Let n denote a positive integer relatively prime to 10. Let the period of 1/n be a · b with b > 1. Break the repeating block of a · b digits up into b sub blocks, each of length a, and let B(n, a, b) denote the sum of these b blocks. In 1836, E. Midy proved that if p is a prime greater than 5, and the period of 1/p is 2 · a, then B(p, a, 2) = 10 − 1. In 2004, B. Ginsberg [2] showed that if p is...

New nonlattice sphere packings in dimensions 20, 22, and 44–47 that are denser than the best previously known sphere packings were recently discovered. We extend these results, showing that the density of many sphere packings in dimensions just below a power of 2 can be doubled using orthogonal binary codes. This produces new dense sphere packings in Rn for n = 25, 26, . . . , 31 and 55, 56, . ...

We introduce a fast algorithm for determining the linear complexity and the minimal polynomial of a sequence with period p over GF(q) , where p is an odd prime, q is a prime and a primitive root modulo p; and its two generalized algorithms. One is the algorithm for determining the linear complexity and the minimal polynomial of a sequence with period pq over GF(q), the other is the algorithm fo...

We study the sequences generated by neuronal recurrence equations of the form x(n) = 1[ ∑h j=1 ajx(n − j) − θ]. From a neuronal recurrence equation of memory size h which describes a cycle of length ρ(m)× lcm(p0, p1, . . . , p−1+ρ(m)), we construct a set of ρ(m) neuronal recurrence equations whose dynamics describe respectively the transient of length O(ρ(m)×lcm(p0, . . . , pd)) and the cycle o...

where a, ai, a2> ■ ■ ■ , a* are given rational integers. The purpose of this paper is to investigate the periodicity of such sequences with respect to a rational integral modulus m. Carmichaelî has studied the period for a modulus m whose prime divisors exceed k and are prime to ak. In this paper, I give a solution to the problem without restriction on m. If m is prime to ak the sequence (1) is...