We obtain sufficient conditions to know if given a positive even integer number and a set of positive integer numbers being all even or all odd, such a number can be expressed as sum of two elements of this set. As consequence we obtain a result which, when applied to the prime numbers set, would prove Goldbach's Conjecture provided that certain conditions are satisfied. These hypothesis include Prime Consecutive Conjecture, which is a generalized form of Twin Prime Conjecture. In addition, we extend these results to sets of positive real numbers, even for two different sets. We also obtain a recurrent approximation of pi(x) for enough large real x, being pi the distribution function of the prime number set, which uses whichever expression of x as product of enough large factors. We also state this approximation in a more general context, give upper and lower bounds for the error, and show that this approximation is asymptotically equivalent to pi(x).
{"title":"Advances in additive number theory","authors":"José Alfonso López Nicolás","doi":"10.5269/bspm.51233","DOIUrl":"https://doi.org/10.5269/bspm.51233","url":null,"abstract":"We obtain sufficient conditions to know if given a positive even integer number and a set of positive integer numbers being all even or all odd, such a number can be expressed as sum of two elements of this set. As consequence we obtain a result which, when applied to the prime numbers set, would prove Goldbach's Conjecture provided that certain conditions are satisfied. These hypothesis include Prime Consecutive Conjecture, which is a generalized form of Twin Prime Conjecture. In addition, we extend these results to sets of positive real numbers, even for two different sets. We also obtain a recurrent approximation of pi(x) for enough large real x, being pi the distribution function of the prime number set, which uses whichever expression of x as product of enough large factors. We also state this approximation in a more general context, give upper and lower bounds for the error, and show that this approximation is asymptotically equivalent to pi(x).","PeriodicalId":44941,"journal":{"name":"Boletim Sociedade Paranaense de Matematica","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41588631","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}
H. Ramane, Daneshwari Patil, K. Ashoka, B. Parvathalu
The spectral polynomial of a graph is the characteristic polynomial of its adjacency matrix. Spectral polynomial of the splice and links of complete graph and star have been obatined recently in the literature. In this paper we generalize these results using the concept of equitable partition.
{"title":"On spectral polynomial of splices and links of graphs","authors":"H. Ramane, Daneshwari Patil, K. Ashoka, B. Parvathalu","doi":"10.5269/bspm.51691","DOIUrl":"https://doi.org/10.5269/bspm.51691","url":null,"abstract":"The spectral polynomial of a graph is the characteristic polynomial of its adjacency matrix. Spectral polynomial of the splice and links of complete graph and star have been obatined recently in the literature. In this paper we generalize these results using the concept of equitable partition.","PeriodicalId":44941,"journal":{"name":"Boletim Sociedade Paranaense de Matematica","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47047631","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 $S$ be a discrete semigroup and $T$ be a left multiplier operator on $S$. A new product on $S$ defined by $T$ creates a new induced semigroup $S _{T} $. In this paper, we show that if $T$ is bijective, then the first module cohomology groups $ HH_{ell^1(E)}^{1}(ell^1(S), ell^{infty}(S))$ and $ HH_{ell^1(E_{T})}^{1}(ell^1({S_{T}}), ell^{infty}(S_{T})) $ are equal, where $E$ and $E_{T}$ are sets of idempotent elements in $S$ and $S _{T}$, respectively. Which in particular means that $ell^1(S)$ is weak $ell^1(E)$-module amenable if and only if $ell^1(S_T)$ is weak $ell^1(E_T)$-module amenable. Finally, by giving an example, we show that the condition of bijectivity for $T$, is necessary.
