Volume 10, Issue 1

A theorem of S. Warschawski on the derivative of a holomorphic function mapping conformally the circle onto a simply-connected domain bounded by the piecewise-Lyapunov Jordan curve is extended to domains with a non-Jordan boundary having interior cusps of a certain type.

Statical and dynamical two-dimensional models of a prismatic elastic shell are constructed. The existence and uniqueness of solutions of the corresponding boundary and initial boundary value problems are proved, the rate of approximation of the solution of a three-dimensional problem by the vector-function restored from the solution of a two-dimensional one is estimated.

We consider the vanishing problem for higher cohomology groups on certain infinite-dimensional complex spaces: good branched coverings of suitable projective spaces and subvarieties with a finite free resolution in a projective space P ( V ) (e.g. complete intersections or cones over finitedimensional projective spaces). In the former case we obtain the vanishing result for H 1 . In the latter case the corresponding results are only conditional for sheaf cohomology because we do not have the corresponding vanishing theorem for P ( V ).

Starting from the exponential, some classes of analytic functions of the derivative operator are studied, including pseudo-hyperbolic and pseudo-circular functions. Some formulas related to operational calculus are deduced, and the important role played in such a context by Hermite–Kampé de Fériet polynomials is underlined.

The correspondence between non-Noether symmetries and bi-Hamiltonian structures is discussed. We show that in regular Hamiltonian systems the presence of the global bi-Hamiltonian structure is caused by the symmetry of the space of solution. As an example, the well-known bi-Hamiltonian realization of the Korteweg–de Vries equation is discussed.

For the differential equation where 𝑝 𝑖 ∈ 𝐿 loc (𝑅 + ; 𝑅 + ), τ 𝑖 ∈ 𝐶(𝑅 + ; 𝑅 + ), τ 𝑖 (𝑡) ≤ 𝑡 for 𝑡 ∈ 𝑅 + , , optimal integral conditions for the oscillation of all solutions are established.

This is the sequel of a paper where we introduced an intrinsic ho-motopy theory and homotopy groups for simplicial complexes. We study here the relations of this homotopy theory with the well-known homology theory of simplicial complexes. Also, our investigation is aimed at applications in image analysis. A metric space 𝑋 representing an image, has a structure of simplicial complex at each resolution ε > 0, and the corresponding combinatorial homology groups give information on the image. Combining the methods developed here with programs for automatic computation of combinatorial homology might open the way to realistic applications.

We introduce the notion of (pre)crossed module in a semiabelian category, and establish equivalences between internal reflexive graphs and precrossed modules, and between internal categories and crossed modules.

The set of cochain operations defining Steenrod ⌣ 𝑖 -products in the bar construction 𝐵𝐶*(𝑋) is defined in terms of surjection operad. This structure extends a Homotopy G-algebra structure which defines only ⌣.

We say that a subset 𝑋 of the plane 𝐑 2 is an 𝑜𝑡-set if any three points of 𝑋 form an obtuse triangle. Some properties of 𝑜𝑡-sets are investigated. It is shown that no finite 𝑜𝑡-subset of 𝐑 2 is maximal, but there exists a countable maximal 𝑜𝑡-subset of 𝐑 2 . Several related problems are formulated and discussed.

The Cauchy type singular integral equation is investigated when the line of integration is the union of a countable number of disconnected segments. The equation is equivalently reduced to an equation with double periodic kernel. Effective solutions having integrable singularities at the end-points of the line of integration are obtained.

In the weighted Lebesgue space with variable exponent the boundedness of the Calderón–Zygmund operator is established. The variable exponent 𝑝(𝑥) is assumed to satisfy the logarithmic Dini condition and the exponent β of the power weight ρ (𝑥) = |𝑥 – 𝑥 0 | β is related only to the value 𝑝(𝑥 0 ). The mapping properties of Cauchy singular integrals defined on the Lyapunov curve and on curves of bounded rotation are also investigated within the framework of the above-mentioned weighted space.

Two explicit guard functions 𝐾 𝑗 = 𝐾 𝑗 ( δ 𝑧 ), 𝑗 = 1, 2, are obtained, which depend on the distance δ 𝑧 between 𝑧 and the nearest point of the integer lattice in the complex plane, such that δ 𝑧 𝐾 1 ( δ 𝑧 ) ≤ | σ (𝑧)|𝑒 –π|𝑧| 2 /2 ≤ δ 𝑧 𝐾 2 ( δ 𝑧 ), 𝑧 ∈ ℂ, where σ (𝑧) stands for the Weierstraß σ -function. This result is used to improve the circular truncation error upper bound in the 𝑞-th order Whittaker-type derivative sampling for the Leont'ev functions space .

The paper deals with the existence of solutions for singular higher order differential equations with Sturm–Liouville or (𝑝, 𝑛 – 𝑝) right focal boundary conditions or the (𝑛 – 𝑝, 𝑝) left focal boundary conditions. Right-hand sides of differential equations may be singular in the zero values of all their phase variables. The proofs are based on the regularization and sequential techniques.

We define a generalized Cesáro sequence space and consider it equipped with the Luxemburg norm under which it is a Banach space, and we show that it is locally uniformly rotund.