Pub Date : 2024-02-25DOI: 10.56947/gjom.v16i1.1419
Ilias Elmouki, S. Abdelalim
Wedderburn and Hasse theorems have always fascinated many algebraists as there are just very few versions of their proofs and that are somehow not easy to follow because they contain other results inside. In this note, we aim to look at their proofs again but nowwith more simplified versions that include sufficient details in order to let them be understandable for most readers in the subjects of finite fields and elliptic curves. In the proof of Wedderburn’s theorem, we succeed to show how it is equivalent to define either a stabilizer or centralizer of non-central elements and that helps to consider the equation on the order of group by using interchangeably the index of its subgroup and the order of the orbit. As for Hasse’s theorem, we provide a short proof in just two paragraphs with a recall of the most important results that have been used, and then, we state three of its version with an explanation of how this all leads in the end to the need of an algorithm like the one of René Schoof in 1985.
{"title":"A note on Wedderburn and Hasse theorems","authors":"Ilias Elmouki, S. Abdelalim","doi":"10.56947/gjom.v16i1.1419","DOIUrl":"https://doi.org/10.56947/gjom.v16i1.1419","url":null,"abstract":"Wedderburn and Hasse theorems have always fascinated many algebraists as there are just very few versions of their proofs and that are somehow not easy to follow because they contain other results inside. In this note, we aim to look at their proofs again but nowwith more simplified versions that include sufficient details in order to let them be understandable for most readers in the subjects of finite fields and elliptic curves. In the proof of Wedderburn’s theorem, we succeed to show how it is equivalent to define either a stabilizer or centralizer of non-central elements and that helps to consider the equation on the order of group by using interchangeably the index of its subgroup and the order of the orbit. As for Hasse’s theorem, we provide a short proof in just two paragraphs with a recall of the most important results that have been used, and then, we state three of its version with an explanation of how this all leads in the end to the need of an algorithm like the one of René Schoof in 1985.","PeriodicalId":421614,"journal":{"name":"Gulf Journal of Mathematics","volume":"33 8","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140433310","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}
Pub Date : 2024-02-25DOI: 10.56947/gjom.v16i1.1368
A. Harfash, Ghufran A. Al-Musawi
We conducted a mathematical investigation on a system of interconnected Cahn-Hilliard equations featuring a logarithmic potential, nondegenerate mobility, and homogeneous Neumann boundary conditions. This system emerges from a model depicting the phase separation of a binary liquid mixture in a thin film. Assuming certain conditions on the initial data, we successfully established the existence, uniqueness, and stability estimates for the weak solution. Our approach involved initially replacing the logarithmic potential with a smooth counterpart, resulting in the regularization of the original problem (Q) into a regularized problem (Qε). Utilizing the Faedo-Galerkin method and compactness arguments, we demonstrated the existence and uniqueness of a solution for (Qε). Subsequently, by letting ε approach zero, we attained the existence of a solution for the original problem (Q). Additionally, we addressed higher regularity aspects of the weak solutions for both (Q) and (Qε). Employing the standard regularity theory for elliptic problems and introducing additional assumptions regarding the domain's boundary and the initial data, we established that the weak solutions belong to higher-order Sobolev spaces.
