Chowla's (inverse) problem is a deduction of linear independence over the rationals of circular functions at rational arguments from L.(1, x) ≠ 0, while determinant expressions for the (relative) class number of (subfields of) a cyclotomic field are referred to as the Maillet-Demyanenko determinants. In Wang, Chakraborty and Kanemitsu (to appear), Chowla's problem and Maillet-Demyanenko determinants (CPMD) in the case of Bernoulli polynomial entries (odd part) are unified as different-looking expressions of the (relative) class number on the grounds of the base change formula for periodic Dirichlet series, Dedekind determinant and the Euler product. Our aim here is to show that the genesis of the new theory of discrete Fourier transform as well as the Dedekind determinant is the characters of a finite Abelian group and its convolution map, thus revealing that CPMD boils down to analysis of the class number by group characters. We settle the case of Clausen function (log sine function) entries (even part) as an example. Other cases are similar.
{"title":"DETERMINANT EXPRESSION FOR THE CLASS NUMBER OF AN ABELIAN NUMBER FIELD","authors":"Quan YANG, Nianliang WANG, Shigeru KANEMITSU","doi":"10.2206/kyushujm.77.237","DOIUrl":"https://doi.org/10.2206/kyushujm.77.237","url":null,"abstract":"Chowla's (inverse) problem is a deduction of linear independence over the rationals of circular functions at rational arguments from L.(1, x) ≠ 0, while determinant expressions for the (relative) class number of (subfields of) a cyclotomic field are referred to as the Maillet-Demyanenko determinants. In Wang, Chakraborty and Kanemitsu (to appear), Chowla's problem and Maillet-Demyanenko determinants (CPMD) in the case of Bernoulli polynomial entries (odd part) are unified as different-looking expressions of the (relative) class number on the grounds of the base change formula for periodic Dirichlet series, Dedekind determinant and the Euler product. Our aim here is to show that the genesis of the new theory of discrete Fourier transform as well as the Dedekind determinant is the characters of a finite Abelian group and its convolution map, thus revealing that CPMD boils down to analysis of the class number by group characters. We settle the case of Clausen function (log sine function) entries (even part) as an example. Other cases are similar.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136367758","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}
The paper [S. Shimomura, Kyushu J. Math. 76 (2022), 43-99] contains an incorrect Stokes graph. The amendment to the Stokes graph replaces the phase shifts of asymptotic solutions in the main theorems.
报纸[S]。Shimomura, Kyushu J. Math. 76(2022)， 43-99]包含一个不正确的Stokes图。对Stokes图的修正取代了主要定理中渐近解的相移。
{"title":"CORRIGENDUM: ELLIPTIC ASYMPTOTIC REPRESENTATION OF THE FIFTH PAINLEVÉ TRANSCENDENTS","authors":"Shun SHIMOMURA","doi":"10.2206/kyushujm.77.191","DOIUrl":"https://doi.org/10.2206/kyushujm.77.191","url":null,"abstract":"The paper [S. Shimomura, Kyushu J. Math. 76 (2022), 43-99] contains an incorrect Stokes graph. The amendment to the Stokes graph replaces the phase shifts of asymptotic solutions in the main theorems.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":"186 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136367965","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}
A probabilistic proof by means of the adelic formulation is given to the classical mean-value theorem for [0, 1]-valued multiplicative arithmetical functions f. Then a more general mean-value theorem is derived for composite functions φ(f) of f with (semi-)continuous functions φ.
{"title":"PROBABILISTIC PROOF OF THE MEAN-VALUE THEOREM FOR [0,<i> </i>1]-VALUED MULTIPLICATIVE FUNCTIONS BY MEANS OF THE ADELIC FORMULATION","authors":"Hiroshi SUGITA","doi":"10.2206/kyushujm.77.355","DOIUrl":"https://doi.org/10.2206/kyushujm.77.355","url":null,"abstract":"A probabilistic proof by means of the adelic formulation is given to the classical mean-value theorem for [0, 1]-valued multiplicative arithmetical functions f. Then a more general mean-value theorem is derived for composite functions φ(f) of f with (semi-)continuous functions φ.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136367979","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}
Weakly holomorphic modular forms for modular groups are holomorphic away from the cusp. We study a certain family of weakly holomorphic modular forms and the locations of their zeros. We prove that all of the zeros in the standard fundamental domain for the modular group lie on a lower boundary arc, providing conditions.
