results for au:Huang_J in:math

- Sep 08 2017 math.AP arXiv:1709.01983v2The 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.).
- Sep 05 2017 math.AP arXiv:1709.00933v1We 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.
- Aug 30 2017 math.NA arXiv:1708.08558v2Some 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.
- Aug 29 2017 math.NA arXiv:1708.08427v1In 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.
- Aug 29 2017 math.OC arXiv:1708.07930v1In 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.
- Aug 29 2017 math.CA arXiv:1708.07959v2In 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.
- Aug 24 2017 math.PR arXiv:1708.07115v2We 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.
- Aug 14 2017 math.GR arXiv:1708.03550v1Let $G$ 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:1708.00960v1Under 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.
- Aug 03 2017 math.AP arXiv:1708.00773v2Inspired 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.
- Jul 27 2017 math.OC arXiv:1707.08421v1This 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.
- Jul 18 2017 math.RT arXiv:1707.04707v1The 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.
- Jul 10 2017 math.CO arXiv:1707.02002v1A 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:1706.05473v1We 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.
- Jun 19 2017 math.OC arXiv:1706.05355v1Online 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.
- May 18 2017 math.NA arXiv:1705.06101v1In 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.
- May 02 2017 math.AP arXiv:1705.00260v1In 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.
- Apr 28 2017 math.PR arXiv:1704.08334v2Consider 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.
- Apr 25 2017 math.CO arXiv:1704.06867v1An 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.
- Cooperative Robust Output Regulation Problem for Discrete-Time Linear Time-Delay Multi-Agent SystemsApr 03 2017 math.OC arXiv:1703.10771v2In 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.
- Mar 31 2017 math.OC arXiv:1703.10359v1In 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.
- Mar 30 2017 math.OC arXiv:1703.09854v1Shunt 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.
- Mar 29 2017 math.OC arXiv:1703.09415v1In 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.
- Mar 28 2017 math.NA arXiv:1703.09136v1In 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.
- Mar 08 2017 math.NA arXiv:1703.02265v1In 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.
- Mar 06 2017 math.RT arXiv:1703.01100v1Let $\mathfrak{g}$ 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.
- Mar 02 2017 math.PR arXiv:1703.00137v1We 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:1702.08374v1The 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.
- 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:1612.06306v2We 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:1611.05576v2An 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.
- Nov 07 2016 math.CO arXiv:1611.01251v3Let 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:1609.09022v3We 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:1607.00682v2We 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.
- 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:1605.01606v2In 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.
- 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:1601.07071v2In this paper, we study the leader-following consensus problem for a class of uncertain nonlinear multi-agent systems under jointly connected directed switching networks. The uncertainty includes constant unbounded parameters and external disturbances. We first extend the recent result on the adaptive distributed observer from global asymptotical convergence to global exponential convergence. Then, by integrating the conventional adaptive control technique with the adaptive distributed observer, we present our solution by a distributed adaptive state feedback control law. Our result is illustrated by the 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 $j\geq2$.