Víctor Álvarez

The homological reduction method for computing cocyclic Hadamard matrices

An alternate method for constructing (Hadamard) cocyclic matrices over a finite group G is described. Provided that a homological model for G is known, the homological reduction method automatically generates a full basis for 2-cocycles over G (including 2-coboundaries). From these data, either an exhaustive or a heuristic search for Hadamard cocyclic matrices is then developed. The knowledge of an explicit basis for 2-cocycles which includes 2-coboundaries is a key point for the designing of the heuristic search. It is worth noting that some Hadamard cocyclic matrices have been obtained over groups G for which the exhaustive searching techniques are not feasible. From the computational-cost point of view, even in the case that the calculation of such a homological model is also included, comparison with other methods in the literature shows that the homological reduction method drastically reduces the required computing time of the operations involved, so that even exhaustive searches succeeded at orders for which previous calculations could not be completed. With aid of an implementation of the method in MATHEMATICA, some examples are discussed, including the case of very well-known groups (finite abelian groups, dihedral groups) for clarity.

A genetic algorithm for cocyclic hadamard matrices

A genetic algorithm for finding cocyclic Hadamard matrices is described. Though we focus on the case of dihedral groups, the algorithm may be easily extended to cover any group. Some executions and examples are also included, with aid of Mathematica 4.0.

Necesidades del profesorado de la Universidad de Sevilla respecto a la enseñanza que requiere el EEES

Resumen En el presente artículo exponemos una parte de los resultados de una investigación llevada a cabo por el ICE de la Universidad de Sevilla. Una de las metas que se perseguían con esa pesquisa era determinar las principales necesidades de formación y apoyo institucional, percibidas por los docentes de la Universidad Hispalense, respecto a las demandas de cambios metodológicos en la docencia contenidas en los principios programáticos del Espacio Europeo de Educación Superior (EEES). Utilizando procedimientos de corte cualitativo (grupos de discusión) y cuantitativo (encuesta) se obtuvieron las opiniones de una amplia muestra del profesorado de las cinco áreas de conocimiento de la universidad sobre carencias percibidas respecto a los cuatro ámbitos básicos de la actuación docente: planificación de la docencia, desarrollo de la enseñanza, evaluación y tutoría de los aprendizajes. Los datos obtenidos señalan dificultades (y necesidades de formación) relacionadas primordialmente con el manejo de las competencias profesionales a enseñar en el nuevo contexto: identificación de las mismas, selección adecuada de contenidos/temarios necesarios para su adquisición y conocimiento/manejo de técnicas de evaluación de competencias.

Calculating cocyclic hadamard matrices in mathematica : exhaustive and heuristic searches

We describe a notebook in Mathematica which, taking as input data a homological model for a finite group G of order |G| = 4t, performs an exhaustive search for constructing the whole set of cocyclic Hadamard matrices over G. Since such an exhaustive search is not practical for orders 4t ≥28, the program also provides an alternate method, in which an heuristic search (in terms of a genetic algorithm) is performed. We include some executions and examples.

A mathematica notebook for computing the homology of iterated products of groups

Let G be a group which admits the structure of an iterated product of central extensions and semidirect products of abelian groups Gi (both finite and infinite). We describe a Mathematica 4.0 notebook for computing the homology of G, in terms of some homological models for the factor groups Gi and the products involved. Computational results provided by our program have allowed the simplification of some of the formulae involved in the calculation of Hn(G). Consequently the efficiency of the method has been improved as well. We include some executions and examples.

Determinants of (–1,1)-matrices of the skew-symmetric type: a cocyclic approach

Abstract An n by n skew-symmetric type (-1; 1)-matrix K =[ki;j ] has 1’s on the main diagonal and ±1’s elsewhere with ki;j =-kj;i . The largest possible determinant of such a matrix K is an interesting problem. The literature is extensive for n ≡ 0 mod 4 (skew-Hadamard matrices), but for n ≡ 2 mod 4 there are few results known for this question. In this paper we approach this problem constructing cocyclic matrices over the dihedral group of 2t elements, for t odd, which are equivalent to (-1; 1)-matrices of skew type. Some explicit calculations have been done up to t =11. To our knowledge, the upper bounds on the maximal determinant in orders 18 and 22 have been improved.

Rooted Trees Searching for Cocyclic Hadamard Matrices over D4t

A new reduction on the size of the search space for cocyclic Hadamard matrices over dihedral groups D 4t is described, in terms of the so called central distribution . This new search space adopt the form of a forest consisting of two rooted trees (the vertices representing subsets of coboundaries) which contains all cocyclic Hadamard matrices satisfying the constraining condition. Experimental calculations indicate that the ratio between the number of constrained cocyclic Hadamard matrices and the size of the constrained search space is greater than the usual ratio.

An Algorithm for Computing Cocyclic Matrices Developed over Some Semidirect Products

An algorithm for calculating a set of generators of representative 2-cocycles on semidirect product offinite abelian groups is constructed, in light of the theory over cocyclic matrices developed by Horadam and de Launey in [7,8]. The method involves some homological perturbation techniques [3,1], in the homological correspondent to the work which Grabmeier and Lambe described in [12] from the viewpoint of cohomology. Examples of explicit computations over all dihedral groups D4t are given, with aid of MATHEMATICA.

Embedding cocylic D-optimal designs in cocylic Hadamard matrices

A method for embedding cocyclic submatrices with “large” determinants of orders 2t in certain cocyclic Hadamard matrices of orders 4t is described (t an odd integer). If these determinants attain the largest possible value, we are embedding D-optimal designs. Applications to the pivot values that appear when Gaussian elimination with complete pivoting is performed on these cocyclic Hadamard matrices are studied.

Algebra Structures on the Comparison of the Reduced Bar Construction and the Reduced W-Construction

For a simplicial augmented algebra K, Eilenberg–Mac Lane constructed a chain map . They proved that g is a reduction (homology isomorphism) and conjectured that it is also the injection of a contraction (special homotopy equivalence). The contraction is followed at once by using homological perturbation techniques. If K is commutative, Eilenberg–Mac Lane proved that g is a morphism of DGA-algebras. The present article is devoted to proving that f and φ satisfy certain multiplicative properties (weaker than g) and showing how they can be used for computing in an economical way the homology of twisted cartesian products of two Eilenberg–Mac Lane spaces.