Learning Bayesian networks from data by the incremental compilation of new network polynomials

Probabilistic graphical models are a huge research field in artificial intelligence nowadays. The scope of this work is the study of directed graphical models for the representation of discrete distributions. Two of the main research topics related to this area focus on performing inference over graphical models and on learning graphical models from data. Traditionally, the inference process and the learning process have been treated separately, but given that the learned models structure marks the inference complexity, this kind of strategies will sometimes produce very inefficient models. With the purpose of learning thinner models, in this master thesis we propose a new model for the representation of network polynomials, which we call polynomial trees. Polynomial trees are a complementary representation for Bayesian networks that allows an efficient evaluation of the inference complexity and provides a framework for exact inference. We also propose a set of methods for the incremental compilation of polynomial trees and an algorithm for learning polynomial trees from data using a greedy score+search method that includes the inference complexity as a penalization in the scoring function.

Differential elimination by differential specialization of Sylvester style matrices

Differential resultant formulas are defined, for a system $\cP$ of $n$ ordinary Laurent differential polynomials in $n-1$ differential variables. These are determinants of coefficient matrices of an extended system of polynomials obtained from $\cP$ through derivations and multiplications by Laurent monomials. To start, through derivations, a system $\ps(\cP)$ of $L$ polynomials in $L-1$ algebraic variables is obtained, which is non sparse in the order of derivation. This enables the use of existing formulas for the computation of algebraic resultants, of the multivariate sparse algebraic polynomials in $\ps(\cP)$, to obtain polynomials in the differential elimination ideal generated by $\cP$. The formulas obtained are multiples of the sparse differential resultant defined by Li, Yuan and Gao, and provide order and degree bounds in terms of mixed volumes in the generic case.

