81 results
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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. 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
- Subjects
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
4. 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
5. 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
6. 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 $$ {\Delta}_{\rho}\left(x;{\zeta}_3\right)=\Big\{{\displaystyle \begin{array}{ll}O\left(x\log x\right)& \left(\rho =1\right),\\ {}O\left({x}^{2/3+\rho /3+\varepsilon}\right)& \left(1/2
- Published
- 2018
7. Quadratic Interaction Estimate for Hyperbolic Conservation Laws: an Overview
- Author
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Stefano Modena
- Subjects
Statistics and Probability ,Conservation law ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,01 natural sciences ,Prime (order theory) ,Interaction time ,Combinatorics ,Quadratic equation ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
In a joint work with S. Bianchini [8] (see also [6, 7]), we proved a quadratic interaction estimate for the system of conservation laws $$ \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
8. Regularity of Maximum Distance Minimizers
- Author
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Yana Teplitskaya
- Subjects
Statistics and Probability ,Class (set theory) ,Applied Mathematics ,General Mathematics ,Pipeline (computing) ,010102 general mathematics ,01 natural sciences ,Steiner tree problem ,010101 applied mathematics ,Set (abstract data type) ,Combinatorics ,symbols.namesake ,Compact space ,Tangent lines to circles ,symbols ,Hausdorff measure ,Point (geometry) ,0101 mathematics ,Mathematics - Abstract
We study properties of sets having the minimum length (one-dimensional Hausdorff measure) in the class of closed connected sets Σ ⊂ ℝ2 satisfying the inequality max yϵM dist (y, Σ) ≤ r for a given compact set M ⊂ ℝ2 and given r > 0. Such sets play the role of the shortest possible pipelines arriving at a distance at most r to every point of M where M is the set of customers of the pipeline. In this paper, it is announced that every maximum distance minimizer is a union of finitely many curves having one-sided tangent lines at every point. This shows that a maximum distance minimizer is isotopic to a finite Steiner tree even for a “bad” compact set M, which distinguishes it from a solution of the Steiner problem (an example of a Steiner tree with infinitely many branching points can be found in a paper by Paolini, Stepanov, and Teplitskaya). Moreover, the angle between these lines at each point of a maximum distance minimizer is at least 2π/3. Also, we classify the behavior of a minimizer Σ in a neighborhood of any point of Σ. In fact, all the results are proved for a more general class of local minimizers.
- Published
- 2018
9. Bounded Remainder Sets
- Author
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V. G. Zhuravlev
- Subjects
Statistics and Probability ,Sequence ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Torus ,Space (mathematics) ,01 natural sciences ,Bounded operator ,010101 applied mathematics ,Combinatorics ,Distribution function ,Bounded function ,0101 mathematics ,Remainder ,Klein bottle ,Mathematics - Abstract
The paper considers the category ( $$ \mathcal{T} $$ , S, X) consisting of mappings S : $$ \mathcal{T} $$ −→ $$ \mathcal{T} $$ of spaces $$ \mathcal{T} $$ with distinguished subsets X ⊂ $$ \mathcal{T} $$ . Let rX (i, x0) be the distribution function of points of an S-orbit x0, x1 = S(x0), . . . , xi−1 = Si−1(x0) getting into X, and let δX (i, x0) be the deviation defined by the equation rX (i, x0) = aX i + δX (i, x0), where aX i is the average value. If δX (i, x0) = O(1), then such sets X are called bounded remainder sets. In the paper, bounded remainder sets X are constructed in the following cases: (1) the space $$ \mathcal{T} $$ is the circle, torus, or the Klein bottle; (2) the map S is a rotation of the circle, a shift or an exchange mapping of the torus; (3) X is a fixed subset X ⊂ $$ \mathcal{T} $$ or a sequence of subsets depending on the iteration number i = 0, 1, 2, . . .. Bibliography: 27 titles.
- Published
- 2017
10. On the Lattice of Subvarieties of the Wreath Product of the Variety of Semilattices and the Variety of Semigroups with Zero Multiplication
- Author
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A. V. Tishchenko
- Subjects
Statistics and Probability ,Semigroup ,High Energy Physics::Lattice ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Integer lattice ,01 natural sciences ,Upper and lower bounds ,Exponential function ,010101 applied mathematics ,Combinatorics ,Wreath product ,Lattice (order) ,0101 mathematics ,Mathematics - Abstract
It is known that the monoid wreath product of any two semigroup varieties that are atoms in the lattice of all semigroup varieties may have a finite as well as an infinite lattice of subvarieties. If this lattice is finite, then as a rule it has at most eleven elements. This was proved in a paper of the author in 2007. The exclusion is the monoid wreath product Sl w N 2 of the variety of semilattices and the variety of semigroups with zero multiplication. The number of elements of the lattice L(Sl w N 2) of subvarieties of Sl w N 2 is still unknown. In our paper, we show that the lattice L(Sl w N 2) contains no less than 33 elements. In addition, we give some exponential upper bound of the cardinality of this lattice.
- Published
- 2017
11. Some Results of the Theory of Exponential R-Groups
- Author
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M. G. Amaglobeli and T. Bokelavadze
- Subjects
Statistics and Probability ,Group (mathematics) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,State (functional analysis) ,01 natural sciences ,010101 applied mathematics ,Combinatorics ,Mathematics::Group Theory ,Nilpotent ,Euclidean domain ,0101 mathematics ,Variety (universal algebra) ,Algebraic number ,Nilpotent group ,Abelian group ,Mathematics - Abstract
This paper is devoted to the study of groups from the category M of R-power groups. We examine problems on the commutation of the tensor completion with basic group operations and on the exactness of the tensor completion. Moreover, we introduce the notion of a variety and obtain a description of abelian varieties and some results on nilpotent varieties of A-groups. We prove the hypothesis on irreducible coordinate groups of algebraic sets for the nilpotent R-groups of nilpotency class 2, where R is a Euclidean ring. We state that the analog to the Lyndon result for the free groups (see [10]) holds in this case, whereas the analog to the Myasnikov–Kharlampovich result fails.The paper is dedicated to partial R-power groups which are embeddable to their A-tensor completions. The free R-groups and free R-products are described with usual group-theoretical free constructions.
