Lattice constellations and codes from quadratic number fields

We propose new classes of linear codes over integer rings of quadratic extensions of Q, the field of rational numbers. The codes are considered with respect to a Mannheim metric, which is a Manhattan metric modulo a two-dimensional (2-D) grid. In particular, codes over Gaussian integers and Eisenstein-Jacobi integers are extensively studied. Decoding algorithms are proposed for these codes when up to two coordinates of a transmitted code vector are affected by errors of arbitrary Mannheim weight. Moreover, we show that the proposed codes are maximum-distance separable (MDS), with respect to the Hamming distance. The practical interest in such Mannheim-metric codes is their use in coded modulation schemes based on quadrature amplitude modulation (QAM)-type constellations, for which neither the Hamming nor the Lee metric is appropriate.

Asymptotics of number fields and the Cohen-Lenstra heuristics

We study the asymptotics conjecture of Malle for dihedral groups Dl of order 2l, where l is an odd prime. We prove the expected lower bound for those groups. For the upper bounds we show that there is a connection to class groups of quadratic number fields. The asymptotic behavior of those class groups is predicted by the Cohen-Lenstra heuristics. Under the assumption of this heuristic we are able to prove the expected upper bounds.

A functional equation related to the product in a quadratic number field

"Vegeu el resum a l'inici del document del fitxer adjunt."

Arithmetical problems in number fields, abelian varieties and modular forms

Number theory, a fascinating area in mathematics and one of the oldest, has experienced spectacular progress in recent years. The development of a deep theoretical background and the implementation of algorithms have led to new and interesting interrelations with mathematics in general which have paved the way for the emergence of major theorems in the area. This report summarizes the contribution to number theory made by the members of the Seminari de Teoria de Nombres (UB-UAB-UPC) in Barcelona. These results are presented in connection with the state of certain arithmetical problems, and so this monograph seeks to provide readers with a glimpse of some specific lines of current mathematical research.

The Helmholtzian Equation: Stabilizing Mechanisms for Quadratic Nonlinear Fields

Hexagonal Resonant Triad patterns are shown to exist as stable solutions of a particular type of nonlinear field where no cubic field nonlinearity is present. The zero ‘dc’ Fourier mode is shown to stabilize these patterns produced by a pure quadratic field nonlinearity. Closed form solutions and stability results are obtained near the critical point, complimented by numerical studies far from the critical point. These results are obtained using a neural field based on the Helmholtzian operator. Constraints on structure and parameters for a general pure quadratic neural field which supports hexagonal patterns are obtained.

Dénombrement dans les empilements apolloniens généralisés et distribution angulaire dans les extensions quadratiques imaginaires.

Cette thèse traite de deux thèmes principaux. Le premier concerne l'étude des empilements apolloniens généralisés de cercles et de sphères. Généralisations des classiques empilements apolloniens, dont l'étude remonte à la Grèce antique, ces objets s'imposent comme particulièrement attractifs en théorie des nombres. Dans cette thèse sera étudié l'ensemble des courbures (les inverses des rayons) des cercles ou sphères de tels empilements. Sous de bonnes conditions, ces courbures s'avèrent être toutes entières. Nous montrerons qu'elles vérifient un principe local-global partiel, nous compterons le nombre de cercles de courbures plus petites qu'une quantité donnée et nous nous intéresserons également à l'étude des courbures premières. Le second thème a trait à la distribution angulaire des idéaux (ou plutôt ici des nombres idéaux) des corps de nombres quadratiques imaginaires (que l'on peut voir comme la distribution des points à coordonnées entières sur des ellipses). Nous montrerons que la discrépance de l'ensemble des angles des nombres idéaux entiers de norme donnée est faible et nous nous intéresserons également au problème des écarts bornés entre les premiers d'extensions quadratiques imaginaires dans des secteurs.

On T-Semisimplicity of Iwasawa Modules and Some Computations with Z3-Extensions

Thesis (Ph.D.)--University of Washington, 2016-08

Phase portraits of quadratic polynomial vector fields having a rational first integral of degree 3

In this paper, we classify all the global phase portraits of the quadratic polynomial vector fields having a rational first integral of degree 3. (C) 2008 Elsevier Ltd. All rights reserved.

A new number field construction of the lattice E8

Known number theoretical constructions of the lattice E8 use the cyclotomic fields Q(ζ15), Q(ζ20), and Q(ζ24). In this work, an infinite family of Abelian number fields yielding rotated versions of the lattice E 8 is exhibited. © 2012 The Managing Editors.

Exact algorithms for p-adic fields and epsilon constant conjectures

We develop several algorithms for computations in Galois extensions of p-adic fields. Our algorithms are based on existing algorithms for number fields and are exact in the sense that we do not need to consider approximations to p-adic numbers. As an application we describe an algorithmic approach to prove or disprove various conjectures for local and global epsilon constants.

PERIODIC PERTURBATION of QUADRATIC SYSTEMS WITH TWO INFINITE HETEROCLINIC CYCLES

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Periodic orbits for a class of reversible quadratic vector field on R-3

For a class of reversible quadratic vector fields on R-3 we study the periodic orbits that bifurcate from a heteroclinic loop having two singular points at infinity connected by an invariant straight line in the finite part and another straight line at infinity in the local chart U-2. More specifically, we prove that for all n is an element of N, there exists epsilon(n) > 0 such that the reversible quadratic polynomial differential systemx = a(0) + a(1y) + a(3y)(2) + a(4Y)(2) + epsilon(a(2x)(2) + a(3xz)),y = b(1z) + b(3yz) + epsilon b(2xy),z = c(1y) +c(4az)(2) + epsilon c(2xz)in R-3, with a(0) < 0, b(1)c(1) < 0, a(2) < 0, b(2) < a(2), a(4) > 0, c(2) < a(2) and b(3) is not an element of (c(4), 4c(4)), for epsilon is an element of (0, epsilon(n)) has at least n periodic orbits near the heteroclinic loop. (c) 2007 Elsevier B.V. All rights reserved.

Lattices from subfields of Q (xi n)

In this paper we present a method for evaluating the center density of algebraic lattices from subfields of Q(xi n), where n is a positive integer. This method allows to reproduce rotated versions of dense lattices in some dimensions. Constellations on algebraic lattices with high packing density have been proposed for use in communications in Gaussian channels and also in Rayleigh fading channels in case they have high diversity.

The Hasse-Minkowski Theorem

The Hasse-Minkowski theorem concerns the classification of quadratic forms over global fields (i.e., finite extensions of Q and rational function fields with a finite constant field). Hasse proved the theorem over the rational numbers in his Ph.D. thesis in 1921. He extended the research of his thesis to quadratic forms over all number fields in 1924. Historically, the Hasse-Minkowski theorem was the first notable application of p-adic fields that caught the attention of a wide mathematical audience. The goal of this thesis is to discuss the Hasse-Minkowski theorem over the rational numbers and over the rational function fields with a finite constant field of odd characteristic. Our treatments of quadratic forms and local fields, though, are more general than what is strictly necessary for our proofs of the Hasse-Minkowski theorem over Q and its analogue over rational function fields (of odd characteristic). Our discussion concludes with some applications of the Hasse-Minkowski theorem.