Search

In this paper, we describe the technique of extremely monotonic functions in the theory of computable linear orders. The basic definitions of extremely monotonic functions and their generalizations are given, and a number of their basic properties and applications are presented.

The categoricity spectrum of a computable structure S is the set of all Turing degrees capable of computing isomorphisms among arbitrary computable presentations of S. The degree of categoricity of S is the least degree in the categoricity spectrum of S. This paper is a survey of results on categoricity spectra and degrees of categoricity for computable structures. We focus on the results about degrees of categoricity for linear orders and Boolean algebras. We construct a new series of examples of degrees of categoricity for linear orders.

The paper characterizes linear operators converting column majorization into weak, directional, and strong majorizations. An example of a linear converter from weak, directional, and strong majorizations to column majorization preserving none of these majorizations is provided.

In this paper, the length of the group algebra of a dihedral group in the modular case is computed under the assumption that the order of the group is a power of two. Various methods for studying the length of a group algebra in the modular case are considered. It is proved that the length of the group algebra of a dihedral group of order 2k+1 over an arbitrary field of characteristic 2 is equal to 2k.

The paper presents a matrix representation of filter banks corresponding to spline wavelets with shifted supports. The matrix form of the decomposition and reconstruction filters simplifies the representation of nonuniform nonstationary wavelet transforms built on nonuniform grids on a finite interval. Such a representation of filters is used, for example, in constructing error-correcting codes.

The paper considers parallel preconditioned iterative methods in Krylov subspaces for solving systems of linear algebraic equations with large sparse symmetric positive-definite matrices resulting from grid approximations of multidimensional problems. For preconditioning, generalized block algorithms of symmetric successive over-relaxation or incomplete factorization type with matching row sums are used. Preconditioners are based on variable-triangular matrix factors with multiple alternations in triangular structure. For three-dimensional grid algebraic systems, methods are based on nested factorizations, as well as on two-level iterative processes. Successive approximations in Krylov subspaces are computed by applying a family of conjugate direction algorithms with various orthogonality and variational properties, including preconditioned conjugate gradient, conjugate residual, and minimal error methods.

A finite computational process using arithmetic operations only is called a rational algorithm. Presently, no rational algorithm for checking the congruence of arbitrary complex matrices A and B is known. The situation can be different if both A and B belong to a special matrix class. For instance, there exist rational algorithms for the cases where both matrices are Hermitian, unitary, or accretive. This paper proposes a rational algorithm for checking whether two involutive matrices A and B are congruent.

We study the Gnedin–Kingman graph, which corresponds to Pieri’s rule for the monomial basis {Mλ} in the algebra QSym of quasisymmetric functions. The paper contains a detailed announcement of results concerning the classification of indecomposable semifinite harmonic functions on the Gnedin–Kingman graph. For these functions, we also establish a multiplicativity property, which is an analog of the Vershik–Kerov ring theorem.

The following general variant of deterministic “Hats” game is analyzed. Several sages wearing colored hats occupy the vertices of a graph, the kth sage can have hats of one of h(k) colors. Each sage tries to guess the color of his own hat merely on the basis of observing the hats of his neighbors without exchanging any information. A predetermined guessing strategy is winning if it guarantees at least one correct individual guess for every assignment of colors. For complete graphs and cycles, the problem of describing the function h(k) for which the sages win is solved in the present paper. A “theory of constructors,” i.e., a collection of theorems demonstrating how one can construct new graphs for which the sages win is developed. A new game “Rook check ” equivalent to the Hats game on a 4-cycle is introduced and completely analyzed.

The goal of the paper is to study vertices of degree 6 of minimal and contraction critical 6-connected graph, i.e., a 6-connected graph that looses 6-connectivity both upon removal and upon contraction of any edge. It is proved that if x and z are adjacent vertices of degree 6, then x and z have at least 4 common neighbors. In addition, a detailed description of the neighborhood of the set {x, z} is given. An infinite series of examples of minimal and contraction critical 6-connected graphs for which the fraction of vertices of degree 6 equals $$ \frac{11}{17} $$ is constructed.

