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":"39 1","pages":"29-38"},"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":"39 1","pages":"165-166"},"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":"39 1","pages":"1-14"},"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":"38 1","pages":"171-188"},"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":"38 1","pages":"207-225"},"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":"38 1","pages":"159-170"},"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":"38 1","pages":"189-199"},"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":"38 1","pages":"313-313"},"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/1407938673
E. Casta, ñeda-Alvarado, J. Sánchez-Martínez
. A continuum means a nonempty, compact and connected metric space. Given a continuum X , the symbols F n ð X Þ and C 1 ð X Þ denotes the hyperspace of all subsets of X with at most n points and the hyperspace of subcontinua of X , respectively. If n > 1, we consider the quotient spaces SF n 1 ð X Þ ¼ F n ð X Þ = F 1 ð X Þ and C 1 ð X Þ = F 1 ð X Þ obtained by shrinking F 1 ð X Þ to a point in F n ð X Þ and C 1 ð X Þ , re-spectively. In this paper, we study the continua X such that SF n 1 ð X Þ is homeomorphic to C 1 ð X Þ = F 1 ð X Þ and we analyze when the spaces F n ð X Þ and SF n 1 ð X Þ are homeomorphic to some sphere.
。连续统是指一个非空的、紧的、连通的度量空间。给定连续体X，符号fn ð X Þ和c1 ð X Þ分别表示X的所有最多n个点的子集的超空间和X的次连续体的超空间。如果n > 1，我们考虑商空间SF n 1 ð X Þ¼F n ð X Þ = F 1 ð X Þ和C 1 ð X Þ = F 1 ð X Þ分别将F 1 ð X Þ缩小到F n ð X Þ和C 1 ð X Þ中的一个点。本文研究了连续体X使SF n 1 ð X Þ同胚于C 1 ð X Þ = F 1 ð X Þ，并分析了空间F n ð X Þ与SF n 1 ð X Þ同胚于某球的情况。
{"title":"Spheres, symmetric products, and quotient of hyperspaces of continua","authors":"E. Casta, ñeda-Alvarado, J. Sánchez-Martínez","doi":"10.21099/tkbjm/1407938673","DOIUrl":"https://doi.org/10.21099/tkbjm/1407938673","url":null,"abstract":". A continuum means a nonempty, compact and connected metric space. Given a continuum X , the symbols F n ð X Þ and C 1 ð X Þ denotes the hyperspace of all subsets of X with at most n points and the hyperspace of subcontinua of X , respectively. If n > 1, we consider the quotient spaces SF n 1 ð X Þ ¼ F n ð X Þ = F 1 ð X Þ and C 1 ð X Þ = F 1 ð X Þ obtained by shrinking F 1 ð X Þ to a point in F n ð X Þ and C 1 ð X Þ , re-spectively. In this paper, we study the continua X such that SF n 1 ð X Þ is homeomorphic to C 1 ð X Þ = F 1 ð X Þ and we analyze when the spaces F n ð X Þ and SF n 1 ð X Þ are homeomorphic to some sphere.","PeriodicalId":44321,"journal":{"name":"Tsukuba Journal of Mathematics","volume":"38 1","pages":"75-84"},"PeriodicalIF":0.7,"publicationDate":"2014-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.21099/tkbjm/1407938673","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67830052","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。
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