Deformations of the Tracy-Widom distribution

In random matrix theory, the Tracy-Widom (TW) distribution describes the behavior of the largest eigenvalue. We consider here two models in which TW undergoes transformations. In the first one disorder is introduced in the Gaussian ensembles by superimposing an external source of randomness. A competition between TW and a normal (Gaussian) distribution results, depending on the spreading of the disorder. The second model consists of removing at random a fraction of (correlated) eigenvalues of a random matrix. The usual formalism of Fredholm determinants extends naturally. A continuous transition from TW to the Weilbull distribution, characteristic of extreme values of an uncorrelated sequence, is obtained.

Robust Mixture Modelling Using the t Distribution

Normal mixture models are being increasingly used to model the distributions of a wide variety of random phenomena and to cluster sets of continuous multivariate data. However, for a set of data containing a group or groups of observations with longer than normal tails or atypical observations, the use of normal components may unduly affect the fit of the mixture model. In this paper, we consider a more robust approach by modelling the data by a mixture of t distributions. The use of the ECM algorithm to fit this t mixture model is described and examples of its use are given in the context of clustering multivariate data in the presence of atypical observations in the form of background noise.

Cavity QED analog of the harmonic-oscillator probability distribution function and quantum collapses

We establish a connection between the simple harmonic oscillator and a two-level atom interacting with resonant, quantized cavity and strong driving fields, which suggests an experiment to measure the harmonic-oscillator's probability distribution function. To achieve this, we calculate the Autler-Townes spectrum by coupling the system to a third level. We find that there are two different regions of the atomic dynamics depending on the ratio of the: Rabi frequency Omega (c) of the cavity field to that of the Rabi frequency Omega of the driving field. For Omega (c)

Equilibrium self-assembly of colloids with distinct interaction sites: Thermodynamics, percolation, and cluster distribution functions

We calculate the equilibrium thermodynamic properties, percolation threshold, and cluster distribution functions for a model of associating colloids, which consists of hard spherical particles having on their surfaces three short-ranged attractive sites (sticky spots) of two different types, A and B. The thermodynamic properties are calculated using Wertheim's perturbation theory of associating fluids. This also allows us to find the onset of self-assembly, which can be quantified by the maxima of the specific heat at constant volume. The percolation threshold is derived, under the no-loop assumption, for the correlated bond model: In all cases it is two percolated phases that become identical at a critical point, when one exists. Finally, the cluster size distributions are calculated by mapping the model onto an effective model, characterized by a-state-dependent-functionality (f) over bar and unique bonding probability (p) over bar. The mapping is based on the asymptotic limit of the cluster distributions functions of the generic model and the effective parameters are defined through the requirement that the equilibrium cluster distributions of the true and effective models have the same number-averaged and weight-averaged sizes at all densities and temperatures. We also study the model numerically in the case where BB interactions are missing. In this limit, AB bonds either provide branching between A-chains (Y-junctions) if epsilon(AB)/epsilon(AA) is small, or drive the formation of a hyperbranched polymer if epsilon(AB)/epsilon(AA) is large. We find that the theoretical predictions describe quite accurately the numerical data, especially in the region where Y-junctions are present. There is fairly good agreement between theoretical and numerical results both for the thermodynamic (number of bonds and phase coexistence) and the connectivity properties of the model (cluster size distributions and percolation locus).

Probability distribution functions

This technical report describes the PDFs which have been implemented to model the behaviours of certain parameters of the Repeater-Based Hybrid Wired/Wireless PROFIBUS Network Simulator (RHW2PNetSim) and Bridge-Based Hybrid Wired/Wireless PROFIBUS Network Simulator (BHW2PNetSim).

Modified Discrete PSO to Increase the Delivered Energy Probability in Distribution energy Systems

This paper proposes a PSO based approach to increase the probability of delivering power to any load point by identifying new investments in distribution energy systems. The statistical failure and repair data of distribution components is the main basis of the proposed methodology that uses a fuzzyprobabilistic modeling for the components outage parameters. The fuzzy membership functions of the outage parameters of each component are based on statistical records. A Modified Discrete PSO optimization model is developed in order to identify the adequate investments in distribution energy system components which allow increasing the probability of delivering power to any customer in the distribution system at the minimum possible cost for the system operator. To illustrate the application of the proposed methodology, the paper includes a case study that considers a 180 bus distribution network.

