We prove a fairly general inequality that estimates the number of lattice points in a ball of positive radius in general position in a Euclidean space. The bound is uniform over lattices induced by a matrix having a bounded operator norm.
{"title":"On the number of lattice points in a ball","authors":"Jeffrey D. Vaaler","doi":"10.46298/cm.11119","DOIUrl":"https://doi.org/10.46298/cm.11119","url":null,"abstract":"We prove a fairly general inequality that estimates the number of lattice\u0000points in a ball of positive radius in general position in a Euclidean space.\u0000The bound is uniform over lattices induced by a matrix having a bounded\u0000operator norm.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41773419","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 (M, g, e −f dv) be a smooth metric measure space. We consider local gradient estimates for positive solutions to the following elliptic equation ∆ f u + au log u + bu = 0 where a, b are two real constants and f be a smooth function defined on M. As an application, we obtain a Liouville type result for such equation in the case a < 0 under the m-dimensions Bakry-Émery Ricci curvature.
{"title":"Gradient estimates for a nonlinear elliptic equation on a smooth metric measure space","authors":"Xiaoshan Wang, Linfen Cao","doi":"10.46298/cm.10951","DOIUrl":"https://doi.org/10.46298/cm.10951","url":null,"abstract":"Let (M, g, e −f dv) be a smooth metric measure space. We consider local gradient estimates for positive solutions to the following elliptic equation ∆ f u + au log u + bu = 0 where a, b are two real constants and f be a smooth function defined on M. As an application, we obtain a Liouville type result for such equation in the case a < 0 under the m-dimensions Bakry-Émery Ricci curvature.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41867795","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 $mathds{k}$ be a real quadratic number field. Denote by $mathrm{Cl}_2(mathds{k})$ its $2$-class group and by $mathds{k}_2^{(1)}$ (resp. $mathds{k}_2^{(2)}$) its first (resp. second) Hilbert $2$-class field. The aim of this paper is to study, for a real quadratic number field whose discriminant is divisible by one prime number congruent to $3$ modulo 4, the metacyclicity of $G=mathrm{Gal}(mathds{k}_2^{(2)}/mathds{k})$ and the cyclicity of $mathrm{Gal}(mathds{k}_2^{(2)}/mathds{k}_2^{(1)})$ whenever the rank of $mathrm{Cl}_2(mathds{k})$ is $2$, and the $4$-rank of $mathrm{Cl}_2(mathds{k})$ is $1$.
{"title":"Cyclicity of the 2-class group of the first Hilbert 2-class field of some number fields","authors":"A. Azizi, M. Rezzougui, A. Zekhnini","doi":"10.46298/cm.10983","DOIUrl":"https://doi.org/10.46298/cm.10983","url":null,"abstract":"Let $mathds{k}$ be a real quadratic number field. Denote by\u0000$mathrm{Cl}_2(mathds{k})$ its $2$-class group and by $mathds{k}_2^{(1)}$\u0000(resp. $mathds{k}_2^{(2)}$) its first (resp. second) Hilbert $2$-class field.\u0000The aim of this paper is to study, for a real quadratic number field whose\u0000discriminant is divisible by one prime number congruent to $3$ modulo 4, the\u0000metacyclicity of $G=mathrm{Gal}(mathds{k}_2^{(2)}/mathds{k})$ and the\u0000cyclicity of $mathrm{Gal}(mathds{k}_2^{(2)}/mathds{k}_2^{(1)})$ whenever the\u0000rank of $mathrm{Cl}_2(mathds{k})$ is $2$, and the $4$-rank of\u0000$mathrm{Cl}_2(mathds{k})$ is $1$.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46032598","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 aim of this paper is to study theCPE (Critical Point Equation) on some paracontact metric manifolds.First, we prove that if a para-Sasakian metric satisfies the CPE,then it is Einstein with constant scalar curvature -2n(2n+1). Next,we prove that if $(kappa,mu)$-paracontact metric satisfies theCPE, then it is locally isometric to the product of a flat$(n+1)$-dimensional manifold and $n$-dimensional manifold ofnegative constant curvature$-4$.
