results for au:Huang_J in:math

- 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.
- Jan 17 2017 math.GR arXiv:1701.03969v1We 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.
- Jan 03 2017 math.OC arXiv:1701.00196v1This 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.
- Dec 21 2016 math.PR arXiv:1612.06437v1This 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.
- Dec 20 2016 math.PR arXiv:1612.06306v1We 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 $N^{-1}\ll\eta\ll t$
- Dec 15 2016 math.PR arXiv:1612.04763v1The 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).
- Nov 18 2016 math.NA arXiv:1611.05576v1An efficient 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 further reduced into a symmetric positive definite system of Poisson type. Furthermore an empirical choice of the parameter used in the splitting is found. By comparing the number of iterations and CPU time of different solvers in several numerical experiments, our method is shown to convergent with a rate independent of the mesh size and the nonlinearity parameter and the computational cost is nearly linear.
- Nov 07 2016 math.CO arXiv:1611.01251v1Let 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.
- Oct 26 2016 math.PR arXiv:1610.07727v1A 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.
- Sep 29 2016 math.PR arXiv:1609.09022v2We 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.
- Aug 16 2016 math.NA arXiv:1608.04060v1This 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.
- Jul 15 2016 math.PR arXiv:1607.03998v1We 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 $\epsilon>0$.
- Jul 05 2016 math.PR arXiv:1607.00682v1In this paper we study 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.
- Jun 16 2016 math.CA arXiv:1606.04776v1This 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.
- May 06 2016 math.OC arXiv:1605.01606v1In this paper, we consider the cooperative robust output regulation problem for a class of heterogeneous linear uncertain minimum-phase multi-agent systems by an output-based event-triggered distributed control law. First, we convert the cooperative robust output regulation into the cooperative robust stabilization problem of a well defined augmented system via the distributed internal model principle. Second, we design an output-based event-triggered distributed control law to stabilize the augmented system, which leads to the solvability of the cooperative robust output regulation problem of the original plant. We also show that our distributed triggering mechanism is able to prevent the Zeno behavior for all time. Finally, we illustrate our design by an example.
- Apr 26 2016 math.OC arXiv:1604.07261v1The 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.
- A right triangular billiard system is equivalent to the system of two colliding particles confined in a one-dimensional box. In spite of their seeming simplicity, no definite conclusion has been drawn so far concerning their ergodic properties. To answer this question, we transform the dynamics of the right triangular billiard system to a piecewise map and analytically prove the broken ergodicity. The mechanism leading to the broken ergodicity is discussed, and some numerical evidence corroborating our conclusion is provided.
- We are motivated by the question that for which class of right-angled Artin groups (RAAG's), the quasi-isometry classification coincides with commensurability classification. This is previously known for RAAG's with finite outer automorphism groups. In this paper, we identify two classes of RAAG's, where their outer automorphism groups are allowed to contain adjacent transvections and partial conjugations, hence infinite. If $G$ belongs to one of these classes, then any other RAAG $G'$ is quasi-isometric to $G$ if and only if $G'$ is commensurable to $G$. We also show that in this case, there exists an algorithm to determine whether two RAAG's are quasi-isometric by looking at their defining graphs. Compared to the finite out case, as well as the previous quasi-isometry rigidity results for symmetric spaces, thick Euclidean buildings and mapping class groups, the main issue we need to deal with here is the reconstruction map may not have nice properties as before, or may not even exist. We introduce a deformation argument, as well as techniques from cubulation to deal with this issue.
- Feb 19 2016 math.RA arXiv:1602.05646v1For Leavitt path algebras, we show that whereas removing sources from a graph produces a Morita equivalence, removing sinks gives rise to a recollement situation. In general, we show that for a graph $E$ and a finite hereditary subset $H$ of $E^0$ there is a recollement $$\xymatrix L_K(E/\overline H) \rModd \ar[r] & \ar@<3pt>[l] \ar@<-3pt>[l] L_K(E) \rModd \ar[r] & \ar@<3pt>[l] \ar@<-3pt>[l] L_K(E_H) \rModd .$$ We record several corollaries.