- Published
- 2016
12. Graph-Links: Nonrealizability, Orientation, and Jones Polynomial
- Author
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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
13. On Algorithmic Methods of Analysis of Two-Colorings of Hypergraphs
- Author
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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
14. sp-Groups and Their Endomorphism Rings
- Author
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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
15. Cliques and Constructors in 'Hats' Game. I
- Author
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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
16. On Vertices of Degree 6 of Minimal and Contraction Critical 6-Connected Graph
- Author
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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} $$ is constructed.
- Published
- 2021
17. A Motivic Segal Theorem for Pairs (Announcement)
- Author
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A. Tsybyshev
- Subjects
Statistics and Probability ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Order (ring theory) ,Codimension ,Space (mathematics) ,01 natural sciences ,Weak equivalence ,010305 fluids & plasmas ,Combinatorics ,Morphism ,Scheme (mathematics) ,0103 physical sciences ,Sheaf ,Perfect field ,0101 mathematics ,Mathematics - Abstract
In order to provide a new, more computation-friendly, construction of the stable motivic category SH(k), V. Voevodsyky laid the foundation of delooping motivic spaces. G. Garkusha and I. Panin based on joint works with A. Ananievsky, A. Neshitov, and A. Druzhinin made that project a reality. In particular, G. Garkusha and I. Panin proved that for an infinite perfect field k and any k-smooth scheme X, the canonical morphism of motivic spaces $$ {C}_{\ast } Fr(X)\to {\Omega}_{{\mathrm{\mathbb{P}}}^1}^{\infty }{\sum}_{{\mathrm{\mathbb{P}}}^1}^{\infty}\left({X}_{+}\right) $$ is a Nisnevich locally group-completion. In the present paper, a generalization of that theorem is established to the case of smooth open pairs (X,U), where X is a k-smooth scheme and U is its open subscheme intersecting each component of X in a nonempty subscheme. It is claimed that in this case the motivic space C*Fr((X,U)) is a Nisnevich locally connected, and the motivic space morphism $$ {C}_{\ast } Fr\left(\left(X,U\right)\right)\to {\Omega}_{{\mathrm{\mathbb{P}}}^1}^{\infty }{\sum}_{{\mathrm{\mathbb{P}}}^1}^{\infty}\left(X/U\right) $$ is Nisnevich locally weak equivalence. Moreover, it is proved that if the codimension of S = X−U in each component of X is greater than r ≥ 0, then the simplicial sheaf C*Fr((X,U)) is locally r-connected.
- Published
- 2021
18. Symmetry-Based Approach to the Problem of a Perfect Cuboid
- Author
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Ruslan Sharipov
- Subjects
Statistics and Probability ,Reduction (recursion theory) ,Cuboid ,Mathematics::Number Theory ,Applied Mathematics ,General Mathematics ,Diophantine equation ,010102 general mathematics ,Diagonal ,Computer Science::Computational Geometry ,01 natural sciences ,010305 fluids & plasmas ,Combinatorics ,symbols.namesake ,Parallelepiped ,Euler brick ,Face (geometry) ,0103 physical sciences ,Physics::Atomic and Molecular Clusters ,symbols ,0101 mathematics ,Symmetry (geometry) ,Computer Science::Databases ,Mathematics - Abstract
A perfect cuboid is a rectangular parallelepiped in which the lengths of all edges, the lengths of all face diagonals, and also the lengths of spatial diagonals are integers. No such cuboid has yet been found, but their nonexistence has also not been proved. The problem of a perfect cuboid is among unsolved mathematical problems. The problem has a natural S3-symmetry connected to permutations of edges of the cuboid and the corresponding permutations of face diagonals. In this paper, we give a survey of author’s results and results of J. R. Ramsden on using the S3 symmetry for the reduction and analysis of the Diophantine equations for a perfect cuboid.
- Published
- 2020
19. Trace and Commutators of Measurable Operators Affiliated to a Von Neumann Algebra
- Author
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A. M. Bikchentaev
- Subjects
Statistics and Probability ,Trace (linear algebra) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Space (mathematics) ,01 natural sciences ,010305 fluids & plasmas ,Combinatorics ,symbols.namesake ,Operator (computer programming) ,Von Neumann algebra ,0103 physical sciences ,Isometry ,symbols ,0101 mathematics ,Mathematics - Abstract
In this paper, we present new properties of the space L1(M, τ) of integrable (with respect to the trace τ ) operators affiliated to a semifinite von Neumann algebra M. For self-adjoint τ-measurable operators A and B, we find sufficient conditions of the τ -integrability of the operator λI −AB and the real-valuedness of the trace τ (λI − AB), where λ ∈ ℝ. Under these conditions, [A,B] = AB − BA ∈ L1(M, τ) and τ ([A,B]) = 0. For τ -measurable operators A and B = B2, we find conditions that are sufficient for the validity of the relation τ ([A,B]) = 0. For an isometry U ∈ M and a nonnegative τ -measurable operator A, we prove that U − A ∈ L1(M, τ) if and only if I − A, I − U ∈ L1(M, τ). For a τ -measurable operator A, we present estimates of the trace of the autocommutator [A∗,A]. Let self-adjoint τ -measurable operators X ≥ 0 and Y be such that [X1/2, YX1/2] ∈ L1(M, τ). Then τ ([X1/2, YX1/2]) = it, where t ∈ ℝ and t = 0 for XY ∈ L1(M, τ).
