By combining the telescoping method with Cassini–like formulae, we evaluate, in closed forms, four classes of sums about products of two arctangent functions with their argument involving Pell and Pell–Lucas polynomials. Several infinite series identities for Fibonacci and Lucas numbers are deduced as consequences.
{"title":"Arctangent Series on Pell and Pell-Lucas Polynomials via Cassini-Like Formulae","authors":"Dongwei Guo, Wenchang Chu","doi":"10.61091/um118-01","DOIUrl":"https://doi.org/10.61091/um118-01","url":null,"abstract":"By combining the telescoping method with Cassini–like formulae, we evaluate, in closed forms, four classes of sums about products of two arctangent functions with their argument involving Pell and Pell–Lucas polynomials. Several infinite series identities for Fibonacci and Lucas numbers are deduced as consequences.","PeriodicalId":49389,"journal":{"name":"Utilitas Mathematica","volume":"85 2","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139629328","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 G=(V(G),E(G)) be a graph with p vertices and q edges. A graph G of size q is said to be odd graceful if there exists an injection λ:V(G)→0,1,2,…,2q−1 such that assigning each edge xy the label or weight |λ(x)–λ(y)| results in the set of edge labels being 1,3,5,…,2q−1. This concept was introduced in 1991 by Gananajothi. In this paper, we examine the odd graceful labeling of the W-tree, denoted as WT(n,k).
假设 G=(V(G),E(G)) 是一个有 p 个顶点和 q 条边的图。如果存在注入λ:V(G)→0,1,2,......,2q-1，给每条边 xy 赋上标签或权重|λ(x)-λ(y)|，结果边标签集为 1,3,5,......,2q-1，那么大小为 q 的图 G 称为奇数优美图。这一概念由 Gananajothi 于 1991 年提出。在本文中，我们将研究 W 树的奇数优美标签，记为 WT(n,k)。
{"title":"Odd Graceful Labeling of W -Tree W T ( n , k ) and its Disjoint Union","authors":"Abaid ur Rehman Virk, A. Riasat","doi":"10.61091/um118-05","DOIUrl":"https://doi.org/10.61091/um118-05","url":null,"abstract":"Let G=(V(G),E(G)) be a graph with p vertices and q edges. A graph G of size q is said to be odd graceful if there exists an injection λ:V(G)→0,1,2,…,2q−1 such that assigning each edge xy the label or weight |λ(x)–λ(y)| results in the set of edge labels being 1,3,5,…,2q−1. This concept was introduced in 1991 by Gananajothi. In this paper, we examine the odd graceful labeling of the W-tree, denoted as WT(n,k).","PeriodicalId":49389,"journal":{"name":"Utilitas Mathematica","volume":"231 4","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139628746","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}
Mankagna Albert Diompy, Alhousseynou Ba, Andé Souleye Diabang
A module M over a commutative ring is termed an SCDF-module if every Dedekind finite object in σ[M] is finitely cogenerated. Utilizing this concept, we explore several properties and characterize various types of SCDF-modules. These include local SCDF-modules, finitely generated $SCDF$-modules, and hollow SCDF-modules with Rad(M)=0≠M. Additionally, we examine QF SCDF-odules in the context of duo-ring.
{"title":"On S C D F -Modules","authors":"Mankagna Albert Diompy, Alhousseynou Ba, Andé Souleye Diabang","doi":"10.61091/um118-06","DOIUrl":"https://doi.org/10.61091/um118-06","url":null,"abstract":"A module M over a commutative ring is termed an SCDF-module if every Dedekind finite object in σ[M] is finitely cogenerated. Utilizing this concept, we explore several properties and characterize various types of SCDF-modules. These include local SCDF-modules, finitely generated $SCDF$-modules, and hollow SCDF-modules with Rad(M)=0≠M. Additionally, we examine QF SCDF-odules in the context of duo-ring.","PeriodicalId":49389,"journal":{"name":"Utilitas Mathematica","volume":"141 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139628874","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}
Mankagna Albert Diompy, Ousseynou Bousso, Remy Diaga Diouf, Oumar Diankha
In this paper, we utilize the σ category to introduce EKFN-modules, which extend the concept of the EKFN-ring. After presenting some properties, we demonstrate, under certain hypotheses, that if M is an EKFN-module, then the following equivalences hold: the class of uniserial modules coincides with the class of cu-uniserial modules; EKFN-modules correspond to the class of locally noetherian modules; and the class of CD-modules is a subset of the EKFN-modules.
