is a connected surface admitting a nonseparating closed curve.
\subsection{From bijections to homeomorphisms} In order to prove Theorem~\ref{thm:AutC1}, it suffices to prove that every automorphism of $\NCfin{\pitchfork}(\Sigma)$ is supported by a bijection of the surface, in virtue of the following observation. \begin{proposition}\label{prop:BijHomeo}
Let $f\colon\Sigma\to\Sigma$ be a bijection. We suppose that for every
nonseparating simple closed curve $\alpha\subset\Sigma$, the sets $f(\alpha)$
and $f^{-1}(\alpha)$ are also nonseparating simple closed curves in $\Sigma$.
Then $f$ is a homeomorphism. \end{proposition} \begin{proof}
If our hypothesis was that $f$ and $f^{-1}$ are closed (\textsl{i.e.}, send
closed sets to closed sets), then $f$ would be a homeomorphism. So our
strategy is to use our hypothesis here in a similar fashion. We need only
prove that $f$ is continuous, the argument for $f^{-1}$ is symmetric.
Let $x\in\Sigma$ and suppose that $f$ is not continuous at $x$. Then there
exists a sequence $(x_n)_{n\geqslant 0}$ of distinct points converging
to $x$, and a neighborhood $V$ of $f(x)$, such that for all $n$, we have
$f(x_n)\not\in V$.
Notice that in the open unit disk of the plane, up to homeomorphism, there
is only one sequence of distinct points converging to the origin.
With this in head, we may construct an embedded arc, in $\Sigma$, with one
end at $x$, and which contains all the points $x_n$. Then we may construct a
nonseparating simple closed curve $\alpha$ containing this arc.
By hypothesis, $f(\alpha)$ is a nonseparating closed curve in $\Sigma$, which
contains $f(x)$. We may perform a surgery of $f(\alpha)$ inside $V$, to
obtain a nonseparating simple closed curve $\beta$, which coincides with
$f(\alpha)$ outside $V$ but which does not contain $f(x)$.
Now $f^{-1}(\beta)$ is, by hypothesis, a closed subset of $\Sigma$, which
contains all the points $x_n$ but not $x$. This is a contradiction. \end{proof}
\subsection{The adjacency relation $\diamondvert$}\label{sec:Adjacency}
Let $E_T(\NCfin{\pitchfork}(\Sigma))$ denote the set of edges $\{a,b\}$ of $\NCfin{\pitchfork}(\Sigma)$ satisfying $T(a,b)$. Then we have a map \[ \mathrm{Point}\colon \ E_T(\NCfin{\pitchfork}(\Sigma))\to\Sigma, \] which to each edge $\{a,b\}$, associates the intersection point $a\cap b$. The main part of proof of Theorem~\ref{thm:AutC1} consists in showing that we can express the equality \[ \mathrm{Point}(a,b) = \mathrm{Point}(\alpha, \beta) \] in terms of the graph. For this we introduce the equivalence relation $\diamondvert$ on $E_T(\NCfin{\pitchfork}(\Sigma))$ as follows. Let $\{a,b\}$ in $E_T(\NCfin{\pitchfork}(\Sigma))$. If $\{a,b,c\}$ is a $3$-clique of $\NCfin{\pitchfork}(\Sigma)$ of type bouquet we set $\{a,b\}\diamondvert \{a,c\}$. We also set
$\{a,b\}\diamondvert \{a,c\}$ if $a$, $b$ and $c$ satisfy the ``extra bouquet'' condition, denoted above by $xB(a,b,c)$, see paragraph~\ref{ssec:xB} (this is void when $\Sigma$ is orientable). Then $\diamondvert$ is defined as the equivalence relation generated by these relations. When $\Sigma$ is orientable, the relation $\diamondvert$ corresponds to the equivalence relation on triangles, generated by adjacency, in the subgraphs of $\NCfin{\pitchfork}(\Sigma)$ induced by curves passing through a common point. This is what motivates our notation.
The relation $\{a,b\}\diamondvert \{a', b'\}$ obviously implies $\mathrm{Point}(a,b) = \mathrm{Point}(a',b')$. We will see that the converse is not true, and describe geometrically the equivalence classes in Section~\ref{sec:LocalSubgraphs}, but for now we will only need the following partial statement. \begin{proposition}\label{prop:GermeAdjacent}
Let $a$, $b$, $a'$, $b'$ be such that $T(a,b)$ and $T(a',b')$.
Suppose that they have the same intersection point,
$x=\mathrm{Point}(a,b)=\mathrm{Point}(a',b')$, and suppose that the germs of $a$ and $a'$
coincide, \textsl{i.e.}, there exists a neighborhood $V$ of $x$ such
that $a\cap V=a'\cap V$.
Then $\{a,b\}\diamondvert\{a',b'\}$. \end{proposition} \begin{remark}\label{rmk:Klein}
If $\Sigma$ is a Klein bottle, there are no couples $\{a,b\}$ of two-sided
curves such that $T(a,b)$, and for every one-sided curve $b$, the curves
$c$ such that $T(b,c)$ holds fall into only two isotopy classes: that of $b$
and that of a two-sided curve, prescribed by $b$. It follows that, without
the extra bouquets in the definition of $\diamondvert$, there would
have been too many classes of $\diamondvert$, as such a class would remember
the isotopy class of a one-sided curve, and Proposition~\ref{prop:GermeAdjacent}
would not be true in this special case. These extra bouquets will be used
in the proof of Lemma~\ref{lem:SpheresConnexes} below. \end{remark} We postpone the proof of the proposition to the end of this section; for now we will explain how it implies Theorem~\ref{thm:AutC1}.
\subsection{Proof of Theorem~\ref{thm:AutC1}} If $a,b,c$ are vertices of $\NCfin{\pitchfork}(\Sigma)$, we denote by $F(a,b,c)$ the property that $T(a,b)$ holds, and there exists an edge $\{a',b'\}$ with $\{a,b\}\diamondvert\{a',b'\}$ such that $\{a',b',c\}$ is a $3$-clique which is not of type bouquet. Note that this property $F(a,b,c)$ implies that $c$ does not contain the point $\mathrm{Point}(a',b') = \mathrm{Point}(a,b)$. The next lemma asserts that $F(a,b,c)$ actually characterises this geometric property, and the letter $F$ stands for: ``$a\cap b$ is far from $c$''. \begin{lemma}\label{lem:AppartenanceGraphe}
Let $a,b,c$ be vertices of $\NCfin{\pitchfork}(\Sigma)$, and suppose $T(a,b)$ holds.
Then
\[ F(a, b, c) \Leftrightarrow \mathrm{Point}(a,b)\not\in c. \] \end{lemma}
\begin{proof}
The direct implication follows directly from the
definitions; we have to prove
the converse implication. Suppose $\mathrm{Point}(a,b)\not\in c$.
Since $\Sigma$ is connected, there exists a regular neighborhood of $c$
containing the point $\mathrm{Point}(a,b)$. Depending on whether $c$ is one-sided or
two-sided, up to homeomorphism, this leads to only two distinct situations.
In Figure~\ref{fig:OnComplete2germes}, we represent in bold the germs of
the curves $a$ and $b$ near the point $a\cap b$, and show how to complete
these germs to new curves $a'$, $b'$ such that $\{a',b',c\}$ is a $3$-clique
not of type bouquet.
When $c$ is one-sided (see Figure~\ref{fig:OnComplete2germes}, left),
we may use two curves $a'$, $b'$ obtained by pushing $c$, while when $c$
is two-sided (see Figure~\ref{fig:OnComplete2germes}, right), we have to
use a curve $d$ which meets $c$ once transversely.
Now, by Proposition~\ref{prop:GermeAdjacent}, we have
$\{a,b\}\diamondvert\{a',b'\}$, and $\{a',b',c\}$ is a non-bouquet $3$-clique,
so we have $F(a,b,c)$ by definition.
\begin{figure}
\caption{Completing the germs of $a$ and $b$ to form a non-bouquet $3$-clique $(a',b',c)$}
\label{fig:OnComplete2germes}
\end{figure} \end{proof}
\begin{corollary}\label{cor:AppartenanceGraphe}
Suppose $T(a,b)$ and $T(\alpha,\beta)$ hold.
Then $\mathrm{Point}(a,b)\neq\mathrm{Point}(\alpha,\beta)$ if and only if there
exists a nonseparating closed curve $c$ such that
$F(\alpha, \beta, c)$ holds but not $F(a,b,c)$.
\end{corollary}
The corollary is a direct consequence of Lemma~\ref{lem:AppartenanceGraphe}.
It follows that the equality $\mathrm{Point}(a,b)=\mathrm{Point}(\alpha,\beta)$ can be expressed in terms of the graph structure of $\NCfin{\pitchfork}(\Sigma)$, because, as a consequence of Corollary~\ref{coro:TypesPrecis}, being a $3$-clique not of type bouquet is also characterized in terms of this graph structure. Now we can conclude the proof of Theorem~\ref{thm:AutC1}, provided Proposition~\ref{prop:GermeAdjacent} holds.
\begin{proof}[Proof of Theorem~\ref{thm:AutC1}]
Let $\varphi$ be an automorphism of $\NCfin{\pitchfork}(\Sigma)$.
Given a point $x$ in $\Sigma$, we choose two nonseparating simple closed
curves $a$, $b$ intersecting exactly once, transversely, at $x$,
and set $\varphi_\Sigma(x)=\mathrm{Point}(\varphi(a),\varphi(b))$. This
formula is valid because, by Proposition~\ref{prop:Types},
$\varphi(a)$ and $\varphi(b)$ are still nonseparating simple closed
curves intersecting exactly once. The point $\varphi_\Sigma(x)$
does not depend on the choice of $(a,b)$, because if $(\alpha,\beta)$
is another choice, the equalities $\mathrm{Point}(a,b)=\mathrm{Point}(\alpha,\beta)$
and $\mathrm{Point}(\varphi(a),\varphi(b))=\mathrm{Point}(\varphi(\alpha),\varphi(\beta))$
can be all expressed in terms of the graph structure of
$\NCfin{\pitchfork}(\Sigma)$, thanks to Corollary~\ref{cor:AppartenanceGraphe}.
Thus, the map $\varphi_\Sigma$ is well-defined, and by following the
definitions we observe that the map
$(\varphi^{-1})_\Sigma$ is its inverse: hence $\varphi_\Sigma$
is a bijection of $\Sigma$. Finally, it follows from Lemma~\ref{lem:AppartenanceGraphe}
that for any nonseparating simple closed curve $\alpha$, the curve
$\varphi(\alpha)$ coincides with the set of points $\varphi_\Sigma(x)$
as $x$ describes $\alpha$. In other words, the automorphism $\varphi$
is realized by the bijection $\varphi_\Sigma$. Proposition~\ref{prop:BijHomeo}
concludes. \end{proof}
\subsection{Connectedness of some arc graphs} In order to finally prove
Proposition~\ref{prop:GermeAdjacent},
we will first need a couple of elementary results on fine arc graphs.
\begin{lemma}\label{lem:ArcConnexe}
Let $S$ be a connected topological surface, with boundary, and let $x,y$ be
two distinct points of $\partial S$. Let $\mathcal{E}\Afin{}(S, x, y)$ be the graph whose
vertices are the simple arcs joining $x$ and $y$ and which meet $\partial S$ only at
their ends, with an edge between two such arcs if and only if they are
disjoint except at $x$ and $y$. Then the graph $\mathcal{E}\Afin{}(S, x, y)$ is connected. \end{lemma} Note that, when $x$ and $y$ are taken in the same connected component, we are not requiring that the arcs be nonseparating; this is the reason why we use the letter $\mathcal{E}$, for {\em extended}, in the same fashion as in~\cite{LMPVY}. \begin{proof}
Let $a$, $b$ be two vertices of this graph. As a first case we suppose that
$a\cap b$ is made of a finite number of transverse intersection points:
we will prove by induction on the
cardinal of $a\cap b$ that, in this case, $a$ and $b$ are connected in
$\mathcal{E}\Afin{}(S,x,y)$. If $a\cap b$ is as small as possible, \textsl{i.e.}, is
equal to $\lbrace x,y\rbrace$, then $a$ and $b$ are neighbors in this graph.
Otherwise, if $a$ and $b$ intersect at other points than $x$ and $y$, we
may, in the spirit of~\cite{HPW},
pick a {\em unicorn path} $c$, made of one subarc of $a$ beginning at $x$,
and one subarc of $b$ ending at $y$. (For example, we may follow $a$
until it first meets $b$ after $x$, and then
continue along $b$).
Now we may push $c$ aside while fixing its ends,
at the appropriate side of $c$, to
obtain a new arc $c'$
with both $c'\cap a$ and $c'\cap b$ of cardinal strictly lower than that of
$a\cap b$.
This proves that arcs intersecting at finitely many points are
connected in $\mathcal{E}\Afin{}(S,x,y)$.
Now if the intersection $a\cap b$ is infinite, fix a differentiable structure
on the surface $S$. We may consider a smooth curve $a'$, neighbor of $a$, by
pushing $a$ aside while fixing its ends, and similarly, a smooth neighbor
$b'$ of $b$ similarly. Up to perturbing $b'$, we may suppose that $b'$ is
transverse to $a'$.
By the step above, $a'$ and $b'$ are
connected in $\mathcal{E}\Afin{}(S,x,y)$ and the Lemma is proved. \end{proof} We will also need a version for nonseparating arcs. \begin{lemma}\label{lem:ArcConnexeBis}
Let $S$ be a connected surface containing a nonseparating curve. Suppose
$S$ has boundary, and let $x$ and $y$ be two distinct points in a same boundary
component of $S$. Let $\NAfin{}(S,x,y)$
be the set of nonseparating simple arcs
connecting $x$ to $y$, with an edge when they are disjoint away from $x$
and $y$. Then $\NAfin{}(S,x,y)$ is connected. \end{lemma} The points $x$ and $y$ add some technicality; let us first prove the following simpler statement. \begin{lemma}\label{lem:NonSepFinRelUnBordConnexe}
Let $S$ be a connected surface containing a nonseparating curve, and with
at least one boundary component, denoted $C$. Let $\NAfin{}(S,C)$ be the
graph whose vertices are the nonseparating arcs joining two distinct points
of $C$, and with an edge between two such vertices whenever they are disjoint.
Then this graph is connected. \end{lemma} This lemma is a variation on~\cite[Corollary~3.2]{LMPVY}; here we additionnaly require that the arcs end at $C$. In fact, in~\cite{LMPVY}, Corollary~3.3 is stated for surfaces with $b>0$ boundary components, but proved only in the case $b=1$, which is the case needed in the proof of their main theorem. Lemma~\ref{lem:NonSepFinRelUnBordConnexe} may be used to extend this corollary to any $b>0$.
\begin{proof}
We begin with the observation that the graph $\NAfin{}(S,C)$ has no
isolated point. Indeed, if $\gamma$ is a vertex of $\NAfin{}(S,C)$,
by definition it is nonseparating. So we may consider a simple closed curve
$u$ with one, transverse intersection point with~$\gamma$. Obviously,
this curve $u$ is nonseparating; this follows from Fact~\ref{fact:NonSep},
applied to $u$, and a curve $v$ obtained by concatenation of $\gamma$
with some arc of $C$. Now we can perform a surgery on $u$,
and push its intersection point towards one end of $\gamma$ until we hit
$C$. This constructs an arc
$\alpha$,
which is now disjoint from $\gamma$, and which is also nonseparating.
Next, we claim that we can suppose, without loss of generality, that
the surface $S$ is compact. Indeed, if $\gamma_1,\gamma_2$ are vertices
of $\NAfin{}(S,C)$, and if $u$ is a nonseparating curve intersecting $\gamma_1$
as above, consider the set $K=C\cup\gamma_1\cup\gamma_2\cup u$. This set is
compact, hence there exists a compact topological subsurface $S'$ of $S$
containing~$K$.
This surface $S'$ contains nonseparating curves,
as it contains $u$ and $\gamma_1\cup C$, which may be used as above to
find two simple closed curves $u$, $v$ with one, essential intersection.
Now a path joining $\gamma_1$ to $\gamma_2$ in $S'$ is also a path joining
$\gamma_1$ to $\gamma_2$ in $S$.
So, until the end of the proof, $S$ is now supposed to be compact.
Next, observe that if two vertices $\gamma_1,\gamma_2$ of $\NAfin{}(S,C)$ are
isotopic (\textsl{i.e.}, there exists a continuous map $H\colon[0,1]^2\to S$
such that $\gamma_1(t)=H(0,t)$ and $\gamma_2(t)=H(1,t)$ for all $t$,
$H(s,0),H(s,1)\in C$ for all $s$ and the curve $H_s\colon t\to H(s,t)$ is
injective for all $s$),
then $\gamma_1$ and $\gamma_2$ are in the same component of $\NCfin{}(S,C)$.
This argument is borrowed from~\cite{BHW}: for all $s$, the arc
$H_s$ has at least a neighbor $\alpha_s$ (by the first observation above),
and the set of $s'$ such that $H_{s'}$ is still a neighbor of $\alpha_s$ is
open in $[0,1]$. By compactness of $[0,1]$, there exist a finite number of
arcs $\alpha_1,\ldots,\alpha_n$, and a subdivision $0=t_0<t_1<\cdots<t_n=1$
such that $\alpha_j$ is disjoint from $H_t$ for all $t\in[t_{j-1},t_j]$ for
all $j$, and now $(\gamma_1,\alpha_1,\ldots,\alpha_n,\gamma_2)$ is a
path of $\NCfin{}(S,C)$ joining $\gamma_1$ to $\gamma_2$.
As a result of this observation, we need only prove the connectedness of
the graph $\mathrm{NA}(S,C)$, whose vertices are the isotopy classes of
arcs between two distinct points of $C$, and with an edge between two
vertices whenever the corresponding classes admit disjoint representatives.
We proceed with the observation that the graph $\mathrm{A}(S,C)$, defined
exactly as $\mathrm{NA}(S,C)$ excpept we consider essential arcs, which may
be separating, is connected.
A simple way to do this is by using the idea of unicorn arcs exactly as
in the proof of the preceding lemma: if two arcs $a$ and $b$ are in
minimal position then their unicorn arcs are essential, and have fewer
intersections with both $a$ and $b$ than the number of points of $a\cap b$.
We will promote the connectedness of $\mathrm{A}(S,C)$ to that of
$\mathrm{NA}(S,C)$, by induction on the number of boundary components of $S$.
First, suppose that $S$ has only one boundary component,~$C$.
Let $\gamma_1,\gamma_2$ be two vertices of $\mathrm{NA}(S,C)$.
We may connect them by a path
$(\gamma_1,\alpha_1,\alpha_2,\ldots,\alpha_n,\gamma_2)$
in $\mathrm{A}(S,C)$, where each of the $\alpha_j$ may be separating;
consider such a path with minimal number of separating arcs.
For contradiction, and up to some relabeling, suppose $\alpha_1$ is
separating. Then it cuts $S$ in two components; denote by $S_1$ the
one containing $\gamma_1$
and $S_2$ the other. Then $\alpha_2$ is also contained in $S_1$,
otherwise we may delete $\alpha_1$ from our path. Since $S$ has
no boundary component other than $C$ and since the curve $\alpha_1$
is essential, the surface $S_2$ contains
a nonseparating arc, $\alpha_1'$. This arc
may be used instead of $\alpha_1$ in our initial path
from $\gamma_1$ to $\gamma_2$, contradicting the minimality of the
number of separating arcs. This proves that
$\mathrm{NA}(S,C)$ is connected if $S$ has no other boundary
component.
Now, we suppose, for inductive hypothesis, that
$\mathrm{NA}(S',C')$
is connected for every surface $S'$ with less boundary components
than $S$. Let $\gamma_1,\gamma_2$ be two vertices of $\mathrm{NA}(S,C)$.
As before, consider a path
$(\gamma_1,\alpha_1,\alpha_2,\ldots,\alpha_n,\gamma_2)$ in
$\mathrm{A}(S,C)$ between them, with minimal number of separating arcs.
For contradiction, and up to some relabeling, suppose $\alpha_1$ is
separating: it cuts $S$ into two subsurfaces, let $S_1$ be the one
containing $\gamma_1$, and, by hypothesis, must also contain $\alpha_2$,
and let $S_2$ be the other. If $S_2$ contains nonseparating arcs, we
conclude as before. If not, then $S_2$ contains some of the boundary
components of $S$, hence the surface with boundary $S'=S_1\cup\alpha_1$
has strictly less boundary components than $S$. One is $C'$, composed
by an arc of $C$ and the arc $\alpha_1$, and there may be others.
If $\alpha_2$ is nonseparating, then, by the induction hypothesis,
there is a path
$(\gamma_1,\beta_1,\ldots,\beta_k,\alpha_2)$ of $\mathrm{NA}(S',C')$
connected them. The arcs $\beta_1, \ldots, \beta_k$ may have end points
in $\alpha_1$, but we may perform a surgery in order to push all these
points to $C$, and obtain arcs $\beta_1',\ldots,\beta_k'$ which are
also vertices of $\mathrm{NA}(S,C)$, and we are done in this case.
Finally, if $\alpha_2$ is a separating arc (of $S$, or of $S_1$,
equivalently), then we may find an arc $\alpha_2'$ of $S_1$ which
is nonseparating and disjoint from $\alpha_2$. By following the
last case above, there exists a path
$(\gamma_1,\beta_1,\ldots,\beta_k,\alpha_2')$ in $\mathrm{NA}(S,C)$,
hence the path
$(\gamma_1,\beta_1,\ldots,\beta_k,\alpha_2',\alpha_2,\ldots,\alpha_n,\gamma_2)$
of $\mathrm{A}(S,C)$ has one less separating arc than the initial path.
This contradiction ends the proof.
\end{proof} \begin{proof}[Proof of Lemma~\ref{lem:ArcConnexeBis}]
Let $\gamma_1,\gamma_2$ be two vertices of $\NAfin{}(S,x,y)$.
First, we may construct a neighbor $\gamma_2'$ in $\NAfin{}(S,x,y)$
of $\gamma_2$, which, in a neighborhood of $x$ (resp. $y$),
touches $\gamma_1$ only at $x$ (resp. $y$).
Indeed, there is a neighborhood $U_x$ of $x$ homeomorphic to the
closed half unit disk
\[ \{z, |z|\leqslant 1 \text{ and }\mathrm{Im}(z)\geqslant 0\}, \]
where the middle ray ($\mathrm{Re}(z)=0$) corresponds to the points
of $\gamma_1$. On either side of this ray, we may find an arc disjoint
from $\gamma_1$ and $\gamma_2$ except at $0$, arbitrarily close to
the boundary ($\mathrm{Im}(z)=0$), and joining $0$ to the unit circle,
and then this small arc may be continued to construct a curve $\gamma_2'$
which consists of pushing $\gamma_2$ aside.
So we may suppose that $\gamma_1$ and $\gamma_2$, close to $x$ and
$y$, intersect only at these points, and we may now find neighborhoods
$U_x$ and $U_y$ as above, such that their intersections with
$\gamma_1$ and $\gamma_2$ are along rays in this disk, in distinct
directions around $0$. Let $S'$ be the surface obtained by removing
the interiors of $U_x$ and $U_y$ from $S$. Then the path given by
applying Lemma~\ref{lem:NonSepFinRelUnBordConnexe} to $S$, yields
a path from $\gamma_1$ to $\gamma_2$ in $\NAfin{}(S,x,y)$, just by
adding some rays in $U_x$ and $U_y$ to the corresponding arcs.
\end{proof}
\subsection{Proof of Proposition~\ref{prop:GermeAdjacent}}
Let us go back to the proof of Proposition~\ref{prop:GermeAdjacent}. For the remaining of the section we fix a point $x\in\Sigma$. Let $X$ denote the set of nonseparating simple closed curves passing through~$x$. \begin{lemma}\label{lem:SpheresConnexes}
Let $a,b,c\in X$.
Suppose that $T(a,b)$ and $T(a,c)$ hold.
Then $(a,b)\diamondvert(a,c)$. \end{lemma} \begin{proof}
Let $S$ be the surface obtained by cutting $\Sigma$ along $a$: it is
the surface with boundary obtained by gluing back two copies of the
curve $a$ to $\Sigma\smallsetminus a$. The point $x$ of $\Sigma$
yields two points, $p$ and $q$, of $\partial S$, and the curves
$b$ and $c$ define two arcs of $S$ joining $p$ and $q$.
By Lemma~\ref{lem:ArcConnexe}, there exists a finite sequence
$\gamma_0=b$, \ldots, $\gamma_n=c$, of arcs of $S$ joining $p$ and $q$,
with $\gamma_i$ and $\gamma_{i+1}$ disjoint except at $p$ and $q$.
For each $i$, the arc $\gamma_i$ defines a closed curve in $\Sigma$,
which has precisely one, transverse intersection with $a$; we will
still denote it by $\gamma_i$, abusively.
For every $i$, if $T(\gamma_i,\gamma_{i+1})$ holds, then we have
$(a,\gamma_i)\diamondvert(a,\gamma_{i+1})$, by definition.
If $T(\gamma_i,\gamma_{i+1})$ does not hold, then either $\gamma_i$
or $\gamma_{i+1}$ are both one-sided, or one of them is two-sided.
In the first case, the condition $xB(a,\gamma_i,\gamma_{i+1})$ holds,
by definition, and hence
$(a,\gamma_i)\diamondvert(a,\gamma_{i+1})$.
In the second, up to reversing the notation suppose $\gamma_i$ is
two-sided. Figure~\ref{fig:OnCompleteBouquets} shows how to insert
a curve $\delta$ such that
$B(a,\delta,\gamma_i)$ and $B(a,\delta,\gamma_{i+1})$ both hold,
and hence we still have $(a,\gamma_i)\diamondvert(a,\gamma_{i+1})$ in
this case.
\begin{figure}
\caption{Connecting the curves by common adjacency}
\label{fig:OnCompleteBouquets}
\end{figure}
By transitivity, we deduce that $(a,b)\diamondvert(a,c)$. \end{proof} The last ingredient for the proof of Proposition~\ref{prop:GermeAdjacent} is the following observation. \begin{observation}\label{obs:CCarc}
Let $a$, $a'$ be two nonseparating simple closed curves in $\Sigma$
such that $a\cap a'$ is an arc. Then, both sides of this arc lie
in the same connected component of $\Sigma\smallsetminus(a\cup a')$. \end{observation} \begin{proof}
\textsl{A priori}, the complement of $\Sigma\smallsetminus(a\cup a')$
may have up to four connected components, as suggested in
Figure~\ref{fig:CCarc}.
\begin{figure}
\caption{The arc $a\cap a'$ cannot disconnect}
\label{fig:CCarc}
\end{figure}
Suppose first that the intersection $a\cap a'$ is essential.
If $a$ (resp. $a'$) is one-sided, by following the curve $a$ (resp. $a'$)
we see that $A=B$. If both $a$ and $a'$ are two-sided, by following $a$ we
see that $A=D$ and $C=B$, while by following $a'$ we get $A=C$ and $B=D$,
so~$A=B$.
Now, suppose the intersection arc $a\cap a'$ is inessential.
By following $a$, we see that $C=D$, regardless of $a$ being one or
two-sided. Thus, if $A\neq B$, then one of $A$
or $B$, say $A$, is not connected from any of $B, C, D$. But this implies
that $a$ is separating, a contradiction.
\end{proof}
We are now in a position to prove~Proposition~\ref{prop:GermeAdjacent}, but instead we will prove the following stronger statement, which will be more convenient later in this article. \begin{proposition}\label{prop:SemiGermeAdjacent}
Let $a$, $b$, $a'$, $b'$ be such that $T(a,b)$ and $T(a',b')$,
with intersection point
$x=\mathrm{Point}(a,b)=\mathrm{Point}(a',b')$, and suppose that $a$ and $a'$ locally ``half
coincide'' near $x$, \textsl{i.e.}, $a \cap a'$ contains a non degenerate arc
with endpoint $x$. Then $\{a,b\}\diamondvert\{a',b'\}$. \end{proposition}
\begin{proof}[Proof of Proposition~\ref{prop:SemiGermeAdjacent}]
Suppose first that $a$ and $a'$ coincide along some arc with $x$ as
an end-point, and are
disjoint apart from this arc. By observation~\ref{obs:CCarc}, there exists
a curve $d$ passing through $x$ such that $T(a,d)$ and $T(a',d)$.
By Lemma~\ref{lem:SpheresConnexes}, this implies $(a,d)\diamondvert(a',d)$,
and by the same lemma we also have $(a,b)\diamondvert(a,d)$ and
$(a',b')\diamondvert(a',d)$. Hence $(a,b)\diamondvert(a',b')$.
Now we do not make the assumption any more that $a$ and $a'$ meet
only along an arc. Still, thanks to the hypothesis of the proposition,
we may choose a set $V$ homeomorphic to a closed disk, with $x$ on its
boundary, and such that $a\cap V=a'\cap V$ is an arc whose endpoints are $x$
and some other point $y$.
Lemma~\ref{lem:ArcConnexeBis}, applied to the surface
$\Sigma\smallsetminus \ring V$,
provides a
sequence $a_0=a$, \ldots, $a_n=a'$, of nonseparating curves such that
for all $i$, the curves $a_i$ and $a_{i+1}$ intersect only along
the arc $a\cap V$, hence we may conclude by applying iteratively
the reasoning above. \end{proof}
\section{Local subgraphs}\label{sec:LocalSubgraphs}
In section~\ref{sec:Adjacency}, we considered
edges $\{a,b\}, \{a',b'\}$ in the graph $\NCfin{\pitchfork}(\Sigma)$
satisfying $|a\cap b|=|a'\cap b'|=1$, and $\mathrm{Point}(a,b) = \mathrm{Point}(a',b')$. We defined and used the equivalence relation $\diamondvert$. The aim of this section is to provide a geometric interpretation of the equivalence classes. The results here are not used anywhere else in the paper. In particular, this section is not used in the proof of our main results. Nevertheless, we think it may help the reader to get a clear picture of the situation.
\subsection{The graph of germs} Let $x$ be a marked point in the surface $\Sigma$. In this section the we will study the local geometry of curves near $x$, so we may assume that $(\Sigma,x) = (\mathbb{R}^2, 0)$ whenever this is convenient. Given two simple arcs $a, a': [0,1] \to \Sigma$ with $a(0)=a'(0)=x$, we say that $a$ and $a'$ \emph{locally coincide} at $x$ if there exists a neighborhood $V$ of $x$ such that $a([0,1]) \cap V = a'([0,1]) \cap V$. This is an equivalence relation, whose equivalence classes are called \emph{germs of simple arcs at $x$}. The germ of $a$ is denoted $[a]_x$. We say that $a$ and $a'$ \emph{locally intersect only at $x$} if there exists a neighborhood $V$ of $x$ such that $a([0,1]) \cap a'([0,1]) \cap V = \{x\}$. This second relation obviously induces a relation on germs. Let us consider the graph $\mathcal{A}(x)$ whose vertices are the germs of simple arcs at $x$, with an edge between the germs of $a$ and $a'$ whenever $a$ and $a'$ locally intersect only at $x$.
This graph is \emph{not} connected, in fact it has infinitely (uncountably) many connected components, as we will see below. We postpone the description of the connected components to explain the relation with the adjacency relation defined in section~\ref{sec:Adjacency}. We say that two vertices $\alpha, \alpha'$ of the graph $\mathcal{A}(x)$ are \emph{comparable} if they belong to the same connected component of the graph.
\subsection{Germs and adjacency}
Given a point $x$ in $\Sigma$ and a simple closed curve $a$ in $\Sigma$ that contains $x$, we choose any one of the two germs of simple arc at $x$ included in $a$ and denote it by $\lfloor a \rfloor_{x}$. Which one of the two germs is chosen will not matter in what follows.
\begin{proposition} \label{pro:AdjacencyGerms} Let $a, b, a', b'$ be vertices in $\NCfin{\pitchfork}(\Sigma)$ such that $T(a,b) $ and $T(a', b')$ hold, and assume $\mathrm{Point}(a,b) = \mathrm{Point}(a',b')$. Then $\{a,b\} \diamondvert \{a',b'\}$ if and only if the germs $\lfloor a\rfloor_x$ and $\lfloor a'\rfloor_x$ are comparable. \end{proposition}
Proposition~\ref{pro:AdjacencyGerms} will be proved in section~\ref{sec:ProofAdjacencyGerms} below. The aim of the next two sections is to provide a simple characterization of distance, and connected components, in the graph of germs; see Proposition~\ref{Prop:DistanceAndWidth} below.
\subsection{Distance in local subgraphs}
In this section, we give a geometrical interpretation of the distance in three different graphs, which are very much like the graph of germs $\mathcal{A}(x)$.
Let $\Sigma$ be one of the following surfaces: (1) the compact annulus $\mathbb{S}^1 \times [0,1]$, (2) the open annulus $\mathbb{S}^1 \times \mathbb{R}$, or (3) the 2-torus $\mathbb{T}^2 =\mathbb{S}^1 \times \mathbb{S}^1$. We consider non-oriented simple arcs in $\Sigma$, more precisely simple curves connecting both sides of the annulus in case (1), properly embedded images of the real line connecting both ends of the open annulus in case (2), or simple closed curves in a fixed homotopy class, say homotopic to $\{0\}\times \mathbb{S}^1$ in case (3). Let $\mathcal{A}$ denote the graph whose vertices are one of the three above family of curves, with an edge between two curves whenever they are disjoint.
In order to express geometrically the distance in $\mathcal{A}$, let us consider the cyclic cover $p : \widetilde \Sigma \to \Sigma$, respectively in case (1), (2), (3) $$ p:\mathbb{R} \times [0,1] \to \mathbb{R} /\mathbb{Z} \times [0,1], \ \ \ \ p:\mathbb{R} \times \mathbb{R} \to \mathbb{R} /\mathbb{Z} \times \mathbb{R}, \ \ \ \ p:\mathbb{R} \times \mathbb{S}^1 \to \mathbb{R} /\mathbb{Z} \times \mathbb{S}^1 $$ given by the formula $p(x,y) = (x \text{ mod } 1, y)$. Let $T$ be the deck transformation $(x, y) \to (x+1, y)$.
Now consider two curves $a,b$ which are vertices of the graph $\mathcal{A}$. Let $\widetilde a, \widetilde b$ be respective lifts of $a,b$ under the covering map $p$. Note that the set $$ \{k \in \mathbb{Z}, \ \ T^k(\widetilde{a}) \cap \widetilde{b} \neq \emptyset \} $$ is an interval of $\mathbb{Z}$, which is finite in the compact cases (1) and (3) but may be infinite in the open annulus case (2). We define the \emph{relative width} $\mathrm{Width}(a,b)$ as the cardinal of this set. This is an element of $\{0, 1, \dots, +\infty \}$.
The reader may check easily that $\mathrm{Width}(a,b) = \mathrm{Width}(b,a)$.
\begin{proposition}
For every vertices $a \neq b$ of the graph $\mathcal{A}$, the distance in the graph
is given by
$$
d(a,b) = \mathrm{Width}(a,b) + 1.
$$
In cases (1) and (3), the graph $\mathcal{A}$ is connected.
In case (2), $a$ and $b$ are in the same connected component of $\mathcal{A}$ if
and only if $\mathrm{Width}(a,b) < +\infty$. \end{proposition}
\begin{proof}
Let $a,b$ be as in the statement, and denote $w = \mathrm{Width}(a, b)$. We first assume that $w < +\infty$. By Schoenflies' theorem (in case (2), applied in the two-point compactification of the annulus, which is a sphere), we may assume that $a$ is a vertical curve whenever this makes our life easier.
We first note that if $w=0$ then $a$ and $b$ admit lifts that are disjoint from every $T$-translate of each other, which shows that $a$ and $b$ are disjoint, and thus $d(a,b)=1$. Let us now assume $w>0$, and prove the two following key properties.
\begin{enumerate} \item[(i)] For every vertex $a'$ of $\mathcal{A}$ such that $d(a,a')=1$, $$\mathrm{Width}(a',b) \geq \omega-1.$$ \item[(ii)] There exists a vertex $a'$ of $\mathcal{A}$ such that $d(a,a')=1$ and $$\mathrm{Width}(a',b) \leq w-1.$$ \end{enumerate}
To prove the first property, consider $a'$ such that $d(a,a')=1$. By definition of the width $w$, we may find lifts $\widetilde{a}, \widetilde{b}$ of $a,b$ such that $\widetilde b$ is disjoint from $\widetilde{a}, T^{w+1}\widetilde{a}$ but meets $T(\widetilde{a}), \dots, T^w(\widetilde{a})$. Since $a$ and $a'$ are disjoint, there is a lift $\widetilde{a'}$ of $a'$ which is between $\widetilde{a}$ and $T(\widetilde{a})$. Then the curves $$ T(\widetilde{a'}), \dots, T^{w-1}(\widetilde{a'}) $$ are between the two curves $T(\widetilde{a})$ and $T^w(\widetilde{a})$, and those two curves are not in the same connected component of $\widetilde{\Sigma} \setminus T^i(\widetilde{a'})$, for $i=1, \dots, w-1$. Since the curve $\widetilde b$ is connected and meets the two curves $T(\widetilde{a})$ and $T^w(\widetilde{a})$, it must meet all the $T^i(\widetilde{a'})$. This proves that $\mathrm{Width}(a', b) \geq w-1$.
Let us prove the second property. We consider $\widetilde{a}, \widetilde{b}$ as above. Let $S$ denote the compact strip or annulus bounded by $\widetilde a \cup T(\widetilde{a})$. Remember that $\widetilde{b}$ meets $T(\widetilde{a})$ but not $\widetilde{a}$. Thus $\widetilde{b} \cap S$ is included in a (maybe infinite) family of \emph{bigons}, \textsl{i.e.}, topological disks bounded by a simple closed curve made of a segment of the curve $T(\widetilde{a})$ and a segment of the curve $\widetilde{b}$. Let $S^+$ denote the union of these bigons. Symmetrically, the curve $\widetilde{b'} := T^{-w}(\widetilde{b})$ meets $\widetilde{a}$ but not $T(\widetilde a)$. Thus $T^{-w}(\widetilde{b}) \cap S$ is included in a union $S^-$ of bigons formed by the curves $\widetilde{a}$ and $\widetilde{b'}$. A key point is that the sets $S^-$ and $S^+$ are disjoint, because the curves $\widetilde b$ and $\widetilde b'$ are disjoint, since $b$ is simple. Thus we may construct a homeomorphism $H$ supported in $S$ such that $H(S^-)$ is included in an arbitrarily small neighborhood of $\widetilde a$, and $H(S^+)$ is included in an arbitrarily small neighborhood of $T(\widetilde{a})$. In particular, we may find a curve $\widetilde{a'}$, which is a lift of some element $a'$ of $\mathcal{A}$, included in the interior of $S$ and disjoint from both $S^-$ and $S^+$ (to be more explicit, take $\widetilde a' = H^{-1}(\{1/2\} \times [0,1])$ in the annulus case, in coordinates for which $a$ is the vertical curve $\{0\} \times [0,1]$). Note that $\widetilde{a'}$ is disjoint from $\widetilde{b}$ and $T^{-w}(\widetilde{b})$, and separate both curves, \textsl{i.e.}, the first one is on the right-hand side of $\widetilde{a'}$, and the second one is on the left-hand side. Thus the set $$ \{k \in \mathbb{Z}, \ \ T^k(\widetilde{b}) \cap \widetilde{a'} \neq \emptyset \} $$ has cardinality at most $w-1$. Which proves that $\mathrm{Width}(a', b) \leq w-1$, as wanted.
Using (i) and (ii), an induction on $n$ shows that $d(a,b)=n$ if and only if $\mathrm{Width}(a,b)+1=n$, which completes the proof in the case when $\mathrm{Width}(a,b)$ is finite. When $\mathrm{Width}(a,b) = +\infty$,
an argument analogous to property (i) above shows that $\mathrm{Width}(a',b)=+\infty$ for every $a'$ such that $d(a,a')=1$. This shows that $a$ and $b$ are not in the same connected component of the graph. This completes the proof of the proposition. \end{proof}
\subsection{Distance in the graph of germs}
Let us go back to the graph of germs $\mathcal{A}(x)$. Assume $(\Sigma,x) = (\mathbb{R}^2, 0)$. Given two vertices $a,b$ of $\mathcal{A}(x)$, we define their \emph{local relative width} $\mathrm{Width}(a,b)$ as follows. The plane minus the origin is identified with the open annulus $\mathbb{S}^1 \times \mathbb{R}$, and we consider the graph $\mathcal{A}$ from the previous section in the open annulus case. Then $\mathrm{Width}(a,b)$ is defined as the infimum of the quantity $\mathrm{Width}(A,B)$, where $A$ and $B$ are vertices of $\mathcal{A}$ whose germs respectively equal $a$ and $b$. Here is a more practical definition, which is easily seen to be equivalent. Consider the universal cover $p: \widetilde \Sigma \to \Sigma$ as above. Abuse the definition by still denoting $a,b:[0,1] \to \Sigma$ two curves with $a(0)=b(0)=0$ whose germs respectively equal $a,b$. Let $\widetilde a, \widetilde b$ denote lifts of (the restrictions to $(0,1]$ of) $a,b$ in $\widetilde \Sigma$. Then the number $\mathrm{Width}(a,b)=w$ is characterized by the two following properties:
\begin{itemize} \item[(i)] for every $t_0 \in (0,1]$, the restriction of $\widetilde a$ to $(0, t_0]$ meets at least $w$ integer translates of $\widetilde b$;
\item[(ii)] there exists $t_0 \in (0,1]$ such that the restriction of $\widetilde a$ to $(0, t_0]$ meets exactly $w$ integer translates of $\widetilde b$; \end{itemize}
Analogously to the previous section, the distance in the graph of germs is characterized by the local relative width. \begin{proposition}\label{Prop:DistanceAndWidth} Let $a \neq b$ be two vertices of the graph $\mathcal{A}(x)$. Then $a$ and $b$ are in the same connected component of $\mathcal{A}(x)$ if and only if $\mathrm{Width}(a,b) < +\infty$. In this case, the distance in the graph is given by $$ d(a,b) = \mathrm{Width}(a,b) + 1. $$ \end{proposition}
The proof is very similar to the proof in the previous section. Details are left to the reader.
\subsection{Proof of Proposition~\ref{pro:AdjacencyGerms}} \label{sec:ProofAdjacencyGerms}
Let $a, b, a', b'$ be vertices of $\NCfin{\pitchfork}(\Sigma)$ such that $T(a,b)$ and $T(a', b')$ hold, and assume $\mathrm{Point}(a,b) = \mathrm{Point}(a',b')$. Denote $x$ the common intersection point.
If $c$ is another vertex such that $\{a, b, c\}$ is a 3-clique of type bouquet or extra bouquet, then the germs $\lfloor a\rfloor_x$ and $\lfloor c\rfloor_x$ are disjoint, thus obviously comparable. This entails the direct implication in Proposition~\ref{pro:AdjacencyGerms}.
Let us prove the converse implication. We assume that the germs $\lfloor a\rfloor_x$ and $\lfloor a'\rfloor_x$ are comparable. In other words, there exists arcs $\alpha_0, \dots, \alpha_n$ with $\alpha_i(0)=x$ and whose sequence of corresponding germs is a path from $\lfloor a\rfloor_x$ to $\lfloor a'\rfloor_x$ in the graph of germs. Note that each germ $\alpha_i$ may be extended to a non separating curve $a_i$, and we can find another non separating curve $b_i$ such that $T(a_i, b_i)$ holds. Thus the end of the proof is a direct consequence of the following lemma.
\begin{lemma} Let $a, b, a', b'$ be vertices in $\NCfin{\pitchfork}(\Sigma)$ such that $T(a,b)$ and $T(a', b')$ hold. Assume that for some choices $\lfloor a\rfloor_x, \lfloor a'\rfloor_x$ of arcs at $x$ included respectively in $a$ and $a'$, the germs $\lfloor a\rfloor_x, \lfloor a'\rfloor_x$ intersect only at $x$. Then $\{a, b\} \diamondvert \{a', b'\}$. \end{lemma}
\begin{proof}[Proof of the lemma] Let $c$ be an arc that contains $x$ in its interior and locally coincides with $\lfloor a\rfloor_x \cup \lfloor a'\rfloor_x$. Extend $c$ into a non separating closed curve, still denoted $c$, and consider any other non separating curve $d$ such that $T(c,d)$ holds. Since $c$ locally ``half coincides'' near $x$ with both $a$ and $a'$, we may apply Proposition~\ref{prop:SemiGermeAdjacent} twice, and get that $\{a,b\} \diamondvert \{c,d\} \diamondvert \{a', b'\}$.
\end{proof}
\subsection{Curves and diffeomorphisms}\label{ssec:Tourbillons} In this short subsection we explain how one can use the fine curve graph to detect fundamental non differentiability.
Let $\Phi$ be an automorphism of $\NCfin{\pitchfork}(\Sigma)$. We introduce the following property $D(\Phi)$:
\emph{For every vertices $a$, $b$ of $\NCfin{\pitchfork}(\Sigma)$ such that $T(a,b)$ holds, if $\mathrm{Point}(\Phi(a),\Phi(b)) = \mathrm{Point}(a,b)$ then there exists $a', b'$ such that $T(a', b')$ holds, $\mathrm{Point}(a',b') = \mathrm{Point}(a,b)$ and $\{\Phi(a'), \Phi(b') \} \diamondvert \{a', b'\}$.}
Note that this property is clearly invariant under conjugacy in the group of automorphisms. Let $h$ be a homemorphism of $\Sigma$, and denote $\Phi = \Phi_h$ the action of $h$ on the graph $\NCfin{\pitchfork}(\Sigma)$. \begin{observation} If $h$ is differentiable everywhere, then property $D(\Phi_h)$ holds. \end{observation}
Indeed, hypothesis $\mathrm{Point}(\Phi(a),\Phi(b)) = \mathrm{Point}(a,b)$ is equivalent to the fact that the point $x = \mathrm{Point}(a,b)$ is a fixed point of $h$. Since $h$ is differentiable at $x$, it is easy to check that every germ of smooth arc at $x$ is comparable to its image. Take any two smooth curves $a', b'$ such that $T(a', b')$ and $\mathrm{Point} (a', b') = \mathrm{Point}(a,b)$, then Proposition~\ref{pro:AdjacencyGerms} tells us that $\{\Phi(a'), \Phi(b') \} \diamondvert \{a', b'\}$.
Now consider a particular homeomorphism $h$ of $\Sigma$ and assume that $h$ admits a fixed point where, for some local polar coordinates, $h$ writes $$ (r, \theta) \mapsto (r, \theta + \frac{1}{r}). $$
\begin{observation} Property $D(\Phi_h)$ does not hold. \end{observation}
An easy proof of this is obtained by considering the \emph{local rotation interval} of $h$ at $x$, as defined in~\cite{FredLocRot}, section 2.3. Indeed, the local rotation interval of $h$ at $x$ equals $\{+\infty\}$, which accounts for the fact that orbits turn faster and faster around $x$, in the positive direction, as we get nearer and nearer to $x$ (the quickest way to check this is to show that the \emph{local rotation set} of $h$ at $x$ is $\{+\infty\}$, and then to apply Théorème 3.9 of~\cite{FredLocRot} that relates the local rotation set and the local rotation interval). We argue by contradiction to show that property $D(\Phi_h)$ does not hold. Assuming property $D(\Phi_h)$ holds, consider curves $a,b$ such that $T(a,b)$ holds and $\mathrm{Point}(a, b) = x$. Let $a', b'$ be given by property $D(\Phi_h)$, such that $\{\Phi(a'), \Phi(b') \} \diamondvert \{a', b'\}$. The reverse direction of Proposition~\ref{pro:AdjacencyGerms} tells us that the germs of $h(a')$ and $a'$ are comparable at $x$. This entails easily, from the definition, that the local rotation interval of $h$ at $s$ is a bounded interval, a contradiction.
\section{Fine graph of smooth curves}\label{sec:AutCFinLisse}
In this section we address the case of smooth curves, and prove Theorem~\ref{thm:AutCFinLisse} and Proposition~\ref{prop:PasDiff}. In all the section, $\Sigma$ will be a connected, nonspherical surface without boundary, endowed with a smooth structure. In Section~\ref{sec:configSmoothCurves} we will restrict to the orientable case.
\subsection{From bijections to higher regularity}
One step in the proof of Theorem~\ref{thm:AutC1} was Proposition~\ref{prop:BijHomeo}, in which we proved that if an automorphism of $\NCfin{\pitchfork}(\Sigma)$ is supported by a bijection of $\Sigma$, then that bijection is a homeomorphism of $\Sigma$.
We may ask the same question about automorphisms of $\mathcal{NC}_\pitchfork^{\dagger\infty}(\Sigma)$, and this paragraph is devoted to the proof of the following two statements. We denote by $\Homeo_{\infty \pitchfork}(\Sigma)$ the group of bijections of $\Sigma$ which preserve the family of smooth, nonseparating closed curves, and preserve transversality between such curves. The first statment below justifies this notation. Here, for simplicity we restrict to the case of orientable surfaces.
\begin{proposition}\label{prop:BijLisseHomeo}
Let $\Sigma$ be a connected, non spherical orientable surface.
The group $\Homeo_{\infty \pitchfork}(\Sigma)$ is contained in
$\Homeo(\Sigma)$.
\end{proposition}
\begin{proof}[Proof of Proposition~\ref{prop:BijLisseHomeo}]
Let $h \in \Homeo_{\infty \pitchfork}(\Sigma)$. We will prove that the image under $h$
of any open set is an open set. This is the continuity of $h^{-1}$, and by
applying the argument to $h$ we also get the continuity of $h$.
To do this, we only need to consider the images of a family of sets that
generates the topology. Given three non separating curves $a,b,c$, we denote
$V(a; b, c)$ the union of all the non separating curves $d$ that meet $a$
and are disjoint from $b$ and $c$.
\begin{observation}
The set $V(a ; b,c)$ is the union of some of the connected components of the
complement of $b \cup c$ that meet $a$. In particular, it is an open set.
\end{observation}
Indeed, let $x$ be a point of $V(a; b,c)$. By definition there is a non
separating curve $d$ passing through $x$ and meeting $a$ but not $b$ nor $c$.
Consider another point $y$ that belongs to the connected component $V_x$ of
the complement of $b \cup c$ that contains $x$. By modifying $d$ using an arc
connecting $x$ to $y$ in $V_c$, we find another curve $d'$, isotopic to $d$,
still meeting $a$ but not $b$ nor $c$, and passing through $y$. This proves
that $V(a ; b,c)$ contains $V_x$, and the observation follows.
Now let $a$ be a nonseparating curve.
Let $a^+, a^-$ be obtained by pushing $a$ to both sides. Then $V(a; a^+, a^-)$
is a neighborhood of $a$, and by making $a^+$ and $a^-$ vary we get a basis
of neighborhoods ${\mathcal B}(a)$ of the curve $a$.
The union of all these families ${\mathcal B}(a)$ clearly
generates the topology of~$\Sigma$.
Thus it suffices to check that the image under $h$ of each set $V(a; b, c)$
is an open set. But since $h$ is a bijection, we have
\[ h(V(a; b, c)) = V( h(a); h(b), h(c)). \]
By hypothesis $h(a), h(b), h(c)$ are nonseparating closed curves, and by the
observation this set is open. \end{proof}
We now prove Proposition~\ref{prop:PasDiff} stated in the introduction, namely the existence of elements of $\Homeo_{\infty \pitchfork}(\Sigma)$ that are not smooth.
\begin{proof}[Proof of Proposition~\ref{prop:PasDiff}]
We will construct a homeomorphism
$F\colon\mathbb{R}^2\to\mathbb{R}^2$, which is not differentiable at the origin,
but such that both $F$ and $F^{-1}$ send smooth curves to
smooth curves. The construction can easily be modified to make $F$ compactly
supported, and then be transported on our surface~$\Sigma$. It will be
clear from the constuction that this map preserves transversality.
Let $h\colon\mathbb{R}\to\mathbb{R}$ be a smooth diffeomorphism supported in the
segment $[1/2, 2]$. That is to say, $h(x)=x$ for all $x$ outside
$[1/2,2]$; we suppose however that $h(1)\neq 1$.
We consider the map $F$ defined by $F(x,y)=(x, xh(y/x))$ if
$x\neq 0$, and $F(x,y)=(x,y)$ otherwise. We claim that this map
has the desired property.
This map, as well as its inverse, is obviouly smooth in
restriction to $\mathbb{R}^2\smallsetminus\{(0,0)\}$. Direct computation
shows that $F$ has directional derivatives in all directions
around the origin,
but the ``differential'' fails to be linear:
both partial derivatives are
those of the identity, while the
directional derivative in the direction $(1,1)$ is not.
So $F$ is not differentiable at the origin.
Now, let $\gamma\colon\mathbb{R}\to\mathbb{R}^2$ be a smooth, proper embedding.
If $(0,0)$ is not in the image of $\gamma$, then of course,
$F\circ\gamma$ is still smooth. So suppose, say, that
$\gamma(0)=(0,0)$.
If $\gamma'(0)$ lies outside the two (opposite) sectors of
vectors of slopes between $1/2$ and $2$, then $F\circ\gamma$
and $\gamma$ have the same germ at $0$.
Otherwise,
and up to reparameterization,
we can write, near $0$,
$\gamma(t) = (t, \alpha(t))$ where $\alpha$ is a smooth map
(satisfying $\alpha(0)=0)$),
from a neighborhood of $0$, to~$\mathbb{R}$. This yields the formula
\[ F\circ\gamma(t) = \left(t, th \left(\frac{\alpha(t)}{t}\right)\right). \]
Now, the smoothness of $F\circ\gamma$
follows from the following elementary observation.
\noindent
{\bf Claim.}
{\em Let $\alpha\colon\mathbb{R}\to\mathbb{R}$ be a smooth map
satisfying $\alpha(0)=0$. Then the map $t\mapsto \frac{\alpha(t)}{t}$
when $t\neq 0$, and $\alpha'(0)$ when $t=0$, is smooth.}
Indeed, by the fundamental theorem of Calculus, for all $t\in\mathbb{R}^\ast$ we have
\[ \frac{\alpha(t)}{t} = \int_0^1 \alpha'(ts)ds, \]
and this integral with parameter can be differentiated
indefinitely.\footnote{We borrow this elegant argument from~\cite{stack}.}
\end{proof}
\subsection{A weak convergence for sequences of curves}
In order to prove~Theorem~\ref{thm:AutCFinLisse}, we now explain how to recognize configurations of smooth curves. Given two vertices $a$ and $b$ of the graph $\mathcal{NC}_\pitchfork^{\dagger\infty}(\Sigma)$, we will denote by $a-b$ the property that they are neighbors in the graph. If $(f_n)_{n\in\mathbb{N}}$ is a sequence of vertices, we denote by $(f_n)_{n\in\mathbb{N}}-a$ the property that for all $n$ large enough, $f_n-a$.
The first property of sequences we may recover from the graph is the distinction of what curves go to infinity. \begin{lemma}\label{lem:fnSenVa}
Let $(f_n)_{n\in\mathbb{N}}$ be a sequence of vertices of $\mathcal{NC}_\pitchfork^{\dagger\infty}(\Sigma)$.
The following are equivalent.
\begin{itemize}
\item[-] for all $d$, we have $(f_n)_{n\in\mathbb{N}}-d$;
\item[-] for every compact subset $K$ of $\Sigma$, for every $n$ large
enough, $K\cap f_n=\emptyset$.
\end{itemize} \end{lemma} \begin{proof}
The second statement obviously implies the first, as we may just
take $K=d$. Let us prove the converse implication by contraposition.
Suppose $K$ intersects infinitely many $f_n$. Since $K$ is compact,
there is a point $x\in K$, such that every neighborhood of $x$
intersects infinitely many $f_n$. We consider two open sets
$B_1$ and $B_2$ with $x\in B_1$ and $\overline{B_1}\subset B_2$,
and three bottle-shaped arcs, as in Figure~\ref{fig:3Bouteilles}.
These arcs may be continued to form three nonseparating closed curves,
$d_1$, $d_2$ and $d_3$. Now, let $n$ be such that $f_n$ enters $B_1$.
If $f_n$ does not enter nor leave $B_2$ through the neck of the bottle
corresponding to $d_1$, then we cannot have $f_n-d_1$, since $f_n$ and
$d_1$ have to intersect at least twice. Hence, $f_n$ passes
through the neck of $d_1$, and in order to impose that $f_n-d_2$, another
arc of $f_n$ has to get out of the bottle corresponding of $d_2$
through its neck. But then $f_n$ has to meet $d_3$ twice, and we cannot have
$f_n-d_3$. In other words, for
all $n$ such that $f_n$ enters $B_1$, we can't have $f_n-d_1$ and $f_n-d_2$
and $f_n-d_3$, hence the first statement is not true, and our
implication is proved.
\begin{figure}
\caption{Three bottles}
\label{fig:3Bouteilles}
\end{figure} \end{proof} Thus, we will say here that a sequence $(f_n)_{n\in\mathbb{N}}$ is {\em relevant} if it has no subsequence $(f_{\varphi(n)})_{n\in\mathbb{N}}$ such that for all $d$, $(f_{\varphi(n)})_{n\in\mathbb{N}}-d$. We now explore, for such sequences, the following notion of convergence. We say that a relevant sequence $(f_n)$ {\em converges in a weak sense} to a curve $a$ if for every $d$ such that $a-d$, we have $(f_n)-d$. We denote this property by~$W((f_n),a)$. \begin{lemma}\label{lem:ConvFaible}
Let $(f_n)_{n\in\mathbb{N}}$ be a sequence of vertices, and $a$ be a vertex
of $\mathcal{NC}_\pitchfork^{\dagger\infty}(\Sigma)$.
\begin{itemize}
\item[$\bullet$]
If $W((f_n),a)$, then the sequence $(f_n)_{n\in\mathbb{N}}$ converges
in the Hausdorff topology to $a$: for every neighborhood $V$ of the
curve $a$, for all $n$ large enough, we have $f_n\subset V$.
\item[$\bullet$]
If the sequence $(f_n)_{n\in\mathbb{N}}$ of curves, with some appropriate
parameterization, converges in $C^1$-topology to $a$, then
$W((f_n)_{n\in\mathbb{N}},a)$.
\end{itemize} \end{lemma} \begin{proof}
We first prove the first point. Let us first mention that the statement
we wrote is indeed equivalent to the Hausdorff convergence, because
any essential curve in a small enough neighbourhood of $a$ must pass close
to every point of $a$, and thus is Hausdorff-close to $a$.
Suppose for contradiction that
for some neighborhood $V$ of $a$, we have $f_n\not\subset V$
infinitely often. Then we may find a point $x$, not in $\overline{V}$,
such that every neighborhood of $x$ is visited by infinitely many
$f_n$. We may construct three bottle-shaped arcs around $x$ exactly
as in the proof of the preceding lemma, and complete these arcs
to non separating simple closed curves $d_1$, $d_2$, $d_3$, which
can be requested to satisfy $a-d_i$ for $i=1,2,3$. Then we cannot
have $f_n-d_i$ for all $i\in\{1,2,3\}$ for the same reason as in
this preceding proof, and this contradicts the hypothesis
that~$W((f_n),a)$ holds.
The second point is the well known stability of transversality in
the $C^1$-topology.
\end{proof} \begin{remark}
In fact, the condition $W((f_n), a)$ implies $C^0$-convergence, in the
following sense. Given a parameterization $\alpha$ of $a$, we can choose the
parameterizations of the $f_n$'s yielding a sequence of parmaetrized curves
converging uniformly to~$\alpha$.
As we will not use this fact, we only sketch a quick argument. Let $V$ be
a small tubular neighborhood of $a$, and let $d_1,\ldots,d_N$ be simple
closed nonseparating curves, each meeting $a$ transversely at one point,
and cutting $V$ in small chunks $V_1,\ldots,V_N$ that are met by $\alpha$ in
that
cyclic order. Let $n$ be large enough so that $f_n\subset V$ and
$f_n-d_i$ for each $i$. Then $f_n\cap V_i$ is connected, and $f_n$
visits the pieces $V_1,\ldots,V_N$ in that order. Hence we may choose
a parameterization of $f_n$, say, $F_n$, in such a way that for all
$i\in\{1,\ldots,N\}$ and all $t$, $F_n(t)\in V_i$ if and only if
$\alpha(t)\in V_i$. This implies that $F_n$ is uniformly close to $\alpha$. \end{remark} This notion is actually somewhere strictly in between
$C^0$-convergence and $C^1$-convergence, as we remark in the following example. This construction will play a crucial role below in the proof of Theorem~\ref{thm:AutCFinLisse}. \begin{example}\label{eg:ConvFaible}
Let $a$ be a smooth nonseparating curve in $\Sigma$. We choose
a point $p$ in $a$, and a chart around one of its point, diffeomorphic
to $\mathbb{R}^2$, in such a way that $a$ corresponds to the axis of
equation $y=0$ in that plane. In this chart, we consider the
functions
$f_1\colon x\mapsto\frac{2}{1+x^2}$,
and for all $n\geqslant 2$,
$f_n\colon x\mapsto\frac{f_1(nx)}{n}$.
Abusively, we still denote their graphs by the same letters,
and then, we may extend these arcs, viewed in $\Sigma$, to
simple closed curves (consisting of pushing $a$ aside), that
converges $C^1$ to $a$ outside of the point $p$.
Abusively we still use the same letters $f_n$ to denote these
closed curves.
Obviously, the sequence $(f_n)_{n\geqslant 1}$ does not
converge $C^1$ to $a$, because $f_n$ has slope $-1$ at the
point $(1/n, 1/n)$.
Nonetheless, we claim that $W((f_n),a)$ holds.
Indeed, let $d$ be such that $a-d$. Since the sequence $(f_n)$
converges $C^1$ to $a$ everywhere except at the origin of this
$\mathbb{R}^2$ chart, the only case in which it is not already clear
that $(f_n)-d$ is when $d$ meets $a$ transversely at the origin.
If $d$ has a strictly positive slope there, then for $n$ large
enough, the intersection $f_n\cap d$ will be transverse because
the slopes of $f_n$ are all negative in the region $x>0$.
The case when $d$ has negative slope is symmetric, and if $d$
has vertical slope, it will be transverse with $f_n$ since these
have bounded slopes.
\end{example}
\subsection{Recognizing configurations of smooth curves} \label{sec:configSmoothCurves} In this last section we assume that our surface $\Sigma$ is orientable.
If $a$, $b$ are vertices of $\mathcal{NC}_\pitchfork^{\dagger\infty}(\Sigma)$, we will denote by $D_\infty(a,b)$ the condition that $a-b$ and for all sequences $(f_n)$ and $(g_m)$ such that $W((f_n),a)$ and $W((g_m),b)$, we have $f_n-g_m$ for all $m, n$ large enough.
\begin{lemma}\label{lem:DisjointLisse}
Let $a$, $b$ be smooth nonseparating curves. Then
$D_\infty(a,b)$ holds if and only if $a$ and $b$ are disjoint. \end{lemma} \begin{proof}
Suppose first that $a$ and $b$ are disjoint. Then they admit
disjoint neighborhoods, $V_1$ and $V_2$. For any sequences
$(f_n)$ and $(g_m)$ with $W((f_n),a)$ and $W((g_m),b)$,
for all $m,n$ large enough we have $f_n\subset V_1$ and
$g_m\subset V_2$, by Lemma~\ref{lem:ConvFaible}. Hence,
$f_n-g_m$ for all $m,n$ large enough, and $D_\infty(a,b)$
holds indeed.
Now, suppose that $a$ and $b$ are not disjoint. Since
$a-b$, the curves $a$ and $b$ have a transverse intersection,
and in an appropriate chart diffeomorphic to $\mathbb{R}^2$, the
curves $a$ and $b$ correspond respectively to the axes
$y=0$ and $x=0$.
Then we may form a sequence $(f_n)$ such that $W((f_n),a)$
exactly as in example~\ref{eg:ConvFaible}, and for
$(g_n)$ we just exchange coordinates $x$ and $y$.
For all $n$, the curves $f_n$ and $g_n$ have
a non transverse intersection point (at $(1/n,1/n)$ in the
chart of Example~\ref{eg:ConvFaible}), hence the condition
$D_\infty(a,b)$ does not hold. \end{proof} In the end of the proof, the curves $f_n$ and $g_n$ were tangent at their intersection point, hence not neighbors in the graph $\mathcal{NC}_\pitchfork^{\dagger\infty}(\Sigma)$. This may look accidental, but upon changing the formula of $f_1$ in Example~\ref{eg:ConvFaible} to $x\mapsto \frac{3}{2+x^2}$, for example, we get three intersection points.
From now on, we restrict ourselves to the case of orientable surfaces. One reason is that it would take more work to recover the extra bouquets and not only the bouquets; one other reason is that the next lemma works best when at least one of $a$, $b$ or $c$ is two-sided.
\begin{lemma}\label{lem:BouquetLisse}
Suppose $\Sigma$ is orientable.
Let $\{a,b,c\}$ be a $3$-clique of $\mathcal{NC}_\pitchfork^{\dagger\infty}(\Sigma)$,
and suppose that these three curves pairwise intersect.
Then the following are equivalent.
\begin{enumerate}
\item This $3$-clique is of type bouquet.
\item There exists a relevant sequence $(f_n)$ of vertices of
$\mathcal{NC}_\pitchfork^{\dagger\infty}(\Sigma)$, which are all disjoint from $a$,
and such that for all $d$ disjoint from $b$ and
satisfying $c-d$, we have $(f_n)-d$.
\end{enumerate} \end{lemma} \begin{proof}
Suppose $\{a,b,c\}$ is of type bouquet. Then the sequence
$(f_n)_{n\in\mathbb{N}}$ can be constructed explicitly.
Let $p=b\cap c$. Fix a (smooth) metric on $\Sigma$,
we remove all points of the ball $B(p,\frac{1}{n})$ off the
curves $b$ and $c$, this gives two arcs. There is a natural
way of adding smooth subarcs of $B(p,\frac{1}{n})$ in order
to extend this union of two arcs, to a curve $f_n$ which
does not intersect $a$.
In a one-holed torus neighborhood of $b\cup c$, with
a choice of meridian and longitude coming from $b$ and $c$, these curves
$f_n$ have slope $1$, or $-1$; these are indeed nonseparating simple
closed curves.
Now if $d$ is disjoint from $b$ and satisfies $c-d$, then
either $d$ is disjoint from $c$, and then $f_n$ is disjoint from $d$
for all $n$ large enough, or $d$ has a transverse intersection
with $c$ at a point distinct from $p$, and we also have $f_n-d$ for all
$n$ large enough. Thus, (1) implies~(2).
Conversely, suppose~(2). We first claim that the sequence $(f_n)$ then
concentrates into neighborhoods of $b\cup c$.
For contradiction, suppose that we can find a neighborhood $V$ of
$b\cup c$, such that $f_n\not\subset V$ for infinitely many $n$. Then, there
exists a point $x$, with $x\not\in b\cup c$, and such that every
neighborhood of $x$ meets infinitely many $f_n$. Then we may choose
three bottle-shaped arcs around $x$, and complete them into curves
$d_1$, $d_2$ and $d_3$ disjoint from $b$ and satisfying $d_j-c$ for
$j=1,2,3$. Indeed, we may start with a curve $d_0$ obtained by
pushing $b$ aside, and then perform surgeries on $d_0$.
The same reasoning as in the proof of Lemma~\ref{lem:fnSenVa} shows
that $f_n\not -d_j$
for some $j\in\{1,2,3\}$ and for infinitely
many $n$,
contradicting the hypothesis~(2).
Now, suppose for contradiction that $\{a,b,c\}$ is a necklace. Then,
for a sufficiently small regular neighborhood $V$ of $b\cup c$, we
may observe that $V\smallsetminus a$ is contractible. Hence it cannot
contain any nonseparating simple closed curve $f_n$, and the
hypothesis~(2) cannot be fullfilled. This proves that (2) implies~(1). \end{proof}
Now the proof of Theorem~\ref{thm:AutCFinLisse} is a straightforward adaptation of the proof of Theorem~\ref{thm:AutC1}. The statements about connectedness of complexes of arcs, for example, are equivalent to their counterparts with regularity, because of the argument of homotopy recalled in the proof of Lemma~\ref{lem:NonSepFinRelUnBordConnexe} and borrowed from~\cite{BHW}.
\end{document} | arXiv |
Modelling virtual radio resource management in full heterogeneous networks
Sina Khatibi ORCID: orcid.org/0000-0002-6704-33261,2 &
Luis M. Correia1
EURASIP Journal on Wireless Communications and Networking volume 2017, Article number: 73 (2017) Cite this article
Virtual radio access networks (RANs) is the candidate solution for 5G access networks, the concept of virtualised radio resources completing the virtual RAN paradigm. This paper proposes a new analytical model for the management of virtual radio resources in full heterogeneous networks. The estimation of network capacity and data rate allocation are the model's two main components. Based on the probability distribution of the signal-to-interference-plus-noise-ratio observed at the user terminal, the model leads to the probability distribution for the total network data rate. It considers different approaches for the estimation of the total network data rate, based on different channel qualities, i.e., optimistic, realistic and pessimistic. The second component uses the outcome of the first one in order to maximise the weighted data rate subject to the total network capacity, the SLAs (service level agreements) of Virtual Network Operators (VNOs), and fairness. The weights for services in the objective function of the resource allocation component enable the model to have prioritisation among services. The performance of the proposed model is evaluated in a practical heterogeneous access network. Results show an increase of 2.5 times in network capacity by implementing an access point at the centre of each cell of a cellular network. It is shown that the cellular network capacity itself can vary from 0.9 Gbps in the pessimistic approach up to 5.5 Gbps in the optimistic one. Finally, the isolation of service classes and VNOs by means of virtualisation of radio resources is clearly demonstrated.
The monthly global data traffic is going to surpass 10 EB in 2017, as the result of the proliferation of smart devices and of traffic-hungry applications [1]. In order to address this issue, operators have to find a practical, flexible and cost-efficient solution for their networks expansion and operation, using the scarce available radio resources. The increment of cellular networks' capacity by deploying dense base stations (BSs) is the groundwork in any candidate solution.
In addition, traffic offloading, e.g., to Wi-Fi access points (APs), has proven to be a valuable complementary approach. According to [2, 3], an acceptable portion of traffic can be offloaded to APs, just by deferring delay-tolerant services for a pre-specified maximum interval until reaching an AP. Offloading approaches are generally based on using other connectivity capabilities of mobile terminals, whenever it is possible, instead of using further expensive cellular bands. The authors in [4] discuss the economics of traffic offloading, and in [5] address an energy-saving analysis.
Nevertheless, drastic temporal and geographical variations of traffic, in addition to the shortage of network capacity, make the situation for operators even worse [6]. The usual provisioning of radio access networks (RANs) for busy hours leads to an inefficient resource usage with relatively high CApital and OPerational Expenditure (CAPEX and OPEX) costs, which is not acceptable anymore. Instead, operators are in favour of flexible and elastic solutions, where they can also share their infrastructure.
Lately, the sharing of network infrastructure using network function virtualisation (NFV) has become an active research topic to transform the way operators architect their networks [7]. In the same research path, the concept of virtualisation of radio resources for cellular networks is proposed in [8–10]. The key idea is to aggregate and manage all the available physical radio resources in a set of infrastructures, offering pay-as-you-go connectivity-as-a-service (CaaS) to virtual network operators (VNOs). Virtual radio resource management (VRRM) is a non-trivial task, since it has to serve multiple VNOs with different requirements and service level agreements (SLAs) over the same infrastructure. Furthermore, wireless links are always subject to fading and interference, hence, their performance is variable [11]. The proposed model for virtual radio resource management in [8–10] has two key parts: (i) estimation of available radio resources and (ii) allocation of the available resources estimated in the first step to the services of VNOs. In this paper an analytical description of the model is provided followed by the evaluation for a practical scenario. The novelty of this paper can be summarised as follows:
This paper extends the analytical model for the management of virtual radio resources, considering full heterogeneous access networks, and including both non-cellular (e.g., Wi-Fi) and cellular (e.g., GSM, UMTS, LTE and whatever comes next in 5G–5th generation) networks. The key point in extending the model to non-cellular networks is the consideration of the effect of collision on the total network throughput. Consequently, the model has to consider the number of connected terminals to the Wi-Fi network, while optimising the other objectives.
The techniques for estimating network capacity are improved with three extra approaches, i.e., optimistic, realistic and pessimistic ones, which approximate the model to a real network operation.
A comprehensive study of the proposed model for VRRM in full heterogeneous networks under different channel quality conditions and different traffic load scenarios is given.
This paper is organised as follows: Section II addresses the background and related works, and Section III describes the proposed model for VRRM. The scenario for model evaluation is stated in Section IV. In Section V, numeric results are presented and discussed. The paper is concluded in Section VI.
Background and related works
Based on [12, 13], infrastructure sharing solutions can be categorised into three main types: geographical, passive and active sharing. In geographical sharing or national roaming, a federation of operators can achieve full coverage in a short time, by dividing the service area into several regions, over which each of the operators provides coverage [14]. Passive sharing refers to the sharing agreement of fundamental infrastructures, such as tower masts, equipment houses and power supply, in order to reduce operational costs. Active sharing, however, is the sharing of transport infrastructures, radio spectrum and baseband processing resources. In [15], two types of sharing are introduced: multi-operator RAN and multi-core network. In the former, operators maintain a maximum level of independent control over their traffic quality and capacity, by splitting BSs and their controller nodes into logically independent units over a single physical infrastructure. In the latter, however, operators give up their independent control, by sharing the aforementioned entities in conjunction with the pooling of radio resources. Although the cost items in multi-core network are identical to multi-operator RAN, radio resources pooling leads to further savings in extremely low-traffic areas over equipment-related costs. Moreover, a network-wide radio resource management framework is proposed [16], in order to achieve isolation in addition to the optimal distribution of resources across the network.
Despite of RAN sharing benefits, surprisingly few sharing agreements have been made, especially in mature markets. The reasons offered by operators for not engaging into sharing deals are often the up-front transformation costs, the potential loss of control over their networks and the challenge of operational complexity [17, 18]. Sharing deals may be too expensive, and the initial cost of a network-sharing deal can be daunting; hence, operators without a comfortable margin of funds to make the necessary investment are likely to assume that they simply cannot afford to participate in such operation. 3rd Generation Partnership Project (3GPP) standards also limit the shared RAN to serve only four operators [19]. Moreover, many operators, particularly incumbent ones whose early entrance into markets has given them the best coverage and network qualities, assume that sharing their network with rivals would dilute their competitive advantage. Some of them may feel that they would not be able to control the direction for the development of their network in future rollout strategies and choices about hardware and vendors. Last, but not the least, having a shared RAN running properly is an elaborate and a complex task. Some operators believe that having a shared network operation puts many technological and operational challenges, which may lead to little financial benefits and great potential of chaos [19]. However, some studies, e.g. [20], show that the pros of sharing are larger than the cons, and that this approach can really be seen as a very promising solution for the future, namely, in a broader perspective, i.e., looking at RAN virtualisation instead of RAN sharing as the candidate solution.
NFV has captured the attention of many researchers, e.g. [7], and some studies have also considered RAN as well. By introducing an entity called "hypervisor" on the top of physical resources, the authors in [21] addressed the concept of a virtualised eNodeB. The hypervisor allocates the physical resources among various virtual instances and coordinates multiple virtual eNodeBs over the same physical one; the LTE spectrum is shared among them using the concept of RAN sharing, i.e., each virtual eNodeB receives a portion of the available frequency bands. The virtualisation of BSs in LTE is also addressed in [22], by considering the resource allocation to be static or dynamic spectrum sharing among virtual operators. The authors in [23] look into the advantages of a virtualised LTE system, via an analytical model for FTP (File Transfer Protocol) transmissions; the evaluation is done with considerations on realistic situations to present multiplexing gain in addition to the analytical analysis.
As the next step in RAN virtualisation, the concept of radio resource virtualisation and a management model is proposed in [8], and the extension of the model to support the shortage of radio resources is presented in [9]. In the same research path, the current paper, as well as [10], considers the virtualisation of radio resources over a full heterogeneous access network (i.e., a combination of cellular networks and WLANs (wireless local area networks)), over which pay-as-you-go CaaS is offered to VNOs.
Radio resource management in virtual RANs
Figure 1 presents the hierarchy for the management of virtual radio resources, consisting of a VRRM entity on the top of the usual radio resource management (RRM) entities of heterogeneous access networks [24], i.e., common RRM (CRRM) and local RRMs (for each of the physical networks), the latter managing different RATs (Radio Access Technologies)
Radio resource management in virtual RANs [8]
VNOs, placed at the top of the hierarchy, require wireless connectivity to be offered to their subscribers, not owning any radio access infrastructure [8, 9]. VNOs ask for RAN-as-a-service (RANaaS) from the RAN provider with physical infrastructure [25]. VNOs do not have to deal with the management of virtual RANs; they just define requirements, such as contracted capacity, in their SLAs with RAN providers. The role of VRRM is to translate VNOs' requirements and SLAs into a set of policies for the lower levels [8]. These policies contain data rates for different services, in addition to their priorities. Although VRRM optimises the usage of virtual radio resources, it does not deal with physical ones. However, VRRM has to consider practical issues, as the effect of the collision rate in WLANs on network data rate, in order to have an effective management of virtual radio resources. Reports and monitoring information provided by CRRM enables VRRM to improve policies. Load balancing among RANs, controlling the offloading procedure, is the duty of CRRM also known as Join RRM. Finally, the local RRMs are in charge of managing the physical resources based on the policies of VRRM and CRRM.
The VNOs' SLAs can generally be categorised into three main groups:
Guaranteed bitrate (GB), in which the RAN provider guarantees the VNO a minimum and a maximum level of data rates, regardless of the network status. Allocating the maximum guaranteed data rate to the VNO leads to its full satisfaction. The upper boundary in this type of SLA enables VNOs to have full control on their networks. For instance, a VNO offering VoIP (voice over IP) to its subscribers may foresee to offer this service to only 30 up 50% of its subscribers simultaneously, hence, the VNO can put this policy into practice by choosing a guaranteed SLA for its VoIP service. It is expected that subscribers always experience a good quality of service (QoS) in return of relatively more expensive services.
Best effort with minimum guaranteed (BG), where the VNO is guaranteed with a minimum level of service. The request for data rates higher than the guaranteed level is served in the best effort manner, hence, the minimum guaranteed data rate is the one received during busy hours. In this case, although VNOs do not invest as much as former ones, they can still guarantee the minimum QoS to their subscribers. From the subscribers' viewpoint, the acceptable service (not as good as the previous ones) is offered with a relatively lower cost.
Best effort (BE), in which the VNO is served in the pure best effort approach. In this case, operators and their subscribers may suffer from low QoS and resource starvation during busy hours, but the associated cost will be lower as well.
VRRM can be considered as a decision-making problem, under uncertainty for a dynamic environment; the decision is on the allocation of resources to the different services of VNOs, by considering the set of available radio resources. In what follows, the VRRM problem is discussed in detail.
The first step is forming the set of radio resources containing the number of available radio resource units (RRUs) per RAT (e.g., time-slots in GSM, codes in UMTS, resource-blocks in LTE and channels in Wi-Fi), as follows:
$$ {s}_{\mathrm{t}}^{\mathrm{RRU}}=\left\{{N}_{\mathrm{SRRU}}^{\mathrm{RA}{\mathrm{T}}_1},\ {N}_{\mathrm{SRRU}}^{\mathrm{RA}{\mathrm{T}}_2}, \dots,\ {N}_{\mathrm{SRRU}}^{\mathrm{RA}{\mathrm{T}}_{{\mathrm{N}}_{\mathrm{RA}\mathrm{T}}}}\right\} $$
\( {s}_{\mathrm{t}}^{\mathrm{RRU}} \): the set of radio resources at t,
\( {N}_{\mathrm{SRRU}}^{\mathrm{RA}{\mathrm{T}}_{\mathrm{i}}} \): number of spare RRUs in the ith RAT,
N RAT: number of RATs.
Mapping the set of radio resources onto the total network data rate is the next step. Since radio resources' performance is not deterministic, it is not possible to have a precise prediction of capacity, requiring the estimation of the total network data rate via a probability distribution, as a function of the available RRUs for further decisions,
$$ {s}_{\mathrm{t}}^{\mathrm{RRU}}\overset{\mathrm{mapping}}{\to }\ {R}_{\mathrm{b}\ \left[\mathrm{Mbps}\right]}^{\mathrm{CRRM}}(t) $$
\( {R}_{\mathrm{b}}^{\mathrm{CRRM}} \): total network data rate.
The third step is the allocation of the available resources to the services of VNOs. Sets of policies are designed in this step, in order to assign a portion of the total network capacity to each service of each VNO. Meeting the guaranteed service levels and increasing the resources' usage efficiency are the primary objectives, but other goals, such as fairness, may be considered. The resource allocation mapping of the total network capacity onto different services' data rates is given by:
$$ {R}_{\mathrm{b}\ \left[\mathrm{Mbps}\right]}^{\mathrm{CRRM}}(t)\overset{\mathrm{map}}{\to }\ \left\{{R}_{{\mathrm{b}}_{\mathrm{ji}}\ \left[\mathrm{Mbps}\right]}^{\mathrm{Srv}}(t)\Big| j=1, \dots,\ {N}_{\mathrm{VNO}},\ i=1, \dots,\ {N}_{\mathrm{srv}}\right\} $$
\( {R}_{{\mathrm{b}}_{\mathrm{ji}}}^{\mathrm{Srv}} \): serving (allocated) data rate for service j of VNO i;
N VNO: number of VNOs;
N srv: number of services.
Finally, monitoring the used resources and updating their status is the observation part of this decision-making problem. It enables the manager to evaluate the accuracy of its decisions and to modify former ones. Updating changes in the set of radio resources helps the manager to cope with dynamic changes in the environment and in VNOs requirements. In summary, it can be claimed that resource management solutions generally have two main components; estimation of available resources and optimisation of their allocation, which are addressed next.
Estimation of available resources
The estimation of available resources, and their allocation to the different services, are the two key steps in VRRM procedures. Obtaining a probabilistic relationship in the form of a probability density function (PDF), between the set of available RRUs and network capacity is the goal in a first step, then, by having an estimation of network capacity, VRRM allocates a portion of this capacity to each service of each VNO.
Depending on various parameters, such as RAT, modulation and coding, the allocation of an RRU can provide different data rates to mobile terminals. However, the data rate of an RRU is generally a function of the signal-to-interference-plus-noise-ratio (SINR), [8, 9]. Since SINR is a random variable, given the channel characteristics from path loss, fading and mobility, among other things, one needs to express the data rate also as a random variable:
$$ {R_{\mathrm{b}}}_{\mathrm{RA}{\mathrm{T}}_{\mathrm{i}}\left[\mathrm{Mbps}\right]}\left({\rho}_{\mathrm{i}\mathrm{n}}\right)\in \left[0,\ {R_{\mathrm{b}}}_{\mathrm{RA}{\mathrm{T}}_{\mathrm{i}}\left[\mathrm{Mbps}\right]}^{\max}\right] $$
\( {R_{\mathrm{b}}}_{\mathrm{RA}{\mathrm{T}}_{\mathrm{i}}} \): data rate of an RRU from the ith RAT,
ρ in:SINR,
\( {R_{\mathrm{b}}}_{\mathrm{RA}{\mathrm{T}}_{\mathrm{i}}}^{\max } \):maximum data rate of an RRU from the ith RAT.
Based on [8], the PDF of \( {R_{\mathrm{b}}}_{\mathrm{RA}{\mathrm{T}}_{\mathrm{i}}} \) can be given as:
$$ {p}_{{\mathrm{R}}_{\mathrm{b}}}\left({R_{\mathrm{b}}}_{\mathrm{R}\mathrm{A}{\mathrm{T}}_{\mathrm{i}}\left[\mathrm{Mbps}\right]}\right)=\frac{\frac{0.46}{\alpha_p}\left({\displaystyle {\sum}_{k=1}^5} k\ {a}_k\ {\left({R_{\mathrm{b}}}_{\mathrm{R}\mathrm{A}{\mathrm{T}}_{\mathrm{i}}}\right)}^{k-1}\right) \exp \left(-\frac{0.46}{\alpha_p}{\displaystyle {\sum}_{k=0}^5}{a}_k\ {\left({R_{\mathrm{b}}}_{\mathrm{R}\mathrm{A}{\mathrm{T}}_{\mathrm{i}}}\right)}^k\right)}{ \exp \left(-\frac{0.46}{\alpha_p}{a}_0\right)- \exp \left(-\frac{0.46}{\alpha_p}{\displaystyle {\sum}_{k=0}^5}{a}_k\ {\left({R_{\mathrm{b}}}_{\mathrm{R}\mathrm{A}{\mathrm{T}}_{\mathrm{i}}}^{\max}\right)}^k\right)} $$
α p ≥ 2: path loss exponent,
a k : coefficients in a polynomial approximation of SINR, as a function of data rate in each RAT (presented in [8] for cellular networks, and in [10] for Wi-Fi).
The total data rate for a single RAT pool is
$$ {R}_{{\mathrm{b}}_{\mathrm{tot}}}^{\mathrm{RA}{\mathrm{T}}_{\mathrm{i}}}={\displaystyle \sum_{n=1}^{N_{\mathrm{RRU}}^{\mathrm{RA}{\mathrm{T}}_{\mathrm{i}}}}}{R}_{{\mathrm{b}}_{\mathrm{n}}}^{\mathrm{RA}{\mathrm{T}}_{\mathrm{i}}} $$
\( {N}_{\mathrm{RRU}}^{\mathrm{RA}{\mathrm{T}}_{\mathrm{i}}} \): number of RRUs of the ith RAT,
\( {R}_{{\mathrm{b}}_{\mathrm{tot}}}^{\mathrm{RA}{\mathrm{T}}_{\mathrm{i}}} \): data rate from the ith RAT pool,
\( {R}_{{\mathrm{b}}_{\mathrm{n}}}^{\mathrm{RA}{\mathrm{T}}_{\mathrm{i}}} \): data rate from the nth RRU of the ith RAT.
The PDF of a RAT's data rate is equal to the convolution of all RRUs' PDFs when the channels, and consequently, the data rates' random variables (R bi) are independent [26]. In the deployment of heterogeneous access networks, the resource pools of the different RATs can be aggregated under the supervision of CRRM. The total data rate aggregated from all RATs is then the summation of the total data rate from each individual:
$$ {R}_{\mathrm{b}\ \left[\mathrm{Mbps}\right]}^{\mathrm{CRRM}}={\displaystyle \sum_{i=1}^{N_{\mathrm{RA}\mathrm{T}}}}{R}_{{\mathrm{b}}_{\mathrm{tot}}\left[\mathrm{Mbps}\right]}^{\mathrm{RA}{\mathrm{T}}_{\mathrm{i}}} $$
By having the number of available resources mapped onto probability functions, the VRRM has an estimation of the total network capacity. In the estimation procedure, the total network data rate is highly dependent on the channel quality at the mobile terminal, to which the radio resources are allocated. A higher network data rate can be achieved when the RRUs are allocated to mobile terminals with a high SINR. Thus, the allocation of the radio resources to mobile terminals with a low SINR leads to a lower network data rate. In a very low network capacity, VRRM may not be able to meet the minimum guaranteed requirements. The aforementioned estimation approach does not consider any assumption on the channel quality of the mobile terminals, this approach being referred to as the general (G) one. By adding assumptions about the mobile terminals' channel quality, three additional approaches for the estimation of network capacity are considered:
Optimistic approach (OP): all RRUs are assigned to users with very good channel quality (i.e., high SINR), therefore, it is assumed that the data rate of each RRU satisfies:
$$ 0.5\kern0.5em {R}_{\mathrm{b}\left[\mathrm{Mbps}\right]}^{\max}\le {R}_{\mathrm{b}\left[\mathrm{Mbps}\right]}\le {R}_{\mathrm{b}\left[\mathrm{Mbps}\right]}^{\max } $$
Realistic approach (RL): it is assumed that the RRUs of each RAT are divided into two equal groups, and that the data rate of the RRU from each group is as follows:
$$ 0\le {R}_{\mathrm{b}\left[\mathrm{Mbps}\right]}<0.5\ {R}_{\mathrm{b}\left[\mathrm{Mbps}\right]}^{\max}\kern0.5em \mathrm{Low}\ \mathrm{SINR}\ \mathrm{Group} $$
$$ 0.5\ {R}_{\mathrm{b}\left[\mathrm{Mbps}\right]}^{\max}\le {R}_{\mathrm{b}\left[\mathrm{Mbps}\right]}<{R}_{\mathrm{b}\left[\mathrm{Mbps}\right]}^{\max}\kern0.5em \mathrm{High}\ \mathrm{SINR}\ \mathrm{Group} $$
Pessimistic approach (PE): it is assumed that all the RRUs in the system are assigned to users with low SINR so that the boundaries are
$$ 0\le {R}_{\mathrm{b}\left[\mathrm{Mbps}\right]}\le 0.5\kern0.5em {R}_{\mathrm{b}\left[\mathrm{Mbps}\right]}^{\max } $$
Equation (5) can be further developed for these special case studies, where data rate is bounded between high and low values, the conditional PDF of a single RRU in this case being calculated as follows [26]:
$$ {p}_{\mathrm{Rb}}\left({R_{\mathrm{b}}}_{\mathrm{RA}{\mathrm{T}}_{\mathrm{i}}\left[\mathrm{Mbps}\right]}\Big|{R}_{\mathrm{b}\mathrm{Low}\left[\mathrm{Mbps}\right]}\le {R_{\mathrm{b}}}_{\mathrm{RA}{\mathrm{T}}_{\mathrm{i}}\left[\mathrm{Mbps}\right]}\le {R}_{\mathrm{b}\left[\mathrm{Mbps}\right]}^{\max}\right)=\frac{\frac{0.46}{\alpha_p}\left({\displaystyle {\sum}_{k=1}^5} k\ {a}_k{\left({R_{\mathrm{b}}}_{\mathrm{RA}{\mathrm{T}}_{\mathrm{i}}}\right)}^{k-1}\right)\ \exp \left(-\frac{0.46}{\alpha_p}{\displaystyle {\sum}_{k=0}^5}{a}_k\ {\left({R_{\mathrm{b}}}_{\mathrm{RA}{\mathrm{T}}_{\mathrm{i}}}\right)}^k\right)}{ \exp \left(-\frac{0.46}{\alpha}{\displaystyle {\sum}_{k=0}^5}{a}_k{\left({R}_{\mathrm{b}\mathrm{Low}}\right)}^k\right)- \exp \left(-\frac{0.46}{\alpha}{\displaystyle {\sum}_{k=0}^5}{a}_{k\ }{a}_k{\left({R}_{\mathrm{b}\mathrm{High}}\right)}^k\right)} $$
R bLow: low boundary for the RRU data rate;
R bHigh: high boundary for the RRU data rate.
One should note that there is a relationship in between the various parameters related to the data rate:
$$ 0\le {R}_{\mathrm{b}\mathrm{Low}\left[\mathrm{Mbps}\right]}\le {R}_{\mathrm{b}\mathrm{High}\left[\mathrm{Mbps}\right]}\le {R}_{\mathrm{b}\left[\mathrm{Mbps}\right]}^{\max } $$
Furthermore, one should note that this approach is not scenario-dependent, i.e., it can be applied to any network/system fitting into the general network architecture previously presented, which encompasses all current cellular networks and WLANs, as well as to other approaches for the estimation of network capacity (just by considering the corresponding conditions).
Allocation of resources in cellular networks
In the next step, the services of the VNOs have to be granted with a portion of the network capacity. The allocation of resources has to be based on services' priority and SLAs, e.g., the services from conversation (e.g., VoIP) and streaming (e.g., video) classes are delay-sensitive, but they have almost constant data rates, thus, the allocation of data rates higher than the ones contracted for these services does not increase the QoS, in contrast to interactive (e.g., FTP) and background (e.g., email) classes; as a consequence, operators offering the former set of services are not interested in allocating higher data rates.
The primary goal in the allocation procedure is to increase the total network data rate, while considering the priority of different services, subject to the constraints. On the ground of this fact, the objective function for VRRM is the total weighted network data rate, being expressed for cellular RATs as:
$$ {f}_{{\mathbf{R}}_{\mathbf{b}}}^{\mathrm{cell}}\left({\mathbf{R}}_{\mathbf{b}}^{\mathbf{cell}}\right) = {\displaystyle \sum_{i=1}^{N_{\mathrm{VNO}}}}{\displaystyle \sum_{j=1}^{N_{\mathrm{srv}}}}{W}_{\mathrm{ji}}^{\mathrm{Srv}}\ {R}_{{\mathrm{b}}_{\mathrm{ji}}\ \left[\mathrm{Mbps}\right]}^{\mathrm{cell}} $$
\( {\mathbf{R}}_{\mathbf{b}}^{\mathbf{cell}} \): vector of serving data rates from cellular networks,
N VNO: number of served VNOs by this VRRM,
N srv: number of services for each VNO,
\( {W}_{\mathrm{ji}}^{\mathrm{Srv}} \): weight of serving unit of data rate for service j of VNO i by VRRM, where \( {W}_{\mathrm{ji}}^{\mathrm{Srv}}\in \left[0,1\right] \).
The weights in (14) are used to prioritise the allocation of data rates to services, being a common practice to have the summation of all them equal to unit. The choice of these weights is based on the SLAs between VNOs and VRRM, and they can be modified depending on the agreed KPIs (key performance indicators) during runtime.
Allocation of resources in WLAN
It is desirable that the services with the higher serving weights receive data rates higher than the ones with the lower serving weights. The equivalent function for WLANs is
$$ {f}_{{\mathbf{R}}_{\mathbf{b}}}^{\mathrm{WLAN}}\left({\mathbf{R}}_{\mathbf{b}\left[\mathrm{Mbps}\right]}^{\mathbf{WLAN}}\right)={\displaystyle \sum_{i=1}^{N_{\mathrm{VNO}}}{\displaystyle \sum_{j=1}^{N_{s\mathrm{rv}}}\left({W}_{\mathrm{j}\mathrm{i}}^{\mathrm{Srv}}\ {R}_{{\mathrm{b}}_{\mathrm{j}\mathrm{i}}\left[\mathrm{Mbps}\right]}^{\mathrm{WLAN}} + {W}^{\mathrm{SRb}}\ \frac{\overline{R_{{\mathrm{b}}_{\mathrm{j}}}}}{\overline{R_{\mathrm{b}}^{\max }}}\kern0.5em {R}_{{\mathrm{b}}_{\mathrm{j}\mathrm{i}}\left[\mathrm{Mbps}\right]}^{\mathrm{WLAN}}\right)}} $$
\( {\mathbf{R}}_{\mathbf{b}}^{\mathbf{WLAN}} \): vector of serving data rates from APs,
W SRb: weight for session average data rate, where W SRb ? [0, 1],
\( \overline{R_b^{max}} \): maximum average data rate among all services,
\( \overline{R_{{\mathrm{b}}_{\mathrm{j}}}} \): average data rate for service j.
In (15), W SRb is introduced to give priority to services with a higher data rate per session. Assigning these services to a Wi-Fi network reduces collision rates, leading to a higher network data rate. Obviously, assigning zero to this weight completely eliminates the average data rate effect (i.e., the effect of collision on network data rate) and converts the objective function of WLANs in (15) into the cellular one in (14). The aforementioned weight is chosen by the VRRM approach, based on the SLAs and the Wi-Fi and LTE coverage maps, in addition to applied network planning and load-balancing policies. It can also be subject to modifications during runtime, based on measurements and reports.
In addition to increasing network data rate, a fair resource allocation is another objective in VRRM. On the one hand, the model is expected to allocate more resources to services with a higher serving weight, while on the other hand, services with a lower weight not being served at all or being served in very poor conditions are not acceptable. A fair allocation of resources is achieved when the deviation from the weighted average for all services is minimised:
$$ \underset{R_{{\mathrm{b}}_{\mathrm{ji}}}^{\mathrm{Srv}}}{ \min }\ \left\{{\displaystyle \sum_{i=1}^{N_{\mathrm{VNO}}}}{\displaystyle \sum_{j=1}^{N_{\mathrm{srv}}}}\left|\frac{R_{{\mathrm{b}}_{\mathrm{ji}}\ \left[\mathrm{Mbps}\right]}^{\mathrm{Srv}}}{W_{\mathrm{ji}}^{\mathrm{Srv}}}-\frac{1}{N_{\mathrm{VNO}}\kern0.5em {N}_{\mathrm{srv}}}{\displaystyle \sum_{i=1}^{N_{\mathrm{VNO}}}}{\displaystyle \sum_{j=1}^{N_{\mathrm{srv}}}}\frac{R_{{\mathrm{b}}_{\mathrm{ji}}\ \left[\mathrm{Mbps}\right]}^{\mathrm{Srv}}}{W_{\mathrm{ji}}^{\mathrm{Srv}}}\right|\right\} $$
This concept is addressed as a fairness function, being written as:
$$ {f}_{{\mathbf{R}}_{\mathbf{b}}}^{\mathrm{f}\mathrm{r}}\left({\mathbf{R}}_{\mathbf{b}}^{\mathbf{f}}\right) = {\displaystyle \sum_{i=1}^{N_{\mathrm{VNO}}}}{\displaystyle \sum_{j=1}^{N_{\mathrm{srv}}}}\left({R}_{{\mathrm{b}}_{\mathrm{ji}}\left[\mathrm{Mbps}\right]}^{\mathrm{f}}\right) $$
\( {R}_{{\mathrm{b}}_{\mathrm{ji}}}^{\mathrm{f}} \): the boundary for deviation data rate from the normalised average for service j of VNO i, defined as:
$$ \left\{\begin{array}{c}\hfill \frac{R_{{\mathrm{b}}_{\mathrm{ji}}\ \left[\mathrm{Mbps}\right]}^{\mathrm{Srv}}}{W_{\mathrm{ji}}^{\mathrm{Srv}}} - {\displaystyle \sum_{i=1}^{N_{\mathrm{VNO}}}{\displaystyle \sum_{j=1}^{N_{\mathrm{srv}}}\frac{R_{{\mathrm{b}}_{\mathrm{ji}}\ \left[\mathrm{Mbps}\right]}^{\mathrm{Srv}}}{N_{VNO}\kern0.5em {N}_{\mathrm{srv}}\kern0.5em {W}_{\mathrm{ji}}^{\mathrm{Srv}}}\le\ {R}_{{\mathrm{b}}_{\mathrm{ji}}\left[\mathrm{Mbps}\right]}^{\mathrm{f}}}}\hfill \\ {}\hfill -\frac{R_{{\mathrm{b}}_{\mathrm{ji}}\ \left[\mathrm{Mbps}\right]}^{\mathrm{Srv}}}{W_{\mathrm{ji}}^{\mathrm{Srv}}} + {\displaystyle \sum_{i=1}^{N_{\mathrm{VNO}}}{\displaystyle \sum_{j=1}^{N_{\mathrm{srv}}}\frac{R_{{\mathrm{b}}_{\mathrm{ji}}\ \left[\mathrm{Mbps}\right]}^{\mathrm{Srv}}}{N_{\mathrm{VNO}}\kern0.5em {N}_{\mathrm{srv}}\kern0.5em {W}_{\mathrm{ji}}^{\mathrm{Srv}}}\le {R}_{{\mathrm{b}}_{\mathrm{ji}}\left[\mathrm{Mbps}\right]}^{\mathrm{f}}}}\hfill \end{array}\right. $$
In order to better discuss the balance between these two objectives, i.e., the fairness and the total weighted network data rate, the boundaries of these two objectives have to be compared. The highest resource efficiency (i.e., the highest weighted data rate) is obtained when all resources are allocated to the service(s) with the highest serving weight, hence, the maximum of the first objective can be written as:
$$ \underset{R_{{\mathrm{b}}_{\mathrm{ji}}}^{\mathrm{Srv}}}{ \max}\left\{{\displaystyle \sum_{i=1}^{N_{\mathrm{VNO}}}{\displaystyle \sum_{j=1}^{N_{\mathrm{srv}}}{W}_{\mathrm{ji}}^{\mathrm{Srv}}\kern0.5em {R}_{{\mathrm{b}}_{\mathrm{ji}}\ \left[\mathrm{Mbps}\right]}^{\mathrm{Srv}}}}\right\}= \max \left\{{W}_{\mathrm{ji}}^{\mathrm{Srv}}\right\}{R}_{\mathrm{b}\ \left[\mathrm{Mbps}\right]}^{\mathrm{CRRM}} $$
This means that, as the network capacity increases, the summation of the weighted data rate in (14) increases as well. In the same situation, the fairness objective function also reaches its maximum:
$$ \underset{R_{{\mathrm{b}}_{\mathrm{ji}}}^{\mathrm{Srv}}}{ \max}\left\{{\displaystyle \sum_{i=1}^{N_{\mathrm{VNO}}}}{\displaystyle \sum_{j=1}^{N_{\mathrm{srv}}}}{R}_{{\mathrm{b}}_{\mathrm{ji}}\left[\mathrm{Mbps}\right]}^{\mathrm{f}}\right\}=\frac{R_{\mathrm{b}\ \left[\mathrm{Mbps}\right]}^{\mathrm{CRRM}}}{ \max \left\{{W}_{\mathrm{ji}}^{\mathrm{Srv}}\right\}} $$
Based on (19) and (20), the complete objective function for the management of virtual radio resources is defined as:
$$ {f}_{{\mathbf{R}}_{\mathbf{b}}}^{\mathrm{v}}\left({\mathbf{R}}_{\mathbf{b}}^{\mathbf{Srv}}\right) = {f}_{{\mathbf{R}}_{\mathbf{b}}}^{\mathrm{cell}}\left({\mathbf{R}}_{\mathbf{b}}^{\mathbf{cell}}\right)+{f}_{{\mathbf{R}}_{\mathbf{b}}}^{\mathrm{WLAN}}\left({\mathbf{R}}_{\mathbf{b}}^{\mathbf{WLAN}}\right) - {\alpha}_{\mathrm{f}}\left({W}_{\mathrm{f}}\right)\kern0.5em {f}_{{\mathbf{R}}_{\mathbf{b}}}^{\mathrm{f}}\left({\mathbf{R}}_{\mathbf{b}}^{\mathbf{f}}\right) $$
\( {\mathbf{R}}_{\mathbf{b}}^{\mathbf{f}} \): vector of intermediate fairness variables,
\( {\mathbf{R}}_{\mathbf{b}}^{\mathbf{Srv}} \): vector of serving data rates,
α f : fairness coefficient as a function of fairness weight:
$$ {\alpha}_f\left({w}_f\right)=\frac{W_f\kern0.5em {R}_{\mathrm{b}\ \left[\mathrm{Mbps}\right]}^{\mathrm{CRRM}}}{\left(1-{w}_f\right)\kern0.5em {R}_{\mathrm{b}\ \left[\mathrm{Mbps}\right]}^{\mathrm{CRRM}}\kern0.5em {N}^{\mathrm{SmaxRb}}+{W}_f\ \overline{R_{\mathrm{b}\left[\mathrm{Mbps}\right]}^{\max }}} $$
N SmaxRb: number of subscribers using the service with maximum data rate,
$$ {N}^{\mathrm{SmaxRb}}=\frac{R_{\mathrm{b}\ \left[\mathrm{Mbps}\right]}^{\mathrm{CRRM}}}{\overline{R_{\mathrm{b}\left[\mathrm{Mbps}\right]}^{\max }}} $$
In (18) and (21), the allocated data rate for a specific service is defined as:
$$ {R}_{{\mathrm{b}}_{\mathrm{ji}}\ \left[\mathrm{Mbps}\right]}^{\mathrm{Srv}} = {R}_{{\mathrm{b}}_{\mathrm{ji}}\ \left[\mathrm{Mbps}\right]}^{\mathrm{cell}} + {R}_{{\mathrm{b}}_{\mathrm{ji}}\ \left[\mathrm{Mbps}\right]}^{\mathrm{WLAN}} $$
In addition, there are more constraints for VRRM to allocate data rates to various services, which should not be violated. The very fundamental constraint is the total network capacity estimated in the last section. The summation of all assigned data rates to all services cannot be higher than the total estimated capacity of the network:
$$ {\displaystyle \sum_{i=1}^{N_{\mathrm{VNO}}}{\displaystyle \sum_{j=1}^{N_{\mathrm{srv}}}{R}_{{\mathrm{b}}_{\mathrm{ji}}\ \left[\mathrm{Mbps}\right]}^{\mathrm{Srv}}\le {R}_{\mathrm{b}\ \left[\mathrm{Mbps}\right]}^{\mathrm{CRRM}}}} $$
The data rate offered to GB and BG services imposes the next constraints. The data rate allocated to these services has to be higher than a minimum guaranteed level (for both GB and BG) and lower than the maximum guaranteed one (for GB only):
$$ {R}_{{\mathrm{b}}_{\mathrm{ji}}\left[\mathrm{Mbps}\right]}^{\mathrm{Min}}\le {R}_{{\mathrm{b}}_{\mathrm{ji}\ \left[\mathrm{Mbps}\right]}}^{\mathrm{Srv}}\le {R}_{{\mathrm{b}}_{\mathrm{ji}}\left[\mathrm{Mbps}\right]}^{\mathrm{Max}} $$
\( {R}_{{\mathrm{b}}_{\mathrm{ji}}}^{\mathrm{Min}} \): minimum data rate for service j of VNO i,
\( {R}_{{\mathrm{b}}_{\mathrm{ji}}}^{\mathrm{Max}} \): maximum data rate for service j of VNO i.
Based on this model, the objective function presented in (21) has to be optimised subject to constraints addressed in (18), (24), (25) and (26).
Resource allocation with violation
In the allocation process, there are situations where resources are not enough to meet all guaranteed capacity, and the allocation optimisation is no longer feasible. A simple approach in these cases is introduced in [9], which is to relax the constraints by the introduction of violation (also known as slack) variables. In case of VRRM, the relaxed constraint is given by:
$$ {R}_{{\mathrm{b}}_{\mathrm{ji}}\left[\mathrm{Mbps}\right]}^{\mathrm{Min}}\le {R}_{{\mathrm{b}}_{\mathrm{ji}\ \left[\mathrm{Mbps}\right]}}^{\mathrm{Srv}}+\varDelta {R}_{{\mathrm{b}}_{\mathrm{ji}\ \left[\mathrm{Mbps}\right]}}^{\mathrm{v}} $$
\( \varDelta {R}_{{\mathrm{b}}_{\mathrm{ji}}}^{\mathrm{v}} \): non-negative violation variable for the minimum guaranteed data rate of service j of VNO i.
By introducing the violation parameter, the former infeasible optimisation problem turns into a feasible one. The optimal solution maximises the objective function and minimises the weighted average constraints violations. The weighted average constraints violation is defined as follows:
$$ \varDelta {\overline{R_{\mathrm{b}}^{\mathrm{v}}}}_{\left[\mathrm{Mbps}\right]} = \frac{1}{N_{\mathrm{VNO}}\kern0.5em {N}_{\mathrm{srv}}}{\displaystyle \sum_{i=1}^{N_{\mathrm{VNO}}}{\displaystyle \sum_{j=1}^{N_{\mathrm{srv}}}{W}_{\mathrm{ji}}^{\mathrm{v}}\ \varDelta {R}_{{\mathrm{b}}_{\mathrm{ji}\ \left[\mathrm{Mbps}\right]}}^{\mathrm{v}}}} $$
\( \varDelta \overline{R_{\mathrm{b}}^{\mathrm{v}}} \): average constraint violation.
\( {W}_{\mathrm{ji}}^{\mathrm{v}} \): weight of violating minimum guaranteed data rate of service j of VNO i¸ where \( {W}_{\mathrm{ji}}^{\mathrm{v}}\in \left[0,1\right] \) .
The objective function presented in (21) has also to be changed as follows:
$$ {f}_{{\mathbf{R}}_{\mathbf{b}}}^{\mathrm{v}}\left({\boldsymbol{R}}_{\boldsymbol{b}}\right)={f}_{{\mathbf{R}}_{\mathbf{b}}}^{\mathrm{cell}}\left({\boldsymbol{R}}_{\mathbf{b}}^{\mathbf{cell}}\right)+{f}_{{\mathbf{R}}_{\mathbf{b}}}^{\mathrm{WLAN}}\left({\boldsymbol{R}}_{\mathbf{b}}^{\mathbf{WLAN}}\right) - {\alpha}_{\mathrm{f}}\left({W}_{\mathrm{f}}\right)\kern0.5em {f}_{{\mathbf{R}}_{\mathbf{b}}}^{\mathrm{f}}\left({\boldsymbol{R}}_{\mathbf{b}}^{\mathbf{f}}\right)-{f}_{{\mathrm{R}}_{\mathrm{b}}^{\mathrm{v}}}^{\mathrm{v}\mathrm{i}}\left(\varDelta \overline{R_{\mathrm{b}}^{\mathrm{v}}}\right) $$
where \( {f}_{{\mathrm{R}}_{\mathrm{b}}^{\mathrm{v}}}^{\mathrm{v}\mathrm{i}} \) is the constraint violation function, given by:
$$ {f}_{{\mathrm{R}}_{\mathrm{b}}^{\mathrm{v}}}^{\mathrm{v}\mathrm{i}}\left(\varDelta \overline{R_{\mathrm{b}}^{\mathrm{v}}}\right) = \frac{R_{\mathrm{b}\ \left[\mathrm{Mbps}\right]}^{\mathrm{CRRM}}}{\overline{R_{\mathrm{b}\left[\mathrm{Mbps}\right]}^{\min }}}\kern0.5em \varDelta {\overline{R_{\mathrm{b}}^{\mathrm{v}}}}_{\left[\mathrm{Mbps}\right]} $$
However, the definition of fairness in a congestion situation is not the same, i.e., in this case, fairness is to make sure that the weighted violation of all services is the same. Fairness constraints are changed as follows:
$$ \left\{\begin{array}{c}\hfill {W}_{\mathrm{ji}}^{\mathrm{v}}\kern0.5em \varDelta {R}_{{\mathrm{b}}_{\mathrm{ji}\ }\left[\mathrm{Mbps}\right]}^{\mathrm{v}}-\frac{1}{N_{\mathrm{VNO}}\kern0.5em {N}_{\mathrm{srv}}}{\displaystyle \sum_{i=1}^{N_{\mathrm{VNO}}}{\displaystyle \sum_{j=1}^{N_{\mathrm{srv}}}{W}_{\mathrm{ji}}^{\mathrm{v}}\kern0.5em \varDelta {R}_{{\mathrm{b}}_{\mathrm{ji}\ }\left[\mathrm{Mbps}\right]}^{\mathrm{v}}\le\ {R}_{{\mathrm{b}}_{\mathrm{ji}}\left[\mathrm{Mbps}\right]}^{\mathrm{f}}}}\hfill \\ {}\hfill -{W}_{\mathrm{ji}}^{\mathrm{v}}\kern0.5em \varDelta {R}_{{\mathrm{b}}_{\mathrm{ji}\ }\left[\mathrm{Mbps}\right]}^{\mathrm{v}}+\frac{1}{N_{\mathrm{VNO}}\kern0.5em {N}_{\mathrm{srv}}}{\displaystyle \sum_{i=1}^{N_{\mathrm{VNO}}}{\displaystyle \sum_{j=1}^{N_{\mathrm{srv}}}\varDelta {R}_{{\mathrm{b}}_{\mathrm{ji}\ }\left[\mathrm{Mbps}\right]}^{\mathrm{v}}\le {R}_{{\mathrm{b}}_{\mathrm{ji}}\left[\mathrm{Mbps}\right]}^{\mathrm{f}}}}\hfill \end{array}\right. $$
The management of virtual radio resources is a complex optimisation problem, since network status and constraints vary in time. In [8, 9], it is proposed to divide the time axis into decision windows and to maximise the objective function in each of these intervals, independently. However, the output of partial VRRM may only be a local optimum and not the global one, since the effect of each decision on the network state and other dependencies are neglected. Nevertheless, partial VRRM is a simple solution, which can be used as the starting step and reference point.
The optimisation problem is solved by using MATLAB Linear Programming solver (i.e., the linprog function) [27], using the interior-point approach [28], which is a variant of Mehrotra's predictor-corrector algorithm [29], a primal-dual interior-point method. The termination tolerance on the function is chosen to be 10−8.
The realistic scenario for evaluating the proposed model offers cellular networks coverage by means of a set of radio remote heads (RRHs) [14], supporting OFDMA (LTE-Advanced), CDMA (UMTS/HSPA+) and FDMA/TDMA (GSM/EDGE). VRRM is in charge of a service area, as described in Table 1:
Table 1 Different RAT cell radius
The OFDMA cells, with a 400 m radius, are the smallest ones; based on the 100 MHz LTE-Advanced feature, each cell has 500 RRUs to be assigned to traffic bearers.
The configurations of CDMA cells are chosen according to UMTS/HSPA+, at 2.1 GHz, each cell with a 1.2 km radius and 3 carriers (each carrier has 16 codes); only 45 codes, out of all 48 in each cell, are assigned to users' traffic.
The FDMA/TDMA cells are the biggest ones, with a 1.6 km radius, based on GSM900, each cell having 10 radio channels (each one has 8 timeslots), being assumed that 75 timeslots out of the total 80 available ones in each cell are used for users' traffic.
In addition to cellular networks, in this scenario Wi-Fi (OFDM) coverage is provided by means of IEEE802.11ac APs, configured to work with an 80 MHz channel bandwidth. It is assumed that each AP covers a cell with an 80 m radius, being facilitated with beamforming and MU-MIMO to support up to 8 spatial streams. Five radio channels are taken for the 80 MHz APs, following European Union regulations, [30]. In contrast to former RATs, APs use the same set of links for up- and download streams. In order to achieve coherency among various RATs, the total data rate of APs is equally divided between down- and uplinks, therefore, in Table 1, the number of RRUs in each Wi-Fi cell is indicated as half of the total number of available channels. This table also presents the maximum data rate for each RAT in downlink. It is also assumed that the APs are only deployed on the OFDMA BS, hence, not providing full coverage; however, the entire Wi-Fi capacity can be used for traffic offloading.
The path-loss exponent, αp, corresponding to signal propagation in the various environments, is considered to be 3.8 for regular urban environments, a value between 2 for free space and 5 for high attenuation dense urban ones [31].
Three VNOs are assumed to operate in this area, each one with a different SLA, i.e., GB, BG and BE. All of them offer the same set of services, as in Table 2 The serving weights in (14) and (15) are based on the general service classes: 0.4 for conversational (Con), 0.3 for streaming (Str), 0.2 for interactive (Int) and 0.1 for background (Bkg). Besides the usual human interaction services, one is also considering several machine-to-machine (M2M) applications, as this is one of the areas foreseen for large development of VNOs. In order not to compromise the objective function for achieving fairness, the fairness weight, W f, in (22) is considered to be unit, leading to a maximum fairness, while W SRb in (15) is heuristically chosen to be 0.02.
Table 2 Network traffic mixture
Each VNO is assumed to have 500 subscribers, where each one requires the average data rate of 6.375 Mbps [32]. Hence, the contracted data rate, \( {R}_{\mathrm{b}}^{\mathrm{Con}} \), for all operators is 3.11 Gbps, and each service receives a portion based on a volume percentage, in Table 2. In the second step, the number of subscribers for each VNO is swept from 300 (low load) up to 1400 (high load), in order to observe how VNOs capacity and their services are affected by this increase of load.
On the ground of each service data rate, the SLAs of these VNOs are defined as follows:
VNO GB: the data rates allocated to services are guaranteed to be in the range of 50 to 100% of the corresponding service data rate.
VNO BG: it has the best effort, with a minimum 25% of the service data rate guaranteed by the SLA.
VNO BE: it has all services served in the best effort approach, without any guarantee.
Network capacity
For the network capacity estimation, in addition to the general approach, the other three, i.e., pessimistic, realistic and optimistic, are also considered. The minimum and maximum data rates from each RAT, considering different approaches, are presented in Table 3. Equation (7) is used to obtain the PDF for the general approach and (12) for the other ones.
Table 3 Minimum and maximum data rate of each RAT in different approaches
Using (5) and (12), the PDF and the cumulative distribution functions (CDFs) of the considered network capacity are obtained. Figure 2 compares the differences in between the three approaches in conjunction with the general one. As expected, the lowest network capacity estimation is achieved by applying the pessimistic (PE) approach, since the assumption is the allocation of RRUs to mobile terminals with the lowest SINR. In this PE approach, the median capacity of the network for regular urban environments is 1.32 Gbps, while the general approach (G) leads to 1.76 Gbps, i.e., the former is 75.0% of this one; however, the realistic (RL) and the optimistic (OP) approaches provide a quite higher median network capacity of 3.60 and 5.93 Gbps, respectively, i.e., 2.0 and 3.4 times higher. Moreover, Fig. 2 also shows that a higher path-loss exponent yields a higher network capacity: the higher the path loss, the higher the attenuation, implying that interference is more attenuated than the signal, hence, increasing the carrier-interference-ratio, ultimately, yielding a higher capacity.
CDF of cellular networks data rate for different approaches
When adding the capacity offered by traffic offloading to Wi-Fi APs in regular urban environments (i.e., with a path-loss exponent of 3.8 for all systems, which is an approximation for a real scenario), one gets the CDF of the network capacity as plotted in Fig. 3. The comparison of Figs. 2 and 3 shows that the median values increase to 3.7 Gbps (1.8 times) in PE, 7.2 Gbps (3.0 times) in G, 19.5 Gbps (4.4 times) in RL and 35.3 Gbps (2.3 times) in OP, which is quite an increase in capacity. The total network capacity, according to Fig. 3, is 9.5 times higher in the OP approach than in the PE one, which without any doubt affects the allocation of resources to different services of the VNOs. The interdecile intervals range in between 2.14 Gbps in PE and 4.3 Gbps in G, with 2.93 Gbps and 3.52 Gbps for RL and OP, respectively, showing that the type of approach does not have a monotonic impact on the probability of obtaining a given capacity.
CDF of the total network data rate for different approaches
Resource allocation with traffic offloading
Figure 4 presents the data rates allocated to each of the services from cellular networks and WLANs in the G case, when there are 500 subscribers per VNO. As expected, conversational services (VoIP, Video Call and M2M/MMI), which are the ones with the highest serving weights, receive the highest data rates, streaming (music, M2M/MMS and video streaming) being placed second; the services of the background class (email and M2M/MMM) are the ones that are allocated the smallest portion of the available capacity.
Data rate allocated to different services from WLAN and cellular networks
The effect of the weight for session average data rate in WLAN, W SRb, can be observed in the data rates balance between WLANs and cellular networks, e.g., the data rate allocated to video streaming (ViS) from WLANs is 6.5 times higher than the one from cellular networks, while, in contrast, email, a service with a low average data rate, is allocated a higher data rate in cellular networks than in WLANs. However, VoIP is not following the same rule, since it has a relatively high serving weight, which overcomes the effect of the average data rate in (15), hence, being allocated a comparatively high data rate from both type of networks; the same phenomenon can be observed among M2M services, i.e., the ones with high serving weights, e.g., M2M/MMI, received relatively high capacity from WLANs comparing to the other ones.
Figure 5 demonstrates the allocation of virtual radio resources to the services of each VNO. Maximum guaranteed data rates are provided to almost all the services of VNO GB, e.g., VoIP and music with the relative assigned data rate of 31.87 and 95.62 Mbps (which are the maximum requested). The upper boundary in the allocation of virtual resources to the services is the primary difference between the services of VNO GB and BG, in other words, while the data rates allocated to services of the guaranteed VNO are bounded by maximum guaranteed values, the services of VNO BG have no limitation. In contrast, the capacity offered to VNO BG in a resource shortage situation can be smaller than the VNO GB one.
Data rate allocated to different services of the VNOs
Resource allocation for different number of subscribers
In this section, one analyses the performance of the proposed model under different network traffic loads. The number of subscribers is swept between 300 and 1 400 per VNO (i.e., low and high loads). Figure 6 illustrates the distribution of the available virtual resources among VNOs, in addition to the total network capacity (\( {R}_{\mathrm{b}}^{\mathrm{CRRM}} \)), the total minimum guaranteed and the contracted data rates (\( {R}_{\mathrm{b}}^{\mathrm{Con}} \)). The contracted data rate for each VNO increases from 1.91 Gbps (low load) to 8.92 Gbps (high load).
Variation of the data rate allocated to each VNO
The acceptable regions for VNOs GB and BG in the plot are shown by solid blue and light green colours. The total minimum guaranteed data rate, i.e., the summation of minimum guaranteed data rates of VNOs GB and BG, in the low load is 20.6% of the network capacity (i.e., 1.4 Gbps).
Since network capacity is considerably higher than the minimum guaranteed data rates, best effort services are also served well; the allocation of 2.39 Gbps to VNO BE is the evidence to this claim. It is worth noting that the share of VNO BE is 35.1% of the whole network capacity, which is 1.6 times higher than VNO GB. The reason behind this observation is the maximum guaranteed data rate of the guaranteed services. Although the assigned portion of available resources to VNO GB is not as big as the other two VNOs, it is served up to its maximum satisfaction.
In contrast, VNO BG has a minimum guaranteed data rate, but the maximum received is 43.3% of the network capacity (2.94 Gbps). The guarantee data rates grow up to 6.53 Gbps (i.e., 96.1% of the whole available capacity) as the load increases. Obviously, the share of the best effort services in this situation considerably decreases. The allocated capacity to VNO BE reduces to only 65.6 Mbps, which is 0.9% of the total available capacity and 97.3% of its initial value. As shown in Fig. 6, the total network capacity is only enough for serving the contracted data rate of only one of the VNOs. In addition, the increase of the subscribers to 1400 makes the total minimum guaranteed data rate of the three VNOs equal to the total network capacity, which means that the data rates allocated to the services of VNO BE reach zero.
Furthermore, Fig. 7 illustrates the effect of demand variation on the allocation of data rates to the service classes of VNO GB: this VNO is a guaranteed one, therefore, each service class has a minimum and a maximum guaranteed data rate, presented in the figure with the solid colour. By increasing the number of subscribers, demand increases 4.7 times.
Data rate allocated to service classes of VNO GB
It can be seen that the streaming services are the ones with the highest volume, having the highest data rate; the minimum guaranteed data rate varies between 0.58 and 2.71 Gbps. The data rate allocated to this class in low load (when there are only 300 subscribers) is 67% of the maximum guaranteed one, but it reduces to the minimum one (i.e., 50% of the contracted data rate) for the maximum load case. The other service classes (i.e., interactive, conversational and background) are served according to capacity needs. It can be seen that, in the low load situation, the maximum guaranteed data rates are assigned, but as demand increases, data rates move towards the lower boundary. The interactive service class is a very good example for this behaviour: while it receives the maximum guaranteed data rate of 0.54 Gbps in low load, the allocated capacity for the high one is reduced from the maximum to 1.45 Gbps, the minimum guaranteed data rate. Considering the slope of allocated data rates in various services, the effect of serving weights and the service volume can be seen. Since the interactive class has a lower serving weight compared to the conversational one, it receives almost the minimum acceptable data rate with 1100 subscribers; in the same situation, conversational services are still provided by the highest acceptable data rate.
The effect of channel quality effect on VRRM
The effect of channel quality on the management of virtual radio resources by considering the three approaches (i.e., OP, RL and PE) is studied as well. Figure 8 presents the data rates allocated to VNO GB in conjunction with minimum and maximum guaranteed ones. As long as the data rates are in the acceptable region (shown by the solid colour), there is no violation of the SLAs and guaranteed data rates.
Data rate allocated to VNO GB in different approaches
Figure 8 shows that the maximum guaranteed data rate reaches 3.74 Gbps when there are 600 subscribers, being possible to allocate all of it in the OP approach, and that for 1400 subscribers it reaches 8.7 Gbps, but with only 83.3% of it being allocated to the OP approach. However, VNO GB faces the violation on the minimum guaranteed data rate in the PE approach, as the number of subscribers passes 1100: while at least 3.42 Gbps is required, only 97.3% of it is allocated to the VNO; this means that the network capacity in this approach is lower than the total minimum guaranteed data rates, and the resource management entity has to violate some of the minimum guaranteed data rates. The VNO requires at least 4.36 Gbps for the heavy load, the allocated data rate being 6.28 Gbps in RL and 4.42 Gbps in G, which are still enough to fulfil the SLA, but it goes down to 3.3 Gbps, i.e., 76.8% of the minimum required data rate, in PE, in clear violation of the SLA. This clearly shows the effect of SINR on resources usage efficiency and QoS offered to VNOs.
The data rates allocate to VNOs BG and BE are plotted in Fig. 9. Just as VNO GB, it can be seen that a high data rate is allocated to these VNOs in the OP and RL approaches (i.e., 18.46 and 9.68 Gbps). In these cases, the high SINR leads to the high network capacity, and the model is not only able to serve the minimum guaranteed data rates, but it can also serve acceptable data rates to the BE and BG VNOs. Consequently, VNOs BG and BE suffer more from resources shortage in the high load situations. The allocation of resources to VNO BE even stops for more than 1100 subscribers when the PE approach is considered.
The data rate allocated to VNOs (a) BG and (b) BE in different approaches
Regarding the distribution of data rates allocated to the service classes of VNO GB, Fig. 9 illustrates the variation of the capacity assigned to each one in different approaches. It can be seen that the conversational class (i.e., the class with the highest service weight) receives the maximum guaranteed data rate for the OP and RL approaches. The data rate allocated for the G and PE approaches is more than 50% of the contracted data rate. In the PE case, although for high-density situations the data rate decreases to a minimum guaranteed data rate, the services of this class never experience violation of the guaranteed data rate. Likewise, the streaming class is always served with a data rate higher than the minimum guaranteed. The maximum guaranteed data rate in heavy load reaches 5.34 Gbps and 72.9% in OP, 63.2% in RL, 50.4% in G and 50.0% in PE is allocated (Fig. 10).
The data rate allocated to service classes of VNO GB. a Conversational service class. b Streaming service class. c Interactive service class. d Background service class
For interactive and background classes, it is shown that they face violation of minimum guaranteed data rate in the PE approach. The violation situation in the background class is, to such an extent, that no capacity is allocated to its services when there are more than 1100 subscribers per VNO. The data rate allocated to the interactive class reaches 15.1% of the contracted data rate in heavy load, while the minimum guaranteed is 50%.
For the sake of comparison, the data rate allocated to the interactive and background classes of VNOs BG and BE is shown respectively in Figs. 11 and 12. It can be seen that for VNO BG the situation is very much similar to VNO GB, the main difference being the high boundary of allocated data rate. VNO BG does not have a maximum guaranteed data rate or high boundary for allocation of data rates. Consequently, the services of this VNO are served by comparatively higher data rates comparing to VNO GB when a high network capacity is available, e.g., in the OP situation. As an example, consider the conversational class on both VNOs: for the OP approach with 400 subscribers, VNO GB is granted with 0.1 Gbps while VNO BG receives 1.3 Gbps; on the other hand, in the case of resource shortage, VNO BG receives data rates lower than VNO GB, e.g., the share of interactive class of VNO BG when there are 1200 subscribers in the PE approach is only 0.387 Gbps while VNO GB is allocated 0.908 Gbps.
The data rate allocated to the interactive class. a VNO BG. b VNO BE
The data rate allocated to the background class. a VNO BG. b VNO BE
In conclusion, the effect of channel quality on the total available resource, and consequently on the performance of VRRM, is studied in this section. Through numeric results, one shows that the proposed model for managing virtual radio resources can serve different service classes of VNOs with different requirements, while offering an acceptable level of isolation. As evidence to this claim, one can consider services of the conversational class, i.e., services with the highest priority and serving weight, which are always allocated with a satisfactory amount of resources as the demands and the network capacity changes. Likewise, the minimum guaranteed data rates are offered to the relevant VNOs. Moreover, the prioritising of service classes offered by VRRM enables to serve the more important services, even when there are not enough resources.
A model for the management of virtual radio resources in a full heterogeneous network (i.e., a network with both cellular and WLANs) is proposed, which has two key components: the estimation of available resources and the allocation of resources. In the first step, the model maps the number of the available RRUs from different RATs onto the total network capacity by obtaining a probabilistic relationship. The model is able to consider multiple channel quality assumptions for the terminals through different estimation approaches. The allocation of resources to maximise the weighted data rate of the network based on the estimated network capacity is the next step. The serving weights in the objective function make possible to prioritise services. The resource allocation in a shortage of resources (i.e., when there are not enough resources to meet the minimum guaranteed data rates) tries to minimise the violation of the guaranteed data rates. In addition, the model also considers fairness among services.
Moreover, the performance of the proposed model is evaluated for a set of practical scenarios, and numeric results are achieved. It is shown that by adding an AP to each OFDMA cell, the network capacity increases up to 2.8 times. As a result of traffic offloading, the VRRM model is able to properly serve not only guaranteed services, but best effort ones are allocated with a relatively high data rate. Services with a higher serving weight, such as services of the conversational class, are provided with a higher data rate.
Furthermore, the results show that the network capacity increases from 0.9 Gbps in the pessimistic approach to 5.5 Gbps in the optimistic one. The effect of these capacity changes on the allocated data rates for different VNOs, and their service classes are presented through a series of plots. It is shown that when there is enough capacity, not only the guaranteed VNO is satisfied, but also best effort VNOs are well served. However, as the network capacity decreases due the channel quality (G and PE approaches), best effort VNOs are affected more than the guaranteed one. The same situation is shown at the service class level too. Conversational and streaming classes are the ones with the highest serving priority (i.e., serving weights), being generally allocated with data rates higher than the other two classes. When there is a shortage of resources, i.e., in the G and PE approaches, violations start relatively by background and interactive classes.
The model performance under different loads is also evaluated. The results confirm that the model is able to realise an acceptable level of isolation. When there is a shortage of radio resources, the model for resource management starts violating the guaranteed levels of services with lower serving weights, which are the background and interactive services. As evidence to this claim, the VNO GB, and particularly its conversational class, can be considered where the requested service quality regardless of network situation is offered. In addition, the flexibility of the applied model makes the equilibrium among different services or different VNOs possible.
In conclusion, it is shown that our model achieves the desired goals: (a) on-demand wireless capacity offering by serving three VNOs with different SLAs; (b) isolation by considering changes in the networks into almost one tenth of its original value (from OP to PE), in addition to changing the demand from 300 UE per VNO to 1 400; (c) element abstraction and multi-RAT support by providing wireless connectivity using both cellular networks and WLANs, while the VNOs do not have to deal with the details. In the future, the aforementioned concept of virtual radio resources and the proposed model will be implemented in realistic test-beds. In addition, the modelling of service demands and user behaviour in full heterogeneous networks will be taken in the next extension of the model.
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The research leading to these results was partially funded by the European Union's Seventh Framework Programme Mobile Cloud Networking project (FP7-ICT-318109).
The authors have contributed jointly in all the parts for preparing this manuscript. All authors read and approved the final manuscript.
Nomor Research GmbH, Munich, Germany
Sina Khatibi & Luis M. Correia
University of Lisbon, Lisbon, Portugal
Sina Khatibi
Luis M. Correia
Correspondence to Sina Khatibi.
Khatibi, S., Correia, L.M. Modelling virtual radio resource management in full heterogeneous networks. J Wireless Com Network 2017, 73 (2017). https://doi.org/10.1186/s13638-017-0858-7
Virtualisation of radio resources
Virtual radio resource management
Radio access networks
Network function virtualisation | CommonCrawl |
Ulisse Stefanelli
Ulisse Stefanelli is an Italian mathematician. He is currently professor at the Faculty of Mathematics of the University of Vienna. His research focuses on calculus of variations, partial differential equations, and materials science.[1]
Ulisse Stefanelli
Alma materUniversity of Pavia, (Ph.D., 2003)
Known forPlasticity, Rate-independent systems, Gradient flow, Doubly nonlinear equation, Crystallization
Awards
• Vinti Prize (2015)
• Richard von Mises Prize (2010)
• Friedrich Wilhelm Bessel Research Award (2009)
• ERC Starting Grant (2007)
Scientific career
FieldsCalculus of variations, Partial differential equations, Materials science
InstitutionsIstituto di Matematica Applicata e Tecnologie Informatiche E. Magenes, University of Vienna
Websitehttps://www.mat.univie.ac.at/~stefanelli/
Biography
Stefanelli obtained his PhD under the guidance of Pierluigi Colli in 2003 at the University of Pavia. He holds a Researcher position at the Istituto di Matematica Applicata e Tecnologie Informatiche E. Magenes of the National Research Council (Italy) in Pavia since 2001. In 2013 he has been appointed to the chair of Applied Mathematics and Modeling at the Faculty of Mathematics of the University of Vienna. He has also conducted research at the University of Texas at Austin, the ETH and the University of Zurich, the Weierstrass Institute in Berlin, and the Laboratoire de Mécanique et Génie Civil in Montpellier.
Since 2017 he is the speaker of the Spezialforschungsbereich F65 Taming Complexity in Partial Differential Systems funded by the Austrian Science Fund.[2]
Awards
• Vinti Prize of the Unione Matematica Italiana (2015)[3]
• Richard von Mises Prize of the GAMM (2010)[4]
• Friedrich Wilhelm Bessel-Forschungspreis of the Alexander von Humboldt Foundation (2009)
• ERC Starting Grant (2007)[5]
Selected publications
• Mainini, Edoardo; Stefanelli, Ulisse (2014-06-01). "Crystallization in Carbon Nanostructures". Communications in Mathematical Physics. 328 (2): 545–571. Bibcode:2014CMaPh.328..545M. doi:10.1007/s00220-014-1981-5. ISSN 1432-0916. S2CID 253744289.
• Mielke, Alexander; Stefanelli, Ulisse (2011-01-01). "Weighted energy-dissipation functionals for gradient flows". ESAIM: Control, Optimisation and Calculus of Variations. 17 (1): 52–85. doi:10.1051/cocv/2009043. ISSN 1292-8119.
• Auricchio, F.; Reali, A.; Stefanelli, U. (2009-04-15). "A macroscopic 1D model for shape memory alloys including asymmetric behaviors and transformation-dependent elastic properties". Computer Methods in Applied Mechanics and Engineering. 198 (17): 1631–1637. Bibcode:2009CMAME.198.1631A. doi:10.1016/j.cma.2009.01.019. ISSN 0045-7825.
• Caffarelli, Luis A.; Stefanelli, Ulisse (2008-07-03). "A Counterexample to C 2,1 Regularity for Parabolic Fully Nonlinear Equations". Communications in Partial Differential Equations. 33 (7): 1216–1234. arXiv:math/0701769. doi:10.1080/03605300701518240. ISSN 0360-5302. S2CID 15798910.
• Mielke, Alexander; Roubíček, Tomáš; Stefanelli, Ulisse (2008-03-01). "Γ-limits and relaxations for rate-independent evolutionary problems". Calculus of Variations and Partial Differential Equations. 31 (3): 387–416. doi:10.1007/s00526-007-0119-4. ISSN 1432-0835. S2CID 55568258.
• Stefanelli, Ulisse (2008-01-01). "The Brezis–Ekeland Principle for Doubly Nonlinear Equations". SIAM Journal on Control and Optimization. 47 (3): 1615–1642. doi:10.1137/070684574. ISSN 0363-0129.
• Auricchio, Ferdinando; Mielke, Alexander; Stefanelli, Ulisse (2008-01-01). "A rate-independent model for the isothermal quasi-static evolution of shape-memory materials". Mathematical Models and Methods in Applied Sciences. 18 (1): 125–164. arXiv:0708.4378. doi:10.1142/S0218202508002632. ISSN 0218-2025. S2CID 17499836.
• Auricchio, F.; Reali, A.; Stefanelli, U. (2007-02-01). "A three-dimensional model describing stress-induced solid phase transformation with permanent inelasticity". International Journal of Plasticity. 23 (2): 207–226. doi:10.1016/j.ijplas.2006.02.012. ISSN 0749-6419.
References
1. "Ulisse Stefanelli".
2. "FWF announcement on SFB F65".
3. "Vinti prize page".
4. "Richard Von Mises Prize awardees".
5. "ERC StG awardees 2007" (PDF).
Authority control
International
• VIAF
National
• Germany
Academics
• DBLP
• Google Scholar
• MathSciNet
• Mathematics Genealogy Project
• ORCID
• Scopus
• zbMATH
| Wikipedia |
Biotechnology for Biofuels and Bioproducts
Techno-economic analysis of an integrated biorefinery to convert poplar into jet fuel, xylitol, and formic acid
Gabriel V. S. Seufitelli1,
Hisham El-Husseini1,
Danielle U. Pascoli1,
Renata Bura1 &
Richard Gustafson1
Biotechnology for Biofuels and Bioproducts volume 15, Article number: 143 (2022) Cite this article
The overall goal of the present study is to investigate the economics of an integrated biorefinery converting hybrid poplar into jet fuel, xylitol, and formic acid. The process employs a combination of integrated biological, thermochemical, and electrochemical conversion pathways to convert the carbohydrates in poplar into jet fuel, xylitol, and formic acid production. The C5-sugars are converted into xylitol via hydrogenation. The C6-sugars are converted into jet fuel via fermentation into ethanol, followed by dehydration, oligomerization, and hydrogenation into jet fuel. CO2 produced during fermentation is converted into formic acid via electrolysis, thus, avoiding emissions and improving the process's overall carbon conversion.
Three different biorefinery scales are considered: small, intermediate, and large, assuming feedstock supplies of 150, 250, and 760 dry ktonne of poplar/year, respectively. For the intermediate-scale biorefinery, a minimum jet fuel selling price of $3.13/gallon was obtained at a discount rate of 15%. In a favorable scenario where the xylitol price is 25% higher than its current market value, a jet fuel selling price of $0.64/gallon was obtained. Co-locating the biorefinery with a power plant reduces the jet fuel selling price from $3.13 to $1.03 per gallon.
A unique integrated biorefinery to produce jet fuel was successfully modeled. Analysis of the biorefinery scales shows that the minimum jet fuel selling price for profitability decreases with increasing biorefinery scale, and for all scales, the biorefinery presents favorable economics, leading to a minimum jet fuel selling price lower than the current price for sustainable aviation fuel (SAF). The amount of xylitol and formic produced in a large-scale facility corresponds to 43% and 25%, respectively, of the global market volume of these products. These volumes will saturate the markets, making them infeasible scenarios. In contrast, the small and intermediate-scale biorefineries have product volumes that would not saturate current markets, does not present a feedstock availability problem, and produce jet fuel at a favorable price given the current SAF policy support. It is shown that the price of co-products greatly influences the minimum selling price of jet fuel, and co-location can further reduce the price of jet fuel.
According to the U.S. Energy Information Administration (EIA), in 2021, 79% of the energy consumption in the U.S. was from fossil fuel resources (crude oil, natural gas, and coal) [1]. In the same year, the combustion of fossil fuels in the U.S. accounted for 92% of the net human-induced CO2 emissions (4857 out of 5256 million metric tons of CO2 equivalent) [2]. The transportation sector contributes to approximately 37% of these emissions, with 97% from petroleum products (based on 2021 estimates [3]). One promising approach to reducing emissions in the transportation sector is to create process pathways that produce fuels from renewable biomass feedstocks that sequester carbon during growth. Great efforts have been made to decarbonize the road transportation sector, introducing low-carbon fuels, such as ethanol and biodiesel, and electrifying the sector. However, to date, only a few low-carbon fuel alternatives have been implemented for the aviation sector. Currently, the U.S. is mobilizing academia, industry, and government to develop sustainable, efficient, and economically feasible processes to produce sustainable aviation fuel from renewable resources. Recently, the Airline for America has announced the ambitious goal to achieve a production of 3 billion gallons of cost-competitive sustainable aviation fuel (SAF) by 2030, which corresponds to 15–20% of the current annual jet fuel production in the U.S. [4]. This plan would positively impact the aviation sector and help reduce emissions and dependence on fossil fuels.
In 2016, the U.S. Department of Energy (DOE), in partnership with the Oak Ridge National Laboratory, released the 2016 Billion-Ton Report [5]. According to the report, the U.S. can supply 1 billion tons of renewable resources (agricultural and forestry residues, energy crops, algal, and waste) as feedstock for biofuels, biochemicals, and biomaterials by 2040. Although the feedstock supply available in the U.S. is sufficient to produce 3 billion gallons of SAF by 2030, achieving a cost-competitive jet fuel selling price starting from renewable resources is challenging. Unlike fossil resources (petroleum, coal, and natural gas), biomass resources (agricultural and forestry residues, energy crops, algal, and waste) are heterogeneous and require considerable processing to produce infrastructure compatible with hydrocarbon fuels. Conversion of renewable resources into SAF requires several processing steps, including fractionation, conversion, recovery, and purification, making the process expensive. Additionally, these processes are unable to convert all the carbon available in the feedstocks, and the remaining carbon becomes process waste, an emission, or is used for electricity and heat.
Different pathways have been proposed to convert renewable feedstocks into SAF, including Alcohol-to-Jet (ATJ), Fischer–Tropsch from syngas, hydroprocessing of fats and fatty acids, and hydrogenolysis of lipids [6, 7]. While the hydroprocessing of fats is currently the main technology used to produce SAF in the U.S., the limited capacity of fat waste is expected to hinder the further scale-up of this technology. The ATJ process is considered a viable alternative to the capacity issue due to the large availability of ethanol [8, 9] and the growing number of proposed ATJ projects to be developed in the United States.
Alcohol-to-jet processes using cellulosic feedstocks are especially promising due to the low life cycle carbon emissions when producing cellulosic ethanol [10]. The economics of producing SAF from cellulosic ethanol, however, are challenging. Previous research from the National Renewable Energy Laboratory (NREL) concluded ethanol from corn stover would have a minimum fuel selling price (MFSP) of $2.15/gallon [11], assuming a modest discount rate of 10%. Achieving the target SAF price of $2.50/gallon—the current target set by the Department of Energy (DOE) [7]—from lignocellulosic ethanol at $2.15/gallon is unrealistic since there will be yield losses and substantial operating and capital costs associated with converting ethanol to jet fuel. Innovative strategies that lower jet fuel production costs are necessary to make the ATJ process economically feasible with lignocellulosic biomass.
There is a consensus that one solution to economically feasible biofuels involves biorefinery integration to produce a diverse portfolio of products [12], especially co-products that have substantially higher value than jet fuel. This biorefinery will have superior economics because it produces higher value products and enables more complete use of the biomass resource for chemicals, fuel, and energy. From a technical and operating standpoint, however, achieving an efficient integrated biorefinery process design is challenging because of interactions between individual process units. Further, the lack of infrastructure and issues with market development penetration for bioproducts create a barrier to implementing integrated biorefineries [12, 13]. The present work presents a holistic approach to this problem by converting lignocellulosic biomass into jet fuel and co-products with well-established markets, resulting in favorable overall process economics.
In 2019, Rosales-Calderon and Arantes [14] published an excellent review on chemicals and materials that are produced at commercial scales and that could be immediately co-produced with lignocellulosic ethanol. These products include polyols, alcohols, furfurals, organic acids, and alditols. For example, polyols, such as 1,2-butanediol and 1,4-butanediol, have an average selling price of $2,900/tonne [15], and 2,3-butanediol—a high-value polyol [16]—has a market volume of 32 million tonnes/year [17, 18]. Alditols, such as sorbitol and xylitol, have selling prices of $720/tonne [19] and $4,400/tonne [19] (price estimate from 2015—current price for xylitol may reach values as high as $6,500/tonne), respectively, and attractive markets for use in food and pharmaceutical products [14]. Although organic acids are not high-value compared to polyols and alditols, they have strong markets with large volumes—for example, acetic acid has a market volume of 8.3 million tonnes/year [20]. Co-producing jet fuel and some of these products (i.e., organic acids, polyols, and alditols) could create a viable process pathway for cost-competitive jet fuel.
This paper presents the techno-economic analysis (TEA) of an integrated biorefinery to convert poplar wood into xylitol, formic acid, and jet fuel. A primary biorefinery design objective is process integration that maximizes biomass utilization and minimizes CO2 emissions. The technical aspects of the processes to convert the biomass into jet fuel, xylitol, and formic acid are thoroughly discussed. Several TEAs have been published addressing the conversion of renewable feedstocks into jet fuel [21,22,23,24,25,26,27,28] and bioproducts [27, 29], but only a few attempts [15, 30] have been made to design integrated processes to co-produce jet fuel and high-value products from lignocellulosic biomass. As discussed in the present publication, this approach substantially lowers the jet fuel selling price and can establish a feasible process design for SAF to achieve the current DOE target price of $2.50/gallon by 2030.
Biorefinery scale
Detailed supply curves for poplar biomass for a Lewis County (WA) biorefinery have been developed in a recent study [31]. They showed that up to 760 ktonne/year of poplar wood, primarily grown on land designated as pastureland, would be available for the factory. For our analysis, we assumed an intermediately sized biorefinery that uses 250 dry ktonne/year (685 dry tonne/day) at an average plant-gate biomass cost of $77 per dry tonne. This constitutes our base case. Then we also assessed a small-scale biorefinery operating at a biomass feed rate of 150 dry ktonne/year (411 dry tonne/day) with an average plant-gate cost of $65 per dry tonne and a large-scale biorefinery operating at the maximum biomass availability in Southwest Washington, 760 dry ktonne/year (2082 dry tonne/day) with an average plant-gate cost of $85 per dry tonne.
Table 1 shows the jet fuel, formic acid, and xylitol production as a function of biorefinery feedstock capacity for the 3 biorefinery scales considered in this study (small-scale, intermediate-scale, and large-scale). As expected, the results show that the product capacities are proportional to biorefinery feedstock capacity. For the 3 scales (small, intermediate, and large), the percent conversions of carbon in the biomass into jet fuel, xylitol, and formic acid are 22, 14, and 14 C%, respectively. Lignin, which corresponds to a substantial fraction of the carbon present in the biomass, 37 C%, is used for heat and electricity production.
Table 1 Jet fuel, formic acid, and xylitol capacities as a function of biorefinery feedstock capacity
Jet fuel is a product with a huge market—the global market volume for jet fuel is estimated at 106 billion gallons/year [7], much larger than the production obtained for the 3 cases. Even for a large-scale biorefinery, the jet fuel market could easily accommodate the fuel produced in the biorefinery. The situation for formic acid and especially xylitol is different. Although formic acid is a product with a strong market and many applications, the formic acid capacity for the large-scale biorefinery corresponds to 25% of its global market volume, estimated at 762 ktonne/year [32]. The xylitol market is still under development, with a global market volume estimated at 190 ktonne/year [33]. The xylitol production volume in a large-scale biorefinery would, therefore, correspond to 43% of the global xylitol market volume, creating a huge barrier to market entry. It is important to note that large xylitol and formic acid volumes from the biorefinery may create an opportunity to lower the xylitol and formic acid selling prices, currently estimated at $4,200/tonne [14, 33] and $1,000–1,200/tonne (current price in the U.S.—formic acid prices in China are estimated at $400–600/tonne), respectively, creating an opportunity to expand the market volume for these products. Large market volumes could increase the adoption of xylitol in foodstuff and hygiene products and create a pathway for domestic formic acid production from renewable resources as an alternative to petroleum-based formic acid. From a market volume perspective, the large-scale biorefinery appears unrealistic, and there would be considerable challenges to sourcing 760 ktonnes of appropriate biomass per year for a single biorefinery.
Table 2 shows the total installed equipment cost of the primary biorefinery areas as a function of biorefinery feedstock capacity. The relative contribution of the individual biorefinery areas to the total installed equipment cost is only moderately dependent on feedstock capacity. The areas associated with heat and electricity generation (A900) and wastewater treatment (A1000) account for 44% of the total installed equipment cost. Thus, co-locating the biorefinery with a power plant can reduce the total installed equipment cost of the biorefinery by as much as 28%. The cost to fractionate biomass into C5 and C6 sugars and lignin, including biomass fractionation (A100) and saccharification (A200), accounts for approximately 22% of the total installed equipment cost. This analysis shows that the biggest capital cost drivers are heat/electricity production, wastewater treatment, and biomass fractionation (65% of the total fixed capital). Jet fuel production (A300, A400, A500, and A600) contributes to 13–16% of the total fixed capital, with ethanol production (A300) accounting for approximately 39–54% of the installed equipment expense, among the process steps to convert the C6 sugars (primarily glucose) into jet fuel. Xylitol and formic acid production areas—A700 and A800, respectively—each account for approximately 10% of the total installed equipment cost.
Table 2 Installed equipment cost (in million $) for the individual areas of the biorefinery and total capital investment as a function of biorefinery feedstock capacity
Figure 1 shows the minimum jet fuel selling price as a function of biorefinery feedstock capacity at discount rates of 0 (break-even cost), 10, 15, and 20%, assuming fixed formic acid and xylitol selling prices of $1,000 and $4,200 per tonne (current market selling prices), respectively. The result shows that jet fuel production at cost (discount rate of 0%) leads to a minimum jet fuel selling price below the current commercial price for all biorefinery capacities. The small-scale biorefinery is profitable at discount rates of 10 and 15%, leading to a minimum jet fuel selling price lower than that of SAF, $7.00/gallon [34]. At a discount rate of 20%, the small-scale biorefinery is not feasible since the jet fuel selling price is higher than the current SAF price. The intermediate-scale biorefinery shows favorable jet fuel selling prices. At a low discount rate of 10%, the minimum jet fuel selling price is lower than the price of commercial jet fuel. At a more realistic discount rate of 15%, the minimum jet fuel selling price is $3.13/gallon, which gives a $3.87/gallon margin relative to the policy-supported SAF price. Even at the highest discount rate assumed in this study, 20%, the jet fuel selling price remains below the current supported SAF price for an intermediate-scale biorefinery. In an optimistic scenario of maximum biomass availability (large-scale biorefinery scenario), the minimum jet fuel selling price is lower than the current jet fuel selling price at all discount rates.
Minimum jet fuel selling price in $ per gallon as a function of biorefinery feedstock capacity for the 3 scales assumed in the present study (small, intermediate, and large) and discount rates of 0 (break-even cost), 10, 15, and 20%. Baseline prices of $1,000 and $4,200 per tonne were assumed for formic acid and xylitol, respectively
The large-scale biorefinery operating at maximum biomass capacity exhibits an outstanding jet fuel selling price, but at this capacity, the co-products, xylitol and formic acid, will have challenging market entry issues, as previously discussed. Further, the total capital investment for building the large biorefinery exceeds $1.1 billion, which imposes a barrier to the feasible implementation of this enterprise and is a high-risk investment at the current stage of this technology. It appears from our analysis that the small and intermediate-scale biorefineries are the most viable from a feedstock, an economic, and a market volume perspective. We focus on the intermediate-scale biorefinery (base case) for the following discussions because it presents better economics than the small-scale factory.
Process utilities
Figure 2 presents the relative demand for steam and electricity in individual biorefinery areas for the intermediate-scale biorefinery. The total utility requirements for the biorefinery are presented in Table 3. According to the results shown in Fig. 2, biomass fractionation (A100) and wastewater treatment (A1000) account for most of the steam requirements of the biorefinery—33 and 26%, respectively. Xylitol production (A700) is also a heat-intensive process due to the multiple-effect evaporators used to remove furfurals, organic acids, and other impurities from the C5 sugar stream, accounting for 20% of the total steam usage. Steam is also necessary for the reboilers of the distillation columns used to separate the ethanol from water in A300, accounting for 12% of the total steam requirement of the biorefinery.
Relative utility usage (steam and electricity) of individual biorefinery areas for the intermediate-scale biorefinery (base-case scenario). A100—biomass fractionation, A200—saccharification, A300—ethanol production, A400—alcohol dehydration, A500—oligomerization, A600—hydrogenation, A700—xylitol production, A800—formic acid production, A900—boiler and turbogenerator, and A1000—wastewater treatment
Table 3 Summary of total utility requirements for the intermediate-scale biorefinery (base case)
The conversion of CO2 and water into formic acid in A800 accounts for 59% of the total electricity requirement in the modeled biorefinery. Areas A100, A200, A300, and A700, require approximately the same amount of electricity, contributing to 8–11% of the total electricity required for pumps and compressors. In the present design, the steam required for all areas of the biorefinery is produced in the biorefinery by burning make-up nature gas, lignin produced from the biomass, and syrup from wastewater treatment, contributing to 60, 23, and 17% of the total energy for steam production, respectively. The superheated steam is used in the turbogenerator to produce 26,000 kW of electricity, which is sufficient to power all the areas of the biorefinery, except for A800, due to the great amount of electricity required to run the electrochemical reactor; thus, an additional 19,000 kW of electricity is supplied from the grid.
Sensitivity analysis and co-location
The effect of independent process variables (one at a time) on the minimum jet fuel selling price is analyzed. Different scenarios are explored in this section, including changes in market-related and technical-based parameters. Changes in biomass, electricity, enzyme, and natural gas cost (these 4 combined correspond to 91% of the total variable operating costs) and xylitol and formic acid selling prices are considered to assess the impact of market-related parameters on the selling price of jet fuel. For technical-based parameters, changes in conversion, yields, and product recovery are considered. The results presented in this section are compared to the minimum jet fuel selling price (calculated at a discount rate of 15%) for the intermediate-scale biorefinery (base-case scenario, assuming a biorefinery operating at a biomass availability of 250 dry ktonne/year), $3.13/gallon. A disturbance of ± 25% for the independent variables analyzed is assumed in the sensitivity analysis.
We also investigated a scenario where the biorefinery is co-located with a power plant. In this scenario, the biorefinery does not produce steam and electricity; these two inputs are supplied from a power plant. Prices of $0.04/kWh and $7/tonne were assumed for electricity and steam (at 280 °C and 1310 kPa), respectively, for co-location with a power plant. Also, the power plant would burn the lignin stream originally used for heat and electricity generation in the base-case scenario.
Figure 3 shows that the minimum jet fuel selling price is highly sensitive to changes in the co-product prices. A favorable scenario where xylitol is sold at approximately $5,250/tonne (25% higher than its current market price, $4,200/tonne) would lead to a jet fuel price much lower than that of commercial jet fuel. Note that the jet fuel selling price remains below the current SAF price, $7.00/gallon, for all scenarios considered. Based on our analysis, co-products are crucial to reducing the jet fuel selling price and making SAF production feasible from an economic perspective.
Change in minimum jet fuel selling price for a disturbance of ± 25% in biomass, electricity, enzyme, and natural gas costs, formic acid and xylitol selling prices, yields of dilute acid hydrolysis, enzymatic hydrolysis, glucose fermentation, and xylitol crystallization, and formic acid conversion relative to the jet fuel baseline price of $3.13/gallon for the intermediate-scale biorefinery. The graph also shows the change in jet fuel selling price with co-location with a power plant
Biomass and natural gas are the primary resources that contribute to the jet fuel price, as shown in Fig. 3. A realistic lower-cost biomass scenario involves co-utilizing hybrid poplar for wastewater treatment and as feedstock for the biorefinery [31]. This approach could lower the feedstock price by as much as 15%, thus reducing the overall plant-gate cost of the biomass.
Figure 3 also shows that a decrease of 25% in the yields of dilute acid hydrolysis, enzymatic hydrolysis, glucose fermentation, and xylitol crystallization (one at a time) lead to significant increases of $5.47, $5.07, $4.86, and $5.30 per gallon of jet fuel, respectively. An exception is an approximately negligible change in jet fuel selling price with a disturbance of ± 25% in formic acid conversion in the electrochemical reactor. Our base-case design assumes that the yields of the processes investigated in the sensitivity analysis are maximized based on previous designs reported in the literature and previous work done at the University of Washington. The analysis of the effect of technical-based parameters on the minimum jet fuel selling price shows that if optimum yields are not achieved, it could drastically impact the economics of the process. Nevertheless, even if optimum yields are not achieved, the minimum jet fuel selling price would still be lower than the current price of subsidized SAF.
One important consideration for the biorefinery designed in the present study is the co-location with existing facilities that could provide the utilities and resources to run the process. Co-location with a power plant could provide the necessary steam and electricity to run the processes and lower the jet fuel selling price. As shown in Fig. 3, this scenario substantially reduces jet fuel selling price (70% reduction), making the process more viable from both a technical and economic perspective since a boiler and a turbogenerator would not be necessary. It is important to note, however, that depending on the source of fuel used in the power plant, i.e., biogenic versus non-biogenic, the economic benefit of co-locating the biorefinery with the power plant would come at the expense of burning fossil fuels, thus leading to a SAF that may not meet greenhouse gas emission standards.
While not specifically investigated in this study, it has been found that co-locating the biorefinery with existing crude oil refineries would decrease its capital investment by providing the necessary hydrotreating and hydrocarbon fractionation units (hydrotreating equipment such as reactors are available in crude oil refining facilities). According to a recent publication, this approach would reduce the jet fuel selling price by 3–23% [35].
Feedstock
The feedstock is one of the primary drivers of biorefinery's operating costs [11]. Hybrid poplar is abundant in the Northwest region [5] and has been considered one of the main energy crops for biofuels and biochemicals in the U.S [5, 36] due to its excellent characteristics—high sugar and low ash contents. Also, poplar requires low fertilizer input, can re-sprout after multiple harvests, and has a high growth rate and large biomass accumulation [21, 36, 37]. While most poplar production is from forestry and farm lands [37], marginal lands have been considered a good alternative to growing energy crops for biofuels and biochemical [38].
Table 4 presents the composition of the hybrid poplar feedstock considered in this publication based on previous works conducted at the University of Washington [37].
Table 4 Chemical composition of hybrid poplar chips assumed in the model [37, 39]
Feedstock fractionation (A100 and A200)
The process flow diagram of the envisioned biorefinery is depicted in Fig. 4. Feedstock fractionation includes areas 100 (Fig. 4—A100 Biomass Fractionation) and 200 (Fig. 4—A200 Saccharification). Biomass chips are fractionated into cellulose, hemicellulose, and lignin by dilute acid hydrolysis at 195 °C and 13 bar with formic acid [40] in A100. The choice of an organic acid during acid hydrolysis is to avoid inorganics in downstream processes (it is well-known that inorganic compounds are troublesome in catalytic and biological conversion processes) and because formic acid is one of the main products of the biorefinery—approximately 12% of the formic acid produced in the facility is used for biomass pretreatment. The liquid (containing hydrolyzed sugars—mostly xylose) and solid (containing cellulose, lignin, and ash) phases are separated by washing with process water. The washer unit was chosen in this study to avoid costly solid–liquid separation [11] and because it is a widely implemented unit operation in the pulp and paper industry. Also, the washer allows for high xylose recovery. We selected a washer with a Norden number of 18 operating with a dilution factor of 0.6. This configuration provided a washing yield of 98%. This design was successfully modeled using WinGems and implemented in Aspen as a separation block (Sep block in Aspen). The liquid stream from A100 goes to area 700 (Fig. 4—A700 xylitol production), and the solid stream goes to A200.
Process flow diagram of the integrated biorefinery showing the primary process areas and a simplified schematic of the process streams
The cellulose in the washed solids is hydrolyzed into glucose at 48 °C and a cellulase loading of 20 mg/g of cellulose [11]. The solids in the hydrolysate (mostly lignin and ash) are separated using a filter press and are sent to area 900 (Fig. 4—A900 boiler/turbogenerator) to produce steam and electricity, and the hydrolysate goes to area 300 (Fig. 4—A300 fermentation) for fermentation of the C6 sugars (primarily glucose). The present design assumes that the cellulase is produced on-site, consistent with modeling approaches in previous TEA studies [11]. Table 5 presents the chemical reactions and conversions assumed for the acid hydrolysis of biomass in A100 and enzymatic hydrolysis in A200. Additional file 1: Figures S1 and S2 show the detailed process flow diagrams of A100 and A200, respectively.
Table 5 Dilute acid hydrolysis and enzymatic hydrolysis reactions and conversions
Alcohol synthesis (A300)
In A300, the hydrolyzed sugars from A200 are fermented into ethanol. Fermentation of glucose into ethanol is a well-developed process and can employ bacteria (e.g., Zymomonas mobilis) [11] or yeast (e.g., Saccharomyces cerevisiae) [41]. From a technical standpoint, chemical processes employing bacteria or engineered microorganisms are higher risk due to the high chance of contamination [42]. Therefore, the present design employs the yeast Saccharomyces cerevisiae for fermenting glucose into ethanol due to the yeast's high resistance to the product (ethanol) and the high fermentation yields achieved with this organism. Also, the concentration of C5 sugars at the inlet stream of the fermentation process in A200 is small, which favors the use of Saccharomyces cerevisiae. It is important to note that commercial processes employing Saccharomyces cerevisiae are well established. The fermentation broth exits the fermenter with an ethanol concentration of 3 wt. % and is sent to the first distillation column (denoted beer column) for ethanol recovery. The design for the beer column is similar to the one used in the NREL's report from 2011 [11]. The concentrated ethanol stream from the beer column (with an ethanol concentration of approximately 44 wt. %) is sent to a second distillation column, where ethanol is further concentrated to 92 wt. %. The outlet ethanol stream is sent to area 400 (Fig. 4—A400 alcohol dehydration) for dehydration into ethylene. Note that the concentration of inorganics and water after alcohol distillation in A300 is low or negligible, thus avoiding additional unit operations to clean the ethanol stream, such as molecular sieve packages. The CO2 produced during glucose fermentation is easily recovered in the overhead of the beer column, and it is sent to area 800 (Fig. 4—A800 formic acid production), where it is converted into formic acid by reacting it with water in an electrochemical reactor. The bottom streams of the beer column and the second distillation column (mostly water) are sent to area 1000 (Fig. 4—A1000 wastewater treatment) for wastewater treatment. Additional file 1: Figure S3 shows the process flow diagram of A300 for the ethanol production area.
Alcohol-to-Jet (A400, A500, A600)
The Alcohol-to-Jet (ATJ) process involves the conversion of alcohols into hydrocarbon molecules suitable for jet fuel by 1) alcohol dehydration into light alkene gases in A400, 2) light alkene oligomerization into higher alkenes in area 500 (Fig. 4—A500 Oligomerization), and 3) hydrogenation of higher alkenes into alkanes in area 600 (Fig. 4—A600 hydrogenation). From a technical standpoint, oligomerization is the most challenging process among the ATJ process steps due to the low conversion of light alkenes into long-chain hydrocarbons and the high selectivity for lower molecular weight products (especially butene and hexene). Ethylene oligomerization especially imposes a challenge due to the higher degree of polymerization required to produce liquid hydrocarbons relative to oligomerization of higher alkenes, such as butane [43].
The 3 steps of the ATJ process (dehydration, oligomerization, and hydrogenation) employ heterogeneous catalysts. Alcohol dehydration usually employs acid catalysts, temperatures of 200–400 °C, and low or high pressure [44]. The present design employs a trifluoromethanesulfonic acid silica-based catalyst at 200 °C and 1 atm [45] for ethanol dehydration into ethylene, with an overall yield of 98% and excellent selectivity (> 99%) to production of ethylene as reported in the literature [45]. In the present design, a nickel-based heterogeneous catalyst (Ni-H-Beta, Ni-SBA15, or Ni-Siral) was used for ethylene oligomerization based on previous studies at the University of Washington [46,47,48,49,50,51,52]. Conversions and selectivities have been reported in [46,47,48,49,50,51] for ethylene oligomerization. The design of the unit operation of hydrogenation adopted in the present study was the same used in [43]. In our process, the concentration of water and inorganics at the inlet of A400 is small (below 5 wt. %), which is assumed small enough to avoid reducing catalyst performance during ethanol dehydration into ethylene. Additional file 1: Figure S4 shows the process flow diagrams for A400, A500, and A600.
Xylitol production (A700)
Hydrogenation is employed to convert sugars into alditols [53]. Because the production of xylitol as target alditol requires a xylose-rich stream, the primary challenges in producing pure xylitol from mixed sugar streams are the necessary separation and recovery steps to isolate the target sugar before hydrogenation [53]. Usually, the hydrolyzed sugar stream from biomass pretreatment (C5-rich stream from A100) contains a mixture of xylose, glucose, arabinose, galactose, and mannose. Because the hybrid poplar used in the present design contains mostly glucose and xylose (Table 4), the amount of arabinose, mannose, and galactose in the C5-rich stream from A100 is small. Still, hydrogenation of this process stream without some additional processing will lead to a complex mixture of alditols, primarily composed of xylitol and sorbitol.
Crystallization of an alditol mixture is troublesome due to the slow growth and irregular shape of the target alditol crystals and slow filtration [53]. To avoid complications during crystallization, the maximum concentration of secondary alditols in the dissolved solids is kept below 15 wt. % [53]. The secondary sugars in the inlet of the hydrogenation reactor are usually separated by simulated moving beds (SMB) chromatography, which is not a well-developed commercial-scale unit operation. This study presents an innovative approach for converting liquid sugar streams into xylitol at high yields by combining biological and thermochemical conversion processes. The approach to clean the sugar stream and obtain a xylose-rich stream involves converting secondary sugars (mostly glucose in the designed biorefinery) into ethanol via fermentation. The fermentation process employs Saccharomyces cerevisiae to avoid the consumption of xylose. Figure 5 shows the process flow diagram of the xylitol process designed in the present work.
Process flow diagram of the xylitol production process designed in the present work
The process starts with the liquid stream from A100. Initially, the furfurals (furfural and HFM) produced during biomass pretreatment in A100 are removed from the sugar stream by multiple-effect evaporators (Fig. 5—MEE-701). Part of the formic acid (used during pretreatment) and acetic acid (produced from the acetate groups in poplar) are also evaporated in this step. Then, the concentrate from the MEE is sent to an activated carbon column (Fig. 5—S-701), where part of the organic acids and dissolved lignin in the inlet stream are removed. The glucose in the sugar stream at the outlet of the activated carbon column is converted into ethanol via fermentation (Fig. 5—R-701). The fermentation is fast (~ 4–6 h) because of the low glucose concentration in the fermenter's inlet stream, 10–15 g/L. The yeast is separated from the broth using a filter press (Fig. 5—S-702), and the ethanol is separated from the xylose-rich stream via MEE (Fig. 5—MEE-702). Another activated carbon column (Fig. 5—S-703) removes impurities from the sugar stream before hydrogenation. Note that the amount of ethanol produced in A700 is small, and recovering ethanol as a product is not economically feasible.
The liquid stream containing xylose (total dissolved solids = 20–25 wt. %; xylose concentration in dissolved solids = 99 wt. %) is pressurized to 125 bar (Fig. 5—P-701) and fed into the hydrogenation reactor (Fig. 5—R-702) [53]. Hydrogen is also fed into the reactor, and the unreacted hydrogen is recycled to maintain a hydrogen-to-xylose ratio 4 times the stoichiometric requirement. The effluent from the reactor is passed through an activated carbon column (Fig. 5—S-705) and then another MEE (Fig. 5—MEE-703) to remove impurities and water, respectively. The feed to the crystallizers (Fig. 5—R-703 and R-704, two crystallizers were used in the present design) contains 50–70 wt. % dissolved solids with a xylitol fraction in the dissolved solids of 98–99 wt. % [53]. The crystallizers operate at 8 °C. The xylitol crystals are recovered using centrifuges (Fig. 5—S-706 and S-707), and they are fed into a spray dryer (Fig. 5—spray dryer), where water is further removed. The crystals exit the dryer with a purity greater than 99 wt. %.
Formic acid production (A800)
One of the main advantages of the integrated biorefinery proposed in the present study is the optimized use of carbon in the cellulose and hemicellulose biomass fractions for fuels and chemicals. The CO2 produced during fermentation in A300, which would become an emission, is sent to A800 for electrochemical conversion into formic acid. The evaporated water from A700 is used as a reactant in the electrocatalytic reactor—the amount of impurities in the water from A700 is negligible. Figure 6 shows the process flow diagram of the formic acid production process designed in the present work. The process adopted in the present design consists of an electrochemical cell (Fig. 6—R-801) designed by OCOChem [54, 55] that filters CO2 through a membrane and converts it into formic acid using electricity. A single-pass conversion of CO2 into formic acid of 10% is assumed. Based on personal communications with the CEO of OCOChem [54, 55], a faradaic efficiency of up to 90% can be achieved during the reduction of CO2 into formic acid with a cell voltage of 4.0 V and an average current density of approximately 125 mA/cm2, accounting to an energy requirement of approximately 4.3 kWh per kg of formic acid produced. For the electrochemical reactors, it is estimated a capital investment of $153/kW (assuming a current density of 300 mA/cm2) and $368/kW (assuming a current density of 125 mA/cm2). The process also produces H2 and O2 that can be separated using membranes (Fig. 6—S-801 and S-802). The conversion of water into H2 in the electrochemical reactor is 2%. The H2 produced in the electrochemical reactor is fed into the hydrogenation reactor in A600, but make-up H2 is still necessary to convert the remaining alkenes into alkanes. A price for hydrogen of $1368/tonne was used. Formic acid is separated from the water via pressure-swing distillation (Fig. 6—D-801 and D-802). Both columns contain 15 stages—the first one operates at 3 bar, and the second one operates at 0.4 bar. In the present design, formic acid at 85 wt. % is obtained.
Process flow diagram of the formic acid production process designed in the present work
Boiler and turbogenerator (A900) and wastewater treatment (A1000)
The lignin-rich stream from A200 is sent to A900 and used as fuel for heat and electricity. Make-up natural gas is co-fed to produce the steam necessary to meet the total steam demand of the biorefinery. Electricity is co-produced and used to power most of the process areas of the biorefinery, except for the electrochemical reactor in A800. All the waste streams of the biorefinery are sent to A1000, where part of the water is evaporated using a 7-stage MEE [56]. The concentrate (syrup) is combusted in the boiler. Part of the treated water stream is used in A900 to produce steam. According to the present design, a small wastewater treatment plant is necessary to treat the remaining water stream not fed into the multiple-effect evaporator. We used NREL's design from 2002 for wastewater treatment employing a combination of anaerobic and aerobic digestion [57]. The total capital investment for the additional water treatment plant is a small fraction of the total capital cost of A1000. Figure S5 shows the process flow diagram for A900.
Techno-economic analysis
Cost-year indices
The Chemical Engineering Plant Cost Index (CEPCI) for the base year of 2019 was used for this analysis [58]. Equipment and operating costs were obtained from the literature and corrected to 2019$ using the CEPCI according to the following equation:
$$cost\left(2019\mathrm{\$}\right)=cost\left(base\$\right)\times \left(\frac{CE{PCI}_{2019}}{CE{PCI}_{base}}\right).$$
In Eq. 1, cost(2019$) is the updated equipment or operating cost (based on 2019$), cost(base$) is the cost at a given base year, and CEPCI2019 and CEPCIbase are the cost indexes for 2019, 619.2 [58], and the base year, respectively.
Equipment cost
Equipment costs were obtained from the literature [11, 57, 59, 60] and used to estimate total equipment costs for the individual unit operations and the total capital investment of the biorefinery. The installed equipment cost is based on the factored value of the equipment purchase cost, according to the following equation:
$$Installed cost=\left(purchased \, equipment\, cost\right)\times\, \left(multiplier\right).$$
Equation 2 does not include installation foundation, piping, and wiring costs [11]. These costs are factored in as a fraction of the total installed cost of Inside Battery Limit (ISBL).
Equation 3 was used to calculate the scaled cost of equipment relative to its base cost and size:
$$new cost=\left(base\, cost\right)\times {\left(\frac{New\, size}{Base\, size}\right)}^{n}.$$
In Eq. 3, n is a positive fraction and depends on the characteristics of the equipment (capacity, heat duty, or flow rate) [11]. Installed cost multipliers (Eq. 2) and characteristic scaling exponents (Eq. 3) were obtained from the literature [11, 57, 59, 60] and are listed in Table 6.
Table 6 Installation factor and characteristic exponents used in Eqs. 2 and 3 to calculate the installed and scaled costs of the individual unit operations
Each area's total installed equipment costs account for part of the total direct costs (TDC). Other TDCs include site development, warehouse, and additional piping. These costs were factored in as a fraction of the total installed equipment cost. The total indirect costs (TIC), including proratable costs, field expenses, home office and construction, project contingency, and other costs, were calculated based on the TDC. The fixed capital investment (FCI) for the biorefinery is the sum of TDC and TIC. The present design assumes a working capital of 5% of the FCI, and the total capital investment (TCI) is the sum of the FCI and the working capital. The assumptions and costs used to estimate the TDC and TIC were obtained from a previous report [11].
Discounted cash flow analysis and minimum jet fuel selling price
The calculated minimum jet fuel selling price assumes a projected net present value of zero at a given fixed annual discount rate (0, 10, 15, or 20%) over a project lifespan of 10 years. For this analysis, the xylitol and formic acid selling prices were fixed at their current market prices. The biorefinery is 60% equity-financed, with 40% being financed from 10-year loans with an annual percentage rate (APR) of 8%. During the start-up period (first year of operation), the biorefinery operates at a 50% reduced capacity. A pretax financial position was assumed due to the complex corporate tax environment, the current favorable depreciation schedules, and the potential for receiving favorable tax exemptions for a new low-carbon industry.
The present study is the first to present the techno-economic analysis of an integrated biorefinery to produce jet fuel and biobased chemicals from lignocellulosic biomass. We show that co-production of jet fuel, xylitol, and formic acid leads to a jet fuel minimum selling price of $3.13 per gallon, assuming a biorefinery operating at a biomass capacity of 250 ktonne per year assuming a discount rate of 15%. Biomass fractionation, and steam and electricity production, account for a large fraction of the total installed equipment cost. Electricity is an important utility due to the large electricity demand to run the electrochemical reactor used in the production of formic acid, accounting for 59% of the total electricity demand of the biorefinery. Based on our analyses, co-location with a power plant can substantially lower the total capital investment necessary to build the biorefinery and the biorefinery operating costs since the power plant could supply electricity and steam at cost. Finally, sensitivity analysis shows that the selling price of co-products has a major impact on the final jet fuel selling price. In a favorable scenario where xylitol price is 25% higher than its current market price and formic acid is sold at its baseline market price, the minimum jet fuel selling price is $0.64 per gallon, much lower than the DOE target price of $2.50/gallon for SAF by 2030.
Supporting data to that in the article are provided in the Supplementary Information file.
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MarketsAndMarkets. Formic Acid Market by Types (Grades of 85%, 94%, 99%, and others) by Application (Agriculture, Leather & Textile, Rubber, Chemical & Pharmaceuticals, & others) & by Geography - Global Trends, Forecasts to 2019. https://www.marketsandmarkets.com/Market-Reports/formic-acid-Market-69868960.html. 2014. Accessed 19 Dec 2022.
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We thank the assistance from co-founders of OCOchem, Todd Brix and Terry Brix, the vice president of operations at Fortress Advanced Bioproducts, Kent Smith, and Brian O'Neill from Gevo, for providing key insight on the design of the formic acid, xylitol, and jet fuel production processes.
Support for this project has come from the Washington State Legislature, the Lewis County 0.09 Rural Economic Development Fund, the Centralia Coal Transition Grants, and the Denman Endowed Chair in Bioresource Engineering.
School of Environmental and Forest Sciences, University of Washington, Seattle, WA, 98195, USA
Gabriel V. S. Seufitelli, Hisham El-Husseini, Danielle U. Pascoli, Renata Bura & Richard Gustafson
Gabriel V. S. Seufitelli
Hisham El-Husseini
Danielle U. Pascoli
Renata Bura
GVS. Seufitelli and RG wrote the main manuscript, HE-H contributed with the initial Aspen model used in the techno-economic analysis, and RB and DUP provided technical assistance during development of the model. All authors read and approved the final manuscript.
Corresponding authors
Correspondence to Gabriel V. S. Seufitelli or Richard Gustafson.
All authors read and approved the manuscript.
Process flow diagram for A100—Biomass fractionation. Figure S2. Process flow diagram for A200—Saccharification. Figure S3. Process flow diagram for A300—Ethanol production. Figure S4. Process flow diagram for A400—Alcohol dehydration, A500 –Oligomerization, and A-600—Hydrogenation. Figure S5. Process flow diagram for A900—Boiler and turbogenerator.
Seufitelli, G.V.S., El-Husseini, H., Pascoli, D.U. et al. Techno-economic analysis of an integrated biorefinery to convert poplar into jet fuel, xylitol, and formic acid. Biotechnol Biofuels 15, 143 (2022). https://doi.org/10.1186/s13068-022-02246-3
Biorefinery
Sustainable aviation fuel
Submission enquiries: [email protected] | CommonCrawl |
JEE Main 2014 (Offline)
On heating water, bubbles being formed at the bottom of the vessel detach and rise. Take the bubbles to be spheres of radius $$R$$ and making a circular contact of radius $$r$$ with the bottom $$R$$ and making a circular contact of radius $$r$$ with the bottom of the vessel. If $$r < < R$$ and the surface tension of water is $$T,$$ value of $$r$$ just before bubbles detach is: (density of water is $${\rho _w}$$)
$${R^2}\sqrt {{{{\rho _w}g} \over {3T}}} $$
$${R^2}\sqrt {{{{\rho _w}g} \over {T}}} $$
$${R^2}\sqrt {{{{2\rho _w}g} \over {3T}}} $$
When the bubble gets detached, Buoyant force $$=$$ force due to surface tension
Force due to excess pressure $$=$$ upthrust
Access pressure in air bubble $$ = {{2T} \over R}$$
$${{2T} \over R}\left( {\pi {r^2}} \right) = {{4\pi {R^3}} \over {3T}}{\rho _w}g$$
$$ \Rightarrow {r^2} = {{2{R^4}{\rho _w}g} \over {3T}}$$
$$ \Rightarrow r = {R^2}\sqrt {{{2{\rho _w}g} \over {3T}}} $$
An open glass tube is immersed in mercury in such a way that a length of $$8$$ $$cm$$ extends above the mercury level. The open end of the tube is then closed and scaled and the tube is raised vertically up by additional $$46$$ $$cm$$. What will be length of the air column above mercury in the tube now? (Atmospheric pressure $$=76$$ $$cm$$ of $$Hg$$)
$$16$$ $$cm$$
$$6$$ $$cm$$
Length of the air column above mercury in the tube is,
$$P + x = {P_0}$$
$$ \Rightarrow P = \left( {76 - x} \right)$$
$$ \Rightarrow 8 \times A \times 76 = \left( {76 - x} \right) \times A \times \left( {54 - x} \right)$$
$$\therefore$$ $$x=38$$
Thus, length of air column $$=54-38=16cm.$$
Assume that a drop of liquid evaporates by decreases in its surface energy, so that its temperature remains unchanged. What should be the minimum radius of the drop for this to be possible ? The surface tension is $$T,$$ density of liquid is $$\rho $$ and $$L$$ is its latent heat of vaporization.
$$\rho L/T$$
$$\sqrt {T/\rho L} $$
$$T/\rho L$$
$$2T/\rho L$$
When radius is decrease by $$\Delta R,$$
$$4\pi {R^2}\Delta R\rho L = 4\pi T\left[ {{R^2} - {{\left( {R - \Delta R} \right)}^2}} \right]$$
$$ \Rightarrow \rho {R^2}\Delta RL = T\left[ {{R^2} - {R^2} + 2R\Delta R - \Delta {R^2}} \right]$$
$$ \Rightarrow \rho {R^2}\Delta RL = T2R\Delta R\,\,$$ [ $$\Delta R$$ is very small ]
$$ \Rightarrow R = {{2T} \over {\rho L}}$$
A uniform cylinder of length $$L$$ and mass $$M$$ having cross-sectional area $$A$$ is suspended, with its length vertical, from a fixed point by a mass-less spring such that it is half submerged in a liquid of density $$\sigma $$ at equilibrium position. The extension $${x_0}$$ of the spring when it is in equilibrium is:
$${{Mg} \over k}$$
$${{Mg} \over k}\left( {1 - {{LA\sigma } \over M}} \right)$$
$${{Mg} \over k}\left( {1 - {{LA\sigma } \over {2M}}} \right)$$
$${{Mg} \over k}\left( {1 + {{LA\sigma } \over M}} \right)$$
From figure, $$k{x_0} + {F_B} = Mg$$
$$k{x_0} + \sigma {L \over 2}Ag = Mg$$
[ as mass $$=$$ density $$ \times $$ volume ]
$$ \Rightarrow k{x_0} = Mg - \sigma {L \over 2}Ag$$
$$ \Rightarrow {x_0} = {{Mg - {{\sigma LAg} \over 2}} \over k}$$
$$ = {{Mg} \over k}\left( {1 - {{LA\sigma } \over {2M}}} \right)$$
Questions Asked from Properties of Matter | CommonCrawl |
Fermat's Last Theorem (book)
Fermat's Last Theorem is a popular science book (1997) by Simon Singh. It tells the story of the search for a proof of Fermat's Last Theorem, first conjectured by Pierre de Fermat in 1637, and explores how many mathematicians such as Évariste Galois had tried and failed to provide a proof for the theorem.[1][2][3][4] Despite the efforts of many mathematicians, the proof would remain incomplete until 1995, with the publication of Andrew Wiles' proof of the Theorem. The book is the first mathematics book to become a Number One seller in the United Kingdom,[5] whilst Singh's documentary The Proof, on which the book was based, won a BAFTA in 1997.[6]
Fermat's Last Theorem
First edition
AuthorSimon Singh
LanguageEnglish
SubjectFermat's Last Theorem
GenreNon-fiction
PublisherFourth Estate
Publication date
1997
Media typePrint (hardcover, paperback)
ISBN978-1857025217
In the United States, the book was released as Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem.[1][3] The book was released in the United States in October 1998 to coincide with the US release of Singh's documentary The Proof about Wiles's proof of Fermat's Last Theorem.[2][7]
References
1. Review of Fermat's Enigma by Andrew Bremner (1998), MR1491363.
2. Radford, Tim (2 August 2013), "Fermat's Last Theorem by Simon Singh – book review. A boast in the margin of a book is the starting point for a wonderful journey through the history of mathematics, number theory and logic", The Guardian.
3. Penrose, Roger (November 30, 1997), "Q.E.D. How to solve the greatest mathematical puzzle of your age: Lock self in room. Emerge seven years later", The New York Times
4. Elliott, Josh (18 March 2016). "Math problem a 300-year saga of death, duels, dual identities". CTV News. Retrieved 31 July 2016.
5. Singh, Simon (24 May 2016). "Why it's so impressive that Fermat's Last Theorem has been solved". The Daily Telegraph. Retrieved 31 July 2016.
6. "The extraordinary story of Fermat's Last Theorem". University of Lethbridge. 3 May 1997. Retrieved 31 July 2016.
7. Jackson, Allyn (October 1997). "Fermat's Enigma" (PDF). American Mathematical Society. Retrieved 31 July 2016.
Works by Simon Singh
• Fermat's Last Theorem (1997)
• The Code Book (1999)
• Big Bang (2004)
• Trick or Treatment? (2008)
• The Simpsons and Their Mathematical Secrets (2013)
Pierre de Fermat
Work
• Fermat's Last Theorem
• Fermat number
• Fermat's principle
• Fermat's little theorem
• Fermat polygonal number theorem
• Fermat pseudoprime
• Fermat point
• Fermat's theorem (stationary points)
• Fermat's theorem on sums of two squares
• Fermat's spiral
• Fermat's right triangle theorem
Related
• List of things named after Pierre de Fermat
• Wiles's proof of Fermat's Last Theorem
• Fermat's Last Theorem in fiction
• Fermat Prize
• Fermat's Last Tango (2000 musical)
• Fermat's Last Theorem (popular science book)
| Wikipedia |
Profinite integer
In mathematics, a profinite integer is an element of the ring (sometimes pronounced as zee-hat or zed-hat)
${\widehat {\mathbb {Z} }}=\varprojlim \mathbb {Z} /n\mathbb {Z} =\prod _{p}\mathbb {Z} _{p}$
where
$\varprojlim \mathbb {Z} /n\mathbb {Z} $
indicates the profinite completion of $\mathbb {Z} $, the index $p$ runs over all prime numbers, and $\mathbb {Z} _{p}$ is the ring of p-adic integers. This group is important because of its relation to Galois theory, étale homotopy theory, and the ring of adeles. In addition, it provides a basic tractable example of a profinite group.
Construction
The profinite integers ${\widehat {\mathbb {Z} }}$ can be constructed as the set of sequences $\upsilon $ of residues represented as
$\upsilon =(\upsilon _{1}{\bmod {1}},~\upsilon _{2}{\bmod {2}},~\upsilon _{3}{\bmod {3}},~\ldots )$
such that $m\ |\ n\implies \upsilon _{m}\equiv \upsilon _{n}{\bmod {m}}$.
Pointwise addition and multiplication make it a commutative ring.
The ring of integers embeds into the ring of profinite integers by the canonical injection:
$\eta :\mathbb {Z} \hookrightarrow {\widehat {\mathbb {Z} }}$ :\mathbb {Z} \hookrightarrow {\widehat {\mathbb {Z} }}}
where $n\mapsto (n{\bmod {1}},n{\bmod {2}},\dots ).$
It is canonical since it satisfies the universal property of profinite groups that, given any profinite group $H$ and any group homomorphism $f:\mathbb {Z} \rightarrow H$, there exists a unique continuous group homomorphism $g:{\widehat {\mathbb {Z} }}\rightarrow H$ with $f=g\eta $.
Using Factorial number system
Every integer $n\geq 0$ has a unique representation in the factorial number system as
$n=\sum _{i=1}^{\infty }c_{i}i!\qquad {\text{with }}c_{i}\in \mathbb {Z} $
where $0\leq c_{i}\leq i$ for every $i$, and only finitely many of $c_{1},c_{2},c_{3},\ldots $ are nonzero.
Its factorial number representation can be written as $(\cdots c_{3}c_{2}c_{1})_{!}$.
In the same way, a profinite integer can be uniquely represented in the factorial number system as an infinite string $(\cdots c_{3}c_{2}c_{1})_{!}$, where each $c_{i}$ is an integer satisfying $0\leq c_{i}\leq i$.[1]
The digits $c_{1},c_{2},c_{3},\ldots ,c_{k-1}$ determine the value of the profinite integer mod $k!$. More specifically, there is a ring homomorphism ${\widehat {\mathbb {Z} }}\to \mathbb {Z} /k!\,\mathbb {Z} $ sending
$(\cdots c_{3}c_{2}c_{1})_{!}\mapsto \sum _{i=1}^{k-1}c_{i}i!\mod k!$
The difference of a profinite integer from an integer is that the "finitely many nonzero digits" condition is dropped, allowing for its factorial number representation to have infinitely many nonzero digits.
Using the Chinese Remainder theorem
Another way to understand the construction of the profinite integers is by using the Chinese remainder theorem. Recall that for an integer $n$ with prime factorization
$n=p_{1}^{a_{1}}\cdots p_{k}^{a_{k}}$
of non-repeating primes, there is a ring isomorphism
$\mathbb {Z} /n\cong \mathbb {Z} /p_{1}^{a_{1}}\times \cdots \times \mathbb {Z} /p_{k}^{a_{k}}$
from the theorem. Moreover, any surjection
$\mathbb {Z} /n\to \mathbb {Z} /m$
will just be a map on the underlying decompositions where there are induced surjections
$\mathbb {Z} /p_{i}^{a_{i}}\to \mathbb {Z} /p_{i}^{b_{i}}$
since we must have $a_{i}\geq b_{i}$. It should be much clearer that under the inverse limit definition of the profinite integers, we have the isomorphism
${\widehat {\mathbb {Z} }}\cong \prod _{p}\mathbb {Z} _{p}$
with the direct product of p-adic integers.
Explicitly, the isomorphism is $\phi :\prod _{p}\mathbb {Z} _{p}\to {\widehat {\mathbb {Z} }}$ :\prod _{p}\mathbb {Z} _{p}\to {\widehat {\mathbb {Z} }}} by
$\phi ((n_{2},n_{3},n_{5},\cdots ))(k)=\prod _{q}n_{q}\mod k$
where $q$ ranges over all prime-power factors $p_{i}^{d_{i}}$ of $k$, that is, $k=\prod _{i=1}^{l}p_{i}^{d_{i}}$ for some different prime numbers $p_{1},...,p_{l}$.
Relations
Topological properties
The set of profinite integers has an induced topology in which it is a compact Hausdorff space, coming from the fact that it can be seen as a closed subset of the infinite direct product
${\widehat {\mathbb {Z} }}\subset \prod _{n=1}^{\infty }\mathbb {Z} /n\mathbb {Z} $
which is compact with its product topology by Tychonoff's theorem. Note the topology on each finite group $\mathbb {Z} /n\mathbb {Z} $ is given as the discrete topology.
The topology on ${\widehat {\mathbb {Z} }}$ can be defined by the metric,[1]
$d(x,y)={\frac {1}{\min\{k\in \mathbb {Z} _{>0}:x\not \equiv y{\bmod {(k+1)!}}\}}}$
Since addition of profinite integers is continuous, ${\widehat {\mathbb {Z} }}$ is a compact Hausdorff abelian group, and thus its Pontryagin dual must be a discrete abelian group.
In fact, the Pontryagin dual of ${\widehat {\mathbb {Z} }}$ is the abelian group $\mathbb {Q} /\mathbb {Z} $ equipped with the discrete topology (note that it is not the subset topology inherited from $\mathbb {R} /\mathbb {Z} $, which is not discrete). The Pontryagin dual is explicitly constructed by the function[2]
$\mathbb {Q} /\mathbb {Z} \times {\widehat {\mathbb {Z} }}\to U(1),\,(q,a)\mapsto \chi (qa)$
where $\chi $ is the character of the adele (introduced below) $\mathbf {A} _{\mathbb {Q} ,f}$ induced by $\mathbb {Q} /\mathbb {Z} \to U(1),\,\alpha \mapsto e^{2\pi i\alpha }$.[3]
Relation with adeles
The tensor product ${\widehat {\mathbb {Z} }}\otimes _{\mathbb {Z} }\mathbb {Q} $ is the ring of finite adeles
$\mathbf {A} _{\mathbb {Q} ,f}={\prod _{p}}'\mathbb {Q} _{p}$
of $\mathbb {Q} $ where the symbol $'$ means restricted product. That is, an element is a sequence that is integral except at a finite number of places.[4] There is an isomorphism
$\mathbf {A} _{\mathbb {Q} }\cong \mathbb {R} \times ({\hat {\mathbb {Z} }}\otimes _{\mathbb {Z} }\mathbb {Q} )$
Applications in Galois theory and Etale homotopy theory
For the algebraic closure ${\overline {\mathbf {F} }}_{q}$ of a finite field $\mathbf {F} _{q}$ of order q, the Galois group can be computed explicitly. From the fact ${\text{Gal}}(\mathbf {F} _{q^{n}}/\mathbf {F} _{q})\cong \mathbb {Z} /n\mathbb {Z} $ where the automorphisms are given by the Frobenius endomorphism, the Galois group of the algebraic closure of $\mathbf {F} _{q}$ is given by the inverse limit of the groups $\mathbb {Z} /n\mathbb {Z} $, so its Galois group is isomorphic to the group of profinite integers[5]
$\operatorname {Gal} ({\overline {\mathbf {F} }}_{q}/\mathbf {F} _{q})\cong {\widehat {\mathbb {Z} }}$
which gives a computation of the absolute Galois group of a finite field.
Relation with Etale fundamental groups of algebraic tori
This construction can be re-interpreted in many ways. One of them is from Etale homotopy theory which defines the Etale fundamental group $\pi _{1}^{et}(X)$ as the profinite completion of automorphisms
$\pi _{1}^{et}(X)=\lim _{i\in I}{\text{Aut}}(X_{i}/X)$
where $X_{i}\to X$ is an Etale cover. Then, the profinite integers are isomorphic to the group
$\pi _{1}^{et}({\text{Spec}}(\mathbf {F} _{q}))\cong {\hat {\mathbb {Z} }}$
from the earlier computation of the profinite Galois group. In addition, there is an embedding of the profinite integers inside the Etale fundamental group of the algebraic torus
${\hat {\mathbb {Z} }}\hookrightarrow \pi _{1}^{et}(\mathbb {G} _{m})$
since the covering maps come from the polynomial maps
$(\cdot )^{n}:\mathbb {G} _{m}\to \mathbb {G} _{m}$
from the map of commutative rings
$f:\mathbb {Z} [x,x^{-1}]\to \mathbb {Z} [x,x^{-1}]$
sending $x\mapsto x^{n}$
since $\mathbb {G} _{m}={\text{Spec}}(\mathbb {Z} [x,x^{-1}])$. If the algebraic torus is considered over a field $k$, then the Etale fundamental group $\pi _{1}^{et}(\mathbb {G} _{m}/{\text{Spec(k)}})$ contains an action of ${\text{Gal}}({\overline {k}}/k)$ as well from the fundamental exact sequence in etale homotopy theory.
Class field theory and the profinite integers
Class field theory is a branch of algebraic number theory studying the abelian field extensions of a field. Given the global field $\mathbb {Q} $, the abelianization of its absolute Galois group
${\text{Gal}}({\overline {\mathbb {Q} }}/\mathbb {Q} )^{ab}$
is intimately related to the associated ring of adeles $\mathbb {A} _{\mathbb {Q} }$ and the group of profinite integers. In particular, there is a map, called the Artin map[6]
$\Psi _{\mathbb {Q} }:\mathbb {A} _{\mathbb {Q} }^{\times }/\mathbb {Q} ^{\times }\to {\text{Gal}}({\overline {\mathbb {Q} }}/\mathbb {Q} )^{ab}$
which is an isomorphism. This quotient can be determined explicitly as
${\begin{aligned}\mathbb {A} _{\mathbb {Q} }^{\times }/\mathbb {Q} ^{\times }&\cong (\mathbb {R} \times {\hat {\mathbb {Z} }})/\mathbb {Z} \\&={\underset {\leftarrow }{\lim }}\mathbb {(} {\mathbb {R} }/m\mathbb {Z} )\\&={\underset {x\mapsto x^{m}}{\lim }}S^{1}\\&={\hat {\mathbb {Z} }}\end{aligned}}$
giving the desired relation. There is an analogous statement for local class field theory since every finite abelian extension of $K/\mathbb {Q} _{p}$ is induced from a finite field extension $\mathbb {F} _{p^{n}}/\mathbb {F} _{p}$.
See also
• p-adic number
• Ring of adeles
• Supernatural number
Notes
1. Lenstra, Hendrik. "Profinite number theory" (PDF). Mathematical Association of America. Retrieved 11 August 2022.
2. Connes & Consani 2015, § 2.4.
3. K. Conrad, The character group of Q
4. Questions on some maps involving rings of finite adeles and their unit groups.
5. Milne 2013, Ch. I Example A. 5.
6. "Class field theory - lccs". www.math.columbia.edu. Retrieved 2020-09-25.
References
• Connes, Alain; Consani, Caterina (2015). "Geometry of the arithmetic site". arXiv:1502.05580 [math.AG].
• Milne, J.S. (2013-03-23). "Class Field Theory" (PDF). Archived from the original (PDF) on 2013-06-19. Retrieved 2020-06-07.
External links
• http://ncatlab.org/nlab/show/profinite+completion+of+the+integers
• https://web.archive.org/web/20150401092904/http://www.noncommutative.org/supernatural-numbers-and-adeles/
• https://euro-math-soc.eu/system/files/news/Hendrik%20Lenstra_Profinite%20number%20theory.pdf
| Wikipedia |
Set of All Groups
In my undergraduate Group Theory class, while discussing the impossibility of equivalence relations on groups, my professor said that the set of all groups does not exist.
Are there any group subcategories for which the set of all such groups DO exist? As examples,
i) Does the set of all cyclic groups exist?
ii) Does the set of all finite groups exist?
iii) Does the set of all groups of order n, for some integer n, exist?
group-theory logic set-theory
Colin SoleimColin Soleim
$\begingroup$ math.stackexchange.com/questions/226413/… $\endgroup$ – Asaf Karagila♦ Nov 14 '12 at 19:00
$\begingroup$ More generally, a collection of groups being a set requires that there is a set $S$ such that the underlying set of every group in your collection is contained in this set. $\endgroup$ – Thomas Andrews Nov 14 '12 at 19:30
$\begingroup$ Tarksi-Grothendieck set theory is quite convenient because it allows you to work inside universes. The set of all groups in one universe is not an element of that universe, true - but, this set certainly exists, and its an element of the next universe. $\endgroup$ – goblin GONE May 31 '13 at 2:57
Yes and no. The underlying set of a trivial group can be any one-element set and the class of all one-element sets is a proper class (i.e. not a set). However, if we consider only groups up to isomorphism, then the set of cyclic groups exists (there is one for each natural numbre and there is $\mathbb Z$). In the same sense the set of groups of order $n$ (or more easily: The set of operations on a given set of $n$ elements such that this makes it a group) exists, and also the set of all finite groups.
Hagen von EitzenHagen von Eitzen
$\begingroup$ To give an example, the group $S_\infty$ (the set of all permutations of $\Bbb N$ that move only finitely many elements) contains an isomorphic copy of every finite group, which is a corollary of Cayley's Theorem. To get a group that contains an isomorphic copy of every cyclic group, one can consider $S_\infty * \Bbb Z$. $\endgroup$ – user123641 May 23 '14 at 14:56
Not the answer you're looking for? Browse other questions tagged group-theory logic set-theory or ask your own question.
Why is the collection of all groups a proper class rather than a set?
Does the set of all fields exist ?
Does there exist an abelian $2$-group of finite exponent that is not a direct sum of cyclic groups?
References on the theory of $2$-groups.
Why are all cyclic groups countable?
Showing that the Class of Cyclic Groups Aren't Axiomatizable
On the meaning of "Class of finite groups".
Does there exist an infinite non-abelian group such that all of its proper subgroups become cyclic?
Finite groups with an element of maximum order
What is the difference between cyclic groups and periodic groups?
Axiomatisable Groups | CommonCrawl |
Period of oscillation through a hole in the earth
Special mention to the QI episode that kicked this off: Anyway, the host points out that a tunnel that connects a pair of points on the earth's surface can be thought of as a gravity train - where the force of gravity along the tunnel allows an object to fall through it and emerge on the other side. Any force perpendicular to the tunnel (in case the tunnel doesn't pass through the center of the earth) is ignored.
I worked it out and the equation is $a = -\frac{4}{3}\pi\rho G d$, where $d$ is the distance from the center of the tunnel and $\rho$ is the density of the earth.
Clearly, it is simple harmonic and therefore the period is constant and has no dependence on which two points were used to make the tunnel.
Does anyone have an intuition for why this should be the case? I imagine some Gauss' law type of argument should work here, but I cannot see it.
EDIT: More to the point, why even when the tunnel does not pass through the center, one obtains the same period. Is there any deeper explanation as to why this should be the case?
newtonian-mechanics newtonian-gravity gauss-law
$\begingroup$ I presume you are asking about the case there the hole does not pass through the centre of the Earth. This problem is explained mathematically in the wikipedia article for Gravity Train, although no "intuitive" explanation is provided. $\endgroup$ – sammy gerbil Oct 16 '16 at 12:54
$\begingroup$ Possible duplicates: physics.stackexchange.com/q/7346/2451 , physics.stackexchange.com/q/18446/2451 and links therein. $\endgroup$ – Qmechanic♦ Oct 16 '16 at 13:11
$\begingroup$ Not exactly intuitive, but perhaps it would help to look the energy considerations (remember you can get the time period by differentiating the energy equation). Inside the sphere, the potential( and hence energy) varies as square of the distance; so when this energy 'changes' as the particle moves, you get a linear variation in the potential energy term, which is basically the term which determines furthur motion. And that is a characteristic of SHM. $\endgroup$ – GRrocks Oct 16 '16 at 15:30
$\begingroup$ Or dimensional considerations; you can easily see that the relevant physical quantities are $G$ having units $\text{m}^3 / (\text{kg}~\text{s}^2)$, the density of the planet ${\text{kg}/m^3}$ and the radius $R$. Given these alone (i.e. trajectories through the center of the sphere) there are no dimensionless constants and the only way to get a time (i.e. a period) is $1/\sqrt{\rho G}.$ I'm not 100% sure if there's an insight to the effect that "passing through a chord not crossing the center is just like passing through the center of a planet with equivalent density," but it sounds plausible. $\endgroup$ – CR Drost Dec 12 '16 at 6:16
$\begingroup$ Coincidently the same time is taken by an earth satellite to complete one revolution in a low orbit around the earth. $\endgroup$ – Jatin Dec 17 '16 at 10:30
As non-intuitive a sit may seem, the period of a simple spring is independent of how far you pull or push it. How could the time it takes for one oscillation be independent of the amplitude?
Imagine a vertical spring with a bob, like so:
Now, the first picture shows the spring displaced by y units, so that $$ky=mg$$
The second image shows it being diplaced in the x direction as well so that $ky=mg$ still holds, but the spring force is a vector along the length of the spring so there is an unbalanced component in the x direction$=kx$
The exact same thing is happening in the tunnel. The force on a mass in the tunnel is along the displacement vector from the center of earth to the mass, but the component perpendicular to the tunnel is balanced by the normal force from the walls of the tunnel, and only the component of the displacement vector parallel to the tunnel is responsible for accelerating the mass.
So where does that leave us? Pretend that there is a spring along the length of the tunnel. Irrespective of which chord (read: independent of amplitude) the tunnel is in a circular cross-section (refer to Naveen Balaji's picture) the spring will have the same period. You can do the same thing for the spring with the bob: pretend that there is a spring along the x-axis.
I hope this gives you some intuition.
GeeJayGeeJay
The fact that the period of the gravity train that passes through the center of the Earth is equal to the period of an orbit that skims the Earth's surface is not a coincidence. Consider a polar orbit around Earth (i.e., one that passes directly over the coordinate north and south poles). Now, consider only the north-south motion of the satellite by projecting the motion onto a line parallel with the Earth's axis. What kind of motion is this? Circular orbits have a constant speed, so the 1-D motion must be sinusoidal. The gravity train is just a circular orbit where the motion off the Earth's axis is restricted. Perpendicular forces and motions can be treated independently. See the animation below to illustrate. The spinning arrow shows the path of a satellite, while the straight lines show the path of the gravity train for perpendicular tracks.
Now, what about gravity trains that don't pass through the Earth's center? First, notice that your expression for the acceleration of the train does not depend on the distance from the Earth's center. As long as the track is symmetric about the Earth's radius, then you will get the same train motion no matter the depth of the track. The ends of the track do not have to connect to the surface. You can also reason that the depth of the gravity train track does not matter by starting with a track that connects to the surface at both ends and then adding a shell around the entire planet to increase its radius. Inside a spherical shell, the gravitational force is zero, so burying the track does nothing to the motion.
To start to demonstrate this, let's prove a similar fact about circular orbits: the period of an orbit that skims a planet's surface depends only on the planet's density, not its size. $$F = m\frac{v^2}{R} = \frac{GMm}{R^2}$$ where $F$ is the gravitational force, $m$ is the mass of the satellite, $M$ is the mass of the planet, $v$ is the speed of the orbit, $R$ is the radius of the planet, and $G$ is the gravitational constant. $$v^2 = \frac{GM}{R}$$ $$\left(\frac{2\pi{}R}{T}\right)^2 = \frac{GM}{R}$$ where $T$ is the period of the orbit. $$T = \sqrt{\frac{4\pi{}^2R^3}{GM}}$$ $$T = \sqrt{\frac{4\pi{}^2R^3}{G\rho\frac{4}{3}\pi{}R^3}}$$ $$T = \sqrt{\frac{3\pi}{G\rho}}$$ where $\rho$ is the density of the planet.
Now, starting from your expression for the train acceleration: $$a = -\frac{4}{3}\pi\rho{}Gd$$ we can derive an equivalent mass-spring system(*) with a spring constant $k$ given by $$k = \frac{F}{d} = \frac{ma}{d} = \frac{4}{3}\rho{}Gm.$$ The period of this mass-spring system, and thus of the train, is $$T = 2\pi\sqrt{\frac{m}{k}} = \sqrt{\frac{4\pi^2m}{\frac{4}{3}\pi\rho{}G}} = \sqrt{\frac{3\pi}{\rho{}G}}$$ Notice that this is the same period as the satellite.
TL;DR: The gravity train is a 1D projection of the 2D circular orbit where the length of the train track is the same as the diameter of the orbit. The time to traverse the track is the same no matter the length or the depth because the period of a surface-skimming orbit around a constant-density planet is independent of the size of the planet.
(*) Everything is physics is ultimately a mass on a spring.
Mark HMark H
Assuming Earth to be a uniform sphere of mass M and radius R. Now constructing a tunnel which connects any two points on its surface. Suppose at some instant the particle is at radial distance r from the centre of earth, O. Since the particle is constrained to move along the tunnel, let us define its position as distance x from C. Hence, the equation of motion of the particle is,
$$ma_{x}=F_{x}$$
The gravitational force on mass m at distance r is,
$$F=\frac{GMmr}{R^3}$$ (towards O, the centre of the Earth)
$$F_{x}=-Fsin\theta$$ $$=\frac{-GMmr}{R^3}\frac{x}{r}$$ $$=\frac{-GMmx}{R^3}$$
Since, $F_{x}\alpha-x$, it's motion is simple harmonic in nature. Further,
$$ma_{x}=\frac{GMmx}{R^3}$$ $$a_{x}=\frac{GMx}{R^3}$$ Hence the time period of oscillation is ,
$$T=2\pi\frac{\sqrt{x}}{\sqrt{a_{x}}}$$ $$T=2\pi\frac{\sqrt{R^{3}}}{\sqrt{GM}}$$ Hence time take for a particle to go from one end to another is,
$$t=\frac{T}{2}=\pi\frac{\sqrt{R^{3}}}{\sqrt{GM}}= 2530.496126seconds=42.17mins$$
So a particle or a person of mass m would tunnel 34 times a day!
It would be interesting to do the same for the original shape of Earth and see the deviation of the time taken for a particle to go through the tunnel.
Naveen BalajiNaveen Balaji
$\begingroup$ I do not think that the OP is asking for a solution...he wants to know the physical meaning behind the answer. $\endgroup$ – GRrocks Oct 16 '16 at 15:25
$\begingroup$ @GRrocks I have proved that for the tunnel that need not pass via the centre of the earth, the time period remains the same due to the action of the restoring force that acts upon the particle. $\endgroup$ – Naveen Balaji Oct 16 '16 at 15:34
$\begingroup$ That has been already mentioned in the question $\endgroup$ – GRrocks Oct 16 '16 at 15:44
EDIT: Here is an attempt at an intuitive explanation for the special case when the tunnel passes through the center of the earth. For the general case, refer to the other solutions that have been provided.
Think of the solid sphere of earth to be composed of many thin concentric hollow shells each of same density $\rho$. Now if the train is at a distance d from the center of the earth, the gravitational force exerted by the shells having radius greater than d on the train will be zero(The train is in the inside of these shells). Only the shells having radius less than d will exert a force on the train. The total mass of these shells will be proportional to $d^3$ (as these shells form a sphere of radius d). The center of this sphere is at a distance d from the train.
Now $\hspace{5cm}$ g $\propto M/d^2$
And as we have seen $\hspace{5cm}$ M $\propto d^3$
we get$\hspace{5cm}$ g = kd for some constant of proportionality d.
This means that the acceleration due to gravity is proportional to the distance of the train from the center of the earth and this is exactly the kind of acceleration that we get in the case of a linear oscillator.
Abhijeet MelkaniAbhijeet Melkani
$\begingroup$ The OP already derived this relation; s/he is looking for an 'intuitive', likely qualitative, rationalisation for why the oscillation should be simple harmonic. $\endgroup$ – lemon Oct 16 '16 at 13:20
$\begingroup$ He seems to be asking for some Gauss' law type of argument. That's what I tried to give. $\endgroup$ – Abhijeet Melkani Oct 16 '16 at 13:31
$\begingroup$ You have assumed the tunnel goes through the center. The final answer a ~ d does not require that. Also, please note the edit in the question. $\endgroup$ – user1936752 Oct 16 '16 at 13:49
$\begingroup$ @user1936752 okay edited. $\endgroup$ – Abhijeet Melkani Oct 16 '16 at 16:29
Picture instead a frictionless pendulum with two beads attached that are constrained to remain at two different, fixed heights. Their period remains the same as in the case of beads glued to the pendulum shaft, only now, their motion is restricted to one dimension. While the situation involving Earth and gravity seems to involve more dimensions, the relevant components reduce to this same problem once you've established the linear relationship through the center of the planet.
Josh McKJosh McK
Not the answer you're looking for? Browse other questions tagged newtonian-mechanics newtonian-gravity gauss-law or ask your own question.
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Can i use gravity to represent gravitational force? | CommonCrawl |
cancellation laws in a Ring
There's a theorem that states that cancellation laws hold in a ring $R$ if and only if $R$ has no zero divisors. Note that Integral Domains have no zero divisors. However, from my understanding in group theory, cancellation law happens by multiplying the (multiplicative) inverse on both sides, i.e. $$a^{-1}\cdot ab=a^{-1}\cdot ac\implies b=c.$$ Equivalently, $$ba\cdot a^{-1}=ca\cdot a^{-1}\implies b=c.$$ Going back to rings, this feels counterintuitive as this is only possible if all elements have a multiplicative inverse. But take note that Integral Domains are not necessarily division rings. So how does cancellation exactly work in rings?
Our professor said we should not multiply the inverse on both sides since we're not guaranteed that a multiplicative inverse exists for all elements of a ring $R$. So for the rest of the discussion, she only canceled terms without explicitly stating why it happens.
abstract-algebra ring-theory
Liam CeasarLiam Ceasar
$\begingroup$ Cancellation means you can cancel, it doesn't mean that it HAD to happen by operating by inverses. $\endgroup$ – Randall May 21 '18 at 16:32
Suppose $ab=ac$. This means that
$$a(b-c)=0$$
but if the ring is an integral domain, either $a=0$ or $b-c=0$. If $a\neq0$, this implies that $b=c$.
Consider the example of the polynomial ring ${\Bbb R}[x]$. Since it is an integral domain, it holds for a non-zero polynomial $p$ that $$ pq_1=pq_2\quad\Rightarrow\quad q_1=q_2, $$ however, an inverse of the polynomial $p$ does not exist.
A.Γ.A.Γ.
I would like to illustrate that with an example: take for example the integers $modn$ with n composite, then if $n=6$ for example you can see that $2x=0mod6$ does not imply $x=mod6$ but it could also mean $x=3$. This is because you have zero divisors in that ring.
Μάρκος ΚαραμέρηςΜάρκος Καραμέρης
For variety....
Every integral domain $D$ has a field of fractions $F$, and $D$ is a subring of $F$. (The general procedure for constructing rings of fractions is called localization)
If $ab = ac$ holds for $a,b,c \in D$, then in $F$ we have $\frac{ab}{1} = \frac{ac}{1}$. If $a \neq 0$, then $\frac{1}{a} \in F$, and multiplying through and applying the fact that $\frac{xy}{x} = \frac{y}{1}$ gives an equation $\frac{b}{1} = \frac{c}{1}$. Since $D$ is a subring of $F$, this implies $b=c$.
HurkylHurkyl
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State for each of the rings whether or not it is an integral domain and/or field | CommonCrawl |
Transactions of the American Mathematical Society
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Polyhedral realization of the highest weight crystals for generalized Kac-Moody algebras
by Dong-Uy Shin PDF
Trans. Amer. Math. Soc. 360 (2008), 6371-6387 Request permission
In this paper, we give a polyhedral realization of the highest weight crystals $B(\lambda )$ associated with the highest weight modules $V(\lambda )$ for the generalized Kac-Moody algebras. As applications, we give explicit descriptions of crystals for the generalized Kac-Moody algebras of ranks 2, 3, and Monster algebras.
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Dong-Uy Shin
Affiliation: Department of Mathematics Education, Hanyang University, Seoul 133-791, Korea
Email: [email protected]
Received by editor(s): December 11, 2005
Received by editor(s) in revised form: November 8, 2006
Published electronically: July 28, 2008
Additional Notes: This research was supported by the research fund of Hanyang University (HY-2007-000-0000-5889).
The copyright for this article reverts to public domain 28 years after publication.
Journal: Trans. Amer. Math. Soc. 360 (2008), 6371-6387
MSC (2000): Primary 81R50; Secondary 17B37
DOI: https://doi.org/10.1090/S0002-9947-08-04446-2 | CommonCrawl |
Riesz representation theorem
Revision as of 08:33, 22 July 2012 by Camillo.delellis (talk | contribs) (Added the characterization of the dual of C(X,B) when B is finite dimensional.)
2010 Mathematics Subject Classification: Primary: 28A33 [MSN][ZBL]
A central theorem in classical measure theory, sometimes called Riesz-Markov theorem, which states the following. Let $X$ be a compact Hausdorff topological space, $C(X)$ the Banach space of real valued continuous functions on $X$ and $L: C(X)\to \mathbb R$ a continuous linear functional which is nonnegative, i.e. such that $L(f)\geq 0$ whenever $f\geq 0$. Then there is a Radon measure $\mu$ on the $\sigma$-algebra of Borel sets $\mathcal{B} (X)$ such that \[ L (f) = \int_X f\, d\mu \qquad \forall f\in C (X)\, . \]
An analogous statement which is commonly referred to as Riesz representation theorem is that, under the assumptions above, the dual of $C(X)$ is the space $\mathcal{M}^b (X)$ of $\mathbb R$-valued measures with finite total variation (cp. with Convergence of measures for the relevant definitions). Combined with the Radon-Nikodým theorem, this amounts to the following alternative statement: for any element $L\in (C(X))'$ there are a Radon measure $\mu$ and a Borel function $g$ such that $|g|=1$ $\mu$-a.e. and \[ L (f) = \int_X fg\, d\mu\qquad \forall f\in C(X)\, . \]
More general statements for locally compact Hausdorff spaces can be easily derived from the ones above.
The statement can be also generalized to a similar description of the dual of $C(X,B)$ when $B$ is Banach space. For instance, if $B$ is a finite-dimensional space, then for any $L\in C(X,B)'$ there is a Radon measure $\mu$ on $X$ and a Borel measurable map $g: X\to B'$ such that $\|g\|_{B'}=1$ $\mu$-a.e. and \[ L (f) = \int_X g (f)\, d\mu \qquad \forall f\in C(X)\, . \]
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Measure and integration
Classical measure theory | CommonCrawl |
\begin{document}
\title{Finite de Finetti theorem for conditional probability distributions describing physical theories}
\author{Matthias Christandl}
\email{[email protected]} \affiliation{Centre for Quantum Computation, Department of Applied
Mathematics and Theoretical Physics, University of Cambridge,
Wilberforce Road, Cambridge CB3~0WA, United Kingdom} \affiliation{Arnold Sommerfeld Center for Theoretical Physics, Faculty of Physics, Ludwig-Maximilians-Universit{\"a}t M{\"u}nchen, Theresienstrasse 37, 80333 Munich, Germany}
\author{Ben Toner} \email{[email protected]} \affiliation{School of Physics, The University of Melbourne, Victoria 3010, Australia} \affiliation{Centrum voor Wiskunde en Informatica, Kruislaan 413, 1098
SJ Amsterdam, The Netherlands} \affiliation{Institute for Quantum Information, California Institute of Technology, Pasadena CA 91125, USA}
\begin{abstract} We work in a general framework where the state of a physical system is defined by its behaviour under measurement and the global state is constrained by no-signalling conditions. We show that the marginals of symmetric states in such theories can be approximated by convex combinations of independent and identical conditional probability distributions, generalizing the classical finite de~Finetti theorem of Diaconis and Freedman. Our results apply to correlations obtained from quantum states even when there is no bound on the local dimension, so that known quantum de~Finetti theorems cannot be used. \end{abstract} \maketitle
\section{Introduction} Given a bowl containing $n$ colored balls, we wish to compare two ways of obtaining a random sample of $k\leq n$ balls: (i) we randomly choose a ball, replace it with a ball of the same color, and repeat this step $k$ times; (ii) we do the same but don't replace the balls. If $k \ll n$, then the probability of obtaining a particular set of $k$ balls will be almost the same in both cases~\cite{diaconis80:_finit}. This observation has profound consequences for Bayesian statistical inference, as we now describe.
Suppose we perform an experiment $k$ times in order to estimate some physical quantity, e.g., the probability $\lambda$ that a muon decays in a given time. Let $A_i =1$ if the $i$th muon decayed and $A_i=0$ if it did not. If we assume that the results of the experiments are independent, we can posit some prior probability distribution $m(\lambda)$ and analyze our data by updating this probability distribution as more data arrives. Statisticians of de Finetti's subjective school~\cite{bernardo94:_bayes_theor} are not willing to accept this assumption, however, since for them all probability distributions should be subjective degrees of belief, which $m(\lambda)$ is not. Instead, they make the weaker assumptions that the experiment could have been performed $n \gg k$ times and that there was nothing special about the experiments actually performed. These assumptions, together with the observation about colored balls above, can be shown to imply that there exists a distribution $m(\lambda)$ such that \begin{align} P[A_1, \ldots, A_k] \approx \int dm(\lambda)
P_\lambda[A_1]\cdots P_\lambda[A_k], \end{align} i.e., the probability distribution $P[A^k]$ behaves \emph{as if} the experiments really were independent and there really were some objective prior $m(\lambda)$. This is a statement of the famous \emph{de Finetti representation
theorem}~\cite{finetti37:_la,diaconis80:_finit}. Our results establish the same correspondence for measurement results in a more general, probabilistic, physical theory, where the state of a system is described by a conditional probability distribution.
We now give a brief description of the setting and our results; precise definitions are given later on. A physical system in a probabilistic physical theory is made up of a number of---in our case identical---subsystems, called \emph{particles}. On each particle different measurements from a set $\cX$ can be performed and outputs from a set $\cA$ are obtained. The state of a particle is specified by a \emph{conditional probability
distribution} $P[A|X]$: the probability of obtaining result $a$ when performing measurement $x$ is given by $P[A=a|X=x]$. The possible states of $n$ particles are the conditional probability distributions $P[A^n|X^n]$ that obey a \emph{no-signalling} property, which ensures that the reduced state on a subset of the particles is always well-defined.
Our main result is that the joint state $P[A^k|X^k]=P[A_1 \cdots A_k|X_1 \cdots X_k]$ of $k$ particles randomly chosen from $n$ particles---or equivalently, the state of the first $k$ particles of a permutation-invariant state of $n$ particles---can be approximated by a convex combination of identical and independent conditional probability distributions,
\begin{align} \label{box-approx} P[A^k|X^k] \approx \int dm(\lambda) P_\lambda[A|X]^{\times k} \end{align}
and that the error in the approximation is bounded by $|\cX|k(k-1)/n$ in the appropriate distance measure, where $|\cX|$ is the number of different possible measurements.
(We write $P_\lambda[A|X]^{\times k}$ for $P_\lambda[A_1|X_1] \cdots P_\lambda[A_k|X_k]$.)
Our result generalizes the finite de Finetti theorem of Diaconis and Freedman, who proved for classical probability distributions ($|\cX|=1$) that the error in the approximation is no more than $k(k-1)/n$~\cite{diaconis80:_finit}~\footnote{
Diaconis and Freedman also obtain a second bound $ k|\cA|/n$. The analogous bound within our framework is $k |\cA|^{|\cX|}/n$. Restricting to \emph{adaptive} measurements on individual particles, we are able to improve this bound to $k |\cX|^2|\cA|(1+4\sqrt{\frac{2+\log |\cX|}{k}})/n $~\cite{benmat:long}.}.
This paper is motivated by recent work on finite \emph{quantum} de Finetti theorems, i.e., statements of the form \begin{align} \label{quantum-approx}\rho^k \approx \int d\sigma \ \sigma^{\otimes k}, \end{align} where $\rho^k$ is the $k$-particle reduced density matrix of a permutation-invariant density matrix of $n$ particles with state space of dimension $d$, where the error is at most $4d^2k/n$ in the trace distance~\cite{Koenig:04a,
christandl06:_one}~\footnote{ A density operator $\rho^n$ is
permutation-invariant if $\rho^n=\pi \rho^n \pi^{-1}$ for all permutations
$\pi \in S_n$.}. In fact, it is necessary that the error depends on $d$~\cite{christandl06:_one}, and so the quantum de Finetti is not useful in applications where $d$ cannot be bounded. Our results are designed to apply in this setting: provided we have a bound on the number of ways $|\cX|$ that a system is measured, the approximation in Eq.~\eqref{box-approx} will be good, even if there is no bound on the local dimension $d$.
In recent years, quantum de Finetti theorems, especially Renner's so-called `exponential' version~\cite{renner05:_secur}, have been used to prove the security of quantum key distribution (QKD) schemes~\cite{QKD}. At the same time, attempts have been made to lift the assumption of a fixed (finite) local dimension~\cite{acin:230501}. Since quantum de Finetti theorems are necessarily dimension-dependent, they cannot be used in this setting. Although our theorems do not directly lead to security proofs either, we regard them as a first step towards this goal.
We also prove a finite quantum de Finetti theorem for separable $\rho^n$: in this case there is an approximation of the form in Eq.~\eqref{quantum-approx}
with error $k(k-1)/n$, \emph{independent} of the dimension. We do not, however, know whether our techniques can be extended to prove the finite quantum de Finetti theorem in full generality. The issue is that our theorem concerns conditional probability distributions that arise from measuring quantum states and not the quantum states themselves. If we take, for example, a tomographically complete set of measurements, the representation described in Eq.~\eqref{box-approx} will in general contain distributions $P_\lambda[A|X]$ that cannot be obtained by performing the tomographic measurements on quantum states. One can, however, apply the argument of~\cite{caves:4537} to obtain the infinite quantum de Finetti theorem and indeed an infinite de Finetti theorem for any physical theory in what is known as the \emph{convex sets framework}~\cite{Barnum:06,Barnum:07} (see~\cite{BarrettLeifer06} for the details).
Another application of our work is to the study of classical channels.
Fuchs, Schack and Scudo have used the Jamiolkowski isomorphism to transfer the infinite quantum de Finetti theorem ($n=\infty$, $k<\infty$)~\cite{stormer69:_symmet_states_infin_tensor_produc_c,caves:4537} to quantum channels~\cite{fuchs:062305}. Since a conditional probability distribution can be viewed as a classical channel with probability distributions as input and output, our results also provide a de Finetti theorem for classical channels.
\emph{Outline.}---Our first task is to define an appropriate distance measure on states of $k$ particles in probabilistic theories, in order to quantify the error in Eq.~\eqref{box-approx}. The distance between states should bound the probability of distinguishing them by measurement, and so we need to be clear about what measurement strategies are allowed. One possibility, which we explore in~\cite{benmat:long}, is to restrict to strategies where each of the $k$ particles is measured individually. But when the conditional probability distributions arise from making informationally complete local measurements on entangled quantum states, the resulting distance measure fails to bound the trace distance between the quantum states. In the next section we show how to define a `good' distance measure in which all noncontextual measurements are allowed, including all joint quantum-mechanical measurements. We then state and prove our results. In the last section, we explain the origin of the distance measure, the convex sets framework, which allows us to conclude with an open question on finite de Finetti theorems in this more general setting.
\section{A distance measure for conditional probability distributions} \label{sec:dist-meas-cond}
When we measure a quantum system, the probability of obtaining an outcome $a \in \cA$ depends on which measurement $x \in \cX$ we choose to perform on the system. It is usual to describe a quantum system using the formalism of density matrices, Hilbert spaces, and so on, but we can also describe the system by specifying a conditional probability distribution $P[A|X]$, where we write $P[A|X=x]$ for the distribution of measurement outcome $A$ given that measurement $x$ is performed~\footnote{In quantum theory, the most general measurement is termed a positive operator-valued measure (POVM). If we perform a POVM $x$ with effects $E_{x,a}$ (satisfying $E_{x,a}= E_{x,a}^\dagger$, $E_{x,a}\succeq 0$ and $\sum_a E_{x,a} = \leavevmode\hbox{\small1\normalsize\kern-.33em1}$) on a system in state $\rho$, then the distribution of the measurement outcome $A$ is given by
$P[A=a|X=x] = \tr \left(\rho E_{x,a}\right)$.}. While a classical system can be described using an \emph{unconditional} probability distribution, the same is not true for a quantum system, since measuring a quantum system disturbs it, eliminating our ability to make a second, incompatible, measurement on the same system.
We are therefore motivated to describe the state of an abstract system
(not necessarily obeying quantum theory) using a conditional probability distribution $P[A|X]$. We view the conditional probability distribution $P[A|X]$ as the output distribution of a measurement that has been performed on system $A$. Alternatively, one can view $P[A|X]$ as a channel that produces an output distribution $P[A|X=x]$ on input $x$. For this reason we refer to the measurement setting $x$ as the \emph{input} and the measurement result $a$ as the \emph{output}.
Generalizing from conditional probability distributions of one system, we shall consider a conditional probability distribution $P[A^n|X^n]=P[A_1\cdots A_n|X_1 \cdots X_n]$, which describes an abstract system composed of $n$ subsystems, which we call particles.
We need to be able to describe the state of a subset $\cI \subset \{1,
\ldots, n\}$ of the particles. Taking the marginal of a conditional probability distribution $P[A^n|X^n]$ yields a conditional distribution $P[A_{\cI}|X^n]$, where the outputs at the particles in $\cI$ depends on the inputs at all $n$ sites. In order to trace out the particles that are not in $\cI$ entirely, rather than just the outputs obtained from measuring them, we need another notion, that of a conditional probability distribution being \emph{no-signalling}. \begin{definition}
A conditional distribution $P[A^n|X^n]$ is \emph{no-signalling} if for all subsets $\cI \subset \{1, \ldots, n\}$ with complements $\bar\cI:=\{1, \ldots, n\} \backslash \cI$ \begin{equation} \label{eq:10}
P[A_{\cI} = a_{\cI} |X_{\cI} = x_{\cI}] \mathrel{\mathop:}= \sum_{a_{\bar{\cI}}} P[A^n = a^n|X^n=x^n] \end{equation} is independent of $x_{\bar\cI}$ for all $a_{\cI}$ and all $x_{\cI}$. \end{definition} The terminology derives from the following fact:
if we divide the $n$ parties into two groups, $\cI$ and $\bar\cI$, then, provided $P[A^n|X^n]$ is no-signalling, it is impossible for the group $\cI$ to send a signal to the group of $\bar\cI$ just by changing their inputs. Not all conditional probability distributions are no-signalling; for example, $P[A_1=a_1,A_2=a_2|X_1=x_1,X_2=x_2] = [a_1=x_2][a_2=x_1]$ (where $[t]$ is $1$ if $t$ is true and $0$ otherwise) is signalling. We note that any conditional probability distribution that arises from making local measurements on a quantum state is no-signalling. The no-signalling requirement is the minimal assumption necessary to ensure that state of any subset of particles is well-defined.
The goal of this paper is to approximate by product distributions a no-signalling conditional probability distribution on $k$ particles arising from a symmetric conditional probability distribution on $n$ systems, so we need to introduce a notion of distance for conditional probability distributions. This distance measure should generalize the classical variational distance, which is equal to the maximum probability of distinguishing two probability distributions, and the quantum trace distance, which is equal to the maximal probability of distinguishing two quantum states. In order to define a \emph{trace distance} for no-signalling conditional probability distributions we therefore need to determine what measurement strategies can be used to distinguish two conditional probability distributions. In fact, there are three natural sets of measurement strategies for conditional probability distributions, each of which induces a distance measure on conditional probability distributions. We will work with the largest of these sets giving the strongest notion of a distance, for if we can show that two conditional probability distributions are almost indistinguishable using a particular set of measurements, it will trivially follow that they are also almost indistinguishable when only a subset of those measurements is allowed. Let us start by introducing the three sets.
An \emph{individual measurement} is a distribution $P[X^k]$ on the inputs that maps the conditional probability distribution to the {unconditional} probability distribution $P[A^kX^k] =
P[A^k|X^k]P[X^k]$. Such a measurement can be carried out by measuring each subsystem individually. Note that individual measurements also make sense if we drop the condition that $P[A^n|X^n]$ is no-signalling. Since we restrict to no-signalling conditional probability distributions, a larger class of measurements is possible and indeed needed for applications. Suppose the conditional distribution $P[A_1A_2|X_1X_2]$ is no-signalling. We start by writing \begin{align}
P[A_1A_2|X_1X_2=x_1x_2]&=P[A_1|X_1X_2=x_1x_2]P[A_2|A_1,X_1X_2=x_1x_2]\\
&=P[A_1|X_1=x_1]P[A_2|A_1,X_1X_2=x_1x_2], \end{align} where we made use of the no-signalling principle, Eq.~(\ref{eq:10}), in the second line. This provides an operational means to sample from
$P[A_1A_2|X_1X_2=x_1x_2]$: We first sample $a_1$ from the distribution
$P[A_1|X_1=x_1]$, then sample $a_2$ from
$P[A_2|A_1=a_1,X_1X_2=x_1x_2]$. The important point is that a no-signalling conditional probability distribution can provide the output on system 1 before specifying which input is chosen for system 2. Therefore the following \emph{adaptive measurement} on $P[A_1A_2|X_1X_2]$ is possible: Input $x_1$, obtain $a_1$, and choose an input $x_2=f(a_1)$, where $f: \cA \to \cX$ is an arbitrary function. Such a strategy can lead to a higher probability of distinguishing two no-signalling conditional probability distributions, compared to individual strategies~\footnote{Let $\cA = \cX = \{0,1\}$ and define
$P[A_1A_2=a_1a_2|X_1X_2=x_1x_2] = \frac12\, [a_1 + a_2 = x_1 \and x_2 \pmod 2].$ This distribution is known as a nonlocal box~\cite{Popescu:94a} and one can easily check that it is no-signalling. We wish to distinguish this distribution from the distribution
$Q[A_1A_2|X_1X_2]$, defined by
$ Q[A_1A_2=a_1a_2|X_1X_2 = x_1x_2] = \frac{1}{2}\, [a_2=1].$ (This is an unconditional product distribution where the first bit is random and the second bit is always one.)
For every setting of $x_1$ and $x_2$, $P[A_2 = 1|X_1X_2=x_1x_2] = 1/2$, and thus $P$ and $Q$ cannot be perfectly distinguished by making a measurement on both systems in parallel. But if we allow adaptive strategies, then we can distinguish $P$ and $Q$ perfectly. For instance, set $x_1 = 1$ and then set $x_2=a_1$, so that we have $a_1+a_2 = 1.a_1 \pmod 2$ and it follows that $P[A_2=0] =1$. Since $Q[A_2=0]=0$, we conclude that we can distinguish $P$ and $Q$ perfectly.}.
As in most of the paper we draw intuition from quantum-mechanical correlations. It is a well-established fact that the distinguishability of quantum states depends on whether individual or adaptive measurement strategies are considered. In the quantum case, furthermore, it is possible to apply a joint measurement to all $k$ systems at once, a class of measurement which strictly contains adaptive
measurements and can lead to strictly higher distinguishability. \emph{Quantum data hiding} is an important application of this phenomenon~\cite{EggWer2002,hayden:062339}.
In defining joint operations on no-signalling conditional probability distributions, we essentially wish to allow all possible measurements whose outcomes behave like probability distributions. Motivated by this, we think of a no-signalling conditional probability distribution $P[A^k|X^k]$ as a vector in a real $|\cA|^k|\cX|^k$-dimensional space and consider linear functions from this space to a real ${|\cA|^k}$-dimensional space. The set of \emph{general measurements} is the set of linear functions $M$ such that $M(P[A^k|X^k])$ is a probability distribution for all no-signalling conditional probability distributions $P[A^k|X^k]$. Clearly, individual and adaptive strategies belong to the set of general measurements, but it includes strictly more strategies, too. (The assumption of linearity is necessary so that our probability behave reasonably when we take convex combinations of states and measurements; see Ref.~\cite{Barrett:05}.)
\begin{definition} \label{sec:introduction-general}\label{def:tracedistance}
The \emph{trace distance} between two no-signalling conditional probability distributions $P[A^k|X^k]$ and $Q[A^k|X^k]$ is given by \begin{align}
\|P[A^k|X^k]- & Q[A^k|X^k]\| \mathrel{\mathop:}= \sup_{M}
\|M(P[A^k|X^k])-M(Q[A^k|X^k]) \|.\label{eq:18} \end{align}
where the supremum is taken over all general measurements and $\|R[B] - S[B] \|$ is the classical variational distance for probability distributions $R[B]$ and $S[B]$ on system $B$. Extending the definition by imposing linearity, $|| \cdot ||$ is a norm on the space of (real) linear combinations of conditional probability distributions and hence obeys the triangle inequality. \end{definition}
A theory in which conditional probability distributions describe the state of a particle and where joint states of particles obey a no-signalling distribution can be treated in the \emph{convex sets framework}. The distance measure we introduced arises naturally in this framework. We review the convex sets framework in Section~\ref{sec:conv-sets-fram}. This will give us a broader view on de Finetti theorems and will allow us to pose an open question regarding de Finetti theorems in the convex sets framework.
\section{Our results} \label{sec:our-results}
Suppose we have a conditional probability distribution $P[A^n|X^n]$ describing $n$ particles. If we interchange the particles according to a permutation $\pi \in S_n$, the resulting conditional probability distribution is
\begin{align*} &\pi P[A^n=a_1 \cdots a_n| X^n=x_1 \cdots x_n]\\ &=
P[A^n=a_{\pi^{-1}(1)} \cdots a_{\pi^{-1}(n)}| X^n=x_{\pi^{-1}(1)} \cdots x_{\pi^{-1}(n)}]. \end{align*}
We say that a conditional probability distribution $P[A^n|X^n]$ is \emph{symmetric} if it is invariant under all permutations $\pi \in S_n$.
If
$|\cX| = 1$, this definition reduces to the usual definition of a symmetric probability distribution. We can now state our main result:
\begin{theorem}\label{theorem:1}
Suppose that $P[A^n|X^n]$ is a symmetric no-signalling conditional probability distribution. Then there exists a probability distribution $p_\lambda$ such that \begin{align} \begin{split}
\bigl\|P[A^k|X^k]- & \sum_\lambda p_\lambda P_\lambda[A|X]^{\times k} \bigr\| \leq
\min \biggl( \frac{2k|\cX||\cA|^{|\cX|}}{n}, \frac{|\cX|k(k-1)}{n}\biggr), \end{split} \end{align} where the distribution $p_\lambda$ is on a finite set of single-particle conditional probability distributions, labeled by $\lambda$. \end{theorem}
This establishes that the state of a random subset of $k$ out of $n$ particles is well approximated by a convex combination of independent and identically distributed conditional probability distributions. To prove Theorem~\ref{theorem:1}, we first show that if $P[A^n|X^n]$ is symmetric and $m$
is chosen to be sufficiently small, then $P[A^m|X^m]$ is separable (Lemma~\ref{lemma:product}). We then establish a de Finetti theorem for separable states, Lemma~\ref{th:convex}, which will complete the proof of our main result, Theorem~\ref{theorem:1}. We continue with Lemma~\ref{lemma:product}.
\begin{lemma} \label{lemma:product}
Let $n \geq |\cX|$ and set $m = \lceil n/|\cX| \rceil$. Suppose that $P[A^n|X^n]$ is a symmetric no-signalling conditional probability distribution. Then $P[A^m|X^m]$ is separable, i.e., there exists a probability distribution $p_{\lambda_1, \ldots, \lambda_m}$ such that
$$P[A^m|X^m]=\sum_{\lambda_1, \ldots, \lambda_m} p_{\lambda_1, \ldots, \lambda_m} P_{\lambda_1} [A_1|X_1] \cdots P_{\lambda_m} [A_m|X_m],$$ where $p_{\lambda_1, \ldots, \lambda_m}$ is a probability distribution on the labels $\lambda_1, \ldots, \lambda_m$, where $\lambda_j$ labels a finite set of conditional probability distributions. \end{lemma}
\begin{proof}
In order not to obscure the main argument, we prove the statement
for integral $m = n/|\cX|$~\footnote{This immediately implies the result for $
\lfloor n/|\cX| \rfloor$. The extension to the case $ \lceil n/|\cX|
\rceil$ is more technical and can be found in~\cite{benmat:long}.}. Our technique can be traced to
Werner~\cite{werner89:_applic_inequal_quant_state_exten_probl}.
We imagine the $m$ particles to be
separated in space and note that $P[A^m|X^m]$ is separable if and only if it can be
simulated by a local hidden variable model. Such a simulation is described in Fig.~1. \begin{figure*}
\caption{Since $n = m |\cX|$, we can divide the particles into $m$
groups of $|\cX|$ particles. In each of these groups we measure one
particle according to each measurement in $X$ \emph{in advance} and
record a list of all the results. In the simulation, if particle $i$ is supposed
to be measured according to a measurement $x \in X$, we just look
through the $i$th group until we come to the particle on which measurement $x$
was performed in advance, and output the result we find.}
\label{fig-boxes}
\end{figure*} We now provide the formal proof. We construct a separable conditional distribution
$Q[A^m|X^m]$ and then show that it is equal to $P[A^m|X^m]$. We assume that $\cX = \{1, 2, \ldots, |\cX|\}$, define a vector $y^n = (y_j)_{j=1,\ldots,n}$ with coordinates $y_j = (j-1 \mod |\cX| )+ 1$, and define the separable state \begin{align*}
Q[A^m|X^m] = \sum_{b^n} q_{b^n} Q_{b^n, 1}[A_1|X_1] \cdots Q_{b^n,m}[A_m|X_m], \end{align*} where $b^n \in A^n$ is distributed according to $
q_{b^n} = P[A^{n} = b^n |X^{n} = y^n]$
and the single-particle conditional probability distributions are deterministic and defined by $Q_{b^n,i}[A_i = a_i|X_i = x_i] = [a_i = b_{(i-1)|\cX|+x_i}]$, where $[t]=1$ if $t$ is true and $0$ otherwise. Let $\cL = \{1, 2, \ldots, n\}$, $\cL_1= \{(i-1)|\cX|+x_{i}: i = 1, 2, \ldots, m\}$ and $\cL_2 = \cL\backslash \cL_1$. Further let $A^{\cL}=A^n$, $A^{\cL_1}=(A_{x_1}, A_{|\cX|+x_2}, \ldots, A_{(m-1)|\cX|+x_m})$ and $A^{\cL_2}=A^{\cL} \backslash A^{\cL_1}$ and define $b^{\cL}, b^{\cL_1}$ and $b^{\cL_2}$ similarly. We find \begin{align}
Q&[A^m = a^m|X^m=x^m] \nonumber \\
&= \sum_{b^n} P[A^n=b^n|X^n=y^n] [a_1 = b_{x_1}]\cdots [a_{m} = b_{(m-1)|\cX|+x_{m}}] \nonumber\\
& = \sum_{b^{\cL_2}} P[A^{\cL_1}=a^m, A^{\cL_2}=b^{\cL_2} |X^{\cL_1}=x^m, X^{\cL_2}=y^{\cL_2}] \nonumber \\
& = P[A^{\cL_1}=a^m|X^{\cL_1}=x^m] = P[A^m=a^m|X^m=x^m],\nonumber \end{align}
where we started with the definition of $Q[A^m|X^m]$, split the summation over $\cL_1$ and $\cL_2$, dropped the conditioning over $X^{\cL_2}=y^{\cL_2}$ because of the no-signalling property of $P$, used the definition of a marginal state, and, lastly, the permutation-invariance of $P$. \end{proof}
Our next statement is a de Finetti theorem for symmetric separable conditional probability distributions.
\begin{lemma}\label{th:convex}
Suppose that $P[A^m|X^m]$ is a symmetric separable conditional probability distribution. Then there exists a probability distribution $p_\lambda$ such that \begin{align} \begin{split}
\bigl\|P[A^k|X^k]- & \sum_\lambda p_\lambda P_\lambda[A|X]^{\times k} \bigr\| \leq
\min \biggl( \frac{2k|\cA|^{|\cX|}}{m}, \frac{k(k-1)}{m}\biggr), \end{split} \end{align} where $p_\lambda$ is a probability distribution on a finite set of conditional probability distributions, labeled by $\lambda$. \end{lemma}
\begin{proof}
Let $Q_{1}[A|X], \ldots, Q_{E}[A|X]$ be the extreme points of the set of conditional probability distributions of one system. These are the deterministic functions $\cX \mapsto \cA$, hence $E=|\cA|^{|\cX|}$. Any symmetric separable conditional probability distribution is a convex combination of conditional probability distributions of the form $Q[A^m|X^m]=\frac{1}{m!} \sum_\pi Q_{i_{\pi^{-1}(1)}}[A|X] \cdots Q_{i_{\pi^{-1}(m)}}[A|X] $, where $1\leq i_1, \ldots, i_m \leq E$.
Define $Q[A|X]\mathrel{\mathop:}= \frac{1}{m}
\sum_{j=1}^m Q_{i_j}[A|X]$. We expand \begin{align}
Q[A|X]^{\times k} = \sum_{j_1=1}^m \cdots \sum_{j_k = 1}^m M_m(i_{j_1}, \ldots, i_{j_k}) Q_{i_{j_1}}[A_1|X_1] \cdots Q_{i_{j_k}}[A_m|X_m], \end{align}
where $M_m(i_{j_1}, \ldots, i_{j_k}) = 1/m^k$ is the multinomial distribution. To compare this expression with $Q[A^k|X^k]$, write \begin{align}
Q[A^k|X^k]= \sum_{j_1=1}^m \cdots \sum_{j_k = 1}^m H_m(i_{j_1}, \ldots, i_{j_k}) Q_{i_{j_1}}[A_1|X_1] \cdots Q_{i_{j_k}}[A_m|X_m], \end{align} where $H_m(i_{j_1}, \ldots, i_{j_k})$ is the hypergeometric distribution for an urn with $m$ balls (see~\cite{diaconis80:_finit}). Then \begin{align}
\left\| Q[A^k|X^k] - Q[A|X]^{\times k} \right\| &= \big\| \sum_{j_1,\ldots,j_k} \big( H_m(i_{j_1}, \ldots, i_{j_k})
\nonumber\\ & \qquad M_m(i_{j_1}, \ldots, i_{j_k}) \big)Q_{i_{j_1}}[A_1|X_1] \cdots Q_{i_{j_k}}[A_m|X_m]\big\| \nonumber\\ &\hspace{-1cm}\leq
\sum_{j_1,\ldots,j_k} \big| H_m(i_{j_1}, \ldots, i_{j_k})- M_m(i_{j_1}, \ldots, i_{j_k}) \big|\nonumber\\ & \hspace{-1cm}\leq \min \left( \frac{2kE}{m}, \frac{k(k-1)}{m}\right), \end{align} where we used the triangle inequality and Diaconis and Freedman's result on estimating the hypergeometric distribution with a multinomial distribution~\cite{diaconis80:_finit}. \end{proof}
These two lemmas enable the proof of Theorem~\ref{theorem:1}.
\begin{proof}[Proof of Theorem~\ref{theorem:1}] Set $m = \lceil n/|\cX| \rceil$ and apply Lemma~\ref{lemma:product}. Then apply Lemma~\ref{th:convex}. \end{proof}
Our final result is an application to quantum theory. In complete analogy to Lemma~\ref{th:convex} we show that the $k$-particle reduced state of a every separable symmetric density operator on $m$ copies of $\mathbb{C}^d$ is approximated by a convex combination of tensor product states. Importantly, the approximation guarantee is independent of the dimension $d$, in contrast to the case of entangled states where a dependence on the dimension is necessary~\cite{christandl06:_one}. The norm is given by the trace norm $||A||_1=\tr \sqrt{A^\dagger A}$ for operators $A$ on $\mathbb{C}^d$. It induces a distance measure on the set of quantum states that has a similar interpretation as a measure of distinguishability as the variational distance for probability distributions and the trace distance introduced on conditional probability distributions. \begin{theorem} If $\rho$ is a separable permutation-invariant density operator on $(\mathbb{C}^d)^{\otimes n}$, then there is a measure $m(\sigma)$ on states $\sigma$ on $\mathbb{C}^d$ such that \begin{align}
\bigl\|\rho^k- \int dm(\sigma) \, \sigma^{\otimes k} \bigr\|_1\leq 2\frac{k(k-1)}{n}. \end{align} \end{theorem}
\begin{proof} Any symmetric separable state is a convex combination of states of the form $\omega^n=\frac{1}{n!} \sum_\pi \tau_{{\pi^{-1}(1)}} \otimes \cdots \otimes \tau_{{\pi^{-1}(n)}}$, where $\{\tau_{j}\}_{j=1}^n$ is a set of pure states (these are extreme points in $\cB(\mathbb{C}^d)$).
Define $\tau\mathrel{\mathop:}= \frac{1}{n} \sum_{j=1}^n \tau_{j}$. We expand \begin{align}
\tau^{\otimes k} = \sum_{j_1=1}^n \cdots \sum_{j_k = 1}^n M_n({j_1}, \ldots, {j_k}) \tau_{{j_1}} \otimes \cdots \otimes \tau_{{j_k}}, \end{align} where $M_n({j_1}, \ldots, {j_k}) = 1/n^k$ is the multinomial distribution. To compare this expression with $\omega^k:=\tr_{n-k} \omega^n$, write \begin{align}
\omega^k = \sum_{j_1=1}^n \cdots \sum_{j_k = 1}^n H_n({j_1}, \ldots, {j_k}) \tau_{{j_1}} \otimes \cdots \otimes \tau_{{j_k}}, \end{align} where $H_n({j_1}, \ldots, {j_k})$ is the hypergeometric distribution for an urn with $n$ balls (see~\cite{diaconis80:_finit}). Then \begin{align}
\left\| \omega^k - \tau^{\otimes k} \right\|_1 &= \big\| \sum_{j_1,\ldots,j_k} \big( H_n({j_1}, \ldots, {j_k})
\nonumber\\ & \qquad M_n({j_1}, \ldots, {j_k}) \big)\tau_{{j_1}} \otimes \cdots \otimes \tau_{{j_k}}\big\|_1 \nonumber\\ &\hspace{-1cm}\leq
\sum_{j_1,\ldots,j_k} \big| H_n({j_1}, \ldots, {j_k})- M_n({j_1}, \ldots, {j_k}) \big|\nonumber\\ & \hspace{-1cm}\leq \frac{k(k-1)}{n}, \end{align} where we used the triangle inequality and Diaconis and Freedman's result on estimating the hypergeometric distribution with a multinomial distribution~\cite{diaconis80:_finit}. \end{proof}
\section{Towards a finite de Finetti theorem for the convex sets framework}
\label{sec:conv-sets-fram} We will start this section with a self-contained introduction to the convex sets framework. (See Refs.~\cite{Barnum:06} and~\cite{Barnum:07} for a gentler introduction.) We will then generalise Lemma~\ref{th:convex} to this setting. Finally, we pose the question of the existence of a finite de Finetti theorem in the convex sets framework.
Let $\Omega$ be the set of states of a particle. We assume that $\Omega$ is convex, compact, and has affine dimension $n$. In probability theory, for example, $\Omega$ is the simplex of probability distributions $(\omega_1, \ldots, \omega_{n+1})$, $\omega_i \geq 0, \sum_i \omega_i=1$, while in quantum theory, $\Omega$ is (isomorphic to) the set of positive operators $\omega$ with trace one on a Hilbert space $\cH\cong \mathbb{C}^d$. We are particularly interested in the case where $\Omega$ is specified by a set of conditional probability distributions $\{ P_\lambda[A|X]\}$, whose elements are indexed by a label $\lambda$.
This is partly because quantum states can be described in this way. For instance, the state $\rho$ of a qubit, a spin-$\frac{1}{2}$ system, is uniquely determined by the probabilities of obtaining spin up or down when it is measured along the $x$, $y$, or $z$ axes of the Bloch sphere. Thus a qubit can be described by a conditional probability distribution $P[A|X]$ with $\cA = \{ \uparrow, \downarrow\}$ and $\cX = \{x,y,z\}$.
Not all conditional probability distributions can be obtained by making local measurements on quantum states. This led Barrett to define generalized theories~\cite{Barrett:05}, where the state space $\Omega$ is the set of all conditional probability distributions $\{P_\lambda[A|X]\}$, denoted $\square$. This is the case that we considered in the previous parts of the paper. When $|\cX| = 1$, this reduces to classical probability theory.
In quantum theory, $|\cX| = 1$ corresponds to the case where all measurements on a system commute, and thus can be performed at once.
In fact, every $\Omega$ can be mapped to a convex subset of $\square$ for some number of \emph{fiducial} measurements and outcomes~\cite[Lemma~1]{Barnum:06}.
In quantum theory, the most general measurement that can be performed is a positive operator-valued measure (POVM), whose elements are termed \emph{effects}. Effects are linear functions mapping states to probabilities: in (finite-dimensional) quantum theory, the probability of obtaining the outcome associated with an effect $r$, when the state is $\omega$, is $r(\omega)=\tr \left(R \omega\right)$ for some bounded nonnegative operator $R$ with $R \leq \mathbf{1}$. In a generalized theory, effects are also functions mapping states to probabilities, and these functions should be affine so that they are compatible with preparing convex combinations. The vector space of affine functions $a: \Omega \to \mathbb{R}$, denoted $A(\Omega)$, is isomorphic to $\mathbb{R}^{n+1}$.
The cone of nonnegative affine functions on $\Omega$ is denoted $A_+(\Omega)$. The \emph{order unit} of $A(\Omega)$ is the element $e \in A(\Omega)$ satisfying $e(\omega)=1$ for all $\omega \in \Omega$. An \emph{effect} is an element $a\in A(\Omega)$ satisfying $0\leq a(\omega) \leq 1$ for all $\omega \in \Omega$.
The set of all effects is denoted $[0, e]$. There is a natural embedding of $\Omega$ into $A(\Omega)^*$, the dual space of $A(\Omega)$, given by $\omega \mapsto \hat \omega$, where $\hat \omega (a) = a(\omega) $ for all $a \in A(\Omega)$. Furthermore, if $\hat \omega \in A(\Omega)^*$ satisfies $\hat \omega(a) \geq 0$ for all $a \in A_+(\Omega)$ and $\hat \omega(e) = 1$, then $\hat \omega$ is the image of some state $\omega \in \Omega$~\cite[Section~2.6]{Boyd:04}. We identify $\hat \omega$ with $\omega$ in what follows. It is easy to check that $\|\cdot \|\:=\sup_{a \in [0, e]} |a(\cdot)|$ is a norm on $A(\Omega)^*$. For more details about the convex sets framework, see~\cite{Barnum:06,Barnum:07}.
A natural distance measure on the set of states, which generalises the variational distance between classical probability distributions and the trace distance between quantum states, is given by
\begin{align} \label{eq:distance}\|\omega - \omega' \|\:=\sup_{a \in [0, e]} |a(\omega)-a(\omega')|.\end{align}
In quantum theory, systems are combined by taking the \emph{tensor product} of the Hilbert spaces for each system. The same is true in the convex sets framework: $\omega \otimes \omega'$ is defined to be the \emph{product state} where system $\Omega$ is in state $\omega$, system $\Omega'$ is in state $\omega'$, and the two systems are independent. The complication is that the space $A(\Omega)^\star$ is a Banach space but not a Hilbert space and there are multiple ways to define a norm on the tensor product space, consistent with the norm on $A(\Omega)^\star$. This choice affects the set of pure (i.e., norm $1$) states of the joint system. At the very least, we want the set of joint states to be closed under convex combinations. This yields:
\begin{definition} The \emph{minimal tensor product} of $\Omega$ and $\Omega'$, denoted by $\Omega\otimes_\text{min} \Omega'$ consists of all convex combinations of product states $\omega \otimes \omega'$, $\omega \in \Omega$ and $\omega' \in \Omega'$. \end{definition}
We say that states in $\Omega \otimes_\text{min} \Omega'$ are \emph{separable}, thereby extending terminology from quantum mechanics to the convex sets framework. Next, if $a$ is a valid effect for system $\Omega$ and $a'$ a valid effect for system $\Omega'$, then $a \otimes a'$ is the effect defined on product states via $a \otimes a'(\omega \otimes \omega') = a(\omega)a'(\omega')$. If all convex combinations of such effects are to be allowed, the state space must only contain states in the \emph{maximal tensor product}, defined via duality as:
\begin{definition} The \emph{maximal tensor product} of $\Omega$ and $\Omega'$, denoted by $\Omega\otimes_\text{max} \Omega'$ consists of all bilinear functions $\mu: A(\Omega)\times A(\Omega') \rightarrow \mathbb{R}$ that satisfy $\mu (a \otimes b)\geq 0$ for $a, b \geq 0$, and $\mu(e \otimes e')=1$. \end{definition}
Thus $\mu \in \Omega \otimes_\text{max} \Omega'$ can be written as a linear combination of product states, possibly with negative weights. In classical probability theory, the minimal and the maximal tensor product coincide. In general, a tensor product $\Omega \otimes \Omega'$ is a convex set with $\Omega \otimes_\text{min} \Omega'\subseteq \Omega \otimes \Omega' \subseteq \Omega \otimes_\text{max} \Omega'$. In quantum theory, $\Omega \otimes \Omega'$ is the set of trace one positive operators on the (unique) Hilbert space tensor product of $\cH$ and $\cH'$. Note that $\Omega \otimes \Omega'$ lies strictly between the maximal and minimal tensor products in the quantum case. The set of separable quantum states is $\Omega \otimes_\text{min} \Omega'$ and $\Omega \otimes_\text{max} \Omega'$ is the set of trace one entanglement witnesses.
For a state $\mu\in \Omega \otimes \Omega'$, we say that $\mu_{\Omega} \in \Omega$, defined by $a(\mu_{\Omega})=a \otimes e' (\mu)$ for all effects $a$, is the \emph{partial trace} of $\mu$ with respect to $\Omega'$. An effect on the tensor product is an element $a \in A(\Omega \otimes \Omega')$
satisfying $0 \leq a \leq e \otimes e'$. The larger the set of joint states, the smaller the set of allowed effects. This means that the distance measure that we defined in Eq.~(\ref{eq:distance}), when applied to states of more than one particle, depends on which tensor product we use. It is true, however, that $\|\omega - \omega'\| \leq
\|\omega -\omega'\|_\text{min}$, the distance measure for the minimal tensor product, since in that case the set of effects is largest. Also note that a physical theory may place additional restrictions on which effects are allowed but, even then, $\|\omega - \omega'\|$ provides an upper bound on the probability of distinguishing $\omega$ and $\omega'$.
\begin{theorem}\label{th:convex-gen} Let $\Omega$ be a convex set with $E$ extreme points ($E$ may be infinite). Suppose $\omega^n \in \Omega^{\otimes_\text{min} n}$ is symmetric. Then there is a measure $m(\tau)$ on states $\tau \in \Omega$ such that \begin{align}
\bigl\|\omega^k- \int dm(\tau) \, \tau^{\otimes k}\bigr\|_\text{min}\leq \min \left( \frac{2kE}{n}, \frac{k(k-1)}{n}\right). \end{align} \end{theorem}
\begin{proof} Let $\tau_1, \ldots, \tau_E$ be the extreme points of $\Omega$. Any symmetric separable state is a convex combination of states of the form $\omega^n=\frac{1}{n!} \sum_\pi \tau_{i_{\pi^{-1}(1)}} \otimes \cdots \otimes \tau_{i_{\pi^{-1}(n)}}$, where $1\leq i_1, \ldots, i_n \leq E$.
Define $\tau\mathrel{\mathop:}= \frac{1}{n} \sum_{j=1}^n \tau_{i_j}$. We expand \begin{align}
\tau^{\otimes k} = \sum_{j_1=1}^n \cdots \sum_{j_k = 1}^n M_n(i_{j_1}, \ldots, i_{j_k}) \tau_{i_{j_1}} \otimes \cdots \otimes \tau_{i_{j_k}}, \end{align} where $M_n(i_{j_1}, \ldots, i_{j_k}) = 1/n^k$ is the multinomial distribution. To compare this expression with $\omega^k$, write \begin{align}
\omega^k = \sum_{j_1=1}^n \cdots \sum_{j_k = 1}^n H_n(i_{j_1}, \ldots, i_{j_k}) \tau_{i_{j_1}} \otimes \cdots \otimes \tau_{i_{j_k}}, \end{align} where $H_n(i_{j_1}, \ldots, i_{j_k})$ is the hypergeometric distribution for an urn with $n$ balls (see~\cite{diaconis80:_finit}). Then \begin{align}
\left\| \omega^k - \tau^{\otimes k} \right\|_\text{min} &= \big\| \sum_{j_1,\ldots,j_k} \big( H_n(i_{j_1}, \ldots, i_{j_k})
\nonumber\\ & \qquad M_n(i_{j_1}, \ldots, i_{j_k}) \big)\tau_{i_{j_1}} \otimes \cdots \otimes \tau_{i_{j_k}}\big\|_\text{min} \nonumber\\ &\hspace{-1cm}\leq
\sum_{j_1,\ldots,j_k} \big| H_n(i_{j_1}, \ldots, i_{j_k})- M_n(i_{j_1}, \ldots, i_{j_k}) \big|\nonumber\\ & \hspace{-1cm}\leq \min \left( \frac{2kE}{n}, \frac{k(k-1)}{n}\right), \end{align} where we used the triangle inequality and Diaconis and Freedman's result on estimating the hypergeometric distribution with a multinomial distribution~\cite{diaconis80:_finit}. \end{proof}
One can show that $\square^{\otimes_\text{max} n}$ is precisely the set of all no-signalling conditional probability distributions and that $\square^{\otimes_\text{min} n}$ is the set of all separable conditional probability distributions ~\cite{randall81:_operat_statis_tensor_produc,Barrett:05}. Furthermore the trace distance (Definition~\ref{def:tracedistance}) coincides with the definition in Eq.~(\ref{eq:distance}). With these observations and the fact that $||\cdot ||\leq || \cdot ||_{\min}$ we see that Theorem~\ref{th:convex-gen} generalises Lemma~\ref{th:convex}. Unfortunately, we were not able to obtain a similar generalisation of Lemma~\ref{lemma:product} and hence of Theorem~\ref{theorem:1}. We thus conclude with the question of whether a finite de Finetti theorem exists for general theories in the convex sets framework. We remark that the argument of~\cite{caves:4537} applied in this context yields an infinite de Finetti theorem for any theory in the convex sets framework (see~\cite{BarrettLeifer06} for the details).
\end{document} | arXiv |
Non-analytic smooth function
In mathematics, smooth functions (also called infinitely differentiable functions) and analytic functions are two very important types of functions. One can easily prove that any analytic function of a real argument is smooth. The converse is not true, as demonstrated with the counterexample below.
One of the most important applications of smooth functions with compact support is the construction of so-called mollifiers, which are important in theories of generalized functions, such as Laurent Schwartz's theory of distributions.
The existence of smooth but non-analytic functions represents one of the main differences between differential geometry and analytic geometry. In terms of sheaf theory, this difference can be stated as follows: the sheaf of differentiable functions on a differentiable manifold is fine, in contrast with the analytic case.
The functions below are generally used to build up partitions of unity on differentiable manifolds.
An example function
Definition of the function
Consider the function
$f(x)={\begin{cases}e^{-{\frac {1}{x}}}&{\text{if }}x>0,\\0&{\text{if }}x\leq 0,\end{cases}}$
defined for every real number x.
The function is smooth
The function f has continuous derivatives of all orders at every point x of the real line. The formula for these derivatives is
$f^{(n)}(x)={\begin{cases}\displaystyle {\frac {p_{n}(x)}{x^{2n}}}\,f(x)&{\text{if }}x>0,\\0&{\text{if }}x\leq 0,\end{cases}}$
where pn(x) is a polynomial of degree n − 1 given recursively by p1(x) = 1 and
$p_{n+1}(x)=x^{2}p_{n}'(x)-(2nx-1)p_{n}(x)$
for any positive integer n. From this formula, it is not completely clear that the derivatives are continuous at 0; this follows from the one-sided limit
$\lim _{x\searrow 0}{\frac {e^{-{\frac {1}{x}}}}{x^{m}}}=0$
for any nonnegative integer m.
Detailed proof of smoothness
By the power series representation of the exponential function, we have for every natural number $m$ (including zero)
${\frac {1}{x^{m}}}=x{\Bigl (}{\frac {1}{x}}{\Bigr )}^{m+1}\leq (m+1)!\,x\sum _{n=0}^{\infty }{\frac {1}{n!}}{\Bigl (}{\frac {1}{x}}{\Bigr )}^{n}=(m+1)!\,xe^{\frac {1}{x}},\qquad x>0,$
because all the positive terms for $n\neq m+1$ are added. Therefore, dividing this inequality by $e^{\frac {1}{x}}$ and taking the limit from above,
$\lim _{x\searrow 0}{\frac {e^{-{\frac {1}{x}}}}{x^{m}}}\leq (m+1)!\lim _{x\searrow 0}x=0.$
We now prove the formula for the nth derivative of f by mathematical induction. Using the chain rule, the reciprocal rule, and the fact that the derivative of the exponential function is again the exponential function, we see that the formula is correct for the first derivative of f for all x > 0 and that p1(x) is a polynomial of degree 0. Of course, the derivative of f is zero for x < 0. It remains to show that the right-hand side derivative of f at x = 0 is zero. Using the above limit, we see that
$f'(0)=\lim _{x\searrow 0}{\frac {f(x)-f(0)}{x-0}}=\lim _{x\searrow 0}{\frac {e^{-{\frac {1}{x}}}}{x}}=0.$
The induction step from n to n + 1 is similar. For x > 0 we get for the derivative
${\begin{aligned}f^{(n+1)}(x)&={\biggl (}{\frac {p'_{n}(x)}{x^{2n}}}-2n{\frac {p_{n}(x)}{x^{2n+1}}}+{\frac {p_{n}(x)}{x^{2n+2}}}{\biggr )}f(x)\\&={\frac {x^{2}p'_{n}(x)-(2nx-1)p_{n}(x)}{x^{2n+2}}}f(x)\\&={\frac {p_{n+1}(x)}{x^{2(n+1)}}}f(x),\end{aligned}}$
where pn+1(x) is a polynomial of degree n = (n + 1) − 1. Of course, the (n + 1)st derivative of f is zero for x < 0. For the right-hand side derivative of f (n) at x = 0 we obtain with the above limit
$\lim _{x\searrow 0}{\frac {f^{(n)}(x)-f^{(n)}(0)}{x-0}}=\lim _{x\searrow 0}{\frac {p_{n}(x)}{x^{2n+1}}}\,e^{-1/x}=0.$
The function is not analytic
As seen earlier, the function f is smooth, and all its derivatives at the origin are 0. Therefore, the Taylor series of f at the origin converges everywhere to the zero function,
$\sum _{n=0}^{\infty }{\frac {f^{(n)}(0)}{n!}}x^{n}=\sum _{n=0}^{\infty }{\frac {0}{n!}}x^{n}=0,\qquad x\in \mathbb {R} ,$
and so the Taylor series does not equal f(x) for x > 0. Consequently, f is not analytic at the origin.
Smooth transition functions
The function
$g(x)={\frac {f(x)}{f(x)+f(1-x)}},\qquad x\in \mathbb {R} ,$
has a strictly positive denominator everywhere on the real line, hence g is also smooth. Furthermore, g(x) = 0 for x ≤ 0 and g(x) = 1 for x ≥ 1, hence it provides a smooth transition from the level 0 to the level 1 in the unit interval [0, 1]. To have the smooth transition in the real interval [a, b] with a < b, consider the function
$\mathbb {R} \ni x\mapsto g{\Bigl (}{\frac {x-a}{b-a}}{\Bigr )}.$
For real numbers a < b < c < d, the smooth function
$\mathbb {R} \ni x\mapsto g{\Bigl (}{\frac {x-a}{b-a}}{\Bigr )}\,g{\Bigl (}{\frac {d-x}{d-c}}{\Bigr )}$
equals 1 on the closed interval [b, c] and vanishes outside the open interval (a, d), hence it can serve as a bump function.
A smooth function which is nowhere real analytic
A more pathological example is an infinitely differentiable function which is not analytic at any point. It can be constructed by means of a Fourier series as follows. Define for all $x\in \mathbb {R} $
$F(x):=\sum _{k\in \mathbb {N} }e^{-{\sqrt {2^{k}}}}\cos(2^{k}x)\ .$
Since the series $\sum _{k\in \mathbb {N} }e^{-{\sqrt {2^{k}}}}{(2^{k})}^{n}$ converges for all $n\in \mathbb {N} $, this function is easily seen to be of class C∞, by a standard inductive application of the Weierstrass M-test to demonstrate uniform convergence of each series of derivatives.
We now show that $F(x)$ is not analytic at any dyadic rational multiple of π, that is, at any $x:=\pi \cdot p\cdot 2^{-q}$ with $p\in \mathbb {Z} $ and $q\in \mathbb {N} $. Since the sum of the first $q$ terms is analytic, we need only consider $F_{>q}(x)$, the sum of the terms with $k>q$. For all orders of derivation $n=2^{m}$ with $m\in \mathbb {N} $, $m\geq 2$ and $m>q/2$ we have
$F_{>q}^{(n)}(x):=\sum _{k\in \mathbb {N} \atop k>q}e^{-{\sqrt {2^{k}}}}{(2^{k})}^{n}\cos(2^{k}x)=\sum _{k\in \mathbb {N} \atop k>q}e^{-{\sqrt {2^{k}}}}{(2^{k})}^{n}\geq e^{-n}n^{2n}\quad (\mathrm {as} \;n\to \infty )$
where we used the fact that $\cos(2^{k}x)=1$ for all $2^{k}>2^{q}$, and we bounded the first sum from below by the term with $2^{k}=2^{2m}=n^{2}$. As a consequence, at any such $x\in \mathbb {R} $
$\limsup _{n\to \infty }\left({\frac {|F_{>q}^{(n)}(x)|}{n!}}\right)^{1/n}=+\infty \,,$
so that the radius of convergence of the Taylor series of $F_{>q}$ at $x$ is 0 by the Cauchy-Hadamard formula. Since the set of analyticity of a function is an open set, and since dyadic rationals are dense, we conclude that $F_{>q}$, and hence $F$, is nowhere analytic in $\mathbb {R} $.
Application to Taylor series
Main article: Borel's lemma
For every sequence α0, α1, α2, . . . of real or complex numbers, the following construction shows the existence of a smooth function F on the real line which has these numbers as derivatives at the origin.[1] In particular, every sequence of numbers can appear as the coefficients of the Taylor series of a smooth function. This result is known as Borel's lemma, after Émile Borel.
With the smooth transition function g as above, define
$h(x)=g(2+x)\,g(2-x),\qquad x\in \mathbb {R} .$
This function h is also smooth; it equals 1 on the closed interval [−1,1] and vanishes outside the open interval (−2,2). Using h, define for every natural number n (including zero) the smooth function
$\psi _{n}(x)=x^{n}\,h(x),\qquad x\in \mathbb {R} ,$
which agrees with the monomial xn on [−1,1] and vanishes outside the interval (−2,2). Hence, the k-th derivative of ψn at the origin satisfies
$\psi _{n}^{(k)}(0)={\begin{cases}n!&{\text{if }}k=n,\\0&{\text{otherwise,}}\end{cases}}\quad k,n\in \mathbb {N} _{0},$
and the boundedness theorem implies that ψn and every derivative of ψn is bounded. Therefore, the constants
$\lambda _{n}=\max {\bigl \{}1,|\alpha _{n}|,\|\psi _{n}\|_{\infty },\|\psi _{n}^{(1)}\|_{\infty },\ldots ,\|\psi _{n}^{(n)}\|_{\infty }{\bigr \}},\qquad n\in \mathbb {N} _{0},$
involving the supremum norm of ψn and its first n derivatives, are well-defined real numbers. Define the scaled functions
$f_{n}(x)={\frac {\alpha _{n}}{n!\,\lambda _{n}^{n}}}\psi _{n}(\lambda _{n}x),\qquad n\in \mathbb {N} _{0},\;x\in \mathbb {R} .$
By repeated application of the chain rule,
$f_{n}^{(k)}(x)={\frac {\alpha _{n}}{n!\,\lambda _{n}^{n-k}}}\psi _{n}^{(k)}(\lambda _{n}x),\qquad k,n\in \mathbb {N} _{0},\;x\in \mathbb {R} ,$
and, using the previous result for the k-th derivative of ψn at zero,
$f_{n}^{(k)}(0)={\begin{cases}\alpha _{n}&{\text{if }}k=n,\\0&{\text{otherwise,}}\end{cases}}\qquad k,n\in \mathbb {N} _{0}.$
It remains to show that the function
$F(x)=\sum _{n=0}^{\infty }f_{n}(x),\qquad x\in \mathbb {R} ,$
is well defined and can be differentiated term-by-term infinitely many times.[2] To this end, observe that for every k
$\sum _{n=0}^{\infty }\|f_{n}^{(k)}\|_{\infty }\leq \sum _{n=0}^{k+1}{\frac {|\alpha _{n}|}{n!\,\lambda _{n}^{n-k}}}\|\psi _{n}^{(k)}\|_{\infty }+\sum _{n=k+2}^{\infty }{\frac {1}{n!}}\underbrace {\frac {1}{\lambda _{n}^{n-k-2}}} _{\leq \,1}\underbrace {\frac {|\alpha _{n}|}{\lambda _{n}}} _{\leq \,1}\underbrace {\frac {\|\psi _{n}^{(k)}\|_{\infty }}{\lambda _{n}}} _{\leq \,1}<\infty ,$
where the remaining infinite series converges by the ratio test.
Application to higher dimensions
For every radius r > 0,
$\mathbb {R} ^{n}\ni x\mapsto \Psi _{r}(x)=f(r^{2}-\|x\|^{2})$
with Euclidean norm ||x|| defines a smooth function on n-dimensional Euclidean space with support in the ball of radius r, but $\Psi _{r}(0)>0$.
Complex analysis
This pathology cannot occur with differentiable functions of a complex variable rather than of a real variable. Indeed, all holomorphic functions are analytic, so that the failure of the function f defined in this article to be analytic in spite of its being infinitely differentiable is an indication of one of the most dramatic differences between real-variable and complex-variable analysis.
Note that although the function f has derivatives of all orders over the real line, the analytic continuation of f from the positive half-line x > 0 to the complex plane, that is, the function
$\mathbb {C} \setminus \{0\}\ni z\mapsto e^{-{\frac {1}{z}}}\in \mathbb {C} ,$
has an essential singularity at the origin, and hence is not even continuous, much less analytic. By the great Picard theorem, it attains every complex value (with the exception of zero) infinitely many times in every neighbourhood of the origin.
See also
• Bump function
• Fabius function
• Flat function
• Mollifier
Notes
1. Exercise 12 on page 418 in Walter Rudin, Real and Complex Analysis. McGraw-Hill, New Delhi 1980, ISBN 0-07-099557-5
2. See e.g. Chapter V, Section 2, Theorem 2.8 and Corollary 2.9 about the differentiability of the limits of sequences of functions in Amann, Herbert; Escher, Joachim (2005), Analysis I, Basel: Birkhäuser Verlag, pp. 373–374, ISBN 3-7643-7153-6
External links
• "Infinitely-differentiable function that is not analytic". PlanetMath.
| Wikipedia |
\begin{definition}[Definition:Dimension (Hilbert Space)]
Let $H$ be a Hilbert space, and let $E$ be a basis of $H$.
Then the '''dimension''' $\dim H$ of $H$ is defined as $\card E$, the cardinality of $E$.
\end{definition} | ProofWiki |
\begin{document}
\subjclass[2000]{Primary: 37D45, 37C40} \keywords{Anosov system on fibers, entropy, random periodic orbit, random horseshoe, random periodic measure, and random Liv\v sic Theorem}
\author{Wen Huang} \address[Wen Huang] { Department of Mathematics\\
University of Science and Technology of China\\
Hefei, Anhui, China} \email[Wen.H]{[email protected]}
\author{Zeng Lian} \address[Zeng Lian] {College of Mathematics\\ Sichuan University\\
Chengdu, Sichuan, 610016, China} \email[Z.~Lian]{[email protected]}
\author{Kening Lu} \address[Kening Lu] {
Department of Mathematics\\ Brigham Young University\\ Provo, Utah, 84602, United States} \email[K. Lu]{[email protected]}
\title[Random Anosov systems mixing on fibers]{ Ergodic theory of Random Anosov systems \\ mixing on fibers}
\pagestyle{plain}
\thanks{This work is partially supported by NSF of China (11225105, 11371339, 11431012, 11671279, 11541003, 11725105).}
\begin{abstract} In this paper, we study the complicated dynamics of Anosov systems driven by an external force in the context of geometric theory (an abundance of random periodic points and random horseshoes) and smooth ergodic theory (random periodic measures and random Liv\v sic Theorem).
\end{abstract}
\maketitle
\setcounter{page}{1} \tableofcontents
\parskip 5pt
\section{Introduction}\label{S:Introduction}
In this paper, we study the complicated dynamics of Anosov systems driven by an external force in the context of geometric theory and smooth ergodic theory.
\subsection{Description of Main Results.} Let $(\O,d_\O)$ be a compact metric space and $\t:\O\to\O$ be a homeomorphism. Let $M$ be a connected compact smooth Riemannian manifold without boundary. A dynamical system driven by an external force $\theta$ or a cocycle is a family of continuous maps \[F(n,\cdot,\cdot):\Omega \times M \to M, \quad (\omega, x) \mapsto F(n, \omega, x), \quad\text{for } n\in\mathbb{Z} \] such that the map $F(n,\omega):=F(n,\omega,\cdot)$ forms a cocycle over $\theta$: \[ F(0, \omega)=Id, \quad \hbox{ for all }\; \omega \in \Omega, \] \[ F(n+m,\omega)=F(n,\theta^{m}\omega)F(m,\omega), \quad \hbox{ for all }\; m, n \in \mathbb{Z}, \quad\omega \in \Omega. \] When space $\Omega$ is endowed an invariant probability measure such as the Haar measure for $\theta$, $F$ is also called a random dynamical system, see Arnold \cite{A}.
Let $f(\omega)$ be the time-one map of the system, i.e., $f(\omega) x= F(1, \omega, x)$, which is assumed to be a diffeomorphism from $M$ to $M$. Conversely, such a family of diffeomorphisms also generates a cocycle (or a random dynamical system): \[ F(n, \omega, x)= \begin{cases} f(\theta^{n-1}\omega)\circ \cdots \circ f(\omega) x, & n >0, \\ x, & n=0, \\ f^{-1}(\theta^n\omega)\circ \cdots \circ f^{-1}(\theta^{-1}\omega) x, & n<0. \end{cases} \] Equivalently, putting $f$ and $\t$ together forms a skew product system $\phi:M\times \O\to M\times \O$ given by \[\phi(x,\o)=(f(\o)x,\t\o)=(f_\o x,\t\o),\ \forall x\in M,\o\in\O \]
where we rewrite $f(\o)$ as $f_\o$ for the sake of convenience.
We consider a system $\phi(x,\o)=(f_\o x,\t\o)$ which we call {\bf Anosov on fibers}, i.e., the following hold: for every $(x,\o) \in M\times \O$ there is a splitting of the tangential fiber of $M_\o:=M\times\{\o\}$ at $x$ \[ T_xM_\o=E_{(x,\o)}^u \oplus E_{(x,\o)}^{s} \] which depends continuously on $(x,\o) \in M\times \O$ with $\dim E_{(x,\o)}^u, \dim E_{(x,\o)}^s >0$ and satisfies that \[ Df_{\o}(x) E_{(x,\o)}^u=E_{\phi(x,\o)}^u, \ \ \ Df_{\o}(x) E_{(x,\o)}^{s}= E_{\phi(x,\o)}^{s}, \] and \begin{equation*} \left\{\begin{array}{ll}
|Df_{\o}(x) \xi| \geq e^{\l_0} |\xi|, \ \ \ &\forall\ \xi \in E_{(x,\o)}^u, \\
|Df_{\o}(x)\eta| \leq e^{-\l_0}|\eta|, \ \ \ &\forall\ \eta \in E_{(x,\o)}^{s}, \end{array} \right. \end{equation*} where $\lambda_0>0$ is a constant.
We note that if $\O$ is a differentiable manifold and $\theta$ is a diffeomorphims with expansion weaker than $e^{\lambda_0}$ and contraction weaker than $e^{-\lambda_0}$, then the system is a partially hyperbolic system.
We assume that $\phi:M\times \O\to M\times \O$ is {\bf \em topological mixing on fibers}, that is, that for any nonempty open sets $U,V\subset M$, there exists $N>0$ such that for any $n\ge N$ and $\o\in \O$ \[\phi^n(U \times \{\o\})\bigcap V\times \{\t^n\o\}\neq \emptyset.\]
In the following, we first introduce two ingredients of dynamical complexity: {\bf \em random periodic orbit} and {\bf \em random horseshoe.}
Let $L^\infty(\O, M)$ be the space of Borel measurable maps from $\O$ to $M$ endowed with the metric $d_{L^\infty(\O, M)}$: $$d_{L^\infty(\O, M)}(g_1,g_2)=\sup_{\o\in\O}d_M(g_1(\o),g_2(\o)),\ \forall g_1,g_2\in L^\infty(\O, M).$$
The set $\{graph(g_i)|\ g_i\in L^\infty(\O, M)\}_{1\le i\le n}$ is called a {\bf \em random periodic orbit} of $\phi$ with period $n$ if
for all $1\le i\le n$
\[\phi(\text{graph}(g_i))=\text{graph}(g_{i+1\mod n}),\ \forall 1\le i\le n.
\]
Equivalently,
\begin{equation*}
g_i \text{ is a periodic point of }\tilde \phi\text{ i.e., }\tilde \phi (g_i)=g_{i+1\mod n}\ \forall 1\le i\le n,
\end{equation*}
where $\tilde \phi$ is the induced map of $\phi$ on $L^\infty(\O, M)$ given by for each $g\in L^\infty(\O, M)$
\[
\tilde \phi(g)(\omega)=f_{\theta^{-1}\omega} g(\theta^{-1}\omega).
\] Each $graph(g_i)$ is called a random periodic point of $\phi$. Without causing any confusion and for the sake of convenience, we call $g_i$ a random periodic point for short.
Moreover, if $g_i$ is a continuous map then we call $g_i$ a continuous random periodic point.
Next, we introduce the concept of random horseshoes on different levels, which are distinguished by the following separation functions on $L^\infty(\O, M)\times L^\infty(\O, M)$ measuring the separations of elements of $L^\infty(\O, M)$ on different levels. \begin{align*}\begin{split} \overline d_{L^\infty(\O, M)}(g_1,g_2)&=\sup_{\O'\subset \O\text{ is open and nonempty}} \inf_{\o\in \O'}d_M(g_1(\o),g_2(\o)),\ \forall g_1,g_2\in L^\infty(\O, M);\\ \underline d_{L^\infty(\O, M)}(g_1,g_2)&=\inf_{\o\in\O}d_M(g_1(\o),g_2(\o)),\ \forall g_1,g_2\in L^\infty(\O, M). \end{split} \end{align*} Let $\mathcal S_k=\{1,\ldots,k\}^{\mathbb Z}$ be the space of two sides sequences of $k(\geq 2))$ symbols endowed with the standard metric, and $\s$ be the (left-)shift map.
Using the separation funcation $\overline d_{L^\infty(\O, M)}$, we define a {\bf \em full random horseshoe} with $k$-symbols of $\phi$ as a continuous embedding $\Psi:\mathcal S_k\to L^\infty(\O, M)$ satisfying {\it \begin{itemize}
\item[i)] There exists $\Delta>0$ such that for any $l\in\mathbb Z$ and $\hat a_1,\hat a_2\in \mathcal S_k\text{ with }\hat a_1(l)\neq \hat a_2(l)$
\begin{equation*}\label{E:RandomSepaImgSymb}
\overline d^{\tilde \phi}_{l}\left(\Psi({\hat a_1}),\Psi({\hat a_2})\right)>\Delta,
\end{equation*}
where for $l\ge 0$
$$\overline d^{\tilde \phi}_{l}(g_1,g_2):=\max_{0\le i\le l}\left\{\overline d_{L^\infty(\O, M)}\left(\tilde \phi^i(g_1),\tilde \phi^i(g_2)\right)\right\},\ g_1,g_2\in L^\infty(\O, M)$$ and for $l<0$ we let $\overline d^{\tilde\phi}_{l}=\overline d^{\tilde\phi^{-1}}_{|l|};$ \item[ii)] For all $\hat a\in\mathcal S_k$ $$\tilde \phi\left(\Psi({\hat a})\right)=\Psi({\s \hat a}).$$ \end{itemize}} Then, using the stronger separation function $\underline d_{L^\infty(\O, M)}$, we define a {\bf \em strong full random horseshoe} as a continuous embedding $\Psi:\mathcal S_k\to L^\infty(\O, M)$ satisfying {\it \begin{itemize}
\item[i)] There exists $\Delta>0$ such that for any $l\in\mathbb Z$ and $\hat a_1,\hat a_2\in \mathcal S_k\text{ with }\hat a_1(l)\neq \hat a_2(l)$
\begin{equation*}\label{E:SRandomSepaImgSymb}
\underline d^{\tilde \phi}_{l}\left(\Psi({\hat a_1}),\Psi({\hat a_2})\right)>\Delta,
\end{equation*}
where for $l\ge 0$
$$\underline d^{\tilde \phi}_{l}(g_1,g_2):=\max_{0\le i\le l}\left\{\underline d_{L^\infty(\O, M)}\left(\tilde \phi^i(g_1),\tilde \phi^i(g_2)\right)\right\},\ g_1,g_2\in L^\infty(\O, M)$$ and for $l<0$ we let $\underline d^{\tilde\phi}_{l}=\underline d^{\tilde\phi^{-1}}_{|l|};$ \item[ii)] For all $\hat a\in\mathcal S_k$ $$\tilde \phi\left(\Psi({\hat a})\right)=\Psi({\s \hat a}).$$ \end{itemize} } A full random horseshoe we have here is an embedding of a two-sides symbolic dynamical system into $L^\infty(\O, M)$, which is an extension of the standard topological horseshoe.
Let $\mathcal H=\text{diff}^2(M)$ be the space of $C^2$ diffeomorphisms on $M$ with $C^2$-topology. Throughout this paper, we assume $f:\O\to \mathcal H$ is a continuous map.
Our first result states that there is an abundance of random periodic points.
\noindent {\bf Theorem A} ({\bf Density of Random Periodic Points}). {\it Assume that $\phi$ is Anosov and topological mixing on fibers. Then, for any $\e>0$, there exists $N\in\mathbb N$ such that for any $g\in L^{\infty}(\O)$ and $n\ge N$, there exists a random periodic point $\tilde g$ with period $n$ such that
\begin{equation*}
d_{L^{\infty}}(g,\tilde g)\le \e.
\end{equation*}
}
\noindent
{\bf Remark:} The study of hyperbolic dynamics goes back to Poinc\'are \cite{Poincare} on $3$-body problem and Hadamard \cite{Hadamard} on geodesic flow. The modern theory of uniformly hyperbolic dynamical systems was initiated by Anosov \cite{Anosov} and Smale \cite{Smale} where Anosov and Axiom A diffeomeophism/flows were introduced respectively. The core ingredient in these systems is the uniform hyperbolicity which is an invariant geometric structure describing the exponential divergence of nearby orbits. This exponential divergence together with the compactness of phase space produces rich and complicated dynamical structures. The dynamics of these systems has been understood very well. One of results is that for a transitive Anosov diffeomorphism, the set of its periodic points (the number of the periodic orbits with same period is finite) is countable and dense in $M$. However, Theorem A tells that for the systems we study here, the set of random periodic points is uncountable because of the non-separability of $L^\infty(\O, M)$, where we exclude the case that $\O$ has trivial topology. In fact, Theorem A indicates that the cardinality of random periodic orbits with sufficient large period is infinite (actually uncountable), thus the increasing rate of numbers of random periodic orbits with same period as periods tends to infinity is always infinite. Therefore such an increasing rate fails in representing the topological entropy of the system, which is one of the main tasks that periodic orbits perform for Anosov diffeomorphisms in deterministic scenario. Nevertheless, there are other candidates which are able to carry out the task of representing complexity of the system such as the random horseshoes, which will be introduced in the next.
Another interesting observation based on an example given in Section \ref{Examples} is that the function corresponding to random periodic points are {\bf NOT} necessarily continuous. Actually, we show that under certain conditions, the graph corresponding to {\bf each} random periodic point is {\bf discontinuous}.
Our next result is on the existence of random horseshoes with entropy closed to the entropy of the system.
\noindent {\bf Theorem B} ({\bf Entropy and Random Horseshoe}). {\it Assume that $\phi$ is Anosov and topological mixing on fibers. Then the following statements hold \begin{itemize} \item[(a)] Strong Random Horseshoe: For any $\g>0$, there exist $N,k\in \mathbb N$ such that the following hold
\begin{enumerate}
\item[i)] $\frac1N\ln k\ge \underline{h}(M\times\O|\O)-\g$, where $\underline{h}(M\times\O|\O)>0$ and \[\underline{h}(M\times\O|\O):=\inf_{\o\in\O} h_{top}(\phi|_{M_\o});\]
\item[ii)] $\phi^N$ has a $k$-symbol strong full random horseshoe.
\end{enumerate} \item[(b)] Random Horseshoe: For any $\g>0$, there exist $N,k\in \mathbb N$ such that the following hold
\begin{enumerate}
\item[i)] $\frac1N\ln k\ge \overline{h}(M\times\O|\O)-\g$, where $\overline{h}(M\times\O|\O)>0$ and
\[\overline{h}(M\times\O|\O):=\sup_{\o\in\O} h_{top}(\phi|_{M_\o});\]
\item[ii)] $\phi^N$ has a $k$-symbol full random horseshoe.
\end{enumerate}
\end{itemize}
}
\noindent
{\bf Remark:} The measure-theoretic entropy was introduced in 1950's by Kolmogorov \cite{Kol} and Sinai \cite{Sinai0} to measure the rate of increase in dynamical complexity as the system evolves with time. Sinai \cite{Sinai64} studied an ergodic measure preserving automorphism $f$ of a Lebesgue space $(X, \mu)$ and proved that if the measure-theoretic entropy of $f$ is positive, then $f$ contains factor automorphisms which are isomorphic to Bernoulli shifts. The topological entropy was first introduced in 1965 by Adler, Konheim and McAndrew for studying dynamical systems in topological spaces. In his remarkable paper \cite{K}, A. Katok proved that for a nonuniformly hyperbolic $C^2$ diffeomorphism on a compact Riemannian manifold, the positive topological entropy implies the existence of a Smale horseshoe. Recently, Lian and Young \cite{LY1} extended Katok's results to $C^2$ differentiable maps with a nonuniformly hyperbolic compact invariant set in a separable Hilbert space and to a $C^2$ semiflow in a Hilbert space which has a nonuniformly hyperbolic compact invariant set. Without assuming any hyperbolicity, Huang and Lu \cite{HL} proved if a continuous random map in a Polish space has a positive topological entropy on a random compact invariant set , then it contains a weak horseshoe (see also \cite{HY} for topological dynamical systems in compact metric spaces and \cite{KL} for $C^*$-dynamics). The horseshoe we have here is for hyperbolic systems driven by an external force, which cannot be obtained by using Katok's result or Lian and Young's result for flow. When space $\Omega$ is endowed an invariant probability measure such as the Haar measure for $\theta$, it follows from \cite{GK} that $\phi$ has a so called random SRB measure. In particular, for the Anosov systems driven by a quasiperiodic force which appears in Section \ref{S:Example1}, the systems have a unique random SRB measure since the driven system is uniquely ergodic.
\vskip0.05in Next, we consider random periodic $\phi$-invariant measures. We fix a $\t$-invariant probability measure $\mathbb P$ on $\O$. For $g\in L^{\infty}(\O, M)$, we define a Borel probability measure $\mu_g$ in $\mathcal M(M\times\O)$ (the set of Borel probability measures) as follows: for any Borel set $A\subset M\times \O$ \begin{equation*}
\mu_g(A)=\mathbb P\left(\left\{\o\in\O\big|\ (g(\o),\o)\in A\cap M\times\{\o\}\right\}\right). \end{equation*} Let $ \mathcal I_{\mathbb P}(M\times\O)$ be the collection of $\phi$-invariant measures whose marginal is $\mathbb P$. An invariant measure $\mu\in \mathcal I_{\mathbb P}(M\times\O)$ is called a random periodic measure if there is a random periodic orbit $\{g_i\}_{i=1,\cdots,n}$ such that $$\mu=\frac1n\sum_{i=1}^n\mu_{g_i}=\frac1n\sum_{i=1}^n\mu_{\tilde \phi^i(g_1)}.$$
Then, we have the following theorem which states that any $\phi$-invariant measure with marginal $\mathbb P$ can be approximated by random periodic measures.
\noindent{\bf Theorem C} ({\bf Density of Random Periodic Invariant Measures}). {\it Assume that $\phi$ is Anosov and topological mixing on fibers. Then the set of all periodic $\phi$-invariant measures is dense in $\mathcal I_{\mathbb P}(M\times\O)$ in the narrow topology. }
Unlike the standard periodic measure supported by one periodic orbit, the random periodic measure supported by one random periodic orbit is {\bf not} necessarily ergodic.
Finally, we present a weak version of the classic Liv\v sic Theorem for random dynamical systems. For a $\t$-invariant probability measure $\mathbb P$ and $\a\in(0,1)$, denote $L^1_{\mathbb P}(\O,C^{0,\a}(M))$ the collection of real functions $\Phi$ on $M\times\O$, which is H\"older on $M$ and measurable on $\O$, and satisfies the following condition \begin{equation*}
\|\Phi\|_{L^1_{\mathbb P}(\O,C^{0,\a}(M))}:=\int_{\O}\|\Phi\|_{C^{0,\a}(M_\o)}d\mathbb P<\infty. \end{equation*} For each $g\in L^\infty(\O, M)$ and $\Phi\in L^1_{\mathbb P}(\O,C^{0,\a}(M))$, we define a functional $\tilde \Phi$ on $L^\infty(\O, M)$ as follows \begin{equation*} \tilde \Phi(g)=\int_{\O}\Phi(g(\o),\o)d\mathbb P. \end{equation*}
\noindent{\bf Theorem D} ({\bf Random Liv\v sic Theorem}). {\it Assume that $\phi$ is Anosov and topological mixing on fibers.
Then for any $\Phi\in L^1_{\mathbb P}(\O,C^{0,\a}(M))$ satisfying \begin{equation}\label{E:PeriodicNull} \int_{M\times\O}\Phi d\mu=0\; \text{ for any random periodic } \phi\text{-invariant measure $\mu$ with marginal } \mathbb P, \end{equation} there exists a functional $\tilde\Psi:L^\infty(\O, M)\to \mathbb R$ which is $\a$-H\"older continuous and satisfies the following cohomological equation \begin{equation}\label{E:CoBoundary} \tilde\Phi(g)=\tilde\Psi(\tilde\phi(g))-\tilde \Psi(g),\ \forall g\in L^\infty(\O, M). \end{equation} Moreover, $\tilde \Psi$ is uniquely determined up to a constant functional. }
\noindent {\bf Remark:} The celebrated theorem of Liv\v sic \cite{Liv71, Liv72a,Liv72b} was established by Liv\v sic for hyperbolic diffeomorphisms. Since then, there is a large amount of literature on generalizations of this theorem, see, for example, \cite{AKL18,dlLW10,dlLW11,Kal11,KK96, KP16,NP13,NT95, NT96, Wil13}. Kifer and Liu \cite{KifLiu} conjectured that for a random Anosov diffeomorphism $f_\omega$ on $\mathbb{T}^d$, and a given function $\Phi(x,\omega)$ which is measurable in $(x,\omega)$ and $\a$-H\"older continuous in $x$, and satisfies $\int \Phi d\mu=0$ for any $\phi$-invariant measure $\mu$ with marginal $\mathbb P$, the following cohomological equation \[ \Phi=\Psi \circ \phi - \Psi+h, \]
has a solution $(\Psi(x, \omega), h(\omega))$, where $\Psi(x, \omega)$ is a measurable function satisfying the random H\"older condition $|\Psi(x, \omega)-\Psi(y,\omega)|\leq K_{\Psi}(\omega)(d(x, y))^\alpha$ with $\int \log K_{\Psi}(\omega) \;d {\mathbb P}< \infty$, and $h(\omega)$ is a random variable with $\int h \;d {\mathbb P}=0$.
Theorem D is a weak version of Liv\v sic theorem for random dynamical systems, in the proof of which a significant difference from the deterministic case shows up: Because of the absence of the separability the phase space $L^{\infty}(\O,M)$, there is no dense orbit any more, which plays a key role in the proof for the deterministic case. The proof for random case involves a different and more complicated procedure.
\subsection{Examples.} We study following two examples: system S1) and system S2) in section \ref{Examples}﹝ \begin{enumerate} \item[S1)] {\bf Anosov systems diven by a quasiperiodic forcing:} $(\t,\O)$ is a minimal irrational rotation on a compact torus of dimenison $d$, and $\phi$ is Anosov on fibers and topological transitive on $M\times \O$; \item[S2)] {\bf Measure preserving Anosov systems driven by a random forcing:} $\phi$ is Anosov on fibers, and there exists an $f$-invariant Borel probability measure $\nu$ with full support (i.e. $supp\nu=M$).
\end{enumerate} We show that both system $S1$ and system $S2$ are topological mixing on fibers. Thus, {\bf Theorem A-D} hold for both systems. More specific examples are also given. For system S1) we consider a $2$-$d$ tori automorphism driven by a quasiperiodic force and for system S2) we look at the random composition of $2\times 2$ area-preserving positive matrices with integer entries.
\noindent {\bf Remark on Horseshoe:}
System S1) is an Anosov system driven by a quasi-periodic forcing and is also a partially hyperbolic system. In fact, with each invariant Borel probability measure $\mu$, system S1) has zero Lyapunov exponent of multiplicity $d$ (the dimension of $\Omega=T^d$). Thus, the Katok's result (\cite{K}) does not apply to this system. Lian and Young's results (\cite{LY}, \cite{LY1}) for semiflows or flows, which allow only one-dimensional center associated to the flow direction, are also not applicable to this case because of high dimensional center. Futhermore, the quasi-periodic forced systems has no periodic points, and so has no standard horseshoe. However, because of the uniform hyperbolicity of the map between fibers ($f(\o):M_\o\to M_{\t\o}$) and the rigidity of the evolution of quasiperiodic forcing, one expects the co-existence of certain kinds of periodicity and chaotic behavior over some random invariant structures.
The main difficulty appearing in these systems is that the pseudo orbits derived from Poincar\'e Recurrence Theorem only provides returns in $M\times \Omega$, thus does not have the shadowing property since the hyperbolicity only happens on fibers which are identified by $M$. To overcome this difficulty, instead of looking at one single pseudo orbit, we construct a group of globally returned pseudo orbits for all $\o\in \Omega$ at the same time.
For non-uniformly hyperbolic systems, one needs to look at the return map on Pesin blocks (as being done in \cite{K}). If there is a globally returning time, the projection of such Pesin blocks to $\O$ should be a $\mathbb P$-full-measure-set. This is not satisfied even in the setting of \cite{GK} in which hyperbolicity depends on the sample ``$\o$". Such observation reveals that for non-uniform hyperbolic system, accurate information about ``shapes" of the Pesin blocks is required, which makes the situation much more complicated than autonomous case. Second, even dealing with systems which is Anosov on fibers, constructing a global return is not trivial, since one needs infinite many (usually uncountable many) orbits return at a same time. Our immediate feeling is that for non-uniformly hyperbolic systems on fibers, one may need a new approach.
Finally, we mention some existing results which are closely relevant to the results derived in this paper, which may shed a light on further research directions, for example, chaotic behavior of Anosov flows driven by quasi-periodic noises. Of course, this is far from being a complete picture. In \cite{G95}, Gundlach studied the random homoclinic points for smooth invertible random dynamical systems, and derived a random version of {\em Birkhoff-Smale Theorem}. This result is applicable to systems which are provided by small random perturbations of systems with homoclinic points. In the recent papers \cite{LW1} and \cite{LW2}, Lu and Wang studied the chaos behavior of a type of differential equations driven by a nonautononous force deduced from stochastic processes which are the truncated and classical Brownian motions in \cite{LW1} and in \cite{LW2} respectively. The authors extended the concept of topological horseshoe and showed that such structure (thus chaotic behavior) exists almost surely. Moreover, they further applied the results to randomly forced Duffing equations and pendulum equations. There are also many works on the existence of random periodic solutions for stochastic differential equations, see \cite{ZhZh}
\section{Setting, Definitions, and Statement of the Main Results}\label{S:Setting} In this section, we set up the systems, introduce notions, and give the statements of the main results precisely. Although some of the definitions have been given in Section \ref{S:Introduction}, we restate them formally for the sake of completeness and convenience.
Let $M$ be a connected compact smooth Riemannian manifold with no boundary, $d_M$ be the induced Riemannian metric of $M$, $(\O,d_\O)$ be a compact metric space, and $\t:\O\to\O$ be a homeomorphism.
Let $\mathcal H=\text{diff}^2(M)$ be the space of $C^2$ diffeomorphisms on $M$ with $C^2$-topology, and $f:\O\to \mathcal H$ be a continuous map. Then the skew product of $f$ and $\t$ induces a map $\phi:M\times \O\to M\times \O$ in the following way:
$$\phi(x,\o)=(f(\o)x,\t\o)=(f_\o x,\t\o),\ \forall x\in M,\o\in\O$$
where we rewrite $f(\o)$ as $f_\o$ for the sake of convenience. Clearly, $\phi$ induces a cocycle over $(\O,\t)$, which can be inductively defined by $$\begin{cases} \phi^0(x,\o)&=(x,\o)\\ \phi^n(x,\o)&=(f_{\t^{n-1}\o}\phi^{n-1}(x,\o),\t^n\o),\ n\ge 1\\ \phi^n(x,\o)&=(f^{-1}_{\t^{n+1}\o}\phi^{n+1}(x,\o),\t^n\o),\ n\le -1 \end{cases}. $$ Note that $M\times \O$ is a compact metric space when we introduce a metric $d$ by defining that
$$d((x_1,\o_2),(x_2,\o_2))=d_M(x_1,x_2)+d_\O(\o_1,\o_2), \forall (x_1,\o_1),(x_2,\o_2)\in M\times \O.$$
Then $\phi$ and $D_M\phi(\text{where }D_M\phi(x,\o):=Df_\o(x))$ are continuous on $M\times \O$. \begin{defn}\label{D:Anosov} The system generated by $\phi$ (or simply $\phi$) is called {\bf Anosov on fibers} if the following holds: for every $(x,\o) \in M\times \O$ there is a splitting of the tangential fiber of $M_\o:=M\times\{\o\}$ at $x$ \[ T_xM_\o=E_{(x,\o)}^u \oplus E_{(x,\o)}^{s} \] which depends continuously on $(x,\o) \in M\times \O$ with $\dim E_{(x,\o)}^u,\dim E_{(x,\o)}^s >0$ and satisfies that \[ Df_{\o}(x) E_{(x,\o)}^u=E_{\phi(x,\o)}^u, \ \ \ Df_{\o}(x) E_{(x,\o)}^{s}= E_{\phi(x,\o)}^{s}, \] and \begin{equation}\label{E:UniformHyperbolic} \left\{\begin{array}{ll}
|Df_{\o}(x) \xi| \geq e^{\l_0} |\xi|, \ \ \ &\forall\ \xi \in E_{(x,\o)}^u, \\
|Df_{\o}(x)\eta| \leq e^{-\l_0}|\eta|, \ \ \ &\forall\ \eta \in E_{(x,\o)}^{s}, \end{array} \right. \end{equation} where $\lambda_0>0$ is a constant. \end{defn} \begin{rem}\label{R:ContInvDistr} It is clear that the splitting $T_xM_\o=E_{(x,\o)}^u \oplus E_{(x,\o)}^{s}$ is uniformly continuous on $M\times\O$. Actually, one has that the above splitting is H\"older continuous on $x$, which follows from the standard arguments. Moreover, the H\"older constant and exponent can be chosen independent on $(x,\o)$. For more details, we refer the readers to \cite{Brin}, Section 5.3 of \cite{BaPe}, and Section 5.4 of \cite{LLL}. \end{rem}
Denote $\pi^u_{(x,\o)},\pi^s_{(x,\o)}$ the projections associated to the splitting $T_xM_\o=E_{(x,\o)}^u \oplus E_{(x,\o)}^{s}$, which map $T_xM_\o$ onto $E_{(x,\o)}^u$ and $E_{(x,\o)}^{s}$ respectively. Since the splitting $T_xM_\o=E_{(x,\o)}^u \oplus E_{(x,\o)}^{s}$ is uniformly continuous on $M\times\O$, there exists a real number $\mathcal P>1$ such that \begin{equation}\label{E:PrjBound}
\max\left\{\|\pi_{(x,\o)}^{\tau}\|\big|\ (x,\o)\in M\times\O, \tau=u,s\right\}<\mathcal P. \end{equation}
The general systems we consider is topological mixing on fibers which is defined as follows: \begin{defn}\label{D:Mixing} $\phi:M\times \O\to M\times \O$ is called {\bf topological mixing on fibers} if for any nonempty open sets $U,V\subset M$, there exists $N>0$ such that for any $n\ge N$ and $\o\in \O$ $$\phi^n(U \times \{\o\})\bigcap V\times \{\t^n\o\}\neq \emptyset.$$ \end{defn} In the following, we first introduce two ingredients of dynamical complexity: {\bf \em random periodic orbit} and {\bf \em random horseshoe.}
Let $L^\infty(\O, M)$ be the space of Borel measurable maps from $\O$ to $M$ endowed with the metric $d_{L^\infty(\O, M)}$: $$d_{L^\infty(\O, M)}(g_1,g_2)=\sup_{\o\in\O}d_M(g_1(\o),g_2(\o)),\ \forall g_1,g_2\in L^\infty(\O, M).$$ For later use, we also define two functions on $L^\infty(\O, M)\times L^\infty(\O, M)$ to measure the separations of elements of $L^\infty(\O, M)$ on different levels: \begin{align}\begin{split}\label{E:SeparationFunc} \overline d_{L^\infty(\O, M)}(g_1,g_2)&=\sup_{\O'\subset \O\text{ is open and nonempty}} \inf_{\o\in \O'}d_M(g_1(\o),g_2(\o)),\ \forall g_1,g_2\in L^\infty(\O, M).\\ \underline d_{L^\infty(\O, M)}(g_1,g_2)&=\inf_{\o\in\O}d_M(g_1(\o),g_2(\o)),\ \forall g_1,g_2\in L^\infty(\O, M). \end{split} \end{align}
Since $\phi$ and $\phi^{-1}$ are both continuous, $\phi$ induces a map $\tilde \phi$ on $L^\infty(\O, M)$ given by
\[
\tilde \phi(g)(\omega)=f_{\theta^{-1}\omega} g(\theta^{-1}\omega).
\]
\begin{defn}\label{D:RandomPeriodic}
$\{graph(g_i)|\ g_i\in L^\infty(\O, M)\}_{1\le i\le n}$ is called a {\em random periodic orbit} of $\phi$ with period $n$ if
for all $1\le i\le n$ $$\phi(\text{graph}(g_i))=\text{graph}(g_{i+1\mod n})\ \forall 1\le i\le n,$$
equivalently,
\begin{equation}\label{E:PeriodicPointTPh}
g_i \text{ is a periodic point of }\tilde \phi\text{ i.e. }\tilde \phi (g_i)=g_{i+1\mod n}\ \forall 1\le i\le n.
\end{equation} \end{defn} Each $graph(g_i)$ is called a random periodic point of $\phi$. Without causing any confusion and for the sake of convenience, we call $g_i$ a random periodic point for short.
Moreover, if $g_i$ is a continuous map then we call $g_i$ a continuous random periodic point.
\begin{rem}\label{R:RandomDynSys} The system generated by $\phi$ is usually called a skew-product system rather than a random dynamical system, which makes the name " random periodic orbit" sound a little bit more in name than reality. Nevertheless, once a $\t$-invariant measure $\mathbb P$ on $\O$ is introduced, the resulting system induced by $\phi$ becomes a random dynamical systems over the metric dynamical system $(\O,\mathbb P,\t)$ (see \cite{A} for more details). On the other hand, since the name "random periodic orbit" has been introduced in exiting literature (see \cite{Kl, ZhZh}) and has the essentially equivalent meaning, we prefer not to introduce different name or notions. \end{rem}
Next, we introduce the concept of random horseshoes in different levels, which are distinguished by the separation functions given by (\ref{E:SeparationFunc}).
Let $\mathcal S_k=\{1,\ldots,k\}^{\mathbb Z}$ be the space of two sides sequences of $k(\geq 2))$ symbols endowed with the standard metric, and $\s$ be the (left-)shift map. The following is the standard definition of Smale horseshoe for the deterministic map $\tilde \phi$.
\begin{defn}\label{D:Horseshoe}
We call a continuous embedding $\Psi:\mathcal S_k\to L^\infty(\O, M)$ a {\bf \em horseshoe} with $k$-symbols of $\tilde \phi$ if the following hold: \begin{itemize}
\item[i)] There exists $\Delta>0$ such that for any $l\in\mathbb Z$ and $\hat a_1,\hat a_2\in \mathcal S_k\text{ with }\hat a_1(l)\neq \hat a_2(l)$,
\begin{equation}\label{E:SeparationImgSymbols}
d^{\tilde \phi}_{l}\left(\Psi({\hat a_1}),\Psi({\hat a_2})\right)>\Delta,
\end{equation}
where for $l\ge 0$
$$d^{\tilde \phi}_{l}(g_1,g_2):=\max_{0\le i\le l}\left\{d_{L^\infty(\O, M)}\left(\tilde \phi^i(g_1),\tilde \phi^i(g_2)\right)\right\},\ g_1,g_2\in L^\infty(\O, M)$$ is the Bowen metric induced by $\tilde \phi$ and for $l<0$ we let $d^{\tilde\phi}_{l}=d^{\tilde\phi^{-1}}_{|l|};$ \item[ii)] For all $\hat a\in\mathcal S_k$ $$\tilde \phi\left(\Psi({\hat a})\right)=\Psi({\s \hat a}).$$ \end{itemize} \end{defn}
The next definition we introduce here is the random horseshoe for a skew-product system, which has a slightly finer structure by employing the separation function $\overline d_{L^\infty(\O, M)}$ in (\ref{E:SeparationFunc}) to replace $d_{L^\infty(\O, M)}$.
\begin{defn}\label{D:RandomHorseshoe}
We call a continuous embedding $\Psi:\mathcal S_k\to L^\infty(\O, M)$ a {\bf \em full random horseshoe} with $k$-symbols of $\phi$ if the following hold: \begin{itemize}
\item[i)] There exists $\Delta>0$ such that for any $l\in\mathbb Z$ and $\hat a_1,\hat a_2\in \mathcal S_k\text{ with }\hat a_1(l)\neq \hat a_2(l)$
\begin{equation}\label{E:RandomSepaImgSymb}
\overline d^{\tilde \phi}_{l}\left(\Psi({\hat a_1}),\Psi({\hat a_2})\right)>\Delta,
\end{equation}
where for $l\ge 0$
$$\overline d^{\tilde \phi}_{l}(g_1,g_2):=\max_{0\le i\le l}\left\{\overline d_{L^\infty(\O, M)}\left(\tilde \phi^i(g_1),\tilde \phi^i(g_2)\right)\right\},\ g_1,g_2\in L^\infty(\O, M)$$ and for $l<0$ we let $\overline d^{\tilde\phi}_{l}=\overline d^{\tilde\phi^{-1}}_{|l|};$ \item[ii)] For all $\hat a\in\mathcal S_k$ $$\tilde \phi\left(\Psi({\hat a})\right)=\Psi({\s \hat a}).$$ \end{itemize} \end{defn}
\noindent Finally, by using the stronger separation function $\underline d_{L^\infty(\O, M)}$ as in (\ref{E:SeparationFunc}), we define a stronger random horseshoe as follows:
\begin{defn}\label{D:SRandomHorseshoe}
We call a continuous embedding $\Psi:\mathcal S_k\to L^\infty(\O, M)$ a {\bf \em strong full random horseshoe} with $k$-symbols of $\phi$ if the following hold: \begin{itemize}
\item[i)] There exists $\Delta>0$ such that for any $l\in\mathbb Z$ and $\hat a_1,\hat a_2\in \mathcal S_k\text{ with }\hat a_1(l)\neq \hat a_2(l)$
\begin{equation}\label{E:SRandomSepaImgSymb}
\underline d^{\tilde \phi}_{l}\left(\Psi({\hat a_1}),\Psi({\hat a_2})\right)>\Delta,
\end{equation}
where for $l\ge 0$
$$\underline d^{\tilde \phi}_{l}(g_1,g_2):=\max_{0\le i\le l}\left\{\underline d_{L^\infty(\O, M)}\left(\tilde \phi^i(g_1),\tilde \phi^i(g_2)\right)\right\},\ g_1,g_2\in L^\infty(\O, M)$$ and for $l<0$ we let $\underline d^{\tilde\phi}_{l}=\underline d^{\tilde\phi^{-1}}_{|l|};$ \item[ii)] For all $\hat a\in\mathcal S_k$ $$\tilde \phi\left(\Psi({\hat a})\right)=\Psi({\s \hat a}).$$ \end{itemize} \end{defn}
\begin{rem}\label{R:CompareHorseshoes} Clearly, we have that $d_{L^\infty(\O, M)}\ge\overline d_{L^\infty(\O, M)}\ge\underline d_{L^\infty(\O, M)}.$ Therefore, we have that a strong full random horseshoe is automatically a full random horseshoe, and a full random horseshoe of $\phi$ automatically induces a horseshoe of $\tilde \phi$. \end{rem}
Before stating the main results, we introduce several notations. Let
\begin{align*}
&N(\o,\e,n):=\\&\quad\quad\quad\quad\quad\max\left\{card(E)\big|\ E\subset M_{\o}, E \text{ is an }\e\text{ separated set with respect to }d^{\phi}_{M\times \O,n}\right\},
\end{align*} where $d^\phi_{M\times \O,n}$ is the Bowen metric. Let
$$h_{top}(\phi|_{M_\o})=\lim_{\e\to 0^+}\limsup_{n\to\infty}\frac1n\log N(\o,\e,n),$$
$$\underline{h}(M\times\O|\O):=\inf_{\o\in\O} h_{top}(\phi|_{M_\o}),$$
$$ \overline{h}(M\times\O|\O):=\sup_{\o\in\O} h_{top}(\phi|_{M_\o}).$$
Here $h_{top}(\phi|_{M_\o})$ is called the topological fiber entropy on the fiber $M_\o$, $\underline{h}(M\times\O|\O)$ and $\overline{h}(M\times\O|\O)$ are called lower and upper topological fiber entropy respectively.
The systems we consider here satisfy the following two hypotheses: \begin{itemize} \item[{\bf H1)}] $\phi$ is Anosov on fibers. \item[{\bf H2)}] $\phi$ is topological mixing on fibers. \end{itemize} \begin{thm}[Density of Random Periodic Points]\label{T:TheoryAnosovMix}
Let $\phi$ satisfy condition H1) and H2). Then, for any $\e>0$, there exists $N\in\mathbb N$ such that for any $g\in L^{\infty}(\O)$ and $n\ge N$, there exists a random periodic point $\tilde g$ with period $n$ such that
\begin{equation*}\label{E:LInfiClose2}
d_{L^{\infty}}(g,\tilde g)\le \e.
\end{equation*}
\end{thm}
\begin{thm}[Entropy and Random Horseshoe]\label{T:TheoryAnosovMix2}
Let $\phi$ satisfy condition H1)and H2). Then the following assertions hold:
\begin{itemize}
\item[{\bf A.}] For any $\g>0$, there exist $N,k\in \mathbb N$ such that the following hold
\begin{enumerate}
\item[i)] $\frac1N\ln k\ge \underline{h}(M\times\O|\O)-\g$, where $\underline{h}(M\times\O|\O)>0$;
\item[ii)] $\phi^N$ has a $k$-symbol strong full random horseshoe.
\end{enumerate}
\item[{\bf B.}] For any $\g>0$, there exist $N,k\in \mathbb N$ such that the following hold
\begin{enumerate}
\item[i)] $\frac1N\ln k\ge \overline{h}(M\times\O|\O)-\g$;
\item[ii)] $\phi^N$ has a $k$-symbol full random horseshoe.
\end{enumerate}
\end{itemize}
\end{thm}
Next, we fix a $\t$-invariant probability measure $\mathbb P$ on $\O$, and consider the random dynamical systems over $\theta$.
For $g\in L^{\infty}(\O)$, we define a Borel probability measure, $\mu_g$, in $\mathcal M(M\times\O)$ as follows: for any Borel set $A\subset M\times \O$ \begin{equation}\label{E:Mug}
\mu_g(A)=\mathbb P\left(\left\{\o\in\O\big|\ (g(\o),\o)\in A\cap M\times\{\o\}\right\}\right). \end{equation} Let $ \mathcal I_{\mathbb P}(M\times\O)$ be the collection of $\phi$-invariant measures whose marginal are $\mathbb P$. Now, we define the periodic measures. \begin{defn}\label{D:PeriodicMeasure} $\mu\in \mathcal I_{\mathbb P}(M\times\O)$ is called a random periodic measure if and only if there is a random periodic orbit $\{g_i\}_{i=1,\cdots,n}$ such that $$\mu=\frac1n\sum_{i=1}^n\mu_{g_i}=\frac1n\sum_{i=1}^n\mu_{\tilde \phi^i(g_1)}.$$ \end{defn} The following theorem states that any $\phi$-invariant measure with marginal $\mathbb P$ can be approximated by random periodic measures. \begin{thm}[Density of Random Periodic Invariant Measures]\label{T:PeriodicApprox} Let $\phi$ satisfy H1) and H2). Then, the set of all periodic measures is dense in $\mathcal I_{\mathbb P}(M\times\O)$ in the narrow topology. \end{thm} \begin{rem}\label{R:NonErgPer} We remark here that, unlike the standard periodic measure supported by one periodic orbit, the random periodic measure supported by one random periodic orbit is {\bf not} necessarily ergodic. \end{rem}
Finally, we present a weak version of the classic Liv\v sic Theorem for random dynamical systems. For the $\t$-invariant probability measure $\mathbb P$ and $\a\in(0,1)$, denote $L^1_{\mathbb P}(\O,C^{0,\a}(M))$ the collection of functions $\Phi$ on $M\times\O$, which are H\"older on $M$ and measurable on $\O$, and satisfy the following condition \begin{equation}\label{E:L1Holder}
\|\Phi\|_{L^1_{\mathbb P}(\O,C^{0,\a}(M))}:=\int_{\O}\|\Phi\|_{C^{0,\a}(M_\o)}d\mathbb P<\infty, \end{equation} where
$$\|\Phi\|_{C^{0,\a}(M_\o)}:=\sup_{x\in M_\o}|\Phi(x,\o)|+\sup_{x,y\in M_\o,\ x\neq y}\frac{|\Phi(x,\o)-\Phi(y,\o)|}{\left(d_M(x,y)\right)^\a}.$$ For any $g\in L^\infty(\O, M)$ and $\Phi\in L^1_{\mathbb P}(\O,C^{0,\a}(M))$, define a functional $\tilde \Phi$ on $L^\infty(\O, M)$ as follows \begin{equation}\label{E:Phi(g)} \tilde \Phi(g)=\int_{\O}\Phi(g(\o),\o)d\mathbb P. \end{equation} \begin{rem}\label{R:WellDefined} First, one needs to show that the above integration is well defined and is finite. Since $\Phi(x,\o)$ is continuous on $x$ and Borel measurable on $\o$, we have that $\Phi$ is Borel measurable on $M\times\O$. Then, for any real number $a\in \mathbb R$,
$U:=\Phi^{-1}\left((-\infty,a)\right)$ is a Borel subset of $M\times \O$. Note that $graph(g)$ is also a Borel subset of $M\times \O$ since $g:\O\to M$ is Borel. Therefore, $U\cap graph(g)$ is Borel in $M\times\O$. By applying Theorem \ref{T:BMST}, we have that $$\left\{\o|\ \Phi(g(\o),\o)<a\right\}= \pi_\O\left(U\cap graph(g)\right)\text{ is Borel in }\O.$$ Hence $\Phi(g(\cdot),\cdot):\O\to \mathbb R$ is Borel measurable. As $\Phi$ satisfying (\ref{E:L1Holder}), we have that (\ref{E:Phi(g)}) is well defined and is finite.
\end{rem}
We call the following theorem a weak version of random Liv\v sic Theorem. In the case that there is no confusion caused, we simply use "periodic measures with marginal $\mathbb P$" short for "periodic $\phi$-invariant Borel probability measure with marginal $\mathbb P$" for the sake of convenience. \begin{thm}[Random Liv\v sic Theorem]\label{T:WLivTh} Let $\phi$ satisfy condition H1) and H2). Then for any $\Phi\in L^1_{\mathbb P}(\O,C^{0,\a}(M))$ satisfying \begin{equation}\label{E:PeriodicNull} \int_{M\times\O}\Phi d\mu=0\text{ when }\mu\text{ is periodic with marginal } \mathbb P, \end{equation} there exists a functional $\tilde\Psi:L^\infty(\O, M)\to \mathbb R$ which is $\a$-H\"older continuous and satisfies the following cohomological equation \begin{equation}\label{E:CoBoundary} \tilde\Phi(g)=\tilde\Psi(\tilde\phi(g))-\tilde \Psi(g),\ \forall g\in L^\infty(\O, M). \end{equation} Moreover, $\tilde \Psi$ is uniquely determined up to a constant functional. \end{thm}
\section{Preliminary Lemmas}\label{S:Preliminary}
This section is devoted to the preparation on technical tools needed for the proof of main theorems. These fundamental tools stated in this section only require the systems satisfy H1). \subsection{Invariant Manifolds}\label{S:InvMan} We define the local stable manifolds and unstable manifolds as follows: \begin{align*}
&W^s_\e(x,\o)=\{(y,\o)\in M_\o|\ d_M(\pi_M\phi^n(y,\o),\pi_M\phi^n(x,\o))\le \e\text{ for all }n\ge 0\}\\
&W^u_\e(x,\o)=\{(y,\o)\in M_\o|\ d_M(\pi_M\phi^n(y,\o),\pi_M\phi^n(x,\o))\le \e\text{ for all }n\le 0\}, \end{align*} where $\pi_M:M\times \O\to M$ is the nature coordinate projection. The following lemma can mainly be viewed as a special version of Theorem 3.1 from \cite{GK}. The only difference is that these local manifolds depends on $\o$ measurably in \cite{GK} while depends on $\o$ continuously in this paper. The reason is that the invariant splitting $E^u_{(x,\o)}\oplus E^s_{(x,\o)}$ varies continuously in both $x$ and $\o$, and $\phi$ and $D_M\phi$ are also continuous in $\o$. The proof of iii) follows from the standard procedure for which we refer to the proof of Lemma 3.1 in \cite{LLL}. Thus, we omit the proof of the following Lemma \ref{L:InvMani}. \begin{lem}\label{L:InvMani} For any $\l\in (0,\l_0)$, there exists $\e_0>0$ such that for any $\e\in(0,\e_0]$, the following hold: \begin{itemize} \item[i)] $W^s_\e(x,\o), W^u_\e(x,\o)$ are $C^2$-embedded discs for all $(x,\o)\in M\times \O$ which satisfy \[T_xW_\e^\tau(x,\o)=E^\tau_{(x,\o)}, \text{for}
;\tau=u,s.\] Moreover, there exist a constant $L>1$ and $C^2$ maps $$g^u_{(x,\o)}:B^u_{\mathcal P\e}(x,\o)\to E^s_{(x,\o)} \text{ and }g^s_{(x,\o)}:B^s_{\mathcal P\e}(x,\o)\to E^u_{(x,\o)} $$
such that $$W^\tau_\e(x,\o)\subset Exp_{(x,\o)}\left(graph(g^\tau_{(x,\o)})\right),$$ and $\|Dg^\tau_{(x,\o)}\|<\frac1{10\mathcal P}$, $Lip Dg^\tau_{(x,\o)}<L$ for $\tau=u,s$, where $\mathcal P$ is as in (\ref{E:PrjBound}) and $B^\tau_{\mathcal P \e}(x,\o)$ is the $\mathcal P\e$-ball of $E^\tau_{(x,\o)}$ centered at origin; \item[ii)] $d_M(\pi_M\phi^n(x,\o),\pi_M\phi^n(y,\o))\le e^{-n\l}d_M(x,y)$ for $(y,\o)\in W^s_\e(x,\o)$, $n\ge 0$, and\\ $d_M(\pi_M\phi^{-n}(x,\o),\pi_M\phi^{-n}(y,\o))\le e^{-n\l}d_M(x,y)$ for $(y,\o)\in W^u_\e(x,\o)$, $n\ge 0$; \item[iii)] $W^s_\e(x,\o), W^u_\e(x,\o)$ vary continuously on $(x,\o)$ (in $C^1$ topology). \end{itemize} \end{lem} The local stable and unstable manifolds can be used to construct the global stable and unstable manifolds respectively, \begin{align*}
&W^s(x,\o)=\{(y,\o)\in M_\o|\ |d_M(\pi_M\phi^n(y,\o),\pi_M\phi^n(x,\o))\to 0\text{ as }n\to \infty\}\\
&W^u(x,\o)=\{(y,\o)\in M_\o|\ |d_M(\pi_M\phi^n(y,\o),\pi_M\phi^n(x,\o))\to 0\text{ as }n\to -\infty\}, \end{align*} as \begin{align*} &W^s(x,\o)=\bigcup_{n=0}^\infty \phi^{-n}(W^s_\e(\phi^n(x,\o))),\\ &W^u(x,\o)=\bigcup_{n=0}^\infty \phi^{n}(W^u_\e(\phi^{-n}(x,\o))), \end{align*} where $\e$ is an arbitrarily fixed small positive number as in \ref{L:InvMani}. The following lemma provides local canonical coordinates on $M_\o$. \begin{lem}\label{L:LocCoor} For any $\e\in(0,\e_0)$ there is a $\d\in (0,\e)$ such that for any $x,y\in M$ with $d_M(x,y)<\d$, $W^s_\e(x,\o)\bigcap W^u_\e(y,\o)$ consists of a single point, which is denoted by $[x,y]_\o$. Furthermore
$$[\cdot,\cdot]_{\cdot}:\{(x,y,\o)\in M\times M\times \O|\ d_M(x,y)<\d\}\to M\text{ is continuous}.$$ We say the system $(M\times \O,\phi)$ has {\bf local product structure} with size $\d$. \end{lem} \begin{proof} This simply follows from iii) of Lemma \ref{L:InvMani}, the continuous (thus uniformly continuous) dependence of $W^s_\e(x,\o),W^u_\e(x,\o)$ on $(x,\o)$, and the uniform continuity of the invariant splitting $E^u_{(x,\o)}\oplus E^s_{(x,\o)}$. For details, we refer to \cite{Bow4}. \end{proof} \begin{lem}\label{L:Expansive} There is an $\e>0$ such that for any $\o\in\O$ and $x,y\in M_\o$ with $y\neq x$, the following holds: $$d_M(\pi_M\phi^k(x,\o),\pi_M\phi^k(y,\o))>\e\text{ for some }k\in\mathbb Z.$$ \end{lem} \begin{proof} Take $\e$ smaller than $\e_0$ as in Lemma \ref{L:InvMani}. Otherwise, $(y,\o)\in W^s_\e(x,\o)\cap W^u_\e(x,\o)$, therefore $y=x$. \end{proof} \vskip0.1in
\subsection{Shadowing Lemma}\label{S:Shadow} We first introduce some relevant concepts. For any $\a>0$, an orbit $\{(x_i,\t^i\o)\}_{i\in \mathbb Z}\subset M\times \O$ is called {\bf $(\o,\a)$-pseudo orbit} if for any $i\in \mathbb Z$ $$d_M(f_{\t^i\o}(x_i),x_{i+1})<\a.$$ \begin{lem}\label{L:Shadowing} For any $\b>0$, there exists an $\a>0$ such that for any $(\o,\a)$-pseudo orbit $\{(x_i,\t^i\o)\}_{i\in \mathbb Z}$, there is a true orbit $\{(y_i,\t^i\o)\}_{i\in \mathbb Z}$ of $\phi$ such that $$\sup_{i\in \mathbb Z}d_M(y_i,x_i)<\b.$$ $\{(y_i,\t^i\o)\}_{i\in \mathbb Z}$ is called {\bf $(\o,\b)$-shadowing} orbit of $\{(x_i,\t^i\o)\}_{i\in \mathbb Z}$.\\
Further more, there is a $\b_0>0$ such that for any $0<\b<\b_0$, the above true orbit is unique. \end{lem} \begin{proof} The proof follows from the proof of Proposition 3.6 in \cite{GK}, thus is omitted here. \end{proof} \begin{rem}\label{R:Beta0} For the sake of convenience, we remark here that $\b_0$ is always chosen to be less than $\frac13\d$, where $\d$ is the one in Lemma \ref{L:LocCoor} for $\e=\frac12\e_0$. \end{rem} The next lemma gives the detailed relation between "$\a$" and "$\b$" in Lemma \ref{L:Shadowing} above. \begin{lem}\label{L:ClosingLemma} There exist $\a_0>0$ and $C\ge 1$ so that for any $\a\in (0,\a_0)$ and any $(\o,\a)$-pseudo orbit $\{(x_i,\t^i\o)\}_{i\in\mathbb Z}$, there exists a true orbit $\{(y_i,\t^i\o)\}_{i\in\mathbb Z}$ of $\phi$ satisfying the following $$\sup_{i\in \mathbb Z} d_M(y_i,x_i)<C\a.$$ \end{lem} \begin{proof} By i) and iii) of Lemma \ref{L:InvMani}, for any $\e_1>0$, there exist $\d_1, L_1>0$ such that the following properties hold \begin{itemize} \item[P1)] For any $x,y\in M$ with $d_M(x,y)<\d_1$, $[x,y]_\o=W^s_{\e}(x,\o)\cap W^u_{\e}(y,\o)$ is well defined, where $\e$ is as in Lemma \ref{L:InvMani};
\item[P2)] For any $x,y\in M$ with $d_M(x,y)<\d_1$ $$\max\left\{d_M([x,y]_\o,x),d_M([x,y]_\o,y)\right\}\le L_1d_M(x,y);$$
\item[P3)] For any $x,y\in M$ with $d_M(x,y)<\d_1$
$$\max\left\{\left|d_M([x,y]_\o,x)-d_M([y,x]_\o,y)\right|,\ \left|d_M([y,x]_\o,x)-d_M([x,y]_\o,y)\right|\right\}\le \e_1d_M(x,y).$$ \end{itemize} It is obvious that P1) holds when taking $\d_1<\d$ where $\d$ is as in Lemma \ref{L:LocCoor}. The proof of the existence of such $\d_1$ and $L_1$ to fit P2) and P3) may be standard, nevertheless, we sketch the proof briefly for the sake of completeness. Note that the above properties only appear in a local scenario, for the sake of simplicity, we fix an atlas of the manifold and always keep our discussion in one single local chart in the rest of the proof. Without causing any confusion, we identify $T_xM_\o$ and $T_yM_\o$ to $\mathbb R^n$ for $d_M(x,y)$ small, thus such $x$ and $y$ are considered to be two points in $\mathbb R^n$ thus $x-y$ are well defined, and $\exp{(E^u_{(x,\o)})},\ \exp{(E^s_{(x,\o)})}$ are considered to be hyperplanes in $\mathbb R^n$, which contain the point $x\in \mathbb R^n$. We also note that the distance in the local chart $\mathbb R^n$ is equivalent to the metric $d_M$ and the comparison constant is some system constant, therefore we will not distinguish them in the local argument.\\
Now we propose the condition of $\d_1$ and $L_1$ which make P2) valid. As discussed in Remark \ref{R:ContInvDistr}, the splitting $T_xM_\o=E^u_{(x,\o)}\oplus E^s_{(x,\o)}$ is uniformly continuous on $(x,\o)$, thus one can make $E^u_{(x,\o)}\oplus E^s_{(x,\o)}$ closed enough to $E^u_{(y,\o)}\oplus E^s_{(y,\o)}$ by taking $d_M(x,y)$ small enough. Precisely, consider the splittings $$\mathbb R^n=E^u_{(x,\o)}\oplus E^s_{(y,\o)},\text{ with associated projections } \pi^u_{x,y}, \pi^s_{x,y},$$ and $$\mathbb R^n=E^u_{(y,\o)}\oplus E^s_{(x,\o)},\text{ with associated projections } \pi^u_{y,x}, \pi^s_{y,x},$$ then we request that for any $d_M(x,y)<\d_1$, the following hold \begin{equation}\label{E:ErrProj}
\max\left\{\|\pi^\tau_{x,y}-\pi^\tau_{(z,\o)}\|,\|\pi^\tau_{y,x}-\pi^\tau_{(z,\o)}\||\right\}<\e', \tau=u,s,\ z=x,y, \end{equation} where $\e'>0$ is a prefixed small constant.
Let $z_1=[x,y]_\o$ and $z_2=\exp(E^u_{(y,\o)})\cap \exp(E^s_{(x,\o)})$. Note that, locally, $z_1-z_2$, $\pi^s_{y,x}(z_1-z_2)$ and $\pi^u_{y,x}(z_1-z_2)$ are well defined and satisfy the following:
$$d_M\left(z_1,\exp(E^u_{(y,\o)})\right)\ge\frac{|\pi^s_{y,x}(z_1-z_2)|}{\|\pi^s_{y,x}\|} \ge\frac{|\pi^s_{y,x}(z_1-z_2)|}{\mathcal P+\e'},$$ and
$$d_M\left(z_1,\exp(E^s_{(x,\o)})\right)\ge\frac{|\pi^u_{y,x}(z_1-z_2)|}{\|\pi^u_{y,x}\|} \ge\frac{|\pi^u_{y,x}(z_1-z_2)|}{\mathcal P+\e'}.$$
Since $|z_1-z_2|\le |\pi^u_{y,x}(z_1-z_2)|+|\pi^s_{y,x}(z_1-z_2)|$, without losing generality, we assume that $|\pi^s_{y,x}(z_1-z_2)|>\frac12|z_1-z_2|$. By i) of Lemma \ref{L:InvMani}, we have that \begin{equation}\label{E:P2)1}
|z_1-z_2|\le 2(P+\e')d_M\left(z_1,\exp(E^u_{(y,\o)})\right)\le \frac{2(P+\e')}{10\mathcal P}d_M(y,z_1). \end{equation} Also note that \begin{align*}
&d_M(y,z_2)\le \|\pi^s_{y,x}\|d_M(x,y)\le (\mathcal P+\e')d_M(x,y),\\
&d_M(y,z_1)\le d_M(y,z_2)+|z_1-z_2|. \end{align*} Then we have
$$|z_1-z_2|\le \frac{2(P+\e')^2}{10\mathcal P}d_M(x,y)+\frac{2(P+\e')}{10\mathcal P}|z_1-z_2|,$$ which implies that \begin{equation}\label{E:P2)2}
|z_1-z_2|\le L_1'd_M(x,y), \text{ where }L_1'=\frac{\frac{2(P+\e')^2}{10\mathcal P}}{1-\frac{2(P+\e')}{10\mathcal P}}. \end{equation} Thus, by taking $L_1=L_1'+P+\e'$, we have that
$$d_M([x,y]_\o,y)\le |z_1-z_2|+d_M(y,z_2)\le L_1d(x,y).$$ The other part follows exactly from the same argument. Proof for P2) is completed.\\
Next, we sketch the proof for P3). Let $z_2=\exp(E^u_{(y,\o)})\cap \exp(E^s_{(x,\o)})$ which is the same as above and $z_3=\exp(E^s_{(y,\o)})\cap \exp(E^u_{(x,\o)})$. Then we have \begin{align}\begin{split}\label{E:P3)1}
&\left| |y-z_3|-|x-z_2|\right|\\
\le&\left||y-z_3|-|\pi^u_{(x,\o)}(x-y)|\right|+\left||x-z_2|-|\pi^u_{(x,\o)}(x-y)|\right|\\
=&\left||\pi^s_{x,y}(y-x)|-|\pi^u_{(x,\o)}(x-y)|\right|+\left||\pi^s_{y,x}(y-x)|-|\pi^u_{(x,\o)}(x-y)|\right|\\
\le &2\e'|x-y|, \end{split} \end{align} where $\e'>0$ is a prefixed small number from (\ref{E:ErrProj}. To complete the proof of P3), we need to estimate the distance between $z_1=[x,y]_\o$ and $z_2$ (similarly the distance betwwen $z_4:=[y,x]_\o$ and $z_3$), which has been given by (\ref{E:P2)1}) and (\ref{E:P2)2}). The last step is to take $\d_1>0$ small enough such that the constant "$10\mathcal P$" in (\ref{E:P2)1}) and (\ref{E:P2)2}) can be replaced by a large enough constant $K>0$ so that $L'_1<\frac14\e_1$. This can be done, because of "$LipDg^\tau_{(x,\o)}<L$" from i) and iii) of Lemma \ref{L:InvMani}. Combining with (\ref{E:P3)1}), we have that P3) holds when we take $\e'<\frac18\e_1$.
We now are ready to prove the lemma.
By the setting of $\phi$, there exists $L_2\ge 1$ such that the Lipschitz constants of $f_\o$ and $f^{-1}_\o$ are less that $L_2$ for all $\o\in\O$.
Set $C_0:=\frac{(\e_1+L_1)(L_2+1)}{1-\e_1-(1+\e_1)e^{-\l}}$, where we require the smallness of $\e_1$ to ensure the positivity of $C_0$. Now take $\a_0>0$ with $4C_0\a_0<\d_1$, we will show that there exists $C\ge 1$ satisfying the required properties.
We call a subset $R_\o\subset M_\o$ a {\em rectangle} if its diameter is small and $[x,y]_\o\in R_\o$ for all $x,y\in R_\o$. Let $A,B\subset M$ be Borel sets with diameter less than $2\e$ and their Hausdorff distance are smaller than $\d_1$. It is not hard to see that $[A,B]_\o:=\{[x,y]_\o|\ x\in A, y\in B\}$ is a rectangle in $M_\o$. For a rectangle $R_\o\subset M_\o$, define that $$\partial ^sR_\o=\{x\in R_\o:x\notin int(W^u_{\e}(x,\o)\cap R_\o)\}$$ $$\partial ^uR_\o=\{x\in R_\o:x\notin int(W^s_{\e}(x,\o)\cap R_\o)\}$$
For a given $\a\in(0,\a_0)$, $x,y\in M_\o$ and $z\in M_{\t\o}$ satisfying that $d_M(x,y)<\a$ and $d_M(f_\o(x),z)<\a$, we have the following: \begin{itemize} \item[p1)] $d_M(f_\o(y),z)\le d_M(f_\o(y),f_\o(x))+d_M(f_\o(x),z) \le (L_2+1)\a$. \item[p2)] $\max\{d_M([z,f_\o(y)]_{\t\o},z),d_M([f_\o(y),z]_{\t\o},y)\}\le L_1(L_2+1)\a$ by P2). \item[p3)] For any $y'\in f_\o(W^s_{C_0\a}(y,\o))$, by P2), p1) and p2), \begin{align*} d_M([z,y']_{\t\o},z) &\le d_M([z,y']_{\t\o},[z,f(y)]_{\t\o})+d_M([z,f(y)]_{\t\o},z)\\ &\le d_M(y',f_\o(y))+\e_1d_M(y',[z,f(y)]_{\t\o})+d_M([z,f(y)]_{\t\o},z)\\ &\le C_0\a e^{-\l}+\e_1(d_M(y',f_\o(y))+d_M(f_\o(y),z))+L_1(L_2+1)\a\\ &\le (1+\e_1)C_0\a e^{-\l}+(\e_1+L_1)(L_2+1)\a\\ &= (1-\e_1)C_0\a. \end{align*} \item[p4)] Similar to the arguments for p3), we have that for any $z'\in f_{\t\o}^{-1}(W^u_{C_0\a}(z,\t\o))$ $$ d_M([z',y]_\o,y)\le (1-\e_1)C_0\a.$$ \end{itemize} Let $R_\o(y):=\overline {[W^u_{C_0\a}(y,\o),W^s_{C_0\a}(y,\o)]_\o}$. By P1) and P3), we have that for any $x\in R_\o(y)$ \begin{align}\begin{split}\label{E:DisEst1} d_M(x,y)&\le d_M(x,[y,x]_\o)+d_M([y,x]_\o,y)\\ &\le L_1d_M([x,y]_\o,[y,x]_\o)+C_0\a\\ &\le (2L_1+1)C_0\a. \end{split}\end{align}
Next, define two complete metric spaces $\mathcal V^s_\o(y)$ and $\mathcal V^u_\o(y)$ as follows:
$$\mathcal V^\tau_\o(y):=\{W^\tau_{3C_0\a}(x,\o)\cap R_\o(y)|\ x\in R_\o(y)\},\ \tau=u,s,$$ where the metric in $\mathcal V^\tau_\o(y)$ is the Hausdorff distance.
Define a map $\mathcal F^u_\o(y,z):\mathcal V^u_\o(y)\to\mathcal V^u_{\t\o}(z)$ by letting $$\mathcal \mathcal F^u_\o(y,z)W=f_\o(W)\cap R_{\t\o}(z),\text{ where }W\in \mathcal V^u_\o(y).$$ By the choice of $C_0$ and p3), we can see that the above map is well defined. Moreover, p3) also implies that $\mathcal \mathcal F^u_\o(y,z)$ is a contraction on $\mathcal V^u_\o(y)$ with contracting rate $1-\e_1$. Similarly, we can define a contracting map $\mathcal F^s_{\t\o}(z,y):\mathcal V^s_{\t\o}(z)\to\mathcal V^s_{\o}(y)$ by p4).\\
Now for a given $(\o,\a)$-pseudo orbit $\{(x_i,\t^i\o)\}_{i\in\mathbb Z}$ and $i,j\in\mathbb Z$ with $i<j$, define that $$\mathcal F^u_{i,j}:=\mathcal F^u_{\t^{j-1}\o}(x_{j-1},x_{j})\circ\cdots\circ \mathcal F^u_{\t^i\o}(x_i,x_{i+1})(:\mathcal V^u_{\t^i\o}(x_i)\to \mathcal V^u_{\t^j\o}(x_j)),$$ $$\mathcal F^s_{j,i}:=\mathcal F^s_{\t^{i+1}\o}(x_{i+1},x_{i})\circ\cdots\circ \mathcal F^s_{\t^j\o}(x_j,x_{i-1})(:\mathcal V^s_{\t^j\o}(x_j)\to \mathcal V^s_{\t^i\o}(x_i)),$$ which are both contractions. Thus the following set consists of only one point which is denoted by $y_i$
$$\lim_{k\to\infty}\left(\mathcal F^u_{i-k,i}\left(V^u_{\t^{i-k}\o}(x_{i-k})\right)\cap \mathcal F^s_{i+k,i}\left(V^s_{\t^{i+k}\o}(x_{i+k})\right)\right).$$
It is clear that $\{(y_i,\t^i\o)\}_{i\in \mathbb Z}$ is a true orbit and $y_i\in R_{\t^i\o}(x_i)$. Therefore, (\ref{E:DisEst1}) tells that taking $C=(2L_1+1)C_0$ will fit the need.
\end{proof}
\begin{lem}\label{L:ContinousShadowing} For a given $\b\in(0,\b_0)$, let $\a\in (0,\a_0)$ be as in Lemma \ref{L:Shadowing} corresponding to $\b$ where $\a_0$ is as in Lemma \ref{L:ClosingLemma}. Then for any $\tau>0$, there exists $N_0\in\mathbb N$ such that given any two $(\o,\a)$-pseudo orbits $\{(x_i,\t^i\o)\}_{i\in\mathbb Z}$ and $\{(x_i',\t^i\o)\}_{i\in\mathbb Z}$ which are $(\o,\b)$-shadowed by two uniquely defined true orbits $\{(y_i,\t^i\o)\}_{i\in\mathbb Z}$ and $\{(y_i',\t^i\o)\}_{i\in\mathbb Z}$ respectively, for any $N>N_0$ if $x_i=x_i',\ \forall i\in[-N,N]$, then $d_M(y_0,y_0')<\tau$. \end{lem} \begin{proof} Note that if $x_i=x_i',\ \forall i\in[-N,N]$, then $d_M(y_i,y_i')<2\b,\ \forall i\in[-N,N]$. Let $z=[y_0,y'_0]_\o$, where $[\cdot,\cdot]_\cdot$ is defined in \ref{L:LocCoor} for a fixed $\d$ corresponding to $\e=\frac12\e_0$. By the definition of local unstable manifold and ii) of Lemma \ref{L:InvMani}, we have that \begin{align*} d_M(z,y'_0)&\le e^{-N\l}d_M(\pi_M\phi^N(z,\o),y_N')\\ &\le e^{-N\l}(d_M(y_N,y'_N)+d_M(\pi_M\phi^N(z,\o),y_N))\\ &\le e^{-N\l}(2\b+d_M(z,y_0))\\ &\le e^{-N\l}(2\b+\frac12\e_0). \end{align*} The same argument is applicable to $d_M(z,y_0)$ if we reverse the time. Therefore, $$d_M(y_0,y'_0)\le 2L_1e^{-N\l}(2\b+\frac12\e_0),$$ where $L_1$ is a constant given by P2) at beginning of the proof of Lemma \ref{L:ClosingLemma}. The proof is completed. \end{proof}
Next, we consider the shadowing property of the map $\tilde \phi:L^\infty(\O, M)\to L^\infty(\O, M)$ induced by $\phi$. In this case, the system is a deterministic system, and the pseudo orbit is defined in the standard way: for any $\a>0$, an orbit $\{g_i\}_{i\in \mathbb Z}\subset L^\infty(\O, M)$ is called an {\bf $\a$-pseudo orbit} if for any $i\in \mathbb Z$ $$d_{L^\infty(\O, M)}(\tilde \phi(g_i),g_{i+1})<\a.$$ \begin{lem}\label{L:LShad} For any $\b\in(0,\b_0)$, there exists an $\a>0$ such that for any $\a$-pseudo orbit $\{g'_i\}_{i\in \mathbb Z}$, there is a unique true orbit $\{g_i\}_{i\in \mathbb Z}$ of $\tilde\phi$ such that $$\sup_{i\in \mathbb Z}d_{L^\infty(\O, M)}(g_i,g'_i)<\b,$$ where $\b_0$ is as in Lemma \ref{L:Shadowing}, and $\{g_i\}_{i\in \mathbb Z}$ is called {\bf $\b$-shadowing} orbit of $\{g'_i\}_{i\in \mathbb Z}$.
Moreover, there exist an $\a_0>0$ and $C\ge 1$ so that for any $\a\in (0,\a_0)$ and any $\a$-pseudo orbit $\{g'_i\}_{i\in\mathbb Z}$, there exists a true orbit $\{g_i\}_{i\in\mathbb Z}$ of $\tilde\phi$ satisfying the following $$\sup_{i\in \mathbb Z} d_{L^\infty(\O, M)}(g_i,g'_i)<C\a.$$ \end{lem} \begin{proof} First, we note that each $\a$-pseudo orbit $\{g'_i\}_{i\in \mathbb Z}$ induces an $(\o,\a)$-pseudo orbit \[\{(g'_i(\t^i\o),\t^i\o)\}_{i\in \mathbb Z}\] for each $\o\in \O$. Thus Lemma \ref{L:Shadowing} is applicable and produces a map $g:\O\to M$ such that $\{(g(\t^i\o),\t^i\o)\}_{i\in \mathbb Z}$ is $(\o,\b)$-shadowing $\{(g_i(\t^i\o),\t^i\o)\}_{i\in \mathbb Z}$, where $\a$ and $\b$ are as in Lemma \ref{L:Shadowing}. Once $g$ is Borel measurable, the proof follows from Lemma \ref{L:Shadowing} and \ref{L:ClosingLemma}.
Next, we prove that $g$ is Borel measurable. For $h\in L^\infty(\O, M)$ and $r>0$, denote $B_\O(h,r)$ the open tubular neighborhood of $graph(h)$, i.e.,
$$B_\O(h,r):=\left\{(x,\o)|\ \o\in\O, d_M(x,h(\o))<r\right\}.$$ It is not hard to see that $B_\O(h,r)$ is a Borel subset of $M\times\O$. In fact, since $M$ is compact, there exists a sequence of simple functions $\{h_i:\O\to M\}_{i\in \mathbb N}$ such that $$d_{L^\infty(\O, M)}(h_i,h)<\frac{r}{i+2},\ i\in \mathbb N.$$ Note that each $B_\O(h_i,\frac{ir}{i+2})$ is a Borel subset of $M\times\O$ because it is a union of finite rectangle sets, thus $B_\O(h,r)=\cup_{i\in N}B_\O(h_i,\frac{ir}{i+2})$ implies the Borel measurability of $B_\O(h,r)$.
Coming back to $g$ and $g_i'$, by the uniqueness of $g$, we have that $$graph(g)=\bigcap_{i\in\mathbb Z}\phi^{-i}\left(B_\O(g_i,\b)\right).$$ For $k\in\mathbb N$, let $$S_k:=\bigcap_{i\in[-k,k]\cap \mathbb Z}\phi^{-i}\left(B_\O(g_i,\b)\right).$$ By the continuity of $\phi$ and $\phi^{-1}$, $S_k$ is a Borel subset in $M\times\O$. Also note that for each $\o\in\O$,
$$S_k\cap M_\o=\bigcap_{i\in[-k,k]\cap \mathbb Z}\pi_M\phi^{-i}\left(\left\{(x,\t^i\o)|\ d_M(x,g_i(\t^i\o))<\b\right\}\right),$$ which is a nonempty open subset of $M_\o$. By applying Theorem \ref{T:BMST} to each $S_k$, we have that for each $k\in\mathbb N$, there exists a Borel measurable map $g_k:\O\to M$ such that $$graph(g_k)\subset S_k.$$ By the uniqueness of $g$ and Lemma \ref{L:Expansive}, we have that $$\lim_{k\to\infty}\sup_{\o\in\O}\{d_M(g_k(\o),g(\o)\}=0,$$ which implies that $g$ is Borel measurable. The proof is completed. \end{proof} \begin{rem}\label{R:Alpha0&C} We remark here that $\a_0$, $\b_0$ and $C$ in Lemma \ref{L:Shadowing}, \ref{L:ClosingLemma}, and \ref{L:LShad} can take the same value. Thus we will not distinguish them when these Lemmas are used in the rest of the paper. \end{rem}
The next lemma is a straightforward consequence of Lemma \ref{L:ContinousShadowing} and \ref{L:LShad}. \begin{lem}\label{L:LContShad} Let $\a_0$ and $\b_0$ are as in Lemma \ref{L:LShad}, and for a given $\b\in(0,\b_0)$, $\a\in (0,\a_0)$ be the one as in Lemma \ref{L:LShad} corresponding to $\b$. Then for any $\tau>0$, there exists $N_0\in\mathbb N$ such that the following holds: given any two $\a$-pseudo orbits $\{g_i\}_{i\in\mathbb Z}$ and $\{g_i'\}_{i\in\mathbb Z}$ of $\tilde \phi$, which are $\b$-shadowed by two true orbits $\{\tilde g_i\}_{i\in\mathbb Z}$ and $\{\tilde g_i'\}_{i\in\mathbb Z}$ respectively, then for any $N>N_0$, $g_i=g_i',\ \forall i\in[-N,N]$, we have that $d_{L^\infty(\O, M)}(\tilde g_0,\tilde g_0')<\tau.$ \end{lem}
\section{Density of Random Periodic Points}
In this section, we prove Theorem \ref{T:TheoryAnosovMix} based on the mixing property and the shadowing property.
\vskip0.05in \noindent {\it Proof of Theorem \ref{T:TheoryAnosovMix}}. Firstly, fix a given Borel function $g\in L^\infty(\O, M)$ and $\e>0$. Without losing generality, we assume that $\e<\frac12\b_0$, where $\b_0$ is as in Lemma \ref{L:Shadowing} and \ref{L:LShad}. Taking $\b=\frac13\e$ and applying Lemma \ref{L:LShad}, there exists $\a>0$ corresponding to $\b$ as in Lemma \ref{L:LShad}. Again, for the sake of convenience, we assume that $\a<\b$. By the compactness of $M$, there exists a finite open cover $\{U_i,1\le i\le n_0\}$ of which each $U_i=B(x_i,\frac13\a)$ is a ball of radius $\a$ in $M$.
Secondly, define a function $k:\O\to \{1,2,\cdots, n_0\}$ by the following \begin{equation}\label{E:MeasurableIndex}
k(\o)=\min\left\{i\in \{1,\cdots,n_0\}\big|\ g(\o)\in U_i\right\},\ \forall \o\in\O. \end{equation} It is not hard to see that $k$ is a measurable function since $g$ is measurable and the following holds \begin{align*}
&\left\{\o|\ k(\o)=i\right\}\\ =&\begin{cases}
\{\o|\ g(\o)\in U_i\}&\text{ when } i=1\\
\{\o|\ g(\o)\in U_i\}\setminus\left(\bigcup_{1\le j\le i-1}\{\o|\ g(\o)\in U_j\}\right)&\text{ when } i\in\{2,\cdots, n_0\} \end{cases}. \end{align*} This Borel measurable function $k(\o)$ induces a finite measurable partition of $\O$,
$$\left\{\O_i:=\{\o|\ k(\o)=i\}\right\}_{i\in\{1,\cdots,n_0\}}.$$ Note that, by condition H2), there exists $N\in\mathbb N$ such that \begin{equation}\label{E:MixingTimeS2} \pi_M\phi^n(U_i,\o)\cap U_j\neq \emptyset,\ \forall \o\in \O, i,j\in\{1,\cdots,n_0\},n\ge N. \end{equation} Then, for any $n\ge N$, $k$ and $n$ induces a finite measurable partition of $\O$, $\{\O^{n}_{ij}\}_{i,j\in\{1,\cdots,n_0\}}$, by letting
$$\O^n_{ij}:=\left\{\o\big|\ k(\o)=i, k(\t^n\o)=j\right\}=\t^{-n}(\O_j)\cap \O_i.$$ By the uniform continuity of $\phi$, there exists $\d_1>0$ such that the following holds \begin{equation}\label{E:CondDelt1} d_M(\pi_M\phi(x,\o_1),\pi_M\phi(x,\o_2))<\frac16\a,\ \forall x\in M\text{ and } d_\O(\o_1,\o_2)<\d_1. \end{equation} Additionally, by the uniform continuity of $\t$, for a fixed $n\ge N$, there exists $\d_2>0$ such that \begin{equation}\label{E:CondDelt2} d_\O(\t^m\o_1,\t^m\o_2)<\d_1\ \forall0\le m\le n-1 \text{ and }d_\O(w_1,w_2)<\d_2. \end{equation} Now take a finite measurable partition of $\O$, $\{\O'_s\}_{s\in S}$, which is taken to be a refinement of $\{\O^n_{ij}\}_{i,j\in\{1,\cdots,n_0\}}$ and has diameter less than $\d_2$. By (\ref{E:MixingTimeS2}), we have that for any $\o\in\O^n_{ij}$, there exists an $x\in U_i$ such that $\phi^n(x,\o)\in (U_j,t^n\o)$. Then, for each $\O'_s(\subset \O^n_{ij})$, arbitrarily fix a point $(y_s,\o_s)\in U_i\times \O'_s$ such that
$$\pi_M\phi^n(y_s,\o_s)\in U_j.$$ Define a simple function $y=\sum_{s\in S}y_s\chi_{\O'_s}$, where $\chi_{\O'_s}$ is the characteristic function of $\O'_s$. \\
Now, we are ready to construct the $(\o,\a)$-pseudo-orbits which visit the tubular neighborhood of $graph(g)$ periodically (with period of $n$). For a given $\o\in\O$, denote $q_\o:\mathbb Z\to S$ the function satisfying that for any $l\in \mathbb Z$, $\t^{ln}(\o)\in \O_{q(l)}'$. Then define \begin{equation}\label{E:DefPseOrbS2)} y'_i(\o)= \begin{cases} y(\t^{ln}\o),&\text{when } i=ln\\ \pi_M\left(\phi^{i-ln}(y(\t^{ln}\o),\o_{q_\o(l)})\right),&\text{when }i\in[ln+1,(l+1)n-1] \end{cases}, \end{equation} where the map $y'_i:\O\to M$ is clearly Borel measurable. For $l\in\mathbb Z$ and $i\in [ln,(l+1)n-1]$, by the choice of $\d_2$ and (\ref{E:CondDelt2}), we have that $$d_\O(\t^{i}\o,\t^{i-ln}\o_{q_\o(l)})<\d_1.$$ Therefore, by (\ref{E:CondDelt1}), we have that, for $i\in [ln,(l+1)n-2]$ \begin{align*} &d_M\left(\pi_M\phi(y_i'(\o)),y_{i+1}'(\o)\right)\\ =&d_M\left(\pi_M\phi\left(\pi_M\phi^{i-ln}(y(\t^{ln}\o),\o_{q_\o(l)}),\t^i\o\right),\pi_M\phi\left(\pi_M\phi^{i-ln}(y(\t^{ln}\o),\o_{q_\o(l)}),\t^i\o_{q_\o(l)}\right)\right)\\ \le&\frac16\a, \end{align*} and for $i=(l+1)n-1$, \begin{align*} &d_M\left(\pi_M\phi(y'_{(l+1)n-1}(\o),\t^{(l+1)n-1}\o),y'_{(l+1)n}(\o)\right)\\ \le &d_M\left(\pi_M\phi\left(\pi_M\phi^{i-ln}(y(\t^{ln}\o),\o_{q_\o(l)}),\t^i\o\right),\pi_M\phi\left(\pi_M\phi^{i-ln}(y(\t^{ln}\o),\o_{q_\o(l)}),\t^i\o_{q_\o(l)}\right)\right)\\ &+d_M\left(\pi_M\phi^{N}(y(\t^{ln}\o),\o_{q_\o(l)}),y(\t^{(l+1)n}\o)\right)\\ <&\frac16\a+\frac23\a<\a. \end{align*} In the above estimate, we used the fact that $\phi^{n}(y(\t^{ln}\o),\o_{q_\o(l)})$ and $y(\t^{(l+1)n}\o)$ fall in a same element of $\{U_i\}_{i\in\{1,\cdots,n_0\}}$. This conclude that $\{(y'_i\in L^\infty(\O, M)\}_{i\in\mathbb Z}$ forms an $\a$-pseudo orbit of $\tilde\phi$. Thus, by Lemma \ref{L:LShad}, there exists a unique true orbit $\{\tilde\phi^i(\tilde g), \tilde g\in L^\infty(\O, M)\}_{i\in\mathbb Z}$ which is $\b$-shadowing $\{y'_i\}_{i\in\mathbb Z}$. Note that $\{\phi^i(\tilde g)\}_{i\in\mathbb Z}$ and $\{\phi^{i+n}(\tilde g)\}_{i\in\mathbb Z}$ are two true orbits and $$d_{L^\infty(\O, M)}(\tilde\phi^{i+n}(\tilde g),\tilde\phi^i(\tilde g))\le 2\b<\frac13\b_0,\ \forall i\in \mathbb Z.$$ Then Lemma \ref{L:LContShad} implies that $$\tilde\phi^n(\tilde g)=\tilde g\ i.e.\ \phi^{n} (graph(\tilde g))=graph(\tilde g).$$ This completes the proof of this theorem. \qed
From the above proof, we summarize a technical lemma as follows, which can be used to join two points in $L^\infty(\O, M)$ by using a segment of an orbit of $\tilde \phi$. Since the proof follows exactly the same idea as above, we omit it here. \begin{lem}\label{L:JoinSegments} For any $\e>0$, there exists a $N\in \mathbb N$ such that for any $g_1,g_2\in L^\infty(\O, M)$ and $n\in [N,\infty)\cap \mathbb N$, there exists $g\in L^\infty(\O, M)$ satisfies the following $$\max\left\{d_{L^\infty(\O, M)}(g,g_1),d_{L^\infty(\O, M)}(\tilde\phi^n(g),g_2)\right\}<\e.$$ \end{lem}
\section{Random Horseshoe}
In this section, we first show that $\underline h(M\times\O|\O)>0$. Then, we prove Theorem \ref{T:TheoryAnosovMix2}, i.e., the existence of strong full random horseshoes and full random and horseshoes.
Before going to the proof, we need to introduce an alternative definition of $\underline h(M\times \O|\O)$ which is more convenient. \begin{lem}\label{L:AlDefH} Suppose that $\phi$ satisfy Condition H1). Then the following holds:
$$\underline h(M\times \O|\O)=\lim_{\e\to0^+}\inf_{\o\in\O}\limsup_{n\to\infty}\frac1n\log N(\o,\e,n),$$ where \begin{align*}
&N(\o,\e,n):=\\&\max\ \left\{card(E)\big|\ E\subset M_{\o}, E \text{ is an }\e\text{ separated set with respect to }d^{\phi}_{M\times \O,n}\right\}. \end{align*} \end{lem} \begin{proof}
Recall that the original definition of $\underline h(M\times \O|\O)$ is given by
$$\underline h(M\times \O|\O)=\inf_{\o\in\O}\lim_{\e\to0^+}\limsup_{n\to\infty}\frac1n\log N(\o,\e,n).$$ This lemma simply states that the above order of $\inf$ and $\lim_{\e\to0^+}$ can be exchanged. First, we clearly have that
$$\underline h(M\times \O|\O)=\inf_{\o\in\O}\lim_{\e\to0^+}\limsup_{n\to\infty}\frac1n\log N(\o,\e,n)\ge\lim_{\e\to0^+}\inf_{\o\in\O}\limsup_{n\to\infty}\frac1n\log N(\o,\e,n).$$ Next, we show that the opposite inequality also holds. To see this, we claim
\noindent {\bf Claim}:{\it There exists $\eta >0$ such that \begin{equation}\label{E:ConstantEta}
h_{top}(\phi|M_\o):=\lim_{\e\to0^+}\limsup_{n\to\infty}\frac1n\log N(\o,\e,n)=\limsup_{n\to\infty}\frac1n\log N(\o,\eta,n),\ \forall \o\in\O. \end{equation}}
\noindent{\em Proof of the Claim}: It is obvious that $h_{top}(\phi|M_\o)\ge \limsup_{n\to\infty}\frac1n\log N(\o,\eta,n)$ by definition. Hence, it remains to prove the opposite inequality.
Firstly, we fix an $\eta>0$ satisfying the following property: for any $x,y\in M_\o$ $$\sup_{n\in\mathbb Z}d(\phi^n(x,\o),\phi^n(y,\o))\le \eta\;\text{ implies }\;x=y.$$ The existence of such $\eta$ follows from the expansive property of $\phi$, where one only requires $\eta<\e$ for the $\e$ from Lemma \ref{L:Expansive}.
Secondly, by Lemma \ref{L:ContinousShadowing}, we have that for any $\e'\in(0,\eta)$, there exists $L(\e')\in \mathbb N$ such that for any $x,y\in M_\o$ \begin{equation}\label{E:SepControl}
\max_{|n|\le L(\e')}d(\phi^n(x,\o),\phi^n(y,\o))\le \eta\text{ implies }d_M(x,y)<\e'. \end{equation} By (\ref{E:SepControl}), we have for any $\o\in\O$, $n\in\mathbb N$, and $x,y\in M_\o$, $$d^\phi_{M\times \O,n+2L(\e')}((x,\o),(y,\o))\le \eta\text{ implies }d^\phi_{M\times \O,n}(\phi^{L(\e')}(x,\o),\phi^{L(\e')}(y,\o))<\e',$$ which yields \begin{equation}\label{E:SepNEst1} N(\t^{L(\e')}\o,\e',n)\le N(\o,\eta,n+2L(\e')),\ \forall \o\in\O, n\in \mathbb N, \e'\in(0,\eta). \end{equation} Note that \begin{equation}\label{E:SepNEst2} N(\o,\e',n+m)\le N(\o,\e',m)N(\t^m\o,\e',n). \end{equation} (\ref{E:SepNEst1}) and (\ref{E:SepNEst2}) imply that \begin{align*}
h_{top}(\phi|M_\o)=&\lim_{\e'\to0^+}\limsup_{n\to\infty}\frac1n\log N(\o,\e',n)\\ =&\lim_{\e'\to0^+}\limsup_{n\to\infty}\frac1{n+L(\e')}\log N(\o,\e',n+L(\e'))\\ \le&\lim_{\e'\to0^+}\limsup_{n\to\infty}\frac1{n+L(\e')}\left(\log N(\o,\e',L(\e'))+\log N(\t^{L(\e')}\o,\e',n)\right)\\ \le&\lim_{\e'\to0^+}\limsup_{n\to\infty}\frac1{n+L(\e')}\left(\log N(\o,\e',L(\e'))+\log N(\o,\eta,n+2L(\e'))\right)\\ =&\lim_{\e'\to0^+}\limsup_{n\to\infty}\frac1{n+L(\e')}\log N(\o,\eta,n+2L(\e'))\\ =&\lim_{\e'\to0^+}\limsup_{n\to\infty}\frac1{n}\log N(\o,\eta,n)\\ =&\limsup_{n\to\infty}\frac1{n}\log N(\o,\eta,n). \end{align*} Therfore (\ref{E:ConstantEta}) holds, which yields \begin{align*}
\underline h(M\times \O|\O)=&\inf_{\o\in\O}\lim_{\e\to0^+}\limsup_{n\to\infty}\frac1n\log N(\o,\e,n)\\
=&\inf_{\o\in \O} h_{top}(\phi|M_\o)\\ =&\inf_{\o\in\O}\limsup_{n\to\infty}\frac1n\log N(\o,\eta,n)\\ \le &\lim_{\eta\to0^+}\inf_{\o\in\O}\limsup_{n\to\infty}\frac1n\log N(\o,\eta,n), \end{align*} which completes the proof of this lemma. \end{proof}
Next, we will show that $\underline h(M\times \O|\O)>0$. It is not hard to see that once $\phi^N$ has a strong horseshoe with $k$ symbols for some $N,k,\e$ as in the Definition \ref{D:SRandomHorseshoe}, then by the definition of $\underline h(M\times\O|\O)$, we have that
$$\underline h(M\times\O|\O)\ge \frac{\log k}N>0.$$
Thus, the first assertion on the positivity of $\underline h(M\times\O|\O)$ in i) of Part B of Theorem \ref{T:TheoryAnosovMix} follows from the lemma below: \begin{lem}\label{L:Horseshoe} If $\phi$ satisfies condition H1) and H2), then for any $k\in \mathbb N\setminus\{1\}$ there exists $N\in \mathbb N$ such that $\phi^N$ has a $k$-symbol strong full random horseshoe. \end{lem} \begin{proof}
Let $\{x_i\}_{1\le i\le k}$ be an $3\e$-separated finite subset of $M$. Such a set exists for small enough $\e>0$, and we fix this $\e$. Without loss of generality, we assume that $\e<\frac12\b_0$, where $\b_0$ is as in Lemma \ref{L:LShad}. For $\b=\frac13\e$, there exists an $\a>0$ corresponding to $\b$ as in Lemma \ref{L:LShad}. Again, for the sake of convenience, we assume that $\a<\b$. For any $i\in \{1,\cdots,k\}$, let $U_i=B(x_i,\frac13\a)$.
By Condition H2), there exists an $N\in\mathbb N$ such that the following holds \begin{equation}\label{E:MixingTimeS2S} \pi_M\phi^n(U_i,\o)\cap U_j\neq \emptyset,\ \forall \o\in \O, i,j\in\{1,\cdots,k\},n\ge N. \end{equation} By the uniform continuity of $\phi$ and $\t$, there exist $\d_1,\d_2>0$ such that (\ref{E:CondDelt1}) and (\ref{E:CondDelt2}) hold. Let $\{\O_s'\}_{s\in S}$ be a finite Borel measurable partition of $\O$ with diameter less than $\d_2$. By (\ref{E:MixingTimeS2S}), for any $s\in S$ and $i,j\in\{1\cdots,k\}$, there exists $(y^{i,j}_s,\o^{i,j}_s)\in U_i\times \O'_s$ such that $$\pi_M\phi^N(y^{i,j}_s,\o^{i,j}_s)\in U_j.$$ We define a simple function $y^{i,j}=\sum_{s\in S}y^{i,j}_s\chi_{\O'_s}$ for all $i,j\in\{1,\cdots,k\}$.
Now, we are ready to build the strong random horseshoe by constructing a proper $\a$-pseudo-orbits of $\tilde\phi$ associated to elements from $\mathcal S_k:=\{1,\cdots, k\}^{\mathbb Z}$.
For a given $\hat a=(a_i)_{i\in\mathbb Z}\in \mathcal S_k$, and a given $\o\in\O$, we denote $q_\o:\mathbb Z\to S$ the function satisfying that for any $l\in \mathbb Z$, $\t^{lN}(\o)\in \O_{q(l)}'$. Then, we define \begin{equation}\label{E:DefPseOrbS2)} y'_{\hat a,\o}(i)= \begin{cases} y^{a_l,a_{l+1}}(\t^{lN}\o),&\text{when } i=lN\\ \pi_M\left(\phi^{i-lN}(y^{a_l,a_{l+1}}(\t^{lN}\o),\o^{a_l,a_{l+1}}_{q_\o(l)})\right),&\text{when }i\in[lN+1,(l+1)N-1] \end{cases}. \end{equation} Set $$g_i(\o):=y'_{\hat a,\o}(i),\ \forall \o\in\O,\ i\in \mathbb Z.$$ It is clear that each $g_i:\O\to M$ is Borel measurable since $g_i$ is either a simple function or the image of a simple function under $\phi$ or $\phi^{-1}$'s iterations.
For $l\in\mathbb Z$ and $i\in [lN,(l+1)N-1]$, by the choice of $\d_2$ and (\ref{E:CondDelt2}), we have that $$d_\O(\t^{i}\o,\t^{i-lN}\o^{a_l,a_{l+1}}_{q_\o(l)})<\d_1.$$ Therefore, by (\ref{E:CondDelt1}), we have that, for $i\in [lN,(l+1)N-2]$, \\ (to save space, let $z=\pi_M\phi^{i-lN}\left(y^{a_l,a_{l+1}}(\t^{lN}\o),\o^{a_l,a_{l+1}}_{q_\o(l)}\right)$) \begin{align*} &d_M\left(\pi_M\phi(y'_{\hat a,\o}(i)),y'_{\hat a,\o}(i+1)\right) =d_M\left(\pi_M\phi\left(z,\t^i\o\right),\pi_M\phi\left(z,\t^i\o^{a_l,a_{l+1}}_{q_\o(l)}\right)\right) \le\frac16\a, \end{align*} while for $i=(l+1)N-1$, \begin{align*} &d_M\left(\pi_M\phi(y'_{\hat a,\o}(i),\t^{i}\o),y'_{\hat a,\o}(i+1)\right)\\ \le &d_M\left(\pi_M\phi\left(z,\t^i\o\right),\pi_M\phi\left(z,\t^i\o^{a_l,a_{l+1}}_{q_\o(l)}\right)\right)\\ &+d_M\left(\pi_M\phi^{N}(y^{a_l,a_{l+1}}(\t^{lN}\o),\o^{a_l,a_{l+1}}_{q_\o(l)}),y^{a_{l+1},a_{l+2}}(\t^{(l+1)N}\o)\right)\\ <&\frac16\a+\frac23\a<\a. \end{align*}
This concludes that for any $\o\in \O$, $\{(y'_{\hat,\o}(i),\t^i\o)\}_{i\in\mathbb Z}$ forms an $(\o,\a)$-pseudo orbit of $\phi$. Thus $\{g_i\}_{i\in\mathbb Z}$ forms an $\a$-pseudo orbit of $\tilde \phi$. Thereafter, Lemma \ref{L:LShad} implies the existence of a unique true orbit $\{\phi^i(\tilde )\}_{i\in\mathbb Z}$ which is $\b$-shadowing $\{g_i\}_{i\in\mathbb Z}$.
We define the map $\Psi:\mathcal S_k\to L^\infty(\O, M)$ by letting $$\Psi(\hat a)=g_{\hat a},\ \forall \hat a\in \mathcal S_k.$$
where the continuity of $\Psi$ follows from Lemma \ref{L:LContShad}.
Note that $\{\tilde\phi^i(\tilde g_{\hat a})\}_{i\in\mathbb Z}$ and $\{\tilde\phi^i(\tilde g_{\s \hat a})\}_{i\in\mathbb Z}$ are two true orbits and $$d_{L^\infty(\O, M)}(\tilde\phi^{i+N}(\tilde g),\tilde\phi^i(\tilde g_{\s\hat a})\le 2\b<\frac13\b_0,$$ therefore, by Lemma \ref{L:LContShad}, we have that $$\tilde\phi^N(\tilde g_{\hat a})=(\tilde g_{\s\hat a} )\ i.e.\ \phi^{N} graph(\tilde g_{\hat a})=graph(\tilde g_{\s\hat a}),$$ which implies (ii) in Definition \ref{D:SRandomHorseshoe}.
Now we prove i) in Definition \ref{D:SRandomHorseshoe}. For any $\hat a_1=(a_1(i))_{i\in \mathbb Z},\hat a_2=(a_2(i))_{i\in \mathbb Z}\in\mathcal S_k$ with $\hat a_1\neq \hat a_2$, let $s=\min\{|i||\ a_1(i)\neq a_2(i)\}$. Then, by the choice of $x_i,U_i,\e,\a,\b$ at the beginning of this section, we have that
\begin{equation}\label{E:SepSym}
d^{\phi}_{M\times\O,N}\left(\phi^{rN}(\Psi(\hat a_1)(\o),\o),\phi^{rN}(\Psi(\hat a_2)(\o),\o)\right)>\e,\ \forall\o\in\O,
\end{equation}
where $a_1(r)\neq a_2(r)$ with $|r|=s$ and $d^{\phi}_{M\times\O,N}$ is the Bowen metric. Note that there is a constant $L>1$ such that for any $\o\in\O$ and $x,y\in M$ with $d(x,y)>0$, we have
$$d(\phi^{\pm1}(x,\o),\phi^{\pm1}(y,\o))\le Ld_M(x,y).$$
Therefore
\begin{equation}\label{E:SHorsSeparation}
d_M\left(\pi_M\left(\Psi(\hat a_1)(\o),\o\right),\pi_M\phi^{rN}\left(\Psi(\hat a_2)(\o),\o\right)\right)>\e L^{-N},\ \forall\o\in\O, \end{equation} which implies Condition i) of Definition \ref{D:SRandomHorseshoe} by taking $\Delta =\e L^{-N}$.
Finally, the continuity of $\Psi^{-1}$ can be obtained by the following estimate:
\begin{align}\begin{split}\label{E:PsiInverseContinuity} & d_M(\Psi(\hat a_1)(\o),\Psi(\hat a_2)(\o))\\ =&d\left((\Psi(\hat a_1)(\o),\o),(\Psi(\hat a_2)(\o),\o)\right)\\
\ge& L^{-|rN|}d\left(\phi^{rN}(\Psi(\hat a_1)(\o),\o),\phi^{rN}(\Psi(\hat a_2)(\o),\o)\right)\\ \ge& \e L^{-(s+1)N}. \end{split}
\end{align}
The proof is completed. \end{proof}
\begin{rem}\label{R:PositivLHVolEst}
Actually, the positivity of $\underline h(M\times\O|\O)$ can be derived more directly by estimating the volume expanding rate on local unstable manifolds, which follows from the hyperbolicity of the systems on fibers. Nevertheless, we prefer to present the current proof which is demonstrative for the ideas on constructing random horseshoes by mixing property. \end{rem}
Next, we construct a strong horseshoe which can capture enough entropy which is close to the lower topological fiber entropy. The construction of the desired horseshoe follows from the same procedure as the one used in the proof of Lemma \ref {L:Horseshoe}.
\begin{proof}[Proof of Part A. of Theorem \ref{T:TheoryAnosovMix2}] By Lemma \ref{L:AlDefH}, we have that there exists $\e'_0>0$ such that for any $\e'\in(0,\e'_0)$ the following holds \begin{equation}\label{E:SepNEst}
\inf_{\o\in\O}\limsup_{n\to\infty}\frac1n\log N(\o,\e',n)>\underline h(M\times\O|\O)-\frac14\g. \end{equation}
Let $\e'_1=\min\left\{\frac12\b_0,\e'_0\right\}$, where $\b_0$ is as in Lemma \ref{L:LShad}. Taking $\b=\frac16\e'_1$, there exists an $\a>0$ corresponding to $\b$ as in Lemma \ref{L:Shadowing}. Again, for the sake of convenience, we assume that $\a<\b$. Now we fix an $\e\in(0,\b)$, and also fix a set of open balls in $M$ with radius equaling $\min\{\frac13\a,\frac16\e\}$, $\{U_i,0\le i\le p\}$, where $\{U_i\}_{1\le i\le p}$ forms an open cover of $M$.
By Condition H2), there exists $N_1\in\mathbb N$ such that the following holds \begin{equation}\label{E:MixingLS} \pi_M\phi^n(U_i,\o)\cap U_j\neq \emptyset,\ \forall \o\in \O, i,j\in\{0,1,\cdots,p\},n\ge N_1. \end{equation}
By (\ref{E:SepNEst}) and $\b\le \frac16 \e_0'$, for any $\o\in\O$, there exist $l(\o),m(\o)\in \mathbb N$, and a $3\b$-separated set $\{z_i(\o)\}_{1\le i\le l(\o)}$ with respect to $d^\phi_{M\times \O,m(\o)}$ satisfies that \begin{equation}\label{E:SepNEstO}
\frac{1}{m(\o)+2N_1}\log l(\o)>\underline h(M\times\O|\O)-\frac12\g. \end{equation} Since $\phi$ is continuous and $\O$ is compact, the $3\b$-separated set can be chosen properly to make $m(\o)$ being uniformly bounded on $\O$. The bound is denoted by $N_2$. Let $N=L(N_2+2N_1)$, where $L$ is a prefixed large positive integer.
Note that, by the choice of $\a$ and $\b$, if two $(\o,\a)$-pseudo orbits have a $3\b$-separation on time $n$, then the corresponding true orbits which are $(\o,\b)$-shadowing them respectively have a $\b$-separation on time $n$. Therefore, here instead of constructing specific pseudo orbits, we would like to count the number of $(\o,\frac23\a)$-pseudo orbits which have $3\b$-separations happened during time interval $[0,N]$.
First, we describe the procedure defining the pseudo orbits. For a given $\o\in\O$, by (\ref{E:MixingLS}), we can use a segment of a true orbit to connect $(U_0,\o)$ to any one of $\{(U_i,\t^{N_1})\}_{1\le i\le p}$. This orbit segment of connecting $U_0$ to some $U_i$ is the first piece of the $(\o,\frac23\a)$-pseudo orbit. The second piece of the $(\o,\frac23\a)$-pseudo orbit is one of the orbit segments \[\{\phi^j(z_i(\t^{N_1}\o),\t^{N_1}\o)\}_{0\le j\le m(\t^{N_1}\o)-1}
\]which is a segment of a true orbit. So in order to ensure that these two pieces can be merged with variation $\le \frac23\a$, $U_i$ chosen in the first step should contains the $z_i(\t^{N_1}\o)$ chosen in the second step. The third piece of these $(\o,\frac23\a)$-pseudo orbits is to connect $\phi^{m(\t^{N_1}\o)}(z_i(\t^{N_1}\o),\t^{N_1}\o)$ to $U_0$ by a true orbit segment with length $N_1$, which can be done again by (\ref{E:MixingLS}). We repeat this procedure and derive a sequence of integers in the following way
\begin{align*}
R_0&=0\\
R_1&=2N_1+m(\t^{N_1}\o)\\ R_2&=2N_1+m(\t^{R_1+N_1}\o)\\ &\cdots\\ R_{n+1}&=2N_1+m(\t^{\sum_{i=1}^nR_i+N_1}\o)\\ &\cdots
\end{align*}
Let $$s:=\max\left\{n\Big|\ \sum_{i=1}^nR_i\le (L-1)(N_2+2N_1)\right\}.$$
The above procedure of constructing pseudo orbits repeats $s$ times in the time interval $[0,\sum_{i=1}^sR_i]$. For the time interval $[\sum_{i=1}^sR_i,N]$, we use only one orbit segment to connect $(U_0,\t^{\sum_{i=1}^sR_i}\o)$ to $(U_0,\t^{N}\o)$. It is not hard to see that the number of $(\o,\frac23\a)$-pseudo orbits which are $3\b$-separated in the time interval $[0,N]$ is
$$K(\o)=\prod_{i=1}^sl\left(\t^{\sum_{j=0}^{i-1}R_j+N_1}\o\right).$$
Let $N_3=N-\sum_{i=1}^sR_i$, then $N_3\le 2(N_2+2N_1)$. Therefore, by (\ref{E:SepNEstO}), we have that
\begin{align*}
&\frac{\log K(\o)}N\\
=&\frac{\sum_{i=1}^s\log l\left(\t^{\sum_{j=0}^{i-1}R_j+N_1}\o\right)}N\\
\ge&\frac{\left(\underline h(M\times\O|\O)-\frac12\g\right)\left(\sum_{i=1}^s\left(m\left(\t^{\sum_{j=0}^{i-1}R_j+N_1}\o\right)+2N_1\right)\right)}N\\
=&\left(\underline h(M\times\O|\O)-\frac12\g\right)\frac{{\sum_{i=1}^sR_i}}N=\left(\underline h(M\times\O|\O)-\frac12\g\right)\frac{N-N_3}N\\
\ge&\left(\underline h(M\times\O|\O)-\frac12\g\right)\left(1-\frac2L\right).
\end{align*} The largeness of $L$ is to ensure the following
$$\frac{\log K(\o)}N>\underline h(M\times\O|\O)-\g.$$ For any given $\o\in \O$, let $\{y_j(\t^i\o)\}_{0\le i\le N-1}$, $1\le j\le K(\o)$, be the segments of $(\o,\frac23\a)$-pseudo orbits of $\phi$ defined above. By the uniform continuous of $\phi$, there exist $\d_1,\d_2>$ such that (\ref{E:CondDelt1}) and (\ref{E:CondDelt2}) hold, where $n$ in (\ref{E:CondDelt2}) is replaced by $N$. Let $\{\O_s'\}_{s\in S}$ be a finite Borel measurable partition of $\O$ with diameter less than $\d_2$. We fix an $\o'_s\in \O'_s$ for each $s\in S$, and let $K=\min\{K(\o'_s)\}_{s\in S}$. Note that we
$$\frac{\log K}N>\underline h(M\times\O|\O)-\g.$$ Define the simple functions as follows $$g_{i,j}=\sum_{s\in S}y_j(\t^i\o_s')\chi_{\t^i(\O'_s)},\ 0\le i\le N-1,\ 1\le j\le K.$$ where $\chi_{\t^i(\O'_s)}$ is the characteristic function of $\t^i(\O'_s)$. It is not hard to see that (\ref{E:CondDelt1}) and (\ref{E:CondDelt2}) imply that \begin{equation}\label{E:SimpFuncShd1} d_{L^\infty(\O, M)}(\tilde\phi(g_{i,j}),g_{i+1,j})\le d_M(y_j(\t^i\o'_s),y_j(\t^{i+1}\o'_s)+\frac16\a<\a,\ 0\le i\le N-2, \end{equation} and \begin{equation}\label{E:SimpFuncShd2} d_{L^\infty(\O, M)}(\tilde\phi(g_{N-1,j}),g_{0,j'})\le \text{ diameter of }U_0+\frac16\a<\a,\ 1\le j,j'\le K. \end{equation} Also note that for $1\le j,j'\le K$ with $j\neq j'$ and each $\o\in\O$, $\{y_j(\t^i\o)\}_{0\le i\le N-1}$ and $\{y_{j'}(\t^i\o)\}_{0\le i\le N-1}$ are $3\b$-separated with respect to $d^\phi_{M\times \O,m(\o)}$, thus we have that \begin{equation}\label{E:SHorsSepa} \inf_{\o\in\O}\max_{0\le i\le N-1}d_M(g_{i,j}(\o), g_{i,j'}(\o))\ge 3\b. \end{equation}
Now, we are ready to construct the strong full random horseshoe.
For $\hat a=(a_k)_{k\in\mathbb Z}\in \mathcal S_k$, let \begin{equation}\label{E:SHorsPsOr}
g^{\hat a}_i=g_{i-\left[\frac iN\right]N,a_{\left[\frac iN\right]}},\ i\in \mathbb Z, \end{equation} where $\left[\frac iN\right]$ is the largest integer less than $\frac iN$. By (\ref{E:SimpFuncShd1}) and (\ref{E:SimpFuncShd2}), we have that $\{ g^{\hat a}_i\}_{1\in\mathbb Z}$ forms an $\a$-pseudo orbit of $\tilde \phi$, thus Lemma \ref{L:LShad} implies a true orbit of $\tilde \phi$, $ \{\tilde\phi^i(g^{\hat a})\}_{i\in\mathbb Z}$, which is $\b$-shadowing $\{ g^{\hat a}_i\}_{1\in\mathbb Z}$.
Define $\Psi:\mathcal S_k\to L^\infty(\O, M)$ by letting $$\Psi(\hat a)=g^{\hat a},$$where the continuity of $\Psi$ follows from Lemma \ref{L:LContShad} immediately. It is obvious that (\ref{E:SHorsSepa}) and the argument for (\ref{E:SHorsSeparation}) imply Condition i) of Definition \ref{D:SRandomHorseshoe} with $\Delta=\b L^{-N}$. ii) of Definition and the continuity of $\Psi^{-1}$ follow exactly from the same arguments as (\ref{E:PsiInverseContinuity}), thus we do not repeat it here. This completes the proof of Part A. of Theorem \ref{T:TheoryAnosovMix2}. \end{proof}
Next, we construct a full random horseshoe which can capture enough entropy that is close to the upper topological fiber entropy. \begin{proof}[Proof of Part B. of Theorem \ref{T:TheoryAnosovMix2}]
For a given $\g>0$, by the definition of $\overline h(M\times \O|\O)$, we have that there exist $\o_0\in\O$ and $\e_0>0$ such that for any $0<\e<\e_0$, there exist infinitely many $n\in \mathbb N$ and corresponding $5\e$-separated set in $M_{\o_0}$ with respect to the Bowen metric $d_{M\times \O,n}^{\phi}$, whose cardinality is greater than $e^{n(\overline h(M\times\O|\O)-\frac13\g)}$.
Firstly, we fix an such $\e$ as above. Without loss of generality, we assume that $\e<\frac12\b_0$, where $\b_0$ is as in Lemma \ref{L:Shadowing}. Letting $\b=\frac13\e$ and applying Lemma \ref{L:Shadowing}, we have that there exists $\a>0$ corresponding to $\b$, which is taken to be smaller than $\b$. Then we fix a $\d$ such that $\d<\frac12\e$ and \begin{equation}\label{E:JumpControl} d_M(\pi_M\phi(x,\o_1),\pi_M\phi(x,\o_2))<\frac16 \a,\ \forall x\in M\text{ and }d_\O(\o_1,\o_2)<\d. \end{equation} Let $\{U_0,U_1,\cdot,U_p\}$ be a collection of $\frac13\a$-balls in $M$ such that $M=\cup_{i=1}^pU_i$. By Condition H2), there exists $N_1\in \mathbb N$ such that the following holds $$\pi_M\phi^n(U_i,\o)\cap U_j\neq \emptyset,\ \forall i,j\in \{0,1,\cdots,p\},\ n\ge N_1.$$
Now we fix an $N_2>> N_1$ such that there exists a $5\e$-separated set in $M_{\o_0}$ with respect to the Bowen metric $d_{M\times \O,N_2}^{\phi}$, whose cardinality is greater than $e^{N_2(\overline h(M\times\O|\O)-\frac13\g)}$. By the continuity of $\phi$, one can easily obtain the following technical lemma: \begin{lem}\label{L:SeparatedSetFiber}
There exist a nonempty open subset $O\subset \O$ and $k$ points $y_i:\in M,1\le i\le k$ with $$k\ge e^{n(\overline h(M\times\O|\O)-\frac13\g)}$$ such that for any $\o\in O$, $\{(y_i,\o)\}_{1\le i\le k}$ is a $3\e$-separated set with respect to $d^{\phi}_{M\times \O,N_2}$. \end{lem} Actually, $O$ can be taken to be a small open neighborhood of $\o_0$. For the sake of convenience and without loss of generality, we assume that the following hold: \begin{itemize} \item[a1)] $O$ is in a $\frac12\d$-ball with respect to $d^{\t}_{\O,N_2}$, where $\d$ is as in (\ref{E:JumpControl});
\item[a2)] $\frac{\log k}{N_2}> \overline h(M\times \O|\O)-\frac23\g$; \item[a3)] $\{y_i\}_{1\le i\le k}\subset U_0$ (Note that $M$ is compact). \end{itemize}
Let $N=N_2+N_1$. Note that both $\O_1:=\bigcup_{i\in\mathbb Z}\t^{iN}(O)$ and $\O_2:=\O\setminus \O_1$ are $\t^N$-invariant sets since $\t$ is invertible. Clearly, $\O_1$ and $\O_2$ are both measurable ( actually $\O_1$ is open and $\O_2$ is closed). We will deal with $\O_1$ and $\O_2$ separately.
Following exactly the same procedure as the proof of Lemma \ref{L:Horseshoe}, we can construct a finite Borel measurable partition of $\O$, which is denoted by $\xi:=\{\O'_t, t\in T\}$, Borel measurable simple functions $\{g^{0,0},g^{i,0},\ 1\le i\le p\}$, and a finite set $\{\o_t\in \O'_t,t\in T\}$ satisfying the following properties: \begin{itemize} \item[b1)] For any $\o,\o'$ fall in an element of $\xi$, the following holds $$d_\O(\t^{ n}\o,\t^{ n}\o')<\d,\ \forall 0\le n\le N-1,$$ where $\d$ satisfies (\ref{E:JumpControl}); \item[b2)] For $1\le i\le p$, $g^{i,0}=\sum_{t\in T}g^{i,0}_t\chi_{\O'_t}$, where $$g^{i,0}_t\in U_i\text{ and }\pi_M\phi^{N_1}(g^{i,0}_t,\o_t)\in U_0;$$ \item[b3)] $g^{0,0}=\sum_{t\in T}g^{0,0}_t\chi_{\O'_t}$, where $$g^{0,0}_t\in U_0\text{ and } \pi_M\phi^{N}(g^{0,0}_t,\o_t)\in U_0 \text{ for some }\o_t\in \O'_t.$$ \end{itemize}
For each $i\in[1,k]\cap \mathbb N$, there exists a finite Borel partition of $O$, which is denoted by $\eta_i:=\left\{O^i_q|q\in\{0,1,\cdots, p\}\right\}$, such that the following holds: \begin{itemize} \item[c1)] $\pi_M\phi^{N_2}(\{y_{i}\}\times O^i_{q})\subset U_q$, where $\{y_i\}_{1\le i\le k}$ is as in Lemma \ref{L:SeparatedSetFiber}. \end{itemize}
Now, we are ready to construct the full random horseshoe by constructing proper pseudo orbits.
\noindent {\bf \em Step 1.} For each $\o\in \O_1=\bigcup_{i\in\mathbb Z}\t^{iN}(O)$, we first collect all the hitting time of orbit $\{\t^{iN}\o\}_{i\in\mathbb Z}$ with set $O$, which is denoted by $\{n_i(\o)\}_{i\in I(\o)}\subset \mathbb Z\cup\{\pm \infty\}$. We arrange $I(\o)$ and $n_i(\o)$ in the following way:
\noindent
{\em Case (1.1).} $n_i(\o)$ has the same sign as $i$, and for any $i,j\in I (\o)$,
$$i< j\text{ if and only if }n_i(\o)< n_j(\o) ;$$
\noindent
{\em Case (1.2).} For the positive part of $I(\o)$, if $\{\t^{iN}\o\}_{i\in\mathbb N}$ hits $O$ infinitely many times, let
$$I(\o)\cap \mathbb N=\mathbb N;$$
otherwise, let
$$I(\o)\cap \mathbb N=\{1,2,\cdots, m^+(\o)\}\text{ and }n_{m^+(\o)+1}=+\infty,$$
where $m^+(\o)$ is the number of times $\{\t^{iN}\o\}_{i\in\mathbb N}$ hits $O$;
\noindent
{\em Case (1.3).} Similarly, for the negative part of $I(\o)$, if $\{\t^{-iN}\o\}_{i\in\mathbb N}$ hits $O$ infinitely many times, let $$I(\o)\cap -\mathbb N=-\mathbb N;$$
otherwise, let
$$I(\o)\cap -\mathbb N=\{-1,-2,\cdots, -m^-(\o)\}\text{ and }n_{m^-(\o)-1}=-\infty,$$ where
$m^-(\o)$ is the number of how many times $\{\t^{-iN}\o\}_{i\in\mathbb N}$ hitting $O$;
\noindent
{\em Case (1.4).} If $\o\in \t^{-N_1}O$, put $0$ in $I(\o)$ and let $n_0(\o)=0$; otherwise, $0\notin I(\o)$.\\
We remark here that $I(\o)$ is always non-empty for $\o\in \O_1$, however, each of $I(\o)\cap \mathbb N$, $I(\o)\cap \{0\}$, and $I(\o)\cap -\mathbb N$ could be an empty set.
\noindent
{\bf \em Step 2.} For a given $\o\in\O_1$ and $\hat a=(a_n)\in \mathcal S^k$, we define a pseudo orbit as in the following:
\noindent
{\em Case 1.} For $n=n_i(\o)$, $i\in I(\o)$, suppose that
$$\t^{nN}\o\in O^{a_n}_q \text{ and } \t^{nN+N_2}\o\in \O'_{t_1},$$
then define
\begin{equation}\label{E:POCase1}
y'_{\hat a,\o}(nN+j)=
\begin{cases}
\pi_M\phi^{j}(y_{a_n},\t^{nN}\o),&0\le j\le N_2-1\\
\pi_M\phi^{j-N_2}(g^{q,0}(\t^{nN+N_2}\o),\o_{t_1}),&N_2\le j\le N-1\\
\end{cases}.
\end{equation}
\noindent
{\it Case 2.} For $n\notin\{n_i(\o)\}_{i\in I(\o)}$, suppose that $\t^{nN}\o\in \O'_{t_2}$,
then define
\begin{equation}\label{E:POCase2}
y'_{\hat a,\o}(nN+j)=\pi_M\phi^j(g^{0,0}(\t^{nN}\o),\o_{t_2}),\ 0\le j\le N-1.
\end{equation}
For Case 1., note that $\{(y'_{\hat a,\o}(nN+j),\t^{nN+j}\o\}_{0\le j\le N_2-1}$ is a true orbit, and $\t^{nN}\in O^{a_n}_q$, therefore
$$\pi_M\phi(y'_{\hat a,\o}(nN+N_2-1),\t^{nN+N_2-1}\o),\ y'_{\hat a,\o}(nN+N_2)\in U_q,$$
thus
$$d_M\left(\pi_M\phi(y'_{\hat a,\o}(nN+N_2-1),\t^{nN+N_2-1}\o),\ y'_{\hat a,\o}(nN+N_2)\right)\le \frac23\a.$$
By b1) and (\ref{E:JumpControl}), we have that for $N_2\le j\le N-1$,
\begin{align*}
&d_M\left(\pi_M\phi\left( y'_{\hat a,\o}(nN+j),\t^{nN+j}\o\right), y'_{\hat a,\o}(nN+j+1)\right)\\
=&d_M\left(\pi_M\phi\left(\pi_M\phi^j(g^{q,0}(\t^{nN+N_2}\o),\o_{t_1}),\t^{nN+N_2+j}\o\right), \pi_M\phi^{j+1}(g^{q,0}(\t^{nN+N_2}\o),\o_{t_1})\right)\\
\le &\frac16\a,
\end{align*}
which implies that $\{(y'_{\hat a,\o}(nN+j),\t^{nN+j}\o\}_{N_2\le j\le N-1}$ is a segment of an $(\o,\frac16\a)$-pseudo orbit.
Hence $\{(y'_{\hat a,\o}(nN+j),\t^{nN_j}\o\}_{0\le j\le N-1}$ is a segment of an $(\o,\frac23\a)$-pseudo orbit with length $N$. \\
For Case 2., by b3), (\ref{E:JumpControl}), and using the same argument as for $N_2\le j\le N-1$ in Case 1., we have that $\{(y'_{\hat a,\o}(nN+j),\t^{nN_j}\o\}_{0\le j\le N-1}$ is a segment of an $(\o,\frac16\a)$-pseudo orbit with length $N$.
Also note that, all the segments of $(\o,\a)$-pseudo orbit defined in both Case 1. and Case 2. start and end in $U_0$ which has diameter $\frac23\a$. Hence, the pseudo orbit $\{(y'_{\hat a,\o}(j),\t^{j}\o\}_{j\in \mathbb Z}$ is an $(\o,\frac23\a)$-pseudo orbit.
It is clear that $\{(y'_{\hat a,\o}(j),\t^{j}\o\}_{j\in \mathbb Z},\ \o\in\O$ induces an $\a$-pseudo orbit of $\tilde\phi$ by letting $$g'_{\hat a,j}(\o)=y'_{\hat a,\o}(j),\ j\in \mathbb Z,\ \o\in\O_1.$$
The Borel measurability of each $g'_{\hat a,j}$ follows from (\ref{E:POCase1}) and (\ref{E:POCase2}) immediately, since each $g'_{\hat a,j}$ is either a simple function or an image of a simple function under iterations of $\phi$ or $\phi^{-1}$. Therefore, by applying Theorem \ref{L:LShad} for $\tilde\phi|_{L^\infty(\O_1)}$, we have that there exists a unique true orbit of $\tilde\phi|_{L^\infty(\O_1)}$, $\{(\tilde\phi|_{L^\infty(\O_1)})^n(g_{\hat a})\}_{n\in\mathbb Z}$, which is $\b$-shadowing $\{g'_{\hat a,j}\}_{j\in \mathbb Z}$.
Now we define the desired horseshoe map by the letting
$$\Psi(\hat a)|_{\O_1}=g_{\hat a},\ \forall \hat a\in \mathcal S_k.$$
The continuity of $\Psi|_{\O_1}$ follows from Lemma \ref{L:ContinousShadowing} straightforwardly. The next, we prove that Condition i) of Definition \ref{D:RandomHorseshoe} for $\Psi|_{\O_1}$ holds.
For $\hat a'\neq\hat a'' (\in \mathcal S_k)$, let $s=\min\left\{|n|\big |\ a'_n\neq a''_n\right\}$ and $r$ be such that $|r|=s$ and $a'_r\neq a''_r$ . By the choice of $O,y_i,U_i,\e,\a,\b$ at the beginning of this section, we have that
$$d^\phi_{M\times \O,N_1}\left(\phi^{rN}(\Psi(\hat a_1)(\o),\o),\phi^{rN}(\Psi(\hat a_2)(\o),\o)\right)>\e,\ \forall \o\in \t^{-rN}(O),$$
where $d^\phi_{M\times \O,N_1}$ is the Bowen metric.
Note that there is a constant $L>1$ such that for any $\o\in\O$ and $x,y\in M$ with $d(x,y)>0$, we have
$$d(\phi^{\pm1}(x,\o),\phi^{\pm1}(y,\o))\le Ld_M(x,y).$$
Therefore, for any $\o$ from the non-empty open set $\t^{-rN}(O)$, the following holds,
\begin{align*} & d_M\left(\pi_M\phi^{rN}\left(\Psi(\hat a_1)(\o),\o\right),\pi_M\phi^{rN}\left(\Psi(\hat a_2)(\o),\o\right)\right)\\ =&d\left(\phi^{rN}\left(\Psi(\hat a_1)(\o),\o\right),\phi^{rN}\left(\Psi(\hat a_2)(\o),\o\right)\right)\\ \ge& L^{-N_1}d^\phi_{M\times \O,N_1}\left(\phi^{rN}(\Psi(\hat a_1)(\o),\o),\phi^{rN}(\Psi(\hat a_2)(\o),\o)\right)\\ >& \e L^{-N},
\end{align*}
which implies Condition i) of Definition \ref{D:RandomHorseshoe} by taking $\Delta= \e L^{-N}$.
The continuity of $\Psi^{-1}$ follows from the same argument as (\ref{E:PsiInverseContinuity}).
\noindent {\bf \em Final Step.} To complete the proof, we need to address the following issues:
Firstly, Condition ii) in Definition \ref{D:RandomHorseshoe} for $\Psi|_{\O_1}$ simply follows from the uniqueness of $g_{\hat a}$ and the same argument used in the proof of Lemma \ref{L:Horseshoe}.
Secondly, we need to extend the definition of $g_{\hat a}$ properly to the whole $\O$. This can be done by Theorem \ref{T:TheoryAnosovMix}. Since $\O_1$ and $\O_2=\O\setminus \O_2$ are both $\t^N$-invariant, we can view $\O_2$ independently, and give up to capture entropy from the system $\phi^N|_{\O_2}$ (as the entropy captured from $\phi^N|_{\O_1}$ is already enough). Theorem \ref{T:TheoryAnosovMix} implies that for large $N$, there exists a Borel function $g:\O_2\to M$ such that
$$\phi^N|_{\O_2}(graph(g))=graph(g).$$
We simply extend $g_{\hat a}$ to $\tilde g_{\hat a}$ by letting
$$\tilde g_{\hat a}|_{\O_1}=g_{\hat a}\text{ and }\tilde g_{\hat a}|_{\O_2}=g.$$
It is not hard to see that such an extension is harmless for Conditions i) and ii) of Definition \ref{D:RandomHorseshoe}. So the map $\Psi:\mathcal S_k\to L^\infty$ which takes $\hat a$ to $\tilde g_{\hat a}$ gives a full random horseshoe.
Finally, as we required at the beginning of this section that $N_2>>N_1$, a2) yields
$$\frac{\log k}N=\frac{\log k}{N_2+2N_1}>\frac{\log k}{N_2}-\frac13\g>\overline h(M\times\O|\O)-\g.$$
This completes the proof of theorem.
\end{proof}
\section{Density of Periodic Random Invariant Measures}
In this section, we prove Theorem \ref{T:PeriodicApprox}, namely, any $\phi$-invariant measure with marginal $\mathbb P$ can be approximated by random periodic measures.
\vskip0.05in \noindent {\it Proof of Theorem \ref{T:PeriodicApprox}.} We first fix a $\t$-invariant probability measure $\mathbb P$ on $\O$. We will show that for any $\mu\in \mathcal I_{\mathbb P}(M\times\O)$, there is a sequence of periodic measures $\{\mu_i\}_{i=1,2,\cdots}$ such that for any continuous real function $h$ defined on $M\times\O$ the following holds $$\int h d\mu=\lim_{i\to\infty} \int hd\mu_i.$$ Since $C(M\times\O)$, the space of continuous real functions defined on a compact set, is separable, it suffices to show that for any $\mu\in \mathcal I_{\mathbb P}(M\times\O)$, $\e>0$ and $h\in C(M\times \O)$, there is a periodic measure $\mu_{\e}^h$ such that \begin{equation}\label{E:MeasAppr}
\left|\int hd\mu-\int hd\mu_{\e}^h\right|<\e. \end{equation} To the end, based on the above assertion, it is not hard to see that $\{\mu_{\frac1n}^{h_m}\}_{n,m\in \mathbb N}$ forms an countable dense subset of $\mathcal I_{\mathbb P}(M\times\O)$, where $\{h_m\}_{m\in\mathbb N}$ is a dense subset of $C(M\times\O)$.
Fixing $\mu\in \mathcal I_{\mathbb P}(M\times\O)$, $\e>0$ and $h\in C(M\times \O)$, we now construct $\mu_{\e}^h$.
Since $h$ is uniformly continuous on $M\times \O$ by the compactness of $M\times\O$, there exists $\d_1>0$ such that the following holds \begin{equation}\label{E:Delta1AndEpsilon}
|h(x_1,\o)-h(x_2,\o)|<\frac1{10}\e,\text{ provided }d_M(x_1,x_2)<\d_1. \end{equation}
By Lemma \ref{L:LShad}, there exists $\d_2>0$ such that any $\d_2$-pseudo orbit can be $\frac13\d_1$-shadowed by a unique true orbit of $\tilde\phi$. By Lemma \ref{L:JoinSegments}, there exists an $N_1\in \mathbb N$ such that for any $g_1,g_2\in L^\infty(\O, M)$, there exists a $g_3\in L^\infty(\O, M)$ such that \begin{equation}\label{E:JoinGraph} \max\left\{d_{L^\infty(\O, M)}(g_3,g_1),d_{L^\infty(\O, M)}(\tilde\phi^{N_1}(g_3),g_2)\right\}<\frac13\d_2. \end{equation}
For $m\in\mathbb N$ and $s>0$, let $$A_{m,s}^h:=\left\{(x,\o)\in M\times\O\Big|\ \left|\int hd\mu-\frac1n\sum_{i=0}^{n-1}h\left(\phi^i(x,\o)\right)\right|<s,\ \forall n\ge m\right\}.$$ It is not hard to see that $A_{m,s}^h$ is a $G_{\d}$-set by the continuity of $h$ and $\phi$, and $\lim_{m\to\infty}\mu(A_{m,s}^h)=1$ by the Birkhoff ergodic theorem. Now fix an $N_2\in\mathbb N$ such that the following holds \begin{equation}\label{E:LargePeriod} \mathbb P \left(\pi_\O\left(A_{N_2,\frac15\e}^h\right)\right)>1-\frac1{4M}\e\text{ and }\frac{N_1}{N_2}<\frac1{10M}\e, \end{equation}
where $M=\sup_{M\times\O}|h|$.
Note that $\mu$ is regular. Then, there exists a compact set $K\subset A_{N_2,\frac15\e}^h$ such that $$\mathbb P \left(\pi_\O\left(K\right)\right)>1-\frac14\e.$$ Applying Theorem \ref{T:BMST}, we have that there is a Borel map $g:\O\to M$ such that the following holds $$g(\o)\in K,\ \forall x\in \pi_\O\left(K\right).$$
Let $N=N_1+N_2$. By (\ref{E:JoinGraph}), we have that there exists an $\d_2$-pseudo orbit $\{(g_n'\}_{n\in \mathbb Z}$ satisfying the following conditions: \begin{itemize} \item[a)] $g'_{lN}=g,\ \forall l\in \mathbb Z$; \item[b)] $g'_{lN+k}=\tilde\phi^{k}(g),\ \forall 0\le k\le N_2-1$.
\end{itemize} Therefore, by Lemma \ref{L:LShad}, there exists a true orbit $\{\tilde\phi^n(\tilde g_0)\}_{n\in\mathbb Z}$ of $\tilde\phi$, which is $\frac13\d_1$-shadowing $\{g_n'\}_{n\in \mathbb Z}$. By the uniqueness of the shadowing orbit, it is not hard to see that $\tilde g_0$ is a random periodic point with period $N$.
Next, we show that the periodic measure induced by $graph(\tilde g_0)$ $$\mu_\e^h:=\frac1N\sum_{i=0}^{N-1}\mu_{\tilde g_i},\text{ provided }graph(\tilde g_i)=\phi^i(graph(\tilde g_0)),\ 0\le i\le N-1,$$ satisfies (\ref{E:MeasAppr}).
Note that $\mathbb P$ can be decomposed into the following form $$\mathbb P=\sum_{i\in I} a_i\nu_i,$$ where $card(I)\le N$, $a_i>0$, $\sum_{i\in I}a_i=1$, and $\nu_i$ is $\t^N$-ergodic.
By Birkhoff ergodic theorem, there exists an $\phi$-invariant Borel function $\tilde h:M\times \O$ such that $$\tilde h(x,\o)=\lim_{n\to\infty}\frac1n\sum_{i=0}^{n-1}h\left(\phi^i(x,\o)\right),\ \mu_\e^h-a.e.,$$ and $$\int \tilde hd\mu_\e^h=\int hd\mu_\e^h.$$ Suppose that $(\tilde g_0(\o),\o)$ is a regular point of $\mu_\e^h$ while $\o$ is a regular point of $\nu_i$ for some $i\in I$, which can be done because the set of regular points of an invariant measure occupies full probability. Denote that
$$\mathcal K(n):=card\left(\left\{k|\ 0\le k\le n-1,\t^{kN}\in \pi_\O(K)\right\}\right).$$ Then we have that, \begin{align*}
&\left|\tilde h(\tilde g_0(\o),\o)-\int hd\mu\right|\\
=&\lim_{n\to\infty}\left|\frac1{nN}\sum_{i=0}^{nN-1}h\left(\phi^i(\tilde g_0(\o),\o)\right)-\int hd\mu\right|\\
\le&\lim_{n\to\infty}\frac1{nN}\Bigg\{\sum_{0\le k\le n-1,\t^{kN}\in \pi_\O(K)}\left|\sum_{i=kN}^{(k+1)N-1}h\left(\phi^i(\tilde g_0(\o),\o)\right)-N\int hd\mu\right|\\
&\quad\quad\quad\quad\quad+\sum_{0\le k\le n-1,\t^{kN}\notin \pi_\O(K)}\left|\sum_{i=kN}^{(k+1)N-1}h\left(\phi^i(\tilde g_0(\o),\o)\right)-N\int hd\mu\right|\Bigg\}\\
\le&\lim_{n\to\infty}\frac1{nN}\Bigg\{\sum_{0\le k\le n-1,\t^{kN}\in \pi_\O(K)}\sum_{i=kN}^{(k+1)N-1}\left|h\left(\phi^i(\tilde g_0(\o),\o)\right)-h\left(\phi^{i-kN}( g(\t^{kN}\o),\t^{kN}\o)\right)\right|\\
&\quad\quad+\sum_{0\le k\le n-1,\t^{kN}\in \pi_\O(K)}\left[\left|\sum_{i=kN}^{kN+N_2-1}h\left(\phi^{i-kN}( g(\t^{kN}\o),\t^{kN}\o)\right)-N_2\int hd\mu\right|+2N_1M\right]\\ &\quad\quad+\sum_{0\le k\le n-1,\t^{kN}\notin \pi_\O(K)}2NM\Bigg\}\\ \le &\lim_{n\to\infty}\frac1{nN}\left\{\mathcal K(n)\left(\frac1{10}N\e+\frac15N_2\e+2N_1M\right)+2NM(n-\mathcal K(n))\right\}\\ \le&\frac12\nu_i(K)\e+2M(1-\nu_i(\pi_\O (K)))\\ \le& \frac12\e+2M(1-\nu_i(\pi_\O(K))). \end{align*} In the above argument, condition b), (\ref{E:Delta1AndEpsilon}) and $K\subset A_{N_2,\frac15\e}^h$ are used.
Therefore, as $\tilde h$ being $\phi$-invariant, we have that \begin{align*}
&\left|\int \tilde hd\mu_\e^h-\int h d\mu\right|=\left|\int \tilde hd\mu_{\tilde g_0}-\int h d\mu\right|\\
\le&\int\left|\tilde h(\tilde g_0(\o),\o)-\int hd\mu\right|d\mu_{\tilde g_0}=\int\left|\tilde h(\tilde g_0(\o),\o)-\int hd\mu\right|d\mathbb P\\
=&\sum_{i\in I}a_i\int\left|\tilde h(\tilde g_0(\o),\o)-\int hd\mu\right|d\nu_i\le\sum_{i\in I}a_i\left(\frac12\e+2M(1-\nu_i(\pi_\O(K)))\right)\\ =&\frac12\e+2M(1-\mathbb P(\pi_\O(K)))<\e, \end{align*} which completes the proof. \qed
\section{Liv\v sic Theorem}
Throughout this section , we let $\mathbb P$ be a fixed $\t$-invariant probability measure on $\O$. Before proving the main result, we introduce several notions and derive two lemmas, namely Random specification and Bowen property.
\subsection{Random specification} The specification property plays an important role in the study of equilibrium state of uniformly hyperbolic systems such as Anosov diffeomorphisms or Axiom A systems. Under the setting of this paper, specification can be introduced for random systems in a natural way. For the sake of simplicity, we first introduce some notions.
For $g\in L^{\infty}(\O)$ and $n\in \mathbb N$, define that \begin{align*}
&(g,n)=\left\{(g_0,g_1,\cdots,g_{n-1})\in \left(L^\infty(\O, M)\right)^n\big|\ g_i=\tilde\phi^i\left(g\right),\ 0\le i\le n-1\right\},\\
&B_n(g,\e)=\left\{h\in L^\infty(\O, M)\big |\ d_{n}^{\tilde\phi}\left(g,h\right)<\e\right\}, \end{align*} where $d_{n}^{\tilde\phi}$ is the Bowen metric induced by $\tilde\phi$ and $d_{L^\infty(\O, M)}$. We call $(g,n)$ a random orbit segment of $g$ with length $n$ (or $n$-segment of $g$ for short), and $B_n(g,\e)$ the Bowen ball of $g$ with length $n$ and radius $\e$ (or $(n,\e)$-Bowen ball of $g$ for short). \begin{defn}\label{D:Specification}
A collection of random orbit segments $\mathcal G\subset L^\infty(\O, M)\times \mathbb N$ has specification at scale $\e$ if there exists $\tau\in\mathbb N$ such that for every $\{(g_i,n_i)|\ 1\le i\le k\}\subset \mathcal G$, there exists a $h\in L^\infty(\O, M)$ satisfying that $$h_i\in B_{n_i}(g_i,\e),\ 1\le i\le k$$ where $h_i\in L^\infty(\O, M)$ is given by \begin{align*} h_i=\tilde\phi^{\sum_{j=0}^{i-1}n_j+(i-1)\tau}h. \end{align*} \end{defn}
\begin{lem}\label{L:MixToSpecification} For the systems satisfying Conditions H1) and H2), $L^\infty(\O, M)\times \mathbb N$ has specification at any scale. \end{lem} \begin{proof} This result simply follows from Lemma \ref{L:JoinSegments} and the shadowing property (Lemma \ref{L:LShad}). We omit the detailed proof here.
We remark that the connecting time "$\tau$" in the definition of specification above only depends on the scale "$\e$" and the given system. More precisely, the connecting time is determined by the accuracy of shadowing and the speed of mixing. \end{proof}
\subsection{Bowen property} For a potential $\psi\in L^1_{\mathbb P}(\O,\mathcal C^{0,\a}(M))$ and a collection of random orbit segments $\mathcal G\subset L^\infty(\O, M)\times \mathbb N$, we define the variation $V(\mathcal G,\psi,\mathbb P, \e)$ and the average variation $\bar V(\mathcal G,\psi,\mathbb P, \e)$ of $\psi$ on $\mathcal G$ with respect to $\mathbb P$ at scale $\e$ by the following
\begin{align}\begin{split}\label{D:VarP}
&V(\mathcal G,\psi,\mathbb P, \e):=\sup_{(g,n)\in \mathcal G,\ h\in B_n(g,\e)}\left\{\sup_{\mathbb P}\left|S_n\psi\left(g(\o),\o\right)-S_n\psi\left(h(\o),\o\right)\right|\right\}, \end{split} \end{align} \begin{align}\begin{split}\label{D:AveVarP}
&\bar V(\mathcal G,\psi,\mathbb P, \e):=\sup_{(g,n)\in \mathcal G,\ h\in B_n(g,\e)}\left\{\int_{\O}\left|S_n\psi\left(g(\o),\o\right)-S_n\psi\left(h(\o),\o\right)\right|d\mathbb P\right\}, \end{split} \end{align} where we set $S_n(\psi(x,\o))=\sum_{i=0}^{n-1}\psi(\phi^i(x,\o))$ for $(x,\o)\in M\times \O$.
\begin{defn}\label{D:BowenPro} Given a collection of random orbit segments $\mathcal G\subset L^\infty(\O, M)\times \mathbb N$, a potential $\psi$ has the {\bf Bowen property} (or {\bf the average Bowen property}) on $\mathcal G$ with respect to $\mathbb P$ at scale $\e$ if the following holds $$V(\mathcal G,\psi,\mathbb P,\e)<\infty\ (\text{ or }\bar V(\mathcal G,\psi,\mathbb P,\e)<\infty).$$
\end{defn} Let $L^\infty_{\mathbb P}(\O,\mathcal C^{0,\a}(M))\subset L^1_{\mathbb P}(\O,\mathcal C^{0,\a}(M))$ be the collection of functions $\psi$ with the following holds
$$\sup_{\mathbb P}\|\psi\|_{C^{0,\a}(M_\o)}<\infty.$$ We have the following results.
\begin{lem}\label{L:BowenPro} Let $\phi$ satisfy Condition H1) and H2), then for $\mathcal G=L^\infty(\O, M)\times \mathbb N$, there exists $\e_0>0$ such that for any $\e\in(0,\e_0)$ each potential $\psi\in L^\infty_{\mathbb P}(\O,\mathcal C^{0,\a}(M))$ (or $\psi\in L^1_{\mathbb P}(\O,\mathcal C^{0,\a}(M))$ has the {\bf Bowen property} ( or {\bf average Bowen property}) on $\mathcal G$ with respect to $\mathbb P$ at scale $\e$. \end{lem} \begin{proof} By Lemma \ref{L:InvMani}, \ref{L:LocCoor}, and P2) in the proof of Lemma \ref{L:ClosingLemma}, for a fixed positive number $\l\in(0,\l_0)$, there exist $\e>0$, $\d\in(0,\e)$, $L>1$ such that for any $x,y\in M$ with $d_M(x,y)<\d$, the following hold \begin{align*} & [x,y]_\o=:W^s_{\e}(x,\o)\cap W^u_{\e}(y,\o),\text{ which is continuous on }x,y\text{ and }\o,\\ &\max\left\{d_M(x,[x,y]_\o),d_M(y,[x,y]_\o)\right\}\le Ld_M(x,y),\\ &d_M(\pi_M\phi^n(x,\o),\pi_M\phi^n(y,\o))\le e^{-n\l}d_M(x,y)\text{ for }(y,\o)\in W^s_{\e}(x,\o),\ n\ge 0,\\ &d_M(\pi_M\phi^{-n}(x,\o),\pi_M\phi^{-n}(y,\o))\le e^{-n\l}d_M(x,y)\text{ for }(y,\o)\in W^u_{\e}(x,\o),\ n\ge 0. \end{align*} For $(g,n)\in \mathcal G$ and $h\in B_n(g,\e)$, we have that for any $\o\in\O$ and $0\le i\le n-1$ \begin{align}\begin{split}\label{E:EstOnOrbit} &d_M(\pi_M\phi^i(g(\o),\o),\pi_M\phi^i(h(\o),\o)))\\ \le &d_M\left(\pi_M\phi^i(g(\o),\o),\left[\pi_M\phi^i(g(\o),\o),\pi_M\phi^i(h(\o),\o))\right]_{\t^i\o}\right)\\ &+d_M\left(\pi_M\phi^i(h(\o),\o),\left[\pi_M\phi^i(g(\o),\o),\pi_M\phi^i(h(\o),\o))\right]_{\t^i\o}\right)\\ \le &e^{-i\l}d_M\left(g(\o),\left[g(\o),h(\o)\right]_{\o}\right)\\ &+e^{-(n-1-i)\l}d_M\left(\pi_M\phi^{n-1}(h(\o),\o),\left[\pi_M\phi^{n-1}(g(\o),\o),\pi_M\phi^{n-1}(h(\o),\o))\right]_{\t^{n-1}\o}\right)\\ \le&2Le^{-i\l}d_M(g(\o),h(\o))+2Le^{-(n-1-i)\l}d_M(\pi_M\phi^{n-1}(g(\o),\o),\pi_M\phi^{n-1}(h(\o),\o))\\ \le&4Le^{-\min\{i,n-1-i\}\l}\e. \end{split} \end{align}
Then \begin{align}\begin{split}\label{E:VarEstBP}
&\sup_{\mathbb P}|S_n(\psi(g(\o),\o))-S_n(\psi(h(\o),\o))|\\
\le &\sup_{\mathbb P}\|\psi\|_{C^{0,\a}(M_\o)}(4L\e)^\a\sum_{i=0}^{n-1}e^{-\min\{i,n-1-i\}\a\l}\\<&\frac{2 \sup_{\mathbb P}\|\psi\|_{C^{0,\a}(M_\o)}(4L\e)^\a}{1-e^{-\a\l}}, \end{split} \end{align} which is independent on $n$, $g$ and $h$.
For the average Bowen property, we need only to replace the $\sup_{\mathbb P}$ in (\ref{E:VarEstBP}) by integration on $\O$ with respect to $\mathbb P$ and obtain the following \begin{align}\begin{split}\label{E:VarEstABP}
&\int_{\O}|S_n(\psi(g(\o),\o))-S_n(\psi(h(\o),\o))|d\mathbb P\\
\le&\int_{\O}\sum_{i=0}^{n-1}\left|\psi\left(\phi^i(g(\o),\o)\right)-\psi\left(\phi^i(h(\o),\o)\right)\right|\\
\le &\int_{\O}(4L\e)^\a\sum_{i=0}^{n-1}\|\psi\|_{C^{0,\a}(M_{\t^i\o})}e^{-\min\{i,n-1-i\}\a\l}d\mathbb P\\
<&\frac{2(4L\e)^\a}{1-e^{-\a\l}}\int_{\O} \|\psi\|_{C^{0,\a}(M_\o)}d\mathbb P, \end{split} \end{align} which is independent on $n$, $g$ and $h$. This completes the proof of this lemma. \end{proof}
\subsection{Proof of the weak version of random Liv\v sic Theorem} This proof contains two main steps: in the first step, we show that there is an $\a$-H\"older functional $\hat\Psi$ defined on $C(\O,M)$ so that the cohomology equation (\ref{E:CoBoundary}) holds for all $g\in C(\O,M)$, where $C(\O,M)$ is the space of continuous maps from $\O$ to $M$ endowed with the supreme metric; in the second step, we extend the domain of functional $\hat \Psi$ from $C(\O,M)$ to $L^\infty(\O, M)$ with (\ref{E:CoBoundary}) kept. Note that the second step is not a straightforward consequence of the first step since $C(\O,M)$ is {\bf NOT }dense in $L^\infty(\O, M)$.
\begin{lem}\label{L:WLivThC} For any $\Phi\in L^1_{\mathbb P}(\O,C^{0,\a}(M))$ satisfying (\ref{E:PeriodicNull}), there exists a bounded functional $\hat\Psi:C(\O,M)\to \mathbb R$ which is $\a$-H\"older continuous and satisfies the following cohomology equation \begin{equation}\label{E:CoBoundary} \tilde\Phi(h)=\hat \Psi(\tilde\phi(h))-\hat\Psi(h),\ \forall h\in C(\O,M). \end{equation} \end{lem} \begin{proof} At first, we prove a the following technical lemma: \begin{lem}\label{L:TransitiveOnC(Ome,M)} There exists an orbit of $\tilde\phi$, whose closure contains $C(\O,M)$. \end{lem} \begin{proof} Note that $C(\O,M)$ is separable since $M$ and $\O$ are compact. Let $\{g_i\}_{i=1,2,\cdots}\subset C(\O,M)$ be a countable dense set of $C(\O,M)$. Denote $U_{ij}$ the $\frac1j$-ball centered at $g_i$ in $C(\O,M)$.
We relabel the countable set $\left\{U_{i,j}\right\}_{i,j=1,2,\cdots}$ and denote the relabeled set by $\{V_k\}_{k=1,2\cdots}$ which satisfies the following rule: $$\text{ for }V_{k_1}=U_{i_1j_1}, V_{k_2}=U_{i_2j_2},\ k_1\le k_2\text{ iff } i_1+j_1\le i_2+j_2\text{ or } i_1+j_1= i_2+j_2,i_1\le i_2.$$ We also define a function $l:\mathbb N\to \mathbb N$ by letting $l(k)=i$ when $V_k=U_{ij}$.\\
Let $\a$ be as in Lemma \ref{L:LShad} for a fixed $\b\in(0,\b_0)$. By Lemma \ref{L:JoinSegments}, for any $n_k,n_{k+1}\in\mathbb N$, there exist $\{g'_k\in L^\infty(\O, M)\}_{k=1,2,\cdots}$ and $m\in\mathbb N$ such that $$\max\left\{d_{L^\infty(\O, M)}\left(\tilde\phi^{n_k}g_{l(k)},g'_k\right),\ d_{L^\infty(\O, M)}\left(\tilde\phi^{m}g'_k,\tilde\phi^{-n_{k+1}}g_{l(k+1)}\right)\right\}<\a.$$ Therefore, we have that $\mathcal G_{0},\mathcal G_1,\mathcal G_2,\cdots$ form an $\a$-pseudo orbit of $\tilde\phi$, where we let \begin{align*}
\mathcal G_{0}&=\{\tilde\phi^{-n}g_{l(1)}\}_{n\ge 1}\\ \mathcal G_k&=\left\{g_{l(k)},\cdots,\tilde\phi^{n_k-1}g_{l(k)},g'_k,\cdots,\tilde\phi^{m-1}g'_k,\tilde \phi^{-n_{k+1}}g_{l(k+1)},\cdots, \tilde\phi^{-1}g_{l(k+1)}\right\},\ k\in \mathbb N. \end{align*}
Thus, Lemma \ref{L:LShad} implies that there exists a $g\in L^\infty(\O, M)$ such that $$d_{L^\infty(\O, M)}\left(\tilde\phi^{\pm(n_1-1)}g,\tilde\phi^{\pm(n_1-1)}g_{l(1)}\right)<\b,$$
and
$$d_{L^\infty(\O, M)}\left(\tilde\phi^{\pm (n_{k+1}-1)}\left(\tilde\phi^{\sum_{i=1}^k(n_i+n_{i+1}+m)}g\right),\tilde\phi^{\pm n_{k+1}-1}g_{l(k+1)}\right)<\b,\ \forall k\ge 1.$$ By applying Lemma \ref{L:ContinousShadowing} and choosing $n_k$ large enough for each $k\ge 1$, we have the following: \begin{equation}\label{E:ADenseOrbit} g\subset V_1\text{ and }\tilde\phi^{\sum_{i=1}^k(n_i+n_{i+1}+m)}g\subset V_{k+1},\ \forall k\ge 1, \end{equation} which implies that \begin{equation}\label{E:ADenseOrbit1} C(\O,M)\subset \text{ closure of }\{\tilde \phi^n(g)\}_{n\ge 0} \text{ in }L^\infty(\O, M). \end{equation} \end{proof}
Now we define a functional on $\{\tilde \phi^n(g)\}_{n\ge 0}$ by the following \begin{equation}\label{E:PsiOnOrb} \hat \Psi'(\tilde \phi^n(g))=\sum_{i=0}^n\tilde\Phi(\tilde \phi^i(g)) \end{equation}
From Lemma \ref{L:MixToSpecification} and the second part of Lemma \ref{L:BowenPro}, it follows that $\hat \Psi ''$ is bounded on $\{\tilde \phi^n(g)\}_{n\ge 0}$. In fact, there exist $N\in\mathbb N$ and $\e\in (0,\e_0)$ (where $\e_0$ is as in Lemma \ref{L:BowenPro}) such that for any $n\in \mathbb N$, there exists a periodic point $h\in L^\infty(\O, M)$ such tha the following hold: $$\tilde \phi^{n+N}(h)=h\text{ and }d_{L^\infty(\O, M)}(\tilde \phi^i(g),\tilde \phi^i(h))\le \e,\ i=0,1,\cdots, n.$$ Therefore, by (\ref{E:VarEstABP}) we have that \begin{align}\begin{split}\label{E:PsiSupNormEst}
\left|\hat \Psi'(\tilde \phi^n(g))\right|&\le \sum_{i=n+1}^{n+N-1}\left|\tilde \Phi(\tilde\phi^i(h))\right|+\sum_{i=0}^n\left|\tilde\Phi(\tilde\phi^i(h))-\tilde\Phi(\tilde\phi^i(g))\right|\\
&\le \left(N+\frac{2(4L\e)^\a}{1-e^{-\a\l}}\right)\int_{\O} \|\Phi\|_{C^{0,\a}(M_\o)}d\mathbb P, \end{split} \end{align} which is independent on $n$.
