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### results for au:Huang_J in:math

• This note describes sufficient conditions under which total-cost and average-cost Markov decision processes (MDPs) with general state and action spaces, and with weakly continuous transition probabilities, can be reduced to discounted MDPs. For undiscounted problems, these reductions imply the validity of optimality equations and the existence of stationary optimal policies. The reductions also provide methods for computing optimal policies. The results are applied to a capacitated inventory control problem with fixed costs and lost sales.
• We make a convergence analysis of the standard, harmonic, refined and refined harmonic extraction versions of Jacobi-Davidson SVD (JDSVD) type methods for computing an interior singular triplet of a large matrix $A$. At every iteration of these methods, a correction equation, i.e., inner linear system, is solved approximately by using iterative methods, which leads to four inexact JDSVD type methods, as opposed to the exact methods where inner linear systems are solved exactly. Accuracy of inner iterations critically affects the convergence and overall efficiency of the inexact JDSVD methods. A central problem is how accurately the correction equations should be solved so as to ensure that each of the inexact JDSVD methods can mimic its exact counterpart well, that is, they use almost the same outer iterations to achieve the convergence. In this paper, we prove that each inexact JDSVD method behaves like its exact counterpart if all the inner linear systems are solved with $low\ or\ modest$ accuracy during outer iterations. Based on the theory, we propose practical stopping criteria for inner iterations. Numerical experiments confirm our theory.
• Nov 07 2017 math.PR arXiv:1711.01650v1
We consider the stochastic convection-diffusion equation $\partial_t u(t\,,\bf x) =\nu∆u(t\,,\bf x) + V(t\,,x_1)\partial_x_2u(t\,,\bf x),$for $t>0$ and ${\bf x}=(x_1\,,x_2)\in\mathbb{R}^2$, subject to $\theta_0$ being a nice initial profile. Here, the velocity field $V$ is assumed to be centered Gaussian with covariance structure $\textCov[V(t\,,a)\,,V(s\,,b)]= \delta_0(t-s)\rho(a-b)\qquad\textfor all s,t\ge0 and a,b\in\mathbb{R},$where $\rho$ is a continuous and bounded positive-definite function on $\mathbb{R}$. We prove a quite general existence/uniqueness/regularity theorem, together with a probabilistic representation of the solution that represents $u$ as an expectation functional of an exogenous infinite-dimensional Brownian motion. We use that probabilistic representation in order to study the Itô/Walsh solution, when it exists, and relate it to the Stratonovich solution which is shown to exist for all $\nu>0$. Our a priori estimates imply the physically-natural fact that, quite generally, the solution dissipates. In fact, very often, \beginequation P\left{\sup_|x_1|≤m\sup_x_2∈\mathbbR |u(t\,,\bf x)| = O\left(\frac1\sqrt t\right)\qquad\textas $t\to\infty$ \right}=1\qquad\textfor all $m>0$, \endequation and the $O(1/\sqrt t)$ rate is shown to be unimproveable. Our probabilistic representation is malleable enough to allow us to analyze the solution in two physically-relevant regimes: As $t\to\infty$ and as $\nu\to 0$. Among other things, our analysis leads to a "macroscopic multifractal analysis" of the rate of decay in the above equation in terms of the reciprocal of the Prandtl (or Schmidt) number, valid in a number of simple though still physically-relevant cases.
• We establish a sharp asymptotic formula for the number of rational points up to a given height and within a given distance from a hypersurface. Our main innovation is a bootstrap method that relies on the synthesis of Poisson summation, projective duality and the method of stationary phase. This has surprising applications to counting rational points lying on the manifold; indeed, we are able to prove an analogue of Serre's Dimension Growth Conjecture (originally stated for projective varieties) in this general setup. As another consequence of our main counting result, we obtain an optimal Jarník type theorem for simultaneous approximation on hypersurfaces.
• In 1-bit compressive sensing (1-bit CS) where target signal is coded into a binary measurement, one goal is to recover the signal from noisy and quantized samples. Mathematically, the 1-bit CS model reads: $y = \eta \odot\textrm{sign} (\Psi x^* + \epsilon)$, where $x^{*}\in \mathcal{R}^{n}, y\in \mathcal{R}^{m}$, $\Psi \in \mathcal{R}^{m\times n}$, and $\epsilon$ is the random error before quantization and $\eta\in \mathcal{R}^{n}$ is a random vector modeling the sign flips. Due to the presence of nonlinearity, noise and sign flips, it is quite challenging to decode from the 1-bit CS. In this paper, we consider least squares approach under the over-determined and under-determined settings. For $m>n$, we show that, up to a constant $c$, with high probability, the least squares solution $x_{\textrm{ls}}$ approximates $x^*$ with precision $\delta$ as long as $m \geq\widetilde{\mathcal{O}}(\frac{n}{\delta^2})$. For $m< n$, we prove that, up to a constant $c$, with high probability, the $\ell_1$-regularized least-squares solution $x_{\ell_1}$ lies in the ball with center $x^*$ and radius $\delta$ provided that $m \geq \mathcal{O}( \frac{s\log n}{\delta^2})$ and $\|x^*\|_0 := s < m$. We introduce a Newton type method, the so-called primal and dual active set (PDAS) algorithm, to solve the nonsmooth optimization problem. The PDAS possesses the property of one-step convergence. It only requires to solve a small least squares problem on the active set. Therefore, the PDAS is extremely efficient for recovering sparse signals through continuation. We propose a novel regularization parameter selection rule which does not introduce any extra computational overhead. Extensive numerical experiments are presented to illustrate the robustness of our proposed model and the efficiency of our algorithm.
• We describe the structure of quasiflats in two-dimensional Artin groups. We rely on the notion of metric systolicity developed in our previous work. Using this weak form of non-positive curvature and analyzing in details the combinatorics of tilings of the plane we describe precisely the building blocks for quasiflats in all two-dimensional Artin groups - atomic sectors. This allows us to provide useful quasi-isometry invariants for such groups - completions of atomic sectors, stable lines, and the intersection pattern of certain abelian subgroups. These are described combinatorially, in terms of the structure of the graph defining an Artin group. As immediate consequences we present a number of results concerning quasi-isometric rigidity for the subclass of CLTTF Artin groups. We give a necessary and sufficient condition for such groups to be strongly rigid (self quasi-isometries are close to automorphisms), we describe quasi-isometry groups, we indicate when quasi-isometries imply isomorphisms for such groups. In particular, there exist many strongly rigid large-type Artin groups. In contrast, none of the right-angled Artin groups are strongly rigid by a previous work of Bestvina, Kleiner and Sageev.