Medidas autosemejantes en el plano, momentos y matrices de Hessenberg

La tesis MEDIDAS AUTOSEMEJANTES EN EL PLANO, MOMENTOS Y MATRICES DE HESSENBERG se enmarca entre las áreas de la teoría geométrica de la medida, la teoría de polinomios ortogonales y la teoría de operadores. La memoria aborda el estudio de medidas con soporte acotado en el plano complejo vistas con la óptica de las matrices infinitas de momentos y de Hessenberg asociadas a estas medidas que en la teoría de los polinomios ortogonales las representan. En particular se centra en el estudio de las medidas autosemejantes que son las medidas de equilibrio definidas por un sistema de funciones iteradas (SFI). Los conjuntos autosemejantes son conjuntos que tienen la propiedad geométrica de descomponerse en unión de piezas semejantes al conjunto total. Estas piezas pueden solaparse o no, cuando el solapamiento es pequeño la teoría de Hutchinson [Hut81] funciona bien, pero cuando no existen restricciones falla. El problema del solapamiento consiste en controlar la medida de este solapamiento. Un ejemplo de la complejidad de este problema se plantea con las convoluciones infinitas de distribuciones de Bernoulli, que han resultado ser un ejemplo de medidas autosemejantes en el caso real. En 1935 Jessen y A. Wintner [JW35] ya se planteaba este problema, lejos de ser sencillo ha sido estudiado durante más de setenta y cinco años y siguen sin resolverse las principales cuestiones planteadas ya por A. Garsia [Gar62] en 1962. El interés que ha despertado este problema así como la complejidad del mismo está demostrado por las numerosas publicaciones que abordan cuestiones relacionadas con este problema ver por ejemplo [JW35], [Erd39], [PS96], [Ma00], [Ma96], [Sol98], [Mat95], [PS96], [Sim05],[JKS07] [JKS11]. En el primer capítulo comenzamos introduciendo con detalle las medidas autosemejante en el plano complejo y los sistemas de funciones iteradas, así como los conceptos de la teoría de la medida necesarios para describirlos. A continuación se introducen las herramientas necesarias de teoría de polinomios ortogonales, matrices infinitas y operadores que se van a usar. En el segundo y tercer capítulo trasladamos las propiedades geométricas de las medidas autosemejantes a las matrices de momentos y de Hessenberg, respectivamente. A partir de estos resultados se describen algoritmos para calcular estas matrices a partir del SFI correspondiente. Concretamente, se obtienen fórmulas explícitas y algoritmos de aproximación para los momentos y matrices de momentos de medidas fractales, a partir de un teorema del punto fijo para las matrices. Además utilizando técnicas de la teoría de operadores, se han extendido al plano complejo los resultados que G. Mantica [Ma00, Ma96] obtenía en el caso real. Este resultado es la base para definir un algoritmo estable de aproximación de la matriz de Hessenberg asociada a una medida fractal u obtener secciones finitas exactas de matrices Hessenberg asociadas a una suma de medidas. En el último capítulo, se consideran medidas, μ, más generales y se estudia el comportamiento asintótico de los autovalores de una matriz hermitiana de momentos y su impacto en las propiedades de la medida asociada. En el resultado central se demuestra que si los polinomios asociados son densos en L2(μ) entonces necesariamente el autovalor mínimo de las secciones finitas de la matriz de momentos de la medida tiende a cero. ABSTRACT The Thesis work “Self-similar Measures on the Plane, Moments and Hessenberg Matrices” is framed among the geometric measure theory, orthogonal polynomials and operator theory. The work studies measures with compact support on the complex plane from the point of view of the associated infinite moments and Hessenberg matrices representing them in the theory of orthogonal polynomials. More precisely, it concentrates on the study of the self-similar measures that are equilibrium measures in a iterated functions system. Self-similar sets have the geometric property of being decomposable in a union of similar pieces to the complete set. These pieces can overlap. If the overlapping is small, Hutchinson’s theory [Hut81] works well, however, when it has no restrictions, the theory does not hold. The overlapping problem consists in controlling the measure of the overlap. The complexity of this problem is exemplified in the infinite convolutions of Bernoulli’s distributions, that are an example of self-similar measures in the real case. As early as 1935 [JW35], Jessen and Wintner posed this problem, that far from being simple, has been studied during more than 75 years. The main cuestiones posed by Garsia in 1962 [Gar62] remain unsolved. The interest in this problem, together with its complexity, is demonstrated by the number of publications that over the years have dealt with it. See, for example, [JW35], [Erd39], [PS96], [Ma00], [Ma96], [Sol98], [Mat95], [PS96], [Sim05], [JKS07] [JKS11]. In the first chapter, we will start with a detailed introduction to the self-similar measurements in the complex plane and to the iterated functions systems, also including the concepts of measure theory needed to describe them. Next, we introduce the necessary tools from orthogonal polynomials, infinite matrices and operators. In the second and third chapter we will translate the geometric properties of selfsimilar measures to the moments and Hessenberg matrices. From these results, we will describe algorithms to calculate these matrices from the corresponding iterated functions systems. To be precise, we obtain explicit formulas and approximation algorithms for the moments and moment matrices of fractal measures from a new fixed point theorem for matrices. Moreover, using techniques from operator theory, we extend to the complex plane the real case results obtained by Mantica [Ma00, Ma96]. This result is the base to define a stable algorithm that approximates the Hessenberg matrix associated to a fractal measure and obtains exact finite sections of Hessenberg matrices associated to a sum of measurements. In the last chapter, we consider more general measures, μ, and study the asymptotic behaviour of the eigenvalues of a hermitian matrix of moments, together with its impact on the properties of the associated measure. In the main result we demonstrate that, if the associated polynomials are dense in L2(μ), then necessarily follows that the minimum eigenvalue of the finite sections of the moments matrix goes to zero.

Contribución al estudio del criptoanálisis y diseño de los criptosistemas caóticos

