{"title":"An application of Green's formula of a discrete function: Determination of periodicity moduli. II.","authors":"Hisao Mizumoto","doi":"10.2996/KMJ/1138846121","DOIUrl":"https://doi.org/10.2996/KMJ/1138846121","url":null,"abstract":"Introduction. Recently Opfer published a very interesting result [6] (also cf. [5]) in which he concerned himself with the problem of determining the modulus of a doubly connected domain by means of the difference method. In the present paper we shall consider a corresponding problem for a general multiply connected domain. It is known that for a non-degenerated N-ply connected domain (W^2) there exist N(N—ϊ)/2 quantities which are said to be periodicity moduli of the domain, which are conformally invariant, and which have an important meaning in the function theory. We shall concern ourselves with the problem of determining the system of periodicity moduli by means of the difference method (cf. Theorem 3.1 and Corollaries 2. 4, 3.1). Our method making effective use of Green's formula of a discrete function admits to deal with our problem by a unified principle. Also for a harmonic function u and a discrete harmonic function U on a domain G and a lattice R respectively which are constant on each boundary component of G and R, the monotonicity of the Dirichlet integral DG(u) and the summation SR(U) (see §2. 2) with respect to G and R is effectively utilized (cf. Lemmas 1.1, 2. 4, 2. 5 and 2. 6, and Theorem 2.1). For N=2 our main results (Theorem 3.1 and Corollary 3.1) coincide to Opfer's (Satz 7 of [6]). However even such a special case our method is deferent from his and is more simplified.","PeriodicalId":318148,"journal":{"name":"Kodai Mathematical Seminar Reports","volume":"49 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":"132839179","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":"Deficiencies of an entire algebroid function of finite order","authors":"Tsuneo Sato","doi":"10.2996/KMJ/1138846472","DOIUrl":"https://doi.org/10.2996/KMJ/1138846472","url":null,"abstract":"","PeriodicalId":318148,"journal":{"name":"Kodai Mathematical Seminar Reports","volume":"92 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":"133046853","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 unipotent groups of characteristic p>0","authors":"Tetsuo Nakamura","doi":"10.2996/KMJ/1138846313","DOIUrl":"https://doi.org/10.2996/KMJ/1138846313","url":null,"abstract":"Borel and Springer [1] deal with a unipotent group U defined over a field of prime characteristic p and investigate the conditions of the existence of the one dimensional subgroup to which a given element of the Lie algebra L(U) of U is tangent. They use the lemma (9. 15) (ii) (p. 493) in the proof of the last theorem (9. 16) (ii) (p. 495). In this report we shall show that this lemma is not correct (cf. §2). But a modification of it does not disturb the truth of the theorem. Moreover we want to show that the theorem (9. 16) (iii) in [1] is still true under a weaker assumption (cf. § 1). The notations in this report are the same as in [1].","PeriodicalId":318148,"journal":{"name":"Kodai Mathematical Seminar Reports","volume":"17 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":"133631454","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 certain entire functions which together with their derivatives are prime","authors":"H. Urabe, Chung-Chun Yang","doi":"10.2996/KMJ/1138833579","DOIUrl":"https://doi.org/10.2996/KMJ/1138833579","url":null,"abstract":"Introduction. In studying the factorization of meromorphic functions, we may ask the relationship between the factors of a function and those of its derivatives. A meromorphic function F(z)=f(g(z)) is said to have / and g as left and right factors, respectively, provided that f is meromorphic and g is entire (g may be meromorphic if /is rational). F(z) is said to be prime (pseudoprime, left-prime, right-prime) if every factorization of the above form into factors implies either / is linear or g is linear (either / is rational or g is a polynomial, / is linear whenever g is transcendental, g is linear whenever / is transcendental). When factors are restricted to entire functions, it is called to be a factorization in entire sense. In this paper only entire factors will be considered. We note here it is known ([7]) that, when F is not periodic, then F is prime if F is prime in entire sense. Because of this observation, in this