{"title":"Certain Paracontact Metrics Satisfying the Critical Point Equation","authors":"D. Patra","doi":"10.46298/cm.10549","DOIUrl":"https://doi.org/10.46298/cm.10549","url":null,"abstract":"The aim of this paper is to study theCPE (Critical Point Equation) on some paracontact metric manifolds.First, we prove that if a para-Sasakian metric satisfies the CPE,then it is Einstein with constant scalar curvature -2n(2n+1). Next,we prove that if $(kappa,mu)$-paracontact metric satisfies theCPE, then it is locally isometric to the product of a flat$(n+1)$-dimensional manifold and $n$-dimensional manifold ofnegative constant curvature$-4$.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41523437","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 article, we extend Z. H. Sun's congruences concerning Legendre polynomials P p−1 2 (x) to P p+1 2 (x) for odd prime p, which enables us to deduce some congruences resembling p+1 2 ∑ k=0 4pk + 4k 2 − 1 16 k (2k − 1) 2 (2k k)2 (mod p 2). 이 논문에서 우리는 Z. H. Sun의 르장드르 다항식의 합동식 P p−1 2 (x) 에서 P p+1 2 (x) (단, p는 소수) 까지를 이용해서 이 합동식과 비슷한 합동식 p+1 2 ∑ k=0 4pk + 4k 2 − 1 16 k (2k − 1) 2 (2k k)2 (mod p 2) 을 유도한다.
In this article, we extend Z. H. Sun's congruences concerning Legendre polynomials P P - 1 2 (x) to P P +1 2 (x) for odd prime P;which enables us to deduce some congruences resembling p + 1 2∑k = 0 4pk 16 k (2k + 4k 2 - 1, - 1) 2 (2k k) 2 (mod p 2)。这篇论文中,我们z·h·sun的章吱扭다항식的联合式p 2 (x)在p - 1, p = p + 1 2 (x)段,p是质数)到利用类似联合式的联合式p + 1 2∑k = 0 4pk 16 k (2k + 4k 2 - 1, - 1) 2 (2k k) 2 (mod p 2)诱导的。
{"title":"CONGRUENCES CONCERNING LEGENDRE POLYNOMIALS MODULO p^2","authors":"Aeran Kim","doi":"10.46298/cm.10767","DOIUrl":"https://doi.org/10.46298/cm.10767","url":null,"abstract":"In this article, we extend Z. H. Sun's congruences concerning Legendre polynomials P p−1 2 (x) to P p+1 2 (x) for odd prime p, which enables us to deduce some congruences resembling p+1 2 ∑ k=0 4pk + 4k 2 − 1 16 k (2k − 1) 2 (2k k)2 (mod p 2).\u0000 이 논문에서 우리는 Z. H. Sun의 르장드르 다항식의 합동식 P p−1 2 (x) 에서 P p+1 2 (x) (단, p는 소수) 까지를 이용해서 이 합동식과 비슷한 합동식 p+1 2 ∑ k=0 4pk + 4k 2 − 1 16 k (2k − 1) 2 (2k k)2 (mod p 2) 을 유도한다.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43163509","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 object of this paper is to study generalized φ-recurrent almost Kenmotsu manifolds with characteristic vector field ξ belonging to (k, µ)-nullity distribution. We have showed that these manifolds reduce to Kenmotsu manifolds with scalar curvature-1. Further we establish the relations among the associated 1-forms and proved the conditions under which gradient Ricci almost soliton reduce to gradient Ricci soliton.
{"title":"Almost Kenmotsu Manifolds","authors":"H. Nagaraja, Uppara Manjulamma","doi":"10.46298/cm.10925","DOIUrl":"https://doi.org/10.46298/cm.10925","url":null,"abstract":"The object of this paper is to study generalized φ-recurrent almost Kenmotsu manifolds with characteristic vector field ξ belonging to (k, µ)-nullity distribution. We have showed that these manifolds reduce to Kenmotsu manifolds with scalar curvature-1. Further we establish the relations among the associated 1-forms and proved the conditions under which gradient Ricci almost soliton reduce to gradient Ricci soliton.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48661119","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 manuscripts, we consider the coupled differential-integral equations including the variable-order Caputo fractional operator. To solve numerically these type of equations, we apply the shifted Jacobi-Gauss collocation scheme. Using this numerical method a system of algebraic equations is constructed. We solve this system with a recursive method in the nonlinear case and we solve it in linear case with algebraic formulas. Finally, for the high performance of the suggested method three Examples are illustrated.
