Theorem A.1 in [II19] claimed that a topological CW complex is homotopy equivalent to a smooth CW complex without details. To give a more precise proof, we show a version of Whitney Approximation for a smooth CW complex, which actually enables us to give a concrete proof for Theorem A.1 in [II19].
{"title":"WHITNEY APPROXIMATION FOR SMOOTH CW COMPLEX","authors":"Norio Iwase","doi":"10.2206/kyushujm.76.177","DOIUrl":"https://doi.org/10.2206/kyushujm.76.177","url":null,"abstract":"Theorem A.1 in [II19] claimed that a topological CW complex is homotopy equivalent to a smooth CW complex without details. To give a more precise proof, we show a version of Whitney Approximation for a smooth CW complex, which actually enables us to give a concrete proof for Theorem A.1 in [II19].","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49204437","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}
For each positive integer m, an arbitrary finite non-solvable group acts smoothly on infinitely many standard spheres with exactly m fixed points. However, for a given finite non-solvable group G and a given positive integer m, all standard spheres do not admit smooth actions of G with exactly m fixed points. In this paper, for each of the alternating group A6 on six letters, the symmetric group S6 on six letters, the projective general linear group PGL(2, 9) of order 720, the Mathieu group M10 of order 720, the automorphism group Aut(A6) of A6 and the special linear group SL(2, 9) of order 720, we will give the dimensions of homology spheres whose fixed point sets of smooth actions of the group do not consist of odd numbers of points.
{"title":"REMARKS ON DIMENSION OF HOMOLOGY SPHERES WITH ODD NUMBERS OF FIXED POINTS OF FINITE GROUP ACTIONS","authors":"S. Tamura","doi":"10.2206/kyushujm.74.255","DOIUrl":"https://doi.org/10.2206/kyushujm.74.255","url":null,"abstract":"For each positive integer m, an arbitrary finite non-solvable group acts smoothly on infinitely many standard spheres with exactly m fixed points. However, for a given finite non-solvable group G and a given positive integer m, all standard spheres do not admit smooth actions of G with exactly m fixed points. In this paper, for each of the alternating group A6 on six letters, the symmetric group S6 on six letters, the projective general linear group PGL(2, 9) of order 720, the Mathieu group M10 of order 720, the automorphism group Aut(A6) of A6 and the special linear group SL(2, 9) of order 720, we will give the dimensions of homology spheres whose fixed point sets of smooth actions of the group do not consist of odd numbers of points.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68557027","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 k1, . . . , kr be positive integers. Let q1, . . . , qr be pairwise coprime positive integers with qi > 2 (i = 1, . . . , r ), and set q = q1 · · · qr . For each i = 1, . . . , r , let Ti be a set of φ(qi )/2 representatives mod qi such that the union Ti ∪ (−Ti ) is a complete set of coprime residues mod qi . Let K be an algebraic number field over which the qth cyclotomic polynomial 8q is irreducible. Then, φ(q)/2r numbers r ∏ i=1 dki−1 dzi i (cot π zi )|zi=ai /qi (ai ∈ Ti , i = 1, . . . , r) are linearly independent over K . As an application, a generalization of the Baker–Birch– Wirsing theorem on the non-vanishing of the multiple Dirichlet series L(s1, . . . , sr ; f ) with periodic coefficients at (s1, . . . , sr )= (k1, . . . , kr ) is proven under a parity condition.
