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In this paper, we consider classes of analytical functions that map the unit disk into itself. Functions of these classes can be described in terms of hyperbolic derivative and hyperbolic metric. Under an appropriate choice of the corresponding metrics, these classes are metric spaces. Functions of the hyperbolic classes considered generate composition operators from the Bloch space into classical spaces of analytical functions in the unit disk.

In this paper, we study the Carleman classes in Jordan domains of the complex plane. We obtain a quasianalyticity criterion for the regular Carleman classes, which is universal for all weakly uniform domains. The proof is based on the solution of the Dirichlet problem with an unbounded boundary function and a Beurling result on the estimate of the harmonic measure.

In this paper, we find the principal asymptotic term at infinity of a certain fundamental system of solutions to the equation l2n[y] = λy of order 2n, where l2n is the product of second-order linear differential expressions and λ is a fixed complex number. We assume that the coefficients of these differential expressions are not necessarily smooth but have a prescribed power growth at infinity. The asymptotic formulas obtained are applied for the problem on the defect index of differential operators in the case where l2n is a symmetric (formally self-adjoint) differential expression.

In this paper, we discuss the inverse scattering problem to recover the potential from the scattering data of a class of Schrodinger equations with a nonlinear spectral parameter in the boundary condition. It turns out that for real-valued potential function q(x), the scattering data is defined as in the non-self-adjoint case: the scattering function, the nonreal singular values, and normalization polynomials. Characteristic properties of the spectral data are investigated. The solution of the problem is constructed by using the Gelfand–Levitan–Marchenko procedure. The uniqueness of the algorithm for the potential with given scattering data is proved.

In this paper, we examine inverse problems for initial conditions for the wave and telegraph equations and state uniqueness criteria. Solutions of these problems are constructed in the series form. In the proof of uniform convergence of these series, the problem on small denominators appears. We prove estimates of small denominators separated from zero and obtain asymptotics that allow one to justify the convergence in the class of regular solutions.

This paper is devoted to the problem of classification of integrable nonlinear models with three independent variables. The classification algorithm based on the notion of characteristic Lie rings is applied to a class of two-dimensional lattices of hydrodynamic type. By imposing appropriate cutting-off boundary conditions, we reduce the lattice to a system of hyperbolic equations, which is assumed to be a Darboux integrable system. As a result, we found a new integrable lattice.

In this paper, we study a system of nonlinear equations on a square graph related to the affine algebra A(1)1 . This system is the simplest representative of the class of discrete systems corresponding to affine Lie algebras. We find the Lax representation and construct hierarchies of higher symmetries. In neighborhoods of singular points ⋋ = 0 and ⋋ = ∞, we construct formal asymptotic expansions of eigenfunctions of the Lax pair and, based on these expansions, find series of local conservation laws for the system considered.

In this paper, we construct stability regions (in the linear approximation) of triangular libration points for the planar, restricted, elliptic three-body problem and examine bifurcations that occur when parameters of the system pass through the boundaries of these regions. A new scheme for the construction of stability regions is presented, which leads to approximation formulas describing these boundaries. We prove that on one part of the boundary, the main scenario of bifurcation is the appearance of nonstationary 4π-periodic solutions that are close to a triangular libration point, whereas on the other part, the main scenario is the appearance of quasiperiodic solutions.

In this paper, we discuss approximations of the dynamical quantum Zeno effect by a fixed number of nonselective quantum measurements. A wide class of measurements whose efficiency is close to optimal in the case of two-level systems is found.

The paper evaluates the lengths of group algebras of all groups of orders not exceeding 7 over an arbitrary field.

The paper considers the minimal hyperbolic splines and their properties. Formulas for constructing quadratic splines and the corresponding biorthogonal (dual) functionals are obtained. Numerical results, demonstrating how approximation quality can be improved by using hyperbolic splines and changing control parameters, are presented.