{"title":"ZEROS OF CERTAIN WEAKLY HOLOMORPHIC MODULAR FORMS","authors":"Seiichi HANAMOTO","doi":"10.2206/kyushujm.77.255","DOIUrl":"https://doi.org/10.2206/kyushujm.77.255","url":null,"abstract":"Weakly holomorphic modular forms for modular groups are holomorphic away from the cusp. We study a certain family of weakly holomorphic modular forms and the locations of their zeros. We prove that all of the zeros in the standard fundamental domain for the modular group lie on a lower boundary arc, providing conditions.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":"40 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136367982","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}
On the basis of fractional calculus, the integral of controlled paths along Hölder rough paths is given explicitly as Lebesgue integrals for fractional derivative operators, without using any arguments from a discrete approximation. In this paper, we introduce a backward version of the integral and provide fundamental relations between both integrals from the perspective of the backward representation of the rough integral.
{"title":"BACKWARD REPRESENTATION OF THE ROUGH INTEGRAL: AN APPROACH BASED ON FRACTIONAL CALCULUS","authors":"Yu ITO","doi":"10.2206/kyushujm.77.367","DOIUrl":"https://doi.org/10.2206/kyushujm.77.367","url":null,"abstract":"On the basis of fractional calculus, the integral of controlled paths along Hölder rough paths is given explicitly as Lebesgue integrals for fractional derivative operators, without using any arguments from a discrete approximation. In this paper, we introduce a backward version of the integral and provide fundamental relations between both integrals from the perspective of the backward representation of the rough integral.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":"45-46 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136367740","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}
We introduce a class of discrete dynamical systems that we call virtually expanding. This is an open subset of self-covering maps on a closed manifold which contains all expanding maps and some partially hyperbolic volume-expanding maps. We show that the Perron-Frobenius operator is quasi-compact on a Sobolev space of positive order for such a class of dynamical systems.
{"title":"VIRTUALLY EXPANDING DYNAMICS","authors":"Masato TSUJII","doi":"10.2206/kyushujm.77.291","DOIUrl":"https://doi.org/10.2206/kyushujm.77.291","url":null,"abstract":"We introduce a class of discrete dynamical systems that we call virtually expanding. This is an open subset of self-covering maps on a closed manifold which contains all expanding maps and some partially hyperbolic volume-expanding maps. We show that the Perron-Frobenius operator is quasi-compact on a Sobolev space of positive order for such a class of dynamical systems.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":"40 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136367963","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}
By means of the linearization method and the Carlitz-Sears transformation, the terminating almost poised 3Φ2-series is shown to be always explicitly evaluable with the number of terms being finite, which is independent of the summation limit.
{"title":"TERMINATING ALMOST POISED <i>q</i>-DIXON SUMS","authors":"Wenchang CHU","doi":"10.2206/kyushujm.77.131","DOIUrl":"https://doi.org/10.2206/kyushujm.77.131","url":null,"abstract":"By means of the linearization method and the Carlitz-Sears transformation, the terminating almost poised 3Φ2-series is shown to be always explicitly evaluable with the number of terms being finite, which is independent of the summation limit.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136367968","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}
The Kaneko-Zagier conjecture describes a correspondence between finite multiple zeta values and symmetric multiple zeta values. Its refined version has been established by Jarossay, Rosen and Ono-Seki-Yamamoto. In this paper, we further explain these conjectures through studies of multiple harmonic q-sums. We show that the (generalized) finite/symmetric multiple zeta values are obtained by taking an algebraic/analytic limit of multiple harmonic q-sums. As applications, new proofs of reversal, duality and cyclic sum formulas for the generalized finite/symmetric multiple zeta values are given.