{"title":"Global existence of a pair of coupled Cahn-Hilliard equations with nondegenerate mobility and logarithmic potential","authors":"A. Harfash, Ghufran A. Al-Musawi","doi":"10.56947/gjom.v16i1.1368","DOIUrl":"https://doi.org/10.56947/gjom.v16i1.1368","url":null,"abstract":"We conducted a mathematical investigation on a system of interconnected Cahn-Hilliard equations featuring a logarithmic potential, nondegenerate mobility, and homogeneous Neumann boundary conditions. This system emerges from a model depicting the phase separation of a binary liquid mixture in a thin film. Assuming certain conditions on the initial data, we successfully established the existence, uniqueness, and stability estimates for the weak solution. Our approach involved initially replacing the logarithmic potential with a smooth counterpart, resulting in the regularization of the original problem (Q) into a regularized problem (Qε). Utilizing the Faedo-Galerkin method and compactness arguments, we demonstrated the existence and uniqueness of a solution for (Qε). Subsequently, by letting ε approach zero, we attained the existence of a solution for the original problem (Q). Additionally, we addressed higher regularity aspects of the weak solutions for both (Q) and (Qε). Employing the standard regularity theory for elliptic problems and introducing additional assumptions regarding the domain's boundary and the initial data, we established that the weak solutions belong to higher-order Sobolev spaces.","PeriodicalId":421614,"journal":{"name":"Gulf Journal of Mathematics","volume":"20 2","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140432224","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}
Pub Date : 2024-02-25DOI: 10.56947/gjom.v16i1.1432
J. Ettayb
Let X and Y be two ultrametric Banach spaces over K. Let A,D ∈ B(X,Y) and B,C ∈ B(Y,X) such that ABA=ACA (resp. ACD=DBD and DBA=ACA). In this paper, the operator equation ABA=ACA is studied, and the common operator properties of AC-IY and BA-IX are described. In particular, it is proved that N(IY-AC) is complemented in Y if and only if N(IX-BA) is complemented in X. Moreover, the approach is generalized (i.e., CD=DBD and DBA=ACA) for considering relationships between the properties IY-AC and IX-BD. Finally, several illustrative examples are provided.
设 X 和 Y 是 K 上的两个超对称巴拿赫空间，设 A,D∈B(X,Y) 和 B,C∈B(Y,X) 令 ABA=ACA (resp. ACD=DBD 和 DBA=ACA)。本文研究了算子方程 ABA=ACA，并描述了 AC-IY 和 BA-IX 的共同算子性质。特别是，本文证明了当且仅当 N(IX-BA) 在 X 中被补全时，N(IY-AC) 在 Y 中被补全。此外，本文还推广了考虑 IY-AC 和 IX-BD 性质之间关系的方法（即 CD=DBD 和 DBA=ACA）。最后，还提供了几个示例。
{"title":"Common properties of the operator equations in ultrametric specrtal theory","authors":"J. Ettayb","doi":"10.56947/gjom.v16i1.1432","DOIUrl":"https://doi.org/10.56947/gjom.v16i1.1432","url":null,"abstract":"Let X and Y be two ultrametric Banach spaces over K. Let A,D ∈ B(X,Y) and B,C ∈ B(Y,X) such that ABA=ACA (resp. ACD=DBD and DBA=ACA). In this paper, the operator equation ABA=ACA is studied, and the common operator properties of AC-IY and BA-IX are described. In particular, it is proved that N(IY-AC) is complemented in Y if and only if N(IX-BA) is complemented in X. Moreover, the approach is generalized (i.e., CD=DBD and DBA=ACA) for considering relationships between the properties IY-AC and IX-BD. Finally, several illustrative examples are provided.","PeriodicalId":421614,"journal":{"name":"Gulf Journal of Mathematics","volume":"27 4","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140433149","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}
Pub Date : 2023-11-12DOI: 10.56947/gjom.v15i2.1415
Ismat Beg
We broach the notions of face, core and convex function of a convex mixture set. A version of famous Karush-Kuhn-Tucker theorem is proved. Several examples to illustrate are also given.
{"title":"Core of a convex mixture set and convex functions","authors":"Ismat Beg","doi":"10.56947/gjom.v15i2.1415","DOIUrl":"https://doi.org/10.56947/gjom.v15i2.1415","url":null,"abstract":"We broach the notions of face, core and convex function of a convex mixture set. A version of famous Karush-Kuhn-Tucker theorem is proved. Several examples to illustrate are also given.","PeriodicalId":421614,"journal":{"name":"Gulf Journal of Mathematics","volume":"18 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139279304","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}
Pub Date : 2023-11-12DOI: 10.56947/gjom.v15i2.1226
A. Rekic-Vukovic, N. Okičić
In this paper we consider the space R2 with the river metric d* and different types of convexity of this space. We define W-convex structure in (R2, d*) and we give the complete characterization of the convex sets in this space. We consider some measures of noncompactness and we give the moduli of noncompactness for considered measures on this space.
