We show that if DTIME[2] is not included in DSPACE[2], then, for every set B in PSPACE, all strings x in B of length n can be represented by a string compressed(x) of length at most log(|B|) + O(logn), such that a polynomial-time algorithm, given compressed(x), can distinguish x from all the other strings in B. Modulo the O(logn) additive term, this achieves the information-theoretical optimum ...

We show that the minimum period modulo p of the Bell exponential integers is (pp−1)/(p−1) for all primes p < 102 and several larger p. Our proof of this result requires the prime factorization of these periods. For some primes p the factoring is aided by an algebraic formula called an Aurifeuillian factorization. We explain how the coefficients of the factors in these formulas may be computed.

Let m be a positive integer, let r be a prime such that 2 is a primitive root modulo r, and let q = 2(r−1)r m−1 . In this note a binomial exponential sum over Fq which assumes 3 2 log2 q + 2 distinct values is explicitly evaluated and its value distribution is determined.

Nowadays, critical and complex systems are widely used in our daily life. Modern technology provides great rapid changes in manufacturing these systems, and so continuous increasing in the complexity of their diagnostic systems. However, many intelligent systems and techniques have been developed in this manner. But, till now there is a great complexity appeared to have good performance for dia...

Let p be an odd prime. Assuming the Extended Riemann Hypothesis, we show how to construct O((log p)4(log log p)−3) residues modulo p, one of which must be a primitive root, in deterministic polynomial time. Granting some well-known character sum bounds, the proof is elementary, leading to an explicit algorithm.

This paper contains the application criteria and coordination process for a proposed overcurrent relay in a radial power system with feed from one or multiple sources. This relay uses independent functions to detect faults and to calculate the operation time. Also this relay uses a time element function that allows it to reduce the time relay operation, enhancing the backup protection. Some of ...

Let E be an elliptic curve defined over Q and with complex multiplication by OK , the ring of integers in an imaginary quadratic field K. Let p be a prime of good reduction for E. It is known that E(Fp) has a structure (1) E(Fp) ' Z/dpZ⊕ Z/epZ with uniquely determined dp|ep. We give an asymptotic formula for the average order of ep over primes p ≤ x of good reduction, with improved error term O...

Abstract. We consider the q-ary digital expansion of the first N terms of an exponential sequence a. Using a result due to Kiss und Tichy [8], we prove that the average number of occurrences of an arbitrary digital block in the last c logN digits is asymptotically equal to the expected value. Under stronger assumptions we get a similar result for the first (logN) 3 2 −2 digits, where 2 is a pos...

In 1989, F. R. K. Chung gave a construction for certain directed h-regular graphs of small diameter. Her construction is based on finite fields, and the upper bound on the diameter of these graphs is derived from bounds for certain very short character sums. Here we present two similar constructions that are based on properties of discrete logarithms and exponential functions in residue rings m...

We will study the solution of a congruence, x ≡ g(1/2)ωg(2 n) mod 2n, depending on the integers g and n, where ωg(2) denotes the order of g modulo 2n. Moreover, we introduce an application of the above result to the study of an estimation of exponential sums.