Asynchronous cellular automata and pattern classification

This article designs an efficient two‐class pattern classifier utilizing asynchronous cellular automata (ACAs). The two‐state three‐neighborhood one‐dimensional ACAs that converge to fixed points from arbitrary seeds are used here for pattern classification. To design the classifier, (1) we first identify a set of ACAs that always converge to fixed points from any seeds, (2) each ACA should have at least two but not huge number of fixed point attractors, and (3) the convergence time of these ACAs are not to be exponential. To address the second issue, we propose a graph, coined as fixed point graph of an ACA that facilitates in counting the fixed points. We further perform an experimental study to estimate the convergence time of ACAs, and find there are some convergent ACAs which demand exponential convergence time. Finally, we identify there are 73 (out of 256) ACAs which can be effective candidates as pattern classifier. We use each of the candidate ACAs on some standard datasets, and observe the effectiveness of each ACAs as pattern classifier. It is observed that the proposed classifier is very competitive and performs reliably better than many standard existing classifier algorithms. © 2016 Wiley Periodicals, Inc. Complexity 21: 370–386, 2016     

Fast tensor method for summation of long‐range potentials on 3D lattices with defects

In this paper, we present a method for fast summation of long‐range potentials on 3D lattices with multiple defects and having non‐rectangular geometries, based on rank‐structured tensor representations. This is a significant generalization of our recent technique for the grid‐based summation of electrostatic potentials on the rectangular L × L × L lattices by using the canonical tensor decompositions and yielding the O(L) computational complexity instead of O(L³) by traditional approaches. The resulting lattice sum is calculated as a Tucker or canonical representation whose directional vectors are assembled by the 1D summation of the generating vectors for the shifted reference tensor, once precomputed on large N × N × N representation grid in a 3D bounding box. The tensor numerical treatment of defects is performed in an algebraic way by simple summation of tensors in the canonical or Tucker formats. To diminish the considerable increase in the tensor rank of the resulting potential sum, the ?‐rank reduction procedure is applied based on the generalized reduced higher‐order SVD scheme. For the reduced higher‐order SVD approximation to a sum of canonical/Tucker tensors, we prove the stable error bounds in the relative norm in terms of discarded singular values of the side matrices. The required storage scales linearly in the 1D grid‐size, O(N), while the numerical cost is estimated by O(NL). The approach applies to a general class of kernel functions including those for the Newton, Slater, Yukawa, Lennard‐Jones, and dipole‐dipole interactions. Numerical tests confirm the efficiency of the presented tensor summation method; we demonstrate that a sum of millions of Newton kernels on a 3D lattice with defects/impurities can be computed in seconds in Matlab implementation. The tensor approach is advantageous in further functional calculus with the lattice potential sums represented on a 3D grid, like integration or differentiation, using tensor arithmetics of 1D complexity. Copyright © 2015 John Wiley & Sons, Ltd.     

Structure of random 312‐avoiding permutations

We evaluate the probabilities of various events under the uniform distribution on the set of 312‐avoiding permutations of . We derive exact formulas for the probability that the i^th element of a random permutation is a specific value less than i, and for joint probabilities of two such events. In addition, we obtain asymptotic approximations to these probabilities for large N when the elements are not close to the boundaries or to each other. We also evaluate the probability that the graph of a random 312‐avoiding permutation has k specified decreasing points, and we show that for large N the points below the diagonal look like trajectories of a random walk. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 49, 599–631, 2016     

On the inertia sets of some symmetric sign patterns

A matrix whose entries consist of elements from the set {+, −, 0} is a sign pattern matrix. Using a linear algebra theoretical approach we generalize of some recent results due to Hall, Li and others involving the inertia of symmetric tridiagonal sign matrices.     

谱任意的符号模式矩阵

一个n阶符号模式矩阵A称为是谱任意的,如果对任意的实系数n次首1多项式r(x),在A的定性矩阵类Q(A)中至少存在一个实矩阵B,使得B的特征多项式是r(x),文中证明了当n为奇数时n阶谱任意符号模式矩阵是存在的。     

On the construction of samples for testing approximate optimization methods for estimate calculation algorithms

To validate approximate optimization schemes for estimate calculation algorithms (ECAs), it is necessary to compute the optimal height, which cannot be done in a reasonable amount of time. A variety of samples are built for which the optimal height of the ECAs is known by construction.     

电子结构计算的数值方法与理论

第一原理电子结构计算已成为探索与研究物质机理、理解与预测材料性质的重要手段和工具.虽然第一原理电子结构计算取得了巨大的成功,但是如何利用高性能计算机又快又好地计算大规模体系,如何从数学角度理解电子结构模型的合理性与计算的可靠性和有效性,依然充满各种挑战.基于密度泛函理论的第一原理电子结构计算的核心数学模型为Kohn-Sham方程或相应的Kohn-Sham能量泛函极小问题.近年来,人们分别从非线性算子特征值问题的高效离散及Kohn-Sham能量泛函极小问题的最优化方法设计两个方面对电子结构计算的高效算法设计及分析展开了诸多研究.本文重点介绍我们小组在电子结构计算的方法与理论方面的一些进展,同时简单介绍该领域存在的困难与挑战.     

平方非正的符号矩阵的平方中负元个数

本文给出了一个符号矩阵A当A^2非正时A^2中负元个数N－(A^2)的上,下界,并确定相应的极矩阵。     

Turing vegetation patterns in a generalized hyperbolic Klausmeier model

The formation of Turing vegetation patterns in flat arid environments is investigated in the framework of a generalized version of the hyperbolic Klausmeier model. The extensions here considered involve, on the one hand, the strength of the rate at which rainfall water enters into the soil and, on the other hand, the functional dependence of the inertial times on vegetation biomass and soil water. The study aims at elucidating how the inclusion of these generalized quantities affects the onset of bifurcation of supercritical Turing patterns as well as the transient dynamics observed from an uniformly vegetated state towards a patterned state. To achieve these goals, linear and multiple-scales weakly nonlinear stability analysis are addressed, this latter being inspected in both large and small spatial domains. Analytical results are then corroborated by numerical simulations, which also serve to describe more deeply the spatio-temporal evolution of the emerging patterns as well as to characterize the different timescales involved in vegetation dynamics.