Let $N(G)$ be the set of conjugacy classes sizes of $G$. We prove that if $N(G)=Omegatimes {1,n}$ for specific set $Omega$ of integers, then $Gsimeq Atimes B$ where $N(A)=Omega$, $N(B)={1,n}$, and $n$ is a power of prime.
{"title":"Structure of finite groups with restrictions on the set of conjugacy classes sizes","authors":"I. Gorshkov","doi":"10.46298/cm.9722","DOIUrl":"https://doi.org/10.46298/cm.9722","url":null,"abstract":"Let $N(G)$ be the set of conjugacy classes sizes of $G$. We prove that if\u0000$N(G)=Omegatimes {1,n}$ for specific set $Omega$ of integers, then\u0000$Gsimeq Atimes B$ where $N(A)=Omega$, $N(B)={1,n}$, and $n$ is a power of\u0000prime.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45530861","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}
In this paper we initiate the study of quasi Yamabe soliton on 3-dimensional contact metric manifold with Qvarphi=varphi Q and prove that if a 3-dimensional contact metric manifold M such that Qvarphi=varphi Q admits a quasi Yamabe soliton with non-zero soliton vector field V being point-wise collinear with the Reeb vector field {xi}, then V is a constant multiple of {xi}, the scalar curvature is constant and the manifold is Sasakian. Moreover, V is Killing. Finally, we prove that if M is a 3-dimensional compact contact metric manifold such that Qvarphi=varphi Q endowed with a quasi Yamabe soliton, then either M is flat or soliton is trivial.
{"title":"Quasi Yamabe Solitons on 3-Dimensional Contact Metric Manifolds with Qvarphi=varphi Q","authors":"V. Venkatesha, H. Kumara","doi":"10.46298/cm.9695","DOIUrl":"https://doi.org/10.46298/cm.9695","url":null,"abstract":"In this paper we initiate the study of quasi Yamabe soliton on 3-dimensional\u0000contact metric manifold with Qvarphi=varphi Q and prove that if a\u00003-dimensional contact metric manifold M such that Qvarphi=varphi Q admits a\u0000quasi Yamabe soliton with non-zero soliton vector field V being point-wise\u0000collinear with the Reeb vector field {xi}, then V is a constant multiple of\u0000{xi}, the scalar curvature is constant and the manifold is Sasakian. Moreover,\u0000V is Killing. Finally, we prove that if M is a 3-dimensional compact contact\u0000metric manifold such that Qvarphi=varphi Q endowed with a quasi Yamabe\u0000soliton, then either M is flat or soliton is trivial.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46923967","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 phrase "(co)simplicial (pre)sheaf" can be reasonably interpreted in multiple ways. In this survey we study how the various notions familiar to the author relate to one another. We end by giving some example applications of the most general of these notions.
{"title":"Various notions of (co)simplicial (pre)sheaves","authors":"Timothy Hosgood","doi":"10.46298/cm.10359","DOIUrl":"https://doi.org/10.46298/cm.10359","url":null,"abstract":"The phrase \"(co)simplicial (pre)sheaf\" can be reasonably interpreted in\u0000multiple ways. In this survey we study how the various notions familiar to the\u0000author relate to one another. We end by giving some example applications of the\u0000most general of these notions.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43102193","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}
Here, I study the problem of classification of non-split supermanifolds having as retract the split supermanifold $(M,Omega)$, where $Omega$ is the sheaf of holomorphic forms on a given complex manifold $M$ of dimension $> 1$. I propose a general construction associating with any $d$-closed $(1,1)$-form $omega$ on $M$ a supermanifold with retract $(M,Omega)$ which is non-split whenever the Dolbeault class of $omega$ is non-zero. In particular, this gives a non-empty family of non-split supermanifolds for any flag manifold $Mne mathbb{CP}^1$. In the case where $M$ is an irreducible compact Hermitian symmetric space, I get a complete classification of non-split supermanifolds with retract $(M,Omega)$. For each of these supermanifolds, the 0- and 1-cohomology with values in the tangent sheaf are calculated. As an example, I study the $Pi$-symmetric super-Grassmannians introduced by Yu. Manin.
