. Komori, Matsumoto and Tsumura introduced a zeta function ζ r ( s , (cid:49)) associated with a root system (cid:49) . In this paper, we introduce a q -analogue of this zeta function, denoted by ζ r ( s , a , (cid:49) ; q ) , and investigate its properties. We show that a ‘Weyl group symmetric’ linear combination of ζ r ( s , a , (cid:49) ; q ) can be written as a multiple integral over a torus involving functions ψ s . For positive integers k , functions ψ k can be regarded as q -analogues of the periodic Bernoulli polynomials. When (cid:49) is of type A 2 or A 3 , the linear combinations can be expressed as the functions ψ k , which are q -analogues of explicit expressions of Witten’s volume formula. We also introduce a two-parameter deformation of the zeta function ζ r ( s , (cid:49)) and study its properties. ,
{"title":"ON q-ANALOGUES OF ZETA FUNCTIONS OF ROOT SYSTEMS","authors":"Masakimi Kato","doi":"10.2206/kyushujm.76.451","DOIUrl":"https://doi.org/10.2206/kyushujm.76.451","url":null,"abstract":". Komori, Matsumoto and Tsumura introduced a zeta function ζ r ( s , (cid:49)) associated with a root system (cid:49) . In this paper, we introduce a q -analogue of this zeta function, denoted by ζ r ( s , a , (cid:49) ; q ) , and investigate its properties. We show that a ‘Weyl group symmetric’ linear combination of ζ r ( s , a , (cid:49) ; q ) can be written as a multiple integral over a torus involving functions ψ s . For positive integers k , functions ψ k can be regarded as q -analogues of the periodic Bernoulli polynomials. When (cid:49) is of type A 2 or A 3 , the linear combinations can be expressed as the functions ψ k , which are q -analogues of explicit expressions of Witten’s volume formula. We also introduce a two-parameter deformation of the zeta function ζ r ( s , (cid:49)) and study its properties. ,","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45727784","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 note we are interested in showing a simple property which characterizes linear operators having empty spectrum on complex Banach spaces. Although it is rather well known that such operators exist, the current literature deals with very few such examples,only as an exercise topic. Our characterization says that such operators are in one-to-one correspondence with injective quasinilpotent operators. This means that the purely unbounded property of linear operators can be reduced to the bounded one. In fact, we obtain a simple way to test and construct examples of such operators and then proceed to make observations of some interest from the function-theoretic point of view. For instance, we define an analogue of the order in the theory of entire functions and show the existence of operators of all order types.
{"title":"LINEAR OPERATORS WITH EMPTY SPECTRUM","authors":"Morisuke HASUMI, Michio SETO","doi":"10.2206/kyushujm.77.63","DOIUrl":"https://doi.org/10.2206/kyushujm.77.63","url":null,"abstract":"In this note we are interested in showing a simple property which characterizes linear operators having empty spectrum on complex Banach spaces. Although it is rather well known that such operators exist, the current literature deals with very few such examples,only as an exercise topic. Our characterization says that such operators are in one-to-one correspondence with injective quasinilpotent operators. This means that the purely unbounded property of linear operators can be reduced to the bounded one. In fact, we obtain a simple way to test and construct examples of such operators and then proceed to make observations of some interest from the function-theoretic point of view. For instance, we define an analogue of the order in the theory of entire functions and show the existence of operators of all order types.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136367750","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 curve shortening flows on rotational surfaces in ℝ3. We assume that the surfaces have negative Gauss curvatures and that some condition related to the Gauss curvature and the curvature of an embedded curve holds on them. Under these assumptions, we prove that the curve remains a graph over the parallels of the rotational surface along the flow. Also, we prove the comparison principle and the long-time existence of the flow.
