Pub Date : 2022-01-01DOI: 10.59277/mrar.2023.25.75.1.113
Akbar Ali, S. Balachandran, S. Elumalai
"The general Randi´c index of a graph G is defined as Rα(G) = P uv∈E(G)(dudv)α, where du and dv denote the degrees of the vertices u and v, respectively, α is a real number, and E(G) is the edge set of G. The minimum number of edges of a graph G whose removal makes G as acyclic is known as the cyclomatic number and it is usually denoted by ν. A graph with the maximum degree at most 4 is known as a chemical graph. For ν = 0, 1, 2 and α > 1, the problem of finding graph(s) with the minimum general Randi´c index Rα among all n-vertex chemical graphs with the cyclomatic number ν has already been solved. In this paper, this problem is solved for the case when ν ≥ 3, n ≥ 5(ν − 1), and 1 < α < α0, where α0 ≈ 11.4496 is the unique positive root of the equation 4(8α − 6α) + 4α − 9α = 0."
{"title":"ON N-VERTEX CHEMICAL GRAPHS WITH A FIXED CYCLOMATIC NUMBER AND MINIMUM GENERAL RANDI´C INDEX","authors":"Akbar Ali, S. Balachandran, S. Elumalai","doi":"10.59277/mrar.2023.25.75.1.113","DOIUrl":"https://doi.org/10.59277/mrar.2023.25.75.1.113","url":null,"abstract":"\"The general Randi´c index of a graph G is defined as Rα(G) = P uv∈E(G)(dudv)α, where du and dv denote the degrees of the vertices u and v, respectively, α is a real number, and E(G) is the edge set of G. The minimum number of edges of a graph G whose removal makes G as acyclic is known as the cyclomatic number and it is usually denoted by ν. A graph with the maximum degree at most 4 is known as a chemical graph. For ν = 0, 1, 2 and α > 1, the problem of finding graph(s) with the minimum general Randi´c index Rα among all n-vertex chemical graphs with the cyclomatic number ν has already been solved. In this paper, this problem is solved for the case when ν ≥ 3, n ≥ 5(ν − 1), and 1 < α < α0, where α0 ≈ 11.4496 is the unique positive root of the equation 4(8α − 6α) + 4α − 9α = 0.\"","PeriodicalId":49858,"journal":{"name":"Mathematical Reports","volume":"44 1","pages":""},"PeriodicalIF":0.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84704854","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}
Pub Date : 2022-01-01DOI: 10.59277/mrar.2023.25.75.1.23
S. Baraket, Rima Chetouane, F. Mtiri
We study the existence of solutions having singular limits for some four-di-men-sion-al semilinear elliptic problems involving exponential nonlinearity with nonlinear terms with Navier boundary condition. In particular, we extend the result of cite{BBT}.
{"title":"Singular limiting radial solutions for 4-dimensional elliptic problem involving exponentially dominated nonlinearity","authors":"S. Baraket, Rima Chetouane, F. Mtiri","doi":"10.59277/mrar.2023.25.75.1.23","DOIUrl":"https://doi.org/10.59277/mrar.2023.25.75.1.23","url":null,"abstract":"We study the existence of solutions having singular limits for some four-di-men-sion-al semilinear elliptic problems involving exponential nonlinearity with nonlinear terms with Navier boundary condition. In particular, we extend the result of cite{BBT}.","PeriodicalId":49858,"journal":{"name":"Mathematical Reports","volume":"96 1","pages":""},"PeriodicalIF":0.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73417601","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}
Pub Date : 2022-01-01DOI: 10.59277/mrar.2023.25.75.1.187
Zhanmin Zhu
"Let A be a class of right R-modules that is closed under isomorphisms, and let M be a right R-module. Then M is called A -C3 if, whenever N and K are direct summands of M with N ∩K = 0 and K ∈ A , then N ⊕K is also a direct summand of M; M is called an A -C4 module, if whenever M = A⊕B where A and B are submodules of M and A ∈ A , then every monomorphism f : A → B splits. Some characterizations and properties of these classes of modules are investigated. As applications, some new characterizations of semisimple artinian rings, right V-rings, quasi-Frobenius rings and von Neumann regular rings are given."
