{"title":"A geometric condition for smoothability of combinatorial manifolds","authors":"Keiko Kudo, H. Noguchi","doi":"10.2996/KMJ/1138844815","DOIUrl":"https://doi.org/10.2996/KMJ/1138844815","url":null,"abstract":"Let us commence with the terminology. For a complex Y, Y will denote the polyhedron covered by Y and Y will stand for the first barycentric subdivision of Y. We say that a subcomplex X of Y is complete if the intersection of a (closed) simplex of Y and | X is either empty or a simplex of X. A combinatorial manifold is a polyhedron with a distinguished class of simplicial subdivisions which are formal manifolds, [5, p. 825]. For a combinatorial manifold P, the boundary of P is written dP and the interior P—dPΊs written IntP, and a closed combinatorial manifold will be a compact combinatorial manifold without boundary. Let X be a subcomplex of Y where | Y is a combinatorial manifold. (Note that X' is a complete subcomplex of Y'.} Then N(X, Y) denotes the star neighborhood of X in y, that is, the polyhedron consists of simplices of F, which contain simplices of X. It is well known that dN(K', L') (that is, the boundary of the star neighborhood of the first barycentric subdivision of K in the first barycentric subdivision of L) is a closed combinatorial (m—1) -manifold if the polyhedron L is a combinatorial m-manifold without boundary and K is a finite complete subcomplex of L; [4, p. 293]. For convenience, we say that a polyhedron Q is imbedded piecewise linearly in euclidean space R if there are (linear) simplicial subdivisions X and L of Q and R respectively such that X is a subcomplex of L, where it may be assumed without loss of generality that X is a complete subcomplex of L. Now let us explain the condition for smoothability. DEFINITION 1. Let M be a closed combinatorial ^-manifold imbedded piecewise linearly in euclidean (n+r)-space R, r^l. We say that M is in smoothable position in R if the following is satisfied. Let K0 and L0 be simplicial subdivisions of M and R respectively, where K0 is a complete subcomplex of L0. Then there exist piecewise linear homeomorphisms pi. Mί-^dN(Kτ', Z '̂) for each O^a'^r— 1, where MQ=M and for l^z^r, Mτ=pi-ι(Mι-ι) and where Kl and LL are simplicial subdivisions of Mi and dN(Kτ-ι', L l_/) respectively such that Kt is a complete subcomplex of Li. In the text, however, Wτ stands for dN(Kι-ι, Ll-l ) and Lτ will be the subcomplex of L t_/ covering W% for each l<^'^r. Note that M^ is a closed combinatorial ^-manifold, which is combinatorially equivalent to M\", and Wι is a closed combinatorial (n+r—ϊ) -manifold, for each l^i^r, satisfying MtCTFt and WΊD W2 =) D TFr.","PeriodicalId":318148,"journal":{"name":"Kodai Mathematical Seminar Reports","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1963-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127714977","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}
{"title":"On conformal slit mapping","authors":"Yūsaku Komatu","doi":"10.2996/KMJ/1138844813","DOIUrl":"https://doi.org/10.2996/KMJ/1138844813","url":null,"abstract":"Each function w=f(z,t)=e '(z+ ) carries out also a univalent mapping of j^]<l onto a bounded slit domain Bt with a slit Lt. The continuous function κ(t)=e ^ involved in the Lόwner equation shows an interesting behavior if the original slit L=Lt0 is supposed to be analytic. Every slit Lt (0<t<t0) is then also analytic and meets wt = l orthogonally at its endpoint κ(f)=e~~ Let the curvature of Lt at κ(f) be denoted by p(t). The following theorem is known; cf. [3], [9]:","PeriodicalId":318148,"journal":{"name":"Kodai Mathematical Seminar Reports","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1963-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129634610","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}
{"title":"Testing hypotheses for Markov chains when the parameter space is finite","authors":"K. Yoshihara","doi":"10.2996/KMJ/1138844784","DOIUrl":"https://doi.org/10.2996/KMJ/1138844784","url":null,"abstract":"In Chernoff [2], a procedure was presented for the sequential design of experiments where the problem was one of testing a hypothesis. When there were only a finite number of states of nature and a finite number of available experiments, the procedure was shown to be \" asymptotically optimal\" as the cost of sampling approached zero. An analogous procedure can be applied to the problem of testing a hypothesis with respect to a Markov process, and this procedure will also be shown to be \" asymptotically optimal\".","PeriodicalId":318148,"journal":{"name":"Kodai Mathematical Seminar Reports","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1963-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129766786","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}
{"title":"On Fubinian and $C$-Fubinian manifolds","authors":"Y. Tashiro, S. Tachibana","doi":"10.2996/KMJ/1138844787","DOIUrl":"https://doi.org/10.2996/KMJ/1138844787","url":null,"abstract":"present hypersurface satisfying the a C-umbilical hypersurface. A manifold having the same Sasakian structure as a C-umbilical hypersurface in a locally Fubinian manifold will be said to be locally C-Fubinian. The purpose of the present paper is to show some character-istic properties of Fubinian and C-Fubinian manifolds.","PeriodicalId":318148,"journal":{"name":"Kodai Mathematical Seminar Reports","volume":"160 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1963-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124492656","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}