DC Fuzzy Multicriteria Approach to Increase the Probability of Delivering Power in Distribution Networks

This paper proposes a methodology to increase the probability of delivering power to any load point through the identification of new investments. The methodology uses a fuzzy set approach to model the uncertainty of outage parameters, load and generation. A DC fuzzy multicriteria optimization model considering the Pareto front and based on mixed integer non-linear optimization programming is developed in order to identify the adequate investments in distribution networks components which allow increasing the probability of delivering power to all customers in the distribution network at the minimum possible cost for the system operator, while minimizing the non supplied energy cost. To illustrate the application of the proposed methodology, the paper includes a case study which considers an 33 bus distribution network.

Increase of the Delivered Power Probability in Distribution Networks using Pareto DC Programming

A methodology to increase the probability of delivering power to any load point through the identification of new investments in distribution network components is proposed in this paper. The method minimizes the investment cost as well as the cost of energy not supplied in the network. A DC optimization model based on mixed integer non-linear programming is developed considering the Pareto front technique in order to identify the adequate investments in distribution networks components which allow increasing the probability of delivering power for any customer in the distribution system at the minimum possible cost for the system operator, while minimizing the energy not supplied cost. Thus, a multi-objective problem is formulated. To illustrate the application of the proposed methodology, the paper includes a case study which considers a 180 bus distribution network

Corrigendum to "On randomized matching mechanisms" [Economic Theory 8(1996)377-381]

Ma (1996) studied the random order mechanism, a matching mechanism suggested by Roth and Vande Vate (1990) for marriage markets. By means of an example he showed that the random order mechanism does not always reach all stable matchings. Although Ma's (1996) result is true, we show that the probability distribution he presented - and therefore the proof of his Claim 2 - is not correct. The mistake in the calculations by Ma (1996) is due to the fact that even though the example looks very symmetric, some of the calculations are not as ''symmetric.''

Cell size distribution in a random tessellation of space governed by the Kolmogorov-Johnson-Mehl-Avrami model: Grain size distribution in crystallization

The space subdivision in cells resulting from a process of random nucleation and growth is a subject of interest in many scientific fields. In this paper, we deduce the expected value and variance of these distributions while assuming that the space subdivision process is in accordance with the premises of the Kolmogorov-Johnson-Mehl-Avrami model. We have not imposed restrictions on the time dependency of nucleation and growth rates. We have also developed an approximate analytical cell size probability density function. Finally, we have applied our approach to the distributions resulting from solid phase crystallization under isochronal heating conditions

Particle shape and orientation in laser diffraction and static image analysis size distribution analysis of micrometer sized rectangular particles

Laser diffraction (LD) and static image analysis (SIA) of rectangular particles [United States Pharmacopeia, USP30-NF25, General Chapter <776>, Optical Miroscopy.] have been systematically studied. To rule out sample dispersion and particle orientation as the root cause of differences in size distribution profiles, we immobilize powder samples on a glass plate by means of a dry disperser. For a defined region of the glass plate, we measure the diffraction pattern as induced by the dispersed particles, and the 2D dimensions of the individual particles using LD and optical microscopy, respectively. We demonstrate a correlation between LD and SIA, with the scattering intensity of the individual particles as the dominant factor. In theory, the scattering intensity is related to the square of the projected area of both spherical and rectangular particles. In traditional LD the size distribution profile is dominated by the maximum projected area of the particles (A). The diffraction diameters of a rectangular particle with length L and breadth B as measured by the LD instrument approximately correspond to spheres of diameter ØL and ØB respectively. Differences in the scattering intensity between spherical and rectangular particles suggest that the contribution made to the overall LD volume probability distribution by each rectangular particle is proportional to A2/L and A2/B. Accordingly, for rectangular particles the scattering intensity weighted diffraction diameter (SIWDD) explains an overestimation of their shortest dimension and an underestimation of their longest dimension. This study analyzes various samples of particles whose length ranges from approximately 10 to 1000 μm. The correlation we demonstrate between LD and SIA can be used to improve validation of LD methods based on SIA data for a variety of pharmaceutical powders all with a different rectangular particle size and shape.

