2,206 results
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2. Analytic Detection in Homotopy Groups of Smooth Manifolds
- Author
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I. S. Zubov
- Subjects
Statistics and Probability ,Pure mathematics ,Fundamental group ,Homotopy group ,Riemann surface ,Applied Mathematics ,General Mathematics ,Holomorphic function ,General Medicine ,Central series ,Hopf invariant ,symbols.namesake ,Linear differential equation ,symbols ,Element (category theory) ,Mathematics - Abstract
In this paper, for the mapping of a sphere into a compact orientable manifold S n → M , n ⩾ 1 , we solve the problem of determining whether it represents a nontrivial element in the homotopy group of the manifold π n ( M ) πn(M ). For this purpose, we consistently use the theory of iterated integrals developed by K.-T. Chen. It should be noted that the iterated integrals as repeated integration were previously meaningfully used by Lappo-Danilevsky to represent solutions of systems of linear differential equations and by Whitehead for the analytical description of the Hopf invariant for mappings f : S 2 n - 1 → S n , n ⩾ 2 . We give a brief description of Chen’s theory, representing Whitehead’s and Haefliger’s formulas for the Hopf invariant and generalized Hopf invariant. Examples of calculating these invariants using the technique of iterated integrals are given. Further, it is shown how one can detect any element of the fundamental group of a Riemann surface using iterated integrals of holomorphic forms. This required to prove that the intersection of the terms of the lower central series of the fundamental group of a Riemann surface is a unit group.
- Published
- 2022
3. Multiplication of Distributions and Algebras of Mnemofunctions
- Author
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A B Antonevich and T G Shagova
- Subjects
Statistics and Probability ,Classical theory ,Pure mathematics ,Distribution (mathematics) ,General method ,Operator (physics) ,Applied Mathematics ,General Mathematics ,Embedding ,Multiplication ,General Medicine ,Space (mathematics) ,Mathematics - Abstract
In this paper, we discuss methods and approaches for definition of multiplication of distributions, which is not defined in general in the classical theory. We show that this problem is related to the fact that the operator of multiplication by a smooth function is nonclosable in the space of distributions. We give the general method of construction of new objects called new distributions, or mnemofunctions, that preserve essential properties of usual distributions and produce algebras as well. We describe various methods of embedding of distribution spaces into algebras of mnemofunctions. All ideas and considerations are illustrated by the simplest example of the distribution space on a circle. Some effects arising in study of equations with distributions as coefficients are demonstrated by example of a linear first-order differential equation.
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- 2022
4. On Initial-Boundary Value Problem on Semiaxis for Generalized Kawahara Equation
- Author
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A. V. Faminskii and E. V. Martynov
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Statistics and Probability ,media_common.quotation_subject ,Applied Mathematics ,General Mathematics ,General Medicine ,Infinity ,Term (time) ,Nonlinear system ,Applied mathematics ,Boundary value problem ,Uniqueness ,Value (mathematics) ,Mathematics ,media_common - Abstract
In this paper, we consider initial-boundary value problem on semiaxis for generalized Kawahara equation with higher-order nonlinearity. We obtain the result on existence and uniqueness of the global solution. Also, if the equation contains the absorbing term vanishing at infinity, we prove that the solution decays at large time values.
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- 2022
5. Smoothness of Generalized Solutions of the Neumann Problem for a Strongly Elliptic Differential-Difference Equation on the Boundary of Adjacent Subdomains
- Author
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D. A. Neverova
- Subjects
Statistics and Probability ,Smoothness (probability theory) ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Neumann boundary condition ,Boundary (topology) ,Differential difference equations ,General Medicine ,Mathematics - Abstract
This paper is devoted to the study of the qualitative properties of solutions to boundary-value problems for strongly elliptic differential-difference equations. Some results for these equations such as existence and smoothness of generalized solutions in certain subdomains of Q were obtained earlier. Nevertheless, the smoothness of generalized solutions of such problems can be violated near the boundary of these subdomains even for infinitely differentiable right-hand side. The subdomains are defined as connected components of the set that is obtained from the domain Q by throwing out all possible shifts of the boundary Q by vectors of a certain group generated by shifts occurring in the difference operators. For the one dimensional Neumann problem for differential-difference equations there were obtained conditions on the coefficients of difference operators, under which for any continuous right-hand side there is a classical solution of the problem that coincides with the generalized solution. 2 Also there was obtained the smoothness (in Sobolev spaces W k ) of generalized solutions of the second and the third boundary-value problems for strongly elliptic differential-difference equations in subdomains excluding -neighborhoods of certain points. However, the smoothness (in Ho lder spaces) of generalized solutions of the second boundary-value problem for strongly elliptic differential-difference equations on the boundary of adjacent subdomains was not considered. In this paper, we study this question in Ho lder spaces. We establish necessary and sufficient conditions for the coefficients of difference operators that guarantee smoothness of the generalized solution on the boundary of adjacent subdomains for any right-hand side from the Ho lder space.
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- 2022
6. On Spectral and Evolutional Problems Generated by a Sesquilinear Form
- Author
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A. R. Yakubova
- Subjects
Statistics and Probability ,Pure mathematics ,Sesquilinear form ,Applied Mathematics ,General Mathematics ,General Medicine ,Mathematics - Abstract
On the base of boundary-value, spectral and initial-boundary value problems studied earlier for the case of single domain, we consider corresponding problems generated by sesquilinear form for two domains. Arising operator pencils with corresponding operator coefficients acting in a Hilbert space and depending on two parameters are studied in detail. In the perturbed and unperturbed cases, we consider two situations when one of the parameters is spectral and the other is fixed. In this paper, we use the superposition principle that allow us to present the solution of the original problem as a sum of solutions of auxiliary boundary-value problems containing inhomogeneity either in the equation or in one of the boundary conditions. The necessary and sufficient conditions for the correct solvability of boundary-value problems on given time interval are obtained. The theorems on properties of the spectrum and on the completeness and basicity of the system of root elements are proved.
