358 results
Search Results
2. Degrees of Enumerations of Countable Wehner-Like Families
- Author
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I. Sh. Kalimullin and M. Kh. Faizrahmanov
- Subjects
Statistics and Probability ,Class (set theory) ,Degree (graph theory) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,01 natural sciences ,Spectrum (topology) ,010305 fluids & plasmas ,Combinatorics ,0103 physical sciences ,Enumeration ,Countable set ,Family of sets ,0101 mathematics ,Turing ,computer ,Finite set ,computer.programming_language ,Mathematics - Abstract
This paper is a survey of results on countable families with natural degree spectra. These results were obtained by a modification of the methodology proposed by Wechner, who first found a family of sets with the spectrum consisting precisely of nonzero Turing degrees. Based on this method, many researchers obtained examples of families with other natural spectra. In addition, in this paper we extend these results and present new examples of natural spectra. In particular, we construct a family of finite sets with the spectrum consisting of exactly non-K-trivial degrees and also we find new sufficient conditions on $$ {\Delta}_2^0 $$ -degree a, which guarantees that the class {x : x ≰ a} is the degree spectrum of some family. Finally, we give a survey of our recent results on the degree spectra of α-families, where α is an arbitrary computable ordinal.
- Published
- 2021
3. Theoretical Foundations of the Study of a Certain Class of Hybrid Systems of Differential Equations
- Author
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A. D. Mizhidon
- Subjects
Statistics and Probability ,Partial differential equation ,Differential equation ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Dirac (software) ,Equations of motion ,01 natural sciences ,010305 fluids & plasmas ,Mechanical system ,Variational principle ,Hybrid system ,0103 physical sciences ,Applied mathematics ,0101 mathematics ,Eigenvalues and eigenvectors ,Mathematics - Abstract
In this paper, we consider boundary-value problems for a new class of hybrid systems of differential equations whose coefficients contain the Dirac delta-function. Hybrid systems are systems that contain both ordinary and partial differential equations; such systems appear, for example, when equations of motion of mechanical systems of rigid bodies attached to a beam by elastic bonds are derived from the Hamilton–Ostrogradsky variational principle. We present examples that lead to such systems and introduce the notions of generalized solutions and eigenvalues of a boundary-value problem. We also compare results of numerical simulations based on methods proposed in this paper with results obtained by previously known methods and show that our approach is reliable and universal.
- Published
- 2021
4. Simplest Test for the Three-Dimensional Dynamical Inverse Problem (The BC-Method)
- Author
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Mikhail I. Belishev, N. A. Karazeeva, and A. S. Blagoveshchensky
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Statistics and Probability ,Applied Mathematics ,General Mathematics ,Operator (physics) ,010102 general mathematics ,Boundary (topology) ,Function (mathematics) ,Inverse problem ,Positive function ,01 natural sciences ,010305 fluids & plasmas ,Combinatorics ,0103 physical sciences ,Nabla symbol ,0101 mathematics ,Dynamical system (definition) ,Realization (systems) ,Mathematics - Abstract
A dynamical system $$ {\displaystyle \begin{array}{ll}{u}_{tt}-\Delta u-\nabla 1\mathrm{n}\;\rho \cdot \nabla u=0& in\kern0.6em {\mathrm{\mathbb{R}}}_{+}^3\times \left(0,T\right),\\ {}{\left.u\right|}_{t=0}={\left.{u}_t\right|}_{t=0}=0& in\kern0.6em \overline{{\mathrm{\mathbb{R}}}_{+}^3},\\ {}{\left.{u}_z\right|}_{z=0}=f& for\kern0.36em 0\le t\le T,\end{array}} $$ is under consideration, where ρ = ρ(x, y, z) is a smooth positive function; f = f(x, y, t) is a boundary control; u = uf (x, y, z, t) is a solution. With the system one associates a response operator R : f ↦ uf|z = 0. The inverse problem is to recover the function ρ via the response operator. A short representation of the local version of the BC-method, which recovers ρ via the data given on a part of the boundary, is provided. If ρ is constant, the forward problem is solved in explicit form. In the paper, the corresponding representations for the solutions and response operator are derived. A way to use them for testing the BC-algorithm, which solves the inverse problem, is outlined. The goal of the paper is to extend the circle of the BC-method users, who are interested in numerical realization of methods for solving inverse problems.
- Published
- 2021
5. Perturbations of the Continuous Spectrum of a Certain Nonlinear Two-Dimensional Operator Sheaf
- Author
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Denis Borisov
- Subjects
Statistics and Probability ,Pure mathematics ,Dimensional operator ,Applied Mathematics ,General Mathematics ,Operator (physics) ,010102 general mathematics ,Continuous spectrum ,Essential spectrum ,01 natural sciences ,010305 fluids & plasmas ,Bounded function ,0103 physical sciences ,Sheaf ,0101 mathematics ,Laplace operator ,Eigenvalues and eigenvectors ,Mathematics - Abstract
In this paper, we consider the operator sheaf $$ -\Delta +V+\varepsilon {\mathrm{\mathcal{L}}}_{\varepsilon}\left(\lambda \right)+{\lambda}^2 $$ in the space L2(ℝ2), where the real-valued potential V depends only on the first variable x1, e is a small positive parameter, λ is the spectral parameter, $$ {\mathrm{\mathcal{L}}}_{\varepsilon}\left(\lambda \right) $$ is a localized operator bounded with respect to the Laplacian −Δ, and the essential spectrum of this operator is independent of e and contains certain critical points defined as isolated eigenvalues of the operator $$ -\frac{d^2}{dx_1^2}+V\left({x}_1\right) $$ in L2(ℝ). The basic result obtained in this paper states that for small values of e, in neighborhoods of critical points mentioned, isolated eigenvalues of the sheaf considered arise. Sufficient conditions for the existence or absence of such eigenvalues are obtained. The number of arising eigenvalues is determined, and in the case where they exist, the first terms of their asymptotic expansions are found.
- Published
- 2020
6. A New Convexity-Based Inequality, Characterization of Probability Distributions, and Some Free-of-Distribution Tests
- Author
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Lev B. Klebanov and Irina V. Volchenkova
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Statistics and Probability ,Class (set theory) ,Generalization ,Applied Mathematics ,General Mathematics ,Probability (math.PR) ,010102 general mathematics ,Probabilistic logic ,01 natural sciences ,Convexity ,010305 fluids & plasmas ,Interpretation (model theory) ,Character (mathematics) ,Distribution (mathematics) ,0103 physical sciences ,FOS: Mathematics ,Probability distribution ,Applied mathematics ,60E10, 62E10 ,0101 mathematics ,Mathematics - Probability ,Mathematics - Abstract
A goal of the paper is to prove new inequalities connecting some functionals of probability distribution functions. These inequalities are based on the strict convexity of functions used in the definition of the functionals. The starting point is the paper “Cramer–von Mises distance: probabilistic interpretation, confidence intervals and neighborhood of model validation” by Ludwig Baringhaus and Norbert Henze. The present paper provides a generalization of inequality obtained in probabilistic interpretation of the Cramer–von Mises distance. If the equality holds there, then a chance to give characterization of some probability distribution functions appears. Considering this fact and a special character of the functional, it is possible to create a class of free-of-distribution two sample tests.
