Pub Date : 2015-07-01DOI: 10.21099/TKBJM/1438951819
Y. Oka
. The aim of this paper is to give a characterization of the tempered distributions supported by a (Whitney’s) regular closed set in the Euclidean space and the Heisenberg group by means of the heat kernel method. The heat kernel method, introduced by T. Matsuzawa, is the method to characterize the generalized functions on the Euclidean space by the initial value of the solutions of the heat equation.
{"title":"A characterization of the tempered distributions supported by a regular closed set in the Heisenberg group","authors":"Y. Oka","doi":"10.21099/TKBJM/1438951819","DOIUrl":"https://doi.org/10.21099/TKBJM/1438951819","url":null,"abstract":". The aim of this paper is to give a characterization of the tempered distributions supported by a (Whitney’s) regular closed set in the Euclidean space and the Heisenberg group by means of the heat kernel method. The heat kernel method, introduced by T. Matsuzawa, is the method to characterize the generalized functions on the Euclidean space by the initial value of the solutions of the heat equation.","PeriodicalId":44321,"journal":{"name":"Tsukuba Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2015-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.21099/TKBJM/1438951819","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67831525","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2015-07-01DOI: 10.21099/TKBJM/1438951816
Katsuhisa Koshino
Let X be a compact metrizable space and Y be a nondegenerate dendrite with an end point 0. For each continuous function f : X ! Y , we define the hypo-graph # f 1⁄4 6 x AX fxg 1⁄20; f ðxÞ of f , where 1⁄20; f ðxÞ is the unique path from 0 to f ðxÞ in Y . Then we can regard #CðX ;Y Þ 1⁄4 f# f j f : X ! Y is continuousg as a subspace of the hyperspace consisting of non-empty closed sets in X Y equipped with the Vietoris topology. In this paper, we prove that #CðX ;Y Þ is a Baire space if and only if the set of isolated points of X is dense.
设X是一个紧化的可度量空间，Y是一个端点为0的非简并枝晶。对于每一个连续函数f: X !Y，我们定义伪图# f 1 / 4 6 x x x xg 1 / 20;F ðxÞ (F)其中1 / 20;f ðxÞ是Y中从0到f ðxÞ的唯一路径。那么我们可以考虑# c & X;Y Þ 1⁄4 f# f j f: X !Y作为具有Vietoris拓扑的由X Y中的非空闭集组成的超空间的子空间是连续的。本文证明了# c & X;Y Þ是一个Baire空间，当且仅当X的孤立点集是稠密的。
{"title":"The Baire property of certain hypo-graph spaces","authors":"Katsuhisa Koshino","doi":"10.21099/TKBJM/1438951816","DOIUrl":"https://doi.org/10.21099/TKBJM/1438951816","url":null,"abstract":"Let X be a compact metrizable space and Y be a nondegenerate dendrite with an end point 0. For each continuous function f : X ! Y , we define the hypo-graph # f 1⁄4 6 x AX fxg 1⁄20; f ðxÞ of f , where 1⁄20; f ðxÞ is the unique path from 0 to f ðxÞ in Y . Then we can regard #CðX ;Y Þ 1⁄4 f# f j f : X ! Y is continuousg as a subspace of the hyperspace consisting of non-empty closed sets in X Y equipped with the Vietoris topology. In this paper, we prove that #CðX ;Y Þ is a Baire space if and only if the set of isolated points of X is dense.","PeriodicalId":44321,"journal":{"name":"Tsukuba Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2015-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67831461","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2015-07-01DOI: 10.21099/tkbjm/1438951821
Naotsugu Chinen, T. Hosaka
. The review on [1] in Mathematical Reviews points out that the proof of its main result is incorrect. The aim of this paper is to correct the previous paper’s argument and clarify the statement. In [2] it is stated that the proof of [1, Theorem 1.1] is incorrect, i.e., the map f does not satisfy ð(cid:1)Þ r , as claimed on line 4 of the first paragraph on [1, p. 188]. In fact, diam f ð B ð c i k ð x 0 Þ ; 1 ÞÞ ¼ diam a 1 ð B ð x 0 ; 1 ÞÞ 0 0 for each k A N . In this paper, we redefine the map f ¼ 6 k A N f k : ð Y ; r Þ ! ð B n þ 1 ; s Þ , in particular f k : c i k ð B ð x 0 ; k ÞÞ ! B n þ 1 , where let B ð x 0 ; r Þ ¼ f y A X : d ð x 0 ; y Þ a r g for r > 0. Let ð X ; d Þ be a proper CAT ð 0 Þ space and let c : ð X ; d Þ ! ð X ; d Þ be an isometry satisfying that f c i ð x Þ : i A Z g is unbounded (see [1, Theorem 1.1]). Fix a point x 0 of X . For every x A X , let x x : ½ 0 ; d ð x 0 ; x Þ(cid:2) ! X be the geodesic from x 0 to x in ð X ; d Þ . Recall the projection map p 1 : X ! B ð x 0 ; 1 Þ in [1, p. 187] defined by p 1 ð x Þ ¼ x x ð min f d ð x 0 ; x Þ ; 1 gÞ for each x A X .
