Abstract

We discuss some methods for computing the dynamics and interaction of singularities in nonlinear media. We also consider a physical problem (of solidification in a binary alloy), which has some discontinuous limit solutions.

Keywords: Propagation of singularities, solidification in a binary alloy.

MSC: 46F, 35A05, 35K57

Abstract

We prove that for all $0\le\alpha\le 2/3$ $$ \Vert |A|^{\alpha}X-X|B|^{\alpha}\Vert \le 2^{2-\alpha}\Vert X\Vert^{1-\alpha} \Vert AX-XB\Vert^{\alpha}, $$ for all bounded Hilbert space operators $A=A^*$, $B=B^*$ and $X$, as well as $$ \Vert |A|^{\alpha}-|B|^{\alpha}\Vert \le 2^{2-\alpha} \Vert A-B\Vert^{\alpha}, $$ for arbitrary bounded $A$ and $B$.

Keywords: Singular values, three line theorem for operators, unitarily invariant norms.

MSC: 47A30, 47B05, 47B10, 47B15

Abstract

The $\a$-times integrated semigroups, $\a\in\R^-=(-\infty,0]$, are introduced and analyzed as extensions of $0$-integrated semigroups.

Keywords: Integrated semigroups.

MSC: 46F99

Abstract

Contrary to the usual inverse of a square matrix, it is well known that the Moore-Penrose and Drazin inverses of a matrix are not necessarily continuous functions of the elements of the matrix. This paper is a short summary of some results concerning the continuity of the Moore-Penrose and Drazin inverses.

Keywords: Moore-Penrose inverse, Drazin inverse.

MSC: 47A05, 47A53, 15A09

Abstract

Let $H$ be a separable Hilbert space. We write $\sigma (A)$ for the spectrum of $A\in B(H)$, $\sigma_a(A)$ and $\sigma_{ea}(A)$ for the approximate point and the essential approximate point spectrum of $A$. Operator $A\in B(H)$ is quasihyponormal if $\| A^*Ax\| \le \| A^2x\|$ for all $x\in H$. In this paper we show that the approximate point spectrum $\sigma_a$ and the essential approximate point spectrum $\sigma_{ea}$ are continuous in the set of all quasihyponormal operators.

Keywords: Qusihyponormal operator, approximate point spectrum, essential approximate point spectrum.

MSC: 47A10, 47A53

Abstract

In this paper investigation is conducted of various essential spectra of minimal, maximal and intermediate ordinary differential operators in scale of Lebesque spaces $L^p(a,\infty)$, $1\leq p\leq \infty$, obtained by means of relatively small perturbations of differential operators with constant coefficients of order $n$ by differential operators of the same order, which generalizes the results [1--3]. This makes it possible to prove the new analogons of the classical Weyl theorem of invariance of essential spectrum as well as to obtain the precise formulas for calculating essential spectra of various classes of ordinary differential operators in Lebesque spaces $L^p$. In contemporary mathematical literature a few assertions are known as Weyl's theorem (see, for example, survey [4]). The classical Weyl theorem states that if $A$ and $B$ are self-adjoint and $A-B$ is compact then $\sigma_e(A)=\sigma_e(B)$, where $\sigma_e$ is the essential spectrum of an operator. Generalization of Weyl theorem on various essential spectra for closed operators in Banach spaces and special classes of perturbations is dealt with in papers [5--7].

Keywords: Essential spectrum, operator with almost concstant coefficients.

MSC: 47A10, 47A53, 47E05

Abstract

In this paper we give a survey of recent results in the theory of matrix transfomrations between sequence spaces. We shall deal with sequence spaces that are closely related to various concepts of summability, study their topological structures, find their Schauder-bases and determine their $\beta$-duals. Further we give necessary and sufficient conditions for matrix transformations between them.

Keywords: BK spaces, bases, matrix transformations.

MSC: 40H05, 46A45, 47B07

Abstract

Let $l^{p,q}$, $0

Keywords: Mixed-norm sequence space, multiplier, Hausdorff measure of noncompactness.

MSC: 30B10, 47B07

Abstract

Estimates for the measure of noncompactness of bounded subsets of spaces of (Bochner-) integrable functions are obtained, a new class of condensing operators is discussed, and the solvability of a certain operator equation in a Hilbert space is proved.

Keywords: Hilbert space, regular space, Lebesgue space, Bochner space, Sobolev space, embedding operator, measure of noncompactness.

MSC: 47H09, 46E35, 47B07, 47B38

Abstract

The aim of this paper is to analyze the asymptotic behavior at infinity of the integral wavelet transform of somewhat more general elements than $L^2$ functions, namely generalized functions from the space of exponential distributions ${\Cal K}_1'$. We prove both an Abelian and a Tauberian type theorem at infinity for the integral wavelet transform.

Keywords: Integral wavelet transform, generalized functions, Abelian type theorems, Tauberian type theorems.

MSC: 42C05

Abstract

In this note we present some properties of Hausdorff measure of noncompactness on locally bounded topological vector space.

