Simulation of Error Correction Algorithms Using Reed Solomon Codes

The Reed-Solomon codes for multiple-error-correction are examined in this study. The results of a comparison between the conventional Gorenstein-Zierler method and a transform method are discussed, and simple examples are given. Then decoding algorithms are compared in terms of the numerical complexity. Finally the conclusions of the simulation are stated.Computer Scienc

Dynamical Systems

Complex systems are pervasive in many areas of science integrated in our daily lives. Examples include financial markets, highway transportation networks, telecommunication networks, world and country economies, social networks, immunological systems, living organisms, computational systems and electrical and mechanical structures. Complex systems are often composed of a large number of interconnected and interacting entities, exhibiting much richer global scale dynamics than the properties and behavior of individual entities. Complex systems are studied in many areas of natural sciences, social sciences, engineering and mathematical sciences. This special issue therefore intends to contribute towards the dissemination of the multifaceted concepts in accepted use by the scientific community. We hope readers enjoy this pertinent selection of papers which represents relevant examples of the state of the art in present day research. [...

Modular Forms: Constructions & Applications

This thesis combines results of five research papers on the construction and applications of modular forms and their generalisations. We begin by constructing new examples of quantum modular forms of depth two, generalising results of Bringmann, Kaszian, and Milas. To do so, we relate the asymptotics of certain false theta functions of binary quadratic forms to multiple Eichler integrals of theta functions. Quantum modularity of the false theta functions follows from the behaviour of such integrals near the real line. Next, we turn our attention to the asymptotic profile of a certain eta-theta quotient that arises in the partition function of entanglement entropy in string theory. In particular, we generalise methods of Bringmann and Dousse, and Dousse and Mertens, to deal with the meromorphic Jacobi form at hand. Applying Wright's circle method for Jacobi forms we obtain a bivariate asymptotic for the two-variable coefficients of the eta-theta quotient. Thirdly, we investigate the asymptotic behaviour of the generating function of integer partitions whose ranks are congruent to r modulo t, denoted by N(r; t; n). By proving that the series has monotonic increasing coefficients above some bound, we are in a position to apply Ingham's Tauberian theorem. This immediately implies that N(r; t; n) is equidistributed in r for fixed t as n tends to infinity, in turn implying a recent conjecture of Hou and Jagadeeson on a convexity-type result. The following chapter is dedicated to an investigation of traces of cycle integrals of meromorphic modular forms and their relationship to coefficients of harmonic Maass forms. Working on lattices of signature (1,2), we first relate a locally harmonic Maass form to a Siegel theta lift involving the Maass raising operator by explicitly computing the raising of the locally harmonic Maass form, and using the usual unfolding argument for the theta lift. We then borrow techniques of Bruinier, Ehlen, and Yang to compute the theta lift as (up to terms that vanish for certain classes of input functions) the constant term in a q-series involving the coefficients of xi-preimages of unary theta functions as well as theta functions. Since such preimages are harmonic Maass forms, we obtain a description of the traces in terms of coefficients of theta functions and harmonic Maass forms. Choosing a specific lattice related to quadratic forms and noting that the functions determining the constant term can be chosen to have rational coefficients, we obtain a new proof of a recent result of Alfes-Neumann, Bringmann, and Schwagenscheidt. Finally, we investigate the relationship between modular forms and self-conjugate t-core partitions. We obtain the number of self-conjugate 7-cores as a single class number in two ways. The first we show with modularity arguments on the generating function of Hurwitz class numbers. We also provide a complementary combinatorial description to explain the equality. In particular, we construct an explicit map between self-conjugate t-cores and quadratic forms in a given class group. Moreover, we show that the genus of the quadratic forms is unique, and determine the number of preimages of the genus. Using these results, we show an equality between the number of 4-cores and the number of self-conjugate 7-cores on specific arithmetic progressions. Aside from the t = 4 case, we consider whether equalities between t-cores and self-conjugate 2t-1-cores are possible. We show for t = 2,3,5 that they are not, and offer a conjecture and partial results for t > 5

Discrete Time Systems

Discrete-Time Systems comprehend an important and broad research field. The consolidation of digital-based computational means in the present, pushes a technological tool into the field with a tremendous impact in areas like Control, Signal Processing, Communications, System Modelling and related Applications. This book attempts to give a scope in the wide area of Discrete-Time Systems. Their contents are grouped conveniently in sections according to significant areas, namely Filtering, Fixed and Adaptive Control Systems, Stability Problems and Miscellaneous Applications. We think that the contribution of the book enlarges the field of the Discrete-Time Systems with signification in the present state-of-the-art. Despite the vertiginous advance in the field, we also believe that the topics described here allow us also to look through some main tendencies in the next years in the research area

Nonlinear Analysis and Optimization with Applications

Nonlinear analysis has wide and significant applications in many areas of mathematics, including functional analysis, variational analysis, nonlinear optimization, convex analysis, nonlinear ordinary and partial differential equations, dynamical system theory, mathematical economics, game theory, signal processing, control theory, data mining, and so forth. Optimization problems have been intensively investigated, and various feasible methods in analyzing convergence of algorithms have been developed over the last half century. In this Special Issue, we will focus on the connection between nonlinear analysis and optimization as well as their applications to integrate basic science into the real world