Pub Date : 2021-06-21DOI: 10.12697/acutm.2021.25.11
Orhan Dişkaya, H. Menken
This paper examines the Fibonacci quaternion sequence with quadruple-produce components, and demonstrates a golden-like ratio and some identities for this sequence. Its generating and exponential generating functions are given. Along with these, its series and binomial sum formula are established.
{"title":"On the Fibonacci quaternion sequence with quadruple-produce components","authors":"Orhan Dişkaya, H. Menken","doi":"10.12697/acutm.2021.25.11","DOIUrl":"https://doi.org/10.12697/acutm.2021.25.11","url":null,"abstract":"This paper examines the Fibonacci quaternion sequence with quadruple-produce components, and demonstrates a golden-like ratio and some identities for this sequence. Its generating and exponential generating functions are given. Along with these, its series and binomial sum formula are established.","PeriodicalId":42426,"journal":{"name":"Acta et Commentationes Universitatis Tartuensis de Mathematica","volume":"2 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2021-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88011789","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 : 2021-06-21DOI: 10.12697/acutm.2021.25.02
T. Ernst, P. W. Karlsson
We find some new simple hypergeometric formulas in the footsteps of the important article by Gessel and Stanton. These are multiple reduction formulas, multiple summation formulas, as well as multiple transformation formulas for special Kampé de Fériet functions and Appell functions. The hypergeometric summation formulas have special function arguments in Q and parameter values in N or C. The proofs use Pfaff-Kummer transformation, Euler transformation, or an improved form of Slater reversion.
我们从格塞尔和斯坦顿的重要文章中找到了一些新的简单的超几何公式。这些是多个约简公式，多个求和公式，以及特殊kamp de fsamriet函数和Appell函数的多个变换公式。超几何求和公式在Q上有特殊的函数参数，在N或c上有特殊的参数值。证明采用了Pfaff-Kummer变换、Euler变换或Slater反演的改进形式。
{"title":"Corollaries and multiple extensions of Gessel and Stanton hypergeometric summation formulas","authors":"T. Ernst, P. W. Karlsson","doi":"10.12697/acutm.2021.25.02","DOIUrl":"https://doi.org/10.12697/acutm.2021.25.02","url":null,"abstract":"We find some new simple hypergeometric formulas in the footsteps of the important article by Gessel and Stanton. These are multiple reduction formulas, multiple summation formulas, as well as multiple transformation formulas for special Kampé de Fériet functions and Appell functions. The hypergeometric summation formulas have special function arguments in Q and parameter values in N or C. The proofs use Pfaff-Kummer transformation, Euler transformation, or an improved form of Slater reversion.","PeriodicalId":42426,"journal":{"name":"Acta et Commentationes Universitatis Tartuensis de Mathematica","volume":"41 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2021-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88374275","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 : 2021-06-21DOI: 10.12697/acutm.2021.25.09
H. Yildirim, Seda Kilinç Yildirim
We obtain new generalizations of Ostrowski inequality by using generalized Riemann{Liouville fractional integrals. Some special cases are also discussed.
{"title":"On generalized fractional integral inequalities of Ostrowski type","authors":"H. Yildirim, Seda Kilinç Yildirim","doi":"10.12697/acutm.2021.25.09","DOIUrl":"https://doi.org/10.12697/acutm.2021.25.09","url":null,"abstract":"We obtain new generalizations of Ostrowski inequality by using generalized Riemann{Liouville fractional integrals. Some special cases are also discussed.","PeriodicalId":42426,"journal":{"name":"Acta et Commentationes Universitatis Tartuensis de Mathematica","volume":"21 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2021-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80352281","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 : 2021-06-21DOI: 10.12697/acutm.2021.25.10
S. Bouroubi, Ali Debbache
A Thue equation is a Diophantine equation of the form f(x; y) = r, where f is an irreducible binary form of degree at least 3, and r is a given nonzero rational number. A set S of at least three positive integers is called a D13-set if the product of any of its three distinct elements is a perfect cube minus one. We prove that any D13-set is finite and, for any positive integer a, the two-tuple {a, 2a} is extendible to a D13-set 3-tuple, but not to a 4-tuple. Using the well-known Thue equation 2x3 - y3 = 1, we show that the only cubic-triangular number is 1.
