非线性差分方程的多重周期解

用变分方法得到一类非线性差分方程多重周期解的存在性．我们的结果推广了Cai，Yu和Guo[Comput．Math．Appl．，52（2006），1630-1647]的结果，并且这里给出的证明显著地简化了．     

Gronwall-Bellman type nonlinear delay integral inequalities on time scales

In this paper, some Gronwall-Bellman type nonlinear delay integral inequalities on time scales are established, which provide a handy tool in deriving boundedness of solutions of certain delay dynamic equations on time scales. Our results generalize some of the main results in Lipovan (2006) [1], Pachpatte (2000) [2], Ferreira and Torres (2009) [3], Zhang and Meng (2008) [4], Cheung and Ren (2006) [5], Kim (2009) [6], and some of our results unify continuous and discrete analysis in the literature.     

On the stability of rational Runge Kutta methods

This paper is concerned with the stability of rational two-stage Runge Kutta methods for the numerical solution of stiff differential systems. With a stability analysis based on linear diagonal systems of arbitrary dimension, we find necessary and sufficient conditions for the coefficients of a method to be A(α) and A(α, β) stable, extending previous results on this subject given by Hairer [1] and Wambecq [3], [4], [5].     

On strong convergence in real Banach spaces obtained by dropping the Lipshitz condition

We consider an implicit iterative process for two finite families of mappings in a real Banach space and prove strong convergence results without using the Lipschitz condition on mappings. Our results mainly improve and extend the recent results of Chang et al. (2001, 2009, 2007) [1], [2], [3], Cho et al. (2005) [4], Gu (2008) [14], Ofoedu (2006) [9], Schu (1991) [13], Zeng (2003, 2005) [20], [21], and Qin et al. (2008) [11], [12].     

Interval goal programming for S-shaped penalty function

The approach of Jones and Tamiz (1995) [Jones, D.F., Tamiz, M., 1995. Expanding the flexibility of goal programming via preference modeling techniques. Omega 23, 41–48] has been accepted as the most efficient approach in the field of interval goal programming (IGP). Although several modifications to the original approach have been proposed recently [Vitoriano, B., Romero, C., 1999. Extended interval goal programming. Journal of the Operational Research Society 50, 1280–1283; Chang, C.-T., 2006. Mixed binary interval goal programming. Journal of the Operational Research Society 35, 389–396], all of them cannot formulate IGP with an S-shaped penalty function. In order to improve the utility of IGP, we extend the model of Chang (2006) [Chang, C.-T., 2006. Mixed binary interval goal programming. Journal of the Operational Research Society 35, 389–396] to be able to model an S-shaped penalty function. The newly formulated model is more concise and compact than the method of Li and Yu (2000) and it can easily be applied to a decision problem with the S-shaped penalty function. Finally, an illustrative example (i.e. how to build an appropriate E-learning system) is included for demonstrating the usefulness of the proposed model.     

Justification of the Courant-Friedrich's hypothesis in the case of a weak shock. Part I. Presentation of solution to linear problem

As is well known, two solutions of the problem of a supersonic stationary inviscid nonheatconducting gas flow onto a planar infinite wedge are theoretically possible: the solution with a strong shock (the flow speed behind the shock is subsonic) and the solution with a weak shock (the flow speed behind the shock is supersonic). Unlike the well-studied case of a strong shock that is generically unstable [A.M. Blokhin, D.L. Tkachev, L.O. Baldan, Study of the stability in the problem on flowing around a wedge. The case of strong wave, J. Math. Anal. Appl. 319 (2006) 248-277; A.M. Blokhin, D.L. Tkachev, Yu.Yu. Pashinin, Stability condition for strong shock waves in the problem of flow around an infinite plane wedge, Nonlinear Anal. Hybrid Syst. 2 (2008) 1-17], R. Courant and K.O. Friedrichs [R. Courant, K.O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publishers, New York, 1948] assumed that the solution with a weak shock is asymptotically stable by Lyapunov. Presentation of classical solution to the corresponding problem which is found in the present paper is the first step on the way to justification of Courant-Friedrichs hypothesis on linear level.     

Modified block preconditioners for the discretized time-harmonic Maxwell equations in mixed form

In this paper, building on the previous work by Greif and Schötzau [Preconditioners for the discretized time-harmonic Maxwell equations in mixed form, Numer. Linear Algebra Appl. 14 (2007) 281–297] and Benzi and Olshanskii [An augmented lagrangian-based approach to the Oseen problem, SIAM J. Sci. Comput. 28 (2006) 2095–2113], we present the improved preconditioning techniques for the iterative solution of the saddle point linear systems, which arise from the finite element discretization of the mixed formulation of the time-harmonic Maxwell equations. The modified block diagonal and triangular preconditioners considered are based on augmentation with using the symmetric nonsingular weighted matrix. We discuss the spectral properties of the preconditioned matrix in detail and generalize the results of the above-mentioned paper by Greif and Schötzau. Numerical experiments are given to demonstrate the efficiency of the presented preconditioners.     