• The present paper aims at the investigation of the global stability of large solutions to the compressible Navier-Stokes equations in the whole space. Our main results and innovations can be concluded as follows: Under the assumption that the density $\rho(t,x)$ verifies $\rho(0,x)\ge c>0$ and $\sup_{t\ge0}\|\rho(t)\|_{C^\alpha}\le M$ with $\alpha$ sufficiently small, we establish a new mechanism for the convergence of the solution to its associated equilibrium with an explicit decay rate which is as the same as that for the heat equation. The main idea of the proof relies on the basic energy identity, techniques from blow-up criterion and a new estimate for the low frequency part of the solution. We prove the global-in-time stability for the equations, i.e, any perturbed solution will remain close to the reference solution if initially they are close to each other. Our result implies that the set of the smooth and bounded solutions is an open set. Going beyond the close-to-equilibrium setting, we construct the global large solutions to the equations with a class of initial data in $L^p$ type critical spaces. Here the "large solution" means that the vertical component of the velocity could be arbitrarily large initially.
• The French Revolution brought principles of "liberty, equality, and brotherhood" to bear on the day-to-day challenges of governing what was then the largest country in Europe. Its experiments provided a model for future revolutions and democracies across the globe, but this first modern revolution had no model to follow. Using reconstructed transcripts of debates held in the Revolution's first parliament, we present a quantitative analysis of how this system managed innovation. We use information theory to track the creation, transmission, and destruction of patterns of word-use across over 40,000 speeches and more than one thousand speakers. The parliament as a whole was biased toward the adoption of new patterns, but speakers' individual qualities could break these overall trends. Speakers on the left innovated at higher rates while speakers on the right acted, often successfully, to preserve prior patterns. Key players such as Robespierre (on the left) and Abbé Maury (on the right) played information-processing roles emblematic of their politics. Newly-created organizational functions---such as the Assembly's President and committee chairs---had significant effects on debate outcomes, and a distinct transition appears mid-way through the parliament when committees, external to the debate process, gain new powers to "propose and dispose" to the body as a whole. Taken together, these quantitative results align with existing qualitative interpretations but also reveal crucial information-processing dynamics that have hitherto been overlooked. Great orators had the public's attention, but deputies (mostly on the political left) who mastered the committee system gained new powers to shape revolutionary legislation.
• The sequence A000975 in OEIS can be defined by $A_1=1$, $A_{n+1}=2A_n$ if $n$ is odd, and $A_{n+1}=2A_n+1$ if $n$ is even. This sequence satisfies other recurrence relations, admits some closed formulas, and is known to enumerate several interesting families of objects. We provide a new interpretation of this sequence using a binary operation defined by $a\ominus b := -a -b$. We show that the number of distinct results obtained by inserting parentheses in the expression $x_0\ominus x_1\ominus \cdots\ominus x_n$ equals $A_n$, by investigating the leaf depth in binary trees. Our result can be viewed as a quantitative measurement for the nonassociativity of the binary operation $\ominus$.
• We introduce the notion of metrically systolic simplicial complexes. We study geometric and large-scale properties of such complexes and of groups acting on them geometrically. We show that all two-dimensional Artin groups act geometrically on metrically systolic complexes. As direct corollaries we obtain new results on two-dimensional Artin groups and all their finitely presented subgroups: we prove that the Conjugacy Problem is solvable, and that the Dehn function is quadratic. We also show several large-scale features of finitely presented subgroups of two-dimensional Artin groups, lying background for further studies concerning their quasi-isometric rigidity.
• Oct 17 2017 math.GR arXiv:1710.05378v1
Let $\sigma =\{\sigma_{i} | i\in I\}$ be a partition of the set of all primes $\Bbb{P}$ and $G$ a finite group. Let $\sigma (G)=\{\sigma _{i} : \sigma _{i}\cap \pi (G)\ne \emptyset$. A set ${\cal H}$ of subgroups of $G$ is said to be a complete Hall $\sigma$-set of $G$ if every member $\ne 1$ of ${\cal H}$ is a Hall $\sigma _{i}$-subgroup of $G$ for some $i\in I$ and $\cal H$ contains exactly one Hall $\sigma _{i}$-subgroup of $G$ for every $i$ such that $\sigma _{i}\in \sigma (G)$. We say that $G$ is $\sigma$-full if $G$ possesses a complete Hall $\sigma$-set. A complete Hall $\sigma$-set $\cal H$ of $G$ is said to be a $\sigma$-basis of $G$ if every two subgroups $A, B \in\cal H$ are permutable, that is, $AB=BA$. In this paper, we study properties of finite groups having a $\sigma$-basis. In particular, we prove that if $G$ has a a $\sigma$-basis, then $G$ is generalized $\sigma$-soluble, that is, $G$ has a complete Hall $\sigma$-set and for every chief factor $H/K$ of $G$ we have $|\sigma (H/K)|\leq 2$. Moreover, answering to Problem 8.28 in [A.N. Skiba, On some results in the theory of finite partially soluble groups, Commun. Math. Stat., 4(3) (2016), 281--309], we prove the following Theorem A. Suppose that $G$ is $\sigma$-full. Then every complete Hall $\sigma$-set of $G$ forms a $\sigma$-basis of $G$ if and only if $G$ is generalized $\sigma$-soluble and for the automorphism group $G/C_{G}(H/K)$, induced by $G$ on any its chief factor $H/K$, we have either $\sigma (H/K)=\sigma (G/C_{G}(H/K))$ or $\sigma (H/K) =\{\sigma _{i}\}$ and $G/C_{G}(H/K)$ is a $\sigma _{i} \cup \sigma _{j}$-group for some $i\ne j$.
• We consider a class of linear-quadratic-Gaussian mean-field games with a major agent and considerable heterogeneous minor agents in the presence of mean-field interactions. The individual admissible controls are constrained in closed convex subsets $\Gamma_{k}$ of $\mathbb{R}^{m}.$ The decentralized strategies for individual agents and consistency condition system are represented in an unified manner through a class of mean-field forward-backward stochastic differential equations involving projection operators on $\Gamma_{k}$. The well-posedness of consistency system is established in both the local and global cases by the contraction mapping and discounting method respectively. Related $\varepsilon-$Nash equilibrium property is also verified.
• We construct an action of the Hecke algebra $H_n(q)$ on a quotient of the polynomial ring $F[x_1, \dots, x_n]$, where $F = \mathbb{Q}(q)$. The dimension of our quotient ring is the number of $k$-block ordered set partitions of $\{1, 2, \dots, n \}$. This gives a quantum analog of a construction of Haglund-Rhoades-Shimozono and interpolates between their result at $q = 1$ and work of Huang-Rhoades at $q = 0$.