The extraordinary increase of new information technologies, the development of Internet, the electronic commerce, the e-government, mobile telephony and future cloud computing and storage, have provided great benefits in all areas of society. Besides these, there are new challenges for the protection of information, such as the loss of confidentiality and integrity of electronic documents. Cryptography plays a key role by providing the necessary tools to ensure the safety of these new media. It is imperative to intensify the research in this area, to meet the growing demand for new secure cryptographic techniques. The theory of chaotic nonlinear dynamical systems and the theory of cryptography give rise to the chaotic cryptography, which is the field of study of this thesis. The link between cryptography and chaotic systems is still subject of intense study. The combination of apparently stochastic behavior, the properties of sensitivity to initial conditions and parameters, ergodicity, mixing, and the fact that periodic points are dense, suggests that chaotic orbits resemble random sequences. This fact, and the ability to synchronize multiple chaotic systems, initially described by Pecora and Carroll, has generated an avalanche of research papers that relate cryptography and chaos. The chaotic cryptography addresses two fundamental design paradigms. In the first paradigm, chaotic cryptosystems are designed using continuous time, mainly based on chaotic synchronization techniques; they are implemented with analog circuits or by computer simulation. In the second paradigm, chaotic cryptosystems are constructed using discrete time and generally do not depend on chaos synchronization techniques. The contributions in this thesis involve three aspects about chaotic cryptography. The first one is a theoretical analysis of the geometric properties of some of the most employed chaotic attractors for the design of chaotic cryptosystems. The second one is the cryptanalysis of continuos chaotic cryptosystems and finally concludes with three new designs of cryptographically secure chaotic pseudorandom generators. The main accomplishments contained in this thesis are: v Development of a method for determining the parameters of some double scroll chaotic systems, including Lorenz system and Chua’s circuit. First, some geometrical characteristics of chaotic system have been used to reduce the search space of parameters. Next, a scheme based on the synchronization of chaotic systems was built. The geometric properties have been employed as matching criterion, to determine the values of the parameters with the desired accuracy. The method is not affected by a moderate amount of noise in the waveform. The proposed method has been applied to find security flaws in the continuous chaotic encryption systems. Based on previous results, the chaotic ciphers proposed by Wang and Bu and those proposed by Xu and Li are cryptanalyzed. We propose some solutions to improve the cryptosystems, although very limited because these systems are not suitable for use in cryptography. Development of a method for determining the parameters of the Lorenz system, when it is used in the design of two-channel cryptosystem. The method uses the geometric properties of the Lorenz system. The search space of parameters has been reduced. Next, the parameters have been accurately determined from the ciphertext. The method has been applied to cryptanalysis of an encryption scheme proposed by Jiang. In 2005, Gunay et al. proposed a chaotic encryption system based on a cellular neural network implementation of Chua’s circuit. This scheme has been cryptanalyzed. Some gaps in security design have been identified. Based on the theoretical results of digital chaotic systems and cryptanalysis of several chaotic ciphers recently proposed, a family of pseudorandom generators has been designed using finite precision. The design is based on the coupling of several piecewise linear chaotic maps. Based on the above results a new family of chaotic pseudorandom generators named Trident has been designed. These generators have been specially designed to meet the needs of real-time encryption of mobile technology. According to the above results, this thesis proposes another family of pseudorandom generators called Trifork. These generators are based on a combination of perturbed Lagged Fibonacci generators. This family of generators is cryptographically secure and suitable for use in real-time encryption. Detailed analysis shows that the proposed pseudorandom generator can provide fast encryption speed and a high level of security, at the same time. El extraordinario auge de las nuevas tecnologías de la información, el desarrollo de Internet, el comercio electrónico, la administración electrónica, la telefonía móvil y la futura computación y almacenamiento en la nube, han proporcionado grandes beneficios en todos los ámbitos de la sociedad. Junto a éstos, se presentan nuevos retos para la protección de la información, como la suplantación de personalidad y la pérdida de la confidencialidad