This paper determined the components of the generalized curvature tensor for the class of Kenmotsu type and established the mentioned class is {eta}-Einstein manifold when the generalized curvature tensor is flat; the converse holds true under suitable conditions. It also introduced the notion of generalized {Phi}-holomorphic sectional (G{Phi}SH-) curvature tensor and thus found the necessary and sufficient conditions for the class of Kenmotsu type to be of constant G{Phi}SH-curvature. In addition, the notion of {Phi}-generalized semi-symmetric was introduced and its relationship with the class of Kenmotsu type and {eta}-Einstein manifold established. Furthermore, this paper generalized the notion of the manifold of constant curvature and deduced its relationship with the aforementioned ideas. It finally showed that the class of Kenmotsu type exists as a hypersurface of the Hermitian manifold and derived a relation between the components of the Riemannian curvature tensors of the almost Hermitian manifold and its hypersurfaces.
{"title":"Generalized curvature tensor and the hypersurfaces of the Hermitian manifold for the class of Kenmotsu type","authors":"M. Y. Abass, H. M. Abood","doi":"10.46298/cm.10869","DOIUrl":"https://doi.org/10.46298/cm.10869","url":null,"abstract":"This paper determined the components of the generalized curvature tensor for\u0000the class of Kenmotsu type and established the mentioned class is\u0000{eta}-Einstein manifold when the generalized curvature tensor is flat; the\u0000converse holds true under suitable conditions. It also introduced the notion of\u0000generalized {Phi}-holomorphic sectional (G{Phi}SH-) curvature tensor and thus\u0000found the necessary and sufficient conditions for the class of Kenmotsu type to\u0000be of constant G{Phi}SH-curvature. In addition, the notion of\u0000{Phi}-generalized semi-symmetric was introduced and its relationship with the\u0000class of Kenmotsu type and {eta}-Einstein manifold established. Furthermore,\u0000this paper generalized the notion of the manifold of constant curvature and\u0000deduced its relationship with the aforementioned ideas. It finally showed that\u0000the class of Kenmotsu type exists as a hypersurface of the Hermitian manifold\u0000and derived a relation between the components of the Riemannian curvature\u0000tensors of the almost Hermitian manifold and its hypersurfaces.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45767268","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 conjectured since long that for any convex body $K subset mathbb{R}^n$ there exists a point in the interior of $K$ which belongs to at least $2n$ normals from different points on the boundary of $K$. The conjecture is known to be true for $n=2,3,4$. Motivated by a recent results of Y. Martinez-Maure, and an approach by A. Grebennikov and G. Panina, we prove the following: Let a compact smooth $m$-dimensional manifold $M^m$ be immersed in $ mathbb{R}^n$. We assume that at least one of the homology groups $H_k(M^m,mathbb{Z}_2)$ with $k
长久以来，我们推测对于任何凸体$K 子集$ mathbb{R}^n$，在$K$的内部存在一个点，该点至少属于$K$边界上不同点的$2n$法线。这个猜想对于n=2,3,4是成立的。根据Y. Martinez-Maure最近的结果，以及a . grebennikov和G. Panina的一种方法，我们证明了以下问题:设一个紧化光滑$m$维流形$m ^m$浸入$ mathbb{R}^n$中。我们假设在k< M的同调群$H_k(M^ M，mathbb{Z}_2)$中至少有一个不存在。然后在温和的条件下，几乎每条到$M^ M $的法线都包含至少$beta +4$从$M^ M $的不同点来的法线的交点，其中$beta$是$M^ M $的贝蒂数之和。
{"title":"Concurrent normals of immersed manifolds","authors":"G. Panina, D. Siersma","doi":"10.46298/cm.10840","DOIUrl":"https://doi.org/10.46298/cm.10840","url":null,"abstract":"It is conjectured since long that for any convex body $K subset\u0000mathbb{R}^n$ there exists a point in the interior of $K$ which belongs to at\u0000least $2n$ normals from different points on the boundary of $K$. The conjecture\u0000is known to be true for $n=2,3,4$.\u0000 Motivated by a recent results of Y. Martinez-Maure, and an approach by A.\u0000Grebennikov and G. Panina, we prove the following: Let a compact smooth\u0000$m$-dimensional manifold $M^m$ be immersed in $ mathbb{R}^n$. We assume that\u0000at least one of the homology groups $H_k(M^m,mathbb{Z}_2)$ with $k<m$\u0000vanishes. Then under mild conditions, almost every normal line to $M^m$\u0000contains an intersection point of at least $beta +4$ normals from different\u0000points of $M^m$, where $beta$ is the sum of Betti numbers of $M^m$.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44046443","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 $mathbf L_k$ be the holomorphic line bundle of degree $k in mathbb Z$ on the projective line. Here, the tuples $(k_1 k_2 k_3 k_4)$ for which there does not exists homogeneous non-split supermanifolds $CP^{1|4}_{k_1 k_2 k_3 k_4}$ associated with the vector bundle $mathbf L_{−k_1} oplus mathbf L _{−k_2} oplus mathbf L_{−k_3} oplus mathbf L_{−k_4}$ are classified. For many types of the remaining tuples, there are listed cocycles that determine homogeneous non-split supermanifolds. Proofs follow the lines indicated in the paper Bunegina V.A., Onishchik A.L., Homogeneous supermanifolds associated with the complex projective line.neous supermanifolds associated with the complex projective line. J. Math. Sci. V. 82 (1996)3503--3527.
