In this paper, we derive several asymptotic formulas for the sum of d (gcd( m,n )), where d ( n ) is the divisor function and m,n are in Piatetski-Shapiro and Beatty sequences.
{"title":"Divisor Problem for the Greatest Common Divisor of Integers in Piatetski-Shapiro and Beatty Sequences","authors":"Sunanta Srisopha, Teerapat Srichan, Pinthira Tangsupphathawat","doi":"10.1556/314.2023.00024","DOIUrl":"https://doi.org/10.1556/314.2023.00024","url":null,"abstract":"In this paper, we derive several asymptotic formulas for the sum of d (gcd( m,n )), where d ( n ) is the divisor function and m,n are in Piatetski-Shapiro and Beatty sequences.","PeriodicalId":383314,"journal":{"name":"Mathematica Pannonica","volume":"27 6","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134992163","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 𝑛 ∈ ℕ. An element ( x 1 , … , x 𝑛 ) ∈ E n is called a norming point of T ∈ ( n E ) if ‖ x 1 ‖ = ⋯ = ‖ x n ‖ = 1 and | T ( x 1 , … , x n )| = ‖ T ‖, where ( n E ) denotes the space of all continuous n -linear forms on E . For T ∈ ( n E ), we define Norm( T ) = {( x 1 , … , x n ) ∈ E n ∶ ( x 1 , … , x n ) is a norming point of T }. Norm( T ) is called the norming set of T . We classify Norm( T ) for every T ∈ ( 2 𝑑 ∗ (1, w ) 2 ), where 𝑑 ∗ (1, w ) 2 = ℝ 2 with the octagonal norm of weight 0 < w < 1 endowed with .
让𝑛∈ℕ。一个元素x (x = 1, ...𝑛)∈E n是叫a norming point of T∈E (n)如果‖x 1‖=⋯=‖x n‖= 1和| T (x 1 x ..., n) | = T‖‖太空》(n E) denotes哪里,所有挑战n -linear forms on E。为T∈E (n),我们定义规范(T) = {(x 1, ... x, x n)∈E n∶(1、... x n)是a norming point of T了。Norm(T)是一组T。我们classify Norm (T)为每T∈(2𝑑∗(1 w) 2),哪里𝑑∗(1,w) 2 =ℝ2.0和重量之octagonal Norm <w <1 .充满活力。
{"title":"THE NORMING SETS OF L(2d*(1, w)2)","authors":"Sung Guen Kim","doi":"10.1556/314.2023.00022","DOIUrl":"https://doi.org/10.1556/314.2023.00022","url":null,"abstract":"Let 𝑛 ∈ ℕ. An element ( x 1 , … , x 𝑛 ) ∈ E n is called a norming point of T ∈ ( n E ) if ‖ x 1 ‖ = ⋯ = ‖ x n ‖ = 1 and | T ( x 1 , … , x n )| = ‖ T ‖, where ( n E ) denotes the space of all continuous n -linear forms on E . For T ∈ ( n E ), we define Norm( T ) = {( x 1 , … , x n ) ∈ E n ∶ ( x 1 , … , x n ) is a norming point of T }. Norm( T ) is called the norming set of T . We classify Norm( T ) for every T ∈ ( 2 𝑑 ∗ (1, w ) 2 ), where 𝑑 ∗ (1, w ) 2 = ℝ 2 with the octagonal norm of weight 0 < w < 1 endowed with .","PeriodicalId":383314,"journal":{"name":"Mathematica Pannonica","volume":"30 39","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135390275","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 paper, we introduce and study the class of k -strictly quasi-Fredholm linear relations on Banach spaces for nonnegative integer k . Then we investigate its robustness through perturbation by finite rank operators.
{"title":"On k-Strictly Quasi-Fredholm Linear Relations","authors":"Hafsa Bouaniza, Imen Issaoui, Maher Mnif","doi":"10.1556/314.2023.00021","DOIUrl":"https://doi.org/10.1556/314.2023.00021","url":null,"abstract":"In this paper, we introduce and study the class of k -strictly quasi-Fredholm linear relations on Banach spaces for nonnegative integer k . Then we investigate its robustness through perturbation by finite rank operators.","PeriodicalId":383314,"journal":{"name":"Mathematica Pannonica","volume":"205 5","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136318451","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}
We construct an algebra of dimension 2 ℵ0 consisting only of functions which in no point possess a finite one-sided derivative. We further show that some well known nowhere differentiable functions generate algebras, which contain functions which are differentiable at some points, but where for all functions in the algebra the set of points of differentiability is quite small.
