We consider semi-reducible pseudo-Riemannian spaces with algebraic conditions on the Ricci tensor and the Riemann tensor. For almost Einstein and weakly recurrent spaces we find the type of tensor characteristic of semi-reducibility. Semi-reducible almost Einstein spaces and weakly recurrent spaces are divided into types depending on the properties of the vector fields that exist in them by necessity. The study is carried out locally in the tensor form.

In the paper an extreme problem of geometric function theory of a complex variable on the maximum of product of the inner radii on a system of n mutually non-overlapping multiply connected domains Bk containing the fixed points ak, k=1,...,n, located on a hyperbola branches is investigated.

The article is devoted to the basic questions of the theory of 2F-planar mappings of manifolds, which are endowed with a certain type affinor structure. We proved that the class of pseudo-Riemannian spaces with absolutely parallel f-structure is closed with respect to the considered mappings. In addition, under the condition of covariant constancy of the affinor f-structure in the mapped spaces, non-trivial 2F-planar mappings can be of three types: complete and canonical of types I and II. Previously, we studied complete 2F-planar mappings in detail. This article considers the main issues for canonical 2F-planar mappings of the first type. Theorems have been proved that give a regular method that allows for any pseudo-Riemannian space with absolutely parallel f-structure (Vn, gij, Fhi) to either find all spaces (V'n, g'ij, F'hi) onto which Vn admits a canonical 2F-planar mapping of the first type, or to prove that there are no such spaces. In particular, we have shown that a pseudo-Riemannian space with absolutely parallel f-structure, in which there is a concircular or quasi-concircular vector field, admits a non-trivial canonical 2F-planar mapping of the first type.

We obtain first and second variation formulae for minimal surfaces in three-dimensional sub-Riemannian manifolds which are vertical, i.e., perpendicular to the horizontal distribution of the sub-Riemannian structure. We use these formulae to explore the connection between Riemannian and sub-Riemannian properties of a surface. We also describe vertical minimal surfaces of left-invariant sub-Riemannian structures on some three-dimensional Lie groups and find out whether they are stable.

The spectral Sturm-Liouville problem on an equilateral star graph of three edges with the standard conditions at the interior vertex and the Robin conditions at the pendant vertices is considered. It is shown that under the conditions of zero potentials on the edges the asymptotics of the eigenvalues determine the shape of the graph and the constants in the Roben conditions. This method can be applied for the case of more complicated equilateral simple connected graphs.

In the present paper, we study the topology of the punctual Hilbert Schemes of monomial plane curve singularities. Piontkowski described the Euler numbers and the Betti numbers of the Jacobian factors of the same singularities. Applying his method, we generalize his results to the case of punctual Hilbert schemes.

Abelian relations of a curvilinear web are the solutions of the partial differential equation defined by a certain differential operator that we prove to be always "ordinary" and "calibrated". Thus we begin by explaning why the space of solutions of the partial differential equation defined by such an ordinary and calibrated homogeneous linear differential operator (of arbitrary order) is isomorphic to the space of the sections of a certain vector bundle with vanishing covariant derivative for a certain tautological connection. We then apply this result to the case of webs by curves, recovering the upper-bound for the rank of such a web given in [D. Damiano, PhD Thesis, Brown University, 1986] та [D. Damiano, Amer. J. Math. (1983) 105:6, 1325-1345], and defining finally the "curvature" of the web which vanishes iff the web has maximal rank.

In this paper we study a new class of pseudo-differential equations on functions of two $p$-adic variables. It is proved that the correspondent Cauchy problem has a unique solution. Some properties of this solution are studied, in particular, the finite dependence property and an $L^1$-estimate.

We present some regularity results on the gradient of the weak or entropic-renormalized solution $u$ to the homogeneous Dirichlet problem for the quasilinear equations of the form \begin{equation*}\label{p-laplacian_eq} -\div(|\nabla u|^{p-2}\nabla u)+V(x;u)=f, \end{equation*}where $\Omega$ is a bounded smooth domain of $\mathbb R^n$, $V$ is a nonlinear potential and $f$ belongs to non-standard spaces like Lorentz-Zygmund spaces. The results, which have been exposed by the third author in a talk presented in AGMA 2024, the International Scientific Online Conference «Algebraic and geometric methods of analysis», May 27-30, 2024, Ukraine, constitute only a part of results proved in detail in a paper coauthored with I. Ahmed, A. Fiorenza, A. Gogatishvili, A. El Hamidi and J.M. Rakotoson (\cite{AFFGER2023}). Moreover, we collect some well-known and new results of the identication of some interpolation spaces and we enrich some contents with details.

The present work concerns generalized convex sets in the real multi-dimensional Euclidean space, known as weakly 1-convex and weakly 1-semiconvex sets. An open set is called weakly 1-convex (weakly 1-semiconvex) if, through every boundary point of the set, there passes a straight line (a closed ray) not intersecting the set. A closed set is called weakly 1-convex (weakly 1-semiconvex) if it is approximated from the outside by a family of open weakly 1-convex (weakly 1-semiconvex) sets. A point of the complement of a set to the whole space is a 1-nonconvexity (1-nonsemiconvexity) point of the set if every straight line passing through the point (every ray emanating from the point) intersects the set. It is proved that if the collection of all 1-nonconvexity (1-nonsemiconvexity) points corresponding to an open weakly 1-convex (weakly 1-semiconvex) set is non-empty, then it is open. It is also proved that the non-empty interior of a closed weakly 1-convex (weakly 1-semiconvex) set in the space is weakly 1-convex (weakly 1-semiconvex).