In this paper, we examine the solvability of a new class of nonclassical direct problems for multidimensional loaded parabolic equations with Cauchy data. We obtain sufficient conditions for the solvability of the problem; the proof is based on the method of weak approximation. By an example, we demonstrate the application of the theorem proved to the study of inverse problems for multidimensional parabolic equations with Cauchy data.

In this paper, we consider a loaded parabolic equation of a special form in an unbounded domain with Cauchy data. The equation is one-dimensional and its right-hand side depends on the unknown function u(t, x) and traces of this function and its derivatives by the spatial variable at a finite number of different points of space. Such equations appear after the reduction of some identification problems for coefficients of one-dimensional parabolic equations with Cauchy data to auxiliary direct problems. We obtain sufficient conditions of the global solvability and sufficient conditions of the solvability of the problem considered in a small time interval. We search for solutions in the class of sufficiently smooth bounded functions. We examine the uniqueness of the classical solution found and prove the corresponding sufficient conditions. We also obtain an a priori estimate of a solution that guarantees the continuous dependence of the solution on the right-hand side of the equation and the initial conditions.

In this paper, a review of mathematical methods of randomized machine learning is presented.

Systematization of numerical randomized functional algorithms for approximation of solutions to second-kind Fredholm integral equation is performed in this paper. Three types of algorithms are emphasized: projection algorithms, grid algorithms, and projection-grid algorithms. Disadvantages of grid algorithms that require calculation of values of the kernel of integral equations at fixed points are revealed (practically, the kernels of equation have integrable singularities and this calculation is impossible). Thus, for applied problems related to the solution of second-kind Fredholm integral equations, projection or projection-grid randomized algorithms are more convenient.

In this paper, we present results of the development of certain parallel numerical methods for solving three-dimensional evolutionary and stationary problems of diffusion and heat transfer. We present a detailed description of a special, explicit iteration scheme for parabolic equations and discuss a multigrid technology used for solving elliptic equations and implicit schemes for parabolic equations.

During the last two decades, new models were developed in actuarial sciences. Different notions of insurance company ruin (bankruptcy) and other objective functions evaluating the company performance were introduced. Several types of decision (such as dividend payment, reinsurance, investment) are used for optimization of company functioning. Therefore, it is necessary to ensure that the model under consideration is stable with respect to parameter fluctuation and perturbation of underlying stochastic processes. The aim of this paper is the description of methods for investigation of these problems and presentation of recent results concerning some insurance models. Numerical results are also included.

Earlier, a model of a time series with heavy tails constructed from a Gaussian time series was developed. In the present paper, the reverse problem is considered: an estimator of the copula function is built; the copula function is a nonlinear function that maps Gaussian variables to the variables from the Frechet maximum domain of attraction. The statistical properties of this estimator are considered for a stationary time series with a low rate of covariance decay.

We consider continuous-time branching random walks on multidimensional lattices with birth and death of particles at a finite number of lattice points. Such processes are used in numerous applications, in particular, in statistical physics, population dynamics, and chemical kinetics. In the last decade, for various models of branching random walks, a series of limit theorems about the behavior of the process for large times has been obtained. However, it is almost impossible to analyze analytically branching random walks on finite time intervals; so in this paper we present an algorithm for simulating branching random walks and examples of its numerical realization.

This paper is a survey of our last results about solutions to the Dirichlet and Robin boundary problems, the Robin transmission problem for an elliptic quasilinear second-order equation with the constant p– and variable p(x)-Laplacians, as well as to the degenerate oblique derivative problem for elliptic linear and quasilinear second-order equations in a conical bounded n-dimensional domain.

