{"title":"Biharmonic and quasiharmonic degeneracy","authors":"L. O. Chung, L. Sario, Cecilia Y. Wang","doi":"10.2996/KMJ/1138833581","DOIUrl":"https://doi.org/10.2996/KMJ/1138833581","url":null,"abstract":"Among the vast complex of problems on inclusion relations between biharmonic and quasiharmonic null classes of Riemannian manifolds, we consider in the present paper perhaps the most intriguing case: Are there inclusion relations between Of 2c and OξLP? Here H , C, Q, L are the classes of functions which are nonharmonic biharmonic, bounded Dirichlet finite, quasiharmonic, or of finite L norm, respectively; a function u is biharmonic or quasiharmonic according as Δu=Q or Δu=l, with Δ the Laplace-Beltrami operator dδ+δd; for any two classes X, Y of functions, XY stands for XrY, and 0%γ for the class of Riemannian iV-manifolds on which XY—φ. The classes H, Q, and L are not meaningful on Riemann surfaces, but are of great interest on Riemannian manifolds. It is known that both Of 2C and OQLP are strictly contained in Oξc, but whether or not there is an inclusion relation between O^c and OQLP ha-s been an open question. The purpose of the present paper is to show that the answer is in the negative. In particular, for any Λfe2 and any p^l, there exist Riemannian Λf-manifolds which carry QL functions but nevertheless fail to carry HC functions. For any null class 0^ of Riemannian Λf-manifolds, denote by 0 the complementary class. In Nos. 1 and 2, it is readily verified that the classes Of 2C ΓOQLP, O%2cΓλθQLP, and 0%2CΓ0$LP are all nonvoid. The interesting relation is OH^CΓΛOQLVΦΦ, for which we use two approaches, one in Nos. 3-6, the other in Nos. 7-10.","PeriodicalId":318148,"journal":{"name":"Kodai Mathematical Seminar Reports","volume":"56 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129980922","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":"Controlled Galton-Watson process and its asymptotic behavior","authors":"T. Fujimagari","doi":"10.2996/KMJ/1138847159","DOIUrl":"https://doi.org/10.2996/KMJ/1138847159","url":null,"abstract":"1. In a stochastic population process described as a Galton-Watson process each individual splits independently according to a given probability law and new born particles constitute the following generation. In addition to the independence in splitting the law of splitting of each individual depends on neither the generation to which an individual belongs nor the existence of the other individuals of the same generation and is common to all individuals. We shall consider a somewhat generalized Galton-Watson process in the sense that the law of splitting of each individual depends on the total number of individuals of the same generation and the other independence properties are reserved. The object of this note is to study asymptotic behaviors of such processes, although we can hardly obtain any complete results up to now except some partial results. The difficulties in analysing the process will be due to the dependence introduced above from which it no longer holds such as the iteration property of a generating function which plays a fundamental role in GaltonWatson processes. We shall formulate the process under consideration as follows. Let Zn be the size or the total number of individuals which belong to the n-th generation and given a sequence of probability distributions £>(i)= pr(i): rΞ>0}, z=0, 1,2, ••• oo","PeriodicalId":318148,"journal":{"name":"Kodai Mathematical Seminar Reports","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134235528","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":"The axiom of coholomorphic 3-spheres in an almost Tachibana manifold","authors":"S. Yamaguchi","doi":"10.2996/KMJ/1138847324","DOIUrl":"https://doi.org/10.2996/KMJ/1138847324","url":null,"abstract":"§","PeriodicalId":318148,"journal":{"name":"Kodai Mathematical Seminar Reports","volume":"63 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133916482","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 ultrahyperelliptic surfaces","authors":"Mitsuru Ozawa","doi":"10.2996/KMJ/1138845443","DOIUrl":"https://doi.org/10.2996/KMJ/1138845443","url":null,"abstract":"and every an is a simple zero of g(z) and s=0 or 1. Hiromi and Mutδ [1] proved the following result: Assume there exists a nontrivial analytic mapping φ from R into S. Then p=n-r, where r is the order of g(z) and n is an integer. The aim of the present paper is to prove the following THEOREM. If S is an ultrahyperelliptic surface of non-zero finite order into which there is a non-trivial analytic mapping from an ultrahyperelliptic surface R of finite order and with P(7?)=4, then the order of S is a half of an integer. 2. Proof of theorem. For our purpose we need our previous result in [4], which asserts the existence of two functions h(z) and f(z) such that f(z) is meromorphic in |2|<oo and h(z) is a polynomial of degree n in the present situation [1] satisfying","PeriodicalId":318148,"journal":{"name":"Kodai Mathematical Seminar Reports","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131796515","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 a simplified estimate of correlogram for a stationary non-Gaussian process","authors":"Mituaki Huzii","doi":"10.2996/KMJ/1138845495","DOIUrl":"https://doi.org/10.2996/KMJ/1138845495","url":null,"abstract":"","PeriodicalId":318148,"journal":{"name":"Kodai Mathematical Seminar Reports","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130790528","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":"Fixed points of reversible semigroups of nonexpansive mappings","authors":"Theodore Mitchell","doi":"10.2996/KMJ/1138846168","DOIUrl":"https://doi.org/10.2996/KMJ/1138846168","url":null,"abstract":"Takahashi [7, p. 384] proved that if K is a compact convex subset of a Banach space, and S is a left amenable semigroup of nonexpansive self-maps of K, then K contains a common fixed point of S. This theorem generalizes a result of DeMarr [2, p. 1139], who obtained the above implication for the case where S is commutative. In this note, we observe that Takahashi's theorem can be further extended, and the proof slightly simplified, by considering a purely algebraic property that every left amenable semigroup must possess, that of left reversibility. The proof employs suitable modifications of the methods of [2] and [7].","PeriodicalId":318148,"journal":{"name":"Kodai Mathematical Seminar Reports","volume":"44 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130813282","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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