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- 2022
7. On Boundedness of Maximal Operators Associated with Hypersurfaces
- Author
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S E Usmanov and I A Ikromov
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Statistics and Probability ,Pure mathematics ,Mathematics::Algebraic Geometry ,Hypersurface ,Mathematics::Complex Variables ,Applied Mathematics ,General Mathematics ,Mathematics::Classical Analysis and ODEs ,Regular polygon ,Mathematics::Differential Geometry ,General Medicine ,Value (mathematics) ,Mathematics - Abstract
In this paper, we obtain the criterion of boundedness of maximal operators associated with smooth hypersurfaces. Also we compute the exact value of the boundedness index of such operators associated with arbitrary convex analytic hypersurfaces in the case where the height of a hypersurface in the sense of A. N. Varchenko is greater than 2. Moreover, we obtain the exact value of the boundedness index for degenerated smooth hypersurfaces, i.e., for hypersurfaces satisfying conditions of the classical Hartman-Nirenberg theorem. The obtained results justify the Stein-Iosevich-Sawyer hypothesis for arbitrary convex analytic hypersurfaces as well as for smooth degenerated hypersurfaces. Also we discuss some related problems of the theory of oscillatory integrals.
- Published
- 2022
8. Asymptotic Properties of Solutions of Two-Dimensional Differential-Difference Elliptic Problems
- Author
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A. B. Muravnik
- Subjects
Statistics and Probability ,Dirichlet problem ,Pure mathematics ,Generalized function ,Applied Mathematics ,General Mathematics ,Operator (physics) ,Poisson kernel ,Zero (complex analysis) ,Boundary (topology) ,Function (mathematics) ,symbols.namesake ,Elliptic curve ,symbols ,Mathematics - Abstract
In the half-plane {−∞ < x < +∞} × {0 < y < +∞}, the Dirichlet problem is considered for differential-difference equations of the kind $$ {u}_{xx}+\sum_{k=1}^m{a}_k{u}_{xx}\left(x+{h}_{k,}y\right)+{u}_{yy}=0 $$ , where the amount m of nonlocal terms of the equation is arbitrary and no commensurability conditions are imposed on their coefficients a1, . . . , am and the parameters h1, . . . , hm determining the translations of the independent variable x. The only condition imposed on the coefficients and parameters of the studied equation is the nonpositivity of the real part of the symbol of the operator acting with respect to the variable x. Earlier, it was proved that the specified condition (i.e., the strong ellipticity condition for the corresponding differential-difference operator) guarantees the solvability of the considered problem in the sense of generalized functions (according to the Gel’fand–Shilov definition), a Poisson integral representation of a solution was constructed, and it was proved that the constructed solution is smooth outside the boundary line. In the present paper, the behavior of the specified solution as y → +∞ is investigated. We prove the asymptotic closeness between the investigated solution and the classical Dirichlet problem for the differential elliptic equation (with the same boundary function as in the original nonlocal problem) determined as follows: all parameters h1, . . . , hm of the original differential-difference elliptic equation are assigned to be equal to zero. As a corollary, we prove that the investigated solutions obey the classical Repnikov–Eidel’man stabilization condition: the solution stabilizes as y → +∞ if and only if the mean value of the boundary function over the interval (−R,+R) has a limit as R → +∞.
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- 2021
9. Definability of Completely Decomposable Torsion-Free Abelian Groups by Semigroups of Endomorphisms and Groups of Homomorphisms
- Author
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T. A. Pushkova
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Statistics and Probability ,Combinatorics ,Endomorphism ,Applied Mathematics ,General Mathematics ,Torsion (algebra) ,Homomorphism ,Isomorphism ,Abelian group ,Mathematics - Abstract
Let C be an Abelian group. A class X of Abelian groups is called a CE• H-class if for any groups A, B ∈ X, it follows from the existence of isomorphisms E• (A) ≅ E• (B) and Hom(C,A) ≅ Hom(C,B) that there is an isomorphism A ≅ B. In this paper, conditions are studied under which the class $$ {\mathfrak{I}}_{\mathrm{cd}}^{\mathrm{ad}} $$ of completely decomposable almost divisible Abelian groups and class $$ {\mathfrak{I}}_{\mathrm{cd}}^{\ast } $$ of completely decomposable torsion-free Abelian groups A where Ω(A) contains only incomparable types are CE• H-classes, where C is a completely decomposable torsion-free Abelian group.
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- 2021
10. Some Formulas for Ordinary and Hyper Bessel–Clifford Functions Related to the Proper Lorentz Group
- Author
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I. A. Shilin and J. Choi
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Statistics and Probability ,Lorentz group ,Pure mathematics ,symbols.namesake ,Matrix (mathematics) ,Applied Mathematics ,General Mathematics ,symbols ,Space (mathematics) ,Representation (mathematics) ,Bessel function ,Mathematics - Abstract
In this paper, we show that the matrix elements of some Lorentz group representation operators and bases transform operators acting in the representation space may be expressed in terms of the modified Bessel–Clifford functions and their multi-index analogs introduced by the authors via Delerue hyper Bessel functions.
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- 2021
11. Computer Calculation of Green Functions for Third-Order Ordinary Differential Equations
- Author
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I.N. Belyaeva, L. V. Krasovskaya, N.A. Chekanov, and N.N. Chekanova
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Statistics and Probability ,Power series ,business.industry ,Applied Mathematics ,General Mathematics ,Construct (python library) ,Third order ,Software ,Linear differential equation ,Ordinary differential equation ,Applied mathematics ,Gravitational singularity ,business ,Mathematics - Abstract
In this paper, we present a method of computer calculation of Green functions in the form of generalized power series for third-order linear differential equations admitting regular singularities. For specific boundary-value problems, we construct Green functions by using the software proposed.
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- 2021
12. Application of the A. N. Tikhonov Regularization to Restoring Microstructural Characteristics of Hail Clouds
- Author
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L T Sozaeva and A. Kh. Kagermazov
- Subjects
Statistics and Probability ,Applied Mathematics ,General Mathematics ,Inverse problem ,Residual ,law.invention ,Tikhonov regularization ,Distribution function ,law ,Conjugate gradient method ,Applied mathematics ,Minification ,Radar ,Mathematics - Abstract
This paper is devoted to solving the inverse problem of restoring the hydrometeor distribution function from radar measurements. We use the Tikhonov regularization algorithm based on the minimization of the residual functional by the conjugate gradient method.