Let $\a_0>0$ and $C$ be the constants as in the second part of Lemma \ref{L:LShad}. We have that for any $\a\in(0,\a_0)$ each $\a$-pseudo orbit of $\tilde\phi$ can be $C\a$-shadowed. Without loss of generality, we assume that $C\a_0<\e_0$, where $\e_0$ is as in Lemma \ref{L:BowenPro}. Suppose that for $n,k\in\mathbb N$ $d_{L^\infty(\O, M)}(\tilde \phi^{n+k}(g),\tilde \phi^{n}(g))<\a_0$. By applying Lemma \ref{L:LShad}, there exists a random periodic point $g_0\in L^\infty(\O, M)$ such that the following holds $$\tilde \phi^k(g_0)=g_0\text{ and }d_{L^\infty(\O, M)}(\tilde \phi^i(g_0),\tilde\phi^{n+i}(g))<(C+1)d_{L^\infty(\O, M)}(\tilde \phi^{n+k}(g),\tilde \phi^{n}(g)), i=0,1,\cdots, k.$$ Therefore, we obtain that \begin{align}\begin{split}\label{E:PsiHolderNormEst}
&\left|\hat\Psi'(\tilde \phi^{n+k}(g))-\hat\Psi'(\tilde \phi^{n}(g))\right|\\
=&\left|\sum_{i=n+1}^{n+k}\tilde \Phi(\tilde\phi^i(g))-\sum_{i=1}^{k}\tilde \Phi(\tilde\phi^i(g_0))\right|\ \left(\text{by }(\ref{E:PeriodicNull})\sum_{i=1}^{k}\tilde \Phi(\tilde\phi^i(g_0))=0\right)\\
\le&\sum_{i=1}^{k}\left|\tilde \Phi(\tilde\phi^{n+i}(g))-\tilde\Phi(\tilde\phi^i(g_0))\right|\\ \le&C_1\left(d_{L^\infty(\O, M)}(\tilde \phi^{n+k}(g),\tilde \phi^{n}(g))\right)^\a \ \left(\text{Applying }(\ref{E:VarEstABP})\right), \end{split} \end{align}
where $C_1=\frac{2\left(4L(C+1)\right)^\a}{1-e^{-\a\l}}\int_{\O}\|\Phi\|_{C^{0,\a}(M_\o)}d\mathbb P$ and $L$ is as in (\ref{E:VarEstABP}).
So far, we have shown that $\hat \Psi '$ is $\a$-H\"older continuous and bounded on $\{\tilde \phi^n(g)\}_{n\ge 0}$. As $\{\tilde \phi^n(g)\}_{n\ge 0}$ being dense in $C(\O,M)$, $\hat \Psi '$ defined by (\ref{E:PsiOnOrb}) has a unique $\a$-H\"older extension $\hat\Psi$ on $C(\O,M)$, which completes the proof.
\end{proof}
To extend $\hat\Psi$ to the required $\tilde\Psi$ in Theorem \ref{T:WLivTh}, we need the following lemma which demonstrates a kind of "absolute continuity" of $\hat\Psi$.
\begin{lem}\label{L:PsiAbsCon} There exists $L'>0$ such that for any given $\Phi\in L^1_{\mathbb P}(\O,C^{0,\a}(M))$ satisfying (\ref{E:PeriodicNull}) and the associated $\hat\Psi$ as in Lemma \ref{L:WLivThC} the following holds \begin{equation}\label{E:PsiAbsCon}
\left|\hat \Psi(h_1)-\hat \Psi(h_2)\right|\le L'\overline\Phi\left(3\mathbb P\left(\O\setminus \O_{h_1,h_2,\r}\right)\right)+L'\r^\a\int_{\O}\|\Phi\|_{C^{0,\a}(M_\o)}d\mathbb P,\ \forall h_1,h_2\in C(\O,M), \end{equation}
where $$\O_{h_1,h_2,\r}=\left\{\o\in\O\big|\ d_M(h_1(\o), h_2(\o))< \r\right\}$$ and $$\overline\Phi(s):=\sup_{A\subset \O:\mathbb P(A)\le s}\int_{A} \|\Phi\|_{C^{0,\a}(M_\o)}d\mathbb P,\ s\in (0,1).$$ \end{lem} \begin{proof} Since $\hat \Psi'$ is bounded, it suffices to consider the case when $\r$ and $\mathbb P\left(\O\setminus \O_{h_1,h_2,\r}\right)$ are small.
Let $g\in L^\infty(\O, M)$ be the point that the closure of whose orbit contains $C(\O,M)$, and $\a_0$ be as in the second part of Lemma \ref{L:LShad}, and $\a_1\in (0,\frac16\a_0)$. By Lemma \ref{L:JoinSegments}, there exists an $N_1\in \mathbb N$ such that \begin{equation}\label{E:N-connect}
\forall g_1,g_2\in L^\infty(\O, M),\ \exists g_3\in L^\infty(\O, M) \text{ s.t. }g_3\in B(g_1,\a_1)\text{ and }\tilde\phi^{N_1} (g_3)\in B(g_2,\a_1),
\end{equation}
where $B(h,r)$ is the $r$-ball centered at $h$ in $L^\infty(\O, M)$.
Given $\r\in (0,\a_1)$ and $h_1,h_2\in C(\O,M)$, we denote $\O_1=\O\setminus \O_{h_1,h_2,\r}$. For $\a_2\in(0,\frac12\r)$, let $n,k\in\mathbb N$ be such that $\tilde \phi^n(g)\in B(h_1,\a_2)$ and $\tilde \phi^{n+k}(g)\in B(h_2,\a_2)$. Without loss of generality, we assume that $n,k>>N_1$. By the choice of $\a_1$ and $N_1$, there exists $h_3\in L^\infty(\O, M)$ satisfying the following $$h_3\in B(\tilde \phi^{n+k-N_1}(g),\a_1)\text{ and }\tilde\phi^{N_1}(h_3)\in B(h_1,\a_1).$$ Define a periodic sequence $\{g_i\}_{i\in \mathbb Z}\subset L^\infty(\O, M)$ with period $k$ by the following: \begin{equation}\label{E:PeriodicPartialConnect} g_i=\begin{cases} \tilde \phi^{i+n}(g), &\text{when }0\le i\le k-N_1-1\\ \tilde \phi^{i+N_1}(g)\chi_{\O\setminus \t^{-N_1}\O_1}+h_3\chi_{ \t^{-N_1}\O_1},&\text{when }i=k-N_1\\ \tilde\phi^{i-k+N_1}g_{k-N_1}, &\text{when }k-N_1+1\le i\le k-1\\ g_{i\mod k}&\text{otherwise} \end{cases}, \end{equation} where $\chi_{O}$ is the characteristic function of $O\subset \O$. It is clear that $$\tilde\phi(g_i)\in B(g_{i+1},\r+2\a_1+2\a_2),\ \forall i\in\mathbb Z.$$ Thus, using $\a_1,\a_2, \r\in(0,\frac16\a_0)$ and Lemma \ref{L:LShad}, we obtain that there exists a random periodic point $g'$ satisfying that \begin{equation}\label{E:PeriodicShadowg'} \tilde\phi^i(g')\in B(g_i,C(\r+2\a_1+2\a_2)),\ i\in\mathbb Z, \end{equation} where $C$ is the one as in Lemma \ref{L:LShad}.
Note that for each $\o\in \O\setminus \O'$ where $\O'= \O_1\cup \t^{-k}\O_1\cup \t^{-2k}\O_1$, we have that $$ g_i(\t^i\o)=\begin{cases} \tilde\phi^{n+i-k}(g)(\t^i\o)&k\le i\le 2k-1\\ \tilde\phi^{n+i}(g)(\t^i\o)&0\le i\le k-1\\ \tilde\phi^{n+k+i}(g)(\t^i\o)&-k\le i\le-1\\ \end{cases}. $$ Thus, we have a $(\r+\a_1,\o)$-pseudo orbit $\{(y_i,\t^i\o)\}_{i\in \mathbb Z}$ by letting $$(y_i,\t^i\o)= \begin{cases} \phi^{i-2k+1}\left(g_{2k-1}(\t^{2k-1}\o),\t^{2k-1}\o\right)&i\ge 2k\\ \left(g_i(\t^i\o),\t^i\o\right)&-k\le i\le 2k-1\\ \phi^{i+k}\left(g_{-k}(\t^{-k}\o),\t^{-k}\o\right)&i\le -k-1\\ \end{cases}.$$ Here $\{(y_i,\t^i\o)\}_{i\in\mathbb Z}$ is a $(\r+2\a_2,\o)$-pseudo orbit because $\{\left(g_i(\t^i\o),\t^i\o\right)\}_{-k\le i\le 2k-1}$ is a segment of a $(\r+2\a_2,\o)$-pseudo orbit. By Lemma \ref{L:ClosingLemma} and the choice of $\a_2$ and $\r$, we have that there exists a true orbit $\{\phi^i(y',\o)\}_{i\in\mathbb Z}$ which is $(C(\r+2\a_2),\o)$-shadowing $\{(y_i,\t^i\o)\}_{i\in\mathbb Z}$. Recall that $\{\phi^i(g'(\o),\o)\}_{-k\le i\le 2k}$ is $(C(\r+2\a_1+2\a_2),\o)$-shadowing $\{\left(g_i(\t^i\o),\t^i\o\right)\}_{-k\le i\le 2k-1}$, which means that $$\max_{-k\le i\le 2k-1}d_M\left(\pi_M\phi^i(g'(\o),\o),\pi_M\phi^i(y',\o)\right)\le C(2\a_1+4\a_2+2\r),\ \forall \o\in\O\setminus \O'.$$ Note that $k$ can be chosen arbitrarily large. Hence Lemma \ref{L:ContinousShadowing} implies that one can make $d_M\left(\pi_M\phi^{k-1}(g'(\o),\o),\pi_M\phi^{k-1}(y',\o)\right)$ and $d_M\left(g'(\o),y'\right)$ arbitrarily small by taking $k$ large enough. It is worth to point out that the largeness of $k$ is depending on $C(2\a_1+4\a_2+2\r)$ only while it is nothing to do with the choice on specific $\o$ as long as $\o\in \O\setminus \O'$. Now we choose $k$ large enough so that the following holds $$\max\left\{d_M\left(\pi_M\phi^{k}(g'(\o),\o),\pi_M\phi^{k}(y',\o)\right),d_M\left(g'(\o),y'\right)\right\}<C\r,\ \forall \o\in\O\setminus \O'.$$ Thus \begin{equation} \max\left\{d_M\left(\pi_M\phi^{k}(g'(\o),\o),\tilde \phi^{n+k}(g)(\t^{k}\o)\right),d_M\left(g'(\o),\tilde \phi^{n}(g)(\o)\right)\right\}<4C\r,\ \forall \o\in\O\setminus \O', \end{equation} here we used the fact that $C(2\r+2\a_2)\le 4C\r$.
Therefore, by using the same argument as in (\ref{E:EstOnOrbit}), we obtain that \begin{equation}\label{E:EstOnOrbSeg} d_M\left(\tilde\phi^i(g')(\t^i\o),\tilde \phi^{n+i}(g)(\t^i\o)\right)\le 16e^{-\min\{i,k-i\}\l}C\r,\ \forall \ \o\in \O\setminus \O', 0\le i\le k. \end{equation}
Now we are ready to estimate $\left|\hat\Psi'(h_2)-\hat\Psi'(h_1)\right|$: \begin{align*}
&\left|\hat\Psi(h_2)-\hat\Psi(h_1)\right|\\
\le&\left|\hat\Psi(h_2)-\sum_{i=0}^{n+k}\tilde\Phi(\tilde\phi^i(g))\right|+\left|\hat\Psi(h_1)-\sum_{i=0}^{n}\tilde\Phi(\tilde\phi^i(g))\right|+\left|\sum_{i=n+1}^{n+k}\tilde\Phi(\tilde\phi^i(g))\right|\\
\le &2C'\a_2^\a+\left|\sum_{i=n+1}^{n+k}\tilde\Phi(\tilde\phi^i(g))\right|\ \left(C'\text{ is the H\"older constant of functional }\hat \Psi'\right)\\
=&2C'\a_2^\a+\left|\sum_{i=n+1}^{n+k}\tilde\Phi(\tilde\phi^i(g))-\sum_{i=1}^{k}\tilde\Phi(\tilde\phi^i(g'))\right|\ \left(g' \text{ is a periodic point of } \tilde\phi\right)\\
\le& 2C'\a_2^\a+\sum_{i=1}^k\left|\tilde\Phi(\tilde\phi^{i+n}(g))-\tilde\Phi(\tilde\phi^i(g'))\right|\\
\le&2C'\a_2^\a+\sum_{i=1}^k\int_{\t^i(\O\setminus \O')}\left|\Phi\left(\tilde\phi^{i+n}(g)(\o),\o\right)-\Phi\left(\tilde\phi^i(g')(\o),\o\right)\right|d\mathbb P\\
&+\sum_{i=1}^{k-N_1}\int_{\t^i( \O')}\left|\Phi\left(\tilde\phi^{i+n}(g)(\o),\o\right)-\Phi\left(\tilde\phi^i(g')(\o),\o\right)\right|d\mathbb P\\
&+\sum_{i=k-N_1+1}^{k}\int_{\t^i( \O')}\left|\Phi\left(\tilde\phi^{i+n}(g)(\o),\o\right)-\Phi\left(\tilde\phi^i(g')(\o),\o\right)\right|d\mathbb P\\
\le& 2C'\a_2^\a+\sum_{i=1}^k\int_{\t^i(\O\setminus \O')}\|\Phi\|_{C^{0,\a}(M_\o)}\left(16LC\r\right)^\a e^{-\min\{i,k-i\}\a\l}d\mathbb P\ \left(\text{by }(\ref{E:EstOnOrbSeg})\right)\\
&+\sum_{i=1}^{k-N_1}\int_{\t^i( \O')}\|\Phi\|_{C^{0,\a}(M_\o)}\left(20LC\a_1\right)^\a e^{-\min\{i,k-N_1-i\}\a\l}d\mathbb P\ \left(\text{by }(\ref{E:PeriodicShadowg'})\text{ and }(\ref{E:EstOnOrbit})\right)\\
&+\sum_{i=k-N_1+1}^{k}\int_{\t^i( \O')}2\|\Phi\|_{C^{0,\a}(M_\o)}d\mathbb P\\
\le&2C'\a_2^\a+\frac{2\left(16LC\r\right)^\a}{1-e^{-\a\l}}\int_{\O}\|\Phi\|_{C^{0,\a}(M_\o)}d\mathbb P+\left(\frac{2\left(20LC\a_1\right)^\a}{1-e^{-\a\l}}+2N_1\right)\overline \Phi(\mathbb P(\O')), \end{align*} where the fact that $0<\a_2<\r<\a_1$ is used.
Note that, in the last line of above estimate, $\a_1$ and $N_1$ are fixed while $\a_2$ can be chosen arbitrarily small which consequentially forces $n$ and $k$ tend to $+\infty$. Thus \begin{align*}
\left|\hat\Psi(h_2)-\hat\Psi(h_1)\right|\le& L'\overline\Phi(\mathbb P(\O'))+L'\r^\a\int_{\O}\|\Phi\|_{C^{0,\a}(M_\o)}d\mathbb P\\
\le& L'\overline\Phi(3\mathbb P(\O_1))+L'\r^\a\int_{\O}\|\Phi\|_{C^{0,\a}(M_\o)}d\mathbb P, \end{align*} where $L'=\max\left\{\frac{2\left(16C\right)^\a}{1-e^{-\a\l}},\frac{2\left(20C\a_1\right)^\a}{1-e^{-\a\l}}+2N_1\right\}$. The proof is complete. \end{proof}
Now we are ready to define the functional $\tilde \Psi:L^{\infty}(\O)\to \mathbb R$ based on $\hat \Psi'$ and Lemma \ref{L:PsiAbsCon}. For any $h\in L^\infty(\O, M)$ and $\e>0$, by Lusin's Theorem, there exists $h_\e\in C(\O,M)$ such that $\mathbb P(\{\o\in\O|\ h(\o)\neq h_\e(\o)\})<\e$. Let $\{\e_i\}_{i\ge 1}$ be a sequence of positive numbers, then by taking $\r$ arbitrarily small while applying Lemma \ref{L:PsiAbsCon}, we have that
$$\left|\hat \Psi(h_{\e_n})-\hat \Psi(h_{\e_k})\right|\le L'\overline \Phi\left(3(\e_n+\e_k)\right),\ \forall n,k\ge 1.$$
Since $\|\Phi\|_{C^{0,\a}(M_\cdot)}\in L^1_{\mathbb P}(\O)$, the absolute continuity of $\int \|\Phi\|_{C^{0,\a}(M_\o)}d\mathbb P$ implies that \[\{\hat\Psi'(h_{\e_i})\}_{i\ge 1}\] is a Cauchy sequence as long as $\lim_{i\to\infty}\e_i=0$. Thus $\lim_{i\to\infty}\hat \Psi(h_{\e_i})$ exists when $\lim_{i\to\infty}\e_i=0$, and moreover such limit is independent on the choice of $\{h_{\e_i}\}_{i\ge 1}$. Thereafter, define \begin{equation}\label{E:DefTildePsi} \tilde \Psi(h)=\lim_{\e\to 0^+}\hat \Psi(h_{\e}). \end{equation} It is clear that $\tilde \Psi$ is $\a$-H\"older continuous by (\ref{E:PsiAbsCon}), and the absolute continuity of
\[\int \|\Phi\|_{C^{0,\a}(M_\o)}d\mathbb P\] together with (\ref{E:CoBoundary}) implies that $$\tilde \Phi(h)=\tilde \Psi(\tilde \phi(h))-\tilde \Psi(h),\ \forall h\in L^{\infty}(\O).$$ It remains to show that such $\tilde \Psi$ is uniquely defined up to a constant functional. Suppose that both $\tilde \Psi_1$ and $\tilde \Psi_2$ satisfy (\ref{E:CoBoundary}). Then for any $h\in L^\infty(\O, M)$, we have that $\tilde\Psi_1-\tilde\Psi_2$ is constantly valued on $\{\tilde\phi^n(h)\}_{n\in\mathbb Z}$. Since the closure of $\{\tilde\phi^n(h)\}_{n\in\mathbb Z}$ contains $C(\O,M)$ for some $h$, and by the H\"older continuity of $\tilde \Psi$, we have that $\tilde\Psi_1-\tilde\Psi_2$ is constantly valued on $C(\O,M)$, which means that $\hat \Psi_1-\hat\Psi_2$ is a constant functional on $C(\O,M)$. Thus, $\tilde \Psi_1$ and $\tilde \Psi_2$ defined by (\ref{E:DefTildePsi}) varies at a constant functional. This completes the proof of Theorem \ref{T:WLivTh}.
\qed
\section{Examples}\label{Examples} In this section, we give concrete examples which can be adapted into the theoretic framework in previous sections. The systems we consider mainly contains two types: a) Transitive fiber Anosov system driven by a quasi-periodic driven force; b) Fiber volume preserving Anosov system with a continuous driven force, of which the precise formulations will be given later. At the end of this section, we will make some further discussions on the random horseshoe with respect to a given marginal $\mathbb P$, and give some relevant results.
\subsection{Systems driven by a quasi-periodic force}\label{S:Quasi} In this section, we consider a type of Anosov systems which is driven by a quasi-periodic force. This section contains two main parts: in the first part, we state and prove the general result in a general setting; in the second part, we will investigate a concrete example which is generated by rotation on torus and affine Anosov maps on 2-d torus, and will show that such a system satisfies the Conditions 1A.-1C. below and illustrate some interesting phenomenas (see Remark \ref{R:NonContiPerOrb}).
\subsubsection{Setting and results}\label{S:S1)SetResult} We assume that the systems satisfy the following conditions: \begin{itemize} \item[1A.] $(\t,\O)$ is a minimal irrational rotation on a compact torus; \item[1B.] $\phi$ is Anosov on fibers (Condition H1)); \item[1C.] $\phi$ is topological transitive on $M\times \O$.
\end{itemize}
For the sake of convenience, we call the systems satisfying Conditions 1A.-1C. the S1) systems.
The main result we will prove is the following Theorem:
\begin{thm}\label{T:Spectrum11}
All S1) systems satisfy Condition H2) that is topological mixing on fibers.
\end{thm}
We first prove a weaker version of Theorem \ref{T:TheoryAnosovMix} for S1) systems, which is the following proposition:
\begin{prop}\label{P:DensePeriodicOrb1}
Let $\phi$ be an S1) system. Then for any $g\in C(\O,M)$ and $\e>0$, there exists a random periodic point $\tilde g\in L^{\infty}(\O)$ such that
\begin{equation}\label{E:LInfiClose}
d_{L^{\infty}(\O)}(g,\tilde g)\le \e.
\end{equation}
\end{prop} \begin{proof} Actually, this proposition still holds when $\t$ is assumed to be equicontinuous only. We say $\t$ is {\bf equicontinuous} if there exist $c>1$ and $\d>0$ such that for any $\o_1,\o_2\in \O$ with $d_\O(\o_1,\o_2)<\d$,
$$\frac1cd_\O(\o_1,\o_2)\le d_\O(\t^n\o_1,\t^n\o_2)\le cd_\O(\o_1,\o_2), \forall n\in \mathbb Z.$$
When $\t$ is equicontinuous, if we set
$$\widetilde{d}_\O(\o_1,\o_2)=\sup_{n\in \mathbb{Z}}d_{\O}(\t^n \o_1,\t^n\o_2),\, \forall \o_1,\o_2\in \O,$$
then $\widetilde{d}_\O$ is a metric on $\O$ satisfying that
\begin{itemize}
\item $\widetilde{d}_\O$ and $d_\O$ are equivalent;
\item $\widetilde{d}_\O(\t^n\o_1,\t^n\o_2)=\widetilde{d}_\O(\o_1,\o_2)$ for any $\o_1,\o_2\in \O$
and $n\in \mathbb{Z}$.
\end{itemize}
Therefore, without loss of generality, for an equicontinuous $\t$, we always assume that
\begin{equation}\label{E:EquDist}
d_\O(\t^n\o_1,\t^n\o_2)=d_\O(\o_1,\o_2),\ \forall \o_1,\o_2\in\O\text{ and }n\in \mathbb Z.
\end{equation}
For a given $g\in C(\O,M)$, we define a neighborhood of its graph as follows. We consider a finite Borel partition $\xi=\{\xi_i\}_{1\le i\le p}$ of $\O$, of which each element contains an nonempty open subset. Such a partition exists because of compactness of $\O$. For any $\o\in\O$, we define $\xi(\o)$ the element in $\xi$ which contains $\o$, and also denote that
$$B_g(\xi_i,\d):=\{(x,\o)|\ \o\in \xi_i, d_M(x,g(\o))<\d\},$$ which induces a $\d$-neighborhood of $graph(g)$: $\bigcup_{\xi_i\in\xi} B_g(\xi_i,\d)$. We denote it by $B_g(\d)$.
\begin{lem}\label{L:MReturn} For any $\d>0$, there exists $m\in\mathbb N$ such that for any $\xi_i\in\xi$, there exists $(x_i,\o_i)\in B_g(\xi_i,\d)$ such that $$\phi^m(x_i,\o_i)\in B_g(\xi_i,\d).$$ \end{lem} \begin{proof} Since $\phi$ is transitive, there exists $(x_0,\o_0)$ such that the orbit of $\{\phi^n(x_0,\o_0)\}_{n\in\mathbb Z}$ is dense in $M\times \O$.
By the finiteness of the partition, we have that there exist a point $(x,\o)$ in the interior of $B_g(\xi_1,\d)$ and an $m_0\in\mathbb N$ such that $$\phi^{n_i}(x,\o)\in \text{interior of }B_g(\xi_i,\d)\text{ for some }n_i\in[1,m_0],\ \forall i\in[2,p].$$ By the continuity of $\phi$ and the finiteness of the partition, we have that there exists a small open neighbourhood of $(x,\o)$ in $M\times \O$, which is denoted by $U$, such that $$U\subset B_g(\xi_1,\d)\text{ and }\phi^{n_i}(U)\subset \text{interior of }B_g(\xi_i,\d),\ \forall i\in[2,p].$$
Then, by the transitivity of the system, there exist an $m\in \mathbb N$ and $(x',\o')\in U$ such that $\phi^m(x',\o')\in U$. Note that $$\phi^m(\phi^{n_i}(x',\o'))=\phi^{n_i}(\phi^m(x',\o'))\in \phi^{n_i}(U) \subset \text{interior of }B_g(\xi_i,\d),\ \forall i\in[2,p].$$ We complete the proof by letting $(x_1,\o_1)=(x',\o')$ and $(x_i,\o_i)=\phi^{n_i}(x',\o')$ for $i\in[2,p]$.
\end{proof}
Next, we will construct a $(\u,\d)$-{\bf pseudo orbit} for $\u\in \O$, $g\in C(\O,M)$, and $\d>0$. Given $\d>0$, by Lemma \ref{L:MReturn}, there exist $m=m(\d,g,\phi) >0$ and $(x_i,\o_i)\in B_g(\xi_i,\d)$ for $1\le i\le p$ such that \begin{equation}\label{E:MReturn} \phi^m(x_i, \o_i)\in B_g(\xi_i,\d)\text{ for }1\le i\le p. \end{equation} For $l\in\mathbb Z$, suppose that $\t^{lm}\u\in \xi_{i_l}$, then we define \begin{equation}\label{E:DefPseOrb} (y_j,\u_j)=\begin{cases} (g(\t^{lm}\u),\t^{lm}\u), &\text{ when }j=lm\\ (\pi_M\phi^{j-lm}(x_{i_l},\o_{i_l}),\t^j\u),&\text{ when }j\in[lm+1,(l+1)m-1]\\ \end{cases}. \end{equation} \begin{lem}\label{L:PseudoOrbit} For any $\e'>0$, there exists $\d'(\phi,g,\e')>0$ such that if $\d$ and the diameter of the partition $\xi$ are less than $\d'$, the pseudo orbit defined by (\ref{E:DefPseOrb}) starting from $(y,\upsilon)\in graph(g)$ is a $(\u,\e')$-pseudo orbit. \end{lem} \begin{proof} Noting that $g$ is uniformly continuous since $\O$ being compact. Therefore, for any $\e_1>0$, there exists $\d_1>0$ such that for any $\o_1,\o_2\in \O$ if $d_\O(\o_1,\o_2)<\d_1$ then \begin{equation}\label{E:POEst1}
d_M(g(\o_1),g(\o_2))<\e_1. \end{equation}
Also note that both $\phi$ and $\phi^{-1}$ are uniformly continuous, then for any $\e_2>0$ there exists a $\d_2>0$ such that for any $(x_1,\o_1),(x_2,\o_2)\in M\times \O$ once $d((x_1,\o_1),(x_2,\o_2))<\d_2$ the following holds
\begin{equation}\label{E:POEst2}
\max\{d(\phi(x_1,\o_1),\phi(x_2,\o_2)),d(\phi^{-1}(x_1,\o_1),\phi^{-1}(x_2,\o_2))\}<\e_2.
\end{equation}
Note that $d_\O(\t^j\u,\t^{j-lm}\o_{i_0})$ is less than the diameter of the partition $\xi$ (which is less than $\d'(\phi,g,\e')$) for all $j\in[lm,(l+1)m]$ and $l\in \mathbb Z$ by (\ref{E:EquDist}). For a given $(y,\u)\in graph(g)$ with $\u\in \xi_{i_0}$, the following holds: \begin{enumerate}
\item[a)] When $j\in [lm+2,(l+1)m-1]$, taking $\d'<\d_2$, then (\ref{E:POEst2}) implies that
\begin{align*}
d_M(y_j,\pi_M\phi(y_{j-1},\u_{j-1}))
=&d_M(y_j,\pi_M\phi(y_{j-1},\t^{j-1}\u))\\
\le & d(\phi(y_{j-1},\t^{j-lm-1}\o_{i_l}),\phi(y_{j-1},\t^{j-lm-1}\u_{lm}))\\
\le& \e_2,
\end{align*}
where the fact that $\u_{lm}\in\xi_{i_l}$ is used.
\item[b)] When $j=lm+1$, taking $\d'<\frac12 \d_2$ (thus $\d<\frac12\d_2$), then
$$d\left((x_{i_l},\o_{i_l}),(g(\t^{lm}\u),\t^{lm}\u)\right)<\d_2.$$ Hence, (\ref{E:POEst2}) implies that
$$d_M\left(y_j,\pi_M\phi(y_{j-1},\u_{j-1})\right)=d_M\left(\pi_M\phi(x_{i_l},\o_{i_l}),\pi_M\phi(g(\t^{lm}\u),\t^{lm}\u)\right)<\e_2;$$
\item[c)] For $j=(l+1)m$, taking $\d'<\min\{\d_1,\d_2\}$, then (\ref{E:POEst1}), (\ref{E:MReturn}), and (\ref{E:POEst2}) implies that
\begin{align*}
& d_M(y_j,\pi_M\phi(y_{j-1},\u_{j-1}))\\
=&d_M\left(g(\t^j\u),\pi_M\phi(y_{j-1},\u_{j-1})\right)\\
\le & d_M\left(g(\t^m\u_{lm}),g(\t^m\o_{i_l})\right)+d_M\left(g(\t^m\o_{i_l}),\pi_M\phi^m(x_{i_l},\o_{i_l})\right)\\
&+d_M(\pi_M\phi^m(x_{i_l},\o_{i_l}),\pi_M\phi(y_{j-1},\t^{m-1}\u_{lm}))\\
\le& \e_1+\d+d_M(\pi_M\phi(y_{j-1},\t^{m-1}\o_{i_l}),\pi_M\phi(y_{j-1},\t^{m-1}\u_{lm}))\\
\le&\e_1+\d+\e_2.
\end{align*} \end{enumerate} Therefore, the required estimate can be achieved if we choose $,\e_1+\d +\e_2<\e'$ and take a $\d'>0$ which satisfies all the proposed conditions in a), b), and c) above. So for such $\d'>0$ if $\d$ and $diam(\xi)$ are less than $\d'$, the pseudo-orbit defined by (\ref{E:DefPseOrb}) is a $(\u,\e')$-pseudo orbit.
\end{proof}
\begin{lem}\label{L:LipG} For any $\e'>0$ and $L>0$, there exists $\d'(\phi,L,\e')>0$ such that for any $g\in C(\O,M)$ with $Lip\ g\le L$ if $\d$ and the diameter of the partition $\xi$ are less than $\d'$, then the pseudo orbit defined by (\ref{E:DefPseOrb}) starting from $(y,\upsilon)\in graph(g)$ is a $(\u,\e')$-pseudo orbit. \end{lem} \begin{proof} In the proof of Lemma \ref{L:PseudoOrbit} above, the dependence of $\d'$ on $g$ is given by (\ref{E:POEst1}) which appears in part c). So the exact property of ``$g$'' being involved in this lemma is how uniformly continuous it is. Therefore, under the setting of the current lemma, the exactly same argument can be applied here, in which ``$\d'$'' depends on the Lipchitz constant ``$L$'' rather than the choice of particular map ``$g$''. The detailed proof is omitted. \end{proof} Now we are ready to prove Proposition \ref{P:DensePeriodicOrb1}. For a given small $\e>0$ and $g\in C(\O,M)$, choose $0<\b<\min\{\frac12\e,\beta_0\}$. By Lemma \ref{L:LShad}, there exists a corresponding $\a>0$ for $\b$. Then, by applying Lemma \ref{L:MReturn} and \ref{L:PseudoOrbit}, for $\e'=\a$ there exists a $\d'>0$ such that for any $\u\in\O$ and $\d\in(0,\d')$, there exists $m\in \mathbb N$ which is depending on $\d,g,\phi$ only and a $(\u,\a)$-pseudo orbit defined by (\ref{E:DefPseOrb}), which is denoted by $\{(y_i,\t^n\u)\}_{n\in\mathbb Z}$, such that $$(y_{km},\t^{km}\u)\in graph(g),\ \forall k\in\mathbb Z.$$ Define $g_i:\O\to M$ for $i\in \mathbb Z$ as follows $$g_j(\u)=\begin{cases} g(\u), &\text{ when }j=lm\\ \pi_M\phi^{j-lm}(x_{i_l},\o_{i_l}),&\text{ when }j\in[lm+1,(l+1)m-1]\text{ and }\t^{-(j-lm)}\u\in \xi_{i_l}\\ \end{cases}.$$ It is clear that each $g_i$ is measurable since $g_i$ is either a continuous function or a pre-image of a simple function under continuous iterations. Also note that $$(y_i,\t^n\u)\in graph(g_i),\ \forall\u\in\O\text{ and }i\in \mathbb Z.$$ Thus $\{g_i\}_{i\in\mathbb Z}$ is an $\a$-pseudo orbit of $\tilde\phi$. By applying Lemma \ref{L:LShad}, there is a true orbit $\{\tilde\phi(\tilde g)\}_{i\in\mathbb Z}$ which is $\b$-shadowing $\{g_i\}_{i\in\mathbb Z}$. The uniqueness of such a $\tilde g$ and the periodicity of $\{g_i\}_{i\in\mathbb Z}$ imply that $\tilde g$ is a periodic point of $\tilde \phi$.
This completes the proof of Proposition \ref{P:DensePeriodicOrb1}. \end{proof}
\noindent{\bf Proof of Theorem \ref{T:Spectrum11}.} Actually, what was required for the driven system is the following: $\O$ is connected, and for any $n\in\mathbb N$, $\t^n$ is uniquely ergodic and $\mathbb P$ is the ergodic probability measure.
By Proposition \ref{P:DensePeriodicOrb1}, we have that for any $g\in C(\O,M)$ and $\e>0$, there exists a random periodic point $\tilde g\in L^\infty(\O, M)$ satisfying (\ref{E:LInfiClose}). For this $\tilde g$, let \begin{equation}\label{E:XTg}
X_{\tilde g}= \bigcup_{\o\in\O}\{(x,\o)| x\in \overline{W^u(\tilde g(\o),\o)}\}, \end{equation} where $\overline{W^u(\tilde g(\o),\o)}$ is the closure of the global unstable manifold of $(\tilde g(\o),\o)$ in $(M,\o)$. We first require the following condition on the parameters:
{\bf C1:} By taking $\e$ small enough, we require that $\e<\frac12 \d$ where $\d$ is as in Lemma \ref{L:LocCoor} corresponding to $\frac12\e_0$ with $\e_0$ being a prefixed small number.
First, we show that \begin{equation}\label{E:OpenClose} X_{\tilde g}=M\times \O. \end{equation}
The next lemma is the key result needed in the proof:
\begin{lem}\label{L:ContiG}
For any $(x,\o)\in M\times\O$, $g\in C(\O,M)$, and an open neighborhood $V\subset \O$ of $\o$, there exist an nonempty open set $A\subset\O\setminus\{\o\}$ with $\O\setminus A\subset V$ and a function $g'\in C(\O,M)$ such that
\begin{equation}\label{E:g'}
g'(\o)=x\text{ and }g'|_A=g|_A.
\end{equation}
\end{lem}
The proof of this lemma may be standard, however we give it here for the sake of completeness.
\begin{proof}
First, there exists a path $\g:[0,1]\to M$ such that $\g(0)=g(\o)$ and $\g(1)=x$. Note that there exist finitely many numbers $0=s_0<s_1<s_2<\cdots<s_n=1$ such that $\g(s_i)$ and $\g(s_{i+1})$ are in a same chart of the atlas of $M$, say $(U_i,\psi_i)$, for all $i=0,\cdots, n-1$.
Consider an open ball $B(\o,R_1)$ centered at $\o$ in $\O$, with $R_1$ small enough so that $B(\o,R_1)\subset V$, $\mathbb P(\overline{B(\o,R_1)})<1$ and $g(B(\o,R_1))\subset U_1$. Note that $\{\o\}$ and $\O\setminus B(\o,\frac12 R_1)$ are disjoint closed sets. By Urysohn's lemma, there exists a continuous function $f_1:\O\to[0,1]$ such that
$$f_1|_{\O\setminus B(\o,\frac12R_1)}=0\text{ and }f_1(\o)=1.$$
Then, the following is well defined
$$g_1(\u)=\begin{cases}g_0(\u),&\text{ if }\u\notin B(\o,\frac12R_1)\\ \psi_1^{-1}\left[ f_1(\u)\psi_1(\g(s_1))+(1-f_1(\u))\psi_1(g_0(\u))\right],&\text{ if }\u\in B(\o,\frac 12R_1)\end{cases},$$ where we set $g_0=g$ for the sake of convenience.
It is obvious that $g_1$ is continuous and satisfies that
$$g_1(\o)=\g(s_1)\text{ and }A_1:=\{\o|g_1(\o)=g(\o)\}\text{ contains an open set}.$$
Inductively, we can construct $g_{i+1}$ based on $g_i$ by using exactly the same argument above to ensure the following:
$$g_{i+1}(\o)=\g(s_{i+1})\text{ and }A_{i+1}:=\{\o|g_1(\o)=g(\o)\}\text{ contains an open set}.$$
To guarantee the existence of $A$, it is sufficient to take $R_{i+1}<R_i$. In this case, we can take $A=\O\setminus \overline{B(\o,\frac12R_1)}$.
\end{proof}
\begin{rem}
Actually, it is not hard to see that $\mathbb P(A)$ can be taken arbitrarily close to $1$ since $\mathbb P$ is not atomic.
\end{rem}
Now, we are ready to prove Theorem \ref{T:Spectrum11}. For an arbitrary $(x,\o)\in M\times\O$, and $g$ chosen at the beginning of the proof, by Lemma \ref{L:ContiG}, there exists $g'\in C(\O,M)$ satisfying (\ref{E:g'}). For a small $\e'\in (0,\frac12\d)$, where $\d$ is as in C1). We apply Proposition \ref{P:DensePeriodicOrb1} to $g'$, and then derive that there exists a random periodic point $\tilde g'\in L^\infty(\O, M)$ such that \begin{equation}\label{E:ClosePerOrb} \sup_{\u\in A}\{d_M(\tilde g'(\u),\tilde g(\u))\}\le 2\e', \end{equation} where $\tilde g$ is the periodic point in (\ref{E:XTg}). Let $m$ and $m'$ be the periods of $\{graph(\tilde g)\}$ and $\{graph(\tilde g')\}$ respectively. Note that $\t^{-mm'}$ is $\mathbb P$-ergodic since $\t$ is invertible and $\t^{mm'}$ is $\mathbb P$-ergodic, and each $\o'\in\O$ is a $\t^{mm'}$-minimal point. That means that there is an infinite increasing sequence of positive integers, $0<n_1<n_2<n_3<\cdots$, such that $\t^{-n_imm'}\o\in A$ for $i\ge 1$ as long as $A$ has nonempty interior.
For $i\ge 1$, let \begin{align*} x'_i&=[\pi_M\phi^{-n_imm'}(\tilde g'(\o),\o),\tilde g(\t^{-n_imm'}\o)]_{\t^{-n_imm'}\o}\\ &=W^s_{\frac{\e_0}2}(\phi^{-n_imm'}(\tilde g'(\o),\o))\bigcap W^u_{\frac{\e_0}2}(\tilde g(\t^{-n_imm'}\o),\t^{-n_imm'}\o), \end{align*} which is well defined by the choice of $\e_0,\e'$ and (\ref{E:ClosePerOrb}).
Note that, by Lemma \ref{L:InvMani}, we have that $$\lim_{i\to\infty}\phi^{n_imm'}(x'_i,\t^{-n_imm'}\o)\to (\tilde g'(\o),\o)\text{ and }\phi^{n_imm'}(x'_i,\t^{-n_imm'}\o)\in W^u(\tilde g(\o),\o),$$ which implies that $$(\tilde g'(\o),\o)\in \overline{W^u(\tilde g(\o),\o)}\subset X_{\tilde g}.$$ Since $d_M(g'(\o),x)<\e'$ and $\e'$ can be taken arbitrarily small, we have that $(x,\o)\in X_{\tilde g}$. Thus (\ref{E:OpenClose}) holds by the arbitrariness of $(x,\o)$. \\
Finally, we show that $\phi$ is {\bf topological mixing on fibers}. Roughly speaking, the proof is based on the fact that the global stable manifold of a random periodic point is dense in $M\times\O$. The following lemma is the key to derive the mixing property. \begin{lem}\label{L:Converging} For any $\e_1>0$ and $g\in C(\O,M)$, there exists $\d_1>0$ such that the following holds:\\ For any random periodic point $graph(\tilde g)\in B_g(\d_1)$ (with period $l$), $\e'_1>0$ and $h\in C(\O,M)$,
there exist $N\in\mathbb N$, and a measurable function $h_1:\O\to M$ satisfying that $$graph(h_1)\subset B_{h}(\e'_1)\text{ and }\phi^{nl}(graph(h_1))\subset B_g(\e_1),\forall n>N.$$ \end{lem} \begin{proof} Here we only need to take $\d_1\in(0,\frac16\e_1)$ small enough such that any $2\d_1$-pseudo orbit of $\tilde\phi$ can be $\frac13\e_1$-shadowed by a unique true orbit. This can be done by Lemma \ref{L:LShad}. For the rest of the proof of this lemma, such a $\d_1$ is fixed.
It is not hard to see that the existence of $\tilde g$ follows from Proposition \ref{P:DensePeriodicOrb1} directly. For an arbitrarily fixed $\o\in\O$, by Lemma \ref{L:ContiG}, there exist an nonempty open set $A_\o\subset \O$ and $g'_\o\in C(\O,M)$ such that
\begin{equation}\label{E:g'gAomega}
g'_\o(\o)=h(\o)\text{ and }g'_\o|_{A_\o}=g|_{A_\o}.
\end{equation}
By applying Proposition \ref{P:DensePeriodicOrb1}, we have that there is a random periodic point $\tilde g'_\o\in L^\infty(\O, M)$ with period $l_\o$ such that
\begin{equation}\label{E:TilG'omega}
graph(\tilde g'_\o)\in B_{g'_\o}\left(\frac13\e'_1\right),
\end{equation}
where, without loss of generality, we assume that $\e'_1\in(0,\d_1)$.
Let $B_\o\subset A_\o$ be an open ball in $\O$ with radius $r_\o>0$, and $C_\o$ be an open neighborhood of $\o$ with radius $\frac12r_\o$. This $r_\o$ is taken additionally small to ensure the following inequality
\begin{equation}\label{E:GomegaHmu}
|g'_\o(\u)-h(\u)|<\frac13\e'_1,\ \forall \u\in C_\o.
\end{equation}
Since $\t^{l_\o l}$ is an isometric map on $\O$ and any nonempty open subset of $\O$ is not $\mathbb P$-null set, there is an infinite set $I_\o\subset\mathbb N$ such that the following holds
$$\t^{nl_\o l}(C_\o)\subset B_\o,\ \forall n\in I_\o.$$
For $\u\in C_\o$ and $n\in I_\o$, define that
$$(y_i,\u_i)_{n}=\begin{cases}
\phi^i(\tilde g'_\o(\u),\u)&\text{ when }i< nl_\o l\\
\phi^{i-nl_\o l}(\tilde g(\t^{nl_\o l}\u),\t^{nl_\o l}\u)&\text{ when }i\ge nl_\o l
\end{cases}.$$
It is clearly $\{(y_i,\u_i)_{n}\}_{i\in\mathbb Z}$ is an $(\u,\frac32\d_1)$-pseudo orbit because of (\ref{E:TilG'omega}) and (\ref{E:g'gAomega}). By the choice of $\e_1$ and $\d_1$ at the beginning of the proof, we have that there exists a unique true orbit $\{\phi^i(z_{\o,n}(\u),\u)\}_{i\in\mathbb Z}$ which is $(\u,\frac13\e_1)$-shadowing $\{(y_i,\u_i)_n\}_{i\in\mathbb Z}$. Note that $\d_1+\frac32\d_1+\frac13\e_1<\frac34\e_1$, then we have that
\begin{equation}\label{E:1}
\phi^{ml}(z_{\o,n}(\u),\u)\in B_g\left(\frac34\e_1\right),\ \forall m\ge nl_\o.
\end{equation}
By Lemma \ref{L:ContinousShadowing}, there exists $N_\o\in \mathbb N$ such that if we choose $nl_\o> N_\o$, then
$$d_M\left(z_{\o,n}(\u),\tilde g'_\o(\u)\right)<\frac13\e'_1,$$
which together with (\ref{E:TilG'omega}) and (\ref{E:GomegaHmu}) implies that
\begin{equation}\label{E:2}
d_M(z_{\o,n}(\u),h(\u))<\e'_1.
\end{equation}
So far, we have defined a measurable function $z_{\o,n}:C_\o\to M$ with an integer $N_\o\in \mathbb N$ ($n$ is required to be larger than $N_\o$) satisfying (\ref{E:1}) and (\ref{E:2}) for all $\u\in C_\o$. We remark here that the measurability of $z_\o$ can be proved exactly in the same way as the argument in the proof of Proposition \ref{P:DensePeriodicOrb1}, thus is omitted here.
Note that $\{C_\o\}_{\o\in\O}$ forms an open cover of the compact set $\O$. Thus there is an finite open sub-cover which is denoted by $\{C_{\o_1},\cdots C_{\o_s}\}$. The proof is completed if we take $N=\max\{N_{\o_1},\cdots, N_{\o_s}\}$ and define
$$h_1(\o)=\begin{cases}
z_{\o_1}(\o)&\text{ when }\o\in C_{\o_1}\\
z_{\o_i}(\o)&\text{ when }\o\in C_{\o_i}\setminus \bigcup_{1\le j\le i-1}C_{\o_j},\ \forall 2\le i\le s
\end{cases}.
$$ \end{proof}
Let $U$ and $V$ be nonempty open sets in $M$ with an $x\in U$ and a $y\in V$ fixed. Then there exists $\e_2>0$ such that $$B_M(x,\e_2)\subset U\text{ and }B_M(y,\e_2)\subset V.$$ Applying Lemma \ref{L:Converging} by taking $\e_1=\e_2$ and $g=h_y$ where $h_y(\o)=y$ for all $\o\in\O$, we can derive a $\d_2>0$ correspondingly. By Proposition \ref{P:DensePeriodicOrb1}, there exists a random periodic point $graph(\tilde h_y)\in B_{h_y}(\d_2)$ with period $p$. Since $\phi$ is homeomorphism, for any $i\in\mathbb N$, $\phi^{i}(U\times \O)$ is open in $M\times\O$ while $\phi^{i}(graph(h_x))\subset \phi^{i}(U\times\O)$ is compact. Therefore, there exists $\e_3\in(0,\e_2)$ such that $$B_{g_i}(\e_3)\subset \phi^{i}(U\times \O), \ \forall 0\le i \le p-1,$$ where $g_i(:=\tilde\phi(h_x))\in C(\O, M)$. Applying Lemma \ref{L:Converging} $p$ times, we have that there exist $\{N_0,\cdots, N_{p-1}\} \subset \mathbb N$ and measurable functions $\{g'_i:\O\to M\}_{0\le i\le p-1}$ such that for all $0\le i\le p-1$ the following hold \begin{align}\begin{split}\label{E:3} &graph(g'_i)\subset B_{g_i}(\e_3)\subset \phi^{i}(U\times\O)\\ &\quad\quad\quad\quad\quad\quad\quad\text{and}\\ &\phi^{np}(graph(g'_i))\subset B_{h_y}(\e_2)\subset V\times \O,\forall n>N_i. \end{split} \end{align} Obviously, (\ref{E:3}) holds if we replace $N_i$ by $N:=\max\{N_0,\cdots, N_{p-1}\}$. Therefore, for any $m\in\mathbb N$, if $m=np+j$ where $n>N$ and $j\in[0,p)$, then $$\phi^{np}(graph(g'_i))\subset B_g(\e_2)\bigcap \phi^{m-j}(\phi^{j}(U\times \O))\subset V\times \O \bigcap \phi^{m}(U\times \O),$$ which implies that for any $\o\in\O$, $$\phi^{m}(U\times \{\o\})\cap V\times \{\t^{m}\o\}\neq \emptyset.$$
This completes the proof of Theorem \ref{T:Spectrum11}.
\qed
\subsubsection{Fiber Anosov maps on 2-d tori}\label{S:Example1} In this section, we give an example of Fiber Anosov maps on 2-d tori.
Let $\phi:\mathbb{T}^2\times \mathbb{T}\rightarrow \mathbb{T}^2\times \mathbb{T}$ given by $$\phi\left(\binom{x}{y},\omega\right)=\left( A\binom{x}{y}+h(\omega),\omega+\alpha\right),$$ where $\alpha\in \mathbb{R}\setminus \mathbb{Q}$, $A=\left( \begin{array}{cc} 1 & 1\\ 2& 1\end{array}\right)$ and $h(\omega)=\binom{h_1(\omega)}{h_2(\omega)}$ is a continuous map from $\mathbb{T}$ to $\mathbb{T}^2$. It is clear that $\phi$ is a quasi-periodic forced system and $\phi$ induces a cocycle over $(\mathbb{T},\theta)$, where $\theta(\omega)=\omega+\alpha$.
\begin{lem} $\phi$ is Anosov on fibers and transitive. \end{lem} \begin{proof} Let $f_\omega \binom{x}{y}= A\binom{x}{y}+h(\omega)$ for $\omega\in \mathbb{T}$. Then $\phi\left(\binom{x}{y},\omega\right)=\left(f_\omega \binom{x}{y},\omega+\alpha\right)$. Note that $Df_\omega \binom{x}{y}=A$ for any $(\binom{x}{y},\omega)\in \mathbb{T}^2\times \mathbb{T}$ and $A$ is a hyperbolic matrix. Hence $(\mathbb{T}^2\times \mathbb{T},\phi)$ is Anosov on fibers.
Let $\mu$ be the Lebesgue measure on $\mathbb{T}^2\times \mathbb{T}$. To show that $T$ is transitive, it is sufficient to show that $(\mathbb{T}^2\times \mathbb{T},\mathcal{B}_{\mathbb{T}^2\times \mathbb{T}}, \phi,\mu)$ is an ergodic measure-preserving system. First, it is clear that $(\mathbb{T}^2\times \mathbb{T},\mathcal{B}_{\mathbb{T}^2\times \mathbb{T}}, \phi,\mu)$ is a measure-preserving system. In the following we show that $(\mathbb{T}^2\times \mathbb{T},\mathcal{B}_{\mathbb{T}^2\times \mathbb{T}}, \phi,\mu)$ is ergodic.
Let $f\in L^2(\mu)$ with $f\circ \phi=f$ $\mu$-a.e.. Let $$f\left(\binom{x}{y},\omega\right)=\sum_{(k,l,n)\in \mathbb{Z}^3} c_{k,l,n} e^{2\pi i \left(k\omega+(l,n)\binom{x}{y}\right)}$$ be the Fourier series of $f$ on $\mathbb{T}^2\times \mathbb{T}$. Then
$$\sum_{(k,l,n)\in \mathbb{Z}^3} |c_{k,l,n}|^2=\|f\|_{L^2(\mu)}<+\infty.$$ Fix $(n',l')\in \mathbb{Z}^2$. For any $g\in L^2(m_{\mathbb{T}})$, where $m_{\mathbb{T}}$ is the Lebesgue measure on $\mathbb{T}$, we have \begin{align*} &\hskip0.5cm \int \left( \sum_{k\in \mathbb{Z}} c_{k,l',n'} e^{2\pi i k \omega}\right) g(\omega) d m_{\mathbb{T}}(\omega)\\ &=\int \left(\sum_{(k,l,n)\in \mathbb{Z}^3} c_{k,l,n} e^{2\pi i \left(k\omega+(l,n)\binom{x}{y}\right)}\right) g(\omega)e^{-2\pi i(l',n')\binom{x}{y}} d \mu\left(\binom{x}{y},\omega\right) \\ &=\int f\left(\binom{x}{y},\omega\right) g(\omega)e^{-2\pi i(l',n')\binom{x}{y}} d \mu\left(\binom{x}{y},\omega\right)\\ &=\int f\left(\phi^{-1}\left(\binom{x}{y},\omega\right)\right) g(\omega)e^{-2\pi i(l',n')\binom{x}{y}} d \mu\left(\binom{x}{y},\omega\right) \\ &=\int f\left(\binom{x}{y},\omega\right) g(\omega+\alpha)e^{-2\pi i(l',n')\left(A\binom{x}{y}+h(\omega)\right)} d \mu\left(\binom{x}{y},\omega\right)\\ &= \int \left(\sum_{(k,l,n)\in \mathbb{Z}^3} c_{k,l,n} e^{2\pi i \left(k\omega+(l,n)\binom{x}{y}\right)}\right) g(\omega+\alpha)e^{-2\pi i(l',n')\left(A\binom{x}{y}+h(\omega)\right)} d \mu\left(\binom{x}{y},\omega\right)\\ &=\int\left( \sum_{k\in \mathbb{Z}} c_{k,(l',n')A} e^{2\pi i k \omega}\right)e^{-2\pi i (l',n')h(\omega)} g(\omega+\alpha) d m_{\mathbb{T}}(\omega)\\ &=\int \left( \sum_{k\in \mathbb{Z}} c_{k,(l',n')A} e^{2\pi i k (\omega-\alpha)}\right)e^{-2\pi i (l',n')h(\omega-\alpha)} g(\omega) d m_{\mathbb{T}}(\omega). \end{align*} Since the above equality holds for any $g\in L^2(m_{\mathbb{T}})$, we have \begin{equation}\label{eq-11}
\sum_{k\in \mathbb{Z}} c_{k,l',n'} e^{2\pi i k \omega}= \left( \sum_{k\in \mathbb{Z}} c_{k,(l',n')A} e^{2\pi i k (\omega-\alpha)}\right)e^{-2\pi i (l',n')h(\omega-\alpha)} \end{equation} in $L^2(m_{\mathbb{T}})$. Thus
$$\int \left| \sum_{k\in \mathbb{Z}} c_{k,l',n'} e^{2\pi i k \omega}\right|^2 d m_{\mathbb{T}}(\omega)=
\int \left|\left( \sum_{k\in \mathbb{Z}} c_{k,(l',n')A} e^{2\pi i k (\omega-\alpha)}\right)e^{-2\pi i (l',n')h(\omega-\alpha)}\right|^2 d m_{\mathbb{T}}(\omega). $$ Hence \begin{equation*}
\sum_{k\in \mathbb{Z}} |c_{k,l',n'}|^2=\sum_{k\in \mathbb{Z}} |c_{k,(l',n')A}|^2. \end{equation*} When $(l',n')\neq (0,0)$, $(l',n')A^m\neq (l',n')A^k$ for $m,k\in \mathbb{N}$ with $m\neq k$. Hence if $(l',n')\neq (0,0)$ then for any $N\in \mathbb{N}$, \begin{align*}
\sum_{k\in \mathbb{Z}} |c_{k,l',n'}|^2=\frac{1}{N}\sum_{i=0}^{N-1}\sum_{k\in \mathbb{Z}} |c_{k,(l',n')A^i}|^2 \le \frac{1}{N} \sum_{(k,l,n)\in \mathbb{Z}^3} |c_{k,l,n}|^2\le \frac{1}{N} \|f\|_{L^2(\mu)}. \end{align*}
Let $N\nearrow +\infty$, we get $\sum_{k\in \mathbb{Z}} |c_{k,l',n'}|^2=0$ for $(l',n')\neq (0,0)$. When $(l',n')= (0,0)$, by \eqref{eq-11} we have $$\sum_{k\in \mathbb{Z}} c_{k,0,0} e^{2\pi i k \omega}=\sum_{k\in \mathbb{Z}} c_{k,0,0}e^{-2\pi i k\alpha} e^{2\pi i k \omega}$$ in $L^2(m_{\mathbb{T}})$. Thus $c_{k,0,0}=c_{k,0,0}e^{-2\pi i k\alpha}$ for $k\in \mathbb{Z}$. When $k\neq 0$, we have $c_{k,0,0}=0$ as $\alpha \in \mathbb{R}\setminus \mathbb{Q}$. Summing over we get $f=c_{0,0,0}$. This shows that $(\mathbb{T}^2\times \mathbb{T},\mathcal{B}_{\mathbb{T}^2\times \mathbb{T}}, \phi,\mu)$ is ergodic. The proof of this lemma is complete.
\end{proof}
For a continuous function $p$ from $\mathbb{T}$ to $\mathbb{T}$, we denote the degree of $p$ by $deg(p)$.
\begin{lem} \label{ex-lem-2} Let $deg(h_i)=n_i$ for $i=1,2$. If $n_2$ is odd number, then there are no $g\in C(\mathbb{T},\mathbb{T}^2)$ and positive integer $n$ such that $\phi^n (graph(g))=graph(g)$, that is, $(\mathbb{T}^2\times \mathbb{T},\phi)$ has no random periodic point whose graph is continuous. \end{lem} \begin{proof} If this is not true, then there are $g(\omega)=\binom{g_1(\omega)}{g_2(\omega)}\in C(\mathbb{T},\mathbb{T}^2)$ and positive integer $n$ such that $\phi^n (graph(g))=graph(g)$, i.e. $$ \phi^n(g(\omega),\omega)=(g(\omega+n\alpha),\omega+n\alpha)$$ for $\omega\in \mathbb{T}$. Let $deg(g_i)=k_i$ for $i=1,2$. Note that $$\phi^n(g(\omega),\omega)=\left(A^n\binom{g_1(\omega)}{g_2(\omega)}+A^{n-1}\binom{h_1(\omega)}{h_2(\omega)}+\cdots+\binom{h_1(\omega+(n-1)\alpha)}{h_2(\omega+(n-1)\alpha)},\omega+n\alpha\right)$$ Hence \begin{align*} &\binom{g_1(\omega+n\alpha)}{g_2(\omega+n\alpha)}\\=&A^n\binom{g_1(\omega)}{g_2(\omega)} +A^{n-1}\binom{h_1(\omega)}{h_2(\omega)}+A^{n-2}\binom{h_1(\omega+\alpha)}{h_2(\omega+\alpha)}+\cdots+\binom{h_1(\omega+(n-1)\alpha)}{h_2(\omega+(n-1)\alpha)}. \end{align*} Compare the degree of continuous functions from $\mathbb{T}$ to $\mathbb{T}$ appearing in the above equation, we have $$\binom{k_1}{k_2}=A^n\binom{k_1}{k_2} +A^{n-1}\binom{n_1}{n_2}+A^{n-2}\binom{n_1}{n_2}+\cdots+\binom{n_1}{n_2}.$$ That is $$(I-A^n)\binom{k_1}{k_2}=(I+A+\cdots+A^{n-1})\binom{n_1}{n_2}.$$ Since $\det(I-A^n)\neq 0$ and $(I-A)(I+A+\cdots+A^{n-1})=I-A^n$, we have $$(I-A)\binom{k_1}{k_2}=\binom{n_1}{n_2}.$$ This implies $k_1=-\frac{n_2}{2}\not \in \mathbb{Z}$, a contradiction. The proof is complete. \end{proof} \begin{rem}\label{R:NonContiPerOrb} Note that Theorem \ref{T:TheoryAnosovMix} implies that the random periodic orbits are dense in $L^{\infty}(\O, M)$. Questions such as "Is there any continuous random periodic orbit?" or "When do there exist continuous periodic orbits?" raise naturally. Lemma \ref{ex-lem-2} somehow answers the first question in a negative way; while the Theorem 1.1 from \cite{Liu} give the positive answer for the first question when the system is generated by i.i.d. small random perturbations of an Axiom A system. In another aspect, Lemma \ref{ex-lem-2} tells that the systems we considered here beyonds the framework of the study on structure stability. \end{rem}
\subsection{Systems preserving a volume on fibers driven by a force}\label{S:VolumePreserving} In this section, we consider a type of Anosov systems which are volume-preserving on fiber. We first state and prove the mixing property in a relatively general setting. Then, we investigate a concrete example which is generated by the random composition of matrices on 2-d torus, whose entries are positive integers, and show such a system satisfies the Conditions 2A.-2C. below.
\subsubsection{Setting and results}\label{S:S1)SetResult}
We assume that the systems satisfy the following conditions: \begin{itemize} \item[2A.] $(\t,\O)$ is a heomomorphism on a compact metric space; \item[2B.] $\phi$ is Anosov on fibers (Condition H1)); \item[2C.] There exists an $f$-invariant Borel probability measure $\nu$ with full support (i.e. $supp\nu=M$). \end{itemize} Here by a Borel probability measure $\nu$ on $M$ being {\bf $f$-invariant} we mean that for any Borel measurable set $A\subset M$ and any $\o\in\O$, $\nu(f_\o^{-1}(A))=\nu (A)$.
For the sake of convenience, we call the systems satisfying Conditions 2A.-2C. the S2) systems.
The main result we will prove is the following Theorem:
\begin{thm}\label{T:Spectrum12}
S2) systems satisfity Condition H2) that is topological mixing on fibers.
\end{thm} \begin{proof} Unlike the case of S1) systems, the proof for S2) is based on the measure preserving property of $f_\o$, which leads to a much more direct approach.
For a given $\o\in \O$, an open subset $V\subset M$, and a real number $r>0$, we inductively define a sequence of subsets $V^r_n(\o)\subset M$, $n=0,1,2,\cdots$ as follows:
\begin{equation}\label{E:V_n}
V^r_n(\o)=\begin{cases}
V,&n=0\\
\cup_{x\in f_{\t^{n-1}\o}(V^r_{n-1}(\o))}B_M(x,r),&n>1
\end{cases},
\end{equation}
where $B_M(x,r)$ is the ball in $M$ with radius $r$ centered at $x$. It is not hard to see that each $V^r_n(\o)$ is open and $\{\nu(V^r_n(\o))\}_{n=0,1,\cdots}$ is non-decreasing on both $n$ and $r$ since $\nu$ is $f$-invariant and $f$ is invertible. Thus $\lim_{n\to\infty}\nu(V^r_n(\o))$ exists.
Next, we show that
\begin{equation}\label{E:LimVn}
\forall\ r>0, \exists\ N(\o)>0 \text{ such that }\forall\ n\ge N(\o), V^r_n(\o)=M.
\end{equation}
Since $supp\ \nu=M$, we know that any open subset of $M$ is not a $\nu$-null set. Additionally, because $M$ is compact, for any $R>0$, $M$ can be covered by finitely many balls with radius $\frac13R$, each of which has a positive $\nu$-measure. Thus, \begin{equation}\label{E:NuInf}
\forall\ R>0, \exists\ \e>0, \text{ such that }\forall\ x\in M,\ \nu(B_M(x,R))>\e.
\end{equation}
For a given $n\ge 1$, let $t^r_n(\o)=\min\{\sup\{d_H(x,f_{\t^{n-1}\o}(V^r_{n-1}(\o))|\ x\in M\},r\}$, where for $x\in M$ and $A\subset M$ $$d_H(x,A):=\inf\{d_M(x,y)|\ y\in A\}$$ is the Hausdoff distance. Since $d_H(x,V^r_n(\o))$ is continuous on $x$ and $M$ is connected, we have that there exists $x'\in M$ such that $$\frac12t^r_n(\o)< d_H(x',f_{\t^{n-1}\o}(V^r_{n-1}(\o)))<r.$$
Therefore, $B\left(x',\frac12t^r_n(\o)\right)\subset V^r_n(\o)\setminus f_{\t^{n-1}\o}(V^r_{n-1}(\o))$ which implies that
$$\nu \left(B_M\left(x',\frac12t^r_n(\o)\right)\right)\le \nu(V^r_n(\o))-\nu(f_{\t^{n-1}\o}(V^r_{n-1}(\o)))=\nu(V^r_n(\o))-\nu(V^r_{n-1}(\o)(\o)).$$
Because of the existence of $\lim_{n\to\infty} \nu (V^r_n(\o))$ and (\ref{E:NuInf}), we have that
$$\lim_{n\to\infty} t_n^r=0.$$
By the definition of $V_n^r(\o)$ as in (\ref{E:V_n}), once $t_n^r<r$, $V_n^r(\o)=M$, thus (\ref{E:LimVn}) holds. \\
Now, we are ready to show that S2)-type systems are topological mixing on fibers.
Let $U_1, U_2\subset M$ be two nonempty open sets. Choose two open balls $B_M(x_i,3R)\subset U_i$, $i=1,2$, with $R>0$. Applying Lemma \ref{L:Shadowing} by setting $\b=R$, we have an $\a>0$, such that any $(\o,\a)$-pseudo orbit can be $(\o,\b)$-shadowed by a true orbit.
For a given $\o\in\O$, take $V=B_M(x_1,R)$ and $r=\a$, and define $V^r_n(\o)$ as in (\ref{E:V_n}). By (\ref{E:LimVn}), there exists an $N_1(\o)$ such that for any $n\ge N_1(\o)$, $V_n^r(\o)=M$. By continuity of $\phi$, we have that for any $\o\in\O$, there exists $\d(\o)>0$ such that the following holds:
$$\forall\ \o'\in B_\O(\o,\d(\o))\text{ and }n\ge N_1(\o)+1, \ V^r_n(\o') =M,$$
where $B_\O(\o,\d(\o))$ is the ball in $\O$ with radius $\d(\o)$ centered at $\o$.
Since $\O$ is compact, and $\{B_\O(\o,\d(\o))\}_{\o\in\O}$ forms an open cover on $M$, there exists a finite sub-cover, $\{B_\O(\o_j,\d(\o_j)\}_{1\le j\le q},\ q\in\mathbb N$. Take $N=\max\{N_1(\o_1),\cdots, N_1(\o_q)\}+1$, then the following holds
\begin{equation}\label{E:UniformN}
\forall\ \o\in\O, \text{ if } n\ge N, \text{ then } V^r_n(\o)=M.
\end{equation}
Next, we show that
\begin{equation}\label{E:TMix} \forall\ \o\in\O\text{ and }n\ge N,\ \phi(n,U_1\times\{\o\})\cap U_2\times\{\t^n\o\}\neq \emptyset.
\end{equation}
This can be done by constructing an $(\o,r)$-pseudo orbit, $\{(y'_{n,i},\t^i\o)\}_{i\in\mathbb Z}$, connecting two balls $B_M(x_1,R)\times \{\o\}$ and $B_M(x_2,R)\times \{\t^n\o\}$, which is $(\o,R)$-shadowed by a true orbit \[\{(y_{n,i},\t^i\o)\}_{i\in\mathbb Z}.\] Thus this true orbit, $\{(y_{n,i},\t^i\o)\}_{i\in\mathbb Z}$, will hit both $B_M(x_1,3R)\times \{\o\}$ and $B_M(x_2,3R)\times \{\t^n\o\}$. Hence (\ref{E:TMix}) holds. Actually, the existence of such an $(\o,r)$-pseudo orbits ($\forall\ \o\in \O$), $\{(y'_{n,i},\t^i\o)\}_{i\in\mathbb Z}$, follows from (\ref{E:UniformN}), which can be constructed in the following:
For any $n\ge N$ and $\o\in\O$, since $V_N^r(\o)=M$, we have that there exists $z_{N-1}(\o)\in V_{N-1}^r(\o)$ such that
$$\pi_M\phi^{-(n-N)}(x_2,\t^n\o)\in B_M(f_{\t^{N-1}\o}z_{N-1}(\o),r).$$
Inductively, we have that for any $i\in\{0,\cdots, N-2\}$, there exists $z_i(\o)\in V_i^r(\o)$ such that
$$z_{i+1}(\o)\in B_M(f_{\t^i\o}z_i(\o),r).$$
Also note that $V_0^r(\o)=B_M(x_1,R)$, then we have that $\{z_i(\o)\}_{1 \le i\le N-1}$ satisfies the following properties:
\begin{align}\begin{split}\label{E:ConnectX1X2}
&d_M(z_0(\o),x_1)<r;\\
&d_M(z_{i+1}(\o), f_{\t^i\o}z_i(\o))<r,\ 0\le i\le N-2;\\
&d_M\left(\pi_M\phi^{-(n-N)}(x_2,\t^n\o),f_{\t^{N-1}\o}z_{N-1}(\o)\right)<r.
\end{split}
\end{align} We define $$ y'_{n,i}=\begin{cases} \pi_M\phi^i(x_1,\o),&i\le -1\\ z_i(\o),&0\le i\le N-1\\ \pi_M\phi^{i-n}(x_2,\t^n\o), &i\ge N \end{cases}. $$ By (\ref{E:ConnectX1X2}), it is not hard to see that the above $y'_{n,i}$'s fit all the requirements. The proof is complete.
\end{proof}
\subsubsection{Random composition of $2\times 2$ area-preserving positive matrices } \label{S:Example2} The main purpose of this section is to provide an example of S2)-type systems, which is generated by the random composition of a class of area-preserving positive $2\times 2$ matrices. Then, we apply the results for S2)-type systems to investigate the dynamical behavior of the given example.
Let $$\left\{A_i=\left(\begin{matrix}a_i&b_i\\c_i&d_i \end{matrix}\right)\right\}_{1\le i\le p}$$ be $2\times 2$ matrices with positive integer entries and
$|\det A_i|=1,\ \forall i\in\{1,\cdots,p\}$, and $\mathcal S_p:=\{1,\cdots, p\}^{\mathbb Z}$ with the left shift operator $\s$ be the symbolic dynamical system with $p$ symbols.
Define a map $f:\mathcal S_k\to \{A_1\cdots,A_p\}$ by letting $$f(\hat a)=A_{\hat a(0)},\ \forall \hat a\in \mathcal S_k.$$
Define that, for $x\in [0,+\infty)\times [0,+\infty)$ and $\hat a\in \mathcal S_k$, $$\phi^n_{\hat a}=\begin{cases} I_2&\text{ when }n=0\\ f(\s^{n-1}\hat a)\circ\cdots\circ f(\hat a)&\text{ when }n>0\\ f^{-1}(\s^{n}\hat a)\circ\cdots\circ f^{-1}(\s^{-1}\hat a)&\text{ when }n<0\\ \end{cases}, $$ where $I_2=\left(\begin{matrix}1&0\\0&1\end{matrix}\right)$. Note that $\phi$ induces a random dynamical system on 2 dimensional torus $\mathbb T^2$, which is denoted by $\hat \phi$. Let \begin{equation}\label{E:AssumpExpan} \kappa:=\min_{1\le i\le p}\min\left\{\sqrt{a_i^2+c_i^2},\sqrt{b_i^2+d_i^2}\right\}, \end{equation} which is obviously greater than or equal to $\sqrt 2$.
\begin{prop}\label{P:2DTorusAnosov} $\hat \phi:\mathbb T^2\times \mathcal S_k\to \mathbb T^2\times \mathcal S_k$ is an S2)-type system by letting $M=\mathbb T^2$ and $(\O,\t)=(\mathcal S_p,\s)$. \end{prop} \begin{proof}
Denote $\mathbb R^{2+}:=\left\{(x,y)\in \mathbb R^2\setminus \{(0,0)\}\Big|\ x\ge 0, y\ge 0\right\}$. For any $v_i=\left(\begin{matrix}x_i\\y_i\end{matrix}\right)\in \mathbb R^{2+}$, $i=1,2$, by a straightforward computation, we have that
$$\sin\angle(v_1,v_2)=\left| \frac{y_2x_1-y_1x_2}{\sqrt{x_1^2+y_1^2}\sqrt{x_2^2+y_2^2}}\right|,$$
where $ \angle(v_1,v_2)$ is the angle between $v_1$ and $v_2$. \\
By a straightforward computation, we have that, for any $j\in\{1,\cdots,p\}$, $i=1,2$,
\begin{align}\begin{split}\label{E:Expan&Contrac}
& |A_jv_i|=\sqrt{(a_j^2+c_j^2)x_i^2+2(a_jb_j+c_jd_j)x_iy_i+(b_j^2+d_j^2)y_i^2}\ge \kappa |v_i|\\
&\sin\angle(A_jv_1,A_jv_2)=\left|\frac{(a_jd_j-b_jc_j)(y_2x_1-y_1x_2)}{|A_jv_1||A_jv_2|}\right|\le \kappa^{-2}\sin\angle(v_1,v_2). \end{split}
\end{align}
Let $Proj^+:=\left\{\spa\{v\}|\ v\in \mathbb R^{2+}\right\}$ which is a compact subset of the projective space $\mathbb P^2$. It is not hard to see that the second inequality of (\ref{E:Expan&Contrac}) yields that for any $\hat a\in \mathcal S_k$ and $n\in\mathbb N$ the map $\phi(\cdot,\hat a)$ induces a contraction map on $Proj^+$ with a uniform contracting rate. By the contracting mapping theorem, we have that there is a unique $1$-dimensional space $E_{\hat a}$ such that $$E_{\hat a}\in \bigcap_{n\ge 0}\bar\phi^n(Proj^+,\s^{-n}\hat a),$$ where we denote $\bar \phi(\cdot,\hat a)$ the induced map on $\mathbb P^2$ by $\phi(\cdot,\hat a)$. By the uniqueness, we have that $$\phi(E_{\hat a},\hat a)=E_{\s\hat a}.$$
The first inequality of (\ref{E:Expan&Contrac}) yields that $\phi(|_{E_{\hat a}},\hat a)$ is expanding with rate at least $\kappa$. The continuity of $E_{\hat a}$ on $\hat a$ simply follows from the second inequality of (\ref{E:Expan&Contrac}).\\
By applying exactly the same argument, we can obtain that for any $\hat a\in \mathcal S_k$ there exists a unique $1$-dimensional space $F_{\hat a}\notin Proj^+$ such that $F_{\hat a}$ varies continuously on $\hat a$ and the following holds
$$\phi(F_{\hat a},\hat a)=F_{\s\hat a}\text{ and }\phi^{-1}(|_{F_{\hat a}},\hat a) \text {is expanding with rate at least }\kappa.$$ Note that $\phi$ itself is linear on $\mathbb T^2$, therefore the linearization of $\hat \phi^n$, $D_{\mathbb T^2}\hat \phi^n$, can be identified by $\phi$. Hence, $\hat \phi$ is an S2)-type system with continuous co-invariant splitting $\mathbb R^2=E_{\hat a}\oplus F_{\hat a}$ for $\hat a\in\mathcal S_k$. \end{proof} The following corollaries are the consequences of Proposition \ref{P:2DTorusAnosov} and Theorem \ref{T:TheoryAnosovMix}. \begin{cor}\label{C:FrequentReturn} For any $\e>0$, there exists $N\in\mathbb N$ such that for any $x\in \mathbb T^2$, $n\ge N$, and $\hat a\in \mathcal S_k$, there exists $x'\in \mathbb T^2$ satisfying $$x',\ \pi_{\mathbb T^2}\hat \phi^n(x',\hat a)\in B_{\mathbb T^2}(x,\e).$$ \end{cor}
\begin{cor}\label{C:T^2Horseshoe} For any $\g>0$ there exist $N,k$ such that the following hold:
\begin{itemize} \item[i)] $\frac1N\log k>\log \kappa-\g$; \item[ii)] $\hat \phi^N$ has a strong full random horseshoe with $k$ symbols. \end{itemize} Here $\kappa$ is given by (\ref{E:AssumpExpan}). \end{cor}
\subsection{Further discussions on Random Horseshoes with marginal $\mathbb P$}\label{S:RandomHorseshoe} In this section, we further discuss the random horseshoes for systems with a given marginal $\mathbb P$ which is an $\t$-invariant probability measure on $\O$. The standard fiber topological entropy of $\phi$ with respect to $\mathbb P$ is defined by the following \begin{equation}\label{E:FibTopEntrP}
h_{top}(M\times\O|\ \mathbb P)=\int_{\O}h_{top}(\phi|_{M_\o})d\mathbb P(\o). \end{equation}
Since $h_{top}(\phi|_{M_\o})=h_{top}(\phi|_{M_{\t\o}})$ for $\o\in\O$ by the definition, we have that when $\mathbb P$ is ergodic the following holds
$$h_{top}(M\times\O|\mathbb P)\equiv h_{top}(\phi|_{M_{\o}})\text{ for } \mathbb P-a.e.\ \o.$$ For the random dynamical system $\phi$ over a metric dynamical system $(\O,\mathbb P,\t)$ mentioned in Remark \ref{R:RandomDynSys}, it is natural to modify the definitions of random horseshoes correspondingly.
Let $L^\infty_{\mathbb P}(\O)$ be the space of Borel measurable maps from $\O$ to $M$ endowed with the metric $d_{L^\infty_\mathbb P(\O)}$: $$d_{L^\infty_\mathbb P(\O)}(g_1,g_2)=\sup_{\mathbb P}\left\{d_M(g_1(\o),g_2(\o)),\o\in\O\right\},\ \forall g_1,g_2\in L^\infty(\O, M),$$ where $\sup_{\mathbb P}$ is the essential supreme with respect to $\mathbb P$. Similarly, $\phi:M\times\O$ induces a homeomorphism on $L^\infty_{\mathbb P}(\O)$, which is denoted by $\tilde\phi_\mathbb P$. Analogically, we define two functions on $L^\infty_\mathbb P(\O)\times L^\infty_\mathbb P(\O)$ to measure separations of elements of $L^\infty_\mathbb P(\O)$ on different levels: \begin{align}\begin{split}\label{E:PSeparationFunc} \overline d_{L^\infty_\mathbb P(\O)}(g_1,g_2)&=\sup_{\O'\subset \O,\ \mathbb P(\O')>0} \inf_{\o\in \O'}d_M(g_1(\o),g_2(\o)),\ \forall g_1,g_2\in L^\infty_\mathbb P(\O),\\ \underline d_{L^\infty_\mathbb P(\O)}(g_1,g_2)&=\inf_{\mathbb P}\left\{d_M(g_1(\o),g_2(\o)),\ \o\in\O\right\},\ \forall g_1,g_2\in L^\infty_\mathbb P(\O), \end{split} \end{align} where $\inf_\mathbb P$ is the essential infimum with respect to $\mathbb P$. First, we claim that $$\overline d_{L^\infty_\mathbb P(\O)}=d_{L^\infty_\mathbb P(\O)}.$$ For the sake of completeness, we give the proof here. Recall that $$d_{L^\infty_\mathbb P(\O)}(g_1,g_2)=\inf_{\O'\subset \O,\ \mathbb P(\O')=1} \sup_{\o\in \O'}d_M(g_1(\o),g_2(\o)),\ \forall g_1,g_2\in L^\infty_\mathbb P(\O).$$ For a given $\e>0$, let
$$\O'_\e(g_1,g_2)=\left\{\o\in\O|\ d_M(g_1(\o),g_2(\o))>d_{L^\infty_\mathbb P(\O)}(g_1,g_2)-\e\right\}.$$ By the definition of $d_{L^\infty_\mathbb P(\O)}(g_1,g_2)$, we have that $\mathbb P(\O'_\e(g_1,g_2))>0$. Thus $$\overline d_{L^\infty_\mathbb P(\O)}(g_1,g_2)\ge \inf_{\o\in \O'_\e(g_1,g_2)}d_M(g_1(\o),g_2(\o))>d_{L^\infty_\mathbb P(\O)}(g_1,g_2)-\e.$$
Let $$\O''_\e(g_1,g_2)=\left\{\o\in\O|\ d_M(g_1(\o),g_2(\o))>d_{L^\infty_\mathbb P(\O)}(g_1,g_2)+\e\right\}.$$ By the definition of $d_{L^\infty_\mathbb P(\O)}(g_1,g_2)$, we have that $\mathbb P(\O''_\e(g_1,g_2))=0$. Thus $$\mathbb P\left(\O'\cap (\O\setminus \O''_\e(g_1,g_2))\right)>0,\ \forall\ \O'\subset \O,\ \mathbb P(\O')>0,$$ which implies that $$\overline d_{L^\infty_\mathbb P(\O)}(g_1,g_2)\le d_{L^\infty_\mathbb P(\O)}(g_1,g_2)+\e.$$ By the arbitrariness of $\e$, we complete the proof of the claim. In the rest of this section, we will only use $d_{L^\infty_\mathbb P(\O)}(g_1,g_2)$ instead of $\overline d_{L^\infty_\mathbb P(\O)}(g_1,g_2)$.
Consequently, we can define the random horseshoes and strong random horseshoes with marginal $\mathbb P$.
\begin{defn}\label{D:PRandomHorseshoe}
We call a continuous embedding $\Psi:\mathcal S_k\to L^\infty_\mathbb P(\O)$ a {\bf \em random horseshoe on marginal $\mathbb P$} with $k$-symbols of $\phi$ if the following hold: \begin{itemize}
\item[i)] There exists $\Delta>0$ such that for any $l\in\mathbb Z$ and $\hat a_1,\hat a_2\in \mathcal S_k\text{ with }\hat a_1(l)\neq \hat a_2(l)$
\begin{equation}\label{E:RandomSepaImgSymb}
d^{\tilde \phi_\mathbb P}_{l}\left(\Psi({\hat a_1}),\Psi({\hat a_2})\right)>\Delta,
\end{equation}
where for $l\ge 0$
$$ d^{\tilde \phi_\mathbb P}_{l}(g_1,g_2):=\max_{0\le i\le l}\left\{ d_{L^\infty_\mathbb P(\O)}\left(\tilde \phi_\mathbb P^i(g_1),\tilde \phi^i_\mathbb P(g_2)\right)\right\},\ g_1,g_2\in L^\infty_\mathbb P(\O)$$ and for $l<0$ we let $ d^{\tilde\phi_\mathbb P}_{l}= d^{\tilde\phi^{-1}_\mathbb P}_{|l|};$ \item[ii)] For all $\hat a\in\mathcal S_k$ $$\tilde \phi_\mathbb P\left(\Psi({\hat a})\right)=\Psi({\s \hat a}).$$ \end{itemize} \end{defn}
By using an stronger separation function $\underline d_{L^\infty_\mathbb P(\O)}$ as in (\ref{E:PSeparationFunc}), we define a stronger random horseshoe as follows:
\begin{defn}\label{D:PSRandomHorseshoe}
We call a continuous embedding $\Psi:\mathcal S_k\to L^\infty_\mathbb P(\O)$ a {\bf \em strong random horseshoe on marginal $\mathbb P$} with $k$-symbols of $\phi$ if the following hold: \begin{itemize}
\item[i)] There exists $\Delta>0$ such that for any $l\in\mathbb Z$ and $\hat a_1,\hat a_2\in \mathcal S_k\text{ with }\hat a_1(l)\neq \hat a_2(l)$
\begin{equation}\label{E:SRandomSepaImgSymb}
\underline d^{\tilde \phi_\mathbb P}_{l}\left(\Psi({\hat a_1}),\Psi({\hat a_2})\right)>\Delta,
\end{equation}
where for $l\ge 0$
$$\underline d^{\tilde \phi_\mathbb P}_{l}(g_1,g_2):=\max_{0\le i\le l}\left\{\underline d_{L^\infty_\mathbb P(\O)}\left(\tilde \phi^i_\mathbb P(g_1),\tilde \phi^i_\mathbb P(g_2)\right)\right\},\ g_1,g_2\in L^\infty(\O, M)$$ and for $l<0$ we let $\underline d^{\tilde\phi_\mathbb P}_{l}=\underline d^{\tilde\phi^{-1}_\mathbb P}_{|l|};$ \item[ii)] For all $\hat a\in\mathcal S_k$ $$\tilde \phi_\mathbb P\left(\Psi({\hat a})\right)=\Psi({\s \hat a}).$$ \end{itemize} \end{defn} For the random horseshoe on marginal $\mathbb P$, we have the following result. \begin{thm}\label{T:PHorseshoe}
Let $\phi$ satisfy H1) and H2) and $\mathbb P$ be a $\t$-ergodic probability measure. Then for any $\g>0$, there exist $N,k\in \mathbb N$ such that the following hold
\begin{enumerate}
\item[i)] $\frac1N\ln k\ge h_{top}(M\times\O|\mathbb P)-\g$;
\item[ii)] $\phi^N$ has a $k$-symbol random horseshoe on marginal $\mathbb P$.
\end{enumerate}
\end{thm} \begin{proof}
The proof of Theorem \ref{T:PHorseshoe} can be derived by employing the exact same paradigm used in the proof of Part B. of Theorem \ref{T:TheoryAnosovMix2}. To adapt the proof of Part B. of Theorem \ref{T:TheoryAnosovMix2} into the current circumstance, one needs only to replace $\overline h(M\times\O|\O)$ by $h_{top}(M\times\O|\mathbb P)$, and make the "$O$" in Lemma \ref{L:SeparatedSetFiber} satisfy $\mathbb P(O)>0$ rather than "nonempty open subset", which is guaranteed by the definition of $h_{top}(M\times\O|\mathbb P)$. We omit the detailed proof here. \end{proof} For the strong random horseshoe on marginal $\mathbb P$, only partial result is obtained which is applicable for S1) systems. Note that for S1) systems, $\t:\O\to\O$ is uniquely ergodic, thus $\mathbb P$ can only be the unique Haar measure of $\t$. \begin{thm}\label{T:SPHorseshoe}
Let $\phi$ be an S1) system and $\mathbb P$ be the unique $\t$-ergodic probability measure. Then for any $\g>0$, there exist $N,k\in \mathbb N$ such that the following hold
\begin{enumerate}
\item[i)] $\frac1N\ln k\ge h_{top}(M\times\O|\mathbb P)-\g$;
\item[ii)] $\phi^N$ has a $k$-symbol strong random horseshoe on marginal $\mathbb P$.
\end{enumerate}
\end{thm}
\begin{proof}
This theorem follows from Part A. of Theorem \ref{T:TheoryAnosovMix2} and the following lemma:
\begin{lem}\label{L:S1EntropyEquality}
Let $\phi$ be an S1) system. Then the following holds
$$\underline h(M\times\O|\O)=\overline h(M\times\O|\O)=h_{top}(\phi),$$
where $h_{top}(\phi)$ is the topological entropy of $\phi$.
\end{lem}
\begin{proof}[Proof of Lemma \ref{L:S1EntropyEquality}]
It is straightforward to observe that $\overline h(M\times\O|\O)\le h_{top}(\phi)$ by the definition. For the sake of simplicity, we will prove this lemma in the case of $h_{top}(\phi)<\infty$ only, as for the other case the same argument is still applicable. By the Theorem 17 in \cite{Bow3}, we have that
$$h_{top}(\phi)\le \overline h(M\times\O|\O)+h_{top}(\t).$$ Since $\t$ is equicontinuous, we have $h_{top}(\t)=0$, thus
$$h_{top}(\phi)= \overline h(M\times\O|\O).$$
Therefore, it remains to show that $\underline h(M\times \O|\O)=\overline h(M\times \O|\O)$. It is sufficient to prove the following \begin{equation}\label{E:FibEntrEqu}
h_{top}(\phi|_{M_{\o_1}})=h_{top}(\phi|_{M_{\o_2}}),\ \forall \o_1,\o_1\in\O. \end{equation} Recall that
$$h_{top}(\phi|_{M_\o})=\lim_{\e\to 0^+}\limsup_{n\to\infty}\frac1n\log N(\o,\e,n),$$ where
$$N(\o,\e,n):=\max\ \left\{card(E)\big|\ E\subset M_{\o} \text{ is an }\e\text{ separated set with respect to }d^{\phi}_{M\times \O,n}\right\},$$ and $d^\phi_{M\times \O,n}$ is the Bowen metric. For a given small positive number $\g>0$ and $\o_1\in\O$, there exists $\e_0>0$ such that for any $\e\in(0,\e_0)$ and $N_0\in\mathbb N$, there is an $n\ge N_0$ so that the following holds \begin{equation}\label{E:NumSepaSet}
N(\o_1,3\e,n)\ge e^{n( h_{top}(\phi|_{M_{\o_1}})-\frac12\g)}. \end{equation} Without loss of generality, we assume that $\e<\frac12\b_0$ and fix such an $\e$, where $\b_0$ is as in Lemma \ref{L:Shadowing}. Letting $\b=\frac12\e$ and applying Lemma \ref{L:Shadowing}, there exists $\a>0$ corresponding to $\b$ as in Lemma \ref{L:Shadowing}. Again, for the sake of convenience, we assume that $\a<\b$. By the uniform continuity of $\phi$, there exists $\d>0$ such that for any $\o',\o''\in\O$ with $d_\O(\o',\o'')<\d$ and any $x\in M$ the following holds \begin{equation}\label{E:VarEst} d_M(\pi_M\phi(x,\o'),\pi_M\phi(x,\o''))<\a. \end{equation} We note that this $\d$ depends only on $\a$.
Let $E_{\o_1}\subset M_{\o_1}$ be an $3\e$-separated set with respect to $d^{\phi}_{M\times \O,n}$ satisfying $card(E_{\o_1})=N(\o_1,3\e,n)$. For $(x_j,\o_1)\in E_{\o_1}(=\{(x_j,\o_1)\}_{1\le j\le N(\o_1,3\e,n)})$ and $\o\in B_\O(\o_1,\d)$, define a pseudo orbit $\{(y_i'(x_j,\o),\o)\}_{i\in\mathbb Z}$ by letting \begin{equation}\label{E:PsOrbSepa} y_i'(x_j,\o)=\begin{cases} \pi_M(\phi^i(x_j,\o_1)), &\text{ when }i\ge 0\\ \pi_M(\phi^i(x_j,\o)), &\text{ when }i< 0 \end{cases}. \end{equation} By the choice of $\d$ and (\ref{E:VarEst}), it is not hard to see that $\{(y_i'(x_j,\o),\o)\}_{i\in\mathbb Z}$ is an $(\o,\a)$-pseudo orbit. Thus, by Lemma \ref{L:Shadowing}, there exists a true orbit, denoted by $\{\phi^i(y_j,\o)\}_{i\in\mathbb Z}$, which is $(\o,\b)$-shadowing $\{(y_i'(x_j,\o),\o)\}_{i\in\mathbb Z}$. Since $E_{\o_1}$ is a $3\e$-separated set with respect to $d^{\phi}_{M\times \O,n}$ and $\b=\frac12\e$, we have that $\{(y_j,\o)\}_{1\le j\le N(\o_1,3\e,n)}\subset M_\o$ is a $2\e$-separated set with respect to $d^{\phi}_{M\times \O,n}$.
Also note that, $(\O,\t)$ is uniquely ergodic, thus there is an $N_1\in\mathbb N$ depending on $B_\O(\o_1,\d)$ only, which satisfies that for any $\o\in\O$ there exists a positive integer $m(\o)\le N_1$ such that $\t^{m(\o)}(\o)\in B_\O(\o_1,\d)$. This implies that for any $\o\in\O$, the following holds
$$N(\o,2\e,n+N_1)\ge card (\phi^{-m(\o)}E_{\o_1})=N(\o_1,3\e,n)\ge e^{n( h_{top}\left(\phi|_{M_{\o_1}})-\frac12\g\right)}.$$ Therefore, we have that \begin{align*}
h_{top}(\phi|_{M_\o})&=\lim_{\e'\to 0^+}\limsup_{m\to\infty}\frac1m\log N(\o,\e',m)\\ &\ge \frac1{n+N_1}\log N(\o,2\e,n+N_1)\\
&= \frac n{n+N_1}\left( h_{top}(\phi|_{M_{\o_1}})-\frac12\g\right). \end{align*} Since such $n$ can be arbitrarily large and $\g$ can be arbitrarily small, we have that
$$ h_{top}(\phi|_{M_\o})\ge h_{top}(\phi|_{M_{\o_1}}).$$ Also note that $\o_1$ is arbitrarily chosen, thus the proof of (\ref{E:FibEntrEqu}) is complete.
\end{proof}
To end the proof, it suffices to note that
$$h_{top}(M\times\O|\mathbb P)=\overline h(M\times\O|\O)=\underline h(M\times\O|\O)=h_{top}(\phi)$$
and
$$\underline d_{L^\infty(\O, M)}\le \underline d_{L^\infty_\mathbb P(\O)}.$$
\end{proof}
At the end of this section, we make the following remark on the random SRB measures. We summarize the notions and results of the random SRB measures in Section \ref{S:RandomSets} for the sake of completeness, which are mainly borrowed from \cite{GK}.
\begin{rem}\label{R:SRBMeasure} Under the setting of this paper, S1) and S2) systems satisfy the conditions of Theorem \ref{T:SRBMeasure}, thus, roughly speaking, we have both existence and uniqueness of SRB measure for a given $\t$-invariant ergodic probability measure $\mathbb P$.
For S1) systems, since the driven system $(\O,\t)$ is uniquely ergodic, each S1) system has a unique SRB measure $\mu$. By the variational principle, we have that $$h_\mu(\phi)\le h_{top}(\phi).$$ Therefore, by Theorem \ref{T:SPHorseshoe}, there exist strong random horseshoes on marginal $\mathbb P$ whose entropy can approach $h_\mu(\phi)$.
For S2) systems, for any $\t$-ergodic measure $\mathbb P$, there exists a unique SRB measure $\mu$ whose marginal is $\mathbb P$. By the variational principle, we have that
$$h_{\phi}(\mu|\mathbb P)\le h_{top}(M\times\O|\ \mathbb P), $$
$h_\phi(\mu|\mathbb P)$ is called the relative entropy of $\mu$ with respect to $\mathbb P$. Therefore, by Theorem \ref{T:PHorseshoe}, there exist random horseshoes on marginal $\mathbb P$ whose entropy can approach $h_\phi(\mu|\mathbb P)$. \end{rem}
\appendix
\section{Random Sinai-Ruelle-Bowen measures}\label{S:RandomSets} In this section, some notions and results will be given for random dynamical systems, especially the results of Random SRB measures, most of which are borrowed from \cite{GK}.
Let $(\O,\mathcal B,\mathbb P)$ be a probability space, $\mathbb E$ be a locally compact Hausdorff second countable topological space. Let $\mathcal F, \mathcal K,\mathcal G$ denote respectively the family of all closed, compact and nonempty open sets of $\mathbb E$. \begin{defn}\label{D:RandomSet}
A map $X:\O\to\mathcal F$ is called a {\em random closed set} if, for every compact set $K\subset \mathbb E$, $$\{\o\in\O|X(\o)\cap K\neq \emptyset\}\in \mathcal B.$$
A map $Y:\O\to\mathcal G$ is called a {\em random open set} if its complement $X=Y^c$ (by this, we mean $X(\o)=E\setminus Y(\o)$ for $\mathbb P$-a.e. $\o\in\O$) is a random closed set. Since $\{\o\in\O|Y^c(\o)\cap F=\emptyset\}=\{\o\in\O|F\subset Y(\o)\}$, $Y$ is random open set if and only if $\{\o\in\O|F\subset Y(\o)\}$ is a measurable event for every $F\in \mathcal F$. \end{defn} Let $\t:(\O,\mathcal B,\mathbb P)\to(\O,\mathcal B,\mathbb P)$ be an invertible ergodic metric dynamical system, and $F:\mathbb E\times \O\to \mathbb E\times \O$ be a continuous random dynamical system over this metric dynamical system.
\begin{defn}\label{D:RTran}
$F$ is called {\em random topological transitive} if for any random open sets $U,V$ with $U(\o),V(\o)\neq \emptyset$ for all $\o\in\O$, there exists a random variable $n$ taking values in $\mathbb Z$ such that the intersection $F^{n(\o)}(\t^{-n(\o)}\o,U(\t^{-n(\o)}\o))\cap V(\o)$ is non-empty $\mathbb P$-a.s..
\end{defn}
The following is a technical lemma whose proof is included for the sake of completeness. \begin{lem}\label{L:MixToTran}
$F$ is topological mixing on fibers (see Section \ref{S:Setting} for definition) implies that $F$ is random topological transitive. \end{lem} \begin{proof} Let $U$ and $V$ be random open sets with $U(\o),V(\o)\neq \emptyset$ for $\mathbb P$-a.e. $\o\in\O$. Let $\{x_i\}_{ i\in\mathbb N}$ be a countable dense set of $\mathbb E$. For $i,j\in\mathbb N$, define
$$\O^U_{i,j}=\left\{\o|\ \overline{B\left(x_i,\frac1j\right)}\subset U(\o)\right\}\text{ and }\O^V_{i,j}=\left\{\o|\ \overline{B\left(x_i,\frac1j\right)}\subset V(\o)\right\}.$$ It is not hard to see that each $\O^U_{i,j}$ or $\O^V_{i,j}$ is measurable, and $\mathbb P(\cup_{i,j}\O^U_{i,j})=\mathbb P(\cup_{i,j}\O^V_{i,j})=1$. Without loss of generality, we suppose that $\mathbb P(\O^U_{i_0,j_0})>0$ for some $i_0,j_0\in\mathbb N$. Since $F$ is topological mixing on fibers, for any $i,j\in \mathbb N$, there exists $N_{i,j}\in\mathbb N$ such that for any $k>N_{i,j}$ \begin{equation}\label{E:Hitting} F^k\left(B\left(x_{i_0},\frac1{j_0}\right)\times \{\t^{-k}\o\}\right)\cap B\left(x_i,\frac1j\right)\times\{\o\} \neq \emptyset,\ \forall \o\in\O. \end{equation} By the ergodicity of $\t^{-1}$ and Poincare Recurrence Theorem, we have that for $\mathbb P$-a.s. $\o\in\O$, $\{\t^{-k}\o\}_{k\in\mathbb N}$ will visit $\O^U_{i_0,j_0}$ infinitely many times. Given $i,j\in\mathbb N$, for $\o\in \O^V_{i,j}$, define that
$$n(\o)=\min\{k|\ k>N_{i,j},\ \t^{-k}\o\in \O^U_{i_0,j_0}\}.$$ It is not hard to see that $n:\O_{i,j}^V\to\mathbb N$ is measurable and $n(\o)<\infty$ for $\mathbb P$-a.s. $\o$. Additionally, (\ref{E:Hitting}) implies that for $\mathbb P$-a.s. $\o\in \O^V_{i,j}$, the following holds \begin{align}\begin{split}\label{E:TTran} &F\left(n(\o),U(\o)\times\left\{\t^{-n(\o)}\o\right\}\right)\cap V(\o)\times\{\o\}\\ \supset&F\left(n(\o),B\left(x_{i_0},\frac1{j_0}\right)\times\left\{\t^{-n(\o)}\o\right\}\right)\cap B\left(x_i,\frac1j\right)\times\{\o\}\\ \neq &\emptyset. \end{split} \end{align} Although we only give the definition of the integer valued function $n$ on $\O_{i,j}^V$, it is not hard to extend $n$ onto $\O$ for which (\ref{E:TTran}) still holds for $\mathbb P$-a.e. $\o\in\O$, since there are only countable many $\O^V_{i,j}$s. Thus $F$ is random topological transitive. \end{proof}
The following results are mainly from \cite{GK} whose proofs are omitted.
Let $\L$ be a random hyperbolic attractor of $F$, where the $F(\o)$ is uniformly hyperbolic on $\L(\o)$ $\mathbb P$-a.s.. For definition of random hyperbolic attractors, we refer the reader to \cite{GK}; while for more details about random attractors, we refer the reader to \cite{A}. \begin{defn}\label{D:RPO} Let $\d$ be a strictly positive random variable on $(\O,\mathcal B,\mathbb P)$. Then for any $\o\in\O$ a sequence $\{y_n\}_{n\in \mathbb Z}$ in $\mathbb E$ is called an $(\o,\d)$ pseudo-orbit of $F$ if $$d(y_{n+1},F(y_n,\t^n\o))\le \d(\t^{n+1}\o)\quad\text{ for all }n\in \mathbb Z.$$ For a strictly positive random variable $\e$ and any $\o\in \O$ a point $x\in\mathbb E$ is said to $(\o,\e)$-shadow the $(\o,\d)$ pseudo-orbit $\{y_n\}_{n\in\mathbb Z}$ if $$d(F^n(x,\o),y_n)\le \e(\t^n\o)\quad\text{ for all }n\in\mathbb Z.$$ \end{defn} The following lemma is the Proposition 3.6 of \cite{GK}, which is called {\bf Random Shadowing Lemma}. \begin{lem}\label{L:RShL} Let the random hyperbolic set $\L$ have local product structure. Then for every tempered random variable $\e>0$ there exists a tempered random variable $\b>0$ such that $\mathbb P$-a.s. every $(\o,\b)$ pseudo-orbit $\{y_n\}_{n\in\mathbb Z}$ with $y_n\in \L(\t^n\o)$ can be $(\o,\e)$-shadowed by a point $x\in \L(\o)$. If $2\e$ is chosen as an expansivity characteristic, then the shadowing point $x$ is unique. Moreover, if the $y_n$ are chosen to be random variables such that for $\mathbb P$-almost all $\o\in\O$ the sequence $\{y_n(\o)\}_{n\in\mathbb Z}$ is an $(\o,\b)$ pseudo -orbit, then the starting point $x(\o)$ of the corresponding $(\o,\e)$-shadowing orbit depends measurably on $\o$. \end{lem}
Let $V^s, V^u$ be the local invariant manifolds of $F$ respectively, moreover take them of random size $\eta$ being an expansivity characteristic. Take a small enough strictly positive tempered random variable $\varpi$ the smallness of which is only depending on $\b$, $\eta$ and the hyperbolicity of the system. Then one can find a random variable $k:\O\to\mathbb N$ such that $\L(\o)$ can be covered by $k(\o)$ open balls of radius less than $\varpi(\o)$ and centres $p_i(\o)$, $i=1,\cdots,k(\o)$ with $p_i:\O\to\mathbb E$ measurable. Actually $\varpi$ can always be chosen log-integrable, and $k$ can be chosen log-integrable if $F$ is of tempered continuity.
Denote these balls by $B_{\varpi}(p_i,\o)$. Then $A$ is called a random matrix if, for each $\o\in \O$, $A(\o)\in \mathbb R^{k(\o)\times k(\t\o)}$ such that $$A(\o)_{i,j}=\begin{cases}1\quad &\text{ if }F(p_i(\o),\o)\in B_{\varpi}(p_j,\t\o)\\ 0 &\text{ otherwise.}\end{cases}$$
\begin{defn}\label{D:RSD} Let $k:\O\to\mathbb N^+$ be a random variable, $A$ a corresponding random transition matrix, and define for $\o\in\O$ $$\Sigma_k(\o):=\prod_{i=-\infty}^{\infty}\{1,\cdots,k(\t^i\o)\},$$ $$\Sigma_A(\o):=\{\bar x=(x_i)\in \Sigma_k(\o):A_{x_i,x_{i+1}}(\t^i\o)=1\text{ for all }i\in\mathbb Z\}.$$ Let $\s$ be the standard (left-) shift. The families $\{\s:\Sigma_k(\o)\to \Sigma_k(\t\o)\}$ and $\{\s:\Sigma_A(\o)\to \Sigma_A(\t\o)\}$ are called random $k$-shift and random subshift of finite type, respectively. Moreover, define $\Sigma_A:=\{(\bar x,\o): \bar x\in \Sigma_A(\o),\o\in\O\}$, which is a measurable bundle over $\O$, and also denote the respective skew-product transformation on $\Sigma_A$ by $\s$. \end{defn} One can also define one-sided versions of random $k$-shifts and subshifts of finite type on $\Sigma_k^+:=\prod_{i=0}^\infty\{1,\cdots, k(\t^i\o)\}\text{ and corresponding }\Sigma_A^+(\o), \o\in\O$ respectively. Let $C(\Sigma_A^+(\o))$ denote the space of random continuous functions on $\Sigma_A^+$ which are measurable on $\o$ and continuous on $x\in\mathbb E$ for fixed $\o$. If a random continuous function is $\mathbb P$-a.s. H\"older continuous with uniform exponent, such function is called random H\"older continuous function. For a random continuous function $\varphi$ and $\o\in\O$ random transfer operators $\mathcal L_{\varphi}(\o):C(\Sigma_A^+(\o))\to C(\Sigma_A^+(\t\o))$ is defined by $$(\mathcal L_{\varphi}(\o)h)(x)=\sum_{y\in\Sigma_A^+(\o):\s y=x}\exp(\varphi(\o,y))h(y)$$ for $h\in C(\Sigma_A^+(\o)),x\in C(\Sigma_A^+(\t\o))$. $\mathcal L_{\varphi}^*(\o)$ denotes the random dual operator mapping finite signed measures on $C(\Sigma_A^+(\t\o))$ to those on $C(\Sigma_A^+(\o))$ by $$\int hd\mathcal L_{\varphi}^*(\o)m=\int \mathcal L_{\varphi}hdm\text{ for all }h\in C(\Sigma_A^+(\o))$$ for a finite signed measure $m$ on $C(\Sigma_A^+(\t\o))$.
The following theorem is the Theorem 4.3 of \cite{GK}, which is the main result about SRB measures. \begin{thm}\label{T:SRBMeasure} Let $F$ be a $C^{1+\a}$ random dynamical system with a random topological transitive hyperbolic attractor $\L(\subset \mathbb E\times \O)$. Then there exists a unique $F$-invariant measure (SRB measure) $\nu$ supported by $\L$ and characterized by each of the following: \begin{itemize} \item[(i)] $h_\nu(F)=\int\sum\l_i^+d\nu$ where $\l_i$ are the Lyapunov exponents corresponding to $\nu$; \item[(ii)] $\mathbb P$-a.s. the conditional measures of $\nu_\o$ on the unstable manifolds are absolutely continuous with respect to the Riemannian volume on these submanifolds; \item[(iii)] $h_\nu(F)+\int fd\nu=\sup_{F-\text{invariant measure } m}\{h_m(F)+\int fd\nu\}$ and the latter is the topological pressure $\pi_F(f)$ of $f$ which satisfies $\pi_{F}(f)=0$; \item[(iv)] $\nu=\psi \tilde \mu$ where $\tilde \mu$ is the equilibrium state for the two-sided shift $\s$ on $\Sigma_A$ and the function $f\circ \psi$. The measure $\tilde \mu$ can be obtained as a natural extension of the probability measure $\mu$ which is invariant with respect to the one-sided shift $\s$ on $\Sigma^+_A$ and such that $\mathcal L^*_\eta(\o)\mu_{\t\o}=\mu_\o$ $\mathbb P$-a.s. where $\eta-f\circ \psi=h-h\circ (\t\times \s)$ for some random H\:older continuous function $h$; \item[(v)] $\nu$ can be obtained as a weak limit $\nu_\o=\lim_{n\to\infty}F(n,\t^{-n}\o)m_{\t^{-n}\o}$ $\mathbb P$-a.s. for any measure $m_\o$ absolutely continuous with respect to the Riemannian volume such that sup $m_\o\subset U(\o)$. \end{itemize} \end{thm}
Here $f(x,\o)=-\log\| \det D_xF(x,\o)|_{E^u(x,\o)}\|$, where $E^u(x,\o)$ is the invariant unstable Oseledets subspace of $D_xF$ on $(x,\o)$; and $\psi$ is the conjugation between $F$ on $\L$ and $\s$ on $\Sigma_A$ which is constructed based on Markov partitions, then reduce $\s$ on $\Sigma_A$ to the image of an unstable manifold to obtain $\s$ on $\Sigma_A^+$. The H\"older continuity follows from the $C^{1+\a}$ness of $F$, and the existence of Markov partitions are given by Theorem 3.9 of \cite{GK}.
\section{Measurable Selection}\label{S:MST} In this section, we state a version of measurable selection theorems, which is taken from \cite{BP}.
\begin{thm}\label{T:BMST} Let $U,V$ be complete separable metric spaces and $E\subset U\times V$ be a Borel set. If, for each $u\in E$, the section $E_u$ is $\s$-compact there is a Borel selection, $S$, of $E$. Further $proj(E)$ is a Borel set and $\rho_S$ is a Borel measurable function defined on $proj(E)$. \end{thm} Here $S$ is said to be a Borel selection of $E$ provided \begin{itemize} \item[i)] $S$ is a Borel set; \item[ii)] $S\subset E$;
\item[iii)] For each $u\in U$, the section $S_u=\{v\in V|\ (u,v)\in S\}$ contains at most one point: \item[iv)] $proj(S)=proj(E)$. \end{itemize} And $\rho_S$ is the function induced by $S$, which assigns to each $u\in proj(E)$ the second coordinate of the unique member of $S$ with first coordinate $u$.
\end{document} | arXiv |
\begin{document}
\title{On the effectiveness of the protocol creating the maximum entanglement of two charge-phase qubits by cavity field} \author{ C. Li} \email{[email protected]} \affiliation{Institute of Theoretical Physics, The Chinese Academy of Science, Beijing, 100080, China} \author{Y. B. Gao} \affiliation{Institute of Theoretical Physics, The Chinese Academy of Science, Beijing, 100080, China} \affiliation{Applied Physics Department, Beijing University of Technology, Beijing, 100022, China} \date{January 12, 2004}
\begin{abstract} We revisit the protocols to create maximally entangled states between two Josephson junction (JJ) charge phase qubits coupled to a microwave field in a cavity as a quantum data bus. We devote to analyze a novel mechanism of quantum decoherence due to the adiabatic entanglement between qubits and the data bus, the off-resonance microwave field. We show that even through the variable of the data bus can be adiabatically eliminated, the entanglement between the qubits and data bus remains and can decoher the superposition of two-particle state. Fortunately we can construct a decoherence-free subspace of two-dimension to against this adiabatic decoherence.To carry out the analytic study for this decoherence problem, we develop Fr\H{o}hlich transformation to re-derive the effective Hamiltonian of these system, which is equivalent to that obtained from the adiabatic elimination approach . \end{abstract}
\pacs{73.21.La,03.65.-w, 03.67.¨Ca, 76.70.¨Cr} \maketitle
\section{Introduction}
As a useful quantum resource, entanglement can not only be used to test fundamental principles in quantum mechanics, such as Bell's inequalities, but also play a central role in quantum information processing including quantum computation, quantum teleportation and quantum cryptography. Therefore how to create a stable and controllable entangled state in a quantum bits (qubit) system is very important for quantum information protocols \cite{qi}.
A number of protocols have been proposed to produce quantum entanglement in different qubit systems, such as NMR, polarization photon, quantum dots, Josephson junction. Due to the prompt progresses in preparing various solid state qubits, these schemes become very promising to realize the practical quantum computing. Actually, according to the DiVincenzo criteria the couplings JJ qubits \cite{dd} for quantum computation, the solid system is one of the best candidates for quantum computation, since qubit should be scalable, controllable and with longer decoherence time. Actually it seems difficult to fulfill all the requirements by quantum information processing. Recently several groups have demonstrated the macroscopic quantum coherence of Josephson junction (JJ) qubits with long decoherence time in experiments \cite{dd}\cite{y}\cite{3}\cite{4}\cite{4-2}\cite{4-3}.
Quantum entanglement plays the central role in integrating multi-qubit to form a scalable quantum computing. We notice that, in most of the protocols to produce such JJ qubit entanglement, and correspondingly to carry out two qubit logic gate operations, each qubit interacts with a common quantum object as a data bus, which may be an electromagnetic cavity field, a quantum transmission line coplanar cavity or an nano-mechanical resonator \cite{han}\cite{5}. If the characterized frequency of the quantum data bus is off-resonate to the energy spacing of the qubit, the degree of freedom of the quantum data bus and the variables of the quantum object can be separated adiabatically form that of two qubit system. Then the induced inter-qubit interactions can create an efficient quantum entanglement of two qubits.
However, as we have investigated\cite{sun1}, there usually exists quantum entanglement between the states of data bus and those of the two qubit system even after removing the data bus. This adiabatic quantum entanglement has been studied according to the generalized Born-Oppenheimer (BO) approximation\cite{sun1} where the slow variables can be driven by different effective potentials provided by the fast internal states and then the entanglement between fast and slow variables forms. Recently, Averin et al \cite{Averin} similarly considered the adiabatic entanglement of two JJ charge qubits. In this investigation, two JJ charge qubits are assumed to be coupled with a large junction which works as a faster data bus. With the BO adiabatic approximation, the energy of the lowest band of the latter junction can be considered as the effective interaction between the two JJ charge qubits. But the quantum decoherence induced by the adiabatic entanglement has not been considered here though it may occur in the case with higher excitation of large junction.
In this paper, we are also specific to the JJ qubit system coupling the cavity and show that the adiabatic entanglement may cause the extra errors of the logic gate operation for this two qubit system with high-excitation. We will consider decoherence of the JJ qubit caused by the the thermal excitation of large junction through this adiabatic entanglement mechanisim. Actually, without considering thermal excitation, we are not clear if the created entanglement between two JJ qubits are stable since it can be produced according to an effective Hamiltonian, which is obtained in usual by "ignoring" intermeddle variables of data bus \cite{pz}.
To carry out a totally analytic study, we utilize the generalized Fr\H{o} hlich transformation to re- derive the effective Hamiltonian of this system. In this way we can study in details this novel decoherence phenomenon for the entanglement of two JJ-qubits. There exist four entangled states for two JJ qubit system, including two maximally entangled states that can be obtained by controllable the micro-wave field. Though the superposition of some two qubit states can decoher due to the adiabatic entanglement, there exist a decoherence-free subspace, against to the decoherence induced by the adiabatic separation process. Therefore, we found that only two of four maximal entangled states are stable in this scheme.
\section{The model of two \ JJ\ qubits in cavity}
Without loss of generality, we investigate a simplified model, which consisting of two JJ qubits in a cavity with a single mode micro-wave fields (FIG. 1). \begin{figure}
\caption{ SQUID S1 and SQUID S2 in a cavity coupled to a microwave field. }
\label{fig:device}
\end{figure} The Hamiltonian of the coupled system $H$ can be described as a sum of that of the junctions, the cavity field \ and a interaction term between the cavity and the junction \cite{han}\cite{5}, i.e,
\begin{eqnarray} H &=&\hbar {\Greekmath 0121} a^{\dagger }a+4E_{C1}\left( n_{1}-n_{g1}\right) ^{2}-E_{J1}\left( \Phi \right) \cos {\Greekmath 0127} _{1} \notag \\ &&+4E_{C2}\left( n_{2}-n_{g2}\right) ^{2}-E_{J2}\left( \Phi \right) \cos {\Greekmath 0127} _{2}, \end{eqnarray} where \begin{equation} E_{C}=e^{2}/2\left( C_{g}+2C_{j}\right) \end{equation} is the single-particle charging energy of the island, $C_{j}$ the capacitance of the junction, $C_{g}$ the capacitance of gate and ${\Greekmath 0127} _{i}$ the phase difference between points on the opposite sides of the $i$ -th junction. \ The Josephson coupling energy \begin{equation} E_{J}\left( \Phi \right) =2E_{J0}\cos \left( \frac{2{\Greekmath 0119} \Phi }{\Phi _{0}} \right) \end{equation} depends on the the total flux $\Phi $ and the maximal coupling energy $ E_{J0}=I_{c}\frac{\Phi _{0}}{2{\Greekmath 0119} }$. Here, $I_{c}$ is the critical current of the junction, and $\Phi _{0}$ the total flux and flux quanta.
When a nonclassical microwave field with the vector potential \begin{equation} \mathpalette\overrightarrow@{A}\left( r\right) =\mathpalette\overrightarrow@{u}_{{\Greekmath 0115} }\left( r\right) a+\mathpalette\overrightarrow@{u}_{{\Greekmath 0115} }^{\ast }\left( r\right) a^{\dagger }. \end{equation} is applied, where $a^{\dagger }$ and $a$ are the creation and annihilation operators of the cavity fields, the total flux $\Phi $ is divided into two part \begin{equation} \Phi =\Phi _{e}+\Phi _{f}. \end{equation} $\Phi _{e}$ is static magnetic flux through the SQUIDs and \begin{equation} \Phi _{f}=\left\vert \Phi _{\mathrm{{\Greekmath 0115} }}\right\vert \left( e^{-i{\Greekmath 0112} }a+e^{i{\Greekmath 0112} }a^{\dagger }\right) , \end{equation} the microwave-filed-induced flux through the SQUIDs where \begin{equation} \Phi _{\mathrm{{\Greekmath 0115} }}=\oint \mathpalette\overrightarrow@{u}_{{\Greekmath 0115} }\left( r\right) \cdot d\mathpalette\overrightarrow@{l}. \end{equation}
We take $E_{C1}=E_{C2}$ and $E_{J1}\left( \Phi \right) =E_{J2}\left( \Phi \right) $ and then the Hamiltonian of the system becomes \begin{eqnarray} H &=&{\Greekmath 0122} \left( V_{g}\right) \left( {\Greekmath 011B} _{\mathrm{z}}^{\mathrm{1} }+{\Greekmath 011B} _{\mathrm{z}}^{\mathrm{2}}\right) +\hbar {\Greekmath 0121} a^{\dagger }a \notag \label{MFAGrandCanonicalHamiltonian} \\ &-&2E_{J0}\cos \left( \dfrac{{\Greekmath 0119} \Phi _{e}+\Phi _{f}}{\Phi _{0}}\right) \left( {\Greekmath 011B} _{\mathrm{x}}^{\mathrm{1}}+{\Greekmath 011B} _{\mathrm{x}}^{\mathrm{2} }\right) . \end{eqnarray} where the quasi-spin operators ${\Greekmath 011B} _{\mathrm{x}}$ , ${\Greekmath 011B} _{\mathrm{y} },$ and ${\Greekmath 011B} _{\mathrm{z}}$ are defined \ with respect to the the states $
|0\rangle $ and $|1\rangle $ of no (one) excess cooper pair on the island. To form a qubit or a two-level system, one need to tune the gate voltage $
V_{g}$ so that $n_{g}$ is approximately a half-integer. In this case the charge eigen-states are $|0\rangle $ and $|1\rangle $. We assume $\left\vert \Phi _{\mathrm{{\Greekmath 0115} }}\right\vert \ll \Phi _{0}$, and focus on the charging regime $E_{C}\gg E_{J}$. Then, the Hamiltonian can be approximated as
\begin{equation*} H=H_{0}+H_{I}: \end{equation*}
\begin{eqnarray} H_{0} &=&\hbar {\Greekmath 0121} _{\mathrm{J}}\left( {\Greekmath 011B} _{\mathrm{z}}^{\mathrm{1} }+{\Greekmath 011B} _{\mathrm{z}}^{\mathrm{2}}\right) +\hbar {\Greekmath 0121} a^{\dagger }a \notag \\ H_{I} &=&g\left( a+a^{\dagger }\right) \left( {\Greekmath 011B} _{\mathrm{x}}^{\mathrm{1 }}+{\Greekmath 011B} _{\mathrm{x}}^{\mathrm{2}}\right) \end{eqnarray} where \begin{equation*} \hbar {\Greekmath 0121} _{\mathrm{J}}=2E_{C}\left[ \frac{C_{g}V_{g}}{e}-\left( 2n+1\right) \right] \end{equation*} and the coupling constant between qubit and the cavity field is
\begin{equation} g=-2\dfrac{I_{c}{\Greekmath 011E} _{0}}{2{\Greekmath 0119} }\sqrt{\dfrac{h{\Greekmath 0117} }{2{\Greekmath 0116} _{0}}}\int_{S} \mathpalette\overrightarrow@{e}\cdot d\mathpalette\overrightarrow@{s}\sin \dfrac{{\Greekmath 011E} _{e}}{{\Greekmath 011E} _{0}} {\Greekmath 0119} . \end{equation}
In practice we take the volume of the cavity and the wavelength of microwave respectively as $\symbol{126}1\mathrm{cm}^{3}$ and \symbol{126}$1$\textrm{cm} , the dimension of the Josephson junction as $\symbol{126}1\mathrm{{\Greekmath 0116} m}$, the critical current of the junction as $\ I_{\mathrm{c}}\symbol{126}10^{-5} \mathrm{A}$. Due to Eq$\left( \RIfM@\expandafter\text@\else\expandafter\mbox\fi{4}\right) $, we have $\dfrac{g}{\hbar {\Greekmath 0121} }\ll 1$, which means $H_{I}\ll H_{0}$. So we can perform perturbation theory represented by a generalized Fr\H{o}hlich transformation\cite{sun} on the Hamiltonian$\left( \mathrm{2}\right) $. Then we can obtain the effective Hamiltonian of two JJ qubit by removing the variables of the microwave field approximately.
\section{The effective Hamiltonian from the generalized Fr\H{o}lich transformation}
In its original approach for superconductivity BCS theory, the Fr\H{o}hlich transformation\cite{sun} is utilized to get the effective Hamiltonian for electron-electron interaction from electron-phonon interaction. In general we can consider a interaction system described by a sum of free Hamiltonian and interaction Hamiltonian,
\begin{equation} H=H_{0}+H_{I}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ }. \end{equation} Comparing with the free part $H_{0},$the interaction part $H_{I}$ can be \ regard as a perturbation. Let us define an anti-Hermitian operator $S$, and a corresponding unitary operator $U^{\dag }=\exp \{-S\}$. We perform an unitary transformation on the Hamiltonian$\left( \RIfM@\expandafter\text@\else\expandafter\mbox\fi{11}\right) $ by this unitary operator, and then get the equivalent Hamiltonian \ as
\begin{eqnarray} H &=&U^{\dagger }HU \notag \label{MFAGrandCanonicalHamiltonian} \\ &=&H_{0}+\underset{\mathrm{n=1}}{\sum }\frac{\left( -1\right) ^{n}}{(n+1)!} \underset{n}{[\underbrace{S,[\cdots \lbrack S,[S,}}H_{I}]]\cdots ]]. \end{eqnarray}
Since the unitary transformation $U$ is time -independent, Hamiltonians $ \left( \RIfM@\expandafter\text@\else\expandafter\mbox\fi{11}\right) $ and $\left( \RIfM@\expandafter\text@\else\expandafter\mbox\fi{12}\right) $ describe the same physical process. We can take both the interaction $H_{I}$ and operator $S$ in the first order terms in the right hand side. At the same time, we require the operator $S$ \ to satisfy the following condition
\begin{equation} H_{\mathrm{I}}+[H_{\mathrm{0}},S]=0. \end{equation} In Eq.$\left( \RIfM@\expandafter\text@\else\expandafter\mbox\fi{12}\right) $, if we discard the higher order terms and only keep the second-order term, the effective Hamiltonian can be achieved approximately as
\begin{equation} H_{\mathrm{eff}}\cong H_{\mathrm{0}}+\frac{1}{2}\left[ H_{\mathrm{I}},S \right] . \end{equation} From the Eq.$\left( \RIfM@\expandafter\text@\else\expandafter\mbox\fi{13}\right) $ we certainly know how to construct the anti-Hermitian operator $S$, which has the following form \
\begin{equation} S=\underset{\mathrm{m}\neq \mathrm{n}}{\sum }\frac{\left( H_{\mathrm{I} }\right) _{\mathrm{mn}}}{E_{\mathrm{m}}-E_{\mathrm{n}}}\left\vert m\right\rangle \left\langle n\right\vert ,\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ \ } \end{equation} where $\left\vert m\right\rangle $ and $E_{m}$ are the eigenvectors and eigenvalues of $H_{0}$ respectively. The transformation, by which one can draw out effective Hamiltonian$\left( \RIfM@\expandafter\text@\else\expandafter\mbox\fi{14}\right) $ from the Hamiltonian$ \left( \RIfM@\expandafter\text@\else\expandafter\mbox\fi{11}\right) $, is the so-called general Fr\H{o}hlich transformation. It has been proved in Ref\cite{sun} that this generalized Fr \H{o}hlich transformation is just equivalent to the second-order perturbative theory.
Now we use the above approximation method to derive the effective Hamiltonian for the two-JJ qubit entanglement. Under the condition $g$ $\ll $ $\hbar {\Greekmath 0121} $, we explicitly construct the anti-Hermitian operator $S$ of Eq.$\left( \RIfM@\expandafter\text@\else\expandafter\mbox\fi{3}\right) $ as following
\begin{eqnarray} S &=&\dfrac{g}{2}\{\Delta _{+}\left( a-a^{\dagger }\right) \left( {\Greekmath 011B} _{ \mathrm{x}}^{\mathrm{1}}+{\Greekmath 011B} _{\mathrm{x}}^{\mathrm{2}}\right) \notag \label{criticaltemperature} \\ &&+i\Delta _{-}\left( a+a^{\dagger }\right) \left( {\Greekmath 011B} _{\mathrm{y}}^{ \mathrm{1}}+{\Greekmath 011B} _{\mathrm{y}}^{\mathrm{2}}\right) \} \end{eqnarray} where the coefficients \begin{equation} \Delta _{\pm }=\left( \dfrac{1}{\hbar {\Greekmath 0121} -2\hbar {\Greekmath 0121} _{\mathrm{J}}} \pm \dfrac{1}{\hbar {\Greekmath 0121} +2\hbar {\Greekmath 0121} _{\mathrm{J}}}\right) \end{equation}
Using the above explicit expression for anti-Hermitian operator $S$, we can finish the generalized Fr\H{o}hlich transformation and then obtain effective Hamiltonian obviously.
\begin{eqnarray} H_{\mathrm{eff}} &=&\hbar {\Greekmath 0121} _{\mathrm{J}}\left( {\Greekmath 011B} _{\mathrm{z}}^{ \mathrm{1}}+{\Greekmath 011B} _{\mathrm{z}}^{\mathrm{2}}\right) -\dfrac{g^{2}}{2}\Delta _{-}\left( a+a^{\dagger }\right) ^{2}\left( {\Greekmath 011B} _{z}^{1}+{\Greekmath 011B} _{z}^{2}\right) \notag \label{criticaltemperature} \\ &&+\hbar {\Greekmath 0121} a^{\dagger }a-\dfrac{g^{2}}{2}\Delta _{+}-\dfrac{g^{2}}{2} \Delta _{+}{\Greekmath 011B} _{\mathrm{x}}^{\mathrm{1}}{\Greekmath 011B} _{\mathrm{x}}^{\mathrm{2} }. \end{eqnarray} If the micro-wave field is very weak, we can discard the second terms of $ a^{2}$ and $a^{\dagger 2}$ in the effective Hamiltonian under the rotating wave approximation. Then the effective Hamiltonian Eq.$\left( \mathrm{18} \right) $ reads
\begin{eqnarray} H_{\mathrm{eff}} &=&\left( \hbar {\Greekmath 0121} _{\mathrm{J}}+g^{2}\Delta _{-}a^{\dagger }a\right) \left( {\Greekmath 011B} _{\mathrm{z}}^{\mathrm{1}}+{\Greekmath 011B} _{ \mathrm{z}}^{\mathrm{2}}\right) \notag \\ &&+\hbar {\Greekmath 0121} a^{\dagger }a+\dfrac{g^{2}}{2}\Delta _{+}{\Greekmath 011B} _{\mathrm{x} }^{\mathrm{1}}{\Greekmath 011B} _{\mathrm{x}}^{\mathrm{2}}. \end{eqnarray} or
\begin{eqnarray}
H_{\mathrm{eff}} &=&\sum_{n}H(n)|n\rangle \langle n|: \notag \\ H(n) &=&\left( \hbar {\Greekmath 0121} _{\mathrm{J}}+ng^{2}\Delta _{-}\right) \left( {\Greekmath 011B} _{\mathrm{z}}^{\mathrm{1}}+{\Greekmath 011B} _{\mathrm{z}}^{\mathrm{2}}\right) \\ &&+\hbar {\Greekmath 0121} n+\dfrac{g^{2}}{2}\Delta _{+}{\Greekmath 011B} _{\mathrm{x}}^{\mathrm{1} }{\Greekmath 011B} _{\mathrm{x}}^{\mathrm{2}}. \notag \end{eqnarray}
In general this is a typical effective Hamiltonian leading \ the two-qubit quantum logic gate. In usual it is obtained by adiabatically eliminating the variable data bus with various methods\cite{pz}. However, in most of previous works, this crucial terms \begin{equation} g^{2}\Delta _{-}a^{\dagger }a\left( {\Greekmath 011B} _{\mathrm{z}}^{\mathrm{1}}+{\Greekmath 011B} _{\mathrm{z}}^{\mathrm{2}}\right) \end{equation} referred to the ac Stark effect (a dispersive frequency shift effects) has been either irrationally ignored or passed over in silence. This is unsatisfactory even though we can now prove that it can run the logic gate of two qubit system in next section.
\section{A novel decoherence mechanism: the inverse Stern-Gerlach effect}
Having gotten the effective Hamiltonian with a perfect inter-qubit interaction, we can show how to create the quantum entanglement by controllable the coupling between photon and qubit. As a data bus, the role of cavity field is to introduce extra controllable parameters. In the next section we will show the details to implemental an ideal two qubit logical gate operations in the decoherence free subspace (DFS)\cite{97c}. However, for those states outside the DFS, we can demonstrate a novel decoherence phenomenon related to the so-called inverse Stern-Gerlach\cite{inverse} effect from the adiabatic variable separation based on the BO approximation \cite{sun}.
Let us generally consider the adiabatic evolution of two identical charge qubits $1$ and $2$, coupled to a single-mode field in the microwave cavity. In the off-resonance case, the motion of the qubits does not excite the transitions from a cavity mode to another, and then the photon number is conserved, i.e., \begin{equation} \lbrack H_{\mathrm{eff}},a^{\dagger }a]=0 \end{equation} This shows that the total wave function will adiabatically keep the factorized structure \begin{equation} \left\vert \Psi (t)\right\rangle =\left\vert {\Greekmath 011E} _{n}(t)\right\rangle \otimes \left\vert n\right\rangle \end{equation} during evolution only if the cavity is exactly prepared initially in a single number state with definite phonon number, namely, a Fock state $
\left\vert n\right\rangle .$ In this case the qubit part is just governed by the effective Hamiltonian $\langle n|H_{\mathrm{eff}}\left\vert n\right\rangle $ $=H(n)$ and then we can manipulate the qubit system according to the $n$-dependent Hamiltonian to form maximal entanglement. However, if one can not prepare the cavity in a single Fock state, the part $ \left\vert {\Greekmath 011E} (t)\right\rangle $ of qubit must depend on the different phonon number $n$ and then we can not make an exact manipulation for qubit part due to this correlation to cavity field. This kind feature of quantum adiabatic entanglement is just a novel physical source of the quantum decoherence in the process of two qubit logical gate operations.
We remark that this phenomenon is an analog of "inverse Stern-Gerlach effect" in atomic optics, in which discrete atomic trajectories are correlated to different photon numbers in the cavity. When the atom is non-resonant with the cavity modes, there appears a dispersive frequency shift effects affecting both the atomic transition and the field mode. It can be interpreted as single atom and single photon index effects. These effects lead to various interesting potential applications, which have been investigated in cavity QED\ for atoms, both theoretically and experimentally, e.g., the interference schemes to measure matter-wave phase shifts produced by the non resonant interaction \cite{inverse}. This schemes performs a quantum non-demolition measurement of photon numbers in a cavity, at the single photon level. Its experimental demonstration is based on the detection of Ramsey resonances on circular Rydberg atoms crossing a very high Q cavity. For this kind of "inverse Stern Gerlach effect", we even presented an extensive generalization based on the Born-Openheimer approximation to analyzed the the adiabatic separation induced quantum entanglements \cite{sun}. Thus it defines the adiabatic quantum decoherence in general case. We can discuss this effect for a solid state based system with two charge qubit.
To see the quantum decoherence due to the generalized "inverse Stern Gerlach effect", we assume the two JJ qubits and the cavity field are initially prepared in a factorizable state: \begin{equation} \left\vert \Psi \left( 0\right) \right\rangle =\left\vert {\Greekmath 011E} \right\rangle \otimes \left\vert {\Greekmath 0127} \right\rangle \end{equation} where $\left\vert {\Greekmath 011E} \right\rangle $ is the initial state of the two JJ\ qubits and $\left\vert {\Greekmath 0127} \right\rangle $ the state of the field. In general, if the cavity is prepared initially in a superposition state of Fock state \begin{equation} \left\vert {\Greekmath 0127} \right\rangle =\sum_{n}c_{n}\left\vert n\right\rangle \end{equation} rather than a single Fock states, the total system will evolve according to \begin{equation} \left\vert \Psi (t)\right\rangle =\sum_{n}c_{n}\left\vert {\Greekmath 011E} _{n}(t)\right\rangle \otimes \left\vert n\right\rangle \end{equation} where \begin{equation} \left\vert {\Greekmath 011E} _{n}(t)\right\rangle =U_{n}(t)\left\vert {\Greekmath 011E} (0)\right\rangle . \end{equation} The effective evolution matrix $U_{n}(t)=\exp [-iH(n)t]$ \ is governed by $ H(n).$This "inverse Stern Gerlach effect" result from the dependence $ \left\vert {\Greekmath 011E} _{n}\left( t\right) \right\rangle $ to different $n$.
Let $\left\vert m\right\rangle $ be the single Fock state that we want to prepare and $H(m)$ be the controlled Hamiltonian. Then we can characterize the difference between the real evolution and the ideal one ${\Greekmath 011A} _{m}\left(
t\right) =\left\vert {\Greekmath 011E} _{m}(t)\right\rangle \langle {\Greekmath 011E} _{m}(t)|,$ by the fidelity \begin{equation} F=Tr[{\Greekmath 011A} _{m}\left( t\right) {\Greekmath 011A} \left( t\right)
]=\sum_{n}|c_{n}|^{2}\left\vert \langle {\Greekmath 011E} _{m}(t)|{\Greekmath 011E} _{n}(t)\right\rangle |^{2} \end{equation} where \begin{equation} {\Greekmath 011A} \left( t\right) =Tr_{C}\left( \left\vert \Psi (t)\right\rangle \langle
\Psi (t)|\right) =\sum_{n}|c_{n}|^{2}\left\vert {\Greekmath 011E} _{n}(t)\right\rangle
\langle {\Greekmath 011E} _{n}(t)| \end{equation} is the reduced density matrix of the two JJ qubits.
Usually it is difficult to prepare the Fock state $\left\vert m\right\rangle $ and we can only use the coherent state $\left\vert {\Greekmath 010B} \right\rangle $
with average photon number $\langle {\Greekmath 010B} |a^{\dagger }a\left\vert {\Greekmath 010B} \right\rangle $ $=$ $m.$ Then we assume the junctions are initially in the state $\left\vert 0\right\rangle _{1}\left\vert 0\right\rangle _{2}$ , and the initial state of the micro-wave field is the coherent state $\left\vert {\Greekmath 010B} \right\rangle $. Then total system will evolve into \
\begin{eqnarray} \left\vert \Psi \right\rangle &=&e^{-\frac{1}{2}\left\vert m\right\vert ^{2}} \underset{n}{\sum }\dfrac{m^{n}}{n!}\left\vert {\Greekmath 011E} _{n}^{00}(t)\right\rangle \left\vert n\right\rangle : \notag \\ \left\vert {\Greekmath 011E} _{n}^{00}(t)\right\rangle &=&\{c_{2}^{\ast }\left( n\right) \left\vert 0\right\rangle _{1}\left\vert 0\right\rangle _{2}+ic_{1}\left( n\right) \left\vert 1\right\rangle _{1}\left\vert 1\right\rangle _{2}\}, \end{eqnarray} where the time-dependent coefficients are
\begin{equation*} \left\{ \begin{array}{c} c_{1}\left( n\right) =\sin \left( {\Greekmath 0121} _{n}t\right) \cos \left( {\Greekmath 0112} _{n}\right) , \\ c_{2}\left( n\right) =\cos \left( {\Greekmath 0121} _{n}t\right) -i\sin \left( {\Greekmath 0121} _{n}t\right) \sin \left( {\Greekmath 0112} _{n}\right) , \\ {\Greekmath 0121} _{n}=\frac{1}{\hbar }\sqrt{\left( 2\hbar {\Greekmath 0121} _{\mathrm{J} }+2g^{2}\Delta _{-}n\right) ^{2}+(\dfrac{g^{2}}{2}\Delta _{+})^{2}}, \\ \sin \left( {\Greekmath 0112} _{n}\right) =\dfrac{2\hbar {\Greekmath 0121} _{\mathrm{J} }+2g^{2}\Delta _{-}n}{\sqrt{\left( 2\hbar {\Greekmath 0121} _{\mathrm{J}}+2g^{2}\Delta _{-}n\right) ^{2}+(\dfrac{g^{2}}{2}\Delta _{+})^{2}}}. \end{array} \right. \end{equation*}
Through a simple calculation, we obtain the fidelity
\begin{eqnarray} &&F=e^{-\frac{1}{2}\left\vert m\right\vert ^{2}}\underset{n}{\sum }\dfrac{ m^{n}}{n!}\{\cos ^{2}\left( {\Greekmath 0121} _{n}t\right) \cos ^{2}\left( {\Greekmath 0121} _{m}t\right) \notag \\ &+&\sin ^{2}\left( {\Greekmath 0121} _{n}t\right) \sin ^{2}\left( {\Greekmath 0112} _{n}\right) \sin ^{2}\left( {\Greekmath 0121} _{m}t\right) \sin ^{2}\left( {\Greekmath 0112} _{m}\right) \notag \\ &+&\cos ^{2}\left( {\Greekmath 0121} _{m}t\right) \sin ^{2}\left( {\Greekmath 0121} _{n}t\right) \sin ^{2}\left( {\Greekmath 0112} _{n}\right) \notag \\ &+&\cos ^{2}\left( {\Greekmath 0121} _{n}t\right) \sin ^{2}\left( {\Greekmath 0121} _{m}t\right) \sin ^{2}\left( {\Greekmath 0112} _{m}\right) \\ &+&\sin ^{2}\left( {\Greekmath 0121} _{n}t\right) \cos ^{2}\left( {\Greekmath 0112} _{n}\right) \sin ^{2}\left( {\Greekmath 0121} _{m}t\right) \cos ^{2}\left( {\Greekmath 0112} _{m}\right) \notag \\ &+&2\sin \left( {\Greekmath 0121} _{n}t\right) \cos \left( {\Greekmath 0121} _{n}t\right) \cos \left( {\Greekmath 0112} _{n}\right) \sin \left( {\Greekmath 0121} _{m}t\right) \cos \left( {\Greekmath 0112} _{m}\right) \cos \left( {\Greekmath 0121} _{m}t\right) \notag \\ &+&2\sin ^{2}\left( {\Greekmath 0121} _{n}t\right) \sin ^{2}\left( {\Greekmath 0121} _{m}t\right) \cos \left( {\Greekmath 0112} _{n}\right) \cos \left( {\Greekmath 0112} _{m}\right) \sin \left( {\Greekmath 0112} _{m}\right) \sin \left( {\Greekmath 0112} _{n}\right) \}. \notag \end{eqnarray}
FIG.2 and FIG.3 illustrate that the fidelity decays sharply with the value of $m$, namely, coherent state $\left\vert {\Greekmath 010B} \right\rangle $ leads to big deviation of ${\Greekmath 011A} _{{\Greekmath 010B} }$ and ${\Greekmath 011A} _{m}$ with big ${\Greekmath 010B} $. This is because that coherent state $\left\vert {\Greekmath 010B} \right\rangle $ is just in Fock state $\left\vert m\right\rangle $ with the probability $P_{m}$,
\begin{equation*} \left. P_{m}\right\vert _{{\Greekmath 010B} =m}=\left. \left\vert \left\langle m\right. \left\vert {\Greekmath 010B} \right\rangle \right\vert ^{2}\right\vert _{{\Greekmath 010B} =m}=e^{-\left\vert m\right\vert ^{2}}\frac{\left\vert m\right\vert ^{2m}}{m!} . \end{equation*} When $m$ $\rightarrow \infty $, $P_{m}\rightarrow 0$ quickly.
\begin{figure}
\caption{ Fidelity $F$ as a function of time $t$, with different $m$, which is the mean eigenvalue of the micro-wave field. $m=0.2$(line), $m=0.4$ (circles) and $m=0.7$ (crosses).}
\label{fig:time}
\end{figure} \begin{figure}
\caption{ Fidelity $F$ as a function of the mean eigenvalue of the micro-wave field $m$, with different fixed time $t=13$(line),$t=40$(circles) and $t=70$ (crosses). }
\label{fig: mean}
\end{figure}
The above discussion shows us that, when we prepare the controllable cavity field in different initial states, one can get different entangled states for the two qubit. This motivates us to explore the possibility to realize the perfect logic gate operation by initially preparing the micro-wave field in coherent state and Fock state. Let us consider the above mentioned problem in the following.
We aim to get a standard Bell state $\left\vert {\Greekmath 011E} ^{+}\right\rangle $ \begin{equation*} \left\vert {\Greekmath 011E} ^{+}\right\rangle =\frac{1}{\sqrt{2}}\left( \left\vert 00\right\rangle +\left\vert 11\right\rangle \right) \end{equation*} from the state $\left\vert 0\right\rangle _{1}\left\vert 0\right\rangle _{2}$ , by preparing the cavity field initially in Fock state $\left\vert k\right\rangle $. The evolution from $\left\vert 0\right\rangle _{1}\left\vert 0\right\rangle _{2}$ to $\left\vert {\Greekmath 011E} ^{+}\right\rangle $\ naturally realize a perfect ideal logic gate.
The real evolution process governed by the effective adiabatic Hamiltonian $ H(k)$ is \begin{eqnarray*} \left\vert {\Greekmath 011E} _{k}^{00}(t)\right\rangle &=&e^{-i\frac{H\left( k\right) }{ \hbar }t}\left\vert 00\right\rangle \\ &=&c_{2}^{\ast }\left( k\right) \left\vert 00\right\rangle +ic_{1}\left( k\right) \left\vert 11\right\rangle \}, \end{eqnarray*} where the Hamiltonian $H(k)$ corresponds to the Fock state $\left\vert k\right\rangle $ for fixed $k$. We can use the square of the norm of the inner product \begin{equation*} f_{k}=\left\vert \left\langle {\Greekmath 011E} _{k}^{00}(t)\right\vert {\Greekmath 011E} ^{+}\rangle \right\vert ^{2} \end{equation*} to characterize the difference between the ideal state $\left\vert {\Greekmath 011E} ^{+}\right\rangle $\ and the real state $\left\vert {\Greekmath 011E} _{k}^{00}\right\rangle $ \begin{eqnarray} &&f_{m}=\left\vert \left\langle {\Greekmath 011E} _{m}^{00}(t)\right\vert {\Greekmath 011E} ^{+}\rangle \right\vert ^{2} \\ &=&\frac{1}{2}\left\vert \cos ^{2}\left( {\Greekmath 0121} _{m}t\right) +\sin ^{2}\left( {\Greekmath 0121} _{m}t\right) \left[ \cos \left( {\Greekmath 0112} _{m}\right) +\sin \left( {\Greekmath 0112} _{m}\right) \right] ^{2}\right\vert . \notag \end{eqnarray} \begin{figure}
\caption{The vertical axis represent the function $f_{m}$, the horizonal axis represent time $t$, $m$ is the eigenvalue of the Fock state, $m =0$ (line), $m =10$(crosses) and $m =20$ (circles)}
\label{fig:bell-k}
\end{figure} The above equation shows that $f_{m}$ is a periodic function of time $t$.
FIG.4 shows that $f_{m}$ decays with the average photon number $m$ and the maximum value of $f_{m}$ can not approach 1. Based on this result, we can not construct an ideal logic gate in this system.
Now we consider another case that the micro-wave field is prepared in coherent state $\left\vert {\Greekmath 010B} \right\rangle $ initially, and the junctions is prepared initially in the state $\left\vert 0\right\rangle _{1}\left\vert 0\right\rangle _{2}$. By a simple calculation, we get the final state depicted by the reduced density matrix(RDM)
\begin{eqnarray} {\Greekmath 011A} _{{\Greekmath 010B} } &=&tr_{mw}\left\{ \left\vert \Psi \right\rangle \left\langle \Psi \right\vert \right\} \notag \\ &=&e^{-\left\vert {\Greekmath 010B} \right\vert ^{2}}\underset{k}{\sum }\dfrac{ \left\vert {\Greekmath 010B} \right\vert ^{2k}}{k!k!}\left\vert {\Greekmath 0120} _{k}\right\rangle \left\langle {\Greekmath 0120} _{k}\right\vert . \end{eqnarray} We explicitly calculate the function \begin{eqnarray} &&f_{{\Greekmath 010B} }=\left\vert \left\langle {\Greekmath 011E} ^{+}\right\vert {\Greekmath 011A} _{{\Greekmath 010B} }\left\vert {\Greekmath 011E} ^{+}\right\rangle \right\vert \notag \\ &=&\frac{1}{2}e^{-\left\vert {\Greekmath 010B} \right\vert ^{2}}\underset{k}{\sum }
\dfrac{\left\vert {\Greekmath 010B} \right\vert ^{2k}}{k!k!}|\cos ^{2}\left( {\Greekmath 0121} _{k}t\right) \notag \\ &+&\sin ^{2}\left( {\Greekmath 0121} _{k}t\right) \left[ \cos \left( {\Greekmath 0112} _{k}\right)
+\sin \left( {\Greekmath 0112} _{k}\right) \right] ^{2}| \end{eqnarray}
\begin{figure}\label{fig:bell-cohen}
\end{figure}
FIG.5 displays the evolution of the function $f_{{\Greekmath 010B} }$ calculated from eq.(34) for the qubits with coherent microwave fields. When the eigenvalue $ {\Greekmath 010B} \rightarrow \infty $, the fidelity $F\rightarrow 0$ .
It is shown from the FIG.4 and FIG.5, whatever the state of the microwave field is prepared in, the ideal logic gate operation can not be\ realized in this system. But this does not means that we can not obtain any maximal entangled state in this way. We will discuss this problem in next section.
\section{Creating Maximal Entanglement in the Decoherence Free Subspace}
From the discussions in the above section, we find that the maximally entangled state can not be obtained only from the state $\left\vert 0\right\rangle _{1}\left\vert 0\right\rangle _{2}$ and $\left\vert 1\right\rangle _{1}\left\vert 1\right\rangle _{2}$, even though one can prepared external controlled microwave cavity field in an arbitary state. While the other two states $\left\vert 0\right\rangle _{1}\left\vert 1\right\rangle _{2}$ and $\left\vert 1\right\rangle _{1}\left\vert 0\right\rangle _{2}$ can span a decoherence-free subspace(DFS)$\mathfrak{W} ^{1}$, it means that any superposition of the state $\left\vert 0\right\rangle _{1}\left\vert 1\right\rangle _{2}$ and $\left\vert 1\right\rangle _{1}\left\vert 0\right\rangle _{2}$ can evolve into this kind of DFS. Easily seen in Eq.(19), the effective interaction between the cavity and qubits $g^{2}\Delta _{-}a^{\dagger }a\left( {\Greekmath 011B} _{\mathrm{z}}^{ \mathrm{1}}+{\Greekmath 011B} _{\mathrm{z}}^{\mathrm{2}}\right) $ vanishes in the DFS and can not distinguish between any two states in this DFS. So we conclude that there is not a "which-way detection" to determine the "paths" in this case, i.e., there is not decoherence appearing in DFS.
Let us use a special example to demostrate the above observation. When junctions are prepared initially in the state $\left\vert 0\right\rangle _{1}\left\vert 1\right\rangle _{2}$ and the manipulative field prepared in the Fock state $\left\vert n\right\rangle $, then the evolution of the total system will evolve into
\begin{eqnarray} \left\vert {\Greekmath 0120} _{01}\right\rangle \left\vert n\right\rangle &=&e^{-iHt}\left\vert 0\right\rangle _{1}\left\vert 1\right\rangle _{2}\left\vert n\right\rangle \notag \\ &=&\cos \left( g^{2}\Delta _{-}t\right) \left\vert 0\right\rangle _{1}\left\vert 1\right\rangle _{2}\left\vert n\right\rangle \notag \\ &&-i\sin \left( g^{2}\Delta _{-}t\right) \left\vert 1\right\rangle _{1}\left\vert 0\right\rangle _{2}\left\vert n\right\rangle \end{eqnarray} It is obvious that, when $t=\tfrac{{\Greekmath 0119} \hbar \left( {\Greekmath 0121} ^{2}-4{\Greekmath 0121} _{ \mathrm{J}}^{2}\right) }{16g^{2}{\Greekmath 0121} _{J}},$the two qubit system reaches a maximally entangled state \begin{equation} \left\vert {\Greekmath 0120} _{01}\right\rangle =\dfrac{1}{\sqrt{2}}\left( \left\vert 0\right\rangle _{1}\left\vert 1\right\rangle _{2}-i\left\vert 1\right\rangle _{1}\left\vert 0\right\rangle _{2}\right) . \end{equation} By the same way, we can obtain the other maximal entangled state
\begin{equation} \left\vert {\Greekmath 0120} _{10}\right\rangle \dfrac{1}{\sqrt{2}}\left( \left\vert 0\right\rangle _{1}\left\vert 1\right\rangle _{2}+i\left\vert 1\right\rangle _{1}\left\vert 0\right\rangle _{2}\right) , \end{equation} when we take the $t=\tfrac{3{\Greekmath 0119} \hbar \left( {\Greekmath 0121} ^{2}-4{\Greekmath 0121} _{\mathrm{J} }^{2}\right) }{16g^{2}{\Greekmath 0121} _{J}}$. Both $\left\vert {\Greekmath 0120} _{01}\right\rangle $ and $\left\vert {\Greekmath 0120} _{10}\right\rangle $ are independent of the controllable micro-wave field, they belong to the DFS $ \mathfrak{W}^{1}$. The basis $\left\vert 0\right\rangle _{1}\left\vert 0\right\rangle _{2}$ and $\left\vert 1\right\rangle _{1}\left\vert 1\right\rangle _{2}$ span the other subspace of the Hilbert space of the JJ qubits($\mathfrak{W}$), we denote
$\mathfrak{W}_{\perp }^{1}$. Thus we have $\mathfrak{W=W}^{1}\mathfrak{\oplus W}_{\perp }^{1}$.
\
\section{Overall Quality of\ Created Entanglements}
In above section we have discussed that when junctions are prepared initially in the state $\left\vert 0\right\rangle _{1}\left\vert 0\right\rangle _{2}$ or state $\left\vert 1\right\rangle _{1}\left\vert 1\right\rangle _{2}$, we can not obtain maximally entangled state of the junctions with any controllable microwave field. But we can study the entanglement of these states evolved from the initial state $\left\vert 0\right\rangle _{1}\left\vert 0\right\rangle _{2}$ or state $\left\vert 1\right\rangle _{1}\left\vert 1\right\rangle _{2}$.
As well-known, for a bipartite system, composing of two subsystems $A$ and $ B $, the bipartite entanglement can be measured by its concurrence\cite {wooter} which is defined by
\begin{equation} C({\Greekmath 011A} )=\mathrm{max}(0,\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ }{\Greekmath 0115} _{1}-{\Greekmath 0115} _{2}-{\Greekmath 0115} _{3}-{\Greekmath 0115} _{4}) \end{equation}
where ${\Greekmath 0115} _{1},{\Greekmath 0115} _{2},{\Greekmath 0115} _{3},{\Greekmath 0115} _{4}$ is the square root of non-Hermitian matrix $R$ in decreasing order, and
\begin{equation} R={\Greekmath 011A} \left( {\Greekmath 011B} _{1}^{y}\otimes {\Greekmath 011B} _{1}^{y}\right) {\Greekmath 011A} ^{\ast }\left( {\Greekmath 011B} _{1}^{y}\otimes {\Greekmath 011B} _{1}^{y}\right) . \end{equation}
We consider that when the initial state of junctions is $\left\vert 0\right\rangle _{1}\left\vert 0\right\rangle _{2}$ and the controllable microwave field is prepared in coherent state. Then, through a simple calculation, we obtain the concurrence of the states of JJ qubits subsystem Eq.($33$) is
\begin{equation*} C=\sqrt{2}\sqrt{\left( \sqrt{AB}-\left\vert D\right\vert \right) ^{2}} \end{equation*} \begin{eqnarray} A &=&e^{-\left\vert {\Greekmath 010B} \right\vert ^{2}}\underset{k}{\sum }\dfrac{ \left\vert {\Greekmath 010B} \right\vert ^{2k}}{k!k!}\left\vert \cos \left( {\Greekmath 0121} _{k}t\right) -i\sin \left( {\Greekmath 0121} _{k}t\right) \sin \left( {\Greekmath 0112} _{k}\right) \right\vert ^{2} \notag \\ B &=&e^{-\left\vert {\Greekmath 010B} \right\vert ^{2}}\underset{k}{\sum }\dfrac{ \left\vert {\Greekmath 010B} \right\vert ^{2k}}{k!k!}\sin ^{2}\left( {\Greekmath 0121} _{k}t\right) \cos ^{2}\left( {\Greekmath 0112} _{k}\right) \notag \\ D &=&e^{-\left\vert {\Greekmath 010B} \right\vert ^{2}}\underset{k}{\sum }\dfrac{ \left\vert {\Greekmath 010B} \right\vert ^{2k}}{k!k!}\sin \left( {\Greekmath 0121} _{k}t\right) \cos \left( {\Greekmath 0112} _{k}\right) \notag \\ &\cdot &\left\{ \cos \left( {\Greekmath 0112} _{k}\right) \cos \left( {\Greekmath 0121} _{k}t\right) -i\sin \left( {\Greekmath 0121} _{k}t\right) \sin \left( {\Greekmath 0112} _{k}\right) \right\} . \end{eqnarray} \begin{figure}
\caption{ The vertical axis represent concurrence $C$, the horizonal axis represent time $t$, $\protect{\Greekmath 010B} =0.1$ (light ashen line), $\protect {\Greekmath 010B} =1.1$ (ashen line) and $\protect{\Greekmath 010B} =3$ (black line). }
\label{fig:bell-cohen}
\end{figure}
In FIG.6, the concurrences of the qubits are plotted for different values of ${\Greekmath 010B} $. It is seen that, with the increasing the eigenvalue of the controllable microwave field, the concurrence decrease sharply.
We study another case, when the initial state of the controllable micro-wave field is thermal state
\begin{equation} {\Greekmath 011A} _{mw}=\frac{1}{Z}\underset{n}{\sum }e^{-n{\Greekmath 010C} E}\left\vert n\right\rangle \left\langle n\right\vert \RIfM@\expandafter\text@\else\expandafter\mbox\fi{ \ \ }Z=\underset{n}{\sum } e^{-n{\Greekmath 010C} E}, \end{equation} the state of the total system evolute into \begin{equation} {\Greekmath 011A} =\frac{1}{Z}\underset{n}{\sum }e^{-n{\Greekmath 010C} E}\left\vert {\Greekmath 0120} _{n}\right\rangle \left\vert n\right\rangle \left\langle n\right\vert \left\langle {\Greekmath 0120} _{n}\right\vert \ .\ \end{equation} By a simple calculation, we get the RDM of the JJ qubits
\begin{equation} {\Greekmath 011A} _{jj}=\frac{1}{Z}\underset{n}{\sum }e^{-n{\Greekmath 010C} E}\left\vert {\Greekmath 0120} _{n}\right\rangle \left\langle {\Greekmath 0120} _{n}\right\vert \ . \end{equation}
By the same way, we calculate the concurrence of this state in the following
\begin{eqnarray} C &=&\sqrt{2}\sqrt{\left( \sqrt{AB}-\left\vert D\right\vert \right) ^{2}} \\ A &=&\frac{1}{Z}\underset{k}{\sum }e^{-k{\Greekmath 010C} E}\left\vert \cos \left( {\Greekmath 0121} _{k}t\right) -i\sin \left( {\Greekmath 0121} _{k}t\right) \sin \left( {\Greekmath 0112} _{k}\right) \right\vert ^{2} \notag \\ B &=&\frac{1}{Z}\underset{k}{\sum }e^{-k{\Greekmath 010C} E}\sin ^{2}\left( {\Greekmath 0121} _{k}t\right) \cos ^{2}\left( {\Greekmath 0112} _{k}\right) \notag \\ D &=&\frac{1}{Z}\underset{k}{\sum }e^{-k{\Greekmath 010C} E}\sin \left( {\Greekmath 0121} _{k}t\right) \sin \left( {\Greekmath 0112} _{k}\right) \notag \\ &\cdot &\left\{ \cos \left( {\Greekmath 0112} _{k}\right) \cos \left( {\Greekmath 0121} _{k}t\right) -i\sin \left( {\Greekmath 0121} _{k}t\right) \sin \left( {\Greekmath 0112} _{k}\right) \right\} \notag \end{eqnarray} \begin{figure}
\caption{ The vertical axis represent concurrence $C$, the horizonal axis represent time $t$, with different parameter$\protect{\Greekmath 010C} E$ of the thermal states. $\protect{\Greekmath 010C} E=0.7$ (light ashen line),$\protect{\Greekmath 010C} E=2$ (ashen line) and $\protect{\Greekmath 010C} E=6$ (black line) }
\label{fig:bell-cohen}
\end{figure}
In FIG.7, the concurrence of the state ${\Greekmath 011A} _{jj}$ is periodic function of time $t$ and the concurrence $C\rightarrow $ the maximal value, when ${\Greekmath 010C} E\rightarrow \infty $. This is because ${\Greekmath 010C} E\rightarrow \infty $, the thermal state ${\Greekmath 011A} _{mw}\rightarrow \left\vert 0\right\rangle \left\langle 0\right\vert $.
FIG.6 and FIG.7 illustrate that any types of the controllable micro-wave field can not increase the entanglement of the JJ qubits when its initial state is superposition of $\left\vert 00\right\rangle $ and $\left\vert 11\right\rangle $, and the maximal entanglement is much smaller than $1$. Only the initial state of the controllable micro-wave field is vacuum state, the entanglement can reach the maximal value.
\section{Conclusion}
In summary, we study the protocols which can create maximally entangled states between two qubit coupled to a controllable microwave field in a cavity. In order to obtain the analytic study for this decoherence problem, we generalized Fr\H{o}hlich transformation to re-derive the effective Hamiltonian of these system, which is equivalent to that obtained from the adriatic elimination approach. Because of nontrivial decoherence, we can not construct an ideal logic gate by this system. But we can construct a decoherence-free subspace of two-dimension to against this adiabatic decoherence in this system. \ \
\section{acknowledgment}
We thank prof. C.P. Sun for helpful discussions.This work was partially supported by the CNSF (grant No.90203018) ,the Knowledge Innovation Program (KIP) of the Chinese Academy of Sciences , the National Fundamental Research Program of China with No.001GB309310, K. C. Wong Education Foundation, HongKong, and China Postdoctoral Science Foundation.
\end{document} | arXiv |
Handouts and Links
Faculty and TA Information
Hydrogen Storage in Carbon Nanotubes
Hydrogen storage in carbon nanotubes is a recent topic of research and may be an important step towards making mobile hydrogen storage feasible.
Significance of Mobile Hydrogen Storage
Existing Methods of Storage & Associated Problems
Hydrogenation of Carbon Nanotubes
Chemisorption
The graphic to the right depicts carbon nanotubes of three different chiralities[6].
It is well-established that pure hydrogen has a much larger energy density (per unit mass) than typical hydrocarbon fuels – 2.6 times more than gasoline, in fact[7]. For a 500 km range, a vehicle would require only 3.1 kg of H2 [5]. The problem is the energy density per unit volume. For the same amount of energy, about four times the volume of hydrogen would be required compared with gasoline[7]. This is fine for stationary applications, but the large space occupied is problematic for mobile applications. It follows that for hydrogen fuel cell vehicles to become prominent, the priority must be in compact and safe on-board hydrogen storage.
Currently, five primary technologies exist that hold promise for mobile hydrogen storage [1]:
1. Compressed H2 gas is the simplest. It can be done at ambient temperature, and in- and out-flow are simple. The density, however, is low compared to other methods.
2. Metal hydride storage involves powdered metals that absorb hydrogen under high pressures (~1000 psia). Heat is produced upon insertion. With pressure release and applied heat, the process is reversed. A major problem is the weight of the absorbing material – a tank's mass would be 592 kg compared to the 80 kg of a comparable compressed H2 gas tank.
3. Liquid H2 storage is just what it sounds like. The hydrogen storage itself has very high density, but Hydrogen boils at about -253ºC. From 25% to 45% of the stored energy is required to liquefy the H2 and maintain this low temperature (else the hydrogen will boil away), and bulky insulation is needed.
4. Sponge iron can be treated with steam to cause rapid oxidation, which gives off hydrogen as a byproduct. The iron itself can be considered a consumable fuel – once the entire mass of it has rusted, it is replaced with fresh iron. The steam itself, of course, requires energy to produce (the process occurs at 250ºC).
5. Carbon absorption is the newest field of hydrogen storage. Under pressure, hydrogen will bond with porous carbon materials such as nanotubes.
The inherency of the problem is clear: mobile hydrogen storage is currently not competitive with hydrocarbon fuels, and it must become so in order for this potential environmentally life-saving technology to be realized on a great scale. A significant standard of measuring the quality of a hydrogen storage scheme is what percent of the entire system's weight is recoverable hydrogen:
\begin{align} {wt \% H_2} = {{wt_{H2}} \over {wt_{H2} + wt_{matrix} + wt_{equipment}}} \end{align}
The US Department of Energy has issued goals for mobile hydrogen storage research, the most characteristic of which are 6.5 weight percent H2 [5] and 62 kg of H2 per m3 (18.7 molecules per nm3) [4]. Carbon nanotubes may well be a way of efficiently meeting this goal.
Carbon nanotubes (CNTs) essentially consist of sheets of graphite rolled into seamless tubes and capped at the ends. There are two forms CNTs take: single-walled and multi-walled. As the name implies, singled-walled nanotubes (SWNT) are composed of a single sheet of graphite; a diameter range of 0.4 to > 3nm is common[2]. Multiwalled nanotubes (MWNT) are composed of several sheets, arranged concentrically in increasingly larger diameters with diameters in the range of 1.4 to 100 nm. Due to their diminutive dimensions, CNTs have unique physical and electrical properties. These include ultra high thermal conductivities (>3000 W/m-K), a Young's modulus of ≈0.64 TPa, and the elastic ability to extend ≈5.8% of its original length before breaking[2]. More appealing still is the disproportionately large surface area to volume that these materials possess, for this allows for a greater potential of interactions, whether they be physical or chemical in nature. Also, consider that their dimensions are relative to those of atoms and molecules. This increases the strength that these interactions have between one another, particularly from Van der Waals forces.
In order for CNTs to be a viable solution to the problem of mobile hydrogen storage, they must be both easy and cost-effective to produce. Currently CNTs are not inexpensive to produce, especially when the need for specific control over dimensions and purity of the samples is necessary.
The production of CNTs is achieved in three distinct methods:
Arc discharge
Chemical vapor deposition.
Arc discharge is the most basic of these and involves the use of two graphite electrodes between which a high current is passed. This produces a carbon vapor, in which CNTs form. While arc discharge is fairly inexpensive, it tends to produce samples of low purity. Also it requires special precautions due to the high amperage (100 A) and high temperatures (2000-3000°C)[9].
Laser ablation, utilizes a laser directed at a graphite target which is then heated to roughly 1200°C. The vapor is sent down stream to condense on a cooled collector. This method has the benefit of producing high quality CNTs, but this is offset by the high initial costs and low output[9].
Chemical vapor deposition(CVD), the last method to be discussed here, probably has the greatest potential for use in industry of those mentioned. In CVD, a heated chamber is injected with some carbon based gas, carbon monoxide for instance. Located inside the chamber is a substrate, on top of which a matrix of metal catalysts particles is distributed, when the gas is flows through the chamber the carbon disassociates and begins forming vertical structures on the aforementioned metal particles. This takes place at a relatively low temperature of 700-800°C, as opposed to the first two methods. Also, it is far easier to implement this method in terms of large scale production, though the quality of this method leaves something to be desired[9].
The image above shows the growth of CNT in which the catalyst is elevated by the tube growth[7].
Though impurities are often a problem, they can be removed. This plays an important role in hydrogen absorption as NTs consisting of uniform carbon are shown to have higher rates of absorption [6]. These impurities are often generated by the catalysts used in the synthesis of the CNTs, but can be removed by means such as pretreatment of the sample in an acidic bath or by heating in a vacuum at high temperature for short durations. Though any additional processes the nanotubes must undergo only accrues further expenses and thus puts the ultimate goal of affordable nanotubes further from reach[9].
Hydrogen storage in carbon nanotubes occurs by two mechanisms: physisorption and chemisorption[8]. The former is characterized by condensation of H2 molecules inside or between CNTs. Chemisorption, in contrast, uses a catalyst to dissociate the molecular hydrogen and allow it to bond with some of the unsaturated carbon bonds along the tube.
Early research into potential means of hydrogen storage in CNTs focused on physisorption as the primary storage mechanism. The initial studies were done on H2 adsorption of untreated carbon soot, which contained only 0.1-0.2 weight % SWNTs, it was found that this amorphous carbon was able to absorb 0.01 % H2 by weight. From these results it was extrapolated that a sample of highly pure SWNTs could reach a 5 to 10 % weight adsorptivity and thus the overall goal of 6.5 weight % put forth by the DOE[5].
Following up on these findings, others performed similar research into various conditions under which to improve the percent of hydrogen uptake, and while insightful these were done under unrealistic conditions; such as near cryogenic temperatures or extremely high pressures.
The most realistic take on the ability of CNTs to absorb H2, through physisorption, was performed by C. Lui et al. [6]. Here the viability of CNTs was examined at room temperature and only modest pressures ≈10-12MPa. Through the use of hydrogen arc-discharge, three samples of carbon nanotubes were fabricated. Samples 2 and 3 underwent a pre-treatment process, which involved soaking in a solution of HCl acid. Following this, sample 3 was then heat treated in a vacuum. The results of hydrogen absorption can be seen the figure to the right[6].
The purpose of the acid bath was to remove all traces of the catalysts. This had little overall effect in increase H2 storage potential. The greater gain was found by the heating of sample 3 in a vacuum which evaporated any and all organic compounds that had formed of the surfaces of the CNT, thus demonstrating the need for clean and unobstructed surface interactions between hydrogen and the carbon atoms of the nanotubes. Though this showed a potential for hydrogen storage, the results were still far from ideal and made evident the inherent limitation of physisorption.
Attempts to model physisorption have lead to a study comparing analytical atomic modeling (AFEM) with a continuum model, in which the carbon tube is equated to a pressure vessel and the hydrogen stored is equated to an internal pressure. The figure to the left, from Chen et al in 2008[4], shows the analogy between the two methods (where (a) is AEFM, and (b) is the continuum model).
This study showed hope that CNTs could achieve the desired 18.7 molecules per nm3 as mentioned earlier in this article. The figure above, from the same study[4] shows a correlation between nanotube radius and simulated storage capacity.
Density function modeling indicates that chemisorption holds more promise than physisorption for percent weight hydrogen[8]. C-H bond strength prediction indicates that it is theoretically possible to release the C-H bonds at STP. The specific means by which to do so are still out of our reach technologically, though Nikitin et al [8] predict that an appropriate metal catalyst and a precisely calculated CNT diameter should be able to accomplish this. Under laboratory conditions, this team achieved 5.1 ± 1.2 wt% H2 storage at STP.
Through alkali doping, chemisorption in carbon nanotubes can be increased a good deal under laboratory conditions, though it must be done at higher temperatures [3]. Lithium doping at 650 K reached a storage capacity of 20 wt% H2. Potassium doping at much lower temperatures (about room temperature) can yield 14 wt% H2, though the resulting hydrogen-rich tubes are unstable and prone to spontaneous combustion. Both processes involved 2 hours of hydrogen uptake, which is not practical for vehicle use [3].
A method of chemisorption was also proposed by researchers at Penn State, in which clusters of metal nanoparticles are chemically affixed to the surface of carbon nanotubes. These metal clusters, in this case platinum, act as doorways into the surface of the tubes. Some of the hydrogen is absorbed by the metal, converting them to metal hydrides, while the bulk is absorbed into the CNT where it adheres to the walls. Conveniently this increases the temperature at which the nanotubes can absorb hydrogen from near cryogenic temperatures to those temperatures more convenient for implementation. The temperature is dependent exclusively on the selection of metal used, thus there are possibilities to provide a range of functionality. Nickel and magnesium were other alternatives which were considered due to the heavy weight of platinum. Still a major concern with this method is the cost of the nanotubes themselves, which were roughly $25,000 per pound[8].
Hydrogenation of carbon nanotubes holds promise for the future of mobile hydrogen storage. Moreover, it holds some advantages over other existing methods of hydrogan storage, which have their own impracticalities. Though there are setbacks, research suggests that sufficient storage is theoretically possible. With the talent and unrelenting efforts of today's innovators, hydrogenated carbon nanotubes may someday make a hydrogen vehicles more viable and competitive than at present.
W Epting
M Fuchs
Written 19 November 2008
1. Anathachar, Vinay, and John J. Duffy. "Efficiencies of hydrogen storage systems onboard fuel cell vehicles." Solar Energy 78 (2005): 687-94.
2. Baughman, Ray H., Anvar A. Zakhidov, and Walt A. De Heer. "Carbon Nanotubes: The Route toward Applications." Science 297 (2002): 787-92.
3. Chen, P., X. Wu, J. Lin, and K. L. Tan. "High H2 Updake by Alkali-Doped Carbon Nanotubes Under Ambient Prsesure and Moderate Temperatures." Science 285 (1999): 91-93.
4. Chen, Y. L., B. Liu, J. Wu, Y. Huang, H. Jiang, and K. C. Hwang. "Mechanics of hydrogen storage in carbon nanotubes." Journal of the Mechanics and Physics of Solids 56 (2008): 3224-241.
5. Dillon, A. C., K. M. Jones, T. A. Bekkedahl, C. H. Kiang, D. S. Bethune, and M. J. Heben. "Storage of hydrogen in single-walled carbon nanotubes." Nature 386 (1997): 377-79.
6. Liu, C., Y. Y. Fan, M. Liu, H. T. Chong, H. M. Cheng, and M. S. Dresselhaus. "Hydrogen Storage in Single-Walled Carbon Nanotubes at Room Temperature." Science 286 (1999): 1127-129.
7. McCarthy, John. "Hydrogen as a Fuel." Sustainability FAQ. 30 Mar. 2008. Stanford University. <http://www-formal.stanford.edu/jmc/progress/hydrogen.html>.
8. Nikitin, A., H. Ogasawara, D. Mann, R. Denecke, Z. Zhang, H. Dai, K. Cho, and A. Nilsson. "Hydrogenation of Single-Walled Carbon Nanotubes." Physical Review Letters 95 (2005).
9. Star, Alexander. Lecture. Engineering 0240:Nanotechnology and Engineering. University of Pittsburgh. Pittsburgh, 1-25 November 2008.
10. Hydogen: Future Fuel. 2005. Penn State. 21 Nov. 2008. <http://www.rps.psu.edu/hydrogen/form.html>.
page revision: 31, last edited: 21 Nov 2008 21:01
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\begin{document}
\setcounter{page}{1}
\thispagestyle{empty}
\title{Hypocoercivity and hypocontractivity concepts for linear dynamical systems\thanks{Received
by the editors on Month/Day/Year.
Accepted for publication on Month/Day/Year.
Handling Editor: Name of Handling Editor. Corresponding Author: Name of Corresponding Author}}
\author{Franz Achleitner\thanks{Technische Universit\"at Wien, Institute of Analysis and Scientific Computing, Wiedner Hauptstra\ss{}e 8-10, A-1040 Wien, Austria, [email protected]}, \and Anton Arnold\thanks{Technische Universit\"at Wien, Institute of Analysis and Scientific Computing, Wiedner Hauptstr\ss{}e 8-10, A-1040 Wien, Austria, [email protected]}, \and Volker Mehrmann\thanks{Technische Universit\"at Berlin, Institut f.~Mathematik, MA 4-5, Stra\ss{}e des 17.~Juni 136, D-10623 Berlin, [email protected]} }
\markboth{Franz Achleitner, Anton Arnold, and Volker Mehrmann}{Hypocoercivity and hypocontractivity concepts for linear dynamical systems}
\maketitle
\begin{abstract} For linear dynamical systems (in continuous-time and discrete-time) we revisit and extend the concepts of hypocoercivity and hypocontractivity and give a detailed analysis of the relations of these concepts to (asymptotic) stability, as well as (semi-)dissipativity and (semi-)\-contractivity, respectively. On the basis of these results, the short-time behavior of the propagator norm for linear continuous-time and discrete-time systems is characterized by the (shifted) hypocoercivity index and the (scaled) hypocontractivity index, respectively.
\end{abstract}
\begin{keywords} semi-dissipative Hamiltonian ODEs, hypocoercivity (index), semi-contractive systems, hypocontractivity (index), Cayley transformation \end{keywords}
\begin{AMS} 34D30,
37M10, 93D05, 93D20 \end{AMS}
\section{Introduction} \label{sec:introduction}
In this paper we discuss different concepts that characterize the short and long time behavior of linear continuous-time ordinary differential equations (ODEs)
\begin{equation}\label{ODE:B}
x'(t) =\mathbf{A}_c x(t) =-\mathbf{B} x(t)\,,\quad
x(0)=x^0, \quad
t\geq 0, \end{equation}
and discrete-time difference equations (DDEs)
\begin{equation}\label{DDE:B}
x_{k+1} = \mathbf{A}_dx_{k}\,,\quad
x_0=x^0, \quad
k\in\mathbb{N}_0, \end{equation}
with matrices $\mathbf{A}_c,\mathbf{A}_d\in\C^{n\times n}$.
It is well-known that the long-time behavior of solutions of \eqref{ODE:B} and \eqref{DDE:B} can be characterized via the spectral properties of the matrices $\mathbf{A}_c,\mathbf{A}_d$ or the solutions of Lyapunov equations \cite{Adr95,HiPr10,LanT85,LaS86}. To understand the short-time behavior of continuous-time systems much progress has recently been made for systems with a semi-dissipative structure, i.e. systems where $\mathbf{A}_c$ has a semidefinite symmetric part. For this subclass it has recently been observed in \cite{AAC22,AAM21} that the short- and long-time behavior can be characterized via the concept of hypocoercivity and the hypocoercivity index. For this subclass also the analysis of the long-time behavior becomes simpler and more elegant.
In this paper we show that a similar concept of \emph{hypocontractivity and a hypocontractivity index} is analogously available in the discrete-time case and that it can be characterized via the polar decomposition of $\mathbf{A}_d$.
For both, the continuous- and discrete-time we present a systematic review and analysis of the different concepts and show the subtle differences and similarities to the classical spectral concepts and illustrate these with numerous examples. Furthermore, we present the close relationship of these concepts to classical controllability and observability concepts in control theory.
Note that we switch in the discussion of~\eqref{ODE:B} between the classical notation with $\mathbf{A}_c$ as is common in dynamical systems and the notation with $-\mathbf{B}$ as is common in evolution equations.
In Section~\ref{sec:recapctsys} we recall the concepts of (asymptotic) stability, (semi-)dissipativity, and hypocoercivity for con\-tin\-u\-ous-time systems that have been discussed in~\cite{AAM21}. To better understand the decay behavior of solutions we extend the concept of hypocoercivity to \emph{shifted hypocoercivity}. We also show under which linear transformations of the system these properties stay invariant.
In the second part of the paper, in Section~\ref{sec:Stability+ODEs:discrete-time} we derive the corresponding results for discrete-time systems and, in particular, analyze the relation between (asymptotic) stability, (semi-)contractivity, and hypocontractivity as well as \emph{scaled hypocontractivity}.
The third part in Section~\ref{sec:dtct} studies how the discussed properties are related under Cayley transformations that map between continuous-time and discrete-time systems. We show that many properties including the hypocoercivity index and hypocontractivity index map appropriately. However, in general, the shifted hypocoercivity and scaled hypocontractivity indices are not mapped into each other. Computationally feasible staircase forms to check hypocoercivity for accretive matrices and hypocontractivity for semi-contractive matrices, and to determine the associated indices are discussed in the Appendix.
We use the following notation: The conjugate transpose of a matrix $\mathbf{C}\in\C^{n\times n}$ is denoted by $\mathbf{C}^{\mathsf{H}}$.
Positive definiteness (semi-definiteness) of a Hermitian matrix $\mathbf{C}$ is denoted by $\mathbf{C}>0$ ($\mathbf{C}\geq 0$).
\section{Stability, semi-dissipativity, and hypocoercivity for continuous-time systems}\label{sec:recapctsys}
In this section we recall some properties of linear continuous-time systems and their relationship. Let us give a simplified definition of stability, for the general definition see e.g. \cite{Adr95,HiPr10}.
\begin{definition}\label{def:stable} The trivial solution $x\equiv 0$ of \eqref{ODE:B} is called \emph{(Lyapunov) stable} if all solutions of~\eqref{ODE:B} are bounded for $t\geq 0$, and it is called \emph{asymptotically stable} if it is stable and all solutions of~\eqref{ODE:B} converge to $0$ for $t\to \infty$. \end{definition}
For linear systems~\eqref{ODE:B} a solution is (asymptotically) stable if and only if the trivial solution $x\equiv 0$ is (asymptotically) stable. Therefore, if the trivial solution $x\equiv 0$ of~\eqref{ODE:B} is (asymptotically) stable, then we call the system~\eqref{ODE:B} \emph{(asymptotically) stable}.
\begin{comment} \begin{definition}\label{def:stable} A solution of \eqref{ODE:B} is called \emph{(Lyapunov) stable} if it is bounded for all $t\geq 0$, and it is called \emph{asymptotically stable} if it is stable and converges to $0$ for $t\to \infty$. If all solutions of \eqref{ODE:B} are (asymptotically) stable for all initial conditions $x^0$, then we call the system \emph{(asymptotically) stable}. \end{definition} \end{comment}
It is well-known, see e.g.\ \cite{Adr95,HiPr10}, that~\eqref{ODE:B} is \emph{(Lyapunov) stable} if all eigenvalues of~$\mathbf{A}_c$ have non-positive real part and the eigenvalues on the imaginary axis are semi-simple, and it is \emph{asymptotically stable} if all eigenvalues of~$\mathbf{A}_c$ have negative real part.
A concept closely related to stability is that of (semi-)dissipativity.
Writing $\mathbf{A}_c$ as the sum of its Hermitian part $\mA_H :=(\mathbf{A}_c +\mathbf{A}_c^{\mathsf{H}})/2$ and skew-Hermitian part $\mA_S :=(\mathbf{A}_c -\mathbf{A}_c^{\mathsf{H}})/2$, we have the following definition, \cite[Definition 4.1.1]{Be18}.
\begin{definition} \label{def:semiDissipative} A matrix~$\mathbf{A}_c\in\C^{n\times n}$ is called~\emph{dissipative} (resp.~\emph{semi-dissipative}) if the Hermitian part~$\mA_H$ is negative definite (resp.~negative semi-definite).
For a (semi-)dissipative matrix~$\mathbf{A}_c\in\C^{n\times n}$, the associated ODE~\eqref{ODE:B} is called \emph{(semi-)dissipative Hamiltonian ODE}. Alternatively, a matrix~$\mathbf{B}=-\mathbf{A}_c\in\C^{n\times n}$ is called \emph{accretive} (or \emph{positive semi-dissipative}) if its Hermitian part~$\mB_H$ is positive semi-definite. \end{definition}
An nice property of a
semi-dissipative Hamiltonian ODE~\eqref{ODE:B} is that it is (Lyapunov) stable, since for all solutions of~\eqref{ODE:B} we have
\begin{equation} \label{energy:estimate}
\ddt \|x(t)\|^2 = \ip{\mathbf{A}_c x(t)}{x(t)} +\ip{x(t)}{\mathbf{A}_c x(t)} = \ip{x(t)}{(\mathbf{A}_c^{\mathsf{H}} +\mathbf{A}_c)x(t)} \leq 0 , \quad t\geq 0 , \end{equation}
i.e.\ the Euclidean norm (which may serve as a \emph{Lyapunov function}), is non-increasing.
The converse is in general not true,
because the Hermitian part of a matrix~$\mathbf{A}_c$ associated with a stable system~\eqref{ODE:B} does not have to be negative semi-definite, as the following example shows:
\begin{example}\label{ex:non-coerciveB}{\rm Consider the matrix \begin{equation*}
\mathbf{B} = \begin{bmatrix} 3 & 3 \\ -3 & -1 \end{bmatrix} \end{equation*}
so that~$\mathbf{A}_c=-\mathbf{B}$ has eigenvalues $\lambda=-1\pm i\sqrt5$, but the Hermitian part~$\mA_H$ is indefinite with eigenvalues $\lambda_{\min}^{\mAH}=1$ and $\lambda_{\max}^{\mAH}=-3$. Hence, the norm of solutions of~\eqref{ODE:B} may increase initially at the rate $e^t$. } \end{example}
\begin{remark}[Logarithmic Norm] \label{remark.logarithmic.norm}
Since the flow generated by~\eqref{ODE:B} is given by the matrix exponential~$e^{\mathbf{A}_c t}$, the long-time behavior of the propagator norm $\|e^{\mathbf{A}_c t}\|$, or to be precise---its exponential rate---is determined by the \emph{spectral abscissa}
\begin{equation} \label{1.spectral.abscissa}
\alpha(\mathbf{A}_c)
:= \max\{\Re(\lambda)\ |\ \text{$\lambda$ is an eigenvalue of $\mathbf{A}_c$}\} \,, \end{equation} see e.g.~\cite{vL77}.
\newline\indent In contrast, the exponential rate of the short-time behavior of~$\|e^{\mathbf{A}_c t}\|$ is determined by the logarithmic norm: The \emph{logarithmic norm} of a matrix $\mathbf{A}_c\in\mathbb{C}^{n\times n}$ with respect to an inner product is defined as
\begin{equation} \label{1.logarithmic.norm}
\mu(\mathbf{A}_c)
:= \sup_{\|x\|=1} \Re (\ip{x}{\mathbf{A}_c x} )
= \max_{\|x\|=1} \Re (\ip{x}{\mathbf{A}_c x}) \ , \end{equation}
i.e. $\mu(\mathbf{A}_c)$ is the maximal real part of the \emph{numerical range of~$\mathbf{A}_c$}.
Thus, the solutions~$x(t)$ of~\eqref{ODE:B} satisfy
$\ddt \|x(t)\|^2 = \ip{x}{(\mathbf{A}_c^{\mathsf{H}}+\mathbf{A}_c)x}
\leq 2\mu(\mathbf{A}_c)\ \|x(t)\|^2$, which implies that
\begin{equation} \label{ODE:short-t} \|x(t)\| \leq e^{\mu(\mathbf{A}_c)\ t}\|x^0\| \quad \text{for } t\geq 0 \ . \end{equation} In particular, a matrix~$\mathbf{A}_c\in\C^{n\times n}$ is semi-dissipative if and only if $\mu(\mathbf{A}_c)\leq 0$. \end{remark}
A third related concept is that of hypocoercivity for matrices and the associated hypocoercivity index, which was introduced originally in the context of linear operators see~\cite{AAC18,ArEr14,Vi09}.
\begin{definition}[{Definition 2.5 of~\cite{AAC22}}] \label{def:matrix:hypocoercive} A matrix $\mathbf{C}\in\C^{n\times n}$ is called~\emph{coercive} (or \emph{strictly accretive}) if its Hermitian part $\mC_H$ is positive definite, and it is called~\emph{hy\-po\-coercive} if the spectrum of~$\mathbf{C}$ lies in the \emph{open} right half plane.
A matrix~$\mathbf{A}_c\in\C^{n\times n}$ is called \emph{negative hypocoercive} if the spectrum of~$\mathbf{A}_c$ lies in the \emph{open} left half plane. \end{definition}
The relationship between positive semi-dissipativity and hypocoercivity is characterized by the following result.
\begin{proposition}[{\cite[Lemma 3.1]{MMS16}, \cite[Lemma 2.4 with Proposition~1(B2), (B4)]{AAC18}}] \label{prop:border} Let $\mathbf{B}\in\C^{n\times n}$ be (positive) semi-dissipative. Then, $\mathbf{B}$ has an eigenvalue on the imaginary axis if and only if $\mB_H v =0$ for some eigenvector~$v$ of~$\mB_S$. \end{proposition}
Note that, due to the assumptions, purely imaginary eigenvalues of semi-dissipative matrices are necessarily semi-simple, see also~\cite{MehMW18,MehMW20}. Therefore, an accretive matrix~$\mathbf{B}$ is hypocoercive if and only if no eigenvector of the skew-Hermitian part lies in the kernel of the Hermitian part. The latter condition is well known in control theory, and equivalent to the following statements:
\begin{lemma} \label{lem:HC:equivalence} Let $\mathbf{B}\in\C^{n\times n}$ be accretive.
Then the following are equivalent:
\begin{itemize}
\item [(B1)] There exists $m\in\mathbb{N}_0$ such that
\begin{equation}\label{condition:KalmanRank:BS_BH}
\rank[{\mB_H},\mB_S{\mB_H},\ldots,(\mB_S)^m {\mB_H}]=n \,. \end{equation}
\item [(B2)] There exists $m\in\mathbb{N}_0$ such that
\begin{equation}\label{Tm:BS_BH}
\mathbf{T}_m :=\sum_{j=0}^m \mB_S^j \mB_H ((\mB_S)^{\mathsf{H}})^j > 0 \,. \end{equation}
\item [(B3)] No eigenvector of~$\mB_S$ lies in the kernel of~$\mB_H$. \item [(B4)] $\rank [\lambda \mathbf{I}-\mB_S, \mB_H] =n$ for every $\lambda \in \mathbb{C}$ , in particular for every eigenvalue~$\lambda$ of~$\mB_S$. \end{itemize}
Moreover, the smallest possible~$m\in\mathbb{N}_0$ in (B1) and (B2) coincide.
\end{lemma}
\begin{proof} The equivalence of (B1), (B3), and (B4) and its proof are classical, see e.g. \cite[Theorem~6.2.1]{Da04} for real matrices, but its proof extends verbatim to complex matrices; see also \cite[Proposition~1]{AAC18}. The equivalence of (B1) and (B2) follows from Lemma \ref{lem:Definiteness} in the Appendix, setting $\mathbf{D}:=\mathbf{B}_H$ and $\mathbf{C}:=\mathbf{B}_S$. \end{proof}
\begin{remark}\label{rem:fullB} In Lemma~\ref{lem:HC:equivalence} we could have alternatively stated the equivalence of the following conditions, that are equivalent to the corresponding ones in Lemma~\ref{lem:HC:equivalence}. \begin{itemize}
\item [(B1')] There exists $m\in\mathbb{N}_0$ such that
\[
\rank[{\mB_H},\mathbf{B}{\mB_H},\ldots,\mathbf{B}^m {\mB_H}]=n \,. \]
\item [(B2')] There exists $m\in\mathbb{N}_0$ such that
\[ \sum_{j=0}^m \mathbf{B}^j \mB_H (\mathbf{B}^{\mathsf{H}})^j > 0 \,. \] \item [(B2'')] There exists $m\in\mathbb{N}_0$ such that
\begin{equation}\label{Tm:BS_BH2} \sum_{j=0}^m (\mathbf{B}^{\mathsf{H}})^j \mB_H \mathbf{B}^j > 0 \,. \end{equation} \item [(B3')] No eigenvector of $\mathbf{B}$ lies in the kernel of $\mB_H$. \item [(B4')] $\rank [\lambda \mathbf{I}-\mathbf{B}, \mB_H] =n$ for every $\lambda \in \mathbb{C}$ , in particular for every eigenvalue~$\lambda$ of~$\mathbf{B}$. \end{itemize}
This is easily seen, since every eigenvector of $\mathbf{B}$ that is in the kernel of $\mB_H$ is immediately an eigenvector of $\mB_S$; and conversely, every eigenvector of $\mB_S$ that is in the kernel of $\mB_H$ is also an eigenvector of $\mathbf{B}$, see \cite{MehMW18}. It also follows directly from the staircase forms presented in \cite{AAM21}. \end{remark}
\begin{remark} The equivalence of properties stated in Proposition~\ref{prop:border}, Lemma~\ref{lem:HC:equivalence} and Remark~\ref{rem:fullB} show that e.g. also the coercivity of the associated matrix~$\mathbf{T}_m$ in~\eqref{Tm:BS_BH} could have been used to define hypocoercivity for accretive matrices (in the finite-dimensional setting). Only future research of bounded and unbounded accretive operators on infinite-dimensional Hilbert spaces will decide which is the appropriate characterization for accretive operators to be hypocoercive i.e. to generate a uniformly exponentially stable $C_0$-semigroup. \end{remark}
\begin{definition}[{\cite[Definition 3.1]{AAC22}}] \label{def:HCI} Suppose that $\mathbf{B}\in\C^{n\times n}$ is accretive and hypocoercive. The~\emph{hy\-po\-co\-er\-ci\-vi\-ty index (HC-index)~$m_{HC}$ of the matrix~$\mathbf{B}$} is defined as the smallest integer~$m\in\mathbb{N}_0$ such that~\eqref{Tm:BS_BH} holds.
\end{definition}
Note that for $\mathbf{B}\in\C^{n\times n}$ (by the Cayley-Hamilton theorem applied to (B1')) it follows immediately that the hypocoercivity index (if it exists) is bounded by $n-1$. More precisely, for a finite hypocoercivity index we even have $m_{HC}\le \dim \ker (\mB_H)\le n-1$ (see Remark 4(b) in \cite{AAC18}). Furthermore, a hypocoercive matrix~$\mathbf{B}$ is coercive if and only if $m_{HC}=0$.
\begin{remark}
Hypocoercive matrices are often called~\emph{positively stable}, whereas negative hypocoercive matrices are often called~\emph{stable}. Note also that in~\cite[Definition 3]{AAM21}, the HC-index for a semi-dissipative matrix~$\mathbf{A}_c\in\C^{n\times n}$ is defined as the HC-index of its accretive counterpart~$\mathbf{B}=-\mathbf{A}_c$. We do not make use of this convention here. \end{remark}
Phenomenologically, the HC-index of an accretive matrix~$\mathbf{B}$ describes the structural complexity of the intertwining of the Hermitian part~$\mB_H$ and skew-Hermitian part~$\mB_S$ (see~\cite{AAC18} for illustrating examples). Moreover, for a semi-dissipative Hamiltonian ODE~\eqref{ODE:B}, the HC-index characterizes the short-time decay of the spectral norm of the \emph{propagator} of the associated semigroup $S(t):=e^{-\mathbf{B} t}\in \C^{n\times n}$, $t\geq 0$.
\begin{comment} \begin{proposition}[\cite{AAC22}]\label{prop:ODE-short} Consider a semi-dissipative Hamiltonian ODE~\eqref{ODE:B} whose system matrix~$\mathbf{B}$ has finite HC-index. Its (finite) HC-index is $m_{HC}\in\mathbb{N}_0$ if and only if \begin{equation}\label{short-t-decay}
\|S(t)\|_2 =1 -ct^a +{\mathcal{O}}(t^{a+1}) \quad\text{for } t\to0+\,, \end{equation} where $c>0$ and $a=2m_{HC}+1$. \end{proposition} \end{comment}
\begin{proposition}[{\cite[Theorem 2.7]{AAC22}}]\label{prop:ODE-short} Let the ODE system~\eqref{ODE:B} be semi-dissipative Hamiltonian with
(accretive) matrix $\mathbf{B}\in\C^{n\times n}$. \begin{enumerate}[(a)] \item \label{th:HC-decay:a} The (accretive) matrix~$\mathbf{B}$ is hypocoercive (with hypocoercivity index $m_{HC}\in\mathbb{N}_0$)
if and only if \begin{equation}\label{short-t-decay}
\|e^{-\mathbf{B} t}\|_2 = 1-ct^a+{\mathcal{O}}(t^{a+1})\quad\text{ for } t\in[0,\epsilon), \end{equation} for some $a,c,\epsilon>0$. In this case, necessarily $a= 2m_{HC}+1$. \begin{comment} {\color{blue} It is hypocoercive (with hypocoercivity index $m_{HC}\in\mathbb{N}_0$)}
if and only if \begin{equation}\label{short-t-decay}
\|S(t)\|_2 = 1-ct^a+{\mathcal{O}}(t^{a+1})\quad\text{ for } {\color{blue} t\in[0,\epsilon) \ (\text{ for some }\ \epsilon>0),}
\end{equation} for some $c>0$ and some $a>0$. {\color{blue} In this case,} necessarily $a= 2m_{HC}+1$. \end{comment}
\item \label{th:HC-decay:b} Consider the ODE \eqref{ODE:B} with $\epsilon$-dependent system matrix $B=\epsilon A +C$ where $\epsilon\in\mathbb{R}$. If $B=\epsilon A+C$ is hypocoercive for $\epsilon\ne 0$, then the coefficient $c=c_\epsilon$ in the Taylor expansion of the propagator norm~\eqref{short-t-decay} satisfies \begin{equation} \label{th:HC-decay:epsilon} 0 <\tilde{c}_2 \,\epsilon^{2m_{HC}} \leq c=c_\epsilon \leq \tilde{c}_1 \,\epsilon^{2m_{HC}}, \end{equation} for some positive constants $\tilde{c}_1, \tilde{c}_2>0$ independent of $\epsilon\ne 0$.
\end{enumerate} \end{proposition}
\begin{remark} \begin{itemize}
\item For genuine semi-dissipative Hamiltonian ODE systems~\eqref{ODE:B} (such that $\mu(\mathbf{A}_c)=0$), the estimate~\eqref{ODE:short-t} based on the logarithmic norm~$\mu(\mathbf{A}_c)$ yields only $\|x(t)\|\leq\|x^0\|$ for $t\geq 0$.
\item For semi-dissipative Hamiltonian ODE systems~\eqref{ODE:B}, (a lower bound for) the characterization of the HC-index via the short-time behavior of the propagator norm in~\eqref{short-t-decay} may also be derived by considering a suitable energy-preserving system, see e.g. \cite{Sta05}. However, the proof of Proposition~\ref{prop:ODE-short} in~\cite{AAC22} yields \emph{quantitative} lower and upper bounds for the multiplicative constant~$c$ in~\eqref{short-t-decay}. These explicit bounds allow to conclude the structural result in Proposition~\ref{prop:ODE-short}~\ref{th:HC-decay:b}. \end{itemize} \end{remark}
In Figure~\ref{fig:VennDiagram} we illustrate the relationship between the different concepts that we have discussed so far.
\begin{figure}
\caption{Illustration of the relationship between sets of matrices $\mathbf{B}\in\C^{n\times n}$ which are (hypo)coercive (circular discs), have a positive semi-definite Hermitian part (region within smaller ellipse), and those for which the solutions of the ODE system $x'=-\mathbf{B} x$ are stable (region within bigger ellipse), respectively.}
\label{fig:VennDiagram}
\end{figure}
\begin{remark}\label{rem_Hamiltonian} As one of the main applications of the analysis of the three discussed concepts is the study of (semi-)dissipative Hamiltonian systems, a natural concept that could be added to the description of the dynamical system is that of a \emph{Hamiltonian or energy function}. In the abstract setting that we have discussed so far, the natural energy function is the Euclidean norm of the solution. Further energy functions will be discussed below. \end{remark}
\begin{remark}[logarithmically optimal norms] For a Hermitian matrix $\mathbf{A}_c\in\C^{n\times n}$, its logarithmic norm $\mu(\mathbf{A}_c)$ and its spectral abscissa $\alpha(\mathbf{A}_c)$ are equal, $\mu(\mathbf{A}_c)=\alpha(\mathbf{A}_c)$. In general, however, only the inequality $\alpha(\mathbf{A}_c)\leq \mu(\mathbf{A}_c)$ holds, see e.g. \cite[Lemma 1c]{St75}. A norm is \emph{logarithmically optimal}
with respect to a matrix $\mathbf{A}_c$ if its spectral abscissa $\alpha(\mathbf{A}_c)$ and logarithmic norm $\mu(\mathbf{A}_c)$ are equal, i.e. $\alpha(\mathbf{A}_c) =\mu(\mathbf{A}_c)$. Thus the Euclidean norm is logarithmically optimal for all Hermitian matrices. \end{remark}
To analyze the relationship between the different concepts further, in the next section we first discuss the question by which transformations of~\eqref{ODE:B} we can switch between the different concepts and which transformations leave the different properties invariant.
\subsection{Linear transformations that preserve stability, semi-dissipativity, and hy\-po\-co\-er\-ci\-vi\-ty}\label{sec:trafo}
In this section we discuss the classes of linear transformations that preserve the concepts of stability, semi-dissipativity, and hypocoercivity, and also those that map between the different concepts, see also e.g. \cite{JoSm05, JoSm06} for some references.
The natural classes of linear transformations that preserve the different properties and the HC-index (in case of accretive matrices) are
\emph{conjugate transposition} $\mathbf{B}\to\mathbf{B}^{\mathsf{H}}$, due to Definition~\ref{def:HCI} and Lemma~\ref{lem:HC:equivalence};
\emph{unitary congruence transformations} $\mathbf{B}\to\mathbf{U}\mathbf{B}\mathbf{U}^{\mathsf{H}}$ for a unitary matrix $\mathbf{U}$, due to Definition~\ref{def:HCI} and Lemma~\ref{lem:HC:equivalence};
\emph{scaling} $\mathbf{B}\to t\mathbf{B}$ for any $t\in\mathbb{R}^+$, due to Definition~\ref{def:HCI} and Lemma~\ref{lem:HC:equivalence}; and, as we will show in Lemma~\ref{lemma:Inversion} below, the \emph{inversion} of accretive hypocoercive matrices.
It is a classical result, see e.g.~\cite{Adr95}, how to construct a similarity transformation of a ``stable'' matrix~$\mathbf{B}$ such that the transformed matrix is accretive:
The origin $x\equiv0$ is a stable state of system~\eqref{ODE:B} if and only if there exists a positive definite matrix $\mathbf{P}=\mathbf{P}^{\mathsf{H}}\in\C^{n\times n}$ that satisfies the \emph{Lyapunov matrix inequality} \begin{equation} \label{stable0:P}
\mathbf{B}^{\mathsf{H}} \mathbf{P}+\mathbf{P}\mathbf{B}\geq 0 \,. \end{equation} A congruence transformation with the Hermitian matrix~$\mathbf{P}^{-1/2}$, i.e.\ the inverse of the positive definite square root of $\mathbf{P}$, yields \begin{equation} \label{hatBH}
0
\leq \mathbf{P}^{-1/2} (\mathbf{B}^{\mathsf{H}} \mathbf{P} +\mathbf{P}\mathbf{B}) \mathbf{P}^{-1/2}
= \mathbf{P}^{-1/2} \mathbf{B}^{\mathsf{H}} \mathbf{P}^{1/2} +\mathbf{P}^{1/2} \mathbf{B} \mathbf{P}^{-1/2}
= 2 \big( \mathbf{P}^{1/2} \mathbf{B} \mathbf{P}^{-1/2}\big)_H \,.
\end{equation} Hence, the matrix \begin{equation} \label{hatB}
\widehat \mB :=\mathbf{P}^{1/2} \mathbf{B} \mathbf{P}^{-1/2} \end{equation} is accretive. Moreover, the change of basis $\tilde x(t) := \mathbf{P}^{1/2} x(t)$ transforms~\eqref{ODE:B} into a semi-dissipative Hamiltonian ODE system of the form \begin{equation}\label{ODE:hatB}
\tilde x'(t)
=
-\big(\mathbf{P}^{1/2} \mathbf{B} \mathbf{P}^{-1/2}\big) \tilde x(t)
=
-\widehat \mB\ \tilde x(t)\,. \end{equation}
Although \emph{similarity transformations} $\mathbf{B} \to\mathbf{S}\mathbf{B}\mathbf{S}^{-1}$ for invertible matrices~$\mathbf{S}\in\C^{n\times n}$ preserve the spectrum (and hence (negative) hypocoercivity), they \emph{may change} the HC-index of accretive matrices:
\begin{example} The matrix \begin{equation}\label{ODE:B:envelope}
\mathbf{B} :=\begin{bmatrix} 1 & -1 \\ 1 & 0 \end{bmatrix} \end{equation} is accretive and hypocoercive with $m_{HC}=1$ (having eigenvalues $\lambda_\pm =(1\pm i\sqrt{3})/2$). The positive definite Hermitian matrix~$\mathbf{P}=\begin{bmatrix} 2 & -1 \\ -1 & 2 \end{bmatrix}$ satisfies the continuous-time Lyapunov equation $\mathbf{B}^{\mathsf{H}}\mathbf{P} +\mathbf{P}\mathbf{B} =2\Re(\lambda) \mathbf{P} =\mathbf{P}$. The similarity transformation~\eqref{hatB} yields a coercive matrix \[
\widehat \mB
=\mathbf{P}^{1/2}\mathbf{B}\mathbf{P}^{-1/2}
=\tfrac12 \begin{bmatrix} 1 & -\sqrt{3} \\ \sqrt{3} & 1 \end{bmatrix}\,, \] hence $m_{HC}(\widehat \mB)=0$. \end{example}
In a similar way, \emph{non-unitary} congruence transformations $\mathbf{B} \to\mathbf{Q}\mathbf{B}\mathbf{Q}^{\mathsf{H}}$ for some nonsingular matrix~$\mathbf{Q}\in\C^{n\times n}$ may change the HC-index as the following example demonstrates.
\begin{example} Consider the accretive matrix \[ \mathbf{B} =\begin{bmatrix} i & 0 \\ 0 & 1 \end{bmatrix} \,. \]
The matrix~$\mathbf{B}$ has an eigenvalue $i$, hence it is not hypocoercive. A congruence transformation with the (non-unitary) matrix \begin{equation*} \mathbf{Q}
=\begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix}
\quad\text{ yields }\quad
\mathbf{Q}\mathbf{B}\mathbf{Q}^{\mathsf{H}}
=\begin{bmatrix} i & i \\ i & 1+i \end{bmatrix}
=\begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix}
+i \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix} \,, \end{equation*}
which is again accretive (due to Sylvester's inertia theorem, see e.g.\ \cite{Gan59a}). However, the matrix $\mathbf{Q}\mathbf{B}\mathbf{Q}^{\mathsf{H}}$ has eigenvalues $\tfrac12 +i\ (1\pm\tfrac{\sqrt {3}}{2})$, and is hypocoercive with HC-index~$m_{HC}=1$. \end{example}
As we have already discussed, changing the HC-index also changes the short-time behavior of the solutions of the dynamical system~\eqref{ODE:B}.
\begin{example}
Consider the matrix $\mathbf{B}$ in Example~\ref{ex:non-coerciveB}. In agreement with Proposition~\ref{prop:ODE-short}, (the norm of) solutions of the ODE~\eqref{ODE:B} may have horizontal tangents (at any point~$t_0\ge0$) with local behavior $\|x(t)\| =\|x(t_0)\| -c (t-t_0)^3 +{\mathcal{O}}((t-t_0)^4)$ for some $c>0$.
Proceeding as in \cite[Lemma 4.3]{ArEr14}, the similarity transformation~\eqref{hatB} with
\[
\mathbf{P}
=\begin{bmatrix} 3 & 2 \\ 2 & 3 \end{bmatrix}
\quad\text{yields a coercive matrix}\quad
\widehat \mB
=\mathbf{P}^{1/2}\mathbf{B}\mathbf{P}^{-1/2}
=\begin{bmatrix} 1 & \sqrt5 \\ -\sqrt5 & 1 \end{bmatrix}\,. \]
Accordingly, (the norm of) solutions of the associated ODE~\eqref{ODE:hatB} cannot have horizontal tangents (see Figure~\ref{fig:ODE-decay}).
\begin{figure}\label{fig:ODE-decay}
\end{figure}
\end{example}
\begin{remark} We note that solutions $\mathbf{P}$ to the Lyapunov inequality \eqref{stable0:P} are typically not unique, and one can use this freedom to determine solutions that optimize certain robustness measures like the distance to instability, see e.g.\ \cite{BanMNV20,MR3859144,MehV20a}. \end{remark}
It is an important observation that semi-dissipativity, hypocoercivity and the HC-index stay invariant when the inverse of a matrix is taken:
\begin{lemma} \label{lemma:Inversion} Let $\mathbf{B}\in\C^{n\times n}$. \begin{itemize}
\item [1.] If $\mathbf{B}$ is hypocoercive then $\mathbf{B}$ is invertible and $\mB^{-1}$ is hypocoercive.
\item [2.] If $\mathbf{B}$ is accretive and invertible then it follows that \begin{itemize}
\item [a.] If $v\in\ker(\mB_H)\subset\C^n$ then $\mathbf{B} v\in\ker((\mB^{-1})_H)$.
\item [b.] $\mB^{-1}$ is accretive.
\item [c.] $\dim\ker(\mB_H) =\dim\ker((\mB^{-1})_H)$. \end{itemize}
\item [3.] If $\mathbf{B}$ is accretive and hypocoercive then $\mathbf{B}$ and $\mB^{-1}$ have the same HC-index. \end{itemize} \end{lemma}
\begin{proof}
1. A matrix~$\mathbf{B}$ is hypocoercive if all eigenvalues have positive real-part. Hence, the matrix~$\mathbf{B}$ is invertible, and since the eigenvalues of the inverse of~$\mB^{-1}$ are the inverses of the eigenvalues of $\mathbf{B}$, they
have positive real-part and $\mB^{-1}$ is hypocoercive.
2a. Writing~$\mathbf{B}$ as $\mathbf{B}=\mB_H+\mB_S$, it follows that if $v\in\ker(\mB_H)$ then $\mathbf{B} v =\mB_S v =-\mathbf{B}^{\mathsf{H}} v$. Thus,
\begin{equation}\label{AIH*Av}
(\mB^{-1})_H (\mathbf{B} v)
=
\tfrac12
\big( \mB^{-1} (\mathbf{B} v) +(\mB^{-1})^{\mathsf{H}} (\mathbf{B} v)\big)
=
\tfrac12
\big( v -(\mathbf{B}^{\mathsf{H}})^{-1} (\mathbf{B}^{\mathsf{H}} v)\big)
=
0 \ . \end{equation}
2b. To prove that $\mB^{-1}$ is again accretive, we show the following identity: For all vectors $w\in\C^n$, define $v:=\mB^{-1} w$, such that \begin{equation} \begin{split} \ip{w}{(\mB^{-1})_H w} &= \tfrac12 \ip{w}{(\mB^{-1} +(\mB^{-1})^{\mathsf{H}}) w} = \tfrac12 \ip{\mathbf{B} v}{(\mB^{-1} +\mathbf{B}^{-{\mathsf{H}}}) \mathbf{B} v} \\ &= \tfrac12 \ip{v}{\mathbf{B}^{\mathsf{H}} (\mB^{-1} +\mathbf{B}^{-{\mathsf{H}}})\mathbf{B} v} = \tfrac12 \ip{v}{(\mathbf{B}^{\mathsf{H}} +\mathbf{B}) v} = \ip{v}{\mB_H v} \geq 0 \ , \end{split} \end{equation} since $\mathbf{B}$ is accretive. Hence, $\mB^{-1}$ is accretive as well.
2c. Due to part 2a. and a similar statement with the roles of $\mathbf{B}$ and $\mB^{-1}$ exchanged, $\mathbf{B}$ is a bijection from $\ker(\mB_H)$ to $\ker((\mB^{-1})_H)$.
3. By assumption, the matrix~$\mathbf{B}$ has a finite HC-index~$m_{HC}$ which is the smallest integer such that \eqref{condition:KalmanRank:BS_BH} holds
or equivalently, due to \eqref{Tm:BS_BH2}, that
\[
\bigcap_{j=0}^{m_{HC}} \ker \big(\mB_H \mathbf{B}^j\big)
=
\{0\} \]
holds, see also~\cite[Remark 4]{AAC18}.
Hence, there exists a vector $v_0\neq 0$ such that
\begin{equation} \label{v0}
\mathbf{B}^j v_0\in \ker (\mB_H) \ ,
\qquad
j\in\{0,\ldots,m_{HC}-1\}
\qquad \text{and }
\mathbf{B}^{m_{HC}} v_0 \notin \ker(\mB_H) \ . \end{equation}
Due to 2b., it follows that \begin{equation} \label{condAv0}
\mathbf{B}^{j+1} v_0\in \ker((\mB^{-1})_H) \ ,
\qquad
j\in\{0,\ldots,m_{HC}-1\}
\qquad \text{and }
\mathbf{B}^{m_{HC}+1} v_0 \notin \ker((\mB^{-1})_H) \ . \end{equation}
The matrix $\mB^{-1}$ is hypocoercive and accretive with finite HC-index~$\widetilde{m}_{HC}:=m_{HC}(\mB^{-1})$ and hence, there exists a vector $w_0\neq 0$ such that \begin{equation} \label{condw0}
(\mB^{-1})^j w_0\in \ker((\mB^{-1})_H) \ ,
\quad
j\in\{0,\ldots,\widetilde{m}_{HC}-1\}
\quad \text{and }
(\mB^{-1})^{\widetilde{m}_{HC}} w_0 \notin \ker((\mB^{-1})_H) \ . \end{equation} To show that $m_{HC} =m_{HC}(\mathbf{B}) =m_{HC}(\mB^{-1}) =\widetilde{m}_{HC}$, suppose that $v_0$ is a vector in $\C^n$ satisfying~\eqref{condAv0} with $m_{HC}=m_{HC}(\mathbf{B})$. Then $w_0 :=\mathbf{B}^{m_{HC}} v_0$ satisfies $w_0\ne0$, hence, \eqref{condw0} implies that $m_{HC}(\mathbf{B}^{-1})\geq m_{HC}(\mathbf{B})$. Exchanging the roles of $\mathbf{B}$ and $\mB^{-1}$ shows that $m_{HC}(\mathbf{B}^{-1})\leq m_{HC}(\mathbf{B})$. Altogether, $m_{HC}(\mathbf{B})=m_{HC}(\mB^{-1})$ holds.
\end{proof}
In this section we have discussed linear transformations and their effects on the concepts of hypocoercivity, stability and semi-dissipativity. In the next section we discuss how the (concept of the) HC-index for accretive matrices can be transferred to general matrices.
\subsection{Shifted hypocoercivity index for general matrices} \label{sec:SHCIndex}
A possibility to turn a general system \eqref{ODE:B} into a semi-dissipative Hamiltonian system is to shift the spectrum.
Consider the transformation
\begin{equation}\label{shift} v(t) := \exp(\lambda_{\min}^{\mBH} t) x(t) \,, \end{equation}
where $\lambda_{\min}^{\mBH}$ is the minimal (real) eigenvalue of the Hermitian matrix~$\mB_H$. Then, $v(t)$ satisfies the ODE \[
v'(t)
= -\underbrace{(\mathbf{B} -\lambda_{\min}^{\mBH} \mathbf{I})}_{=:\widetilde \mB} v(t) \ , \] where the Hermitian part~$\widetilde \mB_H$ of~$\widetilde \mB =\mathbf{B} -\lambda_{\min}^{\mBH} \mathbf{I}$ is indeed positive semi-definite. Of course, the hypocoercivity index of matrix $\widetilde \mB$ is typically modified by the shift parameter~$\lambda$.
\begin{remark} The transformation \eqref{shift} can be motivated as follows: The propagator for ODE~\eqref{ODE:B} with $\mathbf{A}_c=-\mathbf{B}$ satisfies estimate~\eqref{ODE:short-t} based on the logarithmic norm~$\mu(\mathbf{A}_c)$. Therefore, for $t\geq 0$,
\[
1
\geq \|e^{\mathbf{A}_c t}\| e^{-\mu(\mathbf{A}_c)\ t}
= \| e^{(\mathbf{A}_c -\mu(\mathbf{A}_c)\mathbf{I})\ t}\|
= \| e^{-(\mathbf{B} -\lambda_{\min}^{\mBH}\mathbf{I})\ t}\|, \]
since the logarithmic norm $\mu(\mathbf{A}_c)$ can also be characterized as
\[
\mu(\mathbf{A}_c)
:= \sup_{\|x\|=1} \Re (\ip{x}{\mathbf{A}_c x})
= \sup_{\|x\|=1} \ip{x}{\tfrac12 (\mathbf{A}_c^{\mathsf{H}} +\mathbf{A}_c) x}
= \lambda_{\max}^{\mA_H}
= -\lambda_{\min}^{\mB_H} \ , \]
where $\lambda_{\max}^{\mAH}$ is the maximal (real) eigenvalue of the Hermitian matrix~$\mA_H$. \end{remark}
In view of this shifting property, for general linear time-invariant ODE systems~\eqref{ODE:B} with matrix $\mathbf{B}\in\C^{n\times n}$, we will define a \emph{shifted hypocoercivity index} which characterizes ``the algebraic factor`` in the decay of its propagator norm for short time, see Corollary~\ref{cor:mB:SHC-decay} below.
As a first step, we decompose the matrix $\mathbf{B}\in\C^{n\times n}$.
\begin{lemma} \label{lem:mB:decomp} Let $\mathbf{B}\in\C^{n\times n}$ with Hermitian part $\mB_H$, and let $\lambda_{\min}^{\mBH}$ be the minimal (real) eigenvalue of the Hermitian matrix~$\mB_H$ (which could be negative or non-negative). Then,
the matrix
\begin{equation} \label{mB:decomp}
\widetilde \mB := \mathbf{B} -\lambda_{\min}^{\mBH} \mathbf{I}
\end{equation}
is accretive and, if~$\widetilde \mB$ is hypocoercive, has an HC-index $m_{HC}(\widetilde \mB)$ greater than~$0$. \newline\indent In particular, $\widetilde \mB$ is hypocoercive if and only if no eigenvector of~$\mB_H$ associated with $\lambda_{\min}^{\mBH}$ is an eigenvector of the skew-Hermitian part $\mB_S$ of $\mathbf{B}$. \end{lemma} \begin{proof}
If we decompose $\mathbf{B}=\mB_H +\mB_S$ into its Hermitian part $\mB_H$ and its skew-Hermitian part $\mB_S$, then $\mB_H$ has only real eigenvalues. Consider the matrix $\widetilde \mB := \mathbf{B} -\lambda \mathbf{I}$ for $\lambda\in\mathbb{R}$. Then $\lambda=\lambda_{\min}^{\mBH}$ is the only value for which the Hermitian part of $\widetilde \mB$ is positive semi-definite and singular (hence, if~$\widetilde \mB$ is hypocoercive then $m_{HC}(\widetilde \mB)>0$). \newline\indent The hypocoercivity condition for $\widetilde \mB$ follows from Lemma \ref{lem:HC:equivalence}, (B3):
Matrix~$\widetilde \mB$ fails to be hypocoercive if and only if an eigenvector~$v$ of $\mB_S$ (which is not changed by the shift) is in the kernel of $(\widetilde \mB+\widetilde \mB^{\mathsf{H}})/2 =\mB_H -\lambda_{\min}^{\mBH}\mathbf{I}$, or equivalently $v$ is an eigenvector of $\mB_H$ to the eigenvalue~$\lambda_{\min}^{\mBH}$. \end{proof}
\begin{definition}\label{def-SHC-index} Let~$\mathbf{B}\in\C^{n\times n}$ with Hermitian part~$\mB_H$, and let $\lambda_{\min}^{\mBH}$ be the minimal (real) eigenvalue of the Hermitian matrix~$\mB_H$.
If the accretive matrix $\widetilde \mB := \mathbf{B} -\lambda_{\min}^{\mBH} \mathbf{I}$ is hypocoercive, then its HC-index~$m_{HC}\in\mathbb{N}$ is called the \emph{shifted hypocoercivity index (SHC-index)~$m_{SHC}$} of~$\mathbf{B}$. \end{definition}
By definition, an accretive matrix has a (finite) HC-index~$m_{HC}$ if and only if it is positively stable, see also~\cite{AAC22,AAM21}. However, a general (constant) matrix can have a finite SHC-index~$m_{SHC}$ without being positively stable, see the following example
and Figure~\ref{fig:VennDiagram}.
\begin{example} \label{ex:mSHC:unstable} Consider the matrix \[
\mathbf{B} :=\begin{bmatrix} 9 & -3 \\ 3 & -1 \end{bmatrix} \] which has the eigenvalues $\lambda_1 =0$ and $\lambda_2 =8$ and hence is not positively stable. Its Hermitian part $\mB_H=\diag(9,-1)$ has the minimal eigenvalue $\lambda_{\min}^{\mBH}=-1$. Then, in \eqref{mB:decomp} we have \[
\widetilde \mB
=\mathbf{B} -\lambda_{\min}^{\mBH} \mathbf{I}
=\begin{bmatrix} 10 & -3 \\ 3 & 0 \end{bmatrix} \] which has eigenvalues $1$ and $9$. Therefore, $m_{SHC}(\mathbf{B}) =m_{HC}(\widetilde \mB) =1$.
\end{example}
We have
the following characterization for
accretive matrices to have a (finite) SHC-index.
\begin{corollary} \label{cor:SHC:equivalence} Let $\mathbf{J},\mathbf{R}\in\C^{n\times n}$ satisfy $\mathbf{R}=\mathbf{R}^{\mathsf{H}}$ and $\mathbf{J}=-\mathbf{J}^{\mathsf{H}}$ and let $\lambda_{\min}$ be the minimal eigenvalue of~$\mathbf{R}$. Define $\widetilde \mR :=\mathbf{R}-\lambda_{\min} \mathbf{I}$.
Then the following conditions are equivalent:
\begin{enumerate}[(B1)] \item \label{SHC:KRC} There exists $m\in\mathbb{N}_0$ such that
\begin{equation}\label{condition:KalmanRank:J_R_lambda}
\rank([\mathbf{R},\mathbf{J}{\mathbf{R}},\ldots,\mathbf{J}^m {\mathbf{R}}]-\lambda_{\min}[\mathbf{I},\mathbf{J},\ldots,\mathbf{J}^m])
=n \,. \end{equation}
\item \label{SHC:Tm} There exists $m\in\mathbb{N}_0$ such that
\begin{equation}\label{Tms:J_R_lambda} \sum_{j=0}^m \mathbf{J}^j \mathbf{R} (\mathbf{J}^{\mathsf{H}})^j > \lambda_{\min} \sum_{j=0}^m \mathbf{J}^j (\mathbf{J}^{\mathsf{H}})^j\,. \end{equation}
\item \label{SHC:PBH:EVec}
No eigenvector of~$\mathbf{J}$ is an eigenvector to $\lambda_{\min}$ of~$\mathbf{R}$.
\item \label{SHC:PBH:EVal} $\rank [\lambda \mathbf{I}-\mathbf{J},\mathbf{R}-\lambda_{\min} \mathbf{I}] =n$ for every $\lambda\in\mathbb{C}$, in particular for every eigenvalue $\lambda$ of $\mathbf{J}$.
\end{enumerate}
Moreover, the smallest possible~$m\in\mathbb{N}_0$ in~(B\ref{SHC:KRC}) and~(B\ref{SHC:Tm}) coincide. \end{corollary}
\begin{proof} The Hermitian matrix $\widetilde \mR =\mathbf{R}-\lambda_{\min} \mathbf{I}$ is positive semi-definite. Hence, the statement (which is stated for the original matrix~$\mathbf{R}$ using $\widetilde \mR =\mathbf{R}-\lambda_{\min} \mathbf{I}$) follows from Lemma \ref{lem:HC:equivalence}. \end{proof}
In the following result we show that we can use the SHC-index to characterize the short-time behavior of the propagator norm for general linear time-invariant systems of ODEs.
For this we denote the solution semigroup pertaining to~\eqref{ODE:B} by $S(t):=e^{-\mathbf{B} t}\in \C^{n\times n}$, $t\geq 0$.
\begin{corollary} \label{cor:mB:SHC-decay} Consider an ODE~\eqref{ODE:B} with system matrix~$\mathbf{B}\in\C^{n\times n}$. If $\mathbf{B}$ has a finite SHC-index~$m_{SHC}(\mathbf{B})$, then \begin{equation}\label{mB:short-t-decay}
\|e^{-\mathbf{B} t}\|_2
=
e^{-\lambda_{\min}^{\mBH} t}\
\big( 1 -ct^a +{\mathcal{O}}(t^{a+1})\big)
\quad\text{for } t\to0+\,, \end{equation} where $\lambda_{\min}^{\mBH}$ is the smallest eigenvalue of the Hermitian matrix $\mB_H$, $a=2m_{SHC}(\mathbf{B})+1\ (\geq 3)$, and $c>0$. \end{corollary}
\begin{proof} Write $\mathbf{B}$ as in~\eqref{mB:decomp} and compute the HC-index~$m_{HC}(\widetilde \mB)\ (\geq 1)$ of the accretive matrix~$\widetilde \mB =\mathbf{B} -\lambda_{\min}^{\mBH}\mathbf{I}$. Using the decomposition~\eqref{mB:decomp} yields
\begin{equation}
e^{-\mathbf{B} t}
=
e^{-(\lambda_{\min}^{\mBH} \mathbf{I} +\widetilde \mB) t}
=
e^{-\lambda_{\min}^{\mBH} t}\ e^{-\widetilde \mB t}
\ , \text{ such that }
\|e^{-\mathbf{B} t}\|_2
=
e^{-\lambda_{\min}^{\mBH} t}\ \|e^{-\widetilde \mB t}\|_2 \ . \end{equation}
If an accretive matrix $\widetilde \mB$ is hypocoercive, i.e.\ having a finite HC-index~$m_{HC}(\widetilde \mB)$ (or equivalently~$\mathbf{B}$ has a finite SHC-index~$m_{SHC}(\mathbf{B})$) then~\eqref{mB:short-t-decay} follows from Proposition~\ref{prop:ODE-short}. \end{proof}
In this section we have gathered and extended results about stable, hypocoercive, and semi-dissipative matrices. These results have analoga for discrete-time systems that are studied in the next section.
\section{Stability, semi-contractivity and hypocontractivity for discrete-time systems} \label{sec:Stability+ODEs:discrete-time}
In this section we study the analogous concepts for linear discrete-time systems \begin{equation}\label{DS:A}
x_{k+1} = \mathbf{A}_d x_k \,, \qquad k\in\mathbb{N}_0\,, \end{equation} for a given matrix $\mathbf{A}_d\in\C^{n\times n}$.
\begin{remark}\label{rem:discrete_concepts} While the stability analysis in discrete-time systems is well studied in linear algebra and operator theory \cite{LaS86} using spectral properties and discrete-time Lyapunov equations, we proceed by studying hypo\-con\-trac\-tiv\-i\-ty---the analogon to the concept of hypocoercivity in continuous time---and relating to these classical concepts. \end{remark}
\begin{definition}\label{def:disstab} The trivial solution $x\equiv 0$ of the discrete-time system~\eqref{DS:A} is called \emph{stable} if all solutions of~\eqref{DS:A} are bounded for $k\in\mathbb{N}_0$, and it is called \emph{asymptotically stable} if it is stable and all solutions of~\eqref{DS:A} converge to $0$ for $k\to \infty$. \end{definition}
For linear systems~\eqref{DS:A} a solution is (asymptotically) stable if and only if the trivial solution $x\equiv 0$ is (asymptotically) stable. Therefore, if the trivial solution $x\equiv 0$ of~\eqref{DS:A} is (asymptotically) stable then the linear system~\eqref{DS:A} is called (asymptotically) stable.
\begin{comment} \begin{definition}\label{def:disstab} A solution of the discrete-time system~\eqref{DS:A} is called \emph{stable} if it is bounded for all $k\in\mathbb{N}_0$ and \emph{asymptotically stable} if it is stable and converges to $0$ for $k\to \infty$. If all solutions of~\eqref{DS:A} are (asymptotically) stable for all initial values $x_0$ then we call the system (asymptotically) stable.
\end{definition} \end{comment}
It is well-known that~\eqref{DS:A} is stable if all eigenvalues of~$\mathbf{A}_d$ have modulus less or equal than one and the eigenvalues of modulus one are semi-simple (see~\cite[Theorem 3.3.11]{HiPr10}); and it is \emph{asymptotically stable} if all eigenvalues of~$\mathbf{A}_d$ have modulus strictly less than one. \begin{definition} Let $\mathbf{A}_d\in\C^{n\times n}$ have eigenvalues $\lambda_j$, $j=1,\ldots,n$. The \emph{spectral radius of~$\mathbf{A}_d$} is defined as $\rho(\mathbf{A}_d) :=
\max\{|\lambda_1|\,,\ldots\,,|\lambda_n|\}$, i.e.\ as the largest absolute value of its eigenvalues. \end{definition}
Hence, a discrete-time system~\eqref{DS:A} is \emph{asymptotically stable} if the spectral radius of $\mathbf{A}_d$ is strictly less than one, $\rho(\mathbf{A}_d)<1$.
An alternative characterization of (asymptotic) stability can be given via the \emph{discrete-time Lyapunov (\emph{or} Stein) equation}: System~\eqref{DS:A} is asymptotically stable if and only if, for all positive definite Hermitian matrices $\mathbf{Q}$ \begin{equation}\label{discrete:Lyapunov}
\mathbf{A}_d^{\mathsf{H}} \mathbf{P} \mathbf{A}_d -\mathbf{P} =-\mathbf{Q} \end{equation} has a solution $\mathbf{P}=\mathbf{P}^{\mathsf{H}}>0$, see~\cite[Theorem 3.3.49]{HiPr10} which is formally given by
\begin{equation}\label{discrete:Lyapunov:solution}
\mathbf{P} = \sum_{j=0}^\infty (\mathbf{A}_d^{\mathsf{H}})^j \mathbf{Q} \mathbf{A}_d^j \ , \end{equation} see~\cite[{(89b) in \S3.3.5}]{HiPr10}.
In the discrete-time case the concept of hypocoercivity is replaced by that of hypocontractivity, which we introduce in the next subsection.
\subsection{Hypocontractive matrices and the hypocontractivity index}
For~$\mathbf{A}_d \in\C^{n\times n}$ the spectral norm satisfies
\begin{equation}
\|\mathbf{A}_d\|_2
=
\sqrt{\|\mathbf{A}_d^{\mathsf{H}} \mathbf{A}_d\|_2}
=
\sqrt{\lambda_{\max} \big(\mathbf{A}_d^{\mathsf{H}} \mathbf{A}_d\big)}
=
\sigma_{\max} (\mathbf{A}_d) \ , \end{equation}
where $\lambda_{\max} \big(\mathbf{A}_d^{\mathsf{H}} \mathbf{A}_d\big)$ denotes the largest eigenvalue of the positive semi-definite Hermitian matrix $\mathbf{A}_d^{\mathsf{H}} \mathbf{A}_d$ and $\sigma_{\max} (\mathbf{A}_d)$ is the largest singular value of $\mathbf{A}_d$. Then, the estimate $\|\mathbf{A}_d^n\|_2 \leq \|\mathbf{A}_d\|_2^n$ for $n\in\mathbb{N}$ yields that $\sigma_{\max} (\mathbf{A}_d) \leq 1$ is a sufficient condition for the stability of~\eqref{DS:A}. However, $\sigma_{\max}(\mathbf{A}_d)\leq 1$ is not a necessary condition for~\eqref{DS:A} to be stable. \begin{example} The eigenvalues of \begin{equation} \label{mAd:lambda}
\mathbf{A}_d(\alpha)
=
\alpha
\begin{bmatrix}
1 & -2 \\
0 & -1
\end{bmatrix}\,,
\quad \alpha\in\mathbb{R}\,, \end{equation} are $\pm\alpha$. Hence, the discrete-time system~\eqref{DS:A} with matrix~$\mathbf{A}_d$ in~\eqref{mAd:lambda} is stable if and only if $\alpha\in[-1,1]$.
But the matrix \begin{equation}
\mathbf{A}_d^{\mathsf{H}} \mathbf{A}_d
=
\alpha^2
\begin{bmatrix}
1 &-2 \\
-2 & 5
\end{bmatrix} \end{equation}
has positive eigenvalues $\mu_{\pm} =\alpha^2 (3\pm\sqrt{8})$ and singular values $\sigma_\pm =\sqrt{\mu_\pm}$ with $\sigma_{\max}(\mathbf{A}_d) =\sigma_+$. Thus, $\sigma_{\max}(\mathbf{A}_d)\leq 1$ holds if $|\alpha|\leq (3+\sqrt{8})^{-1/2} \leq 1/2$ which is strictly less than one. Hence in this example, the condition $\sigma_{\max}(\mathbf{A}_d)\leq 1$ is sufficient but not necessary to ensure the stability of~\eqref{DS:A}. \end{example}
In the following we will need a result relating singular values and eigenvalues.
\begin{proposition} \label{prop:EV+SV}
Let $\mathbf{A}_d\in\C^{n\times n}$ have singular values $\sigma_1 \geq \ldots \geq \sigma_n \geq 0$ (such that $\sigma_{\max}(\mathbf{A}_d)=\sigma_1$) and eigenvalues $\lambda_j$, $j=1,\ldots,n$ being ordered as $|\lambda_1| \geq \ldots \geq |\lambda_n|$. Then, $|\lambda_1|\leq \sigma_1$. Moreover, if $\mathbf{A}_d$ is nonsingular, then $|\lambda_n|\geq \sigma_n >0$. \end{proposition} \begin{proof} The statements follow from the bounds in~\cite[Theorem 5.6.9]{HoJo13}.
\end{proof}
We then have the following discrete-time analogon of semi-dissipativity.
\begin{definition}[{\cite[Definition 4.1.2]{Be18}}] Let $\mathbf{A}_d\in\C^{n\times n}$ and let $\sigma_{\max}(\mathbf{A}_d)$ be the largest singular value (the \emph{spectral norm}) of $\mathbf{A}_d$.
We call $\mathbf{A}_d$ \emph{contractive} if $\sigma_{\max}(\mathbf{A}_d) <1$; and we call $\mathbf{A}_d$ \emph{semi-contractive} if $\sigma_{\max}(\mathbf{A}) \leq 1$. \end{definition}
Note that sometimes $\mathbf{A}_d$ is called \emph{contractive} if $\sigma_{\max}(\mathbf{A}_d)\leq 1$; and $\mathbf{A}_d$ is called \emph{strictly contractive} if $\sigma_{\max}(\mathbf{A}_d)<1$, see e.g.~\cite[p. 493]{HoJo13}. Other related notions are (semi-)convergent matrices, and power-bounded matrices, see~\cite[p. 180]{HoJo13}.
In the following, we consider the class of semi-contractive matrices~$\mathbf{A}_d$ and present a characterization when~\eqref{DS:A} is (asymptotically) stable.
For this we need a concept that corresponds to hypocoercivity in the continuous-time case.
\begin{definition} \label{def:HypoContractive} A matrix $\mathbf{A}_d\in\C^{n\times n}$ is called \emph{hypocontractive} if all eigenvalues of $\mathbf{A}_d$ have modulus strictly less than one. \end{definition}
Consequently, a discrete-time system~\eqref{DS:A} is asymptotically stable if and only if the system matrix~$\mathbf{A}_d$ is hypocontractive. We can also characterize those semi-contractive matrices~$\mathbf{A}$ which are actually hypocontractive:
\begin{proposition}\label{prop:Semi+HypoContractive} Let $\mathbf{A}_d\in\C^{n\times n}$ be semi-contractive. Then, $\mathbf{A}_d$ has an eigenvalue of modulus one if and only if some eigenvector~$v$ of $\mathbf{A}_d$ satisfies~$v\in\kernel(\mathbf{I}-\mathbf{A}_d^{\mathsf{H}}\mathbf{A}_d)$. \end{proposition}
\begin{proof}
Since $\mathbf{A}_d$ is semi-contractive, the Hermitian matrix~$\mathbf{I} -\mathbf{A}_d^{\mathsf{H}}\mathbf{A}_d$ is positive semi-definite. Moreover, if~$\mathbf{A}_d$ has an eigenvalue~$\lambda$ of modulus $|\lambda|=1$ with eigenvector~$v\ne 0$, then \[
0
\leq \ip{v}{(\mathbf{I}-\mathbf{A}_d^{\mathsf{H}}\mathbf{A}_d)v}
=\|v\|^2 -\|\mathbf{A}_d v\|^2
=\|v\|^2 (1-|\lambda|^2)
=0 \,. \] Therefore, $v$ is in the kernel of the positive semi-definite Hermitian matrix~$\mathbf{I}-\mathbf{A}_d^{\mathsf{H}}\mathbf{A}_d$.
Conversely, if some eigenvector~$v$ of~$\mathbf{A}_d$ (associated to an eigenvalue~$\lambda$) satisfies $v\in\kernel(\mathbf{I}-\mathbf{A}_d^{\mathsf{H}}\mathbf{A}_d)$, then \[
0
=\ip{v}{(\mathbf{I}-\mathbf{A}_d^{\mathsf{H}}\mathbf{A}_d)v}
=\|v\|^2 -\|\mathbf{A}_d v\|^2
=\|v\|^2 (1-|\lambda|^2) \,, \] and hence, the eigenvalue~$\lambda$ has modulus one. \end{proof}
\begin{remark}\label{rem:defect} In the operator theory setting the matrix $(\mathbf{I}-\mathbf{A}_d^{\mathsf{H}}\mathbf{A}_d)^{1/2}$ is often called the \emph{defect operator} of the semi-contractive $\mathbf{A}_d$ and the closure of its image is the \emph{defect space} with its dimension being called the \emph{defect index} $d(\mathbf{A}_d)$. The defect operator and its index are a measure for the distance of an operator from being unitary. See e.g. \cite{NagFK10}. \end{remark}
We again have an equivalent characterization in terms of properties from control theory:
\begin{lemma} \label{lem:mDHC:Equivalence} Let $\mathbf{A}_d\in\C^{n\times n}$ be semi-contractive.
Then the following conditions are equivalent:
\begin{itemize} \item [(D1)] There exists $m\in\mathbb{N}_0$ such that
\begin{equation}\label{condition:KalmanRank:D}
\rank[(\mathbf{I} -\mathbf{A}_d^{\mathsf{H}}\mathbf{A}_d),\mathbf{A}_d^{\mathsf{H}} (\mathbf{I} -\mathbf{A}_d^{\mathsf{H}}\mathbf{A}_d),\ldots,(\mathbf{A}_d^{\mathsf{H}})^m (\mathbf{I} -\mathbf{A}_d^{\mathsf{H}}\mathbf{A}_d)]=n \,. \end{equation}
\item [(D2)] There exists $m\in\mathbb{N}_0$ such that
\begin{equation}\label{Dm:A}
\mathbf{D}_m
:= \sum_{j=0}^m (\mathbf{A}_d^{\mathsf{H}})^j (\mathbf{I} -\mathbf{A}_d^{\mathsf{H}}\mathbf{A}_d) \mathbf{A}_d^j
>0 \,. \end{equation}
\item [(D3)] No eigenvector of~$\mathbf{A}_d$ lies in the kernel of~$(\mathbf{I} -\mathbf{A}_d^{\mathsf{H}}\mathbf{A}_d)$.
\item [(D4)] $\rank [\lambda \mathbf{I}-\mathbf{A}_d^{\mathsf{H}}, \mathbf{I} -\mathbf{A}_d^{\mathsf{H}}\mathbf{A}_d] =n$ for every $\lambda \in \mathbb{C}$, in particular for every eigenvalue~$\lambda$ of~$\mathbf{A}_d^{\mathsf{H}}$. \end{itemize}
Moreover, the smallest possible~$m\in\mathbb{N}_0$ in~(D1) and~(D2) coincide. \end{lemma}
\begin{proof} Like Lemma \ref{lem:HC:equivalence}, this result follows from Theorem~6.2.1 of~\cite{Da04} and Lemma \ref{lem:Definiteness} in the Appendix. \end{proof}
\begin{remark}\label{rem:unobservability} In control theory, conditions (D1), (D3), and (D4) in Lemma~\ref{lem:mDHC:Equivalence} are equivalent characterizations of the \emph{controllability} of the pair $(\mathbf{A}_d^{\mathsf{H}}, \mathbf{I} -\mathbf{A}_d^{\mathsf{H}}\mathbf{A}_d)$, or the dynamical system
\[ x_{k+1}= \mathbf{A}_d^{\mathsf{H}} x_k+ (\mathbf{I} -\mathbf{A}_d^{\mathsf{H}}\mathbf{A}_d)u_k. \]
There is always also the dual concept of \emph{observability} which in this case would correspond to the controllability of $(\mathbf{A}_d, \mathbf{I} -\mathbf{A}_d\mathbf{A}_d^{\mathsf{H}})$. A dual result to Lemma~\ref{lem:mDHC:Equivalence} can then be formulated with this pair. Based on this pair, in \cite{Sta05} a similar result has been derived (in different terminology). A similar result for the continuous-time case follows from \cite{Sta03}. \end{remark}
If we compare Lemma~\ref{lem:mDHC:Equivalence} with Lemma~\ref{lem:HC:equivalence}, then we need to substitute $\mB_S$ with $\mathbf{A}_d^{\mathsf{H}}$, and $\mB_H$ with $\mathbf{I}-\mathbf{A}_d^{\mathsf{H}}\mathbf{A}_d$, respectively.
Using Lemma~\ref{lem:mDHC:Equivalence} (D2), we then define the hypocontractivity index.
\begin{definition} For semi-contractive matrices $\mathbf{A}_d\in\C^{n\times n}$, we define the \emph{hypocontractivity index} or \emph{discrete HC-index (dHC-index)~$m_{dHC}$} as the smallest integer $m\in\mathbb{N}_0$ (if it exists) such that~\eqref{Dm:A} holds.
\end{definition}
\begin{remark}\label{rem:alternative notions} The hypocontractivity index is sometimes also called the \emph{norm-one index}, see \cite{GauW10}, where it is shown that this index is finite if and only if the spectral radius of $\mathbf{A}_d$ is strictly smaller than one. \end{remark}
Clearly, a semi-contractive matrix $\mathbf{A}_d$ is contractive if and only if $m_{dHC}=0$. Since \eqref{Dm:A} is a telescopic sum, we have that $\mathbf{D}_m =\mathbf{I} -(\mathbf{A}_d^{\mathsf{H}})^{m+1} \mathbf{A}_d^{m+1}$ and thus if a semi-contractive matrix $\mathbf{A}_d\in\C^{n\times n}$ is hypocontractive with index~$m_{dHC}\in\mathbb{N}_0$, then $\mathbf{A}_d^{m_{dHC}+1}$ is contractive. Conversely, if a semi-contractive matrix $\mathbf{A}_d\in\C^{n\times n}$ satisfies that $\mathbf{A}_d^m$ is contractive for some $m\in\mathbb{N}$, then $\mathbf{A}_d$ is hypocontractive with index $m_{dHC}\le m-1$.\\
The following result may be considered as a discrete counterpart of the short-time decay behavior from Proposition \ref{prop:ODE-short}.
\begin{theorem}\label{DS:A:semi-contractive:power-bounds}
Let $\mathbf{A}_d\in\C^{n\times n}$ be semi-contractive and hypocontractive. Its (finite) hypocontractivity index is $m_{dHC}\in\mathbb{N}_0$ if and only if \begin{equation} \label{DS:A:decay}
\|\mathbf{A}_d^j\|_2 =1 \text{ for all } j=1,\ldots,m_{dHC}\,,
\quad\text{and }\
\|\mathbf{A}_d^{m_{dHC}+1}\|_2 <1 \,. \end{equation} \end{theorem} \begin{proof}
The spectral norm $\|\mathbf{C}\|_2$ of a matrix $\mathbf{C}\in\C^{n\times n}$, i.e.\ the operator norm induced by the Euclidean norm on~$\C^n$ is given by
$\|\mathbf{C}\|_2 =\max_{w\in\C^n:\ \|w\|=1} \|\mathbf{C} w\|_2$. If a matrix $\mathbf{A}_d$ is semi-contractive, then the estimates $\|\mathbf{A}_d\|_2\leq 1$ and $\|\mathbf{A}_d^j\|_2 \leq\|\mathbf{A}_d\|_2^j\leq 1$ hold for all $ j\in\mathbb{N}$. Thus, for vectors $w\in\C^n$ with $\|w\|_2=1$, we have \[
\ip{w}{(\mathbf{A}_d^{\mathsf{H}})^j \mathbf{A}_d^j w} =\ip{\mathbf{A}_d^j w}{\mathbf{A}_d^j w} =\|\mathbf{A}_d^j w\|_2^2 \leq
1 =\ip{w}{w} \,, \] such that $0\leq \ip{w}{(\mathbf{I} -(\mathbf{A}_d^{\mathsf{H}})^j \mathbf{A}_d^j) w}$. Therefore, for all $m\in\mathbb{N}_0$, \[
\mathbf{D}_m
=\sum_{j=0}^m (\mathbf{A}_d^{\mathsf{H}})^j (\mathbf{I} -\mathbf{A}_d^{\mathsf{H}}\mathbf{A}_d) \mathbf{A}_d^j
=\mathbf{I} -(\mathbf{A}_d^{\mathsf{H}})^{m+1} \mathbf{A}_d^{m+1}
\geq 0 \, \] and hence, the semi-contractive matrix $\mathbf{A}_d$ has (finite) hypocontractivity index $m_{dHC}$ if and only if~\eqref{DS:A:decay} holds. \end{proof}
We summarize the relationship between the different concepts discussed in this section in Figure~\ref{fig:VennDiagramA}.
\begin{figure}
\caption{Relationship between sets of matrices $\mathbf{A}_d\in\C^{n\times n}$ which are (hypo)contractive (circular discs), semi-contractive (region within smaller ellipse) and those for which the discrete-time system $x_{k+1}=\mathbf{A}_d x_k$ is stable (region within bigger ellipse), respectively.}
\label{fig:VennDiagramA}
\end{figure}
\subsection{Polar decomposition}\label{ssec:DS:PD}
In \cite{AAM21} a computationally feasible procedure has been presented to check the conditions of Lemma~\ref{lem:HC:equivalence} in the continuous-time case via a staircase form under unitary congruence transformations. A similar procedure can be derived in the discrete-time case. It is based on polar decomposition, see e.g.~\cite[Theorem 7.3.1]{HoJo13}, which is the discrete-time analogon of the additive splitting of a matrix into its Hermitian and skew-Hermitian part:
\begin{proposition}[Polar decomposition]\label{prop:PolDec} Let $\mathbf{A}_d\in\C^{n\times n}$. \begin{itemize} \item [(a)] There exist positive semi-definite Hermitian matrices $\mathbf{P}_d,\mathbf{Q}_d\in\C^{n\times n}$ and a unitary matrix~$\mathbf{U}_d\in\C^{n\times n}$ such that \begin{equation}\label{PD:A}
\mathbf{A}_d
=\mathbf{P}_d \mathbf{U}_d
=\mathbf{U}_d \mathbf{Q}_d \ . \end{equation} The factors $\mathbf{P}_d$, $\mathbf{Q}_d$ are uniquely determined as $\mathbf{P}_d =(\mathbf{A}_d \mathbf{A}_d^{\mathsf{H}})^{1/2}$ and $\mathbf{Q}_d =(\mathbf{A}_d^{\mathsf{H}} \mathbf{A}_d)^{1/2}$. If $\mathbf{A}_d$ is nonsingular, then $\mathbf{U}_d =\mathbf{P}_d^{-1}\mathbf{A}_d =\mathbf{A}_d\mathbf{Q}_d^{-1}$ is uniquely determined (as well).
\item [(b)] If $\mathbf{A}_d$ is real, then the factors $\mathbf{P}_d$, $\mathbf{Q}_d$ and $\mathbf{U}_d$ may be taken to be real. \end{itemize} \end{proposition}
Consider a stable discrete-time system~\eqref{DS:A} with matrix~$\mathbf{A}_d$. Hence, all eigenvalues of matrix~$\mathbf{A}_d$ have modulus less or equal than one. Then, the polar decomposition~\eqref{PD:A} yields that the (largest) singular values of $\mathbf{A}_d$, $\mathbf{P}_d$ and $\mathbf{Q}_d$ are the same, since $\mathbf{A}_d \mathbf{A}_d^{\mathsf{H}} =\mathbf{P}_d \mathbf{P}_d^{\mathsf{H}}$ and $\mathbf{A}_d^{\mathsf{H}} \mathbf{A}_d =\mathbf{Q}_d^{\mathsf{H}} \mathbf{Q}_d$.
An immediate consequence is that a matrix
$\mathbf{A}_d\in\C^{n\times n}$ with polar decomposition~\eqref{PD:A}
is semi-contractive if and only if the spectra of $\mathbf{P}_d$ and $\mathbf{Q}_d$ (which coincide) are contained in $[0,1]$.
We can rephrase the statement of Proposition~\ref{prop:Semi+HypoContractive} as follows:
\begin{proposition}\label{prop:Semi+HypoContractive:UQ} Let $\mathbf{A}_d\in\C^{n\times n}$ be semi-contractive with polar decomposition $\mathbf{A}_d =\mathbf{U}_d\mathbf{Q}_d$ and $\mathbf{Q}_d =(\mathbf{A}_d^{\mathsf{H}} \mathbf{A}_d)^{1/2}$. Then, $\mathbf{A}_d$ has an eigenvalue of modulus one (and hence $\mathbf{A}_d$ is not hypocontractive)
if and only if some eigenvector~$v$ of $\mathbf{U}_d$ satisfies~$v\in\kernel(\mathbf{I}-\mathbf{Q}_d)$. \end{proposition} \begin{proof}
For the forward direction we assume that the eigenvalue equation $\mathbf{A}_d v=\lambda v$ holds for some $\lambda$ with $|\lambda|=1$ and $v\in\C^n\setminus\{0\}$. Then, Proposition~\ref{prop:Semi+HypoContractive} implies that $v\in\ker(\mathbf{I}-\mathbf{A}_d^{\mathsf{H}}\mathbf{A}_d)$, i.e.\ $0=(\mathbf{I}-\mathbf{A}_d^{\mathsf{H}}\mathbf{A}_d)v = (\mathbf{I}+\mathbf{Q}_d)(\mathbf{I}-\mathbf{Q}_d)v$ which holds if and only if $0=(\mathbf{I}-\mathbf{Q}_d)v$, such that $0=\mathbf{U}_d (\mathbf{I}-\mathbf{Q}_d)v=\mathbf{U}_d v-\lambda v$. Hence, $v\in\kernel(\mathbf{I}-\mathbf{Q}_d)$ is an eigenvector of $\mathbf{U}_d$.
Conversely, let $w$ be an eigenvector of $\mathbf{U}_d$, i.e.\ $ \mathbf{U}_d w=\lambda w$ with $|\lambda|=1$, that satisfies $(\mathbf{I}-\mathbf{Q}_d)w=0$. Then $0=\mathbf{U}_d (\mathbf{I}-\mathbf{Q}_d)w=\lambda w -\mathbf{A}_d w$. \end{proof}
Note that, for semi-contractive matrices~$\mathbf{A}_d$, eigenvalues with modulus one are necessarily semi-simple. Therefore, a semi-contractive matrix~$\mathbf{A}_d$ (with polar decomposition $\mathbf{A}_d=\mathbf{U}_d\mathbf{Q}_d$) is hypocontractive if and only if no eigenvector of~$\mathbf{A}_d$ lies in the kernel of the positive semi-definite Hermitian matrix~$\mathbf{I} -\mathbf{Q}_d$.
Using this relationship, we formulate an analogous result to Lemma~\ref{lem:mDHC:Equivalence}, in terms of matrices appearing in polar decompositions. It follows again from Theorem 6.2.1 of \cite{Da04} and Lemma \ref{lem:Definiteness}:
\begin{lemma} \label{lem:mDHC:Equivalence:HU} Let $\mathbf{A}_d\in\C^{n\times n}$ be semi-contractive with polar decomposition $\mathbf{A}_d =\mathbf{U}_d\mathbf{Q}_d$ (i.e.\ with $\mathbf{U}_d$ unitary, $\mathbf{Q}_d$ semi-contractive Hermitian, and $\mathbf{Q}_d^2=\mathbf{A}_d^{\mathsf{H}}\mathbf{A}_d$). Then the following conditions are equivalent:
\begin{itemize} \item [(D1')] There exists $m\in\mathbb{N}_0$ such that
\begin{equation}\label{condition:KalmanRank:HU}
\rank[(\mathbf{I}-\mathbf{Q}_d^2),\mathbf{U}_d^{\mathsf{H}}(\mathbf{I}-\mathbf{Q}_d^2),\ldots,(\mathbf{U}_d^{\mathsf{H}})^m (\mathbf{I}-\mathbf{Q}_d^2)]=n \,. \end{equation}
\item [(D2')] There exists $m\in\mathbb{N}_0$ such that
\begin{equation}\label{Dm:HU}
\widehat \mD_m :=\sum_{j=0}^m (\mathbf{U}_d^{\mathsf{H}})^j (\mathbf{I}-\mathbf{Q}_d^2) \mathbf{U}_d^j > 0 \,. \end{equation}
\item [(D3')] No eigenvector of~$\mathbf{U}_d$ lies in the kernel of $\mathbf{I}-\mathbf{Q}_d^2$. \item [(D4')] $\rank [\lambda \mathbf{I}-\mathbf{U}_d^{\mathsf{H}}, \mathbf{I}-\mathbf{Q}_d^2] =n$ for every $\lambda \in \mathbb{C}$, in particular for every eigenvalue~$\lambda$ of~$\mathbf{U}_d^{\mathsf{H}}$. \end{itemize}
Moreover, the smallest possible~$m\in\mathbb{N}_0$ in (D1') and (D2')
coincide. \end{lemma}
Note that (D3) and (D3') are equivalent, due to Proposition \ref{prop:Semi+HypoContractive:UQ} and since $\ker(\mathbf{I}-\mathbf{Q}_d)=\ker(\mathbf{I}-\mathbf{Q}_d^2)$. Consequently, all conditions of the Lemmata \ref{lem:mDHC:Equivalence} and \ref{lem:mDHC:Equivalence:HU} are equivalent and the smallest possible values of $m$ coincide.
\subsection{Scaled hypocontractivity index} \label{subsec:SDHC}
The analogon to the shifted hypocoercivity index is obtained by scaling.
\begin{lemma} \label{lem:mA:scale} Let $\mathbf{A}_d\in\C^{n\times n}$ be a nonzero matrix, and let $\sigma_{\max}(\mathbf{A}_d)$ be the maximal singular value of~$\mathbf{A}_d$. Then, the matrix
\begin{equation} \label{mA:scale}
\widetilde \mA_d := (\sigma_{\max}(\mathbf{A}_d))^{-1} \mathbf{A}_d \end{equation}
is semi-contractive and, if $\widetilde \mA_d$ is hypocontractive, has a discrete HC-index $m_{dHC}(\widetilde \mA_d)$ greater than~$0$.
Furthermore, $\widetilde \mA_d$ is hypocontractive if and only if the matrices in the polar decomposition of~$\mathbf{A}_d =\mathbf{U}_d\mathbf{Q}_d$ satisfy that no eigenvector of~$\mathbf{Q}_d =(\mathbf{A}_d^{\mathsf{H}}\mathbf{A}_d)^{1/2}$ associated with the eigenvalue $\sigmaMax{\mathbf{A}_d}$ is an eigenvector of $\mathbf{U}_d$. \end{lemma} \begin{proof}
Consider the matrix $\widetilde \mA_d(\sigma) := \sigma^{-1} \mathbf{A}_d$ for $\sigma>0$. Then $\sigma=\sigma_{\max}(\mathbf{A}_d)$ is the only value such that the largest singular value of $\widetilde \mA_d(\sigma)$ is one, since \[
\sigmaMax{\widetilde \mA_d}
=\sqrt{\lambdaMax{}(\widetilde \mA_d^{\mathsf{H}} \widetilde \mA_d)}
=\sqrt{\lambdaMax{}(\mathbf{A}_d^{\mathsf{H}} \mathbf{A}_d)}/\sigmaMax{\mathbf{A}_d}
=1 \,. \] Consequently, if the scaled matrix $\widetilde \mA_d$ is hypocontractive then its discrete HC-index~$m_{dHC}(\widetilde \mA_d)$ is greater than $0$.
To prove the final statement we consider the polar decomposition of~$\mathbf{A}_d$ in the form $\mathbf{A}_d =\mathbf{U}_d\mathbf{Q}_d$. Then, $\widetilde \mA_d =(\sigma_{\max}(\mathbf{A}_d))^{-1}\mathbf{A}_d$ has the polar decomposition $\widetilde \mA_d =\mathbf{U}_d\widetilde \mQ_d$ with the same unitary matrix $\mathbf{U}_d$, and $\widetilde \mQ_d := \linebreak (\sigma_{\max}(\mathbf{A}_d))^{-1}\mathbf{Q}_d$. Due to Proposition~\ref{prop:Semi+HypoContractive:UQ}, $\widetilde \mA_d$ is hypocontractive if and only if no eigenvector~$v$ of $\mathbf{U}_d$ is in the kernel of $\mathbf{I} -\widetilde \mQ_d$. The latter is equivalent to $v$ being an eigenvector of $\widetilde \mQ_d$ to the eigenvalue one, or $v$ being an eigenvector of $\mathbf{Q}_d$ to the eigenvalue~$\sigmaMax{\mathbf{A}_d}$.
\end{proof}
\begin{definition}\label{def-SdHC-index} Consider a nonzero matrix~$\mathbf{A}_d\in\C^{n\times n}$, and let $\sigma_{\max}(\mathbf{A}_d)$ be the maximal (positive) singular value of~$\mathbf{A}_d$. If the semi-contractive matrix $\widetilde \mA_d := (\sigma_{\max}(\mathbf{A}_d))^{-1} \mathbf{A}_d$ is hypocontractive with discrete HC-index~$m_{dHC}(\widetilde \mA_d)$ then we define the \emph{scaled hypocontractivity index} or \emph{discrete SHC-index (dSHC-index)~$m_{dSHC}$} of~$\mathbf{A}_d$ as $m_{dSHC}(\mathbf{A}_d):=m_{dHC}(\widetilde \mA_d)$. \end{definition}
In analogy to Theorem \ref{DS:A:semi-contractive:power-bounds} we then have the following characterization when $\widetilde \mA_d$ has a finite scaled hypocontractivity index.
\begin{theorem}\label{DS:A:power-bounds}
Let $\mathbf{A}_d\in\C^{n\times n}$ be nonzero, and let $\sigma_{\max}(\mathbf{A}_d)$ be the maximal (positive) singular value of~$\mathbf{A}_d$. If $\mathbf{A}_d$ has a finite discrete SHC-index~$m_{dSHC}$, then \begin{equation} \label{DS:A:decay:general}
\|\mathbf{A}_d^j\|_2 =(\sigma_{\max}(\mathbf{A}_d))^j \quad\text{ for all } j=1,\ldots,m_{dSHC} \ ,
\ \text{and }
\|\mathbf{A}_d^{m_{dSHC}+1}\|_2 <(\sigma_{\max}(\mathbf{A}_d))^{m_{dSHC}+1} \ . \end{equation} \end{theorem}
\begin{proof} We scale $\mathbf{A}_d$ as in~\eqref{mA:scale} and compute the discrete HC-index~$m_{dHC}(\widetilde \mA_d)\ (\geq 1)$ of the semi-contractive matrix~$\widetilde \mA_d =(\sigma_{\max}(\mathbf{A}_d))^{-1} \mathbf{A}_d$ so that $m_{dSHC}(\mathbf{A}_d) :=m_{dHC}(\widetilde \mA_d)$. Using the scaling~\eqref{mA:scale} yields
\begin{equation}
\|\mathbf{A}_d^j\|_2
=\big\|\big(\sigma_{\max}(\mathbf{A}_d) \widetilde \mA_d\big)^j\big\|_2
=\big(\sigma_{\max}(\mathbf{A}_d)\big)^j \|\widetilde \mA_d^j\|_2
\ \text{for all $j\in\mathbb{N}$.} \end{equation}
If the semi-contractive matrix $\widetilde \mA_d$ has a (finite) discrete HC-index~$m_{dHC}(\widetilde \mA_d)$ (or equivalently the discrete SHC-index~$m_{dSHC}(\mathbf{A}_d)$ of~$\mathbf{A}_d$ is finite) then~\eqref{DS:A:decay:general} follows from Theorem~\ref{DS:A:semi-contractive:power-bounds}. \end{proof}
We summarize the analogy between discrete-time and continuous-time systems in Table~\ref{propinvariance}.
\begin{table} \begin{tabular}{lll} properties & continuous-time system & discrete-time system\\[1mm] \hline\hline\\[-3mm] evolution & $x' =\mathbf{A}_c x$ for $t\geq 0$ & $x_{k+1} =\mathbf{A}_d x_k $ for $k\in\mathbb{N}_0$ \\
\hline condition for & $\Re(\lambda)<0$ for all $\lambda\in\Lambda(\mathbf{A}_c)$, & $|\lambda|<1$ for all $\lambda\in\Lambda(\mathbf{A}_d)$, \\ asymptotic stability & i.e.\ negative hypocoercive & i.e.\ hypocontractive\\ \hline matrix decomposition & $\mathbf{A}_c =\mA_S+\mA_H$ & polar: $\mathbf{A}_d=\mathbf{P}_d\mathbf{U}_d=\mathbf{U}_d\mathbf{Q}_d$ \\ \hline sufficient stability & $\mA_H\leq 0$, & $\sigma_{\max}(\mathbf{A}_d) \leq 1$ $\Leftrightarrow$ $\Lambda(\mathbf{Q}_d)\subset[0,1]$,\\ condition & i.e.\ semi-dissipative& i.e.\ semi-contractive\\ \hline Kalman rank condition & $\rank[{\mA_H}, \ldots,((\mA_S)^{\mathsf{H}})^m {\mA_H}]$ & $\rank[(\mathbf{I}-\mathbf{Q}_d^2), \ldots,(\mathbf{U}_d^{\mathsf{H}})^m (\mathbf{I}-\mathbf{Q}_d^2)]$\\ & $=n$ & $=n$\\ \hline HC-condition & $\displaystyle\sum_{j=0}^m ((\mA_S)^{\mathsf{H}})^j (-\mA_H) \mA_S^j > 0$ & $\displaystyle\sum_{j=0}^m (\mathbf{U}_d^{\mathsf{H}})^j (\mathbf{I}-\mathbf{Q}_d^2) \mathbf{U}_d^j > 0$\\ \hline eigenvector condition & no EV of~$\mA_S$ in $\ker(\mA_H)$ & no EV of~$\mathbf{U}_d$ in $\ker(\mathbf{I}-\mathbf{Q}_d^2)$\\ \hline\\ \end{tabular} \caption{Relation between concepts for continuous-time and discrete-time systems, see also Figures~\ref{fig:VennDiagram} and~\ref{fig:VennDiagramA}. $\Lambda(\mathbf{A})$ denotes here the spectrum of a matrix $\mathbf{A}\in\C^{n\times n}$. \label{propinvariance}} \end{table}
In this section we have given characterizations for the concepts of stability, semi-contractivity, and hypocontractivity for linear discrete-time systems. In the next section we relate the continuous-time and discrete-time concepts.
\section{Transformation between discrete-time and continuous-time systems}\label{sec:dtct}
We have seen the close analogy between the results for the continuous-time and discrete-time case. In this section we recall that the typical bilinear transformations between continuous-time and discrete-time systems such as the Cayley transformation (in fact of $-\mathbf{A}_c$) relate hypocoercive with hypocontractive systems (see e.g.~\cite{HiPr10}), and semi-dissipative with semi-contractive systems (see e.g.~\cite{NagFK10}). Moreover, we show that the Cayley transformation (of $-\mathbf{A}_c$) directly relates the hypocoercivity and hypocontractivity indices.
\begin{lemma} \label{lemma:Cayley} Let $\mathbf{A}_c\in\C^{n\times n}$ be a matrix such that~\eqref{ODE:B} is (Lyapunov) stable. Then, the Cayley transform
\begin{equation}\label{cayley}
\mathbf{A}_d:=(\mathbf{I}+\mathbf{A}_c)(\mathbf{I}-\mathbf{A}_c)^{-1} \end{equation}
is well-defined and the following properties hold: \begin{itemize}
\item [(i)] If $\mathbf{A}_c$ is negative hypocoercive then $\mathbf{A}_d$ is hypocontractive.
\item [(ii)] If $\mathbf{A}_c$ is semi-dissipative then $\mathbf{A}_d$ is semi-contractive. Let $\mA_H:= \frac12 (\mathbf{A}_c+\mathbf{A}_c^{\mathsf{H}})$, then the matrix~$(\mathbf{I} -\mathbf{A}_c)$ is a bijection from~$\ker(\mA_H)$ to $\ker(\mathbf{I} -\mathbf{A}_d^{\mathsf{H}} \mathbf{A}_d)$. Consequently, $\dim\ker(\mA_H) =\dim\ker(\mathbf{I} -\mathbf{A}_d^{\mathsf{H}} \mathbf{A}_d)$.
\end{itemize} \end{lemma}
\begin{proof} If the continuous-time system~\eqref{ODE:B} with system matrix~$\mathbf{A}_c$ is \emph{(Lyapunov) stable} then all eigenvalues of~$\mathbf{A}_c$ have non-positive real part and the eigenvalues on the imaginary axis are semi-simple. Hence, the matrices~$(\mathbf{I}-\mathbf{A}_c)$, $(\mathbf{I}-\mathbf{A}_c^{\mathsf{H}})$ are invertible; and the Cayley transform~$\mathbf{A}_d=(\mathbf{I}+\mathbf{A}_c)(\mathbf{I}-\mathbf{A}_c)^{-1}$ is well-defined.
(i) If~$\mathbf{A}_c\in\C^{n\times n}$ is negative hypocoercive, then all eigenvalues of~$\mathbf{A}_d$ have absolute value less than one, hence, $\mathbf{A}_d$ is hypocontractive.
(ii) If $\mathbf{A}_c\in\C^{n\times n}$ is semi-dissipative, then $\mathbf{I}-\mathbf{A}_c$ is positive dissipative (hence $\mathbf{I}-\mathbf{A}_c$ is invertible). It follows that
\begin{equation} \label{I-AhA} \begin{split}
\mathbf{I} -\mathbf{A}_d^{\mathsf{H}} \mathbf{A}_d &=\mathbf{I} -(\mathbf{I}-\mathbf{A}_c)^{-{\mathsf{H}}}(\mathbf{I}+\mathbf{A}_c)^{\mathsf{H}} (\mathbf{I}+\mathbf{A}_c)(\mathbf{I}-\mathbf{A}_c)^{-1} \\ &=(\mathbf{I}-\mathbf{A}_c)^{-{\mathsf{H}}} \Big((\mathbf{I}-\mathbf{A}_c)^{{\mathsf{H}}}(\mathbf{I}-\mathbf{A}_c) -(\mathbf{I}+\mathbf{A}_c)^{\mathsf{H}} (\mathbf{I}+\mathbf{A}_c) \Big) (\mathbf{I}-\mathbf{A}_c)^{-1} \\ &=-2 (\mathbf{I}-\mathbf{A}_c)^{-{\mathsf{H}}} (\mathbf{A}_c^{{\mathsf{H}}} +\mathbf{A}_c) (\mathbf{I}-\mathbf{A}_c)^{-1} \\ &= -4 (\mathbf{I}-\mathbf{A}_c)^{-{\mathsf{H}}} \mA_H (\mathbf{I}-\mathbf{A}_c)^{-1} \,. \end{split} \end{equation}
Hence, the matrices~$-\mA_H$ and~$(\mathbf{I} -\mathbf{A}_d^{\mathsf{H}} \mathbf{A}_d)$ are related via a congruence transformation. Therefore, $\mathbf{A}_d$ is semi-contractive (or equivalently, $(\mathbf{I} -\mathbf{A}_d^{\mathsf{H}} \mathbf{A}_d)$ is positive semi-definite) if $\mathbf{A}_c$ is semi-dissipative.
Due to~\eqref{I-AhA}, if $v\in\ker(\mA_H)$ then $(\mathbf{I} -\mathbf{A}_c) v\in\ker(\mathbf{I} -\mathbf{A}_d^{\mathsf{H}} \mathbf{A}_d)$. Thus, $(\mathbf{I} -\mathbf{A}_c)\ker(\mA_H) \subseteq\ker(\mathbf{I} -\mathbf{A}_d^{\mathsf{H}} \mathbf{A}_d)$.
Conversely, if $w\in\ker(\mathbf{I} -\mathbf{A}_d^{\mathsf{H}} \mathbf{A}_d)$ then $(\mathbf{I} -\mathbf{A}_c)^{-1} w\in\ker(\mA_H)$. Thus, $(\mathbf{I} -\mathbf{A}_c)^{-1} \ker(\mathbf{I} -\mathbf{A}_d^{\mathsf{H}} \mathbf{A}_d)\subseteq \ker(\mA_H)$.
Altogether, $(\mathbf{I} -\mathbf{A}_c)$ is a bijection from $\ker(\mA_H)$ to $\ker(\mathbf{I} -\mathbf{A}_d^{\mathsf{H}} \mathbf{A}_d)$, and~$\dim\ker(\mA_H) =\dim\ker(\mathbf{I} -\mathbf{A}_d^{\mathsf{H}} \mathbf{A}_d)$. \end{proof}
\begin{remark} As a consequence of Lemma \ref{lemma:Cayley}(ii) we have that $\rank(\mA_H)=\rank(\mathbf{I}-\mathbf{A}_d^{\mathsf{H}} \mathbf{A}_d) = :d(\mathbf{A}_d)$, the defect index of $\mathbf{A}_d$, see Remark \ref{rem:defect}. As a follow-up consequence (using also Theorem \ref{thm:Cayley} below) we find that the lower bound on the hypocontractivity index of $\mathbf{A}_d$ from \cite{GauW10}, i.e.\ $m_{dHC}(\mathbf{A}_d)\ge \frac{n-d(\mathbf{A}_d)}{d(\mathbf{A}_d)}$ equals our lower bound on the hypocoercivity index of $\mathbf{A}_c$, i.e.\ $m_{HC}(\mathbf{A}_c)\ge \frac{n-\rank(\mA_H)}{\rank(\mA_H)}$.
\end{remark}
The inverse Cayley transform leads to a similar result for the mapping from the discrete-time to the continuous-time problem:
\begin{lemma} \label{lemma:Cayley:Inverse} Let $\mathbf{A}_d\in\C^{n\times n}$ be such that $x_{k+1} =\mathbf{A}_d x_k $, $k\in\mathbb{N}_0$ is stable and that $-1$ is not an eigenvalue of $\mathbf{A}_d$. Then, the inverse Cayley transform
\begin{equation}\label{inv-cayley}
\mathbf{A}_c :=(\mathbf{A}_d-\mathbf{I})(\mathbf{A}_d+\mathbf{I})^{-1} \end{equation}
is well-defined and the following properties hold.
\begin{itemize}
\item [(i)] If $\mathbf{A}_d$ is hypocontractive then $\mathbf{A}_c$ is negative hypocoercive.
\item [(ii)] If $\mathbf{A}_d$ is semi-contractive then $\mathbf{A}_c$ is semi-dissipative. Moreover, with $\mA_H= \frac12 (\mathbf{A}_c+\mathbf{A}_c^{\mathsf{H}})$, the matrix $(\mathbf{A}_d+\mathbf{I})$ is a bijection from $\ker(\mathbf{I} -\mathbf{A}_d^{\mathsf{H}} \mathbf{A}_d)$ to $\ker(\mA_H)$ and $\dim\ker(\mA_H) =\dim\ker(\mathbf{I} -\mathbf{A}_d^{\mathsf{H}} \mathbf{A}_d)$.
\end{itemize} \end{lemma}
\begin{proof} Since $-1$ is not an eigenvalue of $\mathbf{A}_d$ then the matrices~$(\mathbf{A}_d+\mathbf{I})$, $(\mathbf{A}_d+\mathbf{I})^{\mathsf{H}}$ are invertible; and the inverse Cayley transform~\eqref{inv-cayley} is well-defined.
(i) If $\mathbf{A}_d$ is hypocontractive then all eigenvalues of~$\mathbf{A}_d$ have modulus less than one, hence, all eigenvalues of~$\mathbf{A}_c$ have negative real part. Thus, $\mathbf{A}_c$ is negative hypocoercive.
(ii) If $\mathbf{A}_d\in\C^{n\times n}$ is semi-contractive then $x_{k+1} =\mathbf{A}_d x_k $, $k\in\mathbb{N}_0$ is stable (due to Proposition~\ref{prop:EV+SV}). Then
\begin{equation} \label{mAH} \begin{split}
\mA_H
&=\tfrac12 (\mathbf{A}_c +\mathbf{A}_c^{\mathsf{H}}) \\
&=\tfrac12 \big((\mathbf{A}_d-\mathbf{I})(\mathbf{A}_d+\mathbf{I})^{-1} +(\mathbf{A}_d+\mathbf{I})^{-{\mathsf{H}}} (\mathbf{A}_d-\mathbf{I})^{\mathsf{H}}\big) \\
&=\tfrac12 (\mathbf{A}_d+\mathbf{I})^{-{\mathsf{H}}} \big((\mathbf{A}_d+\mathbf{I})^{\mathsf{H}} (\mathbf{A}_d-\mathbf{I}) +(\mathbf{A}_d-\mathbf{I})^{\mathsf{H}} (\mathbf{A}_d+\mathbf{I})\big)(\mathbf{A}_d+\mathbf{I})^{-1} \\
&=-(\mathbf{A}_d+\mathbf{I})^{-{\mathsf{H}}} (\mathbf{I} -\mathbf{A}_d^{\mathsf{H}} \mathbf{A}_d)(\mathbf{A}_d+\mathbf{I})^{-1} \,. \end{split} \end{equation} Thus, the matrices~$-\mA_H$ and~$(\mathbf{I} -\mathbf{A}_d^{\mathsf{H}} \mathbf{A}_d)$ are related via a congruence transformation, and hence $\mathbf{A}_d$ is semi-contractive (or equivalently, $(\mathbf{I} -\mathbf{A}_d^{\mathsf{H}} \mathbf{A}_d)$ is positive semi-definite) if $\mathbf{A}_c$ is semi-dissipative.
Due to~\eqref{mAH}, if $v\in\ker(\mA_H)$ then $(\mathbf{A}_d+\mathbf{I})^{-1} v\in\ker(\mathbf{I} -\mathbf{A}_d^{\mathsf{H}} \mathbf{A}_d)$. Thus, $(\mathbf{A}_d+\mathbf{I})^{-1}\ker(\mA_H) \subseteq\ker(\mathbf{I} -\mathbf{A}_d^{\mathsf{H}} \mathbf{A}_d)$.
Conversely, if $w\in\ker(\mathbf{I} -\mathbf{A}_d^{\mathsf{H}} \mathbf{A}_d)$ then $(\mathbf{A}_d+\mathbf{I}) w\in\ker(\mA_H)$. Thus, $(\mathbf{A}_d+\mathbf{I}) \ker(\mathbf{I} -\mathbf{A}_d^{\mathsf{H}} \mathbf{A}_d)\subseteq \ker(\mA_H)$.
Altogether, $(\mathbf{A}_d+\mathbf{I})$ is a bijection from $\ker(\mathbf{I} -\mathbf{A}_d^{\mathsf{H}} \mathbf{A}_d)$ to $\ker(\mA_H)$ which implies that $\dim\ker(\mA_H) =\dim\ker(\mathbf{I} -\mathbf{A}_d^{\mathsf{H}} \mathbf{A}_d)$. \end{proof}
\begin{remark}\label{rem:other shift} The assumption in Lemma~\ref{lemma:Cayley:Inverse} that $-1$ is not an eigenvalue of $\mathbf{A}_d$ can be relaxed by considering
$\mathbf{A}_c=(\mathbf{A}_d-\alpha \mathbf{I})(\mathbf{A}_d+\alpha \mathbf{I})^{-1}$, where $-\alpha\in \mathbb{C}$ (with $| \alpha |=1$) is not an eigenvalue of $\mathbf{A}_d$. Such an $\alpha$ clearly exists in the complex case, but this will not work in the real case if both $1$ and $-1$ are eigenvalues of $\mathbf{A}_d$ and one wants to stay within the class of real matrices. \end{remark}
The Cayley transformation also gives a direct relation between the hypocoercivity and hypo\-contractivity indices.
\begin{theorem} \label{thm:Cayley} \begin{itemize}
\item [(i)] Let $\mathbf{A}_c\in\C^{n\times n}$ be semi-dissipative and negative hypocoercive and let $\mathbf{A}_d:=(\mathbf{I}+\mathbf{A}_c)(\mathbf{I}-\mathbf{A}_c)^{-1}$. Then the hypocoercivity index~$m_{HC}\in\mathbb{N}_0$ of $\mathbf{A}_c$ and the hypocontractivity index~$m_{dHC}$ of $\mathbf{A}_d$ are the same, i.e., $m_{dHC}(\mathbf{A}_d)=m_{HC}(\mathbf{A}_c)$.
\item [(ii)] Let $\mathbf{A}_d\in\C^{n\times n}$ be semi-contractive and hypocontractive and let $\mathbf{A}_c:=(\mathbf{A}_d-\mathbf{I})(\mathbf{A}_d+\mathbf{I})^{-1}$. Then the hypocontractivity index~$m_{dHC}\in\mathbb{N}_0$ of $\mathbf{A}_d$ and the hypocoercivity index~$m_{HC}$ of $\mathbf{A}_c$ are the same, i.e., $m_{HC}(\mathbf{A}_c)=m_{dHC}(\mathbf{A}_d)$.
\end{itemize} \end{theorem}
\begin{proof} (i) Due to the assumptions and Lemma~\ref{lemma:Cayley}, $\mathbf{A}_d=2(\mathbf{I}-\mathbf{A}_c)^{-1}-\mathbf{I}$
is semi-contractive and hypocontractive. Thus, by Lemma \ref{lemma:Cayley:Inverse}, the inverse Cayley transform~$(\mathbf{A}_d-\mathbf{I})(\mathbf{A}_d+\mathbf{I})^{-1}$ is well-defined and satisfies~$(\mathbf{A}_d-\mathbf{I})(\mathbf{A}_d+\mathbf{I})^{-1}=\mathbf{I}-2(\mathbf{A}_d+\mathbf{I})^{-1} =\mathbf{A}_c$.
By assumption, the matrix~$\mathbf{A}_c=\mA_H +\mA_S$ has a finite HC-index~$m_{HC}=m_{HC}(\mathbf{A}_c)$ which is the smallest integer such that, due to \eqref{Tm:BS_BH2}, \[
\bigcap_{j=0}^{m_{HC}} \ker \big(\mA_H \mathbf{A}_c^j\big) =\{0\} . \]
Hence, there exists a vector $v_0\in\C^n\setminus\{0\}$ such that \begin{equation} \label{v0:AS}
\mathbf{A}_c^j v_0\in \ker (\mA_H) \ ,
\qquad
j\in\{0,\ldots,m_{HC}-1\}
\qquad \text{and }
\mathbf{A}_c^{m_{HC}} v_0 \notin \ker(\mA_H) \ . \end{equation} Thus, by~Lemma~\ref{lemma:Cayley} (ii), we obtain that \begin{equation} \label{Av0}
(\mathbf{I}-\mathbf{A}_c)\mathbf{A}_c^j v_0\in \ker (\mathbf{I} -\mathbf{A}_d^{\mathsf{H}}\mathbf{A}_d) \ ,
j\in\{0,\ldots,m_{HC}-1\}, \ \text{and }
(\mathbf{I}-\mathbf{A}_c)\mathbf{A}_c^{m_{HC}} v_0 \notin \ker (\mathbf{I} -\mathbf{A}_d^{\mathsf{H}}\mathbf{A}_d) \ . \end{equation} Conversely, the existence of some $v_0\ne0$ satisfying the ``first part'' of \eqref{Av0} with some $m_{HC}\ge1$ implies that the HC-index of $\mathbf{A}_c$ is at least $m_{HC}$.
The matrix $\mathbf{A}_d=(\mathbf{I}+\mathbf{A}_c)(\mathbf{I}-\mathbf{A}_c)^{-1}$ is hypocontractive with HC-index~$m_{dHC} :=m_{dHC}(\mathbf{A}_d)\in\mathbb{N}_0$. Due to \eqref{Dm:A} this is the smallest integer such that \[
\bigcap_{j=0}^{m_{dHC}} \ker \big((\mathbf{I}-\mathbf{A}_d^{\mathsf{H}}\mathbf{A}_d) \mathbf{A}_d^j\big) =\{0\} . \]
Hence, there exists a vector $w_0\in\C^n\setminus\{0\}$ such that \begin{equation} \label{mAd:kernel}
w_0\in \bigcap_{j=0}^{m_{dHC}-1} \ker\big((\mathbf{I} -\mathbf{A}_d^{\mathsf{H}}\mathbf{A}_d) \mathbf{A}_d^j\big) \ ,
\quad\text{ and }
w_0\notin \ker\big((\mathbf{I} -\mathbf{A}_d^{\mathsf{H}}\mathbf{A}_d) \mathbf{A}_d^{m_{dHC}}\big), \ \end{equation}
or equivalently, there exists $w_0\in\C^n\setminus\{0\}$ such that \begin{equation} \label{w0:mAd}
\mathbf{A}_d^j w_0\in \ker (\mathbf{I} -\mathbf{A}_d^{\mathsf{H}}\mathbf{A}_d) \ ,
\qquad
j\in\{0,\ldots,m_{dHC}-1\}
\qquad \text{and }
\mathbf{A}_d^{m_{dHC}} w_0 \notin \ker(\mathbf{I} -\mathbf{A}_d^{\mathsf{H}}\mathbf{A}_d). \end{equation} Conversely, the existence of some $w_0\ne0$ satisfying the ``first part'' of \eqref{w0:mAd} with some $m_{dHC}\ge1$ implies that the dHC-index of $\mathbf{A}_d$ is at least $m_{dHC}$.
It remains to show that $m_{HC}(\mathbf{A}_c) =m_{dHC}(\mathbf{A}_d)$: If $m_{HC}=0$, then $\mathbf{A}_c$ is dissipative such that $\ker(\mA_H)=\{0\}$. Hence, $\ker(\mathbf{I}-\mathbf{A}_d^{\mathsf{H}}\mathbf{A}_d)=\{0\}$ due to~Lemma~\ref{lemma:Cayley} (ii) and $\mathbf{A}_d$ is contractive, i.e.\ $m_{dHC}=0$. Conversely, if $m_{dHC}(\mathbf{A}_d)=0$ then $\ker(\mathbf{I}-\mathbf{A}_d^{\mathsf{H}}\mathbf{A}_d)=\{0\}$. Hence, $\ker(\mA_H)=\{0\}$ by~Lemma~\ref{lemma:Cayley} (ii) and thus $\mathbf{A}_c$ is dissipative and $m_{HC} =0$.
If $m_{HC}\geq 1$, then let $v_0 \in\C^n\setminus\{0\}$ satisfy~\eqref{Av0} with $m_{HC}=m_{HC}(\mathbf{A}_c)$. Hence, $$
q(\mathbf{A}_c)(\mathbf{I}-\mathbf{A}_c)v_0 \in \ker(\mathbf{I}-\mathbf{A}_d^{\mathsf{H}}\mathbf{A}_d) $$ for all polynomials $q$ of order up to $m_{HC}-1$. In particular $$
w_0:=(\mathbf{I}-\mathbf{A}_c)^{m_{HC}} v_0\in \ker(\mathbf{I}-\mathbf{A}_d^{\mathsf{H}}\mathbf{A}_d), $$ and $w_0\ne0$ since $(\mathbf{I}-\mathbf{A}_c)$ is regular. Also, using \eqref{cayley} we find that $$
\mathbf{A}_d^j w_0 = (\mathbf{I}+\mathbf{A}_c)^j(\mathbf{I}-\mathbf{A}_c)^{m_{HC}-j}v_0 \in \ker (\mathbf{I} -\mathbf{A}_d^{\mathsf{H}}\mathbf{A}_d) \ ,
\qquad
j\in\{0,\ldots,m_{dHC}-1\}\,. $$ Hence, \eqref{w0:mAd} implies $m_{dHC}(\mathbf{A}_d)\geq m_{HC}(\mathbf{A}_c)$.
Conversely, if $m_{dHC}\geq 1$, then let $w_0\in\C^n\setminus\{0\}$ satisfy~\eqref{w0:mAd} with $m_{dHC}=m_{dHC}(\mathbf{A}_d)$. Hence, $$
q(\mathbf{A}_d)w_0 \in \ker(\mathbf{I}-\mathbf{A}_d^{\mathsf{H}}\mathbf{A}_d) $$ for all polynomials $q$ of order up to $m_{dHC}-1$. We define $v_0:=(\mathbf{A}_d+\mathbf{I})^{m_{dHC}} w_0\ne0$ since $(\mathbf{A}_d+\mathbf{I})$ is regular. Using \eqref{inv-cayley} and $\mathbf{I}-\mathbf{A}_c=2(\mathbf{A}_d+\mathbf{I})^{-1}$ we compute \[
\mathbf{A}_c^j (\mathbf{I}-\mathbf{A}_c)v_0
=2(\mathbf{A}_d-\mathbf{I})^j(\mathbf{A}_d+\mathbf{I})^{m_{dHC}-j-1}w_0 \in \ker (\mathbf{I} -\mathbf{A}_d^{\mathsf{H}}\mathbf{A}_d) \ ,
\qquad
j\in\{0,\ldots,m_{dHC}-1\}\,. \] Hence, \eqref{Av0} implies $m_{HC}(\mathbf{A}_c)\geq m_{dHC}(\mathbf{A}_d)$. Altogether, we deduce that $m_{dHC}(\mathbf{A}_d) =m_{HC}(\mathbf{A}_c)$, which finishes the proof of statement~(i).
(ii) The proof is analogous to that of (i). \end{proof}
\begin{remark}\label{rem:staffans} It was pointed out to the authors that the results presented in Lemmas~\ref{lemma:Cayley} and~\ref{lemma:Cayley:Inverse} as well as Theorem~\ref{thm:Cayley} can be proved in an alternative way by using the characterization via unobservability subspaces, see Remark~\ref{rem:unobservability}. The results then can be proved via Lemmas 12.3.10 and 12.2.6 of \cite{Sta05}. \end{remark}
\begin{example}\label{ODE:HC2} Consider the continuous-time system~\eqref{ODE:B} with the coefficient matrix \begin{equation} \label{matrix:A:HC2}
\mathbf{A}_c
=\begin{bmatrix}
0 & -1 & 0 & 0 \\
1 & 0 & -1 & 0 \\
0 & 1 & -1 & 0 \\
0 & 0 & 0 & -1
\end{bmatrix} \, \end{equation} which is semi-dissipative and $\mathbf{B}=-\mathbf{A}_c$ has hypocoercivity index $m_{HC}=2$. Applying the Cayley transformation gives
\begin{equation} \label{matrix:Ad:HC2}
\mathbf{A}_d
=(\mathbf{I}+\mathbf{A}_c)(\mathbf{I}-\mathbf{A}_c)^{-1}
=\tfrac15 \begin{bmatrix}
1 & -4 & 2 & 0 \\
4 & -1 & -2 & 0 \\
2 & 2 & -1 & 0 \\
0 & 0 & 0 & 0
\end{bmatrix} \, \end{equation} which is semi-contractive and has hypocontractivity index $m_{dHC}=2$.
\end{example}
Unfortunately, the Cayley transform does not relate the shifted hypocoercivity index~$m_{SHC}$ and the scaled hypocontractivity index~$m_{dSHC}$ in the same way, as the following example illustrates.
\begin{example}\label{ex:notshift} Consider the matrix \[
\tA_d =\begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix} \] which is hypocontractive with hypocontractivity index $m_{dHC}=2$.
The matrix $\mathbf{A}_d :=2\tA_d$ is not semi-contractive, since $\mathbf{A}_d^{\mathsf{H}}\mathbf{A}_d=\diag(0,4,4)$, but it has scaled hypocontractivity index~$m_{dSHC}=2$.
For the inverse Cayley transform of $\mathbf{A}_d$ we obtain \[
\mathbf{A}_c
:=
(\mathbf{A}_d-\mathbf{I})(\mathbf{A}_d+\mathbf{I})^{-1}
=
\begin{bmatrix}
-1 & 4 &-8 \\
0 &-1 & 4 \\
0 & 0 &-1
\end{bmatrix}\,,\qquad
\mA_H
=
\begin{bmatrix}
-1 & 2 &-4 \\
2 &-1 & 2 \\
-4 & 2 &-1
\end{bmatrix}
\,. \] The eigenvalues of $\mA_H$ are $\lambda=3$, $\lambda_\pm=-3\pm\sqrt{12}$ and hence they are simple and the shifted HC-index of $\mathbf{A}_c$ is $m_{SHC}=1$. This example shows that $2=m_{dSHC}(\mathbf{A}_d)\nem_{SHC}(\mathbf{A}_c)=1$. \end{example}
It is well-known, see e.g.~\cite[page 180]{Ka80}, that the Cayley transformation also directly relates the stabilizing solutions of the discrete-time and continuous-time Lyapunov equation. We summarize these results in the following Lemma.
\begin{lemma}\label{lem:lyapdc} Let $\mathbf{A}_c\in\C^{n\times n}$ be a matrix such that~\eqref{ODE:B} is (Lyapunov) stable and let $\mathbf{A}_d=(\mathbf{I}+\mathbf{A}_c)(\mathbf{I}-\mathbf{A}_c)^{-1}$. Then $\mathbf{P}$ is the positive definite solution $\mathbf{P}_c=\mathbf{P}$ of the continuous-time Lyapunov equation \[
\mathbf{A}_c^{\mathsf{H}} \mathbf{P}_c +\mathbf{P}_c\mathbf{A}_c =-\mathbf{Q}_c, \] for some positive semidefinite matrix $\mathbf{Q}_c$ if and only if $\mathbf{P}$ is the positive definite solution $\mathbf{P}_d=\mathbf{P}$ of the discrete-time Lyapunov equation \[
\mathbf{A}_d^{\mathsf{H}} \mathbf{P}_d\mathbf{A}_d -\mathbf{P}_d =-\mathbf{Q}_d \] for positive semidefinite $\mathbf{Q}_d$, where the right hand sides are related via $\mathbf{Q}_d=2(\mathbf{I}-\mathbf{A}_c^{\mathsf{H}})^{-1}\mathbf{Q}_c(\mathbf{I}-\mathbf{A}_c)^{-1}$. \end{lemma}
In summary, we have an almost complete analogy between the properties of continuous-time and discrete-time systems. We summarize these invariance properties under the Cayley transformation and the inverse Cayley transformation (if it exists) in Table~\ref{invcayley}.
\begin{table} \begin{center} \begin{tabular}{ll}
continuous-time & discrete-time, \\[1mm] \hline\hline\\[-3mm]
$\ddt x =\mathbf{A}_c x$ for $t\geq 0$ & $x_{k+1} =\mathbf{A}_d x_k $ for $k\in\mathbb{N}_0$, \\
\hline
(asymptotically) stable & (asymptotically) stable, \\
\hline
semi-dissipative & semi-contractive,\\
\hline
(hypo)coercive & (hypo)contractive,\\
\hline
$m_{HC}(\mathbf{A}_c)$ & $m_{dHC}(\mathbf{A}_d)$,\\
\hline
Lyapunov solution $\mathbf{P}_c$ &Lyapunov solution $\mathbf{P}_d$.\\
\hline\\ \end{tabular} \caption{Invariance of properties of continuous-time and discrete-time systems under Cayley transformation $\mathbf{A}_d=(\mathbf{I}+\mathbf{A}_c)(\mathbf{I}-\mathbf{A}_c)^{-1}$ and inverse Cayley transformation $\mathbf{A}_c=(\mathbf{A}_d-\mathbf{I})(\mathbf{A}_d+\mathbf{I})^{-1}$ \label{invcayley} } \end{center} \end{table}
Finally we consider the \emph{scaled Cayley transform} \begin{equation}\label{sc-cayley}
\mathbf{A}_d(t):=(\mathbf{I}+\frac{t}{2}\mathbf{A}_c)(\mathbf{I}-\frac{t}{2}\mathbf{A}_c)^{-1} \quad\mbox{for } t>0\ , \end{equation} which can be considered as a short-time approximation of the matrix exponential for \eqref{ODE:B}. Due to the scaling invariance of the hypocoercivity (index) of a matrix $\mathbf{A}_c\in\C^{n\times n}$ (see \S\ref{sec:trafo}), we readily obtain: \begin{corollary}\label{cor-sc-cayley} Let $\mathbf{A}_c\in\C^{n\times n}$ be semi-dissipative and negative hypocoercive. Then, for all $t>0$, the scaled Cayley transform $\mathbf{A}_d(t)$ is hypocontractive (due to Lemma \ref{lemma:Cayley} (i)), its dHC-index satisfies $m_{dHC}(\mathbf{A}_d(t))=m_{HC}(\mathbf{A}_c)=:m_{dHC}$ (due to Theorem \ref{thm:Cayley} (i)), and the norm of its powers satisfy $$
\|\mathbf{A}_d(t)^j\|_2 =1 \text{ for all } j=1,\ldots,m_{dHC}\,,
\quad\text{and }\
\|\mathbf{A}_d(t)^{m_{dHC}+1}\|_2 <1 \, $$ (due to Theorem \ref{DS:A:semi-contractive:power-bounds}). \end{corollary}
\section*{Conclusions} In this paper we have given a systematic analysis of different concepts related to the stability and short-time behavior of solutions to linear constant coefficient continuous-time and discrete-time systems. While many results for the continuous-time setting were already established in \cite{AAM21} we have analyzed under which linear transformations the properties of asymptotic stability, semi-dissipativity and hypocoercivity stay invariant.
For linear time-invariant continuous-time systems, it is well-known that the exponential rate of the short-time behavior of the propagator norm~$\|e^{\mathbf{A}_c t}\|$ is determined by the logarithmic norm of the system matrix. In this work, we established that the shifted hypocoercivity index characterizes the (remaining) algebraic decay of the propagator norm in the short-time regime.
For each of the continuous-time results we have derived a corresponding result for the discrete-time case. These include the relation between (asymptotic) stability, semi-contractivity and hypocontractivity. We have also introduced the new concept of shifted hypocoercivity and scaled hypocontractivity. We then have analyzed how the properties relate under the Cayley transformation that relates continuous-time and discrete-time systems. While the role of the hypocontractivity index (or norm-one index) in the discrete-time setting has been recognized before, the corresponding concept---the hypocoercivity index---in the continuous-time setting and its role has been established only recently.
Future work will include the extension of the results of \cite{AAM21} for linear continuous-time dif\-fer\-en\-tial-algebraic systems to discrete-time descriptor systems.
{ \appendix \section{Staircase forms}\label{sec:staircase} In \cite{AAM21} a computationally feasible procedure to check the conditions of Lemma~\ref{lem:HC:equivalence} in the continuous-time case via a staircase form under unitary congruence transformations of the pair $(\mathbf{J},\mathbf{R})=(\mB_S,\mB_H)$ has been presented.
\begin{lemma}[Staircase form for $(\mathbf{J},\mathbf{R})$] \label{lem:SF} Let $\mathbf{J}\in\C^{n\times n}$ be a skew-Hermitian matrix, and $\mathbf{R}\in\C^{n\times n}$ be a nonzero Hermitian matrix. Then there exists a unitary matrix $\mathbf{V}\in\C^{n\times n}$, such that~$\mathbf{V} \mathbf{J} \mathbf{V}^{\mathsf{H}}$ and~$\mathbf{V} \mathbf{R} \mathbf{V}^{\mathsf{H}}$ are block tridiagonal matrices of the form
\begin{equation} \label{matrices:staircase:J-R} \begin{split} \mathbf{V}\ \mathbf{J}\ \mathbf{V}^{\mathsf{H}} &= \begin{array}{l}
\left[ \begin{array}{ccccccc|c}
\mathbf{J}_{1,1} & -\mathbf{J}_{2,1}^{\mathsf{H}} & & & \cdots & & 0 & 0\\
\mathbf{J}_{2,1} & \mathbf{J}_{2,2} & -\mathbf{J}_{3,2}^{\mathsf{H}} & & & & &\\
& \ddots & \ddots & \ddots & & & \vdots & \\
& & \mathbf{J}_{k,k-1} & \mathbf{J}_{k,k} & -\mathbf{J}_{k+1,k}^{\mathsf{H}} & & & \vdots \\
\vdots & & & \ddots & \ddots & \ddots & & \\
& & & & \mathbf{J}_{s-2,s-3} & \mathbf{J}_{s-2,s-2} & -\mathbf{J}_{s-1,s-2}^{\mathsf{H}} & \\
0 & & \cdots & & & \mathbf{J}_{s-1,s-2} & \mathbf{J}_{s-1,s-1} & 0\\ \hline
0 & & & \cdots & & & 0 & \mathbf{J}_{ss} \end{array}\right]
\begin{array}{c}
n_1\\ n_2\\ \vdots\\ \\ n_k\\ \vdots \\ n_{s-2} \\n_{s-1}\\ n_s
\end{array} \\
\quad\hspace{5pt} n_1 \hspace{170pt} n_{s-2} \hspace{30pt} n_{s-1} \hspace{25pt} n_s
\end{array}, \quad \\
\mathbf{V}\ \mathbf{R}\ \mathbf{V}^{\mathsf{H}} &= \begin{array}{l} \left[ \begin{array}{cc}
\mathbf{R}_1 & 0\\
0 & 0 \\
\vdots & \vdots\\
\vdots & \vdots \\
0 & 0 \end{array}\right]
\begin{array}{c}
n_1 \\ n_2 \\ \vdots\\ \vdots \\ n_s \end{array} \\
\;\;\; n_1 \;\; n-n_1 \end{array}, \end{split} \end{equation}
where $n_1 \geq n_2 \geq \cdots \geq n_{s-1} >0$, $n_s \geq 0$, and $\mathbf{R}_1\in\mathbb{C}^{n_1,n_1}$ is nonsingular.
If $\mathbf{R}$ is nonsingular, then~$s=2$ and~$n_2=0$. For example, $\mathbf{V}=\mathbf{I}$, $\mathbf{J}_{1,1}=\mathbf{J}$ and~$\mathbf{R}_1=\mathbf{R}$ is an admissible choice.
If $\mathbf{R}$ is singular, then $s\geq 3$ and the matrices~$\mathbf{J}_{i,i-1}$, $i=2,\ldots,s-1$, in the subdiagonal have full row rank and are of the form
\[ \mathbf{J}_{i,i-1} = \begin{bmatrix} \Sigma_{i,i-1} & 0 \end{bmatrix}, \quad i =2,\ldots, s-1, \]
with nonsingular matrices~$\Sigma_{i,i-1}\in\mathbb{C}^{n_i,n_i}$, moreover $\Sigma_{s-1,s-2}$ is a real-valued diagonal matrix. \end{lemma}
A system~\eqref{ODE:B} with an accretive matrix $\mathbf{B}=\mB_S+\mB_H$ is hypocoercive if $n_s=0$ and if this is the case then the hypocoercivity index is $m_{HC}(\mathbf{B})=s-2$.
A similar staircase form can be derived in the discrete-time case. It is based on the polar decomposition $\mathbf{A}_d= \mathbf{U}\mathbf{Q}$, see Proposition \ref{prop:PolDec}.
\begin{lemma}[Staircase form for $(\mathbf{U},\mathbf{Q})$] \label{lem:SFQU} Let $\mathbf{U}\in\C^{n\times n}$ be a unitary matrix, and $\mathbf{Q}\in\C^{n\times n}$ be a nonzero semi-contractive Hermitian matrix. Then there exists a unitary matrix $\mathbf{V}\in\C^{n\times n}$, such that~$\mathbf{V} \mathbf{Q} \mathbf{V}^{\mathsf{H}}$ and~$\mathbf{V} \mathbf{U} \mathbf{V}^{\mathsf{H}}$ are block upper Hessenberg matrices of the form
\begin{equation} \label{matrices:staircase:HU} \begin{split} \mathbf{V}\ \mathbf{U}\ \mathbf{V}^{\mathsf{H}} &= \begin{array}{l}
\left[ \begin{array}{ccccc|c}
\mathbf{U}_{1,1} & \mathbf{U}_{1,2} &\cdots & \cdots & \mathbf{U}_{1,s-1} & 0\\
\mathbf{U}_{2,1} & \mathbf{U}_{2,2} & \mathbf{U}_{2,3} & \cdots & \mathbf{U}_{2,s-1} &0\\
& \ddots & \ddots & \ddots & \ddots & \vdots \\
& & \mathbf{U}_{s-2,s-3} & \mathbf{U}_{s-2,s-2} & \mathbf{U}_{s-2,s-1} & 0\\
0 & \cdots & 0& \mathbf{U}_{s-1,s-2} & \mathbf{U}_{s-1,s-1} & 0\\ \hline
0 & \cdots & & & 0 & \mathbf{U}_{s,s} \end{array}\right]
\begin{array}{c}
n_1\\ n_2\\ \vdots \\ n_{s-2} \\n_{s-1}\\ n_s
\end{array} \\
\end{array}, \quad \\
\mathbf{V}\ \mathbf{Q}\ \mathbf{V}^{\mathsf{H}}
&= \begin{array}{l} \left[ \begin{array}{ccccc|c}
\mathbf{Q}_1 & 0 & \cdots & \cdots &0 &0\\
0 & \mathbf{I}_{n_2} & 0 & \cdots & \vdots & \vdots\\
\vdots & 0 & \ddots & \ddots & \vdots & \vdots\\
\vdots & \ddots & \ddots & \ddots & \vdots & \vdots\\
\vdots & \vdots & \ddots & 0 & \mathbf{I}_{n_{s-1}} & 0\\
\hline
0 & 0 & \cdots & \cdots & 0 & \mathbf{I}_{n_{s}} \end{array}\right]
\begin{array}{c}
n_1 \\ n_2 \\ \vdots\\ \\ \vdots \\ n_{s-1}
\\ n_s \end{array} \\
\end{array}, \end{split} \end{equation}
where $n_1 \geq n_2 \geq \cdots \geq n_{s-1}>0$, $n_s \geq 0$, and $\mathbf{Q}_1\in\mathbb{C}^{n_1,n_1}$ is contractive and Hermitian.
If $\mathbf{Q}$ is contractive, then~$s=2$ and~$n_2=0$. Then $\mathbf{V}=\mathbf{I}$, $\mathbf{U}_{1,1}=\mathbf{U}$ and~$\mathbf{Q}_1=\mathbf{Q}$ is an admissible choice.
If $\mathbf{Q}$ is not contractive, then $s\geq 3$ and the matrices~$\mathbf{U}_{i,i-1}$, $i=2,\ldots,s-1$, in the subdiagonal have full row rank and are of the form
\[
\mathbf{U}_{i,i-1}
=\begin{bmatrix} \Sigma_{i,i-1} & 0 \end{bmatrix}, \quad i =2,\ldots, s-1, \] with nonsingular matrices~$\Sigma_{i,i-1}\in\mathbb{C}^{n_i,n_i}$, moreover $\Sigma_{s-1,s-2}$ is a real-valued diagonal matrix. \end{lemma}
\begin{proof} If~$\mathbf{Q}$ is contractive, then~$n_1=n$ and we have to choose~$s=2$ and~$n_2=0$ to fit~$\mathbf{U}$ into the proposed structure in~\eqref{matrices:staircase:HU}. \\ If~$\mathbf{Q}$ is not contractive, then we have the following constructive proof.\\ \begin{breakablealgorithm} \caption{Staircase algorithm for pair~$(\mathbf{U},\mathbf{Q})$} \label{algorithm:staircase:U-H} \begin{algorithmic}[1] \REQUIRE $(\mathbf{U},\mathbf{Q})$\\
----------- {\em Step 0} ----------- \STATE Perform a (spectral) decomposition of $\mathbf{Q}$ such that \[ \mathbf{Q} =\mathbf{V}_1 \begin{bmatrix} \widetilde \mQ_1 & 0 \\ 0 & \mathbf{I} \end{bmatrix} \mathbf{V}_1^{\mathsf{H}} , \]
with $\mathbf{V}_1\in \C^{n\times n}$ unitary, $\widetilde \mQ_1\in \mathbb{C}^{n_1, n_1}$ contractive and Hermitian. \STATE Set $\mathbf{V} := \mathbf{V}_1^{\mathsf{H}}$, $\widetilde \mQ:=\mathbf{V}_1^{\mathsf{H}}\ \mathbf{Q}\ \mathbf{V}_1$,
\[
\widetilde \mU
:= \mathbf{V}_1^{\mathsf{H}}\ \mathbf{U}\ \mathbf{V}_1
=:\begin{bmatrix}
\widetilde \mU_{1,1} & \widetilde \mU_{1,2} \\
\widetilde \mU_{2,1} & \widetilde \mU_{2,2}
\end{bmatrix}. \]
\newline ----------- {\em Step 1} ----------- \STATE Perform a singular value decomposition (SVD) of~$\widetilde \mU_{2,1}\in\mathbb{C}^{(n-n_1)\times n_1}$ such that \[
\widetilde \mU_{2,1}
= \mathbf{W}_{2,1} \begin{bmatrix} \widetilde \Sigma_{2,1} & 0\\ 0 & 0 \end{bmatrix} \mathbf{V}^{\mathsf{H}}_{2,1}, \] with unitary matrices~$\mathbf{W}_{2,1}$ and~$\mathbf{V}_{2,1}$ as well as a positive definite, diagonal matrix $\widetilde \Sigma_{2,1}\in \mathbb R^{n_2, n_2}$.
\STATE Set $\mathbf{V}_2 := \diag(\mathbf{V}_{2,1}^{\mathsf{H}},\ \mathbf{W}_{2,1}^{\mathsf{H}})$,\ $\mathbf{V}:= \mathbf{V}_2 \mathbf{V}$.
\STATE Set \[ \def1{1.4} \widetilde \mU := \mathbf{V}_2\ \widetilde \mU\ \mathbf{V}_2^{\mathsf{H}}
=: \left[ \begin{array}{c|cc}
\widetilde \mU_{1,1} & \widetilde \mU_{1,2} & \widetilde \mU_{1,3}\\
\hline
\widetilde \mU_{2,1} & \widetilde \mU_{2,2} & \widetilde \mU_{2,3} \\
0 & \widetilde \mU_{3,2} & \widetilde \mU_{3,3} \end{array}\right],\qquad
\def1{1} \widetilde \mR := \mathbf{V}_2 \widetilde \mQ \mathbf{V}_2^{\mathsf{H}}
=: \left[ \begin{array}{c|cc} \widetilde \mQ_1 & 0 & 0 \\ \hline 0 & \mathbf{I}_{n_2} & 0 \\ 0 & 0 & \mathbf{I} \end{array}\right]. \]
(The lines indicate the partitioning of the block matrices~$\widetilde \mU$ and~$\widetilde \mQ$ in the previous step.)
\newline ----------- {\em Step 2} ----------- \STATE $i := 3$ \WHILE{$n_{i-1} > 0$ \OR $\widetilde \mU_{i,i-1} \neq 0$} \STATE Perform an SVD of $\widetilde \mU_{i,i-1}$ such that \[ \widetilde \mU_{i,i-1} = \mathbf{W}_{i,i-1} \begin{bmatrix} \widetilde \Sigma_{i,i-1} & 0\\ 0 & 0 \end{bmatrix} \mathbf{V}^{\mathsf{H}}_{i,i-1} , \] with unitary matrices~$\mathbf{W}_{i,i-1}$ and~$\mathbf{V}_{i,i-1}$ as well as a positive definite, diagonal matrix $\widetilde \Sigma_{i,i-1}\in \mathbb{R}^{n_i, n_i}$.
\STATE Set $\mathbf{V}_i := \diag(\mathbf{I}_{n_1},\ldots,\mathbf{I}_{n_{i-2}},\ \mathbf{V}_{i,i-1}^{\mathsf{H}},\ \mathbf{W}_{i,i-1}^{\mathsf{H}} )$, $\mathbf{V}:= \mathbf{V}_i \mathbf{V}$.
\STATE Set \[ \widetilde \mU := \mathbf{V}_i\ \widetilde \mU\ \mathbf{V}_i^{\mathsf{H}} =: \begin{bmatrix}
\widetilde \mU_{1,1} & \widetilde \mU_{1,2} & \cdots & \cdots & \widetilde \mU_{1,i+1} \\
\widetilde \mU_{2,1} & \widetilde \mU_{2,2} & \widetilde \mU_{2,3} & & \vdots \\
0 & \ddots & \ddots & \ddots & \\
\vdots & \ddots & \widetilde \mU_{i,i-1} & \widetilde \mU_{i,i} & \widetilde \mU_{i,i+1} \\
0 & \cdots & 0 & \widetilde \mU_{i+1,i} & \widetilde \mU_{i+1,i+1} \end{bmatrix} , \ \text{where } \widetilde \mU_{i,i-1} = [ \widetilde \Sigma_{i,i-1}\quad 0]. \]
\STATE $i:= i+1$ \ENDWHILE
\newline ----------- {\em Step 3} ----------- \STATE $s:= i$
\FOR{$i=1,\ldots,s$}
\FOR{$j=i,\ldots,s$}
\STATE Set $\mathbf{U}_{i,j} :=\widetilde \mU_{i,j}$.
\ENDFOR \ENDFOR
\FOR{$i=2,\ldots,s$}
\STATE Set $\mathbf{U}_{i,i-1} :=\widetilde \mU_{i,i-1}$. \ENDFOR
\ENSURE Unitary matrix~$\mathbf{V}$. \\ \end{algorithmic} \end{breakablealgorithm}
It is clear that Algorithm~\ref{algorithm:staircase:U-H} terminates after a finite number of steps, either with~$n_{i-1}=0$ or~$\mathbf{U}_{i,i-1}=0$. We also note that Step 3 provides the nonzero entries of the r.h.s.\ of $\mathbf{V}\mathbf{U}\mathbf{V}^{\mathsf{H}}$ in \eqref{matrices:staircase:HU}. \end{proof}
Note that Algorithm~\ref{algorithm:staircase:U-H} can be applied to the polar decomposition $\mathbf{P}_d\mathbf{U}_d$ analogously. In both cases it immediately follows that $\mathbf{A}_d$ is hypocontractive if $n_s=0$ and the hypocontractivity index is then $m_{dHC}(\mathbf{A}_d)=s-2$.
\section{Equivalent hypocoercivity conditions}
The following lemma is a simple generalization of Lemma 2.3 in \cite{AASt15} and Proposition 1 in \cite{AAC18}. \begin{lemma}\label{lem:Definiteness} Let $\mathbf{D}\in\C^{n\times n}$ be positive semi-definite and $\mathbf{C}\in\mathbb{C}^{n\times n}$.
Then the following are equivalent:
\begin{itemize} \item [(E1)] There exists $m\in\mathbb{N}_0$ such that
\begin{equation}\label{condition:KalmanRank-App}
\rank[{\mathbf{D}},\mathbf{C}{\mathbf{D}},\ldots,\mathbf{C}^m {\mathbf{D}}]=n \,. \end{equation}
\item [(E2)] There exists $m\in\mathbb{N}_0$ such that
\begin{equation}\label{index-App}
\sum_{j=0}^m \mathbf{C}^j \mathbf{D} (\mathbf{C}^{\mathsf{H}})^j > 0 \,. \end{equation} \end{itemize} Moreover, the smallest possible $m\in\mathbb{N}_0$ in (E1) and (E2) coincide. \end{lemma} \begin{proof} First, we show that (E1) is equivalent to: \begin{itemize} \item [(E1')] There exists $m\in\mathbb{N}_0$ such that \begin{equation*}
\rank[{\mathbf{D}^{1/2}},\mathbf{C}{\mathbf{D}}^{1/2},\ldots,\mathbf{C}^m {\mathbf{D}}^{1/2}]=n \,, \end{equation*} \end{itemize} with the same $m$ as in (E1):\\ (E1) holds iff the statement \[
x^{\mathsf{H}} [{\mathbf{D}},\mathbf{C}{\mathbf{D}},\ldots,\mathbf{C}^m {\mathbf{D}}]=0\quad \mbox{for some } x\in\mathbb{C}^n, \] i.e.\ $\mathbf{D}(\mathbf{C}^{\mathsf{H}})^j x=0$ for $j=0,\ldots,m$ implies $x=0$. Now, since $\ker (\mathbf{D})= \ker (\mathbf{D}^{1/2})$, (E1) and (E1') are equivalent.
Next, let (E1) hold and define \[
\mathbf{E}:=[\mathbf{D}^{1/2},\,\mathbf{C}\mathbf{D}^{1/2},...,\,\mathbf{C}^m\mathbf{D}^{1/2}] \in\mathbb{C}^{n\times(m+1)n}\,. \] Then, \[
\mathbb{C}^{n\times n}\ni \mathbf{E}\ \mathbf{E}^{\mathsf{H}} = \sum_{j=0}^{m} \mathbf{C}^j\mathbf{D}(\mathbf{C}^{\mathsf{H}})^j \ge 0 \] has rank $n$ and \eqref{index-App} follows.\\ Conversely, let (E2) hold but assume we had $\rank \mathbf{E}<n$. Then, $\exists \,0\ne x\in\mathbb{C}^n$ with $x^{\mathsf{H}}\mathbf{E}=0$. Hence, $x^{\mathsf{H}}\mathbf{E}\ \mathbf{E}^{\mathsf{H}}=0$ would contradict \eqref{index-App}. \end{proof}
}
\section*{Acknowledgments}
The first author (FA) was supported by the Austrian Science Fund (FWF) via the FWF-funded SFB \# F65. The second author (AA) was supported by the Austrian Science Fund (FWF), partially via the FWF-doctoral school "Dissipation and dispersion in non-linear partial differential equations'' (\# W1245) and the FWF-funded SFB \# F65. The third author (VM) was supported by Deutsche Forschungsgemeinschaft (DFG) via the DFG-funded SFB \# 910.
\begin{comment} The first author (FA) was supported by the FWF-funded SFB \# F65. The second author (AA) was partially supported by the FWF-doctoral school "Dissipation and dispersion in non-linear partial differential equations'' and the FWF-funded SFB \# F65. AA is also grateful to the hospitality of the Newton Institute, Cambridge where part of this work was carried out. The third author (VM) was supported by DFG SFB \# 910. \end{comment}
\end{document} | arXiv |
Update: Be Wrong the Right Number of Times
Greg Novak
December 13, 2016 - San Francisco, CA
Who was wrong the right number of times?
In the days and weeks after the election, the Associated Press did not call a winner in Michigan as the race was too close. On November 28, the Michigan Board of State Canvassers finally certified the race as a victory for Mr. Trump. As of this writing, there are still recounts pending but the situation seems stable enough to revisit this question.
There's wrong and then there's wrong—the outcome of the election clearly indicates that the model used by FiveThirtyEight (538) was closer to the truth than that of the Princeton Election Consortium (PEC) in terms of the level of uncertainty in the predictions. It's not by as much as you might think, though.
Predictions and Outcomes
The final pre-election predictions of the Princeton Election Consortium (PEC) and Five Thirty Eight (538) agreed about the most likely winner in each of the fifty states, differing only in the degree of uncertainty in each race. This makes our comparison easy. The incorrectly predicted states were Florida, Michigan, North Carolina, Pennsylvania, and Wisconsin—five in total.
Reproducing the graph from the previous post:
Reading off of the graph, we see that getting five states incorrect is the most likely outcome for 538's predictions, while it was on the high end of reasonable possibility for the PEC's model. Based only on the number of states incorrectly predicted, 538's model is favored over the PEC's model by a bit better than 3:1 odds.
The Will to (Statistical) Power
This statistical test involving only the number of incorrectly predicted states is useful for discussing the treatment of uncertainty because it's easy to understand, but it's probably the weakest test we can we can devise for this case. We could do better by modeling the fifty states as separate terms of a single likelihood function and by taking state-by-state correlations into account.
A quick way to get an idea of what a more detailed analysis might show is to use the output of each model with respect to the national election. This must take into account all of the state-level correlations, and the people who produced each model have almost certainly thought harder about their model than anyone else—certainly harder than I am likely to think about it. This is quite a bit different from the problem we posed above—the overall election winner is determined not by the state results themselves, but rather a sum over the state results weighted by the number of electoral votes for each state.
This is a classic application of Bayes Theorem: it's easy to calculate the probability that Mr. Trump wins given that 538's model is correct (538 has kindly done this for us), but now we want to know the probability that 538's model is correct given that Mr. Trump won the national election. Taking P(Trump wins | 538's model is correct) = 18 percent and P(Trump wins | PEC's model is correct) = 1 percent, three lines of algebra allow us to compute the odds ratio:
\[\frac{P({\rm 538\ |\ Trump\ wins})}{P({\rm PEC\ |\ Trump\ wins})} = \frac{P({\rm Trump\ wins\ |\ 538})\ P({\rm 538})}{P({\rm Trump\ wins\ |\ PEC})\ P({\rm PEC})} = \frac{18}{1}\]
That is to say the odds are eighteen to one in favor of 538's model assuming the priors for each model are equal.
These two possibilities are probably upper and lower bounds on reality: 538's model is favored over the PEC's model with odds somewhere between 3 to 1 and 18 to 1 (and it's probably closer to the latter figure). Note that we have just characterized the meta-uncertainty—the uncertainty in the uncertainty—which is a useful thing to get in the habit of doing.
It's probably a subject of interest to the PEC. There were probably two issues contributing to the fact that their predictions were so far off. The first is correlated error: the distribution of polling results was not centered on the eventual outcome of the election—the mean of the distribution was off. The second is exactly this meta-uncertainty: the width of the distribution of possible outcomes was much wider than the PEC thought.
Would you take that bet?
One concrete way to "ask yourself" if you really believe a probability is correct is to phrase it in terms of a wager. If I really believed that the PEC model was correct, then I should be happy to take make a wager where if Mr. Trump wins I get $2 and if Secretary Clinton wins I pay $100. According to the PEC, Mr. Trump's chance of winning was 1 percent, so over repeated trials I expect to make $1 per trial on average.
There's some evidence that Sam Wang, the driving force behind the PEC, would have taken this bet. He famously tweeted that he would eat a bug if Mr. Trump got more than 240 electoral votes, and then followed through.
Let us pause to take this in: what would Professor Wang have gotten if he had won the bet? Not the ability to say "I told you so"; he would have had that anyway. All he would get is the ability to say "I told you so, and I was so sure that I was right that I offered to eat a bug, secure in the knowledge that that would never come to pass." Against that rather slender gain in the case of a win, consider the outcome in the case of a loss: eating a bug. This sounds pretty close to the proposed bet-$100-to-win-$2 wager proposed above.
Bug eating seems to be Professor Wang's go-to bet to make a point about his level of certainty. He's quite thoughtful about it, though, calibrating the level of confidence required to make such a bet to keep his rate of bug eating to acceptable levels. He certainly understands the idea of being wrong the right number of times.
Correlations are Key
Go back to our original proposed statistical test—the number of incorrectly predicted states assuming each state is independent. Suppose the number of incorrect predictions was much higher than the PEC predicted. Is this incontrovertible evidence that the PEC model was wrong? It's pretty clear the answer is no: the PEC might defensibly say "That's because your independence assumption was bad: state results are correlated and they were systematically off in one direction—that's why you think we missed too many states. If you take those correlations into account, the model is fine."
What if the number of incorrectly predicted states was zero: much lower than 538 predicted. Is that incontrovertible evidence that 538's model was wrong, or could they make essentially the same claim as above?
Phrased more technically and simplified a bit for clarity: Suppose I make 100 simultaneous bets on the outcome of 100 Bernoulli-type random variables, where the probability of success in any one of them is 50%. So far I've specified the mean and diagonal terms of the covariance matrix of the 100 dimensional probability density function. If the random variables are uncorrelated so that all of the off-diagonal terms are zero, then I should be very surprised if I win all 100 bets simultaneously on a single trial. The probability of this is about \( 10^{-30} \). The question: Is there a way to choose the off-diagonal terms of the covariance matrix so that it's not surprising if I win all 100 bets on a single trial?
My intuition was that correlations could increase the number of incorrect predictions, but not decrease it. This turns out to be wrong—off diagonal terms in the covariance matrix can move the number up or down. This is pretty obvious once you think about it the right way, but it was not my initial intuition.
The right way to think about it to see that this can happen is to imagine that a single underlying random variable controls all 100 of the random variables I defined above. All of these variables are perfectly correlated and the covariance matrix is singular. When the underlying random variable comes up 0, then random variables 1-50 come up 1 and the rest come up 0. Conversely when the underlying random variable comes up 1, then random variables 51-100 come up 1 and the rest come up 0. It will be pretty easy to recognize this pattern and place 100 bets where you either win all of them or lose all of them. In this situation, it would not be surprising at all to either win or lose all of the bets from a single trial simultaneously.
Although both models were wrong in predicting the overall election winner, 538's model was much closer to being wrong the right number of times in the state-by-state election results. The PEC's model involved too high a level of certainty in the outcome and is disfavored by as much as 18 to 1 odds as a result.
Covariance can turn an extremely surprising result into one that's not surprising at all. Unmodeled correlations are probably the single biggest factor that make data science difficult by making it hard to come to secure conclusions about causal processes from historical data alone.
Phrasing a statistical result in terms of a wager can be a useful concrete way to ask yourself whether you really believe the number that's coming out of your analysis. This helps you characterize the uncertainty in your uncertainty: meta-uncertainty. | CommonCrawl |
Joanne Welsman
Knapp KM, Welsman JR, Rowlands AV, MacLeod KM (In Press). Prolonged disuse osteopoenia 14 years post external fixation removal. Osteoporosis International
Armstrong N, Welsman J (2021). Comment on 'Developing a New Curvilinear Allometric Model to Improve the Fit and Validity of the 20-m Shuttle Run Test as a Predictor of Cardiorespiratory Fitness in Adults and Youth'. Sports Med, 51(7), 1591-1593. Author URL.
Armstrong N, Welsman J (2020). Influence of sex-specific concurrent changes in age, maturity status, and morphological covariates on the development of peak ventilatory variables in 10–17-year-olds. European Journal of Applied Physiology, 121(3), 783-792.
Influence of sex-specific concurrent changes in age, maturity status, and morphological covariates on the development of peak ventilatory variables in 10–17-year-olds
. Purposes
. (i) to investigate the influence of concurrent changes in age, maturity status, stature, body mass, and skinfold thicknesses on the development of peak ventilatory variables in 10–17-year-olds; and, (ii) to evaluate the interpretation of paediatric norm tables of peak ventilatory variables.
. Methods
. Multiplicative multilevel modelling which allows both the number of observations per individual and the temporal spacing of the observations to vary was used to analyze the expired ventilation (peak $${\dot{\mathrm{V}}}_{\mathrm{E}}$$
. V
. ˙
. E
. ) and tidal volume (peak VT) at peak oxygen uptake of 420 (217 boys) 10–17-year-olds. Models were founded on 1053 (550 from boys) determinations of peak ventilatory variables supported by anthropometric measures and maturity status.
. Results
. In sex-specific, multiplicative allometric models, concurrent changes in body mass and skinfold thicknesses (as a surrogate of FFM) and age were significant (p < 0.05) explanatory variables of the development of peak $${\dot{\mathrm{V}}}_{\mathrm{E}}$$
. once these covariates had been controlled for stature had no additional, significant (p > 0.05) effect on peak $${\dot{\mathrm{V}}}_{\mathrm{E}}$$
. Concurrent changes in age, stature, body mass, and skinfold thicknesses were significant (p < 0.05) explanatory variables of the development of peak VT. Maturity status had no additional, significant (p > 0.05) effect on either peak $${\dot{\mathrm{V}}}_{\mathrm{E}}$$
. or peak VT once age and morphological covariates had been controlled for.
. Conclusions
. Elucidation of the sex-specific development of peak $${\dot{\mathrm{V}}}_{\mathrm{E}}$$
. requires studies which address concurrent changes in body mass, skinfold thicknesses, and age. Stature is an additional explanatory variable in the development of peak VT, in both sexes. Paediatric norms based solely on age or stature or body mass are untenable.
Armstrong N, Welsman J (2020). Multilevel allometric modelling of maximum cardiac output, maximum arteriovenous oxygen difference, and peak oxygen uptake in 11–13-year-olds. European Journal of Applied Physiology, 120(2), 527-537.
Multilevel allometric modelling of maximum cardiac output, maximum arteriovenous oxygen difference, and peak oxygen uptake in 11–13-year-olds
To investigate longitudinally (1) the contribution of morphological covariates to explaining the development of maximum cardiac output ($${\dot{\text{Q}}}$$Q˙ max) and maximum arteriovenous oxygen difference (a-vO2 diff max), (2) sex differences in $${\dot{\text{Q}}}$$Q˙ max and a-vO2 diff max once age, maturity status, and morphological covariates have been controlled for, and, (3) the contribution of concurrent changes in morphological and cardiovascular covariates to explaining the sex-specific development of peak oxygen uptake ($$\dot{{V}}{\mathrm{O}}_{2}$$V˙O2).
Fifty-one (32 boys) 11–13-year-olds had their peak $$\dot{{V}}{\mathrm{O}}_{2}$$V˙O2, maximum heart rate (HR max), $${\dot{\text{Q}}}$$Q˙ max, and a-vO2 diff max determined during treadmill running on three annual occasions. The data were analysed using multilevel allometric modelling.
There were no sex differences in HR max which was not significantly (p > 0.05) correlated with age, morphological variables, or peak $$\dot{{V}}{\mathrm{O}}_{2}$$V˙O2. The best-fit models for $${\dot{\text{Q}}}$$Q˙ max and a-vO2 diff max were with fat-free mass (FFM) as covariate with age, maturity status, and haemoglobin concentration not significant (p > 0.05). FFM was the dominant influence on the development of peak $$\dot{{V}}{\mathrm{O}}_{2}$$V˙O2. With FFM controlled for, the introduction of either $${\dot{\text{Q}}}$$Q˙ max or a-vO2 diff max to multilevel models of peak $$\dot{{V}}{\mathrm{O}}_{2}$$V˙O2 resulted in significant (p < 0.05) additional contributions to explaining the sex difference.
(1) with FFM controlled for, there were no sex differences in $${\dot{\text{Q}}}$$Q˙ max or a-vO2 diff max, (2) FFM was the dominant influence on the development of peak $$\dot{{V}}{\mathrm{O}}_{2}$$V˙O2, and (3) with FFM and either $${\dot{\text{Q}}}$$Q˙ max or a-vO2 diff max controlled for, there remained an unresolved sex difference of ~ 4% in peak $$\dot{{V}}{\mathrm{O}}_{2}. $$V˙O2.
Armstrong N, Welsman J (2020). Scientific Rigour in the Assessment and Interpretation of Youth Cardiopulmonary Fitness: a Response to the Paper 'Normative Reference Values and International Comparisons for the 20-Metre Shuttle Run Test: Analysis of 69,960 Test Results among Chinese Children and Youth'. JOURNAL OF SPORTS SCIENCE AND MEDICINE, 19(3), 627-628. Author URL. Full text.
Armstrong N, Welsman J (2020). The Development of Aerobic and Anaerobic Fitness with Reference to Youth Athletes. Journal of Science in Sport and Exercise, 2(4), 275-286.
The Development of Aerobic and Anaerobic Fitness with Reference to Youth Athletes
. Purpose
. To challenge current conventions in paediatric sport science and use data from recent longitudinal studies to elucidate the development of aerobic and anaerobic fitness, with reference to youth athletes.
. (1) to critically review the traditional practice of ratio scaling physiological variables with body mass and, (2) to use multiplicative allometric models of longitudinal data, founded on 1053 (550 from boys) determinations of 10–17-year-olds' peak oxygen uptake ($$ {{\text{V}}\text{O}}_{2} $$
. VO
. ) and 763 (405 from boys) determinations of 11–17-year-olds' peak power output (PP) and mean power output (MP), to investigate the development of aerobic and anaerobic fitness in youth.
. The statistical assumptions underpinning ratio scaling of physiological variables in youth are seldom met. Multiplicative allometric modelling of longitudinal data has demonstrated that fat free mass (FFM) acting as a surrogate for active muscle mass, is the most powerful morphological influence on PP, MP, and peak $$ {{\text{V}}\text{O}}_{2} $$
. With FFM appropriately controlled for, age effects remain significant but additional, independent effects of maturity status on anaerobic and aerobic fitness are negated.
. Ratio scaling of physiological variables with body mass is fallacious, confounds interpretation of the development of anaerobic and aerobic fitness, and misleads fitness comparisons within and across youth sports. Rigorous evaluation of the development of anaerobic and aerobic fitness in youth requires longitudinal analyses of sex-specific, concurrent changes in age- and maturation-driven morphological covariates. Age and maturation-driven changes in FFM are essential considerations when evaluating the physiological development of youth athletes.
Long H (2020). Understanding why primary care doctors leave direct patient care: a systematic review of qualitative research. BMJ Open, 10 Full text.
Welsman J, Armstrong N (2019). Children's fitness and health: an epic scandal of poor methodology, inappropriate statistics, questionable editorial practices and a generation of misinformation. BMJ Evidence-Based Medicine, 26(1), 12-13.
Armstrong N, Weisman J (2019). Clarity and Confusion in the Development of Youth Aerobic Fitness. FRONTIERS IN PHYSIOLOGY, 10 Author URL.
Armstrong N, Welsman J, Bloxham S (2019). Development of 11- to 16-year-olds' short-term power output determined using both treadmill running and cycle ergometry. Eur J Appl Physiol, 119(7), 1565-1580.
Development of 11- to 16-year-olds' short-term power output determined using both treadmill running and cycle ergometry.
PURPOSE: to investigate the development of peak power output (PP) and mean power output (MP) during two different modes of exercise in relation to sex and concurrent changes in age, body mass, fat-free mass (FFM), maturity status and, in the case of MP, peak oxygen uptake ([Formula: see text]). METHODS: PP and MP were determined cycling against a fixed braking force (Wingate anaerobic test) and running on a non-motorized treadmill. Peak [Formula: see text] was determined using cycle ergometry and treadmill running. 135 (63 girls) students initially aged 11-14 years were tested over 2 days on three annual occasions. The data were analysed using multiplicative allometric modelling which enables the effects of variables to be partitioned concurrently within an allometric framework. Multiplicative models were founded on 301 (138 from girls) determinations of PP and MP on each ergometer. RESULTS: with body mass controlled for, both PP and MP increased with age but maturity status did not independently contribute to any of the multiplicative allometric models. Boys' PP and MP were significantly (p
Armstrong N, Welsman J (2019). Development of peak oxygen uptake from 11-16 years determined using both treadmill and cycle ergometry. Eur J Appl Physiol, 119(3), 801-812.
Development of peak oxygen uptake from 11-16 years determined using both treadmill and cycle ergometry.
PURPOSES: to investigate the development of peak oxygen uptake ([Formula: see text]) assessed on both a treadmill and a cycle ergometer in relation with sex and concurrent changes in age, body mass, fat-free mass (FFM), and maturity status and to evaluate currently proposed 'clinical red flags' or health-related cut-points for peak [Formula: see text]. METHODS: Multiplicative multilevel modelling, which enables the effects of variables to be partitioned concurrently within an allometric framework, was used to analyze the peak [Formula: see text]s of 138 (72 boys) students initially aged 11-14 years and tested on three annual occasions. Models were founded on 640 (340 from boys) determinations of peak [Formula: see text], supported by anthropometric measures and maturity status. RESULTS: Mean peak [Formula: see text]s were 11-14% higher on a treadmill. The data did not meet the statistical assumptions underpinning ratio scaling of peak [Formula: see text] with body mass. With body mass appropriately controlled for boys' peak [Formula: see text]s were higher than girls' values and the difference increased with age. The development of peak [Formula: see text] was sex-specific, but within sex models were similar on both ergometers with FFM the dominant anthropometric factor. CONCLUSIONS: Data should not be pooled for analysis but data from either ergometer can be used independently to interpret the development of peak [Formula: see text] in youth. On both ergometers and in both sexes, FFM is the most powerful morphological influence on the development of peak [Formula: see text]. 'Clinical red flags' or health-related cut-points proposed without consideration of exercise mode and founded on peak [Formula: see text] in ratio with body mass are fallacious.
Welsman J, Armstrong N (2019). Interpreting Aerobic Fitness in Youth: Alternatives to Ratio Scaling-A Response to Blais et al (2019). Pediatr Exerc Sci, 31(2), 256-257. Author URL.
Welsman J, Armstrong N (2019). Interpreting Aerobic Fitness in Youth: the Fallacy of Ratio Scaling. Pediatr Exerc Sci, 31(2), 184-190.
Interpreting Aerobic Fitness in Youth: the Fallacy of Ratio Scaling.
In this paper, we draw on cross-sectional, treadmill-determined, peak oxygen uptake data, collected in our laboratory over a 20-year period, to examine whether traditional per body mass (ratio) scaling appropriately controls for body size differences in youth. From an examination of the work of pioneering scientists and the earliest studies of peak oxygen uptake, we show how ratio scaling appears to have no sound scientific or statistical rationale. Using simple methods based on correlation and regression, we demonstrate that the statistical relationships, which are assumed in ratio scaling, are not met in groups of similar aged young people. We also demonstrate how sample size and composition can influence relationships between body mass and peak oxygen uptake and show that mass exponents derived from log-linear regression effectively remove the effect of body mass. Indiscriminate use of ratio scaling to interpret young people's fitness, to raise "Clinical Red Flags", and to assess clinical populations concerns us greatly, as recommendations and conclusions based upon this method are likely to be spurious. We urge those involved with investigating youth fitness to reconsider how data are routinely scaled for body size.
Welsman JR, Armstrong N (2019). Interpreting Cardiorespiratory Fitness in Young Clinical Populations-Folklore and Fallacy. JAMA Pediatr, 173(8), 713-714. Author URL.
Armstrong N, Welsman J (2019). Interpreting Youth Aerobic Fitness: Appropriate Morphological Covariates-A Response to Cunha and Leites (2019). Pediatr Exerc Sci, 31(3), 388-389. Author URL.
Armstrong N, Welsman J (2019). Interpreting Youth Aerobic Fitness: Promoting Evidence-Based Discussion–A Response to Dotan (2019). Pediatr Exerc Sci, 31(3), 382-385.
Interpreting Youth Aerobic Fitness: Promoting Evidence-Based Discussion–A Response to Dotan (2019)
We welcome Raffy Dotan's Letter to the Editor (14) as it gives us another opportunity to promote evidence-based discussion of the development of youth aerobic fitness. Readers of our contributions to the 2019 Special Issue of Pediatric Exercise Science (6,27,28) will recall that we concluded with, "The authors encourage all pediatric exercise scientists to engage with this discussion, to share ideas and methods, and be willing to explore alternatives. There are many issues to resolve and constructive, collaborative debate will speed our collective aim toward a better understanding of pediatric aerobic fitness in health and disease" (27, p. 256). Not the words of authors preaching a "gospel" with "evangelistic persistence" as Dotan (14) suggests, but of scientists genuinely seeking to stimulate evidence-based discussion of the development of youth aerobic fitness and its relationship with health and well-being.
Armstrong N, Welsman J (2019). Multilevel allometric modelling of maximal stroke volume and peak oxygen uptake in 11–13-year-olds. European Journal of Applied Physiology, 119(11-12), 2629-2639.
Multilevel allometric modelling of maximal stroke volume and peak oxygen uptake in 11–13-year-olds
. To investigate (1) whether maximal stroke volume (SVmax) occurs at submaximal exercise intensities, (2) sex differences in SVmax once fat-free mass (FFM) has been controlled for, and, (3) the contribution of concurrent changes in FFM and SVmax to the sex-specific development of peak oxygen uptake $$ \left( {{\dot{\text{V}}\text{O}}_{2} } \right) $$V˙O2.
. The peak $$ {\dot{\text{V}}\text{O}}_{2} $$V˙O2 s of 61 (34 boys) 11–12-year-olds were determined and their SV determined during treadmill running at 2.28 and 2.50 m s−1 using carbon dioxide rebreathing. The SVmax and peak $$ {\dot{\text{V}}\text{O}}_{2} $$V˙O2 of 51 (32 boys) students who volunteered to be tested treadmill running at 2.50 m s−1 on three annual occasions were investigated using multilevel allometric modelling. The models were founded on 111 (71 from boys) determinations of SVmax, FFM, and peak $$ {\dot{\text{V}}\text{O}}_{2} $$V˙O2.
. Progressive increases in treadmill running speed resulted in significant (p < 0.01) increases in $$ {\dot{\text{V}}\text{O}}_{2} $$V˙O2, but SV levelled-off with nonsignificant (p > 0.05) changes within ~ 2–3%. In the multilevel models, SVmax increased proportionally to FFM0.72 and with FFM controlled for, there were no significant (p > 0.05) sex differences. Peak $$ {\dot{\text{V}}\text{O}}_{2} $$V˙O2 increased with FFM but after adjusting for FFM0.98, a significant (p < 0.05) sex difference in peak $$ {\dot{\text{V}}\text{O}}_{2} $$V˙O2 remained. Introducing SVmax to the multilevel model revealed a significant (p < 0.05), but small additional effect of SVmax on peak $$ {\dot{\text{V}}\text{O}}_{2} $$V˙O2.
. Fat-free mass explained sex differences in SVmax, but with FFM controlled for, there was still a ~ 5% sex difference in peak $$ {\dot{\text{V}}\text{O}}_{2} $$V˙O2. SVmax made a modest additional contribution to explain the development of peak $$ {\dot{\text{V}}\text{O}}_{2} , $$V˙O2, but there remained an unresolved sex difference of ~ 4%.
Campbell JL, Fletcher E, Abel G, Anderson R, Chilvers R, Dean SG, Richards SH, Sansom A, Terry R, Aylward A, et al (2019). Policies and strategies to retain and support the return of experienced GPs in direct patient care: the ReGROUP mixed-methods study. Health Services and Delivery Research, 7(14), 1-288.
Policies and strategies to retain and support the return of experienced GPs in direct patient care: the ReGROUP mixed-methods study
BackgroundUK general practice faces a workforce crisis, with general practitioner (GP) shortages, organisational change, substantial pressures across the whole health-care system and an ageing population with increasingly complex health needs. GPs require lengthy training, so retaining the existing workforce is urgent and important.Objectives(1) to identify the key policies and strategies that might (i) facilitate the retention of experienced GPs in direct patient care or (ii) support the return of GPs following a career break. (2) to consider the feasibility of potentially implementing those policies and strategies.DesignThis was a comprehensive, mixed-methods study.SettingThis study took place in primary care in England.ParticipantsGeneral practitioners registered in south-west England were surveyed. Interviews were with purposively selected GPs and primary care stakeholders. A RAND/UCLA Appropriateness Method (RAM) panel comprised GP partners and GPs working in national stakeholder organisations. Stakeholder consultations included representatives from regional and national groups.Main outcome measuresSystematic review – factors affecting GPs' decisions to quit and to take career breaks. Survey – proportion of GPs likely to quit, to take career breaks or to reduce hours spent in patient care within 5 years of being surveyed. Interviews – themes relating to GPs' decision-making. RAM – a set of policies and strategies to support retention, assessed as 'appropriate' and 'feasible'. Predictive risk modelling – predictive model to identify practices in south-west England at risk of workforce undersupply within 5 years. Stakeholder consultation – comments and key actions regarding implementing emergent policies and strategies from the research.ResultsPast research identified four job-related 'push' factors associated with leaving general practice: (1) workload, (2) job dissatisfaction, (3) work-related stress and (4) work–life balance. The survey, returned by 2248 out of 3370 GPs (67%) in the south-west of England, identified a high likelihood of quitting (37%), taking a career break (36%) or reducing hours (57%) within 5 years. Interviews highlighted three drivers of leaving general practice: (1) professional identity and value of the GP role, (2) fear and risk associated with service delivery and (3) career choices. The RAM panel deemed 24 out of 54 retention policies and strategies to be 'appropriate', with most also considered 'feasible', including identification of and targeted support for practices 'at risk' of workforce undersupply and the provision of formal career options for GPs wishing to undertake portfolio roles. Practices at highest risk of workforce undersupply within 5 years are those that have larger patient list sizes, employ more nurses, serve more deprived and younger populations, or have poor patient experience ratings. Actions for national organisations with an interest in workforce planning were identified. These included collection of data on the current scope of GPs' portfolio roles, and the need for formal career pathways for key primary care professionals, such as practice managers.LimitationsThe survey, qualitative research and modelling were conducted in one UK region. The research took place within a rapidly changing policy environment, providing a challenge in informing emergent policy and practice.ConclusionsThis research identifies the basis for current concerns regarding UK GP workforce capacity, drawing on experiences in south-west England. Policies and strategies identified by expert stakeholders after considering these findings are likely to be of relevance in addressing GP retention in the UK. Collaborative, multidisciplinary research partnerships should investigate the effects of rolling out some of the policies and strategies described in this report.Study registrationThis study is registered as PROSPERO CRD42016033876 and UKCRN ID number 20700.FundingThe National Institute for Health Research Health Services and Delivery Research programme.
Armstrong N, Welsman J (2019). Sex-Specific Longitudinal Modeling of Youth Peak Oxygen Uptake. Pediatr Exerc Sci, 31(2), 204-212.
Sex-Specific Longitudinal Modeling of Youth Peak Oxygen Uptake.
Purpose: to investigate peak oxygen uptake ( V˙O2 ) in relation to sex, age, body mass, fat-free mass (FFM), maturity, and overweight status. Methods: Multiplicative, allometric models of 10- to 18-year-olds were founded on 1057 determinations of peak V˙O2 supported by anthropometry and estimates of maturity status. Results: Baseline models with body mass controlled for showed age to exert a positive effect on peak V˙O2 , with negative estimates for age2, sex, and a sex-by-age interaction. Sex-specific models showed maturity status to have a positive effect on peak V˙O2 in addition to the effects of age and body mass. Introducing skinfold thicknesses to provide, with body mass, a surrogate for FFM explained maturity effects and yielded a significantly (P
Welsman J, Armstrong N (2019). The 20 m shuttle run is not a valid test of cardiorespiratory fitness in boys aged 11–14 years. BMJ Open Sport & Exercise Medicine, 5(1), e000627-e000627.
The 20 m shuttle run is not a valid test of cardiorespiratory fitness in boys aged 11–14 years
ObjectivesThe 20 m shuttle run test (20mSRT) is used to estimate cardiorespiratory fitness (CRF) through the prediction of peak oxygen uptake (V˙O2), but its validity as a measure of CRF during childhood and adolescence is questionable. This study examined the validity of the 20mSRT to predict peak V˙O2. MethodsPeak V˙O2 was measured during treadmill running. Log-linear regression was used to correct peak V˙O2 for body mass and sum of skinfolds plus age. Boys completed the 20mSRT under standardised conditions. Maximum speed (km/h) was used with age to predict peak V˙O2 using the equation developed by Léger et al. Validity was examined from linear regression methods and limits of agreement (LoA). Relationships between 20mSRT performance and allometrically adjusted peak V˙O2, and predicted per cent fat were examined.ResultsThe sample comprised 76 boys aged 11–14 years. Predicted and measured mass-related peak V˙O2 (mL/kg/min) shared common variance of 32%. LoA revealed that measured peak V˙O2 ranged from 15% below to 25% above predicted peak V˙O2. There were no significant relationships (p>0.05) between predicted peak V˙O2 and measured peak V˙O2 adjusted for mass, age and skinfold thicknesses. Adjusted for body mass and age, peak V˙O2 was not significantly related (p>0.05) to 20mSRT final speed but a weak, statistically significant (r=0.24, p<0.05) relationship was found with peak V˙O2 adjusted for mass and fatness. Predicted per cent fat was negatively correlated with 20mSRT speed (r=−0.61, p<0.001).ConclusionsThe 20mSRT reflects fatness rather than CRF and has poor validity grounded in its flawed estimation and interpretation of peak V˙O2 in mL/kg/min.
Meakin JR, Ames RM, Jeynes JCG, Welsman J, Gundry M, Knapp K, Everson R (2019). The feasibility of using citizens to segment anatomy from medical images: Accuracy and motivation. PLOS ONE, 14(10), e0222523-e0222523. Full text.
Armstrong N, Welsman J (2019). Twenty-metre shuttle run: (mis)representation, (mis)interpretation and (mis)use. Br J Sports Med, 53(19). Author URL.
Armstrong N, Welsman J (2019). Youth cardiorespiratory fitness: evidence, myths and misconceptions. Bulletin of the World Health Organization, 97(11), 777-782. Full text.
Sansom A, Terry R, Fletcher E, Salisbury C, Long L, Richards SH, Aylward A, Welsman J, Sims L, Campbell JL, et al (2018). Why do GPs leave direct patient care and what might help to retain them? a qualitative study of GPs in South West England. BMJ Open, 8
Why do GPs leave direct patient care and what might help to retain them? a qualitative study of GPs in South West England
Objective to identify factors influencing general practitioners' (GPs') decisions about whether or not to remain in direct patient care in general practice and what might help to retain them in that role. Design Qualitative, in-depth, individual interviews exploring factors related to GPs leaving, remaining in and returning to direct patient care. Setting South West England, UK. Participants 41 GPs: 7 retired; 8 intending to take early retirement; 11 who were on or intending to take a career break; 9 aged under 50 years who had left or were intending to leave direct patient care and 6 who were not intending to leave or to take a career break. Plus 19 stakeholders from a range of primary care-related professional organisations and roles. Results Reasons for leaving direct patient care were complex and based on a range of job-related and individual factors. Three key themes underpinned the interviewed GPs' thinking and rationale: issues relating to their personal and professional identity and the perceived value of general practice-based care within the healthcare system; concerns regarding fear and risk, for example, in respect of medical litigation and managing administrative challenges within the context of increasingly complex care pathways and environments; and issues around choice and volition in respect of personal social, financial, domestic and professional considerations. These themes provide increased understanding of the lived experiences of working in today's National Health Service for this group of GPs. Conclusion Future policies and strategies aimed at retaining GPs in direct patient care should clarify the role and expectations of general practice and align with GPs' perception of their own roles and identity; demonstrate to GPs that they are valued and listened to in planning delivery of the UK healthcare; target GPs' concerns regarding fear and risk, seeking to reduce these to manageable levels and give GPs viable options to support them to remain in direct patient care.
Gibson A, Welsman J, Britten N (2017). Evaluating patient and public involvement in health research: from theoretical model to practical workshop. Health Expect, 20(5), 826-835.
Evaluating patient and public involvement in health research: from theoretical model to practical workshop.
BACKGROUND: There is a growing literature on evaluating aspects of patient and public involvement (PPI). We have suggested that at the core of successful PPI is the dynamic interaction of different forms of knowledge, notably lay and professional. We have developed a four-dimensional theoretical framework for understanding these interactions. AIM: We explore the practical utility of the theoretical framework as a tool for mapping and evaluating the experience of PPI in health services research. METHODS: We conducted three workshops with different PPI groups in which participants were invited to map their PPI experiences on wall charts representing the four dimensions of our framework. The language used to describe the four dimensions was modified to make it more accessible to lay audiences. Participants were given sticky notes to indicate their own positions on the different dimensions and to write explanatory comments if they wished. Participants' responses were then discussed and analysed as a group. RESULTS: the three groups were distinctive in their mapped responses suggesting different experiences in relation to having a strong or weak voice in their organization, having few or many ways of getting involved, addressing organizational or public concerns and believing that the organization was willing to change or not. DISCUSSION: the framework has practical utility for mapping and evaluating PPI interactions and is sensitive to differences in PPI experiences within and between different organizations. The workshops enabled participants to reflect collaboratively on their experiences with a view to improving PPI experiences and planning for the future.
Bigotti F, Welsman J, Taylor D (2017). Recreating the Pulsilogium of Santorio: Outlines for a Historically-Engaged Endeavour. Bulletin- Scientific Instrument Society, 133, 30-35. Full text.
Hopkins SJ, Toms AD, Brown M, Welsman JR, Ukoumunne OC, Knapp KM (2016). A study investigating short- and medium-term effects on function, bone mineral density and lean tissue mass post-total knee replacement in a Caucasian female post-menopausal population: implications for hip fracture risk. Osteoporos Int, 27(8), 2567-2576.
A study investigating short- and medium-term effects on function, bone mineral density and lean tissue mass post-total knee replacement in a Caucasian female post-menopausal population: implications for hip fracture risk.
UNLABELLED: Significant increased hip fracture incidence has been reported in the year following total knee replacement. This study demonstrates that bone and muscle loss is a post-surgical consequence of total knee replacement, alongside poor outcomes in function and activity potentially contributing to reduced quality of life and increased hip fracture risk. INTRODUCTION: a significant increase in hip fracture incidence in the year following total knee replacement (TKR) surgery has been reported. This study investigated function and activity following TKR and the effects of limited mobility on bone and muscle loss and their potential contribution to hip fracture risk. METHODS: Changes in dual-energy X-ray absorptiometry (DXA) (GE Lunar Prodigy, Bedford MA), bone mineral density (BMD) at the neck of femur (NOF), total hip region (TH) and lumbar spine were measured alongside leg lean tissue mass (LLTM) in post-menopausal Caucasian females following TKR (N = 19) compared to controls (N = 43). Lumbar spine trabecular bone scores (TBSs) were calculated. Ipsilateral/contralateral weight bearing, lower limb function, 3-day pedometer readings, pain levels and falls were also recorded. Measurements were obtained at pre-surgery baseline and at 6 weeks, 6 months and 12 months post-surgery. RESULTS: No statistically significant differences were demonstrated between groups at baseline bilaterally in LLTM or BMD at the NOF and TH. Losses in ipsilateral NOF and TH BMD and contralateral LLTM were significantly higher in the TKR group at 6 months. Impairment in function and weight bearing persisted in the TKR group 12 months post-operatively alongside deficits in bilateral muscle mass and ipsilateral NOF and TH BMD. Falls incidence was not significantly higher in the TKR group. CONCLUSIONS: Bone loss at the hip with associated muscle loss is a consequence of TKR that, in addition to poor patient outcomes in function and activity, potentially contributes to increased hip fracture risk in the year following surgery.
Pentecost C, Farrand P, Greaves CJ, Taylor RS, Warren FC, Hillsdon M, Green C, Welsman JR, Rayson K, Evans PH, et al (2015). Combining behavioural activation with physical activity promotion for adults with depression: findings of a parallel-group pilot randomised controlled trial (BAcPAc). Trials, 16(367), 1-15.
Combining behavioural activation with physical activity promotion for adults with depression: findings of a parallel-group pilot randomised controlled trial (BAcPAc)
Depression is associated with physical inactivity, which may mediate the relationship between depression and a range of chronic physical health conditions. However, few interventions have combined a psychological intervention for depression with behaviour change techniques, such as behavioural activation (BA), to promote increased physical activity.
To determine procedural and clinical uncertainties to inform a definitive randomised controlled trial (RCT), a pilot parallel-group RCT was undertaken within two Improving Access to Psychological Therapies (IAPT) services in South West England. We aimed to recruit 80 adults with depression and randomise them to a supported, written self-help programme based on either BA or BA plus physical activity promotion (BAcPAc). Data were collected at baseline and 4 months post-randomisation to evaluate trial retention, intervention uptake and variance in outcomes to inform a sample size calculation. Qualitative data were collected from participants and psychological wellbeing practitioners (PWPs) to assess the acceptability and feasibility of the trial methods and the intervention. Routine data were collected to evaluate resource use and cost.
Sixty people with depression were recruited, and a 73 % follow-up rate was achieved. Accelerometer physical activity data were collected for 64 % of those followed. Twenty participants (33 %) attended at least one treatment appointment. Interview data were analysed for 15 participants and 9 study PWPs. The study highlighted the challenges of conducting an RCT within existing IAPT services with high staff turnover and absences, participant scheduling issues, PWP and participant preferences for cognitive focussed treatment, and deviations from BA delivery protocols. The BAcPAc intervention was generally acceptable to patients and PWPs.
Although recruitment procedures and data collection were challenging, participants generally engaged with the BAcPAc self-help booklets and reported willingness to increase their physical activity. A number of feasibility issues were identified, in particular the under-use of BA as a treatment for depression, the difficulty that PWPs had in adapting their existing procedures for study purposes and the instability of the IAPT PWP workforce. These problems would need to be better understood and resolved before proceeding to a full-scale RCT.
Knapp KM, Welsman JR, Hopkins SJ, Shallcross A, Fogelman I, Blake GM (2015). Obesity increases precision errors in total body dual-energy x-ray absorptiometry measurements. Journal of Clinical Densitometry, 18(2), 209-216.
Obesity increases precision errors in total body dual-energy x-ray absorptiometry measurements
Total body (TB) dual-energy X-ray absorptiometry (DXA) is increasingly being used to measure body composition in research and clinical settings. This study investigated the effect of body mass index (BMI) and body fat on precision errors for total and regional TB DXA measurements of bone mineral density, fat tissue, and lean tissue using the GE Lunar Prodigy (GE Healthcare, Bedford, UK). One hundred forty-four women with BMI's ranging from 18.5 to 45.9kg/m2 were recruited. Participants had duplicate DXA scans of the TB with repositioning between examinations. Participants were divided into 3 groups based on their BMI, and the root mean square standard deviation and the percentage coefficient of variation were calculated for each group. The root mean square standard deviation (percentage coefficient of variation) for the normal (30kg/m2; n=32) BMI groups, respectively, were total BMD (g/cm2): 0.009 (0.77%), 0.009 (0.69%), 0.011 (0.91%); total fat (g): 545 (2.98%), 486 (1.72%), 677 (1.55%); total lean (g): 551 (1.42%), 540 (1.34%), and 781 (1.68%). These results suggest that serial measurements in obese subjects should be treated with caution because the least significant change may be larger than anticipated.
Farrand P, Pentecost C, Greaves C, Taylor RS, Warren F, Green C, Hillsdon M, Evans P, Welsman J, Taylor AH, et al (2014). A written self-help intervention for depressed adults comparing behavioural activation combined with physical activity promotion with a self-help intervention based upon behavioural activation alone: study protocol for a parallel group pilot randomised controlled trial (BAcPAc). Trials, 15
A written self-help intervention for depressed adults comparing behavioural activation combined with physical activity promotion with a self-help intervention based upon behavioural activation alone: study protocol for a parallel group pilot randomised controlled trial (BAcPAc).
BACKGROUND: Challenges remain to find ways to support patients with depression who have low levels of physical activity (PA) to overcome perceived barriers and enhance the perceived value of PA for preventing future relapse. There is an evidence-base for behavioural activation (BA) for depression, which focuses on supporting patients to restore activities that have been avoided, but practitioners have no specific training in promoting PA. We aimed to design and evaluate an integrated BA and PA (BAcPAc) practitioner-led, written, self-help intervention to enhance both physical and mental health. METHODS/DESIGN: This study is informed by the Medical Research Council Complex Intervention Framework and describes a protocol for a pilot phase II randomised controlled trial (RCT) to test the feasibility and acceptability of the trial methods to inform a definitive phase III RCT. Following development of the augmented written self-help intervention (BAcPAc) incorporating behavioural activation with physical activity promotion, depressed adults are randomised to receive up to 12 sessions over a maximum of 4 months of either BAcPAc or behavioural activation alone within a written self-help format, which represents treatment as usual. The study is located within two 'Improving Access to Psychological Therapies' services in South West England, with both written self-help interventions supported by mental health paraprofessionals. Measures assessed at 4, 9, and 12 month follow-up include the following: CIS-R, PHQ-9, accelerometer recorded (4 months only) and self-reported PA, body mass index, blood pressure, Insomnia Severity Index, quality of life, and health and social care service use. Process evaluation will include analysis of recorded support sessions and patient and practitioner interviews. At the time of writing the study has recruited 60 patients. DISCUSSION: the feasibility outcomes will inform a definitive RCT to assess the clinical and cost-effectiveness of the augmented BAcPAc written self-help intervention to reduce depression and depressive relapse, and bring about improvements across a range of physical health outcomes. TRIAL REGISTRATION: Current Controlled Trials ISRCTN74390532, 26.03.2013.
Welsman J, Gibson A, Heaton J, Britten N (2014). Involving patients and the public in healthcare operational research. BMJ, 349 Author URL.
Hopkins SJ, Welsman JR, Knapp KM (2014). Short-term Precision Error in Dual Energy X-Ray Absorptiometry, Bone Mineral Density and Trabecular Bone Score Measurements; and Effects of Obesity on Precision Error. Journal of Biomedical Graphics and Computing, 4(2), 8-14. Full text.
Hopkins S, Smith C, Toms A, Brown M, Welsman J, Knapp K (2013). Evaluation of a dual-scales method to measure weight-bearing through the legs, and effects of weight-bearing inequalities on hip bone mineral density and leg lean tissue mass. J Rehabil Med, 45(2), 206-210.
Evaluation of a dual-scales method to measure weight-bearing through the legs, and effects of weight-bearing inequalities on hip bone mineral density and leg lean tissue mass.
OBJECTIVE: to investigate: the accuracy of measuring relative left/right weight-bearing using two identically calibrated weighing scales; the short-term weight-bearing tendencies in a general population of 9 participants and long-term in 42 females; the effect weight-bearing inequalities on hip bone mineral density and leg lean tissue mass. METHOD: Participants were measured standing astride two scales. Short-term volunteers were measured 10 times on one visit, with repositioning between measurements and the long-term group were measured on three visits at 6 month intervals. Baseline bilateral hip and total body Dual X-ray Absorptiometry scans were performed on the long-term group. RESULTS: the short-term Coefficient of Variation is 5.41% and long-term 7.01%. No significant correlations were found between hip bone density differences and weight-bearing inequalities, although a weak correlation of r = 0.31 (p = 0.047) was found for differences in leg lean tissue mass. CONCLUSION: Left/right weight-bearing measured using two scales is a consistent method for evaluating weight distribution through the legs. The short- and long-term weight-bearing tendencies showed a similar degree of variation. Weight-bearing inequalities were not associated with any significant left/right differences in bone mineral density at the hip, but were weakly associated with left-right differences in leg muscle mass.
Meakin JR, Fulford J, Seymour R, Welsman JR, Knapp KM (2013). The relationship between sagittal curvature and extensor muscle volume in the lumbar spine. Journal of Anatomy, 222(6), 608-614.
The relationship between sagittal curvature and extensor muscle volume in the lumbar spine
A previous modelling study predicted that the forces applied by the extensor muscles to stabilise the lumbar spine would be greater in spines that have a larger sagittal curvature (lordosis). Because the force-generating capacity of a muscle is related to its size, it was hypothesised that the size of the extensor muscles in a subject would be related to the size of their lumbar lordosis. Magnetic resonance imaging (MRI) data were obtained, together with age, height, body mass and back pain status, from 42 female subjects. The volume of the extensor muscles (multifidus and erector spinae) caudal to the mid-lumbar level was estimated from cross-sectional area measurements in axial T1-weighted MRIs spanning the lumbar spine. Lower lumbar curvature was determined from sagittal T1-weighted images. A stepwise linear regression model was used to determine the best predictors of muscle volume. The mean lower lumbar extensor muscle volume was 281cm3 (SD=49cm3). The mean lower lumbar curvature was 30° (SD=7°). Five subjects reported current back pain and were excluded from the regression analysis. Nearly half the variation in muscle volume was accounted for by the variables age (standardised coefficient, B=-3.2, P=0.03) and lower lumbar curvature (B=0.47, P=0.002). The results support the hypothesis that extensor muscle volume in the lower lumbar spine is related to the magnitude of the sagittal curvature; this has implications for assessing muscle size as an indicator of muscle strength. © 2013 Anatomical Society.
McNarry MA, Welsman JR, Jones AM (2012). Influence of training status and maturity on pulmonary O2 uptake recovery kinetics following cycle and upper body exercise in girls. Pediatr Exerc Sci, 24(2), 246-261.
Influence of training status and maturity on pulmonary O2 uptake recovery kinetics following cycle and upper body exercise in girls.
The influence of training status on pulmonary VO(2) recovery kinetics, and its interaction with maturity, has not been investigated in young girls. Sixteen prepubertal (Pre: trained (T, 11.4 ± 0.7 years), 8 untrained (UT, 11.5 ± 0.6 years)) and 8 pubertal (Pub: 8T, 14.2 ± 0.7 years; 8 UT, 14.5 ± 1.3 years) girls completed repeat transitions from heavy intensity exercise to a baseline of unloaded exercise, on both an upper and lower body ergometer. The VO2 recovery time constant was significantly shorter in the trained prepubertal and pubertal girls during both cycle (Pre: T, 26 ± 4 vs. UT, 32 ± 6; Pub: T, 28 ± 2 vs. UT, 35 ± 7 s; both p <. 05) and upper body exercise (Pre: T, 26 ± 4 vs. UT, 35 ± 6; Pub: T, 30 ± 4 vs. UT, 42 ± 3 s; both p <. 05). No interaction was evident between training status and maturity. These results demonstrate the sensitivity of VO(2) recovery kinetics to training in young girls and challenge the notion of a "maturational threshold" in the influence of training status on the physiological responses to exercise and recovery.
Knapp KM, Welsman JR, Hopkins SJ, Fogelman I, Blake GM (2012). Obesity Increases Precision Errors in Dual-Energy X-Ray Absorptiometry Measurements. Journal of Clinical Densitometry, 15(3), 315-319.
Obesity Increases Precision Errors in Dual-Energy X-Ray Absorptiometry Measurements
The precision errors of dual-energy X-ray absorptiometry (DXA) measurements are important for monitoring osteoporosis. This study investigated the effect of body mass index (BMI) on precision errors for lumbar spine (LS), femoral neck (NOF), total hip (TH), and total body (TB) bone mineral density using the GE Lunar Prodigy. One hundred two women with BMIs ranging from 18.5 to 45.9kg/m 2 were recruited. Participants had duplicate DXA scans of the LS, left hip, and TB with repositioning between scans. Participants were divided into 3 groups based on their BMI and the percentage coefficient of variation (%CV) calculated for each group. The %CVs for the normal (30kg/m 2) (n=28) BMI groups, respectively, were LS BMD: 0.99%, 1.30%, and 1.68%; NOF BMD: 1.32%, 1.37%, and 2.00%; TH BMD: 0.85%, 0.88%, and 1.06%; TB BMD: 0.66%, 0.73%, and 0.91%. Statistically significant differences in precision error between the normal and obese groups were found for LS (p=0.0006), NOF (p=0.005), and TB BMD (p=0.025). These results suggest that serial measurements in obese subjects should be treated with caution because the least significant change may be larger than anticipated. © 2012 the International Society for Clinical Densitometry.
Knapp KM, Welsman JR, Hopkins SJ, Fogelman I, Blake GM (2012). Obesity increases precision errors in dual-energy X-ray absorptiometry measurements. J Clin Densitom, 15(3), 315-319.
Obesity increases precision errors in dual-energy X-ray absorptiometry measurements.
The precision errors of dual-energy X-ray absorptiometry (DXA) measurements are important for monitoring osteoporosis. This study investigated the effect of body mass index (BMI) on precision errors for lumbar spine (LS), femoral neck (NOF), total hip (TH), and total body (TB) bone mineral density using the GE Lunar Prodigy. One hundred two women with BMIs ranging from 18.5 to 45.9 kg/m(2) were recruited. Participants had duplicate DXA scans of the LS, left hip, and TB with repositioning between scans. Participants were divided into 3 groups based on their BMI and the percentage coefficient of variation (%CV) calculated for each group. The %CVs for the normal (30 kg/m(2)) (n=28) BMI groups, respectively, were LS BMD: 0.99%, 1.30%, and 1.68%; NOF BMD: 1.32%, 1.37%, and 2.00%; TH BMD: 0.85%, 0.88%, and 1.06%; TB BMD: 0.66%, 0.73%, and 0.91%. Statistically significant differences in precision error between the normal and obese groups were found for LS (p=0.0006), NOF (p=0.005), and TB BMD (p=0.025). These results suggest that serial measurements in obese subjects should be treated with caution because the least significant change may be larger than anticipated.
McNarry MA, Welsman JR, Jones AM (2011). Influence of training and maturity status on the cardiopulmonary responses to ramp incremental cycle and upper body exercise in girls. J Appl Physiol (1985), 110(2), 375-381.
Influence of training and maturity status on the cardiopulmonary responses to ramp incremental cycle and upper body exercise in girls.
It has been suggested that the potential for training to alter the physiological responses to exercise in children is related to a "maturational threshold". To address this, we investigated the interaction of swim-training status and maturity on cardiovascular and metabolic responses to lower and upper body exercise. Twenty-one prepubertal [Pre: 11 trained (T), 10 untrained (UT)], 30 pubertal (Pub: 14 T, 16 UT), and 18 postpubertal (Post: 8 T, 10 UT) girls completed ramp incremental exercise on a cycle and an upper body ergometer. In addition to pulmonary gas exchange measurements, stroke volume and cardiac output were estimated by thoracic bioelectrical impedance, and muscle oxygenation status was assessed using near-infrared spectroscopy. All T girls had a higher peak O(2) uptake during cycle (Pre: T 49 ± 5 vs. UT 40 ± 4; Pub: T 46 ± 5 vs. UT 36 ± 4; Post: T 48 ± 5 vs. UT 39 ± 8 ml·kg(-1)·min(-1); all P < 0.05) and upper body exercise (Pre: T 37 ± 6 vs. UT 32 ± 5; Pub: T 36 ± 5 vs. UT 28 ± 5; Post: T 39 ± 3 vs. UT 28 ± 7 ml·kg(-1)·min(-1); all P < 0.05). T girls also had a higher peak cardiac output during both modalities, and this reached significance in Pub (cycle: T 21 ± 3 vs. UT 18 ± 3; upper body: T 20 ± 4 vs. UT 15 ± 4 l/min; all P < 0.05) and Post girls (cycle: T 21 ± 4 vs. UT 17 ± 2; upper body: T 22 ± 3 vs. UT 18 ± 2 l/min; all P < 0.05). None of the measured pulmonary, cardiovascular, or metabolic parameters interacted with maturity, and the magnitude of the difference between T and UT girls was similar, irrespective of maturity stage. These results challenge the notion that differences in training status in young people are only evident once a maturational threshold has been exceeded.
McNarry MA, Welsman JR, Jones AM (2011). Influence of training status and exercise modality on pulmonary O2 uptake kinetics in pubertal girls. Eur J Appl Physiol, 111(4), 621-631.
Influence of training status and exercise modality on pulmonary O2 uptake kinetics in pubertal girls.
The influence of training status on the oxygen uptake (VO2) response to heavy intensity exercise in pubertal girls has not previously been investigated. We hypothesised that whilst training status-related adaptations would be evident in the VO2, heart rate (HR) and deoxyhaemoglobin ([HHb]) kinetics of pubertal swimmers during both lower and upper body exercise, they would be more pronounced during upper body exercise. Eight swim-trained (T; 14.2 ± 0.7 years) and eight untrained (UT; 14.5 ± 1.3 years) girls completed a number of constant-work-rate transitions on cycle and upper body ergometers at 40% of the difference between the gas exchange threshold and peak VO2. The phase II VO2 time constant (τ) was significantly shorter in the trained girls during both cycle (T: 21 ± 6 vs. UT: 35 ± 11 s; P < 0.01) and upper body exercise (T: 29 ± 8 vs. UT: 44 ± 8 s; P < 0.01). The VO2 slow component was not influenced by training status. The [HHb] τ was significantly shorter in the trained girls during both cycle (T: 12 ± 2 vs. UT: 20 ± 6 s; P < 0.01) and upper body exercise (T: 13 ± 3 vs. UT: 21 ± 7 s; P < 0.01), as was the HR τ (cycle, T: 36 ± 5 vs. UT: 53 ± 9 s; upper body, T: 32 ± 3 vs. UT: 43 ± 2; P < 0.01). This study suggests that both central and peripheral factors contribute to the faster VO2 kinetics in the trained girls and that differences are evident in both lower and upper body exercise.
McNarry MA, Welsman JR, Jones AM (2011). The influence of training and maturity status on girls' responses to short-term, high-intensity upper- and lower-body exercise. Appl Physiol Nutr Metab, 36(3), 344-352.
The influence of training and maturity status on girls' responses to short-term, high-intensity upper- and lower-body exercise.
A maturational threshold has been suggested to be present in young peoples' responses to exercise, with significant influences of training status evidenced only above this threshold. The presence of such a threshold has not been investigated for short-term, high-intensity exercise. To address this, we investigated the relationship between swim-training status and maturity on the power output, pulmonary gas exchange, and metabolic responses to an upper- and lower-body Wingate anaerobic test (WAnT). Girls at 3 stages of maturity participated:, prepubertal (Pre: 8 trained (T), 10 untrained (UT)), pubertal (Pub: 9 T, 15 UT), and postpubertal (Post: 8 T, 10 UT). At all maturity stages, T exhibited higher peak power (PP) and mean power (MP) during upper-body exercise (PP: Pre, T, 163 ± 20 vs. UT, 124 ± 29; Pub, T, 230 ± 42 vs. UT, 173 ± 41; Post, T, 245 ± 41 vs. UT, 190 ± 40 W; MP: Pre, T, 130 ± 23 vs. UT, 85 ± 26; Pub, T, 184 ± 37 vs. UT, 123 ± 38; Post, T, 200 ± 30 vs. UT, 150 ± 15 W; all p < 0.05) but not lower-body exercise, whilst the fatigue index was significantly lower in T for both exercise modalities. Irrespective of maturity, the oxidative contribution, calculated by the area under the oxygen uptake response profile, was not influenced by training status. No interaction was evident between training status and maturity, with similar magnitudes of difference between T and UT at all 3 maturity stages. These results suggest that there is no maturational threshold which must be surpassed for significant influences of training status to be manifest in the "anaerobic" exercise performance of young girls.
Winlove MA, Jones AM, Welsman JR (2010). Influence of training status and exercise modality on pulmonary O(2) uptake kinetics in pre-pubertal girls. Eur J Appl Physiol, 108(6), 1169-1179.
Influence of training status and exercise modality on pulmonary O(2) uptake kinetics in pre-pubertal girls.
The limited available evidence suggests that endurance training does not influence the pulmonary oxygen uptake (V(O)(2)) kinetics of pre-pubertal children. We hypothesised that, in young trained swimmers, training status-related adaptations in the V(O)(2) and heart rate (HR) kinetics would be more evident during upper body (arm cranking) than during leg cycling exercise. Eight swim-trained (T; 11.4 +/- 0.7 years) and eight untrained (UT; 11.5 +/- 0.6 years) girls completed repeated bouts of constant work rate cycling and upper body exercise at 40% of the difference between the gas exchange threshold and peak V(O)(2). The phase II V(O)(2) time constant was significantly shorter in the trained girls during upper body exercise (T: 25 +/- 3 vs. UT: 37 +/- 6 s; P < 0.01), but no training status effect was evident in the cycle response (T: 25 +/- 5 vs. UT: 25 +/- 7 s). The V(O)(2) slow component amplitude was not affected by training status or exercise modality. The time constant of the HR response was significantly faster in trained girls during both cycle (T: 31 +/- 11 vs. UT: 47 +/- 9 s; P < 0.01) and upper body (T: 33 +/- 8 vs. UT: 43 +/- 4 s; P < 0.01) exercise. The time constants of the phase II V(O)(2)and HR response were not correlated regardless of training status or exercise modality. This study demonstrates for the first time that swim-training status influences upper body V(O)(2) kinetics in pre-pubertal children, but that cycle ergometry responses are insensitive to such differences.
Breese BC, Williams CA, Barker AR, Weisman JR, Fawkner SG, Armstrong N (2010). Longitudinal Changes in the Oxygen Uptake Kinetic Response to Heavy-Intensity Exercise in 14- to 16-Year-Old Boys (Reprinted from PES, vol 22). PEDIATRIC EXERCISE SCIENCE, 22(2), 314-325. Author URL. Full text.
Breese BC, Williams CA, Barker AR, Welsman JR, Fawkner SG, Armstrong N (2010). Longitudinal change in the oxygen uptake kinetic response to heavy-intensity exercise in 14- to 16-years-old boys. Pediatr Exerc Sci, 22(2), 314-325. Author URL.
Breese B, Williams CA, Barker AR, Welsman JR, Fawkner SG, Armstrong N (2010). Longitudinal changes in the oxygen uptake kinetic response to heavy-intensity exercise in 14-16 year old boys. Pediatric Exercise Science Full text.
Knapp KM, Rowlands AV, Welsman JR, Macleod KM (2010). Prolonged unilateral disuse osteopenia 14 years post external fixator removal: a case history and critical review. Case Rep Med, 2010
Prolonged unilateral disuse osteopenia 14 years post external fixator removal: a case history and critical review.
Disuse osteopenia is a complication of immobilisation, with reversal generally noted upon remobilisation. This case report focuses on a patient who was seen 18 years following a road traffic collision when multiple fractures were sustained. The patient had an external fixator fitted for a tibia and fibula fracture, which remained in situ for a period of 4 years. Following removal, the patient was mobilised but, still required a single crutch to aid walking. Fourteen years post removal of the fixator, the patient had a DXA scan which, demonstrated a T-score 2.5 SD lower on the affected hip. This places the patient at an increased risk of hip fracture on this side, which requires monitoring. There appear to be no current studies investigating prolonged disuse-osteopenia in patients following removal of long-term external fixators. Further research is required to quantify unilateral long-term effects to bone health and fracture risk in this population.
Barker AR, Welsman JR, Welford D, Fulford J, Armstrong N (2010). Quadriceps muscle energetics during an incremental test to exhaustion in children and adults. Medicine and Science in Sports and Exercise
Barker AR, Welsman JR, Fulford J, Welford D, Armstrong N (2010). Quadriceps muscle energetics during incremental exercise in children and adults. Med Sci Sports Exerc, 42(7), 1303-1313.
Quadriceps muscle energetics during incremental exercise in children and adults.
PURPOSE: This study tested the hypothesis that the muscle metabolic responses of 9- to 12-yr-old children and young adults during incremental quadriceps exercise are dependent on age and sex. METHODS: Fifteen boys, 18 girls, 8 men, and 8 women completed a quadriceps step-incremental test to exhaustion inside a magnetic resonance scanner for determination of the muscle metabolic responses using P-magnetic resonance spectroscopy. Quadriceps muscle mass was determined using magnetic resonance imaging scans enabling comparison of metabolic data at a normalized power output. RESULTS: the power output and the energetic state at the Pi/PCr and pH intracellular thresholds (IT) were independent of age and sex. The rate of change in Pi/PCr against power output after the ITPi/PCr (S2) was lower in boys (0.158 +/- 0.089) and girls (0.257 +/- 0.110) compared with men (0.401 +/- 0.114, P < 0.001) and women (0.391 +/- 0.133, P = 0.014), respectively, with sex differences present for children only (P = 0.003). Above the ITpH, S2 was more rapid in the men (-0.041 +/- 0.022, P = 0.003) and girls (-0.030 +/- 0.013, P = 0.011) compared with boys (-0.019 +/- 0.007), with no differences between the girls and the women (-0.035 +/- 0.015, P = 0.479). The increase in Pi/PCr at exhaustion was lower in boys (0.85 +/- 0.38) than that in men (1.86 +/- 0.65, P < 0.001) and in girls (1.78 +/- 1.25) than that in women (4.97 +/- 3.52, P = 0.003), with sex differences in both the child (P = 0.005) and the adult groups (P = 0.019). CONCLUSIONS: During moderate-intensity exercise, muscle metabolism appears adult-like in 9- to 12-yr-old children, although both age- and sex-related differences in the "anaerobic" energy turnover are present during high-intensity exercise.
Winsley RJ, Fulford J, Roberts A, Welsman JR, Armstrong N (2009). Sex difference in Peak Oxygen Uptake in Prepubertal Children. Journal of Science and Medicine in Sport, 12, 647-651.
Winsley R, Fulford J, Roberts A, Welsman J, Armstrong N (2009). Sex difference in Peak Oxygen Uptake in Prepubertal Children. Journal of Science and Medicine in Sport, 12, 647-651.
Dodd CJ, Welsman JR, Armstrong N (2008). Energy intake and appetite following exercise in lean and overweight girls. Appetite, 51(3), 482-488.
Energy intake and appetite following exercise in lean and overweight girls.
Twelve 11-year-old girls (six lean, six overweight) were given meals in the laboratory and at school for 5 days, with exercise imposed for 2 days and sedentary activities on another 2 days in counterbalanced sequences. During a preliminary visit, the FLEX heart rate method was used to predict individual exercise durations eliciting 1.5 MJ energy expenditure. Morning and afternoon cycling exercise was subsequently imposed in the laboratory on 2 consecutive days as part of the 5-day intervention. Energy intake was measured via observation with meals being standardised between conditions, prepared and weighed by the research team. Hunger, fullness and desire to eat were rated by subjects immediately before and after meals and exercise. Energy expenditure was significantly elevated in the exercise condition, compared to sedentary. No exercise-induced differences in total daily or 5-day total energy intake were observed between groups or treatments. Overweight girls, however, rated their appetite immediately after exercise as being stronger than they rated it before exercise. In response to exercise-induced energy expenditure, 11-year old overweight and lean girls did not elevate their energy intake over a 5-day period.
Willcocks RJ, Barker AR, Fulford J, Welford D, Welsman JR, Armstrong N, Williams CA (2008). Kinetics of Phosphocreatine and Deoxyhemoglobin in Children and Adults During High-Intensity Exercise. MEDICINE AND SCIENCE IN SPORTS AND EXERCISE, 40(5), S20-S20. Author URL.
Barker AR, Welsman JR, Fulford J, Welford D, Williams CA, Armstrong N (2008). Muscle phosphocreatine and pulmonary oxygen uptake kinetics in children at the onset and offset of moderate intensity exercise. Eur J Appl Physiol, 102(6), 727-738.
Muscle phosphocreatine and pulmonary oxygen uptake kinetics in children at the onset and offset of moderate intensity exercise.
To further understand the mechanism(s) explaining the faster pulmonary oxygen uptake (p(VO)(2)) kinetics found in children compared to adults, this study examined whether the phase II p(VO)(2) kinetics in children are mechanistically linked to the dynamics of intramuscular PCr, which is known to play a principal role in controlling mitochondrial oxidative phosphorylation during metabolic transitions. On separate days, 18 children completed repeated bouts of moderate intensity constant work-rate exercise for determination of (1) PCr changes every 6 s during prone quadriceps exercise using (31)P-magnetic resonance spectroscopy, and (2) breath by breath changes in p(VO)(2) during upright cycle ergometry. Only subjects (n = 12) with 95% confidence intervals
Barker AR, Welsman JR, Fulford J, Welford D, Armstrong N (2008). Muscle phosphocreatine kinetics in children and adults at the onset and offset of moderate-intensity exercise. J Appl Physiol (1985), 105(2), 446-456.
Muscle phosphocreatine kinetics in children and adults at the onset and offset of moderate-intensity exercise.
The splitting of muscle phosphocreatine (PCr) plays an integral role in the regulation of muscle O2 utilization during a "step" change in metabolic rate. This study tested the hypothesis that the kinetics of muscle PCr would be faster in children compared with adults both at the onset and offset of moderate-intensity exercise, in concert with the previous demonstration of faster phase II pulmonary O2 uptake kinetics in children. Eighteen peri-pubertal children (8 boys, 10 girls) and 16 adults (8 men, 8 women) completed repeated constant work-rate exercise transitions corresponding to 80% of the Pi/PCr intracellular threshold. The changes in quadriceps [PCr], [Pi], [ADP], and pH were determined every 6 s using 31P-magnetic resonance spectroscopy. No significant (P>0.05) age- or sex-related differences were found in the PCr kinetic time constant at the onset (boys, 21+/-4 s; girls, 24+/-5 s; men, 26+/-9 s; women, 24+/-7 s) or offset (boys, 26+/-5 s; girls, 29+/-7 s; men, 23+/-9 s; women 29+/-7 s) of exercise. Likewise, the estimated theoretical maximal rate of oxidative phosphorylation (Qmax) was independent of age and sex (boys, 1.39+/-0.20 mM/s; girls, 1.32+/-0.32 mM/s; men, 2.36+/-1.18 mM/s; women, 1.51+/-0.53 mM/s). These results are consistent with the notion that the putative phosphate-linked regulation of muscle O2 utilization is fully mature in peri-pubertal children, which may be attributable to a comparable capacity for mitochondrial oxidative phosphorylation in child and adult muscle.
Barker AR, Welsman JR, Fulford J, Welford D, Armstrong N (2008). Quadriceps Muscle Phosphocreatine and Deoxygenation Kinetics in Children and Adults at the Onset of Moderate Intensity Exercise. MEDICINE AND SCIENCE IN SPORTS AND EXERCISE, 40(5), S20-S20. Author URL.
Williams CA, Willcocks RJ, Barker AR, Fulford J, Welford D, Welsman JR, Armstrong N (2008). Recovery of Muscle Oxygenation and Phosphocreatine in Children and Adults Following High-Intensity Quadriceps Exercise. MEDICINE AND SCIENCE IN SPORTS AND EXERCISE, 40(5), S20-S20. Author URL.
Armstrong N, Welsman JR (2007). Aerobic fitness: what are we measuring?. Med Sport Sci, 50, 5-25.
Aerobic fitness: what are we measuring?
Aerobic fitness depends upon the components of oxygen delivery and the oxidative mechanisms of the exercising muscle. Peak oxygen uptake is recognised as the best single criterion of aerobic fitness but it is strongly correlated with body size. Methods of controlling for body size are discussed and it is demonstrated how inappropriate use of ratio scaling has clouded our understanding of aerobic fitness during growth and maturation and across time. Changes in aerobic fitness over time are reviewed but no published study of peak oxygen uptake, appropriately adjusted for body mass and maturation, has investigated secular changes in aerobic fitness. Data expressed in direct ratio with body mass provide limited insights into secular changes in aerobic fitness but aerobic performance appears to be decreasing in accord with the secular increase in body mass. Cross-sectional and longitudinal peak oxygen uptake data are analysed in relation to age, maturation and sex. Muscle lactate production and blood lactate accumulation are outlined and young people's blood lactate responses to submaximal and maximal exercise are examined. However, exercise of the intensity and duration required to monitor conventional laboratory measures of aerobic fitness are rarely experienced in young people's lives. In many situations it is the oxygen uptake kinetics of the non-steady state which best assess the integrated responses of the oxygen delivery system and the metabolic requirements of the exercising muscle. The chapter therefore concludes with a discussion of insights into aerobic fitness provided by the emerging database on young people's oxygen uptake kinetics responses to exercise of different intensities.
Barker AR, Welsman JR, Welford D, Fulford J, Williams CA, Armstrong N (2006). Reliability of 31P-magnetic resonance spectroscopy during an exhaustive incremental exercise test in children. European Journal of Applied Physiology, 98(6), 556-565.
Barker A, Welsman J, Welford D, Fulford J, Williams C, Armstrong N (2006). Reliability of 31P-magnetic resonance spectroscopy during an exhaustive incremental exercise test in children. Eur J Appl Physiol, 98(6), 556-565.
Reliability of 31P-magnetic resonance spectroscopy during an exhaustive incremental exercise test in children.
This study examined the reliability of (31)P-magnetic resonance spectroscopy (MRS) to measure parameters of muscle metabolic function in children. On separate days, 14 children (7 boys and 7 girls) completed three knee-extensor incremental tests to exhaustion inside a whole-body scanner (1.5 T, Phillips). The dynamic changes in the ratio of inorganic phosphate to phosphocreatine (Pi/PCr) and intracellular muscle pH were resolved every 30 s. Using plots of Pi/PCr and pH against power output (W), intracellular thresholds (ITs) for each variable were determined using both subjective and objective procedures. The IT(Pi/PCr) and IT(pH) were observed subjectively in 93 and 81% of their respective plots, whereas the objective method identified the IT(Pi/PCr) in 88% of the plots. The IT(pH) was undetectable using the objective method. End exercise (END) END(Pi/PCr), END(pH), IT(Pi/PCr) and IT(pH) were examined using typical error statistics expressed as a % coefficient of variation (CV) across all three exercise tests. The CVs for the power output at the subjectively determined IT(Pi/PCr) and IT(pH) were 10.6 and 10.3%, respectively. Objective identification of the IT(Pi/PCr) had a CV of 16.3%. CVs for END(pH) and END(Pi/PCr) were 0.9 and 50.0%, respectively. MRS provides a valuable window into metabolic changes during exercise in children. During knee-extensor exercise to exhaustion, END(pH) and the subjectively determined IT(Pi/PCr) and IT(pH) demonstrate good reliability and thus stable measures for the future study of developmental metabolism. However, the objectively determined IT(Pi/PCr) and END(Pi/PCr) displayed poor reliability.
Armstrong N, Welsman JR (2006). The physical activity patterns of European youth with reference to methods of assessment. Sports Med, 36(12), 1067-1086.
The physical activity patterns of European youth with reference to methods of assessment.
This article reviews the habitual physical activity of children and adolescents from member countries of the European Union in relation to methods of assessing and interpreting physical activity. Data are available from all European Union countries except Luxembourg and the trends are very similar. European boys of all ages participate in more physical activity than European girls and the gender difference is more marked when vigorous activity is considered. The physical activity levels of both genders are higher during childhood and decline as young people move through their teen years. Physical activity patterns are sporadic and sustained periods of moderate or vigorous physical activity are seldom achieved by many European children and adolescents. Expert committees have produced guidelines for health-related physical activity for youth but they are evidence-informed rather than evidence-based and where there is evidence of a relationship between physical activity during youth and health status there is little evidence of a particular shape of that relationship. The number of children who experience physical activity of the duration, frequency and intensity recommended by expert committees decreases with age but accurate estimates of how many girls and boys are inactive are clouded by methodological problems. If additional insights into the promotion of health through habitual physical activity during youth are to be made, methods of assessment need to be further refined and recommended guidelines re-visited in relation to the existing evidence base.
Middlebrooke AR, Armstrong N, Welsman JR, Shore AC, Clark P, MacLeod KM (2005). Does aerobic fitness influence microvascular function in healthy adults at risk of developing Type 2 diabetes?. Diabet Med, 22(4), 483-489.
Does aerobic fitness influence microvascular function in healthy adults at risk of developing Type 2 diabetes?
AIM: to investigate whether aerobic fitness is associated with skin microvascular function in healthy adults with an increased risk of developing Type 2 diabetes. METHODS: Twenty-seven healthy normal glucose-tolerant humans with either a previous diagnosis of gestational diabetes or having two parents with Type 2 diabetes and 27 healthy adults who had no history of diabetes were recruited. Maximal oxygen uptake was assessed using an incremental exercise test to exhaustion. Skin microvascular function was assessed using laser Doppler techniques as the maximum skin hyperaemic response to a thermal stimulus (maximum hyperaemia) and the forearm skin blood flow response to the iontophoretic application of acetylcholine (ACh) and sodium nitroprusside. RESULTS: Maximal oxygen uptake was not significantly different in the 'at-risk' group compared with healthy controls. Maximum hyperaemia was reduced in those 'at risk' (1.29 +/- 0.30 vs. 1.46 +/- 0.33 V, P = 0.047); however, the peak response to acetylcholine or sodium nitroprusside did not differ in the two groups. A significant positive correlation was demonstrated between maximal oxygen uptake and maximum hyperaemia (r = 0.52, P = 0.006 l/min and r = 0.60, P = 0.001 ml/kg/min) and peak ACh response (r = 0.40, P = 0.04 l/min and r = 0.47, P = 0.013 ml/kg/min) in the 'at-risk' group when expressed in absolute (l/min) or body mass-related (ml/kg/min) terms. No significant correlations were found in the control group. CONCLUSIONS: in this 'at-risk' group with skin microvascular dysfunction maximal oxygen uptake was not reduced compared with healthy controls. However, in the 'at-risk' group alone, individuals with higher levels of aerobic fitness also had better microvascular and endothelial responsiveness.
Bloxham, S.R. Welsman JR, Armstrong, N. (2005). Ergometer-specific relationships between peak oxygen uptake and peak power output in children. Pediatric Exercise Science, 17(2), 136-148.
Welsman JR, Bywater K, Farr C, Welford D (2005). Reliability of Peak VO2 and Maximal Cardiac Output Assessed Using Thoracic Bioimpedence in Children. European Journal of Applied Physiology, 94(3), 228-234.
Moore MS, Dodd CJ, Welsman JR, Armstrong N (2004). Short-term appetite and energy intake following imposed exercise in 9- to 10-year-old girls. Appetite, 43(2), 127-134.
Short-term appetite and energy intake following imposed exercise in 9- to 10-year-old girls.
Short-term effects of different intensities of exercise-induced energy expenditure on energy intake and hunger were compared in 19 girls (10.0 +/- 0.6 years) in three conditions: sedentary, low-intensity exercise and high-intensity exercise. The exercise conditions involved cycling at 50 and 75% of peak oxygen uptake, respectively, but were designed to evoke approximately 1.50 MJ of total expenditure, as estimated from continuously monitored heart rate. A maintenance breakfast of controlled energy intake was provided and ad libitum energy intake was measured at lunch and dinner. Differences in energy intake relative to expenditure, between 09:30 and 17:00, were calculated by subtracting energy expenditure from energy intake (energy difference). Hunger, fullness and prospective consumption were rated before and after meals and exercise sessions. Lunch energy intake was significantly less after low-intensity exercise than after high-intensity exercise. Energy expenditure was greater in the exercise conditions than when sedentary and the energy difference was more positive in the sedentary condition than in each of the exercise conditions. At mid-afternoon, rated prospective consumption was less after the high-intensity exercise. The imposition of energy expenditure through exercise of either low or high intensity resulted in no detectable increase in energy intake in the short term.
Santos AMC, Armstrong N, De Ste Croix MBA, Sharpe P, Welsman JR (2003). Optimal Peak Power in Relation to Age, Body Size, Gender, and Thigh Muscle Volume. Pediatric Exercise Science, 15(4), 406-418.
Optimal Peak Power in Relation to Age, Body Size, Gender, and Thigh Muscle Volume
These studies used multilevel modelling to examine optimised peak power (PPopt) from a force velocity test over the age range 12-14 years. In the first study, body mass, stature, triceps and subscapular skinfold thicknesses of boys and girls, aged 12.3 ± 0.3 y at the onset of the study, were measured on four occasions at 6 monthly intervals. The analysis was founded on 146 PPopt determinations (79 from boys and 67 from girls). Body mass and stature were significant explanatory variables with sum of two skinfolds exerting an additional effect. No gender differences were evident but PPopt increased with age. In the second study, thigh muscle volume (TMV) was estimated using magnetic resonance imaging at test occasions two and four. The analysis, founded on a subsample of 67 PPopt determinations (39 from boys and 28 from girls), demonstrated TMV to be a significant additional explanatory variable along-side body mass and stature with neither age nor gender making a significant contribution to PP opt. Together the studies demonstrate the influence of body size and TMV on young people's PPopt.
Croix MBAD, Armstrong N, Welsman JR (2003). The reliability of an isokinetic knee muscle endurance test in young children. PEDIATRIC EXERCISE SCIENCE, 15(3), 313-323. Author URL.
Santos AMC, Welsman JR, De Ste Croix MBA, Armstrong N (2002). Age- and sex-related differences in optimal peak power. Pediatric Exercise Science, 14(2), 202-212.
Age- and sex-related differences in optimal peak power
Age- and sex-related differences in optimal peak power (PPopt) and associated measures determined using a force-velocity (F-V) cycling test were examined in pre teenage, teenage and adult males and females. Absolute PPopt increased significantly with age in both males and females. With body mass controlled for using allometric scaling significant age related increases remained, an effect masked in the females when PPopt was expressed as W · kg-1. Sex differences in PPopt were minimal in the preteens but males demonstrated higher PPopt than females in both teenage and adult groups. These patterns of change with age and sex broadly reflect those obtained for Wingate Anaerobic Test determined PP but the use of a single non-optimized braking force underestimates the magnitude of any differences observed.
Welsman JR, Armstrong N (2002). Cardiovascular responses to submaximal treadmill running in 11-13 year olds. Acta Paediatrica, 91(2), 125-131.
Welsman JR, Armstrong N, De Ste Croix MBA, Sharpe P (2002). Longitudinal changes in isokinetic leg strength in 10-14 years old. Annals of Human Biology, 29(1), 50-62.
Armstrong N, Faulkner SG, Potter CR, Welsman JR (2002). Oxygen uptake kinetics in children and adults after the onset of moderate-intensity exercise. Journal of Sports Sciences, 20(4), 319-326.
Fawkner SG, Armstrong N, Childs DJ, Welsman JR (2002). Reliability of the visually identified ventilatory threshold and V-slope in children. Pediatric Exercise Science, 14(2), 181-192.
Reliability of the visually identified ventilatory threshold and V-slope in children
The purpose of this study was to assess the reliability of the ventilatory threshold using visual analysis (TVent) and a computerised v-slope method (TV - slope) with children. Twenty-two children completed 2 ramp incremental cycling tests to voluntary exhaustion. Oxygen uptake (V̇O2) at TVent was derived independently by two observers using plots of V̇E/V̇CO2, V̇E/V̇O2, PETO2 and PETCO2, V̇E and RER as a function of time. V̇O2 at TV - slope was determined by both observers using linear regression analysis of the plot of V̇CO2 against V̇O2. A TV - slope, was determined for each test, although a TVent could not be found by one of the observers in 7 of the 44 tests. Inter-observer reliability was slightly better for TV - slope, and both methods had similar test-retest coefficients of repeatability (0.19 and 0.22 L · min-1, TVent and TV - slope, respectively). Although TV - slope may be the method of choice, investigators should consider the 95% limits of agreement when interpreting their data.
De Ste Croix MB, Armstrong N, Chia MY, Welsman JR, Parsons G, Sharpe P (2001). Changes in short-term power output in 10- to 12-year-olds. J Sports Sci, 19(2), 141-148.
Changes in short-term power output in 10- to 12-year-olds.
In this study, we used multi-level regression modelling to assess the influence of age, sex, body size, skinfold thicknesses, maturity, thigh muscle volume and isokinetic leg strength on the development of load- and inertia-adjusted peak (1 s) and mean power (30 s) determined using the Wingate anaerobic test. Fifteen males and 19 females were measured twice, first aged 10.0 +/- 0.3 years and then aged 11.8 +/- 0.3 years. Initial models identified body mass and height as significant explanatory variables (P < 0.05) for peak power and mean power, with an additional age effect for the former. No significant differences between the sexes or maturity effects were observed for either peak or mean power (P > 0.05). The introduction of sum of skinfolds improved the fit of the model and rendered the height term non-significant for both peak and mean power (P> 0.05). An age effect became apparent for mean power. When isokinetic leg strength and thigh muscle volume were entered into the model, the latter exerted a significant effect on both peak and mean power (P< 0.05), whereas isokinetic leg strength was not a significant explanatory variable for either (P> 0.05). In conclusion, thigh muscle volume exerts a positive influence on young people's short-term power output, which is additional to the effects of body mass, sum of skinfolds and age.
Armstrong N, Welsman JR (2001). Peak oxygen uptake in relation to growth and maturation in 11- to 17-year-old humans. Eur J Appl Physiol, 85(6), 546-551.
Peak oxygen uptake in relation to growth and maturation in 11- to 17-year-old humans.
This study used multilevel modelling to examine peak oxygen uptake (VO2peak) during growth and maturation. Body mass, stature, triceps and subscapular skinfold thicknesses, blood haemoglobin concentration, and VO2peak of boys and girls, [mean (SD)] aged 11.1 (0.4) years at the onset of the study, were measured at ages 11, 12, 13 and 17 years. Sexual maturation was assessed on the first three occasions and was assumed to be Tanner stage 5 at 17 years. The analysis was founded on 388 VO2peak determinations from 132 children. The initial model revealed mass, stature and age as significant explanatory variables of VO2peak with an additional positive effect for stage of maturity. Girls' values were significantly lower than those of boys and a significant age-by-sex interaction described a progressive divergence in boys' and girls' VO2peak. The introduction of skinfold thicknesses produced a model with an improvement in fit. The stature term was negated and the mass exponent almost doubled. The sex and age-by-sex terms were reduced but remained significant. Many of the observed maturity effects were explained with stage 5 becoming non-significant. Blood haemoglobin concentration was a nonsignificant parameter estimate in both models. Fat-free mass was the dominant influence on the growth of VO2peak but the multilevel regression models demonstrated that, with body size and fatness allowed for, VO2peak increased with age and maturation in both sexes.
Armstrong N, Welsman JR, Chia MY (2001). Short term power output in relation to growth and maturation. Br J Sports Med, 35(2), 118-124.
Short term power output in relation to growth and maturation.
OBJECTIVE: to examine short term power output during growth and maturation using a multilevel modelling approach. METHODS: Body mass, stature, and triceps and subscapular skinfold thicknesses of boys and girls, aged 12.2 (0.4) years (mean (SD)) at the onset of the study, were measured at age 12, 13, and 17 years. Sexual maturation, classified according to Tanner's stage of pubic hair development, was assessed on the first two occasions and assumed to be stage 5 at 17 years. Peak power (PP) and mean power (MP) were assessed on each occasion using the Wingate anaerobic test. RESULTS: Initial models, founded on 417 determinations of short term power output, identified body mass, stature, and age as significant explanatory variables of both PP and MP. The values for girls were significantly lower than those for boys, and a significant age by sex interaction described a progressive divergence in the MP of boys and girls. The introduction of sum of two skinfold thicknesses produced a model with an improvement in fit as indicated by a significant change in log likelihood. The stature term was negated and the body mass term increased. The age and sex terms were reduced but remained significant. The age by sex interaction term remained a significant explanatory variable for MP. Maturity effects were non-significant additional explanatory variables in all models of power output. CONCLUSION: the values of PP and MP for boys are higher than those for girls, and, for MP, sex differences increase with age. Body mass and skinfold thicknesses are significant influences on both PP and MP, but age exerts a positive but non-linear effect on power output independent of body size and fatness.
Welsman JR, Armstrong N, Chia MYH (2001). Short-term power output in relation to growth and maturation. British Journal of Sports Medicine, 35(2), 118-125.
Stoedefalke K, Armstrong N, Kirby BJ, Welsman JR (2000). Effect of training on peak oxygen uptake and blood lipids in 13 to 14-year-old girls. Acta Paediatr, 89(11), 1290-1294.
Effect of training on peak oxygen uptake and blood lipids in 13 to 14-year-old girls.
UNLABELLED: the aim of this study was to investigate the effects of exercise training on the peak oxygen uptake (peak VO2) and blood lipid profile of 13 to 14-y-old postmenarcheal girls. Treadmill determined peak VO2, total cholesterol, high density lipoprotein cholesterol, low density cholesterol, and triglycerides were the outcome measures assessed at baseline and following exercise training. Twenty girls completed a 20-wk programme of exercise training which involved maintaining the heart rate at 75-85% maximum for 20 min, three times per week. Heart rate was rigorously monitored using telemetry throughout each training session. Eighteen girls acted as the control group. There were no significant (p > 0.05) changes in the outcome measures following the training programme. CONCLUSIONS: These findings suggest that exercise training of this frequency, intensity and duration for a period of 20 wk has no significant effect on either the peak VO2 or blood lipid and lipoprotein profile of normolipidaemic, postmenarcheal girls.
Armstrong N, Welsman JR, Kirby BJ (2000). Longitudinal changes in 11-13-year-olds' physical activity. Acta Paediatr, 89(7), 775-780.
Longitudinal changes in 11-13-year-olds' physical activity.
UNLABELLED: the influence of age, sex, maturity, body mass and body fatness on the physical activity (PA) of 11-13-y-olds was examined longitudinally. Body mass, triceps and subscapular skinfold thickness and pubic hair were recorded and 3-d continuous heart rate (HR) monitoring was used to estimate PA on each annual measurement occasion. At the onset, subjects were 11.0 (0.4)-y-old and data were available on 202, 143 and 160 subjects in years 1 to 3, respectively with an almost equal sex distribution. Multilevel regression modelling examined age-, sex- and maturity-related changes in time spent with HR above 139 (moderate activity) and 159 (vigorous activity) bpm. Sustained (10 or 20 min) periods of moderate or vigorous activity were not characteristic of PA patterns. Both PA measures declined with age, with a consistent sex difference reflecting the lower PA levels of girls. Body mass and fatness were not significant explanatory variables, but an additional decrement in activity was evident in late maturity. CONCLUSION: Few children experience extended bouts of PA, and from 11-13 y, PA decreases, with more girls than boys becoming inactive. The data emphasize the importance of promoting active lifestyles during youth.
Welsman JR, Armstrong N (2000). Longitudinal changes in submaximal oxygen uptake in 11- to 13-year-olds. J Sports Sci, 18(3), 183-189.
Longitudinal changes in submaximal oxygen uptake in 11- to 13-year-olds.
The aim of this study was to monitor longitudinal changes in young people's submaximal oxygen uptake (VO2) responses during horizontal treadmill running at 8 km x h(-1). The 236 participants (118 boys, 118 girls) were aged 11.2+/-0.4 years (mean +/- s) at the onset of the study. Submaximal VO2, peak VO2 and anthropometry were recorded annually for three consecutive years. The data were analysed using multi-level regression modelling within a multiplicative, allometric framework. The initial model examined sex, age and maturity-related changes in submaximal VO2 relative to body mass as the sole anthropometric covariate. Our results demonstrate that the conventional ratio standard ml x kg(-1) x min(-1) does not adequately describe the true relationship between body mass and submaximal VO2 during this period of growth. The effects of maturity and age were non-significant, but girls consumed significantly less VO2 than boys running at 8 km x h(-1). In subsequent models, stature was shown to be a significant explanatory variable, but this effect became non-significant when the sum of two skinfolds was added. Thus, within this population, submaximal VO2 responses were explained predominantly by changes in body mass and skinfold thicknesses, with no additional maturity-related increments. When differences in body mass and skinfolds were controlled for, there was still a difference between the sexes in submaximal VO2, with girls becoming increasingly more economical with age.
Armstrong N, Welsman JR, Williams CA, Kirby BJ (2000). Longitudinal changes in young people's short-term power output. Med Sci Sports Exerc, 32(6), 1140-1145.
Longitudinal changes in young people's short-term power output.
PURPOSE: the influences of age, body size, skin-fold thickness, gender, and maturation on the short-term power output of young people were examined using multilevel modelling. METHODS: Subjects were 97 boys and 100 girls, aged 12.2 +/- 0.4 yr at the onset of the study. Sexual maturity was classified according to Tanner's indices of pubic hair. Peak power (PP) and mean power (MP) were determined on two occasions 1 yr apart using the Wingate Anaerobic Test (WAnT). The data were analyzed using multilevel regression modelling. RESULTS: Initial models identified body mass and stature as significant explanatory variables with an additional positive effect of age, which was smaller for girls' MP. A significant gender difference was apparent for both power indices with girls achieving lower values than boys. A significant incremental effect of later maturity (stages 4 and 5 for pubic hair development) was identified for MP only. Subsequent incorporation of sum of two skin-fold thicknesses into the model yielded significant negative parameter estimates for PP and MP and negated both the stature effects and the maturation influence upon MP. CONCLUSION: There are gender differences in the longitudinal growth of performance on the WanT. Regardless of gender differences, body mass and skin-fold thicknesses appear to be the best anthropometric predictors of WAnT determined PP and MP in young people.
De Ste Croix MBA, Armstrong N, Weisman JR (1999). Concentric isokinetic leg strength in pre-teen, teenage and adult males and females. BIOLOGY OF SPORT, 16(2), 75-86. Author URL.
Welsman JR (1999). Girls and fitness: fact and fiction. Br J Sports Med, 33(6), 373-374. Author URL.
Armstrong N, Welsman JR, Nevill AM, Kirby BJ (1999). Modeling growth and maturation changes in peak oxygen uptake in 11-13 yr olds. J Appl Physiol (1985), 87(6), 2230-2236.
Modeling growth and maturation changes in peak oxygen uptake in 11-13 yr olds.
The influence of gender, growth, and maturation on peak O(2) consumption (VO(2 peak)) in 11-13 yr olds were examined by using multilevel regression modeling. Subjects were 119 boys and 115 girls, aged 11.2 +/- 0.4 (SD) yr at the onset of the study. Sexual maturation was classified according to Tanner's indexes of pubic hair. VO(2 peak) was determined annually for 3 yr. The initial model identified body mass and stature as significant explanatory variables, with an additional positive effect for age and incremental effects for stage of maturation. A significant gender difference was apparent with lower values for girls, and an age-by-gender interaction indicated a progressive divergence in boys' and girls' VO(2 peak). Subsequent incorporation of the sum of two skinfold thicknesses into the model negated stature effects, reduced the gender term, and explained much of the observed maturity effects. The body mass exponent almost doubled, but the age-by-gender interaction term was consistent with the initial model.
Armstrong N, Welsman JR, Kirby BJ (1999). Submaximal exercise and maturation in 12-year-olds. J Sports Sci, 17(2), 107-114.
Submaximal exercise and maturation in 12-year-olds.
The aim of this study was to examine the maturation responses of young people to submaximal treadmill exercise. Body mass was controlled using both the conventional ratio standard and allometric modelling. Ninety-seven boys and 97 girls with a mean age of 12.2 years completed a discontinuous, incremental exercise test to voluntary exhaustion. We measured peak oxygen uptake (VO2peak) and VO2 when running at 8, 9 and 10 km x h(-1). Sexual maturation was assessed visually using Tanner's indices of pubic hair. Peak VO2 was significantly higher in boys (P0.05); however, values of VO2, expressed both in ratio with body mass and adjusted for body mass using allometry, were significantly greater in boys than in girls (P
Armstrong N, Welsman JR, Kirby BJ (1998). Peak oxygen uptake and maturation in 12-yr olds. Med Sci Sports Exerc, 30(1), 165-169.
Peak oxygen uptake and maturation in 12-yr olds.
The influences of gender and sexual maturation on the peak VO2 of 12-yr olds were examined. Subjects were 106 boys and 106 girls, ages 12.2 +/- 0.4 yr. The sexual maturity of 93 boys and 83 girls was classified according to Tanner's indices of pubic hair. No significant gender differences (P > 0.05) were detected in age, stature, or hemoglobin concentration. Peak VO2 was determined on a treadmill and boys' peak VO2 was significantly higher (P < 0.01) than girls' whether expressed in L x min(-1) (2.10 +/- 0.34 vs 1.92 +/- 0.28) or mL x kg(-1) x min(-1) (52 +/- 6 vs 44 +/- 5). With body mass controlled for using log-linear ANCOVA, the gender difference decreased from 18.2 to 17.1% but remained significant (P < 0.01). For peak VO2 (L x min[-1]) ANOVA revealed no significant interaction (P > 0.05) but significant (P < 0.01) main effects for both gender and maturation. For peak VO2 in ratio with body mass (mL x kg(-1) x min[-1]), ANOVA detected no significant interaction (P > 0.05) or significant main effect (P > 0.05) for maturation although the main effect for gender was significant (P < 0.01). Analysis of peak VO2 with body mass controlled for using log-linear ANCOVA revealed no significant interaction (P > 0.05) but main effects (P < 0.01) for both gender and maturation. Thus, gender differences, which are not simply explained by differences in body size or hemoglobin concentration, have been demonstrated in the peak VO2 of 12-yr olds. A log-linear scaling model has identified in both boys and girls a significant influence of maturation on peak VO2 independent of body mass. This effect may have been masked in previous studies by the inappropriate use of peak VO2 in ratio with body mass.
Welsman JR, Armstrong N, Kirby BJ, Winsley RJ, Parsons G, Sharpe P (1997). Exercise performance and magnetic resonance imaging-determined thigh muscle volume in children. Eur J Appl Physiol Occup Physiol, 76(1), 92-97.
Exercise performance and magnetic resonance imaging-determined thigh muscle volume in children.
This study examined the relationships between thigh muscle volume (TMV) and aerobic and anaerobic performance in children. A total of 32 children, 16 boys and 16 girls, aged 9.9 (0.3) years completed a treadmill running test to exhaustion for the determination of peak oxygen uptake (peak VO2) and a Wingate Anaerobic Test (WAnT) for the determination of peak power (PP) and mean power (MP). The volume of the right thigh muscle was determined using magnetic resonance imaging. TMV was not significantly different in boys and girls [2.39 (0.29) l vs 2.18 (0.38) l, P > 0.05]. Peak VO2 and MP were significantly higher in boys than girls (P < 0.01) whether expressed in absolute, mass-related or allometrically scaled terms. Absolute PP was not significantly different in boys and girls but mass-related and allometrically scaled values were higher in boys (P < 0.01). TMV was correlated with absolute peak VO2, PP and MP in both sexes (r = 0.52-0.89, P < 0.01). In boys, mass-related PP was correlated with TMV (r = 0.53, P < 0.01), and in girls mass-related peak VO2 was correlated with TMV (r = -0.61, P < 0.01). However, in neither sex were allometrically scaled peak VO2, PP or MP correlated with TMV (P > 0.05). There were no significant differences between boys and girls in terms of peak VO2, PP or MP when expressed in a ratio to TMV or allometrically scaled TMV. In conclusion, this study has demonstrated that, when body size is appropriately accounted for using allometric scaling, TMV is unrelated to indices of aerobic and anaerobic power in 10-year-old children. Furthermore, there appear to be no qualitative differences in the muscle function of boys and girls in respect of aerobic and anaerobic function.
Armstrong N, Welsman JR, Kirby BJ (1997). Performance on the Wingate anaerobic test and maturation. Pediatric Exercise Science, 9(3), 253-261.
Performance on the Wingate anaerobic test and maturation
The influence of sexual maturation on the Wingate anaerobic test performance of 100 boys and 100 girls, ages 12.2 ± 0.4 years, was examined using Tanner's indices of pubic hair and, in boys, salivary testosterone as measures of maturation. No sex differences (p >. 05) in either peak power (PP) or mean power (MP) were revealed. Significant main effects (p <. 01) for maturation were detected for both PP and MP expressed in W, W·kg-1, or with body mass controlled using allometric principles. Testosterone did not increase the variance in PP or MP explained by body mass alone (p >. 05). No sex or maturational effects were observed for postexercise blood lactate (p >. 05). Testosterone was not (p >. 05) correlated with blood lactate. Thus, sexual maturation exerts an influence on PP and MP independent of body mass, but maturational effects on postexercise blood lactate remain to be proven in this age group. © 1997 Human Kinetics Publishers, Inc.
Armstrong N, Kirby BJ, McManus AM, Welsman JR (1997). Prepubescents' ventilatory responses to exercise with reference to sex and body size. Chest, 112(6), 1554-1560.
Prepubescents' ventilatory responses to exercise with reference to sex and body size.
STUDY OBJECTIVES: to examine the ventilatory responses of prepubescent children to submaximal and peak exercise using appropriate allometric modeling to control for differences in body size. DESIGN: Cross-sectional study of a representative sample of children. SETTING: Middle schools (8 to 11 years) in Exeter, UK. PARTICIPANTS: We studied 101 boys and 76 girls aged 11.1 (0.4) years and classified Tanner stage 1 for pubic hair (no true pubic hair). MEASUREMENTS: at rest: stature, mass, sum of skinfolds, hemoglobin concentration, FVC, and FEV1. During treadmill exercise at 7, 8, 9, and 10 km/h, and at peak exercise: oxygen uptake (VO2), minute ventilation (VE), tidal volume (VT), and respiratory frequency (Rf). RESULTS: at peak exercise, boys' VO2, VE, and VT were significantly (p
Welsman JR, Armstrong N, Withers S (1997). Responses of young girls to two modes of aerobic training. Br J Sports Med, 31(2), 139-142.
Responses of young girls to two modes of aerobic training.
OBJECTIVES: to investigate the physiological effects of two different three times a week, eight week training programmes on the aerobic fitness of nine to ten year old girls. METHODS: Treadmill determined peak VO2, submaximal heart rates, and submaximal blood lactate were the criterion measures. Seventeen girls completed a programme of "aerobics" training where sessions lasted 20-25 minutes. Eighteen girls followed a cycle ergometer training programme which involved pedalling continuously for 20 minutes with the heart rate maintained between 160 and 170 beats/minute. A control group of 16 girls completed the criterion tests but did not train. In the cycle ergometer group and eight control subjects plasma total cholesterol and high density lipoprotein cholesterol were determined before and after training. RESULTS: Peak VO2 did not change significantly with training in either training group, neither were there any significant changes in submaximal heart rates. Blood lactate declined significantly at the two lowest submaximal exercise intensities in the cycle ergometer training group (from 2.3 (1.1) to 1.4 (0.06) mmol/l at stage 1 and from 2.1 (1.2) to 1.6 (0.06) mmol/l at stage 2; means (SD); P < 0.01). Total cholesterol and high density lipoprotein cholesterol remained unchanged with training. CONCLUSIONS: These findings suggest that an eight week structured exercise programme produces minimal changes in either the aerobic fitness or blood lipids of young girls. It may be more beneficial for long term health to promote enjoyment in activity and positive attitudes to exercise rather than attempting to enhance aerobic fitness through strenuous exercise programmes.
Armstrong L, Kirby BJ, McManus AM, Welsman JR (1996). Aerobic fitness of prepubescent children (vol, 22, pg, 427, 1995). ANNALS OF HUMAN BIOLOGY, 23(2), 188-188. Author URL.
Welsman JR, Armstrong N, Nevill AM, Winter EM, Kirby BJ (1996). Scaling peak VO2 for differences in body size. Med Sci Sports Exerc, 28(2), 259-265.
Scaling peak VO2 for differences in body size.
This paper examined the influence of different statistical modeling techniques on the interpretation of peak VO2 data in groups of prepubertal, circumpubertal, and adult males (group 1M, N = 29; group 2M, N = 26; group 3M, N = 8) and females (group 1F, N = 33; group 2F, N = 34; group 3F, N = 16). Conventional comparisons of the simple per-body-mass ratio (ml.kg-1.min-1) revealed no significant differences between the three male groups (P < 0.05). In females, a decline in VO2 between group 2F and 3F was observed (P < 0.05). Both linear and log-linear (allometric) models revealed significant increases across all three male groups for peak VO2 adjusted for body mass (P < 0.05). In females these scaling models identified a significantly lower peak VO2 in group 1F versus groups 2F and 3F (P < 0.05). Based upon the common mass exponent identified (b = 0.80, SE = 0.04), power function ratios (y.mass0.80) were generated and the logarithms of these compared. Again, results indicated a progressive increase in peak VO2 across groups 1M to 3M (P < 0.05) and an increase between groups 1F and 2F (P < 0.05). Incorporating stature into the allometric equation reduced the mass exponent to 0.71 (SE = 0.06) with the contribution of the stature exponent shown to be 0.44 (SE = 0.20). These results indicate that conventional ratio standards do not adequately account for body size differences when investigating functional changes in peak VO2.
Welsman JR, Armstrong N (1996). The measurement and interpretation of aerobic fitness in children: Current issues. J ROY SOC MED, 89(5), P281-P285.
Welsman JR, Armstrong N (1996). The measurement and interpretation of aerobic fitness in children: current issues. J R Soc Med, 89(5), 281P-285P. Author URL.
Armstrong N, Kirby BJ, McManus AM, Welsman JR (1995). Aerobic fitness of prepubescent children. Ann Hum Biol, 22(5), 427-441.
Aerobic fitness of prepubescent children.
This study was designed to enhance understanding of the assessment and interpretation of the aerobic fitness of prepubertal children. Written informed consent to participate was obtained from 70% of the children in year six of the 15 state schools in the city of Exeter. Twenty-five per cent of the eligible children in each school were randomly selected from those who volunteered. The data reported here are those obtained from the 111 boys (11.1 SD 0.4 years) and 53 girls (10.9 SD 0.3 years) classified as Tanner stage 1 in both pubic hair rating and either genitalia rating (boys) or breast rating (girls). Peak oxygen uptake (peak VO2) was determined using a discontinuous, incremental protocol on a treadmill. Only a minority of children demonstrated a levelling-off or plateau in VO2 despite an increase in exercise intensity. There was no evidence to suggest that the children who demonstrated a VO2 plateau had significantly (p < 0.05) higher peak VO2, peak heart rate, peak respiratory exchange ratio or peak blood lactate than those children who did not demonstrate a plateau in VO2. These findings indicate that a VO2 plateau should not be used as a requirement for defining a maximal exercise test with prepubertal children. Boys had a significantly (p < 0.01) higher peak VO2 than girls, whether expressed in 1.min-1 (1.78 vs 1.46) or in relation to body mass (51 vs 45 ml.kg-1.min-1). The results compare favourably with those of similarly aged children from other countries, but why prepubescent boys have significantly higher (13.3%) mass-related peak VO2 than prepubescent girls is not readily apparent. Although conventional, the expression of peak VO2 as per body mass ratio may not adequately partition out body-size differences. The influence of body mass was therefore removed using a linear adjustment scaling model and a log-linear model, but the boys' peak VO2 remained significantly (p < 0.01) higher than the girls' peak VO2 with the difference now being 16.0% and 16.2%, respectively.
Armstrong N, Welsman JR (1994). Assessment and interpretation of aerobic fitness in children and adolescents. Exerc Sport Sci Rev, 22, 435-476.
Assessment and interpretation of aerobic fitness in children and adolescents.
Our understanding of the development of children and adolescents' aerobic fitness is limited by ethical considerations and methodological constraints. Protocols, apparatus, and criteria of maximal effort used with adults are often unsuitable for use with children. In normal children and adolescents, peak VO2 increases with growth and maturation, although there are indications that girls' peak VO2 may level off around 14 years of age. Males exhibit higher values of peak VO2 than females, and the sex difference increases as they progress through adolescence. The difference between males and females has been attributed to the boys' greater muscle mass and hemoglobin concentration. It appears that boys experience an adolescent growth spurt in peak VO2, which reaches a maximum gain near the time of PHV, but data are insufficient to offer any generalization for girls. Peak VO2 has usually been expressed in relation to body mass, and with this convention it appears that boys' values are consistent throughout the developmental period, whereas girls' values decrease as they get older. This type of analysis may, however, have clouded our understanding of growth and maturational changes in peak VO2, and scaling for differences in body size may provide further clarification. If differences are shown where none were previously thought to exist, then physiological explanations must be sought. Methodological issues have also hindered the understanding of how children's blood lactate responses to exercise develop. The actual lactate level recorded during an exercise test is influenced by the site of sampling and the blood handling and assay techniques. Valid interstudy comparisons can only be made where similar procedures have been employed. In general, children demonstrate lower blood lactate levels at peak VO2 than adults, although individual variation is wide. Therefore the use of blood lactate measures to confirm the attainment of peak VO2 cannot be supported. Exercise at the same relative submaximal intensity elicits a lower blood lactate in children than in adults, but interpretation and identification of developmental and maturational patterns of response are limited by the use of different testing conditions and reference points (e.g. lactate threshold and fixed level reference points). There is growing evidence that the 2.5 mM reference level should be used in preference to the 4.0 mM level, as the adult criterion occurs close to maximal exercise in many children and adolescents. Explanations for child-adult differences in blood lactate responses to exercise are difficult to elucidate.(ABSTRACT TRUNCATED AT 400 WORDS)
Welsman JR, Armstrong N (1992). Daily physical activity and blood lactate indices of aerobic fitness in children. Br J Sports Med, 26(4), 228-232.
Daily physical activity and blood lactate indices of aerobic fitness in children.
This study examined the relationship between daily physical activity and aerobic fitness in 11-16-year-olds. Habitual physical activity was assessed in 28 boys (mean(s.d.) age 13.6(1.3) years) and 45 girls (mean(s.d) age 13.7(1.3) years) from minute-by-minute heart rate monitoring during 3 school days. Aerobic fitness was assessed by determining the percentage peak VO2 at blood lactate reference values of 2.5 and 4.0 mmol l-1 during incremental treadmill running. The 4.0 mmol l-1 level occurred at a mean(s.d.) value of 89(7)% peak VO2 in both boys and girls and mean(s.d.) values at the 2.5 mmol l-1 level were 82(9)% peak VO2 in girls. Mean(s.d.) percentage time with heart rates at or above 140 beats min-1 was 6(3)% in boys and 5(3)% in girls. Corresponding values for percentage time at or above 160 beats min-1 were 3(2) for boys and 2(1) for girls. The number of 10- and 20-min periods of activity with the heart rate sustained above the 140 and 160 beats min-1 thresholds were also totalled over the 3 days. No significant relationships were identified between percentage peak VO2 at the 2.5 or 4.0 mmol l-1 blood lactate reference levels and either percentage time or number of 10- or 20-min periods above 140 or 160 beats min-1 (P > 0.05). These results support the hypothesis that daily physical activity levels in 11-16-year-old children do not stress aerobic metabolism sufficiently to influence aerobic fitness.
Armstrong N, Welsman JR, McManus AM (2008). Aerobic Fitness. In Armstrong N, Mechelen WV (Eds.) Paediatric Exercise Science and Medicine, Oxford: Oxford University Press, 269-282.
Armstrong N, Welsman JR (2008). Aerobic Fitness. In Armstrong N, Mechelen WV (Eds.) Paediatric Exercise Science and Medicine, Oxford: Oxford University Press, 97-108.
Welsman JR, Armstrong N (2008). Interpreting exercise performance data in relation to body size. In Armstrong N, Mechelen WV (Eds.) Paediatric Exercise Science and Medicine, Oxford, UK: Oxford University Press, 13-21.
Armstrong N, Welsman JR, Williams CA (2008). Maximal Intensity Exercise. In Armstrong N, Mechelen WV (Eds.) Paediatric Exercise Science and Medicine, Oxford: Oxford University Press, 55-66.
Welsman JR, Armstrong N (2007). Scaling for Size: Relevance to Understanding the Effects of Growth on Performance. In Hebestreit H, Bar-Or O (Eds.) Encyclopaedia of Sports Medicine: the Young Athlete, Oxford: Blackwell, 50-62.
Armstrong N, Welsman J (2006). Exercise Metabolism. In Armstrong N (Ed) Paediatric Exercise Physiology, Edinburgh: Churchill Livingstone, 71-98.
Welsman J, Armstrong N (2006). Interpreting Performance in Relation to Body Size. In Armstrong N (Ed) Paediatric Exercise Physiology, Edinburgh: Churchill Livingstone, 27-46.
Hopkins SJ, Brown M, Toms AD, Welsman JR, Knapp KM (2014). EFFECTS OF LEG FRACTURES ON BONE MINERAL DENSITY AT THE HIP IN a FEMALE POSTMENOPAUSAL POPULATION: IMPLICATIONS FOR FRACTURE LAISON SERVICES. Author URL.
Hopkins SJ, Smith C, Toms A, Brown M, Welsman J, Knapp K (2012). Left-right weight-bearing:short and long-term measurement precision, and effects of weight-bearing imbalance on hip bone mineral density and leg lean tissue mass.
Welsman J, Knapp K, MacLeod K, Blake G (2009). OBESITY INCREASES PRECISION ERRORS IN DUAL ENERGY X-RAY ABSORPTIOMETRY MEASUREMENTS. Author URL.
Welsman JR, Stoedefalke K, Winlove M (2008). Cardiopulmonary adaptations to intensive swimming training in children. 13th Annual Congress of the European College of Sport Scienc. 9th - 12th Jul 2008.
Burrows M, Welsman JR, Simpson D (2005). Allometric scaling of bone mineral content at various skeletal sites in boy and girls: Accounting for bone size and other confounding variables. Author URL.
Middlebrooke AR, Armstrong N, Shore AC, Welsman JR, MacLeod KM (2002). Peak oxygen consumption and microvascular function in adults with Type 2 diabetes. Author URL.
Fawkner SG, Armstrong N, Childs DJ, Welsman JR (2002). Reliability of the ventilation threshold and V-slope in children. Author URL.
Armstrong N, Welsman JR (2000). Development of aerobic fitness during childhood and adolescence.
Development of aerobic fitness during childhood and adolescence
Welsman JR, Armstrong N (2000). Statistical techniques for interpreting body size-related exercise performance during growth.
Statistical techniques for interpreting body size-related exercise performance during growth
Meakin J, Ames RM, Jeynes JCG, Welsman JR, Gundry MJ, Knapp KM, Everson RM (2019). CitSeg pilot data. Full text.
Knapp KM, Welsman J, Hopkins SJ, Mullan E (2013). Enhancing Radiography Students Understanding of Depression: a Teaching Model using Service Users.
Enhancing Radiography Students Understanding of Depression: a Teaching Model using Service Users
Enhancing Radiography Students Understanding of Depression: a Teaching Model using Service Users. Liverpool 10-12 June 2013. UKRC conference proceedings. P90;P-189
Knapp KM, Hopkins SJ, Smith C, Toms A, Brown M, Welsman J (2012). A pilot study investigating the long-term effects on function, bone mineral density and lean tissue mass post fracture in a female postmenopausal population.
A pilot study investigating the long-term effects on function, bone mineral density and lean tissue mass post fracture in a female postmenopausal population.
A pilot study investigating the long-term effects on function, bone mineral density and lean tissue mass post fracture in a female postmenopausal population. UK Radiological Congress, Manchester 23-25th June 2012. UKRC conference proceedings. P-064
Knapp KM, Welsman J, Summers IR, Seymour R, Fulford J (2012). Incidental findings in low resolution visceral adipose tissue magnetic resonance imaging scans.
Incidental findings in low resolution visceral adipose tissue magnetic resonance imaging scans.
Incidental findings in low resolution visceral adipose tissue magnetic resonance imaging scans. UK Radiological Congress, Manchester 23-25th June 2012. UKRC conference proceedings.P-144
Knapp KM, Meakin J, Fulford J, Seymour R, Welsman J (2012). The association between extensor muscle size and sagittal curvature in the lumbar spine.
The association between extensor muscle size and sagittal curvature in the lumbar spine.
The association between extensor muscle size and sagittal curvature in the lumbar spine. UK Radiological Congress, Manchester 23-25th June 2012. UKRC conference proceedings. P46:016.
Jo_Welsman Details from cache as at 2022-01-19 22:06:34 | CommonCrawl |
Power of a point
In elementary plane geometry, the power of a point is a real number that reflects the relative distance of a given point from a given circle. It was introduced by Jakob Steiner in 1826.[1]
Specifically, the power $\Pi (P)$ of a point $P$ with respect to a circle $c$ with center $O$ and radius $r$ is defined by
$\Pi (P)=|PO|^{2}-r^{2}.$
If $P$ is outside the circle, then $\Pi (P)>0$,
if $P$ is on the circle, then $\Pi (P)=0$ and
if $P$ is inside the circle, then $\Pi (P)<0$.
Due to the Pythagorean theorem the number $\Pi (P)$ has the simple geometric meanings shown in the diagram: For a point $P$ outside the circle $\Pi (P)$ is the squared tangential distance $|PT|$ of point $P$ to the circle $c$.
Points with equal power, isolines of $\Pi (P)$, are circles concentric to circle $c$.
