Pseudo-periodic homeomorphisms and degeneration of Riemann surfaces

We will announce two theorems. The first theorem will classify all topological types of degenerate fibers appearing in one-parameter families of Riemann surfaces, in terms of ``pseudoperiodic'' surface homeomorphisms. The second theorem will give a complete set of conjugacy invariants for the mapping classes of such homeomorphisms. This latter result implies that Nielsen's set of invariants [{\it Surface transformation classes of algebraically finite type}, Collected Papers 2, Birkh\"auser (1986)] is not complete.Comment: 6 page

Word representation of cords on a punctured plane

AbstractIn this paper a purely algebraic condition for a word in a free group to be representable by a simple curve on a punctured plane will be given.As an application, an algorithm for simple closed curves on a punctured plane will be obtained. Our solution is different from any algorithm due to Reinhart [Ann. of Math. 75 (1962) 209], Zieschang [Math. Scand. 17 (1965) 17] or Chillingworth [Bull. London Math. Soc. 1 (1969) 310]. Although the study here will be confined to the case of a plane, similar arguments could be carried out on the 2-sphere. This research was motivated by monodromy problems appearing in Lefschetz fibrations and surface braids. See [Math. Proc. Cambridge Philos. Soc. 120 (1996) 237; Kamada, Braid and Knots Theory in Dimension Four, American Mathematical Society, 2002; Kamada and Matsumoto, in: Proceedings of the International Conference on Knot Theory “Knots in Hellas '98”, World Scientific, 2000, p. 118; Kamada and Matsumoto, Enveloping monoidal quandles, Preprint, 2002; Matsumoto, in: S. Kojima et al. (Eds.), Proc. the 37th Taniguchi Sympos., World Scientific, 1996, p. 123]

A study on pseudo-hypersthene

It should be reported here about the pseudo-hypersthene and honeycombed clinopyroxene found in the basalt lava from Saga Prefecture, northern Kyushu. The results of the petrological study of these minrals are summarrized in the following studies, that is to say, both the relict orthopyroxene in the central part of pseudohypersthene and honeycombed clinopyroxene are as xenocryst. Namely, the orthopyroxene in this pseudo-hypersthene and honeycombed clinopyroxene may be considered to have been crystallized under the plutonic condition than the volcanic condition, and to have originally constituted gabbroic rock or basic nodule

Application of Wald Function to OR and AND Fuzzy Operations in No- Data Problems

Subjective qualifiers of Wald's theory of decision functions are the fuzzy events of the subsequent fuzzy set theory. Wald's notion of subjective qualifiers involves applying integral transforms to convert states of nature into fuzzy events. Probabilities of fuzzy events and arithmetic formulas for fuzzy utility function values are readily derived from Wald's integral transforms. We have applied Zadeh's extension principle to Wald's integral transforms and demonstrated that fuzzy mathematics is effective when applied to multiple subjective probability distributions conjoined by OR and AND operations. In this paper, we focus on no-data problems and construct a fuzzy Bayes' theorem for cases in which a membership function and multiple subjective probability distributions conjoined by OR or AND operations are given. In addition, we devise a formulation for the corresponding decision making problem