矩阵奇异值的极值性质

讨论了矩阵奇异值的问题,利用奇异值分解定理给出了奇异值的极值性质,并用其证明了矩阵论中关于奇异值的一些经典结论.     

ON GUNDERSEN＇S QUESTION FOR THE UNICITY OF MEROMORPHIC FUNCTIONS

The uniqueness of meromoRphic functions that share four values are investigated, some results are obtained to show that if two nonconstant entire functions share three finite values IM, then the functions necessarily share all three values CM.     

Renormalization of multiple zeta values

Multiple zeta values (MZVs) in the usual sense are the special values of multiple variable zeta functions at positive integers. Their extensive studies are important in both mathematics and physics with broad connections and applications. In contrast, very little is known about the special values of multiple zeta functions at non-positive integers since the values are usually undefined. We define and study multiple zeta functions at integer values by adapting methods of renormalization from quantum field theory, and following the Hopf algebra approach of Connes and Kreimer. This definition of renormalized MZVs agrees with the convergent MZVs and extends the work of Ihara–Kaneko–Zagier on renormalization of MZVs with positive arguments. We further show that the important quasi-shuffle (stuffle) relation for usual MZVs remains true for the renormalized MZVs.     

Logarithmic convexity of extended mean values

In this article, the logarithmic convexity of the extended mean values are proved and an inequality of mean values is presented. As by-products, two analytic inequalities are derived. Two open problems are proposed.

     

面向个体优势识别的区间直觉模糊综合评价

针对指标值为区间直觉模糊数的多指标综合评价问题,提出了一种能识别个体优势的综合评价方法。借鉴理想点思想,利用行为主体各指标值与理想点的距离,刻画其状态特征或行为表现;从最有利于行为主体自身,利用数学规划方法,求出各行为主体的权重向量。综合自评值和他评值,对行为主体进行排序和择优。最后,给出了一个应用实例,说明方法的有效性和可行性。     

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.     

Bounded probability properties of Kolmogorov-Smirnov and similar statistics for discrete data

Summary A somewhat general class of situations, that include Kolmogorov-Smirnov type results as special cases, is considered. These situations, which are described in the following sections, are required to have uniquely determined probability properties when the sample values used are from continuous populations of any nature. If the populations sampled are discrete, however, these probability values are not uniquely determined. This paper shows that the values for the continuous case represent bounds for the values that occur in any discrete case. The method used to show that these bound relations hold consists in noting that any discrete data situation can be interpreted as a situation involving the grouping of continuous data. Then bound relationships are established between the values of probabilities for the grouped data situations and the corresponding ungrouped data situations, which are the situations considered for the case of the continuous data. These bounds on probabilities for discrete data cases should be useful for practical applications. In practice, all data are discrete (due to limitations in measurement accuracy).     

Use of the pseudoinverse matrix of a factor pattern in the measurement of factors

Under consideration is the problem of finding factor values. Some theorems are proved, yielding the pseudoinversematrix for a factor pattern obtained by the principal-component and factor analysis methods. The corresponding computational algorithms are developed for finding factor values.     

Biprobabilistic values for bicooperative games

The present paper introduces bicooperative games and develops some general values on the vector space of these games. First, we define biprobabilistic values for bicooperative games and observe in detail the axioms that characterize such values. Following the work of Weber [R.J. Weber, Probabilistic values for games, in: A.E. Roth (Ed.), The Shapley Value: Essays in Honor of Lloyd S. Shapley Cambridge University Press, Cambridge, 1988, pp. 101–119], these axioms are sequentially introduced observing the repercussions they have on the value expression. Moreover, compatible-order values are introduced and there is shown the relationship between these values and efficient values such that their components are biprobabilistic values.     

Axiomatic characterizations of generalized values

In the framework of cooperative game theory, the concept of generalized value, which is an extension of that of value, has been recently proposed to measure the overall influence of coalitions in games. Axiomatizations of two classes of generalized values, namely probabilistic generalized values and generalized semivalues, which extend probabilistic values and semivalues, respectively, are first proposed. The axioms we utilize are based on natural extensions of axioms involved in the axiomatizations of values. In the second half of the paper, special instances of generalized semivalues are also axiomatized.     

