Vol 96, No 1 (2017)

Sharp sufficient conditions on the coefficients of a second-order parabolic equation are examined under which the solution of the corresponding Cauchy problem with a power-law growing initial function stabilizes to zero. An example is presented showing that the found sufficient conditions are sharp. Conditions on the coefficients of a parabolic equation are obtained under which the solution of the Cauchy problem with a bounded initial function stabilizes to zero at a power law rate.

Necessary and sufficient conditions for the weighted boundedness of a class of positive quasilinear two-kernel integral operators of iterated type on the real half-line are given.

Spectra of first-order formulas are studied. The spectrum of a first-order formula is the set of all positive α such that either this formula is true for the random graph G(n, n^−α) with an asymptotic probability being neither 0 nor 1 or the limit does not exist. It is well known that there exists a first-order formula with an infinite spectrum. The minimum number of quantifier alternations in such a formula is found.

Article [1] raised the question of the finiteness of the number of square-free polynomials f ∈ ℚ[h] of fixed degree for which \(\sqrt f \) has periodic continued fraction expansion in the field ℚ((h)) and the fields ℚ(h)(\(\sqrt f \)) are not isomorphic to one another and to fields of the form ℚ(h)\(\left( {\sqrt {c{h^n} + 1} } \right)\), where c ∈ ℚ* and n ∈ ℕ. In this paper, we give a positive answer to this question for an elliptic field ℚ(h)(\(\sqrt f \)) in the case deg f = 3.

A dispersion analysis is conducted for bicompact schemes of fourth-order accuracy in space, namely, for a semidiscrete scheme and a second-order accurate scheme in time. It is shown that their numerical group velocity is positive for all dimensionless wavenumbers. It is proved that the dispersion properties of the bicompact schemes are preserved on highly nonuniform meshes. A comparison reveals that the fourth-order bicompact schemes have a higher spectral resolution than not only other same-order compact schemes, but also some sixth-order ones. Two numerical examples are presented that demonstrate the ability of the bicompact schemes to adequately simulate wave propagation on highly nonuniform meshes over long time intervals.

Suppose that a strongly regular graph Γ with parameters (v, k, λ, μ) has eigenvalues k, r, and s. If the graphs Γ and \(\bar \Gamma \) are connected, then the following inequalities, known as Krein’s conditions, hold: (i) (r + 1)(k + r + 2rs) ≤ (k + r)(s + 1)² and (ii) (s + 1)(k + s + 2rs) ≤ (k + s)(r + 1)². We say that Γ is a Krein graph if one of Krein’s conditions (i) and (ii) is an equality for this graph. A triangle-free Krein graph has parameters ((r² + 3r)², r³ + 3r² + r, 0, r² + r). We denote such a graph by Kre(r). It is known that, in the cases r = 1 and r = 2, the graphs Kre(r) exist and are unique; these are the Clebsch and Higman–Sims graphs, respectively. The latter was constructed in 1968 together with the Higman–Sims sporadic simple group. A.L. Gavrilyuk and A.A. Makhnev have proved that the graph Kre(3) does not exist. In this paper, it is proved that the graph Kre(4) (a strongly regular graph with parameters (784, 116, 0, 20)) does not exist either.

The paper studies the quantity p(n, k, t₁, t₂) equal to the maximum number of edges in a k-uniform hypergraph with the property that the size of the intersection of any two edges lies in the interval [t₁, t₂]. Previously known upper and lower bounds are given. New bounds for p(n, k, t₁, t₂) are obtained, and the relationship between these bounds and known estimates is studied. For some parameter values, the exact values of p(n, k, t₁, t₂) are explicitly calculated.

The paper studies approximation and structural geometric-topological properties of sets in normed and more general (asymmetric) spaces for which there exists a continuous selection for the best and near-best approximation operators. Sufficient conditions on the metric projection of sets which ensure the existence of a continuous selection for this projection are obtained, and the structural properties of such sets are determined. The existence of a continuous selection for the near-best approximation operator on a finite-dimensional space more general than a normed space is investigated. It is shown that the lower semicontinuity of the metric projection is sufficient for the existence of a continuous selection for the near-best approximation operator in the general case.

The definition of Feynman path integrals (Feynman functional integrals) as integrals with respect to a generalized measure, called the Lebesgue–Feynman measure in the paper and being an infinite-dimensional analogue of the classical Lebesgue measure on finite-dimensional Euclidean space, is discussed. This definition, which is a formalization of Feynman’s original definition, is different from those used previously in the mathematical literature. It makes it possible to give a description of the origin of quantum anomaly which is a mathematically correct version of the description given in the book Path Integrals and Quantum Anomalies by K. Fujikawa and H. Suzuki (Oxford, 2004) (and erroneously qualified as wrong in the book Functional Integration: Action and Symmetries by P. Cartier and C. DeWitt-Morette (Cambridge Univ. Press, Cambridge, 2006)).

The communication concerns a theory of global solvability of initial value problem for nonlinear hyperbolic equations with two independent variables that is an immediate analog of a theory of global solvability of ordinary differential equations.

The Fokker–Planck–Kolmogorov equations with a degenerate or partially degenerate diffusion matrix are considered. The distance between probability solutions of these equations with different drift coefficients and different initial conditions is estimated. Sufficient conditions for the existence and uniqueness of probability solutions to nonlinear Fokker–Planck–Kolmogorov equations with a partially degenerate diffusion matrix are established.

The initial stage of the Rayleigh–Bénard convective instability regarded as a nonequilibrium phase transition is reconstructed. Its mechanism is based on spinodal decomposition (diffusion separation).

Interest in the inverse Sturm–Liouville problem is motivated by its numerous applications in mathematics and computational physics. To solve a complete inverse problem, one needs two exact spectra, which are usually not known in experimental spectroscopy. Accordingly, a problem of interest is to restore the potential from a finite set of noisy spectral data. A new variational method for solving inverse spectral problems is proposed, which is based on the regularization of ill-posed problems. The method takes into account the measurement error of the spectrum and restores the potential without using simplifying assumptions that it belongs to a certain functional class. The method has been tested on potentials involving smooth segments and jump discontinuities.

The computation of the H_∞ and H₂ norms of 2D systems (systems whose dynamic state depends on two independent variables) can be reduced to algebraic problems underlain by parametrized linear matrix inequalities, which have to hold for all parameter values. Available methods for solving such problems based on their interpretation in terms of nonnegative polynomials are poorly scaled since the sizes of auxiliary problems grow rapidly. In this paper, solution methods are presented that are simpler and more efficient as compared with well-known results.