In this article, we introduce the concept of co-covering which is the dual of covering concept. Then, we prove several theorems being similar to the theorems that have been de-veloped for the covering concept. For example, we provide the lifting criterion for co-coverings of a topological space X, which helps us to classify them by subgroups of the group of all homeomorphisms of X.

we give the concept of ‘braiding’ for 2-groupoids, and we show that this structure is equivalent tobraided regular, crossed modules.

We investigate (1,λ) ESs using isotropic mutations for optimization in R by means of a theoretical runtime analysis. In particular, a constant offspring-population size λ will be of interest. We start off by considering an adaptation-less (1,2) ES minimizing a linear function. Subsequently, a piecewise linear function with a jump/cliff is considered, where a (1+λ) ES gets trapped, i. e., (at le...

We show how the use of graded alphabets allows one to provide simpler proofs some results on free monoids and Lie algebras. first generalize characterization length distributions circular codes. also that existence a code with given distribution degrees is equivalent an embedding finally give generalization famous result Eastman comma codes odd degree.