The paper presents new nonsingularity conditions for n ×n matrices, which involve a subset S of the index set {1, . . ., n} and take into consideration the matrix sparsity pattern. It is shown that the matrices satisfying these conditions form a subclass of the class of nonsingular $$ \mathcal{H} $$ -matrices, which contains some known matrix classes such as the class of doubly strictly diagonally dominant (DSDD) matrices and the class of Dashnic–Zusmanovich type (DZT) matrices. The nonsingularity conditions established are used to obtain the corresponding eigenvalue inclusion sets, which, in their turn, are used in deriving new inclusion sets for the singular values of a square matrix, improving some recently suggested ones.

The paper is mainly devoted to studying the so-called Dashnic–Zusmanovich type (DZT) matrices, introduced recently. Interrelations among the DZT matrices and related subclasses of the class of nonsingular $$ \mathcal{H} $$ -matrices, namely, the Dashnic–Zusmanovich (DZ) and S-SDD matrices are considered. Upper bounds for the l∞-norms of the inverses to DZT, DZ, and strictly diagonally dominant (SDD) matrices are obtained. A new eigenvalue inclusion set is provided.

The paper studies two numerical characteristics of matrix incidence algebras over finite fields associated with generating sets of such algebras: the minimal cardinality of a generating set and the length of an algebra. Generating sets are understood in the usual sense, the identity of the algebra being considered a word of length 0 in generators, and also in the strict sense, where this assumption is not used. A criterion for a subset to generate an incidence algebra in the strict sense is obtained. For all matrix incidence algebras, the minimum cardinality of a generating set and a generating set in the strict sense are determined as functions of the field cardinality and the order of the matrices. Some new results on the lengths of such algebras are obtained. In particular, the length of the algebra of “almost” diagonal matrices is determined, and a new upper bound for the length of an arbitrary matrix incidence algebra is obtained.

The paper considers preconditioned iterative methods in Krylov subspaces for solving large systems of linear algebraic equations with sparse coefficient matrices arising in solving multidimensional boundary-value problems by finite volume or finite element methods of different orders on unstructured grids. Block versions of the weighted Cimmino methods, based on various orthogonal and/or variational approaches and realizing preconditioning functions for two-level multi-preconditioned semi-conjugate residual algorithms with periodic restarts, are proposed. At the inner iterations between restarts, additional acceleration is achieved by applying deflation methods, providing low-rank approximations of the original matrix and playing the part of an additional preconditioner. At the outer level of the Krylov process, in order to compensate the convergence deceleration caused by restricting the number of the orthogonalized direction vectors, restarted approximations are corrected by using the least squares method. Scalable parallelization of the methods considered, based on domain decomposition, where the commonly used block Jacobi–Schwarz iterative processes is replaced by the block Cimmino–Schwarz algorithm, is discussed. Hybrid programming technologies for implementing different stages of the computational process on heterogeneous multi-processor systems with distributed and hierarchical shared memory are described.

The paper computes the Hilbert basis of the cone constructed from matrices describing generic situations, i.e., such vector subspaces in a finite direct sum of finite-dimensional subspaces that are in generic position with respect to the direct summands.

The paper studies the anticommutativity condition for elements of arbitrary real Cayley–Dickson algebras. As a consequence, the anticommutativity graphs on equivalence classes of such algebras are classified. Under some additional assumptions on the algebras considered, an expression for the centralizer of an element in terms of its orthogonalizer is obtained. Conditions sufficient for this interrelation to hold are provided. Also examples of real Cayley–Dickson algebras in which the centralizer and orthogonalizer of an element are not interrelated in this way are considered.

Let Qn(ℂ) denote the space of all skew-symmetric n × n matrices over the complex field ℂ. The paper characterizes the linear mappings T : Qn(ℂ) → Qn(ℂ) that satisfy the condition per(T(A)) = per(A) for all matrices A ∈ Qn(ℂ) and an arbitrary n > 4.

It is well known that there are remarkable differential equations that can be integrated in CAS, but admit several inequivalent approaches to their description. In the present paper, we discuss remarkable differential equations in another sense, namely, equations for which there exist finite difference schemes that preserve the algebraic properties of solutions exactly. It should be noted that this class of differential equations coincides with the class introduced by Painleve. In terms of the Cauchy problem, a differential equation of this class defines an algebraic correspondence between the initial and final values. For example, the Riccati equation y′ = p(x)y2 + q(x)y + r(x) defines a one-to-one correspondence between the initial and final values of y on the projective line. However, the standard finite difference schemes do not preserve this algebraic property of the exact solution. Moreover, the scheme that defines a one-to-one correspondence between the layers actually describes the solution not only before moving singularities but also after them and preserves algebraic properties of equations, such as the anharmonic ratio. After a necessary introduction (Secs. 1 and 2), we describe such a one-to-one scheme for the Riccati equation and prove its properties mentioned above.

