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...

Input is considered one of the most important factors in acquisition lexical and grammatical skills. has been found to interact with other factors, such as learner cognitive skills circumstances where language heard. Language learning itself sometimes enhance Indeed, intensive contact another boost skills, even instructed settings, immersion programs (bilingual advantage hypothesis). In this pa...

The purpose of the present study is to analyse abstracts related to Applied Linguistics, and more precisely the discourse functions of grammatical subjects and verbs. The corpus consisted of 50 PhD thesis abstracts written on the subject of Applied Linguistics. All of the abstracts were written from 2010 to 2014. The theses from which the abstracts were extracted are available in the ProQuest d...

In this note, it is proved that the necessary and sufficient conditions for the existence of a balanced (q, {3, 4}, λ)-DF over GF (q) are q ≥ 4 and q ≡ 1 (mod 18 gcd(18,λ) ).

We wish to obtain information about BP̃LF ; in fact we can study BGF (which is purely homotopy theoretic) and the fibre GF /P̃LF of χ. In Chapter III we construct a map θ : GF /P̃LF−−→(G/PL) , where G/PL is the space studied in Sullivan’s thesis under the name F/PL, and (G/PL) is the space of all unbased maps from F to G/PL. Theorems 5,6 show that, under suitable conditions, θ is almost a homotopy...

Let R be a right GF -closed ring with finite left and right Gorenstein global dimension. We prove that if I is an ideal of R such that R/I is a semi-simple ring, then the Gorenstein flat dimension of R/I as a right R-module and the Gorenstein injective dimension of R/I as a left R-module are identical. In particular, we show that for a simple module S over a commutative Gorenstein ring R, the G...