Seismic characterization of heterogeneous sedimentary and fractured rocks based on Biot's theory of poroelasticity

Understanding and quantifying seismic energy dissipation, which manifests itself in terms of velocity dispersion and attenuation, in fluid-saturated porous rocks is of considerable interest, since it offers the perspective of extracting information with regard to the elastic and hydraulic rock properties. There is increasing evidence to suggest that wave-induced fluid flow, or simply WIFF, is the dominant underlying physical mechanism governing these phenomena throughout the seismic, sonic, and ultrasonic frequency ranges. This mechanism, which can prevail at the microscopic, mesoscopic, and macroscopic scale ranges, operates through viscous energy dissipation in response to fluid pressure gradients and inertial effects induced by the passing wavefield. In the first part of this thesis, we present an analysis of broad-band multi-frequency sonic log data from a borehole penetrating water-saturated unconsolidated glacio-fluvial sediments. An inherent complication arising in the interpretation of the observed P-wave attenuation and velocity dispersion is, however, that the relative importance of WIFF at the various scales is unknown and difficult to unravel. An important generic result of our work is that the levels of attenuation and velocity dispersion due to the presence of mesoscopic heterogeneities in water-saturated unconsolidated clastic sediments are expected to be largely negligible. Conversely, WIFF at the macroscopic scale allows for explaining most of the considered data while refinements provided by including WIFF at the microscopic scale in the analysis are locally meaningful. Using a Monte-Carlo-type inversion approach, we compare the capability of the different models describing WIFF at the macroscopic and microscopic scales with regard to their ability to constrain the dry frame elastic moduli and the permeability as well as their local probability distribution. In the second part of this thesis, we explore the issue of determining the size of a representative elementary volume (REV) arising in the numerical upscaling procedures of effective seismic velocity dispersion and attenuation of heterogeneous media. To this end, we focus on a set of idealized synthetic rock samples characterized by the presence of layers, fractures or patchy saturation in the mesocopic scale range. These scenarios are highly pertinent because they tend to be associated with very high levels of velocity dispersion and attenuation caused by WIFF in the mesoscopic scale range. The problem of determining the REV size for generic heterogeneous rocks is extremely complex and entirely unexplored in the given context. In this pilot study, we have therefore focused on periodic media, which assures the inherent self- similarity of the considered samples regardless of their size and thus simplifies the problem to a systematic analysis of the dependence of the REV size on the applied boundary conditions in the numerical simulations. Our results demonstrate that boundary condition effects are absent for layered media and negligible in the presence of patchy saturation, thus resulting in minimum REV sizes. Conversely, strong boundary condition effects arise in the presence of a periodic distribution of finite-length fractures, thus leading to large REV sizes. In the third part of the thesis, we propose a novel effective poroelastic model for periodic media characterized by mesoscopic layering, which accounts for WIFF at both the macroscopic and mesoscopic scales as well as for the anisotropy associated with the layering. Correspondingly, this model correctly predicts the existence of the fast and slow P-waves as well as quasi and pure S-waves for any direction of wave propagation as long as the corresponding wavelengths are much larger than the layer thicknesses. The primary motivation for this work is that, for formations of intermediate to high permeability, such as, for example, unconsolidated sediments, clean sandstones, or fractured rocks, these two WIFF mechanisms may prevail at similar frequencies. This scenario, which can be expected rather common, cannot be accounted for by existing models for layered porous media. Comparisons of analytical solutions of the P- and S-wave phase velocities and inverse quality factors for wave propagation perpendicular to the layering with those obtained from numerical simulations based on a ID finite-element solution of the poroelastic equations of motion show very good agreement as long as the assumption of long wavelengths remains valid. A limitation of the proposed model is its inability to account for inertial effects in mesoscopic WIFF when both WIFF mechanisms prevail at similar frequencies. Our results do, however, also indicate that the associated error is likely to be relatively small, as, even at frequencies at which both inertial and scattering effects are expected to be at play, the proposed model provides a solution that is remarkably close to its numerical benchmark. -- Comprendre et pouvoir quantifier la dissipation d'énergie sismique qui se traduit par la dispersion et l'atténuation des vitesses dans les roches poreuses et saturées en fluide est un intérêt