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Let $G$ be a finite group, $N(G)$ be the set of conjugacy classes of the group $G$. In the present paper it is proved $G\simeq L$ if $N(G)=N(L)$, where $G$ is a finite group with trivial center and $L$ is a finite simple group of exceptional Lie type or Tits group.

Let G be a simply connected Chevalley group over an infinite field K, and let $$ \tilde{w} $$ : Gn → G be a word map that corresponds to a nontrivial word w. In 2015, it has been proved that if w = w1w2w3w4 is the product of four words in independent variables, then every noncentral element of G is contained in the image of $$ \tilde{w} $$. A similar result for a word w = w1w2w3, which is the product of three independent words, was obtained in 2019 under the condition that the group G is not of type B2 or G2. In the present paper, it is proved that for a group of type B2 or G2, all elements of the large Bruhat cell B nw0B are contained in the image of the word map $$ \tilde{w} $$, where w = w1w2w3 is the product of three independent words. For a group G of type Ar, Cr, or G2 (respectively, for a group of type Ar) or a group over a perfect field K (respectively, over a perfect field K the characteristic of which is not a bad prime for G) with dim K ≤ 1 (here, dim K is the cohomological dimension of K), it is proved that all split regular semisimple elements (respectively, all regular unipotent elements) of G are contained in the image of $$ \tilde{w} $$, where w = w1w2 is the product of two independent words. Also, for any isotropic (but not necessary split) simple algebraic group G over a field K of characteristic zero, it is shown that for a word map $$ \tilde{w} $$ : G(K)n → G(K), where w = w1w2 is a product of two independent words, all unipotent elements are contained in Im $$ \tilde{w} $$.

We describe the Hochschild cohomology ring for a family of self-injective algebras of tree class $E_6$ in terms of generators and relations. Together with the results of the previous paper, this gives a complete description of the Hochschild cohomology ring for a self-injective algebras of tree class $E_6$., part II of arXiv:1311.4756

In this paper, nontrivial upper bounds on the chromatic numbers of the spaces $$ {\mathrm{\mathbb{R}}}_p^n=\left({\mathrm{\mathbb{R}}}^n, lp\right) $$ with forbidden monochromatic sets are proved. In the case of a forbidden rectangular parallelepiped or a regular simplex, explicit exponential lower bounds on the chromatic numbers are obtained. Exact values of the chromatic numbers of the spaces $$ {\mathrm{\mathbb{R}}}_p^n $$ with a forbidden regular simplex in the case p = ∞ are found. Bibliography: 39 titles.

This paper provides a brief overview of various definitions of multivariate regularly varying functions and some of their applications in probability theory and related fields.

In this paper, the method of estimating bioequivalence, whose main goal is to break the drug concentration curve in the body into components, is considered, implying that this curve is a signal that demonstrates the behavior of the drug. These components are directly related to the main stages of the drug. Denoting the boundaries of these stages, we can, with a minimum of error, compare drugs by the duration and nature of these stages. To isolate the components, methods such as the method of independent components, the window dispersion method, and the study of the variance gamma process will be used.

The paper considers the problem of estimating the signal function from noisy observations using threshold processing of its wavelet expansion coefficients. Under general assumptions about the properties of the noise distribution, the asymptotic order of the optimal threshold is calculated, minimizing the loss function, based on the probability that the maximum error in the wavelet coefficients exceeds a given critical level.

We analyze the role of the Mathisson–Papapetrou equations in establishing the regularities of spingravitational interaction under the conditions of ultrarelativistic motions of spinning particles in the fields of black holes and neutron stars. In recent years, these equations are extensively used in the investigation of spin-gravitational effects. At the same time, it is necessary to perform the detailed analysis of the specific features of ultrarelativistic motions of spinning particles. This is done in the present paper. We apply the representation of the Mathisson–Papapetrou equations in terms of comoving tetrads, which enables us to reveal physical foundations of the specific features of spingravitational effects for spinning particles. We also make conclusions that follow from the obtained new solutions of the Mathisson–Papapetrou equations.

Some approximative characteristics of classes of periodic functions of many variables $$ {L}_{\beta, p}^{\psi }, $$ 1 < p < 1, in a uniform metric are investigated. The first part of the paper is devoted to the construction of estimates of the best orthogonal trigonometric approximations of the mentioned classes in the space L∞. In the second part, we have established the ordinal estimates of the orthoprojective widths of the classes $$ {L}_{\beta, p}^{\psi }, $$ 1 < p < 1, in the same space, as well as the estimates of another approximative characteristic which is close, in a definite meaning, to the orthoprojective width.

