On symmetry reduction and some classes of invariant solutions of the (1+3)-dimensional homogeneous Monge-Ampère equation

We study the relationship between structural properties of the two-dimensional nonconjugate subalgebras of the same rank of the Lie algebra of the Poincaré group P(1,4) and the properties of reduced equations for the (1+3)-dimensional homogeneous Monge-Ampère equation. In this paper, we present some of the results obtained concerning symmetry reduction of the equation under investigation to identities. Some classes of the invariant solutions (with arbitrary smooth functions) are presented.

On tensor products of nuclear operators in Banach spaces

The following result of G. Pisier contributed to the appearance of this paper: if a convolution operator ★f : M(G) → C(G), where $G$ is a compact Abelian group, can be factored through a Hilbert space, then f has the absolutely summable set of Fourier coefficients. We give some generalizations of the Pisier's result to the cases of factorizations of operators through the operators from the Lorentz-Schatten classes Sp,q in Hilbert spaces both in scalar and in vector-valued cases. Some applications are given.

On homotopy nilpotency of some suspended spaces

A homological criterium from [Golasiński, M., On homotopy nilpotency of loop spaces of Moore spaces, Canad. Math. Bull. (2021), 1–12] is applied to investigate the homotopy nilpotency of some suspended spaces. We investigate the homotopy nilpotency of the wedge sum and smash products of Moore spaces M (A, n) with n ≥ 1. The homotopy nilpotency of homological spheres are studied as well.

Generalized φ(Ric)-vector fields in special pseudo-Riemannian spaces

The paper treats pseudo-Riemannian spaces permitting generalized φ(Ric)-vector fields. We study conditions for the existence of such vector fields in conformally flat, equidistant, reducible and Kählerian pseudo-Riemannian spaces. The obtained results can be applied for the construction of generalized φ(Ric)-vector fields that differ from φ(Ric)-vector fields. The research is carried out locally without limitations imposed on a sign of metric tensor.

On the geodesic mappings of pseudo-Riemannian spaces with special supplementary tensor

В роботі досліджуються два псевдоріманових простори, які мають спільні геодезичні лінії. Вимагається виконання умов алгебраїчного та диференціального характеру на тензор Рімана одного з них. А операція опускання індексів та обчислення коваріантної похідної здійснюється відносно метрики та об'єктів зв'язності іншого простору. Для досліджень використовується спеціальний допоміжний тензор. Доведено, що виконання додаткових умов приводить до просторів, що не допускають нетривіальних геодезичних відображень, або простори належать до еквідістантних просторів. Використовуються тензорні методи без обмежень на знак метрики.

On conformally reducible pseudo-Riemannian spaces

The present paper studies the main type of conformal reducible conformally flat spaces. We prove that these spaces are subprojective spaces of Kagan, while Riemann tensor is defined by a vector defining the conformal mapping. This allows to carry out the complete classification of these spaces. The obtained results can be effectively applied in further research in mechanics, geometry, and general theory of relativity. Under certain conditions the obtained equations describe the state of an ideal fluid and represent quasi-Einstein spaces. Research is carried out locally in tensor shape.

Some applications of transversality for infinite dimensional manifolds

We present some transversality results for a category of Frechet manifolds, the so-called MCk - Frechet manifolds. In this context, we apply the obtained transversality results to construct the degree of nonlinear Fredholm mappings by virtue of which we prove a rank theorem, an invariance of domain theorem and a Bursuk-Ulam type theorem.

Galois coverings of one-sided bimodule problems

Applying geometric methods of 2-dimensional cell complex theory, we construct a Galois covering of a bimodule problem satisfying some structure, triangularity and finiteness conditions in order to describe the objects of finite representation type. Each admitted bimodule problem A is endowed with a quasi multiplicative basis. The main result shows that for a problem from the considered class having some finiteness restrictions and the schurian universal covering A', either A is schurian, or its basic bigraph contains a dotted loop, or it has a standard minimal non-schurian bimodule subproblem.

Deformations of circle-valued Morse functions on 2-torus

In this paper we give an algebraic description of fundamental groups of orbits of circle-valued Morse functions on T2 with respect to the action of the group of diffeomorphisms of T2

Геодезичні відображення коспактних квазі-ейнштейнових просторів

The paper treats geodesic mappings of quasi-Einstein spaces with gradient defining vector. Previously the authors defined three types of these spaces. In the present paper it is proved that there are no quasi-Einstein spaces of special type. It is demonstrated that quasi-Einstein spaces of main type are closed with respect to geodesic mappings. The spaces of particular type are proved to be geodesic $D$-symmetric spaces.