- Published
- 2020
7. Examples of Integrable Systems with Dissipation on the Tangent Bundles of Three-Dimensional Manifolds
- Author
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Maxim V. Shamolin
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Statistics and Probability ,Pure mathematics ,Integrable system ,Dynamical systems theory ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Degrees of freedom ,Tangent ,Dissipation ,01 natural sciences ,Force field (chemistry) ,010305 fluids & plasmas ,0103 physical sciences ,Mathematics::Differential Geometry ,0101 mathematics ,Mathematics::Symplectic Geometry ,Mathematics ,Variable (mathematics) - Abstract
In this paper, we prove the integrability of certain classes of dynamical systems on the tangent bundles of four-dimensional manifolds (systems with four degrees of freedom). The force field considered possessed so-called variable dissipation; they are generalizations of fields studied earlier. This paper continues earlier works of the author devoted to systems on the tangent bundles of two- and three-dimensional manifolds.
- Published
- 2020
8. Tailoring a Pair of Pants: The Phase Tropical Version
- Author
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Ilia Zharkov
- Subjects
Statistics and Probability ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Phase (waves) ,01 natural sciences ,010305 fluids & plasmas ,Mathematics - Algebraic Geometry ,0103 physical sciences ,FOS: Mathematics ,Isotopy ,0101 mathematics ,Algebraic Geometry (math.AG) ,Pair of pants ,Mathematics - Abstract
We show that the phase tropical pair-of-pants is (ambient) isotopic to the complex pair-of-pants. This paper can serve as an addendum to the author's joint paper with Ruddat arXiv:2001.08267 where an isotopy between complex and ober-tropical pairs-of-pants was shown. Thus all three versions are isotopic., 10 pages, 8 figures. arXiv admin note: text overlap with arXiv:2001.08267
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- 2020
9. A Short Proof of a Theorem Due to O. Gabber
- Author
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Ivan Panin
- Subjects
Statistics and Probability ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Field (mathematics) ,Regular local ring ,Reductive group ,01 natural sciences ,010305 fluids & plasmas ,Finite field ,Scheme (mathematics) ,0103 physical sciences ,Fraction (mathematics) ,0101 mathematics ,Mathematics - Abstract
A very short proof of an unpublished result due to O. Gabber is given. More exactly, let R be a regular local ring containing a finite field k. Let G be a simply-connected reductive group scheme over k. It is proved that a principal G-bundle over R is trivial if it is trivial over the fraction field of R. This is the mentioned unpublished result due to O. Gabber. In this paper, this result is derived from a purely geometric one, proved in another paper of the author and stated in the Introduction.
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- 2020
10. Commutators of Congruence Subgroups in the Arithmetic Case
- Author
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Nikolai Vavilov
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Statistics and Probability ,Ring (mathematics) ,Multiplicative group ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,General linear group ,Commutative ring ,Type (model theory) ,01 natural sciences ,010305 fluids & plasmas ,0103 physical sciences ,Arithmetic function ,Dedekind cut ,0101 mathematics ,Arithmetic ,Mathematics ,Counterexample - Abstract
In a joint paper of the author with Alexei Stepanov, it was established that for any two comaximal ideals A and B of a commutative ring R, A + B = R, and any n ≥ 3 one has [E(n,R,A),E(n,R,B)] = E(n,R,AB). Alec Mason and Wilson Stothers constructed counterexamples demonstrating that the above equality may fail when A and B are not comaximal, even for such nice rings as ℤ [i]. The present note proves a rather striking result that the above equality and, consequently, also the stronger equality [GL(n,R,A), GL(n,R,B)] = E(n,R,AB) hold whenever R is a Dedekind ring of arithmetic type with infinite multiplicative group. The proof is based on elementary calculations in the spirit of the previous papers by Wilberd van der Kallen, Roozbeh Hazrat, Zuhong Zhang, Alexei Stepanov, and the author, and also on an explicit computation of the multirelative SK1 from the author’s paper of 1982, which, in its turn, relied on very deep arithmetical results by Jean-Pierre Serre and Leonid Vaserstein (as corrected by Armin Leutbecher and Bernhard Liehl). Bibliography: 50 titles.
- Published
- 2020
11. Delone sets in ℝ3: Regularity Conditions
- Author
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N. P. Dolbilin
- Subjects
Statistics and Probability ,Discrete mathematics ,Euclidean space ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Delone set ,01 natural sciences ,Identity (music) ,010305 fluids & plasmas ,Set (abstract data type) ,0103 physical sciences ,Homogeneous space ,Mathematics::Metric Geometry ,0101 mathematics ,Symmetry (geometry) ,Orbit (control theory) ,Link (knot theory) ,Mathematics - Abstract
A regular system is a Delone set in Euclidean space with a transitive group of symmetries or, in other words, the orbit of a crystallographic group. The local theory for regular systems, created by the geometric school of B. N. Delone, was aimed, in particular, to rigorously establish the “local-global-order” link, i.e., the link between the arrangement of a set around each of its points and symmetry/regularity of the set as a whole. The main result of this paper is a proof of the so-called 10R-theorem. This theorem asserts that identity of neighborhoods within a radius 10R of all points of a Delone set (in other words, an (r, R)-system) in 3D Euclidean space implies regularity of this set. The result was obtained and announced long ago independently by M. Shtogrin and the author of this paper. However, a detailed proof remains unpublished for many years. In this paper, we give a proof of the 10R-theorem. In the proof, we use some recent results of the author, which simplify the proof.
- Published
- 2020
12. On Finiteness Conditions in Twisted K-Theory
- Author
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M. A. Gerasimova
- Subjects
Statistics and Probability ,Pure mathematics ,Statement (logic) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Connection (vector bundle) ,Lie group ,Twisted K-theory ,01 natural sciences ,010305 fluids & plasmas ,Elliptic operator ,Mathematics::K-Theory and Homology ,Bundle ,0103 physical sciences ,0101 mathematics ,Special case ,Mathematics - Abstract
The aim of this (mostly expository) article is to show a connection between the finiteness conditions arising in twisted K-theory. There are two different conditions arising naturally in two main approaches to the problem of computing the index of the appropriate family of elliptic operators (the approach of Nistor and Troitsky and the approach of Mathai, Melrose, and Singer). These conditions are formulated absolutely differently, but in some sense they should be close to each other. In this paper, we find this connection and prove the corresponding formal statement. Thereby it is shown that these conditions map to each other. This opens a possibility to synthesize these approaches. It is also shown that the finiteness condition arising in the paper of Nistor and Troitsky is a special case of the finiteness condition that appears in the paper of Emerson and Meyer, where the theorem of Nistor and Troitsky is proved not only for the case of a bundle of Lie groups, but also for the case of a general groupoid.
- Published
- 2020
13. Programmed Control with Probability 1 for Stochastic Dynamical Systems
- Author
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E. V. Karachanskaya
- Subjects
Statistics and Probability ,Dynamical systems theory ,Differential equation ,Process (engineering) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Type (model theory) ,01 natural sciences ,Invariant theory ,010305 fluids & plasmas ,Set (abstract data type) ,Control theory ,0103 physical sciences ,Applied mathematics ,0101 mathematics ,Diffusion (business) ,Mathematics - Abstract
In this paper, we suggest a new type of tasks for control theory for stochastic dynamical systems — programmed control with Probability 1 (PCP1). PCP1 is an application of an invariant theory. We use the PCP1 concept for dynamical processes described by a system of Ito differential equations with jump-diffusion (GSDES). The considered equations include the drift, the diffusion, and the jumps, together or not. Features of our approach are both a wide set of dynamical systems and investigation of such systems for their unique trajectories. Our method is based on the concept of a stochastic first integral (SFI) for GSDES and its equations which author studied before. The purpose of the present paper is to construct a differential equation system (both stochastic and deterministic) using a known set of FIs for the investigating process. Several examples are given.