． 《数学评论》对[1]的评论指出，其主要结果的证明是不正确的。本文的目的是纠正前一篇论文的论点，澄清陈述。在[2]中指出[1，定理1.1]的证明是不正确的，即映射f不满足ð(cid:1)Þ r，如[1,p. 188]第一段第4行所述。实际上，diam f ð B ð c i k ð x 0 Þ;1 ÞÞ¼diam a 1 ð B ð x 0;1 ÞÞ 0 0每k个A N。在本文中，我们重新定义了¼6 k A N f k: ð Y的映射;R Þ !ð n þ 1;s Þ，特别是f k: c i k ð B ð x 0;K ÞÞ !bn + 1，其中设B ð x0;r Þ¼f y A X: d ð X 0;Y Þ a r g为r > 0。令ð X;d Þ是一个合适的CAT ð 0 Þ空间，设c: ð X;D Þ !ð x;d Þ是一个等距，满足f c i ð x Þ: i A Z g是无界的(见[1，定理1.1])。固定x的点x0。对于每一个ax x，设x x: 1 / 2 0;D ð x0;X Þ(cid:2) !X是从X到X的测地线;D Þ。回想一下投影图p 1: X !B ð x0;1 Þ in [1, p. 187]定义为p 1 ð x Þ¼x x ð min f d ð x 0;X Þ;1 gÞ对应每个x A x。
{"title":"Erratum to \"Asymptotic dimension and boundary dimension of proper CAT(0) spaces\"","authors":"Naotsugu Chinen, T. Hosaka","doi":"10.21099/tkbjm/1438951821","DOIUrl":"https://doi.org/10.21099/tkbjm/1438951821","url":null,"abstract":". The review on [1] in Mathematical Reviews points out that the proof of its main result is incorrect. The aim of this paper is to correct the previous paper’s argument and clarify the statement. In [2] it is stated that the proof of [1, Theorem 1.1] is incorrect, i.e., the map f does not satisfy ð(cid:1)Þ r , as claimed on line 4 of the first paragraph on [1, p. 188]. In fact, diam f ð B ð c i k ð x 0 Þ ; 1 ÞÞ ¼ diam a 1 ð B ð x 0 ; 1 ÞÞ 0 0 for each k A N . In this paper, we redefine the map f ¼ 6 k A N f k : ð Y ; r Þ ! ð B n þ 1 ; s Þ , in particular f k : c i k ð B ð x 0 ; k ÞÞ ! B n þ 1 , where let B ð x 0 ; r Þ ¼ f y A X : d ð x 0 ; y Þ a r g for r > 0. Let ð X ; d Þ be a proper CAT ð 0 Þ space and let c : ð X ; d Þ ! ð X ; d Þ be an isometry satisfying that f c i ð x Þ : i A Z g is unbounded (see [1, Theorem 1.1]). Fix a point x 0 of X . For every x A X , let x x : ½ 0 ; d ð x 0 ; x Þ(cid:2) ! X be the geodesic from x 0 to x in ð X ; d Þ . Recall the projection map p 1 : X ! B ð x 0 ; 1 Þ in [1, p. 187] defined by p 1 ð x Þ ¼ x x ð min f d ð x 0 ; x Þ ; 1 gÞ for each x A X .","PeriodicalId":44321,"journal":{"name":"Tsukuba Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2015-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67831565","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2015-07-01DOI: 10.21099/TKBJM/1438951814
Nagatoshi Sasano
{"title":"Lie algebras associated with a standard quadruplet and prehomogeneous vector spaces","authors":"Nagatoshi Sasano","doi":"10.21099/TKBJM/1438951814","DOIUrl":"https://doi.org/10.21099/TKBJM/1438951814","url":null,"abstract":"","PeriodicalId":44321,"journal":{"name":"Tsukuba Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2015-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67831340","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2015-03-01DOI: 10.21099/TKBJM/1429103719
M. Kawashima
{"title":"Evaluation of the dimension of the Q-vector space spanned by the special values of the Lerch function","authors":"M. Kawashima","doi":"10.21099/TKBJM/1429103719","DOIUrl":"https://doi.org/10.21099/TKBJM/1429103719","url":null,"abstract":"","PeriodicalId":44321,"journal":{"name":"Tsukuba Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2015-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67830120","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2015-03-01DOI: 10.21099/TKBJM/1429103722
Setsuo Nagai
{"title":"A characterization of isoparametric hypersurfaces in a sphere with $gle 3$","authors":"Setsuo Nagai","doi":"10.21099/TKBJM/1429103722","DOIUrl":"https://doi.org/10.21099/TKBJM/1429103722","url":null,"abstract":"","PeriodicalId":44321,"journal":{"name":"Tsukuba Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2015-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.21099/TKBJM/1429103722","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67831118","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2015-03-01DOI: 10.21099/TKBJM/1429103718
Ryunosuke Ozawa
. We study the Prohorov distance between the distance matrix distributions of two metric measure spaces. We prove that it is not smaller than 1-box distance between two metric measure spaces and also prove that it is not larger than 0-box distance between two metric measure spaces.