e integridad de los documentos electrónicos. La criptografía juega un papel fundamental aportando las herramientas necesarias para garantizar la seguridad de estos nuevos medios, pero es imperativo intensificar la investigación en este ámbito para dar respuesta a la demanda creciente de nuevas técnicas criptográficas seguras. La teoría de los sistemas dinámicos no lineales junto a la criptografía dan lugar a la ((criptografía caótica)), que es el campo de estudio de esta tesis. El vínculo entre la criptografía y los sistemas caóticos continúa siendo objeto de un intenso estudio. La combinación del comportamiento aparentemente estocástico, las propiedades de sensibilidad a las condiciones iniciales y a los parámetros, la ergodicidad, la mezcla, y que los puntos periódicos sean densos asemejan las órbitas caóticas a secuencias aleatorias, lo que supone su potencial utilización en el enmascaramiento de mensajes. Este hecho, junto a la posibilidad de sincronizar varios sistemas caóticos descrita inicialmente en los trabajos de Pecora y Carroll, ha generado una avalancha de trabajos de investigación donde se plantean muchas ideas sobre la forma de realizar sistemas de comunicaciones seguros, relacionando así la criptografía y el caos. La criptografía caótica aborda dos paradigmas de diseño fundamentales. En el primero, los criptosistemas caóticos se diseñan utilizando circuitos analógicos, principalmente basados en las técnicas de sincronización caótica; en el segundo, los criptosistemas caóticos se construyen en circuitos discretos u ordenadores, y generalmente no dependen de las técnicas de sincronización del caos. Nuestra contribución en esta tesis implica tres aspectos sobre el cifrado caótico. En primer lugar, se realiza un análisis teórico de las propiedades geométricas de algunos de los sistemas caóticos más empleados en el diseño de criptosistemas caóticos vii continuos; en segundo lugar, se realiza el criptoanálisis de cifrados caóticos continuos basados en el análisis anterior; y, finalmente, se realizan tres nuevas propuestas de diseño de generadores de secuencias pseudoaleatorias criptográficamente seguros y rápidos. La primera parte de esta memoria realiza un análisis crítico acerca de la seguridad de los criptosistemas caóticos, llegando a la conclusión de que la gran mayoría de los algoritmos de cifrado caóticos continuos —ya sean realizados físicamente o programados numéricamente— tienen serios inconvenientes para proteger la confidencialidad de la información ya que son inseguros e ineficientes. Asimismo una gran parte de los criptosistemas caóticos discretos propuestos se consideran inseguros y otros no han sido atacados por lo que se considera necesario más trabajo de criptoanálisis. Esta parte concluye señalando las principales debilidades encontradas en los criptosistemas analizados y algunas recomendaciones para su mejora. En la segunda parte se diseña un método de criptoanálisis que permite la identificaci ón de los parámetros, que en general forman parte de la clave, de algoritmos de cifrado basados en sistemas caóticos de Lorenz y similares, que utilizan los esquemas de sincronización excitador-respuesta. Este método se basa en algunas características geométricas del atractor de Lorenz. El método diseñado se ha empleado para criptoanalizar eficientemente tres algoritmos de cifrado. Finalmente se realiza el criptoanálisis de otros dos esquemas de cifrado propuestos recientemente. La tercera parte de la tesis abarca el diseño de generadores de secuencias pseudoaleatorias criptográficamente seguras, basadas en aplicaciones caóticas, realizando las pruebas estadísticas, que corroboran las propiedades de aleatoriedad. Estos generadores pueden ser utilizados en el desarrollo de sistemas de cifrado en flujo y para cubrir las necesidades del cifrado en tiempo real. Una cuestión importante en el diseño de sistemas de cifrado discreto caótico es la degradación dinámica debida a la precisión finita; sin embargo, la mayoría de los diseñadores de sistemas de cifrado discreto caótico no ha considerado seriamente este aspecto. En esta tesis se hace hincapié en la importancia de esta cuestión y se contribuye a su esclarecimiento con algunas consideraciones iniciales. Ya que las cuestiones teóricas sobre la dinámica de la degradación de los sistemas caóticos digitales no ha sido totalmente resuelta, en este trabajo utilizamos algunas soluciones prácticas para evitar esta dificultad teórica. Entre las técnicas posibles, se proponen y evalúan varias soluciones, como operaciones de rotación de bits y desplazamiento de bits, que combinadas con la variación dinámica de parámetros y con la perturbación cruzada, proporcionan un excelente remedio al problema de la degradación dinámica. Además de los problemas de seguridad sobre la degradación dinámica, muchos criptosistemas se rompen debido a su diseño descuidado, no a causa de los defectos esenciales de los sistemas caóticos digitales. Este hecho se ha tomado en cuenta en esta tesis y se ha logrado el diseño de generadores pseudoaleatorios caóticos criptogr áficamente seguros.

