Abstract We propose 𝒮𝒜, η−𝒮𝒜, η−𝒮 𝒜min, and 𝒮𝒜η,δ,ζ−contractions and notions of η−admissibility type b and ηb−regularity in parametric Nb-metric spaces to determine a unique fixed point, a unique fixed circle, and a greatest fixed disc. Further, we investigate the geometry of non-unique fixed points of a self mapping and demonstrate by illustrative examples that a circle or a disc in parametric Nb−metric space is not necessarily the same as a circle or a disc in a Euclidean space. Obtained outcomes are extensions, unifications, improvements, and generalizations of some of the well-known previous results. We provide non-trivial illustrations to exhibit the importance of our explorations. Towards the end, we resolve the system of linear equations to demonstrate the significance of our contractions in parametric Nb−metric space.
{"title":"On unique and non-unique fixed point in parametric Nb−metric spaces with application","authors":"Sudheer Petwal, A. Tomar, M. Joshi","doi":"10.2478/ausm-2022-0019","DOIUrl":"https://doi.org/10.2478/ausm-2022-0019","url":null,"abstract":"Abstract We propose 𝒮𝒜, η−𝒮𝒜, η−𝒮 𝒜min, and 𝒮𝒜η,δ,ζ−contractions and notions of η−admissibility type b and ηb−regularity in parametric Nb-metric spaces to determine a unique fixed point, a unique fixed circle, and a greatest fixed disc. Further, we investigate the geometry of non-unique fixed points of a self mapping and demonstrate by illustrative examples that a circle or a disc in parametric Nb−metric space is not necessarily the same as a circle or a disc in a Euclidean space. Obtained outcomes are extensions, unifications, improvements, and generalizations of some of the well-known previous results. We provide non-trivial illustrations to exhibit the importance of our explorations. Towards the end, we resolve the system of linear equations to demonstrate the significance of our contractions in parametric Nb−metric space.","PeriodicalId":43054,"journal":{"name":"Acta Universitatis Sapientiae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76780212","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 Graph theory has applications in various fields due to offering important tools such as topological indices. Among the topological indices, the Randić index is simple and of great importance. The Randić index of a graph 𝒢 can be expressed as R(G)=∑xy∈Y(G)1τ(x)τ(y) Rleft( G right) = sumnolimits_{xy in Yleft( G right)} {{1 over {sqrt {tau left( x right)tau left( y right)} }}} , where 𝒴(𝒢) represents the edge set and τ(x) is the degree of vertex x. In this paper, considering the importance of the Randić index and applications two-trees graphs, we determine the first two minimums among the two-trees graphs.
图论由于提供了拓扑指标等重要工具，在各个领域都有广泛的应用。在拓扑指标中，兰迪奇指数是一种简单而重要的指标。图𝒢的randici指数可表示为R(G)=∑xy∈Y(G)1τ(x)τ(Y) R left (G right)= sumnolimits ＿xy{in Y left (G right)}1 {{over{sqrt{tauleft (x right) tauleft (Y right)，}其中𝒴(𝒢)表示边集，τ(x)表示顶点x的度。考虑到兰迪奇指数的重要性和二树图的应用，我们确定了二树图中的前两个最小值。}}}
{"title":"Extremal trees for the Randić index","authors":"A. Jahanbani, H. Shooshtari, Y. Shang","doi":"10.2478/ausm-2022-0016","DOIUrl":"https://doi.org/10.2478/ausm-2022-0016","url":null,"abstract":"Abstract Graph theory has applications in various fields due to offering important tools such as topological indices. Among the topological indices, the Randić index is simple and of great importance. The Randić index of a graph 𝒢 can be expressed as R(G)=∑xy∈Y(G)1τ(x)τ(y) Rleft( G right) = sumnolimits_{xy in Yleft( G right)} {{1 over {sqrt {tau left( x right)tau left( y right)} }}} , where 𝒴(𝒢) represents the edge set and τ(x) is the degree of vertex x. In this paper, considering the importance of the Randić index and applications two-trees graphs, we determine the first two minimums among the two-trees graphs.","PeriodicalId":43054,"journal":{"name":"Acta Universitatis Sapientiae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90211843","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 A permutation p of [k] = {1, 2, 3, …, k} is called Layman permutation iff i + p(i) is a Fibonacci number for 1 ≤ i ≤ k. This concept is introduced by Layman in the A097082 entry of the Encyclopedia of Integers Sequences, that is the number of Layman permutations of [n]. In this paper, we will study Layman permutations. We introduce the notion of the Fibonacci complement of a natural number, that plays a crucial role in our investigation. Using this notion we prove some results on the number of Layman permutations, related to a conjecture of Layman that is implicit in the A097083 entry of OEIS.