• The goal of this paper is three-fold. Firstly, we prove that the Cauchy problem for generalized KP-I equation \begineqnarray* u_t+|D_x|^\alpha\partial_xu+\partial_x^-1\partial_y^2u+\frac12\partial_x(u^2)=0,\alpha\geq4 \endeqnarray* is locally well-posed in the anisotropic Sobolev spaces$H^{s_{1},\>s_{2}}(\R^{2})$ with $s_{1}>-\frac{\alpha-1}{4}$ and $s_{2}\geq 0$. Secondly, we prove that the problem is globally well-posed in $H^{s_{1},\>0}(\R^{2})$ with $s_{1}>-\frac{(\alpha-1)(3\alpha-4)}{4(5\alpha+3)}$ if $4\leq \alpha \leq5$. Finally, we prove that the problem is globally well-posed in $H^{s_{1},\>0}(\R^{2})$ with $s_{1}>-\frac{\alpha(3\alpha-4)}{4(5\alpha+4)}$ if $\alpha>5$. Our result improves the result of Saut and Tzvetkov (J. Math. Pures Appl. 79(2000), 307-338.) and Li and Xiao (J. Math. Pures Appl. 90(2008), 338-352.).
• We consider the Cauchy problem for a generalized KdV equation \begineqnarray* u_t+\partial_x^3u+u^7u_x=0, \endeqnarray* with random data on \R. Kenig, Ponce, Vega(Comm. Pure Appl. Math.46(1993), 527-620)proved that the problem is globally well-posed in H^s(\R)$with s> s_crit=\frac314, which is the scaling critical regularity indices. Birnir, Kenig, Ponce, Svanstedt, Vega(J. London Math. Soc. 53 (1996), 551-559.) proved that the problem is ill-posed in the sense that the time of existence T and the continuous dependence cannot be expressed in terms of the size of the data in the H^\frac314-norm. In this present paper, we prove that almost sure local in time well-posedness holds in H^s(\R) with s>\frac17112, whose lower bound is below \frac314. The key ingredients are the Wiener randomization of the initial data and probabilistic Strichartz estimates together with some important embedding Theorems. • Some error analysis on virtual element methods including inverse inequalities, norm equivalence, and interpolation error estimates are presented for polygonal meshes which admits a virtual quasi-uniform triangulation. • In this paper, we apply the hierarchical modeling technique and study some numerical linear algebra problems arising from the Brownian dynamics simulations of biomolecular systems where molecules are modeled as ensembles of rigid bodies. Given a rigid body$p$consisting of$n$beads, the$6 \times 3n$transformation matrix$Z$that maps the force on each bead to$p$'s translational and rotational forces (a$6\times 1$vector), and$V$the row space of$Z$, we show how to explicitly construct the$(3n-6) \times 3n$matrix$\tilde{Q}$consisting of$(3n-6)$orthonormal basis vectors of$V^{\perp}$(orthogonal complement of$V$) using only$\mathcal{O}(n \log n)$operations and storage. For applications where only the matrix-vector multiplications$\tilde{Q}{\bf v}$and$\tilde{Q}^T {\bf v}$are needed, we introduce asymptotically optimal$\mathcal{O}(n)$hierarchical algorithms without explicitly forming$\tilde{Q}$. Preliminary numerical results are presented to demonstrate the performance and accuracy of the numerical algorithms. • In this paper, we focus on a data-driven risk-averse multistage stochastic programming (RMSP) model considering distributional robustness. We optimize the RMSP over the worst-case distribution within an ambiguity set of probability distributions constructed directly from historical data samples. The proposed RMSP is intractable due to the multistage nested minimax structure in its objective function, so we reformulate it into a deterministic equivalent that contains a series of convex combination of expectation and conditional value at risk (CVaR), which can be solved by a customized stochastic dual dynamic programming (SDDP) algorithm in this paper. As the size of collected data samples increases to infinity, we show the consistency of the RMSP with distributional robustness to the traditional multistage stochastic programming. In addition, to test the computational performance of our proposed model and algorithm, we conduct numerical experiments for a risk-averse hydrothermal scheduling problem, the results of which demonstrate the effectiveness of our RMSP framework. • In this paper we consider the limit cycles of the planar system $$\fracddt(x,y)=\mathbf X_n+\mathbf X_m,$$ where$\mathbf X_n$and$\mathbf X_m$are quasi-homogeneous vector fields of degree$n$and$m$respectively. We prove that under a new hypothesis, the maximal number of limit cycles of the system is$1$. We also show that our result can be applied to some systems when the previous results are invalid. The proof is based on the investigations for the Abel equation and the generalized-polar equation associated with the system, respectively. Usually these two kinds of equations need to be dealt with separately, and for both equations, an efficient approach to estimate the number of periodic solutions is constructing suitable auxiliary functions. In the present paper we develop a formula on the divergence, which allows us to construct an auxiliary function of one equation with the auxiliary function of the other equation, and vice versa. • We study the$\beta$analogue of the nonintersecting Poisson random walks. We derive a stochastic differential equation of the Stieltjes transform of the empirical measure process, which can be viewed as a dynamical version of the Nekrasov's equation in [7, Section 4]. We find that the empirical measure process converges weakly in the space of cádlág measure-valued processes to a deterministic process, characterized by the quantized free convolution, as introduced in [11]. For suitable initial data, we prove that the rescaled empirical measure process converges weakly in the space of distributions acting on analytic test functions to a Gaussian process. The means and the covariances are universal, and coincide with those of$\beta$-Dyson Brownian motions with the initial data constructed by the Markov-Krein correspondence. Our proof relies on integrable features of the generators of the$\beta$-nonintersecting Poisson random walks, the method of characteristics, and a coupling technique for Poisson random walks. • Let$G$be a finite group. If$M_n < M_{n-1} < \ldots < M_1 < M_{0}=G $where$M_i$is a maximal subgroup of$M_{i-1}$for all$i=1, \ldots ,n$, then$M_n $($n > 0$) is an \emph$n$-maximal subgroup of$G$. A subgroup$M$of$G$is called \emphmodular if the following conditions are held: (i)$\langle X, M \cap Z \rangle=\langle X, M \rangle \cap Z$for all$X \leq G, Z \leq G$such that$X \leq Z$, and (ii)$\langle M, Y \cap Z \rangle=\langle M, Y \rangle \cap Z$for all$Y \leq G, Z \leq G$such that$M \leq Z$. In this paper, we study finite groups whose$n$-maximal subgroups are modular. • Aug 04 2017 math.GR arXiv:1708.00960v1 Under the assumption that a defining graph of a Coxeter group admits only twists in$\mathbb{Z}_2$and is of type FC, we prove Mühlherr's Twist Conjecture. • Inspired by the work of Burq and Tzvetkov (Invent. math. 173(2008), 449-475.), firstly, we construct the local strong solution to the cubic nonlinear wave equation with random data for a large set of initial data in$H^{s}(M)$with$s\geq \frac{5}{14}$, where M is a three dimensional compact manifold with boundary, moreover, our result improves the result of Theorem 2 in (Invent. math. 173(2008), 449-475.); secondly, we construct the local strong solution to the quintic nonlinear wave equation with random data for a large set of initial data in$H^{s}(M)$with$s\geq\frac{1}{6}$, where M is a two dimensional compact boundaryless manifold; finally, we construct the local strong solution to the quintic nonlinear wave equation with random data for a large set of initial data in$H^{s}(M)$with$s\geq \frac{23}{90}$, where M is a two dimensional