{"title":"SUPERCONGRUENCES OF MULTIPLE HARMONIC <i>q</i>-SUMS AND GENERALIZED FINITE/SYMMETRIC MULTIPLE ZETA VALUES","authors":"Yoshihiro TAKEYAMA, Koji TASAKA","doi":"10.2206/kyushujm.77.75","DOIUrl":"https://doi.org/10.2206/kyushujm.77.75","url":null,"abstract":"The Kaneko-Zagier conjecture describes a correspondence between finite multiple zeta values and symmetric multiple zeta values. Its refined version has been established by Jarossay, Rosen and Ono-Seki-Yamamoto. In this paper, we further explain these conjectures through studies of multiple harmonic q-sums. We show that the (generalized) finite/symmetric multiple zeta values are obtained by taking an algebraic/analytic limit of multiple harmonic q-sums. As applications, new proofs of reversal, duality and cyclic sum formulas for the generalized finite/symmetric multiple zeta values are given.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":"48 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136367976","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}
We consider confluences of Euler-type integrals expressing solutions to Appell's F2 system of hypergeometric differential equations, and study systems of confluent hypergeometric differential equations of rank four of two variables. Our consideration is based on a confluence transforming the abelian group (ℂ×)2 to the Jordan group of size two. For each system obtained by our study, we give its Pfaffian system with a connection matrix admitting a decomposition into four or five parts, each of which is the product of a matrix depending only on parameters and a rational 1-form in two variables. We classify these Pfaffian systems under an equivalence relation. Any system obtained by our study is equivalent to one of Humbert's Ψ1 system, Humbert's Ξ1 system, and the system satisfied by the product of two Kummer's confluent hypergeometric functions.
{"title":"CONFLUENCES OF APPELL'S <i>F</i><sub>2 </sub>SYSTEM OF HYPERGEOMETRIC DIFFERENTIAL EQUATIONS","authors":"Shigeo MUKAI","doi":"10.2206/kyushujm.77.1","DOIUrl":"https://doi.org/10.2206/kyushujm.77.1","url":null,"abstract":"We consider confluences of Euler-type integrals expressing solutions to Appell's F2 system of hypergeometric differential equations, and study systems of confluent hypergeometric differential equations of rank four of two variables. Our consideration is based on a confluence transforming the abelian group (ℂ×)2 to the Jordan group of size two. For each system obtained by our study, we give its Pfaffian system with a connection matrix admitting a decomposition into four or five parts, each of which is the product of a matrix depending only on parameters and a rational 1-form in two variables. We classify these Pfaffian systems under an equivalence relation. Any system obtained by our study is equivalent to one of Humbert's Ψ1 system, Humbert's Ξ1 system, and the system satisfied by the product of two Kummer's confluent hypergeometric functions.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":"100 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136367977","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}
Trisection maps are certain stable maps from closed 4-manifolds to R2. A simpler but reasonable class of trisection maps was introduced by Baykur and Saeki, called a simplified (g, k)-trisection. We focus on the right-left equivalence classes of simplified (2, 0)-trisections. Simplified trisections are determined by their simplified trisection diagrams, which are diagrams on a genus-two surface consisting of simple closed curves of vanishing cycles with labels. The aim of this paper is to study how the replacement of reference paths changes simplified trisection diagrams up to upper-triangular handle-slides. We show that, for a simplified trisection f satisfying a certain condition, there exist at least two simplified (2, 0)-trisections f' and f" such that f , f' and f" are right-left equivalent to each other but their simplified trisection diagrams are not related by automorphisms of a genus-two surface and upper-triangular handle-slides.
{"title":"RIGHT-LEFT EQUIVALENT MAPS OF SIMPLIFIED (2,<i> </i>0)-TRISECTIONS WITH DIFFERENT CONFIGURATIONS OF VANISHING CYCLES","authors":"Nobutaka ASANO","doi":"10.2206/kyushujm.77.299","DOIUrl":"https://doi.org/10.2206/kyushujm.77.299","url":null,"abstract":"Trisection maps are certain stable maps from closed 4-manifolds to R2. A simpler but reasonable class of trisection maps was introduced by Baykur and Saeki, called a simplified (g, k)-trisection. We focus on the right-left equivalence classes of simplified (2, 0)-trisections. Simplified trisections are determined by their simplified trisection diagrams, which are diagrams on a genus-two surface consisting of simple closed curves of vanishing cycles with labels. The aim of this paper is to study how the replacement of reference paths changes simplified trisection diagrams up to upper-triangular handle-slides. We show that, for a simplified trisection f satisfying a certain condition, there exist at least two simplified (2, 0)-trisections f' and f\" such that f , f' and f\" are right-left equivalent to each other but their simplified trisection diagrams are not related by automorphisms of a genus-two surface and upper-triangular handle-slides.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136367759","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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