- Published
- 2020
20. Commutators of Relative and Unrelative Elementary Groups, Revisited
- Author
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Nikolai Vavilov and Zuhong Zhang
- Subjects
Statistics and Probability ,Ring (mathematics) ,Group (mathematics) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Commutator subgroup ,Type (model theory) ,01 natural sciences ,010305 fluids & plasmas ,Combinatorics ,0103 physical sciences ,0101 mathematics ,Commutative property ,Mathematics - Abstract
Let R be any associative ring with 1, let n ≥ 3, and let A,B be two-sided ideals of R. In the present paper, we show that the mixed commutator subgroup [E(n,R,A),E(n,R,B)] is generated as a group by the elements of the two following forms: 1) zij(ab, c) and zij (ba, c), 2) [tij(a), tji(b)], where 1 ≤ i ≠ j ≤ n, a ∈ A, b ∈ B, c ∈ R. Moreover, for the second type of generators, it suffices to fix one pair of indices (i, j). This result is both stronger and more general than the previous results by Roozbeh Hazrat and the authors. In particular, it implies that for all associative rings one has the equality [E(n,R,A),E(n,R,B)] = [E(n,A),E(n,B)], and many further corollaries can be derived for rings subject to commutativity conditions. Bibliography: 36 titles.
- Published
- 2020
21. On Colorings of 3-Homogeneous Hypergraphs in 3 Colors
- Author
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I. A. Akolzin
- Subjects
Statistics and Probability ,Combinatorics ,Homogeneous ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,0101 mathematics ,01 natural sciences ,Value (mathematics) ,010305 fluids & plasmas ,Mathematics - Abstract
In this paper, we examine the value m(n, r) in the Erdős–Hajnal problem. Using various methods, we obtain the estimate 27 ≤ m(3) ≤ 35.
- Published
- 2020
22. Criterion for the Existence of a 1-Lipschitz Selection from the Metric Projection onto the Set of Continuous Selections from a Multivalued Mapping
- Author
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A. A. Vasil’eva
- Subjects
Statistics and Probability ,Selection (relational algebra) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Hausdorff space ,Hilbert space ,Topological space ,Lipschitz continuity ,01 natural sciences ,010305 fluids & plasmas ,Combinatorics ,Set (abstract data type) ,symbols.namesake ,Bounded function ,0103 physical sciences ,symbols ,Paracompact space ,0101 mathematics ,Mathematics - Abstract
Let SF be the set of continuous bounded selections from the set-valued mapping F : T → 2H with nonempty convex closed values; here T is a paracompact Hausdorff topological space, and H is a Hilbert space. In this paper, we obtain a criterion for the existence of a 1-Lipschitz selection from the metric projection onto the set SF in C(T, H).
- Published
- 2020
23. Double Occurrence Words: Their Graphs and Matrices
- Author
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E. M. Kreines, Alexander Guterman, and N. V. Ostroukhova
- Subjects
Statistics and Probability ,Series (mathematics) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Geometric representation ,Structure (category theory) ,Characterization (mathematics) ,Quantitative Biology::Genomics ,01 natural sciences ,010305 fluids & plasmas ,Combinatorics ,Matrix (mathematics) ,Simple (abstract algebra) ,0103 physical sciences ,natural sciences ,0101 mathematics ,Incidence (geometry) ,Mathematics - Abstract
Double occurrence words play an important part in genetics in describing epigenetic genome rearrangements. A useful geometric representation for double occurrence words is provided by the so-called assembly graphs. The paper investigates properties of the incidence matrices that correspond to the assembly graphs. An explicit matrix characterization of the simple assembly graphs of a given structure and a series of constructions, using these graphs and important for genetic investigations, are provided.
- Published
- 2020
24. Nekrasov Type Matrices and Upper Bounds for Their Inverses
- Author
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L. Yu. Kolotilina
- Subjects
Statistics and Probability ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Inverse ,Permutation matrix ,Type (model theory) ,01 natural sciences ,010305 fluids & plasmas ,Combinatorics ,Matrix (mathematics) ,0103 physical sciences ,Bibliography ,0101 mathematics ,Mathematics ,Diagonally dominant matrix - Abstract
The paper considers the so-called P-Nekrasov and {P1, P2}-Nekrasov matrices, defined in terms of permutation matrices P, P1, P2, which generalize the well-known notion of Nekrasov matrices. For such matrices A, available upper bounds on ‖A−1‖∞ are recalled, and new upper bounds for the P-Nekrasov and {P1, P2}-Nekrasov matrices are suggested. It is shown that the latter bound generally improves the earlier bounds, as well as the bound for the inverse of a P-Nekrasov matrix and the classical bound for the inverse of a strictly diagonally dominant matrix. Bibliography: 12 titles.
- Published
- 2020
25. New Classes of Nonsingular Matrices and Upper Bounds for their Inverses
- Author
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L. Yu. Kolotilina
- Subjects
Statistics and Probability ,Class (set theory) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,01 natural sciences ,010305 fluids & plasmas ,law.invention ,Combinatorics ,Invertible matrix ,law ,0103 physical sciences ,Bibliography ,0101 mathematics ,Mathematics - Abstract
The paper introduces new classes of nonsingular matrices, which contain some known subclasses of the class of nonsingular H-matrices, such as the Nekrasov, Q-Nekrasov, {P1, P2}-Nekrasov, and DZ matrices. For matrices in the classes introduced, upper bounds for ‖A−1‖∞ are derived (in a unified manner) and shown to improve the known bounds for matrices from the corresponding subclasses of the nonsingular H-matrices. Bibliography: 20 titles.
