We propose a fast algorithm for computing Jones polynomials of Montesinos links. The Jones polynomial is a useful invariant and Montesinos links are one of the fundamental classes in knot theory. Given the Tait graph of a Montesinos diagram with n edges, our algorithm runs with O(n) additions and multiplications in polynomials of degree O(n), namely in O(n² log n) time.

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In this note, we treat a two-player zero-sum Dynkin game on two stocks driven by geometric Lévy processes, for a given terminal reward cost. Explicit forms for the optimal stopping times and the value of the game are both sought for, under certain conditions. The present note extends a recent result of the author to include a wider class of diffusion processes with jumps. The main result is derived following a decomposition of a stopping game into two standard optimal stopping problems which is due to Yasuda for a standard Brownian motion.

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We introduce a notion of a matched pair of r-discrete groupoids and that of a trinary relation associated with a matched pair. We construct three C^∗-algebras from a trinary relation and study properties of these algebras. The above results are applied to an action of a countable discrete semidirect product group on a topological space with an invariant measure.

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For a finitely generated semigroup S, there exist a finite alphabet X and a surjective homomorphism φ of X⁺ to S. We say that S has a presentation given by X and φ. In this paper, we investigate finitely generated semigroups having a presentation with regular congruence classes or with finite congruence classes.

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We prove that every closed exhaustive vector-valued modular measure on a lattice ordered effect algebra L can be decomposed into the sum of a Lyapunov exhaustive modular measure (i.e. its restriction to every interval of L has convex range) and an ”anti- Lyapunov” exhaustive modular measure. This result extends a Kluvanek-Knowles decomposition theorem for measures on Boolean algebras.

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There is an (immobile) object in a node except for a specified node, with a priori probabilities. A seeker starts at the specified node and examines each node until he finds an object, traveling along edges. Associated with an examination of a node is the examination cost, and associated with a movement from a node to a node is a traveling cost. A strategy for the seeker is an ordering of nodes in which the seeker examines each node. The purpose of the seeker is to find a strategy which minimizes the expected cost. Necessary conditions for a strategy to be optimal are presented. Special cases are solved.

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We consider a class of games which is suggested from a timing problem for putting some kind of farm products on the market. Two players, Player I and II, take possession of the right to put some kind of farm products on the market with even ratio. Each of the players can put the farm products at any time in [0, 1]. The price of them increases over [0,m] ⊂ [0, 1] and decreases over (m, 1] with pass time t so long as both of the players do not sell them, however if one of the two players puts his farm products on the market, the price falls discontinuously and then fluctuates analogously as before. Both players have to put their farm products on the market within the unit interval [0, 1]. In such a situation, each player wishes to put at the optimal time which gives him the highest price, considering opponents action time with each other. This model yields us a certain class of two person non-zero sum infinite games on the unit square.

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