The monograph consists of three parts. Part I is presented here. In this monograph, we develop a new approach (mainly based on papers of the author). Many results are published for the first time here. Chapter 1 is introductory. It provides the necessary background from functional analysis (for completeness). In this monograph, we mostly use weighted HOlder spaces; they are considered in Chap. 2. Chapter 3 plays the key role: in weighted HOlder spaces, we consider estimates of integral operators with homogeneous difference kernels, covering potential-type integrals and singular integrals as well as Cauchy-type integrals and double layer potentials. In Chap. 4, similar estimates in weighted Lebesgue spaces are proved. Integrals with homogeneous difference kernels will play an important role in Part III of the monograph, which will be devoted to elliptic boundary-value problems. They naturally arise in integral representations of solutions of first-order elliptic systems in terms of fundamental matrices or their parametrices. The investigation of boundary-value problems for second-order and higher-order elliptic equations or systems is reduced to first-order elliptic systems.

This paper is a review of results of studies of affine transformations in generalized spaces over real linear algebras over the past 15-20 years.

This paper is devoted to Young diagrams and $$ {L}_n^1 $$-equivariant projections of tensor spaces. We present the theory of representations of finite groups developed according to works of H. Weyl and H. Boerner.

In this paper, we discuss the global aspect of the geometric dynamics of volumetric expansion and its applications to the problem of the existence in the space-time of compact and complete spacelike hypersurfaces and to the global geometry of generalized Robertson–Walker space-times.

In this paper, geometric properties of the six-dimensional sphere with a nearly Kahlerian structure are described.

This paper is a review of some directions in the study of special classes of metric affince spaces, i.e., spaces endowed with a (pseudo)Riemannian metric and a linear connection.

In this paper, we discuss almost complex structures on the sphere S6 and on the products of spheres S3 × S3, S1 × S5, and S2 × S4. We prove that all almost complex Cayley structures that naturally appear from their embeddings into the Cayley octave algebra ℂa are nonintegrable. We obtain expressions for the Nijenhuis tensor and the fundamental form 𝜔 for each gauge of the space ℂa and prove the nondegeneracy of the form d𝜔. We show that through each point of a fiber of the twistor bundle over S6, a one-parameter family of Cayley structures passes. We describe the set of U(2) ×U(2)- invariant Hermitian metrics on S3 × S3 and find estimates of the sectional sectional curvature. We consider the space of left-invariant, almost complex structures on S3 × S3 = SU(2) × SU(2) and prove the properties of left-invariant structures that yield the maximal value of the norm of the Nijenhuis tensor on the set of left-invariant, orthogonal, almost complex structures.

In this paper, we examine topological spaces that can be effectively defined over factor sets modulo equivalences on the set of natural numbers. We formulate a criterion of computable (effective) separability of topological spaces in terms of the approximability of the corresponding algebras by negative (uniformly effectively separated) algebras. We compare negative and positive algebra representations from the standpoint of the structure of the corresponding effective spaces. For effective infinite topological spaces, we prove the existence of their infinite effective compact extensions.

In this paper, we prove the explicit integrability of certain classes of dynamical systems on the tangent bundles of 2- and 3-dimensional spheres in the case where the forces are fields with so-called variable dissipation.

In this paper, we obtain formulas for residues and prove analogs of the argument principle and Rouche theorems for A(z)-analytic functions.

Investigation of various problems of mechanics and mathematical physics is reduced to the solution of second-order linear differential equations with variable coefficients. In 1986, the American mathematician J. Kovacic proposed an algorithm for solution of a second-order linear differential equation in the case where the solution can be expressed in terms of so-called Liouville functions. If a linear second-order differential equation has no Liouville solutions, the Kovacic algorithm also allows one to ascertain this fact. In this paper, we discuss the application of the Kovacic algorithm to the problem of the motion of a heavy body of revolution on a perfectly rough horizontal plane. The existence of Liouville solutions of the problem is examined for the cases where the rolling body is an infinitely thin disk, a disk of finite thickness, a dynamically symmetric torus, a paraboloid of revolution, and a spindle-shaped body.

