Luis Lechuga

Complexity in rational homotopy

Abstract The computation of classical invariants of the rational homotopy type of simply connected spaces is shown to be an NP-hard problem.

The computation of the Betti numbers of an elliptic space is a NP-hard problem

Let S a 1-connected space such that π∗(S) ⊗ Q and H ∗ (S; Q) are both finite-dimensional. Then, using the Sullivan model as a codification for S, we prove that the computation of the Betti numbers of S is a NP-hard problem.  2002 Elsevier Science B.V. All rights reserved.

Computing the Sullivan Milnor-Moore S.S. and the rational LS category of certain spaces

Let (ΛV , d) be the Sullivan model of an elliptic space S and (ΛV , dσ ) be the associated pure model. We give an algorithm, based on Groebner basis computations, that computes the stage lσ = l0(ΛV , dσ ) at which the (Sullivan version of the) Milnor–Moore spectral sequence of (ΛV , dσ ) collapses. When (d − dσ )V ⊂ Λ >lσ V we call S a Ginsburg space. We show that the rational LS category of any Ginsburg space S, cat0(ΛV , d), coincides with that of the associated pure space cat0(ΛV , dσ ). A previous algorithm due to the author computes cat0(ΛV , dσ ). So we obtain an algorithm that determines whether a space is Ginsburg and which in this case computes its rational LS category.  2002 Elsevier Science B.V. All rights reserved.

Quantitative phase imaging techniques for the study of Red Blood Cells

We present a Sagnac interferometer for the analysis of red blood cells (RBC). The system generates two p/2-shifted interferograms. The optical phase was processed using a Vargas-Quiroga algorithm. Related experimental results obtained for RBC are presented.

Measurement of the slope of transparent micro-structures using parallel phase shifting interferometry

In this research we implemented a two-step phase shifting system based on two cyclic-path coupled interferometers for slope measurements of transparent samples. We analyzed the cases of four patterns with shifts of π/2 captured in two shots; the optical phase was processed by a four-step algorithm.

Bounding the rational LS-category of certain spaces via the Hilbert function☆

Abstract Let (ΛV,d) be the Sullivan model of a pure elliptic space S. We give an algorithm, based on the computation of the Hilbert function of certain ideal, that provides a lower and an upper bound for the rational L.–S. category of S,cat0(ΛV,d). When S has null Euler homotopy characteristic, both bounds are sharp.

An extension of Miller's version of the de Rham Theorem with any coefficients

In this paper we present an approximation to the de Rham theorem for simplicial sets with any coeffcients based, using simplicial techniques on Poincares lemma and q-extendability.