note entire factors only need to be considered. Suppose that a transcendental entire function F(z) is prime. Does it follow that its tt-th derivative Fz) is also prime? In general, there is not much that we can really say. For example, take F(z)=e*+z, then F is known to be prime (cf. [5] or [10] etc.), but F'(z)'=e+1 is not prime (F'(z) is pseudo-prime). Further take F(z)=exp [e]+z, then F(z) is prime (cf. [6] or [10]), but F'(z) =e-exp [>]+l is composite (both factors are transcendental). While if we take F(z)=z-e, then F(z) is prime for n=0,1, (F(z)=F(z)). (Note that F(z)=z-exp [>] is prime but F'(z) is not prime, since F'(z) is an even function.) Another interesting example is given by","PeriodicalId":318148,"journal":{"name":"Kodai Mathematical Seminar Reports","volume":"29 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":"128897954","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 the stochastic process of random noise","authors":"T. Kawata","doi":"10.2996/KMJ/1138843604","DOIUrl":"https://doi.org/10.2996/KMJ/1138843604","url":null,"abstract":"If events are arrivals of electrons at the anode of vacuum tube, (l l) represents the noise current. The formulation (1.1) is due to J. L. Doob ί 1 ) Formally, the extensive analysis of random noise was done by S. O Rice, in the case t Ct) is a trigonometric polynomial. The object of this paper is to prove some results, in connection with Rice theory, with X, (t) defined by (1.1) in a rather rigorous way from mathematical view points. The proofs of some known results are contained, because of completeness","PeriodicalId":318148,"journal":{"name":"Kodai Mathematical Seminar Reports","volume":"4 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":"128909486","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":"Some properties of extremal polynomials for the Ilieff conjecture","authors":"D. Phelps, R. S. Rodriguez","doi":"10.2996/KMJ/1138846519","DOIUrl":"https://doi.org/10.2996/KMJ/1138846519","url":null,"abstract":"Let Pn denote the family of complex polynomials each of degree n, with leading coefficient 1, and having all of its roots in J9(0,1), the closed unit disc with center at 0 and radius 1. Let p€Pn have roots zly •••, zn and have roots wu •••, wn-ι. For such p we use I(zj), I(p), and I(Pn) to denote the numbers min {|z$—wk: l^k^n—l},m3x{I(Zj):l^j^n}, and sup {/(/>): psPn} respectively. Then p€Pn is called an extremal polynomial for the Ilieff conjecture if I(p)=I(Pn). With this notation the Gauss-Lucas theorem implies that I(Pn)^2 and the conjecture of Ilieff is that I(p)^l for all psPnWe show that there exist extremal polynomials, that an extremal 'polynomial must have at least one root on each subarc of the unit circle of length i^π, and we find the extremal polynomials for n—Z and 4. We begin with a","PeriodicalId":318148,"journal":{"name":"Kodai Mathematical Seminar Reports","volume":"48 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":"127836198","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":"Almost quaternion structures of the second kind and almost tangent structures","authors":"K. Yano, Mitsue Ako","doi":"10.2996/KMJ/1138846724","DOIUrl":"https://doi.org/10.2996/KMJ/1138846724","url":null,"abstract":"is called an almost quaternion structure and a differentiate manifold with an almost quaternion structure an almost quaternion manifold. If there exists, in an almost quaternion manifold, a system of coordinate neighborhoods with respect to which components of F, G and H are all constant, then the almost quaternion structure is said to be integrable and the almost quaternion manifold is called a quaternion manifold. In a previous paper [8], the present authors studied integrability conditions for almost quaternion structures. A set of three tensor fields F, G and PI of type (1, 1) in a differentiate manifold which satisfy","PeriodicalId":318148,"journal":{"name":"Kodai Mathematical Seminar Reports","volume":"35 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":"134609049","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 note on the cross-norm of the direct product of operator algebra","authors":"M. Takesaki","doi":"10.2996/KMJ/1138844027","DOIUrl":"https://doi.org/10.2996/KMJ/1138844027","url":null,"abstract":"","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":"131315313","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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