In this paper, we examine categorical and cardinal properties of hyperspaces with finite number of components. We prove that the functor Cn : Comp → Comp is not normal, i.e., it does not preserve epimorphisms of continuous mappings. We also discuss the density, the caliber, and the Shanin number of the space Cn(X).

In this paper, we describe the class of all quadratic homeomorphisms of a finite-dimensional simplex and examine the asymptotic behavior of positive and negative trajectories of such homeomorphisms.

It is well known that the study of the geometry and topology of the attainability set of a family of vector fields is one of the main tasks of the qualitative control theory, which is closely related to the geometry of orbits of vector fields. In this paper, we present the authors’ results on the geometry of the attainability set of a family of vector fields: results on the geometry of T -attainability sets and the geometry of orbits of Killing vector fields.

This review paper is concerned with modules in which all automorphisms (endomorphisms) of submodules can be extended to endomorphisms of the entire module. We consider old results and obtain a number of new results. We also consider automorphism-invariant, quasi-injective, and injective modules.

According to modern concepts, the Universe consists of only 5% of baryonic matter (including black holes), 23% of dark matter, and the rest is mysterious dark energy. On the other hand, the movement of the satellites Pioneer 10 and Pioneer 11, which have already left the Solar system, has not revealed any impact of forces other than Newtonian gravity. This casts doubt on the existence of the dark matter and the dark energy. In this paper, we provide a hydrodynamic explanation of well-known cosmological paradoxes. In our research, dark matter forces are replaced with nonlocal friction forces and the dark energy is replaced by centrifugal forces.

The paper is concerned with special questions on the behavior of zeros of sequences of Bernstein polynomials. For a piecewise linear generating function, computer mathematics machinery was used to find the rules controlling the limit behavior of zeros as the number of the Bernstein polynomial unboundedly increases. New problems for theoretical investigations are formulated.

In this paper, we propose an approach for compact storage of big graphs. We propose preprocessing algorithms for graphs of a certain type, which can significantly increase the data density on the disk and increase performance for basic operations with graphs.

In this paper, we construct finite-difference approximations of optimal-control problems involving non-self-adjoint convection-diffusion elliptic equations with discontinuous coefficients and states and examine the convergence of these approximations. Control functions in these problems are the coefficients of the convective-transfer operator in the equation of state and its right-hand side. We study the well-posedness of problems considered. For finite-difference approximations, we obtain estimates of the exactness by the state and the convergence rate by the functional and prove the weak convergence by the control. In addition, we regularize approximations in the Tikhonov sense.

The paper is devoted to an optimal feedback control problem for the Bingham model with periodic conditions on spatial variables. An interpretation of the problem is given in the form of operator inclusion with multivalued right-hand side. On the base of a topological approximation approach to studying hydrodynamics problems, and the degree theory of multivalued vector fields, the existence of solutions of this inclusion is deduced. Then it is proved that among the solutions of the problem in question, there is a solution minimizing a given cost functional.

We construct a stochastic interpretation of a generalized solution of the Cauchy problem for the simplest magneto-hydrodynamics system, namely, a system including the Burgers equation with a pressure due to a magnetic field. The probabilistic representation constructed in the paper can be used for numerical computations.

The paper deals with methods of computation of joint distributions of functionals of the telegraph process and switching diffusions. A switching between two collections of diffusion coefficients happens at Poisson time moments that are independent of the initial diffusions. It is also possible to consider more general switching diffusions when the choice is performed from three or more collections of diffusion coefficients.

In practice, we often encounter situations where a sample size is not defined in advance and can be a random value. In the present paper, we derive second-order Chebyshev–Edgeworth and Cornish–Fisher expansions based of Student’s t- and Laplace distributions and their quantiles for samples with random size of a special kind. This derivation uses a general transfer theorem, which allows us to construct asymptotic expansions for distributions of randomly normalized statistics from the distributions of the considered nonrandomly normalized statistics and of the random size of the underlying sample. Recently, interest in Cornish–Fisher expansions has increased because of study in risk management. Widespread risk measure Value at Risk (VaR) substantially depends on quantiles of the loss function, which is connected with description of investment portfolio of financial instruments.

The paper deals with the limit behavior of a compound Poisson process with switching. The switching is provided by Bernoulli’s random variables. Under a suitable normalization, the limit process is a Brownian motion with switching variance.