- We consider online optimization in the 1-lookahead setting, where the objective does not decompose additively over the rounds of the online game. The resulting formulation enables us to deal with non-stationary and/or long-term constraints , which arise, for example, in online display advertising problems. We propose an on-line primal-dual algorithm for which we obtain dynamic cumulative regret guarantees. They depend on the convexity and the smoothness of the non-additive penalty, as well as terms capturing the smoothness with which the residuals of the non-stationary and long-term constraints vary over the rounds. We conduct experiments on synthetic data to illustrate the benefits of the non-additive penalty and show vanishing regret convergence on live traffic data collected by a display advertising platform in production.
- Feb 02 2016 math.OC arXiv:1602.00414v1We extend a primal-dual fixed point algorithm (PDFP) proposed in [5] to solve two kinds of separable multi-block minimization problems, arising in signal processing and imaging science. This work shows the flexibility of applying PDFP algorithm to multi-block problems and illustrate how practical and fully decoupled schemes can be derived, especially for parallel implementation of large scale problems. The connections and comparisons to the alternating direction method of multiplier (ADMM) are also present. We demonstrate how different algorithms can be obtained by splitting the problems in different ways through the classic example of sparsity regularized least square model with constraint. In particular, for a class of linearly constrained problems, which are of great interest in the context of multi-block ADMM, can be solved by PDFP with a guarantee of convergence. Finally, some experiments are provided to illustrate the performance of several schemes derived by the PDFP algorithm.
- Jan 27 2016 math.OC arXiv:1601.07071v1This paper studies the leader-following consensus problem for a class of nonlinear multi-agent systems with linearly parameterized uncertainty subject to jointly connected switching networks. To deal with the jointly connected switching networks, a distributed adaptive control law utilizing certainty equivalence principle and the adaptive distributed observer is developed. Global convergence of the closed-loop system is established. The design procedure is illustrated by a leader-following consensus problem for a group of Van der Pol oscillators.
- We characterize groups quasi-isometric to a right-angled Artin group $G$ with finite outer automorphism group. In particular all such groups admit a geometric action on a $CAT(0)$ cube complex that has an equivariant "fibering" over the Davis building of $G$.
- Jan 01 2016 math.OC arXiv:1512.09235v1Many problems arising in image processing and signal recovery with multi-regularization can be formulated as minimization of a sum of three convex separable functions. Typically, the objective function involves a smooth function with Lipschitz continuous gradient, a linear composite nonsmooth function and a nonsmooth function. In this paper, we propose a primal-dual fixed-point (PDFP) scheme to solve the above class of problems. The proposed algorithm for three block problems is a fully splitting symmetric scheme, only involving explicit gradient and linear operators without inner iteration, when the nonsmooth functions can be easily solved via their proximity operators, such as $\ell_1$ type regularization. We study the convergence of the proposed algorithm and illustrate its efficiency through examples on fused LASSO and image restoration with non-negative constraint and sparse regularization.
- We present an adaptive online gradient descent algorithm to solve online convex optimization problems with long-term constraints , which are constraints that need to be satisfied when accumulated over a finite number of rounds T , but can be violated in intermediate rounds. For some user-defined trade-off parameter $\beta$ $\in$ (0, 1), the proposed algorithm achieves cumulative regret bounds of O(T^max$\beta$,1--$\beta$) and O(T^(1--$\beta$/2)) for the loss and the constraint violations respectively. Our results hold for convex losses and can handle arbitrary convex constraints without requiring knowledge of the number of rounds in advance. Our contributions improve over the best known cumulative regret bounds by Mahdavi, et al. (2012) that are respectively O(T^1/2) and O(T^3/4) for general convex domains, and respectively O(T^2/3) and O(T^2/3) when further restricting to polyhedral domains. We supplement the analysis with experiments validating the performance of our algorithm in practice.
