The Fock quantization of free fields propagating in cosmological backgrounds is in general not unambiguously defined due to the non-stationarity of the spacetime. For the case of a scalar field in cosmological scenarios it is known that the criterion of unitary implementation of the dynamics serves to remove the ambiguity in the choice of Fock representation (up to unitary equivalence). Here, applying the same type of arguments and methods previously used for the scalar field case, we discuss the issue of the uniqueness of the Fock quantization of the Dirac field in the closed FRW spacetime proposed by D'Eath and Halliwell. * jacq@ciencias.unam.mx † ilda@ubi.pt ‡ mmartinb@fc.ul.pt § jvelhi@ubi.pt 2

Abstract:We describe subspaces of generalized Hessenberg matrices where the determinant is convertible into the permanent by affixing ± signs. An explicit characterization of convertible Hessenberg-type matrices is presented. We conclude that convertible matrices with the maximum number of nonzero entries can be reduced to a basic set.

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Bicontactual regular maps (regular maps with the property that each face meets only two others) were introduced and classified by Wilson in 1985. This property generalizes to hypermaps (cellular embeddings of hypergraphs in compact surfaces) giving rise to three types of bicontactuality, namely, the edge-twin, the vertex-twin, and the alternate (the first two of which are the dual of each other). In 2003 Wilson and Breda classified (up to an isomorphism and duality) the bicontactual regular nonorientable hypermaps, leaving the orientable case open. In this paper we classify the bicontactual regular oriented hypermaps. Introduction.A map is a cellular embedding of a graph in a compact surface. A hypermap is a generalization of a map obtained by allowing edges to connect more than two vertices; in other words, a hypermap is a "cellular embedding" of a hypergraph (or rather, the bipartite graph whose white vertices, black vertices, and edges are the hyperedges, hypervertices, and hypervertex-hyperedge incidence pairs, respectively, of the hypergraph) G in a compact and connected surface S. A hypermap is orientable if the underlying surface S has no boundary and is orientable; is nonorientable if S has no boundary and is nonorientable; and with boundary if S has boundary and the underlying bipartite graph G has no vertices lying on the boundary. In what follows is a brief introduction to hypermaps. For a further and deeper reading on maps we refer the reader to [1, 6, 13] and on hypermaps to [8,10,14]. The notation used in this paper follows that in [2,4,5].An oriented hypermap is an orientable hypermap with a fixed orientation; algebraically it is described by a triple Q = (D; R, L) consisting of a finite set of "abstract" darts D and two permutations R, L of D that generate a transitive group M on(Q) = R, L on D, called the monodromy group of Q; the orbits of R, L, and RL are the hypervertices, hyperedges, and hyperfaces, respectively. Maps are hypermaps satisfying L 2 = 1 1 and when a hypermap is not a map (L 2 = 1) it is called a proper hypermap. Since hypergraphs can be represented by bipartite (multi)graphs, hypermaps H will be viewed as bipartite maps M b (also called Walsh bipartite maps [15]) whose automorphism groups preserve the bipartition (two-set