- Published
- 2020
14. On the Structure of a 3-Connected Graph. 2
- Author
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D. V. Karpov
- Subjects
Statistics and Probability ,Hypergraph ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Structure (category theory) ,Type (model theory) ,01 natural sciences ,010305 fluids & plasmas ,Set (abstract data type) ,Combinatorics ,0103 physical sciences ,Decomposition (computer science) ,Graph (abstract data type) ,0101 mathematics ,Connectivity ,Hyperbolic tree ,Mathematics - Abstract
In this paper, the structure of relative disposition of 3-vertex cutsets in a 3-connected graph is studied. All such cutsets are divided into structural units – complexes of flowers, of cuts, of single cutsets, and trivial complexes. The decomposition of the graph by a complex of each type is described in detail. It is proved that for any two complexes C1 and C2 of a 3-connected graph G there is a unique part of the decomposition of G by C1 that contains C2. The relative disposition of complexes is described with the help of a hypertree T (G) – a hypergraph any cycle of which is a subset of a certain hyperedge. It is also proved that each nonempty part of the decomposition of G by the set of all of its 3-vertex cutsets is either a part of the decomposition of G by one of the complexes or corresponds to a hyperedge of T (G). This paper can be considered as a continuation of studies begun in the joint paper by D. V. Karpov and A. V. Pastor “On the structure of a 3-connected graph,” published in 2011. Bibliography: 10 titles.
- Published
- 2020
15. The Simulation of Finite-Source Retrial Queueing Systems with Collisions and Blocking
- Author
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János Sztrik, Attila Kuki, Ádám Tóth, Tamás Bérczes, and Wolfgang Schreiner
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Statistics and Probability ,Queueing theory ,Mathematical optimization ,Exponential distribution ,Queue management system ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Response time ,Variance (accounting) ,Blocking (statistics) ,01 natural sciences ,010305 fluids & plasmas ,0103 physical sciences ,Orbit (dynamics) ,0101 mathematics ,Random variable ,Mathematics - Abstract
This paper investigates, using a simulation program, a retrial queuing system with a single server which is subject to random breakdowns. The number of sources of calls is finite, and collisions can take place. We assume that the failure of the server blocks the system’s operation such that newly arriving customers cannot enter the system, contrary to an earlier paper where the failure does not affect the arrivals. All the random variables included in the model construction are assumed to be independent of each other, and all times are exponentially distributed except for the service time, which is gamma distributed. The novelty of this analysis is the inspection of the blocking effect on the performance measures using different distributions. Various figures represent the impact of the squared coefficient of the variation of the service time on the main performance measures such as the mean and variance of the number of customers in the system, the mean and variance of the response time, the mean and variance of the time a customer spends in the service, and the mean and variance of the sojourn time in the orbit.
- Published
- 2020
16. The Inverse Ill-Posed Problem of Magnetoencephalography
- Author
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T. V. Zakharova
- Subjects
Statistics and Probability ,Well-posed problem ,Quantitative Biology::Neurons and Cognition ,Series (mathematics) ,medicine.diagnostic_test ,Applied Mathematics ,General Mathematics ,Physics::Medical Physics ,010102 general mathematics ,Stability (learning theory) ,Inverse ,Magnetoencephalography ,Inverse problem ,01 natural sciences ,010305 fluids & plasmas ,Spherical model ,Noise ,0103 physical sciences ,medicine ,Applied mathematics ,0101 mathematics ,Mathematics - Abstract
This paper continues a series of studies dealing with noninvasive preoperative methods for localizing eloquent areas of the human brain. The inverse problem of magnetoencephalography (MEG) is illposed and difficult for both analytical and numerical solutions. An analytical formula is derived for the solution of the forward problem that computes the magnetic field on the surface of the head from the known location and orientation of a current dipole in the low-frequency approximation in the spherical model. In addition, the paper considers the question of stability of solutions of the inverse problem of MEG to the effect of noise. The solution is unstable to the effect of noise on its angular component, but the deviation from the true solution is much less than the noise variance.
- Published
- 2020
17. The Wiener Measure on the Heisenberg Group and Parabolic Equations
- Author
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S. V. Mamon
- Subjects
Statistics and Probability ,Pure mathematics ,Semigroup ,Stochastic process ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Markov process ,01 natural sciences ,Measure (mathematics) ,010305 fluids & plasmas ,Nilpotent ,symbols.namesake ,0103 physical sciences ,Path integral formulation ,Lie algebra ,symbols ,Heisenberg group ,0101 mathematics ,Mathematics - Abstract
In this paper, we study questions related to the theory of stochastic processes on Lie nilpotent groups. In particular, we consider the stochastic process on the Heisenberg group H3(ℝ) whose trajectories satisfy the horizontal conditions in the stochastic sense relative to the standard contact structure on H3 (ℝ). It is shown that this process is a homogeneous Markov process relative to the Heisenberg group operation. There was found a representation in the form of a Wiener integral for a one-parameter linear semigroup of operators for which the Heisenberg sublaplacian generated by basis vector fields of the corresponding Lie algebra L(H3) is producing. The main method of solving the problem in this paper is using the path integrals technique, which indicates the common direction of further development of the results.
- Published
- 2020
18. Products of Commutators on a General Linear Group Over a Division Algebra
- Author
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Nikolai Gordeev and E. A. Egorchenkova
- Subjects
Statistics and Probability ,Group (mathematics) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Center (category theory) ,General linear group ,Field (mathematics) ,01 natural sciences ,010305 fluids & plasmas ,Combinatorics ,0103 physical sciences ,Division algebra ,0101 mathematics ,Word (group theory) ,Mathematics - Abstract
The word maps $$ \tilde{w}:\kern0.5em {\mathrm{GL}}_m{(D)}^{2k}\to {\mathrm{GL}}_n(D) $$ and $$ \tilde{w}:\kern0.5em {D}^{\ast 2k}\to {D}^{\ast } $$ for a word $$ w=\prod \limits_{i=1}^k\left[{x}_i,{y}_i\right], $$ where D is a division algebra over a field K, are considered. It is proved that if $$ \tilde{w}\left({D}^{\ast 2k}\right)=\left[{D}^{\ast },{D}^{\ast}\right], $$ then $$ \tilde{w}\left({\mathrm{GL}}_n(D)\right)\supset {E}_n(D)\backslash Z\left({E}_n(D)\right), $$ where En(D) is the subgroup of GLn(D), generated by transvections, and Z(En(D)) is its center. Furthermore if, in addition, n > 2, then $$ \tilde{w}\left({E}_n(D)\right)\supset {E}_n(D)\backslash Z\left({E}_n(D)\right). $$ The proof of the result is based on an analog of the “Gauss decomposition with prescribed semisimple part” (introduced and studied in two papers of the second author with collaborators) in the case of the group GLn(D), which is also considered in the present paper.