{"title":"Thue's equation as a tool to solve two different problems","authors":"S. Bouroubi, Ali Debbache","doi":"10.12697/acutm.2021.25.10","DOIUrl":"https://doi.org/10.12697/acutm.2021.25.10","url":null,"abstract":"A Thue equation is a Diophantine equation of the form f(x; y) = r, where f is an irreducible binary form of degree at least 3, and r is a given nonzero rational number. A set S of at least three positive integers is called a D13-set if the product of any of its three distinct elements is a perfect cube minus one. We prove that any D13-set is finite and, for any positive integer a, the two-tuple {a, 2a} is extendible to a D13-set 3-tuple, but not to a 4-tuple. Using the well-known Thue equation 2x3 - y3 = 1, we show that the only cubic-triangular number is 1.","PeriodicalId":42426,"journal":{"name":"Acta et Commentationes Universitatis Tartuensis de Mathematica","volume":"11 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2021-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82221232","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 : 2021-06-21DOI: 10.12697/acutm.2021.25.08
Terje Hill, D. Robbins
Let X be a compact Hausdorff space, and let {Ax : x ∈ X} and {Bx : x ∈ X} be collections of Banach algebras such that each Ax is a Bx-bimodule. Using the theory of bundles of Banach spaces as a tool, we investigate the module amenability of certain algebras of Ax-valued functions on X over algebras of Bx-valued functions on X.
{"title":"Module bundles and module amenability","authors":"Terje Hill, D. Robbins","doi":"10.12697/acutm.2021.25.08","DOIUrl":"https://doi.org/10.12697/acutm.2021.25.08","url":null,"abstract":"Let X be a compact Hausdorff space, and let {Ax : x ∈ X} and {Bx : x ∈ X} be collections of Banach algebras such that each Ax is a Bx-bimodule. Using the theory of bundles of Banach spaces as a tool, we investigate the module amenability of certain algebras of Ax-valued functions on X over algebras of Bx-valued functions on X.","PeriodicalId":42426,"journal":{"name":"Acta et Commentationes Universitatis Tartuensis de Mathematica","volume":"17 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2021-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72744355","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 : 2021-01-01DOI: 10.12697/acutm.2021.25.01
X. Krasniqi
Using the Mean Rest Bounded Variation Sequences or the Mean Head Bounded Variation Sequences, we have proved four theorems pertaining to the degree of approximation in sup-norm of a continuous function f by general means τλn;A(f) of partial sums of its Fourier series. The degree of approximation is expressed via an auxiliary function H(t) ≥ 0 and via entries of a matrix whose indices form a strictly increasing sequence of positive integers λ := {λ(n)}∞n=1.
利用中差有界变差序列或中头有界变差序列，我们证明了关于连续函数f在其傅里叶级数的部分和的一般均值τλn; a (f)在其上模的逼近程度的四个定理。逼近的程度通过一个辅助函数H(t)≥0和一个矩阵的条目来表示，该矩阵的索引形成一个严格递增的正整数序列λ:= {λ(n)}∞n=1。
{"title":"On the degree of approximation of continuous functions by a specific transform of partial sums of their Fourier series","authors":"X. Krasniqi","doi":"10.12697/acutm.2021.25.01","DOIUrl":"https://doi.org/10.12697/acutm.2021.25.01","url":null,"abstract":"Using the Mean Rest Bounded Variation Sequences or the Mean Head Bounded Variation Sequences, we have proved four theorems pertaining to the degree of approximation in sup-norm of a continuous function f by general means τλn;A(f) of partial sums of its Fourier series. The degree of approximation is expressed via an auxiliary function H(t) ≥ 0 and via entries of a matrix whose indices form a strictly increasing sequence of positive integers λ := {λ(n)}∞n=1.","PeriodicalId":42426,"journal":{"name":"Acta et Commentationes Universitatis Tartuensis de Mathematica","volume":"15 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83331061","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 : 2020-12-31DOI: 10.12697/ACUTM.2020.24.15
Fumie Nakaoka, Nobuyuki Oda
A set with an operation is a generalization of a topological space. Two types of continuous functions are dened between sets with operations. They are characterized making use of two types of closures and interiors. Homeomorphisms between sets with operations are also characterized. Variants of subspaces, connected spaces and compact spaces are introduced in a set with an operation and some fundamental properties of them are proved.
