We investigate a simple mechanical model in which a particle is projected from a platform moving horizontally with constant speed. This projection model has three variables and includes the constraint between the initial speed and initial angle appeared in the projection with run-up effect in sports. Under the initial condition given by this model, we derive analytically the projection angle at which the flight distance is maximized. The result shows that the present mechanism based on the elementary mechanics explains successfully the well-known angles observed in actual situations.

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The approximated normal random variable which has similar moment properties to the normal random variable is used in the weak approximation of stochastic differential equations. In this paper we describe three types of the approximated normal random variable, namely multi-point distribution, rectangular approximation and polygonal line approximation. And asymptotic stability of the Euler-Maruyama simplified scheme for the approximated normal random variable is studied. We also present some numerical results which demonstrate the stability properties.

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Cyclotomic fields and the relative class numbers have been recognized from the 19th century as important number theorical objects. The theory of cyclotomic fields was developed into a class field theory and Iwasawa theory in the 20th century and later. In this paper, concerning the present conditions of the calculation of relative class numbers, we will summarize the range, technique (determinant, modular arithtmetic, FFT over finite fields, 2-dimentional FFT), computational complexity, and state the future prospects.

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