- Published
- 2019
19. Extremal decomposition of a multidimensional complex space for five domains
- Author
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Yaroslav Zabolotnii and I. V. Denega
- Subjects
Statistics and Probability ,Pure mathematics ,Geometric function theory ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,01 natural sciences ,010305 fluids & plasmas ,symbols.namesake ,Unit circle ,Complex space ,Product (mathematics) ,Green's function ,0103 physical sciences ,Simply connected space ,Decomposition (computer science) ,symbols ,0101 mathematics ,Quadratic differential ,Mathematics - Abstract
The paper is devoted to one open extremal problem in the geometric function theory of complex variables associated with estimates of a functional defined on the systems of non-overlapping domains. We consider the problem of the maximum of a product of inner radii of n non-overlapping domains containing points of a unit circle and the power γ of the inner radius of a domain containing the origin. The problem was formulated in 1994 in Dubinin’s paper in the journal “Russian Mathematical Surveys” in the list of unsolved problems and then repeated in his monograph in 2014. Currently, it is not solved in general. In this paper, we obtained a solution of the problem for five simply connected domains and power γ ∈ (1; 2:57] and generalized this result to the case of multidimensional complex space.
- Published
- 2019
20. Comparison of Asymptotic and Numerical Approaches to the Study of the Resonant Tunneling in Two-Dimensional Symmetric Quantum Waveguides of Variable Cross-Sections
- Author
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M. M. Kabardov, N. M. Sharkova, O. V. Sarafanov, and Boris Plamenevskii
- Subjects
Statistics and Probability ,Helmholtz equation ,Applied Mathematics ,General Mathematics ,Numerical analysis ,010102 general mathematics ,Mathematical analysis ,01 natural sciences ,010305 fluids & plasmas ,Matrix (mathematics) ,Resonator ,0103 physical sciences ,Boundary value problem ,0101 mathematics ,Wave function ,Quantum ,Quantum tunnelling ,Mathematics - Abstract
The waveguide considered coincides with a strip having two narrows of width e. An electron wave function satisfies the Dirichlet boundary value problem for the Helmholtz equation. The part of the waveguide between the narrows serves as a resonator, and conditions for the electron resonant tunneling may occur. In the paper, asymptotic formulas as e → 0 for characteristics of the resonant tunneling are used. The asymptotic results are compared with the numerical ones obtained by approximate calculation of the scattering matrix for energies in the interval between the second and third thresholds. The comparison allows us to state an interval of e, where the asymptotic and numerical approaches agree. The suggested methods can be applied to more complicated models than that considered in the paper. In particular, the same approach can be used for asymptotic and numerical analysis of the tunneling in three-dimensional quantum waveguides of variable cross-sections. Bibliography: 3 titles.
- Published
- 2019
21. On Justification of the Asymptotics of Eigenfunctions of the Absolutely Continuous Spectrum in the Problem of Three One-Dimensional Short-Range Quantum Particles with Repulsion
- Author
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S. B. Levin, A. M. Budylin, and I. V. Baibulov
- Subjects
Statistics and Probability ,Scattering ,Applied Mathematics ,General Mathematics ,Operator (physics) ,010102 general mathematics ,Spectrum (functional analysis) ,Eigenfunction ,Absolute continuity ,01 natural sciences ,010305 fluids & plasmas ,symbols.namesake ,0103 physical sciences ,symbols ,0101 mathematics ,Quantum ,Schrödinger's cat ,Resolvent ,Mathematics ,Mathematical physics - Abstract
The present paper offers a new approach to the construction of the coordinate asymptotics of the kernel of the resolvent of the Schrodinger operator in the scattering problem of three onedimensional quantum particles with short-range pair potentials. Within the framework of this approach, the asymptotics of eigenfunctions of the absolutely continuous spectrum of the Schrodinger operator can be constructed. In the paper, the possibility of a generalization of the suggested approach to the case of the scattering problem of N particles with arbitrary masses is discussed.
- Published
- 2019
22. Duality and Free Measures in Vector Spaces, the Spectral Theory of Actions of Non-Locally Compact Groups
- Author
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Anatoly Vershik
- Subjects
Statistics and Probability ,Pure mathematics ,Measurable function ,Applied Mathematics ,General Mathematics ,Probability (math.PR) ,010102 general mathematics ,Duality (mathematics) ,22D25, 22D40, 28O15, 46A22, 60B11 ,Space (mathematics) ,01 natural sciences ,Measure (mathematics) ,Linear subspace ,Functional Analysis (math.FA) ,010305 fluids & plasmas ,Mathematics - Functional Analysis ,Vector measure ,0103 physical sciences ,FOS: Mathematics ,Locally compact space ,0101 mathematics ,Mathematics - Probability ,Mathematics ,Vector space - Abstract
The paper presents a general duality theory for vector measure spaces taking its origin in the author's papers written in the 1960s. The main result establishes a direct correspondence between the geometry of a measure in a vector space and the properties of the space of measurable linear functionals on this space regarded as closed subspaces of an abstract space of measurable functions. An example of useful new features of this theory is the notion of a free measure and its applications., Comment: 20 pp.23 Ref
- Published
- 2019
23. Performance Simulation of Finite-Source Cognitive Radio Networks with Servers Subjects to Breakdowns and Repairs
- Author
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Hamza Nemouchi and János Sztrik
- Subjects
Statistics and Probability ,Primary channel ,Queueing theory ,business.industry ,Network packet ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,01 natural sciences ,010305 fluids & plasmas ,Task (computing) ,Channel (programming) ,Server ,0103 physical sciences ,0101 mathematics ,Priority queue ,business ,Queue ,Mathematics ,Computer network - Abstract
The present paper deals with the performance evaluation of a cognitive radio network with the help of a queueing model. The queueing system contains two interconnected, not independent sub-systems. The first part is for the requests of the Primary Units (PU). The number of sources is finite, and each source generates high priority requests after a exponentially distributed time. The requests are sent to a single server unit or Primary Channel Service (PCS) with a preemptive priority queue. The service times are assumed to be exponentially distributed. The second sub-system is for the requests of the Secondary Units (SU), which is finite sources system too; the inter-request times and service times of the single server unit or Secondary system Channel Service (SCS) are assumed to be exponentially distributed, respectively. A generated high priority packet goes to the primary service unit. If the unit is idle, the service of the packet begins immediately. If the server is busy with a high priority request, the packet joins the preemptive priority queue. When the unit is engaged with a request from SUs, the service is interrupted and the interrupted low priority task is sent back to the SCS. Depending on the state of the secondary channel, the interrupted job is directed to either the server or the orbit. In case the requests from SUs find the SCS idle, the service starts, and if the SCS is busy, the packet looks for the PCS. In the case of an idle PCS, the service of the low-priority packet begins at the high-priority channel (PCS). If the PCS is busy, the packet goes to the orbit. From the orbit it retries to be served after an exponentially distributed time. The novelty of our investigation is that each server is subject to random breakdowns, in which case the interrupted request is sent to the queue or orbit, respectively. The operating and repair times of the servers are assumed to be generally distributed. Finally, all the random times included in the model construction are assumed to be independent of each other. The main aim of the paper is to analyze the effect of the nonreliability of the servers on the mean and variance of the response time for the SUs by using simulation.
