• Greatest common divisor (GCD) algorithms. We begin with Euclid’s algorithm, and the extended Euclidean algorithm [2, 12]. We will then discuss variations and improvements such as Lehmer’s algorithm [14], the binary algorithms [12], generalized binary algorithms [20], and FFT-based methods. We will also discuss how to adapt GCD algorithms to compute modular inverses and to compute the Jacobi a...

We survey properties of the gcd-sum function and of its analogs. As new results, we establish asymptotic formulae with remainder terms for the quadratic moment and the reciprocal of the gcd-sum function and for the function defined by the harmonic mean of the gcd’s.

The present paper analyses and presents several improvements to the algorithm for finding the (a, b)-pairs of integers used in the k-ary reduction of the right-shift k-ary integer GCD algorithm. While the worst-case complexity of Weber’s “Accelerated integer GCD algorithm” is O (

In this paper we perform a further investigation for r-gcd-sum and r-lcm-sum functions. By making use of the properties generalization Euler’s φ-function, Abel’s identity elementary arguments, derive asymptotic formulas average function, function their generalizations. Moreover, also study sums reciprocals r-gcd r-lcm.

We present a phylogeographic study of at least six reproductively isolated lineages of new world harvester ants within the Pogonomyrmex barbatus and P. rugosus species group. The genetic and geographic relationships within this clade are complex: Four of the identified lineages show genetic caste determination (GCD) and are divided into two pairs. Each pair has evolved under a mutualistic syste...

Purpose To examine the preoperative conjunctival dendritic cell density (DCD), goblet cell density (GCD), and stromal meshwork reflectivity (SMR) in glaucomatous patients undergoing filtration surgery, using in vivo confocal microscopy (IVCM). Methods Sixty-six patients were enrolled. At baseline, IVCM was performed at the site planned for surgery, and was repeated after 12 months at bleb sit...

Granule cell dispersion (GCD) represents a pathological widening of the granule cell layer in the dentate gyrus and it is frequently observed in patients with mesial temporal lobe epilepsy (MTLE). Recent studies in human MTLE specimens and in animal epilepsy models have shown that a decreased expression and functional inactivation of the extracellular matrix protein Reelin correlates with GCD f...

Greatest common divisors and least common multiples of quotients of elements of integral domains have been investigated by Lüneburg and further by Jäger. In this paper we extend these results to invertible fractional ideals. We also lift our earlier study of the greatest common divisor and least common multiple of finitely generated faithful multiplication ideals to finitely generated projectiv...