{"title":"On approximations to some limiting distributions with applications to the theory of sampling inspections by attributes","authors":"H. Makabe","doi":"10.2996/KMJ/1138844855","DOIUrl":"https://doi.org/10.2996/KMJ/1138844855","url":null,"abstract":"In the previous papers [7], [8], [11] and [12], we have discussed on the several approximations to the probability distributions and noted their applications. The purposes of this paper are to continue and extend our discussions, hence this paper which is a continuation of [8] and [11] may be seen as the Part III of them. Poisson approximations to binomial distribution and to Poisson binomial distribution were treated ([2], [3]), but in [2] the expressions of the evaluations for the error term are not quite simple ones. In §2, using the analogous technique to that in [8], we evaluate the errors of the approximation in term of p under some restrictive conditions, and remark that when binomial distributionis replaced by negative binomial distribution, the similar results hold. Based upon these results, we can deal with some of its applications to the sampling inspection theory. Binomial approximation to Poisson binomial distribution was treated by LeCam [6]. In § 3, we shall show that first approximating term of the above approximation can be expressed as the sum of binomial distribution and its difference of the second order. The error terms of the approximations of the approximation are evaluated in terms of the square sum of Δpk where Δpk=pk—pv The evaluation of the error of the normal opproximation to the binomial distribution is given in [3] and [7]. In §4, we shall treat the normal approximation to the Poisson binomial distribution which seems to be not investigated ever. For evaluation of the error of approximation, we utilize the results of our previous paper [7], and obtain the similar expression to the results of [7]. Finally in order to show the applicability of our results in § 1, we shall proceed to some problems on sampling inspections by attributes based on prior distribution, and add some remarks and tables. In [9] and [10], we have also stated some results concerning with those problems from the other view points.","PeriodicalId":318148,"journal":{"name":"Kodai Mathematical Seminar Reports","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1964-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131102203","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":"General treatment of alphabet-message space and integral representation of entropy","authors":"H. Umegaki","doi":"10.2996/KMJ/1138844856","DOIUrl":"https://doi.org/10.2996/KMJ/1138844856","url":null,"abstract":"In this paper we shall clarify a topological structure of the alphabet-message space of the memory channel in information theory, and study the integral representation of entropy amount from a general view point of a certain generalized message space. In order to apply to the general theory of entropy, the present fashion will develop a message space into more general treatment, in which the basic space X will be assumed to be totally disconnected. As will be shown in §2, the alphabet-message space A is a totally disconnected compact space, and in § 3, a kind of theorem relative to sufficiency for a (/-field generated by a partition and a homeoporphism (cf. Theorem 2) and the others (Theorems 3 and 4) are concerned with the semi-continuity of entropy amount which are general form of Breiman's Theorem [1]. Finally, in §4, it will be discussed about the function h(x) found by Parthasarathy [7] whose integral defines the corresponding amount of entropy (cf. Theorem 5). It is also shown that the results in [9] can be generalized (cf. the footnote 2) below). The function h(x) may give useful and interesting tool for the general theory of entropy of measure preserving automorphism or flow over a probabiltiy space.","PeriodicalId":318148,"journal":{"name":"Kodai Mathematical Seminar Reports","volume":"93 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1964-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121441975","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":"Rigidity of projection map and the growth of analytic functions.","authors":"Mitsuru Ozawa","doi":"10.2996/KMJ/1138844858","DOIUrl":"https://doi.org/10.2996/KMJ/1138844858","url":null,"abstract":"for a single-valued meromorphic function F(w) in the punctured disc <70< log |M;|<OO satisfying the condition TP(σ, F)=o(e ). This fact says that f(p) preserves the projection map F0: W— >2B*, that is, f(Pι)=f(Pz) if Fo(p1)=F0(p2)t when f(p) satisfies the desired growth condition. Such a phenomenon was studied non-systematically by the various authors. Excepting the closed surface case, the first one who explained the phenomenon is Selberg [5]. However his ramification theorem in his celebrated theory [4], that is,","PeriodicalId":318148,"journal":{"name":"Kodai Mathematical Seminar Reports","volume":"12 