The study is devoted to analysis of records of rapid coordinated eye movements—saccades. Saccadic eye movements are performed by a human several thousand times a day and have a standard form. Changes in the movement of the eyes, including saccades, are noted in the case of disorders of the central nervous system, middle ear organs, and after a long space flight. Eye movement testing is used in medical practice for diagnosis and control when treating disorders in functioning of the organs of the inner ear, central nervous system, and strabismus. This paper proposes a method for approximating records of saccades with possible presence of pre- and postsaccadic movements. Numerical parameters describing preand postsaccadic movements and asymmetry of a saccade are suggested. The results of a statistical study of forms of saccades in a normal person are given. The statistical study has shown that the difference in forms of saccades is significant.

This paper describes a generalization of a method used for precision calibration of a strapdown INS. This method was proposed at the Laboratory of Control and Navigation and at the Department of Applied Mechanics and Control, Lomonosov Moscow State University. The method is in practice used in specific applications. The purpose of a strapdown INS calibration is to determine the parameters of instrument errors of inertial sensors: accelerometers and gyros (angular rate sensors) to provide compensation in INS navigation mode. It is assumed that the assembled strapdown INS is located on the platform of a test bench. Test bench performs a set of rotations, the choice of which is part of the calibration task. The main source of data used for calibration are the readings of accelerometers and gyros themselves. Two modifications of the proposed method are described. In the first modification, only data from the accelerometers and gyros are used; in the second modification, the angular data provided by the test bench are also used. The estimation algorithms used are Kalman-type filtering algorithms.

We use specialized simulators to train drivers of vehicles difficult to drive. The environment created by such simulator for drivers must be as close to reality as possible. To simulate the overloads that occur during the motion of the simulated object, movable platform stands are used. Usually simulator software is developed for a particular imitation stand, which limits the use of such software for other equipment configuration. In addition, often the sensors in the mechanisms of the platform are either not precise enough or absent altogether, and the geometric parameters of the platform change with time due to deformations due to constant loads. These factors negatively affect the management of the platform and, therefore, the accuracy of dynamic simulation and the quality of coordination with visual and other types of simulation. The purpose of this paper is to show that identification of platform parameters and platform positioning can be constructed without using measurements from internal sensors located in the mechanisms of the platform. To solve the problem, the authors developed an algorithm of semi-automatic identification of parameters of a platform model with a system of video analysis. Upon identification of parameters of a platform it is possible to monitor its angular motions without video analysis. Assessment of platform orientation is performed with angular velocity sensors (AVS) and accelerometers. The use of the suggested algorithms enables quick adaptation of stimulator software to any motion platform.

At present, the theory of wheeled robotic systems is being actively developed. In modeling the motion of wheeled robots, one mostly uses the classical nonholonomic motion model, which does not take into account the slip of deformable wheels. Meanwhile, for robots with deformable wheels, nonholonomic models can be inadequate for the design and analysis of control algorithms. This can be the case for statically unstable balancing robots with coaxial wheels, similar in design with such vehicles as Segway. Thus, modeling the motion of a two-wheeled robot taking into account the possibility of wheels slip and analysis of applicability of simplified models are of interest. Such models can be used to develop new control algorithms in active maneuvering, and for preliminary estimates of robustness of algorithms designed using approximate nonholonomic models. This paper focuses on modeling the motion of balancing robots, on analyzing their steady-state motion, and on possibilities of their stabilization. It is shown that for models with deformable wheels in the steady-state motion the body has a forward pitch. Such a pitch is not found in most nonholonomic models.

The condition of ellipticity of the equilibrium equation plays an important role for the correct description of the mechanical behavior of materials and is a necessary condition for new constitutive relations. Earlier, new deformation measures were proposed to remove correlations between the terms, which dramatically simplifies restoration of constitutive relations from experimental data. One of these new deformation measures is based on the QR-expansion of deformation gradient. In this paper, we study the strong ellipticity condition for hyperelastic material using the QR-expansion of deformation gradient.

In this paper we suggest an explicit finite-difference scheme for numerical simulation of the Cauchy problem with an integro-differential nonlinear equation that describes an oscillatory process with friction and memory (hereditarity), and with the corresponding local initial conditions. The problems of approximation, stability, and convergence of the proposed finite-difference scheme are investigated. The results of computer experiments that implement the proposed numerical scheme, confirming the theoretical estimates obtained in theorems, are given.