{"title":"A functional method on amount of entropy","authors":"H. Umegaki","doi":"10.2996/KMJ/1138844786","DOIUrl":"https://doi.org/10.2996/KMJ/1138844786","url":null,"abstract":"The theory of information, originated by Shannon, was applied in the new subject to investigate the theory of transformation with invariant measure by Kolmogorov and his school, cf. Rokhlin [12]. Recently, Halmos [7] gave a very clarified note relative to their investigations. While, in order to achieving the channel capacity in stationary finite memory channels, cf. Feinstein [6], some important properties of the entropy (the average amount of information) of information sources in these channels were studied by Khinchin [8], Takano [13], Traregradsky [14], Breiman [2], Parthasarathy [11] and others. The basic space of information sources of the channels is the doubly infinite product set A (the messages space) of the alphabet A, which becomes a compact metric space relative to the weak product topology and in which the shift transformation is a homeomorphism on A (so-called the Bernoulli automorphism).","PeriodicalId":318148,"journal":{"name":"Kodai Mathematical Seminar Reports","volume":"47 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1963-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115504322","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}
{"title":"On curvatures of spaces with normal general connections. II.","authors":"T. Ôtsuki","doi":"10.2996/KMJ/1138844788","DOIUrl":"https://doi.org/10.2996/KMJ/1138844788","url":null,"abstract":"In this paper, the author makes a formula (§ 2) related to the curvature tensors of a normal general connection γ and BγB, where B is a tensor field of type (1, 1) satisfying some conditions, making use of the results in a previous paper [14], and then he shows that the formula applied to the case in which γ is a classical affine connection is a generalization of the Gauss' equations in the theory of subspaces of Riemannian geometry (§4). He also shows that regarding the set of general connections as a vector space over the algebra of all tensor fields of type (1, 1), the calculations in connection with the above purpose can be simplified.","PeriodicalId":318148,"journal":{"name":"Kodai Mathematical Seminar Reports","volume":"46 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1963-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123237061","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}
{"title":"On some asymptotic expansion theorems","authors":"T. Aonuma","doi":"10.2996/KMJ/1138844785","DOIUrl":"https://doi.org/10.2996/KMJ/1138844785","url":null,"abstract":"as extensions of the well-known summability theorem (§ 2) and convergence theorems (§3) in this paper. Before proceeding them, we state the following summability theorem and convergence theorems which we are concerned with. THEOREM A. (Bochner and Chandrasekharan [2]). If K(u)£L±(Q, oo), K(u) =o(u~) as U—+CO, /(w)€Lι(0, oo) and K(u) is monotone decreasing in 0^^<oo, then the condition at a point x","PeriodicalId":318148,"journal":{"name":"Kodai Mathematical Seminar Reports","volume":"72 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1963-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115002611","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}
{"title":"A remark on the generalization of Harnack's first theorem","authors":"Yoshikazu Hirasawa","doi":"10.2996/KMJ/1138844760","DOIUrl":"https://doi.org/10.2996/KMJ/1138844760","url":null,"abstract":"and under one of those uniqueness conditions, Harnack's first theorem was extended to the solution of the equation (1. 1). It was the case where the function f(x, u, p) was non-decreasing with respect to u. In the present paper, we consider the case where the function f(x, u, p) has not necessarily the above-mentioned property, and since Harnack's first theorem for solutions of the elliptic differential equation is really based on the continuous dependence of solutions upon the boundary data, we will here treat of this dependence. Regarding the notations used in the present paper, confer the above-cited papers.","PeriodicalId":318148,"journal":{"name":"Kodai Mathematical Seminar Reports","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1963-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121916816","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}
{"title":"On an application of L. Ehrenpreis' method to ordinary differential equations","authors":"J. Kajiwara","doi":"10.2996/KMJ/1138844757","DOIUrl":"https://doi.org/10.2996/KMJ/1138844757","url":null,"abstract":"In 1956 Ehrenpreis [3] considered an application of the sheaf theory to differential equations and gave a criterion for the existence of global solutions of differential equations where the existence of local solutions are assured. We shall apply this method to systems of ordinary and linear differential equations with coefficients meromorphic in a domain D on the plane C of one complex variable z. Let D and Wl be the sheaves of all germs of functions holomorphic and meromorphic in D respectively. Let a^ (j, k=l, 2, •••,/>) be functions meromorphic in D. For any element f=(f,f, , f p ) of 30̂ , we define","PeriodicalId":318148,"journal":{"name":"Kodai Mathematical Seminar Reports","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1963-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121615544","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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