primordial pour obtenir des informations à propos des propriétés élastique et hydraulique des roches en question. De plus en plus d'études montrent que le déplacement relatif du fluide par rapport au solide induit par le passage de l'onde (wave induced fluid flow en anglais, dont on gardera ici l'abréviation largement utilisée, WIFF), représente le principal mécanisme physique qui régit ces phénomènes, pour la gamme des fréquences sismiques, sonique et jusqu'à l'ultrasonique. Ce mécanisme, qui prédomine aux échelles microscopique, mésoscopique et macroscopique, est lié à la dissipation d'énergie visqueuse résultant des gradients de pression de fluide et des effets inertiels induits par le passage du champ d'onde. Dans la première partie de cette thèse, nous présentons une analyse de données de diagraphie acoustique à large bande et multifréquences, issues d'un forage réalisé dans des sédiments glaciaux-fluviaux, non-consolidés et saturés en eau. La difficulté inhérente à l'interprétation de l'atténuation et de la dispersion des vitesses des ondes P observées, est que l'importance des WIFF aux différentes échelles est inconnue et difficile à quantifier. Notre étude montre que l'on peut négliger le taux d'atténuation et de dispersion des vitesses dû à la présence d'hétérogénéités à l'échelle mésoscopique dans des sédiments clastiques, non- consolidés et saturés en eau. A l'inverse, les WIFF à l'échelle macroscopique expliquent la plupart des données, tandis que les précisions apportées par les WIFF à l'échelle microscopique sont localement significatives. En utilisant une méthode d'inversion du type Monte-Carlo, nous avons comparé, pour les deux modèles WIFF aux échelles macroscopique et microscopique, leur capacité à contraindre les modules élastiques de la matrice sèche et la perméabilité ainsi que leur distribution de probabilité locale. Dans une seconde partie de cette thèse, nous cherchons une solution pour déterminer la dimension d'un volume élémentaire représentatif (noté VER). Cette problématique se pose dans les procédures numériques de changement d'échelle pour déterminer l'atténuation effective et la dispersion effective de la vitesse sismique dans un milieu hétérogène. Pour ce faire, nous nous concentrons sur un ensemble d'échantillons de roches synthétiques idéalisés incluant des strates, des fissures, ou une saturation partielle à l'échelle mésoscopique. Ces scénarios sont hautement pertinents, car ils sont associés à un taux très élevé d'atténuation et de dispersion des vitesses causé par les WIFF à l'échelle mésoscopique. L'enjeu de déterminer la dimension d'un VER pour une roche hétérogène est très complexe et encore inexploré dans le contexte actuel. Dans cette étude-pilote, nous nous focalisons sur des milieux périodiques, qui assurent l'autosimilarité des échantillons considérés indépendamment de leur taille. Ainsi, nous simplifions le problème à une analyse systématique de la dépendance de la dimension des VER aux conditions aux limites appliquées. Nos résultats indiquent que les effets des conditions aux limites sont absents pour un milieu stratifié, et négligeables pour un milieu à saturation partielle : cela résultant à des dimensions petites des VER. Au contraire, de forts effets des conditions aux limites apparaissent dans les milieux présentant une distribution périodique de fissures de taille finie : cela conduisant à de grandes dimensions des VER. Dans la troisième partie de cette thèse, nous proposons un nouveau modèle poro- élastique effectif, pour les milieux périodiques caractérisés par une stratification mésoscopique, qui prendra en compte les WIFF à la fois aux échelles mésoscopique et macroscopique, ainsi que l'anisotropie associée à ces strates. Ce modèle prédit alors avec exactitude l'existence des ondes P rapides et lentes ainsi que les quasis et pures ondes S, pour toutes les directions de propagation de l'onde, tant que la longueur d'onde correspondante est bien plus grande que l'épaisseur de la strate. L'intérêt principal de ce travail est que, pour les formations à perméabilité moyenne à élevée, comme, par exemple, les sédiments non- consolidés, les grès ou encore les roches fissurées, ces deux mécanismes d'WIFF peuvent avoir lieu à des fréquences similaires. Or, ce scénario, qui est assez commun, n'est pas décrit par les modèles existants pour les milieux poreux stratifiés. Les comparaisons des solutions analytiques des vitesses des ondes P et S et de l'atténuation de la propagation des ondes perpendiculaires à la stratification, avec les solutions obtenues à partir de simulations numériques en éléments finis, fondées sur une solution obtenue en 1D des équations poro- élastiques, montrent un très bon accord, tant que l'hypothèse des grandes longueurs d'onde reste valable. Il y a cependant une limitation de ce modèle qui est liée à son incapacité à prendre en compte les effets inertiels dans les WIFF mésoscopiques quand les deux mécanismes d'WIFF prédominent à des fréquences similaires. Néanmoins, nos résultats montrent aussi que l'erreur associée est relativement faible, même à des fréquences à laquelle sont attendus les deux effets d'inertie et de diffusion, indiquant que le modèle proposé fournit une solution qui est remarquablement proche de sa référence numérique.

