Vol 98, No 3 (2018)

Spaces of test functions and spaces of distributions (generalized measures) on infinite-dimensional spaces are constructed, which, in the finite-dimensional case, coincide with classical spaces \(\mathscr{D}\) and \(\mathscr{D}'\). These distribution spaces contain generalized Feynman measures (but do not contain a generalized Lebesgue measure, which is not considered in this paper). For broad classes of infinite-dimensional differential equations in distribution spaces, the Cauchy problem has fundamental solutions. These results are much more definitive than those of A.Yu. Khrennikov’s and A.V. Uglanov’s pioneering works.

By introducing the concept of a γ-convex set, a new discrete analogue of Pontryagin’s maximum principle is obtained. By generalizing the concept of the relative interior of a set, an equality-type optimality condition is proved, which is called by the authors the Pontryagin equation.

A mixed problem with acoustic transmission conditions for nonlinear hyperbolic equations with nonlinear dissipation is considered. The existence, uniqueness, and exponential decay of global solutions to this problem with focusing nonlinear sources are proved Additionally, the existence of global solutions and the solution blow-up in a finite time are proved for the case of defocusing nonlinear sources.

A key intractable problem in logical data analysis, namely, dualization over the product of partial orders, is considered. The important special case where each order is a chain is studied. If the cardinality of each chain is equal to two, then the considered problem is to construct a reduced disjunctive normal form of a monotone Boolean function defined by a conjunctive normal form, which is equivalent to the enumeration of irreducible coverings of a Boolean matrix. The asymptotics of the typical number of irreducible coverings is known in the case where the number of rows in the Boolean matrix has a lower order of growth than the number of columns. In this paper, a similar result is obtained for dualization over the product of chains when the cardinality of each chain is higher than two. Deriving such asymptotic estimates is a technically complicated task, and they are required, in particular, for proving the existence of asymptotically optimal algorithms for the problem of monotone dualization and its generalizations.

Structures and objects used in Hamiltonian secondary quantization are discussed. By the secondary quantization of a Hamiltonian system ℋ, we mean the Schrödinger quantization of another Hamiltonian system ℋ₁ for which the Hamiltonian equation is the Schrödinger one obtained by the quantization of the original Hamiltonian system ℋ. The phase space of ℋ₁ is the realification ℍ_R of the complex Hilbert space ℍ of the quantum analogue of ℋ equipped with the natural symplectic structure. The role of a configuration space is played by the maximal real subspace of ℍ.

The first initial boundary value problem for a one-dimensional (in x) Petrovskii parabolic second-order system with constant coefficients in a semibounded (in x) domain with a nonsmooth lateral boundary is proved to have a unique classical solution in certain Hölder classes.

The solution of the Cauchy problem for a nonlinear Schrödinger evolution equation with certain initial data is proved to blow up in a finite time, which is estimated from above. Additionally, lower bounds for the blow-up rate are obtained in some norms.

The method of molecular dynamics was used for modeling the isomerization of a hydrogen bonding network in small water clusters (hexamer and octamer). The collective modes of the particles moving in the clusters were determined by applying principal component analysis. An entropy criterion for phase transitions in water clusters was suggested. This criterion can be used to study phase transitions in weakly bound atomic and molecular clusters.

We classify the types of root systems R in the rings of integers of number fields K such that the Weyl group W(R) lies in the group generated by Aut(K) and multiplications by the elements of K*.

In this paper, we find new identities and asymptotic formulas in the problem of multiplicative partitions, which is an analogue of the multidimensional problem of Dirichlet divisors with the condition of disorder of the factors.

In the paper, Lax pairs for linear Hamiltonian systems of differential equations are found. Besides, first integrals of the system which are obtained from the Lax pairs are investigated.

The classical problem of level crossing by a compound renewal process is considered, which has been extensively studied and has various applications. For the distribution of the first level crossing time, a new approximation is proposed, which is valid under minimal conditions and is obtained by applying a new method. It has a number of advantages over previously known approximations.

We calculate homology groups with certain twisted coefficients of configuration spaces of projective spaces. This completes a calculation of rational homology groups of spaces of odd maps of spheres S^m → S^M, m < M, and of the stable homology of spaces of non-resultant polynomial maps ℝ^m+1 → ℝ^M+1. Also, we calculate the homology of spaces of ℤ_r-equivariant maps of odd-dimensional spheres, and discuss further generalizations.

The Le Bars conjecture (2001) states that the binomial random graph G(n, \(\frac{1}{2}\)) obeys the zero–one law for existential monadic sentences with two first-order variables. This conjecture is disproved. Moreover, it is proved that there exists an existential monadic sentence with a single monadic variable and two first-order variables whose truth probability does not converge.

A new method for entropy-randomized machine learning is proposed based on empirical risk minimization instead of the exact fulfillment of empirical balance conditions. The corresponding machine learning algorithm is shown to generate a family of exponential distributions, and their structure is found.

A method for introducing artificial dissipation coefficients into a numerical algorithm based on the quasi-gasdynamic system of equations is proposed. The method applies to aerodynamic flows with large Mach and Reynolds numbers. Simulation results for the supersonic flow over the X-43 aircraft are presented as an illustration. The pressure distribution over the aircraft surface is obtained, which can be used to calculate the aerodynamic characteristics of X-43.

The set of extremal trajectories is completely described. Their construction is reduced to finding the best routes on a directed graph whose vertices are subsets (boxes) of \(Y\backslash \mathop \cup \limits_S K\left( S \right)\) and whose edges are segments T(S) of the trajectory T that intersect the cones K(S) in the “best way.” The edge length is the deviation of S from T(S). The best routes are ones for which the length of the shortest edge is maximal.