分裂四元数的实表示矩阵的棣莫弗定理

依据分裂四元数的概念,首先,给出了分裂四元数的实表示矩阵形式,将对分裂四元数的研究转化为实数域上四阶矩阵的研究;其次,根据分裂四元数不同的极表示形式,得到了分裂四元数实表示矩阵相应的棣莫弗定理,推广了欧拉公式,并给出了单位分裂四元数的实表示矩阵方程的求根公式.最后,通过算例验证了结论的正确性.     

块H－矩阵的进一步推广

对Ｎａｂｂｅｎ［２］提出的块Ｈ－矩阵做进一步推广，使得非对角块矩阵不必是Ｈｅｒｍｉｔｅ矩阵．但仍保留其基本特征不变．对块Ｈ－矩阵提出块Ｈ－分裂及块相容Ｈ－分裂．证明了矩阵的任意块相容Ｈ－分裂都是收敛分裂．对ＪＯＲ迭代法给出松驰参数的上界．     

广义并行矩阵多分裂松弛算法

求解大型线性代数方程组的并行矩阵多分裂算法讨论的大多为系数矩阵是非奇日矩阵的情况，[2]提出了当系数矩阵是非奇H矩阵时的广义矩阵多分裂松弛算法．对系数矩阵是奇异日矩阵的情况研究较少，本文给出了当系数矩阵G是不可约奇异H矩阵时的齐次线性方程组Gx=0的广义矩阵多分裂松弛算法并讨论其收敛性。     

关于块有限无限方阵的非异性

利用有向图给出了块有限无限方阵的概念 ,得到了该类矩阵的置换相似标准形 ,以及该类矩阵非异性的一个充分必要条件 .     

求解广义绝对值方程的交替牛顿矩阵多分裂方法

本文针对广义绝对值方程,提出了基于牛顿法的矩阵多分裂方法.并在该方法的基础上进一步改进,得到了基于牛顿法的交替矩阵多分裂方法.给出两种算法在一定条件下的全局收敛性,并分析当分裂为H分裂时,基于牛顿法的矩阵多分裂方法的收敛条件.通过数值实验验证了所提出的算法的可行性和有效性.     

矩阵多分裂迭代法的收敛条件

本文给出了求解非奇异线性方程组的矩阵多分裂并行迭代法的一些新的收敛结果.当系数矩阵单调和多分裂序列为弱正则分裂时,得到了几个与已有的收敛准则等价的条件,并且证明了异步迭代法在较弱条件下的收敛性.对于同步迭代,给出了与异步迭代不同且较为宽松的收敛条件.     

完备分配格上的∧ - →型矩阵方程

给出了完备分配格∧ - →型矩阵方程可解的充要条件，并且求出了该类方程的所有极大解     

阶梯矩阵及其一般化在迭代法中的应用

Lu Hao首先给出了阶梯矩阵及其一般性的定义和性质．这类矩阵为迭代法提供了新矩阵分裂的基础．基于此新矩阵类的迭代方法的显著特征是它对于并行计算很容易被实现．应用这一新的分解方法，给出了一般的加速松弛方法（GAOR），而关于AOR方法的一些性质可以被延伸到该新方法中，并针对Hermite正定矩阵进行了新方法收敛性的分析．最后，给出了一些例子来表明新方法的优越性．     

一类弱非线性互补问题的广义模系矩阵多分裂多参数加速松弛迭代方法

本文提出一类求解弱非线性互补问题的广义模系矩阵多分裂多参数加速松弛迭代方法，并给出了系数矩阵为H₊-矩阵时该方法的收敛性分析.数值实验表明新方法是有效的.     

求解-类马尔可夫链方程的逆矩阵

一类马尔可夫链方程的求逆过程进行研究,该求逆是用该类马氏链解析地研究客户关系管理问题的关键环节,研究给出了该类矩阵逆矩阵的解析解.     

A modified iterated operator splitting method

In this paper we give a short overview of some traditional operator splitting methods. Furthermore, we introduce two recently developed methods, namely the additive splitting and the iterated splitting. We analyze the iterated splitting method in detail and give the suitable strategy for the choice of the initial elements in the iterations in order to get higher order discretization.     

Player splitting in extensive form games

By a player splitting we mean a mechanism that distributes the information sets of a player among so-called agents. A player splitting is called independent if each path in the game tree contains at most one agent of every player. Following Mertens (1989), a solution is said to have the player splitting property if, roughly speaking, the solution of an extensive form game does not change by applying independent player splittings. We show that Nash equilibria, perfect equilibria, Kohlberg-Mertens stable sets and Mertens stable sets have the player splitting property. An example is given to show that the proper equilibrium concept does not satisfy the player splitting property. Next, we give a definition of invariance under (general) player splittings which is an extension of the player splitting property to the situation where we also allow for dependent player splittings. We come to the conclusion that, for any given dependent player splitting, each of the above solutions is not invariant under this player splitting. The results are used to give several characterizations of the class of independent player splittings and the class of single appearance structures by means of invariance of solution concepts under player splittings. Received: December 1996/Revised Version: January 2000     

Exponentially Small Splitting for the Pendulum: A Classical Problem Revisited

In this paper, we study the classical problem of the exponentially small splitting of separatrices of the rapidly forced pendulum. Firstly, we give an asymptotic formula for the distance between the perturbed invariant manifolds in the so-called singular case and we compare it with the prediction of Melnikov theory. Secondly, we give exponentially small upper bounds in some cases in which the perturbation is bigger than in the singular case and we give some heuristic ideas how to obtain an asymptotic formula for these cases. Finally, we study how the splitting of separatrices behaves when the parameters are close to a codimension-2 bifurcation point.     