{"title":"Non-split supermanifolds associated with the cotangent bundle","authors":"A. Onishchik","doi":"10.46298/cm.9613","DOIUrl":"https://doi.org/10.46298/cm.9613","url":null,"abstract":"Here, I study the problem of classification of non-split supermanifolds\u0000having as retract the split supermanifold $(M,Omega)$, where $Omega$ is the\u0000sheaf of holomorphic forms on a given complex manifold $M$ of dimension $> 1$.\u0000I propose a general construction associating with any $d$-closed $(1,1)$-form\u0000$omega$ on $M$ a supermanifold with retract $(M,Omega)$ which is non-split\u0000whenever the Dolbeault class of $omega$ is non-zero. In particular, this gives\u0000a non-empty family of non-split supermanifolds for any flag manifold $Mne\u0000mathbb{CP}^1$. In the case where $M$ is an irreducible compact Hermitian\u0000symmetric space, I get a complete classification of non-split supermanifolds\u0000with retract $(M,Omega)$. For each of these supermanifolds, the 0- and\u00001-cohomology with values in the tangent sheaf are calculated. As an example, I\u0000study the $Pi$-symmetric super-Grassmannians introduced by Yu. Manin.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42483634","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}
In this work, the fractional Lie symmetry method is used to find the exact solutions of the time-fractional coupled Drinfeld-Sokolov-Wilson equations with the Riemann-Liouville fractional derivative. Time-fractional coupled Drinfeld-Sokolov-Wilson equations are obtained by replacing the first-order time derivative to the fractional derivatives (FD) of order $alpha$ in the classical Drinfeld-Sokolov-Wilson (DSW) model. Using the fractional Lie symmetry method, the Lie symmetry generators are obtained. With the help of symmetry generators, FCDSW equations are reduced into fractional ordinary differential equations (FODEs) with Erd$acute{e}$lyi-Kober fractional differential operator. Also, we have obtained the exact solution of FCDSW equations and shown the effects of non-integer order derivative value on the solutions graphically. The effect of fractional order $alpha$ on the behavior of solutions is studied graphically. Finally, new conservation laws are constructed along with the formal Lagrangian and fractional generalization of Noether operators. It is quite interesting the exact analytic solutions are obtained in explicit form.
{"title":"Invariance analysis and some new exact analytic solutions of the time-fractional coupled Drinfeld-Sokolov-Wilson equations","authors":"Chauhan Astha, Arora Rajan","doi":"10.46298/cm.9283","DOIUrl":"https://doi.org/10.46298/cm.9283","url":null,"abstract":"In this work, the fractional Lie symmetry method is used to find the exact solutions of the time-fractional coupled Drinfeld-Sokolov-Wilson equations with the Riemann-Liouville fractional derivative. Time-fractional coupled Drinfeld-Sokolov-Wilson equations are obtained by replacing the first-order time derivative to the fractional derivatives (FD) of order $alpha$ in the classical Drinfeld-Sokolov-Wilson (DSW) model. Using the fractional Lie symmetry method, the Lie symmetry generators are obtained. With the help of symmetry generators, FCDSW equations are reduced into fractional ordinary differential equations (FODEs) with Erd$acute{e}$lyi-Kober fractional differential operator. Also, we have obtained the exact solution of FCDSW equations and shown the effects of non-integer order derivative value on the solutions graphically. The effect of fractional order $alpha$ on the behavior of solutions is studied graphically. Finally, new conservation laws are constructed along with the formal Lagrangian and fractional generalization of Noether operators. It is quite interesting the exact analytic solutions are obtained in explicit form.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43657614","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 introduces two new algorithms for Lie algebras over finite fields and applies them to the investigate the known simple Lie algebras of dimension at most $20$ over the field $mathbb{F}_2$ with two elements. The first algorithm is a new approach towards the construction of $mathbb{Z}_2$-gradings of a Lie algebra over a finite field of characteristic $2$. Using this, we observe that each of the known simple Lie algebras of dimension at most $20$ over $mathbb{F}_2$ has a $mathbb{Z}_2$-grading and we determine the associated simple Lie superalgebras. The second algorithm allows us to compute all subalgebras of a Lie algebra over a finite field. We apply this to compute the subalgebras, the maximal subalgebras and the simple subquotients of the known simple Lie algebras of dimension at most $16$ over $mathbb{F}_2$ (with the exception of the $15$-dimensional Zassenhaus algebra).