Keywords: Hausdorff measure of noncompactness, locally bounded space.

MSC: 46 A16, 46A50

Abstract

The result of Krassowska and Sliwa [6] about weak topologies in (DF)-spaces is extended to variuos other classes of locally convex spaces with a fundamental sequence of bounded subsets.

Keywords: Weak topology, locally convex space, (DF)-space, fundamental sequence of bounded subsets.

MSC: 46A04

Abstract

We define and study a class of holomorphic Besov type spaces $B^p$, $0 < p < 1$, on bounded symmetric domains $\Omega$. We show that the dual of holomorphic Besov space $B^p$, $0 < p < 1$, on bounded symmetric domain $\Omega$ can be identified with the Bloch space $\Cal B^{\infty}$.

Keywords: Besov type spaces, dual spaces, Bloch spaces.

MSC: 32A37

Abstract

In this note we will state a new version of Grötzsch's principle and using this principle we will sketch the proof of a generalization of the main inequality. Also, we will announce some related results and briefly explain that one can use new version of the main inequality to study uniqueness property of harmonic mapping in general. More details will be given in a forthcoming paper.

Keywords: Grötzsch principle and Reich-Strebel inequality

MSC: 30C75

Abstract

For a function $f(z)=z+a_2z^2+\dotsb$, analytic in the unit disc, we find $\lambda>0$ such that $|f''(z)|\le\lambda$ implies starlikeness (Mocanu's problem [2]) or convexity. The given results are sharp.

Keywords: Starlike, convex, subordinate.

MSC: 30C45

Abstract

We study the set $K(f)$ of positive numbers $t$ for which the operator $I-t\nabla f$ is contractible, where $f$ is a differentiable function defined on a convex subset of the Hilbert space ($I$ is the identity operator of that Hilbert space). The set $K(f)$ is interesting for a problem of minimization of strongly convex functions when the method of contractible mappings is applied.

Keywords: Contractible operator, strongly convex function.

MSC: 26B25

Abstract

In this work we use function space interpolation to prove some convergence rate estimates for finite difference schemes. We concentrate on a Dirichlet boundary value problem for a second-order linear elliptic equation with variable coefficients in the unit 3-dimensional cube. We assume that the solution to the problem and the coefficients of the equation belong to corresponding Sobolev spaces.

Keywords: Boundary Value Problems (BVP), Finite Difference Schemes (FDS), Sobolev Spaces, Interpolation of Function Spaces, Convergence Rate Estimates.

MSC: 65N15, 46B70

Abstract

In this paper we show how the theory of interpolation of function spaces can be used to establish convergence rate estimates for finite difference schemes. As a model problem we consider the first initial-boundary value problem for the heat equation with variable coefficients in a domain $(0,1)^2\times (0,T]$. We assume that the solution of the problem and the coefficients of equation belong to corresponding Sobolev spaces. Using interpolation theory we construct a fractional-order convergence rate estimate which is consistent with the smoothness of the data.

Keywords: Initial-Boundary Value Problems, Finite Differences, Interpolation of Function Spaces, Sobolev Spaces, Convergence Rate Estimates.

MSC: 65M15, 46B70

Abstract

In the paper [2], R. André-Jeannin studied a class of polynomials $U_{n}(p,q;x)$. In this paper we consider a new class of polynomials $U_{n,m}(p,q;x)$ and determine the coefficients $c_{n,k}(p,q)$ of these introduced polynomials. Also, we define the polynomials $f_{n,m}(p,q;x)$, which are the rising diagonal polynomials of $U_{n,m}(p,q;x)$.

Keywords: Fibonacci polynomilas, Pell polynomials, Fermat polynomilas, Morgan-Voyce polynomials, Chebyshev polynomials.

MSC: 33C55

Abstract

We study the initial value problem for the Davey-Stewartson systems $$ \cases i\partial_t u+c_0\partial_{x_1}^2u+\partial_{x_2}^2 u = c_1|u|^2u+c_2u\partial_{x_1}\varphi, \quad (x,t)\in{\bold R}^3,\\ \partial_{x_1}^2\varphi+c_3\partial_{x_2}^2\varphi = \partial_{x_1}|u|^2,\\ u(x,0) = \phi(x), \endcases $$ where $c_0,c_3\in{\bold R}$, $c_1,c_2\in{\bold C}$, $u$ is a complex valued function and $\varphi$ is a real valued function. The initial data $\phi$ is $\bold C$-valued function on $\bold R^n$, and usually it belongs to some kind of Sobolev type spaces.

Keywords: Elliptic-hyperbolic system, Davey-Stewartson system.

MSC: 35J45, 35L45

Abstract

In this note we start from Reyman and Semenov-Tian-Shansky L-A pair, in order to get L-A pairs for KG and GCG with magnetic field. The resulting matrices have all the symmetries necessary for procedure of algebro-geometric integration described in~[6].

Keywords: L-A pair, Kowalewskaya gyrostat.

MSC: 35Q60, 78A99