- Published
- 2021
13. On the Rate of Stabilization of Solutions to the Cauchy Problem for the Godunov–Sultangazin System with Periodic Initial Data
- Author
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S. A. Dukhnovskii
- Subjects
Statistics and Probability ,Momentum ,Boltzmann kinetic equation ,Exponential stabilization ,Thermodynamic equilibrium ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Initial value problem ,Perturbation (astronomy) ,System of linear equations ,Energy (signal processing) ,Mathematics - Abstract
In this paper, we examine a one-dimensional system of equations for a discrete gas model (the Godunov–Sultangazin system). The Godunov–Sultangazin system is the Boltzmann kinetic equation for a model one-dimensional gas consisting of three groups of particles. In this model, the momentum is preserved whereas the energy is not. We prove the existence of a unique global solution to the Cauchy problem for a perturbation of the equilibrium state with periodic initial data. For the first time, we find the rate of stabilization to the equilibrium state (exponential stabilization).
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- 2021
14. On boundary-value problems for semi-linear equations in the plane
- Author
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Vladimir Gutlyanskiĭ, Vladimir Ryazanov, O.V. Nesmelova, and A.S. Yefimushkin
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Statistics and Probability ,Dirichlet problem ,Sobolev space ,Pure mathematics ,Harmonic function ,Applied Mathematics ,General Mathematics ,Neumann boundary condition ,Hölder condition ,Boundary value problem ,Type (model theory) ,Analytic function ,Mathematics - Abstract
The study of the Dirichlet problem with arbitrary measurable data for harmonic functions in the unit disk 𝔻 is due to the dissertation of Luzin. Later on, the known monograph of Vekua was devoted to boundary-value problems only with Holder continuous data for generalized analytic functions, i.e., continuous complex-valued functions f(z) of the complex variable z = x + iy with generalized first partial derivatives by Sobolev satisfying equations of the form $$ {\partial}_{\overline{z}}f+ af+b\overline{f}=c, $$ where the complexvalued functions a; b, and c are assumed to belong to the class Lp with some p > 2 in smooth enough domains D in ℂ. Our last paper [12] contained theorems on the existence of nonclassical solutions of the Hilbert boundaryvalue problem with arbitrary measurable data (with respect to logarithmic capacity) for generalized analytic functions f : D → ℂ such that $$ {\partial}_{\overline{z}}f=g $$ with the real-valued sources. On this basis, the corresponding existence theorems were established for the Poincare problem on directional derivatives and, in particular, for the Neumann problem to the Poisson equations △U = G ∈ Lp; p > 2, with arbitrary measurable boundary data over logarithmic capacity. The present paper is a natural continuation of the article [12] and includes, in particular, theorems on the existence of solutions for the Hilbert boundary-value problem with arbitrary measurable data for the corresponding nonlinear equations of the Vekua type $$ {\partial}_{\overline{z}}f(z)=h(z)q\left(f(z)\right). $$ On this basis, existence theorems were also established for the Poincar´e boundary-value problem and, in particular, for the Neumann problem for the nonlinear Poisson equations of the form △U(z) = H(z)Q(U(z)) with arbitrary measurable boundary data over logarithmic capacity. The Dirichlet problem was investigated by us for the given equations, too. Our approach is based on the interpretation of boundary values in the sense of angular (along nontangential paths) limits that are a conventional tool of the geometric function theory. As consequences, we give applications to some concrete semi-linear equations of mathematical physics arising from modelling various physical processes. Those results can also be applied to semi-linear equations of mathematical physics in anisotropic and inhomogeneous media.
- Published
- 2021
15. Representation of Weierstrass integral via Poisson integrals
- Author
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Arsen M. Shutovskyi and Vasyl Ye. Sakhnyuk
- Subjects
Statistics and Probability ,Partial differential equation ,Differential equation ,Applied Mathematics ,General Mathematics ,Multiple integral ,Mathematical analysis ,Separation of variables ,Dirac delta function ,Function (mathematics) ,symbols.namesake ,symbols ,Heat equation ,Fourier series ,Mathematics - Abstract
In our research, we have presented a second-order linear partial differential equation in polar coordinates. Considering this differential equation on the unit disk, we have obtained a one-dimensional heat equation. It is well-known that the heat equation can be solved taking into account the boundary condition for the general solution on the unit circle. In our paper, the boundary-value problem is solved using the well-known method called the separation of variables. As a result, the general solution to the boundary-value problem is presented in terms of the Fourier series. Then the expressions for the Fourier coefficients are used to transform the Fourier series expansion for the general solution to the boundary-value problem into the so-called Weierstrass integral, which is represented via the so-called Weierstrass kernel. A representation of the Weierstrass kernel via the infinite geometric series is derived by a way allowing a complicated function to be parameterized via a simplified function. The derivation of the corresponding parametrization is based on two well-known integrals. As a result, a complicated function of the natural argument is represented in the form of a double integral that contains a simplified function of the same natural argument. So, the double-integral representation of the Weierstrass kernel has been derived. To obtain this result, the integral representation of the so-called Dirac delta function is taken into account. The expression found for the Weierstrass kernel is substituted into the expression for the Weierstrass integral. As a result, it was found that the Weierstrass integral can be considered a double-integral that contains the Poisson and conjugate Poisson integrals.
- Published
- 2021
16. Distribution of Functionals of a Brownian Motion with Nonstandard Switching
- Author
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Andrei N. Borodin
- Subjects
Statistics and Probability ,Moment (mathematics) ,Distribution (mathematics) ,Applied Mathematics ,General Mathematics ,Local time ,Process (computing) ,Inverse ,Point (geometry) ,Statistical physics ,Diffusion (business) ,Brownian motion ,Mathematics - Abstract
The standard switching from one set of diffusion coefficients to another one occurs at random times corresponding to the moments of jumps of a Poisson process independent of the initial diffusion. The paper deals with the process of Brownian motion with variance taking one of two values by the switching depending on trajectories of the process. The most attractive from the computational point of view is the moment inverse to local time.
