On Euclid's Algorithm and Elementary Number Theory

Beyond first order logic: From number of structures to structure of numbers: Part II

We study the history and recent developments in nonelementarymodel theory focusing on the framework of abstractelementary classes. We discuss the role of syntax and semanticsand the motivation to generalize first order model theory to nonelementaryframeworks and illuminate the study with concrete examplesof classes of models. This second part continues to study the question of catecoricitytrans...

A New Look at Euclid's Second Proposition

There has been considerable interest during the past 2300 years in comparing different models of geometric computation in terms of their computing power. One of the most well-known results is Mohr's proof in 1672 that all constructions that can be executed with straightedge and compass can be carried out with compass alone. The earliest such proof of the equivalence of models of computation is ...

Euclid's Algorithm, Guass' Elimination and Buchberger's Algorithm

It is known that Euclid’s algorithm, Guass’ elimination and Buchberger’s algorithm play important roles in algorithmic number theory, symbolic computation and cryptography, and even in science and engineering. The aim of this paper is to reveal again the relations of these three algorithms, and, simplify Buchberger’s algorithm without using multivariate division algorithm. We obtain an algorith...

An algorithm for integrated worker assignment, mixed-model two-sided assembly line balancing and bottleneck analysis

This paper addresses a multi-objective mixed-model two-sided assembly line balancing and worker assignment with bottleneck analysis when the task times are dependent on the worker’s skill. This problem is known as NP-hard class, thus, a hybrid cyclic-hierarchical algorithm is presented for solving it. The algorithm is based on Particle Swarm Optimization (PSO) and Theory of Constraints (TOC) an...

On Artin's Conjecture and Euclid's Algorithm in Global Fields