- Published
- 2020
26. Maps that Strongly Preserve λ-Scrambling Matrices
- Author
-
A. M. Maksaev
- Subjects
Statistics and Probability ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Commutative semiring ,01 natural sciences ,010305 fluids & plasmas ,Semiring ,Scrambling ,Set (abstract data type) ,Combinatorics ,Identity (mathematics) ,0103 physical sciences ,Bibliography ,Bijection ,0101 mathematics ,Zero divisor ,Mathematics - Abstract
In this paper, it is proved that for λ > 1, an additive map that strongly preserves the set of λ-scrambling matrices over the Boolean semiring B is a bijection. The general form of such a map over any antinegative commutative semiring with identity and without zero divisors is characterized. Bibliography: 20 titles.
- Published
- 2020
27. Dual Diophantine Systems of Linear Inequalities
- Author
-
V. G. Zhuravlev
- Subjects
Statistics and Probability ,Recurrence relation ,Applied Mathematics ,General Mathematics ,Diophantine equation ,010102 general mathematics ,Order (ring theory) ,01 natural sciences ,010305 fluids & plasmas ,Combinatorics ,Linear inequality ,0103 physical sciences ,0101 mathematics ,Algebraic number ,Mathematics - Abstract
The paper suggests a modified version of the ℒ-algorithm for constructing an infinite sequence of integral solutions of dual systems $$ \mathcal{S} $$ and $$ {\mathcal{S}}^{\ast } $$ of linear inequalities in d + 1 variables consisting of k⊥ and k*⊥ inequalities, respectively, where k⊥ + k*⊥ = d + 1. Solutions are obtained from two recurrence relations of order d + 1. Approximation in the inequality systems $$ \mathcal{S} $$ and $$ {\mathcal{S}}^{\ast } $$ is effected with the Diophantine exponents $$ \frac{d+1-{k}^{\perp }}{k^{\perp }}-\upvarrho $$ and $$ \frac{d+1-{k}^{\ast \perp }}{k^{\ast \perp }}-\upvarrho $$ , respectively, where the deviation ϱ > 0 can be made arbitrarily small by appropriately choosing the recurrence relations. The ℒ-algorithm is based on a method for localizing units in algebraic number fields.
- Published
- 2020
28. Ind-Varieties of Generalized Flags: A Survey
- Author
-
Ivan Penkov and Mikhail V. Ignatyev
- Subjects
Statistics and Probability ,Applied Mathematics ,General Mathematics ,Flag (linear algebra) ,010102 general mathematics ,FLAGS register ,Structure (category theory) ,Vector bundle ,Direct limit ,Characterization (mathematics) ,Space (mathematics) ,01 natural sciences ,Linear subspace ,010305 fluids & plasmas ,Combinatorics ,0103 physical sciences ,0101 mathematics ,Mathematics - Abstract
This paper is a review of results on the structure of homogeneous ind-varieties G/P of the ind-groups G = GL∞(ℂ), SL∞(ℂ), SO∞(ℂ), and Sp∞(ℂ), subject to the condition that G/P is the inductive limit of compact homogeneous spaces Gn/Pn. In this case, the subgroup P ⊂ G is a splitting parabolic subgroup of G and the ind-variety G/P admits a “flag realization.” Instead of ordinary flags, one considers generalized flags that are, in general, infinite chains $$ \mathcal{C} $$ of subspaces in the natural representation V of G satisfying a certain condition; roughly speaking, for each nonzero vector 𝜐 of V , there exist the largest space in $$ \mathcal{C} $$ , which does not contain 𝜐, and the smallest space in $$ \mathcal{C} $$ , which contains v. We start with a review of the construction of ind-varieties of generalized flags and then show that these ind-varieties are homogeneous ind-spaces of the form G/P for splitting parabolic ind-subgroups P ⊂ G. Also, we briefly review the characterization of more general, i.e., nonsplitting, parabolic ind-subgroups in terms of generalized flags. In the special case of the ind-grassmannian X, we give a purely algebraic-geometric construction of X. Further topics discussed are the Bott–Borel–Weil theorem for ind-varieties of generalized flags, finite-rank vector bundles on ind-varieties of generalized flags, the theory of Schubert decomposition of G/P for arbitrary splitting parabolic ind-subgroups P ⊂ G, as well as the orbits of real forms on G/P for G = SL∞(ℂ).
- Published
- 2020
29. Multiple Flag Varieties
- Author
-
E. Yu. Smirnov
- Subjects
Statistics and Probability ,Combinatorics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,0101 mathematics ,Variety (universal algebra) ,Mathematics::Representation Theory ,01 natural sciences ,010305 fluids & plasmas ,Flag (geometry) ,Mathematics - Abstract
This paper is a review of results on multiple flag varieties, i.e., varieties of the form G/P1×· · ·×G/Pr. We provide a classification of multiple flag varieties of complexity 0 and 1 and results on the combinatorics and geometry of B-orbits and their closures in double cominuscule flag varieties. We also discuss questions of finiteness for the number of G-orbits and existence of an open G-orbits on a multiple flag variety.
- Published
- 2020
30. On Thompson’s Conjecture for Finite Simple Exceptional Groups of Lie Type
- Author
-
A. A. Shlepkin, Ilya Gorshkov, Ivan Kaygorodov, and Andrei Kukharev
- Subjects
Statistics and Probability ,Finite group ,Conjecture ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Group Theory (math.GR) ,Center (group theory) ,Type (model theory) ,01 natural sciences ,010305 fluids & plasmas ,Combinatorics ,Set (abstract data type) ,Conjugacy class ,Simple (abstract algebra) ,Simple group ,0103 physical sciences ,FOS: Mathematics ,0101 mathematics ,Mathematics - Group Theory ,Mathematics - Abstract
Let $G$ be a finite group, $N(G)$ be the set of conjugacy classes of the group $G$. In the present paper it is proved $G\simeq L$ if $N(G)=N(L)$, where $G$ is a finite group with trivial center and $L$ is a finite simple group of exceptional Lie type or Tits group.
