The normalized distance Laplacian matrix of a connected graph $ G $, denoted by $ D^{mathcal{L}}(G) $, is defined by $ D^{mathcal{L}}(G)=Tr(G)^{-1/2}D^L(G)Tr(G)^{-1/2}, $ where $ D(G) $ is the distance matrix, the $D^{L}(G)$ is the distance Laplacian matrix and $ Tr(G)$ is the diagonal matrix of vertex transmissions of $ G. $ The set of all eigenvalues of $ D^{mathcal{L}}(G) $ including their multiplicities is the normalized distance Laplacian spectrum or $ D^{mathcal{L}} $-spectrum of $G$. In this paper, we find the $ D^{mathcal{L}} $-spectrum of the joined union of regular graphs in terms of the adjacency spectrum and the spectrum of an auxiliary matrix. As applications, we determine the $ D^{mathcal{L}} $-spectrum of the graphs associated with algebraic structures. In particular, we find the $ D^{mathcal{L}} $-spectrum of the power graphs of groups, the $ D^{mathcal{L}} $-spectrum of the commuting graphs of non-abelian groups and the $ D^{mathcal{L}} $-spectrum of the zero-divisor graphs of commutative rings. Several open problems are given for further work.
{"title":"On normalized distance Laplacian eigenvalues of graphs and applications to graphs defined on groups and rings","authors":"B. Rather, H. A. Ganie, M. Aouchiche","doi":"10.37193/cjm.2023.01.14","DOIUrl":"https://doi.org/10.37193/cjm.2023.01.14","url":null,"abstract":"The normalized distance Laplacian matrix of a connected graph $ G $, denoted by $ D^{mathcal{L}}(G) $, is defined by $ D^{mathcal{L}}(G)=Tr(G)^{-1/2}D^L(G)Tr(G)^{-1/2}, $ where $ D(G) $ is the distance matrix, the $D^{L}(G)$ is the distance Laplacian matrix and $ Tr(G)$ is the diagonal matrix of vertex transmissions of $ G. $ The set of all eigenvalues of $ D^{mathcal{L}}(G) $ including their multiplicities is the normalized distance Laplacian spectrum or $ D^{mathcal{L}} $-spectrum of $G$. In this paper, we find the $ D^{mathcal{L}} $-spectrum of the joined union of regular graphs in terms of the adjacency spectrum and the spectrum of an auxiliary matrix. As applications, we determine the $ D^{mathcal{L}} $-spectrum of the graphs associated with algebraic structures. In particular, we find the $ D^{mathcal{L}} $-spectrum of the power graphs of groups, the $ D^{mathcal{L}} $-spectrum of the commuting graphs of non-abelian groups and the $ D^{mathcal{L}} $-spectrum of the zero-divisor graphs of commutative rings. Several open problems are given for further work.","PeriodicalId":50711,"journal":{"name":"Carpathian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46827457","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
"In this paper we provide some properties and applications of MV-algebras. We prove that a Fibonacci stationary sequence in an MV-algebra gives us an idempotent element. Moreover, taking into account of the representation of a finite MV-algebra, by using Boolean elements of this algebra, we prove that a Fibonacci sequence in an MV-algebra is always stationary. This result is interesting comparing with the behavior of such a sequence on the group $left( mathbb{Z}% _{n},+right) $, where the Fibonacci sequences are periodic, with the period given by the Pisano period. We also give some examples of finite MV-algebras and the number of their idempotent elements. As an application in Coding Theory, to a Boolean algebra it is attached a binary block code and it is proved that, under some conditions, the converse is also true."
{"title":"Some applications of idempotent elements in MV algebras","authors":"C. Flaut","doi":"10.37193/cjm.2023.01.10","DOIUrl":"https://doi.org/10.37193/cjm.2023.01.10","url":null,"abstract":"\"In this paper we provide some properties and applications of MV-algebras. We prove that a Fibonacci stationary sequence in an MV-algebra gives us an idempotent element. Moreover, taking into account of the representation of a finite MV-algebra, by using Boolean elements of this algebra, we prove that a Fibonacci sequence in an MV-algebra is always stationary. This result is interesting comparing with the behavior of such a sequence on the group $left( mathbb{Z}% _{n},+right) $, where the Fibonacci sequences are periodic, with the period given by the Pisano period. We also give some examples of finite MV-algebras and the number of their idempotent elements. As an application in Coding Theory, to a Boolean algebra it is attached a binary block code and it is proved that, under some conditions, the converse is also true.\"","PeriodicalId":50711,"journal":{"name":"Carpathian Journal of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70023834","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
P. Chaipunya, Nantaporn Chuensupantharat, Printaporn Sanguansuttigul
In this paper, we introduce the generalized Ekeland's variational principle in several forms. The general setting of our results includes a graphical metric structure and also employs a generalized $w$-distance. We then applied the proposed variational principles to obtain existence theorems for a class of quasi-equilibrium problems whose constraint maps are induced from the graphical structure. The conditions used in our existence results are based on a very general concept called a convergence class. Finally, we deduce the existence of a generalized Nash equilibrium via its quasi-equilibrium reformulation. A validating example is also presented.
