On Polynomial Relations in the Heisenberg Algebra

Polynomial relations between the generators of the classical and quantum Heisenberg algebras are presented. Some of those relations can have a meaning of the formulas of the normal ordering for the creation/annihilation operators occurred in the method of the second quantization.Comment: 10 pages, Preprint CRN 94/08 and IFUNAM FT 94-41 (corrected), LaTeX The title is changed, the paper is expanded, extra references are added, typos are correcte

Thin shell implies spectral gap up to polylog via a stochastic localization scheme

We consider the isoperimetric inequality on the class of high-dimensional isotropic convex bodies. We establish quantitative connections between two well-known open problems related to this inequality, namely, the thin shell conjecture, and the conjecture by Kannan, Lovasz, and Simonovits, showing that the corresponding optimal bounds are equivalent up to logarithmic factors. In particular we prove that, up to logarithmic factors, the minimal possible ratio between surface area and volume is attained on ellipsoids. We also show that a positive answer to the thin shell conjecture would imply an optimal dependence on the dimension in a certain formulation of the Brunn-Minkowski inequality. Our results rely on the construction of a stochastic localization scheme for log-concave measures.Comment: 33 page

Bounding the norm of a log-concave vector via thin-shell estimates

Chaining techniques show that if X is an isotropic log-concave random vector in R^n and Gamma is a standard Gaussian vector then E |X| < C n^{1/4} E |Gamma| for any norm |*|, where C is a universal constant. Using a completely different argument we establish a similar inequality relying on the thin-shell constant sigma_n = sup ((var|X|^){1/2} ; X isotropic and log-concave on R^n). In particular, we show that if the thin-shell conjecture sigma_n = O(1) holds, then n^{1/4} can be replaced by log (n) in the inequality. As a consequence, we obtain certain bounds for the mean-width, the dual mean-width and the isotropic constant of an isotropic convex body. In particular, we give an alternative proof of the fact that a positive answer to the thin-shell conjecture implies a positive answer to the slicing problem, up to a logarithmic factor.Comment: preliminary version, 13 page

Pareto-Optimal Methods for Gene Ranking

The massive scale and variability of microarray gene data creates new and challenging problems of signal extraction, gene clustering, and data mining, especially for temporal gene profiles. Many data mining methods for finding interesting gene expression patterns are based on thresholding single discriminants, e.g. the ratio of between-class to within-class variation or correlation to a template. Here a different approach is introduced for extracting information from gene microarrays. The approach is based on multiple objective optimization and we call it Pareto front analysis (PFA). This method establishes a ranking of genes according to estimated probabilities that each gene is Pareto-optimal, i.e., that it lies on the Pareto front of the multiple objective scattergram. Both a model-driven Bayesian Pareto method and a data-driven non-parametric Pareto method, based on rank-order statistics, are presented. The methods are illustrated for two gene microarray experiments.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/41339/1/11265_2005_Article_5273219.pd

High-performance compression of visual information - A tutorial review - Part I : Still Pictures

Digital images have become an important source of information in the modern world of communication systems. In their raw form, digital images require a tremendous amount of memory. Many research efforts have been devoted to the problem of image compression in the last two decades. Two different compression categories must be distinguished: lossless and lossy. Lossless compression is achieved if no distortion is introduced in the coded image. Applications requiring this type of compression include medical imaging and satellite photography. For applications such as video telephony or multimedia applications, some loss of information is usually tolerated in exchange for a high compression ratio. In this two-part paper, the major building blocks of image coding schemes are overviewed. Part I covers still image coding, and Part II covers motion picture sequences. In this first part, still image coding schemes have been classified into predictive, block transform, and multiresolution approaches. Predictive methods are suited to lossless and low-compression applications. Transform-based coding schemes achieve higher compression ratios for lossy compression but suffer from blocking artifacts at high-compression ratios. Multiresolution approaches are suited for lossy as well for lossless compression. At lossy high-compression ratios, the typical artifact visible in the reconstructed images is the ringing effect. New applications in a multimedia environment drove the need for new functionalities of the image coding schemes. For that purpose, second-generation coding techniques segment the image into semantically meaningful parts. Therefore, parts of these methods have been adapted to work for arbitrarily shaped regions. In order to add another functionality, such as progressive transmission of the information, specific quantization algorithms must be defined. A final step in the compression scheme is achieved by the codeword assignment. Finally, coding results are presented which compare stateof- the-art techniques for lossy and lossless compression. The different artifacts of each technique are highlighted and discussed. Also, the possibility of progressive transmission is illustrated

Magnetism of a tetrahedral cluster spin-chain

We discuss the magnetic properties of a dimerized and completely frustrated tetrahedral spin-1/2 chain. Using a combination of exact diagonalization and bond-operator theory the quantum phase diagram is shown to incorporate a singlet-product, a dimer, and a Haldane phase. In addition we consider one-, and two-triplet excitations in the dimer phase and evaluate the magnetic Raman cross section which is found to be strongly renormalized by the presence of a two-triplet bound state. The link to a novel tellurate materials is clarified.Comment: 8 pages, 8 figure