Finite difference schemes are widely used in applied mathematics to numerically solve partial differential equations. However, for a given solution scheme, it is usually difficult to evaluate the quality of the underlying finite difference approximation with respect to the inheritance of algebraic properties of the differential problem under consideration. In this paper, we present an appropriate quality criterion of strong consistency for finite difference approximations to systems of nonlinear partial differential equations. This property strengthens the standard requirement of consistency of difference equations with differential ones. We use a verification algorithm for strong consistency, which is based on the computation of difference Grobner bases. This allows for the evaluation and construction of solution schemes that preserve some fundamental algebraic properties of the system at the discrete level. We demonstrate the suggested approach by simulating a Karman vortex street for the two-dimensional incompressible viscous flow described by the Navier–Stokes equations.

A ring is said to be serial if its right and left regular modules are the direct sums of chain modules. The aim of the paper is to give an answer to the following question: for which finite simple groups, the group ring over a given field is serial.

The Hochschild cohomology groups for algebras of semidihedral type, that are contained in the family SD(2B)1 (from the famous K. Erdmann’s classification), are computed. In the calculation, a construction of the minimal bimodule resolution for algebras from the family under discussion, that is defined in the present paper, is used.

The Hochschild cohomology groups for algebras of dihedral type which are contained in the family D(3R) (from the famous K.Erdmann’s classification) are calculated. In the calculation, a construction of the minimal bimodule resolution for algebras from the family under discussion, that is defined in the present paper, is used.

According to a result claimed by Pogorza_ly, selfinjective special biserial algebras can be stably equivalent to stably biserial algebras only, and these two classes coincide. By an example of Ariki, Iijima, and Park, the classes of stably biserial and selfinjective special biserial algebras do not coincide. In these notes based on some ideas from the Pogorzaly paper, a detailed proof is given for the fact that a selfinjective special biserial algebra can be stably equivalent to a stably biserial algebra only. The structure of symmetric stably biserial algebras is analyzed. It is shown that in characteristic other than 2, the classes of symmetric special biserial (Brauer graph) algebras and symmetric stably biserial algebras coincide. Also a proof of the Auslander–Reiten conjecture for special biserial algebras is given.

In this paper, mathematical models of compressible viscoelastic Maxwell, Oldroyd, and Kelvin–Voigt fluids are derived. A model of rotating viscoelastic barotropic Oldroyd fluid is studied. A theorem on strong unique solvability of the corresponding initial-boundary value problem is proved. The spectral problem associated with such a system is studied. Results on the spectrum localization, essential and discrete spectra, and spectrum asymptotics are obtained. In the case where the system is in the weightlessness state and does not rotate, results on multiple completeness and basis property of a special system of elements are proved. In such a case, under the assumption the viscosity is sufficiently large, an expansion of the solution of the evolution problem with respect to a special system of elements is obtained.

In this paper, explicit integral volume formulas for arbitrary compact hyperbolic octahedra with mm2-symmetry are obtained in terms of dihedral angles. Also, we provide an algorithm to compute the volume of such octahedra in spherical spaces.

In this paper, we review the results on topological ∗-algebras S(M), S(M, τ), and LS(M) of measurable, τ -measurable, and locally measurable operators affiliated with the von Neumann algebra M. Also, we consider relations between those algebras for different classes of von Neumann algebras and establish the continuity of operator-valued functions with respect to the local convergence in measure. We describe maximal commutative ∗-subalgebras of the algebra LS(M) as well.

