Pub Date : 2014-12-01DOI: 10.2478/AUPCSM-2014-0007
Uday Chand De, Prajjwal Pal
Abstract The purpose of the present paper is to study generalized M-projectively recurrent manifolds. Some geometric properties of generalized M projectively recurrent manifolds have been studied under certain curvature conditions. An application of such a manifold in the theory of relativity has also been shown. Finally, we give an example of a generalized M-projectively recurrent manifold.
{"title":"On generalized M-projectively recurrent manifolds","authors":"Uday Chand De, Prajjwal Pal","doi":"10.2478/AUPCSM-2014-0007","DOIUrl":"https://doi.org/10.2478/AUPCSM-2014-0007","url":null,"abstract":"Abstract The purpose of the present paper is to study generalized M-projectively recurrent manifolds. Some geometric properties of generalized M projectively recurrent manifolds have been studied under certain curvature conditions. An application of such a manifold in the theory of relativity has also been shown. Finally, we give an example of a generalized M-projectively recurrent manifold.","PeriodicalId":53863,"journal":{"name":"Annales Universitatis Paedagogicae Cracoviensis-Studia Mathematica","volume":"54 1","pages":"77 - 101"},"PeriodicalIF":0.9,"publicationDate":"2014-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83301310","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 : 2014-12-01DOI: 10.2478/AUPCSM-2014-0001
J. Gorowski, Adam Łomnicki
Abstract In this paper a remarkable simple proof of the Gauss’s generalization of the Wilson’s theorem is given. The proof is based on properties of a subgroup generated by element of order 2 of a finite abelian group. Some conditions equivalent to the cyclicity of (Φ(n), ·n), where n > 2 is an integer are presented, in particular, a condition for the existence of the unique element of order 2 in such a group.
{"title":"Simple proofs of some generalizations of the Wilson’s theorem","authors":"J. Gorowski, Adam Łomnicki","doi":"10.2478/AUPCSM-2014-0001","DOIUrl":"https://doi.org/10.2478/AUPCSM-2014-0001","url":null,"abstract":"Abstract In this paper a remarkable simple proof of the Gauss’s generalization of the Wilson’s theorem is given. The proof is based on properties of a subgroup generated by element of order 2 of a finite abelian group. Some conditions equivalent to the cyclicity of (Φ(n), ·n), where n > 2 is an integer are presented, in particular, a condition for the existence of the unique element of order 2 in such a group.","PeriodicalId":53863,"journal":{"name":"Annales Universitatis Paedagogicae Cracoviensis-Studia Mathematica","volume":"141 1","pages":"7 - 14"},"PeriodicalIF":0.9,"publicationDate":"2014-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77458221","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 : 2014-12-01DOI: 10.2478/AUPCSM-2014-0010
D. Cichoń
Abstract A new proof of the Ficken criterion is given together with a comment concerning the known proofs and related results
摘要给出了菲肯准则的一个新的证明，并对已知的证明和相关结果作了评论
{"title":"À la recherche de la preuve perdue: a simple proof of the Ficken theorem","authors":"D. Cichoń","doi":"10.2478/AUPCSM-2014-0010","DOIUrl":"https://doi.org/10.2478/AUPCSM-2014-0010","url":null,"abstract":"Abstract A new proof of the Ficken criterion is given together with a comment concerning the known proofs and related results","PeriodicalId":53863,"journal":{"name":"Annales Universitatis Paedagogicae Cracoviensis-Studia Mathematica","volume":"1 1","pages":"133 - 137"},"PeriodicalIF":0.9,"publicationDate":"2014-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85681008","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 : 2014-12-01DOI: 10.2478/AUPCSM-2014-0005
K. Vijaya, G. Murugusundaramoorthy, M. Kasthuri
Abstract Recently Kanas and Ronning introduced the classes of starlike and convex functions, which are normalized with ƒ(ξ) = ƒ0(ξ) − 1 = 0, ξ (|ξ| = d) is a fixed point in the open disc U = {z ∈ ℂ: |z| < 1}. In this paper we define a new subclass of starlike functions of complex order based on q-hypergeometric functions and continue to obtain coefficient estimates, extreme points, inclusion properties and neighbourhood results for the function class T Sξ(α, β,γ). Further, we obtain integral means inequalities for the function ƒ ∈ T Sξ(α, β,γ).