{"title":"A short note on Layman permutations","authors":"P. Hajnal","doi":"10.2478/ausm-2022-0015","DOIUrl":"https://doi.org/10.2478/ausm-2022-0015","url":null,"abstract":"Abstract A permutation p of [k] = {1, 2, 3, …, k} is called Layman permutation iff i + p(i) is a Fibonacci number for 1 ≤ i ≤ k. This concept is introduced by Layman in the A097082 entry of the Encyclopedia of Integers Sequences, that is the number of Layman permutations of [n]. In this paper, we will study Layman permutations. We introduce the notion of the Fibonacci complement of a natural number, that plays a crucial role in our investigation. Using this notion we prove some results on the number of Layman permutations, related to a conjecture of Layman that is implicit in the A097083 entry of OEIS.","PeriodicalId":43054,"journal":{"name":"Acta Universitatis Sapientiae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85697552","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 work we define the concepts of the coupled orbit and coupled orbitally completeness. After then, using the method of Bollenbacher and Hicks [8], we prove some Caristi type coupled fixed point theorems in coupled orbitally complete metric spaces for a function P : E × E → E. We also give two examples that support our results.
摘要本文定义了耦合轨道和耦合轨道完备性的概念。然后，利用Bollenbacher和Hicks[8]的方法，证明了函数P: E × E→E在耦合轨道完备度量空间中的一些Caristi型耦合不动点定理，并给出了两个支持我们结果的例子。
{"title":"Some results on Caristi type coupled fixed point theorems","authors":"I. Şahin, M. Telci","doi":"10.2478/ausm-2022-0021","DOIUrl":"https://doi.org/10.2478/ausm-2022-0021","url":null,"abstract":"Abstract In this work we define the concepts of the coupled orbit and coupled orbitally completeness. After then, using the method of Bollenbacher and Hicks [8], we prove some Caristi type coupled fixed point theorems in coupled orbitally complete metric spaces for a function P : E × E → E. We also give two examples that support our results.","PeriodicalId":43054,"journal":{"name":"Acta Universitatis Sapientiae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78682248","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 utilize Hardy-Rogers contraction and CJM−contraction in a C*−algebra valued partial metric space to create an environment to establish a fixed point. Next, we present examples to elaborate on the novel space and validate our result. We conclude the paper by solving a boundary value problem and a matrix equation as applications of our main results which demonstrate the significance of our contraction and motivation for such investigations.
{"title":"On existence of fixed points and applications to a boundary value problem and a matrix equation in C*−algebra valued partial metric spaces","authors":"A. Tomar, M. Joshi","doi":"10.2478/ausm-2022-0023","DOIUrl":"https://doi.org/10.2478/ausm-2022-0023","url":null,"abstract":"Abstract We utilize Hardy-Rogers contraction and CJM−contraction in a C*−algebra valued partial metric space to create an environment to establish a fixed point. Next, we present examples to elaborate on the novel space and validate our result. We conclude the paper by solving a boundary value problem and a matrix equation as applications of our main results which demonstrate the significance of our contraction and motivation for such investigations.","PeriodicalId":43054,"journal":{"name":"Acta Universitatis Sapientiae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74922546","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 This article deals with the ratio of normalized Rabotnov function ℝα,β (z) and its sequence of partial sums (ℝα,β)m (z). Several examples which illustrate the validity of our results are also given.