Dry months in the agricultural region of Ribeirão Preto, state of São Paulo-Brazil: an study based on the extreme value theory

The application of the Extreme Value Theory (EVT) to model the probability of occurrence of extreme low Standardized Precipitation Index (SPI) values leads to an increase of the knowledge related to the occurrence of extreme dry months. This sort of analysis can be carried out by means of two approaches: the block maxima (BM; associated with the General Extreme Value distribution) and the peaks-over-threshold (POT; associated with the Generalized Pareto distribution). Each of these procedures has its own advantages and drawbacks. Thus, the main goal of this study is to compare the performance of BM and POT in characterizing the probability of occurrence of extreme dry SPI values obtained from the weather station of Ribeirão Preto-SP (1937-2012). According to the goodness-of-fit tests, both BM and POT can be used to assess the probability of occurrence of the aforementioned extreme dry SPI monthly values. However, the scalar measures of accuracy and the return level plots indicate that POT provides the best fit distribution. The study also indicated that the uncertainties in the parameters estimates of a probabilistic model should be taken into account when the probability associated with a severe/extreme dry event is under analysis.

Irrégularités dans la distribution des nombres premiers et des suites plus générales dans les progressions arithmétiques

Le sujet principal de cette thèse est la distribution des nombres premiers dans les progressions arithmétiques, c'est-à-dire des nombres premiers de la forme $qn+a$, avec $a$ et $q$ des entiers fixés et $n=1,2,3,\dots$ La thèse porte aussi sur la comparaison de différentes suites arithmétiques par rapport à leur comportement dans les progressions arithmétiques. Elle est divisée en quatre chapitres et contient trois articles. Le premier chapitre est une invitation à la théorie analytique des nombres, suivie d'une revue des outils qui seront utilisés plus tard. Cette introduction comporte aussi certains résultats de recherche, que nous avons cru bon d'inclure au fil du texte. Le deuxième chapitre contient l'article \emph{Inequities in the Shanks-Rényi prime number race: an asymptotic formula for the densities}, qui est le fruit de recherche conjointe avec le professeur Greg Martin. Le but de cet article est d'étudier un phénomène appelé le <>, qui s'observe dans les <>. Chebyshev a observé qu'il semble y avoir plus de premiers de la forme $4n+3$ que de la forme $4n+1$. De manière plus générale, Rubinstein et Sarnak ont montré l'existence d'une quantité $\delta(q;a,b)$, qui désigne la probabilité d'avoir plus de premiers de la forme $qn+a$ que de la forme $qn+b$. Dans cet article nous prouvons une formule asymptotique pour $\delta(q;a,b)$ qui peut être d'un ordre de précision arbitraire (en terme de puissance négative de $q$). Nous présentons aussi des résultats numériques qui supportent nos formules. Le troisième chapitre contient l'article \emph{Residue classes containing an unexpected number of primes}. Le but est de fixer un entier $a\neq 0$ et ensuite d'étudier la répartition des premiers de la forme $qn+a$, en moyenne sur $q$. Nous montrons que l'entier $a$ fixé au départ a une grande influence sur cette répartition, et qu'il existe en fait certaines progressions arithmétiques contenant moins de premiers que d'autres. Ce phénomène est plutôt surprenant, compte tenu du théorème des premiers dans les progressions arithmétiques qui stipule que les premiers sont équidistribués dans les classes d'équivalence $\bmod q$. Le quatrième chapitre contient l'article \emph{The influence of the first term of an arithmetic progression}. Dans cet article on s'intéresse à des irrégularités similaires à celles observées au troisième chapitre, mais pour des suites arithmétiques plus générales. En effet, nous étudions des suites telles que les entiers s'exprimant comme la somme de deux carrés, les valeurs d'une forme quadratique binaire, les $k$-tuplets de premiers et les entiers sans petit facteur premier. Nous démontrons que dans chacun de ces exemples, ainsi que dans une grande classe de suites arithmétiques, il existe des irrégularités dans les progressions arithmétiques $a\bmod q$, avec $a$ fixé et en moyenne sur $q$.