On Some Rings of Arithmetical Functions

In this paper, we consider several constructions which from a given B-product * _B lead to another one We shall be interested in finding what algebraic properties of the ring are shared also by the ring . In particular, for some constructions the rings R _B and will be isomorphic and therefore have the same algebraic properties.     

Generalized Quasi-Variational Inequalities Without Continuities

Given a nonempty set and two multifunctions , we consider the following generalized quasi-variational inequality problem associated with X, : Find such that . We prove several existence results in which the multifunction is not supposed to have any continuity property. Among others, we extend the results obtained in Ref. 1 for the case (x(X.     

Strictly Positive Definite Functions on a Real Inner Product Space

If converges for all with all coefficients , then the function is positive definite on H×H for any inner product space H. Set K={k: a _k >0}. We show that is strictly positive definite if and only if K contains the index 0 plus an infinite number of even integers and an infinite number of odd integers.     

Invariance principles for random walks on hypergroups on ℝ2 and ℕ

Let (X _n:n) be i.i.d. with finite variance and values in a hypergroupK:=₊ or and _j=1 ⁿ X _j be the randomized sum of these random variables. It is shown that the processes converge in distribution to a Gaussian process in the caseK=₊, that the processes converge towards a Bessel process on ₊ in the case of polynomial growth of the hypergroupK=₊ or , and that in the case of exponential growth converges towards a Brownian motion asn.     

An Extension of a Class of Iterative Procedures for Nonlinear Quasivariational Inequalities

Consider the convergence of the projection methods based on an extension of a special class of algorithms for the approximation--solvability of the following class of nonlinear quasivariational inequality (NQVI) problems: find an element such that and

On transformations z(t) = y(ϕ(t)) of ordinary differential equations

The paper describes the general form of an ordinary differential equation of the order n + 1 (n 1) which allows a nontrivial global transformation consisting of the change of the independent variable. A result given by J. Aczél is generalized. A functional equation of the form where are given functions, is solved on .     

Existence of Minimizers for Nonconvex Variational Problems with Slow Growth

Consider the minimization problem

Regular Unitarily Invariant Spaces on the Complex Sphere

Let K be a compact space, let X be a closed subspace of C(K), and let be a positive measure on K. The triple is said to be regular if, for any positive function and for any , there exists a function such that on K and . The case where K is the unit sphere in and the subspace X is invariant with respect to the unitary group is investigated. Sufficient spectral conditions and a necessary condition for the regularity of a triple are obtained. Connections with compactness of certain Hankel operators and applications to interpolation problems are presented. Bibliography: 16 titles.     

Convergence Analysis of Some Methods for Minimizing a Nonsmooth Convex Function

In this paper, we analyze a class of methods for minimizing a proper lower semicontinuous extended-valued convex function . Instead of the original objective function f, we employ a convex approximation f _{k + 1} at the kth iteration. Some global convergence rate estimates are obtained. We illustrate our approach by proposing (i) a new family of proximal point algorithms which possesses the global convergence rate estimate even it the iteration points are calculated approximately, where are the proximal parameters, and (ii) a variant proximal bundle method. Applications to stochastic programs are discussed.     

Hoffman’s Least Error Bounds for Systems of Linear Inequalities

Let E be a normed space, and . Let . We give some exact formulas for 7#x03C4;.     

On Singular Functional Differential Inequalities

Classical theorems on differential inequalities [1, 2, 3] are generalized for initial value problems of the kind and where is a singular Volterra operator, is continuous and positive on ]a, b], is a norm in R ⁿ, and [u]₊ and [u]_– are respectively the positive and the negative part of the vector u R ⁿ.     

On the Structure of a Certain Class of Mixed Tsirelson Spaces

We study Banach spaces of the form We call such a space a p-space, p[1,), if for every k the space is isomorphic to _pk and the sequence (p_k) strictly decreases to p. We examine the finite block representability of the spaces _r in a p-space proving that it depends not only on p but also on the sequences (p_k) and (n_k). Assuming that _i n_i ^1/q decreases to 0, where q is the conjugate exponent of p, we prove the existence of an asymptotic biorthogonal system in X and also that c ₀ is finitely representable in X. Moreover we investigate the modified versions of p-spaces proving that, if n_km1/pkm-1/pk_m-1 increases to infinity for a subsequence (n_km) , then ₁ embeds into X. We also investigate complemented minimality for the class of spaces where is either a subsequence of the sequence of Schreier classes ( _n)_{n N} or a subsequence of ( _n)_{n N}.     

On a Binary Diophantine Inequality Involving Prime Numbers

Let 1 < c < 15/14 and N a sufficiently large real number. In this paper we prove that, for all (N, 2N\ A with , the inequality has solutions in primes .     

On Asymptotic Properties of Solutions of Diffusion Equations

In this work the authors study the conditions for the existence of diffusion equations

Spectral Properties of Solutions of the Burgers Equation with Small Dissipation

We study the asymptotic behavior as of the Sobolev norm of the solution to the Cauchy problem for the one-dimensional quasilinear Burgers type equation (It is assumed that the problem is , the boundary conditions are periodic, and .) We show that the locally time-averaged Sobolev norms satisfy the estimate . The estimates obtained as a consequence for the Fourier coefficients justify Kolmogorov's spectral theory of turbulence for the case of the Burgers equation.     

Monotonicity of Average Power Means

A new numerical inequality for average power means is presented. Let and let be a sequence of positive numbers. Consider the operator . We denote by the superposition of these operators. The following assertion is proved: if . Bibliography: 2 titles.     

The Boundary Equivalence for Rings and Matrix Rings over Them

We study into the question of whether some rings and their associated matrix rings have equal decidability boundaries in the scheme and scheme-alternative hierarchies. Let be a decidability boundary for an algebraic system A; w.r.t. the hierarchy H. For a ring R, denote by an algebra with universe . On this algebra, define the operations + and in such a way as to extend, if necessary, the initial matrices by suitably many zero rows and columns added to the underside and to the right of each matrix, followed by ordinary addition and multiplication of the matrices obtained. The main results are collected in Theorems 1-3. Theorem 1 holds that if R is a division or an integral ring, and R has zero or odd characteristic, then the equalities hold for any n1. And if R is an arbitrary associative ring with identity then for any n 1 and i,j { 1,..., n}, where e _ij is a matrix identity. Theorem 2 maintains that if R is an associative ring with identity then . Theorem 3 proves that for any n 1.     

The Cauchy Problem for a Higher Order Telegraph Equation

Using the integral operator that defines a solution of the Cauchy problem for the equation

Scalar Operators Representable as a Sum of Projectors

We study sets there exist n projectors P₁,...,P_n such that . We prove that if n 6, then .     

On the Lax Solution to the Hamilton–Jacobi Equation and Its Generalizations. Part II

This article is a continuation of [J. Math. Sci., 99, No.5, 1541–1547 (2000)] devoted to the validity of the Lax formula (cited in the article of Crandall, Ishii, and Lions [Bull. AMS, 27, No.1, 1–67 (2000)])