Metric mean dimension is a metric invariant of dynamical systems. It is a dynamical analogue of Minkowski dimension of metric spaces. We explain that old ideas of Bowen (1972) can be used for clarifying the local nature of metric mean dimension. We also explain the generalization to R-actions and an illustrating example.
{"title":"REMARK ON THE LOCAL NATURE OF METRIC MEAN DIMENSION","authors":"M. Tsukamoto","doi":"10.2206/kyushujm.76.143","DOIUrl":"https://doi.org/10.2206/kyushujm.76.143","url":null,"abstract":"Metric mean dimension is a metric invariant of dynamical systems. It is a dynamical analogue of Minkowski dimension of metric spaces. We explain that old ideas of Bowen (1972) can be used for clarifying the local nature of metric mean dimension. We also explain the generalization to R-actions and an illustrating example.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46868049","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 show that, based on Grabner’s recent results on modular differential equations satisfied by quasimodular forms, there exist only finitely many normalized extremal quasimodular forms of depth r that have all Fourier coefficients integral for each of r = 1, 2, 3, 4, and partly classifies them, where the classification is complete for r = 2, 3, 4; in fact, we show that there exists no normalized extremal quasimodular forms of depth 4 with all Fourier coefficients integral. Our result disproves a conjecture by Pellarin.
{"title":"EXTREMAL QUASIMODULAR FORMS OF LOWER DEPTH WITH INTEGRAL FOURIER COEFFICIENTS","authors":"Tsudoi Kaminaka, Fumiharu Kato","doi":"10.2206/kyushujm.75.351","DOIUrl":"https://doi.org/10.2206/kyushujm.75.351","url":null,"abstract":"We show that, based on Grabner’s recent results on modular differential equations satisfied by quasimodular forms, there exist only finitely many normalized extremal quasimodular forms of depth r that have all Fourier coefficients integral for each of r = 1, 2, 3, 4, and partly classifies them, where the classification is complete for r = 2, 3, 4; in fact, we show that there exists no normalized extremal quasimodular forms of depth 4 with all Fourier coefficients integral. Our result disproves a conjecture by Pellarin.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47516634","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}
Given positive integers a1, . . . , ak with gcd(a1, . . . , ak) = 1, it is well-known that all sufficiently large n can be represented as a nonnegative integer combination of a1, . . . , ak. The Frobenius Problem is to determine the largest positive integer that is NOT representable as a nonnegative integer combination of given positive integers that are coprime (see [15] for general references). This number is denoted by g(a1, . . . , ak) and often called Frobenius number. The Frobenius Problem has been also known as the Coin Exchange Problem (or Postage Stamp Problem / Chicken McNugget Problem), which
{"title":"WEIGHTED SYLVESTER SUMS ON THE FROBENIUS SET IN MORE VARIABLES","authors":"T. Komatsu, Yuan Zhang","doi":"10.2206/kyushujm.76.163","DOIUrl":"https://doi.org/10.2206/kyushujm.76.163","url":null,"abstract":"Given positive integers a1, . . . , ak with gcd(a1, . . . , ak) = 1, it is well-known that all sufficiently large n can be represented as a nonnegative integer combination of a1, . . . , ak. The Frobenius Problem is to determine the largest positive integer that is NOT representable as a nonnegative integer combination of given positive integers that are coprime (see [15] for general references). This number is denoted by g(a1, . . . , ak) and often called Frobenius number. The Frobenius Problem has been also known as the Coin Exchange Problem (or Postage Stamp Problem / Chicken McNugget Problem), which","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42780849","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 give a characterization of cusp forms of half-integral weight of level four in the plus space in terms of a functional equation of attached L-series.
我们用附加l级数的泛函方程给出了正空间中第四级半积分权的尖峰形式的刻画。
{"title":"ON HECKE L-FUNCTIONS ATTACHED TO CUSP FORMS OF HALF-INTEGRAL WEIGHT","authors":"W. Kohnen","doi":"10.2206/KYUSHUJM.75.125","DOIUrl":"https://doi.org/10.2206/KYUSHUJM.75.125","url":null,"abstract":"We give a characterization of cusp forms of half-integral weight of level four in the plus space in terms of a functional equation of attached L-series.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68557631","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}
In this paper, we study non-singular extensions of Morse functions on closed orientable surfaces. By a non-singular extension of such a Morse function, we mean an extension to a function without critical points on some compact orientable 3-manifold having as boundary the given surface. In 1977, Curley characterized the existence of non-singular extensions of non-singular boundary germs in terms of combinatorics on associated labeled Reeb graphs. We apply Curley’s result to show that every Morse function on a closed orientable (possibly disconnected) surface has a non-singular extension to a 3-manifold that is connected.
{"title":"NON-SINGULAR EXTENSIONS OF MORSE FUNCTIONS ON DISCONNECTED SURFACES","authors":"Kentaro Iwamoto","doi":"10.2206/KYUSHUJM.75.23","DOIUrl":"https://doi.org/10.2206/KYUSHUJM.75.23","url":null,"abstract":"In this paper, we study non-singular extensions of Morse functions on closed orientable surfaces. By a non-singular extension of such a Morse function, we mean an extension to a function without critical points on some compact orientable 3-manifold having as boundary the given surface. In 1977, Curley characterized the existence of non-singular extensions of non-singular boundary germs in terms of combinatorics on associated labeled Reeb graphs. We apply Curley’s result to show that every Morse function on a closed orientable (possibly disconnected) surface has a non-singular extension to a 3-manifold that is connected.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68558005","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}
Given a set X = (X1, X2, . . . , Xm) of pointed spaces, we introduce a family {X(k,l)} of subquotients of X1 × X2 × · · · × Xm . This family extends the family of subspaces of X1 × X2 × · · · × Xm introduced by G. J. Porter and contains the product, the fat wedge, the wedge and the smash product. The (co)homology with field coefficients of X(k,l) is completely determined, which is used to study the group E(X(k,l)) of self-homotopy equivalences of X(k,l). Especially, in the case of X1 = X2 = · · · = Xm = X , we construct a homomorphism (k,l) from the semi-direct product of the m-fold product E(X)m and the symmetric group Sm to E(X(k,l)) and give sufficient conditions for (k,l) to be injective. We apply this result to the case where X = Sn , CPn , or K (Ar , n) with A a subring of Q or a field Z/p, providing an important subgroup of E(X(k,l)).
