Basic results are obtained concerning Galois connections between collections of closure operators (of various types) and collections consisting of subclasses of (pairs of) morphisms in M for an 〈E,M 〉 -category X . In effect, the “lattice” of closure operators on M is shown to be equivalent to the fixed point lattice of the polarity induced by the orthogonality relation between composable pairs...

The tomographic description of a quantum state is formulated in an abstract infinite dimensional Hilbert space framework, the space of the Hilbert-Schmidt linear operators, with trace formula as scalar product. Resolutions of the unity, written in terms of over-complete sets of rank-one projectors and of associated Gram-Schmidt operators taking into account their non-orthogonality, are then use...

A procedure is outlined for adapting Krylov subspace methods to solving approximately the underdetermined linear systems that arise in path following (continuation, homotopy) methods. This procedure, in addition to preserving the usual desirable features of Krylov subspace methods, has the advantages of satisfying orthogonality constraints exactly and of not introducing ill-conditioning through...

A warping operator consists of an invertible axisdeformation applied either in the signal domain or in thecorresponding Fourier domain. Additionally, a warping trans-formation is usually required to preserve the signal energy, thuspreserving orthogonality and being invertible by its adjoint.Initially, the design of such operators has been motivated bythe idea of suitably...

A procedure is outlined for adapting Krylov subspace methods to solving approximately the underdetermined linear systems that arise in path following (continuation, homotopy) methods. This procedure, in addition to preserving the usual desirable features of Krylov subspace methods, has the advantages of satisfying orthogonality constraints exactly and of not introducing ill-conditioning through...