{"title":"OKADA'S THEOREM AND MULTIPLE DIRICHLET SERIES","authors":"Y. Hamahata","doi":"10.2206/kyushujm.74.429","DOIUrl":"https://doi.org/10.2206/kyushujm.74.429","url":null,"abstract":"Let k1, . . . , kr be positive integers. Let q1, . . . , qr be pairwise coprime positive integers with qi > 2 (i = 1, . . . , r ), and set q = q1 · · · qr . For each i = 1, . . . , r , let Ti be a set of φ(qi )/2 representatives mod qi such that the union Ti ∪ (−Ti ) is a complete set of coprime residues mod qi . Let K be an algebraic number field over which the qth cyclotomic polynomial 8q is irreducible. Then, φ(q)/2r numbers r ∏ i=1 dki−1 dzi i (cot π zi )|zi=ai /qi (ai ∈ Ti , i = 1, . . . , r) are linearly independent over K . As an application, a generalization of the Baker–Birch– Wirsing theorem on the non-vanishing of the multiple Dirichlet series L(s1, . . . , sr ; f ) with periodic coefficients at (s1, . . . , sr )= (k1, . . . , kr ) is proven under a parity condition.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68557468","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}
. As has been pointed out by Chakraborty et al (Seeing the invisible: around generalized Kubert functions. Ann. Univ. Sci. Budapest. Sect. Comput. 47 (2018), 185–195), there have appeared many instances in which only the imaginary part—the odd part—of the Lerch zeta-function was considered by eliminating the real part. In this paper we shall make full use of (the boundary function aspect of) the q -expansion for the Lerch zeta-function, the boundary function being in the sense of Wintner (On Riemann’s fragment concerning elliptic modular functions. Amer. J. Math. 63 (1941), 628–634). We may thus refer to this as the ‘Fourier series–boundary q -series’, and we shall show that the decisive result of Yamamoto (Dirichlet series with periodic coefficients. Algebraic Number Theory. Japan Society for the Promotion of Science, Tokyo, 1977, pp. 275–289) on short character sums is its natural consequence. We shall also elucidate the aspect of generalized Euler constants as Laurent coefficients after a brief introduction of the discrete Fourier transform. These are rather remote consequences of the modular relation, i.e. the functional equation for the Lerch zeta-function or the polylogarithm function. That such a remote-looking subject as short character sums is, in the long run, also a consequence of the functional equation indicates the ubiquity and omnipotence of the Lerch zeta-function—and, a fortiori , the modular relation (S. Kanemitsu and H. Tsukada. Contributions to the Theory of Zeta-Functions: the Modular Relation Supremacy. World Scientific, Singapore, 2014). (1.6), the Lerch zeta-function (1.4) is less well known, the existing monograph [ 36 ] notwithstanding. Recently there has been a new representation-theoretic interpretation of the Lerch zeta-function, cf. e.g. [ 35 ]. In the last few decades, the most fundamental and influential works related to the Lerch zeta-function are [ 16 ], [ 43 ], [ 47 ], and [ 71 ], which are partly incorporated in [ 10 ]. We shall describe these toward the end of this section. In the paper [ 9 ] the ubiquity of the Lerch zeta-function, especially the monologarithm (cid:96) 1 ( x ) (1.22) of the complex exponential argument, has been pursued.
． 正如Chakraborty等人(Seeing the invisible: around generalized Kubert functions)所指出的那样。安。大学,科学。布达佩斯。在第47节(2018)，185-195节)中，出现了许多例子，其中通过消除实部只考虑Lerch ζ函数的虚部-奇部。本文充分利用了lach ζ函数的q展开(边界函数方面)，该边界函数是关于椭圆模函数的Wintner (On Riemann’s fragment)意义上的。阿米尔。数学学报，63(1941)，628-634。因此，我们可以把它称为“傅立叶级数-边界q级数”，我们将证明具有周期系数的山本(Dirichlet)级数的决定性结果。代数数论。日本科学促进会，东京，1977年，第275-289页)的短字符和是其自然结果。在简要介绍离散傅里叶变换之后，我们还将阐明广义欧拉常数作为洛朗系数的方面。这些都是模关系的相当远的结果，即，对于勒奇ζ函数或多对数函数的函数方程。从长远来看，像短字符和这样一个看似遥远的主题，也是函数方程的一个结果，这表明了莱奇ζ函数的普遍性和全能性，更重要的是，模关系(S. Kanemitsu和H. Tsukada)。对ζ函数理论的贡献:模关系至上。世界科学，新加坡，2014)。(1.6)， Lerch ζ函数(1.4)不太为人所知，尽管现有的专著[36]。最近出现了一种新的对莱奇ζ函数的表示理论解释，例如[35]。近几十年来，与Lerch ζ函数相关的最基础、最具影响力的工作是[16]、[43]、[47]和[71]，它们部分被纳入[10]。我们将在本节末尾描述这些特性。本文讨论了lach ζ函数的普遍性，特别是复指数参数的单对数(cid:96) 1 (x)(1.22)。