- Published
- 2021
17. Limit Behavior of a Compound Poisson Process with Switching Between Multiple Values
- Author
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Andrei N. Borodin
- Subjects
Statistics and Probability ,Normalization (statistics) ,Bernoulli's principle ,Applied Mathematics ,General Mathematics ,Compound Poisson process ,Process (computing) ,Limit (mathematics) ,Statistical physics ,Random variable ,Finite set ,Brownian motion ,Mathematics - Abstract
The paper deals with the limit behavior of a compound Poisson process with switching between a finite number of sequences of i.i.d. random variables. The switching is provided by Bernoulli’s random variables. Under suitable normalization, the limit process is a Brownian motion with switching variance.
- Published
- 2021
18. Nonasymptotic Analysis of the Lawley–Hotelling Statistic for High-Dimensional Data
- Author
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A. A. Lipatiev and Vladimir V. Ulyanov
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Statistics and Probability ,General linear model ,Clustering high-dimensional data ,Multivariate analysis ,Multivariate analysis of variance ,Sample size determination ,Applied Mathematics ,General Mathematics ,Root test ,Linear regression ,Statistics ,Statistic ,Mathematics - Abstract
We consider the General Linear Model (GLM) which includes multivariate analysis of variance (MANOVA) and multiple linear regression as special cases. In practice, there are several widely used criteria for GLM: Wilks’ lambda, Bartlett–Nanda–Pillai test, Lawley–Hotelling test, and Roy maximum root test. Limiting distributions for the first three mentioned tests are known under different asymptotic settings. In the present paper, we obtain computable error bounds for the normal approximation of the Lawley–Hotelling statistic when the dimension grows proportionally to the sample size. This result enables us to get more precise calculations of the p-values in applications of multivariate analysis. In practice, more and more often analysts encounter situations where the number of factors is large and comparable with the sample size. Examples include medicine, biology (i.e., DNA microarray studies), and finance.
- Published
- 2021
19. Estimation of a Vector-Valued Function in a Stationary Gaussian Noise
- Author
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Valentin Solev
- Subjects
Statistics and Probability ,Applied Mathematics ,General Mathematics ,Gaussian ,Spectral density ,Version vector ,Minimax ,Upper and lower bounds ,symbols.namesake ,Gaussian noise ,symbols ,Applied mathematics ,Vector-valued function ,Mathematics ,Stationary noise - Abstract
In this paper, we construct a lower bound of the minimax risk in the estimation problem when we observe an unknoun pseudo-periodic vector-function in a Gaussian stationary noise with spectral density satisfying the vector version of the Muckenhoupt condition.
- Published
- 2021
20. Oscillation and Nonoscillation Results for the Caputo Fractional q-Difference Equations and Inclusions
- Author
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Saïd Abbas, Mouffak Benchohra, and John R. Graef
- Subjects
Statistics and Probability ,Oscillation ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Mathematics - Abstract
The paper deals with the existence, oscillation, and nonoscillation of solutions to some classes of Caputo fractional q-difference equations and inclusions. The technique of proving employs set-valued analysis, fixed-point theory, and the method of upper and lower solutions.
- Published
- 2021
21. Asymptotic soliton-like solutions to the Benjamin–Bona–Mahony equation with variable coefficients and a strong singularity
- Author
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V. H. Samoilenko and Yuliia Samoilenko
- Subjects
Statistics and Probability ,Nonlinear Sciences::Exactly Solvable and Integrable Systems ,Singularity ,Applied Mathematics ,General Mathematics ,Benjamin–Bona–Mahony equation ,Mathematical analysis ,Structure (category theory) ,Soliton ,WKB approximation ,Variable (mathematics) ,Mathematics ,Term (time) - Abstract
The paper deals with constructing an asymptotic one-phase soliton-like solution to the Benjamin--Bona--Mahony equation with variable coefficients and a strong singularity making use of the non-linear WKB technique. The influence of the small-parameter value on the structure and the qualitative properties of the asymptotic solution, as well as the accuracy with which the solution satisfies the considerable equation, have been analyzed. It was demonstrated that due to the strong singularity, it is possible to write explicitly not only the main term of the asymptotics but at least its first-order term.
- Published
- 2021
22. On the Frequency of a Nonlinear Oscillator
- Author
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L. A. Kalyakin
- Subjects
Statistics and Probability ,Nonlinear oscillators ,Separatrix ,Applied Mathematics ,General Mathematics ,Quantum electrodynamics ,Monotonic function ,Nonlinear Oscillations ,Energy (signal processing) ,Mathematics - Abstract
In the study of nonlinear oscillations, the question on the dependence of the frequency or the period on the energy often arises. In this paper, we find conditions under which the frequency depends on the energy monotonically. In addition, for oscillations near separatrix trajectories, an asymptotics of the period with respect to the energy is constructed.
- Published
- 2021
23. Parametric Resonance in Integrable Systems and Averaging on Riemann Surfaces
- Author
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V. Yu. Novokshenov
- Subjects
Statistics and Probability ,Integrable system ,Applied Mathematics ,General Mathematics ,Riemann surface ,Dynamics (mechanics) ,symbols.namesake ,Nonlinear system ,Amplitude ,Classical mechanics ,Quasiperiodic function ,symbols ,Parametric oscillator ,Adiabatic process ,Mathematics - Abstract
In this paper, we consider adiabatic deformations of Riemann surfaces that preserve the integrability of the corresponding dynamic system, which leads to the appearance of modulated quasiperiodic motions, similar to the effect of parametric resonance. We show that in this way it is possible to control the amplitude and frequency of nonlinear modes. We consider several examples of the dynamics of top-type systems.