The paper proposes a new mathematical FitzHugh—Nagumo model with memory, which describes the propagation of a nerve impulse in a membrane. The model consists of an integro-differential equation with initial conditions (the Cauchy problem). The difference kernel (memory function) of the model equation was chosen as a power function so that it can be rewritten in terms of the fractional derivative. For the Cauchy problem, an explicit finite-difference scheme is constructed, and an investigation of its stability and convergence was performed in computer experiments. The finite-difference scheme was implemented in the Maple computer program, with the help of which the simulation results were visualized, and oscillograms and phase trajectories were obtained.

In this paper, we propose an extended version of the principle of minimizing empirical risk (ER) based on the use of averaging aggregating functions (AAF) for calculating the ER instead of the arithmetic mean. This is expedient if the distribution of losses has outliers and hence risk assessments are biased. Therefore, a robust estimate of the average risk should be used for optimization of the parameters. Such estimates can be constructed by using AAF that are solutions of the problem of minimizing the penalty function for deviating from the mean value. We also propose an iterative reweighting scheme for the numerical solution of the ER minimization problem. We give examples of constructing a robust procedure for estimating parameters in a linear regression problem and a linear separation problem for two classes based on the use of an averaging aggregating function that replaces the α-quantile.

A brief review of the emergence and development of the nonlocal approach to the problem of turbulent diffusion with a discussion of the physical reasons of the nonlocality is given. The main attention is paid to fractional differential operators. In concluding the paper, the author’s original results on applications to the diffusion of cosmic rays in the interstellar galactic medium are presented.

In this paper, we consider the problem of constructing a correction algorithm with the aim of increasing the adaptive properties of the ΣΠ-neuron, relying solely on the structure of the ΣΠ-neuron itself. To build the corrector, the logical-algebraic method of data analysis is used. Comparison of the advantages of the neural network approach and the logical-algebraic method suggests that a combined approach to the organization of the neural network improves its efficiency and allows one to build a set of rules that reveal hidden patterns in a given subject area, thus improving the quality of the recognition system.

Let R be a discrete valuation ring with infinite residue field and X a smooth projective curve over R. Let G be a simple simply-connected group scheme over R and E a principal G-bundle over X. It is proved that E is trivial locally for the Zariski topology on X providing E is trivial over the generic point of X. The main aim of the present paper is to develop a method rather than to get a very strong concrete result.

In order to provide a new, more computation-friendly, construction of the stable motivic category SH(k), V. Voevodsyky laid the foundation of delooping motivic spaces. G. Garkusha and I. Panin based on joint works with A. Ananievsky, A. Neshitov, and A. Druzhinin made that project a reality. In particular, G. Garkusha and I. Panin proved that for an infinite perfect field k and any k-smooth scheme X, the canonical morphism of motivic spaces $$ {C}_{\ast } Fr(X)\to {\Omega}_{{\mathrm{\mathbb{P}}}^1}^{\infty }{\sum}_{{\mathrm{\mathbb{P}}}^1}^{\infty}\left({X}_{+}\right) $$ is a Nisnevich locally group-completion. In the present paper, a generalization of that theorem is established to the case of smooth open pairs (X,U), where X is a k-smooth scheme and U is its open subscheme intersecting each component of X in a nonempty subscheme. It is claimed that in this case the motivic space C*Fr((X,U)) is a Nisnevich locally connected, and the motivic space morphism $$ {C}_{\ast } Fr\left(\left(X,U\right)\right)\to {\Omega}_{{\mathrm{\mathbb{P}}}^1}^{\infty }{\sum}_{{\mathrm{\mathbb{P}}}^1}^{\infty}\left(X/U\right) $$ is Nisnevich locally weak equivalence. Moreover, it is proved that if the codimension of S = X−U in each component of X is greater than r ≥ 0, then the simplicial sheaf C*Fr((X,U)) is locally r-connected.