{"title":"THE BOUNDARY LERCH ZETA-FUNCTION AND SHORT CHARACTER SUMS À LA Y. YAMAMOTO","authors":"Xiaohan-H. Wang, J. Mehta, S. Kanemitsu","doi":"10.2206/kyushujm.74.313","DOIUrl":"https://doi.org/10.2206/kyushujm.74.313","url":null,"abstract":". As has been pointed out by Chakraborty et al (Seeing the invisible: around generalized Kubert functions. Ann. Univ. Sci. Budapest. Sect. Comput. 47 (2018), 185–195), there have appeared many instances in which only the imaginary part—the odd part—of the Lerch zeta-function was considered by eliminating the real part. In this paper we shall make full use of (the boundary function aspect of) the q -expansion for the Lerch zeta-function, the boundary function being in the sense of Wintner (On Riemann’s fragment concerning elliptic modular functions. Amer. J. Math. 63 (1941), 628–634). We may thus refer to this as the ‘Fourier series–boundary q -series’, and we shall show that the decisive result of Yamamoto (Dirichlet series with periodic coefficients. Algebraic Number Theory. Japan Society for the Promotion of Science, Tokyo, 1977, pp. 275–289) on short character sums is its natural consequence. We shall also elucidate the aspect of generalized Euler constants as Laurent coefficients after a brief introduction of the discrete Fourier transform. These are rather remote consequences of the modular relation, i.e. the functional equation for the Lerch zeta-function or the polylogarithm function. That such a remote-looking subject as short character sums is, in the long run, also a consequence of the functional equation indicates the ubiquity and omnipotence of the Lerch zeta-function—and, a fortiori , the modular relation (S. Kanemitsu and H. Tsukada. Contributions to the Theory of Zeta-Functions: the Modular Relation Supremacy. World Scientific, Singapore, 2014). (1.6), the Lerch zeta-function (1.4) is less well known, the existing monograph [ 36 ] notwithstanding. Recently there has been a new representation-theoretic interpretation of the Lerch zeta-function, cf. e.g. [ 35 ]. In the last few decades, the most fundamental and influential works related to the Lerch zeta-function are [ 16 ], [ 43 ], [ 47 ], and [ 71 ], which are partly incorporated in [ 10 ]. We shall describe these toward the end of this section. In the paper [ 9 ] the ubiquity of the Lerch zeta-function, especially the monologarithm (cid:96) 1 ( x ) (1.22) of the complex exponential argument, has been pursued.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68557660","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 the inhomogeneous Dirichlet–boundary value problem for the quadratic nonlinear Schrödinger equations, which is considered as a critical case for the largetime asymptotics of solutions. We present sufficient conditions on the initial and boundary data which ensure asymptotic behavior of small solutions to the equations by using the classical energy method and factorization techniques of the free Schrödinger group.
{"title":"INHOMOGENEOUS DIRICHLET-BOUNDARY VALUE PROBLEM FOR TWO-DIMENSIONAL QUADRATIC NONLINEAR SCHRÖDINGER EQUATIONS","authors":"N. Hayashi, E. Kaikina","doi":"10.2206/kyushujm.74.375","DOIUrl":"https://doi.org/10.2206/kyushujm.74.375","url":null,"abstract":"We consider the inhomogeneous Dirichlet–boundary value problem for the quadratic nonlinear Schrödinger equations, which is considered as a critical case for the largetime asymptotics of solutions. We present sufficient conditions on the initial and boundary data which ensure asymptotic behavior of small solutions to the equations by using the classical energy method and factorization techniques of the free Schrödinger group.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68557763","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 G = exp(g) be an exponential solvable Lie group and Ad(G)⊂ D an exponential solvable Lie group of automorphisms of G. Assume that for every non-∗-regular orbit D · q, q ∈ g, of D= exp(d) in g, there exists a nilpotent ideal n of g containing d · g such that D · q|n is closed in n. We then show that for every D-orbit in g the kernel kerC∗() of in the C-algebra of G is L1-determined, which means that kerC∗() is the closure of the kernel kerL1() of in the group algebra L 1(G). This establishes also a new proof of a result of Ungermann, who obtained the same result for the trivial group D= Ad(G). We finally give an example of a non-closed non-∗-regular orbit of an exponential solvable group G and of a coadjoint orbit O ⊂ g, for which the corresponding kernel kerC∗(πO) in C(G) is not L1-determined.
