Volume 7, Issue 4

Let be an interior mapping of the unit disk, continuous in D 2 and such that the restriction of f to the unit circle S 1 is a locally simple curve γ . Suppose that f ( a ) ≠ a on S 1 and denote by n ( a ) the number of solutions of the equation f (z) = a in D 2 , by μ ( f ) the sum of multiplicities of the critical points of f in , by q ( a ) the angular order of γ with respect to a, and by τ ( γ ) the angular order of γ . It is proved that the Morse formula 2 n ( a ) – μ ( f ) – 2 q ( a ) + τ ( γ ) – 1 = 0 remains correct for a piecewise smooth curve which is not locally simple.

In this paper we investigate the existence of mild solutions, on infinite intervals, to initial value problems for neutral functional differential and integrodifferential inclusions in Banach spaces. We shall rely on the fixed point theorem due to Ma, which is an extension on locally convex topological spaces, of Schaefer's theorem.

It is proved that if a linear operator l : C ([ a , b ], R ) → L ([ a , b ], R ) is nonpositive and for the Cauchy problem u ″( t ) + l ( u )( t ) + q ( t ), u ( a ) = c the theorem on differential inequalities is valid, then l is a Volterra operator.

We prove the boundedness of the Cauchy singular integral operator in special weighted Sobolev and Hölder-Zygmund spaces for large values of the smoothness parameter, which is an integer m ≥ 0, when the underlying contour is piecewise-smooth with angular points and even with cusps. We obtain Fredholm criteria and an index formula for singular integral equations with piecewise-continuous coefficients and complex conjugation in the spaces and provided that the underlying contour has only angular points but no cusps. The Fredholm property and the index turn out to be independent of the smoothness parameter m .

We study multidimensional stochastic equations where x o is an arbitrary initial state, W is a d -dimensional Wiener process and is a measurable diffusion coefficient. We give sufficient conditions for the existence of weak solutions. Our main result generalizes some results obtained by A. Rozkosz and L. Słomiński [Stochastics Stochasties Rep. 42: 199–208, 1993] and T. Senf [Stochastics Stochastics Rep. 43: 199–220, 1993] for the existence of weak solutions of one-dimensional stochastic equations and also some results by A. Rozkosz and L. Słomiński [Stochastic Process. Appl. 37: 187–197, 1991], [Stochastic Process. Appl. 68: 285–302, 1997] for multidimensional equations. Finally, we also discuss the homogeneous case.

We establish the conditions for the partial moduli of continuity, which guarantee the uniform convergence of N -dimensional trigonometric Fourier series of the functions of the so called generalized partial bounded p -variation.

The existence results obtained for the Dirichlet and mixed BVPs for the equation x ″ = f ( t , x , x ′) are extended to BVPs with full nonlinear conditions. The proofs are based on the theorem of Granas, Guenther and Lee, while barrier strips are used to obtain a priori bounds for solutions.

A class of algebraic Poisson structures on R 4 is introduced which contains the well-known Sklyanin algebras. For this class, an effective algebraic method of computing Euler characteristics of Casimir levels is developed which enables us to compute them in the case of Sklyanin algebras. It is also shown that one may associate a finite graph with every generalized Sklyanin algebra. Some related results and open problems are also presented.

An exact solution of the boundary value problems of thermoelastic equilibrium of a homogeneous isotropic rectangular parallelepiped is constructed. The parallelepiped is affected by a stationary thermal field and surface disturbances, in particular, on each side of the rectangular parallelepiped the following parameters are defined: a normal component of the displacement vector and tangential stresses (nonhomogeneous symmetry conditions) or normal stress and tangential stresses (nonhomogeneous antisymmetry conditions). The solution of the problems is constructed in series using the method of separation of variables.

We give a complete description of the class of univalent holomorphic functions with regular quasiconformal extensions for which the Grunsky inequalities are both necessary and sufficient. This concerns the question posed by several authors.

The paper deals with embeddings for sequence spaces with general weights. Our main results clarify in a rather final way the compactness of embeddings between these spaces including estimates for the entropy numbers of such compact embeddings in a generalized nonlimiting case.

Sufficient conditions are established for the oscillation of even order neutral differential equations with deviating arguments of the form where n ≥ 2 is an even integer. These results are illustrated by some examples.

Bausov-Telyakovskii type results on the integrability of trigonometric series are generalized to the case of the Fourier transform from which, in turn, a stronger form of initial results is derived. This is based on the approach to the problem of integrability developed earlier by the author.

We consider a financial market model, where the dynamics of asset prices is given by an R m -valued continuous semimartingale. Using the dynamic programming approach we obtain an explicit description of the variance optimal martingale measure in terms of the value process of a suitable problem of an optimal equivalent change of measure and show that this value process uniquely solves the corresponding semimartingale backward equation. This result is applied to prove the existence of a unique generalized solution of Bellman's equation for stochastic volatility models, which is used to determine the variance-optimal martingale measure.

The modular properties of generalized theta-functions with characteristics are used to build the bases of the cusp form spaces corresponding to eight level quadratic forms.