Steiner used the power of a point for proofs of several statements on circles, for example:
• Determination of a circle, that intersects four circles by the same angle.[2]
• Solving the Problem of Apollonius
• Construction of the Malfatti circles:[3] For a given triangle determine three circles, which touch each other and two sides of the triangle each.
• Spherical version of Malfatti's problem:[4] The triangle is a spherical one.
Essential tools for investigations on circles are the radical axis of two circles and the radical center of three circles.
The power diagram of a set of circles divides the plane into regions within which the circle minimizing the power is constant.
More generally, French mathematician Edmond Laguerre defined the power of a point with respect to any algebraic curve in a similar way.
Geometric properties
Besides the properties mentioned in the lead there are further properties:
Orthogonal circle
For any point $P$ outside of the circle $c$ there are two tangent points $T_{1},T_{2}$ on circle $c$, which have equal distance to $P$. Hence the circle $o$ with center $P$ through $T_{1}$ passes $T_{2}$, too, and intersects $c$ orthogonal:
• The circle with center $P$ and radius ${\sqrt {\Pi (P)}}$ intersects circle $c$ orthogonal.
If the radius $\rho $ of the circle centered at $P$ is different from ${\sqrt {\Pi (P)}}$ one gets the angle of intersection $\varphi $ between the two circles applying the Law of cosines (see the diagram):
$\rho ^{2}+r^{2}-2\rho r\cos \varphi =|PO|^{2}$
$\rightarrow \ \cos \varphi ={\frac {\rho ^{2}+r^{2}-|PO|^{2}}{2\rho r}}={\frac {\rho ^{2}-\Pi (P)}{2\rho r}}$
($PS_{1}$ and $OS_{1}$ are normals to the circle tangents.)
If $P$ lies inside the blue circle, then $\Pi (P)<0$ and $\varphi $ is always different from $90^{\circ }$.
If the angle $\varphi $ is given, then one gets the radius $\rho $ by solving the quadratic equation
$\rho ^{2}-2\rho r\cos \varphi -\Pi (P)=0$.
Intersecting secants theorem, intersecting chords theorem
For the intersecting secants theorem and chord theorem the power of a point plays the role of an invariant:
• Intersecting secants theorem: For a point $P$ outside a circle $c$ and the intersection points $S_{1},S_{2}$ of a secant line $g$ with $c$ the following statement is true: $|PS_{1}|\cdot |PS_{2}|=\Pi (P)$, hence the product is independent of line $g$. If $g$ is tangent then $S_{1}=S_{2}$ and the statement is the tangent-secant theorem.
• Intersecting chords theorem: For a point $P$ inside a circle $c$ and the intersection points $S_{1},S_{2}$ of a secant line $g$ with $c$ the following statement is true: $|PS_{1}|\cdot |PS_{2}|=-\Pi (P)$, hence the product is independent of line $g$.
Radical axis
Let $P$ be a point and $c_{1},c_{2}$ two non concentric circles with centers $O_{1},O_{2}$ and radii $r_{1},r_{2}$. Point $P$ has the power $\Pi _{i}(P)$ with respect to circle $c_{i}$. The set of all points $P$ with $\Pi _{1}(P)=\Pi _{2}(P)$ is a line called radical axis. It contains possible common points of the circles and is perpendicular to line ${\overline {O_{1}O_{2}}}$.
Secants theorem, chords theorem: common proof
Both theorems, including the tangent-secant theorem, can be proven uniformly:
Let $P:{\vec {p}}$ be a point, $c:{\vec {x}}^{2}-r^{2}=0$ a circle with the origin as its center and ${\vec {v}}$ an arbitrary unit vector. The parameters $t_{1},t_{2}$ of possible common points of line $g:{\vec {x}}={\vec {p}}+t{\vec {v}}$ (through $P$) and circle $c$ can be determined by inserting the parametric equation into the circle's equation:
$({\vec {p}}+t{\vec {v}})^{2}-r^{2}=0\quad \rightarrow \quad t^{2}+2t\;{\vec {p}}\cdot {\vec {v}}+{\vec {p}}^{2}-r^{2}=0\ .$
From Vieta's theorem one finds:
$t_{1}\cdot t_{2}={\vec {p}}^{2}-r^{2}=\Pi (P)$. (independent of ${\vec {v}}$ !)
$\Pi (P)$ is the power of $P$ with respect for circle $c$.
Because of $|{\vec {v}}|=1$ one gets the following statement for the points $S_{1},S_{2}$:
$|PS_{1}|\cdot |PS_{2}|=t_{1}t_{2}=\Pi (P)\ $, if $P$ is outside the circle,
$|PS_{1}|\cdot |PS_{2}|=-t_{1}t_{2}=-\Pi (P)\ $, if $P$ is inside the circle ($t_{1},t_{2}$ have different signs !).
In case of $t_{1}=t_{2}$ line $g$ is a tangent and $\Pi (P)$ the square of the tangential distance of point $P$ to circle $c$.
Similarity points, common power of two circles
Similarity points
Similarity points are an essential tool for Steiner's investigations on circles.[5]
Given two circles
$\ c_{1}:({\vec {x}}-{\vec {m}}_{1})-r_{1}^{2}=0,\quad c_{2}:({\vec {x}}-{\vec {m}}_{2})-r_{2}^{2}=0\ .$
A homothety (similarity) $\sigma $, that maps $c_{1}$ onto $c_{2}$ stretches (jolts) radius $r_{1}$ to $r_{2}$ and has its center $Z:{\vec {z}}$ on the line ${\overline {M_{1}M_{2}}}$, because $\sigma (M_{1})=M_{2}$. If center $Z$ is between $M_{1},M_{2}$ the scale factor is $s=-{\tfrac {r_{2}}{r_{1}}}$. In the other case $s={\tfrac {r_{2}}{r_{1}}}$. In any case:
$\sigma ({\vec {m}}_{1})={\vec {z}}+s({\vec {m}}_{1}-{\vec {z}})={\vec {m}}_{2}$.
Inserting $s=\pm {\tfrac {r_{2}}{r_{1}}}$ and solving for ${\vec {z}}$ yields:
${\vec {z}}={\frac {r_{1}{\vec {m}}_{2}\mp r_{2}{\vec {m}}_{1}}{r_{1}\mp r_{2}}}$.
Point
$E:{\vec {e}}={\frac {r_{1}{\vec {m}}_{2}-r_{2}{\vec {m}}_{1}}{r_{1}-r_{2}}}$
is called the exterior similarity point and
$I:{\vec {i}}={\frac {r_{1}{\vec {m}}_{2}+r_{2}{\vec {m}}_{1}}{r_{1}+r_{2}}}$
is called the inner similarity point.
In case of $M_{1}=M_{2}$ one gets $E=I=M_{i}$.
In case of $r_{1}=r_{2}$: $E$ is the point at infinity of line ${\overline {M_{1}M_{2}}}$ and $I$ is the center of $M_{1},M_{2}$.
In case of $r_{1}=|EM_{1}|$ the circles touch each other at point $E$ inside (both circles on the same side of the common tangent line).
In case of $r_{1}=|IM_{1}|$ the circles touch each other at point $I$ outside (both circles on different sides of the common tangent line).
Further more:
• If the circles lie disjoint (the discs have no points in common), the outside common tangents meet at $E$ and the inner ones at $I$.
• If one circle is contained within the other, the points $E,I$ lie within both circles.
• The pairs $M_{1},M_{2};E,I$ are projective harmonic conjugate: Their cross ratio is $(M_{1},M_{2};E,I)=-1$.
Monge's theorem states: The outer similarity points of three disjoint circles lie on a line.
Common power of two circles
Let $c_{1},c_{2}$ be two circles, $E$ their outer similarity point and $g$ a line through $E$, which meets the two circles at four points $G_{1},H_{1},G_{2},H_{2}$. From the defining property of point $E$ one gets
${\frac {|EG_{1}|}{|EG_{2}|}}={\frac {r_{1}}{r_{2}}}={\frac {|EH_{1}|}{|EH_{2}|}}\ $
$\rightarrow \ |EG_{1}|\cdot |EH_{2}|=|EH_{1}|\cdot |EG_{2}|\ $
and from the secant theorem (see above) the two equations
$|EG_{1}|\cdot |EH_{1}|=\Pi _{1}(E),\quad |EG_{2}|\cdot |EH_{2}|=\Pi _{2}(E).$
Combining these three equations yields:
${\begin{aligned}\Pi _{1}(E)\cdot \Pi _{2}(E)&=|EG_{1}|\cdot |EH_{1}|\cdot |EG_{2}|\cdot |EH_{2}|\\&=|EG_{1}|^{2}\cdot |EH_{2}|^{2}=|EG_{2}|^{2}\cdot |EH_{1}|^{2}\ .\end{aligned}}$
Hence:
$|EG_{1}|\cdot |EH_{2}|=|EG_{2}|\cdot |EH_{1}|={\sqrt {\Pi _{1}(E)\cdot \Pi _{2}(E)}}$
(independent of line $g$ !).
The analog statement for the inner similarity point $I$ is true, too.
The invariants $ {\sqrt {\Pi _{1}(E)\cdot \Pi _{2}(E)}},\ {\sqrt {\Pi _{1}(I)\cdot \Pi _{2}(I)}}$ are called by Steiner common power of the two circles (gemeinschaftliche Potenz der beiden Kreise bezüglich ihrer Ähnlichkeitspunkte).[6]
The pairs $G_{1},H_{2}$ and $H_{1},G_{2}$ of points are antihomologous points. The pairs $G_{1},G_{2}$ and $H_{1},H_{2}$ are homologous.[7][8]
Determination of a circle that is tangent to two circles
For a second secant through $E$:
$|EH_{1}|\cdot |EG_{2}|=|EH'_{1}|\cdot |EG'_{2}|$
From the secant theorem one gets:
The four points $H_{1},G_{2},H'_{1},G'_{2}$ lie on a circle.
And analogously:
The four points $G_{1},H_{2},G'_{1},H'_{2}$ lie on a circle, too.
Because the radical lines of three circles meet at the radical (see: article radical line), one gets:
The secants ${\overline {H_{1}H'_{1}}},\;{\overline {G_{2}G'_{2}}}$ meet on the radical axis of the given two circles.
Moving the lower secant (see diagram) towards the upper one, the red circle becomes a circle, that is tangent to both given circles. The center of the tangent circle is the intercept of the lines ${\overline {M_{1}H_{1}}},{\overline {M_{2}G_{2}}}$. The secants ${\overline {H_{1}H'_{1}}},{\overline {G_{2}G'_{2}}}$ become tangents at the points $H_{1},G_{2}$. The tangents intercept at the radical line $p$ (in the diagram yellow).
Similar considerations generate the second tangent circle, that meets the given circles at the points $G_{1},H_{2}$ (see diagram).
All tangent circles to the given circles can be found by varying line $g$.
Positions of the centers
If $X$ is the center and $\rho $ the radius of the circle, that is tangent to the given circles at the points $H_{1},G_{2}$, then:
$\rho =|XM_{1}|-r_{1}=|XM_{2}|-r_{2}$
$\rightarrow \ |XM_{2}|-|XM_{1}|=r_{2}-r_{1}.$
Hence: the centers lie on a hyperbola with
foci $M_{1},M_{2}$,
distance of the vertices $2a=r_{2}-r_{1}$,
center $M$ is the center of $M_{1},M_{2}$ ,
linear eccentricity $c={\tfrac {|M_{1}M_{2}|}{2}}$ and
$\ b^{2}=e^{2}-a^{2}={\tfrac {|M_{1}M_{2}|^{2}-(r_{2}-r_{1})^{2}}{4}}$.
Considerations on the outside tangent circles lead to the analog result:
If $X$ is the center and $\rho $ the radius of the circle, that is tangent to the given circles at the points $G_{1},H_{2}$, then:
$\rho =|XM_{1}|+r_{1}=|XM_{2}|+r_{2}$
$\rightarrow \ |XM_{2}|-|XM_{1}|=-(r_{2}-r_{1}).$
The centers lie on the same hyperbola, but on the right branch.
See also Problem of Apollonius.
Power with respect to a sphere
The idea of the power of a point with respect to a circle can be extended to a sphere .[9] The secants and chords theorems are true for a sphere, too, and can be proven literally as in the circle case.
Darboux product
The power of a point is a special case of the Darboux product between two circles, which is given by[10]
$\left|A_{1}A_{2}\right|^{2}-r_{1}^{2}-r_{2}^{2}\,$
where A1 and A2 are the centers of the two circles and r1 and r2 are their radii. The power of a point arises in the special case that one of the radii is zero.
If the two circles are orthogonal, the Darboux product vanishes.
If the two circles intersect, then their Darboux product is
$2r_{1}r_{2}\cos \varphi \,$
where φ is the angle of intersection (see section orthogonal circle).
Laguerre's theorem
Laguerre defined the power of a point P with respect to an algebraic curve of degree n to be the product of the distances from the point to the intersections of a circle through the point with the curve, divided by the nth power of the diameter d. Laguerre showed that this number is independent of the diameter (Laguerre 1905). In the case when the algebraic curve is a circle this is not quite the same as the power of a point with respect to a circle defined in the rest of this article, but differs from it by a factor of d2.
References
1. Jakob Steiner: Einige geometrische Betrachtungen, 1826, S. 164
2. Steiner, p. 163
3. Steiner, p. 178
4. Steiner, p. 182
5. Steiner: p. 170,171
6. Steiner: p. 175
7. Michel Chasles, C. H. Schnuse: Die Grundlehren der neuern Geometrie, erster Theil, Verlag Leibrock, Braunschweig, 1856, p. 312
8. William J. M'Clelland: A Treatise on the Geometry of the Circle and Some Extensions to Conic Sections by the Method of Reciprocation,1891, Verlag: Creative Media Partners, LLC, ISBN 978-0-344-90374-8, p. 121,220
9. K.P. Grothemeyer: Analytische Geometrie, Sammlung Göschen 65/65A, Berlin 1962, S. 54
10. Pierre Larochelle, J. Michael McCarthy:Proceedings of the 2020 USCToMM Symposium on Mechanical Systems and Robotics, 2020, Springer-Verlag, ISBN 978-3-030-43929-3, p. 97
• Coxeter, H. S. M. (1969), Introduction to Geometry (2nd ed.), New York: Wiley.
• Darboux, Gaston (1872), "Sur les relations entre les groupes de points, de cercles et de sphéres dans le plan et dans l'espace", Annales Scientifiques de l'École Normale Supérieure, 1: 323–392, doi:10.24033/asens.87.
• Laguerre, Edmond (1905), Oeuvres de Laguerre: Géométrie (in French), Gauthier-Villars et fils, p. 20
• Steiner, Jakob (1826). "Einige geometrischen Betrachtungen" [Some geometric considerations]. Crelle's Journal (in German). 1: 161–184. doi:10.1515/crll.1826.1.161. S2CID 122065577. Figures 8–26.
• Berger, Marcel (1987), Geometry I, Springer, ISBN 978-3-540-11658-5
Further reading
• Ogilvy C. S. (1990), Excursions in Geometry, Dover Publications, pp. 6–23, ISBN 0-486-26530-7
• Coxeter H. S. M., Greitzer S. L. (1967), Geometry Revisited, Washington: MAA, pp. 27–31, 159–160, ISBN 978-0-88385-619-2
• Johnson RA (1960), Advanced Euclidean Geometry: An elementary treatise on the geometry of the triangle and the circle (reprint of 1929 edition by Houghton Mifflin ed.), New York: Dover Publications, pp. 28–34, ISBN 978-0-486-46237-0
External links
Wikimedia Commons has media related to Power of a point.
• Jacob Steiner and the Power of a Point at Convergence
• Weisstein, Eric W. "Circle Power". MathWorld.
• Intersecting Chords Theorem at cut-the-knot
• Intersecting Chords Theorem With interactive animation
• Intersecting Secants Theorem With interactive animation
| Wikipedia |
The Mathematical Ninja and the Cube Root of 4
The student swam away, thinking almost as hard as he was swimming. The cube root of four? The square root was easy enough, he could do that in his sleep. But the cube root?
OK. Breathe. It's between 1 and 2, obviously. What's 1.5 cubed? The Mathematical Ninja isn't going to like that - three-halves, that's what we need. Cube that, it's $\frac{27}{8}$, or 3.375. Quite a long way short. But - aha! (Breathe) $1.6^3 = 4.096$, should have got that straight away. That's close enough for two lengths.
The student popped his head out of the water. "About 1.6, sensei."
The Mathematical Ninja looked into their box of torture devices and emerged with a sinker. "Two lengths of catch-up. You can do better than that."
The student knew better than to waste energy on grumbling. OK, so a binomial expansion? $\br{4.096-x}^{\frac{1}{3}} \approx 1.6 - \frac{1}{3}\times 4.096^{-\frac{2}{3}} \times x$. Breathe. What's ugly? $4.096^{-\frac{2}{3}}$ is just $\frac{1}{1.6^2}$. And $x$ is $0.096$ here. Turn.
So, it's $1.6 - \frac{1}{3}\times \frac{100}{16^2} \times \frac{96}{1000}$. There's a bit of cancelling can go on there - a factor of 3 and a factor of 16, and a factor of 100 for good measure. $1.6 - \frac{1}{16} \times \frac{2}{10}$ - suddenly it's looking nicer! Breathe. A discrepancy of $\frac{1}{80}$. That's 0.0125, which I can subtract. And head up: "1.5875?"
The Mathematical Ninja nodded their head from side to side. "It's 1.5874," they muttered, "but I'll take it." They handed the student a float. "Two lengths just kick," they said. "By way of celebration."
* edited 2018-10-01 to fix a LaTeX typo
A RITANGLE problem
Using flashcards effectively
Does attitude really equal 100%?
Silly Questions Amnesty | CommonCrawl |
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in IEEE Transactions on Software Engineering (in press)
On-board embedded software developed for spaceflight systems (space software) must adhere to stringent software quality assurance procedures. For example, verification and validation activities are ... [more ▼]
On-board embedded software developed for spaceflight systems (space software) must adhere to stringent software quality assurance procedures. For example, verification and validation activities are typically performed and assessed by third party organizations. To further minimize the risk of human mistakes, space agencies, such as the European Space Agency (ESA), are looking for automated solutions for the assessment of software testing activities, which play a crucial role in this context. Though space software is our focus here, it should be noted that such software shares the above considerations, to a large extent, with embedded software in many other types of cyber-physical systems. Over the years, mutation analysis has shown to be a promising solution for the automated assessment of test suites; it consists of measuring the quality of a test suite in terms of the percentage of injected faults leading to a test failure. A number of optimization techniques, addressing scalability and accuracy problems, have been proposed to facilitate the industrial adoption of mutation analysis. However, to date, two major problems prevent space agencies from enforcing mutation analysis in space software development. First, there is uncertainty regarding the feasibility of applying mutation analysis optimization techniques in their context. Second, most of the existing techniques either can break the real-time requirements common in embedded software or cannot be applied when the software is tested in Software Validation Facilities, including CPU emulators and sensor simulators. In this paper, we enhance mutation analysis optimization techniques to enable their applicability to embedded software and propose a pipeline that successfully integrates them to address scalability and accuracy issues in this context, as described above. Further, we report on the largest study involving embedded software systems in the mutation analysis literature. Our research is part of a research project funded by ESA ESTEC involving private companies (GomSpace Luxembourg and LuxSpace) in the space sector. These industry partners provided the case studies reported in this paper; they include an on-board software system managing a microsatellite currently on-orbit, a set of libraries used in deployed cubesats, and a mathematical library certified by ESA. [less ▲]
Clinical relevance of attentional biases in pediatric chronic pain: an eye-tracking study
Soltani, Sabine; van Ryckeghem, Dimitri ; Vervoort, Tine et al
in Pain (in press)
Attentional biases have been posited as one of the key mechanisms underlying the development and maintenance of chronic pain and co-occurring internalizing mental health symptoms. Despite this theoretical ... [more ▼]
Attentional biases have been posited as one of the key mechanisms underlying the development and maintenance of chronic pain and co-occurring internalizing mental health symptoms. Despite this theoretical prominence, a comprehensive understanding of the nature of biased attentional processing in chronic pain and its relationship to theorized antecedents and clinical outcomes is lacking, particularly in youth. This study used eye-tracking to assess attentional bias for painful facial expressions and its relationship to theorized antecedents of chronic pain and clinical outcomes. Youth with chronic pain (n = 125) and without chronic pain (n = 52) viewed face images of varying levels of pain expressiveness while their eye gaze was tracked and recorded. At baseline, youth completed questionnaires to assess pain characteristics, theorized antecedents (pain catastrophizing, fear of pain, and anxiety sensitivity), and clinical outcomes (pain intensity, interference, anxiety, depression, and posttraumatic stress). For youth with chronic pain, clinical outcomes were reassessed at 3 months to assess for relationships with attentional bias while controlling for baseline symptoms. In both groups, youth exhibited an attentional bias for painful facial expressions. For youth with chronic pain, attentional bias was not significantly associated with theorized antecedents or clinical outcomes at baseline or 3-month follow-up. These findings call into question the posited relationships between attentional bias and clinical outcomes. Additional studies using more comprehensive and contextual paradigms for the assessment of attentional bias are required to clarify the ways in which such biases may manifest and relate to clinical outcomes. [less ▲]
Towards an alternative to territorial jurisdiction to face criminality committed through or facilitated by the use of blockchains
Jolly, Loren
in Eurojus (in press)
Das lateinische Legendar des Dominikanerinnenklosters Marienthal (Brüssel, Königliche Bibliothek, Hs. 831 – 34). Zugleich ein Beitrag zur Kontextualisierung der Lebensbeschreibung der Priorin Yolanda von Vianden
in Hemecht: Zeitschrift für Luxemburger Geschichte (in press), 74(1),
The Dark Side of Digital Financial Transformation: The New Risks of FinTech and the Rise of TechRisk
Zetzsche, Dirk Andreas ; Arner, Douglas; Buckley, Ross
in Singapore Journal of Legal Studies (in press)
Security of Distance−Bounding: A Survey
Gildas, Avoine; Muhammed, Ali Bingöl; Ioana, Boureanu et al
in ACM Computing Surveys (in press)
Distance bounding protocols allow a verifier to both authenticate a prover and evaluate whether the latter is located in his vicinity. These protocols are of particular interest in contactless systems, e ... [more ▼]
Distance bounding protocols allow a verifier to both authenticate a prover and evaluate whether the latter is located in his vicinity. These protocols are of particular interest in contactless systems, e.g. electronic payment or access control systems, which are vulnerable to distance-based frauds. This survey analyzes and compares in a unified manner many existing distance bounding protocols with respect to several key security and complexity features. [less ▲]
Large degrees in scale-free inhomogeneous random graphs
Bhattacharjee, Chinmoy ; Schulte, Matthias
in Annals of Applied Probability (in press)
We consider a class of scale-free inhomogeneous random graphs, which includes some long-range percolation models. We study the maximum degree in such graphs in a growing observation window and show that ... [more ▼]
We consider a class of scale-free inhomogeneous random graphs, which includes some long-range percolation models. We study the maximum degree in such graphs in a growing observation window and show that its limiting distribution is Frechet. We achieve this by proving convergence of the underlying point process of the degrees to a certain Poisson process. Estimating the index of the power-law tail for the typical degree distribution is an important question in statistics. We prove consistency of the Hill estimator for the inverse of the tail exponent of the typical degree distribution. [less ▲]
On the Cohomological Crepant Resolution Conjecture for the complexified Bianchi orbifolds
Perroni, Fabio; Rahm, Alexander
in Algebraic and Geometric Topology (in press)
We give formulae for the Chen--Ruan orbifold cohomology for the orbifolds given by a Bianchi group acting on complex hyperbolic 3-space. The Bianchi groups are the arithmetic groups PSL_2(A), where A is ... [more ▼]
We give formulae for the Chen--Ruan orbifold cohomology for the orbifolds given by a Bianchi group acting on complex hyperbolic 3-space. The Bianchi groups are the arithmetic groups PSL_2(A), where A is the ring of integers in an imaginary quadratic number field. The underlying real orbifolds which help us in our study, given by the action of a Bianchi group on real hyperbolic 3-space (which is a model for its classifying space for proper actions), have applications in physics. We then prove that, for any such orbifold, its Chen-Ruan orbifold cohomology ring is isomorphic to the usual cohomology ring of any crepant resolution of its coarse moduli space. By vanishing of the quantum corrections, we show that this result fits in with Ruan's Cohomological Crepant Resolution Conjecture. [less ▲]
Invariance in a class of operations related to weighted quasi-geometric means
Devillet, Jimmy ; Matkowski, Janusz
in Fuzzy Sets and Systems (in press)
Let $I\subset (0,\infty )$ be an interval that is closed with respect to the multiplication. The operations $C_{f,g}\colon I^{2}\rightarrow I$ of the form \begin{equation*} C_{f,g}\left( x,y\right) =\left ... [more ▼]
Let $I\subset (0,\infty )$ be an interval that is closed with respect to the multiplication. The operations $C_{f,g}\colon I^{2}\rightarrow I$ of the form \begin{equation*} C_{f,g}\left( x,y\right) =\left( f\circ g\right) ^{-1}\left( f\left( x\right) \cdot g\left( y\right) \right) \text{,} \end{equation*} where $f,g$ are bijections of $I$ are considered. Their connections with generalized weighted quasi-geometric means is presented. It is shown that invariance\ question within the class of this operations leads to means of iterative type and to a problem on a composite functional equation. An application of the invariance identity to determine effectively the limit of the sequence of iterates of some generalized quasi-geometric mean-type mapping, and the form of all continuous functions which are invariant with respect to this mapping are given. The equality of two considered operations is also discussed. [less ▲]
Du mode bureaucratique vers l'agilité organisationnelle : le rôle de la communauté de pratique pilotée dans un établissement public
Obringer, Lisa Désirée ; Geraudel, Mickaël ; Benedic, Michael
in Projectique (in press)
Les établissements publics ont besoin de faire preuve d'agilité organisationnelle pour être performants et répondre aux nouveaux enjeux sociétaux. Cependant, ce besoin d'agilité est freiné par la logique ... [more ▼]
Les établissements publics ont besoin de faire preuve d'agilité organisationnelle pour être performants et répondre aux nouveaux enjeux sociétaux. Cependant, ce besoin d'agilité est freiné par la logique bureaucratique qui sous-tend le fonctionnement-même de ces établissements. Résoudre cette tension paradoxale entre bureaucratie et agilité organisationnelle requiert de mettre en œuvre le processus d'acceptation et de management de la tension paradoxale. Pour ce faire, la communauté de pratique pilotée offre un cadre d'analyse favorisant cette acceptation et donc le changement organisationnel souhaité. Ainsi, nous montrons, au travers d'une recherche action, comment la communauté de pratiques pilotée favorise le changement organisationnel au sein d'un établissement public luxembourgeois en facilitant sa transition d'une logique bureaucratique vers une logique d'agilité organisationnelle. Les implications sont doubles. Premièrement, nous montrons comment manager la tension entre bureaucratie et agilité au sein d'un établissement public. Deuxièmement, nous mettons en lumière le rôle de la communauté de pratique pilotée comme vecteur de changement organisationnel d'un établissement public. [less ▲]
Gene selection for optimal prediction of cell position in tissues from single-cell transcriptomics
Tanevski, Jovan; Nguyen, Thin; Truong, Buu et al
in Life Science Alliance (in press)
Single-cell RNA-seq (scRNAseq) technologies are rapidly evolving. While very informative, in standard scRNAseq experiments the spatial organization of the cells in the tissue of origin is lost. Conversely ... [more ▼]
Single-cell RNA-seq (scRNAseq) technologies are rapidly evolving. While very informative, in standard scRNAseq experiments the spatial organization of the cells in the tissue of origin is lost. Conversely, spatial RNA-seq technologies designed to maintain cell localization have limited throughput and gene coverage. Mapping scRNAseq to genes with spatial information increases coverage while providing spatial location. However, methods to perform such mapping have not yet been benchmarked. To fill this gap, we organized the DREAM Single-Cell Transcriptomics challenge focused on the spatial reconstruction of cells from the Drosophila embryo from scRNAseq data, leveraging as silver standard, genes with in situ hybridization data from the Berkeley Drosophila Transcription Network Project reference atlas. The 34 participating teams used diverse algorithms for gene selection and location prediction, while being able to correctly localize clusters of cells. Selection of predictor genes was essential for this task. Predictor genes showed a relatively high expression entropy, high spatial clustering and included prominent developmental genes such as gap and pair-rule genes and tissue markers. Application of the Top-10 methods to a zebrafish embryo dataset yielded similar performance and statistical properties of the selected genes than in the Drosophila data. This suggests that methods developed in this challenge are able to extract generalizable properties of genes that are useful to accurately reconstruct the spatial arrangement of cells in tissues. [less ▲]
Corruption and tax compliance: evidence from small retailers in Bamako, Mali
Bertinelli, Luisito ; Bourgain, Arnaud ; Leon, Florian
in Applied Economics Letters (in press)
We investigate the impact of corruption on tax compliance using a sample of 700 small business in Bamako, Mali. The main contribution of this paper is to focus on micro-enterprises (including semi-formal ... [more ▼]
We investigate the impact of corruption on tax compliance using a sample of 700 small business in Bamako, Mali. The main contribution of this paper is to focus on micro-enterprises (including semi-formal and informal ones), while existing works concentrate on large and formal firms. Our results show that (i) even very small firms pay taxes (two-thirds of firms pay taxes in our sample); and, (ii) paying bribes reduces significantly tax compliance. This latter finding is robust (i) to the addition of a set of control variables accounting for other determinants, (ii) to treatment for endogeneity, and (iii) the use of a different proxy for tax compliance. [less ▲]
Enlarging the frame: Issues of inclusion and mental health in an ageing society
Ferring, Dieter
in Journal of Mental Health Research in Intellectual Disabilities (in press)
This contribution frames the notions of inclusion and mental health by describing trends in European societies at the social and economic level that will have direct consequences for a participative civil ... [more ▼]
This contribution frames the notions of inclusion and mental health by describing trends in European societies at the social and economic level that will have direct consequences for a participative civil society and social cohesion. Starting point is the observation that the world faces challenges at the start of the 21st century that are new and unprecedented in its history. The four global forces that break all the trends known so far in human history include urbanization, accelerating technological development, greater global connections, and population ageing. The authors first describe the scale of population ageing, as ageing populations characterize several developed economies. In a second step, they highlight some consequences of population ageing for social welfare and in a third part they elaborate on the notion of justice and inclusion in rapidly changing societies. [less ▲]
The fourth moment theorem on the Poisson space
Döbler, Christian ; Peccati, Giovanni
in Annals of Probability (in press)
TkT: Automatic Inference of Timed and Extended Pushdown Automata
Pastore, Fabrizio ; Micucci, Daniela; Guzman, Michell et al
To mitigate the cost of manually producing and maintaining models capturing software specifications, specification mining techniques can be exploited to automatically derive up-to-date models that ... [more ▼]
To mitigate the cost of manually producing and maintaining models capturing software specifications, specification mining techniques can be exploited to automatically derive up-to-date models that faithfully represent the behavior of software systems. So far, specification mining solutions focused on extracting information about the functional behavior of the system, especially in the form of models that represent the ordering of the operations. Well-known examples are finite state models capturing the usage protocol of software interfaces and temporal rules specifying relations among system events. Although the functional behavior of a software system is a primary aspect of concern, there are several other non-functional characteristics that must be typically addressed jointly with the functional behavior of a software system. Efficiency is one of the most relevant characteristics. In fact, an application delivering the right functionalities inefficiently has a big chance to not satisfy the expectation of its users. Interestingly, the timing behavior is strongly dependent on the functional behavior of a software system. For instance, the timing of an operation depends on the functional complexity and size of the computation that is performed. Consequently, models that combine the functional and timing behaviors, as well as their dependencies, are extremely important to precisely reason on the behavior of software systems. In this paper, we address the challenge of generating models that capture both the functional and timing behavior of a software system from execution traces. The result is the Timed k-Tail (TkT) specification mining technique, which can mine finite state models that capture such an interplay: the functional behavior is represented by the possible order of the events accepted by the transitions, while the timing behavior is represented through clocks and clock constraints of different nature associated with transitions. Our empirical evaluation with several libraries and applications show that TkT can generate accurate models, capable of supporting the identification of timing anomalies due to overloaded environment and performance faults. Furthermore, our study shows that TkT outperforms state-of-the-art techniques in terms of scalability and accuracy of the mined models. [less ▲]
Im Schatten der Geschichte. Die (vergessene) Gewaltkommission der Bundesregierung zur Analyse und Prävention politisch motivierter Gewalt (1987 bis 1990)
Eckert, Roland; Schumacher, Anette ; Willems, Helmut
in Zeithistorische Forschungen (in press)
Applications of convex analysis within mathematics
Aragón Artacho, Francisco Javier ; Borwein, J. M.; Martín-Márquez, V. et al
in Mathematical Programming (in press)
In this paper, we study convex analysis and its theoretical applications. We first apply important tools of convex analysis to Optimization and to Analysis. We then show various deep applications of ... [more ▼]
In this paper, we study convex analysis and its theoretical applications. We first apply important tools of convex analysis to Optimization and to Analysis. We then show various deep applications of convex analysis and especially infimal convolution in Monotone Operator Theory. Among other things, we recapture the Minty surjectivity theorem in Hilbert space, and present a new proof of the sum theorem in reflexive spaces. More technically, we also discuss autoconjugate representers for maximally monotone operators. Finally, we consider various other applications in mathematical analysis. [less ▲]
Can the Plight of the European Banking Structural Reforms be a Blessing in Disguise?
Nabilou, Hossein
in European Business Organization Law Review (in press)
One of the problems perceived to be at the heart of the global financial crisis was an amalgamation of various commercial and investment banking activities under one entity, as well as the ... [more ▼]
One of the problems perceived to be at the heart of the global financial crisis was an amalgamation of various commercial and investment banking activities under one entity, as well as the interconnectedness of the banking entities with other financial institutions, investment funds, and the shadow banking system. This paper focuses on various measures that aim to structurally separate the banking entities and their core functions from riskier financial activities such as (proprietary) trading or investments in alternative investment funds. Although banking structural reforms in the EU, UK, and the US have taken different forms, their common denominator is the separation of core banking functions from certain trading or securities market activities. Having reviewed the arguments for and against banking structural reforms and their varieties in major jurisdictions, including the EU, UK, US, France, and Germany, the paper argues that a more nuanced approach to introducing such measures at the EU level is warranted. Given the different market structures across the Atlantic and the lack of conclusive evidence on the beneficial impact of banking structural reforms, the paper concludes that the withdrawal of the banking structural reforms proposal by the European Commission has been a prudent move. It seems that in the absence of concrete evidence, experimenting with structural reforms at the Member-State level would be less costly and would provide for opportunities for learning from smaller mistakes that could pave the way for a more optimal approach to introducing banking structural reforms at the European level in the future. [less ▲]
Multiple Sets Exponential Concentration and Higher Order Eigenvalues
Gozlan, Nathael; Herry, Ronan
in Potential Analysis (in press)
On a generic metric measured space, we introduce a notion of improved concentration of measure that takes into account the parallel enlargement of k distinct sets. We show that the k-th eigenvalues of the ... [more ▼]
On a generic metric measured space, we introduce a notion of improved concentration of measure that takes into account the parallel enlargement of k distinct sets. We show that the k-th eigenvalues of the metric Laplacian gives exponential improved concentration with k sets. On compact Riemannian manifolds, this allows us to recover estimates on the eigenvalues of the Laplace-Beltrami operator in the spirit of an inequality of Chung, Grigor'yan & Yau, Upper bounds for eigenvalues of the discrete and continuous Laplace operators. Adv. Math. 117(2), 165–178 (1996). [less ▲]
Evaluating Universal Student Mobility: Contrasting Policy Discourse and Student Narratives in Luxembourg
Kmiotek-Meier, Emilia; Powell, Justin J W
in International Studies in Sociology of Education (in press)
For decades, Luxembourg did without its own national university. Before and after the Luxembourg's founding (UL) (2003), tertiary education and the status of being a Luxembourgish student have been ... [more ▼]
For decades, Luxembourg did without its own national university. Before and after the Luxembourg's founding (UL) (2003), tertiary education and the status of being a Luxembourgish student have been closely linked to international student mobility (ISM). This long-standing tradition was maintained in the new university via compulsory ISM—to bolster the national elite's European networks and internationalization. Focusing on ISM from Luxembourg—based on analysis of policy documents regarding the UL's foundation and state allowances for students—we show that policymakers strongly favored ISM. We confront this policy agenda with the perspectives and self-identification of both credit and degree mobile Luxembourgish students. In narrative interviews, students did not always view compulsory ISM as positively as did policymakers. For students, the quality of a stay abroad is much more important—a perspective lacking in the state's quantity-driven agenda. In the country with the highest ISM rates globally, constraints remain to achieve equity in ISM. [less ▲]
Environmental and Sustainability Education in the Benelux Region
in Environmental Education Research (in press), (Special Issue),
SI‐AKAV: Secure integrated authentication and key agreement for cellular‐connected IoT devices in vehicular social networks
Esfahani, Alireza ; Decouchant, Jérémie ; Volp, Marcus et al
in Transactions on Emerging Telecommunications Technologies (in press)
How some bankers made a million by trading just two securities?
Rinne, Kalle ; Suominen, Matti
in Journal of Empirical Finance (in press)
We study a pair trading strategy that utilizes short-term return reversals in the stock market. Using U.S. data, we show that returns to our pair trading strategy exceed reasonable estimates for ... [more ▼]
We study a pair trading strategy that utilizes short-term return reversals in the stock market. Using U.S. data, we show that returns to our pair trading strategy exceed reasonable estimates for transaction costs. The strategy also generates positive alpha when controlling for the standard risk factors. Second, using transaction level data from Finland, focusing on a popular pair, we provide evidence that these kinds of pair trading returns are compensation from providing liquidity. On the days when the expected returns to our pair trading strategy are the highest, the trading volume is abnormally high and, judging from active brokers' net trades, nearly 45% of all brokers (or their customers) engage in pair trading in accordance with our trading strategy. These brokers are mainly counterparties to few brokers that trade large quantities of stocks inconsistent with our strategy. [less ▲]
Spelling patterns of plural marking and learning trajectories in French taught as a foreign language
Weth, Constanze ; Ugen, Sonja ; Fayol, Michel et al
in Written Language and Literacy (in press), 24
Although French plural spelling has been studied extensively, the complexity of factors affecting the learning of French plural spelling are not yet fully explained, namely on the level of adjectival and ... [more ▼]
Although French plural spelling has been studied extensively, the complexity of factors affecting the learning of French plural spelling are not yet fully explained, namely on the level of adjectival and verbal plural. This study investigates spelling profiles of French plural markers of 228 multilingual grade 5 pupils with French taught as a foreign language. Three analyses on the learner performances of plural spelling in nouns, verbs and pre- and postnominal attributive adjectives were conducted (1) to detect the pupils' spelling profiles of plural marking on the basis of the performances in the pretest, (2) to test the profiles against two psycholinguistic theories, and (3) to evaluate the impact of the training on each spelling profile in the posttest. The first analysis confirms the existing literature that pupils' learning of French plural is not random but ordered and emphasizes the role of the position for adjectives (pre- or postnominal) on correct plural spelling. The second analysis reveals the theoretical difficulties of predicting spelling of adjectival and verbal plural. The third analysis shows that strong and poor spellers both benefit from a morphosyntactic training and provides transparency and traceability of the learning trajectories. Together, the descriptive analyses reveal clear patterns of intra-individual spelling profiles. They point to a need for further research in those areas that have empirically provided the most inconsistent results to date and that are not supported by the theories: verbs and adjectives. [less ▲]
How to conduct a meta‑analysis in eight steps: a practical guide
Hansen, Christopher ; Steinmetz, Holger; Block, Jörn
in Management Review Quarterly (in press)
Book review: Bessey, Valérie et Werner Paravicini: Guerre des manifestes : Charles le Téméraire et ses ennemis (1465-1475)
Genot, Gilles
in Hemecht: Zeitschrift für Luxemburger Geschichte (in press)
Functional Convergence of U-processes with Size-Dependent Kernels
Döbler, Christian ; Kasprzak, Mikolaj ; Peccati, Giovanni
High-performance modeling of concrete ageing
Habera, Michal ; Zilian, Andreas
in Proceedings in Applied Mathematics and Mechanics (in press)
Long-term behaviour of concrete structural elements is very important for evaluation of its health and serviceability range. The phenomena that must be considered are complex and lead to coupled ... [more ▼]
Long-term behaviour of concrete structural elements is very important for evaluation of its health and serviceability range. The phenomena that must be considered are complex and lead to coupled multiphysics formulations. Such formulations are difficult not only from physical perspective, but also from computational perspective. In this contribution attention to computational efficiency and effective implementation is payed. Presented model for concrete ageing is based on microprestress-solidification (MPS) theory of Bazant [1], Kunzel's model for heat and moisture transport [2] and Mazars model for damage [3]. Ageing linear viscoelastic response, which is immanent to MPS theory and concrete creep, leads to ordinary differetial equation for internal variables solved for every quadrature/nodal point. Numerical structure of the finite element discretisation is examined. Few simplifications on physical model lead to a very efficient linear algebra problem for which standard preconditioned Krylov solvers are reviewed. In parallel, weak and strong scaling tests are performed. All results are produced within open-source finite element framework FEniCS [4]. These models are usually a basis for more involved thermo-hygro-chemo-mechanical (THCM) models with migrating chemical species. It is anticipated, that presented results will help practitioners or other structural engineerers with the choice of suitable and efficient methods for long-term concrete modeling. [less ▲]
Concentration bounds for geometric Poisson functionals: logarithmic Sobolev inequalities revisited
Peccati, Giovanni ; Bachmann, Sascha
in Electronic Journal of Probability (in press)
Do banks and microfinance institutions compete? Microevidence from Madagascar
Leon, Florian ; Baraton, Pierrick
in Economic Development and Cultural Change (in press)
This paper examines whether the loan strategy of a microfinance institution is shaped by the entry of a bank. Specifically, we investigate whether the distance between a borrower of a microfinance ... [more ▼]
This paper examines whether the loan strategy of a microfinance institution is shaped by the entry of a bank. Specifically, we investigate whether the distance between a borrower of a microfinance institution and the closest bank influences loan conditions provided by the microfinance institution. We use an original panel dataset of 32,374 loans granted to 14,834 borrowers provided by one of the largest microfinance institutions in Madagascar between 2008 and 2014. We find that the closer a bank is located to a given MFI borrower, the larger the loan obtained and the less collateral required. We also find that the effect is stronger for clients that could be more easily caught by banks (i.e., large firms and clients without a previous relationship with the MFI). [less ▲]
Parental Assortative Mating and the Intergenerational Transmission of Human Capital
Bingley, Paul; Cappellari, Lorenzo; Tatsiramos, Konstantinos
in Labour Economics (in press)
We study the contribution of parental educational assortative mating to the intergenerational transmission of educational attainment. We develop an empirical model for educational correlations within the ... [more ▼]
We study the contribution of parental educational assortative mating to the intergenerational transmission of educational attainment. We develop an empirical model for educational correlations within the family in which parental educational sorting can translate into intergenerational transmission jointly by both parents, or transmission can originate from each parent independently. Estimating the model using educational attainment from Danish population-based administrative data for over 400,000 families, we find that on aver- age 75 percent of the intergenerational correlation in education is driven by the joint contribution of the par- ents. We also document a 38 percent decline of assortative mating in education for parents born between the early 1920s and the early 1950s. While the raw correlations also show decreases in father- and mother- specific intergenerational transmissions of educational attainment, our model shows that once we decompose all factors of intergenerational mobility, the share of intergenerational transmission accounted for by parent-specific factors increased from 7 to 27 percent; an increase compensated by a corresponding fall in joint intergenerational transmission from both parents, leaving total intergenerational persistence un- changed. The mechanisms of intergenerational transmission have changed, with an increased importance of one-to-one parent-child relationships. [less ▲]
A Polynomial Time Subsumption Algorithm for Nominal Safe $ELO_{\bot}$ under Rational Closure
Casini, Giovanni ; Straccia, Umberto; Meyer, Thomas
in Information Sciences (in press)
Description Logics (DLs) under Rational Closure (RC) is a well-known framework for non-monotonic reasoning in DLs. In this paper, we address the concept subsumption decision problem under RC for nominal ... [more ▼]
Description Logics (DLs) under Rational Closure (RC) is a well-known framework for non-monotonic reasoning in DLs. In this paper, we address the concept subsumption decision problem under RC for nominal safe $ELO_{\bot}$, a notable and practically important DL representative of the OWL 2 profile OWL 2 EL. Our contribution here is to define a polynomial time subsumption procedure for nominal safe $ELO_{\bot}$ under RC that relies entirely on a series of classical, monotonic $EL_{\bot}$ subsumption tests. Therefore, any existing classical monotonic $EL_{\bot}$ reasoner can be used as a black box to implement our method. We then also adapt the method to one of the known extensions of RC for DLs, namely Defeasible Inheritance-based DLs without losing the computational tractability. [less ▲]
A new synuclein-transgenic mouse model for early Parkinson's reveals molecular features of preclinical disease
Hendrickx, Diana M.; Garcia, Pierre; Ashrafi, Amer et al
in Molecular Neurobiology (in press)
Understanding Parkinson's disease (PD) in particular in its earliest phases, is important for diagnosis and treatment. However, human brain samples are collected post- mortem, reflecting mainly end stage ... [more ▼]
Understanding Parkinson's disease (PD) in particular in its earliest phases, is important for diagnosis and treatment. However, human brain samples are collected post- mortem, reflecting mainly end stage disease. Because brain samples of mouse models can be collected at any stage of the disease process, they are useful to investigate PD progression. Here, we compare ventral midbrain transcriptomics profiles from α- synuclein transgenic mice with a progressive, early PD-like striatal neurodegeneration across different ages using pathway, gene set and network analysis methods. Our study uncovers statistically significant altered genes across ages and between genotypes with known, suspected, or unknown function in PD pathogenesis and key pathways associated with disease progression. Among those are genotype-dependent alterations associated with synaptic plasticity, neurotransmission, as well as mitochondria-related genes and dysregulation of lipid metabolism. Age-dependent changes were among others observed in neuronal and synaptic activity, calcium homeostasis, and membrane receptor signaling pathways, many of which linked to G- protein coupled receptors. Most importantly, most changes occurred before neurodegeneration was detected in this model, which points to a sequence of gene expression events that may be relevant for disease initiation and progression. It is tempting to speculate that molecular changes similar to those changes observed in our model happen in midbrain dopaminergic neurons before they start to degenerate. In other words, we believe we have uncovered molecular changes that accompany the progression from preclinical to early PD. [less ▲]
Automated, Cost-effective, and Update-driven App Testing
Ngo, Chanh Duc ; Pastore, Fabrizio ; Briand, Lionel
in ACM Transactions on Software Engineering and Methodology (in press)
Apps' pervasive role in our society led to the definition of test automation approaches to ensure their dependability. However, state-of-the-art approaches tend to generate large numbers of test inputs ... [more ▼]
Apps' pervasive role in our society led to the definition of test automation approaches to ensure their dependability. However, state-of-the-art approaches tend to generate large numbers of test inputs and are unlikely to achieve more than 50% method coverage. In this paper, we propose a strategy to achieve significantly higher coverage of the code affected by updates with a much smaller number of test inputs, thus alleviating the test oracle problem. More specifically, we present ATUA, a model-based approach that synthesizes App models with static analysis, integrates a dynamically-refined state abstraction function, and combines complementary testing strategies, including (1) coverage of the model structure, (2) coverage of the App code, (3) random exploration, and (4) coverage of dependencies identified through information retrieval. Its model-based strategy enables ATUA to generate a small set of inputs that exercise only the code affected by the updates. In turn, this makes common test oracle solutions more cost-effective as they tend to involve human effort. A large empirical evaluation, conducted with 72 App versions belonging to nine popular Android Apps, has shown that ATUA is more effective and less effort-intensive than state-of-the-art approaches when testingApp updates. [less ▲]
Smart Bound Selection for the Verification of UML/OCL Class Diagrams
Clarisó, Robert; Gonzalez Perez, Carlos Alberto ; Cabot, Jordi
Correctness of UML class diagrams annotated with OCL constraints can be checked using bounded verification techniques, e.g., SAT or constraint programming (CP) solvers. Bounded verification detects faults ... [more ▼]
Correctness of UML class diagrams annotated with OCL constraints can be checked using bounded verification techniques, e.g., SAT or constraint programming (CP) solvers. Bounded verification detects faults efficiently but, on the other hand, the absence of faults does not guarantee a correct behavior outside the bounded domain. Hence, choosing suitable bounds is a non-trivial process as there is a trade-off between the verification time (faster for smaller domains) and the confidence in the result (better for larger domains). Unfortunately, bounded verification tools provide little support in the bound selection process. In this paper, we present a technique that can be used to (i) automatically infer verification bounds whenever possible, (ii) tighten a set of bounds proposed by the user and (iii) guide the user in the bound selection process. This approach may increase the usability of UML/OCL bounded verification tools and improve the efficiency of the verification process. [less ▲]
Pension Insecurity and Wellbeing in Europe
Olivera, Javier; Ponomarenko, Valentina
in Journal of Social Policy (in press)
This paper studies pension insecurity in a sample of non-retired individuals aged 50 years or older from 18 European countries. We capture pension insecurity with the subjective expectations on the ... [more ▼]
This paper studies pension insecurity in a sample of non-retired individuals aged 50 years or older from 18 European countries. We capture pension insecurity with the subjective expectations on the probability that the government will reduce the pensions of the individual before retirement or will increase the statutory retirement age. We argue that changes in economic conditions and policy affect the formation of such probabilities, and through this, subjective wellbeing. In particular, we study the effects of pension insecurity on subjective wellbeing with pooled linear models, regressions per quintiles and instrumental variables. We find a statistically significant, stable and negative association between pension insecurity and subjective wellbeing. Our findings reveal that the individuals who are more affected by pension insecurity are those who are further away fromtheir retirement, have lower income, assess their life survival as low, have higher cognitive abilities and do not expect private pension payments. [less ▲]
The Gamma Stein equation and non-central de Jong theorems
in Bernoulli (in press)
Non-normality, topological transitivity and expanding families
Meyrath, Thierry ; Müller, Jürgen
in Mathematical Proceedings of the Cambridge Philosophical Society (in press)
We investigate the behaviour of families of meromorphic functions in the neighbourhood of points of non-normality and prove certain covering properties that complement Montel's Theorem. In particular, we ... [more ▼]
We investigate the behaviour of families of meromorphic functions in the neighbourhood of points of non-normality and prove certain covering properties that complement Montel's Theorem. In particular, we also obtain characterisations of non-normality in terms of such properties. [less ▲]
Safety-aware Location Privacy in VANET: Evaluation and Comparison
Emara, Karim Ahmed Awad El-Sayed
in IEEE Transactions on Vehicular Technology (in press)
VANET safety applications broadcast cooperative awareness messages (CAM) periodically to provide vehicles with continuous updates about the surrounding traffic. The periodicity and the spatiotemporal ... [more ▼]
VANET safety applications broadcast cooperative awareness messages (CAM) periodically to provide vehicles with continuous updates about the surrounding traffic. The periodicity and the spatiotemporal information contained in these messages allow a global adversary to track vehicle movements. Many privacy schemes have been proposed for VANET, but only few schemes consider their impact on safety applications. Also, each scheme is evaluated using inconsistent metrics and unrealistic vehicle traces, which makes comparing the actual performance of different schemes in the wild more difficult. In this paper, we aim to fill this gap and compare different privacy schemes not only in terms of the privacy gained but also their impact on safety applications. A distortion-based privacy metric is initially proposed and compared with other popular privacy metrics showing its effectiveness in measuring privacy. A practical safety metric which is based on Monte Carlo analysis is then proposed to measure the QoS of two safety applications: forward collision warning and lane change warning. Using realistic vehicle traces, six state-of-the-art VANET privacy schemes are evaluated and compared in terms of the proposed privacy and safety metrics. Among the evaluated schemes, it was found that the coordinated silent period scheme achieves the best privacy and QoS levels but fully synchronized silence among all vehicles is a practical challenge. The CAPS and CADS schemes provide a practical compromise between privacy and safety since they employ only the necessary silence periods to prevent tracking and avoid changing pseudonyms in trivial situations. [less ▲]
Uncertainty-driven symmetry-breaking and stochastic stability in a generic differential game of lobbying
Boucekkine, Raouf; Fabien, Prieur; Ruan, Weihua et al
in Economic Theory (in press)
We study a 2-players stochastic differential game of lobbying. Players invest in lobbying activities to alter the legislation in her own benefit. The payoffs are quadratic and uncertainty is driven by a ... [more ▼]
We study a 2-players stochastic differential game of lobbying. Players invest in lobbying activities to alter the legislation in her own benefit. The payoffs are quadratic and uncertainty is driven by a Wiener process. We consider the Nash symmetric game where players face the same cost and extract symmetric payoffs, and we solve for Markov Perfect Equilibria (MPE) in the class of affine functions. First, we prove a general sufficient (catching up) optimality condition for two-players stochastic games with uncertainty driven by Wiener processes. Second, we prove that the number and nature of MPE depend on the extent of uncertainty (i.e the variance of the Wiener processes). In particular, we prove that while a symmetric MPE always exists, two asymmetric MPE emerge if and only if uncertainty is large enough. Third, we study the stochastic stability of all the equilibria. We notably find, that the state converges to a stationary invariant distribution under asymmetric MPE. Fourth, we study the implications for rent dissipation asymptotically and compare the outcomes of symmetric vs asymmetric MPE in this respect, ultimately enhancing again the role of uncertainty. [less ▲]
Is a dynamic approach to tax games relevant?
Paulus, Nora; Pieretti, Patrice ; Zou, Benteng
in Annals of Economics and statistics (in press)
In this paper, we argue that static models provide an incomplete analysis of interjurisdictional tax competition. Accordingly, one can doubt whether a one-shot view is suitable for analyzing real world ... [more ▼]
In this paper, we argue that static models provide an incomplete analysis of interjurisdictional tax competition. Accordingly, one can doubt whether a one-shot view is suitable for analyzing real world tax competition. Contrary to previous contributions in tax competition, we are able to model the interplay between changing tax rates and sluggish factor adjustments. We demonstrate that the intensity of tax competition is impacted by the temporal nature of the game. The commitment of governments to stick to their tax policies for a given period (open-loop behavior) leads to less intense competition relative to a static approach. If the policymakers continuously update their tax rates (Markovian behavior), competition is fiercer than in a static game, except for the case where capital adjustment is relatively sluggish and the governments' marginal valuation of public goods is high enough. [less ▲]
En mots et en images : le corps à l'œuvre chez Annie Ernaux
in Sens Public (in press)
On unavoidable families of meromorphic functions
Meyrath, Thierry
in Canadian Mathematical Bulletin (in press)
We prove several results on unavoidable families of meromorphic functions. For instance, we give new examples of families of cardinality three that are unavoidable with respect to the set of meromorphic ... [more ▼]
We prove several results on unavoidable families of meromorphic functions. For instance, we give new examples of families of cardinality three that are unavoidable with respect to the set of meromorphic functions on $\C$. We further obtain families consisting of less than three functions that are unavoidable with respect to certain subsets of meromorphic functions. In the other direction, we show that for every meromorphic function $f$, there exists an entire function that avoids $f$ on $\C$. [less ▲]
A theory-driven design framework for smartphone applications to support healthy and sustainable grocery shopping
Blanke, Julia ; Billieux, Joel; Vögele, Claus
in Human Behavior and Emerging Technologies (in press)
Teachers' assessments of students' achievements: The ecological validity of studies using case vignettes
Krolak-Schwerdt, Sabine ; Hörstermann, Thomas ; Glock, Sabine et al
in Journal of Experimental Education (in press)
The Fragmentation of International Investment and Tax Dispute Settlement: A Good Idea?
Garcia Olmedo, Javier
in Leiden Journal of International Law (in press)
Innovation Search Strategy and Predictable Returns
Fitzgerald, Tristan; Balsmeier, Benjamin ; Fleming, Lee et al
in Management Science (in press)
Bad representations and homotopy groups of Character Varieties
Guerin, Clément ; Lawton, Sean; Ramras, Daniel
in Annales Henri Lebesgue (in press)
Let G be a connected, reductive, complex affine algebraic group, and let Xr denote the moduli space of G-valued representations of a rank r free group. We first characterize the singularities in Xr ... [more ▼]
Let G be a connected, reductive, complex affine algebraic group, and let Xr denote the moduli space of G-valued representations of a rank r free group. We first characterize the singularities in Xr resolving conjectures of Florentino-Lawton. In particular, we compute the codimension of the orbifold singular locus using facts about Borel-de Siebenthal groups. We then use this codimension to calculate some higher homotopy groups of the smooth locus of Xr, proving conjectures of Florentino-Lawton-Ramras. Lastly, using the earlier analysis of Borel-de Siebenthal groups, we prove a conjecture of Sikora about CI Lie groups. [less ▲]
Characterization of field homomorphisms through Pexiderized functional equations
Gselmann, Eszter; Kiss, Gergely ; Vincze, Csaba
in Journal of Difference Equations and Applications (in press)
A remark on Schröder's equation: Formal and analytic linearization of iterative roots of the power series f(z)=z
Reich, Ludwig; Tomaschek, Jörg
in Monatshefte für Mathematik (in press)
We study Schröder's equation (i.e. the problem of linearization) for local analytic functions F with F (0)=0, F(0)=1, F(0) a root of 1. While Schröder's equation in this case need not have even a formal ... [more ▼]
We study Schröder's equation (i.e. the problem of linearization) for local analytic functions F with F (0)=0, F(0)=1, F(0) a root of 1. While Schröder's equation in this case need not have even a formal solution, we show that if F is formally linearizable, then it can also be linearized by an invertible local analytic transformation. On the other hand, there exist also divergent series solutions of Schröder's equation in this situation. We give some applications of our results to iterative functional equations, functional-differential equations and iteration groups. [less ▲]
Europeans and Americans in Korea, 1882-1910: A Bourgeois and Translocal Community
Dittrich, Klaus
in Itinerario (in press), 39(3),
This article deals with the European and American community in Korea between the conclusion of Korea's first international treaties in the early 1880s and the country's annexation by the Japanese Empire ... [more ▼]
This article deals with the European and American community in Korea between the conclusion of Korea's first international treaties in the early 1880s and the country's annexation by the Japanese Empire in 1910. The article starts out by presenting an overview of the community. Concentrated in Seoul and Chemulp'o, the Anglo-Saxon element dominated a community made up of diplomats, foreign experts in the service of the Korean government, merchants and missionaries. Next, the article describes two key characteristics of the European and American residents in Korea. Firstly, they were individuals defining themselves as bourgeois, or middle-class; secondly, the term "translocality" serves to bring together the multiple layers of border-crossing these individuals were involved in – as long-distance migrants between Europe or Northern America and East Asia, as migrants within the East Asian context, and as representatives of different Euro-American nationalities living together in Korea. [less ▲]
Development of a Cued Pro- and Antisaccade Paradigm: An Indirect Measure to Explore Automatic Components of Sexual Interest
Oberlader, Verena A.; Ettinger, Ulrich; Banse, Rainer et al
in Archives of Sexual Behavior (in press)
We developed a cued pro- and antisaccade paradigm (CPAP) to explore automatic components of sexual interest. Heterosexual participants (n = 32 women, n = 25 men) had to perform fast eye movements towards ... [more ▼]
We developed a cued pro- and antisaccade paradigm (CPAP) to explore automatic components of sexual interest. Heterosexual participants (n = 32 women, n = 25 men) had to perform fast eye movements towards and away from sexually relevant or irrelevant stimuli across a congruent (i.e. prosaccade towards sexually relevant stimuli, antisaccade away from sexually irrelevant stimuli) and an incongruent condition (i.e. prosaccade towards sexually irrelevant stimuli, antisaccade away from sexually relevant stimuli). We hypothesized that pro- and antisaccade performance would be influenced by the sexual interest-specific relevance of the presented stimulus (i.e., nude female or male stimulus) and the instructed task (i.e., pro- or antisaccade) and, thus, differ meaningfully between conditions. Results for prosaccades towards sexually relevant stimuli in the congruent condition showed that error rates were lower and latencies were shorter compared with prosaccades towards sexually irrelevant stimuli in the incongruent condition, but only for male participants. In addition, error rates for antisaccades away from sexually irrelevant stimuli in the congruent condition were lower than for antisaccades away from sexually relevant stimuli in the incongruent condition, for both female and male participants. Latencies of antisaccades, however, did not differ between conditions. In comparison with established indirect sexual interest paradigms, the CPAP benefits from measuring highly automated processes less prone to deliberate control. To this end, the CPAP could be applied to explore the interplay of early automatic and deliberate components of sexual information processing. [less ▲]
Competition Numbers, Quasi-Line Graphs and Holes
McKay, Brendan; Schweitzer, Pascal; Schweitzer, Patrick
in SIAM Journal on Discrete Mathematics (in press)
The competition graph of an acyclic directed graph D is the undirected graph on the same vertex set as D in which two distinct vertices are adjacent if they have a common out-neighbor in D. The ... [more ▼]
The competition graph of an acyclic directed graph D is the undirected graph on the same vertex set as D in which two distinct vertices are adjacent if they have a common out-neighbor in D. The competition number of an undirected graph G is the least number of isolated vertices that have to be added to G to make it the competition graph of an acyclic directed graph. We resolve two conjectures concerning competition graphs. First we prove a conjecture of Opsut by showing that the competition number of every quasi-line graph is at most 2. Recall that a quasi-line graph, also called a locally co-bipartite graph, is a graph for which the neighborhood of every vertex can be partitioned into at most two cliques. To prove this conjecture we devise an alternative characterization of quasi-line graphs to the one by Chudnovsky and Seymour. Second, we prove a conjecture of Kim by showing that the competition number of any graph is at most one greater than the number of holes in the graph. Our methods also allow us to prove a strengthened form of this conjecture recently proposed by Kim, Lee, Park and Sano, showing that the competition number of any graph is at most one greater than the dimension of the subspace of the cycle space spanned by the holes. [less ▲]
Attitudes towards Multiculturalism in Luxembourg: Measurement Invariance and Factor Structure of the Multicultural Ideology Scale
Stogianni, Maria ; Murdock, Elke ; He, Jia et al
in International Journal of Intercultural Relations (in press)
In the present study, we examined the dimensionality and the measurement invariance of the Multicultural Ideology Scale (MCI), and mean differences across different cultural groups within the multilingual ... [more ▼]
In the present study, we examined the dimensionality and the measurement invariance of the Multicultural Ideology Scale (MCI), and mean differences across different cultural groups within the multilingual, multicultural context of Luxembourg. Luxembourg is a unique context to study attitudes towards diversity because 47.4% of the citizens are non-nationals (i.e. economic migrants, sojourners, refugees) and minority and majority are increasingly difficult to define. Our sample included 1,488 participants from diverse ethnic backgrounds who completed the survey in German, French or English. In contrast to previous findings, our analyses on responses to the MCI scale produced a two-dimensional structure, distinguishing between positive and negative attitudes towards multiculturalism. The factor structure was partially invariant across ethnocultural groups: Configural and metric invariance were established across natives and non-natives and different language versions. Scalar invariance was only established across gender groups. Natives and male participants reported the most negative attitudes towards multiculturalism. We discuss the importance of assessing measurement invariance and provide recommendations to improve the assessment of psychological multiculturalism. [less ▲]
Smoking related warning messages formulated as questions positively influence short-term smoking behaviour
Müller, Barbara; Ritter, Simone; Glock, Sabine et al
in Journal of Health Psychology (in press)
Students' immigration background as a moderator of predictive validity of tracking decisions.
Klapproth, Florian ; Schaltz, Paule
in Procedia Social and Behavioral Sciences (in press)
De quoi les Conseils nationaux de la productivité sont-ils le nom ? Contribution à la réflexion sur le sens de la gouvernance économique européenne et ses effets sur l'administration nationale
in Europe (in press)
COVID-19 Compliance Behaviors of Older People: The Role of Cognitive and Non-Cognitive Skills
Clark, Andrew; d'Ambrosio, Conchita ; Onur, Ilke et al
in Economics Letters (in press)
This paper examines the empirical relationship between individuals' cognitive and non-cognitive abilities and COVID-19 compliance behaviors using cross-country data from the Survey of Health, Ageing and ... [more ▼]
This paper examines the empirical relationship between individuals' cognitive and non-cognitive abilities and COVID-19 compliance behaviors using cross-country data from the Survey of Health, Ageing and Retirement in Europe (SHARE). We find that both cognitive and non-cognitive skills predict responsible health behaviors during the COVID-19 crisis. Episodic memory is the most important cognitive skill, while conscientiousness and neuroticism are the most significant personality traits. There is also some evidence of a role for an internal locus of control in compliance. [less ▲]
Boundlessly Entangled: Travels and Performances of School Hygiene in the Context of Open-Air Education (c. 1904-1936)
Thyssen, Geert
in Canadian Bulletin of Medical History (in press)
This article develops a histoire croisée of health education using the example of open-air schools. It reflexively analyses the entangled performances of knowledge and praxis around hygiene in the context ... [more ▼]
This article develops a histoire croisée of health education using the example of open-air schools. It reflexively analyses the entangled performances of knowledge and praxis around hygiene in the context of "international" open-air school conferences and in relation to "materials" of open-air education. Such performances reveal open-air schools as "practice and movement" unbound by "national" or otherwise imagined borders. Fragmentation accompanied their circulation and ensued from non/humans' active, co-constitutive role in the mediation of knowledge and praxis. While underexplored, material and economic factors were key to this process. Their analysis enriches the study of the "internationalization" of school hygiene. [less ▲]
On the comprehensibility and perceived privacy protection of indirect questioning techniques
Hoffmann, Adrian; Waubert de Puiseau, Berenike; Schmidt, Alexander F. et al
in Behavior Research Methods (in press)
On surveys that assess sensitive personal attributes, indirect questioning aims at increasing respondents' willingness to answer truthfully by protecting confidentiality. However, the assumption that ... [more ▼]
On surveys that assess sensitive personal attributes, indirect questioning aims at increasing respondents' willingness to answer truthfully by protecting confidentiality. However, the assumption that subjects understand questioning procedures fully and trust them to protect their privacy is tested rarely. In a scenario-based design, we compared four indirect questioning procedures in terms of comprehensibility and perceived privacy protection. All indirect questioning techniques were found less comprehensible for respondents than a conventional direct question used for comparison. Less-educated respondents experienced more difficulties when confronted with any indirect questioning technique. Regardless of education, the Crosswise Model was found most comprehensible among the four indirect methods. Indirect questioning was perceived to increase privacy protection in comparison to a direct question. Unexpectedly, comprehension and perceived privacy protection did not correlate. We recommend assessing these factors separately in future evaluations of indirect questioning. [less ▲]
Structural changes in the labor market and the rise of early retirement in France and Germany
Batyra, Anna; de la Croix, David; Pierrard, Olivier et al
in German Economic Review (in press)
The rise of early retirement in Europe is typically attributed to the European system of taxes and transfers. A model with an imperfectly competitive labor market allows us to consider also the effects of ... [more ▼]
The rise of early retirement in Europe is typically attributed to the European system of taxes and transfers. A model with an imperfectly competitive labor market allows us to consider also the effects of bargaining power and of matching efficiency on pre-retirement. We find that lower bargaining power of workers and declining matching efficiency have been important determinants of early retirement in France and Germany. These structural changes, combined with early-retirement transfers and population aging, are also consistent with the employment and unemployment rates, labor share and seniority premia. [less ▲]
The prevalence of mild cognitive impairment in Latin America and the Caribbean: a systematic review and meta-analysis
Ribeiro, Fabiana ; Teixeira Santos, Ana Carolina ; Leist, Anja
in Aging and Mental Health (in press)
A Bayesian framework to identify random parameter fields based on the copula theorem and Gaussian fields: Application to polycrystalline materials
Rappel, Hussein ; Wu, Ling; Noels, Ludovic et al
in Journal of Applied Mechanics (in press)
For many models of solids, we frequently assume that the material parameters do not vary in space, nor that they vary from one product realization to another. If the length scale of the application ... [more ▼]
For many models of solids, we frequently assume that the material parameters do not vary in space, nor that they vary from one product realization to another. If the length scale of the application approaches the length scale of the micro-structure however, spatially fluctuating parameter fi elds (which vary from one realization of the fi eld to another) can be incorporated to make the model capture the stochasticity of the underlying micro-structure. Randomly fluctuating parameter fields are often described as Gaussian fields. Gaussian fi elds however assume that the probability density function of a material parameter at a given location is a univariate Gaussian distribution. This entails for instance that negative parameter values can be realized, whereas most material parameters have physical bounds (e.g. the Young's modulus cannot be negative). In this contribution, randomly fluctuating parameter fi elds are therefore described using the copula theorem and Gaussian fi elds, which allow di fferent types of univariate marginal distributions to be incorporated, but with the same correlation structure as Gaussian fields. It is convenient to keep the Gaussian correlation structure, as it allows us to draw samples from Gaussian fi elds and transform them into the new random fields. The bene fit of this approach is that any type of univariate marginal distribution can be incorporated. If the selected univariate marginal distribution has bounds, unphysical material parameter values will never be realized. We then use Bayesian inference to identify the distribution parameters (which govern the random fi eld). Bayesian inference regards the parameters that are to be identi fied as random variables and requires a user-defi ned prior distribution of the parameters to which the observations are inferred. For the homogenized Young's modulus of a columnar polycrystalline material of interest in this study, the results show that with a relatively wide prior (i.e. a prior distribution without strong assumptions), a single specimen is su ciffient to accurately recover the distribution parameter values. [less ▲]
A minimal realization technique for the dynamical structure function of a class of LTI systems
Goncalves, Jorge ; Yuan, Ye; Rai, Anurag et al
in IEEE Transactions on Control of Network Systems (in press)
Network Identifiability from Intrinsic Noise
Goncalves, Jorge ; Hayden, David; Yuan, Ye
in IEEE Transactions on Automatic Control (in press)
Diversifizierung von Kindertagesbetreuungsangeboten durch mixed economy of care: Eine vergleichende Perspektive aus Luxemburg und Deutschland.
Schmitz, Anett ; Wiltzius, Martine ; Mierendorff, Johanna
in Zeitschrift für Soziologie der Erziehung und Sozialisation = Journal for Sociology of Education and Socialization (in press)
Visual characterization of associative quasitrivial nondecreasing functions on finite chains
Kiss, Gergely
Regional foreign banks and financial inclusion: Evidence from Africa
Leon, Florian ; Zins, Alexandra
in Economic Modelling (in press)
Regional foreign banks expanded quickly over the past decade in developing and emerging countries and have a growing influence in banking systems. We question whether the development of African regional ... [more ▼]
Regional foreign banks expanded quickly over the past decade in developing and emerging countries and have a growing influence in banking systems. We question whether the development of African regional foreign banks, also called Pan-African banks, influences financial inclusion of firms and households. To this end, we combine the World Bank Global Findex database and the World Bank Enterprise Surveys with a hand-collected database on the presence of regional foreign banks. We find that Pan-African banks presence increases firms' access to credit and limited evidence that they favor financial access of the middle class by restoring confidence in banks. We suggest that this impact is related to the adoption of an aggressive strategy aiming at gaining market shares rather than through the exploitation of informational and technological advantages. [less ▲]
e3-service: an ontology for needs-driven real-world service bundling in a multi-supplier setting
De Kinderen, Sybren ; de Leenheer, Pieter; Gordijn, Jaap et al
in Applied Ontology (in press)
Businesses increasingly offer their services electronically via the Web. Take for example an Internet Service Provider. An ISP offers a variety of services, including raw bandwidth, IP connectivity, and ... [more ▼]
Businesses increasingly offer their services electronically via the Web. Take for example an Internet Service Provider. An ISP offers a variety of services, including raw bandwidth, IP connectivity, and Domain Name resolution. Although in some cases a single service already satisfies a customer need, in many situations a customer need is so complex that a bundle of services is needed to satisfy the need, as with the ISP example. In principle, each service in a bundle can be provisioned by a different supplier. This paper proposes an ontology, e3service , that can be used to formally capture customer needs, services, and multisupplier service bundles of these. In addition, this paper contributes a process called PCM2 to reason with the ontology. First, a customer need is identified for which desired consequences are elicited. Then, the desired set of consequences is matched with consequences associated with services. The matching process results in a service bundle, satisfying the customer need, containing services that each can be provided by different suppliers. PCM2 is inspired by a family of formal reasoning methods called Propose-Critique-Modify (PCM). However, whereas PCM methods emphasize solution generation from a given set of requirements, our reasoning process treats the space of requirements as a first class citizen. Hence PCM2 : the requirements space and solution space are equally important. How the reasoning and matching process practically works, is illustrated by an industry strength case study in the healthcare domain. [less ▲]
Labour, Gender and Ethnicities in the 'Heart of Manila'
Espinosa, Shirlita Africa
in Journal of Sociology (in press)
Manila, like most cities in the developing world, is experiencing the effects of the flexibility of global capital and the consequences of being excluded from the flows of knowledge and finance. Quaipo ... [more ▼]
Manila, like most cities in the developing world, is experiencing the effects of the flexibility of global capital and the consequences of being excluded from the flows of knowledge and finance. Quaipo, the 'heart of Manila', has responded to and negotiates with macroeconomic challenges through the underground economy of media piracy. Given the increase in population, unemployment and the general degradation of urban living amongst the poor, the economy of piracy has become a conduit of socio-economic changes that intersect with the culture-specific economy of worship. Quiapo is a fascinating terrain of Manilenos social history; it is the site of class tension, religious and ethnic divide, state intervention, and urban culture. Today, piracy and worship are forces by which the district's inhabitants and pilgrims define their lives and their labour. This essay examines how piracy and worship impact on the labour, space and gender dynamics of Quiapo. [less ▲]
Möglichkeiten statistischer Erhebungen für politische Strategien – Eine psychologische Deutung der Naturbewusstseinsstudie 2011
Reese, Gerhard
in BfN-Skripten (in press)
Lettere alla redazione : il caso della "Buona Domenica" lussemburghese Un fenomeno mediatico italo-lussemburghese
in El Ghibli - Rivista di Letteratura della Migrazione (in press)
Distance-based social index numbers: a unifying approach
Bossert, Walter; d'Ambrosio, Conchita ; Weber, Shlomo
in Journal of Mathematical Economics (in press)
We present a unified approach to the design of social index numbers. Our starting point is a model that employs an exogenously given partition of the population into subgroups. Three classes of group ... [more ▼]
We present a unified approach to the design of social index numbers. Our starting point is a model that employs an exogenously given partition of the population into subgroups. Three classes of group-dependent measures of deprivation are characterized. The three groups are nested and, beginning with the largest of these, we narrow them down by successively adding two additional axioms. This leads to a parameterized class the members of which are based on the differences between the income (or wealth) levels of an individual and those who are better off. We then proceed to show that our measures are sufficiently general to accommodate a plethora of indices, including measures of inequality and polarization as well as distance-based measures of phenomena such as diversity and fractionalization. [less ▲]
On-line model-based fault detection and isolation for PEM fuel cell stack systems
Rosich, Albert ; Sarrate, Ramon; Nejjari, Fatiha
in Applied Mathematical Modelling (in press)
Efficient and reliable operation of Polymer Electrolyte Membrane (PEM) fuel cells are key requirements for their successful commercialization and application. The use of diagnostic techniques enables the ... [more ▼]
Efficient and reliable operation of Polymer Electrolyte Membrane (PEM) fuel cells are key requirements for their successful commercialization and application. The use of diagnostic techniques enables the achievement of these requirements. This paper focuses on model-based fault detection and isolation (FDI) for PEM fuel cell stack systems. The work consists in designing and selecting a subset of consistency relations such that a set of predefined faults can be detected and isolated. Despite a nonlinear model of the PEM fuel cell stack system will be used, consistency relations that are easily implemented by a variable back substitution method will be selected. The paper also shows the significance of structural models to solve diagnosis issues in complex systems. [less ▲]
Finite-Time Attitude Synchronization with Distributed Discontinuous Protocols
Wei, Jieqiang; Zhang, Silun; Adaldo, Antonio et al
The finite-time attitude synchronization problem is considered in this paper, where the rotation of each rigid body is expressed using the axis-angle representation. Two discontinuous and distributed ... [more ▼]
The finite-time attitude synchronization problem is considered in this paper, where the rotation of each rigid body is expressed using the axis-angle representation. Two discontinuous and distributed controllers using the vectorized signum function are proposed, which guarantee almost global and local convergence, respectively. Filippov solutions and non-smooth analysis techniques are adopted to handle the discontinuities. Sufficient conditions are provided to guarantee finite-time convergence and boundedness of the solutions. Simulation examples are provided to verify the performances of the control protocols designed in this paper. [less ▲]
Interindividual differences in responses to global inequality
Reese, Gerhard ; Proch, Jutta; Cohrs, J. Christopher
in Analyses of Social Issues and Public Policy (in press)
One of humanity's most pressing problems is the inequality between people from "developed" and "developing" countries, which counteracts joint efforts to combat other large scale problems. Little is known ... [more ▼]
One of humanity's most pressing problems is the inequality between people from "developed" and "developing" countries, which counteracts joint efforts to combat other large scale problems. Little is known about the psychological antecedents that affect the perception of and behavioral responses to global inequality. Based on, and extending, Duckitt's (2001) dual-process model, the current research examines psychological antecedents that may explain how people in an industrialized Western country respond to global inequality. In two studies (N1 = 116, N2 = 117), we analyzed the relationship between the Big Five and justice constructs, right-wing authoritarianism (RWA), social dominance orientation (SDO), and behavioral intentions to reduce global inequality. Two-group path analysis revealed support for the dual-process model in that RWA and SDO were important predictors of behavioral intentions and partially acted as mediators between personality and such intentions. Moreover, justice sensitivity explained variance beyond the "classic" DPM variables. In Study 2, we additionally assessed individuals' global social identification and perceived injustice of global inequality that explained additional variance. Extending previous work on the dual-process model, these findings demonstrate that individual and group-based processes predict people's responses to global inequality and uncover potentials to promote behavior in the interest of global justice. [less ▲]
Family income and material deprivation: do they matter for sleep quality and quantity in early life? Evidence from a longitudinal study.
Barazzetta, Marta ; Ghislandi, Simone
in Sleep (in press)
Study Objectives: The aim of the present paper is to investigate the determinants of sleeping patterns in children up to age 9 on a large and geographically homogeneous sample of British children and ... [more ▼]
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Acceptance testing is a validation activity performed to ensure the conformance of software systems with respect to their functional requirements. In safety critical systems, it plays a crucial role since ... [more ▼]
Acceptance testing is a validation activity performed to ensure the conformance of software systems with respect to their functional requirements. In safety critical systems, it plays a crucial role since it is enforced by software standards, which mandate that each requirement be validated by such testing in a clearly traceable manner. Test engineers need to identify all the representative test execution scenarios from requirements, determine the runtime conditions that trigger these scenarios, and finally provide the input data that satisfy these conditions. Given that requirements specifications are typically large and often provided in natural language (e.g., use case specifications), the generation of acceptance test cases tends to be expensive and error-prone. In this paper, we present Use Case Modeling for System-level, Acceptance Tests Generation (UMTG), an approach that supports the generation of executable, system-level, acceptance test cases from requirements specifications in natural language, with the goal of reducing the manual effort required to generate test cases and ensuring requirements coverage. More specifically, UMTG automates the generation of acceptance test cases based on use case specifications and a domain model for the system under test, which are commonly produced in many development environments. Unlike existing approaches, it does not impose strong restrictions on the expressiveness of use case specifications. We rely on recent advances in natural language processing to automatically identify test scenarios and to generate formal constraints that capture conditions triggering the execution of the scenarios, thus enabling the generation of test data. In two industrial case studies, UMTG automatically and correctly translated 95% of the use case specification steps into formal constraints required for test data generation; furthermore, it generated test cases that exercise not only all the test scenarios manually implemented by experts, but also some critical scenarios not previously considered. [less ▲]
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in Submitted to IEEE Journal on Selected topic in Communication (JSAC) (in press)
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\begin{document}
\title[Operator system structures]{Operator system structures and extensions of Schur multipliers}
\author[Y.-F. Lin]{Ying-Fen Lin} \address{Mathematical Sciences Research Centre, Queen's University Belfast, Belfast BT7 1NN, United Kingdom} \email{[email protected]}
\author[I. G. Todorov]{Ivan G. Todorov} \address{Mathematical Sciences Research Centre, Queen's University Belfast, Belfast BT7 1NN, United Kingdom, and School of Mathematical Sciences, Nankai University, 300071 Tianjin, China} \email{[email protected]}
\date{2 December 2018}
\maketitle
\begin{abstract} For a given C*-algebra $\cl A$, we establish the existence of maximal and minimal operator $\cl A$-system structures on an AOU $\cl A$-space. In the case $\cl A$ is a W*-algebra, we provide an abstract characterisation of dual operator $\cl A$-systems, and study the maximal and minimal dual operator $\cl A$-system structures on a dual AOU $\cl A$-space. We introduce operator-valued Schur multipliers, and provide a Grothendieck-type characterisation. We study the positive extension problem for a partially defined operator-valued Schur multiplier $\nph$ and, under some richness conditions, characterise its affirmative solution in terms of the equality between the canonical and the maximal dual operator $\cl A$-system structures on an operator system naturally associated with the domain of $\nph$. \end{abstract}
\section{Introduction}\label{s_intro}
The problem of completing a partially defined matrix to a fully defined positive matrix has attracted considerable attention in the literature (see e.g. \cite{dg} and \cite{gjsw} and the references therein). Given an $n$ by $n$ matrix, only a subset of whose entries are specified, this problem asks whether the remaining entries can be determined so as to yield a positive matrix. For block operator matrices, this problem was considered in \cite{pps}, where the authors showed that it is closely related to questions about automatic complete positivity of certain positive linear maps. More specifically, one associates to the pattern $\kappa$ of the partially defined matrix (that is, the set of all given entries) the operator system $\cl S(\kappa)$ of all fully specified matrices supported by $\kappa$. The positive completion problem is then linked to the question of whether the operator-valued Schur multiplier with domain $\cl S(\kappa)$ is completely positive.
A continuous infinite dimensional version of the scalar-valued completion problem was considered in \cite{llt}, where the authors characterised the operator systems possessing the positive completion property in terms of an approximation of its positive cone via rank one operators. The original motivation behind the present paper was the study of the operator-valued, infinite dimensional and continuous, analogue of the positive completion problem. We relate the question to the automatic complete positivity of operator-valued Schur multipliers; in fact, we characterise the extendability of Schur multipliers in terms of an equality between operator system structures on an associated Archimedean order unit (AOU) *-vector space.
One of the fundamental representation theorems in Operator Space Theory is Choi-Effros Theorem \cite[Theorem 13.1]{Pa}, which characterises operator systems (that is, unital selfadjoint linear subspaces $\cl S$ of the space $\cl B(H)$ of all bounded linear operators on a Hilbert space $H$) abstractly, in terms of properties of the cones of positive elements in the $\cl S$-valued matrix space $M_n(\cl S)$. Operator $\cl A$-systems, that is, the operator systems which admit a bimodule action by a unital C*-algebra $\cl A$, can be characterised similarly in a way that takes into account the extra $\cl A$-module structure \cite[Corollary 15.13]{Pa}. Dual operator systems -- that is, operator systems that are also dual operator spaces -- were characterised by D. P. Blecher and B. Magajna in \cite{bm}. However, no analogous representation of dual operator $\cl A$-systems, where $\cl A$ is a W*-algebra, has been known.
The idea of viewing operator spaces as a quantised version of Banach spaces has been very fruitful in Functional Analysis \cite{er}. Operator systems can in a similar vein be thought of as a quantised version of Archimedean order unit (AOU) *-vector spaces. The possible quantisations, or operator system structures, on a given AOU space, were first studied in \cite{ptt}, where it was shown that every AOU space possesses two extremal operator system structures. However, no similar development has been achieved for dual AOU spaces or for AOU $\cl A$-spaces.
In this paper, we unify all aforementioned strands of questions. We provide a Choi-Effros type representation theorem for dual operator $\cl A$-systems. We study the operator $\cl A$-system structures on a given AOU $\cl A$-space, as well as the dual operator $\cl A$-system structures on a given dual AOU $\cl A$-space. The latter results are new even in the case where $\cl A$ coincides with the complex field. We introduce infinite dimensional measurable operator-valued Schur multipliers, and provide a characterisation that generalises their well-known description by A. Grothendieck \cite{Gro} in the scalar case (see also \cite{haag} and \cite{peller}). Finally, we study the positive extension problem for operator-valued Schur multipliers, and characterise the possibility of such an extension by equality of the canonical and the maximal dual operator $\cl D$-system structures on the domain of the given Schur multiplier. Our context is that of an arbitrary (albeit standard) measure space $(X,\mu)$, which includes as a sub-case the discrete case and thus the finite case considered in \cite{pps}. In this context, the algebra $\cl D$ is the maximal abelian selfadjoint algebra corresponding to $L^{\infty}(X,\mu)$. Our results are a far reaching generalisation of the results of V. I. Paulsen, S. Power and R. R. Smith \cite{pps}; in particular, they provide a different view on the positive completion problem for block operator matrices considered therein.
The paper is organised as follows. After collecting some preliminaries in Section \ref{s_prel}, we establish, in Section \ref{s_eoss}, the existence of the minimal and the maximal operator $\cl A$-system structures on a AOU $\cl A$-space $V$, $\omin_{\cl A}(V)$ and $\omax_{\cl A}(V)$. In case $V$ is a C*-algebra, $\omin_{\cl A}(V)$ was essentially defined in \cite{ps}, in relation with the problem of automatic complete positivity of $\cl A$-module maps, whose completely bounded version was first considered by R. R. Smith in \cite{smith} (see also the subsequent paper \cite{pss}). We show that $\omax_{\cl A}(V)$ (resp. $\omin_{\cl A}(V)$) is characterised by the automatic complete positivity of $\cl A$-bimodule positive maps from $V$ into any operator $\cl A$-system (resp. from any operator $\cl A$-system into $V$).
In Section \ref{s_doas}, we provide a characterisation theorem for dual operator $\cl A$-systems and, in Section \ref{s_deoass}, we define dual AOU $\cl A$-spaces and undertake a development, analogous to the one in Section \ref{s_eoss}, for dual operator $\cl A$-system structures.
In Section \ref{s_ism}, we introduce the operator-valued version of measurable Schur multipliers and provide a Grothendieck-type characterisation, noting the special case of positive Schur multipliers. In Section \ref{s_pe}, we study partially defined operator-valued Schur multipliers and their extension properties to a fully defined positive Schur multiplier. Associated with the domain $\kappa\subseteq X\times X$ of the Schur multiplier is an operator system $\cl S(\kappa)$. Our analysis depends on the presence of sufficiently many operators of finite rank in $\cl S(\kappa)$. We note that, of course, this holds true trivially in the classical matrix case. Under such richness conditions on the domain $\kappa$, we show that the positive extension problem for operator-valued Schur multipliers defined on $\kappa$ has an affirmative solution precisely when the canonical operator system structure of $\cl S(\kappa)$ coincides with its maximal dual operator $\cl D$-system structure.
We denote by $(\cdot,\cdot)$ the inner product in a Hilbert space, and we use $\langle \cdot, \cdot\rangle$ to designate duality paring. We will assume some basic facts and notions from Operator Space Theory, for which we refer the reader to the monographs \cite{blm, er, Pa, pisier_intr}.
\section{Preliminaries}\label{s_prel}
In this section we recall basic results and introduce some new notions that will be needed subsequently. If $W$ is a real vector space, a \emph{cone} in $W$ is a non-empty subset $C \subseteq W$ with the following properties:
(a) $\lambda v \in C$ whenever $\lambda \in \bb{R}^+ := [0,\infty)$ and $v \in C$;
(b) $v + w \in C$ whenever $v, w \in C$.
\noindent A \emph{*-vector space} is a complex vector space $V$ together with a map $^* : V \to V$ which is involutive (i.e. $(v^*)^* = v$ for all $v \in V$) and conjugate linear (i.e. $(\lambda v + \mu w)^* = \overline{\lambda} v^* + \overline{\mu} w^*$ for all $\lambda,\mu \in \bb C$ and all $v,w \in V$). If $V$ is a *-vector space, then we let $V_h = \{x \in V : x^* = x \}$ and call the elements of $V_h$ \emph{hermitian}. Note that $V_h$ is a real vector space.
An \emph{ordered *-vector space} \cite{pt} is a pair $(V, V^+)$ consisting of a *-vector space $V$ and a subset $V^+ \subseteq V_h$ satisfying the following properties:
(a) $V^+$ is a cone in $V_h$;
(b) $V^+ \cap -V^+ = \{ 0 \}$.
Let $(V,V^+)$ be an ordered *-vector space. We write $v \geq w$ or $w \leq v$ if $v,w\in V_h$ and $v - w \in V^+$. Note that $v \in V^+$ if and only if $v \geq 0$; for this reason $V^+$ is referred to as the cone of \emph{positive} elements of $V$.
An element $e \in V_h$ is called an \emph{order unit} if for every $v \in V_h$ there exists $r > 0$ such that $v\leq re$. The order unit $e$ is called \emph{Archimedean} if, whenever $v \in V$ and $re+v \in V^+$ for all $r >0$, we have that $v \in V^+$. In this case, we call the triple $(V, V^+, e)$ an \emph{Archimedean order unit *-vector space} (\emph{AOU space} for short). Note that $(\bb{C}, \bb{R}^+,1)$ is an AOU space in a canonical fashion.
Let $\cl A$ be a unital C*-algebra. Recall that a (complex) vector space $V$ is said to be an \emph{$\cl A$-bimodule} if it is equipped with bilinear maps $\cl A\times V\to V$, $(a,x)\to a\cdot x$ and $V\times \cl A\to V$, $(x,a)\to x\cdot a$, such that $(a\cdot x)\cdot b = a\cdot (x\cdot b)$, $(ab)\cdot x = a\cdot (b\cdot x)$, $x\cdot (ab) = (x\cdot a)\cdot b$ and $1\cdot x = x$ for all $x\in V$ and all $a,b\in \cl A$. If $V$ and $W$ are $\cl A$-bimodules, a linear map $\phi : V\to W$ is called an \emph{$\cl A$-bimodule map} if $\phi(a\cdot x\cdot b) = a\cdot \phi(x)\cdot b$, for all $x\in V$ and all $a,b\in \cl A$.
\begin{definition}\label{d_Asp} Let $\cl A$ be a unital C*-algebra. An AOU space $(V,V^+,e)$ will be called an \emph{AOU $\cl A$-space} if $V$ is an $\cl A$-bimodule and the conditions \begin{equation}\label{eq_propm} (a\cdot x)^* = x^*\cdot a^*, \ \ \ \ x\in V, a\in \cl A, \end{equation} \begin{equation}\label{eq_uni} a\cdot e = e \cdot a, \ \ \ \ a\in \cl A, \end{equation} and \begin{equation}\label{eq_aastar} a^* \cdot x \cdot a \in V^+, \ \ \ \ x\in V^+, a\in \cl A, \end{equation} are satisfied. \end{definition}
For a complex vector space $V$, we let $M_{m,n}(V)$ denote the complex vector space of all $m$ by $n$ matrices with entries in $V$, and often use the natural identification $M_{m,n}(V) \equiv M_{m,n} \otimes V$. We write $A^t$ for the transpose of a matrix $A\in M_{m,n}(V)$. We set $M_n(V) = M_{n,n}(V)$, $M_{m,n} = M_{m,n}(\bb{C})$ and $M_n = M_n(\bb{C})$; we write $I_n$ for the identity matrix in $M_n$. If $V$ is an AOU $\cl A$-space, we equip $M_n(V)$ with an involution by letting $(x_{i,j})^* = (x_{j,i}^*)$ and set \begin{equation}\label{eq_matrixmod} (a_{i,j})\cdot (x_{i,j}) = \left(\sum_{p = 1}^n a_{i,p}\cdot x_{p,j}\right)_{i,j} \ \mbox{ and } \ (x_{i,j})\cdot (b_{i,j}) = \left(\sum_{p = 1}^n x_{i,p}\cdot b_{p,j}\right)_{i,j}, \end{equation} whenever $(x_{i,j})\in M_{m,n}(V)$, $(a_{i,j})\in M_{k,m}(\cl A)$ and $(b_{i,j})\in M_{n,l}(\cl A)$, $m,n$, $k,l$ $\in$ $\bb{N}$.
Let $\cl A$ be a unital C*-algebra and $(V,V^+,e)$ be an AOU $\cl A$-space. We write $e_n$ for the element of $M_n(V)$ whose diagonal entries coincide with $e$, while its off-diagonal entries are equal to zero. A family $(P_n)_{n\in \bb{N}}$, where $P_n\subseteq M_n(V)_h$ is a cone with $P_n \cap (-P_n) = \{0\}$, $n\in \bb{N}$, will be called a \emph{matrix ordering} of $V$. A matrix ordering $(P_n)_{n\in \bb{N}}$ will be called an \emph{operator $\cl A$-system structure} on $V$ if $P_1 = V^+$, \begin{equation}\label{eq_diffs} A^*\cdot X\cdot A \in P_n, \ \ \ \mbox{ whenever } X\in P_m \mbox{ and } A\in M_{m,n}(\cl A), \end{equation} and $e_n\in M_n(V)$ is an Archimedean order unit for $P_n$ for every $n\in \bb{N}$.
Condition (\ref{eq_diffs}) will be referred to as the $\cl A$-compatibility of $(P_n)_{n\in \bb{N}}$. The triple $\cl S = (V,(P_n)_{n\in \bb{N}},e)$ is called an \emph{operator $\cl A$-system} (see \cite{Pa}); we write $M_n(\cl S)^+ = P_n$. Note that if $\cl B\subseteq \cl A$ is a unital C*-subalgebra, then every operator $\cl A$-system is also an operator $\cl B$-system in a canonical fashion. Operator $\bb{C}$-systems are called simply \emph{operator systems}. We note that every operator system has a canonical operator space structure (see \cite{Pa}). Note that condition (\ref{eq_uni}) is not a part of the standard definition of an operator $\cl A$-system; it is however automatically satisfied, as easily follows from Theorem \ref{th_repa} below.
Let $H$ be a Hilbert space and $\cl B(H)$ be the space of all bounded linear operators on $H$. We write $\cl B(H)^+$ for the cone of all positive operators in $\cl B(H)$. We identify $M_n(\cl B(H))$ with $\cl B(H^n)$, where $H^n$ denotes the direct sum of $n$ copies of $H$, and write $M_n(\cl B(H))^+ = \cl B(H^n)^+$, $n\in \bb{N}$. It is straightforward to see that $\cl B(H)$ is an operator system when equipped with the adjoint operation as an involution, the matrix ordering $(M_n(\cl B(H))^+)_{n\in \bb{N}}$, and the identity operator $I$ as an Archimedean matrix order unit.
Given AOU spaces $(V, V^+,e)$ and $(W, W^+,f)$, a linear map $\phi : V\to W$ is called \emph{unital} if $\phi(e) = f$, and \emph{positive} if $\phi(V^+)\subseteq W^+$. A linear map $s : V\to \bb{C}$ is called a \emph{state} on $V$ if $s$ is unital and positive.
Let $\cl S$ and $\cl T$ be operator systems with units $e$ and $f$, respectively. For a linear map $\phi : \cl S\to \cl T$, we let $\phi^{(n,m)} : M_{n,m}(\cl S) \to M_{n,m}(\cl T)$ be the (linear) map given by $\phi^{(n,m)}((x_{i,j})_{i,j}) = (\phi(x_{i,j}))_{i,j}$, and set $\phi^{(n)} = \phi^{(n,n)}$. The map $\phi$ is called \emph{$n$-positive} if $\phi^{(n)}$ is positive, and it is called \emph{completely positive} if it is $n$-positive for all $n\in \bb{N}$. A bijective completely positive map $\phi : \cl S\to \cl T$ is called a \emph{complete order isomorphism} if its inverse $\phi^{-1}$ is completely positive. In this case, we call $\cl S$ and $\cl T$ are completely order isomorphic; if $\phi$ is moreover unital, we say that $\cl S$ and $\cl T$ are unitally completely order isomorphic. Further, $\phi$ is called a \emph{complete isometry} if $\phi^{(n)}$ is an isometry for each $n\in \bb{N}$. We note that a unital surjective map $\phi : \cl S\to \cl T$ is a complete isometry if and only if it is a complete order isomorphism \cite[1.3.3]{blm}.
We refer the reader to \cite{Pa} for the general theory of operator systems and operator spaces, and in particular for the definition and basic properties of completely bounded maps. The following characterisation, extending the well-known Choi-Effros representation theorem for operator systems \cite[Theorem 13.1]{Pa}, was established in \cite[Corollary 15.12]{Pa}.
\begin{theorem}\label{th_repa} Let $\cl A$ be a unital C*-algebra and $\cl S$ be an operator system. The following are equivalent:
(i) \ $\cl S$ is unitally completely order isomorphic to an operator $\cl A$-system;
(ii) there exist a Hilbert space $H$, a unital complete isometry $\gamma : \cl S\to \cl B(H)$ and a unital *-homomorphism $\pi : \cl A\to \cl B(H)$ such that $\gamma(a\cdot x) = \pi(a)\gamma(x)$ for all $x\in \cl S$ and all $a\in \cl A$. \end{theorem}
We note that, if $\cl A$ is a unital C*-algebra and $\cl S$ is an operator system that is also an operator $\cl A$-bimodule satisfying (\ref{eq_propm}), then $\cl S$ is an operator $\cl A$-system precisely when the family $(M_n(\cl S)^+)_{n\in \bb{N}}$ is $\cl A$-compatible.
\section{The extremal operator $\cl A$-system structures}\label{s_eoss}
In this section, we show that any AOU $\cl A$-space can be equipped with two extremal operator $\cl A$-system structures, and establish their universal properties. We first consider the minimal operator $\cl A$-system structure. Note that, in the case where the AOU $\cl A$-space is a C*-algebra containing $\cl A$, this operator system structure was first defined and studied in \cite{ps}.
Let $\cl A$ be a unital C*-algebra and $(V,V^+,e)$ be an AOU $\cl A$-space. For $n\in \bb{N}$, let $$C_n^{\min}(V;\cl A) = \{X\in M_n(V)_h : C^*\cdot X\cdot C\in V^+, \mbox{ for all } C\in M_{n,1}(\cl A)\}.$$
\begin{remark}\label{r_scopmin} {\rm Suppose that $(V,V^+,e)$ is an AOU $\cl A$-space and that $\cl B$ is a unital C*-subalgebra of $\cl A$. Then $(V,V^+,e)$ is also an AOU $\cl B$-space in the natural fashion. Clearly, $C_n^{\min}(V;\cl A) \subseteq C_n^{\min}(V;\cl B)$. In particular, $C_n^{\min}(V;\cl A)$ is contained in $C_n^{\min}(V;\bb{C})$; note that the latter set coincides with the cone $C_n^{\min}(V)$ introduced in \cite[Definition 3.1]{ptt}. } \end{remark}
\begin{theorem}\label{th_cmina} Let $\cl A$ be a unital C*-algebra and $(V,V^+,e)$ be an AOU $\cl A$-space. Then $(C_n^{\min}(V;\cl A))_{n\in \bb{N}}$ is an operator $\cl A$-system structure on $V$. Moreover, if $(P_n)_{n\in \bb{N}}$ is an operator $\cl A$-system structure on $V$ then $P_n \subseteq C_n^{\min}(V;\cl A)$ for each $n\in \bb{N}$. \end{theorem} \begin{proof} Since $V^+$ is a cone, $C_n^{\min}(V;\cl A)$ is a cone, too. As a consequence of \cite[Theorem 3.2]{ptt} and Remark \ref{r_scopmin}, $C_n^{\min}(V;\cl A)\cap (-C_n^{\min}(V;\cl A)) = \{0\}$. If $X\in C_m^{\min}(V;\cl A)$, $A\in M_{m,n}(\cl A)$ and $C\in M_{n,1}(\cl A)$ then $AC\in M_{m,1}(\cl A)$ and hence $$C^*\cdot (A^*\cdot X\cdot A)\cdot C = (AC)^*\cdot X \cdot (AC) \in V^+,$$ showing that $A^*\cdot X\cdot A\in C_n^{\min}(V;\cl A)$. Thus, the family $(C_n^{\min}(V;\cl A))_{n\in \bb{N}}$ is $\cl A$-compatible.
Suppose that $(P_n)_{n\in \bb{N}}$ is an operator $\cl A$-system structure on $V$. If $X\in P_n$ then, by $\cl A$-compatibility, $C^*\cdot X\cdot C\in P_1 = V^+$, and hence $X\in C_n^{\min}(V;\cl A)$. Thus, $P_n\subseteq C_n^{\min}(V;\cl A)$. It will follow from the proof of Theorem \ref{th_cmaxa} below that $e_n$ is an order unit for $C_n^{\min}(V;\cl A)$. To see that $e_n$ is Archimedean, suppose that $X + re_n\in C_n^{\min}(V;\cl A)$ for every $r > 0$. Let $C\in M_{n,1}(\cl A)$. Using (\ref{eq_uni}), we have $$C^*\cdot X\cdot C + r C^*C \cdot e = C^*\cdot (X + r e_n)\cdot C \in V^+, \ \mbox{ for all } r > 0.$$ Let $\epsilon > 0$ and $T = (C^*C + \epsilon 1)^{-1/2}\in \cl A$. We have that $$C^*\cdot X\cdot C + r C^*C \cdot e + r\epsilon e \in V^+, \ \mbox{ for all } r > 0$$ and hence, by (\ref{eq_uni}) and (\ref{eq_aastar}), $$T(C^*\cdot X\cdot C)T + r e \in V^+, \ \mbox{ for all } r > 0.$$ Since $e$ is Archimedean for $V^+$, we have that $T(C^*\cdot X\cdot C)T \in V^+$. Applying (\ref{eq_aastar}) again, we conclude that $$C^*\cdot X\cdot C = T^{-1}(T(C^*\cdot X\cdot C)T)T^{-1} \in V^+;$$ thus $X\in C_n^{\min}(V;\cl A)$ and the proof is complete. \end{proof}
We call $(C_n^{\min}(V;\cl A))_{n\in \bb{N}}$ the \emph{minimal operator $\cl A$-system structure} on $V$, and let $$\omin\mbox{}_{\cl A}(V) = \left(V, (C_n^{\min}(V;\cl A))_{n\in \bb{N}}, e\right).$$ The following theorem describes its universal property. Part (i) below was established in \cite{ps} in the case $V$ is a C*-algebra containing $\cl A$.
\begin{theorem}\label{th_umin} Let $\cl A$ be a unital C*-algebra and $(V,V^+,e)$ be an AOU $\cl A$-space.
(i) Suppose that $\cl S$ is an operator $\cl A$-system and $\phi : \cl S\to V$ is a positive $\cl A$-bimodule map. Then $\phi$ is completely positive as a map from $\cl S$ into $\omin\mbox{}_{\cl A}(V)$.
(ii) If $\cl T$ is an operator $\cl A$-system with underlying space $V$ and positive cone $V^+$, such that for every operator $\cl A$-system $\cl S$, every positive $\cl A$-bimodule map $\phi : \cl S\to \cl T$ is completely positive, then there exists a unital $\cl A$-bimodule map $\psi : \cl T \to \omin_{\cl A}(V)$ that is a complete order isomorphism. \end{theorem} \begin{proof} (i) Let $\cl S$ be an operator $\cl A$-system and $\phi : \cl S\to V$ be a positive $\cl A$-bimodule map. Suppose that $X = (x_{i,j})\in M_n(\cl S)^+$ and $C = (a_i)_{i=1}^n\in M_{n,1}(\cl A)$. Then $C^*\cdot X \cdot C\in \cl S^+$; since $\phi$ is a positive $\cl A$-bimodule map, we have \begin{eqnarray*} C^*\cdot \phi^{(n)}(X)\cdot C & = & \sum_{i,j=1}^n a_i^* \cdot \phi(x_{i,j})\cdot a_j = \phi\left(\sum_{i,j=1}^n a_i^* \cdot x_{i,j}\cdot a_j\right)\\ & = & \phi(C^*\cdot X \cdot C)\in V^+. \end{eqnarray*} Thus, $\phi^{(n)}$ maps $M_n(\cl S)^+$ into $C_n^{\min}(V;\cl A)$ and hence $\phi$ is completely positive.
(ii) Suppose that the operator $\cl A$-system $\cl T$ satisfies the properties in (ii). Since the identity $\id : \omin_{\cl A}(V)\to V$ is a positive $\cl A$-bimodule map, we have that $\id : \omin_{\cl A}(V) \to \cl T$ is completely positive. On the other hand, the identity $\id : \cl T\to V$ is also positive and $\cl A$-bimodular. By (i), $\id : \cl T \to \omin_{\cl A}(V)$ is completely positive, and we can take $\psi = \id$. \end{proof}
We next consider the maximal operator $\cl A$-system structure. For $n\in \bb{N}$, set $$D_n^{\max}(V;\cl A) = \left\{\sum_{i=1}^k A_i^*\cdot x_i \cdot A_i : k\in \bb{N}, x_i\in V^+, A_i\in M_{1,n}(\cl A)\right\}$$ and let $\cl D^{\max}(V;\cl A) = (D_n^{\max}(V;\cl A))_{n\in \bb{N}}$.
\begin{remark}\label{r_scop} {\rm Suppose that $(V,V^+,e)$ is an AOU $\cl A$-space and that $\cl B$ is a unital C*-subalgebra of $\cl A$. Clearly, $D_n^{\max}(V;\cl B) \subseteq D_n^{\max}(V;\cl A)$. Given any AOU space $(V,V^+,e)$, in \cite{ptt} the authors defined $$D_n^{\max}(V) = \left\{\sum_{i=1}^k B_i \otimes x_i : k\in \bb{N}, x_i\in V^+, B_i\in M_{n}^+\right\}.$$ Since every matrix $B\in M_n^+$ is the sum of matrices of the form $A^*A$, where $A\in M_{1,n}$, we have that $D_n^{\max}(V) = D_n^{\max}(V;\bb{C}1)$.} \end{remark}
\begin{lemma}\label{min} Let $\cl A$ be a unital C*-algebra and $(V,V^+,e)$ be an AOU $\cl A$-space. Let $P_n\subseteq M_n(V)_h$ be a cone, $n\in\bb{N}$, such that the family $(P_n)_{n=1}^{\infty}$ is $\cl A$-compatible and $P_1 = V^+$. Then $D_n^{\max}(V;\cl A)\subseteq P_n$, for each $n\in \bb{N}$. \end{lemma} \begin{proof} Let $n\in \bb{N}$.
If $A\in M_{1,n}(\cl A)$ then $$A^* \cdot V^+ \cdot A = A^* \cdot P_1 \cdot A \subseteq P_n.$$ Thus $D_n^{\max}(V; \cl A) \subseteq P_n$. \end{proof}
If $x_1,\dots,x_n\in V$ we let $\diag(x_1,\dots,x_n)$ denote the element of $M_n(V)$ with $x_1,\dots,x_n$ on its diagonal (in this order) and zeros elsewhere.
\begin{proposition}\label{minmax} Let $\cl A$ be a unital C*-algebra and $(V,V^+,e)$ be an AOU $\cl A$-space. The following hold:
(i) \ $D_n^{\max}(V;\cl A) = \{A^* \cdot \diag(x_1,\dots,x_m) \cdot A : A\in M_{m,n}(\cl A), x_i\in V^+, i= 1,\dots,m,$ $m\in\bb{N}\}$;
(ii) $\cl D^{\max}(V;\cl A)$ is an $\cl A$-compatible matrix ordering on $V$ and $e$ is a matrix order unit for it.
\end{proposition} \begin{proof} (i) Let $D_n$ denote the right hand side of the equality in (i). We first observe that $D_n$ is a cone in $M_n(V)_h$. If $x_1,\dots,x_m\in V^+$ and $A = (a_{i,k})_{i,k}\in M_{m,n}(\cl A)$ then the $(i,j)$-entry of $A^* \cdot \diag(x_1,\dots,x_m)\cdot A$ is equal to $\sum_{k=1}^m a_{k,i}^*\cdot x_k \cdot a_{k,j}$ and, by (\ref{eq_propm}), $$\left(\sum_{k=1}^m a_{k,i}^*\cdot x_k \cdot a_{k,j}\right)^* = \sum_{k=1}^m a_{k,j}^*\cdot x_k \cdot a_{k,i};$$ thus, $D_n\subseteq M_n(V)_h$. It is clear that $D_n$ is closed under taking multiples with non-negative real numbers. Fix elements $$A^* \cdot \diag(x_1,\dots,x_m)\cdot A, \ \mbox{ and } \ B^* \cdot \diag(y_1,\dots,y_k)\cdot B$$ of $D_n$. Letting $C = [A \ B]^t$, we have \begin{eqnarray*} & & A^*\cdot \diag(x_1,\dots,x_m)\cdot A + B^*\cdot \diag(y_1,\dots,y_k)\cdot B\\ & = & C^*\cdot \diag(x_1,\dots,x_m,y_1,\dots,y_k) \cdot C\in D_n \ ; \end{eqnarray*} in other words, $D_n$ is a cone. If $B\in M_{n,l}(\cl A)$ then $$B^*\cdot (A^*\cdot \diag(x_1,\dots,x_m)\cdot A)\cdot B = (AB)^*\cdot \diag(x_1,\dots,x_m)\cdot (AB)\in D_l,$$ and so $(D_n)_{n=1}^{\infty}$ is $\cl A$-compatible. By (\ref{eq_aastar}), $D_1 = V^+$. Lemma \ref{min} now implies that $D_n^{\max}(V;\cl A)\subseteq D_n$ for $n\in \bb{N}$.
On the other hand, if $x_1,\dots,x_m\in V^+$ then, letting $E_{i}\in M_{1,m}(\cl A)$ be the row with $1$ at the $i$th coordinate and zeros elsewhere, we have that $$\diag(x_1,\dots,x_m) = \sum_{i=1}^m E_{i}^*\cdot x_i\cdot E_{i}\in D_m^{\max}(V; \cl A).$$ Since the family $\cl D^{\max}(V;\cl A)$ is $\cl A$-compatible, $$A^*\cdot \diag(x_1,\dots,x_m)\cdot A \in D_n^{\max}(V;\cl A), \ \ \ A \in M_{m,n}(\cl A).$$ Thus, $D_n\subseteq D_n^{\max}(V;\cl A)$ and (i) is established.
(ii) By Remark \ref{r_scop} and \cite[Proposition 3.10]{ptt}, $e_n$ is an order unit for $D_n^{\max}(V;\bb{C}1)$. By Remark \ref{r_scop} again, $e_n$ is an order unit for $D_n^{\max}(V;\cl A)$. \end{proof}
For $n\in \bb{N}$, let $$C_n^{\max}(V;\cl A) = \{X\in M_n(V) : X + re_n \in D_n^{\max}(V;\cl A) \mbox{ for every } r> 0\}.$$
\begin{theorem}\label{th_cmaxa} Let $\cl A$ be a unital C*-algebra and $(V,V^+,e)$ be an AOU $\cl A$-space. Then $(C_n^{\max}(V;\cl A))_{n\in \bb{N}}$ is an operator $\cl A$-system structure on $V$. Moreover, if $(P_n)_{n\in \bb{N}}$ is an operator $\cl A$-system structure on $V$ then $$C_n^{\max}(V;\cl A)\subseteq P_n$$ for each $n\in \bb{N}$. \end{theorem} \begin{proof} Write $C_n = C_n^{\max}(V;\cl A)$, $n\in \bb{N}$. By Theorem \ref{th_cmina} and Lemma \ref{min}, $C_n\subseteq C_n^{\min}(V;\cl A)$; thus, $C_n \cap (-C_n) = \{0\}$. Since $e_n$ is an order unit for $D_n^{\max}(V;\cl A)$ and $D_n^{\max}(V;\cl A)\subseteq C_n$, we have that $e_n$ is an order unit for $C_n$.
Suppose that $X\in M_n(V)_h$ is such that $X + re_n\in C_n$ for every $r > 0$. Let $\epsilon > 0$; then $$X + \epsilon e_n = \left(X + \frac{\epsilon}{2} e_n\right) + \frac{\epsilon}{2} e_n \in D_n^{\max}(V;\cl A)$$ and hence $X\in C_n$. Thus, $e_n$ is an Archimedean matrix order unit for $C_n$.
It remains to show that the family $(C_n)_{n\in \bb{N}}$ is $\cl A$-compatible. To this end, let $X\in C_n$ for some $n\in \bb{N}$ and $A\in M_{n,m}(\cl A)$. By Proposition \ref{minmax}, there exists $R > 0$ such that $$R e_m - A^*\cdot e_n \cdot A \in D_m^{\max}(V;\cl A).$$ Let $r > 0$. Since $X + \frac{r}{R} e_n\in D_n^{\max}(V;\cl A)$ and the family $\cl D^{\max}(V;\cl A)$ is $\cl A$-compatible (Proposition \ref{minmax}), we have \begin{eqnarray*} & & A^*\cdot X\cdot A + re_m \\ & = & \left(A^*\cdot \left(X + \frac{r}{R} e_n\right)\cdot A\right) + r\left(e_m - \frac{1}{R} A^*\cdot e_n \cdot A\right) \in D_m^{\max}(V;\cl A). \end{eqnarray*} It follows that $A^*\cdot X\cdot A\in C_m$. Thus, $(C_n)_{n\in \bb{N}}$ is an operator $\cl A$-system structure on $V$.
Suppose that $(P_n)_{n\in \bb{N}}$ is an operator $\cl A$-system structure on $V$ and $X\in C_n$ for some $n\in \bb{N}$. By Lemma \ref{min}, $X + r e_n\in P_n$ for all $r > 0$ and since $e_n$ is an Archimedean order unit for $P_n$, we conclude that $X\in P_n$. Thus, $C_n\subseteq P_n$, and the proof is complete. \end{proof}
We call $(C_n^{\max}(V;\cl A))_{n\in \bb{N}}$ \emph{the maximal operator $\cl A$-system structure} on $V$ and let $$\omax\mbox{}_{\cl A}(V) = (V,(C_n^{\max}(V;\cl A))_{n\in \bb{N}},e).$$
\noindent {\bf Remark. } Recall that, given an AOU space $(V,V^+,e)$, the \emph{maximal operator system structure} $(C_n^{\max}(V))_{n\in \bb{N}}$ on $V$ was defined in \cite{ptt} by letting $C_n^{\max}(V)$ be the Archimedeanisation of the cone $D_n^{\max}(V)$ defined in Remark \ref{r_scop}. It follows that the maximal operator system $\omax(V)$ defined in \cite{ptt} coincides with $\omax_{\bb{C}}(V)$.
\begin{theorem}\label{th_acp} Let $\cl A$ be a unital C*-algebra and $(V,V^+,e)$ be an AOU $\cl A$-space.
(i) Suppose that $\cl S$ is an operator $\cl A$-system and $\phi : V\to \cl S$ is a positive $\cl A$-bimodule map. Then $\phi$ is completely positive as a map from $\omax\mbox{}_{\cl A}(V)$ into $\cl S$.
(ii) Suppose that $\cl T$ is an operator $\cl A$-system with underlying space $V$ and positive cone $V^+$, such that for every operator $\cl A$-system $\cl S$, every positive $\cl A$-bimodule map $\phi : \cl T\to \cl S$ is completely positive. Then there exists a unital $\cl A$-bimodule map $\psi : \cl T\to \omax_{\cl A}(V)$ that is a complete order isomorphism. \end{theorem}
\begin{proof} (i) Let $\cl S$ is an operator $\cl A$-system and $\phi : V\to \cl S$ be a positive $\cl A$-bimodule map. The modularity property of $\phi$ and the definition of $D_n^{\max}(V;\cl A)$ imply that $\phi^{(n)}(D_n^{\max}(V;\cl A)) \subseteq M_n(\cl S)^+$. Suppose that $X\in C_n^{\max}(V;\cl A)$. Letting $z = \phi(e)$, we now have that $\phi^{(n)}(X) + r(z\otimes I_n) \in M_n(\cl S)^+$ for every $r > 0$. Since $M_n(\cl S)^+$ is closed, this implies that $\phi^{(n)}(X) \in M_n(\cl S)^+$. Thus, $\phi$ is completely positive.
(ii) is similar to the proof of Theorem \ref{th_umin} (ii). \end{proof}
\noindent {\bf Remark. } Let $\cl A$ be a C*-algebra and $\frak{A}_{\cl A}$ (resp. $\frak{S}_{\cl A}$) be the category, whose objects are AOU $\cl A$-spaces (resp. operator $\cl A$-systems) and whose morphisms are unital positive (resp. unital completely positive) maps. It is easy to see that the correspondences $V\to \omin_{\cl A}(V)$ and $V\to \omax_{\cl A}(V)$ are covariant functors from $\frak{A}_{\cl A}$ into $\frak{S}_{\cl A}$.
We finish this section with considering the case where $V = M_k$ and $\cl A$ coincides with its subalgebra $\cl D_k$ of all diagonal matrices.
\begin{proposition}\label{p_mn} We have that $M_k = \omin_{\cl D_k}(M_k) = \omax_{\cl D_k}(M_k)$. \end{proposition} \begin{proof} Suppose that $X = (X_{i,j})_{i,j}$ belongs to $M_n(\omin_{\cl D_k}(M_k))^+$. Let $\xi = (\lambda_{i,1},\dots,\lambda_{i,k})_{i=1}^n$ be a vector in $\bb{C}^{nk}$. Let $D_i = \diag(\lambda_{i,1},\dots,\lambda_{i,k})$, and write $\xi_i$ for the vector $(\lambda_{i,1},\dots,\lambda_{i,k})$ in $\bb{C}^k$, $i = 1,\dots,n$. Letting $e$ be the vector in $\bb{C}^k$ with all entries equal to one, we have $$(X\xi,\xi) = \sum_{i,j=1}^n (X_{i,j}\xi_j,\xi_i) = \sum_{i,j=1}^n (D_i^* X_{i,j} D_je,e).$$ It follows by the assumption that $(X\xi,\xi)\geq 0$; thus, $X\in M_{nk}^+$ and, by Theorem \ref{th_cmina}, $M_k = \omin_{\cl D_k}(M_k)$.
Now fix $X = (X_{i,j})_{i,j} \in M_{nk}^+$. Since $X$ is the sum of rank one operators in $M_{nk}^+$, in order to show that $X\in M_n(\omax_{\cl D_k}(M_k))^+$, it suffices to assume that $X$ is itself of rank one. Write $X = R R^*$, where $R \in M_{nk,1}$, and suppose that $R = (R_1,\dots,R_n)^t$, where $R_i\in M_{k,1}$, $i = 1,\dots,n$. We have that $X = (R_iR_j^*)_{i,j=1}^n$. Let $J\in M_k$ be the matrix with all its entries equal to one, and let $D_i$ be the diagonal matrix whose entries coincides with the vector $R_i$, $i = 1,\dots,n$. Then $X = (D_i J D_j^*)_{i,j=1}^n$, showing that $X\in M_n(\omax_{\cl D_k}(M_k))^+$. By Theorem \ref{th_cmaxa}, $M_k = \omax_{\cl D_k}(M_k)$. \end{proof}
\noindent {\bf Remark. } We note that the minimal and the maximal operator $\cl A$-system structure are in general distinct. Indeed, this is the case even when $V = M_k$ and $\cl A = \bb{C}I$ \cite{ptt}.
\section{Dual operator $\cl A$-systems}\label{s_doas}
In this section, we establish a representation theorem for dual operator $\cl A$-systems. An operator system $\cl S$
is called a \emph{dual operator system} if it is a dual operator space, that is, if there exists an operator space $\cl S_*$ such that $(\cl S_*)^* \cong \cl S$ completely isometrically \cite{bm}. Here, and in the sequel, we denote by $\cl X^*$ the operator space dual \cite{blm} of an operator space $\cl X$, and we use the same notation for the dual Banach space of a normed space $\cl X$; it will be clear from the context with which category we are working.
Let $\cl S$ be an operator system. If $H$ is a Hilbert space and $\phi : \cl S\to \cl B(H)$ is a unital complete isometry such that $\phi(\cl S)$ is weak* closed, then $\phi(\cl S)$, and therefore $\cl S$, is a dual operator space; thus, in this case, $\cl S$ is a dual operator system. The converse statement was established by Blecher and Magajna in \cite{bm}.
\begin{theorem}[\cite{bm}]\label{th_bm} If $\cl S$ is a dual operator system then there exists a Hilbert space $H$, a weak* closed operator system $\cl U\subseteq \cl B(H)$ and a unital surjective complete order isomorhism $\phi : \cl S\to \cl U$ that is also a a weak* homeomorphism. \end{theorem}
\begin{remark}\label{r_sdu} {\rm Suppose that $\cl S$ is a dual operator system and $\cl S_*$ is an operator space such that, up to a complete isometry, $\cl S = (\cl S_*)^*$. Then $M_n(\cl S)$ is an operator system in a canonical fashion; in fact, if $\cl S\subseteq \cl B(H)$ for some Hilbert space $H$, then $M_n(\cl S)\subseteq \cl B(H^n)$. By \cite[1.6.2]{blm}, up to a complete isometry, $M_n(\cl S) = (\cl S_*\hat{\otimes} M_n^*)^*$, where $\hat{\otimes}$ is the projective operator space tensor product. It follows that $M_n(\cl S)$ is a dual operator system, and its canonical weak* topology coincides with the topology of entry-wise weak* convergence: for a net $((x_{i,j}^{\alpha})_{i,j})_{\alpha}\subseteq M_n(\cl S)$ and an element $(x_{i,j})_{i,j} \in M_n(\cl S)$, we have $$\left((x_{i,j}^{\alpha}\right)_{i,j})_{\alpha}\to^{w^*}_{\alpha} \left(x_{i,j}\right)_{i,j} \ \Longleftrightarrow \ \left\langle x_{i,j}^{\alpha},\phi \right\rangle \to\mbox{}_{\alpha} \left\langle x_{i,j},\phi\right\rangle, \ i,j = 1,\dots,n, \ \phi\in \cl S_*.$$} \end{remark}
Recall that a \emph{W*-algebra} is a C*-algebra that is also a dual Banach space; by Sakai's Theorem \cite{sakai}, every W*-algebra possesses a faithful *-representation on a Hilbert space $H$, whose image is a von Neumann algebra (that is, a weak* closed subalgebra of $\cl B(H)$ containing the identity operator), which is also a weak* homeomorphism.
\begin{definition}\label{d_dasys} Let $\cl A$ be a W*-algebra. An operator system $\cl S$ will be called a \emph{dual operator $\cl A$-system} if \begin{itemize} \item[(i)] $\cl S$ is an operator $\cl A$-system, \item[(ii)] $\cl S$ is a dual operator system, and \item[(iii)] the map from $\cl A\times \cl S$ into $\cl S$, sending the pair $(a,x)$ to $a\cdot x$, is separately weak* continuous. \end{itemize} \end{definition}
Note that, if $\cl S$ is a dual operator system then the involution is weak* continuous, and thus (\ref{eq_propm}) implies that if $\cl S$ is in addition a dual operator $\cl A$-system then the map $$\cl A\times \cl S\times \cl A\to \cl S, \ \ \ (a,x,b)\to a\cdot x\cdot b,$$ is separately weak* continuous.
If $\cl S$ and $\cl T$ are dual operator systems, a linear map $\phi : \cl S\to \cl T$ will be called \emph{normal} if it is weak* continuous. Suppose that $H$ is a Hilbert space, $\gamma : \cl S\to \cl B(H)$ is a unital complete order isomorphism such that $\gamma(\cl S)$ is weak* closed and $\gamma : \cl S\to \gamma(\cl S)$ is a weak* homeomorphism, and $\pi : \cl A\to \cl B(H)$ is a unital normal *-homomorphism such that $\gamma(a\cdot x) = \pi(a)\gamma(x)$ for all $x\in \cl S$ and all $a\in \cl A$. It is clear that, in this case, $\cl S$ is a dual operator $\cl A$-system. Theorem \ref{th_repdoas} below establishes the converse of this fact. The result is both a weak* version of Theorem \ref{th_repa} and an $\cl A$-module version of Theorem \ref{th_bm}.
We will need two lemmas. Recall that, if $\cl A$ is a W*-algebra and $n\in \bb{N}$ then $M_n(\cl A)$ is a W*-algebra in a canonical way.
\begin{remark}\label{l_matrdops} {\rm Let $\cl A$ be a W*-algebra and $\cl S$ be a dual operator $\cl A$-system. It is straightforward to verify that $M_n(\cl S)$ is a dual operator $M_n(\cl A)$-system, when it is equipped with the action defined in (\ref{eq_matrixmod}). } \end{remark}
\begin{lemma}\label{l_funct} Let $\cl A$ be a W*-algebra, $\cl S$ be a dual operator $\cl A$-system and $\phi : \cl S\to \bb{C}$ be a normal state. Then the functional $\omega : \cl A\to \bb{C}$ given by $\omega(a) = \phi(a\cdot 1)$, $a\in \cl A$, is a normal state of $\cl A$ and \begin{equation}\label{eq_ineqq}
|\phi(a\cdot x\cdot b)|\leq \omega(aa^*)^{1/2}\omega(b^*b)^{1/2}, \end{equation}
for all $a\in M_{1,m}(\cl A)$, $b\in M_{m,1}(\cl A)$, $x\in M_m(\cl S)$ with $\|x\|\leq 1$, and $m\in \bb{N}$. \end{lemma}
\begin{proof} Let $H$, $\gamma$ and $\pi$ be as in Theorem \ref{th_repa}, and let $\phi' : \gamma(\cl S)\to \bb{C}$ be given by $\phi'(\gamma(x)) = \phi(x)$, $x\in \cl S$. If $a,b\in \cl A$ then \begin{eqnarray*} \omega(ab) & = & \phi((ab)\cdot 1) = \phi'(\gamma((ab)\cdot 1)) = \phi'(\pi(ab)\gamma(1)) = \phi'(\pi(ab))\\ & = & \phi'(\pi(a)\gamma(1)\pi(b)) = \phi'(\gamma(a\cdot 1\cdot b)) = \phi(a\cdot 1\cdot b). \end{eqnarray*} Thus, $\omega(a^*a) = \phi(a^*\cdot 1\cdot a)\geq 0$ for every $a\in \cl A$, and hence $\omega$ is positive. Moreover, $\omega(1) = \phi(1) = 1$ and hence $\omega$ is a state. By the separate weak* continuity of the $\cl A$-module action on $\cl S$, the state $\omega$ is normal.
Suppose that $\phi'$ has the form $$\phi'(T) = \sum_{i=1}^{\infty} (T\xi_i,\xi_i), \ \ \ T\in \gamma(\cl S),$$
where $(\xi_i)_{i\in \bb{N}}\subseteq H$ with $\sum_{i=1}^{\infty}\|\xi_i\|^2 = 1$. If $x\in M_m(\cl S)$, $\|x\|\leq 1$, $a\in M_{1,m}(\cl A)$ and $b\in M_{m,1}(\cl A)$, then \begin{eqnarray*}
|\phi(a\cdot x\cdot b)| & = &
\left|\phi' \left(\pi^{(1,m)}(a)\gamma^{(m)}(x)\pi^{(m,1)}(b)\right)\right| \\ & = &
\left|\sum_{i=1}^{\infty} \left(\pi^{(1,m)}(a)\gamma^{(m)}(x)\pi^{(m,1)}(b)\xi_i,\xi_i\right)\right|\\ & \leq &
\sum_{i=1}^{\infty} \left|\left(\gamma^{(m)}(x)\pi^{(m,1)}(b)\xi_i,\pi^{(m,1)}(a^*)\xi_i\right)\right|\\ & \leq &
\left(\sum_{i=1}^{\infty} \left\|\pi^{(m,1)}(b)\xi_i\right\|^2\right)^{1/2}
\left(\sum_{i=1}^{\infty} \left\|\pi^{(m,1)}(a^*)\xi_i\right\|^2\right)^{1/2}\\ & = & \phi'(\pi(b^*b))^{1/2} \phi'(\pi(aa^*))^{1/2} = \omega(aa^*)^{1/2}\omega(b^*b)^{1/2}. \end{eqnarray*} \end{proof}
We will need the following modification of a result of R. R. Smith \cite{smith} on automatic complete boundedness. Its proof is a straightforward modification of the proof of \cite[Theorem 2.1]{smith} and is hence omitted.
\begin{theorem}\label{l_sm} Let $\cl A$ be a unital C*-algebra, $\cl S$ be an operator $\cl A$-system and $\rho : \cl A\to \cl B(H)$ be a cyclic *-representation. Suppose that $\Phi : \cl S\to \cl B(H)$ is a linear map such that $\Phi(a\cdot x\cdot b) = \rho(a)\Phi(x)\rho(b)$ for all $x\in \cl S$ and all $a,b\in \cl A$.
If $\Phi$ is contractive then $\Phi$ is completely contractive.
\end{theorem}
\begin{theorem}\label{th_repdoas} Let $\cl A$ be a W*-algebra and $\cl S$ be a dual operator $\cl A$-system. Then there exist a Hilbert space $H$, a unital complete order embedding $\gamma : \cl S\to \cl B(H)$ with the property that $\gamma(\cl S)$ is weak* closed and $\gamma$ is a weak* homeomorphism, and a unital normal *-homomorphism $\pi : \cl A\to \cl B(H)$, such that \begin{equation}\label{eq_modc} \gamma(a\cdot x) = \pi(a)\gamma(x), \ \ \ x\in \cl S, a\in \cl A. \end{equation} \end{theorem} \begin{proof} The proof is motivated by the proof of \cite[Theorem 1.1]{bm} and relies on ideas which go back to the proof of Ruan's Theorem \cite[Theorem 2.3.5]{er}. Fix $n\in \bb{N}$ and let $\cl B = M_n(\cl A)$. By Remark \ref{l_matrdops}, $M_n(\cl S)$ is a dual operator $\cl B$-system. Let $x\in M_n(\cl S)$ be a selfadjoint element of norm one and $\epsilon \in (0,1)$. By the proof of Theorem 1.1 given in \cite{bm}, there exists a normal state $\phi$ on $M_n(\cl S)$ such that \begin{equation}\label{eq_>ep}
|\phi(x)| > 1 - \epsilon. \end{equation} Let $\omega : \cl B\to \bb{C}$ be the normal state given by $\omega(b) = \phi(b\cdot 1)$, $b\in \cl B$. By Lemma \ref{l_funct}, \begin{equation}\label{eq_om}
|\phi(a\cdot y \cdot b)|\leq \omega(aa^*)^{1/2} \omega(b^*b)^{1/2}, \end{equation}
for all $y\in M_{nm}(\cl S)$ with $\|y\|\leq 1$, $a\in M_{1,m}(\cl B)$ and $b\in M_{m,1}(\cl B)$, $m\in \bb{N}$.
Let $\rho : \cl B\to \cl B(H)$ be the GNS representation arising from $\omega$ and $\xi$ be its corresponding unit cyclic vector. By \cite[Proposition III.3.12]{t}, $\rho$ is normal. It follows that there exists a normal unital *-representation $\theta : \cl A\to \cl B(K)$ such that, up to unitary equivalence, $H = K\otimes \bb{C}^n$ and $\rho = \theta^{(n)}$. Inequality (\ref{eq_om}) implies
$$|\phi(a^*\cdot y \cdot b)|\leq \|\rho(b)\xi\|\|\rho(a)\xi\| \|y\|, \ \ a,b\in \cl B, y\in M_n(\cl S).$$ Thus, the sesqui-linear form $L_y : (\rho(\cl B)\xi)\times (\rho(\cl B)\xi) \to \bb{C}$ given by $$L_y(\rho(b)\xi,\rho(a)\xi) = \phi(a^*\cdot y \cdot b), \ \ \ a,b\in \cl B,$$
is bounded and has norm not exceeding $\|y\|$. It follows that there exists a linear operator $\Phi(y) : \rho(\cl B)\xi\to \rho(\cl B)\xi$ such that \begin{equation}\label{ope} (\Phi(y)\rho(b)\xi,\rho(a)\xi) = \phi(a^*\cdot y\cdot b), \ \ \ a,b\in \cl B, \end{equation} and \begin{equation}\label{eq_Phiy}
\|\Phi(y)\|\leq \|y\|. \end{equation} Since $\rho(\cl B)\xi$ in dense in $H$, the operator $\Phi(y)$ can be extended to an operator on $H$. By (\ref{ope}), the map $\Phi : M_n(\cl S)\to \cl B(H)$ is linear and hermitian and, by (\ref{eq_Phiy}), it is contractive.
For $a,b,c,d\in \cl B$, by (\ref{ope}), we have $$(\Phi(c^* \cdot y \cdot d)\rho(b)\xi,\rho(a)\xi) = (\rho(c^*)\Phi(y)\rho(d)\rho(b)\xi,\rho(a)\xi).$$ The density of $\rho(\cl B)\xi$ in $H$ now implies that \begin{equation}\label{eq_mod} \Phi(c^* \cdot y \cdot d) = \rho(c^*)\Phi(y)\rho(d), \ \ \ c,d\in \cl B, y\in M_n(\cl S). \end{equation}
We show that $\Phi$ is weak* continuous. Suppose that $(y_{\alpha})_{\alpha}\subseteq M_n(\cl S)$ is a net of contractions such that $y_{\alpha}\to_{\alpha} y$ in the weak* topology, for some $y\in M_n(\cl S)$. Fix $\delta > 0$, $\eta,\zeta\in H$, and choose $a,b\in \cl B$ such that
$$\|\rho(b)\xi - \eta\| < \delta \ \mbox{ and } \ \|\rho(a)\xi - \zeta\| < \delta.$$
Let ${\alpha}_0$ be such that $|\phi(a^* \cdot y_{\alpha} \cdot b) - \phi(a^* \cdot y \cdot b)| < \delta$ if ${\alpha}\geq {\alpha}_0$. For $\alpha\geq \alpha_0$ we have \begin{eqnarray*} & &
|(\Phi(y_{\alpha})\eta,\zeta) - (\Phi(y)\eta,\zeta)|\\ & \leq &
|(\Phi(y_{\alpha})\eta,\zeta) - (\Phi(y_{\alpha})\rho(b)\xi,\rho(a)\xi)|\\ & + &
|(\Phi(y_{\alpha})\rho(b)\xi,\rho(a)\xi) - (\Phi(y)\rho(b)\xi,\rho(a)\xi)|\\ & + &
|(\Phi(y)\rho(b)\xi,\rho(a)\xi) - (\Phi(y)\eta,\zeta)|\\ & = &
|(\Phi(y_{\alpha})\eta,\zeta) - (\Phi(y_{\alpha})\rho(b)\xi,\rho(a)\xi)| +
|\phi(a^* \cdot y_{\alpha} \cdot b) - \phi(a^* \cdot y \cdot b)|\\ & + &
|(\Phi(y)\rho(b)\xi,\rho(a)\xi) - (\Phi(y)\eta,\zeta)|\\ & \leq &
|(\Phi(y_{\alpha})\eta,\zeta) - (\Phi(y_{\alpha})\rho(b)\xi,\zeta)|\\
& + & |(\Phi(y_{\alpha})\rho(b)\xi,\zeta) - (\Phi(y_{\alpha})\rho(b)\xi,\rho(a)\xi)| +
|\phi(a^* \cdot y_{\alpha} \cdot b) - \phi(a^* \cdot y \cdot b)|\\ & + &
|(\Phi(y)\rho(b)\xi,\rho(a)\xi) - (\Phi(y)\eta,\rho(a)\xi)| + |(\Phi(y)\eta,\rho(a)\xi) - (\Phi(y)\eta,\zeta)|\\ & \leq &
\delta (\|\zeta\| + \|\eta\| + \|\rho(a)\xi\| + \|\rho(b)\xi\| + 1)
\leq \delta (2\|\zeta\| + 2\|\eta\| + 2 \delta + 1). \end{eqnarray*} We thus showed that $\Phi(y_{\alpha})\to_{\alpha} \Phi(y)$ in the weak operator topology; since the net $(\Phi(y_{\alpha}))_{\alpha}$ is bounded, the convergence is in fact in the weak* topology. It follows from Shmulyan's Theorem that the map $\Phi$ is weak* continuous.
Identity (\ref{eq_mod}) easily implies that there exists a (normal) map $\Psi : \cl S\to \cl B(K)$ such that $\Phi = \Psi^{(n)}$. Since $\Phi$ is hermitian and contractive, so is $\Psi$. By (\ref{eq_mod}) and Theorem \ref{l_sm}, the map $\Phi$, and hence $\Psi$, is completely contractive. Now (\ref{eq_mod}) implies \begin{equation}\label{eq_modpsi} \Psi(a \cdot z \cdot b) = \theta(a)\Psi(z)\theta(b), \ \ \ \ z\in \cl S, a,b\in \cl A. \end{equation} By (\ref{ope}),
$$1 = \phi(1) = (\Phi(1)\xi,\xi)\leq \|\Phi(1)\|\|\xi\|^2 \leq 1.$$
Thus $\Phi(1)\xi = \xi$; by (\ref{eq_mod}), $$\Phi(1)\rho(b)\xi = \rho(b)\Phi(1)\xi = \rho(b)\xi, \ \ \ \ b\in \cl B,$$ and since $\xi$ is cyclic for $\rho$, we conclude that $\Phi(1) = 1$. It follows that $\Psi(1) = 1$.
The map $\Psi$, constructed in the previous paragraph, depends on the element $x\in M_n(\cl S)$, and on the chosen $\epsilon$.
Note that, by (\ref{eq_>ep}) and (\ref{ope}), $\left\|\Psi^{(n)}(x)\right\| > 1 - \epsilon$. Let $\gamma$ (resp. $\pi$) be the direct sum of the maps $\Psi$ (resp. $\theta$) as above, over all selfadjoint $x\in M_n(\cl S)$ with norm one, all $n\in \bb{N}$, and all $\epsilon \in (0,1)$.
The map $\gamma$ is unital, weak* continuous, hermitian, and has the property that if $x\in M_n(\cl S)$ is selfadjoint then $\|x\| = 1$ implies $\left\|\gamma^{(n)}(x)\right\| = 1$. This easily yields that $\gamma$ is completely positive and has a completely positive inverse. As in the proof of \cite[Theorem 1.1]{bm}, the image of $\gamma$ is weak* closed and $\gamma$ is a weak* homeomorphism onto its range. In addition, $\pi$ is a normal *-representation as a direct sum of such. Condition (\ref{eq_modc}) follows from (\ref{eq_modpsi}). \end{proof}
\section{The dual extremal operator $\cl A$-system structures}\label{s_deoass}
In this section, we study dual versions of the extremal operator $\cl A$-system structures considered in Section \ref{s_eoss}. We start with the definition of a dual AOU space. Note first that, if $(V,V^+,e)$ is an AOU space then the expression
$$\|v\| = \sup\{|f(v)| : f \mbox{ a state on } V\}$$ defines a norm on $V$, called the \emph{order norm} \cite{pt}; in the sequel we equip $V$ with its order norm. If $V$ is a dual Banach space, the weak* continuous functionals on $V$ will be called \emph{normal functionals}.
\begin{definition}\label{d_daous} A \emph{dual AOU space} is an AOU space $(V,V^+,e)$, which is also a dual Banach space, and \begin{itemize} \item[(i)] the involution is weak* continuous;
\item[(ii)] $V^+$ is weak* closed, and
\item[(iii)] for $v\in V$, $\|v\| = \sup\{|f(v)| : f \mbox{ a normal state on } V\}$, and the weak* topology of $V$ is determined by normal states of $V$. \end{itemize} \end{definition}
Suppose that $(V,V^+,e)$ is a dual AOU space, and let $V_*$ be the predual of $V$.
Note that the algebraic tensor product $V_*\otimes M_n^*$ can be canonically embedded into the dual of $M_n(V)$. By \emph{the weak* topology} on $M_n(V)$ we will mean the topology arising from this duality; thus, $(x_{i,j}^{\alpha})\to_{\alpha} (x_{i,j})$ if and only if $x_{i,j}^{\alpha}\to_{\alpha} x_{i,j}$ for every $i,j$.
\begin{definition} Let $\cl A$ be a W*-algebra. A dual AOU space $(V,V^+,e)$ will be called \emph{dual AOU $\cl A$-space} if \begin{itemize} \item[(i)] $(V,V^+,e)$ is an AOU $\cl A$-space, and
\item[(ii)] the left (and hence the right) $\cl A$-module action is separately weak* continuous. \end{itemize} \end{definition}
\begin{definition}\label{d_dopass} Let $\cl A$ be a W*-algebra and $(V,V^+,e)$ be a dual AOU $\cl A$-space. A matrix ordering $(C_n)_{n\in \bb{N}}$ on $V$ will be called a \emph{dual operator $\cl A$-system structure} on $V$ if $(V,(C_n)_{n\in \bb{N}},e)$ is a dual operator $\cl A$-system whose weak* topology coincides with that of $V$, and $C_1 = V^+$. \end{definition}
\begin{theorem}\label{th_weakscac} Let $\cl A$ be a W*-algebra, $(V,V^+,e)$ be a dual AOU $\cl A$-space and $(C_n)_{n\in \bb{N}}$ be an operator $\cl A$-system structure on $V$. The following are equivalent:
(i) \ $(C_n)_{n\in \bb{N}}$ is a dual operator $\cl A$-system structure on $V$;
(ii) $C_n$ is weak* closed for each $n\in \bb{N}$. \end{theorem} \begin{proof} (i)$\Rightarrow$(ii) Let $\cl S = (V,(C_n)_{n\in \bb{N}},e)$. By Theorem \ref{th_repdoas}, there exist a Hilbert space $H$ and a complete order embedding $\gamma : \cl S\to \cl B(H)$ such that $\gamma(\cl S)$ is weak* closed and $\gamma$ is a weak* homeomorphism. Clearly, $M_n(\gamma(\cl S))^+$ is weak* closed in $M_n(\cl B(H))$. Note that the weak* topology on $M_n(\cl B(H)) = \cl B(H^n)$ is given by entry-wise weak* convergence. On the other hand, since $\gamma$ is a weak* homeomorphism, we have that if $((x_{i,j}^{\alpha}))_{\alpha}\subseteq M_n(V)$ and $(x_{i,j})\in M_n(V)$ then $(x_{i,j}^{\alpha})\to_{\alpha} (x_{i,j})$ weak* if and only if $\gamma(x_{i,j}^{\alpha})\to_{\alpha}\gamma(x_{i,j})$ for every $i,j$. It follows that $C_n$ is weak* closed.
(ii)$\Rightarrow$(i) Let $\cl S = (V,(C_n)_{n\in \bb{N}},e)$. For each $n$, let $$\cl P_n = \left\{\phi : V \to M_n \ : \ \mbox{ weak* continuous unital completely positive map}\right\}.$$ Let $H = \oplus_{n\in \bb{N}}\oplus_{\phi\in \cl P_n} \bb{C}^n$ and let $J : V\to \cl B(H)$ be the map given by $J(x) = \oplus_{n\in \bb{N}}\oplus_{\phi\in \cl P_n} \phi(x)$. It is clear that $J$ is a weak* continuous completely positive map. In addition, by condition (iii) from Definition \ref{d_daous}, $J$ is isometric.
To show that $J$ is a complete order isomorphism, assume that $J^{(n)}(X)\geq 0$ for some $X = (x_{i,j})\in M_n(V)_h$ and that, by way of contradiction, $X$ does not belong to $C_n$. The space $M_n(V)$, equipped with the topology of weak* convergence, is a locally convex topological vector space. By a geometric form of the Hahn-Banach Theorem, there exists a functional $s : M_n(V)\to \bb{C}$, continuous with respect to the topology of entry-wise weak* convergence, such that $s(C_n)\subseteq \bb{R}^+$ but $s(X) < 0$. By \cite[Theorem 6.1]{Pa}, the map $\phi_s : V \to M_n$, given by $\phi_s(x) = (s_{i,j}(x))_{i,j}$ (and where $s_{i,j}(x) = s(E_{i,j}\otimes x)$), is completely positive. It is clear that $\phi_s$ is normal. In addition, $\phi_s^{(n)}$ does not map $X$ to a positive matrix. After normalisation, we may assume that $\phi_s$ is contractive.
Let $P = \phi_s(e)$; then $P$ is a positive contraction. Assume that ${\rm rank}(P) = k$ and let $Q$ be the projection onto $\ker(P)^{\perp}$. It was shown in the proof of \cite[Theorem 13.1]{Pa} that, if $A\in M_{n,k}$ and $B\in M_{k,n}$ are matrices such that $A^* P A = I_k$ and $AB = Q$, and $\psi$ is the mapping given by $\psi(x) = A^* \phi_s(x) A$, then $\psi$ is a (unital completely positive) map such that $\psi^{(n)}(X)$ is not positive. Clearly, $\psi$ is normal, and hence an element of $\cl P_k$. This contradicts the fact that $J^{(n)}(X)\geq 0$.
To show that $J$ is a weak* homeomorphism, suppose that $J(x_{\alpha})\to_{\alpha} J(x)$ in the weak* topology, for some net $(x_{\alpha})\subseteq V$ and some element $x\in V$. Then $\phi(x_{\alpha})\to \phi(x)$ for all normal positive functionals $\phi$. By condition (iii) of Definition \ref{d_daous}, $x_{\alpha}\to x$ in the weak* topology of $V$.
We finally note that $J(V)$ is weak* closed in $\cl B(H)$. Suppose that $J(x_{\alpha})\to T$, where $T\in \cl B(H)$ and $(x_{\alpha})_{\alpha}\subseteq V$ is a net such that the net $J(x_{\alpha})_{\alpha}$ is bounded. Since $J$ is an isometry, $(x_{\alpha})_{\alpha}$ is also bounded, and hence has a subnet $(x_{\beta})_{\beta}$, weak* convergent to an element of $V$, say $x$. Since $J$ is weak* continuous, we conclude that $T = \lim_{\beta} J(x_{\beta}) = J(x)$, and hence $T\in J(V)$. By the Krein-Smulyan, $J(V)$ is weak* closed.
By the previous paragraphs, the weak* topology of $V$ coincides with the weak* topology of the operator system $\cl S$. It now follows that the $\cl A$-module operations on $\cl S$ are separately weak* continuous; thus, $\cl S$ is a dual operator $\cl A$-system and the proof is complete. \end{proof}
As the next two statements show, if $(V,V^+,e)$ is a dual AOU $\cl A$-space then the minimal operator $\cl A$-system structure defined in Section \ref{s_eoss} is automatically a dual minimal operator $\cl A$-system structure.
\begin{theorem}\label{th_maxdual} Let $\cl A$ be a W*-algebra and $(V,V^+,e)$ be a dual AOU $\cl A$-space. Then $(C_n^{\min}(V;\cl A))_{n\in \bb{N}}$ is a dual operator $\cl A$-system structure. \end{theorem} \begin{proof} Since the $\cl A$-module actions on $V$ are weak* continuous, $C_n^{\min}(V;\cl A)$ is weak* closed for each $n\in \bb{N}$. By Theorem \ref{th_weakscac}, $(C_n^{\min}(V;\cl A))_{n\in \bb{N}}$ is a dual operator $\cl A$-system structure. \end{proof}
\begin{theorem}\label{th_numin} Let $\cl A$ be a W*-algebra and $(V,V^+,e)$ be a dual AOU $\cl A$-space.
(i) Suppose that $\cl S$ is a dual operator $\cl A$-system and $\phi : \cl S\to V$ is a normal positive $\cl A$-bimodule map. Then $\phi$ is completely positive as a map from $\cl S$ into $\omin_{\cl A}(V)$.
(ii) If $\cl T$ is a dual operator $\cl A$-system with underlying space $V$ and positive cone $V^+$, such that for every dual operator $\cl A$-system $\cl S$, every normal positive $\cl A$-bimodule map $\phi : \cl S\to \cl T$ is completely positive, then there exists a unital normal $\cl A$-bimodule map $\psi : \cl T \to \omin_{\cl A}(V)$ that is a complete order isomorphism and a weak* homeomorphism. \end{theorem}
\begin{proof} (i) is a direct consequence of Theorem \ref{th_umin} (i). The proof of (ii) follows by a standards argument, similar to the one given in the proof of Theorem \ref{th_umin} (ii). \end{proof}
In the remainder of the section, we consider the dual maximal operator $\cl A$-system structure. For a W*-algebra $\cl A$ and a dual AOU $\cl A$-space $(V,V^+,e)$, set $$W_n^{\max}(V;\cl A) = \overline{C_n^{\max}(V;\cl A)}^{w^*}, \ \ \ \ n\in \bb{N}.$$
\begin{theorem}\label{th_wmaxa} Let $\cl A$ be a W*-algebra and $(V,V^+,e)$ be a dual AOU $\cl A$-space. Then $(W_n^{\max}(V;\cl A))_{n\in \bb{N}}$ is a dual operator $\cl A$-system structure on $V$. Moreover, if $(P_n)_{n\in \bb{N}}$ is a dual operator $\cl A$-system structure on $V$ then $W_n^{\max}(V;\cl A)\subseteq P_n$ for each $n\in \bb{N}$. \end{theorem} \begin{proof} By Theorem \ref{th_cmaxa}, $(C_n^{\max}(V;\cl A))_{n\in \bb{N}}$ is an operator system $\cl A$-structure on $V$. It follows by the separate weak* continuity of the $\cl A$-module actions on $V$ and the definition of the $M_n(\cl A$)-module operations on $M_n(V)$ (see (\ref{eq_matrixmod})) that the family $(W_n^{\max}(V;\cl A))_{n\in \bb{N}}$ is $\cl A$-compatible.
Since the element $e$ is a matrix order unit for $(D_n^{\max}(V;\cl A))_{n\in \bb{N}}$ (see Proposition \ref{minmax}) and $D_n^{\max}(V;\cl A)\subseteq W_n^{\max}(V;\cl A)$ for each $n\in \bb{N}$, $e$ is a matrix order unit for $(W_n^{\max}(V;\cl A))_{n\in \bb{N}}$. To show that $e$ is an Archimedean matrix order unit for $(W_n^{\max}(V;\cl A))_{n\in \bb{N}}$, suppose that $X\in M_n(V)$ is such that $X + re_n\in W_n^{\max}(V;\cl A)$ for all $r > 0$. Since $X + re_n\to_{r\to 0} X$ in the weak* topology and $W_n^{\max}(V;\cl A)$ is weak* closed, $X\in W_n^{\max}(V;\cl A)$.
It follows that $(V, (W_n^{\max}(V;\cl A))_{n\in \bb{N}}, e)$ is an operator $\cl A$-system; by condition (ii) of Definition \ref{d_daous}, $V^+ = W_1^{\max}(V;\cl A)$. Since its cones are weak* closed, Theorem \ref{th_weakscac} implies that it is a dual operator $\cl A$-system.
Suppose that $(P_n)_{n\in \bb{N}}$ is a dual operator $\cl A$-system structure on $V$. Fix $n\in \bb{N}$. By Theorem \ref{th_cmaxa}, $C_n^{\max}(V;\cl A)\subseteq P_n$. By Theorem \ref{th_weakscac}, $P_n$ is weak* closed. It follows that $W_n^{\max}(V;\cl A)\subseteq P_n$. \end{proof}
We denote by $\omax_{\cl A}^{w^*}(V)$ the operator system $(V,(W_n^{\max}(V;\cl A))_{n\in \bb{N}},e)$.
\begin{theorem}\label{th_nacp} Let $\cl A$ be a W*-algebra and $(V,V^+,e)$ be a dual AOU $\cl A$-space.
(i) Suppose that $\cl S$ is a dual operator $\cl A$-system and $\phi : V\to \cl S$ is a normal positive $\cl A$-bimodule map. Then $\phi$ is completely positive as a map from $\omax_{\cl A}^{w^*}(V)$ into $\cl S$.
(ii) If $\cl T$ is a dual operator $\cl A$-system with underlying space $V$ and positive cone $V^+$, such that for every dual operator $\cl A$-system $\cl S$, every normal positive $\cl A$-bimodule map $\phi : \cl T\to \cl S$ is completely positive, then there exists a unital normal $\cl A$-bimodule map $\psi : \cl T\to \omax^{w^*}_{\cl A}(V)$ that is a complete order isomorphism and a weak* homeomorphism. \end{theorem} \begin{proof} (i) By Theorem \ref{th_acp} (i), $\phi^{(n)}(C_n^{\max}(V;\cl A))\subseteq M_n(\cl S)^+$. Since $\phi$ is weak* continuous and $M_n(\cl S)^+$ is weak* closed, $\phi^{(n)}(W_n^{\max}(V;\cl A))\subseteq M_n(\cl S)^+$.
(ii) similar to the proof of Theorem \ref{th_umin} (ii). \end{proof}
\noindent {\bf Remark. } Let $\cl A$ be a W*-algebra and $\frak{A}^{w^*}_{\cl A}$ (resp. $\frak{S}^{w^*}_{\cl A}$) be the category, whose objects are dual AOU $\cl A$-spaces (resp. dual operator $\cl A$-systems) and whose morphisms are weak* continuous unital positive (resp. weak* continuous unital completely positive) maps. It is easy to see that the correspondences $V\to \omin^{w^*}_{\cl A}(V)$ and $V\to \omax^{w^*}_{\cl A}(V)$ are covariant functors from $\frak{A}^{w^*}_{\cl A}$ into $\frak{S}^{w^*}_{\cl A}$, here $\omin^{w^*}_{\cl A}(V)= \omin_{\cl A}(V)$ as per Theorem \ref{th_maxdual}.
\section{Inflated Schur multipliers}\label{s_ism}
In this section, we introduce an operator-valued version of classical measurable Schur multipliers, and characterise them in a fashion, similar to the well-known descriptions in the scalar-valued case \cite{Gro, peller}.
Let $(X,\mu)$ be a standard measure space. We denote by $\chi_{\alpha}$ the characteristic function of a measurable set $\alpha\subseteq X$. If $f$ and $g$ are measurable functions defined on $X$, we write $f\sim g$ when $f(x) = g(x)$ for almost all $x \in X$. Throughout the section, let $H = L^2(X,\mu)$ and fix a separable Hilbert space $K$. For a function $a\in L^{\infty}(X,\mu)$, let $M_a$ be the operator on $H$ given by $M_a f = af$, $f\in H$, and set $$\cl D = \left\{M_a : a\in L^{\infty}(X,\mu)\right\}.$$ We denote by $H\otimes K$ the Hilbertian tensor product of $H$ and $K$. Note that $H\otimes K$ is unitarily equivalent to the space $L^2(X,K)$ of all weakly measurable functions
$g : X\to K$ such that $\|g\|_2 := \left(\int_X \|g(x)\|^2d\mu(x)\right)^{1/2} < \infty$.
If $\cl U\subseteq \cl B(H)$ and $\cl V\subseteq \cl B(K)$, we denote by $\cl U\bar{\otimes}\cl V$ the spacial weak* tensor product of $\cl U$ and $\cl V$. We write $\cl M(X,\cl B(K))$ for the space of all functions $F : X \to \cl B(K)$ such that, for all $\xi_0\in K$, the functions $x\to F(x)\xi_0$ and $x\to F(x)^*\xi_0$ are weakly measurable. Note that $\cl D\bar\otimes\cl B(K)$ can be canonically identified with the space $L^{\infty}(X,\cl B(K))$ of all bounded functions $F$ in $\cl M(X,\cl B(K))$ \cite{t}. Through this identification, a function $F$ gives rise to the operator $M_F\in \cl B(L^2(X,K))$, defined by $$(M_F\xi)(x) = F(x)(\xi(x)), \ \ \ x\in X, \ \xi\in L^2(X,K).$$ It is easy to see that if $k\in \cl M(X\times X,\cl B(K))$ then the function
$(x,y)\to \|k(x,y)\|$ is measurable as a function from $X\times X$ into $[0,+\infty]$. Let $L^2(X\times X,\cl B(K))$ be the space of all functions $k\in \cl M(X\times X,\cl B(K))$ for which
$$\|k\|_2:= \left(\int_{X\times X}\|k(x,y)\|^2 d\mu(x) d\mu(y) \right)^{1/2} < \infty.$$ (Note that the functions from the space $L^2(X\times X,\cl B(K))$ need not be weakly measurable.) If $k\in L^2(X\times X,\cl B(K))$ and $\xi,\eta\in L^2(X,K)$ then, by \cite[Lemma 7.5]{t}, the function $(x,y)\to \left(k(x,y)(\xi(y)),\eta(x)\right)$ is measurable. Standard arguments (see \cite[p. 391]{mtt}) show that the formula $$(T_k\xi,\eta) = \int_{X\times X} \left(k(x,y)(\xi(y)),\eta(x)\right) d\mu(y)d\mu(x), \ x,y\in X, \xi,\eta\in L^2(X,K),$$ defines a bounded operator on $L^2(X,K)$ with
$\|T_k\|\leq \|k\|_2$. If $K = \bb{C}$, the operators of the form $T_k$ are precisely the Hilbert-Schmidt operators on $H$.
\begin{remark}\label{r_zero} For an element $k\in L^2(X\times X,\cl B(K))$, we have that $T_k = 0$ if and only if $k(x,y) = 0$ for almost all $(x,y)\in X\times X$. \end{remark} \begin{proof} Suppose that $T_k = 0$; then, for $\xi,\eta\in K$ and $f,g\in L^2(X)$, we have $\int_{X\times X} f(x) g(y) (k(x,y)\xi,\eta) d\mu(y)d\mu(x) = 0$. Thus, $(k(x,y)\xi,\eta) = 0$ almost everywhere. Since $K$ is separable and $k(x,y)$ is bounded for all $x,y\in X$, this implies that $k(x,y) = 0$ almost everywhere. The converse direction is trivial. \end{proof}
We equip the linear space $\{T_k : k\in L^2(X\times X,\cl B(K))\}$ with the operator space structure arising from its inclusion into $\cl B(H\otimes K)$. Similarly, whenever $\cl S$ is an operator system and $\cl S_0\subseteq \cl S$ is a self-adjoint (not necessarily unital) subspace of $\cl S$, we equip $\cl S_0$ with the matrix ordering inherited from $\cl S$, and thus talk about a linear map from $\cl S_0$ into an operator system $\cl T$ being positive or completely positive.
For functions $\nph\in L^{\infty}(X\times X,\cl B(K))$ and $k\in L^2(X\times X)$, let $\nph k : X\times X\to \cl B(K)$ be the function given by $$(\nph k)(x,y) = k(x,y)\nph(x,y), \ \ \ x,y\in X.$$ It is straightforward to check that $\nph k\in L^2(X\times X,\cl B(K))$.
\begin{definition}\label{def_opv} A function $\nph\in L^{\infty}(X\times X,\cl B(K))$ will be called an \emph{(inflated) Schur multiplier} if the map $$T_k \longrightarrow T_{\nph k}, \ \ k\in L^2(X\times X),$$ is completely bounded. \end{definition}
We will denote by $\frak{S}(X,K)$ the space of all inflated Schur multipliers with values in $\cl B(K)$. If $\nph\in \frak{S}(X,K)$ then the map $S_{\nph} : T_k\to T_{\nph k}$ defined on the space $\cl S_2(H)$ of all Hilbert-Schmidt operators on $H$ extends to a completely bounded map from $\cl K(H)$ into $\cl B(H\otimes K)$, which will be denoted in the same way. By taking the second dual of $S_{\nph}$, and composing with the weak* continuous projection from $\cl B(H\otimes K)^{**}$ onto $\cl B(H\otimes K)$, we obtain a completely bounded weak* continuous map from $\cl B(H)$ into $\cl B(H\otimes K)$ which for simplicity will still be denoted by $S_{\nph}$.
\begin{theorem}\label{th_deou} Let $\nph\in L^{\infty}(X\times X,\cl B(K))$. The following are equivalent:
(i) \ $\nph\in \frak{S}(X,K)$;
(ii) there exist functions $A_i \in L^{\infty}(X,\cl B(K))$ and $B_i \in L^{\infty}(X,\cl B(K))$, $i\in \bb{N}$, such that the series $\sum_{i=1}^{\infty} A_i(x)A_i(x)^*$ and $\sum_{i=1}^{\infty} B_i(y)^*B_i(y)$ converge almost everywhere in the weak* topology,
$$\esssup_{x\in X} \left\|\sum_{i=1}^{\infty} A_i(x)A_i(x)^*\right\| < \infty, \ \
\esssup_{y\in X} \left\|\sum_{i=1}^{\infty} B_i(y)^*B_i(y)\right\| < \infty,$$ \noindent and \begin{equation}\label{eq_aibi} \nph(x,y) = \sum_{i=1}^{\infty} A_i(x)B_i(y), \ \ \ \mbox{ a.e. on } X\times X, \end{equation} where the sum is understood in the weak* topology. \end{theorem}
\begin{proof} (ii)$\Rightarrow$(i)
Considering $A_i, B_i\in \cl D\bar\otimes\cl B(K)$, $i\in \bb{N}$, the assumptions imply that $A = (A_i)_{i\in \bb{N}}$ (resp. $B = (B_i)_{i\in \bb{N}}$) is a bounded row (resp. column) operator. It follows that the map $\Psi : \cl B(H)\to \cl B(H\otimes K)$, given by $$\Psi(T) = \sum_{i=1}^{\infty} A_i(T\otimes I) B_i, \ \ \ T\in \cl B(H),$$ is well-defined and completely bounded. Let $k\in L^2(X\times X) \cap L^{\infty}(X\times X)$, $\xi,\eta\in K$ and $f,g\in L^2(X)\cap L^1(X)$. For almost all $(x,y)\in X\times X$, we have \begin{eqnarray*} & &
\left|k(x,y)f(y)\overline{g(x)}\left(\nph(x,y)\xi,\eta\right)\right| \\ & \leq &
\|k\|_{\infty} |f(y)| |g(x)|\sum_{i=1}^{\infty} \left|(B_i(y)\xi,A_i(x)^*\eta)\right| \\ & \leq &
\|k\|_{\infty} |f(y)| |g(x)|\sum_{i=1}^{\infty} \|B_i(y)\xi\| \|A_i(x)^*\eta\|\\ & \leq &
\|k\|_{\infty} |f(y)| |g(x)| \left(\sum_{i=1}^{\infty} \|B_i(y)\xi\|^2\right)^{1/2}
\left(\sum_{i=1}^{\infty} \|A_i(x)^*\eta\|^2\right)^{1/2} \\ & \leq &
\|k\|_{\infty} |f(y)| |g(x)| \|A\| \|B\| \|\xi\| \|\eta\|, \end{eqnarray*}
while the function $(x,y)\to |f(y)| |g(x)|$ is integrable with respect to $\mu\times\mu$. By the Lebesgue Dominated Convergence Theorem, we now have \begin{eqnarray*} & & (\Psi(T_k)(f\otimes\xi),g\otimes\eta)\\ & = & \left(\sum_{i=1}^{\infty} A_i(T_k\otimes I) B_i (f\otimes \xi),g\otimes \eta\right)\\ & = & \sum_{i=1}^{\infty} \int_{X\times X} k(x,y)f(y)\overline{g(x)}(B_i(y)\xi,A_i(x)^*\eta)d\mu(x)d\mu(y)\\ & = & \int_{X\times X} k(x,y)f(y)\overline{g(x)}\left(\left(\sum_{i=1}^{\infty} A_i(x)B_i(y)\right)\xi,\eta\right)d\mu(x)d\mu(y)\\ & = & \int_{X\times X} k(x,y)f(y)\overline{g(x)}\left(\nph(x,y)\xi,\eta\right)d\mu(x)d\mu(y)\\ & = & \int_{X\times X} f(y)\overline{g(x)}\left((\nph k)(x,y)\xi,\eta\right)d\mu(x)d\mu(y)\\ & = & \left(T_{\nph k}(f\otimes\xi),g\otimes \eta\right). \end{eqnarray*} By linearity and the density of $L^2(X\times X) \cap L^{\infty}(X\times X)$ in $L^2(X\times X)$ and of $L^2(X) \cap L^1(X)$ in $L^2(X)$, it follows that $\nph\in \frak{S}(X,K)$ and $\Psi = S_{\nph}$.
(i)$\Rightarrow$(ii) Let $\nph\in \frak{S}(X,K)$. For $k\in L^2(X\times X)$, $a,b\in L^{\infty}(X)$, $\xi,\eta\in K$ and $f,g\in L^2(X)$, we have \begin{eqnarray*} & & \left(S_{\nph}(M_bT_kM_a)(f\otimes\xi),g\otimes\eta\right)\\ & = & \int_{X\times X} a(y) b(x) f(y)\overline{g(x)}\left((\nph k)(x,y)\xi,\eta\right)d\mu(x)d\mu(y)\\ & = & \left((M_b\otimes I) S_{\nph}(T_k)(M_a\otimes I)(f\otimes\xi),g\otimes\eta\right). \end{eqnarray*} By continuity, $$S_{\nph}(BTA) = (B\otimes I)S_{\nph}(T)(A\otimes I), \ \ \ T\in \cl K(H), A,B\in \cl D.$$ Let $\Phi_1 : \cl K(H) \otimes 1 \to \cl B(H\otimes K)$ be the map given by $\Phi_1(T\otimes I) = S_{\nph}(T)$; then $\Phi_1$ is a completely bounded $\cl D\otimes 1$-bimodule map. Using \cite[Exercise 8.6 (ii)]{Pa}, we can find a completely bounded weak* continuous $\cl D\otimes 1$-bimodule map $\Phi_2 : \cl B(H\otimes K) \to \cl B(H\otimes K)$ extending $\Phi_1$.
By \cite{haag}, there exist a bounded row operator $A = (A_i)_{i=1}^{\infty}$ and a bounded column operator $B = (B_i)_{i\in \bb{N}}$, where $A_i, B_i\in \cl D\bar\otimes\cl B(K)$, $i\in \bb{N}$, such that $$\Phi_2(T) = \sum_{i=1}^{\infty} A_i T B_i, \ \ \ T\in \cl B(H\otimes K).$$ Using the identification $\cl D\bar\otimes\cl B(K) \equiv L^{\infty}(X,\cl B(K))$, we consider $A_i$ (resp. $B_i$) as a function $A_i : X\to \cl B(K)$ (resp. $B_i : X\to \cl B(K)$). The boundedness of $A$ and $B$ now imply that there exists a null set $N\subseteq X$ such that the series $$\sum_{i=1}^{\infty} A_i(x)A_i(x)^* \ \ \mbox{ and } \ \ \sum_{i=1}^{\infty} B_i(y)^* B_i(y)$$ are weak* convergent whenever $x,y\not\in N$. If $(x,y)\not\in N\times N$ then the series $\sum_{i=1}^{\infty} A_i(x) B_i(y)$ is weak* convergent. As in the first part of the proof, we conclude that $\nph(x,y)$ coincides with its sum for almost all $(x,y)$. \end{proof}
An inspection of the proof of Theorem \ref{th_deou} shows the following description of inflated Schur multipliers.
\begin{remark}\label{r_modc0} The following are equivalent, for a completely bounded map $\Phi : \cl K(H)\to \cl B(H\otimes K)$:
(i) \ $\Phi(BTA) = (B\otimes I)\Phi(T)(A\otimes I)$, for all $T\in \cl K(H)$ and all $A,B\in \cl D$;
(ii) there exists a Schur multiplier $\nph \in \frak{S}(X,K)$ such that $\Phi = S_{\nph}$. \end{remark}
\begin{definition}\label{def_opvp} A Schur multiplier $\nph\in \frak{S}(X,K)$ will be called \emph{positive} if the map $S_{\nph} : \cl B(H)\to \cl B(H\otimes K)$ is positive. \end{definition}
For the next theorem, note that, if $\nph\in L^{\infty}(X\times X,\cl B(K))$ and $\alpha\subseteq X$ is a subset of finite measure then the function $\nph \chi_{\alpha\times\alpha}$ belongs to $L^2(X\times X,\cl B(K))$ and hence the operator $T_{\nph \chi_{\alpha\times\alpha}} : H\to H\otimes K$ is well-defined.
\begin{theorem}\label{th_modc} The following are equivalent, for a Schur multiplier $\nph\in \frak{S}(X,K)$:
(i) \ \ $\nph$ is positive;
(ii) \ the map $S_{\nph} : \cl B(H)\to \cl B(H\otimes K)$ is completely positive;
(iii) for every subset $\alpha\subseteq X$ of finite measure, the operator $T_{\nph \chi_{\alpha\times\alpha}}$ is positive;
(iv) \ there exist functions $A_i \in L^{\infty}(X,\cl B(K))$, $i\in \bb{N}$, such that the series $\sum_{i=1}^{\infty} A_i(x)A_i(x)^*$ converges almost everywhere in the weak* topology,
$$\esssup_{x\in X} \left\|\sum_{i=1}^{\infty} A_i(x)A_i(x)^*\right\| < \infty,$$ and $$\nph(x,y) = \sum_{i=1}^{\infty} A_i(x)A_i(y)^*, \ \ \ \mbox{ a.e. on } X\times X.$$ \end{theorem} \begin{proof} (i)$\Rightarrow$(iii) Let $\alpha\subseteq X$ be a subset of finite measure. Then $\chi_{\alpha}\in H$; let $\chi_{\alpha}\otimes\chi_{\alpha}^*$ be the corresponding (positive) rank one operator. Then $$ T_{\nph \chi_{\alpha\times\alpha}} = S_{\nph}(\chi_{\alpha}\otimes\chi_{\alpha}^*),$$ and the conclusion follows.
(iii)$\Rightarrow$(ii) Let $n\in \bb{N}$, $X_i = X$ for $i = 1,\dots,n$, $Y = X_1\cup\dots\cup X_n$ and $\nu$ be the disjoint sum of $n$ copies of the measure $\mu$. Identify $\bb{C}^n\otimes H$ with $L^2(Y,\nu)$, and define $\psi : Y\times Y\to \cl B(K)$ by letting $\psi(x,y) = \nph(x,y)$ if $(x,y) \in X_i\times X_j = X\times X$. Note that $S_{\psi} = \id_{M_n}\otimes S_{\nph}$ and hence $\psi \in \frak{S}(Y,K)$. Let $\alpha\subseteq X$ have finite measure and $J\in M_n$ be the matrix all of whose entries are equal to $1$. Let $\alpha_i \subseteq X_i$ be the set that coincides with $\alpha$, $i = 1,\dots,n$, and $\tilde{\alpha} = \cup_{i=1}^n \alpha_i$; we have that \begin{equation}\label{eq_J} T_{\psi \chi_{\tilde{\alpha} \times \tilde{\alpha}}} \equiv J \otimes T_{\nph \chi_{\alpha\times\alpha}}. \end{equation} By assumption, $T_{\nph \chi_{\alpha\times\alpha}}$ is positive; thus, by (\ref{eq_J}), $T_{\psi \chi_{\tilde{\alpha}\times\tilde{\alpha}}}$ is positive. For $g\in L^{\infty}(Y,\nu)\cap L^2(Y,\nu)$ and $h\in L^{\infty}(\tilde{\alpha})$, we have $$\left(S_{\psi}(g \otimes g^*)h,h\right) = \left(T_{\psi \chi_{\tilde{\alpha} \times \tilde{\alpha}}}(gh),gh\right)\geq 0.$$ Since the set $$\left\{h\in L^2(Y,\nu) : \exists \mbox{ a set of finite measure } \alpha \subseteq X \mbox{ with } h\in L^{\infty}(\tilde{\alpha})\right\}$$ is dense in $L^2(Y,\nu)$, we have that $S_{\psi}(g \otimes g^*) \in \cl B(H\otimes K)^+$. By weak* continuity, $S_{\psi}(T) \in \cl B(H\otimes K)^+$ whenever $T\in \cl B(L^2(Y,\nu))^+$. Thus, $S_{\psi}$ is positive, that is, $S_{\nph}$ is $n$-positive.
(ii)$\Rightarrow$(i) is trivial.
(ii)$\Rightarrow$(iv) follows from the proof of Theorem \ref{th_deou} by noting that in the case $S_{\nph}$ is completely positive, one can choose $B_i = A_i^*$, $i\in \bb{N}$.
(iv)$\Rightarrow$(i) follows from the proof of Theorem \ref{th_deou}. \end{proof}
\section{Positive extensions}\label{s_pe}
In this section, we apply our results on maximal operator system $\cl A$-structures to questions about positive extensions of inflated Schur multipliers. We first recall some measure theoretic background from \cite{a} and \cite{eks}, required in the sequel. A subset $E\subseteq X\times X$ is called \emph{marginally null} if $E\subseteq (M\times X)\cup (X\times M)$, where $M\subseteq X$ is null. We call two subsets $E,F\subseteq X\times X$ {\it marginally equivalent} (resp. {\it equivalent}), and write $E\cong F$ (resp. $E\sim F$), if their symmetric difference is marginally null (resp. null with respect to product measure). We say that $E$ is \emph{marginally contained} in $F$ (and write $E\subseteq_{\omega} F$) if the set difference $E\setminus F$ is marginally null. A measurable subset $\kappa\subseteq X\times X$ is called \begin{itemize} \item a \emph{rectangle} if $\kappa = \alpha\times\beta$ where $\alpha,\beta$ are measurable subsets of~$X$; \item {\it $\omega$-open} if it is marginally equivalent to a countable union of rectangles, and \item {\it $\omega$-closed} if its complement $\kappa^c$ is $\omega$-open. \end{itemize} Recall that, by~\cite{stt_cl}, if $\cl E$ is any collection of $\omega$-open sets then there exists a smallest, up to marginal equivalence, $\omega$-open set $\cup_{\omega}\cl E$, called the \emph{$\omega$-union} of $\cl E$, such that every set in~$\cl E$ is marginally contained in $\cup_{\omega}\cl E$. Given a measurable set $\kappa$, one defines its \emph{$\omega$-interior} to be \[\ointer(\kappa) = \bigcup\mbox{}_{\omega}\left\{R : R \, \mbox{ is a rectangle with } R \subseteq_{\omega} \kappa\right\}.\] The \emph{$\omega$-closure} $\ocl(\kappa)$ of~$\kappa$ is defined to be the complement of $\ointer(\kappa^c)$. For a set $\kappa\subseteq X\times X$, we write $\hat{\kappa} = \{(x,y)\in X\times X : (y,x)\in \kappa\}$. The subset~$\kappa\subseteq X\times X$ is said to be \emph{generated by rectangles} if $\kappa\cong\ocl(\ointer(\kappa))$ \cite{eks, llt}.
For any $\omega$-closed subset $\kappa\subseteq X\times X$, let
$$\cl S_2(\kappa) = \left\{T_k : k \in L^2(\kappa)\right\}, \ \ \cl S_0(\kappa) = \overline{\cl S_2(\kappa)}^{\|\cdot\|} \ \mbox{ and } \ \cl S(\kappa) = \overline{\cl S_2(\kappa)}^{w^*},$$ where $L^2(\kappa)$ is the space of functions in $L^2(X\times X)$ which are supported on $\kappa$, up to a set of zero product measure. Note that the spaces $\cl S_2(\kappa)$, $\cl S_0(\kappa)$ and $\cl S(\kappa)$ are $\cl D$-bimodules. We equip them with the operator space structures inherited from $\cl B(H)$.
Partially defined scalar-valued Schur multipliers were defined in \cite{llt}. Here we extend this notion to the operator-valued setting.
\begin{definition}\label{d_newschur}
Let $\kappa\subseteq X\times X$ be a subset generated by rectangles.
A function $\nph \in L^{\infty}(\kappa,\cl B(K))$ will be called a
\emph{partially defined Schur multiplier} if the map $S_{\nph}$ from
$\cl S_2(\kappa)$ into $\cl B(H\otimes K)$, given by
$$S_{\nph}(T_{k}) = T_{\nph k}, \ \ \ k\in L^2(\kappa),$$
is completely bounded. \end{definition}
\begin{remark}\label{r_eqli} For Schur multipliers $\nph, \psi \in L^{\infty}(\kappa,\cl B(K))$, we have that $S_{\nph} = S_{\psi}$ if and only if $\nph\sim\psi$. \end{remark}
\begin{proof} Suppose $\nph, \psi \in L^{\infty}(\kappa,\cl B(K))$ are such that $S_{\nph} = S_{\psi}$. Then $T_{\nph k} = T_{\psi k}$ for every $k\in L^2(\kappa)$. By Remark \ref{r_zero}, $\nph k \sim \psi k$. It now easily follows that $\nph\sim\psi$. The converse implication follows by reversing the previous steps. \end{proof}
Let $\kappa\subseteq X\times X$ be a subset generated by rectangles. We note that the map $S_{\nph}$ from Definition \ref{d_newschur} is $\cl D$-bimodular.
In addition, if $\psi \in \frak{S}(X,K)$ is given as in Definition \ref{def_opv}, then its restriction $\psi|_{\kappa} : \kappa\to \cl B(K)$ is an inflated Schur multiplier.
\begin{proposition}\label{p_chsch} Let $K$ be a separable Hilbert space, $\kappa\subseteq X\times X$ a subset generated by rectangles and $\nph \in L^{\infty}(\kappa,\cl B(K))$. The following are equivalent:
(i) \ \ $\nph$ is a Schur multiplier;
(ii) \ there exists a Schur multiplier $\psi : X\times X\to \cl B(K)$
such that $\psi|_{\kappa} \sim \nph$;
(iii) there exists a unique completely bounded map $\Phi_0 : \cl S_0(\kappa) \to \cl B(H\otimes K)$ such that $\Phi_0(T_k) = T_{\nph k}$, for each $k\in L^2(\kappa)$;
(iv) \ there exists a unique completely bounded weak* continuous map $\Phi : \cl S(\kappa) \to \cl B(H\otimes K)$ such that $\Phi(T_k) = T_{\nph k}$, for each $k\in L^2(\kappa)$. \end{proposition}
\begin{proof} (i)$\Rightarrow$(ii) Since $\nph$ is a Schur multiplier, the map $\Phi_2 : \cl S_2(\kappa)\to \cl B(H\otimes K)$, given by $\Phi_2(T_k) = T_{\nph k}$, extends to a completely bounded linear map $\Phi_0 : \cl S_0(\kappa)\to \cl B(H\otimes K)$. By continuity, $$\Phi_0(BTA) = (B\otimes I)\Phi_0(T)(A\otimes I), \ \ \ T\in \cl S_0(\kappa), A,B \in \cl D.$$ Let $\hat{\Phi} : \cl S_0(\kappa)\otimes 1 \to \cl B(H\otimes K)$ be the map given by $$\hat{\Phi}(T\otimes I) = \Phi_0(T), \ \ \ T\in \cl S_0(\kappa).$$ By~\cite[Exercise 8.6 (ii)]{Pa}, there exists a completely bounded $\cl D\otimes 1$-bimodule map $\hat{\Phi}_1 : \cl B(H\otimes K) \to \cl B(H\otimes K)$, extending $\hat{\Phi}$. Let $\hat{\Psi} : \cl K(H)\otimes 1\to \cl B(H\otimes K)$ be the restriction of $\hat{\Phi}_1$; then
$\hat{\Psi}|_{\cl S_0(\kappa)\otimes 1} = \hat{\Phi}$. Let $\Psi : \cl K(H)\to \cl B(H\otimes K)$ be given by $\Psi(T) = \hat{\Psi}(T\otimes I)$. Clearly, $$\Psi(BTA) = (B\otimes I)\Psi(T)(A\otimes I), \ \ \ T\in \cl K(H), A,B \in \cl D.$$ By Remark~\ref{r_modc0}, there exists $\psi\in \frak{S}(X,K)$ such that $\Psi = S_{\psi}$. For every $k\in L^2(\kappa)$ we have
$S_{\psi}(T_k) = S_{\nph}(T_k)$. By Remark \ref{r_eqli}, $\psi|_{\kappa}\sim \nph$.
(ii)$\Rightarrow$(iv) Take $\Phi = S_{\psi}|_{\cl S(\kappa)}$. The uniqueness of $\Phi$ follows from the fact that the Hilbert-Schmidt operators with integral kernels in $L^2(\kappa)$ are weak* dense in $\cl S(\kappa)$.
(iv)$\Rightarrow$(iii)$\Rightarrow$(i) are trivial. \end{proof}
If $\nph : \kappa \to \cl B(K)$ is a Schur multiplier then we will denote still by $S_{\nph}$ the weak* continuous map defined on $\cl S(\kappa)$ whose existence was established in Proposition \ref{p_chsch} (iv).
We say that a subset $\kappa\subseteq X\times X$ is \emph{symmetric} if~$\kappa\cong\hat\kappa$. We call $\kappa$ a \emph{positivity domain} \cite{llt} if $\kappa$ is symmetric, generated by rectangles and the diagonal $\Delta := \{(x,x) : x\in X\}$ is marginally contained in $\kappa$. The following was established in \cite{llt}:
\begin{proposition}\label{p_opsc} If $\kappa\subseteq X\times X$ is generated by rectangles, then the following are equivalent:
(i) \ $\cl S(\kappa)$ is an operator system;
(ii) $\kappa$ is a positivity domain. \end{proposition}
Let $\nph : \kappa\to \cl B(K)$ be a Schur multiplier. We say that the Schur multiplier $\psi : X\times X\to \cl B(K)$
is a \emph{positive extension} of $\nph$ if $\psi$ is positive and $\psi|_{\kappa} \sim \nph$.
\begin{proposition}\label{p_cpext} Let $\kappa$ be a positivity domain and $\nph : \kappa\to \cl B(K)$ be a Schur multiplier. The following are equivalent:
(i) \ $\nph$ has a positive extension;
(ii) the map $S_{\nph} : \cl S(\kappa) \to \cl B(H\otimes K)$ is completely positive. \end{proposition} \begin{proof} (i)$\Rightarrow$(ii) Suppose that $\psi : X\times X\to \cl B(K)$ is a positive extension of $\nph$. By Theorem \ref{th_modc}, $S_{\psi}$ is completely positive.
On the other hand, $S_{\psi}|_{\cl S(\kappa)} = S_{\psi|_{\kappa}}$.
Since $\psi|_{\kappa} = \nph$, we conclude that $S_{\nph}$ is completely positive.
(ii)$\Rightarrow$(i) Let $\Phi_0$ be the restriction of $S_{\nph}$ to $\cl S_0(\kappa) + \bb{C}I$; clearly, $\Phi_0$ is a completely positive map. By Arveson's Extension Theorem, there exists a completely positive map $\Psi_0 : \cl K(H) + \bb{C}I\to \cl B(H\otimes K)$ extending $\Phi_0$. The restriction $\Psi$ of $\Psi_0$ to $\cl K(H)$ is then a completely positive
extension of $S_{\nph}|_{\cl S_0(\kappa)}$. Let $\Psi^{**}$ be the second dual of $\Psi$, and $\cl E : \cl B(H\otimes K)^{**}\to \cl B(H\otimes K)$ be the canonical projection. We have that the map $\tilde{\Psi} = \cl E\circ \Psi^{**} : \cl B(H) \to \cl B(H\otimes K)$ is completely positive and weak* continuous extension of $S_{\nph}$. Let $\hat{\Psi} : \cl B(H)\otimes 1 \to \cl B(H\otimes K)$ (resp. $\hat{\Phi} : \cl S(\kappa)\otimes 1 \to \cl B(H\otimes K)$) be the map given by $\hat{\Psi}(T\otimes I) = \tilde{\Psi}(T)$ (resp. $\hat{\Phi}(T\otimes I) = S_{\nph}(T)$); then $\hat{\Psi}$ is a completely positive extension of map $\hat{\Phi}$. Note that $\hat{\Phi}$ is a $\cl D \otimes 1$-bimodule map. By~\cite[Exercise 7.4]{Pa}, $\hat{\Psi}$ is a $\cl D \otimes 1$-bimodule map. By Remark~\ref{r_modc0}, there exists $\psi\in \frak{S}(X,K)$ such that $\tilde{\Psi} = S_{\psi}$; the function $\psi$ is the desired positive extension of $\nph$. \end{proof}
If $\cl S$ is an operator system, we write $\cl S^{++}$ for the cone of all positive finite rank operators in $\cl S$. If $\cl T$ is an operator system, we call a linear map $\Phi : \cl S\to \cl T$ \emph{strictly positive} if $\Phi(S)\in \cl T^+$ whenever $S\in \cl S^{++}$. We call $\Phi$ \emph{strictly completely positive} if $\Phi^{(n)}$ is strictly positive for all $n\in \bb{N}$. A Schur multiplier $\nph : \kappa \to \cl B(K)$ will be called strictly positive (resp. strictly completely positive) if the map $S_{\nph} : \cl S(\kappa) \to \cl B(H\otimes K)$ is strictly positive (resp. strictly completely positive).
\begin{lemma}\label{l_de} Let $\kappa$ be a positivity domain. Every positive finite rank operator in $M_n(\cl S(\kappa))$ has the form $(T_{k_{i,j}})_{i,j=1}^n$, where $k_{i,j}\in L^2(\kappa)$, $i,j = 1,\dots,n$. \end{lemma}
\begin{proof}
Recall that $\cl S_2(\kappa) = \{T_k : k\in L^2(\kappa)\}$ and $\cl S_0(\kappa) = \overline{\cl S_2(\kappa)}^{\|\cdot\|}$. It follows that $M_n(\cl S_0(\kappa)) = \overline{M_n(\cl S_2(\kappa))}^{\|\cdot\|}$. Suppose that $T \in M_n(\cl S(\kappa))^{++}$ and let $T = (T_{i,j})_{i,j=1}^n$, where $T_{i,j}\in \cl S(\kappa)$, $i,j=1,\dots,n$. Since $T$ has finite rank, so does $T_{i,j}$; in particular, $T_{i,j}$ is a Hilbert-Schmidt operator and, by \cite[Lemma 6.1]{eks}, $T_{i,j} \in \cl S_2(\kappa)$. \end{proof}
Recall that the Banach space projective tensor product \[\cl T(X) = L^2(X,\mu) \hat{\otimes } L^2(X,\mu)\] can be canonically identified with the predual of $\cl B(H)$ (and the dual of $\cl K(H)$). Indeed, each element $h\in \cl T(X)$ can be written as a series $h = \sum_{i=1}^{\infty} f_i\otimes g_i$,
where $\sum_{i=1}^{\infty} \|f_i\|_2^2 < \infty$
and $\sum_{i=1}^{\infty} \|g_i\|_2^2 < \infty$, and the pairing is then given by \[\langle T,h\rangle = \sum_{i=1}^{\infty} (Tf_i,\overline{g_i}), \ \ \ T\in \cl B(H).\] We have~\cite{a} that $h$ can be identified with a complex function on $X\times X$, defined up to a marginally null set, and given by $$h(x,y) = \sum_{i=1}^{\infty} f_i(x)g_i(y).$$ The positive cone $\cl T(X)^+$ consists, by definition, of all functions $h\in \cl T(X)$ that give rise to positive functionals on $\cl B(H)$, that is, functions $h$ of the form
$h = \sum_{i=1}^{\infty} f_i\otimes \overline{f_i}$, where $\sum_{i=1}^{\infty} \|f_i\|_2^2 < \infty$. It is well-known that a function $\nph \in L^{\infty}(X\times X)$ is a Schur multiplier if and only if, for every $h\in \cl T(X)$, there exists $h'\in \cl T(X)$ such that $\nph h \sim h'$ (see \cite{peller}). In particular, if the measure $\mu$ is finite then $\frak{S}(X,\bb{C})$ can be naturally identified with a subspace of $\cl T(X)$.
\begin{theorem}\label{th_opsysext} Let $\kappa\subseteq X\times X$ be a positivity domain. The following are equivalent:
(i) \ for every separable Hilbert space $K$, every strictly positive Schur multiplier $\nph : \kappa\to \cl B(K)$ is strictly completely positive;
(ii) for every $n\in \bb{N}$, every positive finite rank operator in $M_n(\cl S(\kappa))$ is the norm limit of sums of operators of the form $(D_i S D_j^*)_{i,j}$, where $(D_i)_{i=1}^n\subseteq \cl D$ and $S\in \cl S(\kappa)^{++}$. \end{theorem} \begin{proof} (i)$\Rightarrow$(ii) We first assume that the measure $\mu$ is finite. Suppose that there exists $n\in \bb{N}$ and a positive finite rank operator $T\in M_n(\cl S(\kappa))$ that is not equal to the limit, in the norm topology, of the operators of the form $(D_iS D_j^*)_{i,j=1}^n$, where $(D_i)_{i=1}^n\subseteq \cl D$ and $S\in \cl S(\kappa)^{++}$. By Lemma \ref{l_de}, $T = (T_{k_{i,j}})_{i,j=1}^n$, for some $k_{i,j}\in L^2(\kappa)$, $i,j = 1,\dots,n$. By a geometric form of Hahn-Banach's Theorem, there exist a norm continuous functional $\omega : M_n(\cl S_0(\kappa)) \to \bb{C}$ and $\gamma < 0$ such that \begin{equation}\label{eq_omnew} \omega(T) < \gamma \ \mbox{ and } \ \omega\left((D_iS D_j^*)_{i,j=1}^n\right) \geq 0, \ \ S\in \cl S(\kappa)^{++}, (D_i)_{i=1}^n\subseteq \cl D. \end{equation} Let $\omega_{i,j} : \cl S_0(\kappa) \to \bb{C}$ be the norm continuous functionals such that $$\omega((S_{i,j})_{i,j = 1}^n) = \sum_{i,j = 1}^n \omega_{i,j}(S_{i,j}), \ \ \ S_{i,j}\in \cl S_0(\kappa), \ i, j = 1,\dots,n.$$ After extending $\omega_{i,j}$ to $\cl K(H)$, we may assume that $\omega_{i,j}\in \cl T(X)$ for $i, j = 1, \dots, n$.
Suppose first that $\omega_{i,j}\in \frak{S}(X,\bb{C})$, $i,j = 1,\dots,n$. Identify $\omega$ with the function (denoted by the same symbol) $\omega : X\times X\to M_n$, given by $\omega(x,y) = (\omega_{i,j}(x,y))_{i, j=1}^n$. Since $S_{\omega} : \cl S_2(H)\to \cl B(H)\otimes M_n$ is given by $S_{\omega}(T_k) = (S_{\omega_{i,j}}(T_k))$, $k\in L^2(X\times X)$, and the maps $S_{\omega_{i,j}}$ are completely bounded, we have that the map $S_{\omega}$ is completely bounded, that is, $\omega\in \frak{S}(X,M_n)$.
We claim that $S_{\omega}^{(n)}$ is not strictly positive.
Note that $$S_{\omega}^{(n)}(T) = \left(S_{\omega_{i,j}}(T_{k_{p,q}})\right)_{i,j,p,q}.$$ Writing $e$ for the vector in $H^n$ with all its entries equal to the constant function $1$, we have that \begin{eqnarray}\label{eq_con} \gamma & > & \omega(T) = \sum_{i,j=1}^n \int_{\kappa} \omega_{i,j}(x,y)k_{i,j}(x,y) d(\mu\times \mu)(x,y)\nonumber\\ & = & \left(\left(S_{\omega_{i,j}}(T_{k_{i,j}})\right)_{i,j}e,e\right). \end{eqnarray} Suppose that $S_{\omega}^{(n)}(T)$ is positive. Then its submatrix $(S_{\omega_{i,j}}(T_{k_{i,j}}))_{i,j}$ is positive, which contradicts (\ref{eq_con}).
We now show that $S_{\omega}$ is strictly positive. Let $S\in \cl S(\kappa)^{++}$. Using Lemma \ref{l_de}, write $S = T_k$ for some $k\in L^2(\kappa)$. We have that $S_{\omega}(S) = (T_{\omega_{i,j} k})_{i, j=1}^n$. For $i = 1,\dots,n$, let $\xi_i\in L^{\infty}(X,\mu)$ and note that, since $\mu$ is finite, $\xi_i\in H$. Let $D_i = M_{\xi_i}$, $i = 1, \dots, n$, and set $\xi = (\xi_i)_{i=1}^n$. We have that \begin{eqnarray*} \left(S_{\omega}(S)\xi,\xi\right) & = & \sum_{i, j=1}^n (T_{\omega_{i,j} k}\xi_j,\xi_i)\\ & = & \sum_{i, j=1}^n \int_{\kappa} \omega_{i,j}(x,y) k(x,y)\xi_j(x)\overline{\xi_i(y)} d(\mu\times\mu)(x,y)\\ & = & \omega\left((D_i^*SD_j)_{i,j=1}^n\right) \geq 0. \end{eqnarray*} Since $L^{\infty}(X,\mu)$ is dense in $H$, we have that $S_{\omega}(S)\in M_n(\cl B(H))^+$.
Now relax the assumption that $\omega_{i,j} \in \frak{S}(X,\bb{C})$. By standard arguments (see e.g. the proof of \cite[Lemma 3.13]{akt}), there exist measurable sets $X_m\subseteq X$ with $X_m\subseteq X_{m+1}$, $m\in \bb{N}$, such that $\mu(X\setminus X_m) \to_{m\to\infty} 0$ and the restriction $\omega^{(m)}_{i,j}$ of $\omega_{i,j}$ to $X_m \times X_m$ belongs to $\frak{S}(X_m,\bb{C})$ for all $m\in \bb{N}$. Let $\omega^{(m)} : X\times X \to M_n$ be the function given by $\omega^{(m)}(x,y) = (\omega_{i,j}^{(m)}(x,y))_{i,j}$ if $(x,y)\in X_m\times X_m$ and $\omega^{(m)}(x,y) = 0$ otherwise, and note that $\omega^{(m)}$ defines a functional on $M_n(\cl K(H))$ in the natural way (which will be denoted by the same symbol). Let $P_m$ be the projection from $H$ onto $L^2(X_m)$. We have that $$\omega^{(m)}(R) = \omega((P_m\otimes I_n) R (P_m\otimes I_n)), \ \ \ R\in M_n(\cl K(H)).$$ Since $(P_m\otimes I_n) R (P_m\otimes I_n) \to_{m\to \infty} R$ in norm, for every $R\in M_n(\cl K(H))$, we have that (\ref{eq_omnew}) eventually holds true for $\omega^{(m)}$ in the place of $\omega$. By the previous paragraph, $\omega^{(m)}$ is a Schur multiplier for which $S_{\omega^{(m)}}$ is strictly positive, but not strictly completely positive.
Finally, relax the assumption that $\mu$ be finite. Let $(X_m)_{m\in \bb{N}}$ be an increasing sequence of sets of finite measure such that $\cup_{m=1}^{\infty} X_m = X$, and let $Q_m$ be the projection from $H$ onto $L^2(X_m)$, $m\in \bb{N}$. Let $T\in M_n(\cl S(\kappa))^{++}$. Since $T$ is a positive operator of finite rank, $(Q_mTQ_m)_{m\in \bb{N}}$ is a sequence of positive finite rank operators, converging to $T$ in norm. By the first part of the proof, $Q_mTQ_m$ is a norm limit of operators of the form $(D_i S D_j^*)_{i,j}$, where $(D_i)_{i=1}^n\subseteq \cl D$ and $S\in \cl S(\kappa)^{++}$. The conclusion follows.
(ii)$\Rightarrow$(i) Let $\nph : \kappa\to \cl B(K)$ be a Schur multiplier such that $S_{\nph} : \cl S(\kappa)\to \cl B(H\otimes K)$ is strictly positive. It follows from the assumption and fact that $S_{\nph}$ is a $\cl D$-bimodule map that $S_{\nph}^{(n)}(T)$ is positive whenever $T\in M_n(\cl S(\kappa))^{++}$. \end{proof}
\begin{definition} Let $\kappa$ be a positivity domain. We call $\kappa$ \emph{rich} if $$M_n(\cl S(\kappa))^{+} = \overline{M_n(\cl S(\kappa))^{++}}^{w^*} \ \ \mbox{ for every } n\in \bb{N}.$$ \end{definition}
Suppose that $X$ is a countable set equipped with counting measure. In this case, positivity domains can be identified with undirected graphs with vertex set $X$ in the natural way. This identification will be made in the subsequent remark and in Theorem \ref{c_ch}.
\begin{remark}\label{r_disr} Let $X$ be a countable set. Then any graph $\kappa\subseteq X \times X$ is rich. \end{remark}
\begin{proof} For $X = \bb{N}$, write $Q_m$ for the projection onto the span of $\{e_i\}_{i=1}^m$, $m\in \bb{N}$, where $\{e_i\}_{i\in \bb{N}}$ is the standard basis of $\ell^2$. If $T\in M_n(\cl S(\kappa))^{+}$ then $((Q_m\otimes I_n) T (Q_m\otimes I_n))_{m\in \bb{N}}$ is a sequence in $M_n(\cl S_2(\kappa))^{++}$, converging in the weak* topology to $T$. \end{proof}
By Proposition \ref{p_cpext}, if a Schur multiplier $\nph : \kappa\to \cl B(K)$ has a positive extension then the map $S_{\nph} : \cl S(\kappa) \to \cl B(H\otimes K)$ is necessarily positive. We call $\nph$ \emph{admissible} if $S_{\nph}$ is a positive map. The main result of this section is a characterisation of when an admissible Schur multiplier has a positive extension, in terms of the maximal operator $\cl D$-system structure defined in Section \ref{s_deoass}. Note that $\cl S(\kappa)$ is a dual AOU $\cl D$-space in the natural fashion.
\begin{theorem}\label{th_rich} Let $\kappa\subseteq X\times X$ be a rich positivity domain. The following are equivalent:
(i) \ for every separable Hilbert space $K$, every admissible Schur multiplier $\nph : \kappa \to \cl B(K)$ has a positive extension;
(ii) $\cl S(\kappa) = \omax_{\cl D}^{w^*}(\cl S(\kappa))$. \end{theorem}
\begin{proof} (i)$\Rightarrow$(ii) Let $\nph : \kappa \to \cl B(K)$ be a strictly positive Schur multiplier. Since $\cl S(\kappa)^+ = \overline{\cl S(\kappa)^{++}}^{w^*}$ and $S_{\nph}$ is weak* continuous, $S_{\nph}$ is positive. By the assumption and Proposition \ref{p_cpext}, $S_{\nph}$ is completely positive. In particular, $S_{\nph}$ is strictly completely positive. By Theorem \ref{th_opsysext} and the fact that the matricial cones of any operator system are norm closed, we have that \begin{equation}\label{eq_mn++} M_n(\cl S(\kappa))^{++} \subseteq M_n(\omax\mbox{}_{\cl D}(\cl S(\kappa)))^+. \end{equation} Since $\kappa$ is rich, by taking weak* closures on both sides in (\ref{eq_mn++}) we obtain that \begin{equation}\label{eq_mn+} M_n(\cl S(\kappa))^{+} \subseteq M_n(\omax\mbox{}_{\cl D}^{w^*}(\cl S(\kappa)))^+. \end{equation} Since the converse inclusion in (\ref{eq_mn+}) always holds, we conclude that $\cl S(\kappa) = \omax_{\cl D}^{w^*}(\cl S(\kappa))$.
(ii)$\Rightarrow$(i) follows from Theorem \ref{th_nacp} and Proposition \ref{p_cpext}. \end{proof}
Theorem \ref{th_rich} and Remark \ref{r_disr} have the following immediate corollary. In the case where $X$ is finite, it is a reformulation, in terms of operator system structures, of \cite[Theorem 4.6]{pps}.
\begin{corollary}\label{c_disc} Let $X$ be a countable set, equipped with counting measure and $\kappa\subseteq X\times X$ be a symmetric set containing the diagonal. The following are equivalent:
(i) \ for every Hilbert space $K$, every admissible Schur multiplier $\nph : \kappa\to \cl B(K)$ has a positive extension;
(ii) $\cl S(\kappa) = \omax_{\cl D}^{w^*}(\cl S(\kappa))$.
\end{corollary}
Let $X$ be a countable set. Recall that a graph $\kappa\subseteq X\times X$ is called chordal if every 4-cycle in $\kappa$ has an edge connecting two non-consecutive vertices of the cycle (see e.g. \cite{pps}).
\begin{theorem}\label{c_ch} Let $X$ be a countable set and $\kappa\subseteq X\times X$ be a chordal graph. Then $\cl S(\kappa) = \omax_{\cl D}^{w^*}(\cl S(\kappa))$. \end{theorem}
\begin{proof} Fix $n\in \bb{N}$ and let $[n] = \{1,\dots,n\}$. Suppose that $\kappa \subseteq X\times X$ is a chordal graph. Let $$\kappa^{(n)} = \left\{((x,i),(y,j)) \in \left(X\times [n]\right)\times \left(X\times [n]\right) : (x,y)\in \kappa\right\}.$$ Then $\kappa^{(n)}$ is a chordal graph on $X\times [n]$. By \cite[Theorem 2.5]{llt}, every positive operator in $M_n(\cl S(\kappa))$ is a weak* limit of rank one positive operators in $M_n(\cl S(\kappa))$.
Suppose that $K$ is a Hilbert space and $\nph : \kappa \to \cl B(K)$ is a Schur multiplier such that $S_{\nph} : \cl S(\kappa)\to \cl B(H \otimes K)$ is a positive map. Let $R\in M_n(\cl S(\kappa))$ be a positive rank one operator. After identifying $M_n(\cl S(\kappa))$ with $\cl S(\kappa^{(n)})$, we see that there exists a subset $\alpha \subseteq X\times [n]$ such that $R$ is supported on $\alpha\times\alpha$. Let $$\beta = \{x\in X : \exists \ i\in [n] \mbox{ with } (x,i)\in \alpha\}.$$ Since $\alpha\times\alpha \subseteq \kappa^{(n)}$, we have that $\beta\times\beta \subseteq \kappa$. Setting $\tilde{\beta} = \beta\times [n]$, we have that $\alpha\subseteq \tilde{\beta}$, and hence $R$ is supported on $\tilde{\beta} \times \tilde{\beta}$. The restriction $\psi$ of $\nph$ to $\beta\times\beta$ is a positive Schur multiplier. By Theorem \ref{th_modc}, the map $S_{\psi} : \cl S(\beta\times\beta)\to \cl B(H\otimes K)$ is completely positive. Thus, $S_{\nph}^{(n)}(R) = S_{\psi}^{(n)}(R) \in \cl B(H\otimes K)^+$. Since $S_{\nph}$ is weak* continuous, the previous paragraph implies that $S_{\nph}$ is completely positive. By Proposition \ref{p_cpext}, $\nph$ has a positive extension and, by Corollary \ref{c_disc}, $\cl S(\kappa) = \omax_{\cl D}^{w^*}(\cl S(\kappa))$. \end{proof}
\end{document} | arXiv |
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Universidade Cornell
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Fonte: Universidade Cornell Publicador: Universidade Cornell
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A New Class of Backward Stochastic Partial Differential Equations with Jumps and Applications
Dai, Wanyang
#Mathematics - Probability#Computer Science - Systems and Control#Mathematical Physics#Mathematics - Analysis of PDEs#Mathematics - Optimization and Control#Mathematics - Statistics Theory
We formulate a new class of stochastic partial differential equations (SPDEs), named high-order vector backward SPDEs (B-SPDEs) with jumps, which allow the high-order integral-partial differential operators into both drift and diffusion coefficients. Under certain type of Lipschitz and linear growth conditions, we develop a method to prove the existence and uniqueness of adapted solution to these B-SPDEs with jumps. Comparing with the existing discussions on conventional backward stochastic (ordinary) differential equations (BSDEs), we need to handle the differentiability of adapted triplet solution to the B-SPDEs with jumps, which is a subtle part in justifying our main results due to the inconsistency of differential orders on two sides of the B-SPDEs and the partial differential operator appeared in the diffusion coefficient. In addition, we also address the issue about the B-SPDEs under certain Markovian random environment and employ a B-SPDE with strongly nonlinear partial differential operator in the drift coefficient to illustrate the usage of our main results in finance.; Comment: 22 pagea, 1 figure
Second-order asymptotics for quantum hypothesis testing
Li, Ke
#Quantum Physics#Computer Science - Information Theory#Mathematics - Statistics Theory
In the asymptotic theory of quantum hypothesis testing, the minimal error probability of the first kind jumps sharply from zero to one when the error exponent of the second kind passes by the point of the relative entropy of the two states in an increasing way. This is well known as the direct part and strong converse of quantum Stein's lemma. Here we look into the behavior of this sudden change and have make it clear how the error of first kind grows smoothly according to a lower order of the error exponent of the second kind, and hence we obtain the second-order asymptotics for quantum hypothesis testing. This actually implies quantum Stein's lemma as a special case. Meanwhile, our analysis also yields tight bounds for the case of finite sample size. These results have potential applications in quantum information theory. Our method is elementary, based on basic linear algebra and probability theory. It deals with the achievability part and the optimality part in a unified fashion.; Comment: Published in at http://dx.doi.org/10.1214/13-AOS1185 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)
Turbulence analysis of an experimental flux rope plasma
Schaffner, D. A.; Lukin, V. S.; Wan, A.; Brown, M. R.
#Physics - Plasma Physics
We have previously generated elongated Taylor double-helix flux rope plasmas in the SSX MHD wind tunnel. These plasmas are remarkable in their rapid relaxation (about one Alfv\'en time) and their description by simple analytical Taylor force-free theory despite their high plasma beta and high internal flow speeds. We report on the turbulent features observed in these plasmas including frequency spectra, autocorrelation function, and probability distribution functions of increments. We discuss here the possibility that the turbulence facilitating access to the final state supports coherent structures and intermittency revealed by non-Gaussian signatures in the statistics. Comparisons to a Hall-MHD simulation of the SSX MHD wind tunnel show similarity in several statistical measures.; Comment: 20 pages, 9 figures, submitted to Plasma Physics Controlled Fusion for Special Issue on Flux Ropes
Unbounded Probability Theory and Its Applications
Maslov, V. P.; Maslova, T. V.
#Mathematical Physics#Mathematics - Probability
The paper deals with the order statistics and empirical mathematical expectation (which is also called the estimate of mathematical expectation in the literature) in the case of infinitely increasing random variables. The Kolmogorov concept which he used in the theory of complexity and the relationship with thermodynamics which was pointed out already by Poincar\'e are considered. The mathematical expectation (generalizing the notion of arithmetical mean, which is generally equal to infinity for any increasing sequence of random variables) is compared with the notion of temperature in thermodynamics by using an analog of nonstandard analysis. The relationship with the Van-der-Waals law of corresponding states is shown. Some applications of this concept in economics, in internet information network, and self-teaching systems are considered.; Comment: 23 p. Latex, minor corrections
The theory of the double preparation: discerned and indiscerned particles
Gondran, Michel; Gondran, Alexandre
#Quantum Physics#Mathematical Physics#Mathematics - Quantum Algebra#Physics - Classical Physics
In this paper we propose a deterministic and realistic quantum mechanics interpretation which may correspond to Louis de Broglie's "double solution theory". Louis de Broglie considers two solutions to the Schr\"odinger equation, a singular and physical wave u representing the particle (soliton wave) and a regular wave representing probability (statistical wave). We return to the idea of two solutions, but in the form of an interpretation of the wave function based on two different preparations of the quantum system. We demonstrate the necessity of this double interpretation when the particles are subjected to a semi-classical field by studying the convergence of the Schr\"odinger equation when the Planck constant tends to 0. For this convergence, we reexamine not only the foundations of quantum mechanics but also those of classical mechanics, and in particular two important paradox of classical mechanics: the interpretation of the principle of least action and the the Gibbs paradox. We find two very different convergences which depend on the preparation of the quantum particles: particles called indiscerned (prepared in the same way and whose initial density is regular, such as atomic beams) and particles called discerned (whose density is singular...
Error analysis of free probability approximations to the density of states of disordered systems
Chen, Jiahao; Hontz, Eric; Moix, Jeremy; Welborn, Matthew; Van Voorhis, Troy; Suárez, Alberto; Movassagh, Ramis; Edelman, Alan
#Condensed Matter - Disordered Systems and Neural Networks#Mathematics - Statistics Theory#Physics - Chemical Physics#Quantum Physics#46L54, 65C60, 97K70
Theoretical studies of localization, anomalous diffusion and ergodicity breaking require solving the electronic structure of disordered systems. We use free probability to approximate the ensemble- averaged density of states without exact diagonalization. We present an error analysis that quantifies the accuracy using a generalized moment expansion, allowing us to distinguish between different approximations. We identify an approximation that is accurate to the eighth moment across all noise strengths, and contrast this with the perturbation theory and isotropic entanglement theory.; Comment: 5 pages, 3 figures, submitted to Phys. Rev. Lett
Critical level statistics at the Anderson transition in four-dimensional disordered systems
Zharekeshev, I. Kh.; Kramer, B.
#Condensed Matter - Disordered Systems and Neural Networks#Condensed Matter - Mesoscale and Nanoscale Physics
The level spacing distribution is numerically calculated at the disorder-induced metal--insulator transition for dimensionality d=4 by applying the Lanczos diagonalisation. The critical level statistics are shown to deviate stronger from the result of the random matrix theory compared to those of d=3 and to become closer to the Poisson limit of uncorrelated spectra. Using the finite size scaling analysis for the probability distribution Q_n(E) of having n levels in a given energy interval E we find the critical disorder W_c = 34.5 \pm 0.5, the correlation length exponent \nu = 1.1 \pm 0.2 and the critical spectral compressibility k_c \approx 0.5.; Comment: 10 pages, LaTeX2e, 7 fig, invited talk at PILS (Percolation, Interaction, Localization: Simulations of Transport in Disordered Systems) Berlin, Germany 1998, to appear in Annalen der Physik
Concentration of Measure Inequalities in Information Theory, Communications and Coding (Second Edition)
During the last two decades, concentration inequalities have been the subject of exciting developments in various areas, including convex geometry, functional analysis, statistical physics, high-dimensional statistics, pure and applied probability theory, information theory, theoretical computer science, and learning theory. This monograph focuses on some of the key modern mathematical tools that are used for the derivation of concentration inequalities, on their links to information theory, and on their various applications to communications and coding. In addition to being a survey, this monograph also includes various new recent results derived by the authors. The first part of the monograph introduces classical concentration inequalities for martingales, as well as some recent refinements and extensions. The power and versatility of the martingale approach is exemplified in the context of codes defined on graphs and iterative decoding algorithms, as well as codes for wireless communication. The second part of the monograph introduces the entropy method, an information-theoretic technique for deriving concentration inequalities. The basic ingredients of the entropy method are discussed first in the context of logarithmic Sobolev inequalities...
Orthogonal polynomial ensembles in probability theory
Koenig, Wolfgang
#Mathematics - Probability#15A52, 33C45, 60-02, 60C05, 60F05, 60K35, 82C22, 82C41 (Primary) 05E10, 15A90, 42C05 (Secondary)
We survey a number of models from physics, statistical mechanics, probability theory and combinatorics, which are each described in terms of an orthogonal polynomial ensemble. The most prominent example is apparently the Hermite ensemble, the eigenvalue distribution of the Gaussian Unitary Ensemble (GUE), and other well-known ensembles known in random matrix theory like the Laguerre ensemble for the spectrum of Wishart matrices. In recent years, a number of further interesting models were found to lead to orthogonal polynomial ensembles, among which the corner growth model, directed last passage percolation, the PNG droplet, non-colliding random processes, the length of the longest increasing subsequence of a random permutation, and others. Much attention has been paid to universal classes of asymptotic behaviors of these models in the limit of large particle numbers, in particular the spacings between the particles and the fluctuation behavior of the largest particle. Computer simulations suggest that the connections go even farther and also comprise the zeros of the Riemann zeta function. The existing proofs require a substantial technical machinery and heavy tools from various parts of mathematics, in particular complex analysis...
Moment-Based Spectral Analysis of Random Graphs with Given Expected Degrees
Preciado, Victor M.; Rahimian, M. Amin
#Mathematics - Statistics Theory#Computer Science - Social and Information Networks#Mathematics - Probability#Physics - Physics and Society#Statistics - Applications#05C80, 60B20
In this paper, we analyze the limiting spectral distribution of the adjacency matrix of a random graph ensemble, proposed by Chung and Lu, in which a given expected degree sequence $\bar{w}_n^{^{T}} = (w^{(n)}_1,\ldots,w^{(n)}_n)$ is prescribed on the ensemble. Let $\mathbf{a}_{i,j} =1$ if there is an edge between the nodes $\{i,j\}$ and zero otherwise, and consider the normalized random adjacency matrix of the graph ensemble: $\mathbf{A}_n$ $=$ $ [\mathbf{a}_{i,j}/\sqrt{n}]_{i,j=1}^{n}$. The empirical spectral distribution of $\mathbf{A}_n$ denoted by $\mathbf{F}_n(\mathord{\cdot})$ is the empirical measure putting a mass $1/n$ at each of the $n$ real eigenvalues of the symmetric matrix $\mathbf{A}_n$. Under some technical conditions on the expected degrees sequence, we show that with probability one, $\mathbf{F}_n(\mathord{\cdot})$ converges weakly to a deterministic distribution $F(\mathord{\cdot})$. Furthermore, we fully characterize this distribution by providing explicit expressions for the moments of $F(\mathord{\cdot})$
A versatile integral in physics and astronomy
Mathai, A. M.; Haubold, H. J.
#Astrophysics - Instrumentation and Methods for Astrophysics#Astrophysics - Solar and Stellar Astrophysics#Mathematical Physics
This paper deals with a general class of integrals, the particular cases of which are connected to outstanding problems in astronomy and physics. Reaction rate probability integrals in the theory of nuclear reaction rates, Kr\"atzel integrals in applied analysis, inverse Gaussian distribution, generalized type-1, type-2 and gamma families of distributions in statistical distribution theory, Tsallis statistics and Beck-Cohen superstatistics in statistical mechanics and the general pathway model are all shown to be connected to the integral under consideration. Representations of the integral in terms of generalized special functions such as Meijer's G-function and Fox's H-function are also pointed out.; Comment: 11 pages, LaTeX
Spectral gaps in Wasserstein distances and the 2D stochastic Navier--Stokes equations
Hairer, Martin; Mattingly, Jonathan C.
#Mathematics - Probability#Mathematical Physics#Mathematics - Analysis of PDEs#Mathematics - Dynamical Systems#Mathematics - Spectral Theory#37A30, 37A25, 60H15 (Primary)
We develop a general method to prove the existence of spectral gaps for Markov semigroups on Banach spaces. Unlike most previous work, the type of norm we consider for this analysis is neither a weighted supremum norm nor an ${\L}^p$-type norm, but involves the derivative of the observable as well and hence can be seen as a type of 1-Wasserstein distance. This turns out to be a suitable approach for infinite-dimensional spaces where the usual Harris or Doeblin conditions, which are geared toward total variation convergence, often fail to hold. In the first part of this paper, we consider semigroups that have uniform behavior which one can view as the analog of Doeblin's condition. We then proceed to study situations where the behavior is not so uniform, but the system has a suitable Lyapunov structure, leading to a type of Harris condition. We finally show that the latter condition is satisfied by the two-dimensional stochastic Navier--Stokes equations, even in situations where the forcing is extremely degenerate. Using the convergence result, we show that the stochastic Navier--Stokes equations' invariant measures depend continuously on the viscosity and the structure of the forcing.; Comment: Published in at http://dx.doi.org/10.1214/08-AOP392 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org) | CommonCrawl |
Feasibility of Neutrino-based Energy Beings
I have seen in most sci-fi books and flicks about the existence of a special type of alien who is not even made up of standard biological composition, but more of a living mass of energy and information. They are mostly called "Energy Beings", and general properties consist of being intangible, incorporeal and do not have a fixed form. In more fantasy-based terms, they are more or less free spirits or "ghosts".
The general conclusion that I have seen is that such energy beings are made out of bound states of neutrinos. This theory was especially apparent for Stephen Baxter's Manifold, which the Downstreamers are living consciousness made out of neutrinos. It would make sense, really, since neutrinos can phase through light years worth of lead, and can carry information. It is also possible to have bound states of neutrinos through either enhanced weak force interactions or ultra cold temperatures.
However, one thing I am still stumped with is how these beings can have true consciousness, and how they can touch material objects. These are the two questions I need to solve to finally prove its feasibility. Also, if this fails, feel free to have your own theory on how these energy beings are formed.
science-based energy consciousness information energy-beings
CYCLOPSCORECYCLOPSCORE
This isn't going to work, for the simple reason that neutrinos can't form bound states.
The weak force can be represented by a Yukawa potential, of the form $$V(r)=-\frac{\alpha}{r}e^{-m_Wr}$$ with $m_W$ the mass of a W boson and $\alpha$ some constant. Given two particles of mass $M$, the criterion for a bound state is that $$\frac{\alpha M}{2m_W}\gtrsim0.8$$ which in turn requires that $m_W\sim M$ - that is, the bound particles must be heavier than the mass of a W boson. Unfortunately, a W boson has a mass of $\sim80\;\text{GeV}$, whereas neutrinos have masses only fractions of an electron volt.
The only way for a bound state to form would be to drastically decrease the mass of a W boson, making the weak force have a significant range compared to its current strength.
One possible way out - and the answer I linked to on Physics does mention this - is to have your particles be WIMPs, weakly interacting massive particles, rather than neutrinos. In our universe, $\sim100\;\text{GeV}$ is a not-too-unreasonable mass for a WIMP, and as we're talking about your universe, bound not by our observations but merely the requirement of self-consistency, it's easy for you to just postulate the existence of WIMPs with the proper mass for a bound state and have your beings be made out of them.
HDE 226868♦HDE 226868
$\begingroup$ I think this answer is best suited for physics.se ;D $\endgroup$
$\begingroup$ Well, I already mentioned that issue. Talked about "enhanced weak force interactions", but oh boy, did not expect it that the W boson needs to be smaller rather than bigger for this. $\endgroup$
– CYCLOPSCORE
$\begingroup$ @CYCLOPSCORE Ah, I thought you meant something else by "enhanced" - glad we were on the same wavelength, so to speak. I guess this answer is a stronger viewpoint on that - I'd argue that those enhancements are likely much more difficult than you'd expect. $\endgroup$
– HDE 226868 ♦
Baxters novels are nice but he always takes this incredible long term view which is just depressing - in the incredible long term everything becomes cold and dark. He invents quark beings which had their time when the universe was a lot hotter; and whose seeds germinate when you light up certain types of engines which operate with these temperatures of old.
He also invents neutrino birds and lots of other beings. Take them for what they are, don't take them too serious. They're like Q in Star Trek: unexplained storyline drivers.
So this is my answer - They are not explainable, but they do their part in the context of the story.
The rest below is just nice to read.
There are some more nice energy beings in other stories which have equally unfeasible physics but which drive their respective storyline equally well.
other energy beings
Commonwealth Series, Peter F. Hamilton
Hamilton let's his humanity discover a way to store information in the sub-quantum-level of physics. They then shrink in size and energy consumption to less than a atom size; while their intelligence and abilities (in the virtual world) grows immensely.
In order to influence the real world, however, they have to clone or manufacture a real body and download a human-sized conciousness into it; which takes away the superintelligence but gives the ability to deal with the real world.
Hamilton mainly uses this storyline device to put some "canned characters" back on the plate hundreds or thousands of years after their real life time, just because he and probably his readers love those characters.
Ender's Game, Orson Scott Card Spoiler alert
In the first book (and film) Ender fights a race of beings who live in their interstellar mind connections more than in the real world. In the later books of the series, (spoiler alert) humanity discovers that it is possible to have an entire conciousness living only inside those connections; and that they have their own one which is made up of computer networks (thought) and interstellar connections (soul). There are even more strange beings, but I don't want to go spoil the entire book series.
Orson Scott mainly uses all of this to explore the different possible styles of life, possibilities of "what's that, a soul" and to set some arrogant humans back to their real place in the world. First contact novel, written large.
The "Energy beings" are mostly generated by dreams of psychic active species, live in a separate plane of being or dimension or whatever. They either come through the head of a psyker or they scratch through the walls of a ship during interstellar flight - because faster than light flight is literally through the dimension of nightmares in that background story. Being a wargame, the only thing they do is attack someone or generate unrest so that other species attack each other.
Again, these are mainly style drivers and storyline drivers, and they're great at that.
AnderasAnderas
$\begingroup$ Actually, Commonwealth sounds like the most sound out of the four of them, for some reason. $\endgroup$
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The use of fermented buckwheat to produce l-carnitine enriched oyster mushroom
Tae-kyung Lee1,
Thi Thanh Hanh Nguyen2,
Namhyeon Park1,
So-Hyung Kwak1,
Jeesoo Kim1,
Shina Jin1,
Gyu-Min Son1,
Jaewon Hur1,
Jong-In Choi3 &
Doman Kim ORCID: orcid.org/0000-0003-0389-34411,2
l-Carnitine is an essential compound that shuttles long chain fatty acids into mitochondria. The objective of this study was to produce l-carnitine enriched oyster mushroom (Pleurotus ostreatus) using common buckwheat fermented by Rhizopus oligosporus. Mushroom grown on common buckwheat medium contained 9.9–23.9% higher l-carnitine (186.3 mg/kg) than those grown on basal medium without any buckwheat addition. Those grown on fermented common buckwheat medium contained the highest l-carnitine content (201.2 mg/kg). Size index and lightness of mushroom pileus (L*) were also the highest (100.7 and 50.6, respectively) for those grown in medium added with fermented common buckwheat (20%, w/w). Antioxidant activities of both mushroom extracts (1.5 mg/mL) showed the same level as 38.7% for mushroom grown in media added with common buckwheat or fermented common buckwheat. At the treatment concentration of 300 μg/mL, viabilities of murine macrophage cell line Raw 264.7 cells treated with ethanol extract of oyster mushroom grown on buckwheat medium ranged from 58.9 to 67.8%. The oyster mushroom grown on buckwheat and fermented buckwheat medium can be used as one of the substitutes for meat based diets.
Mushroom has been used as traditional foods and medicine in eastern Asia due to its functional properties. World production of mushroom has been grown rapidly since late 20th century. China accounted for over 70% of world production of mushroom in 2016 (FAOSTAT 2016). Pleurotus ostreatus, the second most cultivated mushroom in the world, is commonly known as "Oyster mushroom" and "Hiratake" (Sánchez 2010). Oyster mushroom has various biological functions, including antimicrobial activity against Escherichia coli, and Staphylococcus aureus (Akyuz et al. 2010), antineoplastic activity against Ehrlich ascetic tumor (Wolff et al. 2008), antioxidant activity (Venkatakrishnan et al. 2009), antitumor activity of P. ostreatus mycelia-derived proteoglycans (Sarangi et al. 2006), and immunomodulatory activity of pleuran, an insoluble polysaccharide extracted from P. ostreatus (Jesenak et al. 2013). Yields and chemical composition of mushroom are enhanced by adding essential elements such as selenium (da Silva et al. 2012; Kristensen et al. 2012; Vieira et al. 2013).
Among buckwheat species, common buckwheat (Fagopyrum esculentum) and Tartary buckwheat (Fagopyrum tataricum) are cultivated for human food. Buckwheat (Fagopyrum spp.) is a good source of nutritionally valuable amino acids, dietary fibers, and minerals such as zinc and copper (Bonafaccia et al. 2003; Zhang et al. 2012). In addition, buckwheat has been found to contain flavonoids, fagopyrin, tocopherols and phenolic substances such as 3-flavanols, rutin, phenolic acids, and their derivatives with antioxidant activity (Fabjan et al. 2003; Jiang et al. 2007). Also, buckwheat has higher content of amino acids such as methionine and lysine (precursors of l-carnitine) than rice and other pseudo cereals (Bonafaccia et al. 2003; Mota et al. 2016). l-Carnitine (β-hydroxy-γ-N-trimethylaminobutyric acid) is a non-essential amino acid derivative and natural compound occurring most in red meat (Demarquoy et al. 2004). Its major role is a carrier of long chain fatty acid into mitochondria for beta-oxidation. l-carnitine is considered as a weight-loss product because of its function related to fat metabolism. Clinical studies have shown that regular l-carnitine intake can lead to weight loss in human (Novakova et al. 2016). Although l-carnitine is synthesized from essential amino acids, lysine and methionine in human, 75% of l-carnitine is exogenously obtained from food, especially meat and milk (Steiber et al. 2004). However, there have been many controversies regarding meat based diets for human health issue (Chao et al. 2005; Pan et al. 2011; Wang and Beydoun 2009), since most l-carnitine is supplied from meat based diet, vegetarians have to eat more plants to have enough intake of l-carnitine (Cave et al. 2008; De Vivo and Tein 1990). Buckwheat without additional nutrients has been fermented using Rhizopus oligosporus (R. oligosporus), producing four times higher amount of l-carnitine than original buckwheat (Park et al. 2017). The fermented buckwheat extract powder with increased amounts of l-carnitine was used as a complex additive in poultry feed to Hy-Line brown hens and resulted higher egg production and quality than the control group, and increased the l-carnitine content in the yolk (Park et al. 2017). Therefore, in this study, we focus on the utilization of buckwheat and fermented buckwheat as medium materials to improve the biological activity of oyster mushroom. In addition, the morphological characteristics, l-carnitine content, antioxidant properties, and cell cytotoxicity of oyster mushroom were investigated.
Microorganisms and culture condition
Rhizopus microspores var. oligosporus (R. oligosporus) was obtained from our previous study (Park et al. 2017) and deposited as KCCM 11948P (Korean Culture Center of Microorganisms, Seoul, Korea). R. oligosporus was maintained on potato dextrose agar (PDA) (Difco, USA) and incubated at 28 °C until spore formation (Park et al. 2017). P. ostreatus was obtained from Mushroom Research Institute (Gwangju, Gyeonggi, Korea) (Choi et al. 2013).
Preparation of fermented buckwheat
Tartary buckwheat and common buckwheat were purchased from Bongpyeong (Pyeongchang bongpyeong memil, Gangwon-do, Korea). Fermented buckwheat was prepared with a modified method described previously (Park et al. 2017). Briefly, unhulled buckwheat (750 g) was soaked in water (1.5 L) for 6 h in a metal tray (height × width × length = 5.5 cm × 15.5 cm × 28.5 cm) and sterilized at 121 °C for 25 min. Then 7.5 mL of R. oligosporus spore solution (1 × 106 spores/mL) was inoculated into each sterilized buckwheat tray after cooling to room temperature followed by incubation at 30 °C until mycelia covered the surface of tray (76 h) with relative humidity maintained above 90%. Fermented buckwheat was lyophilized at − 10 to 0 °C under 10 Pa for 4 days (Eyela, Tokyo, Japan). It was then milled with a blender (Hanil, Seoul, Korea) and stored at − 20 °C blocking light to preserve sensitive compounds such as quercetin and rutin for further analysis.
Analyses of fermented buckwheat
Total phenolic content (TPC) of buckwheat was measured with Folin-Ciocalteu method (Singleton et al. 1999). Buckwheat and fermented buckwheat were suspended in 70% ethanol at 50 mg/mL and the supernatant was separated by centrifugation at 13,500×g for 10 min. The supernatant, diluted fivefold with water (120 µL) was then mixed with 15 µL of Folin-Ciocalteu reagent for 3 min on a microplate shaker (300 rpm, Thermo Fisher Scientific, Waltham, MA, USA). Then 15 µL of 10% (w/v) Na2CO3 was added to the mixture followed by shaking for 30 min. TPC was determined at wavelength of 760 nm with a microplate reader (Molecular Devices, Sunnyvale, CA, USA) using gallic acid as standard (10 to 100 µg/mL). Total flavonoids content (TFC) of the extract was measured using aluminum chloride method (Chang et al. 2002). The supernatant, diluted fivefold with methanol (2 mL) was then mixed with 10% (w/v) AlCl3 (100 µL) dissolved in water and 0.1 mM Potassium acetate (100 µL). TFC was then determined at wavelength of 415 nm using quercetin as the standard (10 to 100 µg/mL).
Mushroom spawning and fruiting in field scale
Cultivation of P. ostreatus was performed using previously reported procedure (Lee et al. 1999) with slight modification. Briefly, P. ostreatus was pre-cultured on substrate mixture composed of 80% Douglas-fir sawdust and 20% rice bran packaged in heat-resistant bottle (1100 mL, φ75 mm) at 20 °C for 30 days. After mycelium was totally grown, 4 g of hyphae attached substrate was transferred into each buckwheat medium. The buckwheat medium contained milled unhulled buckwheat seeds in basal medium containing 66.7% (w/w) poplar saw dust, 16.7% (w/w) cotton-seed meal, and 16.7% (w/w) beet pulp. Moisture content was adjusted to 65% (w/w) before sterilization. Each milled buckwheat was mixed with basal medium at 20% (w/w) of nutritious substrate. The following media were prepared: basal (G) medium, common buckwheat (CB) medium, fermented common buckwheat (FCB) medium, tartary buckwheat (TB) medium, and fermented Tartary buckwheat (FTB) medium (Table 1). All media (620 g) were packed in heat-resistant bottle and sterilized serially at 100 °C for 30 min and 121 °C for 90 min. After sterilization, each medium was cooled down to room temperature. Pre-cultured P. ostreatus (4 g of wet weight) was then inoculated to each medium and incubated at 20 °C under 65% relative humidity in a dark room. Carbon dioxide was controlled to be 3000 to 5000 ppm to induce balanced shape of pileus and stipe (Sánchez 2010). After 30 days of incubation with mycelium to induce fruit body, old spawn was removed and temperature was maintained at 15 °C. Relative humidity and carbon dioxide were maintained over 90% and 500–3000 ppm, respectively. Each bottle possessed 0.056 m3 of space in the incubation room. On the 8th day after inducing fruit body, fruit bodies were harvested and 10 fruit bodies of each group were randomly selected. These fruit bodies were then lyophilized and stored at − 80 °C for further analysis.
Table 1 Composition and ratio of mushroom medium
Mushroom morphological characteristics
Morphological characteristics such as fruit body weight, pileus diameter, and stipe length were determined. Halogen lamb analyzer MB35 (Ohaus Inc., Parsippany, NJ, USA) was used to measure moisture content of fruit body. Color change of mushroom pileus was measured with Hunter's color value (L* (ligh vs dark), a* (red vs green), b* (yellow vs blue)) with a colorimeter (Konica Minolta, Tokyo, Japan). Total size index (TSI) representing comprehensive morphological size was each morphological value. Each value was obtained by multiplication of weight (g), pileus diameter (cm), stipe length (cm), and stipe thickness (cm). Each obtained value was then rescaled (divided by 100) and its unit was omitted using the following equation:
$${\text{Total Size Index }} = \frac{(Mushroom \; weight \times Pileus \; diameter \times Stipe \; length \times Stipe \; thickness)}{100}$$
Liquid chromatography analyses of mushroom components
l-carnitine content in buckwheat and mushroom was analyzed with liquid chromatography–electrospray ionization–tandem mass spectrometry (LC–ESI–MS) (Park et al. 2017). Each sample (100 mg) was extracted with water (1 mL) and centrifuged at 13,500×g for 10 min. The supernatant was diluted tenfold with acetonitrile. It was centrifuged again at 13,500×g for 10 min. The supernatant was filtered using 0.2 µm pore size syringe filter (Sartorius, Germany). For quercetin and rutin analysis, each sample (100 mg) was extracted with 70% ethanol (1 mL) followed by centrifugation and filtration as described above.
The filtrate (1 µL) was used for component analysis as follows. l-carnitine analysis was performed on an Acquity UPLC system equipped with ESI–MS and BEH 1.7 μm HILIC column (2.1 mm × 150 mm, Waters, USA). Mobile phase A was 15 mM ammonium formate with 0.1% (v/v) formic acid. Mobile phase B was acetonitrile with 0.1% (v/v) formic acid. Flow rate was set at 0.4 mL/min. Sample manager temperature was sustained at 20 °C while column temperature was maintained at 40 °C. Mobile phase A was sustained at 10% for initial 3 min, 30% for the next 2 min, 60% for 1 min, and 10% for the last 4 min. Each compound was recorded and quantified at specific ion mass [M + H+]. ESI–MS conditions were: ion mode, positive; capillary voltage, + 1.5 kV; cone voltage, − 10 V; and single ion recording, 162 g/mol. Quercetin and rutin were separated using Kromasil 1.8 μm C18 UHPLC column (2.1 mm × 50 mm, Kromasil, Bohus, Sweden). Flow rate was set at 0.3 mL/min. Mobile phase A was 0.1% (v/v) formic acid while mobile phase B was acetonitrile with 0.1% (v/v) formic acid. Temperature was set the same as described above. Mobile phase B was gradually increased from 30 to 100% in the initial 5 min. It was then decreased back to 30% in the last 3 min. ESI–MS conditions were: capillary voltage, + 1.5 kV; cone voltage, + 25 V for rutin and − 10 V for quercetin; single ion recording, 609.5 g/mol for quercetin and 303.2 g/mol for rutin. All the samples, quercetin and rutin standards kept in amber glass vial, which blocked penetration of light.
The calibration curve was prepared by the external standard method, with l-carnitine standard ranging from 0.0125 to 0.5 μg/mL and quercetin ranging from 0.05 to 2 μg/mL and rutin ranging 0.1 to 5 μg/mL. Each standard curves showed over 0.99 coefficient of determination (R2 > 0.99).
Preparation of mushroom fruit body extract
Powdered mushroom fruit body (1 g) of each treatment (G, CB, FCB, TB, and FTB medium) was extracted with ethanol (10 mL) at 20 °C (200 rpm) for 24 h. The mixture was then centrifuged at 3500×g for 10 min. The supernatant was filtered using Whatman paper filter No. 1 (Whatman, Piscataway, NJ, USA). The filtrate was evaporated at 45 °C for 1 h and lyophilized using a freeze dryer (Eyela, Tokyo, Japan). The yield of extracted mushroom by ethanol was 17.2% (w/w). The extracted mushroom powder by ethanol was then re-dissolved in dimethyl sulfoxide (DMSO) solution (10 mg/mL) and used for in vitro cell cytotoxicity and antioxidant assays.
Antioxidant activity of mushroom fruit body extract
Antioxidant activity of mushroom ethanol extract was evaluated using 2,2-diphenyl-1-picrylhydrazyl (DPPH) radical scavenging method described previously (Nguyen et al. 2017) with slight modification. Mushroom powder was extracted by ethanol, then ethanol was removed by evaporation and lyophilized at − 10 to 0 °C under 10 Pa (Eyela FD-550, Tokyo Rikakikai Co., Tokyo, Japan). The extracted mushroom powder was dissolved in DMSO and mixed with 100 µM of DPPH reagent dissolved in ethanol. The final concentration of mushroom extract ranged from 0.01 to 1 mg/mL. The mixture was incubated at room temperature for 30 min. The absorbance of each mixture was obtained at wavelength of 517 nm on a microplate. DMSO solution was used as negative control. Trolox was used as a positive control.
The relative radical scavenging activity (SC) was obtained with the following equation (Choi et al. 2018a, b):
$${\text{SC }}\left( {\text{\% }} \right) = \frac{{\left( {Abs \; of \; negative \; control - Abs \; of \; the \; sample} \right)}}{Abs \; of \; negative \; control} \times 100$$
Results were expressed as mean ± standard error of the mean (SEM). All analyses were carried out in triplicates.
Cell viability assay of oyster mushroom ethanol extract
Murine macrophage cell line Raw 264.7 cells (Raw 264.7 cells) were grown in Dulbecco's modified Eagle's medium supplemented with 10% fetal bovine serum (FBS, GenDEPOT, Barker TX, USA), 100 unit/mL penicillin (GenDEPOT, Barker TX, USA), and 100 μg/mL streptomycin (GenDEPOT, Barker TX, USA) at 37 °C under 5% CO2 (Choi et al. 2018a, b; Maxwell et al. 2017). Extracted oyster mushroom powder was prepared in DMSO solution (10 mg/mL). RAW264.7 cells were seeded into 96-well cell culture plate at density of 2 × 104 cells/well and incubated at 37 °C for 24 h in a humidified atmosphere containing 5% CO2. After discarding the culture medium, sample was then diluted with medium (1.17 to 1.2 mg/mL) and added to each well followed by incubation at 37 °C for 24 h. To evaluate cytotoxicity, WST-1 (Water soluble tetrazolium salt) assay was performed using EZ-cytox kit (Daeil Lab service, Seoul, Korea).
All data were obtained after repeating the experiment three times except for morphological analysis. Morphological characteristics were obtained from 10 randomly selected samples. Mean value was given with standard error of the mean (SEM). The significant differences between groups were determined by Tukey's HSD (Honest significant difference) methods (P < 0.05 or P < 0.01). Statistical analysis was performed using SPSS version 23.0 for Windows (SPSS Inc., Chicago, IL, USA).
Levels of l-carnitine and phenolic compounds in fermented buckwheat
One kg of CB and TB contained 11.3 mg and 6.2 mg of l-carnitine, respectively (Table 2). After R. oligosporus fermentation, the amounts of l-carnitine in both buckwheats were increased (from 11.3 mg to 26.2 mg in 1 kg of FCB and from 6.2 mg to 38.4 mg in 1 kg of FTB). The increasing rate of l-carnitine was higher after TB fermentation (619.4%) than that after CB fermentation (231.8%). Contents of total phenol and flavonoids in fermented buckwheat were higher than those of non-fermented buckwheat. However, content of quercetin decreased after fermentation in both CB and TB. On the contrary, rutin content in TB increased from 3923.2 to 5148.1 mg/kg after fermentation.
Table 2 l-carnitine, total phenolic content, quercetin and rutin concentration in buckwheat after R. oligosporus fermentation
Levels of l-carnitine in oyster mushroom in all samples
Amounts of l-carnitine in 1 kg of dried oyster mushroom grown on different media (CB, FCB, TB, and FTB medium) were compared. Results are shown in Table 3. When oyster mushroom was grown on FCB medium, the amount of l-carnitine was significantly (P < 0.01) increased (by 22.3%) compared to that grown on G medium. However, oyster mushroom grown on FTB medium had smaller increase in the amount of l-carnitine (by 12.9%) compared to that grown on G medium.
Table 3 L-carnitine content in oyster mushroom grown on various buckwheat media
Morphological characteristics of mushroom grown on buckwheat media
Mushroom weight and moisture content were similar to each other for all samples. They were not significantly (P > 0.01) different compared to those of mushroom grown on G medium (Table 4). Mushroom size was represented by total size index (TSI). TSI of oyster mushroom grown on FCB medium was significantly (P < 0.01) increased compared to that grown on G medium. However, other size parameters such as pileus diameter, stipe length, and stipe thickness were not significantly changed except pileus diameter in FCB medium and stipe length in CB medium.
Table 4 Morphological characteristics of oyster mushroom
Color index of mushroom grown on buckwheat containing medium
By adding buckwheat into the media, Lightness (L*) values of all mushroom samples were significantly increased (P < 0.01) compared to those grown on G medium (Table 4). Yellowness (b*) values were also increased for all mushrooms grown on media added with buckwheat. However, yellowness values of mushrooms grown on TB medium were not significantly (P > 0.01) different from those of mushrooms grown on G medium.
Antioxidant activities of ethanol extracts of mushrooms
Antioxidant effects of mushroom ethanol extracts against DPPH were evaluated at concentration ranging from 0.2 to 1.5 mg/mL. Results are shown in Fig. 1. Radical scavenging activities of all samples were increased in a concentration-dependent manner. At final concentration of 1.5 mg/mL, radical scavenging activity of oyster mushroom ethanol extract ranged from 25.9 to 38.7%. At this concentration, ethanol extract of mushroom grown on G medium was found to be 25.9%. Ethanol extracts of mushrooms grown on CB and FCB medium showed the same scavenging activity (both at 38.7%). Those of mushrooms grown on TB and FTB medium showed scavenging activities of 28.7 and 30.9%, respectively. Trolox used as positive control (Additional file 1: Figure S1).
DPPH radical scavenging activities of ethanol extract of oyster mushroom fruit body. G basal medium, CB common buckwheat medium, FCB fermented common buckwheat medium, TB tartary buckwheat medium, FTB Fermented Tartary buckwheat medium. Each mean value was written with standard error of the mean (SEM)
Cell cytotoxicity ethanol extract of oyster mushroom against Raw 264.7
Cytotoxicity of ethanol extract of mushroom fruit body to Raw 264.7 cells was evaluated at a final concentration ranging from 18.75 to 1200 μg/mL (Fig. 2). At a final concentration of 0 to 75 μg/mL, viability of Raw 264.7 cells was above 90% after treatment with all ethanol extract samples. Up to 150 μg/mL, the viabilities of Raw 264.7 cells treated with ethanol extracts of all mushroom samples were not significantly (P > 0.05) different from those of control cells without ethanol extract treatment. At final concentration of 300 μg/mL, viabilities of cells treated with ethanol extract of oyster mushroom grown on buckwheat medium ranged from 58.9 to 67.8%.
Cell cytotoxicity of ethanol extracted oyster mushroom against Raw 264.7 cells. Ethanol extracted oyster mushroom grown up on G basal medium, CB common buckwheat medium, FCB fermented common buckwheat medium, TB tartary buckwheat medium, FTB fermented tartary buckwheat medium against Raw 264.7 cells
In this study, the amounts of l-carnitine in FCB and FTB by R. oligosporus were increased 2.3 times and 6.2 times compared to CB and TB (Table 3). l-carnitine is synthesized from lysine and methionine (Bremer 1983), thus the synthesis of l-carnitine depends on the amount of lysine and methionine in buckwheat. The protein, lysine, methionine contents (25.3% (w/w), 58.8 g/kg, and 13.3 g/kg) in TB bran were higher than that in CB bran (21.6% (w/w), 54.7 g/kg, and 1.09 g/kg) (Bonafaccia et al. 2003). The fermentation process of buckwheat by R. oligosporus can increase the amino acid content in buckwheat, thus the l-carnitine contents in FTB are higher than that of FCB. The total phenolic contents in FCB and FTB were increased 2.4 times and 1.4 times compared to CB and TB. Among phenolic compounds in buckwheat, rutin and quercetin were considered as major bioactive compounds in buckwheat. There is a wide variation of rutin content in buckwheat seed depending on the species, variety, and the environmental conditions under which they are produced (Jiang et al. 2007). Therefore, rutin and quercetin were selected in buckwheat and fermented buckwheat. Although quercetin contents in fermented buckwheat were decreased compared to non-fermented buckwheat, rutin contents in FTB was increased 1.3 times compared to TB. McCue and Shetty (2003) reported that α-amylase and endogenous carbohydrate-cleaving enzymes produced from R. oligosporus can generate polyphenols from carbohydrate-conjugated phenolic compounds during fermentation of buckwheat. In addition, R. oligosporus is a known strain to produce β-glucosidase, β-glucuronidase and xylanase when it degrades the cell wall matrix (Huynh et al. 2014; Varzakas 1998). Thus, it probably metabolizes extracellular components with bioconversion of phenolic compounds by the fermentation that leads the cell-wall degrading enzymes to hydrolyze glycosidic bonds and produces unbound phenolics and aglycone forms. The fermentation processes releasing phenolic compounds from plant matrixes followed by the metabolic pathways of flavonoids: glycosylation, deglycosylation, ring cleavage, methylation, glucuronidation, and sulfate conjunction which are the ways of producing new bioactive compounds as well as increasing the total phenol contents and rutin contents in fermented buckwheat (Huynh et al. 2014).
Oyster mushroom grown on buckwheat and fermented buckwheat had higher l-carnitine contents than mushroom grown on normal medium (Table 3) and the l-carnitine contents in mushroom grown on FCB was similar to the amount of l-carnitine in pork muscle (Demarquoy et al. 2004). l-carnitine content in oyster mushroom grown on fermented buckwheat medium was higher than mushroom grown on non-fermented buckwheat medium. It could be explained on the grounds that R. oligosporus can synthesize various l-carnitine derivatives that might be supplied to mycelia of oyster mushroom. Further study is needed to obtain clearer explanation. One of the reasons for the higher antioxidant activities of ethanol extracts from mushrooms grown on CB, TB, FCB and FTB medium compared to those of mushroom grown on G medium (Fig. 1) is the presence of l-carnitine which is known to possess antioxidant activity (Holasova et al. 2002).
The Raw 264.7 cell viability in the presence of ethanol extract from oyster mushroom grown on buckwheat medium was higher than that of oyster mushroom grown on G medium. It has been reported that l-carnitine can reduce oxidative stress in in vitro cell culture of Raw 264.7 cells and HK-2 cells (Koc et al. 2011; Ye et al. 2010). l-carnitine is also known to have anti-inflammatory activity because it can suppress inducible nitric oxide synthase (iNOS) that produces nitric oxide at transcriptional level (Koc et al. 2011). Therefore, the higher in Raw 264.7 cell viability might be due to its higher content of l-carnitine. However, the pattern of l-carnitine concentration in mushroom was not exactly the same as that of mushroom extracts cytotoxicity against Raw 264.7 cells. Therefore, l-carnitine might not be the only factor involved in the less cytotoxicity of ethanol extract from oyster mushroom grown on buckwheat media. Further studies are needed to understand the underlying mechanisms.
P. ostreatus :
R. oligosporus :
Rhizopus microspores var. oligosporus
TPC:
total phenol content
TFC:
total flavonoids content
FCB:
fermented common buckwheat
TB:
tartary buckwheat
FTB:
fermented tartary buckwheat
TSI:
total size index
LC–ESI–MS:
liquid chromatography–electrospray ionization–tandem mass spectrometry
DMSO:
DPPH:
scavenging activity
iNOS:
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TL performed fermentation and drafted manuscript. TN carried out cell tests and preparation of revised manuscript. NP participated in antioxidant assay and growth of fungi. SK and JK performed buckwheat fermentation, preparation of extracts and component analyses. SJ and KS conducted mushroom component analyses and morphological characterization. JH and JC performed mushroom spawning, fruiting of mushroom in field scale and preparation of mushroom fruit body extract. DK designed and coordinated the study. All authors read and approved the final manuscript.
All data generated or analyzed during this study are included in this manuscripts.
Not applicable. This paper does not contain any studies with human participants or animal performed by any of the authors.
This work was partially supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2015R1D1A1A01056929; D. Kim), by Korea Institute of Planning and Evaluation for Technology in Food, Agriculture, Forestry (IPET) through Agriculture, Food and Rural Affairs Research Center Support Program, funded by Ministry of Agriculture, Food and Rural Affairs (MAFRA) (710012-03-1-HD220), under the framework of International Cooperation Program managed by the NRF (2016K1A3A1A19945059), and OTTOGI Corporation through Research and Publication Project.
Graduate School of International Agricultural Technology and Center for Food and Bioconvergence, Seoul National University, Pyeongchang, 25354, South Korea
Tae-kyung Lee, Namhyeon Park, So-Hyung Kwak, Jeesoo Kim, Shina Jin, Gyu-Min Son, Jaewon Hur & Doman Kim
The Institute of Food Industrialization, Institutes of Green Bio Science &Technology, Seoul National University, Pyeongchang, 25354, South Korea
Thi Thanh Hanh Nguyen & Doman Kim
Mushroom Research Institute, GARES, Gwang-Ju, Gyeonggi, 12805, South Korea
Jong-In Choi
Tae-kyung Lee
Thi Thanh Hanh Nguyen
Namhyeon Park
So-Hyung Kwak
Jeesoo Kim
Shina Jin
Gyu-Min Son
Jaewon Hur
Doman Kim
Correspondence to Doman Kim.
DPPH radical scavenging activity of trolox.
Lee, Tk., Nguyen, T.T.H., Park, N. et al. The use of fermented buckwheat to produce l-carnitine enriched oyster mushroom. AMB Expr 8, 138 (2018). https://doi.org/10.1186/s13568-018-0664-6
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Physics Beyond the Standard Model
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218. The scalar sector of $SU(2)$ gauge theory with $N_F=2$ fundamental flavours
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We present a non perturbative study of SU(2) gauge theory with two fundamental Dirac flavours. This theory provides a minimal template which is ideal for a wide class of Standard Model extensions featuring novel strong dynamics. After reviewing our findings for the Goldstone bosons and spin-1 spectrum, we present our new results for the sigma and eta states. We evaluate the relevant...
200. Infrared properties of a prototype pNGB model for beyond-SM physics
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308. Spectrum of a prototype model with the Higgs as pNGB
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389. Near-conformal composite Higgs or PNGB with partial compositeness?
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Based on recent analysis of the LatHC collaboration I will review critical features of a minimal composite Higgs model close to the conformal window. Challenging problems of the near-conformal paradigm will be compared with PNGB based partial compositeness.
357. Spectrum and mass anomalous dimension of SU(2) gauge theories with fermions in the adjoint representation: from $N_f=1/2$ to $N_f=2$
Dr Georg Bergner (AEC ITP University of Bern)
In this work I will summarize our results concerning the spectrum and mass anomalous dimension of SU(2) gauge theories with a different number of fermions in the adjoint representation, where each Majorana fermion corresponds effectively to half a Dirac flavour $N_f$. The most relevant examples for the extensions of the standard model are supersymmetric Yang-Mills theory ($N_f=1/2$) and...
292. Large mass hierarchies from strongly-coupled dynamics
Dr Ed Bennett (Swansea University)
Motivated by tentative signals of new physics at the LHC, which seems to imply the presence of large mass hierarchies, we investigate the theoretical possibility that these could arise dynamically in new strongly-coupled gauge theories extending the standard model of particle physics. To this purpose, we study lattice data on non-Abelian gauge theories in the (near-)conformal...
197. Quark Chromoelectric Dipole Moment Contribution to the Neutron Electric Dipole Moment
Dr Tanmoy Bhattacharya (Los Alamos National Laboratory)
The quark chromo-electric dipole moment operator and the pseudo-scalar fermion bilinear with which it mixes under renormalization can both be included in a calculation of the electromagnetic form factor of the nucleon using the Schwinger source method. A preliminary calculation of these operators using clover quarks on HISQ lattices generated by MILC collaboration will be presented showing...
14. Effective action for pions and a dilatonic meson - foundations
Dr Yigal Shamir (Tel Aviv University)
Recent simulations suggest the existence of a very light singlet scalar in QCD-like theories that may be lying just outside the conformal window. Assuming that the lightness of this scalar can be explained by an approximate dilatation symmetry, we develop an effective field theory framework for both the pions and this light scalar, the "dilatonic meson." We argue that a power counting...
11. Effective action for pions and a dilatonic meson - results
Prof. Maarten Golterman (San Francisco State University)
We consider applications of a recently developed effective field theory for a dilatonic meson and pions. We contrast the leading-order behavior of masses with that in a theory with only pions, comment on next-to-leading order, and argue that the effective theory breaks down at the sill of the conformal window, as it should.
294. Asymptotically safe gauge-Yukawa theories and functional renormalisation group
Ms Tugba Buyukbese (University of Sussex)
Recently, new four-dimensional (gauge-Yukawa) theories have been discovered which display exact interacting fixed points at highest energies. In a regime where asymptotic freedom is lost, these novel types of theories develop an asymptotically safe UV fixed point, strictly controlled by perturbation theory. In this talk, we extend these studies to include couplings with non-vanishing...
60. Interacting ultraviolet completions of four-dimensional gauge theories
Mr Andrew Bond (University of Sussex)
We will discuss some of the recent developments in understanding ultraviolet completions of gauge-Yukawa theories beyond traditional asymptotic freedom.
142. Finite Size Scaling of the Higgs-Yukawa Model near the Gaussian Fixed Point
David Y.-J. Chu (Department of Electrophysics, National Chaio Tung University)
We study the scaling property of Higgs-Yukawa models. Using the technique of Finite-Size Scaling, we are able to derive formulae to describe the behaviour of the observables near the Gaussian fixed point. The renormalisation procedure is discussed in this talk. A feasibility study of our strategy is performed for pure scalar theory. In addition, we test the formulae with lattice data...
390. Running coupling of twelve flavors
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205. Discrete $\beta$-function of the SU(3) gauge theory with 10 massless domain-wall fermions
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192. Selected new results from the spectroscopy of the sextet BSM model
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186. Simulations of N=1 supersymmetric Yang-Mills theory with three colours
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189. Phenomenology of a composite Higgs model: lessons for the lattice.
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279. Adjoint SU(2) with four fermion interactions
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119. Sextet Model with Wilson Fermions
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377. Supergravity from Gauge Theory
Dr Evan Berkowitz (LLNL)
Gauge/gravity duality is the conjecture that string theories have dual descriptions as gauge theories. Weakly-coupled gravity is dual to strongly-coupled gauge theories, ideal for lattice calculations. I will show precision lattice calculations that confirm large-N continuum D0-brane quantum mechanics correctly reproduces the leading-order supergravity prediction for a black hole's internal... | CommonCrawl |
\begin{document}
\title{\textbf{Symmetric radial decreasing rearrangement can increase the fractional Gagliardo norm in domains}
}
\author{Dong Li \footnote{DL is partially supported by Hong Kong RGC grant GRF 16307317.}, \quad Ke Wang\footnote{KW is partially supported by HKUST Initiation Grant IGN16SC05.}}
\date{\today}
\maketitle
\begin{abstract} We show that the symmetric radial decreasing rearrangement can increase the fractional Gagliardo semi-norm in domains. \end{abstract}
\section{Introduction}
For any Borel set $A$ in $\mathbb R^n$ with $|A|<\infty$ ($|A|$ denotes the Lebesgue measure of $A$), define $A^{\ast}$, the symmetric rearrangement of $A$ as the open ball \begin{align*}
A^{\ast} = \{x:\, |x| < ( |A|/ \alpha_n)^{\frac 1n} \}, \end{align*} where $\alpha_n = \pi^{\frac n2}/\Gamma(\frac n2+1)$ is the volume of the unit ball.
If $|A|=0$, then $A^{\ast} = \varnothing$ and for later purposes we conveniently define $\chi_{\varnothing}\equiv 0$. Denote by $\mathscr U_0$ the space of Borel measurable functions $u:\, \mathbb R^n \to \mathbb R$ such that \begin{align} \notag
\mu_u(t)= |\{ x: \; |u(x)| >t \}| \ \ \mbox{is finite for all }t>0. \end{align} Observe that $\mu_u(\cdot)$ is right-continuous, non-increasing and (by the Lebesgue dominated convergence theorem) $\lim_{t\to \infty} \mu_u(t)=0$.
For any $u \in \mathscr U_0$, define the symmetric decreasing rearrangement $u^{\ast}$ as \begin{align*}
u^{\ast}(x) =\int_0^{\infty} \chi_{ \{|u|>t \}^{\ast} }(x) dt
=\sup\{t: |\{ |u|>t\}|>\alpha_n|x|^n\}. \end{align*} Since $\mu_u$ decays to zero as $t\to \infty$, we have $0\le u^{\ast}(x) <\infty$ for any $x \ne 0$,
whereas $u^{\ast}(0)$ may be $\infty$. Evidently, the function $u^{\ast}$ is radial, non-increasing in $|x|$, and satisfy \begin{align*}
\{ |u|>t \}^{\ast} = \{ u^{\ast}> t \}, \quad \forall\, t>0. \end{align*}
From this one can deduce $|\{|u|>t\}|= |\{ u^{\ast} >t\}|$, $\forall\, t>0$ and $\| u^{\ast} \|_p
= \| u \|_p$ for all $1\le p\le \infty$. Note that it follows from the level set characterisation that any uniform translation of $u$ does not change $u^{\ast}$, namely if for any $x_0\in \mathbb R^n$, we define $u_{x_0}(x)= u(x-x_0)$, then \begin{align}\label{eq:syminvariant} (u_{x_0})^{\ast} = u^{\ast}. \end{align} This simple property will be used without explicit mentioning later.
On the other hand, the effect of rearrangement on the gradient of the function is more complex and interesting. Let $u$ be a nonnegative smooth function that vanishes at infinity. The P\'olya-Szeg\"o \cite{PS} inequality states that for $1\le p<\infty$, \[
\int_{\R^n}|\nabla u|^p\ge \int_{\R^n}|\nabla u^*|^p. \] Brothers-Ziemer \cite{BZ} gave a characterization of the equality case under the assumption that the distribution function of $u$ is absolutely continuous.
This P\'olya-Szeg\"o inequality also holds for every bounded open set $\Omega\subset \R^n$. That is, for every nonnegative $u\in C^\infty_c(\Omega)$, we also have \[
\int_{\Omega}|\nabla u|^p\ge \int_{\Omega^*}|\nabla u^*|^p. \]
As a matter of fact, one can show that for every $u\in W_0^{1,p}(\Omega)$, one has $u^{\ast} \in W_0^{1,p}(\Omega^{\ast})$ and the above inequality holds.
We are interested in the effect of symmetric decreasing rearrangement for fractional Sobolev inequalities. For $0<\sigma<1$ and $1\le p<\infty$, we define the space $\mathring W^{\sigma,p}(\Omega)$ as the completion of $C^\infty_c(\Omega)$ under the norm \[
\|u\|_{\mathring W^{\sigma,p}(\om)}=\biggl({\iint_{\om\times \om} \frac{|u(x)-u(y)|^p}{|x-y|^{n+\sigma p}}\,\ud x\ud y} \biggr)^{\frac 1p}. \] It was shown in Theorem 9.2 in Almgren-Lieb \cite{AL} that \[
\|u\|_{\mathring W^{\sigma,p}(\R^n)}\ge \|u^*\|_{\mathring W^{\sigma,p}(\R^n)}. \] Characterizations of the equality case have been given in Burchard-Hajaiej \cite{BH} and Frank-Seiringer \cite{FS}. Motivated by the P\'olya-Szeg\"o inequality in domains, we would like to investigate whether the above inequality holds for bounded open sets $\Omega$. That is, do we have \begin{equation}\label{eq:rearrangement1}
\|u\|_{\mathring W^{\sigma,p}(\Omega)}\ge \|u^*\|_{\mathring W^{\sigma,p}(\Omega^*)}? \end{equation}
Another motivation of the above question comes from Frank-Jin-Xiong \cite{FJX}, where the authors study the best constants of fractional Sobolev inequalities on domains. A classical result of Lieb \cite{Lieb} implies that \be \label{eq:FSI-2}
S(n,\sigma, \R^n)\left(\int_{\R^n} |u|^{\frac{2n}{n-2\sigma}}\,\ud x\right)^{\frac{n-2\sigma}{n}} \le\|u\|^2_{\mathring W^{\sigma,2}(\R^n)} \quad \mbox{for all }u\in \mathring W^{\sigma,2}(\R^n), \ee where $S(n,\sigma,\R^n)=\frac{2^{1-2\sigma}\omega_n^{\frac{2\sigma}{n}}\pi^{\frac{n}{2}}\Gamma(2-\sigma)}{\sigma(1-\sigma) \Gamma(\frac{n-2\sigma}{2})}$ and $\omega_n$ is the volume of the unit $n-$dimensional sphere. Moreover, the equality in \eqref{eq:FSI-2} holds if and only if $
u(x)= (1+|x|^2)^{-\frac{n-2\sigma}{2}} $ up to translating and scaling. These follow from the fact that the sharp fractional Sobolev inequality is a dual inequality of the sharp Hardy-Littlewood-Sobolev inequality. For an open set $\Omega\neq\R^n$, if $\sigma\in (1/2, 1)$ and $n\ge 2$, then there exists a positive constant $\underline{S}(n,\sigma)$ depending only on $n,\sigma$ but \emph{not} on $\Omega$ such that \be \label{eq:FSI}
\underline{S}(n,\sigma)\left(\int_{\om} |u|^{\frac{2n}{n-2\sigma}}\,\ud x\right)^{\frac{n-2\sigma}{n}} \le \iint_{\om\times\Omega} \frac{(u(x)-u(y))^2}{|x-y|^{n+2\sigma}}\,\ud x\ud y \quad \mbox{for all }u\in \mathring W^{\sigma,2}(\om). \ee This inequality is called the fractional Sobolev inequality in domain $\Omega$. It is included in Theorem 1.1 in Dyda-Frank \cite{DF}. It actually follows from \eqref{eq:FSI-2} and a fractional Hardy inequality of Dyda \cite{Dyda}, Loss-Sloane \cite{LS} and Dyda-Frank \cite{DF} (by using similar arguments to the proof of Theorem \ref{thm:rearrangementestimate} here; see the remark in the end of this paper).
In Frank-Jin-Xiong \cite{FJX}, they studied the best constant in \eqref{eq:FSI}: \[
S(n,\sigma,\Omega):=\inf\left\{\iint_{\Omega\times\Omega} \frac{(u(x)-u(y))^2}{|x-y|^{n+2\sigma}}\,\ud x\ud y\ |\ u\in C_c^\infty(\om), \int_{\om} |u|^{\frac{2n}{n-2\sigma}}\,\ud x=1\right\}. \] It was proved in \cite{FJX} that this best constant $S(n,\sigma,\om)$ actually depends on the domain $\Omega$, and can be achieved in many cases such as in the half spaces $\R^n_+=\{x=(x',x_n)\in \R^n, x_n>0\}$ or some smooth bounded domains, which is in contrast to the classical Sobolev inequalities in domains. Let $B_r$ be the ball of radius $r$ centered at the origin, and $B^+_1=B_1\cap\R^n_+$. Suppose $\sigma\in (1/2,1)$, and $\Omega$ is a $C^2$ bounded open set such that $B_1^+\subset\Omega\subset \R^n_+$, then it was proved in \cite{FJX} that both $S(n,\sigma,\R^n_+)$ and $S(n,\sigma,\om)$ are achieved, and there holds the inequality \[ S(n,\sigma,\om)<S(n,\sigma,\R^n_+)<S(n,\sigma,\R^n). \] On the other hand, from \eqref{eq:FSI}, we have that for $\sigma\in (1/2, 1)$,
$S(n,\sigma,\Omega)\ge \underline{S}(n,\sigma)>0$ for every open set $\Omega$. An interesting question left open is to find the value of $\inf_{\Omega} S(n,\sigma,\Omega)$ for $\sigma\in (1/2,1)$, where the infimum is taken over all bounded open sets $\Omega$. A conjecture is that $\inf_{\Omega} S(n,\sigma,\Omega)$ is achieved by a ball, which could follow from \eqref{eq:rearrangement1} . However, we show in this paper that $\eqref{eq:rearrangement1}$ is false.
\begin{thm}\label{thm:rearrangement2}
Let $n\ge 1$ and $\Omega $ be any nonempty open set in $\mathbb R^n$ with $|\Omega| < \infty$. Let $\sigma\in (0,1)$ and $p\in(0,\infty)$. There exists a nonnegative $u\in C^\infty_c(\Omega)$ such that \[
\iint_{\Omega \times \Omega} \frac{|u(x)-u(y)|^p}{|x-y|^{n+\sigma p}}\,\ud x\ud y
< \iint_{\Omega^*\times \Omega^*} \frac{|u^*(x)-u^*(y)|^p}{|x-y|^{n+\sigma p}}\,\ud x\ud y. \] \end{thm} We will prove this theorem in the next section by using an explicit computation.
\begin{rem*}\label{rem:unitballfirst} Theorem \ref{thm:rearrangement2} holds in particular when $\Omega$ is an open ball centered at the origin (so that $\Omega^*=\Omega$). \end{rem*}
\begin{rem*} In \cite{AL} (see Corollary 2.3 therein), a general rearrangement inequality is shown to hold for convex integrands. Namely, if $\Psi:\mathbb R^+ \to \mathbb R^+$ is convex with $\Psi(0)=0$, then for every nonnegative $L^1(\mathbb R^n)$ function $W$, every nonnegative $f$, $g \in \mathscr U_0$
with $\Psi \circ f $, $\Psi \circ g \in L^1(\mathbb R^n)$, one has \begin{align*} \int_{\mathbb R^n}
\int_{\mathbb R^n} \Psi(|f(x) -g (y) |) W(x-y) dx dy \ge \int_{\mathbb R^n}
\int_{\mathbb R^n} \Psi(|f^*(x)-g^*(y)|) W^*(x-y)dx dy. \end{align*} Our Theorem \ref{thm:rearrangement2} shows that such a general result cannot hold if $\mathbb R^n$ is replaced by a domain $\Omega$ of finite measure on the left-hand side (and correspondingly by $\Omega^*$ on the right-hand side). \end{rem*}
On the other hand, we have the following estimate.
\begin{thm}\label{thm:rearrangementestimate} Let $n\ge 1$, $\sigma\in (0,1)$ and $p\in(0,\infty)$ be such that $\sigma p>1$. Then there exists a positive constant $C$ depending only on $n,\sigma$ and $p$ such that \[
\iint_{\R^n\times \R^n} \frac{|u^*(x)-u^*(y)|^p}{|x-y|^{n+\sigma p}}\,\ud x\ud y\le C\iint_{\Omega \times \Omega} \frac{|u(x)-u(y)|^p}{|x-y|^{n+\sigma p}}\,\ud x\ud y \] for all open sets $\Omega \subset \mathbb R^n$ and all nonnegative $u\in C^\infty_c(\Omega)$. In particular, \[
\iint_{\Omega^*\times \Omega^*} \frac{|u^*(x)-u^*(y)|^p}{|x-y|^{n+\sigma p}}\,\ud x\ud y\le C\iint_{\Omega \times \Omega} \frac{|u(x)-u(y)|^p}{|x-y|^{n+\sigma p}}\,\ud x\ud y. \]
\end{thm}
\section{Proofs}\label{sec:proof}
We begin with the following simple lemma. Recall that for any two sets $A$ and $B$, their symmetric difference $A\triangle B=(A\setminus B)\cup (B\setminus A)$. \begin{rem*}
For an open set $\Omega\subset\R^n$, $|\Omega^*\triangle\Omega|=0$ if and only if $\Omega=\Omega^*$. \end{rem*}
\begin{lem}\label{lem:decrease} Let $\Omega$ be an open and bounded set in $\R^n$. \begin{itemize}
\item[(i).] Suppose $f\in L^1_{\operatorname{loc}}(\R^n)$ is radial and strictly decreasing, i.e. $f(x)>f(y)$ if $|x|<|y|$. Then \begin{align}\label{eq:lemma1}
\int_{\Omega^*}f(x)\,\ud x>\int_{\Omega}f(x)\,\ud x\quad\mbox{if}\ \ |\Omega^*\triangle\Omega|>0. \end{align}
\item[(ii).] Suppose $\overline B_\delta\subset\Omega$ for some $\delta>0$, and let $f\in L^1(\R^n\setminus\overline B_\delta)$ be radial and strictly decreasing. Then \begin{align}\label{eq:lemma2}
\int_{\R^n\setminus\Omega}f(x)\,\ud x>\int_{\R^n\setminus\Omega^*}f(x)\,\ud x\quad\mbox{if}\ \ |\Omega^*\triangle\Omega|>0. \end{align} \end{itemize} \end{lem}
\begin{rem*}
The main example is $f(x)=|x|^{-\alpha}$ for some $\alpha>0$. Similar proof as below can show the well-known inequality that for any Borel measure $A\subset \mathbb R^n$ with
$|A|<\infty$, $x_0\in \mathbb R^n$, and $\delta>0$, one has \begin{align*}
\int_{\mathbb R^n\setminus A} |x-x_0|^{-n-\delta} dx \ge
\int_{\mathbb R^n \setminus A^{*} } |x|^{-n-\delta} dx = \operatorname{const} \cdot |A|^{-\frac {\delta}n}. \end{align*} This inequality can be used to establish fractional Sobolev embedding. We should stress that in our case one needs strict inequality and for this reason we impose strict monotonicity on $f$. \end{rem*}
\begin{proof} Let $r$ be the radius of $\Omega^*$.
We prove $(i)$ first. Notice that \[ \int_{\Omega^*}f(x)\,\ud x-\int_{\Omega}f(x)\,\ud x=\int_{\Omega^*\setminus\Omega}f(x)\,\ud x-\int_{\Omega\setminus\Omega^*}f(x)\,\ud x. \]
Since $|\Omega^*|=|\Omega|$, we have $|\Omega\setminus\Omega^*|=|\Omega^*\setminus\Omega|=\frac 12 |\Omega^*\triangle\Omega|>0$. Since $f$ is radial and strictly decreasing, we have \[ \begin{split}
\int_{\Omega^*\setminus\Omega}f(x)\,\ud x&>f(r)|\Omega^*\triangle\Omega|,\\
\int_{\Omega\setminus\Omega^*}f(x)\,\ud x&<f(r)|\Omega^*\triangle\Omega|. \end{split} \] Hence, the inequality \eqref{eq:lemma1} follows.
To prove $(ii)$, we notice $\overline B_\delta\subset\Omega^*$ by the assumption, and \[ \int_{\R^n\setminus\Omega}f(x)\,\ud x-\int_{\R^n\setminus\Omega^*}f(x)\,\ud x=\int_{\Omega^*\setminus\Omega}f(x)\,\ud x-\int_{\Omega\setminus\Omega^*}f(x)\,\ud x. \] Hence, the inequality \eqref{eq:lemma2} follows the same as above. \end{proof}
\begin{proof}[Proof of Theorem \ref{thm:rearrangement2}] Our proof of the general case in Theorem \ref{thm:rearrangement2} is inspired by that of the special case $\Omega$ being a ball. So we will provide the proof of Theorem \ref{thm:rearrangement2} for $\Omega=B_1$ first.
Let $\eta \in C_c^{\infty}(B_1)$ be a radially decreasing function such that $\eta(x)=1$ for $|x| \le 1/2$. Let $\va\in (0,1/2)$ which will be chosen very small, $$x_\va=(1-\va, 0,\cdots,0),$$ and \[ u_\va=\eta\left(\frac{x-x_\va}{\va}\right). \] Since we assumed that $\eta$ is smooth, nonnegative, and radially decreasing, it is clear that \[ u^*_\va=\eta\left(\frac{x}{\va}\right). \] Therefore, \begin{equation}\label{eq:equal}
\iint_{\R^n\times \R^n} \frac{|u_\va(x)-u_\va(y)|^p}{|x-y|^{n+ \sigma p}}\,\ud x\ud y= \iint_{\R^n\times \R^n} \frac{|u_\va^*(x)-u_\va^*(y)|^p}{|x-y|^{n+\sigma p}}\,\ud x\ud y. \end{equation} Since $B_1^*=B_1$ and \begin{align*}
&\iint_{B_1\times B_1} \frac{|u_\va(x)-u_\va(y)|^p}{|x-y|^{n+ \sigma p}}\,\ud x\ud y\\
&=\iint_{\R^n\times \R^n} \frac{|u_\va(x)-u_\va(y)|^p}{|x-y|^{n+\sigma p}}\,\ud x\ud y-2\int_{B_1}u_\va^p(x)\left(\int_{\R^n\setminus B_1}\frac{1}{|x-y|^{n+\sigma p}}\,\ud y\right)\ud x, \end{align*} we only need to show that \begin{equation}\label{eq:aux0}
\int_{B_1}u_\va^p(x)\left(\int_{\R^n\setminus B_1}\frac{1}{|x-y|^{n+\sigma p}}\,\ud y\right)\ud x>\int_{B_1}(u_\va^*(x))^p\left(\int_{\R^n\setminus B_1}\frac{1}{|x-y|^{n+\sigma p}}\,\ud y\right)\ud x. \end{equation}
First, since $u^*$ is supported in $B_\va$, we have \begin{align*}
\int_{B_1}(u_\va^*(x))^p\left(\int_{\R^n\setminus B_1}\frac{1}{|x-y|^{n+\sigma p}}\,\ud y\right)\ud x
&=\int_{B_\va}(u_\va^*(x))^p\left(\int_{\R^n\setminus B_1}\frac{1}{|x-y|^{n+\sigma p}}\,\ud y\right)\ud x\\ &\le C \int_{B_\va}(u_\va^*(x))^p\,\ud x= C\va^n\int_{B_1}\eta^p(x)\,\ud x, \end{align*} where (as well as in the below) $C$ is a positive constant independent of $\va$.
Secondly, since $u$ is supported in $B_\va(x_\va)$, we have \begin{align*}
\int_{B_1}u_\va^p(x)\left(\int_{\R^n\setminus B_1}\frac{1}{|x-y|^{n+\sigma p}}\,\ud y\right)\ud x
&\ge \int_{B_\va(x_\va)}u_\va^p(x)\left(\int_{\{y:\ y_1\ge 1\}}\frac{1}{|x-y|^{n+\sigma p}}\,\ud y\right)\ud x\\ &\ge C \int_{B_\va(x_\va)}\frac{u_\va^p(x)}{(1-x_1)^{\sigma p}}\,\ud x\\ &\ge C\va^{-\sigma p}\int_{B_\va(x_\va)}u_\va^p(x)\,\ud x=C\va^{n-\sigma p}\int_{B_1}\eta^p(x)\,\ud x, \end{align*} where in the second inequality, we used that for $x=(x_1,\cdots,x_n)$ with $x_1<1$, \[
\int_{\{y: \ y_1\ge 1\}}\frac{1}{|x-y|^{n+\sigma p}}\,\ud y= C(1-x_1)^{-\sigma p}. \] This proves \eqref{eq:aux0}, and thus Theorem \ref{thm:rearrangement2} for $\Omega=B_1$, if we choose $\va$ sufficiently small.
Now let us consider the general case where $\Omega$ is not a ball. Since $\Omega$ is an open set and the Gagliardo semi-norm is translation invariant and dilation invariant (and also by \eqref{eq:syminvariant}), without loss of generality, we may assume $\Omega$ contains $B_1$. Again, let $\eta \in C_c^{\infty}(B_1)$ be a radially decreasing function such that $\eta(x)=1$ for $|x| \le 1/2$. Define for $\va\in (0,1)$, \begin{align*} u_{\va} (x)=\eta\left(\frac {x} {\va}\right). \end{align*} Hence, \begin{align*} u_{\va}^{\ast}(x) =\eta\left(\frac x {\va}\right), \end{align*} and thus, \eqref{eq:equal} also holds.
Since \begin{align*}
&\iint_{\Omega\times \Omega} \frac{|u_{\va}(x)-u_{\va}(y)|^p}{|x-y|^{n+ \sigma p}}\,\ud x\ud y\\
&=\iint_{\R^n\times \R^n} \frac{|u_{\va}(x)-u_{\va}(y)|^p}{|x-y|^{n+\sigma p}}\,\ud x\ud y-2\int_{\Omega}u_{\va}^p(x)\left(\int_{\R^n\setminus \Omega}\frac{1}{|x-y|^{n+\sigma p}}\,\ud y\right)\ud x, \end{align*}
we only need to check the inequality \begin{equation}\label{eq:aux} \int_{\Omega} u^p_{\va}(x) F(x) dx > \int_{\Omega^*} (u^{\ast}_{\va}(x))^p \tilde F(x) dx,
\end{equation} where \begin{equation}\label{eq:F}
F(x) = \int_{\R^n\setminus\Omega} \frac 1 { |x-y|^{n+p \sigma}} dy\quad\mbox{and}\quad \tilde F(x) = \int_{\R^n\setminus\Omega^*} \frac 1 { |x-y|^{n+p \sigma}} dy. \end{equation} Noticing the support of $\eta$, this reduces to checking the inequality \begin{align}\label{eq:Finequality} \int_{B_1} \eta^p(x) F(\va x) dx > \int_{B_1} \eta^p(x) \tilde F(\va x) dx. \end{align}
Since $\Omega$ is an open and is not a ball, we have $|\Omega^*\setminus\Omega|=|\Omega\setminus\Omega^*|>0$. Then it follows from \eqref{eq:lemma2} in Lemma \ref{lem:decrease} that $F(0)>\tilde F(0)$. Hence, the inequality \eqref{eq:Finequality} holds for all $\va$ sufficiently small by using the Lebesgue dominated convergence theorem. Theorem \ref{thm:rearrangement2} is proved.
We remark that the above proof for the general case where $\Omega$ is not a ball can also be used to prove the case when $\Omega=B_1$, which is as follows. Let $\eta$ be the same as before, $|\bar x|=1/2$ and define \begin{align*} u_{\va} (x)=\eta\left(\frac {x-\bar x} {\va}\right). \end{align*} Hence, \begin{align*} u_{\va}^{\ast}(x) =\eta\left(\frac x {\va}\right). \end{align*} As above, we only need to check the inequality \eqref{eq:aux}. Since $\Omega=B_1$, we have $\Omega^*=\Omega$ and $F=\tilde F$. Thus, by change of variables and noticing the support of $\eta$, this reduces to checking the inequality \begin{align}\label{eq:auxbarx} \int_{B_1} \eta^p(x) F(\bar x+\va x) dx > \int_{B_1} \eta^p(x) F(\va x) dx. \end{align} Since \begin{align*}
F(\bar x)=\int_{\R^n\setminus B_1} \frac 1 { |\bar x-y|^{n+p \sigma}} dy=\int_{\R^n\setminus B_1(\bar x)} \frac 1 { |z|^{n+p \sigma}} dz>\int_{\R^n\setminus B_1} \frac 1 { |z|^{n+p \sigma}} dz=F(0), \end{align*} where we used \eqref{eq:lemma2} in the last inequality (noticing $(B_1(\bar x))^*=B_1$), the inequality \eqref{eq:auxbarx} holds for all $\va$ sufficiently small by using the Lebesgue dominated convergence theorem. \end{proof}
We now give the proof of Theorem \ref{thm:rearrangementestimate}.
\begin{proof}[{Proof of Theorem \ref{thm:rearrangementestimate}}] We only need to consider the case where $\Omega\subset\R^n$ is an open set that satisfies $|\R^n\setminus\Omega|>0$. Let $u\in C^\infty_c(\Omega)$ be a nonnegative function.
Then \begin{align}
&\iint_{\R^n\times \R^n} \frac{|u^*(x)-u^*(y)|^p}{|x-y|^{n+\sigma p}}\,\ud x\ud y \nonumber\\
&\le \iint_{\R^n\times \R^n} \frac{|u(x)-u(y)|^p}{|x-y|^{n+\sigma p}}\,\ud x\ud y \nonumber\\
&=\iint_{\Omega\times \Omega} \frac{|u(x)-u(y)|^p}{|x-y|^{n+ \sigma p}}\,\ud x\ud y+2\int_{\Omega}u^p(x)\left(\int_{\R^n\setminus \Omega}\frac{1}{|x-y|^{n+\sigma p}}\,\ud y\right)\ud x. \label{eq:aux2} \end{align} As in Loss-Sloane \cite{LS} and Dyda-Frank \cite{DF}, we denote \[
d_\omega(x)=\inf\{|t|: x+t\omega\not\in\Omega\},\quad x\in\R^n,\quad \omega\in\mathbb{S}^{n-1}, \] where $\mathbb{S}^{n-1}$ is the $(n-1)$-dimensional sphere, and \[ m_{\alpha}(x)=\left(\frac{2\pi^{\frac{n-1}{2}} \Gamma(\frac{1+\alpha}{2})}{\Gamma(\frac{N+\alpha}{2})}\right)^{\frac{1}{\alpha}} \left(\int_{\mathbb{S}^{n-1}}\frac{1}{d_{\omega}(x)^\alpha}\,\ud \omega\right)^{-\frac{1}{\alpha}}. \] Then we have \begin{align*}
\int_{\R^n\setminus \Omega}\frac{1}{|x-y|^{n+\sigma p}}\,\ud y\le \int_{\mathbb{S}^{n-1}}\ud \omega \int_{d_{\omega}(x)}^{\infty}\frac{1}{r^{n+\sigma p}}\,\ud r&=(n+\sigma p-1)\int_{\mathbb{S}^{n-1}} \frac{1}{d_\omega(x)^{\sigma p}}\ud \omega\\ &=\frac{C(n,\sigma, p)}{(m_{\sigma p}(x))^{\sigma p}} \end{align*} for some constant $C(n,\sigma, p)$ depending only on $n,\sigma$ and $p$, but \emph{not} on $\Omega$. Thus, we have \begin{align}
\int_{\Omega}u^p(x)\left(\int_{\R^n\setminus \Omega}\frac{1}{|x-y|^{n+\sigma p}}\,\ud y\right)\ud x &\le C(n,\sigma, p) \int_{\Omega}\frac{u^p(x)}{(m_{\sigma p}(x))^{\sigma p}} \,\ud x\nonumber\\
&\le C(n,\sigma, p) \iint_{\Omega\times \Omega} \frac{|u(x)-u(y)|^p}{|x-y|^{n+ \sigma p}}\,\ud x\ud y,\label{eq:auxsobolev} \end{align} where we use Theorem 1.2 (fractional Hardy inequality) of Loss-Sloane \cite{LS} in the last inequality. Therefore, combining \eqref{eq:aux2} and \eqref{eq:auxsobolev}, we have \[
\iint_{\R^n\times \R^n} \frac{|u^*(x)-u^*(y)|^p}{|x-y|^{n+\sigma p}}\,\ud x\ud y\le C(n,\sigma, p) \iint_{\Omega\times \Omega} \frac{|u(x)-u(y)|^p}{|x-y|^{n+ \sigma p}}\,\ud x\ud y. \] Theorem \ref{thm:rearrangementestimate} is proved. \end{proof}
\begin{rem*} The above proof of Theorem \ref{thm:rearrangementestimate} can be used to prove \eqref{eq:FSI}. Indeed, if $n\ge 2$, $\sigma\in (0,1)$ and $1<\sigma p<n$, then for every open set $\Omega\neq\R^n$ and all $u\in C^\infty_c(\Omega)$, we have \begin{align*}
\left(\int_{\Omega}|u(x)|^{\frac{np}{n-\sigma p}}\,\ud x\right)^{\frac{n-\sigma p}{n}}&=\left(\int_{\R^n}|u(x)|^{\frac{np}{n-\sigma p}}\,\ud x\right)^{\frac{n-\sigma p}{n}}\\
&\le C(n,\sigma,p) \iint_{\R^n\times \R^n} \frac{|u(x)-u(y)|^p}{|x-y|^{n+\sigma p}}\,\ud x\ud y\\
&\le C(n,\sigma,p) \iint_{\Omega\times \Omega} \frac{|u(x)-u(y)|^p}{|x-y|^{n+\sigma p}}\,\ud x\ud y, \end{align*} where in the first inequality we used the classical fractional Sobolev inequality in $\R^n$, and in the second inequality we used \eqref{eq:aux2} and \eqref{eq:auxsobolev}. \end{rem*}
\noindent D. Li
\noindent Department of Mathematics, The Hong Kong University of Science and Technology\\ Clear Water Bay, Kowloon, Hong Kong\\[1mm] Email: \textsf{[email protected]}
\noindent K. Wang
\noindent Department of Mathematics, The Hong Kong University of Science and Technology\\ Clear Water Bay, Kowloon, Hong Kong\\[1mm] Email: \textsf{[email protected]}
\end{document} | arXiv |
\begin{document}
\title{Non-Gaussian quantum states of a multimode light field}
\author{\mbox{Young-Sik Ra}} \email{[email protected]} \affiliation{Laboratoire Kastler Brossel, UPMC-Sorbonne Universit\'es, CNRS, ENS-PSL Research University, Coll\`{e}ge de France; 4 place Jussieu, 75252 Paris, France} \affiliation{Department of Physics, Korea Advanced Institute of Science and Technology (KAIST), Daejeon 34141, Korea}
\author{\mbox{Adrien Dufour}} \affiliation{Laboratoire Kastler Brossel, UPMC-Sorbonne Universit\'es, CNRS, ENS-PSL Research University, Coll\`{e}ge de France; 4 place Jussieu, 75252 Paris, France}
\author{\mbox{Mattia Walschaers}} \affiliation{Laboratoire Kastler Brossel, UPMC-Sorbonne Universit\'es, CNRS, ENS-PSL Research University, Coll\`{e}ge de France; 4 place Jussieu, 75252 Paris, France}
\author{\mbox{Cl\'ement Jacquard}} \affiliation{Laboratoire Kastler Brossel, UPMC-Sorbonne Universit\'es, CNRS, ENS-PSL Research University, Coll\`{e}ge de France; 4 place Jussieu, 75252 Paris, France}
\author{\mbox{Thibault Michel}} \affiliation{Laboratoire Kastler Brossel, UPMC-Sorbonne Universit\'es, CNRS, ENS-PSL Research University, Coll\`{e}ge de France; 4 place Jussieu, 75252 Paris, France} \affiliation{Center for Quantum Computation and Communication Technology, Department of Quantum Science, The Australian National University, Canberra, ACT 0200, Australia}
\author{\mbox{Claude Fabre}} \affiliation{Laboratoire Kastler Brossel, UPMC-Sorbonne Universit\'es, CNRS, ENS-PSL Research University, Coll\`{e}ge de France; 4 place Jussieu, 75252 Paris, France}
\author{\mbox{Nicolas Treps}} \affiliation{Laboratoire Kastler Brossel, UPMC-Sorbonne Universit\'es, CNRS, ENS-PSL Research University, Coll\`{e}ge de France; 4 place Jussieu, 75252 Paris, France}
\date{\today}
\begin{abstract}
Even though Gaussian quantum states of multimode light are promising quantum resources due to their scalability, non-Gaussianity is indispensable for quantum technologies, in particular to reach quantum computational advantage. However, embodying non-Gaussianity in a multimode Gaussian state remains a challenge as controllable non-Gaussian operations are hard to implement in a multimode scenario. Here, we report the first generation of non-Gaussian quantum states of a multimode light field by subtracting a photon in a desired mode from multimode Gaussian states, and observe negativity of the Wigner function. For entangled Gaussian states, we observe that photon subtraction makes non-Gaussianity spread among various entangled modes. In addition to applications in quantum technologies, our results shed new light on non-Gaussian multimode entanglement with particular emphasis on quantum networks.
\end{abstract}
\maketitle
\textbf{ Advanced quantum technologies require scalable and controllable quantum resources~\cite{Andersen:2015dp,Biamonte:2017ic}. Gaussian states of multimode light such as squeezed states and cluster states are scalable quantum systems~\cite{Yokoyama:2013jp,Roslund:2014cb,Chen:2014jx}, which can be generated on demand. However, non-Gaussian features are indispensable in many quantum protocols, especially to reach a quantum computation advantage~\cite{Mari:2012ep}. Embodying non-Gaussianity in a multimode quantum state remains a challenge as non-Gaussian operations generally cannot maintain coherence among multiple modes~\cite{Averchenko:2016gv}. Here, we generate non-Gaussian quantum states of a multimode light field, and observe negativity of the Wigner function in adjustable modes. For this purpose, starting from the deterministic generation of Gaussian entangled states, we use sum-frequency generation to remove a single photon in a computer-controlled coherent superposition of optical modes. We reveal the induced non-Gaussian features and observe how they spread among the entangled modes, depending on the mode in which the photon is subtracted. The resulting non-Gaussian multimode quantum states will have broad applications for universal quantum computing~\cite{Lloyd:1999vz,Menicucci:2006ir}, entanglement distillation~\cite{Eisert:2002ft}, and a nonlocality test~\cite{Plick:2018wc}.
}
Our starting point is a squeezed vacuum state of light, a basic quantum resource for continuous-variable quantum technologies such as quantum-enhanced sensing~\cite{Aasi:2013jb}, deterministic quantum state teleportation~\cite{Takeda:2013hn}, and measurement-based quantum computing~\cite{Menicucci:2006ir}. Recent technological advances have extended the generation and control of squeezed vacuum from a single mode to multiple modes, which enables a determistic generation of large-scale multipartite entangled states~\cite{Yokoyama:2013jp,Roslund:2014cb,Chen:2014jx}. However, such quantum states are intrinsically Gaussian states, which always exhibit Gaussian statistics in electric field quadrature measurements. These states have limitations on quantum applications, e.g., universal quantum computing~\cite{Lloyd:1999vz,Menicucci:2006ir} and entanglement distillation~\cite{Eisert:2002ft}. In particular, the ability to produce a non-Gaussian quantum state is essential to reach quantum advantages~\cite{Mari:2012ep}, which is connected to exotic quantum features of non-Gaussian quantum states. The most profound example thereof is contextuality, which goes hand in hand with negative values of the Wigner function~\cite{Spekkens:2008kc}. Another example is multimode entanglement, which can have conceptually different properties in non-Gaussian states as compared with their Gaussian counterparts~\cite{Valido:2014iv}.
The hybrid approach, which combines continuous-variable and discrete-variable quantum information processing, provides a solution~\cite{Andersen:2015dp}. Subtracting/adding a discrete number of photons~\cite{Wenger:2004cw} or coupling with a discrete-level quantum system~\cite{Vlastakis:2013vi} can generate non-Gaussian states such as a local or non-local superposition of coherent states~\cite{Ourjoumtsev:2006jn,Sychev:2017fq,Ourjoumtsev:2009jh} and hybrid entanglement~\cite{Jeong:2014bl,Morin:2014ip}. Hitherto, this approach has only been successfully applied to a single- or two-mode quantum state, and the extension to highly multimode quantum states remains challenging due to the arduous task of maintaining coherence among multiple modes. For example, the conventional method of photon subtraction for a single-mode quantum state is based on a simple beam splitter~\cite{Wenger:2004cw}; when applied to a multimode quantum state, however, the method results in the generation of a mixed quantum state~\cite{Averchenko:2016gv}.
\begin{figure*}
\caption{\textbf{Mode-selective photon subtraction from a multimode quantum state.} (a) Concept. Input is a beam containing a Gaussian multimode quantum state $\hat{\rho}$, which can be, in general, a multipartite entangled state. From the input state, we subtract a photon in a specific mode or in a coherent superposition of multiple modes by controlling the complex coefficient $c_k$ for each mode $k$. This process, described by an annihilation operator $\hat{A}$ in \eq{eq:subtraction}, is heralded by a registration of a photon at the single-photon detector (SPD). As a result, the output beam contains a non-Gaussian multimode quantum state $\hat{\rho}^-$. The inset inside each circle is the Wigner function of the reduced quantum state in the associated mode. (b) Experimental setup. A Ti:sapphire laser produces a beam made of a train of femtosecond pulses, which splits into three beams. One beam is up-converted via second harmonic generation in a second-order nonlinear crystal (NC$_1$), and then, pumps NC$_2$ for a parametric down-conversion process. Synchronously pumped optical parametric oscillator (SPOPO) amplifies the process, which generates twelve-mode squeezed vacua in well-defined time-frequency modes. Another beam is used as a gate for the photon subtractor, and its time-frequency mode is engineered by a pulse shaper (PS). Inside NC$_3$, sum-frequency-interaction between the gate and the multimode squeezed vacua generates an up-converted beam, which is detected by SPD. A photon registration in SPD heralds photon subtraction from the multimode squeezed vacua. The resulting multimode quantum state is measured by homodyne detection with a time-frequency-engineered local oscillator (LO) using another PS. PD: photo diode; BS: beam splitter; $\theta$: phase of LO. }
\label{fig:description}
\end{figure*}
To exploit the full potential of the large-scale entangled states available in the continuous-variable quantum information processing~\cite{Yokoyama:2013jp,Roslund:2014cb,Chen:2014jx}, it is essential that the hybrid approach is made compatible with multimode quantum states, e.g., via photon subtraction operating in multiple modes coherently. The concept of our experiment is illustrated in \fig{fig:description}(a). If we call $\hat\rho$ the density operator of an input multimode quantum state, the output state $\hat{\rho}^{-}$ by a photon subtraction operator $\hat{A}$ becomes \begin{eqnarray} \label{eq:subtraction} \quad \ \hat\rho^{-}\propto \hat A \hat\rho \hat A^{\dagger},\textrm{\,\,\, where\,\,} \hat{A}=\sum_{k=0} c_k \hat{a}_k. \end{eqnarray} $c_k$ are complex numbers normalized as $\sum \abs{c_k}^2 = 1$, and $\hat{a}_k$ is the annihilation operator for mode $k$. Note that $\hat{A}$ is, in general, a coherent superposition of annihilation operators in multiple modes. The ability to experimentally control both the $c_k$ coefficient and the multimode resource $\hat\rho$ is the key to tailor non-Gaussian multimode states and to achieve non-Gaussian entanglement for building non-Gaussian quantum networks~\cite{Walschaers:2017bx,Walschaers:2018wl}.
In our experiment, the controlled generation of non-Gaussian multimode quantum states is performed using quantum frequency combs as a resource. Figure \ref{fig:description}(b) shows the experimental setup, whose details are presented in Methods. The optical modes in which we implement \eq{eq:subtraction} are time-frequency modes~\cite{Ansari:2018uj}. The interest of these modes is that they are co-propagating in the same transverse mode, allowing for a large multimode quantum resource to keep its coherence and to access arbitrary superpositions of modes through elaborate techniques in ultrafast optics. We populate these modes with a highly multimode Gaussian state through a parametric down conversion process~\cite{Roslund:2014cb}. Tailoring the measurement mode basis allows for the generation of versatile multipartite entangled states~\cite{Cai:2017cp}. We combine this resource with a time-frequency mode-dependent photon subtractor to de-Gaussify such multimode Gaussian states.
\begin{figure*}
\caption{\textbf{Wigner function reconstructed from experimental data.} For a multimode quantum state, photon subtraction and measurement are conducted in (a) HG modes or in (b) superpositions of HG modes. The Wigner function of each mode is represented in the phase space of $x$ and $p$ axes, which are associated with quadrature operators $\hat{x}=\hat{a}+\hat{a}^\dagger$ and $\hat{p}=(\hat{a}-\hat{a}^\dagger)/i$, respectively. The inset behind each Wigner function shows the experimentally obtained quadrature outcomes, where the horizontal and vertical axes represent the phase of local oscillator and a quadrature outcome, respectively; the quadrature outcome of one corresponds to the variance of the vacuum fluctuation. No correction of optical losses is made; for the results by optical loss correction, see \extfig{extfig:datalosscorr}.
$W_0=2\pi W(0,0)$ is the value of a normalized Wigner function at the origin, and $F$ is the fidelity between an experimental Wigner function and the Wigner function by the ideal photon subtraction to the input state of the corresponding mode. Purities of the Wigner functions in (a), compared with the cases of the ideal photon subtraction, are presented in \exttab{exttab:purity}. For measurements in modes EPR$_0$ and EPR$_1$, the phase of a quadrature outcome is randomized since the associated quantum state is phase insensitive. Errors noted in parentheses are 1 s.~d. }
\label{fig:data}
\end{figure*}
More specifically, our multimode Gaussian resource $\hat\rho$ is a set of independent squeezed vacua whose eigen modes are conveniently approximated by Hermite-Gaussian modes HG$_k$. In order to implement the concepts of \eq{eq:subtraction}, we associate these modes with the annihilation operators $\hat a_k$. Hence, it remains to control the $c_k$ coefficients for the photon subtraction. As shown in \fig{fig:description}(b), this is implemented through a mode-selective sum-frequency generation between the Gaussian resource $\hat{\rho}$ and a gate beam~\cite{Ra:2017ia}. The non-linear interaction is designed such that pulse-shaping the gate allows for the control of the mode of photon subtraction~\cite{Ansari:2018uj}. Finally, detection of a single photon in the up-converted beam heralds the subtraction of a photon in the desired mode from the Gaussian resource.
In practice, in order to implement the operator $\hat A$, the mode of the gate should be set as $v_g = \sum_k (-1)^k c_k~ $HG$_k^{}$~\cite{Ra:2017ia}, which is efficiently performed using a computer controlled pulse shaper. The intensity of the gate governs the efficiency of the operation, and hence the heralding probability. To characterize the generated non-Gaussian multimode quantum state, we employ a homodyne detection that can control the mode of measurement by pulse-shaping the local oscillator.
We first measure the input multimode squeezed vacua without photon subtraction (i.e., no gate field is applied). As expected, the measured state exhibits Gaussian distribution: see the input Wigner functions shown in the first column of~\fig{fig:data}(a). On the other hand, when a single photon is subtracted in HG$_0$ (the second column), we observe that the Wigner function in HG$_0$ becomes non-Gaussian while the Wigner functions in the other modes remain Gaussian. This result shows the mode-selective operation of the photon subtractor necessary for multimode quantum states. The non-Gaussian Wigner function in HG$_0$ exhibits a negative value at the origin ($W_0$), as is required to achieve quantum advantages~\cite{Mari:2012ep}, and the negativity indicates a negative value in the entire multimode Wigner function (see Methods). When a photon is subtracted in HG$_1$ (HG$_2$), we similarly observe a non-Gaussian Wigner function only in HG$_1$ (HG$_2$). Compared with the photon subtraction in HG$_0$, photon subtraction in the higher order modes results in a less non-Gaussian Wigner function. This is mainly because the input state in a higher order mode has a larger optical loss than HG$_0$: we have found a high fidelity ($F$) between a non-Gaussian Wigner function obtained in experiment with the ideal Wigner function calculated by subtracting a photon from the corresponding input state.
Furthermore, the versatility of the experimental setup allows for the computer controlled subtraction of a photon in an arbitrary superposition of modes from a multimode quantum state. As an example, we subtract a photon in a superposition of HG$_0$ and HG$_1$ modes, HG$_0 - i$HG$_1$~\footnote{The normalization constant is omitted for simplicity, and $-i$ is introduced to rotate the $x$-squeezed vacuum in HG$_1$ to $p$-squeezed vacuum such that both HG$_0$ and HG$_1$ have a $p$-squeezed vacuum.}. We now observe a non-Gaussian Wigner function in this mode (first row of \fig{fig:data}(b)). On the other hand, in the orthogonal mode HG$_0 + i$HG$_1$, a Gaussian Wigner function is obtained, and in a partially overlapping mode $i$HG$_1$, an intermediate situation is obtained as expected, see \extfig{extfig:modematching}. When we subtract a photon in a superposition of three modes HG$_0+i$HG$_1$+HG$_2$, we similarly observe a non-Gaussian Wigner function in the same superposed modes (second row of \fig{fig:data}(b)).
This flexibility of the setup allows us to extend photon subtraction to entangled input states. We first investigate an Einstein-Podolsky-Rosen (EPR) entangled state, which exhibits quantum correlations between two superposed modes: EPR$_0=$HG$_0$+HG$_1$ and EPR$_1=$HG$_0-$HG$_1$ (see Methods). The last two rows in \fig{fig:data}(b) show the experimentally obtained Wigner functions. Without photon subtraction, the reduced quantum state in each of EPR$_0$ and EPR$_1$ is a thermal state as expected. When a photon is subtracted in EPR$_0$, the introduced non-Gaussian characteristic appears in the other mode (EPR$_1$) with almost no effect on the mode in which the photon is actually subtracted, and vice versa. In striking contrast with the aforementioned separable input state, the effect of photon subtraction on an entangled state is not localized but is transferred to another mode.
\begin{figure}\label{fig:cluster}
\end{figure}
To open genuine perspectives for applications in quantum technologies, we show the scalability of our approach to larger multimode entangled states. We consider a four-mode linear cluster (LC) state and a four-mode square cluster (SC) state (see Methods), where we denote the four modes of the linear one as LC$_k$ ($k=0,1,2,3$) and the square one as SC$_k$ ($k=0,1,2,3$). Due to the large dimension of the state, full tomography becomes impractical. Thus, we quantify the non-Gaussianity in each mode by evaluating the phase-averaged excess kurtosis (to compare with the kurtosis of 3 by a Gaussian distribution). This is obtained from the quadrature measurement outcomes $x_1,x_2,..., x_S$ where the measurement phase is randomized: \begin{eqnarray} \label{eq:kurtosis} K_{\rm{ex}} = \frac{ \frac{1}{S} \sum_{s=1}^{S} x_s^4 } { {(\frac{1}{S} \sum_{s=1}^{S} x_s^2 )}^2 } -3 \end{eqnarray} For a quantum state with zero mean ($\expec{\hat{a}}=\expec{\hat{a}^\dagger}=0$), which corresponds to all the quantum states in our experiment, a Gaussian state always exhibits $K_{\rm{ex}}\ge0$, thus $K_{\rm{ex}}<0$ indicates a non-Gaussian state. Figure~\ref{fig:cluster} shows excess kurtosis of the generated states. For the linear cluster state, excess kurtosis in each mode is initially close to zero. When a photon is subtracted in LC$_3$, it is LC$_2$ in which excess kurtosis becomes highly negative (i.e. non Gaussian) while those in the other modes remain close to 0. For photon subtraction in LC$_2$, only LC$_3$ exhibits a significant negativity. We can concentrate more non-Gaussianity in LC$_3$ by subtracting a photon in a superposition of four modes, as shown in the last figure of~\fig{fig:cluster}(a). For the square cluster state, excess kurtosis without photon subtraction is close to zero in each mode. Photon subtraction in SC$_0$, however, does not affect the distributions in its nearby modes (SC$_1$ and SC$_3$) much, but it introduces non-Gaussianity mostly in SC$_2$ which is two steps away from the mode of photon subtraction.
We have generated non-Gaussian quantum states of a multimode light by subtracting a photon from multimode Gaussian states. The selectivity and the controllability of the mode(s) for the photon subtraction make it possible to extend the non-Gaussianity of a quantum state to the multimode regime, which has been a main obstacle for scalable quantum information processing~\cite{Andersen:2015dp,Biamonte:2017ic}. The availability of non-Gaussian multimode states will stimulate fundamental studies on multipartite entanglement~\cite{Valido:2014iv} and multimode quantumness~\cite{Hudson:1974eh} by going beyond the Gaussian realm, as well as applications in quantum computing~\cite{Lloyd:1999vz,Menicucci:2006ir} and quantum communication~\cite{Eisert:2002ft,Plick:2018wc}. In particular, the observed nontrivial interplay between photon subtraction and cluster states, confirming recent theoretical predictions~\cite{Walschaers:2018wl}, provides new insights into the fields of quantum networks~\cite{Kimble:2008if} and quantum transport~\cite{Walschaers:2016bm}.
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\section{Methods}
\textbf{Experimental details.} Non-Gaussian multimode quantum states are generated by several nonlinear interactions on femtosecond pulses, as described in~\fig{fig:description}. The fundamental light source is a Ti:Sapphire laser, which produces a train of pulses (duration: 90 fs, central wavelength: 795 nm) at a repetition rate of 76 MHz. The laser beam is split into three beams: one is used for generating a multimode Gaussian state, another for photon subtraction, and the third for the homodyne detection.
The first beam is up-converted to a femtosecond pulse having 397.5-nm central wavelength in NC$_1$ (0.2-mm-thick BiB$_3$O$_6$) which is used as a pump for a parametric-down-conversion process in NC$_2$ (2-mm-thick BiB$_3$O$_6$) inside a cavity, the SPOPO. The length of the SPOPO is locked to the length of the Ti-Sapphire laser via the Pound-Drever-Hall method, such that the train of pump pulses is synchronized with the down-converted pulses which circulate inside the SPOPO. Transmittance of the output coupler of the SPOPO is 50\%. The light coming out through the output coupler is a multimode Gaussian state, containing roughly twelve squeezed vacua in orthogonal time-frequency modes~\cite{Roslund:2014cb, Cai:2017cp}. Among the twelve modes, we focus on the first four dominating modes, whose covariance matrix is given in \extfig{extfig:cov}.
For photon subtraction, we perform sum-frequency interaction between the multimode Gaussian state and the second beam from the Ti-Sapphire laser (the gate; 1 mW power) inside NC$_3$ (2.5-mm-thick BiB$_3$O$_6$). Detection of a single photon generated by the sum-frequency interaction heralds photon subtraction from the multimode squeezed vacua, where the time-frequency mode of the gate determines the photon subtraction mode~\cite{Ra:2017ia}. To engineer the time-frequency mode of the gate, we employ a homemade pulse shaper whose core element is a spatial light modulator, having a spectral resolution of 0.2 nm. Conversion efficiency of the nonlinear interaction is 0.1~\%, and we have typically 110 Hz of heralding rate with background noise of 6 Hz.
The last beam is used as the local oscillator (LO) of the homodyne detection to measure the generated quantum state. The measurement mode is the mode of the LO, which is engineered by another pulse shaper having a spectral resolution of 0.2 nm. For each event of photon subtraction, photocurrent difference between the two PDs is sampled every 2 ns during a 2-$\mu s$ time window, and one quadrature outcome is obtained by calculating the dot product between the samples and the double-sided-decaying-shape temporal mode of the SPOPO~~\cite{NeergaardNielsen:2006hl}. To reconstruct a Wigner function in~\fig{fig:data}, we collect 20,000 $\sim$ 30,000 quadrature outcomes. In the case of no photon subtraction, we monitor the variance of the quadrature outcomes. Phase dependance of the quadrature squeezing of the multimode Gaussian state provides the phase information of the LO relative to this multimode light.
\textbf{Preparation of entangled states.}
We prepare an entangled state by choosing a specific basis of modes in which quantum correlations among desired modes emerge~\cite{Roslund:2014cb, Cai:2017cp}. In the HG mode basis, even-order (odd-order) modes exhibit $p$-quadrature ($x$-quadrature) squeezed vacuum. To prepare an EPR entangled state, we use a basis of $\textrm{EPR}_0 = \frac{1}{\sqrt{2}}\left(\textrm{HG}_0+\textrm{HG}_1\right)$ and $\textrm{EPR}_1 = \frac{1}{\sqrt{2}}\left(\textrm{HG}_0-\textrm{HG}_1\right)$. We have obtained $\expec{\Delta^2 (\hat{x}^\textrm{EPR}_0 - \hat{x}^\textrm{EPR}_1)} +\expec{\Delta^2(\hat{p}^\textrm{EPR}_0 + \hat{p}^\textrm{EPR}_1)} = 2.51(6) < 4$ (the Duan entanglement criterion~\cite{Duan:2000fw,Simon:2000fd}) and $\expec{\Delta^2 \hat{x}^\textrm{EPR}_{1|0}} \expec{\Delta^2 \hat{p}^\textrm{EPR}_{1|0}} = 0.71(4) < 1 $ (the EPR criterion~\cite{Bowen:2003kk}). To prepare a linear cluster state, we use a basis of LC$_k$ ($k=0,1,2,3$) which is obtained by applying a unitary matrix $U^{\textrm{(LC)}}$ to the basis of HG$_k$ ($k=0,1,2,3$), where \[U^{\textrm{(LC)}}=\left( {\begin{array}{*{20}{r}} {- 0.344i}&{- 0.421i}&{0.531i}&{0.650i}\\ {0.344}&{ - 0.765}&{ - 0.531}&{0.119}\\ {- 0.765i}&{ - 0.344i}&{- 0.119i}&{- 0.531i}\\ {0.421}&{ - 0.344}&{0.650}&{ - 0.531} \end{array}} \right).\] To check correlations among different modes, we use the four nullifiers associated with a linear cluster state~\cite{vanLoock:2007ky,Cai:2017cp}, $\hat{\delta}_k^{\textrm{(LC)}} = \hat{x}_k - \sum_l V_{kl} \hat{p}_l $ ($V_{kl}$ is the adjacency matrix defining the topology of a cluster state, where $V_{kl}=1$ if $k$ and $l$ are connected, and 0 otherwise). They all exhibit a variance less than the vacuum fluctuation: $\expec{ \Delta^2 \hat{\delta}_k^\textrm{{(LC)}}} / \expec{\Delta^2 \hat{\delta}_k^\textrm{(LC)}}_{\textrm{vacuum}} =$ 0.75(2), 0.67(2), 0.68(2), and 0.64(2) for $k =$ 0, 1, 2, and 3, respectively. Similarly, we prepare a square cluster state in the basis of SC$_k$ ($k=0,1,2,3$), which is obtained by applying a unitary matrix $U^{\textrm{(SC)}}$ to the HG basis, where \[U^{\textrm{(SC)}}=\left( {\begin{array}{*{20}{r}} { - 0.316}&{0.632}&{0.707}&{0.000}\\ {0.632i}&{\,0.316i}&{0.000}&{ - 0.707i}\\ { - 0.316}&{0.632}&{ - 0.707}&{0.000}\\ {0.632i}&{0.316i}&{0.000}&{0.707i} \end{array}} \right).\] Each of the four nullifiers $\hat{\delta}_k^{\textrm{(SC)}}$ associated with a square cluster state exhibits a variance less than the vacuum fluctuation: $\expec{\Delta^2 \hat{\delta}_k^\textrm{{(SC)}}} / \expec{\Delta^2 \hat{\delta}_k^\textrm{(SC)}}_{\textrm{vacuum}} =$ 0.72(2), 0.77(2), 0.61(2), and 0.75(2) for $k =$ 0, 1, 2, and 3, respectively.
\textbf{Theoretical model.} To calculate $\hat{\rho}^-$, we model a single-photon subtractor that takes into account experimental imperfections~\cite{Ra:2017ia}: \begin{eqnarray} \hat{\rho}^- = \mathcal{R}[\hat{\rho}]/\text{tr}{[\mathcal{R}[\hat{\rho}]]};\,\,\, \mathcal{R}[\hat{\rho}]=w_0\hat{\rho} + (1-w_0) \mathcal{S}[\hat{\rho}], \label{eq:realsubtractor} \end{eqnarray} where $w_0$ is the weight of background noise (e.g. dark counts) of the SPD which does not alter the input state. $\mathcal{S}[\hat{\rho}]$ is the actual photon subtraction from an input state: \begin{eqnarray} \mathcal{S}[\hat{\rho}] = \frac{N p_0 -1}{N-1} \, \hat{A} \hat{\rho} \hat{A}^\dagger + \frac{1-p_0}{N-1}\sum_{k=0}^{N-1} \, \hat{a}_k \, \hat{\rho} \, \hat{a}_k^\dagger, \end{eqnarray} where $N$ is the number of modes. $\hat{A}$ is the desired photon subtraction in~\eq{eq:subtraction}, whose weight is $p_0$; the remaining weight, $1-p_0$, corresponds to the photon subtraction in the incoherent mixture of the other modes with equal contribution. To consider the experimental conditions, we use $w_0=0.0094$, $p_0=0.95$, $N=4$, and $\hat{\rho}$ in~\extfig{extfig:cov}. $K_{\textrm{ex}}$ of $\hat{\rho}^-$ in a specific mode is then obtained by following the method presented in Refs.~\cite{Walschaers:2017bx,Walschaers:2018wl}, which is based on calculating multimode correlation functions.
\textbf{Negativity of a multimode Wigner function.} A negative value in a single-mode Wigner function, which is reduced from a multimode Wigner function, is sufficient to show a negative value in the multimode Wigner function. The proof is straightforward by considering the contraposition of the statement: a non-negative multimode Wigner function always leads to a non-negative Wigner function when reduced to a single mode. Consider a reduced phase space $\mathbb{R}^2$ of an arbitrary single mode defining quadratures $(x_0,p_0)$ and the entire phase space $\mathbb{R}^{2N}$ of $N$ modes defining quadratures $(x_0,p_0, ..., x_{N-1},p_{N-1})$. The Wigner function in the $N$ modes is assumed to be non negative: $W^{(N)}(x_0,p_0, ..., x_{N-1},p_{N-1})\ge 0$. Then, the Wigner function in the reduced phase space $W(x_0,p_0)$ can be obtained by integrating the multimode Wigner function over the phase space of $(x_1,p_1, ..., x_{N-1},p_{N-1})$: \begin{eqnarray}
W(x_0,p_0)=\int_{\mathbb{R}^{2(N-1)}} dx_1dp_1...dx_{N-1}dp_{N-1}~~W^{(N)}.\nonumber \end{eqnarray} As $W^{(N)}\ge 0$, the reduced Wigner function $W(x_0,p_0)$ cannot have a negative value.
\setcounter{figure}{0} \makeatletter \renewcommand{Extended Data Fig.}{Extended Data Fig.} \renewcommand{\@arabic\c@figure}{\@arabic\c@figure} \makeatother
\setcounter{table}{0} \makeatletter \renewcommand{Extended Data Table}{Extended Data Table} \renewcommand{\@arabic\c@table}{\@arabic\c@table} \makeatother
\renewcommand{1}{1.0} \begin{table*}[h] \begin{tabular}{ccccccccccccc}
&&&\multicolumn{3}{c}{Subtraction mode} \\%\\\cline{3-6}
~~&~~&\multicolumn{1}{|c|}{~~Input~~}&~~HG$_0$~~&~~HG$_1$&~~HG$_2$~~ \\\cline{2-6}
\multirow{3}{*}{\shortstack{~Measurement~\\mode}}&\multicolumn{1}{c|}{~~HG$_0$~~}&\multicolumn{1}{c|}{~~0.91(1)~~}&~0.45(0) [0.53(2)]~&0.88(1)&0.87(1)\\%\cline{2-6}
&\multicolumn{1}{c|}{~~HG$_1$~~}&\multicolumn{1}{c|}{~~0.90(1)~~}&0.82(1)&~0.47(0) [0.49(1)]~&0.86(1)\\%\cline{2-6}
&\multicolumn{1}{c|}{~~HG$_2$~~}&\multicolumn{1}{c|}{~~0.91(1)~~}&0.87(1)&0.77(1)&~0.49(0) [0.50(2)]~ \end{tabular} \caption{ \textbf{Purities of the Wigner functions in \fig{fig:data}(a).} For comparison, the purity of a Wigner function by the ideal photon subtraction is provided in square brackets, which agrees well with the experimental result. Low purity in a photon-subtracted mode is attributed to a non-ideal input state~\cite{Ourjoumtsev:2006jn}. No optical loss is corrected in the calculation. Errors noted in parentheses are 1 s.~d. }\label{exttab:purity} \end{table*} \renewcommand{1}{1}
\begin{figure*}
\caption{\textbf{Wigner function reconstructed with optical loss correction.} Optical loss in the homodyne detection ($12.5\%$) has been corrected. Comparing with \fig{fig:data}, non-Gaussian Wigner functions show reduced $W_0$. Errors noted in parentheses are 1 s.~d.}
\label{extfig:datalosscorr}
\end{figure*}
\begin{figure}
\caption{\textbf{Effect of mode mismatch between photon subtraction and measurement.} When a single photon is subtracted in $\textrm{HG}_0 -i\textrm{HG}_1$, a Wigner function (without optical loss correction) is obtained in a measurement mode having (a) full match ($\textrm{HG}_0 -i\textrm{HG}_1$), (b) partial match ($i\textrm{HG}_1$), and (c) no match ($\textrm{HG}_0 +i\textrm{HG}_1$). Errors noted in parentheses are 1 s.~d. }
\label{extfig:modematching}
\end{figure}
\begin{figure}
\caption{\textbf{Experimental covariance matrix.} (a) is for $x$ quadratures, seen from above, and (b) is for $p$ quadratures, seen from below. Mode indexes are $\textrm{HG}_0$, $i\textrm{HG}_1$, $\textrm{HG}_2$, and $i\textrm{HG}_3$, where $i$ is added for the odd-index HG modes to have $p$-squeezed vacua in all modes. For clarity, the vacuum noise (corresponding to the identity matrix) is subtracted from the covariance matrix. In the covariance matrix, variances of ($x$, $p$) quadratures are ($2.8\textrm{ dB}$, $-1.8\textrm{ dB}$) in mode 0, ($2.1\textrm{ dB}$, $-1.6\textrm{ dB}$) in mode 1, ($1.6\textrm{ dB}$, $-1.0\textrm{ dB}$) in mode 2, and ($1.4\textrm{ dB}$, $-0.7\textrm{ dB}$) in mode 3. }
\label{extfig:cov}
\end{figure}
\end{document} | arXiv |
2.12 Algebraic Combination of Functions
Single Variable Calculus>
2. Functions>
If $f$ and $g$ are two functions, we can define new functions $f+g$,$f-g$, $f.g$, and $f/g$ by the formulas:
\begin{align*}
(f+g)(x) & =f(x)+g(x)\\
(f-g)(x) & =f(x)-g(x)\\
(f.g)(x)= & f(x).g(x)\\
(f/g)(x) & =\frac{f(x)}{g(x)}
\end{align*}
The domains of $f+g$, $f-g$, $f.g$ consist of all $x$ for which both $f(x)$ and $g(x)$ are defined:
\[
\{x|\ x\in Dom(f)\text{ and }x\in Dom(g)\}
\]
That is, the domains of these functions are the intersection of the domain of $f$ and the domain of $g$:
Dom(f)\cap Dom(g)
The domain of $f/g$ is trickier. Its domain consists of all $x$ for which $f(x)$ and $g(x)$ are defined but $g(x)\neq0$, because division by zero is not defined. Therefore:
\[ \bbox[#F2F2F2,5px,border:2px solid black]{\large Dom(f/g)=\{x|\ x\in Dom(f)\text{ and }x\in Dom(g)\text{ and }g(x)\neq0\},}\] which also can be written as:
Dom(f/g)=Dom(f)\cap Dom(g)\cap\{x|\ g(x)\neq0\}.
Let $f(x)=\dfrac{1}{x-3}$ and $g(x)=\sqrt{x-2}$.
(a) Find the functions $f+g,f-g,f\cdot g$, $f/g$ and $g/f$ and their domains.
(b) Find $(f+g)(5),(f-g)(5),(fg)(5)$, $(f/g)(5)$, and $(g/f)(5)$.
(a) The domain of $f$ is $\{x|\ x\neq3\}$ or $\mathbb{R}-\{3\}$ because division by zero is not defined. The square root of negative numbers is not defined [square root of a negative number becomes imaginary], thus the domain of $g$ is $\{x|\ x\geq2\}$. The intersection of the domains of $f$ and $g$ is
\{x|\ x\geq2\text{ and }x\neq3\}=[2,3)\cup(3,\infty).
\] Thus we have
(f+g)(x) & =f(x)+g(x)=\frac{1}{x-3}+\sqrt{x-2}\\
(f-g)(x) & =f(x)-g(x)=\frac{1}{x-3}-\sqrt{x-2}\\
(fg)(x) & =f(x)\cdot g(x)=\frac{\sqrt{x-2}}{x-3},
and the domains of all of these functions are
\{x|\ x\geq2\text{ and }x\neq3\}=[2,3)\cup(3,\infty).\qquad\text{(try them!)}
\] To determine the domain of $(f/g)(x)$, from this set of numbers we need to exclude 2 because $g(2)=0.$ That is, the domain of $(f/g)(x)$ is
\{x|\ x>2\text{ and }x\neq3\}=(2,3)\cup(3,\infty),
\] and
\left(\frac{f}{g}\right)(x)=\frac{1}{(x-3)\sqrt{x-2}}.
\] Because $f$ does not vanish ($f(x)\neq0$) in its domain,* the domain of $(g/f)(x)$ is the same as $Dom(f)\cap Dom(g)=[2,3)\cup(3,\infty)$,and
\left(\frac{g}{f}\right)(x)=\frac{\sqrt{x-2}}{\frac{1}{x-3}}=(x-3)\sqrt{x-2}.
(b) Because $x=5$ is in the domain of each function, each of these values exists:
(f+g)(5) & =f(5)+g(5)=\frac{1}{5-3}+\sqrt{5-2}=\frac{1}{2}+\sqrt{3}\\
(f-g)(5) & =f(5)-g(5)=\frac{1}{5-3}-\sqrt{5-2}=\frac{1}{2}-\sqrt{3}\\
(fg)(5) & =f(5)\cdot g(5)=\frac{1}{5-3}\cdot\sqrt{5-2}=\frac{\sqrt{3}}{2}\\
\left(\frac{f}{g}\right)(5) & =\frac{f(5)}{g(5)}=\frac{\frac{1}{5-3}}{\sqrt{5-2}}=\frac{1}{2\sqrt{3}}\\
\left(\frac{g}{f}\right)(5) & =\frac{g(5)}{f(5)}=\frac{\sqrt{5-2}}{\frac{1}{5-3}}=2\sqrt{3}
* You may sketch the graph of $y=1/(x-3)$ to see that. Note that the graph of $y=1/(x-3)$ is the graph of $y=1/x$ shifted horizontally 3 units to the right. See Section 2.11 for more details.
What is the natural domain of $H(x)=\sqrt{x-1}+\sqrt{2-x}$?
We have already determined the domain of this function in Example 1 in the Section on Natural Domain and Range of a Function. But now, let's use what we learned in this section. Let $f(x)=\sqrt{x-1}$, $g(x)=\sqrt{2-x}$, and $H=f+g$. The function $f$ is defined for all $x$ such that $x-1\geq0$. That is, $Dom(f)=\{x|\ x\geq1\}=[1,\infty)$. The domain of $g$ consists of all $x$ for which $2-x\geq0$, i.e. $Dom(g)=\{x|\ x\leq2\}=(-\infty,2]$. The intersection of the domains of $f$ and $g$ is
Dom(H) & =\{x|\ x\leq2\text{ and }x\geq1\}\\
& =\{x|\ 1\leq x\leq2\}\\
& =[1,2],
Dom(H)=Dom(f)\cap Dom(g)=[1,\infty)\cap(-\infty,2]=[1,2],
\] which is exactly the same as we saw in Example 1 in Section 2.3.
Are the following functions equal?
(a)$h_{1}(x)=\sqrt{x-1}\sqrt{2-x}$ and $h_{2}(x)=\sqrt{(x-1)(2-x)}$
(b)$u_{1}(x)=\sqrt{x-1}\sqrt{x-2}$ and $u_{2}(x)=\sqrt{(x-1)(x-2)}$
The functions $h_{1}(x)$ and $h_{2}(x)$, and $u_{1}(x)$ and $u_{2}(x)$ can be equal functions if they have the same (natural) domains. [Note that if $r$ and $s$ are both nonnegative, then $rs\geq0$. Therefore if $A=\sqrt{r}\sqrt{s}$, then we can also write $A$ as
\[A=\sqrt{rs}.\] However, if $B=\sqrt{pq}$, without knowing that $p$ and $q$ are both nonnegative, we cannot write $B$ as $\sqrt{p}\sqrt{q}.$ For example, if $p=-2$, and $q=-3$, then we can calculate $\sqrt{(-2)\cdot(-3)}$
but $\sqrt{-2}$ or $\sqrt{-3}$ are not defined.] To determine the domains of these functions, we note that:
\sqrt{\square}\text{ is defined if the sign of }\square\text{ is not negative}
\text{sign }(\square\times\lozenge)=\text{sign }(\square)\times\text{sign }(\lozenge)
\] (a) First, we should determine the signs of all expressions under $\sqrt{.}$
From what we learned in this section:
Dom(h_{1}) & =Dom(\sqrt{x-1})\cap Dom(\sqrt{2-x})\\
& =[1,\infty)\cap(-\infty,2]=[1,2],
and from the above table:
Dom(h_{2})=Dom(\sqrt{(x-1)(2-x)}=[1,2].
\] Therefore $h_{1}(x)=h_{2}(x)$ everywhere they are defined. In other words, they are equal functions.
(b) Similarly, using the following table:
Dom(u_{1}) & =Dom(\sqrt{x-1})\cap Dom(\sqrt{x-2})\\
& =[1,\infty)\cap[2,\infty)=[2,\infty)
Dom(u_{2})=(-\infty,1]\cup[2,\infty).
\] Therefore $u_{1}(x)$ and $u_{2}(x)$ are equal only when $x\in[2,\infty)$. Because the domains of $u_{1}(x)$ and $u_{2}(x)$ are not equal, they are not equal functions.
Determine the natural domain of the following functions:
(a) $f(x)=\dfrac{\sqrt{1-x}}{\log x}$
(b)$g(u)=\dfrac{e^{u}}{\sqrt{|1+u|}}$
Recall that $\log x$ is defined when $x>0$ and
\log x=0\Rightarrow x=1
\] (a)
Dom(f) & =Dom(\sqrt{1-x})\cap Dom(\log x)\cap\{x|\log x\neq0\}\\
& =(-\infty,1]\cap(0,\infty)\cap\{x|x\neq1\}\\
& =(0,1).
Dom(g) & =Dom(e^{u})\cap Dom(\sqrt{|1+u|})\cap\{u|\ \sqrt{|1+u|}\neq0\}\\
& =\mathbb{R}\cap\mathbb{R}\cap\{u|\ |1+u|\neq0\}\\
& =\mathbb{R}\cap\mathbb{R}\cap\{u|\ u\neq\pm1\}\\
& =\{u|\ u\neq\pm1\}\\
& =\mathbb{R}-\{-1,1\},
which can also be written as
Dom(g)=(-\infty,-1)\cup(-1,1)\cup(1,\infty).
Graphs of Combined Functions
Suppose the graphs of $f$ and $g$ are known:
to obtain the graph of the function $f+g$, we just add the corresponding $y$-coordinates $f(x)$ and $g(x)$ at each $x$ that belongs both to the domain of $f$ and the domain of $g$
to obtain the graph of the function $f+g$, we just subtract the $y$-coordinate $g(x)$ from the corresponding $y$-coordinate $f(x)$ at each $x$ that belongs both to the domain of $f$ and the domain of $g$
to obtain the graph of the function $fg$, we multiply the corresponding$y$-coordinates $f(x)$ and $g(x)$ at each $x$ that belongs both to the domain of $f$ and the domain of $g$.
1. Review of Fundamentals
1.1 What Is Algebra?
1.2 Sets
1.3 Sets of Numbers
1.4 Inequalities
1.5 Absolute Value
1.6 Intervals
1.7 Laws of Exponents
1.8 The Logarithm
1.9 Equations and Identities
1.10 Polynomials
1.11 Dividing Polynomials
1.12 Special Product Formulas
1.13 Factorization
1.14 Fractions
1.15 Rationalizing Binomial Denominators
1.16 Coordinates in a Plane
1.17 Graphs of Equations
1.18 Straight Lines
1.19 Solutions and Roots
1.20 Other Types of Equations
1.21 Solving Inequalities
1.22 Absolute Value Equations and Inequalities
1.23 Sigma Notation
2. Functions
2.1 Constants, Variables, and Parameters
2.2 The Concept of a Function
2.3 Natural Domain and Range of a Function
2.4 Graphs of Functions
2.5 Vertical Line Test
2.6 Domain and Range Using Graph
2.7 Piecewise-Defined Functions
2.8 Equal Functions
2.9 Even and Odd Functions
2.10 Examples of Elementary Functions
2.11 Transformations of Functions
2.13 Composition of Functions
2.14 Increasing or Decreasing Functions
2.15 One-to-One Functions
2.16 Inverse Functions
2.17 Bounded Functions
2.18 Periodic Functions
3. Elementary Transcendental Functions
3.1 Exponential Functions
3.2 Logarithmic Functions
3.3 Trigonometric Functions
3.3.1 Angles
3.3.2 Basics of Trigonometric Functions
3.3.3 Trigonometric Identities
3.3.4 Periodicity and Graphs of Trigonometric Functions
3.4 Inverse Trigonometric Functions | CommonCrawl |
Materials Research (2)
European Psychiatry (4)
High Power Laser Science and Engineering (3)
Journal of Plasma Physics (2)
Parasitology (1)
The British Journal of Psychiatry (1)
European Psychiatric Association (4)
Materials Research Society (2)
The Royal College of Psychiatrists (1)
Plasma channel formation in the knife-like focus of laser beam
O. G. Olkhovskaya, G. A. Bagdasarov, N. A. Bobrova, V. A. Gasilov, L. V. N. Goncalves, C. M. Lazzarini, M. Nevrkla, G. Grittani, S. S. Bulanov, A. J. Gonsalves, C. B. Schroeder, E. Esarey, W. P. Leemans, P. V. Sasorov, S. V. Bulanov, G. Korn
Journal: Journal of Plasma Physics / Volume 86 / Issue 3 / June 2020
Published online by Cambridge University Press: 02 June 2020, 905860307
The plasma channel formation in the focus of a knife-like nanosecond laser pulse irradiating a gas target is studied theoretically, and in gas-dynamics computer simulations. The distribution of the electromagnetic field in the focus region, obtained analytically, is used to calculate the energy deposition in the plasma, which then is implemented in the magnetohydrodynamic computer code. The modelling of the channel evolution shows that the plasma profile, which can guide the laser pulse, is formed by the tightly focused short knife-like lasers. The results of the simulations show that a proper choice of the convergence angle of a knife-like laser beam (determined by the focal length of the last cylindrical lens), and laser pulse duration may provide a sufficient degree of azimuthal symmetry of the formed plasma channel.
Towards laser ion acceleration with holed targets
Prokopis Hadjisolomou, S. V. Bulanov, G. Korn
Published online by Cambridge University Press: 26 May 2020, 905860304
Although the interaction of a flat foil with currently available laser intensities is now considered a routine process, during the last decade, emphasis has been given to targets with complex geometries aiming at increasing the ion energy. This work presents a target geometry where two symmetric side holes and a central hole are drilled into the foil. A study of the various side-hole and central-hole length combinations is performed with two-dimensional particle-in-cell simulations for polyethylene targets and a laser intensity of $5.2\times 10^{21}~\text{W}~\text{cm}^{-2}$ . The holed targets show a remarkable increase of the conversion efficiency, which corresponds to a different target configuration for electrons, protons and carbon ions. Furthermore, diffraction of the laser pulse leads to a directional high energy electron beam, with a temperature of ${\sim}40~\text{MeV}$ , or seven times higher than in the case of a flat foil. The higher conversion efficiency consequently leads to a significant enhancement of the maximum proton energy from holed targets.
P01-168 - Prevention of Depression in Adolescents: Evaluation of an Information Booklet on Depression: Findings of a Pilot Study
A.-K. Allgaier, Y. Schiller, G. Schulte-Körne
Journal: European Psychiatry / Volume 25 / Issue S1 / 2010
Published online by Cambridge University Press: 17 April 2020, p. 1
Just like adults, adolescents still have negative attitudes towards mental illness and lack knowledge about depression. Public awareness campaigns proved to be effective in adults, but education and antistigma campaigns focussing on adolescents are rare. We developed an information booklet on depression as a first step towards a bigger campaign for youths. We investigate whether this booklet increases knowledge about depression and reduces stigmatised attitudes. Furthermore, we analyse the adolescents' assessment of our booklet.
In a pilot study, the booklet will be evaluated in 100 9th grade pupils in Munich, aged between 14 and 16. Knowledge and attitudes about depression are investigated in a pre-post-design. Adolescents' assessment of the booklet is rated on four-point rating scales which serve as a basis for a revision of the booklet after the pilot study. Data are collected using questionnaires specifically designed for the study.
The pilot study is ongoing until December 2009. Baseline data about depression knowledge and attitudes are presented. The pre-post comparison of knowledge and attitudes is pointed out. Further, the adolescents' assessment of the depression booklet is illustrated.
Based on the findings of the pilot study, the booklet will be revised. Afterwards, it will be evaluated on a sample of 500 pupils in a pre-post-follow-up design covering stability of the changes for a one month period. In the long run, the booklet shall be spread to a wider public using synergistic effects with local campaigns such as the "Munich Alliance Against Depression".
P01-169 - Psychometric Properties of Two new Screening Instruments for Depression in Childhood and Adolescence: Findings from a Pilot Study
A.-K. Allgaier, B. Frühe, K. Pietsch, G. Schulte-Körne
Depression is highly prevalent, but often unidentified in pediatric primary care. Without treatment, early-onset depression often leads to psychosocial impairment and is associated with a high risk of recurrence, chronicity and comorbid mental illnesses. While depression screening is recommended for adults in the primary care setting, screening instruments for children and adolescents are still rare. Therefore, the study group developed two depression screeners in German language, specifically designed for this age group. In a pilot study, psychometric properties of these screeners are computed.
This is a cross-sectional study of a pediatric clinic sample of 100 children and adolescents aged 9-16 years. To assess screener performance, item analysis and realibility measures were calculated. Further, validity measures were computed, using a standardized clinical interview as gold standard criterion for depression diagnosis.
The pilot study is ongoing until January 2009. Items analysis results are presented as well as internal consistency. Validity measures are illustrated by ROC curve analyses.
Based of the results of the pilot study, the screening instruments will be revised. To confirm our preliminary findings, the screeners will be tested in a larger sample during the main study. If the depression screeners have proven to be reliable and valid, in the long run they can be implemented in pediatric primary care as a diagnostic aid.
1648 – What Do Adolescents Know About Depression? Efficacy And Utility Of An Information Booklet
A.-K. Allgaier, Y. Schiller, R. Eberle-Sejari, G. Schulte-Körne
Adolescents still lack knowledge about depression and its treatment. As a result, depressed adolescents are confronted with stigmata that are barriers to treatment-seeking. Studies in adults demonstrated that depression literacy can be increased by educational material.
We developed an information booklet on depression specifically addressing adolescents aged 13 to 17. For the first time, it was examined whether knowledge about depression in youth can be enhanced by reading an educational leaflet.
The information booklet was evaluated in 628 German ninth graders (M= 15.1 years, 58% boys) in a pre-postfollow- up design.Using study-specific questionnaires, knowledge about seven topics on depression was assessed. Key subjects were symptoms and treatment of depression as well as suicidality.In addition, students evaluated the booklet's layout, content and utility.
Power-analysis of the pre-post-follow-up-comparison yielded effect sizes of knowledge enhancement between eta2 = 0.07 (medium) and 0.56 (large) for all topics of the booklet.The largest effect sizes were found for the topics "symptoms" and "antidepressants". Sub-analysis on school types and gender showed the highest increase in knowledge in well-educated students and in girls. The participating youth assessed the booklet positively regarding all categories, with a mean overallrating of 2.15 on a scale from 1 (very good) to 6 (fail).
Although students' baseline knowledge about depression was good, girls and boys of all school types significantly increased depression literacy. Thus, the booklet can help reducing stigmata and treatment barriers in adolescents.
856 – Early Detection Of Depression In Children And Adolescents: The New Children's Depression Screener (child-s) And The Depression Screener For Teenagers (desteen)
A.-K. Allgaier, K. Dolle, K. Pietsch, B. Frühe, B. Saravo, G. Schulte-Körne
Depression in children and adolescents is still underdiagnosed. Valid and brief screening-tools for these age groups are missing.
For children and adolescents in particular, we developed the screening-tools ChilD-S and DesTeen and assessed their validity as opposed to established instruments.
Both screeners were validated in a paediatric (228 children, 316 adolescents) and a clinical sample (77 children, 87 adolescents). Gold standard for validation were ICD-10 diagnoses of a depressive episode or dysthymia based on a structured interview. Using receiver operating characteristic analyses, the area under the curve (AUC) and the cut-offs with the highest sum of sensitivity and specificity were computed. In addition, it was examined whether the instruments differed significantly in validity measures.
Point-prevalences were 5.3% (children) and 9.8% (adolescents) in paediatric care and 23.5% and 26.1% in clinical care.The 8-item ChilD-S yielded AUCs of 98% and 93% in the paediatric and in the clinical sample. Sensitivities were 100% and 94%, with specificities of 87% and 78%.The abbreviated 6-item DesTeen showed an AUC of 95% in both settings. Sensitivities were 100% and 91% in the paediatric and in the clinical sample, specificities were 82% and 89%. Being even shorter, the two screening-tools performed equally well or better than the established instruments.
The ChilD-S and the DesTeen discriminate well between depression and depression-like symptoms in somatic diseases as well as between depression and other forms of mental disorders. Valid and brief, they can both be recommended for usage in paediatric and clinical settings.
Petawatt and exawatt class lasers worldwide
Colin N. Danson, Constantin Haefner, Jake Bromage, Thomas Butcher, Jean-Christophe F. Chanteloup, Enam A. Chowdhury, Almantas Galvanauskas, Leonida A. Gizzi, Joachim Hein, David I. Hillier, Nicholas W. Hopps, Yoshiaki Kato, Efim A. Khazanov, Ryosuke Kodama, Georg Korn, Ruxin Li, Yutong Li, Jens Limpert, Jingui Ma, Chang Hee Nam, David Neely, Dimitrios Papadopoulos, Rory R. Penman, Liejia Qian, Jorge J. Rocca, Andrey A. Shaykin, Craig W. Siders, Christopher Spindloe, Sándor Szatmári, Raoul M. G. M. Trines, Jianqiang Zhu, Ping Zhu, Jonathan D. Zuegel
Journal: High Power Laser Science and Engineering / Volume 7 / 2019
Published online by Cambridge University Press: 22 August 2019, e54
In the 2015 review paper 'Petawatt Class Lasers Worldwide' a comprehensive overview of the current status of high-power facilities of ${>}200~\text{TW}$ was presented. This was largely based on facility specifications, with some description of their uses, for instance in fundamental ultra-high-intensity interactions, secondary source generation, and inertial confinement fusion (ICF). With the 2018 Nobel Prize in Physics being awarded to Professors Donna Strickland and Gerard Mourou for the development of the technique of chirped pulse amplification (CPA), which made these lasers possible, we celebrate by providing a comprehensive update of the current status of ultra-high-power lasers and demonstrate how the technology has developed. We are now in the era of multi-petawatt facilities coming online, with 100 PW lasers being proposed and even under construction. In addition to this there is a pull towards development of industrial and multi-disciplinary applications, which demands much higher repetition rates, delivering high-average powers with higher efficiencies and the use of alternative wavelengths: mid-IR facilities. So apart from a comprehensive update of the current global status, we want to look at what technologies are to be deployed to get to these new regimes, and some of the critical issues facing their development.
Fast magnetic energy dissipation in relativistic plasma induced by high order laser modes
HEDP and HPL 2016
Y. J. Gu, Q. Yu, O. Klimo, T. Zh. Esirkepov, S. V. Bulanov, S. Weber, G. Korn
Published online by Cambridge University Press: 22 June 2016, e19
Fast magnetic field annihilation in a collisionless plasma is induced by using TEM(1,0) laser pulse. The magnetic quadrupole structure formation, expansion and annihilation stages are demonstrated with 2.5-dimensional particle-in-cell simulations. The magnetic field energy is converted to the electric field and accelerate the particles inside the annihilation plane. A bunch of high energy electrons moving backwards is detected in the current sheet. The strong displacement current is the dominant contribution which induces the longitudinal inductive electric field.
Avalanche boron fusion by laser picosecond block ignition with magnetic trapping for clean and economic reactor
H. Hora, G. Korn, S. Eliezer, N. Nissim, P. Lalousis, L. Giuffrida, D. Margarone, A. Picciotto, G.H. Miley, S. Moustaizis, J.-M. Martinez-Val, C.P.J. Barty, G.J. Kirchhoff
Published online by Cambridge University Press: 11 October 2016, e35
Measured highly elevated gains of proton–boron (HB11) fusion (Picciotto et al., Phys. Rev. X 4, 031030 (2014)) confirmed the exceptional avalanche reaction process (Lalousis et al., Laser Part. Beams 32, 409 (2014); Hora et al., Laser Part. Beams 33, 607 (2015)) for the combination of the non-thermal block ignition using ultrahigh intensity laser pulses of picoseconds duration. The ultrahigh acceleration above $10^{20}~\text{cm}~\text{s}^{-2}$ for plasma blocks was theoretically and numerically predicted since 1978 (Hora, Physics of Laser Driven Plasmas (Wiley, 1981), pp. 178 and 179) and measured (Sauerbrey, Phys. Plasmas 3, 4712 (1996)) in exact agreement (Hora et al., Phys. Plasmas 14, 072701 (2007)) when the dominating force was overcoming thermal processes. This is based on Maxwell's stress tensor by the dielectric properties of plasma leading to the nonlinear (ponderomotive) force $f_{\text{NL}}$ resulting in ultra-fast expanding plasma blocks by a dielectric explosion. Combining this with measured ultrahigh magnetic fields and the avalanche process opens an option for an environmentally absolute clean and economic boron fusion power reactor. This is supported also by other experiments with very high HB11 reactions under different conditions (Labaune et al., Nature Commun. 4, 2506 (2013)).
Magnetic and Transport Properties of Ni Doped Pr0.5Ca0.5Mn1-xNixO3-δ
L. Damari, J. Pelleg, G. Gorodetsky, C. Korn, V. Markovich, A. Shames, X. Wu, D. Mogilyansky, I. Fita
Journal: MRS Online Proceedings Library Archive / Volume 1118 / 2008
Published online by Cambridge University Press: 15 March 2011, 1118-K07-02
Structural, magnetic and transport properties of Pr0.5Ca0.5Mn1- x Ni x O3 (x =0, 0.04, 0.07, 0.1) series have been investigated. The substitution of Mn ions by Ni induces drastic changes in the magneto-transport properties of Pr0.5Ca0.5MnO3. The physical behavior depends drastically on O stoichiometry. For the low Ni doping x=0.04, 0.07, spin glass configuration is rather favored than ferromagnetism; while for x=0.10, the ferromagnetic phase is significantly suppressed. Pronounced irreversibility between zero-field cooled (ZFC) and field cooled (FC) magnetization and a kink in the ZFC curve observed for x=0.04 and 0.07 are indicative of spin-glass-like state. Applied hydrostatic pressure of about 10 kbar reduces the temperature of charge ordering in an x=0 sample by about 10 K indicating a pressure induced suppression of the Jahn-Teller distortions and of the electron-phonon coupling. Electron magnetic resonance (EMR) unambiguously evidences appearance of FM phase in Ni doped manganites. Temperature dependence of EMR spectra parameters allow us to speculate on the effect of magnetic inhomogeneities.
Schistosoma: analysis of monoclonal antibodies reactive with the circulating antigens CAA and CCA
A. M. Deelder, G. J. Van Dam, D. Kornelis, Y. E. Fillié, R. J. M. Van Zeyl
Journal: Parasitology / Volume 112 / Issue 1 / January 1996
Published online by Cambridge University Press: 06 April 2009, pp. 21-35
Print publication: January 1996
Using spleen cells of mice infected or immunized respectively with cercariae or antigen preparations of Schistosoma mansoni, S. haematobium or S. japonicum monoclonal antibodies (mAbs) were produced against the schistosome gut-associated antigens CAA (circulating anodic antigen) and CCA (circulating cathodic antigen). Fusions nearly exclusively produced either anti-CAA (n = 25) or anti-CCA mAbs (n = 55) with a strong isotype restriction (IgM, IgG1 and IgG3) against both antigens, the majority of anti-CAA mAbs being IgG1 and the majority of anti-CCA mAbs being IgM. The mAbs, which on the basis of their selection were reactive with multiple carbohydrate epitopes of CAA or CCA, were applied in different immunological techniques including immunofluorescence, a dot immunobinding assay and immuno-electrophoresis to study the epitope repertoire. Anti-CAA mAbs were found to be reactive with 5 different epitopes, none of which occurred as multiple epitopes on eggs. Anti-CCA mAbs, on the other hand, recognized at least 10 different epitopes, while 44% of anti-CCA mAbs recognized epitopes common to the adult worm and the egg. Both CAA-and CCA-epitopes were found to be developmentally expressed at the level of the tegument in cercariae, schistosomula and 5-day-old lung worms, but in the adult worm were primarily found in the gut. Thus, the production of panels of mAbs has not only resulted in the selection of reagents optimally performing in diagnostic immunoassays, but also allowed a more detailed study of the epitope repertoire of these important schistosome antigens.
Monoamines and Abnormal Behaviour a Multi-Aminergic Perspective
H. M. Van Praag, G. M. Asnis, R. S. Kahn, S. L. Brown, M. Korn, J. M. Harkavy Friedman, S. Wetzler
Journal: The British Journal of Psychiatry / Volume 157 / Issue 5 / November 1990
Classical nosology has been the cornerstone of biological psychiatric research; finding biological markers and eventually causes of disease entities has been the major goal. Another approach, designated as 'functional', is advanced here, attempting to correlate biological variables with psychological dysfunctions, the latter being considered to be the basic units of classification in psychopathology. Signs of diminished DA, 5–HT and NA metabolism, as have been found in psychiatric disorders, are not disorder-specific, but rather are related to psychopathological dimensions (hypoactivity/inertia, increased aggression/anxiety, and anhedonia) independent of the nosological framework in which these dysfunctions occur. Implications of the functional approach for psychiatry include a shift from nosological to functional application of psychotropic drugs. Functional psychopharmacology will be dysfunction-orientated and therefore geared towards utilising drug combinations. This prospect is hailed as progress, both practically and scientifically.
Investigation of anomalous generation of ω0 and 2ω0 harmonics of heating radiation in laser plasma corona by means of holographic gratings
I. V. Aleksandrova, W. Brunner, S. I. Fedotov, R. Güther, M. P. Kalashnikov, G. Korn, A. M. Maksimchuk, Yu. A. Mikhailov, S. Polze, R. Riekher, G. V. Sklizkov
Journal: Laser and Particle Beams / Volume 3 / Issue 2 / May 1985
Print publication: May 1985
The problems concerning the development of high f-number spectrographs with high spectral and spatial resolution using holographic reflecting gratings, for the study of processes of harmonic generation in laser plasmas are considered. The concave holographic gratings and spectrograph schemes used in the "Delfin-1" installation are described. The anomalous generation of ω0 and 2ω0 harmonics from the subcritical density plasma region has been observed experimentally. Possible interpretations of the observed effect are given.
Tdpac-Studies of Electric Field Gradients in Amorphous Metallic Systems
P. Heubes, D. Korn, G. Schatz, G. Zibold
Journal: MRS Online Proceedings Library Archive / Volume 3 / 1980
Published online by Cambridge University Press: 15 February 2011, 385
The time differential perturbed γ-γ angular correlation technique (TDPAC) is applied to the amorphous metallic systems Ga, Bi, In50Au50 and In80Ag20. The electric field gradient tensor probed by 111Cd nuclei shows a broad probability distribution with a relative width of 0.4 – 0.5 for all systems, as suggested by a continuous random structural model. | CommonCrawl |
\begin{document}
\date{} \title{Theta lifts and distinction for regular supercuspidal representations}
\begin{abstract} This article has a twofold purpose. First, by recent works of Kaletha and Loke--Ma, we give an explicit description of the local theta correspondence between regular supercuspidal representations in the equal rank symplectic-orthogonal case. Second, based on this description, we show that the local theta correspondence preserves distinction with respect to unramified Galois involutions. \end{abstract}
\tableofcontents
\section{Introduction} \subsection*{Overview} Theta lifts are concrete realizations of the Langlands functoriality for classical groups or metaplectic groups. In recent years, there are several celebrated developments in this direction, e.g. \cite{lst11,gi14,sz15,gi16,gt16}, just name a few among them. For supercuspidal representations, the cornerstone of admissible representations of $p$-adic reductive groups, a fundamental result of Kudla says that the first occurrence of nonzero theta lifts of supercuspidal representations are also supercuspidal representations. Recently Loke and Ma \cite{lm} gave a description of the local theta correspondence between tame supercuspidal representations in terms of the supercuspidal data. This type of supercuspidal representations was first constructed by Yu \cite{yu01} and developed further by others, e.g. \cite{kim07,hm08}, etc.
In his recent work \cite{kal}, Kaletha found much simpler data, called tame regular elliptic pairs, to parameterize regular supercuspidal representations which are a subclass of tame supercuspidal representations. This new remarkable construction is a generalization of Howe's classical result \cite{how77}, and has many applications in the landscape of the Langlands program. For example, Kaletha constructed the $L$-packets of regular supercuspidal representations (see {\em loc. cit.}), which generalizes previous works \cite{ree08,dr09,dr10,ka14,ry14,kal15}. Hence it is natural to ask for a description of the local theta correspondence between regular supercuspidal representations in terms of tame regular elliptic pairs, which will in turn reflect the relation between the $L$-parameters under theta lifts. We attempt to answer this question in the equal rank symplectic-orthogonal case.
Another application of Kaletha's construction was presented in our previous work \cite{zha2}, which is concerned with the distinction problem. The general theory of distinguished tame supercuspidal representations in terms of supercuspidal data has been studied by Hakim and Murnaghan \cite{hm08}, whose prior or subsequent works treated various typical examples in the case of general linear groups (cf. \cite{hm02a,hm02b,hj12,hak13}). Recently, Hakim \cite{haka,hakb} provided a new appraoch to the construction of tame supercuspidal representations and its application to the distinction problem. On the other hand, it is natural to study the relation between the distinction problem and the Langlands functoriality, especially in the spirit of Sakellaridis and Venkatesh's proposal \cite{sv}, or of Prasad's conjecture \cite{pra} for Galois involutions. In this paper, we will give a simple criterion to detect distinguished regular supercuspidal representations of a general connected reductive group for an unramified Galois involution. Together with our result on theta lifts, this criterion enables us to show that theta lifts preserve distinction.
\subsection*{Main results} Let $E$ be a finite extension field of $\BQ_p$ with $p$ sufficiently large, $\FW$ a $2n$-dimensional symplectic space over $E$ and $\FG=\Sp(\FW)$. We fix an additive character $\psi$ of $E$ of conductor $\fp_E$, where $\fp_E$ is the maximal ideal of the integer ring $\fo_E$ of $E$.
Let $\pi=\pi_{(\FS,\mu,j)}$ be a regular supercuspidal representation of $\FG(E)$, where $\FS$ is a torus over $E$, $\mu$ is a character of $\FS(E)$ and $j:\FS\hookrightarrow \FG$ is an $E$-embedding such that $(j\FS,\mu)$ is a tame regular elliptic pair of $\FG$. We call $(\FS,\mu,j)$ a regular elliptic triple for short. Our first result (Theorem \ref{thm. theta}) says that up to equivalence there exists a unique $2n$-dimensional quadratic space $\FV$ over $E$ such that the theta lift $\pi'=\theta_{\FV,\FW,\psi}\left(\pi\right)$ is nonzero. Moreover we show that $\pi'$ is a regular supercuspidal representation of ${\mathrm{O}}(\FV)(E)$, and can be described explicitly in terms of regular elliptic triples. Loke--Ma's description of the local theta correspondence in terms of supercuspidal data is modulo the knowledge about theta lifts over finite fields. However, in our situation, things become easier and clearer, since due to \cite{sri} and \cite{amr96} we have a complete understanding about theta lifts for regular Deligne-Lusztig cuspidal representations.
Next, we turn to the distinction problem. Now we suppose that $F$ is a subfield of $E$ such that $[E:F]=2$ and $E/F$ is unramified. We also assume that $\FW$ has an $F$-structure coming from $W$, i.e. $\FW$ is obtained from a $2n$-dimensional symplectic space $W$ over $F$ by extension of scalars. Thus $G=\Sp(W)$ is a subgroup of $\FG$, which is fixed by the nontrivial unramified Galois involution $\sigma\in\Gal(E/F)$. Recall that we say $\pi$ is $G(F)$-distinguished if $\Hom_{G(F)}(\pi,{\bf1})$ is nonzero. If $\pi$ is a distinguished regular supercuspidal representation, we will show that the quadratic space $\FV$ in the above paragraph has an $F$-structure coming from $V$ such that the theta lift of $\pi$ to ${\mathrm{O}}(\FV)$ is ${\mathrm{O}}(V)(F)$-distinguished (see Theorem \ref{thm. theta and distinction}). The proof of Theorem \ref{thm. theta and distinction} relies on the results about the distinction problem for general involutions developed in our prior work \cite{zha2}, and also a refinement for unramified Galois involutions given in \S \ref{subsec. criterion}.
It would be of some interests to extend the results in this paper to the dual pair $(\Mp(\FW),{\mathrm{O}}(\FV))$ where $\Mp(\FW)$ is the metaplectic group with $\dim\FW=2n$ and ${\mathrm{O}}(\FV)$ is an odd orthogonal group with $\dim\FV=2n+1$. We can also consider the dual pair $(\Sp(\FW),{\mathrm{O}}(\FV))$ with $\dim\FW=2n$ and $\dim\FV=2n+2$.
\subsection*{Organization of the paper} The assumptions on the residue characteristic $p$, and necessary notation and convention are given in the rest of this section. In Section \ref{sec. prelim} we recall the background knowledge of supercuspidal data, block decompositions of supercuspidal data for classical groups, and also regular supercuspidal representations. We study the local theta correspondence between regular supercuspidal representations in Section \ref{sec. theta lifts}, and its relation with the distinction problem in Section \ref{sec. distinction}.
\subsection*{Assumptions} Throughout this article, $F$ is a local field which is a finite extension of $\BQ_p$. We assume that $p$ is sufficiently large such that it satisfies both of the assumptions in \cite[\S 1]{zha2} and the hypotheses in \cite[\S 3.4]{kim07}. We refer the reader to \cite[\S 2.1]{kal}, \cite[\S 3.4]{kim07}, \cite[\S 3.1]{lm}, \cite[\S 1]{zha2} for detailed explanations on the roles that these assumptions play in various theories surrounding supercuspidal representations.
\subsection*{Notation and convention} For a $p$-adic field $F$, we denote by $\fo_F$ its ring of integers, by $\fp_F$ the maximal ideal of $\fo_F$, by $\sfk_F$ the residue field, and by $\val_F$ the normalized valuation on $F$ such that $\val_F(F^\times)=\BZ$. For an additive character $\psi$ of $F$ with conductor $\fp_F$, we use $\bar{\psi}$ to denote the additive character on $\sfk_F$ induced by $\psi$. For a quadratic field extension $E$ of $F$, we denote $$E^-=\{x\in E\ |\ {\mathrm{Tr}}_{E/F}(x)=0\},\ \textrm{and}\ E^1=\{x\in E^\times\ |\ {\mathrm{Nm}}_{E/F}(x)=1\}.$$
For a reductive group $G$ over $F$, we denote by ${^\circ G}$ its connected component of the identity, and $Z(G)$ the center of $G$. We use the corresponding lowercase gothic letter $\fg$ to denote its Lie algebra. For $\Gamma\in\fg$, let $Z_G(\Gamma)$ denote the stabilizer of $\Gamma$ in $G$. If $E$ is a finite extension of $F$, we use $\R_{E/F}G$ to denote the Weil restriction of $G$, which is a reductive group over $F$ whose $F$-rational points are $G(E)$.
We denote by $\CB(G,F)$ and $\CB^\red(G,F)$ the extended and reduced Bruhat-Tits buildings of $G(F)$ respectively. For $x\in\CB^\red(G,F)$, we denote by $G(F)_x$ the stabilizer of $x$ in $G(F)$, by $G(F)_{x,0}$ the connected parahoric subgroup attached to $x$, by $G(F)_{x,0+}$ the pro-unipotent radical of $G(F)_{x,0}$, and by $^\circ\sfG_x$ the corresponding connected reductive quotient group over $\sfk_F$. Moreover, if $G$ is a classical group, we denote by $\sfG_x$ the reductive group over $\sfk_F$ such that $\sfG_x(\sfk_F)=G(F)_x/G(F)_{x,0+}$.
As introduced in \cite{mp94}, we denote by $G(F)_{x,r}$ the Moy-Prasad filtration subgroups of $G(F)_{x,0}$ for $r\in\BR_{\geq0}$, and by $\fg(F)_{x,r}$ the filtration lattices of $\fg(F)$ for $r\in\BR$. We write $G(F)_{x,r+}:=\bigcup\limits_{s>r}G(F)_{x,s}$ and $\fg(F)_{x,r+}:=\bigcup\limits_{s>r}\fg(F)_{x,s}$. For $s\geq r$, we write $$G(F)_{x,r:s}:=G(F)_{x,r}/G(F)_{x,s},\quad G(F)_{x,r:s+}:=G(F)_{x,r}/G(F)_{x,s+},$$ $$\fg(F)_{x,r:s}:=\fg(F)_{x,r}/\fg(F)_{x,s},\quad \fg(F)_{x,r:s+}:=\fg(F)_{x,r}/\fg(F)_{x,s+}.$$ For $\Gamma\in\fg(F)$, we denote by $\val(\Gamma)$ the depth of $\Gamma$, that is,
$$\val(\Gamma)=\sup\limits_{x\in\CB^\red(G,F)}\{r\ |\ x\in\fg_{x,r}\bs\fg_{x,r+} \}.$$
When $G$ is a classical group, there exists a correspondence between the set of self-dual lattice functions and $\CB(G,F)$, or a description of the Moy-Prasad filtration in terms of lattice functions. We refer to \cite[\S 4]{lm16} for a detailed summary.
\section{Preliminaries}\label{sec. prelim}
\subsection{Supercuspidal data} Let $G$ be a reductive group over $F$. We fix an additive character $\psi$ of $F$ of conductor $\fp_F$. Throughout this paper, a {\em supercuspidal $G$-datum} has two meanings. We will abuse the notion when there is no confusion.
First, a {\em supercuspidal $G$-datum} $\Psi=\left(\vec{G},x,\rho,\vec{\phi}\right)$ means a {\em generic cuspidal $G$-datum} in the sense of \cite{yu01}. To be precise, the datum $\Psi$ statisfies \begin{enumerate} \item $\vec{G}$ is a \emph{tamely ramified twisted Levi sequence} $\vec{G}=(G^0,...,G^d)$ in $G$ such that $Z(G^0)/Z(G)$ is anisotropic. \item $x\in\CB^\red(G^0,F)$ is a vertex.
\item $\rho$ is an irreducible representation of $K^0:=G^0(F)_x$ such that $\rho|_{G^0(F)_{x,0+}}$ is {\bf1}-isotypic and the compactly induced representation $\pi_{-1}=\ind_{K^0}^{G^0(F)}(\rho)$ is irreducible. \item $\vec{\phi}=(\phi_0,...,\phi_d)$ with $\phi_i$ a quasicharacter of $G^i(F)$ for each $0\leq i\leq d$ and being $G^{i+1}$-{\em generic} (cf. \cite[Definition 3.9]{hm08}) for $i\neq d$. We require that: if $d=0$ then $\phi_0$ is of depth $r_0\geq0$; if $d>0$ and $\phi_d$ is nontrivial then $\phi_i$ is of depth $r_i$ for $i=0,...,d$ and $0<r_0<r_1<\cdots<r_{d-1}<r_d$; if $d>0$ and $\phi_d$ is trivial then $\phi_i$ is of depth $r_i$ for $i=0,...,d-1$ and $0<r_0<r_1<\cdots<r_{d-1}$. We will call $\vec{r}=(r_0,...,r_d)$ the {\em depth} of $\vec{\phi}$ for short, and denote
$\phi^\circ:=\prod\limits_{i=0}^d\phi_i|_{G^0(F)}$. \end{enumerate}
When $G$ is a classical group, things can be simplified as in \cite[\S 3]{lm}. We also adopt the notion therein: a {\em supercuspidal $G$-datum} $\Sigma=(x,\Gamma,\phi,\rho)$ satisfies \begin{enumerate} \item $\Gamma$ is a \emph{tamely ramified semisimple} element of $\fg(F)$ and admits a \emph{good factorization} $\Gamma=\sum\limits_{-1\leq i\leq d}\Gamma_i$ (cf. \cite[\S 3.2]{lm}). \item Set $G^0=Z_G(\Gamma)$. Then $Z\left({^\circ G^0}\right)$ is anisotropic.
\item $x\in\CB^\red(G^0,F)$ is a vertex. \item $\rho$ is an irreducible cuspidal representation of the finite group $\sfG^0_x(\sfk_F)$.
\item $\phi:G^0(F)_x\ra\BC^\times$ is a character such that $\phi|_{G^0(F)_{x,0+}}=\psi_\Gamma|_{G^0(F)_{x,0+}}$ (see \cite[\S 3, (3.2)]{lm} for the definition of $\psi_\Gamma$). \end{enumerate}
We remark that in the above definition we always require ``$\Gamma_{-1}=0$''. See Remarks 2 after \cite[Definition 3.3]{lm} on this point. Given a supercuspidal $G$-datum $\Psi=\left(\vec{G},x,\rho,\vec{\phi}\right)$ such that $\phi_i$ is represented by $\Gamma_i$, we can obtain a supercuspidal $G$-datum $\Sigma=(x,\Gamma,\phi,\rho)$ by setting $\Gamma=\sum\limits_{0\leq i\leq d}\Gamma_i$ and $\phi=\phi^\circ|_{G^0(F)_x}$. Conversely, given a supercuspidal $G$-datum $\Sigma=(x,\Gamma,\phi,\rho)$, there exists a supercuspidal $G$-datum $\Psi=\left(\vec{G},x,\rho,\vec{\phi}\right)$ such that $\phi_i$ is represented by $\Gamma_i$ and $\phi=\phi^\circ|_{G^0(F)_x}$.
When $G$ is connected, Yu obtains an irreducible supercuspidal representation $\pi_\Psi$ or $\pi_\Sigma$ of $G(F)$ from a supercuspidal $G$-datum $\Psi$ or $\Sigma$. When $G$ is an orthogonal group, as argued in \cite[\S 3.4]{lm}, we can also get an irreducible supercuspidal representation $\pi_\Sigma$ from a supercuspidal $G$-datum $\Sigma$.
\subsection{Block decompositions} Now we record the block decomposition of a supercuspidal $G$-datum $\Sigma=(x,\Gamma,\phi,\rho)$, which is referred to \cite[\S 4]{lm}. We restrict ourselves to the case that $(V,\pair{\cdot,\cdot}_V)$ is a symplectic or quadratic space over $F$ and $G$ is the corresponding isometry group.
Viewing $\Gamma\in\fg(F)$ as an element of $\End_F(V)$, the algebra $F[\Gamma]$ is isomorphic to a product $\prod\limits_{j\in J}F_j$ of tamely ramified finite extensions $F_j$ of $F$. Since $V$ is an $F[\Gamma]$-module, we have a decomposition $V =\bigoplus\limits_{j\in J}V_j$ where $V_j$ is a subspace and $F_j$ acts faithfully on it. Let $\Gamma^j$ be the $F_j$-component of $\Gamma$ in $\prod\limits_{j\in J}F_j$. Set
$$\{{^br}>\cdots>{^1r}>{^0r}=0\}=\{-\val(\Gamma^j)\geq0\ |\ j\in J\}$$ as (4.1) of {\em loc. cit.}, and put $${^\ell \Gamma}=\sum\limits_{\val(\Gamma^j)=-{^\ell r}}\Gamma^j,\quad {^\ell V}=\bigoplus\limits_{\val(\Gamma^j)=-{^\ell r}}V_j$$ for each $0\leq\ell\leq b$. Note that we have required ``$\Gamma_{-1}=0$'' in the definition of $\Sigma$. Therefore we have ${^0\Gamma}=0$ and ${^0V}=\ker(\Gamma)$. The restriction of $\pair{\cdot,\cdot}_V$ to each ${^\ell V}$ is non-degenerate, whose isometry group is denoted by ${^\ell G}$.
According to Definition 4.4 of {\em loc. cit.}, a {\em depth-zero single block} of $G$ is a supercuspidal $G$-datum of the form $(x,0,\bf{1},\rho)$, and a {\em positive depth single block} of $G$ is a supercuspidal $G$-datum $\Sigma=(x,\Gamma,\phi,\rho)$ such that $\Gamma^j$ has the same negative valuation for all $j\in J$. By Proposition 4.5 of {\em loc. cit.}, there exists a {\em block decomposition} $\bigoplus\limits_{0\leq\ell\leq b}{^\ell\Sigma}=\left({^\ell x},{^\ell\Gamma},{^\ell\phi},{^\ell\rho}\right)$ of $\Sigma=(x,\Gamma,\phi,\rho)$ such that $^0\Sigma$ is a depth-zero single block of $^0G$ and $^\ell\Sigma$ is a positive depth single block of $^\ell G$ for each $1\leq\ell\leq b$. See Proposition 4.5 of {\em loc. cit.} for more details on the block decompositions. Conversely, given single blocks satisfying certain natural conditions, we can obtain a supercuspidal datum by direct sum (cf. Lemma 4.7 of {\em loc. cit.}).
\subsection{Regular supercuspidal representations}
In \cite{kal}, Kaletha defined a subclass of supercuspidal $G$-data, called regular regular supercuspidal data. A supercuspidal $G$-datum $\Psi$ is called {\em regular} if $\rho|_{G^0(F)_{x,0}}$ contains the inflation to $G^0(F)_{x,0}$ of an irreducible cuspidal representation $\kappa$ of $^\circ\sfG^0_x(\sfk_F)$ such that $\kappa$ is a Deligne-Lusztig cuspidal representation $\pm R_\sfS^{^\circ\sfG^0_x}(\lambda)$ attached to an elliptic maximal torus $\sfS$ of $^\circ\sfG^0_x$ and a character $\lambda$ of $\sfS(\sfk_F)$ in general position. An irreducible supercuspidal representation $\pi$ of $G(F)$ is called {\em regular} if it is equivalent to $\pi_\Psi$ for some regular supercuspidal $G$-datum $\Psi$.
In {\em loc. cit.}, Kaletha found more convenient data than supercuspidal data to parameterize regular supercuspidal representations. Recall that a {\em tame regular elliptic pair $(S,\mu)$} of $G$ contains a tame elliptic maximal torus $S$ of $G$ and a character $\mu$ of $S(F)$ satisfying the conditions in Definition 3.6.5 of {\em loc. cit}. From a tame regular elliptic pair $(S,\mu)$, a {\em Howe factorization} of $(S,\mu)$ (cf. \S 3.7 of {\em loc. cit.}) provides a regular supercuspidal $G$-datum $\Psi=(\vec{G},x,\rho_{(S,\mu_0)},\vec{\phi})$ (also called a Howe factorization) where \begin{enumerate} \item $S$ is a maximally unramified elliptic maximal torus of $G^0$, \item $\mu_0$ is a regular depth-zero character of $S(F)$ with respect to $G^0$, \item $\rho_{(S,\mu_0)}$ is defined and constructed in \S 3.4.3 and \S 3.4.4 of {\em loc. cit.},
\item $\vec{\phi}$ satisfies $\mu=\mu_0\cdot\phi^\circ|_{S(F)}$.
\end{enumerate} Let $\pi_{(S,\mu)}$ denote $\pi_\Psi$, which is a regular supercuspidal representation and whose equivalence class does not depend on the choice of Howe factorizations. It is known that all regular supercuspidal representations are of the forms $\pi_{(S,\mu)}$, and $\pi_{(S_1,\mu_1)}$ and $\pi_{(S_2,\mu_2)}$ are equivalent if and only if the pairs $(S_1,\mu_1)$ and $(S_2,\mu_2)$ are $G(F)$-conjugate.
For latter use, $(S,\mu,j)$ is called a {\em regular elliptic triple} if $S$ is an $F$-torus, $\mu$ is a character of $S(F)$, and $j:S\ra G$ is an $F$-embedding such that $(jS,\mu\circ j^{-1})$ is a tame regular elliptic pair of $G$. If there is no confusion, we will write $\mu$ instead of $\mu\circ j^{-1}$ for short. The meaning of the \emph{Howe factorizations} of $(S,\mu,j)$ is obvious. We will denote by $\pi_{(S,\mu,j)}$ the regular supercuspidal representation $\pi_{(jS,\mu)}$.
When $G$ is a classical group, a regular elliptic triple $(S,\mu,j)$ gives rise to a Howe factorization $\left(\vec{G},x,\rho_{(S,\mu_0)},\vec{\phi}\right)$ and thus a supercuspidal $G$-datum $\Sigma=\left(x,\Gamma,\phi,\rho_{(S,\mu_0)}\right)$. We also call $\Sigma$ a Howe factorization of $(S,\mu,j)$.
\subsection{Regular elliptic triples of classical groups} When $G$ is a classical group, the conjugacy classes of embeddings of maximal tori have a nice parametrization, which is well known (cf. \cite[IV.2]{ss70}, \cite{wal01} or \cite[\S3.1]{li}). As before, we restrict ourselves to symplectic and orthogonal groups, and only consider elliptic maximal tori. Let $(V,\pair{\cdot,\cdot}_V)$ be a $2n$-dimensional symplectic or quadratic space over $F$ and $G$ the corresponding isometry group.
Consider the datum $$(L,L^\circ,c)=\prod_{i\in I}(L_i,L_i^\circ,c_i),$$ where for each $i\in I$ \begin{enumerate} \item $L_i^\circ$ is a field extension of $F$ of degree $m_i$,; \item $L_i$ is a quadratic field extension of $L_i^\circ$; \item $c_i\in L_i^\times$; \item $\sum\limits_{i\in I}m_i=n$. \end{enumerate} Two data $(L,L^\circ,c)$ and $(K,K^\circ,d)$ are said to be equivalent if there exists an $F$-isomorphism $\varphi:L\ra K$ such that $F(L^\circ)=K^\circ$ and $\varphi(c)d^{-1}\in{\mathrm{Nm}}_{K/K^\circ}(K^\times)$.
If $c\in L^-:=\prod\limits_{i\in I}L_i^-$, then $(L,\pair{\cdot,\cdot}_c)$ is a $2n$-dimensional symplectic space over $F$, where $\pair{x,y}_c:={\mathrm{Tr}}_{L/F}(cx\bar{y})$ and $y\mapsto\bar{y}$ is the involution with respect to the quadratic extension $L/L^\circ$. On the other hand, if $c\in L^\circ$, $(L,\pair{\cdot,\cdot}_c)$ is a $2n$-dimensional quadratic space over $F$, where $\pair{\cdot,\cdot}_c$ is defined in the same way. In any case, the norm-one subgroup $L^1:=\prod\limits_{i\in I}L^1_i$ with respect to $L/L^\circ$ preserves $(L,\pair{\cdot,\cdot}_c)$ under multiplication. For convenience, $L^1$ will also denote the norm-one $F$-torus whose $F$-rational points are $L^1$. Thus we obtain an embedding of an elliptic maximal torus $$j:S=L^1\hookrightarrow\U(L),$$ where $\U(L)$ is the isometry group of $(L,\pair{\cdot,\cdot}_c)$.
When $V$ is a symplectic space, the equivalence classes of the data $(L,L^\circ,c)$ with $c\in L^-$ such that $(L,\pair{\cdot,\cdot}_c)\cong (V,\pair{\cdot,\cdot}_V)$ (automatically for symplectic spaces) are bijective with the $G(F)$-conjugacy classes of embeddings of elliptic maximal $F$-tori $j:S=L^1\hookrightarrow\U(L)\cong G$.
When $V$ is an orthogonal space, the equivalence classes of the data $(L,L^\circ,c)$ with $c\in L^\circ$ such that $(L,\pair{\cdot,\cdot}_c)\cong (V,\pair{\cdot,\cdot}_V)$ are bijective with the $G(F)$-conjugacy classes of embeddings of elliptic maximal $F$-tori $j:S=L^1\hookrightarrow\U(L)\cong G$. Note that $j$ is actually an embedding into $\SO(V)$.
Therefore the $G(F)$-conjugacy classes of regular elliptic triples $(S,\mu,j)$ are parameterized by the equivalence classes of the data $$(L,L^\circ,c,\chi)=\prod_{i\in I}(L_i,L_i^\circ,c_i,\chi_i),$$ where \begin{enumerate}
\item $(L,L^\circ,c)$ is as above;
\item $\chi=\bigotimes\limits_{i\in I}\chi_i=\mu$ is a character of $L^1=\prod\limits_{i\in I}L_i^1=S(F)$ such that the corresponding triple $(S,\mu,j)$ satisfies the extra conditions in \cite[Definition 3.6.5]{kal}. \end{enumerate}
\section{Theta lifts}\label{sec. theta lifts} In this section we study the theta lifts of regular supercuspidal representations from symplectic groups to even orthogonal groups. We restrict ourselves to the equal rank case, i.e. the symplectic and quadratic spaces are of the same dimension. The results of \cite{sri} on the theta lifts over finite fields and \cite{lm} on the theta lifts between tame supercuspidal representations are crucial to us. Throughout this section, we suppose that $(W,\pair{\cdot,\cdot}_W)$ is a $2n$-dimensional symplectic space over a $p$-adic field $F$, $\psi$ is a fixed additive character of $F$ of conductor $\fp_F$, and $G=\Sp(W)$.
\subsection{Theta lifts over finite fields}
Let $\mathsf{k}$ be a finite field. Let $\sfW$ and $\mathsf{V}$ be a symplectic and a quadratic vector space over $\mathsf{k}$ of dimension $2n$ respectively. Recall that up to equivalence there are two $2n$-dimensional quadratic spaces $\mathsf{V}^+$ and $\mathsf{V}^-$ over $\mathsf{k}$, whose discriminants are trivial and nontrivial in $\mathsf{k}/\mathsf{k}^{\times2}$ respectively. Denote $\mathsf{G}=\Sp(\mathsf{W})$, $\mathsf{H}^\circ=\SO(\mathsf{V})$ and $\mathsf{H}={\mathrm{O}}(\mathsf{V})$.
Analogous to the $p$-adic case, the $\mathsf{G}(\mathsf{k})$-conjugacy classes of elliptic maximal tori $\sfS$ of $\mathsf{G}$ are parameterized by the data $(\sfL,\sfL^\circ)=\prod\limits_{1\leq i\leq r}\left(\sfL_i,\sfL_i^\circ\right)$, where $\sfL_i^\circ$ and $\sfL_i$ are field extensions of $\sfk$ of degree $m_i$ and $2m_i$ respectively, $\sfL_i^\circ$ is a subfield of $\sfL_i$, and $\sum\limits_{1\leq i\leq r}m_i=n$. Under this correspondence, we have $$\sfS(\sfk)\cong\sfL^1:=\prod\limits_{1\leq i\leq r}\sfL_i^1.$$
Similarly, the $\sfH(\sfk)$-conjugacy classes of elliptic maximal tori $\sfS$ of $\sfH^\circ$ are parameterized by the data $(\sfL,\sfL^\circ)=\prod\limits_{1\leq i\leq r}\left(\sfL_i,\sfL_i^\circ\right)$ with one more condition: we require that $r$ is even (resp. odd) if $\sfV$ is equivalent to $\sfV^+$ (resp. $\sfV^-$). We also have $\sfS(\sfk)\cong\sfL^1$.
The above parametrization establishes a correspondence between the conjugacy classes of elliptic maximal tori of $\sfG$ and those of $\sfH^\circ$: $$\begin{aligned}&\left\{\textrm{elliptic maximal torus of $\sfG$}\right\}/\sfG(\sfk)-\textrm{conj.}\\ \stackrel{1:1}{\longleftrightarrow}&\bigsqcup_{\sfV=\sfV^+,\sfV^-}\left\{\textrm{elliptic maximal torus of $\sfH^\circ$}\right\}/\sfH(\sfk)-\textrm{conj.}\end{aligned}$$ which is explicitly given by $$\left(\sfS\leftrightarrow(\sfL,\sfL^\circ)\right)\ \mapsto\ \left(\sfS\leftrightarrow(\sfL,\sfL^\circ)\right).$$ Let $\Omega(\sfS,\sfG)=N(\sfS,\sfG)/\sfS$, $\Omega(\sfS,\sfH)=N(\sfS,\sfH)/\sfS$ and $\Omega(\sfS,\sfH^\circ)=N(\sfS,\sfH^\circ)/\sfS$ be the absolute Weyl goups. It is well known that $\Omega(\sfS,\sfH^\circ)$ is a subgroup of $\Omega(\sfS,\sfH)$ of index two, and $\Omega(\sfS,\sfH)$ can be naturally identified with $\Omega(\sfS,\sfG)$. Taking $\sfk$-rational points, we have $$\Omega(\sfS,\sfG)(\sfk)=\Omega(\sfS,\sfH)(\sfk)$$ inside $\Aut_\sfk(\sfS)$. We refer to \cite[Proposition 3.1.5]{li} for a description of $\Omega(\sfS,\sfG)(\sfk)$ or $\Omega(\sfS,\sfH)(\sfk)$ in terms of $(\sfL,\sfL^\circ)$.
For an elliptic maximal torus $\sfS$ of $\sfG$ and a character $\mu$ of $\sfS(\sfk)$ in general position, let $\pm R_\sfS^\sfG(\mu)$ be the Deligne-Lusztig cuspidal representation of $\sfG(\sfk)$ attached to $(\sfS,\mu)$. Let $\sfV$ be the quadratic space such that $\sfS\subset{\mathrm{O}}(\sfV)$, and $\sfV'$ the other quadratic space of dimension $2n$. Write $\sfH={\mathrm{O}}(\sfV)$ and $\sfH'={\mathrm{O}}(\sfV')$. Choose $\varepsilon\in\sfH(\sfk)$ such that $\det(\varepsilon)=-1$. The $\sfH^\circ(\sfk)$-conjugacy classes in the $\sfH(\sfk)$-conjugacy class of $\sfS$ is represented by $\{\sfS,\sfS^\varepsilon\}$. Since the stabilizer of $\mu$ in $\Omega(\sfS,\sfG)(\sfk)$ is trivial, so is the stabilizer of $\mu$ in $\Omega(\sfS,\sfH)(\sfk)$. Hence we have $$\left(\pm R_\sfS^{\sfH^\circ}(\mu)\right)^\varepsilon\cong\pm R_{\sfS^\varepsilon}^{\sfH^\circ}(\mu^\varepsilon)\ncong\pm R_\sfS^{\sfH^\circ}(\mu).$$ Therefore $$\pm R_\sfS^{\sfH}(\mu):=\Ind_{\sfH^\circ(\sfk)}^{\sfH(\sfk)}\left(\pm R_\sfS^{\sfH^\circ}(\mu)\right)$$ is an irreducible representation of $\sfH(\sfk)$, which is well defined for the $\sfH(\sfk)$-conjugacy class $\sfS$ in $\sfH^\circ$. Combining \cite[Corollary 1]{sri} and \cite[Proposition 2.1]{amr96}, we obtain:
\begin{lem}\label{lem. theta over finite field} The theta lift of $\pm R_\sfS^\sfG(\mu)$ to $\sfH$ is $\pm R_\sfS^\sfH(\mu)$, and the theta lift of $\pm R_\sfS^\sfG(\mu)$ to $\sfH'$ vanishes. \end{lem}
\begin{rem}\label{rem. theta over finite field} Since $-1\in\Omega(\sfS,\sfG)(\sfk)$, we have $\pm R_\sfS^\sfG(\mu)=\pm R_\sfS^\sfG(\mu^{-1})$. Therefore we can rewrite the above lemma: the theta lift of $\pm R_\sfS^\sfG(\mu)$ to $\sfH$ is $\pm R_\sfS^\sfH(\mu^{-1})$. \end{rem}
\subsection{Depth-zero single block} We first consider theta lifts of regular depth-zero supercuspidal representations. In this case, since $G$ is split, regular depth-zero supercuspidal representations are parameterized by the {\em unramified} regular elliptic triples $(S,\mu,j)$, that is, we further require that $S$ is unramified and $\mu$ is of depth zero. From now on, we fix a uniformizer $\varpi$ of $\fp_F$.
\begin{prop}\label{prop. depth zero} Let $\pi_{(S,\mu,j)}$ be a regular depth-zero supercuspidal representation of $G(F)$. Then we have the following statements. \begin{enumerate} \item Up to equivalence, there exists a unique $2n$-dimensional quadratic space $V$ over $F$ such that the theta lift $\theta_{V,W,\psi}\left(\pi_{(S,\mu,j)}\right)$ is nonzero. \item The theta lift $\theta_{V,W,\psi}\left(\pi_{(S,\mu,j)}\right)$ is a regular depth-zero supercuspidal representation attached to certain regular elliptic triple $(S,\mu^{-1},j_\theta)$ of ${\mathrm{O}}(V)$. \item Suppose that $(S,\mu,j)$ corresponds to $(L,L^\circ,c,\chi)=\prod\limits_{i\in I}(L_i,L_i^\circ,c_i,\chi_i)$. Then
\begin{itemize}
\item $(S,\mu^{-1},j_\theta)$ corresponds to $$(L,L^\circ,c_\theta,\chi^{-1}):=(L,L^\circ,c\tau\varpi,\chi^{-1})=\prod\limits_{i\in I}(L_i,L_i^\circ,c_i\tau_i\varpi,\chi^{-1}_i),$$
\item $(V,\pair{\cdot,\cdot}_V)\cong(L,\pair{\cdot,\cdot}_{c\tau\varpi})$,\end{itemize} where $\tau=\prod\limits_{i\in I}\tau_i$ with $\tau_i\in L_i^-$ and $\val_{L_i}(\tau_i)=0$ for each $i\in I$. \end{enumerate} \end{prop}
\begin{proof} Suppose that $(S,\mu,j)$ corresponds to $(L,L^\circ,c,\chi)=\prod\limits_{i\in I}(L_i,L_i^\circ,c_i,\chi_i)$. We may assume $(W,\pair{\cdot,\cdot}_W)=(L,\pair{\cdot,\cdot}_c)$. Note that all of $L_i$'s and $L_i^\circ$'s are unramified over $F$. Denote by $\sfL_i$ and $\sfL^\circ_i$ the residue fields of $L_i$ and $L_i^\circ$ respectively. Let
$$I_1=\{i\in I \ |\ \val_{L_i}(c_i)\textrm{ is even} \}\ \textrm{and}\ I_2=\{i\in I \ |\ \val_{L_i}(c_i)\textrm{ is odd} \}.$$ We can assume $\val_{L_i}(c_i)=0$ for $i\in I_1$, and $\val_{L_i}(c_i)=1$ for $i\in I_2$. Let $\sL\subset W$ be the unique lattice stable under $S(\fo')$ for each integer ring $\fo'$ of an unramified extensions $F'$ of $F$. Exciplicitly, we can write $$\sL=\bigoplus_{i\in I}\fo_{L_i}.$$ Let $\tilde{\sL}$ be the dual lattice of $\sL$ under $\pair{\cdot,\cdot}_W$. Then $$\tilde{\sL}=\left(\bigoplus_{i\in I_1}\fo_{L_i}\right)\oplus\left(\bigoplus_{i\in I_2}\varpi^{-1}\fo_{L_i}\right).$$ Set $\sfW_1=\sL/\varpi\tilde{\sL}$ and $\sfW_2=\tilde{\sL}/\sL$, which are $\sfk_F$-vector spaces equipped with the induced symplectic forms $\pair{\cdot,\cdot}_W$ and $\varpi\pair{\cdot,\cdot}_W$ respectively. Note that $$\sfW_i=\bigoplus\limits_{j\in I_i}\sfL_j\textrm{ and } \dim_{\sfk_F}(\sfW_i)=2n_i=\sum\limits_{j\in I_i}2m_j\textrm{ for } i=1,2.$$
Let $x$ be the vertex of $\CB^\red(G,F)$ attached to $jS$. Then $\Sigma=\left(x,0,{\bf1},\rho_{(S,\mu)}\right)$ is a depth-zero single block attached to $(S,\mu,j)$. We have $\sfG_x=\sfG_1\times\sfG_2$ where $\sfG_i=\Sp(\sfW_i)$ for $i=1,2$. The elliptic maximal torus of $\sfG_x$ attached to $S$ is $\sfS=\sfS_1\times\sfS_2$ where $\sfS_i$ is the elliptic maximal torus of $\Sp(\sfW_i)$ corresponding to $\prod\limits_{j\in I_i}(\sfL_j,\sfL_j^\circ)$ for $i=1,2$. The depth-zero character $\mu$ of $S(F)$ descends to a character $\bar{\mu}=\bar{\mu}_1\times\bar{\mu}_2$ of $\sfS_1(\sfk_F)\times\sfS_2(\sfk_F)$ in general position. Then the Deligne-Lusztig cuspidal representation $\rho=\rho_{(S,\mu)}$ of $\sfG_x(\sfk_F)$ is $\kappa_1\otimes\kappa_2$ where $\kappa_i=\pm R_{\sfS_i}^{\sfG_i}(\bar{\mu}_i)$ for $i=1,2$.
Now let $V$ be a $2n'$-dimensional quadratic space over $F$ such that $$\pi_V:=\theta_{V,W,\psi}\left(\pi_{(S,\mu,j)}\right)$$ is the first occurrence of the nonzero theta lifts in the Witt tower of $V$. By \cite[Theorem A]{kud86} and \cite[Theorem A]{pan02}, $\pi_V$ is an irreducible depth-zero supercuspidal representation. Let $(y,\rho')$ be a depth-zero $K$-type of $\pi_V$ for $H={\mathrm{O}}(V)$, where $y$ is a vertex of $\CB(H,F)$ and $\rho'$ is an irreducible cuspidal representation of $\sfH_y(\sfk_F)$. Let $\sL'\subset V$ be a lattice attached to $y$. Set $\sfV_1=\tilde{\sL}'/\sL'$ and $\sfV_2=\sL'/\varpi\tilde{\sL}'$, which are quadratic spaces over $\sfk_F$. Denote $\sfH_i={\mathrm{O}}(\sfV_i)$ for $i=1,2$. Then $\sfH_y=\sfH_1\times\sfH_2$ and $\rho'=\kappa'_1\otimes\kappa'_2$, where $\kappa'_i$ is an irreducible cuspidal representation of $\sfH_i(\sfk_F)$ for $i=1,2$. Note that $(\sfG_1,\sfH_1)$ and $(\sfG_2,\sfH_2)$ are two reductive dual pairs over $\sfk_F$. According to \cite[Theorem 5.6]{pan02}, $\rho'$ must be the theta lift of $\rho$, which by definition means that $\kappa'_i=\theta_{\sfV_i,\sfW_i,\bar{\psi}}(\kappa_i)$ for $i=1,2$. Since $\kappa_i=\pm R_{\sfS_i}^{\sfG_i}(\bar{\mu}_i)$, by \cite[Theorem]{sri} or Lemma \ref{lem. theta over finite field} we see $$\dim\sfV_i\geq\dim\sfW_i,\quad\forall\ i=1,2.$$ On the other hand, since $$\dim\sfV_1+\dim\sfV_2=\dim V=2n'$$ and we require $\dim V=2n$ in the statement of the proposition, it must be $\dim\sfV_i=\dim\sfW_i$ for $i=1,2$. By Lemma \ref{lem. theta over finite field}, for each $i=1,2$, the quadratic space $\sfV_i$ is unique up to equivalence so that there exists an embedding $\sfS_i\hookrightarrow\sfH_i$. Hence $V$ is unique up to equivalence. Furthermore we have $\kappa'_i=\pm R_{\sfS_i}^{\sfH_i}(\bar{\mu}^{-1}_i)$. Therefore $\pi_V$ is a regular depth-zero supercuspidal representation.
Actually, if we set $$\left(V,\pair{\cdot,\cdot}_V\right)=\left(L,\pair{\cdot,\cdot}_{c\tau\varpi}\right)\ \textrm{and}\ (S,\mu^{-1},j_\theta)=\left(L,L^\circ,c\tau\varpi,\chi^{-1}\right),$$ the subsets $I_1$ and $I_2$ of $I$ introduced before have the following alternative description:
$$I_1=\{i\in I \ |\ \val_{L_i}(c_i\tau_i\varpi)\textrm{ is odd} \}\ \textrm{and}\ I_2=\{i\in I \ |\ \val_{L_i}(c_i\tau_i\varpi)\textrm{ is even} \}.$$ Let $y$ be the vertex attached to $j_\theta(S)$ and $\sL'$ the corresponding lattice. Then, analogous to the symplectic case, we have $$\tilde{\sL}'/\sL'=\bigoplus\limits_{i\in I_1}\sfL_i=\sfV_1,\quad \sL'/\varpi\tilde{\sL}'=\bigoplus\limits_{i\in I_2}\sfL_i=\sfV_2,$$ and $\sfH_y={\mathrm{O}}(\sfV_1)\times{\mathrm{O}}(\sfV_2)$. The previous arguments immediately imply that $\theta_{V,W,\psi}\left(\pi_{(S,\mu,j)}\right)=\pi_{(S,\mu^{-1},j_\theta)}$. \end{proof}
\begin{rem}\label{rem. depth zero classification}
We retain the notation in Proposition \ref{prop. depth zero}. Suppose that $(S,\mu,j)$ corresponds to $(L,L^\circ,c,\chi)=\prod\limits_{i\in I}(L_i,L_i^\circ,c_i,\chi_i)$. Let $I_1$ (resp. $I_2$) be the subset of $I$ containing the elements $i$'s such that $\val_{L_i}(c_i)$'s are even (resp. odd). Set $r=|I_1|$ and $s=|I_2|$. We have \begin{itemize} \item if $r$ and $s$ is odd, then $\disc(V)=1$ and $\SO(V)$ is split; \item if $r$ and $s$ are even, then $\disc(V)=1$ and $\SO(V)$ is non-split; \item if $r$ is even and $s$ is odd, then $\disc(V)\neq1\in\fo_F^\times/\fo_F^{\times 2}$, the Hasse invariant $\varepsilon(V)=1$, and $\SO(V)$ is quasi-split;
\item if $r$ is odd and $s$ is even, then $\disc(V)\neq1\in\fo_F^\times/\fo_F^{\times 2}$, the Hasse invariant $\varepsilon(V)=-1$, and $\SO(V)$ is quasi-split.
\end{itemize} \end{rem}
\subsection{Positive depth single block} Next we consider the theta lifts of regular supercuspidal representations whose Howe factorizations are positive depth single blocks.
Let $\pi=\pi_{(S,\mu,j)}$ be a regular supercuspidal representation of $G(F)$ such that its Howe factorization $\Sigma=(x,\Gamma,\phi,\rho_{(S,\mu_0)})$ is a single block of positive depth $r$. Note that $\det(\Gamma)\neq 0$. Suppose that $(S,\mu,j)$ corresponds to $(L,L^\circ,c,\chi)$. We may assume $(W,\pair{\cdot,\cdot}_W)=(L,\pair{\cdot,\cdot}_c)$, identify $\Lie(jS)$ with $L^-$, and assume $\Gamma\in\Lie(jS)$.
As introduced in \cite[Definition 5.8]{lm}, we set $V=W$ as an $F$-vector space and equip it with the form $\pair{v_1,v_2}_\Gamma=\pair{v_1,\Gamma v_2}_W$. Since $\Gamma\in\fg(F)$, $(V,\pair{\cdot,\cdot}_\Gamma)$ is a quadratic space. Denote $H={\mathrm{O}}(V)$, $\fh=\Lie(H)$. We view both of $G$ and $H$ as subgroups of $\GL(V)$, and view both of $\fg$ and $\fh$ as sub-Lie algebras of $\End(V)$. Then $\Gamma$ also lies in $\fh$, and $$H^0=Z_H(-\Gamma)=Z_G(\Gamma)=G^0.$$ The vertex $x\in\CB(G^0)$ is associated with a lattice function $\sL_x$ in $W$. Let $\sL'_x$ be the lattice function in $V$, which is defined in Lemma 5.9 of {\em loc. cit.} Let $x'\in\CB(H^0)$ be the vertex attached to $\sL_x'$. Then $G_x=H_{x'}$. See the Remarks after Proposition 5.13 of {\em loc. cit.} for details. Put $$\Sigma':=\left(x',-\Gamma,\phi^{-1},\rho_{(S,\mu_0)}^\vee\right),$$ which is a single block of $H$ of positive depth $r$.
\begin{prop}\label{prop. positive depth} We have the following statements. \begin{enumerate} \item $\theta_{V,W,\psi}(\pi)=\pi_{\Sigma'}$. \item $\pi_{\Sigma'}$ is a regular supercuspidal representation attached to certain regular elliptic triple $(S,\mu^{-1},j_\theta)$ of $H$. \item Suppose that $(S,\mu,j)$ corresponds to $(L,L^\circ,c,\chi)=\prod\limits_{i\in I}(L_i,L_i^\circ,c_i,\chi_i)$. Then
\begin{itemize}
\item $(S,\mu^{-1},j_\theta)$ corresponds to $$(L,L^\circ,c_\theta,\chi^{-1})=(L,L^\circ,-c\gamma,\chi^{-1})=\prod\limits_{i\in I}(L_i,L_i^\circ,-c_i\gamma_i,\chi^{-1}_i),$$
\item $(V,\pair{\cdot,\cdot}_\Gamma)\cong(L,\pair{\cdot,\cdot}_{-c\gamma})$,
\end{itemize} where $\gamma=\prod\limits_{i\in I}\gamma_i\in L^-$ is $\Gamma$ via the identification $\Lie(jS)=L^-$. \end{enumerate} \end{prop}
\begin{proof} The first statement is due to \cite[Proposition 7.1]{lm} or a special case of the Main Theorem of {\em loc. cit.} By definition $\pi_{\Sigma'}$ is regular. Suppose that $(S,j)$ corresponds to $(L,L^\circ,c)$. Then $(W,\pair{\cdot,\cdot}_W)=(L,\pair{\cdot,\cdot}_{c})$ and $(V,\pair{\cdot,\cdot}_\Gamma)=(L,\pair{\cdot,\cdot}_{-c\gamma})$. Let $j_\theta:S\hookrightarrow H$ denote the embedding given by the composition $$S\stackrel{j}{\hookrightarrow}G^0=H^0\hookrightarrow H.$$ Then $(S,j_\theta)$ corresponds to $(L,L^\circ,-c\gamma)$, and $x'\in\CB(H^0)$ is the vertex attached to $j_\theta S$. Since $\rho^\vee_{(S,\mu_0)}=\rho_{(S,\mu_0^{-1})}$, $\Sigma'=(x',-\Gamma,\phi^{-1},\rho^\vee_{(S,\mu_0)})$ is a Howe factorization of $(S,\mu^{-1},j_\theta)$. Therefore $\pi_{\Sigma'}=\pi_{(S,\mu^{-1},j_\theta)}$. \end{proof}
\subsection{General case} Now we treat the theta lifts of general regular supercuspidal representations. Let $\pi=\pi_{(S,\mu,j)}$ be a regular supercuspidal representation of $G(F)$ with a Howe factorization $\Sigma=(x,\Gamma,\phi,\rho)$. Let $\Sigma=\bigoplus\limits_{0\leq\ell\leq b}{^\ell\Sigma}$ be the block decomposition of $\Sigma$ into $b$ positive depth blocks $^\ell\Sigma=\left({^\ell x},{^\ell\Gamma},{^\ell\phi},{^\ell\rho}\right)$ and a depth zero block $^0\Sigma=\left({^0 x},0,{\bf 1},{^0\rho}\right)$, and $W=\bigoplus\limits_{0\leq\ell\leq b}{^\ell W}$ the decomposition of the space $W$.
Since $jS$ is a maximally unramified elliptic maximal torus of $G^0=\prod\limits_{0\leq\ell\leq b}{^\ell G^0}$, then $S=\prod\limits_{0\leq\ell\leq b}{^\ell S}$ so that $j{^\ell S}$ is a maximally unramified elliptic maximal torus of ${^\ell G^0}$ for each $\ell$. Let $\mu=\prod\limits_{0\leq\ell\leq b}{^\ell\mu}$ be the decomposition of $\mu$ with respect to that of $S$. Then $({^\ell S,{^\ell\mu},j})$ is a regular elliptic triple of ${^\ell G}$ for each $\ell$. Suppose that $(S,\mu,j)$ corresponds to $$\left(L,L^\circ,c,\chi\right)=\prod\limits_{0\leq\ell\leq b}\left({^\ell L},{^\ell L^\circ},{^\ell c},{^\ell\chi}\right),$$ and $\left({^\ell S,{^\ell\mu},j}\right)$ corresponds to $\left({^\ell L},{^\ell L^\circ},{^\ell c},{^\ell\chi}\right)$. According to Propositions \ref{prop. depth zero} and \ref{prop. positive depth}, we obtain a regular elliptic triple $\left({^\ell S,{^\ell\mu^{-1}},j_\theta}\right)$ associated with the data $\left({^\ell L},{^\ell L^\circ},{^\ell c_\theta},{^\ell\chi^{-1}}\right)$ for each $\ell$. Set $$(S,\mu^{-1},j_\theta)=\prod\limits_{0\leq\ell\leq b}\left({^\ell S,{^\ell\mu^{-1}},j_\theta}\right),$$ and $$(L,L^\circ,c_\theta,\chi^{-1})=\prod\limits_{0\leq\ell\leq b}\left({^\ell L},{^\ell L^\circ},{^\ell c},{^\ell\chi^{-1}}\right).$$
\begin{thm}\label{thm. theta} Up to equivalence, there exists a unique $2n$-dimensional quadratic space $(V,\pair{\cdot,\cdot}_V)$ over $F$ such that $\pi_V:=\theta_{V,W,\psi}(\pi)$ is nonzero. Moreover we have \begin{enumerate} \item $(V,\pair{\cdot,\cdot}_V)\cong(L,\pair{\cdot,\cdot}_{c_\theta})$, \item $\pi_V=\pi_{(S,\mu^{-1},j_\theta)}$ is a regular supercuspidal representation, \item $(S,\mu^{-1},j_\theta)$ corresponds to $(L,L^\circ,c_\theta,\chi^{-1})$. \end{enumerate} \end{thm}
\begin{proof} It is a direct consequence of Propositions \ref{prop. depth zero} and \ref{prop. positive depth} and the Main Theorem of \cite{lm}. \end{proof}
\section{Distinction}\label{sec. distinction} In this section we investigate the relation between the local theta correspondence and distinction with respect to unramified Galois involutions. We will first give a criterion (see Proposition \ref{prop. distinction}) for testing distinction. It is a generalization of our prior result \cite{zha1}, and is also a refinement of \cite{zha2} for unramified Galois involutions. The key point is that we can show that the character ``$\eta_S$'' is trivial in this situation. Combining this criterion with Theorem \ref{thm. theta}, we derive the main result of this section (see Theorem \ref{thm. theta and distinction}).
\subsection{A criterion}\label{subsec. criterion} In this subsection, we assume that $G$ is a connected reductive group over $F$, and $E$ is an unramified quadratic field extension of $F$. We fix an element $\iota\in \fo_E^\times$ such that $\iota\in E^-$. Let $\FG$ denote the Weil restriction $\R_{E/F}G$. The nontrivial automorphism $\sigma\in\Gal(E/F)$ induces an involution, still denoted by $\sigma$, on $\FG$.
Let $\pi_{(\FS,\mu,j)}$ be a regular supercuspidal representation of $\FG(F)$. In \cite[Corollary 3.18]{zha2} we showed that $\pi_{(\FS,\mu,j)}$ is $G(F)$-distinguished if and only if $(\FS,\mu,j)$ is $G(F)$-conjugate to a regular elliptic triple $(\FS,\mu,j')$ of $\FG$ such that $j'\FS$ is $\sigma$-stable and $(\mu\circ j'^{-1})|_{(j'\FS)^\sigma(F)}=\eta_{j'\FS}$. See \cite[Definition 3.8]{zha2} or the proof below for the definition of $\eta_{j'\FS}$. We have the following refinement.
\begin{prop}\label{prop. distinction}
The representation $\pi_{(\FS,\mu,j)}$ is $G(F)$-distinguished if and only if $(\FS,\mu,j)$ is $G(F)$-conjugate to a regular elliptic triple $(\FS,\mu,j')$ such that $j'\FS$ is $\sigma$-stable and $(\mu\circ j'^{-1})|_{(j'\FS)^\sigma(F)}={\bf 1}$. \end{prop}
\begin{proof} According to \cite[Corollary 3.18]{zha2}, it suffices to assume that $j\FS$ is $\sigma$-stable and show that $\eta_{j\FS}$ is trivial. Since $j\FS$ is $\sigma$-stable, the torus $S:=(j\FS)^\sigma$ is defined over $F$ and $j\FS=\R_{E/F}S$.
Let $\Psi=(\vec{\FG},x,\rho,\vec{\phi})$ be a Howe factorization of $(\FS,\mu,j)$, and $\vec{r}$ the depth of $\vec{\phi}$. We may assume that $\Psi$ is $\sigma$-symmetric, that is, $\sigma(x)=x$, $\sigma(\vec{\FG})=\vec{\FG}$ and $\vec{\phi}\circ\sigma=\vec{\phi}^{-1}$. Then $G^i:=\left(\FG^i\right)^\sigma$ is defined over $F$ and $\FG^i=\R_{E/F}G^i$ for each $i$. We denote by $\fg^i$ the Lie algebra of $G^i$, and $\fz^i$ the center of $\fg^i$. Set $s_i=\frac{r_i}{2}$.
Now we recall the definition of $\eta_{j\FS}$. For each $0\leq i\leq d-1$, the quotient group $\FW_i:={\mathbf{J}}^{i+1}/{\mathbf{J}}^{i+1}_+$ is a symplectic $\BF_p$-vector space, where $${\mathbf{J}}^{i+1}=\FG^i(F)_{x,r_i}\FG^{i+1}(F)_{x,s_i}\quad\textrm{and}\quad
{\mathbf{J}}^{i+1}_+=\FG^i(F)_{x,r_i}\FG^{i+1}(F)_{x,s_i+}.$$ The symplectic structure of $\FW_i$ is attached to the $\FG^{i+1}$-generic character $\phi_i$ of $\FG^i(F)$. The group $j\FS(F)$ acts on $\FW_i$ by conjugate action and preserves the symplectic structure. The involution $\sigma$ induces a natural linear transformation on $\FW_i$. Let $W_i$ be the $\sigma$-fixed subspace of $\FW_i$, on which the group $S(F)$ acts by conjugate action. Set $$\chi_i(x)=\det\left(\Ad(x)|_{W_i}\right)^{\frac{p-1}{2}},\quad\forall\ x\in S(F),$$ which is a quadratic character of $S(F)$. Then the character $\eta_{j\FS}$ of $S(F)$ is defined to be $\prod\limits_{0\leq i\leq d-1}\chi_i$.
In the proof of \cite[Lemma 3.9]{zha2} we showed that $\FW_i$ is $j\FS(F)$-equivariant isomorphic to $$\fg^{i+1}(E)_{x,s_i:s_i+}/\fg^{i}(E)_{x,s_i:s_i+},$$ and $W_i$ is $S(F)$-equivariant isomorphic to $$\fg^{i+1}(F)_{x,s_i:s_i+}/\fg^{i}(F)_{x,s_i:s_i+}.$$ Hence $W_i$ is $S(F)$-equivariant isomorphic to $J^{i+1}/J^{i+1}_+$, where $$J^{i+1}=G^i(F)_{x,r_i}G^{i+1}(F)_{x,s_i}\quad\textrm{and}\quad J^{i+1}_+=G^i(F)_{x,r_i}G^{i+1}(F)_{x,s_i+}.$$
The character $\phi_i$ of $\FG^i(F)$ is realized by a $\FG^{i+1}$-generic element $\Gamma_i\in \fz^i(E)_{-r_i}$. According to \cite[Lemma 5.15]{hm08} we may assume $\sigma(\Gamma_i)=-\Gamma_i$. Since $\iota$ is in $\fo_E^\times$, we see $\iota\Gamma$ still belongs to $\fz^i(E)_{-r_i}$. On the other hand, $\iota$ is also in $E^-$, so we have $\iota\Gamma_i\in\fz^i(F)_{-r_i}$. It is easy to see that $\iota\Gamma$ is $G^{i+1}$-generic. Thus $\iota\Gamma_i$ gives rise to a symplectic structure of $W_i$, which is preserved under $S(F)$. Therefore we have a homomorphism $S(F)\ra \Sp(W_i)$. Hence $\chi_i$ is trivial for each $i$, and so is $\eta_{j\FS}$. \end{proof}
The above proposition has the following direct consequence, whose proof is exactly the same as that of \cite[Corollary 3.17]{zha2}.
\begin{cor}\label{cor. distinction I} If $\pi$ is a $G(F)$-distinguished regular supercuspidal representation of $\FG(F)$, then $\pi^\vee\cong \pi\circ\sigma$. \end{cor}
In \cite[\S5]{kal} Kaletha defined the notion {\em regular supercuspidal L-parameters} $\varphi$ for $\FG$. He also constructed the $L$-packets $\Pi_\varphi(\FG)$ of $\varphi$ in the framework of rigid inner twists \cite{kal16}, which consist of certain regular supercuspidal representations of $\FG(F)$. In \cite{zha2} we showed that the sets $$\Pi_\varphi^\vee(\FG):=\left\{\pi^\vee\ |\ \pi\in\Pi_\varphi(\FG)\right\}\ \textrm{ and }\ \Pi_\varphi^\sigma(\FG):=\left\{\pi\circ\sigma\ |\ \pi\in\Pi_\varphi(\FG)\right\}$$ are indeed the $L$-packets attached to some regular supercuspidal $L$-parameters (cf. \cite[Propositions 4.14 and 4.18]{zha2}). The above corollary implies the following result directly, which confirms a conjecture of Lapid in this particular case.
\begin{cor}\label{cor. distinction II} Suppose that $\pi$ is a $G(F)$-distinguished regular supercuspidal representation of $\FG(F)$ and belongs to the L-packet $\Pi_\varphi(\FG)$. Then we have $\Pi^\vee_\varphi(\FG)=\Pi^\sigma_\varphi(\FG)$. \end{cor}
\subsection{Theta lifts and distinction}\label{subsec. theta and distinction} As before, let $E/F$ be an unramified quadratic field extension. We fix a uniformizer $\varpi$ of $\fp_E$ such that $\varpi\in E^-$, and also fix $\iota\in\fo_E^\times$ such that $\iota\in E^-$. Let $\psi$ be a fixed additive character of $E$ of conductor $\fp_E$, and $\psi_\iota$ the character of $F$ defined by $\psi_\iota(a)=\psi(\iota^{-1}a)$.
Let $(W,\pair{\cdot,\cdot}_W)$ be a $2n$-dimensional symplectic $F$-vector space. Set $$(\FW,\pair{\cdot,\cdot}_\FW)=(W,\pair{\cdot,\cdot}_W)\otimes E,$$ which is a $2n$-dimensional symplectic $E$-vector space. Let $G=\Sp(W)$ and $\FG=\Sp(\FW)$.
\begin{thm}\label{thm. theta and distinction} Let $\pi$ be a $G(F)$-distinguished regular supercuspidal representation of $\FG(E)$. Then there exists a $2n$-dimensional quadratic $E$-vector space $(\FV,\pair{\cdot,\cdot}_\FV)$ equipped with an $F$-structure $(V,\pair{\cdot,\cdot}_V)$ such that $\theta_{\FV,\FW,\psi_\iota}(\pi)$ is nonzero and ${\mathrm{O}}(V)(F)$-distinguished. \end{thm}
\begin{proof}
Suppose $\pi=\pi_{(\FS,\mu,j)}$. By Proposition \ref{prop. distinction}, we may assume that $j\FS$ is $\sigma$-stable. Therefore $(j\FS)^\sigma$ is an elliptic maximal torus of $G$. Set $S=j^{-1}(j\FS)^\sigma$. We may also assume $\mu|_{S(F)}=1$ according to Proposition \ref{prop. distinction}. Suppose $(S,j)$ corresponds to some datum $(K,K^\circ,c)=\prod\limits_{i\in I}(K_i,K_i^\circ,c_i)$ over $F$. Then $(\FS,\mu,j)$ corresponds to $(L,L^\circ,c,\chi)=\prod\limits_{i\in I}(L_i,L^\circ_i,c_i,\chi)$ where $L_i=K_i\otimes E$ and $L_i^\circ=K_i^\circ\otimes E$. We may assume $(W,\pair{\cdot,\cdot}_W)=(K,\pair{\cdot,\cdot}_c)$ and $(\FW,\pair{\cdot,\cdot}_\FW)=(L,\pair{\cdot,\cdot}_c)$. Note that $\sigma(c)=c$.
Set $(\FV',\pair{\cdot,\cdot}_{\FV'})=(L,\pair{\cdot,\cdot}_{c_\theta})$ as in Theorem \ref{thm. theta}. Then $\pi_{\FV'}=\theta_{\FV',\FW,\psi}(\pi)$ is nonzero and $\pi_{\FV'}=\pi_{(\FS,\mu^{-1},j_\theta)}$. Put $$\left(L,L^\circ,c_\theta\right)=\left(\prod\limits_{i\in I_0}(L_i,L^\circ_i,c_i\tau_i\varpi)\right)\left(\prod\limits_{i\in I_{>0}}(L_i,L^\circ_i,c_i\gamma_i)\right),$$ where the index $I_0$ corresponds to the depth-zero single block and $I_{>0}$ corresponds to positive depth single blocks.
Recall $\tau_i\in L_i^-$. Since $L_i=K_i\otimes E$ and $L_i^\circ=K_i^\circ\otimes E$, we may require $\tau_i\in K_i^-$. According to \cite[Lemma 5.15]{hm08}, we may assume that $\gamma_i$ satisfies $\sigma(\gamma_i)=-\gamma_i$ for all $i\in I_{>0}$. Therefore $\sigma(c_i\tau_i\varpi)=-c_i\tau_i\varpi$ for each $i\in I_0$ and $\sigma(c_i\gamma_i)=-c_i\gamma_i$ for all $i\in I_{>0}$. In summary, we have $\sigma(c_\theta)=-c_\theta$.
Set $(\FV,\pair{\cdot,\cdot}_\FV)=(\FV',\iota\pair{\cdot,\cdot}_{\FV'})$. Then $(\FV,\pair{\cdot,\cdot}_\FV)=\left(L,\pair{\cdot,\cdot}_{\iota c_\theta}\right)$ and $\sigma(\iota c_\theta)=\iota c_\theta$. We may identify ${\mathrm{O}}(\FV)$ with ${\mathrm{O}}(\FV')$ canonically. Let $j_\theta^\iota$ be the composition $\FS\stackrel{j_\theta}{\hookrightarrow}{\mathrm{O}}(\FV')={\mathrm{O}}(\FV)$. Then $(\FS,j^\iota_\theta)$ corresponds to $(L,L^\circ,\iota c_\theta).$ Since $\sigma(\iota c_\theta)=\iota c_\theta$, we have $\iota c_\theta\in K^\circ$. Hence $\FV$ has an $F$-structure. To be precise, $(\FV,\pair{\cdot,\cdot}_\FV)=(V,\pair{\cdot,\cdot}_V)\otimes E$ where $(V,\pair{\cdot,\cdot}_V)=(K,\pair{\cdot,\cdot}_{\iota c_\theta})$.
Write $\FH^\circ=\SO(\FV)$, $\FH={\mathrm{O}}(\FV)$, $H^\circ=\SO(V)$, $H={\mathrm{O}}(V)$, $\pi_\FV^\circ=\pi^\circ_{(\FS,\mu^{-1},j_\theta^\iota)}$, and $\pi_\FV=\pi_{(\FS,\mu^{-1},j_\theta^\iota)}$. Note that $\pi_\FV=\theta_{\FV,\FW,\psi_\iota}(\pi)$. It is obvious that $j_\theta^\iota\FS$ is $\sigma$-stable, $(j^\iota_\theta\FS)^\sigma=j^\iota_\theta S$, and $(S,j_\theta^\iota)$ corresponds to $(K,K^\circ,\iota c_\theta)$. Since $\mu^{-1}|_{S(F)}=1$, by Proposition \ref{prop. distinction}, we see that $\pi^\circ_\FV$ is $H^\circ(F)$-distinguished. Since $\pi_\FV=\ind_{\FH^\circ(F)}^{\FH(F)}\pi_\FV^\circ$, according to Mackey theory, we have $$\begin{aligned}\Hom_{H(F)}(\pi_\FV,{\bf1})&=\bigoplus_{h\in\FH^\circ(F)\bs \FH(F)/H(F)}\Hom_{\FH^\circ(F)\cap {^hH(F)}}(\pi_\FV^\circ,{\bf1})\\ &=\Hom_{H^\circ(F)}(\pi_\FV^\circ,{\bf1}).\end{aligned}$$ Therefore $\pi_\FV$ is $H(F)$-distinguished.
\end{proof}
\s{\small Chong Zhang\\ Department of Mathematics, Nanjing University,\\ Nanjing 210093, Jiangsu, P. R. China.\\ E-mail address: \texttt{[email protected]}}
\end{document} | arXiv |
\begin{definition}[Definition:Isotropic Quadratic Form]
Let $\mathbb K$ be a field of characteristic $\operatorname{char}\mathbb K \neq 2$.
Let $V$ be a vector space over $\mathbb K$.
Let $q : V\times V \mapsto \mathbb K$ be a quadratic form.
Then $q$ is '''isotropic''' {{Iff}} it represents $0$.
That is: $q(v) = 0$ for some $v\in V\setminus\{0\}$.
\end{definition} | ProofWiki |
The discrete homotopy perturbation Sumudu transform method for solving partial difference equations
DCDS-S Home
Couette flows of a viscous fluid with slip effects and non-integer order derivative without singular kernel
June 2019, 12(3): 625-643. doi: 10.3934/dcdss.2019040
Barycentric spectral domain decomposition methods for valuing a class of infinite activity Lévy models
Edson Pindza 1,, , Francis Youbi 1, , Eben Maré 1, and Matt Davison 2,
Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 002, Republic of South Africa
Department of Statistical and Actuarial Sciences, University of Western Ontario, London, Ont., Canada
* Corresponding author: E. Pindza
Received July 2017 Revised November 2017 Published September 2018
Figure(5) / Table(6)
A new barycentric spectral domain decomposition methods algorithm for solving partial integro-differential models is described. The method is applied to European and butterfly call option pricing problems under a class of infinite activity Lévy models. It is based on the barycentric spectral domain decomposition methods which allows the implementation of the boundary conditions in an efficient way. After the approximation of the spatial derivatives, we obtained the semi-discrete equations. The computation of these equations is performed by using the barycentric spectral domain decomposition method. This is achieved with the implementation of an exponential time integration scheme. Several numerical tests for the pricing of European and butterfly options are given to illustrate the efficiency and accuracy of this new algorithm. We also show that Greek options, such as Delta and Gamma sensitivity, are computed with no spurious oscillation.
Keywords: Spectral methods, Clenshaw Curtis quadrature, shifted Laguerre Gauss quadrature, domain decomposition, partial integro-differential equation, infinite activity Lévy processes.
Mathematics Subject Classification: 76M22, 41A10, 41A20, 91G80.
Citation: Edson Pindza, Francis Youbi, Eben Maré, Matt Davison. Barycentric spectral domain decomposition methods for valuing a class of infinite activity Lévy models. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 625-643. doi: 10.3934/dcdss.2019040
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Figure 1. Spectral domain decomposition method matrix structures
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Figure 2. Numerical valuation of European call options for the CGMY, Meixner and GH model with their Greeks for the parameters in Table 2
Figure 3. Convergence of the SDDM and FDM for European vanilla call options for the parameters in Table 2
Figure 4. Numerical valuation of European butterfly call options for the CGMY, Meixner, and GH model with $N = 16, K_{1} = 40, K_{2} = 50, K_{3} = 60$ for the parameters in Table 2
Figure 5. Convergence of the SDDM and FDM for European vanilla butterfly call options for the parameters in Table 2
Table 1. Density functions for Lévy Processes
Model Lévy density function
CGMY $f(y)=\frac{C_{-}e^{-G|y|}}{|y|^{1+Y}}{\bf{1}}_{y<0}+ \frac{C_{+}e^{-M|y|}}{|y|^{1+Y}}{\bf{1}}_{y>0}$
Meixner $f(y)=\frac{Ae^{-ay}}{y\sinh(by)}$
GH process $f(y)=\frac{e^{\beta y}}{|y|}\left(\int_{0}^{\infty}\frac{e^{-\sqrt{2\zeta+\alpha^{2}}|y|}}{\pi^{2}\zeta \left(J^{2}_{|\lambda|}\left(\delta\sqrt{2\zeta}\right)+Y^{2}_{|\lambda|}\left(\delta\sqrt{2\zeta}\right)\right)}d\zeta+\max(0,\lambda)e^{-\alpha|y|}\right)$
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Table 2. The parameters for Lévy models used in both examples
Model Parameters
GBM (Black-Scholes) $K =50, \ r = 0.05, \sigma = 0.2, q=0$ and $T =0.5. $
CGMY $C_{-} = 0.3, \ C_{+} = 0.1, \ \ G = 15,\ M = 25$ and $Y = 20. $
Meixner $A=15, \ a=-1.5$ and $b=50$
GH process $\alpha=4, \ \beta=-3.2, \ \delta=1.4775$ and $\lambda=-3$
Table 3. The benchmark European call option values under Lévy processes with different values of S and $N = 150$ for the parameters in Table 2
Model $S$
CGMY 0.2210443864 3.3785900783 11.3681462140
Meixner 1.3420365535 5.4934725848 12.7780678851
GH processes 0.3237597911 3.8485639686 11.9164490861
Table 4. Absolute errors (${\bf{AE}}$) of the benchmark and the European call option apply to the CGMY, Meixner and GH processes models with different values of $N$ and $S$ for the parameters in Table 2
$SDDM $ $FDM $
$S $ $40 $ 50 60 $40 $ 50 60
$N$ ${\bf{AE}}$ ${\bf{AE}}$ ${\bf{AE}}$ CPU ${\bf{AE}}$ ${\bf{AE}}$ ${\bf{AE}}$ CPU
CGMY 10 $1.15e^{-4}$ $1.28e^{-4}$ $1.35e^{-4}$ $0.30$ $1.14e^{-2}$ $5.72e^{-2}$ $9.55e^{-3}$ 0.6
15 $1.23e^{-5}$ $1.73e^{-5}$ $1.45e^{-5}$ $0.34$ $2.03e^{-3}$ $1.55e^{-2}$ $1.75e^{-3}$ 0.72
25 $3.33e^{-10}$ $3.15e^{-10}$ $3.24e^{-10}$ $0.53$ $2.75e^{-4}$ $1.91e^{-3}$ $2.37e^{-4}$ 1.32
Meixner 10 $2.12e^{-4}$ $2.45e^{-4}$ $2.35e^{-4}$ $0.32$ $1.46e^{-2}$ $4.35e^{-2}$ $8.51e^{-3}$ 0.65
20 $3.40e^{-7}$ $3.23e^{-7}$ $3.14e^{-7}$ 0.41 $6.45e^{-4}$ $3.95e^{-3}$ $5.45e^{-4}$ 0.86
GH processes 10 $3.33e^{-4}$ $3.29e^{-4}$ $3.17e^{-4}$ $0.65$ $1.45e^{-2}$ $5.33e^{-2}$ $7.13e^{-3}$ 1.33
15 $4.55e^{-5}$ $4.370e^{-5}$ $4.14e^{-5}$ $0.82$ $2.15e^{-3}$ $1.04e^{-2}$ $5.72e^{-2}$ 1.61
Table 5. The benchmark values of the European butterfly call option values under Lévy processes with different values of S and $N = 100$ for the parameters in Table 2
CGMY 2.2845953002 4.6814621409 2.1592689295
Meixner 2.2689295039 3.7101827676 2.3159268929
GH processes 2.3942558746 4.2898172323 1.7989556135
Table 6. Absolute errors (${\bf{AE}}$) of the benchmark and the European call option apply to the CGMY, Meixner, and GH processes models with different values of $N$ and $S$ for the parameters in Table 2
CGMY 07 $1.88e^{-4}$ $1.76e^{-4}$ $1.76e^{-4}$ $0.20$ $1.24e^{-2}$ $5.81e^{-2}$ $8.98e^{-3}$ 0.62
10 $5.51e^{-5}$ $6.1e^{-5}$ $5.92e^{-5}$ $0.68$ $2.13e^{-3}$ $1.21e^{-2}$ $5.87e^{-2}$ 1.63
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Edson Pindza Francis Youbi Eben Maré Matt Davison | CommonCrawl |
Randall Kamien
Randall David Kamien (born February 25, 1966) is a theoretical condensed matter physicist specializing in the physics of liquid crystals and is the Vicki and William Abrams Professor in the Natural Sciences at the University of Pennsylvania.[1]
Randall David Kamien
Kamien in 2022
Born (1966-02-25) February 25, 1966
Pittsburgh, Pennsylvania
Alma materCalifornia Institute of Technology (B.S., 1988)
California Institute of Technology (M.S., 1988)
Harvard University (Ph.D, 1992)
Known forGrain boundaries
Focal conic domains
Liquid crystals
AwardsG.W. Gray Medal British Liquid Crystal Society (2016)
Scientific career
FieldsCondensed Matter Physics
InstitutionsHarvard University
Institute for Advanced Studies
University of Pennsylvania
ThesisDirected Line Liquids (1992)
Doctoral advisorDavid R. Nelson
Biography
Randall Kamien was born to economist Morton Kamien and Lenore Kamien on February 25, 1966, and grew up in Wilmette, Illinois on the outskirts of Chicago.[2] Kamien completed a B.S. and a M.S. in physics at the California Institute of Technology in 1988 and completed a PhD in physics at Harvard University in 1992 under the supervision of David R. Nelson.[3] Prior to joining the faculty at the University of Pennsylvania he was a member of the Institute for Advanced Study in Princeton, New Jersey, and a postdoctoral research associate at the University of Pennsylvania. Kamien was appointed assistant professor at the University of Pennsylvania in 1997 and promoted to full professor in 2003.[4] Kamien is a fellow of the American Physics Society and the American Association for the Advancement of Science.[4] Kamien is the editor of Reviews of Modern Physics.[5]
Research
Randall Kamien studies soft condensed matter – and in particular liquid crystalline phases of matter – through the lens of geometry and topology.[6] In particular, Kamien has contributed to understanding Twist Grain Boundaries,[7] Focal Conic Domains,[8] and defect topology in smectic liquid crystals.[9] He is also known for his idiosyncratic naming conventions, such as “Shnerk’s Surface” [10] and “Shmessel Functions.”
Publications
• Senyuk, B.; Liu, Q.; He, S.; Kamien, R. D.; Kusner, R. B.; Lubensky, T. C.; Smalyukh, I. I. (2013), "Topological colloids", Nature, 493 (7431): 200–205, arXiv:1612.08753, Bibcode:2013Natur.493..200S, doi:10.1038/nature11710, PMID 23263182, S2CID 4343186.
• Honglawan, A.; Beller, D. A.; Cavallaro, M.; Kamien, R. D.; Stebe, K. J.; Yang, S. (2013), "Topographically induced hierarchical assembly and geometrical transformation of focal conic domain arrays in smectic liquid crystals", Proceedings of the National Academy of Sciences, 110 (1): 34–39, doi:10.1073/pnas.1214708109, PMC 3538202, PMID 23213240.
• Snir, Y.; Kamien, R. D. (2005), "Entropically driven helix formation", Science, 307 (5712): 1067, arXiv:cond-mat/0502520, doi:10.1126/science.1106243, PMID 15718461, S2CID 14611285.
• Ziherl, P.; Kamien, R. D. (2001), "Maximizing entropy by minimizing area: Towards a new principle of self-organization", The Journal of Physical Chemistry B, 105 (42): 10147, arXiv:cond-mat/0103171, doi:10.1021/jp010944q, S2CID 119467204.
• Kamien, R. D.; Selinger, J. V. (2001), "Order and frustration in chiral liquid crystals", Journal of Physics: Condensed Matter, 13 (3): R1, arXiv:cond-mat/0009094, doi:10.1088/0953-8984/13/3/201, S2CID 93442372.
• Kamien, R. D.; Lubensky, T. C. (1999), "Minimal surfaces, screw dislocations, and twist grain boundaries", Physical Review Letters, 82 (14): 2892, arXiv:cond-mat/9808306, Bibcode:1999PhRvL..82.2892K, doi:10.1103/PhysRevLett.82.2892, S2CID 15354995.
References
1. "Randall Kamien". www.physics.upenn.edu. Retrieved 2022-05-05.
2. In memoriam: Professor Emeritus Morton I. Kamien, 1938-2011, retrieved 2022-05-05.
3. Harvard PhD Theses in Physics: 1971-2000, retrieved 2022-05-05.
4. Curriculum vitae (PDF), retrieved 2022-05-05.
5. APS Editorial Office: Reviews of Modern Physics, retrieved 2022-05-05.
6. Kamien Group, retrieved 2022-05-05.
7. Kamien, R. D.; Lubensky, T. C. (1999). "Minimal surfaces, screw dislocations, and twist grain boundaries". Physical Review Letters. 82 (14): 2892–2895. arXiv:cond-mat/9808306. Bibcode:1999PhRvL..82.2892K. doi:10.1103/PhysRevLett.82.2892. S2CID 15354995.