让G = exp (G)是一个指数可解李集团和广告(G)⊂D指数可解李群同构的G .假定每一个非∗常规轨道D·q q∈G, G D = exp (D),存在一个幂零理想n含有D·G这样的G D·q | n n关闭。然后,我们表明,每D-orbitG内核柯尔克∗()G的C-algebra L1-determined,这意味着柯尔克∗()关闭内核kerL1()的组代数1 L (G)。这也建立了Ungermann对平凡群D= Ad(G)的相同结果的一个新的证明。最后给出了指数可解群G的非闭非∗正则轨道和伴随轨道O∧G的一个例子，它们在C(G)中对应的核kerC∗(πO)不是l1确定的。
{"title":"L1-DETERMINED PRIMITIVE IDEALS IN THE C∗-ALGEBRA OF AN EXPONENTIAL LIE GROUP WITH CLOSED NON-∗-REGULAR ORBITS","authors":"Junko Inoue, J. Ludwig","doi":"10.2206/kyushujm.74.127","DOIUrl":"https://doi.org/10.2206/kyushujm.74.127","url":null,"abstract":"Let G = exp(g) be an exponential solvable Lie group and Ad(G)⊂ D an exponential solvable Lie group of automorphisms of G. Assume that for every non-∗-regular orbit D · q, q ∈ g, of D= exp(d) in g, there exists a nilpotent ideal n of g containing d · g such that D · q|n is closed in n. We then show that for every D-orbit in g the kernel kerC∗() of in the C-algebra of G is L1-determined, which means that kerC∗() is the closure of the kernel kerL1() of in the group algebra L 1(G). This establishes also a new proof of a result of Ungermann, who obtained the same result for the trivial group D= Ad(G). We finally give an example of a non-closed non-∗-regular orbit of an exponential solvable group G and of a coadjoint orbit O ⊂ g, for which the corresponding kernel kerC∗(πO) in C(G) is not L1-determined.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68557088","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 twisted Alexander polynomial is an invariant of the pair of a knot and its group representation. Herein, we introduce a digraph obtained from an oriented knot diagram, which is used to study the twisted Alexander polynomial of knots. In this context, we show that the inverse of the twisted Alexander polynomial of a knot may be regarded as the matrix-weighted zeta function that is a generalization of the Ihara–Selberg zeta function of a directed weighted graph.
{"title":"TWISTED ALEXANDER POLYNOMIAL AND MATRIX-WEIGHTED ZETA FUNCTION","authors":"H. Goda","doi":"10.2206/kyushujm.74.211","DOIUrl":"https://doi.org/10.2206/kyushujm.74.211","url":null,"abstract":". The twisted Alexander polynomial is an invariant of the pair of a knot and its group representation. Herein, we introduce a digraph obtained from an oriented knot diagram, which is used to study the twisted Alexander polynomial of knots. In this context, we show that the inverse of the twisted Alexander polynomial of a knot may be regarded as the matrix-weighted zeta function that is a generalization of the Ihara–Selberg zeta function of a directed weighted graph.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68557386","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 some finite discrete groups such as semi-direct products of finite cyclic groups by their automorphisms, the corresponding group and subgroup C-algebras, and their K-theory. Consequently, we obtain several isomorphism classification theorems of such groups by their group C-algebras and K-theory.
{"title":"THE C* -ALGEBRAS OF SEMI-DIRECT PRODUCTS OF FINITE CYCLIC GROUPS BY K-THEORY","authors":"Takahiro Sudo","doi":"10.2206/kyushujm.74.223","DOIUrl":"https://doi.org/10.2206/kyushujm.74.223","url":null,"abstract":"We study some finite discrete groups such as semi-direct products of finite cyclic groups by their automorphisms, the corresponding group and subgroup C-algebras, and their K-theory. Consequently, we obtain several isomorphism classification theorems of such groups by their group C-algebras and K-theory.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68556947","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}
{"title":"THE MEAN SQUARE OF THE LOGARITHMIC DERIVATIVE OF THE SELBERG ZETA FUNCTION FOR COCOMPACT DISCRETE SUBGROUPS","authors":"Y. Aoki","doi":"10.2206/kyushujm.74.353","DOIUrl":"https://doi.org/10.2206/kyushujm.74.353","url":null,"abstract":"","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68557738","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 tc M ( M ) ≤ 2 cat ( M ) for a finite simplicial complex M . For example, we have tc M ( S n ∨ S m ) = 2 for any positive integers n and m .
． 我们秀那油漆tc (M)≤2 (M) for afinite simplicial情结。为了操作,我们有tc (S n∨M)为任何积极integers n和M = 2。
{"title":"UPPER BOUND FOR MONOIDAL TOPOLOGICAL COMPLEXITY","authors":"Norio Iwase, Mitsunobu Tsutaya","doi":"10.2206/kyushujm.74.197","DOIUrl":"https://doi.org/10.2206/kyushujm.74.197","url":null,"abstract":". We show that tc M ( M ) ≤ 2 cat ( M ) for a finite simplicial complex M . For example, we have tc M ( S n ∨ S m ) = 2 for any positive integers n and m .","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68557307","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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