{"title":"Distance between metric measure spaces and distance matrix distributions","authors":"Ryunosuke Ozawa","doi":"10.21099/TKBJM/1429103718","DOIUrl":"https://doi.org/10.21099/TKBJM/1429103718","url":null,"abstract":". We study the Prohorov distance between the distance matrix distributions of two metric measure spaces. We prove that it is not smaller than 1-box distance between two metric measure spaces and also prove that it is not larger than 0-box distance between two metric measure spaces.","PeriodicalId":44321,"journal":{"name":"Tsukuba Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2015-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.21099/TKBJM/1429103718","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67830104","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2015-03-01DOI: 10.21099/TKBJM/1429103720
H. Ichimura
For an imaginary abelian field K , Hasse [3, Satz 45] obtained a criterion for the relative class number to be odd in terms of the narrow class number of the maximal real subfield Kþ and the prime numbers which ramify in K , by using the analytic class number formula. In [4], we gave a refined version (1⁄4 ‘‘D-decomposed version’’) of Satz 45 by an algebraic method. In this paper, we give one more algebraic proof of the refined version.
{"title":"Refined version of Hasse's Satz 45 on class number parity","authors":"H. Ichimura","doi":"10.21099/TKBJM/1429103720","DOIUrl":"https://doi.org/10.21099/TKBJM/1429103720","url":null,"abstract":"For an imaginary abelian field K , Hasse [3, Satz 45] obtained a criterion for the relative class number to be odd in terms of the narrow class number of the maximal real subfield Kþ and the prime numbers which ramify in K , by using the analytic class number formula. In [4], we gave a refined version (1⁄4 ‘‘D-decomposed version’’) of Satz 45 by an algebraic method. In this paper, we give one more algebraic proof of the refined version.","PeriodicalId":44321,"journal":{"name":"Tsukuba Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2015-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67830176","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2015-03-01DOI: 10.21099/tkbjm/1429103725
H. Brunotte
{"title":"Corrigendum to \"Periodicity and eigenvalues of matrices over quasi-max-plus algebras\", Tsukuba J. Math., 37 (2013), pp. 51–71","authors":"H. Brunotte","doi":"10.21099/tkbjm/1429103725","DOIUrl":"https://doi.org/10.21099/tkbjm/1429103725","url":null,"abstract":"","PeriodicalId":44321,"journal":{"name":"Tsukuba Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2015-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67831167","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2014-07-01DOI: 10.21099/TKBJM/1407938670
Y. Kuratomi
A module M is said to be quasi-continuous if it is extending with the condition ðC3Þ (cf. [7], [10]). In this paper, by using the notion of a G-extending module which is defined by E. Akalan, G. F. Birkenmeier and A. Tercan [1], we introduce a generalization of quasi-continuous ‘‘a GQC(generalized quasicontinuous)-module’’ and investigate some properties of GQCmodules. Initially we give some properties of a relative ejectivity which is useful in analyzing the structure of G-extending modules and GQC-modules (cf. [1]). And we apply them to the study of direct sums of GQC-modules. We also prove that any direct summand of a GQC-module with the finite internal exchange property is GQC. Moreover, we show that a module M is G-extending modules with ðC3Þ if and only if it is GQC-module with the finite internal exchange property.
如果模M在ðC3Þ (cf.[7]，[10])条件下扩展，则称其为拟连续模。本文利用E. Akalan, G. F. Birkenmeier和a . Tercan[1]所定义的g扩展模的概念，引入了拟连续的推广“广义拟连续模”，并研究了gqc模的一些性质。本文首先给出了相对射射的一些性质，这些性质对分析g -扩展模和gqc -模的结构是有用的(参见[1])。并将其应用于gqc模的直接和的研究。我们还证明了具有有限内交换性质的GQC模块的任何直接和都是GQC。更进一步，我们证明了一个模M是具有有限内交换性质的gqc模，当且仅当它是具有有限内交换性质的gqc模时，我们用ðC3Þ证明了它是g扩展模。
{"title":"Goldie extending modules and generalizations of quasi-continuous modules","authors":"Y. Kuratomi","doi":"10.21099/TKBJM/1407938670","DOIUrl":"https://doi.org/10.21099/TKBJM/1407938670","url":null,"abstract":"A module M is said to be quasi-continuous if it is extending with the condition ðC3Þ (cf. [7], [10]). In this paper, by using the notion of a G-extending module which is defined by E. Akalan, G. F. Birkenmeier and A. Tercan [1], we introduce a generalization of quasi-continuous ‘‘a GQC(generalized quasicontinuous)-module’’ and investigate some properties of GQCmodules. Initially we give some properties of a relative ejectivity which is useful in analyzing the structure of G-extending modules and GQC-modules (cf. [1]). And we apply them to the study of direct sums of GQC-modules. We also prove that any direct summand of a GQC-module with the finite internal exchange property is GQC. Moreover, we show that a module M is G-extending modules with ðC3Þ if and only if it is GQC-module with the finite internal exchange property.","PeriodicalId":44321,"journal":{"name":"Tsukuba Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2014-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.21099/TKBJM/1407938670","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67829222","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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