Simulation from Wishart Distributions with Eigenvalue Constraints

Abstract

This article provides an efficient algorithm for generating a random matrix according to a Wishart distribution, but with eigenvalues constrained to be less than a given vector of positive values. The procedure of Odell and Feiveson provides a guide, but the modifications here ensure that the diagonal elements of a candidate matrix are less than the corresponding elements of the constraint vector, thus greatly improving the chances that the matrix will be acceptable. The Normal hierarchical model with vector outcomes and the multivariate random effects model provide motivating applications.     

Inverse acoustical characterization of natural jute sound absorbing material by the particle swarm optimization method

For modeling of jute as acoustic material, knowledge of its non-acoustical parameters like porosity, tortuosity, air flow resistivity, thermal and viscous characteristic lengths is a prime requisite. Measurement of these non-acoustical parameters is not straightforward and involves a dedicated measurement setup. So in order to overcome this issue, the inverse acoustical characterization can be used. In this paper, the particle swarm optimization method (PSO) is used as an optimization method. This method estimates the non-acoustical parameters of jute material in felt form by minimizing the error between experimental and theoretical sound absorption data. In this work, the impedance prediction models for fibrous materials like Johnson–Champoux–Allard model with rigid and limp frame and Garai–Pompoli model is used for sound absorption coefficient calculation by the transfer matrix method along with the PSO. The inverse estimated non-acoustical parameters for jute material are then compared with estimated and experimentally measured parameters for jute felts. Using these inversely predicted parameters, sound absorption of multilayer sound absorbers is also studied.     

折叠式角锥棱镜谐振腔的偏振特性理论与实验研究

建立了角锥棱镜的偏振模型,分析了角锥棱镜对入射线偏振光的偏振态的影响,分析表明角锥棱镜的单个入射区域存在特殊的偏振方向,该方向的线偏振光被角锥棱镜反射后仍然是线偏振光。利用角锥棱镜的该特性设计了腔内存在线偏振光振荡的角锥棱镜激光谐振腔,能够使用电光开关调Q,在振动环境中具有较好的抗失调稳定性。研制了激光器样机并进行了实验,在重复频率为10 Hz时,输出激光单脉冲能量大于300 mJ,光束质量因子β值在2左右,实验结果也表明角锥棱镜的棱和顶点并不影响输出激光的光束质量。     

Generating orthogonal matrix polynomials satisfying second order differential equations from a trio of triangular matrices

The method developed in [A.J. Durán, F.A. Grünbaum, Orthogonal matrix polynomials satisfying second order differential equations, Int. Math. Res. Not. 10 (2004) 461–484] led us to consider matrix polynomials that are orthogonal with respect to weight matrices W(t) of the form , , and (1−t)^α(1+t)^βT(t)T^*(t), with T satisfying T^′=(2Bt+A)T, T(0)=I, T^′=(A+B/t)T, T(1)=I, and T^′(t)=(−A/(1−t)+B/(1+t))T, T(0)=I, respectively. Here A and B are in general two non-commuting matrices. We are interested in sequences of orthogonal polynomials (P_n)_n which also satisfy a second order differential equation with differential coefficients that are matrix polynomials F₂, F₁ and F₀ (independent of n) of degrees not bigger than 2, 1 and 0 respectively. To proceed further and find situations where these second order differential equations hold, we only dealt with the case when one of the matrices A or B vanishes.The purpose of this paper is to show a method which allows us to deal with the case when A, B and F₀ are simultaneously triangularizable (but without making any commutativity assumption).     

Cayley digraphs with normal adjacency matrices

In this paper we give a criterion for the adjacency matrix of a Cayley digraph to be normal in terms of the Cayley subset S. It is shown with the use of this result that the adjacency matrix of every Cayley digraph on a finite group G is normal iff G is either abelian or has the form for some non-negative integer n, where Q₈ is the quaternion group and is the abelian group of order 2ⁿ and exponent 2.     

矩阵方程AX=B的双反对称最佳逼近解

本文主要讨论下而两个问题并得到相关结果:问题Ⅰ:给定A ∈ R~(k×n),B ∈ R~(k×n),求X ∈ BASR~(n×n),使得AX=B.问题Ⅱ:给定X* ∈R~(n×n),求X使得‖X-X~*‖=minX∈S_E‖X-X~*‖,其中S_E是问题Ⅰ的解集合,‖·‖是Frobenius范数.通过对上述问题的讨论给出了问题Ⅰ解存在的充分必要条件和其解的一般表达式同时给出了问题Ⅱ的解,算法,和数值例子.     

Divisibility properties of power LCM matrices by power GCD matrices on gcd-closed sets

Let e and n be positive integers and S={x₁,…,x_n} a set of n distinct positive integers. For x∈S, define . The n×n matrix whose (i,j)-entry is the eth power (x_i,x_j)^e of the greatest common divisor of x_i and x_j is called the eth power GCD matrix on S, denoted by (S^e). Similarly we can define the eth power LCM matrix [S^e]. Bourque and Ligh showed that (S)∣[S] holds in the ring of n×n matrices over the integers if S is factor closed. Hong showed that for any gcd-closed set S with |S|≤3, (S)∣[S]. Meanwhile Hong proved that there is a gcd-closed set S with max_x∈S{|G_S(x)|}=2 such that (S)?[S]. In this paper, we introduce a new method to study systematically the divisibility for the case max_x∈S{|G_S(x)|}≤2. We give a new proof of Hong’s conjecture and obtain necessary and sufficient conditions on the gcd-closed set S with max_x∈S{|G_S(x)|}=2 such that (S^e)|[S^e]. This partially solves an open question raised by Hong. Furthermore, we show that such factorization holds if S is a gcd-closed set such that each element is a prime power or the product of two distinct primes, and in particular if S is a gcd-closed set with every element less than 12.     

一般矩阵函数的不等式

本文研究一般矩阵函数不等式.运用一般矩阵函数的两个公式和lp-范数的性质,结出了两个矩阵运算的一般矩阵函数不等式.     

SIMS of organic layers with unknown matrix parameters: Locating the interface in dual beam argon gas cluster depth profiles

Four simple methods are evaluated to determine their accuracies for establishing the interface location in secondary ion mass spectrometry intensity depth profiles of organic layers where matrix effects have not been measured. Accurate location requires the separate measurement of each ion's matrix factor. This is often not possible, and so estimates using matrix-less methods are required. Six pure organic material interfaces are measured using many secondary ions to compare their locations from the four methods with those from full evaluation with matrix terms. For different secondary ions, matrix effects cause the apparent interface positions to vary over 20 nm. The shifts in the intensity profiles on going from a layer of P into a layer of Q are in the opposite direction to that for going from Q into P, so doubling layer thickness errors. The four methods are as follows: M1, use of the median interface position in the intensity profiles for the five lightest ions for 15 ≤ m/z ≤ 150; M2, extrapolation of the position for each ion to m/z = 0 for ions with m/z ≤ 150; M3, as M2 but for m/z ≤ 300; and M4, the extreme positions for all m/z ≤ 100. Comparison with the location using matrix terms shows their ranking, from best to worst, to be M4, M3, M1, and M2 with average errors of 10%, 12%, 14%, and 17%, respectively, of the profile interface full widths at half maximum. Use of pseudo-molecular ions is very much poorer, exceeding 50%, and should be avoided.     

内含柱面透镜的象散腔

本文用矩阵光学的方法研究了内含柱面透镜的象散腔，给出了其稳定图，此图清楚地显示出由于象散作而使用腔稳定工作区域大大缩小，本文还给了显示象散特征的稳定脸膜参数计算公式，并作了数值计算。