Sumset phenomenon in countable amenable groups

Jin proved that whenever A and B are sets of positive upper density in Z, A+B is piecewise syndetic. Jin's theorem was subsequently generalized by Jin and Keisler to a certain family of abelian groups, which in particular contains Zd. Answering a question of Jin and Keisler, we show that this result can be extended to countable amenable groups. Moreover we establish that such sumsets (or — depending on the notation — “product sets”) are piecewise Bohr, a result which for G=Z was proved by Bergelson, Furstenberg and Weiss. In the case of an abelian group G, we show that a set is piecewise Bohr if and only if it contains a sumset of two sets of positive upper Banach density.     

Models of thermodiffusion in 1D

The goal of the paper is to study the Cauchy problem for 1D models of thermodiffusion. We explain qualitative properties of solutions. In particular, we show which part of the model has a dominant influence on well‐posedness, propagation of singularities, L^p ? L^q decay estimates on the conjugate line, and on the diffusion phenomenon. Copyright © 2013 John Wiley & Sons, Ltd.     

On the cut‐off phenomenon for the transitivity of randomly generated subgroups

Consider K ≥ 2 independent copies of the random walk on the symmetric group S_N starting from the identity and generated by the products of either independent uniform transpositions or independent uniform neighbor transpositions. At any time $nin mathbb{N}$, let G_n be the subgroup of S_N generated by the K positions of the chains. In the uniform transposition model, we prove that there is a cut‐off phenomenon at time N ln(N)/(2K) for the non‐existence of fixed point of G_n and for the transitivity of G_n, thus showing that these properties occur before the chains have reached equilibrium. In the uniform neighbor transposition model, a transition for the non‐existence of a fixed point of G_n appears at time of order $N^{1+frac{2}{K}}$ (at least for K ≥ 3), but there is no cut‐off phenomenon. In the latter model, we recover a cut‐off phenomenon for the non‐existence of a fixed point at a time proportional to N by allowing the number K to be proportional to ln(N). The main tools of the proofs are spectral analysis and coupling techniques. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2012     

Multigrid Solution of a Lavrentiev-Regularized State-Constrained Parabolic Control Problem

A mesh-independent, robust, and accurate multigrid scheme to solve a linear state-constrained parabolic optimal control problem is presented. We first consider a Lavrentiev regularization of the state-constrained optimization problem. Then, a multigrid scheme is designed for the numerical solution of the regularized optimality system. Central to this scheme is the construction of an iterative pointwise smoother which can be formulated as a local semismooth Newton iteration. Results of numerical experiments and theoretical two-grid local Fourier analysis estimates demonstrate that the proposed scheme is able to solve parabolic state-constrained optimality systems with textbook multigrid efficiency.     

高速运动裂纹扩展和分叉现象的近场动力学数值模拟

首先介绍了近场动力学的基本理论,然后以两个实例分析了高速运动裂纹的扩展及分叉现象.分析了近场动力学参数(邻域半径、相邻节点距)及外部参数(材料的弹性模量、密度、温度改变量)等对裂纹分叉的速度和角度的影响并进行了对比分析,数值结果表明:随着邻域半径的增大,裂纹传播速度逐渐减少而裂纹分叉角度逐渐增加;随着相邻节点间距的增加,裂纹的传播速度逐渐减少而裂纹分叉角度也逐渐减少;裂纹分叉长度偏向于弹性模量小和密度大的材料;裂纹传播速度随着弹性模量差值的增大而增大,随着密度差值的减小而增大,同时随着外界温度改变量的增大而减少.近场动力学能自发地模拟裂纹扩展和分叉,不需要借助任何外部准则,不需要预先设置裂纹扩展路径,因此它具有天然的优势.     

RBF‐based discrete sliding mode control for robust tracking of uncertain time‐delay systems with input nonlinearity

In this article, a control scheme combining radial basis function neural network and discrete sliding mode control method is proposed for robust tracking and model following of uncertain time‐delay systems with input nonlinearity. The proposed robust tracking controller guarantees the stability of overall closed‐loop system and achieves zero‐tracking error in the presence of input nonlinearity, time‐delays, time‐varying parameter uncertainties, and external disturbances. The salient features of the proposed controller include no requirement of a priori knowledge of the upper bound of uncertainties and the elimination of chattering phenomenon and reaching phase. Simulation results are presented to demonstrate the effectiveness of the proposed scheme. © 2015 Wiley Periodicals, Inc. Complexity 21: 194–201, 2016     

Finite valued feedback laws and piecewise classical solutions

Often, in engineering literature, we find control systems in which the open loop inputs are piecewise constant and take values in a finite set. Such open loop inputs cause the system to have fairly regular solutions. On the other hand, when acting in closed loop, feedback laws taking values in a finite set may not be reinterpreted as open loop inputs of the considered type. In fact, pathological behaviours such as the accumulation of discontinuities may appear (Zeno phenomenon). We give some conditions which can be used as tools for building finite valued feedback laws not causing such pathological behaviours.     

The Dirichlet-Jordan test and multidimensional extensions

If is a foliation of an open set by smooth -dimensional surfaces, we define a class of functions , supported in , that are, roughly speaking, smooth along and of bounded variation transverse to . We investigate geometrical conditions on that imply results on pointwise Fourier inversion for these functions. We also note similar results for functions on spheres, on compact 2-dimensional manifolds, and on the 3-dimensional torus. These results are multidimensional analogues of the classical Dirichlet-Jordan test of pointwise convergence of Fourier series in one variable.

     

STABILITY OF EQUILIBRIA OF NEMATIC LIQUID CRYSTALLINE POLYMERS

We provide an analytical study on the stability of equilibria of rigid rodlike nematic liquid crystalline polymers (LCPs) governed by the Smoluchowski equation with the Maier-Saupe intermolecular potential. We simplify the expression of the free energy of an orientational distribution function of rodlike LCP molecules by properly selecting a coordinate system and then investigate its stability with respect to perturbations of orientational probability density. By computing the Hessian matrix explicitly, we are able to prove the hysteresis phenomenon of nematic LCPs: when the normalized polymer concentration b is below a critical value b*(6.7314863965), the only equilibrium state is isotropic and it is stable; when b*< b < 15/2, two anisotropic (prolate) equilibrium states occur together with a stable isotropic equilibrium state. Here the more aligned prolate state is stable whereas the less aligned prolate state is unstable. When b > 15/2, there are three equilibrium states: a stable prolate state, an unstable isotropic state and an unstable oblate state.