- Published
- 2021
24. Methods for Studying the Stability of Linear Periodic Systems Depending on a Small Parameter
- Author
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A. S. Belova, Liliya Sunagatovna Ibragimova, and M. G. Yumagulov
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Statistics and Probability ,Linear differential equation ,Applied Mathematics ,General Mathematics ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,Linear operators ,Applied mathematics ,Monodromy matrix ,Perturbation theory ,Stability (probability) ,Mathematics ,Linear dynamical system - Abstract
In this paper, we consider systems of linear differential equations with periodic coefficients depending on a small parameter. We propose new approaches to the problem of constructing a monodromy matrix that lead to new effective formulas for calculating multipliers of the system studies. We present a number of applications in problems of the perturbation theory of linear operators, in the analysis of stability of linear differential equations with periodic coefficients, in the problem of constructing the stability domains of linear dynamical systems, etc.
- Published
- 2021
25. Solution of the Problem of Equality and Conjugacy of Words in a Certain Class of Artin Groups
- Author
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N. B. Bezverkhnyaya and V. N. Bezverkhnii
- Subjects
Statistics and Probability ,Mathematics::Group Theory ,Class (set theory) ,Pure mathematics ,Conjugacy class ,Applied Mathematics ,General Mathematics ,Structure (category theory) ,Computer Science::Computational Geometry ,Mathematics - Abstract
This paper defines Artin groups with m-gon structure and proves that for m > 3 in this class the problems of equality and conjugacy of words are solvable.
- Published
- 2021
26. On the Artin Semigroups
- Author
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V. N. Bezverkhnii and A. E. Ustyan
- Subjects
Statistics and Probability ,Mathematics::Group Theory ,Pure mathematics ,Class (set theory) ,Semigroup ,Applied Mathematics ,General Mathematics ,Conjugacy problem ,Embedding ,Type (model theory) ,Word (group theory) ,Injective function ,Mathematics - Abstract
This paper introduces the concept of the Artin semigroup of a large (extra large) type. We prove their injective embedding in the corresponding Artin groups of large (extra large) type and the solvability of the word conjugacy problem in this class of semigroups.
- Published
- 2021
27. On Massive Subsets in the Space of Finitely Generated Groups of Diffeomorphisms of the Line and the Circle in the Case of C(1) Smoothness
- Author
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L. A. Beklaryan
- Subjects
Statistics and Probability ,Combinatorics ,Smoothness (probability theory) ,Intersection ,Applied Mathematics ,General Mathematics ,Line (geometry) ,Structure (category theory) ,Countable set ,Point (geometry) ,Orbit (control theory) ,Space (mathematics) ,Mathematics - Abstract
Among the finitely generated groups of diffeomorphisms of the line and the circle, groups that act freely on the orbit of almost every point of the line (circle) are allocated. The paper is devoted to the study of the structure of the set of finitely generated groups of orientation-preserving diffeomorphisms of the line and the circle of C(1) smoothness with a given number of generators and the property noted above. It is shown that such a set contains a massive subset (contains a countable intersection of open everywhere dense subsets). Such a result for finitely generated groups of orientation-preserving diffeomorphisms of the circle, in the case of C(2) smoothness, was obtained by the author earlier.
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- 2021
28. Algebraic Geometry over Algebraic Structures. VIII. Geometric Equivalences and Special Classes of Algebraic Structures
- Author
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E. Yu. Daniyarova, Alexei Myasnikov, and V. N. Remeslennikov
- Subjects
Statistics and Probability ,Noetherian ,Pure mathematics ,Class (set theory) ,Series (mathematics) ,Algebraic structure ,Applied Mathematics ,General Mathematics ,Algebraic geometry ,Invariant (mathematics) ,Mathematics - Abstract
This paper belongs to our series of works on algebraic geometry over arbitrary algebraic structures. In this one, there will be investigated seven equivalences (namely: geometric, universal geometric, quasi-equational, universal, elementary, and combinations thereof) in specific classes of algebraic structures (equationally Noetherian, qω-compact, uω-compact, equational domains, equational co-domains, etc.). The main questions are the following: (1) Which equivalences coincide inside a given class K, which do not? (2) With respect to which equivalences a given class K is invariant, with respect to which it is not?
- Published
- 2021
29. The Group of Quotients of the Semigroup of Invertible Nonnegative Matrices Over Local Rings
- Author
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V. V. Nemiro
- Subjects
Statistics and Probability ,Pure mathematics ,Invertible matrix ,law ,Group (mathematics) ,Semigroup ,Applied Mathematics ,General Mathematics ,Local ring ,Quotient ,law.invention ,Mathematics - Abstract
In this paper, we prove that for a linearly ordered local ring R with 1/2 the group of quotients of the semigroup of invertible nonnegative matrices Gn(R) for n ≥ 3 coincides with the group GLn(R).
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- 2021
30. Enumerative Combinatorics of XX0 Heisenberg Chain
- Author
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N. M. Bogoliubov
- Subjects
Statistics and Probability ,Pure mathematics ,Integrable system ,Chain (algebraic topology) ,Applied Mathematics ,General Mathematics ,Zero (complex analysis) ,Boundary value problem ,Limit (mathematics) ,Lattice (discrete subgroup) ,Enumerative combinatorics ,Exponential function ,Mathematics - Abstract
In the present paper, the enumeration of a certain class of directed lattice paths is based on the analysis of dynamical correlation functions of the exactly solvable XX0 model. This model is the zero anisotropy limit of one of the basic models of the theory of integrable systems, the XXZ Heisenberg magnet. It is demonstrated that the considered correlation functions under different boundary conditions are the exponential generating functions of various types of paths, in particular, Dyck and Motzkin paths.
