Catalan generating functions for bounded operators

In this paper, we study the solution of the quadratic equation TY2−Y+I=0 where T is a linear and bounded operator on a Banach space X. We describe the spectrum set and the resolvent operator of Y in terms of the ones of T. In the case that 4T is a power-bounded operator, we show that a solution (named Catalan generating function) of the above equation is given by the Taylor series C(T):=∑n=0∞CnTn, where the sequence (Cn)nis the well-known Catalan numbers sequence. We express C(T) by means of an integral representation which involves the resolvent operator (λT)−1. Some particular examples to illustrate our results are given, in particular an iterative method defined for square matrices T which involves Catalan number

Optimisation methods in structural systems reliability

Imperial Users onl

Computation of eigenvalues in proportionally damped viscoelastic structures based on the fixed-point iteration

Linear viscoelastic structures are characterized by dissipative forces that depend on the history of the velocity response via hereditary damping functions. The free motion equation leads to a nonlinear eigenvalue problem characterized by a frequency-dependent damping matrix. In the present paper, a novel and efficient numerical method for the computation of the eigenvalues of linear and proportional or lightly non-proportional viscoelastic structures is developed. The central idea is the construction of two complex-valued functions of a complex variable, whose fixed points are precisely the eigenvalues. This important property allows the use of these functions in a fixed-point iterative scheme. With help of some results in fixed point theory, necessary conditions for global and local convergence are provided. It is demonstrated that the speed of convergence is linear and directly depends on the level of induced damping. In addition, under certain conditions the recursive method can also be used for the calculation of non-viscous real eigenvalues. In order to validate the mathematical results, two numerical examples are analyzed, one for single degree-of-freedom systems and another for multiple ones.The authors gratefully acknowledge the support of the Polytechnic University of Valencia under programs PAID 02-11-1828 and 05-10-2674 and of the National Science and Research Council of Canada.Lázaro Navarro, M.; Pérez Aparicio, JL.; Epstein, M. (2012). Computation of eigenvalues in proportionally damped viscoelastic structures based on the fixed-point iteration. Applied Mathematics and Computation. 219(8):3511-3529. https://doi.org/10.1016/j.amc.2012.09.026S35113529219

Numerical Algorithms for Polynomial Optimisation Problems with Applications

In this thesis, we study tensor eigenvalue problems and polynomial optimization problems. In particular, we present a fast algorithm for computing the spectral radii of symmetric nonnegative tensors without requiring the partition of the tensors. We also propose some polynomial time approximation algorithms with new approximation bounds for nonnegative polynomial optimization problems over unit spheres. Furthermore, we develop an efficient and effective algorithm for the maximum clique problem

Unsupervised learning of Arabic non-concatenative morphology

Unsupervised approaches to learning the morphology of a language play an important role in computer processing of language from a practical and theoretical perspective, due their minimal reliance on manually produced linguistic resources and human annotation. Such approaches have been widely researched for the problem of concatenative affixation, but less attention has been paid to the intercalated (non-concatenative) morphology exhibited by Arabic and other Semitic languages. The aim of this research is to learn the root and pattern morphology of Arabic, with accuracy comparable to manually built morphological analysis systems. The approach is kept free from human supervision or manual parameter settings, assuming only that roots and patterns intertwine to form a word. Promising results were obtained by applying a technique adapted from previous work in concatenative morphology learning, which uses machine learning to determine relatedness between words. The output, with probabilistic relatedness values between words, was then used to rank all possible roots and patterns to form a lexicon. Analysis using trilateral roots resulted in correct root identification accuracy of approximately 86% for inflected words. Although the machine learning-based approach is effective, it is conceptually complex. So an alternative, simpler and computationally efficient approach was then devised to obtain morpheme scores based on comparative counts of roots and patterns. In this approach, root and pattern scores are defined in terms of each other in a mutually recursive relationship, converging to an optimized morpheme ranking. This technique gives slightly better accuracy while being conceptually simpler and more efficient. The approach, after further enhancements, was evaluated on a version of the Quranic Arabic Corpus, attaining a final accuracy of approximately 93%. A comparative evaluation shows this to be superior to two existing, well used manually built Arabic stemmers, thus demonstrating the practical feasibility of unsupervised learning of non-concatenative morphology