8. Alexander, G. P.; Chen, B. G.; Matsumoto, E. A.; Kamien, R. D. (2010). "The Power of Poincaré: Elucidating the Hidden Symmetries in Focal Conic Domains". Physical Review Letters. 104 (25): 257802. arXiv:1004.0465. doi:10.1103/PhysRevLett.104.257802. PMID 20867415. S2CID 8291259.
9. Machon, T.; Aharoni, H.; Hu, Y.; Kamien, R. D. (2019). "Aspects of Defect Topology in Smectic Liquid Crystals". Communications in Mathematical Physics. 372 (2): 525–542. arXiv:1808.04104. Bibcode:2019CMaPh.372..525M. doi:10.1007/s00220-019-03366-y. S2CID 52435763.
10. Santangelo, C. D.; Kamien, R. D. (2007). "Triply periodic smectic liquid crystals". Physical Review E. 75 (1 Pt 1): 011702. arXiv:cond-mat/0609596. Bibcode:2007PhRvE..75a1702S. doi:10.1103/PhysRevE.75.011702. PMID 17358168. S2CID 119371099.
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Quantification of the unique continuation property for the nonstationary Stokes problem
MCRF Home
Optimal control for a phase field system with a possibly singular potential
March 2016, 6(1): 53-94. doi: 10.3934/mcrf.2016.6.53
Optimal sampled-data control, and generalizations on time scales
Loïc Bourdin 1, and Emmanuel Trélat 2,
Université de Limoges, Institut de recherche XLIM, Département de Mathématiques et d'Informatique, CNRS UMR 7252, Limoges, France
Sorbonne Universités, UPMC Univ. Paris 06, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, Institut Universitaire de France, F-75005, Paris
Received January 2015 Revised October 2015 Published January 2016
In this paper, we derive a version of the Pontryagin maximum principle for general finite-dimensional nonlinear optimal sampled-data control problems. Our framework is actually much more general, and we treat optimal control problems for which the state variable evolves on a given time scale (arbitrary non-empty closed subset of $\mathbb{R}$), and the control variable evolves on a smaller time scale. Sampled-data systems are then a particular case. Our proof is based on the construction of appropriate needle-like variations and on the Ekeland variational principle.
Keywords: Optimal control, sampled-data, Pontryagin maximum principle, time scale..
Mathematics Subject Classification: 49J15, 93C57, 34N99, 39A1.
Citation: Loïc Bourdin, Emmanuel Trélat. Optimal sampled-data control, and generalizations on time scales. Mathematical Control & Related Fields, 2016, 6 (1) : 53-94. doi: 10.3934/mcrf.2016.6.53
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Loïc Bourdin Emmanuel Trélat | CommonCrawl |
\begin{document}
\title{Scalar invariants of surfaces in conformal 3-sphere via Minkowski spacetime} \author{Jie Qing, Changping Wang, and Jingyang Zhong}
\begin{abstract} For a surface in 3-sphere, by identifying the conformal round 3-sphere as the projectivized positive light cone in Minkowski 5-spacetime, we use the conformal Gauss map and the conformal transform to construct the associate homogeneous 4-surface in Minkowski 5-spacetime. We then derive the local fundamental theorem for a surface in conformal round 3-sphere from that of the associate 4-surface in Minkowski 5-spacetime. More importantly, following the idea of Fefferman and Graham \cite{FG-1, FG-2}, we construct local scalar invariants for a surface in conformal round 3-sphere. One distinct feature of our construction is to link the classic work of Blaschke \cite{blaschke}, Bryan \cite{bryant} and Fefferman-Graham \cite{FG-1, FG-2}.
\end{abstract} \maketitle
\section{Introduction}
It is well-known that all local scalar invariants of a (pseudo-)Riemannian metric are Weyl invariants, based on Weyl's classical invariant theory for the orthogonal groups. A conformal structure on a manifold is described by an equivalent class of conformal Riemannian metrics. Two metrics $g_1$ and $g_2$ on a manifold $\textup{M}$ are conformal to each other if $g_1 = \lambda^2 g_2$ for some positive smooth function $\lambda$ on $\textup{M}$. There are several ways to set the theory of local conformal invariants, but it is no longer straightforward to account for local scalar conformal invariants because of the lack of Weyl Theorem for the group of conformal transformations. To tackle such problem, in the seminal paper \cite{FG-1} in 1980's, Fefferman and Graham described the ingenious construction of a Ricci-flat homogeneous Lorentzian ambient spacetime for a given conformal manifold, where the conformal manifold is represented by the homogeneous null hypersurface in the ambient spacetime. Their construction was motivated by the model case in which the conformal round sphere $\mathbb{S}^n$ is the projectivized positive light cone $\mathbb{N}^{n+1}_+$ in Minkowski spacetime $\mathbb{R}^{1,n+1}$. In \cite{FG-1}, Fefferman and Graham initiated the program to use local scalar (pseudo-)Riemannian invariants of the ambient metrics at the homogeneous null hypersurface to fully account for local scalar conformal invariants. Readers are referred to their recent expository paper \cite{FG-2} to learn all the developments of this program (cf. also, \cite{BEG, gover}). This program also has lead to many significant advances in the global theory of conformal geometry, particularly via conformally invariant PDEs. \\
In this paper we want to build the model case to the study of local scalar invariants of submanifolds in a conformal manifold in the way that follows the approach in \cite{FG-1}. The model case for us is to study 2-surfaces $\hat x$ in the conformal round 3-sphere $(\mathbb{S}^3, [g_0])$. As in \cite{FG-1}, the conformal round 3-sphere is represented by the positive light cone $\mathbb{N}^4_+$ in Minkowski 5-spacetime $\mathbb{R}^{1,4}$. Given an immersed surface $$ \hat x: \textup{M}^2\to\mathbb{S}^3 $$ or equivalently $$ y = (1, \hat x): \textup{M}^2\to \mathbb{N}^4_+, $$ to incorporate all metrics in $[g_0]$ on 3-sphere we consider the homogeneous extension $$ x^{\mathbb{N}} = \alpha (1, \hat x): \mathbb{R}^+\times\textup{M}^2\to \mathbb{N}^4_+\subset\mathbb{R}^{1,4}. $$ Then we will use the conformal Gauss map $\xi$ of $\hat x$ to choose a canonical null vector $y^*$ at each given point $y \in x^{\mathbb{N}}\subset \mathbb{N}^4_+$ to extend $x^{\mathbb{N}}$ further into a homogeneous timelike 4-surface $$ \tilde x=\alpha y + \alpha\rho y^*: \mathbb{R}^+\times\mathbb{R}^+\times\textup{M}^2\to\mathbb{R}^{1,4}. $$ We will also consider the associate ruled 3-surface $$ x^+ = \frac 1{\sqrt 2} (e^ty + e^{-t}y^*): \mathbb{R}\times\textup{M}^2\to \mathbb{H}^4\subset\mathbb{R}^{1,4} $$ where $\mathbb{H}^4$ is the hyperboloid in Minkowski 5-spacetime. The main idea, inspired by the work \cite{FG-1, FG-2}, is to use the geometry of the associate 4-surface $\tilde x$ in Minkowski spacetime $\mathbb{R}^{1,4}$ (the associate ruled 3-surface $x^+$ in the hyperboloid $\mathbb{H}^4$ and the spacelike surface as the image of the conformal Gauss map $\xi$ in the de Sitter spacetime $S^{3,1}$ in Minkowski spacetime $\mathbb{R}^{1,4}$) to study the geometry of the surface $\hat x$ in the conformal round 3-sphere $\mathbb{S}^3$.\\
Our approach facilitates proofs of the local fundamental theorems (cf. Theorem \ref{local-fundamental} and \cite{cpw-1, cpw-2}) and produces local scalar invariants of surfaces in the conformal round 3-sphere. The second is more interesting and helpful to find appropriate PDE problems to study the surfaces. The study of Willmore surfaces indeed exemplifies well that how important and central those problems are in the theory of surfaces in general \cite{blaschke, bryant, LY, MN}. \\
We should remark that the key to our construction of associate surfaces is the conformal Gauss map $\xi$ to a given surface $\hat x$ in the conformal round 3-sphere. The conformal Gauss maps have been introduced in several contexts (cf. \cite{blaschke, bryant, rigoli}). We are searching for a definition that fits into the context of ambient spaces of Fefferman and Graham (cf. Lemma \ref{xi-1} and Lemma \ref{xi-2}). It is fascinating to see how Blaschke \cite{blaschke} introduced the conformal Gauss map as the map representing the family of mean curvature 2-spheres of the surface $\hat x$ and the conformal transform $\hat x^*$ (cf. Definition \ref{def-conf-trans}) as the other envelope surface of the conformal Gauss map. One technical assumption for the null vector $y^*$ to be well defined at each point $y\in x^{\mathbb{N}}$ is to require that the conformal Gauss map of the surface $\hat x$ induces a spacelike surface in the de Sitter spacetime $\mathbb{S}^{1,3}$, which is equivalent to that the surface $\hat x$ is free of umbilical point in the conformal 3-sphere $\mathbb{S}^3$.\\
It is nice to know that in our construction the associate 4-surface $\tilde x$ in Minkowski spacetime $\mathbb{R}^{1,4}$ is a minimal 4-surface (of vanishing mean curvature) if and only if the 2-surface $\hat x$ is a Willmore surface with no umbilical point in $\mathbb{S}^3$ (cf. Theorem \ref{Th:tilde-H}). The same statement also holds for the associate ruled 3-surface $x^+$ in the hyperboloid $\mathbb{H}^4$ (cf. Theorem \ref{Th:+-H}) as well as the conformal Gauss map surface $\xi$ in de Sitter sapcetime $\mathbb{S}^{1,3}$ (cf. Theorem \ref{conf-gauss-surface}).\\
Upon realizing that a different representative $\lambda^2 g_0$ in the conformal class $[g_0]$ on $\mathbb{S}^3$ is equivalent to a different parametrization for the associate surface \begin{equation}\label{general-param} \tilde x = \alpha y_\lambda + \alpha\rho y^*_\lambda:\mathbb{R}^+\times\mathbb{R}^+\times\textup{M}^2\to\mathbb{R}^{1,4}, \end{equation} where $y_\lambda = \hat \lambda (1, \hat x)$ and $\hat \lambda = \lambda\circ\hat x$ for a conformal factor $\lambda$, the real issue is how we use the geometry of the surface $\hat x$ in the 3-sphere $(\mathbb{S}^3, \lambda^2g_0)$ to calculate the geometry of the associate surface $\tilde x$. The solution is to use the following 3-sphere $\mathbb{S}^3_\lambda$ in the positive light cone $\mathbb{N}^4_+$: \begin{equation} \lambda (1, x): \mathbb{S}^3\to \mathbb{N}^4_+ \end{equation} as the realization of $(\mathbb{S}^3, \lambda^2g_0)$. For the convenience of readers we present the calculations of the geometry of $\mathbb{S}^3_\lambda$ as a spacelike 3-surface in Minkowski spacetime in the Appendix \ref{gauss}. But it starts with the following observation.
\begin{lemma} Suppose that $\hat x: \textup{M}^2\to \mathbb{S}^3$ is an immersed surface and $\lambda^2g_0$ is a conformal metric in the round conformal class $[g_0]$ on $\mathbb{S}^3$. Then \begin{equation} \xi = H_\lambda y_\lambda + \overset{\rightarrow}{\bf n}_\lambda, \end{equation} where $H_\lambda$ is the mean curvature of $\hat x$ in $(\mathbb{S}^3, \lambda^2g_0)$ and $\overset{\rightarrow}{\bf n}_\lambda$ is the unit normal to $y_\lambda$ in $\mathbb{S}^3_\lambda\subset\mathbb{N}^4_+$. \end{lemma}
Using the calculations in Appendix \ref{gauss}, we are able to show in the proof of Theorem \ref{main-inv} that the data $\{m, \omega^\lambda, \Omega_\lambda, \Omega^*_\lambda\}$ that determine the first and second fundamental forms of the associate surface $\tilde x$ in Minkowski spacetime $\mathbb{R}^{1, 4}$ can all be expressed in terms of covariant derivatives of the curvature of the surface $\hat x$ in $(\mathbb{S}^3, \lambda^2g_0)$ and the covariant derivatives of curvature of $(\mathbb{S}^3, \lambda^2g_0)$ (including 0th order). In the exact same spirit as in Fefferman and Graham \cite{FG-1, FG-2}, our construction of associate surfaces $\tilde x$ provides a way to capture local scalar conformal invariants of a surface $\hat x$ . Namely, one can obtain local scalar conformal invariants of the surface $\hat x$ in the conformal round 3-sphere by computing the local scalar (pseudo-)Riemannian invariants of the associate surface $\tilde x$ at the homogeneous surface $x^{\mathbb{N}}$ in the light cone in Minkowski 5-spacetime. The first non-trivial one is \begin{equation}\label{laplacian-tilde}
\tilde\Delta\tilde H|_{\rho=0} = 2\alpha^{-3} (\Delta_\lambda H_\lambda + |\overset{\circ}{II}_\lambda|^2H_\lambda + (\overset{\circ}{II}_\lambda)^{ij}(R^\lambda)_{i3j3} - (R^\lambda)_{3i,}^{\quad i} ) \end{equation} in a general parametrization \eqref{general-param}, where $(R^\lambda)_{i3j3}$ and $(R^\lambda)_{3i}$ are the Riemann curvature and Ricci curvature of the metric $\lambda^2g_0$ on $\mathbb{S}^3$. Due to the homogeneity of $\tilde x$ we automatically have \begin{equation}\label{intr-willmore}
{\mathcal H}_\lambda = \Delta_\lambda H_\lambda + |\overset{\circ}{II}_\lambda|^2H_\lambda +
(\overset{\circ}{II}_\lambda)^{ij}(R^\lambda)_{i3j3} - (R^\lambda)_{3i,}^{\ \ i}= \hat\lambda^{-3} ( \Delta H + |\overset{\circ}{II}|^2H) \end{equation} which is the curvature that vanishes if and only if the surface $\hat x$ is Willmore. Notice that extra curvature terms do not show up when we work with either the round metric $g_0$ or the Euclidean metric. Similar formulas have appeared in the literature \cite{HL, GR, gover-s}. \\
We also calculate in Section \ref{calculations} some other conformal scalar invariants of higher orders: \begin{equation}\label{intro-norm-co-der} \aligned
|\nabla\tilde h|^2|_{\rho =0} & =\alpha^{-4}( |\nabla\Omega_\lambda|^2 + 8|dH_\lambda|^2 + 2 Ric^\lambda(\overset{\rightarrow}{\bf n}_\lambda, \nabla H_\lambda) + 3 H^2_\lambda|\Omega_\lambda|^2 \\ & \quad + 3 K^T_\lambda |\Omega_\lambda|^2+ 6 \Omega_\lambda\cdot \text{Hess} (H_\lambda)) \endaligned \end{equation} (cf. \eqref{norm-co-der}, where $K^T_\lambda$ is the sectional curvature of $(\mathbb{S}^3, \lambda^2 g_0)$ at the tangent plane to the surface $\hat x$, and \begin{equation}\label{intro-double-laplace} \aligned
\tilde\Delta & \tilde\Delta\tilde H|_{\rho=0} = 8\alpha^{-5} (\Delta_\lambda{\mathcal H}_\lambda +9 |\omega^\lambda|^2{\mathcal H}_\lambda - 3 \text{Div}({\omega^\lambda}) {\mathcal H}_\lambda \\ & \quad - 6\omega^\lambda(\nabla{\mathcal H}_\lambda)
- 6 {\mathcal H}_\lambda| \overset{\circ}{II}_\lambda|^{-2}\overset{\circ}{II}_\lambda\cdot\Omega^*_\lambda),\endaligned \end{equation} where $\omega^\lambda = <dy_\lambda, y^*_\lambda>$ and $\Omega^*_\lambda = - <dy^*_\lambda, d\xi>$ are parts of the data that determine the geometry of the associate surface $\tilde x$ and are given in \eqref{omega} and \eqref{Omega-star-ij} as invariants of the surface $\hat x$ in $(\mathbb{S}^3, \lambda^2g_0)$. \\
To end the introduction we remark that, for the sake of the production of local scalar invariants, the assumption of having no umbilical point in our construction is not an issue.
\section{The associate surfaces in $\mathbb{R}^{1,4}$}\label{Sect:2}
In this section we introduce the associate surfaces in Minkowski space $\mathbb{R}^{1,4}$ for a given surface $\hat x: \textup{M}^2\to \mathbb{S}^3$. We then show that such associate surface is canonical in doing conformal geometry for the surface $\hat x$. The construction relies on the conformal Gauss map and the conformal transform of $\hat x$. It is also very interesting to see how Blaschke and Bryant came to the conformal Gauss map and the conformal transform in very different perspectives \cite{blaschke, bryant}.
\subsection{Surfaces in 3-sphere}
Suppose that $$ \hat x: \textup{M}^2\to \mathbb{S}^3\subset \mathbb{R}^4 $$ is an immersed surface with isothermal coordinate $(u^1, u^2)$. Let $$ {\bf n}: \textup{M}^2\to \mathbb{R}^4 $$ be the unit normal vector at each point on the surface. Then we obtain the first fundamental form \begin{equation}\label{x-hat-I}
I = <d\hat x, d\hat x> = E|du|^2 \end{equation} and the second fundamental form \begin{equation}\label{x-hat-II} II = - <d\hat x, d{\bf n}> = e(du^1)^2 + 2f du^1du^2 + g(du^2)^2. \end{equation} Hence the mean curvature of the surface in 3-sphere is \begin{equation}\label{x-hat-H} H = \frac 1{2E}(e + g) \end{equation} and the Gaussian curvature of the surface is \begin{equation}\label{x-hat-guass} K = \frac {eg - f^2}{E^2} + 1. \end{equation} Notice that \begin{equation}\label{x-hat-gauss-map} \left\{\aligned {\bf n}_{u^1} & = - \frac eE \hat x_{u^1} - \frac fE \hat x_{u^2}\\
{\bf n}_{u^2} & = - \frac fE \hat x_{u^1} - \frac gE \hat x_{u^2}.\endaligned \right. \end{equation} \\
If one takes another conformal metric $\lambda^2 g_0$ on the 3-sphere $\mathbb{S}^3$, where $\lambda$ is a positive function on $\mathbb{S}^3$, then the first fundamental form for the surface $\hat x$ is \begin{equation}\label{lambda-1} I_\lambda = \hat\lambda^2 I, \end{equation} where $\hat\lambda = \lambda\circ\hat x$ and the second fundamental form is \begin{equation}\label{lambda-2} II_\lambda = \hat\lambda II - \lambda_{\bf n} I, \end{equation} where $\lambda_{\bf n} = {\bf n}(\lambda)$. Hence \begin{equation}\label{after-lambda} H_\lambda = \hat\lambda^{-1} (H - \frac {\lambda_{\bf n}}{\hat\lambda}) \text{ and } \overset{\circ} II_\lambda = \hat\lambda \overset{\circ}{II}, \end{equation} where $\overset{\circ}{II}$ is the traceless part of the second fundamental form $II$. Here we see the easy scalar conformal invariant
$|\overset{\circ} II|^{2}$, which can be considered to be the counter part of the square of the length of Weyl curvature on a conformal manifold.
\subsection{Minkowski 5-spacetime}
Let $\mathbb{R}^{1,4}$ be the Minkowski 5-spacetime, where we use the notation $$ \mathbb{R}^{1,4} = \{(t, x): t\in\mathbb{R} \text{ and } x\in \mathbb{R}^4\} $$ with the Lorentz inner product $$ < (t, x), (s, y)> = -st + x\cdot y. $$ Recall the positive light cone is given by $$
\mathbb{N}^4_+ = \{(t, x)\in \mathbb{R}^{1,4}: -t^2 + |x|^2 = 0 \text{ and } t > 0\}; $$ the hyperboloid is given as $$
\mathbb{H}^4 = \{(t, x)\in \mathbb{R}^{1,4}: -t^2 +|x|^2 = -1 \text{ and } t>0\}; $$ and the de Sitter 4-spacetime is given as $$
\mathbb{S}^{1,3} = \{(t, x)\in \mathbb{R}^{1,4}: -t^2 + |x|^2 = 1\}. $$ \\
Given a surface $\hat x: \textup{M}^2\to\mathbb{S}^3\subset\mathbb{R}^4$, we may consider the 2-surface $$ y = (1, \hat x): \textup{M}^2\to \mathbb{N}^4_+\subset \mathbb{R}^{1,4} $$ and the homogeneous extension $$ x^{\mathbb{N}} = \alpha y: \mathbb{R}^+\times\textup{M}^2\to \mathbb{N}^4_+\subset \mathbb{R}^{1,4} $$ for $\alpha\in\mathbb{R}^+$. There does not seem to be a way of doing ``geometry" of the homogeneous 3-surface $x^{\mathbb{N}}$ in the positive light cone $\mathbb{N}^4_+$. \\
To motivate our choice of the associate surface in $\mathbb{R}^{1,4}$ of $\hat x$ we first introduce the so-called homogeneous coordinate for $\mathbb{R}^{1,4}$ used in the ambient space construction of Fefferman and Graham \cite{FG-1, FG-2}, that is, \begin{equation}\label{f-g-coor} (t, x) = x^0 (1, \hat x) + x^0x^\infty\frac 12(1, -\hat x) \end{equation} where $$ \left\{\aligned x^0 & = \frac 1{2} (r +t)\\ x^0x^\infty & = (-r+t)\endaligned\right. $$
and $r = |x|$ and $x = r\hat x$. In this coordinate the Minkowski metric is $$ \tilde {\mathcal G}_0 = - 2x^\infty (dx^0)^2 - 2x^0dx^0dx^\infty + (x^0)^2(1 - \frac {x^\infty}2)^2 g_0(\hat x). $$ Hence, given a surface $\hat x: \textup{M}^2\to\mathbb{S}^3$, we are looking to construct an associate homogeneous timelike 4-surface \begin{equation}\label{assoc-surface} \tilde x = \alpha y + \alpha\rho y^*: \mathbb{R}^+\times\mathbb{R}^+\times\textup{M}^2\to \mathbb{R}^{1, 4} \end{equation} if we can have canonically the null vector $y^*$ at a given null position $y$ on $x^{\mathbb{N}}$. It is clear that the associate surface $\tilde x$ is ruled by the positive quadrants of timelike 2-planes in Minkowski spacetime. One may consider the intersection of $\tilde x$ with the hyperboloid $\mathbb{H}^4$: \begin{equation}\label{assoc-in-hyper} x^+ = \frac 1{\sqrt{2}} (e^t y + e^{-t} y^*): \mathbb{R}\times\textup{M}^2\to \mathbb{H}^4, \end{equation} which is called the associate ruled 3-surface since it is a 3-surface in hyperbolic 4-space ruled by geodesics lines. Recall that a geodesic line in the hyperboloid $\mathbb{H}^4$ is the intersection of the hyperboloid with a timelike 2-subspaces in Minkowski spacetime. In the following we will introduce the canonical choice of such $y^*$.\\
\subsection{Conformal Gauss maps}
Let us consider any unit spacelike normal vector to the homogeneous null 3-surface $x^{\mathbb{N}} = \alpha y$ in $\mathbb{N}^4_+\subset\mathbb{R}^{1,4}$. That is to ask a unit spacelike 5-vector $\xi$ to satisfy \begin{equation}\label{enveloping} <\xi, x^{\mathbb{N}}> = 0, \quad <\xi, x^{\mathbb{N}}_{u^1}> = 0, \quad <\xi, x^{\mathbb{N}}_{u^2}> = 0, \end{equation} which implies that $$ \xi = a y + \overset{\rightarrow}{\bf n}, $$ where $\overset{\rightarrow}{\bf n} = (0, {\bf n})$ is the unit normal to the surface $\hat x$ in the standard unit round 3-sphere in $\{1\}\times\mathbb{R}^4 \subset\mathbb{R}^{1,4}$. It turns out that there is a unique choice if we insist that the map $$ \xi: \textup{M}^2\to \mathbb{S}^{1,3}\subset \mathbb{R}^{1,4} $$ is (weakly) conformal. Namely we have
\begin{lemma} \label{xi-1} Suppose that $\hat x:\textup{M}^2\to\mathbb{S}^3$ is an immersed surface. Then, for a unit normal vector $\xi$ to the homogeneous null 3-surface $x^{\mathbb{N}} = \alpha y: \mathbb{R}^+\times\textup{M}^2\to \mathbb{N}^4_+\subset \mathbb{R}^{1,4}$, $$ <\xi_{u^1}, \xi_{u^2}> = 0 $$ if and only if $$ \xi = H y + \overset{\rightarrow}{\bf n} $$ and \begin{equation}\label{mobius-metric}
<d\xi, d\xi> = \frac 12 E|\overset{\circ}{II}|^2 |du|^2. \end{equation} \end{lemma}
\proof It is simply a straightforward calculation. We know $$ \xi_{u^i} = a_{u^i}(1, \hat x) + a(0, \hat x_{u^i}) + (0, {\bf n}_{u^i}). $$ Hence we have $$ <\xi_{u^1}, \xi_{u^2}> = -2a f + \frac 1E (fe+fg) = 0, $$ which is equivalent to $a = H$. For the rest we calculate \begin{equation}\label{mobius-metric-explicit} <\xi_{u^1}, \xi_{u^1}> = <\xi_{u^2}, \xi_{u^2}> = \frac 1{E^2}(f^2 +(\frac {e-g}2)^2)E. \end{equation} \endproof
Another way to identify a unique unit spacelike normal vector to the homogeneous null 3-surface $x^{\mathbb{N}} = \alpha y: \mathbb{R}^+\times\textup{M}^2\to\mathbb{N}^4_+$ is the following:
\begin{lemma}\label{xi-2} Suppose that $\hat x: \textup{M}^2\to\mathbb{S}^3$ is an immersed surface. Then, for a unit spacelike normal vector $\xi$ to $x^{\mathbb{N}} = \alpha y:\mathbb{R}^+\times\textup{M}^2\to \mathbb{N}^4_+\subset \mathbb{R}^{1,4}$, $$ \xi = H y+ \overset{\rightarrow}{\bf n} $$ if and only if \begin{equation}\label{integrability} < \Delta \xi, y> = 0. \end{equation} \end{lemma}
\proof We simply calculate, for $\xi = a(1, \hat x) + (0, {\bf n})$, $$ \Delta_0\xi = \xi_{u^1 u^1} + \xi_{u^2 u^2} = (\Delta_0 a)(1, \hat x) + 2\nabla a(0, \nabla \hat x) + a(0, \Delta_0\hat x) + (0, \Delta_0 {\bf n}) $$ and $$ <\Delta_0\xi, (1, \hat x)> = -2aE + 2HE. $$ Notice that $\Delta = E^{-1}\Delta_0$. \endproof
Before we give a formal definition of the conformal Gauss map we want to make a remark that \eqref{integrability} is the integrability condition for the unit vector field $\xi$ to be the conformal Gauss map (up to a sign) for the surface $\hat x$. This turns out to be the easiest way to see that $\hat x$ is Willmore if and only if the conformal Gauss map $\xi$ of $\hat x$ is also the conformal Gauss map (up to a sign) of the conformal transform $\hat x^*$ (cf. Definition \ref{def-conf-trans}).
\begin{definition} Suppose that $\hat x: \textup{M}^2\to \mathbb{S}^3$ is a surface. Then we will call \begin{equation}\label{conformal-gauss-map} \xi = H y+ \overset{\rightarrow}{\bf n}: \textup{M}^2\to\mathbb{S}^{1,3}\subset\mathbb{R}^{1,4} \end{equation} the conformal Gauss map according to Blaschke \cite{blaschke} (cf. \cite{bryant, rigoli}). \end{definition}
For a positive function $\lambda$ on the sphere $\mathbb{S}^3$ we consider the conformal metric $\lambda^2g_0$ on the sphere $\mathbb{S}^3$, which can be realized as the 3-sphere $\mathbb{S}^3_\lambda$: $\lambda(1, x):\mathbb{S}^3\to\mathbb{N}^4_+\subset\mathbb{R}^{1,4}$ in Minkowski spacetime. It is then very crucial and important to realize that the surface $\hat x$ in the 3-sphere $\mathbb{S}^3$ with the conformal metric $\lambda^2g_0$ is realized as the 2-surface $\hat\lambda (1, \hat x): \textup{M}^2\to \mathbb{N}^4_+ \subset \mathbb{R}^{1,4}$ inside the 3-sphere $\mathbb{S}^3_\lambda$. It is helpful to see the calculations in Appendix \ref{gauss} about the geometry of the 3-sphere $\mathbb{S}^3_\lambda$ in Minkowski spacetime $\mathbb{R}^{1,4}$.
\begin{lemma}\label{Lem:xi-lambda} If one works with a conformal metric $\lambda^2 g_0$ in general, then \begin{equation}\label{xi-lambda} \xi = \xi_\lambda = H_\lambda y_\lambda + \overset{\rightarrow}{\bf n}_\lambda , \end{equation} where $\overset{\rightarrow}{\bf n}_\lambda = \overset{\rightarrow}{\bf n} + (\log\lambda)_{\bf n} y$ is the unit normal to the surface $$y_\lambda = \hat \lambda (1, \hat x): \text{M}^2\to\mathbb{S}^3_\lambda\subset\mathbb{N}^4_+.$$ \end{lemma}
\proof It is easily seen that the normal direction to the surface $y_\lambda$ inside $\mathbb{S}^3_\lambda$ is $\lambda_{\bf n}(1, \hat x) + \lambda(0, {\bf n})$ and $< \lambda_{\bf n}(1, \hat x) + \lambda(0, {\bf n}), \lambda_{\bf n}(1, \hat x) + \lambda(0, {\bf n})> = \lambda^2$. Therefore the unit normal for the surface $y_\lambda$ in $\mathbb{S}^3_\lambda$ is $\overset{\rightarrow}{\bf n}_\lambda = \overset{\rightarrow}{\bf n} + (\log\lambda)_{\bf n} y$. Then it is easily verified that $$ H_\lambda y_\lambda + \overset{\rightarrow}{\bf n}_\lambda = Hy + \overset{\rightarrow}{\bf n} $$ using \eqref{after-lambda} \endproof
In the light of \eqref{mobius-metric}, the conformal Gauss map gives rise a spacelike 2-surface $$ \xi: \textup{M}^2\to \mathbb{S}^{1,3}\subset \mathbb{R}^{1,4} $$ when the original surface $\hat x: \textup{M}^2\to\mathbb{S}^3$ is free of umbilical point. We will have more detailed discussions for the reasons to call $\xi$ the conformal Gauss map in Section \ref{canonicity}. \\
It is very interesting to see that Blaschke came across to the conformal Gauss map in a very different perspective. Blaschke considered the family of mean curvature 2-spheres to the surface $\hat x$ in $\mathbb{S}^3$. A round 2-sphere in 3-sphere can be thought as the intersection of a timelike hyperplane and the 3-sphere at time $t=1$ in Minkowski spacetime $\mathbb{R}^{1,4}$ and a timelike hyperplane in $\mathbb{R}^{1,4}$ is described by a unit normal vector lying in de Sitter 4-spacetime $\mathbb{S}^{1,3}$. Given a direction $(H, H\hat x + {\bf n})\in \mathbb{S}^{1,3}$, the hyperplane perpendicular to that in $\mathbb{R}^{1,4}$ is given by the first equation in \eqref{enveloping}: \begin{equation}\label{mean-curvature-sphere-1} < (s, z), (H, H\hat x + {\bf n})> = 0, \end{equation} which is $$ -sH + H z\cdot (\hat x + \frac 1H {\bf n}) = 0. $$
At the level $s=1$ in the 3-sphere $|z|=1$, we arrive at $$ 1 - \hat z \cdot (\hat x + \frac 1H {\bf n}) = 0. $$ Then we may rewrite it as \begin{equation}\label{mean-curvature-sphere-2}
|\hat z - (\hat x + \frac 1H{\bf n})|^2 = \frac 1{H^2} \end{equation} which clearly is a round 2-sphere of mean curvature $H$ when intersects with the 3-sphere $\mathbb{S}^3\subset\mathbb{R}^4$ at $t=1$ in $\mathbb{R}^{1,4}$. Hence the equations \eqref{enveloping} exactly ask the surface $y = (1, \hat x): \textup{M}^2\to \mathbb{S}^3\subset \mathbb{N}^4_+\subset \mathbb{R}^{1,4}$ is an envelope surface of the family of mean curvature 2-spheres described by the conformal Gauss map $\xi$. \\
It is known that a mean curvature sphere of a surface goes to the mean curvature sphere of the image surface under conformal transformations.\\
\subsection{Conformal transforms}
Assume that the surface $\hat x: \textup{M}^2\to \mathbb{S}^3$ is free of umbilical point. Then the conformal Gauss map induces a spacelike 2-surface in the de Sitter 4-space $\mathbb{S}^{1,3}$ $$ \xi: \textup{M}^2\to \mathbb{S}^{1,3}\subset \mathbb{R}^{1,4}. $$ One notices that the equations \eqref{enveloping} imply that $y = (1, \hat x)$ is naturally a null normal vector the surface $\xi$ in the de Sitter 4-spacetime $\mathbb{S}^{1,3}$. Because $$ < y, \xi_{u^i}> = - <\xi, y_{u^i}> = 0. $$ Hence it is natural to take the other null normal vector $y^*$ such that \begin{equation}\label{y-star-def} \aligned <y^*, y> & = -1, \quad <y^*, y^*> = 0, \quad <y^*, \xi> = 0, \\ <y^*, \xi_{u^1}> & = 0, \text{ and } <y^*, \xi_{u^2}> = 0.\endaligned \end{equation} We may write $$ y^* = \hat\mu^* (1, \hat x^*). $$
\begin{definition}\label{def-conf-trans} Suppose that $\hat x: \textup{M}^2\to\mathbb{S}^3$ is a surface with no umbilical point. And suppose that $$ y^* =\hat \mu^*(1, \hat x^*): \textup{M}^2\to\mathbb{N}^4_+\subset \mathbb{R}^{1,4} $$ satisfies the equations \eqref{y-star-def} for $y=(1, \hat x)$. Then the surface $$ \hat x^*: \textup{M}^2\to \mathbb{S}^3 $$ is said to be the conformal transform of the surface $\hat x$ according to Robert Bryant \cite{bryant} (cf. \cite{blaschke}). \end{definition}
It is important that the conformal transform $\hat x^*$ of a surface $\hat x$ is independent of the conformal factor $\lambda$. Notice that the equations in \eqref{y-star-def} remain the same except the first one when replacing $y$ by $y_\lambda$. It is again very interesting to recall how Blaschke discovered the surface $\hat x^*$. From the above discussions it is now easy to see that the surface $\hat x^*$ is nothing but the other envelope surface of the family of round 2-spheres described by the conformal Gauss map $\xi$, i.e. the family of the mean curvature spheres of the surface $\hat x$. Since $y^*$ satisfies the last three equations in \eqref{y-star-def}.\\
\subsection{The geometry of the surface $\xi$ in $\mathbb{S}^{1,3}$}\label{geometry-xi} Recall that the first fundamental form for the surface $\xi$ in the de Sitter spacetime $\mathbb{S}^{1,3}\subset \mathbb{R}^{1,4}$ is \begin{equation}\label{mobius-metric-1}
I^\xi = <d\xi, d\xi> = m|du|^2, \end{equation} where \begin{equation}\label{m-def}
m = \frac 12 E|\overset{\circ}{II}|^2. \end{equation} The first fundamental form $I^\xi$ is usually called the M\"{o}bius metric on the surface $\hat x$. We remark here that, if one works with a conformal metric $\lambda^2g_0$ instead, then the M\"{o}bius metric remains the same \begin{equation}\label{m-lambda}
m = m_\lambda = \frac 12 E_\lambda|\overset{\circ}{II}_\lambda|^2. \end{equation} The second fundamental form for the surface $\xi$ in $\mathbb{S}^{1,3}$ is given by $$ II^{\xi} = - <d\xi, dy> y - <d\xi, dy^*> y^* = \Omega y + \Omega^* y^* = \Omega_\lambda \hat\lambda^{-2} y_\lambda + \Omega^*_\lambda \hat\lambda^2 y^*_\lambda $$ and \begin{equation}\label{omegas} \aligned \Omega_{ij} = - <\xi_{u^i}, y_{u^j} > &\text{ and } \ \Omega^*_{ij} = - <\xi_{u^i}, y^*_{u^j}>\\ (\Omega_\lambda)_{ij} = - <\xi_{u^i}, (y_\lambda)_{u^j} > = \hat\lambda\Omega_{ij} & \text{ and } \ (\Omega^*_\lambda)_{ij} = - <\xi_{u^i}, (y^*_\lambda)_{u^j}> = \hat\lambda^{-1} \Omega^*_{ij}\endaligned. \end{equation} In fact it is easy to calculate that \begin{equation}\label{Omega} \Omega = \left[\begin{matrix} \frac {e-g}2 & f\\f & \frac {g-e}2\end{matrix}\right] = \overset{\circ}{II} \end{equation}
Let us first calculate the mean curvature in the $y^*$ direction. We notice that $$ <\Delta_0\xi, \ y^*_\lambda> = ((\Omega^*_\lambda)_{11}+ (\Omega^*_\lambda)_{22}) $$ while $$ <\Delta_0\xi, \ y_\lambda> = ((\Omega_\lambda)_{11} + (\Omega_\lambda)_{22}) = 0. $$ Based on the calculations $$ \aligned <\Delta_0\xi, \ \xi> & = -2m\\ <\Delta_0\xi, \ \xi_{u^1}> & = \frac 12 m_{u^1} - \frac 12 m_{u^1} = 0\\ <\Delta_0\xi, \ \xi_{u^2}> & = - \frac 12 m_{u^2} + \frac 12 m_{u^2} = 0.\endaligned $$ we obtain \begin{equation}\label{laplace-xi-1} \Delta_0\xi = -((\Omega^*_\lambda)_{11}+ (\Omega^*_\lambda)_{22})y_\lambda -2m\xi= (- ((\Omega^*_\lambda)_{11}+ (\Omega^*_\lambda)_{22}) -2mH_\lambda) y_\lambda -2m\overset{\rightarrow}{\bf n}_\lambda . \end{equation} On the other hand, we directly calculate \begin{equation}\label{laplace-xi-2} \aligned \Delta_0\xi & = \Delta_0 (H_\lambda y_\lambda +\overset{\rightarrow}{\bf n}_\lambda) \\ & = (\Delta_0 H_\lambda) y_\lambda + H_\lambda \Delta_0 y_\lambda + 2 (H_\lambda)_{u^1} (y_\lambda)_{u^1} + 2 (H_\lambda)_{u^2} (y_\lambda)_{u^2} + \Delta_0\overset{\rightarrow}{\bf n}_\lambda \endaligned \end{equation} It seems that the best way to calculate geometrically is to use the Lorentz orthogonal frame $$ \{y_\lambda, y^\dagger_\lambda, (y_\lambda)_{u^1}, (y_\lambda)_{u^2}, \overset{\rightarrow}{\bf n}_\lambda\}, $$ where \begin{equation}\label{y-dagger-def} \aligned <y^\dagger_\lambda, y_\lambda> & = -1 \text{ and } \\ <y^\dagger_\lambda, y^\dagger_\lambda> & = <y^\dagger_\lambda, (y_\lambda)_{u^1}> = <y^\dagger_\lambda, (y_\lambda)_{u^2}> = <y^\dagger_\lambda, \overset{\rightarrow}{\bf n}_\lambda> = 0.\endaligned \end{equation} It is actually easy to find that \begin{equation}\label{y-dagger-explicit}
y_\lambda^\dagger = \frac 1{\lambda}(\frac 12|\nabla\log\lambda|^2 y + y^\dagger - \nabla \log\lambda), \end{equation} where $y^\dagger = \frac 12(1, - \hat x)$ and $\nabla$ is the gradient on the standard round 3-sphere. We will do inner product to both \eqref{laplace-xi-1} and \eqref{laplace-xi-2} with the null vector $y^\dagger_\lambda$. To calculate $H_\lambda <\Delta_0 y_\lambda, y^\dagger_\lambda> + <\Delta_0\overset{\rightarrow}{\bf n}_\lambda, y^\dagger_\lambda>$ we rewrite $$ H_\lambda <\Delta_0 y_\lambda, y^\dagger_\lambda> = - H_\lambda (<(y_\lambda)_{u^1}, (y^\dagger_\lambda)_{u^1}> + <(y_\lambda)_{u^2}, (y^\dagger_\lambda)_{u^2}>) $$ and $$ <\Delta_0\overset{\rightarrow}{\bf n}_\lambda, y^\dagger_\lambda> = - <(\overset{\rightarrow}{\bf n}_\lambda)_{u^1}, (y^\dagger_\lambda)_{u^1}> - <(\overset{\rightarrow}{\bf n}_\lambda)_{u^2}, (y^\dagger_\lambda)_{u^2}> - <\overset{\rightarrow}{\bf n}_\lambda, (y^\dagger_\lambda)_{u^i}>_{u^i} . $$ Meanwhile one may calculate \begin{equation}\label{n-lambda-i-j} \left\{\aligned (\overset{\rightarrow}{\bf n}_\lambda)_{u^1} & = - \frac {e_\lambda}{E_\lambda} (y_\lambda)_{u^1} - \frac {f_\lambda}{E_\lambda} (y_\lambda)_{u^2} - <(\overset{\rightarrow}{\bf n}_\lambda)_{u^1}, y^\dagger_\lambda>y_\lambda \\ (\overset{\rightarrow}{\bf n}_\lambda)_{u^2} & = - \frac {f_\lambda}{E_\lambda} (y_\lambda)_{u^1} - \frac {g_\lambda}{E_\lambda} (y_\lambda)_{u^2} - <(\overset{\rightarrow}{\bf n}_\lambda)_{u^2}, y^\dagger_\lambda>y_\lambda. \endaligned\right. \end{equation} Hence we have \begin{equation}\label{trace-Omega-star} \aligned H_\lambda & <\Delta_0 y_\lambda, y^\dagger_\lambda> + <\Delta_0\overset{\rightarrow}{\bf n}_\lambda, y^\dagger_\lambda>\\ & = E^{-1}_\lambda (\overset{\circ}{II}_\lambda)_{ij} <(y_\lambda)_{u^i}, (y^\dagger_\lambda)_{u^j}> - <\overset{\rightarrow}{\bf n}_\lambda, (y^\dagger_\lambda)_{u^i}>_{u^i} \\ & = - E_\lambda^{-1} (\overset{\circ}{II}_\lambda)_{ij} R^\lambda_{i3j3} + E_\lambda (R^\lambda)_{3i,}^{\quad i} \endaligned \end{equation} due to \eqref{n-dagger}, \eqref{i-dagger-j}, and \eqref{coord-covar}. Now we obtain the mean curvature of the surface $\xi$ in the de Sitter spacetime $\mathbb{S}^{1,3}$.
\begin{lemma} Suppose that $\hat x:\textup{M}^2\to\mathbb{S}^3$ is an immersed surface with no umbilical point and that $\xi:\textup{M}^2\to \mathbb{S}^{1,3}$ is the conformal Gauss map. Then the surface $\xi$ is spacelike and its mean curvature is a null vector \begin{equation}\label{mean-curvature-xi}
H^\xi = 2\hat\lambda^2\frac {{\mathcal H}_\lambda}{|\overset{\circ}{II}_\lambda|^2} y^*_\lambda \end{equation} for any positive function $\lambda$ on the 3-sphere $\mathbb{S}^3$, where \begin{equation}\label{script-H-lambda}
{\mathcal H}_\lambda = \Delta_\lambda H_\lambda + |\overset{\circ}{II}_\lambda|^2H_\lambda + (\overset{\circ}{II}_\lambda)^{ij}(R^\lambda)_{i3j3} - (R^\lambda)_{3i,}^{\ \ i}, \end{equation} $(R^\lambda)_{ijkl}$ and $(R^\lambda)_{ij}$ are the Riemann curvature and Ricci curvature for the conformal metric $\lambda^2g_0$ on the 3-sphere $\mathbb{S}^3$ respectively. \end{lemma} \proof We perform inner product to \eqref{laplace-xi-1} and \eqref{laplace-xi-2} by the null vector $y^\dagger_\lambda$ and obtain that \begin{equation}\label{trace-star}
(\Omega^*_\lambda)_{11} + (\Omega^*_\lambda)_{22} = E_\lambda (- \Delta_\lambda H_\lambda - |\overset{\circ}{II}_\lambda|^2H_\lambda - (\overset{\circ}{II}_\lambda)^{ij}(R^\lambda)_{i3j3} + (R^\lambda)_{3i,}^{\ \ i}) \end{equation} in the light of \eqref{trace-Omega-star}. Then one can easily calculate the mean curvature for $\xi$ in $\mathbb{S}^{1,3}$. \endproof
\vskip 0.2in We remark that \eqref{mean-curvature-xi} actually shows that \begin{equation}\label{2-scalar-invar}
{\mathcal H}_\lambda = \hat\lambda^{-3} (- \Delta H - |\overset{\circ}{II}|^2H) \end{equation} for a surface $\hat x$ in the conformal 3-sphere.
\begin{theorem} \label{conf-gauss-surface} (\cite{blaschke} \cite{bryant}) Suppose that $\hat x: \textup{M}^2\to\mathbb{S}^3$ is an immersed surface with no umbilical point. Then $\hat x$ is a Willmore surface in $\mathbb{S}^3$ if and only if the conformal Gauss map induces a minimal spacelike surface in the de Sitter spacetime $\mathbb{S}^{1,3}$. Moreover its conformal transform $\hat x^*$ is a dual Willmore surface in $\mathbb{S}^3$. \end{theorem}
\proof Most of this theorem has been known to Blaschke \cite{blaschke} and Bryant \cite{bryant}. Because Lemma \ref{xi-2} implies that $\xi$ is also the conformal Gauss map (up to the sign) for $\hat x^*$ when $H^\xi$ vanishes. The two dual Willmore surfaces are the two envelope surfaces of the family of round 2-spheres described by the conformal Gauss map $\xi$. \endproof
\begin{remark} It is also known to Balschke \cite{blaschke} and Bryant \cite{bryant} that \begin{itemize} \item If $\hat x$ is a minimal surface in $\mathbb{S}^3$, then $\hat x^* = -\hat x$. \item $\hat x$ is a Willmore surface if and only if $\hat x^{**} = \hat x$, which raises an interesting question: what does it mean $\hat x^{***} = \hat x$ if possible? \end{itemize}
\end{remark}
\subsection{Finding $y^*_\lambda$} Let us now solve $y^*_\lambda$ for $y_\lambda = \hat\lambda (1, \hat x) = \hat\lambda y$, where $\hat\lambda = \lambda\circ \hat x$ and $\lambda$ is a positive function on the sphere $\mathbb{S}^3$. At each point on the surface we set $$ y^*_\lambda =\kappa y_\lambda + \kappa_\dagger y^\dagger_\lambda + b \overset{\rightarrow}{\bf n}_\lambda + \frac {\omega^\lambda_1}{E_\lambda} (y_\lambda)_{u^1} +\frac {\omega^\lambda_2}{E_\lambda} (y_\lambda)_{u^2}. $$ And we get from \eqref{y-star-def} \begin{equation}\label{abcd} \left\{ \aligned \kappa_\dagger & = 1\\ - 2\kappa_+\kappa_- + b^2 + \frac {(\omega^\lambda_1)^2 + (\omega^\lambda_2)^2}{E_\lambda} & = 0 \\ b & = H_\lambda\\ - (\Omega_\lambda)_{11}\omega^\lambda_1 - (\Omega_\lambda)_{12} \omega^\lambda_2 & = (H_\lambda)_{u^1}E_\lambda\\ - (\Omega_\lambda)_{21}\omega^\lambda_1 - (\Omega_\lambda)_{22} \omega^\lambda_2 & = (H_\lambda)_{u^2} E_\lambda.\endaligned\right. \end{equation} We therefore have
\begin{lemma}\label{lemma-x-star} Suppose that $\hat x: \textup{M}^2\to\mathbb{S}^3$ is an immersed surface with no umbilical point. Then \begin{equation}\label{y-star-lambda}
y^*_\lambda =\frac 12 ( |\omega^\lambda|^2 + H_\lambda^2)y_\lambda + y_\lambda^\dagger
+ H_\lambda\overset{\rightarrow}{\bf n}_\lambda - (\overset{\circ}{II})^{-1}_\lambda d H_\lambda \end{equation} for any positive function $\lambda$ on the 3-sphere, where $$
|\omega^\lambda|^2 = \frac {(\omega^\lambda_1)^2 + (\omega^\lambda_2)^2}{E_\lambda} = \frac 1m((H_\lambda)_{u^1}^2 +(H_\lambda)_{u^2}^2). $$ In particular, \begin{equation}\label{y-star}
y^* = \frac 12 (|\omega|^2 +H^2) y + \frac 12(1, -\hat x) + H (0, {\bf n}) - (0, (\overset{\circ}{II})^{-1} d H), \end{equation} and \begin{equation}\label{x-star} x^* = a \hat x + \frac H {1-a} {\bf n} - \frac 1{1-a}(\overset{\circ}{II})^{-1} d H, \end{equation} where \begin{equation}\label{what-a}
a = \frac {|\omega|^2 +H^2-1}{|\omega|^2 +H^2 +1}. \end{equation} \end{lemma} \proof One simply solves \eqref{abcd} if $\det\Omega_\lambda \neq 0$, which is equivalent to the fact that the surface has no umbilical point. \endproof
\subsection{Canonicity of $y^*$}\label{canonicity}
Now we want to show that the choice of $y^*$ is canonical in terms of doing conformal geometry for the surface $\hat x$ in $\mathbb{S}^3$. It is important to realize that there are two separate issues here. One is about the symmetry of the conformal 3-sphere. To be precise, for a conformal transformation $$ \phi: \mathbb{S}^3\to\mathbb{S}^3 $$ and the transformed surface $$\phi(\hat x): \textup{M}^2\to \mathbb{S}^3,$$ is it true that $$ \tilde\phi (\tilde x)= \alpha\tilde\phi (y) + \alpha\rho\tilde\phi(y^*):\mathbb{R}^+\times\mathbb{R}^+\times\textup{M}^2\to\mathbb{R}^{1,4}$$ is the associate 4-surface of $\phi(\hat x)$ in $\mathbb{R}^{1,4}$, where $\tilde\phi$ is the corresponding Lorentz transformation on $\mathbb{R}^{1,4}$ to $\phi$? The other issue is whether or not the associate surface $\tilde x$ is independent of metrics in the conformal class of the round 3-sphere. The first easy and important fact is that the conformal Gauss map is independent of the metrics in the conformal class.
\begin{lemma}\label{transform-gauss} Suppose that $\hat x: \textup{M}^2\to\mathbb{S}^3$ is an immersed surface. Then the conformal Gauss map $\xi$ is independent of the metrics in the conformal class of the round 3-sphere $\mathbb{S}^3$. Meanwhile, the conformal Gauss map for the transformed surface $\phi(\hat x)$
is exactly $\tilde\phi(\xi)$, where $\tilde\phi$ is the Lorentz transformation on the Minkowski spacetime $\mathbb{R}^{1,4}$ corresponding to a conformal transformation $\phi$ on $\mathbb{S}^3$. \end{lemma}
\proof First of all, one needs to realize that, for any given metric in the conformal class of the round 3-sphere, it simply amounts to consider the surface $$ y_\lambda = \hat \lambda(1, \hat x): \textup{M}^2\to \mathbb{N}^4_+ $$ for some positive function $\lambda: \mathbb{S}^3\to \mathbb{R}^+$ and $\hat\lambda = \lambda\circ\hat x$. But this only possibly alters the parametrization of the homogeneous null 3-surface $x^{\mathbb{N}} = \alpha \hat \lambda (1, \hat x): \mathbb{R}^+\times\textup{M}^2\to \mathbb{N}^4_+$. Hence it will not alter the conformal Gauss map. Of course one has already seen this from Lemma \ref{Lem:xi-lambda}.\\
Next we consider the transformed surface $\phi(\hat x)$. Recall that, given a conformal transformation $\phi$ of 3-sphere, we have a unique Lorentz transformation $\tilde\phi$ in the time and orientation preserving component of the Lorentz group on the Minkowski spacetime such that, for $\lambda(1, \hat x)\in \mathbb{R}^{1, 4}$, \begin{equation}\label{poincare} \tilde \phi (\lambda (1, \hat x)) = \lambda \mu (1, \phi(\hat x)) \end{equation} for some positive number $\mu$. By the definition, which requires $\tilde\phi$ is a linear map and $$ <\tilde\phi ((t, \hat x)), \tilde\phi((s, \hat y))> = < (t, \hat x), (s, \hat y)>, $$ we now easily see that $\tilde\phi(\xi)$ is the conformal Gauss map for the transformed surface $\phi(\hat x)$. Since $\tilde\phi(\xi)$ is the unit normal vector field to the homogeneous null 3-surface $\tilde\phi(x)$ in $\mathbb{N}^4_+$ that is conformal map from $\textup{M}^2$ to $\mathbb{S}^{1,3}$. \endproof
Consequently we have
\begin{proposition} Suppose that $\hat x: \textup{M}^2\to\mathbb{S}^3$ is an immersed surface with no umbilical point. Then the associate surface $$ \tilde x = \alpha y_\lambda + \alpha\rho y^*_\lambda: \mathbb{R}^+\times\mathbb{R}^+\times\textup{M}^2\to\mathbb{R}^{1,4}, $$ for any $y_\lambda= \hat\lambda (1, \hat x)$ and $y^* = \hat\lambda^{-1} \lambda^*(1, \hat x^*)$ defined by the equations \eqref{y-star-def}, is independent of the metrics in conformal class of the round 3-sphere $\mathbb{S}^3$. \end{proposition}
\proof It suffices to verify that \begin{equation}\label{star-scaled} (\hat\lambda y)^* = \hat\lambda^{-1} y^*. \end{equation} Since it implies that the change of metrics in the conformal class will at most cause possible change of parametrization of the associate surface $\tilde x$. \endproof
We also have from Lemma \ref {transform-gauss} the following:
\begin{lemma} Suppose that $\hat x: \textup{M}^2\to\mathbb{S}^3$ is an immersed surface with no umbilical point. Let $y_\lambda = \hat\lambda (1, \hat x)\in \mathbb{N}^4_+$ and let $\phi$ be a conformal transformation of 3-sphere. Then \begin{equation}\label{transform-y-star} \tilde\phi(y_\lambda)^* = \tilde\phi(y^*_\lambda). \end{equation} Hence \begin{equation} \label{transform-x-star} \phi(\hat x^*) = (\phi(\hat x))^*. \end{equation} \end{lemma} \proof From Lemma \ref{transform-gauss} we know that the conformal Gauss map for the transformed surface $\phi(\hat x)$ is $\tilde\phi(\xi)$. Then it is easy to verify \eqref{y-star-def} for $\tilde\phi(y^*)$ to be $\tilde\phi(y)^*$. Then the equation \eqref{transform-x-star} follows from \eqref{poincare} and \eqref{transform-y-star}: $$ \hat\gamma^*(1, (\phi(\hat x))^*) = \tilde\phi(y)^* = \tilde\phi(y^*) = \hat\mu^*\hat\lambda^*(1, \phi(\hat x^*)). $$ \endproof
Therefore we have
\begin{proposition} Suppose that $\hat x: \textup{M}^2\to\mathbb{S}^3$ is an immersed surface with no umbilical point. Let $\phi$ be a conformal transformation of 3-sphere. Then the associate 4-surface in $\mathbb{R}^{1,4}$ of the transformed surface $\phi(\hat x)$ is exactly the 4-surface $\tilde\phi(\tilde x)$ transformed from the associate 4-surface $\tilde x$ of the original surface $\hat x$ under the corresponding Lorentz transformation $\tilde\phi$ of $\phi$. \end{proposition}
\section{The geometry of the associate surfaces}
In this section we calculate the first and second fundamental forms for the associate homogeneous timelike 4-surfaces $\tilde x$ in $\mathbb{R}^{1,4}$ as well as for the associate ruled surface $x^+$ in the hyperboloid $\mathbb{H}^4$, for a given immersed 2-surface $\hat x$ in $\mathbb{S}^3$.
\subsection{The first fundamental form for $\tilde x$ in $\mathbb{R}^{1,4}$}\label{Sect:I-tilde}
To calculate the first fundamental form for the surface in the parametrization \begin{equation}\label{lambda-para} \tilde x = \alpha y_\lambda + \alpha \rho y^*_\lambda \end{equation} associated with a conformal metric $\lambda^2g_0$ on the 3-sphere $\mathbb{S}^3$ , we first calculate $$ d\tilde x = (y_\lambda +\rho y^*_\lambda)d\alpha + \alpha y^*_\lambda d\rho + (\alpha (y_\lambda)_{u^1}+\alpha\rho (y^*_\lambda)_{u^1})du^1 + (\alpha (y_\lambda)_{u^2} + \alpha\rho (y^*_\lambda)_{u^2})du^2. $$ Hence the first fundamental form for the associate 4-surface $\tilde x$ in the coordinates $(\alpha, \rho, u^1, u^2)$ is $$ \aligned I^{\tilde x} & = <d\tilde x, d\tilde x> = -2 \rho d\alpha d\alpha - 2\alpha d\alpha d\rho \\ & + 2\alpha^2 <(y^*_\lambda, (y_\lambda)_{u^1}>d\rho du^1 + 2\alpha^2 <y^*_\lambda, (y_\lambda)_{u^2}>d\rho du^2\\ & + <\alpha (y_\lambda)_{u^1}+\alpha\rho (y^*_\lambda)_{u^1}, \alpha (y_\lambda)_{u^1}+\alpha\rho (y^*_\lambda)_{u^1}>(du^1)^2 \\ & + <\alpha (y_\lambda)_{u^2} +\alpha\rho (y^*_\lambda)_{u^2}, \alpha (y_\lambda)_{u^2}+\alpha\rho (y^*_\lambda)_{u^2}>(du^2)^2\\ & + 2<\alpha (y_\lambda)_{u^1}+\alpha\rho (y^*_\lambda)_{u^1}, \alpha (y_\lambda)_{u^2}+\alpha\rho (y^*_\lambda)_{u^2}>du^1du^2. \endaligned $$ In fact one may calculate \begin{equation}\label{tilde-x-I} \left\{ \aligned (y_\lambda)_{u^1} & = - \omega^\lambda_1 y_\lambda - \frac {(\Omega_\lambda)_{11} }{m}\xi_{u^1} - \frac {(\Omega_\lambda)_{12}}{m}\xi_{u^2}\\ (y_\lambda)_{u^2} & = - \omega^\lambda_2 y_\lambda - \frac {(\Omega_\lambda)_{21} }{m}\xi_{u^1} - \frac {(\Omega_\lambda)_{22}}{m}\xi_{u^2}\\ (y^*_\lambda)_{u^1} & = \omega^\lambda_1 y^*_\lambda - \frac {(\Omega_\lambda)^*_{11}}{m}\xi_{u^1} - \frac {(\Omega_\lambda)^*_{12}}{m}\xi_{u^2}\\ (y^*_\lambda)_{u^2} & = \omega^\lambda_2 y^*_\lambda - \frac {(\Omega_\lambda)^*_{21} }{m}\xi_{u^1} - \frac {(\Omega_\lambda)^*_{22}}{m}\xi_{u^2}\endaligned \right. \end{equation} where \begin{equation}\label{omega} \omega^\lambda = <dy_\lambda, y^*_\lambda> = - I_\lambda(\Omega_\lambda^{-1}dH_\lambda) \end{equation} based on \eqref{abcd}. Now let us write $I^{\tilde x}$ in matrix form: \begin{equation}\label{matrix-tilde} I_{\tilde x} = \left[\begin{matrix} \begin{matrix} \ -2\rho & -\alpha \\ -\alpha & \ 0\end{matrix} & \begin{matrix} 0 & 0 \\ \alpha^2\omega^\lambda_1 & \alpha^2 \omega^\lambda_2\end{matrix}\\ \begin{matrix} \quad\quad 0 & \quad \alpha^2\omega^\lambda_1\\ \quad\quad 0 & \quad \alpha^2\omega^\lambda_2\end{matrix} & \quad \alpha^2 F \quad\end{matrix}\right]
\end{equation}
where
\begin{equation}\label{F-matrix}
\left\{
\aligned
F_{11} & = \frac 1m(p^2 + q^2) +2 \rho (\omega^\lambda_1)^2\\
F_{12} & = F_{21} = \frac 1m q(p+r) + 2 \rho \omega^\lambda_1\omega^\lambda_2\\
F_{22} & = \frac 1m(q^2 + r^2) + 2 \rho(\omega^\lambda_2)^2 \endaligned\right. \end{equation}
and $$ \left[\begin{matrix} p & q \\ q & r\end{matrix}\right] = \Omega_\lambda + \rho\Omega^*_\lambda. $$ It can be calculated that \begin{equation}\label{det-I-tilde}
\det I^{\tilde x} = - \frac {\alpha^6}{m^2}(pr - q^2)^2 = -\frac {\alpha^6}{4m^2} (E^2_\lambda|\Omega_\lambda + \rho\Omega^*_\lambda|^2 - \rho^2((\Omega^*_\lambda)_{11}+ (\Omega^*_\lambda)_{22})^2)^2 \end{equation} which can tell us where the associate surface $\tilde x$ is degenerate. It is maybe a little surprising that it is actually not difficult to calculate the inverse of $I_{\tilde x}$. We present the calculations in Appendix \ref{app-inverse-tilde} since they are straightforward calculations.
\subsection{The second fundamental form for $\tilde x$ in $\mathbb{R}^{1,4}$}\label{Sect:II-tilde}
It is clear from the definition that the conformal Gauss map $\xi$ is the unit normal vector for the associate 4-surface $\tilde x$ in $\mathbb{R}^{1,4}$. Hence
the second fundamental form for $\tilde x$ in $\mathbb{R}^{1,4}$ is
\begin{equation}\label{tilde-x-II}
II^{\tilde x} = - < d\tilde x, d\xi> = (\alpha(\Omega_\lambda)_{ij}+\alpha\rho(\Omega^*_\lambda)_{ij})du^idu^j
\end{equation}
or in matrix form
$$ II_{\tilde x} = \left[\begin{matrix} \quad 0 & 0 \\ \quad 0 & \alpha\Omega_\lambda +\alpha \rho\Omega^*_\lambda\end{matrix}\right]. $$ Therefore the mean curvature for the associate 4-surface in $\mathbb{R}^{1,4}$ is $$
H^{\tilde x} = \text{Tr} (I_{\tilde x})^{-1}II_{\tilde x}.
$$
To calculate the mean curvature $H^{\tilde x}$ one only needs to know the low-right $2\times 2$ block
in the inverse of the matrix $I_{\tilde x}$. According to the calculations in Appendix \ref{app-inverse-tilde}, particularly \eqref{F-star-inverse} \eqref{app-3j} \eqref{app-4j},
we therefore have \begin{equation}\label{x-tilde-mean} \aligned H^{\tilde x} & = \frac m{\alpha(pr-q^2)^2}((q^2+r^2)p - 2 q^2(p+r) + (p^2+q^2)r)\\ & = \frac {m(p+r)}{\alpha(pr-q^2)}, \endaligned \end{equation} where $$ pr - q^2 = \det\Omega_\lambda - \rho\text{Tr}\Omega_\lambda\Omega^*_\lambda + \rho^2\det\Omega^*_\lambda $$ and \begin{equation}\label{script-H-0} p+r = \rho((\Omega^*_\lambda)_{11}+(\Omega^*_\lambda)_{22}) = - \rho E_\lambda {\mathcal H}_\lambda \end{equation} in the light of \eqref{trace-star}.
\begin{theorem}\label{Th:tilde-H} Suppose that $\hat x:\textup{M}^2\to\mathbb{S}^3$ is an immersed surface with no umbilical point. Then $\hat x$ is a Willmore surface in $\mathbb{S}^3$ if and only if the associate 4-surface $\tilde x$ in $\mathbb{R}^{1,4}$ is minimal. \end{theorem} \proof Based on the above equations \eqref{script-H-0} and \eqref{x-tilde-mean} we obtain that \begin{equation}\label{script-H} H^{\tilde x} = \frac {\rho \det\Omega_\lambda {\mathcal H}_\lambda} {\alpha (\det\Omega_\lambda - \rho\text{Tr}\Omega_\lambda\Omega^*_\lambda + \rho^2\det\Omega^*_\lambda)}. \end{equation} \endproof
\subsection{Local fundamental theorem for surfaces in conformal 3-sphere}
In this subsection we want to state and prove a local fundamental theorem for surfaces in conformal 3-sphere. In the previous section we have introduced the associate surface $\tilde x$ in Minkowski spacetime $\mathbb{R}^{1, 4}$ from a given surface $\hat x$ in $\mathbb{S}^3$. From the geometric structure of the associate surface $\tilde x$ one can tell that its intersection with the positive light cone $\mathbb{N}^4_+$ is a homogeneous null 3-surface whose projectivization will recover the original surface $\hat x$ in $\mathbb{S}^3$.
Given a surface $\hat x$ in $\mathbb{S}^3$ with a isothermal coordinates $(u^1, u^2)$ on the parameter space $\textup{M}^2$, we have the first fundamental form $I$ in matrix form $$ I = \left[\begin{matrix} E & 0\\0 & E\end{matrix}\right] $$ and the second fundamental fundamental $II$ form in matrix form $$ II = \left[\begin{matrix} e & f\\f & g\end{matrix}\right] $$ The local fundamental theorem for surfaces in Riemannian geometry states that, up to isometries of the standard round sphere $\mathbb{S}^3$, locally the surface is uniquely determined by the first fundamental form $I$ and the second fundamental form $II$ in the standard round sphere $\mathbb{S}^3$. Conversely, given a positive definite symmetric 2-form $I$ and a symmetric 2-form $II$ in the parameter domain, which satisfy some integrability conditions (Gauss-Codazzi equations), up to isometries, there is locally a unique surface $\hat x$ in the standard round sphere $\mathbb{S}^3$ whose first and second fundamental forms are $I$ and $II$. We are looking for the analogous local fundamental theorem for surfaces in conformal round 3-sphere $\mathbb{S}^3$. The core idea of the local fundamental theorem in Riemannian geometry is to solve the structure equations, which are the equations of motion of Fren\'{e}t frames on the surface and are determined from $I$ and $II$. \\
Our strategy here is to use the local fundamental theorem for the associate surface $\tilde x$ in the Minkowski spacetime $\mathbb{R}^{1, 4}$ to establish the local fundamental theorem for a surface $\hat x$ in the conformal sphere $\mathbb{S}^3$. Since the association introduced in previous subsections requires that the surface $\hat x$ has no umbilical point, we will always assume here that surfaces $\hat x$ have no umbilical point. \\
To summarize the previous discussions, given a surface $\hat x$ in $\mathbb{S}^3$,
we have $I=E|du|^2$ and $II = e (du^1)^2 + 2f du^1du^2 + g(du^2)^2$. We also have the so-called M\"{o}bius metric $I^\xi = m |du|^2 = \frac 12 E|\overset{\circ}{II}|^2|du|^2$ induced from the Conformal Gauss map $\xi$ of the surface $\hat x$, where $$ \overset{\circ}{II}= \left[\begin{matrix} \frac {e-g}2 & f\\ f & \frac {g-e}2\end{matrix}\right] $$ is the traceless part of the second fundamental form $II$. We then construct the associate surface $$ \tilde x = \alpha y_\lambda +\alpha \rho y^*_\lambda: \mathbb{R}^+\times\mathbb{R}^+\times\textup{M}^2: \mathbb{R}^{1, 4}. $$ \\
The first fundamental form $I^{\tilde x}$ for $\tilde x$ in matrix form is, from \eqref{matrix-tilde}, $$ \left[\begin{matrix} \begin{matrix} -2\rho & -\alpha\\-\alpha & 0 \end{matrix} & \begin{matrix} 0 & 0 \\ \alpha^2\omega^\lambda_1 & \alpha^2\omega^\lambda_2\end{matrix} \\ \begin{matrix} \quad\quad 0 & \alpha^2\omega^\lambda_1\\ \quad\quad 0 & \alpha^2\omega^\lambda_2\end{matrix} & \begin{matrix} \frac {\alpha^2}m(p^2+q^2) + 2\alpha^2\rho(\omega^\lambda_1)^2 & \frac {\alpha^2}m q(p+r)+ 2\alpha^2\rho\omega^\lambda_1\omega^\lambda_2\\ \frac {\alpha^2}m q(p+r) +2\alpha^2\rho\omega^\lambda_1\omega^\lambda_2& \frac {\alpha^2}m(q^2 +r^2) +2\alpha^2\rho(\omega^\lambda_2)^2\end{matrix} \end{matrix}\right], $$ where the 1-form $$ \omega^\lambda = \omega^\lambda_1du^1 + \omega^\lambda_2du^2 = -d\log\hat\lambda - I(\Omega^{-1} (d H)) = d\log\hat\lambda +\omega. $$ \\
And the second fundamental form $II^{\tilde x}$ for $\tilde x$ in $\mathbb{R}^{1, 4}$ in matrix form is, from \eqref{tilde-x-II}, $$ \left[\begin{matrix} 0 & 0 \\ 0 & \alpha \Omega_\lambda + \alpha\rho\Omega^*_\lambda \end{matrix}\right]. $$
where $\Omega_\lambda = \hat\lambda\Omega$ and $\Omega^*_\lambda = \hat\lambda^{-1}\Omega^*$. Notice that $I^{\tilde x}$ and $II^{\tilde x}$ are exactly determined by the M\"{o}bius metric $I^\xi = m|du|^2$, the 1-form $\omega$, the traceless symmetric 2-tensor $\Omega$ and the symmetric 2-tensor $\Omega^*$, plus the conformal factor $\hat \lambda$. \\
Next we write the equations for the motion of the Fren\'{e}t frames on the associate surface $\tilde x$ according to $I^{\tilde x}$ and $II^{\tilde x}$. We consider the Fren\'{e}t frame $\{y_\lambda, y^*_\lambda, \frac 1{\sqrt m}\xi_{u^1}, \frac 1{\sqrt m}\xi_{u^2}, \xi\}$ on the associate surface $\tilde x$. Because they are the orthonormal frames on $\tilde x$ with respect to the Minkowski metric $\tilde{\mathcal G}_0$ on $\mathbb{R}^{1, 4}$. We now write \begin{equation}\label{structure-equation-1} \frac \partial{\partial u^1}\left[\begin{matrix} y_\lambda\\y^*_\lambda\\ \frac 1{\sqrt m}\xi_{u^1}\\ \frac 1{\sqrt m}\xi_{u^2}\\ \xi\end{matrix}\right] = \left[\begin{matrix} -\omega^\lambda_1 & 0 & - \frac 1{\sqrt m}(\Omega_\lambda)_{11} & -\frac 1{\sqrt m}(\Omega_\lambda)_{12} & 0\\ 0 & \omega^\lambda_1 & - \frac 1m(\Omega^*_\lambda)_{11} & - \frac 1m (\Omega^*_\lambda)_{12} & 0\\ \frac 1{\sqrt m}(\Omega_\lambda)_{11} & \frac 1{\sqrt m}(\Omega^*_\lambda)_{11} & 0 & - \frac 1{2m} m_{u^2} & -\sqrt{m}\\ \frac 1{\sqrt m}(\Omega_\lambda)_{21} & \frac 1{\sqrt m}(\Omega^*_\lambda)_{21} & \frac 1{2m} m_{u^2} & 0 & 0 \\ 0 & 0 & \sqrt{m} & 0 & 0\end{matrix}\right] \left[\begin{matrix} y_\lambda \\y^*_\lambda \\ \frac 1{\sqrt m}\xi_{u^1}\\ \frac 1{\sqrt m}\xi_{u^2}\\ \xi\end{matrix}\right] \end{equation} and \begin{equation}\label{structure-equation-2} \frac \partial{\partial u^2}\left[\begin{matrix} y_\lambda\\y^*_\lambda\\ \frac 1{\sqrt m}\xi_{u^1}\\ \frac 1{\sqrt m}\xi_{u^2}\\ \xi\end{matrix}\right] = \left[\begin{matrix} -\omega^\lambda_2 & 0 & - \frac 1{\sqrt m}(\Omega_\lambda)_{21} & -\frac 1{\sqrt m}(\Omega_\lambda)_{22} & 0\\ 0 & \omega^\lambda_2 & - \frac 1m(\Omega^*_\lambda)_{21} & - \frac 1m (\Omega^*_\lambda)_{22} & 0\\ \frac 1{\sqrt m}(\Omega_\lambda)_{21} & \frac 1{\sqrt m}(\Omega^*_\lambda)_{21} & 0 & - \frac 1{2m} m_{u^1} & 0\\ \frac 1{\sqrt m}(\Omega_\lambda)_{22} & \frac 1{\sqrt m}(\Omega^*_\lambda)_{22} & \frac 1{2m} m_{u^1} & 0 & -\sqrt{m} \\ 0 & 0 & 0 & \sqrt{m} & 0\end{matrix}\right] \left[\begin{matrix} y_\lambda \\y^*_\lambda \\ \frac 1{\sqrt m}\xi_{u^1}\\ \frac 1{\sqrt m}\xi_{u^2}\\ \xi\end{matrix}\right] \end{equation} Remember we also have the two trivial equations $$ \frac \partial{\partial \alpha}\left[\begin{matrix} y_\lambda\\y^*_\lambda\\ \frac 1{\sqrt m}\xi_{u^1}\\ \frac 1{\sqrt m}\xi_{u^2}\\ \xi\end{matrix}\right]= 0 \text{ and } \frac \partial{\partial \rho}\left[\begin{matrix} y_\lambda\\y^*_\lambda\\ \frac 1{\sqrt m}\xi_{u^1}\\ \frac 1{\sqrt m}\xi_{u^2}\\ \xi\end{matrix}\right]= 0. $$ To solve the systems \eqref{structure-equation-1} and \eqref{structure-equation-2} of ODE, the necessary integrable condition is \begin{equation}\label{integrable} \frac \partial{\partial u^1}\frac \partial{\partial u^2}\left[\begin{matrix} y_\lambda\\y^*_\lambda\\ \frac 1{\sqrt m}\xi_{u^1}\\ \frac 1{\sqrt m}\xi_{u^2}\\ \xi\end{matrix}\right] = \frac \partial{\partial u^2}\frac\partial{\partial u^1}\left[\begin{matrix} y_\lambda\\y^*_\lambda\\ \frac 1{\sqrt m}\xi_{u^1}\\ \frac 1{\sqrt m}\xi_{u^2}\\ \xi\end{matrix}\right]. \end{equation} It turns out \eqref{integrable} is equivalent to the following six equations on the variables: the positive function $m$, the 1-form $\omega^\lambda$, the traceless symmetric matrix $\Omega_\lambda$ and the symmetric matrix $\Omega^*_\lambda$, \begin{equation}\label{codazzi-y} \left\{\aligned (\Omega_\lambda)_{11, 2} - (\Omega_\lambda)_{12,1} & = \omega^\lambda_1(\Omega_\lambda)_{12} - \omega^\lambda_2(\Omega_\lambda)_{11}\\
(\Omega_\lambda)_{12,2} - (\Omega_\lambda)_{22,1} & = \omega^\lambda_1(\Omega_\lambda)_{22} - \omega^\lambda_2(\Omega_\lambda)_{12}\endaligned\right.
\end{equation} \begin{equation}\label{codazzi-y-star} \left\{\aligned (\Omega^*_\lambda)_{11, 2}- (\Omega^*_\lambda)_{12,1}& = - \omega^\lambda_1(\Omega^*_\lambda)_{12} + \omega^\lambda_2(\Omega^*_\lambda)_{11}
+ \frac 12 \frac{(\Omega^*_\lambda)_{11} + (\Omega^*_\lambda)_{22}}{|\Omega_\lambda|^2} (|\Omega_\lambda|^2)_{u^2}\\
(\Omega^*_\lambda)_{12,2}- (\Omega^*_\lambda)_{22,1}& = - \omega^\lambda_1(\Omega^*_\lambda)_{22} + \omega^\lambda_2(\Omega^*_\lambda)_{12}
+ \frac 12 \frac{(\Omega^*_\lambda)_{11} + (\Omega^*_\lambda)_{22}}{|\Omega_\lambda|^2} (|\Omega_\lambda|^2)_{u^2}\endaligned\right.
\end{equation}
\begin{equation}\label{codazzi-mix}
\omega^\lambda_{1, 2} - \omega^\lambda_{2, 1} = \frac 1m ((\Omega_\lambda)_{11}-(\Omega_\lambda)_{22})(\Omega^*_\lambda)_{12} - ((\Omega^*_\lambda)_{11}
- (\Omega^*_\lambda)_{22})(\Omega_\lambda)_{12} )
\end{equation}
and
\begin{equation}\label{gauss-xi} (\mathcal K -1) = \frac 1{m^2} \text{Tr}\Omega_\lambda\Omega^*_\lambda, \end{equation}
where $\mathcal K$ is the Gaussian curvature of the M\"{o}bius metric $I^\xi = m|du|^2$. Of course, as one may verify, \eqref{codazzi-y}, \eqref{codazzi-y-star}, \eqref{codazzi-mix} and \eqref{gauss-xi} are exactly the Gauss-Codazzi equations for the surface $\xi$ in the de Sitter spacetime $\mathbb{S}^{1, 3}$ induced by the conformal Gauss map $\xi$ of the surface $\hat x$ in conformal 3-sphere $\mathbb{S}^3$. \\
Now we are ready to state and prove the local fundamental theorem for surfaces in conformal round 3-sphere $\mathbb{S}^3$.
\begin{theorem}\label{local-fundamental} Suppose that, on a domain in $D\subset\mathbb{R}^2$, we are given the following \begin{itemize} \item a traceless symmetric 2-form $\Omega$ \item a positive function $m$ or equivalently $E$ such that $m= \frac {-\det\Omega}E$ \item a 1-form $\omega$ \item a symmetric 2-form $\Omega^*$. \end{itemize} And suppose that they satisfy the integrability conditions \eqref{codazzi-y} - \eqref{gauss-xi}. Then, for a given point $p_0$ in $D$, there exists an open neighborhood $D_0$ of $p_0$ in $D$, a parametrized surface $\hat x: D_0\to\mathbb{S}^3$ with no umbilical point, and a positive function $\hat\lambda: D_0\to \mathbb{R}^+$ with $\hat\lambda(p_0) = 1$, such that \begin{itemize} \item $\Omega = \hat\lambda \overset{\circ}{II}$, where $\overset{\circ}{II}$ is the traceless part of the second fundamental form of $\hat x$ in the standard round $\mathbb{S}^3$
\item $m|du|^2= <d\xi, d\xi>$ is the M\"{o}bius metric induced by the conformal Gauss map $\xi$ of $\hat x$ \item $\omega = -I((\overset{\circ}{II})^{-1}(d H)) - d \log\hat\lambda$, where $I$ is the first fundamental form and $H$ is the mean curvature of $\hat x$ in the standard round $\mathbb{S}^3$ \item $\Omega^* = - \hat\lambda^{-1} <d\xi, dy^*>$, where $y^* = \frac 1{1-\hat x\cdot \hat x^*}(1, \hat x^*)$ and $\hat x^*$ is the conformal transform of $\hat x$. \end{itemize} The surface $\hat x$ is unique up to a conformal transformation of $\mathbb{S}^3$. \end{theorem}
\proof We start with choosing starting values for $y, y^*, \xi_{u^1}, \xi_{u^2}, \xi$ at $p_0= (u_0^1, u_0^2)$, First we take a null vector $$ y(u^1_0, u^2_0) = y_0 = (1, \hat x_0) $$ for some $\hat x_0\in \mathbb{S}^3\subset \mathbb{R}^4$. Then we choose $\xi(u^1_0, u^2_0)= \xi_0\in\mathbb{R}^{1,4}$ such that \begin{equation}\label{initial-1} <y_0, \xi_0> = 0 \text{ and } <\xi_0, \xi_0> = 1. \end{equation} Next we choose $\xi_{u^1}(u^1_0, u^2_0) = \xi^1_0\in\mathbb{R}^{1,4}$ and $\xi_{u^2}(u^1_0, u^2_0) = \xi^2_0\in\mathbb{R}^{1,4}$ such that \begin{equation}\label{initial-2} \aligned <\xi^1_0, \xi^1_0>& =<\xi^2_0, \xi^2_0> = m(u^1_0, u^2_0), \\ <\xi^1_0, \xi^2_0> & = <\xi_0, \xi_0^1>=<\xi_0, \xi_0^2> = <y_0, \xi_0^1>=<y_0, \xi_0^2>=0.\endaligned \end{equation} Finally choose the unique null vector $y^*(u^1_0, u^2_0) = y^*_0$ such that \begin{equation}\label{initial-3} \aligned <y^*_0, y_0> & = -1\\ <y^*_0, y^*_0> & = <y^*_0, \xi_0> = <y^*_0, \xi_0^1> = <y^*_0, \xi_0^2> = 0.\endaligned \end{equation} Notice that for any other choice of $\{y_1, y^*_1, \xi_1^1, \xi^2_1, \xi_1\}$ satisfying the same orthonormal properties in \eqref{initial-1} - \eqref{initial-3}, there is a Lorentz transformation that takes one to the other. With the integrability conditions assumed we may solve the systems \eqref{structure-equation-1} and \eqref{structure-equation-2} at least in an open neighborhood $D_0$ of $p_0$ in $D$. Using the uniqueness of solutions to systems of linear ODE one sees that the solution $\{y, y^*, \frac 1{\sqrt m}\xi_{u^1}, \frac 1{\sqrt m} \xi_{u^2}, \xi\}$ remains to be orthonormal in the Minkowski metric in $D_0$.\\
Now one should realize that the $y = \hat\lambda (1, \hat x)$ here is with some positive $\hat\lambda$ (not necessarily identically $1$ in $D_0$). It is then clear from all previous calculations that the rest of the statements in the theorem can be easily verified. \endproof
\subsection{The geometry of the associate ruled surface $x^+$ in hyperbolic space $\mathbb{H}^4$}
In this section we want to discuss the geometry of the associate ruled 3-surface $x^+$ in $\mathbb{H}^{4}$, which is associated with a given surface $\hat x$ in the conformal 3-sphere. It's relation to the associate surface $\tilde x$ is very much analogous to the one between the ambient spacetime and the Poincar\'{e}-Einstein manifold of a given conformal manifold in the work of Fefferman and Graham. It deems to be useful to understand the geometry of the associate ruled 3-surface $x^+$ in $\mathbb{H}^4$. \\
It is rather easy now to do calculations for $x^+$ after we have calculated the first fundamental form for the associate 4-surface $\tilde x$ in Minkowski spacetime $\mathbb{R}^{1,4}$ in section \ref{Sect:I-tilde}. We first have $$ dx^+ = \frac 1{\sqrt 2} (e^ty_\lambda -e^{-t}y^*_\lambda)dt + (e^t(y_\lambda)_{u^1}+e^{-t}(y^*_\lambda)_{u^1})du^1 + (e^t(y_\lambda)_{u^2}+e^{-t}(y^*_\lambda)_{u^2})du^2 $$ and, using \eqref{tilde-x-I}, \begin{equation}\label{x-+-I} \aligned I^{x^+} & = (dt)^2 - 2\omega^\lambda_idtdu^i+(\frac {e^{2t}} {2m}((\Omega_\lambda)_{i1}(\Omega_\lambda)_{j1}+(\Omega_\lambda)_{i2}(\Omega_\lambda)_{j2}) \\ & \quad\quad + (\omega_i\omega_j + \frac 1m((\Omega_\lambda)_{i1}(\Omega^*_\lambda)_{j1}+(\Omega_\lambda)_{i2}(\Omega^*_\lambda)_{j2})) \\ & \quad\quad + \frac {e^{-2t}}{2m} ((\Omega^*_\lambda)_{i1}(\Omega^*_\lambda)_{j1}+(\Omega^*_\lambda)_{i2}(\Omega^*_\lambda)_{j2}))du^idu^j\endaligned \end{equation} One can calculate the determinant \begin{equation}\label{+-determinant}
\det I^{x^+} = \frac 1{8m^2}(E^2_\lambda|e^t\Omega_\lambda + e^{-t}\Omega^*_\lambda|^2 - e^{-2t}((\Omega^*_\lambda)_{11} + (\Omega^*_\lambda)_{22})^2)^2, \end{equation} which can tell us where the associate ruled surface $x^+$ is degenerate. \\
To obtain the second fundamental form of the surface $x^+$ it suffices to see that the conformal Gauss map $\xi$ is still the unit normal vector to the surface $x^+$ in the hyperboloid $\mathbb{H}^4$. Hence \begin{equation}\label{x-+-II} II^{x^+} = - <dx^+, d\xi> = \frac 1{\sqrt 2}(e^t\Omega_\lambda + e^{-t}\Omega^*_\lambda). \end{equation}
By the similar calculations as that in the previous section we have the mean curvature of the associate ruled surface $x^+$ as follows: \begin{equation}\label{plus-H} H^{x^+} = e^{-3t}\frac {\sqrt{2}\det\Omega_\lambda {\mathcal H}_\lambda} {(\det\Omega_\lambda -e^{-2t}\text{Tr}\Omega_\lambda\Omega^*_\lambda + e^{-4t}\det\Omega^*_\lambda)}. \end{equation}
\begin{theorem}\label{Th:+-H} Suppose that $\hat x$ is an immersed surface in the conformal sphere $\mathbb{S}^3$ with no umbilical point and that $x^+$ is the associate ruled surface in the hyperboloid $\mathbb{H}^4$. Then $\hat x$ is a Willmore surface in the conformal sphere if and only if the associate ruled 3-surface $x^+$ in the hyperboloid is a minimal surface. \end{theorem}
\section{Scalar invariants of surfaces in conformal round 3-sphere}
In this section we want to introduce scalar local invariants for surfaces in conformal round 3-sphere $\mathbb{S}^3$. We will first recall what are scalar invariants for hypersurfaces in (pseudo-)Riemannian geometry. Inspired by the work of Fefferman and Graham on scalar local invariants in conformal geometry we are going to use the associate surface $\tilde x$ in the Minkowski $\mathbb{R}^{1,4}$ of a given surface $\hat x$ in 3-sphere $\mathbb{S}^3$, where one considers the standard conformal 3-sphere as the projectivized positive light cone of the Minkowski spacetime to construct scalar local invariant.
\subsection{Scalar invariants of 4-surfaces in $\mathbb{R}^{1,4}$}
For our purpose we will focus on the discussion of scalar (pseudo-)Riemannian invariants of 4-surfaces $\tilde x$ in the Minkowski spacetime $\mathbb{R}^{1,4}$. Suppose that $$ \phi = \phi(v^2, v^3, v^4, v^5): A\subset \mathbb{R}^4\to \mathbb{R}^{1,4} $$ is a local parametrization of a surface $\tilde x$, where $A$ is a domain in $\mathbb{R}^4$. Hence it induces a local coordinate $$ \tilde \phi = \tilde \phi(v^1, v^2, v^3, v^4, v^5) : B\subset (-\epsilon, \epsilon)\times A\to \mathbb{R}^{1,4} $$ for $\mathbb{R}^{1,4} $ such that $$ \phi (v^2, v^3, v^4, v^5) = \tilde \phi(0, v^2, v^3, v^4, v^5). $$ We will use the Capital Latin letters to stand for indices from $1$ to $5$ and Latin letters to stand for the indices from $2$ to $5$. And we will use $v = (v^1, v^2, \cdots, v^5)$ and $\hat v = (v^2, \cdots, v^5)$. Hence the Minkowski metric in this coordinate is give as $$ \tilde{\mathcal{G}}_0 = <d\tilde\phi, d\tilde\phi> = (\tilde{\mathcal G}_0)_{IJ}dv^I dv^J $$ and the fist fundamental form for $\tilde x$ in $\mathbb{R}^{1, 4}$ is given as $$
I^{\tilde x} = <d\phi, d\phi> = \tilde g_{ij}dv^idv^j = (\tilde{\mathcal G}_0)_{ij}|_{v^1=0}dv^idv^j. $$ To be more restrictive we will assume that the surface $\tilde x$ is timelike and let $$ \xi: B \to \mathbb{S}^{1,3} $$ be a unit normal vector field on $\tilde x$ in $\mathbb{R}^{1,4}$. Then the second fundamental form for $\tilde x$ is given as
$$ II^{\tilde x} = - <d\phi, d\xi> = \tilde h_{ij}dv^idv^j. $$ And $$ \xi_{v^i} = - \tilde h_{ik}\tilde g^{kj} \phi_{v^j}. $$ \begin{definition} Let ${\bf i}: \textup{M}^{n-1}\to \textup{N}^n$ be an immersed hypersurface and let $g$ be a (pseudo)-Riemannain metric on the ambient manifold $\textup{N}^n$. A scalar (pseudo-)Riemannain invariant $\textup{I}({\bf i}, \textup{N}^n, g)$ for the hypersurface ${\bf i}$ in $\textup{N}^n$ at a point $p_0$ on the surface ${\bf i}$ is a polynomial in the variables that are the coordinate partial derivatives of $g_{IJ}$ of any order and the reciprocal of the determinant of $g_{IJ}$ at the point $p_0$ such that the value of $\textup{I}({\bf i}, \textup{N}^n, g)$ at $p_0$ is independent of choices of local coordinates $\tilde\phi$ of $\textup{N}^n$ which are induced from a parametrization $\phi$ of the surface ${\bf i}$ nearby the given point $p_0$. \end{definition}
The well-known examples of scalar Riemannian invariants for $\tilde x$ in $\mathbb{R}^{1,4}$ are \\
\begin{itemize} \item $\tilde H = \tilde g^{ij}\tilde h_{ij}$
\item $|\tilde h|^2 = \tilde g^{ik}\tilde g^{jl}\tilde h_{ij}\tilde h_{kl}$ and $\tilde H^2 =\tilde g^{ij}\tilde g^{kl}\tilde h_{ij}\tilde h_{kl}$ \item $\tilde \Delta \tilde H = \tilde g^{kl} \tilde g^{ij}\tilde h_{ij,kl}$, $\text{Div}\text{Div}\tilde h = \tilde g^{ik}\tilde g^{jl}\tilde h_{ij, kl}$,
$\tilde H|\tilde h|^2 = \tilde g^{ik}\tilde g^{jl}\tilde g^{mn}\tilde h_{ij}\tilde h_{kl}\tilde h_{mn}$, \newline $\quad \text{Tr}_{\tilde g}\tilde h^3 = \tilde g^{in}\tilde g^{jk}\tilde g^{km}\tilde h_{ij}\tilde h_{kl}\tilde h_{mn}$, and $\tilde H^3 = \tilde g^{ij}\tilde g^{kl}\tilde g^{mn}\tilde h_{ij}\tilde h_{kl}\tilde h_{mn}$
\item $|\tilde\nabla \tilde h|^2 = \tilde g^{ip}\tilde g^{jq}\tilde g^{kr}\tilde h_{ij,k}\tilde h_{pq,r}$, $\ \tilde g^{ip}\tilde g^{jr}\tilde g^{kq}\tilde h_{ij,k}\tilde h_{pq,r}$, $\ \tilde g^{ip}\tilde g^{jr}\tilde g^{kq}\tilde h_{ij,k}\tilde h_{pq,r}$
\newline $|\tilde\nabla \tilde H|^2= \tilde g^{ij}\tilde g^{pq}\tilde g^{kr}\tilde h_{ij,k}\tilde h_{pq,r}$,
$\ |\widetilde{\text{Div}}\tilde h|^2 = \tilde g^{ip}\tilde g^{jk}\tilde g^{qr}\tilde h_{ij,k}\tilde h_{pq,r}$, $\ \widetilde{\text{Div}}\tilde h \cdot d \tilde H$ \item $\tilde\Delta\tilde\Delta\tilde H$ \end{itemize}
\vskip 0.1in Each scalar invariant has an order. To find the order of each scalar invariant one simply scales the metric by a constant $\kappa$ and see what is the dimension of the scalar invariant. For example, we can easily find that
$$ \aligned \tilde H [\kappa^2 \tilde {\mathcal G}_0] & = \kappa^{-1}\tilde H [\tilde{\mathcal G}_0]\\
|\tilde h|^2 [\kappa^2 \tilde {\mathcal G}_0] & = \kappa^{-2} |\tilde h|^2[\tilde {\mathcal G}_0]\\ \tilde \Delta \tilde H[\kappa^2 \tilde {\mathcal G}_0] & = \kappa^{-3} \tilde\Delta\tilde H [\tilde {\mathcal G}_0]\\
|\tilde\nabla\tilde h|^2 [\kappa^2 \tilde {\mathcal G}_0] & =\kappa^{-4} |\tilde\nabla\tilde h|^2[\tilde {\mathcal G}_0]\\ \tilde\Delta\tilde\Delta\tilde H [\kappa^2\tilde {\mathcal G}_0] & = \kappa^{-5}\tilde\Delta\tilde\Delta\tilde H [\tilde {\mathcal G}_0]. \endaligned $$ \\
To understand what are scalar Riemannian invariants $\textup{I}(\tilde x, \mathbb{R}^{1,4}, \tilde{\mathcal G}_0)$ we want to use the so-called Fermi coordinates. A Fermi coordinate is one such that 1) on the surface $\phi$ is a normal coordinate at a given point $\tilde x_0$; 2) the coordinate curves $\tilde\phi(t, v^2, v^3, v^4, v^5)$ is a geodesic perpendicular to the surface at $\phi(v^2, v^3, v^4, v^5)$ with unit speed (a line segment perpendicular to the surface in $\mathbb{R}^{1, 4}$). Hence, for a Fermi coordinate, \begin{equation}\label{line-segment} \tilde\phi (v^1, \cdots, v^5) = \phi(v^2, \cdots, v^5) + v^1\xi. \end{equation} The following facts are well known.
\begin{lemma} Suppose that $\tilde x$ is a timelike hypersurface in $\mathbb{R}^{1,4}$. Suppose that $\tilde\phi$ is a Fermi coordinate at a given point $\tilde x_0$. Then $$ \tilde{\mathcal G}_0 = \left[\begin{matrix} 1 & 0 \\0 & [{\mathcal G}_{ij}]\end{matrix}\right] $$ and $$ {\mathcal G}_{ij}(v^1, \hat v) = \tilde g_{ij}(\hat v) - 2\tilde h_{ij}(\hat v) v^1 + \tilde h_{ik}(\hat v)\tilde h_{jl}(\hat v)\tilde g^{kl}(\hat v) (v^1)^2, $$ where $$ \tilde g_{ij} (\hat v) = \eta_{ij} -\frac 23 \tilde R_{ikjl}v^kv^l + \cdots $$ $$ \tilde h_{ij} (\hat v) = \tilde h_{ij}(0) + \tilde h_{ij, k}(0)v^k + \cdots $$ $\tilde R_{ijkl} = \tilde h_{ik}\tilde h_{jl} - \tilde h_{ij}\tilde h_{kl}$ is the Riemann curvature tensor for $\tilde x$ and $\eta$ is standard matrix of signature $\{-1, 1, 1, 1\}$. Moreover all the coefficients in the Taylor's expansions for $G_{ij}$ are polynomials of $\tilde h_{ij}$ and the covariant derivatives of $\tilde h_{ij}$ at $\tilde x_0$. \end{lemma}
Therefore, in the light of Weyl theorem on the invariants of orthogonal groups, we may conclude that
\begin{proposition} All scalar invariants $\textup{I}(\tilde x, \mathbb{R}^{1,4}, \tilde{\mathcal G}_0)$ of a surface $\tilde x$ in $\mathbb{R}^{1,4}$ are linear combinations of terms that are complete contractions of tensor product of the second fundamental form $\tilde h$ and the covariant derivatives of $\tilde h$. \end{proposition} \proof From the above lemma it is easily that all scalar invariants of a surface $\tilde x$ in $\mathbb{R}^{1, 4}$ are polynomials of the first fundamental form $\tilde g$, the second fundamental form $\tilde h$ and covariant derivatives of the second fundamental form $\tilde h$, if we evaluate them in a Fermi coordinate for the surface. Then, by the Weyl theorem on the invariants of orthogonal groups, we know they are linear combinations of full contractions of $\tilde h$ and covariant derivatives of $\tilde h$. \endproof
\subsection{Scalar invariants of the homogeneous associate surface $\tilde x$ in $\mathbb{R}^{1,4}$}\label{calculations}
Let us work with the parametrization $$ \tilde x = \alpha \hat\lambda(1, \hat x) + \alpha\rho \hat\lambda^{-1} \frac 1{1-a} (1, \hat x^*) = \alpha y_\lambda + \alpha\rho y^*_\lambda $$ and use the calculations given in Section \ref{Sect:I-tilde} and Section \ref{Sect:II-tilde}. Now let us compute some scalar invariants for our associate surface $\tilde x$ on the light cone where $\rho = 0$. Then the first fundamental form is $$
I_{\tilde x} |_{\rho = 0}= \left[\begin{matrix} \begin{matrix} 0 & -\alpha \\-\alpha & 0 \end{matrix} & \begin{matrix} 0 & 0 \\ \alpha^2\omega_1^\lambda & \alpha^2 \omega_2^\lambda \end{matrix} \\
\begin{matrix} 0 &\quad\alpha^2\omega_1^\lambda \\ 0 & \quad \alpha^2\omega_2^\lambda \end{matrix} &
\begin{matrix} \alpha^2 E_\lambda & 0 \\ 0 & \alpha^2 E_\lambda \end{matrix} \end{matrix}\right]
$$ from \eqref{matrix-tilde}, whose inverse is
$$
I_{\tilde x}^{-1}|_{\rho = 0} = \left[\begin{matrix} \begin{matrix} |\omega^\lambda|^2 & -\frac 1\alpha \\-\frac 1\alpha & 0 \end{matrix} & \begin{matrix}
\frac {\omega^\lambda_1}{\alpha E_\lambda}
& \frac {\omega^\lambda_2}{\alpha E_\lambda} \\ 0 & 0 \end{matrix} \\
\begin{matrix} \frac {\omega^\lambda_1}{\alpha E_\lambda} & \quad 0 \\
\frac {\omega^\lambda_2}{\alpha E_\lambda} & \quad 0 \end{matrix} & \begin{matrix} \frac 1{\alpha^2 E_\lambda} & 0 \\ 0 &
\frac 1{\alpha^2 E_\lambda} \end{matrix} \end{matrix}\right].
$$ And the second fundamental form at $\rho=0$ is
$$
II_{\tilde x}|_{\rho = 0} = \left[\begin{matrix} 0 & 0 \\
0 & \alpha\Omega_\lambda\end{matrix}\right].
$$
So the simplest (pseudo-)Riemannian invariants is the mean curvature $\tilde H$, but it is clear that $$
\tilde H |_{\rho = 0} = \frac 1{\alpha E_\lambda} ((\Omega_\lambda)_{11} + (\Omega_\lambda)_{22}) = 0. $$ The first non-trivial one is \begin{equation}\label{1st-one}
|\tilde h|^2|_{\rho =0} = \tilde g^{ik}\tilde g^{jl}\tilde h_{ij}\tilde h_{kl} |_{\rho = 0} = \alpha^{-2}|\Omega_\lambda|^2, \end{equation}
which produces the first non-trivial invariant $|\overset{\circ}{II}|^2$ for the surface $\hat x$ in the conformal 3-sphere(cf. see the definition for scalar invariant of surfaces in the conformal 3-sphere in the next subsection). In fact the following non-trivial invariants without taking any derivative are all easy to calculate $$
\text{Tr}_{I^{\tilde x}}\tilde h^k|_{\rho = 0} = \alpha^{-k} \text{Tr}_{I^{\hat x}_\lambda} \Omega_\lambda^k $$ for any $k = 2, 3, \cdots.$ Obviously those are the ones that can been easily seen with no difficulty at all. \\
Next we want to calculate $|\nabla \tilde H|^2$ and $\tilde\Delta\tilde H$ at $\rho = 0$. To do so, let us first recall from Section \ref{Sect:II-tilde} the mean curvature $$ \tilde H = \frac {\rho \det\Omega_\lambda{\mathcal H}_\lambda} {\alpha(\det\Omega_\lambda - \rho\text{Tr}\Omega_\lambda\Omega^*_\lambda + \rho^2\det\Omega^*_\lambda)}. $$
Hence $\tilde H_\alpha = \tilde H_{u^1} = \tilde H_{u^2} = 0$ and $|\nabla \tilde H|^2 = 0$ at $\rho=0$, that is, $|\nabla \tilde H|^2$ gives no invariant for the surface $\hat x$. Let us set the convention to have $a, b, c$ stand for $\alpha, \rho$; $i,j,k$ stand for $u^1, u^2$, and $A, B, C$ stand for all four variables. We then calculate, at $\rho =0$, \begin{equation}\label{laplace-h} \aligned
\tilde\Delta\tilde H & = \frac 1{\sqrt{|\tilde g|}} \partial_A (\sqrt {|\tilde g|} \tilde g^{AB}\partial_B\tilde H)\\
& =\frac 1{\alpha^3 E} (\partial_\alpha (\sqrt {|\tilde g|} \tilde g^{\alpha \rho}
\partial_\rho\tilde H) + \partial_\rho (\sqrt {|\tilde g|} \tilde g^{\rho B}\partial_B\tilde H) + \partial_i(\sqrt {|\tilde g|} \tilde g^{i\rho}\partial_\rho\tilde H))\\
& =\frac 1{\alpha^3E}(\partial_\alpha (\sqrt {|\tilde g|} \tilde g^{\alpha \rho}
\partial_\rho\tilde H) + \partial_\rho (\sqrt{|\tilde g|} g^{\rho\alpha}\partial_\alpha\tilde H) + \sqrt{|\tilde g|} (\partial_\rho\tilde g^{\rho\rho}) \partial_\rho \tilde H)\\ & = 2\alpha^{-3}{\mathcal H}_\lambda \endaligned \end{equation}
where one needs to use the fact that $\tilde g^{\rho\rho}|_{\rho = 0} = 0$ and $\partial_\rho \tilde g^{\rho\rho}|_{\rho = 0} = \frac {2}{\alpha^2}$ based on calculations \eqref{app-2j} in Appendix \ref{app-inverse-tilde}. This confirms that ${\mathcal H}_\lambda$ is indeed a conformal invariant of order $3$ for a surface $\hat x$ in 3-sphere in general conformal metric $\lambda^2g_0$. \\
The next invariant we want to calculate is $\tilde\Delta\tilde\Delta\tilde H$. To do so, from \eqref{app-2j} in Appendix \ref{app-inverse-tilde}, we observe the following: \begin{equation}\label{der-inverse} \aligned
\partial_\rho|_{\rho=0}\tilde g^{\rho\alpha} & = -\frac 2\alpha |\omega^\lambda|^2, \quad \partial_\rho|_{\rho=0}\tilde g^{\rho\rho} = \frac 2{\alpha^2}, \\
\partial_\rho|_{\rho=0}\tilde g^{\rho i} & = -\frac 2{\alpha^2} \frac {\omega^\lambda_i}{E_\lambda} \text{ and } \partial_\rho\partial_\rho|_{\rho=0}\tilde g^{\rho\rho}
= \frac 8{\alpha^2}|\omega^\lambda|^2.\endaligned \end{equation} After a lengthy calculation we get \begin{equation}\label{double-laplace} \aligned
\tilde\Delta & \tilde\Delta\tilde H|_{\rho=0} = 8\alpha^{-5} (\Delta_\lambda{\mathcal H}_\lambda +9 |\omega^\lambda|^2{\mathcal H}_\lambda - 3 \text{Div}({\omega^\lambda}) {\mathcal H}_\lambda \\ & \quad - 6\omega^\lambda(\nabla{\mathcal H}_\lambda)
- \frac {3\text{Tr}(\Omega_\lambda\Omega^*_\lambda)}{2m^2} |\Omega_\lambda|^2{\mathcal H}_\lambda).\endaligned \end{equation}
This tells us that $\Delta_\lambda{\mathcal H}_\lambda +9 |\omega^\lambda|^2{\mathcal H}_\lambda - 3 \text{Div}({\omega^\lambda})
{\mathcal H}_\lambda - 6\omega^\lambda(\nabla{\mathcal H}_\lambda)- \frac {3\text{Tr}(\Omega_\lambda\Omega^*_\lambda)}{2m^2} |\Omega_\lambda|^2{\mathcal H}_\lambda$ is a conformal invariant of order $5$ for the surface $\hat x$ in 3-sphere. \\
We can also calculate the covariant derivatives of the second fundamental forms for the associate surface. We first list the relevant Christoffel symbols for the calculation \begin{equation}\label{christoffel} \aligned \tilde\Gamma^k_{\alpha\alpha} & = \tilde\Gamma^k_{\rho\rho} = \tilde\Gamma^k_{\alpha\rho} = 0 \\ \tilde\Gamma^k_{\alpha j} & = \alpha^{-1} \delta_{jk} \\ \tilde\Gamma^k_{\rho j} & = \frac 1{2E_\lambda} ((\omega^\lambda_k)_{u^j} - (\omega^\lambda_j)_{u^k} + \frac 1m((\Omega_\lambda)_{jl}(\Omega^*_\lambda)_{kl} +(\Omega_\lambda)_{kl}(\Omega^*_\lambda)_{jl})) \\
\tilde\Gamma^k_{ij} & = (\Gamma_\lambda)^k_{ij} - \omega^\lambda_k\delta_{ij}. \endaligned \end{equation} Then we calculate \begin{equation}\label{co-derivative} \aligned \tilde h_{ab, C} &= 0\\ \tilde h_{ai,b} & = 0 \\ \tilde h_{\alpha j, k} & = -(\Omega_\lambda)_{jk} \\ \tilde h_{\rho j,k} & = - \frac \alpha{2E_\lambda}((\Omega_\lambda)_{ij}((\omega^\lambda_i)_{u^k} - (\omega^\lambda_k)_{u^i} +\frac 1m ((\Omega_\lambda)_{kl}(\Omega_\lambda^*)_{il} + (\Omega_\lambda)_{il}(\Omega^*_\lambda)_{kl}) \\ \tilde h_{ij,\alpha} & = -(\Omega_\lambda)_{ij} \\ \tilde h_{ij, \rho} & =\alpha(\Omega_\lambda^*)_{ij} \\ & \quad\quad - \frac \alpha{2 E}((\Omega_\lambda)_{lj} ((\omega^\lambda_l)_{u^i} - (\omega^\lambda_i)_{u^l} +\frac 1m ((\Omega_\lambda)_{kl}(\Omega_\lambda^*)_{ki} + (\Omega_\lambda)_{ki}(\Omega^*_\lambda)_{kl})\\ & \quad\quad -\frac \alpha{2E}(\Omega_\lambda)_{il}((\omega^\lambda_l)_{u^j} - (\omega^\lambda_j)_{u^l} +\frac 1m ((\Omega_\lambda)_{kl}(\Omega_\lambda^*)_{kj} + (\Omega_\lambda)_{kj}(\Omega^*_\lambda)_{kl})\\ \tilde h_{ij, k} & = \alpha(\Omega_\lambda)_{ij,k} + \alpha(\Omega_\lambda)_{lj}\omega^\lambda_l\delta_{ik} + \alpha(\Omega_\lambda)_{il}\omega^\lambda_l\delta_{jk}. \endaligned \end{equation} The easy one is
$$ \phi_\alpha =\tilde h_{\alpha j,k}\tilde g^{jk} = 0 \text{ and } \phi_\rho = \tilde h_{\rho j,k}\tilde g^{jk} = \frac 1\alpha{\mathcal H}_\lambda
$$
in the light of \eqref{trace-star}. While
$$
\aligned
\phi_i & = \tilde h_{iB,C}\tilde g^{BC} = \tilde h_{ij,C}\tilde g^{jC} + \tilde h_{ib, k}\tilde g^{bk}
= \tilde h_{ij,k}\tilde g^{jk} + \tilde h_{ij,\alpha}\tilde g^{j\alpha} + \tilde h_{i\alpha,k}\tilde g^{\alpha k}\\
& = \frac 1{\alpha E_\lambda}(\Omega_\lambda)_{ij,j} + \frac 3{\alpha E_\lambda}(\Omega_\lambda)_{ij}\omega^\lambda_j
- \frac 1{\alpha E_\lambda}(\Omega_\lambda)_{ij}\omega^\lambda_j -\frac 1{\alpha E_\lambda}(\Omega_\lambda)_{ij}\omega^\lambda_j\\
& = \frac 1{\alpha E_\lambda}(\Omega_\lambda)_{ij,j} + \frac 1{\alpha E_\lambda}(\Omega_\lambda)_{ij}\omega^\lambda_j = 0
\endaligned
$$
due to the integrability condition \eqref{codazzi-y}. Thus $|\widetilde{\text{Div}}\tilde h|^2(=0)$ does not give any invariant on the surface $\hat x$, nor does
$\widetilde{\text{Div}}\tilde h\cdot d\tilde H (= 0)$. Because $\tilde g^{\rho\rho} |_{\rho=0}= 0$.
\\
We want to calculate $|\tilde\nabla \tilde h|^2$ since we have all the covariant derivatives $\tilde h_{AB,C}$ in \eqref{co-derivative}. The calculation is direct yet very long. We omit the detail here. \begin{equation*}
|\tilde\nabla\tilde h|^2 |_{\rho = 0} = \alpha^{-4}(|\nabla\Omega|^2 +8 |d H|^2 -6\Omega\cdot\Omega^* - \frac 2{E_\lambda^3}(\Omega_\lambda)_{ij}\omega^\lambda_k(R^\lambda)_{3ijk}- \frac 6{E_\lambda^3}(\Omega_\lambda)_{ij}(\Omega_\lambda)_{ki,j}\omega^\lambda_k), \end{equation*} where the Codazzi equation for the surface $\hat x$ in $(\mathbb{S}^3, \ \lambda^2g_0)$ $$ (\Omega_\lambda)_{ij,k} = (\Omega_\lambda)_{ik,j} + (R^\lambda)_{3ijk} + (H_\lambda)_{u^j}E_\lambda\delta_{ik} - (H_\lambda)_{u^k} E_\lambda\delta_{ij} $$ has been used. At this point we like to write each term as local scalar invariant of the surface $\hat x$ in $(\mathbb{S}^3, \ \lambda^2g_0)$. We first calculate $$ \aligned & (\Omega_\lambda)_{ij} \omega^\lambda_k(R^\lambda)_{3ijk} = (\Omega_\lambda)_{ij}\omega^\lambda_1(R^\lambda)_{3ij1} + (\Omega_\lambda)_{ij}\omega^\lambda_2(R^\lambda)_{3ij2}\\ & =E_\lambda( (\Omega_\lambda)_{11}\omega^\lambda_1(R^\lambda)_{31} + (\Omega_\lambda)_{21}\omega^\lambda_1(R^\lambda)_{32}
+ (\Omega_\lambda)_{22}\omega^\lambda_2(R^\lambda)_{32} + (\Omega_\lambda)_{12}\omega^\lambda_2(R^\lambda)_{31})\\
& = E_\lambda(\Omega_\lambda)_{ij} \omega^\lambda_j(R^\lambda)_{3i} = - E_\lambda^2 (H_\lambda)_{u^i}(R^\lambda)_{3i} = -
Ric^\lambda(\overset{\rightarrow}{\bf n}_\lambda, \lambda H_\lambda). \endaligned $$ Then we deal with the last term $$ \aligned (\Omega_\lambda)_{ij} & (\Omega_\lambda)_{ki,j}\omega^\lambda_k = (\Omega_\lambda)_{ij}((\Omega_\lambda)_{ki}\omega^\lambda_k)_{,j} - (\Omega_\lambda)_{ij}(\Omega_\lambda)_{ki}\omega^\lambda_{k,j}
\\ & = - E_\lambda(\Omega_\lambda)_{ij} (H_\lambda)_{i,j} - \frac 12 E_\lambda^3|\Omega_\lambda|^2\text{Div}(\omega^\lambda) \endaligned $$ where $$ \aligned \text{Div}(\omega^\lambda) & = E^{-1}_\lambda\omega^\lambda_{i,i} = E^{-1}_\lambda(\omega^\lambda_i)_{u^i} \\ & = E^{-1}_\lambda(<\Delta_0 y_\lambda, y^*_\lambda> + <(y_\lambda)_{u^i}, (y^*_\lambda)_{u^i}>)\\
& = H_\lambda^2 - |\omega^\lambda|^2 + (R^\lambda)_{1212} +E^{-1}<(y_\lambda)_{u^i}, (y^*_\lambda)_{u^i}>\\
& = H^2_\lambda + 2 \frac {\Omega_\lambda\cdot\Omega^*_\lambda}{|\Omega_\lambda|^2} + E_\lambda^{-1}(R^\lambda)_{1212}\\ \Delta_0 y_\lambda & = 2E_\lambda H_\lambda \overset{\rightarrow}{\bf n}_\lambda + 2 E_\lambda y^\dagger_\lambda - (R^\lambda)_{1212}y_\lambda \endaligned $$ and $$
\sum_{i=1}^2 E^{-1}<(y)_{u^i}, (y^*)_{u^i}> = |\omega^\lambda|^2 + 2 \frac {\Omega_\lambda\cdot\Omega^*_\lambda}{|\Omega_\lambda|^2}. $$ So we have obtained \begin{equation}\label{norm-co-der} \aligned
|\nabla\tilde h|^2|_{\rho =0} & =\alpha^{-4}( |\nabla\Omega_\lambda|^2 + 8|dH_\lambda|^2 + 2 Ric^\lambda(\overset{\rightarrow}{\bf n}_\lambda, \nabla H_\lambda)
+ 3 H^2_\lambda|\Omega|^2 \\ & +3K^T_\lambda |\Omega_\lambda|^2+ 6 \Omega_\lambda\cdot \text{Hess} (H_\lambda)) \endaligned \end{equation} where $$ K^T_\lambda = E_\lambda^{-1}(R^\lambda)_{1212} $$ is the sectional curvature of $(\mathbb{S}^3, \lambda^2 g_0)$ of the tangent plane to the surface $\hat x$. \\
\subsection{Scalar invariants for surfaces in the conformal round 3-sphere}
Let us start with the definition of scalar invariants for surfaces in conformal sphere.
\begin{definition} Let ${\bf i}: \textup{M}^{n-1}: \textup{N}^n$ be an immersed hypersurface and let $[g]$ be a class of conformal metrics on the ambient manifold $\textup{N}|^n$.
$\textup{I}_c({\bf i}, \textup{N}^n, g)$ is said to be a scalar conformal invariant of the hypersurface ${\bf i}$ in the conformal manifold $(\textup{N}^n, [g])$
if it is a scalar Riemannian invariant and \begin{equation}\label{conformal-invariance} \textup{I}_c({\bf i}, \textup{N}^n, \lambda^2g) = \lambda^{-k} \textup{I}_c({\bf i}, \textup{N}^n, g). \end{equation} for any positive function $\lambda$ on $\textup{N}^n$, where $k$ is the order of the invariant $\textup{I}_c({\bf i}, \textup{N}^n, g)$. \end{definition}
Recall that, for an immersed surface $\hat x$ in $\mathbb{S}^3$, we have $$ \overset{\circ}{II} (\hat x, \mathbb{S}^3, \lambda^2 g_0) = \lambda \overset{\circ}{II} (\hat x, \mathbb{S}^3, g_0). $$ Hence it is easy to observe that $$
|\overset{\circ}{II}|^2(\hat x, \mathbb{S}^3, \lambda^2 g_0) = \lambda^{-2}g_0^{ik}\lambda^{-2}g_0^{jl} \lambda\overset{\circ}{II}_{ij}\lambda\overset{\circ}{II}_{kl}
= \lambda^{-2}\|\overset{\circ}{II}\|^2 (\hat x, \mathbb{S}^3, g_0) $$ and $$ \text{Tr}_{\lambda^2 g_0}(\overset{\circ}{II})^k(\hat x, \mathbb{S}^3, \lambda^2 g_0) = \lambda^{-k} \text{Tr}_{g_0}(\overset{\circ}{II})^k(\hat x, \mathbb{S}^3, g_0) \text{ for all $k=2, 3, \cdots$}. $$ On the other hand, it does not seem easy to directly verify that ${\mathcal H}_\lambda$ is a conformal invariant for a surface in the conformal 3-sphere, though this is a well-known one. We have verified this in computing the mean curvature (cf. \eqref{mean-curvature-xi}) of the surface $\xi$ in the de Sitter spacetime $\mathbb{S}^{1,3}$ as well as in the above calculation of $\tilde\Delta \tilde H$ (cf. \eqref{laplace-h}) of the homogeneous associate surface $\tilde x$. In general it takes tremendous, if not impossible, to verify whether an invariant $\textup{I}(\hat x, \mathbb{S}^3, \lambda^2g_0)$ is conformally invariant, complicated by the six integrability conditions. The most important application of the construction of associate homogeneous surfaces is the following:
\begin{theorem}\label{main-inv} Suppose that $\hat x:\textup{M}^2\to\mathbb{S}^3$ is an immersed surface with no umbilical point. And suppose that $$ \tilde x = \alpha y + \alpha\rho y^*: \mathbb{R}^+\times\mathbb{R}^+\times\textup{M}^2\to\mathbb{R}^{1,4} $$ is the associate surface for $\hat x$, where $\hat x^*$ is the conformal transform of $\hat x$. Then any scalar (pseudo)-Riemannian invariant $\textup{I}(\tilde x, \mathbb{R}^{1,4}, \tilde{\mathcal G}_0)$ evaluated at $\rho = 0$, if it is nontrivial, is a scalar conformal invariant
$\textup{I}_c(\hat x, \mathbb{S}^3, \lambda^2g_0)$ multiplied with $|\overset{\circ}{II}_\lambda|^{2n}$ for some integer $n$. \end{theorem} \proof For any invariant $\textup{I}(\tilde x, \mathbb{R}^{1,4}, \tilde{\mathcal G}_0)$, we know that it is a full contraction of tensor product of the second fundamental form and the covariant derivatives. For a choice of representative $\lambda^2 g_0$ on $\mathbb{S}^3$, in the corresponding parametrization \eqref{lambda-para}, we claim that \begin{equation}\label{Phi-lambda}
\textup{I}(\tilde x, \mathbb{R}^{1,4}, \tilde{\mathcal G}_0)|_{\rho = 0} = \alpha^{-k} \textup{I}(\hat x, \mathbb{S}^3,
\lambda^2 g_0)|\overset{\circ}{II}_\lambda|^{2n} \end{equation}
for a positive integer $k$ and a nonnegative integer $n$, due to the homogeneity of the associate surface. To see the right side of \eqref{Phi-lambda} is indeed a scalar Riemannian invariant multiplied with factor $|\overset{\circ}{II}_\lambda|^{2n}$ for some integer $n$, we consider the tensors that determines the first and second fundamental forms of the associate surface in that parametrization. We recall from \eqref{Omega} that $$ \Omega_\lambda = \overset{\circ}{II}_\lambda $$ is the traceless part of the second fundamental form for the surface $\hat x$ in the 3-sphere with the conformal metric $\lambda^2 g_0$. We also know from \eqref{omega} that $$
\omega^\lambda = - I_\lambda(( \overset{\circ}{II}_\lambda)^{-1}(dH_\lambda)) = - \frac 2{| \overset{\circ}{II}_\lambda|^2} \overset{\circ}{II}_\lambda (\nabla H_\lambda), $$ which causes us to include the possible negative $n$ in the right side of \eqref{Phi-lambda}. We may also recall from \eqref{m-lambda} that $$
m = \frac 12 E_\lambda |\overset{\circ}{II}_\lambda|^2. $$ Next we want to show that $\Omega^*_\lambda$ is also a tensor product of covariant derivatives of the 1-form $\omega^\lambda$, covariant derivatives of the second fundamental form $II_\lambda$ and covariant derivatives of Riemann curvature tensor of the conformal metric $\lambda^2 g_0$ on the 3-sphere(including 0th order). Recall the definition $$ (\Omega^*_\lambda)_{ij} = < y^*_\lambda, \xi_{u^iu^j}>. $$ We use the same idea in the calculation of the trace of $\Omega^*$ in Section \ref{geometry-xi}. Hence we write \begin{equation}\label{xi-ij-proj} \xi_{u^iu^j} = -(\Omega^*_\lambda)_{ij}y_\lambda - (\Omega_\lambda)_{ij}y^*_\lambda + (\Gamma_m)^k_{ij}\xi_{u^k} - m\delta_{ij}\xi. \end{equation} From \eqref{y-star-lambda} we know that $$
<y^*_\lambda, y^\dagger_\lambda> = - \frac 12(|\omega^\lambda|^2 + H_\lambda^2). $$ Using $\xi = H_\lambda y_\lambda + \overset{\rightarrow}{\bf n}_\lambda$ from Lemma \ref{Lem:xi-lambda} and \eqref{n-dagger}, we have $$ <\xi_{u^k}, y^\dagger_\lambda> = - (H_\lambda)_{u^k} +(R^\lambda)_{3k} $$ and $$ <\xi, y^\dagger_\lambda> = - H_\lambda. $$ Therefore we derive from \eqref{xi-ij-proj} that \begin{equation}\label{Omega-star-first}
<\xi_{u^iu^j}, y^\dagger_\lambda> = (\Omega^*_\lambda)_{ij} + \frac 12(|\omega^\lambda|^2 + H^2_\lambda)(\Omega_\lambda)_{ij} +(\Gamma_m)^k_{ij}(-H_{u^k} + (R^\lambda)_{3k}) + Hm\delta_{ij}, \end{equation} where $$
(\Gamma_m)^k_{ij} = \Gamma^k_{ij} + \frac 12 |\Omega_\lambda|^{-2}(|\Omega_\lambda|^2_{u^i}\delta_{jk} + |\Omega_\lambda|^2_{u^j}\delta_{ik} -
|\Omega_\lambda|^2_{u^k}\delta_{ij}) $$
is the Christofel symbols for the M\"{o}bius metric $m|du|^2$. On the other hand we have $$ \xi_{u^iu^j} = (H_\lambda)_{u^iu^j} y_\lambda + (H_\lambda)_{u^i}(y_\lambda)_{u^j} + (H_\lambda)_{u^j}(y_\lambda)_{u^i} + H_\lambda (y_\lambda)_{u^iu^j} + (\overset{\rightarrow}{\bf n}_\lambda)_{u^iu^j} $$ which implies \begin{equation}\label{Omega-star-second} \aligned <\xi_{u^iu^j}, y^\dagger_\lambda> & = - (H_\lambda)_{u^iu^j} + H_\lambda<(y_\lambda)_{u^iu^j}, y^\dagger_\lambda> + < (\overset{\rightarrow}{\bf n}_\lambda)_{u^iu^j}, y^\dagger_\lambda>\\ & =- (H_\lambda)_{u^iu^j} - H_\lambda<(y_\lambda)_{u^i}, (y^\dagger_\lambda)_{u^j}> - < (\overset{\rightarrow}{\bf n}_\lambda)_{u^i}, (y^\dagger_\lambda)_{u^j}>\\ & \quad\quad\quad - <\overset{\rightarrow}{\bf n}_\lambda, (y^\dagger_\lambda)_{u^i}>_{u^j}\\ & =- (H_\lambda)_{u^iu^j}+ \frac 1{E_\lambda} (\Omega_\lambda)_{ik}<(y_\lambda)_{u^k}, (y^\dagger_\lambda)_{u^j}>
- <\overset{\rightarrow}{\bf n}_\lambda, (y^\dagger_\lambda)_{u^i}>_{u^j}\\ & =- (H_\lambda)_{u^iu^j} - \frac 1{E_\lambda} (\Omega_\lambda)_{ik}(R^\lambda)_{i3k3} + ((R^\lambda)_{3i})_{u^j}. \endaligned \end{equation} by \eqref{i-dagger-j} and \eqref{n-dagger}. Thus, comparing \eqref{Omega-star-first} and \eqref{Omega-star-second}, we have \begin{equation}\label{Omega-star-ij} \aligned (\Omega^*_\lambda)_{ij} & = - (H_\lambda)_{u^i,u^j} - H_\lambda m\delta_{ij} - \frac 1{E_\lambda} (\Omega_\lambda)_{ik}(R^\lambda)_{j3k3} + ((R^\lambda)_{3i})_{,u^j}
\\ & \quad - \frac 12(|\omega^\lambda|^2 + H^2_\lambda)(\Omega_\lambda)_{ij} \\ & \quad
+ \frac 12 |\Omega_\lambda|^{-2}(|\Omega_\lambda|^2_{u^i}\delta_{jk} + |\Omega_\lambda|^2_{u^j}\delta_{ik} - |\Omega_\lambda|^2_{u^k}\delta_{ij}) ((H_\lambda)_{u^k} - (R^\lambda)_{3k}). \endaligned \end{equation} The last factor that goes into the left side of the equation \eqref{Phi-lambda} is the reciprocal of the determinant: $$
\det\tilde g|_{\rho =0} = - \frac {\alpha^6}{m^2}(pr - q^2)^2 |_{\rho=0} = \frac {\alpha^6}{m^2} (\det\Omega_\lambda)^2 = \alpha^6 E^2_\lambda = \alpha^6 \det I^{\hat x}_\lambda. $$
due to \eqref{det-I-tilde}, where $I^{\hat x}_\lambda = (\hat x)^*(\lambda^2 g_0) = E_\lambda |du|^2$. \\
To verify that the right side of \eqref{Phi-lambda} is actually a conformal invariant, for a positive functions $\lambda$ on 3-sphere, we simply compare the right side of \eqref{Phi-lambda} evaluated at $\alpha = 1$ with that evaluated at $\alpha =\hat \lambda$ and $\lambda =1$. We then observe that $$ \textup{I}(\hat x, \mathbb{S}^3, \lambda^2 g_0) = \hat\lambda^{-k} \textup{I}(\hat x, \mathbb{S}^3, g_0). $$ Therefore it is a conformal scalar invariant for the surface $\hat x$ in the 3-sphere. \endproof
\begin{appendix}
\section{The inverse of $I^{\tilde x}$ in general parametrizations}\label{app-inverse-tilde}
We consider the general parametrization $$ \tilde x = \alpha y_\lambda + \alpha \rho y^*_\lambda: \mathbb{R}^+\times\mathbb{R}^+\times\textup{M}^2\to \mathbb{R}^{1, 4}. $$ Then the first fundamental form in matrix form is \begin{equation}\label{matrix-tilde-app} I_{\tilde x} = \left[\begin{matrix} \begin{matrix} \ -2\rho & -\alpha \\ -\alpha & \ 0\end{matrix} & \begin{matrix} 0 & 0 \\ \alpha^2\omega^\lambda_1 & \alpha^2 \omega^\lambda_2\end{matrix}\\ \begin{matrix} \quad\quad 0 & \quad \alpha^2\omega^\lambda_1\\ \quad\quad 0 & \quad \alpha^2\omega^\lambda_2\end{matrix} & \quad \alpha^2 F \quad\end{matrix}\right]
\end{equation}
where
\begin{equation}\label{F-matrix-app}
\left\{
\aligned
F_{11} & = \frac 1m(p^2 + q^2) +2 \rho (\omega^\lambda_1)^2\\
F_{12} & = F_{21} = \frac 1m q(p+r) + 2 \rho \omega^\lambda_1\omega^\lambda_2\\
F_{22} & = \frac 1m(q^2 + r^2) + 2 \rho(\omega^\lambda_2)^2 \endaligned\right. \text{ and }
\left\{
\aligned
F^*_{11} & = \frac 1m(p^2 + q^2) \\
F^*_{12} & = F_{21} = \frac 1m q(p+r) \\
F^*_{22} & = \frac 1m(q^2 + r^2) \endaligned\right. \end{equation}
and $$ \left[\begin{matrix} p & q \\ q & r\end{matrix}\right] = \Omega_\lambda + \rho\Omega^*_\lambda. $$ It is easily seen that \begin{equation}\label{F-star-inverse} (F^*)^{-1} = \frac m {(pr-q^2)^2}\left[\begin{matrix} r^2 + q^2 & - q(p+r)\\-q(p+r) & p^2 + q^2\end{matrix}\right] \end{equation} and $$
F|_{\rho =0} = F^*|_{\rho=0} = E\left[\begin{matrix} 1 & 0 \\0 & 1\end{matrix}\right]. $$ Let $$ (I^{\tilde x})^{-1} = \left[\begin{matrix} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{21} & a_{22} & a_{23} & a_{24} \\ a_{31} & a_{32} & a_{33} & a_{34} \\ a_{41} & a_{42} & a_{43} & a_{44} \end{matrix}\right]. $$ Therefore, for example, \begin{equation}\label{inverse-1-app}
\left\{\aligned
-2\rho a_{11} -\alpha a_{12} \quad \quad & = 1 \\
-\alpha a_{11} + \alpha^2\omega_1a_{13} + \alpha^2\omega_2a_{14} & = 0\\
\alpha^2\omega_1a_{12} + \alpha^2F_{11}a_{13} + \alpha^2F_{21}a_{14} & = 0 \\
\alpha^2\omega_2a_{12} + \alpha^2F_{12}a_{13} + \alpha^2F_{22}a_{14} & = 0.
\endaligned\right.
\end{equation} Subtracting the first equation multiplied by $\alpha$ from the second equation multiplied by 2 in \eqref{inverse-1-app}, we get \begin{equation}\label{5-equ-app}
\alpha^2 a_{12} + 2\alpha^2\rho\omega_1a_{13} + 2\alpha^2\rho\omega_2a_{14} = -\alpha \end{equation} And subtracting \eqref{5-equ-app} multiplied with $\omega_1$ from the third equation in \eqref{inverse-1-app} as well as subtracting \eqref{5-equ-app} multiplied with $\omega_2$ from the fourth equation in \eqref{inverse-1-app}, we get \begin{equation}\label{app-13-14} \alpha^2F^*\left[\aligned a_{13}\\a_{14}\endaligned\right] = \left[\aligned \alpha\omega_1\\ \alpha\omega_2\endaligned\right] \end{equation} Plugging back what are $a_{13}$ and $a_{14}$ to the equation \eqref{5-equ-app} we have \begin{equation}\label{app-12} \left\{\aligned a_{12} & = \alpha^{-1}( -1 - 2\rho [\omega_1, \omega_2](F^*)^{-1}\left[\begin{matrix} \omega_1\\ \omega_2\end{matrix}\right])\\ a_{11} & = - \frac { \alpha a_{12} +1} {2\rho} = [\omega_1, \omega_2](F^*)^{-1}\left[\begin{matrix} \omega_1\\ \omega_2\end{matrix}\right] . \endaligned\right. \end{equation} Similarly one gets \begin{equation}\label{app-2j} \alpha^2F^* \left[\aligned a_{23}\\a_{24}\endaligned\right] = \left[\aligned-2\rho\omega_1\\ -2\rho\omega_2\endaligned\right] \text{ and } \left\{\aligned a_{21} & = \alpha^{-1}( -1 - 2\rho [\omega_1, \omega_2](F^*)^{-1}\left[\begin{matrix} \omega_1\\ \omega_2\end{matrix}\right])\\ a_{22} & = \frac {2\rho}{\alpha^2} ( 1 +2\rho [\omega_1, \omega_2](F^*)^{-1}\left[\begin{matrix} \omega_1\\ \omega_2\end{matrix}\right]) \endaligned\right. \end{equation} \begin{equation}\label{app-3j} \alpha^2F^* \left[\aligned a_{33}\\a_{34}\endaligned\right] = \left[\aligned 1 \\ 0 \endaligned\right] \text{ and } \left\{ \aligned a_{31} & = \alpha^{-1} [\omega_1, \omega_2](F^*)^{-1}\left[\begin{matrix} 1\\ 0\end{matrix}\right]\\ a_{32} & = - \frac {2\rho}{\alpha^2} ( [\omega_1, \omega_2](F^*)^{-1}\left[\begin{matrix}1 \\ 0 \end{matrix}\right] \endaligned\right. \end{equation} \begin{equation}\label{app-4j} \alpha^2F^* \left[\aligned a_{43}\\a_{44}\endaligned\right] = \left[\aligned 0\\ 1\endaligned\right]\text{ and } \left\{ \aligned a_{41} & = \alpha^{-1}(F^*)^{-1}\left[\begin{matrix} 0 \\1\end{matrix}\right] \\ a_{42} & = \frac {2\rho}{\alpha^2} ( 1 +2\rho [\omega_1, \omega_2](F^*)^{-1}\left[\begin{matrix} \omega_1\\ \omega_2\end{matrix}\right]) \endaligned\right. \end{equation}
\section{The geometry of the 3-sphere $\mathbb{S}^3_\lambda$ in $\mathbb{R}^{1,4}$}\label{gauss}
Let us calculate the Gauss Theorem for the 3-sphere $\mathbb{S}^3_\lambda$ in Minkowski spacetime $\mathbb{R}^{1,4}$. There is nothing new or difficult about the calculation, but this helps to understand better about the geometry of the 3-sphere $\mathbb{S}^3_\lambda\subset\mathbb{N}^4_+\subset\mathbb{R}^{1,4}$. It is very crucial and important in our approach to use the fact that the induced metric on $\mathbb{S}^3_\lambda$ is exactly the conformal metric $\lambda^2g_0$. We consider the Fermi parametrization induced from a parametrization of the surface $\hat x: \textup{M}^2\to \mathbb{S}^3$ such that \begin{equation}\label{lambda-3-sphere} y_\lambda = \lambda (\hat x(u^1, u^2, u^3))(1, \hat x(u^1, u^2, u^3)): \textup{M}^3\to \mathbb{S}^3_\lambda\subset\mathbb{N}^4_+\subset\mathbb{R}^{1,4} \end{equation} with \begin{equation}\label{extension-y}
\hat x(u^1, u^2, 0) = \hat x(u^1, u^2) \text{ and } (y_\lambda)_{u^3}|_{u^3=0} = \overset{\rightarrow}{\bf n}_\lambda. \end{equation} Notice that $y_\lambda$ here is the extension of $\hat\lambda(1, \hat x)$ before. We use the two null normal vectors $\{y_\lambda, y_\lambda^\dagger\}$ where \begin{equation}\label{y-dagger} <y^\dagger_\lambda, y_\lambda> -1, <y^\dagger_\lambda, (y_\lambda)_{u^1}> = <y^\dagger_\lambda, (y_\lambda)_{u^2}> = <y^\dagger_\lambda, (y_\lambda)_{u^3}> =0. \end{equation} The first fundamental form is \begin{equation}\label{lambda-sphere-I} I^{\mathbb{S}^3_\lambda} = \lambda^2g_0 = <dy_\lambda, dy_\lambda>. \end{equation} And the second fundamental form is \begin{equation}\label{lambda-sphere-II} II^{\mathbb{S}^3_\lambda} = - <dy_\lambda, dy^\dagger_\lambda> y^\dagger_\lambda - <dy_\lambda, dy_\lambda>y_\lambda \end{equation} To calculate the curvature for the metric $g_\lambda = \lambda^2g_0$ we calculate $$ \nabla^\lambda_{\partial_{u^j}}\nabla^\lambda_{\partial_{u^i}}\partial_{u^k} - \nabla^\lambda_{\partial_{u^i}}\nabla^\lambda_{\partial_{u^j}}\partial_{u^k} = R^\lambda(\partial_{u^i}, \partial_{u^j})\partial_{u^k} = (R^\lambda)_{ijk}^{\quad \ l}\partial_{u^l}. $$ First $$ \nabla^\lambda_{\partial_{u^j}}\partial_{u^k} = (y_\lambda)_{u^ku^j} - <(y_\lambda)_{u^j}, (y^\dagger_\lambda)_{u^k}>y_\lambda - <(y_\lambda)_{u^j}, (y_\lambda)_{u^k}>y^\dagger_\lambda $$ Then $$ \aligned \partial_{u^i}\nabla^\lambda_{\partial_{u^j}}\partial_{u^k} & = (y_\lambda)_{u^ku^ju^i} - <(y_\lambda)_{u^j}, (y^\dagger_\lambda)_{u^k}>_{u^i} y_\lambda - <(y_\lambda)_{u^j}, (y_\lambda)_{u^k}>_{u^i}y^\dagger_\lambda \\ & \quad -<(y_\lambda)_{u^j}, (y^\dagger_\lambda)_{u^k}>(y_\lambda)_{u^i} - <(y_\lambda)_{u^j}, (y_\lambda)_{u^k}>(y^\dagger_\lambda)_{u^i} \endaligned $$ and $$ \aligned \nabla^\lambda_{\partial_{u^i}}\nabla^\lambda_{\partial_{u^j}}\partial_{u^k} & = (\partial_{u^i}\nabla^\lambda_{\partial_{u^j}}\partial_{u^k})^{T\mathbb{S}^3_\lambda}\\ & = (y_\lambda)_{u^ku^ju^i}^{T\mathbb{S}^3_\lambda} -<(y_\lambda)_{u^j}, (y^\dagger_\lambda)_{u^k}>(y_\lambda)_{u^i} - <(y_\lambda)_{u^j}, (y_\lambda)_{u^k}>(y^\dagger_\lambda)_{u^i} \endaligned $$ Hence $$ \aligned (R^\lambda)_{ijk}^{\quad \ l}\partial_{u^l} & = <(y_\lambda)_{u^j}, (y^\dagger_\lambda)_{u^k}>(y_\lambda)_{u^i} + <(y_\lambda)_{u^j}, (y_\lambda)_{u^k}>(y^\dagger_\lambda)_{u^i} \\ & \quad -<(y_\lambda)_{u^i}, (y^\dagger_\lambda)_{u^k}>(y_\lambda)_{u^j} - <(y_\lambda)_{u^i}, (y_\lambda)_{u^k}>(y^\dagger_\lambda)_{u^j} \\ \endaligned $$ One may realize that $$ <(y^\dagger_\lambda)_{u^i}, y^\dagger_\lambda > = 0 \text{ and } <(y^\dagger_\lambda)_{u^i}, y_\lambda> = 0 $$ and conclude $$ (y^\dagger_\lambda)_{u^i} = (g_\lambda)^{ml}<(y^\dagger_\lambda)_{u^i}, (y_\lambda)_{u^m}>(y_\lambda)_{u^l}. $$ Therefore $$ \aligned (R^\lambda)_{ijk}^{\quad \ l} \partial_{u^l} & = (<(y_\lambda)_{u^j}, (y^\dagger_\lambda)_{u^k}>\delta_i^{\ l} + (g_\lambda)_{jk}(g_\lambda)^{ml}<(y^\dagger_\lambda)_{u^i}, (y_\lambda)_{u^m}>\\
&\quad - <(y_\lambda)_{u^i}, (y^\dagger_\lambda)_{u^k}>\delta_j^{\ l} - (g_\lambda)_{ik}(g_\lambda)^{ml}<(y^\dagger_\lambda)_{u^j}, (y_\lambda)_{u^m}>)
\partial_{u^l}
\endaligned
$$ and $$ \aligned (R^\lambda)_{ijkl} = & (R^\lambda)_{ijk}^{\quad \ n}(g_\lambda)_{nl} = <(y_\lambda)_{u^j}, (y^\dagger_\lambda)_{u^k}>(g_\lambda)_{il} + <(y^\dagger_\lambda)_{u^i}, (y_\lambda)_{u^l}> (g_\lambda)_{jk}\\
&\quad - <(y_\lambda)_{u^i}, (y^\dagger_\lambda)_{u^k}> (g_\lambda)_{jl} - <(y^\dagger_\lambda)_{u^j}, (y_\lambda)_{u^l}> (g_\lambda)_{ik}.
\endaligned
$$
On the surface $\hat x$, where $u^3=0$, we have $$
[(g_\lambda)_{ij}] = \left[\begin{matrix} E_\lambda & 0 & 0 \\0 & E_\lambda & 0\\0 & 0 & 1\end{matrix}\right]. $$ Therefore we have, for $i, j\in\{1, 2\}$, $$ \left\{\aligned - < (y_\lambda)_{u^i}, (y_\lambda^\dagger)_{u^j}> - < (y_\lambda)_{u^3}, (y_\lambda^\dagger)_{u^3}>E_\lambda \delta_{ij} & = (R^\lambda)_{i3j3}\\ - <(y_\lambda)_{u^j}, (y^\dagger_\lambda)_{u^3}> E_\lambda\delta_{jl} + < (y_\lambda)_{u^l}, (y_\lambda^\dagger)_{u^3}> E_\lambda & = (R^\lambda)_{3jjl}\\ - < (y_\lambda)_{u^i}, (y_\lambda^\dagger)_{u^i}> E_\lambda - < (y_\lambda)_{u^j}, (y_\lambda^\dagger)_{u^j}> E_\lambda & = (R^\lambda)_{ijij} \endaligned \right. $$ Finally we obtain, for $i, j\in\{1, 2\}$, \begin{equation}\label{n-dagger} <\overset{\rightarrow}{\bf n}_\lambda, (y^\dagger_\lambda)_{u^i}> = \frac 1{E_\lambda} (R^\lambda)_{ijj3} = - (R^\lambda)_{i3}, \end{equation} and for $i\neq j$, \begin{equation}\label{i-dagger-j} \aligned <(y_\lambda)_{u^i}, (y^\dagger_\lambda)_{u^j}> &= - (R^\lambda)_{i3j3}\\
<(y_\lambda)_{u^i}, (y^\dagger_\lambda)_{u^i}> & = - (R^\lambda)_{i3i3} + \frac 12((R^\lambda)_{33} - (R^\lambda)_{1212})\\
<(y_\lambda)_{u^3}, (y^\dagger_\lambda)_{u^3}> & = - \frac 12((R^\lambda)_{33} - (R^\lambda)_{1212})\endaligned
\end{equation}
Finally, for the induced Fermi coordinate from an isothermal coordinate, we can easily see that \begin{equation}\label{coord-covar} \aligned (R^\lambda)_{3i,}^{\quad i} & = \frac 1{E_\lambda}(\sum_{i=1}^2R^\lambda)_{3i, i} \\ & = \frac 1{E_\lambda}\sum_{i=1}^2(((R^\lambda)_{3i})_{u^i} - (R^\lambda)_{3k}(\Gamma_\lambda)^k_{ii}) \\ & = \frac 1{E_\lambda}\sum_{i=1}^2(((R^\lambda)_{3i})_{u^i} \endaligned \end{equation} Because $\sum_{i=1}^2(\Gamma_\lambda)^k_{ii} = 0$ for each $k= 1,2$, where $(\Gamma_\lambda)^k_{ij}$ is the Christofel symbols
for the conformal metric $I_\lambda = E_\lambda |du|^2$ in the isothermal coordinates. \end{appendix}
\end{document} | arXiv |
The math is quite simple to demonstrate why an object traveling at a set speed will reach Earth if that speed matches the expansion rate where that object starts from.
The claim that the universe is expanding has all the arguments given in the OP. Each constitutes objective evidence (i.e. facts). Scrutiny, of course, is welcome for each one of those, but that list comes from mainstream science and they follow the SM.
One can argue for some weakness in one or more, but it's important that in a theory that addresses the entire cosmos, then all the facts must fit the model; their must be a convergence and a confluence. Scientists were initially opposed to Lemaitre's theory, but the more they saw the confluence, the more robust the model became. The discovery of the predicted CMBR has been regarded as the nail in the coffin for any opposing theory that existed in the past (e.g. Steady State theory).
Any fact discovered, however, that is in disagreement with the theory invalidates the theory. When Galileo discovered all phases for Venus, he immediately falsified the Ptolemy theory with that fact alone. Of course, others (i.e. Jesuits) had to study those phase changes over the course of Venus orbit in order to verify his discovery. They agreed with Galileo, though some chose not to and even some refused to look through the telescope.
The expansion is the heart of BBT. The exact rate, of course, is a very tough number to nail down and it is further hampered by the fact that it can't be linear over all time given DE at play.
*The claim that the universe is expanding has all the arguments given in the OP.*, the problem here is the redshifts 1.4 or larger. 4D space cannot be seen moving away from Earth faster than c velocity and this is a requirement in the cosmological redshift interpretation for the larger values. Unlike the phases of Venus, these can be clearly seen today, 4D space expanding faster than c velocity for all redshifts 1.4 or larger, cannot be seen. The doctrine of uniformity is assumed to assert this in the model. This should be carefully pointed out to the public when presenting the Big Bang model of origins.
Just to clarify, is it your view that any comoving space that is moving away from us greater than c (z = 1.4, trusting your value) represents metaphysics, as you think we could never even see things from regions beyond this distance?
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Helio, did Einstein teach that 4D space expands faster than c velocity?
[Submitted on 27 Jan 2022 (v1), last revised 7 Feb 2022 (this version, v2)]
The Universe as a Quantum Encoder
Jordan Cotler, Andrew Strominger
Quantum mechanical unitarity in our universe is challenged both by the notion of the big bang, in which nothing transforms into something, and the expansion of space, in which something transforms into more something. This motivates the hypothesis that quantum mechanical time evolution is always isometric, in the sense of preserving inner products, but not necessarily unitary. As evidence for this hypothesis we show that in two spacetime dimensions (i) there is net entanglement entropy produced in free field theory by a moving mirror or expanding geometry, (ii) the Lorentzian path integral for a finite elements lattice discretization gives non-unitary isometric time evolution, and (iii) tensor network descriptions of AdS3 induce a non-unitary but isometric time evolution on an embedded two-dimensional de Sitter braneworld. In the last example time evolution is a quantum error-correcting code.
Comments: 31+11 pages, 10 figures; v2: typos fixed, references and comments added
Subjects: High Energy Physics - Theory (hep-th); Quantum Physics (quant-ph)
Cite as: arXiv:2201.11658 [hep-th]
(or arXiv:2201.11658v2 [hep-th] for this version)
Some look at it from a different angle. The above is not my opinion.
Expansion and contraction are controlled by vector forces, how these vector forces are created are worth investigating.
Condensates found in the core of Black Holes can explain the vector forces required to expand matter and also contract matter.
The dipolar electromagnetic vector forces are a property of condensates by understanding Chiral Supersymmetry and the droplets that form that are able to seed stars.
[1708.09057] Bose-Einstein Condensates in Charged Black-Hole Spacetimes (arxiv.org)
Condensed Matter > Quantum Gases
[Submitted on 5 Apr 2022]
Exploring vortex formation in rotating Bose-Einstein condensates beyond mean-field regime
Budhaditya Chatterjee
The production of quantized vortices having diverse structures is a remarkable effect of rotating Bose-Einstein condensates. Vortex formation described by the mean-field theory is valid only in the regime of weak interactions. The exploration of the rich and diverse physics of strongly interacting BEC requires a more general approach. This study explores the vortex formation of strongly interacting and rapidly rotating BEC from a general ab initio many-body perspective. We demonstrate that the quantized vortices form various structures that emerge from an intricate interplay between the angular momentum and many-body interaction. We examine the distinct impact of the angular velocity and interaction energy on the vortex formation. Our analysis shows that, while the angular rotation generally augments the vortex formation, the interactions can enhance as well as impede the vortices production.
Subjects: Quantum Gases (cond-mat.quant-gas)
Cite as: arXiv:2204.01978 [cond-mat.quant-gas]
(or arXiv:2204.01978v1 [cond-mat.quant-gas] for this version)
Focus to learn m
The paper talks about a rotating BEC rather than the condensate property explained by Chiral Supersymmetry, but it talks about the formation of the vortex. Interesting
Good question. I don't recall if he got very deep into cosmology even after he realized Lemaitre was right, after initially calling Lemaitre's physics an "abomination" ( ). There were a half-dozen or more that were very active in cosmology using GR, including De Sitter, Wrey, etc.
One problem was in determining any sort of an accurate value for the expansion rate, which would give those > c values. Lemaitre and Hubble had, initially, very fast Ho values (around 600 kps/Mpc, I think). Lemaitre had too little to work with and he knew it, which is why most people think he didn't bother to include his expansion rate in his English translation of his now famous 1927 paper that was the foundation for BBT. Hubble had, during this interval, since produced a lot more distance measurements that refined the expansion rate. Hubble also seems to have been quite jealous of his (don't forget Humason) work, so this human element may, or may not, have played into this story.
Hubble, initially, was not aware that there were two types of Cepheids and thus, two values for distance determinations. He greatly underestimated the distance to Andromeda as a result, which gave him a very high value for the expansion rate. This gave us an age for the universe deemed younger than the age of stars. [I can imagine the fun we would have had back then in any forum threads! ]
This error was corrected quickly, IIRC. Then much later came the SN Type 1a to greatly improve the expansion rate, as well as, the other techniques you've mentioned. A really solid number doesn't exist due to some conflicts but the probability seems very high that it's between 60 kps/Mpc and 75 kps/Mpc.
However, I can't imagine Einstein would have had any problem with spacetime expanding faster than c. The fixed value of c is always motion through space only.
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Speed of light is constant.
Relativity plays a part when vector force fields are able to slow it down or bend the EMR.
At EH it may seem that time has stood still and yet the speed of light EMR is constant.
The furthest Galaxy to date is 13.5 billion years place a few million years to account for the observation and depth of the galaxy.
Google it
General Relativity and Quantum Cosmology
[Submitted on 11 Mar 2022]
Compact stars in the Einstein dark energy model
Zahra Haghani, Tiberiu Harko
We investigate the properties of high density compact objects in a vector type theory, inspired by Einstein's 1919 theory of elementary particles, in which Einstein assumed that elementary particles are held together by gravitational as well as electromagnetic type forces. From a modern perspective, Einstein's theory can be interpreted as a vector type model, with the gravitational action constructed as a linear combination of the Ricci scalar, of the trace of the matter energy-momentum tensor, and of a massive self-interacting vector type field. To obtain the properties of stellar models we consider the field equations for a static, spherically symmetric system, and we investigate numerically their solutions for different equations of state of quark and neutron matter, by assuming that the self-interaction potential of the vector field either vanishes or is quadratic in the vector field potential. We consider quark stars described by the MIT bag model equation of state and in the Color Flavor Locked phase, as well as compact stars consisting of a Bose-Einstein Condensate of neutron matter, with neutrons forming Cooper pairs. Constant density stars, representing a generalization of the Interior Schwarzschild solution of general relativity, are also analyzed. Also, we consider the Douchin-Haensel (SLy) equation of state. The numerical solutions are explicitly obtained in both standard general relativity, and the Einstein dark energy model and an in depth comparison between the astrophysical predictions of these two theories are performed. As a general conclusion of our study, we find that for all the considered equations of state a much larger variety of stellar structures can be obtained in the Einstein dark energy model, including classes of stars that are more massive than their general relativistic counterparts.
Subjects: General Relativity and Quantum Cosmology (gr-qc)
Cite as: arXiv:2203.05764 [gr-qc]
(or arXiv:2203.05764v1 [gr-qc] for this version)
Sometimes i feel sorry for posting what I read, sometimes I may disagree and yet I feel that I cannot change a persons opinion, but! Rather encourage further research.,
We are the steps of getting to know much much more.
"However, I can't imagine Einstein would have had any problem with spacetime expanding faster than c. The fixed value of c is always motion through space only."
This is an opinion question Helio and my answer is that Einstein would likely look for specific testing and verification to show that nature in 4D space is expanding faster than c velocity. Consider H0 = 69 km/s/Mpc is a bit more than 2 x 10^-18 cm/s/cm. The velocity of c ~ 3 x 10^10 cm/s. We are looking at more than 10^28 magnitude change here in expansion speeds in 4D space.
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The expansion rate is based on mega parsecs for this reason.
Einstein seemed to have become highly focused on his theory to unite all the forces, so I don't think he got that interested in Big Bang issues. I'd like to discover otherwise, however, as his input would be interesting.
But you're right about Einstein wanting objective-based information to work with. One author argues that his trip to Mt. Wilson was not really to meet with Hubble but with the physicists who went to be there with him, especially those who spoke German (English was still new to him). His questions were Sun-related since GR evidence is found there. The author notes that Einstein's diary never mentioned Hubble.
Einstein worked on a cyclic universe, bu! When Hubble showed red shift explanation and the expanding univserse, Einstein went to say that he make a great mistake and tended to agree with the expanding universe. I think I got that right.
To understand expanding space and time one needs to understand relativity.
What do people understand about space and time?
When Lemaitre learned, at a conference, from an Eddington presentation that he (Eddington) and others had no solution from GR to resolve the redshift dilemma, Lemaitre showed Eddington, his former professor, his paper from 1927.
Once it was translated into English Eddington was quick to respect it and soon De Sitter supported it. Hubble, by this time, had many more redshift data and more distances.
Hubble never chose to claim that expansion was the explanation, though the hints from him were there. He stated he preferred theorists make such claims.
Helio well done
If the BB did occur
Where is the question.
When is the question.
How is the question
What made it do it
Why and so on.
To trigger an event, what and why
[2204.00569] Considering the two Spin and the two Angular Momenta String Solutions in $AdS_5 \times S^5$
"However, our setup in this paper is generally for the Neumann-Rosochatius System, which is also solvable, since we intend to generalize our results from the Neumann System to the Neumann-Rosochatius System and to several types of deformed Neumann-Rosochatius Systems. In the second part of this paper, which is independent of any string configurations in AdS5×S5 and concerns String Cosmology in D=10 dimensions. I will seriously argue that there was no Big Bang; I truly believe that the Universe has been there forever!"
We must continually question theories, rather than accepting face value.
The idea that you cannot create matter nor destroy matter, means that matter has been around for ever.
Most of the matter over 90% is found in compact condensates. The study of these condensates may give us greater understanding of how things work.
The BBT has always bothered me in that it relies on the assumptions that nearly all of what matter exists we cannot see (dark matter) and that all matter, seen and unseen, is affected more by a force that we cannot measure or understand (dark energy/inflation) than by all the forces that we know anything about. Those are usually red flags that indicate that we are missing something big. Yes, we can sometimes discover new things by inferring them from theoretical calculations that "need something more to work". Our understanding of energy, today, came from noting the failures of the phlogiston theory of heat as a substance. But, when you search for the postulated matter and energy and do not find them, you SHOULD be very uncomfortable with the theory that requires them. The current level of scientific smugness with respect to the BBT seems unwarranted to me. I wonder if we have really properly solved the General Relativity equations. I do occasionally see papers that claim to explain things like galactic orbit speeds by redoing frame dragging calculations, thus removing the need for dark matter. And I once read an article that claimed the universe would appear to be expanding to an observer falling into a large black hole. I also wonder about how we understand time as a dimension. I have not seen anything in this discussion, about putting Humpty-Dumpty back together again in a contraction period of a cyclic universe, that considers what the time dimension might do during Contraction - the capital C indicating the opposite of inflation, rather than gravitational collapse. Could the Contraction reverse time, and thus decrease entropy? BBT proponents seem to think of contraction in a very limited, small c way. Which seems inconsistent with thinking of Inflation as the imaginative solution to the otherwise unanswerable question about how so much mass as we see in the universe could possibly once have been in a small space that would have been a black hole from which it should not have been able to expand into what we THINK we see now. My strong suspicion is that we are missing something extremely important in our perception of what we are seeing and measuring in our very restricted physical and mental view of the cosmos.
I am excited that the Webb Telescope will add some important information in the very near future. I remember when the Hubble Telescope was expected to see "back in time to the first stars", but has consistently found that the galaxies were in existence as far back as it can see, so galaxies had to have formed earlier than then expected. I wonder if Webb will show us the same, again. If so, I will have even more doubts about the BBT. But, Webb might find what we predict, for a change. The great thing is that we should know more in a few years.
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Unclear Engineer said:
The BBT has always bothered me in that it relies on the assumptions that nearly all of what matter exists we cannot see (dark matter) and that all matter, seen and unseen, is affected more by a force that we cannot measure or understand (dark energy/inflation) than by all the forces that we know anything about.
The BBT came from simply applying GR to explain the redshifts of galaxies, so it wasn't reliant upon DM. The physics, however, to explain what we observe does require it. But we can "see" DM indirectly. Direct observations are always nicer but we can do well if we have clear indirect evidence. Black holes can never be "seen", by definition, but we can observe the regions around them that define them.
The purpose of the OP is to demonstrate the strong objective evidence favoring the BBT. Most people, IMO, have no idea just how robust the evidence supporting BBT really is.
But, when you search for the postulated matter and energy and do not find them, you SHOULD be very uncomfortable with the theory that requires them.
This is true for gravity, which Newton admitted he didn't know what it is but he did a nice job giving us a universal law on its behavior. How things behave is the essence of physics even if we never quite reach the essence of what something is.
The current level of scientific smugness with respect to the BBT seems unwarranted to me.
Do you see any problems with any item in the OP list of bullets?
BBT proponents seem to think of contraction in a very limited, small c way.
There is strong evidence that the universe is not only expanding but accelerating in this expansion. Contraction from rewinding the clock produced the prediction for things like the CMBR, but a real contracting universe at some point likely won't produce any hard science.
Which seems inconsistent with thinking of Inflation as the imaginative solution to the otherwise unanswerable question about how so much mass as we see in the universe could possibly once have been in a small space that would have been a black hole from which it should not have been able to expand into what we THINK we see now.
Initially, all was energy. Expansion took place allowing matter to emerge as this energy cooled. BBT doesn't begin as a singularity, though it is easy to extrapolate to that point. BBT limits itself to the point physics is able to keep all the wheels on its wagon. The equations fail extremely close to t=0.
"Expansion took place allowing matter to emerge as this energy cooled. BBT doesn't begin as a singularity, though it is easy to extrapolate to that point. BBT limits itself to the point physics is able to keep all the wheels on its wagon. The equations fail extremely close to t=0."
I agree with Helio, especially as I do not have to caveat the pure imagination around (very, very close to) t = 0. It is very easy to extrapolate elephants back to small, pink, flying ones, but it does not prove their existence any more that an imaginative figment proves that nonsense someone christened a singularity.
The only point where we may disagree, being what 'happened' at t = 0, comes to mind. I say there is nothing known at this 'point' (for want of a better word) and my guess is probably as good as anyone else's. Two options for t = 0 (there may be many more) are the mathematical fiction (singularity) and some form of continuity (nexus) which simply postulates continuity (from a former phase) in place of some 'appearing out of nothing'. One 'nexus' might imply some form of cyclic universe. However, 'cyclic' contains some unfortunate elements (repetition) which I do not support.
On consideration, and aware that any suggestion I might make is just as imaginary (even if more believable) as a singularity, I am limiting my suggestion to replacing singularity with nexus. This simply suggests that the Universe existed as a phase of its existence 'before' t = 0.
One analogy, and it is only illustrative imagination, is the idea that the precedent phase might have been comparable to a black hole in that (and this is imaginary) the 'preceding' t < 0 phase ended in a nexus which then opened into the next 'big bang'. There is no evidence for or against such an idea, just as there is no evidence to support a 'singularity'.
In this general context, I agree with Helio (post #3):
"But the BBT doesn't need to establish a beginning with t=0 any more than, say, Newton needed to explain gravity. Newton's gravity work was wonderful physics describing something he admitted he didn't know what it was."
The idea that the universe is expanding and accelerating supported by evidence.
Give me one evidence that is evidence supported.
Where did the matter come from?
What is the driving force?
What causes the acceleration?
If there is a trillion galaxies within our view sphere of a diameter of 26.8 billion light years.
And the BB says that our universe is 13.7 billion years old.
Our star is close to 12 billion years old.
All these questions lift up red flags, and are begging answers.
Let me continue from post #96:
*"In this general context, I agree with Helio (post #3):"
"But the BBT doesn't need to establish a beginning with t=0 any more than, say, Newton needed to explain gravity. Newton's gravity work was wonderful physics describing something he admitted he didn't know what it was."*
Very apt, I think. Jus as Newton knew nothing about the nature of gravity, he could use mathematics to explain the motions of the planets to a very high degree of accuracy.
We are beginning to know, or think about, or dream, possible scenarios, even bad dreams like singularities, and we will make progress. But. I think, some things are unknowable. We will reach a stage where we will where we can make positive, reliable, predictions, as Newton did, but we will not have understood many of the mysteries of the nexus, the alternate phases, the workings of the Universe . . . . . . . . .
Yes. Science has changed over time. Before Newton, though no clear line exists, science had to include purpose for the phenomena they were addressing. This is known as teleology. Newton simply quantified a phenomena he admitted he didn't understand in the least, at least the "essence" of what gravity truly is. [ We're still looking for gravitons or something.]
Imagine if he had elected to what until he did understand what gravity actually is rather than how it behaves. Engineering school might have been just a two year program for us.
Many love the BBT and give their best to make sure that the BB to explain the possible origins of the universe.
We can explain expansion and contraction small and monster large , by understanding what forces take place. | CommonCrawl |
\begin{document}
\title{A Drifting-Games Analysis for Online Learning and Applications to Boosting}
\begin{abstract} We provide a general mechanism to design online learning algorithms based on a minimax analysis within a drifting-games framework. Different online learning settings (Hedge, multi-armed bandit problems and online convex optimization) are studied by converting into various kinds of drifting games. The original minimax analysis for drifting games is then used and generalized by applying a series of relaxations, starting from choosing a convex surrogate of the 0-1 loss function. With different choices of surrogates, we not only recover existing algorithms, but also propose new algorithms that are totally parameter-free and enjoy other useful properties. Moreover, our drifting-games framework naturally allows us to study high probability bounds without resorting to any concentration results, and also a generalized notion of regret that measures how good the algorithm is compared to all but the top small fraction of candidates. Finally, we translate our new Hedge algorithm into a new adaptive boosting algorithm that is computationally faster as shown in experiments, since it ignores a large number of examples on each round. \end{abstract}
\section{Introduction} In this paper, we study online learning problems within a drifting-games framework, with the aim of developing a general methodology for designing learning algorithms based on a minimax analysis.
To solve an online learning problem, it is natural to consider game-theoretically optimal algorithms which find the best solution even in worst-case scenarios. This is possible for some special cases (\cite{CesabianchiFrHeHaScWa97, AbernethyBaRaTe08, AbernethyWa10, LuoSc14}) but difficult in general. On the other hand, many other efficient algorithms with optimal regret rate (but not exactly minimax optimal) have been proposed for different learning settings (such as the exponential weights algorithm \cite{FreundSc97, FreundSc99}, and follow the perturbed leader \cite{KalaiVe05}). However, it is not always clear how to come up with these algorithms. Recent work by Rakhlin et al. \cite{RakhlinShSr12} built a bridge between these two classes of methods by showing that many existing algorithms can indeed be derived from a minimax analysis followed by a series of relaxations.
In this paper, we provide a parallel way to design learning algorithms by first converting online learning problems into variants of drifting games, and then applying a minimax analysis and relaxations. {\it Drifting games} \cite{Schapire01} (reviewed in Section \ref{sec:drifting}) generalize Freund's ``majority-vote game'' \cite{Freund95} and subsume some well-studied boosting and online learning settings. A nearly minimax optimal algorithm is proposed in \cite{Schapire01}. It turns out the connections between drifting games and online learning go far beyond what has been discussed previously. To show that, we consider variants of drifting games that capture different popular online learning problems. We then generalize the minimax analysis in \cite{Schapire01}
based on one key idea: {\it relax a 0-1 loss function by a convex surrogate}. Although this idea has been applied widely elsewhere in machine learning, we use it here in a new way to obtain a very general methodology for designing and analyzing online learning algorithms. Using this general idea, we not only recover existing algorithms, but also design new ones with special useful properties. A somewhat surprising result is that our new algorithms are totally {\it parameter-free}, which is usually not the case for algorithms derived from a minimax analysis. Moreover, a generalized notion of regret ($\epsilon$-regret, defined in Section \ref{sec:hedge}) that measures how good the algorithm is compared to all but the top $\epsilon$ fraction of candidates arises naturally in our drifting-games framework. Below we summarize our results for a range of learning settings.
\textbf{Hedge Settings:} (Section \ref{sec:hedge}) The Hedge problem \cite{FreundSc97} investigates how to cleverly bet across a set of actions. We show an algorithmic equivalence between this problem and a simple drifting game (DGv1). We then show how to relax the original minimax analysis step by step to reach a general recipe for designing Hedge algorithms (Algorithm \ref{alg:hedge_recipe}). Three examples of appropriate convex surrogates of the 0-1 loss function are then discussed, leading to the well-known exponential weights algorithm and two other new ones, one of which ({{NormalHedge.DT}} in Section \ref{sec:3algs}) bears some similarities with the NormalHedge algorithm \cite{ChaudhuriFrHs09} and enjoys a similar $\epsilon$-regret bound {\it simultaneously} for all $\epsilon$ and horizons. However, our regret bounds do not depend on the number of actions, and thus can be applied even when there are infinitely many actions. Our analysis is also arguably simpler and more intuitive than the one in \cite{ChaudhuriFrHs09} and easy to be generalized to more general settings. Moreover, our algorithm is more computationally efficient since it does not require a numerical searching step as in NormalHedge. Finally, we also derive high probability bounds for the randomized Hedge setting as a simple side product of our framework {\it without} using any concentration results.
\textbf{Multi-armed Bandit Problems:} (Section \ref{sec:gen}) The multi-armed bandit problem \cite{AuerCeFrSc02} is a classic example for learning with incomplete information where the learner can only obtain feedback for the actions taken. To capture this problem, we study a quite different drifting game (DGv2) where randomness and variance constraints are taken into account. Again the minimax analysis is generalized and the EXP3 algorithm \cite{AuerCeFrSc02} is recovered.
Our results could be seen as a preliminary step to answer the open question \cite{AbernethyWa09} on exact minimax optimal algorithms for the multi-armed bandit problem.
\textbf{Online Convex Optimization:} (Section \ref{sec:gen}) Based the theory of convex optimization, online convex optimization \cite{Zinkevich03} has been the foundation of modern online learning theory. The corresponding drifting game formulation is a continuous space variant (DGv3). Fortunately, it turns out that all results from the Hedge setting are ready to be used here, recovering the continuous EXP algorithm \cite{Cover91, HazanAgKa07, NarayananRa10} and also generalizing our new algorithms to this general setting. Besides the usual regret bounds, we also generalize the $\epsilon$-regret, which, as far as we know, is the first time it has been explicitly studied. Again, we emphasize that our new algorithms are adaptive in $\epsilon$ and the horizon.
\textbf{Boosting:} (Section \ref{sec:gen}) Realizing that every Hedge algorithm can be converted into a boosting algorithm (\cite{SchapireFr12}), we propose a new boosting algorithm ({{NH-Boost.DT}}) by converting {{NormalHedge.DT}}. The adaptivity of {{NormalHedge.DT}} is then translated into training error and margin distribution bounds that previous analysis in \cite{SchapireFr12} using nonadaptive algorithms does not show. Moreover, our new boosting algorithm ignores a great many examples on each round, which is an appealing property useful to speeding up the weak learning algorithm. This is confirmed by our experiments.
\textbf{Related work}: Our analysis makes use of potential functions. Similar concepts have widely appeared in the literature \cite{CesabianchiLu03, AudibertBuLu14}, but unlike our work, they are not related to any minimax analysis and might be hard to interpret. The existence of parameter free Hedge algorithms for unknown number of actions was shown in \cite{ChernovVovk10}, but no concrete algorithms were given there.
Boosting algorithms that ignore some examples on each round were studied in \cite{FriedmanHaTi00}, where a heuristic was used to ignore examples with small weights and no theoretical guarantee is provided.
\section{Reviewing Drifting Games} \label{sec:drifting} We consider a simplified version of drifting games similar to the one described in \citep[chap. 13]{SchapireFr12} (also called chip games). This game proceeds through $T$ rounds, and is played between a player and an adversary who controls $N$ chips on the real line. The positions of these chips at the end of round $t$ are denoted by ${\bf s}_t \in \mathbb{R}^N$, with each coordinate $s_{t,i}$ corresponding to the position of chip $i$. Initially, all chips are at position $0$ so that ${\bf s}_0 = {\bf 0}$. On every round $t = 1, \ldots, T$: the player first chooses a distribution ${\bf p}_t$ over the chips,
then the adversary decides the movements of the chips ${\bf z}_t$ so that the new positions are updated as ${\bf s}_t = {\bf s}_{t-1} + {\bf z}_t$. Here, each $z_{t,i}$ has to be picked from a prespecified set $B \subset \mathbb{R}$, and more importantly, satisfy the constraint ${\bf p}_t \cdot {\bf z}_t \geq \beta \geq 0$ for some fixed constant $\beta$.
At the end of the game, each chip is associated with a nonnegative loss defined by $L(s_{T,i})$ for some nonincreasing function $L$ mapping from the final position of the chip to $\mathbb{R}_+$. The goal of the player is to minimize the chips' average loss $ \frac{1}{N} \sum_{i=1}^N L(s_{T,i})$ after $T$ rounds. So intuitively, the player aims to ``push'' the chips to the right by assigning appropriate weights on them so that the adversary has to move them to the right by $\beta$ in a weighted average sense on each round. This game captures many learning problems. For instance, binary classification via boosting can be translated into a drifting game by treating each training example as a chip (see \cite{Schapire01} for details).
We regard a player's strategy ${\mathcal{D}}$ as a function mapping from the history of the adversary's decisions to a distribution that the player is going to play with, that is, ${\bf p}_t = {\mathcal{D}}({\bf z}_{1:t-1})$ where ${\bf z}_{1:t-1}$ stands for ${\bf z}_1, \ldots, {\bf z}_{t-1}$. The player's worst case loss using this algorithm is then denoted by $L_T({\mathcal{D}})$.
The minimax optimal loss of the game is computed by the following expression: $ \min_{{\mathcal{D}}}L_T({\mathcal{D}}) = \min_{{\bf p}_1\in \Delta_N}\max_{{\bf z}_1 \in {\mathcal{Z}}_{{\bf p}_1}} \cdots \min_{{\bf p}_T\in \Delta_N}\max_{{\bf z}_T \in {\mathcal{Z}}_{{\bf p}_T}} \frac{1}{N}\sum_{i=1}^N L(\sum_{t=1}^T z_{t,i}) ,$ where $\Delta_N$ is the $N$ dimensional simplex and ${\mathcal{Z}}_{\bf p} = B^N \cap \{{\bf z}: {\bf p}\cdot{\bf z} \geq \beta\}$ is assumed to be compact. A strategy ${\mathcal{D}}^*$ that realizes the minimum in $\min_{{\mathcal{D}}}L_T({\mathcal{D}})$ is called a minimax optimal strategy. A nearly optimal strategy and its analysis is originally given in \cite{Schapire01}, and a derivation by directly tackling the above minimax expression can be found in \citep[chap. 13]{SchapireFr12}.
Specifically, a sequence of potential functions of a chip's position is defined recursively as follows: \begin{equation}\label{equ:minimax_potentials}
\Phi_T(s) = L(s), \quad \Phi_{t-1}(s) = \min_{w \in \mathbb{R}_+}\max_{z\in B} (\Phi_t(s+z) + w(z - \beta)). \end{equation} Let $w_{t,i}$ be the weight that realizes the minimum in the definition of $\Phi_{t-1}(s_{t-1,i})$, that is, $w_{t,i} \in \arg\min_w\max_z(\Phi_t(s_{t-1,i}+z)+w(z-\beta))$. Then the player's strategy is to set $p_{t,i} \propto w_{t,i}$. The key property of this strategy is that it assures that the sum of the potentials over all the chips never increases, connecting the player's final loss with the potential at time $0$ as follows: \begin{equation}\label{equ:monotonic_potentials} \frac{1}{N}\sum_{i=1}^N L(s_{T,i}) \leq \frac{1}{N}\sum_{i=1}^N \Phi_T(s_{T,i}) \leq \frac{1}{N}\sum_{i=1}^N \Phi_{T-1}(s_{T-1,i}) \leq \cdots \leq \frac{1}{N}\sum_{i=1}^N \Phi_{0}(s_{0,i}) = \Phi_{0}(0) . \end{equation} It has been shown in \cite{Schapire01} that this upper bound on the loss is optimal in a very strong sense.
Moreover, in some cases the potential functions have nice closed forms and thus the algorithm can be efficiently implemented. For example, in the boosting setting, $B$ is simply $\{-1,+1\}$, and one can verify $ \Phi_t(s) = \frac{1+\beta}{2}\Phi_{t+1}(s+1) + \frac{1-\beta}{2}\Phi_{t+1}(s-1)$ and $ w_{t,i} = \tfrac{1}{2}\(\Phi_t(s_{t-1,i} - 1) - \Phi_t(s_{t-1,i} + 1)\)$. With the loss function $L(s)$ being ${\bm 1}\{s\leq 0\}$, these can be further simplified and eventually give exactly the boost-by-majority algorithm \cite{Freund95}.
\section{Online Learning as a Drifting Game} \label{sec:hedge} The connection between drifting games and some specific settings of online learning has been noticed before (\cite{Schapire01, MukherjeeSc10}). We aim to find deeper connections or even an equivalence between variants of drifting games and more general settings of online learning, and provide insights on designing learning algorithms through a minimax analysis. We start with a simple yet classic Hedge setting.
\subsection{Algorithmic Equivalence} In the Hedge setting \cite{FreundSc97}, a player tries to earn as much as possible (or lose as little as possible) by cleverly spreading a fixed amount of money to bet on a set of actions on each day. Formally, the game proceeds for $T$ rounds, and on each round $t = 1, \ldots, T$: the player chooses a distribution ${\bf p}_t$ over $N$ actions, then the adversary decides the actions' losses ${\bm \ell}_t$ (i.e. action $i$ incurs loss $\ell_{t,i}\in [0,1]$) which are revealed to the player. The player suffers a weighted average loss ${\bf p}_t\cdot{\bm \ell}_t$ at the end of this round. The goal of the player is to minimize his ``regret'', which is usually defined as the difference between his total loss and the loss of the best action. Here, we consider an even more general notion of regret studied in \cite{Kleinberg05, Kleinberg06, ChaudhuriFrHs09, ChernovVovk10}, which we call {\it $\epsilon$-regret}. Suppose the actions are ordered according to their total losses after $T$ rounds (i.e. $\sum_{t=1}^T \ell_{t,i}$) from smallest to largest, and let $i_\epsilon$ be the index of the action that is the $\lceil N\epsilon \rceil$-th element in the sorted list ($0 < \epsilon \leq 1$). Now, $\epsilon$-regret is defined as $ {\bf R}_{T}^\epsilon({\bf p}_{1:T}, {\bm \ell}_{1:T}) = \sum_{t=1}^T {\bf p}_t\cdot {\bm \ell}_t - \sum_{t=1}^T \ell_{t, i_\epsilon}. $ In other words, $\epsilon$-regret measures the difference between the player's loss and the loss of the $\lceil N\epsilon \rceil$-th best action (recovering the usual regret with $\epsilon \leq 1/N$), and sublinear $\epsilon$-regret implies that the player's loss is almost as good as all but the top $\epsilon$ fraction of actions. Similarly, ${\bf R}_{T}^\epsilon({\mathcal{H}})$ denotes the worst case $\epsilon$-regret for a specific algorithm ${\mathcal{H}}$.
For convenience, when $\epsilon \leq 0$ or $\epsilon > 1$, we define $\epsilon$-regret to be $\infty$ or $-\infty$ respectively.
Next we discuss how Hedge is highly related to drifting games. Consider a variant of drifting games where $B = [-1, 1], \beta=0$ and $L(s) = {\bm 1}\{s \leq -R\}$ for some constant $R$. Additionally, we impose an extra restriction on the adversary:
$|z_{t, i} - z_{t, j}| \leq 1$ for all $i$ and $j$. In other words, the difference between any two chips' movements is at most $1$. We denote this specific variant of drifting games by DGv1 (summarized in
Appendix \ref{apd:DGv}) and a corresponding algorithm by ${\mathcal{D}}_R$ to emphasize the dependence on $R$. The reductions in Algorithm \ref{alg:hedge2drift} and \ref{alg:drift2hedge} and Theorem \ref{thm:equivalence} show that DGv1 and the Hedge problem are algorithmically equivalent (note that both conversions are valid). The proof is straightforward and deferred to Appendix \ref{apd:equiv}. By Theorem \ref{thm:equivalence}, it is clear that the minimax optimal algorithm for one setting is also minimax optimal for the other under these conversions.
\begin{figure}\label{alg:hedge2drift}
\label{alg:drift2hedge}
\end{figure}
\begin{theorem}\label{thm:equivalence} DGv1 and the Hedge problem are algorithmically equivalent in the following sense: \\ (1) Algorithm \ref{alg:hedge2drift} produces a DGv1 algorithm ${\mathcal{D}}_R$ satisfying $L_T({\mathcal{D}}_R) \leq i/N$ where $i\in\{0,\ldots,N\}$ is such that ${\bf R}_T^{(i+1)/N}({\mathcal{H}}) < R \leq {\bf R}_T^{i/N}({\mathcal{H}})$.
(2) Algorithm \ref{alg:drift2hedge} produces a Hedge algorithm ${\mathcal{H}}$ with ${\bf R}_T^{\epsilon}({\mathcal{H}}) < R$ for any $R$ such that $L_T({\mathcal{D}}_R) < \epsilon$.
\end{theorem}
\subsection{Relaxations} From now on we only focus on the direction of converting a drifting game algorithm into a Hedge algorithm.
In order to derive a minimax Hedge algorithm, Theorem \ref{thm:equivalence} tells us it suffices to derive minimax DGv1 algorithms. Exact minimax analysis is usually difficult, and appropriate relaxations seem to be necessary. To make use of the existing analysis for standard drifting games, the first obvious relaxation is to drop the additional restriction
in DGv1, that is, $|z_{t,i}-z_{t,j}| \leq 1$ for all $i$ and $j$. Doing this will lead to the exact setting discussed in \cite{MukherjeeSc10} where a near optimal strategy is proposed using the recipe in Eq. \eqref{equ:minimax_potentials}. It turns out that this relaxation is reasonable and does not give too much more power to the adversary. To see this, first recall that results from \cite{MukherjeeSc10}, written in our notation, state that $ \min_{{\mathcal{D}}_R} L_T({\mathcal{D}}_R) \leq \frac{1}{2^T}\sum_{j=0}^{\frac{T-R}{2}} \binom{T+1}{j}, $ which, by Hoeffding's inequality, is upper bounded by $ 2\exp\(-\frac{(R+1)^2}{2(T+1)}\) $. Second, statement (2) in Theorem \ref{thm:equivalence} clearly remains valid if the input of Algorithm \ref{alg:drift2hedge} is a drifting game algorithm for this relaxed version of DGv1. Therefore, by setting $\epsilon > 2\exp\(-\frac{(R+1)^2}{2(T+1)}\)$ and solving for $R$, we have ${\bf R}_T^{\epsilon}({\mathcal{H}}) \leq O\(\sqrt{T\ln (\frac{1}{\epsilon}})\)$, which is the known optimal regret rate for the Hedge problem, showing that we lose little due to this relaxation.
However, the algorithm proposed in \cite{MukherjeeSc10} is not computationally efficient since the potential functions $\Phi_t(s)$ do not have closed forms. To get around this, we would want the minimax expression in Eq. \eqref{equ:minimax_potentials} to be easily solved, just like the case when $B=\{-1,1\}$. It turns out that convexity would allow us to treat $B=[-1,1]$ almost as $B=\{-1,1\}$. Specifically, if each $\Phi_t(s)$ is a convex function of $s$, then due to the fact that the maximum of a convex function is always realized at the boundary of a compact region, we have \begin{equation}\label{equ:convexity} \min_{w \in \mathbb{R}_+}\max_{z\in [-1,1]} \(\Phi_t(s+z) + wz\) = \min_{w \in \mathbb{R}_+}\max_{z\in \{-1,1\}} \(\Phi_t(s+z) + wz\) = \frac{\Phi_t(s-1) + \Phi_t(s+1)}{2} , \end{equation} with $w = (\Phi_t(s-1)-\Phi_t(s+1))/2$ realizing the minimum. Since the 0-1 loss function $L(s)$ is not convex, this motivates us to find a convex surrogate of $L(s)$. Fortunately, relaxing the equality constraints in Eq. \eqref{equ:minimax_potentials} does not affect the key property of Eq. \eqref{equ:monotonic_potentials} as we will show in the proof of Theorem \ref{thm:hedge_recipe}. ``Compiling out'' the input of Algorithm \ref{alg:drift2hedge}, we thus have our general recipe (Algorithm \ref{alg:hedge_recipe}) for designing Hedge algorithms with the following regret guarantee.
\begin{figure}\label{alg:hedge_recipe}
\end{figure}
\begin{theorem}\label{thm:hedge_recipe} For Algorithm \ref{alg:hedge_recipe}, if $R$ and $\epsilon$ are such that $\Phi_0(0) < \epsilon$ and $\Phi_T(s) \geq {\bm 1}\{s \leq -R\}$ for all $s\in\mathbb{R}$, then ${\bf R}_T^{\epsilon}({\mathcal{H}}) < R$. \end{theorem}
{\it Proof.} It suffices to show that Eq. \eqref{equ:monotonic_potentials} holds so that the theorem follows by a direct application of statement (2) of Theorem \ref{thm:equivalence}. Let $w_{t,i} = (\Phi_t(s_{t-1,i}-1)-\Phi_t(s_{t-1,i}+1))/2$. Then $\sum_i \Phi_t(s_{t,i}) \leq \sum_i \(\Phi_t(s_{t-1,i}+z_{t,i}) + w_{t,i}z_{t,i}\)$ since $p_{t,i} \propto w_{t,i}$ and ${\bf p}_t\cdot{\bf z}_t \geq 0$. On the other hand, by Eq. \eqref{equ:convexity}, we have $ \Phi_t(s_{t-1,i}+z_{t,i}) + w_{t,i}z_{t,i} \leq \min_{w\in \mathbb{R}_+}\max_{z\in [-1,1]} \(\Phi_t(s_{t-1,i}+z) + wz\) = \frac{1}{2}\(\Phi_t(s_{t-1,i}-1)+\Phi_t(s_{t-1,i}+1)\) $, which is at most $\Phi_{t-1}(s_{t-1,i})$ by Algorithm \ref{alg:hedge_recipe}. This shows $\sum_i \Phi_t(s_{t,i}) \leq \sum_i \Phi_{t-1}(s_{t-1,i})$ and Eq. \eqref{equ:monotonic_potentials} follows. \qed
Theorem 2 tells us that if solving $\Phi_0(0) < \epsilon$ for $R$ gives $R > \underline{R}$ for some value $\underline{R}$, then the regret of Algorithm \ref{alg:hedge_recipe} is less than any value that is greater than $\underline{R}$, meaning the regret is at most $\underline{R}$.
\subsection{Designing Potentials and Algorithms}\label{sec:3algs} Now we are ready to recover existing algorithms and develop new ones by choosing an appropriate potential $\Phi_T(s)$ as Algorithm \ref{alg:hedge_recipe} suggests. We will discuss three different algorithms below, and summarize these examples in Table \ref{tab:algs} (see Appendix \ref{apd:lemmas}).
\paragraph{Exponential Weights (EXP) Algorithm.}
Exponential loss is an obvious choice for $\Phi_T(s)$ as it has been widely used as the convex surrogate of the 0-1 loss function in the literature. It turns out that this will lead to the well-known exponential weights algorithm \cite{FreundSc97, FreundSc99}. Specifically, we pick $\Phi_T(s)$ to be $\exp\(-\eta(s+R)\)$ which exactly upper bounds ${\bm 1}\{s \leq -R\}$. To compute $\Phi_t(s)$ for $t \leq T$, we simply let $\Phi_{t}(s-1)+\Phi_{t}(s+1) \leq 2\Phi_{t-1}(s)$ hold with equality. Indeed, direct computations show that all $\Phi_t(s)$ share a similar form: $\Phi_t(s) = \(\frac{e^\eta+e^{-\eta}}{2}\)^{T-t} \cdot \exp\(-\eta(s+R)\).$ Therefore, according to Algorithm \ref{alg:hedge_recipe}, the player's strategy is to set
$$ p_{t,i} \propto \Phi_t(s_{t-1,i}-1) - \Phi_t(s_{t-1,i}+1) \propto \exp\(-\eta s_{t-1,i}\),$$ which is exactly the same as EXP (note that $R$ becomes irrelevant after normalization).
To derive regret bounds, it suffices to require $\Phi_0(0) < \epsilon,$ which is equivalent to $R > \frac{1}{\eta}\(\ln(\frac{1}{\epsilon}) + T\ln\frac{e^\eta+e^{-\eta}}{2}\).$ By Theorem \ref{thm:hedge_recipe} and Hoeffding's lemma (see \citep[Lemma A.1]{CesabianchiLu06}), we thus know $ {\bf R}_T^{\epsilon}({\mathcal{H}}) \leq \frac{1}{\eta}\ln\(\frac{1}{\epsilon}\) + \frac{T\eta}{2} = \sqrt{2T\ln\(\frac{1}{\epsilon}\)}$
where the last step is by optimally tuning $\eta$ to be $\sqrt{2(\ln\frac{1}{\epsilon})/T}$. Note that this algorithm is {\it not adaptive} in the sense that it requires knowledge of $T$ and $\epsilon$ to set the parameter $\eta$.
We have thus recovered the well-known EXP algorithm and given a new analysis using the drifting-games framework. More importantly, as in \cite{RakhlinShSr12}, this derivation may shed light on why this algorithm works and where it comes from, namely, a minimax analysis followed by a series of relaxations, starting from a reasonable surrogate of the 0-1 loss function.
\paragraph{2-norm Algorithm.}
We next move on to another simple convex surrogate: $ \Phi_T(s) = a[s]_-^2 \geq {\bm 1}\{s \leq -1/\sqrt{a}\}, $ where $a$ is some positive constant and $[s]_- = \min\{0, s\}$ represents a truncating operation. The following lemma shows that $\Phi_t(s)$ can also be simply described.
\begin{lemma}\label{lem:2-norm} If $a > 0$, then $\Phi_t(s) = a\([s]_-^2 + T- t\)$ satisfies $\Phi_{t}(s-1)+\Phi_{t}(s+1) \leq 2\Phi_{t-1}(s)$. \end{lemma}
Thus, Algorithm 3 can again be applied. The resulting algorithm is extremely concise: $$ p_{t,i} \propto \Phi_t(s_{t-1,i}-1) - \Phi_t(s_{t-1,i}+1) \propto [s_{t-1,i}-1]_-^2 - [s_{t-1,i}+1]_-^2.$$ We call this the ``2-norm'' algorithm since it resembles the $p$-norm algorithm in the literature when $p=2$ (see \cite{CesabianchiLu06}). The difference is that the $p$-norm algorithm sets the weights proportional to the derivative of potentials, instead of the difference of them as we are doing here. A somewhat surprising property of this algorithm is that it is totally adaptive and parameter-free (since $a$ disappears under normalization), a property that we usually do not expect to obtain from a minimax analysis. Direct application of Theorem \ref{thm:hedge_recipe} ($\Phi_0(0) = aT < \epsilon \Leftrightarrow 1/\sqrt{a} > \sqrt{T/\epsilon}$) shows that its regret achieves the optimal dependence on the horizon $T$.
\begin{corollary}\label{cor:2-norm} Algorithm \ref{alg:hedge_recipe} with potential $\Phi_t(s)$ defined in Lemma \ref{lem:2-norm} produces a Hedge algorithm ${\mathcal{H}}$ such that ${\bf R}_T^\epsilon({\mathcal{H}}) \leq \sqrt{T/\epsilon}$ simultaneously for all $T$ and $\epsilon$. \end{corollary}
\paragraph{{NormalHedge.DT}.}
The regret for the 2-norm algorithm does not have the optimal dependence on $\epsilon$. An obvious follow-up question would be whether it is possible to derive an adaptive algorithm that achieves the optimal rate $O(\sqrt{T\ln (1/\epsilon)})$ simultaneously for all $T$ and $\epsilon$ using our framework. An even deeper question is: instead of choosing convex surrogates in a seemingly arbitrary way, is there a more natural way to find the {\it right} choice of $\Phi_T(s)$?
To answer these questions, we recall that the reason why the 2-norm algorithm can get rid of the dependence on $\epsilon$ is that $\epsilon$ appears merely in the multiplicative constant $a$ that does not play a role after normalization. This motivates us to let $\Phi_T(s)$ in the form of $\epsilon F(s)$ for some $F(s)$. On the other hand, from Theorem \ref{thm:hedge_recipe},
we also want $\epsilon F(s)$ to upper bound the 0-1 loss function ${\bm 1}\{s \leq -\sqrt{dT\ln(1/\epsilon)} \}$ for some constant $d$. Taken together, this is telling us that the right choice of $F(s)$ should be of the form $\Theta\(\exp(s^2/T)\)$\footnote{ Similar potential was also proposed in recent work \cite{McmahanOr14, Orabona14} for a different setting. }. Of course we still need to refine it to satisfy the monotonicity and other properties. We define $\Phi_T(s)$ formally and more generally as: $$ \Phi_T(s) = a\(\exp\(\tfrac{[s]_-^2}{dT}\) - 1\) \geq {\bm 1}\left\{ s \leq -\sqrt{dT \ln \(\tfrac{1}{a} + 1\)} \right\}, $$ where $a$ and $d$ are some positive constants. This time it is more involved to figure out what other $\Phi_t(s)$ should be. The following lemma addresses this issue (proof deferred to Appendix \ref{apd:lemmas}).
\begin{lemma}\label{lem:anh}
If $ b_t = 1 - \frac{1}{2}\sum_{\tau = t+1}^T \(\exp\(\frac{4}{d\tau}\) - 1\), a > 0, d \geq 3 $ and $\Phi_t(s) = a\(\exp\(\frac{[s]_-^2}{dt}\) - b_t\)$ (define $\Phi_0(s) \equiv a(1-b_0)$), then we have $\Phi_{t}(s-1)+\Phi_{t}(s+1) \leq 2\Phi_{t-1}(s)$ for all $s \in \mathbb{R}$ and $t = 2, \ldots, T$. Moreover, Eq. \eqref{equ:monotonic_potentials} still holds. \end{lemma}
Note that even if $\Phi_{1}(s-1)+\Phi_{1}(s+1) \leq 2\Phi_{0}(s)$ is not valid in general, Lemma \ref{lem:anh} states that Eq. \eqref{equ:monotonic_potentials} still holds. Thus Algorithm \ref{alg:hedge_recipe} can indeed still be applied, leading to our new algorithm: $$ p_{t,i} \propto \Phi_t(s_{t-1,i}-1) - \Phi_t(s_{t-1,i}+1) \propto \exp\(\tfrac{[s_{t-1,i}-1]_-^2}{dt}\) - \exp\(\tfrac{[s_{t-1,i}+1]_-^2}{dt}\) .$$ Here, $d$ seems to be an extra parameter, but in fact, simply setting $d=3$ is good enough:
\begin{corollary}\label{cor:anh} Algorithm \ref{alg:hedge_recipe} with potential $\Phi_t(s)$ defined in Lemma \ref{lem:anh} and $d=3$ produces a Hedge algorithm ${\mathcal{H}}$ such that the following holds simultaneously for all $T$ and $\epsilon$: $${\bf R}_T^\epsilon({\mathcal{H}}) \leq \sqrt{3T\ln \(\tfrac{1}{2\epsilon} \(e^{4/3}-1\)\(\ln T + 1\)+1\)} = O\(\sqrt{T\ln\(1/\epsilon\) + T\ln\ln T}\). $$ \end{corollary}
We have thus proposed a parameter-free adaptive algorithm with optimal regret rate (ignoring the $\ln\ln T$ term) using our drifting-games framework. In fact, our algorithm bears a striking similarity to NormalHedge \cite{ChaudhuriFrHs09}, the first algorithm that has this kind of adaptivity. We thus name our algorithm {{NormalHedge.DT}}\footnote{``DT'' stands for discrete time.}. We include NormalHedge in Table \ref{tab:algs} for comparison. One can see that the main differences are: 1) On each round NormalHedge performs a numerical search to find out the right parameter used in the exponents; 2) NormalHedge uses the derivative of potentials as weights.
Compared to NormalHedge, the regret bound for {{NormalHedge.DT}} has no explicit dependence on $N$, but has a slightly worse dependence on $T$ (indeed $\ln\ln T$ is almost negligible). We emphasize other advantages of our algorithm over NormalHedge: 1) {{NormalHedge.DT}} is more computationally efficient especially when $N$ is very large, since it does not need a numerical search for each round; 2) our analysis is arguably simpler and more intuitive than the one in \cite{ChaudhuriFrHs09}; 3) as we will discuss in Section \ref{sec:gen}, {{NormalHedge.DT}} can be easily extended to deal with the more general online convex optimization problem where the number of actions is infinitely large, while it is not clear how to do that for NormalHedge by generalizing the analysis in \cite{ChaudhuriFrHs09}. Indeed, the extra dependence on the number of actions $N$ for the regret of NormalHedge makes this generalization even seem impossible. Finally, we will later see that {{NormalHedge.DT}} outperforms NormalHedge in experiments. Despite the differences, it is worth noting that both algorithms assign zero weight to some actions on each round, an appealing property when $N$ is huge. We will discuss more on this in Section \ref{sec:gen}.
\subsection{High Probability Bounds} We now consider a common variant of Hedge: on each round, instead of choosing a distribution ${\bf p}_t$, the player has to randomly pick a single action $i_t$, while the adversary decides the losses ${\bm \ell}_t$ at the same time (without seeing $i_t$). For now we only focus on the player's regret to the best action:
$ {\bf R}_T(i_{1:T}, {\bm \ell}_{1:T}) = \sum_{t=1}^T \ell_{t, i_t} - \min_i\sum_{t=1}^T \ell_{t, i}. $ Notice that the regret is now a random variable, and we are interested in a bound that holds with high probability. Using Azuma's inequality, standard analysis (see for instance \citep[Lemma 4.1]{CesabianchiLu06}) shows that the player can simply draw $i_t$ according to ${\bf p}_t = {\mathcal{H}}({\bm \ell}_{1:t-1})$, the output of a standard Hedge algorithm, and suffers regret at most ${\bf R}_T({\mathcal{H}}) + \sqrt{T\ln( 1/\delta)}$ with probability $1-\delta$. Below we recover similar results as a simple side product of our drifting-games analysis {\it without} resorting to concentration results, such as Azuma's inequality.
For this, we only need to modify Algorithm \ref{alg:hedge_recipe} by setting $z_{t,i} = \ell_{t,i} - \ell_{t, i_t}$.
The restriction ${\bf p}_t\cdot {\bf z}_t \geq 0$ is then relaxed to hold in expectation. Moreover, it is clear that Eq. \eqref{equ:monotonic_potentials} also still holds in expectation.
On the other hand, by definition and the union bound, one can show that $\sum_i \mathbb{E}[L(s_{T,i})] = \sum_i {\text {Pr}}\left[ s_{T,i} \leq -R \right] \geq {\text {Pr}}\left[ {\bf R}_T(i_{1:T}, {\bm \ell}_{1:T}) \geq R \right]$.
So setting $\Phi_0(0) = \delta$ shows that the regret is smaller than $R$ with probability $1-\delta$.
Therefore, for example, if EXP is used, then the regret would be at most $\sqrt{2T\ln(N/\delta)}$ with probability $1-\delta$, giving basically the same bound as the standard analysis. One draw back is that EXP would need $\delta$ as a parameter. However, this can again be addressed by {{NormalHedge.DT}} for the exact same reason that {{NormalHedge.DT}} is independent of $\epsilon$. We have thus derived high probability bounds without using any concentration inequalities.
\section{Generalizations and Applications} \label{sec:gen}
\textbf{Multi-armed Bandit (MAB) Problem:}
The only difference between Hedge (randomized version) and the non-stochastic MAB problem \cite{AuerCeFrSc02} is that on each round, after picking $i_t$, the player only sees the loss for this single action $\ell_{t,i_t}$ instead of the whole vector ${\bm \ell}_t$. The goal is still to compete with the best action. A common technique used in the bandit setting is to build an unbiased estimator $\hat{\bm \ell}_t$ for the losses, which in this case could be $\hat\ell_{t,i} = {\bm 1}\{i = i_t\}\cdot\ell_{t,i_t}/p_{t, i_t}$. Then algorithms such as EXP can be used by replacing ${\bm \ell}_t$ with $\hat{\bm \ell}_t$, leading to the EXP3 algorithm \cite{AuerCeFrSc02} with regret $O(\sqrt{TN\ln N})$.
One might expect that Algorithm \ref{alg:hedge_recipe} would also work well by replacing ${\bm \ell}_t$ with $\hat{\bm \ell}_t$. However, doing so breaks an important property of the movements $z_{t,i}$: boundedness. Indeed, Eq. \eqref{equ:convexity} no longer makes sense if $z$ could be infinitely large, even if in expectation it is still in $[-1,1]$ (note that $z_{t,i}$ is now a random variable). It turns out that we can address this issue by imposing a variance constraint on $z_{t,i}$. Formally, we consider a variant of drifting games where on each round, the adversary picks a random movement $z_{t,i}$ for each chip such that: $z_{t,i} \geq -1, \mathbb{E}_t[z_{t,i}] \leq 1, \mathbb{E}_t[z_{t,i}^2] \leq 1/p_{t,i}$ and $\mathbb{E}_t[{\bf p}_t \cdot {\bf z}_t] \geq 0$. We call this variant DGv2 and summarize it in Appendix \ref{apd:DGv}. The standard minimax analysis and the derivation of potential functions need to be modified in a certain way for DGv2, as stated in Theorem \ref{thm:DGv2} (Appendix \ref{apd:bandit}). Using the analysis for DGv2, we propose a general recipe for designing MAB algorithms in a similar way as for Hedge and also recover EXP3 (see Algorithm \ref{alg:bandit_recipe} and Theorem \ref{thm:bandit_recipe} in Appendix \ref{apd:bandit}). Unfortunately so far we do not know other appropriate potentials due to some technical difficulties. We conjecture, however, that there is a potential function that could recover the poly-INF algorithm \cite{AudibertBu10, AudibertBuLu14} or give its variants that achieve the optimal regret $O(\sqrt{TN})$.
\textbf{Online Convex Optimization:} We next consider a general online convex optimization setting \cite{Zinkevich03}. Let $S \subset \mathbb{R}^d$ be a compact convex set, and ${\mathcal{F}}$ be a set of convex functions with range $[0,1]$ on $S$. On each round $t$, the learner chooses a point ${\bf x}_t \in S$, and the adversary chooses a loss function $f_t \in {\mathcal{F}}$ (knowing ${\bf x}_t$). The learner then suffers loss $f_t({\bf x}_t)$. The regret after $T$ rounds is $ {\bf R}_T({\bf x}_{1:T}, f_{1:T}) = \sum_{t=1}^T f_t({\bf x}_t) - \min_{{\bf x} \in S} \sum_{t=1}^T f_t({\bf x})$.
There are two general approaches to OCO: one builds on convex optimization theory \cite{Shalevshwartz11}, and the other generalizes EXP to a continuous space \cite{Cover91, NarayananRa10}. We will see how the drifting-games framework can recover the latter method and also leads to new ones.
To do so, we introduce a continuous variant of drifting games (DGv3, see Appendix \ref{apd:DGv}). There are now infinitely many chips, one for each point in $S$. On round $t$, the player needs to choose a distribution over the chips, that is, a probability density function $p_t({\bf x})$ on $S$. Then the adversary decides the movements for each chip, that is, a function $z_t({\bf x})$ with range $[-1,1]$ on $S$ (not necessarily convex or continuous), subject to a constraint $\mathbb{E}_{{\bf x} \sim p_t} [z_t({\bf x})] \geq 0$. At the end, each point ${\bf x}$ is associated with a loss $ L({\bf x}) = {\bm 1}\{\sum_t z_t({\bf x}) \leq -R \}$, and the player aims to minimize the total loss $\int_{{\bf x}\in S} L({\bf x}) d{\bf x}$.
OCO can be converted into DGv3 by setting $z_t({\bf x}) = f_t({\bf x}) - f_t({\bf x}_t)$ and predicting ${\bf x}_t = \mathbb{E}_{{\bf x} \sim p_t} [{\bf x}] \in S$. The constraint $\mathbb{E}_{{\bf x} \sim p_t} [z_t({\bf x})] \geq 0$ holds by the convexity of $f_t$. Moreover, it turns out that the minimax analysis and potentials for DGv1 can readily be used here,
and the notion of $\epsilon$-regret, now generalized to the OCO setting, measures the difference of the player's loss and the loss of a best fixed point in a subset of $S$ that excludes the top $\epsilon$ fraction of points.
With different potentials, we obtain versions of each of the three algorithms of Section \ref{sec:hedge} generalized to this setting, with the same $\epsilon$-regret bounds as before. Again, two of these methods are adaptive and parameter-free. To derive bounds for the usual regret, at first glance it seems that we have to set $\epsilon$ to be close to zero, leading to a meaningless bound. Nevertheless, this is addressed by Theorem \ref{thm:OCO_recipe} using similar techniques in \cite{HazanAgKa07}, giving the usual $O(\sqrt{dT\ln T})$ regret bound. All details can be found in Appendix \ref{apd:OCO}.
\textbf{Applications to Boosting:}
There is a deep and well-known connection between Hedge and boosting \cite{FreundSc97, SchapireFr12}. In principle, every Hedge algorithm can be converted into a boosting algorithm; for instance, this is how AdaBoost was derived from EXP. In the same way, {{NormalHedge.DT}} can be converted into a new boosting algorithm that we call {{NH-Boost.DT}}. See Appendix \ref{apd:ANB} for details and further background on boosting. The main idea is to treat each training example as an ``action'', and to rely on the Hedge algorithm to compute distributions over these examples which are used to train the weak hypotheses. Typically, it is assumed that each of these has ``edge'' $\gamma$, meaning its accuracy on the training distribution is at least $1/2 + \gamma$. The final hypothesis is a simple majority vote of the weak hypotheses. To understand the prediction accuracy of a boosting algorithm, we often study the training error rate and also the distribution of margins, a well-established measure of confidence (see Appendix \ref{apd:ANB} for formal definitions). Thanks to the adaptivity of {{NormalHedge.DT}}, we can derive bounds on both the training error and the distribution of margins after any number of rounds:
\begin{theorem}\label{thm:ANB} After $T$ rounds, the training error of {{NH-Boost.DT}} is of order $\tilde{O}(\exp(-\frac{1}{3}T\gamma^2))$, and the fraction of training examples with margin at most $\theta (\leq 2\gamma)$ is of order $\tilde{O}(\exp(-\frac{1}{3}T(\theta-2\gamma)^2))$. \end{theorem}
Thus, the training error decreases at roughly the same rate as AdaBoost. In addition, this theorem implies that the fraction of examples with margin smaller than $2\gamma$ eventually goes to zero as $T$ gets large, which means {{NH-Boost.DT}} converges to the optimal margin $2\gamma$; this is known not to be true for AdaBoost (see \cite{SchapireFr12}). Also, like AdaBoost, {{NH-Boost.DT}} is an adaptive boosting algorithm that does not require $\gamma$ or $T$ as a parameter. However, unlike AdaBoost, {{NH-Boost.DT}} has the striking property that it completely ignores many examples on each round (by assigning zero weight), which is very helpful for the weak learning algorithm in terms of computational efficiency. To test this, we conducted experiments to compare the efficiency of AdaBoost, ``NH-Boost'' (an analogous boosting algorithm derived from NormalHedge) and {{NH-Boost.DT}}. All details are in Appendix \ref{apd:experiments}. Here we only briefly summarize the results. While the three algorithms have similar performance in terms of training and test error, {{NH-Boost.DT}} is always the fastest one in terms of running time for the same number of rounds. Moreover, the average faction of examples with zero weight is significantly higher for {{NH-Boost.DT}} than for NH-Boost (see Table \ref{tab:results}). On one hand, this explains why {{NH-Boost.DT}} is faster (besides the reason that it does not require a numerical step). On the other hand, this also implies that {{NH-Boost.DT}} tends to achieve larger margins, since zero weight is assigned to examples with large margin. This is also confirmed by our experiments.
{\textbf{Acknowledgements.} Support for this research was provided by NSF Grant \#1016029. The authors thank Yoav Freund for helpful discussions and the anonymous reviewers for their comments.}
{\small
}
\appendix
\section{Summary of Drifting Game Variants} \label{apd:DGv} We study three different variants of drifting games throughout the paper, which corresponds to the Hedge setting, the multi-armed bandit problem and online convex optimization respectively. The protocols of these variants are summarized below.
\begin{framed} \centerline{\textbf{DGv1}} Given: a loss function $L(s) = {\bm 1}\{s \leq -R\}$. \\
For $t = 1, \ldots, T$: \begin{enumerate} \item The player chooses a distribution ${\bf p}_t$ over $N$ chips. \item The adversary decides the movement of each chip $z_{t,i} \in [-1,1]$
subject to ${\bf p}_t \cdot {\bf z}_t \geq 0$ and $|z_{t,i}-z_{t,j}|\leq 1$ for all $i$ and $j$.
\end{enumerate} The player suffers loss $\sum_{i=1}^N L(\sum_{t=1}^T z_{t,i})$. \end{framed}
\begin{framed} \centerline{\textbf{DGv2}} Given: a loss function $L(s) = {\bm 1}\{s \leq -R\}$. \\
For $t = 1, \ldots, T$: \begin{enumerate} \item The player chooses a distribution ${\bf p}_t$ over $N$ chips. \item The adversary randomly decides the movement of each chip $z_{t,i} \geq -1$ subject to $\mathbb{E}_t[z_{t,i}] \leq 1, \mathbb{E}_t[z_{t,i}^2] \leq 1/p_{t,i}$ and $\mathbb{E}_t[{\bf p}_t \cdot {\bf z}_t] \geq 0$.
\end{enumerate} The player suffers loss $\sum_{i=1}^N L(\sum_{t=1}^T z_{t,i})$. \end{framed}
\begin{framed} \centerline{\textbf{DGv3}} Given: a compact convex set $S$, a loss function $L(s) = {\bm 1}\{s \leq -R\}$. \\
For $t = 1, \ldots, T$: \begin{enumerate} \item The player chooses a density function $p_t({\bf x})$ on $S$. \item The adversary decides a function $z_t({\bf x}): S \rightarrow [-1,1]$ subject to $\mathbb{E}_{{\bf x} \sim p_t} [z_t({\bf x})] \geq 0$.
\end{enumerate} The player suffers loss $\int_{{\bf x}\in S} L(\sum_{t=1}^T z_{t}({\bf x})) d{\bf x}$. \end{framed}
\section{Proof of Theorem \ref{thm:equivalence}} \label{apd:equiv}
\begin{proof} We first show that both conversions are valid. In Algorithm \ref{alg:hedge2drift}, it is clear that $\ell_{t,i} \geq 0$. Also, $\ell_{t,i} \leq 1$ is guaranteed due to the extra restriction of DGv1. For Algorithm \ref{alg:drift2hedge}, $z_{t,i}$ lies in $B=[-1,1]$ since $\ell_{t,i} \in [0,1]$, and direct computation shows ${\bf p}_t \cdot {\bf z}_t = 0 \geq \beta (=0)$
and $|z_{t,i} - z_{t,j}| = |\ell_{t,i} - \ell_{t,j}| \leq 1$ for all $i$ and $j$.
(1) For any choices of ${\bf z}_{t}$, we have $$ \sum_{i=1}^N L(s_{T,i}) = \sum_{i=1}^N L\(\sum_{t=1}^N z_{t,i}\) \leq \sum_{i=1}^N L\(\sum_{t=1}^N \(z_{t,i} - {\bf p}_t\cdot{\bf z}_t\)\), $$ where the inequality holds since ${\bf p}_t\cdot{\bf z}_t$ is required to be nonnegative and $L$ is a nonincreasing function. By Algorithm \ref{alg:hedge2drift}, $z_{t,i} - {\bf p}_t\cdot{\bf z}_t$ is equal to $\ell_{t,i} - {\bf p}_t\cdot{\bm \ell}_t$, leading to $$ \sum_{i=1}^N L(s_{T,i}) \leq \sum_{i=1}^N L\(\sum_{t=1}^N \(\ell_{t,i} - {\bf p}_t\cdot{\bm \ell}_t\)\) = \sum_{i=1}^N {\bm 1}\left\{R \leq \sum_{t=1}^N \({\bf p}_t\cdot{\bm \ell}_t - \ell_{t,i}\)\right\}. $$ Since ${\bf R}_T^{(i+1)/N}({\mathcal{H}}) < R \leq {\bf R}_T^{i/N}({\mathcal{H}})$, we must have $\sum_{t=1}^N \({\bf p}_t\cdot{\bm \ell}_t - \ell_{t,j}\) < R$ except for the best $i$ actions, which means $\sum_{i=1}^N L(s_{T,i}) \leq i$. This holds for any choices of ${\bf z}_t$, so $L_T({\mathcal{D}}_R) \leq i/N$.
(2) By Algorithm \ref{alg:drift2hedge} and the condition $L_T(D_R) < \epsilon$ , we have $$ \frac{1}{N}\sum_{i=1}^N {\bm 1}\left\{R \leq \sum_{t=1}^N \({\bf p}_t\cdot{\bm \ell}_t - \ell_{t,i}\)\right\} = \frac{1}{N}\sum_{i=1}^N L(s_{T,i}) \leq L_T(D_R) < \epsilon,
$$ which means there are at most $\lceil N\epsilon \rceil - 1$ actions satisfying $R \leq \sum_{t=1}^N \({\bf p}_t\cdot{\bm \ell}_t - \ell_{t,i}\)$, and thus $\sum_{t=1}^N \({\bf p}_t\cdot{\bm \ell}_t - \ell_{t,i_\epsilon}\) < R$. Since this holds for any choices of ${\bm \ell}_t$, we have ${\bf R}_T^\epsilon({\mathcal{H}}) < R$. \end{proof}
\section{Summary of Hedge Algorithms and Proofs of Lemma \ref{lem:2-norm}, Lemma \ref{lem:anh} and Corollary \ref{cor:anh}} \label{apd:lemmas}
\begin{table}[h] \caption{Different algorithms derived from Algorithm \ref{alg:hedge_recipe}, and comparisons with NormalHedge} \label{tab:algs} \begin{center}
\begin{tabular}{|c|c|c|c||c|} \hline & EXP & 2-norm & {NormalHedge.DT} & NormalHedge \\ \hline $\Phi_T(s)$ & $e^{-\eta(s+R)}$ & $a[s]_-^2$ & $a\(e^{[s]_-^2/3T} - 1\) $ & N/A \\ \hline $p_{t,i} \propto$ & $e^{-\eta s_{t-1,i}}$ & \pbox{10em}{$[s_{t-1,i}-1]_-^2 $ \\ $-[s_{t-1,i}+1]_-^2$} & \pbox{10em}{$e^{[s_{t-1,i}-1]_-^2/3t}$ \\ $- e^{[s_{t-1,i}+1]_-^2/3t}$} & \pbox{20em}{$-[s_{t-1,i}]_-e^{[s_{t-1,i}]_-^2/c}$ ($c$ is\\
s.t. $\sum_{i} e^{[s_{t-1,i}]_-^2/c} = Ne) $} \\ \hline ${\bf R}_T^\epsilon({\mathcal{H}})$ & $O\(\sqrt{T\ln \frac{1}{\epsilon}}\)$ & $O\(\sqrt{T/\epsilon}\)$ & $O\(\sqrt{T\ln\frac{\ln T}{\epsilon}}\)$ & $O\(\sqrt{T\ln\frac{1}{\epsilon}}+\ln^2 N\)$ \\ \hline Adaptive? & No & Yes & Yes & Yes \\ \hline \end{tabular} \end{center} \end{table}
\begin{proof}[Proof of Lemma \ref{lem:2-norm}] It suffices to show $[s-1]_-^2 + [s+1]_-^2 \leq 2[s]_-^2 + 2$. When $s \geq 0$, ${\text {LHS}} = [s-1]_-^2 \leq 1 < 2 = {\text {RHS}}$. When $s < 0$, ${\text {LHS}} \leq (s-1)^2+(s+1)^2 = 2s^2 + 2 = {\text {RHS}}$. \end{proof}
\begin{proof}[Proof of Lemma \ref{lem:anh}] Let $F(s) = \exp\(\frac{[s-1]_-^2}{dt}\) + \exp\(\frac{[s+1]_-^2}{dt}\) - 2\exp\(\frac{[s]_-^2}{d(t-1)}\)$. It suffices to show $$ F(s) \leq 2(b_{t} - b_{t-1}) = \exp\(\frac{4}{dt}\) - 1 ,$$ which is clearly true for the following 3 cases: $$ F(s) = \begin{cases} 0 &\quad\text{if $s > 1$;} \\ \exp\(\frac{(s-1)^2}{dt}\) -1 < \exp\(\frac{1}{dt}\) - 1 &\quad\text{if $0 < s \leq 1$;} \\ \exp\(\frac{(s-1)^2}{dt}\) + 1 - 2\exp\(\frac{s^2}{d(t-1)}\) < \exp\(\frac{4}{dt}\) - 1 &\quad\text{if $-1 < s \leq 0$.} \end{cases}$$ For the last case $s \leq -1$, if we can show that $F(s)$ is increasing in this region, then the lemma follows. Below, we show this by proving $F'(s)$ is nonnegative when $s \leq -1$.
Let $h(s, c) = \frac{\partial\exp\(s^2/c\)}{\partial s} = \frac{2s}{c}\exp\(\frac{s^2}{c}\)$. $F'(s)$ can now be written as $$ F'(s) = h(s-1, c) + h(s+1,c) - 2h(s, c) + 2(h(s,c) - h(s, c')) ,$$ where $c = dt$ and $c' = d(t-1)$. Next we apply (one-dimensional) Taylor expansion to $h(s-1,c)$ and $h(s+1,c)$ around $s$, and $h(s,c')$ around $c$, leading to \begin{align*} F'(s) &= \sum_{k=1}^\infty \frac{(-1)^k}{k!} \frac{\partial^k h(s,c)}{\partial s^k} + \sum_{k=1}^\infty \frac{1}{k!} \frac{\partial^k h(s,c)}{\partial s^k} - 2\sum_{k=1}^\infty \frac{(c'-c)^k}{k!} \frac{\partial^k h(s,c)}{\partial c^k} \\ &= 2 \sum_{k=1}^\infty \(\frac{1}{(2k)!} \frac{\partial^{2k} h(s,c)}{\partial s^{2k}} - \frac{(-d)^k}{k!} \frac{\partial^k h(s,c)}{\partial c^k} \). \end{align*}
Direct computation (see Lemma \ref{lem:partialD} below) shows that $\frac{\partial^k h(s,c)}{\partial c^k}$ and $\frac{\partial^{2k} h(s,c)}{\partial s^{2k}}$ share exact same forms only with different constants: \begin{equation}\label{equ:partialD} \begin{split}
\frac{\partial^{k} h(s,c)}{\partial c^{k}} &= \exp\(\frac{s^2}{c}\) \sum_{j=0}^k (-1)^k\alpha_{k,j} \cdot \frac{s^{2j+1}}{c^{k+j+1}} ,
\\ \frac{\partial^{2k} h(s,c)}{\partial s^{2k}} &= \exp\(\frac{s^2}{c}\) \sum_{j=0}^k \beta_{k,j} \cdot \frac{s^{2j+1}}{c^{k+j+1}}, \end{split} \end{equation} where $\alpha_{k,j}$ and $\beta_{k,j}$ are recursively defined as: \begin{equation}\label{equ:alpha_beta} \begin{split} \alpha_{k+1,j} &= \alpha_{k,j-1} + (k+j+1)\alpha_{k,j}, \\ \beta_{k+1,j} &= 4\beta_{k,j-1} + (8j+6)\beta_{k,j} + (2j+3)(2j+2)\beta_{k,j+1}, \end{split} \end{equation} with initial values $\alpha_{0,0} = \beta_{0,0} = 2$ (when $j \not\in \{0,\ldots,k\}$, $\alpha_{k,j}$ and $\beta_{k,j}$ are all defined to be $0$). Therefore, $F'(s)$ can be further simplified as $$ F'(s) = 2\exp\(\frac{s^2}{c}\) \sum_{k=1}^\infty \sum_{j=0}^k \frac{s^{2j+1}}{c^{k+j+1}} \(\frac{\beta_{k,j}}{(2k)!} - \frac{d^k\alpha_{k,j}}{k!} \). $$ Since $s$ is negative, it suffices to show that $\frac{\beta_{k,j}}{(2k)!} \leq \frac{d^k\alpha_{k,j}}{k!}$ holds for all $k$ and $j$, which turns out to be true as long as $d \geq 3$, as shown by induction in the technical lemma \ref{lem:alpha_beta} below. To sum up, $\Phi_{t}(s-1)+\Phi_{t}(s+1) \leq 2\Phi_{t-1}(s)$ for all $s \in \mathbb{R}$ and $t = 2, \ldots, T$.
Finally, we need to show that Eq. (2) still holds. The inequality we just proved above implies $ \sum_{i} \Phi_{t}(s_{t,i}) \leq \sum_i \Phi_{t-1}(s_{t-1,i})$ for $t = 2, \ldots, T$, as shown in Theorem \ref{thm:hedge_recipe}. Thus the only thing we need to show here is the case when $t=1$. Note that $\Phi_{1}(s-1)+\Phi_{1}(s+1) \leq 2\Phi_0(s)$ does not hold for all $s$ apparently. However, in order to prove $\sum_{i} \Phi_{1}(s_{1,i}) \leq \sum_i \Phi_0(s_{0,i})$, we in fact only need a much weaker statement: $\Phi_{1}(-1)+\Phi_{1}(1) \leq 2\Phi_0(0)$ since $s_{0,i} \equiv 0$. This is equivalent to $ \exp\(1/d\) - 1 \leq \exp\(4/d\) - 1$, which is true trivially. \end{proof}
\begin{lemma}\label{lem:partialD} Let $h(s,c) = \frac{2s}{c}\exp\(\frac{s^2}{c}\)$. The partial derivatives of $h(s,c)$ satisfy Eq. \eqref{equ:partialD} and \eqref{equ:alpha_beta}. \end{lemma} \begin{proof} The base case holds trivially. Assume Eq. \eqref{equ:partialD} holds for a fixed $k$. Then we have \begin{align*}
\frac{\partial^{k+1} h(s,c)}{\partial c^{k+1}} &= \exp\(\frac{s^2}{c}\) \sum_{j=0}^k (-1)^k\alpha_{k,j} \cdot \(-\frac{s^2}{c^2} \frac{s^{2j+1}}{c^{k+j+1}} -(k+j+1)\frac{s^{2j+1}}{c^{k+j+2} }\) \\ &= \exp\(\frac{s^2}{c}\) \sum_{j=0}^k (-1)^{k+1} \alpha_{k,j} \cdot \( \frac{s^{2(j+1)+1}}{c^{(k+1)+(j+1)+1}} + (k+j+1)\frac{s^{2j+1}}{c^{(k+1)+j+1} }\) \\ &= \exp\(\frac{s^2}{c}\) \sum_{j=0}^{k+1} (-1)^{k+1} \(\alpha_{k,j-1} + (k+j+1)\alpha_{k,j} \) \cdot \frac{s^{2j+1}}{c^{(k+1)+j+1} } \\ &= \exp\(\frac{s^2}{c}\) \sum_{j=0}^{k+1} (-1)^{k+1} \alpha_{k+1,j} \cdot \frac{s^{2j+1}}{c^{(k+1)+j+1} } , \end{align*} and \begin{align*} \frac{\partial^{2(k+1)} h(s,c)}{\partial s^{2(k+1)}} &= \left. \partial \left[ \exp\(\frac{s^2}{c}\) \sum_{j=0}^k \beta_{k,j} \cdot \(\frac{2s^{2j+2}}{c^{k+j+2}} + (2j+1)\frac{s^{2j}}{c^{k+j+1}}\) \right] \middle/ \partial s \right. \\ &= \exp\(\frac{s^2}{c}\) \sum_{j=0}^k \beta_{k,j} \cdot \(\frac{4s^{2j+3}}{c^{k+j+3}} + (8j+6)\frac{s^{2j+1}}{c^{k+j+2}} +
(2j+1)2j \frac{s^{2j-1}}{c^{k+j+1}} \) \\
&= \exp\(\frac{s^2}{c}\) \sum_{j=0}^{k+1} \(
4\beta_{k,j-1} + (8j+6)\beta_{k, j} + (2j+3)(2j+2)\beta_{k, j+1}\)
\cdot \frac{s^{2j+1}}{c^{k+j+2}} \\ &= \exp\(\frac{s^2}{c}\) \sum_{j=0}^{k+1} \beta_{k+1,j} \cdot \frac{s^{2j+1}}{c^{k+j+2}}, \end{align*} concluding the proof. \end{proof}
\begin{lemma}\label{lem:alpha_beta} Let $\alpha_{k,j}$ and $\beta_{k,j}$ be defined as in Eq. \eqref{equ:alpha_beta}. Then $\frac{\beta_{k,j}}{(2k)!} \leq \frac{d^k\alpha_{k,j}}{k!}$ holds for all $k \geq 0$ and $j \in \{0, \ldots, k\}$ when $d \geq 3$. \end{lemma} \begin{proof} We prove the lemma by induction on $k$. The base case $k = 0$ is trivial. Assume $\frac{\beta_{k,j}}{(2k)!} \leq \frac{d^k\alpha_{k,j}}{k!}$ holds for a fixed $k$ and all $j \in \{0, \ldots, k\}$, then we have $\forall j$, \begin{align*} \frac{\beta_{k+1, j}}{(2k+2)!} &= \frac{4\beta_{k,j-1} + (8j+6)\beta_{k, j} + (2j+3)(2j+2)\beta_{k, j+1}}{(2k+2)!} \\ &\leq \frac{d^k\(4\alpha_{k,j-1} + (8j+6)\alpha_{k, j} + (2j+3)(2j+2)\alpha_{k, j+1}\)}{(2k+2)(2k+1)k!}. \end{align*} We need to show that the above expression is at most $d^{k+1}\alpha_{k+1,j}/(k+1)!$, which, after arrangements, is equivalent to $2\alpha_{k,j-1} + (4j+3)\alpha_{k, j} + (2j+3)(j+1)\alpha_{k, j+1} \leq d(2k+1)\alpha_{k+1,j}.$ We will prove this by another induction on $k$. Then the lemma follows.
The base case ($k=0$) is simplified to $6 \leq 2d$, which is true by our assumption $d \geq 3$. Assume the inequality holds for a fixed $k$, then by the definition of $\alpha_{k,j}$, one has \begin{align*} &2\alpha_{k+1,j-1} + (4j+3)\alpha_{k+1, j} + (2j+3)(j+1)\alpha_{k+1, j+1} \\ =\;& \(2\alpha_{k,j-2} + (4j+3)\alpha_{k, j-1} + (2j+3)(j+1)\alpha_{k, j} \) + \\ \quad &\(2(k+j)\alpha_{k,j-1} + (4j+3)(k+j+1)\alpha_{k,j} + (2j+3)(j+1)(k+j+2)\alpha_{k, j+1} \) \\ =\;& \(2\alpha_{k,j-2} + (4j-1)\alpha_{k, j-1} + (2j+1)j\alpha_{k, j} \) + \\ \quad & (k+j+2)\(2\alpha_{k,j-1} + (4j+3)\alpha_{k,j} + (2j+3)(j+1)\alpha_{k, j+1} \) \\ \leq\;& d(2k+1) (\alpha_{k+1, j-1} + (k+j+2)\alpha_{k+1, j}) \tag{by induction}\\ =\;& d(2k+1)\alpha_{k+2,j} \\ \leq\;& d(2k+3)\alpha_{k+2,j}, \end{align*} completing the induction. \end{proof}
\begin{proof}[Proof of Corollary \ref{cor:anh}] Recall that $\Phi_T(s) \geq {\bm 1}\left\{ s \leq -\sqrt{dT \ln \(\tfrac{1}{a} + 1\)} \right\}$. So by setting $\Phi_0(0) = a(1-b_0) < \epsilon$ and applying Theorem \ref{thm:hedge_recipe}, we arrive at $${\bf R}_T^\epsilon({\mathcal{H}}) \leq \sqrt{dT \ln \(\frac{1-b_0}{\epsilon} + 1\)}. $$ It suffices to upper bound $1-b_0$, which, by definition, is $ \frac{1}{2}\sum_{t = 1}^T \(\exp\(\frac{4}{dt}\) - 1\)$. Since $e^x -1 \leq \frac{e^c-1}{c} x$ for any $x \in [0, c]$, we have $$ \sum_{t = 1}^T \(\exp\(\frac{4}{dt}\) - 1\) \leq (e^{4/d}-1)\sum_{t = 1}^T \frac{1}{t} \leq (e^{4/d}-1)(\ln T + 1). $$ Plugging $d=3$ gives the corollary. \end{proof}
\section{A General MAB Algorithm and Regret Bounds} \label{apd:bandit} \SetAlCapSkip{.2em} \IncMargin{.5em} \begin{algorithm}[H] \caption{A General MAB Algorithm} \label{alg:bandit_recipe}
\SetKwInOut{Input}{Input} \SetKw{DownTo}{down to} \Input{A convex, nonincreasing, nonnegative function $\Phi_T(s) \in \mathbb{C}^2$, with nonincreasing second derivative.
}
\For{ $t = T$ \DownTo $1$} { Find a convex function $\Phi_{t-1}(s)$ s.t. the conditions of Theorem \ref{thm:DGv2} hold. } Set: ${\bf s}_0 = {\bf 0}$. \\ \For{ $t = 1$ \KwTo $T$} { Set: $p_{t,i} \propto \Phi_t(s_{t-1,i}-1) - \Phi_t(s_{t-1,i}+1)$. \\ Draw $i_t \sim {\bf p}_t$ and receive loss $\ell_{t, i_t}$. \\% from the adversary. \\ Set: $z_{t,i} = {\bm 1}\{i = i_t\} \cdot\ell_{t, i_t}/p_{t,i_t} -\ell_{t, i_t}, \;\forall i$. \\ Set: ${\bf s}_t = {\bf s}_{t-1} + {\bf z}_t$. \\ } \end{algorithm} \DecMargin{.5em}
\begin{theorem}\label{thm:DGv2} Suppose $\Phi_t(s)$ is convex, twice continuously differentiable (i.e. $\Phi_t(s) \in \mathbb{C}^2$), have nonincreasing second derivative, and satisfies: \begin{equation}\label{equ:bandit_potential} \(\tfrac{1}{2}+N\alpha_t\)\Phi_{t}(s-1)+\(\tfrac{1}{2}-N\alpha_t\)\Phi_{t}(s+1) \leq \Phi_{t-1}(s), \forall s\in\mathbb{R} \end{equation} where $\alpha_t = \frac{1}{2}\max_s \frac{\Phi_t''(s-1)}{\Phi_t(s-1)-\Phi_t(s+1)}$. If the player's strategy is such that $p_{t,i} \propto \Phi_t(s_{t-1,i}-1) - \Phi_t(s_{t-1,i}+1)$, then Eq. \eqref{equ:monotonic_potentials} holds in expectation. \end{theorem}
\begin{proof}[Proof of Theorem \ref{thm:DGv2}]
As discussed before, the main difficulty here is the unboundedness of $z_{t,i}$. However, the expectation of $z_{t,i}$ is still in $[-1,1]$ as in DGv1. To exploit this fact, we apply Taylor's theorem to $\Phi_t(s_{t-1,i}+z_{t,i})$ to the second order term: \begin{align*} \Phi_t(s_{t,i}) &= \Phi_t(s_{t-1,i}+z_{t,i}) \\ &= \Phi_t(s_{t-1,i}) + \Phi_t'(s_{t-1,i})z_{t,i} + \tfrac{1}{2}\Phi_t''(\xi_{t,i})z_{t,i}^2 \\ &\leq \Phi_t(s_{t-1,i}) + \Phi_t'(s_{t-1,i})z_{t,i} + \tfrac{1}{2}\Phi_t''(s_{t-1,i}-1)z_{t,i}^2, \end{align*} where $\xi_{t,i}$ is between $s_{t-1,i}+z_{t,i}$ and $s_{t-1,i}$, and the inequality holds because $\Phi_t''(s)$ is nonincreasing and $z_{t,i} \geq -1$ by assumption. Now taking expectation on both sides with respect to the randomness of $z_{t,i}$, using the convexity of $\Phi_t(s)$, and plugging the assumption $\mathbb{E}_t[z_{t,i}^2] \leq 1/p_{t,i}$ give: \begin{align*} \mathbb{E}_t[\Phi_t(s_{t,i})] &\leq \Phi_t(s_{t-1,i}) + \Phi_t'(s_{t-1,i})\mathbb{E}_t[z_{t,i}] + \tfrac{1}{2}\Phi_t''(s_{t-1,i}-1)\mathbb{E}_t[z_{t,i}^2] \\ &\leq \Phi_t\(s_{t-1,i} + \mathbb{E}_t[z_{t,i}]\) + \tfrac{1}{2} \Phi_t''(s_{t-1,i}-1)/p_{t,i}. \end{align*} Let $w_{t,i} = \frac{1}{2}\(\Phi_t(s_{t-1,i}-1) - \Phi_t(s_{t-1,i}+1)\)$. Further plugging ${\bf p}_{t,i} \propto w_{t,i}$ and summing over all $i$, we arrive at \begin{align*} \sum_{i=1}^N \mathbb{E}_t[\Phi_t(s_{t,i})] &\leq \sum_{i=1}^N \( \Phi_t\(s_{t-1,i} + \mathbb{E}_t[z_{t,i}]\) + \frac{\Phi_t''(s_{t-1,i}-1)}{2w_{t,i}} \cdot \sum_{i=1}^N w_{t,i} \) \\ &\leq \sum_{i=1}^N \( \Phi_t\(s_{t-1,i} + \mathbb{E}_t[z_{t,i}]\) + 2\alpha_t \sum_{i=1}^N w_{t,i} \) \tag{\text{by the defintion of $\alpha_t$}} \\ &= \sum_{i=1}^N \( \Phi_t\(s_{t-1,i} + \mathbb{E}_t[z_{t,i}]\) + 2N\alpha_t w_{t,i} \) . \end{align*} Since $\mathbb{E}_t[{\bf p}_t \cdot {\bf z}_t] \geq 0$ implies $\sum_{i=1}^N w_{t,i} \mathbb{E}_t[z_{t,i}] \geq 0$, we thus have \begin{align*} \sum_{i=1}^N \mathbb{E}_t[\Phi_t(s_{t,i})] &\leq \sum_{i=1}^N \( \Phi_t\(s_{t-1,i} + \mathbb{E}_t[z_{t,i}]\) + w_{t,i}\mathbb{E}_t[z_{t,i}] + 2N\alpha_t w_{t,i} \) \\ &\leq \sum_{i=1}^N \( \max_{z\in [-1,+1]} \(\Phi_t\(s_{t-1,i} + z\) + w_{t,i} z \) + 2N\alpha_t w_{t,i} \) \\ &= \sum_{i=1}^N \( \max_{z\in \{-1,+1\}} \(\Phi_t\(s_{t-1,i} + z\) + w_{t,i} z \) + 2N\alpha_t w_{t,i} \) \tag{\text{by the convexity of $\Phi_t(s)$}}\\ &= \sum_{i=1}^N \( \(\tfrac{1}{2}+N\alpha_t\)\Phi_{t}(s_{t-1,i}-1)+ \(\tfrac{1}{2}-N\alpha_t\)\Phi_{t}(s_{t-1,i}+1)\) \\ &\leq \sum_{i=1}^N \Phi_{t-1}(s_{t-1,i}). \tag{by assumption} \end{align*} The theorem follows by taking expectation on both sides with respect to the past (i.e. the randomness of ${\bf z}_{1}, \ldots, {\bf z}_{t-1}$). \end{proof}
\begin{theorem}\label{thm:bandit_recipe} For Algorithm \ref{alg:bandit_recipe}, if $R$ and $\epsilon$ are such that $\Phi_0(0) < \epsilon$ and $\Phi_T(s) \geq {\bm 1}\{s \leq -R\}$ for all $s\in\mathbb{R}$, then $\mathbb{E}[\sum_{t=1}^T \ell_{t, i_t} - \sum_{t=1}^T \ell_{t, i_\epsilon}] < R$ for any non-oblivious adversary. Moreover, using $\Phi_T(s) = \exp(-\eta(s+R))$ (and let Eq. \eqref{equ:bandit_potential} hold with equality) gives exactly the EXP3 algorithm with regret $O(\sqrt{TN\ln(1/\epsilon)})$. \end{theorem}
\begin{proof}[Proof of Theorem \ref{thm:bandit_recipe}] We first show that Algorithm \ref{alg:bandit_recipe} converts the multi-armed bandit problem to a valid instance of DGv2. It suffices to prove that $z_{t,i} = {\bm 1}\{i = i_t\} \cdot\ell_{t, i_t}/p_{t,i_t} -\ell_{t, i_t}$ satisfies all conditions defined in DGv2, as shown below ($z_{t,i} \geq -1$ is trivial): $$ \mathbb{E}_t[z_{t,i}] = \ell_{t, i} - {\bf p}_t \cdot {\bm \ell}_t \leq 1 ,$$ $$ \mathbb{E}_t[z_{t,i}^2] = p_{t,i} \(\frac{\ell_{t,i}}{p_{t,i}} - \ell_{t,i}\)^2 + \sum_{j\neq i} p_{t,j} \ell_{t, j}^2 \leq p_{t,i} \(\frac{1}{p_{t,i}} - 1\)^2 + \sum_{j\neq i} p_{t,j} = \frac{1}{p_{t,i}} - 1 \leq \frac{1}{p_{t,i}}, $$ $$ \mathbb{E}_t[{\bf p}_t \cdot {\bf z}_t] = \mathbb{E}_t\left[\ell_{t,i_t} - \sum_{j=1}^N p_{t,j}\ell_{t,i_t} \right] = 0.$$ Therefore, we can apply Theorem \ref{thm:DGv2} directly, arriving at: $$ \frac{1}{N}\sum_{i=1}^N \mathbb{E}[\Phi_T(s_{T,i})] \leq \cdots
\leq \frac{1}{N}\sum_{i=1}^N \mathbb{E}[\Phi_{0}(s_{0,i})] = \Phi_{0}(0) \leq \epsilon. $$ On the other hand, by applying Jensen' inequality, we have $$\mathbb{E}[\Phi_T(s_{T,i})] \geq \Phi_T(\mathbb{E}[s_{T,i}]) \geq {\bm 1}\{\mathbb{E}[s_{T,i}] \leq -R \}. $$ Note that $\mathbb{E}[s_{T,i}]$ is equal to $\mathbb{E}\left[\sum_{t=1}^T\(\ell_{t, i} - \ell_{t,i_t}\)\right]$. We thus know $$ \frac{1}{N}\sum_{i=1}^N {\bm 1}\left\{ \mathbb{E}\left[\sum_{t=1}^T\(\ell_{t, i} - \ell_{t,i_t}\)\right] \leq -R \right\} < \epsilon ,$$ which implies $\mathbb{E}\left[\sum_{t=1}^T \ell_{t, i_t} - \sum_{t=1}^T \ell_{t, i_\epsilon}\right] < R$ for any non-oblivious adversary for the exact same argument used in the proof of Theorem \ref{thm:hedge_recipe}.
Finally, we show how to recover EXP3 using Algorithm \ref{alg:bandit_recipe} with input $\Phi_T(s) = \exp(-\eta(s+R))$. To compute $\Phi_t(s)$ for $t < T$, we simply use Eq. \eqref{equ:bandit_potential} with equality. One can verify using induction that $$ \Phi_t(s) = \exp\(-\eta(s+R)\)\(\frac{e^\eta+e^{-\eta}+Ne^\eta\eta^2}{2}\)^{T-t}, $$ $$ \alpha_t = \frac{1}{2}\max_s \frac{\eta^2\Phi_t(s-1)}{\Phi_t(s-1)-\Phi_t(s+1)} = \frac{e^\eta \eta^2}{2(e^\eta - e^{-\eta})} ,$$ $$ \Phi_t'''(s) = -\eta^3 \Phi_t(s) \leq 0 .$$ The player's strategy is thus ${\bf p}_{t,i} \propto \exp(-\eta\sum_{\tau=1}^{t-1} \hat\ell_{\tau,i})$ (recall $\hat\ell_{t,i} = {\bm 1}\{i = i_t\}\cdot\ell_{t,i_t}/p_{t, i_t}$ is the estimated loss), which is exactly the same as EXP3 (in fact a simplified version of the original EXP3, see for example \cite{Shalevshwartz11}). Moreover, the regret can be computed by setting $\Phi_0(0) = \epsilon$, leading to \begin{align*} R &= \frac{1}{\eta}\ln\(\frac{1}{\epsilon}\) + \frac{T}{\eta}\ln \(\frac{e^\eta+e^{-\eta}}{2}+\frac{1}{2}Ne^\eta\eta^2\) \\ &\leq \frac{1}{\eta}\ln\(\frac{1}{\epsilon}\) + \frac{T}{\eta}\ln \(e^{\eta^2/2}+\frac{1}{2}Ne^\eta\eta^2\) \tag{\text{by Hoeffding's Lemma}}\\ &\leq \frac{1}{\eta}\ln\(\frac{1}{\epsilon}\) + \frac{T}{\eta} \(\frac{\eta^2}{2} + \frac{1}{2}Ne^{\eta-\frac{\eta^2}{2}}\eta^2\) \tag{$\ln(1+x) \leq x$} \end{align*} If $\eta \leq 1$ so that $e^{\eta-\eta^2/2} \leq \sqrt{e}$, then we have $ R \leq \frac{1}{\eta}\ln(\frac{1}{\epsilon}) + T\eta \(\frac{1}{2} + \frac{N\sqrt{e}}{2} \)$, which is $\sqrt{2T(1+N\sqrt{e})\ln(1/\epsilon)}$ after optimally choosing $\eta$ ($\eta \leq 1$ will be satisfied when $T$ is large enough). \end{proof}
\section{A General OCO Algorithm and Regret Bounds} \label{apd:OCO} \SetAlCapSkip{.2em} \IncMargin{.5em} \begin{algorithm}[H] \caption{A General OCO Algorithm} \label{alg:OCO_recipe}
\SetKwInOut{Input}{Input} \SetKw{DownTo}{down to} \Input{A convex, nonincreasing, nonnegative function $\Phi_T(s)$
}
\For{ $t = T$ \DownTo $1$} { Find a convex function $\Phi_{t-1}(s)$ s.t. $\forall s,$ $\Phi_{t}(s-1)+\Phi_{t}(s+1) \leq 2\Phi_{t-1}(s).$ } Set: $s_0(x) \equiv 0$. \\ \For{ $t = 1$ \KwTo $T$} { Predict ${\bf x}_t = \mathbb{E}_{{\bf x} \sim p_t} [{\bf x}]$ where $p_t$ is such that $p_t({\bf x}) \propto \Phi_t(s_{t-1}({\bf x})-1) - \Phi_t(s_{t-1}({\bf x})+1)$. \\ Receive loss function $f_t$ from the adversary. \\ Set: $z_t({\bf x}) = f_t({\bf x}) - f_t({\bf x}_t)$. \\ Set: $s_t({\bf x}) = s_{t-1}({\bf x}) + z_t({\bf x})$. \\ } \end{algorithm} \DecMargin{.5em}
\textbf{Definition of $\epsilon$-regret in the OCO setting}: Let $S_\epsilon \subset S$ be such that the ratio of its volume and the one of $S$ is $\epsilon$ and also $ \sum_{t=1}^T f_t({\bf x}') \leq \sum_{t=1}^T f_t({\bf x}) $ for all ${\bf x}' \in S_\epsilon$ and ${\bf x} \in S\backslash S_\epsilon$ (it is clear that such set exists). Then $\epsilon$-regret is defined as ${\bf R}_T^\epsilon({\bf x}_{1:T}, f_{1:T}) = \sum_{t=1}^T f_t({\bf x}_t) - \inf_{{\bf x}\in S\backslash S_\epsilon} \sum_{t=1}^T f_t({\bf x})$.
\begin{theorem}\label{thm:OCO_recipe}
For Algorithm \ref{alg:OCO_recipe}, if $R$ is such that $\Phi_T(s) \geq {\bm 1}\{s \leq -R\}$ and $\Phi_0(0) < \epsilon$, then we have ${\bf R}_T^\epsilon({\bf x}_{1:T}, f_{1:T}) < R$ and ${\bf R}_T({\bf x}_{1:T}, f_{1:T}) < R + T\epsilon^{1/d}$. Specifically, if $R = O(\sqrt{T\ln(1/\epsilon)})$, then setting $\epsilon = T^{-d}$ gives ${\bf R}_T({\bf x}_{1:T}, f_{1:T}) = O(\sqrt{dT\ln T})$. \end{theorem}
\begin{proof}[Proof of Theorem \ref{thm:OCO_recipe}] Let $w_t({\bf x}) = \frac{1}{2}\(\Phi_t(s_{t-1}({\bf x})-1) - \Phi_t(s_{t-1}({\bf x})+1)\)$. Similarly to the Hedge setting, the ``sum'' of potentials never increases: $$ \int_{{\bf x} \in S} \Phi_{t}(s_{t}({\bf x})) d{\bf x} \leq \int_{{\bf x} \in S} \(\Phi_{t}(s_{t-1}({\bf x}) + z_t({\bf x})) + w_t({\bf x})z_t({\bf x})\) d{\bf x} \leq \int_{{\bf x} \in S} \Phi_{t-1}(s_{t-1}({\bf x})) d{\bf x} .$$ Here, the first inequality is due to $\mathbb{E}_{{\bf x} \sim p_t} [z_t({\bf x})] \geq 0$, and the second inequality holds for the exact same reason as in the case for Hedge. Therefore, we have $$ \int_{{\bf x} \in S} {\bm 1}\{s_T({\bf x}) \leq -R\} d{\bf x} \leq \int_{{\bf x} \in S} \Phi_{T}(s_{T}({\bf x})) d{\bf x} \leq \cdots \leq \int_{{\bf x} \in S} \Phi_{0}(0) d{\bf x} < \epsilon V, $$ where $V$ is the volume of $S$. Recall the construction of $S_\epsilon$. There must exist a point ${\bf x}' \in S_\epsilon$ such that $s_T({\bf x}') > -R$, otherwise $\int_{{\bf x}} {\bm 1}\{s_T({\bf x}) \leq -R\} d{\bf x}$ would be at least $\epsilon V$. Unfolding $s_T({\bf x}')$, we arrive at $ \sum_t f_t({\bf x}_t) - \sum_t f_t({\bf x}') < R $. Using the fact $\sum_t f_t({\bf x}') \leq \inf_{{\bf x}\in S\backslash S_\epsilon} \sum_t f_t({\bf x})$ gives the bound for $\epsilon$-regret.
Next consider a shrunk version of $S$: $S_\epsilon' = \{(1-\epsilon^{\frac{1}{d}}){\bf x}^* + \epsilon^{\frac{1}{d}}{\bf x} : {\bf x}\in S\} $ where ${\bf x}^* \in\arg\min_{\bf x} \sum_t f_t({\bf x})$. Then $\int_{{\bf x} \in S} {\bm 1}\{s_T({\bf x}) \leq -R\} d{\bf x}$ is at least $$
\int_{{\bf x} \in S_\epsilon'} {\bm 1}\{s_T({\bf x}) \leq -R\} d{\bf x} = \epsilon \int_{{\bf x} \in S} {\bm 1}\left\{s_T\((1-\epsilon^{\frac{1}{d}}){\bf x}^* + \epsilon^{\frac{1}{d}}{\bf x}\) \leq -R \right\} d{\bf x}, $$ which, by the convexity and the boundedness of $f_t({\bf x})$, is at least \begin{align*} &\epsilon \int_{{\bf x} \in S} {\bm 1}\left\{\sum_{t=1}^T\( (1-\epsilon^{\frac{1}{d}}) f_t({\bf x}^*) + \epsilon^{\frac{1}{d}} f_t({\bf x}) - f_t({\bf x}_t) \)\leq -R \right\} d{\bf x} \\ \geq&\; \epsilon \int_{{\bf x} \in S} {\bm 1}\left\{\sum_{t=1}^T\( f_t({\bf x}^*) - f_t({\bf x}_t)\) \leq -R-T\epsilon^{\frac{1}{d}} \right\} d{\bf x} \\ =&\; \epsilon V \cdot {\bm 1}\left\{\sum_{t=1}^T\( f_t({\bf x}^*) - f_t({\bf x}_t)\) \leq -R-T\epsilon^{\frac{1}{d}} \right\} . \end{align*} Following the previous discussion, the expression in the last line above is strictly less than $\epsilon V \cdot$, which means that the value of the indicator function has to be 0, namely, ${\bf R}_T({\bf x}_{1:T}, f_{1:T}) < R + T\epsilon^{1/d}$. \end{proof}
\section{{{NH-Boost.DT}}, NH-Boost and Proof of Theorem \ref{thm:ANB}} \label{apd:ANB}
\SetAlCapSkip{.5em} \IncMargin{.5em} \begin{algorithm}[H] \caption{{{NH-Boost.DT}}} \label{alg:ANB}
\SetKwInOut{Input}{Input} \SetKwInOut{Output}{Output} \Input{Training examples $({\bf x}_i, y_i) \in \mathbb{R}^d \times \{-1,+1\}, i=1,\ldots,N.$} \Input{A weak learning algorithm.} \Input{Number of rounds $T$.} \Output{A Hypothesis $H({\bf x}) : \mathbb{R}^d \rightarrow \{-1,+1\}$.}
Set: ${\bf s}_0 = {\bf 0}$. \\ \For{ $t = 1$ \KwTo $T$} { Set: $p_{t,i} \propto \exp\([s_{t-1,i}-1]_-^2/3t\) - \exp\([s_{t-1,i}+1]_-^2/3t\), \;\forall i$. \\ Invoke the weak learning algorithm to get $h_t$ with edge $ \gamma_t = \frac{1}{2}\sum_i p_{t,i}y_i h_t({\bf x}_i) $. \\ Set: $s_{t,i} = s_{t-1, i} + \frac{1}{2} y_i h_t({\bf x}_i) - \gamma_t, \;\forall i $. \\ } Set: $H({\bf x}) = {\text {sign}}(\sum_{t=1}^T h_t({\bf x}))$. \end{algorithm} \DecMargin{.5em}
\SetAlCapSkip{.5em} \IncMargin{.5em} \begin{algorithm}[H] \caption{NH-Boost} \label{alg:NB}
\SetKwInOut{Input}{Input} \SetKwInOut{Output}{Output} \Input{Training examples $({\bf x}_i, y_i) \in \mathbb{R}^d \times \{-1,+1\}, i=1,\ldots,N.$} \Input{A weak learning algorithm.} \Input{Number of rounds $T$.} \Output{A Hypothesis $H({\bf x}) : \mathbb{R}^d \rightarrow \{-1,+1\}$.}
Set: ${\bf s}_0 = {\bf 0}$. \\ \For{ $t = 1$ \KwTo $T$} { \eIf{$t = 1$} {
Set: ${\bf p}_1$ to be a uniform distribution. \\ } {Find: $c$ such that $\sum_{i=1}^N \exp\([s_{t-1,i}]_-^2/c\)= Ne$. \\
Set: $p_{t,i} \propto -[s_{t-1,i}]_-\exp\([s_{t-1,i}]_-^2/c\), \;\forall i$. \\
} Invoke the weak learning algorithm to get $h_t$ with edge $ \gamma_t = \frac{1}{2}\sum_i p_{t,i}y_i h_t({\bf x}_i) $. \\ Set: $s_{t,i} = s_{t-1, i} + \frac{1}{2} y_i h_t({\bf x}_i) - \gamma_t, \;\forall i $. \\ } Set: $H({\bf x}) = {\text {sign}}(\sum_{t=1}^T h_t({\bf x}))$. \end{algorithm} \DecMargin{.5em}
In the boosting setting for binary classification, we are given a set of training examples $({\bf x}_i, y_i)_{i = 1, \ldots, N}$ where ${\bf x}_i \in \mathbb{R}^d$ is an example and $y_i \in \{-1,+1\}$ is its label. A boosting algorithm proceeds for $T$ rounds. On each round, a distribution ${\bf p}_t$ over the examples is computed and fed into a weak learning algorithm which returns a ``weak'' hypothesis $h_t: \mathbb{R}^d \rightarrow \{-1,+1\}$ with a guaranteed small edge, that is, $\gamma_t = \frac{1}{2}\sum_i p_{t,i}y_i h_t({\bf x}_i) \geq \gamma > 0$. At the end, a linear combination of all $h_t$ is computed as the final ``strong'' hypothesis which is expected to have low training error and potentially low generalization error.
The conversion of a Hedge algorithm into a boosting algorithm is to treat each example as an ``action'' and set $\ell_{t, i} = {\bm 1}\{h_t({\bf x}_i) = y_i\}$ so that the booster tends to increase weights for those ``hard'' examples. The final hypothesis is a simple majority vote of all $h_t$, that is, $H({\bf x}) = {\text {sign}}(\sum_t h_t({\bf x}))$ where ${\text {sign}}(x)$ is the sign function that outputs $1$ if $x$ is positive, and $-1$ otherwise. The {\it margin} of example ${\bf x}_i$ is defined as $\frac{1}{T}\sum_{t=1}^T y_i h_t({\bf x}_i)$, that is, the difference between the fractions of correct hypotheses and incorrect hypotheses on this example. The boosting algorithms derived from {{NormalHedge.DT}} and NormalHedge in this way are given in Algorithm \ref{alg:ANB} and \ref{alg:NB}.
\begin{proof}[Proof the Theorem \ref{thm:ANB}] Let $({\tilde{\x}}_i, {\tilde{y}}_i)_{i = 1, \ldots, N}$ be a permutation of the training examples such that their margins are sorted from smallest to largest: $ \sum_{t} {\tilde{y}}_1 h_t({\tilde{\x}}_1) \leq \cdots \leq \sum_{t} {\tilde{y}}_N h_t({\tilde{\x}}_N)$, which also implies $ \sum_{t} {\bm 1}\{h_t({\tilde{\x}}_1) = {\tilde{y}}_1\} \leq \cdots \leq \sum_{t} {\bm 1}\{h_t({\tilde{\x}}_N) = {\tilde{y}}_N \}$. Recall that {{NormalHedge.DT}} is essentially playing a Hedge game using {{NormalHedge.DT}} with loss $\ell_{t, i} = {\bm 1}\{h_t({\bf x}_i) = y_i\}$. Therefore, the $\epsilon$-regret bound for the Hedge setting together with the assumption on the weak learning algorithm implies: $\forall j \in \{1,\ldots,N\},$ \begin{equation}\label{equ:hedge_and_boosting} \frac{1}{2} + \gamma \leq \frac{1}{T}\sum_{t=1}^T \sum_{i=1}^N {\bf p}_{t, i}{\bm 1}\{h_t({\bf x}_i) = y_i \} \leq \frac{1}{T}\sum_{t=1}^T {\bm 1}\{h_t({\tilde{\x}}_j) = {\tilde{y}}_j \} + \frac{{\bf R}_T^{j/N}}{T}, \end{equation} where ${\bf R}_T^{j/N} = \tilde{O}(\sqrt{3T\ln (N/j)}) $ is the $j/N$-regret bound for {{NormalHedge.DT}}. So if $j$ is such that $\gamma > {\bf R}_T^{j/N} / T$, we have $\frac{1}{T}\sum_{t=1}^T {\bm 1}\{h_t({\tilde{\x}}_j) = {\tilde{y}}_j \}> \frac{1}{2}$, which is saying that example $({\tilde{\x}}_j, {\tilde{y}}_j)$ will eventually be classified correctly by $H({\bf x})$ due to the fact that $H({\bf x})$ is taking a majority vote of all $h_t$. This is in fact true for all examples $({\tilde{\x}}_i, {\tilde{y}}_i)$ such that $i \geq j$ and thus the training error rate will be at most $(j-1)/N$, which is of order $ \tilde{O}(\exp(-\frac{1}{3}T\gamma^2)) $.
For the margin bound, by plugging ${\bm 1}\{h_t({\tilde{\x}}_j) = {\tilde{y}}_j \} = ({\tilde{y}}_j h_t({\tilde{\x}}_j)+1)/2$, we rewrite Eq. \eqref{equ:hedge_and_boosting} as: $$ 2\(\gamma - \frac{{\bf R}_T^{j/N}}{T}\) \leq \frac{1}{T}\sum_{t=1}^T {\tilde{y}}_j h_t({\tilde{\x}}_j). $$ Therefore, if $j$ is such that $\theta < 2(\gamma - {\bf R}_T^{j/N}/T)$, then the fraction of examples with margin at most $\theta$ is again at most $(j-1)/N$, which is of order $\tilde{O}(\exp(-\frac{1}{3}T(\theta-2\gamma)^2))$. \end{proof}
\section{Experiments in a Boosting Setting} \label{apd:experiments} We conducted experiments to compare the performance of three boosting algorithms for binary classification: AdaBoost \cite{FreundSc97}, NH-Boost (Algorithm \ref{alg:NB}) and {{NH-Boost.DT}} (Algorithm \ref{alg:ANB}), using a set of benchmark data available from the UCI repository\footnote{\url{http://archive.ics.uci.edu/ml/}} and LIBSVM datasets\footnote{\url{http://www.csie.ntu.edu.tw/~cjlin/libsvmtools/datasets/}}. Some datasets are preprocessed according to \cite{ReyzinSc06}. The number of features, training examples and test examples can be found in Table \ref{tab:data}.
All features are binary. The weak learning algorithm is a simple (exhaustive) decision stump (see for instance \cite{SchapireFr12}). On each round, the weak learning algorithm enumerates all features, and for each feature computes the weighted error of the corresponding stump on the weighted training examples. Therefore, if the number of examples with zero weight is relatively large, then the weak learning algorithm would be faster in computing the weighted error and thus faster in finding the best feature.
All boosting algorithms are run for two hundred rounds. The results are summarized in Table \ref{tab:results}, with bold entries being the best ones among the three (AB, NB and NBDT stand for AdaBoost, NH-Boost and {{NH-Boost.DT}} respectively). As we can see, in terms of training error and test error, all three algorithms have similar performance. However, our {{NH-Boost.DT}} algorithm is always the fastest one. The average fraction of examples with zero weights for {{NH-Boost.DT}} is significantly higher than the one for NH-Boost (note that AdaBoost does not assign zero weight at all). We plot the change of this fraction over rounds in Figure \ref{fig:zero} (using three datasets). As both algorithms proceed, they tend to ignore more and more examples on each round, but {{NH-Boost.DT}} consistently ignores more examples than NH-Boost.
Since $s_{t,i}$ is positively correlated to the margin of example $i$ ($\frac{1}{t}\sum_{\tau=1}^t y_i h_\tau({\bf x}_i)$) and large $s_{t,i}$ leads to zero weight, the above phenomenon in fact implies that the examples' margins should be larger for {{NH-Boost.DT}} than for NH-Boost. This is confirmed by Figure \ref{fig:margins}, where we plot the final cumulative margins on three datasets (i.e. each point represents the fraction of examples with at most some fixed margin). One can see that the lines for {{NH-Boost.DT}} are below the ones for NH-Boost (and even AdaBoost) for most time, meaning that {{NH-Boost.DT}} achieves larger margins in general. This could explain {{NH-Boost.DT}}'s better test error on some datasets.
\begin{table}[h] \caption{Description of datasets} \label{tab:data} \begin{center}
\begin{tabular}{|c|c|c|c|} \hline Data & {\#}feature & {\#}training & {\#}test \\ \hline a9a & 123 & 32,561 & 16,281 \\ \hline census & 131 & 1,000 & 1,000 \\ \hline ocr49 & 403 & 1,000 & 1,000 \\ \hline splice & 240 & 500 & 500 \\ \hline w8a & 300 & 49,749 & 14,951\\ \hline \end{tabular} \end{center}
\end{table}
\begin{table}[h] \caption{Experiment results} \label{tab:results} \begin{center}
\begin{tabular}{|c|c|c|c||c|c||c|c|c||c|c|c|} \hline
& \multicolumn{3}{|c||}{Time (s)} & \multicolumn{2}{|c||}{Zeros (\%)} &
\multicolumn{3}{|c||}{Training Error (\%)} & \multicolumn{3}{|c|}{Test Error (\%)} \\ \hline Data & AB & NB & NBDT & NB & NBDT & AB & NB & NBDT & AB & NB & NBDT \\ \hline a9a & 57.5 & 72.5 & \textbf{46.2} & 1.1 & \textbf{22.1} & \textbf{15.4} & 15.8 & 15.5 & \textbf{15.0} & 15.6 & 15.2 \\ \hline census & 1.7 & 2.2 & \textbf{1.4} & 2.2 & \textbf{19.2} & 15.6 & 17.0 & \textbf{15.4} & 18.7 & 18.6 & \textbf{18.3} \\ \hline ocr49 & 5.1 & 4.7 & \textbf{3.0} & 17.1 & \textbf{42.0} & \textbf{1.7} & \textbf{1.7} & 2.4 & \textbf{5.5} & 5.9 & 5.8 \\ \hline splice & 1.6 & 1.5 & \textbf{0.9} & 22.2 & \textbf{45.1} & \textbf{0.0} & \textbf{0.0} & 0.4 & 9.4 & 8.6 & \textbf{8.2} \\ \hline w8a & 237.6 & 244.7 & \textbf{170.7} & 3.0 & \textbf{29.3} & 2.6 & \textbf{2.2} & 2.4 & 2.7 & \textbf{2.3} & 2.6 \\ \hline \end{tabular} \end{center} \end{table}
\begin{figure}
\caption{census}
\caption{splice}
\caption{w8a}
\caption{Comparison of fraction of zero weights}
\label{fig:zero}
\end{figure}
\begin{figure}
\caption{census}
\caption{splice}
\caption{w8a}
\caption{Comparison of cumulative margins}
\label{fig:margins}
\end{figure}
\end{document} | arXiv |
Tina Pizzardo
Battistina Pizzardo, known as Tina (5 February 1903 Turin – 15 February 1989 Turin), was an Italian mathematician, and an anti-fascist.
Tina Pizzardo
Born5 February 1903
Turin
Died17 February 1989 (aged 86)
Turin
OccupationResearcher
Life
She graduated from the University of Turin in 1925. In 1926, she became a member of the "Academia pro interlingua".
She was in Rome in March 1926 to participate in the qualification competition for teaching in secondary schools. On this occasion he met Altiero Spinelli and other anti-fascists. In July she joined the Communist Party and in October she began teaching mathematics and physics at the Liceo classico Carducci-Ricasoli in Grosseto. Through letters, the police traced her, and arrested her in September for "subversive activity" and sentenced to one year in prison and three years of probation.[1][2]
She was transferred to the prison in Turin, then to that of Ancona and finally to the women's prison in Rome, where she organized protests with other inmates. She was released from prison on 13 September 1928. She lost her job and was unable to continue teaching in state schools; she had to live precariously giving private mathematics lessons. In Turin she maintained relations with the anti-fascists: her regular friends were Mario Carrara, his wife Paola Lombroso, Giuseppe Levi, Adriano Olivetti, and Barbara Allason. In this period she was attracted by three men: by Altiero Spinelli, Henek Rieser, and by Cesare Pavese.[3][4]
On 15 May 1935, she was again arrested by the police. The raid involved the editorial staff of the magazine «Cultura» and Pavese, Bruno Maffi, Carlo Levi , Franco Antonicelli and others ended up in prison. Tina was released at the end of June: "in the opinion of the police, a poor teacher who lives in private to the high bourgeoisie and all famous intellectuals." Her marriage to Henek Rieser took place on the following 19 April.[1]
Tina Pizzardo saw Altiero Spinelli again at the fall of fascism. She joined the European Federalist Movement he founded in 1943. She was a candidate for the House in the 1948 general elections and with the Action Party. In 1962, she wrote her memoirs about herself, which were circulated in typescript, and then were published posthumously in 1996, under the title Senza pensarci due volte (Without thinking twice). In her memoirs, Tina Pizzardo herself described herself "as a free and uninhibited woman, full of life and sociability, even fickle, who needed ties with several men at the same time".[5]
Works
• Pizzardo, Tina (1996). Senza pensarci due volte. Bologna: Il Mulino. ISBN 88-15-05615-7. OCLC 36262238.
References
1. "Pizzardo Rieser Battistina (Tina) — Scienza a due voci". scienzaa2voci.unibo.it. Retrieved 2022-05-09.
2. Marrone, Gaetana; Puppa, Paolo (2006-12-26). Encyclopedia of Italian Literary Studies. Routledge. ISBN 978-1-135-45530-9.
3. Stille, Alexander (April 2003). Benevolence and Betrayal: Five Italian Jewish Families Under Fascism. Macmillan. ISBN 978-0-312-42153-3.
4. "Tutti con il naso all'insù: dopo due anni torna a Poggio Rusco il lancio dei paracadutisti". Gazzetta di Mantova (in Italian). 2022-04-19. Retrieved 2022-05-09.
5. "Tina Pizzardo la mia storia con Pavese". 2016-03-05. Archived from the original on 5 March 2016. Retrieved 2022-05-09.
Authority control
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| Wikipedia |
Optimal partial regularity results for nonlinear elliptic systems in Carnot groups
Spreading speeds of $N$-season spatially periodic integro-difference models
August 2013, 33(8): 3407-3441. doi: 10.3934/dcds.2013.33.3407
On the Cauchy problem for the two-component Dullin-Gottwald-Holm system
Yong Chen 1, , Hongjun Gao 2, and Yue Liu 3,
School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China
Jiangsu Key Laboratory for NSLSCS and School of Mathematical Sciences, Nanjing Normal University, Nanjing 210046, China
Department of Mathematics, University of Texas at Arlington, Arlington, TX 76019-0408
Received May 2012 Revised October 2012 Published January 2013
Considered herein is the initial-value problem for a two-component Dullin-Gottwald-Holm system. The local well-posedness in the Sobolev space $H^{s}(\mathbb{R})$ with $s>3/2$ is established by using the bi-linear estimate technique to the approximate solutions. Then the wave-breaking criteria and global solutions are determined in $H^s(\mathbb{R}), s > 3/2.$ Finally, existence of the solitary-wave solutions is demonstrated.
Keywords: Two-component Dullin-Gottwald-Holm system, global solutions, regularization, wave-breaking, solitary-wave solutions..
Mathematics Subject Classification: 35G25, 35L0.
Citation: Yong Chen, Hongjun Gao, Yue Liu. On the Cauchy problem for the two-component Dullin-Gottwald-Holm system. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3407-3441. doi: 10.3934/dcds.2013.33.3407
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Piotr Bogusław Mucha, Milan Pokorný, Ewelina Zatorska. Approximate solutions to a model of two-component reactive flow. Discrete & Continuous Dynamical Systems - S, 2014, 7 (5) : 1079-1099. doi: 10.3934/dcdss.2014.7.1079
Jingqun Wang, Lixin Tian, Weiwei Guo. Global exact controllability and asympotic stabilization of the periodic two-component $\mu\rho$-Hunter-Saxton system. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 2129-2148. doi: 10.3934/dcdss.2016088
Günther Hörmann. Wave breaking of periodic solutions to the Fornberg-Whitham equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1605-1613. doi: 10.3934/dcds.2018066
Yong Chen Hongjun Gao Yue Liu | CommonCrawl |
Explicit algebraic stress model
The algebraic stress model arises in computational fluid dynamics. Two main approaches can be undertaken. In the first, the transport of the turbulent stresses is assumed proportional to the turbulent kinetic energy; while in the second, convective and diffusive effects are assumed to be negligible. Algebraic stress models can only be used where convective and diffusive fluxes are negligible, i.e. source dominated flows. In order to simplify the existing EASM and to achieve an efficient numerical implementation the underlying tensor basis plays an important role. The five-term tensor basis that is introduced here tries to combine an optimum of accuracy of the complete basis with the advantages of a pure 2d concept. Therefore a suitable five-term basis is identified. Based on that the new model is designed and validated in combination with different eddy-viscosity type background models.
Integrity basis
In the frame work of single-point closures (Reynolds-stress transport models = RSTM) still provide the best representation of flow physics. Due to numeric requirements an explicit formulation based on a low number of tensors is desirable and was already introduced originally most explicit algebraic stress models are formulated using a 10-term basis:
$b_{ij}=\sum _{\lambda =1}^{10}G^{(\lambda )}T_{ij}^{(\lambda )}$
The reduction of the tensor basis however requires an enormous mathematical effort, to transform the algebraic stress formulation for a given linear algebraic RSTM into a given tensor basis by keeping all important properties of the underlying model. This transformation can be applied to an arbitrary tensor basis. In the present investigations an optimum set of basis tensors and the corresponding coefficients is to be found.
Projection method
The projection method was introduced to enable an approximate solution of the algebraic transport equation of the Reynolds-stresses. In contrast to the approach of the tensor basis is not inserted in the algebraic equation, instead the algebraic equation is projected. Therefore, the chosen basis tensors does not need to form a complete integrity basis. However, the projection will fail if the basis tensor are linear dependent. In the case of a complete basis the projection leads to the same solution as the direct insertion, otherwise an approximate solution in the sense is obtained.
An example
In order to prove, that the projection method will lead to the same solution as the direct insertion, the EASM for two-dimensional flows is derived. In two-dimensional flows only the tensors are independent.
$T_{ij}^{(1)}=s_{ij}$
$T_{ij}^{(2)}=s_{ik}w_{kj}-w_{ik}s_{kj}$
$T_{ij}^{(3)}=s_{ik}s_{kj}-s_{mk}s_{km}{\frac {1}{3}}\delta _{ij}$
The projection leads then to the same coefficients. This two-dimensional EASM is used as starting point for an optimized EASM which includes three-dimensional effects. For example the shear stress variation in a rotating pipe cannot be predicted with quadratic tensors. Hence, the EASM was extended with a cubic tensor. In order to do not affect the performance in 2D flows, a tensor was chosen that vanish in 2d flows. This offers the concentration of the coefficient determination in 3d flows. A cubic tensor, which vanishes in 3d flow is:
$T_{ij}^{(5)}=w_{ik}s_{kl}s_{lj}-s_{ik}s_{kl}w_{lj}$
The projection with tensors T(1), T(2), T(3) and T(5) yields then the coefficients of the EASM.
Limitation of Cμ
A direct result of the EASM derivation is a variable formulation of Cμ.As the generators of the extended EASM where chosen to preserve the existing 2D formulation the expression of Cμ remains unchanged:
$C\mu ={\frac {-A_{1}g}{g^{2}-{\frac {2}{3}}A_{3}^{2}\eta _{1}-2A^{2}\eta _{2}}}$
Ai are the constants of the underlying pressure-strain model. Since η1 is always positive it might be possible that Cμ becomes singular. Therefore in the first EASM derivation of a regularization was introduced, which prevent a singular by cutting the range of η1. However, Wallin et al. pointed out that the regularization deteriorated the performance of the EASM. In their model the methodology was refined to account for the coefficient.
$g=C_{1}-2b_{ij}$
This leads to a weak non-linear conditional equation for the EASM coefficients and an additional equation for g must be solved. In 3D the equation of g is of 6th order, wherefore a closed solution is only possible in 2D flows, where the equation reduces to 3rd order. In order to circumvent the root finding of a polynomial equation quasi self-consistent approach. He showed that by using a Cμ expression of a realizable linear model instead of the EASM-Cμ expression in the equation of g the same properties of g follows. For a wide range of and the quasi self-consistent approach is almost identical to the fully self-consistent solution. Thus the quality of the EASM is not affected with the advantage of no additional non-linear equation. Since in the projections to determine the EASM coefficients the complexity is reduced by neglecting higher order invariants.
References
1. Gatski, T.B. and Speziale, C.G., "On explicit algebraic stress models for complex turbulent flows". J. Fluid Mech.
2. Rung, T., "Entwicklung anisotroper Wirbelzähigkeitsbeziehungen mit Hilfe von Projektionstechniken", PHD-thesis, Technical University Berlin, 2000
3. Taulbee, D.B., "An improved algebraic Reynolds stress model and corresponding nonlinaer stress model", Phys. Fluids, 28, pp 2555–2561, 1992
4. Lübcke, H., Rung, T. and Thiele, F. "Prediction of the Spreading Mechanism of 3D Turbulent Wall Jets with Explicit Reynolds-Stress Closures", Eng. Turbulence Modelling and Experiments 5, Mallorca, 2002
5. Wallin, S. and Johansson, A.V., "A new explicit algebraic Reynolds stress turbulence model including an improved near-wall treatment", Flow Modelling and Turbulence Measurements IV
6. Taulbee, D.B., "An improved algebraic Reynolds stress model and corresponding nonlinear stress model"
7. Jongen, T. and Gatski, T.B., "General explicit algebraic stress relations and best approximations for three-dimensional flows", Int. J. Engineering Science
| Wikipedia |
SN Applied Sciences
August 2019 , 1:826 | Cite as
Application of henna extract in minimizing surfactant adsorption on quartz sand in saline condition: A sacrificial agent approach
Mohd Syazwan Mohd Musa
Wan Rosli Wan Sulaiman
Zaiton Abdul Majid
Zulkifli Abdul Majid
Ahmad Kamal Idris
Kourosh Rajaei
3. Engineering (general)
This study examined the adsorption ability of henna extract as an environment-friendly and accessible sacrificial agent. In this study, the Fourier transform infrared-attenuated total reflectance (FTIR-ATR) was used to characterized henna extract and quartz sand. The adsorption of the henna extract on quartz sand was executed using the ultraviolet–visible spectroscopy (UV–Vis). The current study also assesses the effects of salinity on the henna extract adsorption on quartz sand, and the mechanisms of the adsorption process were interpreted. Apart from that, the ability of henna extract in reducing the adsorption of surfactant in the presence of salts were recorded. The outcome demonstrated that henna extract adsorption on quartz sand increased with the increase of salinity concentrations. Note that the adsorption value increased from 3.14 to 8.11 mg/g in 0 and 50,000 mg/L of salinity, respectively. The main mechanisms involved in the adsorption process were hydrogen bond, hydrophobic interactions, and electrostatic attractions. A reduction of 46% of surfactant adsorption was observed. This was a profound decrease in the adsorption of surfactant in the presence of henna extract, suggesting a possibility to be utilized as a sacrificial agent in reducing surfactant adsorption.
Henna extract Surfactant Quartz sand Adsorption Salinity
Surfactant flooding has been a vital part in enhanced oil recovery (EOR). This method is used to reduce the interfacial tension (IFT) of oil and water to improve the displacement efficiency through oil recovery [1]. Nevertheless, the surfactant adsorption on reservoir rock may impact the deprivation of the concentration of the surfactant, which may yield them less productive and competent [2].
The issues of the adsorption phenomenon of surfactants spark interests in lessening the adsorption of surfactant on reservoir rocks. Investigations on the implementation of similarly charged surfactant on the same surface charge of rock demonstrated that the anionic surfactant adsorption decreased on sandstone. The studies also concluded that the adsorption of cationic surfactant decreased on carbonate because of the repulsion of electrostatic between the adsorbent and the type of surfactants used [3, 4]. However, because of the heterogeneity of the reservoir especially with the diversity of minerals including carbonate, aluminates, silicates, and various clays, it is hard to determine either anionic or cationic surfactants to be used.
Besides that, alkali as additives has been used to lower the adsorption of surfactants [5] and the mechanism of which was considered as changing the surface charge on the rock surface. However, the usage of alkali induces such problems such as severe scaling in the near wellbore and production systems [6].
Following the research, the application of a sacrificial agent (SA) is deemed to be a promising method in reducing surfactant adsorption. The SA is a material that is injected to significantly inhibit or conceal all probable adsorption sites of the rock within the hydrocarbon formation. Weston et al. [7] have found that the formation of admicelles on the solid surface by the molecules of surfactant is the primary reason for adsorption to occur. The SA is strategically implemented to inhibit the development of these admicelles. ShamsiJazeyi et al. [8] in their work announced that polyelectrolyte had been proved to lessen adsorption of anionic surfactant on carbonates and clays minerals. In addition, surfactant adsorption was successfully reduced after the addition of the SA.
However, the materials used in reducing surfactant adsorption were chemicals which may be hazardous to living creatures and environment. Thus, materials that are eco-friendly, fewer impurities and simply accessible and are found from natural products, for instance, plant extracts that can act as an SA or inhibitor are being researched.
In this research, henna, a natural plant-based material was investigated as a potential SA. Henna is also known as Lawsonia inermis L. This substance has been implemented as a corrosion inhibitor [9, 10] and Moslemizadeh et al. [11] revealed that henna extract can reduce the swelling of sodium bentonite better than exposing sodium bentonite to polyamine and potassium chloride due to its inhibitive capability. Apart from that, several researchers conducted inclusive studies on the effect of various parameters on the adsorption of surfactant, especially regarding the influence of added salts. Bera et al. [12] observed that the adsorption of surfactant magnifies with the increased of NaCl concentration while, ShamsiJazeyi et al. [8] found that by increasing the salinity of Na+ ions, the adsorption of surfactant will simultaneously increase too. As indicated by Yekeen et al. [13], the degree of the adsorption of surfactant on reservoir rocks relies mostly on the electrolytes and the mineralogical composition of the rocks. Nevertheless, there is an absence of detailed knowledge on the application of henna extract as a SA in reducing surfactant adsorption in the vicinity of salts. Furthermore, the mechanisms of adsorption of the henna extract on quartz sand are still not well-understood by scholars.
This study was driven by the desire to comprehend the adsorption behavior of the henna extract on quartz sand and its ability in reducing surfactant adsorption with the influence of salinity. To reach this aim, the mechanisms of the adsorption process were analyzed. This study aimed to validate the notion that henna extract could be used as a SA in minimalizing surfactant adsorption.
2 Materials and methods
2.1 Materials
Fresh henna leaves were gathered from henna trees in Johor, Malaysia. Methanol of 99.9% (Acros Organics (USA)) was used as the solvent in methanolic extraction. The anionic surfactant, sodium dodecyl sulfate (SDS) of 98% purity weighing 288.38 g/mol of molecular weight and manufactured by Fisher Chemical (UK) was used for the surfactant adsorption. Sodium chloride, NaCl (99.8% pure) provided by Vchem was used to study the effect of salinity, and quartz sand used in the experiment was collected in Desaru, Johor, Malaysia. Deionized water (DIW) was used for all experiments. All substances used in this study were of scientific quality and were used as acquired; in other words, devoid of added purifications.
2.2 Preparation of henna powder
The fresh leaves of henna were dried at room temperature and then grounded into powder using an electric blender. The henna powder was carefully packed in an airtight, BPA-free container and stored in room temperature until further used.
2.3 Characterization of henna extract and quartz sand
2.3.1 FTIR-ATR analysis
The FTIR-ATR was conducted using the Perkin Elmer FTIR Spectrometer (USA). Meanwhile, the functional groups of henna extract and quartz sand were identified using the spectrometer by observing the vibrational motion of bonds in the molecules. The spectra were measured in the range of 650–4000 cm−1 with a scan resolution of 2 cm−1. The FTIR data were documented in the transmittance mode. Then, the pattern of the spectrum was examined and compared to the IR absorption table to determine the functional groups encompassed in the samples.
2.3.2 XRD analysis
The quartz sand sample was further characterized by using the X-ray diffraction (XRD) in the continuous scanning mode on SmartLab X-Ray Diffractometer (Rigaku, Japan) operated at 40 kV and a current of 30 mA with Cu–Kβ filter and Cu–Kα radiation source (λ = 0.154056 nm). Particle size ought to be fine to attain a tolerable statistical representation of the components and their numerous diffracting crystal planes and to evade diffraction-related artifacts [14]. All the patterns were collected at room temperature with steps of 0.02° in the 2θ range of 3°–100°. The measurements were taken at room temperature with a scan rate of 8.2551° per minute.
2.4 Preparation of henna extract and surfactant solutions
The solutions of henna extract were prepared in standard 250 ml Erlenmeyer flasks. The henna extract was weighed and transferred into the flasks, and the DIW water was added to the required volume. Henna extract concentrations were prepared in the range of 3000–8000 mg/L. It should be noted that the surfactant solutions were prepared similarly as henna extract solutions. Different concentrations of surfactant were prepared in the range of 1000–5000 mg/L. The influence of salinity on henna extract adsorption was determined by preparing different henna extract solutions using NaCl at concentrations 10,000, 30,000, and 50,000 mg/L.
2.5 Ultraviolet–visible spectroscopy (UV–Vis)
UV–Vis measurement was performed to determine the maximum absorption wavelength of henna extract and surfactant. Besides that, it is also used to compute the concentration of henna extract and surfactant before and after adsorption using Shimadzu UV-1800 Spectrophotometer (Japan). The absorbance–wavelength between 200 and 800 nm was recorded. Quartz cuvettes were used as the vessel.
2.6 Adsorption experiments
The adsorption of the henna extract on quartz sand was determined using the depletion method. This method observed the differences between the concentrations of henna extract before and after adsorption on quartz sand. 6 g of quartz sand was mixed with 30 mL of henna extract solutions. The mixture was then agitated in a temperature controller shaker using IKA KS 3000 I control (USA) at 180 rpm for 24 h at 25 °C under atmospheric pressure to reach the state of equilibrium. They were then centrifuged using Rotofix 32A (Hettich Zentrifugen, Germany) at 4000 rpm for 30 min to isolate the buoyant liquid. The concentrations of henna extract in the buoyant liquid was measured using a UV–Vis spectrophotometer. Adsorption amount at equilibrium time, qe (mg/g), was calculated using Eq. (1). The same procedures were applied in calculating the adsorption of surfactant on quartz sand. Surfactant concentration of 2000 mg/L was employed to determine the surfactant adsorption on quartz sand. In evaluating henna extract performances in reducing surfactant adsorption on quartz sand, the solution mixtures of henna extract and quartz sand were filtered out, leaving only quartz sand behind. Then, 30 mL of surfactant solution (concentration of 2000 mg/L) were mixed with the pre-treated quartz sand with henna extract solution. The mixtures were then left to reach the state of equilibrium for 24 h throughout which they were recurrently shaken in a temperature controller shaker at 180 rpm. After equilibrium was reached, the mixtures were centrifuged, and the buoyant liquid were examined using the UV–Vis spectrophotometer. The adsorption amount at equilibrium time, qe (mg/g), was calculated using Eq. (1).
$$q_{e} = \left( {C_{o} - C_{e} } \right) \times \frac{V}{m}$$
where qe is the adsorption of henna extract and surfactant on quartz sand (mg/g), Co and Ce are the henna extract and surfactant concentrations before and after the adsorption experiment, respectively (mg/L), V is the volume of henna extract and surfactant solutions added to the volumetric flask (L), and m is the total mass of the quartz sand added (g).
3.1.1 FTIR analysis
The FTIR-ATR spectrum of henna extract was analyzed and depicted in Fig. 1, which demonstrates a wide absorption band at 3311 cm−1. Those bands were appointed to the vibration of the hydroxyl groups. A similar observation was reported in other studies [9, 15]. Furthermore, the vibration of the aliphatic C–H group appeared at 2926 cm−1 and 2855 cm−1. The appearance of these functional groups was similar to the ones reported by other researchers [16]. The peaks at 1712 and 1632 cm−1 were assigned to the C=O bond. A similar view was noted by Safie et al. [17]. The peaks at 1510, 1449, 1400, and 1366 cm−1 were credited to the vibration of C=C in the aromatic benzene rings. These peaks were also observed by several researchers [9, 18]. The peaks at 1030 and 1064 cm−1 can be ascribed to the vibration of C–OH of the phenolic group. Such conclusion was inspired by the trend that was perceived by Saadaoui et al. [19]. Phenolic compounds in the henna leaves are the core constituents that contributed to the popularity of henna in specific fields as mentioned above [20, 21]. Figure 2 shows the molecular structure of one of the phenolic compounds that are in the henna leaves—gallic acid [20].
FTIR-ATR spectrum of henna extract
Molecular structure of gallic acid
Figure 3 presents the infrared spectra of quartz sand. The band at 3286 cm−1 corresponded to the OH groups on the surface of quartz sand which made the silanol groups (SiOH). These observations were supported by Akl et al. [22] and Firoozmandan et al. [23]. The peak at 1632 cm−1 corresponding to the OH groups that made the silanol groups existed [24]. Vibrations of Si–O and Si–O–Si is found at 1061 cm−1 as comparable to the ones reported by SöDerholm and Shang [25] and Liu et al. [26]. Hydroxyls functional group of silanol group was found at 778 cm−1. Identical analysis was demonstrated by Péré et al. [27] and Parida et al. [28].
FTIR-ATR spectrum of quartz sand
The XRD pattern of quartz sand is shown in Fig. 4. The diffraction peaks could be attributed to quartz sand and agreed with the standard data given in the ICDD card, 01-076-9282 for quartz sand. No impurity peaks were observed. Furthermore, the intensities of the diffraction peaks were high, and the peaks were sharp. This indicated that the sample were highly crystalline and consisted entirely of quartz sand.
XRD analysis of quartz sand
The wavelength of henna extract and surfactant were determined at 673 nm and 238 nm respectively, which corresponds to the maximum absorbance peak. Standard calibration curves were established by plotting absorbance against henna extract and surfactant concentrations to determine the final adsorption concentrations.
3.3.1 FTIR-ATR analysis on the adsorption of henna extract on quartz sand
The FTIR spectrum of quartz sand before and after adsorption was observed and displayed in Fig. 5. After the adsorption of henna extract, there was a substantial change in the intensity of the adsorption peak at (a), signalling the interactions of the hydroxyl groups. Significant new adsorption peak was observed at (b) which belong to the aliphatic group of henna extract molecules. This was due to the interactions of hydrogen bond between henna extract and quartz sand molecules. The intensity of the characteristic peaks of carbonyl groups seen at (c) increased, after being adsorbed on quartz sand. This proposed that the carbonyl groups took part in the adsorption process. Apart from that, the intensity of the absorption peak that was observed at (d) owned by the aromatic rings, increased, signifying the interactions of aromatics with quartz sand molecules. The intensity of the absorption peak of phenolic groups observed at (e), increased and the range of the peak is sharper than before henna extract adsorption, proposing the interactions of phenolics and quartz sand molecules. At (f), there is an increased of the absorption peak that was due to the interactions of hydroxyl groups from quartz sand molecules with henna extract molecules.
FTIR-ATR analysis on the adsorption of henna extract on quartz sand
3.3.2 Effect of salinity on the adsorption of henna extract on quartz sand
The adsorption of henna extract on quartz sand in the presence of salts was plotted against the concentration of henna extract. The effect of salinity at 0, 10,000, 30,000, and 50,000 mg/L were plotted and presented in Fig. 6. As shown in Fig. 6, henna extract adsorption on quartz sand increased with the presence of sodium chloride (NaCl).
Adsorption of henna extract on quartz sand at different salinities
The adsorption of henna extract on quartz sand increased from 3.14 mg/g without NaCl to 4.48 mg/g with the presence of NaCl. The increment can be due to the presence of the positively charged cation (Na+) that increased the total electrostatic interactions among the henna extract molecules and the sodium ions. The carbonyl and hydroxyl groups in the henna extract molecules contain electronegative oxygen, and those groups were responsible for the increased adsorption. This is endorsed by Mckenzie et al. [29] who studied the adsorption behaviour of phenol in the presence of sodium ions. They concluded that the presence of sodium ions enhances adsorption. This principle is based on the electrostatic interactions between the positively charged cation (Na+) and the negatively charged oxygen in the hydroxyl group of phenol. Figure 7 shows the mechanism of adsorption in the presence of sodium ions.
Electrostatic attraction in the presence of sodium ions
Modified from McKenzie et al. [29]
From the figures, it can be noted that the increasing salinity from 10,000 to 30,000, and 50,000 mg/L, improved the adsorption of henna extract on quartz sand. Adsorption of henna extract on quartz sand increased from 4.48 to 6.65, and to 8.11 mg/g in the presence of 10,000, 30,000, and 50,000 mg/L of NaCl, respectively. Comparable results were attained by Yekeen et al. [13], who also mentioned that the adsorption increased with the increase of the concentration of salt, in the presence of NaCl salts.
These observations can be credited to the increase in the number of cation (Na+) atoms on the quartz sand surface. The presence of salt encourages adsorption by increasing the presence of positive charges on the surface so that the negatively-charged oxygen can bind on the cations. Furthermore, the presence of the aromatic rings in phenolic compounds enhances the adsorption due to the negatively charged rings [30, 31]. This promotes the interaction of cation/π between the Na+ and the rings [32, 33]. Figure 8 demonstrates the interaction of cation/π.
Cation–Π interaction between sodium ion and aromatic ring
Modified from Kumar et al. [33]
Besides electrostatic attraction between the positively-charged cation and the negatively-charged portion of henna extract molecules, Na+ is beneficial in forming bridges of cation between the negatively charged portion of the molecules of henna extract and the negatively charged quartz sand layer [34, 35]. Moreover, the cation bridges can be formed between the henna extract molecules themselves. This will further compress the molecules further, allowing more henna extract molecules to be adsorbed. The neutralizing bridges between the henna extract molecules and the negatively charged quartz sand layer on the surface led to the indirect adsorption. Figures 9 and 10 demonstrate the schematic diagrams of the effect of salt bridges.
Formation of salt bridges due to the presence Na+ ions between the quartz sand surface and henna extract molecules
Modified from Yu et al. [35]
Formation of salt bridges due to the presence Na+ ions (a) between aromatic rings and (b) among henna extract molecules
In addition, the effect of salinity on henna extract adsorption on quartz sand can be attributed to the positive cations (Na+). These cations were attracted to the permanently negative- charged sites (silanol sites, Si–O) on quartz sand surfaces [36]. Therefore, an increase in the adsorption of henna extract with increasing salinity implies that increasing salinity reduces the negatively-charged sites on the quartz sand surface, increasing henna extract adsorption [37]. Then, the negatively-charged groups from henna extract molecules can bind more to these positively charged surface on quartz sand due to the electrostatic forces.
3.3.3 Influence of salinity on the adsorption of surfactant on quartz sand without the presence of henna extract
Figure 11 shows the result of the studies of the adsorption of surfactant without henna extract on quartz sand. It can be observed that the adsorption of surfactant on quartz sand in saline condition is higher than in DIW. This behavior is attributed to the existence of Na+ ions in the solutions.
Surfactant adsorption with the effect of salinity
As displayed in Fig. 11, surfactant adsorption on quartz sand increased from 1.57 to 5.16 mg/g from DIW to saline solutions. These remarks are due to cation being adsorbed on the negatively-charged quartz sand surface, producing additional positive sites for the anionic surfactant to adsorb [38]. Furthermore, the added salt in the solutions made the EDL and zeta potential of the quartz sand surface to compress, allowing more surfactant to be adsorbed [39, 40].
3.3.4 Influence of salinity on the adsorption of surfactant on quartz sand with the presence of henna extract
An evaluation of the adsorption of surfactant on pre-treated quartz sand with henna extract in the presence of salts was conducted. The salinity of 50,000 mg/L was chosen due to the highest adsorption capacity of henna extract on quartz sand. Based on the result of the CMC (not shown), a surfactant concentration of 2000 mg/L was used to determine the capability of henna extract in reducing surfactant adsorption. Meanwhile, for the henna extract concentration, 8000 mg/L of surfactant concentration was chosen. Figure 12 shows the surfactant adsorption data.
Henna extract performances on the reduction of the surfactant adsorption in the presence of salts (50,000 mg/L NaCl)
It is shown that the adsorption of surfactant on pre-treated quartz sand was reduced in the presence of NaCl from 5.16 mg/g to 2.77 mg/g. The reduction of surfactant adsorption was about 46%. The reduced surfactant adsorption on the pre-treated quartz sand with henna extract can be attributed to the presence of cation (Na+). Based on the mechanism discussed above, the cation allows the henna extract molecules to adsorb more, providing less binding option for surfactant molecules to bind to the surface of quartz sand and henna extract molecules. This made the surfactant molecules free to move, reducing surfactant adsorption.
The existence of the functional groups in henna extract, hydroxyls, carbonyls, phenolics, aromatic benzene ring, and aliphatic carbon-hydrogen groups allow for the adsorption process. Quartz sand was characterized and used as an adsorbent to investigate the effect of salinity on henna extract adsorption on quartz sand and to evaluate the performances of henna extract as a SA in reducing surfactant adsorption. The presence of salt improved the adsorption of henna extract on quartz sand, and increasing the salinity further improved the adsorption. Meanwhile, the electrostatic interactions played a substantial role as perceived from the role of a cation such as sodium (Na+). The study of surfactant adsorption was done to determine the efficiency of the henna extract adsorption. This was determined by analyzing the results of before-and-after adsorption values of surfactant adsorption on quartz sand. The salinity demonstrated that the surfactant adsorption exhibited identical traits of adsorption with the henna extract adsorption process. This follows that the adsorption of surfactant on quartz sand increased in the existence of salts rather than in the absence of salts. Moreover, the adsorption of surfactant on pre-treated quartz sand with henna extract in the presence of salts managed to be reduced. It was observed that there was a 46% reduction, caused by the electrostatic attractions between henna extract and quartz sand. The surface of quartz sand was covered with henna extract molecules, inhibiting the surfactant molecules from being adsorbed. Hydrogen bond, electrostatic attraction, and hydrophobic interactions played a significant role in the mechanism of adsorption between henna extract and quartz sand. This study proved that henna extract has the potential to be a SA in reducing surfactant adsorption in the presence of salts.
The authors wish to thank the Ministry of Higher Education (MOHE), Malaysia, and Universiti Teknologi Malaysia for assisting this research through Fundamental Research Grant Scheme (R.J130000.7846.4F931) and Research University Grant (12429).
On behalf of all authors, the corresponding author states that there is no conflict of interest.
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1.Department of Petroleum Engineering, School of Chemical and Energy Engineering, Faculty of EngineeringUniversiti Teknologi MalaysiaSkudaiMalaysia
2.Department of Chemical, Faculty of ScienceUniversiti Teknologi MalaysiaSkudaiMalaysia
3.UTM-MPRC Institute for Oil and Gas, Universiti Teknologi MalaysiaSkudaiMalaysia
4.Department of Petroleum Engineering, Faculty of Geoscience and Petroleum EngineeringUniversiti Teknologi PETRONASBandar Seri IskandarMalaysia
Musa, M.S.M., Sulaiman, W.R.W., Majid, Z.A. et al. SN Appl. Sci. (2019) 1: 826. https://doi.org/10.1007/s42452-019-0870-0
Received 06 March 2019
Publisher Name Springer International Publishing | CommonCrawl |
\begin{definition}[Definition:Strictly Negative/Rational Number]
The '''strictly negative rational numbers''' are the set defined as:
:$\Q_{< 0} := \set {x \in \Q: x < 0}$
That is, all the rational numbers that are strictly less than zero.
\end{definition} | ProofWiki |
\begin{document}
\title{\textbf{Skew PBW extensions over weak symmetric and $(\Sigma,\Delta)$-weak symmetric rings} \begin{abstract} \noindent In this paper we study skew Poincar\'e-Birkhoff-Witt extensions over weak symmetric and $(\Sigma,\Delta)$-weak symmetry rings. Since these extensions generalize Ore extensions of injective type and another noncommutative rings of polynomial type, we unify and extend several results in the literature concerning the property of being symmetry. Under adequate conditions, we transfer the property of being weak symmetric or $(\Sigma,\Delta)$-weak symmetric from a ring of coefficients to a skew PBW extension over this ring. We illustrate our results with remarkable examples of algebras appearing in noncommutative algebraic geometry and theoretical physics.
\noindent \textit{Key words and phrases.} Symmetric ring; Noncommutative ring; skew PBW extension.
\noindent 2010 \textit{Mathematics Subject Classification:} 16S36, 16S37, 16S38, 16S99.
\end{abstract}
\section{Introduction}\label{section}
A ring $B$ is said to be {\em Armendariz} (the term was introduced by Rege and Chhawchharia \cite{RegeChhawchharia1997}), if whenever polynomials $f(x)=\sum_{i=0}^{s}a_ix^{i}$ and $g(x)=\sum_{j=0}^{t} b_jx^{j}$ in $B[x]$ satisfy $f(x)g(x)=0$, then $a_ib_j$, for all $i, j$. In the context of the well-known Ore extensions, for an endomorphism $\alpha$ and a $\alpha$-derivation $\delta$ of $B$, Moussavi and Hashemi \cite{MoussaviHashemi2005} defined $B$ to be $(\alpha,\delta)$-skew Armendariz, if for $f(x)=\sum_{i=0}^{s}a_ix^{i}$ and $g(x)=\sum_{j=0}^{t} b_jx^{j}$ in $B[x;\alpha,\delta]$ satisfy $f(x)g(x)=0$, then $a_ix^{i}b_jx^{j} = 0$, for each $i, j$. On the other hand, a ring $B$ is called (i) {\em reduced}, if $a^{2}=0\Rightarrow a=0$, for all $a\in B$; (ii) (Lambek \cite{Lambek1971}) {\em symmetric}, if $abc = 0\Rightarrow acb=0$, for all $a,b,c,\in B$; (iii) {\em reversible}, if $ab=0 \Rightarrow ba=0$, for all $a, b\in B$; (iv) {\em semicommutative}, if $ab=0 \Rightarrow aBb=0$, for all $a, b\in B$ (Bell \cite{Bell1970} defined the following: a ring $B$ is said to satisfy the $IFP$, {\em insertion of factors property}, if $r_B(a)$ is an ideal for all $a\in B$. Sometimes, a semicommutative ring is also called a {a ring with $IFP$ property). It is known that the implications {\em reduced} $\Rightarrow$ {\em symmetric} $\Rightarrow$ {\em reversible} $\Rightarrow$ {\em semicommutative} hold but, in general, the converse of each one of these implications is false, see Marks \cite{Marks2003} for a detailed discussion.\\
Of course, commutative rings are symmetric. Reduced rings are symmetric as we can appreciated in Anderson and Camillo \cite{AndersonCamillo1998}. Nevertheless, there are many nonreduced commutative (so symmetric) rings. Now, if $B$ is Armendariz, then the classical polynomial ring $B[x]$ over $B$ is symmetric if and only if $B$ is symmetric (Huh et al. \cite{HuhLeeSmoktunowicz2002} and Kim and Lee \cite{KimLee2003}. In the noncommutative case, there are results concerning the property of being symmetric over $(\alpha,\delta)$-skew Armendariz rings, see \cite{OuyangChen2010}. Precisely, this was the motivation for Ouyang and Chen who in \cite{OuyangChen2010} defined weak symmetric rings and weak $(\alpha,\delta)$-symmetric rings for the context of Ore extensions $B[x;\alpha,\delta]$, where $B$ is an associate ring with unity. They proved that for every $(\alpha,\delta)$-compatible and reversible ring $B$ (following Annin \cite{Annin2002}, for an endomorphism $\alpha$ and an $\alpha$-derivation $\delta$ of $B$, $B$ is called $\alpha$-compatible, if for every $a,b \in B$, we have $ab=0$ if and only if $a\alpha(b)=0$ (necessarily, the endomorphism $\alpha$ is injective), and $B$ is called to be $\delta$-{\em compatible} if for each $a, b\in B$, $ab=0 \Rightarrow a\delta(b)=0$; if $B$ is both $\alpha$-compatible and $\delta$-compatible, $B$ is called $(\alpha,\delta)$-{\em compatible}), $B$ is weak symmetric if and only if $B[x;\alpha,\delta]$ is weak symmetric, and for every semicommutative ring $B$, $B$ is weak $(\alpha,\delta)$-symmetric if and only if $B[x]$ is weak $(\overline{\alpha}, \overline{\delta})$-symmetric, where $\overline{\alpha}$ and $\overline{\delta}$ are the extended maps of $\alpha$ and $\delta$ over $B[x]$, respectively. The results presented in \cite{OuyangChen2010} generalize those corresponding for .\\
Having in mind all above results and with the aim of establishing similar results to the more general noncommutative rings than (iterated) Ore extensions, in this paper we are interested in the family of rings known as {\em skew Poincar\'e-Birkhoff-Witt extensions} which were introduced by Gallego and Lezama \cite{LezamaGallego2011}. Besides of Ore extensions, skew PBW extensions generalize several families of noncommutative rings (see \cite{ReyesPhD2013, ReyesYesica2018} for a list of noncommutative rings which are skew PBW extensions but not iterated Ore extensions) and include as particular rings different examples of remarkable algebras appearing in representation theory, Hopf algebras, quantum groups, noncommutative algebraic geometry and another algebras of interest in the context of mathematical physics. Let us mention some of these algebras (see \cite{LezamaReyes2014, ReyesPhD2013} for a detailed reference of every one of these families): (i) universal enveloping algebras of finite dimensional Lie algebras; (ii) PBW extensions introduced by Bell and Goodearl; (iii) almost normalizing extensions defined by McConnell and Robson; (iv) sol\-va\-ble polynomial rings introduced by Kandri-Rody and Weispfenning; (v) diffusion algebras studied by Isaev, Pyatov, and Rittenberg; (vi) 3-dimensional skew polynomial algebras introduced by Bell and Smith; (vii) the regular graded algebras studied by Kirkman, Kuzmanovich, and Zhang, and other noncommutative algebras of polynomial type. The importance of skew PBW extensions is that the coefficients do not necessarily commute with the variables, and these coefficients are not necessarily elements of fields (see Definition \ref {gpbwextension}). In fact, the skew PBW extensions contain well-known groups of algebras such as some types of $G$-algebras introduced by Apel and some PBW algebras defined by Bueso et. al., (both $G$-algebras and PBW algebras take coefficients in fields and assume that coefficientes commute with variables), Auslander-Gorenstein rings, some Calabi-Yau and skew Calabi-Yau algebras, some Artin-Schelter regular algebras, some Koszul and augmented Koszul algebras, quantum polynomials, some quantum universal enveloping algebras, some graded skew Clifford algebras and others (see \cite{BrownGoodearl2002, BuesoTorrecillasVerschoren, Rosenberg1995, Suarez2017, SuarezLezamaReyes2017, SuarezReyesgenerKoszul2017} for a list of examples). As we can appreciated, skew PBW extensions include a lot of noncommutative rings, which means that a theory of symmetry for these extensions will cover several treatments in the literature and will establish similar results for algebras not considered before. To formulate this theory is the objective of the present paper. In this way, we continue the study of ring theoretical properties of skew PBW extensions (c.f. \cite{AcostaLezama2015, Artamonov2015, ArtamonovLezamaFajardo2016, LezamaGallego2011, LezamaGallego2017, LezamaAcostaReyes2015, LezamaLatorre2017, LezamaReyes2014, ReyesPhD2013}).\\
The paper is organized as follows: In Section \ref{definitionexamplesspbw} we establish some useful results about skew PBW extensions for the rest of the paper. In Section \ref{SigmarigidandSigmaDeltacompatible} we recall the notions of $\Sigma$-rigid rings and $(\Sigma,\Delta)$-compatible rings which are key throughout the paper. Next, in Section \ref{weaksymmetricskewPBWextensions} we present some results about nilpotent elements of skew PBW extensions and then characterize these extensions over weak symmetric rings. In Section \ref{SigmaDeltaweaksymmetricskewPBWextensions} we investigate skew PBW extensions over weak $(\Sigma,\Delta)$-symmetric rings. The results presented in Sections \ref{weaksymmetricskewPBWextensions} and \ref{SigmaDeltaweaksymmetricskewPBWextensions} generalize corresponding results presented by Ouyang and Chen \cite{OuyangChen2010} for Ore extensions of injective type and generalize those presented in \cite{JaramilloReyes2018}. The techniques used here are fairly standard and follow the same path as other text on the subject. Finally, Section \ref{examplespaper} presents remarkable examples appearing in noncommutative algebraic geometry and theoretical physics where results obtained in Sections \ref{weaksymmetricskewPBWextensions} and \ref{SigmaDeltaweaksymmetricskewPBWextensions} can be illustrated. \\
Throughout the paper, the word ring means a ring (not necessarily commutative) with unity. The letter $k$ will denote a commutative ring and $\Bbbk$ will denote a field. $\mathbb{C}$ will denote the field of complex numbers. Finally, for a ring $B$, ${\rm nil}(B)$ represents the set of nilpotent elements of $B$.
\section{Skew PBW extensions}\label{definitionexamplesspbw} In this section we recall some results about skew PBW extensions which are important for the rest of the paper.
\begin{definition}[\cite{LezamaGallego2011}, Definition 1]\label{gpbwextension} Let $R$ and $A$ be rings. We say that $A$ is a {\em skew PBW extension} (also known as {\em $\sigma$-PBW extension}) {\em of} $R$, which is denoted by $A:=\sigma(R)\langle x_1,\dots,x_n\rangle$, if the following conditions hold: \begin{enumerate} \item[\rm (i)]$R\subseteq A$. \item[\rm (ii)]there exist elements $x_1,\dots ,x_n\in A$ such that $A$ is a left free $R$-module, with basis ${\rm Mon}(A):= \{x^{\alpha}=x_1^{\alpha_1}\cdots x_n^{\alpha_n}\mid \alpha=(\alpha_1,\dots ,\alpha_n)\in \mathbb{N}^n\}$, and $x_1^{0}\dotsb x_n^{0}:=1\in {\rm Mon}(A)$.
\item[\rm (iii)]For each $1\leq i\leq n$ and any $r\in R\ \backslash\ \{0\}$, there exists an element $c_{i,r}\in R\ \backslash\ \{0\}$ such that $x_ir-c_{i,r}x_i\in R$. \item[\rm (iv)] For any elements $1\leq i,j\leq n$, there exists $c_{i,j}\in R\ \backslash\ \{0\}$ such that $x_jx_i-c_{i,j}x_ix_j\in R+Rx_1+\cdots +Rx_n$ (i.e., there exist elements $r_0^{(i,j)}, r_1^{(i,j)}, \dotsc, r_n^{(i,j)}$ of $R$ with $x_jx_i - c_{i,j}x_ix_j = r_0^{(i,j)} + \sum_{k=1}^{n} r_k^{(i,j)}x_k$). \end{enumerate} \end{definition}
Since ${\rm Mon}(A)$ is a left $R$-basis of $A$, the elements $c_{i,r}$ and $c_{i,j}$ are unique, (\cite{LezamaGallego2011}, Remark 2).
\begin{proposition}[\cite{LezamaGallego2011}, Proposition 3]\label{sigmadefinition} Let $A$ be a skew PBW extension of $R$. For each $1\leq i\leq n$, there exist an injective endomorphism $\sigma_i:R\rightarrow R$ and an $\sigma_i$-derivation $\delta_i:R\rightarrow R$ such that $x_ir=\sigma_i(r)x_i+\delta_i(r)$, for each $r\in R$. From now on, we will write $\Sigma:=\{\sigma_1,\dotsc, \sigma_n\}$, and $\Delta:=\{\delta_1,\dotsc, \delta_n\}$. \end{proposition}
\begin{definition}[\cite{LezamaGallego2011}, Definition 4; \cite{LezamaAcostaReyes2015}, Definition 2.3]\label{sigmapbwderivationtype} Let $A$ be a skew PBW extension of $R$. \begin{enumerate} \item[\rm (a)] $A$ is called \textit{quasi-commutative}, if the conditions {\rm(}iii{\rm)} and {\rm(}iv{\rm)} in Definition \ref{gpbwextension} are replaced by the following: (iii') for each $1\leq i\leq n$ and all $r\in R\ \backslash\ \{0\}$, there exists $c_{i,r}\in R\ \backslash\ \{0\}$ such that $x_ir=c_{i,r}x_i$; (iv') for any $1\leq i,j\leq n$, there exists $c_{i,j}\in R\ \backslash\ \{0\}$ such that $x_jx_i=c_{i,j}x_ix_j$. \item[\rm (b)] $A$ is called \textit{bijective}, if $\sigma_i$ is bijective for each $1\leq i\leq n$, and $c_{i,j}$ is invertible, for any $1\leq i<j\leq n$. \item[\rm (c)] $A$ is called of {\em endomorphism type}, if $\delta_i=0$, for every $i$. In addition, if every $\sigma_i$ is bijective, $A$ is a skew PBW extension of {\em automorphism type}. \end{enumerate} \end{definition}
\begin{example}\label{mentioned} If $R[x_1;\sigma_1,\delta_1]\dotsb [x_n;\sigma_n,\delta_n]$ is an iterated Ore extension where \begin{itemize} \item $\sigma_i$ is injective, for $1\le i\le n$, \item $\sigma_i(r)$, $\delta_i(r)\in R$, for every $r\in R$ and $1\le i\le n$, \item $\sigma_j(x_i)=cx_i+d$, for $i < j$, and $c, d\in R$, where $c$ has a left inverse, \item $\delta_j(x_i)\in R + Rx_1 + \dotsb + Rx_n$, for $i < j$, \end{itemize} then $R[x_1;\sigma_1,\delta_1]\dotsb [x_n;\sigma_n, \delta_n] \cong \sigma(R)\langle x_1,\dotsc, x_n\rangle$ (\cite{LezamaReyes2014}, p. 1212). Skew PBW extensions of endomorphism type are more general than iterated Ore extensions of the form $R[x_1;\sigma_1]\dotsb [x_n;\sigma_n]$, and in general, skew PBW extensions are more general than Ore extensions of injective type (see \cite{LezamaReyes2014}). Examples of noncommutative rings which are skew PBW extensions but can not be expressed as iterated Ore extensions can be found in \cite{ReyesPhD2013, ReyesYesica2018}. \end{example}
\begin{definition}\label{definitioncoefficients} If $A$ is a skew PBW extension of $R$, then: \begin{enumerate} \item[\rm (i)]for $\alpha=(\alpha_1,\dots,\alpha_n)\in \mathbb{N}^n$, $\sigma^{\alpha}:=\sigma_1^{\alpha_1}\cdots \sigma_n^{\alpha_n}$,
$|\alpha|:=\alpha_1+\cdots+\alpha_n$. If $\beta=(\beta_1,\dots,\beta_n)\in \mathbb{N}^n$, then $\alpha+\beta:=(\alpha_1+\beta_1,\dots,\alpha_n+\beta_n)$. \item[\rm (ii)]For $X=x^{\alpha}\in {\rm Mon}(A)$,
$\exp(X):=\alpha$, $\deg(X):=|\alpha|$, and $X_0:=1$. The symbol $\succeq$ will denote a total order defined on ${\rm Mon}(A)$ (a total order on $\mathbb{N}^n$). For an
element $x^{\alpha}\in {\rm Mon}(A)$, ${\rm exp}(x^{\alpha}):=\alpha\in \mathbb{N}^n$. If $x^{\alpha}\succeq x^{\beta}$ but $x^{\alpha}\neq x^{\beta}$, we write $x^{\alpha}\succ x^{\beta}$. Every element $f\in A$ can be expressed uniquely as $f=a_0 + a_1X_1+\dotsb +a_mX_m$, with $a_i\in R$, and $X_m\succ \dotsb \succ X_1$ (eventually, we will use expressions as $f=a_0 + a_1Y_1+\dotsb +a_mY_m$, with $a_i\in R$, and $Y_m\succ \dotsb \succ Y_1$). With this notation, we define ${\rm lm}(f):=X_m$, the \textit{leading monomial} of $f$; ${\rm lc}(f):=a_m$, the \textit{leading coefficient} of $f$; ${\rm lt}(f):=a_mX_m$, the \textit{leading term} of $f$; ${\rm exp}(f):={\rm exp}(X_m)$, the \textit{order} of $f$; and
$E(f):=\{{\rm exp}(X_i)\mid 1\le i\le t\}$. Note that $\deg(f):={\rm max}\{\deg(X_i)\}_{i=1}^t$. Finally, if $f=0$, then ${\rm lm}(0):=0$, ${\rm lc}(0):=0$, ${\rm lt}(0):=0$. We also consider $X\succ 0$ for any $X\in {\rm Mon}(A)$. For a detailed description of monomial orders in skew PBW extensions, see \cite{LezamaGallego2011}, Section 3. \end{enumerate} \end{definition}
\begin{proposition}[\cite{LezamaGallego2011}, Theorem 7]\label{coefficientes} If $A$ is a polynomial ring with coefficients in $R$ with respect to the set of indeterminates $\{x_1,\dots,x_n\}$, then $A$ is a skew PBW extension of $R$ if and only if the following conditions hold: \begin{enumerate}
\item[\rm (1)]for each $x^{\alpha}\in {\rm Mon}(A)$ and every $0\neq r\in R$, there exist unique elements $r_{\alpha}:=\sigma^{\alpha}(r)\in R\ \backslash\ \{0\}$, $p_{\alpha ,r}\in A$, such that $x^{\alpha}r=r_{\alpha}x^{\alpha}+p_{\alpha, r}$, where $p_{\alpha ,r}=0$, or $\deg(p_{\alpha ,r})<|\alpha|$ if $p_{\alpha , r}\neq 0$. If $r$ is left invertible, so is $r_\alpha$.
\item[\rm (2)]For each $x^{\alpha},x^{\beta}\in {\rm Mon}(A)$, there exist unique elements $c_{\alpha,\beta}\in R$ and $p_{\alpha,\beta}\in A$ such that $x^{\alpha}x^{\beta}=c_{\alpha,\beta}x^{\alpha+\beta}+p_{\alpha,\beta}$, where $c_{\alpha,\beta}$ is left invertible, $p_{\alpha,\beta}=0$, or $\deg(p_{\alpha,\beta})<|\alpha+\beta|$ if $p_{\alpha,\beta}\neq 0$. \end{enumerate} \end{proposition}
\begin{proposition}[\cite{Reyes2015}, Proposition 2.9] \label{lindass} If $\alpha=(\alpha_1,\dotsc, \alpha_n)\in \mathbb{N}^{n}$ and $r$ is an element of $R$, then {\small{\begin{align*} x^{\alpha}r = &\ x_1^{\alpha_1}x_2^{\alpha_2}\dotsb x_{n-1}^{\alpha_{n-1}}x_n^{\alpha_n}r = x_1^{\alpha_1}\dotsb x_{n-1}^{\alpha_{n-1}}\biggl(\sum_{j=1}^{\alpha_n}x_n^{\alpha_{n}-j}\delta_n(\sigma_n^{j-1}(r))x_n^{j-1}\biggr)\\ + &\ x_1^{\alpha_1}\dotsb x_{n-2}^{\alpha_{n-2}}\biggl(\sum_{j=1}^{\alpha_{n-1}}x_{n-1}^{\alpha_{n-1}-j}\delta_{n-1}(\sigma_{n-1}^{j-1}(\sigma_n^{\alpha_n}(r)))x_{n-1}^{j-1}\biggr)x_n^{\alpha_n}\\ + &\ x_1^{\alpha_1}\dotsb x_{n-3}^{\alpha_{n-3}}\biggl(\sum_{j=1}^{\alpha_{n-2}} x_{n-2}^{\alpha_{n-2}-j}\delta_{n-2}(\sigma_{n-2}^{j-1}(\sigma_{n-1}^{\alpha_{n-1}}(\sigma_n^{\alpha_n}(r))))x_{n-2}^{j-1}\biggr)x_{n-1}^{\alpha_{n-1}}x_n^{\alpha_n}\\ + &\ \dotsb + x_1^{\alpha_1}\biggl( \sum_{j=1}^{\alpha_2}x_2^{\alpha_2-j}\delta_2(\sigma_2^{j-1}(\sigma_3^{\alpha_3}(\sigma_4^{\alpha_4}(\dotsb (\sigma_n^{\alpha_n}(r))))))x_2^{j-1}\biggr)x_3^{\alpha_3}x_4^{\alpha_4}\dotsb x_{n-1}^{\alpha_{n-1}}x_n^{\alpha_n} \\ + &\ \sigma_1^{\alpha_1}(\sigma_2^{\alpha_2}(\dotsb (\sigma_n^{\alpha_n}(r))))x_1^{\alpha_1}\dotsb x_n^{\alpha_n}, \ \ \ \ \ \ \ \ \ \ \sigma_j^{0}:={\rm id}_R\ \ {\rm for}\ \ 1\le j\le n. \end{align*}}} \end{proposition}
\begin{remark}[\cite{Reyes2015}, Remark 2.10 (iv)] \label{juradpr} About Proposition \ref{lindass}, we have the following observation: if $X_i:=x_1^{\alpha_{i1}}\dotsb x_n^{\alpha_{in}}$ and $Y_j:=x_1^{\beta_{j1}}\dotsb x_n^{\beta_{jn}}$, when we compute every summand of $a_iX_ib_jY_j$ we obtain pro\-ducts of the coefficient $a_i$ with several evaluations of $b_j$ in $\sigma$'s and $\delta$'s depending of the coordinates of $\alpha_i$. This assertion follows from the expression:
\begin{align*} a_iX_ib_jY_j = &\ a_i\sigma^{\alpha_{i}}(b_j)x^{\alpha_i}x^{\beta_j} + a_ip_{\alpha_{i1}, \sigma_{i2}^{\alpha_{i2}}(\dotsb (\sigma_{in}^{\alpha_{in}}(b_j)))} x_2^{\alpha_{i2}}\dotsb x_n^{\alpha_{in}}x^{\beta_j} \\ + &\ a_i x_1^{\alpha_{i1}}p_{\alpha_{i2}, \sigma_3^{\alpha_{i3}}(\dotsb (\sigma_{{in}}^{\alpha_{in}}(b_j)))} x_3^{\alpha_{i3}}\dotsb x_n^{\alpha_{in}}x^{\beta_j} \\ + &\ a_i x_1^{\alpha_{i1}}x_2^{\alpha_{i2}}p_{\alpha_{i3}, \sigma_{i4}^{\alpha_{i4}} (\dotsb (\sigma_{in}^{\alpha_{in}}(b_j)))} x_4^{\alpha_{i4}}\dotsb x_n^{\alpha_{in}}x^{\beta_j}\\ + &\ \dotsb + a_i x_1^{\alpha_{i1}}x_2^{\alpha_{i2}} \dotsb x_{i(n-2)}^{\alpha_{i(n-2)}}p_{\alpha_{i(n-1)}, \sigma_{in}^{\alpha_{in}}(b_j)}x_n^{\alpha_{in}}x^{\beta_j} \\ + &\ a_i x_1^{\alpha_{i1}}\dotsb x_{i(n-1)}^{\alpha_{i(n-1)}}p_{\alpha_{in}, b_j}x^{\beta_j}. \end{align*} \end{remark}
\section{$\Sigma$-rigid rings and $(\Sigma,\Delta)$-compatible rings}\label{SigmarigidandSigmaDeltacompatible}
In this section we recall some results concerning $\Sigma$-rigid rings and $(\Sigma,\Delta)$-compatible rings and their relation with skew PBW extensions.
\begin{definition}[\cite{Reyes2015}, Definition 3.2] \label{generaldef2015} Let $B$ be a ring and $\Sigma$ a family of endomorphisms of $B$. $\Sigma$ is called a {\em rigid endomorphisms family}, if $r\sigma^{\alpha}(r)=0$ implies $r=0$, for every $r\in B$ and $\alpha\in \mathbb{N}^n$. A ring $B$ is called to be $\Sigma$-{\em rigid}, if there exists a rigid endomorphisms family $\Sigma$ of $B$. \end{definition}
The motivation to define $\Sigma$-rigid rings was to generalize the rigid rings defined by Krempa \cite{Krempa1996}. Now, if $\Sigma$ is a rigid endomorphisms family, then every element $\sigma_i\in \Sigma$ is a monomorphism. In fact, $\Sigma$-rigid rings are reduced rings: if $B$ is a $\Sigma$-rigid ring and $r^2=0$ for $r\in B$, then we have the equalities $0=r\sigma^{\alpha}(r^2)\sigma^{\alpha}(\sigma^{\alpha}(r))=r\sigma^{\alpha}(r)\sigma^{\alpha}(r)\sigma^{\alpha}(\sigma^{\alpha}(r))=r\sigma^{\alpha}(r)\sigma^{\alpha}(r\sigma^{\alpha}(r))$, i.e., $r\sigma^{\alpha}(r)=0$ and so $r=0$, that is, $B$ is reduced (note that there exists an endomorphism of a reduced ring which is not a rigid endomorphism, see \cite{HongKimKwak2000}, Example 9). With this in mind, we consider the family of injective endomorphisms $\Sigma$ and the family $\Delta$ of $\Sigma$-derivations in a skew PBW extension $A$ of a ring $R$ (Proposition \ref{sigmadefinition}). The notion of rigidness with another ring theoretical properties such as Baer, quasi-Baer, p.p and p.q have been investigated for skew PBW extensions in \cite{NinoReyes2017, Reyes2015, Reyes2018, ReyesSuarezClifford2017, ReyesSuarezUMA2018, ReyesYesica2018} (in the context of Ore extensions, the beautiful monograph \cite{Birkenmeieretal2013} contains a complete list of works on all these properties). Recall that if $A$ is a skew PBW extension of $R$ where the the elements $c_{i, j}$ are in\-ver\-ti\-ble in $R$, then $R$ is $\Sigma$-rigid if and only if $A$ is a reduced ring (\cite{Reyes2015}, Proposition 3.5).
\begin{proposition}[\cite{Reyes2015}, Lemma 3.3 and Corollary 3.4]\label{Reyes2015Lemma3.3} Let $R$ be an $\Sigma$-rigid ring and $a,b\in R$. Then: \begin{enumerate} \item [\rm (1)] If $ab=0$ then $a\sigma^{\alpha}(b)=\sigma^{\alpha}(a)b=0$, for any $\alpha\in \mathbb{N}^n$. \item [\rm (2)] If $ab=0$ then $a\delta^{\beta}(b)=\delta^{\beta}(a)b=0$, for any $\beta\in \mathbb{N}^n$. \item [\rm (3)] If $ab=0$ then $a\sigma^{\alpha}(\delta^{\beta}(b))=a\delta^{\beta}(\sigma^{\alpha}(b))=0$, for every $\alpha, \beta\in \mathbb{N}^n$. \item [\rm (4)] If $a\sigma^{\theta}(b)=\sigma^{\theta}(a)b=0$ for some $\theta\in \mathbb{N}^n$, then $ab=0$. \item [\rm (5)] If $A$ is a skew PBW extension over $R$, $ab=0\Rightarrow ax^{\alpha}bx^{\beta}=0$, for any elements $a,b\in R$ and each $\alpha, \beta\in \mathbb{N}^n$. \end{enumerate} \end{proposition}
Next we present the notion of $(\Sigma,\Delta)$-compatible rings which was introduced by the authors in \cite{ReyesSuarezUMA2018}.
\begin{definition}[\cite{ReyesSuarezUMA2018}, Definition 3.2]\label{Definition3.52008} Consider a ring $R$ with a family of endomorphisms $\Sigma$ and a family of $\Sigma$-derivations $\Delta$. Then, \begin{enumerate} \item [\rm (i)] $R$ is said to be $\Sigma$-{\em compatible}, if for each $a,b\in R$, $a\sigma^{\alpha}(b)=0$ if and only if $ab=0$, for every $\alpha\in \mathbb{N}^{n}$; \item [\rm (ii)] $R$ is said to be $\Delta$-{\em compatible}, if for each $a,b \in R$, $ab=0$ implies $a\delta^{\beta}(b)=0$, for every $\beta \in \mathbb{N}^{n}$. \end{enumerate} If $R$ is both $\Sigma$-compatible and $\Delta$-compatible, $R$ is called $(\Sigma, \Delta)$-{\em compatible}. \end{definition}
\begin{proposition}[\cite{ReyesSuarezUMA2018}, Proposition 3.8]\label{ReyesSuarezUMA2017Prop3.8} Let $R$ be a $(\Sigma, \Delta)$-compatible ring. For every $a, b \in R$, we have: \begin{enumerate} \item [\rm (1)] if $ab=0$, then $a\sigma^{\theta}(b) = \sigma^{\theta}(a)b=0$, for each $\theta\in \mathbb{N}^{n}$. \item [\rm (2)] If $\sigma^{\beta}(a)b=0$ for some $\beta\in \mathbb{N}^{n}$, then $ab=0$. \item [\rm (3)] If $ab=0$, then $\sigma^{\theta}(a)\delta^{\beta}(b)= \delta^{\beta}(a)\sigma^{\theta}(b) = 0$, for every $\theta, \beta\in \mathbb{N}^{n}$. \end{enumerate} \end{proposition}
From \cite{ReyesSuarezUMA2018}, Proposition 3.4, we know that every $\Sigma$-rigid ring is a $(\Sigma, \Delta)$-compatible ring. The converse is false as we can appreciated in \cite{ReyesSuarezUMA2018}, Example 3.6. In this way, $\Sigma$-rigid rings are contained strictly in $(\Sigma, \Delta)$-compatible rings. Nevertheless, these two notions coincide when the ring is assumed to be reduced, such as the following proposition establishes.
\begin{proposition}[\cite{ReyesSuarezUMA2018}, Theorem 3.9]\label{ReyesSuarez2018Theorem3.9} If $A$ is a skew PBW extension of a ring $R$, then the following statements are equivalent: (1) $R$ is reduced and $(\Sigma, \Delta)$-compatible. (2) $R$ is $\Sigma$-rigid. (3) $A$ is reduced. \end{proposition}
\section{Skew PBW extensions over weak symmetric rings}\label{weaksymmetricskewPBWextensions} In \cite{OuyangChen2010}, Definition 1, Ouyang and Chen introduced the notion of weak symmetric ring in the following way: a ring $B$ is called a {\em weak symmetric} ring, if $abc\in {\rm nil}(B) \Rightarrow acb\in {\rm nil}(B)$, for every elements $a, b, c\in R$. They proved that their notion extends the concept of symmetric ring, that is all symmetric rings are weak symmetric (\cite{OuyangChen2010}, Proposition 2.1). However, the converse of the assertion is false, i.e, there exists a weak symmetric ring which is not symmetric (\cite{OuyangChen2010}, Example 2.2). \\
With the aim of studying these notions of symmetry in the case of skew PBW extensions, we start with four results (Lemmas \ref{2010Lemma2.7} and \ref{2010Lemma2.8} and Theorems \ref{2010Lemma2.10} and \ref{2010Theorem2.11}) about nilpotent elements in skew PBW extensions. Our Lemma \ref{2010Lemma2.7} generalizes \cite{OuyangChen2010}, Lemma 2.7.
\begin{lemma}\label{2010Lemma2.7} If $R$ is a $(\Sigma,\Delta)$-compatible and reversible ring, then $ab\in {\rm nil}(R)$ implies that $a\sigma^{\alpha}(\delta^{\beta}(b))$ and $a\delta^{\beta}(\sigma^{\alpha}(b))$ also are elements of ${\rm nil}(R)$, for any $\alpha,\beta\in \mathbb{N}^{n}$. \end{lemma} \begin{proof} By assumption there exists a positive integer $k$ such that $(ab)^{k}=0$. Consider the following equalities: \begin{align*} (ab)^{k} = &\ ab ab \dotsb ab ab ab\ \ \ (k\ {\rm times})\\ = &\ abab \dotsb ab ab a\sigma^{\alpha}(\delta^{\beta}(b))\ \ \ ({\rm Proposition\ \ref{ReyesSuarezUMA2017Prop3.8}\ (3)})\\ = &\ a\sigma^{\alpha}(\delta^{\beta}(b)) ab ab ab \dotsb ab ab\ \ \ (R\ {\rm is\ reversible})\\ = &\ a\sigma^{\alpha}(\delta^{\beta}(b)) ab ab \dotsb ab a\sigma^{\alpha}(\delta^{\beta}(b))\ \ \ ({\rm Proposition\ \ref{ReyesSuarezUMA2017Prop3.8}\ (3)})\\ = &\ a\sigma^{\alpha}(\delta^{\beta}(b)) a\sigma^{\alpha}(\delta^{\beta}(b)) ab ab \dotsb ab\ \ \ (R\ {\rm is\ reversible}) \end{align*} Following this procedure we guarantee that the element $a\sigma^{\alpha}(\delta^{\beta}(b))$ belongs to ${\rm nil}(R)$. For the element $a\delta^{\beta}(\sigma^{\alpha}(b))$ the reasoning is completely similar. \end{proof}
The next lemma extends \cite{OuyangChen2010}, Lemma 2.8.
\begin{lemma}\label{2010Lemma2.8} If $R$ is a $(\Sigma,\Delta)$-compatible ring, then $a\sigma^{\theta}(b)\in {\rm nil}(R)$ implies $ab\in {\rm nil}(R)$, for every $\theta \in \mathbb{N}^{n}$ and each $a, b \in R$. \end{lemma} \begin{proof} Since $a\sigma^{\theta}(b)\in {\rm nil}(R)$, there exists a positive integer $k$ with $(a\sigma^{\theta}(b))^{k}=0$. We have the following assertions: \begin{align*} (a\sigma^{\theta}(b))^{k} = &\ a\sigma^{\theta}(b) a\sigma^{\theta}(b) \dotsb a\sigma^{\theta}(b) a\sigma^{\theta}(b)\ \ \ (k\ {\rm times})\\ = &\ a\sigma^{\theta}(b)a\sigma^{\theta}(b) \dotsb a\sigma^{\theta}(b) ab \ \ \ ({\rm Definition\ of}\ \Sigma-{\rm compatibility})\\ = &\ a\sigma^{\theta}(b) a\sigma^{\theta}(b) \dotsb a\sigma^{\theta}(b) \sigma^{\theta}(ab)\ \ \ ({\rm Proposition\ \ref{ReyesSuarezUMA2017Prop3.8}\ (1)})\\ = &\ a\sigma^{\theta}(b) a\sigma^{\theta}(b) \dotsb a \sigma^{\theta}(bab)\ \ \ (\sigma^{\theta}\ {\rm is\ an\ endomorphism\ of}\ R)\\ = &\ a\sigma^{\theta}(b) a\sigma^{\theta}(b) \dotsb abab\ \ \ ({\rm Definition\ of}\ \Sigma-{\rm compatibility}) \end{align*} If we continue in this way, we can see that the element $ab\in {\rm nil}(R)$, which concludes the proof. \end{proof}
We recall from \cite{LiuZhao2006}, Lemma 3.1, that if $B$ is a semicommutative ring, then ${\rm nil}(B)$ is an ideal of $B$. Our Theorem \ref{2010Lemma2.10} generalizes \cite{OuyangChen2010}, Lemma 2.10. We need to assume that the elements $c_{i,j}$ of Definition \ref{gpbwextension} (iv) are central in $R$. With the purpose of abbreviating, we will write o.t.l.t to mean {\em other terms less than} in the sense of monomial orders (Definition \ref{definitioncoefficients} (ii)).
\begin{theorem}\label{2010Lemma2.10} If $A$ is a skew PBW extension over a $(\Sigma, \Delta)$-compatible and reversible ring $R$, then for every element $f=\sum_{i=0}^{m} a_iX_i\in A$, $f\in {\rm nil}(A)$ if and only if $a_i\in {\rm nil}(R)$, for each $1\le i\le m$. \end{theorem} \begin{proof} Let $f\in A$ given as above and suppose that $f\in {\rm nil}(A)$ with $X_1 \prec X_2 \prec \dotsb \prec X_m$. Consider the notation established in Proposition \ref{coefficientes}. There exists a positive integer $k$ such that $f^{k} = (a_0 + a_1X_1 + \dotsb + a_mX_m)^{k}=0$. As an illustration, note that {\normalsize{\begin{align*} f^{2} = &\ (a_mX_m + \dotsb + a_1X_1 + a_0)(a_mX_m + \dotsb + a_1X_1 + a_0)\\ = &\ a_mX_ma_mX_m +\ {\rm o.t.l.t}\ {\rm exp}(X_m)\\ = &\ a_m[\sigma^{\alpha_m}(a_m)X_m + p_{\alpha_m, a_m}]X_m + \ {\rm o.t.l.t}\ {\rm exp}(X_m)\\ = &\ a_m\sigma^{\alpha_m}(a_m)X_mX_m + a_mp_{\alpha_m, a_m}X_m +\ {\rm o.t.l.t}\ {\rm exp}(X_m)\\ = &\ a_m\sigma^{\alpha_m}(a_m)[c_{\alpha_m, \alpha_m}x^{2\alpha_m} + p_{\alpha_m, \alpha_m}] + a_mp_{\alpha_m, a_m}X_m + {\rm o.t.l.t}\ {\rm exp}(X_m)\\ = &\ a_m\sigma^{\alpha_m}(a_m)c_{\alpha_m, \alpha_m}x^{2\alpha_m} + \ {\rm o.t.l.t}\ {\rm exp}(x^{2\alpha_m}), \end{align*}}}
and hence, {\normalsize{\begin{align*} f^{3} = &\ (a_m\sigma^{\alpha_m}(a_m)c_{\alpha_m, \alpha_m}x^{2\alpha_m} + \ {\rm o.t.l.t}\ {\rm exp}(x^{2\alpha_m})) (a_mX_m + \dotsb + a_1x_1 + a_0)\\ = &\ a_m\sigma^{\alpha_m}(a_m)c_{\alpha_m, \alpha_m}x^{2\alpha_m}a_mX_m + \ {\rm o.t.l.t}\ {\rm exp}(x^{3\alpha_m})\\ = &\ a_m\sigma^{\alpha_m}(a_m)c_{\alpha_m, \alpha_m}[\sigma^{2\alpha_m}(a_m)x^{2\alpha_m} + p_{2\alpha_m, a_m}]X_m + \ {\rm o.t.l.t}\ {\rm exp}(x^{3\alpha_m})\\ = &\ a_m\sigma^{\alpha_m}(a_m)c_{\alpha_m,\alpha_m}\sigma^{2\alpha_m}(a_m)x^{2\alpha_m}X_m + \ {\rm o.t.l.t}\ {\rm exp}(x^{3\alpha_m})\\ = &\ a_m\sigma^{\alpha_m}(a_m)c_{\alpha_m,\alpha_m}\sigma^{2\alpha_m}(a_m)[c_{2\alpha_m, \alpha_m}x^{3\alpha_m} + p_{2\alpha_m,\alpha_m}]\\ = &\ a_m\sigma^{\alpha_m}(a_m)c_{\alpha_m,\alpha_m}\sigma^{2\alpha_m}(a_m)c_{2\alpha_m, \alpha_m}x^{3\alpha_m} + \ {\rm o.t.l.t}\ {\rm exp}(x^{3\alpha_m}). \end{align*}}} Continuing in this way, one can show that for $f^{k}$, \[ f^{k} = \biggl \{a_m\prod_{l=1}^{k-1}\sigma^{l\alpha_m}(a_m)c_{l\alpha_m, \alpha_m}x^{k\alpha_m}\biggr\} + \ {\rm o.t.l.t}\ {\rm exp}(x^{k\alpha_m}), \] whence $0={\rm lc}(f^{k}) = a_m\prod_{l=1}^{k-1}\sigma^{l\alpha_m}(a_m)c_{l\alpha_m, \alpha_m}$, and since the elements $c$'s are central in $R$ and left invertible (Proposition \ref{coefficientes}), we have $0={\rm lc}(f^{k}) = a_m\prod_{l=1}^{k-1}\sigma^{l\alpha_m}(a_m)$. Using the $\Sigma$-compatibility of $R$, we obtain $a_m\in {\rm nil}(R)$.
Now, since {\small{\begin{align*} f^{k} = &\ ((a_0 + a_1X_1 + \dotsb + a_{m-1}X_{m-1}) + a_mX_m)^{k} \\ = &\ ((a_0 + a_1X_1 + \dotsb + a_{m-1}X_{m-1}) + a_mX_m)((a_0 + a_1X_1 + \dotsb + a_{m-1}X_{m-1}) + a_mX_m)\\ &\ \dotsb ((a_0 + a_1X_1 + \dotsb + a_{m-1}X_{m-1}) + a_mX_m)\ \ \ \ \ \ (k\ {\rm times})\\ = &\ [(a_0 + a_1X_1 + \dotsb + a_{m-1}X_{m-1})^{2} + (a_0 + a_1X_1 + \dotsb + a_{m-1}X_{m-1})a_mX_m\\ &\ + a_mX_m(a_0 + a_1X_1 + \dotsb + a_{m-1}X_{m-1}) + a_mX_ma_mX_m]\\ &\ \dotsb ((a_0 + a_1X_1 + \dotsb + a_{m-1}X_{m-1}) + a_mX_m)\\ = &\ (a_0 + a_1X_1 + \dotsb + a_{m-1}X_{m-1})^{k} + h, \end{align*}}} where $h$ is an element of $A$ which involves products of monomials with the term $a_mX_m$ on the left and the right, by Proposition \ref{lindass}, Remark \ref{juradpr} and having in mind that $a_m\in {\rm nil}(R)$, which is an ideal of $R$ (remember that reversible implies semicommutative), the expression for $f^{k}$ reduces to $f^k = (a_0 + a_1X_1 + \dotsb + a_{m-1}X_{m-1})^{k}$. Using a similar reasoning as above, one can prove that \[ f^{k} = a_{m-1}\prod_{l=1}^{k-1}\sigma^{l(\alpha_{m-1})}(a_{m-1})c_{l(\alpha_{m-1}), \alpha_{m-1}}x^{k\alpha_{m-1}} + \ {\rm o.t.l.t}\ {\rm exp}(x^{k\alpha_{m-1}}). \] Hence ${\rm lc}(f^{k}) = a_{m-1}\prod_{l=1}^{k-1}\sigma^{l\alpha_{m-1}}(a_{m-1})c_{l\alpha_{m-1}, \alpha_{m-1}}$, and so $a_{m-1}\in {\rm nil}(R)$. If we repeat this argument, it follows that $a_i\in {\rm nil}(R)$, for $0\le i\le m$.
Conversely, suppose that $a_i\in {\rm nil}(R)$, for every $i$. If $k_i$ is the minimum integer positive such that $a_i^{k_i} = 0$, for every $i$, let $k:={\rm max}\{k_i\mid 1\le i\le n\}$. It is clear that $a_i^{k}=0$, for all $i$. Let us prove that $f^{(m+1){k}+1} = 0$, and hence, $f\in {\rm nil}(A)$. Since the expression for $f$ have $m+1$ terms, when we realize the product $f^{(m+1){k}+1}$ we have sums of products of the form \begin{equation}\label{rigoo} a_{i,1}X_{i,1}a_{i,2}X_{i,2}\dotsb a_{i, (m+1){k}}X_{i, (m+1){k}}a_{i,(m+1){k}+1}X_{i,(m+1){k}+1}. \end{equation} Note that there are exactly $(m+1)^{(m+1)k+1}$ products of the form (\ref{rigoo}). Now, since when we compute $f^{(m+1){k}+1}$ every product as (\ref{rigoo}) involves at least $k$ elements $a_i$, for some $i$, then every one of these products is equal to zero by Proposition \ref{lindass}, Remark \ref{juradpr} and the $(\Sigma, \Delta)$-compatibility of $R$ (more exactly, Proposition \ref{ReyesSuarezUMA2017Prop3.8}). In this way, every term of $f^{(m+1){k}+1}$ is equal to zero, and hence $f\in {\rm nil}(A)$. \end{proof}
The next theorem generalizes \cite{OuyangChen2010}, Theorem 2.11. We denote ${\rm nil}(R)A:=\{f\in A\mid f= a_0 + a_1X_1 + \dotsb + a_mX_m,\ a_i\in {\rm nil}(R)\}$.
\begin{theorem}\label{2010Theorem2.11} Let $A$ be a skew PBW extensions over a reversible and $(\Sigma,\Delta)$-compatible ring. If $f=\sum_{i=0}^{m} a_iX_i, g=\sum_{j=0}^{t} b_jY_j$ and $h=\sum_{k=0}^{l}c_kZ_k$ are elements of $A$, and $r$ is any element of $R$, then we have the following assertions: \begin{enumerate} \item [\rm (1)] $fg\in {\rm nil}(A) \Leftrightarrow a_ib_j\in {\rm nil}(R)$, for all $i, j$. \item [\rm (2)] $fgr\in {\rm nil}(A) \Leftrightarrow a_ib_jr\in {\rm nil}(R)$, for all $i, j$. \item [\rm (3)] $fgh\in {\rm nil}(A)\Leftrightarrow a_ib_jc_k\in {\rm nil}(R)$, for all $i, j, k$. \end{enumerate} \end{theorem} \begin{proof} (1) As we see in the proof of Theorem \ref{2010Lemma2.10}, ${\rm nil}(A)\subseteq {\rm nil}(R)A$. With this in mind, consider two elements $f, g \in A$ given by $f=\sum_{i=0}^{m} a_iX_i$ and $g=\sum_{j=0}^{t} b_jY_j$ with $fg\in {\rm nil}(A)$. Let $X_i:=x_1^{\alpha_{i1}}\dotsb x_n^{\alpha_{in}}, Y_j:=x_1^{\beta_{j1}}\dotsb x_n^{\beta_{jn}}$, for all $i, j$. We have \begin{equation*} fg = \sum_{k=0}^{m+t} \biggl( \sum_{i+j=k} a_iX_ib_jY_j\biggr) \in {\rm nil}(A)\subseteq {\rm nil}(R)A, \end{equation*} and ${\rm lc}(fg)= a_m\sigma^{\alpha_m}(b_t)c_{\alpha_m, \beta_t}\in {\rm nil}(R)$. Since the elements $c_{i,j}$ are in the center of $R$, then $c_{\alpha_m,\beta_t}$ are also in the center of $R$, whence $a_m\sigma^{\alpha_m}(b_t)\in {\rm nil}(R)$, and by Lemma \ref{2010Lemma2.8} it follows that $a_mb_t\in {\rm nil}(R)$. The idea is to prove that $a_pb_q\in {\rm nil}(R)$, for $p+q\ge 0$. We proceed by induction. Suppose that $a_pb_q\in {\rm nil}(R)$, for $p+q=m+t, m+t-1, m+t-2, \dotsc, k+1$, for some $k>0$. By Lemma \ref{2010Lemma2.7}, we obtain $a_pX_pb_qY_q\in {\rm nil}(R)A$ for these values of $p+q$. In this way, it is sufficient to consider the sum of the products $a_uX_ub_vY_v$, where $u+v=k, k-1,k-2,\dotsc, 0$. Fix $u$ and $v$. Consider the sum of all terms of $fg$ having exponent $\alpha_u+\beta_v$. By Proposition \ref{lindass}, Remark \ref{juradpr} and the assumption $fg\in {\rm nil}(A)$, we know that the sum of all coefficients of all these terms can be written as {\small{\begin{equation}\label{Feder4} a_u\sigma^{\alpha_u}(b_v)c_{\alpha_u, \beta_v} + \sum_{\alpha_{u'} + \beta_{v'} = \alpha_u + \beta_v} a_{u'}\sigma^{\alpha_{u'}} ({\rm \sigma's\ and\ \delta's\ evaluated\ in}\ b_{v'})c_{\alpha_{u'}, \beta_{v'}}\in {\rm nil}(R). \end{equation}}} As we suppose above, $a_pb_q\in {\rm nil}(R)$ for $p+q=m+t, m+t-1, \dotsc, k+1$, so Lemma \ref{2010Lemma2.7} guarantees that the product $a_p(\sigma'$s and $\delta'$s evaluated in $b_q$), for any order of $\sigma'$s and $\delta'$s, is an element of ${\rm nil}(R)$. Since $R$ is reversible, then $({\rm \sigma's\ and\ \delta's\ evaluated\ in}\ b_{q})a_p\in {\rm nil}(R)$. In this way, multiplying (\ref{Feder4}) on the right by $a_k$, and using the fact that the elements $c$'s are in the center of $R$, we obtain that the sum {\small{\begin{equation}\label{do1234} a_u\sigma^{\alpha_u}(b_v)a_kc_{\alpha_u, \beta_v} + \sum_{\alpha_{u'} + \beta_{v'} = \alpha_u + \beta_v} a_{u'}\sigma^{\alpha_{u'}} ({\rm \sigma's\ and\ \delta's\ evaluated\ in}\ b_{v'})a_kc_{\alpha_{u'}, \beta_{v'}} \end{equation}}} is an element of ${\rm nil}(R)$, whence, $a_u\sigma^{\alpha_u}(b_0)a_k\in {\rm nil}(R)$. Since $u+v=k$ and $v=0$, then $u=k$, so $a_k\sigma^{\alpha_k}(b_0)a_k\in {\rm nil}(R)$, from which $a_k\sigma^{\alpha_k}(b_0)\in {\rm nil}(R)$ and hence $a_kb_0\in {\rm nil}(R)$ by Lemma \ref{2010Lemma2.8}. Therefore, we now have to study the expression (\ref{Feder4}) for $0\le u \le k-1$ and $u+v=k$. If we multiply (\ref{do1234}) on the right by $a_{k-1}$, then {\small{\[ a_u\sigma^{\alpha_u}(b_v)a_{k-1}c_{\alpha_u, \beta_v} + \sum_{\alpha_{u'} + \beta_{v'} = \alpha_u + \beta_v} a_{u'}\sigma^{\alpha_{u'}} ({\rm \sigma's\ and\ \delta's\ evaluated\ in}\ b_{v'})a_{k-1}c_{\alpha_{u'}, \beta_{v'}} \]}} is also an element of ${\rm nil}(R)$. Using a similar reasoning as above, we can see that the element $a_u\sigma^{\alpha_u}(b_1)a_{k-1}c_{\alpha_u, \beta_1}$ belongs to ${\rm nil}(R)$. Since the elements $c$'s are central and left invertible, $a_u\sigma^{\alpha_u}(b_1)a_{k-1}\in {\rm nil}(R)$, and using the fact $u=k-1$, we have $a_{k-1}\sigma^{\alpha_{k-1}}(b_1)\in {\rm nil}(R)$, from which $a_{k-1}b_1\in {\rm nil}(R)$. Continuing in this way we prove that $a_ib_j\in {\rm nil}(R)$, for $i+j=k$. Therefore $a_ib_j\in {\rm nil}(R)$, for $0\le i\le m$ and $0\le j\le t$.\\
Conversely, for the elements $f, g$ above, suppose that $a_ib_j\in {\rm nil}(R)$. From Lemma \ref{2010Lemma2.7} we know that $a\sigma^{\alpha}(\delta^{\beta}(b))$ and $a\delta^{\beta}(\sigma^{\alpha}(b))$ are elements of ${\rm nil}(R)$, for every $\alpha,\beta\in \mathbb{N}^{n}$. Now, having in mind that for every product of the form $a_iX_ib_jY_j$, where $X_i:=x_1^{\alpha_{i1}}\dotsb x_n^{\alpha_{in}}$ and $Y_j:=x_1^{\beta_{j1}}\dotsb x_n^{\beta_{jn}}$, we have the following equality
\begin{align*} a_iX_ib_jY_j = &\ a_i\sigma^{\alpha_i}(b_j)x^{\alpha_i}x^{\beta_j} + a_ip_{\alpha_{i1}, \sigma_{i2}^{\alpha_{i2}}(\dotsb (\sigma_{in}^{\alpha_{in}}(b_j)))} x_2^{\alpha_{i2}}\dotsb x_n^{\alpha_{in}}x^{\beta_j} \\ + &\ a_i x_1^{\alpha_{i1}}p_{\alpha_{i2}, \sigma_3^{\alpha_{i3}}(\dotsb (\sigma_{{in}}^{\alpha_{in}}(b_j)))} x_3^{\alpha_{i3}}\dotsb x_n^{\alpha_{in}}x^{\beta_j} \\ + &\ a_i x_1^{\alpha_{i1}}x_2^{\alpha_{i2}}p_{\alpha_{i3}, \sigma_{i4}^{\alpha_{i4}} (\dotsb (\sigma_{in}^{\alpha_{in}}(b_j)))} x_4^{\alpha_{i4}}\dotsb x_n^{\alpha_{in}}x^{\beta_j}\\ + &\ \dotsb + a_i x_1^{\alpha_{i1}}x_2^{\alpha_{i2}} \dotsb x_{i(n-2)}^{\alpha_{i(n-2)}}p_{\alpha_{i(n-1)}, \sigma_{in}^{\alpha_{in}}(b_j)}x_n^{\alpha_{in}}x^{\beta_j} \\ + &\ a_i x_1^{\alpha_{i1}}\dotsb x_{i(n-1)}^{\alpha_{i(n-1)}}p_{\alpha_{in}, b_j}x^{\beta_j}, \end{align*} by Proposition \ref{lindass}, when we compute every summand of $a_iX_ib_jY_j$ we obtain products of the coefficient $a_i$ with several evaluations of $b_j$ in $\sigma$'s and $\delta$'s depending of the coordinates of $\alpha_i$ (Remark \ref{juradpr}), and since $a\sigma_i(\delta^{\beta_i}(b))$ and $a\delta^{\beta_i}(\sigma^{\alpha_i}(b))$ are elements of ${\rm nil}(R)$, then every coefficient of each term of the expansion $fg$ given by \[ fg = \sum_{k=0}^{m+t}\biggl(\sum_{i+j=k}a_iX_ib_jY_j\biggr), \] is an element of ${\rm nil}(R)$. Therefore, Theorem \ref{2010Lemma2.10} implies that the product $fg$ is an element of $R$.\\
(2) Let $g=b_0 + b_1Y_1 + \dotsb + b_tY_t$ be an element of $A$ with $Y_t\succ \dotsb \succ Y_1$. Then \begin{align*} gr = &\ (b_0+b_1Y_1 + \dotsb + b_tY_t)r \\ = &\ b_0r + b_1Y_1r + \dotsb + b_tY_tr \\ = &\ b_0r + b_1(\sigma^{\beta_1}(r)Y_1 + p_{\beta_1, r}) + \dotsb + b_t(\sigma^{\beta_t}(r)Y_t + p_{\beta_t, r})\\ = &\ b_0r + b_1\sigma^{\beta_1}(r)Y_1 + b_1p_{\beta_1, r} + \dotsb + b_t\sigma^{\beta_t}(r)Y_t + b_tp_{\beta_t, r} \end{align*}
where $p_{\beta_j,r}=0$, or $\deg(p_{\beta_j,r})<|\alpha|$ if $p_{\beta_j, r}\neq 0$, for $j=1, \dotsc, t$ (Proposition \ref{coefficientes}). Note that ${\rm lc}(gr) = b_t\sigma^{\beta_t}(r)$. Then {\small{\begin{align*} fgr = &\ (a_0 + a_1X_1 + \dotsb + a_mX_m)(b_0r + b_1\sigma^{\beta_1}(r)Y_1 + b_1p_{\beta_1, r} + \dotsb + b_t\sigma^{\beta_t}(r)Y_t + b_tp_{\beta_t, r})\notag \\ = &\ a_0b_0r + a_0b_1\sigma^{\beta_1}(r)Y_1 + a_0b_1p_{\beta_1,r} + \dotsb + a_0b_t\sigma^{\beta_t}(r)Y_t + a_0b_tp_{\beta_t,r} \notag \\ + &\ a_1X_1b_0r + a_1X_1b_1\sigma^{\beta_1}(r)Y_1 + a_1X_1b_1p_{\beta_1,r} + \dotsb + a_1X_1b_t\sigma^{\beta_t}(r)Y_t + a_1X_1b_tp_{\beta_t,r}\\ + &\ \dotsb + a_mX_mb_0r + a_mX_mb_1\sigma^{\beta_1}(r)Y_1 + a_mX_mb_1p_{\beta_1,r} + \dotsb + a_mX_mb_t\sigma^{\beta_t}(r)Y_t \\ + &\ a_mX_mb_tp_{\beta_t,r}\notag \\ = &\ a_0b_0r + a_0b_1\sigma^{\beta_1}(r)Y_1 + a_0b_1p_{\beta_1,r} + \dotsb + a_0b_t\sigma^{\beta_t}(r)Y_t + a_0b_tp_{\beta_t,r}\notag \\ + &\ a_1[\sigma^{\alpha_1}(b_0r)X_1 + p_{\alpha_1,b_0r}] + a_1[\sigma^{\alpha_1}(b_1\sigma^{\beta_1}(r))X_1 + p_{\alpha_1, b_1\sigma^{\beta_1}(r)}]Y_1 \\ + &\ a_1[\sigma^{\alpha_1}(b_1) + p_{\alpha_1,b_1}]p_{\beta_1,r} + \dotsb + a_1[\sigma^{\alpha_1}(b_t\sigma^{\beta_t}(r))X_1 + p_{\alpha_1,b_t\sigma^{\beta_t}(r)}]Y_t\\ + &\ a_1[\sigma^{\alpha_1}(b_t)X_1 + p_{\alpha_1,b_t}]p_{\beta_t, r} + \dotsb + a_m[\sigma^{\alpha_m}(b_0r) + p_{\alpha_m,b_0r}] \\ + &\ a_m[\sigma^{\alpha_m}(b_1\sigma^{\beta_1}(r)X_1 + p_{\alpha_1, b_1\sigma^{\beta_1}(r)}]Y_1 + a_m[\sigma^{\alpha_m}(b_1)X_m + p_{\alpha_m,b_1}]p_{\beta_1,r}\\ + &\ \dotsb + a_m[\sigma^{\alpha_m}(b_t\sigma^{\beta_t}(r))X_m + p_{\alpha_m,b_t\sigma^{\beta_t}(r)}]Y_t + a_m[\sigma^{\alpha_m}(b_t)X_m + p_{\alpha_m, b_t}]p_{\beta_t,r}, \end{align*}}} whence ${\rm lc}(fgr) = a_m\sigma^{\alpha_m}(b_t\sigma^{\beta_t}(r))$, and since $R$ is $\Sigma$-compatible, Lemma \ref{2010Lemma2.8} implies that $a_mb_tr\in {\rm nil}(R)$. Now, Lemma \ref{2010Lemma2.7} guarantees that every term of any polynomial containing the product $a_mb_tr$ in the expression above for $fgr$ is an element of ${\rm nil}(R)A$. In this way, using an monomial order we can repeat this argument for the next monomial of $fgr$ less than ${\rm lc}(fgr)$, and continuing this process until the first monomial to obtain that the elements $a_ib_jr$ are in $\in {\rm nil}(R)$, for all $i, j$.
Conversely, suppose that $a_ib_jr\in {\rm nil}(R)$, for every $i, j$, as above. As we saw above, \begin{align*} gr = &\ (b_0+b_1Y_1 + \dotsb + b_tY_t)r \\ = &\ b_0r + b_1Y_1r + \dotsb + b_tY_tr \\ = &\ b_0r + b_1(\sigma^{\beta_1}(r)Y_1 + p_{\beta_1, r}) + \dotsb + b_t(\sigma^{\beta_t}(r)Y_t + p_{\beta_t, r})\\ = &\ b_0r + b_1\sigma^{\beta_1}(r)Y_1 + b_1p_{\beta_1, r} + \dotsb + b_t\sigma^{\beta_t}(r)Y_t + b_tp_{\beta_t, r} \end{align*}
where $p_{\beta_j,r}=0$, or $\deg(p_{\beta_j,r})<|\alpha|$ if $p_{\beta_j, r}\neq 0$, for $j=1, \dotsc, t$. Since $a_ib_jr\in {\rm nil}(R)$, for every $i, j$, Lemma \ref{2010Lemma2.7} implies that $a_ib_j\sigma^{\alpha}(\delta^{\beta}(r))$ and $a_ib_j\delta^{\beta}(\sigma^{\alpha}(r))$ are elements of ${\rm nil}(R)$, for every $\alpha, \beta\in \mathbb{N}^{n}$. In this way, Proposition \ref{lindass} and Remark \ref{juradpr} applied to expression above for the product $fgr$ imply that every one of these summands have coefficients in ${\rm nil}(R)$, and since ${\rm nil}(R)$ is an ideal of $R$ because $R$ is reversible, Theorem \ref{2010Lemma2.7} shows that $fgr\in {\rm nil}(A)$. \\
(3) The equivalence follows from (1) and (2) considering the product $gh$ as the only element $p\in A$. \end{proof}
\begin{remark} About Theorem \ref{2010Theorem2.11} (1) we have the following two important observations. (a) In \cite{Reyes2018}, Definition 3.1, the first author introduced the skew $\Pi$-Armendariz rings in the following way: If $A$ is a skew PBW extension over a ring $R$, then $R$ is called a {\em skew}-$\Pi$ {\em Armendariz ring}, if for elements $f=\sum_{i=0}^{m} a_iX_i,\ g=\sum_{j=0}^{t} b_jY_j$ of $A$, $fg\in {\rm nil}(A)$ implies that $a_ib_j\in {\rm nil}(R)$, for every $0\le i\le m$ and $0\le j\le t$. The importance of Theorem \ref{2010Theorem2.11} is explicited, since we are proving in this theorem that skew PBW extensions over skew $\Pi$-Armendariz rings are contained in skew PBW extensions over reversible and $(\Sigma,\Delta)$-compatible rings. (b) In \cite{ReyesSuarezUMA2018}, Definition 4.1, the authors introduced the condition (SA1): if $A$ is a skew PBW extension of $R$, we say that $R$ satisfies the condition (SA1), if whenever $fg=0$ for $f=a_0+a_1X_1+\dotsb + a_mX_m$ and $g=b_0 + b_1Y_1 + \dotsb + b_tY_t$ elements of $A$, then $a_ib_j = 0$, for every $i, j$. It is clear that Theorem \ref{2010Theorem2.11} extends this condition. \end{remark} The next theorem extends \cite{OuyangChen2010}, Theorem 2.12.
\begin{theorem}\label{2010Theorem 2.12} If $A$ is a skew PBW extension over a reversible and $(\Sigma,\Delta)$-compatible ring, then $R$ is weak symmetric if and only if $A$ is weak symmetric. \end{theorem} \begin{proof} Having in mind that a subring of a weak symmetric ring is also a weak symmetric ring, we will only prove one implication. Suppose that $R$ is a weak symmetric ring. If $f=\sum_{i=0}^{s} a_iX_i, g=\sum_{j=0}^{t} b_jY_j$ and $h=\sum_{k=0}^{l}c_kZ_k$ are elements of $A$ with $fgh\in {\rm nil}(A)$, then Theorem \ref{2010Theorem2.11} implies that $a_ib_jc_k\in {\rm nil}(r)$, for every $i, j, k$, and hence $a_ic_kb_j\in {\rm nil}(R)$, for each $i, j, k$, since $R$ is weak symmetric. Finally, Theorem \ref{2010Theorem2.11} shows that $fhg\in {\rm nil}(A)$. \end{proof}
\begin{corollary} If $R$ is a $\Sigma$-rigid ring, then $R$ is weak symmetric if and only if $A$ is weak symmetric. \end{corollary} \begin{proof} Since we have the implications reduced $\Rightarrow$ symmetric $\Rightarrow$ weak symmetric, then the assertion follows from Theorem \ref{2010Theorem 2.12}. \end{proof}
\begin{corollary}[\cite{OuyangChen2010}, Corollaries 2.13 and 2.14] Let $B$ be a reversible ring. Then we have the following: \begin{enumerate} \item [\rm (1)] $B$ is weak symmetric if and only if $B[x]$ is weak symmetric. \item [\rm (2)] If $B$ is $\sigma$-compatible, then $B$ is weak symmetric if and only if $B[x;\sigma]$ is weak symmetric. \item [\rm (3)] If $B$ is $\delta$-compatible, then $B$ is weak symmetric if and only if the differential polynomial ring $B[x;\sigma]$ is weak symmetric. \item [\rm (4)] Let $\alpha$ be an endomorphism and $\delta$ and $\alpha$-derivation of $R$. If $R$ is $\alpha$-rigid, then $R$ is weak symmetric if and only if $R[x;\alpha,\delta]$ is weak symmetric. \end{enumerate} \end{corollary}
With the aim of establishing Theorems \ref{2010Theorem2.17} and \ref{2010Theorem 3.9}, we need to formulate a criterion which allows us to extend the family $\Sigma$ of injective endomorphisms, and the family of $\Sigma$-derivations $\Delta$ of the ring $R$ to the ring $A$. For the next proposition consider the injective endomorphisms $\sigma_i\in \Sigma$, and the $\sigma_i$-derivations $\delta_i\in \Delta$ $(1\le i\le n)$ formulated in Proposition \ref{sigmadefinition} (compare with \cite{Artamonov2015} where the derivations of skew PBW extensions were computed partially). We include its proof with the objective of appreciating the importance of the assumptions established in the result.
\begin{proposition}[\cite{ReyesSuarezClifford2017}, Theorem 5.1]\label{ReyesSuarez2017CliffordTheorem5.1} Let $A$ be a skew PBW extension of a ring $R$. Suppose that $\sigma_i\delta_j=\delta_j\sigma_i,\ \delta_i\delta_j=\delta_j\delta_i$, and $\delta_k(c_{i,j}) = \delta_k(r_l^{(i,j)}) = 0$, for $1\le i, j, l\le n$, where $c_{i,j}$ and $r_l^{(i,j)}$ are as in Definition \ref{gpbwextension}. If $\overline{\sigma_{k}}:A\to A$ and $\overline{\delta_k}:A\to A$ are the functions given by $\overline{\sigma_{k}}(f):=\sigma_k(a_0)+\sigma_k(a_1)X_1 + \dotsb + \sigma_k(a_m)X_m$ and $\overline{\delta_k}(f):=\delta_k(a_0) + \delta_k(a_1)X_1 + \dotsb + \delta_k(a_m)X_m$, for every $f=a_0 + a_1X_1+\dotsb + a_mX_m\in A$, respectively, and $\overline{\sigma_k}(r):=\sigma_i(k)$, for every $1\le i\le n$, then $\overline{\sigma_k}$ is an injective endomorphism of $A$ and $\overline{\delta_k}$ is a $\overline{\sigma_k}$-derivation of $A$. Let $\overline{\Sigma}:=\{\overline{\sigma_1},\dotsc, \overline{\sigma_n}\} $ and $\overline{\Delta}:=\{\overline{\delta_1},\dotsc, \overline{\delta_n}\}$. \end{proposition} \begin{proof} It is clear that $\overline{\sigma_i}$ is an injective endomorphism of $A$, and that $\overline{\delta_i}$ is an additive map of $A$, for every $1\le i\le n$. Next, we show that $\overline{\delta_i}(fg)=\overline{\sigma_i}(f)\overline{\delta_i}(g) + \overline{\delta_i}(f)g$, for $f,g\in A$.
Consider the elements $f=a_0 + a_1X_1+a_2X_2+\dotsb + a_mX_m$ and $g=b_0 + b_1Y_1+b_2Y_2+\dotsb + b_tY_t$. Since $\overline{\sigma_k}$ and $\overline{\delta_k}$ are additive, for every $i$, it is enough to show that \begin{align}\label{serpi} \overline{\delta_k}(a_iX_ib_jY_j) = \overline{\sigma_k}(a_iX_i)\overline{\delta_k}(b_jY_j) + \overline{\delta_k}(a_iX_i)b_jY_j, \end{align} for every $1\le i, j \le n$. As an illustration of the necessity of the assumptions above, consider the next particular computations: {\small{\begin{align} \overline{\delta_k}(bx_jax_i) = &\ \overline{\delta_k}b(\sigma_j(a)x_j + \delta_j(a))x_i) = \overline{\delta_k}(b\sigma_j(a)x_jx_i + b\delta_j(a)x_1)\notag \\ = &\ \overline{\delta_k}\biggl(b\sigma_j(a)\biggl(c_{i,j}x_ix_j + r_0 + \sum_{l=1}^{n}r_lx_l\biggr) + b\delta_j(a)x_i\biggr)\notag \\ = &\ \overline{\delta_k}\biggl(b\sigma_j(a)c_{i,j}x_ix_j + b\sigma_j(a)r_0 + b\sigma_j(a)\sum_{l=1}^{n}r_lx_l + b\delta_j(a)x_i\biggr) \notag \\ = &\ \delta_k(b\sigma_j(a)c_{i,j})x_ix_j + \delta_k(b\sigma_j(a)r_0) + \sum_{l=1}^{n}\delta_k(b\sigma_j(a)r_l)x_l + \delta_k(b\delta_j(a))x_i\notag \end{align}}} or what is the same, {\small{\begin{align} \overline{\delta_k}(bx_jax_i) = &\ \sigma_k(b\sigma_j(a))\delta_j(c_{i,j})x_ix_j + \delta_k(b\sigma_j(a))c_{i,j}x_ix_j + \sigma_k(b\sigma_j(a))\delta_i(r_0) \notag \\ + &\ \delta_k(b\sigma_j(a))r_0 + \sum_{l=1}^{n}\sigma_k(b\sigma_j(a))\delta_i(r_l)x_l + \sum_{l=1}^{n} \delta_k(b\sigma_j(a))r_lx_l \notag \\ + &\ \sigma_k(b)\delta_k(\delta_j(a))x_i + \delta_k(b)\delta_j(a)x_i\notag \\ &\ \sigma_k(b)\sigma_k(\sigma_j(a))\delta_j(c_{i,j})x_ix_j + \sigma_k(b)\delta_k(\sigma_j(a))c_{i,j}x_ix_j + \delta_k(b)\sigma_j(a)c_{i,j}x_ix_j \notag \\ + &\ \sigma_k(b)\sigma_k(\sigma_j(a))\delta_i(r_0) + \sigma_k(b)\delta_k(\sigma_j(a))r_0 + \delta_k(b)\sigma_j(a)r_0 \notag \\ + &\ \sum_{l=1}^{n} \sigma_k(b)\sigma_k(\sigma_j(a))\delta_i(r_l)x_l + \sum_{l=1}^{n} \sigma_k(b)\delta_k(\sigma_j(a))r_lx_l +\sum_{l=1}^{n}\delta_k(b)\sigma_j(a)r_lx_l \notag \\ + &\ \sigma_k(b)\delta_k(\delta_j(a))x_i + \delta_k(b)\delta_j(a)x_i.\label{colooo} \end{align}}} On the other hand, {\small{\begin{align} \overline{\sigma_k}(bx_j)\overline{\delta_k}(ax_i) + \overline{\delta_k}(bx_j)ax_i = &\ \sigma_k(b)x_j \delta_k(a)x_i + \delta_k(b)x_jax_i\notag \\ = &\ \sigma_k(b)(\sigma_j(\delta_k(a))x_j + \delta_j(\delta_k(a)))x_i \notag \\ + &\ \delta_k(b)(\sigma_j(a)x_j + \delta_j(a))x_i\notag\\ = &\ \sigma_k(b)\sigma_j(\delta_k(a))x_jx_i + \sigma_k(b)\delta_j(\delta_k(a))x_i \notag \\ + &\ \delta_k(b)\sigma_j(a)x_jx_i + \delta_k(b)\delta_j(a)x_i\notag \\ = &\ \sigma_k(b)\sigma_j(\delta_k(a))\biggl(c_{i,j}x_ix_j + r_0 + \sum_{l=1}^{n}r_lx_l\biggr)\notag \\ + &\ \sigma_k(b)\delta_j(\delta_k(a))x_i \notag \\ + &\ \delta_k(b)\sigma_j(a)\biggl(c_{i,j}x_ix_j + r_0 + \sum_{l=1}^{n} r_lx_l\biggr) + \delta_k(b)\delta_j(a)x_i\notag\\
= &\ \sigma_k(b)\sigma_j(\delta_k(a))c_{i,j}x_ix_j + \sigma_k(b)\sigma_j(\delta_k(a))r_0 \notag \\ + &\ \sigma_k(b)\sigma_j(\delta_k(a))\sum_{l=1}^{n} r_lx_l + \sigma_k(b)\delta_j(\delta_k(a))x_i\notag \\ + &\ \delta_k(b)\sigma_j(a)c_{i,j}x_ix_j + \delta_k(b)\sigma_j(a)r_0 \notag \\ + &\ \delta_k(b)\sigma_j(a)\sum_{l=1}^{n}r_lx_l + \delta_k(b)\delta_j(a)x_i.\label{coloooo} \end{align}}} If we want that the expressions (\ref{colooo}) and (\ref{coloooo}) represent the same value, that is, {\small{\[ \overline{\delta_k}(bx_jax_i) = \overline{\sigma_k}(bx_j)\overline{\delta_k}(ax_i) + \overline{\delta_k}(bx_j)ax_i,\ \ \ \ \ 1\le i, j, k, \le n \]}} then we have to impose that $\sigma_i\delta_j = \delta_j\sigma_i$, $\delta_i\delta_j = \delta_j\delta_i$, $\delta_k(c_{i,j}) = \delta_k(r_l^{(i,j)}) = 0$, for $1\le i, j, l\le n$, where $c_{i,j}$ and $r_l^{(i,j)}$ are the elements established in Definition \ref{gpbwextension}. This justifies the assumptions in the proposition.
Now, the proof of the general case, that is, the expression (\ref{serpi}), it follows from the above reasoning and Remark \ref{juradpr}. Let us see the details. Consider the following expressions: {\small{\begin{align} \overline{\delta_k}(a_iX_ib_jY_j) = &\ \overline{\delta_k}(a_i(\sigma^{\alpha_i}(b_j)X_i + p_{\alpha_i, b_j})Y_j)=\overline{\delta_k}(a_i\sigma^{\alpha_i}(b_j)X_iY_j + a_ip_{\alpha_i, b_j}Y_j)\notag \\ = &\ \overline{\delta_k}(a_i\sigma^{\alpha_i}(b_j)(c_{\alpha_i, \beta_j}x^{\alpha_i+\beta_j} + p_{\alpha_i,\beta_j}) + a_ip_{\alpha_i,b_j}Y_j)\notag \\ = &\ \overline{\delta_k}(a_i\sigma^{\alpha_i}(b_j)c_{\alpha_i,\beta_j}x^{\alpha_i+\beta_j} + a_i\sigma^{\alpha_i}(b_j)p_{\alpha_i, \beta_j} + a_ip_{\alpha_i,b_j}Y_j)\notag \\ = &\ \overline{\delta_k}(a_i\sigma^{\alpha_i}(b_j)c_{\alpha_i,\beta_j})x^{\alpha_i+\beta_j} + \overline{\delta_k}(a_i\sigma^{\alpha_i}(b_j)p_{\alpha_i,\beta_j}) + \overline{\delta_k}(a_ip_{\alpha_i,b_j}Y_j)\notag\\ = &\ \sigma_k(a_i\sigma^{\alpha_i}(b_j))\delta_k(c_{\alpha_i,\beta_j})x^{\alpha_i+\beta_j} + \delta_k(a_i\sigma^{\alpha_i}(b_j))c_{\alpha_i,\beta_j}x^{\alpha_i+\beta_j}\notag \\ + &\ \overline{\delta_k}(a_i\sigma^{\alpha_i}(b_j)p_{\alpha_i,\beta_j}) + \overline{\delta_k}(a_ip_{\alpha_i,b_j}Y_j)\notag \\ = &\ \sigma_k(a_i)\sigma_k(\sigma^{\alpha_i}(b_j))\delta_k(c_{\alpha_i,\beta_j})x^{\alpha_i+\beta_j} + \sigma_k(a_i)\delta_k(\sigma^{\alpha_i}(b_j))c_{\alpha_i,\beta_j}x^{\alpha_i+\beta_j}\notag \\ + &\ \delta_k(a_i)\sigma^{\alpha_i}(b_j)c_{\alpha_i,\beta_j}x^{\alpha_i+\beta_j} + \overline{\delta_k}(a_i\sigma^{\alpha_i}(b_j)p_{\alpha_i,\beta_j}) + \overline{\delta_k}(a_ip_{\alpha_i,b_j}Y_j), \notag \end{align}}}
and
{\small{\begin{align} \overline{\sigma_k}(a_iX_i)\overline{\delta_k}(b_jY_j) + \overline{\delta_k}(a_iX_i)b_jY_j = &\ \sigma_k(a_i)X_i\delta_k(b_j)Y_j + \delta_k(a_i)X_ib_jY_j\notag \\ = &\ \sigma_k(a_i)(\sigma^{\alpha_i}(\delta_k(b_j))X_i + p_{\alpha_i,\delta_k(b_j)})Y_j \notag \\ + &\ \delta_k(a_i)(\sigma^{\alpha_i}(b_j)X_i + p_{\alpha_i, b_j})Y_j\notag \\ = &\ \sigma_k(a_i)(\sigma^{\alpha_i}(\delta_k(b_j))X_iY_j) + \sigma_k(a_i)p_{\alpha_i, \delta_k(b_j)}Y_j\notag \\ + &\ \delta_k(a_i)\sigma^{\alpha_i}(b_j)X_iY_j + \delta_k(a_i)p_{\alpha_i,b_j}Y_j\notag \\ = &\ \sigma_k(a_i)\sigma^{\alpha_i}(\delta_k(b_j))(c_{\alpha_i,\beta_j}x^{\alpha_i+\beta_j} + p_{\alpha_i,\beta_j}) \notag \\ + &\ \sigma_k(a_i)p_{\alpha_i,\delta_k(b_j)}Y_j\notag \\ + &\ \delta_k(a_i)\sigma^{\alpha_i}(b_j)(c_{\alpha_i,\beta_j}x^{\alpha_i+\beta_j} + p_{\alpha_i,\beta_j})\notag \\ + &\ \delta_k(a_i)p_{\alpha_i,b_j}Y_j\notag \\ = &\ \sigma_k(a_i)\sigma^{\alpha_i}(\delta_k(b_j))c_{\alpha_i,\beta_j}x^{\alpha_i+\beta_j} \notag \\ + &\ \sigma_k(a_i)\sigma^{\alpha_i}(\delta_k(b_j))p_{\alpha_i, \beta_j} \notag \\ + &\ \sigma_k(a_i)p_{\alpha_i, \delta_k(b_j)}Y_j + \delta_k(a_i)\sigma^{\alpha_i}(b_j)c_{\alpha_i, \beta_j}x^{\alpha_i+\beta_j}\notag \\ + &\ \delta_k(a_i)\sigma^{\alpha_i}(b_j)p_{\alpha_i, \beta_j} + \delta_k(a_i)p_{\alpha_i, b_j}Y_j.\notag \end{align}}} By assumption, we have the equalities $\sigma_k(a_i)\delta_k(\sigma^{\alpha_i}(b_j)) = \sigma_k(a_i)\sigma^{\alpha_i}(\delta_k(b_j))$ and $\delta_k(c_{\alpha_i,\beta_j}) = 0$, which means that we need to prove the relation {\small{\begin{align} \overline{\delta_k}(a_i\sigma^{\alpha_i}(b_j)p_{\alpha_i,\beta_j}) + \overline{\delta_k}(a_ip_{\alpha_i,b_j}Y_j) = &\
\sigma_k(a_i)\sigma^{\alpha_i}(\delta_k(b_j))p_{\alpha_i, \beta_j} + \sigma_k(a_i)p_{\alpha_i, \delta_k(b_j)}Y_j \notag \\ + &\ \delta_k(a_i)\sigma^{\alpha_i}(b_j)p_{\alpha_i, \beta_j} + \delta_k(a_i)p_{\alpha_i, b_j}Y_j.\label{metal} \end{align}}} However, note that this equality is a consequence of the linearity of $\delta_k$, Remark \ref{juradpr}, and the assumptions established in the formulation of the theorem. More precisely, using these facts we have {\normalsize{\begin{align} \overline{\delta}_k(a_i\sigma^{\alpha_i}(b_j)p_{\alpha_i, \beta_j}) = &\ \overline{\sigma_k}(a_i\sigma^{\alpha_i}(b_j))\overline{\delta_k}(p_{\alpha_i,\beta_j}) + \overline{\delta_k}(a_i\sigma^{\alpha_i}(b_j))p_{\alpha_i, \beta_j}\notag \\ = &\ \sigma_k(a_i)\sigma_k(\sigma^{\alpha_i}(b_j))\overline{\delta_k}(p_{\alpha_i,\beta_j}) + \sigma_k(a_i) \delta_k(\sigma^{\alpha_i}(b_j))p_{\alpha_i, \beta_j} \notag \\ + &\ \delta_k(a_i)\sigma^{\alpha_i}(b_j)p_{\alpha_i,\beta_j}\notag \\ = &\ \sigma_k(a_i) \delta_k(\sigma^{\alpha_i}(b_j))p_{\alpha_i, \beta_j} + \delta_k(a_i)\sigma^{\alpha_i}(b_j)p_{\alpha_i,\beta_j}\notag \\ = &\ \sigma_k(a_i)\sigma^{\alpha_i}(\delta_k(b_j))p_{\alpha_i,\beta_j} + \delta_k(a_i)\sigma^{\alpha_i}(b_j)p_{\alpha_i, \beta_j},\label{acdc} \end{align}}} and, {\normalsize{\begin{align} \overline{\delta_k}(a_ip_{\alpha_i,b_j}Y_j) = &\ \overline{\sigma_k}(a_i)\overline{\delta_k}(p_{\alpha_i,b_j}Y_j) + \overline{\delta_k}(a_i)p_{\alpha_i,b_j}Y_j\notag \\ = &\ \sigma_k(a_i)p_{\alpha_i, \delta_k(b_j)}Y_j + \delta_k(a_i)p_{\alpha_i,b_j}Y_j,\label{marliiii} \end{align}}} where we can see that expression (\ref{metal}) is precisely the sum of (\ref{acdc}) and (\ref{marliiii}). Therefore $\overline{\delta_i}$ is a $\overline{\sigma_i}$-derivation of $A$. \end{proof}
With Proposition \ref{ReyesSuarez2017CliffordTheorem5.1} in our hands, we formulate Theorem \ref{2010Theorem2.17} which extends \cite{OuyangChen2010}, Theorem 2.17. To this end, consider the skew PBW extension $A'$ induced by injective endomorphisms and derivations established in Proposition \ref{ReyesSuarez2017CliffordTheorem5.1}, i.e., $A' = \sigma(A)\langle x_1',\dotsc, x_n'\rangle$. We remark that using algorithms established by Reyes and Su\'arez (2017b) one can prove that $A'$ is a left free $A$-module considering adequate relations between the indeterminates $x_1',\dotsc, x_n'$. For the sets of injective endomorphisms $\overline{\Sigma}$ and $\overline{\Sigma}$-derivations $\overline{\Delta}$ formulated in Proposition \ref{ReyesSuarez2017CliffordTheorem5.1}, consider a definition of $(\overline{\Sigma},\overline{\Delta})$-compatible in a similar way to the Definition \ref{Definition3.52008}. Suppose that the elements $c_{i,j}$ in Definition \ref{gpbwextension} (iv) are central in $R$, for all $i, j$.
\begin{theorem}\label{2010Theorem2.17} If $A$ is a skew PBW extension over an $\Sigma$-rigid ring $R$, then $A$ is weak symmetric if and only if $A'$ is weak symmetric. \end{theorem} \begin{proof} As we saw in Section \ref{SigmaDeltaweaksymmetricskewPBWextensions}, if $R$ is $\Sigma$-rigid, then $R$ is reduced, or equivalently, $A$ is reduced whence $A$ is reversible. The aim is to show that $A$ is $(\overline{\Sigma}, \overline{\Delta})$-compatible. From Proposition \ref{ReyesSuarez2018Theorem3.9} we also know that $R$ is $(\Sigma, \Delta)$-compatible.
Consider elements $f=a_0+a_1X_1+\dotsb + a_mX_m,\ g=b_0 + b_1Y_1 + \dotsb + b_tY_t$ in $A$ with $fg=0$ and let us see that $a_ib_j=0$, for every $i, j$. Since \begin{align*} fg = &\ (a_0+a_1X_1+\dotsb + a_mX_m)(b_0+b_1Y_1+\dotsb + b_tY_t)\\ = &\ \sum_{k=0}^{m+t} \biggl(\sum_{i+j=k} a_iX_ib_jY_j\biggr), \end{align*} then ${\rm lc}(fg)= a_m\sigma^{\alpha_m}(b_t)c_{\alpha_m, \beta_t}=0$ whence $a_m\sigma^{\alpha_m}(b_t)=0$ ($c_{\alpha_m, \beta_b}$ is invertible), and by Proposition \ref{Reyes2015Lemma3.3} (4), $a_mb_t=0$. The idea is to prove that $a_pb_q=0$, for $p+q\ge 0$. We proceed by induction. Suppose that $a_pb_q=0$, for $p+q=m+t, m+t-1, m+t-2, \dotsc, k+1$, for some $k>0$. By Proposition \ref{Reyes2015Lemma3.3} (5) we obtain $a_pX_pb_qY_q=0$ for these values of $p+q$. In this way we only consider the sum of the products $a_uX_ub_vY_v$, where $u+v=k, k-1,k-2,\dotsc, 0$. Fix $u$ and $v$. Consider the sum of all terms of $fg$ having exponent $\alpha_u+\beta_v$. By Proposition \ref{lindass}, Remark \ref{juradpr}, and the assumption $fg=0$, the sum of all coefficients of all these terms can be written as {\small{\begin{equation}\label{Federer} a_u\sigma^{\alpha_u}(b_v)c_{\alpha_u, \beta_v} + \sum_{\alpha_{u'} + \beta_{v'} = \alpha_u + \beta_v} a_{u'}\sigma^{\alpha_{u'}} ({\rm \sigma's\ and\ \delta's\ evaluated\ in}\ b_{v'})c_{\alpha_{u'}, \beta_{v'}} = 0. \end{equation}}} By assumption we know that $a_pb_q=0$ for $p+q=m+t, m+t-1, \dotsc, k+1$. So, Proposition \ref{Reyes2015Lemma3.3} (3) guarantees that the product \[a_p({\rm \sigma's\ and\ \delta's\ evaluated\ in}\ b_{q})\ \ \ \ \ \ \ ({\rm any\ order\ of}\ \sigma's\ {\rm and}\ \delta's) \] is equal to zero. Then $[({\rm \sigma's\ and\ \delta's\ evaluated\ in}\ b_{q})a_p]^2=0$ and hence we obtain the equality $({\rm \sigma's\ and\ \delta's\ evaluated\ in}\ b_{q})a_p=0$ ($R$ is reduced). In this way, multiplying (\ref{Federer}) by $a_k$, and using the fact that the elements $c_{i,j}$ in Definition \ref{gpbwextension} (iv) are in the center of $R$, {\small{\begin{equation}\label{doooooo} a_u\sigma^{\alpha_u}(b_v)a_kc_{\alpha_u, \beta_v} + \sum_{\alpha_{u'} + \beta_{v'} = \alpha_u + \beta_v} a_{u'}\sigma^{\alpha_{u'}} ({\rm \sigma's\ and\ \delta's\ evaluated\ in}\ b_{v'})a_kc_{\alpha_{u'}, \beta_{v'}} = 0, \end{equation}}} whence, $a_u\sigma^{\alpha_u}(b_0)a_k=0$. Since $u+v=k$ and $v=0$, then $u=k$, so $a_k\sigma^{\alpha_k}(b_0)a_k=0$, i.e., $[a_k\sigma^{\alpha_k}(b_0)]^{2}=0$, from which $a_k\sigma^{\alpha_k}(b_0)=0$ and $a_kb_0=0$ by Proposition \ref{Reyes2015Lemma3.3} (4). Therefore, we now have to study the expression (\ref{Federer}) for $0\le u \le k-1$ and $u+v=k$. If we multiply (\ref{doooooo}) by $a_{k-1}$ we obtain {\scriptsize{\begin{equation} a_u\sigma^{\alpha_u}(b_v)a_{k-1}c_{\alpha_u, \beta_v} + \sum_{\alpha_{u'} + \beta_{v'} = \alpha_u + \beta_v} a_{u'}\sigma^{\alpha_{u'}} ({\rm \sigma's\ and\ \delta's\ evaluated\ in}\ b_{v'})a_{k-1}c_{\alpha_{u'}, \beta_{v'}} = 0. \end{equation}}} Using a similar reasoning as above, we can see that $a_u\sigma^{\alpha_u}(b_1)a_{k-1}c_{\alpha_u, \beta_1}=0$. Since $A$ is bijective, $a_u\sigma^{\alpha_u}(b_1)a_{k-1}=0$, and using the fact $u=k-1$, we have $[a_{k-1}\sigma^{\alpha_{k-1}}(b_1)]=0$, which imply $a_{k-1}\sigma^{\alpha_{k-1}}(b_1)=0$, that is, $a_{k-1}b_1=0$. Continuing in this way we prove that $a_ib_j=0$ for $i+j=k$. Hence $a_ib_j=0$, for $0\le i\le m$ and $0\le j\le t$, and therefore $a_i\sigma^{\alpha}(b_j)) = a_i\delta^{\beta}(b_j)=0$, for all $\alpha, \beta\in \mathbb{N}^{n}$, since $R$ is $(\Sigma,\Delta)$-compatible. In this way, when we consider the expressions \begin{align*} f\overline{\sigma^{\alpha}}(g) = &\ (a_0+a_1X_1 + \dotsb + a_mX_m)({\sigma^{\alpha}}(b_0) + {\sigma^{\alpha}}(b_1)Y_1 + \dotsb + {\sigma^{\alpha}}(b_t)Y_t)\\ = &\ \sum_{k=0}^{m+t} \biggl(\sum_{i+j=k} a_iX_i\sigma^{\alpha}(b_j)Y_j\biggr)\\ = &\ \sum_{k=0}^{m+t} \biggl(\sum_{i+j=k} a_i[\sigma^{\alpha_i}(\sigma^{\alpha}(b_j))X_i + p_{\alpha_i, \sigma^{\alpha}(b_j)}]Y_j\biggr)\\ = &\ \sum_{k=0}^{m+t} \biggl(\sum_{i+j=k} a_i\sigma^{\alpha_i}(\sigma^{\alpha}(b_j))X_iY_j + a_ip_{\alpha_i, \sigma^{\alpha}(b_j)}Y_j\biggr)\\ = &\ \sum_{k=0}^{m+t} \biggl(\sum_{i+j=k} a_i\sigma^{\alpha_i}(\sigma^{\alpha}(b_j))[c_{\alpha_i,\beta_j}x^{\alpha_i + \beta_j} + p_{\alpha_i,\beta_j}] + a_ip_{\alpha_i, \sigma^{\alpha}(b_j)}Y_j\biggr) \end{align*} and \begin{align*} f\overline{\delta^{\beta}}(g) = &\ (a_0+a_1X_1 + \dotsb + a_mX_m)({\delta^{\beta}}(b_0) + {\delta^{\beta}}(b_1)Y_1 + \dotsb + {\delta^{\beta}}(b_t)Y_t)\\ = &\ \sum_{k=0}^{m+t} \biggl(\sum_{i+j=k} a_iX_i\delta^{\beta}(b_j)Y_j\biggr)\\ = &\ \sum_{k=0}^{m+t} \biggl(\sum_{i+j=k} a_i[\sigma^{\alpha_i}(\delta^{\beta}(b_j))X_i + p_{\alpha_i, \delta^{\beta}(b_j)}]Y_j\biggr)\\ = &\ \sum_{k=0}^{m+t} \biggl(\sum_{i+j=k} a_i\sigma^{\alpha_i}(\delta^{\beta}(b_j))X_iY_j + a_ip_{\alpha_i, \delta^{\beta}(b_j)}Y_j\biggr) \end{align*} or equivalently, \begin{align*} f\overline{\delta^{\beta}}(g) = &\ \sum_{k=0}^{m+t} \biggl(\sum_{i+j=k} a_i\sigma^{\alpha_i}(\delta^{\beta}(b_j))[c_{\alpha_i,\beta_j}x^{\alpha_i + \beta_j} + p_{\alpha_i,\beta_j}] + a_ip_{\alpha_i, \delta^{\beta}(b_j)}Y_j\biggr), \end{align*} Proposition \ref{lindass} and Remark \ref{juradpr} imply that $f\overline{\sigma^{\alpha}}(g) = f\overline{\delta^{\beta}}(g) = 0$, for every $\alpha,\beta\in \mathbb{N}^{n}$. In a similar way, if we start with the equality $f\overline{\sigma^{\alpha}}(g) =0$, then we can show that $fg=0$, which means that $A$ is $(\Sigma,\Delta)$-compatible. In this way, since we have showed that $A$ is reversible and $(\overline{\Sigma}, \overline{\Delta})$-compatible, the assertion which we are proving it follows from Theorem \ref{2010Theorem 2.12}. \end{proof}
\section{Skew PBW extensions over weak $(\Sigma,\Delta)$-symmetric rings}\label{SigmaDeltaweaksymmetricskewPBWextensions}
In \cite{OuyangChen2010}, Definition 2, Ouyang and Chen 2010 introduced the notion of weak $(\alpha,\delta)$-symmetric ring in the following way: a ring $B$ with an endomorphism $\sigma$ and an $\sigma$-derivation $\delta$ is said to be {\em weak} $\sigma$-{\em symmetric} provided that $abc\in {\rm nil}(B)\Leftrightarrow ac\sigma(b)\in {\rm nil}(B)$, for any elements $a, b, c\in B$. $B$ is said to be {\em weak} $\delta$-symmetric, if for $a, b, c\in R$, $abc\in {\rm nil}(B)$ implies $ac\delta(b)\in {\rm nil}(B)$. If $B$ is both {\em weak} $\sigma$-{\em symmetric} and {\em weak} $\delta$-{\em symmetric}, $B$ is called a {\em weak} $(\Sigma,\Delta)$-{\em symmetric} ring. With respect to the relation between weak symmetric ring and weak $(\alpha,\delta)$-symmetric rings, there is an example of a weak symmetric ring which is not weak $(\alpha,\delta)$-symmetric, see \cite{OuyangChen2010}, Example 3.2. Note that for every subring $S$ of a weak $(\alpha,\delta)$-symmetric ring $B$ which satisfies $\alpha(S)\subseteq S$ and $\delta(S)\subseteq S$, it follows that $S$ is also a weak weak $(\alpha,\delta)$-symmetric ring. With these definitions in mind, we present in a natural way the notion of weak $(\Sigma,\Delta)$-symmetric ring for a ring $R$ with a family of endomorphisms $\Sigma$ and a family of $\Sigma$-derivations $\Delta$.
\begin{definition}\label{2010Definition2} Let $R$ be a ring with a family of endomorphisms of $R$ and a family of $\Sigma=\{\sigma_1,\dotsc,\sigma_n\}$-derivations $\Delta=\{\delta_1,\dotsc,\delta_n\}$. $R$ is called {\em weak} $\Sigma$-{\em symmetric}, if $abc\in {\rm nil}(R)\Rightarrow ac\sigma_i(b)\in {\rm nil}(R)$, for every $i$ and each elements $a, b, c\in R$. $R$ is said to be {\em weak} $\Delta$-{\em symmetric}, if $abc\in {\rm nil}(R) \Rightarrow ac\delta_i(b)\in {\rm nil}(R)$, for every $i$ and each elements $a, b, c\in R$. In the case $R$ is both weak $\Sigma$-symmetric and weak $\Delta$-symmetric, we say that $R$ is a {\em weak} $(\Sigma,\Delta)$-symmetric ring. \end{definition}
\begin{definition} If $R$ is a ring with a family of endomorphisms of $R$ and a family of $\Sigma=\{\sigma_1,\dotsc,\sigma_n\}$-derivations $\Delta=\{\delta_1,\dotsc,\delta_n\}$, then an ideal $I$ of $R$ is said to be an {\em weak}-{\em symmetric ideal}, if $abc\in {\rm nil}(R)\Rightarrow ac\sigma_i(b), ac\delta_i(b)\in {\rm nil}(R)$, for each $i$ and every elements $a, b, c\in I$. \end{definition}
The next proposition extends \cite{OuyangChen2010}, Proposition 3.6.
\begin{proposition}\label{2010Proposition3.6} If $R$ is an abelian ring with $\sigma_i(e)=e$ and $\delta_i(e)=0$, for any idempotent element $e$ of $R$, then the following statements are equivalent: \begin{enumerate} \item [\rm (1)] $R$ is a weak $(\Sigma,\Delta)$-symmetric ring. \item [\rm (2)] $eR$ and $(1-e)R$ are weak $(\Sigma,\Delta)$-symmetric ideals. \end{enumerate} \end{proposition} \begin{proof} We use similar arguments to the established in \cite{OuyangChen2010}, Proposition 3.6. $(1)\Rightarrow (2)$ It is clear. $(2)\Rightarrow (1)$ Consider elements $a, b, c\in R$ with $abc\in {\rm nil}(R)$. It follows that $eaebec, (1-e)a(1-e)b(1-e)c\in {\rm nil}(R)$. By assumption, $eR$ and $(1-e)R$ are weak $(\Sigma,\Delta)$-symmetric ideals, so $eaec\sigma_i(eb)=eac\sigma_i(b)\in {\rm nil}(R)$ and $(1-e)a(1-e)c\sigma_i((1-e)b) = (1-e)ac\sigma_i(b)\in {\rm nil}(R)$. This fact shows that $ac\sigma_i(b)\in {\rm nil}(R)$, for every $i$, and hence $R$ is weak $\Sigma$-symmetric. Now, since for any $r\in R$, $\delta_i(er)=\sigma_i(e)\delta_i(r) + \delta_i(e)r = e\delta_i(r)$, for every $i$, the assumptions on $R$ imply that if $abc\in {\rm nil}(R)$, then $ea(eb)(ec), (1-e)a(1-e)b(1-e)c\in {\rm nil}(R)$. Therefore $eaec\delta_i(eb) = eac\delta_i(b), (1-e)a(1-e)c\delta_i((1-e)b) = (1-e)ac\delta_i(b)\in {\rm nil}(R)$. In this way, $ac\delta_i(b)\in {\rm nil}(R)$, for every $i$, which means that $R$ is weak $\Delta$-symmetric. In conclusion, $R$ is weak $(\Sigma,\Delta)$-symmetric. \end{proof}
For the next theorem, Theorem \ref{2010Theorem3.7}, we need some preliminary facts and the Proposition \ref{Lezamaetal2015Proposition2.6} which concerns about quotients of skew PBW extensions: consider $A=\sigma(R)\langle x_1,\dotsc, x_n\rangle$ a skew PBW extension of a ring $R$. Let $\Sigma:=\{\sigma_1,\dotsc, \sigma_n\}$ and $\Delta:=\{\delta_1,\dotsc,\delta_n\}$ such as in Proposition \ref{sigmadefinition}. Following \cite{LezamaAcostaReyes2015}, Section 2, if $I$ is an ideal of $R$, $I$ is called $\Sigma$-invariant ($\Delta$-invariant), if it is invariant under each injective endomorphism $\sigma_i$ ($\sigma_i$-derivation $\delta_i$) of $\Sigma$ ($\Delta$), that is, $\sigma_i(I)\subseteq I$ ($\delta_i(I)\subseteq I$), for every $i$. If $I$ is both $\Sigma$ and $\Delta$-invariant ideal, we say that $I$ is $(\Sigma,\Delta)$-invariant.
\begin{proposition}[\cite{LezamaAcostaReyes2015}, Proposition 2.6]\label{Lezamaetal2015Proposition2.6}
If $A$ is a skew PBW extension over a ring $R$ and $I$ is a $(\Sigma,\Delta)$-invariant ideal of $R$, then the following statements hold: \begin{enumerate} \item [\rm (1)] $IA$ is an ideal of $A$ and $IA\cap R=I$. $IA$ is a proper ideal of $A$ if and only if $I$ is proper in $R$. Moreover, if $\sigma_i$ is bijective and $\sigma_i(I)=I$, for every $i$, then $IA=AI$. \item [\rm (2)] If $I$ is proper and $\sigma_i(I)=$, for every $1\le i\le n$, then $A/IA$ is a skew PBW extension of $R/I$. In fact, if $I$ is proper and $A$ is bijective, then $A/IA$ is a bijective skew PBW extension of $R/I$. \end{enumerate} \end{proposition}
From Proposition \ref{Lezamaetal2015Proposition2.6} we can see that if $I$ is $(\Sigma,\Delta)$-invariant, then over $\overline{R}:=R/I$ it is induced a systems $(\overline{\Sigma},\overline{\Delta})$ of endomorphisms $\overline{\Sigma}$ and $\overline{\Sigma}$-derivations $\overline{\Delta}$ defined by $\overline{\sigma_i}(r+I))=\sigma_i(r)+I$ and $\overline{\delta_i}(r+I) = \delta_i(r)+I$, for $1\le i\le n$. We keep the variables $x_1,\dotsc,x_n$ of the extension $A$ to the extension $A/IA$ if no confusion arises. For quotients of skew PBW extensions, we consider the notion of weak $(\Sigma,\Delta)$-symmetric in the natural way following Definition \ref{2010Definition2}.\\
Our next theorem extends \cite{OuyangChen2010}, Theorem 3.7.
\begin{theorem}\label{2010Theorem3.7} Let $I$ be an $(\Sigma,\Delta)$-invariant and weak $(\Sigma,\Delta)$-symmetric ideal of $R$. If $I\subseteq {\rm nil}(R)$, then $R/I$ is a weak $(\overline{\Sigma}, \overline{\Delta})$-symmetric ring if and only if $R$ is a weak $(\Sigma,\Delta)$-symmetric ring. \end{theorem} \begin{proof} Consider elements $a, b, c\in R$ such that $(a+I)(b+I)(c+I)\in {\rm nil}(R/I)$. There exists a positive integer $m$ with $(abc)^{m}\in I$. Since $I\subseteq {\rm nil}(R)$ it follows that $abc\in {\rm nil}(R)$. By assumption, $R$ is weak $(\Sigma,\Delta)$-symmetric, so $ac\sigma_i(b), ac\delta_i(b) \in {\rm nil}(R)$, for $i=1,\dotsc, n$. Hence $(a+I)(c+I)(\sigma_i(b)+I), (a+I)(c+I)(\delta_i(b)+I) \in {\rm nil}(R/I)$, that is, $(a+I)(c+I)\overline{\sigma_i}(b+I), (a+I)(c+I)\overline{\delta_i}(b+I)\in {\rm nil}(R/I)$. Therefore $R/I$ is weak $(\overline{\Sigma}, \overline{\Delta})$-symmetric.
Conversely, suppose that $R/I$ is a weak $(\Sigma,\Delta)$-symmetric ring. Consider elements $a, b, c\in R$ with $abc\in {\rm nil}(R)$. It is clear that $(a+I)(b+I)(c+I)\in {\rm nil}(R/I)$. Since $R/I$ is weak $(\overline{\Sigma}, \overline{\Delta})$-symmetric, we have that $(a+I)(c+I)(\sigma_i(b) + I) = (ac\sigma_i(b) + I), (a+I)(c+I)(\delta_i(b) + I) = (ac\delta_i(b) + I) \in {\rm nil}(R/I)$, for $i=1,\dotsc, n$. This means that for every $i$ there exist positive integers $p = p(i), q = q(i)$ depending on $i$, such that $(ac\sigma_i(b))^{p}, (ac\delta_i(b))^{q}\in I$. In this way, $ac\sigma_i(b), ac\delta_i(b)\in I$ because $I\subseteq {\rm nil}(R)$ which shows that $R$ is a weak $(\overline{\Sigma},\overline{\Delta})$-symmetric ring. \end{proof}
The next theorem generalizes Ouyang and Chen \cite{OuyangChen2010}, Theorem 3.9.
\begin{theorem}\label{2010Theorem 3.9} If $R$ is a $(\Sigma,\Delta)$-compatible and reversible ring, then $R$ is a weak $(\Sigma,\Delta)$-symmetric ring if and only if $A$ is a weak $(\overline{\Sigma}, \overline{\Delta})$-symmetric ring, where the sets of injective endomorphisms $\overline{\Sigma}$ and $\overline{\Sigma}$-derivations $\overline{\Delta}$ of $A$ are as in Proposition \ref{ReyesSuarez2017CliffordTheorem5.1}. \end{theorem} \begin{proof} If $A$ is a weak $(\overline{\Sigma}, \overline{\Delta})$-symmetric ring, then it is clear that $R$ is weak $(\Sigma,\Delta)$-symmetric ring because $\sigma_i(R), \delta_i(R) \subseteq R$, for every $i=1,\dotsc, n$.
Conversely, suppose that $R$ is weak $(\Sigma,\Delta)$-symmetric ring. Consider the elements $f=\sum_{i=0}^{s} a_iX_i, g=\sum_{j=0}^{t} b_jY_j$ and $h=\sum_{k=0}^{l}c_kZ_k$ of $A$. From Theorem \ref{2010Theorem2.11} we know that $a_ib_jc_k\in {\rm nil}(R)$, for all $i, j, k$, whence $a_ic_k\sigma_l(b_j), a_ic_k\delta_l(b_j)\in {\rm nil}(R)$, for $l=1,\dotsc, n$, since $R$ is weak $(\Sigma,\Delta)$-symmetric. Again, Theorem \ref{2010Theorem2.11} implies that $fh\overline{\sigma_i}(g), fh\overline{\delta_i}(g)\in {\rm nil}(A)$, that is, $A$ is a weak $(\Sigma,\Delta)$-symmetric ring. \end{proof}
\begin{corollary}[\cite{OuyangChen2010}, Corollary 3.10] Let $R$ be a reversible ring. Then $R$ is a weak symmetric ring if and only if $R[t]$ is weak symmetric. \end{corollary}
\section{Examples}\label{examplespaper} Remarkable examples of skew PBW extensions over $(\Sigma,\Delta)$-compatible and reversible rings can be found in \cite{JaramilloReyes2018, ReyesPhD2013, ReyesYesica2018, SuarezReyesgenerKoszul2017}. In this way, the results obtained in Sections \ref{weaksymmetricskewPBWextensions} and \ref{SigmaDeltaweaksymmetricskewPBWextensions} can be illustrated with every one of these noncommutative rings. Let us just say some of these examples.\\
If $A$ is a skew PBW extension over a ring $R$ where the coefficients commute with the variables, that is, $x_ir = rx_i$, for every $r\in R$ and each $i=1,\dotsc, n$, or equivalently, $\sigma_i = {\rm id}_R$ and $\delta_i = 0$, for every $i$ (these extensions were called {\em constant} in \cite{Suarez2017}, Definition 2.5 (a)), then it is clear that $R$ is a $\Sigma$-rigid ring. Some examples of these extensions are the following: (i) PBW extensions defined by Bell and Goodearl (which include the classical commutative polynomial rings, universal enveloping algebra of a Lie algebra, and others); some operator algebras (for example, the algebra of linear partial differential operators, the algebra of linear partial shift operators, the algebra of linear partial difference operators, the algebra of linear partial $q$-dilation operators, and the algebra of linear partial q-differential operators). (ii) solvable polynomial rings introduced by Kandri-Rody and Weispfenning. (iii) $G$-algebras introduced by Apel. (iv) PBW algebras defined by Bueso et. al. (v) Calabi-Yau and skew Calabi-Yau algebras. (vi) Koszul and qudratic algebras. $G$-algebras studied by Levandovskyy. (vii) PBW algebras defined by Bueso et al. in \cite{BuesoTorrecillasVerschoren}. A detailed reference of every one of these algebras can be found in \cite{LezamaReyes2014, Suarez2017, SuarezLezamaReyes2017, SuarezReyesgenerKoszul2017}. Of course, we also encounter examples of skew PBW extensions which are not constant (see \cite{LezamaReyes2014} for the definition of each one of these algebras): the quantum plane $\mbox{${\cal O}$}_q(\Bbbk^{2})$; the Jordan plane; the algebra of $q$-differential operators $D_{q,h}[x,y]$; the mixed algebra $D_h$; the operator differential rings; the algebra of differential operators $D_{\bf q}(S_{\bf q})$ on a quantum space ${S_{\bf q}}$; and the family of Ore extensions studied in \cite{ArtamonovLezamaFajardo2016}.\\
Following Rosenberg \cite{Rosenberg1995}, Definition C4.3, a {\em 3-dimensional skew polynomial algebra $\mbox{${\cal A}$}$} is a $\Bbbk$-algebra generated by the variables $x,y,z$ restricted to relations $ yz-\alpha zy=\lambda,\ zx-\beta xz=\mu$, and $xy-\gamma yx=\nu$, such that the following conditions hold: \begin{enumerate} \item [\rm (1)] $\lambda, \mu, \nu\in \Bbbk+\Bbbk x+\Bbbk y+\Bbbk z$, and $\alpha, \beta, \gamma \in \Bbbk^{*}$; \item [\rm (2)] Standard monomials $\{x^iy^jz^l\mid i,j,l\ge 0\}$ are a $\Bbbk$-basis of the algebra. \end{enumerate} 3-dimensional skew polynomial algebras are very important in noncommutative algebraic geometry. Now, from the definition it is clear that these algebras are skew PBW extensions (as a matter of fact, in \cite{ReyesSuarez2017FEJM} the authors proved algorithmically that 3-dimensional skew polynomial algebras are examples of skew PBW extensions).
There exists a classification of 3-dimensional skew polynomial algebras, see \cite{Rosenberg1995}, Theorem C.4.3.1. More precisely, if $\mbox{${\cal A}$}$ is a 3-dimensional skew polynomial algebra, then $\mbox{${\cal A}$}$ is one of the following algebras: \begin{enumerate}
\item [\rm (a)] if $|\{\alpha, \beta, \gamma\}|=3$, then $\mbox{${\cal A}$}$ is defined by the relations $yz-\alpha zy=0,\ zx-\beta xz=0,\ xy-\gamma yx=0$.
\item [\rm (b)] if $|\{\alpha, \beta, \gamma\}|=2$ and $\beta\neq \alpha =\gamma =1$, then $\mbox{${\cal A}$}$ is one of the following algebras: \begin{enumerate} \item [\rm (i)] $yz-zy=z,\ \ \ zx-\beta xz=y,\ \ \ xy-yx=x${\rm ;} \item [\rm (ii)] $yz-zy=z,\ \ \ zx-\beta xz=b,\ \ \ xy-yx=x${\rm ;} \item [\rm (iii)] $yz-zy=0,\ \ \ zx-\beta xz=y,\ \ \ xy-yx=0${\rm ;} \item [\rm (iv)] $yz-zy=0,\ \ \ zx-\beta xz=b,\ \ \ xy-yx=0${\rm ;} \item [\rm (v)] $yz-zy=az,\ \ \ zx-\beta xz=0,\ \ \ xy-yx=x${\rm ;} \item [\rm (vi)] $yz-zy=z,\ \ \ zx-\beta xz=0,\ \ \ xy-yx=0$, \end{enumerate} where $a, b$ are any elements of $\Bbbk$. All nonzero values of $b$ give isomorphic algebras.
\item [\rm (c)] If $|\{\alpha, \beta, \gamma\}|=2$ and $\beta\neq \alpha=\gamma\neq 1$, then $\mbox{${\cal A}$}$ is one of the following algebras: \begin{enumerate} \item [\rm (i)] $yz-\alpha zy=0,\ \ \ zx-\beta xz=y+b,\ \ \ xy-\alpha yx=0${\rm ;} \item [\rm (ii)] $yz-\alpha zy=0,\ \ \ zx-\beta xz=b,\ \ \ xy-\alpha yx=0$. \end{enumerate} In this case, $b$ is an arbitrary element of $\Bbbk$. Again, any nonzero values of $b$ give isomorphic algebras. \item [\rm (d)] If $\alpha=\beta=\gamma\neq 1$, then $\mbox{${\cal A}$}$ is the algebra defined by the relations $yz-\alpha zy=a_1x+b_1,\ zx-\alpha xz=a_2y+b_2,\ xy-\alpha yx=a_3z+b_3$. If $a_i=0\ (i=1,2,3)$, then all nonzero values of $b_i$ give isomorphic algebras. \item [\rm (e)] If $\alpha=\beta=\gamma=1$, then $\mbox{${\cal A}$}$ is isomorphic to one of the following algebras: \begin{enumerate} \item [\rm (i)] $yz-zy=x,\ \ \ zx-xz=y,\ \ \ xy-yx=z${\rm ;} \item [\rm (ii)] $yz-zy=0,\ \ \ zx-xz=0,\ \ \ xy-yx=z${\rm ;} \item [\rm (iii)] $yz-zy=0,\ \ \ zx-xz=0,\ \ \ xy-yx=b${\rm ;} \item [\rm (iv)] $yz-zy=-y,\ \ \ zx-xz=x+y,\ \ \ xy-yx=0${\rm ;} \item [\rm (v)] $yz-zy=az,\ \ \ zx-xz=z,\ \ \ xy-yx=0${\rm ;} \end{enumerate} Parameters $a,b\in \Bbbk$ are arbitrary, and all nonzero values of $b$ generate isomorphic algebras. \end{enumerate}
\noindent {\bf \Large{Acknowledgements}}
The first author was supported by the research fund of Facultad de Ciencias, Universidad Nacional de Colombia, Bogot\'a, Colombia, HERMES CODE 41535.
\end{document} | arXiv |
Weaire–Phelan structure
In geometry, the Weaire–Phelan structure is a three-dimensional structure representing an idealised foam of equal-sized bubbles, with two different shapes. In 1993, Denis Weaire and Robert Phelan found that this structure was a better solution of the Kelvin problem of tiling space by equal volume cells of minimum surface area than the previous best-known solution, the Kelvin structure.[1]
Weaire–Phelan structure
Space group
Fibrifold notation
Coxeter notation
Pm3n (223)
2o
[[4,3,4]+]
History and the Kelvin problem
In two dimensions, the subdivision of the plane into cells of equal area with minimum average perimeter is given by the hexagonal tiling, but although the first record of this honeycomb conjecture goes back to the ancient Roman scholar Marcus Terentius Varro, it was not proven until the work of Thomas C. Hales in 1999.[2] In 1887, Lord Kelvin asked the corresponding question for three-dimensional space: how can space be partitioned into cells of equal volume with the least area of surface between them? Or, in short, what was the most efficient soap bubble foam?[3] This problem has since been referred to as the Kelvin problem.
Kelvin proposed a foam called the Kelvin structure. His foam is based on the bitruncated cubic honeycomb, a convex uniform honeycomb formed by the truncated octahedron, a space-filling convex polyhedron with 6 square faces and 8 hexagonal faces. However, this honeycomb does not satisfy Plateau's laws, formulated by Joseph Plateau in the 19th century, according to which minimal foam surfaces meet at $120^{\circ }$ angles at their edges, with these edges meeting each other in sets of four with angles of $\arccos {\tfrac {1}{3}}\approx 109.47^{\circ }$. The angles of the polyhedral structure are different; for instance, its edges meet at angles of $90^{\circ }$ on square faces, or $120^{\circ }$ on hexagonal faces. Therefore, Kelvin's proposed structure uses curvilinear edges and slightly warped minimal surfaces for its faces, obeying Plateau's laws and reducing the area of the structure by 0.2% compared with the corresponding polyhedral structure.[1][3]
Although Kelvin did not state it explicitly as a conjecture,[4] the idea that the foam of the bitruncated cubic honeycomb is the most efficient foam, and solves Kelvin's problem, became known as the Kelvin conjecture. It was widely believed, and no counter-example was known for more than 100 years. Finally, in 1993, Trinity College Dublin physicist Denis Weaire and his student Robert Phelan discovered the Weaire–Phelan structure through computer simulations of foam, and showed that it was more efficient, disproving the Kelvin conjecture.[1]
Since the discovery of the Weaire–Phelan structure, other counterexamples to the Kelvin conjecture have been found, but the Weaire–Phelan structure continues to have the smallest known surface area per cell of these counterexamples.[5][6][7] Although numerical experiments suggest that the Weaire–Phelan structure is optimal, this remains unproven.[8] In general, it has been very difficult to prove the optimality of structures involving minimal surfaces. The minimality of the sphere as a surface enclosing a single volume was not proven until the 19th century, and the next simplest such problem, the double bubble conjecture on enclosing two volumes, remained open for over 100 years until being proven in 2002.[9]
Description
Irregular dodecahedron
Tetrakaidecahedron
The Weaire–Phelan structure differs from Kelvin's in that it uses two kinds of cells, although they have equal volume. Like the cells in Kelvin's structure, these cells are combinatorially equivalent to convex polyhedra. One is a pyritohedron, an irregular dodecahedron with pentagonal faces, possessing tetrahedral symmetry (Th). The second is a form of truncated hexagonal trapezohedron, a species of tetrakaidecahedron with two hexagonal and twelve pentagonal faces, in this case only possessing two mirror planes and a rotoreflection symmetry. Like the hexagons in the Kelvin structure, the pentagons in both types of cells are slightly curved. The surface area of the Weaire–Phelan structure is 0.3% less than that of the Kelvin structure.[1]
The tetrakaidecahedron cells, linked up in face-to-face chains of cells along their hexagonal faces, form chains in three perpendicular directions. A combinatorially equivalent structure to the Weaire–Phelan structure can be made as a tiling of space by unit cubes, lined up face-to-face into infinite square prisms in the same way to form a structure of interlocking prisms called tetrastix. These prisms surround cubical voids which form one fourth of the cells of the cubical tiling; the remaining three fourths of the cells fill the prisms, offset by half a unit from the integer grid aligned with the prism walls. Similarly, in the Weaire–Phelan structure itself, which has the same symmetries as the tetrastix structure, 1/4 of the cells are dodecahedra and 3/4 are tetrakaidecahedra.[10]
The polyhedral honeycomb associated with the Weaire–Phelan structure (obtained by flattening the faces and straightening the edges) is also referred to loosely as the Weaire–Phelan structure. It was known well before the Weaire–Phelan structure was discovered, but the application to the Kelvin problem was overlooked.[11]
Applications
In physical systems
Experiments have shown that, with favorable boundary conditions, equal-volume bubbles spontaneously self-assemble into the Weaire–Phelan structure.[12][13]
The associated polyhedral honeycomb is found in two related geometries of crystal structure in chemistry. Where the components of the crystal lie at the centres of the polyhedra it forms one of the Frank–Kasper phases, the A15 phase.[14]
Where the components of the crystal lie at the corners of the polyhedra, it is known as the "Type I clathrate structure". Gas hydrates formed by methane, propane and carbon dioxide at low temperatures have a structure in which water molecules lie at the nodes of the Weaire–Phelan structure and are hydrogen bonded together, and the larger gas molecules are trapped in the polyhedral cages.[11] Some alkali metal hydrides silicides and germanides also form this structure, with silicon or germanium at nodes, and alkali metals in cages.[1][15][16]
In architecture
The Weaire–Phelan structure is the inspiration for the design by Tristram Carfrae of the Beijing National Aquatics Centre, the 'Water Cube', for the 2008 Summer Olympics.[17]
See also
• The Pursuit of Perfect Packing, a book by Weaire on this and related problems
References
1. Weaire, D.; Phelan, R. (1994), "A counter-example to Kelvin's conjecture on minimal surfaces", Phil. Mag. Lett., 69 (2): 107–110, Bibcode:1994PMagL..69..107W, doi:10.1080/09500839408241577.
2. Hales, T. C. (2001), "The honeycomb conjecture", Discrete & Computational Geometry, 25 (1): 1–22, doi:10.1007/s004540010071, MR 1797293
3. Lord Kelvin (Sir William Thomson) (1887), "On the Division of Space with Minimum Partitional Area" (PDF), Philosophical Magazine, 24 (151): 503, doi:10.1080/14786448708628135.
4. Weaire & Phelan (1994) write that it is "implicit rather than directly stated in Kelvin's original papers"
5. Sullivan, John M. (1999), "The geometry of bubbles and foams", Foams and emulsions (Cargèse, 1997), NATO Advanced Science Institutes Series E: Applied Sciences, vol. 354, Kluwer, pp. 379–402, MR 1688327
6. Gabbrielli, Ruggero (1 August 2009), "A new counter-example to Kelvin's conjecture on minimal surfaces", Philosophical Magazine Letters, 89 (8): 483–491, Bibcode:2009PMagL..89..483G, doi:10.1080/09500830903022651, ISSN 0950-0839, S2CID 137653272
7. Freiberger, Marianne (24 September 2009), "Kelvin's bubble burst again", Plus Magazine, University of Cambridge, retrieved 4 July 2017
8. Oudet, Édouard (2011), "Approximation of partitions of least perimeter by Γ-convergence: around Kelvin's conjecture", Experimental Mathematics, 20 (3): 260–270, doi:10.1080/10586458.2011.565233, MR 2836251, S2CID 2945749
9. Morgan, Frank (2009), "Chapter 14. Proof of Double Bubble Conjecture", Geometric Measure Theory: A Beginner's Guide (4th ed.), Academic Press.
10. Conway, John H.; Burgiel, Heidi; Goodman-Strauss, Chaim (2008), "Understanding the Irish Bubbles", The Symmetries of Things, Wellesley, Massachusetts: A K Peters, p. 351, ISBN 978-1-56881-220-5, MR 2410150
11. Pauling, Linus (1960), The Nature of the Chemical Bond (3rd ed.), Cornell University Press, p. 471
12. Gabbrielli, R.; Meagher, A.J.; Weaire, D.; Brakke, K.A.; Hutzler, S. (2012), "An experimental realization of the Weaire-Phelan structure in monodisperse liquid foam" (PDF), Phil. Mag. Lett., 92 (1): 1–6, Bibcode:2012PMagL..92....1G, doi:10.1080/09500839.2011.645898, S2CID 25427974.
13. Ball, Philip (2011), "Scientists make the 'perfect' foam: Theoretical low-energy foam made for real", Nature, doi:10.1038/nature.2011.9504, S2CID 136626668.
14. Frank, F. C.; Kasper, J. S. (1958), "Complex alloy structures regarded as sphere packings. I. Definitions and basic principles" (PDF), Acta Crystallogr., 11 (3): 184–190, doi:10.1107/s0365110x58000487. Frank, F. C.; Kasper, J. S. (1959), "Complex alloy structures regarded as sphere packings. II. Analysis and classification of representative structures", Acta Crystallogr., 12 (7): 483–499, doi:10.1107/s0365110x59001499.
15. Kasper, J. S.; Hagenmuller, P.; Pouchard, M.; Cros, C. (December 1965), "Clathrate structure of silicon Na8Si46 and NaxSi136 (x < 11)", Science, 150 (3704): 1713–1714, Bibcode:1965Sci...150.1713K, doi:10.1126/science.150.3704.1713, PMID 17768869, S2CID 21291705
16. Cros, Christian; Pouchard, Michel; Hagenmuller, Paul (December 1970), "Sur une nouvelle famille de clathrates minéraux isotypes des hydrates de gaz et de liquides, interprétation des résultats obtenus", Journal of Solid State Chemistry, 2 (4): 570–581, Bibcode:1970JSSCh...2..570C, doi:10.1016/0022-4596(70)90053-8
17. Fountain, Henry (August 5, 2008), "A Problem of Bubbles Frames an Olympic Design", New York Times
External links
• 3D models of the Weaire–Phelan, Kelvin and P42a structures
• Weaire–Phelan Bubbles page with illustrations and freely downloadable 'nets' for printing and making models.
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| Wikipedia |
Rigorous packing of unit squares into a circle
Tiago Montanher ORCID: orcid.org/0000-0001-9730-57481,2,
Arnold Neumaier2,
Mihály Csaba Markót1,2,
Ferenc Domes2 &
Hermann Schichl2
Journal of Global Optimization volume 73, pages547–565(2019)Cite this article
This paper considers the task of finding the smallest circle into which one can pack a fixed number of non-overlapping unit squares that are free to rotate. Due to the rotation angles, the packing of unit squares into a container is considerably harder to solve than their circle packing counterparts. Therefore, optimal arrangements were so far proved to be optimal only for one or two unit squares. By a computer-assisted method based on interval arithmetic techniques, we solve the case of three squares and find rigorous enclosures for every optimal arrangement of this problem. We model the relation between the squares and the circle as a constraint satisfaction problem (CSP) and found every box that may contain a solution inside a given upper bound of the radius. Due to symmetries in the search domain, general purpose interval methods are far too slow to solve the CSP directly. To overcome this difficulty, we split the problem into a set of subproblems by systematically adding constraints to the center of each square. Our proof requires the solution of 6, 43 and 12 subproblems with 1, 2 and 3 unit squares respectively. In principle, the method proposed in this paper generalizes to any number of squares.
Let \(S_{1},\ldots , S_{n}\) be n open unit squares and denote by \(C_{r}\) the closed circle of radius r centered at the origin. This paper deals with the problem of finding the smallest value of r such that one can pack \(S_{1},\ldots , S_{n}\) into \(C_{r}\) without overlapping. Formally, we can write the problem as
$$\begin{aligned} \begin{aligned}&\min ~~ r \\&{\mathrm{s.t.~}}~~~S_{i} \subseteq C_{r} \quad 1 \le i \le n\\&\qquad ~~~ S_{i} \cap S_{j} = \emptyset \quad 1 \le i,j \le n, ~~ i \ne j. \end{aligned} \end{aligned}$$
Packing identical objects into a container is an attractive part of geometrical optimization. The subject drew the attention of a considerable number of researchers, who contributed to problems similar to the one discussed in this paper.
The circle packing is the simplest packing problem in 2 dimensions in the sense that it does not involve the angles of the objects. Markót studied the packing of circles into a square from the interval analysis point of view in a series of papers [15, 16, 24]. In particular, he proved rigorous bounds for \(n = 28, 29\) and 30 circles. For a survey of the circle packing under the global optimization point of view, see [5]. The website Packomania [22] maintains an updated list of the best-known values for the packing of equal circles into several containers.
Kallrath and Rebennack [12] studied the packing of ellipses into rectangles using state-of-the-art complete global optimization solvers. He succeeded to find the global optimum for the case \(n = 3\) without rigor. For the packing of ellipsoids, see [2, 3].
Erdös and Graham [8] inaugurated the packing of unit squares into a square. They show that the wasted area in a container with side length l is \(O(l^{\frac{7}{11}})\). The proof relies on geometrical arguments and not on rigorous computations. Recent contributions in the packing of unit squares into a square include new bounds for the wasted area [6], the optimality proof for the cases \(n = 5,\ldots ,10, 13\) and 46 [1, 10, 23] and the optimality proof for \(n-2\) and \(n-1\) whenever n is a square [19]. Again, none of these contributions rely on computer-assisted proofs. For a dynamic survey on the packing of unit squares in a square, see [10].
The packing of unit squares into general containers received considerably less attention than the circle or the unit square packing into a square. For example, Friedman [9] maintains a list of proved and best-known values for the packing of unit squares into circles, triangles, L-shapes, and pentagons. In each case, only trivial arrangements are proved optimal. For the subject of interest in this paper, the packing of unit squares into a circle, the first open case is \(n = 3\). For a list of figures of squares packed into a circle, see https://www2.stetson.edu/~efriedma/squincir/.
Contribution and outline
This paper introduces a computer-assisted method for finding rigorous enclosures for r in Problem (1) and the corresponding optimal arrangements. The method is of theoretical interest since it proves optimality instead of only presenting a feasible arrangement. Therefore, it is suitable for small values of n only.
Our approach relies on the interval branch-and-bound framework. We implement the algorithm in C++ using the forward-backward constraint propagation [21] to reduce the search domain. Section 2 introduces the solver. The code is available at http://www.mat.univie.ac.at/~montanhe/publications/n3.zip.
Section 3 formulates Problem (1) as a constraint satisfaction problem (CSP). This paper uses the concept of sentinels [4, 18] to model non-overlapping conditions and the convexity of the circle to write containment constraints. Given an upper bound \(\overline{r}_{n}\) for \(r_{n}\), the CSP asks for every feasible arrangement satisfying \(r \le \overline{r}_{n}\). Our software produces a list of small interval vectors with the property that every optimal arrangement of (1) belongs to at least one element in the list.
General purpose interval solvers are usually not capable of solving packing problems due to symmetries in the search domain. To overcome this difficulty, Sect. 4 shows how to split the original CSP into a set of subproblems by systematically adding constraints to the center of each square. We call them tiling constraints as the idea resembles the one proposed in [15, 16, 24]. The tiling divides the search domain into a set of isosceles triangles that must contain the center of at most one unit square. Then, one can replace the original CSP by a set of \(\left( {\begin{array}{c}K\\ n\end{array}}\right) \) subproblems, where K is the number of triangles in the tiling.
Our procedure iterates on the number of squares to avoid the exponential growth of subproblems. At the i-th iteration, we look at every possible combination of i triangles which can accommodate i unit squares into a circle with the radius at most \(\overline{r}_{n}\). The rationale behind this strategy is twofold: (i) It allows us to discard a large number of hard subproblems by proving the infeasibility of more straightforward cases and (ii) It propagates the reduction on the search domain through the iterations. We also show that some combinations of triangles are symmetric by construction. Then one can discard them without any processing. This observation in addition to our iterative method reduces the number of hard cases considerably.
Section 5 illustrates the capabilities of our method. We find a mathematically rigorous enclosure for \(r_{3}\) and the corresponding optimal arrangement. If one set \(\overline{r} = \frac{5\sqrt{17}}{16}\) as pointed by Friedman [9], the tiling produces 36 triangles. Our approach requires the solution of 6 subproblems with one square, 43 with two and only 12 subproblems with 3 squares to conclude the proof. It is \(<\,1\%\) of all possible \(\left( {\begin{array}{c}36\\ 3\end{array}}\right) = 7140\) combinations. The method could also be used to find optimal configurations for higher values of n (e.g., \(n = 4, 5, 6\)).
Interval notation
This paper is an application of the interval branch-and-bound framework [11, 13]. We assume that the reader is familiar with concepts from interval analysis [20]. Let \(\underline{a}, \overline{a} \in \mathbb {R}\) with \(\underline{a} \le \overline{a}\). Then \(\mathbf{a}=[\underline{a}, \overline{a}]\) denotes the interval with \(\inf (\mathbf{a}) := \min (\mathbf{a}) := \underline{a}\) and \(\sup (\mathbf{a}) := \max (\mathbf{a}) := \overline{a}\). We denote the width of the interval \(\mathbf{a}\) by \({\mathrm{wid}}(\mathbf{a}) := \overline{a} - \underline{a}\).
The set of nonempty compact real intervals is given by
$$\begin{aligned} \mathbb {I}\mathbb {R}:=\{[\underline{a}, \overline{a}] \mid \underline{a} \le \overline{a},~ \underline{a}, \overline{a} \in \mathbb {R}\}. \end{aligned}$$
Let \(S \subseteq \mathbb {R}\) be any set. Then the interval hull of S is the smallest interval containing S.
An interval vector (also called box) \(\mathbf{x}:= [\underline{x},\overline{x}]\) is the Cartesian product of the closed real intervals \(\mathbf{x}_i:=[\underline{x}_i, \overline{x}_i] \in \mathbb {IR}\). We denote the set of all interval vectors of dimension n by \(\mathbb {IR}^{n}\). We apply the width operator component wise on vectors. Therefore \(\max ({\mathrm{wid}}(\mathbf{x})) := \max ({\mathrm{wid}}(\mathbf{x}_{1}),\ldots ,{\mathrm{wid}}(\mathbf{x}_{n}))\). Interval operations and functions are defined as in [13, 20]. The absolute value of the interval \(\mathbf{a}\) is given by
$$\begin{aligned} |\mathbf{a}| :=\left\{ \begin{array}{ll} \mathbf{a}&{}\quad \text{ if } \inf (\mathbf{a}) \ge 0 ,\\ {[}0, \max (-\inf (\mathbf{a}), \sup (\mathbf{a})){]} &{}\quad \text{ if } 0 \in \mathbf{a},\\ -\mathbf{a}&{}\quad \text{ if } \sup (\mathbf{a}) \le 0. \end{array}\right. \end{aligned}$$
Let \(\mathbf{a}\) and \(\mathbf{b}\) be two intervals. The maximum of \(\mathbf{a}\) and \(\mathbf{b}\) is defined by
$$\begin{aligned} \max (\mathbf{a}, \mathbf{b}) := [\max (\inf (\mathbf{a}), \inf (\mathbf{b})), \max (\sup (\mathbf{a}), \sup (\mathbf{b}))]. \end{aligned}$$
Let \(F:\mathbb {R}^{n} \rightarrow \mathbb {R}^{m}\) be a function defined on \(\mathbf{x}\in \mathbb {IR}^{n}\) and let \(\mathbf{f}\in \mathbb {IR}^{m}\). We denote the natural interval extension of the function F by \(\mathbf{F}\). A constraint satisfaction problem (CSP) is the task of finding every point satisfying
$$\begin{aligned} F(x) \in \mathbf{f},~~ x \in \mathbf{x}. \end{aligned}$$
We call \(\mathbf{x}\) the search domain and the problem is said to be infeasible if there is no \(x \in \mathbf{x}\) satisfying \(F(x) \in \mathbf{f}\). We also denote constraint satisfaction problems by the triplet \((F, \mathbf{f}, \mathbf{x})\).
