{"title":"A CONGRUENCE BETWEEN SYMMETRIC MULTIPLE ZETA-STAR VALUES AND MULTIPLE ZETA-STAR VALUES","authors":"Kento Fujita, Y. Komori","doi":"10.2206/KYUSHUJM.75.149","DOIUrl":"https://doi.org/10.2206/KYUSHUJM.75.149","url":null,"abstract":"","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":"68558229","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 ( X , L ) denote a polarized manifold of dimension five. This study considers the dimension of the global sections of K X + mL with m ≥ 6. In particular, we prove that h 0 ( K X + mL ) ≥ (cid:0) m − 1 5 (cid:1) for any polarized 5-fold ( X , L ) with h 0 ( L ) > 0. Furthermore, we also consider ( X , L ) with h 0 ( K X + mL ) = (cid:0) m − 1 5 (cid:1) for some m ≥ 6 with h 0 ( L ) > 0.
{"title":"ON THE DIMENSION OF THE GLOBAL SECTIONS OF THE ADJOINT BUNDLE FOR POLARIZED 5-FOLDS","authors":"Y. Fukuma","doi":"10.2206/kyushujm.75.211","DOIUrl":"https://doi.org/10.2206/kyushujm.75.211","url":null,"abstract":". Let ( X , L ) denote a polarized manifold of dimension five. This study considers the dimension of the global sections of K X + mL with m ≥ 6. In particular, we prove that h 0 ( K X + mL ) ≥ (cid:0) m − 1 5 (cid:1) for any polarized 5-fold ( X , L ) with h 0 ( L ) > 0. Furthermore, we also consider ( X , L ) with h 0 ( K X + mL ) = (cid:0) m − 1 5 (cid:1) for some m ≥ 6 with h 0 ( L ) > 0.","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":"68557922","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 the cohomology groups of vector bundles on neighborhoods of nonpluriharmonic loci in q-complete Kähler manifolds and in compact Kähler manifolds. Applying our results, we show variants of the Lefschetz hyperplane theorem.
{"title":"COHOMOLOGY ON NEIGHBORHOODS OF NON-PLURIHARMONIC LOCI IN PSEUDOCONVEX KÄHLER MANIFOLDS","authors":"Yuta Watanabe","doi":"10.2206/kyushujm.75.323","DOIUrl":"https://doi.org/10.2206/kyushujm.75.323","url":null,"abstract":"We study the cohomology groups of vector bundles on neighborhoods of nonpluriharmonic loci in q-complete Kähler manifolds and in compact Kähler manifolds. Applying our results, we show variants of the Lefschetz hyperplane theorem.","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":"68558667","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 the fifth Painleve transcendents an asymptotic representation by the Jacobi $mathrm{sn}$-function is presented in cheese-like strips along generic directions near the point at infinity. Its elliptic main part depends on a single integration constant, which is the phase shift and is parametrised by monodromy data for the associated isomonodromy deformation. In addition, under a certain supposition, the error term is also expressed by an explicit asymptotic formula, whose leading term is written in terms of integrals of the $mathrm{sn}$-function and the $vartheta$-function, and contains the other integration constant. Instead of the justification scheme for asymptotic solutions of Riemann-Hilbert problems by the Brouwer fixed point theorem, we begin with a boundedness property of a Lagrangian function, which enables us to determine the modulus of the $mathrm{sn}$-function satisfying the Boutroux equations and to construct deductively the elliptic representation.
{"title":"ELLIPTIC ASYMPTOTIC REPRESENTATION OF THE FIFTH PAINLEVÉ TRANSCENDENTS","authors":"S. Shimomura","doi":"10.2206/kyushujm.76.43","DOIUrl":"https://doi.org/10.2206/kyushujm.76.43","url":null,"abstract":"For the fifth Painleve transcendents an asymptotic representation by the Jacobi $mathrm{sn}$-function is presented in cheese-like strips along generic directions near the point at infinity. Its elliptic main part depends on a single integration constant, which is the phase shift and is parametrised by monodromy data for the associated isomonodromy deformation. In addition, under a certain supposition, the error term is also expressed by an explicit asymptotic formula, whose leading term is written in terms of integrals of the $mathrm{sn}$-function and the $vartheta$-function, and contains the other integration constant. Instead of the justification scheme for asymptotic solutions of Riemann-Hilbert problems by the Brouwer fixed point theorem, we begin with a boundedness property of a Lagrangian function, which enables us to determine the modulus of the $mathrm{sn}$-function satisfying the Boutroux equations and to construct deductively the elliptic representation.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45376876","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 a knot whose minimum crossing number $c(K)$ is even and greater than $30$ is not fertile; there exists a knot $K'$ with crossing number less than $c$ such that $K'$ is not obtained from a minimum crossing number diagram of $K$ by suitably changing the over-under information.