{"title":"A numerical technique for solving variable order time fractional differential-integro equations","authors":"M. Derakhshan","doi":"10.46298/cm.10822","DOIUrl":"https://doi.org/10.46298/cm.10822","url":null,"abstract":"In this manuscripts, we consider the coupled differential-integral equations including the variable-order Caputo fractional operator. To solve numerically these type of equations, we apply the shifted Jacobi-Gauss collocation scheme. Using this numerical method a system of algebraic equations is constructed. We solve this system with a recursive method in the nonlinear case and we solve it in linear case with algebraic formulas. Finally, for the high performance of the suggested method three Examples are illustrated.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43490304","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 dedicated to providing an introduction into multidimensional integer trigonometry. We start with an exposition of integer trigonometry in two dimensions, which was introduced in 2008, and use this to generalise these integer trigonometric functions to arbitrary dimension. We then move on to study the basic properties of integer trigonometric functions. We find integer trigonometric relations for transpose and adjacent simplicial cones, and for the cones which generate the same simplices. Additionally, we discuss the relationship between integer trigonometry, the Euclidean algorithm, and continued fractions. Finally, we use adjacent and transpose cones to introduce a notion of best approximations of simplicial cones. In two dimensions, this notion of best approximation coincides with the classical notion of the best approximations of real numbers.
{"title":"Multidimensional integer trigonometry","authors":"J. Blackman, James Dolan, O. Karpenkov","doi":"10.46298/cm.10919","DOIUrl":"https://doi.org/10.46298/cm.10919","url":null,"abstract":"This paper is dedicated to providing an introduction into multidimensional\u0000integer trigonometry. We start with an exposition of integer trigonometry in\u0000two dimensions, which was introduced in 2008, and use this to generalise these\u0000integer trigonometric functions to arbitrary dimension. We then move on to\u0000study the basic properties of integer trigonometric functions. We find integer\u0000trigonometric relations for transpose and adjacent simplicial cones, and for\u0000the cones which generate the same simplices. Additionally, we discuss the\u0000relationship between integer trigonometry, the Euclidean algorithm, and\u0000continued fractions. Finally, we use adjacent and transpose cones to introduce\u0000a notion of best approximations of simplicial cones. In two dimensions, this\u0000notion of best approximation coincides with the classical notion of the best\u0000approximations of real numbers.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41588888","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 give an overview of universal quadratic forms and lattices, focusing on the recent developments over the rings of integers in totally real number fields. In particular, we discuss indecomposable algebraic integers as one of the main tools.
{"title":"Universal quadratic forms and indecomposables in number fields: A survey","authors":"V. Kala","doi":"10.46298/cm.10896","DOIUrl":"https://doi.org/10.46298/cm.10896","url":null,"abstract":"We give an overview of universal quadratic forms and lattices, focusing on\u0000the recent developments over the rings of integers in totally real number\u0000fields. In particular, we discuss indecomposable algebraic integers as one of\u0000the main tools.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45239918","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 define a new subclass $lambda$-bi-pseudo-starlike functions of $Sigma$ related to shell-like curves connected with Fibonacci numbers and determine the initial Taylor-Maclaurin coefficients $|a_2|$ and $|a_3|$ for $finmathcal{PSL}_{Sigma}^lambda(tilde{p}(z)).$ Further we determine the Fekete-Szeg"{o} result for the function class $mathcal{PSL}_{Sigma}^lambda(tilde{p}(z))$ and for special cases, corollaries are stated which some of them are new and have not been studied so far.
{"title":"On $lambda-$ Pseudo bi-starlike functions related with Fibonacci numbers","authors":"K. Vijaya, G. Murugusundaramoorthy, H. Guney","doi":"10.46298/cm.10870","DOIUrl":"https://doi.org/10.46298/cm.10870","url":null,"abstract":"In this paper we define a new subclass $lambda$-bi-pseudo-starlike functions\u0000of $Sigma$ related to shell-like curves connected with Fibonacci numbers and\u0000determine the initial Taylor-Maclaurin coefficients $|a_2|$ and $|a_3|$ for\u0000$finmathcal{PSL}_{Sigma}^lambda(tilde{p}(z)).$ Further we determine the\u0000Fekete-Szeg\"{o} result for the function class\u0000$mathcal{PSL}_{Sigma}^lambda(tilde{p}(z))$ and for special cases,\u0000corollaries are stated which some of them are new and have not been studied so\u0000far.","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":"46463882","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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