Decompositions of spaces of measures

Let M be the Banach space of sigma-additive complex-valued measures on an abstract measurable space. We prove that any closed, with respect to absolute continuity norm-closed, linear subspace L of M is complemented and describe the unique complement, projection onto L along which has norm 1. Using this fact we prove a decomposition theorem, which includes the Jordan decomposition theorem, the generalized Radon-Nikodym theorem and the decomposition of measures into decaying and non-decaying components as particular cases. We also prove an analog of the Jessen-Wintner purity theorem for our decompositions.

Generalized Laplacian Eigenmaps for Modeling and Tracking Human Motions

This paper presents generalized Laplacian eigenmaps, a novel dimensionality reduction approach designed to address stylistic variations in time series. It generates compact and coherent continuous spaces whose geometry is data-driven. This paper also introduces graph-based particle filter, a novel methodology conceived for efficient tracking in low dimensional space derived from a spectral dimensionality reduction method. Its strengths are a propagation scheme, which facilitates the prediction in time and style, and a noise model coherent with the manifold, which prevents divergence, and increases robustness. Experiments show that a combination of both techniques achieves state-of-the-art performance for human pose tracking in underconstrained scenarios.

Analytically continued generalized continuum distorted waves

The continuum distorted-wave eikonal-initial-state (CDW-EIS) theory of Crothers and McCann (Crothers DSF and McCann JF, 1983 J. Phys. B: At. Mol. Opt. Phys. 16 3229 ) used to describe ionization in ion-atom collisions is generalized (G) to GCDW-EIS, to incorporate the azimuthal ange dependence into the final-state wavefunction. This is accomplished by the analytic continuation of hydrogenic-like wavefunctions from below to above threshold, using parabolic coordinates and quantum numbers including magnetic quantum numbers, thus providing a more complete set of states. At impact energies lower than 25 ke V u^{-1}, the total CDW-EIS ionization cross section falls off, with decreasing energy, too quickly in comparison with experimental data by Crothers and McCann. The idea behind and motivation for the GCDW-EIS model is to improve the theory with respect to experiment, by including contributions from non-zero magnetic quantum numbers. We also therefore incidentally provide a new derivation of the theory of continuum distorted waves for zero magnetic quantum numbers while simultaneously generalizing it.