A wider convergence area for the MSTMAOR iteration methods for LCP

In this paper, based on the Hermitian and skew-Hermitian splitting, we give a generalized modified Hermitian and skew-Hermitian splitting (GMHSS) method to solve singular complex symmetric linear systems, this method has two parameters. We give the semi-convergent conditions, and some numerical experiments are given to illustrate the efficiency of this method.     

关于鞍点问题的广义预处理HSS-SOR交替分裂迭代方法

本文研究了鞍点问题的迭代法. 在白中治,Golub和潘建瑜提出的预处理对称/反对称分裂（PHSS）迭代法的基础上,通过结合GSOR迭代格式,利用两个参数加速,提出了一种广义预处理HSS-SOR交替分裂迭代法,并研究了该方法的收敛性.数值结果表明本文所给方法是有效的.     

Privacy-preserving data splitting: a combinatorial approach

Privacy-preserving data splitting is a technique that aims to protect data privacy by storing different fragments of data in different locations. In this work we give a new combinatorial formulation to the data splitting problem. We see the data splitting problem as a purely combinatorial problem, in which we have to split data attributes into different fragments in a way that satisfies certain combinatorial properties derived from processing and privacy constraints. Using this formulation, we develop new combinatorial and algebraic techniques to obtain solutions to the data splitting problem. We present an algebraic method which builds an optimal data splitting solution by using Gröbner bases. Since this method is not efficient in general, we also develop a greedy algorithm for finding solutions that are not necessarily minimally sized.

     

The modulus‐based matrix splitting algorithms for a class of weakly nonlinear complementarity problems

In this paper, we study a class of weakly nonlinear complementarity problems arising from the discretization of free boundary problems. By reformulating the complementarity problems as implicit fixed‐point equations based on splitting of the system matrices, we propose a class of modulus‐based matrix splitting algorithms. We show their convergence by assuming that the system matrix is positive definite. Moreover, we give several kinds of typical practical choices of the modulus‐based matrix splitting iteration methods based on the different splitting of the system matrix. Numerical experiments on two model problems are presented to illustrate the theoretical results and examine the numerical effectiveness of our modulus‐based matrix splitting algorithms. Copyright © 2016 John Wiley & Sons, Ltd.     

Survivors in leader election algorithms

We consider the number of survivors in a broad class of fair leader election algorithms after a number of election rounds. We give sufficient conditions for the number of survivors to converge to a product of independent identically distributed random variables. The number of terms in the product is determined by the round number considered. Each individual term in the product is a limit of a scaled random variable associated with the splitting protocol. The proof is established via convergence (to 0) of the first-order Wasserstein distance from the product limit. In a broader context, the paper is a case study of a class of stochastic recursive equations. We give two illustrative examples, one with binomial splitting protocol (for which we show that a normalized version is asymptotically Gaussian) and one with uniform splitting protocol.     

Splitting Patterns and Invariants of Quadratic Forms

Let φ be an anisotropic quadratic form over a field F of characteristic not 2. The splitting pattern of φ is defined to be the increasing sequence of nonnegative integers obtained by considering the Witt indices i_W(φ_k) of φ over K where K ranges over all field extensions of F. Restating earlier results by HURRELBRINK and REHMANN , we show how the index of the Clifford algebra of φ influences the splitting pattern. In the case where F is formally real, we investigate how the signatures of φ influence the splitting behaviour. This enables us to construct certain splitting patterns which have been known to exist, but now over much “simpler” fields like formally real global fields or ?(t). We also give a full classification of splitting patterns of forms of dimension less than or equal to 9 in terms of properties of the determinant and Clifford invariant. Partial results for splitting patterns in dimensions 10 and 11 are also provided. Finally, we consider two anisotropic forms φ and φ of the same dimension m with φ ? ? φ ∈ Iⁿ F and give some bounds on m depending on n which assure that they have the same splitting pattern.     

Splitting balanced incomplete block designs with block size 3 × 2

Splitting balanced incomplete block designs were first formulated by Ogata, Kurosawa, Stinson, and Saido recently in the investigation of authentication codes. This article investigates the existence of splitting balanced incomplete block designs, i.e., (v, 3k, λ)‐splitting BIBDs; we give the spectrum of (v, 3 × 2, λ)‐splitting BIBDs. As an application, we obtain an infinite class of 2‐splitting A‐codes. © 2004 Wiley Periodicals, Inc.