In many problems of dynamics, one has to deal with mechanical systems whose configurational spaces are two- or three-dimensional manifolds. For such a system, the phase space naturally coincides with the tangent bundle of the corresponding manifold. Thus, the problem of a flow past a (four-dimensional) pendulum on a (generalized) spherical hinge leads to a system on the tangent bundle of a two- or threedimensional sphere whose metric has a particular structure induced by an additional symmetry group. In such cases, dynamical systems have variable dissipation, and their complete list of first integrals consists of transcendental functions in the form of finite combinations of elementary functions. Another class of problems pertains to a point moving on a two- or three-dimensional surface with the metric induced by the encompassing Euclidean space. In this paper, we establish the integrability of some classes of dynamical systems on tangent bundles of two- and three-dimensional manifolds, in particular, systems involving fields of forces with variable dissipation and of a more general type than those considered previously.

This is an ℝn-counterpart of certain considerations on a similar subject for compact Abelian groups exposed by P. Ivanishvili and the author in 2010. The main difference with that paper is that certain notions and results of measure theory should be invoked in the case of ℝn.

In the paper the wave functional model of a symmetric restriction of the regular Sturm-Liouville operator on an interval is constructed. The model is based upon the notion of the wave spectrum and is constructed according to an abstract scheme, which was proposed earlier. The result of the construction is a differential operator of the second order on an interval, which differs from the original operator only by a simple transformation.

In the paper the asymptotic structure of eigenfunctions of the absolutely continuous spectrum of the scattering problem is described. The case of three one-dimensional quantum particles interacting by repulsive pair potentials with a compact support is considered.

The paper is devoted to the calculation of the Morse index on geodesic lines upon smooth surfaces embedded into the 3D Euclidean space. The interest in this theme is created by the fact that the wave field composed of the surface waves slides along the boundaries guided by the geodesic lines, which, generally speaking, give birth to numerous caustics. The same circumstance takes place in problems of the short-wave diffraction by 3D bodies in the shadowed part of the surface of the body, where the creeping waves arise. Two types of geodesic flows are considered upon the surface when they are generated by a point source and by an initial wave front, for instance, by the light-shadow boundary in the short-wave diffraction by a smooth convex body. The position of the points where geodesic lines meet caustics, i.e., focal points, is found and it is proved that all focal points are simple (not multiple) irrespective of the geometric structure of the caustics arisen. The mathematical techniques in use are based on the complexification of the geometrical spreading problem for a geodesics/rays tube.

In the present note, which is a marginalia to the previous papers by Roozbeh Hazrat, Alexei Stepanov, Zuhong Zhang, and the author, I observe that for any ideals A,B≤R of a commutative ring R and all n ≥ 3 the birelative standard commutator formula also holds in the unrelativized form, as [E(n,A),GL(n,B)] = [E(n,A),E(n,B)] and discuss some obvious corollaries thereof.

A set of additive subgroups σ = (σij), 1 ≤ i, j ≤ n, of a field (or ring) K is called a net of order n over K if σirσrj ⊆ σij for all values of the indices i, r, j. The same system, but without diagonal, is called an elementary net. A full or elementary net σ = (σij) is said to be irreducible if all the additive subgroups σij are different from zero. An elementary net σ is closed if the subgroup E(σ) does not contain new elementary transvections. The present paper is related to a question posed by Y. N. Nuzhin in connection with V. M. Levchuk’s question No. 15.46 from the Kourovka notebook about the admissibility (closure) of elementary net (carpet) σ = (σij) over a field K. Let J be an arbitrary subset of {1, . . . , n}, n ≥ 3, and the cardinality m of J satisfies the condition 2 ≤ m ≤ n − 1. Let R be a commutative integral domain (non-field) with identity, and let K be the quotient field of R. An example of a net σ = (σij) of order n over K, for which the group E(σ) ∩ 〈tij(K) : i, j ∈ J〉 is not contained in the group 〈tij(σij) : i, j ∈ J〉, is constructed.

The Hochschild cohomology algebra for the algebras of dihedral type in the subfamily of the family D(2ℬ), for which the parameter c is equal to 0, are described. The calculation of multiplication in this cohomology algebra, uses the minimal bimodule projective resolution for algebras under consideration, that was constructed in the previous paper of the authors. The obtained results allow to describe the Hochschild cohomology algebra also for algebras with c = 0 in the family D(2$$ \mathcal{A} $$).