In the paper, for a Lame-type system, the inverse problem on recovering the fast wave velocity from the boundary dynamical data (the response operator) is solved. The velocity is determined in the near-boundary domain, the depth of determination being proportional to the observation time. We use the BC-method, which is an approach to inverse problems based on their connections with boundary control theory.

This paper deals with studying some properties of a monogenic function defined on a vector space with values in the Clifford algebra generated by the space. We provide some expansions of a monogenic function and consider its application to study solutions of second-order partial differential equations.

In this paper, we discuss free interpolation in the spaces of entire and analytic finite-order functions in the upper half-plane. A review of problems and basic results related to such problems is given. Solvability criteria are formulated in terms of canonical products of interpolation nodes and in terms of the measure determined by these nodes.

Certain Hardy inequalities in domains of Euclidean space contain sharp but unreachable constants. V. G. Maz’ya and other authors used this fact to improve the corresponding inequalities by adding new integral terms. In this paper, a survey of results in this direction initiated by H. Brezis and M. Marcus is presented. Also, we give some generalizations of Brezis–Marcus-type inequalities to the case of Rellich-type inequalities with weights that are powers of the distance from a point to the boundary of the domain. Generalizations to the case of conformally invariant integral inequalities in simply connected and doubly connected planar hyperbolic domains are discussed.

In this paper, we describe the inhomogeneous Hilbert boundary-value problem of the theory of analytic functions with an infinite index and a boundary condition for a half-plane. The coefficients of the boundary condition are Holder-continuous everywhere except for a finite number of singular points at which the argument of the coefficient function has second-type discontinuities (of a power order with exponent

The paper is a review of the latest results of I. R. Kayumov and S. Ponnusamy on the Bohr inequality. An exact estimate in the strong Bohr inequality is obtained and the Bohr–Rogosinski radius for a certain class of subordinations is examined. All results are exact.

In this paper, we examine the problem on the description in terms of the Laplace transform of functionals from the conjugate space for the Hilbert space of entire functions of n variables constructed by a convex function in ℂn, which depends on the modules of variables and grows at infinity faster than a‖z‖ for any a > 0.

In this paper, we study spectral decompositions with respect to a system of generalized eigenvectors of second-order differential operators on an interval whose resolvents possess convolution representations. We obtain the convolution representation of resolvents of second-order differential operators on an interval with integral boundary conditions. Then, using the convolution generated by the initial differential operator, we construct the Fourier transform. A connection between the convolution operation in the original functional space and the multiplication operation in the space of Fourier transforms is established. Finally, the problem on the convergence of spectral expansions generated by the original differential operator is studied. Examples of convolutions generated by operators are also presented.

In this paper, we analyze conditions under which the spectrum of the Sturm–Liouville operator on a certain smooth curve is localized near a countable number of rays. In the case where the potential is piecewise analytical, an asymptotic of eigenvalues is found for each series localized near the corresponding ray. The result obtained allows one to generalize the well-known formula for the asymptotic of the distribution function of the spectrum stated by Davies in the case of a finite number of localization rays.

In this paper, weight subspaces of the space of analytic functions on a bounded convex domain of the complex plane are considered. Descriptions of spaces that are strongly dual to the inductive and projective limits of uniformly weight spaces of analytic functions in a bounded convex domain D ⊂ ℂ are obtained in terms of the Fourier–Laplace transform. For each normed, uniformly weight space H(D, u), we construct the minimal vector space ℋi(D, u) containing H(D, u) and invariant under differentiation and the maximal vector space ℋp(D, u) contained in H(D, u) and invariant under differentiation. We introduce natural locally convex topologies on these spaces and describe strongly dual spaces in terms of the Fourier–Laplace transform. The existence of representing exponential systems in the space ℋi(D, u) is proved.

