Analytic Functions on Some Riemann Surfaces

Matsumoto, Kikuji

doi:10.1017/s0027763000011119

Cited by 4 publications

(5 citation statements)

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“…Concerning our classes, we remark that the following relations have been obtained: OabC2Ib [6], 0°AB C£>b [4], [5]. I want to thank Professor Oikawa for the stimulating discussions which we have had during the preparation of the present paper.…”

Section: Several Years Ago Kuramochimentioning

confidence: 75%

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Meromorphic functions on certain Riemann surfaces

Ozawa¹

1965

Proc. Amer. Math. Soc.

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Section: Several Years Ago Kuramochimentioning

confidence: 75%

“…Several authors have generalized this elegant result in various ways. In particular Toda and Matsumoto [6] obtained the following generalization:…”

Section: Several Years Ago Kuramochimentioning

confidence: 99%

Meromorphic functions on certain Riemann surfaces

Ozawa¹

1965

Proc. Amer. Math. Soc.

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“…To show that F is a desired Riemann surface. By the same reasoning in [12], F has only one ideal boundary component with positive harmonic measure.…”

Section: Dxdymentioning

confidence: 87%

“…PROPOSITION Construction of the example. Our example can be obtained if we take the slits sufficiently small in the example given in [12], successive ratios ?«, 0 <ζ n = 2^ < 1. Then i?…”

Section: Dxdymentioning

confidence: 99%

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Analytic Functions on Some Riemann Surfaces, II

Matsumoto

1963

Nagoya Mathematical Journal

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To Professor Kinjiro Kunugi on the occasion of his 60th birthday 1. In their paper [12], Toda and the author have concerned themselves in the following THEOREM OF KURAMOCHI. Let R be a hyperbolic Riemann surface of the class OHB(OHD, resp.). Then, for any compact subset K of R such that R-K is connected, R-K as an open Riemann surface belongs to the class O A B(O AΌ ,resp.) (Kuramochi [4]).They have raised there the question as to whether there exists a hyperbolic Riemann surface, which has no Martin or Royden boundary point with positive harmonic measure and has yet the same property as stated in Theorem of Kuramochi, and given a positive answer to the Martin part of this question.The main purpose of this paper is to show that the Royden part is also answered in the positive. In the sequel, we shall investigate covering properties of analytic functions on Riemann surfaces of the class O ADy which was introduced by Kuroda in his paper [6], give an extension of the D part of Theorem of Kuramochi and, using this extension, give an example of a Riemann surface which answers the Royden part of the question in the positive.2. Let R be a Riemann surface and let G be a domain on R with smooth relative boundary dG clustering nowhere in R. For simplicity, we shall call such a domain G a subregion of R. If G admits no non-constant single-valued analytic function with a finite Dirichlet integral and with real part vanishing continuously on its relative boundary 'dG, we say that G belongs to the class SO AD .Let w = f(p) be a non-constant single-valued analytic function in a subregion G with relative boundary dG of a Riemann surface R. We suppose that Received May 10, 1963. Therefore at any ideal boundary point P of R, the cluster set Cκ-κ(f, P)can not be total and hence, by Theorem 2, it reduces to one point, .since the finiteness of the spherical area is derived from the finiteness of the Dirichlet integral. Our proof is now complete. con,m(p) increases as ra-* co, so that ω n .m(P) tends to a harmonic function ω n {p). Now let w tend to infinity. Then ω n (p) decreases and tends to a non-R -Ro, and we say that the harmonic measure of an ideal boundary point P of R is zero or positive according as the first or the second case occurs, respectively.Of course this property of P does not depend on the choice of the exhaustion {R n } of R.Now we have an extension of Theorem of Kuramochi stated in § 1. such that each boundary component of the relative boundary of A rounds the hole of (Ci, C 2 ) that is, According to Nakai, we say that a Riemann surface R is an almost finite Riemann surface or that R is of almost finite genus, if there exists a sequence {An} of annuli in R satisfying the following condition:(5) An^Hn, where {H n } is the totality of handles in R, Further we denote by γ n the closed Jordan curve in A n which divides A n into two annuli A n ,i and A n ,2 such that mod i4«,i = mod A n ,2 = mod Aj2.Nakai's theorem can be derived from the following four propositions. PROPOSITION 4. Let u be an HP-minimal f...

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Inclusion Compounds

Weber¹

2000

Kirk-Othmer Encyclopedia of Chemical Technology

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Inclusion compounds may be classified into three main categories: complexes, cavitates, or clathrates. Typical examples for each class of inclusion compounds are the crown complexes, the calix–cavitates, and the hydroquinone clathrates. All of these compounds have in common the ability to incorporate into the cavities of their own molecules, or within their lattices, other molecules of suitable size, spatially to enfold them, that is to hold them, mostly by physical imprisonment. Materials having the attribute of inclusion have given rise to what is called host–guest chemistry, a typical feature of which is accommodation of a complementary guest species into a concave host framework involving a molecular recognition process such as imaged by the complementarity of a lock and a key. Logically, a new broad area of chemistry grew up on this basis starting in the 1970s. It is called supramolecular chemistry and means the chemistry beyond the molecule. This article surveys the various types of inclusion compounds and their applications.

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Analytic Functions on Some Riemann Surfaces

Cited by 4 publications

References 7 publications

Meromorphic functions on certain Riemann surfaces

Meromorphic functions on certain Riemann surfaces

Analytic Functions on Some Riemann Surfaces, II

Inclusion Compounds

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