Asplund Functions and Projectional Resolutions of the Identity

*Supported by the Grants AV ˇCR 101-97-02, 101-90-03, GA ˇCR 201-98-1449, and by the Grant of the Faculty of Civil Engineering of the Czech Technical University No. 2003.

Boundary-Value Problems for almost Nonlinear Singularly Perturbed Systems of Ordinary Differential Equations

A boundary-value problems for almost nonlinear singularly perturbed systems of ordinary differential equations are considered. An asymptotic solution is constructed under some assumption and using boundary functions and generalized inverse matrix and projectors.

The Space of Differences of Convex Functions on [0, 1]

∗Participant in Workshop in Linear Analysis and Probability, Texas A & M University, College Station, Texas, 2000. Research partially supported by the Edmund Landau Center for Research in Mathematical Analysis and related areas, sponsored by Minerva Foundation (Germany).

Some properties of g- and P-spaces

A γ-space with a strictly positive measure is separable. An example of a non-separable γ−space with c.c.c. is given. A P−space with c.c.c. is countable and discrete.

On the Hyperbolicity Domain of the Polynomial x^n + a1x^(n-1) + 1/4+ an

∗ Partially supported by INTAS grant 97-1644

Some Para-Hermitian Related Complex Structures and Non-existence of Semi-Riemannian Metric on Some Spheres

It is shown that the spheres S^(2n) (resp: S^k with k ≡ 1 mod 4) can be given neither an indefinite metric of any signature (resp: of signature (r, k − r) with 2 ≤ r ≤ k − 2) nor an almost paracomplex structure. Further for every given Riemannian metric on an almost para-Hermitian manifold with the associated 2-form φ one can construct an almost Hermitian structure (under certain conditions, two different almost Hermitian structures) whose associated 2-form(s) is φ.

On The Brill-noether Theory of Spanned Vector Bundles on Smooth Curves

Here we study the integers (d, g, r) such that on a smooth projective curve of genus g there exists a rank r stable vector bundle with degree d and spanned by its global sections.

On the Maximum of a Branching Process Conditioned on the Total Progeny

The maximum M of a critical Bienaymé-Galton-Watson process conditioned on the total progeny N is studied. Imbedding of the process in a random walk is used. A limit theorem for the distribution of M as N → ∞ is proved. The result is trasferred to the non-critical processes. A corollary for the maximal strata of a random rooted labeled tree is obtained.

Some Computational Aspects of the Consistent Mass Finite Element Method for a (semi-)periodic Eigenvalue Problem

We consider a model eigenvalue problem (EVP) in 1D, with periodic or semi–periodic boundary conditions (BCs). The discretization of this type of EVP by consistent mass finite element methods (FEMs) leads to the generalized matrix EVP Kc = λ M c, where K and M are real, symmetric matrices, with a certain (skew–)circulant structure. In this paper we fix our attention to the use of a quadratic FE–mesh. Explicit expressions for the eigenvalues of the resulting algebraic EVP are established. This leads to an explicit form for the approximation error in terms of the mesh parameter, which confirms the theoretical error estimates, obtained in [2].

New Binary Extremal Self-Dual Codes of Lengths 50 and 52

* This work was partially supported by the Bulgarian National Science Fund under Contract No. MM – 503/1995.

On the Uniform Decay of the Local Energy

It is proved in [1],[2] that in odd dimensional spaces any uniform decay of the local energy implies that it must decay exponentially. We extend this to even dimensional spaces and to more general perturbations (including the transmission problem) showing that any uniform decay of the local energy implies that it must decay like O(t^(−2n) ), t ≫ 1 being the time and n being the space dimension.

Alternative Characterization of the Class k-UCV and Related Classes of Univalent Functions

In this paper an alternative characterization of the class of functions called k -uniformly convex is found. Various relations concerning connections with other classes of univalent functions are given. Moreover a new class of univalent functions, analogous to the ’Mocanu class’ of functions, is introduced. Some results concerning this class are derived.

Descriptive Sets and the Topology of Nonseparable Banach Spaces

This paper was extensively circulated in manuscript form beginning in the Summer of 1989. It is being published here for the first time in its original form except for minor corrections, updated references and some concluding comments.

Formation of Singularities for Weakly Non-Linear N×N Hyperbolic Systems

We present some results on the formation of singularities for C^1 - solutions of the quasi-linear N × N strictly hyperbolic system Ut + A(U )Ux = 0 in [0, +∞) × Rx . Under certain weak non-linearity conditions (weaker than genuine non-linearity), we prove that the first order derivative of the solution blows-up in finite time.

Equivariant Embeddings of Differentiable Spaces

Given a differentiable action of a compact Lie group G on a compact smooth manifold V , there exists [3] a closed embedding of V into a finite-dimensional real vector space E so that the action of G on V may be extended to a differentiable linear action (a linear representation) of G on E. We prove an analogous equivariant embedding theorem for compact differentiable spaces (∞-standard in the sense of [6, 7, 8]).