{"title":"Partial sums of the Rabotnov function","authors":"S. Kazımoğlu, E. Deniz","doi":"10.2478/ausm-2022-0017","DOIUrl":"https://doi.org/10.2478/ausm-2022-0017","url":null,"abstract":"Abstract This article deals with the ratio of normalized Rabotnov function ℝα,β (z) and its sequence of partial sums (ℝα,β)m (z). Several examples which illustrate the validity of our results are also given.","PeriodicalId":43054,"journal":{"name":"Acta Universitatis Sapientiae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76441007","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 Let R be a finite commutative ring. We define a co-unit graph, associated to a ring R, denoted by Gnu(R) with vertex set V(Gnu(R)) = U(R), where U(R) is the set of units of R, and two distinct vertices x, y of U(R) being adjacent if and only if x + y ∉ / U(R). In this paper, we investigate some basic properties of Gnu(R), where R is the ring of integers modulo n, for different values of n. We find the domination number, clique number and the girth of Gnu(R).
{"title":"Co-unit graphs associated to ring of integers modulo n","authors":"S. Pirzada, Aaqib Altaf","doi":"10.2478/ausm-2022-0020","DOIUrl":"https://doi.org/10.2478/ausm-2022-0020","url":null,"abstract":"Abstract Let R be a finite commutative ring. We define a co-unit graph, associated to a ring R, denoted by Gnu(R) with vertex set V(Gnu(R)) = U(R), where U(R) is the set of units of R, and two distinct vertices x, y of U(R) being adjacent if and only if x + y ∉ / U(R). In this paper, we investigate some basic properties of Gnu(R), where R is the ring of integers modulo n, for different values of n. We find the domination number, clique number and the girth of Gnu(R).","PeriodicalId":43054,"journal":{"name":"Acta Universitatis Sapientiae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84931536","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 present two formulas for Chern classes (polynomial) of the tensor product of two vector bundles. In the first formula the Chern polynomial of the product is expressed as determinant of a polynomial in a matrix variable involving the Chern classes of the first bundle with Chern classes of the second bundle as coefficients. In the second formula the total Chern class of the tensor product is expressed as resultant of two explicit polynomials. Finally, formulas for the total Chern class of the second symmetric and the second alternating products are deduced.
{"title":"On Chern classes of the tensor product of vector bundles","authors":"Zs. Szilágyi","doi":"10.2478/ausm-2022-0022","DOIUrl":"https://doi.org/10.2478/ausm-2022-0022","url":null,"abstract":"Abstract We present two formulas for Chern classes (polynomial) of the tensor product of two vector bundles. In the first formula the Chern polynomial of the product is expressed as determinant of a polynomial in a matrix variable involving the Chern classes of the first bundle with Chern classes of the second bundle as coefficients. In the second formula the total Chern class of the tensor product is expressed as resultant of two explicit polynomials. Finally, formulas for the total Chern class of the second symmetric and the second alternating products are deduced.","PeriodicalId":43054,"journal":{"name":"Acta Universitatis Sapientiae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81420528","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 article, the main objective is to establish the Grüss-type fractional integral inequalities by employing the Caputo-Fabrizio fractional integral.
摘要本文的主要目的是利用Caputo-Fabrizio分数积分建立gr - ss型分数积分不等式。
{"title":"Grüss-type fractional inequality via Caputo-Fabrizio integral operator","authors":"Asha B. Nale, S. K. Panchal, V. L. Chinchane","doi":"10.2478/ausm-2022-0018","DOIUrl":"https://doi.org/10.2478/ausm-2022-0018","url":null,"abstract":"Abstract In this article, the main objective is to establish the Grüss-type fractional integral inequalities by employing the Caputo-Fabrizio fractional integral.","PeriodicalId":43054,"journal":{"name":"Acta Universitatis Sapientiae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75663406","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 paper, we study two generalizations of dual-complex Lucas-balancing numbers: dual-complex k-Lucas balancing numbers and dual-complex k-Lucas-balancing numbers. We give some of their properties, among others the Binet formula, Catalan, Cassini, d’Ocagne identities.
{"title":"Two generalizations of dual-complex Lucas-balancing numbers","authors":"D. Bród, A. Szynal-Liana, I. Włoch","doi":"10.2478/ausm-2022-0014","DOIUrl":"https://doi.org/10.2478/ausm-2022-0014","url":null,"abstract":"Abstract In this paper, we study two generalizations of dual-complex Lucas-balancing numbers: dual-complex k-Lucas balancing numbers and dual-complex k-Lucas-balancing numbers. We give some of their properties, among others the Binet formula, Catalan, Cassini, d’Ocagne identities.","PeriodicalId":43054,"journal":{"name":"Acta Universitatis Sapientiae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73578920","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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