We consider the problem of constructing birational Darboux coordinates on nilpotent coadjoint orbits of the complex Lie groups SO(N, ℂ) and Sp(N, ℂ). The nilpotent case is the most difficult one. Difficulties arise if the Jordan form of matrices from the orbit under consideration contains Jordan blocks of sizes of different parity. The desired coordinates have been found on orbits consisting of matrices with 1 × 1 and 2 × 2 Jordan blocks. Explicit formulas for them are given in the paper.

This paper continues a series of publications on the shortwave diffraction of the plane wave on prolate bodies of revolution with axial symmetry in the Neumann problem. The approach, which is based on the Leontovich–Fock parabolic equation method for the two parameter asymptotic expansion of the solution, is briefly described. Two correction terms are found for the Fock’s main integral term of the solution expansion in the boundary layer. This solution can be continuously transformed into the ray solution in the illuminated zone and decays exponentially in the shadow zone. If the observation point is in the shadow zone near the scatterer, then the wave field can be obtained with the help of residue theory for the integrals of the reflected field, because the incident field does not reach the shadow zone. The obtained residues are necessary for the unique construction of the creeping waves in the boundary layer of the scatterer in the shadow zone.

In the paper, diffraction problems of a plane wave by smooth, convex, and elongated bodies of revolution are considered within the framework of short-wave approximation (axially symmetric cases). The scattering amplitudes are calculated in the direction of limit rays, and the influence of the elongation of the scatterers on the amplitudes behavior is investigated. Mathematical techniques of our approach are based on the Green’s formulas in the exterior of the scatterers and numerical calculations of the wave field current in the boundary layers in the vicinity of the light-shadow zone. It is established that the elongation of axially symmetric bodies relatively weakly affects the scattering amplitudes of the short-wave asymptotics. The main contribution to the amplitudes is made by the solution of the 2D diffraction problem by a convex, smooth curve in the cross section of the scatterers by a plane containing the rotation axis.

This paper is a complement to the article “Scattering amplitudes in a neighborhood of limit rays in short-wave diffraction by elongated bodies of revolution”. It contains discussions of some points of the article, which worth of more detailed considerations, such as the influence of the integration limits on the computation result of scattering amplitudes and the estimation of permissible values of scattering angle intervals as functions of parameters of the problems.

The paper discusses the de-noising method in a model with an additive Gaussian noise, based on the wavelet-vaguelette decomposition and thresholding of the vaguelette coefficients. For soft and hard thresholding procedures, we derive the values of the thresholds and estimate asymptotically optimal orders of the loss function based on the probabilities of errors in calculating decomposition coefficients.

Signal denoising methods based on the threshold processing of wavelet coefficients are widely used in various application areas. When applying these methods, it is usually assumed that the number of wavelet coefficients is fixed, and the noise distribution is Gaussian. Such a model has been well studied in the literature, and optimal threshold values have been calculated for different signal classes and loss functions. However, in some situations the sample size is not known in advance and is modeled by a random variable. In this paper, we consider a model with a random number of observations contaminated by a Gaussian noise, and study the behavior of the loss function based on the probabilities of errors in calculating wavelet coefficients for a growing sample size.

This paper gives an overview of the existing approaches for modelling high quantiles of dependent spatial data and apply the methods to the bivariate circular case. We also adapt the bootstrap to the situation at hand. Since any data set one might use is finite, the interest lies in estimating a continuous curve as its upper limit (or quantile function). This can be obtained by either a kernel type regression or a fitted parametric model. We also introduce a new, more realistic correction formula for a nonparametric method for estimating the pointwise maximum (called frontier in this setup [8]). An additional common challenge in real-life applications is the dependence among subsequent observations. The theoretical results about the GPD limit of the exceedances beyond high threshold remain valid under mixing-type conditions, called D(un) in the extreme-value literature. However, if one intends to use the bootstrap-based reliability estimators, then they need to be adjusted — e.g., by the block-bootstrap approach in [15]. We estimate the reliability of the estimators by a suitable application of the m out of n bootstrap, which turned out to be suitable for high-quantile estimation. We illustrate the introduced methods for simulated as well as real enzymes data.

