The purpose of this note is to extend Fisher’s exact test in 2 × 2 table to k × l table. Also a simple numerical example is given in 2 × 3 table.

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We consider a class of games which is suggested from the timing problems for putting some kind of farm products on the market. N players take possession of the rights to put some kind of farm products on the market with even ratio. Each of the n 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 as long as none of the n players takes his action, however if one of the n players put his farm products on the market the price falls discontinuously and then fluctuates analogously as before. All 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 n person non-zero sum infinite games.

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This paper is concerned with a problem on optimization in stock price model that allows for jumps in the stochastic processes. We show that the problem is formulated into the optimal stochastic control problem under the criterion of mean square tracking error at the end of the planning horizon. The optimal control and the optimal proportion invested in the index fund which are specified by the product of the market value of risk and the discounted terminal value of bench mark portfolio are derived by the solution to ordinary differential equations associated with an adjoint equation.

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We consider the following transportation problem with both random and fuzzy factors. There exist m supply points and n demand points. For each route between supply point and demand point, unit transportation cost is a random variable according to a normal distribution and existence possibility denoting the preference choosing this route is attached. The probability that the total transportation cost is not greater than the budget F should be not less than the fixed probability level. Under the above setting, we seek transportation pattern minimizing F and maximizing the minimal preference among the routes used in a transportation. Since usually there is no transportation pattern optimizing two objectives at a time, we propose a solution algorithm to find some non-dominated transportation patterns after defining non-domination. Finally we discuss the further research problems.

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We propose a stochastic model based on a continuous-time Markov chain to describe certain features of the displacements performed by single myosin heads along actin filaments during the chemical cycle of an ATP molecule. We obtain an expression of the transient probabilities of the model and determine a continuous approximation of the chain by a Wiener-type diffusion process subject to jumps and restricted between two reflecting boundaries.

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We would like to show a precise review of a part of the work [2]. As materials, we take up Rearrangement, Neumann Problem, Equilibrium Equation, and Chemotaxis System. We also explain the outline of the Sobolev embedding described in the work [1]. Finally we investigate the relation between the Rearrangement and Conditionally Convergent Integrals.

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We study Fredholmness of nonlocal boundary value problems for fourth order elliptic differential-operator equations in UMD Banach spaces. The main condition is given in terms of R-boundedness of some families of bounded operators generated by inverse of the characteristic operator pencil of the equation. Then we prove an isomorphism of the problem on the semi-axis, for some special boundary conditions, in appropriate L_p spaces. This implies maximal L_p-regularity for the problem. We also present some relevant application of obtained abstract results to boundary value problems for fourth order elliptic and quasi-elliptic partial differential equations.

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In [12], Mimura and Tsujikawa presented a sophisticated model for describing the process of pattern formation performed by bacteria. This paper is concerned with numerical simulations for their model. We find out various types of stationary solutions which show good correlation with experimental results due to Budrene-Berg [5, 6].

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We introduce a new definition of nonlinear mapping, which is called a quasi-pseudocontractive type mapping, defined on a nonempty closed convex subset of a real Hilbert space, and we investigate convexity of the set of its fixed points and approximation of its fixed point. We consider an iterative sequence generated by the improved hybrid method, also known as the shrinking projection method, and prove that it converges strongly to a fixed point under some conditions of the constants for quasi-pseudocontractive type Lipschitz mapping.

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The aim of this paper is to present ten research problems of the theory of Gronwall lemmas. These problems are in relation with: concrete Gronwall lemmas, abstract Gronwall lemmas, techniques of proof in concrete Gronwall lemmas, Picard and weakly Picard operator theory, operatorial inequations, differential inequalities, integral inequalities and applications of Gronwall lemmas.

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We prove the following theorem (8): If X is a nowhere hereditarily disconnected homogeneous space metrizable by a complete metric, and X is cleavable over R along every punctured closed connected subset, then X is locally connected. Using this result, we establish the next theorem (Theorem 15): Suppose that X is an infinite homogeneous connected locally compact metrizable space. Suppose also that X is cleavable over R along every punctured closed connected subset. Then X is homeomorphic to the space R of real numbers.

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In this paper we consider hyperalgebras and hyper-coalgebras of type τ , as an analogue of algebras of type τ and coalgebras of type τ (see [10]). Hyperalgebras and hyper-coalgebras are special cases of (F₁, F₂)-systems. Therefore many of the results in this paper will be instances of (F₁, F₂)-system results (see [8]). Nevertheless it is interesting and instructive to consider directly the theory of hyperalgebras and hyper-coalgebras of type τ . The aim of this paper is to create a unified theory of hyperalgebras and hyper-coalgebras which is based on (F₁, F₂)-systems.

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