三维空间中带对角步格路的一个注记

Jr.Stocks讨论了从(0,0,0)到(n,n,n)的带对角步格路的计数问题.本文给出了[4]中主要结果的简单公式,并将其推广到了一般情形.     

Filling the gap between the Gerschgorin and Brualdi theorems

The paper presents new diagonal dominance type nonsingularity conditions for n × n matrices formulated in terms of circuits of length not exceeding a fixed number r ≥ 0 and simple paths of length r in the digraph of the matrix. These conditions are intermediate between the diagonal dominance conditions in terms of all paths of length r and Brualdi’s diagonal dominance conditions, involving all the circuits. For r = 0, the new conditions reduce to the standard row diagonal dominance conditions , i = 1, ..., n, whereas for r = n they coincide with the Brualdi circuit conditions. Thus, they connect the classical Lévy-Desplanques theorem and the Brualdi theorem, yielding a family of sufficient nonsingularity conditions. Further, for irreducible matrices satisfying the new diagonal dominance conditions with nonstrict inequalities, the singularity/nonsingularity problem is solved. Also the new sufficient diagonal dominance conditions are extended to the so-called mixed conditions, simultaneously involving the deleted row and column sums of an arbitrary finite set of matrices diagonally conjugated to a given one, which, in the simplest nontrivial case, reduce to the old-known Ostrowski conditions , i = 1, ..., n, 0 ≤ α ≤ 1. The nonsingularity conditions obtained are used to provide new eigenvalue inclusion sets, depending on r, which, as r varies from 0 to n, serve as a bridge connecting the union of Gerschgorin’s disks with the Brualdi inclusion set. Bibliography: 16 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 334, 2006, pp. 128–148.     

Splitting Formulas for the Higher and Lower Energy Levels of the One-Dimensional Schrödinger Operator [Erratum]

In S. Albeverio, S. Yu. Dobrokhotov, and E. S. Semenov Splitting formulas for the higher and lower energy levels of the one-dimensional Schrödinger operator (Vol. 138, No. 1, January, 2004, pp. 98–106), unfortunately, the paper by S. Yu. Slavyanov (Differ. Equations (1969), 5, No. 2, pp. 313–325) was omitted. In that paper, formula (4) for splitting the lower energy levels was published earlier than in the works cited [3]–[5], [9]. Also, the book in which that formula was also published was omitted: S. Yu. Slavyanov and B. Lay, Special Functions: A Unified Theory Based on an Analysis of Singularities [in Russian], Nevskii Dialekt, St. Petersburg (2002). The authors apologize for this omission.     

Theory of Valuations on Manifolds: A Survey

This is a non-technical survey of a recent theory of valuations on manifolds constructed in [A10], [A11], [AF] and [A12], and actually a guide to this series of articles. We also review some recent related results obtained by a number of people. Received: February 2006, Revision: June 2006, Accepted: June 2006     

三对角矩阵逆的估计

1引言 三对角矩阵出现在很多应用中,例如,在求解常系数微分方程的比值问题,三次样条插值等应用中都会遇到三对角矩阵.因此这类矩阵非常重要,而且也有很多学者致力于这类矩阵的研究.在一些应用中,比如估计条件数和构造稀疏近似逆预条件子,需要计算三对角矩阵的逆,或者估计其逆元素的界.文献[1-7]给出了关于三对角矩阵逆的一些很好的结果,但是,这些结果大都建立在矩阵对角占优的条件之下,这限制了他们的应用.在本文中,我们给出一种一般三对角矩阵逆元素的估计办法.     

关于《完备Brouwer格上伪t-模与蕴涵算子》的注记

举例说明关于伪t-模与蕴涵算子的文献[FSS 132(2002)113]中定理4.1是错误的,此定理还被接连用到[FSS 139(2003)673]及其它文献。本文进而给出此基础性定理成立的一个必要条件。注意,该必要条件是不能由无穷∨-分配伪t-模与无穷∧-分配蕴涵算子的定义引出的。     

A Unified Construction of Two-Dimensional Pooling Designs

Following the work of Chateauneuf et al. [3], a simplified construction of a rowcolumn- front diagonal (RCF) design and a union jack design of a prime order using affine transformations is given.AMS Subject Classification: 05B30, 05C65, 60C05, 62K99.     