- Published
- 2019
24. The Kostrikin Radical and Similar Radicals of Lie Algebras
- Author
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A. Yu. Golubkov
- Subjects
Statistics and Probability ,Pure mathematics ,Degree (graph theory) ,Applied Mathematics ,General Mathematics ,Radical ,010102 general mathematics ,Zero (complex analysis) ,01 natural sciences ,010305 fluids & plasmas ,0103 physical sciences ,Lie algebra ,0101 mathematics ,Algebraic number ,Mathematics - Abstract
The existing notion of the Kostrikin radical as a radical in the Kurosh–Amitsur sense on classes of Mal’tsev algebras over rings with 1/6 is not completely justified. More precisely, to the fullest extent it is true for classes of Lie algebras over fields of characteristic zero and, as shown in the given paper, classes of algebraic Lie algebras of degree not greater than n over rings with 1/n! at all n ≥ 1. Similar conclusions are obtained in the paper also for the Jordan, regular, and extremal radicals constructed analogously to the Kostrikin radical.
- Published
- 2019
25. Continuability and Boundedness of Solutions for a Kind of Nonlinear Delay Integrodifferential Equations of the Third Order
- Author
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Timur Ayhan and Cemil Tunç
- Subjects
Statistics and Probability ,Lyapunov function ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,MathematicsofComputing_NUMERICALANALYSIS ,01 natural sciences ,010305 fluids & plasmas ,Nonlinear system ,symbols.namesake ,Third order ,0103 physical sciences ,symbols ,Applied mathematics ,0101 mathematics ,Mathematics - Abstract
In the paper, we consider a nonlinear integrodifferential equation of the third order with delay. We establish sufficient conditions guaranteeing the global existence and boundedness of the solutions of the analyzed equation. We use the Lyapunov second method to prove the main result. An example is also given to illustrate the applicability of our result. The result of this paper is new and improves previously known results.
- Published
- 2018
26. On Riesz Means of the Coefficients of Epstein’s Zeta Functions
- Author
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O. M. Fomenko
- Subjects
Statistics and Probability ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Generating function ,Type (model theory) ,01 natural sciences ,Omega ,010305 fluids & plasmas ,Riemann zeta function ,Combinatorics ,symbols.namesake ,Riesz mean ,0103 physical sciences ,symbols ,0101 mathematics ,Mathematics - Abstract
Let rk(n) denote the number of lattice points on a k-dimensional sphere of radius $$ \sqrt{n} $$ . The generating function $$ {\zeta}_k(s)=\sum \limits_{n=1}^{\infty }{r}_k(n){n}^{-s},\kern0.5em k\ge 2, $$ is Epstein’s zeta function. The paper considers the Riesz mean of the type $$ {D}_{\rho}\left(x;{\zeta}_3\right)=\frac{1}{\Gamma \left(\rho +1\right)}\sum \limits_{n\le x}{\left(x-n\right)}^{\rho }{r}_3(n), $$ where ρ > 0; the error term Δρ(x; ζ3) is defined by $$ {D}_{\rho}\left(x;{\zeta}_3\right)=\frac{\uppi^{3/2}{x}^{\rho +3/2}}{\Gamma \left(\rho +5/2\right)}+\frac{x^{\rho }}{\Gamma \left(\rho +1\right)}{\zeta}_3(0)+{\Delta}_{\rho}\left(x;{\zeta}_3\right). $$ K. Chandrasekharan and R. Narasimhan (1962, MR25#3911) proved that $$ {\Delta}_{\rho}\left(x;{\zeta}_3\right)=\Big\{{\displaystyle \begin{array}{ll}O\Big({x}^{1/2+\rho /2\Big)}& \left(\rho >1\right),\\ {}{\Omega}_{\pm}\left({x}^{1/2+\rho /2}\right)& \left(\rho \ge 0\right).\end{array}} $$ In the present paper, it is proved that $$ {\Delta}_{\rho}\left(x;{\zeta}_3\right)=\Big\{{\displaystyle \begin{array}{ll}O\left(x\log x\right)& \left(\rho =1\right),\\ {}O\left({x}^{2/3+\rho /3+\varepsilon}\right)& \left(1/2
- Published
- 2018
27. Quadratic Interaction Estimate for Hyperbolic Conservation Laws: an Overview
- Author
-
Stefano Modena
- Subjects
Statistics and Probability ,Conservation law ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,01 natural sciences ,Prime (order theory) ,Interaction time ,Combinatorics ,Quadratic equation ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
In a joint work with S. Bianchini [8] (see also [6, 7]), we proved a quadratic interaction estimate for the system of conservation laws $$ \left\{\begin{array}{l}{u}_t+f{(u)}_x=0,\\ {}u\left(t=0\right)={u}_0(x),\end{array}\right. $$ where u : [0, ∞) × ℝ → ℝn, f : ℝn → ℝn is strictly hyperbolic, and Tot.Var.(u0) ≪ 1. For a wavefront solution in which only two wavefronts at a time interact, such an estimate can be written in the form $$ \sum \limits_{t_j\;\mathrm{interaction}\ \mathrm{time}}\frac{\left|\sigma \left({\alpha}_j\right)-\sigma \left({\alpha}_j^{\prime}\right)\right|\left|{\alpha}_j\right|\left|{\alpha}_j^{\prime}\right|}{\left|{\alpha}_j\right|+\left|{\alpha}_j^{\prime}\right|}\le C(f)\mathrm{Tot}.\mathrm{Var}.{\left({u}_0\right)}^2, $$ where αj and $$ {\alpha}_j^{\prime } $$ are the wavefronts interacting at the interaction time tj, σ(·) is the speed, |·| denotes the strength, and C(f) is a constant depending only on f (see [8, Theorem 1.1] or Theorem 3.1 in the present paper for a more general form). The aim of this paper is to provide the reader with a proof for such a quadratic estimate in a simplified setting, in which: • all the main ideas of the construction are presented; • all the technicalities of the proof in the general setting [8] are avoided.
- Published
- 2018
28. On Short-Wave Diffraction by an Elongated Body. Numerical Experiments
- Author
-
N. Ya. Kirpichnikova, M. M. Popov, and N. M. Semtchenok
- Subjects
Statistics and Probability ,Diffraction ,Field (physics) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Rotational symmetry ,01 natural sciences ,Fock space ,Boundary layer ,Exact solutions in general relativity ,0103 physical sciences ,0101 mathematics ,Convex function ,Asymptotic expansion ,010301 acoustics ,Mathematics - Abstract
The paper is a continuation of previous papers of the authors dealing with the exploration of shortwave diffraction by smooth and strictly convex bodies of revolution (the axisymmetric case). In these problems, the boundary layer method contains two large parameters: one is the Fock parameter M and the second is Λ that characterizes the oblongness of the scatterer. This naturally gives the possibility of using the two-scaled asymptotic expansion, where both M and Λ are regarded as independent. The approximate formulas for the wave field in this situation depend on the mutual strength between the large parameters and may vary. In the paper, we carry out numerical experiments with our formulas, in the case where the Fock analytical solution is in good coincidence with the exact solution of a model problem, in order to examine the influence of the parameter Λ on the wave field. It follows from our numerical experiments that the influence of the oblongness of the scatterer on the wave field is really insignificant if the method of Leontovich–Fock parabolic equation does not meet mathematical difficulties.