1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1964-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116585703","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 for stationary channels","authors":"H. Umegaki","doi":"10.2996/KMJ/1138844857","DOIUrl":"https://doi.org/10.2996/KMJ/1138844857","url":null,"abstract":"The concept of the finite memory channel of Shannon has been formulated in the purely mathematical form by McMillan and Khinchin (Cf. [10]), and established in the present elegant style by Feinstein [6]. Around the finite memory channels as its focal point, there exist various theorems, in which one of the most important results is the theorem of the equality of Cs and Ce of stationary and ergodic capacities. The problem 'whether the equality holds' has been an open question since Khinchin's paper [10]. This equality has been recently proved by many authors: Tsaregradsky [13], Carleson [3], Feinstein [7], Breiman [2], Parthasarathy [11] and others. In this paper we shall describe it in an abstract form. The purpose of this paper is to introduce an abstract characterization of finite or infinite memory channel in which the input space and the output space are compact (totally disconnected) Hausdorff spaces with a pair of fixed homeomorphisms, and in which the channel distribution has a continuous property. In particular, every memory channel has always these properties. The usual memory channels are based upon their message symbols with practical applications. However their symbols may sometimes produce certain troublesome complications for the developments of several mathematical computations of them. The message symbols, in the present construction of the channel, will not be presented, and they will be replaced by sets with the property of the closed-openness (clopen, say). The descriptions will be given only by topological and functional forms, that is, they will be described by topological and Banach spaces methods. The entropy functional H(') (cf. Umegaki [15] and [16]) and the transmission functional 9t( ) are defined over the Banach space of bounded signed regular measures, and they depend upon a clopen partition, or upon a pair of such partitions in the input and output spaces. In § 2, in order to clarify the abstract stationary channel (X, v, Y) defined below, the definition of stationary finite memory channel (A, u, B) will be first stated with respect to the conditions (m l)^(m 5). These conditions will be replaced below by the conditions (C 1)~(C 5) in the channel (X, v, Y), respectively. In § 3, several notations and preliminaries will be given, and in § 4 the stationary channel (X, v, Y)","PeriodicalId":318148,"journal":{"name":"Kodai Mathematical Seminar Reports","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1964-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117052634","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 estimate for semi-linear elliptic differential equations of the second order","authors":"Yoshikazu Hirasawa","doi":"10.2996/KMJ/1138844860","DOIUrl":"https://doi.org/10.2996/KMJ/1138844860","url":null,"abstract":"Concerning the estimate of this sort, Nagumo obtained a result on the assumption that the coefficients aτj(x) satisfy the Lipschitz condition ([4], 2) pp. 211-215, Theorem 2), and thereafter Simoda [5] and the author [3] improved Nagumo's result on the assumption that the coefficients ctij(x) satisfy the Holder condition. As, however, it is desirable from a theoretical point of view, that we have the a priori estimate under the weakest possible condition on the continuity of the coefficients #*/#)» we shall form, in this paper, an a priori estimate of the same type as obtained in the above-cited papers, provided that the coefficients aij(x) satisfy the Dini condition. The Dini condition which we impose on the coefficients aτj(x is more restrictive than usual, but it seems to be considerably general. Our method of proof in this paper is analoguous to one in the previous paper [3], which was composed of Nagumo's one and Cordes' modified results [2]. Therefore, the parts of the proof which can be carried out in the same way as in the previous paper, will often be omitted. In §2, we shall give two definitions concerning Dini functions, and prove two lemmas in regard to the properties of Dini functions given in these two definitions. In § 3, we state the main result of this paper, whose proof is left to § 5. A set of lemmas will be made in §4, and two other results will be proved in §6.","PeriodicalId":318148,"journal":{"name":"Kodai Mathematical Seminar Reports","volume":"269 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1964-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121446104","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":"Certain almost contact hypersurfaces in Euclidean spaces.","authors":"M. Okumura","doi":"10.2996/KMJ/1138844859","DOIUrl":"https://doi.org/10.2996/KMJ/1138844859","url":null,"abstract":"An