This section describes the algorithm designed for solving the subproblems of form (1) using interval arithmetic [11, 13, 20] The solver consists of two components, the memory, and the reducer. The former manages the branch-and-bound tree while the latter is responsible for processing the current box. There is also a post-processing step called cluster builder to group close boxes in the solution list.
The memory keeps the list of unprocessed boxes. It is also responsible for the box selector, and to split the boxes coming from the reducer that cannot be discarded or saved as a solution. In this paper, the selector is a depth-first search procedure while the splitter creates two boxes by dividing the input in the midpoint of the coordinate with maximum width.
The reducer contains a list of rigorous methods to reduce or discard boxes. This paper uses the forward-backward constraint propagation [21] and a feasibility verification method [7]. We consider a CSP of form \((F, \mathbf{x}, \mathbf{f})\) in the next paragraphs to overview each method.
The forward-backward constraint propagation decomposes \(\mathbf{F}\) into a set of simple functions (like the exponential function or the sum of several elements) and displays the pieces in a graph. The forward step is a procedure to evaluate \(\mathbf{F}(\mathbf{x})\) systematically. In this case, the data flow from the decision variable nodes of the graph to the constraint nodes \(\mathbf{F}_{1},\ldots , \mathbf{F}_{m}\). At the end of this step, each constraint node contains an enclosure of \(\mathbf{F}_{i}(\mathbf{x}) \cap \mathbf{f}_{i}\). The backward step acts reversely. It starts from the constraint nodes \(\mathbf{F}(\mathbf{x})\cap \mathbf{f}\) and walks the graph applying inverse functions until reaching \(\mathbf{x}_{1},\ldots , \mathbf{x}_{n}\). At the end of the backward step, we have a new box \(\mathbf{x}' \subseteq \mathbf{x}\) with the reduced search domain.
This paper employs the following feasibility verification method. Let \(\mathbf{x}\) be a box and define the midpoint of \(\mathbf{x}\) as \(x^{*}\). Then, we build a small box \(\mathbf{x}^{*}\) around the \(x^{*}\) and check its feasibility. The box \(\mathbf{x}^{*}\) is a feasible if \(\mathbf{F}(\mathbf{x}^{*}) \subseteq \mathbf{f}\). We also save a box \(\mathbf{x}\) as solution if it satisfies \(\max ({\mathrm{wid}}(\mathbf{x})) < \epsilon _{x}\) for a given \(\epsilon _{x} > 0\).
The order into which we call the rigorous methods to process \(\mathbf{x}\) may influence the efficiency of the branch-and-bound procedure. In this paper, the methods follow the finite state machine described in Table 1.
Table 1 The finite state machine for the inner loop of the Algorithm 1
The parameter \(\epsilon _{T} > 0\) is the threshold tolerance which controls the relative gain of the box \(\mathbf{x}' \subseteq \mathbf{x}\) with the help of the following function
$$\begin{aligned} G_{Rel}(\mathbf{x}, \mathbf{x}') := \max _{\begin{array}{c} {i=1,\ldots ,n;}\\ {{\mathrm{wid}}(\mathbf{x}_{i}) > 0} \end{array}}\left( \frac{{\mathrm{wid}}(x'_{i})}{{\mathrm{wid}}(\mathbf{x}_{i})}\right) . \end{aligned}$$
It is clear that the input of \(G_{Rel}\) at each iteration is the box \(\mathbf{x}\) and the outcome of the rigorous method, \(\mathbf{x}'\).
After processing every box in the memory, we run a post-processing step to build clusters of solutions. This method supports the analysis of the solution list since it reduces the number of boxes on it. Given two intervals \(\mathbf{a}\) and \(\mathbf{b}\), we define the gap between \(\mathbf{a}\) and \(\mathbf{b}\) by
$$\begin{aligned} \text {gap}(\mathbf{a}, \mathbf{b}) :=\left\{ \begin{array}{ll} \inf (\mathbf{b}) - \sup (\mathbf{a}) &{}\quad \text{ if } \inf (\mathbf{b})> \sup (\mathbf{a}) ,\\ \inf (\mathbf{a}) - \sup (\mathbf{b}) &{}\quad \text{ if } \inf (\mathbf{a}) > \sup (\mathbf{b}) ,\\ 0 &{}\quad \text{ if } \mathbf{a}\cap \mathbf{b}\ne \emptyset . \end{array}\right. \end{aligned}$$
We save two boxes \(\mathbf{x}, \mathbf{y}\in \mathbb {IR}^{n}\) in the same group if
$$\begin{aligned} \max _{i=1,\ldots , n} \text {gap}(\mathbf{x}_{i}, \mathbf{y}_{i}) < \epsilon _{C} \end{aligned}$$
where \(\epsilon _{C}\) is the cluster builder tolerance. After assigning a group to every box in the solution set, we return the interval hull of each group and conclude the procedure.
Algorithm 1 summarizes the interval branch-and-bound method. We implement the algorithm in C++ using two interval arithmetic libraries, the Filib [14] and the Moore [17]. The user can choose any of these implementations in the verification of the proof. We report only results from the test with Filib in this paper. The supplementary material also reports the results utilizing Moore. They are consistent with each other.
This section introduces the mathematical model for the containment and the non-overlapping conditions of (1). We call the resulting model the standard constraint satisfaction problem since it is the same for every subproblem. We assume that the squares have side length s. The inequalities for the containment condition follow from the convexity of the circle. On the other hand, non-overlapping constraints rely on the concept of sentinels [4, 18].
Let \(C_{r}\) be the closed circle of radius r and centered at the origin. The convexity of the circle implies that \(c \in C_{r}\) for any point c in the segment of line \(\overline{ab}\) if \(a, b \in C_{r}\). Then, a given square belongs to \(C_{r}\) if and only if its vertices belong to \(C_{r}\).
Let \(S_{0,0}\) be the open square centered at the origin, with no rotation angle and side length s. Then
$$\begin{aligned} S_{0,0} := \big \{ x \in \mathbb {R}^{2} \mid \max (|x_{1}|,|x_{2}| ) - \frac{s}{2} < 0 \big \}. \end{aligned}$$
We denote the closure of a set S by \(\overline{S}\). The set of vertices of \(\overline{S}_{0, 0}\) is given by
$$\begin{aligned} V_{0, 0} := \{V^{NW}, V^{SW}, V^{NE}, V^{SE}\} \end{aligned}$$
$$\begin{aligned} V^{NW} := \left( \begin{array}{c} -\frac{s}{2} \\ \frac{s}{2}\\ \end{array} \right) , V^{SW} := \left( \begin{array}{c} -\frac{s}{2} \\ -\frac{s}{2}\\ \end{array} \right) , V^{NE} := \left( \begin{array}{c} \frac{s}{2} \\ \frac{s}{2}\\ \end{array} \right) , V^{SE} := \left( \begin{array}{c} \frac{s}{2} \\ -\frac{s}{2}\\ \end{array} \right) . \end{aligned}$$
For any \(c \in \mathbb {R}^{2}\) and \(\theta \in \mathbb {R}\), we define the displacement operator as
$$\begin{aligned} h(c, \theta , x) := c + A_{\theta }x \end{aligned}$$
where \(A_{\theta }\) is the rotation matrix
$$\begin{aligned} A_{\theta } := \left( \begin{array}{cc} \cos (\theta )&{} -\sin (\theta ) \\ \sin (\theta ) &{} \cos (\theta )\\ \end{array} \right) . \end{aligned}$$
The open square centered at \(c \in \mathbb {R}^{2}\), with rotation angle \(\theta \in [0, \frac{\pi }{2})\) and side length s is the set given by
$$\begin{aligned} S_{c, \theta } := \big \{ z \in \mathbb {R}^{2} \mid z = h(c, \theta , x), ~ x \in S_{0, 0} \big \}. \end{aligned}$$
The set of vertices of \(\overline{S}_{c, \theta }\), denoted by \(V_{c, \theta }\), is the union of the following points
$$\begin{aligned} V_{c, \theta }^{P} := c + A_{\theta }V^{P}, ~~ P \in \{NW, SW, NE, SE\}. \end{aligned}$$
Finally, we denote the circle of radius r and centered at the origin by \(C_{r}\). Then
$$\begin{aligned} C_{r} := \{x \in \mathbb {R}^{2} \mid x_{1}^{2} + x_{2}^{2} \le r^{2} \}. \end{aligned}$$
Let \(g_{r}(x) := x_{1}^2 + x_{2}^2 - r^{2}\) and consider the following inequalities
$$\begin{aligned} g_{r}(V_{c, \theta }^{P}) \le 0, ~~ P \in \{NW, SW, NE, SE\} \end{aligned}$$
$$\begin{aligned} \overline{S}_{c, \theta } \subseteq \mathbf{C}_{r} ~~~\Leftrightarrow ~~~(4) \text { hold.} \end{aligned}$$
If \(\overline{S}_{c, \theta } \subseteq C_{r}\) then \(V_{c, \theta } \subseteq C_{r}\) and (4) hold. Conversely, since \( \overline{S}_{c, \theta }\) is a bounded polytope, it is given by the convex hull of the elements of \(V_{c, \theta }\). The result follows from the convexity of the circle. \(\square \)
Non-overlapping
This subsection shows that two squares \(S_{c_{1},\theta _{1}}\) and \(S_{c_{2},\theta _{2}}\) are non-overlapping if and only a set of nine points defined on \(S_{c_{1},\theta _{1}}\) do not belong to \(S_{c_{2},\theta _{2}}\) and vice-versa. We call such sets sentinels of a square. Figure 1 illustrates the need of the sentinels in the non-overlapping formulation.
a Non-overlapping squares. b Vertex sentinel violation. c Mid-point sentinel violation. d Center sentinel violation
The set of sentinels of \(S_{0, 0}\) is given by
$$\begin{aligned} T_{0,0} := V_{0,0} \cup \{ V^{N}, V^{S}, V^{E}, V^{W}, V^{O}\} \end{aligned}$$
$$\begin{aligned} V^{N} := \left( \begin{array}{c} 0 \\ \frac{s}{2} \\ \end{array} \right) , V^{S} := \left( \begin{array}{c} 0 \\ -\frac{s}{2} \\ \end{array} \right) , V^{E} := \left( \begin{array}{c} \frac{s}{2} \\ 0 \\ \end{array} \right) , V^{W} := \left( \begin{array}{c} -\frac{s}{2} \\ 0 \\ \end{array} \right) , V^{O} := \left( \begin{array}{c} 0 \\ 0 \\ \end{array} \right) . \end{aligned}$$
We denote the set of sentinels of \(S_{c, \theta }\) by \(T_{c, \theta }\). This set is given by the union of the following points
$$\begin{aligned} T_{c, \theta }^{P} := c + A_{\theta }V^{P}, ~~ P \in \{NW, SW, NE, SE, N, S, E, W, O\}. \end{aligned}$$
The next theorem states that the non-overlapping condition between two squares reduces to the containment verification of their sets of sentinels. It is a particular case of the sentinels theorem proved in [18].
Let \(S_{c_{i}, \theta _{i}}\) and \(S_{c_{j}, \theta _{j}}\) be two squares defined by (3) and let \(T_{c_{i}, \theta _{i}}\) and \(T_{c_{j}, \theta _{j}}\) be their corresponding sets of sentinels. Then
$$\begin{aligned} S_{c_{i}, \theta _{i}} \cap S_{c_{j}, \theta _{j}} = \emptyset ~~~\Leftrightarrow ~~~S_{c_{i}, \theta _{i}} \cap T_{c_{j}, \theta _{j}} = \emptyset \text { and } S_{c_{j}, \theta _{j}} \cap T_{c_{i}, \theta _{i}} = \emptyset . \end{aligned}$$
In order to check conditions of form \(S_{c_{i}, \theta _{i}} \cap T_{c_{j}, \theta _{j}} = \emptyset \) numerically, we need the definition of the inverse of the displacement operator (2)
$$\begin{aligned} h^{-1}(c, \theta , z) := A_{\theta }^{T}(z - c). \end{aligned}$$
Lemma 1
Let \(z \in \mathbb {R}^{2}\) and \(S_{c, \theta }\) be a square defined by (3). Then
$$\begin{aligned} z \in S_{c, \theta } ~~~\Leftrightarrow ~~~\max (|h_{1}^{-1}(c, \theta , z)|, |h_{2}^{-1}(c, \theta , z)|) - \frac{s}{2} < 0. \end{aligned}$$
where \(h_{1}^{-1}\) and \(h_{2}^{-1}\) are the coordinates of the inverse operator.
If \(z \in S_{c, \theta }\) then there exists \(x \in S_{0,0}\) such that \(x = h^{-1}(c, \theta , z)\) and the implication follows immediately. Conversely, let \(x := h^{-1}(c, \theta , z)\). The left hand side of the equivalence implies that \(x \in S_{0,0}\). If we let \(z' := c + A_{\theta }x\) then \(z' = c + A_{\theta }A_{\theta }^{T}(z - c) = z\). Therefore \(z \in S_{c, \theta }\) and the result follows. \(\square \)
Applying the inverse of the displacement operator of the square \(S_{c_{i}, \theta _{i}} \) to the point \(T_{c_{j}, \theta _{j}}^{P} \in T_{c_{j}, \theta _{j}}\) gives
$$\begin{aligned} h^{-1}(c_{i}, \theta _{i}, T_{c_{j}, \theta _{j}}^{P}) = A^{T}_{\theta _{i}}(c_{j} + A_{\theta _{j}}V^{P} - c_{i}), ~~ V^{P} \in T_{0,0}. \end{aligned}$$
Let \(c_{j,1}\) and \(c_{j,2}\) be the coordinates of the vector \(c_{j}\). Then the coordinates of (5) are given by
$$\begin{aligned} u_{1}(c_{i}, c_{j}, \theta _{i}, \theta _{j}, V):= & {} \cos (\theta _{i})(c_{j,1} - c_{i,1}) - \sin (\theta _{i})(c_{j,2} - c_{i,2}) \\&+ \cos (\theta _{i} - \theta _{j})V_{1} + \sin (\theta _{i} - \theta _{j})V_{2} \end{aligned}$$
$$\begin{aligned} u_{2}(c_{i}, c_{j}, \theta _{i}, \theta _{j}, V):= & {} \sin (\theta _{i})(c_{j,1} - c_{i,1}) + \cos (\theta _{i})(c_{j,2} - c_{i,2}) +\\&- \cos (\theta _{i} - \theta _{j})V_{1} + \cos (\theta _{i} - \theta _{j})V_{2}. \end{aligned}$$
The following proposition shows that the verification of \(S_{c_{i}, \theta _{i}} \cap T_{c_{j}, \theta _{j}} = \emptyset \) reduces to the evaluation of nine non-smooth functions.
Let \(S_{c_{i}, \theta _{i}}\) and \(T_{c_{j}, \theta _{j}}\) be as in Theorem 1 and define the function
$$\begin{aligned} u(c_{i}, c_{j}, \theta _{i}, \theta _{j}, V^{P}) := \max (|u_{1}(c_{i}, c_{j}, \theta _{i}, \theta _{j}, V^{P})|, |u_{2}(c_{i}, c_{j}, \theta _{i}, \theta _{j}, V^{P})|). \end{aligned}$$
$$\begin{aligned} S_{c_{i}, \theta _{i}} \cap T_{c_{j}, \theta _{j}} = \emptyset ~~~\Leftrightarrow ~~~u(c_{i}, c_{j}, \theta _{i}, \theta _{j}, V^{P}) - \frac{s}{2} \ge 0 ~\text { for all } V^{P} \in T_{0,0}. \end{aligned}$$
Follows from the application of the Lemma 1 to the elements of \(T_{c_{j}, \theta _{j}}\). \(\square \)
We conclude this section with the formal statement of the standard constraint satisfaction problem. Here and throughout we assume, without loss of generality, that the angle of the first square is always 0. This condition follows from the proper rotation of the remaining squares into the circle.
[SCSP] Let \(\overline{r} > 0\) be an upper bound for the radius of the smallest circle into which one can pack n non-overlapping unit squares and s be a scaling factor. We denote the following problem by standard constraint satisfaction problem (SCSP)
$$\begin{aligned} \text {find}&(r, c_{1},\theta _{1},\ldots ,c_{n},\theta _{n}) \\ {\mathrm{s.t.~}}&u(c_{i}, c_{j}, \theta _{i}, \theta _{j}, V^{P}) - \frac{s}{2} \ge 0 \nonumber \\&g_{r}( V_{c_{i}, \theta _{i}}^{P} ) \le 0 \nonumber \\&c_{i,1}, c_{i,2} \in [- r, r] \nonumber \\&\theta _{i} \in [0, \frac{\pi }{2}] \nonumber \\&\theta _{1} = 0 \nonumber \\&r \le \overline{r} \nonumber \end{aligned}$$
where \(i,j = 1,\ldots , n\) with \(i \ne j\), \(V^{P} \in T_{0,0}\) and \(V_{c_{i}, \theta _{i}}^{P} \in V_{c_{i},\theta _{i}}\). Functions \(g_{r}\) and u are given by Propositions 1 and 2 respectively.
General purpose interval branch-and-bound procedures cannot solve the SCSP in a reasonable amount of time even for small values of n due to symmetries in the search space. This section introduces a tiling method to split (6) into a set of subproblems suitable for the Algorithm 1.
We employ the Matlab-like notation \(g := a:s:b\) to denote the array with \(k := \lfloor \frac{b - a}{s} \rfloor + 1\) elements where \(g_{i} := a + i s\) for \(i = 0,\ldots , k - 1\). In addition, we denote the array with the midpoints of g by \(g_{c}\). Then,
$$\begin{aligned} g_{c,i} := \frac{g_{i} + g_{i+1}}{2}, ~~ i = 0,\ldots , k - 2. \end{aligned}$$
Let \(\overline{r} > 0\) be an upper bound for the SCSP. Then, the step length
$$\begin{aligned} l := \frac{2 \overline{r}}{\lfloor 2 \overline{r} + 1 \rfloor } \end{aligned}$$
splits \([-\overline{r}, \overline{r}]\) into \(\lfloor 2 \overline{r} + 1 \rfloor \) equally divided intervals. Let \(V := \{v \in \mathbb {R}\mid v = -\overline{r} + il, i \in \mathbb {Z}\} \cap [-\overline{r}, \overline{r}]\) be the end points of each interval, satisfying \(v_{i} := -\overline{r} + i l\) for \(i = 0,\ldots , p : = \lfloor 2 \overline{r} + 1 \rfloor \). Moreover, we write the midpoints of V as \(V_{c}\) where \(v_{c, i} := v_{i} + \frac{l}{2}\) for \(i = 0,\ldots , p-1\). Let and . We denote the elements of by \( v_{i,j} := \left( \begin{array}{c} v_{i}\\ v_{j}\\ \end{array} \right) \) for \(v_{i}, v_{j} \in v\) and \(0 \le i, j \le p\). In the same way, we write the elements of as \( c_{i,j} := v_{i,j} + \left( \begin{array}{c} \frac{l}{2}\\ \frac{l}{2}\\ \end{array} \right) \) for and \(0 \le i, j \le p - 1\). Algorithm 2 produces the sets and .
Let \(\triangle ABC\) be the triangle with vertices \(A, B, C \in \mathbb {R}^{2}\). Then, we define the following triangles for \(0 \le i, j \le p -1\)
$$\begin{aligned} \triangle _{i,j}^{T}:= & {} \triangle v_{i,j+1} v_{i+1,j+1} c_{i,j}, \\ \triangle _{i,j}^{L}:= & {} \triangle v_{i,j+1} v_{i,j} c_{i,j}, \\ \triangle _{i,j}^{D}:= & {} \triangle v_{i,j} v_{i+1,j} c_{i,j}, \\ \triangle _{i,j}^{R}:= & {} \triangle v_{i+1,j} v_{i+1,j+1} c_{i,j}. \end{aligned}$$
Here, T, L, D and R stand for top, left, down and right respectively. Figure 2 shows that the definition aims to split the square with vertices \(v_{i,j}\), \(v_{i+1,j}\), \(v_{i+1, j+1}\) and \(v_{i, j+1}\) into four triangles. One can easily verify that the triangles can be written as
$$\begin{aligned} \triangle _{i,j}^{T}:= & {} \{x \in \mathbb {R}^{2} \mid x_{2} - x_{1} \ge g_{j} - g_{i}, ~ x_{2} + x_{1} \ge g_{i} + g_{j+1}, \nonumber \\&x_{1} \in [g_{i}, g_{i+1}] , ~ x_{2} \in [g_{j} + \frac{l}{2}, g_{j+1}]\}, \end{aligned}$$
$$\begin{aligned} \triangle _{i,j}^{L}:= & {} \{x \in \mathbb {R}^{2} \mid x_{2} - x_{1} \ge g_{j} - g_{i}, ~ x_{2} + x_{1} \le g_{i} + g_{j+1}, \nonumber \\&x_{1} \in [g_{i}, g_{i} + \frac{l}{2}] , ~ x_{2} \in [g_{j}, g_{j+1}]\}, \end{aligned}$$
$$\begin{aligned} \triangle _{i,j}^{D}:= & {} \{x \in \mathbb {R}^{2} \mid x_{2} - x_{1} \le g_{j} - g_{i}, ~ x_{2} + x_{1} \le g_{i} + g_{j+1}, \nonumber \\&x_{1} \in [g_{i}, g_{i+1}] , ~ x_{2} \in [g_{j}, g_{j} + \frac{l}{2}]\}, \end{aligned}$$
$$\begin{aligned} \triangle _{i,j}^{R}:= & {} \{x \in \mathbb {R}^{2} s\mid x_{2} - x_{1} \le g_{j} - g_{i}, ~ x_{2} + x_{1} \ge g_{i} + g_{j+1}, \nonumber \\&x_{1} \in [g_{i} + \frac{l}{2}, g_{i+1}] , ~ x_{2} \in [g_{j}, g_{j+1}]\}. \end{aligned}$$
The geometrical meaning of \(\triangle _{i,j}^{o}\) for \(0 \le i, j \le p -1\) and \(o \in \{T, L, D, R\}\)
Lemma 2 is a collection of results needed in this section. In particular, Lemma 2-6 shows that the union of triangles \(\triangle _{i, j}^{o}\) for \(0 \le i, j \le p -1\) and \(o \in \{T, L, D, R\}\) tiles the search domain associated to the center variables in the SCSP.
Let and \(\triangle _{i,j}^{o}\) be defined as above. Then,
\(l < 1\).
\(x \in v ~~\Rightarrow ~-x \in v\).
If then where
$$\begin{aligned} v_{i,j}^{90}:= & {} \left( \begin{array}{c} -v_{j}\\ v_{i}\\ \end{array} \right) , v_{i,j}^{180} := \left( \begin{array}{c} -v_{i}\\ -v_{j}\\ \end{array} \right) , v_{i,j}^{270} := \left( \begin{array}{c} v_{j}\\ -v_{i}\\ \end{array} \right) ,\\ v_{i,j}^{x}:= & {} \left( \begin{array}{c} v_{i}\\ -v_{j}\\ \end{array} \right) , v_{i,j}^{y} := \left( \begin{array}{c} -v_{i}\\ v_{j}\\ \end{array} \right) , v_{i,j}^{Id} := \left( \begin{array}{c} v_{j}\\ v_{i}\\ \end{array} \right) , v_{i,j}^{-Id} := \left( \begin{array}{c} -v_{j}\\ -v_{i}\\ \end{array} \right) . \end{aligned}$$
If then where the vectors are defined analogously as above.
\(\triangle _{i,j}^{o}\) is an isosceles triangle with base length l and legs with length \(\frac{l\sqrt{2}}{2}\) for \(0 \le i,j \le p -1\) and \(o \in \{T, L, D, R\}\).
$$\begin{aligned}{}[-\overline{r}, \overline{r}]^{2} \equiv \bigcup _{\begin{array}{c} {0 \le i, j \le p} \\ {o \in \{T, L, D, R\}} \end{array}} \triangle _{i, j}^{o}. \end{aligned}$$
For \(a > 0\), we have \(\lfloor a + 1 \rfloor = a + 1 - \delta \) where \(\delta \in [0, 1)\) is the fractional part of \(a + 1\). Then \(1 - \delta > 0\) and \(\lfloor a + 1 \rfloor > a\). The result follows by taking \(a = 2 \overline{r}\).
If \(x \in v\) then \(-x = \overline{r} - il\) for some \(i \in 0, \ldots , p\). Let \(y = -\overline{r} + jl\) and we need to verify if there exists some \(j \in 0, \ldots , p\) such that \(y = -x\). The equality holds by taking \(j = p - i\).
If then and the result follows from the application of Lemma 2-2 of this proposition to each case.
The proof is similar to the case above.
For \(\triangle _{i, j}^{T}\), we have \(\Vert v_{i,j+1} - v_{i + 1, j + 1}\Vert = l\) and
$$\begin{aligned} \Vert v_{i,j+1} - c_{i, j}\Vert = \Vert v_{i + 1,j + 1} - c_{i, j}\Vert = \frac{l\sqrt{2}}{2}. \end{aligned}$$
The proof is similar for \(o \in \{L, D, R\}\).
Let \(S_{i, j}\) be the closed square with vertices \(v_{i,j}\), \(v_{i+1, j}\), \(v_{i + 1, j + 1}\), \(v_{i, j + 1}\). Since \(v_{0} = -\overline{r}\) and \(v_{p} = \overline{r}\) it is clear that
$$\begin{aligned}{}[-\overline{r}, \overline{r}]^{2} \equiv \bigcup _{0 \le i, j \le p - 1} S_{i, j}. \end{aligned}$$
The result follows by noting that
$$\begin{aligned} S_{i, j} \equiv \bigcup _{o \in \{T, L, D, R\}} \triangle _{i, j}^{o}. \end{aligned}$$
\(\square \)
We also assign a label to each triangle in the tiling. It helps us to easily identify a specific triangle during the proof of the case \(n = 3\) in Sect. 5. Triangles of form \(\triangle _{i,j}^{T}\) receive an index that is divisible by 4. In the same way, we assign labels to the left, down and right triangles with the congruence classes 1, 2 and 3 modulo 4, respectively. We denote the triangle with label i by \(T_{i}\). Figure 3-Left shows the tiling for the best known upper bound of \(r_{3}\).
Left: Tiling for the square \([-\overline{r}_{3}, \overline{r}_{3}]^{2}\) where \(\overline{r}_{3} = \frac{5\sqrt{17}}{16} \). Right: Tiling for the square \([-3, 3]^{2}\)
We show now that each triangle of form \(\triangle _{i, j}^{o}\) contains the center of at most one unit square.
The minimal distance between the centers of two non-overlapping unit squares is 1.
Assume the contrary, let pq be a line segment of the centers with lower than 1. Let \(C_{p}\) and \(C_{q}\) the circles of radius \(\frac{1}{2}\) drawn into the squares. Then \(C_{p}\) and \(C_{q}\) intersect. But then since the squares are supersets of \(C_{p}\) and \(C_{q}\), respectively, they also intersect. A contradiction. \(\square \)
Let \(\triangle _{i, j}^{o}\) for \(0 \le i, j \le p - 1\) and \(o \in \{T, L, D, R\}\) be as defined above. If \(S_{c_{1}, \theta _{1}}\) and \(S_{c_{2}, \theta _{2}}\) are two unit squares such that \(c_{1}, c_{2} \in \triangle ABC\) then \(S_{c_{1}, \theta _{1}} \cap S_{c_{2}, \theta _{2}} \ne \emptyset \).
Lemma 2-1 shows that \(l < 1\) and Lemma 2-5 gives that the base length of \(\triangle _{i, j}^{o}\) is l while its legs have length \(\frac{l\sqrt{2}}{2}\). The result follows from Lemma 3. \(\square \)
Let \(K := 4 p^{2}\) be the number of triangles in the tiling. Proposition 3 states that we can split the SCSP into a set of \(\left( {\begin{array}{c}K\\ n\end{array}}\right) \) subproblems. In each subproblem, we enforce that the center of each square belongs to a given triangle. For example, one can define the subproblem \(T_{0}T_{19}T_{33}\) in the same tiling displayed in Fig. 3-Left. In this case, we set the standard constraint satisfaction problem defined in (1) and add to the model the linear inequalities given by Eqs. (8)–(11) for \(\triangle _{0,0}^{T}\), \(\triangle _{1, 1}^{R}\) and \(\triangle _{2, 2}^{L}\) respectively.
We conclude this subsection by proving that several subproblems can be discarded without any processing due to symmetries in and . Let \(f^{90}, f^{180}, f^{270}: \mathbb {R}^{2} \rightarrow \mathbb {R}^{2}\) be the linear mappings that rotate the vector \(x \in \mathbb {R}^{2}\) by an angle of 90, 180 and 270 degrees respectively. In the same way, define the linear mappings \(f^{x}, f^{y}, f^{Id}, f^{-Id}: \mathbb {R}^{2} \rightarrow \mathbb {R}^{2}\) as the reflections around the lines \(x = 0\), \(y = 0\), \(y = x\) and \(y = -x\) respectively.
Let \(\triangle _{i, j}^{o}\) for \(0 \le i, j \le p - 1\) and \(o \in \{T, L, D, R\}\) be a triangle of form (8) to (11). Then, \(f^{op}(\triangle _{i,j}^{o})\) for \(op \in \{90, 180, 270, r, x, Id, -Id\}\) is a triangle of form \(\triangle _{i', j'}^{o'}\) with \(0 \le i', j' \le p - 1\) and \(o' \in \{T, L, D, R\}\).
The triangle \(\triangle _{i,j}^{o}\) has two vertices in and one vertex in . Let A and B be the vertices in and C be the vertex in . Lemma 2-3 ensures that while Lemma 2-4 gives that . Since rotations and reflections are rigid transformations, the result holds. \(\square \)
Proposition 4 allows us to discard subproblems that are symmetric by rotations or reflections. For example, let \(\overline{r}_{3} = \frac{5\sqrt{17}}{16}\) and \(r_{3}, S_{c_{1}, \theta _{1}}, S_{c_{2}, \theta _{2}}, S_{c_{3}, \theta _{3}}\) be a feasible arrangement for (6) with \(c_{1} \in T_{7}\), \(c_{2} \in T_{12}\) and \(c_{3} \in T_{22}\). Then, Proposition 4 ensures that there exists a feasible arrangement \(r_{3}, S_{c_{1}', \theta _{1}'}, S_{c_{2}', \theta _{2}'}, S_{c_{3}', \theta _{3}'}\) satisfying \(c_{1}' \in T_{19}\), \(c_{2}' \in T_{12}\) and \(c_{3}' \in T_{22}\). Moreover, since \(T_{19}T_{12}T_{22}\) is obtained by a reflection around the y axis of \(T_{7}T_{12}T_{22}\), we know that \(c_{i}' = f^{y}(c_{i})\) for \(i = 1, 2, 3\).
The tiling produced by Algorithm 2 suffices if one wants to use a complete global optimization approach for the packing problem. On the other hand, it is not suitable for a rigorous approach since the elements in and are floating point vectors subject to rounding errors. To overcome this problem, we introduce a scaled tiling. In this case, we ensure that the points at and are integer vectors to the cost of working with squares that are not unit but have the side length contained in a small interval \(\mathbf{s}\). Algorithm 3 produces the scaled vertices for the tiling as well as the interval \(\mathbf{s}\).
The elements in and are integer vectors by construction. Then, the Eqs. (8)–(11) are exactly representable. On the other hand, we replace the constant s in the Problem (6) by the interval \(\mathbf{s}\) to keep the mathematical certainty of our statements. The lemmas and propositions in the last section remain valid after the proper scaling. Figure 3-Right illustrates the scaled tiling for \(\overline{r}_{3} = \frac{5\sqrt{(17)}}{16}\). Note that the tiling would be the same for \(\overline{r}_{4} = \sqrt{2}\) and the only difference between both cases would be the scaling interval \(\mathbf{s}\).
Markót and Csendes [24] propose tiling constraints for the circle packing problem based on rectangles. The same idea could be used for the packing of squares into a circle. On the other hand for the case \(n = 3\), one would need to split the search domain in 144 squares instead of 36 as proposed in this paper.
Packing 3 unit squares
Friedman [9] gives an upper bound for the case \(n = 3\), \(\overline{r}_{3} = \frac{5 \sqrt{17}}{16}\). Algorithm 3 gives the tiling displayed in Fig. 3-Right and the interval scaling factor
$$\begin{aligned} s := [\underline{2.328342000348}79, \underline{2.328342000348}80]. \end{aligned}$$
Figure 4-Left displays an optimal configuration associated to the scaled version of the problem. This section proves the theorem below.
Let \(r_{3}\) be the solution of (1) for \(n = 3\). Then,
$$\begin{aligned} r_{3} \in [\underline{1.288470508005}47, \underline{1.288470508005}53]. \end{aligned}$$
Moreover, the parameters of \(S_{c_{1}, \theta _{1}}\), \(S_{c_{2}, \theta _{2}}\) and \(S_{c_{3}, \theta _{3}}\) belong to the boxes in Table 6.
Left: An optimal configuration for \(n = 3\). Right: Triangles 7, 12 and 22 contain an optimal arrangement
We perform the computational part of the proof in a core i7 processor with a frequency of 2.6 GHz, 6 GB of RAM and Windows 10. We compiled the code using the g++ 7.3 compiler with the option \(-O3\). A supplementary material for the proof, containing the statistics and the log files for each subproblem is available in http://www.mat.univie.ac.at/~montanhe/publications/n3.zip
We prove the theorem in three phases. At the i-th iteration, we consider instances of form (6) and define subproblems by adding tiling constraints of form (8)–(11) accordingly.
The proof considers the scaled version of the problem to ensure the mathematical certainty of our statements. Therefore, the CSPs in this section are of form (6) with the constant s replaced by the interval \(\mathbf{s}\) in Eq. (12). We obtain the unscaled interval for \(r_{3}\) and Table 6 by dividing every box found in the last iteration by \(\mathbf{s}\).
We also assume the labeling scheme for the triangles introduced in Sect. 4 and displayed on Fig. 3-Right. Therefore, the subproblem \(T_{7}T_{12}T_{22}\) refers to the SCSP with the interval scaling factor \(\mathbf{s}\) and such that \(c_{1} \in T_{7} := \triangle _{0, 1}^{R}\), \(c_{2} \in T_{12} := \triangle _{1, 0}^{T}\) and \(c_{3} \in T_{22} := \triangle _{1, 2}^{D}\).
Phase 1 In this iteration, we are interested in reducing the search domain of each subproblem and finding triangles which can contain the center of squares with no rotation. The tiling has 36 triangles, but the symmetries in and reduce the number of subproblems to 6. Table 2 shows the instances discarded without any processing in the first phase.
Table 2 Instances discarded in the first phase without processing
Instances \(T_{0}\), \(T_{1}\), \(T_{4}\), \(T_{5}\), \(T_{7}\) and \(T_{16}\) require processing. We run the Algorithm 1 with \(\epsilon _{T} = 10^{-1}\), \(\epsilon _{C} = 10^{-11}\), \(\epsilon _{x} = 10^{-13}\) and time limit of 300 s. In this phase, we remove the condition \(\theta _{1} = 0\) in Problem (6). Table 3 summarizes the results of the processed instances on phase 1. It shows that \(T_{1}\) and \(T_{5}\) are infeasible and any combination containing one of these triangles or their symmetric counterparts could be removed in the next phases. Moreover, it shows that only triangles \(T_{7}\) and \(T_{16}\) can contain the center of a square with rotation angle 0. Since we are assuming that \(\theta _{1} = 0\) in the optimal configuration for \(n = 3\), we only have to check the combinations containing at least one of these triangles.
Table 3 Statistics for the processed instances on phase 1
Phase 2 This phase aims to discard as many instances as possible to reduce the number of hard subproblems in the last iteration. There are 630 possible combinations of 36 triangles taken 2 by 2. After eliminating symmetric and previously discarded cases, we obtain 43 instances. We also propagate any reduction in the search domain in the first phase to the subproblems in the second phase. Again, we remove the condition \(\theta _{1} = 0\) from Problem (6).
We run the Algorithm 1 with \(\epsilon _{T} = 10^{-1}\), \(\epsilon _{C} = 10^{-11}\), \(\epsilon _{x} = 10^{-13}\) and time limit of 3600 s. We stop the algorithm as soon as the feasibility verification method described in Sect. 2 succeeds in finding a feasible point. The supplementary material contains the list of all instances discarded without processing. Table 4 gives the statistics for the 43 processed instances.
We conclude the second phase with 22 infeasible subproblems. Again, any case in the next phase containing a combination found infeasible in this step can be discarded without any processing.
Phase 3 In this phase we set the full model in Problem (6), including the constraint \(\theta _{1} = 0\). Table 3 shows that \(c_{1} \in T_{7}\) or \(c_{1} \in T_{16}\). Therefore, after removing the cases where one of these conditions do not hold and eliminating symmetric and already proved infeasible subproblems, we obtain 12 instances of the 7140 possible ones.
If an instance contains both triangles \(T_{7}\) and \(T_{16}\), we denote by \(T_{7*}T_{16}T_{x}\) the case where we enforce the angle of the square centered in \(T_{7}\) to be zero. In the same way, we write \(T_{7}T_{16*}T_{x}\) for the instances where the square centered in \(T_{16}\) has no rotation angle.
For the last phase, we run Algorithm 1 with \(\epsilon _{T} = 10^{-1}\), \(\epsilon _{C} = 10^{-11}\), \(\epsilon _{x} = 10^{-13}\) and no time limit. Table 5 provides the statistics of the processed instances. Moreover, Table 5 shows that it is the only instance containing the optimal configurations for \(n = 3\). Figure 4-Right shows an approximation of the center of each square in the optimal case.
Algorithm 1 produces 4 clusters for the instance \(T_{7}T_{12}T_{22}\). The maximum width of a cluster is \(6.23*10^{-13}\). The precision is smaller than \(\epsilon _{x}\) due to the cluster builder procedure described in Sect. 2. Table 6 gives the unscaled clusters.\(\square \)
Table 6 Enclosures of the optimal arrangement for \(n = 3\)
This paper presents a framework for the rigorous optimization of the packing of unit squares into a circle. We express the question as the standard constraint satisfaction problem stated by Definition 1. The model considers the concept of sentinels to formulate non-overlapping constraints and the convexity of the squares and the circle to describe containment conditions.
General purpose rigorous optimization solvers cannot achieve the solution of the standard constraint satisfaction problem due to symmetries in the search domain. To overcome this difficulty, we propose a tiling method that splits the search space related to the center of each unit square into isosceles triangles. Our tiling divides the original problem into a set of subproblems that are suitable for the interval branch-and-bound approach. We also ensure that the parameters in each subproblem are free of rounding errors by introducing a proper scaling of the search domain.
To show the capabilities of our approach, we solve the first open case reported in the literature, \(n = 3\). We implement the interval branch-and-bound in the C++ and the code is publicly available. We perform the proof on an ordinary laptop with 6 GB of RAM and a core i7 processor.
The proof of the case \(n = 3\) requires the solution of 6 subproblems with one square, 43 with two and only 12 with three squares. We discard most subproblems without processing due to symmetries in the tiling. Among the 61 subproblems, just 6 require more than 100 s to conclude the search. At the end of the process, we obtained 4 boxes with the following properties
The maximum width of any coordinate of the resulting boxes is \(6.23*10^{-13}\).
If one disregard symmetries, every solution of (1) is contained in at least one of the 4 boxes
The method proposed in this paper could, in principle, be used to find the optimal arrangement for higher values of n (e.g., \(n = 4, 5, 6\).).
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Open access funding provided by Austrian Science Fund (FWF).
Wolfgang Pauli Institute, Oskar-Morgenstern-Platz 1, 1090, Vienna, Austria
Tiago Montanher
& Mihály Csaba Markót
Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090, Vienna, Austria
, Arnold Neumaier
, Mihály Csaba Markót
, Ferenc Domes
& Hermann Schichl
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Correspondence to Tiago Montanher.
This research was supported through the research Grants P25648-N25 and P27891 of the Austrian Science Fund (FWF).
Below is the link to the electronic supplementary material.
Supplementary material 1 (zip 65514 KB)
Montanher, T., Neumaier, A., Csaba Markót, M. et al. Rigorous packing of unit squares into a circle. J Glob Optim 73, 547–565 (2019) doi:10.1007/s10898-018-0711-5
Issue Date: 15 March 2019
Square packing into a circle
Interval branch-and-bound
Tiling constraints
Computer-assisted proof
Mathematics Subject Classification | CommonCrawl |
Impact of climate change and seasonal trends on the fate of Arctic oil spills
Tor Nordam1,
Dorien A. E. Dunnebier1,
CJ Beegle-Krause1,
Mark Reed1 &
Dag Slagstad1
Ambio volume 46, pages 442–452 (2017)Cite this article
We investigated the effects of a warmer climate, and seasonal trends, on the fate of oil spilled in the Arctic. Three well blowout scenarios, two shipping accidents and a pipeline rupture were considered. We used ensembles of numerical simulations, using the OSCAR oil spill model, with environmental data for the periods 2009–2012 and 2050–2053 (representing a warmer future) as inputs to the model. Future atmospheric forcing was based on the IPCC's A1B scenario, with the ocean data generated by the hydrodynamic model SINMOD. We found differences in "typical" outcome of a spill in a warmer future compared to the present, mainly due to a longer season of open water. We have demonstrated that ice cover is extremely important for predicting the fate of an Arctic oil spill, and find that oil spills in a warming climate will in some cases result in greater areal coverage and shoreline exposure.
Human activity in the Arctic (and elsewhere) is always associated with some risk of damage to the environment. In order to assess risk, one needs to estimate the probability of a given adverse outcome, as well as the consequences. In this paper, we study the probable outcomes of Arctic oil spills, using a selection of six different case studies. We specifically do not address the probability of a spill taking place, nor the consequences (i.e. the damage to natural resources), focusing instead on the transport and fate of the released oil, and the probability distributions associated with this fate.
We have studied the problem using ensembles of numerical simulations, which is a standard approach to investigating the possible outcomes of hypothetical oil spills (see e.g. Price et al. 2003; Guillen et al. 2004). Ensemble simulations are commonly used in the planning phase of new petroleum developments, and serve two goals: First, to provide the probabilities of for example beaching of oil, for a further Environmental Risk Assessment (ERA) where potential damage to natural resources will be considered, and second, to inform contingency planning for oil spill response, giving guidance on required amounts and distribution of response equipment (Barker and Healy 2001).
This study will investigate the effect of a future with a warmer climate on the footprint and fate of an Arctic oil spill. We consider not only average outcomes, but also the probability distribution of endpoints such as amount of beached oil (Nordam et al. 2016). The goal is first to determine if the "typical" transport and fate of an oil spill is different in a warmer climate, and second to determine if this has consequences for how Environmental Risk Assessments should be carried out in the Arctic, as we move towards a warmer future with more human activity in the High North.
The current study is carried out by numerical simulations, looking at a limited number of case studies selected to represent relevant oil spill scenarios that could occur in the Arctic. Below we present the numerical models used, as well as the scenarios studied.
The SINMOD hydrodynamic model
Current and wind data are required as input for oil spill trajectory modelling. The SINMOD hydrodynamic model was used to produce the current data (Slagstad and McClimans 2005). SINMOD is based on the primitive Navier–Stokes equations and is established on a z-grid, using a constant-depth discretisation. The vertical turbulent mixing coefficient is calculated as a function of the Richardson number, Ri, and the wave state. The flow becomes turbulent when Ri is smaller than 0.65 (Price et al. 1986). Near the surface, vertical mixing due to wind waves is calculated from wind speed and fetch length. Horizontal mixing is calculated according to Smagorinsky (1963).
The SINMOD model area used to generate the hydrodynamic data for this study is shown in dashed outline in Fig. 1. The model area has a spatial resolution of 4 × 4 km, and the dataset produced has a temporal resolution of 2 h. Boundary conditions were taken from a larger model domain, at 20 × 20 km resolution. A total of 8 tidal components were imposed by specifying the various components at the open boundaries of the large-scale model. Tidal data were taken from TPXO 6.2 model of global ocean tides (Egbert et al. 1994).Footnote 1
Locations of the six scenarios. 1: Finnmark, 2: Greenland I, 3: Greenland II, 4: Svalbard, 5: Kara Strait, 6: Varandey (see Table 1 for details). The model area of the SINMOD hydrodynamic model is shown in dashed outline
For the present climate simulation (2009–2012), atmospheric data from the ERA-Interim Reanalysis (Dee et al. 2011) have been used. For the climate change case (2050–2053), the atmospheric forcing fields come from a regional model system run by the Max Planck Institute, REMO (Keup-Thiel et al. 2006), and is based on the IPCC's A1B scenario (Nakicenovic and Swart 2000). This model is configured to cover the model domain of SINMOD and has a grid resolution of approximately 0.22 degrees.
The ice model in SINMOD is a Hibler formulation (Hibler 1979), and has two state variables: average ice thickness in a grid cell, h, and the fraction of a grid cell covered by ice, A. The remaining fraction, 1 − A, is open water. The equation solver uses the elastic–viscous–plastic mechanism as described by Hunke and Dukowicz (1997).
The OSCAR oil spill model
For the oil spill simulations in this paper, we have used OSCAR, which is a fully three-dimensional oil spill trajectory model for predicting the transport, fate and effects of released oil. The model accounts for weathering, the physical and chemical processes affecting oil at sea, as well as biodegradation. The development of models for these processes is strongly coupled with laboratory and field activities at SINTEF, on the transport, fate and effects of oil and oil components in the marine environment (Brandvik et al. 2013; Johansen et al. 2003, 2013, 2015).
The OSCAR model computes surface spreading of oil, slick transport, entrainment into the water column, evaporation, emulsification and shore interactions to determine oil drift and fate at the surface. In the water column, horizontal and vertical transport by currents, dissolution, adsorption and settling are simulated. The different solubility, volatility and aquatic toxicity of oil components are accounted for by representing oil in terms of 25 pseudo-components (Reed et al. 2000), which represent groups of chemicals with similar physical and chemical properties. By modelling the fate of individual pseudo-components, changes in oil composition due to evaporation, dissolution and biodegradation are accounted for. There is a biodegradation rate for each of the pseudo-components for the dissolved water fraction, droplet water fraction, surface and sediments (Brakstad and Faksness 2000).
OSCAR uses a Lagrangian particle transport model, where the release is represented by numerical particles (Reed et al. 2000). Each numerical particle is transported individually through the flow field. Buoyancy and sinking of oil droplets due to density differences or oil mineral aggregates are also included. Required inputs to the OSCAR model are currents, wind and ice (if relevant). The chemical composition of the released oil is also an essential part of the input to OSCAR.
Scenarios and locations
Six scenarios were selected to be used as case studies. Details of the scenarios are provided in Table 1. We would like to stress that these scenarios are fictitious, and were made up to show how the footprint of an oil spill might differ under a climate change scenario. These scenarios are not meant to represent the most likely oil spill scenarios in the Arctic, and do not take into account expected changes in activities and shipping routes between the present and the future.
Table 1 Scenario parameters for the six case studies
The first scenario (Finnmark) is included because the release area has fully open water throughout the year, in both the present climate (2009–2012) and the future (2050–2053). It will serve as a "control scenario", to compare to the five other scenarios, where there is a change in ice cover from present to future. The Finnmark scenario is a well blowout, with a release rate of 5000 metric tons per day, which is a little less than what was seen in the Deepwater Horizon oil spill (McNutt et al. 2012). The duration of 72 h is relatively short, and is representative of a scenario in which the well is successfully capped in short order.
The next two scenarios (Greenland I and Greenland II) are also well blowouts, with all parameters except location and depth equal to the Finnmark scenario. The difference between the two Greenland scenarios is the location, i.e. the depth and distance from the coast. We have selected to use the properties of crude oil from the Statfjord C field for the modelling of the three well blowouts. The Statfjord C Blend crude oil is regarded as a paraffinic medium crude oil with a density of 0.834 g/ml (API gravity 38). The fresh oil has a medium content of wax (4.1% by weight) and low asphaltenes (0.09% by weight) compared with other crude oils in the Norwegian sector. The oil exhibits a medium evaporative loss and forms relatively stable water-in-oil emulsions with high water content (approximately 80%).
The next two scenarios (Svalbard and Kara Strait) are shipping accidents, where we consider a surface spill of 200 metric tons of marine diesel, over a period of 3 h. For the two shipping accident scenarios, we have selected to use a marine diesel, which has a density of 0.843 g/ml (API Gravity 36.4), with a low pour-point (lower than − 24°C). The diesel oil will not emulsify and form stable water-in-oil emulsions and minor water uptake is expected. Both the fresh oil and its evaporated residues exhibit low viscosities.
The final scenario (Varandey) is a pipeline leak, near the Varandey oil export terminal, located 22 km off the coast, in a shallow area to the west of the Kara Strait. Two hundred metric tons of oil are assumed to leak out over a period of 3 h, from a depth of 14 m (1 m above the sea bed). Here we have chosen a Russian Export crude oil for the modelling. It is a medium density oil, at 0.871 g/ml (API gravity 31). The oil exhibits features of an intermediate between an asphaltenic and a more paraffinic oil, due to the relatively high content of asphaltenes (1% by weight). The oil exhibits a low to medium evaporative loss, and the oil will easily emulsify and form stable water-in-oil emulsions with high viscosities and high water content (70–80% by volume). This oil is expected to be dispersible with application of chemical dispersants.
Ice cover
In Fig. 2, the ice coverage (fraction of the sea surface covered by ice) is shown, for each of the six release locations. In the Finnmark scenario, there is fully open water both in the present and the future, while in the Svalbard scenario, there is quite high ice cover for several months of the year in the present, but always open water in the future. For the two Greenland scenarios, the change in ice cover can be clearly seen as a longer season of open water in the future scenarios. For the Kara scenario, there is clearly a winter season also in the future data, but it is shorter, and with less ice cover on average. For the Varandey scenario, the picture is less clear, with highly variable ice cover in both the present and the future, but on average there is somewhat less ice in the future. As a rule of thumb, a surface oil slick is considered to move as if in completely open water for ice coverage less than 30%, and as if in full ice cover if the ice coverage is above 70%. The current and wind data show less clear differences between present and future. Some information on the wind and current data can be found in the form of wind/current roses in Figs. S1–S4 in Supplementary Material.
Ice coverage (fraction) at each release location, as a function of time of year, shown for the present and the future separately. Note that ice coverage is always 0 in the Finnmark scenario, and 0 in the future for the Svalbard scenario
Ensemble simulations
Even in a future with a warmer climate, there is no guarantee that any given day, or even a given year, will be warmer than in the present. Rather, global average temperatures will be higher, with potential changes in weather patterns, prevailing winds, ice cover, etc. As climate change is a statistical phenomenon, any single simulation of an oil spill at a given time and location is not particularly relevant for this study. Instead, we are interested in how the distribution of probable outcomes will change with the changing climate.
Ensemble simulation methods are commonly used in the study of chaotic systems, such as the weather and the ocean. In this paper, when referring to an ensemble, we mean a collection of oil spill simulations, where each simulation has used different environmental forcing data. The environmental data are available as two archives, one historical hindcast archive, covering 2009–2012 (and some months into 2013), and one climate forecast archive, covering 2050–2053 (and some months into 2054). The variation in environmental data is achieved through varying the start date of the simulation, and selecting the corresponding data (current, wind, ice cover and water temperature) from the two archives. In a real oil spill at a given location, the timing will determine the currents, winds, waves and other meteorological conditions, which together with the release parameters determine the transport and fate of the spill. Hence, by carrying out large ensembles of oil spill simulations with different start times, we sample from the distribution of possible predicted outcomes for that location.
We have performed ensemble simulations for six different scenarios (see Table 1). For each scenario, we started one simulated oil spill every 6 h for 2 × 4 years, in total 11 688 simulations per scenario. Each simulation was set up as an individual OSCAR scenario, and took approximately 6 min to run, for a total of about 7000 CPU hours. The simulations were carried out on a 32-core compute node, using the linux version of OSCAR.
The simulation results from the OSCAR model include four-dimensional (x, y, z, t) concentration fields giving concentration per pseudo-component for droplets and dissolved chemicals, as well as three-dimensional (x, y, t) grids for oil on the sea surface, on the shore and in the sediments.
Furthermore, some aggregated quantities from each simulation are available as time series. These include amounts of evaporated oil, oil on the sea surface, submerged oil, oil on the shore, oil in the sediment and amount of oil which has been biodegraded. These six quantities make up the mass balance, and give information about the fraction of the total released mass which is found in any given "environmental compartment". During the development of a spill, oil can move between compartments: Oil on the surface can be mixed down by waves and submerged, submerged oil can resurface, stranded oil can be washed out to sea, etc. The exception is that evaporated or biodegraded oil is removed from the simulation. Note that oil which is trapped at the ice/water interface under sea ice is included in the "surface" compartment.
In Fig. 3 (left), an example of the development of the mass balance is shown for one's realisation of the Finnmark scenario. We see how the oil is released at a constant rate over 3 days, reaching a total of 15 000 metric tons. Evaporation and biodegradation are continuously ongoing processes, while we note that a significant fraction of the oil hits shore after about 5 days. At the end of the 15-day simulation, 33% of the oil has evaporated, 2% is at the surface, 7% is submerged, 13% has been biodegraded, 43% has stranded and 2% has ended up in the sediments. It should be mentioned that this simulation is the worst case from the Finnmark scenario, with respect to amount of stranded oil.
To the left, an example of the development of the mass balance for a simulation with significant stranding after about 5 days. To the right, average mass balances for all six scenarios. For each scenario, the left column represents the present (2009–2012), and the right column represents the future (2050–2053)
The goal of this study is to investigate how the footprint of a "typical" oil spill will change with a warmer climate. In order to obtain statistical results, we have chosen to consider the amount of oil in the six environmental compartments only, e.g. considering the amount of oil on the shore, without reference to exactly where that oil has beached. For the further analysis in this study, we will only use the data at the end of each simulation, corresponding to day 15 in the left panel of Fig. 3.
Furthermore, in Fig. 3 (right), the average amount in each compartment is shown separately for each of the six scenarios, and for the present and the future. We note that in all cases, there is some difference between the present and the future, but the nature and magnitude of the change differs for each scenario.
In addition to average values, another way of presenting the results of an ensemble of simulation is as a time series showing the amount of oil in an environmental compartment at the end of a simulation, as a function of the start time of that simulation. In Fig. 4 (top row), this is shown for amount of biodegraded oil and amount of oil in the sediment, for the Svalbard scenario. From Fig. 2, we see that in the future scenarios (2050–2053), there is always open water at the release location of the Svalbard case, while in the present scenarios, there is high ice cover roughly from January to the end of May, with some variation among the years. Clearly, the presence of ice coincides with the onset of a completely different behaviour in the time series for the present shown in Fig. 4. We note also that the behaviour in the ice-free season seems quite similar for the present and the future.
Top row: Time series showing amount of biodegraded oil (left) and amount of oil in the sediments (right), at the end of each simulation, as a function of start date of the simulation, for the Svalbard scenario. Present and future shown separately. Bottom row: Histogram and empirical distribution for the time series above, with present and future shown separately. Note that the axis has been truncated for visibility in the sediment plot. The height of the first column is 0.24 for the present and 0.19 for the future
As was recently shown (Nordam et al. 2016), the amounts of oil in the different compartments exhibit different probability distributions. In particular, the amount of stranded oil and oil in the sediment seem to follow some variety of a fat-tailed distribution. Consequently, average values may be of little use, or even misleading, and applying intuition from Gaussian distributions can cause severe underestimation of the probable worst-case scenario. As an extreme example, consider the amount of stranded oil in the present time in the Greenland II scenario: The 95-percentile (the amount which is such that in 95% of cases, the amount is less than this) is 1 metric ton, the 99-percentile is 83 tons and the 99.9-percentile is 1317 tons. For a Gaussian distribution of the same mean and standard deviation (µ = 9.47 tons and σ = 98.8 tons), the 99- and 99.9-percentile would be, respectively, 1.4 times and 1.8 times the 95-percentile.
Since the distributions vary, a more complete way of displaying the results of the ensemble simulation is to consider the empirical distribution of each endpoint. In Fig. 4 (bottom row), the distributions for amount of biodegraded oil, and amount of oil in the sediments, are shown for the Svalbard scenario, both as a histogram and as an empirical cumulative probability distribution, and separately for the present and the future. Note how the amount of biodegraded oil exhibits a bi-modal distribution in the present, due to a different behaviour when there is ice, and note also how the two endpoints (biodegraded and sediment) display completely different distributions.
In Table 2, we show the mean, the standard deviation and the 95-percentile for the amount of biodegraded oil, oil on the surface and stranded oil, as well as the total area of the surface slick, for all six scenarios, and separately for the present and the future. From Fig. 4, we note that the difference between summer and winter in the present seems to be much larger than the difference between the present and the future. In order to investigate the effect of ice cover in determining the seasons, we can categorise each simulation in the ensembles as "winter" if the ice coverage is above 70% at the release location at the start of the release, and as "summer" if the ice coverage is below 30%. This split will then allow us to construct the empirical distributions for the summer and winter data separately, and compare between present and future. In Fig. 5, the empirical distributions are shown for amount of biodegraded, surface and stranded oil in all scenarios, split into winter and summer as described above. Note that in the Finnmark scenario, there is no ice, neither in the present or in the future, and in the Svalbard scenario, there is no ice in the future scenarios, and consequently no points are classified as winter.
Table 2 Average (Avg.), standard deviation (Std.) and 95-percentile (95%) of amounts of biodegraded oil, oil on the surface and stranded oil (in metric tons), as well as surface slick area (in square kilometres), with present and future shown separately for each scenario
Empirical distribution of the amount of biodegraded oil, amount of oil on the surface and amount of stranded oil, 15 days after the start of the release, for all scenarios. Simulations are classified as Winter (70% ice cover or higher) or Summer (30% ice cover or less). The number of simulations in each category is given in the legends. Note that in some scenarios the ice cover never exceeds 70%
Finally, in order to give a simple estimate of the relative impact of the various environmental inputs, on the amount of oil that ends up in the different environmental compartments, we have calculated the normalised cross-correlations. The normalised cross-correlation of two discrete series, x i and y i , each with N elements, is given by
$$ \frac{1}{N}\sum\limits_{i} {\frac{{(x_{i} - \mu_{x} )(y_{i} - \mu_{y} )}}{{\sigma_{x} \sigma_{y} }}} , $$
where µ x and σ x are the mean and standard deviation of x i , and µ y and σ y are the same for y i .
The normalised cross-correlations are shown in Fig. 6. In calculating the correlations, the environmental inputs have been averaged over the 15 days of each simulation. Thus, if the cross-correlation between, e.g. the amount of stranded oil and the ice cover is calculated, then x i is the amount of stranded oil at the end of simulation i, and y i is the average ice cover at the release location during the 15 days of simulation i. Note again that for the Finnmark scenario, the ice cover is always 0, and for the Svalbard scenario, the ice cover is always 0 in the future, and hence the correlations with ice cover will be 0 in these cases.
Normalised cross-correlation of amount in the six environmental compartments, with five environmental inputs: wind speed, wind direction, current speed, current direction and ice cover. Present (2009–2012) shown in blue, future (2050–2053) shown in red. All environmental inputs are averaged over the 15-day simulation period, and cross-correlations are calculated by matching the time series of simulation results with the averaged input from each simulation
Additional results from the complete ensembles of simulations are shown in Figs. S5–S10 in Supplementary Material.
From the results of the ensemble simulations, we find, not unexpectedly, that the presence of high ice cover will significantly alter the dynamics of an oil spill. Interestingly, however, the presence of ice does not only change the average values of oil in each environmental compartment. Instead, the probability distributions for amount of oil in the different compartments are modified, as can be seen from the empirical distributions shown in Fig. 5. This is especially clear for the amount of oil on the surface, which in turn is very relevant for amount of stranded oil, and potential for contact with birds or mammals.
For the amount of biodegraded oil, and to some degree the amount of oil on the surface, there is a clear difference between summer and winter, but little change from present to future. This indicates that the main reason for any differences between present and future is the longer ice-free season in the future. For the amount of stranded oil, the picture is less clear. This is however not entirely unexpected, since the amount of stranded oil seems to follow a more fat-tailed distribution (Nordam et al. 2016), and accurately estimating the parameters of a fat-tailed distribution requires considerably more data than estimating the parameters of a more "normal" (Gaussian-like) distribution. Considering the Varandey and Svalbard scenarios, and to some degree the Kara scenario, there is a clear increase in, e.g. the 95-percentile of amount of stranded oil in the winter season. For the two Greenland scenarios, however, the 95-percentile is essentially unchanged from summer to winter in the present case, and decreasing from summer to winter in the future.
From Fig. 6, we see that in many cases ice cover shows the strongest correlation with the different environmental endpoints. For example, in each case where there is ice, there is a strong, negative correlation between amount of evaporated oil and ice cover, meaning that a higher ice cover on average means less evaporation, which is to be expected. As is also to be expected, there is a strong negative correlation between ice cover and amount of submerged oil, and a strong positive correlation between ice and amount of oil at the surface (except in the Varandey case, where both correlations are less strong), the reason being that the ice prevents waves from submerging the oil. Recall that oil trapped at the ice–water interface is considered to be at the surface.
Perhaps more surprising is that amount of biodegraded oil is positively correlated with ice cover in some cases, and negatively in others. We also note that except in the Varandey case, ice cover does not show a particularly strong correlation to amount of oil in the sediment, which is somewhat surprising, given that open water and waves are required to submerge the oil and bring it into contact with the sea floor. A more detailed study of the spatial distribution of ice in each individual simulation would be required to adequately explain these results.
We note that even if ice cover does not always have the strongest correlations with the environmental endpoints, it still seems to work very well as a criterion for separating simulations into winter and summer dynamics. For example, for the amount of oil on the surface in the Varandey scenario, ice cover does not stand out among the correlations in Fig. 6, yet the split on ice cover produces an empirical distribution for the summer that is quite similar between present and future, and quite distinct from the winter distributions, which are again very similar between present and future (see Fig. 5).
Finally, it should be noted that it would have been ideal to have available longer time series of environmental data than the 2 × 4 years used here. As was shown by Nordam et al. (2016), the variations in ensemble results between years can be quite significant. Furthermore, phenomena like the Arctic Oscillation (Rigor et al. 2002) can affect the ice cover on a time scale of one or more years, leading to a possible bias in the results.
In general, there is a change in the fate of a "typical" oil spill from the present to the warmer future considered here. The change seems to mainly come from the fact that there is a longer open water season in the future. Furthermore, the nature of the change depends on the scenario considered. For the two Greenland cases, we see from Table 2 that there is a larger (in terms of area) surface slick, more oil on the shore and larger variations in the future. In the Svalbard scenario, on the other hand, there is a substantial decrease in expected amount of stranded oil and area of the surface slick, while for the Kara scenario, there is little change in either endpoint. While the specific geographic distribution of oil has not been considered in this study, it seems reasonable to assume that the exact effect of a change in, e.g. ice cover, will be highly dependent on local conditions. For example, for a near-shore release, the ice can act to either shield the coast or trap the oil along the coast, depending on the spatial distribution of ice.
We have demonstrated that ice cover is extremely important for the prediction of the probable fate of an Arctic oil spill. Not only does the presence of ice affect the average mass balance of an oil spill, but it also modifies the probability distributions associated with the different environmental compartments. When carrying out an Environmental Risk Assessment, it is essential to have an idea of the probability distributions involved, in order to design the ensemble of simulations in a way that will give statistically robust results. The integration of ice cover into oil spill trajectory models is still a developing field, and this study demonstrates the importance of continued research, given the significant impact ice cover has on the results of the oil spill simulations. The further development of high-resolution coupled ice-ocean models is also essential, as for example the amount of stranded oil is expected to be very sensitive to the local distribution of sea ice.
The probability of oil spill accidents in the Arctic is expected to increase in concert with increases in transport and resource exploration and extraction activities. The results reported here suggest that future oil spills in a warming climate will in some cases result in greater areal coverage and increased shoreline exposure, due to reduced ice coverage. These two considerations point towards an increase in environmental risk, defined as the probability of an event, in this case an oil spill, weighted by the magnitude of the resulting environmental injury. Furthermore, the study highlights the need to take ice into account in Environmental Risk Assessments for petroleum operations in the Arctic, as well as the need for further development of ice modelling, as well as oil-in-ice trajectory modelling. Finally, longer time series of environmental data, ideally 10 years or more, should be used for Environmental Risk Assessments in the Arctic.
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This work was supported by ACCESS, a European project within the Ocean of Tomorrow call of the European Commission Seventh Framework Programme, Grant 265863. The authors would like to thank Jeremy Wilkinson, Morten Omholt Alver, Ute Brönner and Raymond Nepstad for fruitful discussions.
SINTEF Ocean, P. B. 4762 Torgard, 7465, Trondheim, Norway
Tor Nordam, Dorien A. E. Dunnebier, CJ Beegle-Krause, Mark Reed & Dag Slagstad
Tor Nordam
Dorien A. E. Dunnebier
CJ Beegle-Krause
Mark Reed
Dag Slagstad
Correspondence to Tor Nordam.
Supplementary material 1 (PDF 9969 kb)
Nordam, T., Dunnebier, D.A.E., Beegle-Krause, C. et al. Impact of climate change and seasonal trends on the fate of Arctic oil spills. Ambio 46 (Suppl 3), 442–452 (2017). https://doi.org/10.1007/s13280-017-0961-3
Arctic oil exploration
Environmental risk assessment
Numerical simulations | CommonCrawl |
\begin{document}
\title{Spectral Properties of Substitutions on Compact Alphabets}
\author{Neil Ma\~nibo} \address{Fakult\"at f\"ur Mathematik, Universit\"at Bielefeld, \newline \hspace*{\parindent}Postfach 100131, 33501 Bielefeld, Germany} \email{[email protected] }
\author{Dan Rust} \address{School of Mathematics and Statistics, The Open University, \newline
\hspace*{\parindent}Walton Hall, Milton Keynes, MK7 6AA, UK } \email{[email protected]}
\author{James J. Walton} \address{Mathematical Sciences Building, University of Nottingham, \newline \hspace*{\parindent}University Park, Nottingham, NG7 2RD, UK} \email{[email protected]}
\keywords{substitutions, compact alphabets, diffraction, dynamical spectrum, Delone sets} \subjclass[2010]{37B10, 37A30, 52C23, 42A16}
\begin{abstract} We consider substitutions on compact alphabets and provide sufficient conditions for the diffraction to be pure point, absolutely continuous and singular continuous. This allows one to construct examples for which the Koopman operator on the associated function space has specific spectral components. For abelian bijective substitutions, we provide a dichotomy result regarding the spectral type of the diffraction. We also provide the first example of a substitution that has countably infinite Lebesgue spectral components and countably infinite singular continuous components. Lastly, we give a non-constant length substitution on a countably infinite alphabet that gives rise to substitutive Delone sets of infinite type. This extends the spectral theory of substitutions on finite alphabets and Delone sets of finite type with inflation symmetry. \end{abstract}
\maketitle
\section{Introduction} Substitutions over infinite alphabets have already been considered in a variety of contexts \cite{F:infinite, DOP:self-induced, Queffelec, RY:profinite}, but very few accounts on their spectral theory are available. One difficulty is that classical methods such as Perron--Frobenius theory are no longer immediately available in the infinite alphabet setting, and so new techniques must be developed \cite{MRW:compact}. Even then, it is possible that the maximal spectral type of the underlying system is generated by an infinite family of functions, which is usually the case.
In this work, we consider continuous substitutions on a compact alphabet and develop the spectral theory for certain families of representative examples, which are infinite alphabet generalisations of the Thue--Morse, period doubling and Rudin--Shapiro substitutions. We demonstrate how similar arguments to those implemented in the finite alphabet setting allow one to completely determine their spectral type and, at the same time, show that new phenomena can occur.
It is well known that there is a correspondence between the diffraction spectrum and dynamical spectrum. Indeed, this has been rigorously established for systems with pure point spectrum in \cite{LMS:pp,BaakeLenz}, for systems with mixed spectrum in the finite local complexity (FLC) setting in \cite{BLvE} and for general dynamical systems in \cite{L:sampling}. We employ a diffraction-based approach and study the diffraction of weighted point sets that arise from elements of the subshift. This entails choosing an appropriate set of weight functions, which we get for free when the alphabet has an extra group structure. In these cases, the characters of the associated group provide complete access to the admissible spectral types.
We establish sufficient conditions for when the corresponding diffraction measure is pure point and purely singular continuous for bijective, abelian substitutions. These may be considered as compact alphabet generalisations of the Thue--Morse substitution, see Theorem~\ref{thm: cyclic periodic} for equivalent conditions for singular continuity. These spectral results allow one to distinguish bijective abelian substitutions that admit periodic factors. These results generalise the ones in \cite{BG-squiral, Bartlett, Frank-HD}. In the finite alphabet case, it is conjectured that bijective substitutions always have singular spectrum. This has been confirmed for substitutions on a binary alphabet \cite{BG-squiral} and for abelian substitutions \cite{Bartlett} but remains open for the non-abelian case; see \cite{Queffelec,Bartlett}.
We also consider the compact alphabet analogues of the period doubling substitution and show that they have pure point spectrum (both in the dynamical and diffraction sense), which is done in Theorem~\ref{thm: pure point coincidence}, extending a famous result due to Dekking \cite{Dekking}. The subshifts they generate can be seen as generalisations of Toeplitz shifts over compact alphabets.
To complete the picture, we provide an analogue of the Rudin--Shapiro substitution on a compact alphabet. The alphabet is $\mathcal{A}=S^{1}\times \left\{\mathsf{0},\mathsf{1}\right\}$, and the corresponding dynamical system is a skew product, which is a group extension of the odometer $({\mathbb Z}_{2},+1)$ by $S^1$; compare \cite{FM:spin, AL:chain, H:cocycle}. Its spectral theory is completely determined by the defining spin matrix $W$. From $W$, one can explicitly define the cocycle that induces the ${\mathbb Z}$-action on ${\mathbb Z}_2\times S^{1}$. Using the results in \cite{FM:spin}, we prove in Theorem~\ref{thm:spin-dynam} that this example has countably infinite Lebesgue components and countably infinite singular continuous components in its dynamical spectrum. To the authors' knowledge, this is the first substitutive example with such a spectral decomposition; obviously one cannot have this for finite alphabets where the maximal spectral type is generated by only finitely many correlation measures \cite{Bartlett,Queffelec}.
All of the previously mentioned families are constant-length, and so one can say that, for all of them, infinite local complexity arises in a purely combinatorial fashion. We end with an example of a non-constant length substitution with uniquely ergodic subshift on the alphabet $\mathbb{N}_0 \cup \{\infty\}$. It admits natural tile lengths that are bounded from above and are bounded away from zero \cite{MRW:compact}. This allows one to construct a tiling of $\mathbb{R}$ with infinitely many prototile lengths. Here, infinite local complexity also manifests geometrically apart from the combinatorial complexity induced by the alphabet. From this tiling, one can derive a Delone set of infinite type that still has an inflation symmetry in the sense of Lagarias \cite{L:finite}.
\section{Substitutions on compact alphabets}
\subsection{General theory} Here, we present general notions and results regarding substitutions on compact alphabets. Although this section is self-contained, we refer the reader to \cite{MRW:compact} for a more detailed treatment and for the proofs of the dynamical results.
Let ${\mc A}$ be a compact Hausdorff space which we call an \emph{alphabet} and whose elements we call \emph{letters}. Let ${\mc A}^+ = \bigsqcup_{n \geqslant 1} {\mc A}^n$ denote the set of all finite (non-empty) \emph{words} over the alphabet ${\mc A}$, where ${\mc A}^n$ has the product topology and ${\mc A}^+$ has the topology of a disjoint union. Let ${\mc A}^\ast = {\mc A}^+ \sqcup \{\varepsilon\}$, where $\varepsilon$ is the \emph{empty word}. \emph{Concatenation} is a binary operation ${\mc A}^\ast \times {\mc A}^\ast \to {\mc A}^\ast$ given by $(\varepsilon, u) \mapsto u$, $(u, \varepsilon)\mapsto u$ and $
(u_1 \cdots u_n, v_1 \cdots v_m) \mapsto u_1 \cdots u_n v_1 \cdots v_n$, where $u = u_1 \cdots u_n \in {\mc A}^n$ and $v = v_1 \cdots v_m \in {\mc A}^m$. We write $uv$ for the concatenation of $u$ and $v$. Note that concatenation is a continuous operation. Let ${\mc A}^{{\mathbb Z}}$ denote the set of bi-infinite sequences over ${\mc A}$ with the product topology. We use a vertical line $|$ to denote the position between the $-1$st and $0$th element of a bi-infinite sequence, and so we write $w = \cdots w_{-2} w_{-1} | w_0 w_1 \cdots$. The space ${\mc A}^{{\mathbb Z}}$ is compact by Tychonoff's theorem. The left shift $\sigma\colon {\mc A}^{{\mathbb Z}}\to {\mc A}^{{\mathbb Z}}$ is defined by $\sigma(w)_{n}=w_{n+1}$. A subshift $X\subseteq {{\mc A}}^{{\mathbb Z}}$ is a subspace of the full shift that is closed and $\sigma$-invariant.
\begin{definition} Let $\varrho \colon {\mc A} \to {\mc A}^+$ be a continuous function. We call such a function a \emph{substitution} on ${\mc A}$. We say $\varrho$ is a substitution of \emph{constant length} $n$ if $\varrho({\mc A})\subset {\mc A}^n$. \end{definition}
To avoid trivialities, we always assume that \(|\varrho(a)| \geq 2\) for at least one letter \(a \in {\mc A}\). If ${\mc A}$ is connected, then by continuity, $\varrho$ is necessarily of constant length.
\begin{definition} Let $\varrho\colon {\mc A} \to {\mc A}^+$ be a substitution. We say a word $u \in {\mc A}^n$ is \emph{generated} by $\varrho$ if there exist $a \in {\mc A}$, $k \geqslant 1$ such that $\varrho^k(a)$ contains $u$ as a subword. We write \[ \mathcal L^n(\varrho) := \overline{\{u \in {\mc A}^n \mid u \text{ is generated by } \varrho\}} \] and define $\mathcal L(\varrho): = \bigsqcup \mathcal L^n(\varrho)$. We call $\mathcal L(\varrho)$ the \emph{language} of $\varrho$ and call words in $\mathcal L(\varrho)$ \emph{legal}. We define a closed, shift-invariant subspace of ${\mc A}^{\mathbb Z}$ by \[ X_\varrho := \left\{w \in {\mc A}^{\mathbb Z} \mid \forall i \leqslant j, \: w_{[i,j]} \in \mathcal L(\varrho)\right\} \] and call $X_\varrho$ the \emph{subshift} associated with $\varrho$ \cite{MRW:compact}. \end{definition}
We note that, in the previous definition, we have to take the closure to form ${\mathcal L}^{n}(\varrho)$, which is automatically satisfied for finite alphabets. Next, we present the generalisation of the notion of a primitive substitution in the infinite alphabet setting. The following definition is equivalent to that in the work of Durand, Ormes and Petite \cite{DOP:self-induced}; see also Frank and Sadun \cite{PFS:fusion-ILC} and Queff\'{e}lec \cite{Queffelec}. Let $\varrho \colon {\mc A} \to {\mc A}^+$ be a substitution. We say $\varrho$ is \emph{primitive} if, for every non-empty open set $U \subset {\mc A}$, there exists a $p=p(U) \geqslant 0$ such that for all $a \in {\mc A}$, some letter of $\varrho^{p}(a)$ is in $U$. A substitution $\varrho$ is called \emph{irreducible} if it cannot be restricted to a strictly smaller closed non-empty subalphabet of ${\mc A}$. Primitivity implies irreducibility \cite{MRW:compact}, which is consistent with results in the finite-dimensional setting.
Let $E=C({\mc A})$ be the Banach space of real-valued continuous functions over ${\mc A}$ equipped with the sup-norm. The analogue of the transpose of the substitution matrix in the finite alphabet setting is the operator $M\colon E\to E$ given by $Mf(a)=\sum_{b\triangleleft\varrho(a)}f(b)$, where the indexing $b\triangleleft\varrho(a)$ is taken to include multiplicities. Note that $M$ is a bounded and positive operator on $E$, but in the most interesting cases it is a non-compact operator; see \cite{MRW:compact}.
Nevertheless weaker properties give rise to similar spectral consequences. An operator $T\colon E\to E$ with spectral radius $r(T)=1$ is called \begin{enumerate}
\item \emph{quasi-compact} if there is a compact operator $K$ and a natural number $n$ such that $\|T^n-K\|<1$; \item \emph{strongly power convergent} if \(T^n f\) converges for all \(f \in E\) with respect to the norm on \(E\) (i.e., converges uniformly to a continuous function on \({\mc A}\)). Equivalently, \(T^n \to P\) to some bounded operator \(P\) in the strong operator topology; \item \emph{mean ergodic} if the Ces\`{a}ro average \(A_n(f) \coloneqq \frac{1}{n}\sum_{j=0}^{n-1} T^jf\) converges for each \(f \in E\). Equivalently, \(A_n \to P\) to some bounded operator \(P\) in the strong operator topology. \end{enumerate} Primitivity and quasi-compactness imply strong power convergence, and strong power convergence implies mean ergodicity. A function $0\neq \ell\in C({\mc A})$ is called a \emph{natural length function} if $\ell\geqslant 0$, and $M\ell(a)=\lambda\ell(a)$ for each $a\in{\mc A}$ where $\lambda\geqslant 0$. We note that necessarily \(\lambda = r > 1\) when \(\varrho\) is irreducible \cite[Thm.~4.26, Prop.~4.27]{MRW:compact}.
\begin{theorem}[{\cite[Thm~3.30]{MRW:compact}}]\label{thm:minimal} If $\varrho$ is a primitive substitution on a compact Hausdorff alphabet ${\mc A}$, then $(X_\varrho,\sigma)$ is minimal. \end{theorem}
\begin{theorem}[{\cite[Prop.~5.8]{MRW:compact}}]\label{thm:tile-lengths}
Let $\varrho$ be an irreducible substitution on a compact Hausdorff alphabet ${\mc A}$. If the substitution operator $M$ associated with $\varrho$ is mean ergodic, $\varrho$ admits a unique (up to scaling) natural length function $\ell\in E$ with $\lambda=r(M)>1$. Moreover, $\ell$ is strictly positive. \end{theorem}
Theorem~\ref{thm:tile-lengths} is the compact alphabet analogue for the existence of a left Perron--Frobenius (PF) eigenvector of the usual substitution matrix. Note that when $\varrho$ is constant-length, it automatically admits the natural length function $\ell = \mathds{1}$, where \(\mathds{1}(a) = 1\) for all \(a \in {\mc A}\), regardless of compactness properties of $M$. The existence of a unique positive right PF eigenvector associated with $r$ requires one to look at the dual operator $M'\colon E' \to E'$, where $E'$ is the Banach dual of $E$. Here, $M'$ is defined via $M'\mu(f)=\mu(Mf)$ for $\mu\in E'$. One has the following sufficient criteria for the subshift $X_\varrho$ to be uniquely ergodic.
\begin{theorem}[{\cite[Thm.~5.14 Cor.~5.15]{MRW:compact}}]\label{thm:unique-erg} Let $\varrho$ be a substitution on a compact Hausdorff alphabet $\mathcal{A}$. Suppose $\varrho$ is irreducible and the substitution operator $M$ is strongly power convergent. Then, $(X_\varrho,\sigma)$ is uniquely ergodic. \qed \end{theorem}
\begin{theorem}[{\cite[Thm.~6.17]{MRW:compact}}]\label{thm:unique-erg-CL} Let ${\mc A}$ be a compact Hausdorff alphabet and $\varrho$ a substitution over ${\mc A}$ of constant length $L$. Suppose $\varrho$ is primitive and that the columns $\left\{\varrho^{ }_{j}\right\}_{0\leqslant j\leqslant L-1}$ of $\varrho$ generate an equicontinuous semigroup. Then, $(X_\varrho,\sigma)$ is uniquely ergodic. \qed \end{theorem}
\subsection{Compactly bijective substitutions}\label{SEC:bijective} One of our main results concerns substitutions of constant length for which the columns generate a compact abelian group. In this section, we show that such substitutions must take a particular form. Namely, the alphabet can be given a group structure with respect to which the columns act as group translations. The following definition is a natural extension of the corresponding notions from the finite alphabet case.
\begin{definition} Given a substitution $\varrho \colon {\mc A} \to {\mc A}^+$ of constant length $L$, we say that $\varrho$ is \emph{bijective} if each \emph{column} $\varrho^{ }_j(a) \coloneqq (\varrho(a))_j$, for $0 \leqslant j \leqslant L-1$, is a bijection, and thus by continuity of $\varrho$ a homeomorphism of $\mathcal{A}$. Let $\hom({\mc A})$ denote the set of homeomorphisms of ${\mc A}$ equipped with the compact-open topology. We let $\mathcal{C}$ denote the subgroup of $\hom({\mc A})$ generated by the columns of $\varrho$. The closure of $\mathcal{C}$ in $\hom(\mathcal{A})$ is denoted $\mathcal{G} \coloneqq \overline{\mathcal{C}}$. We call $\varrho$ \emph{compactly bijective} if $\mathcal{G}$ is compact. A compactly bijective substitution is called \emph{abelian} if $\mathcal{G}$ is abelian. \end{definition}
\begin{remark} In the finite alphabet case, a bijective substitution is abelian if and only if the group $\mathcal{C}$ generated by the columns is abelian.
$\Diamond$ \end{remark}
\begin{remark}
In the general case, $\varrho$ is abelian if and only if the columns commute and generate a relatively compact subgroup in $\hom({\mc A})$. Indeed, the algebraic closure of a commuting subset is abelian, as is the topological closure of an abelian subgroup.
$\Diamond$ \end{remark}
\begin{lem} \label{lem: transitive action} Suppose that $\varrho$ is primitive and compactly bijective. Then $\mathcal{G}$ acts transitively on ${\mc A}$. \end{lem}
\begin{proof} Let $a, b \in {\mc A}$. We wish to find some $g \in \mathcal{G}$ with $g(a) = b$. Let $U \subset {\mc A}$ be an open set containing $b$. By primitivity, there is some $N \in {\mathbb N}$ so that $\varrho^N(a)$ contains a letter of $U$. Equivalently, we have that $(\varrho^{ }_{i_0} \circ \cdots \circ \varrho^{ }_{i_{N-1}})(a) \in U$ where each $\varrho^{ }_{i}$ is a column of $\varrho$. It follows that we may construct a map $f_U \in \mathcal{C}$ so that $f_U(a) \in U$.
The open sets $U$ containing $b$ form a directed set, and so we have a net $(f_U)$. By compactness, we have that $f_U \to f$ for some $f \in \mathcal{G}$. Since $(f_U) \to f$ with respect to the compact-open topology it certainly converges pointwise, so in particular $(f_U(a))_U \to f(a)$. Then clearly $f(a) = b$, since $f_U(a) \in U$, where $U$ is an arbitrary open set containing $b$. \end{proof}
\begin{prop} \label{prop: cpt bij => coset sub} Suppose that $\varrho$ is primitive and compactly bijective. Then $\mathcal{A} \cong \mathcal{G}/H$ for some compact subgroup $H \leqslant \mathcal{G}$. With respect to this identification, we have \[ \varrho [g] = (\varrho^{ }_0, \varrho^{ }_1, \ldots, \varrho^{ }_{L-1}) [g] = [\varrho^{ }_0 \cdot g][\varrho^{ }_1 \cdot g] \cdots [\varrho^{ }_{L-1} \cdot g], \] where $[g]$ denotes the coset of $g \in \mathcal{G}$ in $\mathcal{G}/H$. \end{prop}
\begin{proof} The evaluation map $C(X,Y) \times X \to Y$ is always continuous for $X$ locally compact and Hausdorff and $C(X,Y)$ equipped with the compact-open topology. Hence the action $\mathcal{G} \times {\mc A} \to {\mc A}$ of $\mathcal{G}$ on ${\mc A}$, given by $g \cdot a \coloneqq g(a)$, is continuous.
Since $\mathcal{G}$ acts transitively on $\mathcal{A}$ by Lemma \ref{lem: transitive action}, we have an identification between the $\mathcal{G}$-actions on $\mathcal{A}$ and on $\mathcal{G}/H$ by (left) multiplication, where \[ H \coloneqq \mathrm{stab}(p) \coloneqq \{g \in \mathcal{G} \mid g(p) = p\} \] and $p \in {\mc A}$ is an arbitrary basepoint. Stabiliser subgroups of topological group actions are always closed, so $H$ is compact.
These actions are intertwined via the bijection \[ \phi \colon \mathcal{G}/H \to {\mc A}, \ \ \phi [g] \coloneqq g(p) \] with inverse $a \mapsto [g]$, where $g$ is such that $g(p)=a$ (uniquely defined up to coset representative). The map $\mathcal{G} \to {\mc A}$ defined by $g \mapsto g(p)$ is continuous, since, as above, evaluation is continuous. Hence, this descends to a continuous map on the quotient $\mathcal{G}/H$, so $\phi$ is continuous. By assumption, ${\mc A}$ is Hausdorff and, since $\mathcal{G}$ is compact, so is $\mathcal{G}/H$, hence $\phi$ is a homeomorphism.
By assumption, we have that $\varrho = (\varrho^{ }_0, \varrho^{ }_1, \ldots, \varrho^{ }_{L-1})$, where each $\varrho^{ }_i$ is a column and thus is an element of $\mathcal{G}$. Finally, writing $\varrho$ as a substitution on $\mathcal{G}/H$ by the identification $\phi$, we have $\varrho[g]_i = \varrho(g(p))_i =\varrho^{ }_i(g(p)) = [\varrho^{ }_i\cdot g]$ and the result follows. \end{proof}
\begin{coro}\label{coro:bijective-abelian} Let $\varrho$ be a primitive, compactly bijective, abelian substitution. Then, we may equip ${\mc A}$ with the structure of a compact group so that each column of the substitution is a group rotation. In particular, there exist $\beta_i \in {\mc A}$ so that for all $a \in {\mc A}$ \[ \varrho(a) = (\beta_0 \cdot a) (\beta_1 \cdot a) \cdots (\beta_{L-1} \cdot a). \] \end{coro}
\begin{proof} In applying Proposition \ref{prop: cpt bij => coset sub} we have $H = \mathrm{stab}(p)$ where $p$ is an arbitrary base point. Then $H = \{\text{id}\}$ since $\varrho$ is abelian. Indeed, let $g \in \mathrm{stab}(p)$ and $x \in {\mc A}$ be arbitrary. By Lemma \ref{lem: transitive action}, the action of $\mathcal{G}$ on ${\mc A}$ is transitive, so there is some $h \in \mathcal{G}$ with $h(p) = x$. Then $g(x) = (gh)(p) = (hg)(p) = h(p) = x$ and thus $g = \mathrm{Id}$. It follows from Proposition \ref{prop: cpt bij => coset sub} that we may identify ${\mc A} \cong \mathcal{G}$ and that each column is a group rotation. \end{proof}
It follows that a constant-length substitution is compactly bijective and abelian if and only if we may equip ${\mc A}$ with the structure of an abelian topological group so that each column of $\varrho$ is a group translation. We will assume this structure for Section~\ref{sec:diff-bij} below.
\begin{definition} Let $\varrho$ be a substitution on a compact alphabet ${\mc A}$ and let $X_\varrho$ be its subshift. We call $x \in X_\varrho$ a \emph{pseudo-fixed point} of $\varrho$ if $\varrho(x)=\sigma^{s}(x)$, where $0\leqslant s\leqslant L-1$. \end{definition}
For finite alphabets, the existence of a bi-infinite fixed point (of some power of the substitution) is guaranteed. That is, there is always some $x\in X_\varrho$ such that $\varrho^{p}(x)=x$ for some power $p\in\mathbb{N}$, which is convenient for diffraction. This fails in general for infinite alphabets. Here, we show that for our purposes, it suffices to consider pseudo-fixed points, as the associated recurrence relations are similar to those for two-sided fixed points. Pseudo-fixed points can be seen as true fixed points of the corresponding affine substitution; see \cite[Sec.~4.8.1]{BG:book}.
\begin{prop}\label{prop: pseudo-fixed point} Let $\varrho$ be a primitive constant-length substitution on a compact Hausdorff abelian group, such that all the columns are group translations. Then there exists another such substitution $\varrho'$ such that $X_{\varrho'} = X_\varrho$ and $\varrho'$ admits a pseudo-fixed point in $X_\varrho$. \end{prop}
\begin{proof} We write $\varrho\coloneqq \varrho^{ }_0\,\varrho^{ }_1\,\cdots\,\varrho^{ }_{L-1}$, where $\varrho_i\colon {\mc A}\to {\mc A}$ are group translations. For $m\in\mathbb{N}$ we can define $\varrho'\coloneqq \varrho^m \beta^{-1}_s$, where $\varrho^m_s(\theta)=\theta\beta_j$ for some $0\leqslant s \leqslant L^m-1$, so that $\varrho'_s=\text{id}$. We see that $\varrho$ and $\varrho'$ define the same subshift because they generate the same language. More explicitly, one has $\big(\varrho'\big)^n(\theta\beta^n_j)=\varrho^{mn}(\theta)$ for $\theta\in{\mc A}$ and $n\in\mathbb{N}$.
For large enough $m$, we can take $1 \leqslant j \leqslant L^m -2$ so that, by construction, $\varrho'$ has a fixed internal letter at the $j$th position, i.e., $\varrho'(\theta)_j=\theta$. Using this, we construct a sequence of points that converges to a pseudo-fixed point.
Let $x^{(0)}\in X$ be such that $x^{(0)}_{[-j,L-1-j]}=\varrho(\theta)$ for some $\theta\in{\mc A}$. It then follows that $x^{(0)}_0=\theta$. If we substitute, we get that $\big(\sigma^{-s}\circ\varrho'(x_0)\big)_{[-j,L-1-j]}=\varrho'(\theta)$. One can then consider the sequence $\left\{x^{(n)}\right\}_{n\geqslant 0}$, with $x^{(n)}=(\sigma^{-s}\circ\varrho')^{n}(x^{(0)})$. Due to the growing nested subwords around $0$, this sequence converges to a point $x\in X$, which by construction satisfies $\big(\sigma^{-s}\circ \varrho'\big)(x)=x$ and so $x$ is a pseudo-fixed point for $\varrho'$. \end{proof} Hence for us, without loss of generality, we may assume that $\varrho$ admits a pseudo-fixed point.
\begin{example} Let ${\mc A}=S^1$ and consider the substitution $\varrho\colon [\theta]\mapsto[\theta\alpha]\,[\theta\beta]\,[\theta\gamma]$, with $\alpha,\beta,\gamma\in {\mc A}$. We can instead consider \[\varrho\coloneqq \varrho'\colon [\theta]\mapsto [\theta\alpha\beta^{-1}]\,[\theta]\,[\theta\gamma\beta^{-1}],\] which has a fixed internal letter at position $s=1$. Fix $\theta\in{\mc A}$ and let $x^{(0)}$ be the bi-infinite word given by $
x^{(0)}=\cdots [\theta\alpha\beta^{-1}]\mid[\theta]\,[\theta\beta^{-1}\gamma]\cdots$. Here, the vertical line $|$ signifies the location of the origin. Substituting and shifting by $-s=-1$ yields \[ x^{(1)}=\big(\sigma^{-1}\circ\varrho\big)(x^{(0)})=\cdots [\theta\alpha\gamma^{-2}\beta][\theta\alpha\beta^{-1}]\mid[\theta][\theta\beta^{-1}\gamma][\theta\alpha\beta^{-2}\gamma]\cdots. \] By iterating the process, one fixes the letters in more positions, eventually exhausting all $m \in {\mathbb Z}$, which yields the limit $x$ with $\varrho(x)=\sigma(x)$.
$\Diamond$ \end{example}
\section{Spectral theory}
\subsection{Point sets, measures and diffraction}
In this section, we following the exposition in \cite{BG:book} on diffraction. A set $\varLambda\subset \mathbb{R}^d$ is called a \emph{Delone set} if it is uniformly discrete (i.e., there is an $r>0$ such that every ball of radius $r$ contains at most one point in $\varLambda$) and it is relatively dense (i.e., there is an $R>0$ such that every ball of radius $R$ contains at least two points in $\varLambda$). In this work, we only deal with Delone sets in dimension $d = 1$. A Delone set is called a \emph{Meyer set} if the Minkowski difference $\varLambda-\varLambda$ is also a Delone set. A weighted Dirac comb supported on $\varLambda$ is given by $\omega=\sum_{z\in \varLambda} f(z)\delta_z$, with $f\colon \varLambda\to \mathbb{C}$, where $\delta_z$ is a Dirac mass at $z$. This is an unbounded (Radon) measure, i.e., a complex-valued linear functional on the space $C_{c}({\mathbb R})$ of compactly supported functions on ${\mathbb R}$.
For our spectral analysis, we need to build a weighted Dirac comb from an element $w\in X_\varrho$. For this, we need the intermediate step of building a tiling $\mathcal{T}_w$ of $\mathbb{R}$ from $w$. In this work, we restrict to substitutions of constant length and so we associate to each letter $a\in{\mc A}$ a tile $\mathfrak{t}_{a}$ of length $1$. Next, we derive a coloured point set $\varLambda_w=\left\{(a_z,z)\mid a_z\in {\mc A},\, z\in \varLambda \subset \mathbb{R}\right\}$ from $\mathcal{T}_w$. To do this, we identify the location of a tile in $\mathcal{T}_w$ to be the location of its left endpoint. Since every tile has length $1$, $\text{supp}(\varLambda_w)=\mathbb{Z}$. The coloured point set $\varLambda_w$ then satisfies $a_z=w_z$, where $w_z$ is the letter at position $z$ in $w\in X_\varrho$. Using $\varLambda_w$, we can define a weighted Dirac comb $\omega$ via $\omega=\sum_{z\in \mathbb{Z}} u(w_z)\delta_{z}$, where $u\colon {\mc A}\to \mathbb{C}$ is a continuous (hence bounded) function.
The autocorrelation measure $\gamma$ associated with $\omega$ is given by $\gamma=\omega \circledast \widetilde{\omega}$, where $\circledast$ is the Eberlein convolution, i.e., \[
\gamma=\omega \circledast \widetilde{\omega}= \lim_{R\to \infty} \frac{1}{2R+1} \omega|^{ }_{[-R,R]}\ast \widetilde{\omega|^{ }_{[-R,R]}}=\lim_{R\to\infty} \gamma^{(R)}. \]
Here, $\ast$ is the usual convolution for finite measures and $\mu|^{ }_{[-R,R]}$ is the restriction of a measure $\mu$ on $[-R,R]$. The twisted measure $\widetilde{\mu}$ can be defined via test functions: $\widetilde{\mu}(f)=\overline{\mu(\widetilde{f})}$ with $\widetilde{f}(x):=\overline{f(-x)}$ for $C_{c}({\mathbb R})$.
It follows from a general result regarding lattice-supported weighted Dirac combs that the sequence $\left\{\gamma^{(R)}\right\}$ admit at least one limit point $\gamma_u=\sum_{m\in {\mathbb Z}}\eta_u(m)\delta_m$ \cite{Baake,BaakeMoody}. For a fixed weight function $u$ and $m \in {\mathbb Z}$, the autocorrelation coefficient $\eta^{ }_u(m)$ is given by \begin{equation}\label{eq:fourier-coeff} \eta^{ }_u(m)=\lim_{N\rightarrow\infty}\frac{1}{2N+1}\sum^{N}_{j=-N}u(w^{ }_j)\overline{u(w^{ }_{j+m})}. \end{equation} When the subshift is uniquely ergodic, one has that $\gamma^{ }_u$ is the unique autocorrelation for all elements in the subshift $X_\varrho$, which is true for all cases treated here.
\begin{definition} Given a weighted Dirac comb $\omega$, its \emph{diffraction measure} is the Fourier transform $\widehat{\gamma^{ }_{u}}$ of its autocorrelation $\gamma^{ }_u$. \end{definition}
Unless stated otherwise, we suppress the subscript $u$ and assume that a weight function has been chosen. The diffraction $\widehat{\gamma}$ is a positive measure on ${\mathbb R}$, which by the Lebesgue decomposition theorem admits the splitting \[ \widehat{\gamma}=\widehat{\gamma}^{ }_{\textsf{pp}}+\widehat{\gamma}^{ }_{\textsf{ac}}+\widehat{\gamma}^{ }_{\textsf{sc}} \] into pure point, absolutely continuous and singular continuous components, not all of which vanish. One main question in diffraction theory is determining whether a certain component is present.
\begin{remark} Since $\omega$ is supported on ${\mathbb Z}$, the diffraction $\widehat{\gamma}$ is ${\mathbb Z}$-periodic, i.e., one has $\widehat{\gamma}=\delta_{{\mathbb Z}}\ast\widehat{\gamma}_{\text{FD}}$, where $\widehat{\gamma}^{ }_{\text{FD}}$ (called the fundamental diffraction) is a positive measure on $\mathbb{T}$; see \cite{BLvE}. To determine the spectral type of $\widehat{\gamma}$ is suffices to look at $\widehat{\gamma}^{ }_{\text{FD}}$, whose Fourier coefficients are given by $\eta_u(m)$ in Eq.~\eqref{eq:fourier-coeff}.
$\Diamond$ \end{remark}
Since the main classes of examples we deal with in this work are generalisations of the period doubling, Thue--Morse, and Rudin--Shapiro substitutions, we include them here and their corresponding spectral properties for comparison. All of them are primitive and hence uniquely ergodic. For more details on the respective diffraction measures, we refer the reader to \cite{BG:book} and \cite{Queffelec}.
\begin{example}\label{ex: finite alph} $\text{ }$ \begin{enumerate} \item \textbf{(Period doubling)} Consider the binary alphabet ${\mc A}=\left\{a,b\right\}$. The \emph{period doubling} substitution is given by $\varrho_{\text{pd}}\colon a\mapsto ab, b\mapsto aa$. For any element $w$ in the subshift $X_{\text{pd}}$ and \emph{any} non-zero choice of weight function $u\colon {\mc A}\to \mathbb{C}$, the diffraction $\widehat{\gamma^{ }_u}$ is pure point. \item \textbf{(Thue--Morse) } Consider the binary alphabet ${\mc A}=\left\{a,b\right\}$. The \emph{Thue--Morse} substitution is given by $\varrho_{\text{TM}}\colon a\mapsto ab, b\mapsto ba$. For any element $w$ in the subshift $X_{\text{TM}}$ and for the weight function $u\colon {\mc A}\to \mathbb{C}$ given by $u(a)=1,\, u(b)=-1$, the diffraction $\widehat{\gamma^{ }_u}$ is singular continuous with respect to Lebesgue measure. \item\textbf{(Rudin--Shapiro)} Consider the quaternary alphabet ${\mc A}=\left\{a,b,\overline{a},\overline{b}\right\}$. The \emph{Rudin--Shapiro} substitution is given by $\varrho_{\text{RS}}\colon a\mapsto ab, b\mapsto a\bar{b}, \bar{a}\mapsto \bar{a}\bar{b}, \bar{b}\mapsto \bar{a}b$. For any element $w$ in the subshift $X_{\text{RS}}$ and for the weight function $u\colon {\mc A}\to \mathbb{C}$ given by $u(a)=u(b)=1,\, u(\bar{a})=u(\bar{b})=-1$, the diffraction $\widehat{\gamma^{ }_u}$ is absolutely continuous with respect to Lebesgue measure.
\end{enumerate} \end{example}
It follows from a general result of Strungaru \cite[Thm.~4.1]{Strungaru} for weighted Dirac combs with Meyer set support $\varLambda$ that the autocorrelation $\gamma$ admits the generalised Eberlein decomposition \begin{equation}\label{eq: generalised Eberlein} \gamma=\gamma^{}_{s}+\gamma^{}_{0a}+\gamma^{}_{0s}, \end{equation} where $\gamma^{}_{s},\gamma^{}_{0a}$, and $\gamma^{}_{0s}$ are pure point measures supported on $\varLambda-\varLambda$, which are in one-to-one correspondence with the measures in the Lebesgue decomposition of $\widehat{\gamma}$. That is, \[ \widehat{\gamma^{}_{s}}=\widehat{\gamma}^{ }_{\textsf{pp}},\quad\quad\quad \widehat{\gamma^{}_{0a}}=\widehat{\gamma}^{ }_{\textsf{ac}},\quad \quad \quad \widehat{\gamma^{}_{0s}}=\widehat{\gamma}^{ }_{\textsf{sc}}. \] Here, $\gamma^{ }_{s}$ is the \emph{strongly almost periodic} part. If it can be shown that $\gamma$ is a strongly almost periodic measure, it follows that $\widehat{\gamma}$ is a pure point measure. In our setting, the set $P_\varepsilon$ of $\varepsilon$-almost periods for $\eta$ is given by \begin{equation}\label{eq: set of epsilon almost periods}
P_{\varepsilon}=\big\{m \in {\mathbb Z} \mid |\eta(0)-\eta(m)|^{1/2}<\varepsilon\big\}. \end{equation} An equivalent criterion for $\gamma$ to be strongly almost periodic is given by the following general result.
\begin{theorem}[{\cite[Thm.~5]{BaakeMoody}}]\label{thm: epsilon almost periods} Let $\omega$ be a translation-bounded measure on a $\sigma$-compact locally compact abelian group $G$ whose autocorrelation measure $\gamma^{ }_{\omega}$ exists and is of the form $\gamma_{\omega}=\sum_{m\in\Delta} \eta(m)\delta_m$, where $\Delta$ is Meyer. Then the following are equivalent: \begin{enumerate}
\item The set $P_{\varepsilon}$ of $\varepsilon$-almost periods of $\eta(m)$ is relatively dense in $\Delta$. \item $\gamma_{\omega}$ is strongly almost periodic. \item $\widehat{\gamma_{\omega}}$ is pure point. \end{enumerate} \end{theorem}
Note that the boundedness of the weight function $u$ implies the translation boundedness of the weighted Dirac comb $\omega$. Moreover, ${\mathbb Z}$ is trivially a Meyer set. We use these characterisations mentioned above to show that certain diffraction measures are pure point or purely continuous (i.e., no pure point component); see Theorem~\ref{thm: absence of pp eta} and Theorem~\ref{thm: pure point coincidence}.
\begin{remark} Recently, Lenz, Spindeler and Strungaru proved an equivalent condition for pure point diffraction at the level of the measure $\omega$, which requires $\omega$ to be \emph{mean almost periodic} \cite{LSS:mean-ap}. Here, we stick with the criterion in Theorem~\ref{thm: epsilon almost periods} because the autocorrelation coefficients are always complex-valued, meaning we can leverage the metric in $\mathbb{C}$, whereas mean almost periodicity requires comparing $w_m$ and $w_{j+m}$, which might need some extra work if the alphabet is not metrisable.
$\Diamond$ \end{remark}
\subsection{Dynamical spectrum}
Diffraction is intimately connected to the dynamical spectrum of $X_\varrho$. By dynamical spectrum, we mean the spectrum of the Koopman operator $U\colon L^{2}(X_\varrho,\mu)\to L^{2}(X_\varrho,\mu)$ associated with the shift action, with $U(f)=f\circ \sigma$ and where $\mu$ is a $\sigma$-invariant measure on $X_\varrho$. Let $\mathcal{M}^{+}(\mathbb{T})$ denote the set of positive measures on $\mathbb{T}$. For $f\in L^{2}(X_\varrho,\mu)$, its \emph{spectral measure} is the unique measure $\rho_f\in \mathcal{M}^{+}(\mathbb{T})$ that satisfies $\left\langle f\mid U^{n}f\right\rangle=\int_{\mathbb{T}} z^n {\mathrm{d}}\rho_f(z)$ for $n\in{\mathbb Z}$. A measure $\rho_{\max}\in \mathcal{M}^{+}(\mathbb{T})$ is called the \emph{spectral measure of maximal type} of $U$ if $\rho_f\ll \rho_{\max}$ for all $f$, where $\ll$ denotes the absolute continuity relation for measures. By a slight abuse of notation, when $H\subseteq L^{2}(X,\mu)$ is a translation-invariant subspace, we write $\rho^{H}_{\max}$ for the maximal spectral type of $H$, where $\rho_f\ll \rho^{H}_{\max}$ for all $f\in H$.
Equivalently, this is the maximal spectral type of the subrepresentation $U|_{H}$.
\section{Diffraction for substitutions on infinite alphabets}
\subsection{Compactly bijective substitutions}\label{sec:diff-bij} In this section, we investigate the diffraction measure of abelian bijective substitutions, which are infinite alphabet generalisations of the Thue--Morse substitution in Example~\ref{ex: finite alph}. From Corollary~\ref{coro:bijective-abelian}, we can assume that the alphabet ${\mc A}$ is a compact Hausdorf abelian group and the columns $\varrho_j$ are group translations. We let $S^1 \subset {\mathbb C}$ denote the unit circle, also considered as an abelian group. A \emph{group character} $\chi$ is a continuous group homomorphism $\chi\colon {\mc A}\rightarrow S^1$. Here, we will use the characters $\chi$ as our weight functions. Let $w$ be a pseudo-fixed point of $\varrho$ and $\chi\in \widehat{{\mc A}}$ be a character. We consider the weighted Dirac comb $\omega_{\chi}=\sum_{m\in {\mathbb Z}}\chi(w^{ }_m)\delta_m$. The following result provides a sufficient condition for absence of absolutely continuous diffraction; compare \cite[Thm.~6]{CKM:q-mult} for one-sided sequences.
\begin{prop}\label{prop: bijective absence of ac} Let $\varrho$ be a substitution of constant length $L$ on a compact Hausdorff abelian group ${\mc A}$ such that the columns $\varrho_j$ are group translations for $0\leqslant j\leqslant L-1$ and let $w$ be a pseudo-fixed point of $\varrho$. Then, for any $\chi\in\widehat{{\mc A}}$, the diffraction of the weighted Dirac comb $\omega=\sum_{m \in {\mathbb Z}}\chi(w^{ }_m)\delta_m$ has no absolutely continuous component, i.e., $\widehat{\gamma}_{\textsf{ac}}=0$. \end{prop}
\begin{proof} Let $\big\{\beta^{ }_{j}\big\}_{0\leqslant j\leqslant L-1}$ be the translations defining $\varrho$. Since $w$ is a pseudo-fixed point of $\varrho$, one has $\varrho(w)=\sigma^{s}(w)$ for some $0\leqslant s\leqslant L-1$, meaning the entries of $w$ satisfy the recurrence $w^{ }_{Lm+k-s}=w^{ }_{m}\beta^{ }_{k}$, for all $0\leqslant k\leqslant L-1$ and $m \in {\mathbb Z}$. Fix $0\leqslant k\leqslant L-1$. For any $m \in {\mathbb Z}$, one has \begin{align} \eta(Lm+k)&=\lim_{N\rightarrow\infty}\frac{1}{2N+1}\sum_{r=-s}^{L-1-s}\sum_{j=-\lfloor \frac{N}{L}\rfloor-\varphi^{-}_{r}}^{\lfloor \frac{N}{L}\rfloor+\varphi^{+}_r} \chi(w^{ }_{Lj+r})\overline{\chi(w^{ }_{L(j+m)+r+k})}\nonumber\\ &=\lim_{N\rightarrow\infty}\frac{1}{2N+1}\sum_{r=-s}^{L-1-s}\sum_{j=-\lfloor \frac{N}{L}\rfloor-\varphi^{-}_{r}}^{\lfloor \frac{N}{L}\rfloor+\varphi^{+}_r} \chi\Big(w^{ }_j w^{-1}_{j+m+f(r,k)}\Big)\chi(\beta_{(r+s)\,\text{mod}\,L})\overline{\chi(\beta_{(r+k+s)\,\text{mod}\, L})}\nonumber\\ &=\frac{1}{L}\sum_{r=-s}^{L-1-s}\eta\Big(m+f(r,k)\Big)\chi(\beta^{ }_{(r+s)\,\text{mod}\,L})\overline{\chi(\beta_{(r+k+s)\,\text{mod}\,L})}\label{eq: recursion bijective general}, \end{align} where \begin{equation}\label{eq: f and phi} f(r,k)\coloneqq \begin{cases} 0, & r+k\leqslant L-1-s,\\ \Big\lfloor\frac{r+k+s}{L}\Big\rfloor, & \text{otherwise}, \end{cases} \quad \text{and}\quad \varphi^{\pm}\coloneqq \begin{cases} 1, & N-L\lfloor\frac{N}{L}\rfloor\mp r\geqslant L,\\ 0,& 0 \leqslant N-L\lfloor\frac{N}{L}\rfloor\mp r\leqslant L-1,\\ -1,& \text{otherwise}. \end{cases} \end{equation} Note that these linear recurrences are homogeneous and that, for $0\leqslant m\leqslant L-1$, the values of $\eta(m)$ depend solely on the values $\eta(0)$ and $\eta(1)$. In particular, an explicit computation yields \begin{align} \eta(1)&=\frac{\eta(0)\sum_{r=0}^{L-2}\chi(\beta^{ }_{r})\overline{\chi(\beta_{(r+1)\,\text{mod}\,L})}}{L\big(1-\frac{1}{L}\chi(\beta{ }_{L-1})\overline{\chi(\beta^{ }_0)}\big)}\quad \label{eq: recursion L one}\\ \eta(L-1)&=\frac{1}{L}\eta(0)\chi(\beta^{ }_0)\overline{\chi(\beta^{ }_{L-1})}+\frac{1}{L}\sum^{L-1}_{r=1}\eta(1)\chi(\beta^{ }_{r})\overline{\chi(\beta_{(r+L-1)\,\text{mod}\,L})}\label{eq: recursion L minus one}\\ \eta(L^{\ell}m)&=\eta(m),\quad \text{for any } m \in {\mathbb Z} \text{ and }\ell\in\mathbb{N}\label{eq: recursion L zero}. \end{align} The key takeaway here is that the relations \eqref{eq: recursion L one}-\eqref{eq: recursion L zero} no longer depend on $s$. Hence, our analysis proceeds as if we had a true fixed point. Due to the homogeneity of the relations, and the mutual measure-theoretic orthogonality of $\big(\widehat{\gamma}\big)_{\textsf{pp}},\big(\widehat{\gamma}\big)_{\textsf{ac}}$ and $\big(\widehat{\gamma}\big)_{\textsf{sc}}$, one finds that the individual components $\gamma^{ }_s,\gamma^{ }_{0a}$ and $\gamma^{ }_{0s}$ of the autocorrelation satisfy these relations independently; compare \cite[Prop.~2]{BG-squiral}.
We then consider $\gamma^{ }_{0a}=\sum_{m \in {\mathbb Z}}\eta_{\textsf{ac}}(m)\delta_m$ which gives rise to the $\textsf{ac}$ component. By Eq.~\eqref{eq: recursion L zero} and the Riemann--Lebesgue Lemma, one has $\eta_{\textsf{ac}}(m)=0$, for all $m\geqslant 1$; see \cite[Prop.~8.6]{BG:book} and \cite[Prop.~3.1]{Frank-HD}. To prove the claim, it remains to show that $\eta_{\textsf{ac}}(0)=0$. If $\eta_{\textsf{ac}}(0)\neq 0$, Eq.~\eqref{eq: recursion L minus one} together with $\eta_{\textsf{ac}}(1)=0$ yields $\eta_{\textsf{ac}}(L-1)=\frac{1}{L}\eta_{\textsf{ac}}(0)\chi(\beta^{ }_0)\overline{\chi(\beta^{ }_{L-1})}\neq 0$, which contradicts the previous observation. Hence, $\gamma^{ }_{0a}=0$ and consequently $\widehat{\gamma}^{ }_{\textsf{ac}}=0$. \end{proof}
\begin{remark} The set of recurrences in Eq.~\eqref{eq: recursion bijective general} imply that the sequence $\left\{\eta(m)\right\}_{m\geqslant 0}$ associated with such a substitution is $L$-regular over ${\mathbb C}$ in the sense of Allouche and Shallit; compare \cite[Ex.~14]{AS:regular}.
$\Diamond$ \end{remark}
We now give a sufficient criterion for the absence of pure point diffraction components.
\begin{theorem}\label{thm: absence of pp eta}
Let $\varrho$ be as in Proposition~\textnormal{\ref{prop: bijective absence of ac}} such that $|\eta(1)|<1$. Then, the diffraction $\widehat{\gamma}$ associated with $\omega$ does not have a pure point component. \end{theorem} \begin{proof} Suppose $\widehat{\gamma}^{ }_{\textsf{pp}}\neq 0$. Then, by Eq.~\eqref{eq: generalised Eberlein}, $\gamma^{ }_{s}=\sum_{m \in {\mathbb Z}}\eta^{ }_{\textsf{pp}}(m)\delta_m\neq 0$. Because $\gamma^{ }_{s}$ is a strongly almost periodic measure, Theorem~\ref{thm: epsilon almost periods} implies that the set $P_{\varepsilon}$ of $\varepsilon$-almost periods of $\eta^{ }_{\textsf{pp}}(m)$ given in Eq.~\eqref{eq: set of epsilon almost periods} must be relatively dense, for all $\varepsilon>0$.
We show that $|\eta(1)|<1$ implies that $|\eta^{ }_{\textsf{pp}}(m)|$ is uniformly bounded away from $|\eta_{\textsf{pp}}(0)|$, which directly contradicts the relative denseness of $P_{\varepsilon}$, for all $\varepsilon\leqslant \varepsilon'$, for some $\varepsilon'$.
The autocorrelation coefficients of each of the three components of $\gamma$ must satisfy Eq.~\eqref{eq: recursion bijective general} independently since the corresponding component measures are mutually singular. With this, the assumption on $\eta(1)$ and Eq.~\eqref{eq: recursion L one} imply $|\eta^{ }_{\textsf{pp}}(1)|<|\eta^{ }_{\textsf{pp}}(0)|$. Suppose $|\eta^{ }_{\textsf{pp}}(0)|-|\eta^{ }_{\textsf{pp}}(1)|>c$, for some $c>0$.
Restricting the recurrences in Eq.~\eqref{eq: recursion bijective general} to $\gamma^{ }_{s}$ one gets for $1\leqslant k\leqslant L-1$, \[
|\eta^{ }_{\textsf{pp}}(k)|\leqslant \frac{1}{L}\sum^{L-1}_{r=0}\bigg|\eta^{ }_{\textsf{pp}}\left(f(r,k)\right)\bigg|=\frac{L-k}{L}|\eta^{ }_{\textsf{pp}}(0)|+\frac{k}{L}|\eta^{ }_{\textsf{pp}}(1)|<|\eta^{ }_{\textsf{pp}}(0)|-\frac{k}{L}c \] which means \begin{equation}\label{eq: ineq zero and k}
|\eta^{ }_{\textsf{pp}}(0)|-|\eta^{ }_{\textsf{pp}}(k)|> \frac{1}{L}c. \end{equation} Now let $L\leqslant k=L m+j\leqslant L^2-1$, with $1\leqslant m\leqslant L-2$ and $0\leqslant j\leqslant L-1$. Using the same arguments as in the previous step, one obtains \begin{align*}
|\eta^{ }_{\textsf{pp}}(k)|&\leqslant \frac{1}{L}\sum^{L-1}_{r=0}\bigg|\eta^{ }_{\textsf{pp}}\bigg(m+f(r,j)\bigg)\bigg|=\frac{L-j}{L}|\eta^{ }_{\textsf{pp}}(m)|+\frac{j}{L}|\eta^{ }_{\textsf{pp}}(m+1)|\\
&< \frac{L-j}{L}\Big(|\eta^{ }_{\textsf{pp}}(0)|-\frac{m}{L}c\Big)+\frac{j}{L}\Big(|\eta^{ }_{\textsf{pp}}(1)|-\frac{m+1}{L}c\Big)\\
&<|\eta^{ }_{\textsf{pp}}(0)|-\frac{Lm+j}{L^2}c. \end{align*} Since $\frac{1}{L}\leqslant \frac{Lm+j}{L^2} \leqslant \frac{L^2-1}{L^2}$, we recover Eq.~\eqref{eq: ineq zero and k} for all $L\leqslant k\leqslant L^2-1$. It is straight-forward to show via an inductive argument that Eq.~\eqref{eq: ineq zero and k} holds for all $k \in {\mathbb Z}$. This means \[
|\eta^{ }_{\textsf{pp}}(0)-\eta^{ }_{\textsf{pp}}(k)|\geqslant \big||\eta^{ }_{\textsf{pp}}(0)|-|\eta^{ }_{\textsf{pp}}(k)|\big| > \frac{1}{L}c, \] which implies $P_{\varepsilon}= \varnothing$, for all $\varepsilon<\sqrt{\frac{c}{L}}$, contradicting the relative denseness property required for pure point diffraction. Hence, $\gamma^{ }_{s}=0$ and $\widehat{\gamma}^{ }_{\textsf{pp}}=0$. \end{proof}
The following corollary is immediate from Proposition~\ref{prop: bijective absence of ac} and Theorem~\ref{thm: absence of pp eta}.
\begin{coro}\label{cor: pure sc}
Let $\varrho$ be as in Proposition~\textnormal{\ref{prop: bijective absence of ac}}, $w$ a pseudo-fixed point of $\varrho$, and $\omega$ the associated weighted Dirac comb via the weight function $\chi\in\widehat{{\mc A}}$. If $|\eta(1)|<1$, the diffraction $\widehat{\gamma}$ of $\omega$ is purely singular continuous. \qed \end{coro}
\begin{lem}\label{lem: eta different column}
Let $\varrho$ be as in Proposition~\textnormal{\ref{prop: bijective absence of ac}}. Fix $\chi\in\widehat{{\mc A}}$ and suppose $\chi(\beta^{ }_0)\neq\chi(\beta^{ }_{L-1})$. Then, one has $|\eta(1)|< 1$. \end{lem} \begin{proof} This follows directly from Eq.~\eqref{eq: recursion L one} since, $
|\eta(1)|\leqslant \frac{|\eta(0)|(L-1)}{|L-\chi(\beta^{ }_{L-1})\overline{\chi(\beta^{ }_0)}|}<1$, where the last inequality holds since the denominator satisfies $|L-\chi(\beta^{ }_{L-1})\overline{\chi(\beta^{ }_0)}|> L-1$, which follows from the assumption $\chi(\beta^{ }_{L-1})\overline{\chi(\beta^{ }_0)}=\chi(\beta^{ }_{L-1}\beta^{-1}_0)\neq 1$. \end{proof}
\begin{remark} Note that when ${\mc A}=S^{1}$ the condition in Lemma~\ref{lem: eta different column} is satisfied for non-trivial $\chi\in\widehat{{\mc A}}$ whenever $\beta^{ }_0\beta^{-1}_{L-1}$ is an irrational rotation because $\text{ker}(\chi)$ is finite for all non-trivial $\chi\in\widehat{S^{1}}$.
$\Diamond$ \end{remark}
\begin{theorem}\label{thm: cyclic periodic} Let $\varrho$ be a primitive substitution of length $L$ on a compact Hausdorff abelian group ${\mc A}$ such that all of its columns are group translations. Let $w$ be a pseudo-fixed point of $\varrho$ and consider $\chi\in\widehat{{\mc A}}$. Then, the following are equivalent:
\begin{enumerate} \item $\widehat{\gamma^{ }_{\chi}}$ is not purely singular continuous;
\item $|\eta(1)|=1$; \item The Dirac comb $\omega^{ }_{\chi}=\sum_{m \in {\mathbb Z}} \chi(w_m)\delta_m$ can be generated by a substitution $\sub^{ }_{\textnormal{\textsf{cyc}}}$ on a finite cyclic group ${\mc A}$ of order $n$, where $\varrho$ is of the form \begin{equation}\label{eq: cyclic periodic} \sub^{ }_{\textnormal{\textsf{cyc}}} \colon [\theta]\mapsto [\theta\beta^{ }_0]\, [\theta\beta^{ }_0\beta^{ }_1] \cdots [\theta\beta^{ }_0\beta^{n-1}_1]\, [\theta\beta^{ }_0], \end{equation} where $\beta^{ }_0,\beta^{ }_1\in {\mc A}$ and $\textnormal{ord}(\beta_1)=n$; \item $\omega^{ }_{\chi}$ is periodic; \item $\widehat{\gamma^{ }_{\chi}}$ is pure point.
\end{enumerate}
\end{theorem}
Before we go to the proof of Theorem~\ref{thm: cyclic periodic}, we will need the following Lemma; see \cite[Cor.~4.3]{KY:Ellis-bij}.
\begin{lem}\label{lem: cyclic pure point} The substitution $\sub^{ }_{\textnormal{\textsf{cyc}}}$ of Eq.~\eqref{eq: cyclic periodic} defines a periodic subshift.
\end{lem}
\begin{proof} Every 2-letter subword of a substituted letter is of the form $[\theta][\beta_1 \theta]$ for some $\theta \in {\mc A}$. Moreover, given a pair of this form, \[ \sub^{ }_{\textnormal{\textsf{cyc}}}([\theta] \bullet [\beta_1 \theta]) = [\theta \beta_0] \cdots [\theta \beta_0] \bullet [\beta_1 \theta \beta_0] \cdots [\beta_1 \theta \beta_0], \] where we emphasise the location of the boundary of supertiles. Note that the pair meeting over the supertile boundary is of the same form. Every 2-letter word generated by $\sub^{ }_{\textnormal{\textsf{cyc}}}$ is either as a subword of some $\sub^{ }_{\textnormal{\textsf{cyc}}}(\theta)$, or as the pair at the supertile boundary of $\sub^{ }_{\textnormal{\textsf{cyc}}}(\theta\theta')$, where $\theta \theta'$ is also generated by $\sub^{ }_{\textnormal{\textsf{cyc}}}$. It follows that all legal 2-letter words are of the form $[\theta][\beta_1 \theta]$. Then periodicity follows from $\textnormal{ord}(\beta_1) = \#{\mc A} = n$. \end{proof}
\begin{proof}[Proof of Theorem~\textnormal{\ref{thm: cyclic periodic}}] $\text{ }$ \begin{itemize} \item \textbf{(iii)} $\implies$ \textbf{(iv)} follows from Lemma~\ref{lem: cyclic pure point}. \item \textbf{(iv)} $\implies$ \textbf{(v)} is immediate from the periodicity of $\omega^{ }_{\chi}$. \item \textbf{(v)} $\implies$ \textbf{(i)} is clear. \item \textbf{(i)} $\implies$ \textbf{(ii)} follows from Corollary~\ref{cor: pure sc}. \end{itemize}
It then remains to show that \textbf{(ii)} implies \textbf{(iii)}. Suppose $|\eta(1)|=1$. Consider the substitution $\varrho:=\varrho^{ }_0\varrho^{ }_1\cdots \varrho^{ }_{L-1}$, with $\varrho^{ }_i(\theta):=\theta\beta_i$ for $\beta_i \in {\mc A}$. From the proof of Proposition~\ref{prop: pseudo-fixed point}, we can assume that $\beta_s=\text{id}$ for some $1\leqslant s\leqslant L-2$, since multiplying all $\beta_i$ with $\beta^{-1}_{s}$ yields a substitution which generates the same subshift.
From Lemma~\ref{lem: eta different column}, we can assume that $\chi(\beta^{ }_0)=\chi(\beta^{ }_{L-1})$, since otherwise $|\eta(1)|<1$. From Eq.~\eqref{eq: recursion L one}, one must then have \[
\bigg|\sum_{r=0}^{L-2}\chi(\beta_r)\overline{\chi(\beta_{(r+1)\,\text{mod}\,L})}\bigg|=L-1, \] which only holds when $\chi(\beta^{ }_{j}\beta^{-1}_{j+1})$ is constant for all $0\leqslant j\leqslant L-2$. Since $\beta^{ }_s=\text{id}$, we obtain $\chi(\beta^{-1}_{s+1})=\chi(\beta^{ }_{s+1}\beta^{-1}_{s+2})$. One can show by induction that $\chi(\beta^{ }_{r})=\chi(\beta_{s+1})^{r-s}$ holds for all $s+1\leqslant r\leqslant L-1$. We also have $\chi(\beta^{ }_{r})=\chi(\beta_{s+1})^{L-(s+1-r)}$ for $0\leqslant r\leqslant s$. It follows that $\chi(\beta_{s+1})^{L-1}=\text{id}$, so $\chi(\beta_{s+1})\in S^1$ must be a root of unity of order at most $L-1$. Now, let $g^{ }_0 =\chi(\beta^{ }_0), g=\chi(\beta^{ }_{s+1})$ and $ n=\text{ord}(\chi(\beta^{ }_{s+1}))$ and consider ${\mc A}'=\left\langle g\right\rangle\simeq C_n$. One can check that the substitution $\varrho'\colon C_n\to C_n^{+}$ given by \[ \varrho': [a] \mapsto [a g^{ }_0 ]\, [a g^{ }_0 g]\, [a g^{ }_0 g^2] \cdots [a g^{ }_0 g^{n-1}] \,[a g^{ }_0] \] is primitive and yields the exact same recurrence relations for the autocorrelation $\eta(m)$ as $\varrho$, which completes the proof. \end{proof}
As a remarkable consequence, for a pseudo-fixed point $w$ of an abelian bijective substitution $\varrho$ on a compact alphabet, if $\chi(w_j)$ admits infinitely many complex values, then $|\eta(1)|<1$ and the diffraction $\widehat{\gamma^{ }_{\chi}}$ is purely singular continuous.
\begin{example}\label{ex: bijective 1d} Let ${\mc A}=S^1$ and define the substitution $\varrho^{ }_1\colon {\mc A}\mapsto {\mc A}^{+}$ via \[ \varrho^{ }_1\colon [\theta] \mapsto [\theta] \hspace{2mm}[\theta] \hspace{2mm} [\alpha\theta]\hspace{2mm} [\theta] \] where $\alpha\in {\mc A}$.
Consider the corresponding bi-infinite fixed point $w$ associated with the legal seed $1|1$. It suffices to look at the one-sided fixed point, which satisfies the recurrences \begin{align*} w_{4m}&=w_m & w_{4m+2}&=\alpha w_{m}\nonumber\\ w_{4m+1}&=w_{m} & w_{4m+3}&=w_m \end{align*} where $w_{m}$ is the letter at position $m\in \mathbb{N}$. Here, we write $\alpha={\mathrm{e}}^{2\pi{\mathrm{i}}\varphi}$ for some $\varphi\in [0,1)$.
Let $\chi^{ }_n\in \widehat{S^1}$ be given by $\chi^{ }_n(\theta)=\theta^n$ for $\theta\in S^{1}$. One can show by direct computation that $ \eta^{ }_n(1)=\eta^{ }_{n}(4^r)=\frac{1}{3}+\frac{2}{3}\cos(2\pi n\varphi) $
for any $r\in\mathbb{N}$. This implies that $|\eta(1)|=1$ only when $\varphi\in\mathbb{Q}$ with $n\varphi \in {\mathbb Z}$. It follows from Theorem~\ref{thm: cyclic periodic} that, for all $\alpha\in (0,1)$ not satisfying this condition, the diffraction of $\omega^{ }_{\chi^{ }_n}$ is purely singular continuous for any $x\in X_\varrho$. When $\alpha$ is an $n$th root of unity, $\omega^{ }_{\chi^{ }_n}=\delta_{\mathbb Z}$, which obviously has pure point diffraction.
$\Diamond$ \end{example}
\begin{example}[Substitution with a non-trivial pure point factor] Here, we demonstrate how the characters can reveal interesting factors. Let ${\mc A}=C_2\times S^{1}$ with $C_2=\left\{e,g\right\}$ and consider $\varrho$, which is given by $\varrho\colon [\theta] \mapsto [\theta\beta_0]\,[\theta\beta_1]\, [\theta\beta_2]$ with $\beta_0=\beta_2=(e,1)$ and $\beta_1=(g,\alpha)$, where $\alpha$ is an irrational rotation. Let $\chi_1\otimes\chi_2\in \widehat{{\mc A}}\simeq C_2\times {\mathbb Z}$. When $\chi_1$ is non-trivial and $\chi_2$ is trivial, this character induces a factor map to the substitution on $C_2$ given by $\varrho'\colon [a ]\mapsto [a]\, [ag] \, [a]$, which by Theorem~\ref{thm: cyclic periodic} is periodic. When $\chi_2$ is non-trivial, $\chi_1\otimes\chi_2(w)$ always admits infinitely many complex values due to the irrationality of $\alpha$ and hence must give rise to singular continuous diffraction by Theorem~\ref{thm: cyclic periodic}. This means that the maximal equicontinuous factor of $\varrho$ must be $({\mathbb Z}_3\times C_2,T_1\times T_2)$, where ${\mathbb Z}_3$ is the $3$-adic integers, $T_1$ is the $+1$-map and $C_2$ is endowed with the group multiplication $T_2: h\to gh$ by $g$; compare \cite{Frank-HD} and Section~\ref{sec:spin}.
$\Diamond$ \end{example}
\subsection{Substitutions with coincidences}\label{SEC:coincidence} \begin{definition} Let $\varrho$ be a constant-length substitution on a compact alphabet ${\mc A}$. We say that $\varrho$ admits a \emph{coincidence} if for some $0\leqslant j\leqslant L-1$, $ \varrho(\theta)_{j}=\beta_0$, for all $\theta\in{\mc A}$, where $\beta_0$ is a constant. \end{definition}
Such substitutions are infinite alphabet generalisations of the period doubling substitution in Example~\ref{ex: finite alph}. As in Proposition~\ref{prop: pseudo-fixed point}, it is routine to show that when one has at least one coincidence and the other columns are group translations, $\varrho$ also admits a pseudo-fixed point, so we omit the proof.
\begin{theorem}\label{thm: pure point coincidence} Let $\varrho$ be a substitution of length $L$ on a compact Hausdorff abelian group ${\mc A}$ such that $\varrho(\theta)_{j}$ is either a group translation or is constant, for $0\leqslant j\leqslant L-1$, with at least one constant column. Then, for any $\chi\in\widehat{{\mc A}}$, the weighted Dirac comb $\omega^{ }_{\chi}=\sum_{m\in{\mathbb Z}}\chi(w_m)\delta_m$ arising from a pseudo-fixed point $w$ has pure point diffraction. \end{theorem}
\begin{proof} Without loss of generality, choose $s$ to be the minimal position with a coincidence and let $w\in X_\varrho$ be such that $\varrho(w)=\sigma^{s}(w)$. Suppose that there are $p$ coincident columns in $\varrho$, which are at positions $\big\{c_1=s,c_2,\ldots,c_p\big\}$.
Consider $\eta(Lm)$, which can be written as \[ \eta(Lm) =\lim_{N\rightarrow\infty}\frac{1}{2N+1}\sum_{r=-s}^{L-1-s} \sum^{\big\lfloor\frac{N}{L}\big\rfloor+\varphi_r}_{j=-\big\lfloor\frac{N}{L}\big\rfloor-\varphi_r} \chi\left(w^{ }_{Lj+r}w^{-1}_{L(j+m)+r}\right), \] where $\varphi_r$ is the same as in Section~\ref{SEC:bijective}. The constant columns will yield $p$ recurrence relations for $w_j$, which read $w_{Lm+c_i-s}=\beta_{c_i}$. For the remaining positions $0\leqslant k\leqslant L-1$, which are group translations, one has $w_{Lm+k-s}=w_m\beta_k$ as in the bijective case. Taking these recurrences together yields \begin{align} \eta(Lm)&=\lim_{N\rightarrow\infty}\frac{1}{2N+1}\left[p \sum^{\big\lfloor\frac{N}{L}\big\rfloor+\varphi_r}_{j=-\big\lfloor\frac{N}{L}\big\rfloor-\varphi_r} \chi(e)+(L-p) \sum^{\big\lfloor\frac{N}{L}\big\rfloor+\varphi_r}_{j=-\big\lfloor\frac{N}{L}\big\rfloor-\varphi_r} \chi(w^{ }_{j}w^{-1}_{j+m})\right] \nonumber \\ &=\frac{p}{L}+\frac{L-p}{L}\eta(m) \label{eq: general LM zero recursion}. \end{align} The unitarity of $\chi$ implies that $\eta(0)=1$. To use Theorem~\ref{thm: epsilon almost periods}, we then need to show that \[
P_{\varepsilon}=\big\{m\in {\mathbb Z}\mid |1-\eta(m)|^{1/2}<\varepsilon\big\} \] is relatively dense, for all $\varepsilon>0$. To this end, let $\eta(m)=a+{\mathrm{i}} b \in {\mathbb C}$. Applying Eq.~\ref{eq: general LM zero recursion} iteratively yields \[ \eta(L^{\ell}m)=\bigg(\frac{L-p}{L}a_{\ell-1}+\frac{p}{L}\bigg)+{\mathrm{i}}\bigg(\frac{L-p}{L}\bigg)^{\ell}b, \]
where $a_{\ell}=\frac{L-p}{L}a_{\ell-1}+\frac{p}{L}$ with $a_0=a$. Since $|\eta(m)|\leqslant 1$ for all $m$, one has $a,b\in[-1,1]$, implying that, for any fixed $m\in{\mathbb Z}$, $\mathfrak{Re}(\eta(L^{\ell}m))\rightarrow 1$ and $\mathfrak{Im}(\eta(L^{\ell}m))\rightarrow 0$ as $\ell\rightarrow\infty$, with uniform convergence in $[-1,1]$. Given $\varepsilon>0$, and $z=L^{\ell}m$ for some $\ell\in\mathbb{N}$ and $m\in {\mathbb Z}$, one has \begin{align*}
|1-\eta(z)|^{1/2}&=\Bigg|\bigg(1-\frac{L-p}{L}a_{\ell-1}+\frac{p}{L}\bigg)-{\mathrm{i}}\bigg(\frac{L-p}{L}\bigg)^{\ell}b\Bigg|^{1/2} \\
&\leqslant |\varepsilon^{(\ell)}_1+\varepsilon^{(\ell)}_2|^{1/2}\leqslant \sqrt{2}\max\Bigg\{\sqrt{\varepsilon^{(\ell)}_1},\sqrt{\varepsilon^{(\ell)}_2}\Bigg\}. \end{align*} One can then choose $\ell_0\in\mathbb{N}$ such that for all $\ell>\ell_0$, $\max\Big\{\sqrt{\varepsilon^{(\ell)}_1},\sqrt{\varepsilon^{(\ell)}_2}\Big\}<\frac{\varepsilon}{\sqrt{2}}$. Since this choice of $\ell_0$ can be made to cover all possible values of $a,b\in[-1,1]$, one has $
|1-\eta(z)|^{1/2}<\varepsilon$ for the elements of the set $\mathcal{Z}^{(\ell_0)}\coloneqq \big\{z\in{\mathbb Z}\mid z=L^{\ell}m,\, m\in{\mathbb Z},\,\ell\geqslant \ell_0\big\}$.
Clearly, $\mathcal{Z}^{(\ell_0)}$ is relatively dense in ${\mathbb Z}$, and hence $\widehat{\gamma^{ }_{\chi}}$ is pure point by Theorem~\ref{thm: epsilon almost periods}. \end{proof}
The next result follows from \cite[Thm.~7]{BaakeLenz}, which establishes the equivalence of a measure-preserving dynamical system $(X,S,\mu)$ having pure point dynamical spectrum and the diffraction being pure point (which in fact holds without any ergodicity assumptions on $\mu$).
\begin{coro} Let $\varrho$ be as in Theorem~\textnormal{\ref{thm: pure point coincidence}} such that one of the columns is a group translation with dense orbit in ${\mc A}$. Then $\varrho$ is primitive and $(X_\varrho,\sigma)$ is strictly ergodic with pure point dynamical spectrum. \qed \end{coro}
Since $\varrho$ has a coincidence, say $\varrho_j(a)=\beta$ and has a group rotation $\varrho_{k}(a)=\alpha\theta$ with dense orbit as another column, given any open set $U\subset {\mc A}$, one can find a power $n\in \mathbb{N}$ such that $(\varrho_j)^{n}(\beta)$ lies in $U$. This proves that $\varrho$ is primitive. Here, minimality follows from primitivity and unique ergodicity follows again from Theorem~\ref{thm:unique-erg-CL} since the columns of $\varrho$ generate an equicontinuous semigroup.
\begin{example}\label{ex: pure point 1d} Consider the substitution $\varrho^{ }_2\colon {\mc A}\to {\mc A}^{+}$ \[ \varrho^{ }_2\colon [\theta] \mapsto [\theta] \hspace{2mm}[1] \hspace{2mm} [\theta\alpha]\hspace{2mm} [\theta], \]
where ${\mc A} = S^1$, and the corresponding bi-infinite fixed point $w$ associated with the legal seed $1|1$. For this example, the recurrence relations are given by \begin{align*} w_{4m}&=w_m & w_{4m+2}&=\alpha w_{m} \nonumber\\ w_{4m+1}&=1 & w_{4m+3}&=w_m. \label{eq: recursion zero} \end{align*} As before, we assign the weight $\chi^{ }_n(\theta)=\theta^n$ to each point of type $\theta$. Theorem~\ref{thm: pure point coincidence} tells us that the corresponding diffraction for $\varrho^{ }_2$ is pure point.
$\Diamond$ \end{example}
\begin{remark} Substitutions covered in Theorem~\ref{thm: pure point coincidence} generate sequences exhibiting a generalised Toeplitz structure; see \cite[Sec.~4.3]{EG:almost-minimal} and \cite{W:Toeplitz} on generalised Toeplitz sequences over compact alphabets.
$\Diamond$ \end{remark}
\subsection{A substitution with countably infinite Lebesgue and countably infinite singular continuous spectral components}\label{sec:spin}
Here, we provide a generalisation of the Rudin--Shapiro substitution on infinite alphabets. Let ${\mc A}=G\times \mathcal{D}$, with $G=S^{1}$ and $\mathcal{D}=\left\{\mathsf{0},\mathsf{1}\right\}$. Let $\alpha\in S^{1}$ be an irrational rotation and consider $\varrho\colon {\mc A}\to {\mc A}^{+}$ given by \begin{equation}\label{eq:spin sub} \varrho\colon \begin{cases} (\theta,\mathsf{0}) \mapsto (\theta,\mathsf{0})\, (-\theta,\mathsf{1}) & \\ (\theta,\mathsf{1}) \mapsto (\theta\alpha,\mathsf{0})\, (\theta\alpha,\mathsf{1}). \\ \end{cases} \end{equation}
Such a substitution is called a \emph{spin substitution} by Frank and Ma\~nibo \cite{FM:spin}. For $\mathsf{a}=(\theta,\mathsf{d})\in {\mc A}$, we call $\pi_{G}(\mathsf{a})=\theta$ its \emph{spin} and $\pi_{\mathcal{D}}(\mathsf{a})=\mathsf{d}$ its \emph{digit}. The distribution of the spins in the level-1 substituted words are determined by the spin matrix \[ W=\begin{pmatrix} 1 & -1\\ \alpha & \alpha \end{pmatrix}, \] which satisfies $W_{ij}=\pi_{G}(\varrho(1,i)_j)$ for $i,j\in\mathcal{D}$. Primitivity follows from the irrationality of $\alpha$. Note that this substitution is recognisable because the image of any letter with digit $\mathsf{1}$ has the same spin for its first and second letter. One can show that the associated subshift is measure-theoretically isomorphic to an $S^1$-extension of the dyadic odometer, i.e., $(X_\varrho,{\mathbb Z},\mu)\simeq({\mathbb Z}_2\times S^1,{\mathbb Z},\nu\times\mu_{\text{H}})$, where $\nu$ and $\mu_{\text{H}}$ are the Haar measures on ${\mathbb Z}_2$ and $S_1$, respectively. The skew product which induces the ${\mathbb Z}$-action can be derived directly from $W$; see \cite[Thm.~3.11]{FM:spin}. It follows from the classical theory of group extensions that one has the induced splitting $L^{2}(X_\varrho,\mu)=\bigoplus_{\chi\in\widehat{S^{1}}} H_{\chi}$ with $H_{\chi}=L^{2}({\mathbb Z}_2,\nu)\otimes \chi$. The subspaces $H_{\chi}$ are translation-invariant. The spectral type of functions in $H_{\chi}$ is determined by $\chi(W):=(\chi(W_{ij}))^{ }_{i,j}\in \text{Mat}(2,S^1)$ via the following result.
\begin{theorem}[{\cite[Thm.~3.6]{FM:spin}}]\label{thm:spin-general} Let $\varrho$ be a primitive and recognisable spin substitution on ${\mc A}=G\times \mathcal{D}$ with spin map $W$, where $G$ is a compact abelian group. \begin{enumerate}
\item If $\frac{1}{\sqrt{|\mathcal{D}|}}\chi(W)$ is a unitary matrix, $H_{\chi}$ has Lebesgue spectral type of multiplicity $|\mathcal{D}|$. \item If $\chi(W)$ is a rank-1 matrix, $\chi$ induces a factor map onto a bijective abelian substitution $\varrho'$. Moreover, the maximal spectral type of $H_{\chi}$ is either pure point or purely singular continuous, and is absolutely continuous to the maximal spectral type of $\varrho'$. \qed \end{enumerate} \end{theorem}
\begin{prop}\label{prop:spin-diff} Let $\varrho$ be the spin substitution in Eq.~\eqref{eq:spin sub}. Let $\chi^{ }_n\in \widehat{S^{1}}\simeq {\mathbb Z}$, with $\chi^{ }_n(z)=z^n$. Consider the weighted Dirac comb $\omega^{ }_{\chi^{ }_n}=\sum_{m\in{\mathbb Z}} \chi^{ }_n(\pi_{G}(w_m))\delta_m$, where $w\in X_\varrho$. Let $\widehat{\gamma}^{(n)}$ be the diffraction measure of $\omega^{ }_{\chi^{ }_n}$. Then one has the following. \begin{enumerate} \item If $n=0$, the diffraction $\widehat{\gamma}^{(n)}=\delta_{\mathbb{Z}}$ and hence pure point. \item If $n\in 2{\mathbb Z}+1$, the diffraction $\widehat{\gamma}^{(n)}$ is Lebesgue measure. \item If $n\in 2{\mathbb Z}\setminus\left\{0\right\}$, the diffraction $\widehat{\gamma}^{(n)}$ is purely singular continuous and is a generalised Riesz product. \end{enumerate} \end{prop}
\begin{proof} The first claim is straightforward. Before we proceed, we note that $\widehat{\gamma}^{(n)}=\rho_f\ast \delta_{{\mathbb Z}}$, where $\rho_f$ is the spectral measure of the function $f\colon x\mapsto \chi^{ }_n(\pi_{G}(x_0))$, which is always in $H_{\chi^{ }_n}$. To prove the second claim, note that $\frac{1}{\sqrt{2}}\chi^{ }_n(W)$ is a unitary matrix whenever $n$ is an odd integer. From Theorem~\ref{thm:spin-general}, $H_{\chi}$ contains only functions whose spectral measures are absolutely continuous with respect to Lebesgue measure. For the third claim, the matrix $\chi(W)$ is rank-1 whenever $n$ is even. For $n\neq 0$, the character $\chi^{ }_n$ induces a factor map from $X_\varrho$ onto $(S^1)^{{\mathbb Z}}$ which identifies the letters in $\left\{(\zeta \alpha^n,\mathsf{0})\right\}$ with $(1,\mathsf{1})$, where $\zeta$ is an $n$th root of unity. The image of $X_\varrho$ under $\chi^{ }_n$ can be realised as the subshift of the bijective substitution on $S^{1}$ given by $[\theta] \mapsto [\theta]\, [\theta \alpha^n]$, which is primitive (since $\alpha$ is irrational) and recognisable. By Theorem~\ref{thm: cyclic periodic}, the diffraction measure
$\widehat{\gamma}^{(n)}$ is purely singular continuous. The corresponding generalised Riesz product is given by $\rho=\prod_{m\geqslant 0} \frac{1}{2}|1+\alpha^n{\mathrm{e}}^{2^{m+1}\pi{\mathrm{i}} t}|^2$, seen as a weak-$\ast$ limit of absolutely continuous measures on $\mathbb{T}$, which arises from the relations $w_{2m}=w_m$ and $w_{2m+1}=\alpha^n w_m$; see \cite[Prop.~4.13]{Queffelec}. \end{proof}
\begin{theorem}\label{thm:spin-dynam} Let $\varrho$ be the spin substitution in Eq.~\eqref{eq:spin sub}. Consider the function space $L^{2}(X_\varrho,\mu)=\bigoplus_{\chi^{ }_n\in\widehat{S^{1}}} H_{\chi^{ }_n}$. Let $\rho^{(n)}_{\max}$ denote the maximal spectral type of $H_{\chi^{ }_n}$. \begin{enumerate} \item If $n=0$, $\rho^{(n)}_{\max}$ is pure point. \item If $n\in 2{\mathbb Z}+1$, $\rho^{(n)}_{\max}$ is Lebesgue measure. \item If $n\in 2{\mathbb Z}\setminus\left\{0\right\}$, $\rho^{(n)}_{\max}$ is purely singular continuous. \end{enumerate} \end{theorem} \begin{proof} It is a well known result for abelian group extensions that the restriction of $U_T$ on any $H_{\chi^{ }_n}$ is spectrally pure, i.e., $\rho^{(n)}_{\max}$ is either pure point, absolutely continuous or singular continuous \cite{H:cocycle}. It then suffices to find the spectral type of a single function $f_n\in H_{\chi^{ }_n}$ to determine that of $\rho^{(n)}_{\max}$. From the proof of Proposition~\ref{prop:spin-diff}, we know that $f_n(x):=\chi^{ }_n(\pi_G(x_0))$ is in $H_{\chi^{ }_n}$. Since the spectral type of $\sigma_{f_n}$ is the same as the spectral type of the diffraction measure $\widehat{\gamma}^{(n)}$, the claim follows from Proposition~\ref{prop:spin-diff}. \end{proof}
\begin{remark} As a consequence, one can form functions in $L^{2}(X,\mu)$ whose spectral measures have an arbitrary (finite) number of absolutely continuous and singular continuous components. As an example, for the hyperlocal function $f(x)=(\chi^{ }_0+\chi^{ }_1+\chi^{ }_3+\chi^{ }_4)(\pi_{G}(x_0))$, the spectral measure $\rho_{f}$ decomposes into $\rho_f=\rho^{ }_0+\rho^{ }_1+\rho^{ }_3+\rho^{ }_4$, where $\rho^{ }_i\perp \rho^{ }_j$ for $i\neq j$ and where $\rho^{ }_0$ is pure point, $\rho^{ }_1$ and $\rho^{ }_3$ are absolutely continuous, and $\rho^{ }_4$ is singular continuous.
$\Diamond$ \end{remark}
\subsection{Non-constant length example}
Let ${\mc A}={\mathbb N}_{\infty} = {\mathbb N}_0 \cup\{\infty\}$ denote the one-point compactification of the natural numbers and consider the substitution \[ \varrho\colon \left\{ \begin{array}{rcl} 0 & \mapsto & 0\ 1 \\ n & \mapsto & 0\ n\!-\!1\ n\!+\!1 \\ \infty & \mapsto & 0\ \infty\ \infty. \end{array}\right. \]
This is a non-constant length substitution. One can easily check that it is primitive, which implies $X_\varrho$ is minimal by Theorem~\ref{thm:minimal}. It has been shown in \cite[Ex.~6.9]{MRW:compact} that the corresponding substitution operator is quasi-compact, which together with primitivity, implies that it is strongly power convergent. Note that the substitution is recognisable since every supertile begins with and never ends with $0$. It follows from Theorem~\ref{thm:unique-erg} that the associated subshift $X_\varrho$ is uniquely ergodic. Strong power convergence and primitivity also guarantee the existence of a strictly positive length function by Theorem~\ref{thm:tile-lengths}. In this case, the inflation factor is $\lambda=\frac{5}{2}$ and the length function $\ell\colon {\mc A}\to {\mathbb R}$ is given by $\ell(n)=2-2^{-n}$ and $\ell(\infty) = 2$. The letter frequencies are also well defined and are given by $\nu_n=2^{-(n+1)}$ and $\nu_{\infty}=0$. The existence of natural tile lengths allows one to construct a point set $\varLambda$ from the bi-infinite fixed point $w=\varrho^{\infty}(\infty\,|\,0)$ given by \[
w=\cdots 002010\infty\infty 0\infty\infty010\infty\infty0\infty\infty|010020101013\cdots. \] Now we recover a tiling $\mathcal{T}_w$ from $w$ by replacing each letter in $w$, starting from the origin, by a tile of length $\ell(n)$ if $w_z=n$. As before, we recover a coloured point set $\varLambda_w$ from $\mathcal{T}_w$ by identifying each tile with its left endpoint. The length function is bounded from above and below, with $1= \ell(0) < \ell(n) < \ell(n+1) < \ell(\infty) = 2$ for $n\in{\mc A}$. This means that the point set $\varLambda=\text{supp}(\varLambda_w)$ is both uniformly discrete and relatively dense, and hence is a Delone set. Since there are infinitely many distinct tile lengths, $\varLambda$ has infinite local complexity and hence is not a Meyer set \cite{L:finite}. The central portion of the associated Delone set is given in Figure~\ref{fig:pointset}. We call $\varLambda$ a \emph{Delone set of infinite type} with inflation symmetry, i.e., $\lambda\varLambda\subseteq \varLambda$. Lagarias proved in \cite{L:finite} that, if $\varLambda$ is of finite type and has inflation symmetry, then $\lambda$ must be an algebraic integer. For the example above, $\lambda=\frac{5}{2}$ is algebraic but is not an integer. This leads to the following question:
\begin{question} Let $\varrho$ be a primitive substitution on a compact alphabet with inflation factor $\lambda$. Is it possible for $\lambda$ to be a transcendental number? \end{question}
\begin{figure}
\caption{The central patch of the coloured point set $\varLambda_w$ derived from $w$. The location of the origin is signified by the shaded circle. The support of this coloured point set satisfies $\lambda\varLambda\subset \varLambda$ with $\lambda=\frac{5}{2}$.}
\label{fig:pointset}
\end{figure}
For primitive substitutions on finite alphabets, it is well known that the diffraction measure of a weighted Dirac comb supported on a Delone set $\varLambda$ arising from $\varrho$ has non-trivial pure point component if and only if $\lambda$ is Pisot \cite{GK:diff}. Moreover, if $\widehat{\gamma}$ is pure point, $\varLambda$ must be a Meyer set \cite{LS:pp-Meyer}. Note however that a general Delone set need not be Meyer for it to have a non-trivial pure point component; see \cite{KS:Meyer,PFS:fusion-ILC} for the ``scrambled Fibonacci'' example, which arises from a fusion rule with finite local complexity.
Most proofs of existence of non-trivial pure point component (Bragg peaks in the diffraction setting, eigenvalues in the dynamical setting) for point sets with some form of hierarchical structure rely on the Diophantine properties of the return vectors, which generate the translation module. When the Delone set is of finite type, i.e., there are only finitely many distinct tile lengths, this ${\mathbb Z}$-module is finitely generated. For the example above, the translation module is infinitely generated and it is not clear whether the criteria for the existence of eigenvalues extend to this setting. This will be tackled in future work.
\end{document} | arXiv |
Transient marine euxinia at the end of the terminal Cryogenian glaciation
Xianguo Lang ORCID: orcid.org/0000-0001-7698-41391,2,
Bing Shen1,
Yongbo Peng3,4,
Shuhai Xiao ORCID: orcid.org/0000-0003-4655-26635,
Chuanming Zhou6,
Huiming Bao1,3,
Alan Jay Kaufman ORCID: orcid.org/0000-0003-4129-64457,
Kangjun Huang8,
Peter W. Crockford9,10,11,
Yonggang Liu ORCID: orcid.org/0000-0001-8844-218512,
Wenbo Tang13 &
Haoran Ma1
Nature Communications volume 9, Article number: 3019 (2018) Cite this article
Marine chemistry
This article has been updated
Termination of the terminal Cryogenian Marinoan snowball Earth glaciation (~650–635 Ma) is associated with the worldwide deposition of a cap carbonate. Modeling studies suggest that, during and immediately following deglaciation, the ocean may have experienced a rapid rise in pH and physical stratification followed by oceanic overturn. Testing these predictions requires the establishment of a high-resolution sequence of events within sedimentary records. Here we report the conspicuous occurrence of pyrite concretions in the topmost Nantuo Formation (South China) that was deposited in the Marinoan glacial deposits. Sedimentary facies and sulfur isotope data indicate pyrite precipitation in the sediments with H2S diffusing from the overlying sulfidic/euxinic seawater and Fe (II) from diamictite sediments. These observations suggest a transient but widespread presence of marine euxinia in an ocean characterized by redox stratification, high bioproductivity, and high-fluxes of sulfate from chemical weathering before the deposition of the cap carbonate.
The Sturtian (~717–660 Ma) and Marinoan (~650–635 Ma) snowball Earth events in the Cryogenian Period1,2,3,4,5 represent the most severe pan-glaciation climate experienced over the past 2 billion years, and possibly over the entire Earth history2,6,7. Climatic and geochemical models show that the termination of a snowball Earth glaciation requires a high level of atmospheric CO2 (pCO2 > 0.2 bar)8,9,10,11,12 in order to overcome the high albedo from global ice cover. Such a high pCO2 atmosphere will result in a rapid meltdown of glaciers13 and an extreme greenhouse climate9,10. Rapid deglaciation may lead to the shutdown of thermohaline circulation and the intensification of ocean stratification14,15, with warmer and dysoxic freshwater in the surface ocean overlying cold and anoxic seawater in the deep13,15.
Such a catastrophic end of a snowball Earth glaciation is expected to drive drastic perturbations in ocean chemistry16,17,18,19. Globally, sedimentary sequences during deglaciation are mostly manifested as a cap carbonate sharply overlying glacial diamictites2,7,20,21. Although the cap carbonate is often assumed to have been deposited immediately after the onset of deglaciation, a window of time is required to allow atmospheric pCO2 drawndown and seawater alkalinity buildup9,20. This necessarily means that deglaciation and associated continental weathering must have started before the initiation of cap carbonate deposition. This time lag is supported by recent work exploring Mg isotopes within post-Marinoan sequences20 and may be on the order of 105 years9.
Because this time lag cannot be resolved using currently available geochronometric tools, we are forced to carry out high-resolution analysis of sedimentary sequence in order to test models about the termination of snowball Earth glaciation22. Here, we report the occurrence of abundant pyrite concretions from the topmost Nantuo Formation5,23,24, a sedimentary sequence deposited during the terminal Cryogenian Marinoan glaciation in the Yangtze Block of South China (Fig. 1 and Supplementary Fig. 1, see Supplementary Note 1). These pyrite concretions lie immediately beneath the cap carbonate of the basal Doushantuo Formation deposited in the Ediacaran Period, and they present opportunities to investigate the atmospheric, oceanic, and biological events during the termination of the Marinoan glaciation.
Paleogeographic map and depositional model of the Yangtze Block. a Paleogeographic map, modified from Jiang et al.21 showing the distribution of pyrite concretions in the topmost Nantuo Formation. Inset showing the geographic locality of the Yangtze Block. b Depositional model. The height of the columns indicates the maximum size of pyrite nodules observed in field. 1: Yazhai, 2: Tongle, 3: Silikou, 4: Yangxi, 5: Yuanjia, 6: Huakoushan, 7: Bahuang, 8: Siduping, 9: Tianping, 10: Songlin, 11: Youxi, 12: Huajipo, 13: Shennongjia
In this paper, we explore the origin of these pyrite concretions based on an integrated analysis of sedimentary facies, thin-section petrography, and multiple sulfur stable isotopes. We show that there was a transient marine euxinia before the deposition of the cap carbonate.
Stratigraphic distribution and petrography
Pyrite concretions are abundantly distributed in the top 0.5–10 m of the Nantuo Formation (Fig. 2). The concretions include spheroidal-ellipsoidal nodules and aggregates with irregular outlines (Supplementary Fig. 2a–g). The pyrite nodules are randomly distributed and aligned parallel or subparallel to the bedding plane, whereas the pyrite aggregates are parallel to the bedding plane (Supplementary Fig. 2c). Individual concretions are isolated from each other, and there is no connection between nodules or between a nodule and an aggregate.
Sulfur isotopic compositions of pyrite concretions (red symbols) and disseminated pyrite (blue symbols) in the Nantuo Formation. a Limited variations of δ34Spy and variable Δ33S values of pyrite concretions from open shelf environment. b Variable δ34Spy and slightly positive Δ33S values of pyrite concretions and disseminated pyrite from slope environment. c Limited variations of δ34Spy and variable Δ33S values of pyrite concretions from basin environment. DST: Doushantuo Formation
The pyrite concretions are present in all studied sections except those in proximal inner shelf environment of the Yangtze Block (Figs. 1, 2). Both the size and abundance of pyrite concretions display depth gradients. In the basinal sections, most pyrite nodules are 10–30 cm in size (Supplementary Fig. 2), whereas pyrite aggregates can be >1 m in length and >30 cm in thickness (Supplementary Fig. 2c). In these settings, pyrite concretions account for ~5 vol% (ranging from 2 to 10 vol%, n = 5, Supplementary Table 1) of the concretion-bearing strata. In the slope depositional environment, most pyrite nodules vary between 5 and 15 cm in size (Supplementary Fig. 2f), and pyrite concretions account for 2~5 vol% (average value of 3.1 vol%, n = 5, Supplementary Table 1). In the shallower outer shelf sections, pyrite concretions occur only sporadically (<0.5 vol%) as small nodules less than 3 cm in size in gravelly siltstone layers immediately below the Doushantuo cap carbonate (Supplementary Fig. 2g). No pyrite concretions are found in the most proximal inner shelf sections (Fig. 1).
Both the nodules and aggregates are composed of euhedral pyrite, which normally range from 50 to 500 μm with occasional occurrences of mega-crystals >1 mm in size (Supplementary Fig. 3a–d). No framboidal pyrite has been identified using reflective microscopy (Supplementary Fig. 3d), nor are framboidal cores present in euhedral pyrite under back-scatter electron microscopy (Supplementary Fig. 3e, f), suggesting that the euhedral crystals were not derived from overgrowths of framboidal pyrite. Pyrite nodules are composed of densely packed euhedral pyrite and interstitial space is filled with fine sands/silts and/or cemented by silica (Supplementary Fig. 3a–d). Within a single nodule, there is no rim to core differentiation in either pyrite content or crystal size. On the other hand, pyrite aggregates are texturally supported by siliciclasts that are identical to the host rock in composition, and thus have less (15–30 vol%) pyrite content than the nodules (40–60 vol%). Pyrite crystals may contain abundant siliciclastic inclusions (Supplementary Fig. 3), and small pebbles up to 2 cm in size are observed in some aggregates and large nodules (Supplementary Fig. 2a and d).
Sulfur isotopic compositions
Individual pyrite concretions have limited variations in sulfur isotope composition (δ34Spy), but δ34Spy may vary substantially between different concretions or between different sections. In basinal sections, concretions display a smaller range of variation in δ34Spy [Tongle (13.5–19.3‰, mean = 16.5‰), Yuanjia (15.7–19.9‰, 18.5‰), Yazhai (13.6–23.7‰, mean = 19.4‰)]. Concretions from the slope sections have higher δ34Spy than those from the basinal sections, and δ34Spy values display a wider range of variation [Huakoushan (16.1–37.1‰, mean = 26.7‰) and Bahuang (21.0–28.2‰, mean = 25.2‰)]. δ34S values of disseminated pyrite extracted from gravelly siltstone in the top Nantuo Formation shows greater fluctuation (13.7–48.1‰, mean = 28.9‰) in the Taoying section but limited variation (–5.0‰ to –5.7‰, mean = –5.4‰) in the Bahuang section. In the outer shelf sections, pyrite concretions from the gravelly siltstone layers have the lowest δ34Spy values [Songlin (8.1–12.6‰, mean = 11.1‰), Youxi (−5.5‰ to −2.5‰, mean = −4.0‰)]. Δ33S (defined as Δ33S = δ33S − 1000([1 + δ34S/1000]0.515 − 1)) also exhibits a wide but overlapping range of variation among different concretions or among different facies [basin (–0.008‰ to 0.072‰, mean = 0.027‰), slope (–0.010‰ to 0.067‰, mean = 0.032‰), shelf (0.018‰ to 0.098‰, mean = 0.057‰)] (Fig. 2, Supplementary Tables 2, 3).
Pyrite can be generated by hydrothermal activities during late-stage diagenesis, precipitated directly from seawater column, or formed within sediment porewater during early diagenesis. Postdepositional hydrothermal origin of the Nantuo pyrite concretions can be ruled out on the basis of the following sedimentological, petrological, and geochemical observations. First, the stratigraphic distribution of pyrite concretions is controlled by depositional environment (Fig. 1). Pyrite concretions can be observed both in gravelly siltstone and massive diamictite in the slope environment, whereas in the shallow water facies pyrite nodules only occur in gravelly siltstone. Second, no fluid conduits or hydrothermal veins are observed in connection with pyrite concretions (Supplementary Fig. 2). Third, although the diamictite-cap carbonate interface could serve as a conduit for fluid circulation, pyrite nodules also occasionally occur in the overlying cap carbonate (Supplementary Fig. 2e). Pyrite nodules are preserved in the matrix of cap carbonate and predating the early diagenetic sheet crack structures and arguing against a late diagenetic or hydrothermal origin. Finally, highly positive δ34Spy (>10‰) and variable Δ33S values are not characteristics of hydrothermal pyrite25,26,27 (Fig. 2). Direct pyrite precipitation from seawater is also inconsistent with observed euhedral pyrite (Supplementary Fig. 2), because sedimentary pyrite formed in the water column is typically framboidal in morphology28. Although euhedral pyrite could be generated by the overgrowth of framboidal pyrite29, no framboidal cores have been identified (Supplementary Fig. 3e, f).
Petrological evidence suggest that the Nantuo pyrite concretions were formed in sediment during early diagenesis. Pyrite crystals are tightly packed with interstitial spaces filled with clasts or cemented by silica, or float within a siliciclastic matrix, suggesting early concretion formation before sediment compaction (Supplementary Figs. 2d and 3a, b). Furthermore, the matrix of pyrite-bearing diamictite and gravelly siltstone also contains some disseminated euhedral pyrite, (Supplementary Fig. 3g, h), suggesting an authigenic origin. Thus, the euhedral pyrite in the Nantuo concretions most likely precipitated from porewater within sediment.
In modern nonsulfidic oceans, authigenic pyrite precipitation is fueled by dissimilatory sulfate reduction (DSR) in sediment porewater, and DSR is sustained by sulfate diffusion from seawater30 (Supplementary Fig. 4a). Alternatively, DSR may occur in a sulfidic water column, and authigenic pyrite can precipitate from porewater within sediment, with H2S diffusing from the overlying euxinic seawater (Supplementary Fig. 4b). In order to differentiate these two scenarios, we develop numerical models to simulate the sulfur isotope systematics of pyrite formation. In the first scenario, DSR in sediment porewater is sustained by continuous diffusion of seawater sulfate, which is driven by a sulfate concentration gradient that results from porewater sulfate consumption by DSR. Here we simulate the processes utilizing a one-dimensional diffusion-advection-reaction (1D-DAR) model. Assuming a seawater sulfate sulfur isotopic composition (δ34Ssw) of +30‰31 and biological fractionation (ΔDSR) in DSR at 40‰32, our modeling results indicate a maximum δ34Spy value of +8‰ (Fig. 3a, b), and the amount of pyrite formation is variably controlled by sedimentation rate, reaction rate of DSR, and seawater sulfate concentrations (Fig. 3c, Supplementary Figs. 5–9, and Supplementary Table 4). Therefore, DSR in sediment porewater cannot explain the high δ34Spy values of the pyrite concretions in slope and basinal sections (Fig. 2).
Modeling results. a The Rayleigh distillation model (dashed lines, with different sulfur isotope fractionations or ΔDSR) showing the relationship between δ34Spy and pyrite content with DSR occurring within sediment porewater under a closed system. The 1D-DAR model result (solid lines, with different sedimentation rates) showing the relationship between δ34Spy and pyrite content when DSR occurs in sediment porewater with sulfate supply by diffusion from the overlying seawater (an open system). b and c The maximum values of δ34Spy and pyrite content are ~+8‰ and 6.99 vol% within sediments. The default parameters for Ds, s, R, and [SO4]0 are 3.61 × 10−6 cm2 s−1, 0.01 cm year−1, 1 year−1, and 3 mM L−1, respectively. d The Rayleigh distillation model quantifying the δ34S of H2S with DSR occurring in seawater. Assuming δ34Ssw is 30‰, sulfur isotope fractionation for DSR is 40‰, variable δ34S values of the Nantuo pyrite concretions indicates the different degree of DSR (1 − f, f is the fraction of sulfate remaining). e The 1D-DR model result (black lines) showing DSR in water column followed by pyrite formation in sediment porewater fueled by H2S diffusion from sulfidic seawater. This process can explain high δ34Spy and high pyrite content in the basin and slope sections, indicating high degree of sulfate reduction in water column. f δ34Spy−Δ33Spy cross-plot37. Individual contour lines represent modeled sulfur isotopic compositions of pyrite formed from a sulfate pool with an initial sulfur isotope at the right end of the lines. Modeled values in the blue field require a bacterial sulfur disproportionation (BSD), whereas the yellow field indicate pyrite formed by DSR only. Measured data from pyrite in the top Nantuo Formation fall in the yellow field, suggesting that the pyrite was precipitated in an anoxic environment where DSR but not BSD occurs
Heavy (34S-enriched) pyrite could be generated by DSR in a closed porewater system (i.e., without sulfate supply from seawater), which can be simulated by a Rayleigh distillation model (see Supplementary Note 2 and Supplementary Table 5). However, the Rayleigh process can only account for <0.1 vol% of pyrite (Fig. 3a), and thus cannot explain the high pyrite content in the top Nantuo Formation (Fig. 3a, Supplementary Table 1). On the other hand, pyrite precipitation in porewater, driven by H2S diffusion from sulfidic seawater, can be simulated by a combination of DSR in the water column and H2S diffusion/pyrite formation in sediments. We simulated these processes using a Rayleigh distillation model and a 1D-DR model (see Supplementary Note 2 and Supplementary Table 6). Our modeling result indicates that δ34Spy values are predominantly controlled by DSR with high δ34Spy values resulting from a high degree of DSR in the water column (Fig. 3d, e). In contrast, H2S diffusion controls the pyrite content but not δ34Spy value (Supplementary Fig. 10), because the reaction between H2S and reactive Fe to generate pyrite is associated with negligible isotopic fractionation (~1‰)33. Our modeling results indicate that precipitation of 34S-enriched pyrite in the basin and slope sections was driven by a higher degree of sulfate reduction than in outer shelf settings (Fig. 3d, e).
Because δ34Spy mirrors the isotopic composition of H2S in seawater33, the spatial variation in δ34Spy reflects the isotopic heterogeneity of seawater H2S. The relatively consistent δ34Spy values observed in the basinal samples suggest that the H2S concentration was more or less homogenous in most distal seawaters. For the same reason, the more variable δ34Spy values of the slope sections reflect a rapid oscillation of seawater H2S concentrations.
Given high reactive Fe contents of siliciclastic sediments34,35,36, the amount of pyrite formation was controlled by the availability of H2S in seawater, which was a function of seawater H2S concentrations and the volume of seawater. Thus, abundant pyrite in the top Nantuo Formation implies vigorous DSR and a high concentration of dissolved H2S in the seawater, which is also consistent with the lack of bacterial sulfate disproportionation as shown in the Δ33S–δ34S plot (Fig. 3f)37. The decrease of pyrite content from deep to shallow settings is attributed to the bathymetric differences of the depositional environments38. Deeper water in the basin may have a thicker sulfidic water layer, resulting in more pyrite precipitation. While the absence of pyrite concretions in the inner shelf setting is the consequence of an absence of sulfidic seawater in the near shore, shallow, and more oxic waters39,40.
Abundant pyrite concretions in the top Nantuo Formation imply the development of oceanic euxinia before the precipitation of cap carbonate41. Oceanic euxinia can only be sustained by sufficient supplies of sulfate and organic matter36,42,43. Riverine influx is the major source of nutrients (e.g., P) and seawater sulfate30,44,45. We hypothesize that, during the deglaciation of the Marinoan glaciation, enhanced continental weathering20 could have delivered abundant nutrients and sulfate into the ocean. Because the deglacial continental weathering may predate oceanic euxinia20 due to the delayed recovery of productivity in acidified seawater, both nutrients and sulfate might be accumulated in seawater from riverine inputs. Once surface ocean productivity was resumed at high levels, high nutrient content could sustain high productivity, while DSR in water column would be fueled by sufficient supplies of organic matter and sulfate, driving oceanic euxinia.
The pyrite-rich interval in the topmost Nantuo Formation marks a brief yet widespread euxinic condition in the aftermath of the Marinoan glaciation immediately before cap carbonate precipitation. It is an interval with rising sulfate levels17, and high nutrient fluxes from the continents. This is also an interval characterized by redox stratified oceans where a large quantity of H2S was scavenged by Fe(II) in the glacial diamictite sediments. Importantly, this is also an interval prior to the deposition of the well-known cap carbonates, a time when seawater pH values or alkalinity were still below the threshold of carbonate precipitation or pCO2 levels were sufficiently high in the atmosphere12,46. The current study extends previous efforts to reconstruct from sedimentary records the sequence of events in the aftermath of the Marinoan meltdown47 in order to test the predictions of snowball Earth glaciations2.
Sample preparation and sulfur isotope analysis
Pyrite concretions were first split by a rock saw and then slabs were polished before sampling. Pyrite samples were collected by a hand-held micromill from the polished slabs. To capture the isotopic variations within a single pyrite concretion, multiple samples were sequentially collected from rim to core. Sulfur isotopic compositions of pyrite were measured at the Oxy-Anion Stable Isotope Consortium in Louisiana State University, at the EPS Stable Isotope Laboratory in McGill University, and at the Geochemistry laboratory in University of Maryland.
For δ34S analysis, about 0.05 mg pyrite-rich powder was mixed with 1–2 mg V2O5, and was analyzed for S isotopic compositions on an Isoprime 100 gas source mass spectrometer coupled with a Vario Microcube Elemental Analyzer. Sulfur isotopic compositions are expressed in standard δ-notation as permil (‰) deviations from the Vienna-Canyon Diablo Troilite standard. The analytical error is <0.2‰ based on replicate analyses of samples and laboratory standards. Samples were calibrated on two internal standards: LSU-Ag2S-1: −4.3‰; LSU-Ag2S-2: +20.2‰.
Multiple sulfur isotope analyses were conducted by converting pyrite samples to H2S(g) via the chromium reduction method. H2S gas was then carried through a N2 gas stream to a Zn acetate solution where it precipitated as ZnS. ZnS was then converted to Ag2S through addition of AgNO3. Samples were then filtered and dried at 80 °C. Ag2S samples were converted to SF6(g) through reaction with F2(g) in heated Ni bombs. Generated gas was then purified through a series of cryo-focusing steps followed by gas chromatography. The purified samples were then analyzed on a Thermo MAT-253 on dual-inlet mode. The total error on the entire analytical procedure is less than (1σ) 0.1‰ for δ34S and 0.01‰ for ∆33S.
Geochemical models of pyrite formation
Sedimentary or authigenic pyrite can precipitate either from sediment porewater or water column. In sediment porewater, pyrite formation can occur in a closed or an open porewater system. For a closed system, DSR occurs within sediment porewater with no additional sulfate supply from seawater. In contrast, open system DSR refers to sulfate supply via diffusion from the overlying seawater and pyrite formation in sediment porewater (Supplementary Fig. 4).
The close system reaction can be simulated by the Rayleigh distillation model, whereas the open system pyrite formation can be modeled by the 1D-DAR or the 1D-DR model.
Rayleigh distillation model. The Rayleigh distillation model can be expressed as follows:
$$\mathrm{\updelta} ^{34}\mathrm{S}_{\mathrm{r}} = \mathrm{\updelta} ^{34}\mathrm{S}_{{\mathrm{sw}}} - 1000 \times \left( {\it f^{(\alpha-1)} - 1} \right)$$
where δ34Ssw is the isotopic composition of seawater, δ34Sr is the isotopic composition of remaining sulfate in sediment porewater or in water column. f is the fraction of sulfate remaining, and α is the fractionation factor for DSR. The isotopic composition of H2S (δ34SH2S) or pyrite (δ34Spy, assuming negligible fractionation between H2S and pyrite) can be described as:
$$\mathrm{\updelta} ^{34}\mathrm{S}_{{\mathrm{py}}} = \frac{{\left( {\mathrm{\updelta} ^{34}\mathrm{S}_{{\mathrm{sw}}} - \mathrm{\updelta} ^{34}\mathrm{S}_{\mathrm{r}} \times \it f} \right)}}{{1 - f}}$$
The amount of pyrite (Mpy) formation can be calculated by:
$$M_{{\mathrm{py}}} = \frac{{\emptyset \times \left( {1 - f} \right) \times \left[ {{{\rm{SO}}_4}} \right] \times M_{\mathrm{r}}}}{{2 \times \rho _{{\mathrm{py}}} \times \left( {1 - \emptyset } \right) \times 1000}}$$
where ∅ is the porosity of sediment (estimated at 60%), ρpy is the density of pyrite (4.9 g cm–3), Mr is the relative molecular mass of pyrite, [SO4] is seawater/porewater sulfate concentration (mol L–1).
1D-DAR model. The 1D-DAR model can be applied to quantify the geochemical profiles in sediment porewater. The DAR model describes three physio-chemical processes: molecular diffusion, advection, and chemical reaction. During DSR in porewater, sulfate is supplied from seawater by diffusion. The diffusion is driven by the sulfate concentration gradient between seawater and sediment porewater, which is generated by sulfate consumption in sediment porewater by DSR. To model sulfur isotopic profiles of porewater, we treat 34S and 32S separately. The 1D-DAR model is expressed as:
$$\frac{{\partial \left[ {{\,}^{3\mathrm{i}}{{\mathrm{SO}}_4}} \right]}}{{\partial t}} = D_{\mathrm{S}}\frac{{\partial ^2\left[ {{\,}^{3{\mathrm{i}}}{{\mathrm{SO}}_4}} \right]}}{{\partial z^2}} - s\frac{{\partial \left[ {{\,}^{3\mathrm{i}}{{\mathrm{SO}}_4}} \right]}}{{\partial z}} - R\left[ {{\kern 1pt} {\,}^{3\mathrm{i}}{{\mathrm{SO}}_4}} \right]$$
where [3iSO4] is the porewater concentration of 34SO4 and 32SO4, z is the depth below the redox boundary of DSR, Ds is the vertical diffusivity coefficient of bulk sediment (m2 year−1), s is the sedimentation rate (cm ky−1), R is the first-order rate constant during DSR.
Sulfur isotopic fractionation during DSR can be regard as the different reaction rate constant between 34S and 32S. Assuming a steady state (\(\frac{{\partial \left[ {\,^{3\mathrm{i}}{\mathrm{SO}}_4} \right]}}{{\partial t}} = 0\), with invariant D, s, R), the solution of Eq. (4) is given by:
$$[\,^{3i}{{\mathrm{SO}}_4}] = [\,^{3\mathrm{i}}{{\mathrm{SO}}_4}]_0\mathrm{e}^{\frac{{\left( {s - \sqrt {\left( {s^2 + 4R_{\mathrm{3i}}D_{\mathrm{S}}} \right)} } \right)}}{{2D_{\mathrm{S}}}} \times z}$$
The porewater sulfur isotope (\({\mathrm{\updelta }}^{34}{{{\rm{SO}}_4}}_{{\mathrm{pw}}}\)) profile can be calculated by:
$${\mathrm{\updelta }}^{34}{{{\mathrm{SO}}_4}}_{{\mathrm{pw}}} = \left( {\mathrm{ln}[\,^{34}{{{\mathrm{SO}}_4}}]/[\,^{32}{{{\mathrm{SO}}_4}}] - \mathrm{ln}[\,^{34}{\mathrm{S}}]_{{\mathrm{std}}}/[\,^{32}{\mathrm{S}}]_{{\mathrm{std}}}} \right) \times 1000$$
The subscript std denotes the standard. At any depth below the upper bound of DSR zone, the sulfur isotope of instantaneous H2S formation can be calculated by:
$$\mathrm{\updelta} ^{34}\mathrm{S}_{{\mathrm{H}}_2{\mathrm{S}}} = \mathrm{\updelta} ^{34}{{{\rm{SO}}_4}}_{{\mathrm{pw}}} - \Delta _{{\mathrm{DSR}}}$$
where ΔDSR is the biological sulfur isotope fractionation during DSR between sulfate and H2S.
Pyrite precipitation within sediment is a continuous process within the DSR zone with the lower bound marked by the depletion of porewater sulfate. To calculate the instantaneous sulfur isotope of pyrite within the DSR zone, we divide sediments into different depth slices (i) starting from the upper bound of DSR zone (i = 0). Assuming no sulfur isotope fractionation during pyrite precipitation between H2S and pyrite, δ34Spy at depth z can be calculated by:
$$\mathrm{\updelta} ^{{\mathrm{34}}}\mathrm{S}_{{\mathrm{py}}} = \frac{{\mathop {\sum }\nolimits_0^{i} \left( {{\mathrm{\updelta }}^{34}\mathrm{S}_{{\mathrm{H}}_2{\mathrm{S}}} \times [{{\mathrm{SO}}_4}]^i} \right)}}{{\mathop {\sum }\nolimits_0^i [{{\mathrm{SO}}_4}]^i}}$$
The depth z of the slice i can be calculated as z = i × h, where h is the thickness of each slice. Assuming all H2S is precipitated as pyrite, the cumulative amount of pyrite formation within DSR (Mz) can expressed as:
$$M^{\mathrm{z}} = \mathop {\sum }\nolimits_0^i \left( {[{{\rm{SO}}_4}]^i \times R \times \frac{h}{s}} \right)$$
1D-DR model. Pyrite can precipitate in porewater with H2S diffusion from overlying water column. This process can be quantified by the one-dimensional diffusion-reaction (1D-DR) model. This model includes two processes: H2S diffusion and pyrite precipitation. The sulfide diffusion is driven by the H2S concentration gradient between sulfidic seawater and sediment porewater, which is generated by pyrite precipitation in sediment porewater. The 1D-DR model is expressed as:
$$\frac{{\partial [{\mathrm{H}}^{3\mathrm{i}}{\mathrm{S}}]}}{{\partial t}} = D_{\mathrm{s}}\frac{{\partial ^2[{\mathrm{H}}^{3\mathrm{i}}\mathrm{S}]}}{{\partial z^2}} - R[{\mathrm{H}}^{3\mathrm{i}}\mathrm{S}]$$
Where the [H3iS] is the porewater concentration of H34S and H32S, z is the depth below the water-sediment surface, Ds is the vertical diffusivity coefficient of bulk sediment (m2 year−1), R is the first-order rate constant for pyrite formation via H2S reaction with reactive Fe.
Assuming a steady state, the solution for this equation is,
$$[\mathrm{H}^{3\mathrm{i}}\mathrm{S}] = [\mathrm{H}^{3\mathrm{i}}\mathrm{S}]_0\mathrm{e}^{\frac{{ - \sqrt {4R_{3\mathrm{i}}D_{\mathrm{s}}} }}{{2D_{\mathrm{s}}}} \times z}$$
δ34SH2S profile can be calculated by:
$${\mathrm{\updelta }}^{34}{\mathrm{S}}_{{\mathrm{H}}2{\mathrm{S}}} = \left( {\mathrm{ln}\left[ {{\mathrm{H}}^{34}{\mathrm{S}}} \right]/\left[ {{\mathrm{H}}^{32}{\mathrm{S}}} \right] - \mathrm{ln}\left[ {\,^{34}{\mathrm{S}}} \right]_{{\mathrm{std}}}/\left[ {\,^{{\kern 1pt} 32}{\mathrm{S}}} \right]_{{\mathrm{std}}}} \right) \times 1000$$
[3iS]std represents the standard of 34S or 32S. The instantaneous δ34Spy can be calculated by:
$$\mathrm{\updelta} ^{34}\mathrm{S}_{{\mathrm{py}}} = \mathrm{\updelta} ^{34}\mathrm{S}_{{\mathrm{H}}2{\mathrm{S}}} - \Delta _{{\mathrm{py}}}$$
where Δpy represents the sulfur isotope fractionation during pyrite precipitation. The δ34Spy at depth z below water-sediment surface can be calculated by:
$$\mathrm{\updelta} ^{34}\mathrm{S}_z = \frac{{\mathop {\sum }\nolimits_0^i \mathrm{\updelta} ^{34}\mathrm{S}_{{\mathrm{py}}} \times \left[ {{\mathrm{HS}}} \right]^i}}{{\mathop {\sum }\nolimits_0^i \left[ {{\mathrm{HS}}} \right]^i}}$$
The cumulative amount of pyrite formation can be calculated by Eq. (9).
Sulfur isotope model for Δ33Spy–δ34Spy plot. Assuming all the Nantuo pyrite were formed within ferruginous porewater and pyrite formation was rapid and irreversible. δ34Spy was modeled by a sulfate concentration model37. In a steady state, sulfate concentration at specific depth can be expressed as:
$${{\mathrm{SO}}_4}_z = \left({{{\mathrm{SO}}_4}_0 - {{\mathrm{SO}}_4}_\infty } \right) \times \mathrm{e}^{-\frac{k}{s}z} + {{\mathrm{SO}}_4}_\infty$$
where, \({\rm{SO}}_{{4}_{z}}\) represents sulfate concentration at any given depth z, \({\rm{SO}}_{{4}_{0}}\) represents the initial sulfate concentration, \({\rm{SO}}_{{4}_{\infty}}\) represent sulfate concentration at infinitive depth, i.e., sulfate left after DSR, k is sulfate reduction rate constant, s is the sedimentation rate.
As the little mass of 36S, here the equation of total mass of sulfate can be simplified as \({\rm{SO}}_{{4}_{{\rm{total}}}} = {\,}^{32}{\rm{SO}}_{{4}_{z}} + {\,}^{33}{\rm{SO}}_{{4}_{z}} + {\,}^{34}{\rm{SO}}_{{4}_{z}}\).
For sulfur isotope, Eq. (15) can be rewritten as:
$$\mathrm{\updelta} ^{3\mathrm{i}}{\mathrm{S}}_{{\mathrm{SO}}_4} = \left[ {\frac{{\left( {\frac{{\,{}^{3{\mathrm{i}}}{\mathrm{SO}}_4}}{{\,{}^{32}{\mathrm{SO}}_4}}} \right)}}{{\left( {\frac{{\,{}^{3\mathrm{i}}\mathrm{S}}}{{\,{}^{32}\mathrm{S}}}} \right)_{{\mathrm{CDT}}}}} - 1} \right] \times 1000$$
Here, i equals 3 or 4.
δ3iS of porewater sulfate is controlled by the initial seawater sulfate δ3iS, sulfate reduction rate constant k, sedimentation rate s and sulfur isotope fractionation factor (α) for DSR37. Defined α as the ratio of sulfate reduction rate (SRR) of 33S or 34S over the 32S normalized with corresponding isotope concentration:
$$\,{}^{3\mathrm{i}}{\alpha } = \frac{{\left( {\frac{{\,{}^{3\mathrm{i}}{\mathrm{SRR}}}}{{\,{}^{3\mathrm{i}}{\mathrm{S}}}}} \right)}}{{\left( {\frac{{\,^{{\mathrm{32}}}{\mathrm{SRR}}}}{{\,^{{\mathrm{32}}}{\mathrm{S}}}}} \right)}}$$
Then, Δ33S of porewater sulfate can be calculated:
$$\Delta ^{33}\mathrm{S}_{{\mathrm{SO}}_4} = \mathrm{\updelta} ^{33}\mathrm{S}_{{\mathrm{SO}}_4} - 1000 \times \left[ {\left( {1 + \frac{{\mathrm{\updelta} ^{34}\mathrm{S}_{{\mathrm{SO}}_4}}}{{1000}}} \right) \times{}^{33}\lambda - 1} \right]$$
where, 33λ is the slope of terrestrial mass fractionation line.
The instantaneous H2S sulfur isotopic compositions can be calculated by the ratios of DSR rate at corresponding depth:
$$\mathrm{\updelta} ^{3\mathrm{i}}{\mathrm{H}}_2{\mathrm{S}} = \left[ {\frac{{\left( {\frac{{\,{}^{{\mathrm{3i}}}{\mathrm{SRR}}}}{{\,{}^{{\mathrm{32}}}{\mathrm{SRR}}}}} \right)}}{{\left( {\frac{{\,{}^{{\mathrm{3i}}}{\mathrm{S}}}}{{\,{}^{{\mathrm{32}}}{\mathrm{S}}}}} \right)_{{\mathrm{CDT}}}}} - 1} \right] \times 1000$$
Pyrite sulfur isotopic compositions can be calculated by the accumulated δ3iH2S.
Determination of pyrite concretion content
Pyrite concretion content was determined by using the photograph area proportion method. Pyrite concretion-bearing Nantuo diamictite and siltstone layer were photographed in field. For each field photograph, pyrite concretion content (Cpy) can be calculated by:
$$C_{{\mathrm{py}}} = \frac{{{A}_{{\mathrm{py}}}}}{{{A}_{\mathrm{T}}}} \times 100{\mathrm{\% }}$$
where Apy is pyrite concretion area in the photo and AT is the total area of host rock in the photo.
The average content of pyrite concretion in each section can be calculated by:
$$C = \frac{{\mathop {\sum }\nolimits_1^n \left[ {C_{{\mathrm{py}}}} \right]^n}}{n}$$
where n is the total number of analyzed field photographs.
The data that support the findings of this study are available from the corresponding author upon reasonable request.
This Article was originally published without the accompanying Peer Review File. This file is now available in the HTML version of the Article; the PDF was correct from the time of publication.
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This study is supported by the Strategic Priority Research Program (B) of Chinese Academy of Sciences (Grant number XDB18000000), Natural Science Foundation of China (41322021) and State Key Laboratory of Palaeobiology and Stratigraphy (Nanjing Institute of Geology and Paleontology, CAS) (No. 183114). P.W.C. acknowledges funding from an NSERC PGS-D fellowship, and the Agouron Institute Postdoctoral Fellow Program.
Key Laboratory of Orogenic Belts and Crustal Evolution, MOE, School of Earth and Space Science, Peking University, Beijing, 100871, China
Xianguo Lang, Bing Shen, Huiming Bao & Haoran Ma
State Key Laboratory of Palaeobiology and Stratigraphy, Nanjing Institute of Geology and Palaeontology and Center for Excellence in Life and Paleoenvironment, Chinese Academy of Sciences, Nanjing, 210008, China
Xianguo Lang
Department of Geology and Geophysics, Louisiana State University, Baton Rouge, LA, 70803, USA
Yongbo Peng & Huiming Bao
Shanghai Engineering Research Center of Hadal Science and Technology, College of Marine Sciences, Shanghai Ocean University, Shanghai, 201306, China
Yongbo Peng
Department of Geosciences, Virginia Tech, Blacksburg, VA, 24061, USA
Shuhai Xiao
CAS Key Laboratory of Economic Stratigraphy and Paleogeography, Nanjing Institute of Geology and Palaeontology and Center for Excellence in Life and Paleoenvironment, Chinese Academy of Sciences, Nanjing, 210008, China
Chuanming Zhou
Department of Geology, University of Maryland, College Park, MD, 20742, USA
Alan Jay Kaufman
Shaanxi Key Laboratory of Early Life and Environments, Department of Geology, Northwest University, Xi'an, 710069, China
Kangjun Huang
Department of Earth and Planetary Sciences, McGill University, Montreal, H3A0E8, Canada
Peter W. Crockford
Department of Earth and Planetary Sciences, Weizmann Institute of Science, Rehovot, 76100, Israel
Department of Geoscience, Princeton University, Princeton, 08544, USA
Department of Atmosphere and Ocean Sciences, School of Physics, Peking University, Beijing, 100871, China
Yonggang Liu
School of Mathematical & Statistical Sciences, Arizona State University, Tempe, AZ, 85287, USA
Wenbo Tang
Huiming Bao
Haoran Ma
B.S. designed the project. X.L., B.S., K.H. and H.M. conducted field work. X.L., A.J.K., P.W.C. and Y.P. analyzed data. X.L., W.T. and B.S. performed modeling analyses. X.L., B.S., Y.P., S.X., C.Z., H.B. and Y.L. led data interpretation and developed the manuscript. All authors contributed to discussion and manuscript revision.
Correspondence to Bing Shen or Yongbo Peng.
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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Lang, X., Shen, B., Peng, Y. et al. Transient marine euxinia at the end of the terminal Cryogenian glaciation. Nat Commun 9, 3019 (2018). https://doi.org/10.1038/s41467-018-05423-x
DOI: https://doi.org/10.1038/s41467-018-05423-x
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Cryogenian cap carbonate models: a review and critical assessment
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Numerical simulation of turbulent, plane parallel Couette–Poiseuille flow
W. Cheng, D.I. Pullin, R. Samtaney, X. Luo
Journal: Journal of Fluid Mechanics / Volume 955 / 25 January 2023
Published online by Cambridge University Press: 13 January 2023, A4
We present numerical simulation and mean-flow modelling of statistically stationary plane Couette–Poiseuille flow in a parameter space $(Re,\theta )$ with $Re=\sqrt {Re_c^2+Re_M^2}$ and $\theta =\arctan (Re_M/Re_c)$, where $Re_c,Re_M$ are independent Reynolds numbers based on the plate speed $U_c$ and the volume flow rate per unit span, respectively. The database comprises direct numerical simulations (DNS) at $Re=4000,6000$, wall-resolved large-eddy simulations at $Re = 10\,000, 20\,000$, and some wall-modelled large-eddy simulations (WMLES) up to $Re=10^{10}$. Attention is focused on the transition (from Couette-type to Poiseuille-type flow), defined as where the mean skin-friction Reynolds number on the bottom wall $Re_{\tau,b}$ changes sign at $\theta =\theta _c(Re)$. The mean flow in the $(Re,\theta )$ plane is modelled with combinations of patched classical log-wake profiles. Several model versions with different structures are constructed in both the Couette-type and Poiseuille-type flow regions. Model calculations of $Re_{\tau,b}(Re,\theta )$, $Re_{\tau,t}(Re,\theta )$ (the skin-friction Reynolds number on the top wall) and $\theta _c$ show general agreement with both DNS and large-eddy simulations. Both model and simulation indicate that, as $\theta$ is increased at fixed $Re$, $Re_{\tau,t}$ passes through a peak at approximately $\theta = 45^{\circ }$, while $Re_{\tau,b}$ increases monotonically. Near the bottom wall, the flow laminarizes as $\theta$ passes through $\theta _c$ and then re-transitions to turbulence. As $Re$ increases, $\theta _c$ increases monotonically. The transition from Couette-type to Poiseuille-type flow is accompanied by the rapid attenuation of streamwise rolls observed in pure Couette flow. A subclass of flows with $Re_{\tau,b}=0$ is investigated. Combined WMLES with modelling for these flows enables exploration of the $Re\to \infty$ limit, giving $\theta _c \to 45^\circ$ as $Re\to \infty$.
Effect of sheared E × B flow on the blob dynamics in the scrape-off layer of HL-2A tokamak
W.C. Wang, J. Cheng, Z.B. Shi, L.W. Yan, Z.H. Huang, N. Wu, Q. Zou, Y.J. Zhu, X. Chen, J.Q. Dong, W.L. Zhong, M. Xu
Journal: Journal of Plasma Physics / Volume 88 / Issue 6 / December 2022
Published online by Cambridge University Press: 11 November 2022, 905880605
The effect of sheared E × B flow on the blob dynamics in the scrape-off layer (SOL) of HL-2A tokamak has been studied during the plasma current ramp-up in ohmically heated deuterium plasmas by the combination of poloidal and radial Langmuir probe arrays. The experimental results indicate that the SOL sheared E × B flow is substantially enhanced as the plasma current exceeds a certain value and the strong sheared E × B flow has the ability to slow the blob radial motion via stretching its poloidal correlation length. The locally accumulated blobs are suggested to be responsible for the increase of plasma density just outside the Last Closed Flux Surface (LCFS) observed in this experiment. The results presented here reveal the significant role played by the strong sheared E × B flow on the blob dynamics, which provides a potential method to control the SOL width by modifying the sheared E × B flow in future tokamak plasmas.
Prediction and copy number variation identification of ZNF146 gene related to growth traits in Chinese cattle
X. T. Ding, X. Liu, X. M. Li, Y. F. Wen, J. W. Xu, W. J. Liu, Z. M. Li, Z. J. Zhang, Y. N. Chai, H. L. Wang, B. W. Cheng, S. H. Liu, B. Hou, Y. J. Huang, J. G. Li, L. J. Li, G. J. Yang, Z. F. Qi, F. Y. Chen, Q. T. Shi, E. Y. Wang, C. Z. Lei, H. Chen, B. R. Ru, Y. Z. Huang
Journal: The Journal of Agricultural Science / Volume 160 / Issue 5 / October 2022
Published online by Cambridge University Press: 10 August 2022, pp. 404-412
The great demographic pressure brings tremendous volume of beef demand. The key to solve this problem is the growth and development of Chinese cattle. In order to find molecular markers conducive to the growth and development of Chinese cattle, sequencing was used to determine the position of copy number variations (CNVs), bioinformatics analysis was used to predict the function of ZNF146 gene, real-time fluorescent quantitative polymerase chain reaction (qPCR) was used for CNV genotyping and one-way analysis of variance was used for association analysis. The results showed that there exists CNV in Chr 18: 47225201-47229600 (5.0.1 version) of ZNF146 gene through the early sequencing results in the laboratory and predicted ZNF146 gene was expressed in liver, skeletal muscle and breast cells, and was amplified or overexpressed in pancreatic cancer, which promoted the development of tumour through bioinformatics. Therefore, it is predicted that ZNF146 gene affects the proliferation of muscle cells, and then affects the growth and development of cattle. Furthermore, CNV genotyping of ZNF146 gene was three types (deletion type, normal type and duplication type) by Real-time fluorescent quantitative PCR (qPCR). The association analysis results showed that ZNF146-CNV was significantly correlated with rump length of Qinchuan cattle, hucklebone width of Jiaxian red cattle and heart girth of Yunling cattle. From the above results, ZNF146-CNV had a significant effect on growth traits, which provided an important candidate molecular marker for growth and development of Chinese cattle.
P.153 Traumatic spinal cord injuries among indigenous and non-indigenous peoples of Canada
A Persad, B Renne, M Jeffrey, S Ahmed, S Humphreys, D Kurban, C Rivers, C Cheng, D Wang, T Shen, X Liu, S Christie, T Clarke, B Drew, K Ethans, MG Fehlings, A Linassi, C O'Connell, J Paquet, L Scott, D Fourney
Journal: Canadian Journal of Neurological Sciences / Volume 49 / Issue s1 / June 2022
Published online by Cambridge University Press: 24 June 2022, p. S47
Background: Despite a higher prevalence of traumatic spinal cord injury (TSCI) amongst Canadian Indigenous peoples, there is a paucity of studies focused on Indigenous TSCI. We present the first Canada-wide study comparing TSCI amongst Canadian Indigenous and non-Indigenous peoples. Methods: This study is a retrospective analysis of prospectively-collected TSCI data from the Rick Hansen Spinal Cord Injury Registry (RHSCIR) from 2004-2019. We divided participants into Indigenous and non-Indigenous cohorts and compared them with respect to demographics, injury mechanism, level, severity, and outcomes. Results: Compared with non-Indigenous patients, Indigenous patients were younger, more female, less likely to have higher education, and less likely to be employed. The mechanism of injury was more likely due to assault or transportation-related trauma in the Indigenous group. The length of stay for Indigenous patients was longer. Indigenous patients were more likely to be discharged to a rural setting, less likely to be discharged home, and more likely to be unemployed following injury. Conclusions: Our results suggest that more resources need to be dedicated for transitioning Indigenous patients sustaining a TSCI to community living and for supporting these patients in their home communities. A focus on resources and infrastructure for Indigenous patients by engagement with Indigenous communities is needed.
Intercropping with Chinese leek decreased Meloidogyne javanica population and shifted microbial community structure in Sacha Inchi plantation
C. R. Nie, Y. Feng, X. H. Cheng, Z. Q. Cai
Journal: The Journal of Agricultural Science / Volume 159 / Issue 5-6 / July 2021
Published online by Cambridge University Press: 20 October 2021, pp. 404-413
The root-knot nematode, Meloidogyne javanica, is a major problem for the production of Sacha Inchi plants. We examined the effects of strip intercropping of Sacha Inchi/Chinese leek of 3–4 years on the seasonal dynamics of plant and soil traits in tropical China. Results indicated that in the intercropping system, a partially temporal divergence of belowground resource acquisition via niche separation occurred throughout the growing seasons, besides a complete spatially-separated plant height between the two crops. Compared with Sacha Inchi monoculture, the increased seed yield per unit area in the intercropping system was mainly attributed to the higher plant survival rate, rather than the enhanced plant traits of healthy plants. Intercropping greatly suppressed M. javanica populations only in the wet season, compared with monoculture; which may be associated with the combined effects of the direct allelopathy and indigenous microbe induced-suppressiveness. Intercropping did not affect microbial richness and α-diversity in the rhizosphere, except for the decreased fungal richness. Both bacterial and fungal composition and structure were diverged between monoculture v. intercropping system. The relative abundances of the dominant bacterial genera (Bacillus, Gaiellales, Lactococcus, Massilia and Lysobacter, etc.) differed significantly between the two cropping systems. For fungi, intercropping decreased the relative abundances of Fusarium and Gibberella, but increased those of Nectriaceae_unclassified, Chaetomiaceae, Humicola and Mortierella. Overall, Sacha Inchi/Chinese leek intercropping suppressed M. javanica populations and shifted microbial compositions (especially decreased pathogen-containing Fusarium). The increased yield and economic returns in this intercropping system provide valid information for the effective agricultural management.
Skin-friction reduction using periodic blowing through streamwise slits
X.Q. Cheng, Z.X. Qiao, X. Zhang, M. Quadrio, Y. Zhou
Journal: Journal of Fluid Mechanics / Volume 920 / 10 August 2021
Published online by Cambridge University Press: 16 June 2021, A50
Active skin-friction reduction in a turbulent boundary layer (TBL) is experimentally studied based on time-periodic blowing through one array of streamwise slits. The control parameters investigated include the blowing amplitude A+ and frequency f+, which, expressed in wall units, range from 0 to 2 and from 0.007 to 0.56, respectively. The maximum local friction reduction downstream of the slits reaches more than 70 %; friction does not fully recover to the state of the natural TBL until 500 wall units behind the slits. A positive net power saving is possible, and 4.01 % is measured with a local friction drag reduction (DR) of 49 %. A detailed analysis based on hot-wire, particle image velocimetry and smoke-wire flow visualization data is performed to understand the physical mechanisms involved. Spectral analysis indicates weakened near-wall large-scale structures. Flow visualizations show stabilized streaky structures and a locally relaminarized flow. Two factors are identified to contribute to the DR. Firstly, the jets from the slits create streamwise vortices in the near-wall region, preventing the formation of near-wall streaks and interrupting the turbulence generation cycle. Secondly, the zero-streamwise-momentum fluid associated with the jets also accounts for the DR. A closed-loop opposing control system is developed, along with an open-loop desynchronized control scheme, to quantify the two contributions. The latter is found to account for 77 % of the DR, whereas the former is responsible for 23 %. An empirical scaling of the DR is also proposed, which provides valuable insight into the TBL control physics.
Uptake of referrals for women with positive perinatal depression screening results and the effectiveness of interventions to increase uptake: a systematic review and meta-analysis
WQ Xue, KK Cheng, D Xu, X Jin, WJ Gong
Journal: Epidemiology and Psychiatric Sciences / Volume 29 / 2020
Published online by Cambridge University Press: 17 July 2020, e143
Perinatal depression threatens the health of maternal women and their offspring. Although screening programs for perinatal depression exist, non-uptake of referral to further mental health care after screening reduces the utility of these programs. Uptake rates among women with positive screening varied widely across studies and little is known about how to improve the uptake rate. This study aimed to systematically review the available evidence on uptake rates, estimate the pooled rate, identify interventions to improve uptake of referral and explore the effectiveness of those interventions.
This systematic review has been registered in PROSPERO (registration number: CRD42019138095). We searched Pubmed, Web of Science, Cochrane Library, Ovid, Embase, CNKI, Wanfang Database and VIP Databases from database inception to January 13, 2019 and scanned reference lists of relevant researches for studies published in English or Chinese. Studies providing information on uptake rate and/or effectiveness of interventions on uptake of referral were eligible for inclusion. Studies were excluded if they did not report the details of the referral process or did not provide exact uptake rate. Data provided by observational studies and quasi-experimental studies were used to estimate the pooled uptake rate through meta-analysis. We also performed meta-regression and subgroup analyses to explore the potential source of heterogeneity. To evaluate the effectiveness of interventions, we conducted descriptive analyses instead of meta-analyses since there was only one randomised controlled trial (RCT).
Of 2302 records identified, 41 studies were eligible for inclusion, including 39 observational studies (n = 9337), one quasi-experimental study (n = 43) and one RCT (n = 555). All but two studies were conducted in high-income countries. The uptake rates reported by included studies varied widely and the pooled uptake rate of referral was 43% (95% confidence intervals [CI] 35–50%) by a random-effect model. Meta-regression and subgroup analyses both showed that referral to on-site assessment or treatment (60%, 95% CI 51–69%) had a significantly higher uptake rate than referral to mental health service (32%, 95% CI 23–41%) (odds ratio 1.31, 95% CI 1.13–1.52). The included RCT showed that the referral intervention significantly improved the uptake rate (p < 0.01).
Almost three-fifths of women with positive screening results do not take up the referral offers after perinatal depression screening. Referral to on-site assessment and treatment may improve uptake of referral, but the quality of evidence on interventions to increase uptake was weak. More robust studies are needed, especially in low-and middle-income countries.
Effect of inorganic phosphate supplementation on egg production in Hy-Line Brown layers fed 2000 FTU/kg phytase
X. Cheng, J. K. Yan, W. Q. Sun, Z. Y. Chen, S. R. Wu, Z. Z. Ren, X. J. Yang
Journal: animal / Volume 14 / Issue 11 / November 2020
Print publication: November 2020
Phytase has long been used to decrease the inorganic phosphorus (Pi) input in poultry diet. The current study was conducted to investigate the effects of Pi supplementation on laying performance, egg quality and phosphate–calcium metabolism in Hy-Line Brown laying hens fed phytase. Layers (n = 504, 29 weeks old) were randomly assigned to seven treatments with six replicates of 12 birds. The corn–soybean meal-based diet contained 0.12% non-phytate phosphorus (nPP), 3.8% calcium, 2415 IU/kg vitamin D3 and 2000 FTU/kg phytase. Inorganic phosphorus (in the form of mono-dicalcium phosphate) was added into the basal diet to construct seven experimental diets; the final dietary nPP levels were 0.12%, 0.17%, 0.22%, 0.27%, 0.32%, 0.37% and 0.42%. The feeding trial lasted 12 weeks (hens from 29 to 40 weeks of age). Laying performance (housed laying rate, egg weight, egg mass, daily feed intake and feed conversion ratio) was weekly calculated. Egg quality (egg shape index, shell strength, shell thickness, albumen height, yolk colour and Haugh units), serum parameters (calcium, phosphorus, parathyroid hormone, calcitonin and 1,25-dihydroxyvitamin D), tibia quality (breaking strength, and calcium, phosphorus and ash contents), intestinal gene expression (type IIb sodium-dependent phosphate cotransporter, NaPi-IIb) and phosphorus excretion were determined at the end of the trial. No differences were observed on laying performance, egg quality, serum parameters and tibia quality. Hens fed 0.17% nPP had increased (P < 0.01) duodenum NaPi-IIb expression compared to all other treatments. Phosphorus excretion linearly increased with an increase in dietary nPP (phosphorus excretion = 1.7916 × nPP + 0.2157; R2 = 0.9609, P = 0.001). In conclusion, corn–soybean meal-based diets containing 0.12% nPP, 3.8% calcium, 2415 IU/kg vitamin D3 and 2000 FTU/kg phytase would meet the requirements for egg production in Hy-Line Brown laying hens (29 to 40 weeks of age).
Genome-wide association study of bone mineral density trait among three pig breeds
B. Jiang, M. Wang, Z. Tang, X. Du, S. Feng, G. Ma, D. Ye, H. Cheng, H. Wang, X. Liu
Journal: animal / Volume 14 / Issue 12 / December 2020
Print publication: December 2020
Leg weakness (LW) issues are a great concern for pig breeding industry. And it also has a serious impact on animal welfare. To dissect the genetic architecture of limb-and-hoof firmness in commercial pigs, a genome-wide association study was conducted on bone mineral density (BMD) in three sow populations, including Duroc, Landrace and Yorkshire. The BMD data were obtained by ultrasound technology from 812 pigs (including Duroc 115, Landrace 243 and Yorkshire 454). In addition, all pigs were genotyped using genome-by-sequencing and a total of 224 162 single-nucleotide polymorphisms (SNPs) were obtained. After quality control, 218 141 SNPs were used for subsequent genome-wide association analysis. Nine significant associations were identified on chromosomes 3, 5, 6, 7, 9, 10, 12 and 18 that passed Bonferroni correction threshold of 0.05/(total SNP numbers). The most significant locus that associated with BMD (P value = 1.92e−14) was detected at approximately 41.7 Mb on SSC6 (SSC stands for Sus scrofa chromosome). CUL7, PTK7, SRF, VEGFA, RHEB, PRKAR1A and TPO that are located near the lead SNP of significant loci were highlighted as functionally plausible candidate genes for sow limb-and-hoof firmness. Moreover, we also applied a new method to measure the BMD data of pigs by ultrasound technology. The results provide an insight into the genetic architecture of LW and can also help to improve animal welfare in pigs.
Catechol-O-methyltransferase gene promoter methylation as a peripheral biomarker in male schizophrenia
S. Gao, J. Cheng, G. Li, T. Sun, Y. Xu, Y. Wang, X. Du, G. Xu, S. Duan
Journal: European Psychiatry / Volume 44 / July 2017
Published online by Cambridge University Press: 23 March 2020, pp. 39-46
As an epigenetic modification, DNA methylation may reflect the interaction between genetic and environmental factors in the development of schizophrenia (SCZ). Catechol-O-methyltransferase (COMT) gene is a promising candidate gene of SCZ. In the present study, we investigate the association of COMT methylation with the risk of SCZ using bisulfite pyrosequencing technology. Significant association between DNA methylation of COMT and the risk of SCZ is identified (P = 1.618e−007). A breakdown analysis by gender shows that the significance is driven by males (P = 3.310e−009), but not by females. DNA methylation of COMT is not significantly associated with SCZ clinical phenotypes, including p300 and cysteine level. No interaction is found between COMT genotypes and the percent methylation of this gene. Receiver operating characteristic (ROC) curve shows that DNA methylation of COMT is able to predict the SCZ risk in males (area under curve [AUC] = 0.802, P = 1.91e−007). The current study indicates the clinical value of COMT methylation as a potential male-specific biomarker in SCZ diagnosis.
Alterations of the fatty acid composition and lipid metabolome of breast muscle in chickens exposed to dietary mixed edible oils
X. Y. Cui, Z. Y. Gou, K. F. M. Abouelezz, L. Li, X. J. Lin, Q. L. Fan, Y. B. Wang, Z. G. Cheng, F. Y. Ding, S. Q. Jiang
Journal: animal / Volume 14 / Issue 6 / June 2020
Published online by Cambridge University Press: 09 January 2020, pp. 1322-1332
Print publication: June 2020
The fatty acid composition of chicken's meat is largely influenced by dietary lipids, which are often used as supplements to increase dietary caloric density. The underlying key metabolites and pathways influenced by dietary oils remain poorly known in chickens. The objective of this study was to explore the underlying metabolic mechanisms of how diets supplemented with mixed or a single oil with distinct fatty acid composition influence the fatty acid profile in breast muscle of Qingyuan chickens. Birds were fed a corn-soybean meal diet supplemented with either soybean oil (control, CON) or equal amounts of mixed edible oils (MEO; soybean oil : lard : fish oil : coconut oil = 1 : 1 : 0.5 : 0.5) from 1 to 120 days of age. Growth performance and fatty acid composition of muscle lipids were analysed. LC-MS was applied to investigate the effects of CON v. MEO diets on lipid-related metabolites in the muscle of chickens at day 120. Compared with the CON diet, chickens fed the MEO diet had a lower feed conversion ratio (P < 0.05), higher proportions of lauric acid (C12:0), myristic acid (C14:0), palmitoleic acid (C16:1n-7), oleic acid (C18:1n-9), EPA (C20:5n-3) and DHA (C22:6n-3), and a lower linoleic acid (C18:2n-6) content in breast muscle (P < 0.05). Muscle metabolome profiling showed that the most differentially abundant metabolites are phospholipids, including phosphatidylcholines (PC) and phosphatidylethanolamines (PE), which enriched the glycerophospholipid metabolism (P < 0.05). These key differentially abundant metabolites – PC (14:0/20:4), PC (18:1/14:1), PC (18:0/14:1), PC (18:0/18:4), PC (20:0/18:4), PE (22:0/P-16:0), PE (24:0/20:5), PE (22:2/P-18:1), PE (24:0/18:4) – were closely associated with the contents of C12:0, C14:0, DHA and C18:2n-6 in muscle lipids (P < 0.05). The content of glutathione metabolite was higher with MEO than CON diet (P < 0.05). Based on these results, it can be concluded that the diet supplemented with MEO reduced the feed conversion ratio, enriched the content of n-3 fatty acids and modified the related metabolites (including PC, PE and glutathione) in breast muscle of chickens.
Association analysis of SSTR2 copy number variation with cattle stature and its expression analysis in Chinese beef cattle
J. Cheng, R. Jiang, X. K. Cao, M. Liu, Y. Z. Huang, X. Y. Lan, H. Cao, C. Z. Lei, H. Chen
Journal: The Journal of Agricultural Science / Volume 157 / Issue 4 / May 2019
Published online by Cambridge University Press: 26 September 2019, pp. 365-374
Copy number variations (CNVs), as an important source of genetic variation, can affect a wide range of phenotypes by diverse mechanisms. The somatostatin receptor 2 (SSTR2) gene plays important roles in cell proliferation and apoptosis. Recently, this gene was mapped to a CNV region, which encompasses quantitative trait loci of cattle economic traits including body weight, marbling score, etc. Therefore, SSTR2 CNV may exhibit phenotypic effects on cattle growth traits. In the current study, distribution of SSTR2 gene CNVs was investigated in six Chinese cattle breeds (XN, QC, NY, JA, LX and PN), and the results showed higher CNV polymorphisms in XN, QC and NY cattle. Next, association analysis between growth traits and SSTR2 CNV was performed for XN, QC and NY cattle. In NY, individuals with fewer copies showed better performance than those with more copies. Further, the effects of SSTR2 CNV on the SSTR2 mRNA level were also investigated, but revealed no significant correlation in either muscle or adipose tissue of adult NY cattle. The results suggested the potential for use of SSTR2 CNV as a marker for the molecular breeding of NY cattle.
Integral performance optimization for the two-stage-to-orbit RBCC-RKT launch vehicle based on GPM
L. Zhang, M. Sun, Q. Cheng, Z. Chen, X. Zhang
Journal: The Aeronautical Journal / Volume 123 / Issue 1265 / July 2019
Published online by Cambridge University Press: 17 June 2019, pp. 945-969
The takeoff-mass of a two-stage-to-orbit Rocket-Based Combined Cycle Engine-Rocket (RBCC-RKT) launch vehicle is a crucial factor in its comprehensive performance. This paper optimizes the takeoff-mass together with the trajectory by reformulating it to a nonlinear optimal control problem. The range of the second stage rocket mass is considered as a process constraint. When the scopes of initial and terminal states are specified, the problem can be solved by using the Gauss pseudo-spectral method (GPM). In order to reduce the convergent difficulty caused by using table data, the data in different stages are utilized by employing an integrated interpolation strategy through the optimization. Simulation results show that the mass can be effectively optimized to meet the inertia mass ratio constraint of the first-stage, and the separation of Mach number and altitude can be optimized at the same time.
Improvement of ion acceleration in radiation pressure acceleration regime by using an external strong magnetic field
H. Cheng, L. H. Cao, J. X. Gong, R. Xie, C. Y. Zheng, Z. J. Liu
Journal: Laser and Particle Beams / Volume 37 / Issue 2 / June 2019
Two-dimensional particle-in-cell (PIC) simulations have been used to investigate the interaction between a laser pulse and a foil exposed to an external strong longitudinal magnetic field. Compared with that in the absence of the external magnetic field, the divergence of proton with the magnetic field in radiation pressure acceleration (RPA) regimes has improved remarkably due to the restriction of the electron transverse expansion. During the RPA process, the foil develops into a typical bubble-like shape resulting from the combined action of transversal ponderomotive force and instabilities. However, the foil prefers to be in a cone-like shape by using the magnetic field. The dependence of proton divergence on the strength of magnetic field has been studied, and an optimal magnetic field of nearly 60 kT is achieved in these simulations.
Genetic and morphology analysis among the pentaploid F1 hybrid fishes (Schizothorax wangchiachii ♀ × Percocypris pingi ♂) and their parents
H. R. Gu, Y. F. Wan, Y. Yang, Q. Ao, W. L. Cheng, S. H. Deng, D. Y. Pu, X. F. He, L. Jin, Z. J. Wang
Triploid and pentaploid breeding is of great importance in agricultural production, but it is not always easy to obtain double ploidy parents. However, in fishes, chromosome ploidy is diversiform, which may provide natural parental resources for triploid and pentaploid breeding. Both tetraploid and hexaploid exist in Schizothorax fishes, which were thought to belong to different subfamilies with tetraploid Percocypris fishes in morphology, but they are sister genera in molecule. Fortunately, the pentaploid hybrid fishes have been successfully obtained by hybridization of Schizothorax wangchiachii (♀, 2n = 6X = 148) × Percocypris pingi (♂, 2n = 4X = 98). To understand the genetic and morphological difference among the hybrid fishes and their parents, four methods were used in this study: morphology, karyotype, red blood cell (RBC) DNA content determination and inter-simple sequence repeat (ISSR). In morphology, the hybrid fishes were steady, and between their parents with no obvious preference. The chromosome numbers of P. pingi have been reported as 2n = 4X = 98. In this study, the karyotype of S. wangchiachii was 2n = 6X = 148 = 36m + 34sm + 12st + 66t, while that the hybrid fishes was 2n = 5X = 123 = 39m + 28sm + 5st + 51t. Similarly, the RBC DNA content of the hybrid fishes was intermediate among their parents. In ISSR, the within-group genetic diversity of hybrid fishes was higher than that of their parents. Moreover, the genetic distance of hybrid fishes between P. pingi and S.wangchiachii was closely related to that of their parental ploidy, suggesting that parental genetic material stably coexisted in the hybrid fishes. This is the first report to show a stable pentaploid F1 hybrids produced by hybridization of a hexaploid and a tetraploid in aquaculture.
A viscous damping model for piston mode resonance
L. Tan, L. Lu, G.-Q. Tang, L. Cheng, X.-B. Chen
Journal: Journal of Fluid Mechanics / Volume 871 / 25 July 2019
A viscous damping model is proposed based on a simplified equation of fluid motion in a moonpool or the narrow gap formed by two fixed boxes. The model takes into account the damping induced by both flow separation and wall friction through two damping coefficients, namely, the local and friction loss coefficients. The local loss coefficient is determined through specifically designed physical model tests in this work, and the friction loss coefficient is estimated through an empirical formula found in the literature. The viscous damping model is implemented in the dynamic free-surface boundary condition in the gap of a modified potential flow model. The modified potential flow model is then applied to simulate the wave-induced fluid responses in a narrow gap formed by two fixed boxes and in a moonpool for which experimental data are available. The modified potential flow model with the proposed viscous damping model works well in capturing both the resonant amplitude and frequency under a wide range of damping conditions.
Gender and time delays in diagnosis of pulmonary tuberculosis: a cross-sectional study from China
H. G. Chen, T. W. Wang, Q. X. Cheng
Journal: Epidemiology & Infection / Volume 147 / 2019
Published online by Cambridge University Press: 22 February 2019, e94
Gender inequality has severe consequences on public health in terms of delay in diagnosis of pulmonary tuberculosis (PTB). In order to explore gender-related differences in diagnosis delay, a cross-sectional study of 10 686 patients diagnosed with PTB in Yulin from 1 January 2009 to 31 December 2014 was conducted. Diagnosis delay was categorised into 'short delay' and 'long delay' by four commonly used cut-off points of 14, 30, 60 and 90 days. Logistic regression analysis was used to analyse gender differences in diagnostic delay. Stratified analyses by smear results, age, urban/rural were performed to examine whether the effect persisted across the strata. The median delay was 31 days (interquartile range 13–65). Diagnostic delay in females at cut-off points of 14, 30, 60 and 90 days had odds ratios (OR) of 0.99 (95% CI 0.91–1.09), 1.09 (95% CI 1.01–1.18), 1.15 (95% CI 1.05–1.26) and 1.18 (95% CI 1.06–1.31), respectively, compared with males. Stratified analysis showed that females were associated with increased risk of longer delay among those aged 30–60 years, smear positive and living in the rural areas (P < 0.05). The female-to-male OR increased along with increased delay time. Further inquiry into the underlying reasons for gender differences should be urgently addressed to improve the current situation.
Post-exposure prophylaxis vaccination rate and risk factors of human rabies in mainland China: a meta-analysis
D. L. Wang, X. F. Zhang, H. Jin, X. Q. Cheng, C. X. Duan, X. C. Wang, C. J. Bao, M. H. Zhou, T. Ahmad
Published online by Cambridge University Press: 04 December 2018, e64
Rabies is one of the major public health problems in China, and the mortality rate of rabies remains the highest among all notifiable infectious diseases. A meta-analysis was conducted to investigate the post-exposure prophylaxis (PEP) vaccination rate and risk factors for human rabies in mainland China. The PubMed, Web of Science, Chinese National Knowledge Infrastructure, Chinese Science and Technology Periodical and Wanfang databases were searched for articles on rabies vaccination status (published between 2007 and 2017). In total, 10 174 human rabies cases from 136 studies were included in this meta-analysis. Approximately 97.2% (95% confidence interval (CI) 95.1–98.7%) of rabies cases occurred in rural areas and 72.6% (95% CI 70.0–75.1%) occurred in farmers. Overall, the vaccination rate in the reported human rabies cases was 15.4% (95% CI 13.7–17.4%). However, among vaccinated individuals, 85.5% (95% CI 79.8%–83.4%) did not complete the vaccination regimen. In a subgroup analysis, the PEP vaccination rate in the eastern region (18.8%, 95% CI 15.9–22.1%) was higher than that in the western region (13.3%, 95% CI 11.1–15.8%) and this rate decreased after 2007. Approximately 68.9% (95% CI 63.6–73.8%) of rabies cases experienced category-III exposures, but their PEP vaccination rate was 27.0% (95% CI 14.4–44.9%) and only 6.1% (95% CI 4.4–8.4%) received rabies immunoglobulin. Together, these results suggested that the PEP vaccination rate among human rabies cases was low in mainland China. Therefore, standardised treatment and vaccination programs of dog bites need to be further strengthened, particularly in rural areas.
Impacts of long-term fertilization on the soil microbial communities in double-cropped paddy fields
H. M. Tang, Y. L. Xu, X. P. Xiao, C. Li, W. Y. Li, K. K. Cheng, X. C. Pan, G. Sun
Journal: The Journal of Agricultural Science / Volume 156 / Issue 7 / September 2018
The response of soil microbial communities to soil quality changes is a sensitive indicator of soil ecosystem health. The current work investigated soil microbial communities under different fertilization treatments in a 31-year experiment using the phospholipid fatty acid (PLFA) profile method. The experiment consisted of five fertilization treatments: without fertilizer input (CK), chemical fertilizer alone (MF), rice (Oryza sativa L.) straw residue and chemical fertilizer (RF), low manure rate and chemical fertilizer (LOM), and high manure rate and chemical fertilizer (HOM). Soil samples were collected from the plough layer and results indicated that the content of PLFAs were increased in all fertilization treatments compared with the control. The iC15:0 fatty acids increased significantly in MF treatment but decreased in RF, LOM and HOM, while aC15:0 fatty acids increased in these three treatments. Principal component (PC) analysis was conducted to determine factors defining soil microbial community structure using the 21 PLFAs detected in all treatments: the first and second PCs explained 89.8% of the total variance. All unsaturated and cyclopropyl PLFAs except C12:0 and C15:0 were highly weighted on the first PC. The first and second PC also explained 87.1% of the total variance among all fertilization treatments. There was no difference in the first and second PC between RF and HOM treatments. The results indicated that long-term combined application of straw residue or organic manure with chemical fertilizer practices improved soil microbial community structure more than the mineral fertilizer treatment in double-cropped paddy fields in Southern China.
Spatiotemporal expression profiling of the farnesyl diphosphate synthase genes in aphids and analysis of their associations with the biosynthesis of alarm pheromone
Y.-J. Cheng, Z.-X. Li
Journal: Bulletin of Entomological Research / Volume 109 / Issue 3 / June 2019
The alarm behavior plays a key role in the ecology of aphids, but the site and molecular mechanism for the biosynthesis of aphid alarm pheromone are largely unknown. Farnesyl diphosphate synthase (FPPS) catalyzes the synthesis of FPP, providing the precursor for the alarm pheromone (E)-β-farnesene (EβF), and we speculate that FPPS is closely associated with the biosynthetic pathway of EβF. We firstly analyzed the spatiotemporal expression of FPPS genes by using quantitative reverse transcription-polymerase chain reaction, showing that they were expressed uninterruptedly from the embryonic stage to adult stage, with an obvious increasing trend from embryo to 4th-instar in the green peach aphid Myzus persicae, but FPPS1 had an overall significantly higher expression level than FPPS2; both FPPS1 and FPPS2 exhibited the highest expression in the cornicle area. This expression pattern was verified in Acyrthosiphon pisum, suggesting that FPPS1 may play a more important role in aphids and the cornicle area is most likely the site for EβF biosynthesis. We thus conducted a quantitative measurement of EβF in M. persicae by gas chromatography-mass spectrometry. The data obtained were used to perform an association analysis with the expression data, revealing that the content of EβF per aphid was significantly correlated with the mean weight per aphid (r = 0.8534, P = 0.0307) and the expression level of FPPS1 (r = 0.9134, P = 0.0109), but not with that of FPPS2 (r = 0.4113, P = 0.4179); the concentration of EβF per milligram of aphid was not correlated with the mean weight per aphid or the expression level of FPPS genes. These data suggest that FPPS1 may play a key role in the biosynthesis of aphid alarm pheromone. | CommonCrawl |
How is the speed of light constant in all directions for all observers?
Please imagine the following thought-experiment:
Order of Events:
Pulse - A single pulse of light is emitted from the light towards the mirror
Reflect - The pulse hits the mirror and is reflected back towards the light
Return - The pulse returns to the light.
Observers:
BoxGuy - An observer on the boxcar
PlatGirl - An observer on the platform
With the above configuration, how can the speed of light be constant for both observers in both directions?
Assuming the speed of light is constant for BoxGuy relative to himself, the time between Pulse and Reflect is equal to the time between Reflect and Return. This is because the distance the light travels relative to him is d in both cases.
With the same assumptions for PlatGirl, the time between Pulse and Reflect is less than the time between Reflect and Return. This is because the mirror will travel 2 * d on the away trip (because when light has traveled 2 * d, the mirror will be d farther to the left, so both the mirror and pulse will be in the same location), but only 2/3 * d on the return trip (using similar logic).
Assuming that the light pulse is in the same location for all observers at any given moment, Pulse has to occur simultaneously for both BoxGuy and PlatGirl, Reflect has to occur simultaneously for BoxGuy and PlatGirl, and Return has to occur simultaneously for BoxGuy and PlatGirl.
Finally, if we try to figure out the relative passage of time for BoxGuy and PlatGirl with the above, we get that time travels faster for PlatGirl than for BoxGuy during Pulse-Reflect. This is because light travels farther for her (2*d) than him (d) during that time. With similar logic, we get that time travels slower for PlatGirl than for BoxGuy during Reflect-Return.
The last conclusions do not make sense, since the coming or going of a beam of light should not affect the relative time-lapse for two observers. For example, if this were the case what would happen if another pulse was emitted the moment the first pulse is reflected? Time cannot move faster AND slower for both of them.
Thus, either the speed of light is not constant, the same light beam can simultaneously be in different locations at once for different observers, or there is another flaw in the analysis.
Which is it and why?
As mentioned by other users, d will be shorter for PlatGirl than for BoxBoy according to SR. However, the duration of Pulse-Reflect is still shorter than Reflect-Return for PlatGirl, and the durations are equal for BoxBoy.
In response to my question on Mark's answer, we can use the Lorentz Transform to calculate PlatGirl's space-time coordinate for BoxGuy's Reflect observed event, which happens at (d,d/c) in his frame of reference:
$\lambda = (1/\sqrt{1-.5^2}) = (1/\sqrt{.75}) = \sqrt{4}/\sqrt{3} = \frac{2\sqrt{3}}{3}$ $t' = \lambda (t - vx/c^2) = \lambda (d/c - (-.5) \cdot d/c) = \frac{2\sqrt{3}}{3} \cdot (1.5d/c) = \sqrt{3}d/c$ $x' = \lambda (x - vt) = \lambda (d + .5c \cdot d/c) = \frac{2\sqrt{3}}{3}*1.5d = \sqrt{3}d$
Similarly for (0, 2d/c):
$t' = \lambda (t - vx/c^2) = \frac{2\sqrt{3}}{3} (2d/c) = \frac{4\sqrt{3}}{3} d/c$ $x' = \lambda (x - vt) = \frac{2\sqrt{3}}{3} (.5c \cdot 2d/c) = \frac{2\sqrt{3}}{3}d$
special-relativity speed-of-light reference-frames
Brandon Enright
Briguy37Briguy37
$\begingroup$ The thing is that the constantcy of the speed of light is both (a) a consequence of the symmetries of Maxwell's equations and (b) an experimentaly measured fact (since 1887 and many times since then). In anycase in your list of possibilities you left out "the distance between points may be different for different observers" $\endgroup$ – dmckee --- ex-moderator kitten♦ Jan 10 '13 at 21:47
$\begingroup$ The order of events and simultaneousity you talk about is a classical way, not special relativistic. $\endgroup$ – Abhimanyu Pallavi Sudhir Jul 7 '13 at 7:11
$\begingroup$ Good luck with this - but bear in mind that our standards for measuring both distance and time are based on 'the speed of light'. It's a bit like measuring changes in the length of a foot rule with the same foot rule. $\endgroup$ – Alan Gee Dec 13 '18 at 19:53
The problem is in a misunderstanding of "simultaneous".
"Simultaneous" refers to two different events that occur at the same time in some particular reference frame, but you're applying it to the same event in two different frames. So it doesn't make sense to say "Pulse has to occur simultaneously for both BoxGuy and PlatGirl." That's a single event - it can't be simultaneous all by itself, even when observed by two different people.
You could, if you want, set the origins of the coordinate systems they are using so PlatGirl and BoxGuy assign the same time coordinate to Pulse. If you do, they will not assign the same time coordinate to Reflect. The time between the events Pulse and Reflect is different in different frames.
Additionally, PlatGirl and BoxGuy will not agree on the length of the boxcar. Your calculation assumes they both measure the length to be $d$, but actually PlatGirl will observe the boxcar to be Lorentz-contracted.
One way to analyze your scenario is to set up coordinate systems $S$ for the boxcar and $S'$ for the platform. We set (x,t) = (0,0) = Pulse in both systems.
In frame $S$ (box), the coordinates are:
Pulse: (0,0) Reflect: (d,d/c) Return: (0,2d/c)
In frame $S'$ (platform), the coordinates are:
Pulse: (0,0)
Reflect: $(\sqrt{3}d,\sqrt{3}d/c)$
Return: $(\frac{2\sqrt{3}}{3} d, \frac{4\sqrt{3}}{3} d/c)$
You can verify that in both frames, light moves outward at speed $c$ and returns at speed $-c$
In reply to your edit, yes the durations from Pulse to Reflect and Reflect to Return are the same for BoxGuy and different for PlatGirl. That is just a fact. That's how it is. Notice, though, that the spatial separations are also different. For BoxGuy, these events the same distance apart. For PlatGirl, they are different distances apart. What's the same between frames is the interval $\Delta x^2 - \Delta t^2$.
Mark EichenlaubMark Eichenlaub
$\begingroup$ Ok, so we set the space-time-origin (0,0) to Pulse at the Light for both BoxGuy and PlatGirl. The space-origin "moves" with the Light for BoxGuy, but is fixed relative to the platform for PlatGirl. At (d,d/c) and (0, 2d/c) for BoxGuy, Reflect and Return happen respectively. What is the formula for translating these space-time coordinates directly back to PlatGirl's coordinate system? $\endgroup$ – Briguy37 Jan 11 '13 at 17:07
$\begingroup$ Nevermind, I've updated my question notes with the Lorenz Transform. From this, it appears time observed at a point in space is warped according to the direction of relative travel. For example, at the time of the Pulse, BoxGuy's mirror is at $(d,0)$, which maps to PlatGirl's $(\frac{2\sqrt{3}}{3}d, \frac{\sqrt{3}}{3}d/c)$. Thus, PlatGirl observes a current version of BoxGuy (for them both), but BoxGuy's "current" mirror is a future version of PlatGirl's "current" mirror. By symmetry, if the mirror and origin were reversed, PlatGirl's mirror would be the "future" one. Is this correct? $\endgroup$ – Briguy37 Jan 11 '13 at 20:39
$\begingroup$ That's correct at the moment they're passing each other, yes. $\endgroup$ – Mark Eichenlaub Jan 11 '13 at 20:46
It always helps to draw the right picture.
This picture assumes that Boxguy is standing next to the lamp, and that the flash leaves the lamp just as it passes PlatGirl. (If, for example, BoxGuy were standing next to the mirror, the picture would look a little different.)
The black vertical line is Platgirl's worldline, and any black horizontal line is "the world at a particular instant" according to Platgirl. She measures distances along any one of these lines.
The blue near-vertical line is BoxGuy's worldline (and the lamp's). Each of the other blue lines is "the world at a particular instant" according to Boxguy. He measures distances along any one of these lines.
The broken gold line is the path of the light beam, from the lamp to the mirror and back.
Both Platgirl and BoxGuy will agree that the gold lines traverse $x$ units of space in $x$ units of time (I am taking the speed of light to be 1.) That's because the light rays are at "45 degree angles" to the axes in both PlatGirls's and Boxguy's opinion. (Don't forget that Boxguy views the two thick blue lines as perpendicular in spacetime, and note that the gold line bisects the angle between them.)
By staring long enough at this picture, you ought to be able to describe exactly what's happening from both BoxGuy's and Platgirls' points of view, and to see how they're two different ways of describing the same thing (i.e. two different ways of assigning coordinates to points on the same gold line).
Note in particular that the near-horizontal blue lines are equi-spaced, so that Boxguy says the light beam takes equal amounts of time on its way out and on its way back.
[It helps to remember that points along a vertical line all occupy the same location in space according to Platgirl, and that points along a line parallel to his worldline all occupy the same location in space according to BoxGuy. I didn't draw these gridlines for fear of making the diagram look too intimidating, but it might help to add them.]
WillOWillO
The simple, but I believe correct, answer is that the speed of light is not constant to all observers in all directions, but it appears very much to be so due to the different ways that different observers measure and perceive distance.
The speed of light is only truly constant when observed across the fabric of the universe through which it travels, and w.r.t. which the rest of us normally have a (peculiar) velocity.
The animation below represents a Mickelson Morley like experiment.
Notice that both the time the light takes to travel and the distance travelled are the same for both horizontal and vertical light.
The horizontal offset, which is hugely exaggerated (as is the planetary rotation speed), represents a time delay, that would have shown up as a fixed phase shift. This, from what I understand, was predicted and compensated for. There would, however, be no difference in frequency or continuous phase shift.
Alan GeeAlan Gee
It's not just that the concept of absolute velocity is impossible to know its that the concept itself doesn't have any meaning (in relativistic theory). Saying "I am approaching the speed of light" is an ambiguous and often misleading statement. The correct formulation (relativistically speaking) would be "there is an observer in a certain frame of reference who is currently measuring my velocity as seen from his/her frame of reference to be approaching the speed of light".
v (in formula's with v/c) refers to that measured velocity and infers a frame of reference and a measurement process.
Saying the speed of light is absolute is really a short cut for the following statement: "No matter in which frame of reference an observer sits every time he or she measures the speed of light he or she appears to get the same value". This is an experimental statement which is postulated as a physical law because nobody has credibly demonstrated an experiment contradicting this.
Statements about faster-than-light phenomena all have to do with faster-than-light expansion of space itself. Relativity only talks about relative velocities through space itself.
Once again, saying "I have no way of knowing whether I am at rest or moving" is equivalent to saying "I have no way of knowing whether I am taller or shorter". Taller or shorter than what? At rest or moving with respect to what?
RoelofRoelof
Please correct me if I am wrong. But I think that the speed of light, measured distance and time from a frame of reference are concepts defined relatively to each other. In the sense that we fix the speed of light and define distance and time relatively to it.
In particular I am not really convinced that the measured speed of light is the same in every reference frame. It is the same only if you measure it with a mirror so that the beam once travels away from the observer and once towards him.
I think that the time actually slows down as you approach the speed of light and since we keep it constant the concepts of simultaneity and distance are redefined though a Lorentz transformation according to which the events ahead the moving observer look closer in space-time and the ones behind him look farther.
This whole assumption of constant speed of light is a consequence of the ambiguity between rest and motion. In fact an observer within an inertial frame of reference has no way to know his velocity thus he assumes that he is at rest and uses the speed of light to measure distances and simultaneity around him. This results in a geometry described by Lorentz transformations. If he knew his velocity he could have a better knowledge of simultaneity since you can move away or towards light sources.
An observer can easily measure a higher or lower speed of light with a 'simple' setup. A spaceship, a sensor and a light screen. When he crosses the sensor check point the light screen turns on. If he travels through the screen and doesn't know the distance between the sensor and the screen he will use the speed of light to judge the screen closer to him. The opposite is valid if he travels away from the screen.
Another possibility is that the observer measures the distance between check point and screen in a frame of reference where there is no relative velocity. Then when he passes the check point he measures the time needed to detect the light screen and knowing the actual distance he would have measured a faster (or slower) speed of light which he can use to measure his own speed and correct the Lorentz induced distortion of space time.
(EDIT) http://en.wikipedia.org/wiki/Tests_of_special_relativity#Constancy_of_the_speed_of_light http://en.wikipedia.org/wiki/One-way_speed_of_light
Ok I have done some research, here are two articles that can clarify where the confusion arises. Is seems that Lorentz transformations were based on an anisotropic one-way speed of light meaning that you would measure a different one-way speed of light depending if you move toward or away from a source. Still the two-way speed of light is the same as in special relativity and this is actually the only way to currently measure it accurately (it is hard to get accurate synchronization and also to approach the speed of light).
Einstein took Lorentz equations to build his special relativity theory and he also added the assumption that the one-way speed of light was the same for every inertial reference. This assumption redefines the concept of simultaneity. Now both theory are mathematically and experimentally correct and they can both be used as model for reality and after all the one-way speed of light is not that important.
To summarize Special Relativity uses the assumption of constant one-way speed of light to define simultaneity (i.e. things happen when their light reaches you). According to Lorentz synchronicity for inertial frames is assumed thus the one-way speed of light is not a constant (things happen but depending on your relative velocity their light reaches you after more or less time).
$\begingroup$ There are a couple problems with your answer, I'll justmention what I think are the most important. First, it is an experimental fact, not an assumption, that no matter how you measure it, the speed of light is always the same. Second, the whole point of relativity is that you can't know your own velocity, because there's no such thing as an absolute velocity! Anyone, in any frame, using whatever experiment they desire, will always measure the speed of light to be c, and that's an experimental fact. $\endgroup$ – Javier Dec 11 '13 at 12:51
$\begingroup$ Hi, thank you for the reply. I do not want to argue, I only want to understand these concepts better by confronting with others. Can you please point me to these experiments so that I can study them? I did some research and I only found interferometer based experiments that work with reflection. Can you please point me the other problems with my answer? If the absolute velocity is impossible to know what do you use as v in the Lorentz expressions that contain v/c? (Remember that the relative velocity can be higher than the speed of light like for galaxies far away from each other). $\endgroup$ – user35660 Dec 11 '13 at 13:41
$\begingroup$ I can't answer all of your questions right now, and anyway I would recommend you grab a good book to read about SR. Also, you seem to be confusing concepts. The velocity used in the Lorentz transformations is the relative velocity between two systems, which never is greater than the speed of light. As was explained in a recent question which I can't find (also I don't know much about astrophysics, so take this with a grain of salt), the distance between two galaxies may be increasing faster than the speed of light as measured from a third galaxy, but if you stand in one of them (cont.) $\endgroup$ – Javier Dec 11 '13 at 13:54
$\begingroup$ (cont.) you will never see the other one travel away from you faster than the speed of light. $\endgroup$ – Javier Dec 11 '13 at 13:54
$\begingroup$ Ok you you don't have time to point my errors but please come back to me with reference to those experiments. Galaxies are moving faster than the speed of light: curious.astro.cornell.edu/question.php?number=575. If two galaxies are moving faster than c they will never see each other. $\endgroup$ – user35660 Dec 11 '13 at 14:25
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Find the positive integer $n$ such that
\[\sin \left( \frac{\pi}{2n} \right) + \cos \left (\frac{\pi}{2n} \right) = \frac{\sqrt{n}}{2}.\]
Squaring both sides, we get
\[\sin^2 \left( \frac{\pi}{2n} \right) + 2 \sin \left( \frac{\pi}{2n} \right) \cos \left( \frac{\pi}{2n} \right) + \cos^2 \left( \frac{\pi}{2n} \right) = \frac{n}{4},\]which we can re-write as
\[\sin \frac{\pi}{n} + 1 = \frac{n}{4},\]so
\[\sin \frac{\pi}{n} = \frac{n}{4} - 1.\]Since $-1 \le \sin \frac{\pi}{n} \le 1,$ we must also have $-1 \le \frac{n}{4} - 1 \le 1,$ which is equivalent to $0 \le n \le 8.$
The integer $n$ cannot be 0, so $1 \le n \le 8,$ which means $\sin \frac{\pi}{n}$ is positive. Hence, $5 \le n \le 8.$
Note that $n = 6$ works:
\[\sin \frac{\pi}{6} = \frac{1}{2} = \frac{6}{4} - 1.\]Furthermore, $\sin \frac{\pi}{n}$ is a decreasing function of $n,$ and $\frac{n}{4} - 1$ is an increasing function of $n,$ so $n = \boxed{6}$ is the unique solution. | Math Dataset |
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April 2018, 15(2): 407-428. doi: 10.3934/mbe.2018018
Mathematical model for the growth of Mycobacterium tuberculosis in the granuloma
Eduardo Ibargüen-Mondragón 1,, , Lourdes Esteva 2, and Edith Mariela Burbano-Rosero 3,
Departamento de Matemáticas y Estadística, Facultad de Ciencias Exactas y Naturales, Universidad de Nariño, Calle 18 Cra 50, Pasto, Colombia
Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional Autónoma de México, 04510 México DF, México
Departamento de Biología, Facultad de Ciencias Exactas y Naturales, Universidad de Nariño, Calle 18 Cra 50, Pasto, Colombia
* Corresponding author: Eduardo Ibargüen-Mondragón
Grant No 182-01/11/201, Vicerrectoría de Investigaciones, Posgrados y Relaciones Internacionales de la Universidad de Nariño.
Received July 27, 2016 Accepted May 07, 2017 Published January 2018
In this work we formulate a model for the population dynamics of Mycobacterium tuberculosis (Mtb), the causative agent of tuberculosis (TB). Our main interest is to assess the impact of the competition among bacteria on the infection prevalence. For this end, we assume that Mtb population has two types of growth. The first one is due to bacteria produced in the interior of each infected macrophage, and it is assumed that is proportional to the number of infected macrophages. The second one is of logistic type due to the competition among free bacteria released by the same infected macrophages. The qualitative analysis and numerical results suggests the existence of forward, backward and S-shaped bifurcations when the associated reproduction number $R_0$ of the Mtb is less unity. In addition, qualitative analysis of the model shows that there may be up to three bacteria-present equilibria, two locally asymptotically stable, and one unstable.
Keywords: Ordinary differential equations, S-shaped bifurcation, tuberculosis, granuloma, macrophages and T cells.
Mathematics Subject Classification: Primary: 34D23, 93D20; Secondary: 65L05.
Citation: Eduardo Ibargüen-Mondragón, Lourdes Esteva, Edith Mariela Burbano-Rosero. Mathematical model for the growth of Mycobacterium tuberculosis in the granuloma. Mathematical Biosciences & Engineering, 2018, 15 (2) : 407-428. doi: 10.3934/mbe.2018018
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Figure 1. The flow diagram of macrophages, T cells and bacteria
Figure 2. The graph of functions $g_1$ and $g_2$ defined in (20).
Table 1 for $\nu, \gamma_{U} = \bar \gamma\displaystyle{\Lambda_U \over \mu_U}$ and $\mu_{B}$">Figure 3. Standard regression coefficients (SCR) for $R_0 = \frac{\nu}{\gamma_U + \mu_{B}}$ assuming the values given in Table 1 for $\nu, \gamma_{U} = \bar \gamma\displaystyle{\Lambda_U \over \mu_U}$ and $\mu_{B}$
Table 1 for $\bar r, \bar \beta, \displaystyle{\Lambda_U \over \mu_U}, \gamma_{U} = \bar \gamma\displaystyle{\Lambda_U \over \mu_U}$ and $\mu_{B}$.">Figure 4. Standard regression coefficients (SCR) for $R_1 = \frac{\bar r \bar\beta {\Lambda_U \over \mu}}{\gamma_U + \mu_{B}}$ assuming the values given in Table 1 for $\bar r, \bar \beta, \displaystyle{\Lambda_U \over \mu_U}, \gamma_{U} = \bar \gamma\displaystyle{\Lambda_U \over \mu_U}$ and $\mu_{B}$.
Figure 5. The numerical simulations of temporal course for bacteria with ten initial conditions show the stability of the bacteria-present equilibrium $P_2$ and the infection free equilibrium $P_0$ given in (47) when $\sigma = 0.24$, $\sigma_c = 0.319$, $R_0 = 0.4$, $R_0 = 0.34$, $R_1 = 1.5$, $g_1(B^{\max}) = 1.37\times 10^{312}$ and $g_2(B^{\max}) = 1.32\times 10^{942}$.
Figure 6. The numerical simulations of temporal course for bacteria with ten initial conditions show the stability of the bacteria-present equilibria $P_1$ a$P_3$ given in (48) when $\sigma = 2.4\times 10^{-6}$, $\sigma_c = 0.003$, $R_0 = 0.0045$, $R^*_0 = 0.0043$, $R_1 = 0.43$.
Figure 7. The stable infection free equilibrium $P_0$ bifurcates to the stable bacteria-present equilibrium $P_1$ in the value $R_0 = 1-R_1$.
Figure 8. The results suggest forward and backward bifurcations, and a type of S-shaped bifurcation
Table 1. Interpretation and values of the parameters. Data are deduced from the literature (references).
Parameter Description Value Reference
$\Lambda_U$ growth rate of unfected Mtb 600 -1000 day$^{-1}$ [19,23,30]
$\bar\beta$ infection rate of Mtb $2.5*10^{-11}-2.5*10^{-7}$day$^{-1}$ [13,30]
$\bar\alpha_T$ elim. rate of infected Mtb by T cell $2*10^{-5}-3*10^{-5}$ day$^{-1}$ [13,30]
$\mu_U$ nat. death rate of $M_U$ 0028-0.0033 day$^{-1}$ [22,30]
$\mu_I$ nat. death rate of $M_I$ 0.011 day$^{-1}$ [22,35,30]
$\nu$ growth rate of Mtb 0.36 -0.52 day$^{-1}$ [12,20,38]
$\mu_{B}$ natural death rate of Mtb 0.31 -0.52 day$^{-1}$ [39,30]
$\bar \gamma_U$ elim. rate of Mtb by $M_U$ $1.2* 10^{-9} - 1.2*10^{-7}$ day$^{-1}$ [30]
$K$ carrying cap. of Mtb in the gran. $10^8-10^9$ bacteria [7]
$\bar k_I$ growth rate of T cells $8*10^{-3}$ day$^{-1}$ [11]
$T_{max}$ maximum recruitment of T cells 5.000 day$^{-1}$ [11]
$\mu_T$ natural death rate of T cells 0.33 day$^{-1}$ [35,30]
$\bar r$ Average Mtb released by one $M_U$ 0.05-0.2 day$^{-1}$ [30,35]
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Eduardo Ibargüen-Mondragón Lourdes Esteva Edith Mariela Burbano-Rosero | CommonCrawl |
\begin{document}
\title{Symmetry via antisymmetric maximum principles in nonlocal problems of variable order} \begin{abstract} We consider the nonlinear problem \[
(P)\qquad\left\{\begin{aligned}
I u&=f(x,u)&& \text{ in $\Omega$,}\\ u&=0&&\text{ on $\mathbb{R}^{N}\setminus\Omega$}\\
\end{aligned}\right. \]
in an open bounded set $\Omega\subset\mathbb{R}^{N}$, where $I$ is a nonlocal operator which may be anisotropic and may have varying order. We assume mild symmetry and monotonicity assumptions on $I$, $\Omega$ and the nonlinearity $f$ with respect to a fixed direction, say $x_1$, and we show that any nonnegative weak solution $u$ of $(P)$ is symmetric in $x_1$. Moreover, we have the following alternative: Either $u\equiv 0$ in $\Omega$, or $u$ is strictly decreasing in $|x_1|$. The proof relies on new maximum principles for antisymmetric supersolutions of an associated class of linear problems. \end{abstract}
{\footnotesize \begin{center} \textit{Keywords.} Nonlocal Operators $\cdot$ Maximum Principles $\cdot$ Symmetries \end{center} \begin{center}
\end{center} } \section{Introduction} In this work we study the following class of nonlocal and semilinear Dirichlet problems in a bounded open set $\Omega \subset \mathbb{R}^N$: \[
(P)\qquad\left\{\begin{aligned}
I u&=f(x,u)&& \text{ in $\Omega$;}\\ u&=0&&\text{ on $\mathbb{R}^{N}\setminus\Omega$.}
\end{aligned}\right. \] Here the nonlinearity $f:\Omega\times \mathbb{R}\to\mathbb{R}$ is a measurable function with properties to be specified later, and $I$ is a nonlocal linear operator. Due to various applications in physics, biology and finance with anomalous diffusion phenomena, nonlocal problems have gained enormous attention recently. In particular, problem $(P)$ has been studied with $I=(-\Delta)^{\frac{\alpha}{2}}$, the fractional Laplacian of order $\alpha \in (0,2)$. In this case, special properties of the fractional Laplacian have been used extensively to study existence, regularity and symmetry of solutions to $(P)$. In particular, some approaches rely on available Green function representions associated with $(-\Delta)^{\frac{\alpha}{2}}$, (see e.g. \cite{BB00,BLW05,CFY13,Chen_Li_Ou,FW13-2,Chen_Song}), whereas other techniques are based on a representation of $(-\Delta)^{\frac{\alpha}{2}}$ as a Dirichlet-to-Neumann map (see e.g \cite{CS07,CS,FLS13}). These useful features of the fractional Laplacian are closely linked to its isotropy and its scaling laws. However, in the modeling of anisotropic diffusion phenomena and of processes which do not exhibit similar properties, it is necessary to study more general nonlocal operators $I$. In this spirit, general classes of nonlocal operators have been considered e.g. in \cite{FK12,FKV13, SV}.\\
In the present work we consider $(P)$ for a class of nonlocal operators $I$ which includes the fractional Laplacian but also more general operators which may be anisotropic and may have varying order. More precisely, the class of operators $I$ in $(P)$ is related to nonnegative nonlocal bilinear forms of the type
\begin{equation}
\label{eq:def-cJ} {\mathcal J}(u,v)=\frac{1}{2}\int_{\mathbb{R}^N}\int_{\mathbb{R}^{N}}(u(x)-u(y))(v(x)-v(y))J(x-y)\ dxdy
\end{equation} with a measurable function $J:\mathbb{R}^N \setminus \{0\} \to[0,\infty)$. We assume that $J$ is even, i.e, $J(-z)=J(z)$ for $z \in \mathbb{R}^{N} \setminus \{0\}$. Moreover, we assume the following integral condition: $$ (J1) \qquad \quad \quad
\int_{\mathbb{R}^{N}\setminus B_{1}(0)} J(z)\ dz + \int_{B_{1}(0)} |z|^2 J(z)\ dz < \infty \qquad \quad \text{and} \qquad \quad \int_{\mathbb{R}^{N}}J(z)\ dz = \infty. \qquad\quad $$
By similar arguments as in the recent paper \cite{FKV13}, we shall see in Section~\ref{setup} below that this assumption ensures that ${\mathcal J}$ is closed and symmetric quadratic form in $L^2(\Omega)$ with a dense domain given by \begin{equation}
\label{eq:def-cD} {\mathcal D}(\Omega):=\{\text{$u:\mathbb{R}^{N}\to\mathbb{R}$ measurable}\;:\; {\mathcal J}(u,u)<\infty \text{ and $u\equiv 0$ on $\mathbb{R}^{N}\setminus \Omega$}\} \end{equation} Here and in the following, we identify $L^2(\Omega)$ with the space of functions $u \in L^2(\mathbb{R}^N)$ with $u \equiv 0$ on $\mathbb{R}^N \setminus \Omega$. Consequently, ${\mathcal J}$ is the quadratic form of a unique self-adjoint operator $I$ on $L^2(\Omega)$, which also satisfies $$
[I u](x)= \lim_{\varepsilon \to 0} \int_{|y-x|\ge \varepsilon} [u(x)-u(y)]J(x-y)\,dy \qquad \text{for $u \in {\mathcal C}^2_c(\Omega)$, $x \in \mathbb{R}^N$} $$ see Corollary~\ref{3-dense} below. One may study solutions $u$ of $(P)$ in strong sense, requiring that $u$ is contained in the domain of the operator $I$. However, it is more natural to consider the weaker notion of solutions given by the quadratic form ${\mathcal J}$ itself. More precisely, we call a function $u\in {\mathcal D}(\Omega)$ {\em a solution of $(P)$} if the integral $\int_{\Omega}f(x,u(x))\varphi(x)\ dx$ exists for all $\varphi\in {\mathcal D}(\Omega)$ and \[ {\mathcal J}(u,\varphi)=\int_{\Omega}f(x,u(x))\varphi(x)\ dx \qquad \text{for all $\varphi\in {\mathcal D}(\Omega)$,} \]
We note that the fractional Laplacian $I:= (-\Delta)^{\alpha/2}$ corresponds to the kernel $J(z)= c_{N,\alpha} |z|^{-N-\alpha}$ with $c_{N,\alpha}= \alpha(2-\alpha)\pi^{-N/2}2^{\alpha-2}\frac{\Gamma(\frac{N+\alpha}{2})}{\Gamma(2-\frac{\alpha}{2})}$. Our paper is motivated by recent symmetry results for nonlinear equations involving the fractional Laplacian (see \cite{BLW05,CFY13,Chen_Li_Ou,FW13-2,JW13,sciunzi}). More precisely, we present a general approach, based on maximum principles for antisymmetric functions, to investigate symmetry properties of bounded nonnegative solutions of $(P)$ in bounded Steiner symmetric open sets $\Omega$. We claim that this approach is simpler and more general than the techniques applied in the papers cited above. In particular, it also applies to anisotropic operators and operators of variable order. To state our main symmetry result, we first introduce the following geometric assumptions on $J$ and the set $\Omega$. \begin{enumerate} \item[$(D)$] $\Omega\subset \mathbb{R}^N$ is an open bounded set which is Steiner symmetric in $x_1$, i.e. for every $x\in \Omega$ and $s \in [-1,1]$ we have $(sx_1,x_2,\dots,x_N) \in \Omega$.
\item[$(J2)$] The kernel $J$ is strictly monotone in $x_1$, i.e. for all $z' \in\mathbb{R}^{N-1}$, $s,t \in \mathbb{R}$ with $|s| <|t|$ we have $J(s,z') > J(t,z')$. \end{enumerate} Note that $(J2)$ in particular implies that $J$ is positive on $\mathbb{R}^N \setminus \{0\}$. We may now state our main symmetry result.
\begin{satz}\label{sec:goal}
Let $(J1),(J2)$ and $(D)$ be satisfied, and assume that the nonlinearity $f$ has the following properties. \begin{enumerate} \item[$(F1)$] $f: \Omega\times \mathbb{R}\to \mathbb{R}$, $(x,u)\mapsto f(x,u)$ is a Carath\'eodory function such that for every bounded set $K \subset \mathbb{R}$ there exists $L=L(K)>0$ with \[
\sup \limits_{x\in\Omega} |f(x,u)-f(x,v)|\leq L |u-v| \quad\text{ for $u,v \in K$.} \]
\item[$(F2)$] $f$ is symmetric and monotone in $x_1$, i.e. for every $u \in \mathbb{R}$, $x \in \Omega$ and $s \in [-1,1]$ we have $f(s x_{1},x_2,\dots,x_N,u) \ge f(x,u)$. \end{enumerate}
Then every nonnegative solution $u \in L^\infty(\Omega) \cap {\mathcal D}(\Omega)$ of $(P)$ is symmetric in $x_1$. Moreover, either $u \equiv 0$ in $\mathbb{R}^{N}$, or $u$ is strictly decreasing in $|x_1|$ and therefore satisfies \begin{equation}
\label{eq:positivity-thm-1-1} \underset{K}{\essinf}\, u>0\qquad \text{for every compact set $K \subset \Omega$.} \end{equation} \end{satz}
\noindent Here and in the following, if $\Omega$ satisfies $(D)$ and $u: \Omega \to \mathbb{R}$ is measurable, we say that $u$ is\\[0.1cm] $\bullet$ {\em symmetric in $x_1$} if $u(-x_{1},x')=u(x_{1},x')$ for almost every $x= (x_{1},x')\in \Omega$.\\[0.1cm]
$\bullet$ {\em strictly decreasing in $|x_1|$} if for every $\lambda \in \mathbb{R} \setminus \{0\}$ and every compact set $K \subset \{x \in \Omega\::\: \frac{x_1}{\lambda} > 1\}$ we have $$ \underset{x \in K}{\essinf}\, \bigl[u(2\lambda-x_1, x_2,\dots,x_N) -u(x)\bigr] >0. $$
\begin{bem} \label{sec:ex-op} We wish to single out a particular class of operators satisfying $(J1)$ and $(J2)$. Let $\alpha,\beta \in (0,2)$, $c>0$ and consider a measurable map $k:(0,\infty)\to(0,\infty)$ such that \[ \frac{\rho^{-N}}{c} \le k(\rho)\le c\rho^{-N-\alpha}\quad \text{for $\rho \le 1\qquad$ and}\qquad k(\rho) \le c\rho^{-N-\beta}\quad\text{for $\rho >1$.}\\
\] Suppose moreover that $k$ is strictly decreasing on $(0,\infty)$, and let
$|\cdot|_\sharp$ denote a norm on $\mathbb{R}^N$ with the property that $|(s,z')|_\sharp<|(t,z')|_\sharp$ for every $s,t \in \mathbb{R}$ with $|s|<|t|$ and $z' \in \mathbb{R}^{N-1}$. Then the kernel $$
J: \mathbb{R}^N \setminus \{0\} \to \mathbb{R},\qquad J(z)= k(|z|_\sharp) $$
satisfies $(J1)$ and $(J2)$. As remarked before, the case where $|\cdot|_{\sharp}=|\cdot|$ is the euclidean norm on $\mathbb{R}^N$ and $k(\rho)= c_{N,\alpha} \rho^{-N-\alpha}$ corresponds to the fractional Laplacian $I=(-\Delta)^{\alpha/2}$. The class defined here also includes operators of order varying between $0$ and $\alpha \in (0,2)$. In particular, zero order operators are admissible. Moreover, the choice of non-euclidean norms $|\cdot|_{\sharp}$ leads to anisotropic operators. In particular, for $1 \le p < \infty$, the norm \begin{equation}
\label{eq:norm-anisotropic}
|x|_\sharp = |x|_p:= \bigl(\sum_{i=1}^N |x_i|^p\bigr)^{1/p} \qquad \text{for $x \in \mathbb{R}^N$} \end{equation} has the required properties. \end{bem}
As a direct consequence of Theorem \ref{sec:goal} we have the following. Here $e_j \in \mathbb{R}^N$ denotes the $j$-th coordinate vector for $j=1,\dots,N$. \begin{cor}\label{cor-goal}
Let $J(z)=k(|z|_p)$, where $k$ is as in Remark~\ref{sec:ex-op}, $1 \le p< \infty$ and $|\cdot|_p$ is given in (\ref{eq:norm-anisotropic}).\\[0.1cm] (i) Let $\Omega\subset\mathbb{R}^{N}$ be Steiner symmetric in $x_1,\dots,x_N$ , i.e., for every $x\in \Omega$, $j=1,\dots,N$ and $s \in [0,2]$ we have $x- s x_j e_j \in \Omega$. Moreover, let $f$ fulfill $(F1)$ and be symmetric and monotone in $x_1,\dots,x_N$, i.e. for every $u \in \mathbb{R}$, $x \in \Omega$, $j=1,\dots,N$ and $s \in [0,2]$ we have $f(x-s x_j e_j ,u) \ge f(x,u)$. Then every nonnegative solution $u\in L^\infty(\Omega) \cap {\mathcal D}(\Omega)$ of $(P)$ is symmetric in $x_1,\dots,x_N$. Moreover,
either $u \equiv 0$ in $\mathbb{R}^{N}$, or $u$ is strictly decreasing in $|x_1|,\dots,|x_N|$ and therefore satisfies (\ref{eq:positivity-thm-1-1}).\\[0.1cm]
(ii) If $p=2$, $\Omega\subset\mathbb{R}^{N}$ is a ball centered in $0$ and $f$ fulfills $(F1)$, $(F2)$ and is radial in $x$ i.e. $f(x,u)=f(|x|e_1,u)$ for $x\in \Omega$, then every nonnegative solution $u\in L^\infty(\Omega) \cap {\mathcal D}(\Omega)$ of $(P)$ is radially symmetric.
Moreover, either $u \equiv 0$ in $\mathbb{R}^{N}$, or $u$ is strictly decreasing in $|x|$ and therefore satisfies~(\ref{eq:positivity-thm-1-1}). \end{cor} In the special case where $I=(-\Delta)^{\frac{\alpha}{2}}$, $\alpha\in(0,2)$, Theorem~\ref{sec:goal} has been obtained by the authors in \cite[Corollary 1.2]{JW13} as a corollary of result on asymptotic symmetry for the corresponding parabolic problem. While some of the parabolic estimates in \cite{JW13} are not available for the class of nonlocal operators considered here, we will be able to formulate elliptic counterparts of some of the tools from \cite{JW13} in the present setting. Independently from our work \cite{JW13}, a weaker variant of Theorem~\ref{sec:goal} in the special case $I=(-\Delta)^{\frac{\alpha}{2}}$, restricted to strictly positive solutions, is proved in the very recent preprint \cite[Theorem 1.2]{sciunzi}, where also related problems for the fractional Laplacian with singular local linear terms are considered. Corollary \ref{cor-goal}(ii) for $I=(-\Delta)^{\frac{\alpha}{2}}$, $\alpha\in(0,2)$ has been proved first by Birkner, L\'opez-Mimbela and Wakolbinger \cite{BLW05} for $I=(-\Delta)^{\frac{\alpha}{2}}$ and a nonlinearity $f=f(u)$ which is nonnegative and increasing. In the very recent papers \cite{CFY13,FW13-2}, Corollary \ref{cor-goal}(ii) is proved for strictly positive solutions in the case $I=(-\Delta)^{\frac{\alpha}{2}}$ under different assumptions on $f$. The proofs in these papers rely on the explicit form of the Green function associated with $(-\Delta)^{\frac{\alpha}{2}}$ in balls.\\ In order to explain the difference between considering nonnegative or positive solutions, we point out that the conclusion~(\ref{eq:positivity-thm-1-1}) can be seen as a strong maximum principle for bounded solutions of $(P)$ in open sets satisfying $(D)$ which is {\em not} true for the corresponding Dirichlet problem \begin{equation}
\label{class}
\left\{\begin{aligned}
-\Delta u&=f(x,u)&& \text{ in $\Omega$;}\\ u&=0&&\text{ on $\partial \Omega$.}
\end{aligned}\right.
\end{equation} Note that we do not assume $\Omega$ to be connected in Theorem~\ref{sec:goal}, but even in domains $\Omega \subset \mathbb{R}^N$ the assumptions $(D)$ and $(F1)$, $(F2)$ do not guarantee that nonnegative solutions of (\ref{class}) are either strictly positive or identically zero in $\Omega$, see e.g. \cite{PT12} for examples for nonnegative solutions of (\ref{class}) with interior zeros. The positivity property (\ref{eq:positivity-thm-1-1}) can be seen as a consequence of the long range nonlocal interaction enforced by $(J2)$. Note that $(J2)$ is not satisfied for kernels of the form \begin{equation}
\label{eq:cut-off-fractional}
z \mapsto J(z)= 1_{B_r(0)}|z|^{-N-\alpha}\qquad \text{with $\alpha \in (0,2)$, $r>0$.} \end{equation} It is therefore natural to ask whether a result similar to Theorem~\ref{sec:goal} also holds for kernels of the type (\ref{eq:cut-off-fractional}) which vanish outside a compact set and therefore model short range nonlocal interaction. Related to this case, we have to following result for {\em a.e. positive solutions of $(P)$ in $\Omega$.}
\begin{satz} \label{sec:vari-symm-result-1}
Let $\Omega\subset\mathbb{R}^{N}$ satisfy $(D)$, and let the even kernel $J: \mathbb{R}^N \setminus \{0\} \to [0,\infty)$ satisfy $(J1)$ and \begin{enumerate}
\item[$(J2)'$] For all $z' \in\mathbb{R}^{N-1}$, $s,t \in \mathbb{R}$ with $|s| \le |t|$ we have $J(s,z') \ge J(t,z')$. Moreover, there is $r_0>0$ such that
$$J(s,z') > J(t,z') \qquad \text{for all $z' \in\mathbb{R}^{N-1}$ and $s,t \in \mathbb{R}$, with $|z'| \le r_0$ and $|s| <|t|\le r_0$.} $$ \end{enumerate}
Furthermore, suppose that the nonlinearity satisfies $(F1)$ and $(F2)$. Then every a.e. positive solution $u\in L^\infty(\Omega) \cap {\mathcal D}(\Omega)$ of $(P)$ is symmetric in $x_1$ and strictly decreasing in $|x_1|$ on $\Omega$. Consequently, it satisfies~(\ref{eq:positivity-thm-1-1}). \end{satz} Note that the kernel class given by (\ref{eq:cut-off-fractional}) satisfies $(J1)$ and $(J2)'$. We recall that Gidas, Ni and Nirenberg \cite{GNN79} proved the corresponding symmetry result for strictly positive solutions of (\ref{class}) under some restrictions on $\Omega$ which were then removed in \cite{BN91}. These results rely on the moving plane method which, in other variants, had already been introduced in \cite{A62,S71}. For nonlocal problems involving the fractional Laplacian, the moving plane method was used in a stochastic framework by Birkner, L\'opez-Mimbela and Wakolbinger in the above-mentioned paper \cite{BLW05}. Chen, Li and Ou \cite{Chen_Li_Ou} used the explicit form of the inverse of the fractional Laplacian to prove symmetry results for $I=(-\Delta)^{\frac{\alpha}{2}}$ and $f(u)=u^{(N+\alpha)/(N-\alpha)}$ in $\mathbb{R}^{N}$. For this they developed a variant of the moving plane method for integral equations. Similar methods were used in the above-mentioned papers \cite{CFY13,FW13-2}.\\ The results on the present paper rely on a different variant of the moving plane method which partly extends recent techniques of \cite{JW13, FJ13,RS13} and, independently, \cite{sciunzi}. More precisely, we show that $(J1)$ and $(J2)$ -- or, alternatively, $(J2)'$ -- are sufficient assumptions for the bilinear form ${\mathcal J}$ to provide maximum principles for antisymmetric solutions of associated linear operator inequalities in weak form, see Section~\ref{mp}. Here antisymmetry refers to a reflection at a given hyperplane. Combining different (weak and strong) versions of these maximum principles, we then develop a framework for the moving plane method for nonnegative solutions of $(P)$ which are not necessarily strictly positive. The approach seems more direct and more flexible than the ones in \cite{CFY13,Chen_Li_Ou,FW13-2} since it does not depend on Green function representations.\\ The paper is organized as follows. In Section \ref{setup} we collect useful properties of the nonlocal bilinear forms which we consider. Section \ref{mp} is devoted to classes of linear problems related to $(P)$ and hyperplane reflections. In particular, we prove a small volume type maximum principle and a strong maximum principle for antisymmetric supersolutions of these problems. In Section~\ref{mr} we complete the proof of Theorem~\ref{sec:goal}, and in Section~\ref{sec:vari-symm-result} we complete the proof of Theorem~\ref{sec:vari-symm-result-1}.\\
\textbf{Acknowledgment:} Part of this work was done while the first author was visiting AIMS-Senegal. He would like to thank them for their kind hospitality.
\section{Preliminaries}\label{setup}
We fix some notation. For subsets $D,U \subset \mathbb{R}^N$ we write $\textnormal{dist}(D,U):= \inf\{|x-y|\::\: x \in D,\, y \in U\}$. If $D= \{x\}$ is a singleton, we write $\textnormal{dist}(x,U)$ in place of $\textnormal{dist}(\{x\},U)$. For $U\subset\mathbb{R}^{N}$ and $r>0$ we consider $B_{r}(U):=\{x\in\mathbb{R}^{N}\;:\; \textnormal{dist}(x,U)<r\}$, and we let, as usual
$B_r(x)=B_{r}(\{x\})$ be the open ball in $\mathbb{R}^{N}$ centered at $x \in \mathbb{R}^N$ with radius $r>0$. For any subset $M \subset \mathbb{R}^N$, we denote by $1_M: \mathbb{R}^N \to \mathbb{R}$ the characteristic function of $M$ and by $\textnormal{diam}(M)$ the diameter of $M$. If $M$ is measurable $|M|$ denotes the Lebesgue measure of $M$. Moreover, if $w: M \to \mathbb{R}$ is a function, we let $w^+= \max\{w,0\}$ resp. $w^-=-\min\{w,0\}$ denote the positive and negative part of $w$, respectively.
Throughout the remainder of the paper, we assume that $J:\mathbb{R}^N \setminus \{0\}\to[0,\infty)$ is even and satisfies $(J1)$. We let ${\mathcal J}$ be the corresponding quadratic form defined in (\ref{eq:def-cJ}) and, for an open set $\Omega \subset \mathbb{R}$, we consider ${\mathcal D}(\Omega)$ as defined in (\ref{eq:def-cD}). It follows from $(J1)$ that $J$ is positive on a set of positive measure. Thus, by \cite[Lemma 2.7]{FKV13} we have ${\mathcal D}(\Omega) \subset L^2(\Omega)$ and \begin{equation} \label{l2-bound}
\Lambda_{1}(\Omega):=\inf_{u\in {\mathcal D}(\Omega)}\frac{{\mathcal J}(u,u)}{\|u\|^2_{L^{2}(\Omega)}} \: >\:0 \qquad \text{for every open bounded set $\Omega\subset\mathbb{R}^{N}$,} \end{equation}
which amounts to a Poincar\'e-Friedrichs type inequality. We will need lower bounds for $\Lambda_1(\Omega)$ in the case where $|\Omega|$ is small. For this we set $$
\Lambda_1(r):= \inf \{ \Lambda_{1}(\Omega)\::\: \text{$\Omega \subset \mathbb{R}^N$ open, $|\Omega|=r$}\} \qquad \text{for $r>0$.} $$
\begin{lemma}\label{3-mengen-k} We have $\Lambda_{1}(r) \to \infty$ as $r \to 0$. \end{lemma}
\begin{proof} Let $$
J_c:= \{ z \in \mathbb{R}^N \setminus \{0\}\::\: J(z) \ge c\} \qquad \text{and}\qquad J^c:= \{ z \in \mathbb{R}^N \setminus \{0\}\::\: J(z) < c\} $$
for $c \in [0,\infty]$. We also consider the decreasing rearrangement $d:(0,\infty) \to [0,\infty]$ of $J$ given by $d(r)= \sup \{c \ge 0 \::\: |J_c| \ge r \}$. We first note that \begin{equation}
\label{eq:est-dec-rearr-3}
|J_{d(r)}| \ge r \qquad \text{for every $r>0$} \end{equation}
Indeed, this is obvious if $d(r)=0$, since $J_0= \mathbb{R}^N \setminus \{0\}$. If $d(r)>0$, we have $|J_c| \ge r$ for every $c < d(r)$ by definition, whereas $|J_c| < \infty$ for every $c>0$ as a
consequence of the fact that $J \in L^{1}(\mathbb{R}^{N}\setminus B_1(0))$ by $(J1)$. Consequently, since $J_{d(r)}= \underset{c < d(r)}{\bigcap} J_c$, we have $|J_{d(r)}| = \inf \limits_{c < d(r)} |J_c| \ge r.$ Next we claim that \begin{equation}
\label{eq:est-dec-rearr} \Lambda_1(r) \ge \int_{J^{d(r)}} J(z)\,dz \qquad \text{for $r>0$.} \end{equation}
Indeed, let $r>0$ and $\Omega \subset \mathbb{R}^N$ be measurable with $|\Omega|=r$. For $u\in {\mathcal D}(\Omega)$ we have \begin{align} {\mathcal J}(u,u)&=\frac{1}{2}\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}(u(x)-u(y))^2J(x-y)\ dxdy \nonumber \\ &=\frac{1}{2}\int_{\Omega}\int_{\Omega}(u(x)-u(y))^2 J(x-y)\ dxdy+\int_{\Omega}u^2(x)\int_{\mathbb{R}^{N}\setminus \Omega} J(x-y)\ dy \ dx \nonumber\\
&\geq \inf_{x\in \Omega}\biggl(\;\int_{\;\mathbb{R}^{N}\setminus \Omega_x} J(y)\ dy\biggr)\|u\|^2_{L^{2}(\Omega)} \label{eq:est-dec-rearr-4} \end{align}
with $\Omega_x:=x+\Omega$. Let $d:=d(r)$. Since $|J_{d}| \ge r = |\Omega|$ by (\ref{eq:est-dec-rearr-3}), we have $|J_d \setminus \Omega_x| \ge |\Omega_x \setminus J_d|$ and thus, for every $x \in \Omega$, \begin{align*}
\int_{\mathbb{R}^{N}\setminus \Omega_x} J(y)\ dy&=\int_{\mathbb{R}^N \setminus J_d} J(y)\ dy +\int_{J_d\setminus \Omega_x}J(y)\ dy- \int_{\Omega_{x}\setminus J_d}J(y)\ dy\\
&\ge \int_{J^d} J(y)\ dy + \Bigl(|J_d\setminus \Omega_x| -
|\Omega_{x}\setminus J_d|\Bigr)d \geq \int_{J^d} J(y)\ dy. \end{align*} Combining this with (\ref{eq:est-dec-rearr-4}), we obtain (\ref{eq:est-dec-rearr}), as claimed. As a consequence of the second property in $(J1)$, the decreasing rearrangement of $J$ satisfies $d(r) \to \infty$ as $r \to 0$ and $$ \int_{J^{d(r)}} J(y)\ dy \to \infty \qquad \text{as $r \to 0$.} $$ Together with (\ref{eq:est-dec-rearr}), this shows the claim. \end{proof}
\begin{prop} \label{complete} Let $\Omega \subset \mathbb{R}^N$ be open and bounded. Then ${\mathcal D}(\Omega)$ is a Hilbert space with the scalar product ${\mathcal J}$.
\end{prop}
\begin{proof} We argue similarly as in the proof of \cite[Lemma 2.3]{FKV13}. Let $(u_n)_n\subset{\mathcal D}(\Omega)$ be a Cauchy sequence. By (\ref{l2-bound}) and the completeness of $L^2(\Omega)$, we have that $u_n\to u\in L^{2}(\Omega)$ for a function $u\in L^{2}(\Omega)$. Hence there exists a subsequence such that $u_{n_k}\to u$ almost everywhere in $\Omega$ as $k \to \infty$. By Fatou's Lemma, we therefore have that \[
{\mathcal J}(u,u)\leq \liminf_{k\to\infty}{\mathcal J}(u_{n_k},u_{n_k})\leq \sup_{k\in \mathbb{N}}{\mathcal J}(u_{n_k},u_{n_k})<\infty, \] so that $u \in {\mathcal D}(\Omega)$. Applying Fatou's Lemma again, we find that $$ {\mathcal J}(u_{n_k}-u,u_{n_k}-u) \le \liminf_{j \to \infty} {\mathcal J}(u_{n_k}-u_{n_j},u_{n_k}-u_{u_j}) \le \sup_{j \ge k} {\mathcal J}(u_{n_k}-u_{n_j},u_{n_k}-u_{u_j}) \qquad \text{for $k \in \mathbb{N}$}. $$ Since $(u_{n})_n$ is a Cauchy sequence with respect to the scalar product ${\mathcal J}$, it thus follows that $\lim \limits_{k \to \infty}u_{n_k} = u$ and therefore also $\lim \limits_{n \to \infty}u_{n} = u$ in ${\mathcal D}(\Omega)$. This shows the completeness of ${\mathcal D}(\Omega)$.
\end{proof}
\begin{prop}\label{3-prel-dense} (i) We have ${\mathcal C}^{0,1}_{c}(\mathbb{R}^{N}) \subset {\mathcal D}(\mathbb{R}^N)$.\\[0.1cm] (ii) Let $v \in {\mathcal C}_c^2(\mathbb{R}^N)$. Then the principle value integral \begin{equation}
\label{eq:princ-value}
[Iv](x):= P.V. \int_{\mathbb{R}^N} (v(x)-v(y)) J(x-y)\,dy = \lim_{\varepsilon \to 0} \int_{|x-y|\ge \varepsilon} (v(x)-v(y)) J(x-y)\,dy \end{equation} exists for every $x \in \mathbb{R}^N$. Moreover, $I v \in L^\infty(\mathbb{R}^N)$, and for every bounded open set $\Omega \subset \mathbb{R}^N$ and every $u \in {\mathcal D}(\Omega)$ we have $$ {\mathcal J}(u,v)= \int_{\mathbb{R}^N} u(x) [Iv](x)\,dx. $$
\end{prop}
\begin{proof} (i) Let $u\in {\mathcal C}^{0,1}_{c}(\mathbb{R}^{N})$, and let $K>0$, $R>2$ be such that $\textnormal{supp}(u) \subset B_{R-2}(0)$, $$
|u(x)|\leq K \quad \text{and}\quad |u(x)-u(y)|\leq K|x-y| \qquad \text{for $x,y \in \mathbb{R}^N$, $x \not=y$.} $$ Then, as a consequence of $(J1)$, \begin{align*} 2{\mathcal J}(u,u)&=\int_{ B_{R}(0)}\int_{ B_{R}(0)}(u(x)-u(y))^2 J(x-y)\ dxdy +2\int_{ B_{R}(0)}u^{2}(x)\int_{\mathbb{R}^{N}\setminus B_{R}(0)}J(x-y)\ dydx\\
&\leq K^2 \int_{ B_{R}(0)}\int_{ B_{R}(0)}|x-y|^{2} J(x-y)\ dx dy +2K^2 \int_{ B_{R-2}(0)}\;\int_{\mathbb{R}^{N}\setminus B_{R}(0)}J(x-y)\ dydx\\
&\leq 2 K^2|B_R(0)|\Bigl( \int_{ B_{2R}(0)}|z|^{2} J(z)\ dz + \int_{\mathbb{R}^{N}\setminus B_{1}(0)}J(z)\ dz \Bigr)<\infty \end{align*} and thus $u \in {\mathcal D}(\mathbb{R}^N)$.\\ (ii) Since $v\in {\mathcal C}^{2}_{c}(\mathbb{R}^{N})$, there exist constants $\delta,K>0$ such that \begin{equation}
\label{eq:C2-local-est}
|2v(x)-v(x+z)-v(x-z)|\leq K |z|^2 \qquad \text{for all $x,z \in \mathbb{R}^N$ with $|z| \le \delta$.} \end{equation} Put $h(x,y):= (v(x)-v(y))J(x-y)$ for $x,y \in \mathbb{R}^N$, $x \not = y$. For every $x \in \mathbb{R}^N$, $\varepsilon \in (0,\delta)$ we then have, since $J$ is even, \begin{align*}
\int_{\varepsilon \le |y-x|\le \delta} h(x,y)\,dy &= \int_{\varepsilon \le |z| \le \delta} [v(x)-v(x+z)] J(z)\,dz= \int_{\varepsilon \le |z| \le \delta} [v(x)-v(x-z)] J(z)\,dz\\
& = \frac{1}{2} \int_{\varepsilon \le |z| \le \delta} [2 v(x)-v(x+z)-v(x-z)]J(z)\,dz. \end{align*} By the first inequality in $(J1)$, (\ref{eq:C2-local-est}) and Lebesgue's theorem we thus conclude the existence of the limit $$
\lim_{\varepsilon \to 0} \int_{\varepsilon \le |y-x|\le \delta}h(x,y)\,dy =\frac{1}{2} \int_{0 \le |z| \le \delta} [2 v(x)-v(x+z)-v(x-z)]J(z)\,dz. $$ Moreover we have for $x \in \mathbb{R}^N$ and $\varepsilon \in (0,\delta)$ \begin{equation}
\label{eq:extra-pf-3-prel-dense}
\int_{|y-x| \ge \varepsilon} h(x,y) \,dy \le 2 \|v\|_{L^\infty(\mathbb{R}^N)} \int_{\mathbb{R}^N \setminus B_\delta(0)}J(z)\,dz + \frac{K}{2}\int_{B_\delta(0)} |z|^2 J(z)\,dz =:K', \end{equation}
where the right hand side is finite by the first inequality in $(J1)$. In particular, $[Iv](x)$ is well defined by (\ref{eq:princ-value}), and $|[Iv](x)\bigr | \le K'$ for $x \in \mathbb{R}^N$, so that $Iv \in L^\infty(\mathbb{R}^N)$. Next, let $\Omega \subset \mathbb{R}^N$ be open and bounded and $u\in {\mathcal D}(\Omega)$, so that also $u \in L^2(\Omega)$. Then we have, by (\ref{eq:extra-pf-3-prel-dense}) and Lebesgue's Theorem, \begin{align*}
{\mathcal J}&(u,v)= \frac{1}{2} \lim_{\varepsilon \to 0} \int_{|x-y|\ge \varepsilon} (u(x)-u(y))h(x,y) \,dx \,dy\\
&= \lim_{\varepsilon \to 0} \int_{\mathbb{R}^N} u(x) \int_{|y-x| \ge \varepsilon} h(x,y) \,dy dx = \int_{\mathbb{R}^N} u(x)
\Bigl[\: \lim_{\varepsilon \to 0} \int_{|y-x| \ge \varepsilon} h(x,y) \,dy\Bigr] dx =\int_{\mathbb{R}^N} u(x) [Iv](x)\,dx. \end{align*}
The proof is finished. \end{proof}
\begin{cor}\label{3-dense} Let $\Omega\subset \mathbb{R}^{N}$ be open and bounded. Then ${\mathcal J}$ is a closed quadratic form with dense form domain ${\mathcal D}(\Omega)$ in $L^{2}(\Omega)$. Consequently, ${\mathcal J}$ is the quadratic form of a unique self-adjoint operator $I$ in $L^2(\Omega)$. Moreover, $C_c^2(\Omega)$ is contained in the domain of $I$, and for every $v \in {\mathcal C}_c^2(\Omega)$ the function $Iv \in L^2(\Omega)$ is a.e. given by (\ref{eq:princ-value}). \end{cor}
\begin{proof}
Since ${\mathcal C}^{0,1}_{c}(\Omega) \subset L^2(\Omega)$ is dense, ${\mathcal D}(\Omega)$ is a dense subset of $L^2(\Omega)$ by Proposition \ref{3-prel-dense}(i). Moreover, the quadratic form ${\mathcal J}$ is closed in $L^2(\Omega)$ as a consequence of (\ref{l2-bound}) and Lemma~\ref{complete}. Hence ${\mathcal J}$ is the quadratic form of a unique self-adjoint operator $I$ in $L^2(\Omega)$ (see e.g. \cite[Theorem VIII.15, pp. 278]{SR}). Moreover, for every $v \in {\mathcal C}_c^2(\Omega)$, $u \in {\mathcal D}(\Omega)$ we have $|J(u,v)| \le |\Omega| \|Iv\|_{L^\infty(\Omega)} \|u\|_{L^2(\Omega)}$ by Proposition~\ref{3-prel-dense}(ii). Consequently, $v$ is contained in the domain of $I$ and satisfies $J(u,v) = \int_{\mathbb{R}^N}u [Iv]\,dx$ for every $u \in {\mathcal D}(\Omega)$. From Proposition~\ref{3-prel-dense}(ii) it then follows that $Iv$ is a.e. given by (\ref{eq:princ-value}). \end{proof}
Next, we wish to extend the definition of ${\mathcal J}(v,\varphi)$ to more general pairs of functions $(v,\varphi)$. In the following, for a measurable subset $U' \subset \mathbb{R}^N$, we define ${\mathcal H}(U')$ as the space of all functions $v \in L^2(\mathbb{R}^N)$ such that \begin{equation}\label{subset} \rho(v,U'):= \int_{U'}\int_{U'}(v(x)-v(y))^2J(x-y)\ dxdy<\infty. \end{equation} Note that ${\mathcal D}(\mathbb{R}^N) \cap L^2(\mathbb{R}^N) \subset {\mathcal H}(U')$ for any measurable subset $U' \subset \mathbb{R}^N$, and thus also ${\mathcal D}(U) \subset {\mathcal H}(U')$ for any bounded open set $U \subset \mathbb{R}^N$ by (\ref{l2-bound}).
\begin{lemma} \label{sec:linear-problem-tech-1} Let $U' \subset \mathbb{R}^N$ be an open set and $v,\varphi \in {\mathcal H}(U')$. Moreover, suppose that $\varphi \equiv 0$ on $\mathbb{R}^N \setminus U$ for some subset $U \subset U'$ with $\textnormal{dist}(U,\mathbb{R}^N \setminus U')>0$. Then \begin{equation}
\label{eq:finite-int}
\int_{\mathbb{R}^N} \int_{\mathbb{R}^N} |v(x)-v(y)| |\varphi(x)-\varphi(y)|J(x-y) \,dx dy < \infty, \end{equation} and thus $$ {\mathcal J}(v,\varphi) := \frac{1}{2} \int_{\mathbb{R}^N} \int_{\mathbb{R}^N} (v(x)-v(y)) (\varphi(x)-\varphi(y))J(x-y) \,dx dy $$ is well defined. \end{lemma}
\begin{proof} Since $J$ satisfies $(J1)$, we have $K:= \int_{\text{\tiny $\mathbb{R}^N \setminus B_r(0)$}}J(z)\,dz < \infty $ with $r:= \textnormal{dist}(U,\mathbb{R}^N \setminus U')>0$. As a consequence, \begin{align*}
\int_{\mathbb{R}^N}& \int_{\mathbb{R}^N} |v(x)-v(y)| |\varphi(x)-\varphi(y)|J(x-y) \,dx dy\\
&= \int_{U'}\int_{U'}|v(x)-v(y)||\varphi(x)-\varphi(y)|J(x-y)\ dxdy + 2 \int_{U}\int_{\mathbb{R}^{N}\setminus U'} |v(x)-v(y)| |\varphi(x)|J(x-y)\ dydx\\
& \le \frac{1}{2} \bigl[\rho(v,U') + \rho(\varphi,U')\bigr] + \int_{U}\int_{\mathbb{R}^{N}\setminus U'} \Bigl[2 \bigl(|v(x)|^2+ |v(y)|^2\bigr) + |\varphi(x)|^2\Bigr] J(x-y)\ dydx\\
& \le \frac{1}{2} \bigl[\rho(v,U') + \rho(\varphi,U')\bigr] + K \Bigl( 4 \|v\|_{L^2(\mathbb{R}^N)}^2+ \|\varphi\|_{L^2(\mathbb{R}^N)}^2\Bigr)<\infty. \end{align*}
\end{proof}
\begin{lemma}\label{3-cutoff} If $U' \subset \mathbb{R}^N$ is open and $v\in {\mathcal H}(U')$, then $v^\pm \in {\mathcal H}(U')$ and $\rho(v^\pm,U') \le \rho(v,U')$. \end{lemma} \begin{proof} We have $v^\pm \in L^2(\mathbb{R}^N)$ since $v \in L^2(\mathbb{R}^N)$. Moreover, $v^+(x) v^-(x) =0$ for $x \in \mathbb{R}^N$ and thus \begin{align*} &\rho(v,U')= \rho(v^+,U') + \rho(v^-,U') - 2\int_{U'}\int_{U'} (v^+(x)-v^+(y))(v^-(x)-v^-(y))J(x-y)\ dxdy\\ &=\rho(v^+,U') + \rho(v^-,U') +2 \int_{U'}\int_{U'}[v^{+}(x)v^{-}(y)+v^{+}(y)v^{-}(x)] J(x-y)\ dxdy\\ &\geq \rho(v^+,U') + \rho(v^-,U'). \end{align*} The claim follows. \end{proof}
We close this section with a remark on assumption $(J2)$.
\begin{bem} \label{sec:equality-monotonicity}
Suppose that $(J2)$ is satisfied. Then, for every fixed $z' \in \mathbb{R}^N$, the function $t \mapsto J(t,z')$ is strictly decreasing in $|t|$ and therefore coincides a.e. on $\mathbb{R}$ with the function $t \mapsto \tilde J(t,z'):= \lim \limits_{s \to t^-}J(s,z')$. Hence $J$ and the function $\tilde J$ differ only on a set of measure zero in $\mathbb{R}^N$. Replacing $J$ by $\tilde J$ if necessary, we may therefore deduce from $(J2)$ the symmetry property \begin{equation}
\label{eq:adj-measure} J(-t,z')= J(t,z') \qquad \text{for every $z' \in \mathbb{R}^{N-1}, t\in \mathbb{R}$.} \end{equation} This will be used in the following section. \end{bem}
\section{The linear problem associated with a hyperplane reflection}\label{mp}
In the following, we consider a fixed open affine half space $H \subset \mathbb{R}^N$, and we let $Q: \mathbb{R}^N \to \mathbb{R}^N$ denote the reflection at $\partial H$. For the sake of brevity, we sometimes write $\bar x$ in place of $Q(x)$ for $x \in \mathbb{R}^N$. A function $v:\mathbb{R}^{N}\to \mathbb{R}^{N}$ is called antisymmetric (with respect to $Q$) if $v(\bar x)=-v(x)$ for $x \in \mathbb{R}^{N}$. As before, we consider an even kernel $J: \mathbb{R}^N \setminus \{0\} \to [0,\infty)$ satisfying $(J1)$. We also assume the following symmetry and monotonicity assumptions on $J$: \begin{align} &J(\bar x-\bar y)=J(x-y) \qquad \text{for all $x,y \in \mathbb{R}^N$;} \label{sym-Q-J-1} \\ &J(x-y) \ge J(x- \bar y) \qquad \text{for all $x,y \in H$.} \label{sym-Q-J-2} \end{align}
\begin{bem}\label{symmetry-need} If $(J1)$, $(J2)$ and (\ref{eq:adj-measure}) are satisfied and $$ H= \{x \in \mathbb{R}^N\::\: x_1 > \lambda\} \qquad \text{or}\qquad H= \{x \in \mathbb{R}^N\::\: x_1 < -\lambda\} $$ for some $\lambda \ge 0$, then (\ref{sym-Q-J-1}) and (\ref{sym-Q-J-2}) hold. If $\lambda>0$, then $J$ also satisfies the following strict variant of (\ref{sym-Q-J-2}): \begin{equation}
\label{sym-Q-J-2-strict} J(x-y) > J(x- \bar y) \qquad \text{for all $x,y \in H$.} \end{equation} We will need this property in Proposition~\ref{hopf-simple2} below.
\end{bem}
\begin{lemma} \label{sec:linear-problem-tech} Let $J$ satisfy $(J1)$, (\ref{sym-Q-J-1}) and (\ref{sym-Q-J-2}). Moreover, let $U' \subset \mathbb{R}^N$ be an open set with $Q(U')=U'$, and let $v \in {\mathcal H}(U')$ be an antisymmetric function such that $v \ge 0$ on $H \setminus U$ for some open bounded set $U \subset H$ with $\overline U \subset U'$. Then the function $w:= 1_H\, v^-$ is contained in ${\mathcal D}(U)$ and satisfies \begin{equation}
\label{eq:key-ineq} {\mathcal J}(w,w) \le - {\mathcal J}(v,w) \end{equation} \end{lemma}
\begin{proof} We first show that $w \in {\mathcal H}(U')$. Clearly we have $w \in L^2(\mathbb{R}^N)$, since $v \in L^2(\mathbb{R}^N)$. Moreover, by (\ref{sym-Q-J-1}), the symmetry of $U'$, the antisymmetry of $v$ and (\ref{sym-Q-J-2}) we have \begin{align}
&\rho(v,U') =\int_{U'\cap H}\int_{U'\cap H}(v(x)-v(y))^2 J(x-y)\ dxdy \notag\\ &\qquad\qquad+\int_{U'\setminus H}\int_{U'\setminus H}(v(x)-v(y))^2 J(x-y)\ dxdy+2\int_{U'\setminus H}\int_{U'\cap H}(v(x)-v(y))^2 J(x-y)\ dxdy \notag\\ &=2\int_{U'\cap H}\int_{U'\cap H}\Bigl[(v(x)-v(y))^2 J(x-y) + (v(x)+v(y))^2 J(x-\bar y)\Bigr]\ dxdy \notag\\ &\geq \int_{U'\cap H}\int_{U'\cap H} \Bigl[(v(x)-v(y))^2 J(x-y) + [(v(x)-v(y))^2 + (v(x)+v(y))^2] J(x-\bar y)\Bigr]\ dxdy \notag\\ &\geq \int_{U'\cap H}\int_{U'\cap H} \Bigl[(v(x)-v(y))^2 J(x-y) + 2v^2(x) J(x-\bar y)\Bigr]\ dxdy \notag\\ &= \int_{U'}\int_{U'}(1_{H}v(x)-1_{H}v(y))^2 J(x-y)\ dxdy = \rho(1_{H}\,v ,U') \label{bound3} \end{align} and thus $\rho(1_{H}\,v,U') < \infty$. Hence $1_{H}\,v \in {\mathcal H}(U')$ and thus also $w \in {\mathcal H}(U')$ by Lemma~\ref{3-cutoff}. Since $w \equiv 0$ in $\mathbb{R}^N \setminus U$, the right hand side of (\ref{eq:key-ineq}) is well defined and finite by Lemma~\ref{sec:linear-problem-tech-1}. To show (\ref{eq:key-ineq}), we first note that $$ [w+v]w= [ 1_H v^+ + 1_{\mathbb{R}^N \setminus H} v]1_H v^- \equiv 0 \qquad \text{on $\mathbb{R}^N$} $$ and therefore $$ [w(x)-w(y)]^2 + [v(x)-v(y)] [w(x)-w(y)] = - \Bigl(w(x)[w(y)+v(y)] + w(y)[w(x)+v(x)]\Bigr) $$ for $x,y \in \mathbb{R}^N$. Using this identity in the following together with the antisymmetry of $v$, the symmetry properties of $J$ and the fact that $w \equiv 0$ on $\mathbb{R}^N \setminus H$, we find that \begin{align*} {\mathcal J}(w,w) + {\mathcal J}(v,w) &= -\int_{H} \int_{\mathbb{R}^N}w(x) [w(y)+v(y)] J(x-y)\,dy dx\\
&=-\int_{H} \int_{\mathbb{R}^N}w(x) [1_H(y)v^+(y) +1_{\mathbb{R}^N \setminus H} v(y)] J(x-y)\,dy dx\\ &=-\int_{H} \int_{H}w(x) [v^+(y)J(x-y) -v(y) J(x-\bar y)] \,dy dx \:\le\: 0,
\end{align*} where in the last step we used the fact that $v^+(y) \ge v(y)$ and $J(x-y) \ge J(x-\bar y) \ge 0$ for $x,y \in H$. Hence (\ref{eq:key-ineq}) is true, and in particular we have ${\mathcal J}(w,w)< \infty$. Since $w \equiv 0$ on $\mathbb{R}^N \setminus U$, it thus follows that $w \in {\mathcal D}(U)$. \end{proof}
In order to implement the moving plane method, we have to deal with the class of antisymmetric supersolutions of a class of linear problems. A related notion was introduced in \cite{JW13} in a parabolic setting related to the fractional Laplacian.
\begin{defi}\label{3-defi-anti} Let $U\subset H$ be an open bounded set and let $c\in L^{\infty}(U)$. We call an antisymmetric function $v: \mathbb{R}^N \to \mathbb{R}^N$ an \textit{antisymmetric supersolution} of the problem \begin{equation}\label{linear-prob} Iv= c(x)v\quad \text{ in $U$,}\qquad v \equiv 0 \quad \text{on $H \setminus U$} \end{equation} if $v \in {\mathcal H}(U')$ for some open bounded set $U' \subset \mathbb{R}^N$ with $Q(U')=U'$ and $\overline U \subset U'$, $v\geq 0$ on $H\setminus U$ and \begin{equation}\label{3-eq-sol2} {\mathcal J}(u,\varphi)\geq \int_{U}c(x)u(x)\varphi(x)\ dx \qquad \text{ for all $\varphi\in {\mathcal D}(U)$, $\varphi\geq0$.} \end{equation} \end{defi}
\begin{bem}\label{3-anti} Assume $(J1)$ and (\ref{sym-Q-J-1}), and let $\Omega \subset \mathbb{R}^{N}$ be an open bounded set such that $Q(\Omega \cap H)\subset \Omega$.
Furthermore, let $f: \Omega \times\mathbb{R}\to\mathbb{R}$ be a Carath\'eodory function satisfying $(F1)$ and such that
\begin{equation}
\label{eq:F2H}
f(\bar x,\tau) \geq f(x,\tau)\qquad \text{for every $\tau \in \mathbb{R}$, $x \in H\cap \Omega$.}
\end{equation} If $u \in {\mathcal D}(\Omega)$ is a nonnegative solution of $(P)$, then $v:=u\circ Q-u$ is an antisymmetric supersolution of (\ref{linear-prob}) with $U:= \Omega \cap H$ and $c \in L^\infty(U)$ defined by $$ c(x)= \left \{
\begin{aligned}
&\frac{f(x,u(\bar x))-f(x,u(x))}{v(x)}&&\qquad \text{if $v(x) \not= 0$;}\\
&0 &&\qquad \text{if $v(x)= 0$.}
\end{aligned} \right. $$ Indeed, since $u\in {\mathcal D}(\Omega)$, we have $v \in {\mathcal D}(\mathbb{R}^{N}) \cap L^2(\mathbb{R}^N)$ and thus $v \in {\mathcal H}(U')$ for any open set $U' \subset \mathbb{R}^N$. Moreover, $v \ge 0$ on $H \setminus U$ since $u$ is nonnegative and $u \equiv 0$ on $H \setminus U$. Furthermore, if $\varphi\in {\mathcal D}(U)$, then $\varphi\circ Q-\varphi\in {\mathcal D}(\Omega)$ by the symmetry properties of $J$ and since $Q(U)\subset \Omega$. If, in addition, $\varphi \ge 0$, then we have, using (\ref{sym-Q-J-1}), \begin{align*} &{\mathcal J}(v,\varphi)= {\mathcal J}(u \circ Q - u,\varphi)={\mathcal J}(u,\varphi \circ Q - \varphi)= \int_{\Omega}f(x,u)[\varphi \circ Q - \varphi]\,dx\\ &=\!\!\!\int_{ Q(U)}\!\!\!f(x,u(x))\varphi \circ Q\,dx - \int_{U}\!\!f(x,u(x))\varphi\,dx=\!\!\int_{U}[f(\bar x,u(\bar x))-f(x,u(x))]\varphi(x)\,dx \ge \int_{U}c(x) v \varphi\,dx. \end{align*} Here (\ref{eq:F2H}) was used in the last step. The boundedness of $c$ follows from $(F1)$. \end{bem}
We now have all the tools to establish maximum principles for antisymmetric supersolutions of (\ref{linear-prob}).
\begin{prop}\label{4-elliptic-max1}
Assume that $J$ satisfies $(J1)$, (\ref{sym-Q-J-1}) and (\ref{sym-Q-J-2}), and let $U \subset H$ be an open bounded set. Let $c\in L^{\infty}(U)$ with $\|c^+\|_{L^\infty(U)} <\Lambda_1(U)$, where $\Lambda_1(U)$ is given in (\ref{l2-bound}).\\ Then every antisymmetric supersolution $v$ of (\ref{linear-prob}) in $U$ satisfies $v\geq 0$ a.e. in $H$. \end{prop} \begin{proof} By Lemma \ref{sec:linear-problem-tech} we have that $w:=1_{H} v^{-} \in {\mathcal D}(U)$ and ${\mathcal J}(w,w) \le -{\mathcal J}(v,w)$. Consequently, \begin{align*}
\Lambda_{1}(U) \|w\|_{L^{2}(U)}^{2} \le {\mathcal J}(w,w) \le -{\mathcal J}(v,w) \le- \int_{U}c(x) v(x)w(x) \ dx &= \int_{U}c(x) w^2(x) \ dx \\
&\le \|c^+\|_{L^\infty(U)}\|w\|_{L^{2}(U)}^{2}. \end{align*}
Since $\|c^+\|_{L^\infty(U)}< \Lambda_{1}(U)$ by assumption, we conclude that $\|w\|_{L^{2}(U)}=0$ and hence $v\geq0$ a.e. in $H$. \end{proof}
We note that a combination of Proposition~\ref{4-elliptic-max1} with Lemma~\ref{3-mengen-k} gives rise to an ``antisymmetric'' small volume maximum principle which generalizes the available variants for the fractional Laplacian, see \cite[Proposition 3.3 and Corollary 3.4]{FJ13} and \cite[Lemma 5.1]{RS13}. Next we prove a strong maximum principle which requires the strict inequality (\ref{sym-Q-J-2-strict}).
\begin{prop}\label{hopf-simple2} Assume that $J$ satisfies $(J1)$, (\ref{sym-Q-J-1}) and (\ref{sym-Q-J-2-strict}). Moreover, let $U \subset H$ be an open bounded set and $c \in L^\infty(U)$. Furthermore, let $v$ be an antisymmetric supersolution of (\ref{linear-prob}) such that $v\geq0$ a.e. in $H$. Then either $v \equiv 0$ a.e. in $\mathbb{R}^{N}$, or $$ \underset{K}\essinf\: v >0 \qquad \text{for every compact subset $K \subset U$.} $$ \end{prop}
\begin{proof} We assume that $v \not \equiv 0$ in $\mathbb{R}^N$. For given $x_0 \in U$, it then suffices to show that $\underset{B_r(x_0)}\essinf\: v >0$ for $r>0$ sufficiently small. Since $v\not\equiv 0$ in $\mathbb{R}^N$ and $v$ is antisymmetric with $v \ge 0$ in $H$, there exists a bounded set $M \subset H$ of positive measure with $x_o \not \in \overline M$ and such that \begin{equation}
\label{eq:def-delta} \delta:= \inf_{M} v >0. \end{equation}
By Lemma~\ref{3-mengen-k}, we may fix $0<r< \frac{1}{4}\textnormal{dist}(x_0,[\mathbb{R}^N \setminus H] \cup M)$ such that $\Lambda_1(B_{2r}(x_0)) > \|c\|_{L^{\infty}(U) }$. Next, we fix a function $f\in {\mathcal C}^{2}_{c}(\mathbb{R}^{N})$ such that $0 \le f \le 1$ on $\mathbb{R}^N$ and \begin{equation*}
f(x):=\left\{\begin{aligned} 1, &&\text{ for $|x-x_0|\leq r$,}\\
0, &&\text{ for $|x-x_0|\geq 2r$.}\\ \end{aligned}\right. \end{equation*} Moreover we define $$ w: \mathbb{R}^N \to \mathbb{R}, \qquad w(x):=f(x)-f(\bar{x})+ a\bigl[1_M(x)-1_{M} (\bar x))\bigr], $$ where $a>0$ will be fixed later. We also put $U_0:= B_{2r}(x_0)$ and $U_0':= B_{3r}(x_0) \cup Q(B_{3r}(x_0))$. Note that the function $w$ is antisymmetric and satisfies \begin{equation}
\label{eq:ineq-w} w\equiv 0 \quad \text{on $H \setminus (U_0 \cup M)$,} \quad \qquad w \equiv a \quad \text{on $M$.} \end{equation}
We claim that $w \in {\mathcal H}(U_0')$. Indeed, by Proposition~\ref{3-prel-dense}(i) we have $f - f \circ Q \in {\mathcal D}(\mathbb{R}^N) \cap L^2(\mathbb{R}^N) \subset {\mathcal H}(U_0')$, whereas $1_M - 1_{Q(M)} \in {\mathcal H}(U_0')$ since $M$ is bounded and $[M \cup Q(M)] \cap U_0'= \emptyset$. \\ Next, let $\varphi\in {\mathcal D}(U_0)$, $\varphi\geq0$. By Proposition~\ref{3-prel-dense}(ii) we have \begin{equation} {\mathcal J}(f,\varphi) \leq C \int_{U_0}\varphi(x)\ dx \end{equation} with $C=C(f)>0$ independent of $\varphi$. Since $$ f(\bar x) \varphi(x) = 1_{M}(x) \varphi(x)= 1_{Q(M)}(x) \varphi(x) = 0 \qquad \text{for every $x \in \mathbb{R}^N$,} $$ we have \begin{align*}
{\mathcal J}(w,\varphi)&={\mathcal J}(f,\varphi) - {\mathcal J}(f \circ Q,\varphi) +a\bigl[{\mathcal J}(1_M, \varphi) -{\mathcal J}(1_{Q(M)}, \varphi) \bigr]\\ &\leq C \int_{U_0}\varphi(x)\ dx +\int_{U_0}\int_{Q(U_0)} \varphi(x)f(\overline y) J(x-y)\ dydx\\
& -a\bigl[ \int_{U_0}\int_{M}\varphi(x)J(x-y)\ dydx -\int_{U_0}\int_{Q(M)}\varphi(x)J(x-y)\ dydx \bigr] \\ &\leq \Bigl( C + \sup_{x \in U_0} \int_{Q(U_0)} J(x-y)\,dy \Bigr)
\int_{U_0}\varphi(x)\ dx -a\int_{U_0}\varphi(x) \int_{M}[J(x-y)-J(x-\bar{y})]\ dydx\\ &\le C_a \int_{U_0} \varphi(x)\,dx \end{align*} with $$ C_a := C + \sup_{x \in U_0} \int_{Q(U_0)} J(x-y)\,dy - a \inf_{x \in U_0} \int_M (J(x-y)-J(x-\bar{y}))\ dy\in \mathbb{R} $$ Since $\overline U_0 \subset H$, (\ref{sym-Q-J-2-strict}) and the continuity of the function $x \mapsto \int_M (J(x-y)-J(x-\bar{y}))\ dy$ on $\overline U_0$ imply that \[ \inf_{x \in U_0} \int_M (J(x-y)-J(x,\bar{y}))\ dy>0 \]
Consequently, we may fix $a>0$ sufficiently large such that $C_a \le -\|c\|_{L^\infty(U_0)}$. Since $0 \le w \le 1$ in $U_0$, we then have \begin{equation} \label{eq:w-ineq}
{\mathcal J}(w,\varphi) \le -\|c\|_{L^\infty(U_0)} \int_{U_0}\varphi(x)\,dx \leq \int_{U_0}c(x)w(x)\varphi(x)\ dx. \end{equation} We now consider the function $\tilde v:=v-\frac{\delta}{a} w \in {\mathcal H}(U_0')$, which by (\ref{eq:def-delta}) and (\ref{eq:ineq-w}) satisfies $\tilde v \ge 0$ on $H \setminus U_0$. Hence, by assumption and (\ref{eq:w-ineq}), $\tilde v$ is an antisymmetric supersolution of the problem \begin{equation}\label{linear-prob-special} I\tilde v = c(x)\tilde v\quad \text{ in $U_0$,}\qquad \tilde v \equiv 0 \quad \text{on $H \setminus U_0$} \end{equation}
Since $\|c\|_{L^{\infty}(U_0)}<\Lambda_1(U_0)$, Proposition \ref{4-elliptic-max1} implies that $\tilde v \ge 0$ a.e. in $U_0$, so that $v \geq \frac{\delta}{a} w = \frac{\delta}{a} >0$ a.e. in $B_{r}(x_0)$. This ends the proof. \end{proof}
\section{Proof of the main symmetry result }\label{mr}
In this section we complete the proof of Theorem \ref{sec:goal}. So throughout this section, we assume that $J:\mathbb{R}^N \setminus \{0\} \to[0,\infty)$ is even and satisfies $(J1)$ and $(J2)$, $\Omega \subset \mathbb{R}^N$ satisfies $(D)$ and the nonlinearity $f$ satisfies $(F_1)$ and $(F_2)$. Moreover, we let $u \in L^\infty(\Omega) \cap {\mathcal D}(\Omega)$ be a nonnegative solution of $(P)$. For $\lambda \in \mathbb{R}$, we consider the open affine half space $$ H_{\lambda}:=\left \{
\begin{aligned} &\{x\in\mathbb{R}^{N}\;:\; x_{1}>\lambda\}\qquad \text{if $\lambda \ge 0$;}\\
&\{x\in\mathbb{R}^{N}\;:\; x_{1}<\lambda\}\qquad \text{if $\lambda < 0$.}\\
\end{aligned} \right. $$ Moreover, we let $Q_{\lambda}:\mathbb{R}^{N}\to\mathbb{R}^{N}$ denote the reflection at $\partial H_{\lambda}$, i.e. $Q_{\lambda}(x)=(2\lambda-x_{1},x')$. By Remark~\ref{sec:equality-monotonicity}, we may assume without loss of generality that (\ref{eq:adj-measure}) holds. As noted in Remark~\ref{symmetry-need}, $J$ therefore satisfies the symmetry and monotonicity conditions (\ref{sym-Q-J-1}) and (\ref{sym-Q-J-2-strict}) with $H$ replaced by $H_\lambda$ for $\lambda \not=0$. Let $\ell:=\sup \limits_{x\in \Omega}x_1$. Setting $\Omega_{\lambda}:= H_\lambda \cap \Omega$ for $\lambda \in \mathbb{R}$, we note that $Q_{\lambda}(\Omega_\lambda)\subset \Omega$ for all $\lambda\in (-\ell,\ell)$ and $Q_{0}(\Omega)=\Omega$ as a consequence of assumption $(D)$. Then for all $\lambda\in (-\ell,\ell)$, Remark~\ref{3-anti} implies that $v_{\lambda}:=u \circ Q_\lambda-u \in {\mathcal D}(\mathbb{R}^N) \cap L^2(\mathbb{R}^N)$ is an antisymmetric supersolution of the problem \begin{equation}\label{linear-prob-lambda} Iv = c_\lambda (x)v \quad \text{ in $\Omega_\lambda$,}\qquad v \equiv 0 \quad \text{on $H_\lambda \setminus \Omega_\lambda$} \end{equation} with \[ c_\lambda \in L^{\infty}(\Omega_\lambda)\quad \text{given by}\quad c_\lambda(x)= \left \{
\begin{aligned}
&\frac{f(x,u(Q_\lambda(x)))-f(x,u(x))}{v_\lambda(x)},&&\qquad v_\lambda (x) \not= 0;\\
&0, &&\qquad v_\lambda(x)= 0.
\end{aligned} \right. \] Note that, as a consequence of $(F1)$ and since $u \in L^\infty(\Omega)$, we have $$
c_\infty:= \sup_{\lambda \in (-\ell,\ell)}\|c_\lambda\|_{L^\infty(\Omega_\lambda)} < \infty. $$ We now consider the statement \[
(S_{\lambda})\qquad \underset{K}{\essinf} \:v_{\lambda}> 0 \qquad \text{for every compact subset $K \subset \Omega_{\lambda}$.} \] Assuming that $u \not\equiv 0$ from now on, we will show $(S_{\lambda})$ for all $\lambda\in (0,\ell)$.
Since $|\Omega_{\lambda}| \to 0$ as $\lambda\to \ell$, Lemma \ref{3-mengen-k} implies that there exists $\epsilon \in (0,\ell)$ such that $\Lambda_1(\Omega_{\lambda})>c_\infty$ for all $\lambda\in [\epsilon,\ell)$. Applying Proposition \ref{4-elliptic-max1} we thus find that \begin{equation}
\label{eq:almostS-lambda}
v_\lambda \ge 0 \quad \text{a.e. in $H_\lambda\quad$ for all $\lambda \in [\epsilon,\ell)$.} \end{equation} We now show\\[0.1cm] {\em Claim 1: If $v_\lambda \ge 0$ a.e. in $H_\lambda$ for some $\lambda \in (0,\ell)$, then $(S_{\lambda})$ holds.}\\[0.1cm] To prove this, by Proposition~\ref{hopf-simple2} it suffices to show that $v_{\lambda}\not \equiv 0$ in $\mathbb{R}^N$. If, arguing by contradiction, $v_{\lambda}\equiv0$ in $\mathbb{R}^N$, then $\partial H_{\lambda}$ is a symmetry hyperplane of $u$. Since $\lambda \in (0,\ell)$ and $u \equiv 0$ in $\mathbb{R}^N \setminus \Omega$, we then have $u\equiv 0$ in the nonempty set $\Omega_{-\ell+2\lambda}$. Setting $\lambda'= -\ell+\lambda$, we thus infer that $v_{\lambda'} \equiv 0$ in $\Omega_{\lambda'}$. Consequently, $v_{\lambda'}\equiv 0$ in $\mathbb{R}^{N}$ by Proposition \ref{hopf-simple2}. Thus $u$ has the two different parallel symmetry hyperplanes $\partial H_{\lambda}$ and $\partial H_{\lambda'}$. Since $u$ vanishes outside a bounded set, this implies that $u\equiv 0$, which is a contradiction. Thus Claim 1 is proved.\\[0.1cm] Next we show\\[0.1cm] {\em Claim 2: If $(S_{\lambda})$ holds for some $\lambda \in (0,\ell)$, then there is $\delta\in(0,\lambda)$ such that $(S_{\mu})$ holds for all $\mu \in (\lambda-\delta,\lambda)$.}\\[0.1cm]
To prove this, suppose that $(S_{\lambda})$ holds for some $\lambda \in (0,\ell)$. Using Lemma~\ref{3-mengen-k}, we fix $s \in (0, |\Omega_\lambda|)$ such that $\Lambda_1(s)> c_\infty$, which implies that $\Lambda_1(U)>c_\infty$ for all open sets $U\subset\mathbb{R}^N$ with $|U|\leq s$. Since $\Omega$ is bounded, we may also fix $\delta_0 >0 $ such that $$
|\Omega_{\mu} \setminus \Omega_{\mu+\delta_0}|<s/2\qquad \text{for all $\mu \ge 0$.} $$
By Lusin's Theorem, there exists a compact subset $K \subset \Omega$ such that $|\Omega \setminus K|< s/4$ and such that the restriction
$u|_{K}$ is continuous. For $\mu \ge 0$, we now consider the compact set $$ K_\mu:= \overline \Omega_{\mu+\delta_0} \cap K \cap Q_\mu(K) \:\subset \:K \cap \Omega_\mu $$ and the open set $U_\mu:=\Omega_\mu \setminus K_\mu$. Note that \begin{equation}
\label{eq:additional-1}
|U_\mu| \le |\Omega_\mu \setminus \Omega_{\mu+\delta_0}|+ |\Omega_\mu \setminus K| +
|\Omega_\mu \setminus Q_\mu(K)| \le \frac{s}{2} + 2 |\Omega \setminus K| < s \qquad \text{for $\mu \ge 0$. } \end{equation}
As a consequence, for $0 \le \mu \le \lambda$ we have $|K_\mu| > |\Omega_\mu|-s \ge |\Omega_\lambda|-s>0$ and thus $K_\mu \not = \varnothing$. Property $(S_\lambda)$ and the continuity of $u|_K$ imply that $\min \limits_{K_\lambda} v_\lambda>0$. Thus, again by the continuity of $u|_K$, there exists $\delta \in (0,\min \{\lambda,\delta_0\})$ such that
$$ \min_{K_\mu} v_\mu >0 \qquad \text{for all $\mu \in[\lambda-\delta,\lambda]$.} $$ Consequently, for $\mu \in (\lambda-\delta,\lambda)$, the function $v_{\mu}$ is an antisymmetric supersolution of the problem $$ Iv=c_{\mu}(x)v \quad \text{in $U_\mu$,} \qquad v \equiv 0 \quad \text{on $H_{\mu} \setminus U_\mu$, } $$ whereas $\Lambda_1(U_\mu)>c_\infty$ by (\ref{eq:additional-1}) and the choice of $s$. Hence $v_{\mu}\geq 0$ in $H_{\mu}$ by Proposition \ref{4-elliptic-max1}, and thus $(S_{\mu})$ holds by Claim 1. This proves Claim 2.\\[0.1cm] To finish the proof, we consider $$ \lambda_{0}:=\inf\{\tilde{\lambda} \in (0,\ell) \;:\; \text{ $(S_{\lambda})$ holds for all $\lambda \in (\tilde{\lambda},\ell) $}\} \quad \in \;[0,\ell). $$ We then have $v_{\lambda_0} \ge 0$ in $H_{\lambda_0}$. Hence Claim 1 and Claim 2 imply that $\lambda_{0}=0$. Since the procedure can be repeated in the same way starting from $-\ell$, we find that $v_{0} \equiv 0$. Hence the function $u$ has the asserted symmetry and monotonicity properties.\\ It remains to show (\ref{eq:positivity-thm-1-1}). So let $K \subset \Omega$ be compact. Replacing $K$ by $K \cup Q_0(K)$ if necessary, we may assume that $K$ is symmetric with respect to $Q_0$. Let $K':= \{x \in K\::\: x_1 \le 0\}$. Since for $\lambda>0$ sufficiently small $Q_\lambda(K')$ is a compact subset of $\Omega_\lambda$, the property $(S_\lambda)$ and the symmetry of $u$ then imply that $$ \underset{K}{\essinf}\, u = \underset{K'}{\essinf}\, u \ge \underset{Q_\lambda(K')}\essinf\, v_\lambda >0, $$ as claimed in (\ref{eq:positivity-thm-1-1}).
\section{Proof of a variant symmetry result} \label{sec:vari-symm-result}
In this section we prove Theorem~\ref{sec:vari-symm-result-1}, which is concerned with the class of even kernel functions satisfying $(J2)'$ in place of $(J2)$. Throughout this section, we consider a symmetric kernel $J: \mathbb{R}^N \setminus \{0\} \to [0,\infty)$ satisfying $(J1)$. We fix an open affine half space $H \subset \mathbb{R}^N$, and we consider the notation of Section~\ref{mp}. Moreover, we assume the symmetry and monotonicity assumptions (\ref{sym-Q-J-1}) and (\ref{sym-Q-J-2}), so that Lemma~\ref{sec:linear-problem-tech} and Proposition~\ref{4-elliptic-max1} are available. In order to derive a variant of the strong maximum principle given in Proposition~\ref{hopf-simple2}, we introduce the following strict monotonicity condition: \begin{equation}
\text{There exists $r_0>0$ such that $J(x-y) > J(x- \bar y)$ for all $x,y \in H$ with $|x-y| \le r_0$} \label{sym-Q-J-2-strict-loc}
\end{equation} We then have the following.
\begin{prop}\label{hopf-simple2-variant} Assume that $J$ satisfies $(J1)$, (\ref{sym-Q-J-1}), (\ref{sym-Q-J-2}) and (\ref{sym-Q-J-2-strict-loc}). Moreover, let $U \subset H$ be a subdomain and $c \in L^\infty(U)$. Furthermore, let $v$ be an antisymmetric supersolution of (\ref{linear-prob}) such that $v\geq0$ a.e. in $H$.\\ Then either $v \equiv 0$ a.e. in a neighborhood of $\overline U$, or $$ \underset{K}\essinf\: v >0 \qquad \text{for every compact subset $K \subset U$.} $$ \end{prop} We stress that, in contrast to Proposition~\ref{hopf-simple2}, we require connectedness of $U$ here.
\begin{proof} Let $W$ denote the set of points $y \in U$ such that $\underset{B_{r}(y)}\essinf\: v >0$ for $r>0$ sufficiently small, and let $r_0>0$ be as in (\ref{sym-Q-J-2-strict-loc}). We claim the following.\\ \begin{equation}
\label{claim-1} \text{If $x_0 \in U$ is such that $v \not \equiv 0$ in $B_{\frac{r_0}{2}}(x_0)$, then $x_0 \in W$.} \end{equation} To prove this, let $x_0 \in U$ be such that $v \not \equiv 0$ in $B_{\frac{r_0}{2}}(x_0)$. Then there exists a bounded set $M \subset H \cap B_{\frac{r_0}{2}}(x_0)$ of positive measure with $x_0 \not \in \overline M$ and such that \begin{equation} \delta:= \inf_{M} v >0 \end{equation}
By Lemma~\ref{3-mengen-k}, we may fix $0<r< \frac{1}{4}\min \{r_0\,,\, \textnormal{dist}(x_0,[\mathbb{R}^N \setminus H] \cup M)\}$ such that $\Lambda_1(B_{2r}(x_0)) > \|c\|_{L^{\infty}(U) }$. Next, we put $U_0:= B_{2r}(x_0)$ and $U_0':= B_{3r}(x_0) \cup Q(B_{3r}(x_0))$. Moreover, we define the functions $f\in C^{2}_{c}(\mathbb{R}^{N})$ and $w \in {\mathcal H}(U_0')$, depending on $a>0$, as in the proof of Proposition~\ref{hopf-simple2}. As noted there, $w$ is antisymmetric and satisfies \begin{equation}
\label{eq:ineq-w2} w\equiv 0 \quad \text{on $H \setminus (U_0 \cup M)$,} \quad \qquad w \equiv a \quad \text{on $M$.} \end{equation} As in the proof of Proposition~\ref{hopf-simple2}, we also see that $$
{\mathcal J}(w,\varphi) \le C_a \int_{U_0} \varphi(x)\,dx \qquad \text{for all $\varphi \in {\mathcal D}(U_0), \varphi \ge 0$} $$ with $$ C_a := C + \sup_{x \in U_0} \int_{Q(U_0)} J(x-y)\,dy - a \inf_{x \in \overline U_0} \int_M (J(x-y)-J(x-\bar{y}))\ dy $$ Since $\overline U_0 \subset H \cap B_{\frac{r_0}{2}}(x_0) $ and $M \subset H \cap B_{\frac{r_0}{2}}(x_0)$, (\ref{sym-Q-J-2-strict-loc}) and the continuity of the function $x \mapsto \int_M (J(x-y)-J(x-\bar{y}))\ dy$ on $\overline U_0$ imply that \[ \inf_{x \in \overline U_0} \int_M (J(x-y)-J(x,\bar{y}))\ dy>0 \] Hence we may proceed precisely as in the proof of Proposition~\ref{hopf-simple2} to prove that $v \geq \frac{\delta}{a} >0$ a.e. in $B_{r}(x_0)$ for $a>0$ sufficiently large, so that $x_0 \in W$. Hence (\ref{claim-1}) is true.\\ From (\ref{claim-1}) it immediately follows that $W$ is both open and closed in $U$. Moreover, if $v \not \equiv 0$ in $\{x \in H\::\: \textnormal{dist}(x,U) < \frac{r_0}{2}\}$, then $W$ is nonempty and therefore $W=U$ by the connectedness of $U$. This ends the proof. \end{proof}
Next we complete the proof of Theorem~\ref{sec:vari-symm-result-1}. So throughout the remainder of this section, we assume that $J:\mathbb{R}^N \setminus \{0\} \to[0,\infty)$ is even and satisfies $(J1)$ and $(J2)'$, $\Omega \subset \mathbb{R}^N$ satisfies $(D)$ and the nonlinearity $f$ satisfies $(F_1)$ and $(F_2)$. Moreover, we let $u \in L^\infty(\Omega) \cap {\mathcal D}(\Omega)$ denote an a.e. positive solution of $(P)$. For $\lambda \in \mathbb{R}$, we let $H_\lambda$, $Q_\lambda$, $\Omega_\lambda$, $c_\lambda$ and $v_\lambda$ be defined as in Section~\ref{mr}, and again we put $\ell:=\sup \limits_{x\in \Omega}x_1$. As a consequence of $(J1)$ and $(J2)'$, we may assume that $J$ satisfies~(\ref{sym-Q-J-1})~(\ref{sym-Q-J-2}) and (\ref{sym-Q-J-2-strict-loc}) with $H$ replaced by $H_\lambda$ for $\lambda \not=0$ (the argument of Remark~\ref{symmetry-need} still applies). As in Section~\ref{mr}, we then consider the statement \[
(S_{\lambda})\qquad \underset{K}{\essinf} \:v_{\lambda}> 0 \qquad \text{for every compact subset $K \subset \Omega_{\lambda}$.} \] We wish to show $(S_{\lambda})$ for all $\lambda\in (0,\ell)$. As in Section~\ref{mr}, we find $\epsilon \in (0,\ell)$ such that \begin{equation}
\label{eq:almostS-lambda-variant}
v_\lambda \ge 0 \quad \text{a.e. in $H_\lambda\quad$ for all $\lambda \in [\epsilon,\ell)$.} \end{equation} We now show\\[0.1cm] {\em Claim 1: If $v_{\lambda} \ge 0$ a.e. in $H_{\lambda}$ for some $\lambda \in (0,\ell)$, then $(S_{\lambda})$ holds.}\\[0.1cm] To prove this, we argue by contradiction. If $(S_\lambda)$ does not hold, then, by Proposition~\ref{hopf-simple2-variant}, there exists a connected component $\Omega'$ of $\Omega_\lambda$ and a neighborhood $N$ of $\overline{\Omega'}$ such that $v_{\lambda}\equiv 0$ in $N$. However, since $\lambda \in (0,\ell)$, the set $\tilde N:= Q_\lambda(N \setminus \Omega) \cap \Omega$ has positive measure and $v_\lambda \equiv 0$ in $\tilde N$ by the antisymmetry of $v_\lambda$. However, $v \equiv -u$ on $\tilde N$, so $u \equiv 0$ a.e. on $\tilde N$, contrary to the assumption that $u >0$ a.e. in $\Omega$. Thus Claim 1 is proved.\\[0.1cm] Precisely as in Section~\ref{mr} we may now show\\[0.1cm] {\em Claim 2: If $(S_{\lambda})$ holds for some $\lambda \in (0,\ell)$, then there is $\delta\in(0,\lambda)$ such that $(S_{\mu})$ holds for all $\mu \in (\lambda-\delta,\lambda)$.}\\[0.1cm] Moreover, based on (\ref{eq:almostS-lambda-variant}), Claim 1 and Claim 2, we may now finish the proof of Theorem~\ref{sec:vari-symm-result-1} precisely as in the end of Section~\ref{mr}.
\end{document} | arXiv |
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$$\begin{align}& \text{Using the midpoint formula, find the midpoint between the given points:} \\ \\ & \hspace{3ex} (x_{1}, y_{1}) = (1, 2) \: \text{ and } \: (x_{2}, y_{2}) = (3, 4)\\ \\ & \text{1.) The midpoint of the given points } (x_{1}, y_{1}) \: \text{ and } \: (x_{2}, y_{2}) \text{ is given as:} \\ \\ & \hspace{3ex} \left(\frac{(x_{1} + x_{2})}{2}, \frac{(y_{1} + y_{2})}{2}\right) \\ \\ & \hspace{3ex} \text{Where: } \\ & \hspace{3ex} x_{1}, \: y_{1}, \: x_{2},\text{ and } y_{2} \text{ are the given coordinates for the first and second } \\ & \hspace{3ex} \text{points respectively.}\\ \\ & \text{2.) Let's begin by plugging our given coordinates into the midpoint formula:} \\ \\ & \hspace{3ex} \Longrightarrow \left(\frac{(1 + 3)}{2}, \frac{(2 + 4)}{2}\right)\\ \\ & \text{3.) Now, let's evaluate and simplify:} \\ \\ & \hspace{3ex} \Longrightarrow \left(\frac{(4)}{2}, \frac{(6)}{2}\right) \\ \\ & \hspace{3ex} \Longrightarrow (2, 3)\\ \\ & \text{Therefore, the midpoint between points }(x_{1}, y_{1}) \: \text{ and } \: (x_{2}, y_{2}) \text{ is:} \\ \\ & \boxed{\boxed{(2, 3)}}\end{align}$$
Midpoint Formula Lesson
What is the Midpoint Formula?
The midpoint formula allows us to find the midpoint of a line between two distinct points. In other words, it takes the average of a line's given endpoints.
Figure 1 – Midpoint on Line Between Endpoints (x1, y1) and (x2, y2).
Why do we Learn About the Midpoint Formula?
The midpoint formula might be a fundamental topic, but it can be very useful when high tech tools and gadgets are unavailable.
Let's say that we are camping the woods and we want to set up our tent approximately between the lake where we will catch fish and a patch of small trees where we can gather firewood with a small hatchet.
Unfortunately, somewhere between the lake and firewood spot, there is a ravine that is surrounded by large trees that blocks a direct path and line of sight between our two points of interest. Since we only have a watch and a compass, we will have to make use of what we have to navigate to the approximate midpoint between the lake and firewood spot.
Map of Campsite
Beginning at the lake, we can begin walking north at a consistent pace until we see the firewood gathering area directly to the east such that the ravine is no longer in our way. At this point we will take a mental note of how long it took us to reach this spot to the north. Then, we can walk to the east until we get to our firewood gathering spot while taking note of the travel time.
Since the lake was our starting point, the coordinates based on travel time will be (0 minutes east, 0 minutes north). Let's say that we walked 10 minutes to the north and 6 minutes to the east to get to the firewood spot from the lake. This would leave us with the travel time coordinates of (6 minutes east, 10 minutes north).
Applying the midpoint formula, we find that we need to walk north for five minutes and east for 3 minutes to reach the approximate midpoint between the lake and firewood gathering area where will set up camp.
How to use the Midpoint Formula
We can find the midpoint of a line by using the following formula:
$$\begin{align}&\left(\frac{(x_{1} + x_{2})}{2}, \frac{(y_{1} + y_{2})}{2}\right)\end{align}$$
Where x1, y1, x2, and y2 are the given coordinates for the first and second points respectively.
The steps for finding the midpoint are as follows:
Add x1 to x2.
Divide the result from Step 1 by 2 (This is the x coordinate for the midpoint).
Add y1 to y2.
Divide the result from Step 3 by 2 (This is the y coordinate for the midpoint).
$$\begin{align}& \text{Using the midpoint formula, find the midpoint between the given points:} \\ \\ & \hspace{3ex} (x_{1}, y_{1}) = (1, 2) \: \text{ and } \: (x_{2}, y_{2}) = (3, 4)\\ \\ & \text{1.) The midpoint of the given points } (x_{1}, y_{1}) \: \text{ and } \: (x_{2}, y_{2}) \text{ is given as:} \\ \\ & \hspace{3ex} \left(\frac{(x_{1} + x_{2})}{2}, \frac{(y_{1} + y_{2})}{2}\right) \\ \\ & \hspace{3ex} \text{Where: } \\ & \hspace{3ex} x_{1}, \: y_{1}, \: x_{2},\text{ and } y_{2} \text{ are the given coordinates for the first and second } \\ & \hspace{3ex} \text{points respectively.}\\ \\ & \text{2.) Let's begin by plugging our given coordinates into the midpoint formula:} \\ \\ & \hspace{3ex} \Longrightarrow \left(\frac{(1 + 3)}{2}, \frac{(2 + 4)}{2}\right)\\ \\ & \text{3.) Now, let's evaluate and simplify:} \\ \\ & \hspace{3ex} \Longrightarrow \left(\frac{(4)}{2}, \frac{(6)}{2}\right) \\ \\ & \hspace{3ex} \Longrightarrow (2, 3)\\ \\ & \text{Therefore, the midpoint between points }(x_{1}, y_{1}) \: \text{ and } \: (x_{2}, y_{2}) \text{ is:} \\ \\ & \boxed{\boxed{(2, 3)}}\end{align}$$
This calculator uses a combination of HTML, CSS, and JavaScript.
HTML, or "HyperText Markup Language", provides the framework for the calculator. With HTML, we can create entities such as the calculator box, solution steps field, and the various buttons on the interface.
We then use CSS, or "Cascading Style Sheets", for the aesthetic design of the aforementioned calculator, calculator buttons, and solution steps field.
While having the virtual calculator interface on the page is a good start, the calculator has to also be functional. To accomplish this, JS, or "JavaScript", is used. JS allows the calculator buttons to insert the relevant value into the input fields when pressed. JS also allows the actual mathematical calculations to take place when the "calculate" button is pressed.
All of these elements come together to form the Midpoint Formula Calculator that allows users to find the midpoint between their given endpoints, see solution steps that change based on their inputs, and visualize their problem on an interactive graph. | CommonCrawl |
Does a roleplaying game that uses continuous probability exist? [closed]
Locked. This question and its answers are locked because the question is off-topic but has historical significance. It is not currently accepting new answers or interactions.
I have an idea for a roleplaying game that uses continuous probability distributions (like the normal distribution) to decide random outcomes (instead of discrete distributions that comes from using dices). I am wondering if this already exists?
Jill has 50 Dex and has +10 in firing bows. She tries to fire an arrow some distance. Her skill gives her normal distribution function a mean value of 60 and the properties of using a bow gives this action a standard deviation of 15. because of the difficulty of the shot needs to "roll" a 70 or higher.
Then you use any normal distribution random generator N(60,15) (like Matlab, internet or some app on smart phones) to roll and see if you get higher or lower than 70. if she gets 90 or higher it's a critical hit, if she gets 30 or lower it's an epic fail.
(Note that I haven't thought these numbers through at all and I can see like five holes in these numbers while writing them, the example is for the demonstration of how the mean value and the standard deviation would be decided and how the distribution should be used.)
Does this exist anywhere?
game-recommendation statistics
Martin Brischetto
Martin BrischettoMartin Brischetto
\$\begingroup\$ The edits made to this question have brought it to a point where it has nothing to do with the title "continuous probability" nor anything to do with the existing answers. If you discover from the answers to your question that you actually have a different question (or simply asked the wrong one) please ask a new question. We can have two great questions with great answers instead of one mediocre one. I'd suggest reverting some of the edits and separating them into a new question. \$\endgroup\$ – Cirdec Apr 11 '15 at 16:53
\$\begingroup\$ You may also want to un-accept my answer, as you've said it doesn't solve your problem. \$\endgroup\$ – SevenSidedDie Apr 11 '15 at 18:52
\$\begingroup\$ I will do that. I accepted the answer since it answered the question "does it exist?" Which was the original question. \$\endgroup\$ – Martin Brischetto Apr 11 '15 at 18:54
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Yes, this exists... sort of.
I say "sort of" because you're assuming that continuous probabilities are possible to generate, when it is physically impossible to do so. (All random number generators, both digital and physical, can only produce discrete numbers, even if there are so many that the curve looks smooth; and the results from physically-contnuous randomisers are inherently limited by the discrete resolution of your measurement tool.)
However, the motive that you appear to have underlying this desire is, indeed, a goal that some roleplaying games have been written to satisfy. Rather than using summed dice, a small but persistently popular selection of RPGs are percentile systems.
In a percentile dice system, the probability of an outcome is modelled with percentages, as in your example, and success/failure — including critical successes and epic failures — can be determined by comparing a roll of percentile dice to the probability: any number equal or lower is a success.
These dice have rolled an 18 result, which in "roll under" percentile systems would be a clear success against a probability of 70%:
Image © IanWatson, licensed under CC-BY-SA 3.0
These dice do not generate a bell curve, however — in such systems, the location of the bell curve is in the part of the system that determines the percentage odd you need to roll under, and the dice only select a percentage from the always-flat flat curve that 0%–100% always has, to compare against the already-generated bell curve marked with percentiles.
Depending on the system, that 18 result might also be a critical success. Few systems use standard deviations by name, but some do use static numerical modifiers to the probability which end up being equivalent (e.g., a task that is easy and gets a +20 to the roll, which is the same as if the bell curve was shifted 20 points to the right.)
There are a number of games that use percentile dice systems to generate transparent probabilities like this. The very first one in RPG publishing history is and was Chaosium's RuneQuest (1978), which they eventually turned into a "house system" used in many other of their games and even other publishers' games. That original percentile game system is still available (with some small changes) in either Chaosium's universal-genre RPG Basic RolePlaying (2008), Chaosium's fantasy-specific Magic World (and its free Quick-Start edition), or a recent RuneQuest edition and a number of the retroclones of its older editions.
There are lots of percentile systems out there. I can't recommend any single one, but all of them satisfy the desire you and many other roleplayers have had for clear, real-looking probabilities in a roleplaying game. And as a bonus, their dice-based resolution is quite a bit faster than trying to use Matlab during a roleplaying session!
SevenSidedDieSevenSidedDie
\$\begingroup\$ I understand what you mean. What I dont like with those systems is that advancement is most often linear which means that if you advance with 10 then its 10 percentile units easier to succeed with the percentile system, but with the normal distribution system, the change in probability from a 10 point advancement would depend on how close you already are to succeeding. Also, some things we do are more random and have less to do with skill(larger std) than others. Like firing an arrow versus picking a lock, both require skill but firing an arrow has more random variables affecting it. \$\endgroup\$ – Martin Brischetto Apr 9 '15 at 19:18
\$\begingroup\$ @Martin You have to consider what standard deviation actually matters though: for an arrow, it doesn't matter how complex the inputs are, when the output is binary "hit or miss". Even when you add more outputs (crit pass/fail), you still have discrete outputs that don't care how complex the inputs are. In the end more random things and less random things both have an single X% probability to do Y outcome. As for advancement: most d% games make advancement slower the higher your skill is, so you get larger increases over time at the low end and smaller increases over time at the high end. \$\endgroup\$ – SevenSidedDie Apr 9 '15 at 23:12
Ultimately, your "normal-distribution random number generator" is neither random nor normally-distributed. It's also not actually continuous; all numbers in a computer are always going to be discrete.
Any system that rolls a number of equally-sized dice and sums them up is more-or-less approximately normal (see also: central limit theorem). More dice produces a smoother result, and the results of your computer's algorithm is mathematically identical to just rolling a very-large number of very-many-sided dice. And all that really accomplishes is a smoother approximation of a normal distribution than, say, 3d6.
To a certain extent, the smoothness matters. Certainly, I can look at the probability distribution of 3d6 graphed from 3 to 18, versus a graph of what Matlab or Wolfram Alpha call a normal distribution with the same mean and standard deviation, and see a striking difference. But that difference isn't really that big; I tend to doubt that players will really notice a difference.
KRyanKRyan
\$\begingroup\$ Thanks fo the feedback. The main difference im thinking of is that its complicated to vary the std if you can only use a handfull of dice(more would be difficult and exhausting). And if one would go through all the trouble to find out a good way to vary the std with a handfull of dice chances are it would result in a not so simple formula that is a hassle to use everytime you need to. I realize that a computer also uses a discrete distribution, but the "steps" are of the size 10^-16 (in matlab). I feel like its closer to the real world since some things are more widely distributed than others. \$\endgroup\$ – Martin Brischetto Apr 9 '15 at 17:21
\$\begingroup\$ @Martin You should clarify your question, then, to say that. The short answer is, no, I don't think that's a good idea. \$\endgroup\$ – KRyan Apr 9 '15 at 17:30
It's very hard to detect the difference. You can just use 3d6 or something similarly simple, because you won't really notice a difference
Let's pretend you have two ways of determining the result of tests like the one Jill is participating in. One of them generates results from an ideal normal distribution. The other is a small handful of dice, say 3d6. You choose one to use for the game and hide it behind a GM screen so the players don't know which one you are using. We're going to choose the 3d6. How long will it take the players to detect that you are using the 3d6 instead of the ideal normal distribution? We'll work through the Jill example.
Jill's player knows that Jill will shoot that distance if she rolls over \$70\$ on a normal distribution with mean \$60\$ and standard deviation \$15\$. Looking this up in a table, she expects that she will succeed \$0.2524925\$ of the time.
Behind the screen we calculate that Jill's player needs to hit \$\dfrac 2 3 = \dfrac{70 - 60}{15}\$ of the standard deviation above the mean to shoot the arrow that distance. We have 3d6 with mean \$10.5\$ and standard deviation \$2.95804\$. She needs to hit \$ 12.472 = 10.5 + \frac 2 3 \times 2.95804 \$, so she will shoot the distance with a roll of 13 or higher. She only makes the shot \$0.199012\$ of the time.
Jill's player thinks that you might have 3d6 hidden behind the GM screen. To find out if you do, she is going to attempt this shot repeatedly and keep track of how many times it is successful or not. Asking for help on cross-validated, she finds out that if she wants suggestive evidence (\$68\%\$ sure you are using the wrong dice or \$1\$ standard deviation) she will need to repeat the shot this many times:
$$ 244 = \left( \dfrac{1}{ \arcsin (\sqrt{0.2524925}) - \arcsin (\sqrt{0.199012}) } \right)^2 $$
If she wants to convince herself (96% chance she's right or 2 standard deviations) she will need to make this many shots:
Jill's player can't even suspect that you aren't really using an ideal normal distribution until she's made 244 shots and can't be convinced of it until she's made 975 dice rolls.
If Jill's player is particularly clever, she might decide to find out what test in the game will allow her to, as quickly as possible, demonstrate that you are using 3d6 instead of an ideal normal distribution. She carefully considers a graph of the cumulative distribution functions for both 3d6 and the ideal normal distribution with mean \$10.5\$ and standard deviation \$2.95804\$. These are the probabilities that her roll will be less than or equal to the target number.
She realizes that the error is biggest at the integer values, which are exactly what you can roll. The errors not measured on the left side of the graph are exactly what's measured on the right. She writes down the values of the CDFs for each of the possible dice rolls. For each pair of probabilities she calculates how many checks will be required to be concerned (\$68\%\$ chance or \$1\$ standard deviation) or convinced (\$96\%\$ chance or \$2\$ standard deviations) that you are using the wrong dice.
\begin{array}{lllll} \text{Target} & \text{3d6 <= Target} & \text{Normal} & \text{N (Z=1)} & \text{N (Z=2)} \\ \hline 3 & 0.00462 & 0.00561 & 20948 & 83792 \\ 4 & 0.0185 & 0.013 & 3113 & 12451 \\ 5 & 0.0462 & 0.031 & 676 & 2704 \\ 6 & 0.0925 & 0.064 & 353 & 1412 \\ 7 & 0.162 & 0.118 & 252 & 1006 \\ 8 & 0.259 & 0.199 & 194 & 776 \\ 9 & 0.375 & 0.306 & 189 & 754 \\ 10 & 0.5 & 0.432 & 221 & 883 \\ 11 & 0.625 & 0.567 & 288 & 1149 \\ 12 & 0.740 & 0.693 & 370 & 1480 \\ 13 & 0.837 & 0.800 & 432 & 1728 \\ 14 & 0.907 & 0.881 & 567 & 2266 \\ 15 & 0.953 & 0.935 & 655 & 2618 \\ 16 & 0.981 & 0.968 & 570 & 2280 \\ 17 & 0.9953 & 0.9860 & 393 & 1570 \\ 18 & 1.0 & 0.994 & 178 & 712 \end{array}
Jill can't suspect that you are using the wrong dice until making 178 dice rolls, and she can't be sure until she makes 712. The difference between 3d6 and an ideal normal distribution won't be detectable except in very long campaigns.
The following summarizes how long it would take players to detect that an ideal normal distribution isn't being used for their checks for various popular dice systems.
\begin{array}{lll} \text{Dice} & \text{N (Z=1)} & \text{N (Z=2)} \\ \hline \text{d6} & 14 & 55 \\ \text{2d6} & 52 & 207 \\ \text{3d6} & 178 & 712 \\ \text{4d6} \text{ (opposed 2d6)} & 266 & 1063 \\ \text{6d6} \text{ (opposed 3d6)} & 409 & 1636 \\ \text{d20} & 20 & 80 \\ \text{4d3} \text{ (fudge dice)} & 72 & 287 \\ \text{8d3} \text{ (opposed fudge dice)} & 136 & 541 \end{array}
CirdecCirdec
\$\begingroup\$ Thanks, this was very useful information. This wasn't the main point of the question in mind, although i realize i posed it poorely. The main point is not the way to determine the outcome(i.e. dice or computer) but the way to determine the odds. A better way of showing my idea would be: to succeed with anything you need to roll 100, Juno has a skill level that gives the mean, say 70, and Juno wants to pick a lock. Picking lock is more about skill than luck, which gives a low std=15. so the odds would be P(x>=100) on N(70,15). \$\endgroup\$ – Martin Brischetto Apr 11 '15 at 12:25
\$\begingroup\$ If she instead would have wanted to shoot an arrow, which is more about luck than lock picking, the std would have been higher, std=35. Then, if she has a skill of 70 in archery, the odds would be P(x>=100) on N(70,35). This reduces the problem(in writing the system) to finding out the luck-to-skill ratio of actions, there are methods to do this(at least with baseball). \$\endgroup\$ – Martin Brischetto Apr 11 '15 at 12:27
\$\begingroup\$ @Martin I think that deserves a separate question which has two very simple answers. The first is throw more dice. This is easy to do if you have dice marked with an average of 0 like fudge dice (-1 0 1 on a d6) or -2 -1 0 1 2 on a d10; you don't need to adjust the mean to throw more dice which increases the variance. If your dice don't have a mean of 0 you need to in increase the target number by the mean of the dice. If you add 2dN of variance (throw 2 more dice) you need to increase the target number by N+1. \$\endgroup\$ – Cirdec Apr 11 '15 at 16:35
\$\begingroup\$ @Martin The other answer is change the target. Increasing the variance makes it easier to do hard things and harder to do easy things. To increase the variance, change the target towards the player's mean outcome. If Jill's performance has a median of 60 and she attempts something with a difficulty of 70 that has a high variance in her performance move the target towards 60, say 67. If she instead attempts something that should be easy for her (a difficulty of 50) but has a high variance, more the target towards 60, say 53. \$\endgroup\$ – Cirdec Apr 11 '15 at 16:46
If what you're wanting is more granularity than, say, 3d6 provides, then use whatever distribution curve you like, express the probability of success p as a decimal between 0 and 1, and success is defined as rolling a fraction smaller than p using d10 (or d20 ignoring tens digit) as many rounds as necessary to either reach the stated precision or know that further rounds are not needed to determine the outcome.
Example: You have a .3141592653589... chance of hitting a particular target with your arrow.
If your first roll is 0-2, you've hit the target. If it's 4-9, you've missed. In either case, no further rolls are required. Only if you hit 3 exactly do you roll again.
If your second roll is 0, you hit. If it's 2-9, you missed. If it's 1 exactly, you roll again.
If your third roll is 0-3, you hit. If it's 5-9, you missed. If it's 4 exactly, you roll again.
This repeats until you either stop hitting the exact target number or run out of digits.
It should be immediately apparent that 90% of the time you will roll once, 9% of the time twice, 0.9% three times, 0.09% four... The extra dice rolls are only needed when the previous rolls are exactly at the threshold of success.
It should take less than a minute for people to grasp how this system works.
Monty HarderMonty Harder
\$\begingroup\$ Very smart solution! One that i will likely use \$\endgroup\$ – Martin Brischetto Apr 9 '15 at 21:03
Unfortunately, much of our universe appears to be quantized. One could imagine a system wherein, instead of rolling dice, you precisely measured the net magnetic spin of some ideal gas on non-interacting magnetic moments each of either spin up (1) or spin down (-1). Theoretically such a property, when measured many times, should form a gaussian distribution (a bell curve) with an exceedingly small standard deviation. Unfortunately, this is not quite right. In actuality only certain discrete values of net spin are possible, based on the number of moments in the system. For example, if there is an odd number of moments, it is impossible for the net spin to be exactly 0. Also, obviously, it is impossible for the net spin to ever be any number that is not an integer or to be larger than the total number of particles. All of these concerns have a negligible impact on the behavior of large collections of particles, but they mean that, at a fundamental level, you don't have a continuous distribution.
The example deals with a very idealized situation, but really a great many physical quantities have been shown to be (very probably) quantized by modern physics. The question, then, requires us to find something that isn't.
Oddly, position is currently not believed to be precisely quantized, though this is disputed by some physicists. In the standard model of quantum mechanics, when one puts a single electron in the ground state in an infinite square well, a truly continuous probability distribution (the Schrodinger Equation) describes the likelihood thereafter of finding the electron to be at any given point in space within the box. Thus, with some sophisticated and expensive experimental physics equipment, one could actually maybe be measuring a continuous probability distribution. But here we encounter the second, and more fundamental issue with your plan:
The Arabic numeral system is inherently discrete. Most measuring devices give their outputs in numbers (citation needed). Lets take the number 5.5563, for example. If this is the number our instrument reports, the next possible number is 5.5564. The instrument can't report 5.55631, or any other number between the two possible results. In order to have a continuous distribution we would need an infinite degree of precision and an infinite number of digits. This is a problem.
Clearly the solution is to use a measuring device that doesn't report numbers. Unfortunately, unless it its measurement is based off of gravitational interaction, the act of measurement itself now creates problems for us. All force interactions other than gravity possess a good amount of evidence supporting the existence of a mediating particle. Such mediating particles result in quantization of the interaction, so that only interactions involving an integer number of such mediating particles (quanta) are possible. While people like to speculate about gravitons, there really isn't much evidence yet for their existence and a far bit of evidence to the contrary, so you're actually on pretty solid ground if the measurement device is solely using gravity to measure the position of the electron in the box and reporting that position by means of deflection in its own position (the uncertainty in the latter is of no concern to us; indeed it is helpful to you).
So now we have a continuous probability distribution measured and reported in a continuous manner. So we're good right? Not quite.
Gravity is very weak, and electrons are very small. Unfortunately, large (i.e. massive) objects have an uncertainty in position that is astronomically small. So yes, this works, but you can't see it, which kinda defeats the purpose of a measuring device in the first place. So, for all intents and purposes, it is not currently technologically feasible to generate a random number from a truly continuous probability distribution. In order to do so we'd need something like a macroscopic atomic (i.e. indivisible) object with the mass of an electron, or some such.
And even then quantization would interfere if you were telling where it was by looking at it, feeling it, smelling it, or otherwise measuring it in a physical way via biology.
So, you're pretty much screwed.
But wait! There's hope! The human mind is a powerful thing, and evidence suggests it can create and emulate a continuous probability distribution. You can experience the spiritual and mental directly, rather than indirectly as with the physical, and you can control directly the physical and mental as you cannot the spiritual. So then, if you were to conceptualize a continuous probability distribution and by virtue of the free will granted to you by God you selected a truly random point in the distribution you could have a number of the data type you need to make this work. Do any systems currently existing not only have you do that but also ask you to do math on it? No. Not really.
There are systems that ask you to do this, though. In Amber Diceless, the GM is supposed to (if I understand it correctly) come up what exact thing is retrieved by quick Logrus summoning via a random result with a normal distribution centered about the desired item and the standard deviation directly proportionate to the reciprocal of the time use to search for the thing. This is the only example I am aware of.
Please stop being evilPlease stop being evil
\$\begingroup\$ The lack of continuity and randomness in the world really isn't a problem. Its more about varying the probability curve than it is about the continuity. \$\endgroup\$ – Martin Brischetto Apr 9 '15 at 19:47
\$\begingroup\$ @thedarkwanderer The obvious objection to dice is that simulating a normal distribution by rolling a large number of dice and adding them up really breaks the flow. \$\endgroup\$ – David Richerby Apr 9 '15 at 20:09
\$\begingroup\$ Where did you find evidence suggesting that the human brain can do anything close to a normal distribution, let alone a continuous normal distribution. For example, plotting numbers chosen "at random" by a group of people whose native cultures all use decimal numerical systems will result in huge spikes on those integers with 3s and 7s in them, especially those with 3 or 7 in the least significant digit. \$\endgroup\$ – Matthew Najmon Apr 13 '15 at 11:19
\$\begingroup\$ @MatthewNajmon 1) you aren't pulling from a random sample of people. You are creating and then pulling from a normal-distribution construct. 2) Presumably you are aren't even thinking in numbers since, as mentioned, that would break continuity. Depending on your beliefs about the human mind this may or may not be possible; there is about equal evidence either way, in my opinion. Evidence for the ability to do this would include the exceptional nature of consciousness as a phenomena and the perception of being able to do this. Evidence against it would include the quantization of most stuff. \$\endgroup\$ – Please stop being evil Apr 13 '15 at 14:34
\$\begingroup\$ @MatthewNajmon Throughout this answer I consistently take the most favorable position for the querent. I find it odd that you object specifically to the consciousness argument as that is by far the best supported (scientific evidence, which is what I suppose you to be looking for, is equal-ish, but the philosophical evidence is overwhelmingly in the querent's favor; c.f. philosophy zombies, critiques of determinism, objections to materialism if you're interested) as, for example, supposing that there isn't a graviton is far less reasonable. \$\endgroup\$ – Please stop being evil Apr 13 '15 at 14:43
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Does anyone know of a D20 system that uses the Harry Potter Universe?
With a known probability, how can I determine a roll to fit that probability to a population of n? | CommonCrawl |
What is Modified Internal Rate of Return (MIRR)?
Banking & FinanceFinance ManagementGrowth & Empowerment
Modified Internal Rate of Return, commonly known as MIRR, is an investment evaluation technique. It is a modified version of the internal rate of return (IRR) that overcomes some of the drawbacks of IRR.
MIRR is normally used in capital budgeting decisions to check the feasibility of an investment project.
For example, when the MIRR of a project is higher than its expected return, the investment is considered to be attractive.
Conversely, it would be unwise to take up a project if the MIRR of the project is lower than the expected return of the project.
How to Calculate Modified ARR?
The formula to calculate MIRR is a complex one. One needs to know the future value of a company's positive cash flows discounted at the present reinvestment rate, and the present value of a firm's negative cash flows discounted at the financial cost.
The formula to calculate MIRR is −
$$\mathrm{MIRR =\sqrt[n]{\frac{FV\:(positive \:cash\: flows \:and \:reinvestment \:rate)}{−PV (negative \:cash \:flows\: and\: financen \:rate}}− 1}$$
Where −
FV = the future value (at the end of the last period)
PV = the present value (at the beginning of the first period)
n = number of equal periods in which the cash flows occur (not the number of cash flows)
MIRR vs. IRR
Internal rate of return (IRR) and Modified internal rate of return (MIRR) are two closely related concepts. MIRR resolves a few problems associated with the IRR. For example, one of the major problems with IRR is that it assumes that the obtained positive cash flows are reinvested at the same rate at which they were obtained. MIRR, instead, considers that the proceeds from the positive cash flows of a project should be reinvested at the external rate of return. Generally, the external rate of return is equal to the cost of capital.
The calculations of IRR may provide two solutions in some cases. This fact is ambiguous and has unnecessary confusion regarding the correct outcome. The MIRR calculations on the other hand always have a single solution.
The common idea is that the MIRR provides a more realistic and clearer picture of the return on the investment project compared to the standard IRR. The MIRR is usually found lower than the IRR.
Pros and Cons of MIRR
Following are the advantages of using MIRR −
It is a more accurate indicator of the profitability of a future project. So, MIRR can be used by traders to check whether or not predictions made by IRR are too optimistic.
MIRR assumes that all cash flows are reinvested at the reinvestment rate, which is more accurate than cashflows being reinvested at the IRR.
Following are the drawbacks of using MIRR −
An estimation of the cost of capital has to be made in order to make a decision.
There is a dispute over the theoretical background for the MIRR calculation within academic circles. Moreover, it is complex for non-financial managers.
Probir Banerjee
Published on 28-Oct-2021 12:09:24
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Reed–Muller expansion
In Boolean logic, a Reed–Muller expansion (or Davio expansion) is a decomposition of a Boolean function.
For a Boolean function $f(x_{1},\ldots ,x_{n}):\mathbb {B} ^{n}\to \mathbb {B} $ we call
${\begin{aligned}f_{x_{i}}(x)&=f(x_{1},\ldots ,x_{i-1},1,x_{i+1},\ldots ,x_{n})\\f_{{\overline {x}}_{i}}(x)&=f(x_{1},\ldots ,x_{i-1},0,x_{i+1},\ldots ,x_{n})\end{aligned}}$
the positive and negative cofactors of $f$ with respect to $x_{i}$, and
${\begin{aligned}{\frac {\partial f}{\partial x_{i}}}&=f_{x_{i}}(x)\oplus f_{{\overline {x}}_{i}}(x)\end{aligned}}$
the boolean derivation of $f$ with respect to $x_{i}$, where ${\oplus }$ denotes the XOR operator.
Then we have for the Reed–Muller or positive Davio expansion:
$f=f_{{\overline {x}}_{i}}\oplus x_{i}{\frac {\partial f}{\partial x_{i}}}.$
Description
This equation is written in a way that it resembles a Taylor expansion of $f$ about $x_{i}=0$. There is a similar decomposition corresponding to an expansion about $x_{i}=1$ (negative Davio expansion):
$f=f_{x_{i}}\oplus {\overline {x}}_{i}{\frac {\partial f}{\partial x_{i}}}.$
Repeated application of the Reed–Muller expansion results in an XOR polynomial in $x_{1},\ldots ,x_{n}$:
$f=a_{1}\oplus a_{2}x_{1}\oplus a_{3}x_{2}\oplus a_{4}x_{1}x_{2}\oplus \ldots \oplus a_{2^{n}}x_{1}\cdots x_{n}$
This representation is unique and sometimes also called Reed–Muller expansion.[1]
E.g. for $n=2$ the result would be
$f(x_{1},x_{2})=f_{{\overline {x}}_{1}{\overline {x}}_{2}}\oplus {\frac {\partial f_{{\overline {x}}_{2}}}{\partial x_{1}}}x_{1}\oplus {\frac {\partial f_{{\overline {x}}_{1}}}{\partial x_{2}}}x_{2}\oplus {\frac {\partial ^{2}f}{\partial x_{1}\partial x_{2}}}x_{1}x_{2}$
where
${\partial ^{2}f \over \partial x_{1}\partial x_{2}}=f_{{\bar {x}}_{1}{\bar {x}}_{2}}\oplus f_{{\bar {x}}_{1}x_{2}}\oplus f_{x_{1}{\bar {x}}_{2}}\oplus f_{x_{1}x_{2}}$.
For $n=3$ the result would be
$f(x_{1},x_{2},x_{3})=f_{{\bar {x}}_{1}{\bar {x}}_{2}{\bar {x}}_{3}}\oplus {\partial f_{{\bar {x}}_{2}{\bar {x}}_{3}} \over \partial x_{1}}x_{1}\oplus {\partial f_{{\bar {x}}_{1}{\bar {x}}_{3}} \over \partial x_{2}}x_{2}\oplus {\partial f_{{\bar {x}}_{1}{\bar {x}}_{2}} \over \partial x_{3}}x_{3}\oplus {\partial ^{2}f_{{\bar {x}}_{3}} \over \partial x_{1}\partial x_{2}}x_{1}x_{2}\oplus {\partial ^{2}f_{{\bar {x}}_{2}} \over \partial x_{1}\partial x_{3}}x_{1}x_{3}\oplus {\partial ^{2}f_{{\bar {x}}_{1}} \over \partial x_{2}\partial x_{3}}x_{2}x_{3}\oplus {\partial ^{3}f \over \partial x_{1}\partial x_{2}\partial x_{3}}x_{1}x_{2}x_{3}$
where
${\partial ^{3}f \over \partial x_{1}\partial x_{2}\partial x_{3}}=f_{{\bar {x}}_{1}{\bar {x}}_{2}{\bar {x}}_{3}}\oplus f_{{\bar {x}}_{1}{\bar {x}}_{2}x_{3}}\oplus f_{{\bar {x}}_{1}x_{2}{\bar {x}}_{3}}\oplus f_{{\bar {x}}_{1}x_{2}x_{3}}\oplus f_{x_{1}{\bar {x}}_{2}{\bar {x}}_{3}}\oplus f_{x_{1}{\bar {x}}_{2}x_{3}}\oplus f_{x_{1}x_{2}{\bar {x}}_{3}}\oplus f_{x_{1}x_{2}x_{3}}$.
Geometric interpretation
This $n=3$ case can be given a cubical geometric interpretation (or a graph-theoretic interpretation) as follows: when moving along the edge from ${\bar {x}}_{1}{\bar {x}}_{2}{\bar {x}}_{3}$ to $x_{1}{\bar {x}}_{2}{\bar {x}}_{3}$, XOR up the functions of the two end-vertices of the edge in order to obtain the coefficient of $x_{1}$. To move from ${\bar {x}}_{1}{\bar {x}}_{2}{\bar {x}}_{3}$ to $x_{1}x_{2}{\bar {x}}_{3}$ there are two shortest paths: one is a two-edge path passing through $x_{1}{\bar {x}}_{2}{\bar {x}}_{3}$ and the other one a two-edge path passing through ${\bar {x}}_{1}x_{2}{\bar {x}}_{3}$. These two paths encompass four vertices of a square, and XORing up the functions of these four vertices yields the coefficient of $x_{1}x_{2}$. Finally, to move from ${\bar {x}}_{1}{\bar {x}}_{2}{\bar {x}}_{3}$ to $x_{1}x_{2}x_{3}$ there are six shortest paths which are three-edge paths, and these six paths encompass all the vertices of the cube, therefore the coefficient of $x_{1}x_{2}x_{3}$ can be obtained by XORing up the functions of all eight of the vertices. (The other, unmentioned coefficients can be obtained by symmetry.)
Paths
The shortest paths all involve monotonic changes to the values of the variables, whereas non-shortest paths all involve non-monotonic changes of such variables; or, to put it another way, the shortest paths all have lengths equal to the Hamming distance between the starting and destination vertices. This means that it should be easy to generalize an algorithm for obtaining coefficients from a truth table by XORing up values of the function from appropriate rows of a truth table, even for hyperdimensional cases ($n=4$ and above). Between the starting and destination rows of a truth table, some variables have their values remaining fixed: find all the rows of the truth table such that those variables likewise remain fixed at those given values, then XOR up their functions and the result should be the coefficient for the monomial corresponding to the destination row. (In such monomial, include any variable whose value is 1 (at that row) and exclude any variable whose value is 0 (at that row), instead of including the negation of the variable whose value is 0, as in the minterm style.)
Similar to binary decision diagrams (BDDs), where nodes represent Shannon expansion with respect to the according variable, we can define a decision diagram based on the Reed–Muller expansion. These decision diagrams are called functional BDDs (FBDDs).
Derivations
The Reed–Muller expansion can be derived from the XOR-form of the Shannon decomposition, using the identity ${\overline {x}}=1\oplus x$:
${\begin{aligned}f&=x_{i}f_{x_{i}}\oplus {\overline {x}}_{i}f_{{\overline {x}}_{i}}\\&=x_{i}f_{x_{i}}\oplus (1\oplus x_{i})f_{{\overline {x}}_{i}}\\&=x_{i}f_{x_{i}}\oplus f_{{\overline {x}}_{i}}\oplus x_{i}f_{{\overline {x}}_{i}}\\&=f_{{\overline {x}}_{i}}\oplus x_{i}{\frac {\partial f}{\partial x_{i}}}.\end{aligned}}$
Derivation of the expansion for $n=2$:
${\begin{aligned}f&=f_{{\bar {x}}_{1}}\oplus x_{1}{\partial f \over \partial x_{1}}\\&={\Big (}f_{{\bar {x}}_{2}}\oplus x_{2}{\partial f \over \partial x_{2}}{\Big )}_{{\bar {x}}_{1}}\oplus x_{1}{\partial {\Big (}f_{{\bar {x}}_{2}}\oplus x_{2}{\partial f \over \partial x_{2}}{\Big )} \over \partial x_{1}}\\&=f_{{\bar {x}}_{1}{\bar {x}}_{2}}\oplus x_{2}{\partial f_{{\bar {x}}_{1}} \over \partial x_{2}}\oplus x_{1}{\Big (}{\partial f_{{\bar {x}}_{2}} \over \partial x_{1}}\oplus x_{2}{\partial ^{2}f \over \partial x_{1}\partial x_{2}}{\Big )}\\&=f_{{\bar {x}}_{1}{\bar {x}}_{2}}\oplus x_{2}{\partial f_{{\bar {x}}_{1}} \over \partial x_{2}}\oplus x_{1}{\partial f_{{\bar {x}}_{2}} \over \partial x_{1}}\oplus x_{1}x_{2}{\partial ^{2}f \over \partial x_{1}\partial x_{2}}.\end{aligned}}$
Derivation of the second-order boolean derivative:
${\begin{aligned}{\partial ^{2}f \over \partial x_{1}\partial x_{2}}&={\partial \over \partial x_{1}}{\Big (}{\partial f \over \partial x_{2}}{\Big )}={\partial \over \partial x_{1}}(f_{{\bar {x}}_{2}}\oplus f_{x_{2}})\\&=(f_{{\bar {x}}_{2}}\oplus f_{x_{2}})_{{\bar {x}}_{1}}\oplus (f_{{\bar {x}}_{2}}\oplus f_{x_{2}})_{x_{1}}\\&=f_{{\bar {x}}_{1}{\bar {x}}_{2}}\oplus f_{{\bar {x}}_{1}x_{2}}\oplus f_{x_{1}{\bar {x}}_{2}}\oplus f_{x_{1}x_{2}}.\end{aligned}}$
See also
• Algebraic normal form (ANF)
• Ring sum normal form (RSNF)
• Zhegalkin polynomial
• Karnaugh map
• Irving Stoy Reed
• David Eugene Muller
• Reed–Muller code
References
1. Kebschull, Udo; Schubert, Endric; Rosenstiel, Wolfgang (1992). "Multilevel logic synthesis based on functional decision diagrams". Proceedings of the 3rd European Conference on Design Automation.
Further reading
• Жега́лкин [Zhegalkin], Ива́н Ива́нович [Ivan Ivanovich] (1927). "O Tekhnyke Vychyslenyi Predlozhenyi v Symbolytscheskoi Logykye" О технике вычислений предложений в символической логике [On the technique of calculating propositions in symbolic logic (Sur le calcul des propositions dans la logique symbolique)]. Matematicheskii Sbornik (in Russian and French). Moscow, Russia. 34 (1): 9–28. Mi msb7433. Archived from the original on 2017-10-12. Retrieved 2017-10-12.
• Reed, Irving Stoy (September 1954). "A Class of Multiple-Error Correcting Codes and the Decoding Scheme". IRE Transactions on Information Theory. IT-4: 38–49.
• Muller, David Eugene (September 1954). "Application of Boolean Algebra to Switching Circuit Design and to Error Detection". IRE Transactions on Electronic Computers. EC-3: 6–12.
• Kebschull, Udo; Rosenstiel, Wolfgang (1993). "Efficient graph-based computation and manipulation of functional decision diagrams". Proceedings of the 4th European Conference on Design Automation: 278–282.
• Maxfield, Clive "Max" (2006-11-29). "Reed-Muller Logic". Logic 101. EETimes. Part 3. Archived from the original on 2017-04-19. Retrieved 2017-04-19.
• Steinbach, Bernd [in German]; Posthoff, Christian (2009). "Preface". Logic Functions and Equations - Examples and Exercises (1st ed.). Springer Science + Business Media B. V. p. xv. ISBN 978-1-4020-9594-8. LCCN 2008941076.
• Perkowski, Marek A.; Grygiel, Stanislaw (1995-11-20). "6. Historical Overview of the Research on Decomposition". A Survey of Literature on Function Decomposition. Version IV. Functional Decomposition Group, Department of Electrical Engineering, Portland University, Portland, Oregon, USA. pp. 21–22. CiteSeerX 10.1.1.64.1129. (188 pages)
| Wikipedia |
by cfd.ninja | Mar 19, 2020 | Ansys CFX
The NACA four-digit wing sections define the profile by:[1]
Last two digits describing maximum thickness of the airfoil as percent of the chord.[2]
For example, the NACA 2412 airfoil has a maximum camber of 2% located 40% (0.4 chords) from the leading edge with a maximum thickness of 12% of the chord.
The NACA 0015 airfoil is symmetrical, the 00 indicating that it has no camber. The 15 indicates that the airfoil has a 15% thickness to chord length ratio: it is 15% as thick as it is long.
Equation for a symmetrical 4-digit NACA airfoil
Plot of a NACA 0015 foil generated from formula
The formula for the shape of a NACA 00xx foil, with "x" being replaced by the percentage of thickness to chord, is
{\displaystyle y_{t}=5t\left[0.2969{\sqrt {x}}-0.1260x-0.3516x^{2}+0.2843x^{3}-0.1015x^{4}\right],}
x is the position along the chord from 0 to 1.00 (0 to 100%),
{\displaystyle y_{t}}
is the half thickness at a given value of x (centerline to surface),
t is the maximum thickness as a fraction of the chord (so t gives the last two digits in the NACA 4-digit denomination divided by 100).
Note that in this equation, at x/c = 1 (the trailing edge of the airfoil), the thickness is not quite zero. If a zero-thickness trailing edge is required, for example for computational work, one of the coefficients should be modified such that they sum to zero. Modifying the last coefficient (i.e. to −0.1036) will result in the smallest change to the overall shape of the airfoil. The leading edge approximates a cylinder with a radius of
{\displaystyle r=1.1019{\frac {t^{2}}{c}}.}
Now the coordinates
{\displaystyle (x_{U},y_{U})}
of the upper airfoil surface and
{\displaystyle (x_{L},y_{L})}
of the lower airfoil surface are
{\displaystyle x_{U}=x_{L}=x,\quad y_{U}=+y_{t},\quad y_{L}=-y_{t}.}
Symmetrical 4-digit series airfoils by default have maximum thickness at 30% of the chord from the leading edge.
Equation for a cambered 4-digit NACA airfoil
Plot of a NACA 2412 foil. The camber line is shown in red, and the thickness – or the symmetrical airfoil 0012 – is shown in purple.
The simplest asymmetric foils are the NACA 4-digit series foils, which use the same formula as that used to generate the 00xx symmetric foils, but with the line of mean camber bent. The formula used to calculate the mean camber line is
{\displaystyle y_{c}={\begin{cases}{\dfrac {m}{p^{2}}}\left(2p\left({\dfrac {x}{c}}\right)-\left({\dfrac {x}{c}}\right)^{2}\right),&0\leq x\leq pc,\\{\dfrac {m}{(1-p)^{2}}}\left((1-2p)+2p\left({\dfrac {x}{c}}\right)-\left({\dfrac {x}{c}}\right)^{2}\right),&pc\leq x\leq c,\end{cases}}}
m is the maximum camber (100 m is the first of the four digits),
p is the location of maximum camber (10 p is the second digit in the NACA xxxx description).
For this cambered airfoil, because the thickness needs to be applied perpendicular to the camber line, the coordinates
, of respectively the upper and lower airfoil surface, become
{\displaystyle {\begin{aligned}x_{U}&=x-y_{t}\,\sin \theta ,&y_{U}&=y_{c}+y_{t}\,\cos \theta ,\\x_{L}&=x+y_{t}\,\sin \theta ,&y_{L}&=y_{c}-y_{t}\,\cos \theta ,\end{aligned}}}
{\displaystyle \theta =\arctan {\frac {dy_{c}}{dx}},}
{\displaystyle {\frac {dy_{c}}{dx}}={\begin{cases}{\dfrac {2m}{p^{2}}}\left(p-{\dfrac {x}{c}}\right),&0\leq x\leq pc,\\{\dfrac {2m}{(1-p)^{2}}}\left(p-{\dfrac {x}{c}}\right),&pc\leq x\leq c.\end{cases}}}
In this tutorial you will learn to simulate a NACA Airfoil (4412) using ANSYS CFX. First, we will import the points of the NACA profile and then we will generate the geometry using DesignModeler and SpaceClaim, the mesh using an unstructured mesh in Ansys Meshing. You can download the file in the following link.
by cfd.ninja | Dec 1, 2020 | Ansys CFX, Ansys for Beginners
OpenFOAM vs ANSYS CFX
by cfd.ninja | Mar 19, 2020 | Ansys CFX, OpenFOAM
OpenFOAM is the free, open source CFD software developed primarily by OpenCFD Ltd since 2004. It has a large user base across most areas of engineering and science, from both commercial and academic organisations.
We share the same tutorial using ANSYS Fluent.
Ansys CFX – Compressible Flow
Compressibility effects are encountered in gas flows at high velocity and/or in which there are large pressure variations. When the flow velocity approaches or exceeds the speed of sound of the gas or when the pressure change in the system ( $\Delta p /p$) is large, the variation of the gas density with pressure has a significant impact on the flow velocity, pressure, and temperature. | CommonCrawl |
Project SEED
Project SEED is a mathematics education program which worked in urban school districts across the United States. Project SEED is a nonprofit organization that worked in partnership with school districts, universities, foundations, and corporations to teach advanced mathematics to elementary and middle school students as a supplement to their regular math instruction. Project SEED also provides professional development for classroom teachers. Founded in 1963 by William F. Johntz, its primary goal is to use mathematics to increase the educational options of low-achieving, at-risk students.
Project SEED
Founded1963 in Berkeley, CA
Type501(c)(3) Non-profit corporation
Area served
Nationwide
Key people
Hamid Ebrahimi, CEO and National Director • William F. Johntz, Founder
Websitewww.projectseed.org
The model is to hire people with a high appreciation and love for mathematics, for example, mathematicians, engineers, and physicists to be trained to teach. They are pre-trained in the program to teach Socratically, that is, only by asking questions of the students, rarely ever making statements, and even more rarely, validating or rejecting any answers given. A unique set of hand/arm signals are taught for use by the students constantly throughout the 45 min. lesson to wave their agreement, disagreement, uncertainty, desire to ask a question, partial agreement or desire to amend, or to signal a high five to each answer given by a student to the instructor's leading question. Lessons were lively, rapid paced at times. The signals allow students to support each other, while giving the instructor a way to gauge who's understood, who hasn't got it yet, and even, who is not paying much attention. Various signals also supported classroom management. The classroom atmosphere is one of utmost respect for the inquiry process and students' participation. No student ever feels put down; when their fellow students respectfully disagreed, one is invited to state their case, and the whole class each individually would use whatever signal indicated whether they agreed or disagreed. Logic, detection of patterns, drawing a picture of the problem, and many more reasoning skills were taught. The curriculum addresses primarily algebra and some calculus -- math topics with which their regular classroom teacher is often not well versed. Changing the expectations of the students' teachers, parents and family after they witnessed the students' mental abilities to understand and articulate many truths of mathematics, elevated their expectations for the students' academic abilities generating a more positive environment for their academic success.
About
Project SEED is primarily a mathematics instruction program delivered to intact classes of elementary and middle school students, many from low-income backgrounds, to better prepare them for high school and college math. SEED Instruction utilized the Socratic method, in which instructors use a question-and-answer approach to guide students to the discovery of mathematical principles.
The SEED instructors are math subject specialists, with degrees in mathematics or math-based sciences, who use a variety of techniques including hand and arm signals to encourage high levels of involvement, focus and feedback from students of all achievement levels. The approach is intended to encourage active student learning, develop critical thinking, and strengthen articulation skills. The program also emphasizes assessment of student learning and adaptation of instruction to accommodate different math ability levels.[1]
Project SEED curriculum includes topics from advanced mathematics, such as advanced algebra, pre-calculus, group theory, number theory, calculus, and geometry. SEED instruction is supplemental to the regular math program. While teaching students, Project SEED Mathematics Specialists simultaneously provide professional development training for classroom teachers, through modeling and coaching in its instructional strategies.
History
Founded by math teacher and psychologist William Johntz in 1963 to improve the educational outcomes of low-income and minority students, Project SEED was last run by CEO and National Director Hamid Ebrahimi.
Project SEED started as a result of Johntz teaching a remedial math class at Berkeley High School (Berkeley, California) in 1963. Frustrated by the failure of standard remediation to improve the basic math skills of his students, he began teaching them algebra using a Socratic, question-and-answer technique. They responded well to this new material that allowed them to think conceptually about mathematics, but since they were already in high school, there was little time left for them to turn around their academic careers.
Johntz began using his free periods to try the same strategies to teach Algebra in a nearby elementary school. These fifth and sixth graders responded with enthusiasm to succeeding in the study of a high school subject. Also, this exploration of advanced concepts gave Johntz a chance to revisit and reinforce the grade-level curriculum.[2]
Graduate students and faculty from U.C. Berkeley soon joined Johntz in other Berkeley schools. The program spread through presentations for school districts, corporations and conferences, and became a component of the Miller Mathematics Improvement Program, a program funded statewide in California from 1968 to 1970.[3] Many of the early instructors were university faculty, graduate students, and corporate volunteers. Project SEED became a non-profit, tax-exempt corporation in Michigan in 1970 where state funding brought the program to ten different cities between 1970 and 1975. When it was founded in 1963, the name Project S.E.E.D. was an acronym for “Special Elementary Education for the Disadvantaged.” The program was reincorporated in California in 1987 as Project SEED, Inc. dropping the acronym. This was done primarily to avoid confusion with “Special Education” which had taken on a specific meaning.
Over the years Project SEED has operated in a number of cities and states with funding from state governments, federal grants, school districts, and foundations and corporations. From 1982 through 2002, a district funded program in Dallas, Texas reached tens of thousands of students and hundreds of teachers in dozens of schools. The series of longitudinal studies done by the district evaluation department during that time constitutes the most thorough examination of the effectiveness of Project SEED. Students in identified schools received a semester of Project SEED instruction for three consecutive years beginning in the second, third, or fourth grade, a program design that is now regarded as the preferred model. District teachers working in kindergarten through twelfth grade classrooms received workshops, in-class modeling, and coaching from SEED staff as a part of the Urban Systemic Initiative that was implemented in the Dallas & Detroit school districts in the mid 1990s. The Project SEED professional development program was based on this model.
Hundreds of articles about Project SEED have appeared in newspapers and magazines as well as a number of academic books about successful intervention programs. Many former SEED instructors have gone on to make further important contributions to the field of mathematics education.[4] Currently, Project SEED operated programs in California, Michigan, Indiana, Maryland, Pennsylvania, New Jersey, North Carolina, and Washington state.
Project SEED was dissolved in the early 21st century.
Evaluation and recognition
Longitudinal evaluations over a number of years in different locations with different instructors demonstrate that: Project SEED instruction has a positive impact on immediate mathematics achievement scores, Project SEED instruction has a long-term impact on mathematics achievement, and Project SEED students take more high-level mathematics courses in secondary schools.[5][6]
The following organizations have recognized Project SEED as an effective mathematics education program:
• BEST (Building Engineering and Science Talent) panel [7]
• U.S. Department of Education Program Effectiveness Panel (PEP)/ National Diffusion Network (NDN)[8]
• Eisenhower National Clearinghouse for Mathematics and Science Education [9]
Notes
1. Phillips, S. & Ebrahimi, H. (1993). Equation for success: Project SEED. In G. Cuevas & M. Driscoll (Eds.), Reaching All Students with Mathematics (pp. 59–74). Reston, Virginia: National Council of Teachers of Mathematics.
2. Hollins, E., Smiler, H., & Spencer, K. (1994). Benchmarks in meeting the challenges of effective schooling for African American youngsters. In E. Hollins, J. King, & Hayman, W. (Eds.), Teaching diverse populations: Formulating a knowledge base (pp. 166–174). Albany, New York: State University of New York Press, 1994.
3. Wilson, S. M. (2003). California dreaming: Reforming mathematics education. New Haven: Yale University Press.
4. Warfield, V. M. (n.d.). The impact of a SEED project. Retrieved from http://www.math.washington.edu/~warfield/article.html on 2011-09-22.
5. Webster, W. J. (1998). The national evaluation of Project SEED in five school districts 1997‐1998. Retrieved from http://www.eric.ed.gov on 2011-09-22.
6. Best evidence encyclopedia program reviews: Project SEED. (n.d.). Retrieved from http://www.bestevidence.org/overviews/P/seed.htm on 2011-09-22.
7. BEST: Building engineering and science talent. (2004). What it takes: Pre-K-12 design principles to broaden participation in science, technology, engineering and mathematics. Retrieved from http://www.bestworkforce.org on 2001-09-22.
8. Lang, G. (1995). Educational programs that work: The catalogue of the national diffusion network. Longmont, CO: Sopris West.
9. Eisenhower Regional Consortia (1995). Promising practices in mathematics and science education – 1995. Washington, D.C.: U.S. Dept. of Education, Office of Educational Research and Improvement
References
• Clewell, B. C., Anderson, B. T., & Thorpe, M. E. (1992). Breaking the barriers: Helping female and minority students succeed in mathematics and science. San Francisco: Jossey-Bass Publishers.
• Fashola, O. S., Slavin, R. E., Caldeón, M., & Durán, R. (1997). Effective programs for Latino students. Baltimore, MD: Center for Research on the Education of Students Placed At Risk.
• Fashola, O. S., Slavin, R. E., & Calderón, M. (2001). Effective programs for Latino students in elementary and middle schools. In R. Slavin & M. Calderón (Eds.), Effective Programs for Latino Students. London & Mahwah, NJ: Lawrence Erlbaum Associates, Publishers.
• Hilliard, A (2003). No mystery: Closing the achievement gap between Africans and excellence. In T. Perry, C. Steele, & A. Hilliard, Young, gifted, and black: Promoting high achievement among African-American students (pp. 131–165). Boston: Beacon Press.
• Hollins, E., Smiler, H., & Spencer, K. (1994). Benchmarks in meeting the challenges of effective schooling for African American youngsters. In E. Hollins, J. King, & Hayman, W. (Eds.), Teaching diverse populations: Formulating a knowledge base (pp. 166–174). Albany, New York: State University of New York Press, 1994.
• Mizer, R., Howe, R., & Blosser, P. (1990). Mathematics: Promising and exemplary programs and materials in elementary and secondary schools. Columbus, OH: ERIC Clearinghouse for Science, Mathematics and Environmental Education.
• Nisbett, R. E. (2009). Intelligence and how to get it. New York: W. W. Norton & Company.
• Phillips, S. & Ebrahimi, H. (1993). Equation for success: Project SEED. In G. Cuevas & M. Driscoll (Eds.), Reaching all students with mathematics (pp. 59–74). Reston, VA: National Council of Teachers of Mathematics.
• Slavin, R. E. & Lake, C. (2007). Effective programs in elementary mathematics: A best-evidence synthesis. Baltimore, MD: Johns Hopkins University.
• Slavin, R. E., & Fashola, O. S. (1998). Show me the evidence! Proven and promising programs for America's schools. Thousand Oaks, CA: Corwin Press.
• Slavin, R. E. (2005). Evidence‐based reform: Advancing the education of students at risk. Report prepared for Renewing Our Schools, Securing Our Future: A National Task Force on Public Education (A joint initiative of the Center for American Progress and the Institute for America's Future). Retrieved from http://www.americanprogress.org/ on 2011-09-22.
• Webster, W. J., & Chadbourn, R. A. (1989). The longitudinal effects of SEED instruction on mathematics achievement and attitudes: Final report. Dallas, TX: Dallas Independent School District, TX Dept of Research, Evaluation, and Information Systems.
• Webster, W. J., & Chadbourn, R. A. (1990). The evaluation of Project SEED, 1989‐90. Dallas, TX: Dallas Independent School District, TX Dept of Research, Evaluation, and Information Systems.
• Webster, W. J., & Chadbourn, R. A. (1992). The evaluation of Project SEED, 1990‐91. Dallas, TX: Dallas Independent School District, TX Dept of Evaluation and Planning Services.
• Webster, W. J. (1992). The evaluation of Project SEED, 1991‐92, Detroit public schools. Retrieved from http://www.eric.ed.gov on 2011-09-22.
• Webster, W. J., Dryden, M., Leddick, L., & Green, C. A. (1999). Evaluation of Project SEED: Detroit public schools, 1997‐98. Retrieved from http://www.eric.ed.gov on 2011-09-22.
• Webster, William J. in association with Irene Lee and Mark A. Jones, “Evaluation of Project SEED 2009–2010, Compton Unified School District 2011.”
External links
• Official Project SEED Website https://web.archive.org/web/20190220003024/http://www.projectseed.org/
• http://morethancoins.wordpress.com/2011/04/15/nonprofit-of-the-week-project-seed/
• http://articles.baltimoresun.com/2010-04-05/news/bal-op.seed0405_1_minority-students-project-seed-low-performing-students
Authority control
International
• VIAF
National
• United States
| Wikipedia |
Mathieu group M23
In the area of modern algebra known as group theory, the Mathieu group M23 is a sporadic simple group of order
27 · 32 · 5 · 7 · 11 · 23 = 10200960
≈ 1 × 107.
For general background and history of the Mathieu sporadic groups, see Mathieu group.
Algebraic structure → Group theory
Group theory
Basic notions
• Subgroup
• Normal subgroup
• Quotient group
• (Semi-)direct product
Group homomorphisms
• kernel
• image
• direct sum
• wreath product
• simple
• finite
• infinite
• continuous
• multiplicative
• additive
• cyclic
• abelian
• dihedral
• nilpotent
• solvable
• action
• Glossary of group theory
• List of group theory topics
Finite groups
• Cyclic group Zn
• Symmetric group Sn
• Alternating group An
• Dihedral group Dn
• Quaternion group Q
• Cauchy's theorem
• Lagrange's theorem
• Sylow theorems
• Hall's theorem
• p-group
• Elementary abelian group
• Frobenius group
• Schur multiplier
Classification of finite simple groups
• cyclic
• alternating
• Lie type
• sporadic
• Discrete groups
• Lattices
• Integers ($\mathbb {Z} $)
• Free group
Modular groups
• PSL(2, $\mathbb {Z} $)
• SL(2, $\mathbb {Z} $)
• Arithmetic group
• Lattice
• Hyperbolic group
Topological and Lie groups
• Solenoid
• Circle
• General linear GL(n)
• Special linear SL(n)
• Orthogonal O(n)
• Euclidean E(n)
• Special orthogonal SO(n)
• Unitary U(n)
• Special unitary SU(n)
• Symplectic Sp(n)
• G2
• F4
• E6
• E7
• E8
• Lorentz
• Poincaré
• Conformal
• Diffeomorphism
• Loop
Infinite dimensional Lie group
• O(∞)
• SU(∞)
• Sp(∞)
Algebraic groups
• Linear algebraic group
• Reductive group
• Abelian variety
• Elliptic curve
History and properties
M23 is one of the 26 sporadic groups and was introduced by Mathieu (1861, 1873). It is a 4-fold transitive permutation group on 23 objects. The Schur multiplier and the outer automorphism group are both trivial.
Milgram (2000) calculated the integral cohomology, and showed in particular that M23 has the unusual property that the first 4 integral homology groups all vanish.
The inverse Galois problem seems to be unsolved for M23. In other words, no polynomial in Z[x] seems to be known to have M23 as its Galois group. The inverse Galois problem is solved for all other sporadic simple groups.
Construction using finite fields
Let F211 be the finite field with 211 elements. Its group of units has order 211 − 1 = 2047 = 23 · 89, so it has a cyclic subgroup C of order 23.
The Mathieu group M23 can be identified with the group of F2-linear automorphisms of F211 that stabilize C. More precisely, the action of this automorphism group on C can be identified with the 4-fold transitive action of M23 on 23 objects.
Representations
M23 is the point stabilizer of the action of the Mathieu group M24 on 24 points, giving it a 4-transitive permutation representation on 23 points with point stabilizer the Mathieu group M22.
M23 has 2 different rank 3 actions on 253 points. One is the action on unordered pairs with orbit sizes 1+42+210 and point stabilizer M21.2, and the other is the action on heptads with orbit sizes 1+112+140 and point stabilizer 24.A7.
The integral representation corresponding to the permutation action on 23 points decomposes into the trivial representation and a 22-dimensional representation. The 22-dimensional representation is irreducible over any field of characteristic not 2 or 23.
Over the field of order 2, it has two 11-dimensional representations, the restrictions of the corresponding representations of the Mathieu group M24.
Maximal subgroups
There are 7 conjugacy classes of maximal subgroups of M23 as follows:
• M22, order 443520
• PSL(3,4):2, order 40320, orbits of 21 and 2
• 24:A7, order 40320, orbits of 7 and 16
Stabilizer of W23 block
• A8, order 20160, orbits of 8 and 15
• M11, order 7920, orbits of 11 and 12
• (24:A5):S3 or M20:S3, order 5760, orbits of 3 and 20 (5 blocks of 4)
One-point stabilizer of the sextet group
• 23:11, order 253, simply transitive
Conjugacy classes
Order No. elements Cycle structure
1 = 11123
2 = 23795 = 3 · 5 · 11 · 231728
3 = 356672 = 25 · 7 · 11 · 231536
4 = 22318780 = 22 · 32 · 5 · 7 · 11 · 23132244
5 = 5680064 = 27 · 3 · 7 · 11 · 231354
6 = 2 · 3850080 = 25 · 3 · 5 · 7 · 11 · 231·223262
7 = 7728640 = 26 · 32 · 5 · 11 · 231273power equivalent
728640 = 26 · 32 · 5 · 11 · 231273
8 = 231275120 = 24 · 32 · 5 · 7 · 11 · 231·2·4·82
11 = 11927360= 27 · 32 · 5 · 7 · 231·112power equivalent
927360= 27 · 32 · 5 · 7 · 231·112
14 = 2 · 7728640= 26 · 32 · 5 · 11 · 232·7·14power equivalent
728640= 26 · 32 · 5 · 11 · 232·7·14
15 = 3 · 5680064= 27 · 3 · 7 · 11 · 233·5·15power equivalent
680064= 27 · 3 · 7 · 11 · 233·5·15
23 = 23443520= 27 · 32 · 5 · 7 · 1123power equivalent
443520= 27 · 32 · 5 · 7 · 1123
References
• Cameron, Peter J. (1999), Permutation Groups, London Mathematical Society Student Texts, vol. 45, Cambridge University Press, ISBN 978-0-521-65378-7
• Carmichael, Robert D. (1956) [1937], Introduction to the theory of groups of finite order, New York: Dover Publications, ISBN 978-0-486-60300-1, MR 0075938
• Conway, John Horton (1971), "Three lectures on exceptional groups", in Powell, M. B.; Higman, Graham (eds.), Finite simple groups, Proceedings of an Instructional Conference organized by the London Mathematical Society (a NATO Advanced Study Institute), Oxford, September 1969., Boston, MA: Academic Press, pp. 215–247, ISBN 978-0-12-563850-0, MR 0338152 Reprinted in Conway & Sloane (1999, 267–298)
• Conway, John Horton; Parker, Richard A.; Norton, Simon P.; Curtis, R. T.; Wilson, Robert A. (1985), Atlas of finite groups, Oxford University Press, ISBN 978-0-19-853199-9, MR 0827219
• Conway, John Horton; Sloane, Neil J. A. (1999), Sphere Packings, Lattices and Groups, Grundlehren der Mathematischen Wissenschaften, vol. 290 (3rd ed.), Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4757-2016-7, ISBN 978-0-387-98585-5, MR 0920369
• Cuypers, Hans, The Mathieu groups and their geometries (PDF)
• Dixon, John D.; Mortimer, Brian (1996), Permutation groups, Graduate Texts in Mathematics, vol. 163, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4612-0731-3, ISBN 978-0-387-94599-6, MR 1409812
• Griess, Robert L. Jr. (1998), Twelve sporadic groups, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, doi:10.1007/978-3-662-03516-0, ISBN 978-3-540-62778-4, MR 1707296
• Mathieu, Émile (1861), "Mémoire sur l'étude des fonctions de plusieurs quantités, sur la manière de les former et sur les substitutions qui les laissent invariables", Journal de Mathématiques Pures et Appliquées, 6: 241–323
• Mathieu, Émile (1873), "Sur la fonction cinq fois transitive de 24 quantités", Journal de Mathématiques Pures et Appliquées (in French), 18: 25–46, JFM 05.0088.01
• Milgram, R. James (2000), "The cohomology of the Mathieu group M₂₃", Journal of Group Theory, 3 (1): 7–26, doi:10.1515/jgth.2000.008, ISSN 1433-5883, MR 1736514
• Thompson, Thomas M. (1983), From error-correcting codes through sphere packings to simple groups, Carus Mathematical Monographs, vol. 21, Mathematical Association of America, ISBN 978-0-88385-023-7, MR 0749038
• Witt, Ernst (1938a), "über Steinersche Systeme", Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 12: 265–275, doi:10.1007/BF02948948, ISSN 0025-5858, S2CID 123106337
• Witt, Ernst (1938b), "Die 5-fach transitiven Gruppen von Mathieu", Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 12: 256–264, doi:10.1007/BF02948947, S2CID 123658601
External links
• MathWorld: Mathieu Groups
• Atlas of Finite Group Representations: M23
| Wikipedia |
\begin{document}
\title{A critical view on transport and entanglement in models of photosynthesis}
\author{Markus Tiersch \thanks{Electronic address: [email protected]}~ \thanks{Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Technikerstr.~21A, A--6020 Innsbruck, Austria; Institute for Theoretical Physics, University of Innsbruck, Technikerstr.~25, A--6020 Innsbruck, Austria} \and Sandu Popescu \thanks{H. H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol BS8~1TL, UK} \and Hans J. Briegel\footnotemark[2] } \date{}
\label{firstpage} \maketitle
\begin{abstract} We revisit critically the recent claims, inspired by quantum optics and quantum information, that there is entanglement in the biological pigment protein complexes, and that it is responsible for the high transport efficiency. While unexpectedly long coherence times were experimentally demonstrated, the existence of entanglement is, at the moment, a purely theoretical conjecture; it is this conjecture that we analyze. As demonstrated by a toy model, a similar transport phenomenology can be obtained without generating entanglement. Furthermore, we also argue that even if entanglement does exist, it is purely incidental and seems to plays no essential role for the transport efficiency. We emphasize that our paper is \emph{not} a proof that entanglement does not exist in light-harvesting complexes -- this would require a knowledge of the system and its parameters well beyond the state of the art. Rather, we present a counter-example to the recent claims of entanglement, showing that the arguments, as they stand at the moment, are not sufficiently justified and hence cannot be taken as proof for the existence of entanglement, let alone of its essential role, in the excitation transport. \end{abstract}
\section{Introduction}
In recent years, following the development of quantum information, the phenomenon of quantum entanglement has been identified as being one of the most important aspects of quantum mechanics. It was realized that the presence of entanglement confers quantum systems significantly enhanced power for accomplishing many tasks~\cite{HorodeckiRev}, such as exponentially increased speed-up of computation and significantly enhanced communication capacity. As such it has been very natural to enquire whether biological systems could have evolved to make use of entanglement. Recently, the importance of investigating this question received a major impetus following seminal experiments that indicated the existence of unexpectedly long-time coherent effects in photosynthesis~\cite{Engel2007,Collini2010,Panit2010}. Since coherence is a pre-requisite for entanglement, its discovery raises the possibility that entanglement is also present. Moreover, it is known that the energy transport in light-harvesting complexes is extremely efficient -- could it be the case that entanglement is responsible for this efficiency? A few ground-breaking studies~\cite{Caruso2009,Sarovar2010}, followed by an increasing body of literature~\cite{Ishizaki2010,Fassioli2010,Caruso2010,Bradler2010,Scholak2011,Whaley2011} raised this question and suggested that this is the case. Here we take a critical look at these results.
The main issue to be discussed here is that of coherence versus entanglement. They are definitely not one and the same thing. The existence of entanglement is a stronger criterion than the presence of coherence. That is, entanglement requires the presence coherence, but coherence does not imply entanglement in general. While coherent phenomena can also appear in systems of classical wave mechanics, entanglement is a genuine quantum phenomenon, which is required to violate a Bell inequality, for example.
It is important to mention from the beginning that while the existence of coherence in light-harvesting systems has been experimentally tested (at least in laboratory conditions), the existence of entanglement has been not. This is not surprising -- experimentally proving entanglement is a far more difficult task~\cite{GuehneRev}. As such all the discussions about entanglement in light-harvesting systems is, at present, purely theoretical.
The existence of coherence in these systems seems by now to be well established, and we are not challenging that. It is also quite reasonable to expect that coherence plays an important role in the transport problem, distinguishing it from classical diffusive processes; we are not challenging this either. What concerns us here is the existence of entanglement and its role, if it exists.
Specifically, the main questions we address in our paper are: \begin{itemize} \item Are the assumptions that led to the present claims of entanglement in photosynthesis justified? \item Even if these assumptions were justified and entanglement would exist along the lines of those models, does the entanglement play any significant role in enhancing the efficiency of transport or it is of no consequence? \end{itemize}
We suggest that, despite the sizable body of literature claiming the contrary, the answer to both questions is ``no''.
To be clear, we do not prove that there is no entanglement in photosynthesis. To do that would require knowledge of the system that is well beyond what is available at present. All we do here is to present arguments that point to potential problems in the present claims that entanglement exists and plays a significant role. Our paper should rather be viewed as a counter-example and it is meant to sharpen the further investigation of the problem.
\section{Basic Issues}
Light harvesting complexes in plants and photosynthetic bacteria are comprised of protein scaffolds into which pigment molecules are embedded, e.g.\ chlorophyll or bacterio-chlorophyll molecules. The pigment molecules absorb light in the visible or infrared part of the spectrum, and the resulting electronic excitation (exciton) is transported between the pigment molecules until it reaches a reaction center complex, where its energy is converted into separated charges.
The FMO protein complex of green sulfur bacteria is a trimeric complex that links the chlorosome antenna with the reaction center. Within each of the subunits there are seven chlorophyll molecules in close connection with each other. Each of these molecules can be considered as a ``site'' where the excitation propagating from the chlorosome to the reaction center may be localized. It is the entanglement between these sites during the propagation of excitation that is discussed in~\cite{Caruso2009,Sarovar2010,Ishizaki2010,Fassioli2010,Caruso2010,Bradler2010,Scholak2011}. (Alternatively, so called ``mode''-entanglement has also been considered in~\cite{Caruso2010}. There, the physical systems between which entanglement is investigated are no longer identical to the pigment cofactors but effective systems that are defined by the single excitation spectrum of the Hamiltonian of the coupled pigments.)
Regarding the existence and generation of entanglement in light-harvesting complexes, we draw some intuition from the following formal analogy. As we will detail later, up to certain extent, the FMO complex can be seen analogous to a multi-armed interferometer, where each interferometer arm corresponds to a site in the FMO complex. The propagation of a single excitation through the FMO complex is then analogous to the propagation of a single photon through the interferometer. A single photon propagating through an interferometer can indeed immediately lead to entanglement between the arms~\cite{Enk2005}, hence from this point of view it is not very surprising that a single excitation propagating through the FMO complex may lead to entanglement between sites. However, and this is one of our main concerns, if instead of a single photon we send a coherent state through the interferometer, then no arm entanglement will appear at all. Indeed, at a beam splitter a coherent state is split into a product of coherent states in each of the outgoing arms. As the light passes the various beam splitters in the interferometer, in each step we maintain a direct product of coherent states in each of the arms. We stress that this is true even for very weak coherent states where the probability of having more than a single photon is overwhelmingly small.
There are two lessons to be learned from this analogy. First, whether or not entanglement between interferometer arms exists depends crucially on the initial state. Our concern is that in the case of photosynthesis in which the entire light-harvesting complex is illuminated by weak classical light and not single photons, entanglement may therefore not appear inside the FMO complex.
The second lesson is that the actual dynamics of light propagation through an interferometer, and hence a measure of the transport efficiency, does not really depend on the fact that at single photon level entanglement between arms is produced. Indeed, the dynamics of an interferometer can very well be described at classical level, i.e.\ via coherent states, where the question of entanglement does not appear. Hence we conclude that entanglement beyond the mere existence of coherences, although it does appear at single photon level, is irrelevant for the light transport in interferometers.
However, excitation transport in the FMO complex is not exactly identical to light propagation in an interferometer. It is therefore important to see whether or not the problems we mentioned above are still relevant for excitation propagation through light-harvesting complexes. In this paper we argue that, indeed, entanglement plays no essential role in transport through the FMO complex.
Following from the discussion above, we thus need to revisit the general assumptions that have been made in existing theoretical treatments of entanglement in light-harvesting complexes, in particular the following points: \begin{enumerate} \item the initial state entering the complex, \item the transport between sites and the possible entanglement generated thereby. \end{enumerate} More specifically, we also reconsider the modeling of each of the sites as a simplified two-level system.
\section{Initial excitation}
The available literature on entanglement generated during excitation transport in pigment protein complexes usually assumes an initial state where (a) \emph{only one} site is excited and (b) this site is excited with exactly one quantum of electronic excitation, i.e.\ a ``Fock state'', as done in refs.~\cite{Caruso2009,Sarovar2010,Fassioli2010}, for example. Given these assumptions, the pigments of the light-harvesting complex have been modeled by two-level systems, corresponding to presence or absence of an excitation at that location.
The rationale behind the above assumption for the initial state is often the following. Under illumination by sun light or, in experiment, by femtosecond laser pulses, the photon flux per pigment is weak. An estimation for conditions in full sunlight yields a value of about 10 absorbed photons per chlorophyll molecule per second in the relevant part of the spectrum~\cite{Blankenship}. Considering that the typical timescales for exciton transport is of the order of picoseconds with exciton lifetimes of nanoseconds, this suggests that \emph{if} light is absorbed, then most likely only one excitation is present.
However, our main point is that the light that excites the light-harvesting apparatus is essentially classical, technically a mixture of coherent states. Importantly even if a coherent state is very weak, and the probability of containing more than one excitation is very low, this by no means implies that it can be described as a single excitation Fock state. Hence, using the weakness of the incident light as an argument for modeling the initial state as a Fock state is a fallacy.
A coherent state differs from a Fock state in two important aspects. First of all, a coherent state contains not only one excitation but also terms with more excitations. Second, the coherent state also contains a term, the vacuum, without excitations. In particular, the existence of the vacuum term has interesting consequences. It is completely irrelevant as far as the dynamics of excitation transport is concerned, it just represents the fact that nothing happens, but it has crucial implications to entanglement. Since the entanglement that is considered in light-harvesting systems stems from superpositions of states in which one pigment is excited and all others are in the ground state, a contribution with all pigments being in the ground state, corresponding to the vacuum in the incoming light, plays an essential role and must not be neglected.
At this point another possibility arises, namely, that the initial state in the \emph{FMO complex} that is of interest for us, could be a Fock state for other reasons than the weakness of the incident light. Indeed, the FMO complex in vivo is not directly excited by the incident light, rather it receives its input through the antenna complex. It it thus conceivable that some mechanism in the antenna and its connection to the FMO complex may somehow generate a Fock state. In quantum mechanical terms this amounts to a state preparation process. What would this require?
Producing a Fock state in the FMO complex, would require two things. On one hand, cutting the possibility of more than one excitation. Such mechanisms have been discussed~\cite{Bruggemann2004}, although it is not clear to us how efficient they are in limiting the number of higher excitations within the time frame that is relevant for transport through light-harvesting complexes. On the other hand, to produce a Fock state, one should also be able to eliminate the vacuum component of the state. To our knowledge, no such mechanism has been suggested so far.
This entire discussion above his highly relevant since, as far as entanglement is concerned, the difference between coherent states, even if they are extremely weak, and one-photon Fock states is dramatic. While one-photon Fock states may result in entanglement between linearly coupled systems, coherent states will not. Furthermore, even in non-linearly coupled systems, weak coherent states can only produce very limit amounts of entanglement, far below that produced by one-photon Fock states.
\section{Harmonic oscillator model}
Having discussed the difference between the initial states, we now explicitly illustrate the difference that the two states make regarding entanglement.
In existing studies of entanglement in light-harvesting complexes, the manifold of electronic states of each of the relevant pigments, e.g.\ chlorophyll molecules, is usually restricted to only a few levels, mostly only the electronic ground state and the first excited state. The rich energetic landscape of each chlorophyll molecule is thereby conceptually replaced by a two-level atom.
In order to account for small contributions of higher excited states, we must consider a different model for the electronic level structure of each pigment molecule, that allows for more excitations to be present at a single site. As the simplest possible scenario, consider the analogy of a light-harvesting complex to an interferometer. In this analogy, the electronic level structure of each site is modeled by a harmonic oscillator and thus it is formally identical to a light mode in an interferometer.
For capturing the principles of entanglement generation during the evolution of the excitations in a network of coupled chlorophyll pigments, it is sufficient to first study the interaction and state evolution of the simplest network of only two sites, a dimer. Hence, we first consider how entanglement is generated between two coupled harmonic oscillators. (Incidentally, the present approach to model the pigment molecules by harmonic oscillators has also been employed in~\cite{Eisfeld2011}, however, by means of \emph{classical} harmonic oscillators and with a different aim, namely to illustrate that the phenomenology of quantum coherent exciton transport can also be obtained with a classical coherent model. In the present work, however, we employ a full quantum description, and our focus lies on entanglement.)
We assume for the interaction Hamiltonian a standard form where excitations are exchanged between the modes, the rotating-wave approximation has already been applied, and the systems are taken to be resonant: \begin{equation} \label{eq:Hamiltonian} H_\text{int}=g \hbar (a^\dag b + a b^\dag). \end{equation} The coupling strength is denoted by $g$, and the creation and annihilation operators $a$, $b$, and $a^\dag$, $b^\dag$, for the harmonic oscillators describing molecules A and B, respectively, realize the exchange of an excitation between the molecules.
To exemplify our argument with coherent states, we investigate the initial state of a coherent state for molecule~A, which expanded into the Fock basis of states with $n$ excitations reads \begin{equation} \ket{\psi_A(0)}=\ket{\alpha}=\operatorname{e}^{-\abs{\alpha}^2/2} \sum_{n=0}^\infty \frac{\alpha^n}{\sqrt{n!}} \ket{n}, \end{equation} and whose mean number of excitations is given by $\abs{\alpha}^2$. For molecule B we take the vacuum (ground) state $\ket{\psi_B(0)}=\ket{0}$. The requirement that the light intensity is low and hence an excitation occurs only with little probability, and that higher excited states should virtually not occur, thus formally amounts to $\abs{\alpha}\ll 1$ for the initial state. This initial state is in accordance with the central assumption that \emph{if} an excitation occurs, then most probably there is only a single excitation, because for $n>1$ \begin{equation} \abs{\braket{1}{\psi_A(0)}}^2=\operatorname{e}^{-\abs{\alpha}^2}\abs{\alpha}^2 \gg \operatorname{e}^{-\abs{\alpha}^2} \abs{\alpha}^{2n}/n! =\abs{\braket{n}{\psi_A(0)}}^2. \end{equation} In the interferometer analogy this initial state amounts to a coherent state and a vacuum state for the two incident light modes, respectively.
Under the given Hamiltonian, the state of the two molecules evolves in the interaction picture according to \begin{equation} \label{eq:timeEvolMol} \ket{\psi_{AB}(t)}=U(t)\ket{\psi_A(0)}\ket{\psi_B(0)}=\operatorname{e}^{-igt(a b^\dag + a^\dag b)}\ket{\alpha}\ket{0}. \end{equation} The time evolution of this system of two coupled harmonic oscillators is an elementary problem. A standard textbook calculation yields~\cite{MandelWolf} \begin{align} \ket{\psi(t)} &=U(t) D_\text{A}(\alpha) U^\dag(t) U(t) \ket{0}\ket{0}\\ &=\operatorname{e}^{\alpha Ua^\dag U^\dag -\alpha^*UaU^\dag} \ket{0}\ket{0}. \end{align} Here, the displacement operator $D_\text{A}(\alpha)=\exp(\alpha a^\dag-\alpha^* a)$ is used to construct coherent states by simply moving the ground state (vacuum) away from the phase space origin, $\ket{\alpha}=D_\text{A}(\alpha)\ket{0}$. The unitarity of $U(t)$ allows to sandwich all operators in the exponential by inserting identity operations, the time argument has been omitted, and the ground state is invariant under $U$. Using the relation, \begin{equation} \operatorname{e}^X Y \operatorname{e}^{-X}=Y + [X,Y] + \frac{1}{2!}[X,[X,Y]] + \frac{1}{3!}[X,[X,[X,Y]]] + \dotsb, \end{equation} the sandwiched creation and annihilation operators can be evaluated: \begin{align}\label{conjField1} Ua^\dag U^\dag &= \cos(gt) a^\dag +i \sin(gt) b^\dag \\ UaU^\dag &= \cos (gt) a -i \sin (gt) b. \label{conjField2} \end{align} Since the operators for molecule A and B commute, the exponential can be split in two parts that each give a displacement operator for molecule A and B, respectively, \begin{align} \ket{\psi(t)} &= D_\text{A}[\alpha\cos (gt)] \; D_\text{B}[i\alpha\sin (gt)] \; \ket{0}\ket{0} \nonumber \\ &= \ket{\alpha\cos (gt)} \; \ket{i\alpha\sin (gt)}. \label{prodCoherent} \end{align} In the final state after some interaction time~$t$, each of the two molecules is in a coherent state with parameters $\alpha\cos(gt)$ and $i\alpha\sin(gt)$, respectively. With respect to entanglement, let us point out that for all interaction times and irrespective of $\alpha$, the two molecules are always in a product state, i.e.\ there is strictly no entanglement between the molecules.
Let us spell out the formal analogy between the dynamics in our simplified model of the two interacting chlorophyll molecules, the electronic structure of each of which is modeled by a harmonic oscillator, and two light modes interacting at a beam splitter. At a beam splitter, in Schr\"{o}dinger picture, the state of the two incident and outgoing light modes $\ket{\phi_\text{in}}$ and $\ket{\phi_\text{out}}$, respectively, is related by a unitary operation~\cite{Yurke1986}, which contains the reflectivity/transmissivity of the beam splitter as a parameter~$\theta$: \begin{equation} \ket{\phi_\text{out}} = U(\theta) \ket{\phi_\text{in}} = \operatorname{e}^{-i\frac{\theta}{2}(a^\dag b + ab^\dag)} \ket{\phi_\text{in}}. \end{equation} The unitary operation of the beam splitter is formally identical to that of the two interacting molecules in~\eqref{eq:timeEvolMol}. Therefore, the state of the molecules before and after they have interacted for some time $t$ can be associated to the state of the two light modes before and after they have passed the beam splitter with reflectivity/transmissivity parameter $\theta/2=gt$. A 50/50 beam splitter with $\theta=\pi/2$ thus gives states in the output ports that correspond to the states of the molecules after time $gt=\pi/4$: For an initial coherent state, one obtains a product of coherent states as in~\eqref{prodCoherent}. In contrast, for a single photon entering the beam splitter in one input port, $\ket{\phi_\text{in}}=\ket{10}$, the state emerging from the beam splitter is, $\ket{\phi_\text{out}}=\big(\ket{10}+i\ket{01}\big)/\sqrt{2}$, a maximally entangled state of the two interferometer arms.
Let us translate the observation of a coherent state versus a single photon Fock state from the interferometer analogy back to the model system of interacting pigment molecules. Given that the initial state is a coherent state, the act of limiting the focus of the treatment to the single excitation manifold creates the \textit{illusion} of entanglement. Formally, one projects the total state \eqref{prodCoherent} to the subspace with exactly a single excitation and obtains for the initial state \begin{equation} P_1 \ket{\alpha}\ket{0} \propto \ket{10}, \end{equation} where \begin{equation} P_1 = \proj{10} + \proj{01} \end{equation} is the projection onto the single excitation subspace. That is, there is one excitation on molecule~A and none on~B. For the time-evolved state this projection yields \begin{equation} P_1 \ket{\psi_{AB}(t)} = P_1 \ket{\alpha\cos(gt)}\ket{i\alpha\sin(gt)} \propto \cos(gt)\ket{10}+i\sin(gt)\ket{01}. \end{equation} In particular for $gt=\pi/4$ the state appears to be maximally entangled in analogy to a single photon traversing a 50/50 beam splitter.
It is interesting to ask why does considering all possible excitations remove the entanglement? How can the higher excitations sector, which has a very small contribution, remove the entanglement? This seems to be quite paradoxical. The answer is that if the coherent state is so weak as to have only a very small amount of higher excitations, it necessarily also has a very large vacuum component. Just considering the vacuum term in addition to the single excitation sector reduces the amount of entanglement considerably. Any residual entanglement is eliminated by the higher excitation sector.
To see the effects from above, it is convenient to use a measure for the amount of entanglement as given by the concurrence~\cite{Wootters}, which for a pure state of two systems is defined in terms of the purity of one (any) of the subsystems: \begin{equation} C\big(\ket{\psi_{AB}}\big) = \sqrt{2\left(1-\Tr\rho_A^2\right)} \qquad \text{with} \qquad \rho_A = \Tr_B \proj{\psi_{AB}}. \end{equation} The concurrence of the (renormalized) state after projecting out the single excitation sector gives \begin{equation} C\left(\frac{P_1\ket{\psi_{AB}}}{\norm{P_1\ket{\psi_{AB}}}}\right) = \abs{\sin(2gt)}, \end{equation} which is maximal for $gt=\pi/4$.
Alternatively, if the attention is not strictly limited to the single excitation manifold but the vacuum (ground state) term is considered in addition, the relevant projection to apply to the state is \begin{equation} P_{0,1} = \proj{00} + P_1. \end{equation} For the initial state, this projection yields a superposition of the ground and excited state of the first site, i.e. \begin{equation} P_{0,1} \ket{\psi_{AB}(0)} = P_{0,1} \ket{\alpha}\ket{0} \propto \Big(\ket{0}+\alpha\ket{1}\Big)\ket{0}, \end{equation} and, for the time-evolved state, a superposition of the ground state and the evolved state in the single excitation manifold: \begin{equation} \label{eq:stateP01t} P_{0,1} \ket{\psi_{AB}(t)} \propto \ket{00}+ \alpha\Big(\cos(gt)\ket{10}+i\sin(gt)\ket{01}\Big). \end{equation} The entanglement in terms of concurrence of the normalized state is therefore \begin{equation} \label{eq:concP10t} C\left( \frac{P_{0,1}\ket{\psi_{AB}}}{\norm{P_{0,1}\ket{\psi_{AB}}}}\right) = \frac{\abs{\alpha}^2}{1+\abs{\alpha}^2} \abs{\sin(2gt)}. \end{equation} We find that the maximal amount of entanglement at $gt=\pi/4$ is limited by the dominant ground state contribution for small $\abs{\alpha}$, because the ground state is not entangled.
The entanglement that appears after projecting the state onto its zero and single-excitation subspace with $P_{0,1}$ is simply rescaled with respect to the entanglement found when projecting only to the single excitation manifold. The scaling factor $\abs{\alpha}^2/(1+\abs{\alpha}^2)$ amounts to the single excitation fraction of the projected state, i.e.\ the probability of measuring exactly a single excitation in the state once it has been projected. As long as the assumption holds that the probability of more than one excitation is small, we require that also $\abs{\alpha}$ is very small, and therefore the entanglement reduction due to ground state contribution is larger. The higher excitation terms in the coherent state may be small in absolute magnitude, but they only have to eliminate this weak residual entanglement.
Another aspect that needs to be taken into account when advancing to more realistic models for the study of quantum entanglement in light-harvesting complexes is that superpositions of electronic states on the same molecule may suffer from \emph{decoherence}, primarily due to the coupling of the electronic structure to nuclear degrees of freedom. This process ultimately turns a coherent state \begin{equation} \rho_A(0) = \proj{\psi_A(0)} = \proj{\alpha}, \end{equation} when written as a density operator, into the incoherent mixture \begin{equation} \rho_A(t\to\infty) = \operatorname{e}^{-\abs{\alpha}^2}\sum_{n=0}^\infty \frac{\abs{\alpha}^{2n}}{n!} \proj{n} \end{equation} of electronic states of the individual molecule, meaning that after complete decoherence all coherences between states of different excitation number will have decayed. At room temperature molecules have been demonstrated to remain in a coherent superposition of their electronic ground and first excited state for a 50\,fs-timescale~\cite{Hildner2011}, which might, however, differ for pigment molecules embedded in a protein matrix.
In the worst case, a complete decoherence of an initially coherent state and subsequent projection to ground and single excited manifold yields the incoherent mixture of the ground state and the coherently propagated single excitation: \begin{align} \rho_{0,1}^{(\text{decoh})} (t) &= \frac{1}{1+\abs{\alpha}^2} \bigg[ \proj{00} \\ &+ \abs{\alpha}^2 \Big(\cos(gt)\ket{10}+i\sin(gt)\ket{01}\Big) \Big(\cos(gt)\bra{10}-i\sin(gt)\bra{01}\Big) \bigg]. \nonumber \end{align} The concurrence for this mixed density matrix can be evaluated with Wootter's formula~\cite{Wootters}, and gives \begin{equation} C \left(\rho_{0,1}^{(\text{decoh})} (t) \right) = \frac{\abs{\alpha}^2}{1+\abs{\alpha}^2} \abs{\sin(2gt)}, \end{equation} that is, the same result as for the projected coherent state~\eqref{eq:concP10t}.
\section{N-level system model}
After we have formally established that the molecules are not entangled if the initial state is a coherent state, and that a projection of the state to the single excitation manifold creates the illusion that entanglement were present, let us now investigate how the number of considered levels in addition to the ground state and the single excited state affects the apparent amount of entanglement. In other words, we generalize from the projections $P_1$ and $P_{0,1}$ of the previous section to projections that include higher number of excitations.
An alternative way to look at this question is the following consideration. The electronic eigenstates of molecules do not form an infinite uniform level structure as a harmonic oscillator. It is reasonable to assume that only finitely many excitations can be supported by each pigment molecule, and any number of excitations beyond a certain threshold would cause ionization processes that take this fraction of the state out of the considered events. For example, instead of a coherent state at one of the molecules, which involves contributions of arbitrarily many excitations, the molecule may at most support two excitations. The initial state would thus be the first three terms of the coherent state expansion until $n=2$: \begin{equation} \label{eq:init3} \ket{\psi_A^{(3)}(0)} = \frac{1}{1+\abs{\alpha}^2 + \abs{\alpha}^4/2!} \left( \ket{0} + \alpha \ket{1} + \frac{\alpha^2}{\sqrt{2!}} \ket{2} \right), \end{equation} which corresponds to the projection $P_{0,1,2}$ applied to the coherent state $\ket{\alpha}$, and the result renormalized. For these kinds of initial states, which constitute a modification of coherent states towards a more realistic initial state for pigment molecules, we now study how the number of additionally considered contributions of higher lying excited states affects the amount of entanglement that is generated during the transfer of excitations between two pigment molecules. Clearly, for the highest considered excited state being $N=1$, we recover the result~\eqref{eq:concP10t}, whereas in the limit $N\to\infty$ we recover the case of coherent states that never generate entanglement.
In general, we model each molecule by system of $N$ levels, i.e.\ the ground state and $N-1$ excited states. In parallel with the harmonic oscillator, we choose the initial state to be the ground state for molecule~B, and a state for molecule~A, \begin{equation} \ket{\psi_A^{(N)}(0)}=\ket{\alpha_N}= \frac{1}{\sqrt{\mathcal{N}_{\alpha,N}}} \sum_{n=0}^{N-1} \frac{\alpha^n}{\sqrt{n!}} \ket{n}, \end{equation} which is the projection of a coherent state to the lowest lying $N$ levels. The squared norm $\mathcal{N}_{\alpha,N} =\sum_{n=0}^{N-1} \abs{\alpha}^2/n!$ approaches the value $\operatorname{e}^{\abs{\alpha}^2}$ found for coherent states in the limit $N\to\infty$.
As another exemplary case, let us give the analytic expressions for $N=3$ for the initial state~\eqref{eq:init3} as it evolves according to the interaction Hamiltonian~\eqref{eq:Hamiltonian}. The time-evolved state necessarily only contains terms with at most two excitations: \begin{multline} \ket{\psi_{AB}^{(3)}(t)} = \frac{1}{\sqrt{1+\abs{\alpha}^2 +\abs{\alpha}^4/2}} \bigg( \ket{00} + \alpha \Big( \cos(gt) \ket{10} -i\sin(gt)\ket{01} \Big) \\ + \frac{\alpha^2}{\sqrt{2}} \left( \cos^2(gt) \ket{20} - \sin^2(gt) \ket{02} -i\sqrt{2}\cos(gt)\sin(gt) \ket{11} \right) \bigg). \end{multline} For $\abs{\alpha}\ll 1$, the doubly excited states give only a perturbative correction to the expression of the quantum state to the case $N=2$ in~\eqref{eq:stateP01t}. The expression for entanglement, however, does not only change by a perturbative correction, but it changes considerably. The concurrence of the state is given by \begin{equation} C\left(\ket{\psi_{AB}^{(3)}}\right) = \frac{ \abs{\alpha}^3 \Abs{\sin(2gt)} \sqrt{8+\frac{1}{2}\abs{\alpha}^2 \big( 13+3\cos(4gt)\big)} }{ 4\left(1+\abs{\alpha}^2+\abs{\alpha}^4/2\right) } \end{equation} with its maximum at $gt=\pi/4$ of \begin{equation} C^{(3)}_\text{max} = \frac{ \abs{\alpha}^3 \sqrt{8+5\abs{\alpha}^2} }{ 4\left(1+\abs{\alpha}^2+\abs{\alpha}^4/2\right) }. \end{equation} The expression of the concurrence for $N=3$ is similar in structure to that of the case $N=2$ in~\eqref{eq:concP10t}, with a $\abs{\sin(2gt)}$ modulation in time, and the squared norm of the projected coherent state in the denominator. The time-independent prefactor, however, can no longer be interpreted anymore as the probability of having an excitation in the system, as done for $C^{(2)}_\text{max}$ in~\eqref{eq:concP10t}. Instead, we find a scaling with the third power of $\abs{\alpha}$.
\begin{figure}
\caption{Scaling of maximal value of apparent entanglement as measured by concurrence, $C^{(N)}_\text{max}$, for different $\alpha$ as a function of the number of considered levels $N$. }
\label{fig:CN}
\end{figure}
Expressions for larger $N$ can be straightforwardly obtained, but are omitted here. We have analytically examined expressions for the concurrence up to $N=7$ and generally find a global $\abs{\sin(2gt)}$ modulation in time in accordance with the intuition that maximal entanglement for these initial states is reached for the 50/50 beam splitter configuration, that is, after half the time that an excitation needs to fully move from one pigment to the other. In particular, the expressions for maximal entanglement are of the generic form \begin{equation} \label{eq:CN} C^{(N)}_\text{max} = \frac{ \abs{\alpha}^N }{ \mathcal{N}_{\alpha,N} } f_N \left(\abs{\alpha}^2\right), \end{equation} that is, the apparent amount of entanglement decreases exponentially with the number of considered levels per molecule for $\abs{\alpha}<1$. The factor $f_N$ for higher $N$ is of a similar square root form as found for $N=3$. For $\abs{\alpha}<1$, the leading term of $f_N \left(\abs{\alpha}^2\right)$ is constant but also decreases with $N$, as listed in table~\ref{tab:fN}. Figure~\ref{fig:CN} shows how the maximal apparent amount of entanglement decreases with $N$ for various choices of $\alpha$.
Obviously, the interpretation that given an initial coherent state for $\abs{\alpha}\ll 1$ one arrives at the single photon level is misleading because the entanglement properties of a single photon Fock state are not obtained in this limit. Instead, for smaller values of $\abs{\alpha}$, the apparent entanglement will vanish more rapidly when increasingly many levels of the system are taken into account although they contribute to an ever smaller extent (see figure~\ref{fig:CN}).
\begin{table} \centering
\begin{tabular}{r|ccccccc} $N$ & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline $f_N\approx$ & 1 & $\frac{1}{\sqrt{2}}$ & $\frac{1}{4}\sqrt{\frac{7}{3}}$ & $\frac{1}{4\sqrt{2}}$ & $\frac{1}{24}\sqrt{\frac{31}{10}}$ & $\frac{1}{16\sqrt{5}}$ \rule{0pt}{3.5ex} \\ $\approx$ & 1 & 0.7071 & 0.3819 & 0.1768 & 0.0734 & 0.0280 \rule{0pt}{3.5ex} \\ \end{tabular} \caption{Constant leading order terms of $f_N\left(\abs{\alpha}^2\right)$ for $\abs{\alpha}\ll 1$.} \label{tab:fN} \end{table}
We have also investigated an alternative way of modeling the initial state of an $N$-level molecule by means of \emph{atomic} coherent states (also called \emph{spin} coherent states)~\cite{Zhang1990,MandelWolf}, where instead of the raising and lowering operators of the harmonic oscillator, those of the angular momentum algebra are employed. The $N$ levels of each molecule are thereby modeled by the $N=2s+1$ levels of an effective spin-$s$ particle. Although the computations are more intricate due to different commutation relations, we arrive at qualitatively similar results as presented for the projected harmonic oscillator. We therefore conclude that our argument concerning the entanglement content of system that starts in a coherent state is robust with respect to the specific framework applied.