A perfect cuboid is a rectangular parallelepiped in which the lengths of all edges, the lengths of all face diagonals, and also the lengths of spatial diagonals are integers. No such cuboid has yet been found, but their nonexistence has also not been proved. The problem of a perfect cuboid is among unsolved mathematical problems. The problem has a natural S3-symmetry connected to permutations of edges of the cuboid and the corresponding permutations of face diagonals. In this paper, we give a survey of author’s results and results of J. R. Ramsden on using the S3 symmetry for the reduction and analysis of the Diophantine equations for a perfect cuboid.

In this paper, we discuss stability and instability criteria for extremal surfaces of a special functional, which is a linear combination of an area-type functional and the functional of volumetric force density. Extremals of such functionals can serve as models of physically equilibrium tents or liquids in potential gravitational fields, so that the problem of their stability or instability is very topical. Our results are based on various geometric properties of surfaces; they are obtained by methods developed by V. M. Miklyukov and V. A. Klyachin.

In this paper, we consider the sine-Gordon equation with a high-frequency parametric pumping and a weak dissipative force. We examine the class of π-kink-type solutions that are soliton solutions of the nonperturbed sine-Gordon equation. In contrast to stable 2π-kinks, these solutions are unstable. We prove that the time of decaying of π-kinks due to small perturbations is proportional to the cube of the inverse period of fast oscillations of the parametric pumping. We derive a two-time asymptotic expansion of a solution of the boundary-value problem and analyze evolution of a wave packet whose leading term has the form of a π-kink. Numerical simulations of solutions confirm the good qualitative agreement with asymptotic expansions.

This paper is devoted to a new statement and the study of direct and inverse spectral problems for small linear oscillations of orthotropic plates that carry concentrated masses at a finite set of points, which, in turn, are connected to a stationary base by elastic springs with known stiffness coefficients.

Stream ciphers form one of two large classes of ciphers with private keys in classical cryptography. In this paper, we introduce the concept of a quantum stream cipher. Special types of quantum stream ciphers were proposed earlier by numerous researchers. We prove a general result on the nonexistence of an unconditionally strong quantum stream cipher if the length of a message is much longer than the length of a key. We analyze individual and collective attacks against a quantum stream cipher. A relationship between the problem of guessing the key by the opponent and the problem of distinguishing of random quantum states is established.

This paper is a survey of results on the relationship between gauge fields and infinitedimensional equations for parallel transport that contain the Levy Laplacian or the divergence associated with this Laplacian. Also we analyze the deterministic case where parallel transports are operator-valued functionals on the space of curves and the case of the Malliavin calculus where (stochastic) parallel transports are operator-valued Wiener functionals.

In the paper, we consider thermal states of two particles with spin 1/2 (qubits) located in an inhomogeneous transverse magnetic field and interacting according to the Heisenberg XY -model. We introduce the concepts of magnitude and direction of asymmetry of the entropy of a state and the magnitude and asymmetry of a flow of locally transmitted information. We show that for the system considered, the asymmetry of entropy is directed from the particle in a weaker magnetic field toward the particle in a stronger magnetic field, and this direction coincides with the direction of the excess flow of locally transmitted information. We also demonstrate that this asymmetry direction is consistent with the direction of the excess flow of locally available information: measurements over the particle in a weaker magnetic field provide a greater level of locally available information than measurements over the particle in a stronger magnetic field.

In this paper, we examine properties of the tensor powers of quantum mappings Φ. In particular, we review positivity properties of unitary and nonunitary qubit mappings Φ⊗2. For arbitrary finite-dimensional systems, we present the relationship between the positive and completely positive divisibility of dynamical mappings $$ {\Phi}_t^{\otimes 2} $$ and Φt. A criterion of annihilation of entanglement by an arbitrary qubit mapping Φ⊗2 is found.

A design of an industrial concentration plant based on the method of selective retardation of condensation in an external laser field was earlier proposed. In this paper, we review recent results on optimization criteria (objective functions) proposed for the search for optimal parameters of this scheme, related to the geometry and operational characteristics of the plant. As constraints in the optimization problem, we consider the influence of the condensation rate and the resonance and nonresonance absorption in a separation cell on its geometry. Separating cells play the role of absorbers in laser resonators.