A rack of order n is a binary operation B on a set X of cardinality n, such that right multiplication is an automorphism. More precisely, (X,B) is a rack provided that the map x 7→ x B y is a bijection for all y ∈ X, and (x B y) B z = (x B z) B (y B z) for all x, y, z ∈ X. The paper provides upper and lower bounds of the form 2cn 2 on the number of isomorphism classes of racks of order n. Simil...

By a quasi-permutation matrix we mean a square matrix over the complex field C with non-negative integral trace. Thus every permutation matrix over C is a quasipermutation matrix. For a given finite group G, let p(G) denote the minimal degree of a faithful permutation representation of G (or of a faithful representation of G by permutation matrices), let q(G) denote the minimal degree of a fai...

We develop the technique of reduced word manipulation to give a range of results concerning reduced words and permutations more generally. We prove a broad connection between pattern containment and reduced words, which specializes to our previous work for vexillary permutations. We also analyze general tilings of Elnitsky’s polygon, and demonstrate that these are closely related to the pattern...

Evolutionary algorithms solve problems by simulating the evolution of a population candidate solutions. We focus on evolving permutations for ordering such as traveling salesperson problem (TSP), well assignment quadratic (QAP) and largest common subgraph (LCS). propose cycle mutation, new mutation operator whose inspiration is well-known crossover operator, concept permutation cycle. use fitne...

In this work we present a new approach to crossover operator in the genetic evolution of neural networks. The most widely used evolutionary computation paradigm for neural network evolution is evolutionary programming. This paradigm is usually preferred due to the problems caused by the application of crossover to neural network evolution. However, crossover is the most innovative operator with...

The aim of this paper is to show the influence of genetic operators such as crossover and mutation on the performance of a genetic algorithm (GA). The GA is applied to the multi-objective permutation flowshop problem. To achieve our goal an experimental study of a set of crossover and mutation operators is presented. A measure related to the dominance relations of different non-dominated sets, ...

In this paper, a two-phase hybrid particle swarm optimization algorithm (PRHPSO) is proposed for the permutation flow-shop scheduling problem (PFSP) with the minimizing makespan measure. The smallest position value (SPV) rule is used for encoding the particles that enable PSO for suitable PFSP, and the NEH and Tabu search algorithms are used for initializing the particles. In the first phase, t...

We study de Rham cohomology for various differential calculi on finite groups G up to order 8. These include the permutation group S3, the dihedral group D4 and the quaternion group Q. Poincaré duality holds in every case, and under some assumptions (essentially the existence of a top form) we find that it must hold in general. A short review of the bicovariant (noncommutative) differential cal...