On convergent numerical algorithms for unsymmetric collocation

In this paper, we are interested in some convergent formulations for the unsymmetric collocation method or the so-called Kansa’s method. We review some newly developed theories on solvability and convergence. The rates of convergence of these variations of Kansa’s method are examined and verified in arbitrary–precision computations. Numerical examples confirm with the theories that the modified Kansa’s method converges faster than the interpolant to the solution; that is, exponential convergence for the multiquadric and Gaussian radial basis functions (RBFs). Some numerical algorithms are proposed for efficiency and accuracy in practical applications of Kansa’s method. In double–precision, even for very large RBF shape parameters, we show that the modified Kansa’s method, through a subspace selection using a greedy algorithm, can produce acceptable approximate solutions. A benchmark algorithm is used to verify the optimality of the selection process.     

Gauss-Green cubature and moment computation over arbitrary geometries

We have implemented in Matlab a Gauss-like cubature formula over arbitrary bivariate domains with a piecewise regular boundary, which is tracked by splines of maximum degree p (spline curvilinear polygons). The formula is exact for polynomials of degree at most 2n−1 using N∼cmn² nodes, 1≤c≤p, m being the total number of points given on the boundary. It does not need any decomposition of the domain, but relies directly on univariate Gauss-Legendre quadrature via Green’s integral formula. Several numerical tests are presented, including computation of standard as well as orthogonal moments over a nonstandard planar region.     

Randomized broadcast in networks

In this paper we study the rate at which a rumor spreads through an undirected graph. This study has two important applications in distributed computation: in simple, robust and efficient broadcast protocols, and in the maintenance of replicated databases.     

海量数据导入与导出MATLAB的有效方法

本文结合实例介绍了几种将海量统计数据快速、准确地导入与导出MATLAB系统的方法。克服了以往在编程窗口直接键入原始数据进行处理的诸多问题,因此对保证统计分析结果的准确性与加速MATLAB在统计分析领域的应用有着很重要的意义。     

Spatial chaotic structure of attractors of reaction-diffusion systems

The dynamics described by a system of reaction-diffusion equations with a nonlinear potential exhibits complicated spatial patterns. These patterns emerge from preservation of homotopy classes of solutions with bounded energies. Chaotically arranged stable patterns exist because of realizability of all elements of a fundamental homotopy group of a fixed degree. This group corresponds to level sets of the potential. The estimates of homotopy complexity of attractors are obtained in terms of geometric characteristics of the potential and other data of the problem.

     

一种形式化验证方法:模型检验

模型检验作为一种形式化验证方法,近年来在各种硬件、软件设计中得到了广泛应用．文中首先介绍了描述系统行为的Kripke结构和描述系统性质的CTL逻辑,然后介绍了模型检验中常用的两种算法：标记算法和基于固定点的算法,最后介绍了为避免内存爆炸而引入的符号模型检验技术．     

Ito 5阶和7阶mKdV方程新的周期解

§ 1　IntroductionIn recent years,a research of direct seeking for exactly solutions of nonlinear waveequations (NL WEs) has attracted much attention because of the availability of symboliccom- putation.Many methods,such as the homogeneous balance method[1 ] ,extended tanh-function method[2 ] ,Jocobi elliptic function expansion m ethod[3] ,are developed for thepurpose.In [4 ]the cn- function method,dn- function method and cs- function method wereproposed and applied to some NL WEs.It is sh…     

Enhanced logical stochastic resonance under periodic forcing

It was demonstrated recently that noise in an optimal window allows a bistable system to operate reliably as reconfigurable logic gates (Murali et al., 2009) [1], as well as a memory device (Kohar and Sinha, 2012) [11]. Namely, in a range of moderate noise, the system can operate flexibly, both as a NAND/AND/OR/NOR gate and a Set Reset latch. Here we demonstrate how the width of the optimal noise window can be increased by utilizing the constructive interplay of noise and periodic forcing, namely noise in conjunction with a periodic drive yields consistent logic outputs for all noise strengths below a certain threshold. Thus we establish that in scenarios where noise level is below the minimum threshold required for logical stochastic resonance (or stochastic resonance in general), we can add a periodic forcing to obtain the desired effects. Lastly, we also show how periodic forcing reduces the switching time, leading to faster operation of devices and lower latency effects.     

Sub-exponential graph coloring algorithm for stencil-based Jacobian computations

Partial differential equations can be discretized using a regular Cartesian grid and a stencil-based method to approximate the partial derivatives. The computational effort for determining the associated Jacobian matrix can be reduced. This reduction can be modeled as a (grid) coloring problem. Currently, this problem is solved by using a heuristic approach for general graphs or by developing a formula for every single stencil. We introduce a sub-exponential algorithm using the Lipton–Tarjan separator in a divide-and-conquer approach to compute an optimal coloring. The practical relevance of the algorithm is evaluated when compared with an exponential algorithm and a greedy heuristic.     

“数学实验”在解决几何问题中的应用

阐述"数学实验"在解决几何问题时的可视化优点,列举"数学实验"的工具,给出具体实例,说明"数学实验"在分析、解决几何问题中的有效途径.