{"title":"Homogeneous non-split superstrings of odd dimension 4","authors":"M. Bashkin","doi":"10.46298/cm.9843","DOIUrl":"https://doi.org/10.46298/cm.9843","url":null,"abstract":"Let $mathbf L_k$ be the holomorphic line bundle of degree $k in mathbb Z$ on the projective line. Here, the tuples $(k_1 k_2 k_3 k_4)$ for which there does not exists homogeneous non-split supermanifolds $CP^{1|4}_{k_1 k_2 k_3 k_4}$ associated with the vector bundle $mathbf L_{−k_1} oplus mathbf L _{−k_2} oplus mathbf L_{−k_3} oplus mathbf L_{−k_4}$ are classified. For many types of the remaining tuples, there are listed cocycles that determine homogeneous non-split supermanifolds. Proofs follow the lines indicated in the paper Bunegina V.A., Onishchik A.L., Homogeneous supermanifolds associated with the complex projective line.neous supermanifolds associated with the complex projective line. J. Math. Sci. V. 82 (1996)3503--3527.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46871931","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 classify the left-invariant sub-Riemannian structures on the unique five-dimensional simply connected two-step nilpotent Lie group with two-dimensional commutator subgroup; this 5D group is the first twostep nilpotent Lie group beyond the three-and five-dimensional Heisenberg groups. Alongside, we also present a classification, up to automorphism, of the subspaces of the associated Lie algebra (together with a complete set of invariants).
{"title":"On the classification of sub-Riemannian structures on a 5D two-step nilpotent Lie group","authors":"R. Biggs, Odirile Ntshudisane","doi":"10.46298/cm.10550","DOIUrl":"https://doi.org/10.46298/cm.10550","url":null,"abstract":"We classify the left-invariant sub-Riemannian structures on the unique five-dimensional simply connected two-step nilpotent Lie group with two-dimensional commutator subgroup; this 5D group is the first twostep nilpotent Lie group beyond the three-and five-dimensional Heisenberg groups. Alongside, we also present a classification, up to automorphism, of the subspaces of the associated Lie algebra (together with a complete set of invariants).","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44639134","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}
Any complex-analytic supermanifold whose retract is diffeomorphic to the complex projective superline (superstring) $CP^{1|4}$ is, up to a diffeomorphism, either a member of a 1-parameter family or one of 9 exceptional supermanifolds. I singled out the homogeneous of these supermanifolds and described Lie superalgebras of vector fields on them.
{"title":"Homogeneous superstrings with retract $CP^{1|4}$","authors":"M. Bashkin","doi":"10.46298/cm.9842","DOIUrl":"https://doi.org/10.46298/cm.9842","url":null,"abstract":"Any complex-analytic supermanifold whose retract is diffeomorphic to the complex projective superline (superstring) $CP^{1|4}$ is, up to a diffeomorphism, either a member of a 1-parameter family or one of 9 exceptional supermanifolds. I singled out the homogeneous of these supermanifolds and described Lie superalgebras of vector fields on them.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49014841","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}
All supermanifolds whose retract $T^{m|n}$ is determined by the trivial bundle of rank $n$ over the torus $T^m$ are $overline 0$-homogeneous if and only if $T^{m|n}$ is homogeneous.
{"title":"Supermanifolds corresponding to the trivial vector bundle over torus","authors":"M. Bashkin","doi":"10.46298/cm.9844","DOIUrl":"https://doi.org/10.46298/cm.9844","url":null,"abstract":"All supermanifolds whose retract $T^{m|n}$ is determined by the trivial bundle of rank $n$ over the torus $T^m$ are $overline 0$-homogeneous if and only if $T^{m|n}$ is homogeneous.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46926207","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 give simple extension and uniqueness theorems for restricted additive and logarithmic functional equations.