- Nov 26 2015 math.RT arXiv:1511.07965v2We define Lie algebra cohomology associated with the half-Dirac operators for representations of rational Cherednik algebras and show that it has property described in the Casselman-Osborne Theorem by establishing a version of the Vogan's conjecture for the half-Dirac operators. Moreover, we study the relationship between Lie algebra cohomology and Dirac cohomology in analogy of the representations for semisimple Lie algebras.
- This paper considers the problem of charging station pricing and plug-in electric vehicles (PEVs) station selection. When a PEV needs to be charged, it selects a charging station by considering the charging prices, waiting times, and travel distances. Each charging station optimizes its charging price based on the prediction of the PEVs' charging station selection decisions and the other station's pricing decision, in order to maximize its profit. To obtain insights of such a highly coupled system, we consider a one-dimensional system with two competing charging stations and Poisson arriving PEVs. We propose a multi-leader-multi-follower Stackelberg game model, in which the charging stations (leaders) announce their charging prices in Stage I, and the PEVs (followers) make their charging station selections in Stage II. We show that there always exists a unique charging station selection equilibrium in Stage II, and such equilibrium depends on the charging stations' service capacities and the price difference between them. We then characterize the sufficient conditions for the existence and uniqueness of the pricing equilibrium in Stage I. We also develop a low complexity algorithm that efficiently computes the pricing equilibrium and the subgame perfect equilibrium of the two-stage Stackelberg game.
- Nov 25 2015 math.RT arXiv:1511.07618v1The first part (Sections 1-6) of this paper is a survey of some of the recent developments in the theory of Dirac cohomology, especially the relationship of Dirac cohomology with (g,K)-cohomology and nilpotent Lie algebra cohomology; the second part (Sections 7-12) is devoted to understanding the unitary elliptic representations and endoscopic transfer by using the techniques in Dirac cohomology. A few problems and conjectures are proposed for further investigations.
- Nov 10 2015 math.AP arXiv:1511.02430v1In this paper, we investigate the Cauchy problem for the higher-order KdV-type equation \begineqnarray* u_t+(-1)^j+1\partial_x^2j+1u + \frac12\partial_x(u^2) = 0,j∈N^+,x∈\mathbfT= [0,2\pi \lambda) \endeqnarray* with low regularity data and $\lambda\geq 1$. Firstly, we show that the Cauchy problem for the periodic higher-order KdV equation is locally well-posed in $H^{s}(\mathbf{T})$ with $s\geq -j+\frac{1}{2},j\geq2.$ By using some new Strichartz estimate and some new function spaces, we also show that the Cauchy problem for the periodic higher-order KdV equation is ill-posed in $H^{s}(\mathbf{T})$ with $s<-j+\frac{1}{2},j\geq2$ in the sense that the solution map is $C^{3}.$ The result of this paper improves the result of \citeH with $j\geq2$.
- Oct 30 2015 math.GR arXiv:1510.08493v1We give a necessary and sufficient condition for a 2-dimensional or a three-generator Artin group $A$ to be (virtually) cocompactly cubulated, in terms of the defining graph of $A$.
- Oct 22 2015 math.PR arXiv:1510.06390v1We consider the bulk eigenvalue statistics of Laplacian matrices of large Erdős-Rényi random graphs in the regime $p \geq N^{\delta}/N$ for any fixed $\delta >0$. We prove a local law down to the optimal scale $\eta \gtrsim N^{-1}$ which implies that the eigenvectors are delocalized. We consider the local eigenvalue statistics and prove that both the gap statistics and averaged correlation functions coincide with the GOE in the bulk.