compact manifold with boundary. • This paper studies the event-triggered cooperative global robust output regulation problem for a class of nonlinear multi-agent systems via a distributed internal model design. We show that our problem can be solved practically in the sense that the ultimate bound of the tracking error can be made arbitrarily small by adjusting a design parameter in the proposed event-triggered mechanism. Our result offers a few new features. First, our control law is robust against both external disturbances and parameter uncertainties, which are allowed to belong to some arbitrarily large prescribed compact sets. Second, the nonlinear functions in our system do not need to satisfy the global Lipchitz condition. Thus our systems are general enough to include some benchmark nonlinear systems that cannot be handled by existing approaches. Finally, our control law is a specific distributed output-based event-triggered control law, which lends itself to a direct digital implementation. • The purpose of this erratum and addendum is to correct the errors in [1]. It consists of five components: 1. Lemma 7.1 and Proposition 7.2 are wrong and discarded; 2. A new proof of existence$\lambda(\xi)$in (7.1) without Proposition 7.2; 3. Definition of a new bijection in Theorem 5.2 and a proof by a new technique; 4. A new proof of Theorem 5.5 based on the new bijection in Theorem 5.2; 5. Correction to the list of exceptional simple pairs in Proposition 3.1. The main results of [1] remain true as stated. We also add a final remark on generalization. • We provide a formula for the generating series of the zeta function$Z(X,t)$of symmetric powers$Sym^n X$of varieties over finite fields. This realizes$Z(X,t)$as an exponentiable motivic measure whose associated Kapranov motivic zeta function takes values in$W(R)$the big Witt ring of$R=W(\mathbb{Z})$. We apply our formula to compute$Z(Sym^n X,t)$in a number of explicit cases. Moreover, we show that all$\lambda$-ring motivic measures have zeta functions which are exponentiable. In this setting, the formula for$Z(X,t)$takes the form of a MacDonald formula for the zeta function. • A close relation between hitting times of the simple random walk on a graph, the Kirchhoff index, resistance-centrality, and related invariants of unicyclic graphs is displayed. Combining with the graph transformations and some other techniques, sharp upper and lower bounds on the cover cost (resp. reverse cover cost) of a vertex in an$n$-vertex unicyclic graph are determined. All the corresponding extremal graphs are identified, respectively. • Jun 20 2017 math.GR arXiv:1706.05473v1 We prove that Artin groups from a class containing all large-type Artin groups are systolic. This provides a concise yet precise description of their geometry. Immediate consequences are new results concerning large-type Artin groups: biautomaticity; existence of$EZ$-boundaries; the Novikov conjecture; descriptions of finitely presented subgroups, of virtually solvable subgroups, and of centralizers for infinite order elements; the Burghelea conjecture and the Bass conjecture; existence of low-dimensional models for classifying spaces for some families of subgroups. • Online estimation of electromechanical oscillation parameters provides essential information to prevent system instability and blackout and helps to identify event categories and locations. We formulate the problem as a state space model and employ the extended Kalman filter to estimate oscillation frequencies and damping factors directly based on data from phasor measurement units. Due to considerations of communication burdens and privacy concerns, a fully distributed algorithm is proposed using diffusion extended Kalman filter. The effectiveness of proposed algorithms is confirmed by both simulated and real data collected during events in State Grid Jiangsu Electric Power Company. • This article discusses a framework to support the design and end-to-end planning of fixed millimeter-wave networks. Compared to traditional techniques, the framework allows an organization to quickly plan a deployment in a cost-effective way. We start by using LiDAR data---basically, a 3D point cloud captured from a city---to estimate potential sites to deploy antennas and whether there is line-of-sight between them. With that data on hand, we use combinatorial optimization techniques to determine the optimal set of locations and how they should communicate with each other, to satisfy engineering (e.g., latency, polarity), design (e.g., reliability) and financial (e.g., total cost of operation) constraints. The primary goal is to connect as many people as possible to the network. Our methodology can be used for strategic planning when an organization is in the process of deciding whether to adopt a millimeter-wave technology or choosing between locations, or for operational planning when conducting a detailed design of the actual network to be deployed in a selected location. • In the paper, we present a high order fast algorithm with almost optimum memory for the Caputo fractional derivative, which can be expressed as a convolution of$u'(t)$with the kernel$(t_n-t)^{-\alpha}$. In the fast algorithm, the interval$[0,t_{n-1}]$is split into nonuniform subintervals. The number of the subintervals is in the order of$\log n$at the$n$-th time step. The fractional kernel function is approximated by a polynomial function of$K$-th degree with a uniform absolute error on each subinterval. We save$K+1$integrals on each subinterval, which can be written as a convolution of$u'(t)$with a polynomial base function. As compared with the direct method, the proposed fast algorithm reduces the storage requirement and computational cost from$O(n)$to$O((K+1)\log n)$at the$n$-th time step. We prove that the convergence rate of the fast algorithm is the same as the direct method even a high order direct method is considered. The convergence rate and efficiency of the fast algorithm are illustrated via several numerical examples. • We study discrete$\beta$-ensembles as introduced in [17]. We obtain rigidity estimates on the particle locations, i.e. with high probability, the particles are close to their classical locations with an optimal error estimate. We prove the edge universality of the discrete$\beta$-ensembles, i.e. for$\beta\geq 1$, the distribution of extreme particles converges to the Tracy-Widom$\beta$distribution. As far as we know, this is the first proof of general Tracy-Widom$\beta$distributions in the discrete setting. A special case of our main results implies that under the Jack deformation of the Plancherel measure, the distribution of the lengths of the first few rows in Young diagrams, converges to the Tracy-Widom$\beta$distribution, which answers an open problem in [39]. Our proof relies on Nekrasov's (or loop) equations, a multiscale analysis and a comparison argument with continuous$\beta$-ensembles. • In this article, we consider the equivariant Schrödinger map from$\Bbb H^2$to$\Bbb S^2$which converges to the north pole of$\Bbb S^2$at the origin and spatial infinity of the hyperbolic space. If the energy of the data is less than$4\pi$, we show that the local existence of Schrödinger map. Furthermore, if the energy of the data sufficiently small, we prove the solutions are global in time. • Consider the stochastic heat equation$\dot{u}=\frac12 u"+\sigma(u)\xi$on$(0\,,\infty)\times\mathbb{R}$subject to$u(0)\equiv1$, where$\sigma:\mathbb{R}\to\mathbb{R}$is a Lipschitz (local) function that does not vanish at$1$, and$\xi$denotes space-time white noise. It is well known that$u$has