We study the adiabatic limit in hyperbolic Ginzburg–Landau equations which are Euler–Lagrange equations for the Abelian Higgs model. Solutions of Ginzburg–Landau equations in this limit converge to geodesics on the moduli space of static solutions in the metric determined by the kinetic energy of the system. According to heuristic adiabatic principle, every solution of Ginzburg–Landau equations with sufficiently small kinetic energy can be obtained as a perturbation of some geodesic. A rigorous proof of this result was proposed recently by Palvelev.

A guaranteed deterministic problem setting of super-replication with discrete time is considered: the aim of hedging of a contingent claim is to ensure coverage of the possible payout under an option contract for all admissible scenarios. These scenarios are given by means of compacts given a priori, which depend on the prehistory of prices: the increments of the (discounted) price at each moment of time must lie in the corresponding compacts. The reference probability, common for financial mathematics, is not needed. In the current framework we arrive at a control problem under uncertainty with discrete time, which has a game-theoretic interpretation. The capital, which will be necessary at some moment in time to cover the contingent liability during the time interval up to expiration, satisfies Bellman–Isaacs equations for both the pure and the mixed strategies. The stochastic description of price dynamic therefore arises when considering mixed strategies of the “market,” such that the conditional distributions of price increments given price history have supports contained in the corresponding compacts. In the present paper, under the assumption of no trading constraints, we prove that if there are no arbitrage opportunities, then the equilibrium holds for mixed strategies with the conditional distribution of price increments concentrated in a finite set, and we show that the optimal strategies of the “market” can be found among the “risk-neutral” strategies.

The paper extends a classical result on the convergence of the Krawtchouk polynomials to the Hermite polynomials. We provide a uniform asymptotic expansion in terms of Hermite polynomials and obtain explicit expressions for a few first terms of this expansion. The research is motivated by the study of ergodic sums of the Pascal adic transformation. Bibliography: 10 titles.

Various approaches for data storing and processing are investigated in the article. New algorithm to find paths in a huge graph is introduced.

In this paper we propose an algorithm for finding subgraphs with adjusted properties of large social networks. The description of computational experi-ment which confirms the effectiveness of the proposed algorithm is given.

An improved and corrected version of the functional limit theorem is proved establishing weak convergence of random walks generated by compound doubly stochastic Poisson processes (compound Cox processes) to Lévy processes in the Skorokhod space under more realistic moment conditions. As corollaries, theorems are proved on convergence of random walks with jumps having finite variances to Lévy processes with variance-mean mixed normal distributions, in particular, to stable Lévy processes, generalized hyperbolic and generalized variance-gamma Lévy processes.

In this paper, a computerized model for morphological analysis of languages with word formation based on affixation processes is proposed. The main idea consists in defining structural patterns of words and corresponding lists of suffixes. First, a detailed description of a stemming algorithm, its modification, and the technique of determining grammatical characteristics of word forms are given. The next part of this work focuses on the application of the proposed algorithms for the French language. Finally, some results of execution of these algorithms are provided.

We study the spectral problem for a two-dimensional Hartree type operator with smooth selfaction potential. We find

asymptotic eigenvalues and eigenfunctions and construct an asymptotic expansion for quantum averages near

the lower boundaries of spectral clusters.

We consider the eigenvalue problem for a two-dimensional perturbed resonance oscillator. The role of perturbation is played by an integral Hartree type nonlinearity, where the selfaction potential depends on the distance between points and has logarithmic singularity. We obtain asymptotic eigenvalues near the upper boundaries of spectral clusters appeared near eigenvalues of the unperturbed operator.

In this paper, we prove that the cardinality of the set of all precomplete classes for definite automata is continuum.

In this work, all the dessins d’enfants with no more than 4 edges are listed and their Belyi pairs are computed. In order to enumerate all dessins, the technique of matrix model computations was used. The total number of dessins is 134; among them 77 are spherical, 53 of genus 1, and 4 of genus 2. The automorphism groups of all the dessins are also found. Dessins are listed by the number of edges. Dessins with the same number of edges are ordered lexicographically by their lists of 0-valencies. The corresponding matrix model for any list of 0-valencies is given and computed. Complex matrix models for dessins with 1–3 edges are used. For the dessins with 4 edges, we use Hermitian matrix model.

We obtain central limit type theorems for the total number of edges in the generalized random graphs with random vertex weights under different moment conditions on the distributions of the weights. © 2016 Springer Science+Business Media New York.

A polynomial with exactly two critical values is called a generalized Chebyshev polynomial (or Shabat polynomial). A polynomial with exactly three critical values is called a Zolotarev polynomial. Two Chebyshev polynomials f and g are called Z-homotopic if there exists a family pα, α(Formula presented.) [0, 1], where p0 = f, p1 = g, and pα is a Zolotarev polynomial if α(Formula presented.) (0, 1). As each Chebyshev polynomial defines a plane tree (and vice versa), Z-homotopy can be defined for plane trees. In this work, we prove some necessary geometric conditions for the existence of Z-homotopy of plane trees, describe Z-homotopy for trees with five and six edges, and study one interesting example in the class of trees with seven edges. © 2015 Springer Science+Business Media New York

We consider the class of gradient-like diffeomorphisms possessing an attractor and a repeller separated by a circle on a 2-sphere. For any diffeomorphism in this class we construct a stable arc connecting it with the source-sink system.

In this paper we conduct a comparative analysis of the powers of the two-sample Kolmogorov–Smirnov and Anderson–Darling tests under various alternatives using simulation. We consider two examples. In the first example the alternatives to the standard normal distribution are the distributions of the so-called contaminated normal model. We study the influence of a small contamination with a positive shift on the powers of the test. In the second example the alternatives are the logistic and the Laplace distributions, which are symmetric and differ in shape from the normal distribution having a larger kurtosis coefficient and heavier tails.

We compute the class Wn−4(Formula presented.), which is Poincaré dual to the first Stiefel–Whitney class for the variety (Formula .presented.) in terms of the natural cell decomposition of (Formula presented.) © 2015 Springer Science+Business Media New York.

Since the first optical instruments have been invented, an idea that the visible image of an object under observation depends on tools of observation became commonly assumed in physics. A way of formalizing this idea in mathematics is the construction that assigns to an arbitrary object A in a category K its envelope Env^Ω_ Φ A in a given class Ω of morphisms (a class of representations) with respect to a given class of morphisms (a class of observation tools) Φ. It turns out that if we take a sufficiently wide category of topological algebras as K, then each choice of the classes Ω and Φ defines a “projection of functional analysis into geometry”, and the standard “geometric disciplines”, like complex geometry, differential geometry, and topology, become special cases of this construction. This gives a formal scheme of “categorical construction of geometries” with many interesting applications, in particular, “geometric generalizations of the Pontryagin duality” (to the classes of noncommutative groups). In this paper, we describe this scheme in topology and in differential geometry.