Ritz and harmonic Ritz values and the convergence of FOM and GMRES

The Ritz and harmonic Ritz values are approximate eigenvalues, which can be computed cheaply within the FOM and GMRES Krylov subspace iterative methods for solving non‐symmetric linear systems. They are also the zeros of the residual polynomials of FOM and GMRES, respectively. In this paper we show that the Walker–Zhou interpretation of GMRES enables us to formulate the relation between the harmonic Ritz values and GMRES in the same way as the relation between the Ritz values and FOM. We present an upper bound for the norm of the difference between the matrices from which the Ritz and harmonic Ritz values are computed. The differences between the Ritz and harmonic Ritz values enable us to describe the breakdown of FOM and stagnation of GMRES. Copyright © 1999 John Wiley & Sons, Ltd.     

Diversity and Popularity in Organizations and Communities

This paper examines social groupings whose structure depends only on the distribution of ability to attract attention. When people‘s attention is a scarce resource, those individuals who are rated highest by a large number of other individuals will have to ration their attention, resulting in constraints on the social structure of the group. The incidence of popular individuals by that definition reflects the extent to which individuals agree on who their highest-rated colleague is. We propose three basic distributions or ways to generate the matrix of perceived ability so as to yield popularity profiles that can be parametrically adjusted to match observations. We demonstrate that each of these assumption sets leads to a slightly different correlation between the value of the assumption‘s parameter and the set of observable popularity patterns. Since popularity, in real life, often determines such things as power, centrality, over-utilization and perhaps reduced accessibility, having more realistic ways of representing it is important for modeling and understanding virtual organizations and communities.     

Multidimensional analogs of direct and converse Jackson and Bernshtein theorems and their generalizations

We obtain an instance of the multidimensional analogs of the classical direct and converse Jackson and Bernshtein-Vallée-Poussin theorems (a version stronger than those obtained by S. M. Nikol’skii) and significant generalizations of these theorems to polynomial approximations of functions of real variables from Nikol’skii and Besov spaces on bounded domains with Lipschitzian boundary. The direct theorem concerning the Nikol’skii spaces was previously obtained by Yu. A. Brudnyi and generalized to joint polynomial approximations of functions and their derivatives by B. N. Konovalov. Translated fromMatematicheskie Zametki, Vol. 67, No. 4, pp. 608–615, April, 2000.     

Polytopes and arrangements: Diameter and curvature

We introduce a continuous analogue of the Hirsch conjecture and a discrete analogue of the result of Dedieu, Malajovich and Shub. We prove a continuous analogue of the result of Holt and Klee, namely, we construct a family of polytopes which attain the conjectured order of the largest total curvature.     

E—matrices and Several Necessary and Sufficient Conditions

AER-x.,Aiscalledanonnegative(nonPositive)matrixifau>O(a,jO(aiirtO),markedA>O(AO(i#j),auO,wecallp(A)=max{IA,l,lA,l'.-.,lAnl},(whereA,,A2,..',A"areeigenva1ueofA)asspectralradiusofA.LemmalIfAe7'.,thenareanonnegativesquarematrxBandapositivenumberksuchthatA=B-kI(whereIistheidentitymatrixofordern).ProofLetbti=…     

Lebesgue测度等测包及其应用

利用Lebesgue测度的等测内核和等测外包讨论内外测度的一系列性质,给出外测度有限可加的一个充要条件,该条件改进了原有外测度有限可加性相关问题的结果.     

微言要义之数列的上下极限

本文分别从语义,数学和教学三个方面讨论上下极限与极限概念之间的差别和联系,以求厘清差别,透彻理解上下极限概念的本质.     

Singular quasilinear and Hessian equations and inequalities

We solve the existence problem in the renormalized, or viscosity sense, and obtain global pointwise estimates of solutions for quasilinear and Hessian equations with measure coefficients and data, including the following model problems:

     

Three- and Four-Dimensional Einstein-like Manifolds and Homogeneity

The aim of this paper is to study three- and four-dimensional Einstein-like Riemannian manifolds which are Ricci-curvature homogeneous, that is, have constant Ricci eigenvalues. In the three-dimensional case, we present the complete classification of these spaces while, in the four-dimensional case, this classification is obtained in the special case where the manifold is locally homogeneous. We also present explicit examples of four-dimensional locally homogeneous Riemannian manifolds whose Ricci tensor is cyclic-parallel (that is, are of type A) and has distinct eigenvalues. These examples are invalidating an expectation stated by F. Podestá and A. Spiro, and illustrating a striking contrast with the three-dimensional case (where this situation cannot occur). Finally, we also investigate the relation between three- and four-dimensional Einstein-like manifolds of type A and D'Atri spaces, that is, Riemannian manifolds whose geodesic symmetries are volume-preserving (up to sign).     

