A rearrangement estimate for the generalized multilinear fractional integrals

We study the boundedness of generalized multilinear fractional integrals. An O’Neil type inequality for a k-linear integral operator is proved. Using an O’Neil type inequality for a k-linear integral operator, we obtain a pointwise rearrangement estimate of generalized multilinear fractional integrals. By way of application we prove a Sobolev type theorem for these integrals. __________ Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 48, No. 3, pp. 577–585, May–June, 2007.     

Complex powers of degenerate differential operators connected with the Klein-Gordon-Fock operator

We develop a theory of complex powers of the generalized Klein-Gordon-Fock operator

$m^2 - \square - i\lambda \frac{{\partial ^2 }}{{\partial x_1^2 }},\lambda > 0.$

. The negative powers of this operator are realized as potential-type integrals with nonstandard metrics, while positive powers inverse to negative ones are realized as approximative inverse operators.

     

A note on generalized connectedness

This note is devoted to study the preservation of connectedness under the basic operators in generalized topological spaces. Some characterizations of generalized connectedness are given. As an application, we generalize some results in topological spaces. Supported in part by the NSFC (No. 10571151, 10671173).     

Correctness of Cauchy-type problems for abstract differential equations with fractional derivatives

We prove the uniform correctness of a Cauchy-type problem with two fractional derivatives and a bounded operator A. We propose a criterion for the uniform correctness of unbounded operator A.     

Necessary and sufficient conditions for the boundedness of rough B-fractional integral operators in the Lorentz spaces

In this paper, the necessary and sufficient conditions are found for the boundedness of the rough B-fractional integral operators from the Lorentz spaces Lp,s,γ to Lq,r,γ, 1<p<q<∞, 1?r?s?∞, and from L1,r,γ to Lq,∞,γ≡WLq,γ, 1<q<∞, 1?r?∞. As a consequence of this, the same results are given for the fractional B-maximal operator and B-Riesz potential.     

Integral representation of velocity in a flow with separation simulated by a vortex sheet

The goal of this paper is to present a new integral operator that defines the velocity of a streamline flow in terms of the intensity of a free vortex sheet. This operator is important for numerical simulation of flows.     

A semiexplicit method for numerical solution of functional differential algebraic equations

We study systems with delay effect that contain additional algebraic relations. We propose semiexplicit numerical methods of the Rosenbrock type. We prove the solvability of equations of a numerical model and estimate the order of the global error. The chosen parameters provide the third order of the error.     

On the Norms of Singular Integral Operators on Contours with Intersections

The exact bounds are obtained for the norm of the singular integral operator S on the family of rays originating at the same point. These bounds, with the use of the localization technique, are then extended to the essential norm of S on piecewise smooth curves with finitely many points of self intersection. Submitted: April 13, 2007. Accepted: September 19, 2007.     

The averaging method and the asymptotic behavior of solutions to differential inclusions

We prove one version of the first Bogolyubov theorem for differential inclusions with multivalued mappings that satisfy certain one-sided constraints. We study the dependence of solutions to differential inclusions on the parameters.     

Stability of the generalized alternative cauchy equation

Let G be a commutative semigroup and letL be a complete Archimedean Riesz Space. Suppose thatF: G → L satisfies for somee ∈ L ₊ the inequality

Structure theorems for generalized random processes

Generalized random processes are classified by various types of continuity. Representation theorems of a generalized random process on {M _p } on a set with arbitrary large probability, as well as representations of a correlation operator of a generalized random process on {M _p } and L ^r (R), r > 1, are given. Especially, Gaussian generalized random processes are proven to be representable as a sum of derivatives of classical Gaussian processes with appropriate growth rate at infinity. Examples show the essence of all the proposed assumptions. In order to emphasize the differences in the concept of generalized random processes defined by various conditions of continuity, the stochastic differential equation y′(ω; t) = f(ω; t) is considered, where y is a generalized random process having a point value at t = 0 in the sense of Lojasiewicz. This paper was supported by the project Functional analysis, ODEs and PDEs with singularities, No. 144016, financed by the Ministry of Science, Republic of Serbia.     

On a pythagorean-type functional equation with exponent 3 involving finite fields

We solve the functional equationf(x ³ +y ³) =f(x)³ +f(y)³ for maps of a finite field into itself.     

A note on the generalized rank reduction

We give constructive necessary and sufficient conditions for the validity of the generalized rank reduction formula of Ouellette [12]:

On a convex body with odd hadwiger number

We construct a convex body K ⊃ ℝ³ such that the maximum number of mutually nonoverlapping translates of K which touch K is 15. The work is in part supported by grant no. T043556 of the Hungarian National Science Foundation (OTKA).     

Collapsing monoids consisting of permutations and constants

In this paper we determine all collapsing transformation monoids that contain at least one unary constant operation and whose nonconstant operations are permutations. Furthermore, we find an infinite family of transformation monoids that consist of at least three unary constant operations and some permutations for which the corresponding monoidal intervals are 2-element chains. This research is supported by Hungarian National Foundation for Scientific Research grant nos. T 37877 and K 60148.     

On the maximum number of points at least one unit away from each other in the unit <Emphasis Type="Italic">n</Emphasis>-cube

We present a very short survey of known results and many new estimates and results on the maximum number of points that can be chosen in the n-dimensional unit cube so that every distance between them is at least 1. Research was supported by Slovak national grant VEGA 1/3839/06.     

On pyramids and reducedness

A convex body K in ℝ ^d is said to be reduced if the minimum width of each convex body properly contained in K is strictly smaller than the minimum width of K. We study the question of Lassak on the existence of reduced polytopes of dimension larger than two. We show that a pyramid of dimension larger than two with equal numbers of facets and vertices is not reduced. This generalizes the main result from [8].      

On the universal A.S. central limit theorem

Let (X _k ) be a sequence of independent r.v.’s such that for some measurable functions gk : R ^k → R a weak limit theorem of the form

Polytopes passing through circles

Suppose a convex body wants to pass through a circular hole in a wall. Does its ability to do so depend on the thickness of the wall? In fact in most cases it does, and in this paper we present a sufficient criterion for a polytope to allow an affirmative answer to the question.     

Real hypersurfaces in complex two-plane Grassmannians with commuting normal Jacobi operator

We give a complete classification of -invariant real hypersurfaces in complex two-plane Grassmannians G ₂(C ^m+2) with commuting normal Jacobi operator . The first author was supported by MCYT-FEDER grant BFM 2001-2871-C04-01, the second author by grant Proj. No. KRF-2006-351-C00004 from Korea Research Foundation and the third author by grant Proj. No. R14-2002-003-01001-0 from Korea Research Foundation, Korea 2006 and Proj. No. R17-2007-006-01000-0 from KOSEF.