Volume 3, Issue 4

Sufficient conditions are given for the existence of oscillatory proper solutions of a differential equation with quasiderivatives L n y = f ( t , L 0 y , . . . , L n –1 y ) under the validity of the sign condition f ( t , x 1 , . . . , x n ) x 1 ≤ 0, f ( t , 0, x 2 , . . . , x n ) = 0 on .

In this short note we show that Stanley–Reisner rings of simplicial complexes, which have had a “dramatic application” in combinatorics [Hibi, Algebraic Combinatorics on Convex Polytopes, Carslaw Publications, 1992, p. 41], possess a rigidity property in the sense that they determine their underlying simplicial complexes.

Criteria for weak and strong two-weighted inequalities are obtained for integral transforms with positive kernels.

The classical change-point problem in modern terms, i.e., the mode-change problem, is stated for sufficiently general set-indexed random processes, namely for random measures. A method is shown for solving this problem both in the general form and for the intensity of compound Poisson random measures. The results obtained are novel for the change-point problem, too.

The correct formulation of a Darboux type multidimensional problem for second-order hyperbolic systems is investigated. The correct formulation of such a problem in the Sobolev space is proved for temporal type surfaces on which the boundary conditions of a Darboux type problem are given.

The estimate of the modulus of smoothness of an even function of several variables with quasiconvex Fourier coefficients obtained in this paper extends one result of S. A. Telyakovski.