{"title":"SUBQUOTIENTS OF A FINITE PRODUCT AND THEIR SELF-HOMOTOPY EQUIVALENCES","authors":"Hiroshi Kihara, Nobuyuki Oda","doi":"10.2206/KYUSHUJM.75.129","DOIUrl":"https://doi.org/10.2206/KYUSHUJM.75.129","url":null,"abstract":"Given a set X = (X1, X2, . . . , Xm) of pointed spaces, we introduce a family {X(k,l)} of subquotients of X1 × X2 × · · · × Xm . This family extends the family of subspaces of X1 × X2 × · · · × Xm introduced by G. J. Porter and contains the product, the fat wedge, the wedge and the smash product. The (co)homology with field coefficients of X(k,l) is completely determined, which is used to study the group E(X(k,l)) of self-homotopy equivalences of X(k,l). Especially, in the case of X1 = X2 = · · · = Xm = X , we construct a homomorphism (k,l) from the semi-direct product of the m-fold product E(X)m and the symmetric group Sm to E(X(k,l)) and give sufficient conditions for (k,l) to be injective. We apply this result to the case where X = Sn , CPn , or K (Ar , n) with A a subring of Q or a field Z/p, providing an important subgroup of E(X(k,l)).","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68558068","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}
. From Lauricella’s hypergeometric functions in n variables, we construct a special type of Pfaffian equations of rank n + 1 called Okubo type differential equations. For each homogenized Okubo type differential equation, we construct special n + 1 coordinate functions called flat coordinate functions and an ( n + 1 ) -tuple of homogeneous functions called a potential vector.
{"title":"OKUBO TYPE DIFFERENTIAL EQUATIONS DERIVED FROM HYPERGEOMETRIC FUNCTIONS FD","authors":"Mitsuo Kato","doi":"10.2206/KYUSHUJM.75.1","DOIUrl":"https://doi.org/10.2206/KYUSHUJM.75.1","url":null,"abstract":". From Lauricella’s hypergeometric functions in n variables, we construct a special type of Pfaffian equations of rank n + 1 called Okubo type differential equations. For each homogenized Okubo type differential equation, we construct special n + 1 coordinate functions called flat coordinate functions and an ( n + 1 ) -tuple of homogeneous functions called a potential vector.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68557603","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 Cauchy problem for the semilinear complex Ginzburg–Landau type equation is considered in homogeneous and isotropic spacetime. Global solutions and their asymptotic behaviours for small initial data are obtained. The non-existence of non-trivial global solutions is also shown. The effects of spatial expansion and contraction are studied through the problem.
{"title":"EXISTENCE AND NON-EXISTENCE OF GLOBAL SOLUTIONS FOR THE SEMILINEAR COMPLEX GINZBURG-LANDAU TYPE EQUATION IN HOMOGENEOUS AND ISOTROPIC SPACETIME","authors":"Makoto Nakamura, Y. Sato","doi":"10.2206/kyushujm.75.169","DOIUrl":"https://doi.org/10.2206/kyushujm.75.169","url":null,"abstract":"The Cauchy problem for the semilinear complex Ginzburg–Landau type equation is considered in homogeneous and isotropic spacetime. Global solutions and their asymptotic behaviours for small initial data are obtained. The non-existence of non-trivial global solutions is also shown. The effects of spatial expansion and contraction are studied through the problem.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68558298","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 study modular properties of theta functions of binary quadratic forms with congruence condition and compute their values at arbitrary cusps.
研究了具有同余条件的二元二次型函数的模性质，并计算了它们在任意顶点处的值。
{"title":"ON THETA FUNCTIONS OF BINARY QUADRATIC FORMS WITH CONGRUENCE CONDITION","authors":"Masanari Kida, Ryota Okano, Ken Yokoyama","doi":"10.2206/KYUSHUJM.75.41","DOIUrl":"https://doi.org/10.2206/KYUSHUJM.75.41","url":null,"abstract":"We study modular properties of theta functions of binary quadratic forms with congruence condition and compute their values at arbitrary cusps.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68558735","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}
. Let E k ( z ) be the normalized Eisenstein series of weight k for the full modular group SL ( 2 , Z ) . Let a > 0 be an even integer. In this paper we completely determine when the zeros of E k interlace with the zeros of E k + a . This generalizes a result of Nozaki on the interlacing of zeros of E k and E k + 12 .
{"title":"INTERLACING OF ZEROS OF EISENSTEIN SERIES","authors":"Trevor Griffin, Nathan Kenshur, Abigail Price, Bradshaw Vandenberg-Daves, Hui Xue, Daozhou Zhu","doi":"10.2206/kyushujm.75.249","DOIUrl":"https://doi.org/10.2206/kyushujm.75.249","url":null,"abstract":". Let E k ( z ) be the normalized Eisenstein series of weight k for the full modular group SL ( 2 , Z ) . Let a > 0 be an even integer. In this paper we completely determine when the zeros of E k interlace with the zeros of E k + a . This generalizes a result of Nozaki on the interlacing of zeros of E k and E k + 12 .","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68558105","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：