- Published
- 2021
31. Quantum Equation of Motion and Two-Loop Cutoff Renormalization for 𝜙3 Model
- Author
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A. V. Ivanov and N. V. Kharuk
- Subjects
High Energy Physics - Theory ,Statistics and Probability ,Background field method ,Applied Mathematics ,General Mathematics ,FOS: Physical sciences ,Equations of motion ,Renormalization ,Momentum ,High Energy Physics - Theory (hep-th) ,Regularization (physics) ,Cutoff ,Effective action ,Quantum ,Mathematics ,Mathematical physics - Abstract
We present two-loop renormalization of $\phi^3$-model effective action by using the background field method and cutoff momentum regularization. In this paper, we also study a derivation of the quantum equation of motion and its application to the renormalization., Comment: LaTeX, 15 pages, 3 figures; The work has been published three years ago. In this version we have made some corrections and added calculations for the counterterm
- Published
- 2021
32. Schlesinger Transformations for Algebraic Painlevé VI Solutions
- Author
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Raimundas Vidunas and A. V. Kitaev
- Subjects
Statistics and Probability ,Pure mathematics ,Polynomial ,Applied Mathematics ,General Mathematics ,Computation ,Mathematics::Classical Analysis and ODEs ,Ode ,Hypergeometric distribution ,Nonlinear Sciences::Exactly Solvable and Integrable Systems ,Pullback ,Algebraic function ,Gravitational singularity ,Algebraic number ,Mathematics - Abstract
Schlesinger (S) transformations can be combined with a direct rational (R) pull-back of a hypergeometric 2 × 2 system of ODEs to obtain $$ {RS}_4^2 $$ -pullback transformations to isomonodromic 2 × 2 Fuchsian systems with 4 singularities. The corresponding Painleve VI solutions are algebraic functions, possibly in different orbits under Okamoto transformations. The paper demonstrates direct computations (involving polynomial syzygies) of Schlesinger transformations that affect several singular points at once, and presents an algebraic procedure of computing algebraic Painleve VI solutions without deriving full RS-pullback transformations.
- Published
- 2021
33. Notes on Functional Integration
- Author
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A. V. Ivanov
- Subjects
Statistics and Probability ,Algebra ,Applied Mathematics ,General Mathematics ,Product (mathematics) ,Path integral formulation ,Loop space ,Functional derivative ,Functional integration ,Orthonormal basis ,Space (mathematics) ,Special class ,Mathematics - Abstract
The paper is devoted to the construction of an “integral” on an infinite-dimensional space, combining the approaches proposed previously and at the same time the simplest. A new definition of the construction and study its properties on a special class of functionals is given. An introduction of a quasi-scalar product, an orthonormal system, and applications in physics (path integral, loop space, functional derivative) are proposed. In addition, the paper contains a discussion of generalized functionals.
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- 2021
34. On Lacunas in the Spectrum of the Laplacian with the Dirichlet Boundary Condition in a Band with Oscillating Boundary
- Author
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Denis Borisov
- Subjects
Statistics and Probability ,Applied Mathematics ,General Mathematics ,Operator (physics) ,Mathematical analysis ,Spectrum (functional analysis) ,Boundary (topology) ,Function (mathematics) ,symbols.namesake ,Amplitude ,Dirichlet boundary condition ,symbols ,Flat band ,Laplace operator ,Mathematics - Abstract
In this paper, we consider the Laplace operator in a flat band whose lower boundary periodically oscillates under the Dirichlet boundary condition. The period and the amplitude of oscillations are two independent small parameters. The main result obtained in the paper is the absence of internal lacunas in the lower part of the spectrum of the operator for sufficiently small period and amplitude. We obtain explicit upper estimates of the period and amplitude in the form of constraints with specific numerical constants. The length of the lower part of the spectrum, in which the absence of lacunas is guaranteed, is also expressed explicitly in terms of the period function and the amplitude.
- Published
- 2021
35. Generalized Interpolation Problem of the Korevaar–Dixon Type
- Author
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R. A. Gaisin
- Subjects
Statistics and Probability ,Class (set theory) ,Applied Mathematics ,General Mathematics ,Entire function ,Convergence (routing) ,Applied mathematics ,Type (model theory) ,Exponential type ,Interpolation ,Mathematics - Abstract
In this paper, we study the generalized interpolation problem in the class of entire functions of exponential type defined by a certain majorant from the convergence class.
- Published
- 2021
36. Interpolation by Series of Exponential Functions Whose Exponents Are Condensed in a Certain Direction
- Author
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S. V. Popenov and S. G. Merzlyakov
- Subjects
Statistics and Probability ,Pure mathematics ,Series (mathematics) ,Applied Mathematics ,General Mathematics ,media_common.quotation_subject ,Holomorphic function ,Infinity ,Convolution ,Exponential function ,Set (abstract data type) ,Interpolation space ,Interpolation ,Mathematics ,media_common - Abstract
For complex interpolation nodes, we study the problem of interpolation by series of exponential functions whose exponents form a set, which is condensed at infinity in a certain direction. We obtain a criterion for all sets of nodes from a special class. For arbitrary sets of nodes, we obtain a necessary condition for the solvability of a more general problem of interpolation by functions that can be represented as Radon integrals of an exponential function over a set of exponents. The paper also contains well-known results on interpolation, which, in particular, allow studying the multipoint holomorphic Vallee Poussin problem for convolution operators.
- Published
- 2021
37. Order Versions of the Hahn–Banach Theorem and Envelopes. II. Applications to Function Theory
- Author
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E. B. Khabibullina, A. P. Rozit, and Bulat N. Khabibullin
- Subjects
Statistics and Probability ,Pure mathematics ,Mathematics::Complex Variables ,Applied Mathematics ,General Mathematics ,Convex set ,Zero (complex analysis) ,Holomorphic function ,Hahn–Banach theorem ,Space (mathematics) ,Complex space ,Convex cone ,Mathematics ,Meromorphic function - Abstract
In this paper, we consider the problem on the existence of the upper (lower) envelope of a convex cone or, more generally, a convex set for functions on the projective limit of vector lattices with values in the completion of the Kantorovich space or on the extended real line. We propose vectorial, ordinal, and topological dual interpretations of the existence conditions for such envelopes and the method of constructing it. Applications to the problem on the existence of a nontrivial (pluri)subharmonic and/or (pluri)harmonic minorant for functions in domains of a finite-dimensional real or complex space are considered. We also propose general approaches to problems on the nontriviality of weight classes of holomorphic functions, describing zero (sub)sets for such classes of holomorphic functions, and to the problem of representing a meromorphic function as a ratio of holomorphic function from a given weight class.