odd-dimensional differentiate manifold M is said to have an almost contact structure or to be an almost contact manifold if the structural group of its tangent bundle is reducible to the product of a unitary group with the 1-dimensional identity group [3]. Recently Sasaki and Hatakeyama [4, 5] proved that an almost contact structure is equivalent to the existence of a set of tensor fields φ, ξ, -η of the type (1, 1), (1, 0) and (0, 1) satisfying the following five conditions:","PeriodicalId":318148,"journal":{"name":"Kodai Mathematical Seminar Reports","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1964-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129287149","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 theorems on time change and killing of Markov processes","authors":"M. Nagasawa, Ken-iti Sato","doi":"10.2996/KMJ/1138844812","DOIUrl":"https://doi.org/10.2996/KMJ/1138844812","url":null,"abstract":"It is known that a Markov process is transformed to another Markov process by its continuous non-negative additive functional φt through time change or killing. υ On the other hand, φt is determined by an excessive function u(a)=Ma[φζ-o]. Moreover, if the Green measure of the process is expressed by g(a, b)m(db) and if the process satisfies some additional conditions, then u has the Riesz representation: u(ά)=$g(a, b)n(dV) with some measure n. These results are found in the works of Hunt [4], Volkonskii [13] and Meyer [9] under a general setup and in McKeanTanaka [7] in a concrete case. We want to study what meaning the measure n or m has for the process obtained through time change or killing. In the course of the study we need various generalizations of the resolvent equation and we are compelled to give a unified form in their treatments which is given in §2. In §3 we state construction theorems of processes by time change and killing and give some lemmas concerning (sub) invariant measures. Further, it is proved that the terminal measure of the killed process is represented by K (defined in § 2) and that a measure n is the terminal measure of the killed process with initial measure n if and only if n is an invariant measure of the process obtained through time change. In §4, Gλa and K λ a, defined in §2, are represented using a kernel function gϊ(a, b) under some regularity conditions for the Green kernel ga(a, b). In § 5 we prove that the Riesz measure n is a subinvariant measure of the process obtained through time change by the corresponding additive functional and give some sufficient conditions for the measure to be invariant. And also the meanings of n for killed process are discussed. In order to obtain a necessary and sufficient condition for the measure n to be invariant, we need some considerations on the adjoint process of the process obtained through time change or killing, which is given in § 6. The necessary and sufficient condition is stated in §7. The adjoints of the processes are also treated in [4], [8], [11] and [13].","PeriodicalId":318148,"journal":{"name":"Kodai Mathematical Seminar Reports","volume":"253 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":"123610293","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":"Picard's theorem on some Riemann surfaces","authors":"Mitsuru Ozawa","doi":"10.2996/KMJ/1138844816","DOIUrl":"https://doi.org/10.2996/KMJ/1138844816","url":null,"abstract":"In the present paper we shall establish the Picard theorem on some Riemann surfaces with automorphisms. Here we shall adopt a special method based on the Schottky theorem and the most far-reaching method due to Nevanlinna-Selberg. We shall roughly say that a class of meromorphic functions is exceptional if its any member has unreasonably many exceptional values. This nomenclature has no meaning in some cases when we impose the conditions guaranteeing the presence of an essential singularity or some growth conditions. The most important and well-known example of the exceptional class is that of functions of bounded type in z <1. Anyhow it is important to determine and to study the exceptional class in the various cases. In order to investigate and to determine the number of Picard's exceptional values and the exceptional class of functions it is necessary to prove the existence of the fundamental functions in some cases. The functions play an essential role in the respective cases. We shall make free use of the notations in [4], [6] and [7]. Any quantities in [7] and in [4] are distinguished from those in [6] by the subscripts A and P, respectively. In a way we shall give some remarks on the general value distribution theory, especially on the general defect relation.","PeriodicalId":318148,"journal":{"name":"Kodai Mathematical Seminar Reports","volume":"6 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":"131728247","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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