On Functional Decomposition of Multivariate Polynomials with Differentiation and Homogenization

In this paper, we give a theoretical analysis for the algorithms to compute functional decomposition for multivariate polynomials based on differentiation and homogenization which are proposed by Ye, Dai, Lam (1999) and Faug$\mu$ere, Perret (2006, 2008, 2009). We show that a degree proper functional decomposition for a set of randomly decomposable quartic homogenous polynomials can be computed using the algorithm with high probability. This solves a conjecture proposed by Ye, Dai, and Lam (1999). We also propose a conjecture such that the decomposition for a set of polynomials can be computed from that of its homogenization with high probability. Finally, we prove that the right decomposition factors for a set of polynomials can be computed from its right decomposition factor space. Combining these results together, we prove that the algorithm can compute a degree proper decomposition for a set of randomly decomposable quartic polynomials with probability one when the base field is of characteristic zero, and with probability close to one when the base field is a finite field with sufficiently large number under the assumption that the conjeture is correct

A Complete Characterization of the Gap between Convexity and SOS-Convexity

Our first contribution in this paper is to prove that three natural sum of squares (sos) based sufficient conditions for convexity of polynomials, via the definition of convexity, its first order characterization, and its second order characterization, are equivalent. These three equivalent algebraic conditions, henceforth referred to as sos-convexity, can be checked by semidefinite programming whereas deciding convexity is NP-hard. If we denote the set of convex and sos-convex polynomials in $n$ variables of degree $d$ with $\tilde{C}_{n,d}$ and $\tilde{\Sigma C}_{n,d}$ respectively, then our main contribution is to prove that $\tilde{C}_{n,d}=\tilde{\Sigma C}_{n,d}$ if and only if $n=1$ or $d=2$ or $(n,d)=(2,4)$. We also present a complete characterization for forms (homogeneous polynomials) except for the case $(n,d)=(3,4)$ which is joint work with G. Blekherman and is to be published elsewhere. Our result states that the set $C_{n,d}$ of convex forms in $n$ variables of degree $d$ equals the set $\Sigma C_{n,d}$ of sos-convex forms if and only if $n=2$ or $d=2$ or $(n,d)=(3,4)$. To prove these results, we present in particular explicit examples of polynomials in $\tilde{C}_{2,6}\setminus\tilde{\Sigma C}_{2,6}$ and $\tilde{C}_{3,4}\setminus\tilde{\Sigma C}_{3,4}$ and forms in $C_{3,6}\setminus\Sigma C_{3,6}$ and $C_{4,4}\setminus\Sigma C_{4,4}$, and a general procedure for constructing forms in $C_{n,d+2}\setminus\Sigma C_{n,d+2}$ from nonnegative but not sos forms in $n$ variables and degree $d$. Although for disparate reasons, the remarkable outcome is that convex polynomials (resp. forms) are sos-convex exactly in cases where nonnegative polynomials (resp. forms) are sums of squares, as characterized by Hilbert.Comment: 25 pages; minor editorial revisions made; formal certificates for computer assisted proofs of the paper added to arXi