{"title":"GENERALIZATIONS OF C3 MODULES AND C4 MODULES","authors":"Zhanmin Zhu","doi":"10.59277/mrar.2023.25.75.1.187","DOIUrl":"https://doi.org/10.59277/mrar.2023.25.75.1.187","url":null,"abstract":"\"Let A be a class of right R-modules that is closed under isomorphisms, and let M be a right R-module. Then M is called A -C3 if, whenever N and K are direct summands of M with N ∩K = 0 and K ∈ A , then N ⊕K is also a direct summand of M; M is called an A -C4 module, if whenever M = A⊕B where A and B are submodules of M and A ∈ A , then every monomorphism f : A → B splits. Some characterizations and properties of these classes of modules are investigated. As applications, some new characterizations of semisimple artinian rings, right V-rings, quasi-Frobenius rings and von Neumann regular rings are given.\"","PeriodicalId":49858,"journal":{"name":"Mathematical Reports","volume":"130 ","pages":""},"PeriodicalIF":0.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72494971","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}
Pub Date : 2022-01-01DOI: 10.59277/mrar.2023.25.75.1.1
S. Shu
"Let Mn be an n-dimensional (n ≥ 3) complete smooth connected and oriented hypersurface in a real space form Mn+1(c) (c = 0, 1, −1) with constant quasiGauss-Kronecker curvature and two distinct principal curvatures. Denoting by H the mean curvature, |A| 2 the squared norm of the second fundamental form and Kq the quasi-Gauss-Kronecker curvature of Mn , we obtain some characterizations of S k (a)×Rn−k or S k (a)×S n−k ( √ 1 − a 2) or S k (a)×Hn−k (− √ 1 + a 2) in terms of H, |A| 2 and Kq, where 1 ≤ k ≤ n−1 and S k (a) is the k-dimensional sphere with radius a"
{"title":"HYPERSURFACES WITH CONSTANT QUASI-GAUSS-KRONECKER CURVATURE IN Mn+1(c)","authors":"S. Shu","doi":"10.59277/mrar.2023.25.75.1.1","DOIUrl":"https://doi.org/10.59277/mrar.2023.25.75.1.1","url":null,"abstract":"\"Let Mn be an n-dimensional (n ≥ 3) complete smooth connected and oriented hypersurface in a real space form Mn+1(c) (c = 0, 1, −1) with constant quasiGauss-Kronecker curvature and two distinct principal curvatures. Denoting by H the mean curvature, |A| 2 the squared norm of the second fundamental form and Kq the quasi-Gauss-Kronecker curvature of Mn , we obtain some characterizations of S k (a)×Rn−k or S k (a)×S n−k ( √ 1 − a 2) or S k (a)×Hn−k (− √ 1 + a 2) in terms of H, |A| 2 and Kq, where 1 ≤ k ≤ n−1 and S k (a) is the k-dimensional sphere with radius a\"","PeriodicalId":49858,"journal":{"name":"Mathematical Reports","volume":"452 1","pages":""},"PeriodicalIF":0.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75113780","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}
Pub Date : 2022-01-01DOI: 10.59277/mrar.2023.25.75.1.179
Elham Gholami, J. Rad, A. Tehranian
"For a graph G and an integer k ≥ 2, let f : V (G) → P({1, 2, ..., k}) be a function. If for each vertex v ∈ V (G) such that f(v) = ∅ we have ∪u∈N(v)f(u) = {1, 2, ..., k}, then f is called a k-rainbow dominating function (or simply kRDF) of G. The weight of a kRDF f is defined as w(f) = P v∈V (G) |f(v)|. The minimum weight of a kRDF of G is called the k-rainbow domination number of G, and is denoted by γrk(G). An independent k-rainbow dominating function (IkRDF) is a kRDF f with the property that {v : f(v) ̸= ∅} is an independent set. The minimum weight of an IkRDF of G is called the independent k-rainbow domination number of G, and is denoted by irk(G). A graph G is k-rainbow domination stable if the k-rainbow domination number of G remains unchanged under removal of any vertex. Likewise, a graph G is independent k-rainbow domination stable if the independent k-rainbow domination number of G remains unchanged under removal of any vertex. In this paper, we prove that determining whether a graph is k-rainbow domination stable or independent k-rainbow domination stable is NP-hard even when restricted to bipartite or planar graphs, thus answering a question posed in [11]."