- Published
- 2020
31. Word Maps of Chevalley Groups Over Infinite Fields
- Author
-
E. A. Egorchenkova
- Subjects
Statistics and Probability ,Applied Mathematics ,General Mathematics ,Image (category theory) ,010102 general mathematics ,Cohomological dimension ,Type (model theory) ,Unipotent ,01 natural sciences ,010305 fluids & plasmas ,Combinatorics ,Group of Lie type ,Algebraic group ,0103 physical sciences ,Perfect field ,0101 mathematics ,Word (group theory) ,Mathematics - Abstract
Let G be a simply connected Chevalley group over an infinite field K, and let $$ \tilde{w} $$ : Gn → G be a word map that corresponds to a nontrivial word w. In 2015, it has been proved that if w = w1w2w3w4 is the product of four words in independent variables, then every noncentral element of G is contained in the image of $$ \tilde{w} $$. A similar result for a word w = w1w2w3, which is the product of three independent words, was obtained in 2019 under the condition that the group G is not of type B2 or G2. In the present paper, it is proved that for a group of type B2 or G2, all elements of the large Bruhat cell B nw0B are contained in the image of the word map $$ \tilde{w} $$, where w = w1w2w3 is the product of three independent words. For a group G of type Ar, Cr, or G2 (respectively, for a group of type Ar) or a group over a perfect field K (respectively, over a perfect field K the characteristic of which is not a bad prime for G) with dim K ≤ 1 (here, dim K is the cohomological dimension of K), it is proved that all split regular semisimple elements (respectively, all regular unipotent elements) of G are contained in the image of $$ \tilde{w} $$, where w = w1w2 is the product of two independent words. Also, for any isotropic (but not necessary split) simple algebraic group G over a field K of characteristic zero, it is shown that for a word map $$ \tilde{w} $$ : G(K)n → G(K), where w = w1w2 is a product of two independent words, all unipotent elements are contained in Im $$ \tilde{w} $$.
- Published
- 2020
32. On the Chromatic Numbers Corresponding to Exponentially Ramsey Sets
- Author
-
A. A. Sagdeev
- Subjects
Statistics and Probability ,Simplex ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,01 natural sciences ,010305 fluids & plasmas ,Exponential function ,Combinatorics ,Parallelepiped ,Exponential growth ,0103 physical sciences ,Bibliography ,Chromatic scale ,Monochromatic color ,0101 mathematics ,Mathematics - Abstract
In this paper, nontrivial upper bounds on the chromatic numbers of the spaces $$ {\mathrm{\mathbb{R}}}_p^n=\left({\mathrm{\mathbb{R}}}^n, lp\right) $$ with forbidden monochromatic sets are proved. In the case of a forbidden rectangular parallelepiped or a regular simplex, explicit exponential lower bounds on the chromatic numbers are obtained. Exact values of the chromatic numbers of the spaces $$ {\mathrm{\mathbb{R}}}_p^n $$ with a forbidden regular simplex in the case p = ∞ are found. Bibliography: 39 titles.
- Published
- 2020
33. Estimates of the best orthogonal trigonometric approximations and orthoprojective widths of the classes of periodic functions of many variables in a uniform metric
- Author
-
V Viktoriya Shkapa, M M Hanna Hanna Vlasyk, and V Iryna Zamrii
- Subjects
Statistics and Probability ,Approximations of π ,BETA (programming language) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Space (mathematics) ,01 natural sciences ,010305 fluids & plasmas ,Periodic function ,Combinatorics ,0103 physical sciences ,Metric (mathematics) ,0101 mathematics ,Trigonometry ,computer ,Mathematics ,computer.programming_language - Abstract
Some approximative characteristics of classes of periodic functions of many variables $$ {L}_{\beta, p}^{\psi }, $$ 1 < p < 1, in a uniform metric are investigated. The first part of the paper is devoted to the construction of estimates of the best orthogonal trigonometric approximations of the mentioned classes in the space L∞. In the second part, we have established the ordinal estimates of the orthoprojective widths of the classes $$ {L}_{\beta, p}^{\psi }, $$ 1 < p < 1, in the same space, as well as the estimates of another approximative characteristic which is close, in a definite meaning, to the orthoprojective width.
- Published
- 2020
34. Unrelativized Standard Commutator Formula
- Author
-
Nikolai Vavilov
- Subjects
Statistics and Probability ,Marginalia ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Commutator (electric) ,Commutative ring ,01 natural sciences ,010305 fluids & plasmas ,law.invention ,Combinatorics ,law ,0103 physical sciences ,0101 mathematics ,Mathematics - Abstract
In the present note, which is a marginalia to the previous papers by Roozbeh Hazrat, Alexei Stepanov, Zuhong Zhang, and the author, I observe that for any ideals A,B≤R of a commutative ring R and all n ≥ 3 the birelative standard commutator formula also holds in the unrelativized form, as [E(n,A),GL(n,B)] = [E(n,A),E(n,B)] and discuss some obvious corollaries thereof.