Context tree selection: A unifying view

Context tree models have been introduced by Rissanen in [25] as a parsimonious generalization of Markov models. Since then, they have been widely used in applied probability and statistics. The present paper investigates non-asymptotic properties of two popular procedures of context tree estimation: Rissanen’s algorithm Context and penalized maximum likelihood. First showing how they are related, we prove finite horizon bounds for the probability of over- and under-estimation. Concerning over-estimation, no boundedness or loss-of-memory conditions are required: the proof relies on new deviation inequalities for empirical probabilities of independent interest. The under-estimation properties rely on classical hypotheses for processes of infinite memory. These results improve on and generalize the bounds obtained in Duarte et al. (2006) [12], Galves et al. (2008) [18], Galves and Leonardi (2008) [17], Leonardi (2010) [22], refining asymptotic results of Bühlmann and Wyner (1999) [4] and Csiszár and Talata (2006) [9].     

The Singularity/Nonsingularity Problem for Matrices Satisfying Diagonal Dominance Conditions in Terms of Directed Graphs

The paper considers the singularity/nonsingularity problem for matrices satisfying certain conditions of diagonal dominance. The conditions considered extend the classical diagonal dominance conditions and involve the directed graph of the matrix in question. Furthermore, in the case of the so-called mixed diagonal dominance, the corresponding conditions are allowed to involve both row and column sums for an arbitrary finite set of matrices diagonally conjugated to the original matrix. Conditions sufficient for the nonsingularity of quasi-irreducible matrices strictly diagonally dominant in certain senses are established, as well as necessary and sufficient conditions of singularity/nonsingularity for weakly diagonally dominant matrices in the irreducible case. The results obtained are used to describe inclusion regions for eigenvalues of arbitrary matrices. In particular, a direct extension of the Gerschgorin (r = 1) and Ostrowski-Brauer (r = 2) theorems to r ≥ 3 is presented. Bibliography: 18 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 309, 2004, pp. 40–83.     

Minimum rank problems

A graph describes the zero-nonzero pattern of a family of matrices, with the type of graph (undirected or directed, simple or allowing loops) determining what type of matrices (symmetric or not necessarily symmetric, diagonal entries free or constrained) are described by the graph. The minimum rank problem of the graph is to determine the minimum among the ranks of the matrices in this family; the determination of maximum nullity is equivalent. This problem has been solved for simple trees [P.M. Nylen, Minimum-rank matrices with prescribed graph, Linear Algebra Appl. 248 (1996) 303-316, C.R. Johnson, A. Leal Duarte, The maximum multiplicity of an eigenvalue in a matrix whose graph is a tree, Linear and Multilinear Algebra 46 (1999) 139-144], trees allowing loops [L.M. DeAlba, T.L. Hardy, I.R. Hentzel, L. Hogben, A. Wangsness. Minimum rank and maximum eigenvalue multiplicity of symmetric tree sign patterns, Linear Algebra Appl. 418 (2006) 389-415], and directed trees allowing loops [F. Barioli, S. Fallat, D. Hershkowitz, H.T. Hall, L. Hogben, H. van der Holst, B. Shader, On the minimum rank of not necessarily symmetric matrices: a preliminary study, Electron. J. Linear Algebra 18 (2000) 126-145]. We survey these results from a unified perspective and solve the minimum rank problem for simple directed trees.     

On the distribution of square-full integers

Let A(x) A(x) be the number of square-full integers \leqq x \leqq x and let D(x) \Delta(x) be the error term in the asymptotic formula for A(x) A(x) . Under the Riemann hypothesis, we show that D(x) << x^[12/85]+e \Delta(x)\ll x^{{12\over 85}+\varepsilon} . This improves the earlier results of Zhu and Yu [17], Cao [4, II], Liu [9] and Wu [16], which requires [ 1/7 ] 1\over 7 in place of [ 12/85 ] 12\over 85 .     

Modeling attitudes towards uncertainty and risk through the use of choquet integral

The aim of this paper is to present in a unified framework a survey of some results related to Choquet Expected Utility (CEU) models, a promising class of models introduced separately by Quiggin [35], Yaari [48] and Schmeidler [40, 41] which allow to separate attitudes towards uncertainty (or risk) from attitudes towards wealth, while respecting the first order stochastic dominance axiom.     

A note on the split common fixed-point problem for quasi-nonexpansive operators

Based on the very recent work by Censor and Segal (2009) [1], and inspired by Xu (2006) [9], Zhao and Yang (2005) [10], and Bauschke and Combettes (2001) [2], we introduce and analyze an algorithm for solving the split common fixed-point problem for the wide class of quasi-nonexpansive operators in Hilbert spaces. Our results improve and develop previously discussed feasibility problems and related algorithms.