- Oct 20 2015 math.OC arXiv:1510.05380v2In this paper, we study the cooperative output regulation problem for the discrete linear time-delay multi-agent systems by distributed observer approach. In contrast with the same problem for continuous-time linear time-delay multi-agent systems, the problem has two new features. First, in the presence of time-delay, the regulator equations for discrete-time linear systems are different from those for continuous-time linear systems. Second, under the standard assumption on the connectivity of the communication graph, a distributed observer for any continuous-time linear leader system always exists. However, this is not the case for discrete-time systems, and the behavior of a distributed observer is much more complicated. Thus, we will first study the existence of the discrete distributed observer, and then present the solvability of the problem by distributed dynamic output feedback control law.
- With the rapid growth of mobile traffic demand, a promising approach to relieve cellular network congestion is to offload users' traffic to small-cell networks. In this paper, we investigate how the mobile users (MUs) can effectively offload traffic by taking advantage of the capability of dual-connectivity, which enables an MU to simultaneously communicate with a macro base station (BS) and a small-cell access point (AP) via two radio-interfaces. Offloading traffic to the AP usually reduces the MUs' mobile data cost, but often at the expense of suffering increased interferences from other MUs at the same AP. We thus formulate an optimization problem that jointly determines each MU's traffic schedule (between the BS and AP) and power control (between two radio-interfaces). The system objective is to minimize all MUs' total cost, while satisfying each MU's transmit-power constraints through proper interference control. In spite of the non-convexity of the problem, we design both a centralized algorithm and a distributed algorithm to solve the joint optimization problem. Numerical results show that the proposed algorithms can achieve the close-to-optimum results comparing with the ones achieved by the LINGO (a commercial optimization software), but with significantly less computational complexity. The results also show that the proposed adaptive offloading can significantly reduce the MUs' cost, i.e., save more than 75% of the cost without offloading traffic and 65% of the cost with a fixed offloading.
- Sep 08 2015 math.RT arXiv:1509.01755v3We prove a generalization of Harish-Chandra's character orthogonality relations for discrete series to arbitrary Harish-Chandra modules for real reductive Lie groups. This result is an analogue of a conjecture by Kazhdan for $\mathfrak p$-adic reductive groups proved by Bezrukavnikov, and Schneider and Stuhler.
- Sep 04 2015 math.PR arXiv:1509.00897v3In this paper we study the linear stochastic heat equation, also known as parabolic Anderson model, in multidimension driven by a Gaussian noise which is white in time and it has a correlated spatial covariance. Examples of such covariance include 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, lower and upper exponential growth indices in terms of a variational quantity. The last part of the paper is devoted to study the phase transition property of the Anderson model.
- Aug 20 2015 math.OC arXiv:1508.04505v1This paper studies the cooperative global robust stabilization problem for a class of nonlinear multi-agent systems. The problem is motivated from the study of the cooperative global robust output regulation problem for the class of nonlinear multi-agent systems in normal form with unity relative degree which was studied recently under the conditions that the switching network is undirected and some nonlinear functions satisfy certain growth condition. We first solve the stabilization problem by using the multiple Lyapunov functions approach and the average dwell time method. Then, we apply this result to the cooperative global robust output regulation problem for the class of nonlinear systems in normal form with unity relative degree under directed switching network, and have removed the conditions that the switching network is undirected and some nonlinear functions satisfy certain growth condition.
- Aug 19 2015 math.OC arXiv:1508.04207v1In this paper, we study the cooperative robust output regulation problem for linear uncertain multi-agent systems with both communication delay and input delay by the distributed internal model approach. The problem includes the leader-following consensus problem of linear multi-agent systems with time-delay as a special case. We first generalize the internal model design method to systems with both communication delay and input delay. Then, under a set of standard assumptions, we have obtained the solution of the problem via both the state feedback control and the output feedback control. In contrast with the existing results, our results apply to general uncertain linear multi-agent systems, accommodate a large class of leader signals, and achieve the asymptotic tracking and disturbance rejection at the same time.
