Vol 99, No 3 (2019)

The problem of homogenizing the diffusion equation in a domain perforated along an (n – 1)-dimensional manifold with dynamic boundary conditions on the boundary of the perforations is studied. A homogenized model is constructed that is a transmission problem for the diffusion equation with the transmission conditions containing a term with memory. A theorem on the convergence of solutions of the original problem to the solution of the homogenized one is proved.

Filters and ultrafilters (maximal filters) of a space whose measurable structure is specified by a family of sets that is closed under finite intersections and contains the empty set and an ambient set (unity of the space) are studied. Necessary and sufficient conditions for filters of this space to be maximal are formulated in terms of sets—elements of a dual family, which are called quasi-neighborhoods. These conditions agree with ones known in the theory of Stone spaces, but cover a number of other cases, for example, the situation when the initial set is equipped with a topology (the case of open ultrafilters) or with a family of closed sets of a topological space (i.e., a closed topology in the sense of P.S. Aleksandrov). A key role in these constructions is played by the topology on a space of ultrafilters defined by analogy with the case of a Stone space. Additionally, the case when the above-mentioned measurable space is equipped with a topology admitting a conceptual analogy with the topology used to construct the Wallman extension is considered. As a result, we obtain a bitopological space with comparable topologies, one of which is Hausdorff and the other generates a compact \({{T}_{1}}\)-space. The conditions are indicated under which the topologies coincide, thus yielding a (zero-dimensional) compact set, and the conditions are given under which the topologies differ, thus defining a nondegenerate bitopological space. In the case where the family of sets defining a measurable structure is separable, it is shown that the initial ambient set can be embedded in the above bitopological space in the form of an everywhere dense subset.

This paper is devoted to partial exponential \(MR\)-groups that are isomorphically embeddable in their tensor R-completions. As a consequence, the free \(MR\)-groups and free \(MR\)-products are described in terms of usual group-theoretical free constructions.

The k-variable fragment of first-order logic on graphs is considered. It is proved that, for \(\alpha \leqslant \frac{1}{{k - 1}},\) the random graph \(G(n,{{n}^{{ - \alpha }}})\) obeys the zero–one law with respect to this logic. Moreover, for every \(\varepsilon > 0\), there exists \(\alpha \in \left( {\frac{1}{{k - 1}},\frac{1}{{k - 1}} + \varepsilon } \right)\) such that \(G(n,{{n}^{{ - \alpha }}})\) does not obey the law.

An equivalence theorem is proved for the following conditions: the periodicity of continued fractions of generalized type for key elements of hyperelliptic field \(L\), the existence of nontrivial \(S\)-units in \(L\) for sets \(S\) consisting two valuations of degree one, and the existence of a torsion of certain type in the Jacobian variety associated with hyperelliptic field \(L\). In practice, this theorem allows using continued fractions of generalized type to effectively search for fundamental \(S\)-units of hyperelliptic fields. We give an example of the hyperelliptic field of genus 3, which shows all three equivalent conditions in the indicated theorem.

We study the complexity of an infinite family of graphs \({{H}_{n}} = {{H}_{n}}({{G}_{1}},{{G}_{2}}, \ldots ,{{G}_{m}})\) that are discrete Seifert foliations over a given graph H on m vertices with fibers \({{G}_{1}},{{G}_{2}}, \ldots ,{{G}_{m}}.\) Each fiber G_i = \({{C}_{n}}({{s}_{{i,1}}},{{s}_{{i,2}}},...,{{s}_{{i,{{k}_{i}}}}})\) of this foliation is a circulant graph on n vertices with jumps \({{s}_{{i,1}}},{{s}_{{i,2}}}, \ldots ,{{s}_{{i,{{k}_{i}}}}}.\) The family of discrete Seifert foliations is sufficiently large. It includes the generalized Petersen graphs, I-graphs, Y-graphs, H-graphs, sandwiches of circulant graphs, discrete torus graphs, and other graphs. A closed-form formula for the number \(\tau (n)\) of spanning trees in H_n is obtained in terms of Chebyshev polynomials, some analytical and arithmetic properties of this function are investigated, and its asymptotics as \(n \to \infty \) is determined.

The problem of variational data assimilation for a nonlinear evolutionary model is formulated as an optimal control problem to simultaneously find unknown parameters and the initial state of the model. The response function is treated as a functional of the optimal solution found as a result of assimilation. The sensitivity of the functional to observational data is studied. The gradient of the functional with respect to observations is associated with the solution of a nonstandard problem involving a system of direct and adjoint equations. The solvability of the nonstandard problem is studied using the Hessian of the original cost function. An algorithm for calculating the gradient of the response function with respect to observations is formulated and justified.

For the first time, the theory of strongly continuous cosine operator functions (COF) has been applied to study the correct solvability of boundary value problems for second-order linear differential equations in a Banach space (elliptic case). The correct solvability of the Cauchy problem (hyperbolic case) is usually formulated in COF terms. The conditions on the order of COF growth are specified under which the Dirichlet boundary value problem is correct on a finite interval. An integral representation of the solution and its sharp estimate are given.

Hamiltonian mechanics determined by time-dependent random Hamiltonian functions is called randomized Hamiltonian mechanics. The corresponding Hamiltonian systems are called random. Feynman formulas for random Hamiltonian systems are obtained. It is shown that these formulas describe solutions of a Hamiltonian equation whose Hamiltonian is the mean value of the random Hamiltonian function. Analogous results are obtained for random quantum systems (which are known to be infinite-dimensional random Hamiltonian systems). The random quantum Hamiltonians can be used to describe the so-called open quantum systems (which are part of some lager quantum systems).