Abstract For X, Y ∈ Mn,m, it is said that X is g-tridiagonal majorized by Y (and it is denoted by X ≺gt Y) if there exists a tridiagonal g-doubly stochastic matrix A such that X = AY. In this paper, the linear preservers and strong linear preservers of ≺gt are characterized on Mn,m.
{"title":"G-tridiagonal majorization on Mn,m","authors":"Ahmad Mohammadhasani, Y. Sayyari, Mahdi Sabzvari","doi":"10.2478/cm-2021-0027","DOIUrl":"https://doi.org/10.2478/cm-2021-0027","url":null,"abstract":"Abstract For X, Y ∈ Mn,m, it is said that X is g-tridiagonal majorized by Y (and it is denoted by X ≺gt Y) if there exists a tridiagonal g-doubly stochastic matrix A such that X = AY. In this paper, the linear preservers and strong linear preservers of ≺gt are characterized on Mn,m.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":"29 1","pages":"395 - 405"},"PeriodicalIF":0.0,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43011153","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this note we establish a necessary and sufficient condition for solvability of the homogeneous Riemann boundary problem with infinity index on a rectifiable open curve. The index of the problem we deal with considers the influence of the requirement of the solutions of the problem, the degree of non-smoothness of the curve at the endpoints as well as the behavior of the coefficient at these points.
{"title":"A note on the solvability of homogeneous Riemann boundary problem with infinity index","authors":"J. Bory‐Reyes","doi":"10.2478/cm-2021-0033","DOIUrl":"https://doi.org/10.2478/cm-2021-0033","url":null,"abstract":"Abstract In this note we establish a necessary and sufficient condition for solvability of the homogeneous Riemann boundary problem with infinity index on a rectifiable open curve. The index of the problem we deal with considers the influence of the requirement of the solutions of the problem, the degree of non-smoothness of the curve at the endpoints as well as the behavior of the coefficient at these points.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":"29 1","pages":"527 - 534"},"PeriodicalIF":0.0,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43579926","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The aim of the present note is to derive an integral transform I=∫0∞xs+1e-βx2+γxMk,v(2ζx2)Jμ(χx)dx,I = int_0^infty {{x^{s + 1}}{e^{ - beta x}}^{2 + gamma x}{M_{k,v}}} left( {2zeta {x^2}} right)Jmu left( {chi x} right)dx, involving the product of the Whittaker function Mk,ν and the Bessel function of the first kind Jµ of order µ. As a by-product, we also derive certain new integral transforms as particular cases for some special values of the parameters k and ν of the Whittaker function. Eventually, we show the application of the integral in the propagation of hollow higher-order circular Lorentz-cosh-Gaussian beams through an ABCD optical system (see, for details [13], [3]).
{"title":"An integral transform and its application in the propagation of Lorentz-Gaussian beams","authors":"A. Belafhal, E. M. E. Halba, T. Usman","doi":"10.2478/cm-2021-0030","DOIUrl":"https://doi.org/10.2478/cm-2021-0030","url":null,"abstract":"Abstract The aim of the present note is to derive an integral transform I=∫0∞xs+1e-βx2+γxMk,v(2ζx2)Jμ(χx)dx,I = int_0^infty {{x^{s + 1}}{e^{ - beta x}}^{2 + gamma x}{M_{k,v}}} left( {2zeta {x^2}} right)Jmu left( {chi x} right)dx, involving the product of the Whittaker function Mk,ν and the Bessel function of the first kind Jµ of order µ. As a by-product, we also derive certain new integral transforms as particular cases for some special values of the parameters k and ν of the Whittaker function. Eventually, we show the application of the integral in the propagation of hollow higher-order circular Lorentz-cosh-Gaussian beams through an ABCD optical system (see, for details [13], [3]).","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":"29 1","pages":"483 - 491"},"PeriodicalIF":0.0,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43035820","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The object of the present paper is to study some types of semisymmetry conditions on two classes of almost Kenmotsu manifolds. It is shown that a (k, µ)-almost Kenmotsu manifold satisfying the curvature condition Q · R = 0 is locally isometric to the hyperbolic space ℍ2n+1(−1). Also in (k, µ)-almost Kenmotsu manifolds the following conditions: (1) local symmetry (∇R = 0), (2) semisymmetry (R·R = 0), (3) Q(S, R) = 0, (4) R·R = Q(S, R), (5) locally isometric to the hyperbolic space ℍ2n+1(−1) are equivalent. Further, it is proved that a (k, µ)′-almost Kenmotsu manifold satisfying Q · R = 0 is locally isometric to ℍn+1(−4) × ℝn and a (k, µ)′--almost Kenmotsu manifold satisfying any one of the curvature conditions Q(S, R) = 0 or R · R = Q(S, R) is either an Einstein manifold or locally isometric to ℍn+1(−4) × ℝn. Finally, an illustrative example is presented.