{"title":"CURVE SHORTENING FLOWS ON ROTATIONAL SURFACES GENERATED BY MONOTONE CONVEX FUNCTIONS","authors":"Naotoshi FUJIHARA","doi":"10.2206/kyushujm.77.179","DOIUrl":"https://doi.org/10.2206/kyushujm.77.179","url":null,"abstract":"In this paper, we study curve shortening flows on rotational surfaces in ℝ3. We assume that the surfaces have negative Gauss curvatures and that some condition related to the Gauss curvature and the curvature of an embedded curve holds on them. Under these assumptions, we prove that the curve remains a graph over the parallels of the rotational surface along the flow. Also, we prove the comparison principle and the long-time existence of the flow.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136367755","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 central binomial series at negative integers are expressed as a linear combination of values of certain two polynomials. We show that one of the polynomials is a special value of the bivariate Eulerian polynomial and the other polynomial is related to the anti-diagonal sum of poly-Bernoulli numbers. As an application, we prove Stephan's observation from 2004.
{"title":"REMARKABLE RELATIONS BETWEEN THE CENTRAL BINOMIAL SERIES, EULERIAN POLYNOMIALS, AND POLY-BERNOULLI NUMBERS, LEADING TO STEPHAN'S OBSERVATION","authors":"Beáta BÉNYI, Toshiki MATSUSAKA","doi":"10.2206/kyushujm.77.149","DOIUrl":"https://doi.org/10.2206/kyushujm.77.149","url":null,"abstract":"The central binomial series at negative integers are expressed as a linear combination of values of certain two polynomials. We show that one of the polynomials is a special value of the bivariate Eulerian polynomial and the other polynomial is related to the anti-diagonal sum of poly-Bernoulli numbers. As an application, we prove Stephan's observation from 2004.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136367966","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 purpose of this article is to describe the singularities of one-dimensional oscillatory integrals, whose phases have a certain singularity, in the form of an asymptotic expansion. In the case of the Laplace integral, an analogous result is also given.
{"title":"ASYMPTOTIC EXPANSION OF OSCILLATORY INTEGRALS WITH SINGULAR PHASES","authors":"Joe KAMIMOTO, Hiromichi MIZUNO","doi":"10.2206/kyushujm.77.319","DOIUrl":"https://doi.org/10.2206/kyushujm.77.319","url":null,"abstract":"The purpose of this article is to describe the singularities of one-dimensional oscillatory integrals, whose phases have a certain singularity, in the form of an asymptotic expansion. In the case of the Laplace integral, an analogous result is also given.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136367749","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 using the property of generalized pseudo-anti-commuting, Ric Φ + Φ Ric = f Φ, for real hypersurfaces in the complex quadric Qm = SOm+2/SO2SOm, we give a complete classification of Ricci-Bourguignon soliton Hopf real hypersurfaces in the complex quadric Qm. Then, as an application, we show a complete classification of Hopf Ricci-Bourguignon gradient solitons in the complex quadric Qm.
{"title":"RICCI-BOURGUIGNON AND GRADIENT SOLITONS ON REAL HYPERSURFACES IN THE COMPLEX QUADRIC","authors":"Hyunjin LEE, Eunmi PAK, Young Jin SUH","doi":"10.2206/kyushujm.77.331","DOIUrl":"https://doi.org/10.2206/kyushujm.77.331","url":null,"abstract":"By using the property of generalized pseudo-anti-commuting, Ric Φ + Φ Ric = f Φ, for real hypersurfaces in the complex quadric Qm = SOm+2/SO2SOm, we give a complete classification of Ricci-Bourguignon soliton Hopf real hypersurfaces in the complex quadric Qm. Then, as an application, we show a complete classification of Hopf Ricci-Bourguignon gradient solitons in the complex quadric Qm.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136367975","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 construct the Capelli elements Ck(u) = C(1k)(u) (k = 1, . . . , n) with a parameter u for the symplectic Lie algebras and orthogonal Lie algebras. They correspond to factorial Schur functions with parameter u attached to the column partitions (1k). We also give explicit formulas for Ck(u) arising from the expansion of Cn(u) = C(1n)(u) with respect to the parameter u. We use the Jacobi-Trudi formula for the factorial Schur functions Rλ(x; u) to construct the higher Capelli elements Cλ(u). They are expressed as determinants of matrices whose entries are Capelli elements Ck(u) attached to the column partitions.