- Aug 19 2015 math.OC arXiv:1508.04206v1The cooperative output regulation problem of linear multi-agent systems was formulated and studied by the distributed observer approach in [20, 21]. Since then, several variants and extensions have been proposed, and the technique of the distributed observer has also been applied to such problems as formation, rendezvous, flocking, etc. In this chapter, we will first present a more general formulation of the cooperative output regulation problem for linear multi-agent systems that includes some existing versions of the cooperative output regulation problem as special cases. Then, we will describe a more general distributed observer. Finally, we will simplify the proof of the main results by more explicitly utilizing the separation principle and the certainty equivalence principle.
- Aug 10 2015 math.CO arXiv:1508.01688v5The Catalan number $C_n$ enumerates parenthesizations of $x_0*\dotsb*x_n$ where $*$ is a binary operation. We introduce the modular Catalan number $C_{k,n}$ to count equivalence classes of parenthesizations of $x_0*\dotsb*x_n$ when $*$ satisfies a $k$-associative law generalizing the usual associativity. This leads to a study of restricted families of Catalan objects enumerated by $C_{k,n}$ with emphasis on binary trees, plane trees, and Dyck paths, each avoiding certain patterns. We give closed formulas for $C_{k,n}$ with two different proofs. For each $n\ge0$ we compute the largest size of $k$-associative equivalence classes and show that the number of classes with this size is a Catalan number.
- Aug 04 2015 math.PR arXiv:1508.00252v2This paper studies the linear stochastic partial differential equation of fractional orders both in time and space variables $\left(\partial^\beta + \frac{\nu}{2} (-\Delta)^{\alpha/2} \right) u(t,x)= \lambda u(t,x) \dot{W}(t,x)$, where $\dot W$ is a general Gaussian noise and $\beta\in (1/2, 2)$, $\alpha\in (0, 2]$. The existence and uniqueness of the solution, the moment bounds of the solution are obtained by using the fundamental solutions of the corresponding deterministic counterpart represented by the Fox H-functions. Along the way, we obtain some new properties of the fundamental solutions.
- Jul 03 2015 math.OC arXiv:1507.00664v2This paper provides conditions under which total-cost and average-cost Markov decision processes (MDPs) can be reduced to discounted ones. Results are given for transient total-cost MDPs with transition rates whose values may be greater than one, as well as for average-cost MDPs with transition probabilities satisfying the condition that there is a state such that the expected time to reach it is uniformly bounded in a weighted supremum norm for all initial states and stationary policies. In particular, these reductions imply sufficient conditions for the validity of optimality equations and the existence of stationary optimal policies for MDPs with undiscounted total cost and average-cost criteria. When the state and action sets are finite, these reductions lead to linear programming formulations and complexity estimates for MDPs under the aforementioned criteria.
- Jun 16 2015 math.PR arXiv:1506.04670v1We study the propagation of high peaks (intermittency front) of the solution to a stochastic heat equation driven by multiplicative centered Gaussian noise in $\mathbb{R}^d$. The noise is assumed to have a general homogeneous covariance in both time and space, and the solution is interpreted in the senses of the Wick product. We give some estimates for the upper and lower bounds of the propagation speed, based on a moment formula of the solution. When the space covariance is given by a Riesz kernel, we give more precise bounds for the propagation speed.
- This paper studies a partial order on the general linear group GL(V) called the absolute order, derived from viewing GL(V) as a group generated by reflections, that is, elements whose fixed space has codimension one. The absolute order on GL(V) is shown to have two equivalent descriptions, one via additivity of length for factorizations into reflections, the other via additivity of fixed space codimensions. Other general properties of the order are derived, including self-duality of its intervals. Working over a finite field F_q, it is shown via a complex character computation that the poset interval from the identity to a Singer cycle (or any regular elliptic element) in GL_n(F_q) has a strikingly simple formula for the number of chains passing through a prescribed set of ranks.