{"title":"First module cohomology group of induced semigroup algebras","authors":"M. Miri, E. Nasrabadi, Kianoush Kazemi","doi":"10.5269/bspm.51414","DOIUrl":"https://doi.org/10.5269/bspm.51414","url":null,"abstract":"Let $S$ be a discrete semigroup and $T$ be a left multiplier operator on $S$. A new product on $S$ defined by $T$ creates a new induced semigroup $S _{T} $. In this paper, we show that if $T$ is bijective, then the first module cohomology groups $ HH_{ell^1(E)}^{1}(ell^1(S), ell^{infty}(S))$ and $ HH_{ell^1(E_{T})}^{1}(ell^1({S_{T}}), ell^{infty}(S_{T})) $ are equal, where $E$ and $E_{T}$ are sets of idempotent elements in $S$ and $S _{T}$, respectively. Which in particular means that $ell^1(S)$ is weak $ell^1(E)$-module amenable if and only if $ell^1(S_T)$ is weak $ell^1(E_T)$-module amenable. Finally, by giving an example, we show that the condition of bijectivity for $T$, is necessary.","PeriodicalId":44941,"journal":{"name":"Boletim Sociedade Paranaense de Matematica","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43596099","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 article includes different properties of the independence and domination (total domination, independent domination, co-independent domination) number of the complete z-ray root and Jahangir graphs. Also, the inverse domination number of these graphs of variant dominating sets (total dominating, independent dominating, co-independent dominating) is determined.
{"title":"Independence and inverse domination in complete z-ary tree and Jahangir graphs","authors":"A. Omran, Essam El-Seidy","doi":"10.5269/bspm.53123","DOIUrl":"https://doi.org/10.5269/bspm.53123","url":null,"abstract":"This article includes different properties of the independence and domination (total domination, independent domination, co-independent domination) number of the complete z-ray root and Jahangir graphs. Also, the inverse domination number of these graphs of variant dominating sets (total dominating, independent dominating, co-independent dominating) is determined.","PeriodicalId":44941,"journal":{"name":"Boletim Sociedade Paranaense de Matematica","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42907908","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 current investigation, we study a certain family of analytic and bi-univalent functions with respect to symmetric conjugate points defined in the open unit disk $U$ and find an upper bounds for the second Hankel determinant $H_{2}(2)$ of the functions belongs to this class.
{"title":"Upper bound to second Hankel determinant for a family of bi-univalent functions","authors":"A. Wanas","doi":"10.5269/bspm.51332","DOIUrl":"https://doi.org/10.5269/bspm.51332","url":null,"abstract":"In the current investigation, we study a certain family of analytic and bi-univalent functions with respect to symmetric conjugate points defined in the open unit disk $U$ and find an upper bounds for the second Hankel determinant $H_{2}(2)$ of the functions belongs to this class.","PeriodicalId":44941,"journal":{"name":"Boletim Sociedade Paranaense de Matematica","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42755212","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 existence of infinitely many solutions for a class of impulsive fractional boundary value problems with $p$-Laplacian with Neumann conditions is established. Our approach is based on recent variational methods for smooth functionals defined on reflexive Banach spaces. One example is presented to demonstrate the application of our main results.
{"title":"Infinitely many solutions for a class of fractional boundary value problem with $p$-Laplacian with impulsive effects","authors":"M. Abolghasemi, S. Moradi","doi":"10.5269/bspm.47913","DOIUrl":"https://doi.org/10.5269/bspm.47913","url":null,"abstract":"The existence of infinitely many solutions for a class of impulsive fractional boundary value problems with $p$-Laplacian with Neumann conditions is established. Our approach is based on recent variational methods for smooth functionals defined on reflexive Banach spaces. One example is presented to demonstrate the application of our main results.","PeriodicalId":44941,"journal":{"name":"Boletim Sociedade Paranaense de Matematica","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42880656","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 work, we are interested at the existence of nontrivial solutions for a nonlinear elliptic problems with resonance part and mixed boundary conditions. Our approach is variational and is based on the well known Landesman-Laser type conditions.