对于图G，且k≥2，设f: V (G)→P({1,2，…， k})是一个函数。如果对于每个顶点v∈v (G)使得f(v) =∅我们有∪u∈N(v)f(u) ={1,2，…， k}，则f称为G的k-彩虹支配函数(或简称为kRDF)。一个kRDF f的权值定义为w(f) = P v∈v (G) |f(v)|。G的kRDF的最小权值称为G的k-彩虹支配数，用γrk(G)表示。一个独立的k-彩虹支配函数(IkRDF)是一个具有{v: f(v) h =∅}是一个独立集合的kRDF。G的IkRDF的最小权值称为G的独立k-彩虹支配数，用irk(G)表示。如果图G的k-彩虹控制数在移除任意顶点后保持不变，则图G是k-彩虹控制稳定的。同样，如果图G的独立k-彩虹控制数在移除任意顶点后保持不变，则图G是独立k-彩虹控制稳定的。在本文中，我们证明了判定一个图是k-彩虹控制稳定还是独立k-彩虹控制稳定是np困难的，即使限制在二部图或平面图上，从而回答了[11]中提出的一个问题。
{"title":"ON THE INDEPENDENT RAINBOW DOMINATION STABLE GRAPHS","authors":"Elham Gholami, J. Rad, A. Tehranian","doi":"10.59277/mrar.2023.25.75.1.179","DOIUrl":"https://doi.org/10.59277/mrar.2023.25.75.1.179","url":null,"abstract":"\"For a graph G and an integer k ≥ 2, let f : V (G) → P({1, 2, ..., k}) be a function. If for each vertex v ∈ V (G) such that f(v) = ∅ we have ∪u∈N(v)f(u) = {1, 2, ..., k}, then f is called a k-rainbow dominating function (or simply kRDF) of G. The weight of a kRDF f is defined as w(f) = P v∈V (G) |f(v)|. The minimum weight of a kRDF of G is called the k-rainbow domination number of G, and is denoted by γrk(G). An independent k-rainbow dominating function (IkRDF) is a kRDF f with the property that {v : f(v) ̸= ∅} is an independent set. The minimum weight of an IkRDF of G is called the independent k-rainbow domination number of G, and is denoted by irk(G). A graph G is k-rainbow domination stable if the k-rainbow domination number of G remains unchanged under removal of any vertex. Likewise, a graph G is independent k-rainbow domination stable if the independent k-rainbow domination number of G remains unchanged under removal of any vertex. In this paper, we prove that determining whether a graph is k-rainbow domination stable or independent k-rainbow domination stable is NP-hard even when restricted to bipartite or planar graphs, thus answering a question posed in [11].\"","PeriodicalId":49858,"journal":{"name":"Mathematical Reports","volume":"5 ","pages":""},"PeriodicalIF":0.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72440278","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}
Pub Date : 2022-01-01DOI: 10.59277/mrar.2023.25.75.1.63
Wenlin Huang
"We define p-critical kG-modules, and prove that the Green correspondence induces a bijection between the isomorphism classes of indecomposable p-critical kG-modules and those of indecomposable p-critical kH -modules."
{"title":"\"ON p -CRITICAL MODULES AND THE GREEN CORRESPONDENCE\"","authors":"Wenlin Huang","doi":"10.59277/mrar.2023.25.75.1.63","DOIUrl":"https://doi.org/10.59277/mrar.2023.25.75.1.63","url":null,"abstract":"\"We define p-critical kG-modules, and prove that the Green correspondence induces a bijection between the isomorphism classes of indecomposable p-critical kG-modules and those of indecomposable p-critical kH -modules.\"","PeriodicalId":49858,"journal":{"name":"Mathematical Reports","volume":"29 1","pages":""},"PeriodicalIF":0.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75756140","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}
Pub Date : 2022-01-01DOI: 10.59277/mrar.2023.25.75.1.123
P. Pandey
"In this article, the unified difference and quasi qudratization method are developed and discussed to obtain an approximate numerical solution for the Troesch’s problem, a nonlinear differential equation and corresponding boundary value problems. The degree of accuracy in numerical solution by the proposed method is good and comparable to other existing methods in literature for the range of values of parameter in Troesch’s problem."
{"title":"\"AN UNIFIED DIFFERENCE METHOD FOR THE NUMERICAL SOLUTION OF THE TROESCH’S BOUNDARY VALUE PROBLEM\"","authors":"P. Pandey","doi":"10.59277/mrar.2023.25.75.1.123","DOIUrl":"https://doi.org/10.59277/mrar.2023.25.75.1.123","url":null,"abstract":"\"In this article, the unified difference and quasi qudratization method are developed and discussed to obtain an approximate numerical solution for the Troesch’s problem, a nonlinear differential equation and corresponding boundary value problems. The degree of accuracy in numerical solution by the proposed method is good and comparable to other existing methods in literature for the range of values of parameter in Troesch’s problem.\"","PeriodicalId":49858,"journal":{"name":"Mathematical Reports","volume":"44 1","pages":""},"PeriodicalIF":0.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72533135","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}
Pub Date : 2022-01-01DOI: 10.59277/mrar.2023.25.75.1.85
Shoutao Guo, Xiaoyan Yang
"Adjoint preenvelopes and adjoint precovers are defined in the category of functors by replacing the functor Hom with ⊗. We investigate the existence and basic properties of adjoint preenvelopes and adjoint precovers. The F-projective (F-injective, F-flat) functors introduced by Mao are characterized in terms of adjoint preenvelopes and adjoint precovers. We obtain relationships among adjoint preenvelopes, adjoint precovers, preenvelopes and precovers."