Portraying urban diversity patterns through exploratory data analysis

This thesis analyzes the complexity of the urban system, being described with multiple variables that represent the environmental, economic, and social characters of the city. The portrayal of the urban diversity and its relationship with a better response of the city to disturbances, hence to its sustainability, is the main motivation of the study. Certainly, this thesis aims to provide theoretical knowledge through the application of statistical and computational methodologies that are developed progressively in its chapters. Beginning with the introduction, which draws the city as an abstract urban system and reviews the concepts and measures of diversity within the theoretical frameworks of sustainability, urban ecology, and complex systems theory. Afterward, the city of Barcelona is introduced as the case study: it is constituted by a set of districts and represented by an information system that contains temporal measurements of multiple environmental, economic, and social variables. A first approach to the sustainability of the city is made with the entropy of information as a measure of the urban system's diversity. But the fundamental contribution of the thesis focuses on the application of loratory Multivariate Analysis (EMA) to the urban system: Principal Component Analysis (PCA), Multiple Factorial Analysis (MFA), and Hierarchical Cluster Analysis (HCA). From this EMA approach, diversity is analyzed by identifying the similarity -or dissimilarity- between the different parts that make up the urban system. Some other techniques based on computer science and physics are proposed to evaluate the temporal transformation of the urban system, understood as a three-dimensional data cloud that gradually deforms. Differentiated characters and distinctive functions of districts are identifiable in the EMA application to the case study. Moreover, the temporal dependency of the dataset reveals information about the district's differentiation or homogenization trends. Finally, the conclusions of the most relevant results are presented and some future lines of research are proposed.Esta tesis analiza la complejidad del sistema urbano, descrito con múltiples variables que representan las características ambientales, económicas y sociales de la ciudad. La motivación fundamental para emprender este estudio consiste en describir la diversidad de la ciudad y su relación con una mejor respuesta a perturbaciones y amenazas, y por lo tanto, a su sostenibilidad. La tesis plantea aportar conocimiento teórico mediante la aplicación de metodologías estadísticas y computacionales que se desarrollan progresivamente en sus capítulos. En la introducción se presenta la abstracción de la ciudad como un sistema urbano, y se hace una revisión de los conceptos y medidas de la diversidad dentro de los marcos teóricos de la sostenibilidad, la ecología urbana y la teoría de los sistemas complejos. Posteriormente, se introduce el sistema urbano de la ciudad de Barcelona, constituido por un conjunto de distritos y representado mediante un sistema de información que contiene mediciones temporales de múltiples variables ambientales, económicas y sociales. Se hace una primera aproximación a la sostenibilidad de la ciudad empleando la entropía de la información como medida de diversidad del sistema urbano. Pero el aporte fundamental de la tesis se centra en la aplicación del Análisis Exploratorio Multivariante (EMA) en el sistema urbano: Análisis de Componentes principales (PCA), Análisis Factorial Múltiple (MFA) y Análisis de Agrupamiento Jerárquico (HCA). Desde dicho enfoque se analiza la diversidad identificando la similaridad -o disimilaridad- entre las distintas partes que componen el sistema urbano. Se plantean también algunas de las técnicas de las ciencias de la computación y la física para evaluar la transformación temporal del sistema urbano, entendido como una nube de datos tridimensionales que se deforma gradualmente. En el análisis del estudio de caso se identifican características diferenciadas y funciones distintivas de los distritos. Además, la dependencia temporal del conjunto de datos revela información sobre las tendencias de diferenciación u homogeneización de los distritos. Finalmente, se exponen las conclusiones de los resultados más relevantes y se enuncian algunas líneas futuras de investigaciónesPostprint (published version

A Review of Recent Developments in the Numerical Solution of Stochastic Partial Differential Equations (Stochastic Finite Elements)

The present review discusses recent developments in numerical techniques for the solution of systems with stochastic uncertainties. Such systems are modelled by stochastic partial differential equations (SPDEs), and techniques for their discretisation by stochastic finite elements (SFEM) are reviewed. Also, short overviews of related fields are given, e.g. of mathematical properties of random fields and SPDEs and of techniques for high-dimensional integration. After a summary of aspects of stochastic analysis, models and representations of random variables are presented. Then mathematical theories for SPDEs with stochastic operator are reviewed. Discretisation-techniques for random fields and for SPDEs are summarised and solvers for the resulting discretisations are reviewed, where the main focus lies on series expansions in the stochastic dimensions with an emphasis on Galerkin-schemes

Algebraic Relaxations and Hardness Results in Polynomial Optimization and Lyapunov Analysis

This thesis settles a number of questions related to computational complexity and algebraic, semidefinite programming based relaxations in optimization and control.Comment: PhD Thesis, MIT, September, 201