- We generalize the Hopf algebras of free quasisymmetric functions, quasisymmetric functions, noncommutative symmetric functions, and symmetric functions to certain representations of the category of all finite Coxeter systems and its dual category. We investigate their connections with the representation theory of 0-Hecke algebras of finite Coxeter systems. Restricted to type B and D we obtain dual graded modules and comodules over the corresponding Hopf algebras in type A.
- We consider the uniform random $d$-regular graph on $N$ vertices, with $d \in [N^\alpha, N^{2/3-\alpha}]$ for arbitrary $\alpha > 0$. We prove that in the bulk of the spectrum the local eigenvalue correlation functions and the distribution of the gaps between consecutive eigenvalues coincide with those of the Gaussian Orthogonal Ensemble.
- May 25 2015 math.AP arXiv:1505.05995v1In this paper, we investigate the Cauchy problem for the Ostrovsky equation \begineqnarray* \partial_x\left(u_t-\beta \partial_x^3u +\frac12\partial_x(u^2)\right) -\gamma u=0, \endeqnarray* in the Sobolev space $H^{-3/4}(\R)$, $\beta>0,\gamma >0$. From the result of \citeIM1,IM3,Tsu, we know that $s=-\frac{3}{4}$ is the critical Sobolev index for the Ostrovsky equation. By using some modified Besov spaces, we prove that the Cauchy problem for the Ostrovsky equation with positive dispersion is locally well-posed in $H^{-3/4}(\R).$ The new ingredient that we introduce in this paper is Lemmas 2.1-2.2 which are used to overcome the difficulty caused by the singularity of the phase function at the zero point.
- May 21 2015 math.PR arXiv:1505.04924v1This paper studies the nonlinear one-dimensional stochastic heat equation driven by a Gaussian noise which is white in time and which has the covariance of a fractional Brownian motion with Hurst parameter 1/4\textlessH\textless1/2 in the space variable. The existence and uniqueness of the solution u are proved assuming the nonlinear coefficient is differentiable with a Lipschitz derivative and vanishes at 0. In the case of a multiplicative noise, that is the linear equation, 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 moments of the solution.
- We consider the adjacency matrix of the ensemble of Erdős-Rényi random graphs which consists of graphs on $N$ vertices in which each edge occurs independently with probability $p$. We prove that in the regime $pN \gg 1$ these matrices exhibit bulk universality in the sense that both the averaged $n$-point correlation functions and distribution of a single eigenvalue gap coincide with those of the GOE. Our methods extend to a class of random matrices which includes sparse ensembles whose entries have different variances.
- Apr 10 2015 math.AP arXiv:1504.02172v1This paper is devoted to the Cauchy problem for the stochastic generalized Benjamin-Ono equation. By using the Bourgain spaces and Fourier restriction method and the assumption that $u_{0}$ is $\mathcal{F}_{0}$-measurable, we prove that the Cauchy problem for the stochastic generalized Benjamin-Ono equation is locally well-posed for the initial data $u_{0}(x,w)\in L^{2} (\Omega; H^{s}(\R))$ with $s\geq\frac{1}{2}-\frac{\alpha}{4}$, where $0< \alpha \leq 1.$ In particular, when $u_{0}\in L^{2}(\Omega; H^{\frac{\alpha+1}{2}}(\R))\cap L^{\frac{2(2+3\alpha)}{\alpha}}(\Omega; L^{2}(\R))$, we prove that there exists a unique global solution $u\in L^{2}(\Omega; H^{\frac{\alpha+1}{2}}(\R))$ with $0< \alpha \leq 1.$
- Mar 24 2015 math.AP arXiv:1503.06265v1In this paper, we investigate the Cauchy problem for a higher order shallow water type equation \begineqnarray* u_t-u_txx+\partial_x^2j+1u-\partial_x^2j+3u+3uu_x-2u_xu_xx-uu_xxx=0, \endeqnarray* where $x\in \mathbf{T}=\mathbf{R}/2\pi$ and $j\in N^{+}.$ Firstly, we prove that the Cauchy problem for the shallow water type equation is locally well-posed in $H^{s}(\mathbf{T})$ with $s\geq -\frac{j-2}{2}$ for arbitrary initial data. By using the $I$-method, we prove that the Cauchy problem for the shallow water type equation is globally well-posed in $H^{s}(\mathbf{T})$ with $\frac{2j+1-j^{2}}{2j+1}<s\leq 1.$ Our results improve the result of A. A. Himonas, G. Misiolek (Communications in partial Differential Equations, 23(1998), 123-139;Journal of Differential Equations, 161(2000), 479-495.)
