On the Solvability of the Multidimensional Version of the First Darboux Problem for a Model Second-Order Degenerating Hyperbolic Equation

A multidimensional version of the first Darboux problem is considered for a model second-order degenerating hyperbolic equation. Using the technique of functional spaces with a negative norm, the correct formulation of this problem in the Sobolev weighted space is proved.     

The solution of orthogonal Procrustes problems for a family of orthogonally invariant norms

The orthogonal Procrustes problem involves finding an orthogonal matrix which transforms one given matrix into another in the least-squares sense, and thus it requires the minimization of the Frobenius matrix norm. We consider, the solution of this problem for a family of orthogonally invariant norms which includes the Frobenius norm as a special case.     

Banach-Saks property in Marcinkiewicz spaces

We catalogue all Marcinkiewicz function and sequence spaces with the Banach-Saks property and present necessary and sufficient conditions for a wide subclass of spaces to possess the p-Banach-Saks property, 1<p<∞. We apply our results to several open problems.     

The widest continuous integral

An approach to a definition of an integral, which differs from definitions of Lebesgue and Henstock-Kurzweil integrals, is considered. We use trigonometrical polynomials instead of simple functions. Let V be the space of all complex trigonometrical polynomials without the free term. The definition of a continuous integral on the space V is introduced. All continuous integrals are described in terms of norms on V. The existence of the widest continuous integral is proved, the explicit form of its norm is obtained and it is proved that this norm is equivalent to the Alexiewicz norm. It is shown that the widest continuous integral is wider than the Lebesgue integral. An analog of the fundamental theorem of calculus for the widest continuous integral is given.     

Concentration inequalities for random fields via coupling

We present a new and simple approach to concentration inequalities in the context of dependent random processes and random fields. Our method is based on coupling and does not use information inequalities. In case one has a uniform control on the coupling, one obtains exponential concentration inequalities. If such a uniform control is no more possible, then one obtains polynomial or stretched-exponential concentration inequalities. Our abstract results apply to Gibbs random fields, both at high and low temperatures and in particular to the low-temperature Ising model which is a concrete example of non-uniformity of the coupling.      

An expansion in the exponent for compound binomial approximations

The purpose of this paper is two-fold. First, we introduce a new asymptotic expansion in the exponent for the compound binomial approximation of the generalized Poisson binomial distribution. The dependence of its accuracy on the symmetry and shifting of distributions is investigated. Second, for compound binomial and compound Poisson distributions, we present new smoothness estimates, some of which contain explicit constants. Finally, the ideas used in this paper enable us to prove new precise bounds in the compound Poisson approximation. Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 1, pp. 67–110, January–March, 2006.     

谱约束下对称正交对称矩阵束的最佳逼近

讨论了对称正交对称矩阵的广义逆特征值问题，得到了通解表达式和最佳解的表达式。     

赋Orlicz范数的Musielak-Orlicz序列空间的Gateaux可微点与Frechet可微点

该文给出赋0rlicz范数的Musielak-Orlicz序列空间中Gateaux可微点(光滑点)与Frechet可微点(强光滑点)的判定准则．在此基础上推出了该空间具有光滑性或强光滑性的充分必要条件．     

实二次函数域基本单位的范

作为Ｈ．Ｆ．Ｔｒｏｔｔｅｒ［３］中一个结果的模拟，本文给出实二次函数域Ｋ中基本单位ε的范为Ｎ（ε）＝ｇ的一些充分必要条件，其中ｇ为Ｋ的常数域Ｆｑ的非零乘法群Ｆｑ＾＊的生成元（２├ｑ）。利用图论的语言，本文还给出了Ｎ（ε）＝ｇ的一些充分条件，这些条件对于数域也同样地适用，从而推广了Ｔｒｏｔｔｅｒ［３］和前人工作中关于充分条件的结果。     

ON MATRIX UNITARILY INVARIANT NORM CONDITION NUMBER

1.IntroductionSince1984,severalchinesemathematicianshaveobtailledmanyresultsboutmatrixoperatornormcondition.umbe.[ll'12'18].Twokindsmatrixconditionnllmbers[9]are:(1)IfAECoxsisnonsingular,thellunlberK.(A)~IIAll.llA--'if.iscalledtheor--normconditionnumberof…