continuous sample functions; as a result,$\lim_{t\downarrow0}u(t\,,x)= 1$almost surely for every$x\in\mathbb{R}$. The corresponding fluctuations are also known: For every fixed$x\in\mathbb{R}$,$t\mapsto u(t\,,x)$looks locally like a fixed multiple of fractional Brownian motion (fBm) with index$1/4$. In particular, an application of Fubini's theorem implies that, on an$x$-set of full Lebesgue measure, the short-time behavior of the peaks of the random function$t\mapsto u(t\,,x)$are governed by the law of the iterated logarithm for fBm, up to possibly a suitable normalization constant. By contrast, the main result of this paper claims that, on an$x$-set of full Hausdorff dimension, the short-time peaks of$t\mapsto u(t\,,x)$follow a non-iterated logarithm law, and that those peaks contain a rich multifractal structure a.s. Large-time variations of these results were predicted in the physics literature a number of years ago and proved very recently in Khoshnevisan, Kim and Xiao (2016). To the best of our knowledge, the short-time results of the present paper are observed here for the first time. • An oriented graph$G^\sigma$is a digraph without loops or multiple arcs whose underlying graph is$G$. Let$S\left(G^\sigma\right)$be the skew-adjacency matrix of$G^\sigma$and$\alpha(G)$be the independence number of$G$. The rank of$S(G^\sigma)$is called the skew-rank of$G^\sigma$, denoted by$sr(G^\sigma)$. Wong et al. [European J. Combin. 54 (2016) 76-86] studied the relationship between the skew-rank of an oriented graph and the rank of its underlying graph. In this paper, the correlation involving the skew-rank, the independence number, and some other parameters are considered. First we show that$sr(G^\sigma)+2\alpha(G)\geqslant 2|V_G|-2d(G)$, where$|V_G|$is the order of$G$and$d(G)$is the dimension of cycle space of$G$. We also obtain sharp lower bounds for$sr(G^\sigma)+\alpha(G),\, sr(G^\sigma)-\alpha(G)$,$sr(G^\sigma)/\alpha(G)$and characterize all corresponding extremal graphs. • This paper considers an invariant of modules over a finite-dimensional Hopf algebra, called the critical group. This generalizes the critical groups of complex finite group representations studied by Benkart, Klivans, Reiner and Gaetz. A formula is given for the cardinality of the critical group generally, and the critical group for the regular representation is described completely. A key role in the formulas is played by the greatest common divisor of the dimensions of the indecomposable projective representations. • In this paper, we study the cooperative robust output regulation problem for discrete-time linear multi-agent systems with both communication and input delays by distributed internal model approach. We first introduce the distributed internal model for discrete-time multi-agent systems with both communication and input delays. Then, we define so-called auxiliary system and auxiliary augmented system. Finally, we solve our problem by showing, under some standard assumptions, that if a distributed state feedback control or a distributed output feedback control solves the robust output regulation problem of the auxiliary system, then the same control law solves the cooperative robust output regulation problem of the original multi-agent systems. • In this paper, we first present an adaptive distributed observer for a discrete-time leader system. This adaptive distributed observer will provide, to each follower, not only the estimation of the leader's signal, but also the estimation of the leader's system matrix. Then, based on the estimation of the matrix S, we devise a discrete adaptive algorithm to calculate the solution to the regulator equations associated with each follower, and obtain an estimated feedforward control gain. Finally, we solve the cooperative output regulation problem for discrete-time linear multi-agent systems by both state feedback and output feedback adaptive distributed control laws utilizing the adaptive distributed observer. • Shunt FACTS devices, such as, a Static Var Compensator (SVC), are capable of providing local reactive power compensation. They are widely used in the network to reduce the real power loss and improve the voltage profile. This paper proposes a planning model based on mixed integer conic programming (MICP) to optimally allocate SVCs in the transmission network considering load uncertainty. The load uncertainties are represented by a number of scenarios. Reformulation and linearization techniques are utilized to transform the original non-convex model into a convex second order cone programming (SOCP) model. Numerical case studies based on the IEEE 30-bus system demonstrate the effectiveness of the proposed planning model. • In this paper, we study a class of stochastic time-inconsistent linear-quadratic (LQ) control problems with control input constraints. These problems are investigated within the more general framework associated with random coefficients. This paper aims to further develop a new methodology, which fundamentally differs from those in the standard control (without constraints) theory in the literature, to cope with the mathematical difficulties raised due to the presence of input constraints. We first prove that the existence of an equilibrium solution is equivalent to the existence of a solution to some forward-backward stochastic differential equations with constraints. Under convex cone constraint, an explicit solution to equilibrium for mean-variance portfolio selection can be obtained and proved to be unique. Finally, some examples are discussed to shed light on the comparison between our established results and standard control theory. • In this paper, we will introduce a new heterogeneous fast multipole method (H-FMM) for 2-D Helmholtz equation in layered media. To illustrate the main algorithm ideas, we focus on the case of two and three layers in this work. The key compression step in the H-FMM is based on a fact that the multipole expansion for the sources of the free-space Green's function can be used also to compress the far field of the sources of the layered-media or domain Green's function, and a similar result exists for the translation operators for the multipole and local expansions. The mathematical error analysis is shown rigorously by an image representation of the Sommerfeld spectral form of the domain Green's function. As a result, in the H-FMM algorithm, both the "multipole-to-multipole" and "local-to-local" translation operators are the same as those in the free-space case, allowing easy adaptation of existing free-space FMM. All the spatially variant information of the domain Green's function are collected into the "multipole-to-local" translations and therefore the FMM becomes "heterogeneous". The compressed representation further reduces the cost of evaluating the domain Green's function when computing the local direct interactions. Preliminary numerical experiments are presented to demonstrate the efficiency and accuracy of the algorithm with much improved performance over some existing methods for inhomogeneous media. Furthermore, we also show that, due to the equivalence between the complex line image representation and Sommerfeld integral representation of layered media Green's function, the new algorithm can be generalized to multi-layered media with minor modification where details for compression formulas, translation operators, and bookkeeping strategies will be addressed in a subsequent paper. • In this paper, we consider the initial-boundary value problem for the time-dependent Maxwell--Schrödinger equations in the Coulomb gauge. We first prove