- A 0-Hecke algebra is a deformation of the group algebra of a Coxeter group. Based on work of Norton and Krob--Thibon, we introduce a tableau approach to the representation theory of 0-Hecke algebras of type A, which resembles the classic approach to the representation theory of symmetric groups by Young tableaux and tabloids. We extend this approach to type B and D, and obtain a correspondence between the representation theory of 0-Hecke algebras of type B and D and quasisymmetric functions and noncommutative symmetric functions of type B and D. Other applications are also provided.
- Dec 16 2014 math.PR arXiv:1412.4193v2We consider the eigenvalues and eigenvectors of small rank perturbations of random $N\times N$ matrices. We allow the rank of perturbation $M$ increases with $N$, and the only assumption is $M=o(N)$. In both additive and multiplicative perturbation models, we prove rigidity results for the outliers of the perturbed random matrices. Based on the rigidity results we derive the empirical distribution of outliers of the perturbed random matrices. We also compute the appropriate projection of eigenvectors corresponding to the outliers of the perturbed random matrices, which are approximate eigenvectors of the perturbing matrix. Our results can be regarded as the extension of the finite rank perturbation case to the full generality up to $M=o(N)$.
- Nov 05 2014 math.AP arXiv:1411.0890v1In this paper, we investigate the Cauchy problem for the Ostrovsky equation \begineqnarray* \partial_x\left(u_t-\beta \partial_x^3u +\frac12\partial_x(u^2)\right) -\gamma u=0, \endeqnarray* in the Sobolev space $H^{-3/4}(\R)$. Here $\beta>0(<0)$ corresponds to the positive (negative) dispersion of the media, respectively. P. Isaza and J. Mejı́a (J. Diff. Eqns. 230(2006), 601-681; Nonli. Anal. 70(2009), 2306-2316), K. Tsugawa (J. Diff. Eqns. 247(2009), 3163-3180) proved that the problem is locally well-posed in $H^s(\R)$ when $s>-3/4$ and ill-posed when $s<-3/4$. By using some modified Bourgain spaces, we prove that the problem is locally well-posed in $H^{-3/4}(\R)$ with $\beta <0$ and $\gamma>0.$ The new ingredient that we introduce in this paper is Lemmas 2.1-2.6.
- Let $G$ and $G'$ be two right-angled Artin groups (RAAG). We show they are quasi-isometric iff they are isomorphic, under the assumption that $Out(G)$ and $Out(G')$ are finite. If only $Out(G)$ is finite, then $G'$ is quasi-isometric $G$ iff $G'$ is isomorphic to a finite index subgroup of $G$. In this case, we give an algorithm to determine whether $G$ and $G'$ are quasi-isometric by looking at their defining graphs.
- We show that every $n$-quasiflat in a $n$-dimensional $CAT(0)$ cube complex is at finite Hausdorff distance from a finite union of $n$-dimensional orthants. Then we introduce a class of cube complexes, called \em weakly special cube complexes and show that quasi-isometries between their universal coverings preserve top dimensional flats. We use this to establish several quasi-isometry invariants for right-angled Artin groups. Some of our arguments also extend to $CAT(0)$ spaces of finite geometric dimension. In particular, we give a short proof of the fact that a top dimensional quasiflat in a Euclidean buildings is Hausdorff close to finite union of Weyl cones, which was previously established in several other authors by different methods.