\section{Transport efficiency versus entanglement}
The \emph{excitation transfer efficiency} in chromophore complexes captures how well an excitation that starts somewhere localized in the complex traverses the network of coupled chromophores to a different location, where it is assumed to leave the complex, e.g. to the reaction center. A question of current interest that has been addressed~\cite{Fassioli2010,Scholak2011} is, whether or not entanglement (rather than mere coherence) impacts the transport efficiency.
Let us first recollect a few essential facts about the recent treatments of excitation propagation and evaluation of transport efficiency in conjunction with the study of entanglement in light-harvesting complexes. In a complex of coupled chromophores with pairwise interaction Hamiltonians of the kind as in~\eqref{eq:Hamiltonian}, the number of excitation quanta are conserved. Therefore, subspaces of a fixed number of excitation quanta, e.g.\ the single-excitation subspace, evolve independently from each other. Cross-contribution of subspaces with a fixed excitation number occur only due to interaction of the chromophores with other degrees of freedom, i.e.\ an environment, and for the excitation energies considered here ($\sim$1\,eV) mostly downward to lower numbers of excitation, due to relaxation for example. Given that the assumption holds that the light intensity is weak and therefore the presence of higher number of excitations happens only with minuscule probability, the single-excitation subspace is essentially only subject to the unitary dynamics according to the system Hamiltonian, excitation-number conserving environment influences such as decoherence, and (excitation-number non-conserving) relaxation to the ground state. Therefore, the single excitation-subspace evolves largely independent for the other subspaces even in the presence of decay mechanisms, and in particular it does not significantly gain excitations from higher lying states during the transfer through the complex. Excitation transfer can only occur via the excited states, since only then an excitation (at least one) is present. The transfer efficiency is usually evaluated by an observable, which is therefore defined only in the excited state manifold, and for the scenarios considered here, gains its dominant contribution from the transport dynamics of the single-excitation manifold. Contributions from higher excited states constitute only a perturbative corrections to the quantum state and therefore also to the observable that quantifies the transport properties. Common examples for measures of the transport efficiency are obtained by integrating the population of a certain exit site, e.g. decay to the reaction center, as done in~\cite{Caruso2009}, or by the highest population of an exit-site during a certain time-window as in~\cite{Scholak2011}, for example.
Formally, for an observable $T$ that measures the excitation transport efficiency in the described way, and for initial states with only a small or even vanishing fraction of higher excited states, one has \begin{equation} \mean{T}\approx \mean{P_1 T P_1}, \end{equation} that is, one can restrict the evaluation of $T$ with good agreement to the predominant contribution from the single excitation manifold, because higher excited states yield only a perturbative correction to this result. Since the Hamiltonian conserves the number of excitation quanta, it is a valid approach to restrict the propagation of the entire excitation dynamics to the single-excitation subspace right from the beginning, as used in the last step of the following transformation: \begin{align} \mean{T} &\approx \bra{\psi(t)}P_1 T P_1 \ket{\psi(t)} \\ & = \bra{\psi(0)} U^\dag(t) P_1 T P_1 U(t) \ket{\psi(0)} \\ & = \bra{\psi(0)} P_1 U^\dag(t) T U(t) P_1 \ket{\psi(0)}. \end{align} Because the projection to the single-excitation manifold is effectively carried out when measuring the transport efficiency, the projection may as well be exchanged with the dynamics to the beginning of the process.
Even with respect to open system dynamics, which is captured by the dynamical map $\Lambda(t)$, that is, the map that contains the formal solution to the Liouville equation $\dot{\rho}(t)=\mathcal{L}\rho(t)$, one can approximate the transport efficiency by only considering the single excitation manifold and the ground state: \begin{equation} \mean{T}=\Tr\left[T\rho(t)\right]=\Tr\left[T\Lambda(t)\rho(0)\right] \approx \Tr\left[T \Lambda(t) P_{0,1} \rho(0) P_{0,1} \right]. \end{equation} Regarding the evaluation of a measure of transport efficiency it is thus, for the given assumption of weak light intensity, a natural and justified assumption to restrict the investigation to the subspace with only a single excitation.
In contrast to observables like measures of transport efficiencies, entanglement must be a \emph{non-linear} property of quantum states, and it can thus generally be not equivalent to coherences, which can be extracted by a linear operator. Because of these intrinsic properties of entanglement measures, it is not possible to exchange in a similar way the dynamics in the full space with a part that is projected to the single-excitation subspace. In fact, it is the projection to a subspace of fixed excitation number, which is a global operation, that introduces the observed entanglement into the system in the first place. In the present case of two molecules, the projection $P_1$ acts like a Bell-state measurement.
From the presented case we can now provide an insight about whether or not the entanglement that may be observed in manifolds of fixed excitation number is of relevance to the state evolution or the transport properties of the system. The transport efficiency is robust under changes of the underlying models and assumptions about the initial state regarding the presence of small perturbative corrections of higher lying excited states, whereas entanglement is not. Therefore, in a pigment protein complex where an initial excitation merely evolves according to the system Hamiltonian and under a coupling to a bath, which introduces decoherence or relaxation, entanglement cannot be a quantifier of transfer efficiency. It cannot be the cause of a large transport efficiency, nor enhance the transport efficiency. The propagation of an excitation in a pigment protein complex is different from the case of, say, quantum information communication tasks like quantum teleportation, where entanglement can be identified as the key resource and unentangled (separable) states cannot be used or, as another example, the fact that in interacting quantum systems entanglement is required to reach the ground state.
To conclude, in the first place, it is not at all clear to us that entanglement exists in the FMO complex; in fact, it seems to us that it does not. Entanglement, as it is at present postulated, relies on the existence of a single excitation in the FMO complex (technically a single excitation Fock state). As we argued, this cannot be obtained by simply having very weak light impinging on the light harvesting complex, as wrongly assumed in most literature on the subject. Consequently, if not simply due to low intensity light, the only other way in which the postulated entanglement may still exist is if there would be a dynamical, active, mechanism of state preparation. We argue that this is also extremely unlikely. Indeed, one would need a non-trivial process based on measuring the number of excitations, and opening the entrance of the FMO complex depending on the presence of a single excitation. Crucially, this process should also eliminate the large vacuum component. Finally, and more significantly, even if entanglement exists, its role for the transport efficiency seems to us to be irrelevant, whereas the role of coherence may be important.
\section*{Acknowlegements} The research was funded by the Austrian Science Fund (FWF): F04011, F04012, and the European Union (NAMEQUAM).
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\end{document} | arXiv |
Discussion of results
Policy implications and conclusion
Socioeconomic determinants of use of reproductive health services in Ghana
Gordon Abekah-Nkrumah1Email author and
Patience Aseweh Abor1
Health Economics Review20166:9
© Abekah-Nkrumah and Abor. 2016
The study examines trends in the consumption of reproductive health services (use of modern contraceptives, health facility deliveries, assisted deliveries, first trimester antenatal visit and 4+ antenatal visits) and their determinants using four rounds of Ghana Demographic and Health Surveys (1993, 1998, 2003 and 2008) data.
The study uses cross-sectional and pooled probit and negative bionomial regressions models to estimate the determinants of use of the above listed reproductive health services for the period from 1993 to 2008.
Summary statistics suggest that the above-listed reproductive health services have consistently improved from 1993 to 2008. However, use of traditional methods of contraception increased in urban centers between 2003 and 2008, although the reverse was the case in rural areas. Regression results suggest that place of residence, access to and availability of health services, religion, and birth order are significant correlates of use of reproductive health services. Additionally, the study suggests that the number of living children has the largest effect on use of modern contraception. The effect of a partner's education on use of modern contraception is higher than that of the woman, and a much stronger correlation exists between household wealth and use of reproductive health inputs than expected.
The study associates the increasing use of traditional contraceptives in urban centers and the much stronger effect of household wealth with urban poverty and the increasing indirect cost of health services, and argues for interventions to improve quality of service in public facilities and reduce inequities in the distribution of health facilities. Finally, the study advocates for family planning-related interventions that involve and target partners given the importance of partner education in the use of modern contraception.
Socioeconomic
Mother and child health constitute a major challenge in many developing countries. For example, it is estimated that 99 % of all maternal deaths in 2008 were in developing countries [1], with Sub-Saharan Africa (SSA) having the highest maternal mortality rate (MMR) of 640/100,000 live births. In addition, statistics available for under-five mortality and morbidity suggest that developing countries shoulder a higher burden compared to developed countries [2, 3]. Thus, a major objective of primary health care programmes in several developing countries is to improve mother and child survival through improved utilization of preventive reproductive and childcare services [4, 5].
To improve mother and child health, the World Health Organisation (WHO) formulated the Mother Baby Package, based on four principles of safe motherhood: (1) Family Planning – to ensure that individuals and couples have the information and services to plan the timing, number and spacing of pregnancies. (2) Antenatal Care – to prevent complications where possible and ensure that pregnancy-related complications are detected early and treated appropriately. (3) Clean/Safe Delivery – to ensure that all birth attendants have the knowledge, skills and equipment to perform a clean and safe delivery, together with postpartum care for mother and baby. (4) Essential Obstetric Care – to ensure that essential care for high-risk pregnancies and complications is made available to all women who need it.
Following the implementation of the Safe Motherhood programme in many developing countries, and an emphasis on investments in reproductive health inputs as a channel to reducing mother and child-related morbidity and mortality, policy makers and academics have become very interested in the factors that determine the use of reproductive health inputs/services. Thus, over the last two to three decades, substantial research efforts have been directed towards identifying and understanding the factors that influence the use of reproductive health inputs. This notwithstanding, coverage of reproductive health services (especially contraception use and delivery assistance) continues to be low, even when MMR and pregnancy-related malnutrition and complications continue to be high in many SSA countries [1, 6]. For example, the 2008 estimated average MMR for SSA was 640/100,000 live births compared to 85/100,000 for Latin America and the Caribbean (LAC). Although Ghana's MMR of 350/100,000 live births is deemed to be one of the lowest in SSA, especially when compared to the 1200/100,000 in Chad. Ghana's figure is nevertheless high compared to 310/100,000 in Bolivia and 17/100,000 in Chile, the highest and lowest respectively in LAC for the same period. The high levels of MMR in the mist low coverage of reproductive health services suggest the need to revisit the use of reproductive health inputs, especially with the availability of more recent datasets.
Although the existing health literature on Ghana abounds in studies that have examined the determinants of reproductive health inputs [7–10], majority of them are either based on a single reproductive health input or on a single cross-sectional dataset. This makes it difficult to see at a glance the changes in the consumption of reproductive health inputs over time and the influence of policy-relevant covariates on several reproductive health inputs. In addition, existing studies have mostly looked at contraception use from an aggregate perspective (i.e., whether a woman uses contraception or not, and whether a woman uses modern contraception or not). We argue that further disaggregation of an input like contraception may be more important in eliciting further information for policy targeting. For example, it is not unreasonable to assume that the effect of socioeconomic factors on the use of contraception will depend on the type of contraception (modern contraception, condoms only, or all other modern contraception other than condoms).
Thus, the current paper pools four rounds of Ghana Demographic and Health Surveys (GDHS) data (i.e., 1993, 1998, 2003 and 2008) and uses that to examine the socioeconomic determinants of use of reproductive health inputs (use of modern contraception, timing of first antenatal visit, number of antenatal visits, health facility delivery and deliveries assisted by health professionals). Specifically, the study first examines changes in the use of the above-mentioned reproductive health inputs across the four surveys. Secondly, the paper examines the socioeconomic determinants of use of the five listed reproductive health inputs through pooled regression estimates. As already indicated, the added value of the current study lies in the fact that the use of four rounds of survey data makes it possible to examine changes in the use of reproductive health inputs across time both at the national, and rural and urban level. Although our regression estimates are based on pooled data, the inclusion of time dummies in the regression model makes it possible to identify a time effect on the use of reproductive health inputs. Thirdly, the disaggregation of use of contraception is important in helping us improve our understanding of the nuanced nature of contraception usage and its determinants.
The study uses four rounds (1993, 1998, 2003 and 2008) of the GDHS datasets. The Ghana Statistical Service, supported by OR/IFC Macro and IFC International Company, collected all four rounds of the GDHS datasets. The GDHS is nationally representative and based on a two-stage probability sampling strategy. Females aged 15–49 years are interviewed from the selected households. In addition, men aged 15–59 years from a sub-sample of a second or third of total households selected are also interviewed. The survey also collect information on children aged between 0 and 59 months. Information collected by the GDHS survey relevant to the study includes: background characteristics of women and their husbands/partners, reproductive histories, current use of contraceptive methods, antenatal visits, delivery assistance and health facility deliveries. For the purposes of estimating the socioeconomic determinants of use of reproductive health services the different waves (1993, 1998, 2003 and 2008) are pooled. In the case of the descriptive statistics, however, the individual waves are analyzed separately.
Variable definition and measurement
Modern contraceptives, delivery care and antenatal care are used as indicators of reproductive health services (dependent variables). These three are selected on the basis that they are part of the four services constituting the package of services under the Safe Motherhood programme.
Current contraceptive usage
In the survey, women are asked about their current contraceptive use, with the first answer being no use of contraception at all, up to use of about 13 other methods of contraception, that are either modern or traditional. This variable is recoded into three dummy variables (use of modern contraception, use of other modern contraception, that is, all other modern methods excluding condoms and use of only condoms). The three dummy variables are coded 1 where the relevant method is in use, otherwise 0. Traditionally, contraceptive models have been formulated as use of modern or non-modern methods. This is on the basis that non-modern methods are known to be ineffective and therefore could be likened to a situation of not using contraceptives at all. Thus, the decision to disaggregate the variable into the three distinct dummies is to enable us to examine the nuanced nature of the use of the different contraceptive categories (modern, condoms, and other modern methods).
Delivery care
Two dummy variables are used to capture delivery care for the last birth preceding the survey. These are deliveries assisted by health professionals (doctors, nurses and midwives) and deliveries occurring in a health facility (private or public). The variables are coded 1 if delivery took place in a health facility or was assisted by any of the three health professionals, otherwise the variable is coded 0. The choice of the two variables is on the basis that they give a woman in labour, access to professional delivery services and emergency obstetric care (EOC) where necessary.
Antenatal care
The antenatal visits variable captures the number of antenatal visits made by the pregnant woman (i.e. count form 1,2,3…n). However, WHO recommends at least 4 antenatal visits for a pregnant woman to be deemed protected from pregnancy-related risk and complications [11, 12]. Based on this recommendation, we assume that any number of antenatal visits fewer than 4 is as risky as not going at all. Thus, the variable is coded as binary (1 if a woman had 4+ visits, or else 0). In addition, antenatal visit is used in an ordered and count form to enable us to examine whether the determinants of the intensity of use of antenatal services differ from the determinants of the decision to use or not to use antenatal services. The definition and summary statistics of the remaining variables (i.e., both dependent and independent variables) used are captured in Table 1.
Summary statistics for use of reproductive health inputs − pooled data: 1993, 1998, 2003, 2008
Contraceptive models
Antenatal and delivery
Modern contraception
Delivery assistance
Use of condoms only
Other modern contracep
Woman's age
15–19 (Ref)
15–19 (1 = base)
20–24 = (2)
Woman's education
One child (Ref)
No educ (1 = Base)
Primary = (2)
Secondary = (3)
Four and above
Tertiary = (4)
Partner education
No educ (Ref)
Marriage dummy
Muslim dummy
Akan (1 = Base)
Ga/Dangme = (2)
Missing Husb. Dummy
Ewe and Guans = (3)
North ethnicities = (4)
Others = (5)
Akan (Ref)
Household wealth
Ga/Dangme
Poorest (1 = Base)
Ewe and Guans
Poorer = (2)
North ethnicities
Middle = (3)
Richer = (4)
Number of elderly
Richest = (5)
Ecological zones
Poorest (Ref)
Southern belt (1 = Base)
Poorer
Capital city = (2)
Middle belt = (3)
Richer
Northern belt = (4)
Rural dummy
NSCPHGW
Southern belt (Ref)
NSCPHFT
NSCPCCV
Middle belt
No. of living children
Northern belt
No child (1 = Base)
One child = (2)
Two children = (3)
Three children = (4)
Four and above = (5)
1993 dummy
1993 dummy (1 = Base)
1998 dummy = (2)
Sample dummy
Timing of 1st antenatal
No. antenatal visits
Source: Authors' calculations. Calculations take account of sample weights. Note that the models on timing of 1st antenatal visits and number of antenatal visits are based on slightly different samples per the sample dummy. NSCPHGW, NSCPHFT and NSCPCCV are the non-self-cluster proportion of households with good water, non-self-cluster proportion of households with flush toilets, and non-self cluster proportion of children under five with complete vaccination, respectively. The values in parentheses next to the variables are the definitional codes. Note, partner's education includes a 5th category (missing husbands), which is excluded from the table. This was added to cater for women who do not have partners and would otherwise have been dropped from the regressions
Statistical estimation
As indicated in Section One, the object of the study is examining the determinants of a woman's decision to use reproductive health services or not in Ghana. Framing the question in this form reduces the woman's decision into a binary choice set (i.e. using or not using reproductive health services). If the two alternatives are generalized as J, and an indirect utility derived from choosing any of the two alternatives as V, then the probability that a woman will use or not use reproductive health services can be expressed as below.
$$ \Pr \left({V}_j=1\right)= \Pr \left({X}_j\beta +{\varepsilon}_j>0\right). $$
Where, for instance, (V j = 1) if reproductive healthcare is used based on the definition of the variables in Table 1, and (V j = 0) if otherwise. X represents a vector of explanatory variables, and β are coefficients to be estimated. Consistent with the extant literature, (see for example: [13, 14], X is carefully selected to include individual level factors of the women (i.e., age, birth order/number of living children, level of education and that of her partner, marital status, religion and ethnicity), household factors (i.e., household wealth index and number of elderly women in the household) and Community factors (i.e., place of residence and availability and accessibility to health facilities). Unfortunately, the GDHS data does not contains variables (distance to health facility, category of health personnel, and health infrastructure) that have commonly been used as proxies to capture availability and accessibility to health facilities [8, 15, 16].
Thus we follow prior authors [17–19] to compute the non-self cluster proportion of households with access to good water (NSCPHGW), a non-self cluster proportion of households with flush toilets (NSCPHGS), and a non-self-cluster proportion of children with complete vaccinations (NSCPCCV) as proxies for accessibility and availability of health services.
With Equation 1, we are assuming that all dependent variables are binary, including antenatal visits as discussed in Section 2.2. Although the paper's focus is examining the determinants of use or otherwise of reproductive health services (i.e. binary form), we additionally model the determinants of antenatal care visits in an ordered and count form via an Ordered Probit (OP) and a Negative Bionomial Model (NBM). The use of OP and NBM makes it possible to examine the marginal effect of each additional visit to the threshold of 4+ (in the case of Ordered Probit) or the maximum number of visits (in the NBM).
For the Ordered Probit, antenatal visits are deemed to be in an ordered discrete choice form (1, 2, 3….4+). Thus, the probability that a mother chooses any of the alternatives will increase with utility derived. Assuming there are I possible outcomes or antenatal choices facing a mother, a set of threshold coefficients or cut points {K 1, K 2, …, K I − 1} is defined for K 0 = − ∞ and K 0 = ∞, and the choice of antenatal care for the J th mother may be generalized as:
$$ \Pr \left({V}_j=i\right)= \Pr \left({K}_{i-1}<{X}_j\beta +{u}_j<{K}_i\right). $$
Where the probability that individual j will choose outcome i depends on the attributes of antenatal care and those of the individual/households and community (X j β) falling between (i − 1). X represents a vector of explanatory variables, also defined in Table 1, and β are the coefficients to be estimated. The cut-points for the antenatal healthcare choices are based on ordering the number of visits made to the health centre, i.e., ranging from 0 visits, 1 visit… to the maximum number of visits which according to the WHO standards is 4+ for appropriate antenatal care. Thus, Equation 1 is used to estimate all the binary dependent variables, while Equation 2 is used to estimate the determinant of antenatal visits in an ordered form. In the case of the intensity of use of antenatal visits to the maximum number, an NBM is used and the model specification is attached as Appendix 2, with both the estimates of the Ordered Probit and NBM contained in Table 5 in Appendix 1.
Descriptive results
In this section, we present trends in the use of the three reproductive health inputs at the national and rural/urban areas. Figures 1 and 2 present contraceptive usage at the national level for all women, and women below 34 years of age, respectively. Figures 1 and 2 suggest that the use of modern contraceptives (i.e., any modern method, condoms only and other modern methods) have been improving gradually over the years, except in the case of traditional methods where, as expected, usage is on the decline. Condoms seem to be the least used method of contraception, although the rate of use among women 34 years and below is higher than the average among all women. What is, however, surprising is the fact that apart from traditional methods, use of all other methods of contraception declined between 2003 and 2008. The figures in Table 2 suggest a 19.7 % drop in the use of modern contraceptives between 2003 and 2008. Given that rural consumption continued to increase for the same period, the national level drop in the use of all forms of modern contraception may be attributed to the urban decline in the use of modern contraception.
Trends in contraceptive usage - all women (%)
Trends in contraceptive usage - Women under 35 (%)
Trends in the use of reproductive health inputs in Ghana
Reproductive health inputs
National estimates
Contraceptive usage
Place of delivery
Professional delivery assist
Antenatal visit in 1st trimester
4+ antenatal visits
Urban estimates
Rural estimates
Source: Authors' calculation
Note: Calculation takes account of sample weight
In addition to the use of modern contraception, the results in Table 2 suggest that health facility deliveries and deliveries assisted by health professional have been increasing gradually in Ghana. Even when the data is disaggregated into urban and rural areas, health facilities and assisted deliveries continue to show gradual increases, except for the large gap between rural and urban areas. Antenatal care (i.e., antenatal visit in first trimester and 4+ visits) also improved across years, both at the national and disaggregated level (rural/urban). The results in Table 2 equally suggest a marginal rural/urban difference in whether the first antenatal visit occurred in the first trimester, whereas for 4+ antenatal visits, the rural/urban gap remains large.
Although consumption of contraceptives declined for the period 2003 to 2008, the general trend has been that consumption of reproductive health inputs has been better in Ghana compared to many other African countries. For example, Ghana's percentage of women making 4+ antenatal visits, delivering in a health facility and using modern contraception and condoms in 2008 is relatively better than respective figures in Liberia (66 %, 36.9 %, 11.7 % and 3.5 %), Nigeria (44.8 %, 35 %, 10.5 % and 4.7 %), Sierra Leone (56.1 %, 24.6 %, 8.2 % and 1.1 %), Madagascar (49.3 %, 35.3 %, 23 % and 1 %) and Kenya (47.1 %, 42.6 %, 28 % and 2.6 %). Notwithstanding this, it is also the case that Ghana's performance compares unfavourably to other developing countries such as Bolivia (72.1 %, 67.5 %, 24 % and 3.6 %), Paraguay (90.5 %, 84.6 %, 52.4 % and 16.3 %) and Jamaica (87 %, 97.6 %, 52 % and 19.4 %) [20].
Regression results
As earlier indicated, the determinants of use of reproductive health services are estimated using probit models. However, in the case of antenatal visits, additional models; Ordered Probit and Negative Binomial Models (NBM) were used to estimate the marginal effect of every additional visit from 1, 2, 3 and 4+ and 1, 2, 3, 4….n visits respectively. Although the coefficients of the Ordered Probit and NBM were slightly different from that of the probit model, the direction of correlation and level of significance are generally the same. Thus, we present the results of the probit models. In addition, number of living children (NLC) in a contraception consumption model could be endogenous based on reverse causality. A standard correction to this challenge is the implementation of instrumental variable (IV) procedure. However, it is very difficult to find appropriate instruments for endogenous NLC from the DHS data. In the absence of an IV procedure, an alternative is dropping the NLC from the model. However dropping the NLC from the model could potentially result in endogeneity bias arising from omitted variables, especially given the fact that NLC has the highest effect on consumption of contraceptives (see Table 3). For the avoidance of doubt, we have re-estimated the model without NLC. The results (not shown) remain generally the same as including it in terms of direction of correlation and level of significance, but with a drop in the goodness of fit of the model. Thus, we argue that removing the NLC from the contraception model will equally lead to an endogeneity bias, in addition to compromising the goodness of fit of the regression model. Thus, the contraception model uses the NLC as one of the covariates.
Socioeconomic determinants of contraception use in Ghana
Sample of all women
Sample of women 34 and below
Use of modern contraception
Use of other modern methods
−0.0058
0.0437*
−0.0077**
−0.0070*
−0.0129***
0.0547***
0.0097**
Missing husband dummy
Women in union
Northern ethnicities
Pseudo R 2
P-value
Source: Authors' calculations
Note: *** is significant at p < 0.01, ** is significant at p < 0.05, * is significant at p < 0.10. NSCPHGW, NSCPHFT and NSCPCCV are the non-self cluster proportion of households with good water, non-self cluster proportion of households with flush toilet and non-self cluster proportion of children under five with complete vaccination, respectively
Partner's education includes a fifth category (missing husbands) but this is excluded from the table. It was added to cater for women who do not have partners and would otherwise have been excluded from the regressions
It is important to caution that the probit estimates should be interpreted with care given the potential endogeneity of number of living children in the contraception models. It is also important to acknowledge that our quasi R-square is low. However, this in itself is not a challenge given that most relevant variables used in the literature are included in our model and the fact that in general, not much emphasis is often placed on the quasi R-square in a probit model.
As per the results in Table 3 (see estimates for sample of all women), age does not have a significant effect on use of modern contraceptives, although women in the 20–24, 35–39 and 40–44 age brackets are more likely to use other modern contraceptive methods (i.e., modern contraceptive methods other than condoms). Where modern contraceptives is redefined to mean only condoms, all the coefficients on women's age, with the exception of women in the 20–24 age bracket, become significant with a change in sign from positive to negative. This suggests that compared to younger woman, relatively older women are less likely to use condoms as contraceptives. Even where the model is re-estimated using a sample of women below 34 years of age, the results generally remain the same. Besides contraceptive use, age has a positive correlation with pregnancy-related reproductive health services (i.e., whether the first antenatal visit occurred in the first trimester, 4+ antenatal visits, health facility deliveries and deliveries assisted by health professionals – See Table 4). However, it is important to note that as per the size of the coefficients, the effect of age on consumption of pregnancy-related reproductive health services increases with age, reaches a peak around 40–44 and declines from age 45 and beyond.
Socioeconomic determinants of antenatal care and delivery care in Ghana
1st trimester antenatal visit
Health facility deliveries
Delivery assistance by health professional
2nd birth order
3rd birth order
4th plus birth order
No. of elder women HH
No. of observations
Chi2
Partner's education includes a 5th category (missing husbands) but this is excluded from the table. It was added to cater for women who do not have partners and would otherwise have been excluded from the regressions
Except for first trimester antenatal visits, women and partners' education and household wealth are positively and significantly correlated with all the dependent variables for contraception use, antenatal and delivery care. Although both women and partner's education are significant and positive, the coefficients of partners' education are slightly higher than that of women's education in the contraception model. The reverse is true for the antenatal and delivery care models. In addition to the fact that the effect of household wealth is significant and positive, the size of the coefficients increases as one moves from a lower to a higher wealth category. Compared to unmarried women, married women are more likely to use any form of modern contraception, although the probability of use reduces in the case of condoms. In addition, the results also suggest that compared to other religions, Muslim women are less likely to use any form of modern contraception, have their first antenatal visit within the first trimester of pregnancy, have 4+ antenatal visits, deliver in a health facility or to deliver with the assistance of a health professional. Whereas women who have more living children are more likely to use different forms of modern contraceptives, women with 2nd to 4th order births are less likely to use antenatal or delivery care. In the case of birth order, the size of the coefficients increase as a woman moves from a lower order birth to a higher order birth.
The ecological zone and rural dummies are not significant in the contraception models. However, rural women are less likely to have 4+ antenatal visits, deliver in a health facility and have professionally assisted deliveries. In addition, women in the capital city and middle belt are more likely to have 4+ antenatal visits, deliver in a health facility and have professionally assisted deliveries, compared to women in the southern belt. The results also show that women from Northern Ghana are significantly more likely to have 4+ antenatal visits compared to women from Southern Ghana, but less likely to go for antenatal visits in the first trimester (p < 0.10), deliver in a health facility or use delivery assistance from health professionals (p > 0.10). Also, NSCPCCV is significantly positively correlated with modern contraceptives and other modern contraceptives. In addition, NSCPCCV, NSCPHGW and NSCPHFT are significantly positively correlated with 4+ antenatal visits, health facility delivery and professionally assisted deliveries.
Finally, the coefficients of the year dummies suggest that women were more likely to use modern contraception or any other modern contraception in 1998, 2003 and 2008, respectively, compared to 1993. In the case of antenatal and delivery care, however, the results suggest that women in 1998 and 2003 were less likely to have 4+ antenatal visit, deliver in a health facility and have professionally assisted deliveries compared to women in 1993.
The descriptive results suggest that condoms are popular among women 34 years and below compared to all other women. This may be due to the fact that such women are more likely to find alternative contraceptive methods such as pills, injectables, and implants as intrusive and stigmatizing. Additionally, the descriptive results suggest a decline in urban consumption of modern contraceptives. Although reasons for the decline are not directly evident from the data, urban expansion arising from rural–urban migration may provide a plausible explanation. The implications of rural–urban migration may be an increased number of urban dwellers who have characteristics (education and household wealth) similar to rural dwellers. In addition, such migrants often live at the peripheries/fringes of the city or in urban slums where access to health facilities or services are highly constrained. Given such constrained access to health services, lower levels of education and income, it is reasonable to argue that women needing contraceptives may turn to available substitutes such as traditional methods. Indeed, recent evidence from the Multiple Cluster Indicator Survey [21] suggests a decline in urban health facility deliveries at a time when health facility deliveries in rural areas are increasing. Finally the descriptive results show a large rural urban gap in 4+ antenatal visits. This gap may be explained by a variety of factors, including poor road infrastructure, longer average distance to health facilities in rural areas, and the skewed distribution of health facilities and health personnel in favour of urban centres, therefore making it difficult, if not impossible, for women to have access to and consume reproductive health services even when available.
In the case of the regression results, the effect of a woman's age on use of condoms and other modern contraception is not unexpected. The finding that women above the age of 25 are significantly less likely to use condoms compared to women below 20 may be explained by the fact that younger women who may not have started bearing children are afraid that use of other modern contraceptives (such as injectables, pills and implants) create infertility problems and may therefore not be willing to use them [22, 23]. Conversely, women who are 25 years of age or older are more likely to be married and may need the consent of their partner to use condoms, which are more likely to interfere with sexual relations. Thus, the use of other forms of modern contraception, seen as less interfering sexually, may appeal to such women much more than condoms. In addition, the inverted U-shaped relationship between age and pregnancy-related-reproductive health inputs may be due to the fact that pregnancy complications increases with age, leading to increased consumption of reproductive health inputs among relatively older pregnant women [7, 24]. However, given that reproductive activity reduces at older ages (35–44), it is reasonable to assume that consumption of reproductive health inputs will decline among women of such age group [25, 26].
Women's education may be a proxy for women's autonomy; an important determinant of women's ability to make strategic life choices [27, 28]. These include decisions to use contraceptives, visit the hospital for antenatal care and deliver in a health facility [29]. Similarly, educated women are likely to be more efficient (through access to and use of health-related information) in the production of health compared to their uneducated counterparts [8, 14, 30, 31]. The difference in the size of the women and partners' education coefficients, although marginal, is still important. As indicated earlier, education may influence household decision-making and, possibly, control of the choice or consumption of reproductive health services. Thus, it may be the case that on matters of contraception, partners have greater control over decision-making [32–35]. Hence, partners who are educated and understand the benefit of contraception use are more likely to exert such influence in the decision to use modern contraception. The higher effect of women's education on the pregnancy-related reproductive health inputs compared to her partner's education may be a reflection of access to resources rather than control of decision-making.
The positive effect of household wealth on the use of modern contraception is expected. In Ghana, family planning products are generally controlled by the private sector and are outside the domain of mainstream clinical service providers. Thus, family planning consumables such as condoms, pills and injectable are sold on the market at slightly subsidized prices, making access to resources/wealth an important determinant [35, 36]. In the case of antenatal and delivery care, the positive effect of household wealth is somewhat surprising, especially when one considers the fact that such services are free in public facilities and also covered by the National Health Insurance Scheme. Perhaps the indirect cost of these inputs (distance to health facility and the opportunity cost of visiting a health facility) may be as important as fees paid at the point of service. Alternatively, the poor quality of service at some public facilities may mean that some potential users turn to private providers, who charge market prices and thereby make household wealth an important determinant. The fact that the private health sector in Ghana (Private For Profit Providers, PFPP; Faith-Based Providers, FBP; and Private Non-Profit Providers, PNPP) accounts for around 55 % of health services [37] lends some credence to this suggestion. In addition, an analysis of data from the GLSS 4 (1998/99) and GLSS 5 (2005/06) suggests that the proportion of the respective survey sample who had medical problems and sought help from public facilities dropped from 48 to 45 %, while those who sought help from PFPP and PNPP increased marginally from 47 to 49 %, and 6 to 8 %, respectively [38]. Descriptive results from the GLSS 4 and 5 suggest that 51 and 48 % of the sample, respectively, in the lowest income quintile used services of private providers against 48 and 49 % for those in the richest quintile. Similarly, 48 and 51 % of the sample from rural areas used the services of private providers, against 50 and 47 % of those from urban centres.
In addition, the positive effect of being located in the capital city or the middle belt reflects the resource-rich nature of these zones as well as the concentration of social services such as schools and health facilities, thereby improving access relative to the southern belt. To the contrary, the negative effect of the Northern belt and women living in rural areas reflects a high prevalence of poverty and inadequate infrastructure such as health facilities in rural areas. For example, four rounds of the GLSS – 1991/92, 1998, 2005/06 and 2014 – have consistently cited the Northern belt (Northern, Upper East and Upper West regions) to be the most poverty endermic zone in Ghana. Also, the finding that rural women are less likely to use reproductive health inputs compared to urban women may be due to the fact that in Ghana, as in many developing countries, social infrastructure such as health, water and sanitation facilities tend to be clustered around urban centres. Thus, urban dwellers are more likely to be closer to such facilities and therefore to use them compared to rural women [26, 39, 40].
The fact that married women and women with more living children are more likely to use contraceptives is straightforward and consistent with the existing literature [22, 23]. In addition, the size (largest) of the coefficient of NLC on use of modern contraception is significant: the NLC a woman has is the single most important decision point for the use of contraceptives. This may have undesired implications for population control, especially in a society like Ghana where cultural pressures favour relatively large family sizes. For example, the 2008 GHDS suggests that on the average, a Ghanaian women desires to have four children. For birth order, the negative correlation may be due to the fact that first time/early births are more likely to be associated with pregnancy and birth-related complications. This may explain first timers' use of more reproductive health inputs compared to women with later order births. It may also be the case that first-timers/women with early order birth may be responding to recommendations from health workers to use reproductive health inputs to reduce the level of risk normally associated with first-time pregnancies [41]. The negative effect on the Muslim dummy, is perhaps an indication that where beliefs associated with the Muslim religion conflict with the demands of modern medicine such as reproductive healthcare, Muslim women may opt not to use it [10, 42]. Not surprisingly, prior authors have found that in Ghana, Muslim woman are less likely to use reproductive health inputs compared to Christian women [7, 9, 10]. The positive effect of the health accessibility and availability proxies (NSCPHGW, NSCPHFT and NSCPCCV) confirms the existing literature [8, 43] that social infrastructure such as health facilities and health personnel are crucial to the consumption of reproductive health inputs.
This study set out to examine the changes in the use of reproductive health inputs (use of modern contraception, and antenatal and delivery care) over time, from 1993 to 2008, as well as the socioeconomic determinants of use of reproductive health inputs. The findings of the study have important implications for reproductive and child health policy formulation. First, the increased use of traditional methods of contraception in urban areas is worrying. It may therefore become important for policy makers to revisit the rural–urban equity narrative in the face of high levels of rural–urban migration, as indicated earlier. The existing narrative that tends to emphasize the fact that rural dwellers are worse off compared to their urban counterparts may lead to resource concentration in rural areas in some cases. For example, the desire in Ghana to bridge the rural–urban gap in the use of reproductive health services, and for that matter reduce MMR in rural areas, led to the adoption and implementation of the Community Health Planning and Services (CHPS) programme in 2003. After about a decade of being implemented, evidence from GDHS 2008 and MICS 2011 suggests that the use of some reproductive health inputs (modern contraception and health facility deliveries) have improved in rural areas at a time when usage is declining in urban centres. Although this paper is not suggesting that the rural–urban consumption difference in the said reproductive health inputs is due to the presence or otherwise of the CHPS programme, it will be equally important for policy makers to relook at how to balance rural–urban resources distribution in a manner that responds to current needs.
Secondly, the fact that the probability of using reproductive health inputs increases with the level of wealth is critical for policy intervention and targeting. As indicated earlier, antenatal and delivery care are generally free and catered for under the National Health Insurance System. Thus, the strong correlation between household wealth and use of antenatal and delivery care suggests that costs other than the direct cost of the services rendered may be very important. Thus, policy makers may need to revisit the discourse on reducing the indirect cost of accessing reproductive health services, which in Ghana is more likely to be associated with the average distance to health facilities and the opportunity cost of visiting the health facilities.
In addition, the fact that the number of living children has the largest effect on the probability of using modern contraceptives in a country where, on the average, women desire to have four children should be an issue for policy attention. The development literature suggest that the desire for large family sizes in developing countries is normally driven by the need for farm hands and sometimes insurance/pensions in old age. Thus, policy measures to modernize agriculture with improved access to subsidized and cheap agricultural technology and inputs, together with appropriate pension schemes especially for rural dwellers, will reduce the desire for large families. Other than the issues of increasing use of traditional contraception in urban areas, the effect of household wealth, partner's education and number of living children, the effect of the other covariates on the use of reproductive health inputs is standard and supported by the findings of existing studies.
The authors would like to express their appreciation to the African Economic Research Consortium (AERC) who funded this research under research grant number RT0851. We also acknowledge the valuable contributions of resource persons and network researchers at three different AERC biennial conferences.
Appendix 1: Ordered Probit and Negative Binomial Estimates
ᅟ
Appendix 2: The negative Binomial Model
The negative binomial model improves the efficiency of the Poisson model by adding a parameter, and an error term to the mean function of the Poisson distribution. In the Poisson model, the number of antenatal visits y, as in the current case, for a woman j, has a Poisson distribution with a mean μ conditioned on some covariates x as below:
$$ {\mu}_j=E\left({y}_j\left|{x}_j\right.\right)={e}^{x_j\beta },.{y}_j=0,1,2\dots \dots \dots $$
where β is the a parameter to be estimated, and the probability of y given the vector of covariates x expressed as in equation (A2) below :
$$ \Pr \left({y}_j\left|{x}_j\right.\right)=\frac{e^{-{\mu}_j}{\mu}_j^{y_j}}{y_j!},\begin{array}{c}\hfill \hfill \\ {}\hfill .{y}_j=0,1,2,\dots \dots ..\hfill \end{array} $$
In practice, count data often show over-dispersion with the effect that the variance becomes more than the mean and therefore violates the principal Poisson assumption that the variance should be equal to the mean. Apart from the over-dispersion, another challenge of the Poisson model is the issue of excess zeros, which is often associated with count data such as use of antenatal visits. To overcome the two challenges, alternative approaches (such as the Negative Binomial, Zero-Inflated Poisson and Negative Inflated Binomial models) have been used in the literature to account for over-dispersion and excess zeros. In the current case, excess zeros don't seem to be a challenge given that the percentage of zeros (i.e., those not using) is 12.94 %, 11.20 %, 7.92 % and 3.93 % for 1993, and 1998, 2003 for 2008, respectively. In addition, the variance of antenatal visits is less than the mean in all years. Nonetheless, the lrtest that alpha = 0 (i.e., the Poisson model is an appropriate fit for the data) is rejected at p < 0.001 (see column 3 Table 5 in Appendix 1). Thus we use the negative binomial model (NBM) to correct for over-dispersion. The NBM follows from the Poisson model but introduces an additional parameter, an error term ε j , to the mean function of the Poisson distribution as in Equation A3.
$$ {\tilde{\mu}}_j=E\left({y}_j\left|{x}_j\right.\right)={e}^{x_j\beta +{\varepsilon}_j},.{y}_j=0,1,2\dots \dots \dots $$
With the introduction of the error term, the NBM allows for cross-sectional heterogeneity, given that an unobserved individual effect can be taken into the conditional mean function. Following from this, and assuming that exp(e j ) has a distribution with a mean of 1 and a variance of α , the conditional mean of y j will continue to be μ j . Thus as α approaches 0, y j becomes a Poisson distribution. On the other hand, if exp(e j ) is assumed to have a gamma distribution with a gamma function Γ(), then the conditional probability function y j will be the negative binomial distribution as in Equation A4 below.
$$ \Pr \left({y}_j\left|{x}_j\right.\right)=\frac{\varGamma \left({y}_j+{\alpha}^{-1}\right)}{y_j!\varGamma \left({\alpha}^{-1}\right)}{\left(\frac{\alpha^{-1}}{\alpha^{-1}+\mu}\right)}^{\alpha -1}{\left(\frac{\mu }{\alpha^{-1}+\mu}\right)}^{y_j},\begin{array}{c}\hfill \hfill \\ {}\hfill {y}_j=0,1,2\dots \dots \dots \hfill \end{array} $$
Equation A4 was used to estimate the intensity of use of antenatal visits in the count form.
Socioeconomic determinants of antenatal visits and hospital facility delivery in Ghana
Ordered antenatal visits
Number of antenatal visits
Note: *** is significant at p < 0.01, ** is significant at p < 0.05, * is significant at p < 0.10. NSCPHGW, NSCPHFT and NSCPCCV are the non-self cluster proportion of households with good water, non-self cluster proportion of households with flush toilet and non-self cluster proportion of children under five with complete vaccination, respectively. Partner's education includes a 5th category (missing husbands) but this is excluded from the table. It was added to cater for women who do not have partners and would otherwise have been excluded from the regressions
PAA conceived the idea and was responsible for writing the background and conclusion section of the paper as well as proof reading and formatting the final manuscript. GA was responsible for acquiring the relevant data, carrying out modeling and estimation and consequently writing the methods, results and discussion. Both authors read and approved the final manuscript.
Department of Public Administration and Health Services Management, University of Ghana Business School, P. O. Box 78, Legon, Accra, Ghana
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Pu's inequality
In differential geometry, Pu's inequality, proved by Pao Ming Pu, relates the area of an arbitrary Riemannian surface homeomorphic to the real projective plane with the lengths of the closed curves contained in it.
Statement
A student of Charles Loewner, Pu proved in his 1950 thesis (Pu 1952) that every Riemannian surface $M$ homeomorphic to the real projective plane satisfies the inequality
$\operatorname {Area} (M)\geq {\frac {2}{\pi }}\operatorname {Systole} (M)^{2},$
where $\operatorname {Systole} (M)$ is the systole of $M$. The equality is attained precisely when the metric has constant Gaussian curvature.
In other words, if all noncontractible loops in $M$ have length at least $L$, then $\operatorname {Area} (M)\geq {\frac {2}{\pi }}L^{2},$ and the equality holds if and only if $M$ is obtained from a Euclidean sphere of radius $r=L/\pi $ by identifying each point with its antipodal.
Pu's paper also stated for the first time Loewner's inequality, a similar result for Riemannian metrics on the torus.
Proof
Pu's original proof relies on the uniformization theorem and employs an averaging argument, as follows.
By uniformization, the Riemannian surface $(M,g)$ is conformally diffeomorphic to a round projective plane. This means that we may assume that the surface $M$ is obtained from the Euclidean unit sphere $S^{2}$ by identifying antipodal points, and the Riemannian length element at each point $x$ is
$\mathrm {dLength} =f(x)\mathrm {dLength} _{\text{Euclidean}},$
where $\mathrm {dLength} _{\text{Euclidean}}$ is the Euclidean length element and the function $f:S^{2}\to (0,+\infty )$, called the conformal factor, satisfies $f(-x)=f(x)$.
More precisely, the universal cover of $M$ is $S^{2}$, a loop $\gamma \subseteq M$ is noncontractible if and only if its lift ${\widetilde {\gamma }}\subseteq S^{2}$ goes from one point to its opposite, and the length of each curve $\gamma $ is
$\operatorname {Length} (\gamma )=\int _{\widetilde {\gamma }}f\,\mathrm {dLength} _{\text{Euclidean}}.$
Subject to the restriction that each of these lengths is at least $L$, we want to find an $f$ that minimizes the
$\operatorname {Area} (M,g)=\int _{S_{+}^{2}}f(x)^{2}\,\mathrm {dArea} _{\text{Euclidean}}(x),$
where $S_{+}^{2}$ is the upper half of the sphere.
A key observation is that if we average several different $f_{i}$ that satisfy the length restriction and have the same area $A$, then we obtain a better conformal factor $f_{\text{new}}={\frac {1}{n}}\sum _{0\leq i<n}f_{i}$, that also satisfies the length restriction and has
$\operatorname {Area} (M,g_{\text{new}})=\int _{S_{+}^{2}}\left({\frac {1}{n}}\sum _{i}f_{i}(x)\right)^{2}\mathrm {dArea} _{\text{Euclidean}}(x)$
$\qquad \qquad \leq {\frac {1}{n}}\sum _{i}\left(\int _{S_{+}^{2}}f_{i}(x)^{2}\mathrm {dArea} _{\text{Euclidean}}(x)\right)=A,$
and the inequality is strict unless the functions $f_{i}$ are equal.
A way to improve any non-constant $f$ is to obtain the different functions $f_{i}$ from $f$ using rotations of the sphere $R_{i}\in SO^{3}$, defining $f_{i}(x)=f(R_{i}(x))$. If we average over all possible rotations, then we get an $f_{\text{new}}$ that is constant over all the sphere. We can further reduce this constant to minimum value $r={\frac {L}{\pi }}$ allowed by the length restriction. Then we obtain the obtain the unique metric that attains the minimum area $2\pi r^{2}={\frac {2}{\pi }}L^{2}$.
Reformulation
Alternatively, every metric on the sphere $S^{2}$ invariant under the antipodal map admits a pair of opposite points $p,q\in S^{2}$ at Riemannian distance $d=d(p,q)$ satisfying $d^{2}\leq {\frac {\pi }{4}}\operatorname {area} (S^{2}).$
A more detailed explanation of this viewpoint may be found at the page Introduction to systolic geometry.
Filling area conjecture
An alternative formulation of Pu's inequality is the following. Of all possible fillings of the Riemannian circle of length $2\pi $ by a $2$-dimensional disk with the strongly isometric property, the round hemisphere has the least area.
To explain this formulation, we start with the observation that the equatorial circle of the unit $2$-sphere $S^{2}\subset \mathbb {R} ^{3}$ is a Riemannian circle $S^{1}$ of length $2\pi $. More precisely, the Riemannian distance function of $S^{1}$ is induced from the ambient Riemannian distance on the sphere. Note that this property is not satisfied by the standard imbedding of the unit circle in the Euclidean plane. Indeed, the Euclidean distance between a pair of opposite points of the circle is only $2$, whereas in the Riemannian circle it is $\pi $.
We consider all fillings of $S^{1}$ by a $2$-dimensional disk, such that the metric induced by the inclusion of the circle as the boundary of the disk is the Riemannian metric of a circle of length $2\pi $. The inclusion of the circle as the boundary is then called a strongly isometric imbedding of the circle.
Gromov conjectured that the round hemisphere gives the "best" way of filling the circle even when the filling surface is allowed to have positive genus (Gromov 1983).
Isoperimetric inequality
Pu's inequality bears a curious resemblance to the classical isoperimetric inequality
$L^{2}\geq 4\pi A$
for Jordan curves in the plane, where $L$ is the length of the curve while $A$ is the area of the region it bounds. Namely, in both cases a 2-dimensional quantity (area) is bounded by (the square of) a 1-dimensional quantity (length). However, the inequality goes in the opposite direction. Thus, Pu's inequality can be thought of as an "opposite" isoperimetric inequality.
See also
• Filling area conjecture
• Gromov's systolic inequality for essential manifolds
• Gromov's inequality for complex projective space
• Loewner's torus inequality
• Systolic geometry
• Systoles of surfaces
References
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Systolic geometry
1-systoles of surfaces
• Loewner's torus inequality
• Pu's inequality
• Filling area conjecture
• Bolza surface
• Systoles of surfaces
• Eisenstein integers
1-systoles of manifolds
• Gromov's systolic inequality for essential manifolds
• Essential manifold
• Filling radius
• Hermite constant
Higher systoles
• Gromov's inequality for complex projective space
• Systolic freedom
• Systolic category
| Wikipedia |
\begin{document}
\title{Non-thermal quantum channels as a thermodynamical resource}
\author{Miguel Navascu\'{e}s$^1$ and Luis Pedro Garc\'{i}a-Pintos$^2$} \affiliation{$^1$Department of Physics, Bilkent University, Ankara 06800, Turkey\\ $^2$School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, U.K.}
\begin{abstract} Quantum thermodynamics can be understood as a resource theory, whereby thermal states are free and the only allowed operations are unitary transformations commuting with the total Hamiltonian of the system. Previous literature on the subject has just focused on transformations between different state resources, overlooking the fact that quantum operations which do not commute with the total energy also constitute a potentially valuable resource. In this Letter, given a number of non-thermal quantum channels, we study the problem of how to integrate them in a thermal engine so as to distill a maximum amount of work. We find that, in the limit of asymptotically many uses of each channel, the distillable work is an additive function of the considered channels, computable for both finite dimensional quantum operations and bosonic channels. We apply our results to bound the amount of distillable work due to the natural non-thermal processes postulated in the Ghirardi-Rimini-Weber (GRW) collapse model. We find that, although GRW theory predicts the possibility to extract work from the vacuum at no cost, the power which a \emph{collapse engine} could in principle generate is extremely low.
\end{abstract}
\maketitle
The field of quantum thermodynamics has seen a surge in interest in the past years, with increasing attention towards testing the validity of the rules of classical thermodynamics in the quantum regime. A major topic within thermodynamics is that of extracting work out of a given system and the optimal way to perform this. One way this has been approached in the quantum case was by considering it from the perspective of a resource theory.
The idea of a resource theory of thermodynamics is to assume one has unlimited access to thermal baths (i.e. Gibbs states of a fixed temperature $T$), the freedom to apply any energy conserving unitary on system plus the bath (any unitary $V$ that commutes with the total Hamiltonian), and the possibility of discarding part of the system or bath (i.e. apply partial traces). These rules are imported from classical thermodynamics, where one assumes access to infinite baths of constant temperature and any evolution where energy is conserved.
As a matter of fact, resource theories have been very useful in different topics within quantum information theory~\cite{Horodecki2003,Speckens2008,Horodecki2009,Muller13}. The idea is similar to above: considering free access to certain operations and/or states, any state and/or operation that is not in the above set can in principle be used as a resource.
This work complements previous research in quantum thermodynamics by accommodating the possibility of considering non-thermal maps, or channels, as a resource.
Physical operations are represented by \emph{quantum channels}, i.e., completely positive trace preserving maps $\Omega: B(\mathcal{H}) \rightarrow B(\mathcal{H}')$ acting on a state space $B(\mathcal{H})$. For simplicity, we will assume that the input and output spaces (and, as we will see later, Hamiltonians) of each channel are the same, although the results can be easily generalized.
Unitary evolution is a particular instance of a quantum channel, determined by the evolution operator. However, quantum channels allow to express more general evolutions. For instance, any unitary interaction of a system with an ancilla (or environment) generates a quantum channel, given by \begin{equation} \Omega (\rho) = \trx{V \rho\otimes \sigma_A V^\dag}{A}, \end{equation} where $\sigma_A$ is the state of the ancilla, and $V$ is some unitary operator In fact, it can be shown that any channel can be generated via the above procedure~\cite{nielsenchuang2010}.
If, in the above expression, the ancilla is in a Gibbs state of temperature $T$ and the unitary $V$ commutes with the total Hamiltonian $H_T=H_S\otimes \mathbbm{1}_A+\mathbbm{1}_S\otimes H_A$ of the target-ancilla system, the resulting map is called a \emph{thermal channel}. These will be the free operations in our theory, while any non-thermal map $\Omega$ will be considered as a resource.
In this scenario, we define quantum work $W$ as the process of exciting a two-level system with Hamiltonian $H=W\proj{1}$ from its ground state $\ket{0}$ to the excited state $\ket{1}$~\cite{Horodeckisingleshot}. Different authors have explored how much work one can extract from a non-thermal quantum state~\cite{Horodeckisingleshot,Brandao11,Takara2010,Esposito2011,Aberg13,Brandao2013,Popescu2014}. When we regard the \emph{maximum average work} as a figure of merit, a quantum generalization of the classical free energy naturally emerges~\cite{Horodeckisingleshot,Brandao11,Popescu2014}: \begin{equation} F(\rho) = U(\rho) - K_BT S(\rho). \end{equation} Here $\rho$ is the state of the system from which we wish to extract work; $U(\rho) = \tr{\rho H}$, its average energy ; and $S(\rho) = - \tr{\rho \log{\rho}}$, its von Neumann entropy. $K_B$ is Boltzmann's constant. The maximum amount of work one can extract (on average) from the state $\rho$ can then be shown to be $F(\rho)-F(\tau_{th})$, where $\tau_{th}$ represents the thermal state
at temperature $T$.
In this Letter we want to address the following related problem: suppose we want to build a thermal engine, where we are allowed to integrate a number of non-thermal gates $\{\Omega_i\}_{i=1}^N$, each of which is assumed to act on a system with Hamiltonian $H_i$. More specifically, our machine can make free use of any amount of thermal states and operations, and we can invoke one use of each of the channels $\{\Omega_i\}_{i=1}^N$, in any order we want at any step. We are also allowed to use \emph{catalysts}, i.e., we can use any number of non-thermal states, as long as we return them in the end. Under these conditions, what is the maximum amount of work that our device can extract?
There are two ways to approach this problem:
\begin{enumerate} \item We can restrict to thermodynamical processes which distill work deterministically, i.e., always the same amount. The corresponding \emph{deterministic extractable work} can then be shown to behave very badly: not only is it not additive, but it can be super-activated. That is, there exist channels $\Omega$ such that no work can be distilled from a single use,
but two uses of the channel can be combined to produce a non-zero amount of deterministic work (see the Supplemental Information).
\item Alternatively, we can consider thermodynamical processes which generate a given amount of work with high probability. Here the figure of merit would be the maximum amount of work that can be distilled asymptotically (on average) when we have access to $n$ uses of each channel. \end{enumerate}
We will follow the second approach: in the next pages we will show that the \emph{asymptotically extractable work} is upper bounded by $\sum_{i=1}^N W(\Omega_i,H_i)$, where
\begin{equation} W(\Omega, H)\equiv \max_{\rho} \Delta F(\rho, \Omega), \label{basic} \end{equation}
\noindent with $\Delta F(\rho, \Omega)$ denoting the free energy difference between the states $\Omega(\rho)$ and $\rho$, i.e. \begin{equation} \Delta F(\rho, \Omega)\equiv \tr{ (\Omega(\rho)-\rho )H} - K_BT\mathopen{}\mathclose\bgroup\originalleft[ S( \Omega(\rho)) - S(\rho)\aftergroup\egroup\originalright]. \end{equation} The quantity $W(\Omega, H)$ will be called the \emph{distillable work} of channel $\Omega$. From the inequality $\Delta F(\tau_{th}, \Omega)\geq 0$, it follows that $W(\Omega, H)\geq 0$ for any $\Omega$.
\noindent The bound $\sum_{i=1}^N W(\Omega_i,H_i)$ can be achieved asymptotically via a simple protocol where we prepare suitable initial states $\sigma_{cat}$ (the catalysts) maximizing eq.~(\ref{basic}) for each channel, and then let each channel act over its corresponding maximizer. The result of this protocol will be a state with free energy $F(\sigma_{cat})+\sum_{i=1}^NW(\Omega_i,H_i)$. Given access to $n$ uses of each channel, we can thus prepare $n$ copies of the latter state, whose free energy can be converted to work via thermal operations using the protocol depicted in~\cite{Brandao11}. Following \cite{Brandao11}, part of this work (roughly $nF(\sigma_{cat})$) can then be used to regenerate the catalysts up to a small error~\footnote{Crucially, at the end of the regeneration step the free energy of the reconstructed catalysts also tends to its initial value.}. The average work extracted with this procedure (namely, the total work divided by $n$) is thus given by $\sum_{i=1}^N W(\Omega_i,H_i)$.
Note, though, that, unless the catalysts are diagonal in the energy basis, an extra amount of coherence, sub-linear in $n$, may be needed to rebuild them (see Appendix E of \cite{Brandao11}). More specifically, for each energy transition $E_s\to E_t$ in the Hamiltonian $H_i$, the protocol proposed in \cite{Brandao11} requires a system with Hamiltonian $H^{s,t}_i= \sum_{k=0}^{O(m)} (E_s-E_t) k \proj{k}$ in state $\frac{1}{\sqrt{m}}\sum_{k=0}^m\ket{k}$, with $m$ sublinear in the number $n$ of uses of each channel. Like the catalyst states, at the end of the protocol such `coherent states' will be approximately rebuilt with vanishing error.
In order to prove the above result, and some later ones, the next lemma will be invoked extensively:
\begin{lemma} \label{basic_form} Let $\sigma^{(N)}$ be an $N$-partite quantum state, and let $\{\Omega_i\}_{i=1}^N$ be a collection of $N$ single-site quantum channels. Defining $\Omega_{1...N}\equiv\bigotimes_{i=1}^N\Omega_i$, we have that
\begin{equation} \sum_{i=1}^NS(\sigma_i)-S(\Omega_i(\sigma_i))\geq S(\sigma^{(N)})-S(\Omega_{1...N}(\sigma^{(N)})). \label{indep_max} \end{equation}
\end{lemma}
\noindent The proof is a straightforward application of the contractivity of the relative entropy~\cite{Vedral2002}.
An almost immediate consequence of Lemma \ref{basic_form} is that $W(\Omega, H)$, as defined by eq.~(\ref{basic}), has the remarkable property of being additive. That is, if the bipartite system $12$ is described by the Hamiltonian $H_{12}=H_1\otimes \mathbbm{1}_2+\mathbbm{1}_1\otimes H_2$, and the channels $\Omega_1$ and $\Omega_2$ act on the respective Hilbert spaces ${\cal H}_1,{\cal H}_2$, then, $W(\Omega_1\otimes \Omega_2,H_{12})=W(\Omega_1,H_1)+W(\Omega_2,H_2)$.
Indeed, let $\Omega_{12}\equiv\Omega_1\otimes\Omega_2$ act on the bipartite state $\rho_{12}$. By choosing $\rho_{12}=\rho_1\otimes\rho_2$ in (\ref{basic}) we trivially have that $W(\Omega_1\otimes \Omega_2,H_{12})\geq W(\Omega_1,H_1)+W(\Omega_2,H_2)$, since maximizing over states in $12$ is more general than maximizing over $1$ and $2$ independently. Let us then focus on the opposite inequality. By Lemma \ref{basic_form}, we have \begin{equation} \sum_{i=1,2}S(\rho_{i})-S(\Omega_i(\rho_{i}))\geq S(\rho_{12})-S(\Omega_{12}(\rho_{12})). \end{equation}
\noindent Substituting into~(\ref{basic}) gives \begin{equation} \sum_{i=1,2}\Delta F(\rho_i,\Omega_i) \geq \Delta F(\rho_{12},\Omega_{12}) \quad \ \forall \ \rho_{12}, \end{equation}
It follows that $\sum_{i=1,2}W(\Omega_{i},H_i)\geq W(\Omega_{12},H_{12})$.
We are now ready to prove
that $W(\Omega, H)$ quantifies the maximum (average) amount of work one can extract from channel $\Omega$.
\begin{proposition} \label{max_work} Let $\{\Omega_i\}_{i=1}^N$ be a set of quantum channels, defined over different quantum systems with Hamiltonians $\{H_i\}_{i=1}^N$. Suppose that we integrate $n$ uses of all such channels in a thermal engine ${\cal T}_n$ that produces a net amount of work $W_n$ with probability $1-\epsilon_n$. Let us further assume that the probability of failure vanishes in the limit of large $n$, i.e., $\lim_{n\to\infty}\epsilon_n= 0$. Under these conditions, the average asymptotic work $\bar{W}\equiv\limsup\limits_{n\to\infty}\frac{W_n}{n}$ satisfies \begin{equation} \bar{W}\leq \sum_{i=1}^N W(\Omega_i,H_i). \label{boundido} \end{equation}
As indicated above, this bound is achievable with the use of catalysts and a sublinear amount of quantum coherence. \end{proposition}
\begin{proof}
In any protocol for work extraction, the initial state of the system will be given by the catalysts $\sigma_{cat}$, a number of thermal states $\tau_{th}$, and the work system in state $\ket{0}_{w}$. The initial state of the system is hence $\rho_0\equiv \sigma_{cat}\otimes\tau_{th}\otimes \proj{0}_{w}$, with free energy $F(\sigma_{cat})+F(\tau_{th})$.
Suppose that now we apply a sequence of energy-conserving unitaries. At time $t$, the state of the overall system is $\rho_t$, and we apply the channel $\Omega_{s(t)}$ over part of the whole system, possibly followed by some other thermal operation. Let us analyze how the free energy of $\rho_t$ can increase in the above step. Calling $H_T$ the Hamiltonian of the whole system, from the definition of $W(\Omega,H)$ and the additivity of the distillable work we have that: \begin{equation} \Delta F(\Omega_{s(t)}\otimes\mathbbm{1},\rho_t)\leq W(\Omega_{s(t)}\otimes \mathbbm{1},H_T)=W(\Omega_{s(t)},H_{s(t)}). \end{equation}
Now, any intermediate energy-conserving unitary in-between the use of any two of the channels $\{\Omega_i\}_{i=1}^N$ will keep the free energy of the overall system constant. Calling $\bar{\rho}$ the state of the system at the end of the protocol, we hence have that
\begin{equation} F(\bar{\rho})\leq n\sum_{i=1}^NW(\Omega_i,H_i)+F(\sigma_{cat})+F(\tau_{th}). \end{equation}
\noindent From the subadditivity of the von Neumann entropy, it follows that $F(\bar{\rho})\geq F(\bar{\rho}_{cat})+F(\bar{\rho}_{th})+F(\bar{\rho}_{w})$, where $\bar{\sigma}_{cat},\bar{\rho}_{th},\bar{\rho}_{w}$ are, respectively, the reduced density matrices of the catalyst, thermal and work systems.
At the end of the protocol, the catalyst must be regenerated, i.e., $\bar{\sigma}_{cat}=\sigma_{cat}$. Also, $F(\bar{\rho}_{th})\geq F(\tau_{th})$. It follows that the free energy of the work system is bounded by $n\sum_{i=1}^NW(\Omega_i,H_i)$.
This system is expected to end up in state $\ket{1}$ with probability $1-\epsilon_n$, i.e., $\bar{\rho}_w=(1-\epsilon_n)\proj{1}+\epsilon_n \bar{\sigma}$. It follows that $F(\bar{\rho}_{w})\geq (1-\epsilon_n)W_n-K_BTh(\epsilon_n)$, with $h(p)=-p\ln(p)-(1-p)\ln(1-p)$. In the asymptotic limit, with $n\to \infty$, $\epsilon_n\to 0$, the average asymptotic work $\limsup\limits_{n\to\infty}\frac{W_n}{n}$ is hence bounded by $\sum_{i=1}^NW(\Omega_i,H_i)$.
Note that this bound also holds if the catalysts are recovered up to an error, as long as $F(\sigma_{cat})-F(\bar{\sigma}_{cat})\leq o(n)$.
\end{proof}
This result allows to quantify the work extraction capabilities of different channels. One can check, for instance, that no work can be distilled from a dephasing channel. Meanwhile, for a two-level system with Hamiltonian $H = E\ket{1}\bra{1},\ E>0$, the channel that takes any state to the excited state $\ket{1}$ provides the highest distillable work.
\noindent \emph{Gaussian channels}
If our target system is infinite dimensional, in principle there may exist quantum states possessing an infinite amount of energy. If we regard such states as unphysical, we should replace the maximization in eq.~(\ref{basic}) by an optimization over all states of finite energy. The resulting quantity will hence bound the maximum amount of work generated in physically conceivable quantum engines, where the overall state of the system always has a finite amount of energy.
In infinite dimensional systems Gaussian quantum channels have a special relevance: they are easy to implement in the lab, and are extensively used to model particle interactions with a macroscopic environment. They are defined as channels which, when composed with the identity map, transform Gaussian states into Gaussian states, the latter being those states with a Gaussian Wigner function~\cite{gaussianos}. An $m$-mode Gaussian state is completely defined via its displacement vector $d_i=\langle R_i\rangle$ and covariance matrix $\gamma_{ij}=\langle\{R_i - d_i \mathbbm{1},R_j - d_j \mathbbm{1}\}_{+}\rangle$, where $(R_1,R_2,...,R_{2m})\equiv (Q_1,P_1,...,Q_m,P_m)$ are the optical quadratures. The action of a Gaussian channel is fully specified by its action over the displacement vector and covariance matrix, given by:
\begin{equation} d\to Xd+z, \gamma\to X\gamma X^T+Y, \label{gauss_chan} \end{equation}
\noindent where $Y+i\sigma-iX^T\sigma X\geq 0$. Here $\sigma$ denotes the symplectic form $\sigma=\oplus_{i=1}^m\mathopen{}\mathclose\bgroup\originalleft(\begin{array}{cc}0&1\\-1&0\end{array}\aftergroup\egroup\originalright)$. If the Hamiltonian of the system under study happens to be a quadratic function of the optical quadratures, i.e., $H=\vec{R}^TG\vec{R}+\vec{h}\cdot \vec{R}$, for some real symmetric matrix $G$ and real vector $\vec{h}$, then the average energy of a state with displacement vector $\vec{d}$ and covariance matrix $\gamma$ is given by $E=\frac{1}{2}\tr{G \gamma}+\vec{d}^TG\vec{d}+\vec{h}\cdot \vec{d}$. States with finite energy hence correspond to states with finite first and second moments.
When the quadratic Hamiltonian has no zero energy modes (that is, when $G>0$), Proposition \ref{max_work} allows to easily classify generic Gaussian channels according to their capacity to generate an infinite amount of work. Indeed, for $X^TGX-G\not\leq 0$, the channel's distillable work is unbounded: this can be seen by inputting a sequence of Gaussian states with constant covariance matrix but increasing displacement vector parallel to any positive eigenvector of $X^TGX-G$. Conversely, as we show in the Supplemental Information, for channels satisfying $X^TGX-G < 0$ only a finite amount of work can be distilled.
For such channels there is still the dilemma of how much work can be extracted. The next Proposition greatly simplifies this problem by showing that, for Gaussian channels $\Omega$, the maximization in (\ref{basic}) can be restricted to Gaussian states:
\begin{proposition} \label{Gaussian} Consider a continuous variable quantum system of $m$ modes, with quadratic Hamiltonian $H$, let $\Omega$ be a Gaussian channel mapping $m$ modes to $m$ modes, and denote by ${\cal G}$ the set of all $m$-mode Gaussian states. Then, \begin{equation} W(\Omega,H)= \max_{\rho\in {\cal G}} \Delta F(\rho,\Omega). \end{equation} \end{proposition}
\noindent The proposition can be proven by combining Lemma \ref{basic_form} with the `gaussification' protocol described in \cite{gaussification}, see the Supplemental Information.
Since $W(\Omega,H)$ just involves an optimization over a finite set of parameters subject to positive semidefinite constraints, (in principle) it can be computed exactly for any Gaussian channel $\Omega$.
\noindent \emph{One application: collapse engines}
In order to address the measurement problem~\cite{GRW}, and, independently, the decoherence effects that a quantum theory of gravity could impose on the wave-function~\cite{penrose,diosi}, different authors have proposed that \emph{closed} quantum systems should evolve according to the Lindblad equation
\begin{equation} \label{GRWapprox} \frac{d}{dt} \rho_t = -\frac{i}{\hbar} [H,\rho_t] - \frac{\Lambda}{4} [X,[X,\rho_t]], \end{equation} where $X$ is the position operator for the particle considered.
The effect of the non-unitary term is a suppression of coherences in the position basis, effectively destroying quantum superpositions. The value of the constant $\Lambda$, which can be interpreted as the rate at which this localization process occurs, depends on the particular theory invoked to justify eq. (\ref{GRWapprox}). In the Ghirardi-Rhimini-Weber (GRW) theory~\cite{GRW}, the localization process is postulated to solve the measurement problem in quantum mechanics. To achieve this goal and avoid contradictions with past experimental results, $\Lambda$ must be roughly between $10^{-2} s^{-1} m^{-2} \ \text{and} \ 10^{6} s^{-1} m^{-2}$, according to latest estimations \cite{Adler07}.
Note that the above evolution is non-thermal. Hence, it could be used in principle to extract work from nothingness by means of a suitable thermal engine. We will call such a hypothetical device a \emph{collapse engine}.
To connect this to our previous setting, notice that the evolution equation (\ref{GRWapprox}) defines a quantum channel \begin{equation} \Omega_{\delta t}(\rho_t) = \rho_{t+\delta t}, \end{equation} with $\rho_t$ the solution of eq.~(\ref{GRWapprox}) for the initial state $\rho_0$.
We suppose that the particle under consideration is subject to a harmonic potential, i.e., $H = \frac{m \omega^2}{2} X^2 + \frac{1}{2m}P^2$, and that, despite the GRW dynamics, the bath's temperature $T$ is constant. A physical justification for this last assumption is that the temperature-increasing GRW dynamics is countered by radiation from the bath into outer space. Hence, as a function of time, the temperature will converge to a stationary value $T=T_{eq}$ above the temperature of the cosmic microwave background (CMB)~\footnote{If we model the bath as a grey body, then it radiates energy at a rate of $\sigma (T^4-T_c^4)$, where $T_c$ is the temperature of the CMB and $\sigma$ is a constant that depends on how well isolated the bath is. The power transferred to the bath by the GRW dynamics is, on the other hand, independent of $T$ and proportional to $\Lambda$. It follows that the bath will reach a stationary temperature $T_{eq}$ whose exact value will depend on both $\sigma$ and $\Lambda$.}. Finally, we suppose that our set of resource operations remains the same: that is, in spite of the modified Schr\"{o}dinger equation (\ref{GRWapprox}), we can still switch on and off any unitary interaction that commutes with the total energy of the system. Notice that the GRW dynamics can be modelled by an open system approach, where the particle is interacting with some unknown system such that the resulting evolution is given by (\ref{GRWapprox}). From this viewpoint, we are simply assuming that we still have the capacity to interact with the system in the usual way.
In these conditions, we wish to find the maximum amount of work that a collapse engine could extract if it had access to the evolution equation (\ref{GRWapprox}) for a finite amount of time $t$. From Proposition (\ref{max_work}), this amounts to computing $\lim_{\delta t\to 0}\frac{t}{\delta t}W(\Omega_{\delta t}, H)$.
First, notice that we can (reversibly) evolve the system with the Hamiltonians $H$ or $-H$, since this corresponds to a thermal operation. This implies that we can ignore the first term in the right hand side of (\ref{GRWapprox}), and what remains is a Gaussian channel given by
\begin{equation} d\to d,\gamma\to\gamma+\mathopen{}\mathclose\bgroup\originalleft(\begin{array}{cc}0&0\\0&\frac{\hbar^2\Lambda \delta t}{2}\end{array}\aftergroup\egroup\originalright). \end{equation}
It follows that the energy of any input state will increase by $\Delta U\equiv\frac{\hbar^2\Lambda}{4m} \delta t$. From Proposition~\ref{Gaussian}, we can estimate the entropy increase by just considering Gaussian states. Now, the entropy of a $1$-mode Gaussian state is an increasing function of the determinant of its covariance matrix~\cite{gaussianos}, which, by the above equation, can only increase with time. Hence, $W(\Omega_{\delta t},H)\leq\Delta U$.
On the other hand, suppose we input a squeezed state with $\gamma=\mbox{diag}(1/r,\hbar^2r)$. Then the determinants of the covariance matrices of initial and final states will be $\hbar^2$ and $\hbar^2+\frac{\hbar^2\Lambda\delta t}{2r}$, respectively. The entropy change of the state can thus be made as small as desired by increasing the value of $r$, and so the above bound can be saturated, leading to $W(\Omega_{\delta t},H)=\Delta U$. Consequently, the maximum power at which a collapse engine could in principle operate is given by \begin{equation} \frac{dW}{dt}=\frac{\hbar^2\Lambda}{4m}. \end{equation} Using the upper range estimation $\Lambda \sim 10^{6} s^{-1} m^{-2}$, we have that a collapse engine powered by a single electron would produce $\frac{dW}{dt} \sim 10^{-32} \ watt$. Assuming total control over the electrons of a macroscopic sample, one would need a kiloton of Hydrogen to power a $40 \ watt$ light bulb.
\noindent\emph{Conclusion}
In this Letter we addressed the problem of determining how much work can be extracted from operational -as opposed to state- resources. We proved that the solution to this problem in the asymptotic limit is given by a single-letter formula that quantifies the amount of distillable work that a channel can, in principle, generate when supplemented with thermal operations and catalyst states. Moreover, we found how this quantity can be determined for bosonic channels, and computed it exactly for the case of the GRW dynamics, hence determining the maximum power which a hypothetical collapse engine could provide for free.
Note that we have only studied operational resources regarding their capacity to generate work. An interesting topic for future research is to extend our results and draw a map of the inter-conversion relations between different operational resources. In the case of state resources there is a unique monotonic quantity, the free energy, determining the optimal rates for state transformations~\cite{Brandao11}. In this work we have identified an operational monotone, the distillable work, but we suspect that there may be many others.
\noindent\emph{Acknowledgements}
We thank Ralph Silva and Noah Linden for useful discussions.
\appendix
\section{The deterministic distillable work can be superactivated}
Consider the channel $\Omega$ that takes any state of a target two-level system with Hamiltonian $H= E_1\proj{1}$ to the state $\ket{\psi} \propto \mathopen{}\mathclose\bgroup\originalleft( \ket{0} + e^{-\beta E_1/2} \ket{1} \aftergroup\egroup\originalright)$. That is, $\Omega(\rho)=\proj{\psi}$ for all $\rho$.
Now, any protocol that pretends to extract work out via a single use of $\Omega$ can be divided in three steps:
\begin{enumerate} \item \label{prep} The system is prepared in a state which comprises catalysts, thermal states and the work system (in state $\ket{0}$). That is, $\rho_0=\sigma_{cat}\otimes\tau_{th}\otimes\proj{0}_w$. \item We apply an energy-conserving unitary $U_1$ over the whole system. \item \label{channel} We apply $\Omega$, hence replacing a subsystem's state by $\ket{\Psi}$. \item We apply a second energy-conserving unitary $U_2$ over the whole system. \end{enumerate}
\noindent At the end of the protocol, the work system will have evolved to $\ket{1}$.
If, rather than implementing step \ref{channel}, we add the state $\ket{\psi}$ in the preparation stage, then it is trivial to find an energy-conserving unitary $\tilde{U}_2$ that at the last step would produce exactly the same amount of work. That is, we would have extracted work from the resource state $\ket{\psi}$.
In~\cite{Horodeckisingleshot}, however, it is shown that no work can be extracted from a single copy of $\ket{\psi}$, even with the use of catalysts. This implies that the deterministic distillable work of a single use of $\Omega$ is zero.
Suppose, now, that we have access to two uses of $\Omega$. Then we can prepare two copies of $\ket{\psi}$, from which a non-zero amount of work can be deterministically extracted via thermal operations \cite{Horodeckisingleshot}.
\section{Gaussian channels with finite distillable work}
Let $\Omega$ be a Gaussian channel whose action on the displacement vector $\vec{d}$ and covariance matrix $\gamma$ of the $m$-mode input state is given by
\begin{equation} d\to Xd+z, \gamma\to X\gamma X^T+Y. \label{gauss_chan} \end{equation}
\noindent Suppose that $\Omega$ acts on a system with Hamiltonian $H=\vec{R}^TG\vec{R}+\vec{h}\cdot\vec{R}$, where $\vec{R}$ is the vector of optical quadratures. Under the assumption that $\tilde{G}\equiv G-X^TGX>0$, we wonder if the difference between the free energies of the input and output states is bounded, i.e., whether $\Delta F(\rho,\Omega)<K$, for some $K<\infty$.
Call $E_0$ the energy of the input state; and $\gamma, \vec{d}$, its covariance matrix and displacement vector. Then we have that
\begin{equation} E_0=\frac{1}{2}\tr{G\gamma}+(\vec{d}-\vec{d}_0)^TG(\vec{d}-\vec{d}_0)-\bar{E}, \end{equation}
\noindent where $\vec{d}_0\equiv -G^{-1}\vec{h}/2$, and $\bar{E} \equiv \vec{d}_0^TG\vec{d}_0$. Hence, defining $\mu_{\min}>0$ ($\mu_{\max}>0$) to be the minimum (maximum) eigenvalue of $G$ we have that
\begin{equation}
\frac{E_0+\bar{E}}{\mu_{\max}}\leq\mathopen{}\mathclose\bgroup\originalleft(\frac{1}{2}\tr{\gamma}+\|\vec{d}-\vec{d}_0\|^2\aftergroup\egroup\originalright)\leq \frac{E_0+\bar{E}}{\mu_{\min}}, \end{equation}
\noindent and consequently
\begin{align}
&\tr{\gamma}\leq O(E_0),\|\vec{d}\|\leq O(\sqrt{E_0}),\nonumber\\
&\frac{1}{2}\tr{\gamma}+\|\vec{d}-\vec{d}_0\|^2\geq O(E_0). \label{lower} \end{align}
We can now bound the energy difference between the input and output states. First, note that $\Delta E\equiv E_f-E_0$ can be written as:
\begin{equation} \Delta E=-\frac{1}{2}\tr{\gamma \tilde{G}}-(\vec{d}-\vec{d}_0)^T\tilde{G}(\vec{d}-\vec{d}_0)+O(\vec{d}). \end{equation}
\noindent Defining $\lambda_{\min}>0$ to be the smallest eigenvalue of $\tilde{G}$, we thus arrive at
\begin{align}
\Delta E&&\leq -\lambda_{\min}\mathopen{}\mathclose\bgroup\originalleft(\frac{1}{2}\tr{\gamma}+\|\vec{d}-\vec{d}_0)\|^2\aftergroup\egroup\originalright)+O(\vec{d})\nonumber\\ &&\leq-O(E_0)+O(\sqrt{E_0})=-O(E_0). \end{align}
Let us now bound the entropy of the input state: by the subadditivity of the von Neumann entropy, $S(\rho)$ is bounded from above by $\sum_{i=1}^mS(\rho_i)$, where $\rho_{i}$ denotes the reduced density matrix of each mode $i$. $S(\rho_i)$, in turn, is bounded by the von Neumman entropy of the Gaussian state with the same first and second moments as $\rho_i$, i.e., a Gaussian state with covariance matrix $\gamma_i$. Note that $\sum_{i=1}^m\tr{\gamma_i}=\tr{\gamma}\leq O(E_0)$, where the last inequality is due to eq. (\ref{lower}). In particular, $\tr{\gamma_i}\leq O(E_0)$ for $i=1,...,m$.
\noindent The entropy of a 1-mode Gaussian state with covariance matrix $\tilde{\gamma}$ is given by
\begin{equation} (N+1)\log(N+1)-N\log(N)= O(\log(N)), \end{equation}
\noindent where $N=\sqrt{\det(\tilde{\gamma})}/\hbar\leq \tr{\tilde{\gamma}}/2\hbar$ (this last inequality reflects the fact that the geometric mean of $\tilde{\gamma}$'s two eigenvalues is smaller than their arithmetic mean). It follows that
\begin{equation} S(\rho)\leq O(\log(E_0)). \end{equation}
Putting all together, we have that
\begin{align} \Delta F(\rho, \Omega) &\leq \Delta E+K_BTS(\rho)&\nonumber\\ &\leq -O(E_0)+O(\log(E_0)).& \end{align}
The last expression cannot thus take arbitrarily large values, and so the distillable work of channel $\Omega$ is bounded.
\section{Proof of Proposition 2} Proposition 2 follows straightforwardly from the following Lemma:
\begin{lemma} Let $\rho$ be an arbitrary state with finite first and second moments, and let $\rho_G$ be the unique Gaussian state with the same first and second moments. Then, for any Gaussian channel $\Omega$, we have that
\begin{equation} S(\rho_G)-S(\Omega(\rho_G))\geq S(\rho)-S(\Omega(\rho)). \label{paramecio} \end{equation}
\end{lemma}
\begin{proof}
Let the action of $\Omega$ over the displacement vector and covariance matrix be given by eq. (\ref{gauss_chan}). Since von Neumann entropies remain the same after a unitary transformation, without loss of generality we will assume that $\rho$'s displacement vector is null. Similarly, we will take $z=0$ in the channel description (\ref{gauss_chan}) of $\Omega$. Now, let $U_n$ be the $n$-system `Gaussification' transformation described in \cite{gaussification}. Calling $\tilde{\rho}^{(n)}=U_n\rho^{\otimes n}U_n^\dagger$, we have that
\begin{eqnarray} L_n &\equiv& \sum_{i=1}^nS(\tilde{\rho}_i)-S(\Omega(\tilde{\rho}_i)) \nonumber\\ &\geq& S(\tilde{\rho}^{(n)})-S(\Omega^{\otimes n}(\tilde{\rho}^{(n)}) )\noindent\\ &=& S(\rho^{\otimes n})-S(\Omega^{\otimes n}(\rho^{\otimes n}))\nonumber\\ &=&n\{S(\rho)-S(\Omega(\rho))\}, \label{inter1} \end{eqnarray}
\noindent where the first inequality is due to Lemma 1 in the main text, and the equality follows from the fact that, for all states $\sigma$,
\begin{equation} U_n\Omega^{\otimes n}(\sigma)U_n^\dagger=\Omega^{\otimes n}(U_n\sigma U_n^\dagger). \end{equation}
\noindent This identity follows from three observations: 1) any Gaussian channel with $z=0$ is the result of applying a symplectic unitary $V_S$ over the target system and an ancillary Gaussian state $\omega$ with zero displacement vector; 2) $n$ copies of $\omega$ are invariant with respect to a Gaussification operation $U^A_n$; 3) $[V_S^{\otimes n},U_n']=0$, where $U_n'=U_n\otimes U^A_n$ represents the Gaussification of $n$ copies of the system target-ancilla.
Since the von Neumann entropy is continuous in trace norm with respect to collections of states with finite second moments, by \cite{gaussification}, we end up with
\begin{eqnarray} \lim_{n\to\infty}\frac{L_n}{n} &= &\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^nS(\tilde{\rho}_i)-S(\Omega(\tilde{\rho}_i))\nonumber\\ &=&S(\rho_G)-S(\Omega(\rho_G)). \label{inter2} \end{eqnarray}
\noindent The lemma hence follows from (\ref{inter1}) and (\ref{inter2}).
\end{proof}
Now, let $\rho$ be an arbitrary state with finite energy (and thus finite first and second moments), and let $\rho_G$ be the Gaussian state with the same first and second moments. Then, $\rho$ and $\rho_G$ have the same average energy, and since $\Omega$ is a Gaussian channel the same is true for $\Omega(\rho)$ and $\Omega(\rho_G)$. However, by the previous Lemma, the entropic change is bigger for $\rho_G$, and so $\Delta F(\rho_G,\Omega)\geq \Delta F(\rho,\Omega)$, proving Proposition 2.
\end{document} | arXiv |
# Fourier transform and its applications
The Fourier transform is based on the concept of complex exponentials, which are mathematical functions that oscillate sinusoidally. A signal can be represented as a sum of complex exponentials, each with its own frequency and amplitude. This representation is called the frequency domain representation of the signal.
The Fourier transform is defined as follows:
$$
F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-j\omega t} dt
$$
where $F(\omega)$ is the frequency domain representation of the signal $f(t)$, and $\omega$ is the angular frequency.
The inverse Fourier transform is used to convert the frequency domain representation back to the time domain representation:
$$
f(t) = \frac{1}{2\pi}\int_{-\infty}^{\infty} F(\omega) e^{j\omega t} d\omega
$$
The Fourier transform has several important applications in spectral analysis, including:
- Analyzing the frequency content of a signal
- Separating different frequency components in a signal
- Identifying periodic components in a signal
- Designing filters to remove unwanted frequency components
## Exercise
Instructions:
Consider the following signal:
$$
f(t) = \sin(2\pi t) + \sin(4\pi t)
$$
1. Calculate the frequency domain representation of the signal using the Fourier transform.
2. Find the amplitude and phase of the frequency components.
### Solution
1. The frequency domain representation of the signal can be calculated using the Fourier transform:
$$
F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-j\omega t} dt = \int_{-\infty}^{\infty} (\sin(2\pi t) + \sin(4\pi t)) e^{-j\omega t} dt
$$
2. The amplitude and phase of the frequency components can be found by analyzing the frequency domain representation:
- For the first frequency component, $\omega = 2\pi$, the amplitude is 1, and the phase is 0.
- For the second frequency component, $\omega = 4\pi$, the amplitude is 1, and the phase is 0.
# Python programming basics for spectral analysis
To begin with, it is important to understand the basic syntax and data structures in Python. Some of the key concepts include:
- Variables and data types: Python supports various data types, such as integers, floating-point numbers, strings, and booleans.
- Control structures: Python uses conditionals (if, elif, else) and loops (for, while) to control the flow of execution.
- Functions: Functions are reusable blocks of code that can be defined and called by their name.
- Modules: Python provides a standard library of modules that offer a wide range of functionality.
Before diving into spectral analysis, it is important to familiarize yourself with the basics of Python programming. This will enable you to write efficient and well-structured code for spectral analysis tasks.
# Numpy library: data manipulation and processing
The Numpy library includes the following key data structures:
- Arrays: Numpy arrays are multidimensional arrays that can store numerical data of any data type. They are the primary data structure for numerical computations in Numpy.
- Matrices: Numpy matrices are a special type of array that is specifically designed for linear algebra operations.
The Numpy library also provides a wide range of functions for data manipulation and processing, including:
- Arithmetic operations: Numpy supports basic arithmetic operations, such as addition, subtraction, multiplication, and division, for arrays and matrices.
- Array manipulation: Numpy offers functions for reshaping arrays, slicing arrays, and concatenating arrays.
- Element-wise operations: Numpy provides functions for performing element-wise operations on arrays, such as exponentiation, logarithms, and trigonometric functions.
## Exercise
Instructions:
1. Import the Numpy library.
2. Create a Numpy array with the elements 1, 2, 3, 4, and 5.
3. Calculate the sum of the elements in the array.
4. Create a Numpy matrix with the elements 1, 2, 3, 4, 5, 6, 7, 8, 9.
5. Calculate the determinant of the matrix.
### Solution
1. Import the Numpy library:
```python
import numpy as np
```
2. Create a Numpy array:
```python
array = np.array([1, 2, 3, 4, 5])
```
3. Calculate the sum of the elements in the array:
```python
sum = np.sum(array)
```
4. Create a Numpy matrix:
```python
matrix = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
```
5. Calculate the determinant of the matrix:
```python
determinant = np.linalg.det(matrix)
```
# Data visualization in spectral analysis
Some of the key features of Matplotlib include:
- Line plots: Matplotlib can create line plots to visualize the relationship between two variables.
- Scatter plots: Matplotlib can create scatter plots to visualize the distribution of data points.
- Histograms: Matplotlib can create histograms to visualize the distribution of a continuous variable.
- Images: Matplotlib can create images to visualize two-dimensional data.
## Exercise
Instructions:
1. Import the Matplotlib library.
2. Create a line plot of the sine function from -π to π.
3. Create a scatter plot of random data points.
4. Create a histogram of random data points.
### Solution
1. Import the Matplotlib library:
```python
import matplotlib.pyplot as plt
```
2. Create a line plot of the sine function:
```python
x = np.linspace(-np.pi, np.pi, 100)
y = np.sin(x)
plt.plot(x, y)
plt.xlabel('x')
plt.ylabel('y')
plt.title('Sine Function')
plt.show()
```
3. Create a scatter plot of random data points:
```python
x = np.random.rand(50)
y = np.random.rand(50)
plt.scatter(x, y)
plt.xlabel('x')
plt.ylabel('y')
plt.title('Scatter Plot')
plt.show()
```
4. Create a histogram of random data points:
```python
data = np.random.rand(100)
plt.hist(data, bins=20)
plt.xlabel('x')
plt.ylabel('Frequency')
plt.title('Histogram')
plt.show()
```
# Signal processing using the Numpy library
The Numpy library provides a wide range of functions for signal processing, including:
- Convolution: Numpy can perform convolution to combine two signals or filters.
- Correlation: Numpy can calculate the correlation between two signals.
- Filtering: Numpy can apply filters to remove unwanted frequency components from a signal.
## Exercise
Instructions:
1. Create a signal with a frequency of 1 Hz and a sampling rate of 100 Hz.
2. Create a filter with a cutoff frequency of 50 Hz.
3. Apply the filter to the signal.
### Solution
1. Create a signal with a frequency of 1 Hz and a sampling rate of 100 Hz:
```python
time = np.linspace(0, 1, 100)
signal = np.sin(2 * np.pi * 1 * time)
```
2. Create a filter with a cutoff frequency of 50 Hz:
```python
filter_cutoff = 50
filter_order = 2
filter_coefficients = signal.butter(filter_order, filter_cutoff, btype='low', analog=False)
```
3. Apply the filter to the signal:
```python
filtered_signal = signal.filtfilt(*filter_coefficients)
```
# Applications of Fourier transform in signal processing
The Fourier transform has numerous applications in signal processing, including:
- Filtering: The Fourier transform can be used to design filters that remove unwanted frequency components from a signal.
- Spectral analysis: The Fourier transform can be used to analyze the frequency content of a signal.
- Image processing: The Fourier transform can be used to perform various operations on images, such as filtering, compression, and feature extraction.
- Speech analysis: The Fourier transform can be used to analyze the frequency content of speech signals, which can be used for tasks such as speech recognition and emotion recognition.
## Exercise
Instructions:
Consider the following signal:
$$
f(t) = \sin(2\pi t) + \sin(4\pi t)
$$
1. Calculate the frequency domain representation of the signal using the Fourier transform.
2. Design a filter to remove the 2 Hz frequency component from the signal.
3. Apply the filter to the signal.
### Solution
1. Calculate the frequency domain representation of the signal using the Fourier transform:
```python
import numpy as np
time = np.linspace(0, 1, 100)
signal = np.sin(2 * np.pi * 2 * time) + np.sin(2 * np.pi * 4 * time)
frequency_domain_representation = np.fft.fft(signal)
```
2. Design a filter to remove the 2 Hz frequency component from the signal:
```python
filter_cutoff = 2
filter_order = 2
filter_coefficients = signal.butter(filter_order, filter_cutoff, btype='low', analog=False)
```
3. Apply the filter to the signal:
```python
filtered_signal = signal.filtfilt(*filter_coefficients)
```
# Frequency analysis of signals
The Numpy library provides several functions for frequency analysis, including:
- Fast Fourier transform: Numpy can perform the fast Fourier transform to convert a signal from the time domain to the frequency domain.
- Power spectral density: Numpy can calculate the power spectral density of a signal, which is the average power of the signal across all frequencies.
## Exercise
Instructions:
1. Create a signal with a frequency of 1 Hz and a sampling rate of 100 Hz.
2. Calculate the fast Fourier transform of the signal.
3. Calculate the power spectral density of the signal.
### Solution
1. Create a signal with a frequency of 1 Hz and a sampling rate of 100 Hz:
```python
time = np.linspace(0, 1, 100)
signal = np.sin(2 * np.pi * 1 * time)
```
2. Calculate the fast Fourier transform of the signal:
```python
frequency_domain_representation = np.fft.fft(signal)
```
3. Calculate the power spectral density of the signal:
```python
power_spectral_density = np.abs(frequency_domain_representation)**2 / len(time)
```
# Window functions and their applications
Some common window functions include:
- Rectangular window: The rectangular window is the simplest window function, which simply multiplies the signal by a rectangular pulse.
- Hamming window: The Hamming window is a cosine-based window function that reduces the spectral leakage at the edges of the signal.
- Hanning window: The Hanning window is a cosine-based window function that is similar to the Hamming window, but has a slightly different shape.
## Exercise
Instructions:
1. Create a signal with a frequency of 1 Hz and a sampling rate of 100 Hz.
2. Apply a Hamming window to the signal.
3. Calculate the fast Fourier transform of the windowed signal.
### Solution
1. Create a signal with a frequency of 1 Hz and a sampling rate of 100 Hz:
```python
time = np.linspace(0, 1, 100)
signal = np.sin(2 * np.pi * 1 * time)
```
2. Apply a Hamming window to the signal:
```python
window = np.hamming(len(time))
windowed_signal = signal * window
```
3. Calculate the fast Fourier transform of the windowed signal:
```python
frequency_domain_representation = np.fft.fft(windowed_signal)
``` | Textbooks |
\begin{document}
\begin{abstract} Let $k$ be a number field. For $\mathcal{H}\rightarrow \infty$, we give an asymptotic formula for the number of algebraic integers of absolute Weil height bounded by $\mathcal{H}$ and fixed degree over $k$. \end{abstract}
\title{Counting algebraic integers of fixed degree and bounded height}
\section{Introduction} \label{intro}
Let $k$ be a number field of degree $m$ over $\mathbb{Q}$. We count the number of algebraic integers $\beta$ of degree $e$ over $k$ and bounded height. Here and in the rest of the article, by height we mean the multiplicative height $H$ on the affine space $\overline{\Q}^n$ (see \cite{BombGub}, 1.5.6).
For positive rational integers $n$ and $e$, and a fixed algebraic closure $\overline{\ks}$ of $k$, let $$ k(n,e)=\{\bo{\beta}\in \overline{\ks}^n : [k(\bo{\beta}):k]=e \}, $$ where $k(\bo{\beta})$ is the field obtained by adjoining all the coordinates of $\bo{\beta}$ to $k$. By Northcott's Theorem \cite{Northcott1949} any subset of $k(n,e)$ of uniformly bounded height is finite. Therefore, for any subset $S$ of $k(n,e)$ and $\mathcal{H}> 0$, we may introduce the following counting function $$
N(S,\mathcal{H})= |\left\lbrace \bo{\beta}\in S: H(\bo{\beta})\leq \mathcal{H} \right\rbrace|. $$ The counting function $N(k(n,e),\mathcal{H})$ has been investigated by various people. The best known and one of the earliest is a result of Schanuel \cite{Schanuel1979} who gave an asymptotic formula for $N(k(n,1),\mathcal{H})$. The first who dropped the restriction of the coordinates to lie in a fix number field was Schmidt. In \cite{Schmidt1993}, he found upper and lower bounds for $N(k(n,e),\mathcal{H})$ and in \cite{schmidt1995} he gave an asymptotic formula for $N(\mathbb{Q}(n,2),\mathcal{H})$. Shortly afterwards, Gao \cite{Gao1995} found the asymptotics for $N(\mathbb{Q}(n,e),\mathcal{H})$, provided $n>e$. Later Masser and Vaaler \cite{Masser2007} established an asymptotic estimate for $N(k(1,e),\mathcal{H})$. Finally, Widmer \cite{Widmer2009} proved an asymptotic formula for $N(k(n,e),\mathcal{H})$ for arbitrary number fields $k$, provided $n>5e/2+5+2/\me$. However, for general $n$ and $e$ even the correct order of magnitude for $N(k(n,e),\mathcal{H})$ remains unknown.
In this article we are interested in counting integral points, i.e., points $\bo{\beta}\in \overline{\ks}^n$, whose coordinates are algebraic integers. Let $\mathcal{O}_k$ and $\mathcal{O}_{\overline{\ks}}$ be, respectively, the ring of algebraic integers in $k$ and $\overline{\ks}$. We introduce $$ \mathcal{O}_k(n,e)=k(n,e)\cap \mathcal{O}_{\overline{\ks}}^n=\{\bo{\beta}\in \mathcal{O}_{\overline{\ks}}^n:[k(\bo{\beta}):k]=e\}. $$ Possibly, the first asymptotic result (besides the trivial cases $\mathcal{O}_\mathbb{Q}(n,1)=\mathbb{Z}^n$) can be found in Lang's book \cite{Lang1983}. Lang states, without proof, \begin{equation*} N(\mathcal{O}_k(1,1),\mathcal{H})=\gamma_k \mathcal{H}^m \left( \log \mathcal{H} \right)^q + O\left( {\mathcal{H}}^{m} \left( \log \mc{H} \right)^{q-1} \right), \end{equation*} where $m=[k:\mathbb{Q}]$, $q$ is the rank of the unit group of the ring of integers $\mathcal{O}_k$, and $\gamma_k$ is an unspecified positive constant, depending on $k$. More recently, Widmer \cite{Widmer2013} established the following asymptotic formula \begin{alignat}1\label{Okne} N(\mathcal{O}_k(n,e),\mathcal{H})=\sum_{i=0}^{t} D_i\mathcal{H}^{\men}\left(\log \mathcal{H}^{\me n}\right)^i+O\left(\mathcal{H}^{\men-1}(\log \mathcal{H})^{t}\right), \end{alignat} provided $e=1$ or $n>e+C_{e,m}$, for some explicit $C_{e,m}\leq 7$. Here $t=e(q+1)-1$, and the constants $D_i=D_i(k,n,e)$ are explicitly given. Widmer's result is fairly specific in the sense that he works only with the absolute multiplicative Weil height $H$. On the other hand, the methods used in \cite{Widmer2013} are quite general and powerful, and can probably be applied to handle other heights (such as the heights used by Masser and Vaaler in \cite{Masser2007} to deduce their main result). As mentioned in \cite{Widmer2013} this might lead to multiterm expansions as in (\ref{Okne}) for $N(\mathcal{O}_k(1,e),\mathcal{H})$.
However, for the moment, such generalizations of (\ref{Okne}) are not available, and thus the work \cite{Widmer2013} does not provide any results in the case $n=1$ and $e>1$.
But Chern and Vaaler in \cite{Chern2001}, proved an asymptotic formula for the number of monic polynomials in $\mathbb{Z}[x]$ of given degree and bounded Mahler measure. Theorem 6 of \cite{Chern2001} immediately implies the following result \begin{equation}\label{chern} N(\mathcal{O}_\mathbb{Q}(1,e),\mathcal{H})= C_{e} \mathcal{H}^{e^2}+ O\left( \mc{H}^{e^2-1} \right), \end{equation} for some explicitly given positive real constant $C_e$. Theorem \ref{thmint} extends Chern and Vaaler's result to arbitrary ground fields $k$.
For positive rational integers $e$ we define $$ C_{\mathbb{R},e} = 2^{e-M} \left( \prod_{l=1}^{M}\left( \frac{2l}{2l+1} \right)^{e-2l} \right) \frac{e^M}{M!}, $$ with $M=\lfloor \frac{e-1}{2} \rfloor$, and $$ C_{\mathbb{C},e} = \pi^e \frac{e^e}{\left( e! \right)^2}. $$ And, finally, let \begin{equation}\label{const}
C^{(e)}_{k}= \frac{e^{2q+1} 2^{se} m^{q}}{q!\left( \sqrt{|\Delta_k|}\right)^{e}}C_{\mathbb{R},e}^r C_{\mathbb{C},e}^s, \end{equation} where $m=[k:\mathbb{Q}]$, $r$ is the number of real embeddings of $k$, $s$ the number of pairs of complex conjugate embeddings, $q=r+s-1$, and $\Delta_k$ denotes the discriminant of $k$. As usual, here and in the rest of this article, the empty product is understood to be 1.
For non-negative real functions $f(X),g(X),h(X)$ and $X_0\in \mathbb{R}$ we write
$f(X)=g(X)+O(h(X))$ as $X\geq X_0$ tends to infinity if there is $C_0$ such that $|f(X)-g(X)|\leq C_0 h(X)$ for all $X \geq X_0$.
\begin{theorem}\label{thmint} Let $e$ be a positive integer, and let $k$ be a number field of degree $m$ over $\mathbb{Q}$. Then, as $\mathcal{H}\geq 2$ tends to infinity, we have \begin{equation*}\label{mainest} N(\mathcal{O}_k(1,e),\mathcal{H})=C^{(e)}_{k}\mc{H}^{\me^2} \left( \log \mc{H} \right)^{q} +\left\lbrace \begin{array}{ll} O\left( \mc{H}^{\me^2} \left( \log \mc{H} \right)^{q-1} \right), & \mbox{ if $q\geq 1$, }\\ O\left(\mc{H}^{e(\me-1)} \mc{L} \right), & \mbox{ if $q=0$, } \end{array} \right. \end{equation*} where $\mc{L}=\log \mathcal{H}$ if $(m,e)=(1,2)$ and 1 otherwise. The implicit constant in the error term depends only on $m$ and $e$. \end{theorem}
Let us mention two simple examples. The number of algebraic integers $\alpha$ quadratic over $\mathbb{Q}(\sqrt{2})$ with $H(\alpha)\leq \mathcal{H}$ is $$ 32\mathcal{H}^{8} \log \mathcal{H}+O(\mathcal{H}^8). $$ In case $e=3$, we have $$ 108\sqrt{2} \mathcal{H}^{18}\log \mathcal{H} + O(\mathcal{H}^{18}) $$ algebraic integers $\alpha$ cubic over $\mathbb{Q}(\sqrt{2})$ with $H(\alpha)\leq \mathcal{H}$.
Our approach is similar to the one used to obtain (\ref{chern}) above, because we count monic polynomials in $\mathcal{O}_k[X]$, but this is not a straightforward generalization of Theorem 6 of \cite{Chern2001}. In fact, in \cite{Chern2001} the estimate on the number of monic polynomials in $\mathbb{Z}[x]$ is obtained from a counting lattice points theorem, which is formulated only for the standard lattice $\mathbb{Z}^n$ (\cite{Chern2001}, Lemma 24). Our proof relies on a new counting theorem for points of an arbitrary lattice in definable sets in an o-minimal structure \cite{Barroero2012}. Moreover, our proof is fairly short, and more straightforward than the approach of \cite{Widmer2013}, but to the expense that we do not get a multiterm expansion.
In \cite{Masser2007}, Masser and Vaaler observed that the limit for $ \mathcal{H}\rightarrow \infty$ of $$ \frac{N(k(1,e),\mathcal{H}^{\frac{1}{e}})}{N(k(e,1),\mathcal{H})} $$ is a rational number. Moreover, they asked if this can be extended to some sort of reciprocity law, i.e., whether $$ \lim_{\mathcal{H} \rightarrow \infty}\frac{N(k(n,e),\mathcal{H}^{\frac{1}{e}})}{N(k(e,n),\mathcal{H}^{\frac{1}{n}})}\in \mathbb{Q}. $$ If we consider only the first term in (\ref{Okne}), and combine it with Theorem \ref{thmint} we see that $$ \lim_{\mathcal{H} \rightarrow \infty} \frac{N(\mathcal{O}_k(1,e),\mathcal{H}^{\frac{1}{e}})}{N(\mathcal{O}_k(e,1),\mathcal{H})}=e \left(\frac{C_{\mathbb{R},e}}{2^e}\right)^r \left(\frac{C_{\mathbb{C},e}}{\pi^e}\right)^s $$ is a rational number depending only on $e$, $r$ and $s$. As Masser and Vaaler did, one can ask again whether $$ \lim_{\mathcal{H} \rightarrow \infty}\frac{N(\mathcal{O}_k(n,e),\mathcal{H}^{\frac{1}{e}})}{N(\mathcal{O}_k(e,n),\mathcal{H}^{\frac{1}{n}})}\in \mathbb{Q}. $$
\section{Counting monic polynomials} \label{sect2}
In this section we see how our problem translates to counting monic polynomials of fixed degree that assume a uniformly bounded value under a certain real valued function called $M^k$, defined using the Mahler measure.
Recall we fixed a number field $k$ of degree $m$ over $\mathbb{Q}$ and $\mathcal{O}_k$ is its ring of integers. Let $\sigma_1, \dots , \sigma_r$ be the real embeddings of $k$ and $\sigma_{r+1}, \dots , \sigma_{m}$ be the strictly complex ones, indexed in such a way that $\sigma_{j}=\overline{\sigma}_{j+s}$ for $j=r+1 , \dots , r+s$. Therefore, $r$ and $s$ are, respectively, the number of real and pairs of conjugate complex embeddings of $k$ and $m=r+2s$. We put $d_i=1$ for $i=1, \dots ,r$ and $d_i=2$ for $i=r+1, \dots ,r+s$ and fix a positive integer $e$. Let us recall the definition of the Mahler measure.
\begin{definition} If $f=z_0X^d+z_1X^{d-1}+\cdots +z_d \in \mathbb{C}[X]$ is a non-zero polynomial of degree $d$ with roots $\alpha_1,\ldots , \alpha_d$, the Mahler measure of $f$ is defined to be \begin{equation*}
M(f)=|z_0| \prod_{i=1}^d \max\left\lbrace 1,|\alpha_i|\right\rbrace. \end{equation*} Moreover, we set $M(0)=0$. \end{definition} We see $M$ as a function $\mathbb{C}[X] \rightarrow [0,\infty)$ and define \begin{equation*} \begin{array}{cccl}\label{defmk} M^k:& k[X]& \rightarrow & [0,\infty) \\
& f & \mapsto & \prod_{i=1}^{r+s}M(\sigma_i(f))^{\frac{d_i}{m}}, \end{array} \end{equation*} where $\sigma_i$ acts on the coefficients of $f$. Note that, for every $\alpha \in \mathcal{O}_k$, \begin{equation}\label{mahlheight}
M^k(X-\alpha)=\prod_{i=1}^{r+s}\max \left\lbrace 1,|\sigma_i(\alpha)| \right\rbrace^{\frac{d_i}{m}}=H(\alpha). \end{equation}
In fact, if $\alpha \in \mathcal{O}_k$ then $|\alpha|_v\leq 1$ for every non-archimedean place $v$ of $k$.
Moreover, the Mahler measure is multiplicative by definition, i.e., $$ M(fg)=M(f)M(g), $$ and one can see that $$ M^k(fg)=M^k(f)M^k(g), $$ for every $f,g \in k[X]$.
For some positive integer $e$ and some $\mathcal{H}>0$, we define $\mathcal{M}^k(e,\mathcal{H})$ to be the set of monic $f \in \mathcal{O}_k[X]$ of degree $e$ and $M^k(f)\leq \mathcal{H}$. It is easy to see that $\mathcal{M}^k(e,\mathcal{H})$ is finite for all $\mathcal{H}$. The following theorem gives an estimate for its cardinality.
\begin{theorem}\label{mainthm} For every $\mathcal{H}_0>1$ there exists a $D_0$ such that, for every $\mathcal{H} \geq \mathcal{H}_0$, \begin{equation}\label{asym}
\left| \left|\mathcal{M}^k(e,\mathcal{H})\right|- \frac{C^{(e)}_{k}}{e^{q+1}}\mc{H}^{\me} \left( \log \mc{H} \right)^{q} \right| \leq \left\lbrace \begin{array}{ll} D_0 \mc{H}^{\me} \left( \log \mc{H} \right)^{q-1} , & \mbox{ if $q\geq 1$, }\\ D_0 \mc{H}^{\me-1} , & \mbox{ if $q=0$, } \end{array} \right. \end{equation} where $q=r+s-1$. The constant $D_0$ depends only on $\mathcal{H}_0$, $m$ and $e$. \end{theorem}
Note that our constant $C^{(e)}_{k}$ defined in (\ref{const}), is bounded if we fix $m$ and $e$ and we let $k$ vary among all number fields of degree $m$. This implies that there exists a real constant $C^{(m,e)}$, depending only on $m$ and $e$, such that $ \left|\mathcal{M}^k(e,\mathcal{H})\right|$ is bounded from above by \begin{equation}\label{asym2} C^{(m,e)}\mc{H}^{\me} \left( \log \mc{H} +1 \right)^{q} , \end{equation} for every $\mathcal{H} \geq 1$.
We prove Theorem \ref{mainthm} later and for the rest of this section we derive Theorem \ref{thmint} from Theorem \ref{mainthm}. We follow the line of Masser and Vaaler \cite{Masser2007}.
Now we want to restrict to monic $f$ irreducible over $k$. Let $ \widetilde{\mathcal{M}}^k(e,\mathcal{H})$ be the set of polynomials in $\mathcal{M}^k(e,\mathcal{H})$ that are irreducible over $k$.
\begin{corollary}\label{maincor} For every $\mathcal{H}_0>1$ there exists an $F_0$ such that, for every $\mathcal{H} \geq \mathcal{H}_0$, \begin{equation}\label{asym3}
\left| \left|\widetilde{\mathcal{M}}^k(e,\mathcal{H})\right|- \frac{C^{(e)}_{k}}{e^{q+1}}\mc{H}^{\me} \left( \log \mc{H} \right)^{q} \right| \leq \left\lbrace \begin{array}{ll} F_0 \mc{H}^{\me} \left( \log \mc{H} \right)^{q-1} , & \mbox{ if $q\geq 1$, }\\ F_0 \mc{H}^{\me-1} \mathcal{L} , & \mbox{ if $q=0$, } \end{array} \right. \end{equation} where $\mc{L}=\log \mathcal{H}$ if $(m,e)=(1,2)$ and 1 otherwise. The constant $F_0$ depends again only on $\mathcal{H}_0$, $m$ and $e$. \end{corollary}
\begin{proof} For $e=1$ there is nothing to prove. Suppose $e>1$. We show that, up to a constant, the number of all monic reducible $f \in \mathcal{O}_k[X]$ of degree $e$ with $M^k(f)\leq \mc{H}$ is not larger than the right hand side of (\ref{asym}), except for the case $(m,e)=(1,2)$.
Consider all $f=gh \in \mathcal{M}^k(e,\mathcal{H})$ with $g,h\in \mathcal{O}_k[X]$ monic of degree $a$ and $b$ respectively, with $0<a\leq b<e$ and $a+b=e$. We have $1 \leq M^k(g),M^k(h) \leq \mathcal{H}$ because $g$ and $h$ are monic. Thus, there exists a positive integer $l$ such that $2^{l-1}\leq M^k(g)<2^l$. Note that $l$ must satisfy \begin{equation}\label{ineqk} 1\leq l \leq \frac{\log \mc{H}}{\log 2}+1\leq 2 \log \mc{H} +1. \end{equation} Since $M^k$ is multiplicative, $$ M^k(h)=\frac{M^k(f)}{M^k(g)}\leq 2^{1-l} \mc{H}. $$ Using (\ref{asym2}) and noting that $2^l\leq 2 \mathcal{H}$, we can say that there are at most $$ C^{(m,a)}\left(2^l\right)^{m a } \left( \log 2^l+1 \right)^{q} \leq C^{(m,a)}\left(2^l\right)^{m a } \left( \log \mathcal{H} +2 \right)^{q} $$ possibilities for $g$ and $$ C^{(m,b)}\left(2^{1-l} \mc{H}\right)^{m b}\left( \log \left(2^{1-l} \mc{H} \right)+1 \right)^{q} \leq C^{(m,b)}\left(2^{1-l} \mc{H}\right)^{m b}\left( \log \mc{H} +2 \right)^{q} $$ possibilities for $h$. Therefore, we have at most \begin{align} \label{red} C' \mc{H}^{m b} 2^{m l(a-b)} \left( \log \mc{H} +2 \right)^{2q} \end{align} possibilities for $gh$ with $M^k (gh)\leq \mathcal{H}$ and $2^{l-1}\leq M^k(g)<2^l$, where $C'$ is a real constant. Since there are only finitely many choices for $a$ and $b$ we can take $C'$ to depend only on $m$ and $e$.
If $a=b=\frac{e}{2}$ then (\ref{red}) is $$ C'\mc{H}^{m \frac{e}{2}} \left( \log \mc{H} +2 \right)^{2q}. $$ Summing over all $l$, $1\leq l \leq \lfloor 2 \log \mc{H} \rfloor+1$ (recall (\ref{ineqk})), gives an extra factor $2\log \mc{H} +1 $. Therefore, when $a=b$, there are at most $$ C'\mc{H}^{\frac{m e}{2}}\left(2 \log \mc{H} +2 \right)^{2q+1} $$ possibilities for $f=gh$, with $M^k(f)\leq \mathcal{H}$. If $(m,e)\neq (1,2)$, this has smaller order than the right hand side of (\ref{asym}), since $m e >2$ implies $\frac{m e}{2}< m e-1$. In the case $(m,e)= (1,2)$ we get $C' \mathcal{H} \left(2 \log \mc{H} +2 \right)$ and we need an additional logarithm factor.
In the case $a <b$, summing $2^{m l(a-b)}$ over all $l$, $1\leq l \leq \lfloor 2 \log \mc{H} \rfloor+1 =L$, we get $$ \sum_{l=1}^L \left(2^{m(a-b)}\right)^l \leq \sum_{l=1}^{L}2^{-l}\leq 1. $$ Thus, recalling $b\leq e-1$, when $a<b$, there are at most $$ C'' \mc{H}^{m(e-1)} \left( \log \mc{H} +2 \right)^{2q} $$ possibilities for $f=gh$, with $M^k(f)\leq \mathcal{H}$, where again $C''$ depends only on $m$ and $e$. This is again not larger than the right hand side of (\ref{asym}). \end{proof}
For the last step of the proof we link such monic irreducible polynomials with their roots.
\begin{lemma} \label{lemmapoly}
An algebraic integer $\beta$ has degree $e$ over $k$ and $H( \beta ) \leq \mathcal{H}$ if and only if it is a root of a monic irreducible polynomial $f \in \mathcal{O}_k[X]$ of degree $e$ with $M^k(f)\leq \mathcal{H}^e$
\end{lemma}
\begin{proof} Suppose $f\in \mathcal{O}_k[X]$ is a monic irreducible polynomial of degree $e$ and $\beta$ is one of its roots, i.e., $\beta$ is an algebraic integer with $[k(\beta):k]=e$ and minimal polynomial $f$ over $k$. We claim that $$ M^k(f)=H(\beta)^e. $$
The function $M^k$ is independent of the field $k$ and we can define an absolute $M^{\overline{\mathbb{Q}}}$ over $\overline{\mathbb{Q}}[X]$ that, restricted to any $k[X]$, equals $M^k$. To see this one can simply imitate the proof of the fact that the Weil height is independent of the field containing the coordinates (see \cite{BombGub}, Lemma 1.5.2).
Suppose $f=(X-\alpha_1)\cdots (X-\alpha_e)$. Since the $\alpha_i$ are algebraic integers, by (\ref{mahlheight}), we have $$ M^{\overline{\mathbb{Q}}}(X-\alpha_i)=M^{\mathbb{Q}(\alpha_i)}(X-\alpha_i)=H(\alpha_i), $$ and the $\alpha_i$ have the same height because they are conjugate (see \cite{BombGub}, Proposition 1.5.17). Moreover, by the multiplicativity of $M^k$ we can see that $$ M^k(f)=M^{\overline{\mathbb{Q}}}(f)=\prod_{i=1}^e M^{\overline{\mathbb{Q}}}(X-\alpha_i)= H(\alpha_j)^e, $$ for any $\alpha_j$ root of $f$. \end{proof}
Lemma \ref{lemmapoly} implies that $ N(\mathcal{O}_k(1,e),\mathcal{H})= e \left|\widetilde{\mathcal{M}}^k(e,\mathcal{H}^e) \right|$ because there are $e$ different $\beta $ with the same minimal polynomial $f$ over $k$. Therefore, by (\ref{asym3}), we have that for every $\mathcal{H}_0>1$ there exists a $C_0$, depending only on $\mathcal{H}_0$, $m$ and $e$, such that for every $\mathcal{H}\geq \mathcal{H}_0$, \begin{equation*}
\left| N(\mathcal{O}_k(1,e),\mathcal{H}) - C^{(e)}_{k} \mc{H}^{\me^2} \left( \log \mc{H} \right)^{q} \right| \leq \left\lbrace \begin{array}{ll} C_0 \mc{H}^{\me^2} \left( \log \mc{H} \right)^{q-1} , & \mbox{ if $q\geq 1$, }\\ C_0 \mc{H}^{e(\me-1)} \mathcal{L} , & \mbox{ if $q=0$, } \end{array} \right. \end{equation*} where $\mc{L}=\log \mathcal{H}$ if $(m,e)=(1,2)$ and 1 otherwise. We get Theorem \ref{thmint} by choosing $\mathcal{H}_0=2$.
\section{A counting principle}
In this section we introduce the counting theorem that will be used to prove Theorem \ref{mainthm}. The principle dates back to Davenport \cite{Davenport1951} and was developed by several authors. In a previous work \cite{Barroero2012} the author and Widmer formulated a counting theorem that relies on Davenport's result and uses o-minimal structures. The full generality of Theorem 1.3 of \cite{Barroero2012} is not needed here as we are going to count lattice points in semialgebraic sets. \begin{definition} Let $N$, $M_i$, for $i=1, \dots ,N$, be positive integers. A semialgebraic subset of $\mathbb{R}^n$ is a set of the form $$ \bigcup_{i=1}^{N}\bigcap_{j=1}^{M_i} \{ \boldsymbol{x} \in \mathbb{R}^n : f_{i,j}(\boldsymbol{x}) \ast_{i,j} 0 \}, $$ where $f_{i,j} \in \mathbb{R}[X_1, \dots , X_n]$ and the $\ast_{i,j}$ are either $<$ or $=$. \end{definition}
A very important feature of semialgebraic sets is the fact that this collection of subsets of the Euclidean spaces is closed under projections. This is the well known Tarski-Seidenberg principle.
\begin{theorem}[\cite{Bierstone88}, Theorem 1.5]\label{tarski} Let $A \in \mathbb{R}^{n+1}$ be a semialgebraic set, then $\pi(A)\in \mathbb{R}^n$ is semialgebraic, where $\pi :\mathbb{R}^{n+1}\rightarrow \mathbb{R}^n$ is the projection map on the first $n$ coordinates. \end{theorem}
Let $S \subseteq \mathbb{R}^{n+n'}$, for a $\bo{t}\in \mathbb{R}^{n'}$ we call $S_{\bo{t}}=\{ \boldsymbol{x} \in \mathbb{R}^n: (\boldsymbol{x},\bo{t})\in S\}$ the fiber of $S$ above $\bo{t}$. Clearly, if $S$ is semialgebraic also the fibers $S_{\bo{t}}$ are semialgebraic. If so, we call $S$ a semialgebraic family.
Let $\Lambda$ be a lattice of $\mathbb{R}^n$, i.e., the $\mathbb{Z}$-span of $n$ linearly independent vectors of $\mathbb{R}^n$. Let $\lambda_i=\lambda_i(\Lambda)$ for $i=1,\ldots,n$ be the successive minima of $\Lambda$ with respect to the zero centered unit ball $B_0(1)$, i.e., for $i=1,...,n$ \begin{alignat*}1 \lambda_i=\inf\{\lambda:B_0(\lambda)\cap\Lambda \text{ contains $i$ linearly independent vectors}\}. \end{alignat*} The following theorem is a special case of Theorem 1.3 of \cite{Barroero2012}.
\begin{theorem}\label{counttheorem} Let $Z\subset \mathbb{R}^{n+n'}$ be a semialgebraic family and suppose the fibers $Z_{\bo{t}}$ are bounded. Then there exists a constant $c_Z \in \mathbb{R}$, depending only on the family, such that, for every $\bo{t} \in \mathbb{R}^{n'}$, \begin{equation*}\label{eqcount}
\left| |Z_{\bo{t}}\cap \Lambda|-\frac{\textup{Vol}(Z_{\bo{t}})}{\det \Lambda} \right|\leq \sum_{j=0}^{n-1}c_{Z}\frac{V_j(Z_{\bo{t}})}{\lambda_1\cdots \lambda_j}, \end{equation*} where $V_j(Z_{\bo{t}})$ is the sum of the $j$-dimensional volumes of the orthogonal projections of $Z_{\bo{t}}$ on every $j$-dimensional coordinate subspace of $\mathbb{R}^n$ and $V_0(Z_{\bo{t}})=1$. \end{theorem}
\section{A semialgebraic family}
In this section we introduce the family we want to apply Theorem \ref{counttheorem} to.
We see the Mahler measure as a function of the coefficients of the polynomial. We fix $n>0$ and define $M:\mathbb{R}^{n+1}$ or $\mathbb{C}^{n+1} \rightarrow [0,\infty)$ such that $$ M(z_0,\ldots ,z_n)= M(z_0 X^n + \dots + z_n). $$ These two functions satisfy the definition of bounded distance function in the sense of the geometry of numbers, i.e., \begin{enumerate} \item $M$ is continuous; \item $M(\bo{z})=0$ if and only if $\bo{z}=\bo{0}$;
\item $M(w\bo{z})=|w|M(\bo{z})$, for any scalar $w \in \mathbb{R}$ or $\mathbb{C}$. \end{enumerate}
Properties (2) and (3) are obvious from the definition, while continuity was proved already by Mahler (see \cite{Mahler1961}, Lemma 1).
Let $M_1$ be the monic Mahler measure function, i.e., $M_1(\boldsymbol{z})=M(1,\boldsymbol{z})$ for $\boldsymbol{z} \in \mathbb{R}^n$ or $\mathbb{C}^n$.
In the following we consider the complex monic Mahler measure as a function \begin{equation*} \begin{array}{cccc}\label{Mahlerf} M_1: & \mathbb{R}^{2n} & \rightarrow & \mathbb{R} \\
& \left( x_1,\ldots ,x_{2n}\right) & \mapsto & M\left(X^n + (x_1+ix_2)X^{n-1}+ \cdots +x_{2n-1}+ix_{2n}\right). \end{array} \end{equation*}
We fix positive integers $n, m ,r,s$ with $m=r+2s$ and $d_1 ,\dots d_{r+s}$ such that $d_i=1$ for $i=1, \dots ,r$ and $d_i=2$ for $i=r+1, \dots ,r+s$.
We define \begin{equation}\label{defZ} Z=\left\lbrace (\bo{x}_1, \ldots , \bo{x}_{r+s},t) \in \left( \mathbb{R}^n \right)^r \times \left( \mathbb{R}^{2n} \right)^s \times \mathbb{R} : \prod_{i=1}^{r+s} M_1(\bo{x}_i)^{d_i}\leq t \right\rbrace. \end{equation} Here $\boldsymbol{x}_i \in \mathbb{R}^{d_i n}$ and $M_1(\bo{x}_i)$ is the real or the complex monic Mahler measure respectively if $i=1, \dots ,r$ or $i=r+1, \dots ,r+s$.
We want to count lattice points in the fibers $Z_t\subseteq \mathbb{R}^{mn}$ using Theorem \ref{counttheorem}, therefore we need to show that $Z$ is a semialgebraic set and that the fibers $Z_t$ are bounded.
\begin{lemma}
The set $Z$ defined in (\ref{defZ}) is semialgebraic.
\end{lemma}
\begin{proof} Recall the definition of $Z$. To each $\boldsymbol{x}_i\in \mathbb{R}^{d_in}$ corresponds a monic polynomial $f_i$ of degree $n$ with real (for $i=1,\dots r$) or complex (for $i=r+1,\dots r+s$) coefficients. Let $S$ be the set of points $$ \left( \bo{x}_1, \ldots , \bo{x}_{r+s},t, t_1 ,\dots ,t_{r+s}, \bo{\alpha}^{(1)},\bo{\beta}^{(1)}, \dots ,\bo{\alpha}^{(r+s)},\bo{\beta}^{(r+s)} \right) $$ in $\mathbb{R}^{n(r+2s)+1+r+s+2n(r+s)}$, with $\bo{\alpha}^{(i)}, \bo{\beta}^{(i)} \in \mathbb{R}^n$, such that \begin{itemize} \item $\bo{\alpha}^{(i)} $ and $ \bo{\beta}^{(i)}$ are, respectively, the vectors of the real and the imaginary parts of the $n$ roots of $f_i$, for every $i=1 ,\dots ,r+s$; \item $\prod_{l=1}^{n}\max \left\lbrace 1, \left(\alpha_l^{(i)}\right)^2+\left( \beta_l^{(i)}\right)^2 \right\rbrace=t_i^2$ and $t_i \geq 0$, for every $i=1 ,\dots ,r+s$; \item $\prod_{i=1}^{r+s}t_i^{d_i} \leq t$. \end{itemize} It is clear that the set $S$ is defined by polynomial equalities and inequalities. In fact, the first condition is enforced by the fact that the coordinates of $\boldsymbol{x}_i$ are the images of $\bo{\alpha}^{(i)}$ and $ \bo{\beta}^{(i)}$ under the appropriate symmetric functions, which are polynomials. The second and the third conditions are also clearly obtained by polynomial equalities and inequalities. Therefore, $S$ is a semialgebraic set. The claim follows after noting that $Z$ is nothing but the projection of $S$ on the first $n(r+2s)+1$ coordinates and applying the Tarski-Seidenberg principle (Theorem \ref{tarski}). \end{proof}
By Lemma 1.6.7 of \cite{BombGub}, there exists a positive real constant $\gamma\leq 1$ such that $$
\gamma |\boldsymbol{z}|_\infty \leq M(\boldsymbol{z}), \text{ for every $\boldsymbol{z} \in \mathbb{R}^{n+1}$ or $\mathbb{C}^{n+1}$}, $$
where, if $\boldsymbol{z}=(z_0, \dots ,z_n) \in \mathbb{R}^{n+1}$ or $\mathbb{C}^{n+1}$, $|\boldsymbol{z}|_\infty=\max \left\lbrace |z_0| ,\ldots ,|z_n| \right\rbrace$ is the usual $\max$ norm. Clearly we have, for $\boldsymbol{x} \in \mathbb{R}^n$ \begin{equation}\label{ineqnorm}
N(\boldsymbol{x}):=\gamma |(1,\boldsymbol{x})|_\infty \leq M_1(\boldsymbol{x}) \end{equation} in the real case and, for the complex case, \begin{equation}\label{ineqnorm2}
N(\boldsymbol{x}):= \gamma |(1,\boldsymbol{x})|_\infty \leq \gamma |(1,\boldsymbol{z})|_\infty \leq M_1(\boldsymbol{z})= M_1(\boldsymbol{x}) , \end{equation} where $\boldsymbol{x}=(x_1, \dots ,x_{2n}) \in \mathbb{R}^{2n}$ and $\boldsymbol{z}=(x_1 + i x_2 ,\dots , x_{2n-1} + i x_{2n})$.
Recall that, by the definition, the monic Mahler measure function assumes values greater than or equal to 1, therefore, if $(\bo{x}_1, \ldots , \bo{x}_{r+s}) \in Z_t$ then $M_1(\boldsymbol{x}_i)^{d_i} \leq t$ for every $i$. Thus, $ |\boldsymbol{x}_i|^{d_i}_\infty \leq \frac{t}{\gamma^{d_i}}$ and this means that $Z_t$ is bounded for every $t \in \mathbb{R}$.
Now we can apply Theorem \ref{counttheorem} to the family $Z$. If we set $Z(T)=Z_T$, we have \begin{equation}\label{extim}
\left| |Z(T)\cap \Lambda|-\frac{\textup{Vol}(Z(T))}{\det \Lambda} \right|\leq \sum_{j=0}^{{mn}-1}C \frac{V_j(Z(T))}{\lambda_1\cdots \lambda_j}, \end{equation} for every $T \in \mathbb{R}$, where $\Lambda$ is a lattice in $\mathbb{R}^{mn}$ and $C$ is a real constant independent of $\Lambda$ and $T$. Recall that $V_j(Z(T))$ is the sum of the $j$-dimensional volumes of the orthogonal projections of $Z(T)$ on every $j$-dimensional coordinate subspace of $\mathbb{R}^{mn}$ and $V_0(Z(T))=1$.
\section{Proof of Theorem \ref{mainthm}}
We fix a number field $k$ of degree $m$ over $\mathbb{Q}$. The ring of integers $\mathcal{O}_k$ of $k$, embedded into $\mathbb{R}^{r+2s}$ via $\sigma=(\sigma_1 ,\dots ,\sigma_{r+s})$, is a lattice of full rank. We embed $(\mathcal{O}_k)^n$ in $\mathbb{R}^{mn}$ via $\bo{a}\mapsto (\sigma_1 (\bo{a}) ,\dots ,\sigma_{r+s}(\bo{a}))$, where the $\sigma_i$ are extended to $k^n$. We want to count lattice points of $\Lambda=(\mathcal{O}_k)^n$ inside $Z(T)$.
\begin{lemma}\label{lemdet}
We have $$
\det \Lambda =\left( 2^{-s} \sqrt{|\Delta_k|}\right)^{n}, $$ and its first successive minimum is $\lambda_1 \geq 1$.
\end{lemma}
\begin{proof} This is a special case of Lemma 5 of \cite{Masser2007}. \end{proof}
Now we need to calculate the volume of $Z(T)$. We do something more general. Suppose we have $r+s$ continuous functions $f_i : \mathbb{R}^{n_i} \rightarrow [1, \infty)$, $i=1, \dots ,r+s$ where $1\leq n_i \leq d_i n$ for every $i$. We define \begin{equation}\label{rball} Z_i(T) =\{\boldsymbol{x} \in \mathbb{R}^{ n_i}: f_i(\boldsymbol{x})\leq T\}, \end{equation} for every $i=1,\dots ,r+s$. Suppose that, for every $i$, there exists a polynomial $p_i(X) \in \mathbb{R}[X]$ of degree $n_i$ such that the volume of $Z_i(T)$ is $p_i\left( T \right) $ for every $T\geq 1$. Let $C_i$ be the leading coefficient of $p_i$. Moreover, let $$ \widetilde{Z}(T)=\left\lbrace (\bo{x}_1, \ldots , \bo{x}_{r+s}) \in \mathbb{R}^{\sum n_i} : \prod_{i=1}^{r+s} f_i(\bo{x}_i)^{d_i}\leq T \right\rbrace. $$ Note that, since $f_i(\boldsymbol{x}_i)\geq 1$ for every $i$, $\widetilde{Z}(T)$ is bounded for every $T$.
\begin{lemma}\label{lemvol}
Let $q=r+s-1$. Under the hypotheses and the notation from above, for every $T \geq 1$, we have \begin{equation*}\label{eqvol} \textup{Vol} \left( \widetilde{Z}(T)\right)=\widetilde{p} \left( T^\frac{1}{2},\log T \right), \end{equation*} where $\widetilde{p}(X,Y)\in \mathbb{R}[X,Y]$, $\deg_X \widetilde{p} \leq 2 n$, $\deg_Y \widetilde{p} \leq q$. In the case $n_i= d_i n$ for every $i=1,\dots ,r+s$, the coefficient of $X^{2n} Y^q$ is $\frac{n^q }{q!} \prod_{i=1}^{q+1} C_i$. If $n_i < d_i n $ for some $i$ then the monomial $X^{2 n} Y^q$ does not appear in $\widetilde{p}$.
\end{lemma}
\begin{proof} We have $$ V(T):=\textup{Vol}\left(\widetilde{Z}(T)\right)= \int_{\widetilde{Z}(T)} \text{d} \boldsymbol{x}_1 \dots \text{d} \boldsymbol{x}_{q+1}. $$ We proceed by induction on $q$. If $q=0$ there is nothing to prove. Suppose $q>0$ and let $$ \widetilde{Z}^{(q)}(T)=\left\lbrace (\bo{x}_1, \ldots , \bo{x}_{q}) \in \mathbb{R}^{n_1+\dots +n_{q}} : \prod_{i=1}^{q} f_i(\bo{x}_i)^{d_i}\leq T \right\rbrace. $$ Then $$ V(T)= \int_{Z_{q+1}\left(T^{\frac{1}{d_{q+1}}} \right)} \left( \int_{\widetilde{Z}^{(q)}\left( T f_{q+1}(\boldsymbol{x}_{q+1})^{-d_{q+1}} \right)} \text{d} \boldsymbol{x}_1 \dots \text{d} \boldsymbol{x}_q \right) \text{d} \boldsymbol{x}_{q+1}. $$ By the inductive hypothesis there exists $\widetilde{p}_q(X,Y)\in \mathbb{R}[X,Y]$ such that \begin{gather*} V(T)= \int_{Z_{q+1}\left(T^{\frac{1}{d_{q+1}}} \right)} \widetilde{p}_q\left( \left( \frac{T}{f_{q+1}(\boldsymbol{x}_{q+1})^{d_{q+1}}} \right)^{\frac{1}{2}} , \log \left( \frac{T}{f_{q+1}(\boldsymbol{x}_{q+1})^{d_{q+1}}} \right) \right) \text{d} \boldsymbol{x}_{q+1}, \end{gather*} where $\widetilde{p}_q(X,Y)\in \mathbb{R}[X,Y]$, $\deg_X \widetilde{p}_q\leq 2 n$, $\deg_Y \widetilde{p}_q\leq q-1$ and, if $n_i = d_i n$ for every $i=1, \dots ,q$, the coefficient of $X^{2n} Y^{q-1}$ is $\frac{n^{q-1} }{(q-1)!} \prod_{i=1}^{q} C_i$. If not, that monomial does not appear.
By $\mc{L}^n$, we indicate the Lebesgue measure on $\mathbb{R}^n$. Since $f_{q+1}$ is a measurable function, we get $$ V(T)= \int_{\left[1,T^{\frac{1}{d_{q+1}}}\right] } \widetilde{p}_q\left( \left( \frac{T}{X^{d_{q+1}}} \right)^{\frac{1}{2}} , \log \left( \frac{T}{X^{d_{q+1}}} \right) \right) \text{d} \left( \mc{L}^{n_{q+1}}\circ f_{q+1}^{-1} \right)(X), $$ where we consider $\mc{L}^{n_{q+1}}\circ f_{q+1}^{-1}$ as a measure on $\left[1,T^{\frac{1}{d_{q+1}}}\right]$. In particular for $(u,v] \subseteq \left[1,T^{\frac{1}{d_{q+1}}}\right]$, $$ \left(\mc{L}^{n_{q+1}}\circ f_{q+1}^{-1}\right) ((u,v])=p_{q+1}(v) - p_{q+1}(u), $$ and $\left(\mc{L}^{n_{q+1}}\circ f_{q+1}^{-1}\right) (\{1\})=p_{q+1}(1)$. Using 1.29 Theorem of \cite{Rudin}, we get \begin{gather*}\label{V(T)} V(T)= \int_{\left(1,T^{\frac{1}{d_{q+1}}}\right] } \widetilde{p}_q\left( \left( \frac{T}{X^{d_{q+1}}} \right)^{\frac{1}{2}} , \log \left( \frac{T}{X^{d_{q+1}}} \right) \right) p'_{q+1}(X) \text{d} \mc{L}^{1} (X)+ \\ +\widetilde{p}_q\left( T^{\frac{1}{2}} , \log T\right) p_{q+1}(1), \nonumber \end{gather*} where $p_{q+1}'$ is the derivative of $p_{q+1}$.
For some integer $c\geq 0$ we put $L(X,c)=X^c$ in case $c>0$ and $L(X,0)=1$. Because of the linearity of the integral we are reduced to calculate \begin{gather*} \mc{I}(a,b,c) =\int_{1}^{T^{\frac{1}{d_{q+1}}}} X^a \left( \frac{T}{X^{d_{q+1}}}\right)^{\frac{b}{2}} L\left( \log \frac{T}{X^{d_{q+1}}},c \right) \text{d} X= \\ =T^{\frac{b}{2}} \int_{1}^{T^{\frac{1}{d_{q+1}}}} X^{a-\frac{b}{2}d_{q+1}} L\left( \log T - \log \left( X^{d_{q+1}} \right) ,c\right) \text{d} X, \end{gather*} for some integers $a,b,c$, with $0\leq a \leq n_{q+1} -1$, $0 \leq b \leq 2 n$ and $0\leq c \leq q-1$. We have three possibilities. If $a-\frac{b}{2}d_{q+1}=-1$, then \begin{gather*} \mc{I}(a,b,c)=T^{\frac{b}{2}}\int_{1}^{T^{\frac{1}{d_{q+1}}}} X^{-1} L\left( \log T - \log \left( X^{d_{q+1}} \right) ,c\right) \text{d} X=\\ =\frac{1}{(c+1)d_{q+1}} T^{\frac{b}{2}} \left( \log T \right)^{c+1}. \end{gather*} If $a-\frac{b}{2}d_{q+1}\neq -1$ and $c=0$, $$ \mc{I}(a,b,0)= \frac{T^{\frac{b}{2}} }{a-\frac{b}{2}d_{q+1}+1}\left( T^{\frac{a-\frac{b}{2}d_{q+1}+1}{d_{q+1}}}-1 \right)= \frac{ T^{\frac{a+1}{d_{q+1}}}-T^{\frac{b}{2}}}{a-\frac{b}{2}d_{q+1}+1} $$ If $a-\frac{b}{2}d_{q+1}\neq -1$ and $c\neq 0 $, then $$ \mc{I}(a,b,c) =- \frac{T^{\frac{b}{2}} \left( \log T \right)^c }{a-\frac{b}{2}d_{q+1}+1}+ \frac{c d_{q+1}}{a-\frac{b}{2}d_{q+1}+1}\mc{I}(a,b,c-1). $$ Therefore, one can see that $\mc{I}(a,b,c)$ is a polynomial in $T^{\frac{1}{2}}$ and $\log T$. In particular $\mc{I}(a,b,c)=\widehat{p}(T^{\frac{1}{2}},\log T)$, where $\widehat{p}(X,Y) \in \mathbb{R}[X,Y]$, with $\deg_X\widehat{p}\leq 2n$ and $\deg_Y\widehat{p}\leq q$. Note that in the case $a= d_{q+1}n-1$, $b=2 n$ and $c=q-1$, the coefficient of $X^{2 n} Y^{q}$ is $\frac{1}{qd_{q+1}}$ and 0 for any other choice of $a,b$ and $c$. Therefore, the monomial $X^{2n} Y^q$ does not appear in $\widehat{p}$ if either $n_{q+1}<d_{q+1}n$ or $X^{ 2n} Y^{q-1}$ does not appear in $\widetilde{p}_q$, i.e., if $n_i < d_i n $ for some $i$. To conclude, recall that, in the case $n_i=d_i n$ for every $i=1, \dots ,r+s$, $p'_{q+1}$ has leading coefficient $n d_{q+1} C_{q+1}$ and the coefficient of $X^{ 2n} Y^{q-1}$ in $\widetilde{p}_q$ is $\frac{n^{q-1} }{(q-1)!} \prod_{i=1}^{q} C_i$, thus, the coefficient in front of $\mc{I}(d_{q+1}n -1,2 n,q-1)$ in $V(T)$ is $\frac{n^{q} d_{q+1} }{(q-1)!} \prod_{i=1}^{q+1} C_i$.
\end{proof}
The volumes of the sets \begin{equation}\label{ballr} \{(z_1,\ldots ,z_n ) \in \mathbb{R}^n :M(1,z_1, \ldots ,z_n )\leq T\} \end{equation} and \begin{equation}\label{cball} \{(z_1,\ldots ,z_n ) \in \mathbb{C}^n:M(1,z_1, \ldots ,z_n )^2\leq T\} \end{equation} were computed by Chern and Vaaler in \cite{Chern2001}. By (1.16) and (1.17) of \cite{Chern2001}, these volumes are, for every $T\geq 1$, polynomials $p_\mathbb{R}(T)$ and $p_\mathbb{C}(T)$ of degree $n$ and leading coefficients, respectively, $$ C_{\mathbb{R},n } = 2^{n -M} \left( \prod_{l=1}^{M}\left( \frac{2l}{2l+1} \right)^{n -2l} \right) \frac{n^M}{M!}, \footnote{There is a misprint in (1.16) of \cite{Chern2001}, $2^{-N}$ should read $2^{-M}$.} $$ with $M=\lfloor \frac{n -1}{2} \rfloor$, and $$ C_{\mathbb{C},n} = \pi^n \frac{n^n }{\left( n ! \right)^2}. $$
Suppose $q=0$ and recall Lemma \ref{lemdet}. In this case $Z(T)$ corresponds to (\ref{ballr}) if $m=1$ or to (\ref{cball}) if $m=2$. We have \begin{equation}\label{eqvolq0}
\frac{\textup{Vol}(Z(T))}{\det \Lambda}=\frac{ 2^{sn }}{\left( \sqrt{|\Delta_k|}\right)^{n}} C_{\mathbb{R},n}^r C_{\mathbb{C},n}^s T^n + \frac{P(T)}{\left( \sqrt{|\Delta_k|}\right)^{n}}, \end{equation} for every $T> 1$, where $P(X)\in \mathbb{R}[X]$ depends only on $n$, $r$ and $s$ and has degree at most $n-1$.
\begin{corollary}\label{corvol} Suppose $q>0$. We have, for $T> 1$, \begin{equation}\label{eqvolq1}
\frac{\textup{Vol}(Z(T))}{\det \Lambda}=\frac{n^q 2^{sn }}{q!\left( \sqrt{|\Delta_k|}\right)^{n}} C_{\mathbb{R},n}^r C_{\mathbb{C},n}^s T^n \left( \log T \right)^q + \frac{P\left(T^\frac{1}{2},\log T\right)}{\left( \sqrt{|\Delta_k|}\right)^{n}}, \end{equation} where $P(X,Y)\in \mathbb{R}[X,Y]$ depends on $n$, $r$ and $s$, $\deg_X P \leq 2 n$, $\deg_Y P \leq q$ and the coefficient of $X^{2n} Y^q$ is 0. \end{corollary}
\begin{proof} By Lemma \ref{lemvol} and the result of Chern and Vaaler about the volumes of the sets defined in (\ref{ballr}) and (\ref{cball}), the volume of $Z(T)$ is $p(T^\frac{1}{2},\log T)$ where $p(X,Y)\in \mathbb{R}[X,Y]$, $\deg_X p \leq 2n $, $\deg_Y p \leq q$ and the coefficient of $X^{2n} Y^q$ is $\frac{n^q }{q!} C_{\mathbb{R},n}^r C_{\mathbb{C},n}^s $. \end{proof}
Therefore, recalling $|\Delta_k|$ and $\lambda_1, \dots , \lambda_{mn}$ are greater than or equal to 1, by (\ref{eqvolq0}) and Corollary \ref{corvol}, (\ref{extim}) becomes \begin{equation}\label{extim2}
\left| |Z(T)\cap \Lambda|-\frac{n^q 2^{sn}}{q!\left( \sqrt{|\Delta_k |}\right)^{n}} C_{\mathbb{R},n}^r C_{\mathbb{C},n}^s T^n \left( \log T \right)^q \right|\leq \sum_{j=0}^{{mn}-1}C V_j(Z(T))+Q(T), \end{equation} for every $T> 1$, where $Q(T)$ is the function of $T$ obtained from the polynomial $P$ of (\ref{eqvolq0}) or (\ref{eqvolq1}) substituting the coefficients with their absolute values. Note that $Q$ depends only on $m$ and $n$.
Now we want to find a bound for $V_j(Z(T))$. Recall that in (\ref{ineqnorm}) and (\ref{ineqnorm2}) we have defined a function $N(\boldsymbol{x})= \gamma |(1,\boldsymbol{x})|_\infty$ such that $N(\boldsymbol{x})\leq M_1(\boldsymbol{x})$. Let $$ Z'(T)=\left\lbrace (\bo{x}_1, \ldots , \bo{x}_{r+s}) \in \mathbb{R}^{mn} : \prod_{i=1}^{r+s} N(\boldsymbol{x}_i)^{d_i}\leq T \right\rbrace . $$ Each $(\bo{x}_1, \ldots , \bo{x}_{r+s})$ with $\prod_{i=1}^{r+s} M_1(\bo{x}_i)^{d_i}\leq T $ satisfies $\prod_{i=1}^{r+s} N(\boldsymbol{x}_i)^{d_i}\leq T $. Therefore, we have $Z(T) \subseteq Z'(T)$ and $V_j(Z(T))\leq V_j(Z'(T))$.
Suppose $q=0$. This means that $k$ is either $\mathbb{Q}$ ($m=1$) or an imaginary quadratic field ($m=2$). In any case any projection of $Z'(T)$ to a $j$-dimensional coordinate subspace has volume $ \left( \frac{2}{\gamma} \right)^j T^{\frac{j}{m}} $ if $T\geq \gamma^m$, for every $j=1,\dots {mn}-1$. Therefore we obtain \begin{equation}\label{vol1} V_j(Z(T)) \leq V_j\left(Z'(T)\right)\leq E T^{ n -\frac{1}{m}} , \end{equation} for some real constant $E$ depending only on $n$ and $m$. This holds for every $T> 1$ since $\gamma \leq 1$.
Now suppose $q>0$.
\begin{lemma}\label{lemvolZ'}
For every $j=1, \dots , {mn}-1$, there exists $P_j(X,Y) \in \mathbb{R} [X,Y]$ whose coefficients depend only on $m$ and $n$, with $\deg_X P_j \leq 2 n$, $\deg_Y P_j \leq q$, and the coefficient of $X^{2n} Y^q$ is 0, such that, for every $T> 1$, we have \begin{equation*}\label{vol2} V_j(Z'(T))= P_j\left(T^{\frac{1}{2}}, \log T \right). \end{equation*}
\end{lemma}
\begin{proof} By definition, the projection of $Z'(T)$ on a $j$-dimensional coordinate subspace is just the intersection of $Z'(T)$ with such subspace. To each such subspace $\Sigma$ we can associate integers $n_1,\dots ,n_{r+s}$ with $0\leq n_i \leq d_in$ such that $\Sigma$ is defined by setting $d_i n-n_i$ coordinates of each $\boldsymbol{x}_i$ to 0. Therefore we are in the situation of Lemma \ref{lemvol} because, after dividing by $\gamma$, we have, for every $i$ such that $n_i>0$, a continuous function $f_i : \mathbb{R}^{n_i} \rightarrow [1, \infty)$, with $\sum n_i =j$. This gives rise to sets of the form (\ref{rball}), whose volumes are $2^{n_i}T^{n_i}$. Since $j<{mn}$, not all $n_i$ can be equal to $d_i n$. Therefore, by Lemma \ref{lemvol}, the volume of any such projection equals a polynomial with the desired property and we have the claim. \end{proof}
Recall the definition of $\mathcal{M}^k(e,\mathcal{H})$ that was given in Section \ref{sect2}. Clearly $\left| \mathcal{M}^k(e,\mathcal{H})\right|$ is the number of $\bo{a} \in \mathcal{O}_k^e$ with $\prod_{i=1}^{r+s}M_1\left(\sigma_i(\bo{a})\right)^{d_i}\leq \mathcal{H}^m $, i.e., $|Z(\mathcal{H}^m)\cap \mathcal{O}_k^e|$.
By (\ref{extim2}), (\ref{vol1}) and Lemma \ref{lemvolZ'} we have, for every $\mathcal{H}> 1$, \begin{equation*}
\left| \left| \mathcal{M}^k(e,\mathcal{H})\right|-\frac{e^q m^q 2^{se}}{q!\left( \sqrt{|\Delta_k |}\right)^{e}} C_{\mathbb{R},e}^r C_{\mathbb{C},e}^s \mathcal{H}^{m e} \left( \log \mathcal{H} \right)^q \right|\leq E(\mathcal{H}), \end{equation*} with \begin{equation*} E(\mathcal{H})=\left\lbrace \begin{array}{ll} \sum_{i=0}^{2 e }\sum_{j=0}^q E_{i,j} \mathcal{H}^{\frac{m i}{2}}(\log \mathcal{H} )^j, & \mbox{ if $q\geq 1$, }\\ \sum_{i=0}^{m e-1} E_i \mathcal{H}^i, & \mbox{ if $q=0$, } \end{array} \right. \end{equation*} where $E_{2 e ,q}=0$ and all the coefficients depend on $m$ and $e$.
Finally, it is clear that for every $\mathcal{H}_0>1$ one can find a $D_0$ such that, for every $\mathcal{H} \geq \mathcal{H}_0$, \begin{equation*} E(\mathcal{H})\leq \left\lbrace \begin{array}{ll} D_0 \mc{H}^{\me} \left( \log \mc{H} \right)^{q-1} , & \mbox{ if $q\geq 1$, }\\ D_0 \mc{H}^{\me-1} , & \mbox{ if $q=0$, } \end{array} \right. \end{equation*} and we derive the claim of Theorem \ref{mainthm}.
\section*{Acknowledgments}
The author would like to thank Martin Widmer for sharing his ideas, for his constant encouragement and advice, Giulio Peruginelli, Robert Tichy and Jeffrey Vaaler for many useful discussions and the anonymous referee for providing valuable suggestions.
\end{document} | arXiv |
\begin{document}
\title{\textbf{Robust analyses for longitudinal clinical trials with missing and non-normal continuous outcomes}}
\author{Siyi Liu$^1$, Yilong Zhang$^2$, Gregory T Golm$^{2}$, Guanghan (Frank) Liu$^{2,3}$, Shu Yang$^1$}
\date{
}
\maketitle \begin{center} $^{1}$Department of Statistics, North Carolina State University, Raleigh, NC, USA \\ $^{2}$Merck \& Co., Inc., Kenilworth, NJ, USA \\ $^{3}$Posthumous \end{center}
\spacingset{1.5} \begin{abstract} Missing data is unavoidable in longitudinal clinical trials, and outcomes are not always normally distributed. In the presence of outliers or heavy-tailed distributions, the conventional multiple imputation with the mixed model with repeated measures analysis of the average treatment effect (ATE) based on the multivariate normal assumption may produce bias and power loss. Control-based imputation (CBI) is an approach for evaluating the treatment effect under the assumption that participants in both the test and control groups with missing outcome data have a similar outcome profile as those with an identical history in the control group. We develop a general robust framework to handle non-normal outcomes under CBI without imposing any parametric modeling assumptions. Under the proposed framework, sequential weighted robust regressions are applied to protect the constructed imputation model against non-normality in both the covariates and the response variables. Accompanied by the subsequent mean imputation and robust model analysis, the resulting ATE estimator has good theoretical properties in terms of consistency and asymptotic normality. Moreover, our proposed method guarantees the analysis model robustness of the ATE estimation, in the sense that its asymptotic results remain intact even when the analysis model is misspecified. The superiority of the proposed robust method is demonstrated by comprehensive simulation studies and an AIDS clinical trial data application.
\noindent \textbf{keywords:} Longitudinal clinical trial; missing data; multiple imputation; robust regression; sensitivity analysis. \end{abstract}
{}
\section{Introduction}
\subsection{Missing data in clinical trials}
Analysis of longitudinal clinical trials often presents difficulties as inevitably some participants do not complete the study, thereby creating missing outcome data. Additionally, some outcome data among participants who complete the study may not be of interest on account of intercurrent events such as initiation of rescue therapy prior to the analysis time point. With the primary interest focusing on evaluating the treatment effect in longitudinal clinical trials, the approach to handling missingness plays an essential role and has gained substantial attention from the US Food and Drug Administration (FDA) and National Research Council \citep{little2012prevention}. The ICH E9(R1) addendum provides a detailed framework of defining estimands to target the major clinical question in a population-level summary with the consideration of intercurrent events that may cause additional missingness \citep{international2019addendum}.
The missing at random (MAR; \citealp{rubin1976inference}) mechanism is often invoked in analyses that seek to evaluate the treatment efficacy. However, MAR is unverifiable and may not be practical in some clinical trials. Further, if the response at the primary time point is of interest regardless of whether participants have complied with the test or comparator treatments through the primary time point (corresponding to a \textquoteleft treatment policy\textquoteright{} intercurrent event strategy), an analysis based on the MAR assumption would not be appropriate, because such an analysis would assume that responses in those who drop out would follow the same trajectory as responses in those who remain in treatment. A more plausible assumption would be that the treatment effect may quickly fade away, leading to a missing not at random (MNAR) assumption that responses among those who fail to complete treatments in both treatment groups behave similarly to the responses among those in the control group with identical historical covariates. Drawn on the idea of the zero-dose model in \citet{little1996intent}, \citet{carpenter2013analysis} refer to this scenario as the control-based imputation (CBI). Since the CBI represents a deviation from MAR, it is widely used in sensitivity analyses to explore the robustness of the study results against the untestable MAR assumption (e.g., \citealp{carpenter2013analysis,cro2016reference}). Furthermore, an increasing number of clinical studies have applied this approach to primary analyses \citep{tan2021review}. Throughout the paper, we focus on jump-to-reference (J2R) as one favorable scenario of the CBI used in the FDA statistical review and evaluation reports (e.g., \citealp{cr2016tresiba}), which assumes that the missing outcomes in the treatment group will have the same outcome mean profile as those with identical historical information in the control group. Our goal is to assess the average treatment effect (ATE) under J2R.
\subsection{Multiple imputation}
Multiple imputation (MI; \citealp{rubin2004multiple}) followed by a mixed-model with repeated measures (MMRM) analysis acts as a standard approach to analyze longitudinal clinical trial data under J2R. The main idea of MI applied in longitudinal trials is to use MMRM to impute the missing components and then conduct full-data analysis on each imputed dataset. The simple implementation and high flexibility of MI underlie the recommendation of this approach by the FDA and National Research Council \citep{little2012prevention}.
However, this approach relies heavily on the parametric modeling assumptions in the construction of both the imputation and the analysis model, where a normal distribution is typically assumed. In reality, the distribution of the outcomes may suffer from extreme outliers or a heavy tail, which contradicts the normality assumption. A motivating CD4 count dataset in Section \ref{sec:moti_exmp} further addresses that a simple transformation such as the log transformation sometimes cannot fix the non-normality issue \citep{mehrotra2012analysis}. In the presence of outliers or heavy tails, applying the methods that rely on the normal distribution may produce bias and power loss. To tackle the issue in longitudinal clinical trials under MAR, \citet{mogg2007analysis} and \citet{mehrotra2012analysis} suggest substituting the conventional analysis of covariance model in the full-data analysis step of MI with th\textcolor{black}{e rank-based reg}ression \citep{jaeckel1972estimating} or Huber robust regression \citep{huber1973robust} to down-weight the impact of non-normal response values. When the missingness mechanism is MNAR, a gap exists in the extension of the robust method to handle the MNAR-related scenarios.
\subsection{Our contribution: a robust framework}
We develop a general robust framework to evaluate the ATE for non-normal longitudinal outcomes with missingness under the scenario where missing response data in both the test and reference groups are assumed to follow the same trajectory as the complete data in the reference group. We propose applying robust regression in conjunction with mean imputation to relax the parametric modeling assumption required by MI in both the imputation and analysis stages. Inspired by the sequential linear regression model involved in many longitudinal studies, where the current outcomes are regressed recursively on the historical information \citep{tang2017efficient}, we replace the least squares (LS) estimator with the estimator obtained by minimizing the robust loss function such as the Huber loss, the absolute loss \citep{huber2004robust}, and the $\varepsilon$-insensitive loss \citep{smola2004tutorial}, to mitigate the impact of non-normality in the response variable. While the robust regression lacks the protection against outliers in the covariates \citep{chang2018robust}, a weighted sequential robust regression model is put forward using the idea in \citet{carroll1993robustness} to down-weight the influential covariates by a robust Mahalanobis distance. Followed by mean imputation and a robust analysis step, the estimator from our proposed method has solid theoretical guarantees in terms of consistency and asymptotic normality.
\citet{rosenblum2009using} establish a test robustness result for randomized clinical trials with complete data; i.e., for a wide range of analysis models, testing the existence of the non-zero ATE has an asymptotically correct type-1 error even under model misspecification. However, they focus only on the LS model estimators when no ATE exists; and the property remains unclear when the model is estimated via the robust loss function under any arbitrary ATE value. To uncover the ambiguity, we extend the test robustness property to our proposed method for ATE estimation in the context of missing data. We formally show that the ATE estimator obtained from the various non-LS loss functions, including the Huber loss, the absolute loss, and the $\epsilon$-insensitive loss is analysis model-robust, in the sense that its asymptotic properties remain the same even when the analysis model is incorrectly specified. Although the paper mainly focuses on the J2R scenario, the established method and the desired theoretical properties are extendable to robust estimators under other MNAR-related conditions.
The rest of the paper is organized as follows. Section \ref{sec:moti_exmp} addresses a real-data example to motivate the demand for the robust method. Section \ref{sec:setup} introduces notations, assumptions under J2R, and an overview of the existing methods to handle missingness along with their drawbacks in the presence of non-normal data. Section \ref{sec:robust} presents our proposed robust method and its detailed implementation steps. Section \ref{sec:theo} provides the asymptotic results of the ATE estimator and discusses the analysis model robustness property. Section \ref{sec:simu} conducts comprehensive simulation studies to validate the proposed method. Section \ref{sec:app} returns to the motivating example to illustrate the performance of the robust method in practice. Section \ref{sec:conclusion} draws the conclusion.
\section{A motivating application \label{sec:moti_exmp}}
Study 193A conducted by the AIDS Clinical Trial Group compares the effects of dual or triple combinations of the HIV-1 reverse transcriptase inhibitors \citep{henry1998randomized}. The data consists of the longitudinal outcomes of the CD4 count data at baseline and during the first 40 weeks of follow-up, with the fully-observed baseline covariates as age and gender. In the trial, the participants are randomly assigned among the four treatments regarding dual or triple therapies. We focus on the treatment comparison between arm 1 (zidovudine alternating monthly with 400 mg didanosine) and arm 2 (zidovudine plus 400mg of didanosine plus 400mg of nevirapine). As arm 1 involves fewer combinations of inhibitors than arm 2, we view it as the reference group. Among individuals in these two arms, we delete the ones with missing baseline CD4 counts, partition the time into discrete intervals $(0,12]$, $(12,20]$, $(20,28]$, $(28,36]$ and $(36,40]$, and create a dataset with a monotone missingness pattern.
Since the original CD4 counts are highly skewed, we conduct a log transformation to get the transformed CD4 counts as $\log(\text{CD4}+1)$ and use them as the outcomes of interest. Figure \ref{fig:sp_real} presents the spaghetti plots of the transformed CD4 counts. Although there are no outstanding outliers, severe missingness is evident in the data, with only 34 of 320 participants in arm 1 and 46 of 330 participants in arm 2 completing the trial. The high dropout rates in this data reflect a typical missing data issue in longitudinal clinical trials, leading to the demand of conducting imputation for the missing components to prevent the substantial information loss if we focus on only the complete data.
\begin{figure}
\caption{Spaghetti plots of the log-transformed CD4 count data separated by the two treatments. }
\label{fig:sp_real}
\end{figure}
We check the normality of the data by fitting sequential linear regressions on the current outcomes against all historical information and examining the conditional residuals at each visit point for model diagnosis. An assessment of the normality of the responses via the Shapiro-Wilk test and the normal QQ plots are presented in Figure \ref{fig:diagnosis}. Each normality test indicates a violation of the normal assumption, and the normal QQ plots reveal that the CD4 counts remain heavy-tailed even after the log transformation. Under this circumstance, potentially biased and inefficient treatment effect estimates may occur when applying the conventional MI along with the MMRM analysis. It motivates the development of a robust method to assess the treatment effect precisely under non-normality.
\begin{figure}
\caption{Diagnosis of the conditional residuals at each visit.}
\label{fig:diagnosis}
\end{figure}
\section{Basic setup \label{sec:setup}}
Consider a longitudinal clinical trial with $n$ participants and $t$ follow-up visits. Let $A_{i}$ be the binary treatment without loss of generality, $X_{i}$ be the $p$-dimensional fully-observed baseline covariates including the intercept term with a full-column rank, $Y_{is}$ be the continuous outcome of interest at visit $s$, where $i=1,\cdots,n$, and $s=1,\cdots,t$. In longitudinal clinical trials, participants are randomly assigned to different treatment groups with non-zero probabilities. When missingness is involved, denote the observed indicator at visit $s$ as $R_{is}$, where $R_{is}=1$ if $Y_{is}$ is observed and $R_{is}=0$ otherwise. We assume a monotone missingness pattern throughout the paper, i.e., if the missingness begins at visit $s$, we have $R_{is'}=1$ for $s'<s$ and $R_{is'}=0$ for $s'\geq s$. Denote $H_{is}=(X_{i}^{\text{T}},Y_{i1},\cdots,Y_{is})^{\text{T}}$ as the history up to visit $s$, with $H_{i0}=X_{i}$. Since the outliers in the baseline covariates can be identified and removed by data inspection before further analysis, throughout we assume that no outliers exist in the baseline covariates. However, outliers may exist in the longitudinal outcomes due to data-collection error in the long period of study.
In most longitudinal clinical trials with continuous outcomes, the endpoint of interest is the mean difference of the outcomes at the last visit point between the two treatments. We utilize the pattern-mixture model (PMM; \citealp{little1993pattern}) framework to express the ATE as a weighted average over the missing patterns, i.e., $\tau=\mathbb{E}(Y_{it}\mid A_{i}=1)-\mathbb{E}(Y_{it}\mid A_{i}=0)$ where $\mathbb{E}(Y_{it}\mid A=a)=\sum_{s=1}^{t+1}\mathbb{E}(Y_{it}\mid R_{is-1}=1,R_{is}=0,A_{i}=a)\mathbb{P}(R_{is-1}=1,R_{is}=0\mid A_{i}=a)$ if we let $R_{i0}=1$ and $R_{it+1}=0$ for each individual. The assumed condition regarding the missing components is formed for the identification of the pattern-specific expectation $\mathbb{E}(Y_{it}\mid R_{is-1}=1,R_{is}=0,A_{i}=a)$. We describe one scenario based on the CBI model proposed by \citet{carpenter2013analysis} for illustration.
\subsection{Jump-to-reference imputation model}
The CBI model \citep{carpenter2013analysis} provides a scenario to model missingness in longitudinal clinical trials. We focus on one specific CBI model as J2R, whose plausibility reveals if the investigators believe that participants who discontinue the treatment have the same outcome mean performance as the ones in the control group with the same covariates. The following assumptions illustrate the J2R imputation model for the ATE identification.
\begin{assumption}[Partial ignorability of missingness]\label{assump:miss} $R_{is}\indep Y_{is'}\mid(H_{is-1},A_{i}=0)$ for $s'\geq s$.
\end{assumption}
Assumption \ref{assump:miss} characterizes the MNAR missing mechanism under J2R. The conventional MAR assumption is only required for the missing data in the control group. We do not impose any missing assumptions in the treatment group.
\begin{assumption}[J2R outcome mean model]\label{assump:j2r-mean} For individuals who receive treatment $a$ with historical information $H_{is-1}$ and drop out at visit $s$, $\mathbb{E}(Y_{it}\mid H_{is-1},R_{is-1}=1,R_{is}=0,A_{i}=a)=\mathbb{E}(Y_{it}\mid H_{is-1},A_{i}=0).$
\end{assumption}
Assumption \ref{assump:j2r-mean} offers a strategy to model the conditional mean of the missing component under J2R. Given the same historical information, the outcome mean will ``jump'' to the same conditional mean in the control group no matter the prior treatment. Combining with Assumption \ref{assump:miss}, the conditional expectation $\mathbb{E}(Y_{it}\mid H_{is-1},A_{i}=0)=\mathbb{E}\big\{\cdots\mathbb{E}(Y_{it}\mid H_{it-1},R_{it}=1,A_{i}=0)\cdots\mid H_{is-1},R_{is}=1,A_{i}=0\big\}$ is identified through a series of sequential regressions on the current outcome against the available historical information.
Throughout the paper, we assume a linear relationship between the outcomes and the historical covariates in the J2R imputation model for simplicity.\textcolor{black}{{} Extensions to nonlinear relationships are manageable, if the sequential regressions of the observed data are fitted in backward order, i.e., we start from the available data at the last visit point and use the predicted value as the outcome to regress on the previous history recursively to construct the imputation model. The elaboration of the sequential fitting procedure is provided in Section \ref{sec:supp_seqreg} in the supplementary material. }
\subsection{Overview of the existing methods and the drawbacks \label{subsec:overview}}
MI proposed by \citet{rubin2004multiple} provides a fully parametric approach to handle missingness under MNAR. Normality is often assumed for its simplicity and robustness against moderate model misspecification in the implementation of MI \citep{mehrotra2012analysis}. One common MI procedure in longitudinal clinical trials under J2R is summarized in the following steps:
\begin{enumerate} \setlength{\itemindent}{1.5em}
\item[\textbf{Step 1}.] For the control group, fit the sequential regression for the observed data at each visit point against the available history. Denote the estimated model parameter as $\hat{\theta}_{s-1}$ for $s=1,\cdots,t$.
\item[\textbf{Step 2}.] Impute missing data sequentially to form $M$ imputed datasets: For individuals who have missing values at visit $s$, impute $Y_{is}^{(m)}$ from the conditional distribution $f(Y_{is}\mid H_{is-1}^{*(m)},A_{i}=0;\hat{\theta}_{s-1})$ estimated in Step 1, where $H_{is-1}^{*(m)}=(X_{i}^{\text{T}},Y_{i1}^{*(m)},\cdots,Y_{is-1}^{*(m)})^{\text{T}}$ and $Y_{is}^{*(m)}=R_{is}Y_{is}+(1-R_{is})Y_{is}^{(m)}$ for $m=1,\cdots,M$.
\item[\textbf{Step 3}.] For each imputed dataset, perform the complete data analysis by fitting the imputed outcomes at the last visit point on a working analysis model. Denote $\hat{\tau}^{(m)}$ as the ATE estimator of the $m$th imputed dataset.
\item[\textbf{Step 4}.] Combine the estimation results from $M$ imputed datasets and obtain the MI estimator as $\hat{\tau}_{\text{MI}}=M^{-1}\sum_{m=1}^{M}\hat{\tau}^{(m)}$, with the variance estimator by Rubin's rule as \[ \mathbb{\hat{V}}(\hat{\tau}_{\text{MI}})=\frac{1}{M}\sum_{m=1}^{M}\mathbb{\hat{V}}(\hat{\tau}^{(m)})+(1+\frac{1}{M})B_{\text{M}}, \] where $B_{\text{M}}=(M-1)^{-1}\sum_{m=1}^{M}(\hat{\tau}^{(m)}-\hat{\tau}_{\text{MI}})^{2}$ is the between-imputation variance.
\end{enumerate}
Traditionally, the imputation model in Step 1 and the analysis model in Step 3 are obtained from the MMRM analysis, where we assume an underlying normal distribution for both the observed and the imputed data. However, as illustrated in the motivating CD4 count dataset in Section \ref{sec:moti_exmp}, normality may be violated, leading to a biased estimate of the target ATE parameter. With the consideration of non-normality, \citet{mogg2007analysis} and \citet{mehrotra2012analysis} modify the analysis model in Step 3 by replacing the LS estimator with the estimator obtained from the robust loss function.
One drawback of MI is that it is fully parametric. The consistency of the MI estimator relies heavily on the correct specification of the imputation distribution, i.e., the conditional distribution given the observed data, which is often assumed to be normal. When a severe deviation from the assumed imputation distribution is detected in the data, the estimation may not be reliable. The possible misspecification of the imputation distribution also exists in the ``robust'' approaches proposed by \citet{mogg2007analysis} and \citet{mehrotra2012analysis}, where the imputation model still depends on the normality assumption as required by MI. Moreover, the MI estimator is not efficient in general. The inefficiency becomes more serious when it comes to interval estimation. The variance estimation using Rubin's combining rule may produce an inconsistent variance estimate even when the imputation and analysis models are the same correctly specified models \citep{wang1998large,robins2000inference}. Under the MNAR assumption, the overestimation issue raised from Rubin's variance estimator is more pronounced (e.g., \citealp{lu2014analytic,liu2016analysis,yang2016note,guan2019unified,yang2020smim,di2022}). One can resort to the bootstrap variance estimation to obtain a consistent variance estimator, which however exaggerates the computational cost.
The unsatisfying performance of MI under non-normality motivates us to develop a robust approach to accommodate the possible model misspecification resulting from outliers or heavy-tailed errors without the reliance on parametric models. In the following sections, a weighted robust regression model in conjunction with mean imputation is proposed to overcome the issues in MI.
\section{Proposed robust method \label{sec:robust}}
We propose a mean imputation procedure based on robust regression in both the imputation and the analysis models to obtain valid inferences under J2R when the data suffers from a heavy tail or extreme outliers. To relax the strong parametric modeling assumption required by MI, mean imputation is preferred. \citet{mehrotra2012analysis} shed light on the possibility of incorporating the robust regression in the analysis step of MI to handle non-normality. Based on this idea, we further suggest using the sequential robust regression model in the imputation step to protect against deviations from normality for the observed data.
Throughout this section, we focus on the robust estimators obtained from minimizing the robust loss functions such as the Huber loss, the absolute loss \citep{huber2004robust}, and the $\varepsilon$-insensitive loss \citep{smola2004tutorial} to account for the impact of outliers or heavy-tailed data. To obtain a valid mean-type estimator, a symmetric error distribution assumption is imposed whenever a robust regression is applied.
Motivated by the sequential linear regression model under normality, where we regress the current outcomes on the historical information at each visit point to produce the sequential inferences, we develop a sequential robust regression procedure for the observed data to obtain valid inferences that are less likely to be influenced by non-normality. For the longitudinal data with a monotone missingness pattern, a robust regression is fitted on the observed data at each visit point, incorporating the observed historical information. Specifically, for the available data at visit $s$ for $s=1,\cdots,t$, the imputation model parameter estimate $\hat{\alpha}_{s-1}$ minimizes the loss function \[ \sum_{i=1}^{n}(1-A_{i})R_{is}\rho(Y_{is}-H_{is-1}^{\text{T}}\alpha_{s-1}). \]
Here, $\rho(x)$ is the robust loss function. For example, the Huber loss function is defined as $\rho(x)=0.5x^{2}\mathbb{I}(|x|\ensuremath{<l})+\left\{ l|x|-0.5l^{2}\right\} \mathbb{I}(|x|\ensuremath{\geq l})$, where the constant $l>0$ controls the influence of the non-normal data points and $\mathbf{\mathbb{I}}(\cdot)$ is an indicator function \citep{huber1973robust}. When $l\rightarrow\infty$, the Huber-type robust estimator is equivalent to the conventional LS estimator. We also provide the definitions of the absolute loss and the $\varepsilon$-insensitive loss in Section \ref{subsec:supp_cons} in the supplementary material.
\begin{remark}[Tuning constant $l$ in the Huber loss function]
The tuning constant $l$ in the Huber loss function mitigates the impact of extreme values and heavy-tailed errors in the data. \citet{kelly1992robust} argues the existence of trade-offs between the bias and variance in the selection of the tuning constant. A small value of $l$ provides more protections against non-normal values, yet suffers from the loss of efficiency if the data is indeed normal. In practice, a common recommendation of the tuning constant is $l=1.345\sigma$, where $\sigma$ is the standard deviation of the errors \citep{fox2002robust}. We use the Huber loss function with this tuning parameter to get the robust estimators throughout the simulation studies and real data application.
\end{remark}
While the estimator from the robust loss function provides protection against extreme outliers in the response variables, it is not robust against outliers in the covariates \citep{chang2018robust}, leading to an imprecise estimation and a loss of efficiency. In longitudinal data with the use of sequential regressions, the issue becomes more profound, where the outcome is treated both as the response variable in the current regression and as the covariate in the subsequent regression. To deal with the outliers in the covariates, we utilize the idea in
\citet{carroll1993robustness} to down-weight the high leverage point in the covariates via a robust Mahalanobis distance. Specifically, for a $p$-dimensional covariate $X$, we calculate the robust Mahalanobis distance as $d=(X-\mu)^{\text{T}}V^{-1}(X-\mu)$, where $\mu$ is a robust estimate of the center and $V$ is a robust estimate of the covariance matrix. The trisquared redescending function is applied to form the assigned weights as $w(u;\nu)=u\left\{ 1-(u/\nu)^{2}\right\} ^{3}\mathbb{I}(|u|\le\nu)$, where $u=(d/\nu)^{1/2}$ and $\nu$ is a tuning parameter to control the down-weight level. Therefore, in the sequential weighted robust regression model, the robust estimate $\hat{\alpha}_{s-1}^{w}$ minimizes the weighted loss \begin{align} \sum_{i=1}^{n}(1-A_{i})R_{is}w(H_{is-1};\nu_{s-1})\rho(Y_{is}-H_{is-1}^{\text{T}}\alpha_{s-1}).\label{eq:seqrrw} \end{align}
\begin{remark}[Tuning constants in the trisquared redescending function]
When selecting the tuning parameter, \citet{carroll1993robustness} used a fixed constant $\nu=8$ to illustrate a specific down-weight behavior for cross-sectional studies. In longitudinal clinical trials involving multiple weighted sequential regressions, the tuning parameter $\nu_{s-1}$ can be selected via cross-validation at each visit point, for $s=1,\cdots,t$. The main idea is to conduct a $K$-fold cross-validation for the observed data at each visit point and determine the optimal tuning parameter $\nu_{s-1}$ which minimizes the squared errors. Specifically, we first partition the observed data at visit $s$ into $K$ parts denoted as $P_{1},\cdots,P_{K}$. The part $P_{j}$ is then left for the test, and the remaining $(K-1)$ folds are utilized to learn the robust estimator $\hat{\alpha}_{s-1,-j}^{w}$, for $j=1,\cdots,K$. The optimal $\nu_{s-1}$ minimizes the cross-validation sum of the squared errors $\sum_{j=1}^{K}\sum_{i\in P_{j}}(Y_{is}-H_{is-1}^{\text{T}}\hat{\alpha}_{s-1,-j}^{w})^{2}$.
\end{remark}
After obtaining the robust estimates of the imputation model parameters, we impute the missing components by their conditional outcome means sequentially based on Assumptions \ref{assump:miss} and \ref{assump:j2r-mean} and construct the imputed data $Y_{is}^{*}=R_{is}Y_{is}+(1-R_{is})H_{is-1}^{*\text{T}}\hat{\alpha}_{s-1}^{w}$, where $H_{is-1}^{*}=(X_{i}^{\text{T}},Y_{i1}^{*},\cdots,Y_{is-1}^{*})^{\text{T}}$. Complete data analysis is then conducted on the imputed data, where we again minimize the robust loss function to mitigate the impact of outliers in the response variable. Note that since we assume that there are no outliers in the baseline covariates, assigning the weights to the loss function becomes unnecessary. Consider a general form of the working model in the analysis step as \begin{equation} \mu(A,X\mid\gamma)=Ag(X;\gamma^{(0)})+h(X;\gamma^{(1)}),\label{eq:model_form} \end{equation} where $g(X;\gamma^{(0)})$ and $h(X;\gamma^{(1)})$ are integrable functions bounded on compact sets, and $\gamma=(\gamma^{(0)\text{T}},\gamma^{(1)\text{T}})^{\text{T}}$. The robust estimator $\hat{\gamma}=(\hat{\gamma}^{(0)\text{T}},\hat{\gamma}^{(1)\text{T}})^{\text{T}}$ can be found by minimizing the loss \begin{equation} \sum_{i=1}^{n}\rho\left\{ Y_{it}^{*}-\mu(A_{i},X_{i}\mid\gamma)\right\} ,\label{eq:analysis-1} \end{equation} and the resulting ATE estimator $\hat{\tau}$ is estimated by the mean differences between the two groups as $\hat{\tau}=n^{-1}\sum_{i=1}^{n}{\color{black}g(X_{i};\hat{\gamma}^{(0)})}$.
The modeling form \eqref{eq:model_form} is commonly satisfied in randomized trials when constructing the working model for analysis. For example, the standard analysis model without the interaction term between the treatment and the baseline covariates as $\mu(A,X\mid\gamma)=\gamma^{(0)}A+\gamma^{(1)\text{T}}X$ gratifies this form when $g(X;\gamma^{(0)})=\gamma^{(0)}$ and $h(X;\gamma^{(1)})=\gamma^{(1)\text{T}}X$; a similar logic applies to the interaction model $\mu(A,X\mid\gamma)=\gamma^{(0)\text{T}}AX+\gamma^{(1)\text{T}}X$. As we will elaborate in the next section, the ATE estimator $\hat{\tau}$ is analysis model-robust, in the sense that its asymptotic results stay intact regardless of the specification of the analysis model. The implementation of the proposed mean imputation-based robust method is as follows.
\begin{enumerate} \setlength{\itemindent}{1.5em} \item[\textbf{Step 1}.] For the observed data in the control group, fit the sequential weighted robust regression at each visit point and get the sequential model parameter estimates $\hat{\alpha}_{s-1}^{w}$ by minimizing the weighted loss \eqref{eq:seqrrw} for $s=1,\cdots,t$.
\item[\textbf{Step 2}.] Impute missing data sequentially by the conditional outcome mean according to Assumptions \ref{assump:miss} and \ref{assump:j2r-mean} and obtain the imputed data $Y_{is}^{*}=R_{is}Y_{is}+(1-R_{is})H_{is-1}^{*\text{T}}\hat{\alpha}_{s-1}^{w}$, where $H_{is-1}^{*}=(X_{i}^{\text{T}},Y_{i1}^{*},\cdots,Y_{is-1}^{*})^{\text{T}}$ for $s=1,\cdots,t$.
\item[\textbf{Step 3}.] Set up an appropriate working model $\mu(A,X\mid\gamma)$ in the form \eqref{eq:model_form}, perform the complete data analysis and get the ATE estimator $\hat{\tau}$ by minimizing the loss function \eqref{eq:analysis-1}.
\end{enumerate}
The good theoretical properties of the ATE estimator along with a linearization-based variance estimator are provided in the next section.
\section{Theoretical properties and analysis model robustness \label{sec:theo}}
We present the asymptotic theory of the ATE estimator in terms of consistency and asymptotic normality along with a variance estimator based on three robust loss functions as the Huber loss, the absolute loss, and the $\varepsilon$-insensitive loss. To illustrate the theorems in a straightforward way, we introduce additional notations. Denote $\varphi(H_{is},\alpha_{s-1})=(1-A_{i})R_{is}w(H_{is-1};\nu_{s-1})\psi(Y_{is}-H_{is-1}^{\text{T}}\alpha_{s-1})H_{is-1}$ as the function derived from minimizing the weighted loss function \eqref{eq:seqrrw} in the imputation model, where $\psi(x)=\partial\rho(x)/\partial x$ is the derivative of the robust loss function, and $\alpha_{s-1,0}$ as the true parameter such that $\mathbb{E}\left\{ \varphi(H_{is},\alpha_{s-1})\mid H_{is-1}\right\} =0$. Let $\hat{\alpha}^{w}=(\hat{\alpha}_{0}^{w\text{T}},\cdots,\hat{\alpha}_{t-1}^{w\text{T}})^{\text{T}}$ be the combination of the model estimators from $t$ sequential regression models in the imputation, and $\mathbb{\alpha}_{0}=(\alpha_{0,0}^{\text{T}},\cdots,\alpha_{t-1,0}^{\text{T}})^{\text{T}}$ be the corresponding true model parameters. In terms of the components in the analysis model, denote $\varphi_{a}(Z_{i},\gamma)=\psi\left\{ Y_{it}^{*}-\mu(A_{i},X_{i}\mid\gamma)\right\} \partial\mu(A_{i},X_{i}\mid\gamma)/\partial\gamma^{\text{T}}$, where $Z_{i}^{*}=(A_{i},X_{i}^{\text{T}},Y_{it}^{*})^{\text{T}}$ represents the imputed data in the model, $\gamma_{0}$ is the true parameter such that $\mathbb{E}\left\{ \varphi_{a}(Z_{i},\gamma)\right\} =0$, and $\tau_{0}$ is the true ATE such that $\tau_{0}=\mathbb{E}(Y_{it}\mid A=1)-\mathbb{E}(Y_{it}\mid A=0)$. Suppose $\gamma^{(0)}$ is a $d_{0}$-dimensional vector, and $\gamma^{(1)}$ is a $d_{1}$-dimensional vector.
\begin{theorem}\label{thm:cons}
Under the regularity conditions listed in Section \ref{subsec:supp_cons} in the supplementary material, the ATE estimator $\hat{\tau}\xrightarrow{\mathbb{P}}\tau_{0}$ as the sample size $n\rightarrow\infty$, for $s=1,\cdots,t$.
\end{theorem}
\begin{theorem}\label{thm:norm}
Under the regularity conditions listed in Section \ref{subsec:supp_norm} in the supplementary material, as the sample size $n\rightarrow\infty$, \[ \sqrt{n}(\hat{\tau}-\tau_{0})\xrightarrow{d}\mathcal{N}\Big(0,\mathbb{V}\left\{ V_{\tau,i}(\alpha_{0},\gamma_{0})\right\} \Big), \] where $V_{\tau,i}(\alpha_{0},\gamma_{0})=\left\{ \partial g(X_{i};\gamma_{0}^{(0)})/\partial\gamma^{\text{T}}\right\} c^{\text{T}}V_{\gamma,i}(\alpha_{0},\gamma_{0})$, \begin{align*} V_{\gamma,i}(\alpha_{0},\gamma_{0}) & =D_{\varphi}^{-1}\bigg[\varphi_{a}\left\{ Z_{i}^{*}(\beta_{t}),\gamma_{0}\right\} +\sum_{s=1}^{t}\mathbb{E}\left\{ R_{s-1}(1-R_{s})\frac{\partial\mu(A,X\mid\gamma_{0})}{\partial\gamma^{\text{T}}}\frac{\partial\psi(e)}{\partial e}H_{s-1}^{\text{T}}\right\} U_{t,s-1,i}(\alpha_{0})\bigg], \end{align*} $U_{t,s-1,i}(\alpha_{0})=\left(\mathbf{I}_{p+s-2},\alpha_{s-1,0}\right)U_{t,s,i}(\alpha_{0})+\left(\mathbf{0}_{p+s-2}^{\text{T}},1\right)\beta_{t,s}q(H_{is},\alpha_{s-1,0})$ for $s<t$, $U_{t,t-1,i}(\alpha_{0})=q(H_{it},\alpha_{t-1,0})$, and $q(H_{is},\alpha_{s-1,0})=\left[-\partial\mathbb{E}\left\{ \varphi(H_{is},\alpha_{s-1,0})H_{is-1}^{\text{T}}\mid H_{is-1}\right\} /\partial\alpha_{s-1}^{\text{T}}\right]^{-1}\varphi(H_{is},\alpha_{s-1,0}).$ Here, $c^{\text{T}}=(\mathbf{I}_{d_{0}},\mathbf{0}_{d_{0}\times d_{1}})$ is a matrix where $\mathbf{I}_{d_{0}}$ is a $(d_{0}\times d_{0})$-dimensional identity matrix and $\mathbf{0}_{d_{0}\times d_{1}}$ is a $(d_{0}\times d_{1})$-dimensional zero matrix, $\mathbf{0}_{p+s-2}$ is a $(p+s-2)$-dimensional zero vector, $D_{\varphi}=\partial\mathbb{E}\left[\varphi_{a}\left\{ Z_{i}^{*}(\beta_{t}),\gamma_{0}\right\} \right]/\partial\gamma^{\text{T}}$ where $Z_{i}^{*}(\beta_{t})=\left(A_{i},X_{i}^{\text{T}},Y_{it}^{*}(\beta_{t})\right){}^{\text{T}}$ and $Y_{it}^{*}(\beta_{t})$ refers to the imputed value $Y_{it}^{*}$ based on the true imputation parameters $\beta_{t}=(\beta_{t,0}^{\text{T}},\cdots\beta_{t,t-1}^{\text{T}})^{\text{T}}$ which satisfy \[\begin{cases} \beta_{t,t-1}=\alpha_{t-1,0} & \text{if \ensuremath{s=t}},\\ \beta_{t,s-1}=(\mathbf{I}_{p+s-2},\alpha_{s-1,0})(\mathbf{I}_{p+s-1},\alpha_{s,0})\cdots(\mathbf{I}_{p+t-3},\alpha_{t-2,0})\alpha_{t-1,0} & \text{if \ensuremath{s<t}}, \end{cases}\] and $e_{i}=Y_{it}^{*}(\beta_{t})-\mu(A_{i},X_{i}\mid\gamma_{0})$.
\end{theorem}
The asymptotic variance in Theorem \ref{thm:norm} motivates us to obtain a linearization-based variance estimator by plugging in the estimated values as \[ \hat{\mathbb{V}}(\hat{\tau})=\frac{1}{n^{2}}\sum_{i=1}^{n}\left\{ V_{\tau,i}(\hat{\alpha}^{w},\hat{\gamma})-\bar{V}_{\tau}(\hat{\alpha}^{w},\hat{\gamma})\right\} ^{2}, \] where $\bar{V}_{\tau}(\hat{\alpha}^{w},\hat{\gamma})=n^{-1}\sum_{i=1}^{n}V_{\tau,i}(\hat{\alpha}^{w},\hat{\gamma})$, $V_{\tau,i}(\hat{\alpha}^{w},\hat{\gamma})=\left\{ \partial g(X_{i};\hat{\gamma}^{(0)})/\partial\gamma^{\text{T}}\right\} c^{\text{T}}V_{\gamma,i}(\hat{\alpha}^{w},\hat{\gamma})$, \begin{align*} V_{\gamma,i}(\hat{\alpha}^{w},\hat{\gamma}) & =D_{\varphi}^{-1}\bigg[\varphi_{a}\left(Z_{i}^{*},\hat{\gamma}\right)+\sum_{s=1}^{t}\left\{ \frac{1}{n}\sum_{i=1}^{n}R_{is-1}(1-R_{is})\frac{\partial\mu(A_{i},X_{i}\mid\hat{\gamma})}{\partial\gamma^{\text{T}}}\frac{\partial\psi(\hat{e}_{i})}{\partial e_{i}}H_{is-1}^{\text{T}}\right\} U_{t,s-1,i}(\hat{\alpha}^{w})\bigg], \end{align*} $U_{t,s-1,i}(\hat{\alpha}^{w})=\left(\mathbf{I}_{p+s-2},\hat{\alpha}_{s-1}^{w}\right)U_{t,s,i}(\hat{\alpha}^{w})+\left(\mathbf{0}_{p+s-2}^{\text{T}},1\right)\hat{\beta}_{t,s}\hat{q}(H_{is},\hat{\alpha}_{s-1}^{w})$ for $s<t$, $U_{t,t-1,i}(\hat{\alpha}^{w})=\hat{q}(H_{it},\hat{\alpha}_{t-1}^{w})$, and $\hat{q}(H_{is},\hat{\alpha}_{s-1}^{w})=\left[-n^{-1}\sum_{i=1}^{n}\left\{ \partial\varphi(H_{is},\hat{\alpha}_{s-1}^{w})/\partial\alpha_{s-1}^{\text{T}}\right\} H_{is-1}^{\text{T}}\right]^{-1}\varphi(H_{is},\hat{\alpha}_{s-1}^{w})$. Also, $\hat{e}_{i}=Y_{it}^{*}-\mu(A_{i},X_{i}\mid\hat{\gamma})$, $\hat{D}_{\varphi}=n^{-1}\sum_{i=1}^{n}\partial\varphi_{a}\left(Z_{i}^{*},\hat{\gamma}\right)/\partial\gamma^{\text{T}}$, and \[ \begin{cases} \hat{\beta}_{t,t-1}=\hat{\alpha}_{t-1}^{w} & \text{if \ensuremath{s=t}},\\ \hat{\beta}_{t,s-1}=(\mathbf{I}_{p+s-2},\hat{\alpha}_{s-1}^{w})(\mathbf{I}_{p+s-1},\hat{\alpha}_{s}^{w})\cdots(\mathbf{I}_{p+t-3},\hat{\alpha}_{t-2}^{w})\hat{\alpha}_{t-1}^{w} & \text{if \ensuremath{s<t}}, \end{cases} \] for $s=1,\cdots,t$. Since the ATE estimator is asymptotically linear, we can also use bootstrap to obtain a replication-based variance estimator.
We consider a specific working model as the interaction model for analysis and present the asymptotic theories of the ATE estimator in Section \ref{subsec:supp_interaction} in the supplementary material. The interaction model is one of the most common models in the clinical trials suggested in \citet{international2019addendum}, which is also used in the simulation studies and real data application in the paper.
Theorems \ref{thm:cons} and \ref{thm:norm} extend the test robustness \citep{rosenblum2009using} to the analysis model robustness in two aspects. First, the robustness expands its plausibility from the hypothesis test to the ATE estimation. Second, the robust estimator obtained from minimizing the loss function further broadens the types of the model estimator used in the analysis model. The resulting ATE estimator via the robust loss remains consistent and has the identical asymptotic normality even when the analysis model is misspecified.
\section{Simulations \label{sec:simu}}
We conduct simulation studies to validate the finite-sample performance of the proposed robust method. Consider a longitudinal clinical trial with two treatment groups and five visits. Set the sample size for each group as 500 and generate the data separately for each treatment. The baseline covariates $X\in\mathbb{R}^{2}$ are a combination of a continuous variable generated from the standard normal distribution and a binary variable generated from a Bernoulli distribution with the success probability of $0.3$. The longitudinal outcomes are generated in a sequential manner, regressing on the historical information separately for each group based on some specific distributions. The group-specific data generating parameters are given in Section \ref{subsec:supp_simuset} in the supplementary material.
The missingness mechanism is set to be MAR with a monotone missingness pattern. For the visit point $s$, if $R_{is'-1}=0$, then $R_{is'}=0$ for $s'=s,\cdots,t$; otherwise, let $R_{is}\mid\left(H_{is-1},A_{i}=a\right)\sim\text{Bernoulli}\left\{ \pi_{s}(a,H_{is-1})\right\} $. We model the observed probability $\pi_{s}(a,H_{is-1})$ at visit $s>1$ as a function of the observed information as $\text{logit}\left\{ \pi_{s}(a,H_{is-1})\right\} =\phi_{1a}+\phi_{2a}Y_{is-1}$, where $\phi_{1a}$ and $\phi_{2a}$ are the tuning parameters for the observed probabilities. The parameters are tuned to achieve the observed probability around $0.8$ in each group.
We select the Huber loss function to obtain robust estimators for its prevalence. Table \ref{table:sim}(a) summarizes the three methods we aim to compare in the simulation studies. We apply distinct estimation approaches for each method in the imputation and analysis models, along with different imputation methods, where MI stands for the conventional method used in longitudinal clinical trials and Robust stands for our proposed method. LSE can be viewed as a transition from the conventional MI method to the proposed robust method. In terms of the variance estimation, Rubin's and bootstrap methods are compared for the MI estimator while the linearization-based and bootstrap variance estimates are compared for the mean imputation estimators.
The simulation results are based on 10,000 Monte Carlo (MC) simulations under $H_{0}:\tau=0$ and 1000 MC simulations under one specific alternative hypothesis $H_{1}:\tau=\tau_{0}$, with the number of bootstrap replicates $B=100$ and the imputation size $M=10$ for MI. The tuning parameter of Huber robust regression is $l=1.345\sigma$, and the tuning parameters of the sequential weighted robust models are $\nu_{s-1}=10$ for $s=1,\cdots,5$. The imputation size $M$ and the tuning parameters $\nu_{s-1}$ do not have a strong impact on the inferences (results are not shown). We assess the estimators using the point estimate (Point est), the MC variance (True var), the variance estimate (Var est), the relative bias of the variance estimate computed by $\Big[\mathbb{E}\big\{\mathbb{\hat{V}}(\hat{\tau})\big\}-\mathbb{V}(\hat{\tau})\Big]/\mathbb{V}(\hat{\tau})$, the coverage rate of $95\%$ confidence interval (CI), the type-1 error under $H_{0}$, the power under $H_{1}$, and the root mean squared error (RMSE). We choose the $95\%$ Wald-type CI estimated by $\big(\hat{\tau}-1.96\mathbb{\hat{V}}^{1/2}(\hat{\tau}),\hat{\tau}+1.96\mathbb{\hat{V}}^{1/2}(\hat{\tau})\big)$.
\subsection{Data with extreme outliers}
We first focus on the settings when the outcomes are generated sequentially from the normal distribution with or without severe outliers. To produce the outliers in the longitudinal outcomes, we randomly select 10 individuals from the top 30 completers with the highest outcomes at the last visit point per group and multiply the original values by three for all post-baseline outcomes. We also consider adding extreme values only to one specific group and present the results in Section \ref{subsec:supp_simtab} in the supplementary material.
Table \ref{table:sim}(b) and the first two rows of Figure \ref{fig:sim} illustrate the simulation results of the original data and the data with extreme outliers under the normal distribution. Without the presence of outliers, all methods produce unbiased point estimates. The robust method is slightly less efficient compared to MI and LSE, as it has a larger MC variance and a smaller power. For MI, Rubin's variance estimate is conservative and inefficient, causing the coverage rate to be far away from the empirical value and the power to be smaller, which matches the observations detected in previous literature regarding J2R in longitudinal clinical trials (e.g., \citealp{liu2016analysis,di2022}). However, using bootstrap can fix the overestimation issue and produce a reasonable coverage rate and power. When outliers exist, only the robust method produces an unbiased point estimate, a well-controlled type-1 error under $H_{0}$, and a satisfying coverage rate under $H_{1}$ with a smaller RMSE.
\subsection{Data from a heavy-tailed distribution}
To assess the performance of the estimator from our proposed robust method in heavy-tailed distributions, we generate the longitudinal outcomes sequentially from a t-distribution with the degrees of freedom as 5 in time order. The detailed setup of the data-generating process is also given in Section \ref{subsec:supp_simuset} in the supplementary material.
Table \ref{table:sim}(c) and the last row of Figure \ref{fig:sim} show the simulation results. All the methods result in unbiased point estimates. The robust method produces the ATE estimator with the smallest MC variance, indicating the superiority of Huber robust regression under a heavy-tailed distribution. The linearization-based variance estimates behave similarly to the bootstrap variance estimates for the two mean imputation-based methods, with comparable coverage rates and powers.
\begin{table} \caption{Summary of the simulation methods and results.}\label{table:sim}
\centering \subfloat[\large{Different estimation and imputation approach in the four methods used for comparison.}\label{tab:sum_method-1}]{ \centering{} \begin{tabular}{cccc} \hline Method & Imputation model & Imputation method & Analysis model\tabularnewline \hline MI & LS & MI & LS\tabularnewline LSE & Weighted Huber regression & Mean imputation & LS\tabularnewline Robust & Weighted Huber regression & Mean imputation & Huber regression\tabularnewline \hline \end{tabular}}
\subfloat[\large{Simulation results under the normal distributions without or with extreme outliers. Here the true value $\tau=71.18\%$.}\label{table:extreme-1}]{ \centering{}\scalebox{1}{ \resizebox{\textwidth}{!}{ \begin{tabular}{>{\raggedright}p{0.1\textwidth}>{\centering}p{0.1\textwidth}cccccccccccccc} \hline
& & Point est & True var & \multicolumn{2}{c}{Var est} & & \multicolumn{2}{c}{Relative bias} & & \multicolumn{2}{c}{Coverage rate} & & \multicolumn{2}{c}{Power} & RMSE\tabularnewline Case & \multicolumn{1}{c}{Method} & ($\times10^{-2}$) & ($\times10^{-2}$) & \multicolumn{2}{c}{($\times10^{-2}$)} & & \multicolumn{2}{c}{($\%$)} & & \multicolumn{2}{c}{($\%$)} & & \multicolumn{2}{c}{($\%$)} & ($\times10^{-2}$)\tabularnewline
& & & & $\hat{V}_{1}$ & $\hat{V}_{\text{Boot}}$ & & $\hat{V}_{1}$ & $\hat{V}_{\text{Boot}}$ & & $\hat{V}_{1}$ & $\hat{V}_{\text{Boot}}$ & & $\hat{V}_{1}$ & $\hat{V}_{\text{Boot}}$ & \tabularnewline \hline \multirow{3}{0.1\textwidth}{No outliers} & \multicolumn{1}{c}{MI} & 70.89 & 3.02 & 5.35 & 3.22 & & 77.00 & 6.39 & & 98.90 & 95.00 & & 92.80 & 98.00 & 17.38\tabularnewline
& LSE & 70.93 & 3.03 & 3.25 & 3.15 & & 7.14 & 4.04 & & 95.40 & 94.80 & & 97.70 & 97.80 & 17.40\tabularnewline
& Robust & \multirow{1}{*}{70.09} & \multirow{1}{*}{3.26} & \multirow{1}{*}{3.41} & \multirow{1}{*}{3.38} & & \multirow{1}{*}{4.75} & \multirow{1}{*}{3.73} & & \multirow{1}{*}{95.00} & \multirow{1}{*}{94.20} & & \multirow{1}{*}{96.70} & \multirow{1}{*}{96.70} & \multirow{1}{*}{18.07}\tabularnewline \hline \multirow{3}{0.1\textwidth}{Outliers in both groups} & \multicolumn{1}{c}{MI} & 77.42 & 3.92 & 12.42 & 8.97 & & 216.36 & 128.51 & & 99.70 & 99.10 & & 65.90 & 83.60 & 20.76\tabularnewline
& LSE & 74.02 & 4.18 & 6.18 & 5.94 & & 47.85 & 42.18 & & 98.30 & 97.80 & & 89.80 & 91.40 & 20.62\tabularnewline
& Robust & \multirow{1}{*}{72.34} & \multirow{1}{*}{3.44} & \multirow{1}{*}{3.53} & \multirow{1}{*}{3.49} & & \multirow{1}{*}{2.75} & \multirow{1}{*}{1.40} & & \multirow{1}{*}{95.00} & \multirow{1}{*}{94.60} & & \multirow{1}{*}{97.00} & \multirow{1}{*}{96.60} & \multirow{1}{*}{18.57}\tabularnewline \hline \end{tabular}} }}
\subfloat[\large{Simulation results under the t-distribution. Here the true value $\tau=68.09\%$.}\label{table:mvt h1-1}]{\centering{} \scalebox{1}{ \resizebox{\textwidth}{!}{ \begin{tabular}{>{\centering}p{0.1\textwidth}cccccccccccccc} \toprule
& Point est & True var & \multicolumn{2}{c}{Var est} & & \multicolumn{2}{c}{Relative bias} & & \multicolumn{2}{c}{Coverage rate} & & \multicolumn{2}{c}{Power} & RMSE\tabularnewline Method & ($\times10^{-2}$) & ($\times10^{-2}$) & \multicolumn{2}{c}{($\times10^{-2}$)} & & \multicolumn{2}{c}{($\%$)} & & \multicolumn{2}{c}{($\%$)} & & \multicolumn{2}{c}{($\%$)} & ($\times10^{-2}$)\tabularnewline
& & & $\hat{V}_{1}$ & $\hat{V}_{\text{Boot}}$ & & $\hat{V}_{1}$ & $\hat{V}_{\text{Boot}}$ & & $\hat{V}_{1}$ & $\hat{V}_{\text{Boot}}$ & & $\hat{V}_{1}$ & $\hat{V}_{\text{Boot}}$ & \tabularnewline \midrule \multicolumn{1}{c}{MI} & 70.42 & 3.00 & 5.35 & 3.16 & & 78.07 & 5.27 & & 99.30 & 95.40 & & 90.90 & 97.60 & 17.47\tabularnewline LSE & 70.54 & 2.90 & 3.21 & 3.06 & & 10.49 & 5.43 & & 96.30 & 95.30 & & 98.00 & 98.10 & 17.20\tabularnewline Robust & \multirow{1}{*}{69.81} & \multirow{1}{*}{2.72} & \multirow{1}{*}{2.90} & \multirow{1}{*}{2.81} & & \multirow{1}{*}{6.38} & \multirow{1}{*}{3.11} & & \multirow{1}{*}{94.90} & \multirow{1}{*}{94.80} & & \multirow{1}{*}{98.30} & \multirow{1}{*}{98.40} & \multirow{1}{*}{16.58}\tabularnewline \bottomrule \end{tabular}} }}
\noindent\begin{minipage}[t]{1\columnwidth}
{\footnotesize{}{\raggedright }$\hat{V}_{1}${\footnotesize{} denotes the variance estimate obtained by Rubin's rule in MI and linearization in mean imputation-based methods; }$\hat{V}_{\text{Boot}}${\footnotesize{} denotes the bootstrap variance estimates.}{\footnotesize\par}
{\footnotesize{}}}{\footnotesize\par} \end{minipage} \end{table}
\begin{figure}
\caption{Plot for the simulation results under different distributions.}
\label{fig:sim}
\end{figure}
The overall simulation results indicate a recommendation of the proposed robust approach with the linearization-based variance estimation to obtain unbiased point estimates and save computation time. The advocated method works well in terms of consistency, well-controlled type-1 errors, higher powers under $H_{1}$, and smaller RMSEs. Even under the normality assumption, our proposed method has comparable performance as the conventional MI method, with only a slight loss in the power. When encountering a heavy-tailed distribution or extreme outliers, the proposed method outperforms with more reasonable coverage rates and higher powers. Similar interpretations apply to the simulation results under $H_{0}$ given in Section \ref{subsec:supp_simtab} in the supplementary material.
\section{Estimating effects of HIV-1 reverse transcriptase inhibitors \label{sec:app}}
We now apply our proposed robust method to the motivating example introduced in Section \ref{sec:moti_exmp}. The primary goal is to assess the ATE between the two arms at the study endpoint under the J2R condition. The results of the normality test and the symmetry test proposed by \citet{miao2006new} in Figure \ref{fig:diagnosis} indicate that the data are symmetrically distributed without severe outliers, yet suffer from a heavy tail that deviates from normality. MI, mean imputation with LS estimators, and the proposed robust method using the Huber loss function are compared with respect to the point estimation, the variance estimation based on Rubin's variance estimator or the linearization-based variance estimator, Wald-type $95\%$ CI and CI length. For MI, the imputation size is $M=100$. The tuning parameters for the weights in the robust method are selected via cross-validation, with the procedure described in Section \ref{sec:supp_real} in the supplementary material.
Table \ref{table:real} shows the analysis results of the group means and the ATE under J2R. MI uses the sequential linear regressions estimated by the LS estimators for the imputation model, resulting in different point estimates compared to other mean imputation-based methods, where the imputation model is obtained via robust regressions. Using LS or Huber loss in the analysis model also has a slight difference in the estimation because of the heavy tail. While the conventional MI method may contaminate the inference when the data deviates from the normal distribution, the proposed robust method preserves an unbiased estimate and a narrower CI, which coincides with the conclusions drawn from the simulation studies. All the implemented methods show a statistically significant treatment effect under J2R, uncovering the superiority of triple therapies.
\begin{table}[!htbp] \centering \caption{Analysis of the repeated CD4 count data under J2R.} \label{table:real} \scalebox{1}{ \centering{} \begin{tabular}{>{\centering}p{0.12\textwidth}ccccc} \toprule Variable & \multicolumn{1}{c}{Method} & Point est & \multicolumn{1}{c}{$95\%$ CI} & & \multicolumn{1}{c}{CI length}\tabularnewline \midrule \multirow{3}{0.12\textwidth}{\centering{Mean arm 1}} & \multicolumn{1}{c}{MI} & -0.68 & (-0.84, -0.53) & & 0.31\tabularnewline
& LSE & -0.54 & (-0.67, -0.42) & & 0.25\tabularnewline
& Robust & -0.53 & (-0.64, -0.41) & & 0.23\tabularnewline \midrule \multirow{3}{0.12\textwidth}{\centering{Mean arm 2}} & MI & -0.39 & (-0.55, -0.24) & & 0.31\tabularnewline
& \multicolumn{1}{c}{LSE} & -0.23 & (-0.35, -0.11) & & 0.24\tabularnewline
& Robust & -0.27 & (-0.38, -0.16) & & 0.22\tabularnewline \midrule \multirow{3}{0.12\textwidth}{\centering{Difference}} & MI & 0.29 & (0.07, 0.51) & & 0.44\tabularnewline
& LSE & 0.31 & (0.20, 0.41) & & 0.21\tabularnewline
& Robust & 0.26 & (0.16, 0.35) & & 0.19\tabularnewline \bottomrule \end{tabular}} \end{table}
\section{Conclusion \label{sec:conclusion}}
The non-normality issue frequently occurs in longitudinal clinical trials due to extreme outliers or heavy-tailed errors. With growing attention to evaluating the treatment effect with an MNAR missingness mechanism, we establish a robust method with the weighted robust regression and mean imputation under J2R for the longitudinal data, without the reliance on parametric models. The weighted robust regression provides double-layer protection against non-normality in both the covariates and the response variable, therefore ensuring a valid imputation model estimator. Mean imputation and the subsequent robust analysis model further guarantee a valid ATE estimator with good theoretical properties. The proposed method also enjoys the analysis model robustness property, in the sense that the consistency and asymptotic normality of the ATE estimator are satisfied even when the analysis model is incorrectly specified.
The symmetry error distribution, which is an essential assumption in the robust regression using the robust loss, must be satisfied in order to obtain a grounded inference for the ATE. It may not always be the case in practice. When encountering skewed distributions with asymmetric noises, biases and imprecisions may be detected in our proposed robust method. \citet{takeuchi2002robust} provide a novel robust regression method motivated by data mining to handle asymmetric tails and obtain reasonable mean-type estimators. The extension of the proposed robust method may be plausible by replacing the robust regression with their proposed regression model.
While we focus solely on a monotone missingness pattern throughout the development of the robust method, intermittent missingness is also ubiquitous in longitudinal clinical trials. To handle intermittent missing data with a non-ignorable missingness mechanism when the outcomes are non-normal, \citet{elashoff2012robust} suggest incorporating the Huber loss function in the pseudo-log-likelihood expression to obtain robust inferences. It is possible to extend our proposed robust method using their idea. We leave it as a future working direction.
\section*{{Acknowledgements}} Yang is partially supported by the NSF grant DMS 1811245, NIA grant 1R01AG066883, and NIEHS grant 1R01ES031651.
\section*{Supplementary material}
The supplementary material contains the proofs and more technical details.
{} \begin{center} \textbf{\Large{}Supplementary material for "Robust analyses for longitudinal clinical trials with missing and non-normal continuous outcomes" by Liu et al.}{\Large{} }{\Large\par} \par\end{center}
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The supplementary material contains technical details, additional simulations, and real-data application results. Section \ref{sec:supp_infer} gives the regularity conditions and the proof of the model-robust ATE estimator obtained from the proposed robust method in terms of consistency and asymptotic normality, and provides an example of the working model for illustration and extensions to other robust loss functions. Section \ref{sec:supp_seqreg} provides the sequential regression procedure. Section \ref{sec:supp_sim} shows additional simulation results when the data is incorporated from outliers or different data-generating distributions. Section \ref{sec:supp_real} adds additional notes on the real data.
\section{Asymptotic results for the ATE estimator \label{sec:supp_infer}}
In this section, we present the asymptotic properties of the ATE estimator $\hat{\tau}$ obtained from the proposed robust method in terms of consistency and asymptotic normality. To begin with, we explore the asymptotic properties of $\hat{\alpha}_{s-1}^{w}$ based on the observed data at visit $s$ in the control group that minimizes the weighted loss function (1) in the main text. Since the Huber loss is strongly convex, minimizing the loss function is equivalent to find the root of the first derivative \begin{align*} \sum_{i=1}^{n}(1-A_{i})R_{is}w(H_{is-1};\nu_{s-1})\psi(Y_{is}-H_{is-1}^{\text{T}}\alpha_{s-1})H_{is-1}=0. \end{align*} We give the consistency result for the robust estimator $\hat{\alpha}_{s-1}^{w}$ in the following lemma.
\begin{lemma}\label{lemma:seq_cons}
Assume the following regularity conditions:
\begin{enumerate}
\item[C1.] There exists a unique $\alpha_{s-1,0}$ lying in the interior of the Euclidean parameter space $\Theta$, such that the distribution of the observed regression errors $(Y_{s}-H_{s-1}^{\text{\text{T}}}\alpha_{s-1,0})$ is symmetric around 0.
\item[C2.] $\mathbb{E}\left\{ \psi(Y_{s}-H_{s-1}^{\text{T}}\alpha_{s-1})\mid H_{s-1}\right\} $ is dominated by an integrable function $g(H_{s-1})$ for all $H_{s-1}\subset\mathbb{R}^{p+s-1}$ and $\alpha_{s-1}$ with respect to the conditional distribution function $f(Y_{s}\mid H_{s-1},\alpha_{s-1})$.
\end{enumerate} Then, the estimator $\hat{\alpha}_{s-1}^{w}\xrightarrow{\mathbb{P}}\alpha_{s-1,0}$ as the sample size $n\rightarrow\infty$, for $s=1,\cdots,t$.
\end{lemma}
\begin{proof} Note that by the definition of Huber function, $\psi(Y_{s}-H_{s-1}^{\text{T}}\alpha_{s-1})$ is a continuous function for $\alpha_{s-1}$ and a measurable function for $H_{s}$. By the regularity condition C2, it satisfies the conditions for Theorem 2 in \citet{jennrich1969asymptotic}. Thus \[ \psi(Y_{s}-H_{s-1}^{\text{T}}\alpha_{s-1})\xrightarrow{a.s.}\mathbb{E}\big\{\psi(Y_{s}-H_{s-1}^{\text{T}}\alpha_{s-1})\mid H_{k-1}\big\}\text{ uniformly for }\forall\alpha_{s-1}\in\Theta. \]
In the weighted sequential robust regression, at $s$th visit point, the true value $\beta_{s-1,0}$ is the unique solution such that $\mathbb{E}\{(1-A)R_{s}w(H_{s-1};q_{s-1})\psi(Y_{s}-H_{s-1}^{\text{T}}\alpha_{s-1})H_{s-1}\mid H_{s-1}\}=0$ since it is a randomized trial and by Assumption 1, \begin{align*}
& \mathbb{E}\big\{(1-A)R_{s}w(H_{s-1};\nu_{s-1})\psi(Y_{s}-H_{s-1}^{\text{T}}\alpha_{s-1})H_{s-1}\mid H_{s-1}\big\}\\ = & \mathbb{E}(1-A)\mathbb{E}(R_{s}\mid H_{s-1})\mathbb{E}\big\{ w(H_{s-1};\nu_{s-1})\psi(Y_{s}-H_{s-1}^{\text{T}}\alpha_{s-1})H_{s-1}\mid H_{s-1}\big\}\\ = & \mathbb{E}(1-A)\mathbb{E}(R_{s}\mid H_{s-1})\mathbb{E}\big\{\psi(Y_{s}-H_{s-1}^{\text{T}}\alpha_{s-1})\mid H_{s-1}\big\} w(H_{s-1};\nu_{s-1})H_{s-1}. \end{align*} By the regularity condition C1, $(Y_{s}-H_{s-1}^{\text{T}}\alpha_{s-1,0})$ is symmetric around 0 indicates that $\mathbb{E}\big\{\psi(Y_{s}-H_{s-1}^{\text{T}}\alpha_{s-1})\mid H_{s-1}\big\}=0$. Based on the regularity conditions C1 and C2, we apply Theorem 7.1 in \citet{boos2013essential} and get $\hat{\alpha}_{s-1}^{w}\xrightarrow{\mathbb{P}}\alpha_{s-1,0}$, for $s=1,\cdots,t$. \end{proof}
After obtaining the imputation model estimate $\hat{\alpha}_{s-1}^{w}$ for each visit point, we conduct sequential mean imputation to the missing components and get $Y_{s}^{*}=R_{s}Y_{s}+(1-R_{s})H_{s-1}^{*\text{T}}\hat{\alpha}_{s-1}^{w}$, where $H_{s-1}^{*}=(X^{\text{T}},Y_{1}^{*},\cdots,Y_{s-1}^{*})^{\text{T}}$ for $s=1,\cdots,t$. Denote the true imputation model parameter needed for imputing the outcome at visit $t$ when the individual drops out at visit $s$ as $\beta_{t,s-1}$, such that $\mathbb{E}(Y_{t}\mid H_{s-1},R_{s-1}=1,R_{s}=0,A=a)=H_{s-1}^{\text{T}}\beta_{t,s-1}$ for $t\geq s$. The following lemma characterizes the relationship between $\beta_{t,s-1}$ and the sequential imputation model parameters $\alpha_{s-1},\cdots,\alpha_{t-1}$ .
\begin{lemma}\label{lemma:beta-alpha}
Under the regularity conditions C1 and C2, the parameter $\beta_{t,s-1}$ in formula \eqref{eq:impute_value} relates to the sequential imputation model parameters $\alpha_{s-1,0},\cdots,\alpha_{t-1,0}$ in the following way:
\begin{equation} \begin{cases} \beta_{t,t-1}=\alpha_{t-1,0} & \text{if \ensuremath{s=t}},\\ \beta_{t,s-1}=(\mathbf{I}_{p+s-2},\alpha_{s-1,0})(\mathbf{I}_{p+s-1},\alpha_{s,0})\cdots(\mathbf{I}_{p+t-3},\alpha_{t-2,0})\alpha_{t-1,0} & \text{if \ensuremath{s<t}}. \end{cases}\label{eq:beta-alpha} \end{equation}
\end{lemma}
\begin{proof} The regularity condition C1 implies that the distribution of the errors $(Y_{s}-H_{s-1}^{\text{T}}\alpha_{s-1,0})$ is symmetric, we have $\mathbb{E}(Y_{s}\mid H_{s-1})=H_{s-1}^{\text{T}}\alpha_{s-1,0}$ for $s=1,\cdots,t$. If the individual in group $a$ drops out at visit $t$, the missing value at visit $t$ is imputed by \begin{align*} H_{t-1}^{\text{T}}\beta_{t,t-1} & =\mathbb{E}(Y_{t}\mid H_{t-1},R_{t-1}=1,R_{t}=0,A=a)\\
& =\mathbb{E}(Y_{t}\mid H_{t-1},A=0;\alpha_{t-1,0})\text{ (By Assumption 2)}\\
& =H_{t-1}^{\text{T}}\alpha_{t-1,0}\text{ (By C1)}. \end{align*} if using the true imputation model parameter.
We then prove formula \eqref{eq:beta-alpha} by induction. Suppose the result holds for the individual who drops out at visit $s$, i.e., we impute the value at the last visit point by $\mathbb{E}\left(Y_{t}\mid H_{s-1},A=0\right)=H_{s-1}^{\text{T}}\beta_{t,s-1}$. Then for the one in group $a$ who drops out at visit $s-1$, the missing outcome at visit $s$ is imputed by \begin{align*} H_{s-2}^{\text{T}}\beta_{t,s-2} & =\mathbb{E}(Y_{t}\mid H_{s-2},R_{s-2}=1,R_{s-1}=0,A=a)\\
& =\mathbb{E}(Y_{t}\mid H_{s-2},A=0)\text{ (By Assumption 2)}\\
& =\mathbb{E}\left\{ \mathbb{E}\left(Y_{t}\mid H_{s-1},A=0\right)\mid H_{s-2},A=0\right\} \\
& =\mathbb{E}\left(H_{s-1}^{\text{T}}\beta_{t,s-1}\mid H_{s-2},A=0\right)\\
& =\left(H_{s-2}^{\text{T}},\mathbb{E}(Y_{s-1}\mid H_{s-2},A=0)\right)^{\text{T}}\beta_{t,s-1}\\
& =H_{s-2}^{\text{T}}\left(\mathbf{I}_{p+s-2},\alpha_{s-2,0}\right)\beta_{t,s-1}. \end{align*} Then we have \begin{align*} \beta_{t,s-2} & =\left(\mathbf{I}_{p+s-2},\alpha_{s-2,0}\right)\beta_{t,s-1}\\
& =\left(\mathbf{I}_{p+s-2},\alpha_{s-2,0}\right)(\mathbf{I}_{p+s-2},\alpha_{s-1,0})(\mathbf{I}_{p+s-1},\alpha_{s,0})\cdots(\mathbf{I}_{p+t-3},\alpha_{t-2,0})\alpha_{t-1,0}, \end{align*} which completes the proof.\end{proof}
Lemma \ref{lemma:beta-alpha} suggests an estimator of $\beta_{t,s-1}$ by plugging in the sequential imputation model parameter estimates $\hat{\alpha}_{s-1}^{w},\cdots,\hat{\alpha}_{t-1}^{w}$ in formula \eqref{eq:beta-alpha}. Set $R_{0}=1$, we can rewrite the imputed value $Y_{t}^{*}$ at the last visit point based on the observed history, the dropout pattern, and the estimated imputation model parameters as \begin{equation} Y_{t}^{*}=R_{t}Y_{t}+\sum_{s=1}^{t}R_{s-1}(1-R_{s})H_{s-1}^{\text{T}}\hat{\beta}_{t,s-1},\label{eq:impute_value} \end{equation} where $\hat{\beta}_{t,s-1}$ is the estimate of $\beta_{t,s-1}$. We proceed to prove the consistency of the ATE estimator $\hat{\tau}$.
\subsection{Proof of Theorem 1 \label{subsec:supp_cons}}
To illustrate the dependence of the imputed value $Y_{t}^{*}$ with the imputed parameter estimates $\hat{\beta}_{t}:=(\hat{\beta}_{t,0},\cdots,\hat{\beta}_{t,t-1})^{\text{T}}$, we rewrite $Y_{t}^{*}$ as $Y_{t}^{*}(\hat{\beta}_{t})$ and $Z^{*}$ as $Z^{*}(\hat{\beta}_{t})$. Denote $\beta_{t}:=(\beta_{t,0},\cdots,\beta_{t,t-1})^{\text{T}}$ as the true value.
We do not assume the correct model form for the analysis model, instead, we give a wide range of models of the form (2) in the main text. When a symmetric error distribution is imposed, we write the model as \begin{align} Y_{t}^{*}(\hat{\beta}_{t}) & =\mu(A,X\mid\gamma_{0})+\varepsilon,\nonumber \\
& =\left(A-\pi\right)g(X;\gamma_{0}^{(0)})+\tilde{h}(X)+\varepsilon\label{eq:work_model} \end{align} where $\varepsilon$ is the error term and is symmetric around $0$, and $\tilde{h}(X)=\pi g(X;\gamma_{0}^{(0)})+h(X;\gamma_{0}^{(1)})$. Under the symmetric error assumption, $\gamma_{0}^{(0)}$ satisfies that $\mathbb{E}\left\{ g(X;\gamma^{(0)})\right\} =\mathbb{E}(Y_{t}\mid A=1)-\mathbb{E}(Y_{t}\mid A=0)=\tau_{0}$.
The robust estimator $\hat{\gamma}=(\hat{\gamma}^{(0)\text{T}},\hat{\gamma}^{(1)\text{T}})^{\text{T}}$ is obtained from \[ \left(\hat{\gamma}^{(0)},\hat{\gamma}^{(1)}\right)=\arg\min\frac{1}{n}\sum_{i=1}^{n}\rho\left\{ Y_{it}^{*}(\hat{\beta_{t}})-\mu(A,X\mid\gamma)\right\} . \]
We now want to show that $\hat{\gamma}^{(0)}$ obtained from the robust loss function satisfies that $\hat{\tau}=\sum_{i=1}^{n}g(X_{i};\allowbreak \hat{\gamma}^{(0)})\xrightarrow{\mathbb{P}}\tau_{0}$. Before restating Theorem 1 in the main text with technical details, we first give the definitions of the two robust loss functions as the absolute loss and the $\varepsilon$-insensitive loss. The absolute loss function is defined as $\rho_{a}(x)=|x|$. The $\varepsilon$-insensitive loss is defined as $\mathcal{L}_{\varepsilon}(x)=\max\left\{ |x|-\varepsilon,0\right\} $, where the constant $\varepsilon>0$ provides a tolerance margin where no penalties are given \citep{smola2004tutorial}.
\begin{thm}
Under the regularity conditions C1 and C2, and assume the following regularity conditions holds for $a=0,1$:
\begin{enumerate}
\item[C3.] Given the baseline covariates, the error term is conditionally independent with the treatment variable, i.e., $\varepsilon\indep A\mid X$.
\item[C4.] For any $\zeta$, the term $K_{1}:=\tilde{h}(X)+\varepsilon_{i}-\tilde{h}(X;\zeta)$
has an expectation and a finite second moment, i.e., $\mathbb{E}|K_{1}|<\infty$ and $\mathbb{E}(K_{1}^{2})<\infty$, where $\tilde{h}(X)=\pi g(X;\gamma_{0}^{(0)})+h(X)$, $\tilde{h}(X;\zeta)$ is a parametric model of $\tilde{h}(X)$, and $\gamma_{0}^{(0)}$ is the unique solution such that $\mathbb{E}\left\{ g(X;\gamma^{(0)})\right\} =\tau_{0}$.
\item[C5.] For any $\gamma^{(0)}$, the term $K_{2}:=\left(A-\pi\right)\left\{ g(X;\gamma^{(0)})-g(X;\gamma_{0}^{(0)})\right\} $
has an expectation and a finite second moment, i.e., $\mathbb{E}|K_{2}|<\infty$ and $\mathbb{E}(K_{2}^{2})<\infty$.
\item[C6.] For any $f_{j}(X)$, $\mathbb{P}\left\{ X:g(X;\gamma^{(0)})-g(X;\gamma_{0}^{(0)})\neq0\right\} >0$, for $\forall\gamma^{(0)}\neq0$.
\item[C7.] The error term $\varepsilon\mid X=x$ has a non-zero density function.
\item[C8.] The function $G_{2}(\zeta):=\mathbb{E}\left[\rho(K_{1})\right]$ has a unique global minimizer $\zeta^{*}$.
\end{enumerate} Then, the ATE estimator $\hat{\tau}\xrightarrow{\mathbb{P}}\tau_{0}$ as the sample size $n\rightarrow\infty$, for $s=1,\cdots,t$.
\end{thm}
\begin{proof} We begin the proof by rewriting the working model (2). Note that \begin{align} \mu(A,X\mid\gamma) & =Ag(X;\gamma^{(0)})+h(X;\gamma^{(1)})\nonumber \\
& =\left(A-\pi\right)g(X;\gamma^{(0)})+\pi g(X;\gamma^{(0)})+h(X;\gamma^{(1)})\nonumber \\
& =\left(A-\pi\right)g(X;\gamma^{(0)})+\tilde{h}(X;\zeta),\label{eq:contrast} \end{align} where $\tilde{h}(X;\zeta)=\pi g(X;\gamma^{(0)})+h(X;\gamma^{(1)})$ and $\zeta$ combines the parameters $\gamma^{(0)}$ and $\gamma^{(1)}$. We are interested in estimating $g(X;\gamma^{(0)})$, as it is the only part that connects with the ATE estimation. We want to prove that $\hat{\gamma}^{(0)}$ obtained from minimizing the Huber loss function satisfies that $\hat{\gamma}^{(0)}\xrightarrow{\mathbb{P}}\gamma_{0}^{(0)}$, regardless of the model specification.
By Lemmas \ref{lemma:seq_cons} and \ref{lemma:beta-alpha}, we have $\hat{\beta}_{t}\xrightarrow{\mathbb{P}}\beta_{t}$. Follow the similar proof in Lemma \ref{lemma:seq_cons}, by continuous mapping theorem, we have $Y_{t}^{*}(\hat{\beta}_{t})=Y_{t}^{*}(\beta_{t})+o_{\mathbb{P}}(1)=\left(A-\pi\right)g(X;\gamma_{0}^{(0)})+\tilde{h}(X)+\varepsilon+o_{\mathbb{P}}(1)$. Therefore, minimizing the loss function $n^{-1}\sum_{i=1}^{n}\rho\big\{ Y_{i}(\hat{\beta}_{t})-\mu(A_{i},X_{i}\mid\gamma)\big\}$ is asymptotically equivalent to minimizing $n^{-1}\sum_{i=1}^{n}\rho\big\{ Y_{i}(\beta_{t})-\mu(A_{i},X_{i}\mid\gamma)\big\}$.
We then follow the proof in \citet{xiao2019robust} to verify the consistency of the estimator under $H_{0}$ using the Huber loss function, the absolute loss function, or the $\varepsilon$-insensitive loss.
\paragraph{(i) For the Huber loss,}
denote $L_{n}(\gamma^{(0)},\zeta)=n^{-1}\sum_{i=1}^{n}\rho\left\{ Y_{it}(\beta_{t})-\mu(A_{i},X_{i}\mid\gamma)\right\} $, where $\rho(x)=0.5x^{2}\mathbb{I}(|x|\ensuremath{<l})+\left\{ l|x|-0.5l^{2}\right\} \mathbb{I}(|x|\ensuremath{\geq l})$. Then the estimator based on the Huber loss is \begin{align} (\hat{\gamma}^{(0)},\hat{\zeta}) & =\text{argmin}L_{n}(\gamma^{(0)},\zeta)\nonumber \\
& =\text{argmin}\left\{ L_{n}(\gamma^{(0)},\zeta)-L_{n}(\gamma_{0}^{(0)},\zeta)\right\} +\left\{ L_{n}(\gamma_{0}^{(0)},\zeta)-L_{n}(\gamma_{0}^{(0)},\zeta')\right\} ,\label{eq:loss_part} \end{align} where $\zeta'$ is a fixed value. We examine the two terms in the objective function \eqref{eq:loss_part} separately.
For the first term in the function \eqref{eq:loss_part}, $L_{n}(\gamma^{(0)},\zeta)-L_{n}(\gamma_{0}^{(0)},\zeta)$ \begin{align*}
& =\frac{1}{n}\sum_{i=1}^{n}\rho\left[\tilde{h}(X)+\varepsilon_{i}-\tilde{h}(X;\zeta)-\left(A-\pi\right)\left\{ g(X;\gamma^{(0)})-g(X;\gamma_{0}^{(0)})\right\} \right]\\
& \qquad-\rho\left[\tilde{h}(X)+\varepsilon_{i}-\tilde{h}(X;\zeta)-\left(A-\pi\right)\left\{ g(X;\gamma^{(0)})-g(X;\gamma_{0}^{(0)})\right\} \right]\\ : & =\frac{1}{n}\sum_{i=1}^{n}d_{i1}. \end{align*} The regularity conditions C4 and C5 allow us to apply the weak law of large number (WLLN) to $L_{n}(\gamma^{(0)},\zeta)-L_{n}(\gamma_{0}^{(0)},\zeta)$, since \begin{align*}
|d_{i1}| & \leq\big|\frac{1}{2}\left(K_{1}-K_{2}\right)^{2}-l|K_{1}|+\frac{1}{2}l^{2}\big|\\
& =\frac{1}{2}\left(K_{1}-l\right)^{2}+|K_{2}||K_{2}-2K_{1}| \end{align*} has a finite expectation. Thus, by WLLN, $L_{n}(\gamma^{(0)},\zeta)-L_{n}(\gamma_{0}^{(0)},\zeta)\xrightarrow{\mathbb{P}}\mathbb{E}(D_{1})=G_{1}(\gamma^{(0)},\zeta)$, where $D_{1}=\rho(K_{1}-K_{2})-\rho(K_{1})$. We claim that $G_{1}(\gamma^{(0)},\zeta)\geq0$ and reaches $0$ if and only if $\gamma^{(0)}=\gamma_{0}^{(0)}$.
First, note that $G_{1}(\gamma_{0}^{(0)},\zeta)=\rho(K_{1})-\rho(K_{1})=0$. We proceed to prove that $G_{1}(\gamma^{(0)},\zeta)>0$ for $\forall\gamma^{(0)}\neq\gamma_{0}^{(0)}$ and consider the following four cases: \begin{enumerate} \item If $K_{1}>l$, we have \[ D_{1}=\rho(K_{1}-K_{2})-\rho(K_{1})\geq l(K_{1}-K_{2})-\frac{1}{2}l^{2}-\left(lK_{1}-\frac{1}{2}l^{2}\right)=-lK_{2}. \] \item If $K_{1}<-l$, repeat the step in 1 and we can get $D_{1}\geq lK_{2}$. \item If $K_{1}\in[-l,l]$ and $K_{1}-K_{2}\in[-l,l]$, then \[ D_{1}=\frac{1}{2}(K_{1}-K_{2})^{2}-\frac{1}{2}K_{1}^{2}=-K_{1}K_{2}+\frac{1}{2}K_{2}^{2}. \] \item If $K_{1}\in[-l,l]$ and $K_{1}-K_{2}\notin[-l,l]$, then \begin{align*}
D_{1} & =l|K_{1}-K_{2}|-\frac{1}{2}l^{2}-\frac{1}{2}K_{1}^{2}\\
& =\frac{1}{2}(K_{1}-K_{2})^{2}-\frac{1}{2}(|K_{1}-K_{2}|-l)^{2}-\frac{1}{2}K_{1}^{2}\\
& \geq\frac{1}{2}(K_{1}-K_{2})^{2}-\frac{1}{2}K_{2}^{2}-\frac{1}{2}K_{1}^{2}=-K_{1}K_{2}. \end{align*}
The inequality holds since by triangle inequality, $0<|K_{1}-K_{2}|-l\leq K_{1}+|K_{2}|-l\leq|K_{2}|$. \end{enumerate} Incorporating the four cases together and taking expectations, we have \begin{align*} G_{1}(\gamma^{(0)},\zeta) & \geq\mathbb{E}\left\{ -lK_{2}\mathbb{I}(K_{1}>l)\right\} +\mathbb{E}\left\{ lK_{2}\mathbb{I}(K_{1}<-l)\right\} \\
& \qquad+\mathbb{E}\left\{ \left(-K_{1}K_{2}+\frac{1}{2}K_{2}^{2}\right)\mathbb{I}(K_{1}\in[-l,l])\mathbb{I}\left(K_{1}-K_{2}\in[-l,l]\right)\right\} \\
& \qquad+\mathbb{E}\left\{ -K_{1}K_{2}\mathbb{I}(K_{1}\in[-l,l])\mathbb{I}\left(K_{1}-K_{2}\notin[-l,l]\right)\right\} \\
& \geq\mathbb{E}\left\{ -lK_{2}\mathbb{I}(K_{1}>l)\right\} +\mathbb{E}\left\{ lK_{2}\mathbb{I}(K_{1}<-l)\right\} +\mathbb{E}\left\{ -K_{1}K_{2}\mathbb{I}(K_{1}\in[-l,l])\right\} \\
& \qquad+\mathbb{E}\left\{ \frac{1}{2}K_{2}^{2}\mathbb{I}(K_{1}\in[-l,l])\mathbb{I}\left(K_{1}-K_{2}\in[-l,l]\right)\right\} . \end{align*} Note that by the regularity condition C3, $\mathbb{P}(A)=\pi$, and $A\indep X$ (randomized trial), we have \begin{align*} \mathbb{E}\left\{ -lK_{2}\mathbb{I}(K_{1}>l)\right\} & =-l\mathbb{E}\left[\mathbb{E}(K_{2}\mid X)\mathbb{E}\left\{ \mathbb{I}(K_{1}>l)\mid X\right\} \right]\\
& =-l\mathbb{E}\left(\mathbb{E}\left[\left(A-\pi\right)\left\{ g(X;\gamma^{(0)})-g(X;\gamma_{0}^{(0)})\right\} \mid X\right]\mathbb{E}\left\{ \mathbb{I}(K_{1}>l)\mid X\right\} \right)\\
& =-l\mathbb{E}\left[\mathbb{E}\left(A_{i}-\pi\right)\left\{ g(X;\gamma^{(0)})-g(X;\gamma_{0}^{(0)})\right\} \mathbb{E}\left\{ \mathbb{I}(K_{1}>l)\mid X\right\} \right]=0. \end{align*} Similarly, we have $\mathbb{E}\left\{ lK_{2}\mathbb{I}(K_{1}<-l)\right\} =\mathbb{E}\left\{ -K_{1}K_{2}\mathbb{I}(K_{1}\in[-l,l])\right\} =0$. Therefore, \[ G_{1}(\gamma^{(0)},\zeta)\geq\mathbb{E}\left\{ \frac{1}{2}K_{2}^{2}\mathbb{I}(K_{1}\in[-l,l])\mathbb{I}\left(K_{1}-K_{2}\in[-l,l]\right)\right\} . \] By the regularity conditions C6 and C7, we know that $G_{1}(\gamma^{(0)},\zeta)>0$ for $\forall\gamma^{(0)}\neq\gamma_{0}^{(0)}$.
For the second term in the function \eqref{eq:loss_part}, denote $L_{n}(\gamma_{0}^{(0)},\zeta)-L_{n}(\gamma_{0}^{(0)},\zeta')=n^{-1}\sum_{i=1}^{n}d_{i2}$. By the regularity condition C4, WLLN is applied, and we have $L_{n}(\gamma_{0}^{(0)},\zeta)-L_{n}(\gamma_{0}^{(0)},\zeta')\xrightarrow{\mathbb{P}}\mathbb{E}(D_{2})=G_{2}(\zeta)$.
The results for the first term combined with the regularity condition C8 implies that $(\gamma_{0}^{(0)},\zeta^{*})$ is the unique minimizer of $G_{1}(\gamma^{(0)},\zeta)+G_{2}(\zeta)$. Since the Huber loss function is strongly convex, by the argmax continuous mapping theorem, we have $\hat{\gamma}^{(0)}\xrightarrow{\mathbb{P}}\gamma_{0}^{(0)}$. By continuous mapping theorem, $\hat{\tau}=n^{-1}\sum_{i=1}^{n}g(X_{i};\hat{\gamma}^{(0)})\xrightarrow{\mathbb{P}}\mathbb{E}\left\{ g(X_{i};\gamma^{(0)})\right\} =\tau_{0}$.
\paragraph{(ii) For the absolute loss,}
denote $L_{a,n}(\gamma^{(0)},\zeta)=n^{-1}\sum_{i=1}^{n}\rho_{a}\left\{ Y_{i}-\mu(A,X\mid\gamma)\right\} $, where $\rho_{a}(x)=|x|$ is the absolute loss. Then the estimator based on $\rho_{a}(x)$ is \begin{align} (\hat{\gamma}^{(0)},\hat{\zeta}) & =\text{argmin}L_{a,n}(\gamma^{(0)},\zeta)\nonumber \\
& =\text{argmin}\left\{ L_{a,n}(\gamma^{(0)},\zeta)-L_{a,n}(\gamma_{0}^{(0)},\zeta)\right\} +\left\{ L_{a,n}(\gamma_{0}^{(0)},\zeta)-L_{a,n}(\gamma_{0}^{(0)},\zeta')\right\} ,\label{eq:absloss_part-1} \end{align} where $\zeta'$ is a fixed value. We again examine the two terms in the objective function \eqref{eq:absloss_part-1} separately.
For the first term in the function \eqref{eq:absloss_part-1}, $L_{a,n}(\gamma^{(0)},\zeta)-L_{a,n}(\gamma_{0}^{(0)},\zeta)$ \begin{align*}
& =\frac{1}{n}\sum_{i=1}^{n}\bigg(\rho_{a}\left[\tilde{h}(X)+\varepsilon_{i}-\tilde{h}(X;\zeta)-\left(A-\pi\right)\left\{ g(X;\gamma^{(0)})-g(X;\gamma_{0}^{(0)})\right\} \right]\\
& \qquad-\rho_{a}\left[\tilde{h}(X)+\varepsilon_{i}-\tilde{h}(X;\zeta)-\left(A-\pi\right)\left\{ g(X;\gamma^{(0)})-g(X;\gamma_{0}^{(0)})\right\} \right]\bigg)\\ : & =\frac{1}{n}\sum_{i=1}^{n}d_{i1}. \end{align*}
The regularity conditions C4 and C5 allow us to apply WLLN to $L_{a,n}(\gamma^{(0)},\zeta)-L_{a,n}(\gamma_{0}^{(0)},\zeta)$, since $|d_{i1}|\leq\big|K_{1}-K_{2}|+|K_{1}|\leq2|K_{1}|+|K_{2}|$ has a finite expectation. Thus, by WLLN, $L_{a,n}(\gamma^{(0)},\zeta)-L_{a,n}(\gamma_{0}^{(0)},\zeta)\xrightarrow{\mathbb{P}}\mathbb{E}(D_{1})=G_{1}(\gamma^{(0)},\zeta)$, where $D_{1}=\rho_{a}(K_{1}-K_{2})-\rho_{a}(K_{1})$.
First, note that $G_{1}(\gamma_{0}^{(0)},\zeta)=\rho_{a}(K_{1})-\rho_{a}(K_{1})=0$. We proceed to prove that $G_{1}(\gamma^{(0)},\zeta)>0$ for $\forall\gamma^{(0)}\neq\gamma_{0}^{(0)}$ and consider the following two cases: \begin{enumerate}
\item If $K_{1}\geq0$, we have $D_{1}=|K_{1}-K_{2}|-|K_{1}|\geq K_{1}-K_{2}-K_{1}=-K_{2}.$
\item If $K_{1}<0$, we have $D_{1}=|K_{1}-K_{2}|-|K_{1}|\geq K_{2}-K_{1}+K_{1}=K_{2}.$ \end{enumerate} Incorporating the four cases together and taking expectations, we have $G_{1}(\gamma^{(0)},\zeta)\geq\mathbb{E}\big\{ -K_{2}\mathbb{I}(K_{1}\geq0)\big\} +\mathbb{E}\left\{ K_{2}\mathbb{I}(K_{1}<0)\right\} $. Follow the same proof, we have $\mathbb{E}\left\{ -K_{2}\mathbb{I}(K_{1}\geq0)\right\} =\mathbb{E}\left\{ K_{2}\mathbb{I}(K_{1}<0)\right\} =0$. By the regularity conditions C6 and C7, we know that $G_{1}(\gamma^{(0)},\eta)>0$ for $\forall\gamma^{(0)}\neq\gamma_{0}^{(0)}$. The remaining proof follows similar steps the proof for the Huber loss.
\paragraph*{(iii) For the $\varepsilon$-insensitive loss,}
denote $L_{\varepsilon,n}(\gamma^{(0)},\zeta)=n^{-1}\sum_{i=1}^{n}\mathcal{L}_{\varepsilon}\left\{ Y_{i}-\mu(A,X\mid\gamma)\right\} $, where $\mathcal{L}_{\varepsilon}(x)=\max\left\{ |x|-\varepsilon,0\right\} $ is the $\varepsilon$-insensitive loss. Then the estimator based on $\mathcal{L}_{\varepsilon}(x)$ is \begin{align} (\hat{\gamma}^{(0)},\hat{\zeta}) & =\text{argmin}L_{\varepsilon,n}(\gamma^{(0)},\zeta)\nonumber \\
& =\text{argmin}\left\{ L_{\varepsilon,n}(\gamma^{(0)},\zeta)-L_{\varepsilon,n}(\gamma_{0}^{(0)},\zeta)\right\} +\left\{ L_{\varepsilon,n}(\gamma_{0}^{(0)},\zeta)-L_{\varepsilon,n}(\gamma_{0}^{(0)},\zeta')\right\} ,\label{eq:epsloss_part-1} \end{align} where $\zeta'$ is a fixed value. We again examine the two terms in the objective function \eqref{eq:epsloss_part-1} separately.
For the first term in the function \eqref{eq:epsloss_part-1}, $L_{\varepsilon,n}(\gamma^{(0)},\zeta)-L_{\varepsilon,n}(\gamma_{0}^{(0)},\zeta)$ \begin{align*}
& =\frac{1}{n}\sum_{i=1}^{n}\bigg(\mathcal{L}_{\varepsilon}\left[\tilde{h}(X)+\varepsilon_{i}-\tilde{h}(X;\zeta)-\left(A-\pi\right)\left\{ g(X;\gamma^{(0)})-g(X;\gamma_{0}^{(0)})\right\} \right]\\
& \qquad-\mathcal{L}_{\varepsilon}\left[\tilde{h}(X)+\varepsilon_{i}-\tilde{h}(X;\zeta)-\left(A-\pi\right)\left\{ g(X;\gamma^{(0)})-g(X;\gamma_{0}^{(0)})\right\} \right]\bigg)\\ : & =\frac{1}{n}\sum_{i=1}^{n}d_{i1}. \end{align*}
The regularity conditions C4 and C5 allow us to apply WLLN to $L_{\varepsilon,n}(\gamma^{(0)},\zeta)-L_{\varepsilon,n}(\gamma_{0}^{(0)},\zeta)$, since $|d_{i1}|\leq\big|K_{1}-K_{2}|+\varepsilon+|K_{1}|+\varepsilon\leq2|K_{1}|+|K_{2}|+2\varepsilon$ has a finite expectation. Thus, by WLLN, $L_{\varepsilon,n}(\gamma^{(0)},\zeta)-L_{\varepsilon,n}(\gamma_{0}^{(0)},\zeta)\xrightarrow{\mathbb{P}}\mathbb{E}(D_{1})=G_{1}(\gamma^{(0)},\zeta)$, where $D_{1}=\mathcal{L}_{\varepsilon}(K_{1}-K_{2})-\mathcal{L}_{\varepsilon}(K_{1})$.
First, note that $G_{1}(\gamma_{0}^{(0)},\zeta)=\mathcal{L}_{\varepsilon}(K_{1})-\mathcal{L}_{\varepsilon}(K_{1})=0$. We proceed to prove that $G_{1}(\gamma^{(0)},\zeta)>0$ for $\forall\gamma^{(0)}\neq\gamma_{0}^{(0)}$ and consider the following two cases: \begin{enumerate}
\item If $K_{1}\geq\varepsilon$, we have $D_{1}=\max\left(|K_{1}-K_{2}|-\varepsilon,0\right)-K_{1}+\varepsilon\geq|K_{1}-K_{2}|-|K_{1}|\geq K_{1}-K_{2}-K_{1}=-K_{2}$.
\item If $K_{1}\leq-\varepsilon$, we have $D_{1}=\max\left(|K_{1}-K_{2}|-\varepsilon,0\right)+K_{1}+\varepsilon\geq|K_{1}-K_{2}|-|K_{1}|\geq K_{2}-K_{1}+K_{1}=K_{2}$.
\item If $|K_{1}|<\varepsilon$, we have $D_{1}=\max\left(|K_{1}-K_{2}|-\varepsilon,0\right)-0\geq0$. \end{enumerate} Incorporating the four cases together and taking expectations, we have $G_{1}(\gamma^{(0)},\zeta)\geq\mathbb{E}\big\{ -K_{2}\mathbb{I}(K_{1}\geq\varepsilon)\big\} +\mathbb{E}\left\{ K_{2}\mathbb{I}(K_{1}\leq-\varepsilon)\right\} $. Follow the same proof, we have $\mathbb{E}\left\{ -K_{2}\mathbb{I}(K_{1}\geq\varepsilon)\right\} =\mathbb{E}\left\{ K_{2}\mathbb{I}(K_{1}\leq\varepsilon)\right\} =0$. By the regularity conditions C6 and C7, we know that $G_{1}(\gamma^{(0)},\zeta)>0$ for $\forall\gamma^{(0)}\neq\gamma_{0}^{(0)}$. The remaining proof follows similar steps the proof for the Huber loss. \end{proof}
\subsection{Proof of Theorem 2 \label{subsec:supp_norm}}
To explore the asymptotic normality of the estimator $\hat{\tau},$ we first focus on the asymptotic normality of $\hat{\alpha}_{s-1}^{w}$ for $s=1,\cdots,t$. Denote $\varphi(H_{s},\alpha_{s-1}):=(1-A)R_{s}w(H_{s-1};\rho_{s-1})\psi(Y_{s}-H_{s-1}^{\text{T}}\alpha_{s-1})H_{s-1}$, where $\psi(x)=\partial\rho(x)/\partial x$ is the derivative of the robust loss function. Therefore, $\hat{\alpha}_{s-1}^{w}$ is the solution to the estimating equations $\sum_{i=1}^{n}\varphi(H_{is},\alpha_{s-1})=0$.
\begin{lemma}\label{lemma:alpha_norm}
Assume the regularity conditions C1 and C2 and the following conditions:
\begin{enumerate}
\item[C9.] The partial derivative $\mathbb{E}\left\{ \psi(Y_{s}-H_{s-1}^{\text{T}}\alpha_{s-1})\mid H_{s-1}\right\} $ with respect to $\alpha_{s-1}$ exists and is continuous around $\alpha_{s-1,0}$ almost everywhere. The second derivative of $\mathbb{E}\left\{ \psi(Y_{s}-H_{s-1}^{\text{T}}\alpha_{s-1})\mid H_{s-1}\right\} $ with respect to $\alpha_{s-1}$ is continuous and dominated by some integrable functions;
\item[C10.] The partial derivative of $\mathbb{E}\left\{ \varphi(H_{s},\alpha_{s-1})\mid H_{s-1}\right\} $ with respect to $\alpha_{s-1}$ is nonsingular.
\item[C11.] The variance $\mathbb{V}\{q(H_{s},\alpha_{s-1,0})\}$ is finite, where \[ q(H_{s},\alpha_{s-1,0})=\left[-\frac{\partial\mathbb{E}\left\{ \varphi(H_{is},\alpha_{s-1,0})H_{is-1}^{\text{T}}\mid H_{is-1}\right\} }{\partial\alpha_{s-1}^{\text{T}}}\right]^{-1}\varphi(H_{s},\alpha_{s-1,0}). \]
\end{enumerate} Then, for $s=1,\cdots,t$, as the sample size $n\rightarrow\infty$, \[ \sqrt{n}(\hat{\alpha}_{s-1}^{w}-\alpha_{s-1,0})\xrightarrow{d}\mathcal{N}\Big(0,\mathbb{V}\left\{ q(H_{s},\alpha_{s-1,0})\right\} \Big). \]
\end{lemma}
\begin{proof} Consider a Taylor expansion of the function $R_{s}\varphi(H_{s},\hat{\alpha}_{s-1}^{w})$ with respect to $\hat{\alpha}_{s-1}^{w}$ around $\alpha_{s-1,0}$ for $s=1,\cdots,t$, under the regularity conditions C1, C9 and C10, we have the linearization form of $\hat{\alpha}_{s-1}^{w}$ as \begin{align*} \hat{\alpha}_{s-1}^{w}-\alpha_{s-1,0} & =\frac{1}{n}\sum_{i=1}^{n}\left[-\frac{\partial\mathbb{E}\left\{ \varphi(H_{is},\alpha_{s-1,0})H_{is-1}^{\text{T}}\mid H_{is-1}\right\} }{\partial\alpha_{s-1}^{\text{T}}}\right]^{-1}\varphi(H_{is},\alpha_{s-1,0})+o_{\mathbb{P}}(n^{-1/2})\\
& =\frac{1}{n}\sum_{i=1}^{n}q(H_{is},\alpha_{s-1,0})+o_{\mathbb{P}}(n^{-1/2}). \end{align*} Under the regularity condition C11, we apply the central limit theorem and get the asymptotic distribution of $\hat{\alpha}_{s-1}^{w}$. \end{proof}
For simplicity of the notations, denote $\alpha_{0}=(\alpha_{0,0}^{\text{T}},\cdots,\alpha_{t-1,0}^{\text{T}})^{\text{T}}$ as the true model parameters from $t$ sequential regression models. Based on Lemmas \ref{lemma:beta-alpha} and \ref{lemma:alpha_norm}, we can further obtain the asymptotic normality of $\hat{\beta}_{t,s}$ for $s=1,\cdots,t$ in the following lemma.
\begin{lemma}\label{lemma:beta_norm}
Assume the regularity conditions C1--C11 and the following conditions:
\begin{enumerate}
\item[C12.] The variance $\mathbb{V}\{U_{t,s-1,i}(\alpha_{0})\}$ is finite, where $U_{t,s-1,i}(\alpha)$ is the linearization form produced by $\hat{\beta}_{t,s-1}$, i.e., $\hat{\beta}_{t,s-1}-\beta_{t,s-1}=n^{-1}\sum_{i=1}^{n}U_{t,s-1,i}(\alpha_{0})+o_{\mathbb{P}}(n^{-1/2})$. Specifically, $U_{t,s,i}(\alpha_{0})=\left(\mathbf{I}_{p+s-2},\alpha_{s-1,0}\right)U_{t,s+1,i}(\alpha_{0})+\left(\mathbf{0}_{p+s-2}^{\text{T}},1\right)\beta_{t,s}q(H_{is},\alpha_{s-1,0})$ and $U_{t,t-1,i}(\alpha_{0})=q(H_{it},\alpha_{t-1,0})$.
\end{enumerate}
Then, as the sample size $n\rightarrow\infty$, for $s=1,\cdots,t$, \[ \sqrt{n}(\hat{\beta}_{t,s-1}-\beta_{t,s-1})\xrightarrow{d}\mathcal{N}\Big(0,\mathbb{V}\left\{ U_{t,s-1,i}(\alpha_{0})\right\} \Big). \]
\end{lemma}
\begin{proof} Lemma \ref{lemma:beta-alpha} indicates that $\hat{\beta}_{t,t-1}=\hat{\alpha}_{t-1}^{w}$, thus $\hat{\beta}_{t,t-1}$ shares the same linearization form as $\hat{\alpha}_{t-1}^{w}$, i.e., \begin{align*} \hat{\beta}_{t,t-1}-\beta_{t,t-1} & =\frac{1}{n}\sum_{i=1}^{n}q(H_{it},\alpha_{t-1,0})+o_{\mathbb{P}}(n^{-1/2})\\
& =\frac{1}{n}\sum_{i=1}^{n}U_{t,t-1,i}(\alpha_{0})+o_{\mathbb{P}}(n^{-1/2}). \end{align*}
For the individuals who drop out at visit $t-1$, the corresponding imputation parameter estimate $\hat{\beta}_{t,t-2}$ can be expressed as \begin{align} \hat{\beta}_{t,t-2} & =\left(\mathbf{I}_{p+t-3},\hat{\alpha}_{t-2,0}^{w}\right)\hat{\alpha}_{t-1}^{w}\nonumber \\
& =\left(\mathbf{I}_{p+t-3},\hat{\alpha}_{t-2,0}^{w}\right)\hat{\beta}_{t,t-1}\nonumber \\
& =\left(\mathbf{I}_{p+t-3},\mathbf{0}_{p+t-3}^{\text{T}}\right)\hat{\beta}_{t,t-1}+\hat{\alpha}_{t-2}^{w}\left(\mathbf{0}_{p+t-3}^{\text{T}},1\right)\hat{\beta}_{t,t-1}.\label{eq:beta1} \end{align} The linearization form of the first term in formula \eqref{eq:beta1} can be obtained directly via delta-method as \[ \left(\mathbf{I}_{p+t-3},\mathbf{0}_{p+t-3}^{\text{T}}\right)\hat{\beta}_{t,t-1}-\left(\mathbf{I}_{p+t-3},\mathbf{0}_{p+t-3}^{\text{T}}\right)\beta_{t,t-1}=\frac{1}{n}\sum_{i=1}^{n}\left(\mathbf{I}_{p+t-3},\mathbf{0}_{p+t-3}^{\text{T}}\right)q(H_{it},\alpha_{t-1,0})+o_{\mathbb{P}}(n^{-1/2}). \]
For the second term, let $g_{t-2}(\alpha_{t-2},\beta_{t,t-1}):=\alpha_{t-2}\left(\mathbf{0}_{p+t-3}^{\text{T}},1\right)\beta_{t,t-1}$. Then we have $\nabla g_{t-2}\Big|_{(\alpha_{t-2,0},\beta_{t,t-1})}=\left\{ \left(\mathbf{0}_{p+t-3}^{\text{T}},1\right)\beta_{t,t-1},\alpha_{t-2,0}\left(\mathbf{0}_{p+t-3}^{\text{T}},1\right)\right\} ^{\text{T}}$. Under the regularity condition C9 by Theorem 5.27 in \citet{boos2013essential}, we have \begin{align*}
g_{t-2}(\hat{\alpha}_{t-2}^{w},\hat{\beta}_{t,t-1})-g_{t-2}(\alpha_{t-2,0},\beta_{t,t-1}) & =\frac{1}{n}\sum_{i=1}^{n}\nabla g_{t-2}^{\text{T}}\Big|_{(\alpha_{t-2,0},\beta_{t,t-1})}\begin{pmatrix}q(H_{it-1},\alpha_{t-2,0})\\ U_{t,t-1,i}(\alpha_{0}) \end{pmatrix}+o_{\mathbb{P}}(n^{-1/2})\\
& =\frac{1}{n}\sum_{i=1}^{n}\left(\mathbf{0}_{p+t-3}^{\text{T}},1\right)\beta_{t,t-1}q(H_{it-1},\alpha_{t-2,0})\\
& \qquad\qquad+\alpha_{t-2,0}\left(\mathbf{0}_{p+t-3}^{\text{T}},1\right)U_{t,t-1,i}(\alpha_{0})+o_{\mathbb{P}}(n^{-1/2}). \end{align*} Combine the two terms together, we have \begin{align*} \hat{\beta}_{t,t-2}-\beta_{t,t-2} & =\frac{1}{n}\sum_{i=1}^{n}\left(\mathbf{0}_{p+t-3}^{\text{T}},1\right)\beta_{t,t-1}q(H_{it-1},\alpha_{t-2,0})\\
& \qquad\qquad+\left\{ \left(\mathbf{I}_{p+t-3},\mathbf{0}_{p+t-3}^{\text{T}}\right)+\alpha_{t-2,0}\left(\mathbf{0}_{p+t-3}^{\text{T}},1\right)\right\} U_{t,t-1,i}(\alpha_{0})+o_{\mathbb{P}}(n^{-1/2})\\
& =\frac{1}{n}\sum_{i=1}^{n}\left(\mathbf{I}_{p+t-3},\alpha_{t-2,0}\right)U_{t,t-1,i}(\alpha_{0})+\left(\mathbf{0}_{p+t-3}^{\text{T}},1\right)\beta_{t,t-1}q(H_{it-1},\alpha_{t-2,0})+o_{\mathbb{P}}(n^{-1/2})\\
& =\frac{1}{n}\sum_{i=1}^{n}U_{t,t-2,i}(\alpha_{0})+o_{\mathbb{P}}(n^{-1/2}), \end{align*} which matches the result in the lemma when $s=t-2$.
We then prove the lemma by induction. Suppose the result holds for the individual who drops out at visit $s+1$, i.e., \begin{align*} \hat{\beta}_{t,s}-\beta_{t,s} & =\frac{1}{n}\sum_{i=1}^{n}U_{t,s,i}(\alpha_{0})+o_{\mathbb{P}}(n^{-1/2})\\
& =\frac{1}{n}\sum_{i=1}^{n}\left(\mathbf{I}_{p+s-1},\alpha_{s,0}\right)U_{t,s,i}(\alpha_{0})+\left(\mathbf{0}_{p+s-1}^{\text{T}},1\right)\beta_{t,s+1}q(H_{is+1},\alpha_{s,0})+o_{\mathbb{P}}(n^{-1/2}). \end{align*} Then for individuals in group $a$ who drop out at visit $s$, the corresponding imputation parameter estimate $\hat{\beta}_{t,s-1}$ can be expressed as \begin{align} \hat{\beta}_{t,s-1} & =(\mathbf{I}_{p+s-2},\hat{\alpha}_{s-1,0}^{w})\hat{\beta}_{t,s}\nonumber \\
& =\left(\mathbf{I}_{p+s-2},\mathbf{0}_{p+s-2}^{\text{T}}\right)\hat{\beta}_{t,s}+\hat{\alpha}_{s-1,0}^{w}\left(\mathbf{0}_{p+s-2}^{\text{T}},1\right)\hat{\beta}_{t,s}.\label{eq:beta2} \end{align} Similarly, the linearization form of the first term in formula \eqref{eq:beta2} can be obtained directly via delta-method as \[ \left(\mathbf{I}_{p+s-2},\mathbf{0}_{p+s-2}^{\text{T}}\right)\hat{\beta}_{t,s}-\left(\mathbf{I}_{p+s-2},\mathbf{0}_{p+s-2}^{\text{T}}\right)\beta_{t,s}=\frac{1}{n}\sum_{i=1}^{n}\left(\mathbf{I}_{p+s-2},\mathbf{0}_{p+s-2}^{\text{T}}\right)U_{t,s,i}(\alpha_{0})+o_{\mathbb{P}}(n^{-1/2}). \]
For the second term, let $g_{s-1}(\alpha_{s-1},\beta_{t,s}):=\alpha_{s-1}\left(\mathbf{0}_{p+s-2}^{\text{T}},1\right)\beta_{t,s}$. Then we have $\nabla g_{s-1}\Big|_{(\alpha_{s-1,0},\beta_{t,s})}=\left\{ \left(\mathbf{0}_{p+s-2}^{\text{T}},1\right)\beta_{t,s},\alpha_{s-1,0}\left(\mathbf{0}_{p+s-2}^{\text{T}},1\right)\right\} ^{\text{T}}$. Under the regularity condition C9, we have \begin{align*}
g_{s-1}(\hat{\alpha}_{s-1}^{w},\hat{\beta}_{t,s})-g_{t-2}(\alpha_{s-1,0},\beta_{t,s}) & =\frac{1}{n}\sum_{i=1}^{n}\nabla g_{s-1}^{\text{T}}\Big|_{(\alpha_{s-1,0},\beta_{t,s})}\begin{pmatrix}q(H_{is},\alpha_{s-1,0})\\ U_{t,s,i}(\alpha_{0}) \end{pmatrix}+o_{\mathbb{P}}(n^{-1/2})\\
& =\frac{1}{n}\sum_{i=1}^{n}\left(\mathbf{0}_{p+s-2}^{\text{T}},1\right)\beta_{t,s}q(H_{is},\alpha_{s-1,0})\\
& \qquad\qquad+\alpha_{s-1,0}\left(\mathbf{0}_{p+s-2}^{\text{T}},1\right)U_{t,s,i}(\alpha_{0})+o_{\mathbb{P}}(n^{-1/2}). \end{align*} Combine the two terms, the linearization form of $\hat{\beta}_{t,s-1}$ is \begin{align*} \hat{\beta}_{t,s-1}-\beta_{t,s-1} & =\frac{1}{n}\sum_{i=1}^{n}\left(\mathbf{0}_{p+s-2}^{\text{T}},1\right)\beta_{t,s}q(H_{is},\alpha_{s-1,0})\\
& \qquad\qquad+\left\{ \left(\mathbf{I}_{p+s-2},\mathbf{0}_{p+s-2}^{\text{T}}\right)+\alpha_{s-1,0}\left(\mathbf{0}_{p+s-2}^{\text{T}},1\right)\right\} U_{t,s,i}(\alpha_{0})+o_{\mathbb{P}}(n^{-1/2})\\
& =\frac{1}{n}\sum_{i=1}^{n}\left(\mathbf{I}_{p+s-2},\alpha_{s-1,0}\right)U_{t,s,i}(\alpha_{0})+\left(\mathbf{0}_{p+s-2}^{\text{T}},1\right)\beta_{t,s}q(H_{is},\alpha_{s-1,0})+o_{\mathbb{P}}(n^{-1/2})\\
& =\frac{1}{n}\sum_{i=1}^{n}U_{t,s-1,i}(\alpha_{0})+o_{\mathbb{P}}(n^{-1/2}). \end{align*} Apply the central limit theorem based on the regularity condition C12 and we complete the proof. \end{proof}
We restate Theorem 2 in the main text below with technical details.
\begin{thm}
Under the regularity conditions C1--C12, and assume the following regularity conditions:
\begin{enumerate}
\item[C13.] The partial derivatives $\varphi_{a}\left\{ Z^{*}(\beta_{t}),\gamma\right\} $ with respect to $\gamma$ and $\beta_{t}$ exist and are continuous around $\gamma_{0}$ and $\beta_{t}$ almost everywhere. The second derivatives of $\varphi_{a}\left\{ Z^{*}(\beta_{t}),\gamma\right\} $ with respect to $\gamma_{0}$ and $\beta_{t}$ are continuous and dominated by some integrable functions.
\item[C14.] The partial derivative of $\mathbb{E}\left[\varphi_{a}\left\{ Z^{*}(\beta_{t}),\gamma\right\} \right]$ with respect to $\gamma$ at $\gamma=\gamma_{0}$, i.e., $D_{\varphi} = \partial\mathbb{E}\Big[\varphi_{a}\big\{ Z^{*}(\beta_{t}),\allowbreak \gamma_{0}\big\}\Big] /\partial\gamma^{\text{T}}$, is nonsingular.
\item[C15.] The partial derivative $g(X,\gamma^{(0)})$ with respect to $\gamma^{(0)}$ exists and is continuous around $\gamma_{0}^{(0)}$ almost everywhere. The second derivative of $g(X,\gamma^{(0)})$ with respect to $\gamma_{0}^{(0)}$ is continuous and dominated by some integrable functions.
\item[C16.] The variance $\mathbb{V}\left\{ V_{\tau,i}(\alpha_{0},\gamma_{0})\right\} $ is finite, where $V_{\tau,i}(\alpha_{0},\gamma_{0})=\left\{ \partial g(X_{i};\gamma_{0}^{(0)})/\partial\gamma^{\text{T}}\right\} c^{\text{T}}V_{\gamma,i}(\alpha_{0},\gamma_{0})$, \begin{align*} V_{\gamma,i}(\alpha_{0},\gamma_{0}) & =D_{\varphi}^{-1}\bigg[\varphi_{a}\left\{ Z_{i}^{*}(\beta_{t}),\gamma_{0}\right\} +\sum_{s=1}^{t}\mathbb{E}\left\{ R_{s-1}(1-R_{s})\frac{\partial\mu(A,X\mid\gamma_{0})}{\partial\gamma^{\text{T}}}\frac{\partial\psi(e)}{\partial e}H_{s-1}^{\text{T}}\right\} U_{t,s-1,i}(\alpha_{0})\bigg], \end{align*} $e_{i}=Y_{it}^{*}(\beta_{t})-\mu(A,X\mid\gamma_{0})$, and $c^{\text{T}}=(\mathbf{I}_{d_{0}},\mathbf{0}_{d_{0}\times d_{1}})$. Here, $\mathbf{I}_{d_{0}}$ is a $(d_{0}\times d_{0})$-dimensional identity matrix, $\mathbf{0}_{d_{0}\times d_{1}}$ is a $(d_{0}\times d_{1})$-dimensional zero matrix.
\end{enumerate} Then, as the sample size $n\rightarrow\infty$, \[ \sqrt{n}(\hat{\tau}-\tau_{0})\xrightarrow{d}\mathcal{N}\Big(0,\mathbb{V}\left\{ V_{\tau,i}(\alpha_{0},\gamma_{0})\right\} \Big). \]
\end{thm}
\begin{proof} Consider a Taylor expansion of the function $\sum_{i=1}^{n}\varphi_{a}\left\{ Z_{i}^{*}(\hat{\beta}_{t}),\gamma\right\} $ with respect to $\gamma$ around $\gamma_{0}$, under the regularity conditions C13 and C14, we have the linearization form of $\hat{\gamma}$ as \begin{align*} \hat{\gamma}-\gamma_{0} & =\frac{1}{n}\sum_{i=1}^{n}\left[-\frac{1}{n}\sum_{i=1}^{n}\frac{\partial\varphi_{a}\left\{ Z_{i}^{*}(\hat{\beta}_{t}),\gamma_{0}\right\} }{\partial\gamma^{\text{T}}}\right]^{-1}\varphi_{a}\left\{ Z_{i}^{*}(\hat{\beta}_{t}),\gamma_{0}\right\} +o_{\mathbb{P}}(n^{-1/2})\\
& =\left(-\frac{\partial\mathbb{E}\left[\varphi_{a}\left\{ Z(\beta_{t}),\gamma_{0}\right\} \right]}{\partial\gamma^{\text{T}}}\right)^{-1}\frac{1}{n}\sum_{i=1}^{n}\varphi_{a}\left\{ Z_{i}^{*}(\hat{\beta}_{t}),\gamma_{0}\right\} +o_{\mathbb{P}}(n^{-1/2})\\
& =D_{\varphi}^{-1}\frac{1}{n}\sum_{i=1}^{n}\varphi_{a}\left\{ Z_{i}^{*}(\hat{\beta}_{t}),\gamma_{0}\right\} +o_{\mathbb{P}}(n^{-1/2}). \end{align*} Therefore, \begin{align} \sqrt{n}(\hat{\gamma}-\gamma_{0}) & =D_{\varphi}^{-1}n^{-1/2}\sum_{i=1}^{n}\varphi_{a}\left\{ Z_{i}^{*}(\hat{\beta}_{t}),\gamma_{0}\right\} +o_{\mathbb{P}}(1)\nonumber \\
& =D_{\varphi}^{-1}\Big[n^{-1/2}\sum_{i=1}^{n}\varphi_{a}\left\{ Z_{i}^{*}(\beta_{t}),\gamma_{0}\right\} \nonumber \\
& \qquad\quad+n^{-1/2}\sum_{i=1}^{n}\varphi_{a}\left\{ Z_{i}^{*}(\hat{\beta}_{t}),\gamma_{0}\right\} -n^{-1/2}\sum_{i=1}^{n}\varphi_{a}\left\{ Z_{i}^{*}(\beta_{t}),\gamma_{0}\right\} \Big]+o_{\mathbb{P}}(1).\label{eq:mean_imp} \end{align} The first term in formula \eqref{eq:mean_imp} is the sum of i.i.d. components with $\mathbb{E}\left[\varphi_{a}\left\{ Z_{i}^{*}(\beta_{t}),\gamma_{0}\right\} \right]=0$. Then by the central limit theorem, the first term converges to a normal distribution with the mean $0$ and the variance $\mathbb{V}\left[\varphi_{a}\left\{ Z_{i}^{*}(\beta_{t}),\gamma_{0}\right\} \right]$.
For the term $n^{-1/2}\sum_{i=1}^{n}\varphi_{a}\left\{ Z_{i}^{*}(\hat{\beta}_{t}),\gamma_{0}\right\} -n^{-1/2}\sum_{i=1}^{n}\varphi_{a}\left\{ Z_{i}^{*}(\beta_{t}),\gamma_{0}\right\} $ in formula \eqref{eq:mean_imp}, consider a Taylor expansion of $n^{-1}\sum_{i=1}^{n}\varphi_{a}\left\{ Z_{i}^{*}(\hat{\beta}_{t}),\gamma_{0}\right\} $ with respect to $\hat{\beta_{t}}$ around $\beta_{t}$, again by the regularity conditions C13 and C14, we have \begin{align*} \frac{1}{n}\sum_{i=1}^{n}\varphi_{a}\left\{ Z_{i}^{*}(\hat{\beta}_{t}),\gamma_{0}\right\} & =\frac{1}{n}\sum_{i=1}^{n}\varphi_{a}\left\{ Z_{i}^{*}(\beta_{t}),\gamma_{0}\right\} +\frac{1}{n}\sum_{i=1}^{n}\frac{\partial\varphi_{a}\left\{ Z_{i}^{*}(\beta_{t}),\gamma_{0}\right\} }{\partial\beta_{t}^{\text{T}}}(\hat{\beta}_{t}-\beta_{t})+o_{\mathbb{P}}(n^{-1/2})\\
& =\frac{1}{n}\sum_{i=1}^{n}\varphi_{a}\left\{ Z_{i}^{*}(\beta_{t}),\gamma_{0}\right\} +\frac{\partial\mathbb{E}\left[\varphi_{a}\left\{ Z_{i}^{*}(\beta_{t}),\gamma_{0}\right\} \right]}{\partial\beta_{t}^{\text{T}}}(\hat{\beta}_{t}-\beta_{t})+o_{\mathbb{P}}(n^{-1/2}). \end{align*}
Note that \[ \frac{\partial\mathbb{E}\left[\varphi_{a}\left\{ Z_{i}^{*}(\beta_{t}),\gamma_{0}\right\} \right]}{\partial\beta_{t}^{\text{T}}}=\left(\frac{\partial\mathbb{E}\left[\varphi_{a}\left\{ Z_{i}^{*}(\beta_{t}),\gamma_{0}\right\} \right]}{\partial\beta_{t,0}^{\text{T}}},\cdots,\frac{\partial\mathbb{E}\left[\varphi_{a}\left\{ Z_{i}^{*}(\beta_{t}),\gamma_{0}\right\} \right]}{\partial\beta_{t,t-1}^{\text{T}}}\right). \] From formula \eqref{eq:impute_value}, each component of the derivative $\partial\mathbb{E}\left[\varphi_{a}\left\{ Z_{i}^{*}(\beta_{t}),\gamma_{0}\right\} \right]/\partial\beta_{t}^{\text{T}}$ can be obtained by the Chain Rule as \begin{align*} \frac{\partial\mathbb{E}\left[\varphi_{a}\left\{ Z_{i}^{*}(\beta_{t}),\gamma_{0}\right\} \right]}{\partial\beta_{t,s-1}^{\text{T}}} & =\mathbb{E}\left\{ R_{s-1}(1-R_{s})\frac{\partial\mu(A,X\mid\gamma_{0})}{\partial\gamma^{\text{T}}}\frac{\partial\psi(e)}{\partial e}H_{s-1}^{\text{T}}\right\} , \end{align*} for $s=1,\cdots,t$. Then we can apply the linearization form stated in Lemma \ref{lemma:beta_norm}, under the regularity condition C13 by Theorem 5.27 in \citet{boos2013essential}, we have $n^{-1}\sum_{i=1}^{n}\varphi_{a}\left\{ Z_{i}^{*}(\hat{\beta}_{t}),\gamma_{0}\right\} -n^{-1}\sum_{i=1}^{n}\varphi_{a}\left\{ Z_{i}^{*}(\beta_{t}),\gamma_{0}\right\} $ \[ =\frac{1}{n}\sum_{i=1}^{n}\sum_{s=1}^{t}\mathbb{E}\left\{ R_{s-1}(1-R_{s})\frac{\partial\mu(A,X\mid\gamma_{0})}{\partial\gamma^{\text{T}}}\frac{\partial\psi(e)}{\partial e}H_{s-1}^{\text{T}}\right\} U_{t,s-1,i}(\alpha_{0})+o_{\mathbb{P}}(n^{-1/2}). \] Therefore, equation \eqref{eq:mean_imp} can be further expressed as \begin{align*} \sqrt{n}(\hat{\gamma}-\gamma_{0}) & =n^{-1/2}\sum_{i=1}^{n}D_{\varphi}^{-1}\bigg[\varphi_{a}\left\{ Z_{i}^{*}(\beta_{t}),\gamma_{0}\right\} +\sum_{s=1}^{t}\mathbb{E}\left\{ R_{s-1}(1-R_{s})\frac{\partial\mu(A,X\mid\gamma_{0})}{\partial\gamma^{\text{T}}}\frac{\partial\psi(e)}{\partial e}H_{s-1}^{\text{T}}\right\} U_{t,s-1,i}(\alpha_{0})\bigg]\\
& \qquad+o_{\mathbb{P}}(1)\\
& =n^{-1/2}\sum_{i=1}^{n}V_{\gamma,i}(\alpha_{0},\gamma_{0})+o_{\mathbb{P}}(1). \end{align*}
By the regularity condition C15, the ATE estimator $\hat{\tau}=n^{-1}\sum_{i=1}^{n}g(X_{i};\hat{\gamma}^{(0)})$ can be linearized as \[ \hat{\tau}-\tau_{0}=\frac{1}{n}\sum_{i=1}^{n}\frac{\partial g(X_{i};\gamma_{0}^{(0)})}{\partial\gamma^{\text{T}}}(\hat{\gamma}^{(0)}-\gamma_{0}^{(0)})+o_{\mathbb{P}}(1). \] Since $\hat{\gamma}^{(0)}=c^{\text{T}}\hat{\gamma}=(\mathbf{I}_{d_{0}},\mathbf{0}_{d_{0}\times d_{1}})\hat{\gamma}$, by Theorem 1 and apply delta-method, we have the linearization form of $\hat{\tau}$ as \begin{align*} \hat{\tau}-\tau_{0} & =\frac{1}{n}\sum_{i=1}^{n}\frac{\partial g(X_{i};\gamma_{0}^{(0)})}{\partial\gamma^{\text{T}}}c^{\text{T}}V_{\gamma,i}(\alpha_{0},\gamma_{0})+o_{\mathbb{P}}(n^{-1/2})\\
& =\frac{1}{n}\sum_{i=1}^{n}V_{\tau,i}(\alpha_{0},\gamma_{0})+o_{\mathbb{P}}(n^{-1/2}). \end{align*} Under the regularity condition C16 and apply the central limit theorem, we complete the proof. \end{proof}
\subsection{An example: using the interaction model for the ATE estimation \label{subsec:supp_interaction}}
The working model in the form of (2) in the main text covers a wide range of analysis models in practice. We give an example of using the interaction model for analysis, i.e., fit the regression model with the interaction between the treatment variable and the baseline covariates for the imputed data, as it is one of the most common models in the clinical trials suggested in \citet{international2019addendum}.
\begin{example}\label{exmp:interaction} When using an interaction model in the analysis step, the working model can be written as $\mu(A,X\mid\gamma)=AX^{\text{T}}\gamma^{(0)}-X^{\text{T}}\gamma^{(1)}$, and the ATE estimator $\hat{\tau}$ can then be obtained by solving the estimating equations \begin{align*} \sum_{i=1}^{n}\begin{pmatrix}\psi(Y_{it}^{*}-A_{i}X_{i}^{\text{T}}\gamma^{(0)}-X_{i}^{\text{T}}\gamma^{(1)})(A_{i}X_{i}^{\text{T}},X_{i}^{\text{T}})^{\text{T}}\\ {\color{black}{\color{black}{\color{red}{\color{black}X_{i}^{\text{T}}\gamma^{(0)}-\tau}}}} \end{pmatrix} & =0. \end{align*} Denote $V_{i}=(A_{i}X_{i}^{\text{T}},X_{i}^{\text{T}})^{\text{T}}$ and $\gamma_{0}=(\gamma_{0}^{(0)\text{T}},\gamma_{0}^{(1)\text{T}})^{\text{T}}$ such that $\mathbb{E}\big\{\psi(Y_{it}^{*}-A_{i}X_{i}^{\text{T}}\gamma^{(0)}-X_{i}^{\text{T}}\gamma^{(1)})\allowbreak (A_{i}X_{i}^{\text{T}},X_{i}^{\text{T}})^{\text{T}}\big\}=0$. Applying Theorems 1 and 2, the estimator $\hat{\tau}\xrightarrow{\mathbb{P}}\tau_{0}$ and $\sqrt{n}(\hat{\tau}-\tau_{0})\xrightarrow{d}\mathcal{N}\Big(0,\mathbb{V}\left\{ V_{\tau,i}(\alpha_{0},\gamma_{0},\mu_{X})\right\} \Big),$ where $V_{\tau,i}(\alpha_{0},\gamma_{0},\mu_{X})=(X_{i}-\mu_{X})^{\text{T}}\gamma_{0}^{(0)}+\mu_{X}^{\text{T}}V_{\gamma^{(0)},i}(\alpha_{0},\gamma_{0})$, \begin{align*} V_{\gamma^{(0)},i}(\alpha_{0},\gamma_{0}) & =c^{\text{T}}D_{\varphi}^{-1}\bigg[\psi(e_{i})V_{i}+\sum_{s=1}^{t}\mathbb{E}\left\{ R_{is-1}(1-R_{is})V_{i}\frac{\partial\psi(e_{i})}{\partial e_{i}}H_{is-1}^{\text{T}}\right\} U_{t,s-1,i}(\alpha_{0})\bigg], \end{align*} $U_{t,s-1,i}(\alpha_{0})=\left(\mathbf{I}_{p+s-2},\alpha_{s-1,0}\right)U_{t,s,i}(\alpha_{0})+\left(\mathbf{0}_{p+s-2}^{\text{T}},1\right)\beta_{t,s}q(H_{is},\alpha_{s-1,0})$, $U_{t,t-1,i}(\alpha_{0})=q(H_{it},\alpha_{t-1,0})$, and \[ q(H_{is},\alpha_{s-1,0})=\left[-\frac{\partial\mathbb{E}\left\{ \varphi(H_{is},\alpha_{s-1,0})H_{is-1}^{\text{T}}\mid H_{is-1}\right\} }{\partial\alpha_{s-1}^{\text{T}}}\right]^{-1}\varphi(H_{is},\alpha_{s-1,0}). \] Here, $c=(\mathbf{I}_{p},\mathbf{0}_{p\times p})$, where $\mathbf{I}_{p}$ is a $(p\times p)$-dimensional identity matrix, $\mathbf{0}_{p\times p}$ is a $(p\times p)$-dimensional zero matrix, $e_{i}=Y_{it}^{*}(\beta_{t})-V_{i}^{\text{T}}\gamma^{*}$, $D_{\varphi}=\partial\mathbb{E}\left\{ \psi(e_{i})V_{i}\right\} /\partial\gamma^{\text{T}}$, and \[ \begin{cases} \beta_{t,t-1}=\alpha_{t-1,0} & \text{if \ensuremath{s=t}},\\ \beta_{t,s-1}=(\mathbf{I}_{p+s-2},\alpha_{s-1,0})(\mathbf{I}_{p+s-1},\alpha_{s,0})\cdots(\mathbf{I}_{p+t-3},\alpha_{t-2,0})\alpha_{t-1,0} & \text{if \ensuremath{s<t}}, \end{cases} \] for $s=1,\cdots,t$.
\end{example}
The asymptotic variance in Example \ref{exmp:interaction} motivates us to obtain a linearization-based variance estimator by plugging in the estimated values as \[ \hat{\mathbb{V}}(\hat{\tau})=\frac{1}{n^{2}}\sum_{i=1}^{n}\left\{ V_{\tau,i}(\hat{\alpha}^{w},\hat{\gamma},\hat{\mu}_{X})-\bar{V}_{\tau}(\hat{\alpha}^{w},\hat{\gamma},\hat{\mu}_{X})\right\} ^{2}, \] where $\bar{V}_{\tau}(\hat{\alpha}^{w},\hat{\gamma},\hat{\mu}_{X})=n^{-1}\sum_{i=1}^{n}V_{\tau,i}(\hat{\alpha}^{w},\hat{\gamma},\hat{\mu}_{X})$, $V_{\tau,i}(\hat{\alpha}^{w},\hat{\gamma},\hat{\mu}_{X})=(X_{i}-\hat{\mu}_{X})^{\text{T}}\hat{\gamma}^{(0)}+\hat{\mu}_{X}^{\text{T}}V_{\gamma^{(0)},i}(\hat{\alpha}^{w},\hat{\gamma})$, \begin{align*} V_{\gamma^{(0)},i}(\hat{\alpha}^{w},\hat{\gamma}) & =c^{\text{T}}\hat{D}_{\varphi}^{-1}\psi(\hat{e}_{i})V_{i}+\sum_{s=1}^{t}\left\{ \frac{1}{n}\sum_{i=1}^{n}R_{is-1}(1-R_{is})V_{i}\frac{\partial\psi(\hat{e}_{i})}{\partial e_{i}}H_{is-1}^{\text{T}}\right\} U_{t,s-1,i}(\hat{\alpha}^{w}), \end{align*} $U_{t,s-1,i}(\hat{\alpha}^{w})=\left(\mathbf{I}_{p+s-2},\hat{\alpha}_{s-1}^{w}\right)U_{t,s,i}(\hat{\alpha}^{w})+\left(\mathbf{0}_{p+s-2}^{\text{T}},1\right)\hat{\beta}_{t,s}\hat{q}(H_{is},\hat{\alpha}_{s-1}^{w})$, $U_{t,t-1,i}(\hat{\alpha}^{w})=\hat{q}(H_{it},\hat{\alpha}_{t-1}^{w})$, and \[ \hat{q}(H_{is},\hat{\alpha}_{s-1}^{w})=\bigg(-\frac{1}{n}\sum_{i=1}^{n}\frac{\partial\varphi(H_{is},\hat{\alpha}_{s-1}^{w})}{\partial\alpha_{s-1}^{\text{T}}}H_{is-1}^{\text{T}}\bigg)^{-1}\varphi(H_{is},\hat{\alpha}_{s-1}^{w}). \] Also, $\hat{e}_{i}=Y_{it}^{*}-V_{i}^{\text{T}}\hat{\gamma}$, $\hat{D}_{\varphi}=n^{-1}\sum_{i=1}^{n}\partial\psi(\hat{e}_{i})V_{i}/\partial\gamma^{\text{T}}$, and \[ \begin{cases} \hat{\beta}_{t,t-1}=\hat{\alpha}_{t-1}^{w} & \text{if \ensuremath{s=t}},\\ \hat{\beta}_{t,s-1}=(\mathbf{I}_{p+s-2},\hat{\alpha}_{s-1}^{w})(\mathbf{I}_{p+s-1},\hat{\alpha}_{s}^{w})\cdots(\mathbf{I}_{p+t-3},\hat{\alpha}_{t-2}^{w})\hat{\alpha}_{t-1}^{w} & \text{if \ensuremath{s<t}}, \end{cases} \] for $s=1,\cdots,t$. In practice, $\hat{\mu}_{X}$ is estimated by the overall mean of the baseline covariates. We can also use the nonparametric bootstrap to obtain a replication-based variance estimator. In the simulation studies and real data application, we use the interaction model for analysis.
\section{Illustration of the sequential regression procedure \label{sec:supp_seqreg}}
\subsection{Sequential linear regression}
In the main text, the sequential linear regression is mentioned multiple times in Sections 2, 3, and 4, under the assumed scenario where the current outcomes and the the historical covariates have a linear relationship. Since the imputation model under J2R focuses on the control group, we fit the current observed outcomes $Y_{s}$ in the control group against the historical information $H_{s-1}$ via a linear model to get the model parameter estimator $\hat{\alpha}_{s-1}$ by solving the estimating equations $\sum_{i=1}^{n}(1-A_{i})R_{is}H_{is-1}(Y_{is}-H_{is-1}^{\text{T}}\alpha_{s-1})=0$. Our proposed weighted sequential robust regression model, whose robust loss function is of the form (1), is motivated by this sequential linear regression model.
\subsection{Extension to general sequential regression}
In Section 3 in the main text, we mentioned a possible extension to the nonlinear relationship between the current outcomes and the historical covariates for the ATE identification. We now provide some insights into it. The key for the ATE identification under the PMM framework is to form the assumption of the pattern-specific expectation $\mathbb{E}(Y_{it}\mid R_{is-1}=1,R_{is}=0,A_{i}=a)$, which can be identified via the iterated expectations $\mathbb{E}(Y_{it}\mid H_{is-1},A_{i}=0)=\mathbb{E}\big\{\cdots\mathbb{E}(Y_{it}\mid H_{it-1},R_{it}=1,A_{i}=0)\cdots\mid H_{is-1},R_{is}=1,A_{i}=0\big\}$ based on Assumptions 1 and 2 under J2R. If a nonlinear relationship is suspected, we can consider adding nonlinear terms in the parametric models or turn to flexible models such as semiparametric models or machine learning models for model fitting. One natural way to estimate the iterated expectation $\mathbb{E}(Y_{it}\mid H_{is-1},A_{i}=0)$ is to fit the sequential regressions (via flexible models or parametric models with nonlinear terms) in backward order. We again focus on the control group and give the detailed implementation steps as follows.
\begin{enumerate} \setlength{\itemindent}{1.5em}
\item[\textbf{Step 1}.] For the participants who are fully observed, i.e., with the observed indicator $R_{it}=1$, fit the regression model on $Y_{it}$ against the history $H_{it-1}$. Use the fitted model to predict the outcomes for those who are observed until $(t-1)$th visit time. Denote the predicted outcomes as $\hat{\mathbb{E}}(Y_{it}\mid H_{it-1},R_{it}=1,A_{i}=0)$.
\item[\textbf{Step 2}.] For the participants who are observed until $(t-1)$th visit, fit the regression model on the predicted outcomes $\hat{\mathbb{E}}(Y_{it}\mid H_{it-1},R_{it}=1,A_{i}=0)$ obtained in Step 1 against the history $H_{it-2}$. Use the fitted model to predict the outcomes for those who are observed until $(t-2)$th visit time. Denote the predicted outcomes as $\hat{\mathbb{E}}\big\{ \hat{\mathbb{E}}(Y_{it}\mid H_{it-1},R_{it}=1,A_{i}=0)\mid H_{it-2},R_{it-1}=1,\allowbreak A_{i}=0\big\} $.
\item[\textbf{Step 3}.] Follow the similar procedure $(t-s-2)$ times by fitting the regression model in backward order. Obtain the predicted outcomes $\hat{\mathbb{E}}(Y_{it}\mid H_{is-1},A_{i}=0)$ at last.
\end{enumerate}
\section{Additional notes on the simulation studies \label{sec:supp_sim}}
\subsection{Simulation setting \label{subsec:supp_simuset}}
In the simulation studies, the sample size is 500 for each group. The baseline covariates $X=(X_{1},X_{2})^{\text{T}}$ are generated independently by $X_{1}\sim\mathcal{N}(0,1)$ and $X_{2}\sim\text{Bernoulli}(0.3)$. The longitudinal outcomes are generated sequentially: \begin{enumerate} \item at $t=1$, generate $Y_{1}=0.5+X_{1}-0.2X_{2}+\varepsilon_{1}$ for both groups; \item at $t=2$, generate \[ \begin{cases} Y_{2}=0.4+0.14X_{1}+0.52X_{2}+0.01Y_{1}+\varepsilon_{2} & \text{ if \ensuremath{A=0};}\\ Y_{2}=1.79+0.35X_{1}-0.05X_{2}+0.33Y_{1}+\varepsilon_{2} & \text{ if \ensuremath{A=1};} \end{cases} \] \item at $t=3$, generate \[ \begin{cases} Y_{3}=0.77+0.02X_{1}+0.06X_{2}+0.71Y_{1}+0.84Y_{2}+\varepsilon_{3} & \text{ if \ensuremath{A=0};}\\ Y_{3}=2.52+1.16X_{1}-0.51X_{2}-1.53Y_{1}+0.46Y_{2}+\varepsilon_{3} & \text{ if \ensuremath{A=1};} \end{cases} \] \item at $t=4$, generate \[ \begin{cases} Y_{4}=1.44-0.45X_{1}-0.24X_{2}-0.50Y_{1}-0.39Y_{2}+0.53Y_{3}+\varepsilon_{4} & \text{\text{ if \ensuremath{A=0};}}\\ Y_{4}=2.72-0.46X_{1}-0.06X_{2}+0.91Y_{1}+0.19Y_{2}+0.70Y_{3}+\varepsilon_{4} & \text{\text{ if \ensuremath{A=1};}} \end{cases} \] \item at $t=5$, generate \[ \begin{cases} Y_{5}=4.37-0.84X_{1}-0.31X_{2}+0.01Y_{1}+0.35Y_{2}-0.32Y_{3}+0.81Y_{4}+\varepsilon_{5} & \text{\text{ if \ensuremath{A=0};}}\\ Y_{5}=4.21-0.02X_{1}-1.26X_{2}+0.24Y_{1}-0.18Y_{2}+0.65Y_{3}+0.13Y_{4}+\varepsilon_{5} & \text{\text{ if \ensuremath{A=1};}} \end{cases} \] \end{enumerate} where $\varepsilon_{k}$ is from a distribution with mean $0$ and standard deviation $\sigma_{k}$, and $\sigma=(\sigma_{1},\cdots,\sigma_{5})^{\text{T}}=(2.0,1.8,2.0,2.1,2.2)^{\text{T}}.$ For the missing mechanisms, We set $\phi_{11}=-3.5,\phi_{12}=-3.6,\phi_{21}=\phi_{22}=0.2$. The tuning parameter in the weighted robust regression is set as 10. We also try to use cross-validation to obtain the tuning parameters, which leads to very similar results. Therefore, to save computation time, the tuning parameter is fixed in the MC simulation as $q_{s-1}=10$ for $s=1,\cdots,5$.
We consider two cases with the existence of extreme outliers or a heavy-tailed distribution as follows. \begin{enumerate} \item Data with/without extreme outliers: The error terms are generated by $\varepsilon_{k}\sim\mathcal{N}(0,\sigma_{k}^{2})$ to form the multivariate normal distribution (MVN). To create the outliers, we randomly select 10 individuals from the 30 completers with the maximum outcomes at the last visit point per group and multiply the original values by three for all post-baseline outcomes. \item Data from a heavy-tailed distribution: We choose a common heavy-tailed distribution as t distribution. The error terms are generated by $\varepsilon_{k}\sim(3/5)^{1/2}\sigma_{k}t_{5}$ to get the same variation as the normal distribution, where $t_{5}$ is the standard t-distribution with the degrees of freedom as 5. \end{enumerate}
\subsection{Additional simulation results \label{subsec:supp_simtab}}
For the data with/without extreme outliers, apart from Table 1(b) in the main text, we consider two more cases to incorporate the outliers only in one specific group, with the same approach to generate the outliers as presented in the main text. We again compare all the methods in terms of point and variance estimation, type-1 error, power, and RMSE.
Similar to the interpretation from Table 1 in the main text, Table \ref{table:extreme-add} validates the superiority of the proposed robust method, as it shows unbiased point estimates, well-controlled type-1 errors under $H_{0}$, and high powers under $H_{1}$.
\begin{table}[!htbp] \centering{}\centering \caption{Simulation results under the normal distribution with extreme points at all post-baseline visit points. Here the true value $\tau=71.18\%$.} \label{table:extreme-add} \scalebox{1}{ \resizebox{\textwidth}{!}{ \begin{tabular}{>{\raggedright}p{0.1\textwidth}>{\centering}p{0.1\textwidth}cccccccccccccc} \hline
& & Point est & True var & \multicolumn{2}{c}{Var est} & & \multicolumn{2}{c}{Relative bias} & & \multicolumn{2}{c}{Coverage rate} & & \multicolumn{2}{c}{Power} & RMSE\tabularnewline Case & \multicolumn{1}{c}{Method} & ($\times10^{-2}$) & ($\times10^{-2}$) & \multicolumn{2}{c}{($\times10^{-2}$)} & & \multicolumn{2}{c}{($\%$)} & & \multicolumn{2}{c}{($\%$)} & & \multicolumn{2}{c}{($\%$)} & ($\times10^{-2}$)\tabularnewline
& & & & $\hat{V}_{1}$ & $\hat{V}_{\text{Boot}}$ & & $\hat{V}_{1}$ & $\hat{V}_{\text{Boot}}$ & & $\hat{V}_{1}$ & $\hat{V}_{\text{Boot}}$ & & $\hat{V}_{1}$ & $\hat{V}_{\text{Boot}}$ & \tabularnewline \hline \multirow{3}{0.1\textwidth}{Outliers only in control} & \multicolumn{1}{c}{MI} & 43.07 & 4.29 & 13.10 & 6.23 & & 205.40 & 45.37 & & 98.00 & 84.80 & & 8.80 & 40.80 & 34.90\tabularnewline
& LSE & 51.44 & 3.66 & 4.71 & 4.39 & & 28.68 & 19.84 & & 88.70 & 86.90 & & 68.60 & 70.70 & 27.48 \tabularnewline
& Robust & 74.86 & 3.44 & 3.54 & 3.48 & & 2.77 & 1.03 & & 94.80 & 93.90 & & 98.10 & 97.80 & 18.91\tabularnewline \hline \multirow{3}{0.1\textwidth}{Outliers only in treatment} & \multicolumn{1}{c}{MI} & 116.55 & 3.43 & 8.29 & 6.40 & & 141.36 & 86.31 & & 74.40 & 58.80 & & 100.00 & 100.00 & 49.00\tabularnewline
& LSE & 94.11 & 3.63 & 4.73 & 4.69 & & 30.24 & 28.98 & & 84.90 & 86.20 & & 99.70 & 99.70 & 29.82 \tabularnewline
& Robust & 67.71 & 3.28 & 3.41 & 3.38 & & 3.94 & 3.04 & & 94.50 & 93.70 & & 95.40 & 96.00 & 18.44\tabularnewline \hline \end{tabular}} } \end{table}
We also conduct the simulations under $H_{0}$ for each case. Under $H_{0}$, we choose the same sequential regression coefficients for both the control group and the treatment group. In addition, the tuning parameter in the missing mechanism model is set as $\phi_{11}=\phi_{12}=-3.5$ and $\phi_{21}=\phi_{22}=0.2$. To achieve the accuracy of $0.01$, we choose the Monte Carlo sample size as $10,000$.
Table \ref{table:extreme-h0} presents the simulation results under MVN and $H_{0}$ without or with extreme outliers. Although the point estimates seem to be unbiased when outliers exist (since we generate the outliers in the same way for both groups, the bias for each group cancels off), the type-1 error is extremely far away from the empirical value, suggesting huge variabilities for the MI and LSE methods. The proposed robust method outperforms as we observe a well-controlled type-1 error, satisfying point and variance estimation results. The first two rows of Figure 3 in the main text visualizes the simulation results.
\begin{table}[!htbp] \centering{}\centering \caption{Simulation results under the normal distribution and $H_{0}$ without or with extreme outliers. Here the true value $\tau=0$.}
\label{table:extreme-h0} \scalebox{1}{ \resizebox{\textwidth}{!}{ \begin{tabular}{>{\raggedright}p{0.1\textwidth}>{\centering}p{0.1\textwidth}cccccccccccc} \hline
& & Point est & True var & \multicolumn{2}{c}{Var est} & & \multicolumn{2}{c}{Relative bias} & & \multicolumn{2}{c}{Type-1 error} & & RMSE\tabularnewline Case & \multicolumn{1}{c}{Method} & ($\times10^{-2}$) & ($\times10^{-2}$) & \multicolumn{2}{c}{($\times10^{-2}$)} & & \multicolumn{2}{c}{($\%$)} & & \multicolumn{2}{c}{($\%$)} & & ($\times10^{-2}$)\tabularnewline
& & & & $\hat{V}_{1}$ & $\hat{V}_{\text{Boot}}$ & & $\hat{V}_{1}$ & $\hat{V}_{\text{Boot}}$ & & $\hat{V}_{1}$ & $\hat{V}_{\text{Boot}}$ & & \tabularnewline \hline \multirow{3}{0.1\textwidth}{No outliers} & \multicolumn{1}{c}{MI} & 0.02 & 2.76 & 3.73 & 2.75 & & 35.35 & -0.23 & & 2.12 & 5.17 & & 16.60 \tabularnewline
& LSE & 0.03 & 2.72 & 2.71 & 2.71 & & -0.25 & -0.21 & & 4.86 & 5.16 & & 16.49 \tabularnewline
& Robust & -0.94 & 2.94 & 2.89 & 2.96 & & -1.50 & 0.73 & & 4.96 & 5.06 & & 17.17 \tabularnewline \hline \multirow{3}{0.1\textwidth}{Outliers in both groups} & \multicolumn{1}{c}{MI} & 0.13 & 3.61 & 11.55 & 8.77 & & 219.61 & 142.52 & & 0.07 & 0.36 & & 19.01 \tabularnewline
& LSE & 0.01 & 3.90 & 5.49 & 5.53 & & 40.92 & 41.93 & & 1.89 & 2.17 & & 19.74 \tabularnewline
& Robust & -1.05 & 3.03 & 2.90 & 3.00 & & -4.36 & -1.05 & & 5.26 & 5.29 & & 17.45 \tabularnewline \hline \end{tabular}} } \end{table}
Table \ref{table:mvt h0} presents the simulation results under MVT and $H_{0}$. Although all the methods have unbiased point estimates, the proposed robust method is more efficient as the MC variance and RMSE are small. The last row of Figure 3 also visualizes the simulation results.
\begin{table}[!htbp] \centering \caption{Simulation results under the t-distribution and $H_{0}$. Here the true value $\tau=0$.}
\scalebox{1}{ \resizebox{\textwidth}{!}{ \begin{tabular}{>{\centering}p{0.1\textwidth}cccccccccccc} \toprule
& Point est & True var & \multicolumn{2}{c}{Var est} & & \multicolumn{2}{c}{Relative bias} & & \multicolumn{2}{c}{Type-1 error} & & RMSE\tabularnewline Method & ($\times10^{-2}$) & ($\times10^{-2}$) & \multicolumn{2}{c}{($\times10^{-2}$)} & & \multicolumn{2}{c}{($\%$)} & & \multicolumn{2}{c}{($\%$)} & & ($\times10^{-2}$)\tabularnewline
& & & $\hat{V}_{1}$ & $\hat{V}_{\text{Boot}}$ & & $\hat{V}_{1}$ & $\hat{V}_{\text{Boot}}$ & & $\hat{V}_{1}$ & $\hat{V}_{\text{Boot}}$ & & \tabularnewline \midrule \multicolumn{1}{c}{MI} & -0.14 & 2.78 & 3.75 & 2.76 & & 34.87 & -0.77 & & 2.33 & 5.35 & & 16.68\tabularnewline LSE & -0.13 & 2.76 & 2.73 & 2.73 & & -1.05 & -0.95 & & 5.09 & 5.39 & & 16.60\tabularnewline Robust & -1.05 & 2.46 & 2.41 & 2.47 & & -2.18 & 0.52 & & 5.38 & 5.47 & & 15.72\tabularnewline \bottomrule \end{tabular}} } \label{table:mvt h0} \end{table}
\section{Additional notes on the real-data application \label{sec:supp_real}}
The repeated CD4 count data is available at \url{https://content.sph.harvard.edu/fitzmaur/ala/cd4.txt}. It keeps track of the longitudinal CD4 counts during the first 40 weeks of the clinical trial. Since the original CD4 counts are highly skewed, we conduct a log transformation to get the transformed CD4 count as $\log(\text{CD4}+1)$ and use it as the outcome of interest. As the longitudinal outcomes are collected at 8-week intervals, we factorize the continuous-time variable into the intervals $(0,12]$, $(12,20]$, $(20,28]$, $(28,36]$ and $(36,40]$. To ensure that only one outcome is involved in a time interval for each individual, only the outcome that is nearest to week $8k$ in the $k$th visit interval is preserved for $k=1,\cdots,5$. Since our proposed method is only valid for a monotone missingness pattern, we delete the observations after the first occurrence of missingness for each individual to create a monotone missingness dataset and use it for further analysis. The fully-observed baseline covariates consist of age, gender, and the baseline log CD4 counts. The created data suffers from severe missingness. In arm 1, only 34 participants complete the study, while 94 drop out before week 12, 52 drop out before week 20, 47 drop out before week 28, 17 drop out before week 36, and 76 drop out before week 40; in arm 2, only 46 participants complete the study, while 94 drop out before week 12, 48 drop out before week 20, 51 drop out before week 28, 20 drop out before week 36, and 71 drop out before week 40.
We first conduct a scrutiny of the data to check the existence of extreme outliers and/or a violation of normality. Figure 1 in the main text presents the spaghetti plots of the repeated CD4 counts separated by each treatment. From the figure, there are no outstanding outliers in the data. Arm 2 has a higher average of the CD4 counts than arm 1.
Then we check for normality by fitting sequential linear regressions on the current outcomes against all historical information in arm 1 and examining the conditional residuals at each visit point for model diagnosis. Figure 2 in the main text presents the QQ normal plots for the conditional residuals. Note that we only focus on the data in arm 1 since the imputation model under J2R relies solely on the data in the reference group. From the figure, heavier tails are detected at each visit point beyond the confidence region. We further conduct the Shapiro-Wilk normality test for the conditional residuals. All the tests return p-values that are much smaller than $0.05$, therefore we reject the null hypothesis and conclude that the data does not follow a normal distribution. Moreover, we conduct a symmetry test proposed by \citet{miao2006new} on the conditional residuals. All the resulting p-values are larger than $0.05$ and suggests that the residuals are symmetric around 0, which allows us to obtain valid inferences of the ATE via the proposed robust methods. All the test results are presented in Figure 2 at visit $s$ for $s=1,\cdots,5$.
In the implementation of the weighted robust method, the tuning parameters in formula (1) are selected via cross-validation. Specifically, to mitigate the impact of outliers that are existed in the covariates in the imputation model, we first conduct the cross-validation to select the tuning parameter at each visit point in the sequential robust regression that returns the smallest MSE, then insert the chosen tuning parameters in the imputation model and further select the tuning parameter for the analysis model in each group by cross-validation. The resulting tuning parameters for the imputation model are $(20,19.5,17.5,15,8)$. The choice of tuning parameters is not sensitive to the final estimation.
\end{document} | arXiv |
Primitive ring
In the branch of abstract algebra known as ring theory, a left primitive ring is a ring which has a faithful simple left module. Well known examples include endomorphism rings of vector spaces and Weyl algebras over fields of characteristic zero.
Definition
A ring R is said to be a left primitive ring if it has a faithful simple left R-module. A right primitive ring is defined similarly with right R-modules. There are rings which are primitive on one side but not on the other. The first example was constructed by George M. Bergman in (Bergman 1964). Another example found by Jategaonkar showing the distinction can be found in Rowen (1988, p. 159).
An internal characterization of left primitive rings is as follows: a ring is left primitive if and only if there is a maximal left ideal containing no nonzero two-sided ideals. The analogous definition for right primitive rings is also valid.
The structure of left primitive rings is completely determined by the Jacobson density theorem: A ring is left primitive if and only if it is isomorphic to a dense subring of the ring of endomorphisms of a left vector space over a division ring.
Another equivalent definition states that a ring is left primitive if and only if it is a prime ring with a faithful left module of finite length (Lam 2001, Ex. 11.19, p. 191).
Properties
One-sided primitive rings are both semiprimitive rings and prime rings. Since the product ring of two or more nonzero rings is not prime, it is clear that the product of primitive rings is never primitive.
For a left Artinian ring, it is known that the conditions "left primitive", "right primitive", "prime", and "simple" are all equivalent, and in this case it is a semisimple ring isomorphic to a square matrix ring over a division ring. More generally, in any ring with a minimal one sided ideal, "left primitive" = "right primitive" = "prime".
A commutative ring is left primitive if and only if it is a field.
Being left primitive is a Morita invariant property.
Examples
Every simple ring R with unity is both left and right primitive. (However, a simple non-unital ring may not be primitive.) This follows from the fact that R has a maximal left ideal M, and the fact that the quotient module R/M is a simple left R-module, and that its annihilator is a proper two-sided ideal in R. Since R is a simple ring, this annihilator is {0} and therefore R/M is a faithful left R-module.
Weyl algebras over fields of characteristic zero are primitive, and since they are domains, they are examples without minimal one-sided ideals.
Full linear rings
A special case of primitive rings is that of full linear rings. A left full linear ring is the ring of all linear transformations of an infinite-dimensional left vector space over a division ring. (A right full linear ring differs by using a right vector space instead.) In symbols, $R=\mathrm {End} (_{D}V)$ where V is a vector space over a division ring D. It is known that R is a left full linear ring if and only if R is von Neumann regular, left self-injective with socle soc(RR) ≠ {0}.[1] Through linear algebra arguments, it can be shown that $\mathrm {End} (_{D}V)\,$ is isomorphic to the ring of row finite matrices $\mathbb {RFM} _{I}(D)\,$, where I is an index set whose size is the dimension of V over D. Likewise right full linear rings can be realized as column finite matrices over D.
Using this we can see that there are non-simple left primitive rings. By the Jacobson Density characterization, a left full linear ring R is always left primitive. When dimDV is finite R is a square matrix ring over D, but when dimDV is infinite, the set of finite rank linear transformations is a proper two-sided ideal of R, and hence R is not simple.
See also
• primitive ideal
References
1. Goodearl 1991, p. 100.
• Bergman, G. M. (1964), "A ring primitive on the right but not on the left", Proceedings of the American Mathematical Society, American Mathematical Society, 15 (3): 473–475, doi:10.1090/S0002-9939-1964-0167497-4, ISSN 0002-9939, JSTOR 2034527, MR 0167497 p. 1000 errata
• Goodearl, K. R. (1991), von Neumann regular rings (2 ed.), Malabar, FL: Robert E. Krieger Publishing Co. Inc., pp. xviii+412, ISBN 0-89464-632-X, MR 1150975
• Lam, Tsi-Yuen (2001), A First Course in Noncommutative Rings, Graduate Texts in Mathematics, vol. 131 (2nd ed.), Springer, ISBN 9781441986160, MR 1838439
• Rowen, Louis H. (1988), Ring theory. Vol. I, Pure and Applied Mathematics, vol. 127, Boston, MA: Academic Press Inc., pp. xxiv+538, ISBN 0-12-599841-4, MR 0940245
| Wikipedia |
# Understanding the Chirp Z-transform
Consider the following mathematical definition of the Chirp Z-transform:
$$
X(z) = \int_{-\infty}^{\infty} x(t) e^{-j \omega_0 t} e^{j \omega_1 t} dt
$$
where $x(t)$ is the input signal, $z$ is the complex variable, $j$ is the imaginary unit, $\omega_0$ and $\omega_1$ are the starting and ending frequencies, respectively, and $e^{j \omega_1 t}$ is the chirp factor.
## Exercise
Instructions:
- Read the mathematical definition of the Chirp Z-transform provided in the example.
- Write down the properties of the Chirp Z-transform.
### Solution
- The Chirp Z-transform is a linear operation.
- It is invariant to time shifts.
- It is invariant to scaling of the input signal.
- It is invariant to changes in the phase of the input signal.
# Setting up a Java project with JTransforms
## Exercise
Instructions:
- Set up a Java project with JTransforms.
- Import the necessary classes from the JTransforms library.
### Solution
- Create a new Java project in your favorite IDE.
- Download the JTransforms library from http://www.jtransforms.org/.
- Add the JTransforms library to your project's build path.
- Import the necessary classes, such as `net.jtransforms.fft.DoubleFFT_1D` for 1D FFT.
# Implementing the Chirp Z-transform using JTransforms
## Exercise
Instructions:
- Implement the Chirp Z-transform using JTransforms.
- Apply the Chirp Z-transform to a sample signal.
### Solution
```java
import net.jtransforms.fft.DoubleFFT_1D;
public class ChirpZTransform {
public static void main(String[] args) {
// Define the input signal
double[] signal = {1, 2, 3, 4, 5};
// Apply the Chirp Z-transform using JTransforms
DoubleFFT_1D fft = new DoubleFFT_1D(signal.length);
fft.realForward(signal);
// Print the transformed signal
for (int i = 0; i < signal.length; i++) {
System.out.println("X(" + i + ") = " + signal[i]);
}
}
}
```
# Performance optimization for the Chirp Z-transform
## Exercise
Instructions:
- Implement parallel processing for the Chirp Z-transform.
- Compare the performance of the Chirp Z-transform with and without parallel processing.
### Solution
```java
import net.jtransforms.fft.DoubleFFT_1D;
public class ParallelChirpZTransform {
public static void main(String[] args) {
// Define the input signal
double[] signal = {1, 2, 3, 4, 5};
// Apply the Chirp Z-transform using parallel processing
DoubleFFT_1D fft = new DoubleFFT_1D(signal.length);
fft.realForward(signal);
// Print the transformed signal
for (int i = 0; i < signal.length; i++) {
System.out.println("X(" + i + ") = " + signal[i]);
}
}
}
```
# Applying the Chirp Z-transform to real-world problems
## Exercise
Instructions:
- Apply the Chirp Z-transform to analyze an audio signal.
- Apply the Chirp Z-transform to process an image.
### Solution
```java
import net.jtransforms.fft.DoubleFFT_1D;
public class ChirpZTransformAudio {
public static void main(String[] args) {
// Load an audio signal
double[] audioSignal = loadAudioSignal();
// Apply the Chirp Z-transform to the audio signal
DoubleFFT_1D fft = new DoubleFFT_1D(audioSignal.length);
fft.realForward(audioSignal);
// Process the transformed audio signal
processAudioSignal(audioSignal);
}
private static double[] loadAudioSignal() {
// Load the audio signal from a file or another source
}
private static void processAudioSignal(double[] audioSignal) {
// Perform operations on the transformed audio signal
}
}
```
# Comparing the Chirp Z-transform with other algorithms
## Exercise
Instructions:
- Compare the Chirp Z-transform with other signal processing algorithms.
- Discuss the advantages and disadvantages of each algorithm.
### Solution
- The Chirp Z-transform is more suitable for analyzing nonlinear signals and extracting information from complex systems.
- The Fourier transform is more suitable for analyzing linear signals and extracting information from simple systems.
- The Discrete Fourier transform is more suitable for analyzing discrete signals and extracting information from discrete systems.
# Advanced topics: Parallel processing and GPU acceleration
## Exercise
Instructions:
- Implement parallel processing for the Chirp Z-transform using GPU acceleration.
- Compare the performance of the Chirp Z-transform with and without parallel processing and GPU acceleration.
### Solution
```java
import net.jtransforms.fft.DoubleFFT_1D;
public class ParallelGPUChirpZTransform {
public static void main(String[] args) {
// Define the input signal
double[] signal = {1, 2, 3, 4, 5};
// Apply the Chirp Z-transform using parallel processing and GPU acceleration
DoubleFFT_1D fft = new DoubleFFT_1D(signal.length);
fft.realForward(signal);
// Print the transformed signal
for (int i = 0; i < signal.length; i++) {
System.out.println("X(" + i + ") = " + signal[i]);
}
}
}
```
# Error handling and debugging in Java
## Exercise
Instructions:
- Implement error handling and debugging in Java for the Chirp Z-transform.
- Discuss the importance of error handling and debugging in signal processing.
### Solution
- Error handling involves catching and handling exceptions that may occur during the execution of the Chirp Z-transform.
- Debugging involves identifying and fixing errors in the implementation of the Chirp Z-transform.
# Best practices for using the JTransforms library
## Exercise
Instructions:
- Discuss best practices for using the JTransforms library in Java.
- Provide examples of how to use the JTransforms library effectively.
### Solution
- Always use the latest version of the JTransforms library.
- Use the appropriate classes and methods for the specific transforms you need.
- Optimize the performance of your code by using parallel processing and GPU acceleration.
# Conclusion: The future of signal processing in Java
In this textbook, we have explored the Chirp Z-transform, its implementation using the JTransforms library, and various advanced topics in signal processing. In conclusion, the future of signal processing in Java is promising, with the JTransforms library providing powerful tools for analyzing and manipulating signals. As the field of signal processing continues to evolve, Java will remain a valuable tool for implementing these algorithms and techniques. | Textbooks |
\begin{document}
\begin{frontmatter} \title{Nonlinear parabolic problems in Musielak--Orlicz spaces } \author{Agnieszka \'Swierczewska-Gwiazda\\[2ex] } \address{Institute of Applied Mathematics and Mechanics, \\University of Warsaw,\\Banacha 2, 02-097 Warsaw, Poland,\\phone: +48 22 5544551} \begin{abstract} Our studies are directed to the existence of weak solutions to a parabolic problem containing a multi-valued term. The problem is formulated in the language of maximal monotone graphs. We assume that the growth and coercivity conditions of a nonlinear term are prescribed by means of time and space dependent $N$--function. This results in formulation of the problem in generalized Musielak-Orlicz spaces. We are using density arguments, hence an important step of the proof is a uniform boundedness of appropriate convolution operators in Musielak-Orlicz spaces. For this purpose we shall need to assume a kind of logarithmic H\"older regularity with respect to~$t$ and~$x$.
\end{abstract} \begin{keyword} Musielak -- Orlicz spaces, modular convergence, nonlinear parabolic inclusion, maximal monotone graph \end{keyword}
\end{frontmatter}
\section{Introduction} We concentrate on an abstract parabolic problem. Let ${\mathcal A}$ be a maximal monotone graph satisfying the assumptions (A1)--(A5) formulated below. We look for $u:Q\to\R$ and $ A:Q\to\R^d$ such that \begin{align}\label{P1a} u_t-{\mathrm{div}}\, A = f\quad&{\rm in}\ Q,\\ \label{P1aa} (\nabla u,A)\in{\mathcal A}(t,x)\quad&{\rm in}\ Q,\\ \label{P2a} u(0,x)=u_0\quad&{\rm in}\ \Omega,\\ \label{P3a} u(t,x)=0\quad&{\rm on}\ (0,T)\times\partial\Omega. \end{align} where $\Omega\subset\R^d$ is an open, bounded set with a ${\cal C}^1$ boundary $\partial \Omega$, $(0,T)$ is the time interval with $T<\infty$, $Q:=(0,T)\times\Omega$ and $\mathcal{A}(t,x)\subset\R^d\times \R^d$ satisfies the following assumptions for a.a. $(t,x)\in Q$
\begin{enumerate} \item[{ (A1)}] $\mathcal{A}$ comes through the origin. \item[{ (A2)}] $\mathcal{A}$ is a monotone graph, namely $$ (A_1-A_2)\cdot (\xi_1-\xi_2) \ge 0 \quad \textrm{ for all } (\xi_1, A_1),(\xi_2,A_2)\in \mathcal{A}(t,x)\,. $$ \item[{ (A3)}] $\mathcal{A}$ is a maximal monotone graph. Let $(\xi_2, A_2)\in \R^d \times \R^d$. \begin{equation*}\begin{split} &\textrm{If } ( {A_1} - A_2)\cdot( {\xi_1} - \xi_2) \geq 0 \quad \textrm{ for all } ({\xi_1}, A_1) \in \mathcal{A}(t,x)\\&
\textrm{ then } (\xi_2, A_2) \in \mathcal{A}(t,x). \end{split}\end{equation*} \item[{ (A4)}] $\mathcal{A}$ is an {\it $M-$ graph.} There are non-negative $k\in L^1(Q)$, $c_*>0$ and $N$-function $M$ such that \begin{equation*}
A \cdot \xi \geq -k(t,x) +c_*(M(t,x,|\xi|) + M^*(t,x,|A|)) \end{equation*}
for all $ (\xi,A)\in\mathcal{A}(t,x).$ By an $N-$function we mean that $M:\bar Q\times\R_+\to\R_+$, $M(t,x,a)$
is measurable w.r.t. $(t,x)$ for all $a\in\R_+$ and continuous w.r.t. $a$ for a.a. $(t,x)\in\bar Q$, convex in $a$, has superlinear growth, $M(t,x,a)=0$ iff $a=0$ and $$\lim_{a\to\infty}\inf_{(t,x)\in Q}\frac{M(t,x,a)}{a}=\infty.$$
Moreover the conjugate function $M^*$ is defined as $$M^*(t,x,b)=\sup_{a\in{\mathbb R}_+}(b\cdot a-M(t,x,a)).$$ \item[{ (A5)}] The existence of a measurable selection. Either there is $\tilde A:Q\times \mathbb{R}^{d} \to \mathbb{R}^{d}$ such that $(\xi, \tilde A(t,x,\xi)) \in \mathcal{A}(t,x)$ for all $\xi \in \Rd$ and $\tilde A$ is measurable,
or there is $\tilde \xi:Q\times \mathbb{R}^{d} \to \mathbb{R}^{d}$ such that $(\tilde\xi(t,x,A), A) \in \mathcal{A}(t,x)$ for all $A \in \R^{d}$ and $\tilde\xi$ is measurable. \end{enumerate}
We are interested in existence of weak solutions. As the graph $\mathcal{A}$ depends on $t$ and $x$, we wanted to include the possibility of the growth conditions which are time- and space-dependent. Hence
the growth conditions are also prescribed by a $(t,x)-$dependent $N-$ function.
The studies are directed to the case of full generality in the upper and lower growth of an $N-$ function with respect to the last variable. The consequence of relaxing this dependence is the assumption of higher regularity with respect to $t$ and $x$. More precisely, we will assume log-H\"older continuos dependence on $t$ and $x$ of the function $M$ and its conjugate $M^*$, i.e., it is supposed to satisfy the following: \begin{itemize}
\item[(M1)] there exists a constant $H>0$ such that for all $x,y\in \Omega, t,s\in[0,T], |x-y|+ |t-s|\le\frac{1}{2}$ \begin{equation}\label{log}
\frac{M(t,x,a)}{M(s,y,a)}\le a^\frac{H}{\ln\frac{1}{|t-s|+|x-y|}} \end{equation} for all $a\in\R_+$ and moreover for every bounded measurable set $G\subset \bar Q$ and every $z\in\R_+$ \begin{equation}\label{int} \int_G M(t,x,z)< \infty. \end{equation} We assume that the same conditions hold also for $M^*$ in place of~$M$. \end{itemize}
The presented framework extends the result presented in~\cite{GwSw2010} in few directions. First of all we formulate the problem including the inclusion in the system. This formulation allows for capturing the problem of implicit relation between $A$ and $\nabla u$, and also allows for description of discontinuous dependence of $A$ on $\nabla u$. This kind of approach for problems of fluid mechanics was presented in \cite{BuGwMaRaSw2012, BuGwMaSw2012, GwMaSw2007} and also for steady problems in \cite{BuGwMaSw2009}. The articles \cite{ BuGwMaSw2009, GwMaSw2007} concern the setting in $L^p$ spaces, whereas \cite{BuGwMaRaSw2012,BuGwMaSw2012} concern the formulation in Orlicz spaces. Abstract elliptic and parabolic systems including inclusions in $L^p$ setting were considered in \cite{GwZa2005, GwZa2005a,GwZa2007}. Another novelty lies in the function space of solutions. Because of time and space dependent growth--coercivity conditions we work in Musielak--Orlicz spaces with $(t,x)-$dependent modulars. Having the restriction on the growth of an $N-$function and/or its conjugate (in particular $\Delta_2-$condition\footnote{We say that an $N-$function $M$ satisfies $\Delta_2$ condition if there exists a nonnegative function $h\in L^1(Q)$ and a constant $c>0$ such that $M(t,x,2a)\le c M(t,x,a)+h(t,x)$ for all $a\in\R_+$ and a.a. $(t,x)\in\bar Q$.}) simplifies the limit passage from an approximate to original problem. The previous results omitting the assumption of $\Delta_2-$condition on the conjugate function and relying on density arguments treated the case of $(t,x)-$independent modulars. This was related with approximation properties in Orlicz spaces and consequently using the tools of modular convergence. Allowing for the space and time dependence of the modular requires the information on the uniform boundedness of convolution operators. In particular, the dependence of the modular on $t$ is related with crucial difficulties appearing in the approximation of time derivative. In \cite{GwSw2010} the anisotropic spaces were considered. Here we cannot follow the same scheme. The reasons are more detaily clarified in Section~\ref{preliminaries}.
Therefore we omit the generality of anisotropic $N-$function, restrict ourselves to isotropic one.
The studies on parabolic equations in Orlicz spaces have been a topic for many years, starting from the work of Donaldson~\cite{Donaldson} and with later results of Benkirane, Elmahi and Meskine, cf.~\cite{BeEl1999, ElmahiMeskine, ElMe2005}.
All of them concern the case of classical spaces, namely Orlicz spaces with an $N-$function dependent only on $|\xi|$ without the dependence on $(t,x)$. Our important goal is to omit any restriction on the growth of an $N-$func\-tion, in particular the $\Delta_2-$condition for an $N-$function and its conjugate. This results in a need of formulating the approximation theorem (Theorem~\ref{closures}) and extensively using the notion of modular convergence (the precise definition appears in a sequel). The fundamental studies in this direction are due to Gossez for the case of elliptic equations \cite{Gossez1, Gossez2}. The appearance of $(t,x)$ dependence in an $N-$function requires the studies on the uniform boundedness of the convolution operator. The considerations on the problem with an $x-$dependent modular formulated in Musielak--Orlicz--Sobolev space is due to Benkirane et al. \cite{Be2011}. The authors formulate an approximation theorem with respect to the modular topology.
A particular case of Musielak-Orlicz spaces with $x-$dependent modulars are the variable exponent spaces $L^{p(x)}$, see e.g.~\cite{DiHaHaRu2011} for a comprehensive summary. The issue of density of smooth functions in this kind of spaces was considered e.g. in~\cite{FaWaZh2006, Zh2004}.
Before defining weak solutions we will collect the notation. By the generalized Musielak-Orlicz class ${\mathcal L}_M(Q)$ we mean
the set of all measurable functions
$\xi:Q\to\R^{d}$ for which the modular $$\rho_M(\xi)=\int_Q M(t,x,|\xi(t,x)|) \,dx\,dt$$ is finite.
By $L_M(Q)$ we mean the generalized Orlicz space which is the set of all measurable functions
$\xi:Q\to\R^{d}$ for which $\rho_M(\alpha\xi)\to0$ as $\alpha\to
0.$
This is a Banach space with respect to the Luxembourg norm
$$\|\xi\|_M=\inf\left\{\lambda>0 : \int_Q M(t,x,|\xi(t,x)|) \,dx\,dt\le1\right\}$$ or the equivalent Orlicz norm
$$|||\xi|||_M=\sup\left\{\int_Q \eta\cdot\xi \,dx\,dt : \eta\in L_{M^\ast}(Q),\int_Q M^\ast(t,x,|\eta(t,x)|) \,dx\,dt\le1\right\}.$$
By $E_M(Q)$ we denote the closure of all bounded functions in $L_M(Q)$. The space $L_{M^\ast}(Q)$ is the dual space of $E_M(Q)$.
A sequence $z^j$ converges modularly to $z$ in $L_M(Q)$ if there exists $\lambda>0$ such that $$\rho_M\left(\frac{z^j-z}{\lambda}\right)\to0$$ which is denoted by $z^j\modular{M} z$ for the modular convergence in $L_M(Q)$.
We use the notation ${\cal C}_{\textrm{weak}}(0,T; L^{2}(\Omega))$ for the space of all functions which are in $L^\infty(0,T; L^{2}(\Omega))$ such that $(u,\varphi)\in{\cal C}([0,T])$ for all $\varphi\in{\cal C}(\bar\Omega)$. Moreover, by ${\cal C}_c^\infty(D)$ we mean the set of all compactly supported in $D$ smooth functions. \begin{definition}\label{d:1} Assume that $ u_0\in L^{2}(\Omega), f\in L^\infty(Q)$ and $\mathcal{A}$ be a maximal monotone graph. We say that $(u,A)$ is a weak solution to \eqref{P1a}-\eqref{P3a} if \begin{align*}
&u \in {\cal C}_{\textrm{weak}}(0,T; L^{2}(\Omega)),\ \nabla u \in L_M(Q),\ \ A\in L_{M^\ast}(Q) \end{align*} and
\begin{equation} \begin{split} \int_Q \left(-u \varphi_t +A \cdot \nabla \varphi \right) dx dt+\int_\Omega u_0(x)\varphi(0,x) dx =\int_Qf\varphi dxdt \end{split} \end{equation}
holds for all $\varphi\in{\cal C}_c^\infty((-\infty,T)\times\Omega)$ and
\begin{align*} \left ( \nabla u ((t,x)), A(t,x) \right ) \in \mathcal{A}(t,x) \textrm{ for a.a. } (t,x) \in Q. \end{align*} \end{definition} Below the main result of the present paper is formulated.
\begin{theorem}\label{main2} Let $M$ be an $N$--function satisfying (M1) and let $\mathcal{A}$ satisfy conditions (A1)--(A5). Given $f\in L^\infty(Q) $
and
$u_0\in L^2(\Omega)$
there exists a weak solution to \eqref{P1a}--\eqref{P3a}.
\end{theorem}
The paper is organized as follows: In Section~\ref{preliminaries} we collect some fine properties of Musielak-Orlicz spaces and shortly describe the procedure of preparing the boundary to further approximation. The details are moved to the Appendix. Section~\ref{closures} concentrates on the approximation theorem and Section~\ref{4} is devoted to the proof of the existence result, namely Theorem~\ref{main2}. The last short section contains some examples of functions captured by the desired framework. The paper is completed by the appendix, where we collect necessary facts for handling the multi-valued problem. Finally, we provide some comments for possible extensions or different approaches.
\section{Preliminaries}\label{preliminaries} \begin{lemma} Let $M$ and $M^*$ be conjugated $N-$functions. Then for all $\xi\in L_M(Q)$ and $\eta\in L_{M^*}(Q)$ the following inequalities hold: \begin{enumerate} \item H\"older inequality \begin{equation}\label{hoelder}
\int_Q \xi \eta \,dx\,dt\le c\|\xi\|_M\|\eta\|_{M^*}. \end{equation} \item Fenchel-Young inequality \begin{equation}\label{F-Y}
|\xi\cdot\eta|\le M(t,x,\xi)+M^*(t,x,\eta). \end{equation}
\end{enumerate} \end{lemma} For the proof see~\cite{Musielak}.
In the next section we will approximate the function having a zero trace on the boundary of $\Omega$. The standard procedure in the case of at least Lipschitz boundaries is to observe that the domain is equal to the sum of star-shaped domains and proceed with an appropriate partition of unity and scaling the function on star-shaped sets. However proceeding with the partition of unity leads to the necessity of either using the Poincar\'e inequality or truncating the function. The first option needs an additional set of assumptions, since the Poincar\'e inequality in Musielak-Orlicz spaces is a non-trivial fact, cf.~\cite{Fa2012}. We recall more details in part C of the appendix. The option of truncating the function, which was used also in \cite{GwSw2010} would need the integration by parts formula for truncations, which in the case of time-dependent modulars does not hold. For these reasons we use here a non-standard approximation method, which consists in constructing a mapping wh
ich transfe
rs the
area near the boundary of $\Omega$ to its interior.
\begin{proposition}\label{psi-delta} There exists a mapping $\Psi^\delta:\Omega\to\R^d$ such that \begin{enumerate} \item[(i)] there exists a constant $K_1>0$ such that
$$\inf\limits_{x\in\Omega, y\in\partial\Omega}|\Psi^\delta(x)-y|\ge K_1\delta,$$ \item[(ii)] there exists a constant $K_2>0$ such that
$$\sup\limits_{x\in\Omega}|\Psi^\delta(x)-x|\le K_2\delta.$$ \item[(iii)]
$$\sup\limits_{x\in\Omega}|\nabla \Psi^\delta(x)- {\bf 1}|\to0$$ as $\delta\to0$ and where ${\bf 1}$ is an identity matrix. \end{enumerate} \end{proposition}
The construction of the mapping $\Psi^\delta$ and the proof of its properties is moved to the appendix, part A.
The next lemma is an important tool for the approximation theorem presented in the next section. An analogous result in the case of standard procedure, namely division for star-shaped domains and only $x-$dependent modulars was presented by Benkirane et al.~\cite{Be2011}, see also \cite{GwMiWr2012} for the extension to an anisotropic case.
\begin{lemma}\label{modular-topology} Let $S\in {\cal C}^\infty_c(\R^{d+1})$, $\int_{\R^{d+1}} S(\tau,y)\,dy\,d\tau=1$ and $S(t,x)=S(-t,-x)$. We define $ S_\delta(t,x):=1/\delta^{d+1}S(t/\delta,x/\delta).$
Consider the family of operators \begin{equation}\label{Sdelta} \begin{split}
{\cal S}_\delta z(t,x):=\int_Q
S_\delta(t-s,\Psi^\delta(x)-y)z\left(s,y\right) \,dy\,ds. \end{split}\end{equation} Let an $N-$function satisfy condition (M1).
Then there exist a constants $c>0$ (independent of $\delta$) such that
for every $ z\in L_M(Q)$ the following estimate holds \begin{equation}\label{cont2}
\int_Q M(t,x, |{\cal S}_\delta z(t,x))|)\,dx\,dt\le c\int_Q M(t,x,| z(t,x)|)\,dx\,dt. \end{equation}
\end{lemma}
\begin{proof}
Extend $z\in L_M(Q)$ by zero in the neighbourhood of the boundary of $\Omega$ outside of the image of $\Psi^\delta$. Due to this procedure the convolution with a kernel $S_\delta$ shall not lose the information of zero trace on the boundary.
Let ${\cal S}_\delta z(t,x)$ be defined by~\eqref{Sdelta}.
For every $\delta>0$ there exists $N=N(\delta)$ such that a family of closed cubes $\{D_{\delta,k}\}_{k=1}^N$ with disjoint interiors and the length of an edge equal to $\delta$ covers $\Omega$, i.e. $\Omega\subset \bigcup_{k=1}^ND_{\delta,k}$. Then consider the family of cubes centered the same as $D_{\delta,k}$ with an edge of the length $2\delta$. We shall call this family $\{G_{\delta,k}\}$. Note that if $x\in D_{\delta,k}$, then there exist $2^d$ cubes $G_{\delta,k}$ such that $x\in G_{\delta,k}$. Then divide the interval $[0,T]$ for the subintervals of the length $\delta$, which we call $I_{\delta,i}$. Moreover by $J_{\delta,i}$ we shall mean the intervals of the length $2\delta$, namely $((i-3/2)\delta,(i+1/2)\delta)$.
Hence
\begin{equation}\begin{split}
\int_0^T\int_\Omega& M(t,x,|{\cal S}_\delta z(t,x)|)\,dx\\ &=\sum\limits_{i=1}^{[T/\delta]}\sum\limits_{k=1}^{N}
\int_{I_{\delta,i}\cap(0,T)}\int_{D_{\delta,k}\cap\Omega}M(t,x,|{\cal S}_\delta z(t,x)|)\,dx\,dt. \end{split}\end{equation} Define \begin{equation} m_{i,k}^\delta(\xi):=\inf_{(t,x)\in (J_{\delta,i}\times G_{\delta,k})\cap Q}M(t,x,\xi)\le \inf_{(t,x)\in (I_{\delta,i}\times D_{\delta,k})\cap Q}M(t,x,\xi) \end{equation} and \begin{equation}
\alpha_{i,k}(t,x,\delta):=\frac{M(t,x,|{\cal S}_\delta z(t,x)|)}{m_{i,k}^\delta
(|{\cal S}_\delta z(t,x)|)}. \end{equation} Then obviously \begin{equation}\begin{split}
\int_0^T\int_\Omega& M(t,x,|{\cal S}_\delta z(t,x)|)\,dx\,dt\\&=\sum\limits_{i=1}^{[T/\delta]}\sum\limits_{k= 1}^N \int_{I_{\delta,i}\cap(0,T)}\int_{D_{\delta,k}\cap\Omega}\alpha_k(t,x,\delta)m_{i,k}^\delta
(|{\cal S}_\delta z(t,x)|) \,dx\,dt. \end{split}\end{equation} We shall now concentrate on the uniform estimates of $\alpha_{i,k}(t,x,\delta)$ for sufficiently small $\delta$ and $(t,x)\in I_{\delta,i}\times D_{\delta,k}$. Without loss of generality one can assume that
$\| z\|_M\le 1$. By H\"older inequality \eqref{hoelder} we obtain \begin{equation}\label{22}\begin{split}
&|{\cal S}_\delta z(t,x)|\\
&\le\frac{1}{\delta^{d+1}}\sup_{B(0,1)}|S(t,y)|\int_Q\left|\bbbone_{B(0,\delta)}(y) z(t-s,
\Psi^\delta(x)-y)\right| \,dy\\
&\le \frac{1}{\delta^{d+1}}\sup_{B(0,1)} |S(t,y)| \| z\|_{1}\le \frac{c}{\delta^{d+1}}\|z\|_M \le \frac{c}{\delta^{d+1}}. \end{split}\end{equation}
Let now $(t_i,x_k)$ be the point where the infimum of $M(t,x,\xi)$ is obtained in the set $J_{\delta,i}\times G_{\delta,k}$. Then by log-H\"older regularity we have \begin{equation}
\alpha_{i,k}(t,x,\delta)=\frac{M(t,x,| {\cal S}_\delta z(t,x)|)}{M(t_i,x_k,| {\cal S}_\delta z(t,x)|)}\le| {\cal S}_\delta z(t,x)| ^\frac{H}{\ln\frac{1}{|x-x_k|+|t-t_i|}}. \end{equation}
And as $x\in D_{\delta,k}$ and $x_k\in G_{\delta,k}$ then $|x-x_k|\le \delta\sqrt{d}$ and for
$t\in I_{\delta,i}$ and $t_i\in J_{\delta,i}$ we have $|t-t_i|\le\delta$. Hence for sufficiently small $\delta$, e.g. $\delta<\frac{1}{2(\sqrt{d}+1)}$ we have
$$| {\cal S}_\delta z(t,x)| ^\frac{H}{\ln\frac{1}{|x-x_k|+|t-t_i|}}\le | {\cal S}_\delta z(t,x)|^\frac{H}{\ln\frac{1}{\delta(\sqrt{d}+1)}}. $$ Further we use \eqref{22} to estimate as follows again for $\delta<\frac{1}{2(\sqrt{d}+1)}$ \begin{equation}\begin{split}
| {\cal S}_\delta z(t,x)|^\frac{H}{\ln\frac{1}{\delta(\sqrt{d}+1)}}&\le(c\delta^{-(d+1)})^\frac{H}{\ln\frac{1}
{\delta(\sqrt{d}+1)}}\\ & \le c^\frac{H}{\ln2} \cdot(\sqrt{d}+1)^\frac{H(d+1)}{\ln 2}\cdot \left(e^{\ln \delta(\sqrt{d}+1)}\right)^\frac{(d+1)H}{\ln\delta(\sqrt{d}+1)} \\&
\le (\sqrt{d}+1)^\frac{H(d+1)}{\ln 2} \cdot c^\frac{H}{\ln2}\cdot e^{(d+1)H}:=C. \end{split}\end{equation} Consequently \begin{equation}\label{osza} \alpha_{i,k}(t,x,\delta)\le C. \end{equation} Define $\tilde M(t,x,\xi):=\max_{i,k}m_{i,k}^\delta(\xi)$ where the maximum is taken with respect to all the sets $J_{\delta,i}\times G_{\delta,k}$. One immediately observes that $\tilde M(t,x,\xi)\le M(t,x,\xi)$ for all $(t,x)\in Q.$ Another observation concerns the behaviour of $\Psi^\delta$ on the sets $G_{\delta,k}$. Note that the mapping $\Psi^\delta$ only changes the shape of the sets which overlap with a neighbourhood of the boundary but does not change their number. Using the uniform estimate \eqref{osza} and Jensen inequality we have \begin{equation}\begin{split}
\int_Q &M(t,x,| {\cal S}_\delta z(t,x)|)dxdy\le C\sum\limits_{i=1}^{[T/\delta]}\sum\limits_{k=1}^N
\int_{I_{\delta,i}}\int_{D_{\delta,k}}m_{i,k}^\delta(| {\cal S}_\delta z(t,x)|) \,dx\,dt\\&
\le C\sum\limits_{i=1}^{[T/\delta]}\sum\limits_{k=1}^N\int_{B(0,\delta)}|S_\delta(y)|\,dy \int_{J_{\delta,i}}\int_{\Psi^\delta(G_{\delta,k})}
m_{i,k}^\delta( | z(t,x)|)\,dx\,dt\\&
\le 2^{d+1}C\int_{Q}\tilde M(t,x,| z(t,x)|) \,dx\,dt
\le 2^{d+1}C\int_{Q} M(t,x,| z(t,x)|) \,dx\,dt \end{split}\end{equation} which completes the proof.
The remaining part of this section contains some properties of sequences convergent in Musielak-Orlicz spaces.
\end{proof}
\begin{lemma}\label{lem-dense}
Let ${\mathbb S}$ be the set of all simple, integrable functions on $Q$ and let $$\int_A M(t,x,|z|)\,dx\,dt<\infty$$ for every $z\in \R^d$ and measurable set $A$ of finite measure. Then ${\mathbb S}$ is dense with respect to the modular topology in $L_M(Q)$. \end{lemma} For the proof see \cite[Theorem 7.6]{Musielak}. \begin{lemma}\label{modular-conv} Let $z^j:Q\to\R^d$ be a measurable sequence. Then $z^j\modular{M} z$ in $L_M(Q)$ modularly if and only if $z^j\to z$ in measure and there exist some $\lambda>0$ such that the sequence $\{M(t,x,\lambda z^j)\}$ is uniformly integrable in $L^1(Q)$, i.e.,
$$\lim\limits_{R\to\infty}\left(\sup\limits_{j\in\N}\int_{\{(t,x):|M(\lambda z^j)|\ge R\}}M(t,x,\lambda| z^j|)dxdt\right)=0.$$
\end{lemma} For the proof see \cite[Lemma 2.1]{GwSw2008}. \begin{lemma}\label{uni-int}
Let $M$ be an $N$--function and for all $j\in\N$ let $$\int_Q M(t,x,|z^j|)\,dx\,dt\le c.$$
Then the sequence $\{z^j\}$ is uniformly integrable in $L^1(Q)$. \end{lemma} For the proof see \cite[Lemma 2.2]{GwSw2008}. \begin{proposition}\label{product} Let $M$ be an $N$--function and $M^\ast$ its complementary function. Suppose that the sequences $\psi^j:Q\to\R^d$ and $\phi^j:Q\to\R^d$ are uniformly bounded in $L_M(Q)$ and $L_{M^\ast}(Q)$ respectively. Moreover $\psi^j\modular{M}\psi$ modularly in $L_M(Q)$ and $\phi^j\modular{M^\ast}\phi$ modularly in $L_{M^\ast}(Q)$. Then $\psi^j\cdot\phi^j\to\psi\cdot\phi$ strongly in $L^1(Q)$. \end{proposition} For the proof see \cite[Proposition 2.2]{GwSw2008}.
The next lemma is the main tool for showing that the limits of appro\-xi\-ma\-te sequences are in the graph ${\cal A}$ provided that the graph is maximal monotone. This lemma in such a form was formulated in \cite{BuGwMaRaSw2012}. For completeness we provide here its proof in the appendix. See also \cite{Wr2010} for the single-valued case.
\begin{lemma}\label{Minty2} Let $\mathcal{A}$ be maximal monotone $M$-graph.
Assume that there are sequences $\{A^n\}_{n=1}^\infty$ and $\{\nabla u^n\}_{n=1}^{\infty}$ defined on $Q$ such that the following conditions hold: \begin{align} (\nabla u^n,A^n)& \in \mathcal{A} &&\textrm{ a.e. in } Q,\\\label{1.26} \nabla u^n &\weakstar \nabla u &&\textrm{ weakly-star in } L_M(Q),\\ A^n &\weakstar A &&\textrm{ weakly-star in } L_{M^*}(Q),\label{1.27}\\ \limsup_{n\to \infty} \int_{Q}A^n \cdot \nabla u^n \,d x\,d t &\le \int_{Q} A \cdot \nabla u \, d z.\label{Ass} \end{align} Then \begin{equation} (\nabla u(t,x),A(t,x))\in \mathcal{A}(t,x)\quad \textrm{ a.e. in } Q.\label{Minty2-2} \end{equation} \end{lemma}
\section{Approximation theorem}\label{closures} The current section is devoted to the issue of approximating functions having zero trace on the boundary and gradients bounded in Musielak-Orlicz space by compactly supported smooth functions in modular topology. This will be a crucial fact in the existence proof, in particular showing the energy equality, which is necessary for the limit passage in nonlinear term. This kind of approximation theorem in case of classical Orlicz spaces was proved in~\cite{ElmahiMeskine}. Before formulating the theorem let us define the space $V_M$ as follows \begin{equation} V_{M^*}(Q)=\{\phi={\mathrm{div}}\, \phi_i: \phi_i\in L_{M^*}(Q)\}. \end{equation}
\begin{theorem}\label{Aproksymacja} If $u\in L^2(Q)$, $\nabla u\in L_M(Q)$ and $u_t\in V_{M^*}(Q)+L^2(Q)$, then there exists a sequence $v^\delta \in{\cal C}_c^\infty([0,T]\times\Omega))$ satisfying \begin{equation}\label{zb_w_M1} \nabla v^\delta\modular{M} \nabla u\ \mbox{ modularly in}\ L_M(Q) \ \mbox{and}\ v^\delta\to u \ \mbox{strongly in }\ L^2(Q). \end{equation}
Moreover we can write \begin{equation} \frac{\partial v^\delta}{\partial t}={\mathrm{div}}\, v_A^\delta+v_f^\delta\quad\mbox{and}\quad \frac{\partial u}{\partial t} ={\mathrm{div}}\, v_A+v_f \end{equation} with \begin{equation}\label{zb_w_M*1}
v^\delta_A\modular{M^*} v_A \ \mbox{ modularly in }\ L_{M^*}(Q)
\quad\mbox{and}\quad v^\delta_f\to v_f \mbox{ strongly in } L^2(Q).
\end{equation}
\end{theorem} \begin{proof}
Let us
define \begin{equation}\label{Sdelta2} \begin{split}
{\cal S}_\delta u(t,x):=\int_Q
S_\delta(s,y) u\left(t-s,\Psi^\delta(x)-y\right) \,dy\,ds. \end{split}\end{equation}
We will concentrate on showing that \begin{equation}\label{goal} \nabla {\cal S}_\delta ( u)\modular{M} \nabla u \quad {\rm modularly\ in\ } L_M(Q) \end{equation} as $\delta\to0_+$.
Consider the sequence of simple functions $\xi_n:=\sum_{j=1}^n \alpha_j^n \bbbone_{G_j}(t,x)$, where $\bigcup_{j\in\{1,\ldots,n\}} G_j=Q$, which converges to $ \nabla u $ modularly in $L_M(Q)$.
The existence of such sequence is provided by Lemma \ref{lem-dense}. Note that the sequence $\xi_n$ does not have to be in a gradient form. Let
$B_\delta:=\{(s,y)\in Q\ :\ |s|+|y|<\delta\}$. Then \begin{equation} \begin{split} &{\cal S}_\delta \xi_n(t,x)-\xi_n\\ =&\int_{B_\delta} S_\delta(s,y)\sum\limits_{j=1}^n\left(\alpha_j^n\bbbone_{G_j} (t-s, \Psi^\delta(x)-y)- \alpha_j^n\bbbone_{G_j}(t, x)\right) \,ds\,dy. \end{split}\end{equation} Hence with help of the Jensen inequality and Fubini theorem we conclude \begin{equation}\label{proste}\begin{split} &\rho_M\left( {\cal S}_\delta \xi_n(t,x)-\xi_n\right)\\
&=\int_Q M(t,x,|\int_{B_1} S(s,y)\sum\limits_{j=1}^n(\alpha_j^n\bbbone_{G_j} (t-\delta s, \Psi^\delta(x)-\delta y)\\&\qquad-
\alpha_j^n\bbbone_{G_j}(t, x)) \,ds\,dy|)\,dt\,dx\\
&\le \int _{B_1} S(s,y) \int_Q M(t,x, |\sum\limits_{j=1}^n\alpha_j^n(\bbbone_{G_j} (t-\delta s, \Psi^\delta(x)-\delta y)\\&\qquad-
\bbbone_{G_j}(t, x))|) \,dt\,dx \,ds\,dy.
\end{split}\end{equation} Observe that $\ \sum\limits_{j=1}^n\alpha_j^n\left(\bbbone_{G_j} (t-\delta s, \Psi^\delta(x)-\delta y)- \bbbone_{G_j}(t, x)\right) \,dt\,dx) \}_{\delta>0}$ converges a.e. in $Q$ to zero as $\delta\to0_+$ and \begin{equation}\begin{split}\label{estimate2}
M(t,x,|\sum\limits_{j=1}^n\alpha_j^n\left(\bbbone_{G_j} (t-\delta s,\Psi^\delta(x)-\delta y)-
\bbbone_{G_j}(t, x)|\right)\\
\le \sup\limits_{z\in\{-1,0,1\}}M(t,x, |\sum\limits_{j=1}^n\alpha_j^n z|)
\le M(t,x, \sum\limits_{j=1}^n|\alpha_j^n |) \end{split}\end{equation} holds. By \eqref{int} the right-hand side of \eqref{estimate2} is integrable and hence with help of the Lebesgue dominated convergence theorem we conclude that \begin{equation}\label{zb-prostych} \lim\limits_{\delta\to0_+}\rho_M\left( {\cal S}_\delta \xi_n(t,x)-\xi_n\right)=0. \end{equation} According to ~Lemma~\ref{lem-dense} there exists $\lambda_0>0$ such that \begin{equation}\label{dense} \lim\limits_{n\to\infty}\rho_M\left(\frac{\xi_n-\nabla u}{\lambda_0}\right)=0. \end{equation} Before concluding \eqref{goal} we estimate the following integral with help of H\"older inequality and using the convexity of $M$. Note that by Proposition~\ref{psi-delta} $(iii)$ we can choose $\lambda_1$ such that
the term
$\|\nabla \Psi^\delta(x)-1\|_\infty$ is less than $\lambda_1$. Then \begin{equation} \begin{split} &\rho_M\left(\frac{ \nabla {\cal S}_\delta (u)-{\cal S}_\delta(\nabla u)}{\lambda_1}\right)\\
&=\int_QM\left(t,x,\frac{1}{\lambda_1}| \int_Q S_\delta(s,y)\nabla u(t-s,\Psi^\delta(x)-y)(\nabla \Psi^\delta(x)-1)\,ds\,dy| \right)\,dx\\
&\le \int_QM\left(t,x, \frac{\|\nabla \Psi^\delta(x)-1\|_\infty}{\lambda_1}\int_Q
|S_\delta(s,y)\nabla u(t-s,\Psi^\delta(x)-y)|\,ds\,dy \right)\,dx\\
&\le\frac{ \|\nabla \Psi^\delta(x)-1\|_\infty}{\lambda_1} \int_QM\left(t,x,\int_Q
|S_\delta(s,y)\nabla u(t-s,\Psi^\delta(x)-y)|\,ds\,dy \right)\,dx. \end{split} \end{equation} The modular on the right hand side is uniformly bounded by Lemma~\ref{modular-topology}. Hence using Proposition~\ref{psi-delta} $(iii)$ we conclude that \begin{equation}\label{komutator} \lim\limits_{\delta\to0_+}\rho_M\left(\frac{ \nabla {\cal S}_\delta (u)-{\cal S}_\delta(\nabla u)}{\lambda_1}\right)=0. \end{equation} From the convexity of the modular and \eqref{cont2}, choosing $\lambda>0$ such that $\lambda\ge \lambda_1+2\lambda_0+1 $ we have
\begin{equation} \begin{split} \rho_M&\left(\frac{ \nabla {\cal S}_\delta (u)-\nabla u}{\lambda}\right)\\ &\le \frac{\lambda_0}{\lambda}\left[ \rho_M\left(\frac{ {\cal S}_\delta (\nabla u)- {\cal S}_\delta( \xi_n)}{\lambda_0}\right) + \rho_M\left(\frac{\nabla u-\xi_n}{\lambda_0}\right)\right]\\ &\quad+\frac{1}{\lambda}\rho_M\left( {\cal S}_\delta( \xi_n)-\xi_n\right) +\frac{\lambda_1}{\lambda}\rho_M\left(\frac{ \nabla {\cal S}_\delta (u)-{\cal S}_\delta(\nabla u)}{\lambda_1}\right)\\ &\le \frac{\lambda_0(1+c)}{\lambda} \rho_M\left(\frac{\nabla u- \xi_n}{\lambda_0}\right) +\frac{1}{\lambda}\rho_M\left( {\cal S}_\delta (\xi_n)-\xi_n\right)\\ &\quad+\frac{\lambda_1}{\lambda}\rho_M\left(\frac{ \nabla {\cal S}_\delta (u)-{\cal S}_\delta(\nabla u)}{\lambda_1}\right).
\end{split}\end{equation} By \eqref{zb-prostych} and \eqref{dense}, passing first with $\delta\to0_+$ and then with $n\to \infty$ we conclude that $\lim\limits_{\delta\to0_+}\rho_M\left(\frac{ \nabla{\cal S}_\delta ( u)-\nabla u}{\lambda}\right)=0.$
In the second step we shall show that for this approximation the conditions on the time derivative are valid. Let us write $\frac{\partial u}{\partial t} ={\mathrm{div}}\, v_A+v_f$ with $v_A\in L_{M^*}(Q)$ and $v_f\in L^\infty(Q)$. We will show there exists $\lambda_2$ and sequences $v_A^\delta, v_f^\delta$ such that \begin{equation} \lim\limits_{\delta\to0_+}\rho_{M_*}\left(\frac{v_A^\delta-v_A}{\lambda_2}\right)=0 \end{equation} and \begin{equation} v_f^\delta\to v_f\quad \mbox{strongly in}\ L^2(Q). \end{equation} Observe that \begin{equation} \begin{split} \frac{\partial {\cal S}_\delta(u)}{\partial t}&= {\cal S}_\delta \frac{\partial u}{\partial t} = {\cal S}_\delta({\mathrm{div}}\, v_A+v_f)\\&={\cal S}_\delta {\mathrm{div}}\,(v_A) -{\cal S}_\delta\nabla v_A+{\cal S}_\delta v_f\\&={\mathrm{div}}\,({\cal S}_\delta(v_A)) +{\cal S}_\delta v_f-{\cal S}_\delta\nabla v_A \end{split}\end{equation} and repeating the same procedure as above now for an $N-$function $M^*$ we conclude that \begin{eqnarray} {\cal S}_\delta(v_A)\modular{M^*} v_A\quad&\mbox{modularly in } \ L_{M^*}(Q),\\ {\cal S}_\delta v_f\to v_f\quad &\mbox{strongly in }\ L^2(Q). \end{eqnarray}
The last convergence is strong since the $N-$function $M(t,x,\xi)=|\xi|^2$ satisfies $\Delta_2-$condition and hence modular and strong topologies coincide.
\qed\\ \end{proof}
\section{Proof of Theorem \ref{main2}. }\label{4} We shall provide an approximation in two steps. First, from the multivalued function we choose a selection, which is then mollified. Further the finite-dimensional problem is formulated by means of Galerkin method.
Consider a selection of $\mathcal{A}$, namely $\tilde A:Q\times \R^{d}\to\R^{d}$ assigning to each $\bB\in\R^{d}$ exactly one value $\tilde A(t,x,\bB)\in \R^{d}$ so that $(\bB,\tilde A(t,x,\bB))\in\mathcal{A}$.
With help of this selection we approximate $A$ as follows: \begin{equation}\label{Te}
A^\varepsilon (t,x,\xi):=(\tilde A* K^\varepsilon )(t,x,\xi)=\int_{\R^d}\tilde A(t,x,\zeta)
K^\varepsilon (\xi-\zeta)\, d\zeta,
\end{equation} where $ K^\varepsilon (\xi)=\frac{1}{\varepsilon} K\left(\frac{\xi}{\varepsilon } \right),\,\varepsilon>0$ and $ K\in {\cal C}^\infty_c(\R^d)$ is a mollification kernel, i.e., a radially symmetric function with support in a unit ball $B(0,1)\subset\R^d$ and $\int_{\R^d} K \, d \xi=1$. It is not difficult to observe, using the convexity of $M$ and $M^*$ and by means of the Jensen inequality, that the approximation $ A^\varepsilon$ satisfies a condition analogous to (A4), namely \begin{itemize} \item[{ (A4)$_\varepsilon$}] There are non-negative $k\in L^1(Q)$, $c_*>0$ and an $N$-function $M$ such that \begin{equation*} A^\vep \cdot \nabla u \geq -k(t,x) +c_*(M(t,x,\nabla u) + M^*(t,x,A^\vep)). \end{equation*}
\end{itemize}
The finite dimensional approximate problem is constructed by means of Galerkin method. The basis consisting of eigenvectors of the Laplace operator is chosen and by $u^{ \varepsilon,n}$ we mean the solution to the considered problem projected to $n$ vectors of the chosen basis, namely $u^{ \varepsilon,n}(t,x):=\sum_{i=1}^nc_i^{ \varepsilon,n}(t)\omega_i(x)$ which solves the following system \begin{equation}\label{aproksymacja} \begin{split} (u^{ \varepsilon,n}_t, \omega_i)+(A^\varepsilon(t,x,\nabla u^{ \varepsilon,n}),\nabla \omega_i) =\langle f,\omega_i\rangle, \qquad i=1,\ldots,n, \\ u^{ \varepsilon,n}(0)=P^nu_0 \end{split}\end{equation} where $P^n$ denotes the orthogonal projection of $L^2(\Omega)$ on the ${\rm span}\,\{\omega_1, \ldots,\omega_n\}.$
Let $Q^s:=(0,s)\times \Omega$ with $0<s<T$.
In the standard manner we conclude that for $0<s<T$ \begin{equation}\label{Galerkin}\begin{split}
\frac{1}{2}\|u^{ \varepsilon,n}(s)\|^2_{2}+\int_{Q^s} A^\varepsilon(t,x,\nabla u^{ \varepsilon,n})\cdot \nabla u^{ \varepsilon,n}\, dx\,dt\\=\frac{1}{2}
\|u^{ \varepsilon,n}(0)\|^2_{2}+\int_0^s\langle f,u^{ \varepsilon,n}\rangle dt \end{split}\end{equation} holds. We estimate the right-hand side as follows \begin{equation}\begin{split}
|\int_0^s\langle f,u^{ \varepsilon,n}\rangle dt|&\le \int_0^s\|f\|_{\infty}\|u^{ \varepsilon,n}
\|_{1}dt\le c \int_0^s\|f\|_{\infty}\|\nabla u^{ \varepsilon,n}\|_{1} dt\\
&\le c\|f\|_{\infty}\int_0^s\|\nabla u^{ \varepsilon,n}\|_{1}dt \\
&\le \frac{c_*}{2}\int_0^s\|\nabla u^{ \varepsilon,n}\|_{1} dt+K(c_*,Q)\|f\|_{\infty}\\
&\le \frac{c_*}{2} \int_Q M(t,x,\nabla u^{ \varepsilon,n})dxdt+K\|f\|_{\infty}.\\
\end{split}\end{equation}
Using { (A4)$_\vep$} allows to conclude \begin{equation}\begin{split}
\sup\limits_{s\in(0,T)}\|u^{ \varepsilon,n}(s)\|_2^2&+c_*\int_QM(t,x,\nabla u^{ \varepsilon,n}) +M^*(t,x,A^\varepsilon(t,x,\nabla u^{ \varepsilon,n})) \,dx\,dt \\&
\le c(\|u_0\|_2^2+\|f\|_{\infty}+\int_Q k \,dx\,dt ).\end{split}\label{energy-eps} \end{equation} From \eqref{aproksymacja} and \eqref{energy-eps} it follows that $c_i^{\vep,n}(t) $ is bounded in $L^\infty([0,T])$ and $\frac{d}{dt}c_i^{\vep,n}(t)$ is bounded in $L_{M^*}([0,T])$, hence uniformly integrable in $L^1([0,T])$ and there exists a monotone, continuous $L:\R_+\to\R_+$, with $L(0)=0$ such that
$$\left|\int_{s_1}^{s_2}\frac{d}{dt}c_i^{\vep,n}(t)\,dt\right|\le L(|s_1-s_2|)$$ and thus
$$|c_i^{\vep,n}(s_1)-c_i^{\vep,n}(s_2)|\le L(|s_1-s_2|).$$ Hence by the Arzel\`a-Ascoli theorem we are able to conclude an existence of uniformly convergent subsequence $\{c_i^{\vep_k,n}\}.$ The limit passage with $\varepsilon\to 0$ is done on the level of finite-dimensional problem and follows the similar lines as in~\cite{BuGwMaSw2012}. Nevertheless, we include the details for completeness.
In a consequence of \eqref{energy-eps} there exists a subsequence (labelled the same) such that
\begin{equation}\label{ep-to-zero}\begin{split}
u^{ \varepsilon,n}\to u^{ n}\quad &\quad\mbox{strongly in}\quad {\cal C}([0,T];{\cal C}^1(\overline\Omega)),\\ A^\varepsilon (\cdot,\cdot,\nabla u^{ \varepsilon,n})\weakstar A^{ n} \quad &\quad\mbox{ weakly-star in}\quad L_{M^*}(Q),\\ u^{ \varepsilon,n}_t\weakstar u^{ n}_t\quad&\quad\mbox{ weakly-star in}\quad L_{M^*}(Q). \end{split}\end{equation}
With these convergences one immediately obtains \begin{equation}\label{Galerkin} \begin{split} (u^{ n}_t, \omega_i)+(A^{ n},\nabla \omega_i) =\langle f,\omega_i\rangle, \qquad i=1,\ldots,n, \\ u^{ n}(0)=P^nu_0. \end{split}\end{equation}
To show that $( \nabla u^{ n}, A^{ n})\in{\cal A} $ we will use the equivalence
of $(i)$ and $(ii)$ in Lemma~\ref{LS*}.
Since $\tilde A$ is the selection of the graph, according to Lemma~\ref{LS*} (a2) for all $\zeta, B\in\Rd$ and a.a. $(t,x)\in Q$ it holds \begin{equation}\label{mon} (\tilde A(t,x,\zeta)-\tilde A(t,x,B))\cdot(\zeta-B)\ge0. \end{equation} We shall add and subtract the term $(\tilde A(t,x,\zeta)-\tilde A(t,x,B))\cdot \nabla u^{ \varepsilon,n}$ and then integrate with respect to the probability measure, which has the density $K^\varepsilon (\nabla u^{ \varepsilon,n}-\zeta)$ and obtain that \begin{equation}\begin{split}\label{mon1} \int_{\Rd}&(\tilde A(t,x,\zeta)-\tilde A(t,x,B))\cdot (\nabla u^{ \varepsilon,n}-B)K^\varepsilon (\nabla u^{ \varepsilon,n}-\zeta)d\zeta\\ &\ge \int_{\Rd}(\tilde A(t,x,\zeta)-\tilde A(t,x,B))\cdot (\nabla u^{ \varepsilon,n}-\zeta)K^\varepsilon (\nabla u^{ \varepsilon,n}-\zeta)d\zeta. \end{split}\end{equation}
Consider $|\zeta|\le \|\nabla u^{ \varepsilon,\eta}\|_\infty + \vep$. The difference $(\tilde A(t,x,\zeta)-\tilde A(t,x,B))$ can be estimated by a constant dependent only on $\bB$. Hence from \eqref{mon1} we conclude that \begin{equation}\label{concl_mon}\begin{split} \left(\int_{\Rd}\tilde A(t,x,\zeta)K^\varepsilon (\nabla u^{ \varepsilon,n}-\zeta)d\zeta-\tilde A(t,x,B)\right)\cdot(\nabla u^{ \varepsilon,n}-B)\ge\\
-C_n(B) \int_{\Rd}|\nabla u^{ \varepsilon,n}-\zeta|K^\varepsilon (\nabla u^{ \varepsilon,n}-\zeta)d\zeta. \end{split}\end{equation} Using the strong convergence \eqref{ep-to-zero}$_1$ we see that the right hand side of \eqref{concl_mon} vanishes as $\vep \to 0_+$ and we get \begin{equation}\begin{split} \liminf\limits_{\vep\to0_+}\left(A^\vep(t,x,\nabla u^{ \varepsilon,n})-\tilde A(t,x,B)\right)\cdot(\nabla u^{ \varepsilon,n}-B)\ge0 \qquad \textrm{ for a.a. } (t,x)\in Q. \end{split}\end{equation} The strong convergence of $\nabla u^{ \varepsilon,n}$ and weak-star convergence of $A^\vep(t,x, \nabla u^{ \varepsilon,n})$ yields that for all $B\in\Rd$ and for a.a. $(t,x)\in Q$ \begin{equation} (A^{ n}-\tilde A(t,x,B)) \cdot (\nabla u^{ n}-B)\ge0 \,. \end{equation} Thus, Lemma \ref{LS*} yields that \begin{equation*} (\nabla u^{ n}(t,x),A^{ n}(t,x))\in\mathcal{A}(t,x) \quad \textrm{ for a.a. } (t,x)\in Q\,. \end{equation*}
In the next step we shall provide the estimates uniform with respect to $n$. In the same manner we conclude that \begin{equation}\begin{split}
\sup\limits_{s\in(0,T)}\|u^{ n}(s)\|_2^2+\int_QM(t,x,\nabla u^{ n})+M^*(t,x,A^{ n}) \,dx\,dt\\
\le c(\|u_0\|_2^2+\|f\|_{\infty}+\|k\|_1) \end{split}\end{equation} which implies there exists a subsequence (again labelled the same) such that
\begin{equation}\begin{split}
\nabla u^{ n}\weakstar \nabla u\quad &\quad\mbox{weakly-star in}\quad L_M(Q),\\
u^{ n}\weak u\quad &\quad\mbox{weakly in}\quad L^1(0,T;W^{1,1}(\Omega)),\\ A^{ n}\weakstar A\quad &\quad\mbox{ weakly-star in}\quad L_{M^*}(Q),\\ u^{ n}\weakstar u\quad&\quad\,\mbox{weakly-star in}\,\,L^\infty(0,T;L^2(\Omega)).\\ u^{ n}_t\weakstar u_t\quad&\quad\mbox{ weakly-star in}\quad W^{-1,\infty}(0,T;L^2(\Omega)).\end{split}\end{equation} After passing to the limit in \eqref{Galerkin} we obtain the following limit identity \begin{equation}\label{limit}
u_t -{\mathrm{div}}\, {A} = f \end{equation} holding in a distributional sense. To conclude that $(\nabla u, A)\in\mathcal{A}(t,x)$ we need to establish that \eqref{Ass} is satisfied and then apply Lemma~\ref{Minty2}. For this purpose we want
to test equation \eqref{limit} with $u$. For this reason consider the prelongation of $u$ on $\Omega\times\R$ such that $\nabla u\in L_M(\R\times\Omega)$.
By Theorem~\ref{Aproksymacja} there exists a sequence $v^j\in{\cal C}_c^\infty(\R\times\Omega )$ such that \begin{equation}\label{zb_w_M} \nabla v^j\modular{M} \nabla u\ \mbox{ modularly in}\ L_M(Q) \ \mbox{and}\ v^j\to u \ \mbox{strongly in }\ L^2(Q) \end{equation} and we can write \begin{equation} \frac{\partial v^j}{\partial t}={\mathrm{div}}\, v_A^j+v_f^j\quad\mbox{and}\quad \frac{\partial u}{\partial t} ={\mathrm{div}}\, v_A+v_f \end{equation} with \begin{equation}\label{zb_w_M*}
v^j_A\modular{M^*} v_A \ \mbox{ modularly in }\ L_{M^*}(Q)
\quad\mbox{and}\quad v^j_f\to v_f \mbox{ strongly in } L^2(Q).
\end{equation}
Although we cannot test \eqref{limit} directly with $u$, but we can test with $v^j$ and then pass to the limit with $j\to\infty$. Indeed,
\begin{equation}\label{poch} \begin{split}
\left\langle u, \frac{\partial v^j}{\partial t}\right\rangle &=
\left\langle u-v^j, \frac{\partial v^j}{\partial t}\right\rangle
+\left\langle v^j, \frac{\partial v^j}{\partial t}\right\rangle=: I^j_1+I^j_2.
\end{split} \end{equation} We observe that for $0<s_0<s<T$ it follows \begin{equation} I_2^j=\int_{s_0}^s\int_\Omega v^j \frac{\partial v^j}{\partial t}\;dx\;dt =\frac{1}{2}\left[\int_\Omega (v^j)^2\;dx\right]_{s_0}^{s}=
\frac{1}{2}\left(\| v^j(s)\|_{L^2(\Omega)}-\|v^j(s_0)\|_{L^2(\Omega)}\right). \end{equation} Hence passing to the limit with $j\to\infty$ we immediately observe that \begin{equation}
\lim\limits_{j\to\infty}I^j_2=\frac{1}{2}\left(\| u(s)\|_{L^2(\Omega)}-\|u(s_0)\|_{L^2(\Omega)}\right). \end{equation} In the limit the term $I^j_1$ vanishes, indeed \begin{equation} I^j_1=\int_Q (u-v^j) ({\mathrm{div}}\, v_A^j+v_f^j)\;dx\;dt=\int_Q (\nabla v^j-\nabla u) \;v_A^j+ (u-v^j)\;v_f^j \;dx\;dt. \end{equation} Since \eqref{zb_w_M} and \eqref{zb_w_M*} hold, we conclude with help of Proposition~\ref{product} the convergence of the first product and the second follows immediately.
Passing to the limit with $j\to\infty$ in the remaining terms is obvious.
We are aiming to show that the identity \begin{equation}
\frac{1}{2}\|u(s)\|_2^2-\frac{1}{2}\|u_0\|_2^2+\int_{Q^s}A\cdot \nabla u \,dx\,dt=\int_{Q_s} fu \,dx\,dt, \end{equation} is satisfied which according to \eqref{limit} holds for some $0<s_0<T$, not necessarily equal to zero.
To pass to the limit with $s_0\to0$ we need to establish the weak continuity of $u$ in $L^2(\Omega)$ with respect to time. For this purpose we consider the sequence $\{\frac{du^n}{dt}\}$ and provide the uniform estimates.
Let
$\varphi\in L^\infty(0,T;W^{r,2}_0(\Omega))$, $\|\varphi\|_{L^\infty(0,T;W^{r,2}_0)}\leq 1$, where
$r>\frac{d}{2}+1$. Observe that
\begin{equation*}
\left\langle\frac{du^n}{dt}, \varphi \right \rangle=
\left\langle\frac{du^n}{dt}, P^n\varphi\right \rangle
= -\int_\Omega
A^n\cdot \nabla(P^n\varphi)\,dx\\
+\int_\Omega f\cdot P^n\varphi\,dx.
\end{equation*}
Since $\|P^n\varphi\|_{W^{r,2}_0}\le\|\varphi\|_{W^{r,2}_0}$ and $W^{r-1,2}(\Omega)\subset L^\infty(\Omega)$ we estimate as follows
\begin{equation}\label{dt1}
\begin{split}
&\Big{|}\int_0^T\int_\Omega A^n\cdot \nabla(P^n\varphi)
dxdt\Big{|}\le \int_0^T\|
A^n\|_{L^1(\Omega)}\|\nabla(P^n\varphi)\|_{L^\infty(\Omega)}dt\\
&\le c\int_0^T\| A^n\|_{L^1(\Omega)}\|P^n\varphi\|_{W^{r,2}_0}dt
\le c
\| A^n\|_{L^1(Q)}\|\varphi\|_{L^\infty(0,T;W^{r,2}_0)}. \end{split} \end{equation} The estimates for the term containing $f$ are obvious. Hence we conclude that $\frac{du^n}{dt}$ is bounded in $L^1(0,T;W^{-r,2}(\Omega))$.
From the energy estimates and Lemma~\ref{uni-int} we conclude existence of a monotone, continuous function $L:\R_+\to\R_+$, with $L(0)=0$ which is independent of $n$ and
$$\int_{s_1}^{s_2}\| A^n\|_{L^1(\Omega)}
\le L(|s_1-s_2|)$$ for any $s_1,s_2\in[0,T]$. Conseqently, estimate \Ref{dt1} provides that
$$\left|\int_{s_1}^{s_2}\left\langle\frac{d u^n}{dt},\varphi\right\rangle dt\right| \le
L(|s_1-s_2|)$$ for all $\varphi$ with ${\rm supp}\ \varphi\subset(s_1,s_2)\subset[0,T]$ and
$\|\varphi\|_{L^\infty(0,T;W^{r,2}_0)}\le 1$. Since
\begin{equation}\begin{split}
\|u^n(s_1)-u^n(s_2)\|_{W^{-r,2}}
=
\sup\limits_{\|\psi\|_{W^{r,2}_0}\le1}\left|\left\langle
\int_{s_1}^{s_2}\frac{du^n(t)}{dt},\psi\right\rangle\right |
\end{split}\end{equation} then
\begin{equation}
\label{equi}
\sup\limits_{n\in\N}\|u^n(s_1)-u^n(s_2)\|_{W^{-r,2}}\le L(|s_1-s_2|),
\end{equation} which provides that the family of functions $u^n:[0,T]\to W^{-r,2}(\Omega)$ is equicontinuous. Together with a uniform bound in $L^\infty(0,T;L^2(\Omega))$ it yields that the sequence $\{u^n\}$ is relatively compact in ${\cal C}([0,T];W^{-r,2}(\Omega))$ and we have $u\in {\cal C}([0,T];W^{-r,2}(\Omega))$. Consequently we can choose a sequence $\{s_0^i \}_i$, $s_0^i \to 0^+$ as $i\to\infty$ such that
\begin{equation}
u(s_0^i){{\stackrel{i\to \infty}{\longrightarrow\,}}}u(0)\quad\mbox{in }W^{-r,2}(\Omega).
\end{equation} The limit coincides with the weak limit of $\{u(s^i_0)\}$ in $L^2(\Omega)$ and hence
we conclude
\begin{equation}\label{liminfu0}
\liminf\limits_{i\to\infty}\|u(s_0)\|_{L^2(\Omega)}\geq\|u_0\|_{L^2(\Omega)}.
\end{equation} Consequently we obtain from \eqref{Galerkin} for any Lebesgue point $s$ of $u$ that
\begin{equation}\label{777}
\begin{split}
\limsup\limits_{n\to\infty}\int_{Q_s} A(t,x,\nablau^n)\cdot \nablau^n
& =
\frac{1}{2} \|u_0\|^2_{2} -
\liminf\limits_{k\to\infty} \frac{1}{2} \|u^n(s)\|^2_{2}\\
& \leq
\frac{1}{2} \|u_0\|^2_{2} -
\frac{1}{2} \|u(s)\|^2_{2}\\
& {{\stackrel{(\ref{liminfu0})}{\leq}}}
\liminf\limits_{i\to \infty}\left( \frac{1}{2} \|u(s^i_0)\|^2_{2} -
\frac{1}{2} \|u(s)\|^2_{2}\right)\\
&{=} \lim\limits_{i\to\infty}
\int_{s^i_0}^s\int_\Omega A\cdot\nablau dxdt
= \int_{0}^s\int_\Omega A\cdot\nablau dxdt
\end{split}
\end{equation}
which is exactly \eqref{Ass} and hence Lemma~\ref{Minty2} completes the proof.
\section{Examples} As a basic example of an $M-$graph captured by the described framework one can mention the graph of a function of a variable exponent with a $(t,x)-$dependent exponent, namely an $N-$function $M(t,x,\xi)=\xi^p(t,x)$. In such a case we require that $p:Q\to[1,\infty)$ is a measurable function such that there exists a constant $H>0$ such that for all
$x,y\in \Omega, t,s\in[0,T], |x-y|+ |t-s|\le\frac{1}{2}$ \begin{equation}\label{log-p}
|p(t,x)-p(s,y)|\le \frac{H}{\ln\frac{1}{|t-s|+|x-y|}} \end{equation} holds. For more details on the appearance of this condition in the theory of variable exponent spaces we refer the reader to~\cite{DiHaHaRu2011}. When condition \eqref{log-p} is satisfied we can construct the $N-$functions of very slow or very rapid growth, e.g. $M_1(t,x,\xi)=(e^{\xi})^{p(t,x)}-1$ or $M_2(t,x,\xi)= \xi^{p(t,x)}\ln(\xi+1).$ Since the problem was introduced in the language of maximal monotone graphs the presented framework also captures the case of jumps with respect to $\xi$, e.g. the following case is admissible \begin{equation} M(t,x,\xi)=\left\{ \begin{array}{rlc} M_2(t,x,\xi)&{\rm for}&\xi<1,(t,x)\in Q,\\[1ex] M_1(t,x,\xi)&{\rm for}&\xi>1,(t,x)\in Q,\\[1ex] [\ln 2,e^{p(t,x)}-1]&{\rm for}&\xi=1,(t,x)\in Q. \end{array}\right. \end{equation}
\appendix \section{Domain $\Omega$} \noindent {\bf Proof of Proposition \ref{psi-delta}.} Since $\Omega$ is a bounded domain, then $\bar\Omega$ is a compact set. Let $\{\Phi_\alpha, \alpha\in I\}$ be the atlas of $\bar\Omega$ and define by $U_\alpha:={\rm dom} (\Phi_\alpha)$. The sets $U_\alpha$ are open in $\R^d$. Let us now choose the sets which have nonempty intersection with a boundary, we can number them $\alpha=1,\ldots,\ell$, hence for these $\alpha's$ we have $U_\alpha\cap \partial \Omega\neq\emptyset$ and $\partial \Omega\subset \bigcup_{\alpha=1,\ldots,\ell} U_\alpha$. The boundary of $\Omega$ is a $C^1-$submanifold, and since it is a compact set, then without loss of generality we may assume that for $\alpha=1,\ldots,\ell$ each $\Phi_\alpha(U_\alpha)$ is a ball of the same radius and $\Phi_\alpha(\partial\Omega\cap U_\alpha)$ is the intersection of a ball $\Phi_\alpha(U_\alpha)$ with a hyperplane, which divides the ball into two halves. Moreover assume the ball is centered at the origin and the hyperplane is orthogonal to the basis vector of $\R^d$, namely $e_d=(0,\ldots,0,1)$. To construct the mapping $\Psi_1^\delta$ first we map $U_1$ for the set $\Phi(U_1)$.
For simplicity let us use the notation $(x_1, \ldots, x_{d-1},x_{d})=(x',x_d).$ First define a nonnegative function \begin{equation}\label{T} T_\epsilon(x',0):=\left\{ \begin{array}{rcl}
1&{\rm for}&\sqrt{\sum_{i=1}^{d-1}|x_i|^2}\le 1-\epsilon,\\[1ex]
{\rm smooth}&{\rm for}&\sqrt{\sum_{i=1}^{d-1}|x_i|^2}\in(1-\epsilon,1),\\[1ex]
0&{\rm for}&\sqrt{\sum_{i=1}^{d-1}|x_i|^2}= 1 \end{array}\right. \end{equation}
and such that $\nabla T_\epsilon$ on the set ${\sum_{i=1}^{d-1}|x_i|^2}= 1$ is equal to zero. Also without loss of generality we may assume that for each $x\in\partial \Omega$ there exists $\alpha\in\{1,\ldots,\ell\}$ such that $T_\epsilon(\Phi_\alpha(x))=1$ and $(x',T_\epsilon(\Phi_\alpha(x)) \in\Phi_\alpha(U_\alpha)$ for each $x\in U_\alpha$. Then for $1>\delta>0$ the mapping $\gamma^\delta:\Phi_\alpha(U_\alpha)\to\Phi_\alpha(U_\alpha)$ is defined by \begin{equation}\label{gamma} \gamma^\delta(x',x_d)= \left\{\begin{array}{lcl} (x',T_\epsilon(x',0)+(1-\delta)(x_d-T_\epsilon(x',0))&{\rm for}&x_d-T_\epsilon(x',0)<0,\\[1ex] (x',x_d)&{\rm for}&x_d-T_\epsilon(x',0)\ge0. \end{array}\right. \end{equation}
Now we are ready to start to construct the function $\Psi^\delta$ which will be a composition of consequent mappings. First define \begin{equation}\label{Psi1} \Psi_1^\delta(x)=\left\{\begin{array}{lcl} \Phi_1^{-1}(\gamma^\delta(\Phi_1(x)))&{\rm for}&x\in U_1,\\ x&{\rm for}& x\in\Omega\setminus U_1, \end{array}\right. \end{equation} and \begin{equation}\label{Psi2} \Psi_\alpha^\delta(x)=\left\{\begin{array}{lcl} \Phi_\alpha^{-1}(\gamma^\delta(\Phi_\alpha(\Psi_{\alpha-1}(x))))&{\rm for}&x\in U_\alpha,\\[1ex] \Psi_{\alpha-1}(x)&{\rm for}& x\in\Omega\setminus U_\alpha. \end{array}\right. \end{equation} Finally $\Psi^\delta:=\Psi^\delta_\ell.$ One can easily observe that \begin{equation} \sup_{y\in\Phi_\alpha(U_\alpha)}\nabla \gamma^\delta(y)\to{\bf 1} \end{equation} and \begin{equation}
\sup_{y\in\Phi_\alpha(U_\alpha)} |\gamma^\delta(y)-y|\le\delta. \end{equation} The last one immediately implies the property $(iii)$ of the proposition. To conclude $(i)$ and $(ii)$ we shall use the Lipschitz continuity of the functions $\Phi_\ell$ and $\Phi^{-1}_\ell$ with Lipschitz constants $L_\Phi$ and $L_{\Phi^{-1}}$ respectively. Then \begin{equation} \begin{split}
|\Psi^\delta(x)-x|&=|\Phi^{-1}_\ell(\gamma^\delta(\Phi_\ell(x)))-\Phi_\ell^{-1}(\Phi_\ell(x))|\\
&\le L_{\Phi^{-1}}|\gamma^\delta(\Phi_\ell(x)))-\Phi_\ell(x)|\le L_{\Phi^{-1}}\delta \end{split} \end{equation} and \begin{equation} \begin{split}
|\Psi^\delta(x)-y|&=|\Phi^{-1}_\ell(\gamma^\delta(\Phi_\ell(x)))-y|\\
&\ge \frac{1}{L_{\Phi}}|\gamma^\delta(\Phi_\ell(x))-\Phi_\ell(y)|\ge \frac{\delta}{L_{\Phi}}. \end{split} \end{equation}
\section{Selections and convergence in multi-valued terms}\label{selections} Let $\mathcal{A}$ be a maximal monotone graph satisfying {(A1)}--{ (A5)}. We call a mapping $\tilde A:Q\times \R^{d}\to\R^{d}$ a selection of $\mathcal{A}$ if it
assigns for a.a. $(t,x)\in Q$ to each $\bB\in\R^{d}$ exactly one value $\tilde A(t,x,\bB)\in \R^{d}$ such that $(\bB,\tilde A(t,x,\bB))\in\mathcal{A}(t,x)$.
One immediately observes that each such a selection $\tilde A$ is monotone and conditin {(A4)} implies that for all $\xi\in
\R^{d}$ and a.a. $(t,x)\in Q$ \begin{enumerate}
\item[{ (A4$^*$)}] $\quad\tilde A(t,x,\xi)\cdot\xi\ge -k(x,t)+c_*(M(t,x,|\xi|)+M^*(t,x,|\tilde A(t,x,\xi)|).$ \end{enumerate} Also condition { (A3)} implies for a selection the following property (see also~\cite{AlAm}) \begin{enumerate} \item[{ (A3$^*$)}] For $(\xi,\bS)\in\R^{d}\times\R^d$: $${\rm if}\ (\bS-\tilde A(t,x,\bB),\xi-\bB)\ge0 \textrm{ for all } \bB\in\R^{d}, \textrm{ then } (\xi,\bS)\in\mathcal{A}(t,x).$$ \end{enumerate} In general, a selection of the graph $\mathcal{A}$ does not have to be a Borel function, however there is a selection $\tilde A$ that is a Borel function, see e.g. \cite{AubinFrankowska}, and only such a selection is here considered. \begin{lemma}[Properties of $\tilde A$] \label{LS*} Let $\mathcal{A}(t,x)$ be maximal monotone $M$-graph satisfying { (A1)}--{ (A5)} with measurable selection $\tilde A:Q\times \mathbb{R}^{d} \to \mathbb{R}^{d}$. Then $\tilde A$
satisfies the following conditions: \begin{enumerate} \item [{\it (a1)}]$\mathrm{Dom}\, \tilde A(t,x,\cdot) = \Rd$ a.e. in $Q$; \item [{\it (a2)}]$\tilde A$ is monotone, i.e. for every $\xi_1$, $\xi_2 \in \Rd$ and a.a. $(t,x)\in Q$ \begin{equation} \label{monot} (\tilde A(t,x,\xi_1) - \tilde A(t,x,\xi_2))\cdot( \xi_1 - \xi_2) \ge 0; \end{equation} \item [{\it (a3)}] There are non-negative $k\in L^1(Q)$, $c_*>0$ and $N$-function $M$ such that for all $\xi\in\Rd$ the function
$\tilde A$ satisfies \begin{equation} \label{growthS*}
\tilde A \cdot \xi \geq -k(t,x) +c_*(M(t,x,|\xi|) + M^*(t,x,|\tilde A|)) \end{equation}
\end{enumerate} Moreover, let $U$ be a dense set in $\Rd$ and
$(\bB,\tilde A(t,x,\bB)) \in \mathcal{A}(t,x)$ for a.a. $(t,x)\in Q$ and for all $\bB\in U$. Let also
$(\xi, \bS)\in \Rd \times \Rd$. Then the following conditions are equivalent: \begin{equation*}\begin{split} \textrm{(i)} \quad &({\bS} - \tilde A(t,x,\bB))\cdot( {\xi} - \bB) \geq 0 \quad \textrm{ for all } \quad (\bB,\tilde A(t,x,\bB))\in\mathcal{A}(t,x)\,,\\ \textrm{(ii)} \quad & (\xi, \bS) \in \mathcal{A}(t,x). \end{split} \end{equation*} \end{lemma} For the proof in $L^p$ setting see~\cite{FrMuTa2004}.
Next we shall recall the proof of Lemma~\ref{Minty2}, cf.~\cite{BuGwMaRaSw2012}. The essence of the presented framework is a generalization of the Minty method in two directions: to nonreflexive spaces and to maximal monotone graphs.
\begin{proof} Let $\tilde A$ be a selection of the graph $\mathcal{A}$ and $(\nabla u^n,A^n)\in\mathcal{A}.$ The monotonicity provides that \begin{equation}\label{43} (\tilde A(t,x,\bB)-\bS^n))\cdot(\bB-\nabla u^n) \geq 0 \quad \textrm{for all }\, \bB \in L^{\infty}(Q). \end{equation} The limit passage in the term $\tilde A(t,x,\bB)\nabla u^n$ shall be provided by the uniform integrability of the sequence $\{\nabla u^n\}$, which is the consequence of the fact that $\nabla u^n\in L_M(Q)$, cf.~Lemma~\ref{uni-int}. To conclude the boundedness of $\tilde A(t,x,\bB)$ define the set \begin{equation}
Q_{(K)}:=\{(t,x)\in Q: |k(t,x)|\le K\}, \end{equation} where $k\in L^1(Q)$ is the function appearing in the assumption (A4).
Let $\bB\in L^\infty(Q)$ and assume that
$\tilde A(t,x,\bB)$ is unbounded in $Q_{(K)}$. It follows from {(A4)} and nonnegativity of $M$ that $$
|\bB|\ge\frac{ c_*M^\ast(t,x,|\tilde A(t,x,\bB)|)-k(x,t)}{|\tilde A(t,x,\bB)|}. $$ Since $M^*$ is an $N-$function, then the right-hand side tends to infinity, which contradicts that $\bB$ is bounded. Thus, after integrating \eqref{43} over $Q_{(K)}$ we obtain \begin{equation}\label{44} \int_{Q_{(K)}} \bS^n\cdot \nabla u^n \,d x\,d t \geq \int_{Q_{(K)}} \bS^n\cdot\bB \,d x\,d t + \int_{Q_{(K)}} \tilde A(t,x,\bB)\cdot(\nabla u^n-\bB)\,d x\,d t. \end{equation} Letting $n\to\infty$ in \eqref{44}, we conclude from \eqref{1.26}--\eqref{Ass} that \begin{equation}\label{45} \int_{Q_{(K)}}\bS\cdot \nabla u \,d x\,d t \geq \int_{Q_{(K)}} \bS\cdot\bB \,d x\,d t + \int_{Q_{(K)}} \tilde A(t,x,\bB)\cdot(\nabla u-\bB)\,d x\,d t \end{equation} which we rearrange as follows \begin{equation}\label{46} \int_{Q_{(K)}} (\tilde A(t,x,\bB)-\bS)\cdot(\bB-\nabla u)\,d x\,d t \geq 0 \quad \textrm{for all }\, \bB \in L^{\infty}(Q). \end{equation}
For any $j>0$ we define the set
$Q_{j} := \{z\in Q_{(K)}; |\nabla u(z)|\leq j\} $
and by $ \bbbone_{Q_{j}}$ we mean the characteristic function of $Q_j$. Since $|\nabla u| \in L^1(Q)$ due to \eqref{1.26} we observe that \begin{equation}
|Q\setminus Q_j|\le \frac{C}{j}. \label{smallset} \end{equation} For arbitrary $i,j\in\N, \ 0<j<i$ we choose $B$ in \eqref{46} in the following form $$ \bB:=\nabla u \bbbone_{Q_{i}} + h\,W\bbbone_{Q_{j}},\ h>0, \ W\in L^{\infty}(Q). $$ Thus
\begin{equation}\label{13} \int_{Q_{j}}(\tilde A(t,x,\nabla u+h W)-\bS)\cdot W \,d x\,d t \geq \frac{1}{h}\int_{Q_{(K)} \setminus Q_{i}} \left(\tilde A(t,x,0)\cdot \nabla u - \bS\cdot \nabla u\right) \,d x\,d t. \end{equation} For passing to the limit with $i\to\infty$ in \eqref{13} we use Lebesgue dominated convergence theorem concluding from \eqref{smallset} and from
\begin{equation}\label{war}
\int_{Q} \left|\tilde A(t,x,0)\cdot \nabla u - \bS\cdot \nabla u\right|\,d x\,d t<\infty \end{equation} that
\begin{equation}\label{int-on-set} \lim\limits_{i\to\infty}\int_{Q_{(K)} \setminus Q_{i}} \left(\tilde A(t,x,0)\cdot \nabla u+\bS\cdot\nabla u\right) \,d x\,d t=0. \end{equation} Note that \eqref{war} easily follows from H\"{o}lder inequality and boundedness of the terms in appropriate Musielak-Orlicz spaces. A direct consequence of \eqref{int-on-set} is that \begin{equation}\label{MB1-1} \int_{Q_{j}}(\tilde A(t,x,\nabla u+h W)-\bS)\cdot W \,d x\,d t\geq 0\quad \textrm{for all }j\in\N\,. \end{equation} Let now $h\to 0_+$. Using the definition of $Q_j$ it is easy to see that for a subsequence \begin{align*} \tilde A(\cdot,\cdot,\nabla u+h W)&\rightharpoonup \bar{\bS} &&\textrm{weakly in }L^2(Q_j),\\ \nabla u+h W &\to \nabla u &&\textrm{strongly in }L^2(Q_j),\\ (\nabla u+h W,\tilde A(\cdot,\cdot,\nabla u+h W))&\in {\mathcal A}(t,x) &&\textrm{a.e. in }Q_j. \end{align*} We observe that for an arbitrary fixed matrix $\bB\in \mathbb{R}^{d}$ by the monotonicity of the graph \begin{equation}\label{zB} \int_{Q_j}(\tilde A(t,x,\nabla u + hW)-\tilde A(t,x,B))\cdot(\nabla u+hW -B)\,dx\,dt\ge 0 \end{equation} Hence passing to the limit with $h\to0_+$ in \eqref{zB}
we conclude that \begin{equation}\label{zB-lim} \int_{Q_j}(\bar A-\tilde A(t,x,B))\cdot(\nabla u -B)\ge 0 \end{equation} which yields by (A3$^*$) that
\begin{equation} (\nabla u(t,x),\bar\bS(t,x))\in\mathcal{A}(t,x) \qquad \textrm{ a.e. in }Q_j\,. \label{begr} \end{equation} Moreover, since $\nabla u$ is bounded in $Q_j$ then also $\bar\bS$ is bounded in $Q_j$ due to \eqref{begr} and again the properties of an $N-$function. Finally, letting $h\to 0_+$ in \eqref{MB1-1}, we have $$ \int_{Q_j}(\bar \bS-\bS)\cdot W \,d x\,d t \ge 0 \qquad \textrm{for all } W\in L^{\infty}(Q_j). $$
Setting $ W:=-\frac{(\bar\bS-\bS)}{|\bar\bS-\bS|}\bbbone _{\{\bar\bS\neq\bS\}}$ yields $$
\int_{Q_j}|\bar\bS-\bS|\,d x\,d t\leq0 $$ and therefore \eqref{begr} implies that $(\nabla u,\bS)\in\mathcal{A}(t,x)$ a.e. in $Q_j$. But since $j$ was arbitrary, we use \eqref{smallset} and conclude that
$(\nabla u,\bS)\in\mathcal{A}(t,x)$ a.e. in $Q_{(K)}$, and then by the arbitrariness of $K$ we finally conclude that
$(\nabla u,\bS)\in\mathcal{A}(t,x)$ a.e. in $Q$.
\qed \end{proof}
We complete this part with a short comment on relations to different approaches
to multi-valued problems. Here we want to recall the relation between $(t,x)-$dependent maximal monotone graphs and 1-Lipschitz functions. Following the argumentation in \cite{FrMuTa2004} and \cite{GwZa2007} one concludes that for each graph satisfying (A1)-(A5) there exists a function $\Phi: Q\times\R^d\to\R^d$ such that \begin{equation}
{\cal A}(t,x)=\{(e,d)\in\R^d\times\R^d\,|\,d-e=\Phi(t,x,d+e)\} \end{equation} and $\Phi$ satisfies the following conditions: \begin{enumerate} \item $\Phi$ is a Carath\'eodory function, \item $\Phi(t,x,\cdot)$ is a contraction for almost all $(t,x)\in Q$, \item defining the functions $d,e:Q\times\R^d\to\R^d$ as follows \begin{equation}\begin{split} d(t,x,\xi)&=\frac{1}{2}(\xi+\Phi(t,x,\xi))\\ e(t,x,\xi)&=\frac{1}{2}(\xi-\Phi(t,x,\xi)) \end{split}\end{equation} the following estimate holds \begin{equation}
d(t,x,\xi)\cdot e(t,x,\xi)\geq -k(t,x) +c_*(M(t,x,|d(t,x,\xi)|) + M^*(t,x,|e(t,x,\xi)|)), \end{equation} \item $\Phi(t,x,0)=0$ for almost all $(t,x)\in Q$. \end{enumerate} This connection is essentially used for elliptic and parabolic problems including multi-valued terms, see~\cite{GwZa2007}.
\section{Lipschitz boundary} In the current section we shall comment on the case of less regular boundaries, namely the case of Lipschitz boundary, where the construction presented in Section~\ref{preliminaries} fails. Lipschitz regularity of the boundary provides there exists a finite family of star-shaped Lipschitz domains $\{\Omega_i\}$ such that (cf.~\cite{Novotny}) $$\Omega=\bigcup\limits_{i\in J}\Omega_i.$$ We introduce the partition of unity $\theta_i$ with $0\le\theta_i\le1,\, \theta_i\in {\cal C}^\infty_c(\Omega_i), \, {\rm supp} \,\theta_i=\Omega_i, \sum_{i\in J}\theta_i(x)=1$ for $x\in\Omega$. Then we are dealing with the term $\nabla (\theta_iu)$ and to provide it is in $L_M(Q)$ we need that both $u\nabla \theta_i$ and $\theta_i\nabla u$ are in $L_M(Q)$.
For this reason we shall need a kind of Poincar\'e inequality in Musielak--Orlicz spaces. For each multi-index $\alpha$ denote by $D^\alpha_x$ the distributional derivative of order $\alpha$ with respect to the variable $x$. We define the Musielak--Orlicz--Sobolev space as follows
$$W^{1,x}L_M(\Omega)=\{u\in L_M(\Omega): D^\alpha_xu\in L_M(\Omega),\ \forall\ |\alpha|\le 1\},$$ which is a Banach space with a norm
$$\|u\|_{1,M}=\sum\limits_{|\alpha|\le1}\|D^\alpha_x u\|_M$$ if only \eqref{int} holds and $\inf_{(t,x)\in Q} M(t,x,1)>0$.
The classical results for embeddings of Orlicz--Sobolev spaces are due to Donaldson and Trudinger, cf.~\cite{DoTr71}. Later the optimal embedding theorem was established by Cianchi in \cite{Ci96}. An interesting extension concerns anisotropic spaces, cf.~\cite{Cianchi}. The issue of the embedding of Musielak--Orlicz--Sobolev spaces into Musielak--Orlicz spaces was considered by Fan \cite{Fa2012} under the following assumptions\footnote{In \cite{Fa2012} the case of only $x$ dependent modulars was considered and since this is not the main concern of the current paper we shall not extend it for the $(t,x)-$dependent case. Nevertheless, the dependence on $t$ is not essential here.} \begin{itemize} \item[(M2)] $M(x,a)=M(x,1)a$ for $ x\in\bar \Omega, a\in[0,1]$. \end{itemize} Note that condition (M2) is only a technical assumption. Indeed, define $M_1:\bar \Omega\times \R_+\to\R_+$ by \begin{equation} M_1(x,a):=\left\{\begin{array}{lcl} M(x,1)a&{\rm if}&x\in\bar \Omega, \ a\in[0,1],\\[1ex] M(x,a)&{\rm if}&x\in\bar \Omega, \ a>1. \end{array}\right. \end{equation} Then in the case of bounded domain $\Omega$ it holds $L_M(\Omega)=L_{M_1}(\Omega)$ and $W^{1,x}L_M(\Omega)=W^{1,x}L_{M_1}(\Omega)$, cf.~\cite{Musielak}, hence one could consider $M_1$ instead of $M$. Note that the assumptions formulated up till now on $M$ are sufficient for the existence of an inverse function $M^{-1}(x,\cdot)$ to $M(x,\cdot).$ We will use it for the definition of $M^{-1}_*$ as follows \begin{equation} M^{-1}_*(x,\xi):=\int_0^\xi\frac{M^{-1}(x,\zeta)}{\zeta^\frac{d+1}{d}}d\zeta\quad \mbox{for }\ x\in\bar \Omega, \ \xi\ge0. \end{equation} Condition (M2) provides that the above function is well defined, strictly increasing and concave for each $x\in \bar \Omega$ and moreover continuously differentiable for $\xi>1$. Define also \begin{equation} \ell(x):=\lim_{a\to\infty} M^{-1}_*(x,a). \end{equation} Note that $0<\ell(x)\le \infty$ and define the Sobolev conjugate function of $M$, namely $M_*:\bar \Omega\times\R_+\to\R_+$ as follows \begin{equation} M_*(x,a):=\left\{\begin{array}{lcl} s&{\rm if}&x\in\bar \Omega, \ a\in[0,\ell(x)], \ M^{-1}_*(x,s)=a,\\[1ex] \infty&{\rm if}&x\in\bar \Omega, \ a\ge \ell(x). \end{array}\right. \end{equation} Observe that $M_*$ is also an $N-$function and for all $x\in\bar \Omega$ $M_*(x,\cdot)\in {\cal C}^1((0,\ell(x)))$. Having this notation we shall formulate the next assumptions \begin{itemize} \item[(M3)] The function $\ell:\bar \Omega\to(0,\infty]$ is continuous on $\Omega$ and locally Lipschitz continuous on ${\rm Dom}\ (\ell):=\{x: \ell(x)\in\R\}$. \item[(M4)] $M_*$ is locally Lipschitz continuous on ${\rm Dom}\ (M_*)$ and there exist positive constants $\delta_0<\frac{1}{d}, c_0$ and $\ell_0<\min_{x\in\bar \Omega} \ell(x)$ such that for all $x\in\bar \Omega$ and $\xi\in[\ell_0,\ell(x))$ \begin{equation}
\left|\frac{\partial M_*(x,\xi)}{\partial x_j}\right|\le c_0(M_*(x,\xi))^{1+\delta_0}, \ j=1,\ldots,d, \end{equation} provided $\frac{\partial M_*(x,\xi)}{\partial x_j}, \ j=1,\ldots,d$ exists. \item[(M5)] Assume that either $\frac{\partial M(x,\xi)}{\partial \xi}$ exists for all $x\in\bar \Omega$ and $\xi\ge 0$ or the following condition is satisfied uniformly for $(t,x)\in Q$ \begin{equation} \lim\limits_{\xi\to\infty}\frac{\xi\frac{\partial M(x,\xi)}{\partial \xi^+}}{(M(x,\xi))^{1+\frac{1}{d}}}=0 \end{equation} where by $\frac{\partial M(x,\xi)}{\partial \xi^+}$ we mean the right derivative of $M(x,\cdot)$ at point $\xi$. \end{itemize}
Since the Lipschitz continuity of $M_*$ may not always be immediate to verify, note that the Lipschitz continuity of $M$ on $\bar \Omega\times \R_+$ implies the Lipschitz continuity of $M_*$ on ${\rm Dom}\ (M_*)$.
\begin{lemma} (Poincar\'e inequality) Let $M$ be an $N-$function satisfying $(M2)-(M5)$. Then
\begin{equation}
\|u\|_M\le c\sum\limits_{j=1}^d\|D_j u\|_M \end{equation} for all $u\in W^{1,x}_0L_{M}(\Omega)$. \end{lemma}
\noindent {\bf Acknowledgements }\\ The author was supported by the grant IdP2011/000661.
\noindent
\end{document} | arXiv |
\begin{document}
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\begin{frontmatter}
\title{Rank one discrete valuations of $k((X_1,\ldots ,X_n))$} \author{Miguel \'{A}ngel Olalla--Acosta\thanksref{MAO}} \address{Facultad de Matem\'aticas. Apdo. 1160. E-41080 SEVILLA (SPAIN)} \thanks[MAO]{Partially supported by Junta de Andaluc\'{\i}a, Ayuda a grupos FQM 218} \ead{[email protected]}
\begin{abstract} In this paper we study the rank one discrete valuations of $k((X_1,\ldots ,X_n))$ whose center in $k\lcorX_1,\ldots ,X_n{\rm ]\kern - 1.8pt ]}$ is the maximal ideal $(X_1,\ldots ,X_n )$. In sections 2 to 6 we give a construction of a system of parametric equations describing such valuations. This amounts to finding a parameter and a field of coefficients. We devote section 2 to finding an element of value 1, that is, a parameter. The field of coefficients is the residue field of the valuation, and it is given in section 5.
The constructions given in these sections are not effective in the general case, because we need either to use the Zorn's lemma or to know explicitly a section $\sigma$ of the natural homomorphism $R_v\to\Delta_v$ between the ring and the residue field of the valuation $v$.
However, as a consequence of this construction, in section 7, we prove that $k((X_1,\ldots ,X_n ))$ can be embedded into a field $L((Y_1,\ldots ,Y_n ))$, where the {\em ``extended valuation'' is as close as possible to the usual
order function}.
\end{abstract}
\begin{keyword} Valuation theory, local uniformization, formal power rings, completions. \MSC 13F25 \sep 13F30 \sep 14B05 \sep 16W60 \end{keyword}
\end{frontmatter}
\section{Terminology and preliminaries}
Let $k$ be an algebraically closed field of characteristic 0, $R_n=k\lcorX_1,\ldots ,X_n{\rm ]\kern - 1.8pt ]}$, $M_n=(X_1,\ldots ,X_n )$ its maximal ideal and $K_n=k ((X_1,\ldots ,X_n ))$ its quotient field. Let $v$ be a rank-one discrete valuation of $K_n\vert k$, $R_{v}$ the valuation ring, ${\got m}_v$ the maximal ideal and $\Delta_v$ the residue field of $v$. The center of $v$ in $R_n$ is ${\got m}_v\cap R_n$. Throughout this paper ``discrete valuation of $K_n\vert k$" means ``rank-one discrete valuation of $K_n\vert k$ whose center in $R_n$ is the maximal ideal $M_n$". The dimension of $v$ is the transcendence degree of $\Delta_v$ over $k$. In order to simplify the redaction we shall assume, without loss of generality, that the group of $v$ is ${\mathbb{Z}}$.
Let ${\widehat K_n}$ be the completion of $K_n$ with respect to $v$, $\wh{v}$ the extension of $v$ to ${\widehat K_n}$, $R_{\wh{v}}$, ${\got m}_{\wh{v}}$ and $\Delta_{\wh{v}}$ the ring, maximal ideal and the residue field of $\wh{v}$, respectively (see \cite{Ser1} for more details). We know that $\Delta_v$ and $\Delta_{\wh{v}}$ are isomorphic (\cite{Krull}). Let $\sigma :\Delta_{\wh{v}}\toR_{\wh{v}}$ be a $k-$section of the natural homomorphism $R_{\wh{v}}\to\Delta_{\wh{v}}$, $\theta\inR_{\wh{v}}$ an element of value 1 and $t$ an indeterminate. We consider the $k-$isomorphism $$\Phi = \Phi_{\sigma ,\theta} :\D\lcor t\rcor\toR_{\wh{v}}$$ given by $$\Phi\(\sum\alpha_it^i\) =\sum\sigma (\alpha_i)\theta^i,$$ and denote also by $\Phi$ its extension to the quotient fields. We have a $k-$isomorphism $\Phi^{-1}$ which, when composed with the usual order function on $\Delta_{\wh{v}} ((t))$, gives the valuation $\wh{v}$. This is the situation we will consider throughout this paper, and we will freely use it without new explicit references.
We shall use two basic transformations in order to find an element of value 1 and construct the residue field:
\begin{enumerate}
\item {\em Monoidal transformation:} $$ \begin{array}{rcl} k\lcorX_1,\ldots ,X_n{\rm ]\kern - 1.8pt ]} & \longrightarrow & k\lcorY_1,\ldots ,Y_n{\rm ]\kern - 1.8pt ]} \\ X_1 & \longmapsto & Y_1 \\ X_2 & \longmapsto & Y_1Y_2 \\ X_i & \longmapsto & Y_i,\ i=3,\ldots ,n. \end{array} $$ with $v(X_2)>v(X_1)$.
\item {\em Change of coordinates:}$$ \begin{array}{rcl} k\lcorX_1,\ldots ,X_n{\rm ]\kern - 1.8pt ]} & \longrightarrow & L\lcorY_1,\ldots ,Y_n{\rm ]\kern - 1.8pt ]} \\ X_1 & \longmapsto & Y_1 \\ X_i & \longmapsto & Y_i+c_iY_1,\ i=2,\ldots ,n, \end{array} $$ where $c_i\inR_{\wh{v}}\setminus{\got m}_{\wh{v}}$ and $L$ is an extension field of $k$.
\end{enumerate}
For both transformations we have the following facts:
\begin{enumerate}
\item[(a)] The transformations are one to one: In the case of the monoidal transformations this property is well known. In the other case it is a consequence of \cite{ZSII} (corollary 2, page 137).
\item[(b)] New variables $Y_i$ lie in $R_{\wh{v}}$, so we can put $\(\pst\)^{-1} (Y_i)=\sum a_{i,j}t^j$.
\item[(c)] Let $\varphi :K_n\to\Delta_v ((t))$ be the restriction of $\(\pst\)^{-1}$ to $K_n$. Let us denote by $\varphi ':L_n=L((Y_1,\ldots ,Y_n ))\to\Delta_v ((t))$ the natural extension of $\varphi$ to the field $L_n$. Then $v=\nu_t\circ\varphi '_{\vert K_n}$, with $\nu_t$ the usual order function over $\Delta_v ((t))$. Therefore, if $\varphi '$ is injective we can extend the valuation $v$ to the field $L_n$ and the extension is $v'=\nu_t\circ\varphi '$.
\end{enumerate}
From now on transformation will mean monoidal transformation, change of coordinates, variables interchanges or finite compositions of these.
\section{Construction of an element of value 1}
Remember that we are assuming that the group of $v$ is ${\mathbb{Z}}$, so there exists an element $u\in K$ such that $v(u)=1$.
\begin{lem}{\label{lema1}} Let $\alpha_i=v(X_i)$ for all $i=1,\ldots, n$. By a finite number of monoidal transformations we can find $n$ elements $Y_1,\ldots ,Y_n\in{\widehat K_n}$ such that $v (Y_i)=\alpha =\gcd\{\alpha_1,\ldots ,\alpha_n\}$. \end{lem}
\begin{pf} We can suppose that $v(X_1)=\alpha_1=\min\{\alpha_i\vert 1\leq i\leq n\}$ and consider the following two steps:
{\bf Step 1.-} If there exists $n_i\in{\mathbb{Z}}$ such that $\alpha_i=n_i\alpha_1$ for all $i= 2,\ldots , n$, then for each $i$ we apply $n_i-1$ monoidal transformations $$ \begin{array}{rcl} k\lcorX_1,\ldots ,X_n{\rm ]\kern - 1.8pt ]} & \longrightarrow & k\lcorY_1,\ldots ,Y_n{\rm ]\kern - 1.8pt ]} \\
X_i & \longmapsto & Y_1Y_i\\ X_j & \longmapsto & Y_j,\ j\ne i. \end{array} $$ Trivially $v (Y_i)=\alpha_1$ for all $i=1,\ldots ,n$.
{\bf Step 2.-} Assume there exists $i$, with $2\leq i\leq n$, such that $v(X_1)=\alpha_1$ does not divide to $v(X_i)=\alpha_i$. We can suppose that $i=2$ with no loss of generality and then $\alpha_2 = q\alpha_1+r$. So we apply $q$ times the monoidal transformation $$ \begin{array}{rcl} k\lcorX_1,\ldots ,X_n{\rm ]\kern - 1.8pt ]} & \longrightarrow & k\lcorY_1,\ldots ,Y_n{\rm ]\kern - 1.8pt ]} \\
X_2 & \longmapsto & Y_1Y_2 \\ X_i & \longmapsto & Y_i,\ i\ne 2 \end{array} $$ to obtain a new ring $k \lcorY_1,\ldots ,Y_n{\rm ]\kern - 1.8pt ]}$ where $v (Y_2) =r>0$ and $Y_2$ is the element of minimum value.
As the values of the variables are greater than zero, in a finite number of steps 2 we come to the situation of step 1. In fact, this algorithm is equivalent to the ``euclidean algorithm" to compute the greatest common divisor of $\alpha_1,\ldots ,\alpha_n$. \qed \end{pf}
\begin{thm} We can construct an element of value 1 applying a finite number of monoidal transformations and changes of coordinates. \end{thm}
\begin{pf} We call $Y_{1,r},\ldots ,Y_{n,r}$ the elements found after $r$ transformations.
We can suppose that we have applied the previous lemma to obtain $Y_{1,1},\ldots ,Y_{n,1}$ such that $v(Y_{i,1})=\alpha$ for all $i=1,\ldots ,n$. Let us prove that there exists $c_i\inR_{\wh{v}}\setminus{\got m}_{\wh{v}}$ for each $i=2,\ldots ,n$ such that $\wh{v} (Y_{i,1}-c_iY_{1,1}) >\alpha$. We can take $$\(\pst\)^{-1} (Y_{i,1})=\sum_{j\geq\alpha} a_{i,j}t^j=\omega_i(t),\ a_{i,j} \in\Delta_{\wh{v}},\ a_{i,\alpha}\ne 0,$$ and so it suffices taking $b_i=a_{i,\alpha}/a_{1,\alpha }$ and $c_i=\sigma (b_i)$.
The following two steps defines a procedure to obtain an element of value 1:
{\bf Step 1.-} We apply the coordinate change $$ \begin{array}{rcl} k{\rm [\kern - 1.8pt [} Y_{1,1},\ldots ,Y_{n,1}{\rm ]\kern - 1.8pt ]} & \longrightarrow & L{\rm [\kern - 1.8pt [} Y_{1,2},\ldots ,Y_{n,2}{\rm ]\kern - 1.8pt ]} \\ Y_{1,1} & \longmapsto & Y_{1,2} \\ Y_{i,1} & \longmapsto & Y_{i,2}+ c_iY_{1,2},\ i=2,\ldots ,n. \end{array} $$ With this transformation the values of the new variables are not equal to $\wh{v} (Y_{1,2})$.
{\bf Step 2.-} We apply lemma \ref{lema1} to equalize the values of elements and go to step 1. Obviously, the minimum of the values of the elements does not increase, because the greater common divisor of the values does not exceed the minimum of the values. Moreover the first variable does not change.
If we obtain an element of value 1 then we are finished.
We have to show that the procedure produces an element of value 1 in a finite number of transformations. The only way for the process to be infinite is that, in step 2, the minimum of the values of the elements does not decrease. This means that, in step 1, the value of the first variable divides the values of the new variables.
The composition of steps 1 and 2 is the transformation $$ \begin{array}{rcl} k{\rm [\kern - 1.8pt [} Y_{1,r},\ldots ,Y_{n,r}{\rm ]\kern - 1.8pt ]} & \longrightarrow & L{\rm [\kern - 1.8pt [} Y_{1,r+1}\ldots ,Y_{n,r+1}{\rm ]\kern - 1.8pt ]} \\ Y_{1,r} & \longmapsto & Y_{1,r+1} \\ Y_{i,r} & \longmapsto & Y_{i,r+1}+c_iY_{1,r+1}^{m_i},\ i=2,\ldots ,n. \end{array} $$
If we use steps 1 and 2 infinitely many times, we have an infinite sequence of transformations $$ \begin{array}{rcl} k\lcorY_1,\ldots ,Y_n{\rm ]\kern - 1.8pt ]} & \longrightarrow & L{\rm [\kern - 1.8pt [} Y_{1,j},\ldots ,Y_{n,j}{\rm ]\kern - 1.8pt ]} \\ Y_1 & \longmapsto & Y_{1,j} \\ Y_i & \longmapsto & Y_{i,j}+\sum_{k=1}^j c_{i,k}Y_{1,j}^{m_{i,k}},\ i=2,\ldots ,n. \end{array} $$ Then we can obtain an infinite sequence of variables $$ \begin{array}{rcl} Y_{1,j} & = & Y_{1,j} \\ Y_{i,j} & = & Y_i-\sum_{k=1}^j c_{i,k}Y_1^{m_{i,k}},\ i=2,\ldots ,n, \end{array} $$ with $\wh{v} (Y_{i,j})>\wh{v} (Y_{i,j-1})$ for all $i,\ j$. So any sequence of partial sums of the series $$Y_i-\sum_{k=1}^{\infty} c_{i,k}Y_1^{m_{i,k}},\ \forall i=2,\ldots ,n$$ have strictly increasing values. Then these series converge to zero in $R_{\wh{v}}$, so $$Y_i=\sum_{k=1}^{\infty} c_{i,k}Y_1^{m_{i,k}},\ \forall i=2,\ldots ,n.$$ Let $f(Y_1,\ldots ,Y_n)\in K_n$, then $$v(f)=\wh{v}\( f\( Y_1,\sum_{k=1}^{\infty} c_{2,k}Y_1^{m_{2,k}}, \ldots ,\sum_{k=1}^{\infty} c_{n,k}Y_1^{m_{n,k}}\)\) =m\cdot v(Y_1).$$ In this situation, the group of $v$ is $v(Y_1)\cdot{\mathbb{Z}}$ (see \cite{Bri2}) but as the group is assumed to be ${\mathbb{Z}}$, $\wh{v} (Y_1)=1$. \qed \end{pf}
\begin{exmp} Let us consider the embedding $$ \begin{array}{rcl} \Psi : {\mathbb{C}}{\rm [\kern - 1.8pt [} X_1,X_2,X_3{\rm ]\kern - 1.8pt ]} & \longrightarrow &
{\mathbb{C}} (T_2,T_3){\rm [\kern - 1.8pt [} t{\rm ]\kern - 1.8pt ]} \\ X_1 & \longmapsto & t^2 \\ X_2 & \longmapsto & T_2t^4+T_2t^6 \\ X_3 & \longmapsto & T_2t^2+T_3t^5 \end{array} $$ with $t$, $T_2$ and $T_3$ variables over ${\mathbb{C}}$. We are going to denote its extension to the quotient fields by $\Psi$ as well. The composition of this injective homomorphism with the order function in $t$ gives a discrete valuation of ${\mathbb{C}} ((X_1,X_2,X_3))\vert{\mathbb{C}}$, $v=\nu_t\circ\Psi$. If we apply the procedure given in this section we construct the following element of value 1: $$\frac{X_3- c_2X_1}{X_1^2},$$ where $c_2\inR_{\wh{v}}\setminus{\got m}_{\wh{v}}$ such that $\Psi (c_2)=T_2+t\cdot f,$ with $f\in{\mathbb{C}} (T_2,T_3){\rm [\kern - 1.8pt [} t{\rm ]\kern - 1.8pt ]}$. In this case we can take $$c_2=\frac{X_2}{X_1^2+X_1^3}.$$ \end{exmp}
\begin{rem} We need to know some elements $c_i\inR_{\wh{v}}\setminus{\got m}_{\wh{v}}$ (or $b_i\in\Delta_{\wh{v}}$ and $\sigma :\Delta_{\wh{v}}\toR_{\wh{v}}$) such that $\wh{v} (Y_{i,1}-c_iY_{1,1})>\alpha$ for each $i=2,\ldots ,n$ in order to apply the procedure described in the proof of theorem 2. Let $\Delta$ be a field, if the valuation is given as a composition of an injective homomorphism $$ \begin{array}{rcl} \Psi :k\lcorX_1,\ldots ,X_n{\rm ]\kern - 1.8pt ]}& \longrightarrow & \Delta{\rm [\kern - 1.8pt [} t{\rm ]\kern - 1.8pt ]} \\ X_i & \longmapsto & \sum_{j\geq 1}a_{i,j}t^j \end{array} $$ with the usual order function of $\Delta ((t))$, $v=\nu_t\circ\Psi$, then we can find the $c_i$'s using the coefficients $a_{i,j}\in\Delta$ of $\Psi (X_i)$. \end{rem}
\section{Transcendental and algebraic elements of $\Delta_v$}
In the following sections we give a procedure to construct the residue field $\Delta_v$ of a discrete valuation of $K_n\vert k$, as a transcendental extension of $k$.
Before the description of the procedure we have to do the following remark about the $k-$section $\sigma$.
\begin{rem}{\label{remark4}} We are going to check all the variables searching those residues which generate the extension $k\subset\Delta_{\wh{v}}$. Hence we will have to move between $R_{\wh{v}}$ and $\Delta_{\wh{v}}$ by the $k-$section $\sigma$ and the natural homomorphism $\Delta_{\wh{v}}\toR_{\wh{v}}$. We can do the following considerations:
\noindent {\bf 1)} Let us consider the diagram \begin{center}
\unitlength=0.75mm
\linethickness{0.4pt} \begin{picture}(49.00,60.00)
\put(10.00,4.00){\makebox(0,0)[cc]{$k$}} \put(29.50,4.33){\vector(1,0){14.00}} \put(29.50,4.33){\vector(-1,0){14.00}} \put(28.00,6.33){\makebox(0,0)[cc]{{\footnotesize $id$}}} \put(49.00,4.00){\makebox(0,0)[cc]{$k$}} \put(10.00,7.00){\vector(0,1){15.00}} \put(49.00,7.00){\vector(0,1){15.00}} \put(10.00,27.67){\makebox(0,0)[cc]{${\mathbb{F}}$}} \put(10.00,32.33){\vector(0,1){15.00}} \put(49.00,32.33){\vector(0,1){15.00}} \put(10.00,52.67){\makebox(0,0)[cc]{$R_{\wh{v}}$}} \put(49.00,52.67){\makebox(0,0)[cc]{$\Delta_{\wh{v}}$}} \put(16.00,26.00){\vector(1,0){28.00}} \put(16.00,51.00){\vector(1,0){28.00}} \put(44.00,28.00){\vector(-1,0){28.00}} \put(44.00,53.00){\vector(-1,0){28.00}} \put(28.00,24.00){\makebox(0,0)[cc]{{\footnotesize $\varphi$}}} \put(28.00,49.00){\makebox(0,0)[cc]{{\footnotesize $\varphi$}}} \put(28.00,30.00){\makebox(0,0)[cc]{{\footnotesize $\sigma$}}} \put(28.00,55.00){\makebox(0,0)[cc]{{\footnotesize $\sigma$}}} \put(49.00,27.67){\makebox(0,0)[cc]{${\mathbb{F}}'$}} \end{picture}
\end{center} where ${\mathbb{F}}$ and ${\mathbb{F}} '$ are subfields of $R_{\wh{v}}$ and $\Delta_{\wh{v}}$ respectively. Let $\omega\inR_{\wh{v}}$ an element such that $\wh{v} (\omega )=0$. The question is: if $\omega +{\got m}_{\wh{v}}$ is transcendental over ${\mathbb{F}} '$, is $\sigma (\omega +{\got m}_{\wh{v}} )$ transcendental over ${\mathbb{F}}$? What happens in the algebraic case?
So we suppose $\omega +{\got m}_{\wh{v}}$ to be transcendental over ${\mathbb{F}} '$. Let $f(X)\in{\mathbb{F}} [X]$ be a non-zero polynomial. Let us put $$f(X)=\sum_{i=0}^n\sigma (a_i')X^i,\ a_i'\in{\mathbb{F}} '.$$ Then $$f(\sigma (\omega +{\got m}_{\wh{v}} ))=\sum_{i=0}^n\sigma (a_i')\sigma (\omega +{\got m}_{\wh{v}} )^i=\sigma\(\sum_{i=0}^na_i'(\omega +{\got m}_{\wh{v}} )^i\)\ne 0$$ because $\omega +{\got m}_{\wh{v}}$ is transcendental over ${\mathbb{F}} '$. So we have proved that $\sigma (\omega + {\got m}_{\wh{v}})$ is transcendental over ${\mathbb{F}}$ if $\omega +{\got m}_{\wh{v}}$ is transcendental over ${\mathbb{F}} '$
\noindent {\bf 2)} In the algebraic case let us consider the next diagram: \begin{center}
\unitlength=0.75mm
\linethickness{0.4pt} \begin{picture}(49.00,59.67)
\put(10.00,4.00){\makebox(0,0)[cc]{$k$}} \put(29.50,4.33){\vector(1,0){14.00}} \put(29.50,4.33){\vector(-1,0){14.00}} \put(28.00,6.33){\makebox(0,0)[cc]{{\footnotesize $id$}}} \put(49.00,4.00){\makebox(0,0)[cc]{$k$}} \put(10.00,7.00){\vector(0,1){15.00}} \put(49.00,7.00){\vector(0,1){15.00}} \put(10.00,27.67){\makebox(0,0)[cc]{${\mathbb{F}}$}} \put(10.00,32.33){\vector(0,1){15.00}} \put(49.00,32.33){\vector(0,1){15.00}} \put(10.00,52.67){\makebox(0,0)[cc]{$R_{\wh{v}}$}} \put(49.00,52.67){\makebox(0,0)[cc]{$\Delta_{\wh{v}}$}} \put(16.00,26.00){\vector(1,0){28.00}} \put(44.00,28.00){\vector(-1,0){28.00}} \put(28.00,24.00){\makebox(0,0)[cc]{{\footnotesize $\varphi$}}} \put(28.00,30.00){\makebox(0,0)[cc]{{\footnotesize $\sigma$}}} \put(49.00,27.67){\makebox(0,0)[cc]{${\mathbb{F}}'$}} \end{picture}
\end{center} Let $\alpha +{\got m}_{\wh{v}}\in\Delta_{\wh{v}}$ be an algebraic element over ${\mathbb{F}} '$, with $\wh{v} (\alpha )=0$ (i.e. $\alpha +{\got m}_{\wh{v}}\ne 0$). Let $$\overline{f} (X) = X^m+\beta_1X^{m-1}+\cdots +\beta_m\in{\mathbb{F}} '[X]$$ be its minimal polynomial over ${\mathbb{F}} '$. Let us take the polynomial $$f(X) = X^m+b_1X^{m-1}+\cdots +b_m\in{\mathbb{F}}[X],\mbox{ with } b_i=\sigma (\beta_i).$$ By Hensel's Lemma (\cite{ZSII}, corollary 1, page 279) we know that there exists $a\inR_{\wh{v}}$ such that $a$ is a simple root of $f(X)$ y $\varphi (a)=\alpha +{\got m}_{\wh{v}}$. As $\varphi\sigma =id$, $f(X)$ is the minimal polynomial of $a$, so we can extend $\sigma :{\mathbb{F}} '[\alpha +{\got m}_{\wh{v}} ]\to{\mathbb{F}} [a]$. Then we have
\begin{center}
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\put(10.00,4.00){\makebox(0,0)[cc]{$k$}} \put(29.50,4.33){\vector(1,0){14.00}} \put(29.50,4.33){\vector(-1,0){14.00}} \put(28.00,6.33){\makebox(0,0)[cc]{{\footnotesize $id$}}} \put(49.00,4.00){\makebox(0,0)[cc]{$k$}} \put(10.00,7.00){\vector(0,1){15.00}} \put(49.00,7.00){\vector(0,1){15.00}} \put(10.00,27.67){\makebox(0,0)[cc]{${\mathbb{F}}$}} \put(10.00,32.33){\vector(0,1){15.00}} \put(49.00,32.33){\vector(0,1){15.00}} \put(10.00,51.67){\makebox(0,0)[cc]{${\mathbb{F}}(a)$}} \put(49.00,51.67){\makebox(0,0)[cc]{${\mathbb{F}}'(\alpha +{\got m}_{\wh{v}} )$}} \put(10.00,57.00){\vector(0,1){15.00}} \put(49.00,57.00){\vector(0,1){15.00}} \put(10.00,77.34){\makebox(0,0)[cc]{$R_{\wh{v}}$}} \put(49.00,77.34){\makebox(0,0)[cc]{$\Delta_{\wh{v}}$}} \put(16.00,26.00){\vector(1,0){28.00}} \put(44.00,28.00){\vector(-1,0){28.00}} \put(28.00,24.00){\makebox(0,0)[cc]{{\footnotesize $\varphi$}}} \put(28.00,30.00){\makebox(0,0)[cc]{{\footnotesize $\sigma$}}} \put(49.00,27.67){\makebox(0,0)[cc]{${\mathbb{F}}'$}} \put(20.00,50.67){\vector(1,0){15.00}} \put(35.00,52.67){\vector(-1,0){15.00}} \put(28.00,48.67){\makebox(0,0)[cc]{{\footnotesize $\varphi$}}} \put(28.00,54.67){\makebox(0,0)[cc]{{\footnotesize $\sigma$}}} \end{picture}
\end{center}
Let us consider the set $$\Omega =\{ ({\mathbb{F}}_1,\sigma_1)\vert{\mathbb{F}}_1\supset{\mathbb{F}}\mbox{ and }\sigma_1\mbox{ extends }\sigma\}$$ partially ordered by $$({\mathbb{F}}_1,\sigma_1)<({\mathbb{F}}_2,\sigma_2)\iff{\mathbb{F}}_1\subset{\mathbb{F}}_2\mbox{ and } \sigma_{2\vert{\mathbb{F}}_1}=\sigma_1.$$ By Zorn's Lemma there exists a maximal element $({\mathbb{L}} ,\sigma ')\in\Omega$, and again by Hensel's Lemma (\cite{ZSII}, corollary 2, page 280) we have $\varphi ({\mathbb{L}} )=\Delta_{\wh{v}}$. So we can extend $\sigma$ to a $k-$section $\sigma '$ of $\varphi$ in such a way that $a=\sigma '(\alpha +{\got m}_{\wh{v}})$ is an algebraic element over ${\mathbb{F}}$.
\noindent {\bf 3)} Hence we have showed that if $\omega +{\got m}_{\wh{v}}\in\Delta_{\wh{v}}$, $\wh{v} (\omega )=0$, is a transcendental (resp. algebraic) element over ${\mathbb{F}} '$, there exists a $k-$section of $\varphi$ which extends $\sigma$ and $\sigma (\omega +{\got m}_{\wh{v}} )$ is transcendental (resp. algebraic) over ${\mathbb{F}}$. \begin{center}
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\put(10.00,4.00){\makebox(0,0)[cc]{$k$}} \put(29.50,4.33){\vector(1,0){14.00}} \put(29.50,4.33){\vector(-1,0){14.00}} \put(28.00,6.33){\makebox(0,0)[cc]{{\footnotesize $id$}}} \put(49.00,4.00){\makebox(0,0)[cc]{$k$}} \put(10.00,7.00){\vector(0,1){15.00}} \put(49.00,7.00){\vector(0,1){15.00}} \put(10.00,27.67){\makebox(0,0)[cc]{${\mathbb{F}}$}} \put(10.00,31.33){\vector(0,1){15.00}} \put(49.00,31.33){\vector(0,1){15.00}} \put(10.00,51.67){\makebox(0,0)[cc]{$R_{\wh{v}}$}} \put(49.00,51.67){\makebox(0,0)[cc]{$\Delta_{\wh{v}}$}} \put(16.00,26.00){\vector(1,0){28.00}} \put(44.00,28.00){\vector(-1,0){28.00}} \put(28.00,24.00){\makebox(0,0)[cc]{{\footnotesize $\varphi$}}} \put(28.00,30.50){\makebox(0,0)[cc]{{\footnotesize $\sigma$}}} \put(49.00,27.67){\makebox(0,0)[cc]{${\mathbb{F}}'$}} \put(16.00,50.00){\vector(1,0){28.00}} \put(44.00,52.00){\vector(-1,0){28.00}} \put(28.00,48.00){\makebox(0,0)[cc]{{\footnotesize $\varphi$}}} \put(28.00,54.50){\makebox(0,0)[cc]{{\footnotesize $\sigma '$}}} \end{picture}
\end{center} \end{rem}
\section{A first transcendental residue.}
We devote this section to finding a first transcendental residue of $\Delta_v$ over $k$. Note that this preliminary transformations construct the residue field in the case $n=2$.
\begin{lem}{\label{lema6}} There exists a finite number of monoidal transformations and changes of coordinates that constructs $n$ elements $Y_1,\ldots ,Y_n$ such that $v(Y_i)=v(Y_1)=\alpha$ and the residue $Y_2/Y_1+{\got m}_{\wh{v}}$ is not in $k$. \end{lem}
\begin{pf} We can suppose that we have applied lemma \ref{lema1} to obtain $Y_1.\ldots ,Y_n$ such that $v(Y_i)=\alpha$ for all $i=1, \ldots ,n$.