Magnetoencephalography produces big data. The processing of these data in order to reconstruct signal sources with a given accuracy is an extremely ill-posed problem. The main purpose of this work is an extension of our previous results to spheroid model providing a more accurate solution. In certain models of a human head (spherical and spheroid), the Biot–Savart law and IC analysis give a background for the detailed investigation in order to develop an algorithm for primary motor cortex localization. For a general case of the spheroid model, it is possible to approximately solve the inverse problem, neglecting volume magnetic field near the points of maxima of the magnetic field and taking into account only primary magnetic field. This paper presents a stepwise algorithm to obtain MEG inverse problem solution under the assumptions of discreteness of signal sources, originating from distinct functional brain areas and superficial location of the signal sources.

Animal activity is a descriptor that can be potentially used to track the health and well-being, which is obviously very important to improve the management process and productivity of farms. This paper deals with an animal behavior recognition problem using a chaos theory approach where we adopt such a technique for automatic classification of calves behavioral states. Two main mutually exclusive behaviors are of interest, namely, lying and standing/walking with six possible activities: feeding, drinking water, drinking milk, playing, rumination, and neutral. The time series generated by ear-tags with a 3D-accelerometer after a wavelet denoising transformation and frequency stabilization are treated as representations of the nonlinear dynamical system. The dynamical system of a certain animal state exhibits specific strange attractor in a phase space. A characterization of such an attractor is performed through metric, dynamic, and topological invariants including Lyapunov exponent, correlation dimension, length of a phase trajectory, sum of edges forming a convex hull and others. These measures are used as a feature vector for the subsequent classification. In a cross validation scheme, six classifiers are built on each training set, and the hyper-parameters are optimized using an inner validation set. The classifier that reaches the highest accuracy on the inner validation set is used to classify the outer validation set. It is shown that this approach can be useful at predicting activity states as an alternative methodology for the animal behavior state recognition problem with acceptable classification accuracy. Furthermore it is possible to include this procedure as part of an ensemble method in machine learning where a combination of different models is used.

In this study, we develop a clustering method for multivariate time series data. In practical situations, such problems can arise in finance, economics, control theory, and health science. First, we propose to use a simulation based approximation to the test statistic and develop a method to test if two multivariate time series are coming from same VAR process. Then, the testing method is extended to a group of multivariate time series objects. Finally, a new clustering algorithm is developed using the testing method. The proposed algorithm does not use a predetermined number of clusters and finds the best possible clustering from the data. Empirical studies are provided in this paper, and they establish the accuracy of the algorithm. Finally, as a practical example, the algorithm is implemented to identify activities of different persons from a real-life data obtained from single chest-mounted accelerometers worn by different individuals.

In this paper, we use the theory of generalized inverses to compute the stationary distribution of the (s, S) inventory model directly from the transition matrix of the underlying Markov chain and to provide perturbation bounds. Indeed, approximations are employed to build a tractable model, and statistical methods are used to estimate the unknown parameters and distributions. Hence, the system is subject to perturbations that may cause deviation in the characteristics. The proposed perturbation bound provides a means to estimate the impact of the perturbations on the performance measures of the considered inventory system.

In this paper a closed retrial queueing system is considered with a finite number of customers. If an arriving (primary or secondary) request finds the server busy, two modes are possible: the job is transferred to the orbit (no conflict) or the job under service is interrupted and both of them are transferred to the orbit (conflict). Jobs in the orbit can retry reaching the server after a random time. The unreliable case where the server is subject to breakdown is also investigated. These types of systems can be solved by numerical, asymptotical, and simulation methods. The novelty of the investigations is that it provides a new approach to an algorithmic solution for calculating the steady-state probabilities of the system. With the help of these probabilities the main performance measures can be computed. Several sample examples illustrate the effect of different parameters on the distribution on requests in the system.

This paper is a review of recent results for large random matrices with heavy-tailed entries. First, we outline the development of and some classical results in random matrix theory. We focus on large sample covariance matrices, their limiting spectral distributions, and the asymptotic behavior of their largest and smallest eigenvalues and their eigenvectors. The limits significantly depend on the finite or infiniteness of the fourth moment of the entries of the random matrix. We compare the results for these two regimes which give rise to completely different asymptotic theories. Finally, the limits of the extreme eigenvalues of sample correlation matrices are examined.

