Weak singularities may help periodic solutions to exist

In a periodically forced semilinear differential equation with a singular nonlinearity, a weak force condition enables the achievement of new existence criteria through a basic application of Schauder's fixed point theorem. The originality of the arguments relies in that, contrary to the customary situation in the available references, a weak singularity facilitates the arguments of the proofs.     

Spectral shift function and trace formula

The complete proofs of Krein’s theorem on the spectral shift function and the trace formula are given for a pair of self-adjoint operators such that either (i) their difference is trace-class or (ii) the difference of their resolvents is trace-class. The proofs, essentially due to Krein, is based on Herglotz’s theorem on the boundary value of the analytic functions whose imaginary part is non-negative on the upper half plane, and an almost optimal class of functions are obtained for which the trace formula is valid. Also an alternative method based on Weyl-von Neumann’s theorem for self-adjoint operators, avoiding the complex function theory and inspired by Voiculescu’s work, is given for the first case. Furthermore, some applications of the spectral shift function have been discussed.     

Martingale property of empirical processes

It is shown that for a large collection of independent martingales, the martingale property is preserved on the empirical processes. Under the assumptions of independence and identical finite-dimensional distributions, it is proved that a large collection of stochastic processes are martingales essentially if and only if the empirical processes are also martingales. These two results have implications on the testability of the martingale property in scientific modeling. Extensions to submartingales and supermartingales are given.

     

在ILC上用γγ→Z过程检验非对易时空标度

讨论关于在ILC用γγ到Z过程检验非对易时空能标(原文发在hep-ph/0604115). 在通常时空量子场论中, 由杨氏定理可知一个自旋为1的粒子不可能衰变为两个光子. 但在非对易时空中此过程是允许的. 因此这个过程能作为检验非对易时空的工具. ILC的光子对撞模式能实现这个过程. 如果总亮度能达到500fb^－1, 我们证明对Gamma (Z to gamma gamma)宽度的测量精度将比现有限制(<5.2×10^－5GeV)好3—4个数量级. 对非对易时空能标的检测可高达几个TeV.     

A Glivenko-Cantelli Theorem and Strong Laws for L-Statistics

We prove a weighted Glivenko-Cantelli theorem and apply it to study the rate of convergence in the strong law for L-statistics.     

On a problem of Zygmund

It is proved that there exists an integrable function on [0, 1]² whose integral is nondifferentiable in each direction belonging to a set everywhere dense in [0, 2π] but is strongly differentiable. Translated fromMatematicheskie Zametki, Vol. 64, No. 5, pp. 749–762, November, 1998.     

Helmholtz potentiality conditions for systems of difference-differential equations

For operators corresponding to systems ofμth-order difference-differential equations of neutral type, necessary and sufficient conditions for potentiality with respect to the classical bilinear form are obtained. These results are applied to first-order quasilinear systems. Translated fromMatematicheskie Zametki, Vol. 64, No. 3, pp. 437–442, September, 1998. The author wishes to express his thanks to V. M. Savchin for fruitful discussions and useful remarks.     

On the Seneta Sequences

In this paper, we prove some properties of the Seneta sequences and functions, and in particular we prove a representation theorem in the Karamata sense for the sequences from the Seneta class SOc.     

A remark on the Kochen-Specker theorem and some characterizations of the determinant on sets of Hermitian matrices

In this paper we describe the general forms of all (nonlinear) continuous functionals on the sets of positive definite, positive semi-definite and Hermitian matrices which are multiplicative on the commuting elements. As a consequence, we obtain some new characterizations of the determinant on those classes of matrices.

     

New numerical methods and some applied aspects of the p-regularity theory

The theory of p-regularity is applied to optimization problems and to singular ordinary differential equations (ODE). The special variant of the method of the modified Lagrangian function proposed by Yu.G. Evtushenko for constrained optimization problems with inequality constraints is justified on the basis of the 2-factor transformation. An implicit function theorem is given for the singular case. This theorem is used to show the existence of solutions to a boundary value problem for a nonlinear differential equation in the resonance case. New numerical methods are proposed including the p-factor method for solving ODEs with a small parameter.