{"title":"Graphical Ekeland's variational principle with a generalized $w$-distance and a new approach to quasi-equilibrium problems","authors":"P. Chaipunya, Nantaporn Chuensupantharat, Printaporn Sanguansuttigul","doi":"10.37193/cjm.2023.01.06","DOIUrl":"https://doi.org/10.37193/cjm.2023.01.06","url":null,"abstract":"In this paper, we introduce the generalized Ekeland's variational principle in several forms. The general setting of our results includes a graphical metric structure and also employs a generalized $w$-distance. We then applied the proposed variational principles to obtain existence theorems for a class of quasi-equilibrium problems whose constraint maps are induced from the graphical structure. The conditions used in our existence results are based on a very general concept called a convergence class. Finally, we deduce the existence of a generalized Nash equilibrium via its quasi-equilibrium reformulation. A validating example is also presented.","PeriodicalId":50711,"journal":{"name":"Carpathian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49665277","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
"In this paper, we introduce a new iterative process for solving simultaneously one-parameter nonexpansive semigroup, convex minimization and fixed point problems involving set-valued operators in real Hilbert spaces and establish strong convergence theorems for the proposed iterative process. There is no compactness assumption. Our results improve important recent results."
{"title":"A novel approach for solving simultaneously one-parameter nonexpansive semigroup, convex minimization and fixed point problems involving set-valued operators","authors":"T. Sow, A. Diene, N. Djitté","doi":"10.37193/cjm.2023.01.23","DOIUrl":"https://doi.org/10.37193/cjm.2023.01.23","url":null,"abstract":"\"In this paper, we introduce a new iterative process for solving simultaneously one-parameter nonexpansive semigroup, convex minimization and fixed point problems involving set-valued operators in real Hilbert spaces and establish strong convergence theorems for the proposed iterative process. There is no compactness assumption. Our results improve important recent results.\"","PeriodicalId":50711,"journal":{"name":"Carpathian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41866773","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
"We study the space of all continuous and increasing self-mappings of the real interval $[0,1]$ equipped with the topology of uniform convergence. In particular, we show that most such functions have at least two different fixed points."
{"title":"Most continuous and increasing functions have two different fixed points","authors":"S. Reich, A. Zaslavski","doi":"10.37193/cjm.2023.01.15","DOIUrl":"https://doi.org/10.37193/cjm.2023.01.15","url":null,"abstract":"\"We study the space of all continuous and increasing self-mappings of the real interval $[0,1]$ equipped with the topology of uniform convergence. In particular, we show that most such functions have at least two different fixed points.\"","PeriodicalId":50711,"journal":{"name":"Carpathian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48608167","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
"The purpose of this study is to introduce an inertial algorithm for approximating a solution of the split equality Hammerstein type equation problem in general reflexive real Banach spaces. Strong convergence results are established under the assumption that the associated mappings are monotone and uniformly continuous. The results in this paper generalize and improve many of the existing results in the literature in the sense that the underlying mappings are relaxed from Lipschitz continuous to uniformly continuous and the spaces under consideration are extended from Hilbert spaces to reflexive real Banach spaces with a more general problem which includes the Hammerstein type equation problems."
{"title":"\"Solutions of Split Equality Hammerstein Type Equation Problems in Reflexive Real Banach Spaces\"","authors":"Y. Belay, H. Zegeye, O. A. Boikanyo","doi":"10.37193/cjm.2023.01.03","DOIUrl":"https://doi.org/10.37193/cjm.2023.01.03","url":null,"abstract":"\"The purpose of this study is to introduce an inertial algorithm for approximating a solution of the split equality Hammerstein type equation problem in general reflexive real Banach spaces. Strong convergence results are established under the assumption that the associated mappings are monotone and uniformly continuous. The results in this paper generalize and improve many of the existing results in the literature in the sense that the underlying mappings are relaxed from Lipschitz continuous to uniformly continuous and the spaces under consideration are extended from Hilbert spaces to reflexive real Banach spaces with a more general problem which includes the Hammerstein type equation problems.\"","PeriodicalId":50711,"journal":{"name":"Carpathian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43238444","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A. Abubakar, Kumam Poom, Kitkuan Duangkamon, Padcharoen Anantachai
"In this paper, relaxed and relaxed inertial modified Tseng algorithms for approximating zeros of sum of two monotone operators whose zeros are fixed points or J-fixed points of some nonexpansive-type mappings are introduced and studied. Strong convergence theorems are proved in the setting of real Banach spaces that are uniformly smooth and 2-uniformly convex. Furthermore, applications of the theorems to the concept of J-fixed point, convex minimization, image restoration and signal recovery problems are also presented. In addition, some interesting numerical implementations of our proposed methods in solving image recovery and compressed sensing problems are presented. Finally, the performance of our proposed methods are compared with that of some existing methods in the literature."