{"title":"EKFN-Modules","authors":"Mankagna Albert Diompy, Ousseynou Bousso, Remy Diaga Diouf, Oumar Diankha","doi":"10.61091/um118-03","DOIUrl":"https://doi.org/10.61091/um118-03","url":null,"abstract":"In this paper, we utilize the σ category to introduce EKFN-modules, which extend the concept of the EKFN-ring. After presenting some properties, we demonstrate, under certain hypotheses, that if M is an EKFN-module, then the following equivalences hold: the class of uniserial modules coincides with the class of cu-uniserial modules; EKFN-modules correspond to the class of locally noetherian modules; and the class of CD-modules is a subset of the EKFN-modules.","PeriodicalId":49389,"journal":{"name":"Utilitas Mathematica","volume":"116 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139629155","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 a novel type of convex function known as the refined modified (h,m)-convex function, which is a generalization of the traditional (h,m)-convex function. We establish Hadamard-type inequalities for this new definition by utilizing the Caputo k-fractional derivative. Specifically, we derive two integral identities that involve the nth order derivatives of given functions and use them to prove the estimation of Hadamard-type inequalities for the Caputo k-fractional derivatives of refined modified (h,m)-convex functions. The results obtained in this research demonstrate the versatility of the refined modified (h,m)-convex function and the usefulness of Caputo k-fractional derivatives in establishing important inequalities. Our work contributes to the existing body of knowledge on convex functions and offers insights into the applications of fractional calculus in mathematical analysis. The research findings have the potential to pave the way for future studies in the area of convex functions and fractional calculus, as well as in other areas of mathematical research.
本文介绍了一种新型的凸函数，即精炼修正的（h,m）-凸函数，它是对传统（h,m）-凸函数的概括。我们利用卡普托 k 分数导数为这一新定义建立了哈达玛式不等式。具体地说，我们推导出涉及给定函数 n 阶导数的两个积分等式，并利用它们证明了哈达玛式不等式对改进的（h,m）凸函数的卡普托 k 分导数的估计。这项研究获得的结果证明了精炼修正 (h,m) 凸函数的多功能性，以及卡普托 k 分导数在建立重要不等式方面的实用性。我们的研究为凸函数的现有知识体系做出了贡献，并为分数微积分在数学分析中的应用提供了启示。这些研究成果有可能为凸函数和分数微积分领域以及其他数学研究领域的未来研究铺平道路。
{"title":"Caputo Fractional Derivative Inequalities for Refined Modified ( h , m ) -Convex Functions","authors":"Muhammad Ajmal, Muhammad Rafaqat, Labeeb Ahmad","doi":"10.61091/um118-04","DOIUrl":"https://doi.org/10.61091/um118-04","url":null,"abstract":"This paper introduces a novel type of convex function known as the refined modified (h,m)-convex function, which is a generalization of the traditional (h,m)-convex function. We establish Hadamard-type inequalities for this new definition by utilizing the Caputo k-fractional derivative. Specifically, we derive two integral identities that involve the nth order derivatives of given functions and use them to prove the estimation of Hadamard-type inequalities for the Caputo k-fractional derivatives of refined modified (h,m)-convex functions. The results obtained in this research demonstrate the versatility of the refined modified (h,m)-convex function and the usefulness of Caputo k-fractional derivatives in establishing important inequalities. Our work contributes to the existing body of knowledge on convex functions and offers insights into the applications of fractional calculus in mathematical analysis. The research findings have the potential to pave the way for future studies in the area of convex functions and fractional calculus, as well as in other areas of mathematical research.","PeriodicalId":49389,"journal":{"name":"Utilitas Mathematica","volume":"113 3","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139628951","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 G be a finite solvable group and Δ be the subset of Υ×Υ, where Υ is the set of all pairs of size two commuting elements in G. If G operates on a transitive G – space by the action (υ1,υ2)g=(υg1,υg2); υ1,υ2∈Υ and g∈G, then orbits of G are called orbitals. The subset Δo={(υ,υ);υ∈Υ,(υ,υ)∈Υ×Υ} represents G′s diagonal orbital.The orbital regular graph is a graph on which G acts regularly on the vertices and the edge set. In this paper, we obtain the orbital regular graphs for some finite solvable groups using a regular action. Furthermore, the number of edges for each of a group’s orbitals is obtained.
