Volume 8, Issue 1

The existence of an interior extension with critical points on the boundary is proved for a given closed normal curve and for nonnormal curves of specific kind.

We establish a general fixed point theorem for multivalued maps defined on cones in Banach spaces. Applications to single and multivalued equations are presented.

The Hojman–Lutzky conservation law establishes a certain correspondence between non-Noether symmetries and conserved quantities. In the present paper the extension of the Hojman–Lutzky theorem to singular dynamical systems is carried out.

Optimal sufficient conditions are found in weighted Lorentz spaces for weight functions which provide the boundedness of the Calderón–Zygmund singular integral operator defined on spaces of homogeneous and nonhomogeneous type.

The work of Ambrozie and Vasilescu on perturbations of Fredholm complexes is generalized by discussing the stability theory of Banach space complexes under inessential perturbations.

Necessary and sufficient conditions on weight pairs are found for the validity of a class of weighted exponential inequalities involving certain classical operators. Among the operators considered are the Hardy averaging operator and its variants in one and two dimensions, as well as the Laplace transform. Discrete analogues yield characterizations of weighted forms of Carleman's inequality.

The systems of bases are constructed for the spaces of cusp forms S 2 m (Γ 0 (5), χ m ) and S 2 m (Γ 0 (13), χ m ) for an arbitrary integer m ≥ 2. Formulas are obtained for the number of representations of a positive integers by the direct sums of some quaternary quadratic forms.

A class of totally real surface singularities is introduced for which one can define local topological invariants. The main result yields an algebraic estimate for the self-intersection index of a totally real germ. It is derived from an algebraic upper estimate for the index of a quasi-homogeneous vector field which generalizes the previously known estimates in terms of the so-called Petrovsky numbers.

An explicit exact (non asymptotic) formula is derived for the number of representations of positive integers by the quadratic form . The way by which this formula is derived, gives us a possibility to develop a method of finding the so-called Liouville type formulas for the number of representations of positive integers by positive diagonal quadratic forms in nine variables with integral coefficients.

In the entire functions space consisting of at most second order functions such that their type is less than π q /(2 s 2 ) it is valid the q -order derivative sampling series reconstruction procedure, reading at the von Neumann lattice { s ( m + ni )| ( m , n ) ∈ } via the Weierstrass σ (·) as the sampling function, s > 0. The uniform convergence of the sampling sums to the initial function is proved by the circular truncation error upper bound, especially derived for this reconstruction procedure. Finally, the explicit second and third order sampling formulæ are given.

We establish conditions necessary for the existence of non-constant periodic solutions of non-linear autonomous difference equations with Lipschitzian right-hand sides.

We give a unified approach to inequalities of Calderon–Zygmund type for trigonometric polynomials of several variables based on the Fourier analytic methods. Sharp results are achieved for the full range of admissible parameters p , 0 < p ≤ +∞. The results obtained are applied to the problem of the image of the Fourier transform in the scale of Besov spaces.

A strongly harmonizable continuous time symmetric α -stable process is considered. By using covariations, a Hilbert space is formed from the process elements and used for a purpose of moving average representation and prediction.

The problem of constructing robust optimal in the mean-variance sense trading strategies is considered. The approach based on the notion of sensitivity of a risk functional of the problem w.r.t. small perturbation of asset price model parameters is suggested. The optimal mean-variance robust trading strategies are constructed for one-dimensional diffusion models with misspecified volatility.