{"title":"Starlike functions of complex order involving q-hypergeometric functions with fixed point","authors":"K. Vijaya, G. Murugusundaramoorthy, M. Kasthuri","doi":"10.2478/AUPCSM-2014-0005","DOIUrl":"https://doi.org/10.2478/AUPCSM-2014-0005","url":null,"abstract":"Abstract Recently Kanas and Ronning introduced the classes of starlike and convex functions, which are normalized with ƒ(ξ) = ƒ0(ξ) − 1 = 0, ξ (|ξ| = d) is a fixed point in the open disc U = {z ∈ ℂ: |z| < 1}. In this paper we define a new subclass of starlike functions of complex order based on q-hypergeometric functions and continue to obtain coefficient estimates, extreme points, inclusion properties and neighbourhood results for the function class T Sξ(α, β,γ). Further, we obtain integral means inequalities for the function ƒ ∈ T Sξ(α, β,γ).","PeriodicalId":53863,"journal":{"name":"Annales Universitatis Paedagogicae Cracoviensis-Studia Mathematica","volume":"60 1","pages":"51 - 63"},"PeriodicalIF":0.9,"publicationDate":"2014-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2478/AUPCSM-2014-0005","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72519008","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 : 2014-12-01DOI: 10.2478/AUPCSM-2014-0009
H. Toparkus
Abstract In this paper we consider first-order systems with constant coefficients for two real-valued functions of two real variables. This is both a problem in itself, as well as an alternative view of the classical linear partial differential equations of second order with constant coefficients. The classification of the systems is done using elementary methods of linear algebra. Each type presents its special canonical form in the associated characteristic coordinate system. Then you can formulate initial value problems in appropriate basic areas, and you can try to achieve a solution of these problems by means of transform methods.
{"title":"First-order systems of linear partial differential equations: normal forms, canonical systems, transform methods","authors":"H. Toparkus","doi":"10.2478/AUPCSM-2014-0009","DOIUrl":"https://doi.org/10.2478/AUPCSM-2014-0009","url":null,"abstract":"Abstract In this paper we consider first-order systems with constant coefficients for two real-valued functions of two real variables. This is both a problem in itself, as well as an alternative view of the classical linear partial differential equations of second order with constant coefficients. The classification of the systems is done using elementary methods of linear algebra. Each type presents its special canonical form in the associated characteristic coordinate system. Then you can formulate initial value problems in appropriate basic areas, and you can try to achieve a solution of these problems by means of transform methods.","PeriodicalId":53863,"journal":{"name":"Annales Universitatis Paedagogicae Cracoviensis-Studia Mathematica","volume":"93 1","pages":"109 - 132"},"PeriodicalIF":0.9,"publicationDate":"2014-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90431969","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 : 2014-12-01DOI: 10.2478/AUPCSM-2014-0008
Kamil Kular
Abstract We give an elementary and self-contained proof of the theorem which says that for a semiprime ring commutativity, Lie-nilpotency, and nilpotency of the Lie ring of inner derivations are equivalent conditions
摘要给出了半素环交换性定理的初等自完备证明，证明了内导的Lie-幂零性和Lie-幂零性是等价条件
{"title":"Semiprime rings with nilpotent Lie ring of inner derivations","authors":"Kamil Kular","doi":"10.2478/AUPCSM-2014-0008","DOIUrl":"https://doi.org/10.2478/AUPCSM-2014-0008","url":null,"abstract":"Abstract We give an elementary and self-contained proof of the theorem which says that for a semiprime ring commutativity, Lie-nilpotency, and nilpotency of the Lie ring of inner derivations are equivalent conditions","PeriodicalId":53863,"journal":{"name":"Annales Universitatis Paedagogicae Cracoviensis-Studia Mathematica","volume":"11 1","pages":"103 - 107"},"PeriodicalIF":0.9,"publicationDate":"2014-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78451822","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 : 2014-08-15DOI: 10.2478/aupcsm-2018-0010
Edward Tutaj
Abstract The convex hull of the subgraph of the prime counting function x → π(x) is a convex set, bounded from above by a graph of some piecewise affine function x → (x). The vertices of this function form an infinite sequence of points (ek,π(ek))1∞ $({e_k},pi ({e_k}))_1^infty $ . The elements of the sequence (ek)1∞ shall be called the extremal prime numbers. In this paper we present some observations about the sequence (ek)1∞ and we formulate a number of questions inspired by the numerical data. We prove also two – it seems – interesting results. First states that if the Riemann Hypothesis is true, then ek+1ek=1 ${{{e_k} + 1} over {{e_k}}} = 1$ . The second, also depending on Riemann Hypothesis, describes the order of magnitude of the differences between consecutive extremal prime numbers.