{"title":"Some type of semisymmetry on two classes of almost Kenmotsu manifolds","authors":"D. Dey, P. Majhi","doi":"10.2478/cm-2021-0029","DOIUrl":"https://doi.org/10.2478/cm-2021-0029","url":null,"abstract":"Abstract The object of the present paper is to study some types of semisymmetry conditions on two classes of almost Kenmotsu manifolds. It is shown that a (k, µ)-almost Kenmotsu manifold satisfying the curvature condition Q · R = 0 is locally isometric to the hyperbolic space ℍ2n+1(−1). Also in (k, µ)-almost Kenmotsu manifolds the following conditions: (1) local symmetry (∇R = 0), (2) semisymmetry (R·R = 0), (3) Q(S, R) = 0, (4) R·R = Q(S, R), (5) locally isometric to the hyperbolic space ℍ2n+1(−1) are equivalent. Further, it is proved that a (k, µ)′-almost Kenmotsu manifold satisfying Q · R = 0 is locally isometric to ℍn+1(−4) × ℝn and a (k, µ)′--almost Kenmotsu manifold satisfying any one of the curvature conditions Q(S, R) = 0 or R · R = Q(S, R) is either an Einstein manifold or locally isometric to ℍn+1(−4) × ℝn. Finally, an illustrative example is presented.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":"29 1","pages":"457 - 471"},"PeriodicalIF":0.0,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42715316","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
It is well-known that the antipode $S$ of a commutative or cocommutative Hopf algebra satisfies $S^{2}=operatorname*{id}$ (where $S^{2}=Scirc S$). Recently, similar results have been obtained by Aguiar, Lauve and Mahajan for connected graded Hopf algebras: Namely, if $H$ is a connected graded Hopf algebra with grading $H=bigoplus_{ngeq0}H_n$, then each positive integer $n$ satisfies $left( operatorname*{id}-S^2right)^n left( H_nright) =0$ and (even stronger) [ left( left( operatorname{id}+Sright) circleft( operatorname{id}-S^2right)^{n-1}right) left( H_nright) = 0. ] For some specific $H$'s such as the Malvenuto--Reutenauer Hopf algebra $operatorname{FQSym}$, the exponents can be lowered. In this note, we generalize these results in several directions: We replace the base field by a commutative ring, replace the Hopf algebra by a coalgebra (actually, a slightly more general object, with no coassociativity required), and replace both $operatorname{id}$ and $S^2$ by "coalgebra homomorphisms" (of sorts). Specializing back to connected graded Hopf algebras, we show that the exponent $n$ in the identity $left( operatorname{id}-S^2right) ^n left( H_nright) =0$ can be lowered to $n-1$ (for $n>1$) if and only if $left( operatorname{id} - S^2right) left( H_2right) =0$. (A sufficient condition for this is that every pair of elements of $H_1$ commutes; this is satisfied, e.g., for $operatorname{FQSym}$.)