the global existence of weak solutions to the equations. Next we propose an energy-conserving fully discrete finite element scheme for the system and prove the existence and uniqueness of solutions to the discrete system. The optimal error estimates for the numerical scheme without any time-step restrictions are then derived. Numerical results are provided to support our theoretical analysis. • Let$\mathfrak{g}$be a reductive Lie algebra over$\mathbb{C}$. For any simple weight module of$\mathfrak{g}$with finite-dimensional weight spaces, we show that its Dirac cohomology is vanished unless it is a highest weight module. This completes the calculation of Dirac cohomology for simple weight modules since the Dirac cohomology of simple highest weight modules was carried out in our previous work. We also show that the Dirac index pairing of two weight modules which have infinitesimal characters agrees with their Euler-Poincaré pairing. The analogue of this result for Harish-Chandra modules is a consequence of the Kazhdan's orthogonality conjecture which was settled by the first named author and Binyong Sun. • We investigate the growth of the tallest peaks of random field solutions to the parabolic Anderson models over concentric balls as the radii approach infinity. The noise is white in time and correlated in space. The spatial correlation function is either bounded or non-negative satisfying Dalang's condition. The initial data are Borel measures with compact supports, in particular, include Dirac masses. The results obtained are related to those of Conus, Joseph and Khoshnevisan 2013 and X. Chen 2016, where constant initial data are considered. • Feb 28 2017 math.PR arXiv:1702.08374v1 The main result of this paper is that there are examples of stochastic partial differential equations [hereforth, SPDEs] of the type $$\partial_t u=\frac12∆u +\sigma(u)\eta \qquad\texton (0\,,\infty)\times\mathbb{R}^3$$$ such that the solution exists and is unique as a random field in the sense of Dalang and Walsh, yet the solution has unbounded oscillations in every open neighborhood of every space-time point. We are not aware of the existence of such a construction in spatial dimensions below $3$. En route, it will be proved that there exist a large family of parabolic SPDEs whose moment Lyapunov exponents grow at least sub exponentially in its order parameter in the sense that there exist $A_1,\beta\in(0\,,1)$ such that $\underline\gamma(k) := \liminf_t\to∞t^-1\inf_x∈\mathbbR^3 \log\mathbbE\left(|u(t\,,x)|^k\right) \ge A_1\exp(A_1 k^\beta) \qquad\textfor all k\ge 2.$This sort of "super intermittency" is combined with a local linearization of the solution, and with techniques from Gaussian analysis in order to establish the unbounded oscillations of the sample functions of the solution to our SPDE.
• This paper investigates the task assignment problem for multiple dispersed robots constrained by limited communication range. The robots are initially randomly distributed and need to visit several target locations while minimizing the total travel time. A centralized rendezvous-based algorithm is proposed, under which all the robots first move towards a rendezvous position until communication paths are established between every pair of robots either directly or through intermediate peers, and then one robot is chosen as the leader to make a centralized task assignment for the other robots. Furthermore, we propose a decentralized algorithm based on a single-traveling-salesman tour, which does not require all the robots to be connected through communication. We investigate the variation of the quality of the assignment solutions as the level of information sharing increases and as the communication range grows, respectively. The proposed algorithms are compared with a centralized algorithm with shared global information and a decentralized greedy algorithm respectively. Monte Carlo simulation results show the satisfying performance of the proposed algorithms.
• We show that if a hyperbolic group acts geometrically on a CAT(0) cube complex, then the induced boundary action is hyperfinite. This means that for a cubulated hyperbolic group the natural action on its Gromov boundary is hyperfinite, which generalizes an old result of Dougherty, Jackson and Kechris for the free group case.
• In this paper, we will establish an elliptic local Li-Yau gradient estimate for weak solutions of the heat equation on metric measure spaces with generalized Ricci curvature bounded from below. One of its main applications is a sharp gradient estimate for the logarithm of heat kernels. These results seem new even for smooth Riemannian manifolds.
• This paper considers a class of mean field linear-quadratic-Gaussian (LQG) games with model uncertainty. The drift term in the dynamics of the agents contains a common unknown function. We take a robust optimization approach where a representative agent in the limiting model views the drift uncertainty as an adversarial player. By including the mean field dynamics in an augmented state space, we solve two optimal control problems sequentially, which combined with consistent mean field approximations provides a solution to the robust game. A set of decentralized control strategies is derived by use of forward-backward stochastic differential equations (FBSDE) and shown to be a robust epsilon-Nash equilibrium.
• This paper studies the one-dimensional parabolic Anderson model driven by a Gaussian noise which is white in time and has the covariance of a fractional Brownian motion with Hurst parameter $H \in (\frac{1}{4}, \frac{1}{2})$ in the space variable. We derive the Wiener chaos expansion of the solution and a Feynman-Kac formula for the moments of the solution. These results allow us to establish sharp lower and upper asymptotic bounds for the $n$th moment of the solution.
• We study Dyson Brownian motion with general potential $V$ and for general $\beta \geq 1$. For short times $t = o (1)$ and under suitable conditions on $V$ we obtain a local law and corresponding rigidity estimates on the particle locations; that is, with overwhelming probability, the particles are close to their classical locations with an almost-optimal error estimate. Under the condition that the density of states of the initial data is bounded below and above down to the scale $\eta_* \ll t \ll 1$, we prove a mesoscopic central limit theorem for linear statistics at all scales $N^{-1}\ll\eta\ll t$
• The aim of this short note is to obtain the existence, uniqueness and moment upper bounds of the solution to a stochastic heat equation with measure initial data, without using the iteration method in Chen and Dalang(2015), Chen and Kim(2016).
• An efficient nonlinear multigrid method for a mixed finite element method of the Darcy-Forchheimer model is constructed in this paper. A Peaceman-Rachford type iteration is used as a smoother to decouple the nonlinearity from the divergence constraint. The nonlinear equation can be solved element-wise with a closed formulae. The linear saddle point system for the constraint is reduced into a symmetric positive definite system of Poisson type. Furthermore an empirical choice of the parameter used in the splitting is proposed and the resulting multigrid method is robust to the so-called Forchheimer number which controls the strength of the nonlinearity. By comparing the number of iterations and CPU time of different solvers in several numerical experiments, our multigrid method is shown to convergent with a rate independent of the mesh size and the Forchheimer number and with a nearly linear computational cost.
