The lower bound on the number of n-variable balanced symmetric functions over finite fields GF(p) presented in [1] is improved in this paper.

We prove that the range of a symmetric matrix over GF (2) always contains its diagonal. This is best possible in several ways, for example GF (2) cannot be replaced by any other field.

We determine lower bounds for the number of random vectors in GF (q), chosen uniformly from vectors having k non-zero co-ordinates, needed to obtain a dependent set over GF (q).

Let F be a totally real field, and v a place of F dividing an odd prime p. We study the weight part of Serre’s conjecture for continuous totally odd representations ρ : GF → GL2(Fp) that are reducible locally at v. Let W be the set of predicted Serre weights for the semisimplification of ρ|GFv . We prove that when ρ|GFv is generic, the Serre weights inW for which ρ is modular are exactly the on...

A novel Grammatical Genetic Algorithm, the meta-Grammar Genetic Algorithm (mGGA) is presented. The mGGA borrows a grammatical representation and the ideas of modularity and reuse from Genetic Programming, and in particular an evolvable grammar representation from Grammatical Evolution by Grammatical Evolution. We demonstrate its application to a number of benchmark problems where significant pe...