- Published
- 2019
35. On a Question About Generalized Congruence Subgroups. I
- Author
-
V. A. Koibaev
- Subjects
Statistics and Probability ,Ring (mathematics) ,Group (mathematics) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Zero (complex analysis) ,Field (mathematics) ,Net (mathematics) ,01 natural sciences ,010305 fluids & plasmas ,Integral domain ,Combinatorics ,0103 physical sciences ,Order (group theory) ,0101 mathematics ,Quotient ,Mathematics - Abstract
A set of additive subgroups σ = (σij), 1 ≤ i, j ≤ n, of a field (or ring) K is called a net of order n over K if σirσrj ⊆ σij for all values of the indices i, r, j. The same system, but without diagonal, is called an elementary net. A full or elementary net σ = (σij) is said to be irreducible if all the additive subgroups σij are different from zero. An elementary net σ is closed if the subgroup E(σ) does not contain new elementary transvections. The present paper is related to a question posed by Y. N. Nuzhin in connection with V. M. Levchuk’s question No. 15.46 from the Kourovka notebook about the admissibility (closure) of elementary net (carpet) σ = (σij) over a field K. Let J be an arbitrary subset of {1, . . . , n}, n ≥ 3, and the cardinality m of J satisfies the condition 2 ≤ m ≤ n − 1. Let R be a commutative integral domain (non-field) with identity, and let K be the quotient field of R. An example of a net σ = (σij) of order n over K, for which the group E(σ) ∩ 〈tij(K) : i, j ∈ J〉 is not contained in the group 〈tij(σij) : i, j ∈ J〉, is constructed.
- Published
- 2019
36. A New Subclass of the Class of Nonsingular $$ \mathcal{H} $$-Matrices and Related Inclusion Sets for Eigenvalues and Singular Values
- Author
-
L. Yu. Kolotilina
- Subjects
Statistics and Probability ,Index set (recursion theory) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Type (model theory) ,01 natural sciences ,Square matrix ,010305 fluids & plasmas ,law.invention ,Combinatorics ,Singular value ,Matrix (mathematics) ,Invertible matrix ,law ,0103 physical sciences ,0101 mathematics ,Eigenvalues and eigenvectors ,Mathematics ,Diagonally dominant matrix - Abstract
The paper presents new nonsingularity conditions for n ×n matrices, which involve a subset S of the index set {1, . . ., n} and take into consideration the matrix sparsity pattern. It is shown that the matrices satisfying these conditions form a subclass of the class of nonsingular $$ \mathcal{H} $$ -matrices, which contains some known matrix classes such as the class of doubly strictly diagonally dominant (DSDD) matrices and the class of Dashnic–Zusmanovich type (DZT) matrices. The nonsingularity conditions established are used to obtain the corresponding eigenvalue inclusion sets, which, in their turn, are used in deriving new inclusion sets for the singular values of a square matrix, improving some recently suggested ones.
- Published
- 2019
37. On Dashnic–Zusmanovich (DZ) and Dashnic–Zusmanovich Type (DZT) Matrices and Their Inverses
- Author
-
L. Yu. Kolotilina
- Subjects
Statistics and Probability ,Class (set theory) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Type (model theory) ,01 natural sciences ,010305 fluids & plasmas ,law.invention ,Combinatorics ,Invertible matrix ,law ,0103 physical sciences ,0101 mathematics ,Eigenvalues and eigenvectors ,Mathematics ,Diagonally dominant matrix - Abstract
The paper is mainly devoted to studying the so-called Dashnic–Zusmanovich type (DZT) matrices, introduced recently. Interrelations among the DZT matrices and related subclasses of the class of nonsingular $$ \mathcal{H} $$ -matrices, namely, the Dashnic–Zusmanovich (DZ) and S-SDD matrices are considered. Upper bounds for the l∞-norms of the inverses to DZT, DZ, and strictly diagonally dominant (SDD) matrices are obtained. A new eigenvalue inclusion set is provided.
- Published
- 2019
38. The Structure of Solutions of the Matrix Equation Jn(0)Y + Y ⏉Jn(0) = 0 for Even n
- Author
-
Kh. D. Ikramov
- Subjects
Statistics and Probability ,Direct sum ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Structure (category theory) ,Row and column spaces ,01 natural sciences ,Toeplitz matrix ,010305 fluids & plasmas ,Combinatorics ,Permutation ,0103 physical sciences ,Order (group theory) ,0101 mathematics ,Mathematics - Abstract
It is shown that every solution of the matrix equation in the title of the paper, where n = 2m, can be transformed by a symmetric permutation of its rows and columns to a direct sum of two triangular Toeplitz matrices of order m.
- Published
- 2021
39. Lattice Points in the Four-Dimensional Ball
- Author
-
O. M. Fomenko
- Subjects
Statistics and Probability ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,01 natural sciences ,Omega ,010305 fluids & plasmas ,Riemann zeta function ,Combinatorics ,symbols.namesake ,Riesz mean ,0103 physical sciences ,symbols ,Ball (mathematics) ,0101 mathematics ,Mathematics - Abstract
Let r4(n) denote the number of representations of n as a sum of four squares. The generating function ζ4(s) is Epstein’s zeta function. The paper considers the Riesz mean $$ {D}_{\rho}\left(x;{\zeta}_4\right)=\frac{1}{\Gamma \left(\rho +1\right)}\sum \limits_{n\le x}{\left(x-n\right)}^{\rho }{r}_4(n) $$ for an arbitrary fixed ρ > 0. The error term Δρ(x; ζ4) is defined by $$ {D}_{\rho}\left(x;{\zeta}_4\right)=\frac{\uppi^2{x}^{2+\rho }}{\Gamma \left(\rho +3\right)}+\frac{x^{\rho }}{\Gamma \left(\rho +1\right)}{\zeta}_4(0)+{\Delta}_{\rho}\left(x;{\zeta}_4\right). $$ It is proved that $$ {\Delta}_4\left(x;{\zeta}_4\right)=\Big\{{\displaystyle \begin{array}{ll}O\left({x}^{1/2+\rho +\varepsilon}\right)& \left(1
- Published
- 2018
40. Estimates of Functions, Orthogonal to Piecewise Constant Functions, in Terms of the Second Modulus of Continuity
- Author
-
L. N. Ikhsanov
- Subjects
Statistics and Probability ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Type inequality ,01 natural sciences ,Modulus of continuity ,010305 fluids & plasmas ,Combinatorics ,Range (mathematics) ,Bounded function ,0103 physical sciences ,Piecewise ,Constant function ,0101 mathematics ,Constant (mathematics) ,Mathematics - Abstract
The paper is devoted to the problem of finding the exact constant $$ {W}_2^{\ast } $$ in the inequality ‖f‖ ≤ K ⋅ ω2(f, 1) for bounded functions f with the property $$ \underset{k}{\overset{k+1}{\int }}f(x) dx=0,\kern0.5em k\in \mathrm{\mathbb{Z}}. $$ Our approach allows us to reduce the known range for the desired constant as well as the set of functions involved in the extremal problem for finding the constant in question. It is shown that $$ {W}_2^{\ast } $$ also turns out to be the exact constant in a related Jackson–Stechkin type inequality.