- Published
- 2017
29. Scattering of an Electromagnetic Surface Wave from a Hertzian Dipole by the Edge of an Impedance Wedge
- Author
-
N. Ya. Zhu and Mikhail A. Lyalinov
- Subjects
Statistics and Probability ,Diffraction ,Scattering ,business.industry ,Point source ,Applied Mathematics ,General Mathematics ,020206 networking & telecommunications ,02 engineering and technology ,01 natural sciences ,Wedge (geometry) ,Dipole ,Edge wave ,Optics ,Surface wave ,0103 physical sciences ,0202 electrical engineering, electronic engineering, information engineering ,Wave impedance ,business ,010301 acoustics ,Mathematics - Abstract
In this paper, extensions of the results obtained in our previous paper devoted to the diffraction of waves from a Hertzian dipole (point source) located over an impedance wedge are presented. The surface waves components and the edge wave, produced by a surface wave from a point source coming to the edge, are discussed. The geometric optics laws for the surface waves reflected by and transmitted across the edge are also addressed.
- Published
- 2017
30. Special Representations of the Iwasawa Subgroups of Simple Lie Groups
- Author
-
M. I. Graev and Anatoly Vershik
- Subjects
Statistics and Probability ,Pure mathematics ,Applied Mathematics ,General Mathematics ,Simple Lie group ,010102 general mathematics ,Lie group ,01 natural sciences ,Representation of a Lie group ,Group of Lie type ,Representation theory of SU ,Simple group ,0103 physical sciences ,Fundamental representation ,Maximal torus ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
In the paper, a family of representations of maximal solvable subgroups of the simple Lie groups O(p, q), U(p, q), and Sp(p, q), where 1 ≤ p ≤ q, is introduced. These subgroups are called the Iwasawa subgroups of the corresponding simple groups. The main property of these representations is the existence of nontrivial 1-cohomology with values in the representations. For groups of rank 1, the representations from this family are unitary; for ranks greater than 1, they are nonunitary. The paper continues a series of our previous papers and serves as an introduction to the theory of nonunitary current groups.
- Published
- 2017
31. The Prime Radical of Alternative Rings and Loops
- Author
-
A. V. Gribov
- Subjects
Statistics and Probability ,Ring (mathematics) ,Loop (graph theory) ,Pure mathematics ,Mathematics::Number Theory ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,01 natural sciences ,Prime (order theory) ,0103 physical sciences ,Radical of an ideal ,010307 mathematical physics ,Physics::Chemical Physics ,0101 mathematics ,Connection (algebraic framework) ,Mathematics - Abstract
A characterization of the prime radical of loops as the set of strongly Engel elements was given in our earlier paper. In this paper, some properties of the prime radical of loops are considered. Also a connection between the prime radical of the loop of units of an alternative ring and the prime radical of this ring is given.
- Published
- 2017
32. The BV-Algebra Structure on the Hochschild Cohomology of Local Algebras of Quaternion Type in Characteristic 2
- Author
-
A. A. Ivanov
- Subjects
Statistics and Probability ,Algebraic structure ,Group (mathematics) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Gerstenhaber algebra ,Mathematics::Algebraic Topology ,01 natural sciences ,Noncommutative geometry ,Cohomology ,Algebra ,Mathematics::K-Theory and Homology ,0103 physical sciences ,Equivariant cohomology ,010307 mathematical physics ,0101 mathematics ,Quaternion ,Tame group ,Mathematics - Abstract
This paper is a sequel of the joint paper by the author with S. O. Ivanov, Yu. Volkov, and G. Zhou. In the present paper, the BV -structure, and therefore, the Gerstenhaber algebra structure on the Hochschild cohomology of local algebras of generalized quaternion type is completely described over a field of characteristic 2. The family of algebras under investigation contains group algebras of generalized quaternion groups for which the case of characteristic 2 is the only one where the calculation of Hochschild cohomology and structures on it is a highly nontrivial problem. Also the group algebras of generalized quaternion groups represent classes of Morita-equivalence of tame group blocks from K. Erdmann’s classification. In particular, the BV -structure on the Hochschild cohomology of group algebras of some noncommutative groups is described.
- Published
- 2016
33. Arithmetic of Fuzzy Numbers in Generalized Trapezoidal Form
- Author
-
V. A. Ivanyuk, M. V. Koroteev, and P. V. Terelyanskii
- Subjects
Statistics and Probability ,Decision support system ,Fuzzy classification ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,01 natural sciences ,Fuzzy logic ,010305 fluids & plasmas ,0103 physical sciences ,Fuzzy number ,Fuzzy set operations ,0101 mathematics ,Arithmetic ,Mathematics - Abstract
In this paper, we describe several types of fuzzy arithmetics and discuss their applicability to fuzzy numbers in arbitrary and generalized trapezoidal forms. The methods of calculation of results of arithmetic operations described in the paper can be used for mathematical modeling of economical processes and the creation of decision support systems.
- Published
- 2016
34. To the History of the Appearance of the Notion of the ε-Entropy of an Authomorphism of a Lebesgue Space and (ε,T)-Entropy of a Dynamical System with Continuous Time
- Author
-
D. Z. Arov
- Subjects
Statistics and Probability ,Discrete mathematics ,Dynamical systems theory ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Automorphism ,01 natural sciences ,Separable space ,Compact space ,0103 physical sciences ,Entropy (information theory) ,Standard probability space ,Ergodic theory ,010307 mathematical physics ,Invariant measure ,0101 mathematics ,Mathematics - Abstract
The paper is devoted to the master thesis on “information theory” which was written by the author in 1956–57. The topic was suggested by his advisor A. A. Bobrov (a student of A. Ya. Khinchin and A. N. Kolmogorov), and the thesis was written under the influence of lectures by N. I. Gavrilov (a student of I. G. Petrovskii) on the qualitative theory of differential equations, which included the statement of Birkhoff’s theorem for ergodic dynamical systems. In the thesis, the author used the concept of Shannon entropy in the study of ergodic dynamical systems f(p, t) in a separable compact metric space R with an invariant measure μ (where μ(R) = 1) and introduced the notion of the (ϵ, T)-entropy of a system as a quantitative characteristic of the degree of mixing. In the work, not only partitions of R were considered, but also partitions of the interval (−∞,∞) into subintervals of length T > 0. In particular, f(p, T) was regarded as an automorphism S of X = R, and the (ϵ, T)-entropy is essentially the e-entropy of S. But, despite some “oversights” in the definition of the (ϵ, T)-entropy and many years that have passed, the author decided to publish the corresponding chapter of the thesis in connection with the following: 1) There is a number of papers that refer to this work in the explanation of the history of the concept of Kolmogorov’s entropy. 2) Recently, B. M. Gurevich obtained new results on the ϵ-entropy hϵ(S), which show that for two ergodic automorphisms with equal finite entropies their ϵ-entropies also coincide for all ϵ, but, on the other hand, there are unexpected nonergodic automorphisms with equal finite entropies, but different ϵ-entropies for some ϵ. This shows that the concept of ϵ-entropy is of scientific value.