素数函数x→π(x)的子图的凸包是一个凸集，由上面的某个分段仿射函数x→(x)的图有界，该函数的顶点形成点(ek，π(ek))1∞$({e_k},pi ({e_k}))_1^infty $的无穷序列。数列(ek)1∞的元素称为极值素数。本文给出了关于数列(ek)1∞的一些观察结果，并在数值数据的启发下提出了一些问题。我们还证明了两个——看起来——有趣的结果。首先说明如果黎曼假设成立，那么ek+1ek=1 ${{{e_k} + 1} over {{e_k}}} = 1$。第二种方法，也依赖于黎曼假设，描述了连续极值素数之间差异的数量级。
{"title":"Prime numbers with a certain extremal type property","authors":"Edward Tutaj","doi":"10.2478/aupcsm-2018-0010","DOIUrl":"https://doi.org/10.2478/aupcsm-2018-0010","url":null,"abstract":"Abstract The convex hull of the subgraph of the prime counting function x → π(x) is a convex set, bounded from above by a graph of some piecewise affine function x → (x). The vertices of this function form an infinite sequence of points (ek,π(ek))1∞ $({e_k},pi ({e_k}))_1^infty $ . The elements of the sequence (ek)1∞ shall be called the extremal prime numbers. In this paper we present some observations about the sequence (ek)1∞ and we formulate a number of questions inspired by the numerical data. We prove also two – it seems – interesting results. First states that if the Riemann Hypothesis is true, then ek+1ek=1 ${{{e_k} + 1} over {{e_k}}} = 1$ . The second, also depending on Riemann Hypothesis, describes the order of magnitude of the differences between consecutive extremal prime numbers.","PeriodicalId":53863,"journal":{"name":"Annales Universitatis Paedagogicae Cracoviensis-Studia Mathematica","volume":"66 1","pages":"127 - 151"},"PeriodicalIF":0.9,"publicationDate":"2014-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89609705","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 : 2014-07-16DOI: 10.2478/AUPCSM-2014-0006
B. Piatek
Abstract In this work we consider two hyperconvex diversities (or hyperconvex metric spaces) (X, δX) and (Y, δY ) with nonempty intersection and we wonder whether there is a natural way to glue them so that the new glued diversity (or metric space) remains being hyperconvex. We provide positive and negative answers in both situations.
{"title":"On the gluing of hyperconvex metrics and diversities","authors":"B. Piatek","doi":"10.2478/AUPCSM-2014-0006","DOIUrl":"https://doi.org/10.2478/AUPCSM-2014-0006","url":null,"abstract":"Abstract In this work we consider two hyperconvex diversities (or hyperconvex metric spaces) (X, δX) and (Y, δY ) with nonempty intersection and we wonder whether there is a natural way to glue them so that the new glued diversity (or metric space) remains being hyperconvex. We provide positive and negative answers in both situations.","PeriodicalId":53863,"journal":{"name":"Annales Universitatis Paedagogicae Cracoviensis-Studia Mathematica","volume":"11 1","pages":"65 - 76"},"PeriodicalIF":0.9,"publicationDate":"2014-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78495304","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 : 2013-01-26DOI: 10.1515/AUPCSM-2015-0009
Chokri Abdelkefi, Mongi Rachdi
Abstract In the present paper, we prove weighted inequalities for the Dunkl transform (which generalizes the Fourier transform) when the weights belong to the well-known class Bp. As application, we obtain the Pitt’s inequality for power weights.
{"title":"The class Bp for weighted generalized Fourier transform inequalities","authors":"Chokri Abdelkefi, Mongi Rachdi","doi":"10.1515/AUPCSM-2015-0009","DOIUrl":"https://doi.org/10.1515/AUPCSM-2015-0009","url":null,"abstract":"Abstract In the present paper, we prove weighted inequalities for the Dunkl transform (which generalizes the Fourier transform) when the weights belong to the well-known class Bp. As application, we obtain the Pitt’s inequality for power weights.","PeriodicalId":53863,"journal":{"name":"Annales Universitatis Paedagogicae Cracoviensis-Studia Mathematica","volume":"464 1","pages":"121 - 133"},"PeriodicalIF":0.9,"publicationDate":"2013-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77036770","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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