- Oct 08 2014 math.PR arXiv:1410.1581v1In this paper we consider a general class of second order stochastic partial differential equations on $\mathbb{R}^d$ driven by a Gaussian noise which is white in time and it has a homogeneous spatial covariance. Using the techniques of Malliavin calculus we derive the smoothness of the density of the solution at a fixed number of points $(t,x_1), \dots, (t,x_n)$, $t>0$, assuming some suitable regularity and non degeneracy assumptions. We also prove that the density is strictly positive in the interior of the support of the law.
- The measurement-device-independent quantum key distribution (MDI-QKD) protocol is proposed to remove the detector side channel attacks, while its security relies on the assumption that the encoding systems are perfectly characterized. In contrast, the MDI-QKD protocol based on the Clauser-Horne-Shimony-Holt inequality (CHSH-MDI-QKD) weakens this assumption, which only requires the quantum state to be prepared in the two-dimensional Hilbert space and the devices are independent. In experimental realizations, the weak coherent state, which is always used in QKD systems due to the lack of an ideal single photon source, may be prepared in the high-dimensional space. In this paper, we investigate the decoy-state CHSH-MDI-QKD protocol with $s(3 \le s \le 5)$ intensities, including one signal state and $s-1$ decoy states, and we also consider the finite-size effect on the decoy-state CHSH-MDI-QKD protocol with five intensities. Simulation results show that this scheme is very practical.
- Jul 22 2014 math.AP arXiv:1407.5349v1For periodic initial data with initial density, we establish the global existence and uniqueness of strong and classical solutions for the two-dimensional compressible Navier-Stokes equations with no restrictions on the size of initial data provided the shear viscosity is a positive constant and the bulk one is $\lam=\rho^{\b}$ with $\b>1.$
- Network coding in geometric space, a new research direction also known as Space Information Flow, is a promising research field which shows the superiority of network coding in space over routing in space. Present literatures proved that given six terminal nodes, network coding in space is strictly superior to routing in space in terms of single-source multicast in regular (5+1) model, in which five terminal nodes forms a regular pentagon centered at a terminal node. In order to compare the performance between network coding in space and routing in space, this paper quantitatively studies two classes of network coding in space and optimal routing in space when any terminal node moves arbitrarily in two-dimensional Euclidean space, and cost advantage is used as the metric. Furthermore, the upper-bound of cost advantage is figured out as well as the region where network coding in space is superior to routing in space. Several properties of Space Information Flow are also presented.
- May 09 2014 math.AP arXiv:1405.1863v2In this paper, we consider a complex fluid modeling nematic liquid crystal flows, which is described by a system coupling Navier-Stokes equations with a parabolic Q-tensor system. We first prove the global existence of weak solutions in dimension three. Furthermore, the global well-posedness of strong solutions is studied with sufficiently large viscosity of fluid. Finally, we show a continuous dependence result on the initial data which directly yields the weak-strong uniqueness of solutions.
- We study the Hecke algebra $\H(\bq)$ over an arbitrary field $\FF$ of a Coxeter system $(W,S)$ with independent parameters $\bq=(q_s\in\FF:s\in S)$ for all generators. This algebra is always linearly spanned by elements indexed by the Coxeter group $W$. This spanning set is indeed a basis if and only if every pair of generators joined by an odd edge in the Coxeter diagram receive the same parameter. In general, the dimension of $\H(\bq)$ could be as small as $1$. We construct a basis for $\H(\bq)$ when $(W,S)$ is simply laced. We also characterize when $\H(\bq)$ is commutative, which happens only if the Coxeter diagram of $(W,S)$ is simply laced and bipartite. In particular, for type A we obtain a tower of semisimple commutative algebras whose dimensions are the Fibonacci numbers. We show that the representation theory of these algebras has some features in analogy/connection with the representation theory of the symmetric groups and the 0-Hecke algebras.