Hilbert Functions of Filtered Modules

In this presentation we shall deal with some aspects of the theory of Hilbert functions of modules over local rings, and we intend to guide the reader along one of the possible routes through the last three decades of progress in this area of dynamic mathematical activity. Motivated by the ever increasing interest in this field, our goal is to gather together many new developments of this theory into one place, and to present them using a unifying approach which gives self-contained and easier proofs. In this text we shall discuss many results by different authors, following essentially the direction typified by the pioneering work of J. Sally. Our personal view of the subject is most visibly expressed by the presentation of Chapters 1 and 2 in which we discuss the use of the superficial elements and related devices. Basic techniques will be stressed with the aim of reproving recent results by using a more elementary approach. Over the past few years several papers have appeared which extend classical results on the theory of Hilbert functions to the case of filtered modules. The extension of the theory to the case of general filtrations on a module has one more important motivation. Namely, we have interesting applications to the study of graded algebras which are not associated to a filtration, in particular the Fiber cone and the Sally-module. We show here that each of these algebras fits into certain short exact sequences, together with algebras associated to filtrations. Hence one can study the Hilbert function and the depth of these algebras with the aid of the know-how we got in the case of a filtration.Comment: 127 pages, revised version. Comments and remarks are welcom

Strong exciton binding in quantum structures through remote dielectric confinement

We propose a new type of hybrid systems formed by conventional semiconductor nanostructures with the addition of remote insulating layers, where the electron-hole interaction is enhanced by combining quantum and dielectric confinement over different length scales. Due to the polarization charges induced by the dielectric mismatch at the semiconductor/insulator interfaces, we show that the exciton binding energy can be more than doubled. For conventional III-V quantum wires such remote dielectric confinement allows exciton binding at room temperature.Comment: 4 pages, 3 PostScript figures embedded, best printed in color. Uses RevTex, multicol, and psfig styles. To appear in Phys. Rev. Let

Shape maps for second order partial differential equations

We analyse the singularity formation of congruences of solutions of systems of second order PDEs via the construction of \emph{shape maps}. The trace of such maps represents a congruence volume whose collapse we study through an appropriate evolution equation, akin to Raychaudhuri's equation. We develop the necessary geometric framework on a suitable jet space in which the shape maps appear naturally associated with certain linear connections. Explicit computations are given, along with a nontrivial example

Gamma-Ray Astronomy around 100 TeV with a large Muon Detector operated at Very High Altitude

Measurements at 100 TeV and above are an important goal for the next generation of high energy gamma-ray astronomy experiments to solve the still open problem of the origin of galactic cosmic rays. The most natural experimental solution to detect very low radiation fluxes is provided by the Extensive Air Shower (EAS) arrays. They benefit from a close to 90% duty cycle and a very large field of view (about 2 sr), but the sensitivity is limited by their angular resolution and their poor cosmic ray background discrimination. Above 10 TeV the standard technique for rejecting the hadronic background consists in looking for "muon-poor" showers. In this paper we discuss the capability of a large muon detector (A=2500 m2) operated with an EAS array at very high altitude (>4000 m a.s.l.) to detect gamma-ray fluxes around 100 TeV. Simulation-based estimates of energy ranges and sensitivities are presented.Comment: 4 pages, proceedings of the 30th ICRC, Merida, Mexico, 200

Stochastic Volatility: Univariate and Multivariate Extensions

Stochastic volatility models, aka SVOL, are more difficult to estimate than standard time-varying volatility models (ARCH). Advances in the literature now offer well tested estimators for a basic univariate SVOL model. However, the basic model is too restrictive for many economic and finance applications. The use of the basic model can lead to biased volatility forecasts especially around crucial periods of high volatility. We extend the basic SVOL needs to allow for the leverage effect, through a correlation between observable and variance errors, and fat-tails in the conditional distribution. We develop a Bayesian Markov Chain Monte Carlo algorithm for this extended model. We also provide an algorithm to analyze a multivariate factor SVOL model. The method simultaneously performs finite sample inference and smoothing. We document the performance of the estimator and show why the extensions are warranted. We provide the researcher with a range of model diagnostics, such as the identification of outliers for stochastic volatility models or the assessment of the normality of the conditional distribution. We implement this methodology on a number of univariate financial time series. There is strong evidence of (1) non-normal conditional distributions for most series, and (2) a leverage effect for stock returns. We illustrate the robustness of the results to the choice of the prior distributions. These results have policy implications on decisions based upon prediction of volatility, especially when dealing with tail prediction as in risk management. Les modèles de volatilité stochastique, alias SVOL, sont plus durs à estimer que les modèles traditionnels de type ARCH. La littérature récente offre des estimateurs éprouvés pour un modèle SVOL univarié de base. Ce modèle est trop contraignant pour une utilisation en économie financière. Les prévisions de volatilité qu'il produit peuvent etre biaisées, particulièrement quand la volatilité est élevée. Nous généralisons le modèle de base en y ajoutant des effets de levier par le biais d'une corrélation entre les chocs observables et de variance, et la possibilité de distributions conditionnelles à queues épaisses. Nous développons un algorithme bayésien à chaînes markoviennes de Monte Carlo. Nous développons aussi un algorithme pour l'analyse d'un modèle SVOL multivarié à facteurs. Ces estimateurs permettent une inférence en échantillon fini pour les paramètres et les volatilités. Nous documentons les performances de l'estimateur et montrons que les extensions sont nécessaires. Nous testons la normalité des distributions conditionnelles. Cette méthode est mise en oeuvre sur plusieurs séries financières. Il y a une forte évidence (1) de distributions conditionnelles à queues épaisses, et (2) d'effets de levier pour les actifs financiers. Les résultats sont robustes et ont d'importantes implications sur les décisions fondées sur les prédictions de volatilité, particulièrement pour la gestion de risques.Stochastic volatility, ARCH, MCMC algorithm, leverage effect, risk management, fat-tailed distributions, Volatilité stochastique, ARCH, algorithme MCMC, effets de levier, gestion de risque, distributions à queues épaisses