In 1849, Dirichlet [5] proved that the probability that two positive integers are relatively prime is 1/ζ(2). Later, it was generalized into the case that positive integers has no nontrivial kth power common divisor. In this paper, we further generalize this result: the probability that the gcd of m products of n positive integers is B-smooth is ∏ p>B [ 1 − { 1 − ( 1 − 1 p )n}m] for m ≥ 2. We s...

Abstract. Let a, b be given multiplicatively independent positive integers and let ǫ > 0. In a recent paper written jointly also with Y. Bugeaud we proved the upper bound exp(ǫn) for gcd(a − 1, b − 1); shortly afterwards we generalized this to the estimate gcd(u− 1, v − 1) < max(|u|, |v|), for multiplicatively independent S-units u, v ∈ Z. In a subsequent analysis of those results it turned out...

PURPOSE We analyzed the association of granule cell dispersion (GCD) with memory performance, clinical data and surgical outcome in a series of patients with mesial temporal lobe epilepsy (MTLE) and mesial temporal sclerosis (MTS). METHOD Hippocampal specimens from 54 patients with MTLE (27 patients with right MTLE and 27 with left MTLE) and unilateral MTS, who were separated into GCD and no-...

We present the w constraint combinator that models while loops in Constraint Programming. Embedded in a finite domain constraint solver, it allows programmers to develop non-trivial arithmetical relations using loops, exactly as in an imperative language style. The deduction capabilities of this combinator come from abstract interpretation over the polyhedra abstract domain. This combinator has...

Conflict-directed search algorithms have formed the core of practical, model-based reasoning systems for the last three decades. At the core of many of these applications is a series of discrete constraint optimization problems and a conflict-directed search algorithm, which uses conflicts in the forward search step to focus search away from known infeasibilities and towards the optimal feasibl...

We study the distribution of generalized gcd and lcm functions on average. The function, denoted by [Formula: see text], is greatest text]th power divisor common to text] text]. Likewise, smallest multiple derive asymptotic formulas for average order arithmetic, geometric, harmonic means Additionally, we also deduce with error terms over a set lattice points, thereby generalizing some previous ...

An integral domain without irreducible elements is called an antimatter domain. We give some monoid domain constructions of antimatter domains. Among other things, we show that if D is a GCD domain with quotient …eld K that is algebraically closed, real closed, or perfect of characteristic p > 0, then the monoid domain D[X;Q+] is an antimatter GCD domain. We also show that a GCD domain D is ant...

In this paper we discuss two partially ordered sets and give a connection between max-min identity and GCD-LCM one.