Asymptotics of the partition function for random matrices via Riemann-Hilbert techniques, and applications to graphical enumeration

We study the partition function from random matrix theory using a well known connection to orthogonal polynomials, and a recently developed Riemann-Hilbert approach to the computation of detailed asymptotics for these orthogonal polynomials. We obtain the first proof of a complete large N expansion for the partition function, for a general class of probability measures on matrices, originally conjectured by Bessis, Itzykson, and Zuber. We prove that the coefficients in the asymptotic expansion are analytic functions of parameters in the original probability measure, and that they are generating functions for the enumeration of labelled maps according to genus and valence. Central to the analysis is a large N expansion for the mean density of eigenvalues, uniformly valid on the entire real axis.Comment: 44 pages, 4 figures. To appear, International Mathematics Research Notice

Topological Phenomena in the Real Periodic Sine-Gordon Theory

The set of real finite-gap Sine-Gordon solutions corresponding to a fixed spectral curve consists of several connected components. A simple explicit description of these components obtained by the authors recently is used to study the consequences of this property. In particular this description allows to calculate the topological charge of solutions (the averaging of the $x$-derivative of the potential) and to show that the averaging of other standard conservation laws is the same for all components.Comment: LaTeX, 18 pages, 3 figure

Novel microwave apparatus for breast lesions detection: Preliminary clinical results

This paper presents preliminary results of an innovative microwave imaging apparatus for breast lesions detection. Specially, a Huygens Principle based method is employed to process the microwave signals and to build the respective microwave images. The apparatus has been first tested on phantoms. Next, its performance has been verified through clinical examinations on 22 healthy breasts and on 29 breast having lesions, using as gold standard the output of the radiologist study review obtained using conventional techniques. Specifically, we introduce a metric, which is the ratio between maximum and average of the image intensity (MAX/AVG). We found that MAX/AVG of microwave images can be used for classifying breasts containing lesions. In addition, using MAX/AVG as classification parameter, receiver operating characteristic curves have been empirically determined. Furthermore, for one randomly selected breast having lesion, we have demonstrated that the localisation of the inclusion acquired through microwave imaging is compatible with mammography images

On Soliton-type Solutions of Equations Associated with N-component Systems

The algebraic geometric approach to $N$-component systems of nonlinear integrable PDE's is used to obtain and analyze explicit solutions of the coupled KdV and Dym equations. Detailed analysis of soliton fission, kink to anti-kink transitions and multi-peaked soliton solutions is carried out. Transformations are used to connect these solutions to several other equations that model physical phenomena in fluid dynamics and nonlinear optics.Comment: 43 pages, 16 figure

Large N expansion of the 2-matrix model

We present a method, based on loop equations, to compute recursively all the terms in the large $N$ topological expansion of the free energy for the 2-hermitian matrix model. We illustrate the method by computing the first subleading term, i.e. the free energy of a statistical physics model on a discretized torus.Comment: 41 pages, 9 figures eps

Modelado e impresión 3D de un conjunto mecánico para la práctica del croquizado

En esta ponencia se presentan los trabajos de modelado y de impresión tridimensional de un conjunto mecánico para ser empleado como insumo didáctico en las prácticas de croquizado que realizan los alumnos de la carrera de Ingeniería Mecánica pertenecientes a la cátedra de Diseño Mecánico Asistido del Departamento de Ingeniería de la Universidad Nacional del Sur (UNS). En virtud del inconveniente que se presenta al no contar con una disponibilidad y variedad de conjuntos para que sean utilizados en las prácticas de croquizado, se llevan adelante tareas de reemplazo de los modelos reales por otros realizados digitalmente e impresos en forma tridimensional. Las tareas de croquizado implican un análisis de la forma, proporción y dimensiones de cada una de las piezas que componen un conjunto mecánico, debido a ello resulta necesario disponer de una cantidad suficiente de piezas para que los alumnos puedan manipularlas adecuadamente. Se posee la información completa de los planos de despiece de una corredera y de su lista de materiales, a partir de las cuales se procede a realizar el modelado individual de cada una las piezas que forman el conjunto empleando una versión de prueba del programa SolidWorks. Asimismo, a partir de una pieza base se realiza una minuciosa tarea de verificación de las medidas de ensamble entre las distintas piezas y el cálculo de interferencias. Las piezas se imprimen en 3D por pares para continuar verificando el ensamble y luego siguiendo el orden de ensamblado hasta terminar de imprimir todo el conjunto. Finalmente, los alumnos disponen del modelo impreso en 3D para realizar las siquientes tareas: croquizado de cada una de las piezas con la correspondiente elección de vistas, cortes y detalles, acotación y medición con calibre, incorporación de medidas con tolerancia, empleo de simbología de acabado de superficie y representación de roscas normalizadas. Asimismo, con la utilización del programa los alumnos pueden previsualizar la dinámica del conjunto en forma tridimensional por medio de una animación controlada.This paper presents the modeling and three-dimensional printing work of a mechanical assembly to be used as a didactic input in the sketching practices carried out by the students of the Department of Engineering at the Universidad Nacional del Sur. It has the complete information of the exploded drawings of a slide, from which the individual modeling of each one of the pieces that make up the set is carried out. Students have the model printed in 3D to perform the following tasks: sketching, measurement, incorporation of measures with tolerance, use of surface finish symbology and representation of standardized threads.Facultad de Arquitectura y Urbanism