In this paper, a multi-channel queueing network with input flow of a general structure is considered. The multi-dimensional service process is introduced as the number of customers at network nodes. In the heavy-traffic regime, a functional limit theorem of diffusion approximation type is proved under the condition that the input flows converge to their limits in the uniform topology. A limit Gaussian process is constructed and its correlation characteristics are represented explicitly via the network parameters. A network with nonhomogeneous Poisson input flow is studied as a particular case of the general model, and a correspondent Gaussian limit process is built.

One of the most popular models in survival analysis is the Cox proportional hazards model. This model has been widely used because of its simplicity. Despite its simplicity, the basic problem of the Cox model is its inability to enter unknown risk factors into the model. Some risk factors may affect the survival of a trial unit, but due to time and cost constraints, there is no possibility to measure all of these factors in the form of explanatory variables in the model. In many cases, measuring risk factors is not possible. For entering unknown risk factors into the model, a positive random variable, representative of unknown risk factors, is multiplied in the model. Then a new class of survival models, namely frailty models, is introduced. When survival data are spatially correlated, frailty models do not have good performance for analyzing the data. In these cases, a Gaussian random field would be used as frailty variable such that the spatial correlation of the data can be included in the model. But in some applications, the Gaussian assumption of spatial effects is not realistic. In this paper, we use a spatial survival model with Gaussian and non-Gaussian random effects. Considering the complexity of likelihood function for spatial survival models and the lack of closed form, the frequency approach is very time-consuming for estimation of the model parameters. We use the Bayesian approach and MCMC algorithms for estimating the model parameters. Next, the application of the model is shown in an analysis of a real dataset.

In this paper, we prove general facts on metrically and topologically projective, injective, and flat Banach modules. We prove theorems pointing to the close connection between metric, topological Banach homology and the geometry of Banach spaces. For example, in geometric terms we give a complete description of projective, injective, and flat annihilator modules. We also show that for an algebra with the geometric structure of an - or -space all its homologically trivial modules possess the Dunford–Pettis property.

In this paper, we study universal equivalence of general and special linear groups over fields. We give the following criterion for this relation to hold: two groups Gn(K) and Gm(L) (G = GL, SL, K and L are infinite fields) are universally equivalent if and only if n = m and the fields K and L are universally equivalent.

This paper is devoted to the application aspects of homomorphic cryptography. It provides a description of a fully homomorphic matrix polynomial-based encryption scheme. It also gives the results of practical comparison of fully homomorphic encryption schemes. We consider some special cases of homomorphic encryption allowing computations of a limited number of functions.

In this paper, it is proved that Lie algebras of Chevalley type (An, Bn, Cn, Dn, E6, E7, E8, F4, and G2) over associative commutative rings with 1/2 (with 1/2 and 1/3 in the case of G2) have unique addition. As a corollary of this theorem, we note the uniqueness of addition in semisimple Lie algebras of Chevalley type over fields of characteristic ≠ 2 (≠ 2, 3 in the case of G2).

This paper is a survey of recent results of investigations on varieties of Poisson algebras. We give constructions of varieties of Poisson algebras with extremal properties, we give equivalent conditions for the polynomial codimension growth of a variety of Poisson algebras, we study varieties of Poisson algebras whose ideals of identities contain the identity {x, y} ⋅ {z, t} = 0, and we study the interrelation between such varieties and varieties of Lie algebras.

In this paper, it is proved that a certain power of the topological Jacobson radical for a ring annihilates a left module having topological Krull dimension over this ring. The estimation of this power depends on the topological Krull dimension and the dual topological Krull dimension. A similar estimation for discrete Jacobson radical holds true. Levitzky’s theorem is generalized for topological rings.

In this paper, we consider boundary-value problems with shift and show that such problems are equivalent to boundary-value problems for generalized analytic functions. We interpret the shift as a change of the complex structure on the complex plane with a given closed curve.

In this paper, we obtain differential substitutions for a special class of Liouville-type equations that simplify these equations. We present necessary and sufficient conditions of the existence of y-integrals of the second order. We examine the principal case of necessary and sufficient conditions of the existence of y-integrals.