如果 G 通过作用 (υ1,υ2)g=(υg1,υg2); υ1,υ2∈Υ 和 g∈G 作用于传递 G - 空间，那么 G 的轨道称为轨道。子集 Δo={(υ,υ);υ∈Υ,(υ,υ)∈Υ×Υ} 表示 G′的对角轨道。轨道正则图是 G 规则地作用于顶点和边集的图。本文利用正则作用得到了一些有限可解群的轨道正则图。此外，我们还得到了每个群轨道的边数。
{"title":"On the Orbital Regular Graph of Finite Solvable Groups","authors":"Karnika Sharma, Vijay Kumar Bhat, P. Singh","doi":"10.61091/um118-02","DOIUrl":"https://doi.org/10.61091/um118-02","url":null,"abstract":"Let G be a finite solvable group and Δ be the subset of Υ×Υ, where Υ is the set of all pairs of size two commuting elements in G. If G operates on a transitive G – space by the action (υ1,υ2)g=(υg1,υg2); υ1,υ2∈Υ and g∈G, then orbits of G are called orbitals. The subset Δo={(υ,υ);υ∈Υ,(υ,υ)∈Υ×Υ} represents G′s diagonal orbital.The orbital regular graph is a graph on which G acts regularly on the vertices and the edge set. In this paper, we obtain the orbital regular graphs for some finite solvable groups using a regular action. Furthermore, the number of edges for each of a group’s orbitals is obtained.","PeriodicalId":49389,"journal":{"name":"Utilitas Mathematica","volume":"211 4","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139628783","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}
G. Chartrand, Henry Escuadro, F. Okamoto, Ping Zhang
{"title":"Detectable colorings of graphs","authors":"G. Chartrand, Henry Escuadro, F. Okamoto, Ping Zhang","doi":"10.21136/mb.2005.134214","DOIUrl":"https://doi.org/10.21136/mb.2005.134214","url":null,"abstract":"","PeriodicalId":49389,"journal":{"name":"Utilitas Mathematica","volume":"14 1","pages":"13-32"},"PeriodicalIF":0.0,"publicationDate":"2006-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68443497","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}
{"title":"An optimization theorem with applications in some mathematical programming problems","authors":"C. R. Bector, B. L. Bhatia","doi":"10.5555/14033.14048","DOIUrl":"https://doi.org/10.5555/14033.14048","url":null,"abstract":"","PeriodicalId":49389,"journal":{"name":"Utilitas Mathematica","volume":"26 1","pages":"249-258"},"PeriodicalIF":0.0,"publicationDate":"1984-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71115822","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}
{"title":"The development of a time-domain model for transients on lossy, distorting coaxil cables","authors":"D. Meek","doi":"10.5555/14033.14037","DOIUrl":"https://doi.org/10.5555/14033.14037","url":null,"abstract":"","PeriodicalId":49389,"journal":{"name":"Utilitas Mathematica","volume":"26 1","pages":"33-44"},"PeriodicalIF":0.0,"publicationDate":"1984-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71115705","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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