• Let the symmetric group $\mathfrak{S}_n$ act on the polynomial ring $\mathbb{Q}[\mathbf{x}_n] = \mathbb{Q}[x_1, \dots, x_n]$ by variable permutation. The coinvariant algebra is the graded $\mathfrak{S}_n$-module $R_n := {\mathbb{Q}[\mathbf{x}_n]} / {I_n}$, where $I_n$ is the ideal in $\mathbb{Q}[\mathbf{x}_n]$ generated by invariant polynomials with vanishing constant term. Haglund, Rhoades, and Shimozono introduced a new quotient $R_{n,k}$ of the polynomial ring $\mathbb{Q}[\mathbf{x}_n]$ depending on two positive integers $k \leq n$ which reduces to the classical coinvariant algebra of the symmetric group $\mathfrak{S}_n$ when $k = n$. The quotient $R_{n,k}$ carries the structure of a graded $\mathfrak{S}_n$-module; Haglund et. al. determine its graded isomorphism type and relate it to the Delta Conjecture in the theory of Macdonald polynomials. We introduce and study a related quotient $S_{n,k}$ of $\mathbb{F}[\mathbf{x}_n]$ which carries a graded action of the 0-Hecke algebra $H_n(0)$, where $\mathbb{F}$ is an arbitrary field. We prove 0-Hecke analogs of the results of Haglund, Rhoades, and Shimozono. In the classical case $k = n$, we recover earlier results of Huang concerning the 0-Hecke action on the coinvariant algebra.
• A noteworthy property of many parabolic stochastic PDEs is that they locally linearize (Foondun, Khoshnevisan and Mahboubi (2015), Hairer (2013, 2014), Hairer and Pardoux (2015), Khoshnevisan, Swanson, Xiao and Zhang (2013)). We prove that, by contrast, a large family of stochastic wave equations in dimension one do not possess this important property.
• We study the adjacency matrices of random $d$-regular graphs with large but fixed degree $d$. In the bulk of the spectrum $[-2\sqrt{d-1}+\varepsilon, 2\sqrt{d-1}-\varepsilon]$ down to the optimal spectral scale, we prove that the Green's functions can be approximated by those of certain infinite tree-like (few cycles) graphs that depend only on the local structure of the original graphs. This result implies that the Kesten--McKay law holds for the spectral density down to the smallest scale and the complete delocalization of bulk eigenvectors. Our method is based on estimating the Green's function of the adjacency matrices and a resampling of the boundary edges of large balls in the graphs.
• We prove that the bulk eigenvectors of sparse random matrices, i.e. the adjacency matrices of Erdős-Rényi graphs or random regular graphs, are asymptotically jointly normal, provided the averaged degree increases with the size of the graphs. Our methodology follows [6] by analyzing the eigenvector flow under Dyson Brownian motion, combining with an isotropic local law for Green's function. As an auxiliary result, we prove that for the eigenvector flow of Dyson Brownian motion with general initial data, the eigenvectors are asymptotically jointly normal in the direction $q$ after time $\eta_*\ll t\ll r$, if in a window of size $r$, the initial density of states is bounded below and above down to the scale $\eta_*$, and the initial eigenvectors are delocalized in the direction $q$ down to the scale $\eta_*$.
• In this paper, we study the interactions among interconnected autonomous microgrids, and propose a joint energy trading and scheduling strategy. Each interconnected microgrid not only schedules its local power supply and demand, but also trades energy with other microgrids in a distribution network. Specifically, microgrids with excessive renewable generations can trade with other microgrids in deficit of power supplies for mutual benefits. Since interconnected microgrids operate autonomously, they aim to optimize their own performance and expect to gain benefits through energy trading. We design an incentive mechanism using Nash bargaining theory to encourage proactive energy trading and fair benefit sharing. We solve the bargaining problem by decomposing it into two sequential problems on social cost minimization and trading benefit sharing, respectively. For practical implementation, we propose a decentralized solution method with minimum information exchange overhead. Numerical studies based on realistic data demonstrate that the total cost of the interconnected-microgrids operation can be reduced by up to 13.2% through energy trading, and an individual participating microgrid can achieve up to 29.4% reduction in its cost through energy trading.
• This paper studies the asymptotic properties of the penalized least squares estimator using an adaptive group Lasso penalty for the reduced rank regression. The group Lasso penalty is defined in the way that the regression coefficients corresponding to each predictor are treated as one group. It is shown that under certain regularity conditions, the estimator can achieve the minimax optimal rate of convergence. Moreover, the variable selection consistency can also be achieved, that is, the relevant predictors can be identified with probability approaching one. In the asymptotic theory, the number of response variables, the number of predictors, and the rank number are allowed to grow to infinity with the sample size.
• This paper focuses on the $hp$-version error analysis of a mixed discontinuous Galerkin (DG) method for the linear elasticity problem. We first derive some error estimates for two $L^2$ projection operators in terms of the results in [7,13,23]. Using these estimates and following the techniques in [11], we obtain the $hp$-version error estimates for the solution of the method in energy norm and $L^2$ norm. Finally, we perform several numerical experiments to demonstrate the theoretical results obtained.
• Suppose that $(W,S)$ is a Coxeter system with associated Artin group $A$ and with a simplicial complex $L$ as its nerve. We define the notion of a "standard abelian subgroup" in $A$. The poset of such subgroups in $A$ is parameterized by the poset of simplices in a certain subdivision $L_\oslash$ of $L$. This complex of standard abelian subgroups is used to generalize an earlier result from the case of right-angled Artin groups to case of general Artin groups, by calculating, in many instances, the smallest dimension of a manifold model for $BA$. (This is the "action dimension" of $A$ denoted actdim $A$.) If $H_d(L; \mathbb Z/2)\neq 0$, where $d=\dim L$, then actdim $A \ge 2d+2$. Moreover, when the $K(\pi,1)$-Conjecture holds for $A$, the inequality is an equality.
• We establish the strong comparison principle and strict positivity of solutions to the following nonlinear stochastic heat equation on $\mathbb{R}^d$ $\left(\frac∂∂t -\frac12∆\right) u(t,x) = \rho(u(t,x)) \:\dotM(t,x),$for measure-valued initial data, where $\dot{M}$ is a spatially homogeneous Gaussian noise that is white in time and $\rho$ is Lipschitz continuous. These results are obtained under the condition that $\int_{\mathbb{R}^d}(1+|\xi|^2)^{\alpha-1}\hat{f}(\text{d} \xi)<\infty$ for some $\alpha\in(0,1]$, where $\hat{f}$ is the spectral measure of the noise. The weak comparison principle and nonnegativity of solutions to the same equation are obtained under Dalang's condition, i.e., $\alpha=0$. As some intermediate results, we obtain handy upper bounds for $L^p(\Omega)$-moments of $u(t,x)$ for all $p\ge 2$, and also prove that $u$ is a.s. Hölder continuous with order $\alpha-\epsilon$ in space and $\alpha/2-\epsilon$ in time for any small $\epsilon>0$.