- Published
- 2016
35. Graph-Links: Nonrealizability, Orientation, and Jones Polynomial
- Author
-
V. S. Safina and Denis Petrovich Ilyutko
- Subjects
Statistics and Probability ,Discrete mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Jones polynomial ,Bracket polynomial ,01 natural sciences ,Graph ,Combinatorics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Invariant (mathematics) ,MathematicsofComputing_DISCRETEMATHEMATICS ,Writhe ,Mathematics - Abstract
The present paper is devoted to graph-links with many components and consists of two parts. In the first part of the paper we classify vertices of a labeled graph according to the component they belong to. Using this classification, we construct an invariant of graph-links. This invariant shows that the labeled second Bouchet graph generates a nonrealizable graph-link. In the second part of the work we introduce the notion of an oriented graph-link. We define a writhe number for the oriented graph-link and we get an invariant of oriented graph-links, the Jones polynomial, by normalizing the Kauffman bracket with the writhe number.
- Published
- 2016
36. Descriptive Spaces and Proper Classes of Functions
- Author
-
V. K. Zakharov, A. V. Mikhalev, and T.V. Rodionov
- Subjects
Statistics and Probability ,Mathematical problem ,Measurable function ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Rigid structure ,01 natural sciences ,Algebra ,Constructed language ,0103 physical sciences ,Countable set ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
The remarkable class of measurable functions was introduced by classics of function theory. It has found different applications in various branches of mathematics. However this class turned out too restrictive for solving some natural mathematical problems because it is essentially connected with the property of countability. Therefore, along with it another remarkable class, essentially connected with the property of finiteness, was introduced. It is the class of uniform functions. Measurable functions are described both in the classical Lebesgue–Borel language of preimages and in the quite new language of covers. Uniform functions are described in the language of covers exclusively. Both the families of measurable functions and the families of uniform functions are determined by the rigid structure of their supports (descriptive spaces). For this reason, mathematicians weakened more that once the rigidity of the structure of descriptive spaces at the expense of using the additional property of negligence. The present paper is devoted to a contemporary formalization of the indicated ideas. Some applications of the introduced classes of functions to solving a number of known mathematical problems is traced in the paper.
- Published
- 2016
37. On Algorithmic Methods of Analysis of Two-Colorings of Hypergraphs
- Author
-
A. V. Lebedeva
- Subjects
Statistics and Probability ,Combinatorics ,Hypergraph ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Upper and lower bounds ,Mathematics ,Vertex (geometry) - Abstract
This paper deals with an extremal problem concerning hypergraph colorings. Let k be an integer. The problem is to find the value m k (n) equal to the minimum number of edges in an n-uniform hypergraph not admitting two-colorings of the vertex set such that every edge of the hypergraph contains at least k vertices of each color. In this paper, we obtain upper bounds of m k (n) for small k and n, the exact value of m 4(8), and a lower bound for m 3(7).
- Published
- 2016
38. sp-Groups and Their Endomorphism Rings
- Author
-
Piotr A. Krylov, A. V. Tsarev, and Askar A. Tuganbaev
- Subjects
Statistics and Probability ,Class (set theory) ,абелевы sp-группы ,Endomorphism ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,01 natural sciences ,010305 fluids & plasmas ,Combinatorics ,кольца эндоморфизмов ,0103 physical sciences ,Rank (graph theory) ,0101 mathematics ,Abelian group ,Mathematics - Abstract
sp-Groups form an interesting and informative class of Abelian mixed groups. In this paper, we systematically study self-small sp-groups of finite rank and their endomorphism rings.
- Published
- 2021
39. Modules that are Invariant with Respect to Automorphisms and Idempotent Endomorphisms of Their Hulls and Covers
- Author
-
T. C. Quynh, A. N. Abyzov, and Askar A. Tuganbaev
- Subjects
Statistics and Probability ,Pure mathematics ,Endomorphism ,Applied Mathematics ,General Mathematics ,Mathematics::Rings and Algebras ,010102 general mathematics ,Automorphism ,Mathematical proof ,01 natural sciences ,010305 fluids & plasmas ,Mathematics::K-Theory and Homology ,Mathematics::Quantum Algebra ,Hull ,0103 physical sciences ,Idempotence ,Cover (algebra) ,0101 mathematics ,Invariant (mathematics) ,Mathematics::Representation Theory ,Mathematics - Abstract
The paper contains both known and new results on automorphism-invariant modules, automorphism-coinvariant modules and modules which are invariant or coinvariant with respect to idempotent endomorphisms of its hull and its cover, respectively. The main results are given with proofs.
- Published
- 2021
40. Computable Linear Orders and Limitwise Monotonic Functions
- Author
-
A. N. Frolov and M. V. Zubkov
- Subjects
Statistics and Probability ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Monotonic function ,0101 mathematics ,01 natural sciences ,010305 fluids & plasmas ,Mathematics - Abstract
In this paper, we describe the technique of extremely monotonic functions in the theory of computable linear orders. The basic definitions of extremely monotonic functions and their generalizations are given, and a number of their basic properties and applications are presented.
- Published
- 2021
41. Categoricity Spectra of Computable Structures
- Author
-
Nikolay Bazhenov
- Subjects
Statistics and Probability ,Pure mathematics ,Degree (graph theory) ,Series (mathematics) ,Applied Mathematics ,General Mathematics ,Boolean algebra (structure) ,010102 general mathematics ,Structure (category theory) ,01 natural sciences ,Spectrum (topology) ,010305 fluids & plasmas ,Set (abstract data type) ,symbols.namesake ,ComputingMethodologies_PATTERNRECOGNITION ,Mathematics::Category Theory ,0103 physical sciences ,Index set ,symbols ,0101 mathematics ,Turing ,computer ,computer.programming_language ,Mathematics - Abstract
The categoricity spectrum of a computable structure S is the set of all Turing degrees capable of computing isomorphisms among arbitrary computable presentations of S. The degree of categoricity of S is the least degree in the categoricity spectrum of S. This paper is a survey of results on categoricity spectra and degrees of categoricity for computable structures. We focus on the results about degrees of categoricity for linear orders and Boolean algebras. We construct a new series of examples of degrees of categoricity for linear orders.
- Published
- 2021
42. Converting Column Majorization
- Author
-
Pavel Shteyner
- Subjects
Statistics and Probability ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Mathematical analysis ,Linear operators ,0101 mathematics ,Majorization ,01 natural sciences ,Column (database) ,010305 fluids & plasmas ,Mathematics - Abstract
The paper characterizes linear operators converting column majorization into weak, directional, and strong majorizations. An example of a linear converter from weak, directional, and strong majorizations to column majorization preserving none of these majorizations is provided.
- Published
- 2021
43. Length of the Group Algebra of the Dihedral Group of Order 2k
- Author
-
M. A. Khrystik and O. V. Markova
- Subjects
Statistics and Probability ,Pure mathematics ,Group (mathematics) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Field (mathematics) ,Group algebra ,Power of two ,Dihedral group ,01 natural sciences ,010305 fluids & plasmas ,0103 physical sciences ,Order (group theory) ,0101 mathematics ,Mathematics - Abstract
In this paper, the length of the group algebra of a dihedral group in the modular case is computed under the assumption that the order of the group is a power of two. Various methods for studying the length of a group algebra in the modular case are considered. It is proved that the length of the group algebra of a dihedral group of order 2k+1 over an arbitrary field of characteristic 2 is equal to 2k.