Comment on "From geodesics of the multipole solutions to the perturbed Kepler problem"

In this Comment we explain the discrepancies mentioned by the authors between their results and ours about the in?uence of the gravitational quadrupole moment in the perturbative calculation of corrections to the precession of the periastron of quasielliptical Keplerian equatorial orbits around a point mass. The discrepancy appears to be a consequence of two different calculations of the angular momentum of the orbits.

Perturbed angular correlation study of Ta-181-doped PbTi1-xHfxO3 compounds

In this work, the hyperfine quadrupole interaction at Ta-doped PbTi1-xHfxO3 polycrystalline samples is studied for the first time. Powders with x=0.25, 0.50 and 0.75 were prepared and characterized by X-ray diffraction analysis. Perturbed Angular Correlation (PAC) analyses were done as a function of temperature, using low concentration Ta-181 nuclei as probes. In the ferroelectric and paraelectric phases of these compounds two sites were occupied by the probes. For each site the quadrupole frequency, asymmetry and relative distribution width parameters were obtained as a function of temperature above and below the Curie temperature (T-C). One of these sites was assigned to the regular Ti-Hf site, while the other one was assigned to some kind of defect. The behavior of the hyperfine parameters as a function of temperature was analyzed in terms of a recent published phase diagram and the presence of disorder below and above T-C. For the three compositions measured, the obtained hyperfine parameters present discontinuities which correspond to the ferroelectric-paraelectric phase transition. In both phases it was found broad frequency distributed interactions. The disorder in the electronic distribution would be responsible for the broad line width of the hyperfine interaction. (C) 2012 Elsevier B.V. All rights reserved.

Differential resultants of super essential systems of linear OD-polynomials

The sparse differential resultant dres(P) of an overdetermined system P of generic nonhomogeneous ordinary differential polynomials, was formally defined recently by Li, Gao and Yuan (2011). In this note, a differential resultant formula dfres(P) is defined and proved to be nonzero for linear "super essential" systems. In the linear case, dres(P) is proved to be equal, up to a nonzero constant, to dfres(P*) for the supper essential subsystem P* of P.

A new set of integrals of motion to propagate the perturbed two-body problem

A formulation of the perturbed two-body problem that relies on a new set of orbital elements is presented. The proposed method represents a generalization of the special perturbation method published by Peláez et al. (Celest Mech Dyn Astron 97(2):131?150,2007) for the case of a perturbing force that is partially or totally derivable from a potential. We accomplish this result by employing a generalized Sundman time transformation in the framework of the projective decomposition, which is a known approach for transforming the two-body problem into a set of linear and regular differential equations of motion. Numerical tests, carried out with examples extensively used in the literature, show the remarkable improvement of the performance of the new method for different kinds of perturbations and eccentricities. In particular, one notable result is that the quadratic dependence of the position error on the time-like argument exhibited by Peláez?s method for near-circular motion under the J2 perturbation is transformed into linear.Moreover, themethod reveals to be competitive with two very popular elementmethods derived from theKustaanheimo-Stiefel and Sperling-Burdet regularizations.

Conditional density approximations with mixtures of polynomials

Mixtures of polynomials (MoPs) are a non-parametric density estimation technique especially designed for hybrid Bayesian networks with continuous and discrete variables. Algorithms to learn one- and multi-dimensional (marginal) MoPs from data have recently been proposed. In this paper we introduce two methods for learning MoP approximations of conditional densities from data. Both approaches are based on learning MoP approximations of the joint density and the marginal density of the conditioning variables, but they differ as to how the MoP approximation of the quotient of the two densities is found. We illustrate and study the methods using data sampled from known parametric distributions, and we demonstrate their applicability by learning models based on real neuroscience data. Finally, we compare the performance of the proposed methods with an approach for learning mixtures of truncated basis functions (MoTBFs). The empirical results show that the proposed methods generally yield models that are comparable to or significantly better than those found using the MoTBF-based method.