- Published
- 2018
41. The Normalizer of the Elementary Linear Group of a Module Arising when the Base Ring is Extended
- Author
-
N. H. T. Nhat and T. N. Hoi
- Subjects
Statistics and Probability ,Ring (mathematics) ,Group (mathematics) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Commutative ring ,Rank (differential topology) ,Subring ,01 natural sciences ,Centralizer and normalizer ,Base (group theory) ,Combinatorics ,0103 physical sciences ,010307 mathematical physics ,Ideal (ring theory) ,0101 mathematics ,Mathematics - Abstract
Let S be a commutative ring with 1 and R a unital subring. Let M be a free S-module of rank n ≥ 3. In 1994, V. A. Koibaev described the normalizer of AutS(M) in the group AutR(M). In the present paper, it is proved that the normalizer of the elementary linear group E𝔅(M) in AutR(M) coincides with that of AutS(M), namely, NAutR(M)(E𝔅(M)) = Aut(S/R)⋉AutS(M). If S is free of rank m as an R-module, then NGL(mn,R)(E(n, S)) = Aut(S/R)⋉GL(n, S). Moreover, for any proper ideal A of R, $$ {N}_{GL\left( mn,R\right)}\left(E\left(n,S\right)E\left( mn,R,A\right)\right)={\rho}_A^{-1}\left({N}_{GL\left( mn,R/A\right)}\left(E\left(n,S/ SA\right)\right)\right). $$
- Published
- 2018
42. Limiting profile of solutions of quasilinear parabolic equations with flat peaking
- Author
-
Yevgeniia A. Yevgenieva
- Subjects
Statistics and Probability ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Limiting ,01 natural sciences ,Omega ,Parabolic partial differential equation ,010101 applied mathematics ,Combinatorics ,Domain (ring theory) ,Nabla symbol ,0101 mathematics ,Energy (signal processing) ,Mathematics - Abstract
The paper deals with energy (weak) solutions u (t; x) of the class of equations with the model representative $$ \left(\left|u\right|{p}^{-1}u\right)t-\Delta p(u)=0,\kern0.5em \left(t,x\right)\in \left(0,T\right)\times \varOmega, \varOmega \in {\mathrm{\mathbb{R}}}^n,n\ge 1,p>0, $$ and with the following blow-up condition for the energy: $$ \varepsilon (t):= {\int}_{\Omega}{\left|u\left(t,x\right)\right|}^{p+1} dx+{\int}_0^t{\int}_{\Omega}{\left|{\nabla}_xu\left(\tau, x\right)\right|}^{p+1} dx d\tau \to \infty \mathrm{as}\;t\to T, $$ where Ω is a smooth bounded domain. In the case of flat peaking, namely, under the condition $$ {\displaystyle \begin{array}{cc}\varepsilon (t)\le F\upalpha (t){\upomega}_0{\left(T-t\right)}^{-\upalpha}& \forall t 0,\upalpha >\frac{1}{p+1}, $$ a sharp estimate of the profile of a solution has been obtained in a neighborhood of the blow-up time t = T.
- Published
- 2018
43. Groups in Which the Normal Closures of Cyclic Subgroups Have Bounded Finite Hirsch–Zaitsev Rank
- Author
-
N.N. Semko and Leonid A. Kurdachenko
- Subjects
Statistics and Probability ,Combinatorics ,Applied Mathematics ,General Mathematics ,Bounded function ,010102 general mathematics ,0103 physical sciences ,0101 mathematics ,Invariant (mathematics) ,01 natural sciences ,010305 fluids & plasmas ,Mathematics - Abstract
In this paper, we study generalized soluble groups with restriction on normal closures of cyclic subgroups. A group G is said to have finite Hirsch–Zaitsev rank if G has an ascending series whose factors are either infinite cyclic or periodic and if the number of infinite cyclic factors is finite. It is not hard to see that the number of infinite cyclic factors in each of such series is an invariant of a group G. This invariant is called the Hirsch–Zaitsev rank of G and will be denoted by rhz(G). We study the groups in which the normal closure of every cyclic subgroup has the Hirsch–Zaitsev rank at most b (b is some positive integer). For some natural restrictions we find a function k1(b) such that rhz([G/Tor(G),G/Tor(G)]) ≤ k1(b).
- Published
- 2018
44. A Characterization of the Gaschütz Subgroup of a Finite Soluble Group
- Author
-
S. F. Kamornikov
- Subjects
Statistics and Probability ,Combinatorics ,Finite group ,Group (mathematics) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Characterization (mathematics) ,01 natural sciences ,Mathematics - Abstract
Let H be an $$ \mathfrak{N} $$ -prefrattini subgroup of a soluble finite group G and Δ(G) be its Gaschutz subgroup. In this paper, it is proved that there exist elements x, y ∈ G such that the equality H∩Hx∩Hy = Δ(G) holds.
