Intersection Cohomology of S1-Actions on Pseudomanifolds

For any smooth free action of the unit circle S1 on a smooth manifold M, the Gysin sequence of M is a long exact sequence relating the DeRham Cohomology of M and the orbit space M/S1. If the action is not free then M/S1 is not a smooth manifold but a stratified pseudomanifold, the lenght of M/S1 depending on the number of orbit types; and there is a Gysin sequence relating their intersection cohomologies. The links of the fixed strata in M/S1 are cohomological complex projective spaces, so the conecting homomorphism of this sequences is the multiplication by the Euler class. In this article we extend the above results for any action of S1 on a stratified pseudomanifold X of lenght 1. We use the DeRham-like intersection cohomology defined by means of an unfolding. If the action preserves the local structure, then the orbit space X/S1 is again a stratified pseudomanifold of lenght 1 and has an unfolding. There is a long exact sequence relating the intersection cohomology of X and X/S1 with a third complex $\mathcal{G}$, the Gysin Term, whose cohomology depends on basic cohomological data of two flavours: global and local. Global data concerns the Euler class induced by the action; local information depends on the cohomology of the fixed strata with values on some presheaves.Comment: AMSTeX Article, 23 pages. Keywords and phrases: Intersection Cohomology, Stratified Pseudomanifold

Equivariant intersection cohomology of the circle actions

In this paper, we prove that the orbit space B and the Euler class of an action of the circle S^1 on X determine both the equivariant intersection cohomology of the pseudomanifold X and its localization. We also construct a spectral sequence converging to the equivariant intersection cohomology of X whose third term is described in terms of the intersection cohomology of B.Comment: Final version as accepted in RACSAM. The final publication is available at springerlink.com; Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas, 201

Tensor decomposition with generalized lasso penalties

We present an approach for penalized tensor decomposition (PTD) that estimates smoothly varying latent factors in multi-way data. This generalizes existing work on sparse tensor decomposition and penalized matrix decompositions, in a manner parallel to the generalized lasso for regression and smoothing problems. Our approach presents many nontrivial challenges at the intersection of modeling and computation, which are studied in detail. An efficient coordinate-wise optimization algorithm for (PTD) is presented, and its convergence properties are characterized. The method is applied both to simulated data and real data on flu hospitalizations in Texas. These results show that our penalized tensor decomposition can offer major improvements on existing methods for analyzing multi-way data that exhibit smooth spatial or temporal features

A short review of "DGP Specteroscopy"

In this paper we provide a short review of the main results developed in hep-th/0604086. We focus on linearised vacuum perturbations about the self-accelerating branch of solutions in the DGP model. These are shown to contain a ghost in the spectrum for any value of the brane tension. We also comment on hep-th/0607099, where some counter arguments have been presented.Comment: Minor typos correcte

New Geometric Algorithms for Fully Connected Staged Self-Assembly

We consider staged self-assembly systems, in which square-shaped tiles can be added to bins in several stages. Within these bins, the tiles may connect to each other, depending on the glue types of their edges. Previous work by Demaine et al. showed that a relatively small number of tile types suffices to produce arbitrary shapes in this model. However, these constructions were only based on a spanning tree of the geometric shape, so they did not produce full connectivity of the underlying grid graph in the case of shapes with holes; designing fully connected assemblies with a polylogarithmic number of stages was left as a major open problem. We resolve this challenge by presenting new systems for staged assembly that produce fully connected polyominoes in O(log^2 n) stages, for various scale factors and temperature {\tau} = 2 as well as {\tau} = 1. Our constructions work even for shapes with holes and uses only a constant number of glues and tiles. Moreover, the underlying approach is more geometric in nature, implying that it promised to be more feasible for shapes with compact geometric description.Comment: 21 pages, 14 figures; full version of conference paper in DNA2