Elastic Half-Plane under Random Displacement Excitations on the Boundary

We study in this paper a respond of an elastic half-plane to random boundary excitations. We treat both the white noise excitations and more generally, homogeneous random fluctuations of displacements prescribed on the boundary. Solutions to these problems are inhomogeneous random fields which are however homogeneous with respect to the longitudinal coordinate. This is used to represent the displacements as series expansions involving a complete set of deterministic functions with corresponding random coefficients. We construct the Karhunen-Loève (K-L) series expansion which is based on the eigen-decomposition of the correlation operator. The K-L expansion can be used to calculate the statistical characteristics of other functionals of interest, in particular, the strain and stress tensors and the elastic energy tensor. This work is supported partly by the RFBR Grant N 06-01-00498. I. Shalimova acknowledges the host institute WIAS, Berlin, and the support of DFG, under Grant SA 861/6-1 of 2008.     

积分微分方程组的若干求解公式

提出三类积分微分方程组,利用迭代法及变上限函数的求导法则,论证其可积性,给出这些积分微分方程组的求解公式,是对文献中内容的深化与拓广.     

Does truth-table of linear norm reduce the one-query tautologies to a random oracle?

In our former works, for a given concept of reduction, we study the following hypothesis: “For a random oracle A, with probability one, the degree of the one-query tautologies with respect to A is strictly higher than the degree of A.” In our former works (Suzuki in Kobe J. Math. 15, 91–102, 1998; in Inf. Comput. 176, 66–87, 2002; in Arch. Math. Logic 44, 751–762), the following three results are shown: The hypothesis for p-T (polynomial-time Turing) reduction is equivalent to the assertion that the probabilistic complexity class R is not equal to NP; The hypothesis for p-tt (polynomial-time truth-table) reduction implies that P is not NP; The hypothesis holds for each of the following: disjunctive reduction, conjunctive reduction, and p-btt (polynomial-time bounded-truth-table) reduction. In this paper, we show the following three results: (1) Let c be a positive real number. We consider a concept of truth-table reduction whose norm is at most c times size of input, where for a relativized propositional formula F, the size of F denotes the total number of occurrences of propositional variables, constants and propositional connectives. Then, our main result is that the hypothesis holds for such tt-reduction, provided that c is small enough. How small c can we take so that the above holds? It depends on our syntactic convention on one-query tautologies. In our setting, the statement holds for all c < 1. (2) The hypothesis holds for monotone truth-table reduction (also called positive reduction). (3) Dowd (in Inf. Comput. 96, 65–76, 1992) shows a polynomial upper bound for the minimum sizes of forcing conditions associated with a random oracle. We apply the above result (1), and get a linear lower bound for the sizes.     

Singular integral on bounded strictly pseudoconvex domain

Kytmanov and Myslivets gave a special Cauchy principal value of the singular integral on the bounded strictly pseudoconvex domain with smooth boundary. By means of this Cauchy integral principal value, the corresponding singular integral and a composition formula are obtained. This composition formula is quite different from usual ones in form. As an application, the corresponding singular integral equation and the system of singular integral equations are discussed as well.     

Residues of codimension one singular holomorphic distributions

For a codimension one locally-free singular holomorphic distribution, we give a residue formula in terms of the conormal sheaf given by Pfaffian equations. We also prove a Baum-Bott type residue formula for singular distributions.      

Displacement structure approach to q-adic Chebyshev–Vandermonde-like matrices

In this paper, a new class of so-called q-adic Chebyshev–Vandermonde-like matrices over an arbitrary non-algebraically closed field is introduced. This class generalizes both the ordinary Chebyshev–Vandermonde-like matrices over the complex field studied earlier by Kailath and Olshevsky [T. Kailath, V. Olshevsky, Displacement structure approach to Chebyshev–Vandermonde and related matrices, Integral Equations Operator Theory 22 (1995) 65–92], and the classical q-adic Vandermonde-like matrices with respect to power basis by Yang and Hu [Z.H. Yang, Y.J. Hu, Displacement structure and fast inversion formulas for q-adic Vandermonde-like matrices, J. Comput. Appl. Math. 176 (2005) 1–14]. Three kinds of displacement structures and consequently, three kinds of fast inversion formulas are presented for this class of matrices by using displacement structure theory method, which generalize the corresponding results for Chebyshev–Vandermonde-like and q-adic Vandermonde-like matrices.     

An explicit representation of local times of anticipative processes

In this work we give an explicit representation of the local time of an anticipative Ito process using the stochastic calculus of variations and the theory of the nuclear space-valued stochastic processes     

三元调和函数的几个性质

在几个引理的基础上,利用曲面积分的性质证明了三元调和函数的几个性质.     

Numerical solutions of two non-linear simultaneous equations

In this article, we shall give practical and numerical representations of inverse mappings of two-dimensional mappings (of the solutions of two non-linear simultaneous equations) and show their numerical experiments by using computers. We derive a concrete formula from a very general idea for the representation of the inverse functions.