{"title":"THE DESCRIPTION OF THE CAPELLI ELEMENTS WITH A PARAMETER FOR CLASSICAL LIE ALGEBRAS IN TERMS OF SYMMETRIZED DETERMINANT","authors":"Shotaro KAWATA","doi":"10.2206/kyushujm.77.43","DOIUrl":"https://doi.org/10.2206/kyushujm.77.43","url":null,"abstract":"We construct the Capelli elements Ck(u) = C(1k)(u) (k = 1, . . . , n) with a parameter u for the symplectic Lie algebras and orthogonal Lie algebras. They correspond to factorial Schur functions with parameter u attached to the column partitions (1k). We also give explicit formulas for Ck(u) arising from the expansion of Cn(u) = C(1n)(u) with respect to the parameter u. We use the Jacobi-Trudi formula for the factorial Schur functions Rλ(x; u) to construct the higher Capelli elements Cλ(u). They are expressed as determinants of matrices whose entries are Capelli elements Ck(u) attached to the column partitions.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136367741","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 purpose of this paper is twofold. The first is to give some forms of the second main theorem in Nevanlinna theory for meromorphic mappings from parabolic manifolds intersecting moving targets in general position with truncated counting functions, which are improvements of some recent results. The second is to apply the above forms to the proof of an algebraic dependence theorem for meromorphic mappings on parabolic manifolds sharing moving targets regardless of multiplicity.
{"title":"SECOND MAIN THEOREMS AND ALGEBRAIC DEPENDENCE OF MEROMORPHIC MAPPINGS ON PARABOLIC MANIFOLDS WITH MOVING TARGETS","authors":"Si Duc QUANG, Nguyen Van AN, Pham Duc THOAN","doi":"10.2206/kyushujm.77.203","DOIUrl":"https://doi.org/10.2206/kyushujm.77.203","url":null,"abstract":"The purpose of this paper is twofold. The first is to give some forms of the second main theorem in Nevanlinna theory for meromorphic mappings from parabolic manifolds intersecting moving targets in general position with truncated counting functions, which are improvements of some recent results. The second is to apply the above forms to the proof of an algebraic dependence theorem for meromorphic mappings on parabolic manifolds sharing moving targets regardless of multiplicity.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136367971","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 purpose of this paper is to study the basic properties of a degenerate operator with critical gradient and critical Hardy term, controlled by two free parameters. This operator arises from the critical Caffarelli-Kohn-Nirenberg inequality. We analyze the fundamental solutions in a weighted distributional identity and obtain the Liouville theorem for the Lane-Emden equation with that operator, by using the classification of singular solutions with isolated singularities of the related Poisson problem in a bounded domain containing the origin.
{"title":"QUALITATIVE PROPERTIES FOR ELLIPTIC PROBLEMS WITH CKN OPERATORS","authors":"Huyuan CHEN, Yishan ZHENG","doi":"10.2206/kyushujm.77.385","DOIUrl":"https://doi.org/10.2206/kyushujm.77.385","url":null,"abstract":"The purpose of this paper is to study the basic properties of a degenerate operator with critical gradient and critical Hardy term, controlled by two free parameters. This operator arises from the critical Caffarelli-Kohn-Nirenberg inequality. We analyze the fundamental solutions in a weighted distributional identity and obtain the Liouville theorem for the Lane-Emden equation with that operator, by using the classification of singular solutions with isolated singularities of the related Poisson problem in a bounded domain containing the origin.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136367974","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 determine when the modified Rankin-Cohen bracket of the Eisenstein series of weight two and a Hecke eigenform is again an eigenform.
我们确定权值2和Hecke特征型的爱森斯坦级数的修正Rankin-Cohen括号何时又是特征型。
{"title":"ACTION OF <i>E</i><sub>2 </sub>ON HECKE EIGENFORMS","authors":"Hui XUE, Naveen PRABHATH","doi":"10.2206/kyushujm.77.401","DOIUrl":"https://doi.org/10.2206/kyushujm.77.401","url":null,"abstract":"We determine when the modified Rankin-Cohen bracket of the Eisenstein series of weight two and a Hecke eigenform is again an eigenform.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136367752","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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