A finite class of orthogonal functions generated by Routh-Romanovski polynomials

It is known that some orthogonal systems are mapped onto other orthogonal systems by the Fourier transform. In this article we introduce a finite class of orthogonal functions, which is the Fourier transform of Routh-Romanovski orthogonal polynomials, and obtain its orthogonality relation using Parseval identity.

A parallel implementation of 3D Zernike moment analysis

Zernike polynomials are a well known set of functions that find many applications in image or pattern characterization because they allow to construct shape descriptors that are invariant against translations, rotations or scale changes. The concepts behind them can be extended to higher dimension spaces, making them also fit to describe volumetric data. They have been less used than their properties might suggest due to their high computational cost. We present a parallel implementation of 3D Zernike moments analysis, written in C with CUDA extensions, which makes it practical to employ Zernike descriptors in interactive applications, yielding a performance of several frames per second in voxel datasets about 2003 in size. In our contribution, we describe the challenges of implementing 3D Zernike analysis in a general-purpose GPU. These include how to deal with numerical inaccuracies, due to the high precision demands of the algorithm, or how to deal with the high volume of input data so that it does not become a bottleneck for the system.

Bernstein polynomials in element-free Galerkin method

In the recent decades, meshless methods (MMs), like the element-free Galerkin method (EFGM), have been widely studied and interesting results have been reached when solving partial differential equations. However, such solutions show a problem around boundary conditions, where the accuracy is not adequately achieved. This is caused by the use of moving least squares or residual kernel particle method methods to obtain the shape functions needed in MM, since such methods are good enough in the inner of the integration domains, but not so accurate in boundaries. This way, Bernstein curves, which are a partition of unity themselves,can solve this problem with the same accuracy in the inner area of the domain and at their boundaries.

On a mathematical model arising in MHD perturbed equilibrium for Stellarator devices. A numerical approach.

Abstract?We consider a mathematical model related to the stationary regime of a plasma of fusion nuclear, magnetically confined in a Stellarator device. Using the geometric properties of the fusion device, a suitable system of coordinates and averaging methods, the mathematical problem may be reduced to a two dimensional free boundary problem of nonlocal type, where the corresponding differential equation is of the Grad?Shafranov type. The current balance within each flux magnetic gives us the possibility to define the third covariant magnetic field component with respect to the averaged poloidal flux function. We present here some numerical experiences and we give some numerical approach for the averaged poloidal flux and for the third covariant magnetic field component.

Perfect imaging of three object points with only two analytic lens surfaces in two dimensions

In this work, a new two-dimensional analytic optics design method is presented that enables the coupling of three ray sets with two lens profiles. This method is particularly promising for optical systems designed for wide field of view and with clearly separated optical surfaces. However, this coupling can only be achieved if different ray sets will use different portions of the second lens profile. Based on a very basic example of a single thick lens, the Simultaneous Multiple Surfaces design method in two dimensions (SMS2D) will help to provide a better understanding of the practical implications on the design process by an increased lens thickness and a wider field of view. Fermat?s principle is used to deduce a set of functional differential equations fully describing the entire optical system. The transformation of these functional differential equations into an algebraic linear system of equations allows the successive calculation of the Taylor series coefficients up to an arbitrary order. The evaluation of the solution space reveals the wide range of possible lens configurations covered by this analytic design method. Ray tracing analysis for calculated 20th order Taylor polynomials demonstrate excellent performance and the versatility of this new analytical optics design concept.

Simplified Finite Element Assessment of Beams With and Without Shear Deformation

This paper presents a simplified finite element (FE) methodology for solving accurately beam models with (Timoshenko) and without (Bernoulli-Euler) shear deformation. Special emphasis is made on showing how it is possible to obtain the exact solution on the nodes and a good accuracy inside the element. The proposed simplifying concept, denominated as the equivalent distributed load (EDL) of any order, is based on the use of Legendre orthogonal polynomials to approximate the original or acting load for computing the results between the nodes. The 1-span beam examples show that this is a promising procedure that allows the aim of using either one FE and an EDL of slightly higher order or by using an slightly larger number of FEs leaving the EDL in the lowest possible order assumed by definition to be equal to 4 independently of how irregular the beam is loaded.