In 1997, B. I. Plotkin introduced a concept of geometric equivalence of algebraic structures and posed a question: is it true that every nilpotent torsion-free group is geometrically equivalent to its Mal’cev’s closure? A negative answer in the form of three counterexamples was given by V. V. Bludov and B. V. Gusev in 2007. In the present paper, an infinite series of counterexamples of unbounded Hirsch rank and nilpotency degree is constructed.

The drop of the smoothness of an analytic function compared to the smoothness of its modulus is discussed for the unit ball of ℂn. The paper is devoted to local aspects of the problem.

Earlier, in the study of combinatorial properties of the heat kernel of the Laplace operator with covariant derivative, a diagram technique and matrix formalism were constructed. In particular, the obtained formalism allows one to control the coefficients of the heat kernel, which is useful for calculations. In this paper, we consider a simple case with an Abelian connection in the two-dimensional space. This model allows us to give a mathematical description of the operators and find a relation between these operators and generating functions of numbers.

In this paper, we show that an operator introduced by A. A. Tsvetkov enjoys all the necessary properties of a Q-operator. It is shown that the Q-operator of the XXX spin chain with spin l turns into Tsvetkov’s operator in the continuous limit as l→∞. Bibliography: 18 titles.

As is well known, in the case of the group SL(2, ℝ), the tensor product of two continuous series representations can be decomposed into a direct sum of representations corresponding to the discrete and continuous spectra. The general theory implies the completeness of the system of projections that realize this decomposition. The main purpose of this paper is to check the corresponding relation in the sense of generalized functions. Performing the calculations, we develop a technique for working with projections, in particular, construct operators that realize the unitary equivalence between representations. Our results can be useful in various applications, for example, in calculating 6j-symbols.

International audience; In this paper, we investigate genus 2 algebro-geometric solutions of the AKNS hierarchy equations strictly periodic with respect to the space variable x. In general, genus 2 solutions, which are expressed in terms of two-dimensional Riemann theta functions, are not strictly periodic in x. We show that x-periodic solutions can be obtained by an appropriate choice of a hyperelliptic spectral curve having a structure of a covering of an elliptic curve. For odd-numbered members of the AKNS hierarchy, these solutions can be made periodic also with respect to the corresponding time variables of the AKNS hierarchy, by imposing further restrictions on the structure of the spectral curve. The corresponding solutions are especially interesting from the point of view of potential applications to the study of signal propagation in nonlinear optical fibers.

In the paper, an exact formula for the number of integral points in the system of ellipses related, according to Dirichlet, to an arbitrary imaginary quadratic field is provided. The relation of this formula to an arithmetic Riemann–Roch theorem is discussed. Previously, only nine formulas of such a type have been known. They correspond to the imaginary quadratic fields with the trivial class group.

The paper presents a universal karyon algorithm, applicable to an arbitrary collection of reals α = (α1, . . . , αd), which is a modification of the simplex-karyon algorithm. The main distinction is that instead of a simplex sequence, an infinite sequence T = T0,T1, . . . ,Tn, . . . of d-dimensional parallelohedra Tn appear. Every parallelohedron Tn is obtained from the previous one Tn−1 by differentiation, $$ {\mathbf{T}}_n={\mathbf{T}}_{n-1}^{\sigma n} $$ . The parallelohedra Tn are the karyons of some induced toric tilings. A certain algorithm (ϱ-strategy) for choosing infinite sequences σ|={σ1, σ2, …, σn, …} of differentiations σn is specified. This algorithm ensures the convergence ϱ(Tn) −→ 0 as n → +∞, where ϱ(Tn) denotes the radius of the parallelohedron Tn in the metric ϱ chosen as the objective function. It is proved that the parallelohedra Tn have the minimality property, i.e., the karyon approximation algorithm is the best one with respect to the karyon Tn-norms. Also an estimate for the rate of approximation of real numbers α = (α1, . . . , αd) by multidimensional continued fractions is derived.

The population size has far-reaching effects on the fitness of the population, that, in its turn influences the population extinction or persistence. Understanding the density- and age-dependent factors will facilitate more accurate predictions about the population dynamics and its asymptotic behaviour. In this paper, we develop a rigourous mathematical analysis to study positive and negative effects of increased population density in the classical nonlinear age-structured population model introduced by Gurtin \& MacCamy in the late 1970s. One of our main results expresses the global stability of the system in terms of the newborn function only. We also derive the existence of a threshold population size implying the population extinction, which is well-known in population dynamics as an Allee effect., 20 pages, submitted to J. Math. Sci