On exact and inexact differentials and applications

Considering the important role played by mathematical derivatives in the study of physical–chemical processes, this paper discusses the different possibilities and formulations of this concept and its application. In particular, in Chemical Thermodynamics, we study exact differentials associated with the so-called state functions and inexact differentials associated with path functions. As an application, we discuss the equation of state associated with a gas.     

Ideals and congruences in effect algebras and qmv-algebras

Ideals and sets of ideals of different types on QMV-algebras and the corresponding effect algebras are studied and compared.     

Dual and triple equations and q-orthogonal polynomials

In this paper, we solve dual and triple sequences involving q-orthogonal polynomials. We also introduce and solve a system of dual series equations when the kernel is the q-Laguerre polynomials. Examples are included.     

Navier and Stokes meet Poincare and Dulac

This paper surveys various precise (long-time) asymptotic results for the solutions of the Navier-Stokes equations with potential forces in bounded domains. It turns out that the asymptotic expansion leads surprisingly to a kind of Poincare-Dulac normal form of the Navier-Stokes equations. We will also discuss some related results and a few open issues.     

Duality in mathematics and linear and integer programming

Linear programming (LP) duality is examined in the context of other dualities in mathematics. The mathematical and economic properties of LP duality are discussed and its uses are considered. These mathematical and economic properties are then examined in relation to possible integer programming (IP) dualities. A number of possible IP duals are considered in this light and shown to capture some but not all desirable properties. It is shown that inherent in IP models are inequality and congruence constraints, both of which give on their own well-defined duals. However, taken together, no totally satisfactory dual emerges. The superadditive dual based on the Gomory and Chvátal functions is then described, and its properties are contrasted with LP duals and other IP duals. Finally, possible practical uses of IP duals are considered.The author is indebted to Professor H. B. Griffiths for many stimulating conversations on this topic.     

Internal and external partial difference families and cyclotomy

We introduce the concept of a disjoint partial difference family (DPDF) and an external partial difference family (EPDF), a natural generalization of the much-studied disjoint difference family (DDF), external difference family (EDF) and partial difference set (PDS). We establish properties and indicate connections to other recently-studied combinatorial structures. We show how DPDFs and EPDFs may be formed from PDSs, and present various cyclotomic constructions for DPDFs and EPDFs. As part of this, we develop a unified cyclotomic framework, which yields some known results on PDSs, DDFs and EDFs as special cases.     

C_(pk)写控制图的研究与应用

控制图与过程能力指数Cp(Cpk)都是用来进行质量分析与处理的重要工具。文 [1]谈及控制图的两类错误 (其概率分别为α、β) ,文 [2 ]讨论了Cpk与α的关系 ,但涉及Cpk与控制图检出力 (1- β)关系的文章却不多。本文以 X控制图为例 ,讨论了α、β、1- β、与Cpk的关系 ,得出了联合应用它们会对工序状态有较好的了解 ,对其适当的调整有助於提高产品质量     

Crossings and nestings of matchings and partitions

We present results on the enumeration of crossings and nestings for matchings and set partitions. Using a bijection between partitions and vacillating tableaux, we show that if we fix the sets of minimal block elements and maximal block elements, the crossing number and the nesting number of partitions have a symmetric joint distribution. It follows that the crossing numbers and the nesting numbers are distributed symmetrically over all partitions of , as well as over all matchings on . As a corollary, the number of -noncrossing partitions is equal to the number of -nonnesting partitions. The same is also true for matchings. An application is given to the enumeration of matchings with no -crossing (or with no -nesting).

     

Structure and complexity in economic and social systems

An attempt is made to outline a theory of complexity for economics and social science purposes. A methodology is outlined that realizes complexity as an analytical category, and develops a comprehensive theory of structural forms, structural change and qualitative systems behavior based on these methods. Examples of different nature are given to support the need for such a theory of complexity.     

Normality and boundary behaviour of arbitrary and meromorphic functions along simple curves and applications

In this paper, we establish some theorems giving necessary and sufficient conditions for an arbitrary function defined in the unit disc of the complex plane to have boundary values along classes of an equivalence relation over simple curves. Our results generalize the well-known theorems on asymptotic and angular boundary behaviours of meromorphic functions (Lindelölf-, Lehto–Virtanen- and Seidel–Walsh-type theorems). The obtained results are applied to the study of boundary behaviour of meromorphic functions along curves using P-sequences, as well as in the proof of the uniqueness theorem similar to ?aginjan’s one. The constructed examples of functions show that the results cannot be improved.     

Reduction and realization in toda and volterra

We construct a new symplectic, bi-Hamiltonian realization of the KM-system by reducing the corresponding one for the Toda lattice. The bi-Hamiltonian pair is constructed using a reduction theorem of Fernandes and Vanhaecke. In this paper we also review the important work of Moser on the Toda and KM-systems.