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- 2021
38. Invariant Manifolds of Hyperbolic Integrable Equations and their Applications
- Author
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Ismagil Habibullin and A. R. Khakimova
- Subjects
Statistics and Probability ,Surface (mathematics) ,Pure mathematics ,Integrable system ,Applied Mathematics ,General Mathematics ,Homogeneous space ,Invariant manifold ,Type (model theory) ,Invariant (mathematics) ,Manifold ,Symmetry (physics) ,Mathematics - Abstract
We assign some kind of invariant manifolds to a given integrable PDE (its discrete or semidiscrete variant). First, we linearize the equation around its arbitrary solution u. Then we construct a differential (respectively, difference) equation compatible with the linearized equation for any choice of u. This equation defines a surface called a generalized invariant manifold. In a sense, the manifold generalizes the symmetry, which is also a solution to the linearized equation. In this paper, we concentrate on continuous and discrete models of hyperbolic type. It is known that such kinds of equations have two hierarchies of symmetries, corresponding to the characteristic directions. We have shown that a properly chosen generalized invariant manifold allows one to construct recursion operators that generate these symmetries. It is surprising that both recursion operators are related to different parametrizations of the same invariant manifold. Therefore, knowing one of the recursion operators for the hyperbolic type integrable equation (having no pseudo-constants) we can immediately find the second one.
- Published
- 2021
39. Representing Exponential Systems in Spaces of Analytic Functions
- Author
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K. P. Isaev
- Subjects
Statistics and Probability ,Pure mathematics ,Applied Mathematics ,General Mathematics ,Bounded function ,Convex domain ,Type (model theory) ,Space (mathematics) ,Linear subspace ,Mathematics ,Exponential function ,Analytic function - Abstract
This paper is devoted to representing exponential systems in various subspaces of the space H(D) of functions that are analytic in a bounded convex domain D. We consider two kinds of such subspaces: uniformly weighted spaces H(D,𝜑) and spaces of the type of Carleman classes H(D,ℳ).
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- 2021
40. Coefficients of Exponential Series for Analytic Functions and the Pommiez Operator
- Author
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S. N. Melikhov
- Subjects
Statistics and Probability ,Pure mathematics ,Series (mathematics) ,Applied Mathematics ,General Mathematics ,Entire function ,Bounded function ,Operator (physics) ,Invariant (mathematics) ,Complex plane ,Mathematics ,Analytic function ,Exponential function - Abstract
In this paper, we present results of the existence of a linear continuous right inverse operator for the operator of the representation of analytic functions in a bounded convex domain of the complex plane by series of quasi-polynomials and exponentials. We also present closely related results on the A. F. Leontiev interpolating function and, more generally, on the interpolating functional and the corresponding Pommiez operator. We examine cyclic vectors and closed invariant subspaces of the Pommiez operator in weighted spaces of entire functions.
- Published
- 2021
41. On a Class of Planar Geometrical Curves with Constant Reaction Forces Acting on Particles Moving Along Them
- Author
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S. B. Bogdanova and S. O. Gladkov
- Subjects
Statistics and Probability ,Class (set theory) ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Abscissa ,Function (mathematics) ,Trough (economics) ,symbols.namesake ,Planar ,Reaction ,symbols ,Point (geometry) ,Constant (mathematics) ,Mathematics - Abstract
In this paper, we find the dependence of the reaction force N(y) of a curved trough of arbitrary shape described by a function y(x). Based on the extremum condition dN/dx valid for any point of the abscissa axis, we examine the equation N(y, y', y'') = const whose solutions determine the desired class of curves y(x). We obtain an analytic solution of this equations and perform numerical simulations for various values of parameters. Examples of functions y(x) for which N = const are presented.
- Published
- 2021
42. Application of Resurgent Analysis to the Construction of Asymptotics of Linear Differential Equations with Degeneration in Coefficients
- Author
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M. V. Korovina
- Subjects
Statistics and Probability ,Partial differential equation ,Linear differential equation ,Applied Mathematics ,General Mathematics ,Degenerate energy levels ,Mathematical analysis ,Holomorphic function ,Degeneration (medical) ,Mathematics - Abstract
This paper is a review of results concerning the construction of asymptotics of solutions to degenerate linear differential equations with holomorphic coefficients. We consider both cases of ordinary and partial differential equations.
- Published
- 2021
43. Boundary Displacement Control for the Oscillation Process with Boundary Conditions of Damping Type for a Time Less Than Critical
- Author
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E. I. Moiseev, A. A. Frolov, and A. A. Kholomeeva
- Subjects
Statistics and Probability ,Oscillation ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,C++ string handling ,Boundary (topology) ,Oblique case ,Derivative ,Boundary value problem ,Type (model theory) ,Displacement (vector) ,Mathematics - Abstract
In this paper, we study the problem of boundary control of string oscillations, which is carried out over a period of time less than the critical time. The control is performed by a displacement of one end of the string, whereas at the other end a uniform boundary condition with oblique derivative is given, and the direction of this derivative does not coincide with characteristics. The classical statement of the problem is considered. Necessary and sufficient conditions for the existence of a unique control are found, and the control itself is obtained in an explicit analytical form.
- Published
- 2021
44. On Approximate Solution of Certain Equations
- Author
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A. V. Vasilyev
- Subjects
Statistics and Probability ,Applied Mathematics ,General Mathematics ,Kernel (statistics) ,Bounded function ,Applied mathematics ,State (functional analysis) ,Approximate solution ,Mathematics - Abstract
In this paper, we consider problems of discrete approximation of special integral operators with the Calderon–Zygmund kernel. We introduce discrete spaces and bounded discrete operators acting in these spaces; then we use these operators for the search for approximate solutions of the corresponding equations. We state theorems on the solvability of equations with discrete operators, compare integral operators with their discrete analogs, and obtain estimates of errors of approximate solutions.
