On the distribution of the maximum of a gaussian field with d parameters

Let I be a compact d-dimensional manifold, let X:I\to R be a Gaussian process with regular paths and let F_I(u), u\in R, be the probability distribution function of sup_{t\in I}X(t). We prove that under certain regularity and nondegeneracy conditions, F_I is a C^1-function and satisfies a certain implicit equation that permits to give bounds for its values and to compute its asymptotic behavior as u\to +\infty. This is a partial extension of previous results by the authors in the case d=1. Our methods use strongly the so-called Rice formulae for the moments of the number of roots of an equation of the form Z(t)=x, where Z:I\to R^d is a random field and x is a fixed point in R^d. We also give proofs for this kind of formulae, which have their own interest beyond the present application.Comment: Published at http://dx.doi.org/10.1214/105051604000000602 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Determining DfT Hardware by VHDL-AMS Fault Simulation for Biological Micro-Electronic Fluidic Arrays

The interest of microelectronic fluidic arrays for biomedical applications, like DNA determination, is rapidly increasing. In order to evaluate these systems in terms of required Design-for-Test structures, fault simulations in both fluidic and electronic domains are necessary. VHDL-AMS can be used successfully in this case. This paper shows a highly testable architecture of a DNA Bio-Sensing array, its basic sensing concept, fluidic modeling and sensitivity analysis. The overall VHDL-AMS fault simulation of the system is shown

Critical Behaviour of the Number of Minima of a Random Landscape at the Glass Transition Point and the Tracy-Widom distribution

We exploit a relation between the mean number $N_{m}$ of minima of random Gaussian surfaces and extreme eigenvalues of random matrices to understand the critical behaviour of $N_{m}$ in the simplest glass-like transition occuring in a toy model of a single particle in $N$-dimensional random environment, with $N\gg 1$. Varying the control parameter $\mu$ through the critical value $\mu_c$ we analyse in detail how $N_{m}(\mu)$ drops from being exponentially large in the glassy phase to $N_{m}(\mu)\sim 1$ on the other side of the transition. We also extract a subleading behaviour of $N_{m}(\mu)$ in both glassy and simple phases. The width $\delta{\mu}/\mu_c$ of the critical region is found to scale as $N^{-1/3}$ and inside that region $N_{m}(\mu)$ converges to a limiting shape expressed in terms of the Tracy-Widom distribution

Gloves, Headphones and Soap

This article is a creative reflection on the simple act of washing dishes, encouraging readers to grasp the sensory complexity of acts that are part of our daily life. I have identified five sections of analysis: location, touch, hearing, movement and time perception. To explain these I use different methods including diagrams, vignettes and photographs. Through the use of visual representations I hope to transmit my sensory experience of washing dishes as closely as possible.&nbsp

Restricted Isometries for Partial Random Circulant Matrices

In the theory of compressed sensing, restricted isometry analysis has become a standard tool for studying how efficiently a measurement matrix acquires information about sparse and compressible signals. Many recovery algorithms are known to succeed when the restricted isometry constants of the sampling matrix are small. Many potential applications of compressed sensing involve a data-acquisition process that proceeds by convolution with a random pulse followed by (nonrandom) subsampling. At present, the theoretical analysis of this measurement technique is lacking. This paper demonstrates that the $s$th order restricted isometry constant is small when the number $m$ of samples satisfies $m \gtrsim (s \log n)^{3/2}$, where $n$ is the length of the pulse. This bound improves on previous estimates, which exhibit quadratic scaling

Mean number and correlation function of critical points of isotropic Gaussian fields

Let X = {X(t) : t ∈ R N } be an isotropic Gaussian random field with real values. In a first part we study the mean number of critical points of X with index k using random matrices tools. We obtain an exact expression for the probability density of the eigenvalue of rank k of a N-GOE matrix. We deduce some exact expressions for the mean number of critical points with a given index. In a second part we study attraction or repulsion between these critical points. A measure is the correlation function. We prove attraction between critical points when N > 2, neutrality for N = 2 and repulsion for N = 1. The attraction between critical points that occurs when the dimension is greater than two is due to critical points with adjacent indexes. A strong repulsion between maxima and minima is observed. The correlation function between maxima (or minima) depends on the dimension of the ambient space

Sistemas Produtivos Locais Franceses e Distritos Industriais Italianos: Elementos de uma Problemática na Intersecção da Sociologia e da Economia

The comparative study of Italian industrial districts and of French local productive systems is part of an investigation concerning the economic, social and political determinants which contribute to the appearance and consolidation of a system of production in a given territory, generally based upon small and medium firms.These forms are considered as places where several principles of social organization (internal and external) cross and converge. The industrial district is a melting pot which collects and redistributes social exchanges in the world of labour and extra-labour, of market flows and non-market flows, a privileged place for the game of social interrelations. It is also the product of a specific social structuring.Moreover, the study of industrial districts allows to go back to analytical categories, such as the “local” one, which is a main point in social sciences. Thinking “local” means endeavouring to supersede the readings in macro-economics, to integrate micro-determinants, be they economic, social or political ones.L’étude comparative des districts industrieis italiens et des systèmes productifs locaux français s’inscrit dans la recherche des déterminants économiques, sociaux et politiques qui concourent à 1’émergence et à la consolidation dans un territoire donné d’un système de production, généralement fondé sur des petites et moyennes entreprises.Ces formes d’organisation productive sontvues comme des lieux ou convergent plusieurs principes d’organisation sociale internes et externes à l’entreprise. Le district industriel est un creuset qui collecte et redistribue les échanges sociaux du monde du travail et du hors- travail, de flux marchands et non marchands, un lieu privilégié dujeu des interrelations sociales. Il est aussi le produit d’une structuration sociale spécifique.Leur examen permet de renouer avec certaines catégories analytiques (le “local”) qui paraissaient surannées et fournit des éléments théoriques à la comprehension de phénomènes comme la flexibilité; il représente aussi une tentative de dépasser les modèles de lecture macro-économique afm d’intégrer à l’analyse des determinants d’ordre micro, qu’ils soient économiques, sociaux ou politiques.O estudo comparativo dos distritos industriais italianos e dos sistemas produtivos locais franceses se inscreve na procura dos determinantes econômicos, sociais e políticos que concorrem para a emergência e consolidação num dado território de um sistema de produção, geralmente baseado em pequenas e médias empresas.Estas formas de organização produtiva são vistas como lugares onde convergem diferentes princípios de organização social, internos e externos à empresa. O distrito industrial é um lugar privilegiado que coleta e redistribui os intercâmbios sociais do mundo do trabalho e do extratrabalho, de fluxos mercantis e não-mercantis, um lugar privilegiado do jogo das interrelações sociais que, por sua vez, produz uma estruturação específica das relações sociais.Seu exame permite reatar com certas categorias analíticas (o “local”) que pareciam antiquadas e fornece elementos teóricos para a compreensão de fenômenos como a flexibilidade; ele representa, também, uma tentativa de ultrapassar os modelos de leitura macroeconômica para integrará análise, determinantes de ordem micro, sejam eles econômicos, sociais ou políticos