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

Peculiar Velocities and the Mean Density Parameter

We study the peculiar velocity field inferred from the Mark III spirals using a new method of analysis. We estimate optimal values of Tully-Fisher scatter and zero-point offset, and we derive the 3-dimensional rms peculiar velocity ($\sigma_v$) of the galaxies in the samples analysed. We check our statistical analysis using mock catalogs derived from numerical simulations of CDM models considering measurement uncertainties and sampling variations. Our best determination for the observations is $\sigma_v= (660\pm50) km/s$. We use the linear theory relation between $\sigma_v$, the density parameter $\Omega$, and the galaxy correlation function $\xi(r)$ to infer the quantity $\beta =\Omega^{0.6}/b = 0.60^{+0.13}_{-0.11}$ where $b$ is the linear bias parameter of optical galaxies and the uncertainties correspond to bootstrap resampling and an estimated cosmic variance added in quadrature. Our findings are consistent with the results of cluster abundances and redshift space distortion of the two-point correlation function. These statistical measurements suggest a low value of the density parameter $\Omega \sim 0.4$ if optical galaxies are not strongly biased tracers of mass.Comment: Accepted for publication in MNRAS. 8 pages latex (mn.sty), including 7 figure

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