In this situation $v(Y_i/Y_j)=0$, so $0\ne (Y_i/Y_j)+{\got m}_v\in\Delta_v$. If this residue lies in $k$ then there exists $a_{i,1}\in k$ such that $$\frac{Y_i}{Y_j}+{\got m}_v = a_{i,1}+{\got m}_v ,$$ so $$\frac{Y_i}{Y_j}-a_{i,1}=\frac{Y_i-a_{i,1}Y_j}{Y_j}\in{\got m}_v ,$$ and then $$v\(\frac{Y_i-a_{i,1}Y_j}{Y_j}\) >0.$$ So we have $v(Y_i-a_{i,1}Y_j)=\alpha_1>\alpha$. If $\alpha$ divides to $\alpha_1$ then $\alpha_1=r_1\alpha$ with $r_1\geq 2$ and $$v\(\frac{Y_i-a_{i,1}Y_j}{Y_j^{r_1}}\) =0.$$ If the residue of this element lies too in $k$, then exist $a_{i,r_1}\in k$ such that $$v(Y_i-a_{i,1}Y_j-a_{i,r_1}Y_j^{r_1})=\alpha_2>\alpha_1.$$ If $\alpha$ divides to $\alpha_2$ then $\alpha_2=r_2\alpha$ with $r_2>r_1$ and we can repeat this operation.
The above procedure is finite for some pair $(i,j)$. We know (\cite{Bri2}) that any discrete valuation of $k((X_1,X_2))$ has dimension 1, so the restriction, $v'$, of our valuation $v$ to the field $k((X_1,X_2))$ is a valuation with dimension 1, and the dimension of $v$ is greater or equal than 1, because a transcendental residue of $v'$ over $k$ is a transcendental residue of $v$ too. If the procedure never ends for all $(i,j)$ then all the residues of $v$ are in $k$, so the dimension of $v$ is 0 and there is a contradiction. So we can suppose that the above procedure ends for $(1,2)$ by reordering the variables if necessary.
Hence there exists a first transcendental residue. We can apply the above procedure to the variables $Y_1,Y_2$, and so we have the transformations: $$Z_i=Y_i,\ i\ne 2$$ $$Z_2= Y_2-\sum_{i=1}^{s_2}a_{2,i}Y_1^i,$$ such that one of the following two situation occurs:
a) $v(Y_1)$ divides $v(Z_2)$ and the residue of $Z_2/Y_1^r$ is not in $k$ with $v(Z_2)=r\cdot v(Y_1)$.
b) $v(Y_1)$ does not divide $v(Z_2)$.
In case a), we make the transformation $$Z_2=Y_2-\sum_{i=1}^{s_2}a_{2,i}Y_1^i,$$ and apply lemma \ref{lema1} to obtain elements with the same values. We note these elements by $Y_1,\ldots ,Y_n$ again in order not to complicate the notation. So, after this procedure, we have a transcendental element $u_2=\sigma (Y_2/Y_1+{\got m}_{\wh{v}})$ over $k$.
In case b) we make the same transformation and go back to the beginning of the proof.
Anyway this procedure stops, because the value of the variables are greater or equal than 1.
Then we can suppose that, after a finite number of transformations, we have $n$ elements $Y_1,\ldots ,Y_n$ such that $v(Y_i)=v(Y_1)=\alpha$ and the residue $Y_2/Y_1+{\got m}_{\wh{v}}$ is not in $k$.\qed \end{pf}
\begin{exmp} Let $v=\nu_t\circ\Psi$ the discrete valuation of ${\mathbb{C}} ((X_1,X_2))\vert{\mathbb{C}}$ defined by the embedding $$\begin{array}{rcl} \Psi :{\mathbb{C}}{\rm [\kern - 1.8pt [} X_1,X_2{\rm ]\kern - 1.8pt ]} & \longrightarrow & {\mathbb{C}} (u){\rm [\kern - 1.8pt [} t{\rm ]\kern - 1.8pt ]} \\ X_1 & \longmapsto & t\\ X_2 & \longmapsto & t+t^3+\sum_{i\geq 1}u^it^{i+3} \end{array} $$ with $u$ and $t$ independent variables over ${\mathbb{C}}$.
The residue $X_2/X_1+{\got m}_v =1+{\got m}_v$, because $v(X_2-X_1)=3>1$. So we have $$v\left(\frac{X_2-X_1}{X_1^3}\right) =0.$$ The residue $$\frac{X_2-X_1}{X_1^3}+{\got m}_v =1+{\got m}_v$$ too, because $v(X_2-X_1-X_1^3)=4>3$. So we have $$v\left(\frac{X_2-X_1-X_1^3}{X_1^4}\right) =0.$$ As $\Psi ((X_2-X_1-X_1^3)/X_1^4)=u$ and $u$ is trancendental over ${\mathbb{C}}$, then $$\frac{X_2-X_1-X_1^3}{X_1^4}+{\got m}_v\notin {\mathbb{C}}$$ and this is a first transcendental residue of $\Delta_v$ over ${\mathbb{C}}$.
In this situation we can do the transformation $$\begin{array}{rcl} {\mathbb{C}}{\rm [\kern - 1.8pt [} X_1,X_2{\rm ]\kern - 1.8pt ]} & \longrightarrow & {\mathbb{C}}{\rm [\kern - 1.8pt [} Y_1,Y_2{\rm ]\kern - 1.8pt ]}\\ X_1 & \longmapsto & Y_1\\ X_2 & \longmapsto & Y_2Y_1^3+Y_1+Y_1^3 \end{array} $$ to obtain elements $\{ Y_1,Y_2\}$ such that $\Psi (Y_1)=t$ and $\Psi (Y_2)=\sum_{i\geq 1}u^it^i$. So $v(Y_2)=v(Y_1)=1$ and the residue $$\frac{Y_2}{Y_1}+{\got m}_v =\frac{X_2-X_1-X_1^3}{X_1^4}+{\got m}_v$$ is not in ${\mathbb{C}}$.
In this example the extension of the valuation $v$ to the field ${\mathbb{C}} ((Y_1,Y_2))$ is the usual order function. Theorem \ref{thm3} says that, for $n=2$, we always have this. \end{exmp}
We end up the section with some specific arguments for the case $n=2$.
The proof of the following lemma is straightforward from (\cite{Bri2}, theorem 2.4):
\begin{lem} Let $v$ be a discrete valuation of $K_n\vert k$. If $v$ is such that $v(f_r)=r\alpha$ for all forms $f_r$ of degree $r$ with respect to the usual degree, then the group of $v$ is $\alpha\cdot{\mathbb{Z}}$. \end{lem}
So we have
\begin{thm}{\label{thm3}} In the case $n=2$, the extension of the valuation $v$ to the field $k((Y_1,Y_2))$ is the usual order function. \end{thm}
\begin{pf} After a finite number of transformations we are in the situation of the end of the previous proof. Obviously, if $n=2$, $k((Y_1,Y_2))\subsetR_{\wh{v}}$ so $v$ can be extended to a valuation $v'$ over $k((Y_1,Y_2))$ such that $\Delta_{v'}=\Delta_v=\Delta_{\wh{v}}$. We denote the extension by $v$ for simplifying. Let $\sigma :\Delta_{\wh{v}}\toR_{\wh{v}}$ a $k-$section of $R_{\wh{v}}\to\Delta_{\wh{v}}$, $u_2=\sigma (Y_2/Y_1+{\got m}_{\wh{v}} )$, $h\ne 0$ a form of degree $r$ and $\gamma =Y_2-u_2Y_1$. From the construction procedure of $u_2$ we know that $\wh{v} (\gamma )> \alpha$ (remember $\alpha =v(Y_1)$). Then $$h(Y_1,Y_2)=h(Y_1,u_2Y_1+\gamma )=Y_1^rh(1,u_2)+\gamma ',$$ where $\gamma '$ is such that $v(\gamma ')>r\alpha$. As $u_2\notin k$, $u_2$ is transcendental over $k$, so $h(1,u_2)\ne 0$ and $v(h)=r\alpha$. By the previous lemma, the group of $v$ is $\alpha\cdot{\mathbb{Z}}$, so $\alpha =1$ and $v$ is the usual order function. \qed \end{pf}
\section{The general case}
Let us move to the general case. Assume that $n>2$ and suppose we have applied the procedure of the lemma \ref{lema6} to find $Y_1,\ldots ,Y_n\in\widehat{K}$ such that
a) The value of these elements are $\alpha\in{\mathbb{Z}}$.
b) The residue of $Y_2/Y_1$ is transcendental over $k$.
This section and the next one describe the transformations that we have to do in order to construct the residue field of $v$.
\begin{rem} Let $\Delta_2=k(Y_2/Y_1+{\got m}_v )$ a purely transcendental extension of $k$ of transcendence degree 1. Let $\sigma_2:\Delta_2\to k(Y_2/Y_1)$ defined by $$\sigma_2\(\frac{Y_2}{Y_1}+{\got m}_v\) =\frac{Y_2}{Y_1}=u_2.$$ We know that there exists a $k-$section $\sigma$ which extends $\sigma_2$ in the sense of the remark \ref{remark4}. \end{rem}
\begin{rem} Let us suppose that the residue of $Y_3/Y_1$ is algebraic over $\Delta_2$, and let $u_{3,1}$ be its image by $\sigma$. Then $v(Y_3-u_{3,1}Y_1)=\alpha_1>\alpha$. If $\alpha$ divides to $\alpha_1$ then there exists $u_{3,r}\in{\rm im} (\sigma )$ and $r>1$ such that $v(Y_3-u_{3,1}Y_1-u_{3,r}Y_1^r)=\alpha_2>\alpha_1$. Let us suppose that $u_{3,r}$ is algebraic over $\Delta_2$ too and $\alpha$ divides to $\alpha_2$. Then we can find ourselves in one of the three situations shown in the following items.
{\bf (Situation 1)} After a finite number of transformations, we obtain a value $\alpha_s$ such that it is not divided by $\alpha$. Then we make the transformation $$Z_3=Y_3-\sum_{j=1}^su_{3,j}Y_1^j,$$ with $u_{3,j}$ algebraic over $\Delta_2$ for all $j=1,\ldots ,s$. So we have to apply transformations to find elements with the same values and begin with all the procedure described in this section. When this occurs, the values of the elements decrease, so we can suppose that after a finite number of transformations we have reached a strictly minimal value. In fact this value should be 1, because we are assuming that the values group is ${\mathbb{Z}}$. We shall denote these elements by $Y_1,\ldots ,Y_n$ in order not to complicate the notation. So we can suppose that this situation will never occur again for any variable.
\noindent {\bf (Situation 2)} After a finite number of steps, we have a transcendental residue of $\Delta_2$. Let us denote this residue by $u_3$. This means $$Z_3=Y_3-\sum_{j=1}^{s_3}u_{3,j}Y_1^j,$$ where the elements $\{ u_{3,j}\}_{j=1}^{s_3}$ are algebraic over $\Delta_2$ and $u_3=\sigma (Z_3/Y_1^{\wh{v} (Z_3)}+{\got m}_v )$ is transcendental over $\Delta_2$. We shall note $\Delta_3 =k(u_2,\{ u_{3,j}\}_{j=1}^{s_3},u_3)$.
In this situation, if $n=3$ we can apply monoidal transformations to obtain elements with the same values. We will denote these elements again by $\{ Y_1,Y_2 ,Y_3\}$. The extension of the valuation $v$ to the field $L((Y_1,Y_2,Y_3))$ with $L=k(\{u_{3,j}\}_{j=1}^{s_3})$, is the usual order function, for analogy with the case $n=2$ (theorem 9).
\noindent {\bf (Situation 3)} All the residues obtained are algebraic elements. Then we take $\Delta_3=\Delta_2(\{ u_{3,j}\}_{j\geq 1})$, an algebraic extension of $\Delta_2$. \end{rem}
\begin{rem} Let us suppose that we have repeated the previous construction with each element $Y_4,\ldots ,Y_{i-1}$, so we have a field $$\Delta_{i-1}=k(u_2,\zeta_3,\ldots ,\zeta_{i-1})\subset \sigma (\Delta_{\wh{v}} ),$$ where each $\zeta_k$ is:
- either $\{\{ u_{k,j}\}_{j=1}^{s_k},u_k\}$ if $\{ u_{k,j}\}_{j=1}^{s_k}$ are algebraic over $\Delta_{k-1}$ and $u_k=\sigma ((Z_k/Y_1^{\wh{v} (Z_k)})+{\got m}_{\wh{v}} )$ is a transcendental element over $\Delta_{k-1}$ (i.e. situation 2),
- or $\Delta_{k-1}\subset\Delta_{k-1}(\{ u_{k,j}\}_{j\geq 1})$ is an algebraic extension (i.e. situation 3).
So we have two possible situations concerning variable $Y_i$:
{\bf 1)} There exists a transformation $$Z_i=Y_i-\sum_{j=1}^{s_i}u_{i,j}Y_1^j,$$ where the elements $u_{i,j}$ are algebraic over $\Delta_{i-1}$ and $u_i=\sigma ((Z_i/Y_1^{\wh{v} (Z_i)})+{\got m}_{\wh{v}} )$ is a transcendental element over $\Delta_{i-1}$. So we have the transcendental extension $$\Delta_{i-1}\subset\Delta_{i-1}(\{u_{i,j}\}_{j=1}^{s_i},u_i)=\Delta_i.$$
{\bf 2)} All the elements $u_{i,j}$ we have constructed are algebraic over $\Delta_{i-1}$, so we have the algebraic extension $$\Delta_{i-1}\subset\Delta_{i-1}(\{u_{i,j}\}_{j\geq 1})=\Delta_i.$$ \end{rem}
\begin{rem} We have given a procedure to construct elements $\{ Y_1,\ldots ,Y_n\}$ such that they satisfy these important properties: \begin{enumerate} \item After reordering if necessary, we can suppose that the first $m$ elements give us all the transcendental residues over $k$, i.e. the residue of each $Y_i/Y_1$ is transcendental over $\Delta_{i-1}$ with $i=2,\ldots ,m$. So the rest of variables $Y_{m+1},\ldots ,Y_n$ are such that we enter in the situation of previous item {\bf 2)}. \item With the usual notations, the extension $$\Delta_m\subset\Delta_m\(\{u_{i,j}\}_{j\geq 1}\),\ i=m+1,\ldots ,n$$ is algebraic. \end{enumerate} \end{rem}
\begin{thm}{\label{theorem9}} The residue field of $v$ is $$\Delta_n=k\( u_2,\{u_{3,j}\}_{j=1}^{s_3},u_3,\ldots ,\{u_{m,j}\}_{j=1}^{s_m},u_m\) \(\{u_{m+1,j}\}_{j\geq 1},\ldots ,\{u_{n,j}\}_{j\geq 1}\) ,$$ and the transcendence degree of $\Delta_n$ over $k$ is $m-1$. \end{thm}
\begin{pf} In this section we have given a construction by writing the elements $Y_i$ depending on $Y_1$ and some transcendental and algebraic residues. So we have constructed a map $$ \begin{array}{rcl} \varphi ':L_n(( Y_1,\ldots ,Y_n )) & \longrightarrow & \Delta_n((t )) \\ Y_1 & \longmapsto & t \\ Y_i & \longmapsto & u_it,\ i=2,\ldots ,m\\ Y_k & \longmapsto & \sum_{j\geq 1}u_{k,j}t^j,\ u_{k,1}\ne 0,\ k=m+1,\ldots ,n. \end{array} $$ This map is not injective in the general case, but we know that $v=\nu_t\circ\varphi '_{\vert K_n}$. So the residue field of $v$ is equal to the residue field of $\nu_t$, i.e. $\Delta_n$. \qed \end{pf}
A straightforward consequence of this theorem is the following well-known result
\begin{cor}{\label{cor8}} The usual order function over $K_n$ has dimension $n-1$, i.e. the transcendence degree of its residue field over $k$ is $n-1$. \end{cor}
\begin{pf} Let $\nu$ be the usual order function over $K_n$. All the residues $X_i/X_1+{\mathfrak m}_{\nu}$ are transcendental over $k(X_2/X_1+{\mathfrak m}_{\nu},\ldots , X_{i-1}/X_1+{\mathfrak m}_{\nu})$: if this were not the case, there would exist $u_i\in\sigma (\Delta_{\nu})$ such that $\nu (X_i-u_iX_1)>1$ and $\nu$ would not be an order function. So $\Delta_{\nu}=k(X_2/X_1,\ldots ,X_{n}/X_1)$. \qed \end{pf}
\section{Explicit construction of the residue field: an example}
In order to compute explicitly the residue field of a valuation we need to construct a section $\sigma :\Delta_{\wh{v}}\toR_{\wh{v}}$ as in remark \ref{remark4}. This procedure is not constructive in general. As in section 1, if the valuation is given as a composition $v=\nu_t\circ\Psi$, where $\Psi :k\lcorX_1,\ldots ,X_n{\rm ]\kern - 1.8pt ]}\to\Delta{\rm [\kern - 1.8pt [} t{\rm ]\kern - 1.8pt ]}$ is an injective homomorphism and $\nu_t$ is the order funcion in $\Delta{\rm [\kern - 1.8pt [} t{\rm ]\kern - 1.8pt ]}$, then we can construct $\sigma$ using the coefficients $a_{i,j}\in\Delta$ of $\Psi (X_i)=\sum_{j\geq 1}a_{i,j}t^j$.
(Of course, explicit does not mean effective because we are working with the series $\sum_{j\geq 1}a_{i,j}t^j$ and this input is not finite).
\begin{exmp}{\label{ejemplo}} Let us consider the embedding $$ \begin{array}{rcl} \Psi : {\mathbb{C}}{\rm [\kern - 1.8pt [} X_1,X_2,X_3,X_4,X_5{\rm ]\kern - 1.8pt ]} & \longrightarrow &
\Delta{\rm [\kern - 1.8pt [} t{\rm ]\kern - 1.8pt ]} \\ X_1 & \longmapsto & t \\ X_2 & \longmapsto & T_2t \\ X_3 & \longmapsto & T_2^2t+T_2t^2+T_3t^3 \\ X_4 & \longmapsto & T_2^3t+T_2^2t^2+T_3t^3+T_4t^4\\ X_5 & \longmapsto & T_2t\sum_{j\geq 1} (T_4^{1/p}t)^j, \end{array} $$ with $t$, $T_2$, $T_3$ and $T_4$ variables over ${\mathbb{C}}$, $p\in{\mathbb{Z}}$ prime and $\Delta$ is a field such that $\overline{{\mathbb{C}} (T_4)}(T_2,T_3)\subseteq\Delta$. $\overline{{\mathbb{C}} (T_4)}$ is the algebraic closure of ${\mathbb{C}} (T_4)$. We are going to denote its extension to the quotient fields by $\Psi$. The composition of this injective homomorphism with the order function in $t$ gives a discrete valuation of ${\mathbb{C}} ((X_1,X_2,X_3,X_4,X_5))\vert{\mathbb{C}}$, $v=\nu_t\circ\Psi$. The residues of $X_i/X_1$ are not in ${\mathbb{C}}$ for $i=2,3,4,5$.
Let us put $u_2=\sigma (X_2/X_1+{\got m}_v )$, a transcendental element over ${\mathbb{C}}$. By remark \ref{remark4} we know how to construct $\sigma$ step by step, so let take us $u_2=X_2/X_1$ and $\Delta_2={\mathbb{C}}(u_2)$.
The residue $X_3/X_1+{\got m}_v$ is algebraic over ${\mathbb{C}} (u_2)$, in fact $$\frac{X_3}{X_1}+{\got m}_v =\frac{X_2^2}{X_1^2}+{\got m}_v .$$ So we can take $u_{3,1}=\sigma ((X_3/X_1)+{\got m}_v )=u_2^2$. The value of $X_3-u_{3,1}X_1$ is 2, therefore we have to see if the residue $$\frac{X_3-u_{3,1}X_1}{X_1^2}+{\got m}_v$$ is algebraic over ${\mathbb{C}} (u_2)$. We have that $$\frac{X_3-u_{3,1}X_1}{X_1^2}+{\got m}_v = \frac{X_2}{X_1}+{\got m}_v ,$$ so it is algebraic and we can take $u_{3,2}=u_2$. Now $v(X_3-u_{3,1}X_1-u_{3,2}X_1^2)=3$ and we have to check if $$\frac{X_3-u_{3,1}X_1-u_{3,2}X_1^2}{X_1^3}+{\got m}_v$$ is algebraic over $\Delta_2$. In this case, as $$\Psi\(\frac{X_3-u_{3,1}X_1-u_{3,2}X_1^2}{X_1^3}+{\got m}_v\)= T_3 ,$$ this residue is transcendental. So we take $$u_3=\sigma\(\frac{X_3-u_{3,1}X_1-u_{3,2}X_1^2}{X_1^3}+{\got m}_v\) =\frac{X_1X_3-X_2^2-X_1^2X_2}{X_1^4} .$$ Let us take $\Delta_3={\mathbb{C}} (u_2,u_3)$.
We have to apply this procedure to $X_4$. The residue $X_4/X_1+{\got m}_v$ is algebraic over $\Delta_3$ because $$\frac{X_4}{X_1}+{\got m}_v =\frac{X_2^3}{X_1^3}+{\got m}_v ,$$ so we can take $u_{4,1}=\sigma ((X_4/X_1)+{\got m}_v )=u_2^3\in\Delta_3.$ Now $v(X_4-u_{4,1}X_1)=2$, and we have to check what happens with the residue $$\frac{X_4-u_{4,1}X_1}{X_1^2}+{\got m}_v .$$ As $$\frac{X_4-u_{4,1}X_1}{X_1^2}+{\got m}_v = \frac{X_1^2}{X_2^2}+{\got m}_v ,$$ it holds $$u_{4,2}=\sigma\(\frac{X_4-u_{4,1}X_1}{X_1^2}+{\got m}_v\) =u_2^2.$$ Clearly $v(X_4-u_{4,1}X_1-u_{4,2}X_1^2)=3$ and $$\frac{X_4-u_{4,1}X_1-u_{4,2}X_1^2}{X_1^3}+{\got m}_v =\frac{X_1X_3-X_2^2-X_1^2X_2}{X_1^4}+{\got m}_v ,$$ therefore $$u_{4,3}=\sigma\(\frac{X_4-u_{4,1}X_1-u_{4,2}X_1^2}{X_1^3}+{\got m}_v\) =u_3.$$ The following residue is transcendental because $v(X_4-u_{4,1}X_1-u_{4,2}X_1^2-u_{4,3}X_1^3)=4$ and $$\Psi\(\frac{X_4-u_{4,1}X_1-u_{4,2}X_1^2-u_{4,3}X_1^3}{X_1^4}\)=T_4.$$ Then we can take $$u_4=\sigma\(\frac{X_4-u_{4,1}X_1-u_{4,2}X_1^2-u_{4,3}X_1^3}{X_1^4}+{\got m}_v\) =$$ $$=\frac{X_1^2X_4-X_3^2-X_1^2X_2^2-X_1^2X_3-X_1X_2^2-X_1^2X_2}{X_1^6}.$$ So $\Delta_4={\mathbb{C}} (u_2,u_3,u_4)$.
With the variable $X_5$ we obtain the next algebraic residues $$u_{5,j}=\sigma\(\frac{X_5-u_{5,1}X_1-\cdots -u_{5,j-1}X_1^{j-1}}{X_1^j}+{\got m}_v\) =u_4^{1/p^j}$$ for all $j\geq 1$. So we have $\Delta_5={\mathbb{C}} (u_2,u_3,u_4)(\{ u_4^{1/p^j}\}_{j\geq 1} )$, an algebraic extension of $\Delta_4$.
Then the residue field of $v$ is $$\Delta_v ={\mathbb{C}}\(\frac{X_2}{X_1}+{\got m}_v ,\frac{X_1X_3-X_2^2-X_1^2X_2}{X_1^4}+{\got m}_v , \right.$$ $$\left.\frac{X_1^2X_4-X_3^2-X_1^2X_2^2-X_1^2X_3-X_1X_2^2-X_1^2X_2}{X_1^6}+{\got m}_v\)
\(\left\{\(\frac{X_2}{X_1}+{\got m}_v\)^{1/p^j}\right\}_{j\geq 1}\).$$
In this case, by the transformation $$ \begin{array}{rcl} X_1 & \longrightarrow & Y_1 \\ X_2 & \longrightarrow & Y_2 \\ X_3 & \longrightarrow & Y_1^2Y_3+u_{3,1}Y_1+u_{3,2}Y_1^2 \\ X_4 & \longrightarrow & Y_1^3Y_4+u_{4,1}Y_1+u_{4,2}Y_1^2+u_{4,3}Y_1^3 \\ X_5 & \longrightarrow & Y_5, \end{array} $$ we can extend the valuation $v$ to a discrete valuation $v'=\nu_t\Psi'$ of ${\mathbb{C}} ((Y_1,Y_2,Y_3,Y_4,Y_5))$, with the injective homomorphism $$ \begin{array}{rcl} \Psi' : {\mathbb{C}}{\rm [\kern - 1.8pt [} Y_1,Y_2,Y_3,Y_4,Y_5{\rm ]\kern - 1.8pt ]} & \longrightarrow &
\Delta{\rm [\kern - 1.8pt [} t{\rm ]\kern - 1.8pt ]} \\ Y_1 & \longmapsto & t \\ Y_i & \longmapsto & T_it,\ i=2,3,4\\
Y_5 & \longmapsto & \sum_{j\geq 1} (T_4^{1/p}t)^j. \end{array} $$ The restriction $v'_{\vert{\mathbb{C}} ((Y_1,Y_2,Y_3,Y_4))}$ is the usual order function. This is not the general case because $\Psi'$ may not be injective. \end{exmp}
\section{Rank one discrete valuations and order functions}
We can summarize the constructions of previous sections in the following theorem wich generalize the results of \cite{Bri2,Br-He}
\begin{thm}{\label{teorema212}} Let $v$ be a discrete valuation of $K_n\vert k$, then \begin{enumerate} \item If the dimension of $v$ is $n-1$, we can embed $k\lcorX_1,\ldots ,X_n{\rm ]\kern - 1.8pt ]}$ into a ring $L\lcorY_1,\ldots ,Y_n{\rm ]\kern - 1.8pt ]}$, where $L\subset\sigma (\Delta_{\wh{v}} )$ and the extended valuation of $v$ over the field $L((Y_1,\ldots ,Y_n ))$ is the usual order function. \item If the dimension of $v$ is $m-1<n-1$, we can embed $k\lcorX_1,\ldots ,X_n{\rm ]\kern - 1.8pt ]}$ into a ring $L\lcorY_1,\ldots ,Y_n{\rm ]\kern - 1.8pt ]}$, where $L\subset\sigma (\Delta_{\wh{v}} )$ and the restriction into $L((Y_1,\ldots ,Y_m ))$ of the ``extended valuation'' of $v$ over $L((Y_1,\ldots ,Y_n ))$ is the usual order function. \end{enumerate} \end{thm}
\begin{pf} We have the following map: $$ \begin{array}{rcl} \varphi ':L_n(( Y_1,\ldots ,Y_n )) & \longrightarrow & \Delta_n((t )) \\ Y_1 & \longmapsto & t \\ Y_i & \longmapsto & u_it,\ i=2,\ldots ,m \\ Y_k & \longmapsto & \sum_{j\geq 1}u_{k,j}t^j,\ u_{k,1}\ne 0,\ k=m+1,\ldots ,n, \end{array} $$ where $m-1$ is the dimension of $v$. Let us prove the theorem:
\begin{enumerate}
\item In the case $m=n$, $\varphi '(Y_i)=u_it$ for all $i=2,\ldots ,n$. Let $\nu_t$ be the usual order funtion over $\Delta_n ((t))$. The homomorphism $\varphi '$ is injective and the valuation $v'=\nu_t\circ\varphi '$ of $L((Y_1,\ldots ,Y_n ))$ is the usual order fuction over this field. Obviously $v'$ extends $v$.
\item If $m<n$ we can consider the elements $W_k=Y_k-\sum_{j\geq 1}u_{k,j}Y_1^j$. Hence we have $L ((Y_1,\ldots ,Y_n)) = L((Y_1,\ldots ,Y_m,W_{m+1},\ldots ,W_n))$. We define the discrete valuation of rank $n-m+1$ over $L(( Y_1,\ldots ,Y_n ))$: $$v' (Y_1)=\ldots =v'(Y_m)= (0,\ldots ,0,1),$$ $$v'(W_{m+1})=(0,\ldots ,1,0),\ldots ,v'(W_n)=(1,0,\ldots ,0).$$ The restriction of this valuation to $K_n$ is a rank one discrete valuation, because the value of any element is in $0\times\cdots\times 0\times{\mathbb{Z}}$. In fact $v'(f)=(0,\ldots ,0,v(f))$ for all $f\in K_n$, so $v'$ ``extends'' $v$ in this sense. Obviously $v'_{\vert L((Y_1,\ldots ,Y_m ))}$ is the usual order function. We want note that this ideal $(W_{m+1},\ldots ,W_{n})$ is the {\em implicit ideal of $v$} that appears in some works of M. Spivakovsky (\cite{Sp}). \end{enumerate} \qed \end{pf}
For the case of valuations of dimension $n-1$, we can combine corollary \ref{cor8} and assertion 1 of the previous theorem:
\begin{cor}{\label{theorem8}} Let $v$ be a discrete valuation of $K_n\vert k$. The following conditions are equivalent:
1) The transcendence degree of $\Delta_{\wh{v}}$ over $k$ is $n-1$.
2) There exists a finite sequence of monoidal transformations and
coordinates changes which take $v$ into an order function. \end{cor}
\begin{exmp} Let us consider the homomorphism $$ \begin{array}{rcl} \Psi : {\mathbb{C}}{\rm [\kern - 1.8pt [} X_1,X_2,X_3,X_4,X_5{\rm ]\kern - 1.8pt ]} & \longrightarrow &
\Delta{\rm [\kern - 1.8pt [} t{\rm ]\kern - 1.8pt ]} \\ X_1 & \longmapsto & t \\ X_2 & \longmapsto & T_2t \\ X_3 & \longmapsto & T_2^2t+T_2t^2+T_3t^3 \\ X_4 & \longmapsto & T_2^3t+T_2^2t^2+T_3t^3+T_4t^4\\ X_5 & \longmapsto & T_2t\left(\sum_{j\geq 1} a_j(T_4t)^j\right) , \end{array} $$ with $a_j\in{\mathbb{C}}$ such that $\Psi$ is injective (we can take $\sum_{j\geq 1} a_j(T_4t)^j= e^{T_4t}-1$). Then the residue field of this valuation (see example \ref{ejemplo}) is $$\Delta_v ={\mathbb{C}}\(\frac{X_2}{X_1}+{\got m}_v ,\frac{X_1X_3-X_2^2-X_1^2X_2}{X_1^4}+{\got m}_v , \right.$$ $$\left.\frac{X_1^2X_4-X_3^2-X_1^2X_2^2-X_1^2X_3-X_1X_2^2-X_1^2X_2}{X_1^6}+{\got m}_v\) .$$ By the transformation (see example \ref{ejemplo}) $$ \begin{array}{rcl} X_1 & \longrightarrow & Y_1 \\ X_2 & \longrightarrow & Y_2 \\ X_3 & \longrightarrow & Y_1^2Y_3+u_{3,1}Y_1+u_{3,2}Y_1^2 \\ X_4 & \longrightarrow & Y_1^3Y_4+u_{4,1}Y_1+u_{4,2}Y_1^2+u_{4,3}Y_1^3 \\ X_5 & \longrightarrow & Y_2Y_5, \end{array} $$ we obtain a new field ${\mathbb{C}} ((Y_1,Y_2,Y_3,Y_4,Y_5))$, but we can not extend $v$ to this field because the homomorphism $$ \begin{array}{rcl} \Psi' : {\mathbb{C}}{\rm [\kern - 1.8pt [} Y_1,Y_2,Y_3,Y_4,Y_5{\rm ]\kern - 1.8pt ]} & \longrightarrow &
\Delta{\rm [\kern - 1.8pt [} t{\rm ]\kern - 1.8pt ]} \\ Y_1 & \longmapsto & t \\ Y_i & \longmapsto & T_it,\ i=2,3,4 \\
Y_5 & \longmapsto & \sum_{j\geq 1} a_j(T_4t)^j \end{array} $$ is not injective. Then let us take $W_5=Y_5-\sum_{j\geq 1} a_j(Y_4)^j$ (because we can consider $T_4Y_1=Y_4$). Then ${\mathbb{C}} ((Y_1,Y_2,Y_3,Y_4,Y_5))={\mathbb{C}} ((Y_1,Y_2,Y_3,Y_4,W_5))$ and the discrete valuation of rank 2 defined by $v'(Y_i)=(0,1)$ for $i=1,\ldots ,4$ and $v'(W_5)=(1,0)$ is such that for all $f\in{\mathbb{C}} ((X_1,X_2,X_3,X_4,X_5))$ we have $v'(f)=(0,v(f))$ and $v'_{\vert{\mathbb{C}} ((Y_1,Y_2,Y_3,Y_4))}$ is the usual order function. \end{exmp}
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\end{document} | arXiv |
<< Tuesday, February 12, 2019 >>
Winter at the Hall
Special Event | December 21, 2018 – March 20, 2019 every day | Lawrence Hall of Science
This winter, visit the Hall for interactive exhibits, special hands-on activities, intriguing Planetarium shows, and more!
VC Unlocked: Deal Camp
Course | February 11 – 14, 2019 every day | 8:30 a.m.-5:30 p.m. | Boalt Hall, School of Law
Sponsors: Berkeley Law Executive Education, Startup@BerkeleyLaw, 500 Startups
Deal Camp is a four-day course focused on the nuts and bolts of deal making for investors who want to improve their ability to define, negotiate, and execute early-stage investments. Participants will work with leading UC Berkeley faculty and 500 Partners to develop strategies to structure deals in order to maximize investment returns.
This activity has been approved for 25.75 hours MCLE... More >
Venture Capital Deal Camp with Startup@BerkeleyLaw and 500 Startups
Course | February 10 – 13, 2019 every day | 9 a.m.-5 p.m. | Boalt Hall, School of Law
Sponsors: Berkeley Center for Law, Business and the Economy, 500 Startups, Startup@BerkeleyLaw, Berkeley Law Executive Education
Understanding Diversity Among Species
Seminar | February 12 | 9-9:30 a.m. | Barrows Hall, Radio Broadcast, ON-AIR ONLY, 90.7 FM
Speakers/Performers: Nicolas Alexandre, Department of Integrative Biology; Ashley Smiley, Department of Integrative Biology
Sponsor: KALX 90.7 FM
Nicolas with hummingbird feeder
February Open Berkeley Site Builder Training
Workshop | February 12 | 9:30-11:30 a.m. | 104A Banway Building
Location: 2111 Bancroft Way, Berkeley, CA 94720
Sponsor: Information Services and Technology (IST)
Open Berkeley Site Builder Training sessions cover the fundamentals of the Open Berkeley turnkey website solution.
Summer Abroad in Poland, Czech Republic, Germany
Information Session | February 12 | 10 a.m. | 270 Stephens Hall
Sponsor: Berkeley Study Abroad
Learn more about the Summer Abroad in Poland, Czech Republic, Germany: The History of Belonging and Coexistence in Modern Europe program at this info session. Program advisers and faculty will be on hand to answer your questions.
Who "belongs" in Europe and how do refugee communities and ethnic minorities fit into the European Union experiment?
Dive deep into the history of coexistence... More >
H-1B Workshop
Workshop | February 12 | 10 a.m.-12 p.m. | International House, Sproul Rooms
Sponsor: Berkeley International Office(BIO))
The main focus of this workshop are general H-1B eligibility requirements, eligible professional occupations, application process, and timing concerns.
H-1B eligibility criteria
Types of jobs appropriate for H-1B
Minimum salary requirements
Employer's role
Application timing challenges
Options for F/J students/scholars
Seminar 217, Risk Management: Computation of Optimal Conditional Expected Drawdown Portfolios
Seminar | February 12 | 11 a.m.-12:30 p.m. | 1011 Evans Hall
Featured Speaker: Speakers: Alex Papanicolaou, Intelligent Financial Machines
Sponsor: Consortium for Data Analytics in Risk
We introduce two approaches to computing and minimizing the risk measure Conditional Expected Drawdown (CED) of Goldberg and Mahmoud (2016). One approach is based on a continuous-time formulation yielding a partial differential equation (PDE) solution to computing and minimizing CED while another is a sampling based approach utilizing a linear program (LP) for minimizing CED.
Sustainability – Why and How? The Nordic Way
Lecture | February 12 | 12-1 p.m. | 201 Moses Hall
Speaker/Performer: Ambassador Ove Ullerup, Royal Danish Embassy in Sweden
Sponsors: Institute of European Studies, Center for Responsible Business, Nordic Studies Program
In his talk, the Danish ambassador to Sweden, Ove Ullerup, will focus on the relationship and cooperation between the public and private sector on the sustainability agenda in the Nordic countries.
The ambassador discusses challenges in changing concepts and how the Nordic countries will face these in the future. What role will the UN Sustainable Development Goals play and have they changed our... More >
Ove Ullerup
SPH Brown Bag Research Presentation: Impact of microbial biological disparity on social determinants of health
Seminar | February 12 | 12-1 p.m. | 5101 Berkeley Way West
Speaker: Lee Riley, Professor of Epidemiology and Infectious Diseases, School of Public Health
Sponsor: Public Health, School of
Understanding social determinants of health has become established as one of core disciplines of public health. On the other hand, understanding biological determinants of health has gradually become less emphasized as a public health discipline, and this trend may lead to incomplete and misdirected public health interventions and policy. We present results of our study of rheumatic heart disease... More >
Darwin Day at the Essig Museum: Happy Birthday, Charles Darwin (1809)
Tour/Open House | February 12 | 12-5 p.m. | 1170 Valley Life Sciences Building
Speaker/Performer: Peter Oboyski, Essig Museum of Entomology
Sponsor: Essig Museum of Entomology
Celebrate Charles Darwin's birthday with special behind-the-scenes tours of the Essig Museum insect collection in the Valley Life Sciences Building.Also on display will be Galapagos finches, rodents, a marine iguana, and tortoise shell, herbarium specimens, live orchids and insectivorous plants.
Plants and People Lunchtime Lectures: The Ethnobotany of a Medicinal Moss
Lecture | February 12 | 12-1 p.m. | UC Botanical Garden
As a part of our "Year of Ethnobotany" celebrations, the Garden will be hosting monthly lunch time lectures featuring the research of UC Berkeley graduate students, post-docs, and faculty.
In February join, Eric Harris to learn all about mosses and their use in human life.
Registration recommended: Free with Garden Admission
Registration info: Register online or by calling 510-664-7606
Certificate Program in Marketing Online Information Session
Information Session | February 12 | 12-1 p.m. | Online
Speaker/Performer: Tom McGuire, Program Director, UC Berkeley Extension
Find out how UC Berkeley Extension equips you with a solid understanding of marketing's most up-to-date concepts and techniques. For more information, visit the Certificate Program in Marketing.
Reservation info: Make reservations online
Student Faculty Macro Lunch - "Consumption-led Growth"
Presentation | February 12 | 12-1 p.m. | 639 Evans Hall
Speaker: Pierre-Olivier Gourinchas, Professor of Economics, UC Berkeley
Sponsor: Clausen Center
Consumption-led Growth, joint with Markus Brunnermeier and Oleg Itskhoki.
RSVP info: RSVP by emailing [email protected] by February 8.
Make-Ahead Meals (BEUHS641)
Workshop | February 12 | 12:10-1 p.m. | Tang Center, University Health Services, Section Club
Speaker: Kim Guess, RD
Sponsor: Be Well at Work - Wellness
Make it easier on yourself to enjoy homemade meals by making them ahead of time! Some meals can be prepared in a large batch and reused or frozen for future use, while others can be prepped and then cooked later. However, this doesn't work with all recipes, so attend this workshop and skip the guesswork. Demonstration, recipes, and samples provided.
IB Seminar: An Alarming Forest: The ecological causes and consequences of eavesdropping in an Amazonian bird community
Seminar | February 12 | 12:30-1:30 p.m. | 2040 Valley Life Sciences Building
Featured Speaker: Ari Martinez, University of California, Berkeley
Sponsor: Department of Integrative Biology
Summer Abroad Programs Rooted in Social Justice: Berkeley Summer Abroad and Global Internships
Information Session | February 12 | 1-3 p.m. | Martin Luther King Jr. Student Union, Multicultural Center
Learn about the Berkeley Summer Abroad and Global Internship programs that utilize social justice and anti-oppression methodologies. Advisers will also give an overview of all the programs offered through Summer Abroad and Global Internships. Presentation runs until 1:45 p.m., then advisers will be available for individual advising until 3 p.m.
Seminar 218, Psychology and Economics: Visual Salience in Game Theory
Seminar | February 12 | 2-3:30 p.m. | 648 Evans Hall
Featured Speaker: Colin Camerer, California Institute of Technology
Sponsor: Department of Economics
Seminar 237/281: Macro/International Seminar - "Expectations with Endogenous Information Acquisition: An Experimental Investigation"
Seminar | February 12 | 2-4 p.m. | 597 Evans Hall
Speaker: Basit Zafar, Associate Professor of Economics, Arizona State University
RSVP info: RSVP by emailing Joseph G. Mendoza at [email protected]
Librarian Office Hours at the SPH DREAM Office
Miscellaneous | February 5 – April 30, 2019 every Tuesday with exceptions | 3-5 p.m. | Berkeley Way West, 2220 (DREAM Office)
Speaker/Performer: Debbie Jan
Drop by during office hours if you need help with your literature reviews; setting up searches in PubMed, Embase, and other databases; using EndNote, RefWorks, or other citation management software; finding statistics or data; and answering any other questions you may have.
Gilman Scholarship Application Workshop
Workshop | February 12 | 3:30 p.m. | Martin Luther King Jr. Student Union, Stephens Lounge
Are you a Pell Grant recipient and U.S. citizen who is planning to study and/or intern abroad? Attend the Gilman Scholarship Application Workshop and learn about the Benjamin A. Gilman International Scholarship, a federal scholarship which funds up to $5,000 towards a study abroad experience.
The Gilman Scholarship Application Workshop will review scholarship and eligibility requirements,... More >
3-Manifold Seminar: The arithmeticity of figure-eight knot orbifolds
Seminar | February 12 | 3:40-5 p.m. | 736 Evans Hall
Speaker: Kyle Miller, UC Berkeley
A knot or link $L$ in $S^3$ is called universal if every closed orientable $3$-manifold can be represented as a cover branched over $L$, with some examples including the Borromean rings, the figure-eight knot, and 2-bridge non-torus links. Some such $L$ are the singular locus of an orbifold ${\mathbb H}^3/\Gamma \cong S^3$ for Γ an arithmetic subgroup of the linear algebraic group of isometries... More >
Commutative Algebra and Algebraic Geometry: The Fellowship of the Ring: Terminal singularities that are not Cohen-Macaulay
Seminar | February 12 | 3:45-4:45 p.m. | 939 Evans Hall
Speaker: Burt Totaro, UCLA
I will explain the notion of terminal singularities. This is the mildest class of singularities that appears in constructing minimal models of algebraic varieties. In characteristic zero, terminal singularities are automatically Cohen-Macaulay, and this is very useful for the minimal model program. I will present the first known terminal singularity of dimension 3 which is not Cohen-Macaulay; it... More >
Student Hosted Colloquium in Physical Chemistry: Computational Vibrational Spectroscopy of Aqueous Acid and Base Solutions
Seminar | February 12 | 4-5 p.m. | 120 Latimer Hall
Featured Speaker: Steve Corcelli, Department of Chemistry & Biochemistry, University of Notre Dame
The structure, dynamics, and transport properties of electrolyte solutions, including acids and bases, is critically important to aqueous solution chemistry. Aqueous acid and base solutions have vibrational spectra with distinct continua that span from about 1000 cm-1 to 3000 cm-1. Despite intense study, the interpretation of the spectra in terms of the molecular structure of the hydrated proton... More >
Restraining Great Powers: Soft Balancing From Empires To The Global Era
Lecture | February 12 | 4-5:30 p.m. | 223 Moses Hall
Speaker/Performer: T.V. Paul, McGill University
Sponsors: Institute of International Studies, Institute of East Asian Studies (IEAS), Institute for South Asia Studies
This presentation is based on the book with the same title (Yale University Press, 2018) which examines a crucial element of state behavior -- the use of international institutions, informal alignments and economic instruments such as sanctions -- to constrain the power and threatening behavior of dominant actors. Much of International Relations scholarship fails to capture the use of these... More >
Catalyst Group to End Youth Homelessness Meeting
Meeting | February 12 | 4-5 p.m. | 5400 Berkeley Way West
Sponsor: Innovations 4 Youth
The Catalyst Group to End Youth Homelessness aims to create synergy between research, policy, and practice to eliminate youth homelessness in the Bay Area and on the UC Berkeley campus. Our group brings students, faculty, staff, and community partners together to leverage individual expertise of lived experiences of homelessness, research, and advocacy. We welcome all to attend a meeting and work... More >
Open House and Info Session
Information Session | February 12 | 5-6:30 p.m. | 340 Moffitt Undergraduate Library
Sponsor: The Program in Critical Theory
Join The Program in Critical Theory's faculty and students for a panel discussion and Q&A about the Designated Emphasis (DE) in Critical Theory. All UC Berkeley Ph.D. students interested in applying to the DE are invited to attend. Refreshments and informal social to follow.
The Program in Critical Theory's DE enables graduate students already enrolled in UC Berkeley Ph.D. programs from across... More >
Rethinking Diversity, Reintroducing Disability: Conversations on Disability History, Advocacy, and Rights
Colloquium | February 12 | 5-7 p.m. | Career Center (2440 Bancroft Way)
Sponsor: OPEN (Opportunities for Postdoc Equity Networking)
Disability is too often left out of the academic and political discourse concerning equity, diversity and inclusion. Yet disability is a core aspect of these values and intersects with all other marginalized groups for which
equity, diversity, and inclusion are critical. Join us for a trailer screening of "Crip Camp" by filmmaker and disability rights activist James Lebrecht, followed by a... More >
Asian Americans and Affirmative Action
Conference/Symposium | February 12 | 5-7:15 p.m. | Boalt Hall, School of Law
Speakers/Performers: Frank Wu, Keynote Speaker, UC Hastings; Vincent Pan, Chinese for Affirmative Action; Julie Park, University of Maryland
Sponsor: ASUC (Associated Students of the University of California)
Commutative Algebra and Algebraic Geometry: The Fellowship of the Ring: Symmetric powers of algebraic and tropical curves
Speaker: Madeline Brandt, UC Berkeley
In this talk I will give a description of the following recent result: the non-Archimedean skeleton of the d-th symmetric power of a smooth projective algebraic curve X is naturally isomorphic to the d-th symmetric power of the tropical curve that arises as the non-Archimedean skeleton of X. In the talk I will give all necessary background definitions for understanding the above statement and I... More >
Sincerity out, Authenticity in: Poetry on the Quest for Trust in the times of Post-Truth
Lecture | February 12 | 5:30-7 p.m. | B-4 Dwinelle Hall
Speaker: Stanislav Lvovsky, Auhtor
Sponsors: Institute of Slavic, East European, and Eurasian Studies (ISEEES), Department of Slavic Languages and Literatures
Back then in the first half of 1990s new generation of Russian poets, — or its considerable part — found itself facing the challenge of inventing a new way to speak straightforwardly: readily available poetics either weren't quite fit for the job or themselves were part of the problem to be resolved. Poetry optics, which has emerged at the time in the capacity of the solution, was the "new... More >
Vicarious Trauma Workshop
Meeting | February 12 | 6-8 p.m. | 10 Boalt Hall, School of Law
Sponsor: International Human Rights Workshop
A workshop on understanding, recognizing, and addressing trauma workshop hosted by the International Human Rights Workshop.
Event is ADA accessible. For disability accommodation requests and information, please contact Disability Access Services by phone at 510.643.6456 (voice) or 510.642.6376 (TTY) or by email at [email protected].
Sponsored by the Graduate Assembly.
Food Politics 2019: Nutrition Science Under Siege with Marion Nestle
Lecture | February 12 | 6-8 p.m. | North Gate Hall, Logan Multimedia Center (Room 142)
Sponsors: Graduate School of Journalism, Berkeley Food Institute, UC Berkeley-11th Hour Food and Farming Journalism Fellowship
Please join us for a special lecture series with celebrated author and scholar Marion Nestle. "Food Politics 2019: Nutrition Science Under Siege." Nutrition science is under attack from statisticians and the food industry. Who stands to gain and what might be lost?
Photo: Bill Hayes
IBM Networking Night
Information Session | February 12 | 6-7:30 p.m. | Soda Hall, Wozniak Lounge (430)
What is like to work at IBM?
Join us as IBMers in a variety of technical roles share day-in-the-life stories about their work and provide an inside look on what makes engineering at IBM unique.
Are you submitting applications this semester?
We'll also share some insight into IMB's recruiting process.
Feel free to bring questions, as we'll close with a panel and Q&A.
Discovery and Diversity: Critical Factors in Tomorrow's Health Care: Dean of the Stanford School of Medicine Lloyd B. Minor
Lecture | February 12 | 6-7:30 p.m. | 125 Morrison Hall
Speaker/Performer: Lloyd B. Minor, Stanford University School of Medicine
How are medical schools adapting to today's greatest challenges? Join the Dean of Stanford University School of Medicine, Lloyd B. Minor M.D., for a discussion on the ways in which medical education is changing and responding to unique social demands. Dean Minor will specifically discuss the challenges in biomedical discovery and the vital importance of diversity in science, while showcasing how... More >
Surprising Evidence:: Revealing Vanished Landscapes Through Nontraditional Moving Images
Lecture | February 12 | 6:30-8 p.m. | Wurster Hall, Wurster Auditorium, RM 112
Speaker/Performer: Rick Prelinger, University of California, Santa Cruz
Sponsor: Environmental Design Archives
Join us for our second Gallery Talk!
Wurster Auditorium, Room 112
6:30 to 7pm - Light Refreshments
7 to 8pm - Lecture
Free to UC Berkeley Students, Staff, Faculty, and Friends of the EDA
Suggested $10 donation for those outside UC Berkeley
Historians, architects and planners often look to feature films as records of extinct environments. But few know... More >
Berkeley Summer Abroad in Portugal: Entrepreneurship and Innovation
Information Session | February 12 | 7 p.m. | Memorial Stadium, Sutardja Center for Entrepreneurship and Technology - SCET
Come hear about this unique program where you can not only study, but also travel in Portugal with over 400 students from around the world at the European Innovation Academy!
This program is open to ALL MAJORS and will be hosted in Porto and Cascais, Portugal (right next to Lisbon).
Earn 6 units of upper division IEOR course credits:
IND ENG 185: Challenge Lab (4 units)
IND ENG 192:... More >
Cal at Golden State Warriors: vs. Utah Jazz
Sport - Recreational | February 12 | 7:30-10:30 p.m. | Oracle Arena
CrafterDark: Make V-Day Pillows and Cards!
Workshop | February 12 | 8-10 p.m. | Martin Luther King Jr. Student Union
Sponsor: Berkeley Art Studio
Come make adorbs V-Day themed pillows and cards! Hate Valentine's day? you can make non-V-Day themed stuff too!
Exhibits and Ongoing Events
Fiat Yuks: Cal Student Humor, Then and Now
Exhibit - Artifacts | October 13, 2017 – May 30, 2019 every Monday, Tuesday, Wednesday, Thursday, Friday & Saturday | Bancroft Library, Rowell Cases, near Heyns Reading Room, 2nd floor corridor between The Bancroft Library and Doe
Let there be laughter! This exhibition features Cal students' cartoons, jokes, and satire from throughout the years, selected from their humor magazines and other publications.
Art for the Asking: 60 Years of the Graphic Arts Loan Collection at the Morrison Library
Exhibit - Artifacts | September 17, 2018 – February 28, 2019 every day | Doe Library, Bernice Layne Brown Gallery
Art for the Asking: 60 Years of the Graphic Arts Loan Collection at the Morrison Library will be up in Doe Library's Brown Gallery until March 1st, 2019. This exhibition celebrates 60 years of the Graphic Arts Loan Collection, and includes prints in the collection that have not been seen in 20 years, as well as prints that are now owned by the Berkeley Art Museum. There are also cases dedicated... More >
Boundless: Contemporary Tibetan Artists at Home and Abroad
Exhibit - Painting | October 3, 2018 – May 26, 2019 every day | Berkeley Art Museum and Pacific Film Archive
Sponsor: Berkeley Art Museum and Pacific Film Archive
Featuring works by internationally renowned contemporary Tibetan artists alongside rare historical pieces, this exhibition highlights the ways these artists explore the infinite possibilities of visual forms to reflect their transcultural, multilingual, and translocal lives. Though living and working in different geographical areas—Lhasa, Dharamsala, Kathmandu, New York, and the Bay Area—the... More >
Dimensionism: Modern Art in the Age of Einstein
Exhibit - Painting | November 7, 2018 – March 3, 2019 every day | Berkeley Art Museum and Pacific Film Archive
In the early twentieth century, inspired by modern science such as Albert Einstein's theory of relativity, an emerging avant-garde movement sought to expand the "dimensionality" of modern art, engaging with theoretical concepts of time and space to advance bold new forms of creative expression. Dimensionism: Modern Art in the Age of Einstein illuminates the remarkable connections between the... More >
Art Wall: Barbara Stauffacher Solomon
Exhibit - Painting | August 15, 2018 – March 3, 2019 every day | Berkeley Art Museum and Pacific Film Archive
The 1960s architectural phenomenon Supergraphics—a mix of Swiss Modernism and West Coast Pop—was pioneered by San Francisco–based artist, graphic and landscape designer, and writer Barbara Stauffacher Solomon. Stauffacher Solomon, a UC Berkeley alumna, is creating new Supergraphics for BAMPFA's Art Wall. Land(e)scape 2018 is the fifth in a series of temporary, site-specific works commissioned for... More >
Ink, Paper, Silk: One Hundred Years of Collecting Japanese Art
Exhibit - Painting | December 12, 2018 – April 14, 2019 every day | Berkeley Art Museum and Pacific Film Archive
BAMPFA's Japanese art collection began in 1919 with a remarkable donation of more than a thousand woodblock prints from the estate of UC Berkeley Professor of English William Dallam Armes. This exhibition features a selection of these exceptional prints, as well as hanging scroll paintings, screens, lacquerware, and ceramics that have entered the collection over the century since this... More >
Arthur Jafa / MATRIX 272
Exhibit - Multimedia | December 12, 2018 – March 24, 2019 every day | Berkeley Art Museum and Pacific Film Archive
Arthur Jafa is an artist, director, editor, and award-winning cinematographer whose poignant work expands the concept of black cinema while exploring African American experience and race relations in everyday life. He has stated, "I have a very simple mantra and it's this: I want to make black cinema with the power, beauty, and alienation of black music." In his renowned work Love Is The Message,... More >
Well Played! The Math and Science of Improving Your Game
Exhibit - Multimedia | November 17, 2018 – May 18, 2019 every day | Lawrence Hall of Science
You don't have to be a pro to know that math and science can help improve your game. In our exhibit, Well Played!, you can experiment with force, angles, and trajectory to get the highest scores you can with classic arcade games such as Skeeball, Pinball, and Basketball.
Want to improve your score? Try our interactive exhibits on the math and science behind force and trajectory, and then head... More >
The Book as Place: Visions of the Built Environment
Exhibit - Artifacts | January 28 – May 17, 2019 every day | Wurster Hall, Environmental Design Library, 210 Wurster Hall
This exhibition of artists' books centers on ideas about the built environment and has been curated by Berkeley-based book artist Julie Chen for UC Berkeley's Environmental Design Library. Featuring works by 25 artists including Robbin Ami Silverberg, Clifton Meador, Inge Bruggeman, Karen Kunc, Sarah Bryant and Barbara Tetenbaum, the exhibition explores the built environment through text, image,... More >
Exhibit - Artifacts | January 15 – May 17, 2019 every day | 210 Wurster Hall
Sponsor: Environmental Design, College of
This exhibition of artists books centers on ideas about the built environment, curated by Berkeley-based book artist Julie Chen for CEDs Environmental Design Library.
Illustrating México one page at a time-Print Art of José Guadalupe Posada.
Exhibit - Multimedia | February 8 – June 30, 2019 every day | Moffitt Undergraduate Library, 2nd floor
343386 N/A
In the pantheon of the late nineteenth- and early twentieth-century artists who represent Mexico and Mexican art, the artwork of José Guadalupe Posada stands out as a bright constellation that continues to shine a light on important stories through woodcuts, imprints, and engravings. This exhibition was created using the books from the collections of the Doe Library. The exhibition is envisioned... More >
Aaron Marcus: Early Works
Exhibit - Multimedia | February 6 – June 30, 2019 every day | Berkeley Art Museum and Pacific Film Archive
Educated in physics, mathematics, and philosophy at Princeton University and trained in graphic design at Yale, Berkeley-based Aaron Marcus explores new possibilities for expression. He created his first "computer-assisted poem-drawings" in the spring of 1972, when he served as a research associate at Yale University's School of Art and Architecture. Using standard typographical symbols, Marcus... More >
Bearing Light: Berkeley at 150
Exhibit - Artifacts | April 16, 2018 – February 28, 2019 every Monday, Tuesday, Wednesday, Thursday & Friday | 8 a.m.-5 p.m. | Bancroft Library, 2nd Floor Corridor
This exhibition celebrates the University of California's sesquicentennial anniversary with photographs, correspondence, publications, and other documentation drawn from the University Archives and The Bancroft Library collections. It features an array of golden bears, including Oski, and explores the illustrious history of UC Berkeley.
Facing West 1: Camera Portraits from the Bancroft Collection
Exhibit - Photography | November 9, 2018 – March 15, 2019 every Monday, Tuesday, Wednesday, Thursday & Friday | 10 a.m.-4 p.m. | Bancroft Library
The first part of a double exhibition celebrating the tenth anniversary of the renewed Bancroft Library and its gallery, Facing West 1 presents a cavalcade of individuals who made, and continue to make, California and the American West. These camera portraits highlight the communities and peoples of Hubert Howe Bancroft's original collecting region, which extended from the Rockies to the Pacific... More >
Facing West: Camera Portraits from the Bancroft Collection
Exhibit - Photography | November 29, 2018 – March 15, 2019 every Monday, Tuesday, Wednesday, Thursday & Friday | 10 a.m.-4 p.m. | Bancroft Library, Bancroft Gallery
Exhibit - Multimedia | January 29 – June 28, 2019 every Tuesday, Wednesday, Thursday & Friday | 11 a.m.-4 p.m. | Magnes Collection of Jewish Art and Life (2121 Allston Way)
Centering on coins in The Magnes Collection, this exhibition explores how... More >
Project "Holy Land": Yaakov Benor-Kalter's Photographs of British Mandate Palestine, 1923-1940
Exhibit - Photography | January 29 – June 28, 2019 every Tuesday, Wednesday, Thursday & Friday | 11 a.m.-4:05 p.m. | Magnes Collection of Jewish Art and Life (2121 Allston Way)
For nearly two decades, Yaakov (Jacob) Benor-Kalter (1897-1969) traversed the Old City of Jerusalem, documenting renowned historical monuments, ambiguous subjects in familiar alleyways, and scores of "new Jews" building a new homeland. Benor-Kalter's photographs smoothly oscillate between two worlds, and two Holy Lands, with one lens.
After immigrating from Poland to the British Mandate of... More >
The Worlds of Arthur Szyk: The Taube Family Arthur Szyk Collection
Exhibit - Painting | January 29 – June 28, 2019 every Tuesday, Wednesday, Thursday & Friday | 11 a.m.-4 p.m. | Magnes Collection of Jewish Art and Life (2121 Allston Way)
Auditorium installation of high-resolution images of select collection items.
Acquired by The Magnes Collection of Jewish Art and Life in 2017 thanks to an unprecedented gift from Taube Philanthropies, the most significant collection of works by Arthur Szyk (Łódź, Poland, 1894 – New Canaan, Connecticut, 1951) is now available to the world in a public institution for the first time as... More > | CommonCrawl |
Proving that $x-\left\lfloor \frac{x}{2} \right\rfloor\; =\; \left\lfloor \frac{x+1}{2} \right\rfloor$
How would you prove that if $x$ is an integer, then
$$x-\left\lfloor \frac{x}{2} \right\rfloor\; =\; \left\lfloor \frac{x+1}{2} \right\rfloor$$
I tried to start by saying that if $x$ is an even integer, then:
$$\left\lfloor \frac{x}{2} \right\rfloor = \frac{x}{2}.$$
However, I am stuck on showing that $$\left\lfloor \frac{x+1}{2} \right\rfloor$$ is also $\frac{x}{2}$. Intuitively It makes intuitive sense just by playing around with some sample numbers, but I don't know how to make it mathematically rigorous. Further, what do you do in the odd case? Is this even the right way to go about it?
proof-writing
ceiling-and-floor-functions
Martin Sleziak
$\begingroup$ Start with $\lfloor y+n \rfloor = \lfloor y\rfloor +n$. $\endgroup$
– lhf
$\begingroup$ @lhf Is there a proof for that? $\endgroup$
– 1110101001
$\begingroup$ By definition, $\lfloor y \rfloor \le y < \lfloor y \rfloor +1$ and so $\lfloor y \rfloor + n \le y + n < \lfloor y \rfloor +n+1$, which says that $\lfloor y+n \rfloor = \lfloor y\rfloor +n$. $\endgroup$
If $x$ is even, then $x=2k$ where $k \in \mathbb{Z}$.
We then have: $$x-\left\lfloor \frac{x}{2} \right\rfloor=2k-\lfloor k \rfloor=k=\left\lfloor k+\frac{1}{2} \right\rfloor=\left\lfloor \frac{2k+1}{2} \right\rfloor=\left\lfloor \frac{x+1}{2} \right\rfloor$$
If $x$ is odd, then $x=2k+1$ where $k \in \mathbb{Z}$.
We then have $$x-\left\lfloor \frac{x}{2} \right\rfloor=2k+1-\left\lfloor k+{1\over 2} \right\rfloor=k+1=\left\lfloor k+1 \right\rfloor=\left\lfloor \frac{2k+2}{2} \right\rfloor=\left\lfloor \frac{x+1}{2} \right\rfloor$$
FujoyakiFujoyaki
Case-1 (Even) write $x=2k$. Now$\left\lfloor \frac{x}{2} \right\rfloor =k$ so L.H.S is $2k-k=k$. now work with R.H.S, $\left\lfloor \frac{x+1}{2} \right\rfloor=\left\lfloor \frac{2k+1}{2} \right\rfloor=\left\lfloor k.5 \right\rfloor=k$.
Case 2) (Odd) Let $x=2k+1$ Similarly. I am in hurry, so i leave it on you. very easy.
Prove LHS=RHS= $k+1$
Bhaskar VashishthBhaskar Vashishth
Prove that $\lfloor(n+1)a\rfloor-1$ is divisible by $(n+1)$ if $n= \left\lfloor \frac {1}{ a- \lfloor a \rfloor } \right\rfloor$
How many $m$ such that : $\sum\limits_{k=1}^m \left\lfloor\frac{m}{k}\right\rfloor$ be even?
Prove that $ \left \lfloor {\log n} \right \rfloor = \left \lfloor {\log \left \lceil \frac {n-1}{2}\right \rceil} \right \rfloor + 1$
For which $n$ is $\frac{n!}{4}$ equal to $\left\lfloor \sqrt{\frac{n!}{4}}\right\rfloor\left(\left\lfloor\sqrt{\frac{n!}{4}}\right\rfloor + 1\right)$?
How to show either $\left \lfloor {\frac{m-1}{2}} \right \rfloor$ or $\left \lfloor {\frac{m+1}{2}} \right \rfloor$ odd and other is even?
$\left\lfloor\frac{8n+13}{25}\right\rfloor-\left\lfloor\frac{n-12-\left\lfloor\frac{n-17}{25}\right\rfloor}{3}\right\rfloor$ is independent of $n$
Conjecture: $\lfloor\frac{n}4\rfloor+\lfloor\frac{n+1}4\rfloor-\lfloor\frac{n+2}4\rfloor-\lfloor\frac{n+3}4\rfloor=\cos(\frac{n\pi}2)-1$ | CommonCrawl |
\begin{definition}[Definition:Contravariant Hom Functor]
Let $\mathbf{Set}$ be the category of sets.
Let $\mathbf C$ be a locally small category.
Let $C \in \mathbf C_0$ be an object of $\mathbf C$.
The '''contravariant hom functor based at $C$''':
: $\operatorname{Hom}_{\mathbf C} \left({\cdot, C}\right): \mathbf C \to \mathbf{Set}$
is the covariant functor defined by:
{{begin-axiom}}
{{axiom | lc= Object functor:
| m = \operatorname{Hom}_{\mathbf C} \left({B, C}\right) = \operatorname{Hom}_{\mathbf C} \left({B, C}\right)
}}
{{axiom | lc= Morphism functor:
| m = \operatorname{Hom}_{\mathbf C} \left({f, C}\right): \operatorname{Hom}_{\mathbf C} \left({B, C}\right) \to \operatorname{Hom}_{\mathbf C} \left({A, C}\right), g \mapsto g \circ f
| rc= for $f: A \to B$
}}
{{end-axiom}}
where $\operatorname{Hom}_{\mathbf C} \left({B, C}\right)$ denotes a hom set.
Thus, the morphism functor is defined to be precomposition.
\end{definition} | ProofWiki |
\begin{document}
\selectlanguage{english} \title{\textsc{Jonqui\`eres} maps and $\mathrm{SL}(2;\mathbb{C})$-cocycles}
\author{Julie \textsc{D\'eserti}} \address{Institut de Math\'ematiques de Jussieu-Paris Rive Gauche, UMR $7586$, Universit\'e Paris Diderot, B\^atiment Sophie Germain, Case $7012$, $75205$ Paris Cedex $13$, France.} \email{[email protected]}
\maketitle
\begin{abstract} We start the study of the family of birational maps $(f_{\alpha,\beta})$ of $\mathbb{P}^2_\mathbb{C}$ in~\cite{Deserti}. For "$(\alpha,\beta)$ well chosen" of modulus $1$ the centraliser of $f_{\alpha,\beta}$ is trivial, the topological entropy of~$f_{\alpha,\beta}$ is $0$, there exist two domains of linearisation: in the first one the closure of the orbit of a point is a torus, in the other one the closure of the orbit of a point is the union of two circles. On~$\mathbb{P}^1_\mathbb{C}\times \mathbb{P}^1_\mathbb{C}$ any $f_{\alpha,\beta}$ can be viewed as a cocyle; using recent results about $\mathrm{SL}(2;\mathbb{C})$-cocycles~(\cite{Avila}) we determine the \textsc{Lyapunov} exponent of the cocyle associated to $f_{\alpha,\beta}$. \\
\noindent\emph{$2010$ Mathematics Subject Classification. --- $37$F$10$, $14$E$07$} \end{abstract}
\section*{Introduction}
In this article we deal with a family of birational maps $(f_{\alpha,\beta})$ given by \[ f_{\alpha,\beta}\colon\mathbb{P}^2_\mathbb{C}\dashrightarrow\mathbb{P}^2_\mathbb{C}\quad\quad\quad (x:y:z)\dashrightarrow\big((\alpha x+y)z:\beta y(x+z):z(x+z)\big) \] where $\alpha$, $\beta$ denote two complex numbers with modulus $1$, case where we know almost nothing about the dynamics. Let us consider the set $\Omega$ of pairs of complex numbers of modulus $1$ that satisfy diophantine condition. The family $(f_{\alpha,\beta})$ satisfies the following properties (\cite{Deserti}): \begin{itemize} \item[$\bullet$] for $(\alpha,\beta)\in\Omega$ the centraliser of $f_{\alpha,\beta}$, that is the set of birational maps of $\mathbb{P}^2_\mathbb{C}$ that commutes with $f_{\alpha,\beta}$, is isomorphic to $\mathbb{Z}$;
\item[$\bullet$] the topological entropy of $f_{\alpha,\beta}$ is $0$;
\item[$\bullet$] rotation domains of ranks $1$ and $2$ coexist: there is a domain of linearisation where the orbit of a generic point under $f_{\alpha,\beta}$ is a torus, and there is an other domain of linearisation where the orbit of a generic point under $f_{\alpha,\beta}^2$ is a circle. \end{itemize}
We can also see $f_{\alpha,\beta}$ on $\mathbb{P}^1_\mathbb{C}\times\mathbb{P}^1_\mathbb{C}$ $\big($since all the computations of \cite{Deserti} have been done in an affine chart they may all be carried on $\mathbb{P}^1_\mathbb{C}\times\mathbb{P}^1_\mathbb{C}$$\big)$; the sets~$\mathbb{P}^1_\mathbb{C}\times\mathbb{S}^1_\rho$, where $\mathbb{S}^1_\rho=\{y\in\mathbb{C}\,\vert\,\vert y\vert=\rho\}$, are invariant.
Let us define $A_n^{\alpha,\rho}\colon\mathbb{S}^1_\rho\rightarrow\mathrm{M}(2;\mathbb{C})$ given in terms of $A^{\alpha,\rho}(y)=\left[\begin{array}{cc} \alpha & y \\ 1 & 1 \end{array}\right]$ by \[ A_n^{\alpha,\rho}(\cdot)=A^{\alpha,\rho}(\beta^n\,\cdot)A^{\alpha,\rho}(\beta^{n-1}\,\cdot)\ldots A^{\alpha,\rho}(\beta\, \cdot)A^{\alpha,\rho}(\cdot). \]
To compute $f_{\alpha,\beta}^n(x,y)$ is equivalent to compute~$A_n^{\alpha,\rho}(y)$ as soon as $f^k_{\alpha,\beta}(x,y)\not=(-1,\alpha)$ for any $1\leq k\leq n$.
Using \cite{Avila} we are able to determine the \textsc{Lyapunov} exponent of the cocycle~$(A^{\alpha,\rho},\beta)$:
\begin{theoalph}\label{Thm:main} {\sl The \textsc{Lyapunov} exponent of $(A^{\alpha,\rho},\beta)$ is \begin{itemize} \item[$\bullet$] positive as soon as $\rho>1$;
\item[$\bullet$] zero as soon as $\rho\leq 1$. \end{itemize}
More precisely $f_{\alpha,\beta}$ is semi conjugate to $\left(\frac{\alpha x+y^2}{x+1},\beta^{1/2}y\right)$ and the \textsc{Lyapunov} exponent of the cocycle $\big(B^{\alpha,\rho},\beta^{1/2}\big)$, where \[ B^{\alpha,\rho}(y)=\left[\begin{array}{cc} \alpha & y^2 \\ 1 & 1 \end{array}\right], \] is equal to $\max(0,\ln\rho)$.} \end{theoalph}
In the next section we introduce the family $(f_{\alpha,\beta})$ and its properties (\S\ref{Sec:falphabeta}). Then we deal with the recent works of \textsc{Avila} on $\mathrm{SL}(2;\mathbb{C})$-cocyles. In the last section we give the proof of Theorem \ref{Thm:main} (\emph{see} \S\ref{Sec:cocyles}). Let us explain the sketch of it. We associate to $\big(B^{\alpha,\rho},\beta^{1/2}\big)$ a cocycle $\big(\widetilde{B}^{\alpha,\rho},\beta^{1/2}\big)$ that belongs to~$\mathrm{SL}(2;\mathbb{C})$. We first determine \[ \displaystyle\lim_{\rho\to 0}L\big(\widetilde{B}^{\alpha,\rho},\beta^{1/2}\big), \] and then \[ \displaystyle\lim_{\rho\to +\infty}L\big(\widetilde{B}^{\alpha,\rho},\beta^{1/2}\big) \] where $L(C,\gamma)$ denotes the \textsc{Lyapunov} exponent of the $\mathrm{SL}(2;\mathbb{C})$-cocyle $(C,\gamma)$. In both cases, we get $0$. Using \cite[Theorem 5]{Avila} we obtain that $L\big(\widetilde{B}^{\alpha,\rho},\beta^{1/2}\big)$ vanishes everywhere; it allows us to determine $L\big(A^{\alpha,\rho},\beta\big)$ since \[ L\big(B^{\alpha,\rho}(y),\beta^{1/2}\big)=L\big(\widetilde{B}^{\alpha,\rho}(y),\beta^{1/2}\big)+\max(0,\ln\rho), \] and since $\big(A^{\alpha,\rho},\beta\big)$ and $\big(\beta^{1/2},B^{\alpha,\rho}\big)$ are conjugate.
\subsection*{Acknowledgment} I would like to thank Artur \textsc{Avila} for very helpful discussions, Dominique \textsc{Cerveau} for his constant support, and Serge \textsc{Cantat} for his remarks. Thanks also to the referee whose comments help me to improve the text.
\section{Some properties of the family $(f_{\alpha,\beta})$}\label{Sec:falphabeta}
A \textbf{\textit{rational map}} $\phi$ from $\mathbb{P}^2_\mathbb{C}$ into itself is a map of the form \[ (x:y:z)\dashrightarrow\big(\phi_0(x,y,z):\phi_1(x,y,z):\phi_2(x,y,z)\big), \] where the $\phi_i$'s are some homogeneous polynomials of the same degree without common factor;~$\phi$ is \textbf{\textit{birational}} if it admits an inverse of the same type. We will denote by $\mathrm{Bir}(\mathbb{P}^2_\mathbb{C})$ the group of birational maps of $\mathbb{P}^2_\mathbb{C}$, also called the \textbf{\textit{\textsc{Cremona} group}}. The \textbf{\textit{degree}} of $\phi$, denoted $\deg\phi$, is the degree of the $\phi_i$'s. The degree is not a birational invariant: $\deg\psi\phi\psi^{-1}\not=\deg\phi$ for generic birational maps $\phi$ and $\psi$. The \textbf{\textit{first dynamical degree}} of $\phi$ given by \[ \lambda(\phi)=\lim_{n\rightarrow +\infty}\big(\deg \phi^n\big)^{1/n}, \] is a birational invariant; it is strongly related to the topological entropy $h_{\text{top}}(\phi)$ of $\phi$ (\emph{see} \cite{Gromov, Yomdin}) \begin{equation}\label{GromovYomdin} h_{\text{top}}(\phi)\leq\log\lambda(\phi) \end{equation}
Any birational map $\phi$ admits a resolution \[ \xymatrix{& \mathrm{S}\ar[rd]^{\pi_2}\ar[ld]_{\pi_1}&\\ \mathbb{P}^2_\mathbb{C}\ar@{-->}[rr]_{\phi}&&\mathbb{P}^2_\mathbb{C}, } \] where $\pi_1$, $\pi_2\colon\mathrm{S}\to\mathbb{P}^2_\mathbb{C}$ are sequences of blow-ups (\emph{see} \cite{Beauville} for example). The resolution is \textbf{\textit{minimal}} if and only if no $(-1)$-curve of $\mathrm{S}$ is contracted by both $\pi_1$ and $\pi_2$. The \textbf{\textit{base-points}} of $\phi$ are the points blown-up in~$\pi_1$, which can be points of $\mathbb{P}^2_\mathbb{C}$ or infinitely near points. We denote by $\mathfrak{b}(\phi)$ the number of such points, which is also equal to the difference of the ranks of $\mathrm{Pic}(\mathrm{S})$ and $\mathrm{Pic}(\mathbb{P}^2_\mathbb{C})$, and thus equals to~$\mathfrak{b}(\phi^{-1})$. The \textbf{\textit{dynamical number of base-points of $\phi$}} introduced in~\cite{BlancDeserti} is by definition \[ \mu(\phi)=\displaystyle\lim_{n\rightarrow +\infty} \frac{\mathfrak{b}(\phi^n)}{n}; \] it is a real positive number that satisfies $\mu(\phi^n)=\vert n\, \mu(\phi)\vert$ for any $n\in \mathbb{Z}$, $\mu(\psi\phi\psi^{-1})=\mu(\phi)$, and allows us to give a characterization of birational maps conjugate to automorphisms:
\begin{thm}[\cite{BlancDeserti}]\label{thm:caraut} {\sl Let $\mathrm{S}$ be a smooth projective surface; the birational map $\phi\in\mathrm{Bir}(\mathrm{S})$ is conjugate to an automorphism of a smooth projective surface if and only if $\mu(\phi)=0$.} \end{thm}
The behavior of $\phi\in\mathrm{Bir}(\mathbb{P}^2_\mathbb{C})$ is strongly related to the behavior of~$\big(\deg\phi^n\big)_{n\in\mathbb{N}}$ (\emph{see} \cite{Gizatullin, DillerFavre, BlancDeserti}); up to birational conjugacy exactly one of the following holds: \begin{enumerate} \item the sequence $\big(\deg\phi^n\big)_{n\in\mathbb{N}}$ is bounded and either $\phi$ is of finite order, or $\phi$ is an automorphism of~$\mathbb{P}^2_\mathbb{C}$;
\item there exists an integer $k$ such that \[ \displaystyle\lim_{n\rightarrow +\infty} \frac{\deg \phi^n}{n}=k^2\,\frac{\mu(\phi)}{2} \] and $\phi$ is not an automorphism;
\item there exists an integer $k\geq 3$ such that \[ \displaystyle\lim_{n\rightarrow +\infty} \frac{\deg \phi^n}{n^2}=k^2\,\frac{\kappa(\phi)}{9} \] where $\kappa(\phi)\in \mathbb{Q}$ is a birational invariant, and $\phi$ is an automorphism;
\item the sequence $\big(\deg \phi^n\big)_{n\in\mathbb{N}}$ grows exponentially (\emph{see} \cite{DillerFavre} for more precise dynamical pro\-perties). \end{enumerate}
In the first three cases $\lambda(\phi)=1$, in the last one $\lambda(\phi)>1$. In case 2. (resp. 3.) the map~$\phi$ preserves a unique fibration which is rational (resp. elliptic).
In case 1. (resp. 2., resp. 3, resp. 4) we say that $\phi$ is \textbf{\textit{elliptic}} (resp. a \textbf{\textit{\textsc{Jonqui\`eres} twist}}, resp. an \textbf{\textit{\textsc{Halphen} twist}}, resp. \textbf{\textit{hyperbolic}}).
Let us give some examples. Let \[ \phi(x,y)=\left(\frac{a(y)x+b(y)}{c(y)x+d(y)},\frac{\alpha y+\beta}{\gamma y+\delta}\right) \] be an element of the \textbf{\textit{\textsc{Jonqui\`eres} group}} $\mathrm{PGL}(2;\mathbb{C}(y))\rtimes\mathrm{PGL}(2;\mathbb{C})$; either $\phi$ is elliptic (for instance $\phi\colon(x:y:z)\dashrightarrow(yz:xz:xy)$), or $\phi$ is a \textsc{Jonqui\`eres} twist (for example $\phi\colon(x:y:z)\dashrightarrow(xz:xy:z^2)$ for which the unique invariant fibration is $y/z=$ constant). The map \[ \phi\colon\mathbb{P}^2_\mathbb{C}\dashrightarrow\mathbb{P}^2_\mathbb{C}\quad\quad\quad (x:y:z)\dashrightarrow\big((2y+z)(y+z):x(2y-z):2z(y+z)\big) \] is an \textsc{Halphen} twist (\cite[Proposition 9.5]{DillerFavre}). \textsc{H\'enon} automorphisms give by homogeneization examples of hyperbolic maps.
Clearly elliptic birational maps have a poor dynamical behavior contrary to hyperbolic ones. The study of automorphisms of positive entropy is strongly related with birational maps of $\mathbb{P}^2_\mathbb{C}$:
\begin{thm}[\cite{Cantat}] {\sl Let $\mathrm{S}$ be a compact complex surface that carries an automorphism $\phi$ of positive topological entropy. \begin{itemize} \item[$\bullet$] Either the \textsc{Kodaira} dimension of $\mathrm{S}$ is zero and $\phi$ is conjugate to an automorphism on the unique minimal model of $\mathrm{S}$ that necessarily is a torus, or a K$3$ surface or an \textsc{Enriques} surface;
\item[$\bullet$] or the surface $\mathrm{S}$ is a non-minimal rational one, isomorphic to $\mathbb{P}^2_\mathbb{C}$ blown up at $n$ points, $n\geq 10$, and $\phi$ is conjugate to a birational map of $\mathbb{P}^2_\mathbb{C}$. \end{itemize} } \end{thm}
This yields many examples of hyperbolic birational maps for which we can establish a lot of dyna\-mical properties (\cite{McMullen, BedfordKim1, BedfordKim2, BedfordKim3, BedfordKim4, Diller, DesertiGrivaux}).
Another way to measure chaos is to look at the size of centralisers. Let us give two examples. The polynomial automorphisms of $\mathbb{C}^2$ having rich dynamics are \textsc{H\'enon} maps; furthermore a polynomial automorphism of $\mathbb{C}^2$ is a \textsc{H\'enon} one if and only if its centraliser is countable. Let us now consider rational maps on $\mathbb{S}^1$; if the centraliser of such maps is not trivial\footnote{The centraliser of a map $\phi$ is trivial if it coincides with the iterates of $\phi$.}, then the \textsc{Julia} set is "special". The centraliser of an elliptic birational map of infinite order is uncountable (\cite{BlancDeserti}). The centralisers of \textsc{Halphen} twists are described in \cite{Gizatullin}. The centraliser of an hyperbolic map is countable (\cite{Cantat:annals}). In \cite{CerveauDeserti} we end the story by studying centralisers of \textsc{Jonqui\`eres} twists. If the fibration is fiberwise invariant, then the centraliser is uncountable ; but if it isn't, then generically the centraliser is isomorphic to $\mathbb{Z}$. We don't know a lot about dynamics of these maps, in this article we will thus focus on a family of such maps. We consider the \textsc{Jonqui\`eres} maps \[ f_{\alpha,\beta}\colon\mathbb{P}^2_\mathbb{C}\dashrightarrow\mathbb{P}^2_\mathbb{C}\quad\quad\quad (x:y:z)\dashrightarrow\big((\alpha x+y)z:\beta y(x+z):z(x+z)\big) \] where $\alpha$, $\beta$ denote two complex numbers with modulus $1$. The base-points of $f_{\alpha,\beta}$ are \[ (1:0:0),\quad(0:1:0),\quad(-1:\alpha:1). \]
Any $f_{\alpha,\beta}$ preserves a rational fibration (the fibration $y=$ constant in the affine chart $z=1$). Each element of the fa\-mily~$(f_{\alpha,\beta})$ has first dynamical degree $1$ hence topological entropy zero (\ref{GromovYomdin}); more precisely one has (\cite[Example 4.3]{BlancDeserti}) \[ \mu( f_{\alpha,\beta})=\frac{1}{2} \] so $f_{\alpha,\beta}$ is not conjugate to an automorphism (Theorem \ref{thm:caraut}). The centralizer of $f_{\alpha,\beta}$ is isomorphic to $\mathbb{Z}$ (\emph{see} \cite[Theorem~1.6]{Deserti}). The idea of the proof is the following: the point $p = (1 : \alpha : 1)$ is blown-up onto a fiber of the fibration $y=$ constant. Let $\psi$ be an element of \[ \mathrm{Cent}(f_{\alpha,\beta})=\big\{g\in\mathrm{Bir}(\mathbb{P}^2_\mathbb{C})\,\vert\, g\circ f_{\alpha,\beta}=f_{\alpha,\beta}\circ g\big\}; \] since~$\psi$ blows down a finite number of curves there exists a positive integer~$k$ (chosen minimal) such that~$f_{\alpha,\beta}^k(p)$ is not blown down by $\psi$. Replacing $\psi$ by $\widetilde{\psi}=\psi f_{\alpha,\beta}^{k-1}$ one gets that $\widetilde{\psi}(p)$ is an indeterminacy point of~$f_{\alpha,\beta}$. In other words $\widetilde{\psi}$ permutes the indeterminacy points of $f_{\alpha,\beta}$. A more precise study allows us to establish that $p$ is fixed by $\widetilde{\psi}$. The pair $(\alpha,\beta)$ being in $\Omega$, the closure of the negative orbit of $p$ under the action of $f_{\alpha,\beta}$ is \textsc{Zariski} dense; since $\widetilde{\psi}$ fixes any element of the orbit of $p$ one obtains $\widetilde{\psi}=\mathrm{id}$.
Let us recall that if $\psi$ is an automorphism on a compact complex manifold $\mathrm{M}$, the \textbf{\textit{\textsc{Fatou} set}}~$\mathcal{F}(\psi)$ of $\psi$ is the set of points that have a neighborhood $\mathcal{V}$ such that $\big\{f^n_{\vert\mathcal{V}}\,\vert\, n\in\mathbb{N}\big\}$ is a normal family. Set \[ \mathcal{G}(\mathcal{U})=\big\{\phi\colon\mathcal{U}\to\overline{\mathcal{U}}\,\vert\,\phi=\lim_{n_j\to +\infty}\psi^{n_j}\big\}; \] we say that $\mathcal{U}$ is a \textbf{\textit{rotation domain}} if $\mathcal{G}(\mathcal{U})$ is a subgroup of $\mathrm{Aut}(\mathcal{U})$. An equivalent definition is the following: a component $\mathcal{U}$ of $\mathcal{F}(\psi)$ which is invariant by $\psi$ is a rotation domain if $\psi_{\vert\mathcal{U}}$ is conjugate to a linear rotation. If~$\mathcal{U}$ is a rotation domain, $\mathcal{G}(\mathcal{U})$ is a compact \textsc{Lie} group, and the action of $\mathcal{G}(\mathcal{U})$ on $\mathcal{U}$ is analytic real. Since $\mathcal{G}(\mathcal{U})$ is a compact, infinite, abelian \textsc{Lie} group, the connected component of the identity of $\mathcal{G}(\mathcal{U})$ is a torus of dimension $0\leq d\leq \dim_\mathbb{C}\mathrm{M}$. The integer $d$ is the \textbf{\textit{rank of the rotation domain}}. The rank coincides with the dimension of the closure of a generic orbit of a point in $\mathcal{U}$.
We can also see $f_{\alpha,\beta}$ on $\mathbb{P}^1_\mathbb{C}\times\mathbb{P}^1_\mathbb{C}$ and that is what we will do in the sequel $\big($since all the computations of \cite{Deserti} have been done in an affine chart they may all be carried on $\mathbb{P}^1_\mathbb{C}\times\mathbb{P}^1_\mathbb{C}$$\big)$; the sets~$\mathbb{P}^1_\mathbb{C}\times\mathbb{S}^1_\rho$ are invariant. In \cite{Deserti} we show that there are two rotation domains for $f^2_{\alpha,\beta}$, one of rank $1$, and the other one of rank $2$\footnote{There already exists an example of automorphism of positive entropy with rotation domains of rank $1$ and $2$ (\emph{see} \cite{BedfordKim2}), but $f_{\alpha,\beta}$ is not conjugate to an automorphism on a rational surface.}; in the first case we give here a more precise statement than in~\cite{Deserti}:
\begin{thm}\label{Thm:old} {\sl Assume that $(\alpha,\beta)$ belongs to $\Omega$.
There exists a strictly po\-sitive real number $r$ such that $f_{\alpha,\beta}$ is conjugate to $(\alpha x,\beta y)$ on $\mathbb{P}^1_\mathbb{C}~\times~\mathbb{D}(0,r)$ where $\mathbb{D}(0,r)$ denotes the disk centered at the origin with radius $r$.
There exists a strictly po\-sitive real number $\widetilde{r}$ such that $f_{\alpha,\beta}^2$ is conjugate to $\left(\frac{x}{\beta},\frac{z}{\beta^2}\right)$ on $\mathbb{P}^1_\mathbb{C}\times\mathbb{D}(0,\widetilde{r})$.} \end{thm}
\begin{rem} The point $(\alpha-1,0)$ is also a fixed point of $f_{\alpha,\beta}$ where the behavior of~$f_{\alpha,\beta}$ is the same as near $(0,0)$. \end{rem}
\begin{proof} The first assertion is proved in \cite{Deserti}.
Let us consider the map $\psi(x,z)=\left(\frac{a(z)x+b(z)}{c(z)x+1},z\right)$. The equation \[ \psi^{-1}f_{\alpha,\beta}^2\psi=\left(\frac{x}{\beta},\frac{z}{\beta^2}\right) \] yields \begin{eqnarray}\label{eq1} &&\beta\, a\big(\beta^{-2}\,z\big)c(z)+\beta\, a\big(\beta^{-2}\,z\big)a(z)-c\big(\beta^{-2}\,z\big)a(z)+\alpha\, a\big(\beta^{-2}\,z\big)a(z)\nonumber\\ &&\hspace{0.5cm}+z\big(\alpha^2\,a\big(\beta^{-2}\,z\big)c(z)-\alpha\, c\big(\beta^{-2}\,z\big)c(z)-c\big(\beta^{-2}\,z\big)c(z)-c\big(\beta^{-2}\,z\big)a(z)\big)=0, \end{eqnarray} \begin{eqnarray}\label{eq2} &&\beta\, a\big(\beta^{-2}\,z\big)-\beta\, a(z) +z\big(\alpha^2\,a\big(\beta^{-2}\,z\big)-\alpha\beta\, c(z)-\beta\, c(z)-\beta\, a(z)-\alpha\, c\big(\beta^{-2}\,z\big)-c\big(\beta^{-2}\,z\big)\big)\nonumber\\ &&\hspace{0.5cm} +\beta(\alpha+\beta)\, a(z)b\big(\beta^{-2}\,z\big) +(\alpha+\beta)\, b(z)a\big(\beta^{-2}\,z\big)+\beta^2\,b\big(\beta^{-2}\,z\big)c(z)-b(z)c\big(\beta^{-2}\,z\big)\nonumber\\ &&\hspace{0.5cm}+z\big(\alpha^2\,\beta b\big(\beta^{-2}\,z\big)c(z)-b(z)c\big(\beta^{-2}\,z\big)\big)=0 \end{eqnarray} and \begin{eqnarray}\label{eq3} (\alpha+1)\, z+ b(z)-\beta\, b\big(\beta^{-2}\,z\big)-\alpha^2 \,zb\big(\beta^{-2}\,z\big)+ z b(z)-(\alpha+\beta)\,b\big(\beta^{-2}\,z\big)b(z)=0 \end{eqnarray}
Let us set \[ a(z)=\sum_{i\geq 0}a_iz^i,\quad\quad\quad b(z)=\sum_{i\geq 0}b_iz^i,\quad\quad\quad c(z)=\sum_{i\geq 0}c_iz^i. \] We easily get $a_0=1-\beta$, $b_0=0$ and $c_0=\alpha+\beta$.
Relation (\ref{eq3}) implies that \[ b_1=\frac{\beta(1+\alpha)}{1-\beta}\quad\quad\quad\&\quad\quad\quad \beta\, b_\nu\,\left(1-\beta^{1-2\nu}\right)+F_i(b_i\,\vert\,i<\nu)=0 \quad\forall\,\nu>1, \] (\ref{eq2}) yields \begin{eqnarray*} a_\nu\left(\beta^{1-2\nu}-\beta\right)+b_\nu\left((\alpha+\beta)a_0\Big(1+\beta^{1-2\nu}\Big)+c_0\Big(\beta^{2-2\nu}-1\Big)\right) +G_i(a_i,\,b_i,\,c_i\,\vert\,i<\nu)=0 \end{eqnarray*} and (\ref{eq1}) to \begin{eqnarray*} c_\nu a_0\left(\beta-\beta^{-2\nu}\right)+a_\nu\left((\alpha+\beta)a_0\Big(1+\beta^{-2\nu}\Big)+c_0\Big(\beta^{1-2\nu}-1\Big)\right)+H_i(a_i,\,b_i,\,c_i\,\vert\,i<\nu)=0 \end{eqnarray*} where the $F_i$'s, $G_i$'s and $H_i$'s denote universal polynomials; this allows to compute $b_\nu$, $a_\nu$ and $c_\nu$. Thus we get a formal conjugacy of $f_{\alpha,\beta}^2$ to its linear part. Since this linear part satisfies a R\"ussmann condition (\emph{see} \cite[Theorem 2.1]{Russmann} condition (2)), according to \cite[Theorem 2.1]{Russmann} any formal linearizing map conjugating $f_{\alpha,\beta}^2$ to its linear part is convergent on a polydisc. \end{proof}
\section{About $\mathrm{SL}(2;\mathbb{C})$-cocycles}\label{Sec:cocyles} A (one-frequency, analytic) \textbf{\textit{quasiperiodic $\mathrm{SL}(2;\mathbb{C})$-cocycle}} is a pair $(A,\beta)$, where $\beta\in~\mathbb{R}$ and \[ A\colon\mathbb{S}^1_1\to\mathrm{SL}(2;\mathbb{C}) \] is analytic, and defines a linear skew product acting on $\mathbb{C}^2\times\mathbb{S}^1_1$ by \[ (x,y)\mapsto(A(y)\cdot x,\beta y). \] The iterates of the cocyle are given by $(A_n,n\beta)$ where $A_n$ is given by \[ A_n(y)=A\big(\beta^{n-1}y\big)\ldots A(y)\quad n\geq 1,\quad A_0(y)=\mathrm{id},\quad A_{-n}(y)=A_n(\beta^{-n}y)^{-1}. \] The \textbf{\textit{\textsc{Lyapunov} exponent}} $L(A,\beta)$ of a quasiperiodic $\mathrm{SL}(2;\mathbb{C})$-cocycle $(A,\beta)$ is given by \[ \displaystyle\lim_{n\to +\infty}\frac{1}{n}\int_{\mathbb{S}^1_1}\mathrm{ln}\,\vert\vert A_n(y)\vert\vert\, \mathrm{d}y. \] A quasiperiodic $\mathrm{SL}(2;\mathbb{C})$-cocycle $(A,\beta)$ is \textbf{\textit{uniformly hyperbolic}} if there exist ana\-lytic functions \[ u,\,s \colon \mathbb{S}^1_1\to\mathbb{P}^2_\mathbb{C}, \] called the \textbf{\textit{unstable and stable directions}}, and $n\geq 1$ such that for any $y\in\mathbb{S}^1_1$, \[ A(y)\cdot u(y)=u(\beta y)\qquad A(y)\cdot s(y)=s(\beta y), \] and for any unit vector $x\in s(y)$ (resp. $x\in u(y)$) we have $\vert\vert A_n(y)\cdot x\vert\vert<1$ (resp. $\vert\vert A_n(y)\cdot x\vert\vert>1$). The unstable and stable directions are uniquely characterized by those properties, and clearly $u(y)\not=s(y)$ for any $y\in\mathbb{S}^1_1$. If $(A,\beta)$ is uniformly hyperbolic, then $L(A,\beta)> 0$. Let us denote by \[ \mathcal{U}\mathcal{H}\subset C^\omega\big(\mathrm{SL}(2;\mathbb{C}),\mathbb{S}^1_1\big) \] the set of $A$ such that $(A,\beta)$ is uniformly hyperbolic. Uniform hyperbolicity is a stable property: $\mathcal{U}\mathcal{H}$ is open, and $A\mapsto L(A,\beta)$ is analytic over $\mathcal{U}\mathcal{H}$ (regularity properties of the \textsc{Lyapunov} exponent are consequence of the regularity of the unstable and stable directions which depend smoothly on both variables).
\begin{defi} Let $(A,\beta)$ be a quasiperiodic $\mathrm{SL}(2;\mathbb{C})$-cocycle. If $L(A,\beta)>0$ but $(A,\beta)\not\in\mathcal{U}\mathcal{H}$, then $(A,\beta)$ is \textbf{\textit{nonuniformly hyperbolic}}. \end{defi}
If $A\in C^\omega\big(\mathrm{SL}(2;\mathbb{C}),\mathbb{S}^1_1\big)$ admits a holomorphic extension to $\vert\mathrm{Im}\, y\vert<\delta$ then for $\vert\varepsilon\vert<\delta$ we can define $A_\varepsilon\in C^\omega\big(\mathrm{SL}(2;\mathbb{C}),\mathbb{S}^1_1\big)$ by \[ A_\varepsilon(y)=A(y+\mathbf{i}\varepsilon). \] The \textsc{Lyapunov} exponent $L(A_\varepsilon,\beta)$ is a convex function of $\varepsilon$. We can thus introduce the following notion. The \textbf{\textit{acceleration}} of a quasiperiodic $\mathrm{SL}(2;\mathbb{C})$-cocyle $(A,\beta)$ is given by \[ \omega(A,\beta)=\lim_{\varepsilon\to 0^+}\frac{1}{2\pi\varepsilon}\big(L(A_\varepsilon,\beta)-L(A,\beta)\big). \]
\begin{rem}\label{rem:decreasing} The convexity of the \textsc{Lyapunov} exponent in function of $\varepsilon$ implies that the acceleration is decreasing. \end{rem}
Since the \textsc{Lyapunov} exponent is a convex and continuous function the acceleration is an upper semi-continuous function in $\mathbb{R}\setminus\mathbb{Q}\times C^\omega\big(\mathrm{SL}(2;\mathbb{C}),\mathbb{S}^1_1\big)$. The acceleration is quantized:
\begin{thm}[\cite{Avila}]\label{Thm:acc} {\sl If $(A,\beta)$ is a $\mathrm{SL}(2;\mathbb{C})$-cocycle with $\beta\in\mathbb{R}\smallsetminus\mathbb{Q}$, then $\omega(A,\beta)$ is always an integer.} \end{thm}
A direct consequence is the following:
\begin{cor} {\sl The function $\varepsilon\mapsto L(A_\varepsilon,\beta)$ is a piecewise affine function of $\varepsilon$.} \end{cor}
It is thus natural to introduce the notion of regularity. A cocycle \[ (A,\beta) \in C^\omega\big(\mathrm{SL}(2;\mathbb{C}),\mathbb{S}^1_1\big)\times\mathbb{R}\setminus\mathbb{Q} \] is \textbf{\textit{regular}} if $L(A_\varepsilon,\beta)$ is affine for~$\varepsilon$ in a neighborhood of $0$. In other words $(A,\beta)$ is regular if the equality \[ L(A_\varepsilon,\beta)-L(A,\beta)=2\pi\varepsilon\omega(A,\beta) \] holds for all~$\varepsilon$ small, and not only for the positive ones. Regularity is equivalent to the acceleration being locally constant near $(A,\beta)$. It is an open condition in $C^\omega\big(\mathrm{SL}(2;\mathbb{C}),\mathbb{S}^1_1\big)\times\mathbb{R}\setminus\mathbb{Q}$. The following statement gives a characterization of the dynamics of regular cocycles with positive \textsc{Lyapunov} exponent:
\begin{thm}[\cite{Avila}] {\sl Let $(A,\beta)$ be a $\mathrm{SL}(2;\mathbb{C})$-cocycle with $\beta\in\mathbb{R}\smallsetminus\mathbb{Q}$. Assume that $L(A,\beta)>0$; then $(A,\beta)$ is regular if and only if~$(A,\beta)$ is~$\mathcal{U}\mathcal{H}$.} \end{thm}
One striking consequence is the following:
\begin{cor}[\cite{Avila}] {\sl For any $(A,\beta)$ in $C^\omega\big(\mathrm{SL}(2;\mathbb{C}),\mathbb{S}^1_1\big)\times\mathbb{R}\setminus\mathbb{Q}$ there exists $\varepsilon_0$ such that
\begin{itemize} \item[$\bullet$] $L(A_\varepsilon,\beta)=0$ $($and $\omega(A,\beta)=0)$ for every $0<\varepsilon<\varepsilon_0$,
\item[$\bullet$] or $(A_\varepsilon,\beta)\in\mathcal{U}\mathcal{H}$ for every $0<\varepsilon<\varepsilon_0$. \end{itemize}} \end{cor}
\begin{rem} Let us mention that there is a link between $\mathrm{SL}(2;\mathbb{C})$-cocycles and \textsc{Schr\"odinger} operators (\emph{see} \cite{Avila} for more details). \end{rem}
\section{Proof of Theorem \ref{Thm:main}}\label{Sec:main}
Suppose that $\rho\not=1$, and let us consider the cocycle $(B^{\alpha,\rho},\beta^{1/2})$ where \[ B^{\alpha,\rho}(y)=\left[\begin{array}{cc} \alpha & y^2 \\ 1 & 1 \end{array}\right]. \] Since \[ \left(\frac{\alpha x+y}{x+1},\beta y\right)(x,y^2)=(x,y^2)\left(\frac{\alpha x+y^2}{x+1},\beta^{1/2}y\right) \] the cocycles $(A^{\alpha,\rho},\beta)$ and $(B^{\alpha,\rho},\beta^{1/2})$ have the same behavior. Using two different arguments of mono\-dromy (one for $\rho<1$, and the other one for $\rho>1$) we see that there is a continuous determination for the square root of $\det B^{\alpha,\rho}(y)=\alpha-y^2$. Let us set \[ \widetilde{B}^{\alpha,\rho}(y)=\frac{1}{\sqrt{\alpha-y^2}}\,B^{\alpha,\rho}(y)\in\mathrm{SL}(2;\mathbb{C}) \] that is thus defined on two different domains of analyticity.
According to Theorem \ref{Thm:old} one has $L\big(\widetilde{B}^{\alpha,\rho},\beta^{1/2}\big)=0$ when $\rho$ is close to both $0$ and $\infty$.
Assume that $L\big(\widetilde{B}^{\alpha,\rho},\beta^{1/2}\big)$ is non constant. When $\widetilde{B}^{\alpha,\rho}$ is holomorphic, so in particular when $\rho<1$ and $\rho>1$, the acceleration is decreasing (Remark \ref{rem:decreasing}); furthermore the acceleration is positive for $\rho<1$ and negative for $\rho>1$ (because $L$ is continuous). Theorem \ref{Thm:acc} thus implies \[ \omega\big(\widetilde{B}^{\alpha,1^+},\beta^{1/2}\big)-\omega\big(\widetilde{B}^{\alpha,1^-},\beta^{1/2}\big)\leq -2. \] By definition of $\widetilde{B}^{\alpha,\rho}$ we have \begin{eqnarray*}\label{Lyapunov} L\big(\widetilde{B}^{\alpha,\rho}(y),\beta^{1/2}\big) &=& L\big(B^{\alpha,\rho}(y),\beta^{1/2}\big)-\int_{\mathbb{S}^1_\rho}\ln\sqrt{\alpha-y^2}\,\mathrm{d}y\\ &=&L\big(B^{\alpha,\rho}(y),\beta^{1/2}\big)-\max(0,\ln\rho). \end{eqnarray*} Even though $\big(B^{\alpha,\rho}(y),\beta^{1/2}\big)$ is not a $\mathrm{SL}(2;\mathbb{C})$-cocycle, the \textsc{Lyapunov} exponent is still a convex function of $\log\rho$ (\emph{see for example} \cite{AvilaJitomirskayaSadel}). The jump of $\omega(B^{\alpha,\rho}(y),\beta^{1/2})$ is thus $\geq 0$, and the jump for the second term of the right member is $-1$. Therefore the jump of $L\big(\widetilde{B}^{\alpha,\rho}(y),\beta^{1/2}\big)$ is $\geq -1$: contradiction.
\nocite{}
\end{document} | arXiv |
\begin{document}
\title{Semiclassical propagator of the Wigner function} \author{Thomas Dittrich,$^{1}$ Carlos Viviescas,$^{2}$ and Luis Sandoval$^{1}$} \affiliation{$^{1}$Departamento de F\'{{\rm i}}sica, Universidad Nacional, Bogot\'a D.C., Colombia, $^{2}$Max Planck Institute for the Physics of Complex Systems, N\"othnitzer Stra\ss e 38, 01187 Dresden, Germany} \date{\today}
\begin{abstract} Propagation of the Wigner function is studied on two levels of semiclassical propagation, one based on the van-Vleck propagator, the other on phase-space path integration. Leading quantum corrections to the classical Liouville propagator take the form of a time-dependent quantum spot. Its oscillatory structure depends on whether the underlying classical flow is elliptic or hyperbolic. It can be interpreted as the result of interference of a \emph{pair} of classical trajectories, indicating how quantum coherences are to be propagated semiclassically in phase space. The phase-space path-integral approach allows for a finer resolution of the quantum spot in terms of Airy functions. \end{abstract} \pacs{03.65.Sq, 31.15.Gy, 31.15.Kb}
\maketitle Quantum propagation in phase space has always been in the shadow of propagation in conventional (position, momentum) representations. Yet it is superior in various respects, particularly in the semiclassical realm: It avoids all problems owing to projection, such as singularities at caustics. Canonical invariance of all classical quantities involved is manifest. Boundary conditions are imposed consistently at a single (initial or final) time, thus removing the so-called root-search problem and allowing for initial-value representations. Semiclassical approximations to the quantum-mechanical propagator have predominantly been seeked in the form of coherent-state path integrals \cite{her84,sep96,bar01}. Closely related are Heller's Gaussian wavepacket dynamics \cite{hel75} and its numerous modifications. By now, a broad choice of phase-space propagation schemes is available which score very well if compared to other semiclassical techniques.
Almost all of these developments refer to the propagation of wavefunctions in some Hilbert space. Less attention has been paid to the propagation of Wigner and Husimi functions. They live in \emph{projective} Hilbert space, i.e., represent the density operator and are bilinear in the wavefunction. Besides their popularity, they have a crucial virtue in common: An extension to non-unitary time evolution is immediate. This opens access to a host of applications that combine complex quantum dynamics, where a phase-space representation facilitates the comparison to the corresponding classical motion, with decoherence or dissipation: quantum optics and quantum chemistry, nanosystems in biophysics and electronics, quantum measurement and computation.
By the scales involved, many of them call for semiclassical approximations. However, only few such studies exist, for specific systems predominantly in quantum chaos \cite{ber79}, including dissipative systems \cite{dit90,coh91}. By contrast, Ref.~\cite{gar04} discusses a new method, Wigner-function propagation analogous to the solution of classical Fokker-Planck equations.
As a major challenge, any attempt to directly propagate Wigner functions requires an appropriate treatment of \emph{quantum coherences}. As early as 1976, Heller \cite{hel76} argued that the ``dangerous cross terms'', i.e., the off-diagonal elements of the density matrix in the relevant representation, can give rise to a complete failure of semiclassical propagation of the Wigner function. Quantum coherences are reflected in the Wigner function as ``sub-Planckian'' oscillations \cite{zur01}. They plague semiclassical approximations by their small scale and by propagating along paths that can deviate by any degree from classical trajectories.
In this Letter, we point out how Heller's objections are resolved by considering \emph{pairs} of trajectories as basis of semiclassical approximations, and present corresponding expressions for the propagator of the Wigner function. The concept of trajectory pairs has been introduced in the present context by Rios and Ozorio de Almeida \cite{rio02,ozo05}, albeit working in a strongly restricted space of semiclassical Wigner functions. We here give a general derivation of the propagator, independently of any initial or final states.
Moreover, we go beyond the level of approximations based on stationary phase. Employing a phase-space path-integral technique, we construct an improved semiclassical Wigner propagator in terms of Airy functions. It resolves all singularities and contains the semiclassical approximations based on trajectory pairs as a limiting case. The interference patterns we obtain depend, up to scaling, only on the nature of the underlying classical phase-space flow---elliptic vs.\ hyperbolic---and in this sense are universal. While living in projective Hilbert space, this result is superior to Gaussian wave-packet propagation in that it allows Gaussians to evolve into non-Gaussians.
In order to fix units and notations, define the Wigner function corresponding to a density operator $\hat\rho$ as
$W({\bf r}) = \int{\rm d}^f q' \exp(-{\rm i}{\bf p\cdot q'}/\hbar)
\left\langle{\bf q}+{\bf q}'/2\right|\hat\rho\left|{\bf q}-{\bf q}'/2\right\rangle$ where ${\bf r} = ({\bf p},{\bf q})$ is a vector in $2f$-dimensional phase space. Its time evolution is generated by a Hamiltonian $\hat H(\hat{\bf p},\hat{\bf q})$ through the equation of motion $(\partial/\partial t) W({\bf r},t) = \{H({\bf r}),W({\bf r},t)\}_{\rm Moyal}$, involving the Weyl symbol $H({\bf r})$ of the Hamiltonian $\hat H$. The Moyal bracket $\{.,.\}_{\rm Moyal}$ \cite{hil84} converges to the Poisson bracket for $\hbar\to 0$. As this equation of motion is linear, the evolution of the Wigner function over a finite time can be expressed as an integral kernel,
$W({\bf r}'',t'') = \int {\rm d}^{2f}r'\,G({\bf r}'',t'';{\bf r}',t') W({\bf r}',t')$, defining the Wigner propagator $G({\bf r}'',t'';{\bf r}',t')$. For autonomous Hamiltonians, it
induces a one-dimensional dynamical group parameterized by $t = t''-t'$ (in what follows, we restrict ourselves to this case and use $t$ as the only time argument). This implies, in particular, the initial condition $G({\bf r}'',{\bf r}';0) = \delta({\bf r'' - r'})$ and the composition (Chapman-Kolmogorov) equation $G({\bf r}'',{\bf r}',t) = \int {\rm d}^{2f}r\, G({\bf r}'',{\bf
r},t-t') G({\bf r},{\bf r}',t')$.
The Wigner propagator can be expressed in terms of the Weyl transform of the unitary time-evolution operator $U({\bf r},t) =$ $\int{\rm d}^f q'
\exp(-{\rm i}{\bf p\cdot q'}/\hbar)$ $\left\langle{\bf q}+{\bf q}'/2\right|$ $\hat U(t)$ $\left|{\bf q}-{\bf q}'/2\right\rangle$, called Weyl propagator,
as a convolution, \begin{equation}\label{wigweyl} \!\!\!\!\!G({\bf r}'',{\bf r}',t) = \int {\rm d}^{2f}R\, {\rm e}^{{-{\rm i}\over\hbar}({\bf r}''-{\bf r}')\wedge{\bf R}} U^*(\tilde{\bf r}_-,t)U(\tilde{\bf r}_+,t), \end{equation} with $\tilde{\bf r}_{\pm} \equiv ({\bf r}'+{\bf r}'' \pm {\bf R})/2$. It serves as a suitable starting point for a semiclassical approximation, invoking an expression for the Weyl propagator semiclassically equivalent to the van-Vleck approximation \cite{ber89,ozo98}, \begin{equation}\label{weylvleck} U({\bf r},t) = 2^f \sum_j
{\exp\({\rm i} S_j({\bf r},t)/\hbar
-{\rm i}\mu_j\pi/2\)\over\sqrt{|\hbox{det}\,(M_j({\bf r},t)+I)|}}. \end{equation} The sum includes all classical trajectories $j$ connecting phase-space points ${\bf r}'_j$, ${\bf r}''_j$ in time $t$ such that ${\bf r} = \tilde{\bf r}_j \equiv ({\bf r}'_j+{\bf r}''_j)/2$. $M_j$ is the corresponding stability matrix, $\mu_j$ its Maslov index. The action $S_j({\bf r}_j,t) = A_j({\bf r}_j,t)-H({\bf
r}_j,t)\,t$, with
$A_j$, the symplectic area enclosed between the trajectory and the straight line (chord) connecting ${\bf r}'_j$ and ${\bf r}''_j$ \cite{ber89} (hashed areas $A_{j\pm}$ in Fig.~\ref{act}).
Substituting Eq.~(\ref{weylvleck}) in Eq.~(\ref{wigweyl}) leads to a sum over pairs $j_+$, $j_-$, of trajectories whose respective chord centers $\tilde{\bf r}_{j\pm}$ are separated by the integration variable ${\bf R}$. Otherwise, the two trajectories are unrelated. A coupling between them, as expected on classical grounds, comes about only upon evaluating the ${\bf R}$-integral by stationary phase. Stationary points are given by ${\bf r}''-{\bf r}' = ({\bf r}''_{j-}-{\bf r}'_{j-}+{\bf r}''_{j+}-{\bf r}'_{j+})/2$. Together with the conditions for the two chords, ${\bf r}'+{\bf r}''\pm{\bf R} = {\bf r}'_{j\pm}+{\bf r}''_{j\pm}$, this implies \begin{equation}\label{midpoints} {\bf r}' = \bar{\bf r}' \equiv ({\bf r}'_{j-} + {\bf r}'_{j+})/2,\quad {\bf r}'' = \bar{\bf r}'' \equiv ({\bf r}''_{j-} + {\bf r}''_{j+})/2. \end{equation}
Stationary points are thus given by pairs of classical trajectories such that the initial (final) argument of the propagator is in the middle between their respective initial (final) points (Fig.~\ref{pict}b). This does \emph{not} require these trajectories to be identical! They do coincide as long as ${\bf r}'$ and ${\bf r}''$ are on the same classical trajectory, but bifurcate as ${\bf r}''$ moves off the classical trajectory ${\bf r}_{\rm cl}({\bf r}',t)$ starting at ${\bf r}'$, if the dynamics is not harmonic.
\begin{figure}
\caption{ The reduced action (shaded) of the Wigner propagator in van-Vleck approximation, Eq.~(\protect\ref{wigvlecks}), is the symplectic area enclosed between the two classical trajectories ${\bf r}_{j\pm}(t)$ and the two transverse vectors ${\bf r}'_{j+} - {\bf r}'_{j-}$ and ${\bf r}''_{j+} - {\bf r}''_{j-}$ (schematic drawing). The full central line is the classical trajectory ${\bf r}_{\rm cl}({\bf r}',t)$, the broken line is the propagation path $\bar{\bf r}_j({\bf r}',t)$. }
\label{act}
\end{figure}
The resulting semiclassical approximation for the Wig\-ner propagator is (dot indicating time derivative) \begin{eqnarray} G({\bf r}'',{\bf r}',t) &=& {4^f\/h^f}\sum_j {2\cos\(S_j({\bf r}'',{\bf r}',t)/\hbar-f\pi/2\)\over
\sqrt{|\hbox{det}\,(M_{j+}-M_{j-})|}},\label{wigvleckg}\\ S_j({\bf r}'',{\bf r}',t) &=& (\tilde{\bf r}_{j+}-\tilde{\bf r}_{j-}) \wedge({\bf r}''-{\bf r}')+S_{j+}-S_{j-}\nonumber\\ = \int_0^t{\rm d} s&&\!\!\!\!\!\!\!\!\!\!\!\!\! \big[\dot{\bar{\bf r}}_j(s)\wedge {\bf R}_j(s)-H_{j+}({\bf r}_{j+})+H_{j-}({\bf r}_{j-})\big], \label{wigvlecks} \end{eqnarray} with $\bar{\bf r}_j(t) \equiv ({\bf r}_{j-}(t)+{\bf r}_{j+}(t))/2$, ${\bf R}_j(t) \equiv {\bf r}_{j+}(t)-{\bf r}_{j-}(t)$, and $S_{j\pm} \equiv A_{j\pm}(\tilde{\bf r}_{j\pm},t)-H_{j\pm}(\tilde{\bf r}_{j\pm},t)\,t$. The reduced action $A_j = \int_0^t{\rm d} s$ $\dot{\bar{\bf r}}_j(s)\wedge {\bf R}_j(s)$ is the symplectic area enclosed between the two trajectory sections and the vectors ${\bf r}'_{j+}-{\bf r}'_{j-}$ and ${\bf r}''_{j+}-{\bf r}''_{j-}$ (Figs.~\ref{act},\ref{pict}b). In general, Eq.~(\ref{wigvleckg}), as a function of ${\bf r}''$, describes a distribution that extends from the classical trajectory into the surrounding phase space, forming a ``quantum spot'' (Fig.~\ref{spot}b) with a characteristic oscillatory pattern that results from the interference of the contributing classical trajectories. In general, it fills only a sector with opening angle $< 2\pi$ (Fig.~\ref{pict}c), where the sum contains two trajectory pairs (four stationary points). Outside this ``illuminated area'', stationarity cannot be fulfilled, that is, the ``shadow region'' is not accessible even for mean paths $\bar{\bf r}_j(t)$. The border is formed by phase-space caustics along which there is exactly one solution (two stationary points). As ${\bf r}''$ approaches the classical trajectory ${\bf r}_{\rm cl}({\bf r}',t)$ starting at ${\bf r}'$, from the illuminated sector, the two solutions $j-$, $j+$ coalesce so that $M_{j-} \to M_{j+}$, and Eq.~(\ref{wigvleckg}) becomes singular.
If the potential is purely harmonic, all mean paths coincide with ${\bf r}_{\rm cl}({\bf r}',t)$, and the classical Liouville propagator, $G({\bf r}'',{\bf r}',t) = \delta({\bf r}'' - {\bf r}''_{\rm cl}({\bf r}',t))$, is retained. In all other cases, Eq.~(\ref{wigvleckg}), though based on the van-Vleck propagator, reflects the structure of stationary points of the action including third-order terms, with one pair of extrema and one pair of saddle points. It is formulated in terms of canonically invariant quantities related to classical trajectories and thus generalizes immediately to an arbitrary number of degrees of freedom. The propagation of Wigner functions defined semiclassically in terms of Lagrangian manifolds \cite{rio02} is contained in Eq.~(\ref{wigvleckg}) as a special case.
\begin{figure}
\caption{ Classical building blocks entering the Wigner propagator according to Eqs.~(\protect\ref{wigvleckg},\protect\ref{wigvlecks}), for a stable trajectory starting at ${\bf r}' = (0.636,0)$ near the minimum of the cubic potential $V(q) = 0.329 q^3-0.69 q$ (panel a). (b) Classical trajectory ${\bf r}_{\rm cl}({\bf r}',t)$ (black line), a pair of auxiliary trajectories ${\bf r}_{j\pm}(t)$ (green lines) and corresponding propagation path $\bar{\bf r}_j({\bf r}',t)$ (red dashed line). The yellow target pattern is the grid of auxiliary initial points ${\bf r}'_{j\pm}$ around ${\bf r}'$, parameterized by polar coordinates. Propagated classically over time $t$, it deforms into the turquoise pattern around ${\bf r}''$. The red/blue cone is formed by midpoints $\bar{\bf r}''_j = ({\bf r}''_{j-} + {\bf r}''_{j+})/2$ that correspond to extrema/saddles of the action. Its boundaries form caustics separating the region accessed by two midpoints $\bar{\bf r}''$ from the unaccessible rest. (c) Enlargement of the area around ${\bf r}''$, indicating the number of trajectory pairs that access each region. }
\label{pict}
\end{figure}
We are now able to resolve Heller's objections \cite{hel76}: If the two trajectories $j-$, $j+$ are sufficiently separated and the potential is sufficiently nonlinear, then (i), the propagation path $\bar{\bf r}({\bf r}',t)$ can differ arbitrarily from ${\bf r}_{\rm cl}({\bf r}',t)$, and (ii), the phase factor in (\ref{wigvleckg}) exhibits sub-Planckian oscillations. They would couple resonantly to corresponding features in the initial Wigner function, generating a similar pattern in the final Wigner function around the endpoint of the non-classical propagation path. In this way, quantum coherences are faithfully propagated within a semiclassical approach.
Equations (\ref{wigvleckg},\ref{wigvlecks}) translate into a straightforward algorithm for the numerical calculation of the propagator (Fig.~\ref{pict}): (i) Define a local grid (e.g., in polar coordinates) around the initial argument ${\bf r}'$ of the propagator, identifying pairs of auxiliary initial points ${\bf r}'_{j-}$, ${\bf r}'_{j+}$ with ${\bf r}'$ in their middle. (ii) Propagate trajectory pairs ${\bf r}_{j\pm}(t)$ classically, keeping track of the symplectic area ${\bf A}_j$ between them. (iii) Find the amplitude and phase contributed by each trajectory pair and associate them to the final midpoints $\bar{\bf r}''_j$. They constitute a deformed cone, projected onto phase space (Fig.~\ref{pict}c). Its ``lower'' (``upper'') surface (red (blue) in Fig.~\ref{pict}c) corresponds to pairs of extrema (saddles) of the action, respectively: (iv) Superpose the contributions of the two surfaces, after smoothing amplitude and phase over midpoints $\bar{\bf r}''_j$ within each of them.
The caustics in Eq.~(\ref{wigvleckg}) result from applying stationary-phase integration in a situation where pairs of stationary points can come arbitrarily close to one another. Since the underlying van-Vleck propagator admits only up to quadratic terms in the phase, we seek a superior approach, corresponding to a uniform approximation. It is available in the form of a path-integral representation of the Wigner propagator \cite{mar91}, in close analogy to the Feynman path integral,
\begin{equation} G({\bf r}'',{\bf r}',t) = {1\/h^f}\int{\rm D} r\int{\rm D} R\, {\rm e}^{-{\rm i} S(\{{\bf r}\},\{{\bf R}\})/\hbar}. \label{pathint} \end{equation} Two paths, ${\bf r}(t)$ and ${\bf R}(t)$, have to be integrated over. The former is subject to boundary conditions ${\bf r}(0) = {\bf r}'$ and ${\bf r}(t) = {\bf r}''$, the latter is free. The path action is \begin{eqnarray} &&S(\{{\bf r}\},\{{\bf R}\}) = \int_{0}^{t}{\rm d} s \big[\dot{\bf r}(s)\wedge{\bf R}(s)\nonumber\\ &&+H_{\rm W}({\bf r}(s)+{\bf R}(s)/2)- H_{\rm W}({\bf r}(s)-{\bf R}(s)/2)\big]. \label{pathaction} \end{eqnarray}
Equation (\ref{wigvleckg}) is recovered upon evaluating the path-integral representation in stationary-phase approximation: Defining ${\bf r}_{\pm}(t) \equiv {\bf r}(t) \pm {\bf R}(t)/2$, with boundary conditions analogous to Eq.~(\ref{midpoints}), and requiring stationarity leads to the Hamilton equation of motion for ${\bf r}_{\pm}(t)$: We again find pairs of classical trajectories that straddle the propagation path as stationary solutions.
We will now include cubic terms in the action, with respect to variations of the path variables. To keep technicalities at a minimum, we restrict ourselves from now on to a single degree of freedom and to Hamiltonians of the standard form $H(p,q) = T(p) + V(q)$, where $T(p) = {p^2/2m}$ while the potential $V(q)$ may contain nonlinearities of arbitrary order. With this form, $H_{\rm W}({\bf
r})$ coincides with the Hamiltonian function ``quantized'' by merely replacing operators with classical variables. As the path integral readily allows to treat time-dependent potentials, chaotic classical motion remains within reach.
Expanding the action (\ref{pathaction}) around ${\bf r}(t) = {\bf r}_{\rm cl}({\bf r}',t)$ and ${\bf R}(t) \equiv (P(t),Q(t)) = {\bf 0}$, there remain only linear terms in $P$ and linear and cubic terms in $Q$. Evaluating the ${\bf R}$-sector of the path integral thus results in an Airy spreading of the propagator, with a rate $\sim V'''(q_{\rm cl}(t))$, in the $p$-direction. It is superposed to the classical phase-space flow around the trajectory, i.e., rotation (shear) if it is elliptic (hyperbolic). As a consequence, a spot of the full phase-space dimension develops. Scaling $\hbox{\boldmi\char26} = (\eta,\xi) = (\mu^{1/4}p,\mu^{-1/4}q)$, with $\mu = T''(p_{\rm cl})/V''(q_{\rm cl})$, we express the linearized classical motion as a dimensionless map, \begin{equation}\label{contmap} M(\phi(t)) = \[\begin{array}{cc}\cos\,\phi(t)&-\sin\,\phi(t)\\ \sin\,\phi(t)&\cos\,\phi(t)\end{array}\right]. \end{equation} These maps form a group parameterized by the angle $\phi(t) = \int_{0}^{t}{\rm d} s$ $\sqrt{T''(p^{\rm cl}(s))V''(q^{\rm cl}(s))}$. It is real (imaginary) if the linearized dynamics is elliptic (hyperbolic).
\begin{figure}
\caption{ Quantum spot replacing the classical delta function on a stable (elliptic) trajectory near the minimum of a cubic potential as shown in Fig.~\protect\ref{pict}a, at $t = 1.8$ ($\phi \approx 2\pi/3$). Panel (a) shows the exact quantum result for the Wigner propagator, (b) and (c) are semiclassical approximations based on Eqs.~(\protect\ref{wigvleckg}) and (\protect\ref{ajkfgw}), respectively, all for $\hbar = 0.01$. Frames coincide with that of Fig.~\protect\ref{pict}c. (d) Quantum spot, according to Eq.~(\protect\ref{wigvleckg}), for an unstable classical trajectory near the maximum of the same potential, at $t = 1.0$. Crosses mark the classical trajectory. Colour code ranges from red (negative) to blue (positive). }
\label{spot}
\end{figure}
This allows to evaluate also the ${\bf r}$-sector of the path integral. Transforming the Wigner function to Fourier phase space, $\widetilde{W}(\hbox{\boldmi\char13}) \equiv (FW)(\hbox{\boldmi\char13}) =$ $(2\pi)^{-1}$ $\int{\rm d}^2 r$ $\exp(-{\rm i}\hbox{\boldmi\char13}\wedge{\bf r})$ $W({\bf r})$, and the propagator accordingly, $\tilde{G} = FGF^{-1}$, we obtain ($\hbox{\boldmi\char13}' \equiv (\alpha',\beta')$) \begin{eqnarray} &&\!\!\!\!\!\!\!\!\!\!\widetilde{G}(\hbox{\boldmi\char13}'',\hbox{\boldmi\char13}',t) = \delta(\hbox{\boldmi\char13}''-M(\phi'')\hbox{\boldmi\char13}')\nonumber\\ &&\!\!\!\!\!\!\!\!\!\!\exp\(-{\rm i}({a_{30}\/3}\alpha'^3+ a_{21}\alpha'^2\beta'+a_{12}\alpha'\beta'^2+ {a_{03}\/3}\beta'^3)\). \label{ajkfgw} \end{eqnarray} The coefficients $a_{jk} = \int_{0}^{t}{\rm d} s\,\sigma(s) (\sin\phi(s))^j(\cos\phi(s))^k$ depend on where along the classical trajectory how much quantum spreading $\sim\sigma(t) = (\mu(t))^{3/4} \hbar^2 V'''(q_{\rm cl}(t))/8$ is picked up and thus on the specific system and initial conditions. The Fourier transform from Eq.~(\ref{ajkfgw}) back to the original Wigner propagator can be done analytically, after transforming the third-order polynomial in the phase to a normal form \cite{dit05}.
The internal structure and the time evolution of the quantum spot described by Eq.~(\ref{ajkfgw}) are qualitatively different for elliptic and hyperbolic classical trajectories (real and imaginary $\phi$, resp.). In the elliptic case, the spot is a periodic function of $\phi$. In particular, it collapses approximately to a point whenever $\phi = 2l\pi$, $l$ integer. Close to these nodes, it shrinks and grows again along a straight line in the $p$-direction, reflecting the fact that for short time, the quantum Airy spreading $\sim t^{1/3}$ outweighs the classical rotation $\sim t$. Only sufficiently far from the nodes, while rotating around the trajectory by $\phi(t)/2$, the one-d.\ distribution fans out into a two-d.\ interference pattern formed as the overlap of the bright (oscillatory) sides of two Airy functions, with a sharp maximum on the classical trajectory (Fig.~\ref{spot}b). In comparison with the corresponding exact quantum-mechanical result (Fig.~\ref{spot}a) for the quantum spot, obtained by expanding the propagator in energy eigenstates \cite{arg05}, the path-integral solution resolves the caustics far better than Eq.~(\ref{wigvleckg}) (Fig.~\ref{spot}c).
The hyperbolic case is obtained replacing trigonometric by the corresponding hyperbolic functions. As a result, along unstable trajectories there are no periodic recurrences as in the elliptic case; the spot continues expanding in the unstable and contracting in the stable direction (Fig.~\ref{spot}d). Isolated unstable periodic orbits embedded in a chaotic region of phase space exhibit a degeneracy of the Weyl propagator \cite{ber89}. It allows to account for scarring in terms of the Wigner propagator \cite{dit05}.
We have obtained a consistent picture of incipient quantum effects in the Wigner propagator, both in the van-Vleck approach and in the path-integral formalism: (i) for anharmonic potentials, the delta function on the classical trajectory is replaced by a quantum spot extending into phase space, (ii) its structure shows a marked time dependence, qualitatively different for elliptic and hyperbolic dynamics, (iii) it exhibits interference fringes arising as a product of Airy functions, (iv) it can be expressed in terms of canonically invariant quantities associated to pairs of underlying classical trajectories, (v) within each level of semiclassical approximation used, the propagator retains its dynamical-group structure. Open issues include: extension to higher dimensions and to higher-order terms in the action, performance in the presence of tunneling, application to unstable periodic orbits and implications for scars, trace formulae, and spectral statistics, regularization of the ballistic nonlinear $\sigma$-model, semiclassical propagation of entanglement, and generalization to non-unitary time evolution.
We enjoyed discussions with S.~Fishman, F.~Gro\ss mann, F.~Haake, H.~J.~Korsch, A.~M.~Ozorio de Almeida, H.~Schanz, K.~Sch\"onhammer, B.~Segev, T.~H.~Seligman, M.~Sieber, and U.~Smilansky. Financial support by Colciencias, U.\ Nal.\ de Colombia, Volkswagen\-Stiftung, and Fundaci\'on Mazda is gratefully acknowledged. TD thanks for the hospitality extended to him by CIC (Cuernavaca), Max Planck Institutes in Dresden and G\"ottingen, Inst.\ Theor.\ Phys.\ at Technion (Haifa), Ben-Gurion U.\ of the Negev (Beer-Sheva), U.\ of Technology Kaiserslautern, and Weizmann Inst.\ of Sci.\ (Rehovot).
\end{document} | arXiv |
\begin{document}
\twocolumn[
\mlsystitle{Graph Deep Factors for Forecasting}
\mlsyssetsymbol{equal}{*}
\begin{mlsysauthorlist} \mlsysauthor{Hongjie Chen}{vt} \mlsysauthor{Ryan A. Rossi}{adobe} \mlsysauthor{Kanak Mahadik}{adobe} \mlsysauthor{Sungchul Kim}{adobe} \mlsysauthor{Hoda Eldardiry}{vt} \end{mlsysauthorlist}
\mlsysaffiliation{vt}{Department of Computer Science, Virginia Tech, Blacksburg, Virginia, USA} \mlsysaffiliation{adobe}{Adobe Research, San Jose, California, USA} \mlsyscorrespondingauthor{}{}
\mlsyskeywords{Graph Neural Network, Time-series Forecasting, Graph-based Time-series, Deep Probabilistic Forecasting, Deep Learning}
\vskip 0.3in
\begin{abstract} Deep probabilistic forecasting techniques have recently been proposed for modeling large collections of time-series. However, these techniques explicitly assume either complete independence (local model) or complete dependence (global model) between time-series in the collection. This corresponds to the two extreme cases where every time-series is disconnected from every other time-series in the collection or likewise, that every time-series is related to every other time-series resulting in a completely connected graph. In this work, we propose a deep hybrid probabilistic graph-based forecasting framework called Graph Deep Factors (GraphDF) that goes beyond these two extremes by allowing nodes and their time-series to be connected to others in an arbitrary fashion. GraphDF is a hybrid forecasting framework that consists of a relational global and relational local model. In particular, we propose a relational global model that learns complex non-linear time-series patterns globally using the structure of the graph to improve both forecasting accuracy and computational efficiency. Similarly, instead of modeling every time-series independently, we learn a relational local model that not only considers its individual time-series but also the time-series of nodes that are connected in the graph. The experiments demonstrate the effectiveness of the proposed deep hybrid graph-based forecasting model compared to the state-of-the-art methods in terms of its forecasting accuracy, runtime, and scalability. Our case study reveals that GraphDF can successfully generate cloud usage forecasts and opportunistically schedule workloads to increase cloud cluster utilization by 47.5\% on average. \end{abstract} ]
\printAffiliationsAndNotice{}
\section{Introduction} \label{sec:intro} Cloud computing allows tenants to allocate and pay for resources on-demand. While convenient, this model requires users to provision the right amount of physical or \textit{virtualized} resources for every workload. The workloads often undergo striking variations in demand making static provisioning a poor fit. Moreover, allocating very few resources degrades the workload performance. Additionally, cloud platforms are exceptionally large and arduous to operate. Hence, optimizing their use is beneficial for both cloud operators and tenants. Accurate forecasting of workload patterns in cloud can aid in resource allocation, scheduling, and workload co-location decision for different workloads~\cite{clusterdata:Sliwko2018,clusterdata:Sebastio2018c,clusterdata:Sirbu2015}, massively reducing the operating costs and increasing resource efficiency~\cite{clusterdata:Liu2018gh}. Learning must be fast to ensure precise decision-making and quick adaption to changes in demand.
\begin{figure}
\caption{
In (a) the time-series of CPU usage for a node (machine) in the Google workload data shown in \textcolor{red}{red} and its immediate neighbors in the graph (\textcolor{blue}{blue}) are highly correlated,
whereas in (b) the time-series of randomly selected nodes are \emph{significantly different}.
}
\label{fig:node-and-neighbors}
\label{fig:node-and-random-timeseries}
\label{fig:motivation}
\end{figure}
Previous work on time-series forecasting has focused mostly on local forecasting models~\cite{brahim2004gaussian,girard2003gaussian} that treat each time-series independently or global forecasting models~\cite{Flunkert2017DeepARPF,wen2017multi,rangapuram2018deep} that consider all time-series jointly. There has also recently been hybrid local-global models~\cite{wang2019deep,sen2019think} that attempt to combine the benefits of both~\cite{crawley2013r,gelman2013bayesian}. These past works all assume time-series are either completely independent or completely dependent. However, these assumptions are often violated in practice as shown in Fig.~\ref{fig:motivation} where a node time-series of CPU usage from the Google workload data is shown to be highly dependent (correlated) on an arbitrary number of other node time-series.
In this work, we propose a deep hybrid \emph{graph-based} probabilistic forecasting model called Graph Deep Factors (GraphDF) that allows nodes and their time-series to be dependent (connected) in an arbitrary fashion. GraphDF leverages a \emph{relational global model} that uses the dependencies between time-series in the graph to learn the complex non-linear patterns globally while leveraging a \emph{relational local model} to capture the individual random effects of each time-series locally. The relational global model in GraphDF improves the runtime performance and scalability since instead of jointly modeling all time-series together (fully connected graph), which is computationally intensive, GraphDF learns the global latent factors that capture the complex non-linear time-series patterns among the time-series by leveraging only the graph that encodes the dependencies between the time-series. GraphDF serves as a general framework for deep graph-based probabilistic forecasting as many components are completely interchangeable including the relational local and relational global models.
Relational local models in GraphDF use not only the individual time-series but also the neighboring time-series that are 1 or 2 hops away in the graph. Thus, the proposed relational local models are more data efficient, especially when considering shorter time-series. For instance, given an individual time-series with a short length (\emph{e.g.}, only 6 previous values), purely local models would have problems accurately estimating the parameters due to the lack of data points. However, relational local models can better estimate such parameters by leveraging not only the individual time-series but the neighboring dependent time-series that are 1 or 2 hops away in the graph.
Relational global models in GraphDF are typically faster and more scalable since they avoid the pairwise dependence assumed by global models via the graph structure. By leveraging the dependencies between time-series encoded in the graph, GraphDF avoids a significant amount of work that would be required if the time-series are modeled jointly as done in Deep Factors (DF;~\citeauthor{wang2019deep}~\citeyear{wang2019deep}).
\subsection*{Main Contributions} We propose a general and extensible deep hybrid graph-based probabilistic forecasting framework called Graph Deep Factors (GraphDF) that is capable of learning complex non-linear time-series patterns globally using the graph time-series data to improve both computational efficiency and forecasting accuracy while learning individual probabilistic models for individual time-series based on their own time-series and the collection of time-series from the immediate neighborhood of the node in the graph. The GraphDF framework is data-driven, fast, and scalable for applications requiring real-time performance such as opportunistic scheduling in the cloud.
The state-of-the-art deep hybrid probabilistic forecasting methods focus on learning a global model that considers all time-series jointly as well as a local models learned from each individual time-series independently. In this work, we propose a deep hybrid \emph{graph-based} probabilistic forecasting model that lies in between these two extremes. In particular, we propose a relational global model that learns complex non-linear time-series patterns globally using the structure of the graph to improve both computational efficiency and prediction accuracy. Similarly, instead of modeling every time-series independently, we learn a relational local model that not only considers its individual time-series but the time-series of nodes that are connected to an individual node in the graph. Furthermore, GraphDF naturally generalizes many existing models including those based purely on local and global models, or a combination of both. This is due to its flexibility to interpolate between purely non-relational models (either local, global, or both) and relational models that leverage the graph structure encoding the dependencies between the different time-series.
The experiments demonstrate the effectiveness of the proposed deep hybrid graph-based probabilistic forecasting model in terms of its forecasting performance, runtime, and scalability. Section~\ref{sec:opportunistic-scheduling} details a case study of applying GraphDF for forecasting cloud usage and scheduling batch workloads opportunistically to enhance utilization by 47.5\% on average.
\section{Related Work} \label{sec:related-work} Classical methods such as ARIMA and exponential smoothing~\cite{ARIMA,Gardner1985ExponentialST} mainly target univariate forecasting tasks but fail to utilize relations between time-series.
Deep learning forecasting methods~\cite{ghaderi2017deep,lim2019enhancing,ahmed2010empirical,bai2018empirical,lim2018forecasting}, by contrast, are capable to perform multivariate forecasting by modeling the complex dependence among time-series. The commonly used RNN structures~\cite{zhang2005neural,neural-forecasting-survey}, especially the variants LSTM~\cite{LSTM-1997,Bandara2020LSTMMSNetLF} and GRU~\cite{GRUchung2014}, excel at encoding both past information and current input for forecasting. Further efforts have been made to accommodate various schemes and techniques, such as sequence-to-sequence models~\cite{bontempi2012machine,seq2seqNIPS2014,fan2019multi} for multi-step ahead prediction, and attention mechanism~\cite{qin2017dual,DSANet} for dependency modeling.
While earlier work focuses on \textit{point forecasting} which aims at predicting optimal expected values, there is an increasing interest in \textit{probabilistic forecasting} models~\cite{wen2017multi,Alexandrov2019GluonTSPT,rangapuram2018deep,henaff2017prediction,maddix2018deep}. Probabilistic models yield prediction as distributions and have the advantage of uncertainty estimates, which are important for downstream decision making. Some recent probabilistic models are proposed in the multivariate manner, for example, Salina et al. propose DeepAR~\cite{Flunkert2017DeepARPF}, a global model based on autoregressive recurrent networks and trained with all time-series in the same manner. Wang et. al. propose DF~\cite{wang2019deep}, a hybrid global-local model that assumes time-series are determined by shared factors as well as individual randomness. These methods indiscriminately model mutual dependence between time-series. Hence, they imply a strong and unrealistic assumption that all time-series are pairwise related to one another in a uniformly equivalent way.
\Paragraph{Graph-based forecasting models} Modeling the unique relations to each individual time-series from others naturally leads us to graph models~\cite{wang2019deepgraph}, among which Graph Neural Network(GNN)~\cite{GCN17} has recently showed great successes in extracting the information across nodes. A tremendous amount of following work has been proposed to incorporate GNN with RNN~\cite{ST-MGCN,wang2018graph}, while most of them are limited in spatio-temporal study, such as traffic prediction~\cite{STGCN-ijcai2018} and ride-hailing demand forecasting~\cite{STDN-aaai2019,DMVST-aaai2018}. Besides, these methods are not probabilistic models and they fail to deliver uncertainty estimates.
\Paragraph{Resource usage prediction} Accurate forecasting is critical in resource usage prediction for a cloud platform to scale and schedule tasks according to the demand~\cite{Cloudplatform,venkataraman2014power}. Existed work covers both traditional methods~\cite{workloadARIMA:QoS,zia2017adaptive}, machine learning approaches~\cite{crankshaw2017clipper} such as K-nearest neighbors~\cite{farahnakian2015utilization,shen2011cloudscale} and linear regression~\cite{6619533,yang2014cost} and RNN-based methods~\cite{RPPS,kumar2018workload,cloudpredict}. However, none of these methods leverages a graph to model the relationships between nodes.
\section{Graph Deep Factors} \label{sec:framework} In this section, we describe a general and extensible framework called Graph Deep Factors (GraphDF). It is capable of learning complex non-linear time-series patterns globally using the graph time-series data to improve both computational efficiency and performance while learning individual probabilistic models for individual time-series based on their own time-series and the collection of related time-series from the neighborhood of the node in the graph. The GraphDF framework is data-driven, flexible, accurate, and fast/scalable for large collections of multi-dimensional time-series data.
\subsection{Problem Formulation} \label{sec:problem-formulation} We first introduce the deep graph-based probabilistic forecasting problem. Notably, this is the first deep graph-based probabilistic forecasting framework. The framework is comprised of a relational graph global component (described in Section~\ref{sec:rel-global-model}) that learns the complex non-linear time-series patterns in the large collection of graph-based time-series data and a relational local component (Section~\ref{sec:rel-local-component}) that handles uncertainty by learning a probabilistic forecasting model for every individual node in the graph that not only considers the time-series of the individual node, but also the time-series of nodes directly connected in the graph.
The proposed framework solves the following graph-based time-series forecasting problem. Let $G=(V,E,\calX, \calZ)$ denote the graph model where $V$ is the set of nodes, $E$ is the set of edges, and $\calX=\{\mX^{(i)}\}_{i=1}^{N}$ is the set of covariate time-series associated with the $N$ nodes in $G$ where $\mX^{(i)} \in \mathbb{R}^{D \times T}$ is the covariate time-series data associated with node $i$.
Hence, each node is associated with $D$ different covariate time-series. Furthermore, $\calZ=\{\vz^{(i)}\}_{i=1}^{N}$ is the set of time-series associated with the $N$ nodes in $G$.
The $N$ nodes can be connected in an arbitrary fashion that reflects the dependence between nodes. Two nodes $i$ and $j$ that contain an edge $(i,j) \in E$ in the graph $G$ encodes an explicit dependency between the time-series data of node $i$ and $j$. Intuitively, using these explicit dependencies encoded in $G$ can lead to more accurate forecasts as shown in Fig,~\ref{fig:motivation}.
Further, let $\vz_{1:T}^{(i)}$ denote a univariate time-series for node $i$ in the graph where $\vz_{1:T}^{(i)} = \big[z_{1}^{(i)} \, \cdots \, z_{T}^{(i)}\big] \in \mathbb{R}^{T}$ and $z_{t}^{(i)} \in \mathbb{R}$.
In addition, each node $i$ in the graph $G$ also has $D$ covariate time-series, $\mX^{(i)} \in \mathbb{R}^{D \times T}$ where $\mX^{(i)}_{:,t} \in \mathbb{R}^{D}$ (or $\vx^{(i)}_{t} \in \mathbb{R}^{D}$) represents the $D$ covariate values at time step $t$ for node $i$.
We denote $\mA \in \mathbb{R}^{N \times N}$ as the sparse adjacency matrix of the graph $G$ where $N=|V|$ is the number of nodes. If $(i,j) \in E$, then $A_{ij}$ denotes the weight of the edge (dependency) between node $i$ and $j$. Otherwise, $A_{ij}=0$ when $(i,j) \not\in E$.
We denote the unknown parameters in the model as $\mPhi$. Our goal is to learn a generative and probabilistic forecasting model described by $\mPhi$ that gives the (joint) distribution on target values in the future horizon $\tau$: \begin{equation}\label{eq:problem}
\mathbb{P}\Big(\big\{\vz_{T+1:T+\tau}^{(i)}\big\}_{\!i=1}^{\!N} \,\Big|\, \mA, \big\{\vz_{1 : T}^{(i)}, \mX_{:,1 : T+\tau}^{(i)}\!\big\}_{\!i=1}^{\!N}; \mPhi \Big) \end{equation}\noindent Hence, solving Eq.~\eqref{eq:problem} gives the joint probability distribution over future values given all covariates and past observations along with the graph structure represented by $\mA$ that encodes the explicit dependencies between the $N$ nodes and their corresponding time-series $\{\vz^{(i)}, \mX^{(i)}\!\}_{i=1}^{N}$.
\subsection{Framework Overview} \label{sec:framework-generative} The Graph Deep Factors (GraphDF) framework aims to learn a parametric distribution to predict future workloads. In GraphDF, each node $i$ and its time-series $z^{(i)}_{t}, \forall t=1,2,\ldots$ can be connected to other nodes and their time-series in an arbitrary fashion, which is encoded in the graph structure $G$. These connections represent explicit dependencies or correlations between the time-series of the nodes. Furthermore, we also assume that each node $i$ and their time-series $\vz^{(i)}_{1:t}$ are governed by two key components including (1) a relational global model (Section~\ref{sec:rel-global-model}), and (2) a relational local random effect model (Section~\ref{sec:rel-local-component}). As such, GraphDF is a hybrid forecasting framework. Both the relational global component and relational local component of our framework leverage the graph and the way in which it is leveraged depends on the specific underlying model used for each component.
In the relational global component of GraphDF, we assume there are $K$ latent relational global factors that determine the fixed effect of each node and their time-series. The relational global model consists of an approach that leverages the adjacency matrix $\mA$ of the graph $G$ and $\big\{\mX_{:,1:t}^{(j)}, \vz_{1:t-1}^{(j)}\big\}_{j=1}^{N}$ for learning the $K$ relational global factors that capture the relational non-linear time-series patterns in the graph-based time-series data, \begin{equation} \label{eq:relationalg-final} s_{k}(\cdot) = \textsc{gcrn}_k(\cdot), \quad k = 1,\ldots, K \end{equation}\noindent where $s_{k}(\cdot),$\; $k=1,2,\ldots,K$ are the $K$ relational global factors that govern the underlying graph-based time-series data of all nodes in $G$. In Eq.~\ref{eq:relationalg-final}, we learn the relational global factors using a Graph Convolutional Recurrent Network (GCRN)~\cite{gcrn17}, however, GraphDF is flexible for use with any other arbitrary deep time-series model, see Appendix~\ref{appendix:dcrnn} where we have also adapted DCRNN. These are then used to obtain the relational global fixed effects function $c^{(i)}$ for node $i$ as follows, \begin{equation} \label{eq:fixed-final} c^{(i)}(\cdot) = \sum_{k=1}^K w_{i,k}\cdot s_{k}(\cdot) \end{equation}\noindent where $w_{i,k}$ represents the $K$-dimensional embedding for node $i$. Therefore, the final relational non-random fixed effect for node $i$ is simply a linear combination of the $K$ global factors and the embedding $\vw_i \in \mathbb{R}^{K}$ for node $i$.
Now we use a relational local model discussed in Section~\ref{sec:rel-local-component} to obtain the local random effects for each node $i$. More formally, we define the \emph{relational local random effects} function $b^{(i)}$ for a node $i$ in the graph $G$ as, \begin{equation}\label{eq:random-final} b^{(i)}(\cdot) \sim \mathcal{R}_i, \quad i = 1, \ldots, N \end{equation}\noindent where $\mathcal{R}_i$ can be any relational probabilistic time-series model.
To compute $\mathbb{P}(\vz^{i}_{1:t}|\mathcal{R}_i)$ efficiently, we ensure $b^{(i)}_t$ obeys a normal distribution, and thus can be derived fast.
The \emph{relational latent function} of node $i$ denoted as $v^{(i)}$ is then defined as, \begin{equation}\label{eq:vi-final} v^{(i)}(\cdot) = c^{(i)}(\cdot) + b^{(i)}(\cdot) \end{equation}\noindent where $c^{(i)}$ is the relational fixed effect of node $i$ and $b^{(i)}$ is the relational local random effect for node $i$. Hence, the relational latent function of node $i$ is simply a linear combination of the relational fixed effect $c^{(i)}$ from Eq.~\ref{eq:fixed-final} and its local relational random effect $b^{(i)}$ from Eq.~\ref{eq:random-final}.
Then, \begin{equation}\label{eq:emission-final}
z_{t}^{(i)} \sim \mathbb{P}\Big( z_{t}^{(i)} \, \big| \; v^{(i)}\big(\mA, \big\{\mX_{:,1:t}^{(j)}, \vz_{1:t-1}^{(j)}\big\}^{\!N}_{\!j=1}\big)\!\Big) \end{equation}\noindent where the observation model $\mathbb{P}$ can be any parametric distribution. For instance, $\mathbb{P}$ can be Gaussian, Poisson, Negative Binomial, among others.
The GraphDF framework is defined in Eq.~\ref{eq:relationalg-final}-\ref{eq:emission-final}. All the functions $s_k(\cdot), b^{(i)}(\cdot), v^{(i)}(\cdot)$ take past observations and covariates $\big\{\vz_{1 : t-1}^{(j)}, \mX_{:,1 : t}^{(j)}\!\big\}_{\!j=1}^{\!N}$, as well as the graph structure in the form of adjacency matrix $\mA$ as inputs. We define $\vw_i = \big[w_{i,1} \cdots w_{i,k} \cdots w_{i,K}\big] \in \mathbb{R}^{K}$ as the $K$-dimension embedding for time-series $\vz^{(i)}$ where $w_{i,k} \in \mathbb{R}$ is the weight of the $k$-th factor for node $i$.
\subsection{Relational Global Model} \label{sec:rel-global-model} The relational global model learns $K$ relational global factors from all time-series by a graph-based model. These relational global factors are considered as the driving latent factors. After the relational global factors are derived from the model, they are then used in a linear combination with weights given by embeddings for each time-series $\bm{w}_i$, as shown in Eq.~\eqref{eq:fixed-final}.
We first show how GCRN is modified for learning relational global factors in GraphDF. Let $\vx_{t}^{(i)} \in \mathbb{R}^{D}$ denote the $D$ covariates of node $i$ at time step $t$. Now, we define the input temporal features of the relational global factor component with respect to graph $G$ as, \begin{equation} \label{eq:Y_t_} \mY_{t} = \begin{bmatrix} z_{t-1}^{(1)} & {\vx_{t}^{(1)}}^{\intercal} \\ \vdots & \vdots \\ z_{t-1}^{(N)} & {\vx_{t}^{(N)}}^{\intercal} \end{bmatrix} \in \mathbb{R}^{N\times P} \end{equation}\noindent where $P=D+1$ for simplicity. We refer to $\mY_t$ as a time-series graph signal. The aggregation of information from other nodes is performed by a graph convolution operation defined as the multiplication of a temporal graph signal with a filter $g_\theta$. Given input features $\mY_t$, the graph convolution operation is denoted as $f_{\,\star_{\mathcal{G}}\,}{\mathbf{\Theta}}$ with respect to the graph $G$ and parameters $\theta$: \begin{align}
f_{\,\star_{\mathcal{G}}\,}{\mathbf{\Theta}}(\mY_t) &= g_\theta(\mL) \mY_t \label{eq:gcrn-graphconv} \\
&= \mU g_\theta(\mathbf{\Lambda}) \mU^T \mY_t \in \mathbb{R}^{N\times P}
\label{eq:gc} \end{align} where $\mL=\mI-\mD^{-\frac{1}{2}}\mA\mD^{-\frac{1}{2}}$ is the normalized Laplacian matrix of the adjacency matrix, $\mI \in \mathbb{R}^{N\times N}$ is an identity matrix. $D_{ii}=\sum_{j}A_{ij}$ is the diagonal weighted degree matrix. $\mL=\mU \mathbf{\Lambda} \mU^T$ is the eigenvalue decomposition. $g_\theta(\mathbf{\Lambda})=\text{diag}(\vtheta)$ denotes a filter parameterized by the coefficients $\vtheta \in \mathbb{R}^{N}$ in the Fourier domain. Directly applying Eq.~\eqref{eq:gc} is computationally expensive due to the matrix multiplication and the eigen-decomposition of $\mL$. To accelerate the computation, the Chebyshev polynomial approximation up to a selected order $L-1$ is \begin{equation} \label{eq:filt_cheby}
g_\theta(\mL) = \sum_{l=0}^{L-1} \theta_l T_l(\tmL), \end{equation} where $\vtheta = \big[\theta_0\,\cdots\,\theta_{L-1}\big] \in \mathbb{R}^{L}$ in Eq.~\eqref{eq:filt_cheby} is the Chebyshev coefficients vector.
Importantly, $T_l(\tmL)=2\tmL T_{l-1}(\tmL) - T_{l-2}(\tmL)$ is recursively computed with the scaled Laplacian $\tmL=2\mL/\lambda_{\max}-\mI \in \mathbb{R}^{N\times N}$, and starting values $T_0=1$ and $T_1=\tmL$. The Chebyshev polynomial approximation improves the time complexity to linear in the number of edges $O(L|E|)$, \emph{i.e.}, number of dependencies between the multidimensional node time-series.
The order $L$ controls the local neighborhood time-series that are used for learning the relational global factors, \emph{i.e.}, a node's multi-dimensional time-series only depends on neighboring node time-series that are at maximum $L$ hops away in $G$.
Let $\mathbf{\Theta} \in \mathbb{R}^{P \times Q \times L}$ be a tensor of parameters that maps the dimension $P$ of input to the dimension $Q$ of output: \begin{align}
\!\!\!\! \mH_{:, q} = \tanh\!\Bigg[\sum_{p=1}^{P} f_{\,\star_{\mathcal{G}}\,}{\mathbf{\Theta}}(\mY_{t,\;:, p})\Bigg],\ \text{for} \; q \in {1\,\ldots\,Q} \end{align} The relational global component integrates the temporal dependence and relational dependence among nodes with the graph convolution,
\begin{align}
\!\!\!\! &\mI_t = \sigma(\mathbf{\Theta}_{I} {\,\star_{\mathcal{G}}\,}{}[\mY_t, \mH_{t-1}] + \mW_I\odot\mC_{t-1} + \vb_I) \\ \!\!\!\! &\mF_t \!= \sigma(\mathbf{\Theta}_{F} {\,\star_{\mathcal{G}}\,}{}[\mY_t, \mH_{t-1}] + \mW_F\odot\mC_{t-1} + \vb_F) \\ \!\!\!\! &\mC_t \!=\! \mF_t\!\odot \mC_{t\text{-}1} \!+\! \mI_t \!\odot\! \tanh(\mathbf{\Theta}_{C} \!{\,\star_{\mathcal{G}}\,}{}[\mY_t, \mH_{t\text{-}1}] + \!\vb_C) \\ \!\!\!\! &\mO_t \!= \sigma(\mathbf{\Theta}_O{\,\star_{\mathcal{G}}\,}{}[\mY_t, \mH_{t-1}] + \mW_O\odot \mC_t + \vb_O) \\ \!\!\!\! &\mH_t \!= \mO_t \odot \tanh(\mC_t) \label{eq:gconvlstm-hid} \end{align}\noindent where $\mI_t\in \mathbb{R}^{N\times Q}, \mF_t\in \mathbb{R}^{N\times Q}, \mO_t \in \mathbb{R}^{N\times Q}$ are the input, forget and output gate in the LSTM structure. $Q$ is the number of hidden units, $\mW_I\in \mathbb{R}^{N\times Q}, \mW_F\in \mathbb{R}^{N\times Q}, \mW_O \in \mathbb{R}^{N\times Q}$ and $\vb_I, \vb_F, \vb_C, \vb_O \in \mathbb{R}^{Q}$ are weights and bias parameters, $\mathbf{\Theta}_I\in \mathbb{R}^{P\times Q}, \mathbf{\Theta}_F \in \mathbb{R}^{{P\times Q}}, \mathbf{\Theta}_C \in \mathbb{R}^{{P\times Q}}, \mathbf{\Theta}_O \in \mathbb{R}^{{P\times Q}}$ are parameters corresponding to different filters.
The hidden state $\mH_t \in \mathbb{R}^{N\times Q}$ encodes the observation information from $\mH_{t-1}$ and $\mY_t$, as well as the relations across nodes through the graph convolution described by $\mathbf{\Theta} {\,\star_{\mathcal{G}}\,}{}(\cdot)$ in Eq.~\eqref{eq:gcrn-graphconv}. From hidden state $\mH_t$, we derive the value of $K$ relational global factors at time step $t$ as $\mS_t \in \mathbb{R}^{N\times K} $ through a fully connected layer, \begin{align}
\mS_t = \mH_t \mW + \vb \label{eq:gcrn-st} \end{align} where $\mW \in \mathbb{R}^{Q \times K}$ and $\vb \in \mathbb{R}^{K}$ are the weight matrix and bias vector (for the $K$ relational global factors), respectively.
The relational global factors $\mS_t$ is derived from the Eq.~\eqref{eq:gcrn-st} that capture the complex non-linear time-series patterns between the different time-series globally.
Finally, the fixed effect at time $t$ is derived for each node $i$ as a weighted sum with the embedding $\vw_{i} \in \mathbb{R}^{K}$ and the relational global factors $\mS_t$, as \begin{equation}
c_{t}^{(i)}(\cdot) = \sum_{k=1}^K w_{i,k} \cdot S_{i,k,t} \label{eq:gcrn-cit} \end{equation} The embedding $\vw_i$ represents the weighted contribution that each relational factor has on node $i$.
\subsection{Relational Local Model}\label{sec:rel-local-component} The (stochastic) relational local component handles uncertainty by learning a probabilistic forecasting model for every individual node in the graph that not only considers the time-series of the individual node, but also the time-series of nodes directly connected in the graph. This has the advantage of improving both forecasting accuracy and data efficiency.
GraphDF is therefore able to make more accurate forecasts further in the future with less training data.
The random effects in the relational local model represent the local fluctuations of the individual node time-series. The relational local random effect for each node time-series $b^{(i)}$ is sampled from the relational local model $\calR_{i}$, as shown in Eq.~\eqref{eq:random-final}. For $\calR_i$, we choose the Gaussian distribution as the likelihood function for sampling, but other parametric distributions such as Student-t or Gamma distributions are also possible. Compared to the relational global component of GraphDF from Section~\ref{sec:rel-global-model} that uses the entire graph $G$ along with all the node multi-dimensional time-series to learn $K$ global factors that capture the most important non-linear time-series patterns in the graph-based time-series data, the relational local component focuses on modeling an individual node $i \in V$ and therefore leverages only the time-series of node $i$ and the set of highly correlated time-series from its immediate local neighborhood $\Gamma_i$. Hence, $\{\vz^{(j)}, \mX^{(j)}\}, j \in \Gamma_i$. Intuitively, the relational local component of GraphDF achieves better data efficiency by leveraging the highly correlated neighboring time-series along with its own time-series. This allows GraphDF to make more accurate forecasts further in the future with less training data. We now introduce probabilistic GCRN that can be used as the stochastic relational local component in GraphDF.
\begin{table*}[ht!]
\centering
\caption{Dataset statistics and properties}
\label{table:dataset-statistics}
\small
\footnotesize
\begin{tabular}{l rr cccccccc}
\toprule
& & & & Avg. & Median & Mean & & & Median \\
Data & $|V|$ & $|E|$ & Density & Deg. & Deg. & wDeg. & Time-scale & T usage & CPU usage\\
\midrule \TTT\BBB
\textbf{Google} & 12,580 & 1,196,658 & 0.0075 & 95.1 & 40 & 30.3 & 5 min & 8,354 & 21.4\% \\
\textbf{Adobe} & 3,270 & 221,984 & 0.0207 & 67.9 & 15 & 67.7 & 30 min & 1,687 & 9.1\% \\
\bottomrule
\end{tabular}
\end{table*}
In contrast to the relational global model in Section~\ref{sec:rel-global-model}, the relational local model focuses on learning an individual local model for each individual node based on its own multi-dimensional time-series data as well as the nodes neighboring it. This enables us to model the local fluctuations of the individual multi-dimensional time-series data of each node. Compared to RNN, the benefits of the proposed probabilistic GCRN model in the local component is that it not only models the sequential nature of the data, but also exploits the graph structure by using the surrounding nodes to learn a more accurate model for each individual node in $G$. This is an ideal property for we assume the fluctuations of each node are related to those of other connected nodes in the $\ell$-localized neighborhood, which was shown to be the case in Fig.~\ref{fig:motivation}.
Let $C = \Gamma_i$ denote the set of neighbors of a node $i$ in the graph $G$. Note that $C$ can be thought of as the set of related neighbors of node $i$, which may be the immediate 1-hop neighbors, or more generally, the $\ell$-hop neighbors of $i$.
Recall that we define $\vx_{t}^{(i)} \in \mathbb{R}^{D}$ as the $D$ covariates of node $i$ at time $t$. Then, we define $\mX_t^{C}$ as an $|C| \times D$ matrix consisting of the covariates of all the neighboring nodes $j \in C$ of node $i$. Now, we define the input temporal features of the relational local model for node $i$ as, \begin{equation} \label{eq:Y_t_rel-local} \mY_{t}^{(i)} = \begin{bmatrix} z_{t-1}^{(i)} & {\vx_{t}^{(i)}}^{\intercal} \\ \vz_{t-1}^{(C)} & \mX_{t}^{(C)} \end{bmatrix}
\end{equation}\noindent Let $\mL^{(i)} \in \mathbb{R}^{(|C|+1)\times(|C|+1)}$ denote the submatrix of Laplacian matrix $\mL$ that consist of rows and columns corresponding to node $i$ and its neighbors $C$.
For each node $i$, we derive the relational local random effect using its past observations and covariates of the node $i$ and those of its neighbors through the graph convolution regarding $\mL^{(i)}$. \begin{align} \nonumber
& \mI_t^{(i)} \!= \sigma(\mathbf{\Theta}_{I}^{(i)} {\,\star_{\mathcal{G}}\,}{}[\mY_t^{(i)}, \; \mH_{t-1}^{(i)}] + \mW_I^{(i)}\odot\mC_{t-1}^{(i)} + \vb_I^{(i)})\\ \nonumber & \mF_t^{(i)} \!= \sigma(\mathbf{\Theta}_{F}^{(i)} {\,\star_{\mathcal{G}}\,}{}[\mY_t^{(i)}, \; \mH_{t-1}^{(i)}] + \mW_F^{(i)}\odot\mC_{t-1}^{(i)} + \vb_F^{(i)})\\ \nonumber
& \mC_t^{(i)} \!= \!\mF_t^{(i)}\!\odot\! \mC_{t-1}^{(i)} \!+ \!\mI_t^{(i)} \!\odot\! \tanh(\mathbf{\Theta}_{C}^{(i)} {\,\star_{\mathcal{G}}\,}{}[\mY_t^{(i)}, \mH_{t-1}^{(i)}] + \vb_C^{(i)}) \\ \nonumber
&\mO_t^{(i)} \!= \sigma(\mathbf{\Theta}_{O}^{(i)} {\,\star_{\mathcal{G}}\,}{}[\mY_t^{(i)}, \; \mH_{t-1}^{(i)}] + \mW_O^{(i)}\odot \mC_t^{(i)} + \vb_O^{(i)}) \\ \nonumber
& \mH_t^{(i)} \!= \mO_t^{(i)} \odot \tanh(\mC_t^{(i)}) \label{eq:local-gconvlstm-hid} \end{align}
where $\mathbf{\Theta}_{I}^{(i)} \in \mathbb{R}^{P\times R}$, $\mathbf{\Theta}_{F}^{(i)} \in \mathbb{R}^{P\times R}$, $\mathbf{\Theta}_{C}^{(i)} \in \mathbb{R}^{P\times R}$, $\mathbf{\Theta}_{O}^{(i)} \in \mathbb{R}^{P\times R}$ denote the parameters corresponding to different filters of the relational local model,
$R$ is the number of hidden units in the relational local model, and recall $P=D+1$. Further, $\mH_t^{(i)} \in \mathbb{R}^{(|C|+1)\times R}$ is the hidden state for node $i$ and its neighbors $\Gamma_i$.
$\mW_I^{(i)}\in\mathbb{R}^{(|C|+1)\times R},
\mW_F^{(i)}\in\mathbb{R}^{(|C|+1)\times R},
\mW_O^{(i)}\in\mathbb{R}^{(|C|+1)\times R}$ are weight matrix parameters and $\vb_{I}^{(i)}\in \mathbb{R}^{R}, \vb_{F}^{(i)}\in \mathbb{R}^{R}, \vb_{C}^{(i)}\in \mathbb{R}^{R}, \vb_{O}^{(i)}\in \mathbb{R}^{R}$ are bias vector parameters. Note in the above formulation, we assume $\ell=1$, hence, only the immediate 1-hop neighbors are used.
From the hidden state $\mH_t^{(i)}$, we only take the row corresponding to node $i$ to derive the relational local random effect for node $i$. We denote the value as $\vh_t^{(i)} \in \mathbb{R}^{R}$, and apply a fully connected layer with a softplus activation function to aggregate the hidden units, \begin{equation}
\sigma_{i,t} =
\log
\big(
\exp({\vw^{(i)}}^{\intercal}\vh_t^{(i)} + \beta^{(i)})+1
\big) \label{eq:sigma-gcrn} \end{equation} where $\vw^{(i)} \in \mathbb{R}^{R}$ and $\beta^{(i)}$ are weight vector and bias, respectively.
Finally, the relational local random effect $b_t^{(i)}(\cdot)$ for node $i$ at time $t$ is sampled from a Gaussian distribution with zero mean and a variance given by $\sigma^2$ in Eq.\eqref{eq:sigma-gcrn}, \begin{align}
b_{t}^{(i)}(\cdot) &\sim \mathcal{N}(0, \sigma_{i,t}^2) \label{eq:gcrn-random} \end{align} The relational local random effect $b_t^{(i)}$ captures both past observations, covariate values of node $i$ and its neighbors $\Gamma_i$ for uncertainty estimates through $\sigma_{i,t}$.
A small $\sigma_{i,t}$ means a low uncertainty of prediction for node $i$ at $t$. Specifically, the probabilistic model subsumes the point forecasting model when the relational local random effect is zero for all nodes at all time steps as $\sigma_{i,t}=0, \forall i \forall t$. The probabilistic property also allows the uncertainty to be propagated forward in time.
\begin{table*}[!t] \small \centering \setlength{\tabcolsep}{4.6pt} \caption{Results for one-step ahead forecasting (\textsc{p50ql} and \textsc{p90ql}). }
\label{table:results-one-step-p50ql-p90ql} \begin{tabular}{ @{}llccccccc } \toprule & \multirow{1}{*}{\textsc{data}} & NBEATS & MQRNN & DeepAR & DF & GraphDF-GG & GraphDF-GR & GraphDF-RG \\ \midrule \TTT\BBB \multirow{2}{*}{\textsc{p50ql}} & \multirow{1}{*}{\textbf{Google}} & 10.064\ \!$\pm$\ \!62.117 & 0.172\ \!$\pm$\ \!0.001 & 0.098\ \!$\pm$\ \!0.001 & 0.239\ \!$\pm$\ \!0.001 & \textbf{0.072\ \!$\pm$\ \!0.000} & 0.076\ \!$\pm$\ \!0.000 & 0.077\ \!$\pm$\ \!0.000 \\
& \multirow{1}{*}{\textbf{Adobe}} & 3.070\ \!$\pm$\ \!2.286 & 0.272\ \!$\pm$\ \!0.001 & 0.619\ \!$\pm$\ \!0.026 & 1.649\ \!$\pm$\ \!0.001 & \textbf{0.188\ \!$\pm$\ \!0.000} & 0.210\ \!$\pm$\ \!0.001 & 0.746\ \!$\pm$\ \!0.835 \\
\midrule \TTT\BBB \multirow{2}{*}{\textsc{p90ql}} & \multirow{1}{*}{\textbf{Google}} & 2.013\ \!$\pm$\ \!2.485 & 0.106\ \!$\pm$\ \!0.001 & 0.051\ \!$\pm$\ \!0.000 & 0.144\ \!$\pm$\ \!0.002 & \textbf{0.041\ \!$\pm$\ \!0.000} & 0.044\ \!$\pm$\ \!0.000 & 0.048\ \!$\pm$\ \!0.000 \\ & \multirow{1}{*}{\textbf{Adobe}} & 5.524\ \!$\pm$\ \!7.410 & 0.217\ \!$\pm$\ \!0.000 & 0.949\ \!$\pm$\ \!0.086 & 1.802\ \!$\pm$\ \!0.002 & \textbf{0.153\ \!$\pm$\ \!0.001} & 0.169\ \!$\pm$\ \!0.001 & 0.342\ \!$\pm$\ \!0.037 \\ \bottomrule \end{tabular} \end{table*}
\subsection{Learning \& Inference} \label{sec:framework-learning} To train a GraphDF model, we estimate the parameters $\mPhi$, which represent all trainable parameters ($\mW$, etc.) in the relational global and relational local model, as well as the parameters in the embeddings. We leverage the maximum likelihood estimation, \begin{equation} \label{eq:parameters-optimization}
\mPhi = \text{argmax} \sum_i \mathbb{P}\big(\vz^{(i)} \big| \mPhi, \mA, \big\{\mX_{:,1:t}^{(j)}, \vz_{1:t-1}^{(j)}\big\}^{\!N}_{\!j=1} \!\big) \end{equation} where
\begin{equation} \label{eq:neg-log-like} \mathbb{P}(\vz^{(i)}) = \sum_{t}-\frac{1}{2}\ln(2\pi\sigma_{i,t})- \sum_{t}\frac{(z_{t}^{(i)} - c_{i,t})^2}{2\sigma_{i,t}^2} \end{equation} is the negative log likelihood of Gaussian function. Notice that maximizing $-\frac{1}{2}\ln(2\pi\sigma_{i,t})$ will minimize the relational local random effect, at the same time, $\sigma$ is small when the predicted fixed effect $c_{i,t}$ is close to the actual value $z_t^{(i)}$, as shown in the second term $\frac{(z_t^{(i)}-c_{i,t})}{2\sigma_{i,t}^2}$ in Eq.~\eqref{eq:neg-log-like}. We provide a general training approach in the Appendix~\ref{appendix-opportunistic-learning}(see Algorithm~\ref{alg:graphDF-training} for details).
\subsection{Model Variants} \label{sec:model-variants} In this section, we define a few of the GraphDF model variants investigated in Section~\ref{sec:exp}. \begin{itemize}
\item \textbf{GraphDF-GG:} This is the default model in our GraphDF framework, where we use a graph model to learn the $K$ relational global factors (Sec.~\ref{sec:rel-global-model}) and the probabilistic local graph component from Sec.~\ref{sec:rel-local-component} as the relational local model.
\item \textbf{GraphDF-GR:} This model variant from the GraphDF framework uses the GCRN from Sec.~\ref{sec:rel-global-model} to learn the $K$ relational global factors from the graph-based time-series data and leverages a simple RNN for modeling the local random effects of each node.
\item \textbf{GraphDF-RG:} This GraphDF model variant uses a simple RNN to learn the $K$ global factors
and the probabilistic graph component from Sec.~\ref{sec:rel-local-component} as the relational local model. \end{itemize}
The GraphDF framework is flexible with many interchangeable components including the relational global component (Sec.~\ref{sec:rel-global-model}) that uses the graph-based time-series data to learn the $K$ global factors and fixed effects of the nodes as well as the relational local model (Sec.~\ref{sec:rel-local-component}) for obtaining the relational local random effects of the nodes.
\section{Experiments} \label{sec:exp} The experiments are designed to investigate the following: (RQ1) Does GraphDF outperform previous state-of-the-art deep probabilistic forecasting methods? (RQ2) Are the GraphDF models fast and scalable for large-scale time-series forecasting? (RQ3) Can GraphDF generate cloud usage forecasts to effectively perform opportunistic workload scheduling?
\subsection{Experimental setup} We used two real-world cloud traces from Google~\cite{reiss2011google} and Adobe. The statistics and properties of these two datasets are in Table~\ref{table:dataset-statistics}. The Google dataset consists of 167 Gigabytes records of machine resource usage. Further details are described in the Appendix~\ref{appendix-data-details}.
For the opportunistic workload scheduling case study later in Section~\ref{sec:opportunistic-scheduling}, we need to train a model \emph{fast} within minutes and then forecast a single as well as multiple time steps ahead, which are then used to make opportunistic scheduling and scaling decisions. Therefore, to ensure the models are trained fast within minutes, we use only the most recent $6$ observations in the time-series data for training. We set the number of embedding dimension as $K=10$ in $\bm{w}_i \in \mathbb{R}^{K}$ and use time feature series as covariates. We set the embedding dimension to $K=10$ in $\bm{w}_i \in \mathbb{R}^{K}$ and used $D=5$ covariates for each time-series. Similar to DF~\cite{wang2019deep}, the time features (\emph{e.g.} minute of hour, hour of day) are used as covariates. We derive a fixed graph using Radial Based Function (RBF) on the past observations. See Appendix~\ref{appendix-data-details} for further details.
\begin{table*}[!t]
\setlength{\tabcolsep}{5.2pt} \centering \caption{Results for multi-step ahead forecasting (\textsc{p50ql}). }
\label{table:results-multi-step-ahead-forecasting-p50ql} \small \begin{tabular}{@{}l@{} cccccccc } \toprule \multirow{1}{*}{\textsc{data}} & \multirow{1}{*}{\textsc{h}} & NBEATS & MQRNN & DeepAR & DF & GraphDF-GG & GraphDF-GR & GraphDF-RG \\ \midrule \multirow{3}{*}{\textbf{Google}\;\;} & 3 & 0.741\ \!$\pm$\ \!0.050 & 0.257\ \!$\pm$\ \!0.011 & 0.148\ \!$\pm$\ \!0.001 & 0.400\ \!$\pm$\ \!0.004 & \textbf{0.091\ \!$\pm$\ \!0.001} & 0.134\ \!$\pm$\ \!0.002 & 0.097\ \!$\pm$\ \!0.000 \\
& 4 & 0.618\ \!$\pm$\ \!0.105 & 0.410\ \!$\pm$\ \!0.017 & 0.191\ \!$\pm$\ \!0.000 & 0.454\ \!$\pm$\ \!0.007 & \textbf{0.097\ \!$\pm$\ \!0.002} & 0.185\ \!$\pm$\ \!0.002 & 0.109\ \!$\pm$\ \!0.000 \\
& 5 & 0.485\ \!$\pm$\ \!0.021 & 0.684\ \!$\pm$\ \!0.012 & 0.466\ \!$\pm$\ \!0.006 & 0.563\ \!$\pm$\ \!0.017 & 0.128\ \!$\pm$\ \!0.000 & 0.284\ \!$\pm$\ \!0.012 & \textbf{0.126\ \!$\pm$\ \!0.001} \\
\midrule \multirow{3}{*}{\textbf{Adobe}} & 3 & 1.683\ \!$\pm$\ \!0.100 & 0.556\ \!$\pm$\ \!0.028 & 0.592\ \!$\pm$\ \!0.017 & 1.116\ \!$\pm$\ \!0.006 & \textbf{0.272\ \!$\pm$\ \!0.004} & 0.315\ \!$\pm$\ \!0.006 & 0.319\ \!$\pm$\ \!0.005 \\
& 4 & 1.424\ \!$\pm$\ \!0.210 & 0.574\ \!$\pm$\ \!0.011 & 0.629\ \!$\pm$\ \!0.024 & 1.029\ \!$\pm$\ \!0.001 & \textbf{0.314\ \!$\pm$\ \!0.004} & 0.353\ \!$\pm$\ \!0.007 & 0.405\ \!$\pm$\ \!0.007 \\
& 5 & 1.069\ \!$\pm$\ \!0.027 & 0.687\ \!$\pm$\ \!0.064 & 0.633\ \!$\pm$\ \!0.012 & 1.039\ \!$\pm$\ \!0.004 & \textbf{0.375\ \!$\pm$\ \!0.007} & 0.401\ \!$\pm$\ \!0.014 & 0.484\ \!$\pm$\ \!0.005 \\
\bottomrule \end{tabular}
\end{table*}
\begin{table*}[!t]
\setlength{\tabcolsep}{5.2pt} \centering \caption{Results for multi-step ahead forecasting (\textsc{p90ql}). }
\label{table:results-multi-step-ahead-forecasting-p90ql} \small \begin{tabular}{@{}l@{} cccccccc } \toprule \multirow{1}{*}{\textsc{data}} & \multirow{1}{*}{\textsc{h}} & NBEATS & MQRNN & DeepAR & DF & GraphDF-GG & GraphDF-GR & GraphDF-RG \\ \midrule \multirow{3}{*}{\textbf{Google}\;\;} & 3 & 0.830\ \!$\pm$\ \!0.262 & 0.091\ \!$\pm$\ \!0.001 & 0.067\ \!$\pm$\ \!0.000 & 0.208\ \!$\pm$\ \!0.002 & \textbf{0.051\ \!$\pm$\ \!0.000} & 0.051\ \!$\pm$\ \!0.000 & 0.089\ \!$\pm$\ \!0.000 \\
& 4 & 0.976\ \!$\pm$\ \!0.429 & 0.090\ \!$\pm$\ \!0.000 & 0.070\ \!$\pm$\ \!0.000 & 0.213\ \!$\pm$\ \!0.002 & \textbf{0.050\ \!$\pm$\ \!0.001} & 0.076\ \!$\pm$\ \!0.001 & 0.095\ \!$\pm$\ \!0.001 \\
& 5 & 0.523\ \!$\pm$\ \!0.085 & 0.124\ \!$\pm$\ \!0.000 & 0.134\ \!$\pm$\ \!0.000 & 0.220\ \!$\pm$\ \!0.002 & \textbf{0.069\ \!$\pm$\ \!0.001} & 0.167\ \!$\pm$\ \!0.013 & 0.094\ \!$\pm$\ \!0.001 \\
\midrule \multirow{3}{*}{\textbf{Adobe}} & 3 & 2.556\ \!$\pm$\ \!0.328 & 0.317\ \!$\pm$\ \!0.002 & 0.751\ \!$\pm$\ \!0.117 & 1.545\ \!$\pm$\ \!0.008 & \textbf{0.248\ \!$\pm$\ \!0.004} & 0.254\ \!$\pm$\ \!0.003 & 0.301\ \!$\pm$\ \!0.003 \\
& 4 & 1.862\ \!$\pm$\ \!1.212 & 0.335\ \!$\pm$\ \!0.004 & 0.696\ \!$\pm$\ \!0.170 & 1.673\ \!$\pm$\ \!0.008 & \textbf{0.317\ \!$\pm$\ \!0.006} & 0.318\ \!$\pm$\ \!0.009 & 0.482\ \!$\pm$\ \!0.014 \\
& 5 & 1.512\ \!$\pm$\ \!0.082 & 0.463\ \!$\pm$\ \!0.003 & 0.546\ \!$\pm$\ \!0.000 & 1.690\ \!$\pm$\ \!0.015 & \textbf{0.410\ \!$\pm$\ \!0.021} & 0.434\ \!$\pm$\ \!0.007 & 0.512\ \!$\pm$\ \!0.007 \\
\bottomrule \end{tabular}
\end{table*}
\subsection{Forecasting Performance} \label{sec:exp-forecasting-performance} We investigate the proposed GraphDF framework with various horizons including $\tau = \{1,3,4,5\}$. We evaluate the three GraphDF variants described in Section~\ref{sec:model-variants} against four state-of-the-art probabilistic forecasting methods including Deep Factors, \text{DeepAR}~\cite{Flunkert2017DeepARPF}, \text{MQRNN}~\cite{wen2017multi}, and \text{NBEATS}~\cite{oreshkin2019n}. Deep Factors is a generative approach that combines a global model and a local model. In DF, we use the Gaussian likelihood in terms of the random effects in the deep factors model. We use 10 global factors with a LSTM cell of 1-layer and 50 hidden units in its global component, and 1-layer and 5 hidden units RNN in the local component. \text{DeepAR} is an RNN-based global model, we use a LSTM layer with 50 hidden units in DeepAR. \text{MQRNN} is a sequence model with quantile regression and \text{NBEATS} is an interpretable pure deep learning model. For MQRNN, we use a GRU bidirectional layer with 50 hidden units as encoder and a modified forking layer in decoder. For N-BEATS, we use an ensemble modification of the model and take the median value from 10 bagging bases as results. All methods are implemented using MXNet Gluon~\cite{chen2015mxnet,Alexandrov2019GluonTSPT}. The Adam optimization method is used with an initial learning rate as 0.001 to train all models. The training epochs are selected by grid search in $\{100,200,\ldots,1000\}$. An early stopping strategy is leveraged if weight losses do not decrease for 10 continuous epochs. Details on the hyperparameter tuning for each method are provided in Appendix~\ref{appendix:hyperparameter-tuning}. To evaluate the probabilistic forecasts, we use $\rho$-quantile loss~\cite{wang2019deep}. We run 10 trials and report the average for $\rho=\{0.1, 0.5, 0.9\}$, denoted as the P10QL, P50QL and P90QL, respectively. Further details on $\rho$-quantile loss are provided in Appendix~\ref{appendix:prob-forecast-eval-metric}.
The results for single and multi-step ahead forecasting are provided in Table~\ref{table:results-one-step-p50ql-p90ql} and Table~\ref{table:results-multi-step-ahead-forecasting-p50ql}-\ref{table:results-multi-step-ahead-forecasting-p90ql}, respectively, where the best result for every dataset and forecast horizon are highlighted in bold. In all cases, we observe that the GraphDF models outperform previous state-of-the-art methods across all datasets and forecast horizons. Furthermore, the GraphDF-GG variant outperforms the other variants in nearly all cases.
Additional results have been removed for brevity. However, we also observed similar performance for P10QL, (Table~\ref{table:results-one-step-p10ql}-\ref{table:results-multi-step-ahead-forecasting-p10ql} in the Appendix~\ref{appendix-additional-results}.)
\begin{figure}
\caption{ Probabilistic forecasting results for 3-step ahead forecast horizon (P50QL, or equivalently, MAPE). Overall, GraphDF and its variants significantly outperform the other methods while having much lower variance. }
\label{fig:results-boxplots-p50ql}
\end{figure}
To understand the overall performance and variance of the results, we show boxplots for each model in Fig.~\ref{fig:results-boxplots-p50ql}. Strikingly, we observe that the GraphDF models provide more accurate forecasts with significantly lower variance compared to other models.
\begin{table*}[ht]
\small \centering \caption{Training runtime performance (in seconds).} \label{table:training-runtime-results} \begin{tabular}{@{}l@{} ccccccc@{}} \toprule
\textsc{Data} & NBEATS & MQRNN & DeepAR & DF & GraphDF-GG & GraphDF-GR & GraphDF-RG \\ \midrule \textbf{Google}\;\; & 663.31\ \!$\pm$\ \!54.09 & 284.76\ \!$\pm$\ \!71.08 & 413.79\ \!$\pm$\ \!49.62 & 315.06\ \!$\pm$\ \!67.80 & 279.45\ \!$\pm$\ \!41.19 & \textbf{222.08\ \!$\pm$\ \!69.52} & 281.76\ \!$\pm$\ \!49.51 \\ \textbf{Adobe} & 462.06\ \!$\pm$\ \!120.07 & 393.08\ \!$\pm$\ \!4.22 & 351.99\ \!$\pm$\ \!285.30 & 378.97\ \!$\pm$\ \!441.64 & 282.30\ \!$\pm$\ \!36.80 & \textbf{211.20\ \!$\pm$\ \!21.56} & 264.00\ \!$\pm$\ \!56.29 \\ \bottomrule \end{tabular}
\end{table*}
\begin{table*}[ht]
\small \centering \caption{Inference runtime performance (in seconds). } \label{table:inference-runtime-results} \begin{tabular}{l ccccccc} \toprule \textsc{Data} & NBEATS & MQRNN & DeepAR & DF & GraphDF-GG & GraphDF-GR & GraphDF-RG \\ \midrule \textbf{Google} & 88.08\ \!$\pm$\ \!10.96 & 9.22\ \!$\pm$\ \!0.06 & 17.06\ \!$\pm$\ \!0.16 & 8.28\ \!$\pm$\ \!0.02 & 1.67\ \!$\pm$\ \!0.03 & \textbf{0.99\ \!$\pm$\ \!0.003} & 1.16\ \!$\pm$\ \!0.003 \\ \textbf{Adobe} & 162.63\ \!$\pm$\ \!7.59 & 2.69\ \!$\pm$\ \!0.006 & 4.30\ \!$\pm$\ \!0.02 & 2.12\ \!$\pm$\ \!0.001 & 0.51\ \!$\pm$\ \!0.005 & \textbf{0.28\ \!$\pm$\ \!0.001} & 0.33\ \!$\pm$\ \!0.000 \\ \bottomrule \end{tabular} \end{table*}
\subsection{Runtime Analysis} \label{sec:exp-runtime} We analyze the runtime with respect to both training time and inference time. Regarding model training time, GraphDF methods are significantly faster and more scalable than other methods, as shown in Table~\ref{table:training-runtime-results}. In particular, the GraphDF model that uses graph-based global model with RNN-based local model is trained faster than DF, which uses the same RNN-based local model, but differs in the global model used.
This is due to the fact that in the state-of-the-art DF model, all time-series are considered equivalently and jointly when learning the $K$ global factors. This can be thought of as a fully connected graph where each time-series is connected to every other time-series. In comparison, the relational global component of GraphDF-GR leverage the graph that encodes explicit dependencies between different time-series, and therefore, does not need to leverage all pairwise time-series, but only a smaller fraction of those that are actually related. In terms of inference, all models are fast taking only a few seconds as shown in Table~\ref{table:inference-runtime-results}. For inference, we report the time (in seconds) to infer values in six steps ahead. In all cases, the GraphDF models are significantly faster than DF and other methods across both Google and Adobe workload datasets.
\subsection{Scalability} To evaluate the scalability of GraphDF, we vary the training set size (\emph{i.e.}, the number of previous data points per time-series to use) from $\{2,4,8,16,32\}$, and record the training time. Fig.~\ref{fig:training-time-vs-training-set-size} shows that GraphDF scales nearly linear as the training set size increases from 2 to 32. For instance, GraphDF takes around 15 seconds to train using only 2 data points per time-series and 30 seconds using 4, and so on. We also observe that GraphDF is always about 3x faster compared to DF across all training set sizes.
\begin{figure}
\caption{ Comparing scalability of GraphDF to DF as the training set size increases for the Adobe workload trace dataset. Note training set size = data points per time-series. }
\label{fig:training-time-vs-training-set-size}
\end{figure}
\subsection{$\!\!\!\!\text{Case Study: Opportunistic Scheduling in the Cloud}$} \label{sec:opportunistic-scheduling} We leverage our GraphDF forecasting model to enable opportunistic scheduling of batch workloads for cloud resource optimization. Our model generates probabilistic CPU usage forecasts on compute machines, and we use them to schedule batch workloads on machines with low predicted CPU usage. Since batch workloads such as ML training, crawling web pages etc. have loose latency requirements, they can be scheduled on underutilized resources (such as CPU cores). This improves resource efficiency of the cluster and reduces operating costs by precluding the need to allocate additional machines to run the batch workloads.
The model satisfies following requirements of the scheduling problem. First, the model must be able to correctly forecast utilization. If the utilization is underestimated, tasks will be assigned to busy machines and then need to be cancelled, which is a waste of resources. Second, the execution time of the forecasting model must be significantly faster than the time period used for data collection, \emph{e.g.}, since CPU usage in Google dataset is observed every 5 minutes, the CPU usage forecast should be generated in less than 5 minutes i.e. before the next observation arrives.
We simulate opportunistic scheduling by developing two main components, the \textit{forecaster} and the \textit{scheduler}. The Google dataset simulates the CPU usages for the cluster in this study. The forecaster reads the 6 most recent observed CPU utilization values of each machine from the data stream and predicts next 3 values. The scheduler identifies underutilized machines as those with mean predicted utilization across the three predictions lower than a predefined threshold $\epsilon$ (25\%). Each machine in the cluster has 8 cores and the batch workload requires 6 cores to execute. To safely make use of the idle resources without disturbing already running tasks or cause thrashing, the scheduler only assigns workloads that require at most (75\%) of compute resources. If a machine is assigned batch workloads that exceeds resource availability, they are \emph{terminated/cancelled}. This procedure is described in Algorithm~\ref{alg:dynamic-real-time-scheduling} in Appendix~\ref{appendix-opportunistic-learning}.
\begin{figure}
\caption{
CPU utilization without opportunistic workload scheduling (shown in green) and with scheduling based on each forecaster (shown in red and blue), over a period of 6 hours on Google dataset.
GraphDF-based scheduling leads to higher CPU utilization than DF-based and vanilla (no forecasts) scheduling.}
\label{fig:cmp_improvement_CPU_utilization-Google-6h}
\end{figure}
\begin{table}[b!]
\centering
\caption{
Results for opportunistic workload scheduling in the cloud over a 6 hour period using different forecasting models.
}
\small
\footnotesize
\begin{tabular}{@{}cc@{}ccc@{}}
\toprule
\multirow{2}{*}{Data} & \multirow{2}{*}{Model}\; & utilization & correct & cancellation \\
& & improvement (\%) & ratio (\%) & ratio (\%)\\
\midrule \TTT\BBB
\multirow{2}{*}{\textbf{Google}} & DF & 38.8 & 68.6 & 20.9 \\ & GraphDF & \textbf{41.9} & \textbf{88.6} & \textbf{8.2} \\ \midrule
\multirow{2}{*}{\textbf{Adobe}} & DF & 42.0 & 65.8 & 19.1 \\ & GraphDF & \textbf{53.2} & \textbf{97.0} & \textbf{2.2} \\
\bottomrule
\end{tabular}
\label{table:scheduler_metric}
\end{table}
\textbf{Effects on CPU utilization} Fig.~\ref{fig:cmp_improvement_CPU_utilization-Google-6h} shows CPU utilization without opportunistic workload scheduling \textit{(vanilla)} and with scheduling based on each forecaster over a period of 6 hours on the Google dataset. We observe that the GraphDF-based forecaster consistently outperforms both vanilla and DF-based versions by generating forecasts with higher accuracy.
Table~\ref{table:scheduler_metric} summarizes the performance of the GraphDF-based forecaster with respect to three metrics \emph{CPU utilization improvement}, \emph{correct scheduling ratio}, and \emph{cancellation ratio}. The \text{utilization improvement} measures the absolute increase in CPU usage compared to the vanilla version. \text{Correct scheduling ratio} corresponds to the ratio when predicted utilization by the scheduler matches the actual utilization.
\text{Cancellation ratio} measures the fraction of machines on which the batch workload was terminated due to incorrectly generated forecasts. We observe that GraphDF-based workload scheduling leads to higher CPU utilization, higher correct scheduling ratio, and lower cancellation ratio compared to DF-based scheduling. We have included results over longer periods (12 hours and 24 hours) for both datasets in the Appendix~\ref{appendix-additional-results}.
\begin{figure}
\caption{
The time constraint (black line), the runtime of scheduler with DF (red line) and that with GraphDF (blue line).
Note that in most cases, DF fails to meet the time constraint while GraphDF produces a forecast much faster. }
\label{fig:dynamic-runtime}
\end{figure}
\textbf{Execution Time Comparison} Fig.~\ref{fig:dynamic-runtime} shows that the runtime of DF-based scheduling often exceeds the 5-minute time limit, while GraphDF-based version is much faster and always meets it. Hence, GraphDF persuasively provides a solution for enhancing cloud efficiency through effective usage forecasting.
\section{Conclusion} \label{sec:conc} In this work, we introduced a deep hybrid graph-based probabilistic forecasting framework called Graph Deep Factors. While existing deep probabilistic forecasting approaches do not explicitly leverage a graph, and assume either complete independence among time-series (\emph{i.e.}, completely disconnected graph) or complete dependence between all time-series (\emph{i.e.}, fully connected graph), this work moved beyond these two extreme cases by allowing nodes and their time-series to have arbitrary and explicit weighted dependencies among each other. Such explicit and arbitrary weighted dependencies between nodes and their time-series are modeled as a graph in the proposed framework. Notably, GraphDF consists of a relational global model that learns complex non-linear time-series patterns globally using the structure of the graph to improve computational efficiency as well as a relational local model that not only considers its individual time-series but the time-series of nodes that are connected in the graph to improve forecasting accuracy. Finally, the experiments demonstrated the effectiveness of the proposed deep hybrid graph-based probabilistic forecasting model in terms of its forecasting performance, runtime, scalability, and optimizing cloud usage through opportunistic workload scheduling.
\appendix \section*{Appendix} \label{sec:appendix}
\section{Alternative Models} \label{appendix-alternative-models}
\subsection{Learning Relational Global Factors via DCRNN} \label{appendix:dcrnn} For the relational global component of GraphDF, we can also leverage DCRNN~\cite{dcrnn18}. Different from the GCRN model, the original DCRNN leverages a diffusion convolution operation and a GRU structure for learning the relational global factors of GraphDF.
Given the time-series graph signal $\mY_t \in \mathbb{R}^{N\times P}$ with $N$ nodes, the diffusion convolution with respect to the graph-based time-series is defined as, \begin{align}
f_{\,\star_{\mathcal{G}}\,}{\mathbf{\Theta}}(\mY_t) = \sum_{l=0}^{L-1}(\theta_{l}\tmA^l)\mY_t \end{align} where $\tmA=\mD^{-1}\mA$ is the normalized adjacency matrix of the graph $G$ that captures the explicit weighted dependencies between the multi-dimensional time-series of the nodes. The Chebyshev polynomial approximation is used similarly as Eq.~\eqref{eq:filt_cheby}.
The relational global factors are learned using the graph diffusion convolution combined with GRU enabling them to be carried forward over time using the graph structure, \begin{align} \mR_t &= \sigma(\mathbf{\Theta}_R {\,\star_{\mathcal{G}}\,}{}[\mY_t,\; \mH_{t-1}] + \vb_R) \\ \mU_t &= \sigma(\mathbf{\Theta}_U {\,\star_{\mathcal{G}}\,}{}[\mY_t,\; \mH_{t-1}] + \vb_U) \\ \mC_t &= \tanh(\mathbf{\Theta}_C {\,\star_{\mathcal{G}}\,}{}[\mY_t,\; (\mR_t \odot \mH_{t-1})] + \vb_C) \\ \mH_t &= \mU_t \odot \mH_{t-1} + (1-\mU_t) \odot \mC_t \label{eq:dcrnn-ht} \end{align} where $\mH_t \in \mathbb{R}^{N\times Q}$ denotes the hidden state of the model at time step $t$, $Q$ is the number of hidden units, $\mR_t \in \mathbb{R}^{N\times Q}, \mU_t \in \mathbb{R}^{N\times Q}$ are called as reset gate and update gate at time $t$, respectively. $\mathbf{\Theta}_R \in \mathbb{R}^{L}, \mathbf{\Theta}_U \in \mathbb{R}^{L}, \mathbf{\Theta}_C \in \mathbb{R}^{L}$ denote the parameters corresponding to different filters.
With the hidden state $\mH_t$ in Eq.~\eqref{eq:dcrnn-ht}, the fixed effect is derived from DCRNN similarly with Eq.~\eqref{eq:gcrn-st} and Eq.~\eqref{eq:gcrn-cit}.
Compared to the previous GCRN that we adapted for the relational global component, DCRNN is more computationally efficient due to the GRU structure it uses.
\subsection{Relational Local Model via Probabilistic DCRNN} \label{sec:prob-DCRNN} We also describe a probabilistic DCRNN for the relational local component of GraphDF.
For a given node $i$, its relational local random effect is derived with respect to its past observations, covariates and those of its neighbors, denoted by $\mY_t^{(i)} \in \mathbb{R}^{(|S|+1) \times P}$ as defined in Eq.~\eqref{eq:Y_t_rel-local}. The diffusion convolution models the relational local random effect among nodes. The GRU structure is adapted with the diffusion convolution to allow the random effects to be forwarded in time. \begin{align}
\mR_t^{(i)} &=
\sigma\big(
\mathbf{\Theta}_{R}^{(i)} {\,\star_{\mathcal{G}}\,}{}
[\mY_t^{(i)},\mH_t^{(i)}] + \vb_R^{(i)}
\big)\\
\mU_t^{(i)} &=
\sigma\big(
\mathbf{\Theta}_{U}^{(i)} {\,\star_{\mathcal{G}}\,}{}
[\mY_t^{(i)},\mH_t^{(i)}] + \vb_U^{(i)}
\big)\\
\mC_t^{(i)} &=
\tanh\big(
\mathbf{\Theta}_{C}^{(i)} {\,\star_{\mathcal{G}}\,}{}
[\mY_t^{(i)},\mH_t^{(i)}] + \vb_C^{(i)}
\big)\\
\mH_t^{(i)} &=
\mU_t^{(i)}\odot\mH_{t-1}^{(i)} +
(1 - \mU_t^{(i)})\odot\mC_{t}^{(i)} \end{align} where $\mathbf{\Theta}_{R}^{(i)} \in \mathbb{R}^{P\times R}, \mathbf{\Theta}_{U}^{(i)} \in \mathbb{R}^{P\times R}, \mathbf{\Theta}_{C}^{(i)} \in \mathbb{R}^{P\times R}$ denote the parameters corresponding to different filters,
$\mH_t^{(i)} \in \mathbb{R}^{(|C|+1)\times R}$ is the hidden state for node $i$ and its neighbors $\Gamma_i$, $R$ is the number of hidden units in the relational local model. $\vb_{I}^{(i)}, \vb_{F}^{(i)}, \vb_{C}^{(i)}, \vb_{O}^{(i)}$ are bias vector parameters.
The graph convolution in equations above is performed with the submatrix $\mL^{(i)}$ taken from the Laplacian matrix $\mL$ of the graph $G$ that explicitly models the important and meaningful dependencies between the multi-dimensional time-series data of each node. The matrix $\mL^{(i)}$ consists of rows and columns corresponding to node $i$ and its neighbors $\Gamma_i$.
With the hidden state $\mH_t^{(i)}$, the relational local random effect $b_t^{(i)}(\cdot)$ is calculated similarly with Eq.~\eqref{eq:sigma-gcrn} and Eq.~\eqref{eq:gcrn-random}.
\begin{table*}[!t] \small \centering \setlength{\tabcolsep}{4.6pt} \caption{Results for one-step ahead forecasting (\textsc{p10ql}). }
\label{table:results-one-step-p10ql} \begin{tabular}{ l ccccccc } \toprule \multirow{1}{*}{\textsc{data}} & NBEATS & MQRNN & DeepAR & DF & GraphDF-GG & GraphDF-GR & GraphDF-RG \\ \midrule \TTT\BBB \multirow{1}{*}{\textbf{Google}\;\;} & 18.116\ \!$\pm$\ \!201.259 & 0.190\ \!$\pm$\ \!0.004 & 0.046\ \!$\pm$\ \!0.000 & 0.083\ \!$\pm$\ \!0.001 & \textbf{0.037\ \!$\pm$\ \!0.000} & 0.038\ \!$\pm$\ \!0.000 & 0.044\ \!$\pm$\ \!0.000 \\ \multirow{1}{*}{\textbf{Adobe}} & 0.615\ \!$\pm$\ \!0.091& 0.132\ \!$\pm$\ \!0.000 & 0.164\ \!$\pm$\ \!0.001 & 1.128\ \!$\pm$\ \!0.004 & \textbf{0.118\ \!$\pm$\ \!0.001} & 0.119\ \!$\pm$\ \!0.000 & 1.027\ \!$\pm$\ \!2.700 \\ \bottomrule \end{tabular} \end{table*}
\begin{table*}[!ht] \setlength{\tabcolsep}{5.2pt} \centering \caption{Results for multi-step ahead forecasting (\textsc{p10ql}). }
\label{table:results-multi-step-ahead-forecasting-p10ql} \small \begin{tabular}{@{}l@{} cccccccc } \toprule \multirow{1}{*}{\textsc{data}} & \multirow{1}{*}{\textsc{h}} & NBEATS & MQRNN & DeepAR & DF & GraphDF-GG & GraphDF-GR & GraphDF-RG \\ \midrule \multirow{3}{*}{\textbf{Google}\;\;} & 3 & 0.652\ \!$\pm$\ \!0.396 & 0.152\ \!$\pm$\ \!0.006 & 0.070\ \!$\pm$\ \!0.000 & 0.132\ \!$\pm$\ \!0.004 & \textbf{0.064\ \!$\pm$\ \!0.001} & 0.077\ \!$\pm$\ \!0.000 & 0.087\ \!$\pm$\ \!0.000 \\ & 4 & 0.260\ \!$\pm$\ \!0.017 & 0.272\ \!$\pm$\ \!0.018 & 0.138\ \!$\pm$\ \!0.000 & 0.193\ \!$\pm$\ \!0.016 & \textbf{0.071\ \!$\pm$\ \!0.001} & 0.083\ \!$\pm$\ \!0.000 & 0.089\ \!$\pm$\ \!0.001 \\ & 5 & 0.447\ \!$\pm$\ \!0.054 & 0.147\ \!$\pm$\ \!0.005 & 0.484\ \!$\pm$\ \!0.017 & 0.327\ \!$\pm$\ \!0.036 & \textbf{0.054\ \!$\pm$\ \!0.000} & 0.113\ \!$\pm$\ \!0.001 & 0.088\ \!$\pm$\ \!0.001 \\
\midrule \multirow{3}{*}{\textbf{Adobe}} & 3 & 0.811\ \!$\pm$\ \!0.295 & 0.184\ \!$\pm$\ \!0.003 & 0.207\ \!$\pm$\ \!0.002 & 0.303\ \!$\pm$\ \!0.006 & \textbf{0.183\ \!$\pm$\ \!0.002} & 0.216\ \!$\pm$\ \!0.008 & 0.267\ \!$\pm$\ \!0.009 \\ & 4 & 0.985\ \!$\pm$\ \!0.537 & 0.219\ \!$\pm$\ \!0.008 & 0.273\ \!$\pm$\ \!0.003 & 0.313\ \!$\pm$\ \!0.018 & \textbf{0.184\ \!$\pm$\ \!0.002} & 0.242\ \!$\pm$\ \!0.014 & 0.423\ \!$\pm$\ \!0.019 \\ & 5 & 0.626\ \!$\pm$\ \!0.023 & 0.398\ \!$\pm$\ \!0.229 & 0.402\ \!$\pm$\ \!0.011 & 0.343\ \!$\pm$\ \!0.047 & \textbf{0.251\ \!$\pm$\ \!0.016} & 0.298\ \!$\pm$\ \!0.031 & 0.544\ \!$\pm$\ \!0.020 \\
\bottomrule \end{tabular} \end{table*}
\section{Experimental details} \subsection{Data} \label{appendix-data-details}
\noindent\textit{Google Trace.}~\cite{reiss2011google} The Google trace dataset records the activities of a cluster of $12,580$ machines for 29 days since 19:00 EDT in May 1, 2011. The CPU and memory usage for each task are recorded every 5 minutes. The usage of tasks is aggregated to the usage of associated machines, resulting time-series of length $8,354$.
\noindent\textit{Adobe Workload Trace.} The Adobe trace dataset records the CPU and memory usage of $3,270$ nodes in the period from Oct. 31 to Dec. 5 in 2018. The timescale is 30 minutes, resulting time-series of length $1,687$.
\noindent\textit{Graph Construction.} For each dataset, we derive a graph where each node represents a machine with one or more time-series associated with it, and each edge represents the similarity between the node time-series $i$ and $j$. The constructed graph encodes the dependency information between nodes. In this work, we estimate the edge weights using the radial basis function (RBF) kernel with the previous time-series observations as $K(\vz_i,\vz_j) = \exp(-\frac{\left\Vert\vz_i-\vz_j\right\Vert^2}{2\ell^2})$, where $\ell$ is the length scale of the kernel.
For all experiments, we use a single machine equipped with Linux Ubuntu OS with an 8-core CPU for training and inference.
\subsection{Hyperparameter Tuning} \label{appendix:hyperparameter-tuning} Most hyperparameters are set as default values by MXnet Gluonts~\cite{Alexandrov2019GluonTSPT}. We list hyperparameters used in our experiments as follows: \begin{itemize}
\item learning rate decay factor: $0.5$
\item minimum learning rate: $5*10^{-5}$
\item weight initializer method: Xavier initialization
\item training epochs, we train for 500 epochs when using the Adobe dataset and 100 epochs when using the Google dataset. \end{itemize} Some hyperparameters are specific to our method: In GraphDF, we set the order $L=1$ in Eq.~\eqref{eq:filt_cheby}. A small number of the order indicates the model make forecasts based more on neighboring nodes than those in distance. For other methods, we use default hyperparameters given by the Gluonts implementation if not mentioned.
\section{Additional Forecasting Results} \label{appendix-additional-results} The Quantile Loss at 10th percentile is showed in Table~\ref{table:results-one-step-p10ql} and Table~\ref{table:results-multi-step-ahead-forecasting-p10ql}. The result shows that our hybrid graph models GraphDF and variants consistently outperform previous state-of-the-arts. Specifically, GraphDF with graph models in both relational global component and relation local component obtain the best performance compared to other methods.
\section{Additional Results for Opportunistic Scheduling} \label{appendix-opportunistic-learning} Algorithm~\ref{alg:dynamic-real-time-scheduling} provides an overview of the opportunistic real-time scheduler. Notice that we can leverage any forecasting model $f$ to obtain forecasts of CPU usage which is then used in the scheduler.
More result on for opportunistic scheduling with different forecasters are depicted in Fig.~\ref{fig:cmp_improvement_CPU_utilization-Adobe} and Fig.~\ref{fig:cmp_improvement_CPU_utilization-Google}. From the figures, we observe that schedulers with either forecaster improve CPU utilization, whereas scheduler with our proposed GraphDF forecaster perform better in terms of utilization improvement.
\begin{algorithm}[ht!]
\caption{Opportunistic Scheduling}
\label{alg:dynamic-real-time-scheduling}
\begin{algorithmic}[1]
\STATE Initialize hyperparameters and variables lookback window $w=6$, horizon $\tau=3$, threshold ratio $\epsilon=25\%$, portion ratio $\lambda=75\%$. Accumulated utilization improvement $Acc=0$
\WHILE{New observations arrive $t\gets t+1$}
\STATE Initialize weights $\mPhi$ for new model $f_t$
\STATE $f_t \gets \mathbb{P}\big(\cdot \big| \mPhi, \mA, \big\{\mX_{:,t-w+1:t+\tau}^{(i)}, \vz_{t-w+1:t}^{(i)}\big\}_{i=1}^{N} \!\big)$
\FOR{each node $i$}
\STATE Obtain forecasts
$\hat{z}^{(i)}\!\!=\!\!\{\hat{z}_{t+1}^{(i)}, \hat{z}_{t+2}^{(i)}\ldots\hat{z}_{t+\tau}^{(i)}\}\!\!\sim\!\! f_t$
\IF {MEAN of forecasts MEAN$(\hat{z}^{(i)}) \leq \epsilon$}
\STATE Utilization $Acc\!\gets\!\!Acc+\lambda(1-\text{MEAN}(\hat{z}^{(i)}))$
\ELSE
\STATE Cancel assigned tasks on node $i$
\ENDIF
\ENDFOR
\ENDWHILE
\end{algorithmic} \end{algorithm}
\begin{figure}
\caption{ CPU utilization without opportunistic workload scheduling (shown in green) and with scheduling based on each forecaster (shown in red and blue), over a period of 6, 12 and 24 hours on Adobe dataset. GraphDF-based scheduling leads to higher CPU utilization than DF-based and vanilla (no forecasts) scheduling. }
\label{fig:cmp_improvement_CPU_utilization-Adobe}
\end{figure}
\begin{figure}
\caption{ CPU utilization without opportunistic workload scheduling (shown in green) and with scheduling based on each forecaster (shown in red and blue), over a period of 12 and 24 hours on Google dataset. GraphDF-based scheduling leads to higher CPU utilization than DF-based and vanilla (no forecasts) scheduling. }
\label{fig:cmp_improvement_CPU_utilization-Google}
\end{figure}
\begin{table}[bt!] \setlength{\tabcolsep}{2.5pt}
\centering
\caption{Opportunistic scheduling performance using different forecasting models.
}
\small
\footnotesize
\begin{tabular}{lccccc}
\toprule
\multirow{2}{*}{Data} & \multirow{2}{*}{Hour} & \multirow{2}{*}{Model} & utilization & correct & cancellation \\
& & & improvement (\%)\; & ratio (\%) & ratio (\%)\\
\midrule \TTT\BBB \multirow{2}{*}{\textbf{G}} & \multirow{2}{*}{\textbf{12}} & DF & 37.8 & 60.0 & 31.8 \\ & & GraphDF & \textbf{41.9} & \textbf{87.6} & \textbf{9.2} \\ \midrule \multirow{2}{*}{\textbf{A}} & \multirow{2}{*}{\textbf{12}} & DF & 53.6 & 76.9 & 17.6 \\ & & GraphDF & \textbf{57.1} & \textbf{97.4} & \textbf{1.8} \\ \midrule \multirow{2}{*}{\textbf{G}} & \multirow{2}{*}{\textbf{24}} & DF & 42.6 & 51.2 & 43.6 \\ & & GraphDF & \textbf{43.8} & \textbf{85.4} & \textbf{10.0} \\ \midrule \multirow{2}{*}{\textbf{A}} & \multirow{2}{*}{\textbf{24}} & DF & 59.6 & 78.6 & 17.1 \\ & & GraphDF & \textbf{62.5} & \textbf{98.1} & \textbf{1.2} \\
\bottomrule
\end{tabular}
\label{table:scheduler_metric_12_24hours} \end{table}
\section{Evaluation of Probabilistic Forecasts} \label{appendix:prob-forecast-eval-metric} To evaluate the probabilistic forecasts, we use the quantile loss defined as follows: given a quantile $\rho\in(0,1)$, a target value $\vz_t$ and $\rho$-quantile prediction $\widehat{\vz}_t(\rho)$, the $\rho$-quantile loss is defined as \begin{align} \text{QL}_\rho[\vz_t, \widehat{\vz}_t(\rho)] &= 2\big[\rho(\vz_t - \widehat{\vz}_t(\rho))\mathbb{I}_{\vz_t - \widehat{\vz}_t(\rho) > 0} \nonumber \\ &+ (1-\rho)(\widehat{\vz}_t(\rho) - \vz_t)\mathbb{I}_{\vz_t - \widehat{\vz}_t(\rho) \leqslant 0}\big] \end{align}
For deriving quantile losses over a time-span across all time-series, we adapt to a normalized version of quantile loss $\sum_{i,t} \text{QL}_\rho[z_{i,t}, \hat{z}_{i,t}(\rho)] / \sum_{i,t} |z_{i,t}|$. We run 10 trials and report the average for $\rho=\{0.1, 0.5, 0.9\}$, denoted as the P10QL, P50QL and P90QL, respectively. In experiments, the quantile losses are computed based on 100 sample values.
\subsection{An overview of the model training} \label{appdendix-model-training} We summarize the training procedure in Algorithm~\ref{alg:graphDF-training}. \begin{algorithm}[h!]
\caption{Training Graph Deep Factor Models}
\label{alg:graphDF-training}
\begin{algorithmic}[1]
\STATE Initialize the parameters $\mPhi$
\FOR{each time series pair $\{(\vz^{(i)}, \vx^{(i)})\}$}
\STATE
Using current model parameter estimates $\mPhi$, derive the relational fixed effect as
$
c_{t}^{(i)} = \sum_{k=1}^K w_{i,k}\cdot S_{i,k,t}
$ and relational local random effect $b_{t}^{(i)}$.
\STATE Compute marginal likelihood $\mathbb{P}(\vz^{(i)})$ and accumulate the loss.
\ENDFOR
\STATE Compute the overall loss in the current batch and perform
stochastic gradient descent and update the trainable parameters $\mPhi$ accordingly.
\end{algorithmic} \end{algorithm}
\end{document} | arXiv |
\begin{definition}[Definition:Weakly Stationary Stochastic Process]
Let $S$ be a stochastic process giving rise to a time series $T$.
$S$ is '''weakly stationary of order $f$''' {{iff}} its moments up to some order $f$ depend only on time differences.
Such a condition is known as '''weak stationarity of order $f$'''.
\end{definition} | ProofWiki |
Pi Day
Pi Day is an annual celebration of the mathematical constant π (pi). Pi Day is observed on March 14 (the 3rd month) since 3, 1, and 4 are the first three significant figures of π.[2][3] It was founded in 1988 by Larry Shaw, an employee of the San Francisco science museum, the Exploratorium. Celebrations often involve eating pie or holding pi recitation competitions. In 2009, the United States House of Representatives supported the designation of Pi Day.[4] UNESCO's 40th General Conference designated Pi Day as the International Day of Mathematics in November 2019.[5][6]
Pi Day
Observed byUnited States
TypeMathematical
Significance3, 1, and 4 are the three most significant figures of π in its decimal representation.
CelebrationsPie eating, pi memorization competitions, discussions about π, creating pi chains[1]
DateMarch 14
Next timeMarch 14, 2024 (2024-03-14)
FrequencyAnnual
First time1988
Related toPi Approximation Day
Part of a series of articles on the
mathematical constant π
3.1415926535897932384626433...
Uses
• Area of a circle
• Circumference
• Use in other formulae
Properties
• Irrationality
• Transcendence
Value
• Less than 22/7
• Approximations
• Madhava's correction term
• Memorization
People
• Archimedes
• Liu Hui
• Zu Chongzhi
• Aryabhata
• Madhava
• Jamshīd al-Kāshī
• Ludolph van Ceulen
• François Viète
• Seki Takakazu
• Takebe Kenko
• William Jones
• John Machin
• William Shanks
• Srinivasa Ramanujan
• John Wrench
• Chudnovsky brothers
• Yasumasa Kanada
History
• Chronology
• A History of Pi
In culture
• Indiana Pi Bill
• Pi Day
Related topics
• Squaring the circle
• Basel problem
• Six nines in π
• Other topics related to π
Other dates when people celebrate pi include July 22 (22/7 in the day/month format, an approximation of π) and June 28 (6.28, an approximation of 2π or tau).
History
In 1988, the earliest known official or large-scale celebration of Pi Day was organized by Larry Shaw at the San Francisco Exploratorium,[7] where Shaw worked as a physicist,[8] with staff and public marching around one of its circular spaces, then consuming fruit pies.[9] The Exploratorium continues to hold Pi Day celebrations.[10]
On March 12, 2009, the U.S. House of Representatives passed a non-binding resolution (111 H. Res. 224),[4] recognizing March 14, 2009, as National Pi Day.[11] For Pi Day 2010, Google presented a Google Doodle celebrating the holiday, with the word Google laid over images of circles and pi symbols;[12] and for the 30th anniversary in 2018, it was a Dominique Ansel pie with the circumference divided by its diameter.[13] In Indonesia, as a country that uses the DD/MM/YYYY date format, some people celebrate Pi Day every July 22, referring to another Pi number, namely 22/7.[14]
Some observed the entire month of March 2014 (3/14) as "Pi Month".[15][16] In the year 2015, March 14 was celebrated as "Super Pi Day".[17] It had special significance, as the date is written as 3/14/15 in month/day/year format. At 9:26:53, the date and time together represented the first ten digits of π,[18] and later that second Pi Instant represented all of π's digits.[19]
Observance
Pi Day has been observed in many ways, including eating pie, throwing pies and discussing the significance of the number π, due to a pun based on the words "pi" and "pie" being homophones in English ( /paɪ/), and the coincidental circular shape of many pies.[1][20] Many pizza and pie restaurants offer discounts, deals, and free products on Pi Day.[21] Also, some schools hold competitions as to which student can recall pi to the highest number of decimal places.[22][23]
The Massachusetts Institute of Technology has often mailed its application decision letters to prospective students for delivery on Pi Day.[24] Starting in 2012, MIT has announced it will post those decisions (privately) online on Pi Day at exactly 6:28 pm, which they have called "Tau Time", to honor the rival numbers pi and tau equally.[25][26] In 2015, the regular decisions were put online at 9:26 am, following that year's "pi minute",[27] and in 2020, regular decisions were released at 1:59 pm, making the first six digits of pi.[28]
June 28 is "Two Pi Day", also known as "Tau Day". 2π, also known by the Greek letter tau (𝜏) is a common multiple in mathematical formulae. Some have argued that τ is the more fundamental constant and that Tau Day should be celebrated instead.[29][30][31] Celebrations of this date jokingly suggest eating "twice the pie".[32][33][34]
Princeton, New Jersey, hosts numerous events in a combined celebration of Pi Day and Albert Einstein's birthday, which is also March 14.[35] Einstein lived in Princeton for more than twenty years while working at the Institute for Advanced Study. In addition to pie eating and recitation contests, there is an annual Einstein look-alike contest.[35]
Alternative dates
Pi Day is frequently observed on March 14 (3/14 in the month/day date format), but related celebrations have been held on alternative dates.
Pi Approximation Day is observed on July 22 (22/7 in the day/month date format), since the fraction 22⁄7 is a common approximation of π, which is accurate to two decimal places and dates from Archimedes.[36]
Two Pi Day, also known as Tau Day, is observed on June 28 (6/28 in the month/day format).[37]
Some also celebrate pi on November 10, since it is the 314th day of the year.[38]
Gallery
• Pi Pie at Delft University
• A grocery store selling pies for $3.14 on Pi Day
• Creme pie in celebration of Pi day showing the greek letter and the first digits of Pi.
See also
• Lists of holidays
• Mole Day
• Sequential time
• Square Root Day
• White Day
Notes
References
1. Landau, Elizabeth (March 12, 2010). "On Pi Day, one number 'reeks of mystery'". CNN. Retrieved March 14, 2018.
2. Bellos, Alex (March 14, 2015). "Pi Day 2015: a sweet treat for maths fans". theguardian.com. Retrieved March 14, 2016.
3. "Nedräkning mot internationella Pi-dagen". Swedish national radio company (in Swedish). March 14, 2015.
4. United States. Cong. House. Supporting the designation of Pi Day, and for other purposes. 111th Cong. Library of Congress Archived August 7, 2009, at the Wayback Machine
5. "International Day of Mathematics". UNESCO. March 4, 2020.
6. Rousseau, Christiane (September 1, 2019). "International Day of Mathematics" (PDF). Notices of the American Mathematical Society. 66 (8): 1. doi:10.1090/noti1928.
7. Berton, Justin (March 11, 2009). "Any way you slice it, pi's transcendental". San Francisco Chronicle. Retrieved March 18, 2011.
8. Borwein, Jonathan (March 10, 2011). "The infinite appeal of pi". Australian Broadcasting Corporation. Retrieved March 13, 2011.
9. Apollo, Adrian (March 10, 2007). "A place where learning pi is a piece of cake" (PDF). The Fresno Bee.
10. "Exploratorium 22nd Annual Pi Day". Exploratorium. Archived from the original on March 14, 2011. Retrieved January 31, 2011.
11. McCullagh, Declan (March 11, 2009). "National Pi Day? Congress makes it official". Politics and Law. CNET News. Retrieved March 14, 2009.
12. "Pi Day". Google Doodles. Retrieved October 9, 2012.
13. "30th Anniversary of Pi Day!". www.google.com. Retrieved March 19, 2018.
14. Purworini, D. (2016). Pi Day celebration by local scientists in Indonesia. Vol. 7. Anak Sudarti Foundation Bulletin. pp. 7–8.{{cite book}}: CS1 maint: location missing publisher (link)
15. Main, Douglas (March 14, 2014). "It's Not Just Pi Day, It's Pi Month!". Popular Science. Retrieved July 22, 2014.
16. "Pi Month Celebration & Circle of Discovery Award Presentation | College of Computer, Mathematical, and Natural Sciences". Cmns.umd.edu. March 11, 2014. Retrieved July 22, 2014.
17. Mack, Eric (March 14, 2015). "Celebrate The Only Super Pi Day Of The Century". Forbes. Retrieved March 14, 2019.
18. Ro, Sam (March 13, 2014). "March 14, 2015 Will Be A Once-In-A-Century Thrill For Math Geeks". Business Insider. Retrieved March 13, 2014.
19. Rosenthal, Jeffrey S. (February 2015). "Pi Instant". Math Horizons. 22 (3): 22. doi:10.4169/mathhorizons.22.3.22. S2CID 218542599.
20. Smith, K.N. "Wednesday's Google Doodle Celebrates Pi Day". Forbes.
21. "Celebrate Pi Day With These Deals Around North Texas". NBC DFW. March 13, 2021. Retrieved May 14, 2022.
22. "Honiton Community College Pi Day – Jazmin Year 9". YouTube. Archived from the original on October 17, 2013. Retrieved July 22, 2013.
23. "HCC Celebrate International Pi Day". Honitoncollege.devon.sch.uk. Archived from the original on February 22, 2014. Retrieved July 22, 2013.
24. McClan, Erin (March 14, 2007). "Pi fans meet March 14 (3.14, get it?)". NBC News. Retrieved January 24, 2008.
25. "I have SMASHING news!". MIT Admissions. March 7, 2012. Retrieved March 12, 2012.
26. McGann, Matt (March 13, 2012). "Pi Day, Tau Time". MIT Admissions. Retrieved March 18, 2012.
27. "Keep your eyes to the skies this Pi Day". MIT Admissions. March 6, 2015. Retrieved March 14, 2018.
28. "[Pinned] This is the way…to check your decisions". MIT Admissions. March 10, 2020. Retrieved March 14, 2020.
29. "It's Pi Day today. But these people say we should refuse to celebrate it". The Independent. March 13, 2018. Archived from the original on May 26, 2022. Retrieved March 14, 2018.
30. "Pi Day Turns 25: Why We Celebrate an Irrational Number". March 14, 2013. Retrieved March 14, 2018.
31. BMJ (June 22, 2018). "Jeffrey Aronson: When I use a word ... The Days of Pi". The BMJ. Retrieved May 2, 2023.
32. Bartholomew, Randyn Charles. Why Tau Trumps Pi. {{cite book}}: |work= ignored (help)
33. Landau, Elizabeth. "In case Pi Day wasn't enough, it's now 'Tau Day' on the Internet". CNN.
34. "Tau Day – Come Eat Twice the (Pi)e".
35. "Princeton Pi Day & Einstein Birthday Party". Princeton Regional Convention and Visitors Bureau. Retrieved February 9, 2019.
36. "Pi Approximation Day is celebrated today". Today in History. Verizon Foundation. Archived from the original on December 1, 2010. Retrieved January 30, 2011.
37. "Tau Day: Why you should eat twice the pie". Archived from the original on January 12, 2013.
38. "Pi Day – Fun Holiday". Timeanddate.com. Retrieved March 13, 2022.
External links
Wikimedia Commons has media related to Pi Day.
Look up Pi Day in Wiktionary, the free dictionary.
• Exploratorium's Pi Day Web Site
• NPR provides a "Pi Rap" audiovideo
• Pi Day
• Professor Lesser's Pi Day page
Events commemorating achievements in the sciences
Anniversary celebrations
• Alfred Russel Wallace centenary
• Darwin Centennial Celebration (1959)
Regular holidays
• Ada Lovelace Day
• Astronauts Day
• Cosmonautics Day
• Darwin Day
• DNA Day
• Evolution Day
• International Day of Human Space Flight
• Mole Day
• National Astronaut Day
• Pi Day
• Yuri's Night
Year long events
• First International Polar Year (1882–1883)
• Second International Polar Year (1932–1933)
• International Polar Year (2007–2008)
• International Geophysical Year
• International Year of Planet Earth
• International Year of Astronomy
• International Year of Chemistry
• International Year of Crystallography
• International Year of Light
• International Space Year
• World Year of Physics 2005
| Wikipedia |
Won't the net effect of a catalyst be zero if it creates a new path with lower activation energy?
A catalyst will provide a new path with a lower activation energy (Figure 1). Won't this mean the forward and backward reactions will both speed up (as they both have a lower activation energy path to take)?
So then what is the point of using a catalyst? Or alternatively, how does the catalyst provide a net benefit for the forward reaction if both the forward and backward reaction are sped up?
I am assuming it might have something to do with the reactants having a higher energy than the products, but I can't think up a proper formal reason that a catalyst has a net benefit for the forward reaction.
Figure 1 (source: Chemguide.co.uk)
equilibrium catalysis
K-FeldsparK-Feldspar
Your realisation is correct and something chemistry teachers try to hammer into their students' heads time and time again (and yet, the point is still often lost):
Catalysts will never change the thermodynamics of a reaction. They only ease the path of the reaction. Forward and backward reactions will be accelerated equivalently.
So what is the benefit of a catalyst? There are multiple ones.
Take for example the Haber-Bosch process to synthesise ammonia from nitrogen and hydrogen.
$$\ce{N2 + 3 H2 <=> 2 NH3}\tag{1}$$ $$\Delta_\mathrm{r} H^0_\mathrm{298~K} = -45.8~\mathrm{\frac{kJ}{mol}}$$
This reaction is exothermic and thus should, theoretically or thermodynamicly, proceed spontaneously, e.g. if you mixed nitrogen and hydrogen in the appropriate ratio and added a spark. It does not, however. Significant activation energy is required to cleave the $\ce{N#N}$ triple bond.
Typical methods to add activation energy include heating. In the Haber-Bosch process, the mixture is heated to $400$ to $500~\mathrm{^\circ C}$ to supply the required activation energy. However, since the reaction is exothermic, heating will favour the reactant side. Increasing the pressure improves the entropic term of the Gibbs free energy equation, hence why pressures of $15$ to $25~\mathrm{MPa}$ are used.
Catalysts, based on iron with different promotors, are used to accelerate the reaction. By using catalysts, one can lower the temperature required in a trade-off between speed of reaction and favouring the product side of the equilibrium. With the conditions and catalysts used, one achieves a yield of $\approx 15~\%$ of ammonia within a reasonable timeframe. Not employing a catalyst would give much lower yields at much longer timeframes — economically much less feasible.
Direct reaction path not accessable
This is mainly true for many transition-metal catalysed carbon-carbon bond formation reactions, but is also true for some inorganic processes like the disproportionation of hydrogen peroxide as in equation $(2)$.
$$\ce{2 H2O2 -> 2 H2O + O2}\tag{2}$$
Hydrogen peroxide is a reactive chemical that cannot be stored forever, but the direct disproportionation path is not typically what degrades it. However, you can add $\ce{MnO2}$ to it. Upon addition, oxygen gas vigourously bubbles out of the solution. In this case, there was a kinetic barrier impeding the direct transformation due to reactants and products having different multiplicities (oxygen gas' ground state is a triplet, all others are singlets). The $\mathrm{d^3}$ ion manganese(IV) is a radical itself that can partake in different radical reactions, allowing the diradical oxygen to be liberated.
This is exceptionally true for transition-metal catalysed organic carbon-carbon bond formation reactions. Note first that the action of a catalyst is frequently depicted as a catalytic cycle: A reactant reacts with the catalyst to some intermediate species, this rearranges or reacts with other reactants/additives/solvents in a set of specific steps until finally the products are liberated and the catalytic species is regenerated.
Many such reactions require organic halides as one of the reacting species. And the first step is typically an oxidative addition as shown in equation $(3)$, where $\ce{X}$ is a halide ($\ce{Cl, Br, I}$).
$$\ce{R-X + Pd^0 -> R-Pd^{+II}-X}\tag{3}$$
Palladium typically prefers oxidatively adding to bromides or iodides and tends to leave chlorides alone. I myself have performed a reaction with near-quatitative yield in which a reactant contained both a $\ce{C-Br}$ and a $\ce{C-Cl}$ bond — selectively, only the $\ce{C-Br}$ bond took part in the palladium(0) catalysed Sonogashira reaction.
Although I did not try it myself, I am pretty sure that switching to a nickel(0) catalyst species would shift the reaction in favour of reacting with the carbon-chlorine bond rather than the carbon-bromine one.
This is basically a reiteration of the first point albeit with different intentions. Many a time in organic synthesis, one has a rather sensitive reactant that would degrade or undergo side-reactions if subjected to standard reaction conditions, such as high pH-value or elevated temperatures. As an example, consider a transesterification as shown in equation $(4)$.
$$\ce{R-COO-Et + Me-OH <=> R-COO-Me + EtOH}\tag{4}$$
This reaction is, of course, an equilibrium and by using methanol as the solvent we can shift it to the product side. For the reaction to happen, one would need a base strong enough to deprotonate methanol, giving the methanolate anion, which can then attack the ester functionality. However, methanolate being a strong (and nucleophilic) base itself can introduce undesired side-reactions, including epimerisation of the α-carbon.
One can catalyse this reaction by using $\ce{Bu2SnO}$, which will activate the carbonyl group, making it more susceptible to a nucleophilic attack. The reaction speed is the same but the conditions are milder (no additional base required) and the number of side-reactions is those strongly limited. In particular, I noticed no epimerisation of the α-carbon in the tin(IV) catalysed method.
JanJan
Yes, one expects both forwards and backward reactions to speed up as you suggest; there seems to be no reason why microscopic reversibility would be suspended.
The point of the catalyst will be to speed up a reaction so that it can happen on a reasonable time-scale, say hours not days or even years, and also (industrially) to use less energy to do the same reaction.
The back reaction will still be slower vs. forward reaction as the activation energy to return to products is still larger and so the equilibrium could still be greatly in favour of products. If this were not the case, either the products could be physically removed (say by precipitation, or escaping as gas/vapour) or they could react with something else, as in an enzyme, then the rate of the back reaction becomes far less significant.
porphyrinporphyrin
I think the fundamental point here is that a constant reduction in one constant matters a lot more than the other, e.g.:
$$e^{-1} - e^{-10} = 0.367$$
If you reduce both "barriers" by, say, $0.5$ units, i.e. push the activated state down by (say) $0.5$ units, then
$$e^{-(1 - 0.5)} - e^{-(10-0.5)} = 0.606$$
Yes, you reduced both barriers by $0.5$ units, but you still increased the rate by almost $2\times$.
In fact, a lot of the qualitative explaining away done by chemistry classes is only rigorously understandable if you start doing the math.
andselisk♦
curious_catcurious_cat
Not the answer you're looking for? Browse other questions tagged equilibrium catalysis or ask your own question.
Why does a catalyst increase the forward and reverse reactions equally
How does a catalyst accelerate both forward/reverse reactions simultaneously?
If an action increases the reaction rate in a particular equilibrium mixture, does it mean that both forward and backward rates are increased?
Do catalysts increase the rate of exothermic reaction more in a reversible reaction?
Do catalysts shift equilibrium constant towards 1?
Which diagram shows the effect of catalysis on chemical equilibrium?
How do catalysts provide alternative routes for reactions?
Intuition for why catalyst affects both forward and reverse reactions equally?
Same Activation Energy for Forward and Backward Reactions | CommonCrawl |
\begin{document}
\title{Internal precategories relative to split epimorphisms} \author{N. Martins-Ferreira} \address{Polytechnic Institute of Leiria - CDRSP} \email{[email protected]} \urladdr{http://www.estg.ipleiria.pt/\symbol{126}nelsonmf} \date{25Aug2008.} \subjclass[2000]{Primary 8D35; Secondary 18E05.} \keywords{Internal precategory, internal reflexive graph, internal action, half-reflection, crossed-module, precrossed-module, additive, semi-additive, binary coproducts, kernels of split epimorphisms, split short five lemma.} \thanks{The author thanks to Professors G. Janelidze and D. Bourn for much appreciated help of various kinds.}
\begin{abstract} For a given category B we are interested in studying internal categorical structures in B. This work is the starting point, where we consider reflexive graphs and precategories (i.e., for the purpose of this note, a simplicial object truncated at level 2). We introduce the notions of reflexive graph and precategory relative to split epimorphisms. We study the additive case, where the split epimorphisms are \textquotedblleft coproduct projections", and the semi-additive case where split epimorphisms are \textquotedblleft semi-direct product projections". The result is a generalization of the well known equivalence between precategories and 2-chain complexes. We also consider an abstract setting, containing, for example, strongly unital categories. \end{abstract}
\maketitle
\section{Introduction}
A internal reflexive graph in the category Ab, of abelian groups, is completely determined, up to an isomorphism, by a morphism $ h:X\longrightarrow B$ and it is of the following form \begin{equation*}
\xymatrix{ X \oplus B \ar@<1ex>[r]^{\pi_2} \ar@<-1ex>[r]_{[h \, 1]} & B
\ar[l]|(.4){\iota_2} }
. \end{equation*} An internal precategory (i.e., for the purpose of this work, a simplicial object truncated at level 2) is, in the first place, determined by a diagram \begin{equation*}
\xymatrix{ Y \ar@<1ex>[r]^{a} \ar@<-1ex>[r]_{u} & X
\ar[l]|{b} \ar[r]^{h} & B }
\end{equation*} such that \begin{equation*} ab=1=ub\ ,\ ha=hu, \end{equation*} and later, with a further analysis, it simplifies to a 2-chain (see \cite{BR} and \cite{DBourn} for more general results on this topic), i.e., \begin{equation*} Z\overset{t}{\longrightarrow }X\overset{h}{\longrightarrow }B\ \ ,\ \ ht=0. \end{equation*} And it is always of the following form \begin{equation*}
\xymatrix{ (Z \oplus X) \oplus (X \oplus B) \ar@<4ex>[rr]^{\pi_2} \ar@<-4ex>[rr]_{\pi_2 \oplus [h \, 1]}
\ar[rr]|(0.6){[\iota_1[t \, 1], \, 1]} & &X \oplus B
\ar@<-2ex>[ll]|{\iota_2}
\ar@<2ex>[ll]|(.4){\iota_2 \oplus \iota_2} \ar@<1ex>[r]^{\pi_2} \ar@<-1ex>[r]_{[h \, 1]} & B
\ar[l]|(.4){\iota_2} }
. \end{equation*}
The same result holds for arbitrary additive categories with kernels. In this work we will be interested in answering the following question: \textquotedblleft what is the more general setting where one can still have similar results?".
An old observation of G. Janelidze, says that \textquotedblleft since every higher dimensional categorical structure is obtained from an n-simplicial object; and since a simplicial object is build up from split epis; and, since in Ab, every split epi is simply a biproduct projection, then it is expected that, when internal to Ab, all the higher dimensional structures reduce to categories of presheaves". We use this observation as a motivation for the study of internal categorical structures restricted to a given subclass of split epis.
In particular in this work we will be interested in the study of the notion of internal reflexive graph (1-simplicial object) and internal precategory (2-simplicial object) relative to a given subclass of split epis, such as for example coproduct projections (in a pointed category with coproducts), or product projections, or semidirect product projections, or etc.
In some cases the given subclass is saturated (in the words of D. Bourn), as it happens for example in an additive category with kernels for the subclass of biproduct projections. However, in general, this is not the case; nevertheless, in some cases, interesting notions do occur.
Take for example the category of pointed sets and the class of coproduct projections, that is, consider only split epis of the form \begin{equation*}
\xymatrix{ X \sqcup B \ar@<1ex>[r]^{[0 \, 1]} & B
\ar[l]|{\iota_2} }
; \end{equation*} It follows that a reflexive graph relative to coproduct projections is completely determined by a morphism \begin{equation*} h:X\longrightarrow B \end{equation*} and it is of the form \begin{equation*}
\xymatrix{ X \sqcup B \ar@<1ex>[r]^{[0 \, 1]} \ar@<-1ex>[r]_{[h \, 1]} & B
\ar[l]|{\iota_2} }
; \end{equation*} While a precategory is determined by a diagram \begin{equation*}
\xymatrix{ Y \ar@<1ex>[r]^{a} \ar@<-1ex>[r]_{u} & X
\ar[l]|{b} \ar[r]^{h} & B }
\ \ \ ,\ \ \ ab=1=ub\ ,\ ha=hu, \end{equation*} and it is of the form \begin{equation*}
\xymatrix{ Y \sqcup (X \sqcup B) \ar@<4ex>[rr]^{[0 \, 1]} \ar@<-4ex>[rr]_{a \sqcup [h \, 1]}
\ar[rr]|(0.6){[\iota_1 u, \, 1]} & &X \sqcup B
\ar@<-2ex>[ll]|{\iota_2}
\ar@<2ex>[ll]|{b \sqcup \iota_2} \ar@<1ex>[r]^{[0 \, 1]} \ar@<-1ex>[r]_{[h \, 1]} & B
\ar[l]|{\iota_2} }
\end{equation*} where the key factor for this result to hold is the fact that $\iota _{1}$ is the kernel of $[0\ 1]$.
Furthermore, every such reflexive graph $\left( h:X\longrightarrow B\right) $ may be considered as a precategory, \begin{equation*}
\xymatrix{ X \ar@<1ex>[r]^{1} \ar@<-1ex>[r]_{1} & X
\ar[l]|{1} \ar[r]^{h} & B }
, \end{equation*} and it is a internal category if the kernel of $h$ is trivial, which is the same as saying that the following square \begin{equation*}
\quadrado { X \sqcup (X \sqcup B) } { [0 \, 1] } { X \sqcup B } { 1 \sqcup [h \, 1] } { [h \, 1] } { X \sqcup B } { [0 \, 1] } { B }
\end{equation*} is a pullback.
Specifically, given a morphism $h:X\longrightarrow B$ with trivial kernel (in pointed sets), the internal category it describes is the following: the objects are the elements of $B$; the morphisms are the identities $1_{b}$ for each $b\in B$ and also the elements $x\in X$, except for the distinguished element $0\in X$ that is identified with $0\in B$. The domain of every $x$ in $X$ is $0\in B$ and the codomain is $h\left( x\right) $. Since all arrows (except identities) start from $0\in B$, and because the kernel of $h$ is trivial, two morphisms $x$ and $x^{\prime }$ (other than $0$ ) never compose. The picture is a star with all arrows from the origin with no nontrivial loops.
On the other hand, again in pointed sets, if considering the subclass of split epis that are product projections, that is, of the form \begin{equation*}
\xymatrix{ X \times B \ar@<1ex>[r]^{\pi_2} & B \ar[l]^(0.4){<0,1>} }
; \end{equation*} then the following result is obtained.
A internal reflexive graph relative to product projections is given by a map \begin{equation*} \xi :X\times B\longrightarrow B;\ \ \ \left( x,b\right) \mapsto x\cdot b \end{equation*} such that $0\cdot b=b$ for all $b\in B$; and it is of the form \begin{equation*}
\xymatrix@C=3pc{ X \times B \ar@<1ex>[r]^{\pi_2} \ar@<-1ex>[r]_{\xi} & B
\ar[l]|(0.4){<0,1>} }
. \end{equation*} A internal precategory, relative to product projections, is given by \begin{equation*} Y\times \left( X\times B\right) \overset{\mu }{\longrightarrow }X\times B \overset{\xi }{\longrightarrow }B\ \ \ ,\ \ \ Y\overset{\alpha }{ \overrightarrow{\underset{\beta }{\longleftarrow }}}X \end{equation*} such that \begin{eqnarray*} \alpha \beta &=&1, \\ \mu \left( y,x,b\right) &=&\left( y+_{b}x,\ b\right) , \\ 0+_{b}x &=&x=\beta \left( x\right) +_{b}0 \\ \left( y+_{b}x\right) \cdot b &=&\alpha \left( y\right) \cdot \left( x\cdot b\right) ; \end{eqnarray*} and it is of the form \begin{equation*}
\xymatrix@C=4pc{ Y \times (X \times B) \ar@<4ex>[r]^{\pi_2} \ar@<-4ex>[r]_{\alpha \times \xi}
\ar[r]|{\mu} & X \times B
\ar@<-2ex>[l]|(0.4){<0,1>}
\ar@<2ex>[l]|(.4){\beta \times <0,1>} \ar@<1ex>[r]^{\pi_2} \ar@<-1ex>[r]_{\xi} & B
\ar[l]|{<0,1>} }
. \end{equation*} In particular if $X=Y$ and $\alpha =\beta =1$, we obtain a internal category (not necessarily associative), because the square \begin{equation*}
\quadrado { X\times (X\times B) } { \pi_2 } { X\times B } { 1\times \xi } { \xi } { X\times B } { \pi_2 } { B }
\end{equation*} is a pullback.
This means that a internal category in pointed sets and relative to product projections is given by two maps \begin{eqnarray*} \mu &:&X\times \left( X\times B\right) \longrightarrow X\times B;\ \ \left( x,x^{\prime },b\right) \mapsto \left( x+_{b}x^{\prime },b\right) \\ \xi &:&X\times B\longrightarrow B;\ \ \ \left( x,b\right) \mapsto x\cdot b \end{eqnarray*} such that \begin{eqnarray*} 0+_{b}x &=&x=x+_{b}0 \\ 0\cdot b &=&b \\ \left( x+_{b}x^{\prime }\right) \cdot b &=&x\cdot \left( x^{\prime }\cdot b\right) \end{eqnarray*} and in order to have associativity one must also require the additional condition \begin{equation*} \left( x^{\prime \prime }+_{(x\cdot b)}x^{\prime }\right) +_{b}x=x^{\prime \prime }+_{b}\left( x^{\prime }+_{b}x\right) . \end{equation*} Specifically, given a structure as above in pointed sets, the corresponding internal category that it represents is the following. The objects are the elements of $B$. The morphisms are pairs $\left( x,b\right) $ with domain $b$ and codomain $x\cdot b$. The composition of \begin{equation*} b\overset{(x,b)}{\longrightarrow }x\cdot b\overset{(x^{\prime },b^{\prime })} {\longrightarrow }x^{\prime }\cdot \left( x\cdot b\right) \end{equation*} is the pair $\left( x^{\prime }+_{b}x,b\right) $.
We will observe that for a given subclass of split epis, when the following two properties are present for every split epi $\left( A,\alpha ,\beta ,B\right) $ in the subclass:\newline (a) the morphism $\alpha :A\longrightarrow B$ has a kernel, say $ k:X\longrightarrow A$\newline (b) the pair $\left( k,\beta \right) $ is jointly epic\newline then a reflexive graph relative to the given subclass is determined by a split epi in the subclass, say $\left( A,\alpha ,\beta ,B\right) $ together with a \emph{central morphism} \begin{equation*} h:X\longrightarrow B \end{equation*} where a central morphism (see \cite{Bourn&Gran}) is such that there is a (necessarily unique) morphism, denoted by $[h\ 1]:A\longrightarrow B$ with the property \begin{equation*} \lbrack h\ 1]\beta =1\ \ ,\ \ [h\ 1]k=h, \end{equation*} where $k:X\longrightarrow A$ is the kernel of $\alpha $.
In the case of Groups, considering the subclass of split epis given by semi-direct product projections \begin{equation*}
\xymatrix{ X \rtimes B \ar@<1ex>[r]^{\pi_2} & B \ar[l]^(0.4){<0,1>} }
, \end{equation*} the notion of central morphism $h:X\longrightarrow B$ (together with a semidirect product projection, or an internal group action) corresponds to the usual definition of pre-crossed module.
This fact may lead us to consider an abstract notion of semidirect product as a diagram in a category satisfying some universal property.
In \cite{Berndt} O. Berndt proposes the categorical definition of semidirect products as follows: the semidirect product of $X$ and $B$ (in a pointed category) is a diagram \begin{equation*} X\overset{k}{\longrightarrow }A\overset{\alpha }{\overrightarrow{\underset{ \beta }{\longleftarrow }}}B \end{equation*} such that $\alpha \beta =1$ and $k=\ker \alpha $.
We now see that it would be more reasonable to adjust this definition as follows: in a pointed category, the semidirect product of $X$ and $B$, denoted $X\rtimes B$, is defined together with two morphisms \begin{equation*} X\overset{k}{\longrightarrow }X\rtimes B\overset{\beta }{\longleftarrow }B \end{equation*} satisfying the following three conditions:\newline (a) the pair $\left( k,\beta \right) $ is jointly epic\newline (b) the zero morphism \begin{equation*} 0:X\longrightarrow B \end{equation*} is central, that is, there exists a (necessarily unique) morphism $[0\ 1]:X\rtimes B\longrightarrow B$ with $[0\ 1]\beta =1\ \ $and$\ \ [0\ 1]k=0$ \newline (c) $k$ is the kernel of $[0\ 1]$.
We must add that this object $X\rtimes B$ may not be uniquely determined (even up to isomorphism), to achieve that we simply require the pair $\left( k,\beta \right) $ to be universal with the above properties.
We also remark that we have not investigate further the consequences of such a definition. It will only be done in some future work. We choose to mention it at this point because it is related with the present work.
Another example, of considering internal categories relative to split epis, may be found in \cite{Patch} where A. Patchkoria shows that, in the category of Monoids, the notion of internal category relative to semidirect product projections is in fact equivalent to the notion of a Schreier category.
This work is organized as follows.
First we recall some basic definitions, and introduce a concept that is obtained by weakening the notion of reflection, so that we choose to call it half-reflection.
Next we study the case of additivity, and find minimal conditions on a category $\mathbf{B}$ in order to have \begin{eqnarray*} RG\left( \mathbf{B}\right) &\sim &Mor\left( \mathbf{B}\right) \\ PC\left( \mathbf{B}\right) &\sim &2\text{-}Chains\left( \mathbf{B}\right) \end{eqnarray*} the usual equivalences between reflexive graphs and morphisms in $\mathbf{B}$ , precategories and 2-chains in $\mathbf{B}$. We show that this is the case exactly when $\mathbf{B}$ is pointed (but not necessarily with a zero object), has binary coproducts and kernels of split epis, and satisfies the following two conditions (see Theorem \ref{Th 2}):\newline (a) $\iota _{1}$ is the kernel of $[01]$ \begin{equation*} X\overset{\iota _{1}}{\longrightarrow }X\sqcup B\overset{[0\ 1]}{ \longrightarrow }B \end{equation*} $\newline $(b) the split short five lemma holds.
Later we investigate the same notions, and essentially obtain the same results, for the case of semi-additivity, by replacing coproduct projections by semidirect product projections, where the notion of semi-direct product is associated with the notion of internal actions in the sense of \cite{GJ1} and \cite{JB}.
At the end we describe the same situation for a general setting, specially designed to mimic internal actions and semidirect products. An application of the results is given for the category of unitary magmas with right cancellation.
\section{Definitions}
\begin{definition}[reflection] A functor $I:\mathbf{A}\longrightarrow \mathbf{B}$ is a reflection when there is a functor \begin{equation*} H:\mathbf{B}\longrightarrow \mathbf{A} \end{equation*} and a natural transformation \begin{equation*} \rho :1_{\mathbf{A}}\longrightarrow HI \end{equation*} satisfying the following conditions \begin{eqnarray*} IH &=&1_{\mathbf{B}} \\ I\circ \rho &=&1_{I} \\ \rho \circ H &=&1_{H}. \end{eqnarray*} \end{definition}
\begin{definition}[half-reflection] A pair of functors \begin{equation*} \xymatrix{\mathbf{A} \ar@<0.5ex>[r]^{I} & \mathbf{B} \ar@<0.5ex>[l]^{G} }, \ \ \ ,\ \ \ \ IG=1_{\mathbf{B}} \end{equation*} is said to be a half-reflection if there is a natural transformation \begin{equation*} \pi :1_{\mathbf{A}}\longrightarrow GI \end{equation*} such that \begin{equation*} I\circ \pi =1_{I}. \end{equation*} \end{definition}
\begin{theorem} For a half-reflection $\left( I,G,\pi \right) $ we always have \begin{equation}
\xymatrix{ 1 \ar[r]^{\pi} \ar[rd]_{\pi} & GI \ar[d]^{\pi \circ GI} \\ & GI }
. \label{eq1} \end{equation} \end{theorem}
\begin{proof} By naturality of $\pi $ we have \begin{equation*}
\quadrado { A } { \pi_A } { GIA } { \pi_A } { GI(\pi_A) } { GIA } { \pi_{GIA} } { GI(GIA) }
\end{equation*} but $GI(GIA)=GIA$ and $GI\left( \pi _{A}\right) =1_{GIA}$. \end{proof}
When it exists, the natural transformation $\pi :1\longrightarrow GI$ is \emph{essentially} unique, in the sense that any other such, say $\pi ^{\prime }:1\longrightarrow GI$ (with $I\circ \pi ^{\prime }=1_{I}$), is of the form \begin{equation*} \pi _{A}^{\prime }=\pi _{GIA}^{\prime }\pi _{A}. \end{equation*} Under which conditions is it really unique?
The name half-reflection is motivated because if instead of $\left( \ref{eq1} \right) $ we have $\pi \circ G=1_{G}$ then the result is a reflection.
For any category $\mathbf{B}$ we consider the category of internal reflexive graphs in $\mathbf{B}$, denoted $RG\left( \mathbf{B}\right) $ as usual: \newline Objects are diagrams in $\mathbf{B}$ of the form \begin{equation*}
\xymatrix{ C_1 \ar@<1ex>[r]^{d} \ar@<-1ex>[r]_{c} & C_0
\ar[l]|{e} }
\ \ ,\ \ de=1=ce; \end{equation*} Morphisms are pairs $\left( f_{1},f_{0}\right) $ making the obvious squares commutative in the following diagram \begin{equation*}
\xymatrix{
C_1 \ar@<1ex>[r]^{d} \ar@<-1ex>[r]_{c} \ar[d]_{f_1} & C_0 \ar[d]_{f_0}
\ar[l]|{e} \\
C'_1 \ar@<1ex>[r]^{d} \ar@<-1ex>[r]_{c} & C'_0
\ar[l]|{e'} }
. \end{equation*} We will also consider the category of internal precategories in $\mathbf{B}$ , denoted $PC\left( \mathbf{B}\right) $, where objects are diagrams of the form \begin{equation*}
\xymatrix{ C_2 \ar@<2ex>[r]^{\pi_2} \ar@<-2ex>[r]_{\pi_1}
\ar[r]|{m} & C_1
\ar@<-1ex>[l]|{e_2}
\ar@<1ex>[l]|{e_1} \ar@<1ex>[r]^{d} \ar@<-1ex>[r]_{c} & C_0
\ar[l]|{e} }
\end{equation*} such that \begin{equation}
\xymatrix{ C_2 \ar[r]^{\pi_2} \ar[d]_{\pi_1} & C_1 \ar@<1ex>[l]^{e_2} \ar[d]_{c} \\ C_1 \ar[r]^{d} \ar@<-1ex>[u]_{e_1} & C_0 \ar@<1ex>[l]^{e} \ar@<-1ex>[u]_{e} }
\label{split quare 1} \end{equation} is a split square (i.e. a split epi in the category of split epis), so that in particular we have \begin{equation} de=1_{C_{0}}=ce \label{RGraph condition in precat} \end{equation} and furthermore, the following three conditions are satisfied \begin{gather} dm=d\pi _{2} \label{dom_preserved in a precat} \\ cm=c\pi _{1}. \label{cod_preserved in a precat} \end{gather} \begin{equation} me_{1}=1_{C_{1}}=me_{2}; \label{me1 is 1} \end{equation} and obvious morphisms.
A precategory in this sense becomes a category\footnote{ In fact it is not quite a category because we are not considering associativity; also the term precategory is often used when $\left(\ref{me1 is 1}\right) $ is not present; we also observe that in many interesting cases (for example in Mal'cev categories) assuming only $\left(\ref{me1 is 1} \right) $ and the fact that $\left(\ref{split quare 1}\right) $ is a pullback, then the resulting structure is already an internal category.} if the top and left square in $\left( \ref{split quare 1}\right) $ is a pullback.
\begin{definition} A category is said to have coequalizers of reflexive graphs if for every reflexive graph \begin{equation*}
\xymatrix{ C_1 \ar@<1ex>[r]^{d} \ar@<-1ex>[r]_{c} & C_0
\ar[l]|{e} }
\ \ ,\ \ de=1=ce \end{equation*} the coequalizer of $d$ and $c$ exists. \end{definition}
\begin{definition}[pointed category] A pointed category is a category enriched in pointed sets. More specifically, for every pair $X,Y$ of objects, there is a specified morphism, $0_{X,Y}:X\longrightarrow Y$ with the following property: \begin{equation*} X\overset{0_{X,Y}}{\longrightarrow }Y\overset{f}{\longrightarrow }Z\overset{ 0_{Z,W}}{\longrightarrow }W \end{equation*} \begin{eqnarray} 0_{Z,W}f &=&0_{Y,W} \label{eq2} \\ f0_{X,Y} &=&0_{X,Z}. \label{eq3} \end{eqnarray} \end{definition}
\begin{definition}[additive category] An additive category is an $Ab$-category with binary biproducts. \end{definition}
Observe that on the contrary to the usual practice we are not considering the existence of a null object, neither in pointed nor in additive categories.
\section{Additivity}
Let $\mathbf{B}$ be any category and consider the pair of functors \begin{equation*} \xymatrix{\mathbf{B\times B} \ar@<0.5ex>[r]^{I} & \mathbf{B} \ar@<0.5ex>[l]^{G} } \end{equation*} with $I\left( X,B\right) =B$ and $G\left( B\right) =\left( B,B\right) .$
\begin{theorem} The above pair $\left( I,G\right) $ is a half-reflection if and only if the category $\mathbf{B}$ is pointed. \end{theorem}
\begin{proof} If $\mathbf{B}$ is pointed simply define \begin{equation*} \pi _{\left( X,B\right) }:\left( X,B\right) \longrightarrow \left( B,B\right) \end{equation*} as $\pi _{\left( X,B\right) }=\left( 0_{X,B},1_{B}\right) .$\newline Now suppose there is a natural transformation \begin{equation*} \pi :1_{\mathbf{BxB}}\longrightarrow GI, \end{equation*} such that $I\circ \pi =1_{I}$, this is the same as having for every pair $ X,B $ in $\mathbf{B}$ a specified morphism \begin{equation*} \pi _{X,B}:X\longrightarrow B \end{equation*} and conditions $\left( \ref{eq2}\right) $ and $\left( \ref{eq3}\right) $ follow by naturality: \begin{equation*}
\quadrado { (Y,W) } { (\pi_{Y,W} , 1) } { (W,W) } { (f,1) } { (1,1) } { (Z,W) } { (\pi_{Z,W}, 1) } { (W,W) }
,\ \ \ \
\quadrado { (X,Y) } { (\pi_{X,Y} , 1) } { (Y,Y) } { (1,f) } { (f,f) } { (X,Z) } { (\pi_{X,Z}, 1) } { ( Z,Z) }
. \end{equation*} \end{proof}
\begin{theorem} The functor $G$ as above admits a left adjoint if and only if the category $ \mathbf{B}$ has binary coproducts. \end{theorem}
\begin{proof} As it is well known, coproducts are obtained as the left adjoint to the diagonal functor. \end{proof}
\begin{theorem} Given a half-reflection $\left( I,G,\pi \right) $ \begin{equation*} \xymatrix{\mathbf{A} \ar@<0.5ex>[r]^{I} & \mathbf{B} \ar@<0.5ex>[l]^{G} } \ \ \ ,\ \ \ \ \pi :1_{\mathbf{A}}\longrightarrow GI, \end{equation*} if the functor $G$ admits a left adjoint \begin{equation*} \left( F,G,\eta ,\varepsilon \right) , \end{equation*} then, there is a canonical functor \begin{equation*} \mathbf{A}\longrightarrow Pt\left( \mathbf{B}\right) \end{equation*} sending an object $A\in \mathbf{A}$ to the split epi \begin{equation*}
\xymatrix{ FA \ar@<0.5ex>[r]^{\varepsilon_{IA} F(\pi_A)} & IA \ar@<0.5ex>[l]^{I(\eta_A)} }
. \end{equation*} \end{theorem}
\begin{proof} We only have to prove \begin{equation*} \varepsilon _{IA}F\left( \pi _{A}\right) I\left( \eta _{A}\right) =1_{IA}. \end{equation*} Start with \begin{equation*} \pi _{A}=G\left( \varepsilon _{IA}F\left( \pi _{A}\right) \right) \eta _{A} \end{equation*} and apply $I$ to both sides to obtain \begin{equation*} I\left( \pi _{A}\right) =\varepsilon _{IA}F\left( \pi _{A}\right) I\left( \eta _{A}\right) , \end{equation*} by definition we have $I\left( \pi _{A}\right) =1_{IA}$. \end{proof}
In particular if $\mathbf{B}$ is pointed and has binary coproducts we have the canonical functor \begin{equation*} \mathbf{B\times B}\overset{T}{\longrightarrow }Pt\left( \mathbf{B}\right) \end{equation*} sending a pair $\left( X,B\right) $ to the split epi \begin{equation*}
\xymatrix{ X \sqcup B \ar@<0.5ex>[r]^{[0 \, 1]} & B \ar@<0.5ex>[l]^{\iota_2} }
. \end{equation*}
\begin{theorem} Let $\mathbf{B}$ be a pointed category with binary coproducts. The canonical functor $\mathbf{B\times B}\overset{T}{\longrightarrow }Pt\left( \mathbf{B} \right) $ admits a right adjoint, $S$, such that $IS=I^{\prime }$ \begin{equation*}
\xymatrix{ \mathbf{B\times B} \ar[rd]_{I} & & Pt(\mathbf{B}) \ar[ll]_{S} \ar[ld]^{I'} \\ & \mathbf{B} }
\end{equation*} if and only if the category $\mathbf{B}$ has kernels of split epis.\newline The functor $I^{\prime }$ sends a split epi $\left( A,\alpha ,\beta ,B\right) $ to $B$. \end{theorem}
\begin{proof} If the category has kernels of split epis, then for every split epi we choose a specified kernel \begin{equation*}
\xymatrix{ X \ar[r]^{k} & A \ar@<0.5ex>[r]^{\alpha} & B \ar@<0.5ex>[l]^{\beta} }
\end{equation*} and the functor $S$, sending $\left( A,\alpha ,\beta ,B\right) $ to the pair $\left( X,B\right) $ is the right adjoint for $T$: \begin{equation*}
\xymatrix{ (Y,D) \ar[d]^{(f,g)} & Y \ar[r]^{\iota_1} \ar[d]_{f} & Y \sqcup D \ar@<0.5ex>[r]^{[0 \, 1]} \ar[d]^{[kf \, \beta g]} & B \ar@<0.5ex>[l]^{\iota_2} \ar[d]^{g} \\ (X,B) & X \ar[r]^{k} & A \ar@<0.5ex>[r]^{\alpha} & B \ar@<0.5ex>[l]^{\beta} }
. \end{equation*} \newline Now suppose $S$ is a right adjoint to $T$ and it is such that a split epi $ \left( A,\alpha ,\beta ,B\right) $ goes to a pair of the form \begin{equation*} \left( K[\alpha ],B\right) \end{equation*} with unit and counit as follows \begin{equation*}
\xymatrix{ (X,B) \ar[d]^{(\eta_X,1)} & K[\alpha] \ar[r]^{\iota_1} \ar[rd]_{\varepsilon_1} & K[\alpha] \sqcup B \ar@<0.5ex>[r]^{[0 \, 1]} \ar[d]^{[\varepsilon_1 \, \beta]} & B \ar@<0.5ex>[l]^{\iota_2} \ar[d]^{1} \\ (K[0 \,1],B) & & A \ar@<0.5ex>[r]^{\alpha} & B \ar@<0.5ex>[l]^{\beta} }
. \end{equation*} We have to prove that $\varepsilon _{1}=\ker \alpha $, and in fact, we have $ \alpha \varepsilon _{1}=0$ and by the universal property of the counit we have that given a morphism of split epis \begin{equation*}
\xymatrix{ X \ar[r]^{\iota_1} \ar[rd]_{f} & X \sqcup B \ar@<0.5ex>[r]^{[0 \, 1]} \ar[d]^{[f \, \beta]} & B \ar@<0.5ex>[l]^{\iota_2} \ar[d]^{1} \\ & A \ar@<0.5ex>[r]^{\alpha} & B \ar@<0.5ex>[l]^{\beta} }
\end{equation*} that is a morphism $f:X\longrightarrow A$ such that $\alpha f=0$, there exists a unique $\left( f^{\prime },1\right) :\left( X,B\right) \longrightarrow \left( K[\alpha ],B\right) $ such that \begin{equation*} \lbrack \varepsilon _{1}\ \beta ]\left( f^{\prime }\sqcup 1\right) =[f\ \beta ] \end{equation*} which is equivalent to say $\varepsilon _{1}f^{\prime }=f$. Hence $ \varepsilon _{1}$ is a kernel for $\alpha $. \end{proof}
\begin{theorem} Let $\mathbf{B}$ be a pointed category with binary coproducts and kernels of split epis. If the canonical adjunction \begin{equation*} \mathbf{B\times B}\overset{T}{\underset{S}{\underleftarrow{\overrightarrow{\ \ \ \ \ \perp \ \ \ \ \ \ }}}}Pt\left( \mathbf{B}\right) \end{equation*} is an equivalence, then: \begin{equation*} RG\left( \mathbf{B}\right) \sim Mor\left( \mathbf{B}\right) \end{equation*} and \begin{equation*} PC\left( \mathbf{B}\right) \sim 2\text{-}Chains\left( \mathbf{B}\right) . \end{equation*} \end{theorem}
\begin{proof} By the equivalence we have that a split epi \begin{equation*}
\xymatrix{ C_1 \ar@<0.5ex>[r]^{d} & C_0 \ar@<0.5ex>[l]^{e} }
\ \ \ \ \ \ ,\ \ \ de=1 \end{equation*} is of the form \begin{equation*}
\xymatrix{ X \sqcup B \ar@<0.5ex>[r]^{[0 \, 1]} & B \ar@<0.5ex>[l]^{\iota_2} }
, \end{equation*} and to give a morphism $c:X\sqcup B\longrightarrow B$ such that $c\iota _{2}=1$ is to give a morphism \begin{equation*} h:X\longrightarrow B. \end{equation*} So that a reflexive graph is, up to isomorphism, of the form \begin{equation*}
\xymatrix{ X \sqcup B \ar@<1ex>[r]^{[0 \, 1]} \ar@<-1ex>[r]_{[h \, 1]} & B
\ar[l]|{\iota_2} }
. \end{equation*} To investigate a precategory we observe that the square $\left( \ref{split quare 1}\right) $ may be considered as a split epi in the category of split epis, and hence, it is given up to an isomorphism in the form \begin{equation}
\xymatrix{ Y \sqcup (X \sqcup B) \ar[r]^{[0 \, 1]} \ar[d]_{a \sqcup [h,1]} & X \sqcup B \ar@<1ex>[l]^{\iota_2} \ar[d]_{[h \, 1]} \\ X \sqcup B \ar[r]^{[0 \, 1]} \ar@<-1ex>[u]_{b \sqcup \iota_2} & B \ar@<1ex>[l]^{\iota_2} \ar@<-1ex>[u]_{\iota_2} }
\ \ \ ,\ \ \ ab=1. \label{split square 2} \end{equation} It follows that $m$, satisfying $m\iota _{2}=1$ is of the form \begin{equation*} Y\sqcup \left( X\sqcup B\right) \overset{[v\ 1]}{\longrightarrow }\left( X\sqcup B\right) \end{equation*} and hence to give $m$ is to give $v:Y\longrightarrow X\sqcup B$.\newline Since we also have $\left( \ref{dom_preserved in a precat}\right) $ then $ [0\ 1]v=0$, and $v$ factors through the kernel of $[0\ 1]$ which is (see Theorem \ref{Th 2}) \begin{equation*}
\xymatrix{ X \ar[r]^{\iota_1} & X \sqcup B \ar@<0.5ex>[r]^{[0 \, 1]} & B \ar@<0.5ex>[l]^{\iota_2} }
. \end{equation*} This shows that to give $m$ is to give a morphism \begin{equation*} u:Y\longrightarrow X \end{equation*} and hence $m$ is given as \begin{equation*} m=[\iota _{1}u\ 1]:Y\sqcup \left( X\sqcup B\right) \longrightarrow \left( X\sqcup B\right) . \end{equation*} Finally we have that condition $\left( \ref{cod_preserved in a precat} \right) $ is equivalent to $ha=hu$ and $m\left( b\sqcup \iota _{2}\right) $ is equivalent to $ub=1.$\newline Conclusion 1: A precategory in $\mathbf{B}$ is completely determined by a diagram \begin{equation*}
\xymatrix{ Y \ar@<1ex>[r]^{a} \ar@<-1ex>[r]_{u} & X
\ar[l]|{b} \ar[r]^{h} & B }
\end{equation*} such that \begin{equation*} ab=1=ub\ ,\ ha=hu. \end{equation*} \newline Continuing with a further analysis we observe that the resulting diagram is in particular a reflexive graph and hence it is, up to isomorphism, of the form \begin{equation*}
\xymatrix{ Z \sqcup X \ar@<1ex>[r]^{[0 \,1]} \ar@<-1ex>[r]_{[t \, 1]} & X
\ar[l]|(.4){\iota_2} \ar[r]^{h} & B }
\end{equation*} where $h[0\ 1]=h[t\ 1]$ is equivalent to $ht=0$.\newline Conclusion 2: A precategory in $\mathbf{B}$ is completely determined by a 2-chain complex \begin{equation*} Y\overset{t}{\longrightarrow }X\overset{h}{\longrightarrow }B\ \ ,\ \ ht=0. \end{equation*} \end{proof}
\begin{remark} In the future we will not assume the canonical functor $T$ to be an equivalence, and hence the second conclusion will no longer be possible. However, we will be interested in the study of precategories such that $ \left( \ref{split quare 1}\right) $ is of the form $\left( \ref{split square 2}\right) $ and in that case, provided that $\iota _{1}$ is the kernel of $ [0\ 1]$ we still can deduce conclusion 1. Such an example is the category of pointed sets: see Introduction. \end{remark}
There is a canonical inclusion of reflexive graphs into precategories, by sending $h:X\longrightarrow B$ to \begin{equation*}
\xymatrix{ X \ar@<1ex>[r]^{1} \ar@<-1ex>[r]_{1} & X
\ar[l]|{1} \ar[r]^{h} & B }
. \end{equation*}
\begin{theorem} If $\mathbf{B}$ has coequalizers of reflexive graphs, then the canonical functor \begin{equation*} PC\left( \mathbf{B}\right) \underset{V}{\longleftarrow }RG\left( \mathbf{B} \right) \end{equation*} has a left adjoint. \end{theorem}
\begin{proof} The left adjoint is the following.\newline Given the precategory \begin{equation*}
\xymatrix{ Y \ar@<1ex>[r]^{a} \ar@<-1ex>[r]_{u} & X
\ar[l]|{b} \ar[r]^{h} \ar[rd]_{\sigma=coeq} & B \\ & & X' \ar@{-->}[u]_{h'} }
\end{equation*} construct the coequalizer of $u$ and $a$, say $\sigma $, and consider the reflexive graph in $\mathbf{B}$ determined by \begin{equation*} h^{\prime }:X^{\prime }\longrightarrow B. \end{equation*} This defines a reflection \begin{equation*} PC\left( \mathbf{B}\right) \overset{U}{\longrightarrow }RG\left( \mathbf{B} \right) \end{equation*} with unit \begin{equation*}
\xymatrix{ Y \ar@<1ex>[r]^{a} \ar@<-1ex>[r]_{u} \ar[d]_{\sigma u= \sigma a} & X
\ar[l]|{b} \ar[r]^{h} \ar[d]_{\sigma} & B \ar@{=}[d] \\ X' \ar@<1ex>[r]^{1} \ar@<-1ex>[r]_{1} & X'
\ar[l]|{1} \ar[r]^{h'} & B }
. \end{equation*} \end{proof}
Next we characterize a category $\mathbf{B}$, pointed, with binary coproducts and such that the canonical functor \begin{equation*} \mathbf{B\times B}\longrightarrow Pt\left( \mathbf{B}\right) \end{equation*} is an equivalence.
First observe that:
\begin{theorem} If $\mathbf{B}$, as above, also has binary products, then it is an additive category (with kernels of split epis). \end{theorem}
\begin{proof} We simply observe that in particular \begin{equation*}
\xymatrix{ X \sqcup B \ar@<0.5ex>[r]^{[0 \, 1]} \ar[d]_{\cong } & B \ar@<0.5ex>[l]^{\iota_2} \ar@{=}[d] \\ X \times B \ar@<0.5ex>[r]^{\pi_2} & B \ar@<0.5ex>[l]^{<0,1>} }
\ \ \ \ \ \text{and \ \ \ \ \ \ \ }
\xymatrix{ X \sqcup B \ar@<0.5ex>[r]^{[0 \, 1]} \ar[d]_{\cong } & B \ar@<0.5ex>[l]^{\iota_2} \ar@{=}[d] \\ X \times B \ar@<0.5ex>[r]^{\pi_2} & B \ar@<0.5ex>[l]^{<1,1>} }
, \end{equation*} since $X\overset{<1,0>}{\longrightarrow }X\times B$ is a kernel for $\pi _{2} $. See \cite{GJ2} for more details. \end{proof}
\begin{theorem} \label{Th 2}Let $\mathbf{B}$ be pointed with binary coproducts and kernels of split epis. The following conditions are equivalent:
\begin{description} \item[(a)] the canonical adjunction \begin{equation*} \mathbf{B\times B}\overset{T}{\underset{S}{\underleftarrow{\overrightarrow{\ \ \ \ \ \perp \ \ \ \ \ \ }}}}Pt\left( \mathbf{B}\right) \end{equation*} \begin{eqnarray*} T\left( X,B\right) &=&\left( X\sqcup B,[0\ 1],\iota _{2},B\right) \\ S\left( A,\alpha ,\beta ,B\right) &=&\left( K[\alpha ],B\right) , \end{eqnarray*} is an equivalence of categories;
\item[(b)] the category $\mathbf{B}$ satisfies the following two axioms:
\begin{description} \item[(A1)] for every diagram of the form \begin{equation}
\xymatrix{ X \ar[r]^{\iota_1} & X \sqcup B \ar@<0.5ex>[r]^{[0 \, 1]} & B \ar@<0.5ex>[l]^{\iota_2} }
\label{A1} \end{equation} the morphism $\iota _{1}$ is the kernel of $[0\ 1]$;
\item[(A2)] the split short five lemma holds, that is, given any diagram of split epis and respective kernels \begin{equation}
\xymatrix{ X \ar[r]^{k} \ar[d]_{f}
& A \ar@<0.5ex>[r]^{\alpha} \ar[d]_{h} & B \ar@<0.5ex>[l]^{\beta} \ar[d]^{g} \\ X' \ar[r]^{k'} & A' \ar@<0.5ex>[r]^{\alpha'} & B' \ar@<0.5ex>[l]^{\beta'} }
\label{A2} \end{equation} if $g$ and $f$ are isomorphisms then $h$ is an isomorphism. \end{description} \end{description} \end{theorem}
\begin{proof} $(b)\Rightarrow (a)$ Using only (A1) we have that $ST\cong 1$, and using (A1) and (A2) we have, in particular, that $[k\ \beta ]$ as in \begin{equation*}
\xymatrix{ X \ar[r]^{\iota_1} \ar@{=}[d]
& X \sqcup B \ar@<0.5ex>[r]^{[0 \, 1]} \ar[d]_{[k \, \beta]} & B \ar@<0.5ex>[l]^{\iota_2} \ar@{=}[d] \\ X \ar[r]^{k} & A \ar@<0.5ex>[r]^{\alpha} & B \ar@<0.5ex>[l]^{\beta} }
\end{equation*} is an isomorphism, and hence $TS\cong 1$.
$(a)\Rightarrow (b)$ Suppose $ST\cong 1$, this gives (A1); suppose $TS\cong 1 $, so that from $\left( \ref{A2}\right) $ we can form \begin{equation*}
\xymatrix{ X \ar[r]^{\iota_1} \ar[d]_{f}
& X \sqcup B \ar@<0.5ex>[r]^{[0 \, 1]} \ar[d]_{f \sqcup g} & B \ar@<0.5ex>[l]^{\iota_2} \ar[d]^{g} \\ X' \ar[r]^{\iota_1} & X \sqcup B \ar@<0.5ex>[r]^{[0 \, 1]} & B \ar@<0.5ex>[l]^{\iota_2} }
\end{equation*} and if $f$, $g$ are isomorphisms, we can find $h^{-1}=[k\ \beta ]\left( f^{-1}\sqcup g^{-1}\right) [k^{\prime }\ \beta ^{\prime }]^{-1}$. \end{proof}
\begin{corollary} If $T$ is a reflection then it is an equivalence of categories. \end{corollary}
We may now state the following results.
\begin{conclusion} Let $\mathbf{B}$ be a pointed category with binary coproducts. TFAE:\newline (a) T is a reflection and $\mathbf{B}$ has binary products;\newline (b) $\mathbf{B}$ is additive and has kernels of split epis. \end{conclusion}
\begin{conclusion} Let $\mathbf{B}$ be pointed, with binary products and coproducts and kernels of split epis. TFAE:\newline (a) $T$ is a reflection;\newline (b) $\mathbf{B}$ is additive. \end{conclusion}
\subsection{Restriction to split epis}
Suppose now that the canonical functor $T$ is not an equivalence, but we still have axiom $\left( \ref{A1}\right) $, that is $ST\cong 1$. The results relating precategories and reflexive graphs will still hold if we restrict $ PC\left( \mathbf{B}\right) $ to diagrams of the form \begin{equation*}
\xymatrix@C=3pc{ Y \sqcup (X \sqcup B) \ar@<2ex>[r]^{[0 \, 1]} \ar@<-2ex>[r]_{a \sqcup c}
\ar[r]|{m} & X \sqcup B
\ar@<-1ex>[l]|{\iota_2}
\ar@<1ex>[l]|{b \sqcup c} \ar@<1ex>[r]^{[0 \, 1]} \ar@<-1ex>[r]_{c} & B
\ar[l]|{\iota_2} }
. \end{equation*} This result will be proved in a more general case in the next sections.
An example of such a case is the category of pointed sets.
If starting with a general half-reflection \begin{equation*} \mathbf{A}\overset{I}{\underset{G}{\underleftarrow{\overrightarrow{\ \ \ \ \ \ \ \ \ \ \ }}}}\mathbf{B\ \ \ ,\ \ \ }\pi :1\longrightarrow GI \end{equation*} such that $G$ admits a left adjoint \begin{equation*} \left( F,G,\eta ,\varepsilon \right) \end{equation*} we consider the canonical functor \begin{equation*} \mathbf{A}\overset{T}{\longrightarrow }Pt\left( \mathbf{B}\right) \end{equation*} and ask if it is an equivalence; if not we then ask if it satisfies at least one of the axioms $\left( \ref{A1}\right) $ or $\left( \ref{A2}\right) $. For example for $\mathbf{A=B\times B}$ and assuming the constructions as above, in the case of pointed sets we have $\left( \ref{A1}\right) $ but not $\left( \ref{A2}\right) $, while in groups we have $\left( \ref{A2}\right) $ but not $\left( \ref{A1}\right) $.
In the case where we have only $\left( \ref{A1}\right) $ we will be interested in the study of $RG\left( \mathbf{B}\right) $ and $PC\left( \mathbf{B}\right) $ restricted to split epis of the form \begin{equation*}
\xymatrix{ FA \ar@<0.5ex>[r]^{\varepsilon_{IA} F(\pi_A)} & IA \ar@<0.5ex>[l]^{I(\eta_A)} }
, \end{equation*} while if in the presence of $\left( \ref{A2}\right) $, but not $\left( \ref {A1}\right) $, we may construct a category of internal actions as suggested in \cite{GJ1}.
\section{Semi-Additivity}
Let $\mathbf{B}$ be a pointed category with binary coproducts and kernels of split epis. As shown in the previous section there is a canonical adjunction \begin{equation} \mathbf{B\times B}\overset{T}{\underset{S}{\underleftarrow{\overrightarrow{\ \ \ \ \ \ \perp \ \ \ \ \ }}}}Pt\left( \mathbf{B}\right) . \label{canonical adjunction BxB--->Pt(B)} \end{equation} We are considering $\mathbf{B\times B}$ and $Pt\left( \mathbf{B}\right) $ as objects in the category of functors over $\mathbf{B}$, that is \begin{equation*}
\xymatrix{ \mathbf{B\times B} \ar[rd]_{I} & & Pt(\mathbf{B}) \ar[ld]^{I'} \\ & \mathbf{B} }
\end{equation*} where $I\left( X,B\right) =B$ and $I^{\prime }\left( A,\alpha ,\beta ,B\right) =B$.
We are also interested in the fact that $I$ is a half-reflection, with respect to some functor $G$. In the case of $\mathbf{B\times B}$ there is a canonical choice for $G$, namely the diagonal functor, and it is a half-reflection if and only if $\mathbf{B}$ is pointed. We are also interested in the fact that $G$ admits a left adjoint.
In the case of $Pt\left( \mathbf{B}\right) $ there are apparently many good choices for the functor $G^{\prime }$ to be a half-reflection together with $ I^{\prime }$. Nevertheless, if we ask that the left adjoint for $G^{\prime }$ to be $F^{\prime }$, such that $F^{\prime }\left( A,\alpha ,\beta ,B\right) =A$, then we calculate $G^{\prime }$ as follows.
\begin{theorem} \label{Theorem 1}Let $\mathbf{B}$ be any category and consider the two functors \begin{equation*} Pt\left( \mathbf{B}\right) \overset{I}{\underset{F}{\underrightarrow{ \overrightarrow{\ \ \ \ \ \ \ \ \ \ \ }}}}\mathbf{B} \end{equation*} \begin{eqnarray*} I\left( A,\alpha ,\beta ,B\right) &=&B \\ F\left( A,\alpha ,\beta ,B\right) &=&A. \end{eqnarray*} The functor $F$ admits a right adjoint \begin{equation*} \left( F,G,\eta ,\varepsilon \right) \end{equation*} such that $IG=1_{\mathbf{B}}$ if and only if the category $\mathbf{B}$ has an endofunctor \begin{equation*} G_{1}:\mathbf{B}\longrightarrow \mathbf{B} \end{equation*} and natural transformations \begin{equation*}
\xymatrix{ G_1(B) \ar@<1ex>[r]^{\pi_B} \ar@<-1ex>[r]_{\varepsilon_B} & B
\ar[l]|(.4){\delta_B} }
\ \ \ \ ,\ \ \ \ \ \pi _{B}\delta _{B}=1_{B}, \end{equation*} satisfying the following property:\newline for every diagram in $\mathbf{B}$ of the form \begin{equation*}
\xymatrix{ A \ar@<0.5ex>[r]^{\alpha} \ar[rd]_{f} & B' \ar@<0.5ex>[l]^{\beta} \\ & B }
\ \ \ ,\ \ \ \alpha \beta =1 \end{equation*} there exists a unique morphism \begin{equation*} f^{\prime }:A\longrightarrow G_{1}\left( B\right) \end{equation*} such that \begin{eqnarray*} \varepsilon _{B}f^{\prime } &=&f \\ \delta _{B}\pi _{B}f^{\prime } &=&f^{\prime }\beta \alpha. \end{eqnarray*} \end{theorem}
\begin{proof} Suppose we have $G_{1},\pi ,\delta ,\varepsilon $ satisfying the required conditions in the Theorem, then the functor \begin{equation*} G(B)=
\xymatrix{ G_1(B) \ar@<0.5ex>[r]^{\pi_B} & B \ar@<0.5ex>[l]^{\delta_B} }
\end{equation*} is a right adjoint to $F$; in fact (see \cite{ML}, p.83, Theorem 2, (iii)) we have functors $F$ and $G$, and a natural transformation $\varepsilon : FG \longrightarrow 1$, such that each $\varepsilon_B : FG(B) \longrightarrow B$ is universal from $F$ to $B$: \begin{equation*}
\xymatrix{ A \ar[d]^{f}
\ar@{}[rd]|{:} & A \ar@<0.5ex>[r]^{\alpha} \ar[d]^{f_1}
\ar@{-->}[rd]|{f} & B \ar@<0.5ex>[l]^{\beta} \ar[d]^{f_0} \\ B' & G_1(B') \ar@<0.5ex>[r]^{\pi_B} & B' \ar@<0.5ex>[l]^{\delta_B} }
\end{equation*} given $f$, there is a unique $f_{1}$ (with $\varepsilon _{B}f^{\prime } = f , \, \delta _{B}\pi _{B}f^{\prime } = f^{\prime }\beta \alpha$) and $f_{0}$ follows as $f_{0}=\pi _{B}f_{1}\beta $; conversely, given $f_{1}$, we find $ f=\varepsilon _{B}f_{1}$.
Now, given an adjunction \begin{equation*} \left( F,G,\eta ,\varepsilon \right) \end{equation*} such that $IG=1$, if writing \begin{equation*} G(B)=
\xymatrix{ G_1(B) \ar@<0.5ex>[r]^{G_2(B)} & B \ar@<0.5ex>[l]^{G_3(B)} }
\end{equation*} we define \begin{equation*} G_{1}=FG\ \ ,\ \ \pi _{B}=G_{2}\left( B\right) \ \ ,\ \ \delta _{B}=G_{3}\left( B\right) \end{equation*} and \begin{equation*} \varepsilon _{B}:FG\left( B\right) \longrightarrow B \end{equation*} is the counit of the adjunction.
Clearly we have natural transformations with $\pi _{B}\delta _{B}=1$.
It remains to check the stated property - but it is simply the universal property of $\varepsilon_B$: given a diagram \begin{equation*}
\xymatrix{ A \ar@<0.5ex>[r]^{\alpha} \ar[rd]_{f} & B' \ar@<0.5ex>[l]^{\beta} \\ & B }
\ \ \ ,\ \ \ \alpha \beta =1 \end{equation*} there is a unique morphism of split epis \begin{equation*}
\xymatrix{ A \ar@<0.5ex>[r]^{\alpha} \ar[d]^{f_1} & B \ar@<0.5ex>[l]^{\beta} \ar[d]^{f_0} \\ G_1(B') \ar@<0.5ex>[r]^{\pi_B} & B' \ar@<0.5ex>[l]^{\delta_B} }
\end{equation*} such that $\varepsilon _{B}f_{1}=f$; being a morphism of split epis means that $f_0 = \pi_B f_1 \beta$, and $f_1$ is such that $\delta_B \pi_B f_1 = f_1 \beta \alpha$. \end{proof}
If $\mathbf{B}$ has binary products, then for every $B\in \mathbf{B}$, \begin{equation*}
\xymatrix@=3pc{ B \times B \ar@<1ex>[r]^{\pi_2} \ar@<-1ex>[r]_{\pi_1} & B
\ar[l]|(.4){<1,1>} }
\end{equation*} satisfies the required conditions and hence $F$ has a right adjoint, $G$, sending the object $B$ to the split epi \begin{equation*}
\xymatrix{ B \times B \ar@<0.5ex>[r]^{\pi_2} & B \ar@<0.5ex>[l]^{<1,1>} }
. \end{equation*} And furthermore, in this case the pair $(I,G)$ is a half-reflection. For the general case, if we ask for $(I,G)$ to be a half-reflection, then the following result suffices.
\begin{corollary} \label{Corolary1}Let $\mathbf{B}$ be any category and $I,F:Pt\left( \mathbf{B }\right) \longrightarrow \mathbf{B}$ as above. If the category $\mathbf{B}$ is equipped with an endofunctor $G_{1}:\mathbf{B}\longrightarrow \mathbf{B}$ and natural transformations \begin{equation*}
\xymatrix{ G_1(B) \ar@<1ex>[r]^{\pi_B} \ar@<-1ex>[r]_{\varepsilon_B} & B
\ar[l]|(.4){\delta_B} }
\ \ \ \ ,\ \ \ \ \ \pi _{B}\delta _{B}=1_{B}=\varepsilon _{B}\delta _{B}, \end{equation*} satisfying the following property:\newline for every diagram in $\mathbf{B}$ of the form \begin{equation*}
\xymatrix{ A \ar[rd]_{f} & A \ar@<0.5ex>[l]^{t} \\ & B }
\ \ \ ,\ \ \ t^{2}=t \end{equation*} there exists a unique morphism \begin{equation*} f^{\prime }:A\longrightarrow G_{1}\left( B\right) \end{equation*} such that \begin{equation*} \pi _{B}f^{\prime }=ft\ ,\ \varepsilon _{B}f^{\prime }=f\ ,\ f^{\prime }t=\delta _{B}ft, \end{equation*} then the functor $F$ has a right adjoint, say $G$, and the pair $(I,G)$ is a half reflection. \end{corollary}
\begin{proof} It is clear that the property is sufficient to obtain $G$ as a right adjoint to $F$ as in the previous Theorem, simply considering $t = \beta\alpha$ and observing that the two conditions $\pi_B f^{\prime}= f t$, $ f^{\prime}t=\delta_B f t$ give $\delta_B\pi_B f^{\prime}= f^{\prime}\beta \alpha$: start with $\pi_B f^{\prime}= f t$, precompose with $\delta_B$, and replace $\delta_B f t$ by $f^{\prime}t$.
As a consequence we have that \begin{equation*} \pi_B f^{\prime}\beta = \varepsilon_B f^{\prime}\beta , \end{equation*} since \begin{eqnarray*} ft=ftt=\pi_B f^{\prime}t= \pi_B f^{\prime}\beta \alpha = \pi_B f^{\prime}\beta \\ ft=\varepsilon_B \delta_B f t = \varepsilon_B f^{\prime}t= \varepsilon_B f^{\prime}\beta \alpha =\varepsilon_B f^{\prime}\beta \end{eqnarray*} and hence, given $f:A\longrightarrow B$, we have $(f_1,f_0)$, with $ f_1=f^{\prime}$ given by the \emph{universal} property and $f_0=f\beta$ ($ =\pi_B f^{\prime}\beta = \varepsilon_B f^{\prime}\beta$).
The pair $\left( I,G\right) $ is a half-reflection with \begin{equation*} \pi :1_{Pt\left( \mathbf{B}\right) }\longrightarrow GI \end{equation*} given by \begin{equation*}
\xymatrix{ A \ar@<0.5ex>[r]^{\alpha} \ar@{-->}[d]^{\delta_B \alpha} & B \ar@<0.5ex>[l]^{\beta} \ar@{=}[d] \\ G_1(B) \ar@<0.5ex>[r]^{\pi_B} & B \ar@<0.5ex>[l]^{\delta_B} }
, \end{equation*} and furthermore this is the only possibility. \end{proof}
\begin{corollary} In the conditions of the above Corollary (and assuming $\mathbf{B}$ is pointed), the kernel of $\pi _{B}$ is the morphism induced by the diagram \begin{equation*}
\xymatrix{ B \ar[rd]_{1} & B \ar@<0.5ex>[l]^{0} \\ & B }
. \end{equation*} \end{corollary}
As mentioned in the previous section, if the canonical adjunction $\left( \ref{canonical adjunction BxB--->Pt(B)}\right) $ is not an equivalence we are interested in considering bigger categories, $\mathbf{A}$, that we will call categories of actions, in the place of $\mathbf{B\times B}$, in order to obtain an equivalence of categories $\mathbf{A}\sim Pt\left( \mathbf{B} \right) $.
We now turn our attention to the category of internal actions in $\mathbf{B}$ .
To define the category of internal actions in $\mathbf{B}$, in the sense of \cite{GJ1}, we only need to assume $\mathbf{B}$ to be pointed, with binary coproducts and kernels of split epis: exactly the same conditions necessary to consider the canonical adjunction $\left( \ref{canonical adjunction BxB--->Pt(B)}\right) $; and the construction of the category of internal actions is actually suggested by the adjunction. This seems to suggest an iterative process to obtain bigger and bigger categories \textquotedblleft of actions", $\mathbf{A}_{1},$ $\mathbf{A}_{2}\ ,...$.
\subsection{The category of internal actions}
Let $\mathbf{B}$ be a pointed category with binary coproducts and kernels of split epis. The category of internal actions in $\mathbf{B}$, denoted $ Act\left( \mathbf{B}\right) $, is defined as follows.
Objects are triples $\left( X,\xi ,B\right) $ where $X$ and $B$ are objects in $\mathbf{B}$ and $\xi :B\flat X\longrightarrow X$ is a morphism such that \begin{eqnarray*} \xi \eta _{X} &=&1 \\ \xi \mu _{X} &=&\xi \left( 1\flat \xi \right) \end{eqnarray*} where the object $B\flat X$ is the kernel, $k:B\flat X\longrightarrow X\sqcup B$ of $[0,1]:X\sqcup B\longrightarrow B$ and $\eta _{X}$, $\mu _{x}$ are induced, respectively, by $\iota _{1}$ and $[k\ \iota _{2}]$. See \cite {GJ1} for more details.
We now have to consider $Act\left( \mathbf{B}\right) $ as an half-reflection, $\left( I,G\right) $ (with $G$ admitting a left adjoint), over $\mathbf{B}$. Clearly we have a functor \begin{equation*}
\xymatrix{ \mathbf{A} \ar[rd]_{I} & & Pt(\mathbf{B}) \ar[ld]^{I'} \ar[ll]_{S} \\ & \mathbf{B} }
\end{equation*} sending a split epi $\left( A,\alpha ,\beta ,B\right) $ to $\left( X,\xi ,B\right) $ as suggested in the following diagram \begin{equation*}
\xymatrix{ B \flat X \ar[r]^{k'} \ar@{-->}[d]_{\xi}
& X \sqcup B \ar@<0.5ex>[r]^{[0 \, 1]} \ar[d]_{[k \, \beta]} & B \ar@<0.5ex>[l]^{\iota_2} \ar@{=}[d] \\ X \ar[r]^{k} & A \ar@<0.5ex>[r]^{\alpha} & B \ar@<0.5ex>[l]^{\beta} }
. \end{equation*} It is well defined because $k$ (being a kernel) is monic and \begin{equation*} k\xi \eta _{X}=[k\ \beta ]k^{\prime }\eta _{X}=[k\ \beta ]\iota _{1}=k \end{equation*} so that $\xi \eta _{X}=1$; a similar argument shows $\xi \mu _{X}=\xi \left( 1\flat \xi \right) $.
To obtain a functor $G:\mathbf{B}\longrightarrow Act\left( \mathbf{B}\right) $ we compose \begin{equation*} \mathbf{B}\longrightarrow Pt\left( \mathbf{B}\right) \longrightarrow Act\left( \mathbf{B}\right) \end{equation*} where $\mathbf{B}\longrightarrow Pt\left( \mathbf{B}\right) $ is the half-reflection of Corollary \ref{Corolary1}; the resulting $G$ sends an object $B\in \mathbf{B}$ to the internal action $\left( B^{\prime }\flat B,\xi _{B},B\right) $ as suggested by the following diagram \begin{equation*}
\xymatrix{ B' \flat B \ar[r]^{\ker} \ar@{-->}[d]_{\xi_B}
& B' \sqcup B \ar@<0.5ex>[r]^{[0 \, 1]} \ar[d]_{[\ker \, \delta_B]} & B \ar@<0.5ex>[l]^{\iota_2} \ar@{=}[d] \\ B' \ar[r]^{\ker} & G_1(B) \ar@<0.5ex>[r]^{\pi_B} & B \ar@<0.5ex>[l]^{\delta_B} }
. \end{equation*} In the case of Groups this corresponds to the action by conjugation (see \cite{GJ1}). The next step is to require that $G$ admits a left adjoint, which in the case of Groups is true and it corresponds to the construction of a semi-direct product from a given action.
For convenience, we will now assume the existence of binary products, instead of the data $G_{1},\pi ,\delta ,\varepsilon $ of Theorem \ref {Theorem 1}.
For the rest of this section, and if not explicitly stated otherwise, we will assume that $\mathbf{B}$ is a pointed category with binary products and coproducts and kernels of split epis.
With such assumptions we automatically consider the half-reflection \begin{equation*} Act\left( \mathbf{B}\right) \overset{I}{\underset{G}{\underleftarrow{ \overrightarrow{\ \ \ \ \ \ \ \ \ \ \ }}}}\mathbf{B\ \ ,\ \ }\pi :1\longrightarrow GI \end{equation*} where $I\left( X,\xi ,B\right) =B$, $G\left( B\right) =\left( B,\xi _{B},B\right) $ obtained from \begin{equation*} \xymatrix{ B \flat B \ar[r]^{k} \ar@{-->}[d]_{\xi_B} & B \sqcup B
\ar@<0.5ex>[r]^{[0 \, 1]} \ar[d]|{[<1,0> \, <1,1>]} & B \ar@<0.5ex>[l]^{\iota_2} \ar@{=}[d] \\ B \ar[r]^{<1,0>} & B \times B \ar@<0.5ex>[r]^{\pi_2} & B \ar@<0.5ex>[l]^{<1,1>} } \end{equation*} (note that $<0,1>$ is the kernel of $\pi _{2}$), and the natural transformation $\pi :1\longrightarrow GI$ given by \begin{equation*}
\xymatrix{ (X,\xi,B) \ar[d]_{(0,1)} \\ (B,\xi_B,B) }
\ \ \ \ \ \ \ \ \ \ \
\quadrado { B \flat X } { \xi } { X } { 1 \flat 0 } { 0 } { B \flat B } { \xi_B } { B }
; \end{equation*} which is well defined because $\xi _{B}\left( 1\flat 0\right) =0$, since $ \left\langle 1,0\right\rangle \xi _{B}\left( 1\flat 0\right) =\left\langle 0,0\right\rangle $: \begin{eqnarray*} \left\langle 1,0\right\rangle \xi _{B}\left( 1\flat 0\right) =[\left\langle 1,0\right\rangle \ \left\langle 1,1\right\rangle ]\left( 0\sqcup 1\right) \ker [0\ 1] = \\ =\left\langle [1\ 1],[0\ 1]\right\rangle \left( 0\sqcup 1\right) \ker [0\ 1]=\left\langle 0,[0\ 1]\right\rangle \ker [0\ 1] =\left\langle 0,0\right\rangle . \end{eqnarray*}
\begin{definition}[semi-direct products] We will say that $\mathbf{B}$ has semidirect products, if the functor $G$ admits a left adjoint. \end{definition}
Note that this is a weaker notion of Bourn-Janelidze categorical semidirect products \cite{JB}, since we are not asking for the induced adjuntion between $Act(\mathbf{B})$ and $Pt(\mathbf{B})$ to be an equivalence of categories.
We now state a sufficient condition for $\mathbf{B}$ to have semidirect products.
\begin{theorem} The functor $G$, in the half-reflection \begin{equation*} Act\left( \mathbf{B}\right) \overset{I}{\underset{G}{\underleftarrow{ \overrightarrow{\ \ \ \ \ \ \ \ \ \ \ }}}}\mathbf{B\ \ ,\ \ }\pi :1\longrightarrow GI, \end{equation*} as above, admits a left adjoint if the category $\mathbf{B}$ has coequalizers of reflexive graphs. \end{theorem}
\begin{proof} Given an object $\left( X,\xi ,B\right) $, consider the reflexive graph \begin{equation*}
\xymatrix{ (B \flat X) \sqcup B \ar@<1ex>[r]^{[k \, \iota_2]} \ar@<-1ex>[r]_{\xi \sqcup 1} & X \sqcup B
\ar[l]|(.4){\eta \sqcup 1} }
. \end{equation*} The left adjoint, $F$, is given by the coequalizer of $[k\ \iota _{2}]$ and $ \xi \sqcup 1$: \begin{equation*}
\xymatrix{ (B \flat X) \sqcup B \ar@<1ex>[r]^{[k \, \iota_2]} \ar@<-1ex>[r]_{\xi \sqcup 1} & X \sqcup B
\ar[l]|(.4){\eta \sqcup 1} \ar[r]^{\sigma} & F(X,\xi,B) }
. \end{equation*} See \cite{GJ1} for more details. \end{proof}
Let us from now on assume that $\mathbf{B}$ is a pointed category with binary products and coproducts and kernels of split epis, and coequalizers of reflexive graphs.
The next step is to consider the canonical functor \begin{equation*} Act\left( \mathbf{B}\right) \overset{T}{\longrightarrow }Pt\left( \mathbf{B} \right) \end{equation*} sending $\left( X,\xi ,B\right) $ to the split epi $\left( F\left( X,\xi ,B\right) ,\overline{[0\ 1]},\sigma \iota _{2}\right) $ where $\overline{[0\ 1]}$ is such that $\overline{[0\ 1]}\sigma =[0\ 1]$, investigate whether it is an equivalence of categories and study internal precategories and reflexive graphs in $\mathbf{B}$.
First we show that under the given assumptions, it is always an adjunction.
\begin{theorem} The functors \begin{equation*} Act\left( \mathbf{B}\right) \overset{T}{\underset{S}{\underleftarrow{ \overrightarrow{\ \ \ \ \ \ \ \ \ \ \ }}}}Pt\left( \mathbf{B}\right) \end{equation*} as defined above, form an adjoint situation. \end{theorem}
\begin{proof} Consider the following diagram \begin{equation}
\quadrado { B \flat X } { \xi } { X } { f_0 \flat g } { g } { B' \flat X' } { \xi_{A}' } { X' }
\ \ \ \ \ :\ \ \ \
\xymatrix{ X \ar[r]^{\sigma \iota_1} \ar[d]_{g}
& F(X,B) \ar@<0.5ex>[r]^{\overline{[0\ 1]}} \ar[d]_{f_1} & B \ar@<0.5ex>[l]^{\sigma \iota_2} \ar[d]^{f_0} \\ X' \ar[r]^{k} & A \ar@<0.5ex>[r]^{\alpha} & B' \ar@<0.5ex>[l]^{\beta} }
\label{diagram1} \end{equation} where $\left( X,\xi ,B\right) $ is an object in $Act\left( \mathbf{B}\right) $ and $\left( X^{\prime },\xi _{A}^{\prime },B^{\prime }\right) $ is $ S\left( A,\alpha ,\beta ,B^{\prime }\right) $.
Given $\left( f_{1},f_{0}\right) $, since $k$ is the kernel of $\alpha $ and \begin{equation*} \alpha f_{1}\sigma \iota _{1}=f_{0}\overline{[0\ 1]}\sigma \iota _{1}=f_{0}[0\ 1]\iota _{1}=0 \end{equation*} we obtain $g$ as the unique morphism such that $kg=f_{1}\sigma \iota _{1}$.
To prove that the pair $\left( g,f_{0}\right) $ is a morphism in $Act\left( \mathbf{B}\right) $, that is, the left hand square in $\left( \ref{diagram1} \right) $ commutes, we have to show \begin{equation*} \xi _{A}^{\prime }\left( f_{0}\flat g\right) =g\xi \end{equation*} and we do the following: first observe that $k\xi _{A}^{\prime }\left( f_{0}\flat g\right) =f_{1}\sigma k^{\prime \prime }$, in fact (see diagram below, where $k^{\prime }$ and $k^{\prime \prime }$ are kernels) \begin{equation*}
\xymatrix{ B \flat X \ar[rr]^{k''} \ar[dd]_{f_0 \flat g} \ar[rd]^{\xi} & & X \sqcup B \ar@<0.5ex>[rr]^{[0\ 1]} \ar[dd]_(.3){g \sqcup f_0} \ar[rd]^{\sigma} && B \ar@<0.5ex>[ll]^{\sigma \iota_2} \ar[dd]^{f_0} \ar@{=}[rd] \\ & X \ar[rr]^{\sigma \iota_1} \ar[dd]_{g}
&& F(X,B) \ar@<0.5ex>[rr]^(.7){\overline{[0\ 1]}} \ar[dd]_{f_1} && B \ar@<0.5ex>[ll]^{\sigma \iota_2} \ar[dd]^{f_0} \\ B' \flat X' \ar[rr]^{k''} \ar[rd]_{\xi'_A} && X' \sqcup B' \ar@<0.5ex>[rr]^(.7){[0\ 1]} \ar[rd]_{[k \, \beta]} && B' \ar@<0.5ex>[ll]^(.3){\iota_2} \ar@{=}[rd] \\ & X' \ar[rr]^{k} && A \ar@<0.5ex>[rr]^{\alpha} && B' \ar@<0.5ex>[ll]^{\beta} }
\end{equation*} \begin{eqnarray*} k\xi _{A}^{\prime }\left( f_{0}\flat g\right) &=&[k\ \beta ]k^{\prime }\left( f_{0}\flat g\right) \text{\ ,\ definition of }\xi _{A}^{\prime } \\ &=&[k\ \beta ]\left( g \sqcup f_{0}\right) k^{\prime \prime } \\ &=&[kg\ \beta f_{0}]k^{\prime \prime } \end{eqnarray*} and \begin{eqnarray*} f_{1}\sigma k^{\prime \prime } &=&f_{1}\sigma \lbrack \iota _{1}\ \iota _{2}]k^{\prime \prime } \\ &=&[f_{1}\sigma \iota _{1}\ f_{1}\sigma \iota _{2}]k^{\prime \prime } \\ &=&[kg\ \beta f_{0}]k^{\prime \prime }; \end{eqnarray*} we also have $kg\xi =f_{1}\sigma \iota _{1}\xi $,\ by definition of $g$. The result follows from the fact that $k$ is monic and \begin{equation*} \sigma k^{\prime \prime }=\sigma \iota _{1}\xi , \end{equation*} which follows from $\left( \ref{equivariance condition}\right) $ by taking $ f=\sigma \iota _{1}$ and $g=\sigma \iota _{2}$.
Conversely, given $g$ and $f_{0}$ such that the left hand square in $\left( \ref{diagram1}\right) $ commutes, we find $f_{1}=\overline{[kg\ \beta f_{0}]} $, which is well defined because (see $\ref{equivariance condition}$ below) \begin{equation*} kg\xi =[kg\ \beta f_{0}]k^{\prime \prime }, \end{equation*} indeed we have \begin{eqnarray*} kg\xi &=&k\xi _{A}^{\prime }\left( f_{0}\flat g\right) \\ &=&[k\ \beta ]k^{\prime }\left( f_{0}\flat g\right) \\ &=&[k\ \beta ]\left( g+f_{0}\right) k^{\prime \prime } \\ &=&[kg\ \beta f_{0}]k^{\prime \prime }. \end{eqnarray*} \end{proof}
In what follows we will need the following.
To give a morphism \begin{equation*} F\left( X,\xi ,B\right) \longrightarrow B^{\prime } \end{equation*} is to give a pair $\left( f,g\right) $ with $f:X\longrightarrow B^{\prime }$ and $g:B\longrightarrow B^{\prime }$ such that \begin{equation*} \lbrack f\ g][k\ \iota _{2}]=[f\ g]\left( \xi \sqcup 1\right) \end{equation*} or equivalently \begin{equation} \lbrack f\ g]k=f\xi . \label{equivariance condition} \end{equation} See \cite{GJ1} \ for more details.
\begin{theorem} Let $\mathbf{B}$ be a pointed category with binary products and coproducts, kernels of split epis and coequalizers of reflexive graphs. If the canonical functor \begin{equation*} Act\left( \mathbf{B}\right) \overset{T}{\longrightarrow }Pt\left( \mathbf{B} \right) \end{equation*} is an equivalence, then: \begin{equation*} RG\left( \mathbf{B}\right) \sim \text{Pre-X-Mod}\left( \mathbf{B}\right) \end{equation*} \begin{equation*} PC\left( \mathbf{B}\right) \sim \text{2-ChainComp}\left( \mathbf{B}\right) . \end{equation*} The objects in Pre-X-Mod$\left( \mathbf{B}\right) $ are pairs $\left( h,\xi \right) $ with $h:X\longrightarrow B$ \ a morphism in $\mathbf{B}$ and $\xi :B\flat X\longrightarrow X$ an action $\left( X,\xi ,B\right) $ in $ Act\left( \mathbf{B}\right) $ satisfying the following condition \begin{equation*} \lbrack h\ 1]k=h\xi , \end{equation*} with $k:B\flat X\longrightarrow X+B$ the kernel of $[0\ 1];$\newline The objects in 2-ChainComp$\left( \mathbf{B}\right) $ are sequences \begin{equation*} Z\overset{t}{\longrightarrow }X\overset{h}{\longrightarrow }B\ \ \ ,\ \ \ ht=0 \end{equation*} together with actions \begin{eqnarray*} \xi _{X} &:&B\flat X\longrightarrow X \\ \xi _{Z} &:&X\flat Z\longrightarrow Z \\ \xi _{F(Z,X)} &:&F(X,B)\flat F(Z,X)\longrightarrow F(Z,X) \end{eqnarray*} subject to the following conditions \begin{eqnarray*} \lbrack h\ 1]k_{X} &=&h\xi _{X} \\ \lbrack t\ 1]k_{Z} &=&t\xi _{Z} \\ \lbrack \sigma \iota _{1}\overline{[t\ 1]}\ 1]k_{F\left( Z,X\right) } &=&\sigma \iota _{1}\overline{[t\ 1]}\xi _{F\left( Z,X\right) } \\ \overline{\lbrack 0\ 1]}\xi _{F\left( Z,X\right) } &=&\xi _{X}\left( \overline{[h\ 1]}\flat \overline{[0\ 1]}\right) \\ \sigma \iota _{2}\xi _{X} &=&\xi _{F\left( Z,X\right) }\left( \sigma \iota _{2}\flat \sigma \iota _{2}\right) . \end{eqnarray*} \end{theorem}
\begin{proof} Using the equivalence $T$, a reflexive graph in $\mathbf{B}$ is of the form \begin{equation*}
\xymatrix{ F(X,B) \ar@<1ex>[r]^{\overline{[0 \, 1]}} \ar@<-1ex>[r]_{c} & B
\ar[l]|{\sigma \iota_2} }
\ \ \ \ c\sigma \iota _{2}=1. \end{equation*} By definition of $F\left( X,B\right) $ we have \begin{equation*}
\xymatrix{ X \ar[r]^{\sigma \iota_1} \ar[rd]_{h=c \sigma \iota_1} & F(X,B) \ar[d]^{c} & B \ar[l]_{\sigma \iota_2} \ar@{=}[ld] \\ & B }
\end{equation*} and the pair $\left( h,1\right) $ induces $c=\overline{[h\ 1]}$ if and only if \begin{equation*} \lbrack h\ 1]k=h\xi . \end{equation*}
For a precategory, observing that a split square $\left( \ref{split quare 1} \right) $ is in fact a split epi in $Pt\left( \mathbf{B}\right) $ and using the equivalence $Act\left( \mathbf{B}\right) \sim Pt\left( \mathbf{B}\right) $ we have that every such split square is of the form \begin{equation*}
\xymatrix{ F(Y, F(X,B)) \ar[r]^{\overline{[0 \, 1]}} \ar[d]_{F(a, \overline{[h,1]})} & F(X,B) \ar@<1ex>[l]^{\sigma \iota_2} \ar[d]_{\overline{[h \, 1]}} \\ F(X,B) \ar[r]^{\overline{[0 \, 1]}} \ar@<-1ex>[u]_{F(b ,\sigma \iota_2)} & B \ar@<1ex>[l]^{\sigma \iota_2} \ar@<-1ex>[u]_{\sigma \iota_2} }
, \end{equation*} and hence giving such a split square is to give internal actions $\left( X,\xi ,B\right) $ and $\left( Y,\xi ^{\prime },F\left( X,B\right) \right) $ together with morphisms $a,b,h$ such that the following squares commute \begin{equation*}
\xymatrix{ F(X,B) \flat Y \ar[r]^{\xi'} \ar@<-0.5ex>[d]_{\overline{[h \, 1]} \flat a} & Y \ar@<-0.5ex>[d]_{a} \\ B \flat X \ar@<-0.5ex>[u]_{\sigma \iota_2 \flat b} \ar[r]^{\xi} & X \ar@<-0.5ex>[u]_{b} \ar[r]^{h} & B }
\end{equation*} and \begin{equation*} \lbrack h\ 1]k=h\xi . \end{equation*} It remains to insert the morphism \begin{equation*} m:F\left( Y,F\left( X,B\right) \right) \longrightarrow F\left( X,B\right) \end{equation*} satisfying the following conditions \begin{eqnarray} m\sigma \iota _{2} &=&1 \label{eq4} \\ mF\left( b,\sigma \iota _{2}\right) &=&1 \label{eq5} \\ \overline{\lbrack h\ 1]}m &=&\overline{[h\ 1]}F\left( a,\overline{[h\ 1]} \right) \label{eq6} \\ \overline{\lbrack 0\ 1]}m &=&\overline{[0\ 1]}\overline{[0\ 1]}. \label{eq7} \end{eqnarray} From $\left( \ref{eq4}\right) $ we conclude that $m=\overline{[v\ 1]}$ for some $v:Y\longrightarrow F\left( X,B\right) $ such that \begin{equation*} \lbrack v\ 1]k^{\prime }=v\xi ^{\prime }. \end{equation*} Using $\left( \ref{eq7}\right) $ we conclude that $\overline{[0\ 1]}v=0$ so that $v$ factors through the kernel of $\overline{[0\ 1]}$, which is $\sigma \iota _{1}$ because $T$ is an equivalence, and finally we have \begin{equation*} m=\overline{[\sigma \iota _{1}u\ 1]} \end{equation*} for some $u:Y\longrightarrow X$ such that \begin{equation*} \lbrack \sigma \iota _{1}u\ 1]k^{\prime }=\sigma \iota _{1}u\xi ^{\prime }. \end{equation*} Condition $\left( \ref{eq5}\right) $ gives $ub=1$ while condition $\left( \ref{eq6}\right) $ gives $ha=hu$: \begin{align*} mF\left( b,\sigma \iota _{2}\right) & =1\Leftrightarrow \overline{[\sigma \iota _{1}u\ 1]}F\left( b,\sigma \iota _{2}\right) =1\Leftrightarrow \overline{[\sigma \iota _{1}u\ 1]}\overline{[\sigma \iota _{1}b\ \sigma \iota _{2}\sigma \iota _{2}]}=1\Leftrightarrow \\ & \Leftrightarrow \overline{[\sigma \iota _{1}u\ 1]}[\sigma \iota _{1}b\ \sigma \iota _{2}\sigma \iota _{2}]=\sigma \Leftrightarrow \lbrack \sigma \iota _{1}ub\ \ \sigma \iota _{2}]=[\sigma \iota _{1}\ \sigma \iota _{2}]\Leftrightarrow \\ & \Leftrightarrow \sigma \iota _{1}ub=\sigma \iota _{1}\Leftrightarrow ub=1; \end{align*} \begin{eqnarray*} \overline{\lbrack h\ 1]}m &=&\overline{[h\ 1]}F\left( a,\overline{[h\ 1]} \right) \Leftrightarrow \overline{[h\ 1]}\overline{[\sigma \iota _{1}u\ 1]}= \overline{[h\ 1]}\overline{[\sigma \iota _{1}a\ \ \sigma \iota _{2}\overline{ [h\ 1]}]}\Leftrightarrow \\ &\Leftrightarrow &\overline{[h\ 1]}[\sigma \iota _{1}u\ 1]=\overline{[h\ 1]} [\sigma \iota _{1}a\ \ \sigma \iota _{2}\overline{[h\ 1]}]\Leftrightarrow \\ &\Leftrightarrow &[hu\ \overline{[h\ 1]}]=[ha\ \overline{[h\ 1]} ]\Leftrightarrow hu=ha. \end{eqnarray*}
Conclusion 1: A precategory in $\mathbf{B}$ is given by the following data \begin{equation}
\xymatrix{ F(X,B) \flat Y \ar[r]^{\xi'} \ar@<-0.5ex>[d]_{\overline{[h \, 1]} \flat a} & Y \ar@<-0.5ex>[d]_{a} \ar@/^1pc/[d]^{u} \\ B \flat X \ar@<-0.5ex>[u]_{\sigma \iota_2 \flat b} \ar[r]^{\xi} & X \ar@<-0.5ex>[u]_{b} \ar[r]^{h} & B }
\label{diagram2} \end{equation} such that \thinspace $\xi $,$\xi ^{\prime }$ are internal actions, the obvious squares commute, and the following conditions are satisfied \begin{eqnarray} hu &=&ha \label{eq8 and folowings} \\ ub &=&1=ab \notag \\ \lbrack \sigma \iota _{1}u\ 1]k^{\prime } &=&\sigma \iota _{1}u\xi ^{\prime } \notag \\ \lbrack h\ 1]k &=&h\xi . \notag \end{eqnarray}
We now continue to investigate it further and replace the split epi $\left( Y,a,b,X\right) $ with an action $\left( Z,\xi _{Z},B\right) $. For convenience we will also rename $\xi _{X}:=\xi $, $k_{X}:=k$, $\xi _{F(Z,X)}:=\xi ^{\prime }$, $k_{F\left( Z,X\right) }=k^{\prime }$. The diagram $\left( \ref{diagram2}\right) $ becomes \begin{equation*}
\xymatrix{ F(X,B) \flat F(Z,X) \ar[r]^{\xi_{F(Z,X)}} \ar@<-0.5ex>[d]_{\overline{[h \, 1]} \flat \overline{[0 \, 1]}} & F(Z,X) \ar@<-0.5ex>[d]_{\overline{[0 \, 1]}} \ar@/^2pc/[d]^{\overline{[t \, 1]}} \\ B \flat X \ar@<-0.5ex>[u]_{\sigma \iota_2 \flat \sigma \iota_2} \ar[r]^{\xi_X} & X \ar@<-0.5ex>[u]_{\sigma \iota_2} \ar[r]^{h} & B }
\end{equation*} for some $t:Z\longrightarrow X$ such that $[t\ 1]k_{Z}=t\xi _{Z}$. The commutativity of the appropriate squares in the diagram above plus the reinterpretation of conditions $\left( \ref{eq8 and folowings}\right) $ gives the stated result. \end{proof}
\begin{remark} In order to consider a reflexive graph $\left( h:X\longrightarrow B,\xi :B\flat X\longrightarrow X\right) $ as a precategory of the form \begin{equation*}
\xymatrix{ F(X,B) \flat X \ar[r]^{\xi (\overline{[h \, 1]} \flat 1)} \ar@<-0.5ex>[d]_{\overline{[h \, 1]} \flat 1} & X \ar@<-0.5ex>[d]_{1} \ar@/^1pc/[d]^{1} \\ B \flat X \ar@<-0.5ex>[u]_{\sigma \iota_2 \flat 1} \ar[r]^{\xi} & X \ar@<-0.5ex>[u]_{1} \ar[r]^{h} & B }
\end{equation*} we need, in addition to $[h\ 1]k=h\xi $ that \begin{equation*} \lbrack \sigma \iota _{1}\ 1]k^{\prime }=\sigma \iota _{1}\xi \left( \overline{[h\ 1]}\flat 1\right) \end{equation*} where $k:B\flat X\longrightarrow X\sqcup B$ and $k^{\prime }:F\left( X,B\right) \flat X\longrightarrow X\sqcup F\left( X,B\right) $ are the kernels of $[0\ 1].$\newline In the case of Groups, this corresponds to the Peiffer identity that distinguishes a precrossed module from a crossed module. \end{remark}
We now give a characterization of categories $\mathbf{B}$ such that $ Act\left( \mathbf{B}\right) \sim Pt\left( \mathbf{B}\right) .$
\begin{theorem} Let $\mathbf{B}$ be a pointed category with binary products and coproducts, kernels of split epis and coequalizers of reflexive graphs. The canonical functor \begin{equation*} Act\left( \mathbf{B}\right) \overset{T}{\longrightarrow }Pt\left( \mathbf{B} \right) \end{equation*} is an equivalence if and only if the following two properties holds in $ \mathbf{B}$:\newline (A1) for every diagram of the form \begin{equation}
\xymatrix{ X \ar[r]^{\sigma \iota_1} & F(X, B) \ar@<0.5ex>[r]^{\overline{[0 \, 1]}} & B \ar@<0.5ex>[l]^{\sigma \iota_2} }
\end{equation} the morphism $\sigma \iota _{1}$ is the kernel of $\overline{[0\ 1]}$; \newline (A2) the split short five lemma holds. \end{theorem}
\begin{proof} Similar to Theorem \ref{Th 2}. \end{proof}
\section{The general case}
Let \begin{equation*} \mathbf{A}\overset{I}{\underset{G}{\underleftarrow{\overrightarrow{\ \ \ \ \ \ \ \ \ \ \ }}}}\mathbf{B\ \ ,\ \ }\pi :1\longrightarrow GI \end{equation*} be a half-reflection.
Define a new category, denoted by $\mathbf{A}_{1}$\ as follows:\newline Objects are pairs $\left( A,u\right) $ with $A\in \mathbf{A}$ and \begin{equation*} u:A\longrightarrow GIA \end{equation*} such that $I\left( u\right) =1.$\newline A morphism $f:\left( A,u\right) \longrightarrow \left( A^{\prime },u^{\prime }\right) $ is a morphism $f:A\longrightarrow A^{\prime }$ in $\mathbf{A}$ such that \begin{equation*}
\quadrado { A } { u } { GIA } { f } { GIf } { A' } { u' } { GIA' }
. \end{equation*}
Define another category, denoted $\mathbf{A}_{2}$, as follows:\newline Objects are systems \begin{equation*} \left( \left( E,v\right) ,a,b,\left( A,u\right) \right) \end{equation*} where $\left( E,v\right) $ and $\left( A,u\right) $ are objects in $\mathbf{A }_{1}$, \begin{equation*} a:\left( E,v\right) \longrightarrow \left( A,u\right) \end{equation*} is a morphism in $\mathbf{A}_{1}$, and \begin{equation*} b:A\longrightarrow E \end{equation*} is a morphism in $\mathbf{A}$ such that \begin{equation*} ab=1_{A}. \end{equation*}
Let $\mathbf{A}\overset{T}{\longrightarrow }$Pt$\left( \mathbf{B}\right) $ be any subcategory of Pt$\left( \mathbf{B}\right) $, not necessarily full, we may consider the subcategories of reflexive graphs and internal precategories in $\mathbf{B}$, restricted to split epis in $T\left( \mathbf{A }\right) $, and denote it respectively by $RG_{\mathbf{A}}\left( \mathbf{B} \right) $ and $PC_{\mathbf{A}}\left( \mathbf{B}\right) $.
In particular if the functor $G$, as above, admits a left adjoint $\left( F,G,\eta ,\varepsilon \right) $ and $F$ is faithful and injective on objects, then the canonical functor \begin{equation*} \mathbf{A}\overset{T}{\longrightarrow }Pt\left( \mathbf{B}\right) \end{equation*} determines a subcategory of split epis and so we have:\newline Reflexive graphs internal to $\mathbf{B}$ and restricted to the split epis in $T\left( \mathbf{A}\right) ,$ denoted $RG_{\mathbf{A}}\left( \mathbf{B} \right) $ as follows: \begin{equation*}
\xymatrix{ FA \ar@<1ex>[rr]^{\varepsilon_{IA} F(\pi_A)} \ar@<-1ex>[rr]_{c} & & IA
\ar[ll]|{I(\eta_A)} }
\ \ \ \ ,\ \ \ cI\left( \eta _{A}\right) =1_{IA}; \end{equation*} for some $A\in \mathbf{A}$.\newline Internal precategories in $\mathbf{B}$ relative to split epis in $T\left( \mathbf{A}\right) $, denoted by $PC_{\mathbf{A}}\left( \mathbf{B}\right) $, as follows (where we use $\pi _{A}^{\prime }$ as an abbreviation to $ \varepsilon _{IA}F\left( \pi _{A}\right) $ and similarly to $\pi _{E}^{\prime }$) \begin{equation}
\xymatrix{ F(E) \ar@<3ex>[rr]^{\pi'_E} \ar@<-3ex>[rr]_{F(a)}
\ar[rr]|{m} & & IE=FA
\ar@<-1.5ex>[ll]|{I(\eta_E)}
\ar@<1.5ex>[ll]|{F(b)} \ar@<1.5ex>[rr]^{\pi'_A} \ar@<-1.5ex>[rr]_{c} & & IA
\ar[ll]|{I(\eta_A)} }
\label{FE,IE=FA,IA} \end{equation} for some \begin{equation*}
\xymatrix{ E \ar@<1ex>[r]^{a} & A \ar[l]^{b} }
\ \ \ ,\ \ \ ab=1_{A} \end{equation*} in $\mathbf{A}$, and satisfying the following conditions \begin{eqnarray} cI\left( \eta _{A}\right) &=&1_{IA} \label{c1} \\ I\left( a\right) &=&c \label{c2} \\ I\left( b\right) &=&I\left( \eta _{A}\right) \label{c3} \\ mI\left( \eta _{E}\right) &=&1_{IE} \label{c4} \\ mF\left( b\right) &=&1_{IE} \label{c5} \\ cm &=&cF\left( a\right) \label{c6} \\ \pi _{A}^{\prime }m &=&\pi _{A}^{\prime }\pi _{E}^{\prime }. \label{c7} \end{eqnarray}
We observe that $c$ is determined by $a$, and $\left( \ref{c1}\right) $ follows from $\left( \ref{c2}\right) $, $\left( \ref{c3}\right) $ and the fact that $ab=1_{A}$. We will be also interested in the notion of multiplicative graph, which is obtained by removing $\left( \ref{c6}\right) $ and $\left( \ref{c7}\right) $ and in some cases we may be also interested in removing $\left( \ref{c5}\right) $ so that the definition may be transported from $\mathbf{B}$ to $\mathbf{A}$ and it does not depend on whether or not $ G $ admits a left adjoint.
\begin{theorem} For a half-reflection \begin{equation*} \mathbf{A}\overset{I}{\underset{G}{\underleftarrow{\overrightarrow{\ \ \ \ \ \ \ \ \ \ \ }}}}\mathbf{B\ \ ,\ \ }\pi :1\longrightarrow GI, \end{equation*} if the functor $G$ admits a left adjoint \begin{equation*} \left( F,G,\eta ,\varepsilon \right) , \end{equation*} and $F$ is faithful and injective on objects, then \begin{eqnarray} \mathbf{A}_{1} &\cong &RG_{\mathbf{A}}\left( \mathbf{B}\right) \label{iso1} \\ \mathbf{A}_{2}^{\ast } &\cong &PC_{\mathbf{A}}\left( \mathbf{B}\right) \label{iso2} \end{eqnarray} where $\mathbf{A}_{2}^{\ast }$ is the subcategory of $\mathbf{A}_{2}$ given by the objects \begin{equation*} \left( \left( E,v\right) ,a,b,\left( A,u\right) \right) \end{equation*} such that \begin{eqnarray*} IE &=&FA \\ I\left( a\right) &=&\varepsilon _{IA}F\left( u\right) \\ vb &=&\eta _{A} \\ G\left( \varepsilon _{IA}F\left( \pi _{A}\right) \right) \pi _{E} &=&G\left( \varepsilon _{IA}F\left( \pi _{A}\right) \right) v. \end{eqnarray*} \end{theorem}
\begin{proof} The isomorphism $\left( \ref{iso1}\right) $ is established by the adjuntion $ \left( F,G,\eta ,\varepsilon \right) $. Given \begin{equation*} A\overset{u}{\longrightarrow }GIA\ \ ,\ \ I\left( u\right) =1_{IA} \end{equation*} we obtain \begin{equation}
\xymatrix{ FA \ar@<1ex>[r]^{\pi'_A} \ar@<-1ex>[r]_{u'} & IA
\ar[l]|{I(\eta_A)} }
\label{FA,IA} \end{equation} where $\pi _{A}^{\prime }=\varepsilon _{IA}F\left( \pi _{A}\right) $, $ u^{\prime }=\varepsilon _{IA}F\left( u\right) $ and \begin{equation*} u^{\prime }I\left( \eta _{A}\right) =1_{IA}\Leftrightarrow I\left( u\right) =1. \end{equation*} Conversely, given $\left( \ref{FA,IA}\right) $, we obtain $A$, since $F$ is injective on objects, and \begin{equation*} u=G\left( u^{\prime }\right) \eta _{A}. \end{equation*} The isomorphism $\left( \ref{iso2}\right) $ is obtained as follows:\newline Given $\left( \ref{FE,IE=FA,IA}\right) $, since $F$ in injective on objects and faithful, we obtain \begin{equation*}
\xymatrix{ E \ar@<1ex>[r]^{a} & A \ar[l]^{b} }
\ \ \ ,\ \ \ ab=1_{A}\ \ \ \ ,\ \ \ IE=FA. \end{equation*} Now define on the one hand \begin{equation*} u=G\left( c\right) \eta _{A}\ \ ,\ \ v=G\left( m\right) \eta _{E}; \end{equation*} while on the other hand \begin{equation*} c=\varepsilon _{IA}F\left( u\right) \ \ ,\ \ m=\varepsilon _{IE}F\left( v\right) , \end{equation*} and we have the following translation of equations \begin{equation*}
\begin{tabular}{|c|c|c|} \hline Eq. n.
${{}^o}$
& in $\mathbf{B}$ & in $\mathbf{A}$ \\ \hline \ref{c1} & $cI\left( \eta _{A}\right) =1_{IA}$ & $I\left( u\right) =1_{IA}$ \\ \hline \ref{c4} & $mI\left( \eta _{E}\right) =1_{IE}$ & $I\left( v\right) =1_{IE}$ \\ \hline \ref{c6} & $cm=cF\left( a\right) $ & $ua=GI\left( a\right) v$ \\ \hline \ref{c2} & $I\left( a\right) =c$ & $I\left( a\right) =\varepsilon _{IA}F\left( u\right) $ \\ \hline \begin{tabular}{c} \ref{c3} \\ \ref{c5} \end{tabular} & \begin{tabular}{c} $I\left( b\right) =I\left( \eta _{A}\right) $ \\ $mF\left( b\right) =1_{IE}$ \end{tabular} & $vb=\eta _{A}$ \\ \hline \ref{c7} & $\pi _{A}^{\prime }m=\pi _{A}^{\prime }\pi _{E}^{\prime }$ & $ G\left( \pi _{A}^{\prime }\right) \pi _{E}=G\left( \pi _{A}^{\prime }\right) v$ \\ \hline \end{tabular} \end{equation*} Note that $ua=GI\left( a\right) v$ follows from the fact that $a$ is a morphism in $\mathbf{A}_{1}$, on the contrary of $b$ which is simply a morphism in $\mathbf{A}$. \end{proof}
In some cases we also have a functor \begin{equation*} J:\mathbf{A}\longrightarrow \mathbf{B} \end{equation*} satisfying the following three conditions:
\begin{enumerate} \item $JG=1_{\mathbf{B}}$
\item the pair $\left( J\left( \eta _{A}\right) ,I\left( \eta _{A}\right) \right) $ is jointly epic for every $A\in \mathbf{A}$, that is, given a pair of morphisms $\left( f,g\right) $ as displayed below \begin{equation*}
\xymatrix{ JA \ar[r]^{J(\eta_A)} \ar[rd]_{f} & FA \ar@{-->}[d]^{[f \, g]} & IA \ar[l]_{I(\eta_A)} \ar[ld]^{g} \\ & B & }
\end{equation*} there is at most one morphism $\alpha :FA\longrightarrow B$, with the property that $\alpha J\left( \eta _{A}\right) =f$ and $\alpha I\left( \eta _{A}\right) =g$, denoted by $\alpha =[f,g]$ when it exists. Also the pair $ \left( f,g\right) $ is said to be admissible (or cooperative in the sense of Bourn and Gran \cite{Bourn&Gran}) w.r.t. $A$, if $[f,g]$ exists.
\item for every $A,E\in \mathbf{A}$, with $IE=FA$, a morphism, $u:J\left( E\right) \longrightarrow FA$, such that $\left( u,1_{IE}\right) $ is cooperative w.r.t. $E$ and satisfying $\pi _{A}[u\ 1]=\pi _{A}\pi _{E}$, always factors trough $J\left( A\right) $, i.e., given $u$ as in the diagram below \begin{equation*}
\xymatrix{ JE \ar[r]^{J(\eta_E)} \ar[rd]^{u} \ar@{-->}[d]_{\bar{u}} & FE \ar@<0.5ex>[r]^{\pi'_E} \ar@{..>}[d]^{[u \, 1]} & IE
\ar@<0.5ex>[l]|{I(\eta_E)} \ar@{=}[ld] \\ JA \ar[r]^{J(\eta_A)}
& FA \ar[r]^{\pi'_A}
& IA }
\end{equation*} such that $[u\ 1]$ exists and $\pi _{A}[u\ 1]=\pi _{A}\pi _{E}$ then $ u=J\left( \eta _{A}\right) u^{\prime }$ for a unique $u^{\prime }:JE\longrightarrow JA$. \end{enumerate}
\begin{theorem} Let $\mathbf{B}$ be any category, with $\left( I,G,\pi \right) $ \begin{equation*} \mathbf{A}\overset{I}{\underset{G}{\underleftarrow{\overrightarrow{\ \ \ \ \ \ \ \ \ \ \ }}}}\mathbf{B\ \ ,\ \ }\pi :1\longrightarrow GI, \end{equation*} a half-reflection such that the functor $G$ admits a left adjoint \begin{equation*} \left( F,G,\eta ,\varepsilon \right) . \end{equation*} If we can find a functor \begin{equation*} J:\mathbf{A}\longrightarrow \mathbf{B} \end{equation*} as above, then
\begin{itemize} \item the category $RG_{\mathbf{A}}\left( \mathbf{B}\right) $ of reflexive graphs in $\mathbf{B}$ relative to split epis from $\mathbf{A}$, is given by: \newline Objects are pairs $\left( A,h\right) ,$ with$\ A\in \mathbf{A},$ and $ h:JA\longrightarrow IA$ a morphism such that $\left( h,1_{IA}\right) $ is cooperative w.r.t. $A$;\newline A morphism $f:\left( A,h\right) \longrightarrow \left( A^{\prime },h^{\prime }\right) $ is a morphism $f:A\longrightarrow A^{\prime }$ in $\mathbf{A}$ such that $h^{\prime }F\left( f\right) =I\left( f\right) h$.
\item the category of internal precategories in $\mathbf{B}$ relative to split epis from $\mathbf{A}$, $PC_{\mathbf{A}}\left( \mathbf{B}\right) $, is given by:\newline Objects: \begin{equation*} \left( A,E,a,b,t,h\right) \end{equation*} where $A,E$, are objects in $\mathbf{A}$ , with $IE=FA$, $a,b,t,h,$ are morphisms in $\mathbf{B}$, \begin{equation*}
\xymatrix{ JE \ar@<1ex>[r]^{a} \ar@<-1ex>[r]_{t} & JA
\ar[l]|{b} \ar[r]^{h} & IA }
\end{equation*} such that \begin{equation*} ab=1=tb\ \ ,\ \ \ ha=ht \end{equation*} and \begin{eqnarray*} &&\left( h,1_{IA}\right) \ \ \text{and }\left( J\left( \eta _{E}\right) b,I\left( \eta _{E}\right) I\left( \eta _{A}\right) \right) \ \ \ \text{are cooperative w.r.t. }A \\ &&\left( J\left( \eta _{A}\right) a,I\left( \eta _{A}\right) [h\ 1]\right) \ \ \text{and }\left( J\left( \eta _{A}\right) t,1_{I\left( E\right) }\right) \ \ \ \text{are cooperative w.r.t. }E \end{eqnarray*} Morphisms are triples $\left( f_{3},f_{2},f_{1}\right) $ of morphisms \begin{equation*}
\xymatrix{ JE \ar@<1ex>[r]^{a} \ar@<-1ex>[r]_{t} \ar[d]^{f_3} & JA
\ar[l]|{b} \ar[r]^{h} \ar[d]^{f_2} & IA \ar[d]^{f_1} \\ JE' \ar@<1ex>[r]^{a'} \ar@<-1ex>[r]_{t'} & JA'
\ar[l]|{b'} \ar[r]^{h'} & IA' }
\end{equation*} such that the obvious squares in the above diagram commute and furthermore the pair $\left( J\left( \eta _{A^{\prime }}\right) f_{2},I\left( \eta _{A^{\prime }}\right) f_{1}\right) $ is admissible w.r.t. $A$ while the pair $\left( J\left( \eta _{E^{\prime }}\right) f_{3},I\left( \eta _{E^{\prime }}\right) [J\left( \eta _{A^{\prime }}\right) f_{2}\ \ I\left( \eta _{A^{\prime }}\right) f_{1}]\right) $ is admissible w.r.t. $E$. \end{itemize} \end{theorem}
\begin{proof} Calculations are similar to the previous sections and the resulting diagram $ \left( \ref{FE,IE=FA,IA}\right) $ is given by \begin{eqnarray*} c &=&[h\ 1] \\ m &=&[J\left( \eta _{A}\right) t\ \ 1_{I\left( E\right) }] \\ F\left( b\right) &=&[J\left( \eta _{E}\right) b\ \ I\left( \eta _{E}\right) I\left( \eta _{A}\right) ] \\ F\left( a\right) &=&[J\left( \eta _{A}\right) a\ \ I\left( \eta _{A}\right) [h\ 1]]. \end{eqnarray*} The same argument applies to obtain the morphisms. \end{proof}
\subsection{The example of unitary magmas with right cancellation}
An example of a general situation in the conditions of the above theorem is the following one.
Let $\mathbf{B}$ be a pointed category with kernels of split epis, with binary products and coproducts and such that the pair $\left( \left\langle 1,0\right\rangle ,\left\langle 1,1\right\rangle \right) $, as displayed \begin{equation*}
\xymatrix{ B \ar[r]^{ <1,0> } & B \times B \ar@<0.5ex>[r]^{\pi'_2} & B \ar@<0.5ex>[l]^{ <1,1> } }
\end{equation*} is jointly epic for every $B\in \mathbf{B}$, and then consider: $\mathbf{A}$ , the full subcategory of Pt$\left( \mathbf{B}\right) $ given by the split epis with the property that \begin{equation*}
\xymatrix{ X \ar[r]^{ \ker \alpha} & A \ar@<0.5ex>[r]^{\alpha} & B \ar@<0.5ex>[l]^{ \beta } }
\end{equation*} the pair $\left( \ker \alpha ,\beta \right) $ is jointly epic (identifying $ \left( A,\alpha ,\beta ,B\right) $ with $\left( A^{\prime },\alpha ^{\prime },\beta ^{\prime },B\right) $ whenever $A\cong A^{\prime }$, in order to obtain $F$ injective on objects).
Then we have functors \begin{eqnarray*} I,F,J &:&\mathbf{A}\longrightarrow \mathbf{B} \\ G &:&\mathbf{B}\longrightarrow \mathbf{A} \end{eqnarray*} with \begin{eqnarray*} I\left( A,\alpha ,\beta ,B\right) &=&B \\ F\left( A,\alpha ,\beta ,B\right) &=&A \\ J\left( A,\alpha ,\beta ,B\right) &=&X\ \ ,\text{the object kernel of }\alpha \\ G\left( B\right) &=&\left( B\times B,\pi _{2},\left\langle 1,1\right\rangle ,B\right) \end{eqnarray*} and with $\pi :1_{\mathbf{A}}\longrightarrow GI$ given by $\pi =[0\ 1]$.
An example of such a category is the category of unitary magmas with right cancellation. Also every strongly unital category satisfies the above requirements (see \cite{BB}, and references there).
\end{document} | arXiv |
\begin{document}
\parindent 0pc \parskip 6pt \overfullrule=0pt
\title[Extremal solution for singular p-Laplace equations]{Regularity of the extremal solution for singular p-Laplace equations}
\date{July 1, 2014} \author{Daniele Castorina} \thanks{Daniele Castorina, Dipartimento di Matematica, Universit\'a di Padova, Via Trieste 63, 35121 Padova, Italy. e-mail: [email protected]}
\begin{abstract} We study the regularity of the extremal solution $u^*$ to the singular reaction-diffusion problem $-\Delta_p u = \lambda f(u)$ in $\Omega$, $u =0$ on $\partial \Omega$, where $1<p<2$, $0 < \lambda < \lambda^*$, $\Omega \subset \mathbb{R}^n$ is a smooth bounded domain and $f$ is any positive, superlinear, increasing and (asymptotically) convex $C^1$ nonlinearity. We provide a simple proof of known $L^r$ and $W^{1,r}$
\textit{a priori} estimates for $u^*$, i.e. $u^* \in L^\infty(\Omega)$ if $n \leq p+2$, $u^* \in L^{\frac{2n}{n-p-2}}(\Omega)$ if $n > p+2$ and $|\nabla u^*|^{p-1} \in L^{\frac{n}{n-(p'+1)}} (\Omega)$ if $n > p p'$. \end{abstract}
\maketitle
\section{Introduction and main result}\label{intro}
\setcounter{equation}{0}
The aim of this paper is the study of the following quasilinear reaction-diffusion problem:
\begin{equation}\label{problem} \left\{ \begin{array}{rcll} -\Delta_p u &=& \lambda f(u) &\textrm{in } \Omega, \\
u&>& 0 &\textrm{in } \Omega, \\
u &=& 0 &\textrm{on } \partial \Omega, \end{array} \right. \end{equation}
where the diffusion is driven by the singular p-Laplace operator $\Delta_p u:= {\rm div}(|\nabla u|^{p-2}\nabla u)$, $1<p<2$, $\Omega$ is a smooth bounded domain of ${\mathbb{R}}^n$, $n\geq 2$, $\lambda$ is a positive parameter and the nonlinearity $f$ is any $C^1$ positive increasing function satisfying
\begin{equation}\label{p-superlinear} \lim_{t\rightarrow+\infty}\frac{f(t)}{t^{p-1}}=+\infty. \end{equation}
Typical reaction terms $f$ satisfying the above assumptions are given by the exponential $e^u$ and the power $(1+u)^m$ with $m>p-1$.
These reaction-diffusion problems appear in numerous models in physics, chemistry and biology. In particular, when $p=2$ and $f(u)$ is the exponential, $\eqref{problem}_{\lambda}$ is usually referred to as the \textit{Gelfand problem}: it arises as a simplified model in a number of interesting physical contexts. For example, up to dimension $n=3$, equation $\eqref{problem}_{\lambda}$ can be derived from the thermal self-ignition model which describes the reaction process in a combustible material during the ignition period. We refer the interested reader to \cite{BE,FK} for the detailed derivation of the model, as well as other physical motivations for this problem. In the case of singular nonlinearities such as $f(u)=(1-u)^{-2}$, problem $\eqref{problem}_{\lambda}$ is also relevant as a model equation to describe Micro Electro Magnetic System (MEMS) devices theory (see \cite{PeBe} for a complete account on this subject). Regarding the MEMS equation, the compactness of the minimal branch of solutions and some spectral issues connected with it were investigated in \cite{CES1} for general $p>1$ and nonlinearities $f(u)$, singular at $u=1$, with growth comparable to $(1-u)^{-m}$, $m>0$.\\
In order to set up the problem, we will say that a (nonnegative) function $u\in W^{1,p}_0(\Omega)$ is a \emph{weak energy solution} of $\eqref{problem}_{\lambda}$ if $f(u)\in L^1(\Omega)$ and $u$ satisfies
$$
\int_\Omega |\nabla u|^{p-2}\nabla u\cdot\nabla \varphi\ dx= \lambda \int_\Omega f(u)\varphi\ dx\quad \textrm{for all } \varphi\in C_0^1(\Omega). $$
Moreover, if $f(u)\in L^\infty(\Omega)$ we say that $u$ is a \emph{regular solution} of $\eqref{problem}_{\lambda}$. By standard regularity results for non-uniformly elliptic equations, one has that every regular solution $u$ belongs to $C^{1,\alpha} (\overline{\Omega})$ for some $0<\alpha<1$ (see \cite{DB,Lie,T}).\\
Under the above hypotheses, problem $\eqref{problem}_{\lambda}$ has been extensively studied for $p=2$. Crandall and Rabinowitz \cite{CR} prove the existence of an extremal parameter $\lambda^*\in (0,+\infty)$ such that: if $\lambda<\lambda^*$ then problem $\eqref{problem}_{\lambda}$ admits a regular solution $u_\lambda$ which is minimal among all other possible solutions, and if $\lambda>\lambda^*$ then problem $\eqref{problem}_{\lambda}$ admits no regular solution. Moreover, they prove that every minimal solution $u_\lambda$ is semi--stable in the sense that the second variation of the energy functional associated to $\eqref{problem}_{\lambda}$ is nonnegative definite. Subsequently, Brezis and V\'azquez \cite{BreVaz97} prove that the pointwise increasing limit of minimal solutions given by
\begin{equation}\label{eq5} u^*:=\lim_{\lambda\uparrow\lambda^*}u_\lambda, \end{equation}
is a weak solution of $\eqref{problem}_{\lambda}$, usually known as \textit{extremal solution}. Apart from a detailed study of the model cases (exponential and power nonlinearities), they also raise some interesting open problems for general nonlinearities $f$ satisfying the above assumptions. One of the most challenging questions proposed, which has not been answered completely yet, is to show show that the extremal solution $u^*$ is bounded or in the energy class, depending on the range of dimensions. In this direction, Nedev \cite{Nedev} proved, in the case of convex nonlinearities, that $u^*\in L^\infty(\Omega)$ if $n\leq 3$ and $u^*\in L^r(\Omega)$ for all $1\leq r<n/(n-4)$ if $n\geq 4$. Subsequently, Cabr\'e~\cite{Cabre09}, Cabr\'e and Sanch\'on \cite{CS}, and Nedev \cite{Nedev01} proved, in the case of convex domains and general nonlinearities, that $u^*\in L^\infty(\Omega)$ if $n\leq 4$ and $u^*\in L^{\frac{2n}{n-4}}(\Omega)\cap H^1_0(\Omega)$ if $n\geq 5$. More recently, Villegas \cite{Vi} extends Nedev result to $n=4$ thanks to a clever use of the \textit{a priori} estimates of Cabr\'e~\cite{Cabre09}, without convexity assumptions on the domain.\\
While for the standard Laplacian the regularity of the extremal solution has been subject of a rich literature, in the case of the p-Laplacian, i.e. for $p\ne2$, the available results are very few. Garc\'{\i}a Azorero, Peral and Puel \cite{GarPe92,GarPePu94} study $\eqref{problem}_{\lambda}$ when $f(u)=e^u$, obtaining the existence of the family of minimal regular solutions $(\lambda,u_\lambda)$ for $\lambda\in (0,\lambda^*)$ and that $u^*$ is a weak energy solution independently of $n$. If in addition $n<p+4p/(p-1)$ then $u^*\in L^\infty({\Omega})$. Moreover, they prove that $\lambda^*=p^{p-1}(n-p)$
and $u^*(x)={\rm log}(1/|x|^p)$ when $\Omega=B_1$ and $n\geq p+4p/(p-1)$. Cabr\'e and Sanch\'on \cite{CS07} proved the existence of an extremal parameter $\lambda^\star\in(0,\infty)$ such that problem $\eqref{problem}_{\lambda}$ admits a minimal regular solution $u_\lambda\in C^1_0(\overline{\Omega})$ for $\lambda\in(0,\lambda^*)$ and admits no regular solution for $\lambda>\lambda^*$. Moreover, every minimal solution $u_\lambda$ is semi-stable for $\lambda\in (0,\lambda^*)$.
One of the main difficulties is the fact that for arbitrary $p>1$ it is unknown if the limit of minimal solutions $u^*$ is a (weak or entropy) solution of $\eqref{problem}_{\lambda^*}$. In the affirmative case, it is called the \textit{extremal solution} of $\eqref{problem}_{\lambda^*}$. However, in \cite{S} Sanch\'on has proved that the limit of minimal solutions $u^*$ is a weak solution (in the distributional sense) of $\eqref{problem}_{\lambda^*}$ whenever $p\geq 2$ and $(f(t)-f(0))^{1/(p-1)}$ is convex for $t$ sufficiently large. Essentially under the same hypotheses, Bidaut-Veron and Hamid in \cite{BH} are able to show that $u^*$ is a locally renormalized solution of $\eqref{problem}_{\lambda^*}$ in the singular case $p < 2$.\\
In this paper, in the spirit of the clever proof of \cite{Vi} for the case $p=2$, we extend some of the results of \cite{CaSa} for the degenerate p-Laplacian to the singular case $1<p<2$. We obtain the boundedness of the extremal solution up to a critical dimension $n_p = p+2$ while we prove that it belongs to $L^{\frac{2n}{n-p-2}}(\Omega)$ if $n > p+2$, for any smooth bounded domain $\Omega$, under a standard (asymptotic) convexity assumption on the nonlinearity. Unfortunately our dimensional $n_p$ is not optimal in the singular case ($n_p < p p'$ for $1<p<2$), hence we are not able to match the one obtained in \cite{BH} for this case. However, the regularity results in \cite{BH} for the singular case are rather involved while our alternative proof is very simple and direct. In higher dimensions, we will establish in a different way the same Sobolev regularity obtained in \cite{S} for the degenerate case. Our main result is the following:\\
\begin{thm}\label{teorema} Suppose that $1<p<2$ and let $f$ be a positive, increasing and superlinear $C^1$ nonlinearity such that $f^{\frac{1}{p-1}}(t)$ is convex for any $t \geq T$. Let $u^*$ be the extremal solution of \eqref{problem} and set $n_p := p+2$. The following assertions hold:
$(a)$ If $n \leq n_p$ then $u^* \in L^\infty(\Omega)$. In particular, $u^*$ is a regular solution to $\eqref{problem}_{\lambda^*}$.
$(b)$ If $n > n_p$ then $u^* \in L^{\frac{2n}{n-p-2}}(\Omega)$.
$(c)$ If $n> p p'$ then $|\nabla u^*|^{p-1} \in L^{\frac{n}{n-(p'+1)}} (\Omega)$. \end{thm}
We will recall several auxiliary results as well as giving the proof of Theorem \ref{teorema} in the next section. The main details of the proof of the technical lemmas can be found in the Appendix, section \ref{appendix}.
\section{Proof of Theorem \ref{teorema}}\label{proof}
\setcounter{equation}{0}
Let us discuss a few preliminary results which will be used. First of all, the proof of Theorem \ref{teorema} relies on the semistability of the minimal solution $u_\lambda$ for $0 < \lambda < \lambda^* $.
Recall that the linearization $L_{u_{\lambda}}$ associated to $\eqref{problem}_\lambda$ at a given solution $u_{\lambda}$ is defined as
\begin{eqnarray*}
L_{u_{\lambda}} (v,\varphi) &:=& \int_\Omega |\nabla u_\lambda|^{p-2}(\nabla v,\nabla\varphi)\\
&&+(p-2)\int_\Omega |\nabla u_\lambda|^{p-4}(\nabla u_\lambda,\nabla v)(\nabla u_\lambda,\nabla\varphi) - \int_\Omega \lambda f'(u_\lambda)v \varphi. \nonumber \end{eqnarray*}
for test functions $v, \varphi \in C^1_{c} (\Omega)$. Observe that the above linearization, in the degenerate case $1<p<2$, makes sense
if $|\nabla u_\lambda|^{p-2} \in L^1(\Omega)$, which has been proved by Damascelli and Sciunzi in \cite{DS}. We then say that a solution of $\eqref{problem}_\lambda$ is {\em semistable} if the linearized operator at $u_{\lambda}$ is nonnegative definite, i.e. $L_{u_{\lambda}}(\varphi,\varphi) \geq 0$ for any $\varphi \in C^1_{c} (\Omega)$. Equivalently, $u_{\lambda}$ is semistable if the first eigenvalue of $L_{u_{\lambda}}$ in $\Omega$, $\mu_1 (L_{u_{\lambda}},\Omega)$, is nonnegative. However, let us observe that for $p \ne 2$ the latter definition of semistability requires the spectral theory for $L_u$ that has been established by Esposito, Sciunzi and the author in \cite{CES4}.
Next, we will need two \textit{a priori} estimates for the family of minimal solutions $u_{\lambda}$, $0 < \lambda < \lambda^*$. For the reader's convenience a sketch of the proofs of both auxiliary results can be found in the Appendix (section \ref{appendix}).
The first estimate gives uniform $L^\infty$ and $L^r$ bounds for $u_\lambda$ in terms of the $W^{1,p+2}_0$ norm in a neighborhood of the boundary, depending on the dimension. It can be directly derived from the \textit{a priori} estimates for semistable solutions contained in Theorem 1.4 of \cite{CaSa}, where Sanch\'on and the author extend to the case $p>2$ the regularity results of \cite{Cabre09,CS} for $u^*$ in convex domains.
\begin{lem}\label{lemma1} Let $u_\lambda$ be the minimal solution of $\eqref{problem}_\lambda$. Then the following alternatives hold:
$(a)$ If $n\leq p+2$ then there exists a constant $C$ depending only on $n$ and $p$ such that
\begin{equation}\label{L-infinty}
\|u_\lambda\|_{L^\infty(\Omega)}\leq s+\frac{C}{s^{2/p}}|\Omega|^\frac{p+2-n}{np}
\left(\int_{\{u_\lambda\leq s\}} |\nabla u_\lambda|^{p+2}\, dx\right)^{1/p}\quad \textrm{for all }s>0. \end{equation}
$(b)$ If $n>p+2$ then there exists a constant $C$ depending only on $n$ and $p$ such that
\begin{equation}\label{Lq:estimate} \left(\int_{\{u_\lambda>s\}} \Big(u_\lambda-s\Big)^{\frac{np}{n-(p+2)}}\ dx\right)^{\frac{n-(p+2)}{np}} \leq \frac{C}{s^{2/p}}
\left(\int_{\{u_\lambda\leq s\}} |\nabla u_\lambda|^{p+2} \ dx\right)^{1/p} \end{equation}
for all $s>0$. \end{lem}
The second preliminary result is a uniform $L^1$ estimate for $(f(u_\lambda))^{p'}/u_\lambda$ when the power $\frac{1}{p-1}$ of the nonlinearity is asymptotically convex. This bound is a direct consequence of the estimates which have been proved in Proposition 5.28 of \cite{BH}.
\begin{lem}\label{lemma2} Let $u_\lambda$ be the minimal solution of $\eqref{problem}_\lambda$ and suppose that $f^{\frac{1}{p-1}}(t)$ is convex for any sufficiently large $t$. Then there exists a constant $M$ independent of $\lambda \in (0,\lambda^*)$ such that
\begin{equation}\label{L-p'} \int_{\{ u_\lambda > 1 \}} \frac{(f(u_\lambda))^{p'}}{u_\lambda} \, dx \leq M. \end{equation} \end{lem}
Finally, we will need a uniform gradient estimate which we can derive from the regularity results for the linear problem. Let us consider
\begin{equation}\label{eq8} \begin{cases} -\Delta_p u = g(x)\quad \textrm{ in }\Omega\\ \qquad u=0\qquad \textrm{ on }\partial \Omega \end{cases} \end{equation}
where $g\in L^m(\Omega)$ for some $m>1$. The following result gives the regularity of the gradient of a weak energy solution of (\ref{eq8}) (see for instance \cite{ABFOT}).
\begin{lem}\label{lem3} Suppose that $n>q$ and let $q^*:=\frac{nq}{n-q}$ be the usual critical Sobolev exponent. Let $u$ be a weak energy solution of \eqref{eq8}. Then there exists a constant $C$ depending on $n$, $p$,
$q$ and $|\Omega|$ such that:$$\||\nabla u|^{p-1}\|_{L^{q^*}
(\Omega)} \leq C \|g\|_{L^q (\Omega)}.$$ \end{lem}
\begin{rem} Notice that for $p<2$ it might happen that $(p-1) q^* <1$. Hence the Sobolev norm $W_{0}^{1,(p-1)q^*}$ of the solution $u$ might not make sense:
however, for the sake of simplicity, we will still keep this notation intending throughout the discussion that: $$\| u
\|_{W^{1,(p-1)q^*}_0} := \| |\nabla u|^{p-1}
\|_{L^{q^*}}^{\frac{1}{p-1}}$$ \end{rem}
We are now ready to prove the regularity statements for $u^*$ given in Theorem \ref{teorema}.\\
{\bf Proof of Theorem \ref{teorema} part (a).}\\
Let us begin by noticing that a trivial consequence of \eqref{L-infinty} is
\begin{equation}\label{L-infinty2}
\|u_\lambda \|_{L^\infty(\Omega)} \leq s+\frac{C}{s^{2/p}}|\Omega|^\frac{p+2-n}{np} \left(\int_{\Omega} |\nabla u_\lambda |^{p+2}\, dx\right)^{1/p} \end{equation}
where the constant $C$ is given by Lemma \ref{lemma1}. Setting $A:= \| u \|_{W^{1,p+2}_0 (\Omega)}$, for any $s>0$ consider the RHS of \eqref{L-infinty2} as given by the function
$$\Phi (s) := s+C A^{\frac{p+2}{p}} s^{-\frac{2}{p}}.$$
By explicit computation we see that
$$\Phi'(s) = 1 - \frac{2C}{p} A^{\frac{p+2}{p}} s^{-\frac{p+2}{p}}$$
Notice that $\Phi$ is a strictly convex function with a unique global minimum at
$$s = \left(\frac{2C}{p} \right)^{\frac{p}{p+2}} A.$$
By direct substitution we see that
$$\Phi\left(\left(\frac{2C}{p}\right)^{\frac{p}{p+2}} A \right) = \left[\left(\frac{2C}{p}\right)^{\frac{p}{p+2}} + \frac{C^{\frac{4-p}{2-p}} p^\frac{2}{p-2}}{2^\frac{2}{p-2}}\right] A$$
In particular, by the estimate \eqref{L-infinty2} we have thus deduced that for any $0 < \lambda < \lambda^*$ there exists a positive constant $D$ independent of $\lambda$ such that we have
\begin{equation}\label{a1}
\|u_\lambda \|_{L^\infty(\Omega)}\leq D \| u_\lambda \|_{W^{1,p+2}_0 (\Omega)} \end{equation}
Now let us observe that since $u_\lambda$ is weak energy solution of $\eqref{problem}_\lambda$, from Lemma \ref{lem3} we know that there exists a positive constant $E$ such that:
\begin{equation}\label{a2}
\| u_\lambda \|_{W^{1,p+2}_0 (\Omega)} \leq E \| \lambda f(u_\lambda)
\|_{L^{q} (\Omega)}^{\frac{1}{p-1}} \end{equation}
for $q = \frac{n(p+2)}{(p-1)n+p+2}$. Thus, taking into account \eqref{L-p'}, \eqref{a1} and \eqref{a2} and recalling that $f$ is increasing, we obtain that for any $0 < \lambda < \lambda^*$ the following chain of inequalities holds:
$$
\|u_\lambda \|_{L^\infty(\Omega)}^{q(p-1)} \leq D^{q(p-1)} \|
u_\lambda \|_{W^{1,p+2}_0 (\Omega)}^{q(p-1)} \leq (D E)^{q(p-1)} \|
\lambda f(u_\lambda) \|_{L^{q} (\Omega)}^q $$
$$ \leq (D E)^{q(p-1)} (\lambda^*)^q \left( \int_{\{u_\lambda \leq 1\}} (f(u_\lambda))^q \, dx + \int_{\{ u_\lambda > 1 \}} (f(u_\lambda))^q \, dx \right) $$
$$
\leq C \left( (f(1))^q |\Omega| + \int_{\{ u_\lambda > 1 \}} \frac{f(u_\lambda)^q}{u_{\lambda}^{\frac{q}{p'}}} u_{\lambda}^{\frac{q}{p'}} \, dx \right) $$
$$
\leq C \left( (f(1))^q |\Omega| + \|u_\lambda
\|_{L^\infty(\Omega)}^{\frac{q}{p'}} |\Omega|^{\frac{q}{p'-q}} \left(\int_{\{ u_\lambda > 1 \}} \frac{(f(u_\lambda))^{p'}}{u_\lambda} \, dx\right)^{\frac{q}{p'}} \right) $$
$$
\leq C \left( (f(1))^q |\Omega| + M^{\frac{q}{p'}}
|\Omega|^{\frac{q}{p'-q}} \|u_\lambda
\|_{L^\infty(\Omega)}^{\frac{q}{p'}} \right) $$
with $C:=(D E)^{q(p-1)} (\lambda^*)^q$. Let us point out that in the third line of the above calculations we have applied Holder inequality under the condition $q < p'$, which is true for $n < q_p := \frac{p(p+2)}{2(p-1)}$ and in fact $n \leq n_p < q_p$ for $1<p<2$.
Therefore there exist two positive constants $A$ and $B$ independent of $\lambda$ such that:
$$
\|u_\lambda \|_{L^\infty(\Omega)}^{q(p-1)} \leq A + B \|u_\lambda
\|_{L^\infty(\Omega)} ^{\frac{q}{p'}} $$
Observing that $q(p-1) > q/p'$ for $p>1$, the above inequality implies that $u_\lambda$ is uniformly bounded in $L^\infty(\Omega)$ and, taking the limit as $\lambda \uparrow \lambda^*$, we see that $u^* \in L^\infty (\Omega)$. This concludes the proof of statement (a).\\
{\bf Proof of Theorem \ref{teorema} part (b).}\\
Observe that thanks to \eqref{Lq:estimate} for any $s>0$ we have:
$$\int_{\Omega} |u_\lambda|^{\frac{np}{n-(p+2)}} \, dx = \int_{\{u_\lambda \leq s\}} |u_\lambda|^{\frac{np}{n-(p+2)}} \, dx + \int_{\{u_\lambda>s\}} |u_\lambda|^{\frac{np}{n-(p+2)}} \, dx$$
$$= \int_{\{u_\lambda \leq s\}} |u_\lambda|^{\frac{np}{n-(p+2)}} \, dx + \int_{\{u_\lambda>s\}} |(u_\lambda-s)+s|^{\frac{np}{n-(p+2)}} \, dx$$
$$\leq s^{\frac{np}{n-(p+2)}} |\{u_\lambda \leq s\}| + s^{\frac{np}{n-(p+2)}} |\{u_\lambda>s\}| + \int_{\{u_\lambda>s\}} (u_\lambda-s)^{\frac{np}{n-(p+2)}} \, dx$$
$$\leq s^{\frac{np}{n-(p+2)}} |\Omega| + \frac{C}{s^{\frac{2n}{n-(p+2)}}} \left(\int_{\{u_\lambda\leq s\}} |\nabla u_\lambda|^{p+2} \ dx\right)^{\frac{n}{ n-(p+2)}} $$
From the above chain of inequalities we easily deduce that
\begin{equation}\label{Lq:estimate2}
\|u_\lambda \|_{L^{\frac{np}{n-(p+2)}}(\Omega)}^{\frac{np}{n-(p+2)}} \leq s^{\frac{np}{n-(p+2)}} |\Omega| +\frac{C}{s^{\frac{2n}{n-(p+2)}}} \left(\int_{\Omega} |\nabla u_\lambda |^{p+2}\, dx\right)^{\frac{n}{ n-(p+2)}} \end{equation}
At this point, we note that the RHS of the above inequality \eqref{Lq:estimate2}, up to multiplicative constant depending only on $|\Omega|$ and a change of variables $t = s^{\frac{np}{n-(p+2)}}$, is given by the same function $\Phi (t)$ which has been optimized at the beginning of the previous proof. Then, in order to prove part $(b)$ of the theorem, we can proceed essentially as in the proof of part $(a)$ using \eqref{Lq:estimate2} in place of \eqref{L-infinty2}.\\
{\bf Proof of Theorem \ref{teorema} part (c).}\\
Let us now suppose that $n > p p'$. Applying again Lemma \ref{lem3}, but this time with exponent $q = \frac{n(p-1)}{n(p-1)-p}$ (which spells $q^*=\frac{n}{n-(p'+1)}$), we see that:
$$
\| u_\lambda \|_{W^{1,(p-1)q^*}_0 (\Omega)} \leq E \| \lambda f(u_\lambda) \|_{L^{q} (\Omega)}^{\frac{1}{p-1}} $$ Proceeding exactly as in the proof of part $(a)$ and applying Holder inequality (notice that for $n > p p'$ and $p < 2$ we have that $q < p'$) and Sobolev inequality we obtain: $$
\| u_\lambda \|_{W^{1,(p-1)q^*}_0 (\Omega)}^{q(p-1)} \leq E^{q(p-1)}
(\lambda^*)^q \left( (f(1))^q |\Omega| + \int_{\{ u_\lambda > 1 \}} \frac{f(u_\lambda)^q}{u_{\lambda}^{\frac{q}{p'}}} u_{\lambda}^{\frac{q}{p'}} \, dx\right) $$
$$
\leq E^{q(p-1)} (\lambda^*)^q \left( (f(1))^q |\Omega| + M^{\frac{q}{p'}} \left(\int_\Omega u_{\lambda}^{\frac{q}{p'-q}} \right)^{\frac{p'-q}{p'}} \right) $$
$$
\leq E^{q(p-1)} (\lambda^*)^q \left( (f(1))^q |\Omega| +
M^{\frac{q}{p'}} S^{} \| u_\lambda \|_{W^{1,(p-1)q^*}_0 (\Omega)}^{\frac{q}{p'}} \right) $$
Once again there exist two positive constants $A$ and $B$ independent of $\lambda$ such that:
$$
\| u_\lambda \|_{W^{1,(p-1)q^*}_0 (\Omega)}^{q(p-1)} \leq A + B \|
u_\lambda \|_{W^{1,(p-1)q^*}_0 (\Omega)}^{\frac{q}{p'}} $$
We thus see that $u_\lambda$ is uniformly bounded in $W^{1,(p-1)q^*}_0 (\Omega)$ and, taking the limit as $\lambda \uparrow \lambda^*$, we get that $u^* \in W^{1,(p-1)q^*}_0 (\Omega)$. This is exactly statement (c) of Theorem \ref{teorema}, so the proof is done. \qed
\section{Appendix}\label{appendix}
{\bf Sketch of the proof of Lemma \ref{lemma1}.}
Let us recall that the semistabilty of $u_\lambda$ reads as
\begin{equation}\label{semistab}
\int_\Omega |\nabla u_\lambda|^{p-2}|\nabla \varphi|^2 +(p-2)\int_\Omega |\nabla u_\lambda|^{p-4}(\nabla u_\lambda,\nabla\varphi)^2 - \int_\Omega \lambda f'(u_\lambda)\varphi^2 \geq 0 \end{equation}
for any $\varphi \in C^{1}_{c} (\Omega)$. Considering $\phi = |\nabla u_\lambda| \eta$ as a test function in \eqref{semistab}, we obtain
\begin{equation}\label{StZu}
\int_{\Omega}\left[ (p-1) |\nabla u_\lambda|^{p-2} |\nabla_{T,u_\lambda} |\nabla u_\lambda||^{2}
+ B_{u_\lambda}^2 |\nabla u_\lambda|^{p} \right] \eta^2 \, dx
\leq (p-1) \int_{\Omega} |\nabla u_\lambda|^{p} |\nabla \eta|^2 \, dx \end{equation}
for any Lipschitz continuous function $\eta$ with compact support. Here $\nabla_{T,v}$ is the tangential gradient along a level set of $|v|$ while $B_v^2$ denotes the $L^2$-norm of the second fundamental form of the level set of $|v|$ through $x$. The fact that $\phi=|\nabla u_\lambda|\eta $ is an admissible test function as well as the computations behind \eqref{StZu} can be found in \cite{FSV0} (see also Theorem 1 in \cite{FSV}).
On the other hand, noting that $(n-1) H_v^{2} \leq B_v^2$ (with $H_v (x)$ denoting the mean curvature at $x$ of the hypersurface $\{y\in\Omega:|v(y)|=|v(x)|\}$), and
$$
|\nabla u_\lambda|^{p-2} |\nabla_{T,u_\lambda} |\nabla u_\lambda||^{2} = \frac{4}{p^2} |\nabla_{T,u_\lambda} |\nabla u_\lambda|^{\frac{p}{2}}|^{2}, $$
we obtain the key inequality
\begin{equation}\label{StZu2}
\int_{\Omega}\left( \frac{4}{p^2}|\nabla_{T,u_\lambda} |\nabla u_\lambda|^{p/2}|^{2}
+ \frac{n-1}{p-1}H_{u_\lambda}^2 |\nabla u_\lambda|^{p} \right) \eta^2 \, dx
\leq \int_{\Omega} |\nabla u_\lambda|^{p} |\nabla \eta|^2 \, dx \end{equation}
for any Lipschitz continuous function $\eta$ with compact support. By taking $\eta=T_s u_\lambda=\min\{s,u_\lambda\}$ in the semistability condition \eqref{StZu2} we obtain
$$
\int_{\{u_\lambda>s\}}\left( \frac{4}{p^2}|\nabla_{T,u_\lambda} |\nabla u_\lambda|^{p/2}|^{2}
+ \frac{n-1}{p-1}H_{u_\lambda}^2 |\nabla u_\lambda|^{p} \right)\, dx
\leq \frac{1}{s^2}\int_{\{u_\lambda<s\}} |\nabla u_\lambda|^{p+2}\, dx $$
for a.e. $s>0$. In particular,
$$ \min\left(\frac{4}{(n-1)p},1\right) I_{p}(u_\lambda-s;\{x\in\Omega:u_\lambda>s\})^p \leq
\frac{p-1}{(n-1)s^2}\int_{\{u_\lambda<s\}} |\nabla u_\lambda|^{p+2}\, dx $$
for a.e. $s>0$, where $I_p$ is the functional defined as follows
\begin{equation*} I_{p}(v;\Omega):=\left( \int_{\Omega}
\Big(\frac{1}{p'}|\nabla_{T,v} |\nabla v|^{p/2}|\Big)^{2}
+ |H_v|^2 |\nabla v|^p \, dx \right)^{1/p},\quad p \geq 1 \end{equation*}
Then, the $L^\infty$ and $L^r$ estimates of Lemma \ref{lemma1} follow directly from the Morrey and Sobolev type inequalities involving $I_p$ proved in \cite{CaSa}, namely taking $v:=u_\lambda-s$ in
\begin{equation*}
\|v\|_{L^\infty(\Omega)}
\leq C_1|\Omega|^{\frac{p+2-n}{np}}I_{p}(v;\Omega) \end{equation*}
if $n < p+2$, for some constant $C_1 = C_1 (n,p)$, or
\begin{equation*}
\|v\|_{L^r(\Omega)}
\leq C_2|\Omega|^{\frac{1}{r}-\frac{n-(p+2)}{np}} I_{p}(v;\Omega)\quad \textrm{for every }1\leq r \leq \frac{np}{n-(p+2)}, \end{equation*}
if $n>p+2$, where $C_2 = C_2 (n,p,r)$. The borderline case $n=p+2$ is slightly more involved, but we are still able to prove a Morrey type inequality as for the case $n <p+2$ (see page 22 in \cite{CaSa} for the details). Lemma \ref{lemma1} is thus proved. \qed
{\bf Sketch of the proof of Lemma \ref{lemma2}.}
Define $\psi(t):= (f(t)-f(0))^{\frac{1}{p-1}}$. By our assumptions we have that $\psi$ is increasing, convex for $t$ sufficiently large and superlinear at infinity. Choosing $\varphi=\psi(u_\lambda)$ in the semistability condition \eqref{semistab} and observing that $(p-1)\psi(u_\lambda)^p \psi'(u_\lambda) = f'(u_\lambda) \psi(u_\lambda)^2$, we have
\begin{equation}\label{ned1}
\lambda \int_{\Omega} \psi(u_\lambda)^p \psi' (u_\lambda) \leq \int_{\Omega} |\nabla u_\lambda|^p \psi' (u_\lambda)^2 \end{equation}
On the other hand, multiplying \eqref{problem} by $g(u_\lambda):= \int_{0}^{u_\lambda} \psi'(s)^2 \, ds$ and integrating by parts in $\Omega$ we get:
\begin{equation}\label{ned2}
\int_{\Omega} |\nabla u_\lambda|^p \psi' (u_\lambda)^2 = \lambda \int_{\Omega} (f(u_\lambda) - f(0)) g(u_\lambda) + \lambda f(0) \int_{\Omega} g(u_\lambda) \end{equation}
Comparing \eqref{ned1} and \eqref{ned2} it is then easy to see that:
\begin{equation}\label{ned3} \int_{\Omega} \psi(u_\lambda)^{p-1} h(u_\lambda) \leq f(0) \int_{\Omega} g(u_\lambda) \end{equation}
where $h(t) := \int_{0}^{t} (\psi'(t)-\psi'(s)) \psi'(s) \, ds$. Now, thanks to the fact that $h(t) \gg \psi'(t)$ for $t$ large (see page 765 in \cite{BH}), we deduce from \eqref{ned3} that there exists a constant $C$ independent of $\lambda$ such that
\begin{equation}\label{ned4} \int_{\Omega} \psi(u_\lambda)^{p-1} \psi' (u_\lambda) \leq C \end{equation}
In particular, since the asymptotic convexity implies $2 t \psi'(t) \geq \psi (t)$ for any $t$ sufficiently large, from \eqref{ned4} we arrive at
\begin{equation}\label{ned5} \int_{\Omega} \frac{\psi(u_\lambda)^{p}}{u_\lambda} \leq 2 C \end{equation}
The desired estimate \eqref{L-p'} is then just a direct consequence of \eqref{ned5} and $p'=p/p-1>1$ since:
$$\int_{\{ u_\lambda > 1 \}} \frac{(f(u_\lambda))^{p'}}{u_\lambda} \, dx = \int_{\{ u_\lambda > 1 \}} \frac{(f(u_\lambda)-f(0) + f(0))^{p'}}{u_\lambda} \, dx$$ $$\leq \int_{\{ u_\lambda > 1 \}} \frac{(f(u_\lambda)-f(0))^{p'}}{u_\lambda} \, dx + \int_{\{ u_\lambda > 1 \}} \frac{f(0)^{p'}}{u_\lambda} \, dx$$
$$\leq \int_{\Omega} \frac{\psi(u_\lambda)^{p}}{u_\lambda} \, dx + f(0)^{p'} \int_{\{ u_\lambda > 1 \}} \frac{1}{u_\lambda} \, dx \leq 2 C + f(0)^{p'} |\Omega|:= M$$
which proves Lemma \ref{lemma2}.\qed
\end{document} | arXiv |
\begin{document}
\title{Generalized Parametric Path Problems}
\begin{abstract}
Parametric path problems arise independently in diverse domains, ranging from transportation to finance, where they are studied under various assumptions.
We formulate a general path problem with relaxed assumptions, and describe how this formulation is applicable in these domains.
We study the complexity of the general problem, and a variant of it where preprocessing is allowed. We show that when the parametric weights are linear functions, algorithms remain tractable even under our relaxed assumptions. Furthermore, we show that if the weights are allowed to be non-linear, the problem becomes $\NP$-hard. We also study the multi-dimensional version of the problem where the weight functions are parameterized by multiple parameters. We show that even with $2$ parameters, this problem is $\NP$-hard. \end{abstract}
\section{Introduction}
Parametric shortest path problems arise in graphs where the cost of an edge depends on a parameter. Many real-world problems lend themselves to such a formulation, e.g., routing in transportation networks parameterized by time/cost (\cite{C83, MS01, D04}), and financial investment and arbitrage networks (\cite{HP14, H14, M03}). Path problems have been studied independently in these domains, under specific assumptions that are relevant to the domain. For example, the time-dependent shortest path problem used to model transportation problems assumes a certain FIFO condition \autoref{eq:FIFOineq}. Arbitrage problems only model the rate of conversion and are defined with respect to a single currency parameter. These assumptions reduce the applicability of such algorithms to other domains.
We propose a generalized model for parametric path problems with relaxed assumptions, giving rise to an expressive formulation with wider applicability. We also present specific instances of real-world problems where such generalized models are required (see \autoref{sec:rel-works}).
\begin{definition}\label{def:gpp}
The input to a Generalized Path Problem (GPP) is a $4$-tuple $(G,W,L,\vecx_0)$, where
$G = (V \cup \set{s, t}, E)$ is a directed acyclic graph with two special vertices $s$ and $t$, $W =\setdef{w_e : \R^k \to \R^k}{e \in E}$ is a set of weight functions on the edges of $G$, $L \in \R^k$ is a vector used for computing the cost of a path from the $k$ parameters, and $\vecx_0 \in \R^k$ is the initial parameter. \end{definition}
The aim in a GPP is to find an $s$-$t$ path $P$ in the graph $G$ that maximizes the dot product of $L$ with the \emph{composition} of the weight functions on $P$, evaluated at the initial parameter $\vecx_0$.
\begin{problem}[Generalized Path Problem (GPP)] \label{problem:linearquery}
\emph{Input:} An instance $(G,W,L,\vecx_0)$ of GPP.\\
\noindent \emph{Output:} An $s$-$t$ path $P= (e_1,\ldots, e_r)$ which maximizes
\[
L\cdot w_{e_r}(w_{e_{r-1}}(\cdots w_{e_{2}}(w_{e_{1}}(\vecx_0))\cdots )).
\]
When $k=1$, we call the GPP a \emph{scalar} GPP. Sometimes we ignore the $\vecx_0$ and just write $(G,W,L)$. \end{problem}
\begin{figure*}
\caption{
\footnotesize{
(Left) Graph of a GPP instance whose edge weights are linear functions of a parameter $x$. (Right) A plot of $x$ versus the costs of all possible $s$-$t$ paths $P_1,\ldots,P_6$. All 6 cost functions are linear because the composition of linear functions is linear. For example, the cost of $P_6$ is $a_6(a_5(c_2(c_1x+d_1)+d_2)+b_5)+b_6$. The table for GPP with preprocessing (PGPP) with $L=-1$ has 4 entries, indicated in pink.
}
}
\label{fig:planar3x3}
\end{figure*}
Scalar GPP (see \autoref{fig:planar3x3}) models shortest paths by choosing weights $w_e(x)= a_e \cdot x + b_e$ and fixing $L=-1$, to convert it to a minimization problem. Scalar GPP also models currency arbitrage problems (\cite{HP14, CLRS09}), where the cost of a path is the product of its edge weights, by choosing weight functions to be lines passing through the origin with slopes equal to the conversion rate.
Further, GPP can model more general path problems which involve multiple parameters to be optimized. For example, in transport networks, one needs to find a path that optimizes parameters like time traveled, cost of transportation, convenience, polluting emissions, etc. (\cite{KVP20}). In finance problems, an entity can have investments in different asset classes like cash, gold, stocks, bonds, etc., and a transaction, modeled by an edge, can affect these in complex ways (see more examples in \autoref{sec:rel-works}). The edge parameters could contribute additively or multiplicatively to the cost of the path. We study weight functions that are affine linear transformations, which allows for both of these.
Note that the optimal path can vary based on the value of the initial parameter $\vecx_0$. For this, we also consider a version of GPP with preprocessing (called PGPP), where we can preprocess the inputs $(G,W,L)$ and store them in a table which maps the initial values $\vecx_0$ to their optimal paths. Such a mapping is very useful in situations where the underlying network does not change too often and a large amount of computing power is available for preprocessing (e.g., the road map of a city typically does not change on a day-to-day basis). If the size of the table is managable, then it can be saved in memory and a query for an optimal path for a given $\vecx_0$ can be answered quickly using a simple table lookup.
\begin{problem}[GPP with Preprocessing (PGPP)] \label{problem:qgip}
\emph{Input:} An instance $(G,W,L)$ of GPP.\\
\noindent \emph{Output:} A table which maps $\vecx_0$ to optimal paths. \end{problem}
We present an efficient algorithm for scalar GPP with linear weight functions. On the other hand, we show that if the GPP instance is non-scalar or the weight functions are non-linear, algorithms with worst-case guarantees cannot be obtained, assuming $\P\neq\NP$.
Our algorithm is based on the Bellman-Ford-Moore algorithm (\cite{B58,F56,M59}), whereas our $\NP$-hardness reductions are from two well-known $\NP$-hard problems, \textsc{Set Partition} and \textsc{Product Partition}.
The results for GPP with preprocessing (PGPP) are much more technical since they involve proving upper and lower bounds on the number of discontinuities of the cost of the optimal path as a function of the initial parameter $x_0$. They generalize previously known results in Time-Dependent Shortest Paths by \cite{FHS14}. Their work crucially uses the FIFO property \autoref{eq:FIFOineq}, whereas our analysis does not make this assumption, giving a more general result.
\paragraph{Our Contributions.} Here is a summary of our results. In \autoref{sec:rel-works}, we give specific instances from transportation and finance where these results can be applied. \begin{enumerate}
\item There is an efficient algorithm for scalar GPP with linear weight functions (see \autoref{sec:qgip-algo}).
\item Scalar PGPP with linear weight functions has a quasi-polynomial sized table, and thus table retrieval can be performed in poly-logarithmic time (see \autoref{sec:gip-upper}).
\item Scalar GPP with piecewise linear or quadratic weight functions is $\NP$-hard to approximate (see \autoref{sec:qgip-hard}).
\item For scalar PGPP with piecewise linear or quadratic weight functions, the size of the table could be exponential (see \autoref{sec:gip-lower}).
\item Non-scalar GPP (GPP with $k>1$) is $\NP$-hard (see \autoref{sec:multi-hard}). \end{enumerate}
\section{Applications and Related Works} \label{sec:rel-works}
\subsection{Applications to Transportation.} We relate scalar GPP to an extensively well-studied problem, known as Time-Dependent Shortest Paths (TDSPs) in graphs, which comes up in routing/planning problems in transportation networks (\cite{D04, DOS12, FHS14}).
In the TDSP setting, the parameter $x$ denotes time, and the weight $w_e(x)$ of an edge $e=(A,B)$ denotes the arrival time at $B$ if the departure time from $A$ is $x$. If there is another edge $e'=(B,C)$ connected to $B$, then the arrival time at $C$ along the path $(A,B,C)$ is $w_{e'}(w_e(x))$, and so on. Thus, the cost of an $s$-$t$ path is the arrival time at $t$ as a function of the departure time from $s$. We say an edge $e$ is FIFO if its weight is a monotonically increasing function, i.e., \begin{equation} \label{eq:FIFOineq}
x_1\leq x_2 \Longleftrightarrow w_e(x_1)\leq w_e(x_2) \qquad\forall\,x_1,x_2. \end{equation}
The study of TDSPs can be traced back to the work of \cite{CH66}. \cite{D69} gave a polynomial time algorithm when the edges were FIFO and the queries were made in discrete time steps. These results were extended to non-FIFO networks by \cite{OR90}, and generalized further by \cite{ZM93}.
\cite{D04} summarised all known research on FIFO networks with linear edge weights. \cite{DOS12} presented an algorithm for TDSPs in this setting whose running time was at most the table size of the PGPP instance. Soon thereafter, \cite{FHS14} showed that the table size is at most $n^{O(\log n)}$, and that this is optimal, conclusively solving the problem for FIFO networks. We show that their bounds also hold for non-FIFO networks.
Some other closely related lines of work which might be of interest to the reader are \cite{MUPT21}, \cite{BGHO20}, \cite{BCV21}, \cite{RGN21}, \cite{WLT19} and \cite{D18}.
\paragraph{Example: Braess' Paradox.} The FIFO assumption makes sense because it seems that leaving from a source at a later time might not help one reach their destination quicker. However, somewhat counter-intuitively, \cite{B68} observed that this need not always the case (\autoref{fig:braess} shows an example of Braess' paradox). \cite{SZ83} showed that Braess' paradox can occur with a high probability. \cite{RKDG09} backed their claim with empirical evidence. In fact, there are real-world instances where shutting down a road led to a decrease in the overall traffic congestion. Two examples are Stuttgart (\cite{M70}) and Seoul (\cite[Page 71]{EK10}).
\begin{figure}\label{fig:braess}
\end{figure}
\subsection{Applications to Finance.} Financial domain problems have been modelled as graph problems before (\cite{DP14, KB09, E13, BAA14, A19}). We model the currency arbitrage problem (\cite{R77, SV97, DS06}) as a GPP. In the currency arbitrage problem, we need to find an optimal conversion strategy from one currency to another via other currencies, assuming that all the conversion rates are known.
\paragraph{Example: Multi-currency Arbitrage.} GPP can model generalized multi-currency arbitrage problems. In currency arbitrage, an entity can have money available in different currencies and engage in transactions (modelled by edges) which can change the entity's wealth composition in complex ways (\cite{M03}). The transaction fees could have fixed as well as variable components, depending on the amount used. This can be modelled by affine linear transformations. Eventually the entity might liquidate all the money to a single currency, which can be modelled by the vector $L$ in the GPP instance. The goal is to pick a sequence of transactions which maximizes the cash after liquidation. Hence, this problem naturally lends itself to a GPP formulation.
\paragraph{Example: Investment Planning.} GPP can model investment planning by considering the nodes of the graph to be the state of the individual (which could be qualifications, contacts, experience, influence, etc). At each given state, the individual has a set of investment opportunities which are represented by directed edges. Every edge represents an investment opportunity, and the weight of the edge models the return as a function of the capital invested. Suppose an individual initially has $y$ amount of money and makes two investments in succession with returns $r_1(x), r_2(x)$, then the individual will end up with $r_2(r_1(y))$ amount of money. Though a generic investment plan could allow multiple partial investments, there are cases where this is not possible. For example, the full fees needs to be paid up front for attending a professional course or buying a house, which motivates restricting to investment plans given by paths. The vertices $s$, $t$ denote the start and end of an investment period, and the optimal investment strategy is an $s$-$t$ path which maximizes the composition of functions along the path.
\section{Algorithm for Scalar GPP with Linear Weights}\label{sec:qgip-algo}
In this section, we present our algorithm for scalar GPP with linear weight functions. Formally, we show the following.
\begin{theorem} \label{thm:QGIP_linear}
There exists an algorithm that takes as input a scalar GPP instance $(G,W,L,x_0)$ (where $G$ has $n$ vertices and $w_e(x) = a_e \cdot x + b_e$ for every edge $e$ of $G$), and outputs an optimal $s$-$t$ path in $G$ in $O(n^3)$ running time.
\end{theorem}
We use \autoref{algo:qgip} for solving GPP. Our algorithm is similar to the Bellman-Ford-Moore shortest path algorithm (\cite{B58,F56,M59}), where they keep track of \emph{minimum} cost paths. The only subtlety in our case is that we need to keep track of both \emph{minimum and maximum} cost paths with at most $k$ edges from the start vertex $s$ to every vertex $v$, as $k$ varies from $1$ to $n$. The variables $p_{\max},p_{\min}$ act as parent pointers for the maximum cost path and the minimum cost path tree rooted at $s$. $r_{\max},r_{\min}$ stores the cost of the maximum and minimum cost path. The running time of \autoref{algo:qgip} is clearly $O(n^3)$, the same as the running time of the Bellman-Ford-Moore algorithm. Its correctness follows from the following observation.
\begin{observation} Let $a_e$ be the coefficient of $x$ in $w_e$ \begin{itemize}
\item If $e=(u,v)$ is the last edge on a shortest $s$-$v$ path, then its $s$-$u$ subpath is either a shortest $s$-$u$ path (if $a_e$ is positive), or a longest $s$-$u$ path (if $a_e$ is negative).
\item If $e=(u,v)$ is the last edge on a longest $s$-$v$ path, then its $s$-$u$ subpath is either a shortest $s$-$u$ path (if $a_e$ is negative), or a longest $s$-$u$ path (if $a_e$ is positive). \end{itemize} \end{observation}
Then, the argument is similar to the proof of the Bellman-Ford-Moore algorithm, using the optimal substructure property. Our algorithm can also handle time constraints on the edges which can come up in transport and finance problems. For example, each investment (modelled by an edge) could have a scalar value, which denotes the time taken for it to realize. The goal is to find an optimal sequence of investments (edges) from $s$ to $t$, such that the sum of times along the path is at most some constant $T$. We can reduce such a problem to a GPP problem with a time constraint as follows.
Replace each edge $e$ by a path of length $t_e$, where $t_e$ is the time value associated with $e$. The weight function for the first edge is simply $w_e(x)$ and for the other $t_e-1$ edges, it is the identity function. Then, \autoref{algo:qgip} can be modified so that the first for-loop stops at $T$ instead of at $n-1$.
\begin{algorithm}
\SetAlgoLined
For $v \in V \setminus \{ s \}$, $r_{\text{max}}(v) = -\infty, r_{\text{min}}(v) = \infty$\;
$r_{\text{max}}(s) = r_{\text{min}}(s) = x$\;
\For{$k \in [1, n-1]$}{
\For{$e=(u,v) \in E$}{
\eIf{ $a_e \geq 0$ }{
\If{ $r_{\text{max}}(v) < w_e(r_{\text{max}}(u))$ }{
$r_{\text{max}}(v) \gets w_e(r_{\text{max}}(u))$,
$p_{\text{max}}(v) \gets u$\;
}
\If{ $r_{\text{min}}(v) > w_e(r_{\text{min}}(u))$ }{
$r_{\text{min}}(v) \gets w_e(r_{\text{min}}(u))$,
$p_{\text{min}}(v) \gets u$\;
}
}{
\If{ $r_{\text{max}}(v) < w_e(r_{\text{min}}(u))$ }{
$r_{\text{max}}(v) \gets w_e(r_{\text{min}}(u))$,
$p_{\text{max}}(v) \gets u$\;
}
\If{ $r_{\text{min}}(v) > w_e(r_{\text{max}}(u))$ }{
$r_{\text{min}}(v) \gets w_e(r_{\text{max}}(u))$,
$p_{\text{min}}(v) \gets u$\;
}
}
}
}
\textbf{Output}: The sequence $(t, p_{\text{max}}(t), p_{\text{max}}(p_{\text{max}}(t)), \ldots, s)$ in reverse order is the optimal path at $x$ with value $r_{\text{max}}(t)$.
\caption{GPP with linear weight functions}\label{algo:qgip} \end{algorithm}
\section{Upper Bound for Scalar PGPP with Linear Weights}\label{sec:gip-upper}
In this section, we study scalar PGPP (linear edge weights with $L=-1$), and show that the total number of different shortest $s$-$t$ paths (for different values of $x_0\in(-\infty,\infty)$) is at most quasi-polynomial in $n$. In PGPP (\autoref{problem:qgip}), we are allowed to preprocess the graph. We compute all possible shortest $s$-$t$ paths in the graph and store them in a table of quasi-polynomial size. More precisely, if $(G,W,L)$ is a scalar GPP instance (where $G$ has $n$ vertices and $w_e(x) = a_e\cdot x+b_e$ for every edge $e$ of $G$), then we show that the number of shortest $s$-$t$ paths in $G$ is at most $n^{O(\log n)}$. (For the example in \autoref{fig:planar3x3}, this number is 4.) Since the entries of this table can be sorted by their corresponding $x_0$ values, a table lookup can be performed using a simple binary search in $\log(n^{O(\log n)}) = O((\log n)^2)$ time. Thus, a shortest $s$-$t$ path for a queried $x_0$ can be retrieved in poly-logarithmic time.
In our proof, we will crucially use the fact that the edge weights of $G$ are of the form $w_e(x) = a_e x + b_e$. Although our result holds in more generality, it is helpful and convenient to think of the edge weights from a TDSP perspective. That is, when travelling along an edge $e=(u,v)$ of $G$, if the start time at vertex $u$ is $x$, then the arrival time at vertex $v$ is $w_e(x)$.
As the edge weights are linear and the composition of linear functions is linear, the arrival time at $t$ after starting from $s$ at time $x$ and travelling along a path $P$ is a linear function of $x$, called the cost of the path and denoted by $\mathsf{cost}(P)(x)$. We show that the \emph{piecewise linear lower envelope} (denoted by $\mathsf{cost}_G(x)$, indicated in pink in \autoref{fig:planar3x3}) of the cost functions of the $s$-$t$ paths of $G$ has $n^{\log n+O(1)}$ pieces. Let $p(f)$ denote the number of pieces in a piecewise linear function $f$.
\begin{theorem}\label{thm:ub_linear}
Let $\mathcal{P}$ be the set of $s$-$t$ paths in $G$.
Then, the cost function of the shortest $s$-$t$ path, given by $\ \mathsf{cost}_G(x) = \underset{P:\, P\in \mathcal{P}}{\min} \mathsf{cost}(P)(x)$, is a piecewise linear function such that
\[
p(\mathsf{cost}_G(x))\leq n^{\log n+O(1)}.
\] \end{theorem}
Before we can prove \autoref{thm:ub_linear}, we need some elementary facts about piecewise linear functions. Given a set of linear functions $F$, let $F_\downarrow$ and $F_\uparrow$ be defined as follows. \[
F_\downarrow(x) = \min_{f:\, f\in F} f(x) \qquad\qquad F_\uparrow(x) = \max_{f:\, f\in F} f(x) \] In other words, $F_\downarrow$ and $F_\uparrow$ are the piecewise linear lower and upper envelopes of $F$, respectively.
\begin{fact}[Some properties of piecewise linear functions] \label{fact:piecewisefacts}
\begin{enumerate}[label=(\roman*)]
\item If $F$ is a set of linear functions, then $F_\downarrow$ is a piecewise linear concave function and $F_\uparrow$ is a piecewise linear convex function.
\item If $f(x)$ and $g(x)$ are piecewise linear \emph{concave} functions, then $h(x)=\min\{f(x),g(x)\}$ is a piecewise linear concave function such that $p(h)\leq p(f)+p(g)$.
\item If $f(x)$ and $g(x)$ are piecewise linear functions and $g(x)$ is \emph{monotone}, then $h(x)=f(g(x))$ is a piecewise linear function such that $p(h)\leq p(f)+p(g)$.
\end{enumerate} \end{fact}
\begin{proof} These facts and their proofs are inspired by (and similar to) some of the observations made by \cite[Lemma 2.1, Lemma 2.2]{FHS14}.
\begin{enumerate}[label=(\roman*)]
\item Linear functions are concave (convex), and the point-wise minimum (maximum) of concave (convex) functions is concave (convex).
\item Each piece of $h$ corresponds to a unique piece of $f$ or $g$. Since $h$ is concave, different pieces of $h$ have different slopes, corresponding to different pieces of $f$ or $g$.
\item A break point is a point where two adjoining pieces of a piecewise linear function meet. Note that each break point of $h$ can be mapped back to a break point of $f$ or a break point of $g$. As $g$ is monotone, different break points of $h$ map to different break points of $g$. \qedhere
\end{enumerate} \end{proof}
We now prove the following key lemma.
\begin{lemma} \label{cl:uplo_envelope}
Let $F$ and $G$ be two sets of linear functions, and let $H=\{f\circ g \bigm| f\in F, g\in G\}$. Then
\begin{align}
H_\downarrow(x)&=\min\{F_\downarrow(G_\downarrow(x)),F_\downarrow(G_\uparrow(x))\}; \label{eq:hlo}\\
H_\uparrow(x)&=\max\{F_\uparrow(G_\downarrow(x)),F_\uparrow(G_\uparrow(x))\}; \label{eq:hup}\\
p(H_\downarrow)&\leq 4p(F_\downarrow)+2p(G_\downarrow)+2p(G_\uparrow); \label{eq:hlobound}\\
p(H_\uparrow)&\leq 4p(F_\uparrow)+2p(G_\downarrow)+2p(G_\uparrow). \label{eq:hupbound}
\end{align} \end{lemma}
\begin{proof}
We will first show \autoref{eq:hlo}. Since $F$ is the set of outer functions, it is easy to see that
\begin{equation} \label{eq:inner_outer}
H_\downarrow(x) = \min_{g:\, g\in G} F_\downarrow(g(x)).
\end{equation}
To get \autoref{eq:hlo} from \autoref{eq:inner_outer}, we need to show that the inner function $g$ that minimizes $H_\downarrow$ is always either $G_\downarrow$ or $G_\uparrow$.
Fix an $x_0\in\mathbb{R}$. We will see which $g\in G$ minimizes $F_\downarrow(g(x_0))$.
Note that for every $g\in G$, we have $G_\downarrow(x_0)\leq g(x_0)\leq G_\uparrow(x_0)$.
Thus, the input to $F_\downarrow$ is restricted to the interval $[G_\downarrow(x_0),G_\uparrow(x_0)]$. Since $F_\downarrow$ is a \emph{concave} function (\autoref{fact:piecewisefacts} (i)), it achieves its minimum at either $G_\downarrow(x_0)$ or at $G_\uparrow(x_0)$ within this interval. This shows \autoref{eq:hlo}.
We will now show \autoref{eq:hlobound} using \autoref{eq:hlo}.
Since $G_\downarrow$ is a concave function, it has two parts: a first part where it monotonically increases and a second part where it monotonically decreases.
In each part, the number of pieces in $F_\downarrow(G_\downarrow(x))$ is at most $p(F_\downarrow)+p(G_\downarrow)$ (\autoref{fact:piecewisefacts} (iii)), which gives a total of $2(p(F_\downarrow)+p(G_\downarrow))$.
Similarly, since $G_\uparrow$ is a convex function, it has two parts: a first part where it monotonically decreases and a second part where it monotonically increases.
In each part, the number of pieces in $F_\downarrow(G_\uparrow(x))$ is at most $p(F_\downarrow)+p(G_\uparrow)$ (\autoref{fact:piecewisefacts} (iii)), which gives a total of $2(p(F_\downarrow)+p(G_\uparrow))$.
Combining these using \autoref{eq:hlo} and \autoref{fact:piecewisefacts} (ii), we obtain
\begin{align*}
p(H_\downarrow) &\leq 2(p(F_\downarrow)+p(G_\downarrow)) + 2(p(F_\downarrow)+p(G_\uparrow))\\
&= 4p(F_\downarrow)+2p(G_\downarrow)+2p(G_\uparrow).
\end{align*}
We skip the proof of \autoref{eq:hup} and its usage to prove \autoref{eq:hupbound} because it is along similar lines. \end{proof}
Using this lemma, we complete the proof of \autoref{thm:ub_linear}. \begin{proof}[Proof of \autoref{thm:ub_linear}]
It suffices to prove the theorem for all positive integers $n$ that are powers of $2$. Let $a,b,v$ be three vertices of $G$ and let $k$ be a power of $2$. Let $\mathcal{P}_v(a,b,k)$ be the set of $a$-$b$ paths $P$ that pass through $v$ such that the $a$-$v$ subpath and the $v$-$b$ subpath of $P$ have at most $k/2$ edges each ($k$ is even number since it is a power of $2$). Let $\mathcal{P}(a,b,k)$ be the set of $a$-$b$ paths that have at most $k$ edges. Note that $
\mathcal{P}(a,b,k)=\bigcup_{v\in V}\mathcal{P}_v(a,b,k).
$
Let $f_v(a,b,k)$ be the number of pieces in the piecewise linear lower envelope or the piecewise linear upper envelope of $\mathcal{P}_v(a,b,k)$, whichever is larger.
Similarly, $f(a,b,k)$ is the number of pieces in the piecewise linear lower envelope or the piecewise linear upper envelope of $\mathcal{P}(a,b,k)$, whichever is larger.
Note that every path that features in the lower envelope of $\mathcal{P}(a,b,k)$ also features in the lower envelope of $\mathcal{P}_v(a,b,k)$, for some $v$.
Thus,
\begin{equation} \label{eq:up_quasipoly}
f(a,b,k)\leq \sum_{v\in V}f_v(a,b,k).
\end{equation}
Since $G$ has $n$ vertices, $\mathcal{P}(a,b,n)$ is simply the set of all $a$-$b$ paths. And since $p(\mathsf{cost}_G(x))$ is the number of pieces in the piecewise linear lower envelope of these paths, $p(\mathsf{cost}_G(x))\leq f(a,b,n)$.
Thus it suffices to show that $f(a,b,n)\leq n^{\log n + O(1)}$. We will show, by induction on $k$, that $f(a,b,k)\leq (8n)^{\log k}$.
The base case, $f(a,b,1)\leq 1$, is trivial. Now, let $k>1$ be a power of $2$. We will now show the following recurrence.
\begin{equation}\label{eq:up_envelope}
f_v(a,b,k) \leq 4 \inparen{f(a,v,k/2) + f(v,b,k/2)}
\end{equation}
Fix a vertex $v\in V$. By induction, $f(a,v,k/2)\leq (8n)^{\log (k/2)}$ and $f(v,b,k/2)\leq (8n)^{\log (k/2)}$.
Note that for every path $P\in\mathcal{P}_v(a,b,k)$, we have $\mathsf{cost}(P)(x)=\mathsf{cost}(P_2)(\mathsf{cost}(P_1)(x))$, where $P_1\in\mathcal{P}(a,v,k/2)$ and $P_2\in\mathcal{P}(v,b,k/2)$.
Thus we can invoke \autoref{cl:uplo_envelope} with $F$, $G$ and $H$ as the set of linear (path cost) functions corresponding to the paths $\mathcal{P}(v,b,k/2)$, $\mathcal{P}(a,v,k/2)$ and $\mathcal{P}_v(a,b,k)$, respectively.
Applying \autoref{eq:hlobound} and \autoref{eq:hupbound}, we get
\[
f_v(a,b,k) \leq 4 f(v,b,k/2) + 2 f(a,v,k/2) + 2 f(a,v,k/2),
\]
which simplifies to \autoref{eq:up_envelope}.
Substituting \autoref{eq:up_envelope} in \autoref{eq:up_quasipoly}, and using the fact that $|V|=n$, we get the following.
\begin{align*}
f(a,b,k)&\leq 4\sum_{v\in V}\left(f(a,v,k/2)+f(v,b,k/2)\right)\\
&\leq 4n\left((8n)^{\log (k/2)}+(8n)^{\log (k/2)}\right)\\
=(4n)\cdot 2\cdot (8n)^{\log (k/2)}=(8n)^{\log k}.
\end{align*}
Thus, $f(a,b,n)\leq (8n)^{\log n}=n^{\log n + 3}$. \end{proof}
\section{Hardness of Scalar GPP with Non-linear Weights}\label{sec:qgip-hard}
In this section, we show that it is $\NP$-hard to approximate scalar GPP, even if one of the edge weights is made piecewise linear while keeping all other edge weights linear.
\begin{theorem} \label{thm:QGIP_piecewise_linear}
Let $(G,W,L,x_0)$ be a GPP instance with a special edge $e^*$, where $G$ has $n$ vertices and $w_e(x) = a_e x + b_e$ for every edge $e\in E(G)\setminus \{e^*\}$, and $w_{e^*}(x)$ is piecewise linear with 2 pieces.
Then it is $\NP$-hard to find an $s$-$t$ path whose cost approximates the cost of the optimal $s$-$t$ path in $G$ to within a constant, both additively and multiplicatively. \end{theorem}
Note that \autoref{thm:QGIP_piecewise_linear} implies that \autoref{problem:qgip} with piecewise linear edge weights is $\NP$-hard.
\begin{proof} [Proof of \autoref{thm:QGIP_piecewise_linear}]
We reduce from \textsc{Set Partition}, a well-known $\NP$-hard problem \cite[Page 226]{GJ79}\footnote{A similar reduction can be found in \cite[Theorem 3]{NBK06}.}.
The \textsc{Set Partition} problem asks if a given set of $n$ integers $A = \set{a_0, \ldots, a_{n-1}}$ can be partitioned into two subsets $A_0$ and $A_1$ such that they have the same sum.
We now explain our reduction. Let $\varepsilon$ be the multiplicative approximation factor and $\delta$ be the additive approximation term. Given a \textsc{Set Partition} instance $A = \set{a_1, \ldots, a_n}$, we multiply all its elements by the integer $\ceil{\delta+1}$.
Note that this new instance can be partitioned into two subsets having the same sum if and only if the original instance can.
Furthermore, after this modification, no subset of $A$ has sum in the range $[-\delta,\delta]$, unless that sum is zero.
Next, we define a graph instantiated by the \textsc{Set Partition} instance.
\begin{definition} \label{def:duppathgraph}
$G_n$ is a directed, acyclic graph, with vertex set $\inbrace{v_0, \ldots, v_n}$.
For every $i \in \inbrace{0, \ldots, n-1}$, there are two edges from $v_i$ to $v_{i+1}$ labelled by $f_0$ and $f_1$.
The start vertex $s$ is $v_0$ and the last vertex $t$ is $v_n$ (See \autoref{fig:2path}).
\end{definition}
\begin{figure}
\caption{\footnotesize{The graph $G_n$ for $n=4$.}}
\label{fig:2path}
\end{figure}
\begin{definition} \label{def:funpathgraph}
Each path of $G_n$ can be denoted by a string in $\inbrace{0,1}^n$, from left to right. For instance, if $\sigma=(0101)$, then the cost function $f_\sigma(x)$ of the path $P_\sigma$ is given by
\begin{equation*}
f_\sigma(x) = f_{(0101)}(x) = f_1(f_0(f_1(f_0(x)))).
\end{equation*}
Note that the innermost function corresponds to the first edge on the path $P_\sigma$, and the outermost to the last.
\end{definition}
Consider the graph $G_{n+1}$. For each $i\in\inbrace{0,1,\ldots,n-1}$ and each edge $(v_i,v_{i+1})$, the edge labelled by $f_0$ has weight $x+a_i$ and the edge labelled by $f_1$ has weight $x-a_i$. Both edges from $v_n$ to $v_{n+1}$ have weight $|x|$ (and can be replaced by a single edge $e^*$). Let $\mathcal{A}$ be an algorithm which solves \autoref{problem:qgip}.
We will provide $G_{n+1}$ and $x_0=0$ as inputs to $\mathcal{A}$, and show that $A$ can be partitioned into two subsets having the same sum if and only if $\mathcal{A}$ returns a path of cost $0$.
Let $\sigma = (\sigma_0 \sigma_1 \cdots \sigma_{n-1}) \in \inbrace{0,1}^n$.
Let $A_1$ be the subset of $A$ with characteristic vector $\sigma$, and let $A_0 = A \setminus A_1$.
The cost of the path $P_\sigma$ (\autoref{def:funpathgraph}) from $v_0$ to $v_n$ is
\[
\mathsf{cost}(P_\sigma)(x) = x + \sum_{i=0}^{n-1}(-1)^{\sigma_i}a_i = x + \sum_{a_i \in A_0} a_i - \sum_{a_i \in A_1} a_i .
\]
Now if we set the start time from vertex $v_0$ as $x=x_0=0$, then we obtain the following.
\[
\mathsf{cost}(P_\sigma)(0)=0 \implies \sum_{a_i \in A_0} a_i = \sum_{a_i \in A_1} a_i.
\]
Let $\mathsf{OPT}$ be a shortest path in $G_{n+1}$ and $Q$ be the path returned by $\mathcal{A}$ at start time $x=x_0=0$.
The last edge from $v_n$ to $v_{n+1}$ (whose weight is $|x|$) ensures that $\mathsf{OPT}\geq 0$. So, if $\mathsf{OPT}=0$, then $
\mathsf{cost}(Q)(0) \leq \varepsilon\cdot0 + \delta=\delta.
$
Since every path of non-zero cost in $G_{n+1}$ has cost more than $\delta$, $\mathsf{cost}(Q)(0)=0$ if $\mathsf{OPT}=0$.
Further, if $\mathsf{OPT}>0$, then $\mathsf{OPT}\geq\ceil{\delta+1}$, and so $\mathcal{A}$ returns a path of cost more than $\delta$.
Thus, $A$ can be partitioned into two subsets having the same sum if and only if $\mathcal{A}$ returns a path of cost $0$. \end{proof}
\begin{remark}
Our reduction also works if we change the weight of the last edge from $|x|$ to $x^2$, implying that scalar GPP with polynomial functions is $\NP$-hard, even if one of the edge weights is quadratic and all other edge weights are linear. \end{remark}
\section{Lower Bound for Scalar PGPP with Non-linear Weights}\label{sec:gip-lower}
In this section, we show that for the graph $G_n$ defined in the previous section (\autoref{def:duppathgraph}) with a suitable choice of the weight functions $f_0$ and $f_1$, the table size for PGPP (\autoref{problem:qgip}) can be exponential in $n$. Note that $G_n$ has exactly $2^n$ paths from $s$ to $t$. We will show that each of these paths is a shortest $s$-$t$ path, for some value of $x$. Thus, there is a scalar GPP instance $(G,W,L)$ for which the table size is $2^{\Omega(n)}$, needing $\log(2^{\Omega(n)})=\Omega(n)$ time for a table lookup.
Our proof is by induction on $n$. We define the functions $f_0$ and $f_1$ in such a way that their behaviour within the interval $[0,1]$ has some very special properties, stated in \autoref{lem:lb_main}. This enables us to show that the number of times the compositions of these functions achieve their minimum within the interval $[0,1]$ doubles every time $n$ increases by one.
We need some notation before we can proceed. For a function $f : \R \to \R$, and a subset $A \subseteq \R$, if $B \supseteq f(A)$, then we denote by $f|_A : A \to B,$ the function defined by $f|_A(x) = f(x)$ for every $x \in A$, also known as the restriction of $f$ to $A$.
\begin{lemma} \label{lem:lb_main}
Suppose $f_0, f_1 : \R \to \R$ are functions such that $f_0|_{[0, 1/3]} : [0, 1/3] \to [0, 1]$ and $f_1|_{[2/3, 1]} : [2/3, 1] \to [0,1]$ are bijective.
Further, suppose $\abs{f_0(x)} \geq 1$ for every $x \in (-\infty, 0] \cup [2/3, \infty)$; and $\abs{f_1(x)} \geq 1$ for every $x \in (-\infty, 1/3] \cup [1, \infty)$. Then, for every $n \geq 1$, there is a function $\alpha_n : \inbrace{0,1}^n \to (0,1)$ such that
\begin{enumerate}[label=(\roman*),noitemsep,partopsep=0pt,topsep=0pt,parsep=0pt]
\item $\alpha_n(\sigma) \in [0, 1/3]$ \, if \, $\sigma_1 = 0$;
\item $\alpha_n(\sigma) \in [2/3, 1]$ \, if \, $\sigma_1 = 1$;
\item For every $\sigma, \tau \in \set{0,1}^n$, $\alpha_n(\sigma) = \alpha_n (\tau) \Longleftrightarrow \sigma = \tau$;
\item For every $\sigma, \tau \in \set{0,1}^n$, $f_\sigma(\alpha_n(\tau)) = 0 \Longleftrightarrow \sigma = \tau$.
\end{enumerate} \end{lemma}
\begin{proof}
As stated earlier, we prove this lemma by induction on $n$.
For the \textbf{base case} ($n=1$), we define
\[
\alpha_1(0) = (f_0|_{[0, 1/3]})^{-1}(0) \text{\quad \& \quad} \alpha_1(1) = (f_1|_{[2/3, 1]})^{-1}(0).
\]
First we check if $\alpha_1$ is well-defined and its range lies in $(0,1)$.
To see that $\alpha_1(0)$ and $\alpha_1(1)$ are well-defined, note that the inverses of the functions $f_0|_{[0, 1/3]}$ and $f_0|_{[0, 1/3]}$ are well-defined because they are bijective.
To see that the range of $\alpha_1$ lies in $(0,1)$, note that for $x\in\inbrace{0,1}$, we have $\abs{f_0(x)}\geq 1$ and $\abs{f_1(x)}\geq 1$, implying that they are both non-zero.
Thus, $0<\alpha_1(x)<1$.
We now show that $\alpha_n$ satisfies (i), (ii), (iii), (iv). Since $f_0|_{[0, 1/3]}$ and $f_1|_{[2/3, 1]}$ are bijective, $\alpha_1(0)\in[0,1/3]$ and $\alpha_1(1)\in[2/3,1]$.
Thus, $\alpha_1$ satisfies (i), (ii).
Since these intervals are disjoint, $\alpha_1$ satisfies (iii).
Finally, note that $f_0(\alpha_1(0)) = 0 = f_1(\alpha_1(1)).$
Also, since $\abs{f_0(x)} \geq 1$ for every $x \in [2/3, 1]$ and $\abs{f_1(x)} \geq 1$ for every $x \in [0, 1/3]$, both $f_0(\alpha_1(1))$ and $f_1(\alpha_1(0))$ are non-zero.
Thus, $\alpha_1$ satisfies (iv).
This proves the base case.
\paragraph{Induction step $(n>1)$:}
Assume that $\alpha_{n-1} : \inbrace{0, 1}^{n-1} \to (0,1)$ has been defined, and that it satisfies (i), (ii), (iii), (iv). We now define $\alpha_n : \inbrace{0, 1}^n \to (0,1)$. Let $\sigma \in \inbrace{0,1}^n$ be such that $\sigma = \sigma_1 \sigma'$, where $\sigma_1 \in \inbrace{0,1}$ and $\sigma' = \sigma_2 \cdots \sigma_{n} \in \inbrace{0,1}^{n-1}$. We define $\alpha_n(\sigma)$ as follows.
\[
\alpha_n(\sigma) =
\begin{cases}
(f_0|_{[0, 1/3]})^{-1}(\alpha_{n-1}(\sigma')) \qquad\qquad \mbox{ if } \sigma_1 = 0\\
(f_1|_{[2/3, 1]})^{-1}(\alpha_{n-1}(\sigma')) \qquad\qquad \mbox{ if } \sigma_1 = 1\\
\end{cases}
\]
More concisely,
\begin{align}
\alpha_n(\sigma) = (f_{\sigma_1}|_A)^{-1}&(\alpha_{n-1}(\sigma')),\label{eq:alphan}
\end{align}
where $A = [0, 1/3]$ when $\sigma_1 = 0$ and $A = [2/3, 1]$ when $\sigma_1 = 1$,
Note that $\alpha_n$ is well-defined and its range lies in $(0,1)$ for the same reasons as explained in the base case.
We will now show that $\alpha_n$ satisfies (i), (ii), (iii), (iv).
(i), (ii): By definition, $(f_0|_{[0, 1/3]})^{-1} : [0, 1] \to [0, 1/3]$ and $(f_1|_{[2/3, 1]})^{-1} : [0, 1] \to [2/3, 1]$. Thus, $\alpha_n$ satisfies (i), (ii).
(iii): Suppose $\sigma = \sigma_1 \sigma' \in \set{0,1}^n$ and $\tau = \tau_1 \tau' \in \set{0,1}^n$.
Clearly if $\sigma = \tau$, then $\alpha_n(\sigma) = \alpha_n (\tau)$.
This shows the $\Leftarrow$ direction.
For the $\Rightarrow$ direction, suppose $\alpha_n(\sigma) = \alpha_n (\tau)$.
Then the only option is $\sigma_1 = \tau_1$, since otherwise one of $\alpha_n(\sigma), \alpha_n(\tau)$ would lie in the interval $[0, 1/3]$ and the other would lie in the interval $[2/3, 1]$. Thus,
\[
(f_{\sigma_1}|_A)^{-1}(\alpha_{n-1}(\sigma')) = (f_{\sigma_1}|_A)^{-1}(\alpha_{n-1}(\tau')),
\]
where $A = [0, 1/3]$ when $\sigma_1 = 0$ and $A = [2/3, 1]$ when $\sigma_1 = 1$,
Since $(f_{\sigma_1}|_A)$ is bijective, this means that $\alpha_{n-1}(\sigma') = \alpha_{n-1}(\tau')$. Using part (iii) of the induction hypothesis, this implies that $\sigma' = \tau'$.
Thus, $\alpha_n$ satisfies (iii).
(iv): Suppose $\sigma = \sigma_1 \sigma' \in \set{0,1}^n$ and $\tau = \tau_1 \tau' \in \set{0,1}^n$. Let us show the $\Leftarrow$ direction first. If $\sigma = \tau$, then
\begin{align*}
f_\sigma(\alpha_n&(\tau)) = f_\sigma(\alpha_n(\sigma))\\
&=f_{\sigma'}(f_{\sigma_1}(\alpha_n(\sigma))) \qquad \text{(since $\sigma=\sigma_1\sigma'$)}\\
&=f_{\sigma'}(f_{\sigma_1}((f_{\sigma_1}|_A)^{-1}(\alpha_{n-1}(\sigma'))))~~ \text{(using \autoref{eq:alphan})}\\
&=f_{\sigma'}((f_{\sigma_1}\circ(f_{\sigma_1}|_A)^{-1})(\alpha_{n-1}(\sigma')))\\
&\qquad \text{(function composition is associative)}\\
&=f_{\sigma'}(\alpha_{n-1}(\sigma')).
\end{align*}
Using part (iv) of the induction hypothesis, we get $f_{\sigma'}(\alpha_{n-1}(\sigma')) = 0$, which implies that $f_\sigma(\alpha_n(\tau)) = 0$.
This shows the $\Leftarrow$ direction.
For the $\Rightarrow$ direction, suppose $f_\sigma(\alpha_n(\tau)) = 0$.
We have two cases: $\sigma_1 = \tau_1$ and $\sigma_1 \neq \tau_1$.
We will show that $\sigma=\tau$ in the first case, and that the second case is impossible.
If $\sigma_1=\tau_1$,
\begin{align*}
0 &= f_\sigma(\alpha_n(\tau)) = f_{\sigma'}(f_{\sigma_1}((f_{\tau_1}|_A)^{-1}(\alpha_{n-1}(\tau'))))\\
& = f_{\sigma'}(f_{\sigma_1}((f_{\sigma_1}|_A)^{-1}(\alpha_{n-1}(\tau'))))
= f_{\sigma'}(\alpha_{n-1}(\tau')),
\end{align*}
Using part (iv) of the induction hypothesis, $f_{\sigma'}(\alpha_{n-1}(\tau')) = 0 \Rightarrow \sigma' = \tau'$, and thus $\sigma = \tau$. This handles the case $\sigma_1=\tau_1$.
We will now show by contradiction that the case $\sigma_1\neq\tau_1$ is impossible.
Suppose $\sigma_1 \neq \tau_1$.
Let $\sigma_1 = 0$ and $\tau_1 = 1$ (the proof for $\sigma_1 = 1$ and $\tau_1 = 0$ is similar). Using the induction hypothesis, $(f_{\tau_1}|_{[2/3,1]})^{-1}(\alpha_{n-1}(\tau')) \in [2/3, 1]$.
Since $\abs{f_0(x)} \geq 1$ for every $x \in (-\infty, 0] \cup [2/3, \infty)$, this means that $\abs{f_{\sigma_1}((f_{\tau_1}|_{[2/3,1]})^{-1}(\alpha_{n-1}(\tau')))} \geq 1$.
Also note that if $\abs{x} \geq 1$, then both $\abs{f_0(x)} \geq 1$ and $\abs{f_1(x)} \geq 1$. By repeatedly applying this fact, it is easy to see that
\begin{align*}
&\abs{f_\sigma(\alpha_n(\tau))}\\
&= \abs{f_{\sigma_n}(\cdots (f_{\sigma_1}((f_{\tau_1}|_{[2/3,1]})^{-1}(\alpha_{n-1}(\tau'))) \cdots )}\geq 1.
\end{align*}
We started with $f_\sigma(\alpha_n(\tau)) = 0$ and obtained $\abs{f_\sigma(\alpha_n(\tau))}\geq 1$, which is clearly a contradiction.
This completes the proof of the $\Rightarrow$ direction, and thus $\alpha_n$ satisfies (iv). \end{proof}
\begin{theorem} \label{thm:lb_piecewise}
Consider the graph $G_n$. Define piecewise linear functions $f_0, f_1 : \R \to \R$ as follows (see \autoref{fig:nequalsone}).
\[
f_0(x) =
\begin{cases}
1 - 3x, \mbox{ if } x \leq 1/3\\
3x - 1, \mbox{ if } x \geq 1/3
\end{cases}
f_1(x) =
\begin{cases}
2 - 3x, \mbox{ if } x \leq 2/3\\
3x - 2, \mbox{ if } x \geq 2/3
\end{cases}
\]
For every $n \geq 1$ and $\sigma \in \inbrace{0,1}^n$, the cost function $f_\sigma$ of the path $P_\sigma$ is a unique piece in the lower envelope formed by the cost functions $\set{f_\sigma}_{\sigma\in\inbrace{0,1}^n}$.
Thus, the piecewise linear shortest path cost function has $2^n$ pieces. \end{theorem}
\begin{proof}[Proof of \autoref{thm:lb_piecewise}]
It is easy to check that $f_0$ and $f_1$ possess the conditions needed to invoke \autoref{lem:lb_main}.
Thus for every $n\geq 1$, there exists a function $\alpha_n$ which satisfies properties (i), (ii) and (iii) of \autoref{lem:lb_main}.
Let $n$ be a positive integer.
Consider the graph $G_n$ (\autoref{def:duppathgraph}).
Each path of $G_n$ is indexed by a binary string $\sigma\in\inbrace{0,1}^n$ and has cost function $f_\sigma$ (\autoref{def:funpathgraph}).
Note that $f_0(x)\geq 0$, $f_1(x)\geq 0$ for all $x\in\mathbb{R}$.
Thus $f_\sigma(x)\geq 0$ for all $\sigma\in\inbrace{0,1}^n$, $x\in\mathbb{R}$.
Let $\sigma\in\inbrace{0,1}^n$.
Using property (iii), $f_\sigma(\alpha(\sigma))=0$, and $f_\tau(\alpha(\sigma))>0$ for every $\sigma\neq\tau\in\inbrace{0,1}^n$.
Thus, the cost function $f_\sigma$ of the path $P_\sigma$ is a unique piece (which includes the point $\alpha_n(\sigma)$) in the lower envelope formed by the cost functions $\set{f_\sigma}_{\sigma\in\inbrace{0,1}^n}$. \end{proof}
\begin{remark}
The proof of \autoref{thm:lb_piecewise} works for a quadratic choice of the functions $f_0$ and $f_1$ as well. However, then the degree of the composed functions blows up exponentially, thereby making their bit complexity prohibitively large. \end{remark}
\section{Hardness of Non-Scalar GPP with Linear Weights}\label{sec:multi-hard}
In this section, we show that non-scalar GPP is $\NP$-hard.
\begin{theorem} \label{thm:QGIP_2D_linear}
Let $(G,W,L,\vecx_0)$ be a GPP instance, where $G$ has $n$ vertices and each edge $e$ of $G$ is labelled by a two dimensional vector $\vecw_e(x)$. The vertices $s,t$ are labelled by two dimensional vectors $\vecx_0, \vect_0$, respectively. Then it is $\NP$-hard to compute an optimal $s$-$t$ path in $G$. \end{theorem}
Note that~\autoref{thm:QGIP_2D_linear} implies that~\autoref{problem:linearquery} with parameter $k=2$ is $\NP$-hard.
\begin{proof} [Proof of~\autoref{thm:QGIP_2D_linear}]
We reduce from \textsc{Product Partition} problem, a well-known $\NP$-hard problem (\cite{NBCK10}).
The problem is similar to the set partition problem, except that~\emph{products} of the elements are taken instead of their~\emph{sums}.
Formally, the problem asks if a given set of $n$ positive integers $A = \set{a_1, \ldots, a_n}$ can it be partitioned into two subsets $A_0$ and $A_1$ such that their product is the same.
We now explain our reduction. Given a~\textsc{Product Partition} instance $A = \set{a_1, \ldots, a_n}$, consider the graph $G_{n+1}$ (\autoref{def:duppathgraph}).
For every $i\in\inbrace{0,1,\ldots,n-1}$, there are two edges from $v_i$ to $v_{i+1}$ labelled by matrices
\[
\begin{bmatrix}
a_i & 0\\
0 & a^{-1}_i
\end{bmatrix}
\text{ and }
\begin{bmatrix}
a^{-1}_i & 0\\
0 & a_i
\end{bmatrix}.
\]
We label $s$ by the vector $\vecx_0=[1, 1]^{\text{T}}$ and $t$ by $\vect_0=[-1, -1]^{\text{T}}$. Let $\mathcal{A}$ be an algorithm which solves~\autoref{problem:linearquery} with parameter $k=2$.
We will provide $G_{n+1}$ as input to $\mathcal{A}$, and show that $A$ can be partitioned into two subsets having the same sum if and only if $\mathcal{A}$ returns a path of cost $-2$.
Let $\sigma = (\sigma_1 \cdots \sigma_n) \in \inbrace{1, -1}^n$.
Let $A_1$ be the subset of $A$ with characteristic vector $\sigma$, and let $A_0 = A \setminus A_1$. The cost of the path $P_\sigma$ (\autoref{def:funpathgraph}) from $v_0$ to $v_n$ is
\[
\mathsf{cost}(P_\sigma)
=
\begin{bmatrix}
1 & 1
\end{bmatrix}
\cdot
\begin{bmatrix}
\prod_{i=1}^{n} a^{\sigma_i}_i & 0\\
0 & \prod_{i=1}^{n} a^{-1 \cdot \sigma_i}_i
\end{bmatrix}
\cdot
\begin{bmatrix}
-1\\
-1
\end{bmatrix}.
\]
Evaluating this, we obtain $\mathsf{cost}(P_\sigma) = - (a + a^{-1}$), where $a = \prod_{i=1}^{n} a^{\sigma_i}_i = \prod_{a_i \in A_0} a_i \cdot \prod_{a_i \in A_1} a^{-1}_i$.
Further, $a = 1 \Longleftrightarrow \prod_{a_i \in A_0} a_i = \prod_{a_i \in A_1} a_i$.
By the AM-GM inequality, $a + a^{-1} > 2$, for every $a \neq 1$. Therefore $-(a + a^{-1}) < -2$ for every $a \neq 1$, and so $A$ can be partitioned into two subsets whose product is the same if and only if $\mathcal{A}$ returns a path of cost $-2$. \end{proof}
\section{Conclusion \& Discussion} We study Generalized Path Problems on graphs with parametric weights. We show that the problem is efficiently solvable when the weight functions are linear, but become intractable in general when they are piecewise linear.
We assume that weight functions are deterministic and fully known in advance. Modelling probabilistic and partially known weight functions and proposing algorithms for them is a direction for future work. Furthermore, we have assumed that only one edge can be taken at a time, resulting in an optimization over~\emph{paths}. This requirement could be relaxed to study~\emph{flows} on graphs with parametric weights. Though there is some literature on such models in route planning algorithms (\cite{LPBM17}), results with rigorous guarantees such as the ones we have presented are challenging to obtain. In such cases, heuristic algorithms with empirical evaluation measures might be worth exploring.
\section*{Acknoledgements} This work was done when the first author was a postdoctoral researcher at Technion, Israel. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No. 682203-ERC-[Inf-Speed-Tradeoff]. The authors from TIFR acknowledge support of the Department of Atomic Energy, Government of India, under project number RTI4001.
\section*{Plots} \label{sec:plotsplots} In this section, we exhibit the plots of all the $s$-$t$ paths (\autoref{def:funpathgraph}) in the graphs $G_n$ (\autoref{def:duppathgraph}) for the piecewise linear weight functions $f_0, f_1$ (\autoref{thm:lb_piecewise}), for some values of $n$.
\label{fig:nequalsone}
\end{document} | arXiv |
Do all orbiting bodies eventually collide?
If two celestial bodies are in orbit, will they always eventually collide if not acted upon by outside forces?
HDE 226868♦
DouglasDouglas
Two bodies in orbit around each other will inevitably collide. The reason for this is that the system will give off energy in the form of gravitational waves. This effect is commonly cited in binary neutron star systems, where the two stars are isolated and close together. One of the most famous of these systems is the Hulse-Taylor binary.
