In this paper, some classes of arithmetic circuits are introduced that capture the computational complexity of computing the determinant of matrices with entries either indeterminates or constants from a field. An arithmetic circuit is just like a Boolean circuit, except that all AND and OR gates (with fan-in two) are replaced by gates realizing a multiplication and an addition, respectively, of two polynomials over some indeterminates with coefficients from the field, and the circuit computes a (formal multivariate) polynomial in the obvious sense. An arithmetic circuit is said to be skew if at least one of the inputs of each multiplication gate is either an indeterminate or a constant. Then it is shown that for all square matrices M of dimension q, the determinant of M can be computed by a skew arithmetic circuit of (q20) gates, and is shown that for all skew arithmetic circuits C of size q, the polynomial computed by C can be defined as the determinant of a square matrix M of dimension (q). Thus the size of skew arithmetic circuit is polynomially related to the dimension of square matrices when it is considered to represent multivariate polynomials in both arithmetic circuits and the determinant. The results are extended to some other classes of arithmetic circuits less restricted than skew ones, and by using such an extended result, a difference and a similarity are demonstrated between polynomials represented as the determinant of matrix of relatively small dimension and those polynomials computed by arithmetic formulas and arithmetic circuits of relatively small size and degree.

This paper presents an approximation method for deriving the availability of a parallel redundant system with preventive maintenance (PM) and common-cause failures. The system discussed is composed of two identical units. A single service facility is available for PM and repair. The repair times, the PM times and the failure times except for common-cause failures are all assumed to be arbitrarily distributed. The presented method formulates the problem of the availability analysis of a parallel redundant system as a Markov renewal process which represents the state transitions of one specified unit in the system. This method derives the availability easily and accurately. Further, the availability obtained by this method is exact in a special case.

Based on the semiclassical theory, we deduce the expressions of stimulated absorption, stimulated amplification and threshold by using density matrix equation in the Er3+-doped fibers. Meaningful results have been given and some phenomena occuring in experiments are explained theoretically.