{"title":"\"ADJOINT PREENVELOPES AND ADJOINT PRECOVERS IN THE FUNCTOR CATEGORY\"","authors":"Shoutao Guo, Xiaoyan Yang","doi":"10.59277/mrar.2023.25.75.1.85","DOIUrl":"https://doi.org/10.59277/mrar.2023.25.75.1.85","url":null,"abstract":"\"Adjoint preenvelopes and adjoint precovers are defined in the category of functors by replacing the functor Hom with ⊗. We investigate the existence and basic properties of adjoint preenvelopes and adjoint precovers. The F-projective (F-injective, F-flat) functors introduced by Mao are characterized in terms of adjoint preenvelopes and adjoint precovers. We obtain relationships among adjoint preenvelopes, adjoint precovers, preenvelopes and precovers.\"","PeriodicalId":49858,"journal":{"name":"Mathematical Reports","volume":"35 1","pages":""},"PeriodicalIF":0.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82249385","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}
Pub Date : 2022-01-01DOI: 10.59277/mrar.2023.25.75.1.167
Mohamed Khalifa
"We study commutative ring extensions R ⊂ S in which every ideal of S that is a prime R-submodule of S is prime."
“我们研究交换环扩展R∧S，其中S的每一个理想都是S的素数R子模是素数。”
{"title":"WHEN PRIME SUBMODULES ARE PRIME IDEALS?","authors":"Mohamed Khalifa","doi":"10.59277/mrar.2023.25.75.1.167","DOIUrl":"https://doi.org/10.59277/mrar.2023.25.75.1.167","url":null,"abstract":"\"We study commutative ring extensions R ⊂ S in which every ideal of S that is a prime R-submodule of S is prime.\"","PeriodicalId":49858,"journal":{"name":"Mathematical Reports","volume":"6 1","pages":""},"PeriodicalIF":0.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82535970","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}
Pub Date : 2022-01-01DOI: 10.59277/mrar.2023.25.75.1.153
Huanyin Chen, M. Sheibani
"A ring R is Yaqub nil-clean if a+a3 or a−a3 is nilpotent for all a ∈ R. We prove that a ring R is a Yaqub nil-clean ring if and only if R ∼= R1,R2,R3,R1 ×R2 or R1×R3, where R1/J(R1) is Boolean, R2/J(R2) is a Yaqub ring, R3/J(R3) ∼= Z5 and each J(Ri) is nil, if and only if J(R) is nil and R/J(R) is isomorphic to a Boolean ring R1, a Yaqub ring R2, Z5, R1×R2, or R1×Z5, if and only if for any a ∈ R, there exists e3 = e such that a − e or a + 3e is nilpotent and ae = ea, if and only if R is an exchange Hirano ring. The structure of such rings is thereby completely determined."
{"title":"ON YAQUB NIL-CLEAN RINGS","authors":"Huanyin Chen, M. Sheibani","doi":"10.59277/mrar.2023.25.75.1.153","DOIUrl":"https://doi.org/10.59277/mrar.2023.25.75.1.153","url":null,"abstract":"\"A ring R is Yaqub nil-clean if a+a3 or a−a3 is nilpotent for all a ∈ R. We prove that a ring R is a Yaqub nil-clean ring if and only if R ∼= R1,R2,R3,R1 ×R2 or R1×R3, where R1/J(R1) is Boolean, R2/J(R2) is a Yaqub ring, R3/J(R3) ∼= Z5 and each J(Ri) is nil, if and only if J(R) is nil and R/J(R) is isomorphic to a Boolean ring R1, a Yaqub ring R2, Z5, R1×R2, or R1×Z5, if and only if for any a ∈ R, there exists e3 = e such that a − e or a + 3e is nilpotent and ae = ea, if and only if R is an exchange Hirano ring. The structure of such rings is thereby completely determined.\"","PeriodicalId":49858,"journal":{"name":"Mathematical Reports","volume":"58 1","pages":""},"PeriodicalIF":0.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86568570","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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