在本文中，我们考虑了具有河流度量 d* 的 R2 空间以及该空间的不同凸性类型。我们定义了 (R2, d*) 中的 W 凸结构，并给出了该空间中凸集的完整特征。我们考虑了一些非紧凑性度量，并给出了该空间上所考虑度量的非紧凑性模量。
{"title":"A convexity in R^2 with river metric","authors":"A. Rekic-Vukovic, N. Okičić","doi":"10.56947/gjom.v15i2.1226","DOIUrl":"https://doi.org/10.56947/gjom.v15i2.1226","url":null,"abstract":"In this paper we consider the space R2 with the river metric d* and different types of convexity of this space. We define W-convex structure in (R2, d*) and we give the complete characterization of the convex sets in this space. We consider some measures of noncompactness and we give the moduli of noncompactness for considered measures on this space.","PeriodicalId":421614,"journal":{"name":"Gulf Journal of Mathematics","volume":"103 12","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139279520","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}
Pub Date : 2023-11-12DOI: 10.56947/gjom.v15i2.1373
Jozil Takhirov
All diffusion equations are based on the infinite velocity of potential fields, which leads to well-known paradoxes. Consequently, in non-stationary processes, the evolution of these quantities do not completely obey the above equations due to the lack of parameters in them that take into account the finite rate of potential growth.In the heat conduction theory, numerous generalizations of the Fourier law are used as a remedy for these issues. The article gives a brief overview of generalizations of the Fourier law. Some mathematical issues of well-posed boundary value problems for the Guyer-Krumhansl model are discussed. As an application, a boundary value problem for a general quasilinear equation with periodic boundary conditions is considered. Schauder-type a priori estimates are established and the uniqueness of the solution is proved.
{"title":"A problem with periodic boundary conditions for the non-Fourier heat equation","authors":"Jozil Takhirov","doi":"10.56947/gjom.v15i2.1373","DOIUrl":"https://doi.org/10.56947/gjom.v15i2.1373","url":null,"abstract":"All diffusion equations are based on the infinite velocity of potential fields, which leads to well-known paradoxes. Consequently, in non-stationary processes, the evolution of these quantities do not completely obey the above equations due to the lack of parameters in them that take into account the finite rate of potential growth.In the heat conduction theory, numerous generalizations of the Fourier law are used as a remedy for these issues. The article gives a brief overview of generalizations of the Fourier law. Some mathematical issues of well-posed boundary value problems for the Guyer-Krumhansl model are discussed. As an application, a boundary value problem for a general quasilinear equation with periodic boundary conditions is considered. Schauder-type a priori estimates are established and the uniqueness of the solution is proved.","PeriodicalId":421614,"journal":{"name":"Gulf Journal of Mathematics","volume":"62 5","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139279571","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}
Pub Date : 2023-11-12DOI: 10.56947/gjom.v15i2.1247
Vuk Stojiljković, S. Dragomir
In this paper several tensorial norm inequalities of Ostrowski type for continuous functions of selfadjoint operators in Hilbert spaces have been obtained. Multiple inequalities are obtained with variations due to the convexity properties of the mapping f.
本文获得了希尔伯特空间中自相关算子连续函数的几个奥斯特洛夫斯基式张量规范不等式。由于映射 f 的凸特性，得到的多个不等式都有变化。
{"title":"Differentiable Ostrowski type tensorial norm inequality for continuous functions of selfadjoint operators in Hilbert spaces","authors":"Vuk Stojiljković, S. Dragomir","doi":"10.56947/gjom.v15i2.1247","DOIUrl":"https://doi.org/10.56947/gjom.v15i2.1247","url":null,"abstract":"In this paper several tensorial norm inequalities of Ostrowski type for continuous functions of selfadjoint operators in Hilbert spaces have been obtained. Multiple inequalities are obtained with variations due to the convexity properties of the mapping f.","PeriodicalId":421614,"journal":{"name":"Gulf Journal of Mathematics","volume":"26 3","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139279314","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}
Pub Date : 2023-11-12DOI: 10.56947/gjom.v15i2.1394
C. Choudhury, Carlos Granados
In this paper, we introduce the notion of I2 and I∗2−convergence of double sequences in neutrosophic normed spaces mainly as a generalization of statistical convergence of double sequences. We investigate a few fundamental properties and study the relationship between the two notions. We also introduce and investigate the concept of I2 and I∗2−Cauchy sequence of double sequences and show that the condition (AP2) plays a crucial role to study the interrelationship between them.