• We consider the linear stochastic heat equation on $\mathbb{R}^\ell$, driven by a Gaussian noise which is colored in time and space. The spatial covariance satisfies general assumptions and includes examples such as the Riesz kernel in any dimension and the covariance of the fractional Brownian motion with Hurst parameter $H\in (\frac 14, \frac 12]$ in dimension one. First we establish the existence of a unique mild solution and we derive a Feynman-Kac formula for its moments using a family of independent Brownian bridges and assuming a general integrability condition on the initial data. In the second part of the paper we compute Lyapunov exponents and lower and upper exponential growth indices in terms of a variational quantity.
• This paper is devoted to the investigation of generalized Abel equation $\dot{x}=S(x,t)=\sum^m_{i=0}a_i(t)x^i$, where $a_i\in\mathrm C^{\infty}([0,1])$. A solution $x(t)$ is called a \em periodic solution if $x(0)=x(1)$. In order to estimate the number of isolated periodic solutions of the equation, we propose a hypothesis (H) which is only concerned with $S(x,t)$ on $m$ straight lines: There exist $m$ real numbers $\lambda_1<\cdots<\lambda_m$ such that either $(-1)^i\cdot S(\lambda_i,t)\geq0$ for $i=1,\cdots,m$, or $(-1)^i\cdot S(\lambda_i,t)\leq0$ for $i=1,\cdots,m$. By means of Lagrange interpolation formula, we proves that the equation has at most $m$ isolated periodic solutions (counted with multiplicities) if hypothesis (H) holds, and the upper bound is sharp. Furthermore, this conclusion is also obtained under some weaker geometric hypotheses. Applying our main result for the trigonometrical generalized Abel equation with coefficients of degree one, we give a criterion to obtain the upper bound for the number of isolated periodic solutions. This criterion is "almost equivalent" to hypothesis (H) and can be much more effectively checked.
• In this paper, we consider the event-triggered cooperative robust practical output regulation problem for a class of linear minimum-phase multi-agent systems. We first convert our problem into the cooperative robust practical stabilization problem of a well defined augmented system Based on the distributed internal model approach. Then, we design a distributed event-triggered output feedback control law together with a distributed output-based event-triggered mechanism to stabilize the augmented system, which leads to the solvability of the cooperative robust practical output regulation problem of the original plant. Our distributed control law can be directly implemented in a digital platform provided that the distributed triggering mechanism can monitor the continuous-time output information from neighboring agents. Finally, we illustrate our design by an example.
• The leader-following consensus problem for multiple Euler-Lagrange systems was studied recently by the adaptive distributed observer approach under the assumptions that the leader system is neurally stable and the communication network is jointly connected and undirected. In this paper, we will study the same problem without assuming that the leader system is neutrally stable, and the communication network is undirected. The effectiveness of this new result will be illustrated by an example.
• Let $u=u(t,{\bf x},{\bf p})$ satisfy the transport equation $\frac {\partial u}{\partial t}+\frac {{\bf p}}{p_0}\frac{\partial u}{\partial{\bf x}}=f$, where $f=f(t,\bf x,\bf p)$ belongs to $L^{p}((0,T)\times {\bf R}^{3}\times {\bf R}^{3})$ for $1<p<\infty$ and $\frac {\partial}{\partial t}+\frac {{\bf p}}{p_0}\frac{\partial}{\partial{\bf x}}$ is the relativistic-free transport operator. We show the regularity of $\int_{{\bf R}^{3}}u(t, {\bf x}, {\bf p})d{\bf p}$ using the same method as given by Golse, Lions, Perthame and Sentis. This average regularity is considered in terms of fractional Sobolev spaces and it is very useful for the study of the existence of the solution to the Cauchy problem on the relativistic Boltzmann equation.
• Dynamic spectrum access (DSA) can effectively improve the spectrum efficiency and alleviate the spectrum scarcity, by allowing unlicensed secondary users (SUs) to access the licensed spectrum of primary users (PUs) opportunistically. Cooperative spectrum sharing is a new promising paradigm to provide necessary incentives for both PUs and SUs in dynamic spectrum access. The key idea is that SUs relay the traffic of PUs in exchange for the access time on the PUs' licensed spectrum. In this paper, we formulate the cooperative spectrum sharing between multiple PUs and multiple SUs as a two-sided market, and study the market equilibrium under both complete and incomplete information. First, we characterize the sufficient and necessary conditions for the market equilibrium. We analytically show that there may exist multiple market equilibria, among which there is always a unique Pareto-optimal equilibrium for PUs (called PU-Optimal-EQ), in which every PU achieves a utility no worse than in any other equilibrium. Then, we show that under complete information, the unique Pareto-optimal equilibrium PU-Optimal-EQ can always be achieved despite the competition among PUs; whereas, under incomplete information, the PU-Optimal-EQ may not be achieved due to the mis-representations of SUs (in reporting their private information). Regarding this, we further study the worse-case equilibrium for PUs, and characterize a Robust equilibrium for PUs (called PU-Robust-EQ), which provides every PU a guaranteed utility under all possible mis-representation behaviors of SUs. Numerical results show that in a typical network where the number of PUs and SUs are different, the performance gap between PU-Optimal-EQ and PU-Robust-EQ is quite small (e.g., less than 10% in the simulations).
• In this paper the properties of right invertible row operators, i.e., of 1X2 surjective operator matrices are studied. This investigation is based on a specific space decomposition. Using this decomposition, we characterize the invertibility of a 2X2 operator matrix. As an application, the invertibility of Hamiltonian operator matrices is investigated.
• Let $G$ be a right-angled Artin group with defining graph $\Gamma$ and let $H$ be a finitely generated group quasi-isometric to $G(\Gamma)$. We show if $G$ satisfies (1) its outer automorphism group is finite; (2) $\Gamma$ does not have induced 4-cycle; (3) $\Gamma$ is star-rigid; then $H$ is commensurable to $G$. We show condition (2) is sharp in the sense that if $\Gamma$ contains an induced 4-cycle, then there exists an $H$ quasi-isometric to $G(\Gamma)$ but not commensurable to $G(\Gamma)$. Moreover, one can drop condition (1) if $H$ is a uniform lattice acting on the universal cover of the Salvetti complex of $G(\Gamma)$. As a consequence, we obtain a conjugation theorem for such uniform lattices. The ingredients of the proof include a blow-up building construction in \citecubulation and a Haglund-Wise style combination theorem for certain class of special cube complexes. However, in most of our cases, relative hyperbolicity is absent, so we need new ingredients for the combination theorem.