Numerical analysis of a quadratic matrix equation

The quadratic matrix equation AX²+ BX + C = 0in n x nmatricesarises in applications and is of intrinsic interest as oneof the simplest nonlinear matrix equations. We give a completecharacterization of solutions in terms of the generalized Schurdecomposition and describe and compare various numerical solutiontechniques. In particular, we give a thorough treatment offunctional iteration methods based on Bernoulli’s method.Other methods considered include Newton’s method with exact line searches, symbolic solution and continued fractions.We show that functional iteration applied to the quadraticmatrix equation can provide an efficient way to solve the associated quadratic eigenvalue problem (²A + B + C)x = 0.     

The Genetic Sums' Representation for the Moments of a System of Stieltjes Functions and its Application

This paper deals with the spectral problems for high-order nonsymmetric difference operators. The method of investigation is based on the analysis of the genetic sums formulas for the moments of the operator. The parameters of these sums are shown to be connected with coefficients of the introduced vector Stieltjes continued fraction. The connections with vector orthogonality, Hermite—Padé approximation, and Hankel determinants are investigated. This gives a tool for the analysis of the solution of the direct and inverse spectral problem of the operator. It is applied to the integration of hierarchy of the discrete KdV equations. The existence of a global solution is proved. July 13, 1998. Date revised: July 12, 1999. Date accepted: July 26, 1999.     

基于Samelson逆的向量值连分式的收敛准则

The aim of this work is to give some criteria on the convergence of vector valued continued fractions defined by Samelson inverse. We give a new approach to prove the convergence theory of continued fractions. First, by means of the modified classical backward recurrence relation, we obtain a formula between the m-th and n-th convergence of vector valued continued fractions. Second, using this formula, we give necessary and sufficient conditions for the convergence of vector valued continued fractions.     

Self-complementary magic squares of doubly even orders

A magic square $M$ in which the entries consist of consecutive integers from $1,2,\dots ,n^2$ is said to be self-complementary of order$n$ if the resulting square obtained from $M$ by replacing each entry $i$ by $n^2+1?i$ is equivalent to $M$ (under rotation or reflection). We present a new construction for self-complementary magic squares of order $n$ for each $n\ge 4$, where $n$ is a multiple of $4$.     

Explicit continued fractions with expected partial quotient growth

For let be the continued fraction expansion of . Write

We construct some numbers 's with

     

The Phase Transition in Statistical Models Defined on Farey Fractions

We consider several statistical models defined on the Farey fractions. Two of these models may be regarded as spin chains, with long-range interactions, while another arises in the study of multifractals associated with chaotic maps exhibiting intermittency. We prove that these models all have the same free energy. Their thermodynamic behavior is determined by the spectrum of the transfer operator (Ruelle–Perron–Frobenius operator), which is defined using the maps (presentation functions) generating the Farey tree. The spectrum of this operator was completely determined by Prellberg. It follows that these models have a second-order phase transition with a specific heat divergence of the form C [ ln² ]^–1. The spin chain models are also rigorously known to have a discontinuity in the magnetization at the phase transition.     

交换半群环的分式环

设含幺交换环Ｒ对其乘法子集Ｔ的分式环为ＲＴ，交换幺半群Ｓ在其子半群∑处局部化为Ｓ∑本文证明了Ｒ［Ｓ］对于Ａ的分环式环Ｒ［Ｓ］ＡＭ 构于半群环ＲＴ［Ｓ∑］。     

Multiple ionization of argon in coincidence with projectile ions in 60–120 MeV Si q+-Ar collisions

A projectile ion-recoil ion coincidence technique has been employed to study the multiple ionization and the charge transfer processes in collisions of 60–120 MeV Si ^q+ (q = 4−14) ions with neutral argon atoms. The relative contribution of different ionization channels, namely; direct ionization, electron capture and electron loss leading to the production of slow moving multiply charged argon recoil ions have been investigated. The data reported on the present collision system result from a direct measurement in the considered impact energy for the first time. The total ionization cross-sections for the recoil ions are shown to scale as q ^1.7/E _p ^0.5 , where E _p is the energy in MeV of the projectile and q its charge state. The recoil fractions for the cases of total- and direct ionizations are found to decrease with increasing recoil charge state j. The total ionization fractions of the recoils are seen to depend on q and to show the presence of a ‘shell-effect’ of the target. Further, the fractions are found to vary as 1/j ² upto j = 8+. The average recoil charge state 〈j〉 increases slowly with q and with the number of lost or captured electrons from or into the projectile respectively. The projectile charge changing cross-sections σ _qq′ are found to decrease with increasing q for loss ionization and to increase with q for direct-and capture ionization processes respectively. The physics behind various scaling rules that are found to follow our data for different ionization processes is reviewed and discussed.     

光度法对不同土地利用类型河流在不同水期氮磷分布特性及定量源解析研究

氮和磷是水环境生物生长和繁殖必须的营养成分,影响水体初级生产力的水平,且水体富营养化水平与氮磷形态密切相关,随着水体环境的改变,沉积物会向水体释放氮磷,造成二次污染。同时,对外源氮磷污染来源的贡献进行定量识别,可有效管理和控制水体氮磷污染负荷。毗河和石亭江是沱江的重要支流,影响着长江母亲河的水质。采用钼锑抗分光光度法和连续提取法研究在枯水期和丰水期毗河和石亭江水体和表层沉积物中总氮(TN)、总磷(TP)及各形态氮磷的分布特性,对比不同土地利用类型河流氮磷行为特性及释放风险,并采用Multiple Linear Regression of the Absolute Principal Component Scores(APCS-MLR)受体模型进行氮磷污染源的识别和量化。研究结果表明：(1)研究区水体和表层沉积物中氮磷均处于不同程度的污染水平,水体枯水期TP的主要贡献者是颗粒态无机磷(PIP)和颗粒态有机磷(POP),而丰水期却是颗粒态无机磷(PIP)和溶解态无机磷(DIP),两水期水体中TN的主要贡献者是硝态氮(NO^-₃-N)和有机氮(ON)...