Parametric importance of criteria and intervals of value tradeoff uncertainty in the analysis of multicriteria problems

Multicriteria decision-making problems under bounded (above, below, or from both sides) continuous or discrete criteria are considered. Methods for comparing variants of solutions using the information accumulated in the form of interval estimates of replacing the values of some criteria by the values of others (such replacements are called tradeoffs; in other words, this can be considered as a compensation of the deterioration of some criteria by improving the values of others) are proposed along with simple consistency conditions of such information. The issue of constructing the set of nondominated variants is discussed.     

Perturbation of the SVD in the presence of small singular values

This paper gives SVD perturbation bounds and expansions that are of use when an m × n, m ? n matrix A has small singular values. The first part of the paper gives subspace bounds that are closely related to those of Wedin but are stated so as to isolate the effect of any small singular values to the left singular subspace. In the second part first and second order approximations are given for perturbed singular values. The subspace bounds are used to show that all approximations retain accuracy when applied to small singular values. The paper concludes by deriving a subspace bound for multiplicative perturbations and using that bound to give a simple approximation to a singular value perturbed by a multiplicative perturbation.     

On Diophantine Approximations of Mock Theta Functions of Third Order

The paper gives bounds for the approximation of the values of Ramanujan's Mock Theta functions of third order and more generally of some q-hypergeometric functions by the elements of an algebraic number field. Simultaneous approximations for the values of q-exponential function are also obtained. All the results are given both in the archimedean and p-adic case.     

极大加可约矩阵的谱与周期时间向量的关系

提出了极大加代数上可约矩阵特征值的缺失值及冗余值的概念,得到了相应的定理;对特征值与周期时间向量分量之间的关系作了深入的研究.     

Modularity of Galois traces of class invariants

Zagier showed that the Galois traces of the values of j-invariant at CM points are Fourier coefficients of a weakly holomorphic modular form of weight 3/2 and Bruinier–Funke expanded his result to the sums of the values of arbitrary modular functions at Heegner points. In this paper, we identify the Galois traces of real-valued class invariants with modular traces of the values of certain modular functions at Heegner points so that they are Fourier coefficients of weight 3/2 weakly holomorphic modular forms.     

Exact calculations of extended logical operations on fuzzy truth values

In this paper we propose computationally simple, pointwise formulas for extended t-norms and t-conorms on fuzzy truth values. The complex convolutions of the extended operations are shown to be equivalent to simple pointwise expressions for several special cases. Linear fuzzy truth values are defined and it is shown that the extended Łukasiewicz operations preserve linearity. Since linear fuzzy truth values are common in representing linguistic modifiers, the results simplify fuzzy truth value-based reasoning methods. The results can also be applied immediately to type-2 fuzzy set operations.     

弱Silnikov现象中的全局分支问题

本文考虑在一条同宿于具一对纯虚特征值的鞍－焦点的轨道邻域内的分支问题，证明了在同宿分支值的邻域内，存在着可数无穷多个鞍结点分支值，倍周期分支值和２－脉冲同宿分支值，并且两相邻鞍结点分支值的比趋于常数１。     

The stability of the equilibrium of a pendulum of variable length

The frequencies and modes of parametric oscillations of a pendulum of variable length for values of the modulation index from the smallest to the limit admissible values are investigated. The limits of the resonance zones of the first four oscillation modes are constructed and investigated by analytical and numerical methods, and the fundamental qualitative properties of the higher modes are established. Complete degeneracy of the modes with even numbers, i.e., coincidence of the frequencies of the odd and even eigenmodes for admissible values of the modulation parameter, is proved. The global pattern of the limits of the regions of stability of the lower position of equilibrium is constructed and it is shown that it differs considerably from the Ince–Strutt diagrams. Specific properties of the eigenmodes are established.     

Stability analysis of combustion waves for competitive exothermic reactions using Evans function

We consider travelling wave solutions of a reaction-diffusion system corresponding to an adiabatic two-step competitive exothermic reaction scheme. In such a scheme, a combustion process is assumed to be lumped into two different exothermic reactions. Although the rate constants of the reactions are distinct, both reactions occur simultaneously and feed on the same reactant. The travelling wave solutions are obtained via the shooting-relaxation method. The linear stability analysis is conducted using the Evans function technique and the compound matrix method. Further, threshold values of parameters corresponding to Hopf points are established. It is shown that the system exhibits pulsating behaviour when the parameter values are greater than the threshold values. The onset of instability is found for a broad range of parameter values. Two different numerical methods are then used to obtain solutions from the governing partial differential equations to validate the results.