{"title":"On the square of the antipode in a connected filtered Hopf algebra","authors":"Darij Grinberg","doi":"10.46298/cm.10431","DOIUrl":"https://doi.org/10.46298/cm.10431","url":null,"abstract":"It is well-known that the antipode $S$ of a commutative or cocommutative Hopf\u0000algebra satisfies $S^{2}=operatorname*{id}$ (where $S^{2}=Scirc S$).\u0000Recently, similar results have been obtained by Aguiar, Lauve and Mahajan for\u0000connected graded Hopf algebras: Namely, if $H$ is a connected graded Hopf\u0000algebra with grading $H=bigoplus_{ngeq0}H_n$, then each positive integer $n$\u0000satisfies $left( operatorname*{id}-S^2right)^n left( H_nright) =0$ and\u0000(even stronger) [ left( left( operatorname{id}+Sright) circleft(\u0000operatorname{id}-S^2right)^{n-1}right) left( H_nright) = 0. ] For some\u0000specific $H$'s such as the Malvenuto--Reutenauer Hopf algebra\u0000$operatorname{FQSym}$, the exponents can be lowered.\u0000 In this note, we generalize these results in several directions: We replace\u0000the base field by a commutative ring, replace the Hopf algebra by a coalgebra\u0000(actually, a slightly more general object, with no coassociativity required),\u0000and replace both $operatorname{id}$ and $S^2$ by \"coalgebra homomorphisms\" (of\u0000sorts). Specializing back to connected graded Hopf algebras, we show that the\u0000exponent $n$ in the identity $left( operatorname{id}-S^2right) ^n left(\u0000H_nright) =0$ can be lowered to $n-1$ (for $n>1$) if and only if $left(\u0000operatorname{id} - S^2right) left( H_2right) =0$. (A sufficient condition\u0000for this is that every pair of elements of $H_1$ commutes; this is satisfied,\u0000e.g., for $operatorname{FQSym}$.)","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48856756","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper is devoted to several new results concerning (standard) octonion polynomials. The first is the determination of the roots of all right scalar multiples of octonion polynomials. The roots of left multiples are also discussed, especially over fields of characteristic not 2. We then turn to study the dynamics of monic quadratic real octonion polynomials, classifying the fixed points into attracting, repelling and ambivalent, and concluding with a discussion on the behavior of pseudo-periodic points.
{"title":"Roots and Dynamics of Octonion Polynomials","authors":"Adam Chapman, S. Vishkautsan","doi":"10.46298/cm.9042","DOIUrl":"https://doi.org/10.46298/cm.9042","url":null,"abstract":"This paper is devoted to several new results concerning (standard) octonion\u0000polynomials. The first is the determination of the roots of all right scalar\u0000multiples of octonion polynomials. The roots of left multiples are also\u0000discussed, especially over fields of characteristic not 2. We then turn to\u0000study the dynamics of monic quadratic real octonion polynomials, classifying\u0000the fixed points into attracting, repelling and ambivalent, and concluding with\u0000a discussion on the behavior of pseudo-periodic points.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43303932","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The purpose of this paper is to study Lie-Rinehart superalgebras over characteristic zero fields, which are consisting of a supercommutative associative superalgebra $A$ and a Lie superalgebra $L$ that are compatible in a certain way. We discuss their structure and provide a classification in small dimensions. We describe all possible pairs defining a Lie-Rinehart superalgebra for $dim(A)leq 2$ and $dim(L)leq 4$. Moreover, we construct a cohomology complex and develop a theory of formal deformations based on formal power series and this cohomology.
{"title":"On classification and deformations of Lie-Rinehart superalgebras","authors":"Quentin Ehret, A. Makhlouf","doi":"10.46298/cm.10537","DOIUrl":"https://doi.org/10.46298/cm.10537","url":null,"abstract":"The purpose of this paper is to study Lie-Rinehart superalgebras over\u0000characteristic zero fields, which are consisting of a supercommutative\u0000associative superalgebra $A$ and a Lie superalgebra $L$ that are compatible in\u0000a certain way. We discuss their structure and provide a classification in small\u0000dimensions. We describe all possible pairs defining a Lie-Rinehart superalgebra\u0000for $dim(A)leq 2$ and $dim(L)leq 4$. Moreover, we construct a cohomology\u0000complex and develop a theory of formal deformations based on formal power\u0000series and this cohomology.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48514236","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $mathfrak{n}$ be a maximal nilpotent subalgebra of a simple complex Lie algebra with root system $Phi$. A subset $D$ of the set $Phi^+$ of positive roots is called a rook placement if it consists of roots with pairwise non-positive scalar products. To each rook placement $D$ and each map $xi$ from $D$ to the set $mathbb{C}^{times}$ of nonzero complex numbers one can naturally assign the coadjoint orbit $Omega_{D,xi}$ in the dual space $mathfrak{n}^*$. By definition, $Omega_{D,xi}$ is the orbit of $f_{D,xi}$, where $f_{D,xi}$ is the sum of root covectors $e_{alpha}^*$ multiplied by $xi(alpha)$, $alphain D$. (In fact, almost all coadjoint orbits studied at the moment have such a form for certain $D$ and $xi$.) It follows from the results of Andr`e that if $xi_1$ and $xi_2$ are distinct maps from $D$ to $mathbb{C}^{times}$ then $Omega_{D,xi_1}$ and $Omega_{D,xi_2}$ do not coincide for classical root systems $Phi$. We prove that this is true if $Phi$ is of type $G_2$, or if $Phi$ is of type $F_4$ and $D$ is orthogonal.