{"title":"A Continuum Dimensional Algebra of Nowhere Differentiable Functions","authors":"Jan-Christoph Schlage-Puchta","doi":"10.1556/314.2023.00017","DOIUrl":"https://doi.org/10.1556/314.2023.00017","url":null,"abstract":"We construct an algebra of dimension 2 ℵ0 consisting only of functions which in no point possess a finite one-sided derivative. We further show that some well known nowhere differentiable functions generate algebras, which contain functions which are differentiable at some points, but where for all functions in the algebra the set of points of differentiability is quite small.","PeriodicalId":383314,"journal":{"name":"Mathematica Pannonica","volume":"3 2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135266890","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}
Grätzer and Lakser asked in the 1971 Transactions of the American Mathematical Society if the pseudocomplemented distributive lattices in the amalgamation class of the subvariety generated by 2 n ⊕ 1 can be characterized by the property of not having a * homomorphism onto 2 i ⊕ 1 for 1 < i < n . In this article, this question is answered.
{"title":"A 1971 question of Grätzer and Lakser on pseudocomplemented lattices","authors":"Jonathan David Farley, Dominic van der Zypen","doi":"10.1556/314.2023.00020","DOIUrl":"https://doi.org/10.1556/314.2023.00020","url":null,"abstract":"Grätzer and Lakser asked in the 1971 Transactions of the American Mathematical Society if the pseudocomplemented distributive lattices in the amalgamation class of the subvariety generated by 2 n ⊕ 1 can be characterized by the property of not having a * homomorphism onto 2 i ⊕ 1 for 1 < i < n . In this article, this question is answered.","PeriodicalId":383314,"journal":{"name":"Mathematica Pannonica","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135817134","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}
Over integral domains of characteristics different from 2, we determine all the matrices which are similar to .
在不同于2的特征的积分域上，我们确定了所有与2相似的矩阵。
{"title":"Similarity for 2 × 2 matrices obtained by clockwise “Rotation”","authors":"Grigore Călugăreanu","doi":"10.1556/314.2023.00019","DOIUrl":"https://doi.org/10.1556/314.2023.00019","url":null,"abstract":"Over integral domains of characteristics different from 2, we determine all the matrices which are similar to .","PeriodicalId":383314,"journal":{"name":"Mathematica Pannonica","volume":"79 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136136690","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}
Asymptotic uniform upper density, shortened as a.u.u.d., or simply upper density, is a classical notion which was first introduced by Kahane for sequences in the real line.Syndetic sets were defined by Gottschalk and Hendlund. For a locally compact group 𝐺, a set 𝑆 ⊂ 𝐺 is syndetic, if there exists a compact subset 𝐶 ⋐ 𝐺 such that 𝑆𝐶 = 𝐺. Syndetic sets play an important role in various fields of applications of topological groups and semigroups, ergodic theory and number theory. A lemma in the book of Fürstenberg says that once a subset 𝐴 ⊂ ℤ has positive a.u.u.d., then its difference set 𝐴 − 𝐴 is syndetic.The construction of a reasonable notion of a.u.u.d. in general locally compact Abelian groups (LCA groups for short) was not known for long, but in the late 2000’s several constructions were worked out to generalize it from the base cases of ℤ𝑑 and ℝ𝑑. With the notion available, several classical results of the Euclidean setting became accessible even in general LCA groups.Here we work out various versions in a general locally compact Abelian group 𝐺 of the classical statement that if a set 𝑆 ⊂ 𝐺 has positive asymptotic uniform upper density, then the difference set 𝑆 − 𝑆 is syndetic.