{"title":"Computing subalgebras and $mathbb{Z}_2$-gradings of simple Lie algebras over finite fields","authors":"B. Eick, T. Moede","doi":"10.46298/cm.10193","DOIUrl":"https://doi.org/10.46298/cm.10193","url":null,"abstract":"This paper introduces two new algorithms for Lie algebras over finite fields\u0000and applies them to the investigate the known simple Lie algebras of dimension\u0000at most $20$ over the field $mathbb{F}_2$ with two elements. The first\u0000algorithm is a new approach towards the construction of $mathbb{Z}_2$-gradings\u0000of a Lie algebra over a finite field of characteristic $2$. Using this, we\u0000observe that each of the known simple Lie algebras of dimension at most $20$\u0000over $mathbb{F}_2$ has a $mathbb{Z}_2$-grading and we determine the\u0000associated simple Lie superalgebras. The second algorithm allows us to compute\u0000all subalgebras of a Lie algebra over a finite field. We apply this to compute\u0000the subalgebras, the maximal subalgebras and the simple subquotients of the\u0000known simple Lie algebras of dimension at most $16$ over $mathbb{F}_2$ (with\u0000the exception of the $15$-dimensional Zassenhaus algebra).","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45708605","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 $mathfrak{d}$ be the Lie superalgebra of superderivations of the sheaf of sections of the exterior algebra of the homogeneous vector bundle $E$ over the flag variety $G/P$, where $G$ is a simple finite-dimensional complex Lie group and $P$ its parabolic subgroup. Then, $mathfrak{d}$ is transitive and irreducible whenever $E$ is defined by an irreducible $P$-module $V$ such that the highest weight of $V^*$ is dominant. Moreover, $mathfrak{d}$ is simple; it is isomorphic to the Lie superalgebra of vector fields on the superpoint, i.e., on a $0|n$-dimensional supervariety.
{"title":"Transitive irreducible Lie superalgebras of vector fields","authors":"A. Onishchik","doi":"10.46298/cm.10456","DOIUrl":"https://doi.org/10.46298/cm.10456","url":null,"abstract":"Let $mathfrak{d}$ be the Lie superalgebra of superderivations of the sheaf\u0000of sections of the exterior algebra of the homogeneous vector bundle $E$ over\u0000the flag variety $G/P$, where $G$ is a simple finite-dimensional complex Lie\u0000group and $P$ its parabolic subgroup. Then, $mathfrak{d}$ is transitive and\u0000irreducible whenever $E$ is defined by an irreducible $P$-module $V$ such that\u0000the highest weight of $V^*$ is dominant. Moreover, $mathfrak{d}$ is simple; it\u0000is isomorphic to the Lie superalgebra of vector fields on the superpoint, i.e.,\u0000on a $0|n$-dimensional supervariety.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44022356","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 "curved" super Grassmannian is the supervariety of subsupervarieties of purely odd dimension $k$ in a~supervariety of purely odd dimension $n$, unlike the "usual" super Grassmannian which is the supervariety of linear subsuperspacies of purely odd dimension $k$ in a~superspace of purely odd dimension $n$. The Lie superalgebras of all and Hamiltonian vector fields on the superpoint are realized as Lie superalgebras of derivations of the structure sheaves of certain "curved" super Grassmannians,
{"title":"Action of vectorial Lie superalgebras on some split supermanifolds","authors":"A. Onishchik","doi":"10.46298/cm.10455","DOIUrl":"https://doi.org/10.46298/cm.10455","url":null,"abstract":"The \"curved\" super Grassmannian is the supervariety of subsupervarieties of\u0000purely odd dimension $k$ in a~supervariety of purely odd dimension $n$, unlike\u0000the \"usual\" super Grassmannian which is the supervariety of linear\u0000subsuperspacies of purely odd dimension $k$ in a~superspace of purely odd\u0000dimension $n$. The Lie superalgebras of all and Hamiltonian vector fields on\u0000the superpoint are realized as Lie superalgebras of derivations of the\u0000structure sheaves of certain \"curved\" super Grassmannians,","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42132168","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 (mathcal{N}) be a~(3)-prime near ring and (alpha,beta: mathcal{N}rightarrow mathcal{N}) be endomorphisms. In the present paper we amplify a~few outcomes concerning generalized derivations and two-sided (alpha)-generalized derivations of (3)-prime near rings to generalized ((alpha,beta))-derivations. Cases demonstrating the need of the (3)-primeness speculation are given. When (beta = id_{mathcal{N}}) (resp. (alpha = beta = id_{mathcal{N}})), one can easily obtain the main results of~cite{ref1} (resp.cite{ref5}).
