The model of the torus as a parallelogram in the plane with opposite sides identiied enables us to deene two families of parallel lines and to tessellate the torus, then to associate to each tessellation a toroidal map with an upward drawing. It is proved that a toroidal map has a tessellation representation if and only if its universal cover is 2-connected. Those graphs that admit such an embe...

A complete list of homogeneous operators in the Cowen Douglas class Bn(D) is given. This classification is obtained from an explicit realization of all the homogeneous Hermitian holomorphic vector bundles on the unit disc under the action of the universal covering group of the bi-holomorphic automorphism group of the unit disc.

Chain complexes with a group of operators are a well known tool in algebraic topology, where they arise naturally as the chain complex C∗X̃ of cellular chains of the universal cover X̃ of a reduced CW -complex X. The group of operators here is the fundamental group of X. J.H.C. Whitehead in his classical but little read paper [31] showed that the chain complex C∗(X̃) is useful for the homotopy cla...

For a prime power q and for integers R, η with R > 0, 0 ≤ η ≤ R − 1, let A R,q = (Cni)i denote an infinite sequence of q-ary linear [ni, ni − ri]qR codes Cni with covering radius R and such that the following two properties hold: (a) the codimension ri = Rti + η, where (ti)i is an increasing sequence of integers; (b) the length ni of Ci coincides with f (η) q (ri), where f (η) q is an increasin...