Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ

A higher integrability of the gradient of a solution to the Zaremba problem in a bounded Lipschitz domain is proved for the inhomogeneous p-Laplace equation with drift.

We prove solvability of the Dirichlet problem for the soft Laplacian on a stratified set basing on a modification of the Perron well-known method

We give the rigorous derivation of hydrodynamic equations for an infinite harmonic crystal coupled to the Klein–Gordon field. These equations hold in the hydrodynamic limit, and they should be considered as the analog of the Euler and Navier–Stokes equations for the model under consideration.

The work is devoted to the study of the combinatorial properties of determinism for a family of substitution complexes consisting of quadrangles glued together side-to-side. These properties are useful in constructing algebraic structures with a finite number of defining relations. In particular, this method was used to construct a finitely defined infinite nilsemigroup satisfying the identity x9 = 0. This construction solves the problem of L.N. Shevrin and M.V. Sapir. In this paper, we study the possibility of coloring the entire family of complexes in a finite number of colors, for which the weak determinism property is satisfied: if the colors of the three vertices of a certain quadrilateral are known, then the color of the fourth side is uniquely determined, except in some cases of a special arrangement of the quadrilateral. Even weak determinism is enough to construct a finitely defined nilsemigroup; when using this construction, the proof is reduced in scope. The properties of determinism were studied earlier within the framework of the theory of tessellations; in particular, Kari and Papasoglu constructed a set of square tiles that allowed only aperiodic tessellations of the plane and had determinism: the colors of the two adjacent edges were uniquely determined by the colors of the two remaining edges.

The note provides a new formula for the companion matrix of the superposition of two polynomials over a commutative ring. The results obtained are used to provide a constructive proof of Plans’ theorem for two-bridge knots, which states that the first homology group of an odd-sheeted cyclic covering of a three-dimensional sphere branched over a given knot is the direct sum of two copies of some Abelian group. A similar result is also true for the homology of even-sheeted coverings factored by the reduced homology group of a two-sheeted covering. The structure of the above mentioned Abelian groups is described through Chebyshev polynomials of the second and fourth kind.

In differential equations describing the behavior of continuous media with creep, in accordance with Volterra’s linear theory, applicable to a wide range of materials with amorphous and heterogeneous structure, integral type operators are present. In these equations, the kernel of the integral operator is represented as a sum of exponentials, or as a weakly singular kernel (the Rabotnov function). Obtaining an analytical solution for the equations in question is problematic in some cases, hence the need to develop a numerical method and algorithm for solving such equations, taking into account the memory of the medium in question. To solve these equations, the paper uses the grid-characteristic method and the coordinate splitting method (for multidimensional problems). The approximation and stability of the proposed method are numerically investigated.

We give a simple proof of a recently obtained in [12] result on the completeness of modal logics with the modality that corresponds to the intersection of accessibility relations in a Kripke model. In epistemic logic, this is the so-called distributed knowledge operator. We prove completeness for the logics in the modal languages of two types: one has the modalities □1,...,□n for the relations R1,...,Rn that satisfy a unimodal logic L, and the modality □n+1 for the intersection Rn+1=R1 ∩...∩ Rn; the other language has the modalities □i (i ∈ Σ) for the relations Ri that satisfy the logic L, and, for every nonempty subset of indices I ⊆ Σ, the modality □I for the intersection ∩i∈I Ri. While in [12] the completeness is proved only for the logics over K, KD, KT, K4, S4, and S5, here we give a "uniform" construction that enables us to obtain completeness for the logics with intersection over the 15 so-called "traditional" modal logics KΛ, for Λ ⊆ {D, T, B, 4, 5}. The proof method is based on unravelling of a frame and then taking the Horn closure of the resulting frame.