{"title":"Relaxed modified Tseng algorithm for solving variational inclusion problems in real Banach spaces with applications","authors":"A. Abubakar, Kumam Poom, Kitkuan Duangkamon, Padcharoen Anantachai","doi":"10.37193/cjm.2023.01.01","DOIUrl":"https://doi.org/10.37193/cjm.2023.01.01","url":null,"abstract":"\"In this paper, relaxed and relaxed inertial modified Tseng algorithms for approximating zeros of sum of two monotone operators whose zeros are fixed points or J-fixed points of some nonexpansive-type mappings are introduced and studied. Strong convergence theorems are proved in the setting of real Banach spaces that are uniformly smooth and 2-uniformly convex. Furthermore, applications of the theorems to the concept of J-fixed point, convex minimization, image restoration and signal recovery problems are also presented. In addition, some interesting numerical implementations of our proposed methods in solving image recovery and compressed sensing problems are presented. Finally, the performance of our proposed methods are compared with that of some existing methods in the literature.\"","PeriodicalId":50711,"journal":{"name":"Carpathian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45075763","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Certain bounded linear operators $T$ on a complex Hilbert space $h$ which have 2-isometric liftings $S$ on another space $ka supset h$ are being investigated. We refer also to a more special type of liftings, as well as to those which additionally meet the condition $S^*Sh subset h$. Furthermore we describe other types of dilations for $T$, which are close to 2-isometries. Among these we refer to expansive (concave) operators and also to Brownian unitary dilations. Different matrix representations for such operators are obtained, where matrix entries involve contractive operators.
{"title":"Operators with Brownian unitary dilations","authors":"Laurian Suciu","doi":"10.37193/cjm.2022.03.08","DOIUrl":"https://doi.org/10.37193/cjm.2022.03.08","url":null,"abstract":"Certain bounded linear operators $T$ on a complex Hilbert space $h$ which have 2-isometric liftings $S$ on another space $ka supset h$ are being investigated. We refer also to a more special type of liftings, as well as to those which additionally meet the condition $S^*Sh subset h$. Furthermore we describe other types of dilations for $T$, which are close to 2-isometries. Among these we refer to expansive (concave) operators and also to Brownian unitary dilations. Different matrix representations for such operators are obtained, where matrix entries involve contractive operators.","PeriodicalId":50711,"journal":{"name":"Carpathian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49449420","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
"n this paper, the concept of perturbed metric was introduced within the metric spaces and some fixed point results were established for self-mappings satisfying such contractive conditions, using Picard operators technique and generalized contractions. Moreover, some applications of the main result to continuous data dependence of the fixed points of Picard operators defined on these spaces were presented. Also, the main result of this paper was applied to study the existence and uniqueness of the solution for an integral equation which models an epidemiological problem"
{"title":"Some Fixed Point Results in Spaces with Perturbed Metrics","authors":"A. Branga, I. Olaru","doi":"10.37193/cjm.2022.03.10","DOIUrl":"https://doi.org/10.37193/cjm.2022.03.10","url":null,"abstract":"\"n this paper, the concept of perturbed metric was introduced within the metric spaces and some fixed point results were established for self-mappings satisfying such contractive conditions, using Picard operators technique and generalized contractions. Moreover, some applications of the main result to continuous data dependence of the fixed points of Picard operators defined on these spaces were presented. Also, the main result of this paper was applied to study the existence and uniqueness of the solution for an integral equation which models an epidemiological problem\"","PeriodicalId":50711,"journal":{"name":"Carpathian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43258215","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
"We determine the nonlinear Fokker-Planck equation in one dimension, based on a weighted Tsallis entropy and we derive its associated Lie symmetries. The corresponding Lyapunov functions and Breg- man divergences are also found, together with a formula linking the drift function, the diffusion function and a diffusion constant. We solve the MaxEnt problem associated to the weighted Tsallis entropy."
{"title":"\"Lie symmetries of the nonlinear Fokker-Planck equation based on weighted Tsallis entropy\"","authors":"C. Pripoae, I. Hiricǎ, G. Pripoae, V. Preda","doi":"10.37193/cjm.2022.03.07","DOIUrl":"https://doi.org/10.37193/cjm.2022.03.07","url":null,"abstract":"\"We determine the nonlinear Fokker-Planck equation in one dimension, based on a weighted Tsallis entropy and we derive its associated Lie symmetries. The corresponding Lyapunov functions and Breg- man divergences are also found, together with a formula linking the drift function, the diffusion function and a diffusion constant. We solve the MaxEnt problem associated to the weighted Tsallis entropy.\"","PeriodicalId":50711,"journal":{"name":"Carpathian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44021908","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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