{"title":"A NOTE ON KNOT FERTILITY","authors":"Tetsuya Ito","doi":"10.2206/kyushujm.75.273","DOIUrl":"https://doi.org/10.2206/kyushujm.75.273","url":null,"abstract":"We show that a knot whose minimum crossing number $c(K)$ is even and greater than $30$ is not fertile; there exists a knot $K'$ with crossing number less than $c$ such that $K'$ is not obtained from a minimum crossing number diagram of $K$ by suitably changing the over-under information.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45446401","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 (X) be a compact metric space and (E) be a Banach space. (lip (X, E)) denotes the Banach space of all (E)-valued little Lipschitz functions on (X). We show that (lip (X, E)^{**}) is isometrically isomorphic to Banach space of (E^{**})-valued Lipschitz functions (Lip(X, E^{**})) under several conditions. Moreover, we describe the isometric isomorphism from (lip (X, E)^{**}) to (Lip (X, E^{**})).
{"title":"ON THE SECOND DUAL SPACE OF THE BANACH SPACE OF VECTOR-VALUED LITTLE LIPSCHITZ FUNCTIONS","authors":"Shinnosuke Izumi","doi":"10.2206/kyushujm.75.235","DOIUrl":"https://doi.org/10.2206/kyushujm.75.235","url":null,"abstract":"Let (X) be a compact metric space and (E) be a Banach space. (lip (X, E)) denotes the Banach space of all (E)-valued little Lipschitz functions on (X). We show that (lip (X, E)^{**}) is isometrically isomorphic to Banach space of (E^{**})-valued Lipschitz functions (Lip(X, E^{**})) under several conditions. Moreover, we describe the isometric isomorphism from (lip (X, E)^{**}) to (Lip (X, E^{**})).","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44497193","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 first establish several integral identities. These integrals are of the form [int_0^1 x^{an+b} f(x),dxquad (ain{1,2}, bin{-1,-2})] where $f(x)$ is a single-variable multiple polylogarithm function or $r$-variable multiple polylogarithm function or Kaneko--Tsumura A-function (this is a single-variable multiple polylogarithm function of level two). We find that these integrals can be expressed in terms of multiple zeta (star) values and their related variants (multiple $t$-values, multiple $T$-values, multiple $S$-values etc.), and multiple harmonic (star) sums and their related variants (multiple $T$-harmonic sums, multiple $S$-harmonic sums etc.). Using these integral identities, we prove many explicit evaluations of Kaneko--Yamamoto multiple zeta values and their related variants. Further, we derive some relations involving multiple zeta (star) values and their related variants.
{"title":"EXPLICIT RELATIONS BETWEEN KANEKO-YAMAMOTO TYPE MULTIPLE ZETA VALUES AND RELATED VARIANTS","authors":"Ce Xu, Jianqiang Zhao","doi":"10.2206/kyushujm.76.369","DOIUrl":"https://doi.org/10.2206/kyushujm.76.369","url":null,"abstract":"In this paper we first establish several integral identities. These integrals are of the form [int_0^1 x^{an+b} f(x),dxquad (ain{1,2}, bin{-1,-2})] where $f(x)$ is a single-variable multiple polylogarithm function or $r$-variable multiple polylogarithm function or Kaneko--Tsumura A-function (this is a single-variable multiple polylogarithm function of level two). We find that these integrals can be expressed in terms of multiple zeta (star) values and their related variants (multiple $t$-values, multiple $T$-values, multiple $S$-values etc.), and multiple harmonic (star) sums and their related variants (multiple $T$-harmonic sums, multiple $S$-harmonic sums etc.). Using these integral identities, we prove many explicit evaluations of Kaneko--Yamamoto multiple zeta values and their related variants. Further, we derive some relations involving multiple zeta (star) values and their related variants.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46092052","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 the symmetrized Talagrand inequality that was proved by Fathi and has a connection with the Blaschke-Santalo inequality in convex geometry. As corollaries of our results, we have several refined functional inequalities under some conditions. We also give an alternative proof of Fathi's symmetrized Talagrand inequality on the real line and some applications.