本文给出了限制加法和对数函数方程的简单推广和唯一性定理。
{"title":"Existence and uniqueness theorems for functional equations","authors":"T. Glavosits, Zsolt Kar'acsony","doi":"10.46298/cm.10830","DOIUrl":"https://doi.org/10.46298/cm.10830","url":null,"abstract":"In this paper we give simple extension and uniqueness theorems for restricted\u0000additive and logarithmic functional equations.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44158788","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 the present paper, we give sufficient conditions on the elements of the continued fractions $A$ and $B$ that will assure us that the continued fraction $A^B$ is a transcendental number. With the same condition, we establish a transcendental measure of $A^B.$
{"title":"Transcendental Continued Fractions","authors":"S. Ahallal, A. Kacha","doi":"10.46298/cm.10519","DOIUrl":"https://doi.org/10.46298/cm.10519","url":null,"abstract":"In the present paper, we give sufficient conditions on the elements of the\u0000continued fractions $A$ and $B$ that will assure us that the continued fraction\u0000$A^B$ is a transcendental number. With the same condition, we establish a\u0000transcendental measure of $A^B.$","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48209168","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}
Zahia Ezzourgui, Hafida Saggou, Megdouda Ourbih-Tari, E. Bourennane
This paper studies the stationary analysis of a Markovian queuing system with Bernoulli feedback, interruption vacation, linear impatient customers, strong and weak disaster with the server's repair during the server's operational vacation period. Each customer has its own impatience time and abandons the system as soon as that time ends. When the queue is not empty, the server's operational vacation can be interrupted if the service is completed and the server starts a busy period with a probability q or continues the operational vacation with a probability q. A strong disaster forces simultaneously all present customers (waiting and served) to abandon the system permanently with a probability p but a weak disaster is that all customers decide to be patient by staying in the system, and wait during the repair time with a probability p, where arrival of a new customer can occur. As soon as the repair process of the server is completed, the server remains providing service in the operational vacation period. We analyze this proposed model and derive the probabilities generating functions of the number of customers present in the system together with explicit expressions of some performance measures such as the mean and the variance of the number of customers in the different states, together with the mean sojourn time. Finally, numerical results are presented to show the influence of the system parameters on some studied performance measures.
{"title":"The Markovian Bernoulli queues with operational server vacation, Bernoulli's weak and strong disasters, and linear impatient customers","authors":"Zahia Ezzourgui, Hafida Saggou, Megdouda Ourbih-Tari, E. Bourennane","doi":"10.46298/cm.10404","DOIUrl":"https://doi.org/10.46298/cm.10404","url":null,"abstract":"This paper studies the stationary analysis of a Markovian queuing system with Bernoulli feedback, interruption vacation, linear impatient customers, strong and weak disaster with the server's repair during the server's operational vacation period. Each customer has its own impatience time and abandons the system as soon as that time ends. When the queue is not empty, the server's operational vacation can be interrupted if the service is completed and the server starts a busy period with a probability q or continues the operational vacation with a probability q. A strong disaster forces simultaneously all present customers (waiting and served) to abandon the system permanently with a probability p but a weak disaster is that all customers decide to be patient by staying in the system, and wait during the repair time with a probability p, where arrival of a new customer can occur. As soon as the repair process of the server is completed, the server remains providing service in the operational vacation period. We analyze this proposed model and derive the probabilities generating functions of the number of customers present in the system together with explicit expressions of some performance measures such as the mean and the variance of the number of customers in the different states, together with the mean sojourn time. Finally, numerical results are presented to show the influence of the system parameters on some studied performance measures.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45881875","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 study we find all solutions of the Diophantine equation $B_{n_{1}}+B_{n_{2}}=2^{a_{1}}+2^{a_{2}}+2^{a_{3}}$ in positive integer variables $(n_{1},n_{2},a_{1},a_{2},a_{3}),$ where $B_{n}$ denotes the $n$-th balancing number.
{"title":"On the Diophantine equation $B_{n_{1}}+B_{n_{2}}=2^{a_{1}}+2^{a_{2}}+2^{a_{3}}$","authors":"K. Bhoi, P. Ray","doi":"10.46298/cm.10476","DOIUrl":"https://doi.org/10.46298/cm.10476","url":null,"abstract":"In this study we find all solutions of the Diophantine equation\u0000$B_{n_{1}}+B_{n_{2}}=2^{a_{1}}+2^{a_{2}}+2^{a_{3}}$ in positive integer\u0000variables $(n_{1},n_{2},a_{1},a_{2},a_{3}),$ where $B_{n}$ denotes the $n$-th\u0000balancing number.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44040216","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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