Abstract. We consider the initial value problem for the evolution equations defined by max operators. Though the complexity of expression of general solutions is usually of exponential order, there exist some equations of which order can be reduced to polynomial one using simplification of solution expression. We consider importing an external parameter to the max equations or constructing a system of the max equations. Some concrete examples are reported of which order of solution complexity can be preserved as polynomial.

Abstract. Pair trading is an investment strategy to earn a pro t by taking a position on two assets whose spread (i.e., prices' difference) is expected to converge to a certain level. In this paper, we propose a simple but new formulation for portfolio selection of assets' pairs on the basis of the rst passage time (FPT) of the processes of spreads. We observed that the computed efficient frontier in terms of FPT almost corresponds to the return-based efficient frontier. Besides, obtained portfolios had practically favorable features compared to the conventional single-pair trading strategy.

Abstract. This paper proposes a new statistical testing framework for random number generators. First, we show that the evaluation of “generators” using NIST SP.800-22, which is widely used today, is incomplete. Next, we show that the statistical evaluation of the generator can be interpreted as a “random sampling” evaluation and give some examples of random-sampling-tests. After that, under the assumption that the test will be repeated many times, this paper proposes a new framework of statistical testing for generators where the testing procedure and final judgement rules are clearly defined. Finally, we show some examples using the proposed framework.

Abstract. This article proves that each solution of Yamada’s partial differential equation and thermal equation gives normal vector filed of shapes. First, each problem setting is defined and numerical solutions of the equations are provided using the Finite Element Method. Next, each theorem is given and proved.

Abstract. We propose an algorithm to transform expressions constructed from the operators max, min and negation − into those in a standard form, and implement the algorithm using C programming language. We also apply the program for the initial value problems of evolution equations described by the expressions and discuss the computational efficiency.