{"title":"On ideal convergence of double sequences in neutrosophic normed spaces","authors":"C. Choudhury, Carlos Granados","doi":"10.56947/gjom.v15i2.1394","DOIUrl":"https://doi.org/10.56947/gjom.v15i2.1394","url":null,"abstract":"In this paper, we introduce the notion of I2 and I∗2−convergence of double sequences in neutrosophic normed spaces mainly as a generalization of statistical convergence of double sequences. We investigate a few fundamental properties and study the relationship between the two notions. We also introduce and investigate the concept of I2 and I∗2−Cauchy sequence of double sequences and show that the condition (AP2) plays a crucial role to study the interrelationship between them.","PeriodicalId":421614,"journal":{"name":"Gulf Journal of Mathematics","volume":"13 2","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139279325","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}
Pub Date : 2023-11-12DOI: 10.56947/gjom.v15i2.1439
Khalid Adarbeh, Mohammad Adarbeh
This paper establishes the necessary and sufficient conditions for a bi-amalgamation to inherit the J-clean and weakly J-clean properties in the commutative setting. The new results recover previous works carried on these rings in amalgamations and trivial extensions, also, they are used to provide new examples and counterexamples of J-clean and weakly J-clean rings arise as bi-amalgamations.
{"title":"Bi-amalgamations subject to j-clean and weakly j-clean properties","authors":"Khalid Adarbeh, Mohammad Adarbeh","doi":"10.56947/gjom.v15i2.1439","DOIUrl":"https://doi.org/10.56947/gjom.v15i2.1439","url":null,"abstract":"This paper establishes the necessary and sufficient conditions for a bi-amalgamation to inherit the J-clean and weakly J-clean properties in the commutative setting. The new results recover previous works carried on these rings in amalgamations and trivial extensions, also, they are used to provide new examples and counterexamples of J-clean and weakly J-clean rings arise as bi-amalgamations.","PeriodicalId":421614,"journal":{"name":"Gulf Journal of Mathematics","volume":"39 2","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139279679","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}
Pub Date : 2023-08-22DOI: 10.56947/gjom.v15i1.1384
N. M. Moukala, B. G. R. Bossoto
We define the Schouten-Nijenhuis bracket on the algebra of the module of Kahler differentials. We give the main features of Poisson manifolds by using the universal property of derivation. We prove the equivalence between a Lie-Rinehart algebra structure and a Poisson structure and we recover Lichnerowicz's notion of Poisson manifold. We show that a symplectic Lie-Rinehart algebra structure induce a nondegenerate Poisson structure and conversely.
{"title":"On Lie-Rinehart-Poisson algebras structures","authors":"N. M. Moukala, B. G. R. Bossoto","doi":"10.56947/gjom.v15i1.1384","DOIUrl":"https://doi.org/10.56947/gjom.v15i1.1384","url":null,"abstract":"We define the Schouten-Nijenhuis bracket on the algebra of the module of Kahler differentials. We give the main features of Poisson manifolds by using the universal property of derivation. We prove the equivalence between a Lie-Rinehart algebra structure and a Poisson structure and we recover Lichnerowicz's notion of Poisson manifold. We show that a symplectic Lie-Rinehart algebra structure induce a nondegenerate Poisson structure and conversely.","PeriodicalId":421614,"journal":{"name":"Gulf Journal of Mathematics","volume":"119 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125177436","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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