{"title":"Continuous functions between sets with operations","authors":"Fumie Nakaoka, Nobuyuki Oda","doi":"10.12697/ACUTM.2020.24.15","DOIUrl":"https://doi.org/10.12697/ACUTM.2020.24.15","url":null,"abstract":"A set with an operation is a generalization of a topological space. Two types of continuous functions are dened between sets with operations. They are characterized making use of two types of closures and interiors. Homeomorphisms between sets with operations are also characterized. Variants of subspaces, connected spaces and compact spaces are introduced in a set with an operation and some fundamental properties of them are proved.","PeriodicalId":42426,"journal":{"name":"Acta et Commentationes Universitatis Tartuensis de Mathematica","volume":"176 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2020-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79823539","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 : 2020-12-31DOI: 10.12697/ACUTM.2020.24.18
S. Dragomir
We obtain some Hölder type trace inequalities for operator weighted geometric mean. Some vector inequalities are also given.
得到了算子加权几何均值的一些Hölder型迹不等式。还给出了一些向量不等式。
{"title":"On some Hölder type trace inequalities for operator weighted geometric mean","authors":"S. Dragomir","doi":"10.12697/ACUTM.2020.24.18","DOIUrl":"https://doi.org/10.12697/ACUTM.2020.24.18","url":null,"abstract":"We obtain some Hölder type trace inequalities for operator weighted geometric mean. Some vector inequalities are also given.","PeriodicalId":42426,"journal":{"name":"Acta et Commentationes Universitatis Tartuensis de Mathematica","volume":"66 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2020-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85779921","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 : 2020-12-31DOI: 10.12697/ACUTM.2020.24.13
M. Abbas
By employing Kuratowski's measure of noncompactness together with Sadovskii's fixed point theorem, sufficient conditions for controllability results of Hilfer-Katugampola fractional differential equations in Banach spaces are derived.
{"title":"On the controllability of Hilfer-Katugampola fractional differential equations","authors":"M. Abbas","doi":"10.12697/ACUTM.2020.24.13","DOIUrl":"https://doi.org/10.12697/ACUTM.2020.24.13","url":null,"abstract":"By employing Kuratowski's measure of noncompactness together with Sadovskii's fixed point theorem, sufficient conditions for controllability results of Hilfer-Katugampola fractional differential equations in Banach spaces are derived.","PeriodicalId":42426,"journal":{"name":"Acta et Commentationes Universitatis Tartuensis de Mathematica","volume":"39 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2020-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75843592","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 : 2019-08-09DOI: 10.12697/ACUTM.2019.23.05
S. Dragomir, M. Latif, E. Momoniat
A new identity involving a geometrically symmetric function and a differentiable function is established. Some new Fejér type integral inequalities, connected with the left part of Hermite–Hadamard type inequalities for geometrically-arithmetically convex functions, are presented by using the Hölder integral inequality and the notion of geometrically-arithmetically convexity. Applications of our results to special means of positive real numbers are given.
{"title":"Fejér type integral inequalities related with geometrically-arithmetically convex functions with applications","authors":"S. Dragomir, M. Latif, E. Momoniat","doi":"10.12697/ACUTM.2019.23.05","DOIUrl":"https://doi.org/10.12697/ACUTM.2019.23.05","url":null,"abstract":"A new identity involving a geometrically symmetric function and a differentiable function is established. Some new Fejér type integral inequalities, connected with the left part of Hermite–Hadamard type inequalities for geometrically-arithmetically convex functions, are presented by using the Hölder integral inequality and the notion of geometrically-arithmetically convexity. Applications of our results to special means of positive real numbers are given.","PeriodicalId":42426,"journal":{"name":"Acta et Commentationes Universitatis Tartuensis de Mathematica","volume":"352 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2019-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76584383","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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