{"title":"Study to the Resonance of $ p $-Laplacian problem with mixed boundary","authors":"M. Haddaoui, Hafid Lebrimchi, N. Tsouli","doi":"10.5269/bspm.52640","DOIUrl":"https://doi.org/10.5269/bspm.52640","url":null,"abstract":"In this work, we are interested at the existence of nontrivial solutions for a nonlinear elliptic problems with resonance part and mixed boundary conditions. Our approach is variational and is based on the well known Landesman-Laser type conditions.","PeriodicalId":44941,"journal":{"name":"Boletim Sociedade Paranaense de Matematica","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45774283","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 introduce a new class of mappings called almost nonexpansive mappings in a $b$-metric space. Some characteristics of this class of mappings are discussed. Fixed point and common fixed point results for such mappings are obtained. An application to the Cauchy problem in a Banach space is also shown in this paper.
{"title":"Fixed point results for almost nonexpansive mappings in $b$-metric spaces","authors":"Nilakshi Goswami, N. Haokip","doi":"10.5269/bspm.52759","DOIUrl":"https://doi.org/10.5269/bspm.52759","url":null,"abstract":"The aim of this paper is to introduce a new class of mappings called almost nonexpansive mappings in a $b$-metric space. Some characteristics of this class of mappings are discussed. \u0000Fixed point and common fixed point results for such mappings are obtained. An application to the Cauchy problem in a Banach space is also shown in this paper.","PeriodicalId":44941,"journal":{"name":"Boletim Sociedade Paranaense de Matematica","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46907867","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 work, we study a kind of closure systems (c.s.) that are defined by means of intersections of subsets of a support X with a (fixed) closed set T. These systems (which will be indicated by M(T)-spaces) can be understood as a generalization of the usual relative subspaces. Several results (referred to continuity and to the ordered structure of families of M(T)-spaces) are shown here. In addition, we study the transference of properties from the ``original closure spaces (X,K) to the spaces (X,M(T)). Among them, we are interested mainly in finitariness and in structurality. In this study of transference, we focus our analyisis on the c.s. usually known as abstract logics, and we show some results for them.
{"title":"A note on closure spaces determined by intersections","authors":"Víctor Fernández, Cristian Brunetta","doi":"10.5269/bspm.52790","DOIUrl":"https://doi.org/10.5269/bspm.52790","url":null,"abstract":"In this work, we study a kind of closure systems (c.s.) that are defined by means of intersections of subsets of a support X with a (fixed) closed set T. These systems (which will be indicated by M(T)-spaces) can be understood as a generalization of the usual relative subspaces. Several results (referred to continuity and to the ordered structure of families of M(T)-spaces) are shown here. In addition, we study the transference of properties from the ``original closure spaces (X,K) to the spaces (X,M(T)). Among them, we are interested mainly in finitariness and in structurality. In this study of transference, we focus our analyisis on the c.s. usually known as abstract logics, and we show some results for them.","PeriodicalId":44941,"journal":{"name":"Boletim Sociedade Paranaense de Matematica","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46729390","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 paper is concerned with a second-order abstract viscoelastic equation with time delay and a relaxation function satisfying $ h^{prime}(t)leq -zeta(t) G(h(t))$. Under a suitable conditions, we establish an explicit and general decay rate results of the energy by introducing a suitable Lyaponov functional and some proprieties of the convex functions. Finally, some applications are given. This work generalizes the previous results without time delay term to those with delay.
{"title":"General decay for second-order abstract viscoelastic equation in Hilbert spaces with time delay","authors":"Houria Chellaoua, Y. Boukhatem","doi":"10.5269/bspm.52175","DOIUrl":"https://doi.org/10.5269/bspm.52175","url":null,"abstract":"The paper is concerned with a second-order abstract viscoelastic equation with time delay and a relaxation function satisfying $ h^{prime}(t)leq -zeta(t) G(h(t))$. Under a suitable conditions, we establish an explicit and general decay rate results of the energy by introducing a suitable Lyaponov functional and some proprieties of the convex functions. Finally, some applications are given. This work generalizes the previous results without time delay term to those with delay.","PeriodicalId":44941,"journal":{"name":"Boletim Sociedade Paranaense de Matematica","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48196954","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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