- Published
- 2021
44. Matrix Representation of Filter Banks Corresponding to Spline Wavelets with Shifted Supports
- Author
-
A. A. Makarov, S. V. Makarova, and O. M. Kosogorov
- Subjects
Statistics and Probability ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Matrix representation ,MathematicsofComputing_NUMERICALANALYSIS ,Wavelet transform ,Data_CODINGANDINFORMATIONTHEORY ,Interval (mathematics) ,01 natural sciences ,010305 fluids & plasmas ,Spline (mathematics) ,Wavelet ,Filter (video) ,0103 physical sciences ,0101 mathematics ,Matrix form ,Representation (mathematics) ,Algorithm ,Mathematics - Abstract
The paper presents a matrix representation of filter banks corresponding to spline wavelets with shifted supports. The matrix form of the decomposition and reconstruction filters simplifies the representation of nonuniform nonstationary wavelet transforms built on nonuniform grids on a finite interval. Such a representation of filters is used, for example, in constructing error-correcting codes.
- Published
- 2021
45. Parallel Variable-Triangular Iterative Methods in Krylov Subspaces
- Author
-
V. P. Il’in
- Subjects
Statistics and Probability ,Iterative method ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,MathematicsofComputing_NUMERICALANALYSIS ,01 natural sciences ,Linear subspace ,010305 fluids & plasmas ,Matrix (mathematics) ,Algebraic equation ,Orthogonality ,Factorization ,Conjugate gradient method ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,0103 physical sciences ,Applied mathematics ,0101 mathematics ,Algebraic number ,Mathematics - Abstract
The paper considers parallel preconditioned iterative methods in Krylov subspaces for solving systems of linear algebraic equations with large sparse symmetric positive-definite matrices resulting from grid approximations of multidimensional problems. For preconditioning, generalized block algorithms of symmetric successive over-relaxation or incomplete factorization type with matching row sums are used. Preconditioners are based on variable-triangular matrix factors with multiple alternations in triangular structure. For three-dimensional grid algebraic systems, methods are based on nested factorizations, as well as on two-level iterative processes. Successive approximations in Krylov subspaces are computed by applying a family of conjugate direction algorithms with various orthogonality and variational properties, including preconditioned conjugate gradient, conjugate residual, and minimal error methods.
- Published
- 2021
46. Checking the Congruence of Involutive Matrices
- Author
-
Kh. D. Ikramov
- Subjects
Statistics and Probability ,Class (set theory) ,Complex matrix ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Process (computing) ,01 natural sciences ,Unitary state ,Hermitian matrix ,010305 fluids & plasmas ,Algebra ,Matrix (mathematics) ,Congruence (geometry) ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,0103 physical sciences ,0101 mathematics ,Mathematics - Abstract
A finite computational process using arithmetic operations only is called a rational algorithm. Presently, no rational algorithm for checking the congruence of arbitrary complex matrices A and B is known. The situation can be different if both A and B belong to a special matrix class. For instance, there exist rational algorithms for the cases where both matrices are Hermitian, unitary, or accretive. This paper proposes a rational algorithm for checking whether two involutive matrices A and B are congruent.
- Published
- 2021
47. Semifinite Harmonic Functions on the Gnedin–Kingman Graph
- Author
-
Nikita Safonkin
- Subjects
Statistics and Probability ,Ring (mathematics) ,Pure mathematics ,Mathematics::Combinatorics ,Property (philosophy) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Monomial basis ,01 natural sciences ,010305 fluids & plasmas ,Harmonic function ,0103 physical sciences ,Graph (abstract data type) ,0101 mathematics ,Algebra over a field ,Mathematics::Representation Theory ,Indecomposable module ,Mathematics - Abstract
We study the Gnedin–Kingman graph, which corresponds to Pieri’s rule for the monomial basis {Mλ} in the algebra QSym of quasisymmetric functions. The paper contains a detailed announcement of results concerning the classification of indecomposable semifinite harmonic functions on the Gnedin–Kingman graph. For these functions, we also establish a multiplicativity property, which is an analog of the Vershik–Kerov ring theorem.
- Published
- 2021
48. Cliques and Constructors in 'Hats' Game. I
- Author
-
K. P. Kokhas, V. I. Retinskiy, and Aleksei Latyshev
- Subjects
Statistics and Probability ,Computer Science::Computer Science and Game Theory ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,ComputingMilieux_PERSONALCOMPUTING ,Construct (python library) ,Function (mathematics) ,Basis (universal algebra) ,01 natural sciences ,Graph ,010305 fluids & plasmas ,Combinatorics ,Colored ,0103 physical sciences ,0101 mathematics ,Mathematics - Abstract
The following general variant of deterministic “Hats” game is analyzed. Several sages wearing colored hats occupy the vertices of a graph, the kth sage can have hats of one of h(k) colors. Each sage tries to guess the color of his own hat merely on the basis of observing the hats of his neighbors without exchanging any information. A predetermined guessing strategy is winning if it guarantees at least one correct individual guess for every assignment of colors. For complete graphs and cycles, the problem of describing the function h(k) for which the sages win is solved in the present paper. A “theory of constructors,” i.e., a collection of theorems demonstrating how one can construct new graphs for which the sages win is developed. A new game “Rook check ” equivalent to the Hats game on a 4-cycle is introduced and completely analyzed.
- Published
- 2021
49. On Vertices of Degree 6 of Minimal and Contraction Critical 6-Connected Graph
- Author
-
A. V. Pastor
- Subjects
Statistics and Probability ,Degree (graph theory) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Edge (geometry) ,01 natural sciences ,Graph ,010305 fluids & plasmas ,Combinatorics ,0103 physical sciences ,Fraction (mathematics) ,0101 mathematics ,Contraction (operator theory) ,Connectivity ,Mathematics - Abstract
The goal of the paper is to study vertices of degree 6 of minimal and contraction critical 6-connected graph, i.e., a 6-connected graph that looses 6-connectivity both upon removal and upon contraction of any edge. It is proved that if x and z are adjacent vertices of degree 6, then x and z have at least 4 common neighbors. In addition, a detailed description of the neighborhood of the set {x, z} is given. An infinite series of examples of minimal and contraction critical 6-connected graphs for which the fraction of vertices of degree 6 equals $$ \frac{11}{17} $$ is constructed.
- Published
- 2021
50. On The Solvability of the Cauchy Problem for a Certain Class of Multidimensional Loaded Parabolic Equations
- Author
-
Igor V. Frolenkov and E. N. Kriger
- Subjects
Statistics and Probability ,Class (set theory) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematics::Analysis of PDEs ,Cauchy distribution ,Inverse problem ,01 natural sciences ,Parabolic partial differential equation ,010305 fluids & plasmas ,0103 physical sciences ,Initial value problem ,Applied mathematics ,0101 mathematics ,Mathematics - Abstract
In this paper, we examine the solvability of a new class of nonclassical direct problems for multidimensional loaded parabolic equations with Cauchy data. We obtain sufficient conditions for the solvability of the problem; the proof is based on the method of weak approximation. By an example, we demonstrate the application of the theorem proved to the study of inverse problems for multidimensional parabolic equations with Cauchy data.
- Published
- 2021
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