- Published
- 2018
45. An Approach to Bounding the Spectral Radius of a Weighted Digraph
- Author
-
L. Yu. Kolotilina
- Subjects
Statistics and Probability ,Spectral radius ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Block matrix ,Graph theory ,0102 computer and information sciences ,01 natural sciences ,Square (algebra) ,Combinatorics ,Matrix (mathematics) ,010201 computation theory & mathematics ,Bounding overwatch ,Adjacency matrix ,Nonnegative matrix ,0101 mathematics ,Mathematics - Abstract
The paper suggests a general approach to deriving upper bounds for the spectral radii of weighted digraphs. The approach is based on the generalized Wielandt lemma (GWL), which reduces the problem of bounding the spectral radius of a given block matrix to bounding the Perron root of the matrix composed of the norms of the blocks. In the case of the adjacency matrix of weighted graphs and digraphs where all the blocks are square positive (semi)definite matrices of the same order, the GWL takes an especially nice simple form. The second component of the approach consists in applying known upper bounds for the Perron root of a nonnegative matrix. It is shown that the approach suggested covers, in particular, the known upper bounds of the spectral radius and allows one to describe the equality cases.
- Published
- 2018
46. On the Determinantal Range of Matrix Products
- Author
-
G. Soares and Alexander Guterman
- Subjects
Statistics and Probability ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0211 other engineering and technologies ,021107 urban & regional planning ,02 engineering and technology ,01 natural sciences ,Combinatorics ,Range (mathematics) ,Matrix (mathematics) ,0101 mathematics ,Complex number ,Eigenvalues and eigenvectors ,Mathematics - Abstract
Let matrices A,C ∈ Mn have eigenvalues α1, . . ., αn and γ1, . . . , γn, respectively. The set of complex numbers DC(A) = {det(A−UCU*) : U ∈ Mn, U*U = In} is called the C-determinantal range of A. The paper studies various conditions under which the relation DC(R S) = DC(S R) holds for some matrices R and S.
- Published
- 2018
47. Orthogonality Graphs of Matrices Over Skew Fields
- Author
-
Alexander Guterman and O. V. Markova
- Subjects
Statistics and Probability ,Connected component ,Ring (mathematics) ,Mathematics::Commutative Algebra ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0211 other engineering and technologies ,Skew ,021107 urban & regional planning ,Field (mathematics) ,02 engineering and technology ,01 natural sciences ,Matrix ring ,Combinatorics ,Disjoint union (topology) ,Orthogonality ,Simple (abstract algebra) ,0101 mathematics ,Mathematics - Abstract
The paper is devoted to studying the orthogonality graph of the matrix ring over a skew field. It is shown that for n ≥ 3 and an arbitrary skew field 𝔻, the orthogonality graph of the ring Mn(𝔻) of n × n matrices over a skew field 𝔻 is connected and has diameter 4. If n = 2, then the graph of the ring Mn(𝔻) is a disjoint union of connected components of diameters 1 and 2. As a corollary, the corresponding results on the orthogonality graphs of simple Artinian rings are obtained.
- Published
- 2018
48. Normalizers of Elementary Overgroups of Ep(2, A)
- Author
-
E. Yu. Voronetsky
- Subjects
Statistics and Probability ,Combinatorics ,Involution (mathematics) ,Symplectic group ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,0101 mathematics ,01 natural sciences ,Hermitian matrix ,010305 fluids & plasmas ,Mathematics - Abstract
Let A be an involution ring, e1 , . . . , en be a full system of Hermitian idempotents in A, let every ei generate A as a two-sided ideal, and 2 ∈ A∗. In this paper, the normalizers of the groups Ep(2,A) · E(2,A, I) are calculated under natural assumptions on A, where Ep(2,A) denotes the elementary symplectic group, E(2,A, I) stands for the elementary subgroup of level I.
- Published
- 2018
49. Double Cosets of Stabilizers of Totally Isotropic Subspaces in a Special Unitary Group. I
- Author
-
Ulf Rehmann and N. Gordeev
- Subjects
Statistics and Probability ,Zariski topology ,Symplectic group ,Sesquilinear form ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,01 natural sciences ,Linear subspace ,Combinatorics ,Algebraic group ,0103 physical sciences ,Division algebra ,Coset ,010307 mathematical physics ,0101 mathematics ,Special unitary group ,Mathematics - Abstract
In 2016, the authors considered the decomposition $$ \mathrm{SU}\left(D,h\right)=\underset{i}{\cup }{P}_u{\gamma}_i{P}_{\upsilon } $$ , where SU(D, h) is a special unitary group over a division algebra D with involution, h is a symmetric or skew-symmetric nondegenerate Hermitian form, and Pu, Pυ are stabilizers of totally isotropic subspaces of the unitary space. Since Γ = SU(D, h) is a point group of a classical algebraic group $$ \tilde{\Gamma} $$ , there is the “order of adherence” on the set of double cosets {PuγiPυ}, which is induced by the Zariski topology on $$ \tilde{\Gamma} $$ . In the present paper, the adherence of such double cosets is described for the cases where $$ \tilde{\Gamma} $$ is an orthogonal or a symplectic group (that is, for groups of types Br, Cr, Dr).
- Published
- 2018
50. On Determinability of a Completely Decomposable Torsion-Free Abelian Group by its Automorphism Group
- Author
-
V. K. Vildanov
- Subjects
Statistics and Probability ,Class (set theory) ,Automorphism group ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,01 natural sciences ,010305 fluids & plasmas ,Combinatorics ,Torsion-free abelian group ,0103 physical sciences ,Idempotence ,Rank (graph theory) ,0101 mathematics ,Abelian group ,Mathematics - Abstract
In this paper, we consider the question of determinability of an Abelian group by its automorphism group in the class of completely decomposable torsion-free Abelian groups whose direct summands of rank 1 have idempotent types.
- Published
- 2018
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