{"title":"SYMMETRIZED TALAGRAND INEQUALITIES ON EUCLIDEAN SPACES","authors":"Hiroshi Tsuji","doi":"10.2206/kyushujm.76.119","DOIUrl":"https://doi.org/10.2206/kyushujm.76.119","url":null,"abstract":"In this paper, we study the symmetrized Talagrand inequality that was proved by Fathi and has a connection with the Blaschke-Santalo inequality in convex geometry. As corollaries of our results, we have several refined functional inequalities under some conditions. We also give an alternative proof of Fathi's symmetrized Talagrand inequality on the real line and some applications.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44214419","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 define a new kind of classical digamma function, and establish its some fundamental identities. Then we apply the formulas obtained, and extend tools developed by Flajolet and Salvy to study more general Euler type sums. The main results of Flajolet and Salvy's paper cite{FS1998} are the immediate corollaries of main results in this paper. Furthermore, we provide some parameterized extensions of Ramanujan-type identities that involve hyperbolic series. Some interesting new consequences and illustrative examples are considered.
{"title":"EXTENSIONS OF EULER-TYPE SUMS AND RAMANUJAN-TYPE SUMS","authors":"Ce Xu","doi":"10.2206/kyushujm.75.295","DOIUrl":"https://doi.org/10.2206/kyushujm.75.295","url":null,"abstract":"We define a new kind of classical digamma function, and establish its some fundamental identities. Then we apply the formulas obtained, and extend tools developed by Flajolet and Salvy to study more general Euler type sums. The main results of Flajolet and Salvy's paper cite{FS1998} are the immediate corollaries of main results in this paper. Furthermore, we provide some parameterized extensions of Ramanujan-type identities that involve hyperbolic series. Some interesting new consequences and illustrative examples are considered.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47789758","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 the Hartree-Fock equation and the Hartree-Fock energy functional universally used in many-electron problems. We prove that the set of all critical values of the Hartree-Fock energy functional less than a constant smaller than the first energy threshold is finite. Since the Hartree-Fock equation which is the corresponding Euler-Lagrange equation is a system of nonlinear eigenvalue problems, the spectral theory for linear operators is not applicable. The present result is obtained establishing the finiteness of the critical values associated with orbital energies less than a negative constant and combining the result with the Koopmans' well-known theorem. The main ingredients are the proof of convergence of the solutions and the analysis of the Fr'echet second derivative of the functional at the limit point.
{"title":"FINITENESS OF THE NUMBER OF CRITICAL VALUES OF THE HARTREE-FOCK ENERGY FUNCTIONAL LESS THAN A CONSTANT SMALLER THAN THE FIRST ENERGY THRESHOLD","authors":"Sohei Ashida","doi":"10.2206/kyushujm.75.277","DOIUrl":"https://doi.org/10.2206/kyushujm.75.277","url":null,"abstract":"We study the Hartree-Fock equation and the Hartree-Fock energy functional universally used in many-electron problems. We prove that the set of all critical values of the Hartree-Fock energy functional less than a constant smaller than the first energy threshold is finite. Since the Hartree-Fock equation which is the corresponding Euler-Lagrange equation is a system of nonlinear eigenvalue problems, the spectral theory for linear operators is not applicable. The present result is obtained establishing the finiteness of the critical values associated with orbital energies less than a negative constant and combining the result with the Koopmans' well-known theorem. The main ingredients are the proof of convergence of the solutions and the analysis of the Fr'echet second derivative of the functional at the limit point.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46734870","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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