We investigate the Sandpile Model and Chip Firing Game and an extension of these modelson cycle graphs. The extended model consists of allowing a negative number of chips ateach vertex. We give the characterization of reachable configurations and of fixed pointsof each model. At the end, we give explicit formula for the number of their fixedpoints.

Diel rhythms in feeding and spawning were investigated in Centropages tenuiremis from Xiamen Bay in March to May, 2006. Circular statistics were used to determine the peak time of spawning. The results showed that the feeding activities of females were stably higher at night-time, and there was a remarkable earlier shift in spawning peak time with warmer seawater. Thus, the lag times between peak times of gut pigment content and spawning were shortened with the increase of temperature. It suggested that there was a direct effect of feeding rhythms on egg production variations in copepods, and the seawater temperature would work on the converting time and then influence the spawning peak time. So the effect of temperature cannot be ignored in the investigation of the effects of feeding on egg production.

The purpose of this paper is to investigate a conjecture about the universality of the circular distribution made by Robert Coleman. The algebraic property of the universal distribution is the main ingredient in studying Euler system of Kolyvagin and Rubin. We study the universality of the circular distribution by using the Iwasawa theory and the theory of the Euler systems. The conjecture is a characterization of Euler systems in the case of number field. The results here assert that Euler systems are essentially made out of cyclotomic units.

A survey is made of the mathematical properties of, and the arithmetic relationships between, various distributions on the circle and the sphere. The Brownian motion and angular Gaussian distributions are shown in computer-drawn graphs to bracket the von Mises–Fisher distributions.