Volume 11, Issue 3

It is proved that any closed piecewise-continuous curve can be converted by means of a small deformation to a normal curve such that the initial curve and the obtained one may have an interior extension only simultaneously.

A recursion formula for the coefficients of entire functions which are solutions of linear differential equations with polynomial coefficients is derived. Some explicit examples are developed. The Newton sum rules for the powers of zeros of a class of entire functions are constructed in terms of Bell polynomials.

For a simply connected space 𝐵 the Hirsch model of path fibration is constructed in terms of 𝐵. In particular this means the calculation of loop space cohomology. As an application, the Hirsch model of the fiber of any fibration over 𝐵 is given.

Commencing from a monoidal semigroup 𝐴, we consider the geometry of the space 𝑊(𝐴) of pseudoregular elements. When 𝐴 is a Banachable algebra we show that there exist certain subspaces of 𝑊(𝐴) that can be realized as submanifolds of 𝐴. The space 𝑊(𝐴) contains certain subspaces constituting the Stiefel manifolds of framings for 𝐴. We establish several embedding results for such subspaces, where the relevant maps induce embeddings of associated Grassmann manifolds.

This paper is mainly concerned with the following functional equation where 𝐺 is a locally compact group, 𝐾 a compact subgroup of its morphisms, and μ is a generalized Gelfand measure. It is shown that continuous and bounded solutions of this equation can be expressed in terms of μ -spherical functions. This extends the previous results obtained by Badora (Aequationes Math. 43: 72–89, 1992) on locally compact abelian groups. In the case where 𝐺 is a connected Lie group, we characterize solutions of the equation in question as joint eigenfunctions of certain operators associated to the left invariant differential operators.

We prove that the maximal operator of the Marcinkiewicz mean of integrable two-variable functions is of weak type (1, 1) on bounded two-dimensional Vilenkin groups. Moreover, for any integrable function 𝑓 the Marcinkiewicz mean σ 𝑛 𝑓 converges to 𝑓 almost everywhere.

The notions of a negligible set and of an absolutely nonmeasurable set are introduced and discussed in connection with the measure extension problem. In particular, it is demonstrated that there exist subsets of the plane 𝐑 2 which are 𝑇 2 -negligible and, simultaneously, 𝐺-absolutely nonmeasurable. Here 𝑇 2 denotes the group of all translations of 𝐑 2 and 𝐺 denotes the group generated by {𝑔} ∪ 𝑇 2 , where 𝑔 is an arbitrary rotation of 𝐑 2 distinct from the identity transformation and all central symmetries of 𝐑 2 .

The concept of measurability of real-valued functions with respect to various classes of measures is introduced and the associated notion of an absolutely nonmeasurable function is investigated. A characterization of such functions is given. Also, it is shown that functions produced by the classical Vitali partition of the real line are measurable with respect to the class of all extensions of the Lebesgue measure on this line.

A class of static boundary value problems of thermoelasticity is effectively solved for bodies bounded by coordinate surfaces of generalized cylindrical coordinates ρ , α , 𝑧 ( ρ , α are orthogonal curvilinear coordinates on the plane and 𝑧 is a linear coordinate). Besides in the Cartesian system of coordinates some boundary value thermoelasticity problems are separately considered for a rectangular parallelepiped. An elastic body occupying the domain Ω = { ρ 0 < ρ < ρ 1 , α 0 < α < α 1 , 0 < 𝑧 < 𝑧 1 }, is considered to be weakly transversally isotropic (the medium is weakly transversally isotropic if its nine elastic and thermal characteristics are correlated by one or several conditions) and non-homogeneous with respect to 𝑧.

Abstarct A sufficient condition is given for the existence of a solution to a stochastic differential equation in an arbitrary Banach space. The method is based on the concept of covariance operator and a special construction of the Itô stochastic integral in an arbitrary Banach space.

Let 𝐺 be an open subset of ℂ and let 𝑉 be an arbitrary system of weights on 𝐺. Let 𝐻𝑉 𝑏 (𝐺) and 𝐻𝑉 0 (𝐺) be the weighted locally convex spaces of holomorphic functions with a topology generated by seminorms which are weighted analogues of the supremum norm. In the present article, we characterize the analytic functions inducing multiplication operators and invertible multiplication operators on the spaces 𝐻𝑉 𝑏 (𝐺) and 𝐻𝑉 0 (𝐺) for different systems of weights 𝑉 on 𝐺. A (linear) dynamical system induced by multiplication operators on these spaces is obtained as an application of the theory of multiplication operators.

Let 𝐸 be a real Banach space with norm ‖ · ‖ and let {𝑥 𝑛 } 𝑛≥0 be a generalized nonexpansive sequence in 𝐸 (i.e., ‖ 𝑥 𝑖+1 – 𝑥 𝑗+1 ‖ 2 ≤ ‖𝑥 𝑖 – 𝑥 𝑗 ‖ 2 + ( ε (𝑖+1, 𝑗+1) – ε (𝑖, 𝑗)) 2 for all 𝑖, 𝑗 ≥ 0, where the series of nonnegative terms is convergent). Let . We deal with the mean point of concerning a Banach limit μ . If 𝐸 is reflexive and 𝑑 = 𝑑(0, 𝐾), then we show that and there exists a point 𝑧 0 with ‖ 𝑧 0 ‖= 𝑑 such that . In the sequel, this result is applied to obtain the weak and strong convergence of .

We present a general existence principle which can be used for a large class of singular boundary value problems of the form where 𝑓 satisfies the local Carathéodory conditions on [0, 𝑇] × 𝐷, a set 𝐷 ⊂ is not closed, 𝑓 has singularities in its phase variables on the boundary ∂𝐷 of 𝐷, and 𝑆 is a closed subset in 𝐶 𝑛–1 ([0, 𝑇]). The proof is based on the regularization and sequential techniques. An application of the general existence principle to singular conjugate (𝑝, 𝑛–𝑝) BVPs is also given.

Displacement vectors are represented by combinations of special potentials; singular integral equations of the normal type with zero index are obtained for the first and the second boundary value problem of steady oscillations in the theory of elastic mixtures. It is proved that in the case of positive frequencies the corresponding homogeneous singular integral equations have only a trivial solution.

On the interval [𝑎, 𝑏], we consider the boundary value problem where ℓ : 𝐶([𝑎, 𝑏]; ℝ) → 𝐿([𝑎, 𝑏]; ℝ), 𝑞 ∈ 𝐿([𝑎, 𝑏]; ℝ), and 𝑡 0 ∈]𝑎, 𝑏[. The question on the existence and uniqueness of a nonnegative solution of this problem is studied.