- Published
- 2021
45. Dirichlet Problem for Functions that are Harmonic on a Two-Dimensional Net
- Author
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A. P. Soldatov and L. A. Kovaleva
- Subjects
Statistics and Probability ,Dirichlet problem ,Class (set theory) ,Harmonic function ,Applied Mathematics ,General Mathematics ,Zero (complex analysis) ,Applied mathematics ,Harmonic (mathematics) ,Mathematics::Spectral Theory ,Type (model theory) ,Net (mathematics) ,Mathematics - Abstract
In this paper, we consider the Dirichlet problem for harmonic functions on a two-dimensional complex of a special type. We prove that this problem is a Fredholm problem in the Holder class and its index is zero.
- Published
- 2021
46. Existence of Solutions of Anisotropic Elliptic Equations with Variable Indices of Nonlinearity in ℝn
- Author
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L. M. Kozhevnikova and A. Sh. Kamaletdinov
- Subjects
Statistics and Probability ,Nonlinear system ,Class (set theory) ,Pure mathematics ,Applied Mathematics ,General Mathematics ,Growth rate ,Type (model theory) ,Anisotropy ,Space (mathematics) ,Mathematics ,Variable (mathematics) - Abstract
In this paper, we consider a certain class of anisotropic second-order elliptic equations of divergent type with variable indices of nonlinearities. We examine conditions of the solvability in the whole space ℝn, n ≥ 2. We prove the existence of solutions without restrictions to the growth rate as |x| → ∞.
- Published
- 2021
47. Influence of a Nonstationary Surface Load on an Elastic Porous Half-Space
- Author
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G. V. Fedotenkov and O. N. Peshcherikova
- Subjects
Statistics and Probability ,Surface (mathematics) ,Planar ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Half-space ,Singular integral ,Surface pressure ,Porosity ,Regularization (mathematics) ,Mathematics ,Quadrature (mathematics) - Abstract
In this paper, we develop a method for solving planar nonstationary problems on the influence of surface pressure on an elastic-porous half-space. Using surface influence functions, we obtain an integral relation, which expresses the desired displacements with the surface pressure. A numericalanalytical algorithm is constructed. For calculating singular integrals, we derive special quadrature formulas based on the canonical regularization of integrands.
- Published
- 2021
48. Method of Boundary Integral Equations with Hypersingular Integrals in Boundary-Value Problems
- Author
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A. V. Setukha
- Subjects
Statistics and Probability ,Applied Mathematics ,General Mathematics ,Numerical analysis ,MathematicsofComputing_NUMERICALANALYSIS ,Boundary (topology) ,Singular integral ,Quadrature (mathematics) ,Hadamard transform ,Collocation method ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,Applied mathematics ,Boundary value problem ,Value (mathematics) ,Mathematics - Abstract
In this paper, we formulate mathematical foundations of applications of boundary integral equations with strongly singular integrals understood in the sense of finite Hadamard value to numerical solution of certain boundary-value problems. We describe numerical schemes for solving boundary strongly singular equations based on quadrature formulas and the collocation method. Also, we make references to known results on the mathematical justification of the numerical methods described in the paper.
- Published
- 2021
49. Diagnostic Problem for a Model of a Gyrostabilized Platform
- Author
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Maxim V. Shamolin and E. P. Krugova
- Subjects
Statistics and Probability ,Angular displacement ,Applied Mathematics ,General Mathematics ,ComputerApplications_COMPUTERSINOTHERSYSTEMS ,Gyroscope ,Mathematical proof ,Motion (physics) ,law.invention ,Position (vector) ,law ,Control theory ,Key (cryptography) ,Mathematics - Abstract
This paper is devoted to the study of the motion of a platform maintained on an aircraft in a predetermined position by a system of gyroscopes, which does follows oscillations of the aircraft. Such systems are used for determining the angular position of the aircraft. We briefly review key results on this issue without detailed proofs.
- Published
- 2021
50. Logarithmic Potential and Generalized Analytic Functions
- Author
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O.V. Nesmelova, Vladimir Gutlyanskiĭ, Vladimir Ryazanov, and A.S. Yefimushkin
- Subjects
Statistics and Probability ,Dirichlet problem ,Pure mathematics ,Applied Mathematics ,General Mathematics ,Harmonic (mathematics) ,Unit disk ,Sobolev space ,Riemann hypothesis ,symbols.namesake ,Harmonic function ,symbols ,Neumann boundary condition ,Analytic function ,Mathematics - Abstract
The study of the Dirichlet problem in the unit disk 𝔻 with arbitrary measurable data for harmonic functions is due to the famous dissertation of Luzin [31]. Later on, the known monograph of Vekua [48] has been devoted to boundary-value problems (only with Holder continuous data) for the generalized analytic functions, i.e., continuous complex valued functions h(z) of the complex variable z = x + iy with generalized first partial derivatives by Sobolev satisfying equations of the form 𝜕zh + ah + b $$ \overline{h} $$ = c ; where it was assumed that the complex valued functions a; b and c belong to the class Lp with some p > 2 in smooth enough domains D in ℂ. The present paper is a natural continuation of our previous articles on the Riemann, Hilbert, Dirichlet, Poincar´e and, in particular, Neumann boundary-value problems for quasiconformal, analytic, harmonic, and the so-called A−harmonic functions with boundary data that are measurable with respect to logarithmic capacity. Here, we extend the corresponding results to the generalized analytic functions h : D → ℂ with the sources g : 𝜕zh = g ∈ Lp, p > 2 , and to generalized harmonic functions U with sources G : △U = G ∈ Lp, p > 2. This paper contains various theorems on the existence of nonclassical solutions of the Riemann and Hilbert boundary-value problems with arbitrary measurable (with respect to logarithmic capacity) data for generalized analytic functions with sources. Our approach is based on the geometric (theoretic-functional) interpretation of boundary-values in comparison with the classical operator approach in PDE. On this basis, it is established the corresponding existence theorems for the Poincar´e problem on directional derivatives and, in particular, for the Neumann problem to the Poisson equations △U = G with arbitrary boundary data that are measurable with respect to logarithmic capacity. These results can be also applied to semilinear equations of mathematical physics in anisotropic and inhomogeneous media.
- Published
- 2021
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