Vol 97, No 1 (2018)

Together with the classical plane billiards, topological billiards can be considered, where the motion occurs on a locally flat surface obtained by isometrically gluing together several plane domains along their boundaries, which are arcs of confocal quadrics. A point moves inside each of the domains along straight line segments; when it reaches the boundary of a domain, it passes to another domain. Previously, the author gave a Liouville classification of all topological billiards obtained by gluing along convex boundaries. In the present paper, all topological integrable billiards obtained by gluing along convex or nonconvex boundaries from elementary billiards bounded by arcs of confocal quadrics are classified. For some of such nonconvex topological billiards, the Fomenko–Zieschang invariants (marked molecules W*) for Liouville equivalence are calculated.

Let X be a countable discrete Abelian group containing no elements of order 2. Let α be an automorphism of X. Let ξ₁ and ξ₂ be independent random variables with values in the group X and distributions μ₁ and μ₂. The main result of the article is the following statement. The symmetry of the conditional distribution of the linear form L₂ = ξ₁ + αξ₂ given L₁ = ξ₁ + ξ₂ implies that μ_j are shifts of the Haar distribution of a finite subgroup of X if and only if α satisfies the condition Ker(I + α)= {0}. Some generalisations of this theorem are also proved.

The Sawyer duality principle is obtained for grand Lebesgue spaces on the unit interval, and the Hardy operators are shown to be bounded in these spaces.

The enveloping algebra of the niltriangular subalgebra NΦ(K) of the Chevalley algebra of type A_n−1 is the algebra of niltriangular n × n matrices over K. The enveloping algebras R of other types constructed so far are nonassociative. For classical types, an explicit description of automorphisms of the rings R over any commutative associative ring with an identity is given; in the case where K is a field, all ideals in R are also listed. The enumeration of ideals in R for K = GF(q) leads to a solution of a combinatorial problem concerning ideals of the algebras NΦ(K).

Second-order elliptic differential-difference operators with degeneration in a cylinder associated with closed densely defined sectorial sesquilinear forms in L₂(Q) are considered. These operators are proved to satisfy the Kato conjecture on the square root of an operator.

For Volterra integral equations of the third kind and for Volterra-type integrodifferential equations of the third kind, theorems on the existence of solutions in Sobolev spaces (i.e., regular solutions) are proved. The proofs are based on the theory of boundary value problems for degenerate ordinary differential equations and on the theory of boundary value problems for parabolic equations with a changing evolution direction.

A new statement of a recent theorem of [1, 2] on the maximum number of edges in a hypergraph with forbidden cardinalities of edge intersections is given. This statement is fundamentally simpler than the original one, which makes it possible to obtain important corollaries in combinatorial geometry and Ramsey theory.

A nonlinear perturbed differential system with a linear approximation is considered. An open question has been the validity of a (continual) version of the Perron effect when the set of Lyapunov exponents of all nontrivial solutions (necessarily infinitely extendable on the right) of the corresponding nonlinear perturbed system with a perturbation of arbitrary higher order of smallness in a neighborhood of the origin is measurable, lies entirely on the positive half-line, and has the cardinality of the continuum and even a positive Lebesgue measure. The positive answer to this question is given by the presented theorem, which generally determines an explicit representation of the Lyapunov exponents of all nontrivial solutions to the nonlinear system in terms of their initial values.

One-point commuting difference operators of rank one in the case of hyperelliptic spectral curves are studied. A relationship between such operators and one-dimensional finite-gap Schrödinger operators is investigated. In particular, a discretization of finite-gap Lamé operators is obtained.

An initial–boundary value problem for the heat equation in a three-dimensional domain containing thin cylindrical tubes is considered. The Neumann condition is set on the lateral boundaries of the tubes. The original three-dimensional problem is reduced to a hybrid-dimensional one in which the heat equation in the tubes is replaced by the one-dimensional heat equation in shorter cylinders (subtubes), and the three- and one-dimensional equations are matched on the bases of the subtubes. The difference between the solutions of the original and hybrid-dimensional problems is estimated using two geometric characteristics: the distance between the bases of the tubes and subtubes and the reciprocals of the minimal positive eigenvalues of the Neumann problem for the Laplace operator in the tube cross sections.

A method is proposed for constructing combined shock-capturing finite-difference schemes that localize shock fronts with high accuracy and preserve the high order of convergence in all domains where the computed weak solution is smooth. A particular combined scheme is considered in which a nonmonotone compact scheme with a third-order weak approximation is used as a basis one, while the internal scheme is the second-order accurate (for smooth solutions) monotone CABARET. The advantages of the new scheme are demonstrated using test computations.

The problem of normal waves in a closed (screened) regular waveguide of arbitrary cross section is considered. It is reduced to a boundary value problem for the longitudinal components of the electromagnetic field in Sobolev spaces. The solutions are defined using the variational formulation of the problem. The problem is reduced to the study of an operator function. The properties of the operators involved in the operator function (which are necessary for analyzing its spectral properties) are examined. Theorems are proved concerning the discrete character of the spectrum and the distribution of characteristic numbers of the operator function on the complex plane. The completeness of the system of eigen- and associated vectors of the operator function is investigated. It is proved that the system of eigen- and associated vectors of the operator function is doubly complete with a finite defect.

A one-dimensional equation is presented that generalizes the Burgers equation known in the theory of waves and in turbulence models. It describes the nonlinear evolution of waves in pipes of variable cross section filled with a dissipative medium, as well as in ray tubes, if the approximation of geometric acoustics of an inhomogeneous medium is used. The generalized equation is reduced to the common Burgers equation with a dissipative parameter—the “Reynolds–Goldberg number,” depending on the coordinate. The method for solving statistical problems corresponding to specified characteristics of a noise signal at the input of the system is described. Integral expressions for exact solutions are given for the correlation function and the noise intensity spectrum experiencing nonlinear distortions during propagation in a waveguide. For waves in a dissipative medium, an approximate method of calculating statistical characteristics is given, consisting in finding an auxiliary correlation function and the subsequent nonlinear functional transformation. Solutions have a complicated form, so physical analysis of phenomena requires the numerical methods. For some correlation functions of stationary noise with initial Gaussian statistics and some waveguide systems, it is possible to obtain simple results.

A new approach to the approximation of the effective hull of a multidimensional nonconvex compact set given by a nonlinear mapping is proposed. The effective hull of a nonconvex set is an external estimate of this set that is more accurate than its convex hull. Methods are proposed in the case when the set to be approximated is the image of a compact set of rather high dimension (several hundred variables); moreover, the mapping can be given by a computational module.