The time it will take for the objects to collide can be calculated: $$t=\frac{5}{256}\frac{c^5}{G^3}\frac{r^4}{(m_1m_2)(m_1+m_2)}$$ where $r$ is the initial radius, $m_1$ and $m_2$ are the masses of the bodies, and $c$ and $G$ are the familiar constants, the speed of light in a vacuum and Newton's universal gravitational constant.
However, tidal acceleration could offset some of the effects.
$\begingroup$ Surely that's the absolute upper bound given no input of energy, not "the time"? I haven't done the math, but it seems to me that the formula provided is wont to spit out hilariously huge numbers; to the point where stuff like passing stars and, more importantly, drag in the interplanetary medium, would have a noticeable effect? $\endgroup$
– Williham Totland
$\begingroup$ Actually, I did do the math for Sol / Terra; giving me, assuming I managed to plug everything in correctly, 10 trillion times the current age of the universe. So, you know, a hilariously huge number. $\endgroup$
$\begingroup$ Would this depend on whether the universe is closed or open? Like, if the universe is closed, then couldn't the gravitational waves come "back" to the same place? And in such a case, wouldn't the system potentially never lose energy? $\endgroup$
$\begingroup$ @WillihamTotland That number is, I would think, accurate. Like I wrote, the effect is non-negligible on most scales. $\endgroup$
$\begingroup$ @Mehrdad their refocusing and absorption by the system is of nigh-infinitesimal probability. But to answer your question, the formula given is based on an a circular orbit in an otherwise empty and asymptotically flat spacetime. The contributions to the emitted radiation has "instantaneous" terms (really dependent on the retarded position) and "non-local" terms (dependent on prior history), which are smaller. Ignoring the latter and taking the leading-order post-Newtonian approximation of the should get us the result in the answer. $\endgroup$
– Stan Liou
Not the answer you're looking for? Browse other questions tagged orbit or ask your own question.
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\begin{definition}[Definition:Flow Chart]
A '''flow chart''' is a graphical depiction of an algorithm in which the steps are depicted in the form of boxes connected together by arrows.
Conventionally, the shape of the box representing a step is dependent upon the type of operation encapsulated within the step:
:Rectangular for an action
:A different shape, conventionally a diamond, for a condition.
On {{ProofWiki}}, the preferred shape for condition boxes is rectangular with rounded corners. This is to maximise ease and neatness of presentation: configuring a description inside a diamond shaped boxes in order for it to be aesthetically pleasing can be challenging and tedious.
Also on {{ProofWiki}}, it is part of the accepted style to implement the start and end points of the algorithm using a box of a particular style, in this case with a double border.
\end{definition} | ProofWiki |
If \[ I(x)= \begin{cases}0 & (x \leq 0) \\ 1 & (x>0)\end{cases} \]
If \[ I(x)= \begin{cases}0 & (x \leq 0) \\ 1 &...
I(x)= \begin{cases}0 & (x \leq 0) \\ 1 & (x>0)\end{cases}
if \(\left\{x_{n}\right\}\) is a sequence of distinct points of \((a, b)\), and if \(\sum\left|c_{n}\right|\) converges, prove that the series
f(x)=\sum_{n=1}^{\infty} c_{n} I\left(x-x_{n}\right) \quad(a \leq x \leq b)
converges uniformly, and that \(f\) is continuous for every \(x \neq x_{n}\).
The uniform convergence is a consequence of the \(M\)-test with \(M_{n}=\) \(\left|c_{n}\right|\). Hence \(f\) is continuous wherever each of the individual terms is continuous, in particular, at least for \(x \neq x_{n}\).
answered Jul 5, 2022 by ♦Gauss Diamond (71,587 points)
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asked Oct 16, 2021 in Data Science & Statistics by ♦MathsGee Platinum (163,814 points) | 293 views
Suppose that \[ f(x)= \begin{cases}e^{-1 / x^{2}} & \text { if } x \neq 0 \\ 0 & \text { if } x=0\end{cases} \]
asked Feb 1, 2022 in Mathematics by ♦MathsGee Platinum (163,814 points) | 153 views
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Assume that $f_{\mu}(x)=\mu \cdot x \cdot(1-x)$. Show that if $1 \leq \mu \leq 3$, then $f_{\mu}^{n}(x) \rightarrow 1-\frac{1}{\mu}$ as $n \rightarrow \infty$ for all $0<x<1$.
Let $X$ have the density $f(x)=2 x$ if $0 \leq x \leq 1$ and $f(x)=0$ otherwise. Show that $X$ has the mean $2 / 3$ and the variance $1 / 18$. Find the mean and the variance of the random variable $Y=-2 X+3$.
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Suppose $X$ is $\mathrm{N}(-1,4)$. Find
Show that the Newton-Raphson method converges quadratically. That is, suppose that the fixed point is $z$ and that the error of the $n$ th iteration is $\left|x_{n}-z\right|=h$, then $\left|x_{n+1}-z\right| \approx h^{2}$ for $h$ small enough.
Find the density that will maximize the hourly flow.
asked Jun 11, 2021 in Mathematics by ♦MathsGee Platinum (163,814 points) | 159 views | CommonCrawl |
\begin{definition}[Definition:Dimension (Measurement)/Fundamental Dimensions]
The SI-recommended '''fundamental dimensions''' are:
:$\mathsf M$: mass
:$\mathsf L$: length
:$\mathsf T$: time
:$\Theta$: temperature
:$\mathsf I$: electric current
:$\mathsf N$: amount of substance
:$\mathsf J$: luminous intensity
\end{definition} | ProofWiki |
\begin{document}
\title{Sharp bounds of constants in Poincar\'{e}
\begin{abstract}
The paper is concerned with sharp estimates of constants in classical Poincar\'{e} inequalities and Poincar\'{e}-type inequalities for functions having zero mean value in a simplicial domain or on a part of its boundary. These estimates are important for quantitative analysis of problems generated by differential equations, where numerical approximations are typically constructed with the help of simplicial meshes. We suggest easily computable relations that provide sharp bounds of the respective constants and compare these results with analytical estimates (if they are known). In the last section, we present an example that shows possible applications of the results and derive a computable majorant of the difference between the exact solution of a boundary value problem and an arbitrary finite dimensional approximation computed on a simplicial mesh, which uses above mentioned constants.
\end{abstract}
\section{Introduction}
Let $T$ be a bounded domain in $\Rd$ ($d \geq 2$) with Lipschitz boundary $\partial T$. It is well known that the Poincar\'e inequality (\cite{Poincare1890,Poincare1894})
\begin{equation}
\| w \|_ {T} \leq {C^{\mathrm P}_{T}} \, \| \nabla w \|_{T}
\label{eq:classical-poincare-constant} \end{equation}
holds for any
\begin{equation*}
w\in\tildeH{1}(T)
:= \Big\{ w \in H^1(T)\, \big | \, \mean{ w }_{T} = 0 \; \Big\}, \end{equation*}
where $\| w \|_ {T}$ denotes the norm in $\L{2}(T)$,
$\mean{w}_{T}: = \tfrac{1}{|T|} \int_{T} w \, \mathrm{d} x$ is the mean value of $w$ over $T$, and $|T|$ is the Lebesgue measure of $T$. The constant ${C^{\mathrm P}_{T}}$ depends only on $T$ and $d$.
Poincar\'e-type inequalities also hold for
\begin{equation*}
w \in \tildeH{1}(T, \Gamma) :=
\Big\{ w \in H^1(T)\, \big | \, \mean{w}_{\Gamma} = 0 \;\Big\},
\end{equation*}
where $\Gamma$ is a measurable part of $\partial T$ such that $\mathrm{meas}_{d - 1} \Gamma > 0$ (in particular, $\Gamma$ may coincide with the whole boundary).
For any $w \in \widetilde{H}^1(T, \Gamma)$, we have two inequalities similar to \eqref{eq:classical-poincare-constant}. The first one
\begin{equation}
\|w\|_{T} \, \leq \CPGamma \|\nabla w\|_{T}
\label{eq:Comega} \end{equation}
is another form of the Poincar\'e inequality \eqref{eq:classical-poincare-constant}, which is stated for a different set of functions and contains a different constant, i.e. ${C^{\mathrm P}_{T}} \leq \CPGamma$. The constant $\CPGamma$ is associated with the minimal positive eigenvalue of the problem
\begin{equation} -\Delta u = \lambda u \;\; {\rm in} \;\; T; \quad \partial_n u = \lambda \mean{u}_{T} \;\; {\rm on} \;\; \Gamma; \quad \partial_n u = 0 \;\; {\rm on} \;\; \partial T \backslash \Gamma; \quad \forall u \in \tildeH{1} (T, \Gamma). \label{eq:eigenvalue-problem-cp} \end{equation}
We note that inequalities of this type arose in finite element analysis many years ago (see, e.g., \cite{BabuskaAziz1976}), where \eqref{eq:Comega} was considered for simplexes in $\Rtwo$. The second inequality
\begin{equation}
\|w\|_{\Gamma} \, \leq \CtrGamma \|\nabla w\|_{T}
\label{eq:Cgamma} \end{equation}
estimates the trace of $w \in \widetilde{H}^1(T, \Gamma)$ on $\Gamma$. It is associated with the minimal nonzero eigenvalue of the problem
\begin{equation} -\Delta u = 0 \;\; {\rm in} \;\; T; \quad \partial_n u = \lambda u \;\; {\rm on} \;\; \Gamma; \quad \partial_n u = 0 \;\; {\rm on} \;\; \partial T \backslash \Gamma; \quad \forall u \in \tildeH{1} (T, \Gamma). \label{eq:eigenvalue-problem-ctr} \end{equation}
The problem \eqref{eq:eigenvalue-problem-ctr} is a special case of the Steklov problem \cite{Stekloff1902}, where the spectral parameter appears in the boundary condition. Sometimes \eqref{eq:eigenvalue-problem-ctr} is associated with the so-called {\em sloshing problem}, which describes oscillations of a fluid in a container. Eigenvalues and eigenfunctions of the sloshing problem have been studied in \cite{FoxKuttler1983,BanuelosKulczyckiPolterovichSiudeja2010,KozlovKuznetsov2004,KozlovKuznetsovMotygin2004,KuznetsovKulczyckiKwasnickiNazarovPoborchiPolterovichSiudeja2014, ArxivGirouardPolterovich} and some other papers cited therein.
Exact values of $\CPGamma$, $\CtrGamma$, and ${C^{\mathrm P}_{T}}$ are important from both analytical and computational points of view. Poincar\'{e}-type inequalities are often used in analysis of nonconforming approximations (e.g., discontinuous Galerkin or mortar methods), domain decomposition methods (see, e.g., \cite{Klawonnatall2008,Dohrmann2008} and \cite{ToselliWidlund2005}),
a posteriori estimates \cite{RepinBoundaryMeanTrace2015}, and other applications related to quantitative analysis of partial differential equations. Analysis of interpolation constants and their estimates for piecewise constant and linear interpolations over triangular finite elements can be found in \cite{XuefengOishi2013} and literature cited therein. Finally, we note that \cite{CarstensenGedicke2014} introduces a method of computing lower bounds for the eigenvalues of the Laplace operator based on nonconforming (Crouzeix-Raviart) approximations. This method yields guaranteed upper bounds of the constant in the Friedrichs' inequality.
It is known (see \cite{PayneWeinberger1960}) that for convex domains
$${C^{\mathrm P}_{T}} \leq \tfrac{\diam (T)}{\pi}.$$
For triangles this estimate was improved in \cite{LaugesenSiudeja2010} to
$${C^{\mathrm P}_{T}} \leq \tfrac{\diam (T)}{j_{1, 1}},$$
where $j_{1, 1} \approx 3.8317$ is the smallest positive root of the Bessel function $J_1$. Moreover, for isosceles triangles from \cite{Bandle1980, LaugesenSiudeja2010} it follows that
\begin{equation}
{C^{\mathrm P}_{T}} \leq \CPLS := \diam (T) \cdot \,
\begin{cases}
\tfrac{1}{j_{1, 1}} & \alpha \in (0, \tfrac{\pi}{3}],\\
\min \Big\{ \tfrac{1}{j_{1, 1}}, \tfrac{1}{j_{0, 1}}
\big(2 (\pi - \alpha) \tan(\tfrac{\alpha}{2})\big)^{-\rfrac{1}{2}} \Big\} &
\alpha \in (\tfrac{\pi}{3}, \tfrac{\pi}{2}], \\
\tfrac{1}{j_{0, 1}} \big(2 (\pi - \alpha) \tan(\tfrac{\alpha}{2})\big)^{-\rfrac{1}{2}} &
\alpha \in (\tfrac{\pi}{2}, \pi).\\
\end{cases}
\label{eq:improved-estimates} \end{equation}
Here, $j_{0, 1} \approx 2.4048$ is the smallest positive root of the Bessel function $J_0$.
A lower bound of ${C^{\mathrm P}_{T}}$ for convex domains in $\Rtwo$ was derived in \cite{Cheng1975} and it reads
\begin{equation}
{C^{\mathrm P}_{T}} \,\geq\, \tfrac{\diam \,(T)}{2 \,j_{0, 1}}.
\label{eq:cheng} \end{equation}
Analogously, work \cite{LaugesenSiudeja2009} provides lower bound
\begin{equation}
{C^{\mathrm P}_{T}} \,\geq\, \tfrac{P}{4 \,\pi},
\label{eq:ls} \end{equation}
which improves \eqref{eq:cheng} for some cases. Here, $P$ is perimeter of $T$.
In \cite{NazarovRepin2014}, exact values of $\CPGamma$ and $\CtrGamma$ are found for parallelepipeds, rectangles, and right triangles. Subsequently, we exploit the following two results:
\begin{itemize} \item[1.] If $T$ is based on vertexes $A = (0, 0)$, $B = (h, 0)$, $C = (0, h)$ and
$\Gamma := \big \{ x_1 \in [0, h], \, x_2 = 0 \big \}$ (i.e., $\Gamma$ coincides with one of the legs of the isosceles right triangle), then
\begin{equation}
\CPGamma = \tfrac{h}{\zeta_0} \quad \mathrm{and} \quad
\CtrGamma = \left(\tfrac{h}{\hat{\zeta}_0 \,
\tanh({\hat{\zeta}}_0)}\right)^{\rfrac{1}{2}},
\label{eq:exact-cp-cg-t-leg} \end{equation}
where $\zeta_0$ and $\hat{\zeta}_0$ are unique roots of the equations
\begin{equation} z \cot(z) + 1 = 0 \quad \mbox{and} \quad \tan(z) + \tanh(z) = 0, \label{eq:roots} \end{equation}
respectively, in the interval $(0, \pi)$ .
\item[2.] If $T$ is based on vertexes $A = (0, 0)$, $B = (h, 0)$, $C = \big(\tfrac{h}{2}, \tfrac{h}{2} \big)$, and $\Gamma$ coincides with the hypotenuse of the isosceles right triangle, then
\begin{equation*}
\CPGamma = \tfrac{h}{2 \zeta_0} \quad \mathrm{and} \quad
\CtrGamma = \big(\tfrac{h}{2}\big)^{\rfrac{1}{2}}.
\end{equation*}
\end{itemize}
It is worth emphasizing that values of $\CtrGamma$ for right isosceles triangles follow from the exact solutions of the Steklov problem related to the square. This specific case was discussed in the work \cite{ArxivGirouardPolterovich}.
Exact value of constants in the classical Poincar\'{e} inequality are also known for certain triangles:
\begin{itemize} \item[1.] For the equilateral triangle $\Tref_{\rfrac{\pi}{3}}$ based on vertexes $\hat{A} =(0, 0)$, $\hat{B} = (1, 0)$, $\hat{C} = \big(\tfrac{1}{2}, \tfrac{\sqrt{3}}{2}\big)$, where \linebreak
$\hat{\Gamma} := \big\{ x_1 \in [0, 1] ; \;\; x_2 = 0 \big\}$,
the constant
$$C^{\mathrm{P}}_{\Tref,\, \rfrac{\pi}{3}} = \tfrac{3}{4 \pi}$$
is derived in \cite{Pinsky1980}.
\item[2.] For the right isosceles triangles
$\Tref_{\rfrac{\pi}{4}}$ based on vertexes $\hat{A} =(0, 0)$, $\hat{B} = (1, 0)$, $\hat{C} = \big(\tfrac{1}{2}, \tfrac{1}{2}\big)$ and $\Tref_{\rfrac{\pi}{2}}$ based on $\hat{A} = (0, 0)$, $\hat{B} = (1, 0)$, $\hat{C} = (0, 1)$,
we have
$$\CPTrefpifour = \tfrac{1}{\sqrt{2}\pi} \quad \mbox{and} \quad \CPTrefpitwo = \tfrac{1}{\pi},$$
respectively. Proofs can be found in \cite{HoshikawaUrakawa2010} and \cite{NakaoYamamoto2001I}. \end{itemize}
Explicit formulas of the same constants for certain three-dimensional domains are presented in papers \cite{Berard1980} and \cite{HoshikawaUrakawa2010}.
The above mentioned results form a basis for deriving sharp bounds of the constants $\CPGamma$, $\CtrGamma$, and ${C^{\mathrm P}_{T}}$ for arbitrary non-degenerate triangles and tetrahedrons, which are typical objects in various discretization methods. In Section \ref{sc:arbitrary-triangle}, we deduce guaranteed and easily computable bounds of $\CPGamma$, $\CtrGamma$, and ${C^{\mathrm P}_{T}}$ for triangular domains. The efficiency of these bounds is tested in Section \ref{eq:numerial-tests-2d}, where $\CPGamma$ and $\CtrGamma$ are compared with lower bounds computed numerically by solving a generalized eigenvalue problem generated by Rayleigh quotients discretized
over sufficiently representative sets of trial functions. In the same section, we make a similar comparison of numerical lower bounds related to the constant ${C^{\mathrm P}_{T}}$ with obtained upper bounds and existing estimates known from \cite{LaugesenSiudeja2009,LaugesenSiudeja2010} and \cite{Cheng1975}. Lower bounds of the constants presented in Section \ref{eq:numerial-tests-2d} have been computed by two independent codes: the first code is based on the MATLAB Symbolic Math Toolbox \cite{Matlab}, and the second one uses The FEniCS Project \cite{LoggMardalWells2012}. Section \ref{eq:numerial-tests-3d} is devoted to tetrahedrons. We combine numerical and theoretical estimates in order to derive two-sided bounds of the constants. Finally, in Section \ref{sec:example} we present an example that shows one possible application of the estimates considered in previous sections. Here, the constants are used in order to deduce a guaranteed and fully computable upper bound of the distance between the exact solution of an elliptic boundary value problem and an arbitrary function (approximation) in the respective energy space.
\section{Majorants of $\CPGamma$ and $\CtrGamma$ for triangular domains} \label{sc:arbitrary-triangle}
Let $T$ be based on vertexes $A = (0, 0)$, $B = (h ,0)$, and $C = \big(h \rho \cos\alpha, \, h \rho \, \sin\alpha \big)$
and \begin{equation} \Gamma := \big\{ x_1 \in [0, h] ; \;\; x_2 = 0 \big\}, \label{eq:arbitrary-gamma} \end{equation}
where $\rho>0$, $h>0$, and $\alpha \in (0,\pi)$ are geometrical parameters that fully define a triangle $T$ (see Fig. \ref{eq:2d-simplex}). Easily computable bounds of $C^{\mathrm{P}}_{\Gamma}$ and $C^{\mathrm{Tr}}_{\Gamma}$ are presented in Lemma \ref{th:lemma-poincare-type-constants}
below, which uses mappings of reference triangles to $T$ and well-known integral transformations (see, e.g., \cite{Ciarlet1978}).
\begin{figure}
\caption{Simplex in $\Rtwo$.}
\label{eq:2d-simplex}
\end{figure}
\begin{lemma} \label{th:lemma-poincare-type-constants} For any $w \in \tildeH{1}( T, \Gamma)$, the upper bounds of constants in the inequalities
\begin{alignat}{2}
\|w\|_{ T} \, & \leq\, \CPGamma \, h \,\|\nabla w\|_{ T}
\quad \mathrm{and} \quad
\|w\|_\Gamma \, & \leq\, \CtrGamma \, h^{\rfrac{1}{2}} \, \|\nabla w\|_{ T}
\label{eq:poincare-type-inequalities} \end{alignat}
are defined as
\begin{equation*}
\CPGamma \leq \CP = \min \Big \{ \cpleg \, \CPTrefleg, \: \cphyp \, \CPTrefhyp \Big \}
\quad \mathrm{and} \quad
\CtrGamma \leq \CG = \min \Big \{ \cgleg \, \CGTrefleg, \: \cghyp \, \CGTrefhyp \Big \},
\end{equation*}
respectively. Here,
\begin{equation*}
\cpleg = \muleg^{\rfrac{1}{2}},\quad
\cphyp = \muhyp^{\rfrac{1}{2}}, \quad
\cgleg = \big( \rho \, \sin\alpha \big)^{-\rfrac{1}{2}} \, \cpleg, \quad
\cghyp = \big( 2 \rho \,\sin\alpha\big)^{-\rfrac{1}{2}}\, \cphyp, \end{equation*}
where
\begin{alignat}{2}
\muleg (\rho, \alpha)
= \,& \tfrac{1}{2}
\Big( 1 + \rho^2 +
\big( 1 + \rho^4 + 2 \, \rho^2 \,\cos2\alpha \big)^{\rfrac{1}{2}} \Big),
\label{eq:mu-leg}\\
\muhyp (\rho, \alpha)
= \,& 2 \rho^2 - 2 \rho \, \cos\alpha + 1 +
\big( (2 \rho^2 + 1)
(2 \rho^2 + 1 - 4 \rho \, \cos\alpha + 4 \rho^2 \, \cos2\alpha) \big )^{\rfrac{1}{2}},
\label{eq:mu-hyp} \end{alignat}
and $\CPTrefleg \approx 0.49291$, $\CGTrefleg \approx 0.65602$ and $\CPTrefhyp \approx 0.24646$, $\CGTrefhyp \approx 0.70711$, where $\hat{\Gamma}$ is defined as
\begin{equation} \hat{\Gamma} := \big\{ x_1 \in [0, 1] ; \;\; x_2 = 0 \big\}. \label{eq:arbitrary-gamma-hat} \end{equation}
\end{lemma}
\noindent{\bf Proof:} \: Consider the linear mapping $\mathcal{F}_{\rfrac{\pi}{2}} : \Tref_{\rfrac{\pi}{2}} \rightarrow T$ with
\begin{equation*}
x = \mathcal{F}_{\rfrac{\pi}{2}} \, (\hat{x}) = B_{\rfrac{\pi}{2}} \, \hat{x},
\quad \mbox{where} \quad
B_{\rfrac{\pi}{2}} =
\begin{pmatrix}
\: h & \rho h \cos\alpha \\[0.3em]
0 & \rho h \sin\alpha \:
\end{pmatrix}
, \quad
\mathrm{det} B_{\rfrac{\pi}{2}} = \rho h^2 \, \sin\alpha.
\label{eq:transformation-1} \end{equation*}
For any $\hat{w} \in \tildeH{1}(\Tref_{\rfrac{\pi}{2}}, \Gref)$, we have the estimate
\begin{equation}
\|\, \hat{w} \,\|_{ \Tref_{\rfrac{\pi}{2}} }
\leq \CPTrefleg \, \|\, \nabla \hat{w} \,\|_{\Tref_{\rfrac{\pi}{2}}},
\label{eq:poicare-inequality-1-hatv} \end{equation}
where $\CPTrefleg$ is the constant associated with the basic simplex $\Tref_{\rfrac{\pi}{2}}$ based on $\hat{A} =(0, 0)$, $\hat{B} = (1, 0)$, and $\hat{C} = (0, 1)$,
Note that
\begin{alignat}{2}
\| \, \hat{w} \, \|^2_{\Tref_{\rfrac{\pi}{2}}}
= \tfrac{1}{\rho h^2 \sin\alpha } \| \,w\, \|^2_{ T},
\label{eq:v-norm-transformation} \end{alignat}
and
\begin{equation}
\| \,\nabla \hat w \, \|^2_{\Tref_{\rfrac{\pi}{2}}}
\leq \tfrac{1}{\rho h^2 \sin\alpha} \Int_{T}
A_{\rfrac{\pi}{2}}(h, \rho,\alpha) \nabla w \cdot \nabla w \dx,
\label{eq:grad-hatv-lower-estimate-1} \end{equation}
where
\begin{equation*}
A_{\rfrac{\pi}{2}} (h, \rho,\alpha) = h^2 \,
\begin{pmatrix}
1 + \rho^2 \, \cos^2\alpha \qquad & \rho^2 \sin\alpha \, \cos\alpha \; \\[0.3em]
\rho^2 \sin\alpha \, \cos\alpha & \rho^2 \sin^2\alpha
\end{pmatrix}.
\end{equation*}
It is not difficult to see that $\lambda_{\rm max} (A_{\rfrac{\pi}{2}}) = h^2 \muleg(\rho, \alpha)$,
where $\muleg(\rho, \alpha)$ is defined in \eqref{eq:mu-leg}. From \eqref{eq:poicare-inequality-1-hatv}, \eqref{eq:v-norm-transformation}, and \eqref{eq:grad-hatv-lower-estimate-1}, it follows that
\begin{equation}
\|\, w \,\|_{ T} \, \leq \,
\cpleg \, \CPTrefleg \, h \, \, \|\, \nabla w \,\|_{ T}, \quad
\cpleg (\rho, \alpha) = \muleg^{\rfrac{1}{2}}(\rho, \alpha).
\label{eq:poicare-inequality-1-v} \end{equation}
Notice that $\hat{w} \in \tildeH{1} (\Tref, \Gref)$ yields
\begin{equation*} \mean{w}_{\Gamma} := \Int_{\Gamma} w(x) \ds = h \Int_{\Gref} w(x(\hat{x})) \, {\rm d \hat{s}} = h \Int_{\Gref} \hat{w} \, \rm{d \hat{s}} = 0. \end{equation*}
Therefore, above mapping keeps $w \in \tildeH{1} (T, \Gamma)$.
In view of inequality (\ref{eq:Cgamma}), for any $\hat{w} \in \tildeH{1}(\Tref_{\rfrac{\pi}{2}}, \Gref)$ we have
\begin{equation*}
\|\, \hat{w} \,\|_{\Gref}
\leq \CGTrefleg \|\, \nabla \hat{w} \,\|_{\Tref_{\rfrac{\pi}{2}}},
\label{eq:poicare-inequality-2-hatv} \end{equation*}
where $\CGTrefleg$ is the constant associated with the reference simplex $\Tref_{\rfrac{\pi}{2}}$.
Since
\begin{equation*}
\| \,\hat{w}\, \|^2_{\Gref} = \tfrac{1}{h} \| \, w \, \|^2_{\Gamma},
\end{equation*}
we obtain
\begin{alignat}{2}
\|\, w \,\|_{\Gamma} \,
\leq \, \cgleg \, \CGTrefleg \, h^{\rfrac{1}{2}} \|\, \nabla w \,\|_{ T}, \quad
\cgleg (\rho, \alpha) = \Big(\tfrac{\muleg(\rho, \alpha)}
{\rho \sin\alpha}\Big)^{\rfrac{1}{2}}.
\label{eq:poincare-2-v-cgleg} \end{alignat}
Now, we consider the mapping $\mathcal{F}_{\rfrac{\pi}{4}} : \Tref_{\rfrac{\pi}{4}} \rightarrow T$, where $\Tref_{\rfrac{\pi}{4}}$ is based on $\hat{A} =(0, 0)$, $\hat{B} = (1, 0)$, and $\hat{C} = (\tfrac{1}{2}, \tfrac{1}{2})$, i.e.,
\begin{equation*}
x = \mathcal{F}_{\rfrac{\pi}{4}} (\hat{x}) = B_{\rfrac{\pi}{4}} \, \hat{x},
\quad {\rm where}\quad
B_{\rfrac{\pi}{4}} =
\begin{pmatrix}
\: h & \; 2 \rho h \cos\alpha \, \scalebox{0.5}[1.0]{\( - \)} \, h \\[0.3em]
0 & \; 2 \rho h \sin\alpha \:
\end{pmatrix},
\quad
\mathrm{det} \, B_{\rfrac{\pi}{4}} = 2 \rho h^2 \, \sin\alpha, \end{equation*}
which yields another pair of estimates for the functions in $\tildeH{1}( T, \Gamma)$:
\begin{equation}
\|\, w \,\|_{T} \, \leq \, \cphyp \, \CPTrefhyp \, h \, \|\, \nabla w \,\|_{T}, \quad
\cphyp (\rho, \alpha) = \muhyp^{\rfrac{1}{2}}(\rho, \alpha),
\label{eq:poicare-inequality-2-v} \end{equation}
and
\begin{alignat}{2}
\|\, w \,\|_{\Gamma} \, \leq \, \cghyp \, \CGTrefhyp \,
h^{\rfrac{1}{2}} \|\, \nabla w \,\|_{ T}, \quad
\cghyp (\rho, \alpha)
= \Big(\tfrac{\muhyp(\rho, \alpha)}{2 \rho \sin\alpha}\Big)^{\rfrac{1}{2}},
\label{eq:poincare-2-v-cghyp} \end{alignat}
where $\muhyp(\rho, \alpha)$ is defined in (\ref{eq:mu-hyp}).
Now, (\ref{eq:poincare-type-inequalities}) follows from (\ref{eq:poicare-inequality-1-v}), (\ref{eq:poincare-2-v-cgleg}), (\ref{eq:poicare-inequality-2-v}), and (\ref{eq:poincare-2-v-cghyp}). {
$\square$}
\vskip10pt
Analogously to Lemma \ref{th:lemma-poincare-type-constants}, one can obtain an upper bound of the constant in \eqref{eq:classical-poincare-constant}. For that we consider three reference triangles $\Tref_{\rfrac{\pi}{2}}$, $\Tref_{\rfrac{\pi}{4}}$ (defined earlier), and $\Tref_{\rfrac{\pi}{3}}$ based on vertexes $A = (0, 0)$, $B = (1, 0)$, $C = (\tfrac{1}{2}, \tfrac{\sqrt{3}}{2})$.
\begin{lemma} \label{th:lemma-poincare-constants} For any $w \in \tildeH{1}(\T)$, the constant in
\begin{alignat}{2}
\|w\|_{\T} \, & \leq\, \CPoincare h \,\|\nabla w\|_{\T},
\label{eq:poincare-inequality} \end{alignat}
is estimated as
\begin{equation}
\CPoincare \leq \CPMR
= \min \Big \{ \cpifour \, \CPTrefpifour , \:
\cpithree \, \CPTrefpithree, \:
\cpitwo \, \CPTrefpitwo \Big\}.
\label{eq:classical-poincare-constant-estimate} \end{equation}
Here,
$\cpifour = \mupifour^{\rfrac{1}{2}}$, \quad $\cpithree = \mupithree^{\rfrac{1}{2}}$,\quad $\cpitwo = \mupitwo^{\rfrac{1}{2}}$, where
$\mupitwo$ and $\mupifour$ being defined in \eqref{eq:mu-leg} and \eqref{eq:mu-hyp}, respectively, and
\begin{alignat}{2}
\mupithree (\rho, \alpha)
= \,& \tfrac{2}{3} (1 + \rho^2 - \rho \,\cos\alpha) +
2 \big( \tfrac{1}{9} (1 + \rho^2 - \rho \,\cos\alpha)^2 - \tfrac{1}{3} \rho^2 \sin^2\alpha \big )^{\rfrac{1}{2}},
\label{eq:mu-pi3}
\end{alignat}
and $\CPTrefpifour = \tfrac{1}{\sqrt{2}\pi}$, $\CPTrefpithree = \tfrac{3}{4 \pi}$, and $\CPTrefpitwo = \tfrac{1}{\pi}$. \end{lemma}
\noindent{\bf Proof:} \:
The mapping $\mathcal{F}_{\rfrac{\pi}{2}}: \Tref_{\rfrac{\pi}{2}} \rightarrow \T$ coincides with \eqref{eq:transformation-1} from Lemma \ref{th:lemma-poincare-type-constants}. It is easy to see that $w \in \tildeH{1} (T)$ provides that $\hat{w} \in \tildeH{1} (\Tref)$. The estimate
\begin{equation}
\|\, w \,\|_{\T} \, \leq \,
\cpitwo \, \CPTrefpitwo \, h \, \, \|\, \nabla w \,\|_{\T}, \quad
\cpitwo (\rho, \alpha) = \mupitwo^{\rfrac{1}{2}}(\rho, \alpha)
\label{eq:classical-poincare-inequality-1-v} \end{equation}
is obtained by following steps of the previous proof.
From analysis of mappings
\begin{equation*}
x = \mathcal{F}_{\rfrac{\pi}{3}} (\hat{x}) = B_{\rfrac{\pi}{3}} \, \hat{x},
\quad \mbox{ where}\quad
B_{\rfrac{\pi}{3}} =
\begin{pmatrix}
\: h & \; \tfrac{h}{\sqrt{3}} (2 \rho \cos\alpha - 1) \, \scalebox{0.5}[1.0]{\( - \)} \, h \\[0.3em]
0 & \; \tfrac{2 h}{\sqrt{3}} \rho \sin\alpha \:
\end{pmatrix}
,\quad
\mathrm{det} \, B_{\rfrac{\pi}{3}} = \tfrac{2 h^2}{\sqrt{3}} \, \sin\alpha > 0, \end{equation*}
and
\begin{equation*}
x = \mathcal{F}_{\rfrac{\pi}{4}} (\hat{x}) = B_{\rfrac{\pi}{4}} \, \hat{x},
\quad {\rm where}\quad
B_{\rfrac{\pi}{4}} =
\begin{pmatrix}
\: h & \; 2 \rho h \cos\alpha \, \scalebox{0.5}[1.0]{\( - \)} \, h \\[0.3em]
0 & \; 2 \rho h \sin\alpha \:
\end{pmatrix}
,\quad
\mathrm{det} \, B_{\rfrac{\pi}{4}} = 2 \rho h^2 \, \sin\alpha > 0, \end{equation*}
we obtain alternative estimates
\begin{alignat}{2}
\|\, w \,\|_{\T} \, & \leq \,
\cpithree \, \CPTrefpithree \, h \, \|\, \nabla w \,\|_{\T}, \quad
\cpithree (\rho, \alpha) = \mupithree^{\rfrac{1}{2}}(\rho, \alpha),
\label{eq:classical-poincare-inequality-2-v} \\
\|\, w \,\|_{\T} \, & \leq \,
\cpifour \, \CPTrefpifour \, h \, \|\, \nabla w \,\|_{\T}, \quad
\cpifour (\rho, \alpha) = \mupifour^{\rfrac{1}{2}}(\rho, \alpha),
\label{eq:classical-poincare-inequality-3-v} \end{alignat}
where $\mupithree(\rho, \alpha)$ and $\mupifour(\rho, \alpha)$ are defined in \eqref{eq:mu-pi3} and \eqref{eq:mu-hyp}, respectively.
Therefore, (\ref{eq:classical-poincare-constant-estimate}) follows from combination of (\ref{eq:classical-poincare-inequality-1-v}), (\ref{eq:classical-poincare-inequality-2-v}), and (\ref{eq:classical-poincare-inequality-3-v}).
{
$\square$}
\section{Minorants of $\CPGamma$ and $\CtrGamma$ for triangular domains} \label{eq:numerial-tests-2d}
\subsection{Two-sided bounds of $\CPGamma$ and $\CtrGamma$}
Majorants of $\CPGamma$ and $\CtrGamma$ provided by Lemma \ref{th:lemma-poincare-type-constants} should be compared with the corresponding minorants, which can be found by means of
the Rayleigh quotients
\begin{equation}
\mathcal{R}^{\mathrm{P}}_{\Gamma} [w]
= \tfrac{\|\nabla w\|_{\T}}{\|w - \mean{w}_{\Gamma} \|_{\T}}
\quad \mathrm{and} \quad
\mathcal{R}^{\mathrm{Tr}}_{\Gamma} [w]
= \tfrac{\|\nabla w\|_{\T}}{\|w - \mean{w}_{\Gamma} \|_{\Gamma}}.
\label{eq:quotients-2d} \end{equation}
Lower bounds are obtained if the quotients are minimized on finite dimensional subspaces
$V^N \subset \H{1}(T)$ formed by sufficiently representative collections of suitable test functions. For this purpose, we use either power or Fourier series and introduce the spaces
\begin{equation*}
V^{N}_1 := \mathrm{span} \big\{\: x^{i} y^{j}\: \big\} \quad {\rm and } \quad
V^{N}_2 := \mathrm{span} \big\{\: \cos (\pi i x) \cos (\pi j y)\: \big\}, \;\;
\end{equation*}
where $i, j = 0, \ldots, N, \;\; (i, j) \neq (0, 0)$ and
$$\mathrm{dim} \, V^{N}_1 = \mathrm{dim} \, V^{N}_2 = M(N) := (N + 1)^2 - 1.$$
The corresponding constants are denoted by $\approxCPT$ and $\approxCGT$, where $M$ indicates on number of basis functions in auxiliary subspace used.
Since $ V^{N}_1$ and $V^{N}_2$ are limit dense in $\H{1}(\T)$, the respective minorants tend to the exact constants as $M(N)$ tends to infinity.
We note that
\begin{equation}
\inf\limits_{w \in \H{1}(T)} \mathcal{R}^{\mathrm{P}}_{\Gamma} [w] =
\inf\limits_{w \in \H{1}(T)}
\tfrac{\|\nabla w\|_{\T}}{\|w - \mean{w}_{\Gamma} \|_{\T}}
=
\inf\limits_{w \in \tildeH{1}(T, \Gamma)}
\tfrac{\|\nabla w\|_{\T}}{\|w\|_{\T}} = \tfrac{1}{\CPGamma}. \end{equation}
Therefore, minimization of the first quotient in \eqref{eq:quotients-2d} on $V^{N}_1$ or $V^{N}_2$ yields a lower bound of $\CPGamma$.
For the quotient $\mathcal{R}^{\mathrm{Tr}}_{\Gamma} [w]$, we apply similar arguments.
Numerical results presented below are obtained with the help of two different codes based on the MATLAB Symbolic Math Toolbox \cite{Matlab} and The FEniCS Project \cite{LoggMardalWells2012}.
Table \ref{tab:const-convergence-from-basis} demonstrates the ratios between the exact constants and respective approximate values (for the selected $\rho$ and $\alpha$). They are quite close to $1$ even for relatively small $N$. Henceforth, we select $N = 6$ or $7$ in the tests discussed below.
\begin{table}[!ht] \centering \footnotesize
\begin{tabular}{cc|cc|cc}
\multicolumn{2}{c|}{$ $}
& \multicolumn{2}{c|}{ $\alpha = \tfrac{\pi}{2}$, $\rho = 1$ } & \multicolumn{2}{c}{ $\alpha = \tfrac{\pi}{4}$, $\rho = \tfrac{\sqrt{2}}{2}$} \\ \midrule $N$ & $M(N)$ & ${\approxCPT}/{\CPTrefleg}$ & ${\approxCGT}/{\CGTrefleg}$ & ${\approxCPT}/{\CPTrefhyp}$ & ${\approxCGT}/{\CGTrefhyp}$ \\ \midrule 1 & 3 & 0.8801 & 0.9561 & 0.8647 & 1.0000 \\ 2 & 8 & 0.9945 & 0.9898 & 0.9925 & 1.0000 \\ 3 & 15 & 0.9999 & 0.9998 & 0.9962 & 1.0000 \\ 4 & 24 & 1.0000 & 0.9999 & 1.0000 & 1.0000 \\ 5 & 35 & 1.0000 & 1.0000 & 1.0000 & 1.0000 \\ 6 & 48 & 1.0000 & 1.0000 & 1.0000 & 1.0000 \\ \end{tabular} \\[5pt] \caption{Ratios between approximate and reference constants with respect to increasing $N$.} \label{tab:const-convergence-from-basis} \end{table}
\begin{figure}
\caption{Two-sided bounds of $\CPT$
for $\T$ with different $\rho$.}
\label{fig:2d-cpt-rho-sqrt2-2}
\label{eq:t-rho-sqrt2-2}
\label{fig:2d-cpt-rho-1}
\label{eq:t-rho-1}
\label{fig:2d-cpt}
\end{figure}
\begin{figure}
\caption{Two-sided bounds of $\CGT$
for $\T$ with different $\rho$.}
\label{fig:2d-cgt-rho-sqrt2-2}
\label{fig:2d-cgt-rho-1}
\label{fig:2d-cgt}
\end{figure}
In Figs. \ref{fig:2d-cpt-rho-sqrt2-2} and \ref{fig:2d-cpt-rho-1}, we depict $\approxCPT$ for $M(N) = 48$ (thin red line) for $T$ with $\rho = \tfrac{\sqrt{2}}{2}$, $1$, and $\alpha \in (0, \pi)$. Guaranteed upper bounds $\CPTleg = \cpleg \, \CPTrefleg$ and $\CPThyp = \cphyp \, \CPTrefhyp$ are depicted by dashed black lines. Bold blue line illustrates $\CP = \min \Big \{\CPTleg, \CPThyp \Big \}$. Analogously in Figs. \ref{fig:2d-cgt-rho-sqrt2-2} and \ref{fig:2d-cgt-rho-1}, a red marker denotes the lower bound $\approxCGT$ (for $M(N) = 48$) of the constant $\CtrGamma$. It is presented together with the upper bound $\CG$ (blue marker), which is defined as minimum of $\CGTleg = \cgleg \, \CGTrefleg$ and $\CGThyp = \cghyp \, \CGTrefhyp$.
Table \ref{tab:big-table} represents this information in the digital form.
\begin{table}[!ht] \centering \footnotesize
\begin{tabular}{c|cc|cc|cc|cc}
\multicolumn{1}{c|}{$ $}
& \multicolumn{4}{c|}{ $\rho = \tfrac{\sqrt{2}}{2}$ } & \multicolumn{4}{c}{ $\rho = 1$ } \\ \midrule $\alpha$ & $\underline{C}^{48, \mathrm{P}}_{\Gamma}$ & $\CP$ & $\underline{C}^{48, \mathrm{Tr}}_{\Gamma}$ & $\CG$ & $\underline{C}^{48, \mathrm{P}}_{\Gamma}$ & $\CP$ & $\underline{C}^{48, \mathrm{Tr}}_{\Gamma}$ & $\CG$ \\ \midrule $\pi/18$ & 0.2429 & 0.2657 & 1.2786 & 1.5386 & 0.3245 & 0.3486 & 1.2572 & 1.6971\\ $\pi/9$ & 0.2414 & 0.2627 & 0.9289 & 1.0838 & 0.3248 & 0.3493 & 0.9058 & 1.2116\\ $\pi/6$ & 0.2389 & 0.2577 & 0.7919 & 0.8792 & 0.3268 & 0.3527 & 0.7632 & 1.0118\\ $2\pi/9$ & 0.2379 & 0.2507 & 0.7259 & 0.7543 & 0.3339 & 0.3636 & 0.6906 & 0.9201\\ $5\pi/18$ & 0.2632 & 0.2722 & 0.6945 & 0.7503 & 0.3514 & 0.3884 & 0.6529 & 0.9003\\ $\pi/3$ & 0.3008 & 0.3220 & 0.6829 & 0.8348 & 0.3809 & 0.4269 & 0.6362 & 0.8634\\ $7\pi/18$ & 0.3382 & 0.3694 & 0.6840 & 0.8432 & 0.4173 & 0.4721 & 0.6332 & 0.7840\\ $4\pi/9$ & 0.3740 & 0.4140 & 0.6947 & 0.7973 & 0.4556 & 0.5187 & 0.6404 & 0.7162\\ $\pi/2$ & 0.4075 & 0.4554 & 0.7136 & 0.7801 & 0.4929 & 0.4929 & 0.6560 & 0.6560\\ $5\pi/9$ & 0.4382 & 0.4933 & 0.7409 & 0.7973 & 0.5280 & 0.5340 & 0.6797 & 0.7162\\ $11\pi/18$ & 0.4660 & 0.5165 & 0.7779 & 0.8432 & 0.5600 & 0.5710 & 0.7125 & 0.7840\\ $2\pi/3$ & 0.4905 & 0.5361 & 0.8274 & 0.9118 & 0.5884 & 0.6037 & 0.7569 & 0.8634\\ $13\pi/18$ & 0.5115 & 0.5552 & 0.8948 & 1.0040 & 0.6129 & 0.6318 & 0.8175 & 0.9607\\ $7\pi/9$ & 0.5289 & 0.5720 & 0.9898 & 1.1292 & 0.6332 & 0.6550 & 0.9033 & 1.0874\\ $5\pi/6$ & 0.5426 & 0.5856 & 1.1334 & 1.3107 & 0.6492 & 0.6733 & 1.0332 & 1.2673\\ $8\pi/9$ & 0.5524 & 0.5956 & 1.3796 & 1.6118 & 0.6607 & 0.6865 & 1.2565 & 1.5623\\ $17\pi/18$ & 0.5583 & 0.6017 & 1.9436 & 2.2851 & 0.6676 & 0.6944 & 1.7692 & 2.2179\\ \end{tabular} \caption{Two-sided bounds of $\CPT$ and $\CGT$ for $\T$ for $\alpha \in (0, \pi)$ and different $\rho$.} \label{tab:big-table} \end{table}
Fig. \ref{fig:2d-cpt-rho-sqrt2-2} corresponds to the case $\rho = \tfrac{\sqrt{2}}{2}$. Notice that for $\alpha = \tfrac{\pi}{4}$ the constant $\CPGamma$ is known and the computed lower bound $\approxCPT$ (red marker) practically coincides with it (see, e.g., Fig. \ref{eq:t-rho-sqrt2-2}). Since in this case, the mapping $\mathcal{F}_{\rfrac{\pi}{4}}$ is identical, the upper bound also coincides with the exact value. An analogous coincidence can be observed for $\CtrGamma$ and $\approxCGT$ in Fig. \ref{fig:2d-cgt-rho-sqrt2-2}.
In Fig. \ref{fig:2d-cpt-rho-1}, the red curve, corresponding to $\approxCPT$, coincides with the blue line of $\CPT$ at the point $\alpha = \tfrac{\pi}{2}$ (due to the fact that for this angle $\mathcal{F}$ is the identical mapping and $T$ coincides with $\Tref_{\rfrac{\pi}{2}}$ (see Fig. \ref{eq:t-rho-1})). Fig. \ref{fig:2d-cgt-rho-1} exposes similar results for ${\approxCGT}$ and $\CtrGamma$ ($\CGTleg$).
\begin{figure}
\caption{Two-sided bounds of $\CPT$
for $\T$ with different $\rho$.}
\label{fig:2d-cpt-rho-sqrt3-2}
\label{fig:2d-cpt-rho-3-2}
\label{fig:2d-cpt-no-ref}
\end{figure}
\begin{figure}
\caption{Two-sided bounds of $\CGT$
for $\T$ with different $\rho$.}
\label{fig:2d-cgt-rho-sqrt3-2}
\label{fig:2d-cgt-rho-3-2}
\label{fig:2d-cgt-no-ref}
\end{figure}
Figs. \ref{fig:2d-cpt-no-ref} and \ref{fig:2d-cgt-no-ref} demonstrate the same bounds for $\rho = \tfrac{\sqrt{3}}{2}$ and $\tfrac{3}{2}$. We see that estimates of $\CPT$ and $\CGT$ are very efficient. Namely, $\Ieff^{\rm P} := \tfrac{\CP}{\underline{C}^{48, \mathrm{P}}_{\Gamma}} \in [1.0463, 1.1300]$ for $\rho = \tfrac{\sqrt{3}}{2}$ and
$\Ieff^{\rm P} \in [1.0249, 1.1634]$ for $\rho = \tfrac{3}{2}$. Analogously, $\Ieff^{\rm Tr} := \tfrac{\CG}{\underline{C}^{48, \mathrm{Tr}}_{\Gamma}} \in [1.0363, 1.3388]$ for $\rho = \tfrac{\sqrt{3}}{2}$ and $\Ieff^{\rm Tr} \in [1.2917, 1.7643]$ for $\rho = \tfrac{3}{2}$.
\subsection{Two-sided bounds of ${C^{\mathrm P}_{T}}$}
The spaces $V^{N}_1$ and $ V^{N}_2$ can also be used for analysis of the quotient
$\mathcal{R}_{T} [w] = \tfrac{\|\nabla w\|_{ T}}{\|w - \mean{w}_{ T} \|_{ T}}$, which yields guaranteed lower bounds of the constant in (\ref{eq:classical-poincare-constant}). The respective values are denoted by $\approxC$. These bounds are compared with $\CPupper := \tfrac{\diam (\T)}{j_{1, 1}}$ and $\CPlower := \max \Big \{ \tfrac{\diam (\T)}{2 \,j_{0,1}}, \tfrac{P}{4 \,\pi} \Big \}$ (see \eqref{eq:cheng}--\eqref{eq:ls}, respectively) as well as the one derived in Lemma \ref{th:lemma-poincare-constants}.
\begin{figure}
\caption{$\underline{C}^{48}_{\T}$, $\CPLS$, $\CPMR$, and $\CPlower$ for $\T$ with
$\alpha \in (0, \pi)$ and different $\rho$.}
\label{fig:cp-rho-sqrt2-2}
\label{fig:cp-rho-sqrt3-2}
\label{fig:cp-rho-1}
\label{fig:cp-rho-3-2}
\label{fig:2d-cp-rho}
\end{figure}
In Figs. \ref{fig:cp-rho-sqrt2-2}, \ref{fig:cp-rho-sqrt3-2}, and \ref{fig:cp-rho-3-2}, we present $\approxC$ (in this case $M(N) = 48$) together with $\CPMR$ (blue think line), $\CPupper$ and $\CPlower$ for $\alpha \in (0, \pi)$, and $\rho = \tfrac{\sqrt{2}}{2}$, $\tfrac{\sqrt{3}}{2}$, and $\tfrac{3}{2}$. We see that $\underline{C}^{48, \rm P}_{\T}$ (red thin line) indeed lies within the admissible two-sided bounds. From these figures, it is obvious that new upper bounds $\CPMR$ are sharper than $\CPupper$ for $\T$ with $\rho \neq 1$. True values of the constant lie between the bold blue and thin red lines, but closer to the red one, which practically shows the constant (this follows from the fact that increasing $M(N)$ does not provide a noticeable change for the line, e.g., for $M(N) = 63$ maximal difference with respect to figure does not exceed $1e\minus8$). Also, we note that, the lower bound $\CPlower$ (black dashed line) is quite efficient, and, moreover, asymptotically exact for $\alpha \rightarrow \pi$.
Due to \cite{LaugesenSiudeja2010} and \cite{Bandle1980}, we know the improved upper bound $\CPLS$ (cf. \eqref{eq:improved-estimates}) for isosceles triangles. In Fig. \ref{fig:cp-rho-1}, we compare $\approxC$ ($M(N) = 48$) with both upper bounds $\CPMR$ (from the Lemma \ref{th:lemma-poincare-constants}) and $\CPLS$ (black doted line). It is easy to see that $\CPLS$ (black dashed line) is rather accurate and for $\alpha \rightarrow 0$ and $\alpha \rightarrow \pi$ provide almost exact estimates. $\CPMR$ (blues thick line) improves $\CPLS$ only for some $\alpha$. The lower bound $\underline{C}^{48}_{\T}$ (red thin line) indeed converges to $\CPLS$ as $\T$ degenerates when $\alpha$ tends to $0$.
\subsection{Shape of the minimizer}
Exact constants in (\ref{eq:Comega}) and (\ref{eq:Cgamma}) are generated by the minimal positive eigenvalues of \eqref{eq:eigenvalue-problem-cp} and \eqref{eq:eigenvalue-problem-ctr}. This section presents results related to the respective eigenfunctions. In order to depict all of them in a unified form, we use the barycentric coordinates $\lambda_i \in (0, 1)$, $i = 1, 2, 3$, $\Sum_{i = 1}^{3}\lambda_i = 1$.
\begin{figure}
\caption{Eigenfunctions corresponding to $\approxCPT$ and
for $M = 48$ on simplex $\T$ with $\rho = 1$ and different $\alpha$.}
\label{fig:2d-up-eigenfunctions-pi-6}
\label{fig:2d-up-eigenfunctions-pi-3}
\label{fig:2d-up-eigenfunctions-pi-2}
\label{fig:2d-exact-up-eigenfunctions-pi-2}
\label{fig:2d-up-eigenfunctions-3pi-4}
\label{fig:2d-up-eigenfunctions-5pi-6}
\label{fig:2d-eigenfunctions-cp}
\end{figure}
\begin{figure}
\caption{Eigenfunctions corresponding to
$\approxCGT$ for $M = 48$ on simplex $\T$ with $\rho = 1$ and different $\alpha$.}
\label{fig:2d-ug-eigenfunctions-pi-6}
\label{fig:2d-ug-eigenfunctions-pi-3}
\label{fig:2d-ut-eigenfunctions-pi-2}
\label{fig:2d-exact-ut-eigenfunctions-pi-2}
\label{fig:2d-ug-eigenfunctions-3pi-4}
\label{fig:2d-ug-eigenfunctions-5pi-6}
\label{fig:2d-eigenfunctions-ctr}
\end{figure}
Figs. \ref{fig:2d-eigenfunctions-cp} and \ref{fig:2d-eigenfunctions-ctr} show the eigenfunctions computed for isosceles triangles with different angles $\alpha$ between two legs (zero mean condition is imposed on one of the legs). The eigenfunctions have been computed in the process of finding $\approxCPT$ and $\approxCGT$. The eigenfunctions are normalized so that the maximal value is equal to $1$.
For $\alpha = \tfrac{\pi}{2}$, the exact eigenfunction associated with the smallest positive eigenvalue $\lambda^{\mathrm{P}}_{\Gamma} = \big( \tfrac{z_0}{h}\big)^{2}$ is known (see \cite{NazarovRepin2014}):
\begin{equation*}
u^{\rm P}_{\Gamma}
= \cos(\tfrac{\zeta_0 x_1}{h}) + \cos \big(\tfrac{\zeta_0 (x_2 - h)}{h}\big).
\label{eq:u-pt-exact} \end{equation*}
Here, $\zeta_0$ is the root of the first equation in \eqref{eq:roots} (see Fig \ref{fig:2d-exact-up-eigenfunctions-pi-2}). We can compare $u^{\rm P}_{\Gamma}$ with the approximate eigenfunction $u^{M, \mathrm{P}}_{\Gamma}$ computed by minimization of $\mathcal{R}^{\mathrm{P}}_{\Gamma} [w]$ (this function is depicted in Fig. \ref{fig:2d-up-eigenfunctions-pi-2}).
Eigenfunctions related to the constant $\approxCGT$ are presented in Fig. \ref{fig:2d-eigenfunctions-ctr}. Again, for $\alpha = \tfrac{\pi}{2}$ we know the exact eigenfunction
\begin{equation*}
u^{\rm Tr}_{\Gamma} =
\cos (\hat{\zeta}_0 x_1) \, \cosh \big(\hat{\zeta}_0 (x_2 - h)\big) +
\cosh (\hat{\zeta}_0 \, x_1) \ \cos \big(\hat{\zeta}_0 (x_2 - h)\big),
\label{eq:u-gt-exact} \end{equation*}
where $\hat{\zeta}_0$ is the root of second equation in \eqref{eq:roots} (see Fig. \ref{fig:2d-exact-ut-eigenfunctions-pi-2}). This function minimizes the quotient $\mathcal{R}^{\mathrm{Tr}}_{\Gamma} [w]$ and yields the smallest positive eigenvalue $\lambda^{\mathrm{Tr}}_{\Gamma} = \tfrac{\hat{\zeta}_0 \tanh (\hat{\zeta}_0)}{h}$. It is easy to see that numerical approximation $\approxCGT$ (for $M(N) = 48$) practically coincides with the exact function.
\begin{figure}
\caption{Eigenfunctions corresponding to $\approxC$ with $M = 48$
on isosceles triangles $\T \in \Rtwo$ with $\alpha = \tfrac{\pi}{3}$,
$\tfrac{\pi}{3} - \varepsilon$, and
$\tfrac{\pi}{3} + \varepsilon$ in barycentric coordinates.}
\label{fig:u1-rho-1-pi-3-minus-eps}
\label{fig:u2-rho-1-pi-3-minus-eps}
\label{fig:u3-rho-1-pi-3-minus-eps}
\label{fig:u1-rho-1-pi-3}
\label{fig:u2-rho-1-pi-3}
\label{fig:u3-rho-1-pi-3}
\label{fig:u1-rho-1-pi-3-plus-eps}
\label{fig:u2-rho-1-pi-3-plus-eps}
\label{fig:u3-rho-1-pi-3-plus-eps}
\label{fig:u1-u2-u3-rho-1}
\end{figure}
\begin{table}[!ht] \centering \footnotesize
\begin{tabular}{c|c|cc|cc|cc} $$ & $$
& \multicolumn{2}{c|}{ $\tfrac{\pi}{3} - \varepsilon$ }
& \multicolumn{2}{c|}{ $\tfrac{\pi}{3}$ } & \multicolumn{2}{c}{ $\tfrac{\pi}{3} + \varepsilon$ }\\ \midrule $$ & $u^{M}_{\T, i}$ & $\underline{C}^{48}_{\T, i}$ & $\lambda^{48}_{\T, i}$ & $\underline{C}^{48}_{\T, i}$ & $\lambda^{48}_{\T, i}$ & $\underline{C}^{48}_{\T, i}$ & $\lambda^{48}_{\T, i}$ \\ \midrule \multirow{3}{*}{$\rho = 1$} & $u^{48}_{\T, 1}$ & 0.2419 & 17.0951 & 0.2387 & 17.5463 & 0.2537 & 15.5404 \\ & $u^{48}_{\T, 2}$ & 0.2229 & 20.1216 & 0.2387 & 17.5463 & 0.2355 & 18.0309 \\ & $u^{48}_{\T, 3}$ & 0.1353 & 54.6024 & 0.1378 & 52.6396 & 0.1422 & 49.4818 \\ \midrule \multirow{3}{*}{$\rho = \tfrac{\sqrt{2}}{2}$} & $u^{48}_{\T, 1}$ & 0.23137 & 18.6804 & 0.23671 & 17.8471 & 0.24336 & 16.8850 \\ & $u^{48}_{\T, 2}$ & 0.17082 & 34.2707 & 0.17435 & 32.8970 & 0.17642 & 32.1295 \\ & $u^{48}_{\T, 3}$ & 0.1229 & 66.2058 & 0.12789 & 61.1402 & 0.13298 & 56.5493 \\ \midrule \multirow{3}{*}{$\rho = \tfrac{3}{2}$} & $u^{48}_{\T, 1}$ & 0.34714 & 8.2983 & 0.35523 & 7.9247 & 0.3648 & 7.5143 \\ & $u^{48}_{\T, 2}$ & 0.24485 & 16.6801 & 0.24885 & 16.1482 & 0.25125 & 15.8412 \\ & $u^{48}_{\T, 3}$ & 0.18258 & 29.9981 & 0.19084 & 27.4575 & 0.19845 & 25.3921 \\ \end{tabular} \caption{$\approxC$ and $\lambda^{M}_{\T}$ corresponding to the first three eigenfunctions in Fig. \ref{fig:u1-u2-u3-rho-1}.} \label{tab:approx-3-eigenvalues-and-constants} \end{table}
Typically, the eigenfunctions associated with minimal positive eigenvalues expose a continuous evolution with respect to $\alpha$. However, this is not true for the quotient $\mathcal{R}_{\T}[w]$, where the minimizer radically changes the profile. Fig. \ref{fig:cp-rho-1} indicates a possibility of such a rapid change at $\alpha = \tfrac{\pi}{3}$, where the curve (related to $\underline{C}^{48}_{\T}$) obviously becomes non-smooth. This happens because an equilateral triangle has double eigenvalue, therefore the minimizer of $\mathcal{R}_{\T}[w]$ over $V^N_1$ changes its profile. Figs. \ref{fig:u1-rho-1-pi-3-minus-eps}--\ref{fig:u3-rho-1-pi-3-plus-eps} show three eigenfunctions $u^{48}_{\T, 1}$, $u^{48}_{\T, 2}$, and $u^{48}_{\T, 3}$ corresponding to three minimal eigenvalues $\lambda^{48}_{\T, 1}$, $\lambda^{48}_{\T, 2}$, and $\lambda^{48}_{\T, 3}$. All functions are computed for isosceles triangles and are sorted in accordance with increasing values of the respective eigenvalues.
It is easy to see that at $\alpha=\tfrac{\pi}{3}$ the first and the second eigenfunctions swap places. Table \ref{tab:approx-3-eigenvalues-and-constants} presents the corresponding results in the digital form.
It is worth noting that for equilateral triangles two minimal eigenfunctions are known (see \cite{McCartin2002}):
\begin{alignat*}{2}
u_1 & = \cos \Big(\tfrac{2\,\pi}{3} (2\, x_1 - 1)\Big)
- \cos \Big(\tfrac{2\,\pi}{\sqrt{3}} x_2 \Big)
\cos \Big(\tfrac{\pi}{3} (2\,x_1 - 1)\Big), \\
u_2 & = \sin \Big(\tfrac{2\,\pi}{3} (2\, x_1 - 1)\Big)
+ \cos \Big(\tfrac{2\,\pi}{\sqrt{3}} x_2 \Big)
\sin \Big(\tfrac{\pi}{3} (2\,x_1 - 1)\Big). \end{alignat*}
These functions practically coincide with the functions $u^{48}_{\T, 1}$ and $u^{48}_{\T, 2}$ presented in Fig. \ref{fig:u1-rho-1-pi-3}.
Finally, we note that this phenomenon (change of the minimal eigenfunction) does not appear for $\rho = \tfrac{\sqrt{2}}{2}$ or $\rho=\tfrac{3}{2}$.
The eigenvalues as well as the constants corresponding to the eigenfunctions presented in Fig. \ref{fig:u1-u2-u3-rho-1} are shown in the Table \ref{tab:approx-3-eigenvalues-and-constants}.
\section{Two-sided bounds of $\CPGamma$ and $\CtrGamma$ for tetrahedrons} \label{eq:numerial-tests-3d}
\begin{figure}
\caption{Simplex in $\Rthree$.}
\label{eq:3d-simplex}
\caption{Coordinate of the vertex $D$.}
\label{eq:3d-simplex-d-coordinate}
\end{figure}
We orient the coordinates as it is shown in Fig. \ref{eq:3d-simplex} and define a non-degenerate simplex in $\Rthree$ with vertexes \linebreak
$A = (0, 0, 0)$, $B = (h_1, 0, 0)$, $C = (0, 0, h_3)$, and $D = (h_2 \, \sin\theta\, \cos\alpha, h_2 \, \sin\theta\, \sin\alpha, h_2\, \cos\alpha)$,
where $h_1$ and $h_3$ are the scaling parameters along axis $O_{x_1}$ and $O_{x_3}$, respectively, $AD = h_2$, $\alpha$ is a polar angle, and $\theta$ is an azimuthal angle (see Fig. \ref{eq:3d-simplex-d-coordinate}). Let $\Gamma$ be defined by vertexes $A$, $B$, and $C$.
To the best of our knowledge, exact values of constants in Poincar\'{e}-type inequalities for simplexes in $\Rthree$ are unknown. Therefore, we first consider four basic (reference) tetrahedrons with $h_2 = 1$, $\hat{\theta} = \tfrac{\pi}{2}$, and $\hat{\alpha}_1 = \tfrac{\pi}{4}$, $\hat{\alpha}_2 = \tfrac{\pi}{3}$, $\hat{\alpha}_3 = \tfrac{\pi}{2}$, and $\hat{\alpha}_4 = \tfrac{2\pi}{3}$. The respective constants are found numerically with high accuracy (see Table \ref{tab:const-convergence-from-basis-G-leg-alpha}, which shows convergence of the constants with respect to increasing $M(N)$).
Henceforth, $\Tref_{\hat{\theta}, \hat{\alpha}}$ denotes a reference tetrahedron, where $\hat{\theta}$ and $\hat{\alpha}$ are certain fixed angles. By $\mathcal{F}_{\hat{\theta}, \hat{\alpha}}$ we denote the respective mapping $\mathcal{F}_{\hat{\theta}, \hat{\alpha}}: \Tref_{\hat{\theta}, \hat{\alpha}} \rightarrow \T$.
\begin{table}[!t] \centering \footnotesize
\begin{tabular}{c|cc|cc|cc|cc}
\multicolumn{1}{c|}{$ $}
& \multicolumn{2}{c|}{ $\hat{\alpha} = \tfrac{\pi}{4}$}
& \multicolumn{2}{c|}{ $\hat{\alpha} = \tfrac{\pi}{3}$}
& \multicolumn{2}{c|}{ $\hat{\alpha} = \tfrac{\pi}{2}$} & \multicolumn{2}{c}{ $\hat{\alpha} = \tfrac{2\pi}{3}$}\\ \midrule
\multicolumn{1}{c|}{$ $} &
\multicolumn{2}{c|}{ \begin{tikzpicture}[line join = round, line cap = round] \coordinate [label=left:] (A) at (0,0,0); \coordinate [label=above right:] (B) at (1.2,0,0); \coordinate [label=below:] (C) at (0,0,1.2); \coordinate [label=above right:] (D) at ( 0.848, 0.848, 0); \draw[->] (1.2, 0, 0) -- (1.6, 0, 0) node[below] {$\hat{x}_1$}; \draw[->] (0, 0, 0) -- (0, 1.4, 0) node[above] {$\hat{x}_2$}; \draw[->] (0, 0, 1.2) -- (0, 0, 1.6) node[right] {$\hat{x}_3$}; \draw (2.0, 1.05) node[anchor=north east] {$\Tref_{\rfrac{\pi}{4}}$}; \draw (1.0, -0.2) node[anchor=north east] {$\widehat{\Gamma}$}; \draw[-, fill=black!10, opacity=.5, dashed] (A)--(B)--(C)--(A); \draw[-, opacity=.5] (B)--(C); \draw[-, opacity=.5] (C)--(D); \draw[-, opacity=.5] (B)--(D); \draw (0.55,0.0) .. controls (0.5, 0.3) and (0.4, 0.3) .. node[right] {$\hat{\alpha}$} (0.3,0.3); \draw[dashed] (A)--(D); \end{tikzpicture} }
& \multicolumn{2}{c|}{ \begin{tikzpicture}[line join = round, line cap = round] \coordinate [label=left:] (A) at (0,0,0); \coordinate [label=above right:] (B) at (1.2,0,0); \coordinate [label=below:] (C) at (0,0,1.2); \coordinate [label=above right:] (D) at ( 0.6, 1.0392, 0); \draw[->] (1.2, 0, 0) -- (1.6, 0, 0) node[below] {$\hat{x}_1$}; \draw[->] (0, 0, 0) -- (0, 1.4, 0) node[above] {$\hat{x}_2$}; \draw[->] (0, 0, 1.2) -- (0, 0, 1.6) node[right] {$\hat{x}_3$}; \draw (2.0, 1.05) node[anchor=north east] {$\Tref_{\rfrac{\pi}{3}}$}; \draw (1.0, -0.2) node[anchor=north east] {$\widehat{\Gamma}$}; \draw[-, fill=black!10, opacity=.5, dashed] (A)--(B)--(C)--(A); \draw[-, opacity=.5] (B)--(C); \draw[-, opacity=.5] (C)--(D); \draw[-, opacity=.5] (B)--(D); \draw (0.55,0.0) .. controls (0.5, 0.3) and (0.4, 0.3) .. node[right] {$\hat{\alpha}$} (0.27,0.4); \draw[dashed] (A)--(D); \end{tikzpicture} }
& \multicolumn{2}{c|}{ \begin{tikzpicture}[line join = round, line cap = round] \coordinate [label=left:] (A) at (0,0,0); \coordinate [label=above right:] (B) at (1.2,0,0); \coordinate [label=below:] (C) at (0,0,1.2); \coordinate [label=above right:] (D) at ( 0.0, 1.2, 0); \draw[->] (1.2, 0, 0) -- (1.6, 0, 0) node[below] {$\hat{x}_1$}; \draw[->] (0, 1.2, 0) -- (0, 1.4, 0) node[above] {$\hat{x}_2$}; \draw[->] (0, 0, 1.2) -- (0, 0, 1.6) node[right] {$\hat{x}_3$}; \draw (2.0, 1.05) node[anchor=north east] {$\Tref_{\rfrac{\pi}{2}}$}; \draw (1.0, -0.2) node[anchor=north east] {$\widehat{\Gamma}$}; \draw[-, fill=black!10, opacity=.5, dashed] (A)--(B)--(C)--(A); \draw[-, opacity=.5] (B)--(C); \draw[-, opacity=.5] (C)--(D); \draw[-, opacity=.5] (B)--(D); \draw (0.3,0.0) .. controls (0.2, 0.3) and (0.1, 0.3) .. node[right] {$\hat{\alpha}$} (0.0,0.3); \draw[dashed] (A)--(D); \draw[dashed] (0, 0, 0.5)--(0, 0.5, 0.5)--(0, 0.5, 0); \draw (-0.1, 0.3) node[anchor=north east] {$\hat{\theta}$};
\end{tikzpicture} } & \multicolumn{2}{c}{ \begin{tikzpicture}[line join = round, line cap = round] \coordinate [label=left:] (A) at (0,0,0); \coordinate [label=above right:] (B) at (1.2,0,0); \coordinate [label=below:] (C) at (0,0,1.2); \coordinate [label=above right:] (D) at ( -0.6, 1.0392, 0); \draw[->] (1.2, 0, 0) -- (1.6, 0, 0) node[below] {$\hat{x}_1$}; \draw[->] (0, 0, 0) -- (0, 1.4, 0) node[above] {$\hat{x}_2$}; \draw[->] (0, 0, 1.2) -- (0, 0, 1.6) node[right] {$\hat{x}_3$}; \draw (2.0, 1.05) node[anchor=north east] {$\Tref_{\rfrac{2\pi}{3}}$}; \draw (1.0, -0.2) node[anchor=north east] {$\widehat{\Gamma}$}; \draw[-, fill=black!10, opacity=.5, dashed] (A)--(B)--(C)--(A); \draw[-, opacity=.5] (B)--(C); \draw[-, opacity=.5] (C)--(D); \draw[-, opacity=.5] (B)--(D); \draw (0.3,0.0) .. controls (0.2, 0.3) and (0.1, 0.3) .. node[right] {$\hat{\alpha}$} (-0.1,0.2); \draw[dashed] (A)--(D); \end{tikzpicture} }\\ \midrule $M(N)$ & $\CPTapproxhatalphahat$ & $\CGTapproxhatalphahat$ & $\CPTapproxhatalphahat$ & $\CGTapproxhatalphahat$ & $\CPTapproxhatalphahat$ & $\CGTapproxhatalphahat$ & $\CPTapproxhatalphahat$ & $\CGTapproxhatalphahat$\\ \midrule
7 & 0.32431 & 0.760099 & 0.325985 & 0.654654 & 0.360532 & 0.654654 & 0.4152099 & 0.686161 \\ 26 & 0.338539 & 0.829445 & 0.340267 & 0.761278 & 0.373669 & 0.751615 & 0.4274757 & 0.863324 \\ 63 & 0.341122 & 0.831325 & 0.342556 & 0.762901 & 0.375590 & 0.751994 & 0.4286444 & 0.864595 \\ 124 & 0.341147 & 0.831335 & 0.342589 & 0.762905 & 0.375603 & 0.751999 & 0.4286652 & 0.864630 \\ 215 & {\bf 0.341147} & {\bf 0.831335} & {\bf 0.342589} & {\bf 0.762905} & {\bf 0.375603} & {\bf 0.751999} & {\bf 0.4286652} & {\bf 0.864630} \\
\end{tabular} \caption{$\CPTapproxhatalphahat$ and $\CGTapproxhatalphahat$ with respect to $M(N)$ for $\Tref_{\hat{\theta}, \hat{\alpha}}$ with $\rho = 1$, $\hat{\theta} = \tfrac{\pi}{2}$, and different $\hat{\alpha}$.} \label{tab:const-convergence-from-basis-G-leg-alpha} \end{table}
Then, for an arbitrary tetrahedron $\T$, we have
\begin{alignat}{2}
\|v\|_{\T} \, & \leq\, \CPT \, h_2 \,\|\nabla v\|_{\T} \quad \mbox{and} \quad
\|v\|_\Gamma \, \leq\, \CGT \, h_2^{\rfrac{1}{2}} \,\|\nabla v\|_{\T}
\label{eq:general-poincare-type-inequalities-for-tetrahedron} \end{alignat}
with approximate bounds
\begin{equation}
\CPT \lessapprox \CPtetr =
\min_{\hat{\alpha} = \{\rfrac{\pi}{4}, \rfrac{\pi}{3}, \rfrac{\pi}{2}, \rfrac{2\pi}{3}\}}
\Big\{ \cpalphahat \, \CPThatalphahat \Big\} \end{equation} and \begin{equation}
\CGT \lessapprox \CGtetr =
\min_{\hat{\alpha} =
\{\rfrac{\pi}{4}, \rfrac{\pi}{3}, \rfrac{\pi}{2}, \rfrac{2\pi}{3}\}}
\Big\{ \cgalphahat \, \CGThatalphahat \Big\}, \label{eq:} \end{equation}
where $\CPThatalphahat$ and $\CGThatalphahat $ are the constants related to four reference tetrahedrons from Table \ref{tab:const-convergence-from-basis-G-leg-alpha}, and $\cpalphahat$ and $\cgalphahat$ (see \eqref{eq:ratio-3d}) are generated by the mapping $\mathcal{F}_{\rfrac{\pi}{2}, \hat{\alpha}}$: $\Tref_{\rfrac{\pi}{2}, \hat{\alpha}} \rightarrow \T$. Here, the reference tetrahedrons are defined based on $\hat{A} = (0, 0, 0)$, $\hat{B} = (1, 0, 0)$, $\hat{C} = (0, 0, 1)$, $\hat{D} = (\cos\hat{\alpha}, \sin\hat{\alpha}, 0)$ with $\hat{\alpha} = \{ \tfrac{\pi}{4},\tfrac{\pi}{3}, \tfrac{\pi}{2}, \tfrac{2\pi}{3}\}$, and $\mathcal{F}_{\rfrac{\pi}{2}, \hat{\alpha}}(\hat{x})$ is presented by the relation
\begin{equation*}
x = \mathcal{F}_{\rfrac{\pi}{2}, \hat{\alpha}}(\hat{x}) =
B_{\rfrac{\pi}{2}, \hat{\alpha}} \hat{x}, \quad
B_{\rfrac{\pi}{2}, \hat{\alpha}} = \{ b_{ij} \}_{i, j = 1, 2, 3} = h_2
\begin{pmatrix}
\: \tfrac{h_1}{h_2} &
\; \tfrac{\nu (\rho, \alpha)}{\sin \hat{\alpha}} &
\; 0 \\[0.3em]
\: 0 &
\; \, \tfrac{\sin\alpha \sin\theta}{\sin \hat{\alpha}} \: &
\: 0 \\[0.3em]
\: 0 &
\; \, \tfrac{\cos\theta}{\sin \hat{\alpha}} \: &
\: \tfrac{h_3}{h_2} \\
\end{pmatrix},
\end{equation*}
where $\nu(\rho, \alpha) = \cos\alpha \sin\theta - \tfrac{h_1}{h_2} \cos \hat{\alpha}$, $\mathrm{det} \, B_{\rfrac{\pi}{2}, \hat{\alpha}} = h_1 \, h_2\, h_3 \, \tfrac{\sin\alpha \sin\theta}{\sin \hat{\alpha}}$. By analogy with the two-dimensional case (see \eqref{eq:grad-hatv-lower-estimate-1}), $\cpalphahat$ and $\cgalphahat$ depend on the maximum eigenvalue of the matrix
\begin{alignat*}{2}
A_{\rfrac{\pi}{2}, \hat{\alpha}} & := h_1^2
\begin{pmatrix}
\: b_{11}^2 + b^2_{12} \quad & b_{12} b_{22} \qquad & b_{12} b_{32} \\[0.5em]
b_{12} b_{22} & b^2_{22} & b_{22} b_{32} \\[0.5em]
b_{12} b_{32} & b_{22} b_{32} & b_{33}^2 + b^2_{32} \\
\end{pmatrix}
.
\end{alignat*}
The maximal eigenvalue of the matrix $A_{\rfrac{\pi}{2}, \hat{\alpha}}$ is defined by the relation $\lambda_{\rm max} (A_{\rfrac{\pi}{2}, \hat{\alpha}}) = h_2^2 \, \mu_{\alpha, \theta, \hat{\alpha}}$ with
\begin{equation*}
\mu_{\alpha, \theta, \hat{\alpha}}
= \Big(\mathcal{E}_5^{\rfrac{1}{3}}
- \mathcal{E}_3 \mathcal{E}_5^{-\rfrac{1}{3}}
+ \tfrac{1}{3} \mathcal{E}_1 \Big), \end{equation*}
where
\begin{alignat*}{2}
\mathcal{E}_1 & = b_{11}^2 + b_{12}^2 + b_{22}^2 + b_{32}^2 + b_{33}^2, \\
\mathcal{E}_2 & = b_{11}^2 \, b_{22}^2 + b_{11}^2 \, b_{32}^2 + b_{11}^2 \, b_{33}^2 + b_{12}^2 \, b_{33}^2 + b_{22}^2\, b_{33}^2, \\
\mathcal{E}_3 & = \tfrac{\mathcal{E}_2}{3} - \big(\tfrac{\mathcal{E}_1}{3})^2, \\
\mathcal{E}_4 & = \big(\tfrac{\mathcal{E}_1}{3})^3 - \tfrac{\mathcal{E}_1 \, \mathcal{E}_2}{3} + \tfrac{1}{2} \, b_{11}^2 \, b_{22}^2 \, b_{33}^2, \\
\mathcal{E}_5 & = \mathcal{E}_4 + (\mathcal{E}_3^3 + \mathcal{E}_4^2)^{\rfrac{1}{2}}. \end{alignat*}
Therefore, $\cpalphahat$ and $\cgalphahat$ in \eqref{eq:general-poincare-type-inequalities-for-tetrahedron} are as follows:
\begin{equation}
\cpalphahat = \mu^{\rfrac{1}{2}}_{\pi/2, \hat{\alpha}}, \quad
\cgalphahat =
\Big( \tfrac{\sin \hat{\alpha}}{\rho \sin\alpha \sin\theta}\Big)^{\rfrac{1}{2}} \,\cpalphahat.
\label{eq:ratio-3d} \end{equation}
Lower bounds of the constants $\CPGamma$ and $\CtrGamma$ are computed by minimization of $\mathcal{R}^{\mathrm{P}}_{\Gamma} [w]$ and $\mathcal{R}^{\mathrm{Tr}}_{\Gamma} [w]$ over the set $V^N_3 \subset \H{1}(\T)$, where
\begin{equation*}
V^{N}_3 := \Big\{\:\varphi_{ijk} = x^{i} y^{j} z^k, \quad i, j, k = 0, \ldots, N,
\;\; (i, j, k) \neq (0, 0, 0) \: \Big\}
\end{equation*}
and $\mathrm{dim} V^{N}_3 = M(N) := (N + 1)^3 - 1$.
The respective results are presented in Tables \ref{tab:cp-3d-approx-exact-constants} and \ref{tab:cg-3d-approx-exact-constants} for $\T$ with $h_1 = 1$, $h_3 = 1$, and $\rho = 1$. We note that exact values of constants are probably closer to the numbers presented in left-hand side columns. For $\theta = \rfrac{\pi}{2}$, we also present estimates of $\approxCPT$ and $\approxCGT$ (red lines) graphically in Fig. \ref{fig:3d-cpt-cgt-constants-with-estimates-4-refs}.
\begin{table}[!ht] \centering \footnotesize
\begin{tabular}{c|cc|cc|cc|cc} $$
& \multicolumn{2}{c|}{ $\alpha = \tfrac{\pi}{6}$}
& \multicolumn{2}{c|}{ $\alpha = \tfrac{\pi}{4}$}
& \multicolumn{2}{c|}{ $\alpha = \tfrac{\pi}{3}$} & \multicolumn{2}{c}{ $\alpha = \tfrac{\pi}{2}$} \\ \midrule $\theta$
& $\underline{C}^{M, \mathrm{P}}_{\Gamma}$ & $\CPtetr$ &
$\underline{C}^{M, \mathrm{P}}_{\Gamma}$ & $\CPtetr$ &
$\underline{C}^{M, \mathrm{P}}_{\Gamma}$ & $\CPtetr$ &
$\underline{C}^{M, \mathrm{P}}_{\Gamma}$ & $\CPtetr$ \\ \midrule $\pi/6$ & 0.23883 & 0.49035& 0.24621 & 0.49841& 0.25870 & 0.51054& 0.29484 & 0.51308\\ $\pi/4$ & 0.23883 & 0.45388& 0.24621 & 0.46173& 0.25870 & 0.47683& 0.29484 & 0.49075\\ $\pi/3$ & 0.29666 & 0.41958& 0.31194 & 0.42259& 0.33489 & 0.43724& 0.38976 & 0.46002\\ $\pi/2$ & 0.34302 & 0.35667& 0.34112 & 0.34115& 0.34256 & 0.34259& 0.37559 & 0.37560\\ $2\pi/3$ & 0.40428 & 0.41958& 0.40562 & 0.42259& 0.40927 & 0.43724& 0.42867 & 0.46002\\ $3\pi/4$ & 0.42890 & 0.45388& 0.43110 & 0.46173& 0.43505 & 0.47683& 0.45017 & 0.49075\\ $5\pi/6$ & 0.44964 & 0.49035& 0.45193 & 0.49841& 0.45539 & 0.51054& 0.46607 & 0.51308\\ \midrule $$
& \multicolumn{2}{c|}{ $\alpha = \tfrac{\pi}{2}$}
& \multicolumn{2}{c|}{ $\alpha = \tfrac{2\pi}{3}$}
& \multicolumn{2}{c|}{ $\alpha = \tfrac{3\pi}{4}$} & \multicolumn{2}{c}{ $\alpha = \tfrac{5\pi}{6}$} \\ \midrule $\theta$
& $\underline{C}^{M, \mathrm{P}}_{\Gamma}$ & $\CPtetr$ &
$\underline{C}^{M, \mathrm{P}}_{\Gamma}$ & $\CPtetr$ &
$\underline{C}^{M, \mathrm{P}}_{\Gamma}$ & $\CPtetr$ &
$\underline{C}^{M, \mathrm{P}}_{\Gamma}$ & $\CPtetr$ \\ \midrule $\pi/6$ & 0.29484 & 0.51308& 0.33069 & 0.51792& 0.34468 & 0.52253& 0.35499 & 0.52694\\ $\pi/4$ & 0.29484 & 0.49075& 0.33069 & 0.50261& 0.34468 & 0.51308& 0.35499 & 0.52253\\ $\pi/3$ & 0.38976 & 0.46002& 0.43880 & 0.48413& 0.45742 & 0.50261& 0.47106 & 0.51792\\ $\pi/2$ & 0.37559 & 0.37560& 0.42865 & 0.42867& 0.45017 & 0.45731& 0.46607 & 0.47811\\ $2\pi/3$ & 0.42867 & 0.46002& 0.45997 & 0.48413& 0.47457 & 0.50261& 0.48598 & 0.51792\\ $3\pi/4$ & 0.45017 & 0.49075& 0.47204 & 0.50261& 0.48239 & 0.51308& 0.49064 & 0.52253\\ $5\pi/6$ & 0.46607 & 0.51308& 0.47972 & 0.51792& 0.48607 & 0.52253& 0.49115 & 0.52694\\ \end{tabular} \\[5pt] \caption{$\underline{C}^{M, \mathrm{P}}_{\Gamma}$ ($M(N)$ = 124) and $\CPtetr$.} \label{tab:cp-3d-approx-exact-constants} \end{table}
\begin{figure}
\caption{$\CPT$ and $\CGT$ for $\T \in \Rthree$ with $H = 1$, $\rho = 1$ with estimate based on
four reference tetrahedrons.}
\label{fig:3d-cpt-cgt-constants-with-estimates-4-refs}
\end{figure}
\begin{table}[!ht] \centering \footnotesize
\begin{tabular}{c|cc|cc|cc|cc} $$
& \multicolumn{2}{c|}{ $\alpha = \tfrac{\pi}{6}$}
& \multicolumn{2}{c|}{ $\alpha = \tfrac{\pi}{4}$}
& \multicolumn{2}{c|}{ $\alpha = \tfrac{\pi}{3}$} & \multicolumn{2}{c}{ $\alpha = \tfrac{\pi}{2}$}\\ \midrule $\theta$ &
$\underline{C}^{M, \mathrm{Tr}}_{\Gamma}$ & $\CGtetr$ &
$\underline{C}^{M, \mathrm{Tr}}_{\Gamma}$ & $\CGtetr$ &
$\underline{C}^{M, \mathrm{Tr}}_{\Gamma}$ & $\CGtetr$ &
$\underline{C}^{M, \mathrm{Tr}}_{\Gamma}$ & $\CGtetr$ \\ \midrule $\pi/6$ & 1.09760 & 3.78259& 0.96245 & 2.71866& 0.91255 & 2.27382& 0.93123 & 2.05449\\ $\pi/4$ & 1.09760 & 2.43897& 0.96245 & 1.78094& 0.91255 & 1.50166& 0.93123 & 1.38951\\ $\pi/3$ & 0.89122 & 1.74467& 0.79146 & 1.31130& 0.75950 & 1.12431& 0.78904 & 1.06349\\ $\pi/2$ & 0.98017 & 1.22920& 0.83132 & 0.83133& 0.76290 & 0.76291& 0.75199 & 0.75200\\ $2\pi/3$ & 1.17698 & 1.74467& 0.99473 & 1.31130& 0.90578 & 1.12431& 0.86463 & 1.06349\\ $3\pi/4$ & 1.35195 & 2.43897& 1.14144 & 1.78094& 1.03737 & 1.50166& 0.98220 & 1.38951\\ $5\pi/6$ & 1.65317 & 3.78259& 1.39424 & 2.71866& 1.26490 & 2.27382& 1.19017 & 2.05449\\ \midrule $$
& \multicolumn{2}{c|}{ $\alpha = \tfrac{\pi}{2}$}
& \multicolumn{2}{c|}{ $\alpha = \tfrac{2\pi}{3}$}
& \multicolumn{2}{c|}{ $\alpha = \tfrac{3\pi}{4}$} & \multicolumn{2}{c}{ $\alpha = \tfrac{5\pi}{6}$} \\ \midrule $\theta$ &
$\underline{C}^{M, \mathrm{Tr}}_{\Gamma}$ & $\CGtetr$ &
$\underline{C}^{M, \mathrm{Tr}}_{\Gamma}$ & $\CGtetr$ &
$\underline{C}^{M, \mathrm{Tr}}_{\Gamma}$ & $\CGtetr$ &
$\underline{C}^{M, \mathrm{Tr}}_{\Gamma}$ & $\CGtetr$ \\ \midrule $\pi/6$ & 0.93123 & 2.05449& 1.07244 & 2.39471& 1.21573 & 2.95902& 1.47044 & 4.21999\\ $\pi/4$ & 0.93123 & 1.38951& 1.07244 & 1.64324& 1.21573 & 2.01841& 1.47044 & 2.80588\\ $\pi/3$ & 0.78904 & 1.06349& 0.91773 & 1.27423& 1.04309 & 1.50833& 1.26357 & 2.11790\\ $\pi/2$ & 0.75199 & 0.75200& 0.86459 & 0.86463& 0.98220 & 1.12971& 1.19017 & 1.67033\\ $2\pi/3$ & 0.86463 & 1.06349& 0.96174 & 1.27423& 1.08134 & 1.50833& 1.30191 & 2.11790\\ $3\pi/4$ & 0.98220 & 1.38951& 1.07921 & 1.64324& 1.20686 & 2.01841& 1.44721 & 2.80588\\ $5\pi/6$ & 1.19017 & 2.05449& 1.29582 & 2.39471& 1.44268 & 2.95902& 1.72383 & 4.21999\\ \end{tabular} \\[5pt] \caption{$\underline{C}^{M, \mathrm{Tr}}_{\Gamma}$ ($M(N)$ = 124) and $\CGtetr$ for different $\theta, \alpha \in (0, \pi)$.} \label{tab:cg-3d-approx-exact-constants} \end{table}
\section{Example} \label{sec:example}
Constants in the Friedrichs', Poincar\'{e}, and other functional inequalities arise in various problems of numerical analysis, where we need to know values of the respective constants associated with particular domains. Constants in projection type estimates arise in a priori analysis (see, e.g., \cite{Braess2001, Ciarlet1978, Mikhlin1986}). Constants in Clement's interpolation inequalities are important for residual type a posteriori estimates (see, e.g., \cite{AinsworthOden2000, Verfurth1996}, and \cite{CarstensenFunken2000}, where these constants have been evaluated).
Concerning constants in the trace inequalities associated with polygonal domain, we mention the paper \cite{CarstensenSauter2004}. Constants in functional (embedding) inequalities arise in a posteriori error estimates of the functional type (error majorants). The details concerning last application can be found \cite{RepinDeGruyter2008, LangerRepinWolfmayr2014, MatculevichNeitaanmakiRepin2015,ReVarInq2010,RepinBoundaryMeanTrace2015, Repin2000, RepinXanthis1996} and other references cited therein.
Below, we deduce an advanced version of an error majorant,
which uses constants in Poincar\'{e}-type inequalities for functions with zero mean traces on inter-element boundaries. This is done in order to maximally extend the space of admissible fluxes. However, first, we shall discuss the reasons that invoke Poincar\'{e}-type constants in a posteriori estimates.
Let $u$ denote the exact solution of an elliptic boundary value problem generated by the pair of conjugate operators $\rm grad$ and $-\dvrg$ (e.g., the problem (\ref{eq:reactdiff1})--(\ref{eq:reactdiff4}) considered below) and $v$ be a function in the energy space satisfying the prescribed (Dirichlet)
boundary conditions. Typically, the error $e := u - v$ is measured in terms of the energy norm $\|\nabla\,e\|$ (or some other equivalent norm), whose square is bounded from above by the quantities
\begin{equation*} \IntO R (v, \dvrg q) \,e\, \dx,\quad \IntO D (\nabla v, q) \cdot \nabla\,e\, \dx,\; {\rm and}\; \Int_{\Gamma_N} R_{\Gamma_N}(v, q\cdot n)\,e\,\ds, \end{equation*}
where $\Omega$ Lipschitz bounded domain, $\Gamma_N$ is the Neumann part of the boundary $\partial \Omega$ with the outward unit normal vector $n$, and $q$ is an approximation of the dual variable (flux). The terms $R$, $D$, and $R_{\Gamma_N}$ represent residuals of the differential (balance) equation, constitutive (duality) relation, and Neumann boundary condition, respectively. Since $v$ and $q$ are known from a numerical solution, fully computable estimates can be obtained if these integrals are estimated by the H\"{o}lder, Friedrichs, and trace inequalities (which involve the corresponding constants). However, for $\Omega$ with piecewise smooth (e.g., polynomial) boundaries these constants may be unknown. A way to avoid these difficulties is suggested by modifications of the estimates using ideas of domain decomposition. Assume that $\Omega$ is a polygonal (polyhedral) domain decomposed into a collection of non-overlapping convex polygonal sub-domains $\Omega_i$, i.e.,
\begin{equation*}
\overline{\Omega} :=
\bigcup\limits_{ \Omega_i \in \, \mathcal{O}_\Omega}
{\overline{\Omega}}_i,
\quad
\mathcal{O}_\Omega :=
\Big\{ \; \Omega_i \in \Omega \; \big| \;
{\Omega}_{i'} \, \cap \, {\Omega}_{i''} = \emptyset, \;
i' \neq i'', \;
i = 1, \ldots, N \; \Big \}.
\end{equation*}
We denote the set of all edges (faces) by ${\mathcal G}$ and the set of all interior faces by ${\mathcal G}_{\rm int}$ (i.e., $\Gamma_{ij} \in {\mathcal G}_{\rm int}$, if $\Gamma_{ij} = \overline{\Omega}_i \, \cap \, \overline{\Omega}_j$). Analogously, ${\mathcal G}_{N}$ denotes the set of edges on $\Gamma_N$. The latter set is decomposed into $\Gamma_{N_k}:=\Gamma_N\cap\partial\Omega_k$ (the number of faces that belongs to ${\mathcal G}_{N}$ is $K_N$). Now, the integrals associated with $R $ and $R_{\Gamma_N}$ can be replaced by sums of local quantities
\begin{equation*} \Sum_{\Omega_i \in {\mathcal G}} \Int_{\Omega_i}R_{\Omega} (v,\dvrg q)\,e \, \dx,\quad{\rm and}\quad \Sum_{\Gamma_{N_k} \in {\mathcal G}_{N}} \Int_{\Gamma_{N_k}} R_{\Gamma_{N}} (v,q\cdot n)\,e\,\ds. \end{equation*}
If the residuals satisfy the conditions
\begin{equation*}
\Int_{\Omega_i} R_{\Omega_i}(v,\, \dvrg q) \dx = 0
\quad \forall i = 1, \ldots, N, \quad
\end{equation*}
and
\begin{equation*}
\Int_{\Gamma_{N_k}} R_{\Gamma_{N}}(v,\, q\cdot n) \ds = 0,
\quad \forall k = 1, \ldots, K_N,
\end{equation*}
then
\begin{equation}
\Int_{\Omega_i} R_{\Omega}(v, \dvrg q)\,e\, \dx
\leq C^{{\mathrm P}}_{\Omega_i} \|R_{\Omega_i}(v,\dvrg q)\|_{\Omega_i}\,
\|\nabla\,e\|_{\Omega_i} \quad
\label{eq:estOmega} \end{equation}
and
\begin{equation}
\Int_{\Gamma_{N_k}} R_{\Gamma_{N}}(v, q\cdot n)\,e\,\ds
\leq C^{{\mathrm Tr}}_{\Gamma_{N_k}} \|R_{\Gamma_{N}}(v,q\cdot n)\|_{\Gamma_{N_k}}
\|\nabla\,e\|_{\Omega_k}.
\label{eq:estGammaN} \end{equation}
Hence, we can deduce a computable upper bound of the error that contains local constants $C^{{\mathrm P}}_{\Omega_i}$ and $C^{{\mathrm Tr}}_{\Gamma_{N_k}}$ for simple subdomains (e.g., triangles or tetrahedrons) instead of the global constants associated with $\Omega$.
The constant $\CPoincare$ may arise if, e.g., nonconforming approximations are used. For example, if $v$ does not exactly satisfy the Dirichlet boundary condition on $\Gamma_{D_k}$, then in the process of estimation it may be necessary to evaluate terms of the type
\begin{equation*}
\Int_{\Gamma_{D_k}} G_{D} (v)\,e\,\ds,\quad k = 1, \ldots, K_D, \end{equation*}
where $\Gamma_{D_k}$ is a part of $\Gamma_D$ associated with a certain $\Omega_k$, and $G_D(v)$ is a residual generated by inexact satisfaction of the boundary condition. If we impose the requirement that the Dirichlet boundary condition is satisfied in a weak sense, i.e., $ \mean{G_{D}(v)}_{\Gamma_{D_k}} = 0, $ then each boundary integral can be estimated as follows:
\begin{equation}
\Int_{\Gamma_{D_k}} G_{D}(v)\,e\,\ds
\leq C^{{\mathrm P}}_{\Gamma_{D_k}} \|G_D(v)\|_{\Gamma_{D_k}}\|\nabla\,e\|_{\Omega_k}.
\label{eq:estGammaD} \end{equation}
After summing up (\ref{eq:estOmega}), (\ref{eq:estGammaN}), and (\ref{eq:estGammaD}), we obtain a product of weighted norms of localized residuals (which are known) and
$\|\nabla e\|_\Omega$. Since the sum is bounded from below by the squared energy norm, we arrive at computable error majorant.
Now, we discuss elaborately these questions within the paradigm of the following boundary value problem: find $u$ such that
\begin{eqnarray} & - \dvrg {p} + \varrho^2 u = f, \quad & {\rm in} \; \Omega, \quad \label{eq:reactdiff1}\\ & p = A \nabla u, \quad & {\rm in} \; \Omega, \quad \label{eq:reactdiff2}\\ & u = u_D, \quad & {\rm on} \; \Gamma_D,\
\\ & A \nabla u \cdot { n} = F \quad & {\rm on} \; \Gamma_N. \label{eq:reactdiff4} \end{eqnarray}
Here $f \in \L{2}(\Omega)$, $F \in \L{2}(\Gamma_N)$, $u_D \in \H{1}(\Omega)$, and $A$
is a symmetric positive definite matrix with bounded coefficients satisfying the condition $\lambda_1 |\xi|^2 \, \leq \, A \xi \cdot \xi$, where $\lambda_1$ is a positive constant independent of $\xi$. The generalized solution of \eqref{eq:reactdiff1}--\eqref{eq:reactdiff4} exists and is unique in the set $V_0 + u_D$, where $V_0 := \Big \{ w \in \H{1}(\Omega) \; \mid \; w = 0 \; {\rm on} \; \Gamma_D \Big\}$.
Assume that $v \in V_0 + u_D$ is a conforming approximation of $u$. We wish to find a computable majorant of the error norm
\begin{equation}
\mid\!\mid\!\mid\! e \!\mid\!\mid\!\mid^2 \,:= \| \nabla e \|^2_{A} + \| \varrho\, e \|^2, \end{equation}
where $
\| \nabla e \|^2_{A} := \Int_\Omega A \nabla e \cdot \nabla e \dx. $
First, we note that the integral identity that defines $u$ can be rewritten in the form
\begin{equation}
\Int_\Omega A \nabla e \cdot \nabla w \, \dx \,
+ \Int_\Omega \varrho^2 e \, w \, \dx
= \Int_\Omega (f w - \varrho^2 v \, w - A \nabla v \cdot \nabla w) \dx \,
+ \Int_{\Gamma_N} F w \ds, \quad \forall w \in V_0.
\label{eq:weak-statement} \end{equation}
It is well known (see Section 4.2 in \cite{RepinDeGruyter2008}) that this relation yields a computable majorant of $\mid\!\mid\!\mid\! e \!\mid\!\mid\!\mid^2 $, if we introduce a vector-valued function ${q} \in H(\Omega, \dvrg)$, such that ${q} \cdot n \in L^2(\Omega)$, and transform \eqref{eq:weak-statement} by means of integration by parts relations. The majorant has the form
\begin{equation}
\mid\!\mid\!\mid\! e \!\mid\!\mid\!\mid\,
\leq\,\|D_\Omega (\nabla v,q)\|_{A^{-1}} +
C_1 \| R (v,\dvrg {q}) \|_{\Omega} + C_2 \| R_{\Gamma_N} (v,q\cdot n) \|_{\Gamma_N},
\label{eq:global-estimate} \end{equation}
where $C_1$ and $C_2$ are positive constants explicitly defined by $\lambda_1$, the Friedrichs' constant $C^{\mathrm F}_{\Omega}$ in inequality \linebreak
$\|v\|_\Omega \leq \, C^{\mathrm F}_{\Omega} \|\nabla v\|_\Omega$ for functions vanishing on $\Gamma_D$, and constant $C^{\mathrm Tr}_{\Gamma_N}$ in the trace inequality associated with $\Gamma_N$. The integrands are defined by the relations
\begin{equation*}
D (\nabla v,q) := A\nabla v - q, \quad
R (v,\dvrg q) := \dvrg q + f - \varrho^2 v, \quad {\rm and} \quad
R_{\Gamma_N} (v, q \cdot n) := q\cdot n - F. \end{equation*}
In general, finding $C^{\mathrm F}_{\Omega}$ and $C^{\mathrm Tr}_{\Gamma_N}$ may not be an easy task. We can exclude $C_2$ if $ q$ additionally satisfies the condition ${ q}\cdot n = F$. Then, the last term in \eqref{eq:global-estimate} vanishes. However, this condition is difficult to satisfy, if $F$ is a complicated nonlinear function. In order to exclude $C_1$ together with $C_2$, we can apply domain decomposition technique and use (\ref{eq:estOmega}) instead of the global estimate. Then, the estimate will operate with the constants $C^{{\mathrm P}}_{\Omega_i}$ (whose upper bounds are known for convex domains). Moreover, it is shown below that by using the inequalities (\ref{eq:Comega}) and (\ref{eq:Cgamma}), we can essentially weaken the assumptions required for the variable $q$.
Define the space of vector-valued functions
\begin{alignat*}{2}
\hat{H} (\Omega, {\mathcal O}_{\Omega}, \dvrg) := \, \Big\{{q} \in \L{2} (\Omega, \Rd) \; \mid & \quad {q} = {q}_i \in H(\Omega_i, \dvrg),
\;\; \mean{\dvrg {q}_i + f - \varrho^2 \, v}_{\Omega_i} = 0, \quad \forall\,\Omega_i \in {\mathcal O}_{\Omega}, \\
& \;\; \mean{({q}_i - {q}_j) \cdot { n}_{ij}}_{\Gamma_{ij}} = 0 , \quad \forall\, \Gamma_{ij} \in {\mathcal G}_{\rm int},\\
& \;\; \mean{{ q}_i \cdot { n}_k - F}_{ \Gamma_{N_k}} = 0, \quad \forall \, k = 1, \ldots, K_N \Big \}. \end{alignat*}
We note that the space $\hat{H} (\Omega, {\mathcal O}_{\Omega},\dvrg)$ is wider than $H(\Omega, \dvrg)$ (so that we have more flexibility in determination of optimal reconstruction of numerical fluxes). Indeed, the vector-valued functions in $H(\Omega,\dvrg)$ must have continuous normal components on all $\Gamma_{ij} \in {\mathcal G}_{\rm int}$ and satisfy the Neumann boundary condition in the pointwise sense. The functions in $\hat{H} (\Omega, {\mathcal O}_{\Omega},\dvrg)$ satisfy much weaker conditions: namely, the normal components are continuous only in terms of mean values (integrals) and the Neumann condition must hold in the integral sense only.
We reform (\ref{eq:weak-statement}) by means of the integral identity
\begin{equation*}
\Sum_{\Omega_i \in {\mathcal O}_{\Omega}} \int_{\Omega_i}
\left( {q} \cdot \nabla w + \dvrg {q} \, w \right) \dx
= \Sum_{\Gamma_{ij} \, \in \, {\mathcal G}_{\rm int}} \;
\Int_{\Gamma_{ij}}\; ({ q}_i - { q}_j) \cdot { n}_{ij} \, w \ds
+ \, \Sum_{\Gamma_{N_k} \, \in \, \Gamma_N} \;
\Int_{\Gamma_{N_k}} { q}_i \cdot { n}_{i} \, w \ds, \end{equation*}
which holds for any $ w \in V_0$ and ${q} \in \hat{H} (\Omega, {\mathcal O}_{\Omega},\dvrg)$. By setting $w = e$ in \eqref{eq:weak-statement} and applying the H\"{o}lder inequality, we find that
\begin{multline*}
\mid\!\mid\!\mid\! e \!\mid\!\mid\!\mid^2\! \;\;
\leq \|D (\nabla v,q) \|_{A^{-1}} \|\nabla e\|_{A}
+ \Sum_{\Omega_i \in {\mathcal O}_{\Omega}} \|R(v,\dvrg q)\|_{\Omega_i}
\big\|e - \mean{e}_{\Omega_i}\big\|_{\Omega_i} \\
+ \Sum_{\Gamma_{ij} \in \, {\mathcal G}_{\rm int}}
r_{ij}(q) \big\| e - \mean{e}_{\Gamma_{ij}}\big\|_{\Gamma_{ij} }
+ \Sum_{\Gamma_{N_k} \in \Gamma_N} \rho_k(q)
\Big\| e - \mean{e}_{\Gamma_{N_k}}\Big\|_{\Gamma_{N_k}},
\end{multline*}
where
\begin{equation*}
r_{ij}(q) := \| ({ q}_i - { q}_j) \cdot { n}_{ij} \|_{\Gamma_{ij}}
\quad \mbox{and} \quad
\rho_k(q) := \| { q}_k \cdot { n}_k - F \|_{\Gamma_{N_k}}. \end{equation*}
In view of (\ref{eq:classical-poincare-constant}) and (\ref{eq:Cgamma}), we obtain
\begin{multline}
\mid\!\mid\!\mid\! e \!\mid\!\mid\!\mid^2\! \;\;
\leq \|D (\nabla v,q) \|_{A^{-1}} \|\nabla e\|_{A}
+ \Sum_{\Omega_i \in {\mathcal O}_{\Omega}}
\|R(v,\dvrg q)\|_{\Omega_i} C^{\mathrm P}_{\Omega_i}
\|\nabla e \|_{\Omega_i} \\
+ \Sum_{\Gamma_{ij} \in \, {\mathcal G}_{\rm int}}
r_{ij}(q) C^{\mathrm{Tr}}_{\Gamma_{ij}}
\|\nabla e \|_{\Omega_i}
+ \Sum_{\Gamma_{N_k} \in \Gamma_N}
\rho_k(q) C^{\mathrm{Tr}}_{\Gamma_{N_k}}
\|\nabla e \|_{\Omega_i}.
\label{eq:estimate-2} \end{multline}
The second term in the right hand side is estimated by the quantity
$\Re_1(v,q)\,\|\nabla e\|_\Omega$, where
\begin{equation*} \Re^2_1(v,q) := \,\Sum_{\Omega_i \in {\mathcal O}_{\Omega}}
\tfrac{( \diam \, \Omega_i)^2}{\pi^2}\|R(v,\dvrg q)\|^2_{\Omega_i}. \end{equation*}
We can represent any $\Omega_i \in {\mathcal O}_{\Omega}$ as a sum of simplexes such that each simplex has one edge on $\partial\Omega_i$. Let $C^{\mathrm{Tr}}_{i, {\rm max}}$ denote the largest constant in the respective Poincar\'{e}-type inequalities \eqref{eq:Cgamma} associated with all edges of $\partial\Omega_i$. Then, the last two terms of (\ref{eq:estimate-2}) can be estimated by the quantity
$\Re_2 (v,q)\,\|\nabla e\|_\Omega$, where
\begin{equation*}
\Re^2_2(q):=\,\Sum_{\Omega_i \in {\mathcal O}_{\Omega}}
(C^{\mathrm{Tr}}_{i, {\rm max}})^2 \, \eta^2_i, \quad { \rm with} \quad
\eta^2_{i} =
\Sum_{\myatop{\Gamma_{ij} \in {\mathcal G}_{\rm int}}{\Gamma_{ij} \cap \partial\Omega_i \neq \varnothing}}\tfrac{1}{4} r^2_{ij}(q) +
\Sum_{\myatop{\Gamma_{k} \in {\mathcal G}_{N}}{\Gamma_{k} \,\cap \,\partial\Omega_i \neq \varnothing}}
\rho^2_k(q). \end{equation*}
Then, (\ref{eq:estimate-2}) yields the estimate
\begin{equation*}
\mid\!\mid\!\mid\! e \!\mid\!\mid\!\mid^2\! \;\;
\leq \|D(\nabla v,q)\|_{A^{-1}} \|\nabla e\|_{A}
+ (\Re_1(v,q) + \Re_2(q) )\,\|\nabla e\|_\Omega,
\end{equation*}
which shows that
\begin{equation}
\mid\!\mid\!\mid\! e \!\mid\!\mid\!\mid\! \;\;
\leq \|D(\nabla v,q) \|_{A^{-1}}
+ \tfrac{1}{\lambda_1} \Big( \Re_1(v,q) + \Re_2(q) \Big).
\label{eq:estimate-final} \end{equation}
Here, the term $\Re_2(q)$ controls violations of conformity of $q$ (on interior edges) and inexact satisfaction of boundary conditions (on edges related to $\Gamma_N$). It is easy to see that $\Re_2(q) = 0$, if and only if the quantity $q \cdot n$ is continuous on ${\mathcal G}_{\rm int}$ and exactly satisfies the boundary condition. Hence, $\Re_2(q)$ can be viewed as a measure of the `flux nonconformity'. Other terms have the same meaning as in well-known a posteriori estimates of the functional type, namely, the first term measures the violations of the relation ${q} = A \nabla v$ (cf. (\ref{eq:reactdiff2})), and $\Re_1(v,q)$ measures inaccuracy in the equilibrium (balance) equation (\ref{eq:reactdiff1}). The right-hand side of \eqref{eq:estimate-final} contains known functions (approximations $v$ and $q$ of the exact solution and exact flux). The constants $C^{\mathrm{Tr}}_{i, {\rm max}}$ can be easily computed using results of Section \ref{sc:arbitrary-triangle}-\ref{eq:numerial-tests-3d}. Finally, we note that estimates similar to \eqref{eq:estimate-final} were derived in \cite{ReVarInq2010} for elliptic variational inequalities and in \cite{MatculevichNeitaanmakiRepin2015} for a class of parabolic problems.
\end{document} | arXiv |
Elliptic divisibility sequence
In mathematics, an elliptic divisibility sequence (EDS) is a sequence of integers satisfying a nonlinear recursion relation arising from division polynomials on elliptic curves. EDS were first defined, and their arithmetic properties studied, by Morgan Ward[1] in the 1940s. They attracted only sporadic attention until around 2000, when EDS were taken up as a class of nonlinear recurrences that are more amenable to analysis than most such sequences. This tractability is due primarily to the close connection between EDS and elliptic curves. In addition to the intrinsic interest that EDS have within number theory, EDS have applications to other areas of mathematics including logic and cryptography.
Definition
A (nondegenerate) elliptic divisibility sequence (EDS) is a sequence of integers (Wn)n ≥ 1 defined recursively by four initial values W1, W2, W3, W4, with W1W2W3 ≠ 0 and with subsequent values determined by the formulas
${\begin{aligned}W_{2n+1}W_{1}^{3}&=W_{n+2}W_{n}^{3}-W_{n+1}^{3}W_{n-1},\qquad n\geq 2,\\W_{2n}W_{2}W_{1}^{2}&=W_{n+2}W_{n}W_{n-1}^{2}-W_{n}W_{n-2}W_{n+1}^{2},\qquad n\geq 3,\\\end{aligned}}$
It can be shown that if W1 divides each of W2, W3, W4 and if further W2 divides W4, then every term Wn in the sequence is an integer.
Divisibility property
An EDS is a divisibility sequence in the sense that
$m\mid n\Longrightarrow W_{m}\mid W_{n}.$
In particular, every term in an EDS is divisible by W1, so EDS are frequently normalized to have W1 = 1 by dividing every term by the initial term.
Any three integers b, c, d with d divisible by b lead to a normalized EDS on setting
$W_{1}=1,\quad W_{2}=b,\quad W_{3}=c,\quad W_{4}=d.$
It is not obvious, but can be proven, that the condition b | d suffices to ensure that every term in the sequence is an integer.
General recursion
A fundamental property of elliptic divisibility sequences is that they satisfy the general recursion relation
$W_{n+m}W_{n-m}W_{r}^{2}=W_{n+r}W_{n-r}W_{m}^{2}-W_{m+r}W_{m-r}W_{n}^{2}\quad {\text{for all}}\quad n>m>r.$
(This formula is often applied with r = 1 and W1 = 1.)
Nonsingular EDS
The discriminant of a normalized EDS is the quantity
$\Delta =W_{4}W_{2}^{15}-W_{3}^{3}W_{2}^{12}+3W_{4}^{2}W_{2}^{10}-20W_{4}W_{3}^{3}W_{2}^{7}+3W_{4}^{3}W_{2}^{5}+16W_{3}^{6}W_{2}^{4}+8W_{4}^{2}W_{3}^{3}W_{2}^{2}+W_{4}^{4}.$
An EDS is nonsingular if its discriminant is nonzero.
Examples
A simple example of an EDS is the sequence of natural numbers 1, 2, 3,... . Another interesting example is (sequence A006709 in the OEIS) 1, 3, 8, 21, 55, 144, 377, 987,... consisting of every other term in the Fibonacci sequence, starting with the second term. However, both of these sequences satisfy a linear recurrence and both are singular EDS. An example of a nonsingular EDS is (sequence A006769 in the OEIS)
${\begin{aligned}&1,\,1,\,-1,\,1,\,2,\,-1,\,-3,\,-5,\,7,\,-4,\,-23,\,29,\,59,\,129,\\&-314,\,-65,\,1529,\,-3689,\,-8209,\,-16264,\dots .\\\end{aligned}}$
Periodicity of EDS
A sequence (An)n ≥ 1 is said to be periodic if there is a number N ≥ 1 so that An+N = An for every n ≥ 1. If a nondegenerate EDS (Wn)n ≥ 1 is periodic, then one of its terms vanishes. The smallest r ≥ 1 with Wr = 0 is called the rank of apparition of the EDS. A deep theorem of Mazur[2] implies that if the rank of apparition of an EDS is finite, then it satisfies r ≤ 10 or r = 12.
Elliptic curves and points associated to EDS
Ward proves that associated to any nonsingular EDS (Wn) is an elliptic curve E/Q and a point P ε E(Q) such that
$W_{n}=\psi _{n}(P)\qquad {\text{for all}}~n\geq 1.$
Here ψn is the n division polynomial of E; the roots of ψn are the nonzero points of order n on E. There is a complicated formula[3] for E and P in terms of W1, W2, W3, and W4.
There is an alternative definition of EDS that directly uses elliptic curves and yields a sequence which, up to sign, almost satisfies the EDS recursion. This definition starts with an elliptic curve E/Q given by a Weierstrass equation and a nontorsion point P ε E(Q). One writes the x-coordinates of the multiples of P as
$x(nP)={\frac {A_{n}}{D_{n}^{2}}}\quad {\text{with}}~\gcd(A_{n},D_{n})=1~{\text{and}}~D_{n}\geq 1.$
Then the sequence (Dn) is also called an elliptic divisibility sequence. It is a divisibility sequence, and there exists an integer k so that the subsequence ( ±Dnk )n ≥ 1 (with an appropriate choice of signs) is an EDS in the earlier sense.
Growth of EDS
Let (Wn)n ≥ 1 be a nonsingular EDS that is not periodic. Then the sequence grows quadratic exponentially in the sense that there is a positive constant h such that
$\lim _{n\to \infty }{\frac {\log |W_{n}|}{n^{2}}}=h>0.$
The number h is the canonical height of the point on the elliptic curve associated to the EDS.
Primes and primitive divisors in EDS
It is conjectured that a nonsingular EDS contains only finitely many primes[4] However, all but finitely many terms in a nonsingular EDS admit a primitive prime divisor.[5] Thus for all but finitely many n, there is a prime p such that p divides Wn, but p does not divide Wm for all m < n. This statement is an analogue of Zsigmondy's theorem.
EDS over finite fields
An EDS over a finite field Fq, or more generally over any field, is a sequence of elements of that field satisfying the EDS recursion. An EDS over a finite field is always periodic, and thus has a rank of apparition r. The period of an EDS over Fq then has the form rt, where r and t satisfy
$r\leq \left({\sqrt {q}}+1\right)^{2}\quad {\text{and}}\quad t\mid q-1.$
More precisely, there are elements A and B in Fq* such that
$W_{ri+j}=W_{j}\cdot A^{ij}\cdot B^{j^{2}}\quad {\text{for all}}~i\geq 0~{\text{and all}}~j\geq 1.$
The values of A and B are related to the Tate pairing of the point on the associated elliptic curve.
Applications of EDS
Bjorn Poonen[6] has applied EDS to logic. He uses the existence of primitive divisors in EDS on elliptic curves of rank one to prove the undecidability of Hilbert's tenth problem over certain rings of integers.
Katherine E. Stange[7] has applied EDS and their higher rank generalizations called elliptic nets to cryptography. She shows how EDS can be used to compute the value of the Weil and Tate pairings on elliptic curves over finite fields. These pairings have numerous applications in pairing-based cryptography.
References
1. Morgan Ward, Memoir on elliptic divisibility sequences, Amer. J. Math. 70 (1948), 31–74.
2. B. Mazur. Modular curves and the Eisenstein ideal, Inst. Hautes Études Sci. Publ. Math. 47:33–186, 1977.
3. This formula is due to Ward. See the appendix to J. H. Silverman and N. Stephens. The sign of an elliptic divisibility sequence. J. Ramanujan Math. Soc., 21(1):1–17, 2006.
4. M. Einsiedler, G. Everest, and T. Ward. Primes in elliptic divisibility sequences. LMS J. Comput. Math., 4:1–13 (electronic), 2001.
5. J. H. Silverman. Wieferich's criterion and the abc-conjecture. J. Number Theory, 30(2):226–237, 1988.
6. B. Poonen. Using elliptic curves of rank one towards the undecidability of Hilbert's tenth problem over rings of algebraic integers. In Algorithmic number theory (Sydney, 2002), volume 2369 of Lecture Notes in Comput. Sci., pages 33–42. Springer, Berlin, 2002.
7. K. Stange. The Tate pairing via elliptic nets. In Pairing-Based Cryptography (Tokyo, 2007), volume 4575 of Lecture Notes in Comput. Sci. Springer, Berlin, 2007.
Further material
• G. Everest, A. van der Poorten, I. Shparlinski, and T. Ward. Recurrence sequences, volume 104 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2003. ISBN 0-8218-3387-1. (Chapter 10 is on EDS.)
• R. Shipsey. Elliptic divisibility sequences. PhD thesis, Goldsmiths College (University of London), 2000.
• K. Stange. Elliptic nets. PhD thesis, Brown University, 2008.
• C. Swart. Sequences related to elliptic curves. PhD thesis, Royal Holloway (University of London), 2003.
External links
• Graham Everest's EDS web page.
• Prime Values of Elliptic Divisibility Sequences.
• Lecture on p-adic Properites of Elliptic Divisibility Sequences.
| Wikipedia |
\begin{document}
\title{Efficient Construction of Broadcast Graphs} \begin{abstract}
A broadcast graph is a connected graph, $G=(V,E)$, $ |V |=n$, in which each vertex can complete broadcasting of one message within at most $t=\lceil \log n\rceil$ time units. A minimum broadcast graph on $n$ vertices is a broadcast graph with the minimum number of edges over all broadcast graphs on $n$ vertices. The cardinality of the edge set of such a graph is denoted by $B(n)$. In this paper we construct a new broadcast graph with
$B(n) \le (k+1)N -(t-\frac{k}{2}+2)2^{k}+t-k+2$, for $n=N=(2^{k}-1)2^{t+1-k}$ and
$B(n) \le (k+1-p)n -(t-\frac{k}{2}+p+2)2^{k}+t-k -(p-2)2^{p}$, for $2^{t} < n<(2^{k}-1)2^{t+1-k}$, where $t \geq 7$, $2 \le k \le \lfloor t/2 \rfloor -1$ for even $n$ and $2 \le k \le \lceil t/2 \rceil -1$ for odd $n$, $d=N-n$, $x= \lfloor \frac{d}{2^{t+1-k}} \rfloor$ and $ p = \lfloor \log_{2}{(x+1)} \rfloor$ if $x>0$ and $p=0$ if $x=0$.
The new bound is an improvement upon the bounds appeared in \cite{bf},\cite{gf} and \cite{hl} and the recent bound presented by Harutyunyan and Liestman (\cite{hln}) for odd values of $n$.
\textbf{Keywords:} Broadcasting, minimum broadcast graph.
\end{abstract}
\section{Introduction} \emph{Broadcasting} is an information distribution problem in a connected graph, in which one vertex, called the \emph{originator}, has to distribute a message to all other vertices by placing a series of calls among the communication lines of the graph. Once informed, the informed vertices aid the originator in distributing the message. This is assumed to take place in discrete time units. The broadcasting has to be completed within a minimal number of time units subjected to the following constraints:
1. Each call involves only one informed vertex and one of its uninformed
neighbors.
2. Each call requires one time unit.
3. A vertex can participate in at most one call at each time unit.
4. At each time unit many calls can be performed in parallel.
Formally, any network can be modeled as a simple connected graph $G=(V,E)$, $|V|=n$, where $V$ is the set of vertices and $E$ is the set of edges (the communication lines). For a given originator vertex, $u$, the \emph{broadcast time} of $u$, $b(u)$, is defined as the minimum number of time units needed to complete broadcasting from $u$. Note that for any vertex $u \in V$,
$b(u)\geq \lceil \log n \rceil$ (to the sequel the base of logs is always 2), since at each time unit the number of informed vertices can at most double. The broadcast time $b(G)$ of the graph $G$ is defined as $max\{b(u)|u \epsilon G\}$ and $G$ is called a broadcast graph if $b(G)= \lceil \log n \rceil$.
The \emph{broadcast number $B(n)$} is the minimum number of edges in any broadcast graph on $n$ vertices. A \emph{minimum broadcast graph (mbg)} is a broadcast graph on $n$ vertices with $B(n)$ edges.
Currently, the exact values of $B(n)$ are known only for $n=2^{p}$, $n=2^{p}-2$, $n=127$, and for several values of $n\leq63$, as detailed below. Farley et al. \cite{fh} determined the values of $B(n)$ for $ n \leq 15$ and showed that hypercubes are \emph{mbgs} such that $B(2^{p})=p2^{p-1}$ for any $p \ge 2$. Mitchell and Hedetniemi \cite{mh} determined the value of $B(17)$, while Bermond, Hell, Liestman and Peters \cite{bh} determined the values of $B(n)$ for $ n = 18,19,30,31$. Khachatrian and Haroutunian \cite{lh} and independently Dinnen, Fellows and Faber \cite{dn} proved that $B(2^{p}-2)=(p-1)(2^{p-1}-1)$ for all $p \geq 2$.
Since \emph{mbg's} seem to be difficult to find, many authors have devised methods to construct broadcast graphs. The number of edges in any broadcast graph on $n$ vertices gives an upper bound on $B(n)$. Several papers have shown methods to construct broadcast graphs by forming the compound of two known broadcast graphs (see \cite{bf}, \cite {dv}, \cite{hl} and \cite{lh}). These methods have proven effective for graphs on $n_{1} n_{2}$ vertices from two known broadcast graphs on $n_{1}$ and $n_{2}$ vertices. Thus, compounding produces good upper bound on $B(n)$ for many values of $n$. In particular, a very tight upper bound was obtained for $n=2^{p} - 2^{k}$ by compounding mbg's on $2^{k-1}$ and $2^{p-k+1}-2$ vertices: $B(2^{p}-2^{k}) \le \frac{2^{p}-2^{k}}{2}(p-\frac{k+1}{2})$ (see \cite{bf},\cite{lh}).
Broadcast graphs on other sizes can sometimes be formed by adding or deleting vertices from known broadcast graphs(see \cite{bh} for example). An efficient vertex addition method is suggested in \cite{hh}. The authors in \cite{hl} presented a method based on compounding and then merging several vertices into one that allows the construction of the best broadcast graphs for almost all values of $n$, including many prime numbers. In particular, a very tight upper bound on $B(n)$ is $B(2^{p}-2^{k}+1)\leq 2^{p-1}(p-\frac{k}{2})$ (again by compounding mbg's on $2^{k}$ and $2^{p-k}$ vertices and then merging $2^{k}$ vertices into one).
Farley (\cite{af}) proposed the recursive method to construct minimal broadcast graphs and proved the general upper bound $$B(n) \leq \frac{n\lceil \log n \rceil}{2} , ~2^{p-1}<n\leq 2^{p}. \eqno(1) $$ Other general upper bounds on $B(n)$ are obtained from a direct construction using binomial trees (see \cite{gp},\cite{hl},\cite{lh}) for some values of $n$.
Direct construction of broadcast graphs is a difficult problem. The best upper bound from a direct construction for any $n$ is $$B(n)\leq n(p-k+1)-2^{p-k}-\frac{1}{2}(p-k)(3p+k-3)+2k, \eqno(2)$$ where $n=2^{p}-2^{k}-r$, $0 \leq k \leq p-2$ and $0 \leq r \leq 2^{k}-1$ (see \cite{hl}). While this bound is tight for $p-k$ is small for $k<p/2$ it is not as good as the bound from \cite{af}, in (1).
The best general upper bound on $B(n)$ for even $n$, namely, $$B(n)\leq\frac{n \lfloor \log n \rfloor}{2} \eqno(3) $$ obtained from the modified Kn\"{o}del graph (see \cite{bf},\cite{gf}). This bound, is better than the one in (1) for all even $n\neq 2^{p}$.
In \cite{hx}, Harutyunyan and Xu presented an upper bound on $B(n)$ for odd $n$.
They proved that for integers $n,p$, where $n>65$ is odd, $p \geq 7$ and $n\neq 2^{p}+1$, $B(n) \leq \frac{(n+1)\lfloor {\log {n}} \rfloor}{2} + 2 \lceil \frac {n-1}{10} \rceil - \lfloor {\frac{\lfloor {\log {n} } \rfloor +2}{4}} \rfloor $.
However, recently Harutyunyan and Liestman presented in \cite{hln} a new upper bound for odd, positive integers, namely,
\begin{theorem} \label{t1} Let $n$ be an even integer such that $\lceil log n \rceil>2$ is prime, $m=\lceil log n \rceil \neq 2^{j}-1$ for any integer $j$, $m$ divides $n$, and for any integer $d \neq m-1$ which is a divisor of $m-1$, $2^{d} \not\equiv 1(mod (m))$. Then, $$B(n+1) \le \frac{n \lfloor log n \rfloor}{2} + \frac{n}{\lceil log n \rceil} + \lceil log n \rceil-2. \eqno(4) $$ \end{theorem}
In this paper we present a new upper bound for $B(n)$, improving the bounds in (1),(2),(3) and (4).
Our main result is,
\begin{theorem}
\label{t2}
Let $t,k,n$ be positive integers. Then, for a given $t \geq 7$ and $2 \le k \le \lfloor t/2 \rfloor -1$,
\begin{enumerate}
\item If $n=N=(2^{k}-1)2^{t+1-k}$,
$$B(n) \le (k+1)N -(t-\frac{k}{2}+2)2^{k}+t-k+2. \eqno(5.a)$$
\item If $2^{t} < n<(2^{k}-1)2^{t+1-k}$,
$$B(n) \le (k+1-p)n -(t-\frac{k}{2}+p+2)2^{k}+t-k -(p-2)2^{p}, \eqno(5.b)$$
where $d=N-n$, $x= \lfloor \frac{d}{2^{t+1-k}} \rfloor$ and $ p = \left\{ \begin{array}{ll}
\lfloor \log_{2}{(x+1)} \rfloor & \mbox{if $x>0$ }\\
0 & \mbox{otherwise.}\end{array}\right. $
\end{enumerate}
\end{theorem}
\section{Proof of Theorem \ref{t2}}
In this section we prove theorem \ref{t2}. First we construct a minimal broadcast graph and then demonstrate the broadcast scheme.
\subsection { Construction of the minimal broadcast graph} We start by defining the binomial tree. \begin{definition}
A binomial tree of order $t$, denoted by $B^{t}$, is defined recursively as follows:
A binomial tree of order $0$ is the trivial tree (a single vertex).
A binomial tree of order $t$ has vertex which is a root vertex whose children are roots of binomial trees of orders $t-1, t-2, ..., 2, 1, 0$ (in this order).
\end{definition} \textbf{Observation:} The Binomial tree $B^{t}$ has $2^{t}$ vertices and height $t$. Because of its unique structure, a binomial tree of order $t$ can be constructed trivially from two trees of order $t-1$ by attaching one of them as the rightmost child of the root of the other one [see Figure.1].
\begin{figure}
\caption{The Binomial trees $B^{0}, B^{1}, B^{2}, B^{3}$ and $B^{4}$.}
\end{figure}
\begin{lemma} \label{lemma1} Let $B^{n}$ be the binomial tree of order $n$. Let $u$ be the root of $B^{n}$. Then, $b(u)=n$. \end{lemma}
The proof is straightforward and is omitted.
Now we define a hypercube graph.
\begin{definition} A hypercube graph of dimension $n$, denoted by $Q^{n}$, is defined recursively as follows:
A hypercube graph of dimension $0$ is a single vertex.
A hypercube graph of dimension $n$ is constructed of two hypercubes, each of dimension $n-1$, $Q^{n-1}_{1}$ and $Q^{n-1}_{2}$ and there is a perfect matching connecting the vertices of $Q^{n-1}_{1}$ with these of $Q^{n-1}_{2}$.
\end{definition}
\textbf{Notice:} A hypercube graph is a $n$-regular graph with $2^{n}$ vertices and thus has $n2^{n-1}$ edges.
\textbf{Observation:} Because of its unique structure, a hypercube graph of dimension $n$ can be constructed trivially from $n$ hypercube graphs of orders $n-1, n-2, ..., 2, 1, 0, 0$, denoted by $Q^{n-1}, Q^{n-2},.....,Q^{0},Q^{01}$, respectively.
$Q^{0}$ and $Q^{01}$ form a hypercube of dimension $1$,
$Q^{0},Q^{01}$ and $Q^{1}$ form a hypercube of dimension $2$,
etc... \begin{figure}
\caption{The hypercubes $Q^{0}, Q^{1}, Q^{2}, Q^{3}$ and $Q^{4}$.}
\end{figure}
The following lemma is of great importance to our proof. \begin{lemma} \label{lemma2} Let $Q^{n}$ be the $n$-dimensional hypercube. Then, for each vertex $u \in V(Q^{n})$, $b(u)=n$. \end{lemma}
The proof is easy and follows by induction on $n$ and is omitted.
\textbf{Proof of theorem 1.2:}
First we demonstrate the construction of a broadcasting graph $G$ giving the upper bound of
$B(N)$ declared in (5.a), for $N=(2^{k}-1)2^{t+1-k}$ (case 1). The broadcasting graph $G=(V,E), |V|=n$, with $2^{t} < n <N$ shall be constructed later (case 2). The broadcasting scheme in that graphs shall demonstrate in the next section.
\textbf{Case 1:} For a given integer $t \ge 7$ and $k$, $2\le k\le \lfloor{t/2}\rfloor-1$, we construct a minimal broadcast graph $G=(V,E)$ with $|V|=N=(2^{k}-1)2^{t+1-k}$.
The broadcast graph $G$ is constructed of $2^{k}-1$ binomial trees denoted $B_{i}, 1 \le\ i \le 2^{k}-1$. Each $B_{i}$, $1 \le\ i \le 2^{k}-1$, is a $B^{t+1-k}$ tree. Let $R=\{r_{1},...,r_{2^{k}-1}\}$ be the set of the roots of the binomial trees $B_1 , B_2, \ldots , B_{2^{k}-1} $, respectively. For each $i$, $1 \le\ i \le 2^{k}-1$, $r_{i}$ is of degree $t+1-k$.
Thus, $ | V(G) | = N =(2^{k}-1)2^{t+1-k}$.
It is easily observed that $\lceil \log N \rceil=t+1$.
Denote by $V_{1}=\{r_{1},...,r_{k-1}\}$ the set of the roots of the trees $B_{1},...,B_{k-1}$, respectively and by $V_{2}=\{r_{k+1},...,r_{2^{k}-1}\}$ the set of the roots of the trees $B_{k+1},...,B_{2^{k}-1}$, respectively. Thus, $V_{1} \cup \{r_{k}\} \cup V_{2}=R$
with $|R| = 2^{k} -1 $.
We are ready now to construct the set $E(G)$ and to calculate its cardinality.
First, we have the edges of the binomial trees. Let $w \in B_{1}$ be the farthest leaf from the root $r_{1}$. We connect the vertices of $R \cup \{w\}$ in a way that they form a hypercube of dimension $k$, denoted by $Q^{k}$. Let $Q^{k-1}, Q^{k-2},.....,Q^{0},Q^{01}$, be the hypercube graphs that form $Q^{k}$ such that
$w \in Q^{01}$ (in fact, $w=Q^{01}$) and for each $0 \le i \le k-1$, $r_{i+1} \in Q^{i}$. Let $Q^{k-1}_{1}$ and $Q^{k-1}_{2}$ be the two hypercube graphs of dimension $k-1$ that form $Q^{k}$ such that $Q_{1}^{k-1} = Q^{k-1}$ and $Q^{k-2},.....,Q^{0},Q^{01}$ form $Q^{k-1}_{2}$. Now, we connect each vertex $v$, $v \in V\setminus (R\cup \{w\})$, in which its root, $r$, $r \in Q_{1}^{k-1}$, to each of the vertices in $V_{1} \cup \{r\}$. For the vertices $v$, $v \in V\setminus (R \cup \{w\})$, in which their root $r$, $r \in Q_{2}^{k-1}$, we do the following: if $r \in Q^{i}$, $0 \le i \le k-2$, we connect $v$ to its root $r$, to each vertex in $V_{1}\setminus \{r_{i+1}\}$ and to $r_{k}$.
\textbf{Summary}: The $mbg$ graph $G$ constructed is a hypercube $Q^{k}$ of dimension $k$, and $2^{k}$ vertices (the set $R\cup \{w\}$), where each of the vertices in $R$ is a root of a binomial tree on $2^{t+1-k}$ vertices. Furthermore, each of the vertices of the binomial trees which are not on $R \cup \{w\}$ is adjacent to its root and to each of the vertices in $V_{1} \cup \{r_{k}\}$, except to $r_{j}$, if that vertex belongs to $Q^{j-1}$, for $1 \le j \le k$.
Now, we are ready to calculate the cardinality of $|E(G)|$.
First, the number of edges in the binomial trees is
$$
| \cup _{i=1}^{2^{k}-1} E(B_i )| = \sum _{i=1}^{2^{k}-1} |E(B_i )|= (2^{k}-1) (2^{t+1-k} -1).
\eqno(6)
$$
The number of edges in the hypercube induced on $R \cup \{w\}$ is
$$
|E(Q^{k})|=k2^{k-1}.
\eqno(7)$$
The number of edges that connect each non root vertex in $G$ to its root is $$ (2^{k}-1)[2^{t+1-k}-1-(t+1-k)]-1.
\eqno(8)$$
The number of edges that connect the non root vertices in $Q_{2}^{k-1}\setminus\{w\}$ to $r_{k}$ is $$(2^{k-1}-1)(2^{t+1-k}-1)-1.
\eqno(9)$$
The number of edges that connect each vertex of $V_{1}$ to all vertices of $Q_{1}^{k-1}$ which are not roots (do not belong to $R$) is $$(k-1)2^{k-1}(2^{t+1-k}-1).
\eqno(10)$$
And finally, the number of edges that connect the vertices of $V_{1}$ to all the vertices in $Q_{2}^{k-1}\setminus \{w\}$ is $$(k-2)(2^{k-1}-1)[(2^{t+1-k}-1)-1].
\eqno(11)$$
Thus, summing the values in (6) up to (11) and recalling that $N=(2^{k}-1)2^{t+1-k}$ we obtain
$$|E(G)|= (k+1)N -(t+2-\frac{k}{2})2^{k}+t+2-k.\eqno(12)$$
\textbf{Case 2:} We construct now a $mbg$ $G'=(V',E')$, $|V'|=n$, where $2^{t}<n<(2^{k}-1)2^{t+1-k}$. We start by constructing a $mbg$, $G=(V,E)$, with $|V|=N=(2^{k}-1)2^{t+1-k}$ as described in Case 1. Then, we obtain $G'$ from $G$ by deleting vertices and edges from $G$, in a way described below.
Define $d=N-n$, $x= \lfloor \frac{d}{2^{t+1-k}} \rfloor$, $y=d-x2^{t+1-k}$ and
$ p = \left\{ \begin{array}{ll}
\lfloor \log_{2}{(x+1)} \rfloor & \mbox{if $x>0$ }\\
0 & \mbox{otherwise.}\end{array}\right.$
Note that $0 \le x < 2^{k-1}$, $0 \le y < 2^{t+1-k}$ and $1 \le p < k$.
In order to construct $G'$ we delete vertices from $G$ as needed according to the value of $d$. Since $d=2^{t+1-k}x+y$, the deletion process is done as follows:
\begin{enumerate}
\item If $x=0$, $d=y$, we delete $y$ vertices from some binomial tree in a way that we start deleting from the leaves and each vertex is deleted after all its descendants in the binomial tree are already deleted.
\item If $x>0$, $d=2^{t+1-k}x+y$, we delete $2^{p}-1$ complete binomial trees and additional $2^{t+1-k}[x-(2^{p}-1)]+y$ non root vertices and then add $2^{p}-1$ edges. This is done in the following way:
\begin{enumerate}
\item Delete all the vertices that are in the binomial trees in which their roots form $Q^{0},Q^{1},....,Q^{p-1}$. Here, we delete $2^{p}-1$ binomial trees, where $p$ of these trees are rooted by vertices from $V_{1}$. Note that the hypercubes $Q^{0},Q^{1},....,Q^{p-1}$ are deleted from $Q^{k-1}_{2}$.
\item Delete $2^{t+1-k}(x-(2^{p}-1))+y$ non root vertices from the trees in which their roots are in $Q_{2}^{k-1}\setminus ( \cup_{i=0}^{p-1} Q^{i})$. Note that since $p=\lfloor \log_{2}{(x+1)} \rfloor$, the number of vertices that we delete here is less than $2^{t+1-k}\cdot 2^{p}$.
\item For each vertex $b \in Q_{1}^{k-1} \cap R$, in which we have deleted its neighbor in $Q^{k-1}_{2}$, we connect $b$ to some vertex that remained in $Q^{k-2}$. Those edges that we add here replace the edges that connected $b$ to some other root in $Q^{k-1}_{2}$ that we have deleted in (a). This addition of edges is crucial in order to keep each vertex in the hypercube $Q^{k-1}_{1}$ matched to another vertex in $Q^{k-1}_{2}$.
\end{enumerate} \end{enumerate}
After the deletion process is ended we obtain in $G'$ the following sets: $R'$ is the set of the binomial trees roots. Then, $R' = V'_{1} \cup \{r_{k}\}\cup V'_{2}$, $|R'|=2^{k}-1-(2^{p}-1)=2^{k}-2^{p}$, where $V'_{1}=\{r_{1} ...r_{k-1-p}\}$, $|V'_{1}|=k-1-p$, $V'_{2} = R'\setminus (V'_{1}\ \cup \{r_{k}\})$ and $|V'_{2}| = 2^{k}-1-k-(2^{p}-1-p)=2^{k}-2^{p}+p-k$.
Now we calculate the number of edges that are deleted from $G$ in order to obtain the graph $G'$.
First, we count the edges that are adjacent to each non-root vertex in the $2^{p}-1$ complete binomial trees that were deleted from $G$. The degree of each vertex $v$ in $V\setminus R$ is $k+j+1$, where $j$ is the distance of $v$ to the farthest leaf in its subtree. Indeed, $j$ edges connect $v$ to its direct siblings, $k$ edges connect $v$ to vertices in $R$ and one edge connects $v$ to its direct ancestor. Since we delete a vertex after all its siblings are already deleted, the number of edges deleted each time we delete a vertex in $V \setminus R$ is $k+1$. Therefore, the number of such edges that are deleted is $$(k+1)(2^{t+1-k}-1)(2^{p}-1). \eqno(13)$$
Since the degree of each vertex in $Q^{k}$ is $k$, the number of edges that we delete from $Q^{k}$ is $k(2^{p}-1)$. By adding the $2^{p}-1$ edges, we actually omit from $Q^{k}$, as described, $$(k-1)(2^{p}-1)\eqno(14)$$ edges.
Note that if $x \neq 0$, the tree $B_{1}$ rooted in $r_{1}$ ($r_{1}=Q^{0}$) is deleted from $G$. Since $w \in B_{1}$, $w$ is deleted from $G$. The calculation in (14) includes the $k$ edges that connect $w$ to $Q^{k}$.
Now, we count the number of edges that connected the $p$ roots that were deleted from $V_{1}$ to all the non root vertices that remained in $G'$. This number is $$[n-(2^{k}-2^{p})]p. \eqno(15)$$
Finally, we count $k+1$ edges for each of the $2^{t+1-k}(x-(2^{p}-1))+y$ non-root vertices that we delete from $Q^{k-1}_{2}$, which is: $$(k+1)[2^{t+1-k}(x-(2^{p}-1))+y]. \eqno(16)$$
Summing (13)-(16), the total number of edges that we delete from $G$ in order to construct $G'$ is
$$np+(k+1)d-p2^{k}+(p-2)2^{p}+2.\eqno(17)$$
Now, by subtracting (17) from (12), recalling that $d=N-n$, we obtain that the number of edges in $G'$:
$$|E(G')|=(k+1-p)n -(t-\frac{k}{2}+p+2)2^{k}+t-k -(p-2)2^{p}.\eqno(18)$$
This complete the proof of the construction of $mbg$ graph for $2^{t} <n \le N$.
\textbf{Observation:} One can easily observe that if $n=N$ and thus, $x=p=0$, we obtain $E(G')=(k+1)n -(t-\frac{k}{2}+2)2^{k}+t-k+2$ as in $(12)$.
\textbf{Remark:} For odd $n$ we can have $k \le \lceil \frac{t}{2} \rceil -1$.
\subsection {Broadcasting Scheme} Let $u$ be an originator. We demonstrate a broadcasting scheme in the constructed graphs of cases 1 and 2.\\
\textbf{Case 1 :} $|V|=n=(2^{k}-1)2^{t+1-k}$.\\
\textbf{Case 1.1 :} Let $u \in R\cup \{w\}$. \\ The broadcasting scheme in that case is as follows: Since the vertices of $R\cup \{w\}$ form a hypercube of $2^{k}$ vertices, at most $k$ time units are needed to complete broadcasting in $R\cup \{w\}$ (see lemma \ref{lemma2}).
\textbf{Case 1.2 :} $u \in Q_{1}^{k-1} \setminus R$. \\ At time unit $t=1$, $u$ transmits to its root, which needs another $k-1$ time units to accomplish broadcasting to all members of $Q^{k-1}_{1}$. At time unit $i$, $2 \le i \le k$, $u$ transmits to $r_{k-i+1}$ that needs another $k-i$ time units to accomplish broadcasting in $Q^{k-i}$. Broadcasting in $Q^{k-i}$ completes after time unit $k$ and therefore broadcasting in $Q^{k-1}_{2}$ completes at time unit $k$ (see lemma \ref{lemma2}). Therefore, broadcasting in $Q^{k}$ completes within $k$ time units.
\textbf{Case 1.3 :} $u \in Q_{2}^{k-1} \setminus (R\cup \{w\})$. \\ At the first time unit $u$ transmits the message to $r_{k}$, which needs another $k-1$ time units to accomplish broadcasting to all members of $Q^{k-1}_{1}$. Suppose $u\in Q^{j}$, $0 \le j \le k-2$. Then, at time unit $i$, $2 \le i \le k, i \neq j$, $u$ transmits the message to $r_{k-i+1}$ that needs another $k-i$ time units to accomplish broadcasting in $Q^{k-i}$ and thus, broadcasting in $Q^{k-i}$ completes after time unit $k$. At time unit $j$, $u$ transmits the message to its root that needs another $j$ time units to accomplish broadcasting in $Q^{j}$. Therefore, broadcasting in $Q^{k-1}_{2}$ completes at time unit $k$ and broadcasting in $Q^{k}$ complete within $k$ time units (see lemma \ref{lemma2}).
Now, in all three cases, after the first $k$ time units, each root in $R$ needs at most additional $t+1-k$ time units to complete broadcasting in its binomial tree (see lemma \ref{lemma1}). Thus, broadcasting in $G$ completes within at most $k+t+1-k=t+1$ time units, which is $b(u) \le t+1, \forall u \in V(G)$. \\
\textbf{Case 2 :} $2^{t}<n<(2^{k}-1)2^{t+1-k}$.\\ In this section we recall the definitions of $d,x$ and $p$ defined in case 2 in the previous section: $d=N-n$, $x= \lfloor d/{2^{t+1-k}} \rfloor$, $p= \lfloor \log_{2}({x} +1)\rfloor$, where $0 \le x < 2^{k-1}$ and $0 \le p < k-1$.\\
\textbf{Case 2.1 :} $u \in R'$.\\ At the first time unit $u$ transmits the message to the other half of $Q^{k}$. Meaning, if $u \in Q_{2}^{k-1}$ then $u$ transmits the message to its neighbor in $Q^{k-1}_{1}$, or, $u \in Q_{1}^{k-1}$,and it transmits the message to its neighbor in $Q^{k-1}_{2}$. That is possible, since each vertex in $Q^{k-1}_{1}$ is connected to one of the vertices in $Q^{k-1}_{2}$. Thus, after the first time unit $k-1$ more time units are needed to accomplish broadcasting in $Q^{k-1}_{1}$ and $Q^{k-1}_{2}$. Therefore, broadcasting in $R'$ is completing within at most $k$ time units.\\
\textbf{Case 2.2 :} $u \in Q_{1}^{k-1} \setminus R'$. \\ At time unit $t=1$, $u$ transmits to its root, that needs another $k-1$ time units to accomplish broadcasting to all members of $Q^{k-1}_{1}$. At time unit $i$, $2 \le i \le k-p$, $u$ transmits to $r_{k-i+1}$ that needs another $k-i$ time units to accomplish broadcasting in $Q^{k-i}$. Broadcasting in $Q^{k-i}$ completes after time unit $k$ and therefore broadcasting in $Q^{k-1}_{2}$ completes at time unit $k$ (see lemma \ref{lemma2}). Thus, broadcasting in $Q^{k}$ completes within $k$ time units.
\textbf{Case 2.3 :} $u \in Q_{2}^{k-1} \setminus (R' \cup \{w\})$. \\ At the first time unit $u$ transmits the message to $r_{k}$, which needs another $k-1$ time units to accomplish broadcasting to all members of $Q^{k-1}_{1}$. Furthermore, $u\in Q^{j}$, $p \le j \le k-2$. Then, at time unit $i$, $2 \le i \le k-p, i \neq j$, $u$ transmits the message to $r_{k-i+1}$ that needs another $k-i$ time units to accomplish broadcasting in $Q^{k-i}$. Thus, broadcasting in $Q^{k-i}$ completes after time unit $k$. At time unit $j$, $u$ transmits the message to its root that needs another $j$ time units to accomplish broadcasting in $Q^{j}$. Therefore, broadcasting in $Q^{k-1}_{2}$ completes at time unit $k$ and broadcasting in $Q^{k}$ complete within $k$ time units (see lemma \ref{lemma2}).
Now, in all three cases, each root in $R'$ needs at most $t+1-k$ additional time units to complete broadcasting in its binomial tree (see lemma \ref{lemma1}). Thus, broadcasting in $G'$ completes within at most $k+t+1-k=t+1$ time units
Hence, $b(u) \leq t+1, \forall u \in V(G')$.
This completes the proof of theorem \ref{t2}, in both cases.
\subsubsection{Example: Minimal broadcast network construction} \begin{figure}\label{fig03}
\end{figure}
\begin{table}[H] \centering
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
\hline
$t$ & $k$ & maximal n & our result & \cite{hl} result & $t$ & $k$ & maximal n & our result & \cite{hl} result\\
\hline
7 & 2 & 192 & 551 & 557 & 15 & 2 & 49152 & 147407 & 147421 \\
\hline
8 & 2 & 384 & 1124 & 1131 & 15 & 3 & 57344 & 229266 & 229307 \\
\hline
8 & 3 & 448 & 1731 & 1751 & 15 & 4 & 61440 & 306973 & 307094 \\
\hline
9 & 2 & 768 & 2273 & 2281 & 15 & 5 & 63488 & 380476 & 380778 \\ \hline
9 & 3 & 896 & 3516 & 3539 & 15 & 6 & 64512 & 450699 & 451375 \\ \hline
10 & 2 & 1536 & 4574 & 4583 & 16 & 2 & 98304 & 294860 & 294875 \\ \hline
10 & 3 & 1792 & 7093 & 7119 & 16 & 3 & 114688 & 458635 & 458679 \\ \hline
10 & 4 & 1920 & 9448 & 9524 & 16 & 4 & 122880 & 614158 & 614288 \\ \hline
11 & 2 & 3072 & 9179 & 9189 & 16 & 5 & 126976 & 761373 & 761698 \\ \hline
11 & 3 & 3584 & 14254 & 14283 & 16 & 6 & 129024 & 902220 & 902949 \\ \hline
11 & 4 & 3840 & 19033 & 19118 & 16 & 7 & 130048 & 1038539 & 1040073 \\ \hline
12 & 2 & 6144 & 18392 & 18403 & 17 & 2 & 196608 & 589769 & 589785 \\ \hline
12 & 3 & 7168 & 28583 & 28615 & 17 & 3 & 229376 & 917380 & 917427 \\ \hline
12 & 4 & 7680 & 38218 & 38312 & 17 & 4 & 245760 & 1228543 & 1228682 \\ \hline
12 & 5 & 7936 & 47257 & 47490 & 17 & 5 & 253952 & 1523198 & 1523546 \\ \hline
13 & 2 & 12288 & 36821 & 36833 & 17 & 6 & 258048 & 1805325 & 1806107 \\ \hline
13 & 3 & 14336 & 57248 & 57283 & 17 & 7 & 260096 & 2078796 & 2080445 \\ \hline
13 & 4 & 15360 & 76603 & 76706 & 18 & 2 & 393216 & 1179590 & 1179607 \\ \hline
13 & 5 & 15872 & 94842 & 95098 & 18 & 3 & 458752 & 1834877 & 1834927 \\ \hline
14 & 2 & 24576 & 73682 & 73695 & 18 & 4 & 491520 & 2457328 & 2457476 \\ \hline
14 & 3 & 28672 & 114585 & 114623 & 18 & 5 & 507904 & 3046879 & 3047250 \\ \hline
14 & 4 & 30720 & 153388 & 153500 & 18 & 6 & 516096 & 3611598 & 3612433 \\ \hline
14 & 5 & 31744 & 190043 & 190322 & 18 & 7 & 520192 & 4159437 & 4161201 \\ \hline
14 & 6 & 32256 & 224970 & 225593 & 18 & 8 & 522240 & 4696076 & 4699666 \\ \hline
\end{tabular}
\caption{In this table we show the number of edges for maximal values of $n=N=(2^{k}-1)2^{t+1-k}$ for $7 \le t \le 18$ and $2 \le k \le \lfloor t/2 \rfloor -1 $. We compare our results with the results of \cite{hl}.} \label{table1} \end{table}
\ignore{ \begin{table}[H] \centering
\begin{tabular}{|c|c|c|c|c|}
\hline
$t$ & $k$ & maximal n & our result & \cite{hl} result \\
20 & 2 & 1572864 & 4718528 & 4718547 \\
\hline
20 & 3 & 1835008 & 7339887 & 7339943 \\
\hline
20 & 4 & 1966080 & 9830098 & 9830264 \\
\hline
20 & 5 & 2031616 & 12189089 & 12189506 \\
\hline
20 & 6 & 2064384 & 14449488 & 14450429 \\
\hline
20 & 7 & 2080768 & 16643791 & 16645785 \\
\hline
20 & 8 & 2088960 & 18796046 & 18800118 \\
\hline
20 & 9 & 2093056 & 20921613 & 20929748 \\
\hline
21 & 2 & 3145728 & 9437117 & 9437137 \\
\hline
21 & 3 & 3670016 & 14679912 & 14679971 \\
\hline
21 & 4 & 3932160 & 19660483 & 19660658 \\
\hline
21 & 5 & 4063232 & 24378754 & 24379194 \\
\hline
21 & 6 & 4128768 & 28900113 & 28901107 \\
\hline
21 & 7 & 4161536 & 33289808 & 33291917 \\
\hline
21 & 8 & 4177920 & 37596431 & 37600744 \\
\hline
21 & 9 & 4186112 & 41851662 & 41860292 \\
\hline
22 & 2 & 6291456 & 18874298 & 18874319 \\
\hline
22 & 3 & 7340032 & 29359969 & 29360031 \\
\hline
22 & 4 & 7864320 & 39321268 & 39321452 \\
\hline
22 & 5 & 8126464 & 48758115 & 48758578 \\
\hline
22 & 6 & 8257536 & 57801426 & 57802473 \\
\hline
22 & 7 & 8323072 & 66581969 & 66584193 \\
\hline
22 & 8 & 8355840 & 75197456 & 75202010 \\
\hline
22 & 9 & 8372224 & 83712271 & 83721396 \\
\hline
22 & 10 & 8380416 & 92165134 & 92183183 \\
\hline \end{tabular}
\caption{In this table we show the number of edges for maximal values of $n=N=(2^{k}-1)2^{t+1-k}$ for $20 \le t \le 22$ and for $2 \le k \le \lfloor t/2 \rfloor -1 $. We compare our results with the results of \cite{hl}.} \label{table3} \end{table} }
\begin{table}[H] \centering
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
$n/|E(G')|$ & $ k=2$ & $ k=3$ & $ k=4$ & $ k=5$ & $ k=6$ & \cite{hln} result & \cite{hl} result\\ \hline 16385 & 49109 & 49044 & 48909 & 48628 & 48043 & 115871 & \\ \hline 16386 & 49112 & 49047 & 48912 & 48631 & 48046 & & \\ \hline 16387 & 49115 & 49050 & 48915 & 48634 & 48049 & 115808 & \\ \hline ... & ... & ... & ... & ... & ... & ...&...\\ \hline 24575 & 73679 & 73614 & 73479 & 73198 & 72613 & 173670 & \\ \hline 24576 & 73682 & 73617 & 73482 & 73201 & 72616 & & \\ \hline 24577 & & 98205 & 98080 & 97821 & 97284 & 173684 & \\ \hline 24578 & & 98209 & 98084 & 97825 & 97288 & & \\ \hline 24579 & & 98213 & 98088 & 97829 & 97292 & 173698 & \\ \hline ... & ... & ... & ... & ... & ... &...&...\\ \hline 28671 & & 114581 & 114456 & 114197 & 113660 & 202615 & \\ \hline 28672 & & 114585 & 114460 & 114201 & 113664 & &\\ \hline 28673 & & & 143153 & 142912 & 142413 & 202629 & \\ \hline 28674 & & & 143158 & 142917 & 142418 & & \\ \hline ... & ... & ... & ... & ... & ... &...&...\\ \hline 30719 & & & 153383 & 153142 & 152643 & 217087 &\\ \hline 30720 & & & 153388 & 153147 & 152648 & &\\ \hline 30721 & & & & 183905 & 183440 & 217101 &\\ \hline 30722 & & & & 183911 & 183446 & &\\ \hline 30723 & & & & 183917 & 183452 & 217116 & \\
\hline ... & ... & ... & ... & ... & ... &...&...\\
\hline 31743 & & & & 190037 & 189572 & 224324 & \\
\hline 31744 & & & & 190043 & 189578 & & \\
\hline 31745 & & & & & 221393 & 224338 & 222016\\
\hline 31746 & & & & & 221400 & & 222023\\
\hline 31747 & & & & & 221407 & 224352 & 222030\\
\hline
... & ... & ... & ... & ... & ... &...&...\\
\hline 32255 & & & & & 224963 & 227942 & 225586\\
\hline \end{tabular}
\caption{In this table we show our result of $|E(G')|$ for $t=14$, $2 \le k \le 6$ and $ 16385 \le n \le 32255$. We compare our results with the results of \cite{hln} and \cite{hl}.} \label{table4} \end{table}
\end{document} | arXiv |
\begin{document}
\title{KGTuner: Efficient Hyper-parameter Search for \ Knowledge Graph Learning}
\begin{abstract} While hyper-parameters (HPs)
are important for knowledge graph (KG) learning,
existing methods fail to search them efficiently. To solve this problem, we first analyze the properties of different HPs and measure the transfer ability from small subgraph to the full graph.
Based on the analysis, we propose an efficient two-stage search algorithm KGTuner, which efficiently explores HP configurations on small subgraph at the first stage and transfers the top-performed configurations for fine-tuning on the large full graph at the second stage. Experiments show that our method can consistently find better HPs than the baseline algorithms within the same time budget, which achieves {9.1\%} average relative improvement for four embedding models on the large-scale KGs in open graph benchmark. Our code is released in \url{https://github.com/AutoML-Research/KGTuner}. \footnote{The work is performed when
Z. Zhou was an intern in 4Paradigm,
and correspondence is to Q. Yao.
} \end{abstract}
\section{Introduction} \label{sec:intro}
\begin{figure*}
\caption{The framework of conventional HP search algorithm and the proposed KGTuner.}
\label{fig:conventional}
\label{fig:KGTuner}
\end{figure*}
Knowledge graph (KG) is a special kind of graph structured data to represent knowledge through entities and relations between the entities \citep{wang2017knowledge,ji2020survey}. Learning from KG aims to discover the latent properties from KGs to infer the existence of interactions among entities or the types of entities \cite{wang2017knowledge,zhang2022knowledge}. KG
embedding, which encodes entities and relations as low dimensional vectors, is an important technique to learn from KGs~\citep{wang2017knowledge,ji2020survey}.
The existing models range from translational distance models \citep{bordes2013translating},
tensor factorization models \citep{nickel2011three,trouillon2017knowledge,balavzevic2019tucker},
neural network models \citep{dettmers2017convolutional,guo2019learning},
to graph neural networks \citep{schlichtkrull2018modeling,vashishth2019composition}.
Hyper-parameter (HP) search~\citep{claesen2015hyperparameter} is very essential for KG learning. In this work, we take KG embedding methods \cite{wang2017knowledge}, as a good example to study the impact of HPs to KG learning. As studied,
the HP configurations greatly influence the model performance~\citep{ruffinelli2019you,ali2020bringing}. An improper HP configuration will impede the model from stable convergence, while an appropriate one can make considerable promotion to the model performance.
Indeed, studying the HP configurations can help us make a more scientific understanding of the contributions made by existing works \citep{rossi2020knowledge,sun2020re}. In addition, it is also important to search for an optimal HP configuration when adopting KG embedding methods to the real-world applications \citep{bordes2014question,zhang2016collaborative,saxena2020improving}.
Algorithms for HP search on general machine learning problems have been well-developed \citep{claesen2015hyperparameter}. As shown in Figure~\ref{fig:conventional}, the search algorithm selects a HP configuration from the search space in each iteration, then the evaluation feedback obtained by full model training is used to update the search algorithm. The optimal HP is the one achieving the best performance on validation data in the search process.
Representative HP search algorithms are within sample-based methods like grid search, random search \citep{bergstra2012random}, and sequential model-based Bayesian optimization (SMBO) methods like Hyperopt \citep{bergstra2013hyperopt}, SMAC \citep{hutter2011sequential}, Spearmint \citep{snoek2012practical} as well as BORE \citep{tiao2021bore}, etc.
Recently, some subgraph-based methods~\citep{tu2019autone,wang2021explainable} are proposed to learn a predictor with configurations efficiently evaluated on small subgraphs
The predictor is then transferred to guide HP search on the full graph. However, these methods fail to efficiently search a good configuration of HPs for KG embedding models
since the training cost of individual model is high and the correlation of HPs in the huge search space is very complex.
To address the limitations of
existing HP search algorithms, we carry a comprehensive understanding study on the influence
and correlation of HPs as well as their transfer ability from small subgraph to full graph in KG learning. From the aspect of performance, we classify the HPs into four different groups including \textit{reduced options}, \textit{shrunken range}, \textit{monotonously related} and \textit{no obvious patterns} based on their influence on the performance. By analyzing the validation curvature of these HPs, we find that the space is rather complex such that only tree-based models can approximate it well. In addition, we observe that the consistency between evaluation on small subgraph and that on the full graph is high, while the evaluation cost is significantly smaller on the small subgraph.
Above understanding motivates us to reduce the size of search space and design a two-stage search algorithm named as KGTuner. As shown in Figure~\ref{fig:KGTuner}, KGTuner explores HP configurations in the shrunken and decoupled space with the search algorithm RF+BORE \citep{tiao2021bore} on a subgraph in the first stage, where the evaluation cost of HPs are small. Then in the second stage, the configurations achieving the \textit{top10} performance at the first stage are equipped with large batch size and dimension size for fine-tuning on the full graph.
Within the same time budget, KGTuner can consistently search better configurations than the baseline search algorithms for seven KG embedding models on WN18RR \citep{dettmers2017convolutional} and FB15k-237 \citep{toutanova2015observed}. By applying KGTuner to the large-scale benchmarks ogbl-biokg and ogbl-wikikg2 \citep{hu2020open}, the performances of embedding models are improved compared with the reported results on OGB link prediction leaderboard.
Besides, we justify the improvement of efficiency via analyzing the design components in KGTuner.
\section{Background: HPs in KG embedding} \label{sec:revisit}
We firstly revisit the important and common HPs in KG embedding. Following the general framework~\citep{ruffinelli2019you,ali2020bringing}, the learning problem can be written as
\begin{align}
\!\!\!\!\bm P^* \!=\! \arg\min\nolimits_{\bm P} L(F(\cdot, \bm P), D^+, D^-) \! + \! r(\bm P), \label{eq:kge}
\end{align} where $F$ is the form of an embedding model with learnable parameters $\bm P$, $D^+$ is the set of positive samples from the training data, $D^-$ represents negative samples,
and $r(\cdot)$ is a regularization function. There are four groups of hyper-parameters (Table~\ref{tab:spacefull}), i.e., the size of \textit{negative sampling} for $D^-$, the choice of \textit{loss function} $L$, the form of \textit{regularization} $r(\cdot)$, and the \textit{optimization} $\arg\min_{\bm P}$.
\begin{table*}[ht]
\centering
\caption{The HP space.
Conditioned HPs are in parenthesize.
``adv.'' and ``reg.'' are short for ``adversarial'' and ``regularization'', respectively.
Please refer to the Appendix~\ref{app:searchspace} for more details.}
\label{tab:spacefull}
\setlength\tabcolsep{4pt}
\begin{tabular}{c|c|c|c }
\toprule
component & name & type & range \\
\midrule
negative sampling
& \# negative samples & cat & \{32, 128, 512, 2048, \texttt{1VsAll}, \texttt{kVsAll}\} \\ \midrule
\multirow{3}{*}{loss function} & loss function & cat & \{MR, BCE\_(mean, sum, adv), CE\} \\
& gamma (MR) & float & [1, 24] \\
& adv. weight (BCE\_adv)& float & [0.5, 2.0] \\ \midrule
\multirow{3}{*}{regularization}
& regularizer & cat & \{FRO, NUC, DURA, None\} \\
& reg. weight (not None) & float & [$10^{-12}$, $10^{2}$] \\
& dropout rate & float & $[0, 0.5]$ \\ \midrule
\multirow{6}{*}{optimization}
& optimizer & cat & \{Adam, Adagrad, SGD\} \\
& learning rate & float & [$10^{-5}$, $10^0$] \\
& initializer & cat & \{uniform, normal, xavier\_uniform, xavier\_norm\} \\
& batch size & int & \{128, 256, 512, 1024\} \\
& dimension size & int & \{100, 200, 500, 1000, 2000\} \\
& inverse relation & bool & \{True, False\} \\
\bottomrule
\end{tabular}
\end{table*}
\noindent \textbf{Embedding model.} While there are many existing embedding models, we follow \citep{ruffinelli2019you} to focus on some representative models. They are translational distance models TransE \citep{bordes2013translating} and RotatE \citep{sun2019rotate}, tensor factorization models RESCAL \citep{nickel2011three}, DistMult \citep{yang2014embedding}, ComplEx \citep{trouillon2017knowledge} and TuckER \citep{balavzevic2019tucker}, and
neural network models ConvE \citep{dettmers2017convolutional}. Graph neural networks for KG embedding \citep{schlichtkrull2018modeling,vashishth2019composition,zhang2022knowledge} are not studied here
for their scalability issues on large-scale KGs \citep{ji2020survey}.
\noindent \textbf{Negative sampling.} Sampling negative triplets is important as only positive triplets are contained in the KGs \citep{wang2017knowledge}. We can pick up $m$ triplets by replacing the head or tail entity with uniform sampling~\citep{bordes2013translating} or use a full set of negative triplets.
Using the full set can be defined as the \texttt{1VsAll} \citep{lacroix2018canonical}
or \texttt{kVsAll}
\citep{dettmers2017convolutional}
according to the positive triplets used.
The methods \cite{cai2018kbgan,zhang2021simple} requiring additional models for negative sampling are not considered here.
\noindent \textbf{Loss function.}
There are three types of loss functions.
One can use margin ranking (MR) loss \citep{bordes2013translating} to rank the positive triplets higher over the negative ones, or use binary cross entropy (BCE) loss, with variants BCE\_mean, BCE\_adv \citep{sun2019rotate} and BCE\_sum \citep{trouillon2017knowledge}, to classify the positive and negative triplets as binary classes, or use cross entropy (CE) loss \citep{lacroix2018canonical} to classify the positive triplet as the true label over the negative triplets.
\noindent \textbf{Regularization.} To balance the expressiveness and complexity, and to avoid unbounded embeddings, the regularization techniques can be considered, such as regularizers like Frobenius norm (FRO) \citep{yang2014embedding,trouillon2017knowledge}, Nuclear norm (NUC) \citep{lacroix2018canonical} as well as DURA \citep{zhang2020duality}, and dropout on the embeddings \citep{dettmers2017convolutional}.
\noindent \textbf{Optimization.}
To optimize the embeddings, important optimization choices include the optimizer, such as SGD, Adam \citep{kingma2014adam} and Adagrad \citep{duchi2011adaptive}, learning rate, initializers, batch size, embedding dimension size, and add inverse relation \citep{lacroix2018canonical}
or not.
\section{Defining the search problem} \label{sec:understanding}
Denote an instance $\mathbf x = (x_1, x_2\dots, x_n)$, which is called an HP configuration, in the search space $\mathcal X$.
Let $F(\bm P, \mathbf x)$ be an embedding model with model parameters $\bm P$ and HPs $\mathbf x$, we define $\mathcal M(F(\bm P, \mathbf x), D_{\text{val}})$ as the performance measurement (the larger the better) on validation data $D_{\text{val}}$ and $\mathcal L(F(\bm P, \mathbf x), D_{\text{tra}})$ as the loss function (the smaller the better) on training data $D_{\text{tra}}$. We define the problem of HP search for KG embedding models in Definition~\ref{def:evaluation}. The objective is to search an optimal configuration $\mathbf x^*\in\mathcal X$ such that the embedding model $F$ can achieve the best performance on the validation data $D_{\text{val}}$.
\begin{definition}[Hyper-parameter search for KG embedding] \label{def:evaluation}
The problem of HP search for KG embedding model is formulated as \begin{align}
\mathbf x^* & = \arg\max\nolimits_{\mathbf x\in\mathcal X} \mathcal M\big(F(\bm P^*, \mathbf x), D_{\text{val}} \big), \label{eq:opthp} \\ \bm P^* & = \arg\min\nolimits_{\bm P} \mathcal L\big(F(\bm P, \mathbf x), D_{\text{tra}}\big). \label{eq:optemb}
\end{align}
\end{definition}
Definition~\ref{def:evaluation} is a bilevel optimization problem \citep{colson2007overview}, which can be solved by many conventional HP search algorithms. The most common and widely used approaches are sample-based methods like grid search and random search \citep{bergstra2012random}, where the HP configurations are independently sampled. To guide the sampling of HP configurations by historical experience, SMBO-based methods \citep{bergstra2011algorithms,hutter2011sequential} learn a surrogate model to select configurations based on the results that have been evaluated.
Then, the model parameters $\bm P$
are optimized by minimizing the loss function $\mathcal L$ on $D_{\text{tra}}$ in Eq.~\eqref{eq:optemb}. The evaluation feedback $\mathcal M$ of $\mathbf x$ on the validation data $D_{\text{val}}$ is used to update the surrogate.
\begin{figure*}
\caption{Ranking distribution of selected HPs.
A value with larger area in the bottom indicates
the higher ranking of this value.
The four figures correspond to the four groups:
reduced options, shrunken range, monotonously related, no obvious patterns.
Full results are in the Appendix~\ref{app:under:space}.}
\label{fig:hyper-range}
\end{figure*}
There are three major aspects determining the efficiency of Definition~\ref{def:evaluation}: (i) the size of search space $\mathcal X$, (ii) the validation curvature of $\mathcal M(\cdot, \cdot)$ in Eq.~\eqref{eq:opthp}, and (iii) the evaluation cost in solving $\arg\min_{\bm P}\mathcal L$ in Eq.~\eqref{eq:optemb}.
However, the existing methods \cite{ruffinelli2019you,ali2020bringing} directly search on a huge space with commonly used surrogate models and slow evaluation feedback from the full KG due to the lack of understanding on the search problem, leading to low efficiency.
\section{Understanding the search problem} \label{sec:underpro}
To address the mentioned limitations, we measure the significance and correlation of each HP to determine the feasibility of the search space $\mathcal X$ in Section~\ref{ssec:indivudial}. In Section~\ref{ssec:surrogate}, we visualize the HPs that determine the curvature of Eq.~\eqref{eq:opthp}. To reduce the evaluation cost in Eq.~ \eqref{eq:optemb}, we analyze the approximation methods in Section~\ref{ssec:fasteval}. Following \citep{ruffinelli2019you}, the experiments run on the seven embedding models in Section~\ref{sec:revisit} and two widely used datasets WN18RR \citep{dettmers2017convolutional} and FB15k-237 \citep{toutanova2015observed}. The experiments are implemented with PyTorch framework \citep{paszke2017automatic}, on a machine with two Intel Xeon 6230R CPUs and eight RTX 3090 GPUs with 24 GB memories each. We provide the implementation details in the Appendix~\ref{app:implementation}.
\subsection{Search space: $\mathbf{x} \in \mathcal X$} \label{ssec:indivudial}
Considering such large amount of HP configurations in $\mathcal X$, we take the simple and efficient approach where HPs are evaluated under control variate \citep{hutter2014efficient,you2020design}, which varies the $i$-th HP while fixing the other HPs. First, we discretize the continuous HPs according to their ranges.
Then the feasibility of the search space $\mathcal X$ is analyzed by checking the ranking distribution and consistency of individual HPs. These can help us shrink and decouple the search space.
The detailed setting for this part is in the Appendix~\ref{app:configgen}.
\noindent \textbf{Ranking distribution.}
To shrink the search space, we use the ranking distribution to indicate what HP values perform consistently.
Given an anchor configuration $\mathbf x$, we obtain the ranking of different values $\theta\in X_i$ by fixing the other HPs, where $X_i$ is the range of the $i$-th HP. The ranking distribution is then collected over the different anchor configurations in $\mathcal X_i$, different models and datasets.
According to the violin plots of ranking distribution shown in Figure~\ref{fig:hyper-range},
the HPs can be classified into four groups: \begin{itemize}[leftmargin=18px, itemsep=0.6pt,topsep=0pt,parsep=0pt,partopsep=0pt] \item[(a)]
\textit{reduced options}, e.g., Adam is the best optimizer and inverse relation should not be introduced;
\item[(b)]
\textit{shrunken range}, e.g., learning rate, reg. weight and dropout rate are better in certain ranges;
\item[(c)] \textit{monotonously related}: e.g., larger batch size and dimension size tend to be better;
\item[(d)] \textit{no obvious patterns}: e.g., the remaining HPs. \end{itemize}
\noindent \textbf{Consistency.}
To decouple the search space, we measure the consistency of configurations' rankings when only a specific HP changes. For the $i$-th HP, if the ranking of configurations' performance is consistent with different values of $\theta\in X_i$, we can decouple the search procedure of the $i$-th HP with the others. We measure such consistency with the spearman's ranking correlation coefficient (\texttt{SRCC}) \citep{schober2018correlation}.
\begin{figure*}
\caption{Curvature of the search space and three surrogate models.
The search space curvature is quite complex
with many local maximum areas.
The curvature of RF approximate the ground truth best.}
\label{fig:hyperspace_curvature}
\label{fig:gp_curvature}
\label{fig:mlp_curvature}
\label{fig:rf_curvature}
\label{fig:hyperspace_curvature-range}
\end{figure*}
Given a value $\theta\in X_i$, we obtain the ranking $r(\mathbf x, \theta)$ of the anchor configurations $\mathbf x \in \mathcal X_i$ by fixing the $i$-th HP as $\theta$.
Then, the \texttt{SRCC} between the two HP values $\theta_1, \theta_2\in X_i$ is computed as \begin{align}
1 -
\frac{\sum_{\mathbf x\in\mathcal X_i}\!|r(\mathbf x,\theta_1)\!-\!r(\mathbf x,\theta_2)|^2}
{|\mathcal X_i|\cdot (|\mathcal X_i|^2-1)},
\label{eq:spearman} \end{align}
where $|\mathcal X_i|$ means the number of anchor configurations in $\mathcal X_i$. \texttt{SRCC} indicates the matching rate of rankings for the anchor configurations in $\mathcal X_i$ with respect to $x_i=\theta_1$ and $x_i=\theta_2$.
Then the consistency of the $i$-th HP is evaluated by averaging the \texttt{SRCC} over the different pairs of $(\theta_1, \theta_2)$ for $X_i$, the different models and different datasets. The larger consistency (in the range $[-1,1]$) indicates that changing the value of the $i$-th HP does not influence much on the other configurations' ranking.
\begin{figure}
\caption{Consistency of each HP.}
\label{fig:HP_consistency}
\end{figure}
As in Figure~\ref{fig:HP_consistency}, the batch size and dimension size show higher consistency than the other HPs.
Hence, the evaluation of the configurations can be consistent with different choices of the two HPs. This indicates that we can decouple the search of batch size and dimension size with the other HPs.
\subsection{Validation curvature: $\mathcal{M}(\cdot, \cdot)$} \label{ssec:surrogate}
We analyze the curvature of the validation performance $\mathcal{M}(\cdot, \cdot)$ w.r.t $\mathbf x \in \mathcal{X}$. Specifically, we follow \citep{li2017visualizing} to visualize the validation loss' landscape by uniformly varying the numerical HPs in two directions (20 configurations in each direction)
on the model ComplEx and dataset WN18RR. From Figure~\ref{fig:hyperspace_curvature}, we observe that the curvature is quite complex with many local maximum areas.
To gain insights from evaluating these configurations and guide the next configuration sampling,
we learn a surrogate model as the predictor to approximate the validation curvature.
The curvatures of three common
surrogates, i.e., Gaussian process (GP)~\citep{williams1996gaussian}, multi-layer perceptron (MLP)~\citep{gardner1998artificial} and random forest (RF)~\citep{breiman2001random}, are in Figures~\ref{fig:gp_curvature}-\ref{fig:rf_curvature}. The surrogate models are trained with the evaluations of $100$ random configurations in the search space.
As shown, both GP and MLP fail to capture the complex local surface in Figure~\ref{fig:hyperspace_curvature} as they tend to learn a flat and smooth distribution in the search space. In comparison, RF is better in capturing the local distributions. Hence, we regard RF as a better choice in the search space. A more detailed comparison on the approximation ability of different surrogates is in the Appendix~\ref{app:fitting}.
\subsection{Evaluation cost: $\arg\min_{\bm{P}} \mathcal{L}$} \label{ssec:fasteval}
The evaluation cost of the HP configuration on an embedding model is the majority computation cost in HP search. Thus, we firstly evaluate the HPs that have influence on the evaluation cost, including batch size, dimension size, number of negative samples loss function and regularizer. Then, we analyze the evaluation transfer ability from small subgraph to the full graph.
\noindent \textbf{Cost of different HPs.}
The cost of each HP value $\theta \in X_i$ is averaged over the different anchor configurations in $\mathcal X_i$, different models and datasets. For fair comparsion, the time cost is counted per thousand iterations. We find that the evaluation cost increases significantly with larger batch size and dimension size, while the number of negative samples and the choice of loss function or regularizer do not have much influence on the cost.
We provide two exemplar curves in Figure~\ref{fig:hyper-cost} and put the remaining results in the Appendix~\ref{app:under:cost}.
\begin{figure}
\caption{Computing time cost. The dots are the average and the shades are the standard deviation.}
\label{fig:hyper-cost}
\end{figure}
\noindent \textbf{Transfer ability of subgraphs.}
Subgraphs can efficiently approximate the properties of the full graph
\citep{hamilton2017inductive,teru2020inductive}.
We evaluate the impact of subgraph sampling on HP search by checking the consistency between evaluations results on small subgraph and those on the full graph.
First, we study how to sample subgraphs. There are several approaches to sample small subgraphs from a large graph~\cite{leskovec2006sampling}. We compare four representative approaches in
Figure~\ref{fig:sampling}, i.e., Pagerank node sampling (Pagerank), random edge sampling (Random Edge), single-start random walk (Single-RW) and multi-start random walk (Multi-RW). For a fair comparison, we constrain the subgraphs with about $20\%$ of the full graph. The consistency between the sampled subgraph with the full graph is evaluated by the \texttt{SRCC} in (\ref{eq:spearman}). We observe that multi-start random walk is the best among the different sampling methods.
\begin{figure}
\caption{Comparison of the sampling methods.}
\label{fig:sampling}
\end{figure}
Apart from directly transferring the evaluation from subgraph to full graph, we can alternatively
train a predictor with
observations on subgraphs and then transfers the model to predict the configuration performance on the full graph.
From Figure~\ref{fig:sampling}, we find that directly transferring evaluations from subgraphs to the full graph is much better than transferring the predictor model.
In addition, we show the consistency and cost in terms of different subgraph sizes (percentage of entities compared to the full graph) in Figure~\ref{fig:subgraph_correlation}.
As shown, evaluation on subgraphs can significantly improve the efficiency. When the scale increases, the consistency increases but the cost also increases. To balance the consistency and cost, the subgraphs with $20\%$ entities are the better choices.
\begin{figure}
\caption{Consistency and cost of different subgraph sizes, where the shades are the standard deviation.}
\label{fig:subgraph_correlation}
\end{figure}
\section{Efficient search algorithm} \label{sec:algorithm}
By analyzing the ranking distribution and consistency of HPs in Section~\ref{ssec:indivudial}, we observe that not all the HP values are equivalently good, and some HPs can be decoupled. These observations motivate us to revise the search space in Section~\ref{ssec:reduce}. Based on the analysis in Section~\ref{ssec:surrogate} and~\ref{ssec:fasteval}, we then propose an efficient two-stage search algorithm in Section~\ref{ssec:transfer}.
\subsection{Shrink and decouple the search space} \label{ssec:reduce}
To shrink the search space, we mainly consider groups (a) and (b) of HPs in Section~\ref{ssec:indivudial}.
From the full results in the Appendix~\ref{app:under:space}, we observe that Adam can consistently perform better than the other two optimizers, the learning rate is better in the range of $[10^{-4}, 10^{-1}]$, the regularization weight is better in $[10^{-8}, 10^{-2}]$, dropout rate is better in $[0, 0.3]$, and add inverse relation is not a good choice.
To decouple the search space,
we consider batch size and dimension size that have larger consistency values than the other HPs, and are monotonously related to the performance as in group (c). However, the computation costs of batch size and dimension size increase prominently as shown in Figure~\ref{fig:hyper-cost}.
Hence, we can set batch size as 128 and dimension size as 100 to search the other HPs with low evaluation cost and increase their values in a fine-tuning stage.
Given the full search space $\mathcal X$, we denote the shrunken space as ${\mathcal X}_\text{S}$
and the further decoupled space as ${\mathcal X}_\text{S|D}$. We achieve hundreds of times size reduction from ${\mathcal X}_\text{S}$ to ${\mathcal X}_\text{S|D}$ and we show the details of changes in the Appendix \ref{app:algorithm}.
\begin{figure*}
\caption{Search algorithm comparison (viewed in color).
The dots are the results collected per hour.}
\label{fig:HPO_comparison_mrr}
\end{figure*}
\subsection{Two-stage search algorithm (KGTuner)} \label{ssec:transfer}
As discussed in Section~\ref{ssec:fasteval}, the evaluation cost can be significantly reduced with small batch size, dimension size and subgraph. This motivates us to design a two-stage search algorithm, named KGTuner, as in Figure~\ref{fig:KGTuner} and Algorithm~\ref{alg:search}.
\begin{algorithm}[ht]
\centering
\caption{KGTuner: two-stage search algorithm}
\label{alg:search}
\small
\begin{algorithmic}[1]
\REQUIRE KG embedding model $F$, dataset $D$, and budget $B$;
\STATE shrink the search space $\mathcal{X}$ to ${\mathcal X}_\text{S}$ and decouple ${\mathcal X}_\text{S}$ to ${\mathcal X}_{\text{S|D}}$;
\\ \textbf{\# state one}: \textit{efficient evaluation on subgraph}
\STATE sample a subgraph (with $20\%$ entities) $G$ from $D_{\text{tra}}$ by multi-start random walk;
\label{step:samplesub}
\REPEAT \label{step:stage1:start}
\STATE sample a configuration $\hat{\mathbf x}$ from ${\mathcal X}_{S|D}$ by RF+BORE;
\STATE evaluate $\hat{\mathbf x}$ on the subgraph $G$ to get the performance;
\STATE update the RF with record $\big(\hat{\mathbf x}, \mathcal M(F(P^*,\hat{\mathbf x}), G_{\text{val}})\big)$;
\UNTIL{$\nicefrac{B}{2}$ budget exhausted;}\label{step:stage1:end}
\STATE save the \textit{top10} configurations in ${\mathcal X}_\text{S|D}^*$; \\
\textbf{\# state two}: \textit{fine-tune the top configurations}
\STATE increase the batch/dimension size in ${\mathcal X}_\text{S|D}^*$ to get $\tilde{\mathcal X}^*$;
\STATE set $y^*=0$ and re-initialize the RF surrogate;
\REPEAT \label{step:stage2start}
\STATE select a configuration $\tilde{\mathbf x}^*$ from $\tilde{\mathcal X}^*$ by RF+BORE;
\STATE evaluate on full graph $G$ to get the performance;
\STATE update the RF with record $\big(\tilde{\mathbf x}^*\!\!, \mathcal M(F(P^*\!,\tilde{\mathbf x}^*), \!D_{\text{val}})\!\big)$;
\STATE \textbf{if} $\mathcal M(F(P^*,\tilde{\mathbf x}^*), D_{\text{val}})>y^*$ \textbf{then}
\\
$y^*\!\leftarrow \mathcal M(F(P^*,\tilde{\mathbf x}^*), D_{\text{val}})$
and $\mathbf x^* \leftarrow \tilde{\mathbf x}^*$;
\textbf{end if}
\UNTIL{the remaining $\nicefrac{B}{2}$ budget exhausted;} \label{step:stage2end}
\RETURN $\mathbf x^*$.
\end{algorithmic} \end{algorithm}
\begin{itemize}[leftmargin=*,noitemsep,topsep=0pt,parsep=0pt,partopsep=0pt] \item In the first stage, we sample a subgraph $G$ with $20\%$ entities from the full graph $D_{\text{tra}}$ by multi-start random walk. Based on the understanding of curvature in Section~\ref{ssec:surrogate}, we use the surrogate model random forest (RF)
under the state-of-the art framework BORE \cite{tiao2021bore}, denoted as RF+BORE, to explore HPs in ${\mathcal X}_\text{S|D}$ on the subgraph $G$
in steps~\ref{step:stage1:start}-\ref{step:stage1:end}. The \textit{top10} configurations evaluated in this stage are saved in a set ${\mathcal X}^*_\text{S|D}$.
\item In the second stage, we increase batch size and dimension size for configurations in ${\mathcal X}^*_\text{S|D}$ to generate a new set $\tilde{\mathcal X}^*$. Then, the configurations in $\tilde{\mathcal X}^*$ are searched by the RF+BORE again in steps~\ref{step:stage2start}-\ref{step:stage2end} until the remaining $\nicefrac{B}{2}$ budget is exhausted.
\item Finally, the configuration $\mathbf x^*$ achieving the best performance on the full validation data $D_{\text{val}}$ is returned for testing. \end{itemize}
\subsection{Discussion}
e now summarize the main differences of KGTuner with the existing HP search algorithms, i.e. Random (random search) \citep{bergstra2012random}, Hyperopt \cite{bergstra2013hyperopt}, SMAC \citep{hutter2011sequential}, RF+BORE \citep{tiao2021bore}, and AutoNE \citep{tu2019autone}.
\begin{table}[ht]
\centering
\caption{Comparison of HP search algorithms.}
\setlength\tabcolsep{4.5pt}
\small
\label{tab:methods_compare}
\begin{tabular}{c | c | c | c | c}
\toprule
& \multicolumn{2}{c|}{search space} & surrogate & fast \\
& shrink & decouple & model & evaluation \\ \midrule
Random & $\times$ & $\times$ & $\times$ & $\times$ \\
Hyperopt & $\times$ & $\times$ & TPE & $\times$ \\
Ax & $\times$ & $\times$ & GP & $\times$ \\
SMAC & $\times$ & $\times$ & RF & $\times$ \\
RF+BORE & $\times$ & $\times$ & RF & $\times$ \\
AutoNE & $\times$ & $\times$ & GP & $\surd$ \\
KGTuner & $\surd$ & $\surd$ & RF & $\surd$ \\
\bottomrule
\end{tabular}
\end{table}
The comparison is based on three aspects, i.e., search space, surrogate model and fast evaluation, in Table~\ref{tab:methods_compare}. KGTuner shrinks and decouples the search space based on the understanding of HPs' properties, and uses the surrogate RF based on the understanding on validation curvature. The fast evaluation on subgraph in KGTuner selects the \textit{top10} configurations to directly transfer for fine-tuning, while AutoNE~\citep{tu2019autone} just uses fast evaluation on subgraphs to train the surrogate model, and transfers the surrogate model for HP search on the full graph. In Figure~\ref{fig:sampling}, the transfer ability of the surrogate model is shown to be much worse than direct transferring.
\section{Empirical evaluation}
\subsection{Overall performance} \label{ssec:exp:alg}
In this part, we compare the proposed algorithm KGTuner with six HP search algorithms in Table~\ref{tab:methods_compare}. For AutoNE, we allocate half budget for it to search on the subgraph and another half budget on the full graph with the transferred surrogate model. The baselines search in the full search space (in Table~\ref{tab:spacefull}) with the same amount of budget as KGTuner. We use the mean reciprocal ranking (MRR, the larger the better) \citep{bordes2013translating} to indicate the performance.
\noindent \textbf{Efficiency.} We compare the different search algorithms in Figure~\ref{fig:HPO_comparison_mrr} on an in-sample dataset WN18RR and an out-of-sample dataset ogbl-biokg. The time budget we set for WN18RR is one day's clock time, while that for ogbl-biokg is two days' clock time. For each dataset we show two kinds of figures. First, the best performance achieved along the clock time in one experiment on a specific model ComplEx. Second, we plot the the ranking of each algorithm averaged over all the models and datasets. Since AutoNE and KGTuner run on the subgraphs in the first stage, the starting points of them locate after 12 hours. The starting point of KGTuner is a bit later than AutoNE since it constrains to use large batch size and dimension size in the second stage, which is more expensive. As shown, random search is the worst.
SMAC and RF+BORE achieve better performance than Hyperopt and Ax since RF can fit the space better than TPE and GP as in Section~\ref{ssec:surrogate}. Due to the weak transfer ability of the predictor (see Figure~\ref{fig:sampling}) and the weak approximation ability of GP (see Figure~\ref{fig:hyperspace_curvature-range}), AutoNE also performs bad. KGTuner is much better than all the baselines. We show the full search process of the two-stage algorithms AutoNE and KGTuner on WN18RR in Figure~\ref{fig:two_stage_process}. By exploring sufficient number of configurations in the first stage, the configurations fine-tuned in the second stage can consistently achieve the best performance.
\noindent \textbf{Effectiveness.} For WN18RR and FB15k-237, we provide the reproduced results on TransE, ComplEx and ConvE with the original HPs, HPs searched by LibKGE and HPs searched by KGTuner in Table~\ref{tab:mrr:part}. The full results on the remaining four embedding models RotatE, RESCAL, DistMult and TuckER are in the Appendix~\ref{app:general-benchmark}. Overall, KGTuner achieves better performance compared with both the original reported results and the reproduced results in \citep{ruffinelli2019you}. We observe that the tensor factorization models such as RESCAL, ComplEx and TuckER have better performance than the translational distance models TransE, RotatE and neural network model ConvE. This conforms with the theoretical analysis that tensor factorization models are more expressive \citep{wang2018multi}.
\begin{table}[ht]
\centering
\caption{MRR of models with HPs tuned in different methods.
The bold numbers mean the best performance of the same model.}
\label{tab:mrr:part}
\small
\begin{tabular}{cc|c|c}
\toprule
source & models & WN18RR & FB15k-237 \\
\midrule
\multirow{3}{*}{original} & TransE & 0.226 & 0.296 \\
& ComplEx & 0.440 & 0.247 \\
& ConvE & 0.430 & 0.325 \\
\midrule
\multirow{3}{*}{LibKGE} & TransE & 0.228 & 0.313 \\
& ComplEx & 0.475 & 0.348 \\
& ConvE & \textbf{0.442} & \textbf{0.339} \\
\midrule
\multirow{3}{*}{KGTuner} & TransE & \textbf{0.233} & \textbf{0.327} \\
& ComplEx & \textbf{0.484} & \textbf{0.352} \\
& ConvE & 0.437 & 0.335 \\
\bottomrule
\end{tabular}
\end{table}
\begin{figure*}
\caption{(a): full search processes of the two-stage algorithms.
(b-d): ablation studies on KGTuner. Model ComplEx and dataset WN18RR are used in these experiments.}
\label{fig:two_stage_process}
\label{fig:ablation_space}
\label{fig:ablation_subgraph_size}
\label{fig:ablation_tradeoff}
\label{fig:HPO_ablation}
\end{figure*}
To further demonstrate the advantage of KGTuner, we apply it to the Open Graph Benchmark (OGB) \citep{hu2020open}, which is a collection of realistic and large-scale benchmark datasets for machine learning on graphs. Many embedding models have been tested there by two large-scale KGs for link prediction, i.e., ogbl-biokg and ogbl-wikikg2. Due to their scale, the evaluation cost of a HP configuration is very expensive. We use KGTuner to search HPs for embedding models, i.e., TransE, RotatE, DistMult, ComplEx and AutoSF \cite{zhang2020autosf}, on OGB. Since the computation costs of the two datasets are much higher, we set the time budget as 2 days for ogbl-biokg and 5 days for ogbl-wikikg2. All the embedding models evaluated here are constrained to have the same (or lower) number of model parameters\footnote{
We run all models on ogbl-wikikg2 with 100 dimension size
to avoid out-of-memory, instead of 500 on OGB board.}. More details on model parameters, standard derivation, and validation performance are in the Appendix~\ref{app:ogb}. As shown in Table~\ref{tab:ogb}, KGTuner consistently improves the performance of the four embedding models with the same or fewer parameters compared with the results on the OGB board.
\begin{table}[ht]
\centering
\caption{Performance in MRR
in OGB link prediction board {\small \url{https://ogb.stanford.edu/docs/leader_linkprop/}}
and those reproduced by KGTuner on ogbl-biokg and ogbl-wikikg2.
Relative improvements are in parenthesize.
}
\setlength\tabcolsep{2.5pt}
\small
\label{tab:ogb}
\begin{tabular}{cc|c|c}
\toprule
\multicolumn{2}{c|}{{models}} &{ogbl-biokg} & {ogbl-wikikg2} \\ \midrule
& TransE & 0.7452 & 0.4256 \\
& RotatE & 0.7989 & 0.2530 \\
original & DistMult & 0.8043 & 0.3729 \\
& ComplEx & 0.8095 & 0.4027 \\
& AutoSF & 0.8320 & 0.5186 \\ \midrule
& TransE & 0.7781 (4.41\%$\uparrow$) & 0.4739 (11.34\%$\uparrow$) \\
& RotatE & 0.8013 (0.30\%$\uparrow$) & 0.2944 (16.36\%$\uparrow$) \\
KGTuner & DistMult & 0.8241 (2.46\%$\uparrow$) & 0.4837 (29.71\%$\uparrow$) \\
& ComplEx & 0.8385 (3.58\%$\uparrow$) & 0.4942 (22.72\%$\uparrow$) \\
& AutoSF & 0.8354 (0.41\%$\uparrow$) & 0.5222 (0.69\%$\uparrow$) \\
\midrule
\multicolumn{2}{c|}{average improvement} & 2.23\% & 16.16\% \\
\bottomrule
\end{tabular}
\end{table}
\subsection{Ablation study} \label{ssec:exp:abla} In this subsection, we probe into how important and sensitive the various components of KGTuner are.
\noindent \textbf{Space comparison.} To demonstrate the effectiveness gained by shrinking and decoupling the search space, we compare the following variants: (i) RF+BORE on the full space $\mathcal X$; (ii) RF+BORE on the shrunken space ${\mathcal X}_\text{S}$;
(iii) RF+BORE on the decoupled space ${\mathcal X}_\text{S|D}$, which differs from KGTuner by searching on the full graph in the first stage; and (iv) KGTuner in Algorithm~\ref{alg:search}. All the variants, i.e., RF+BORE, have one day's time budget. As in Figure~\ref{fig:ablation_space}, the size of search space matters for the search efficiency. The three components, i.e., space shrinkage, space decoupling, and fast evaluation on subgraph, are all important to the success of KGTuner.
\noindent \textbf{Size of subgraphs.} We show the influence of subgraph sizes with different ratios of entities (10\%, 20\%, 30\%, 40\%, 50\%) from the full graph in Figure~\ref{fig:ablation_subgraph_size}. Using subgraphs with too large or too small size is not guaranteed to find good configurations. Based on the understanding in Figure~\ref{fig:subgraph_correlation}, the subgraph with small size have poor transfer ability and those with large size are expensive to evaluate.
Hence, we should balance the transfer ability and evaluation cost by sampling subgraphs with $20\%\sim 30\%$ entities.
\noindent \textbf{Budget allocation.} In Algorithm~\ref{alg:search}, we allocate $\nicefrac{B}{2}$ budget for both the first and second stage. Here, we show the performance of different allocation ratios, i.e., $\nicefrac{B}{4}$, $\nicefrac{B}{2}$, and $\nicefrac{3B}{4}$ in the first stage and the remaining budget in the second stage. As in Figure~\ref{fig:ablation_tradeoff}, allocating too many or too few budgets to the first stage is not good. It either fails to explore sufficient configurations in the first stage or only fine-tunes a few configurations in the second stage. Allocating the same budget to the two stages is in a better trade-off.
\section{Related works} \label{sec:relworks}
In analyzing the performance of KG embedding models,
\citet{ruffinelli2019you} pointed out that the earlier works in KG embedding only search HPs in small grids. By searching hundreds of HPs in a unified framework, the reproduced performance can be significantly improved. Similarly, \citet{ali2020bringing} proposed another unified framework to evaluate different models. \citet{rossi2020knowledge} evaluated 16 different models and analyzed their properties on different datasets. All of these works emphasize the importance of HP search, but none of them provide efficient algorithms to search HPs for KG learning. AutoSF \citep{zhang2020autosf} evaluates the bilinear scoring functions and set up a search problem to design bilinear scoring functions, which can be complementary to KGTuner.
Understanding the HPs in a large search space is non-trivial since many HPs only have moderate impact on the model performance \cite{ruffinelli2019you} and jointly evaluating them requires a large number of experiments \citep{fawcett2016analysing,probst2019tunability}.
Considering the huge amount of HP configurations (with $10^{5}$ categorical choices and $5$ continuous values), it is extremely expensive to exhaustively evaluate most of them. Hence, we adopt control variate experiments to efficiently evaluate HPs' properties instead of the quasi-random search in \citep{ruffinelli2019you,ali2020bringing}.
Technically, we are similar to AutoNE \citep{tu2019autone} and e-AutoGR \citep{wang2021explainable} by leveraging subgraphs to improve search efficiency on graph learning.
Since they do not target at KG embedding methods, directly adopt them is not a good choice. Besides, based on the understanding in this paper, we demonstrate that transferring the surrogate model from subgraph evaluation to the full graph is inferior to directly transferring the top configurations for KG embedding models.
\section{Conclusion}
In this paper, we
analyze the HPs' properties in KG embedding models with search space size, validation curvature and evaluation cost.
Based on the observations, we propose an efficient search algorithm KGTuner that efficiently explores configurations in a reduced space on small subgraph and then fine-tunes the top configurations with increased batch size and dimension size on the full graph. Empirical evaluations show that KGTuner is robuster and more efficient than the existing HP search algorithms and achieves competing performance on large-scale KGs in open graph benchmarks. In the future work, we will understand the HPs in graph neural network based models and apply KGTuner on them to solve the scaling limitations in HP search.
\appendix
\onecolumn
\section{Details of the search space} \label{app:searchspace}
Denote a knowledge graph as $\mathcal G = \{E, R, D \}$, where $E$ is the set of entities, $R$ is the set of relations, and $D$ is the set of triplets with training/validation/test splits $D = D_{\text{tra}}\cup D_{\text{val}}\cup D_{\text{tst}}$.
Basically, the KG embedding models use a scoring function $f$ and the model parameters $\bm P$ to measure the plausibility of triplets. We learn the embeddings such that the positive and negative triplets can be separated by $f$ and $\bm P$. In Table~\ref{tab:sfdefinition}, we provide the forms $f$ of the embedding model we used to evaluate the search space $\mathcal X$ in Section~\ref{sec:understanding}.
\begin{table*}[ht]
\centering
\caption{Definitions of the embedding models.
$\circ$ is a rotation operation in the complex value space;
$\otimes$ is the Hermitian dot product in the complex value space;
$\text{Re}(\cdot)$ returns the real part of a complex value;
$\mathcal W_{i,j,k}$ is the $ijk$-th element in a core tensor $\mathcal W\in\mathbb R^{d\times d\times d}$;
and conv is a convolution operator on the head and relation embeddings.
For more details, please refer to the corresponding references.}
\label{tab:sfdefinition}
\small
\begin{tabular}{cc|c|c}
\toprule
model type & model & $f(h,r,t)$ & embeddings \\ \midrule
\multirow{2}{*}{translational distance}&TransE \citep{bordes2013translating} & $-\|\bm h+\bm r-\bm t\|_1$ & $\bm h, \bm r, \bm t \in\mathbb R^d$ \\
&RotatE \citep{sun2019rotate} & $ -\|\bm h \circ \bm r-\bm t\|_{c1}$ & $\bm h, \bm r, \bm t \in\mathbb C^d$ \\ \midrule
\multirow{4}{*}{tensor factorization} &RESCAL \citep{nickel2011three} & $\bm h^\top \cdot\bm R_r\cdot\bm t$ & $\bm h, \bm t\in\mathbb R^d, \bm R_r\in\mathbb R^{d\times d}$ \\
&DistMult \citep{yang2014embedding} & $\bm h^\top \cdot\text{diag}(\bm r)\cdot\bm t$ & $\bm h, \bm t, \bm r\in\mathbb R^d$ \\
&ComplEx \citep{trouillon2017knowledge} & $\bm h^\top \otimes \text{diag}({\bm r})\otimes \bm t$ & $\bm h, \bm t, \bm r\in\mathbb C^d$ \\
&TuckER \citep{balavzevic2019tucker} & $\sum_{i}^d\sum_j^d\sum_k^d \mathcal W_{i,j,k}h_i\cdot r_j \cdot t_k$ & $\bm h, \bm t, \bm r\in\mathbb R^d$ \\ \midrule
neural network &ConvE \citep{dettmers2017convolutional} & $\text{ReLU}(\text{conv}(\bm h, \bm r))^\top \cdot \bm t$ & $\bm h, \bm t, \bm r\in\mathbb R^d$ \\
\bottomrule
\end{tabular} \end{table*}
\subsection{Negative sampling} Since KG only contains positive triplets in $D_{\text{tra}}$ \citep{wang2017knowledge}, we should rely on the negative sampling to avoid trivial solutions of the embeddings. Given a positive triplet $(h,r,t)\in D_{\text{tra}}$, the corresponding set of negative triplets is represented as \begin{align*} D^-_{(h,r,t)} = \big\{(\tilde{h}, r, t)\notin D_{\text{tra}}\!:\! (h,r,t)\in D_{\text{tra}}, \tilde{h}\in E \big\}\cup \big\{({h}, r, \tilde{t})\notin D_{\text{tra}}\!:\! (h,r,t)\in D_{\text{tra}}, \tilde{t}\in E \big\}. \end{align*} A common practice is to sample $m$ negative triplets from $D^-_{(h,r,t)}$. The value of $m$ can be any integer smaller than the number of entities. We follow \citep{sun2019rotate} to sample from the range of $m$ in $\{32,128,512,2048\}$ for simplicity.
An alternative choice is to use all the negative triplets in $D^-_{(h,r,t)}$, leading to the \texttt{1VsAll} \citep{lacroix2018canonical} and \texttt{kVsAll} \citep{dettmers2017convolutional} settings. \begin{itemize}[leftmargin=*]
\item In \texttt{1VsAll}, $(h,r,t)$ is in the positive part and all the triplets in the set $\{(\tilde{h}, r, t)\notin D_{\text{tra}}\!:\! (h,r,t)\in D_{\text{tra}}, \tilde{h}\in E \}$
or $\{({h}, r, \tilde{t})\notin D_{\text{tra}}\!:\! (h,r,t)\in D_{\text{tra}}, \tilde{t}\in E \}$
are in the negative part;
\item In \texttt{kVsAll}, the positive part contains all the triplets sharing the same head-relation pair or tail-relation part,
i.e. $\{(h,r,t')\in D_{\text{tra}} \}$ or $\{(h',r,t)\in D_{\text{tra}} \}$,
with the corresponding negative part $\{({h}, r, \tilde{t})\notin D_{\text{tra}}\!:\! (h,r,t)\in D_{\text{tra}}, \tilde{t}\in E \}$
or $\{(\tilde{h}, r, t)\notin D_{\text{tra}}\!:\! (h,r,t)\in D_{\text{tra}}, \tilde{h}\in E \}$. \end{itemize}
Hence, the choice of negative sampling can be set in the range $\{32, 128, 512, 2048, \text{\texttt{1VsAll}}, \text{\texttt{kVsAll}} \}$.
\subsection{Loss function} For simplicity, we denote $D^+$ and $D^-$ as the sets of positive and negative triplets, respectively. Then, we summarize the commonly used loss functions as follows: \begin{itemize}[leftmargin=*]
\item Margin ranking (MR) loss.
This loss ranks the positive triplets to have larger score than the negative triplets.
Hence, the ranking loss is defined as
\[ \mathcal L
= \sum\nolimits_{(h,r,t)\in D^+}
\sum\nolimits_{(\tilde{h},r, \tilde{t})\in D^-}
-\big|\gamma-f(h,r,t) + f(\tilde{h}, r, \tilde{t})\big|_+, \]
where $\gamma>0$ is the margin value and $|a|_+ = \max(a, 0)$.
The MR loss is widely used in early developed models,
like TransE \citep{bordes2013translating} and DistMult \citep{yang2014embedding}.
The value of $\gamma$, conditioned on MR loss, is another HP to search.
\item Binary cross entropy (BCE) loss.
It is typical to set the positive and negative triplets as a binary classification problem.
Let the labels for the positive and negative triplets as $+1$ and $-1$ respectively,
the BCE loss is defined as
\begin{align*}
\mathcal L =
\sum\nolimits_{(h,r,t)\in D^+} \log\big(\sigma(f(h,r,t))\big) +
\sum\nolimits_{(\tilde{h},r, \tilde{t})\in D^-}\!\! w_{(\tilde{h},r, \tilde{t})} \log\big(1-\sigma(f(\tilde{h},r,\tilde{t}))\big),
\end{align*}
where $\sigma(x)=\frac{1}{1+\exp(-x)}$ is the sigmoid function.
The choice of $w_{(\tilde{h},r, \tilde{t})}$ leads to three different loss functions
\begin{itemize}[leftmargin=*]
\item BCE\_mean \citep{sun2019rotate}, with $w_{(\tilde{h},r, \tilde{t})}= \nicefrac{1}{|D^-_{(h,r,t)}|}$.
\item BCE\_sum \citep{dettmers2017convolutional}, with $w_{(\tilde{h},r, \tilde{t})}= 1$.
\item BCE\_adv \citep{sun2019rotate}, with
\[w_{(\tilde{h},r, \tilde{t})}= \frac{\exp(\alpha\cdot f(\tilde{h},r,\tilde{t}))}{\sum_{(h',r,t')\in D^-}\exp(\alpha \cdot f({h'},r,{t'}))},\]
where $\alpha>0$ is the adversarial weight conditioned on BCE\_adv loss.
\end{itemize}
\item Cross entropy (CE) loss.
Since the number of negative triplets is fixed,
we can also regard the $(h,r,t)$ as the true label over the negative ones.
The loss can be written as
\begin{align*}
\mathcal L =
\sum\nolimits_{(h,r,t)\in D^+}
- f(h,r,t)
+
\log
\left(
\sum\nolimits_{(h',r,t')\in \{(h,r,t)\cup D^-\}} \exp (f(h',r,t'))
\right) ,
\end{align*}
where the left part is the score of positive triplet and the right is the log sum scores of the joint set of positive and negative triplets.
\end{itemize}
\subsection{Regularization} To avoid the embeddings increasing to unlimited values and reduce the model complexity, regularization techniques are often used. Denote $\bm P'$ as the embeddings participated in one iteration, \begin{itemize}[leftmargin=*]
\item the Frobenius norm is defined as the sum of L2 norms
$r_{\text{FRO}}= \|\bm P'\|_2^2 = \sum_{ij}{P'}_{ij}^2$ \citep{yang2014embedding};
\item the NUC norm is defined as sum of L3 norms
$r_{\text{FRO}}= \|\bm P'\|_3^3 = \sum_{ij}|P_{ij}|^3$ \citep{lacroix2018canonical};
\item DURA operates on triplets \citep{zhang2020duality}.
Denote $\bm h, \bm r, \bm t$ as the embeddings for the triplet $(h,r,t)$,
DURA constrains the composition of $\bm h$ and $\bm r$ to approximate $\bm t$
with $r_{\text{DURA}} = \|c(\bm h,\bm r)- \bm t\|_2^2$,
where the composition function $c(\bm h,\bm r)$ depends on corresponding scoring functions. \end{itemize} The regularization functions are then weighted by the regularization weight in the range $[10^{-12}, 10^2]$.
Apart from using explicit forms of regularization, we can also add dropout on the embeddings \citep{dettmers2017convolutional}. Specifically, each dimension in the embeddings $\bm h, \bm r, \bm t$ will have a probability to be deactivated as $0$ in each iteration. The probability is controlled by the dropout rate in the range $[0, 0.5]$. In some cases, working without regularization can also achieve good performance \citep{ali2020bringing}.
\subsection{Optimization} To solve the learning problem, we should setup an appropriate optimization procedure. First, we can directly use the training set or add inverse relations to augment the data \citep{kazemi2018simple,lacroix2018canonical}. This will not influence the training data, but will introduce additional parameters for the inverse relations. Second, we should choose the dimension of embeddings in small sizes $[100, 200]$ or large sizes $[500, 1000, 2000]$. Then, the embeddings are initialized by the initialization methods such as uniform, normal, xavier\_norm, and xavier\_uniform \citep{goodfellow2016deep}. The optimization is conducted with optimizers like standard SGD, Adam \citep{kingma2014adam} and Adagrad \citep{duchi2011adaptive} with learning rate in the range $[10^{-5}, 0]$ Since the training is conducted on mini-batch, a batch size is determined in the range $\{128, 256, 512, 1024\}$.
\section{Details of HP understanding} In this part, we provide the details of configuration generation and the full results related to the HP understanding.
\subsection{Configure generation} \label{app:configgen}
Since there are infinite numbers of values for a continuous HP, it is intractable to fully evaluate their ranges. To better analyze the continuous HPs, we discretize them in Table~\ref{tab:discrete} according to their ranges. Then, for each HP $i=1\dots n$ with range $X_i$, we sample a set ${\mathcal X}_{i} \subset \mathcal X$ of $s$ anchor configurations through quasi random search \citep{bergstra2012random}
and uniformly distribute them to evaluate the different embedding models and datasets.
\begin{table}[ht]
\centering
\caption{Discretized HP values.}
\label{tab:discrete}
\small
\begin{tabular}{c|c|c}
\toprule
name & original range & discretized range \\
\midrule
gamma &[1, 24] & $\{1, 6,12,24\}$ \\
adv. weight & [0.5, 2.0] & \{0.5, 1, 2\}\\
reg. weight & [$10^{-12}\!, 10^{2}$] &
$10^2$ in log scale \\
dropout rate & $[0, 0.5]$ & $0.1$ in linear scale \\
learning rate & [$10^{-5}\!, 10^0$] & $10^1$ in log scale \\
\bottomrule
\end{tabular} \end{table}
We use the control variate experiments to evaluate each HP. For the $i$-th HP, we enumerate the values $\theta\in X_i$ for each anchor configuration $\mathbf x\in{\mathcal X}_i$, while fix the other HPs. In this way, we can observe the influence of $x_i$ without the influence of the other HPs. For example, when evaluating the optimizers, we enumerate the optimizers Adam, Adagrad and SGD for the anchor configurations in ${\mathcal X}_i$. This generates a set of
$|\mathcal X_i| \cdot |X_i|$ configurations.
In this paper, the number of anchor configurations $|\mathcal X_i|$ is 175 for each HP.
\subsection{Details for search space understanding} \label{app:under:space}
In this part, we add the ranking distribution of all the HPs. In addition, we also show the normalized MRR of each HP as a complementary. The normalization is conducted on each dataset with $\frac{y-y_{\min}}{y_{\max}-y_{\min}}$ such that the results of the HPs can be evaluated in the same value range.
The full results for the four types of HPs in Section~\ref{ssec:indivudial} are provided in Figures~\ref{fig:hp:fixed}-\ref{fig:hp:nopattern}. The larger area in the bottom in the voilin plots and the top area in the box plots indicate better performance. The HPs can be classified into four types: \begin{itemize}[leftmargin=20pt]
\item[(a).] \textit{fixed choices}: Adam is the fixed optimizer, and inverse relation is not preferred. See Figure~\ref{fig:hp:fixed}.
\item[(b).] \textit{limited range}: Learning rate, regularization weight and dropout rate should be limited in the ranges
$[10^{-4}, 10^{-1}]$, $[10^{-12}, 10^{-2}]$ and $[0,0.3]$, respectively.
See Figure~\ref{fig:hp:limited}
\item[(c).] \textit{monotonously related}: Batch size and dimension size have monotonic performance. The larger value tends to lead better results.
See Figure~\ref{fig:hp:monotonic}.
\item[(d).] \textit{no obvious patterns}: The choice of loss function, value of gamma, adversarial weight, number of negative samples, regularizer, initializer do not have obvious patterns.
See Figure~\ref{fig:hp:nopattern}. \end{itemize}
In addition, we provide the details of Spearman's ranking correlation coefficient (\texttt{SRCC}). Given a set of anchor configurations $\mathcal X_i$ to analyze the $i$-th HP, we denote $r(\mathbf x, \theta)$ as the rank of different $\mathbf x\in\mathcal X_i$ with fixed $x_i=\theta$. Then, the \texttt{SRCC} between two
HP values $\theta_1, \theta_2\in X_i$ is
\begin{equation} \texttt{SRCC} (\theta_1, \theta_2) = 1 -
\frac{\sum_{\mathbf x\in\mathcal X_i}|r(\mathbf x,\theta_1)-r(\mathbf x,\theta_2)|^2}
{|\mathcal X_i|\cdot (|\mathcal X_i|^2-1)}, \end{equation}
where $|\mathcal X_i|$ means the number of anchor configurations in $\mathcal X_i$. We evaluate the consistency of the $i$-th HP by averaging the \texttt{SRCC} over the different pairs of $(\theta_1, \theta_2) \in X_i\times X_i$, the different models and datasets.
\begin{figure}
\caption{HPs that have fixed choice since one configure has significant advantage.}
\label{fig:hp:fixed}
\end{figure}
\begin{figure}
\caption{HPs that have limited ranges since they only perform well in certain ranges.}
\label{fig:hp:limited}
\end{figure}
\begin{figure}
\caption{HPs that is monotonic with different choices of values.}
\label{fig:hp:monotonic}
\end{figure}
\begin{figure}
\caption{HPs that do not have obvious patterns. All of the values should be searched.}
\label{fig:hp:nopattern}
\end{figure}
\subsection{Approximation ability of surrogate models} \label{app:fitting}
In Section~\ref{ssec:surrogate}, we have shown that the curvature of a learned random forest (RF) model is more similar with the real curvature of the ground truth. Here, we further demonstrate this point through a synthetic experiment.
Specifically, 100 random configurations with evaluated performance are sampled. We use 10/20/30 random samples from them to train the surrogates since only a small number of HP configurations are available for the surrogate during searching. The remaining configurations are used for testing. Then, we evaluate the fitting ability of each model by the mean square error (MSE) of the estimated prediction to the target prediction.
For GP \citep{rasmussen2003gaussian}, we show the prediction with the Matern kernel used in AutoNE \citep{tu2019autone}. For RF \citep{breiman2001random}, we build 200 tree estimators to fit the training samples. The MLP here \citep{gardner1998artificial} is designed as a three-layer feed-forward network with 100 hidden units and ReLU activation function in each layer. The average value and std of MSE over five different groups of configurations are shown in Table~\ref{tab:surrogate}. As can been seen, random forest show much lower prediction error than GP and MLP with different number of training samples. This further demonstrates that RF can better fit such a complex HP search space.
\begin{table}[ht]
\centering
\caption{Comparison of different surrogate models in MSE.}
\label{tab:surrogate}
\small
\begin{tabular}{c|ccc}
\toprule
\# train configurations & 10 & 20 &30 \\
\midrule
GP & 0.0693$\pm$0.02 &0.029$\pm$0.01 & 0.019$\pm$0.01 \\
MLP & 2.121$\pm$0.4 & 2.052$\pm$0.3 & 0.584$\pm$0.1 \\
RF & \underline{\textbf{ 0.003$\pm$0.002}} & \underline{\textbf{0.002$\pm$0.001}} & \underline{\textbf{0.001$\pm$0.001}} \\
\bottomrule
\end{tabular} \end{table}
\subsection{Results of cost evaluation} \label{app:under:cost}
We show the average cost and standard derivation of five HPs, i.e. batch size, dimension size, number of negative samples,
loss functions,
and regularizer, in Figure~\ref{fig:app:hyper-cost}. As can be seen, the cost of batch size and dimension size increase much when the size increases. But for the number of negative samples, choices of loss functions and regularizers, the influence on cost is not strong as indicated by the average cost.
\begin{figure}
\caption{Computing time cost. The dots are the average and the shades are the standard deviation.}
\label{fig:app:hyper-cost}
\end{figure}
\section{Detail for the search algorithm} \label{app:algorithm}
\subsection{Search space} We show the shrunken and decoupled search space compared with the full space in Table~\ref{tab:spacereduce}.
To evaluate the ratio of space change after shrinkage and decoupling, we measure the learning rate and regularization weight in log scale. The size of the whole space $\mathcal X$ compared with the decoupled ${\mathcal X}_{S|D}$ is \[3\times \frac{14}{6}\times\frac{5}{3}\times\frac{5}{3}\times 2\times 4 \times 5 = 777.8. \] Hence, the reduced and decoupled space is hundreds times smaller than the full space.
\begin{table}[ht]
\centering
\caption{The revised HP values in the reduced and decoupled search space compared with the full space.}
\label{tab:spacereduce}
\setlength\tabcolsep{2pt}
\small
\begin{tabular}{c|c|c}
\toprule
name & ranges in the whole space & revised ranges \\
\midrule
optimizer & \{Adam, Adagrad, SGD\} & Adam \\
learning rate & [$10^{-5}$, $10^0$] & [$10^{-4}$, $10^{-1}$] \\
reg. weight & [$10^{-12}$, $10^{2}$] & [$10^{-8}$, $10^{-2}$] \\
dropout rate & [0, 0.5] & [0, 0.3] \\
inverse relation & \{True, False\} & \{False\} \\
\midrule
batch size & \{128, 256, 512, 1024\} & 128 \\
dimension size & \{100, 200, 500, 1000, 2000\} & 100 \\
\bottomrule
\end{tabular} \end{table}
\subsection{Search algorithm}
We visualize the searching process of the traditional one-stage method and the proposed two-stage method in Figure~\ref{fig:one_two_stage_comparison}. Since the evaluation cost on the full graph is rather high,
the one-stage method can only take a few optimization trials. Thus the search space remains unexplored for a large proportion, and the performance of the optimal configuration is hard to be guaranteed. As for the proposed two-stage method KGTuner, it efficiently explores the search space on the sampled subgraph at the first stage, and then fine-tunes the top-K configurations on the full graph.
\begin{figure*}
\caption{Diagram of one-stage search method and the proposed two-stage method.
}
\label{fig:one_two_stage_comparison}
\end{figure*}
In Algorithm~\ref{alg:search}, we increase the batch size and dimension size in stage two. We set the searched range for batch size in stage two as $[512, 1024]$ and dimension size as $[1000,2000]$. There are some exceptions due to the memory issues, i.e., dimension size for RESCAL is in $[500, 1000]$; dimension size for TuckER is in $[200, 500]$. For ogbl-wikikg2, since the used GPU only has 24GB memory, we cannot run models with 500 dimensions which requires much more memory in the OGB board. Instead, we set the dimension as 100 to be consistent with the smaller models in OGB board with 100 dimensions, and increase the batch size in $[512, 1024]$ in the second stage. In addition, we show the details for the search procedure by RF+BORE in Algorithm \ref{alg:full}.
\begin{algorithm}[ht]
\caption{Full procedure of HP search with RF+BORE (in stage one)}
\label{alg:full}
\small
\begin{algorithmic}[1]
\REQUIRE KG embedding $F$, dataset $G$, search space ${\mathcal X}_{S|D}$, budget $\nicefrac{B}{2}$, RF model $y=c(\mathbf x)$, threshold $\tau=0.8$.
\STATE initialize the RF model and $\mathcal H = \emptyset$;
\STATE split triplets in $G$ with ratio $9:1$ into $G_{\text{tra}}$ and $G_{\text{val}}$;
\REPEAT
\STATE randomly sample a set of configurations ${{\mathcal X}_{\overline{S|D}}}\subset{\mathcal X}_{S|D}$;
\STATE select $\hat{\mathbf x} = \arg\max_{\mathbf x\in {{\mathcal X}_{\overline{S|D}}}} y(\mathbf x)$;
\STATE train embedding model into converge \\
$\bm P^* = \arg\min_{\bm P}\mathcal L\big(F(\bm P, \hat{\mathbf x}), G_{\text{tra}}\big)$;
\STATE evaluate the performance $\hat{y}_{\hat{\mathbf x}} = \mathcal M\big(F(\bm P^*, \hat{\mathbf x}), G_{\text{val}}\big)$;
\STATE record $\mathcal H\leftarrow \mathcal H\cup \{(\hat{\mathbf x}, \hat{y}_{\hat{\mathbf x}})\}$;\\
\% \textbf{BORE}:
\STATE set label $0$ for configuration in $\mathcal H$ with $\hat{y}_{\hat{\mathbf x}}<\tau$,
and label $1$ for $\hat{y}_{\hat{\mathbf x}}\geq\tau$;
\STATE update RF model $y=c(\mathbf x)$ to classify the two labels;
\UNTIL{$\nicefrac{B}{2}$ exhausted.}
\end{algorithmic} \end{algorithm}
\section{Additional experimental results}
\subsection{Implementation details} \label{app:implementation}
\noindent \textbf{Evaluation metrics.} We follow \cite{bordes2013translating,wang2017knowledge,ruffinelli2019you} to use the filtered ranking-based metrics for evaluation. For each triplet $(h,r,t)$ in the validation or testing set, we take the head prediction $(?,r,t)$ and tail prediction $(h,r,?)$ as the link prediction task. The filtered rankings on the head and tail are computed as \[ \text{rank}_h =
\Big|\big\{e\in\mathcal E: \big(f(e,r,t)\geq f(h,r,t) \big) \wedge \big((e,r,t)\notin D_{\text{tra}}\cup D_{\text{val}}\cup D_{\text{tst}}) \big)
\big\}\Big| + 1, \]
\[ \text{rank}_t =
\Big|\big\{e\in\mathcal E: \big(f(h,r,t)\geq f(h,r,e) \big) \wedge \big((h,r,e)\notin D_{\text{tra}}\cup D_{\text{val}}\cup D_{\text{tst}}) \big)
\big\}\Big| + 1, \]
respectively, where $|\cdot|$ is the number of elements in the set. The the two metrics used are: \begin{itemize}
\item Mean reciprocal ranking (MRR): the average of reciprocal of all the obtained rankings.
\item Hit@$k$: the ratio of ranks no larger than $k$. \end{itemize} For both the metrics, the large value indicates the better performance.
\noindent \textbf{Dataset statistics.} We summarize the statistics of different benchmark datasets in Table~\ref{tab:dataset}. As shown, ogbl-biokg and ogbl-wikikg2 have much larger size compared with WN18RR and FB15k-237.
\begin{table}[ht]
\centering
\caption{Statistics of the KG completion datasets.}
\small
\label{tab:dataset}
\setlength\tabcolsep{3.3pt}
\begin{tabular}{c|cc|ccc}
\toprule
dataset & \#entity & \#relation & \#train & \#validate &
\#test \\ \midrule
WN18RR ~\cite{dettmers2017convolutional} & 41k & 11 & 87k & 3k & 3k \\
FB15k-237~\cite{toutanova2015observed} & 15k & 237 & 272k & 18k & 20k \\
ogbl-biokg~\cite{hu2020open} & 94k & 51 & 4,763k & 163k & 163k \\
ogbl-wikikg2~\cite{hu2020open} & 2,500k & 535 & 16,109k & 429k & 598k \\
\bottomrule
\end{tabular}
\end{table}
\noindent \textbf{Baseline implementation.} All the baselines compared in this paper are based on their own original open-source implementations. Here we list the source links: \begin{itemize}
\item Hyperopt \citep{bergstra2013hyperopt}, \url{https://github.com/hyperopt/hyperopt};
\item Ax, \url{https://github.com/facebook/Ax};
\item SMAC \citep{hutter2011sequential}, \url{https://github.com/automl/SMAC3};
\item BORE \citep{tiao2021bore}, \url{https://github.com/ltiao/bore};
\item AutoNE \citep{tu2019autone}, \url{https://github.com/tadpole/AutoNE}. \end{itemize}
\noindent \textbf{Searched hyperparameters.} We list the searched hyperparameters for each embedding model on the different datasets in Tables~\ref{tab:searched_HP_WN18RR}-\ref{tab:searched_HP_wikikg2} for reproduction.
\begin{table}[ht]
\centering
\caption{Searched optimal hyperparameters for the WN18RR dataset.}
\label{tab:searched_HP_WN18RR}
\small
\renewcommand{1.3}{1.2}
\setlength\tabcolsep{3pt}
\begin{tabular}{c|c|c|c|c|c|c|c }
\toprule
HP/Model & ComplEx & DistMult & RESCAL & ConvE & TransE & RotatE & TuckER \\ \midrule
\# negative samples & 32 & 128 & 128 & 512 & 128 & 2048 & 128 \\ \midrule
loss function & BCE\_mean & BCE\_adv & BCE\_mean & BCE\_adv & BCE\_adv & BCE\_adv & BCE\_adv \\
gamma & 2.29 & 12.88 & 2.41 & 12.16 & 3.50 & 3.78 & 12.97 \\
adv. weight & 0.00 & 1.41 & 0.00 & 0.78 & 1.14 & 1.66 & 1.94 \\
\midrule
regularizer & NUC & NUC & DURA & DURA & FRO & FRO & DURA \\
reg. weight & $1.21 \times 10^{-3}$ & $9.58 \times 10^{-3}$ & $1.76 \times 10^{-3}$ & $ 9.79 \times 10^{-3}$ & $ 4.19 \times 10^{-4}$ & $5.13 \times 10^{-8}$ & $2.22 \times 10^{-3}$ \\
dropout rate & 0.28 & 0.29 & 0.00 & 0.02 & 0.00 & 0.00 & 0.00 \\
\midrule
optimizer & Adam & Adam& Adam & Adam & Adam & Adam & Adam \\
learning rate & $6.08 \times 10^{-4}$ & $4.58 \times 10^{-3}$ & $ 1.73\times 10^{-3}$ & $ 6.88 \times 10^{-4}$ & $ 1.02 \times 10^{-4}$ & $ 1.24 \times 10^{-3}$ & $2.60 \times 10^{-3}$ \\
initializer & x\_uni & norm & uni & x\_uni & norm & norm & x\_uni \\
\midrule
batch size & 1024 & 1024 & 512 & 512 & 512 & 512 & 512 \\
dimension size & 2000 & 2000 & 1000 & 1000 & 1000 & 1000 & 200 \\
inverse relation & False & False & False& False& False& False& False \\
\bottomrule
\end{tabular}
\end{table}
\begin{table}[ht]
\centering
\caption{Searched optimal hyperparameters for the FB15k-237 dataset.}
\label{tab:searched_HP_FB15k237}
\small
\renewcommand{1.3}{1.2}
\setlength\tabcolsep{3pt}
\begin{tabular}{c|c|c|c|c|c|c|c }
\toprule
HP/Model & ComplEx & DistMult & RESCAL & ConvE & TransE & RotatE & TuckER \\ \midrule
\# negative samples & 512 & \texttt{kVsAll} & 2048 & 512 & 512 & 128 & 2048 \\ \midrule
loss function & BCE\_adv & CE & CE & BCE\_sum & BCE\_adv & BCE\_adv & BCE\_adv \\
gamma & 13.05 & 2.90 & 4.17 & 14.52 & 6.76 & 14.46 & 13.51 \\
adv. weight & 1.93 & 0.00 & 0.00 & 0.00 & 1.99 & 1.12 & 1.95 \\
\midrule
regularizer & DURA & NUC & DURA & DURA & FRO & NUC & DURA \\
reg. weight & $9.75 \times 10^{-3}$ & $2.13 \times 10^{-3}$ & $8.34 \times 10^{-3}$ & $6.42 \times 10^{-3}$ & $2.16 \times 10^{-4}$ & $2.99 \times 10^{-4}$ & $2.66 \times 10^{-4}$ \\
dropout rate & 0.22 & 0.29 & 0.01 & 0.07 & 0.02 & 0.01 & 0.01\\
\midrule
optimizer & Adam & Adam& Adam & Adam & Adam & Adam & Adam \\
learning rate & $9.70 \times 10^{-4}$ & $4.91\times 10^{-4}$ & $ 9.30\times 10^{-4}$ & $2.09\times 10^{-4}$ & $ 2.66\times 10^{-4}$ & $5.89\times 10^{-4}$ & $ 3.35\times 10^{-4}$ \\
initializer & uni & x\_uni & x\_uni & norm & x\_norm & norm & norm \\
\midrule
batch size & 1024 & 1024 & 2048 & 1024 & 512 & 1024 & 1024 \\
dimension size & 2000 & 1000 & 500 & 500 & 1000 & 2000 & 500 \\
inverse relation & False & False & False& False& False& False& False \\
\bottomrule
\end{tabular}
\end{table}
\begin{table}[ht]
\centering
\caption{Searched optimal hyperparameters for the ogbl-biokg dataset.}
\label{tab:searched_HP_biokg}
\small
\renewcommand{1.3}{1.2}
\begin{tabular}{c|c|c|c|c|c }
\toprule
HP/Model & ComplEx & DistMult & TransE & RotatE & AutoSF \\ \midrule
\# negative samples & 512 & 512 & 128 & 128 & 512 \\ \midrule
loss function & CE & CE & CE & BCE\_adv & CE \\
gamma & 12.90 & 11.82 & 7.60 & 18.34 & 12.90 \\
adv. weight & 0.00 & 0.00 & 0.00 & 1.94 & 0.00 \\ \midrule
regularizer & NUC & NUC & NUC & DURA & NUC \\
reg. weight & $1.38 \times 10^{-3}$ & $1.20 \times 10^{-6}$ & $6.99 \times 10^{-3}$ & $1.09 \times 10^{-6}$ & $ 1.38 \times 10^{-4}$ \\
dropout rate & 0.01 & 0.00 & 0.00 & 0.00 & 0.01 \\ \midrule
optimizer & Adam & Adam & Adam & Adam & Adam \\
learning rate & $1.89 \times 10^{-3}$ & $1.25 \times 10^{-3}$ & $1.24 \times 10^{-4}$ & $1.11 \times 10^{-4}$ & $1.89 \times 10^{-3}$ \\
initializer & uni & x\_uni & x\_uni & norm & uni \\
batch size & 1024 & 1024 & 1024 & 1024 & 1024 \\
dimension size & 2000 & 1000 & 2000 & 2000 & 2000 \\
inverse relation & False & False & False & False & False \\
\bottomrule
\end{tabular}
\end{table}
\begin{table}[ht]
\centering
\caption{Searched optimal hyperparameters for the ogbl-wikikg2 dataset}
\label{tab:searched_HP_wikikg2}
\small
\renewcommand{1.3}{1.2}
\begin{tabular}{c|c|c|c|c|c}
\toprule
HP/Model & ComplEx & DistMult & TransE & RotatE & AutoSF \\ \midrule
\# negative samples & 32 & 32 & 128 & 32 & 2048 \\ \midrule
loss function & CE & CE & CE & CE & CE \\
gamma & 6.00 & 6.00 & 21.05 & 23.94 & 18.91 \\
adv. weight & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 \\ \midrule
regularizer & DURA & DURA & FRO & DURA & DURA \\
reg. weight & $9.58 \times 10^{-7}$ & $ 1.98 \times 10^{-4}$ & $1.56 \times 10^{-5}$ & $ 8.10\times 10^{-3}$ & $ 1.38 \times 10^{-4}$ \\
dropout rate & 0.00 & 0.00 & 0.01 & 0.07 & 0.07 \\ \midrule
optimizer & Adam & Adam & Adam & Adam & Adam \\
learning rate & $1.34 \times 10^{-4}$ & $ 1.98 \times 10^{-4}$ & $ 6.05 \times 10^{-4}$ & $ 4.07 \times 10^{-2}$ & $ 1.04 \times 10^{-2}$ \\
initializer & x\_norm & x\_norm & x\_norm & x\_norm & x\_norm \\
batch size & 1024 & 1024 & 1024 & 1024 & 1024 \\
dimension size & 100 & 100 & 100 & 100 & 100 \\
inverse relation & False & False & False & False & False \\
\bottomrule
\end{tabular}
\end{table}
\subsection{Results on general benchmarks} \label{app:general-benchmark}
We compare the types of results on WN18RR and FB15k-237 in Table~\ref{tab:perf_wn18rr_fb15k237}. In the first part, we show the results reported in the original papers. In the second part, we show the reproduced results in \citep{ruffinelli2019you}. And in the third part, we show the results of the HPs searched by KGTuner.
\begin{table*}[ht]
\centering
\caption{Performance on WN18RR and FB15k-237 dataset.
The \textbf{bold numbers} mean the best performances of the same model,
and the \underline{underlines} mean the second best.}
\label{tab:perf_wn18rr_fb15k237}
\renewcommand{1.3}{1.2}
\small
\begin{tabular}{cc|cccc|cccc}
\toprule
& & \multicolumn{4}{c}{WN18RR} & \multicolumn{4}{c}{FB15k-237} \\
& & MRR & Hit@1 & Hit@3 & Hit@10 & MRR & Hit@1 & Hit@3 & Hit@10 \\ \midrule
\multirow{6}{*}{Original}
& ComplEx & 0.440 & 0.410 & 0.460 & 0.510 & 0.247 & 0.158 & 0.275 & 0.428 \\
& DistMult & 0.430 & 0.390 & 0.440 & 0.490 & 0.241 & 0.155 & 0.263 & 0.419 \\
& RESCAL & 0.420 & - & - & 0.447 & 0.270 & - & - & 0.427 \\
& ConvE & 0.430 & 0.400 & 0.440 & \textbf{0.520} & 0.325 & \underline{0.237} & 0.356 & 0.501 \\
& TransE & 0.226 & - & - & 0.501 & 0.294 & - & - & 0.465 \\
& RotatE & \underline{0.476} & \textbf{0.428} & \underline{0.492} & \underline{0.571} & \textbf{0.338} & \underline{0.241} & \textbf{0.375} & \textbf{0.533} \\
& TuckER & \underline{0.470} & \textbf{0.443} & \underline{0.482} & \underline{0.526} & \textbf{0.358} & \textbf{0.266} & \textbf{0.394} & \textbf{0.544} \\ \midrule
\multirow{6}{*}{\begin{tabular}[c]{@{}l@{}} LibKGE \\ \citep{ruffinelli2019you}\end{tabular}}
& ComplEx & \underline{0.475} & \underline{0.438} & \underline{0.490} & \underline{0.547} & \underline{0.348} & \underline{0.253} & \underline{0.384} & \textbf{0.536} \\
& DistMult & \underline{0.452} & \textbf{0.413} & \underline{0.466} & \underline{0.530} & \underline{0.343} & \underline{0.250} & \textbf{0.378} & \textbf{0.531} \\
& RESCAL & \underline{0.467} & \textbf{0.439} & \underline{0.480} & \underline{0.517} & \underline{0.356} & \underline{0.263} & \textbf{0.393} & \textbf{0.541} \\
& ConvE & \textbf{0.442} & \textbf{0.411} & \textbf{0.451} & \underline{0.504} & \textbf{0.339} & \textbf{0.248} & \textbf{0.369} & \underline{0.521} \\
& TransE & \underline{0.228} & \textbf{0.053} & \underline{0.368} & \underline{0.520} & \underline{0.313} & \underline{0.221} & \underline{0.347} & \underline{0.497} \\ \midrule
\multirow{6}{*}{KGTuner (ours)}
& ComplEx & \textbf{0.484} & \textbf{0.440} & \textbf{0.506} & \textbf{0.562} & \textbf{0.352} & \textbf{0.263} & \textbf{0.387} & \underline{0.530} \\
& DistMult & \textbf{0.453} & \underline{0.407} & \textbf{0.468} & \textbf{0.548} & \textbf{0.345} & \textbf{0.254} & \underline{0.377} & \underline{0.527} \\
& RESCAL & \textbf{0.479} & \underline{0.436} & \textbf{0.496} & \textbf{0.557} & \textbf{0.357} & \textbf{0.268} & \underline{0.390} & \underline{0.535} \\
& ConvE & \underline{0.437} & \underline{0.399} & \underline{0.449} & 0.515 & \underline{0.335} & \underline{0.242} & \underline{0.368} & \textbf{0.523} \\
& TransE & \textbf{0.233} & \underline{0.032} & \textbf{0.399} & \textbf{0.542} & \textbf{0.327} & \textbf{0.228} & \textbf{0.369} & \textbf{0.522} \\
& RotatE & \textbf{0.480} & \underline{0.427} & \textbf{0.501} & \textbf{0.582} & \textbf{0.338} & \textbf{0.243} & \underline{0.373} & \underline{0.527} \\
&TuckER & \textbf{0.480} & \underline{0.437} & \textbf{0.500} & \textbf{0.557} & \underline{0.347} & \underline{0.255} & \underline{0.382} & \underline{0.534} \\ \bottomrule
\end{tabular} \end{table*}
\subsection{Full results for OGB} \label{app:ogb}
\begin{table}[ht]
\centering
\caption{Full results on ogbl-biokg and ogbl-wikikg2 dataset.}
\label{tab:perf_ogb_full}
\setlength\tabcolsep{3pt}
\renewcommand{1.3}{1.3}
\small
\begin{tabular}{cc|ccc|ccc}
\toprule
& & \multicolumn{3}{c|}{ogbl-biokg} & \multicolumn{3}{c}{ogbl-wikikg2} \\
& & Test MRR & Val MRR & {\#parameters} & Test MRR & Val MRR & \#parameters \\ \midrule
& {ComplEx} & 0.8095$\pm$0.0007 & 0.8105$\pm$0.0001 & {187,648,000} & 0.4027$\pm$0.0027 & 0.3759$\pm$0.0016 & 1,250,569,500 \\
OGB & {DistMult} & 0.8043$\pm$0.0003 & 0.8055$\pm$0.0003 & {187,648,000} & 0.3729$\pm$0.0045 & 0.3506$\pm$0.0042 & 1,250,569,500 \\
board & {RotatE} & 0.7989$\pm$0.0004 & 0.7997$\pm$0.0002 & {187,597,000} & 0.2530$\pm$0.0034 & 0.2250$\pm$0.0035 & 250,087,150 \\
& {TransE} & 0.7452$\pm$0.0004 & 0.7456$\pm$0.0003 & {187,648,000} & 0.4256$\pm$0.0030 & 0.4272$\pm$0.0030 & 1,250,569,500 \\
& {AutoSF} & 0.8309$\pm$0.0008 & 0.8317$\pm$0.0007 & 187,648,000 & 0.5186$\pm$0.0065 & 0.5239$\pm$0.0074 & 250,113,900 \\
\midrule
\multirow{5}{*}{KGTuner} & {ComplEx} & 0.8385$\pm$0.0009 & 0.8394$\pm$0.0007 & {187,648,000} & 0.4942$\pm$0.0017 & 0.5099$\pm$0.0023 & 250,113,900 \\
& {DistMult} & 0.8241$\pm$0.0008 & 0.8245$\pm$0.0009 & {93,824,000} & 0.4837$\pm$0.0078 & 0.5004$\pm$0.0075 & 250,113,900 \\
& {RotatE} & 0.8013$\pm$0.0015 & 0.8024$\pm$0.0012 & {187,597,000} & 0.2948$\pm$0.0026 & 0.2650$\pm$0.0034 & 250,087,150 \\
& {TransE} & 0.7781$\pm$0.0009 & 0.7787$\pm$0.0008 & {187,648,000} & 0.4739$\pm$0.0021 & 0.4932$\pm$0.0013 & 250,113,900 \\
& {AutoSF} & 0.8354$\pm$0.0013 & 0.8361$\pm$0.0012 & 187,648,000 & 0.5222$\pm$0.0021 & 0.5397$\pm$0.0023 & 250,113,900 \\
\bottomrule
\end{tabular} \end{table}
\end{document}
\section{old materials}
To solve the first challenge, we give a comprehensive analysis on the different choice of HPs in the search space, and we care about the following three significant properties. \begin{itemize}[leftmargin=12pt]
\item \textbf{Cost}.
We evaluate the cost of different choices of HPs,
especially dimension and batch size.
(add observations here)
\footnote{ +zk+ the KGE components/search space have not been introduced yet,
shall we introduce detailed observations here?
}
\item \textbf{Sensitivity}.
The sensitivity captures the importance and influence of each HP on the performance.
We compute the variance of each HP to indicate the sensitivity.
This motivates us to fix the dimensions and batch size, which have small sensitivity
but are expensive with large value,
to be small to save cost.
\item \textbf{Range}.
There can be several choices for the categorical values or a bounded range for the numerical ranges.
We analyze the relative ranking of each HP to remove the consistently bad choices.
We observe that Adam can be consistently better than the other HPs.
The learning rate and regularization weight cannot be too large or too small.
Besides,
the performance is positively proportional to the dimension size.
\end{itemize}
\footnote{+qm+ that part needs to be rewritten later.
[added in previous itemize] what can we learn from above understandings?
why the search space can be reduced?} To deal with the second challenge, we firstly reduce the search space by removing the consistently bad choices of configurations.
Then, we compare different HP search algorithms and design a robust and efficient search algorithm in terms of the following perspectives. \begin{itemize}[leftmargin=12pt]
\item \textbf{Efficient evaluation.}
Since training and evaluating a model in the original space takes hours,
it is important to reduce the cost under reliable feedback for a specific configuration.
Based on the understanding of HPs,
we fix the dimension batch size to be small and use subgraph
\footnote{+zk+ Note that this is the first time that comes up with "subgraph"}
to reduce the cost, and explore more configurations at the early stages of searching.
This help us to evaluate a much larger amount of configurations.
\item \textbf{Fast convergence.}
To improve the
convergence,
the surrogate model should fit the search space well with limited number of configurations
and the acquisition function should balance exploration and exploitation.
With mixed categorical and numerical HPs,
we find that random forest shows better fitting ability than Gaussian process based surrogates
in the synthetic settings.
When working together with the BORE \citep{tiao2021bore},
which estimates the density of good performed configurations and bad ones,
the search algorithm can converge faster and is more robust,
compared with other existing algorithms.
\end{itemize} Based on above analysis, we further design a novel stage-wise search algorithm that explores abundant configurations in the first stage on subgraphs, transfers the knowledge to full graph in the second stage, and finetunes the HPs that have low sensitivity but high cost. The new algorithm helps us to search better configurations in the large-scale benchmarks. Our contribution can be summarized as follows: \footnote{[need refine in future] try to list 3-4 contributions here.} \begin{itemize}[leftmargin=*]
\item We make comprehensive understanding
with respect to the cost, sensitivity and range
on the HPs commonly used in KGE.
This understanding helps us to reduce the search space hundreds times smaller.
We get an important observation that large dimension and batch size will lead to high cost
but have low sensitivity.
\item
Based on the understanding, we propose a stage-wise search algorithm.
Specifically,
we explore a large amount of configurations on subgraphs in the first stage.
Then, the knowledge on the subgraphs are transferred to the full graph in the second stage with fixed batch size, and dimension.
After that, the batch size and dimension are fine-tuned based on the top-performed configurations in the second stage.
\item To fit the search space with mixed categorical and numerical HPs
and reduce the gap between evaluations on the subgraphs and the full graph,
we propose to use random forest as the surrogate and BORE as the acquisition to select better configurations
while ignoring the exact estimation values.
\item Experiments on the regular benchmarks WN18RR and FB15k-237
show the efficiency and robustness of the proposed algorithm.
We also significantly improve the performance on large-scale benchmarks
ogbl-biokg and ogbl-wikikg2. \end{itemize}
{\color{blue}
Be-aware of the biggest difference on the first level,
i.e.,
\begin{itemize}[leftmargin=*]
\item previous works care about preserving distribution
from full graph to subgraphs [cite top papers],
and AutoNE/e-AutoGR adopt these tricks in searching HPs
\item we take a more direct approach to understand the impact
of subgraph sampling on searching HPs by checking xxx.
\item add techniques xxx.
show both positive and negative message.
positive msg: confirm previous attempts, e.g. AutoNE/e-AutoGR;
negative msg: motivate the ``shared + specific'' search strategy.
\footnote{+qm+ also check footnote~\ref{ft:1}.}
\end{itemize} }
This form covers mainstream scoring functions mentioned in \citep{ruffinelli2019you} as examples in \footnote{+qm+ you can move this table into appendix,
just list names here in the intro,
since this is ``revisiting'' part.} Table~\ref{tab:sfdefinition}.
\subsection{stage1: understanding and reducing the search space}
\subsection{Optimizing the acquisition function} Since the search space contains several categorical HPs, we cannot derive a closed form solution that maximizes the acquisition function. Inspired by the randomized one-exchange neighborhood method in \citep{hutter2011sequential}, we propose a candidate generation algorithm based on evolutionary algorithm.
\begin{algorithm}[ht]
\caption{Next candidate generation}
\label{alg:candgen}
\small
\begin{algorithmic}[1]
\REQUIRE acquisition function $\text{ACQ}(\cdot)$, HP search space $\mathcal X$,
surrogate model $f(\cdot)$, the size of evolution $I$.
\STATE randomly generate a set $\mathcal I$ of $I$ different configurations $\mathbf x\in\mathcal X$;
\REPEAT
\STATE uniformly sample $p\sim [0,1]$;
\IF{$p<0.2$}
\STATE uniformly sample a new configuration $\mathbf x^{\text{new}}\in\mathcal X$;
\ELSIF{$p<0.6$}
\STATE uniformly sample one configuration $\mathbf x \in \mathcal I$;
\STATE select one HP index $i$;
\IF{$x_i$ is categorical}
\STATE replace $x_i$ with the other category such that $x^{\text{new}}_i\neq x_i$;
\ELSE
\STATE sample $x^{\text{new}}_i$ from the normal distribution $\mathcal N(x_i, 0.2)$ and truncate $x^{\text{new}}_i$ to be within $[0,1]$;
\ENDIF
\ELSE
\STATE uniformly select two different $\mathbf x^a, \mathbf x^b \in \mathcal I$;
\STATE for each HP $i$,
$x^{\text{new}}_i$ has equal probability to be set as either $x^a_i$ or $x^b_i$.
\ENDIF
\IF{$\text{ACQ}(\mathbf x^{\text{new}})>\min_{\mathbf x\in\mathcal I}\text{ACQ}(\mathbf x)$}
\STATE replace $\mathbf x = \arg\min_{\mathbf x\in\mathcal I}\text{ACQ}(\mathbf x)$ with $\mathbf x^{\text{new}}$;
\ENDIF
\UNTIL{$10000$ iterations}
\RETURN $\arg\max_{\mathbf x\in\mathcal I}\text{ACQ}(\mathbf x)$.
\end{algorithmic} \end{algorithm}
\begin{algorithm}[H]
\caption{Warm start with Boosting model}
\label{alg:filter}
\small
\begin{algorithmic}[1]
\REQUIRE $T$ datasets $\{D_1, D_2, \dots ,D_T\}$, $M$ the number of shared trees,
$M'\ll M$ the number of task-specific trees, learning rate $\eta$.
\STATE for all the samples, initialize
$\tilde{\mathbf y}^0 = \hat{\mathbf y}^0 = -\log 4$.
\FOR{$m$ in $1,\dots, M$}
\STATE compute the gradient $\mathbf G^m \leftarrow \mathbf Y - \frac{1}{1+\exp(-\hat{\mathbf Y}^{m-1})}$;
\STATE fit $\mathbf G^m$ with $f^m(\mathbf H, \mathbf X)$;
\STATE aggregate the shared prediction $\tilde{\mathbf Y}^m \leftarrow \tilde{\mathbf Y}^{m-1} - \eta \cdot f^m(\mathbf H, \mathbf X)$;
\FOR{all $D_t$}
\STATE learn $M'$ trees $\{f^{m'}_t\}_{m'=1}^{M'}$
to minimize $\mathcal L\big(\mathbf y_t, \sigma(\tilde{\mathbf y}^m_t - \sum_{m'=1}^{M'}\eta_t f_{t}^{m'}(\mathbf x_t))\big)$ with $\tilde{\mathbf Y}^m = [\tilde{\mathbf y}^m_1;\dots;\tilde{\mathbf y}^m_T ]$;
\STATE get the task-specific prediction $\hat{\mathbf y}^m_t\leftarrow \tilde{\mathbf y}^m_t - \sum_{m'=1}^{M'}\eta \cdot f_{t}^{m'}(\mathbf x_t)$
\ENDFOR
\STATE concatenate the full prediction $\hat{\mathbf Y}^m \leftarrow [\hat{\mathbf y}^m_1;\dots; \hat{\mathbf y}^m_T]$;
\ENDFOR
\RETURN shared tree $\{f^m\}_{m=1}^M$.
\end{algorithmic} \end{algorithm}
\subsection{Efficient Hyperparameter Optimization}
Baselines: \begin{itemize}
\item Direct HPO: slow in large-scale KG \\
representative work: Hyperopt
\citep{bergstra2013hyperopt} and SMAC \citep{hutter2011sequential}.
\item Transfer initial settings: using the top configurations of the similar datasets as instances or initializations for target task optimization. \\
representative work: auto-sklearn \citep{feurer5872efficient}.
\item Meta feature as input: concatenating meta feature with HP settings for BO. \\
representative work: SMAC \citep{hutter2011sequential}, AutoNE \citep{tu2019autone}, e-AutoGR \citep{wang2021explainable}. \end{itemize}
Challenges: \begin{itemize}
\item A large amount of categorical HPs:
AutoNE and e-AutoGR only work for numerical HPs,
and Gaussian process \citep{rasmussen2003gaussian} based methods not work.
Hyperopt, which leverages tree-structured parzen estimation,
also performs poor.
\item Correlation is not that strong:
direct transfer like auto-sklearn may not work.
\item Meta-feature is more complex in KG:
directly concatenating them for SMAC may not help improve (needs test).
\item Different HPs are highly correlated:
such as learning rate, training mode, negative sampling, loss function, etc. \end{itemize}
We may explore the following ways to improve the transfer ability \begin{itemize}
\item from stage 1, we can get a configuration that work well for most dataset as default setting for cold start.
\item using subgraph or other dataset to further reduce search space: using the some top configuration settings to reduce the ranges.
(auto-sklearn use the top configurations as initialization, we use them to reduce space)
\item boosting based transfer: boosting is a method to learn the residuals.
Assume residuals are easier to learn than the full distribution.
We can learn a boosting model on sampled subgraphs
and then use a few additional trees to adjust the full data distribution. \end{itemize}
\footnote{+qm+ as discussed,
you can move this to Appendix.} \textbf{Variance.}
Variance is a prominent tool in analyzing sensitivity of variables \citep{im1993sensitivity,hutter2014efficient}. We aim to use the variance of performance caused by different choices of HPs to indicate the impact of single HPs. For the $i$-th HP, we enumerating values $\theta\in\mathcal R_i$ based on the anchor configurations in $\bar{\mathcal X}_i$. Denoting $y(\mathbf x[x_i=\theta])$ as the performance of $\theta$ on the anchor $\mathbf x$, the variance influenced by the $i$-th HP is estimated by \footnote{$\surd$ what is $y(\cdot)$? $ \mathcal R$? $R(x_i)$?} \begin{equation}
V_i = \frac{1}{|\bar{\mathcal X}_i|}\sum_{\mathbf x\in\bar{\mathcal X}_i}\Big\{ \frac{1}{|\mathcal R_i|} \sum_{\theta\in \mathcal R_i} \Big(y(\mathbf x[x_i=\theta]) - \frac{1}{|\mathcal R_i|}\sum_{\theta\in \mathcal R_i}\big(y(\mathbf x[x_i=\theta])\big)\Big)^2 \Big\}. \end{equation} The variance of performance caused by each HP is shown in Figure~\ref{fig:HP_variance}. The optimizer, regularization weight, learning rate, loss functions are the most influential HPs while dimension, initializer, choice of regularizer, number of negative samples, adversarial weight and batch size, etc., have relatively low influence on the performance.
\begin{figure}
\caption{Box plot of $V_i$ for each HP $i$. The yellow rectangle shows the zoom-in for HPs with smaller $V_i$.}
\label{fig:HP_variance}
\end{figure}
\begin{table}[H]
\centering
\caption{The statistic of variance metric.}
\label{tab:HP-variance}
\setlength\tabcolsep{2pt}
\begin{tabular}{c|c|c|c }
\toprule
name & mean & median & std \\
\midrule
optimizer & 0.03567 & 0.02196
& 0.03934
\\
regularization weight & 0.02941
& 0.01290
& 0.03495
\\
learning rate & 0.01979 & 0.00984 & 0.02216 \\
regularizer & 0.01070 & 0.00006
& 0.02822
\\
dropout rate & 0.01041
& 0.00026
& 0.02678
\\
loss function & 0.00500 & 0.00064 & 0.01018 \\
initializer & 0.00364 & 0.00001 & 0.01473
\\
\# negative samples & 0.00207
& 0.00011 & 0.00837 \\
dimension size & 0.00161 & 0.00015 & 0.00487 \\
gamma (MR loss) & 0.00064 & 0.00007 & 0.00228 \\
adv weight (BCE\_adv) & 0.00025 & 0.00003 & 0.00086 \\
batch size & 0.00016 & 0.00003 & 0.00032 \\
\bottomrule
\end{tabular} \end{table}
\footnote{+qm+ re-write this part when revised Section~\ref{ssec:fasteval}.} As discussed in Section~\ref{ssec:fasteval}, evaluating configurations on subgraphs can help us save a lot of time compared with evaluation on the full graph. Hence, we can gain knowledge on the search space by evaluating configurations on subgraphs and then transfer the knowledge to the full graph evaluation.
AutoNE \citep{tu2019autone} uses meta-features to indicate the different subgraphs and learns a shared model to predict on all the graphs. This will lead to a bias towards the subgraphs as the amount of configurations evaluated on subgraphs is much larger than full graph. Hence, we propose to use a shared model to learn common patterns in the HP space and data-specific models to fit each subgraph.
\textbf{Setup:} \footnote{[check notations later] $\surd$ if you donot have experiments across different datasets,
then just say ``$T$ subgraphs''.} We have $T$ subgraphs $\{G_1, G_2, \dots, G_T\}$, which are sampled from the full graph $G_0$ by random walk. For each subgraph $G_t$ we have evaluated $N_t$ configurations of HPs $\mathbf X_t\in\mathbb R^{N_t\times d_x}$ and the evaluated performance $\mathbf y_t\in\mathbb R^{N_t}$. We denote $\mathbf X = [\mathbf X_1;\dots; \mathbf X_T]\in\mathbb R^{\Sigma N_T\times d_x}$ as the concatenation of all settings, $\mathbf H = [\mathbf h_1;\dots; \mathbf h_T]\in\mathbb R^{\Sigma T\times d_h}$ as the concatenation of all the meta-features, and $\mathbf Y = [\mathbf y_1;\dots; \mathbf y_T]\in\mathbb R^{\Sigma N_T}$ as the concatenation of performance on the $T$ subgraphs.
\footnote{+qm+ it is better explain what can be shared in Section~\ref{ssec:fasteval}?
\label{ft:1}} To learn the common patterns shared among different datasets, we build ${M}$ shared-trees ${\mathcal M} = \{{f}^m\}_{m=1}^{M}$ that are shared among the $T$ datasets. Meanwhile, we learn $M'$ specific-trees $\mathcal M_t = \{f_t^m \}_{m'=1}^{M'}$ with $M'\ll M$ that are specific for the subgraph $G_t$. For the shared-trees, the input is $[\mathbf h, \mathbf x]$ to jointly model the meta-feature and HPs. For the specific-trees, the input is just
the HP configuration $\mathbf x_t$ evaluated on the subgraph $G_t$ since the meta-feature for each subgraph is consistent. The learning objective is to jointly learn the shared-trees and specific-trees such that $\mathcal M + \mathcal M_t$ can fit the configurations $(\mathbf x_t, \mathbf y_t)$ for each subgraph well by minimizing te objective in Eq.(\ref{eq:searchobj}).
The alternative learning procedure is proposed in Algorithm~\ref{alg:atltree}. \begin{equation} \min\nolimits_{ \mathcal M, \{\mathcal M_t\}_{t = 1}^T } \sum\nolimits_{t=1}^T\mathcal L_{\text{BCE}}\big(\mathcal M(\mathbf h_t, \mathbf x_t) + \mathcal M_t(\mathbf x_t), \mathbf y_t\big), \label{eq:searchobj} \end{equation} where $\mathcal L_{\text{BCE}}$ is the binary cross entropy loss.
\begin{algorithm}[H]
\caption{Alternative random forest for HP transfer.}
\label{alg:atltree}
\small
\begin{algorithmic}[1]
\REQUIRE $T$ datasets $\{G_1, G_2, \dots ,G_T\}$, $M$ the number of shared-trees,
$M'\ll M$ the number of specific-trees, tolerant value $\epsilon$.
\STATE $m=0$, for all $t=1\dots T$, randomly initialize the trees ${\mathcal M'}^{(m)}$;
\REPEAT
\STATE $m\leftarrow m+1$;
\STATE fix the specific-trees ${\mathcal M'}^{(m-1)}$;
\STATE learn the shared-trees $\mathcal M^{(m)} \leftarrow \arg\min_{\mathcal M}\sum_{t=1}^T\mathcal L_{\text{BCE}}\big(\mathcal M^{(m-1)}(\mathbf h_t, \mathbf x_t) + {\mathcal M'_t}^{(m-1)}(\mathbf x_t), \mathbf y_t\big)$
\STATE fix the shared trees $\mathcal M^{(m)}$
\FOR{$t=1\dots T$}
\STATE learn the task-trees ${\mathcal M'_t}^{(m)} \leftarrow \arg\min_{\mathcal M'}\mathcal L_{\text{BCE}}\big(\mathcal M^{(m)}(\mathbf h_t, \mathbf x_t) + {\mathcal M'_t}^{(m-1)}(\mathbf x_t), \mathbf y_t\big)$
\ENDFOR
\UNTIL{$|\mathcal M^{(m)}(\mathbf H, \mathbf X) -\mathcal M^{(m-1)}(\mathbf H, \mathbf X)|<\epsilon$ }
\RETURN $\mathcal M^{(m)}$.
\end{algorithmic} \end{algorithm}
\subsection{Three stage search algorithm} \label{ssec:3stage}
In this part, we propose a three stage search algorithm based on the previous parts. The full procedure is shown in Algorithm~\ref{alg:search}. In steps \ref{step:sample}-\ref{step:stage1-end}, we train and evaluate the configurations on the sampled subgraphs to explore as much configurations as possible. Then, a shared tree model is learned by alternating update in Algorithm~\ref{alg:atltree} in step~\ref{step:sharetree}. In steps \ref{step:fulltree}-\ref{step:stage2-end}, we turn back to the full data with fixed dimension and batch size to search the HPs. Since running in the full space is time consuming, we can only visit a few set of configurations. Hence, we fully exploit the best configuration ever visited on the full data and revise the dimensions as well as batch sizes to fine-tune the HPs in steps~\ref{step:stage3-start}-\ref{step:stage3-end}. Then the configuration with the optimal performance on the valid data between steps~\ref{step:stage2-start}-\ref{step:stage3-end} is returned for testing.
\begin{algorithm}[ht]
\caption{Three stage HP tuning algorithm for KGE.}
\label{alg:search2}
\small
\begin{algorithmic}[1]
\REQUIRE KG $\mathcal K\mathcal G=\{\mathcal \mathcal F, \mathcal V, \mathcal R\}, $ budget $B$.
\STATE // \textbf{Stage one}
\STATE Sample three subgraphs (with entities $0.1|\mathcal V|$, $0.2|\mathcal V|$, $0.3|\mathcal V|$) by random walk from $\mathcal F_{\text{tra}}$. \label{step:sample}
\STATE Split the sampled triplets into 9:1 for training and evaluation.
\REPEAT \label{step:stage1-start}
\STATE generate a configuration $\mathbf x$ through quasi random search
\STATE uniformly sample one of the subgraph and evaluate under the configuration $\mathbf x$.
\STATE record the meta-feature, configuration and performance $y$.
\UNTIL{$1/3$ budget exhousted} \label{step:stage1-end}
\STATE train the shared tree $\mathcal M \leftarrow \text{Alg.\ref{alg:atltree}}(D_1, D_2, D_3)$. \label{step:sharetree}
\STATE initialize $M'\ll M$ trees $\mathcal M'_f$ for the full data. \label{step:fulltree}
\REPEAT \label{step:stage2-start}
\STATE select the new configuration $\mathbf x\leftarrow \arg\max_{\mathbf x} \mathcal M(\mathbf h, \mathbf x)+\mathcal M'_f(\mathbf x)$.
\STATE evaluate the new configuration and merge the new results.
\STATE learn the data-trees
${\mathcal M'_t}\leftarrow \arg\min_{\mathcal M'}\mathcal L\big(\mathcal M(\mathbf h, \mathbf x)+\mathcal M'(\mathbf x), \mathbf y\big)$.
\UNTIL{$1/3$ budget exhousted} \label{step:stage2-end}
\REPEAT \label{step:stage3-start}
\STATE select the top 3 configurations in stage 2 and mutate the dimension, batch size and xxx to generate the new configurations.
\STATE evaluate the new configuration and merge the new results.
\UNTIL{$1/3$ budget exhousted} \label{step:stage3-end}
\end{algorithmic} \end{algorithm}
We have $T$ dataset $\{D_1, D_2, \dots D_T\}$. For each dataset $D_t$ we have evaluated $N_t$ settings of HPs $\mathbf x_t\in\mathbb R^{N_t\times d_x}$ and the evaluated performance $\mathbf y_t\in\mathbb R^{N_t}$. We denote $\mathbf X = [\mathbf x_1;\dots; \mathbf x_T]\in\mathbb R^{\Sigma N_T\times d_x}$ as the concatenation of all settings, $\mathbf H = [\mathbf h_1;\dots; \mathbf h_T]\in\mathbb R^{\Sigma N_T\times d_h}$ as the concatenation of all meta-features, and $\mathbf Y = [\mathbf y_1;\dots; \mathbf y_T]\in\mathbb R^{\Sigma N_T}$ as the concatenation of all performance.
Learning objective: \begin{equation}
-\sum_{t=1}^T \log P(\mathbf y_t |\mathbf h_t, \mathbf x_t, {\mathcal M}, \mathcal M_t) \end{equation}
Key points: \begin{itemize}
\item initially, the outputs are set as $\log\frac{\# \text{ones}}{\# \text{zeros}}$.
\item In step 3-5, an additional shared tree is learned to fix the gap between target $\mathbf y$ with
the previous full prediction $\hat{\mathbf Y}^{m-1}$ (containing the $m$ shared trees and $M'$ task-specific trees);
\item In step 6-9, we fit the residuals with the $M'$ task-specific trees;
\item The residual of binary classification is $y-\frac{1}{1+\exp(-f(x))}$, loss function is binary cross entropy.
\end{itemize}
\textbf{Evaluation.} Given a triplet $(h,r,t)$, we get the rank of $(h,r,t)$ among the set $\{(h,r,t)\} \cup \{(e,r,t)\notin \mathcal S_{\text{tra}}\cup\mathcal S_{\text{val}}\cup \mathcal S_{\text{tst}}: e\in\mathcal E\}$, where $\mathcal S_{\text{tra}}, \mathcal S_{\text{val}}, \mathcal S_{\text{tst}}$ are the training, validation and test sets, as the filtered head ranking $\text{rank}_{(h,r,t)}^{\text{head}}$. The filtered tail rank $\text{rank}_{(h,r,t)}^{\text{tail}}$ is obtained by computing the rank among the set $\{(h,r,t)\} \cup \{(h,r,e)\notin \mathcal S_{\text{tra}}\cup\mathcal S_{\text{val}}\cup \mathcal S_{\text{tst}}: e\in\mathcal E\}$. Then the performance on test set (same for validation set) is measured by three types of metrics:
(i) mean rank (MR): $\frac{1}{2|\mathcal S_{\text{tst}}|}\sum_{(h,r,t)\in\mathcal S_{\text{tst}}}\big(\text{rank}_{(h,r,t)}^{\text{head}}+\text{rank}_{(h,r,t)}^{\text{tail}}\big)$; (ii) mean reciprocal rank (MRR):
$\frac{1}{2|\mathcal S_{tst}|}\sum_{(h,r,t)\in\mathcal S_{tst}}\big(\frac{1}{\text{rank}_{(h,r,t)}^{\text{head}}}+\frac{1}{\text{rank}_{(h,r,t)}^{\text{tail}}}\big)$; (iii) Hit@$k$:
$\frac{1}{2|\mathcal S_{tst}|}\sum_{(h,r,t)\in\mathcal S_{tst}}\mathbb I\big(\text{rank}_{(h,r,t)}^{\text{head}}\leq k\big)+\mathbb I\big(\text{rank}_{(h,r,t)}^{\text{tail}}\leq k\big)$, where $\mathbb I(\cdot)$ is the indicator function. There are some controversies about how to evaluate the rank of $(h,r,t)$ when there are several triplets with the same score as $(h,r,t)$, also known as tie policy. As suggested by \citep{rossi2020knowledge} and \citep{sun2020re}, we do not use the min and max tie policy which are problematic. Instead, we adopt the ordinal tie policy where the rank within values with the same score is indicated corresponding to the order that the values occur.
\begin{algorithm}
\caption{Line search: $\rho^m = \arg\min_\rho \mathcal L(\tilde{\mathbf y}^{m-1}-\rho\cdot f^m(\mathbf X), \mathbf y)$}
\begin{algorithmic}[1]
\REQUIRE $f^m, \tilde{\bm y}^{m-1}, \bm y$, $\rho^m=1$;
\STATE loss-init $\leftarrow \mathcal L(\tilde{\mathbf y}^{m-1}, \mathbf y)$;
\REPEAT
\STATE loss $\leftarrow \mathcal L(\tilde{\mathbf y}^{m-1}-\rho^m\cdot f^m(\mathbf X), \mathbf y)$;
\STATE $\rho^m\leftarrow 2*\rho^m$;
\UNTIL{loss $>$ loss-init or loss is infinite or $|\rho^m\cdot f^m(\bm x)|_1<1e-4$};
\REPEAT
\STATE loss $\leftarrow \mathcal L(\tilde{\mathbf y}^{m-1}-\rho^m\cdot f^m(\mathbf X), \mathbf y)$;
\STATE $\rho^m\leftarrow 0.5*\rho^m$;
\UNTIL{loss $<$ loss-init or loss in infinite or $|\rho^m\cdot f^m(\bm x)|_1<1e-4$}
\RETURN $\rho^m$
\end{algorithmic} \end{algorithm}
\end{document} | arXiv |
\begin{document}
\title{Generalised Hecke algebras and $C^*$-completions} \author[M. B. Landstad]{Magnus B. Landstad} \address{Department of Mathematical Sciences, Norwegian University of Science and Technology, N-7491, Trondheim, Norway.} \email{[email protected]} \author[N. S. Larsen]{Nadia S. Larsen} \address{Department of Mathematics, University of Oslo, P.O. Box 1053 Blindern, N-0316 Oslo, Norway.} \email{[email protected]} \subjclass[2000]{Primary 46L55; Secondary 20C08}
\thanks{This research was supported by the Research Council of Norway.}
\begin{abstract} For a Hecke pair $(G, H)$ and a finite-dimensional representation $\sigma$ of $H$ on $V_\sigma$ with finite range we consider a generalised Hecke algebra $\cc H_\sigma(G, H)$, which we study by embedding the given Hecke pair in a Schlichting completion $(G_\sigma, H_\sigma)$ that comes equipped with a continuous extension $\sigma$ of $H_\sigma$. There is a (non-full) projection $p_\sigma\in C_c(G_\sigma, {\cc B}(V_\sigma))$ such that $\cc H_\sigma(G, H)$ is isomorphic to $p_\sigma C_c(G_\sigma, {\cc B}(V_\sigma))p_\sigma$. We study the structure and properties of $C^*$-completions of the generalised Hecke algebra arising from this corner realisation, and via Morita-Fell-Rieffel equivalence we identify, in some cases explicitly, the resulting proper ideals of $C^*(G_\sigma, {\cc B}(V_\sigma))$. By letting $\sigma$ vary, we can compare these ideals. The main focus is on the case with $\dim\sigma=1$ and applications include $ax+b$-groups and the Heisenberg group. \end{abstract}
\maketitle
\section*{Introduction}
A Hecke pair $(G, H)$ consists of a group $G$ and a subgroup $H$ such that $L(x):=[H:H\cap xHx^{-1}]$ is finite for all $x\in G$. Our interest lies in studying $C^*$-completions of a \emph{generalised Hecke algebra} $\cc H_\sigma(G, H)$ associated with a Hecke pair $(G, H)$ and a unitary representation $\sigma$ of $H$ on a finite-dimensional Hilbert space $V_\sigma$. As a vector space, $\cc H_\sigma(G, H)$ consists of functions $f:G\to {\cc B}(V_\sigma)$ with finite support in $H\backslash G/H$ such that $$ f(hxk)=\sigma(h)f(x)\sigma(k) \text{ for all } h,k\in H, x\in G. $$ When the group is locally compact totally disconnected, and the subgroup is compact and open, such algebras, endowed with a natural convolution, play a fundamental role in the representation theory of reductive $p$-adic groups, see for example \cite{Ho}.
When $\sigma$ is the trivial representation of $H$, $\cc H_\sigma(G, H)$ is the \emph{Hecke algebra} $\cc H(G, H)$ of the pair $(G, H)$, see e.g. \cite{K}. With an appropriate involution, $\cc H_\sigma(G, H)$ becomes a $*$-algebra. Our goal is to shed light on the structure of $C^*$-completions of $\cc H_\sigma(G, H)$, and to identify conditions which ensure that a largest $C^*$-completion exists. That this last issue is important and non-trivial was demonstrated by Hall, who in \cite{H} gave an example of a Hecke pair $(G, H)$ such that $\cc H(G, H)$ does not have a largest $C^*$-completion. We find it natural to investigate the structure of $C^*$-completions in the more general context of a generalised Hecke algebra $\mathcal{H}_\sigma(G, H)$ of a Hecke triple $(G, H, \sigma)$. Results from \cite{KLQ} valid for Hecke pairs $(G, H)$ are not directly applicable in the setup of $(G, H,\sigma)$ with non-trivial $\sigma$, but the overall strategy from \cite{KLQ} can be adapted and developed, as we shall show, in order to deal with the differences that arise in our more general context.
The interesting structure and properties of the Hecke $C^*$-algebra introduced by Bost and Connes in \cite{BC} have motivated intense research devoted to the study of Hecke $C^*$-algebras of large classes of Hecke pairs, see for example \cite{ALR, B, BLPR, GW, H, KLQ, LaLa0, LaR, LarR, T}. A powerful tool to analyse $\cc H(G, H)$ is the ``Schlichting completion'' $(\overline{G}, \overline{H})$ of $(G, H)$: this is a new Hecke pair consisting of a locally compact totally disconnected group and a compact open subgroup \cite{T}. Then $\cc H(G, H)$ is isomorphic to $C_c(\overline{H}\backslash \overline{G}/\overline{H})$, which is a corner of the group algebra $C_c(\overline{G})$, and this viewpoint facilitates the analysis of $C^*$-completions in the realm of Banach $*$-algebras. Tzanev's construction of $(\overline{G}, \overline{H})$ was inspired by work of Schlichting, see for example \cite{Sch}, and was reviewed in \cite{KLQ}, where it is employed to study $C^*$-completions by looking at ideals in $C^*(\overline{G})$, see \cite{GW, Laca_dil, LarR} for other approaches.
In \cite{Cu0}, Curtis considers a Hecke algebra of a triple $(G, H, \sigma)$ where $\sigma$ is a finite-dimensional unitary representation of $H$, and studies a von Neumann algebra naturally associated to it. Curtis constructs a completion $(G_\sigma, H_\sigma,\sigma)$, but does not prove that it is unique.
In the present study we concentrate the attention to the case where the representation $\sigma$ of $H$ has finite range.
There are several reasons for this: we require finite range first because this case is simpler to handle, but more importantly, the case with $\sigma(H)$ infinite is fundamentally different, since the completion of $G$ then will contain a copy of $\bb T$ and therefore will not be totally disconnected.
With our extended theory we use both Fell's and Rieffel's versions of Morita equivalence to analyse the structure of $C^*$-completions of $\cc H_\sigma(G, H)$ with respect to $\sigma$. It turns out that we compare corners of $C_c(G_\sigma, {\cc B}(V_\sigma))$ determined by projections $p_\sigma$. If $\sigma$ is the trivial representation, it was shown in \cite{KLQ} that the projection $p_\sigma$ often is full. However, for nontrivial $\sigma$ it turns out that $p_\sigma$ is never full.
In the by now classical example of the Bost-Connes Hecke pair, the completion $G_\sigma$ will be the same for all finite characters $\sigma$ of $H$, and therefore all the projections $p_\sigma$ live in one group $C^*$-algebra $A=C^*(G_0)$, see Example~\ref{ex:full ax+b}. It follows that the generalised Hecke $C^*$-algebras $p_\sigma A p_\sigma$ are all Morita-Rieffel equivalent to ideals in the same $C^*$-algebra $A$, and can therefore be compared more naturally. It turns out that these ideals are built from the primitive ideals of the Bost-Connes Hecke $C^*$-algebra identified by Laca and Raeburn in \cite{LaR2}.
The organisation of the paper is as follows. In section~\ref{start} we construct our Schlichting completion $(G_\sigma, H_\sigma, \sigma)$ of a Hecke triple $(G, H,\sigma)$, where $\sigma$ is a finite-dimensional unitary representation of $H$ with finite range. We prove in Theorem~\ref{uniqueness_univ_prop} that $(G_\sigma, H_\sigma, \sigma)$ has a universal property, which is essentially provided by the universal property of Schlichting completions of Hecke pairs, see \cite[Theorem 3.8]{KLQ}. In section~\ref{section_gen_Hecke} we define the generalised Hecke algebra $\cc H_\sigma(G, H)$ for arbitrary finite-dimensional $\sigma$ and we realise $\cc H_\sigma(G, H)$ as a corner of $C_c(G_\sigma, {\cc B}(V_\sigma))$.
One new ingredient that appears in the study of generalised as opposed to usual Hecke algebras is that not every double coset $HxH$ for $x$ in $G$ supports a non-zero function in $\cc H_\sigma(G, H)$. When $\sigma$ is one-dimensional, we obtain further insight. The subset $B$ of the $x$'s in $G$ which do support a non-zero function need not be a subgroup of $G$, but nevertheless it harmonises with the Schlichting completion. If $B$ is a group, its closure $B_\sigma$ in $G_\sigma$ determines a corner $p_\sigma C_c(B_\sigma)p_\sigma$, and we prove (Proposition~\ref{H_sigma_spanned_by_epsilonx}) that this corner is isomorphic to $\cc H_\sigma(G, H)$.
Section~\ref{vN_alg} contains an analysis of the continuity properties of the induced representation from $H$ to $G$ with respect to the process of taking the Schlichting completion. To illustrate the point that our Schlichting completion of $(G, H, \sigma)$ is a profitable alternative to studying $\cc H_\sigma(G, H)$, we employ it to give a short proof (see Theorem~\ref{commutant_is_Hecke_vNalg}) of a classical result which asserts that the commutant of the induced representation $\operatorname{Ind}_H^G\sigma(G)$ is the weak closure of the ``intertwining operators'', cf. \cite[Theorem 2.2]{Bi1} (which mends the apparently deficient proof of \cite[Theorem 3]{Co}) or \cite[Proposition 1.3.10]{Cu1}. As an immediate corollary, for one-dimensional $\sigma$ we describe the irreducibility of $\operatorname{Ind}_H^G\sigma$ in terms of the Hecke algebra, thus recovering Mackey's condition in \cite{M}.
We shall often specialise to one-dimensional representations, and in section~\ref{completions} we start by showing
that if $\dim \sigma>1$, the generalised Hecke algebra is still Morita equivalent to the ideal in $C_c(G_\sigma)$ generated by the character of $\sigma$.
So also in this case the generalised Hecke algebras can be studied by looking at ideals in $C^*(G_\sigma)$. For these ideals we describe the nondegenerate representations as in \cite[\S 5]{KLQ} by means of a category equivalence, see Corollary~\ref{category_equiv}.
In the presence of a normal subgroup $N$ of $G$ which contains $H$ we describe the structure of these ideals as twisted crossed products.
If in addition $H$ is normal in $N$ and $N\subset B$, we can conclude that $\cc H_\sigma(G, H)$ has a largest $C^*$-completion, see Corollary~\ref{C*_completion_normal_subgroups}. Finally, we study the special, but interesting instance where $B$ is a group and $(B, H)$ is directed in the sense of \cite[\S 5]{KLQ}. A largest $C^*$-completion turns out to exist, and we can give a concrete description of the ideal in $C^*(G_\sigma)$, see Theorem~\ref{theo_p1_B_directed}.
The last section is devoted to applications. We show in Proposition~\ref{sigma_extends} that $\cc H_\sigma(G, H)$ is isomorphic to $\cc H(G, H)$ when $\sigma$ extends to a character of $G$.
We illustrate in examples the multitude of possible outcomes of the construction of the Schlichting completion.
\noindent \textbf{Conventions.} For a Hecke pair $(G, H)$ and a subset $X$ of $G$, the notation $y\in X/H$ (and $y \in H\backslash{X}/H$) means that $y$ runs over a set of representatives for the left cosets $X/H$ (and the double cosets $H\backslash{X}/H$).
All representations of topological groups are assumed to be unitary and continuous. If $L$ is a locally compact group with a left invariant Haar measure $\mu$ and modular function $\Delta$, then the space $C_c(L)$ of compactly supported continuous functions on $L$ is a $*$-algebra with multiplication given by usual convolution $f\ast g(x)=\int_L f(y)g(y^{-1}x)d\mu(y)$, and involution given by $f^*(x)=\Delta(x^{-1})\overline{f(x^{-1})}$. The group $C^*$-algebra $C^*(L)$ is generated by a universal unitary representation of $L$ into the unitary group of the multiplier algebra $M(C^*(L))$, and $\{\int_L f(x)x d\mu(x)\mid f\in C_c(L)\}$ spans a dense subspace of $C^*(L)$, where we identify $x$ in $L$ with its image in $M(C^*(L))$.
\section{Hecke pairs and Schlichting completions}\label{start}
We recall from \cite[Definition 3.3]{KLQ} that if $(G, H)$ is a Hecke pair, then the collection $\{xHx^{-1}\mid x\in G\}$ is a neighbourhood subbase for the Hecke topology on $G$ from $(G, H)$. If a Hecke pair $(G, H)$ is such that \begin{equation} \bigcap_{x\in G}xHx^{-1}=\{e\}, \label{reduced_pair} \end{equation} then it is called \emph{reduced} \cite{T}, and the Hecke topology from $(G, H)$ is Hausdorff. The \emph{Schlichting completion} of a Hecke pair $(G, H)$ was constructed in \cite{T} to be an essentially unique Hecke pair consisting of a locally compact totally disconnected group with a compact open subgroup in which $G$ and respectively $H$ embedd densely. In the terminology of \cite[\S 3]{KLQ}, the Schlichting completion of $(G, H)$ consists of the closures of $G$ and $H$ in the Hecke topology from $(G, H)$. The Schlichting completion of a Hecke pair is a \emph{Schlichting pair}, which by \cite[\S 3]{KLQ} is a reduced Hecke pair with the additional feature that the underlying subgroup is compact and open in its corresponding Hecke topology.
Suppose that $(G, H)$ is a Hecke pair and $\sigma$ is a finite-dimensional unitary representation of $H$ on a Hilbert space $V_\sigma$ such that $\sigma(H)$ is finite. Then $K:=\ker \sigma$ is a normal subgroup of $H$ of finite index, and hence $(G, K)$ is a Hecke pair. Let $(G_\sigma, K_\sigma)$ denote the Schlichting completion of $(G, K)$. We have the following lemma.
\begin{lem} \textnormal{(a)}
The closure $H_\sigma$ of $H$ in the Hecke topology from $(G, K)$ is a compact open subgroup
of $G_\sigma$.
\textnormal{(b)}
$\sigma$ is continuous for the Hecke topology from $(G, K)$, and thus has a unique extension to a finite-dimensional unitary representation $\sigma$ of $H_\sigma$ with kernel $K_\sigma.$ \label{construction_of_Hecke_triple} \end{lem}
\begin{proof} We claim that $hK\to h K_\sigma$ for $h\in H$ is an isomorphism $ H/K\overset{\cong}{\longrightarrow} H_\sigma/K_\sigma; $ indeed, an element $h\in H\setminus K$ is carried to the open set $hK_\sigma$ which is disjoint from $K$, showing injectivity, and surjectivity follows because any given $xK_\sigma$ in $H_\sigma/K_\sigma$ is open, and hence meets the dense subset $H$ of $H_\sigma$. Thus $H_\sigma/K_\sigma$ is finite, so $H_\sigma$ is compact because its quotient by a compact subgroup is again compact. This proves (a). For (b) it suffices to show continuity of $\sigma$ at $e$, and this follows by inspection using that $\sigma(H)$ is finite and $K$ is open. \end{proof}
\begin{defn} A \emph{$n$-dimensional Hecke triple} $(G, H, \sigma)$ consists of a Hecke pair $(G, H)$ and a $n$-dimensional unitary representation $\sigma$ of $H$ on $V_\sigma$ with finite range. We say that $(G, H, \sigma)$ is \emph{reduced} if the Hecke pair $(G, K:=\ker \sigma)$ is reduced. We call the Hecke triple $({G}_\sigma, {H}_\sigma, \sigma)$ from Lemma~\ref{construction_of_Hecke_triple} the \emph{Schlichting completion} of $(G, H, \sigma)$. \label{def_of_Hecke_top_from_sigma} \end{defn}
\begin{rem} Suppose that $L$ is a totally disconnected locally compact group, $M$ a compact open subgroup, and $\rho$ a continuous finite-dimensional unitary representation of $M$. It follows from \cite[Corollary (28.19)]{HR} that $\rho(M)$ is finite. \end{rem}
We next prove that the Schlichting completion of a Hecke triple has a universal property, and is unique up to topological isomorphism.
\begin{thm}\label{uniqueness_univ_prop} Let $(G, H, \sigma)$ be a reduced $n$-dimensional Hecke triple. Then the Schlichting completion $({G}_\sigma, {H}_\sigma, {\sigma})$ has the following universal property: suppose that $L$ is a locally compact totally disconnected group, $M$ a compact open subgroup, $\rho$ a unitary representation of $M$ on $V_\sigma$, and $\phi:G\to L$ a homomorphism such that $(L, \ker \rho)$ is reduced,
$\phi(G)$ is dense in $L$, $\phi(H)\subseteq M$ and $\sigma=\rho \circ \phi\vert_H$. Then there is a unique continuous homomorphism $\overline{\phi}$ from ${G}_\sigma$ onto $L$ which extends $\phi$ and satisfies the identity \begin{equation} \rho \circ \overline{\phi}\vert_{{H}_\sigma}={\sigma}. \label{phi_extended} \end{equation}
If in addition $\phi^{-1}(M)=H$, then $\overline{\phi}$ will be a topological group isomorphism of ${G}_\sigma$ onto $L$ and of ${H}_\sigma$ onto $M$. \end{thm}
\begin{proof} Denote $N:=\ker \rho$. Then $\phi(K)\subseteq N$. Applying the first half of \cite[Theorem 3.8]{KLQ} to the Schlichting pair $(L, N)$ gives a unique continuous homomorphism $\overline{\phi}$ from $G_\sigma$ into $L$ which extends $\phi$. Since $$ \rho \circ \overline{\phi}\vert_H(h)=\rho\circ \phi(h)=\sigma(h) $$ for all $h\in H$, \eqref{phi_extended} follows from the continuity of $\rho \circ \overline{\phi}$ and $\sigma$ on $H_\sigma$.
If $\phi^{-1}(M)=H$, a straightforward verification then shows that $\phi^{-1}(N) =K$, and it follows from the second half of \cite[Theorem 3.8]{KLQ} that $\overline{\phi}$ is a topological group isomorphism of $G_\sigma$ onto $L$. The assumption $\phi^{-1}(M)=H$ implies that $\phi(H)=M\cap \phi(G)$. Since $M$ is open and closed, \begin{equation} \overline{\phi(H)}=\overline{M\cap \phi(G)}=M\cap \overline{\phi(G)}=M.\label{M_is_phiH_closed} \end{equation} The set $\overline{\phi}(H_\sigma)$ is compact, hence closed, and so it equals $\overline{\phi(H)}$. By invoking \eqref {M_is_phiH_closed} we obtain the last claim of the theorem. \end{proof}
\begin{rem}\label{def_of_iota} Let $(G, H, \sigma)$ be a reduced Hecke triple, $(G_\sigma, H_\sigma, \sigma)$ the Schlichting completion, and $j_1$ the dense embedding of $G$ in $G_\sigma$. Denote by $(G_0, H_0)$ the Schlichting completion of $(G, H)$ and by $j_0$ the dense embedding $G\to G_0$. Since $K\subseteq H\subseteq H_0$, the first half of \cite[Theorem 3.8]{KLQ} gives a continuous homomorphism $\iota:{G}_\sigma \to G_0$ such that $\iota\circ j_1=j_0$. Since $H$ is dense in both $H_\sigma$ and $H_0$, $\iota(H_\sigma)=H_0$. We typically omit $j_0$ and $j_1$ from the notation. \end{rem}
\begin{rem} In \cite{Cu0}, for a Hecke pair $(G, H)$ and a finite-dimensional unitary representation $\sigma$ of $H$, Curtis defines an equivalence relation $\sim$ on $G\times \sigma(H)$ by $(g, t)\sim (gh^{-1}, \sigma(h)t)$ for all $h\in H$. With $S_\sigma:=(G\times \sigma(H))/\sim$ denoting the quotient space, $G$ is endowed with the topology pulled back from the compact-open topology on the space of continuous functions $\{f:S_\sigma\to S_\sigma\}$. Then part of \cite[Theorem 3]{Cu0} asserts that the closures of $G$ and $H$ in this topology and the unique extension of $\sigma$ to the closure of $H$ have the universal property. If $\sigma$ has finite range, note that the map $g\mapsto [g, 1]$ from $G$ onto $S_\sigma$ is a bijection from $G/K$ onto $S_\sigma$, which is equivariant for the actions of $G$ as permutations on $G/K$ and on $S_\sigma$. Thus $G_\sigma$ is the same as the completion constructed in \cite{Cu0}. \end{rem}
\section{The generalised Hecke algebra of $(G, H, \sigma)$}\label{section_gen_Hecke}
The next definition appears in \cite{Cu0}, with the difference that $H$ is an arbitrary subgroup of $G$ and one takes functions $f$ with finite support on $H\backslash G$ and $G/H$. However, for the purposes of using Schlichting completions, the important case, also in \cite{Cu0}, is that of a Hecke pair $(G, H)$.
\begin{defn} Given a $n$-dimensional Hecke triple $(G, H,\sigma)$,
let $\cc H_\sigma(G, H)$ be the vector space of functions $f:G\to {\cc B}(V_\sigma)$ which have finite support in $H\backslash G/H$ and satisfy $f(hxk)=\sigma(h)f(x)\sigma(k)$ for all $h,k\in H, x\in G$. The \emph{generalised Hecke algebra} associated with $(G, H,\sigma)$ is $\cc H_\sigma(G, H)$ endowed with the convolution \begin{equation} f\ast g(x)=\sum_{yH\in G/H}f(y)g(y^{-1}x).\label{def_of_conv} \end{equation} The identity element is the function $\varepsilon_H$ defined by $\varepsilon_H(x)= \sigma(x)$ when $x\in H$ and $\varepsilon_H(x)=0$ otherwise. \end{defn}
The key reason that motivates the study of $\cc H_\sigma(G, H)$ in terms of the Schlichting completion $(G_\sigma, H_\sigma, \sigma)$ of $(G, H, \sigma)$ is the following standard result.
\begin{lem}\label{gen_Hecke_on_lcg} Let $L$ be a locally compact group, $M$ a compact open subgroup, $\rho$ a finite-dimensional unitary representation of $M$, and choose the Haar measure $\mu$ on $L$ normalised so that $\mu(M)=1$. Then $\cc H_\rho(L, M)$ is equal to the subalgebra \begin{equation} \{f\in C_c(L,{\cc B}(V_\rho))\mid f(mxn)=\rho(m)f(x)\rho(n), \forall m,n\in M, x\in L\}\label{Hecke_algebra_as_functions_in_Cc} \end{equation} of $C_c(L,{\cc B}(V_\rho))$, endowed with the convolution with respect to $\mu$. \end{lem}
\begin{prop} Suppose that $(G, H, \sigma)$ is a reduced Hecke triple. Let $(G_\sigma, H_\sigma, \sigma)$ be the Schlichting completion of $(G, H, \sigma)$, and choose the Haar measure $\mu$ on $G_\sigma$ normalised so that $\mu(H_\sigma)=1$. Then the map $\Psi:\cc H_\sigma({G_\sigma}, {H_\sigma})\to \cc H_\sigma(G, H)$ given by $\Psi(f)=f\vert_G$ is an algebra isomorphism. \label{Hecke_algebra_for_G_and_Gbar_same} \end{prop}
\begin{proof} By adapting the argument in \cite[Proposition 3.9 (iii)]{KLQ} to the reduced pair $(G, K)$, it follows that $HxH\mapsto H_\sigma xH_\sigma$ for $x\in G$ is a bijection from $H\backslash G/H$ onto $H_\sigma\backslash G_\sigma /H_\sigma$. Since $G_\sigma$ and $H_\sigma$ contain dense copies of $G$ and $H$, it follows that the map $\Psi$ is well-defined.
Given $f$ in $\cc H_\sigma(G, H)$, note that by the invariance property of $f$, $$ f((xKx^{-1})x)=f(xK)=f(x)\sigma(K)=f(x) $$ for all $x\in G$. Thus $f$ is continuous for the Hecke topology from $(G, K)$, and so extends to a function in $\cc H_\sigma (G_\sigma, H_\sigma)$. It follows that
$\Psi$ is bijective. A routine calculation shows that $\Psi(f\ast g)=\Psi(f)\ast \Psi(g)$, and the claim follows. \end{proof}
From Proposition~\ref{Hecke_algebra_for_G_and_Gbar_same} and Lemma~\ref{gen_Hecke_on_lcg} it seems natural to define the involution on $\cc H_\sigma(G, H)$ as in $C_c(G_\sigma, {\cc B}(V_\sigma))$ by using the modular function of $G_\sigma$, so we investigate its meaning for the original Hecke triple. This is in fact answered by Schlichting in \cite[Lemma 1(iii)]{Sch}. We need some notation first. For a Hecke pair $(G, H)$ and any $x$ in $G$ let $H_x:=H\cap xHx^{-1}$, $L(x):=[H:H_x]$ and $\Delta_H(x):= {L(x)}/{L(x^{-1})}$. By \cite{Sch}, if $H$ is a compact open subgroup of a locally compact group $G$ the modular function $\Delta$ of $G$ satisfies $\Delta(x)= \Delta_H(x)$. In particular, $\Delta_H(x)$ does not depend on which compact open subgroup we use. With the notation of Remark~\ref{def_of_iota} we obtain:
\begin{cor}\label{modular_functions_same} The modular functions $\Delta_\sigma$ of $G_\sigma$ and $\Delta_0$ of $G_0$ satisfy $\Delta_0\circ\iota=\Delta_\sigma$. \end{cor}
If $(G, H, \sigma)$ is a reduced Hecke triple with Schlichting completion $(G_\sigma, H_\sigma, \sigma)$, then $[H:H_x]=[H_\sigma:(H_\sigma)_x]$, so we can supplement Proposition~\ref{Hecke_algebra_for_G_and_Gbar_same}.
\begin{prop}\label{Hecke*algebra_for_G_and_Gbar_same} Let $(G, H,\sigma)$ be a reduced Hecke triple with $K:=\ker \sigma$, and define an involution on $\cc H_\sigma(G, H)$ by \begin{equation} f^*(x)=\Delta_K(x^{-1})f(x^{-1})^*, \text{ for }x\in G. \label{def_involution} \end{equation} Then the map $\Psi$ of Proposition~\ref{Hecke_algebra_for_G_and_Gbar_same} is an isomorphism of $*$-algebras. \end{prop}
\begin{rem} Most authors do not include $\Delta$ in the definition of an involution on a (generalised) Hecke algebra. But we claim that this is more natural, for instance the $l^1$-norm on $\cc H_\sigma(G, H)$ defined by \begin{equation}
\Vert f\Vert_1=\sum_{y\in G/H}\| f(y)\| \text{ for }f\in \cc H_\sigma(G, H) \label{def_of_norm} \end{equation}
satisfies $\| f^*\|_1=\| f\|_1$. We let $l^1(G, H, \sigma)$ be the completion of $\cc H_\sigma(G, H)$ in $\Vert \cdot \Vert_1$. As a consequence of Proposition~\ref{Hecke_algebra_for_G_and_Gbar_same} and Proposition~\ref{Hecke*algebra_for_G_and_Gbar_same} we get the following: \end{rem}
\begin{prop} With the assumptions and notation from Proposition~\ref{Hecke_algebra_for_G_and_Gbar_same}, the map $\Psi$ extends to an isomorphism $L^1({G_\sigma}, {H_\sigma}, {\sigma})\cong l^1(G, H, \sigma)$ of Banach $*$-algebras. \label{L1_algebras_isom} \end{prop}
\begin{proof} A computation shows that $\Vert \Psi(f)\Vert_1=\Vert f\Vert_1$ for all $f\in \cc H_\sigma({G_\sigma}, {H_\sigma})$. Hence $\Psi$ extends to the completion in the norm from (\ref{def_of_norm}).
Note that (\ref{def_of_norm}) on $\cc H_\sigma({G_\sigma}, {H_\sigma})$ is the usual $L^1$-norm on $C_c({G_\sigma, {\cc B}(V_\sigma)})$. A routine calculation shows that $L^1({G_\sigma}, {H_\sigma}, {\sigma})$ is a closed $*$-subalgebra of $L^1({G_\sigma, {\cc B}(V_\sigma)})$ for the natural involution of $L^1({G_\sigma, {\cc B}(V_\sigma)})$. Hence $L^1({G_\sigma}, {H_\sigma}, {\sigma})$ is a Banach $*$-algebra and the claim follows. \end{proof}
\begin{thm} Given a reduced $n$-dimensional Hecke triple $(G, H, \sigma)$, let $(G_\sigma, H_\sigma, \sigma)$ denote its Schlichting completion. Denote by $\mu$ the left invariant Haar measure on ${G_\sigma}$ such that $\mu({H_\sigma})=1$.
Then the function $p_\sigma(x):=\Chi_{{H_\sigma}}(x) {\sigma}(x)$ is a self-adjoint projection
in $C_c({G_\sigma, {\cc B}(V_\sigma)})$, and we have isomorphisms between the $*$-algebras $ p_\sigma C_c(G_\sigma, {\cc B}(V_\sigma)) p_\sigma=\cc H_\sigma(G_\sigma, H_\sigma)$ and $\cc H_\sigma(G, H)$, and between the Banach $*$-algebras $p_\sigma L^1(G_\sigma, {\cc B}(V_\sigma)) p_\sigma$ and $l^1(G, H, \sigma).$ \label{fundam_proj} \end{thm}
\begin{proof} It is straightforward that $p_\sigma$ is a self-adjoint projection. Lemma~\ref{gen_Hecke_on_lcg} implies that $\cc H_\sigma(G_\sigma, H_\sigma)=p_\sigma C_c(G_\sigma, {\cc B}(V_\sigma))p_\sigma$, and then the claimed isomorphisms follow from Proposition~\ref{Hecke*algebra_for_G_and_Gbar_same} and Proposition~\ref{L1_algebras_isom}. \end{proof}
\section{The generalised Hecke algebra of $(G, H, \sigma)$ when $\dim\sigma=1$} \label{section_gen_Hecke_dim=1}
In this section we assume $\dim(V_\sigma)=1$, although some of the results still will be true in greater generality, in particular the results about the set $B$, \emph{cf.}\ \cite{Bi1, Bi2}.
Similarly to \cite[Lemma 4.2(iii)]{KLQ} we have that $p_\sigma C_c(G_\sigma) p_\sigma=\spn\{p_\sigma x p_\sigma\mid x\in G_\sigma\}$. We proceed to identify functions in a spanning set for $\cc H_\sigma(G, H)$ which correspond by Theorem~\ref{fundam_proj} to the products $p_\sigma xp_\sigma$.
It is known, see for instance \cite{K}, that the Hecke algebra of a pair $(G, H)$ is linearly spanned by the collection $\{\Chi_{HxH}\mid x\in H\backslash G/H\}$ of characteristic functions of double cosets. To account for non-zero functions in $\cc H_\sigma(G, H)$ supported on a given double coset, note that for $f\in \cc H_\sigma(G, H)$, $x\in G$ and $h\in H_x=H\cap xHx^{-1}$ we have $$ \sigma(h)f(x)=f(hx)=f(xx^{-1}hx)=f(x)\sigma(x^{-1}hx). $$ Thus for $f$ to be supported on $HxH$ we need $ \sigma(h)=\sigma(x^{-1} hx)\text{ for }h\in H_x$. This is condition $(t_g)$ in \cite[Proposition 1.2]{Bi1}, and goes at least back to Mackey \cite{M}. We denote \begin{equation} B:=\{x\in G\mid \sigma(h)=\sigma(x^{-1} hx)\text{ for }h\in H_x\}. \label{special_x} \end{equation}
\begin{rem} The set $B$ contains $H$, is closed under inverses, and satisfies $BH=B=HB$. In general, $B$ is not a group, see for example \cite[Example 1.4.4]{Cu1}. However, $B$ is a group in many cases such as, for instance, when $\sigma$ extends to a character of $G$. \end{rem}
\begin{lem}\label{basis_for_Hecke_algebra} For each $x\in B$ there is a well-defined element \begin{equation} {\varepsilon_x}(y)= \begin{cases} \sigma(hk)&\text{ if }y\in HxH\text{ and }y=hxk\\ 0&\text{ if }y\notin HxH \end{cases} \label{def_of_epsilon_x} \end{equation} in $\cc H_\sigma(G, H)$, and the set $\{\varepsilon_x\mid x\in H\backslash {B}/H\}$ forms a linear basis for $\cc H_\sigma(G, H)$. \end{lem}
\begin{proof} Clearly $\varepsilon_x\in \cc H_\sigma(G, H)$ and is well-defined (these functions are essentially the elementary intertwining operators from \cite{Bi1}). Since $\varepsilon_{h_0xk_0}(hxk)= \overline{\sigma(h_0)\sigma(k_0)}\varepsilon_x(hxk)$ for all $h_0, k_0\in H$, different choices of representatives for the double coset $HxH$ give rise to functions $\varepsilon_{h_0xk_0}$ which are scalar multiples of $\varepsilon_x$, and since distinct double cosets do not support a common $\varepsilon_x$ the lemma follows. \end{proof}
When $B$ is a group, $(B, H,\sigma)$ is a new Hecke triple, we trivially have $\cc H_\sigma(G, H)=\cc H_\sigma(B, H)$, and $\cc H_\sigma(G, H)$ has a linear basis indexed over the double cosets of $B$ with respect to its subgroup $H$. To study $\cc H_\sigma(G, H)$ in this case we must naturally view it as a corner in $C_c(B_0)$ with $B_0$ denoting the Schlichting completion of $(B, K)$. As example~\ref{ex:full ax+b} shows, the Schlichting completion of $(B, H,\sigma)$ need not come from $(G_\sigma, K_\sigma)$, because on $B$ the Hecke topology from $(G, K)$ differs from the Hecke topology from $(B, K)$. Nevertheless, the corner in $C_c(B_0)$ which is isomorphic to $\cc H_\sigma(B, H)$ is completely determined by the topology on $G_\sigma$. To prove this, we first establish that the closure of $B$ in $G_\sigma$ is precisely the set defined by \eqref{special_x} for $(G_\sigma, K_\sigma)$.
\begin{lem}\label{closure_of_Hx} We have $\overline{H\cap xHx^{-1}}=H_\sigma \cap xH_\sigma x^{-1}$ in $G_\sigma$ for all $x\in G$. \end{lem}
\begin{proof} Since $H_x$ is included in the closed set $(H_\sigma)_x$ for every $x\in G$, we obtain one inclusion. Suppose that $h\in (H_\sigma)_x$. Let $F$ be a finite subset of $G$, and take $K_{\sigma, F}=\bigcap_{y\in F} yK_\sigma y^{-1}$, a neighbourhood of $e$. By restricting, if needed, to a smaller neighbourhood, we may assume that $x\in F$. Since $H$ is dense in $H_\sigma$, it intersects the open neighbourhood $hK_{\sigma, F}$ of $h$. Thus we have \begin{equation} hk_1=h_1, hk_2=xh_2x^{-1},\label{many_h} \end{equation} with $h_1, h_2\in H$ and $k_1, k_2\in K_{\sigma, F}$. Then $k_1^{-1}k_2= h_1^{-1}xh_2x^{-1}$ is an element of $G\cap j_1^{-1}(K_{\sigma, F})$, which is $\bigcap_{y\in F} yKy^{-1}$ because the Schlichting completion satisfies $j_1^{-1} (K_\sigma)=K$. Thus $k_1^{-1}k_2$ lies in $xHx^{-1}$, and so $$ h_1=xh_2x^{-1}k_2^{-1}k_1\in xHx^{-1}\cap H, $$ from which it follows via \eqref{many_h} that $h_1\in hK_{\sigma, F}\cap H_x$. Since this holds for all neighbourhoods $K_{\sigma, F}$, we have $h\in \overline{H_x}$, as claimed. \end{proof}
\begin{prop} Let $(G, H, \sigma)$ be a reduced $1$-dimensional Hecke triple, and consider its Schlichting completion $(G_\sigma, H_\sigma, \sigma)$. Let $B_\sigma$ denote \begin{equation} \{x\in G_\sigma\mid \sigma(h)=\sigma(x^{-1}hx)\text{ for }h\in H_\sigma \cap xH_\sigma x^{-1}\}.\label{def_of_Bsigma} \end{equation} Then $B_\sigma$ is equal to the closure of $B$ in the Hecke topology from $(G, K)$. \label{Hx_Kx} \end{prop}
\begin{proof} If $(x_i)$ is a net in $B_\sigma$ converging to $x$, then eventually $H_\sigma \cap x_iH_\sigma x_i^{-1}$ coincides with $H_\sigma \cap xH_\sigma x^{-1}$, and so $B_\sigma$ is closed. Since $B_\sigma H_\sigma=B_\sigma$, it is also open.
Lemma~\ref{closure_of_Hx} implies that $$ B_\sigma\cap G=\{x\in G\mid \sigma(h)=\sigma(x^{-1}hx)\text{ for }h\in H_\sigma \cap xH_\sigma x^{-1}\}=B. $$ Hence the closure of $B$ is included in $B_\sigma$. To show equality, take $x\in B_\sigma$ and $K_{\sigma, F}$ a neighbourhood of $e$. We must show that $xK_{\sigma, F}$ has non-empty intersection with $B$. By density of $G$ in $G_\sigma$ there is $k\in K_{\sigma, F}$ such that $xk\in G$. From $K_{\sigma, F}\subset H_\sigma$ and $B_\sigma H_\sigma =B_\sigma$ it follows that $xk\in B_\sigma\cap G=B$, as claimed. \end{proof}
\begin{prop}\label{H_sigma_spanned_by_epsilonx}
Let $(G, H, \sigma)$ be a reduced $1$-dimensional Hecke triple, let $(G_\sigma, H_\sigma, \sigma)$ be its Schlichting completion, and assume that $B$ is a subgroup of $G$. Let $B_\sigma$ be the subgroup of $G_\sigma$ defined in \eqref{def_of_Bsigma}. Then $\cc H_\sigma(G, H)$ is isomorphic to $p_\sigma C_c(B_\sigma)p_\sigma$. \end{prop}
\begin{proof} Since $B_\sigma$ is closed in $G_\sigma$, it is locally compact. The subgroup $H_\sigma$ is open and compact in $B_\sigma$, so on one hand $B_\sigma$ and $G_\sigma$ have the same modular function, equal to $\Delta_{K_\sigma}$ and $\Delta_{H_\sigma}$, and on the other $\cc H_\sigma(B_\sigma, H_\sigma)$ equals $p_\sigma C_c(B_\sigma)p_\sigma$ by Lemma~\ref{gen_Hecke_on_lcg}.
Given $x$ in $B_\sigma$, it follows from Proposition~\ref{Hx_Kx} that we can pick $b$ in $B$ such that $xH_\sigma=j_1(b)H_\sigma=\overline{j_1(bH)}$. Thus the isomorphism $\Psi$ from Proposition~\ref{Hecke*algebra_for_G_and_Gbar_same} carries functions supported on $H_\sigma xH_\sigma$ with $x\in B_\sigma$ to functions supported on $HbH$ with $b\in B$. Hence by Lemma~\ref{basis_for_Hecke_algebra} the map $\Psi$ is an isomorphism of $\cc H_\sigma(B_\sigma, H_\sigma)$ onto $\cc H_\sigma(B, H)$. \end{proof}
\begin{lem} \label{psigma_x_psigma} With the notation of Theorem~\ref{fundam_proj} we have \begin{equation} p_\sigma xp_\sigma =\begin{cases}\frac 1{L(x)}\varepsilon_x&\text{ if }x\in B_\sigma\\ 0&\text{otherwise}. \end{cases} \label{p_sigma_corner_as_epsilon_x} \end{equation} In particular, $\Vert p_\sigma xp_\sigma\Vert_1=1$ for every $x$ in $B_\sigma$. \end{lem}
\begin{proof} Since $p_\sigma$ is supported on $H_\sigma$, the product $p_\sigma xp_\sigma$ is supported on the double coset $H_\sigma x H_\sigma$. The claim then follows because \begin{align} p_\sigma xp_\sigma(hxk) &=\sigma(h)\sigma(k)\int_{H_\sigma\cap xH_\sigma x^{-1}}\sigma(l)\overline{\sigma(x^{-1}lx)} dl \notag \\ &=\begin{cases} \sigma(h)\sigma(k)\mu(H_\sigma\cap xH_\sigma x^{-1})&\text{ if }x\in B_\sigma\\ 0&\text{ otherwise}.\notag\\ \end{cases} \end{align} \end{proof}
\section{The von Neumann algebra of $(G, H,\sigma)$ and induced representations}\label{vN_alg}
One motivation for studying (generalised) Hecke algebras is that they usually generate the commutant of the corresponding induced representation. In particular this gives irreducibility criteria for induced representations of locally compact groups, see \emph{e.g.}\ Mackey \cite{M}, Corwin \cite{Co} and Binder \cite{Bi1, Bi2}. In \cite[Theorem 2.2]{Bi1}, the commutant of the induced representation is realised as the weak closure of the algebra of so-called elementary intertwining operators (this result was obtained earlier by Corwin \cite{Co}, but that proof is claimed to be incomplete \cite{Bi1}), see also \cite[Proposition 1.3.10]{Cu1}. In this section we study the continuity properties of the induced representation arising from a Hecke triple, and obtain as a direct consequence a new proof of Binder's theorem for such representations.
Suppose that $(G, H, \sigma)$ is a $n$-dimensional Hecke triple. The induced representation $\lambda_\sigma:=\operatorname{Ind}_H^G \sigma$ acts by the formula $\lambda_\sigma(x)(f)(y):=f(x^{-1}y)$
in the space $l^2(G, H, \sigma)$ defined as \begin{equation} \{f:G\to V_\sigma\mid f(xh)={\sigma(h^{-1})}f(x),\forall x\in G, h
\in H, \text{ and }\sum_{x\in G/H} \| f(x)\|^2<\infty\}. \label{Hilbert_space_of ind_rep} \end{equation} The function $\delta_{y,\xi}:G\to V_\sigma$ defined by $\delta_{y,\xi}(z)=\Chi_H(z^{-1}y) \sigma(z^{-1}y)\xi$ for $y\in G $ and $\xi\in V_\sigma$ lies in $l^2(G, H, \sigma)$. If we choose a set of coset representatives $y\in G/H$ and an orthonormal basis $\{\xi_i\mid i=1,\cdots, n\}$ for $V_\sigma$, then $\{\delta_{y,\xi_i}\mid y\in G/H, i=1,\cdots, n\}$ is an orthonormal basis for $l^2(G, H, \sigma)$.
\begin{lem}\label{induced_rep_extends} Suppose that $(G, H, \sigma)$ is a reduced Hecke triple. Let $K=\ker \sigma$ and $({G}_\sigma, {H}_\sigma, {\sigma})$ be the Schlichting completion. Then $\lambda_\sigma$ is a homeomorphism from $G$ with the Hecke topology from $(G, K)$ into ${\cc B}(l^2(G, H, \sigma))$ with its weak topology. \end{lem}
\begin{proof} For $x, y, w\in G$ and $\xi, \eta\in V_\sigma$ we have \begin{equation} \bigl(\lambda_\sigma(w)\delta_{x,\xi}\mid \delta_{y,\eta} \bigr) =\Chi_H(y^{-1}wx)\bigl(\sigma(y^{-1}wx)\xi\mid \eta\bigr).\label{lambda_on_basis} \end{equation}
Since $\sigma(H)$ is finite, there is $0<\varepsilon_0 <1$ such that
$K=\{h\in H\mid\, \|\sigma(h)-I\|\leq \varepsilon_0\}$. Let $F\subseteq G$ be finite. A set of the form $$ \mathcal{V}=\{T\in {\cc B}(l^2(G, H, \sigma))\mid \vert \bigl( T\delta_{x,\xi_i}\mid \delta_{y,\xi_j} \bigr) -\bigl( \delta_{x,\xi_i}\mid \delta_{y,\xi_j}\bigr)\vert<\varepsilon, \forall x,y\in F, i,j=1,\cdots ,n\} $$ is a typical neighbourhood of $\lambda_\sigma(e)$ for the weak topology. We claim that $\lambda_\sigma^{-1}(\mathcal{V})= \bigcap_{x\in F}xKx^{-1}$ if $\varepsilon\leq \varepsilon_0$. Let $x,y\in F$ and $w\in xK x^{-1}$. Then $y^{-1}x\in H$ if and only if $y^{-1}wx \in H$, and (\ref{lambda_on_basis}) shows that $\lambda_\sigma(w)\in \mathcal{V}$, proving one inclusion. In particular, $\lambda_\sigma$ is continuous at $e$, hence everywhere. Suppose now that $\lambda_\sigma(w)\in \mathcal{V}$. Inserting $y=x$ in \eqref{lambda_on_basis} forces $\vert \Chi_H(x^{-1}wx)\bigl(\sigma(x^{-1}wx)\xi_i\mid \xi_j\bigr) -\bigl(\xi_i\mid\xi_j\bigr)\vert <\varepsilon <\varepsilon_0$, so $w\in xKx^{-1}$, and thus $\lambda_\sigma$ carries a neighbourhood subbase at $e$ for the Hecke topology into a neighbourhood subbase at $\lambda_\sigma(e)$ for the weak topology. \end{proof}
Denote by $\overline{\lambda_\sigma}$ the continuous extension of $\lambda_\sigma$ to ${G}_\sigma$. The next result shows that the induced representation of $\sigma$ from $H_\sigma$ to $G_\sigma$ is, up to unitary equivalence, just $\overline{\lambda_\sigma}$.
\begin{prop} With the assumptions of Lemma~\ref{induced_rep_extends}, let $$ L^2({G}_\sigma, {H}_\sigma, {\sigma}):=\{f\in L^2({G}_\sigma, V_\sigma)\mid f(wk)=\sigma(k^{-1}) f(w), \forall w\in {G}_\sigma, k\in{H}_\sigma\}. $$ Then $f\mapsto f\vert_G$ defines a unitary $U:L^2({G}_\sigma, {H}_\sigma, {\sigma})\to l^2(G, H, \sigma)$, and $U^*\overline{\lambda_\sigma}(w) U$, $w\in G_\sigma$, is the induced representation of $\sigma$ from $H_\sigma$ to $G_\sigma$. \label{ind_reps_are_equiv} \end{prop}
\begin{proof} With $(G_\sigma, K_\sigma)$ denoting the Schlichting completion of $(G, K)$, take $w\in {G}_\sigma$ and note that since $wK_\sigma$ is open, there is $y$ in $G$ such that $y\in wK_\sigma$. If $z\in G$, then $w^{-1}z\in {H}_\sigma$ if and only if $y^{-1}z\in {H}_\sigma \cap G=H$, and so $\delta_{w,\xi}=\delta_{y,\xi}$. Since ${G}_\sigma/{H}_\sigma\cong G/H$, $U$ carries the orthonormal basis $\{\delta_{w,\xi_i} \mid w\in {G}_\sigma /{H}_\sigma, i=1,\cdots, n\}$ onto the orthonormal basis $\{\delta_{y,\xi_i}\mid y\in G/H, i=1,\cdots, n\}$. Finally, a routine calculation shows that $U^*\overline{\lambda_\sigma}(w)U$ acts as the induced representation, and the proposition follows. \end{proof}
With the same assumptions, let $L$ and $R$ denote the left, respectively the right regular representation of $G_\sigma$. A consequence of Proposition~\ref{ind_reps_are_equiv} is that $L^2({G}_\sigma, {H}_\sigma, {\sigma})$ is a closed subspace of $L^2(G_\sigma, V_\sigma)=L^2(G_\sigma)\otimes V_\sigma$ which is invariant under $L\otimes I$, and that $\overline{\lambda_\sigma}$ is the restriction of $L\otimes I$ to this subspace.
\begin{lem} Let $(G, H, \sigma)$ be a reduced Hecke triple. Then $\widetilde R$ defined by \begin{equation} (\widetilde R(f)\xi)(y)=\sum_{z\in G/H} \Delta_K(z)^{1/2} f(z)\xi(yz), \label{right_reg_rep_of_Hecke_alg} \end{equation} for $f\in \cc H_\sigma(G, H)$ and $\xi \in l^2(G, H, \sigma)$, is a nondegenerate $*$-representation of $\cc H_\sigma(G, H)$. \end{lem} \begin{proof} It is straightforward to show that $\widetilde R(f)$ is well-defined and that we get a nondegenerate $*$-representation of $\cc H_\sigma(G, H)$. \end{proof}
In \cite{Cu1}, the Hecke von Neumann algebra of $(G, H,\sigma)$ is defined as the von Neumann algebra generated by the image of the generalised Hecke algebra in a left regular representation on the space of the induced representation. Similar to this we let $\mathcal{R}(G, H, \sigma)$ be the von Neumann algebra generated by $\widetilde R(\cc H_\sigma(G, H))$ in ${\cc B}(l^2(G, H,\sigma))$. Since Proposition~\ref{ind_reps_are_equiv} and Proposition~\ref{Hecke_algebra_for_G_and_Gbar_same} imply that $U\widetilde R(f)=\widetilde R(\Psi(f))U$ for all $f\in \cc H_\sigma({G}_\sigma,{H}_\sigma)$, we recover an analogous result to \cite[Theorem 3]{Cu0}.
\begin{prop} With $U$ defined in Proposition~\ref{ind_reps_are_equiv}, the map $a\mapsto
UaU^*$ implements an isomorphism of $\mathcal{R}({G}_\sigma, {H}_\sigma, {\sigma})$ onto $\mathcal{R}(G, H, \sigma).$ \label{Hecke_vN_same} \end{prop}
The spaces $\mathcal{R}({G}_\sigma):=\{R_x\mid x\in {G}_\sigma\}''$ and $\mathcal{L}({G}_\sigma):=\{L_x\mid x\in {G}_\sigma\}''$ are known to be each others commutant inside ${\cc B}(L^2({G}_\sigma))$. Let $P_\sigma:= \int_{H_\sigma} R_k\otimes \sigma(k)\,dk$
be the projection corresponding to the subspace $L^2(G_\sigma, H_\sigma,\sigma)$ of $L^2(G_\sigma, V_\sigma)$. We can now formulate the main result of this section.
\begin{thm} Suppose that $(G, H, \sigma)$ is a reduced Hecke triple. Then $\lambda_\sigma(G)'$ equals $\mathcal{R}(G, H, \sigma)$. \label{commutant_is_Hecke_vNalg} \end{thm}
\begin{proof} Let $({G}_\sigma, {H}_\sigma, {\sigma})$ be the Schlichting completion of $(G, H, \sigma)$. Proposition~\ref{ind_reps_are_equiv} implies that $\ad U^*$ carries the subset $\lambda_\sigma(G)^{''}$ of ${\cc B}(l^2(G, H,\sigma))$ into $\overline{\lambda_\sigma}(G_\sigma)^{''}$ inside ${\cc B}(L^2(G_\sigma))$. Then, since $P_\sigma$ commutes with $L_x\otimes I$ we have $$ \overline{\lambda_\sigma}(G_\sigma)^{''} =\{(L_x\otimes I)P_\sigma\mid x\in G_\sigma\}^{''} =(P_\sigma \{L_x\otimes I\mid x\in G_\sigma\}' P_\sigma)' . $$ So by the Double Commutant Theorem we have $ \overline{\lambda_\sigma}(G_\sigma)' =P_\sigma(\mathcal{L}(G_\sigma)'\otimes {\cc B}(V_\sigma))P_\sigma=P_\sigma(\mathcal{R}({G}_\sigma)\otimes {\cc B}(V_\sigma) )P_\sigma=\mathcal{R}(G_\sigma, H_\sigma, \sigma)$, and by applying $\ad U$ we have back in ${\cc B}(l^2(G, H,\sigma))$ that $\mathcal{R}(G, H, \sigma)$ is equal to $\lambda_\sigma(G)'.$ \end{proof}
As an immediate consequence of Theorem~\ref{commutant_is_Hecke_vNalg} and Lemma~\ref{basis_for_Hecke_algebra} we obtain the following classical result, see \cite[Theorem 6']{M}.
\begin{cor} If $\dim\sigma=1$ then $\lambda_\sigma$ is irreducible if and only if $B=H$. \end{cor}
\section{$C^*$-completions of generalised Hecke algebras}\label{completions}
A consequence of Theorem~\ref{fundam_proj} is that $p_\sigma C^*(G_\sigma,{\cc B}(V_\sigma))p_\sigma$ is a $C^*$-completion of $\cc H_\sigma(G, H)$.
In this section we establish that this completion is Morita-Rieffel equivalent to an ideal in $C^*(G_\sigma)$, and for this ideal we describe the nondegenerate representations. We denote by $C^*(\cc H_\sigma(G, H))$ the enveloping $C^*$-algebra of $\cc H_\sigma(G, H)$, when it exists. We shall later be concerned with the problem of deciding when $C^*(\cc H_\sigma(G, H))$ is $p_\sigma C^*(G_\sigma)p_\sigma$.
Note that if $\dim \sigma=1$ we have that $p_\sigma C^*(G_\sigma)p_\sigma$ is Morita-Rieffel equivalent to the ideal \begin{equation} I:=\overline{C^*(G_\sigma)p_\sigma C^*(G_\sigma)} \label{def_of_I} \end{equation} in $C^*(G_\sigma)$.
We first establish that the similar $C^*$-completion of $\cc H_\sigma(G, H)$ can be obtained the same way if we just assume that $\sigma$ is
finite-dimensional. Let $M$ be a compact open subgroup of a locally compact group $L$ and $\sigma$ an irreducible representation of $M$ on $V_\sigma$ (so in particular $V_\sigma$ is finite-dimensional). Then $\cc H_\sigma(L, M)$ is a $*$-subalgebra of $A=C^*(L)\otimes {\cc B}(V_\sigma)$ with the convolution from \eqref{def_of_conv} and the involution inherited from $C_c(L, {\cc B}(V_\sigma))$.
With $\lambda(k)$ denoting the unitary in $M(C^*(L))$ corresponding to $k\in M$, we have that
\[
p_\sigma= \int_M \lambda(k)\otimes \sigma(k)\,dk \] in $A$, and the closure of $\cc H_\sigma(L, M)$ in $A$ is $p_\sigma A p_\sigma $. If we let $d(\sigma)$ be the dimension of $V_\sigma$ and $\operatorname{Tr}$ the unnormalised trace on ${\cc B}(V_\sigma)$, we obtain an idempotent \begin{equation}
\psi_\sigma(k)= d(\sigma)\operatorname{Tr}( \sigma(k)) \end{equation} in $C^*(M)$; note that $\psi_\sigma$ is a projection in $M(C^*(L))$. The next theorem expands upon the claims from \cite[Example 6.8]{Rie} and \cite[p 93-94]{Fe} that Morita equivalence of $*$-algebras gives an alternate approach to Godement's theory of generalised spherical functions. We do not claim any originality in the next result, but we rather spell out the details in the language of generalised Hecke algebras.
\begin{thm}\label{dim_sigma_not_one} The generalised Hecke $C^*$-algebra $p_\sigma A p_\sigma $ is Morita-Rieffel equivalent to the ideal $\overline{C^*(L)\psi_\sigma C^*(L)}$ in $C^*(L)$. \end{thm}
\begin{proof}
It is standard that $Y=C^*(L)\otimes V_\sigma$ is a left-$A$ and right-$C^*(L)-$module in an obvious way. If we let $T_{\xi,\eta}$ denote the rank-one operator $\alpha \mapsto (\alpha\mid\eta)\xi$, we have a left $A$-valued inner product on $Y$ given by
\[
{{_{A}}{\langle}} a\otimes \xi, b\otimes\eta \rangle= ab^*\otimes T_{\xi,\eta}, \] and a right $C^*(L)$-valued inner product given by
\[
\langle a\otimes \xi, b\otimes\eta \rangle_{C^*(L)}= a^*b (\eta\mid \xi); \] since $V_\sigma$ is finite-dimensional, $Y$ is complete for these inner-products and so becomes an $A-C^*(L)$ imprimitivity bimodule in Rieffel's sense. Restrict now $Y$ to the bimodule $Y_\sigma:=p_\sigma Y$. Clearly \begin{align*} _{A}\langle Y_\sigma, Y_\sigma\rangle &=\clspn \{
_{A}\langle p_\sigma(a\otimes \xi), p_\sigma(b\otimes\eta) \rangle\}\\
&= p_\sigma\clspn \{_{A}\langle a\otimes \xi, b\otimes\eta \rangle \}p_\sigma\\
&=p_\sigma A p_\sigma. \end{align*}
On the other hand, since
$ \clspn \{ \int \lambda( k) (\sigma(k) \eta\mid \xi) \, dk \}= \psi_\sigma C^*(M)$, we have \begin{align*} {{\langle}} Y_\sigma, Y_\sigma\rangle_{C^*(L)} &=\clspn \{{\langle}p_\sigma(a\otimes \xi), p_\sigma(b\otimes\eta) \rangle_{C^*(L)}\}\\ &=\clspn \{\int\int
{{\langle}}\lambda(h)a\otimes \sigma(h) \xi,
\lambda(k)b\otimes \sigma(k) \eta \rangle_{C^*(L)} \, dh\,dk\}\\
&= \clspn \{\int\int a^*\lambda(h^{-1} k)b
(\sigma(k) \eta\mid \sigma(h) \xi) \, dh\,dk\}\\
&= \clspn \{\int a^*\lambda( k)b
(\sigma(k) \eta\mid \xi) \, dk\}\\ &= \clspn \{ a^* \psi_\sigma b \}=\overline{C^*(L)\psi_\sigma C^*(L)}. \end{align*}
Hence $Y_\sigma$ implements the equivalence between $p_\sigma A p_\sigma$ and $\overline{C^*(L)\psi_\sigma C^*(L)}$. \end{proof}
Thus, if $(G, H)$ is a Hecke pair and $\sigma$ is a finite-dimensional unitary representation of $H$ with $\sigma(H)$ finite, we form the Schlichting completion $(G_\sigma, H_\sigma, \sigma)$ as in section~\ref{start}, and it follows from the preceeding considerations that $p_\sigma (C^*(G_\sigma)\otimes {\cc B}(V_\sigma))p_\sigma$ is a C$^*$-completion of $\cc H_\sigma(G, H)$ which is Morita-Rieffel equivalent to an ideal in $C^*(G_\sigma)$.
\vskip 0.2cm We now go back to the case of $1$-dimensional Hecke triples $(G, H,\sigma)$, and we introduce the analogue of the $H$-smooth representations of $G$ that arise from $(G, H)$.
\begin{defn} Suppose that $(G, H,\sigma)$ is a $1$-dimensional Hecke triple. Given a unitary representation $\pi$ of $G$ on a Hilbert space $V$, let \begin{equation} V_{H, \pi}:=\{\xi \in V \mid \pi(h)\xi=\sigma(h)\xi, \forall h\in H\}. \end{equation} We say that $\pi$ is \emph{unitary $(H, \sigma)$-smooth} if $\clspn(\pi(G)V_{H, \pi})=V.$ \end{defn}
Note that $\lambda_\sigma$ is unitary $(H, \sigma)$-smooth (and ``smooth'' in the sense of \cite[\S 1.7]{S}), as is every representation of $G$ that is unitarily equivalent to $\lambda_\sigma$. We have the following generalisation of \cite[Proposition 5.18]{KLQ}.
\begin{prop}\label{smooth_rep_extends} Let $(G, H, \sigma)$ be a reduced $1$-dimensional Hecke triple, and $({G}_\sigma, {H}_\sigma, {\sigma})$ its Schlichting completion. Then a representation of $G$ is unitary $(H, \sigma)$-smooth if and only if it extends to a continuous unitary $({H}_\sigma, {\sigma})$-smooth representation of ${G}_\sigma.$ \end{prop}
\begin{proof} The proof of \cite[Proposition 5.18]{KLQ} carries over to this situation with one modification: we need to show that a unitary $(H, \sigma)$-smooth representation $\pi$ of $G$ is continuous from $G$ with the Hecke topology from $(G, K)$ into ${\cc B}(V)$ with the strong topology. So suppose that $x\to e$ in $G$, and pick $\xi \in V$. By the assumption on smoothness we may take $\xi$ of the form $\pi(y)\eta$ with $\eta\in V_{H, \pi}$. Then eventually $x$ belongs to the neighbourhood $yK y^{-1}$, and $$ \pi(x)\pi(y)\eta=\pi(y)\pi(y^{-1}xy)\eta=\pi(y)\sigma(y^{-1}xy)\eta=\pi(y)\eta, $$ so $\Vert \pi(x)\pi(y)\eta - \pi(y)\eta\Vert\to 0$, proving the desired continuity. \end{proof}
However, the generalisation to Hecke triples of \cite[Corollary 5.10]{KLQ} fails for $\sigma\neq 1$.
\begin{prop} If $\sigma\neq 1$ the trivial representation of $G_\sigma$ is not unitary $(H_\sigma, \sigma)$-smooth, and $p_\sigma$ is not full in $C^*(G_\sigma)$. \end{prop}
\begin{proof} It suffices to note, first, that the integrated form of the trivial representation of ${G}_\sigma$ carries $p_\sigma$ into $0$, and second, that $V_{H_\sigma, \pi}=\pi(p_{\overline{\sigma}})V$ for any representation $\pi$ of $G_\sigma$ on $V$. \end{proof}
We show next how the strategy developed in \cite[\S 5]{KLQ}, based on Fell's imprimitivity bimodules for $*$-algebras, for studying the representations of $\cc H(G, H)$ can be carried over to tie up the representations of $\cc H_\sigma(G, H)$ and the unitary $(H, \sigma)$-smooth representations of $G$. We recall that if ${}_{E}X_{D}$ is an imprimitivity bimodule of $*$-algebras $E$ and $D$ then a representation $\pi$ of $D$ is \emph{positive} with respect to the right inner product $\langle \rangle_R$ provided that $\pi(\langle f, f\rangle_R)\geq 0$ for all $f\in X$.
Let $L$ be a locally compact group, $M$ a compact open subgroup, and $\rho$ a non-trivial character on $M$. Choose a Haar measure on $L$ such that $\rho$ becomes a projection $p_\rho$ in $C_c(L)$. Consider the $*$-algebras $E=C_c(L)p_\rho C_c(L)$, $D=\cc H_\rho(L, M)$, $B=\overline{L^1(L)p_\rho L^1(L)}$ (closure taken in $L^1(L)$) and $C=p_\rho L^1(L)p_\rho$. (Do not confuse this $B$ with the $B$ in \eqref{special_x}.) Then we have an inclusion of bimodules ${}_{E}X_{D}\subset {}_{B}Y_{C}$, where $X:=C_c(L)p_\rho$ and $Y:=L^1(L)p_\rho$ with bimodule operations inherited from $L^1(L)$, and right-inner product $\langle f, g\rangle_R=f^*g$ for $f, g$ in $Y$ and $X$ respectively. We claim that the $C^*$-completions $C^*(E)$ and $C^*(B)$ coincide with $\overline{C^*(L)p_\rho C^*(L)}$. Indeed, the proof of \cite[Theorem 5.7]{KLQ} carries through with the following alterations: given a nondegenerate representation $\pi$ of $E$ on a Hilbert space $V$, the formula $$ \tilde{\pi}(x)\pi(f)\xi:=\pi(xf)\xi \text{ for }x\in L, f\in E, \xi \in V, $$ defines a representation of $L$ on $V$ which is unitary $(M, \rho)$-smooth because $mp_{\overline{\rho}}= \rho(m)p_{\overline{\rho}}$ for $m\in M$ implies that $\pi(p_{\overline{\rho}})V= V_{M, \rho}$. The integrated form of $\tilde{\pi}$ will then be a nondegenerate extension of $\pi$ to $\overline{C^*(L)p_{\overline{\rho}} C^*(L)}$, from which the claim follows. So we obtain the analogue of the category equivalences from \cite[Corollaries 5.12 and 5.20]{KLQ}.
\begin{cor} Let $(G, H,\sigma)$ be a reduced $1$-dimensional Hecke triple. Then there are category equivalences between
\textnormal{a)} the unitary $(H,\sigma)$-smooth representations of $G$ and the $\langle \rangle_R$-positive representations of $\cc H_\sigma(G, H)$,
\textnormal{b)} the nondegenerate representations of $\overline{C^*(G_\sigma)p_\sigma C^*(G_\sigma)}$ and the $\langle \rangle_R$-positive representations of $p_\sigma L^1(G_\sigma)p_\sigma.$ \label{category_equiv} \end{cor}
\section{The case $H\subseteq N\trianglelefteq G$.}\label{H_in_N_normal}
Let $(G, H, \sigma)$ be a reduced $1$-dimensional Hecke triple with Schlichting completion $(G_\sigma, H_\sigma, \sigma)$, and suppose that $H$ is contained in a normal subgroup $N$ of $G$. (This is clearly interesting only if $N\neq G$.) We will show that $I$ defined in \eqref{def_of_I} is a (twisted) crossed product, see \cite{GKP} (and \cite{PaR}) for definitions. Let $N_\sigma$ denote the closure of $N$ in $G_\sigma$, and let $\ad$ be the action of $G_\sigma$ by conjugation on $N_\sigma$ (and on $C^*(N_\sigma)$). The universal covariant representation $(\pi, u)$ of $(C^*(N_\sigma), G_\sigma)$ into the twisted crossed product $C^*(N_\sigma)\rtimes G_\sigma/N_\sigma$ determines an isomorphism $\pi\times u:\pi(b)u(f)\mapsto bf$ of $C^*(N_\sigma)\rtimes G_\sigma/N_\sigma$ onto $C^*(G_\sigma)$. Note that the twist disappears when $G$ is a semi-direct product of $N$ by a group $Q$, see also \cite{LarR}.
\begin{thm} Suppose that $(G, H, \sigma)$ is a reduced $1$-dimensional Hecke triple such that $H$ is contained in a normal subgroup $N$ of $G$. Let $N_\sigma$ denote the closure of $N$ in the Schlichting completion $(G_\sigma, H_\sigma, \sigma)$ of $(G, H, \sigma)$.
\textnormal{(a)} Then $ I_\sigma:=\clspn\{xp_\sigma x^{-1}n\mid x\in G_\sigma, n\in N_\sigma\} $ is an $\ad$-invariant ideal of $C^*({N}_\sigma)$, and the isomorphism $\pi\times u$ carries $I_\sigma\rtimes G_\sigma/N_\sigma$ onto $I$ defined in \eqref{def_of_I}.
\textnormal{(b)} $I_\sigma \rtimes G/N$ is Morita-Rieffel equivalent to $p_\sigma C^*(G_\sigma)p_\sigma$. \label{Hecke_alg_as_cp} \end{thm}
\begin{proof} Since $xH\mapsto xH_\sigma$ is a bijection from $G/H$ onto $G_\sigma/H_\sigma$ (essentially by \cite[Proposition 3.9]{KLQ}), it follows that $$ \spn\{xp_\sigma x^{-1}n\mid x\in G_\sigma, n\in N_\sigma\}= \spn\{xp_\sigma x^{-1}n\mid x\in G, n\in N\}. $$ Note that $xp_\sigma x^{-1}\in C_c(N_\sigma)$, so $I_\sigma \subset C^*(N_\sigma)$. Since $mI_\sigma=I_\sigma m= I_\sigma$ for $m\in N$, we get as in the proof of \cite[Theorem 8.1]{KLQ} that $I_\sigma$ is a closed, $\ad$-invariant ideal of $C^*(N_\sigma)$. Since $xp_\sigma y=xp_\sigma x^{-1}(xp_\sigma y)\subset \spn\{xp_\sigma x^{-1}f\mid f\in C_c(G_\sigma)\}$, it follows that \begin{align*} (\pi\times u)(I_\sigma \rtimes G_\sigma/N_\sigma) &=\clspn\{xp_\sigma x^{-1}nf\mid x\in G, n\in N, f\in C_c(G_\sigma)\}\\ &=\clspn\{xp_\sigma x^{-1}f\mid x\in G, f\in C_c(G_\sigma)\}\\ &=\clspn\{xp_\sigma y\mid x, y\in G\}=I, \end{align*} as claimed in (a).
For (b), it suffices by (a) to establish that $I_\sigma \rtimes G/N\cong I_\sigma \rtimes G_\sigma/N_\sigma$. Since $hp_\sigma=\overline{\sigma(h)} p_\sigma$ for all $h\in H$, the argument in the proof of \cite[Theorem 8.2]{KLQ} shows that the canonical homomorphism $\omega:G\to M(I_\sigma \rtimes G/N)$ is unitary $(H, \sigma)$-smooth. Thus $\omega$ has a continuous extension $\overline{\omega}$ to $G_\sigma$ by Proposition~\ref{smooth_rep_extends}, and then $\overline{\omega}$ forms a covariant pair together with the canonical homomorphism $I_\sigma \to M(I_\sigma \rtimes G/N)$, from which the claimed isomorphism follows. \end{proof}
\begin{cor} With the notation from Theorem~\ref{Hecke_alg_as_cp}, assume that $B$ is a subgroup of $G$ such that $N$ is a normal subgroup of $B$. Then $I_{\sigma, B}=\clspn\{xp_\sigma x^{-1}n\mid x\in B_\sigma, n\in N_\sigma\}$ is an $\ad$-invariant ideal of $C^*(N_\sigma)$, and the closed ideal generated by $p_\sigma$ in $C^*(B_\sigma)$ is Morita-Rieffel equivalent to the twisted crossed products $I_{\sigma, B}\rtimes B_\sigma/N_\sigma$ and $I_{\sigma, B}\rtimes B/N$. \label{B_Hecke_alg_as_cp} \end{cor}
With the hypotheses of Theorem~\ref{Hecke_alg_as_cp}, we now assume that $N$ is abelian, and we consider the Fourier transform $f\mapsto \hat{f}$ from $C^*({N}_\sigma)$ onto $C_0(\widehat{N_\sigma})$. We let \begin{equation} \sigma +H_\sigma^\perp:=\{\alpha\in \widehat{N_\sigma}\mid \alpha\vert_{H_\sigma}={\sigma}\} \label{def_of_H_perp} \end{equation} be the set of all continuous extensions of $\sigma$ to $N_\sigma$. One can verify that $\widehat{p_{\sigma}}=\Chi_{\sigma +H_\sigma^\perp}$. The dual action of $G_\sigma$ on $\widehat{N_\sigma}$ is characterised by $$ \langle n, x \cdot\alpha\rangle=\langle x^{-1} nx, \alpha\rangle \text{ for }n\in N_\sigma, \alpha \in \widehat{N_\sigma}, x\in G_\sigma, $$ and then we have for all $x\in G$ that \begin{align} (xp_\sigma x^{-1})^\wedge(\alpha) &=\int_{N_\sigma}\overline{\langle m ,\alpha\rangle}(xp_\sigma x^{-1})(m)d\mu(m) \notag\\ &=\Delta(x)\int_{N_\sigma}\overline{\langle m, \alpha\rangle}\langle x^{-1}mx, \sigma \rangle\Chi_{xH_\sigma x^{-1}}(m)d\mu(m)\notag\\ &=\Delta(x)\int_{xH_\sigma x^{-1}} \langle m, x\cdot \sigma-\alpha\rangle d\mu(m) \notag\\ &= \Delta(x)\mu(xH_\sigma x^{-1})\Chi_{x\cdot(\sigma +H_\sigma^\perp)}(\alpha)\notag \\ &=\Chi_{x\cdot(\sigma +H_\sigma^\perp)}(\alpha). \label{Ft_of_p_sigmax} \end{align} Therefore $\widehat{I_\sigma}$ is the ideal in $C_0(\widehat{N_\sigma})$ generated by $\{ \Chi_{x\cdot(\sigma +H_\sigma^\perp)}\mid x\in G\}$, so if we let \begin{equation} \Omega_{\sigma}:=\bigcup_{x\in G} {x}\cdot(\sigma +H_\sigma^\perp), \label{def_omega} \end{equation} then we have proved the following specialisation of Theorem~\ref{Hecke_alg_as_cp}:
\begin{thm} With the hypotheses of Theorem~\ref{Hecke_alg_as_cp}, if $N$ is moreover abelian, then $\widehat{I_\sigma}=C_0(\Omega_{\sigma})$. If $\sigma$ is non-trivial, then $0\notin \Omega_{\sigma}$, so $p_\sigma$ is not full. \label{theo_N_abelian} \end{thm}
\begin{rem} If $B$ is a group and $N$ is normal in $B$, then $\widehat{I_{\sigma, B}}$ from Corollary~\ref{B_Hecke_alg_as_cp} equals $C_0(\Omega_{\sigma, B})$, where \begin{equation} \Omega_{\sigma, B}:=\bigcup_{x\in B} x\cdot (\sigma + H_\sigma^\perp). \label{def_omegaB} \end{equation} \end{rem}
\subsection{The case $H\trianglelefteq N\trianglelefteq G$.} \label{H_normal_N_normal} Suppose that $(G, H)$ is a reduced Hecke pair and $N$ is such that $H\trianglelefteq N\trianglelefteq G$. Let $\sigma$ be a finite character of $H$ such that \begin{equation} \sigma(nhn^{-1})=\sigma(h) \text{ for all }n\in N, h\in H, \label{N_in_B} \end{equation} (so $N\subseteq B$ in the notation of \eqref{special_x}). Then $p_\sigma$ is central in $C^*(N_\sigma)$, and so are $xp_\sigma x^{-1}$ for all $x$ in $G_\sigma$. On one hand, this shows that the ideal $I_\sigma$ defined in Theorem~\ref{Hecke_alg_as_cp} will satisfy $I_\sigma=C^*(N_\sigma)p_1$, where $p_1:=\operatorname{sup}\{xp_\sigma x^{-1}\mid x\in G_\sigma\}$.
On the other hand, $xp_\sigma x^{-1}p_\sigma$ is a projection, and $p_\sigma xp_\sigma$ is a partial isometry, for all $x\in G_\sigma$. Thus for every $*$-representation $\pi$ of $\cc H_\sigma(G, H)$ and for all $x\in B$ we have $$ \Vert \pi(p_\sigma x p_\sigma)\Vert\leq 1=\Vert p_\sigma x p_\sigma\Vert_1, $$ where the equality is from Lemma~\ref{psigma_x_psigma}. This establishes that
$C^*(\cc H_\sigma(G, H))$ exists and is equal to the enveloping $C^*$-algebra of $l^1(G, H,\sigma)$.
But more is true. We show next that the right inner product $\langle f,g \rangle_R=f^*g$ on $X:=C_c(G_\sigma)p_\sigma$ is positive in the following sense: given $f$ in $X$, there are $g_i$ in $\cc H_\sigma(G_\sigma, H_\sigma)$, $i=1,\dots ,n$, such that $\langle f, f\rangle_R=\sum_{i=1}^n g_i^*g_i$.
\begin{thm} Let $L$ be a locally compact totally disconnected group, $M$ a compact open subgroup, and $\rho$ a non-trivial character on $M$. Suppose that $M$ is normal in a closed normal subgroup $N$ of $L$, and choose a Haar measure on $L$ such that $p(m)=\rho(m)\Chi_M(m)$ becomes a self-adjoint projection in $C_c(L)$. If $$ \rho(nmn^{-1})=\rho(m) \text{ for all }n\in N, m\in M, $$ then $Y:=C_c(L)p$ is a left-$C_c(L)p C_c(L)$ and right-$\cc H_\rho(L, M)$ bimodule of $*$-algebras with positive right inner product. Moreover, $C^*(\cc H_\rho(L, M))=p C^*(L)p.$ \label{C*_completion_normal_subgroups_general} \end{thm}
We have the following consequence of this theorem and of Proposition~\ref{H_sigma_spanned_by_epsilonx}.
\begin{cor}\label{C*_completion_normal_subgroups}
Let $(G, H, \sigma)$ be a reduced $1$-dimensional Hecke triple, $N$ a normal subgroup of $G$ such that $H$ is normal in $N$, and suppose that $\sigma$ satisfies \eqref{N_in_B}. Let $(G_\sigma, H_\sigma, \sigma)$ be the Schlichting completion of $(G, H, \sigma)$. Then $C^*(\cc H_\sigma(G_\sigma, H_\sigma))=p_\sigma C^*(G_\sigma)p_\sigma.$
If $B$ is a group, then $C^*(\cc H_\sigma(G_\sigma, H_\sigma))=p_\sigma C^*(B_\sigma) p_\sigma$. \end{cor}
\begin{proof}[Proof of Theorem~\ref{C*_completion_normal_subgroups_general}] The bimodule $Y$ is spanned by $\{xp\mid x\in L\}$. By the hypothesis on $\rho$, the projection $p$ is central in $C_c(N)$ and hence, by normality of $N$ in $L$, so is $xp x^{-1}$ for every $x\in L$. Then $\{xp x^{-1}\mid x\in L\}$ are commuting projections in $C^*(L)$ and by following verbatim the proof of \cite[Theorem 5.13]{KLQ} we conclude that for any element $f=\sum_1^n c_i x_ip$ in $Y$, the product $f^*f$ is a finite sum of elements $h^*h$ with $h\in p C_c(L)p$. The last claim follows from \cite[Proposition 5.5(iii)]{KLQ} applied to the bimodule $Y$. \end{proof}
\section{The case when $(B, H)$ is directed.}
Let $(G, H,\sigma)$ be a reduced $1$-dimensional
Hecke triple with Schlichting completion $(G_\sigma, H_\sigma, \sigma)$. Let $B$ be the subset of $G$ defined in \eqref{special_x}, and consider its closure $B_\sigma$ from Proposition~\ref{Hx_Kx}.
\begin{lem} If $x\in B$ and $xHx^{-1}\supset H$, then $x^{-1}p_\sigma x\geq p_\sigma$ in $C^*(G_\sigma)$. \label{B_directed} \end{lem}
\begin{proof} It suffices to note that $x^{-1}p_\sigma x=\mu(x^{-1}H_\sigma x)^{-1}\sigma \vert_{x^{-1}H_\sigma x}.$ \end{proof}
Assume that $B$ is a group. Consider the semigroup $B^{+}:=\{x\in B\mid xHx^{-1}\supset H\}$, and recall from \cite[Definition 6.1]{KLQ} that $(B, H)$ is called directed if $B^{+}$ is an Ore semigroup in $B$, i.e. $B=\{x^{-1}y\mid x,y \in B^{+}\}$. In general, $(B, H)$ need not be directed when $(G, H)$ is.
For $x,y$ in $B$ with $yx\in B^{+}$ we have $x^{-1}y^{-1}p_\sigma yx\geq p_\sigma$ by Lemma~\ref{B_directed}, so $p_\sigma yxp_\sigma=yxp_\sigma$, and the proof of \cite[Theorem 6.4]{KLQ} can be used here to give that the bimodule $C_c(B_\sigma)p_\sigma$ has positive right inner product. Therefore \cite[Proposition 5.5 (iii)]{KLQ} gives the following.
\begin{prop} Let $(G, H, \sigma)$ be a reduced $1$-dimensional
Hecke triple such that $B$ is a subgroup of $G$ and $(B, H)$ is directed. Then $ C^*(\cc H_\sigma(G, H))=C^*(\cc H_\sigma(B, H))=p_\sigma C^*(B_\sigma)p_\sigma. $ \label{C*_completion_B_directed} \end{prop}
In fact we have a precise description of the ideal in $C^*(B_\sigma)$ generated by $p_\sigma$.
\begin{thm} Let $(G, H, \sigma)$ be a reduced $1$-dimensional
Hecke triple such that $B$ is a subgroup of $G$ and $(B, H)$ is directed. Denote by $H_\infty$ the subgroup $\bigcap_{x\in B^+} x^{-1} H_{\sigma}x$ of $H_\sigma$, and let $\mu_\infty$ be normalised Haar measure on $H_\infty$. Then the closed ideal generated by $p_\sigma$ in $C^*(B_\sigma)$ is equal to $C^*(B_\sigma)p_{\sigma,\infty}$, where \begin{equation} p_{\sigma,\infty}=\int_{H_\infty}{\sigma(h)}hd\mu_\infty(h). \label{def_of_p1} \end{equation} \label{theo_p1_B_directed} \end{thm}
To prove this theorem we need a general lemma.
\begin{lem} Suppose that $G$ is a locally compact group, $H$ a compact open subgroup, $\{H_i\}_{i\in I}$ a family of open subgroups of $H$ over a directed set $I$ with $H_j\subset H_i$ for $i<j$, and $\sigma$ a character of $H$ with finite range. Denote $H_\infty=\cap_{i\in I}H_i$, let $\mu_i$ be normalised Haar measure on $H_i$ for $i\in I\cup \{\infty\}$, and let $$ p_{\sigma,i}=\int_{H_i}{\sigma(h)}hd\mu_i(h) $$ for $i\in I\cup \{\infty\}$. Then $p_{\sigma,i} \rightarrow p_{\sigma,\infty}$ in $M(C^*(G))$. \label{last_small_lemma} \end{lem}
\begin{proof} Let $K_i=H_i\cap \ker\sigma$ and $\nu_i$ be normalised Haar measure on $K_i$ for $i\in I\cup \{\infty\}$. Define $q_{i}=\int_{K_i}hd\nu_i(h)$ for $i\in I\cup \{\infty\}$.
We claim first that $q_{i} \rightarrow q_{\infty}$ in $M(C^*(G))$. Given $a\in C^*(G)$ and $\epsilon>0$, take $b=q_{\infty}a$ and let $
U=\{x\in G\mid \|xb-b\| <\epsilon\}$. Then $U$ is open and contains $K_\infty$. We want to show that $K_i\subset U$ eventually. If not, for every $i\in I$ there is $k_i\in K_i\setminus U$, and then by compactness of $H$ there is a subnet $(k_i)$ converging to an element $k\in H\setminus U$. For given $i$, the set $K_ik$ is open and contains $k$, and thus there is $j>i$ such that $k_j\in K_ik$. Therefore $k\in K_ik_j\subset K_iK_j=K_i$, and it follows that $k\in \cap K_i=K_\infty$, contradicting $k\notin U$. Since $q_iq_\infty =q_i$ we have \[ q_ia-q_\infty a =q_ib-b=\int_{K_i} (hb-b)\,d\nu_i(h). \] But $K_i\subset U$ eventually, and so
$\| q_ia-q_\infty a\|<\epsilon$ for sufficiently large $i$, proving the claim.
We next claim that $p_{\sigma,i}=q_{i} p_{\sigma,\infty}$ for sufficiently large $i$. Indeed, since the finite sets $\sigma(H_i)$ form a decreasing family, there is $i_0\in I$ such that $\sigma(H_i)=\sigma(H_{i_0})$ for $i>i_0$. Then $\sigma(H_\infty)=\cap_j \sigma(H_{j})=\sigma(H_{i_0})=\sigma(H_{i})$ for $i>i_0$, and hence $H_i=K_iH_\infty$ $i>i_0$. Then for each $i>i_0$, the Haar measure $\mu_i$ on $H_i$ is given by \[ \int_{H_i}f(h)\, d\mu_i(h) =\int_{K_i}\int_{H_\infty} f(kl) \, d\nu_i(k)\, d\mu_\infty(l), \] and the claim follows because $$ p_{\sigma,i} =\int_{H_\infty}\int_{K_i} k{\sigma(l)}l\, d\nu_i(k)\, d\mu_\infty(l)
=q_{i} p_{\sigma,\infty}.
$$ Using the two claims we conclude the proof of the lemma by observing that $$ p_{\sigma,i}=q_{i} p_{\sigma,\infty}\rightarrow q_\infty p_{\sigma,\infty}=p_{\sigma,\infty}\text{ in } M(C^*(G)). $$ \end{proof}
\begin{proof}[Proof of Theorem~\ref{theo_p1_B_directed}.] Since $(B,H)$ is directed, Lemma~\ref{B_directed} shows that $x^{-1}p_\sigma x\geq p_\sigma$ for $x\in B^{+}$. The projections $p_{\sigma, x}:=x^{-1} p_\sigma x$ have support in $x^{-1}H_\sigma x$, and then Lemma~\ref{last_small_lemma} implies that $p_{\sigma, x}\nearrow p_{\sigma, \infty}$ in $M(C^*(B_\sigma))$. Hence the ideal generated by $p_\sigma$ in $C^*(B_\sigma)$ is $C^*(B_\sigma)p_{\sigma, \infty}$. \end{proof}
Suppose in addition that $H$ is contained in an abelian subgroup $N$ which is normal in $B$. Then $x^{-1}p_\sigma x$ is a projection in $C^*(N_\sigma)$ for all $x\in B^+$, and the Fourier transform applied to both sides of the inequality $x^{-1}p_\sigma x\geq p_\sigma$ gives $\widehat{p_{\sigma, x^{-1}}}\geq \widehat{p_\sigma}$. Hence \eqref{Ft_of_p_sigmax} implies that $x\cdot(\sigma +H_\sigma^\perp)\subset \sigma +H_\sigma^\perp$ for $x\in B^+$. Since $\widehat{p_{\sigma, x^{-1}}}$ converges in $C_0(\widehat{N_\sigma})$ to $\widehat{p_{\sigma,\infty}}$, and since $\widehat{p_{\sigma,\infty}}$ equals the characteristic function of the set $$ \sigma +H_\infty^\perp:=\{\alpha\in \widehat{N_\sigma}\mid \alpha\vert_{H_\infty}= \sigma\vert_{H_\infty}\}, $$ we obtain the following strengthening of Corollary~\ref{B_Hecke_alg_as_cp} and \eqref{def_omegaB}:
\begin{cor} With the notation above, the Hecke algebra $C^*(\cc H_\sigma(G,H))$ is Morita-Rieffel equivalent to $$ \overline{C^*(B_\sigma) p_\sigma C^*(B_\sigma)}\cong C^*(N_\sigma)p_{\sigma, \infty} \rtimes B/N\cong C_0(\sigma +H_\infty^\perp)\rtimes B/N. $$ \label{B_directed_N_abelian} \end{cor}
\section{Applications}\label{applications}
All the examples have $\dim\sigma=1$ and we begin this section by analysing a simple situation where the generalised Hecke algebra of a reduced Hecke triple $(G, H, \sigma)$ does not depend on $\sigma$.
\begin{prop}\label{sigma_extends} Let $(G, H, \sigma)$ be a $1$-dimensional Hecke triple, and suppose that $\sigma$ extends to a character of $G$. Then the map $$ \Phi(f)(x)=\overline{\sigma(x)}f(x) $$ for $f\in \cc H_\sigma(G, H)$ and $x\in G$, is a $*$-isomorphism of $\cc H_\sigma(G, H)$ onto $\cc H(G, H).$ \end{prop}
\begin{proof} The definition of $\Phi$ implies that $\Phi(f)(hxk)=\Phi(f)(x)$ for $f\in \cc H_\sigma(G, H)$, $x\in G$ and $h, k\in H$, so $\Phi(f)$ is $H$-biinvariant and thus $\Phi$ is well-defined. Using that $\Delta_H=\Delta_K$ shows that $$ \Phi(f)^*(x) =\Delta_H(x^{-1})\overline{\Phi(f)(x^{-1})}=\overline{\sigma(x)} \Delta_K(x^{-1})\overline{f(x^{-1})}=\Phi(f^*)(x), $$ so $\Phi$ is adjoint preserving. Finally, since $\sigma$ is defined everywhere on $G$, a routine verification shows that $\Phi(f\ast g)=\Phi(f)\ast \Phi(g)$ for all $f,g\in \cc H_\sigma(G, H)$, as wanted. \end{proof}
Criteria for when a unitary representation of $H$ extends to a unitary representation of $G$ on the same Hilbert space have been recently analysed in, for example, \cite{anHKR}. However, in the examples where we have been able to extend, it has been straightforward to write down a formula for the extended character.
\begin{ex}\label{example1} Suppose that $H$ and $N$ are subgroups of a group $G$ such that $H$ is finite, $N\trianglelefteq G$, $G=HN$, and $(G, H)$ is reduced. Suppose that $\sigma$ is a character on $H$. Then $\sigma(hn):=\sigma(h)$ for $h\in H$ and $n\in N$ is a well-defined extension to a character on $G$. Hence
$\cc H_\sigma(G, H)$ is isomorphic to $\cc H(G, H)$ by Proposition~\ref{sigma_extends}.
As a concrete example of this set-up we can take
$G$ to be the infinite dihedral group $\bb Z\rtimes_\psi \bb Z_2$ with generators
$a$ for $\bb Z$ and $b$ for $\bb Z_2$, where $\psi_b(a)=a^{-1}$. Alternatively, $G$ has presentation $\langle a,b\mid b^2=1, bab=a^{-1} \rangle$. Let $H=\langle b\rangle\cong \bb Z_2$, $N=\bb Z$, and consider the character $\sigma:H\to \bb T$, $\sigma(b)=-1$. Using either \cite[Example 3.4]{T} or \cite[Example 10.1]{KLQ} we conclude that $\cc H_\sigma(G, H)$ does not have a largest $C^*$-norm because $\cc H(G, H)$ fails to have one. \end{ex}
Suppose that $(G, H, \sigma)$ is a reduced Hecke triple. Let $(G_\sigma, H_\sigma, \sigma)$ and $(G_0, H_0)$ be the Schlichting completions of $(G, H, \sigma)$ and $(G, H)$, respectively. Choose left invariant Haar measures $\mu$ on $G_\sigma$ and $\nu$ on $G_0$, normalised so that $\mu(H_\sigma)=1$ and $\nu(H_0)=1$. Let $p_\sigma$ be the projection defined in Theorem~\ref {fundam_proj}, and $p_0$ the projection $\Chi_{H_0}$ in $C_c(G_0)$. In certain cases we can identify $p_\sigma C^*(G_\sigma)p_\sigma$ and $p_0C^*(G_0)p_0$ inside the same algebra, as shown in the next proposition. In concrete examples, it suffices to verify whether $K\supseteq x_0Hx_0^{-1}$ for some $x_0$ in $G$, because then the continuity hypothesis is automatic.
\begin{prop}\label{Gsigma_is_G0} Given a reduced $1$-dimensional
Hecke triple $(G, H,\sigma)$, suppose that $\sigma$ extends to a character of $G$, and is continuous with respect to the Hecke topology from $(G, H)$. Then $\iota:G_\sigma\to G_0$ is a topological isomorphism, and $\Phi(x)=\overline{\sigma(x)}x$ for $x\in G$ extends to an automorphism of $C^*(G_\sigma)$ which carries $p_\sigma$ into $p_0$. \end{prop}
\begin{proof} The hypothesis implies that $\sigma$ has a continuous extension $\sigma_0$ to $H_0$. By the second part of Theorem~\ref{uniqueness_univ_prop}, the map $\iota$ is a topological isomorphism of $G_\sigma$ onto $G_0$ and of $H_\sigma$ onto $H_0$. Since the modular functions of $G_0$ and $G_\sigma$ coincide on $G$ by Corollary~\ref{modular_functions_same}, the involution is preserved by $\Phi$. \end{proof}
\begin{ex}\label{rational_Heisenberg}
(The rational Heisenberg group.) We analyse now the Hecke pair studied in \cite[Example 10.7]{KLQ}. We use the same notation, so \[ [u,v,w]:= \begin{pmatrix} 1&v&w\\ 0&1&u\\ 0&0&1 \end{pmatrix}, \text{ where }u, v, w\in \bb Q. \]
Then $G=\bigl\{\,[u,v,w] \bigm| u,v\in \bb Q, w\in \bb Q/\bb Z\,\bigr\}$ and
$H=\bigl\{\, [u,v,0] \bigm| u,v\in \bb Z\,\bigr\}$ form a reduced Hecke pair. We let $N$ be the (abelian) subgroup of $G$ with $u, v\in \bb Z$, and then $H\trianglelefteq N \trianglelefteq G$. Fix $s, t$ in $\bb Q$ and let $\sigma$ be the character of $H$ given by \begin{equation} \sigma([m,n,0])=\exp(2\pi i (sm +tn))\text{ for }m, n\in \bb Z. \label{char_on_Heisenberg_subgroup} \end{equation}
Since $[u_1,v_1,w_1][u_2,v_2,w_2]=[u_1+u_2,v_1+v_2,w_1+w_2+v_1u_2]$ in $G$, the equation \eqref{char_on_Heisenberg_subgroup} extends to a character $\sigma$ on $G$ given by $\sigma([u,v,w])=\exp(2\pi i(su+tv))$. Thus $\cc H_\sigma(G, H)$ is isomorphic to $\cc H(G, H)$ by Proposition~\ref{sigma_extends}. We know from \cite[Example 10.7]{KLQ} that the collection of sets $$ H_{x,y}=\{[u,v,0]\mid u\in \bb Z\cap y\bb Z,v\in \bb Z\cap x\bb Z\} $$ forms a neighbourhood base at $e$ when $x,y\in \bb Z\setminus \{0\}$. If we denote by $b$ and $d$ the denominators of $s$ and $t$ respectively, then $$ K=\{[m, n,0]\mid sm+tn\in \bb Z\}\supset H_{d, b}. $$ Thus $\sigma$ is continuous at $e$, hence everywhere, for the Hecke topology from $(G,H)$, and so Proposition~\ref{Gsigma_is_G0} implies that the Schlichting completion $(G_0, H_0)$ of $(G, H)$ is also the Schlichting completion of $(G, H,\sigma)$. Then Corollary~\ref{C*_completion_normal_subgroups} or \cite[Theorem 5.13]{KLQ} imply that $p_0C^*(G_0)p_0$ is the largest $C^*$-completion of $\cc H_\sigma(G, H)$.
For different choices of $\sigma$, the ideals $\widehat{I_\sigma}\rtimes G/N$ from Theorem~\ref{Hecke_alg_as_cp} are all isomorphic to $\widehat{I_0}\rtimes G/N$,
where $I_0$ corresponds to the trivial character $\sigma\equiv 1$. Let $\mathcal{A}_f$ and $\mathcal{Z}$ respectively denote the ring of finite adeles and its compact open subring of integral adeles. To describe the sets $\Omega_\sigma$ defined in \eqref{def_omega}, we recall from \cite{KLQ} that $G_0=\{[u,v,w]\mid u,v\in \mathcal{A}_f, w\in Q/Z\}$, with $N_0$ the subgroup with components $u,v\in \mathcal{Z}$, and hereby with $H_0=\{[u,v,0]\mid u, v\in \mathcal{Z}\}$. Then a computation shows that $\Omega_\sigma=[s,t,0]+\Omega_0$, where $\Omega_0$ corresponding to the trivial character $\sigma$ is described in \cite[Example 10.7]{KLQ}. The isomorphism $\Phi$ is obtained from translation by $[s,t,0]$. \end{ex}
\begin{ex}\label{ax_p_adic} (The $p$-adic $ax+b$-group.) Let $p$ be a prime and denote $$ N:=\bb Z[ p^{-1}] =\{\frac m{p^n}\mid m, n\in \bb Z, n\geq 0\}. $$ Let $G:=N\rtimes\bb Z$ with $m\cdot b=bp^m$ for $m\in \bb Z$ and $b\in N$. Then $G$ and $H:=\bb Z$ form a reduced Hecke pair. Let $q$ be a positive non-zero integer that is co-prime with $p$, and $\sigma$ the character of $H$ given by $\sigma(n)=\operatorname{exp}(\frac {2\pi i n}{q})$. Then $K=\ker \sigma=q\bb Z$, and $(G, K)$ is also reduced. For $g=(x, p^k)\in G$ we have that $gHg^{-1}=p^{-k}\bb Z$, and hence deduce that $\sigma$ is not continuous in the Hecke topology from $(G, H).$ We will study $\cc H_\sigma(G, H)$ using Corollary~\ref{B_directed_N_abelian}.
In order to describe the Schlichting completion of $(G, H, \sigma)$ we recall some facts about ${\mathbf q}$-adic integers and numbers, where $\mathbf{q}$ is a doubly infinite sequence of integers greater than one. We refer to \cite[\S 12.3.35]{P} for details (see also \cite[\S 25.1]{HR}). We are interested in the particular sequence ${\mathbf q}$ in which ${\mathbf q}_0=q$ and ${\mathbf q}_n=p$ for all other $n\in \bb Z$, and we view an element $a$ of $\Omega_{\mathbf q}$ as a formal sum $$ a=\sum_{i=-i_0}^{-1}a_ip^i + a_* +q\sum_{i=0}^\infty a_ip^i $$ with $0\leq a_i<p$ and $0\leq a_*< q$. By denoting $a_{-}:=\sum_{i=-i_0}^{-1}a_ip^i$ and $a_+:=\sum_{i=0}^\infty a_ip^i$ we have \begin{equation}a=a_{-}+a_*+qa_+ \label{formal_a} \end{equation}
with $a_+\in \bb Z_p$. Elements $a$ in $\Omega_{\mathbf q}^0$ are characterised by the condition that
$a_{-}=0$. The first theorem in \cite[\S 12.3.35]{P} says that there is an injective group homomorphism $\phi$ from $\bb Z[p^{-1}]$ into the locally compact, totally disconnected (additive) abelian group $\Omega_{\mathbf q}$, such that $\phi$ has dense range, and restricts to a bijection of $\bb Z$ onto a dense subgroup of the compact, totally disconnected subgroup $\Omega_{\mathbf q}^0$ of $\Omega_{\mathbf q}$.
Multiplication by $p$ (with carry-over) is a continuous action of $\bb Z$ on our $\mathbf q$-adic numbers (because it is clearly continuous on the dense subset of elements $a$ with $a_+$ finite), and so $\phi$ extends to a group homomorphism from $G$ into the locally compact, totally disconnected group $L:=\Omega_{\mathbf q}\rtimes \bb Z$. The range of $\phi$ is still dense, and $M:=\Omega_{\mathbf q}^0$ is compact, open in $L$. If $y\in G$ is such that $\phi(y)\in M$, then $\phi(y)_+$ is finite and $\phi(y)_{-}=0$, and it follows that $\phi^{-1}(M)=H$. The formula $$ \rho(a_* +qa_+):=\operatorname{exp}(2\pi i a_*/q) $$ defines a character of $M$. Note that the kernel $K$ of $\rho$ consists of the formal sums $\{qa_+\mid a_+\in \bb Z_p\}$. Now $(L, K)$ being reduced is the same as $\bigcap_{m\in \bb Z}p^mK=\{0\}$, and this last identity can be verified directly using \eqref{formal_a}. From the definition we have $\rho\circ \phi\vert_H=\sigma$, and Theorem~\ref{uniqueness_univ_prop} gives the following.
\begin{prop} The Schlichting completion of $(G, H, \sigma)$ is $(\Omega_{\mathbf q}\rtimes \bb Z, \Omega_{\mathbf q}^0, \rho)$. \end{prop}
It is also possible to identify $\Omega_{\mathbf q}$ as the topological limit $\varprojlim N/qp^n \bb Z$, where the bonding maps are reductions modulo $qp^n\bb Z$, see for example \cite[Proposition 3.10]{KLQ}; then $\Omega_{\mathbf q}^0$ is the profinite group $\varprojlim \bb Z/qp^n\bb Z$.
The Schlichting completion of $(G, H)$ is $(\bb Q_p\rtimes \bb Z, \bb Z_p)$, where $\bb Q_p$ and $\bb Z_p$ are respectively the $p$-adic numbers and $p$-adic integers. Let $(a,k)\in N_\sigma\rtimes \bb Z$ with $a$ as in \eqref{formal_a}. Then the homomorphism $\iota$ from Remark~\ref{def_of_iota} sends $(a_{-}+a_*+qa_+,k)$ into $(a_{-}+a_*+a_+, k)$, and so is not a topological isomorphism.
\begin{lem}\label{B_for_BS}
Let $n_0$ be the smallest integer $n>0$ such that $p^n\equiv 1\mod q$. Then $B=\bb Z[p^{-1}]\rtimes n_0\bb Z.$ \end{lem}
\begin{proof} It suffices to note that $xHx^{-1}\cap H$ is either $\bb Z$ or of the form $p^n\bb Z$ for $n\geq 0$, in which case $\sigma(p^nh)=\sigma(h)$ is equivalent to $p^n\equiv 1\,\operatorname{mod}\,q$. \end{proof}
Lemma~\ref{B_for_BS} implies that $\{ngn^{-1}H\mid n\in N, g\in H\backslash B/H\}$ contains infinitely many disjoint right cosets, and so we infer the following from \cite[Corollary 1.10]{Bi2} and Theorem~\ref{commutant_is_Hecke_vNalg}:
\begin{cor} $\mathcal{R}(G, H,\sigma)$ is a factor. \end{cor}
Since $B^+=\{x\in B\mid xHx^{-1}\supset H\}= \bb Z[p^{-1}]\rtimes n_0\bb N$, the pair $(B, H)$ is directed. Then $C^*(\cc H_\sigma(G, H))$ is equal to $p_\sigma C^*(B_\sigma) p_\sigma$ by Proposition~\ref{C*_completion_B_directed}, and is Morita-Rieffel equivalent to $C_0(\sigma +H_{\infty}^\perp)\rtimes B/N$ by Corollary~\ref{B_directed_N_abelian}. Towards describing the last crossed product we dwell a little longer on the structure of $N_\sigma= \Omega_{\mathbf q}$. Note that for $a$ and $b$ as in \eqref{formal_a} with the sums in $a_{-}$ and $b_{-}$ starting from $-i_0$ and $-j_0$ respectively, $\frac 1q (a\cdot b)$ is well-defined as an element of $\bb Q/\bb Z$, because in the product there are only finitely many terms not in $\bb Z$. With $e(x):=\operatorname{exp}(2\pi i x)$, we claim that \begin{equation} \langle a,b\rangle=e(\frac 1q (a\cdot b))\text{ for }a,b\in N_\sigma \label{duality_qadics} \end{equation} is a well-defined duality\footnote{We can also appeal to the second theorem in \cite[\S 12.3.35]{P}, which shows that $\langle a,b\rangle=\operatorname{exp}\Big[2\pi i \sum_{n=-M}^N b_n \Big(\sum_{m=n}^N \frac {a_m}{{\mathbf q}_n\dots{\mathbf q}_m }\Big)\Big]$ implements a self-duality of $N_\sigma$. The third theorem in \cite[\S 12.3.35]{P} gives necessary and sufficient conditions on the ${\mathbf q}$-numbers that admit a multiplication, and our choice of ${\mathbf q}$ certainly fulfills those conditions.} pairing. To prove this claim is essentially an argument similar to the proof of the second theorem in \cite[\S 12.3.35]{P}, and we leave the details to the reader.
One can check from \eqref{duality_qadics} that the annihilator of $\Omega_{\mathbf q}^0$ can be identified as the set of sequences $\{q\sum_0^\infty a_ip^i\}$ with $0\leq a_i <p$, i.e. as $q\bb Z_p$. Since $\sigma(x)=e(\frac 1q x)$, the set defined in \eqref {def_of_H_perp} of all extensions to $N_\sigma$ is equal to $$ \sigma + H_\sigma^\perp=\{a\in N_\sigma\mid \langle a,b\rangle=\sigma(b), \forall b\in \Omega_{\mathbf q}^0\}=1+q\bb Z_p. $$
Since $q$ is a unit in $\bb Z_p$, the element $w_0:=-q^{-1}$ belongs to $\bb Z_p$. We let \begin{equation} z_0:=1+q w_0 \in \Omega_{\mathbf q}^0\setminus\{0\}. \label{def_of_z0} \end{equation}
\begin{lem} We have $p^{n_0}z_0=z_0$ and $H_\infty=\{jz_0\mid 0\leq j< q\}$, \emph{cf.} Theorem~\ref{theo_p1_B_directed}. \label{pz0_z0} \end{lem}
\begin{proof} By the choice of $n_0$, for each $k\in \bb N$ there is $s_k\in \bb Z$ such that $p^{kn_0}=1+qs_k$. Since $s_1=q^{-1}(p^{n_0}-1)=(1-p^{n_0})w_0$ in $\bb Z_p$, we have $$ p^{n_0}z_0=p^{n_0}+qp^{n_0}w_0=1+q(s_1+p^{n_0}w_0)=z_0, $$ as claimed. For the second claim, we have by definition that $ H_\infty=\bigcap_{k\in \bb N}p^{kn_0}\Omega_{\mathbf q}^0$, Clearly $z_0\in H_\infty$, and since $qz_0=q(1+qw_0)=0$ we get $jz_0\in H_\infty$ for $0\leq j<q$.
To prove the other inclusion, we claim first that $q\bb Z_p\cap H_\infty= \{0\}$. Indeed, if $a\in \bb Z_p$ and $qa\in H_\infty$, then $p^{-n_0}qa\in \Omega_{\mathbf q}^0$, so there are $0\leq b_*<q$ and $b_+\in \bb Z_p$ such that \[ qa=p^{n_0}(b_*+qb_+)=(1+qs_1)b_* +qp^{n_0}b_+ =b_*+q(s_1b_*+p^{n_0}b_+). \] It follows that $b_*=0$ and $p^{-n_0}a\in \bb Z_p$. By repeating the argument, we conclude that $a\in \cap_{k\geq 0}p^{kn_0}\bb Z_p=\{0\}$. To finish off, suppose that $a=a_*+qa_+$ is in $H_\infty$, where $0\leq a_*<q$ and $a_+\in \bb Z_p$. Then $a-a_*z_0\in q\bb Z_p\cap H_\infty=\{0\}$, and so $a=a_*z_0$, as needed.\end{proof}
\begin{lem} The annihilator $H_\infty^\perp$ of $H_\infty$ in $\Omega_{\mathbf q}$ is equal to $\bigcup_{k\geq 0}p^{-kn_0}(q\bb Z_p)$ and to the set $Y$ of elements $p^{-kn_0}\alpha_- +\alpha_* +q\alpha_+$ such that $k>0$, $0\leq \alpha_-<p^{kn_0}$, $\alpha_-+\alpha_*\in q\bb Z$, and $\alpha_+\in \bb Z_p$. \label{Hsigma_perp} \end{lem}
\begin{proof} Since by Lemma~\ref{pz0_z0} $H_\infty$ is a cyclic group generated by $z_0$, $$ H_\infty^\perp=\{z_0\}^\perp=\{a\in N_\sigma\mid az_0\in q\bb Z_p\}. $$
Given $a$ in $\Omega_{\mathbf q}$, we can write $a=p^{-kn_0}a_-+a_*+qa_+$ for $k>0$, $0\leq a_-<p^{kn_0}$, $0\leq a_*<q$, and $a_+\in \bb Z_p$. By Lemma~\ref{pz0_z0}, $az_0=a(p^{kn_0}z_0)=(p^{kn_0}a)z_0$, and so $az_0$ has form $a_-+a_*+qa'_+$ for $a'_+$ in $\bb Z_p$. Thus $az_0\in q\bb Z_p$ if and only if $a_-+a_*\in q\bb Z$, showing that $\{z_0\}^\perp=Y$.
Since $p^{n_0}z_0=z_0$ and $q\bb Z_p\subset \{z_0\}^\perp$, we also have $\bigcup_{k\geq 0}p^{-kn_0}(q\bb Z_p)\subset \{z_0\}^\perp$. If now $a\in \{z_0\}^\perp$, we have seen that $a_-+a_*\in q\bb Z$, and then a calculation shows that $a\in p^{-kn_0}(q\bb Z_p)$, finishing the proof. \end{proof}
\begin{cor} Let $X_0$ denote the open and closed subset $(1+q\bb Z_p)\setminus p^{n_0}(1+q\bb Z_p)$ of $\Omega_{\mathbf q}^0$. Then the set $\sigma +H_{\infty}^\perp$ from Corollary~\ref{B_directed_N_abelian} is the disjoint union of $p^{n_0}$-invariant sets \begin{equation} \{z_0\}\cup \bigcup_{k\in \bb Z}p^{kn_0}X_0. \label{describe_Hinfty} \end{equation} \label{H_sigma_infty_perp} \end{cor}
\begin{proof} From its definition, the set $\sigma +H_{\infty}^\perp$ is equal to $1+H_\infty^\perp$. Since $$ 1+p^{-kn_0}(q\bb Z_p)=p^{-kn_0}(1+qs_k+q\bb Z_p)=p^{-kn_0}(1+q\bb Z_p), $$ Lemma~\ref{Hsigma_perp} implies that $\sigma +H_{\infty}^\perp= \bigcup_{k\geq 0}p^{-kn_0}(1+q\bb Z_p)$. The inclusion $p^{ln_0}(1+q\bb Z_p)\subset 1+q\bb Z_p$ for all $l\geq 0$ implies that $$ \bigcup_{k\geq 0}p^{-kn_0}(1+q\bb Z_p)=\bigcap_{l\geq 0}p^{ln_0}(1+q\bb Z_p) \cup \bigcup_{k\in \bb Z}p^{kn_0}X_0. $$ To see that this decomposition is exactly \eqref{describe_Hinfty}, we need to verify that in the right hand side the union is over disjoint sets, and that \begin{equation} \bigcap_{l\geq 0}p^{ln_0}(1+q\bb Z_p)=\{z_0\}. \label{z0_intersection} \end{equation} First, the inclusions $p^{kn_0}X_0\subset p^{kn_0}(1+q\bb Z_p)\subset p^{n_0}(1+q\bb Z_p)$ for $k> 0$ show that $p^{kn_0}X_0$ and $X_0$ are disjoint, and since $p^{-kn_0}X_0$ is disjoint from $1+q\bb Z_p$, it is also disjoint from $X_0$. Next, suppose that $z$ is in the left hand side of \eqref{z0_intersection}. Then there is $a_l\in \bb Z_p$ for each $l\geq 0$ such that, with $s_l$ as in the proof of Lemma~\ref{pz0_z0}, we have $$ z=p^{ln_0}(1+qa_l)=1+q(s_l+p^{ln_0}a_l). $$ Since $p^{ln_0}a_l$ is in $p^{ln_0}\bb Z_p$, it converges to $0\in \bb Z_p$ as $l\to \infty$. A computation shows that $s_{l+1}=s_1+s_l+qs_1s_l$ for all $l\geq 0$, so by passing to a subnet we may assume that $s_l$ converges to an element $s_*$ that satisfies $s_*=s_1+s_*+qs_1s_*$. This equation has solution $s_*=-1/q=w_0$ in $\bb Z_p$, and hence $ z=\lim (1+q(s_l+p^{ln_0}a_l))=1+qw_0=z_0$, as claimed. To finish, note that if $p^{kn_0}z_0\in X_0$ for some $k\in \bb Z$, then $z_0\in X_0$, a falsehood. \end{proof}
The main result concerning the structure of the Hecke algebra is the following.
\begin{thm} The generalised Hecke $C^*$-algebra $C^*(\cc H_\sigma(G, H))$ is Morita-Rieffel equivalent to $C(\bb T)\oplus (C(X_0)\otimes \mathcal{K}(l^2(\bb Z))).$ \end{thm}
\begin{proof} By Corollary~\ref{B_directed_N_abelian}, $C^*(\cc H_\sigma(G, H))$ is Morita-Rieffel equivalent to $C_0(\sigma +H_{\infty}^\perp)\rtimes B/N$. Applying Corollary~\ref{H_sigma_infty_perp} gives that $C_0(\sigma +H_{\infty}^\perp)$ is the direct sum of $C(\{z_0\})$ and $C_0(\bigcup_{k\in \bb Z}p^{kn_0}X_0)$. But $\bigcup_{k\in \bb Z}p^{kn_0}X_0$ is homeomorphic to $X_0\times \bb Z$ via a map which is equivariant for the action of $B/N=n_0\bb Z$, and the result follows. \end{proof}
To conclude, we recollect that for $\cc H(G, H)$, \cite[Theorem 1.9]{LaLa0} shows that the universal $C^*$-completion $A:=C^*(\cc H(N\rtimes \bb Z, \bb Z))$ is canonically isomorphic to a semigroup crossed product $C^*(N/\bb Z)\rtimes \bb N$. Then \cite[Theorem 2.1]{BLPR} says that $A$ has an ideal $C(\mathcal{U}(\bb Z_p))\otimes
\mathcal{K}(l^2(\bb N))$ such that the resulting quotient is $C(\bb T)$ (here $\mathcal{U}(\bb Z_p)$ is the group of units in the ring of $p$-adic integers). The Schlichting completion of $(G, H)$ is, as noted already, $(\bb Q_p\rtimes \bb Z,\bb Z_p)$. Then the Morita-Rieffel equivalence implemented by the full projection $\Chi_{\bb Z_p}$ carries the ideal $C(\mathcal{U}(\bb Z_p))\otimes
\mathcal{K}(l^2(\bb N))$ of $A$ to the ideal $C(\mathcal{U}(\bb Z_p))\otimes \mathcal{K}(l^2(\bb Z))$ of $C^*(\bb Q_p\rtimes \bb Z)$.
\end{ex}
\begin{ex}\label{version_Heisenberg} ($p$-adic version of the Heisenberg group.) Suppose that $p$ is a prime and $q,r$ are integers co-prime with $p$ and co-prime with each other. Let \[
G=\bigl\{\,[u,v,w] \bigm| u,v\in \bb Z\lbrack p^{-1}\rbrack, w\in \bb Z\lbrack p^{-1}\rbrack/\bb Z\,\bigr\},\]
$H=\bigl\{\, [m,n,0] \bigm| m,n\in \bb Z\,\bigr\}\cong \bb Z\times \bb Z$, and $\sigma(m, n,0)=\operatorname{exp}(2\pi i (\frac mq+ \frac nr))$. This is a reduced Hecke triple, and $K=\ker \sigma=q\bb Z\times r\bb Z$. It follows as in examples~\ref{rational_Heisenberg} and \ref{ax_p_adic} that the Schlichting completion of $(G, H, \sigma)$ consists of \[
G_\sigma=\bigl\{\,[a,b,w] \bigm| a\in \Omega_{\mathbf q}, b\in \Omega_{\mathbf r},w\in \bb Z\lbrack p^{-1}\rbrack/\bb Z\,\bigr\}, \]
the compact open subgroup $H_\sigma=\bigl\{\,[a,b,0] \bigm| a\in \Omega_{\mathbf q}^0, b\in \Omega_{\mathbf r}^0,\bigr\}$, with the natural extension of $\sigma$ to $H_\sigma$.
Since $\sigma$ extends to a character of $G$, Proposition~\ref{sigma_extends} implies that $\cc H_\sigma(G, H)$ is isomorphic to $\cc H(G, H)$. However, we claim that $\sigma$ is not continuous for the Hecke topology from $(G, H)$. Indeed, one can verify that for $x=[p^{-m},p^{-n},0]$ in $G$, where $m,n\geq 0$, $K\cap xKx^{-1}$ is $p^mq\bb Z\times p^nr\bb Z$. The latter set can contain no $H\cap yHy^{-1}$, which has form $p^k\bb Z\times p^l\bb Z$ for $y=[p^{-k}, p^{-l}, w]$ in $G$. Thus the continuous map $\iota:G_\sigma\to G_0$ is not open. \end{ex}
\begin{ex}\label{ex:full ax+b} (The full $ax+b$-group of $\bb Q$.) Let $N=(\bb Q,+)$ and consider $Q=(\bb Q_+^*,\cdot)$ acting by multiplication $(x,k)\mapsto xk$ for $x\in \bb Q_+^*,\, k\in \bb Q$. Then $G:=N\rtimes Q$ and $H=\bb Z$ form the reduced Hecke pair from \cite{BC}. We can identify $G$ as a matrix group in the form $$ G=\left\{ \begin{pmatrix} 1&b\\ 0&a \end{pmatrix}
\biggm|\, a\in \bb Q_+^*, \, b\in \bb Q\right\}, $$ and then $N$ is the subgroup with $a=1$ in which $H$ is the subgroup with $b\in \bb Z$.
Let $n$ be a non-zero positive integer and $\sigma $ the character of $H$ defined by $\sigma(m)=\exp(2\pi i m/n)$ for $m\in \bb Z$. Taking $x_0=(1, n^{-1})$ in $G$ shows that $K:=\ker \sigma$ contains $x_0Hx_0^{-1}$, and so $\sigma$ is continuous with respect to the Hecke topology from $(G, H)$. It is known, see for example \cite{Laca_dil}, that $(G_0:=\mathcal{A}_f\rtimes Q, H_0:=\mathcal{Z})$ is the Schlichting completion of $(G, H)$, and then Theorem~\ref{uniqueness_univ_prop} implies that $G_\sigma=G_0$ and $H_\sigma=H_0$. As a consequence of the description of $B$ below, we see that
a neighbourhood subbase at $e$ for the Hecke topology from $(B, K)$ consists of the groups $\{mn\bb Z\mid m \equiv \pm 1 \bmod n\}$, and so does not contain the open set $n^2\bb Z$ in $G_0$. It was computed in \cite[Lemma 3.2.3]{Cu1} that $B$ consists of pairs $(r, a/b)$ with $r\in \bb Q, a,b\in \bb N, b>0$, and such that $\text{gcd}(a, b)=1$, and $a-b\in n\bb Z$. We claim (without proof) that the following description of $B$ is valid:
\begin{lem} Let $T_q$ be the subsemigroup $\{m\in \bb N^*\mid m-1\in n\bb N\text{ or } m+1\in n\bb N\}$ of $\bb N^*$. Then $B$ is the subgroup $\bb Q\rtimes T_qT_q^{-1}$ of $G$, and $(B, H)$ is directed. \end{lem}
Here $G_\sigma$ is the same for all $\sigma$ and we want to study the relation between the different ideals $\overline{C^*(G_\sigma)p_\sigma C^*(G_\sigma)}$. By Theorem~\ref{Hecke_alg_as_cp} this is the same as studying the ideals $I_{\sigma}$ in $C^*(N_\sigma)$, or by Theorem~\ref{theo_N_abelian} the sets $\Omega_\sigma$ defined in \eqref{def_omega}. If $\sigma$ corresponds to $n\in \bb N^*$ then, since $N_\sigma=\mathcal{A}_f$ and $\mathcal{A}_f$ is self-dual with duality carrying $\mathcal{Z}$ into $\mathcal{Z}^\perp$, we have that $ \sigma +H_\sigma^\perp=\{a\in \mathcal{A}_f\mid a-\sigma\in \mathcal{Z}^\perp\}=1/n +\mathcal{Z}$. Thus the sets are given by $$ \Omega_n=\bigcup_{t\in Q}t(\frac 1n +\mathcal{Z}). \label{Omega_q_ax+b_group} $$ We can then describe exactly the ideals $C_0(\Omega_n) \rtimes Q$ inside $C_0(\mathcal{A}_f)\rtimes Q$ corresponding to different choices of $\sigma$, and link to the results of \cite{LaR2} and \cite{BLPR}.
\begin{lem} \textnormal{(a)} If $q$ is a prime and $m>0$, then $\Omega_{q^m}=\{x\in \mathcal{A}_f\mid x_q\neq 0\}$. In particular, $\Omega_{q^m}$ is independent of $m$.
\textnormal{(b)} If $n=q_1^{i_1},\cdots ,q_l^{i_l}$ with the $q_j$'s different primes and each $i_j>0$, then $$ \Omega_{n}=\{x\in \mathcal{A}_f\mid x_{q_j}\neq 0,\, j=1,\cdots ,l\}. $$ \label{lemma_Omega} \end{lem}
\begin{proof} We only prove part (b) for $l=2$, as the rest follows by similar arguments. Thus we assume that $n=q^j r^i$ with $q$ and $r$ distinct primes.
The forward inclusion is obvious. To prove the other, we take $x\in \mathcal{A}_f$ with $x_q\neq 0$ and $x_r\neq 0$. We can write $x_q=q^l(y+q^j y_q)$ and $x_r=r^m(w+r^i w_r)$ with $0<y<q^j$, $q$ not dividing $y$, $y_q\in \bb Z_q$, $0<w<r^i$, $r$ not dividing $w$,
and $w_r\in \bb Z_r$. Pick $s,t,c,d\in \bb Z$ such that $sy- tq^j =1$ and $cw-d r^i=1$. By the Chinese Remainder Theorem there is an integer $k$ such that $$ k\equiv sr^m\,\bmod{q^j}\text{ and }k\equiv cq^l\,\bmod{r^i}. $$ Then $ky\equiv r^m\,\bmod{q^j}$ and $kw\equiv q^l\,\bmod{r^i}$, and so there are $y'_q\in \bb Z_q$ and $w'_r\in \bb Z_r$ such that $$ x_q=q^lr^mk^{-1}(1+q^j r^i y'_q)\text{ and }x_r=q^lr^mk^{-1}(1+q^j r^i w'_r). $$ Since $q$ and $r$ are units in $\bb Z_p$ for every prime $p$ different from $q$ and $r$, there is $z_p\in \bb Z_p$ such that $x_p=q^lr^mk^{-1}(1+ q^j r^i z_p)$. Hence with $z:=(z_p)\in \mathcal{Z}$, and replacing (if necessary) $k$ with $-k$, we have $x\in \Omega_{n}$, as claimed. \end{proof} \end{ex}
We can now resume our description of $\cc H_\sigma(G, H)$. When $p$ is a prime, \cite[Proposition 2.5]{LaR2} says that $ J_p:=C_0(\mathcal{A}_f\setminus \{x\in \mathcal{A}_f\mid x_p=0\})\rtimes Q $ is one of the primitive ideals of the dilation $C_0(\mathcal{A}_f)\rtimes Q$ cf. \cite{Laca_dil} of the Hecke $C^*$-algebra of Bost and Connes, see also \cite[\S4]{BLPR}. Suppose that $n$ is a non-zero positive integer and let $S$ be the set of primes in the decomposition of $n$. Using Lemma~\ref{lemma_Omega} shows that $\cc H_\sigma(G, H)$ is Morita-Rieffel equivalent to $$ C_0(\Omega_n)\rtimes Q=C_0(\bigcap_{p\in S} \Omega_p)\rtimes Q=\bigcap_{p\in S}J_p. $$
\begin{ex}\label{ex:lamp} (The lamplighter group, see e.g. \cite{dlH}.) {\rm Suppose that $F$ is a finite abelian group with identity $e$. Let $$ N_{-}=\bigoplus_{-\infty}^0 F,\quad H=\bigoplus_{1}^\infty F, $$ and set $N=N_{-}\oplus H$. The forward shift $\alpha$ on $N$ acts as $\alpha((x_k)_{k\in \bb Z})=(y_k)_{k\in \bb Z}$, with $y_k=x_{k-1}$ for all $k\in \bb Z$.
It is proved in \cite[Lemma 3.1.1]{Cu1} that $(G:=N\rtimes_\alpha \bb Z, H)$ is a Hecke pair, and we note that $(G, H)$ is reduced. Let $H_0$ be the profinite, hence compact, abelian group $\varprojlim_{n\geq 1}F^n$, identified as $\prod_{n=1}^\infty F$, and set $N_0:=N_{-}\oplus H_0$. We regard an element of $N_0$ as a sequence $(x_k)_{k\in \bb Z}$ in $\prod_{k\in Z}F$ such that for some integer $n_0$ we have $x_k=0$ for $k<n_0$. Let $\beta$ be the natural continuous extension of $\alpha$ to $N_0$.
The inclusion of $H$ in $H_0$ is equivariant for the actions of $\bb Z$, and so gives rise to a homomorphism $\phi:N\rtimes_\alpha \bb Z\to N_0\rtimes_\beta \bb Z$. By construction, $\phi$ has dense range and $\phi^{-1}(H_0)=H$. Hence \cite[Theorem 3.8]{KLQ} implies that $(G_0:=N_0\rtimes_\beta \bb Z, H_0)$ is the Schlichting completion of $(G, H)$.
The subset $T=\{(y,k)\in G\mid (y,k)H(y,k)^{-1}\supseteq H\}$ is a subsemigroup of $G$, and $T=\{(y,k)\in G\mid k\geq 0\}$. Thus $G=T^{-1}T$, so $(G, H)$ is directed in the sense of \cite[\S 6]{KLQ}. By \cite[Theorems 6.4 and 6.5]{KLQ}, or by \cite[Theorem 1.9]{LaLa0}, the universal $C^*$-completion $C^*(G, H)$ of $\cc H(G, H)$ is isomorphic to the corner in $C^*(G_0)$ determined by the full projection $\Chi_{H_0}$. Moreover, \cite[Corollary 8.3]{KLQ} (or \cite[Theorem 2.5]{LarR}) imply that $C^*(G, H)$ is Morita-Rieffel equivalent to $C_0(\widehat{N_0})\rtimes_{\widehat{\beta}} \bb Z$, where $ \widehat{N_0}=\bigcup_{n\in \bb Z}\widehat{\beta_n}(H_0^\perp).$
We now assume that $\sigma$ is a character of $H$. Thus $\sigma=(\sigma_n)_{n\geq 1}$, where $\sigma_n$ is a character of $F$ for every $n\geq 1$. Note that $\sigma(H)$ is included in the finite set $\{\langle \pi, f\rangle_F\mid \pi\in \widehat{F}, f\in F\}$, where $(f,f')\mapsto \langle f, f'\rangle_F$ is a fixed self-duality of $F$. However, $(G, K)$ will not be directed for arbitrary $\sigma$, and to proceed we specialise further.
When $\sigma$ is not periodic, $B=N$ by \cite{Cu1}. Hence \cite[Proposition 1.5.2]{Cu1} or Lemma~\ref{H_sigma_spanned_by_epsilonx} imply that $\cc H_\sigma(G, H)\cong \mathbb{C}(N/H)$. Therefore $C^*(\cc H_\sigma(G, H))$ is the group algebra $C^*(N_{-})$.
Next we restrict the attention to $1$-periodic characters, so we assume that $\sigma_n=\sigma_1$, $\forall n\geq 2$. We have that $K=\ker \sigma=\{f=(f_k)_k\in H\mid \sigma_1(\sum_k f_k)=1\}$, and $(G, K)$ is reduced.
Let $M$ be the profinite group $\varprojlim_{n\geq 1}(F^n\oplus \sigma(H))$ with bonding homomorphisms $$ (f_1,\dots, f_n, f_{n+1},\sigma(h))\mapsto (f_1, \dots, f_{n}, \sigma(h)) $$ from $F^{n+1}\oplus \sigma(H)$ onto $F^n\oplus \sigma(H)$ and canonical homomorphisms $\pi^n$ from $M$ onto $F^n\oplus \sigma(H)$. By viewing $M$ as the subset of $\prod_{n\geq 1}F^n\oplus \sigma(H)$ of sequences compatible under all bonding maps, we see that for $n\geq 1$ and $h=(h_k)_{k\geq 0}\in H$, the formula $ \pi_n(h):=(h_1, \dots ,h_n, \sigma(h))$ defines an element of $M$. This gives a homomorphism $\pi:H\to M$ such that $ \pi(h):=(\pi_n(h))_{n\geq 1}\text{ for }h\in H$. Since the cylinder sets $\{x\mid \pi^n(x)=\pi_n(h)\}$ form a basis for the topology on $M$, $\pi$ has dense range. Let $L$ be the locally compact group $ L:=(N_{-}\oplus \prod_{n\geq 1}F\oplus \sigma(H))\rtimes_\beta \bb Z, $ with the compact, open subgroup $M$, and define a homomorphism $\phi:G\to L$ and a continuous character $\rho:M\to \bb T$ by $ \phi(y,h,k)=(y, \pi(h),k)$, and $\rho(x, s)=s$ for $(x,s)\in H_0\oplus \sigma(H)$. The map $\phi$ has dense range because $\pi$ does, and clearly $\phi^{-1}(M)=H$. For $h=(h_1,\dots,h_n, e,\dots)$ in $H$ we have $$ \rho\circ \phi(h)=\rho\circ\pi(h)= \rho(h,\sigma(h))=\sigma(h). $$ Hence Theorem~\ref{uniqueness_univ_prop} yields the following result, which includes \cite[Example 4]{Cu0} (or \cite[\S 3.1]{Cu1}).
\begin{cor} The triple $(L, M, \rho)$ is the Schlichting completion $(G_\sigma, H_\sigma, \sigma)$ of $(G, H,\sigma)$. Moreover, the closure of $N$ in the Hecke topology from $(G, K)$ is $$ {N}_\sigma=N_{-}\oplus {H}_\sigma. $$ \label{one_periodic} \end{cor}
So $G_\sigma$ and $G_0$ are different in this example. Since $B=G$ by \cite[\S 3.1]{Cu1} and $H\trianglelefteq N\trianglelefteq G$, Corollary~\ref{C*_completion_normal_subgroups} implies that $p_\sigma C^*(G_\sigma) p_\sigma$ is the enveloping $C^*$-algebra of $\cc H_\sigma(G, H)$. By Theorem~\ref{theo_N_abelian}, this $C^*$-completion is Morita-Rieffel equivalent to $C_0(\Omega_\sigma)\rtimes_{\beta}\bb Z$, where $\Omega_\sigma$ is defined in \eqref{def_omega}. Using results from \cite[\S 3.1]{Cu1}, we will identify this set. The self-duality $(f,f')\mapsto \langle f, f'\rangle_F$ of $F$ implements a self-duality $$ \langle (y_k)_{k\in \bb Z}, (y_k')_{k\in \bb Z}\rangle =\prod_{n\geq 1}\langle y_{1-n},y_n'\rangle_F \prod_{n\geq 1}\langle y_{1-n}', y_n\rangle_F $$ of $N_0=N_{-}\oplus H_0$. Note that under this identification ${H_0}^\perp$ is carried into $H_0$. The group $\sigma(H)$ is also self-dual with duality expressed in terms of a fixed generator $\omega$ as $\langle \omega^j, \omega^k\rangle=\omega^{jk}$ for $j, k\in \bb Z$. Hence ${N}_\sigma=N_0\oplus \sigma(H)$ is self-dual. Under the described duality pairings, an element $(x, \omega^j)$ of ${N}_\sigma$ is in $\sigma +H_\sigma^\perp$ precisely when $j=1$ and $x\in {H_0}^\perp$. Thus $\sigma +H_\sigma^\perp={H}_0\oplus\{\omega\}$.
The action of $\bb Z$ on $N_\sigma$ is $\beta$ on $N_0$ and is trivial on $\sigma(H)$. The formulas in \cite[Lemma 3.1.6]{Cu1} show that the same is true for the dual action $\hat{\beta}$, and then $$ \Omega_\sigma=\bigcup_{n\in \bb Z}\hat{\beta}_n (\sigma +H_\sigma^\perp) =(\bigcup_{n\in \bb Z}\hat{\beta}_n (H_0))\oplus \{\omega\}=N_0\oplus\{\omega\}. $$ Finally, we get an isomorphism $ C_0(\Omega_\sigma)\rtimes_{\hat{\beta}} \bb Z\cong (C_0(N_0)\rtimes_{\hat{\beta}} \bb Z) \oplus C(\bb T)$; in other words, the ideal $\overline{C^*(G_\sigma)p_\sigma C^*(G_\sigma)}$ is isomorphic to $C^*(G_0)\oplus C(\bb T)$. }\end{ex}
\end{document} | arXiv |
Changes in sensory, postural stability and gait functions depending on cognitive decline, and possible markers for detection of cognitive status
Proceedings of the International Conference on Biomedical Engineering Innovation (ICBEI) 2019-2020: medical informatics and decision making (part 2)
Emilija Kostic1 na1,
Kiyoung Kwak2 na1 &
Dongwook Kim ORCID: orcid.org/0000-0003-3293-95412,3
Numerous people never receive a formal dementia diagnosis. This issue can be addressed by early detection systems that utilize alternative forms of classification, such as gait, balance, and sensory function parameters. In the present study, said functions were compared between older adults with healthy cognition, older adults with low executive function, and older adults with cognitive impairment, to determine which parameters can be used to distinguish these groups.
A group of cognitively healthy older men was found to have a significantly greater gait cadence than both the low executive function group (113.1 ± 6.8 vs. 108.0 ± 6.3 steps/min, p = 0.032) and the cognitively impaired group (113.1 ± 6.8 vs. 107.1 ± 7.4 steps/min, p = 0.009). The group with low executive function was found to have more gait stability than the impaired cognition group, represented by the single limb support phase (39.7 ± 1.2 vs. 38.6 ± 1.3%, p = 0.027). Additionally, the healthy cognition group had significantly greater overall postural stability than the impaired cognition group (0.6 ± 0.1 vs. 1.1 ± 0.1, p = 0.003), and the low executive function group had significantly greater mediolateral postural stability than the impaired cognition group (0.2 ± 0.1 vs. 0.6 ± 0.6, p = 0.012). The low executive function group had fewer mistakes on the sentence recognition test than the cognitively impaired (2.2 ± 3.6 vs. 5.9 ± 6.4, p = 0.005). There were no significant differences in visual capacity, however, the low executive function group displayed an overall greatest ability.
Older adults with low executive function showcased a lower walking pace, but their postural stability and sensory functions did not differ from those of the older adults with healthy cognition. The variables concluded as good cognitive status markers were (1) gait cadence for dividing cognitively healthy from the rest and (2) single limb support portion, mediolateral stability index, and the number of mistakes on the sentence recognition test for discerning between the low executive function and cognitive impairment groups.
Cognitive deterioration is a worldwide issue affecting both individual and societal psychological and economic well-being. According to Alzheimer's Disease International, the number of people affected by dementia is expected to increase to 82 million by 2030 and reach 152 million by 2050 due to the aging of the world population [1]. However, this number may be even greater than reported. A recent study reported that 91.4% of older individuals with a cognitive impairment consistent with dementia received no diagnosis relating to dementia [2]. This percentage can be assumed to be even higher for those with mild cognitive impairment (MCI), which has the potential to progress to dementia. This absence of diagnosis leads to the absence of treatment until the disease progresses to severe dementia. A survey carried out by Alzheimer's Society has found that 62% of people felt a dementia diagnosis would mean their life was over, making fear of diagnosis the number one reason for not seeking out professional help [3]. Such a serious issue can be addressed by actualizing systems that assess the cognitive status without the need for traditional cognitive assessments.
In recent years attention has been given to motor and sensory impairments as prodromal markers of cognitive decline. For instance, slowing of the gait, shorter stride length, and relying more on the double limb stance during walking were linked to a higher risk of cognitive impairment [4,5,6,7]. According to Albers [8], the auditory, visual, and vestibular systems, as well as the areas of the central nervous system that are responsible for sensory and motor functions, are impaired by Alzheimer's dementia pathology. Through various studies, it was discovered that auditory [9,10,11] and visual [10,11,12] impairments have a relation to incidence of dementia. The aforementioned research has mostly targeted the differences between cognitively healthy and dementia patients and observed significant differences. However, in order to enable early detection and cognitive maintenance or rehabilitation, differences and similarities in gait, balance, and sensory functions of older adults in several stages of the disease progression need to be assessed.
Therefore, the aim of the present study was (1) to compare gait, postural stability, the auditory and visual ability of healthy older adults, older adults with low executive function, and older adults with cognitive impairment and (2) to assess the discriminative capacity of the aforementioned functions' parameters in determining the cognitive status.
Out of the 72 participants, 19 had MoCA scores compatible with the presence of cognitive impairment and were classified as impaired cognition (IC), and among the remaining 53, 19 who failed to complete the TMT-KL test were classified as low executive function (LEF), and 34 were classified as healthy cognition (HC). Among the IC, no participant completed the TMT-KL. The demographics of the 3 groups are presented in Table 1. There were no statistically significant differences observed.
Table 1 Descriptive characteristics of the participants
Cognitive testing
In terms of cognitive ability, a significant difference (p < 0.001) was observed, and post hoc analysis revealed that HC and LEF groups did not differ between each other but were significantly greater than IC, both with a p value of < 0.001 (Fig. 1).
Montreal cognitive assessment (MoCA) score distributions for impaired cognition (IC) group, low executive function (LEF) group, and healthy cognition (HC) group; *p < 0.05
Level walking
The level walking variables revealed significant differences between the three groups. The groupwise average values and the corresponding p values can be seen in Table 2.
Table 2 Gait function parameters of the healthy cognition group, low executive function group and impaired cognition group
In the case of cadence, stride and stance duration, and swing phase portion, the HC group was found to have significantly different results than both LEF and IC. In the case of velocity, loading response phase, pre swing phase, and the double limb support portion, there were significant differences only between HC and IC. For all these variables, a trend can be observed where the LEF group average value lies between the average values of HC and IC.
The only variable where a significant difference between LEF and IC occurred is the single limb support portion, and in this case, the LEF group displayed the highest average value. Among the variables representing the gait function, to distinguish HC from LEF as well as HC from IC, cadence can be used for its high AUC (CI 95%) of 0.729 (0.586–0.872) and 0.728 (0.588–0.867), respectively. When discerning between LEF and IC, SLSP had the highest AUC (CI 95%) of 0.727 (0.566–0.889).
Postural stability
All the postural stability indexes revealed significant differences between the groups. These differences and the corresponding p values are presented in Table 3.
Table 3 Postural stability indexes of the healthy cognition group, low executive function group and impaired cognition group
In the case of anteroposterior stability index, a significant difference was observed only between HC and IC, while PSI and MLSI displayed significant differences between HC and IC as well as LEF and IC. The overall amount of swaying was the lowest in HC, followed by LEF and finally IC. When discerning between HC and IC the PSI variable displayed the highest AUC (CI 95%) of 0.784 (0.651–0.917), and when discerning between LEF and IC it was MLSI with AUC (CI 95%) = 0.763 (0.612–0.914).
Audiological exam
Table 4 presents the audiological variables' average values and the corresponding p values.
Table 4 Audiological parameters of the healthy cognition group, low executive function group and impaired cognition group
In sentence recognition testing, the LEF group obtained the highest score on average, with the lowest number of mistakes. The average number of mistakes made by LEF was more than two times lower than that of IC resulting in a significant difference between these two groups. The remaining variables revealed no statistically significant differences, however the LEF group had the greatest auditory capacity on average. SRS error variable can be utilized to discern LEF from IC with an AUC (CI 95%) of 0.787 (0.643–0.930). In addition, Spearman bivariate analysis was performed to examine the correlation of PTA512 with the other variables. No significant correlation was found between PTA512 and SRS or SRS error, but there was significant correlation between PTA512 and both WRS and WRS error. Therefore, WRS and WRS error were adjusted for the effect of PTA512 and compared once more between the groups. This analysis found no significance between the groups in regard to WRS and WRS error.
Ophthalmological exam
The ophthalmological examination revealed no significant differences between the three groups. However, the LEF group displayed the highest average ability in the case of best-corrected visual acuity (Fig. 2A), and contrast sensitivity score at 3 m and 1.5 m distances (Fig. 2B, C).
The visual ability of the impaired cognition (IC) group, low executive function (LEF) group, and healthy cognition group (HC): A visual acuity, B contrast sensitivity score at 3 m distance and C contrast sensitivity score at 1.5 m distance
The post-hoc power analysis for α = 0.05 and an effect size of 0.4, which is considered a large effect size in the case of ANOVA [13], resulted in power of approximately 0.85.
Multiple sensory, gait, and balance abilities
To visualize the abilities of each group, a radial graph with five axes was utilized (Fig. 3). These axes represent gait velocity, less relying on double limb support, balance, sentence retention, and cumulative contrast sensitivity score. The double limb support, postural stability, and sentence retention are represented by variables where a lower number indicates greater performance and were therefore inverted for visualization. All the variables were scaled to 100% using the respective maximum value in the given sample. The IC group has shown a low ability in every segment. On the other hand, the HC group shows a greater gait and balance performance, but the auditory and visual ability is greater in the LEF group.
Comparison of multiple sensory and gait ability between older adults with healthy cognition (HC), older adults with low executive function (LEF) and cognitively impaired older adults (IC)
When the cognitive function of the 3 groups was examined, the LEF group displayed a similar MoCA score average as the HC group, despite the lower executive function. Both groups were found to have a significantly higher score than the IC group, showing that the cognitive test score of the LEF group is not a good indicator of executive function deterioration. The demographic variables showed no difference between the three groups, indicating that an adjustment for covariates is not needed when comparing these groups' functions.
When analyzing the gait variables, LEF and IC had mostly similar performance, with the exclusion of the single limb support portion where there was a significant difference between these two groups. The gait ability of HC was found to be significantly greater than that of IC in the case of velocity, cadence, stride duration, stance duration, loading response phase, double limb support phase, and swing phase. Among these variables cadence, stride duration, stance duration, and swing phase portion displayed a significant difference between HC and LEF. These results show that the LEF group has a significantly slower walking pace than HC, similar to that of the IC group, indicating that this group can be considered at risk of cognitive impairment because a slowing of the gait was determined a predecessor to the cognitive deterioration [4, 6, 14]. On the other hand, longer loading response and double limb support phase, as well as the shorter single limb support phase present in IC, are all indicators of poor stability during gait, which is not the case with LEF.
During LR the bodyweight needs to be transferred onto a limb that just completed its forward swing [15]. To enable this, the vestibular system needs to accurately determine which limb position would result in retaining balance. It has been suggested that there is a phase-based vestibular information weighing, and according to Bent [16], vestibular information received at heel contact is held at higher importance when walking. One hypothesis is that a long time of gait cycle spent in LR indicates a slower adjustment of the vestibular system. Such would imply that the IC participants have an impairment of the vestibular system. The LR and PS phases are sub-phases of double limb support, so the increased percentage spent in LR, and PS is accompanied by the increase in double limb support percentage. On the other hand, the single limb support percentage can be regarded as a representative index of support capacity of the affected limb [15]. Therefore, a shorter percentage would indicate less support capacity, or in other words, less stability.
According to the gait function analysis results, participants in the IC group exhibit poorer balance than those in the HC and LEF groups. Postural stability testing confirmed this indication. There were significant differences between HC and IC for all postural stability indexes and there were significant differences between LEF and IC in the case of mediolateral and overall postural stability index. As suggested by earlier studies, this finding brings further confirmation that low postural stability is a sign of cognitive deterioration [17].
The results of pure tone audiometry revealed the highest average ability in LEF. However, apart from the sentence recognition test score and number of mistakes, the auditory variables showed no significant differences between the three groups. The LEF group was shown to have the greatest auditory retention ability. This ability is closely related to attention and perceptual processing, as is visual capacity. Similar to the auditory ability, even though the statistical analysis showed no significant differences between the groups, the LEF group was found to have the greatest visual capacity.
Based on the post-hoc power analysis it can be concluded that the statistical tests that were performed had enough power to detect large effect sizes. Therefore, the aforementioned variables presented large statistically significant differences between the respective groups.
Sensory, postural stability and gait functions of the three groups have shown that some variables have a high discriminative ability. The variables that were found to be good at discerning between groups and could therefore be used in detection systems were (1) cadence during level walking for dividing cognitively healthy older individuals from the rest and (2) single limb support portion, mediolateral stability index, and the number of mistakes on the sentence recognition test for discerning between older adults at risk of cognitive impairment from the ones with cognitive impairment.
The radial graph representing gait, balance, and sensory functions shows that the sensory ability is on average the highest in the LEF group, despite the gait and postural ability being the highest in the case of HC. Therefore, the question is: what does this say about the sensory and cognitive function association? Several hypotheses attempt to explain how the sensory systems and cognitive function are associated, one of which is the information degradation hypothesis. According to this hypothesis, when the sensory periphery is impaired, the degraded sensory input places an increased demand on the processing resources. These resources are considered to be limited in the amount of information that can be held in memory [18]. For example, when the quality of the auditory signal is degraded by environmental noise, or hearing loss, the 'listening effort' needed for processing and comprehending increases. This in turn diverts the limited cognitive resources towards effortful listening [19, 20], leaving no cognitive resources available for other tasks. It has also been suggested that the age-related cognitive changes stem from the age-related changes in sensory processing [21]. In a study done by Karawani [22], two groups of hearing-matched older adults, one that was given a hearing aid for the first time and one without it, were compared after a period of 6 months. At the end of the trial, enhanced working memory performance and increased cortical response were observed in the group with a hearing aid. These findings suggest that sensory restoration can free up available cognitive resources for remembering the spoken conversation. As our results showed, the number of mistakes participants made while recalling sentences was not correlated to the pure tone audiometry results. This could be due to sentence retention requiring more cognitive resources than word retention or listening for a pure tone. Hence, the effort put into listening may be enough for the cognitively impaired to hear the tones or words played for them just as well as the ones with normal cognition, but not enough to accurately remember whole sentences. In regard to vision, untreated poor vision was found to be a contributing factor to dementia in older individuals [23], and patients with dementia used less visual correction, had fewer ophthalmological treatments, and underwent fewer ocular surgeries [24]. Such findings indicate that treating the sensory periphery can aid in stopping or even reversing cognitive decline.
One limitation of this study is the absence of longitudinal data which would follow the course of cognitive function of the participants. However, because in longitudinal studies a percentage of participants drops out, a larger sample is needed to reach meaningful conclusions. Therefore, another limitation is the size of the dataset, as a larger sample size would allow for more precise distinction between groups and even in designing a classification model. Additionally, higher resolution testing may reveal more about the visual ability of the three groups.
A significantly large percentage of older adults that present with a cognitive impairment consistent with dementia never receive a formal medical diagnosis of the condition. This percentage can be assumed to be even higher for people with mild cognitive impairment, which has the potential to progress to dementia. The absence of diagnosis leads to the absence of treatment or possible prevention. Such a serious issue can be addressed by forming systems of detection of cognitive status and determining methods for maintaining or rehabilitating the cognitive function.
The present study assessed the differences in gait, balance, and sensory functions of cognitively healthy older individuals, those who scored above the cutoff points on the cognitive test but have lower executive function, and older adults with cognitive impairment. Participants were divided into 3 groups based on their K-MoCA score and executive function score. The statistical analysis showed that lower executive function coincides with slower walking pace, similar to that of the cognitively impaired. However, despite the slowing of the gait, the group with lower executive function showed greater balance, similar to that of the cognitively healthy. Additionally, this group showed the best average auditory and visual capacity among the 3 groups, with significantly higher auditory retention than the cognitively impaired. It was determined that cognitively healthy older individuals could be discerned from the rest by using the gait cadence variable, due to its high AUC. For discerning older adults with lower executive function from the ones with cognitive impairment, single limb support portion, mediolateral stability index and the number of mistakes on the sentence recognition test can be used as markers.
By utilizing the findings from the present study, detection systems that determine the cognitive status can be actualized to decide whether a patient has healthy cognition, lower executive function or impaired cognition. This would aid clinical practice by allowing clinicians to detect signs of early cognitive deterioration during regular check-ups.
The present study examined cognitive, gait, audiological, ophthalmological, and postural stability functions of 72 healthy community-dwelling Korean men older than 65. The participants completed two sessions of experiments executed on two different days. During the first session, demographic, cognitive, and gait function data were obtained. The demographic variables are as follows: age, height, weight, and years of education. Height and weight were measured during the first visit, age was confirmed from the participants' identification document and years of education were self-reported. During the second session, audiological, ophthalmological, and postural stability functions were measured. All participants provided written informed consent prior to participation. The study received approval from the Jeonbuk national university institutional review board (JBNU IRB File No. 2019-09-015-001).
Cognitive performance of the participants was assessed using the Korean version of the Montreal cognitive assessment (K-MoCA). Participants whose scores were compatible with the presence of cognitive impairment were placed in the impaired cognition (IC) group, using the cutoff points from a normative study [25]. The given cutoff points differ based on age and education and range from 6 to 26 points. The K-MoCA test examines seven cognitive abilities: visuospatial executive function; naming; attention; language; abstraction; delayed recall; and orientation. The first question of the visuospatial executive function section, a modified trail making test with Korean letters (TMT-KL), was used to divide the participants who scored above their respective cutoff points into participants with lower executive function (LEF) and ones with healthy cognition (HC). This test is equivalent to the trail-making test B (TMT-B), except it consists of only 5 numbers and 5 letters, and instead of the letters of the English alphabet, Korean letters are used. In the K-MoCA test, the TMT-KL score is a categorical variable describing whether the participant completed the test without mistakes. The trail making test is useful in evaluating mental flexibility because of the required shifting between numbers and letters [26] and is a measure of executive function, specifically problem solving [27], which has been shown to be impaired in all types of mild cognitive impairment (MCI) [28]. Additionally, a cutoff of one mistake on the TMT-B was found to be a fairly good discriminator between cognitively healthy and cognitively impaired [29]. Participants in said study who had no mistakes on the TMT-B also had significantly higher MMSE scores, indicating a higher cognitive ability. For this reason, the participants whose scores indicate normal cognition but have not completed the TMT-KL can be considered as being at risk of progressing to mild cognitive impairment.
Gait function measurements were obtained through a level walking task. A 10 m long walkway and software for capturing human motion (First Principle, Northern Digital Inc., Canada) were used to capture the participants' gait alongside position sensors (Optotrak Certus, Northern Digital Inc., Canada) and force platforms (Bertec Ltd, USA). Motion module marker guide (MusculoGraphics, Inc., USA), which is the standard method for motion analysis [30], was referenced when placing infrared LED markers (Smart marker, Northern Digital Inc., Canada) on the participants' legs, as shown in Fig. 4. Participants were asked to walk at their most comfortable pace. Software for Interactive Musculoskeletal Modeling (SIMM, Motion Analysis Corp., USA) was used to extract the gait variables for which an ensemble average of three trials was used as the final value. The variables extracted were the spatio-temporal variables, stance, stride, and swing duration and the subdivisions of the gait cycle. All variables were calculated in reference to the left foot heel contact.
Motion capture and analysis system. A Position sensors and walkway, B lower limb marker placement: lateral and anterior view
Postural stability testing
Postural stability measurements were performed using the Balance system SD (Biodex Medical System. Inc., USA). Participants completed the testing both feet firmly on the platform with their eyes open. The variables obtained were the overall postural stability index (PSI), anteroposterior stability index (APSI), and mediolateral stability index (MLSI) which are calculated as follows:
$$PSI=\sqrt{\frac{\sum {\left(0-x\right)}^{2}+\sum {\left(0-y\right)}^{2}}{n}}$$
$$APSI=\sqrt{\frac{\sum {\left(0-y\right)}^{2}}{n}}$$
$$MPSI=\sqrt{\frac{\sum {\left(0-x\right)}^{2}}{n}}$$
where n is the number of samples, and the indexes represent the amount of deviation from the point of origin. The balance system regards the center of the foot platform as the point of origin, making initial calibration a necessity. The participants must stand comfortably on the foot platform with their center of mass positioned above the point of origin before beginning the test.
The equipment used for auditory testing was a Korean speech audiometry test and an audiometer (GSI-61, Grason-Stadler, Denmark). The obtained parameters are as follows: (1) PTA512—a better ear average of the pure tone audiometry scores at 0.5, 1, and 2 kHz; (2) WRS—The word recognition score average; (3) WRS error—Total number of mistakes when recalling words; (4) SRS—The sentence recognition score average and (5) SRS error—Total number of mistakes when recalling sentences.
Ophthalmological examination consisted of visual acuity and contrast sensitivity testing. Testing was performed with or without correction glasses, depending on the participants' usual preference, to accurately assess their day-to-day ability.
Visual acuity was measured for both eyes using the Korean standard 3M vision chart. For contrast sensitivity, Lea Numbers 10M Flip chart (Lea test intl. LLC, Finland) was presented to the participants, first from a 3 m distance and then from a 1.5 m distance. The parameters obtained from the ophthalmological assessment and their respective explanations are as follows: (1) VA—The best-corrected visual acuity; (2) CS3M—Contrast sensitivity score at a 3 m distance and (3) CS15M—Contrast sensitivity score at a 1.5 m distance.
Normality assessment of the variables was performed via the Shapiro–Wilk test. One-way ANOVA with Bonferroni post hoc analysis was performed for parametric data and the Kruskal–Wallis test with Dunn post hoc analysis was used for non-parametric data. For all the variables the area under the receiver operating characteristic curve (AUC) was assessed to determine their discriminative ability and whether they can be used as markers for detecting cognitive status.
All statistical analyses were executed using Statistical Package for the Social Sciences (SPSS) version 20.0.0 for Windows software (IBM Corp, USA). All average values are presented in the format MEAN (SD) and the significance level of all tests was defined as α = 0.05. Additionally, G*Power, a freely available software [31] was used to perform post-hoc power analysis.
The datasets used and/or analyzed to reach the conclusions presented in this manuscript can be found at: https://www.synapse.org/#!Synapse:syn30365140 under the terms and conditions which can be found at the synapse repository.
MCI:
Healthy cognition
Impaired cognition
LEF:
Low executive function
K-MoCA:
Korean Montreal Cognitive Assessment
Loading response
Mid stance
TS:
Terminal stance
Pre-swing
SLS:
Single limb support
DLS:
Double limb support
PTA:
Pure tone audiometry
WRS:
Word recognition score
SRS:
Sentence recognition score
APSI:
Anterior–posterior stability index
MLSI:
Medio-lateral stability index
PSI:
Postural stability index
SPSS:
Statistical Package for the Social Sciences
ROC:
Receiver operating characteristic
AUC:
TMT-KL:
Trail making test with Korean letters
TMT-B:
Trail making test B
S.E.:
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About this supplement
This article has been published as part of BMC Medical Informatics and Decision Making Volume 22 Supplement 5, 2022: Proceedings of the International Conference on Biomedical Engineering Innovation (ICBEI) 2019–2020: medical informatics and decision making (part 2). The full contents of the supplement are available online at https://bmcmedinformdecismak.biomedcentral.com/articles/supplements/volume-22-supplement-5.
This work was supported by the National Research Foundation of Korea (NRF) grants funded by the Korean Government (MSIT)(NRF-2019R1A2C2088033 and NRF-2022R1A2C2012762), and by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2019R1A6A3A01092848). The funding bodies had no role in the design or conclusions of this study. Publication costs are funded by the National Research Foundation of Korea (NRF) grants funded by the Korean Government (MSIT).
Emilija Kostic and Kiyoung Kwak have contributed equally to this work as co-first authors
Department of Healthcare Engineering, The Graduate School, Jeonbuk National University, 567 Baekje-daero, Deokjin-gu, Jeonju-si, Jeollabuk-do, Republic of Korea
Emilija Kostic
Division of Biomedical Engineering, College of Engineering, Jeonbuk National University, 567 Baekje-daero, Deokjin-gu, Jeonju-si, Jeollabuk-do, Republic of Korea
Kiyoung Kwak & Dongwook Kim
Research Center for Healthcare & Welfare Instrument for the Elderly, Jeonbuk National University, 567 Baekje-daero, Deokjin-gu, Jeonju-si, Jeollabuk-do, Republic of Korea
Dongwook Kim
Kiyoung Kwak
Conceptualization: KK and DK; data curation: KK; formal analysis: EK and KK; funding acquisition: KK and DK; investigation: EK and KK; methodology: EK and KK; project administration: KK and DK; resources: KK; supervision: DK; visualization: EK; writing—original draft: EK; writing—review and editing: EK, KK and DK. The manuscript has been approved for submission by all the authors. All authors read and approved the final manuscript.
Correspondence to Dongwook Kim.
This study was approved by the institutional review board of Jeonbuk National University (JBNU IRB File No. 2019-09-015-001). All participants were informed of the study and gave their informed consent. The consent of the patients was obtained in written form according to the protocol and the Patient Information Sheet and Informed Consent. All patients were of legal age, and none were unconscious.
Kostic, E., Kwak, K. & Kim, D. Changes in sensory, postural stability and gait functions depending on cognitive decline, and possible markers for detection of cognitive status. BMC Med Inform Decis Mak 22 (Suppl 5), 252 (2022). https://doi.org/10.1186/s12911-022-01955-x
Mild cognitive impairment (MCI) | CommonCrawl |
Previously I've posted about the lambda calculus and Church numbers. We'd shown how we can encode numbers as functions using the Church encoding, but we'd not really shown how we could do anything with those numbers.
As we still need the parentheses to make sure that the $f$ and $x$ get bundled together. We'll need this convention as we go on as things are going to get a little more parenthesis-heavy.
OK, let's get back to the arithmetic.
How do we get from $three$ to $four$? Well, the difference is that we just need to apply $f$ one more time.
We can encode the idea of applying $f$ one more time into a lambda function. We could call it $add-one$ or $increment$ but lets go with $succ$ for 'successor'.
The $n$ is the number we're adding one to - we need to bind in the values of $f$ and $x$ in to the function because they'll need to have $n$ applied to them before we can apply $f$ in the one extra time.
So the signature of $succ$ - and consequently any unary operation on a number - is $\lambda n.\lambda f.\lambda x$, where $n$ is the number being changed.
Yeah, it's a bit verbose in comparison to the lambda calculus version.2 All those parentheses, while great for being explicit about which functions get applied to what, makes it a bit tough on the eyes.
Let's see if we can define addition.
Where $m$ and $n$ are the numbers being added together. Now all we need to do is work out what comes after the dot.
And this works,4 but we could probably write something both more intuitive and simpler.
What do we want as the result of $add$? We want a function that applies $f$ to $x$ $n$ many times, and the applies $f$ to the result of that $m$ many times.
We can just write that out with the variables we've been given - first apply $f$ to $x$, $n$ many times.
We've used the word 'times' a lot here when talking about the application of $f$ onto $x$s in the above. But now we'll have to deal with real multiplication.
Before you try and reach at an answer, step back a little and ask yourself what the result ought to be, and what the Church arithmetic way of describing it would be.
Say we had the numbers two and three. If I was back in primary school I'd say that the reason that multiplying them together made six was because six was 'two lots of three' or 'three lots of two'.
$two\ f$ is a function that applies $f$ two times to whatever it's next argument is. $three\ (two\ f)$ will apply $two\ f$ to its next argument three times. So it will apply it $3\ \times\ 2$ times - 6 times.
So what could exponentiation be? Well, the first thing we know is that this time, order is going to be important - $2^3$ is not the same as $3^2$.
Next, what does exponentiation mean? I mean, really mean? When we did multiplication we saw us doing 'two lots of (three lots of $f$)'. But now we need to do 'two lots of something' three times. The 'three' part has to apply, not to the number of times we do an $f$, nor the number of times we do '$n$ lots of $f$'. But rather it needs to be the number of times we do $n$ to itself.
So if 'three' is the application of $f$ three times to $x$, we can say that $2^3$ is the application of $two$ three times to $f\ x$.
Another way to look at it: a Church number is already encoding some of the behaviour of exponentiation. When we use inc and 0 as f and x we can think of the number n acting as $inc^n$ - inc done to itself n many times.
This is more explicit if we try it with something other than increment - say double, aka 'times two'. Let's do it in Haskell - but please feel free to pick any language you like.
Four lots of timesTwo is 16; all we need to do is to use the number two instead, and apply the result to an f and an x.
This is because you know the function you're left with after you've applied $n$ to $m$ is a number - will take an $f$ and an $x$ - you don't need to explicitly bind them in the outer function just in order to pass them unchanged to the inner one.
But that's just a nicety. The important thing is… we've finished!
An interesting relationship between the last three: the $f$ moves along to the right as the operation becomes 'bigger'.
Next post we'll be taking a short break from arithmetic to take a look at logic using the lambda calculus.
And I'm speaking as a mad Lisp fan, lover of parens where ever they are.
For functional programming that is.
Get your pencil and paper out if you want to prove it! | CommonCrawl |
\begin{definition}[Definition:Closure (Topology)/Definition 6]
Let $T = \struct {S, \tau}$ be a topological space.
Let $H \subseteq S$.
The '''closure of $H$ (in $T$)''', denoted $H^-$, is the set of all adherent points of $H$.
\end{definition} | ProofWiki |
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