Euclid’s algorithm for computing the greatest common divisor of 2 numbers is considered to be the oldest proper algorithm known ([10]). This algorithm can be amplified naturally in various ways. The GCD problem for more than two numbers is interesting in its own right. Thus, we can use Euclid’s algorithm recursively to compute the GCD of more than two numbers. Also, we can do a constructive com...

PURPOSE To investigate longitudinal changes in goblet cell density (GCD) in contact lens (CL) wearers who do and do not develop symptoms of dry eye (DE). METHODS Sixty healthy individuals fitted with daily disposable hydrogel CLs and 23 age-balanced non-CL-wearing controls underwent assessment using the 5-item dry eye questionnaire, noninvasive tear film break-up time measurement, ocular surf...

We analyze the long time behavior of a discrete time quantum walk subject to decoherence with a strong spatial dependence, acting on one half of the lattice. We show that, except for limiting cases on the decoherence parameter, the quantum walk at late times behaves sub-ballistically, meaning that the characteristic features of the quantum walk are not completely spoiled. Contrarily to expectat...

Extended gcd calculation has a long history and plays an important role in computational number theory and linear algebra. Recent results have shown that finding optimal multipliers in extended gcd calculations is difficult. We present an algorithm which uses lattice basis reduction to produce small integer multipliers x1, . . . , xm for the equation s = gcd (s1, . . . , sm) = x1s1 + · · · + xm...

The paper presents a new fast k-ary reduction for integer GCD. It enjoys powerful properties and improves on the running time of the quite similar integer GCD algorithm of Kannan et al. Our k-ary reduction also improves on Sorenson's k-ary reductionn14] and thus favorably matches We-ber's algorithmm15]. More generally, the fast k-ary reduction also provides a basic tool for almost all the best ...

Let n1 and n2 be two distinct primes with gcd(n1 − 1, n2 − 1) = 4. In this paper, we compute the autocorrelation values of generalized cyclotomic sequence of order 4. Our results show that this sequence can have very good autocorrelation property. We determine the linear complexity and minimal polynomial of the generalized cyclotomic sequence over GF(q) where q = pm and p is an odd prime. Our r...

Let D be an integral domain. A multiplicative set S of D is an almost splitting set if for each 0 6= d ∈ D, there exists an n = n(d) with dn = st where s ∈ S and t is v-coprime to each element of S. An integral domain D is an almost GCD (AGCD) domain if for every x, y ∈ D, there exists a positive integer n = n(x, y) such that xnD ∩ ynD is a principal ideal. We prove that the polynomial ring D[X...

A heuristic algorithm, GCDHEU, is described for polynomial GCD computation over the integers. The algorithm is based on evaluation at a single large integer value (for each variable), integer GCD computation, and a single-point interpolation scheme. Timing comparisons show that this algorithm is very efficient for most univariate problems and it is also the algorithm of choice for many problems...