{"title":"Rook placements in $G_2$ and $F_4$ and associated coadjoint orbits","authors":"M. V. Ignatev, Matvey A. Surkov","doi":"10.46298/cm.9041","DOIUrl":"https://doi.org/10.46298/cm.9041","url":null,"abstract":"Let $mathfrak{n}$ be a maximal nilpotent subalgebra of a simple complex Lie\u0000algebra with root system $Phi$. A subset $D$ of the set $Phi^+$ of positive\u0000roots is called a rook placement if it consists of roots with pairwise\u0000non-positive scalar products. To each rook placement $D$ and each map $xi$\u0000from $D$ to the set $mathbb{C}^{times}$ of nonzero complex numbers one can\u0000naturally assign the coadjoint orbit $Omega_{D,xi}$ in the dual space\u0000$mathfrak{n}^*$. By definition, $Omega_{D,xi}$ is the orbit of $f_{D,xi}$,\u0000where $f_{D,xi}$ is the sum of root covectors $e_{alpha}^*$ multiplied by\u0000$xi(alpha)$, $alphain D$. (In fact, almost all coadjoint orbits studied at\u0000the moment have such a form for certain $D$ and $xi$.) It follows from the\u0000results of Andr`e that if $xi_1$ and $xi_2$ are distinct maps from $D$ to\u0000$mathbb{C}^{times}$ then $Omega_{D,xi_1}$ and $Omega_{D,xi_2}$ do not\u0000coincide for classical root systems $Phi$. We prove that this is true if\u0000$Phi$ is of type $G_2$, or if $Phi$ is of type $F_4$ and $D$ is orthogonal.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41398185","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We describe the images of multilinear polynomials of arbitrary degree evaluated on the 3×3 upper triangular matrix algebra over an infinite field.
摘要描述了无限域上的任意次多元线性多项式在3×3上三角矩阵代数上的像。
{"title":"The image of multilinear polynomials evaluated on 3 × 3 upper triangular matrices","authors":"Thiago Castilho de Mello","doi":"10.2478/cm-2021-0017","DOIUrl":"https://doi.org/10.2478/cm-2021-0017","url":null,"abstract":"Abstract We describe the images of multilinear polynomials of arbitrary degree evaluated on the 3×3 upper triangular matrix algebra over an infinite field.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":"373 1","pages":"183 - 186"},"PeriodicalIF":0.0,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77979987","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We give the description of Rota–Baxter operators, Reynolds operators, Nijenhuis operators and average operators on 3-dimensional nilpotent associative algebras over ℂ.
{"title":"Rota-type operators on 3-dimensional nilpotent associative algebras","authors":"N.G. Abdujabborov, I. Karimjanov, M. A. Kodirova","doi":"10.2478/cm-2021-0020","DOIUrl":"https://doi.org/10.2478/cm-2021-0020","url":null,"abstract":"Abstract We give the description of Rota–Baxter operators, Reynolds operators, Nijenhuis operators and average operators on 3-dimensional nilpotent associative algebras over ℂ.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":"29 1","pages":"227 - 241"},"PeriodicalIF":0.0,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48097072","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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