{"title":"Kahane’s Upper Density and Syndetic Sets in LCA Groups","authors":"S. R'ev'esz","doi":"10.1556/314.2023.00028","DOIUrl":"https://doi.org/10.1556/314.2023.00028","url":null,"abstract":"Asymptotic uniform upper density, shortened as a.u.u.d., or simply upper density, is a classical notion which was first introduced by Kahane for sequences in the real line.Syndetic sets were defined by Gottschalk and Hendlund. For a locally compact group 𝐺, a set 𝑆 ⊂ 𝐺 is syndetic, if there exists a compact subset 𝐶 ⋐ 𝐺 such that 𝑆𝐶 = 𝐺. Syndetic sets play an important role in various fields of applications of topological groups and semigroups, ergodic theory and number theory. A lemma in the book of Fürstenberg says that once a subset 𝐴 ⊂ ℤ has positive a.u.u.d., then its difference set 𝐴 − 𝐴 is syndetic.The construction of a reasonable notion of a.u.u.d. in general locally compact Abelian groups (LCA groups for short) was not known for long, but in the late 2000’s several constructions were worked out to generalize it from the base cases of ℤ𝑑 and ℝ𝑑. With the notion available, several classical results of the Euclidean setting became accessible even in general LCA groups.Here we work out various versions in a general locally compact Abelian group 𝐺 of the classical statement that if a set 𝑆 ⊂ 𝐺 has positive asymptotic uniform upper density, then the difference set 𝑆 − 𝑆 is syndetic.","PeriodicalId":383314,"journal":{"name":"Mathematica Pannonica","volume":"38 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139340617","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}
We present generalizations of the Pinelis extension of Stolarsky’s inequality and its reverse. In particular, a new Stolarsky-type inequality is obtained. We study the properties of the linear functional related to the new Stolarsky-type inequality, and finally apply these new results in the theory of fractional integrals.
{"title":"A note on the Pinelis extension of Stolarsky’s inequality","authors":"Sanja Varošanec","doi":"10.1556/314.2023.00018","DOIUrl":"https://doi.org/10.1556/314.2023.00018","url":null,"abstract":"We present generalizations of the Pinelis extension of Stolarsky’s inequality and its reverse. In particular, a new Stolarsky-type inequality is obtained. We study the properties of the linear functional related to the new Stolarsky-type inequality, and finally apply these new results in the theory of fractional integrals.","PeriodicalId":383314,"journal":{"name":"Mathematica Pannonica","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135826747","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 paper, we consider the Feuerbach point and the Feuerbach line of a triangle in the isotropic plane, and investigate some properties of these concepts and their relationships with other elements of a triangle in the isotropic plane. We also compare these relationships in Euclidean and isotropic cases.
{"title":"On the Feuerbach Point and Feuerbach Line in the Isotropic Plane","authors":"R. Kolar-Šuper, V. Volenec","doi":"10.1556/314.2023.00016","DOIUrl":"https://doi.org/10.1556/314.2023.00016","url":null,"abstract":"In this paper, we consider the Feuerbach point and the Feuerbach line of a triangle in the isotropic plane, and investigate some properties of these concepts and their relationships with other elements of a triangle in the isotropic plane. We also compare these relationships in Euclidean and isotropic cases.","PeriodicalId":383314,"journal":{"name":"Mathematica Pannonica","volume":"61 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114718836","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}
M. Bin-Saad, T. Ergashev, Dildora A. Ergasheva, A. Hasanov
We define the order of the double hypergeometric series, investigate the properties of the new confluent Kampé de Fériet series, and build systems of partial differential equations that satisfy the new Kampé de Fériet series. We solve the Cauchy problem for a degenerate hyperbolic equation of the second kind with a spectral parameter using the high-order Kampé de Fériet series. Thanks to the properties of the introduced Kampé de Fériet series, it is possible to obtain a solution to the problem in explicit forms.
定义了二重超几何级数的阶，研究了新的合流kampaine de fsamriet级数的性质，建立了满足新kampaine de fsamriet级数的偏微分方程组。利用高阶kampaud de fsamriet级数，求解了一类带谱参数的退化第二类双曲方程的Cauchy问题。由于所介绍的kampedefsamriet系列的性质，有可能以显式形式获得问题的解决方案。
{"title":"Confluent Kampé de Fériet Series Arising in the Solutions of Cauchy Problem for the Degenerate Hyperbolic Equation of the Second Kind with the Spectral Parameter","authors":"M. Bin-Saad, T. Ergashev, Dildora A. Ergasheva, A. Hasanov","doi":"10.1556/314.2023.00015","DOIUrl":"https://doi.org/10.1556/314.2023.00015","url":null,"abstract":"We define the order of the double hypergeometric series, investigate the properties of the new confluent Kampé de Fériet series, and build systems of partial differential equations that satisfy the new Kampé de Fériet series. We solve the Cauchy problem for a degenerate hyperbolic equation of the second kind with a spectral parameter using the high-order Kampé de Fériet series. Thanks to the properties of the introduced Kampé de Fériet series, it is possible to obtain a solution to the problem in explicit forms.","PeriodicalId":383314,"journal":{"name":"Mathematica Pannonica","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122667951","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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