{"title":"On commutativity of 3-prime near-rings with generalized (α; β)-derivations","authors":"Abdelkarim BOUA, Ahmed Abdelwanis","doi":"10.46298/cm.9076","DOIUrl":"https://doi.org/10.46298/cm.9076","url":null,"abstract":"Let (mathcal{N}) be a~(3)-prime near ring and (alpha,beta: mathcal{N}rightarrow mathcal{N}) be endomorphisms. In the present paper we amplify a~few outcomes concerning generalized derivations and two-sided (alpha)-generalized derivations of (3)-prime near rings to generalized ((alpha,beta))-derivations. Cases demonstrating the need of the (3)-primeness speculation are given. When (beta = id_{mathcal{N}}) (resp. (alpha = beta = id_{mathcal{N}})), one can easily obtain the main results of~cite{ref1} (resp.cite{ref5}).","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44369797","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}
We prove the existence of a weak solution to the problem begin{equation*} begin{split} -Delta_{p}u+V(x)|u|^{p-2}u & =f(u,|nabla u|^{p-2}nabla u), u(x) & >0 forall xinmathbb{R}^{N}, end{split} end{equation*} where $Delta_{p}u=hbox{div}(|nabla u|^{p-2}nabla u)$ is the $p$-Laplace operator, $1
我们证明了问题 begin｛方程*｝ begin{split｝-Delta的弱解的存在性_{p}u+V（x）|u|^{p-2}u&=f（u，|nabla u|^｛p-2｝nabla u），u（x）&>0for all xinmathbb｛R｝^｛N｝，end｛split｝end{equipment*｝其中$Delta_{p}u=hbox｛div｝（|nabla u|^｛p-2｝nabla u）$是$p$-拉普拉斯算子，$1＜p＜N$，非线性$f:mathbb｛R｝timesmathbb{R｝^｛N｝rightarrowmathbb｛R}$是连续的，它取决于解的梯度。我们使用一种基于山口定理的迭代技术来证明我们的存在性结果。
{"title":"An existence result for $p$-Laplace equation with gradient nonlinearity\u0000 in $mathbb{R}^N$","authors":"Shilpa Gupta, G. Dwivedi","doi":"10.46298/cm.9316","DOIUrl":"https://doi.org/10.46298/cm.9316","url":null,"abstract":"We prove the existence of a weak solution to the problem begin{equation*}\u0000begin{split} -Delta_{p}u+V(x)|u|^{p-2}u & =f(u,|nabla u|^{p-2}nabla u), \u0000 u(x) & >0 forall xinmathbb{R}^{N}, end{split} end{equation*} where\u0000$Delta_{p}u=hbox{div}(|nabla u|^{p-2}nabla u)$ is the $p$-Laplace operator,\u0000$1<p<N$ and the nonlinearity\u0000$f:mathbb{R}timesmathbb{R}^{N}rightarrowmathbb{R}$ is continuous and it\u0000depends on gradient of the solution. We use an iterative technique based on the\u0000Mountain pass theorem to prove our existence result.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48145521","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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