An Exposition of the Eisenstein Integers

In this thesis, we will give a brief introduction to number theory and prime numbers. We also provide the necessary background to understand how the imaginary ring of quadratic integers behaves. An example of said ring are complex numbers of the form ℤ[ω] = {a+bω ∣ a, b ∈ ℤ} where ω2 + ω + 1 = 0. These are known as the Eisenstein integers, which form a triangular lattice in the complex plane, in contrast with the Gaussian integers, ℤ[i] = {a + bi ∣ a, b ∈ ℤ} which form a square lattice in the complex plane. The Gaussian moat problem, first posed by Basil Gordon in 1962 at the International Congress of Mathematicians in Stockholm [7], asks whether it is possible to walk from the origin to infinity using the Gaussian primes as stepping stones and taking steps of bounded length. Although it has been shown that one cannot walk to infinity on the real number line, taking steps of bounded length and stepping only on the primes, the moat problem for Gaussian and Eisenstein primes remains unsolved. We will provide the necessary background for the reader, then investigate the Eisenstein moat problem

An Exposition of the Eisenstein Integers

In this thesis, we will give a brief introduction to number theory and prime numbers. We also provide the necessary background to understand how the imaginary ring of quadratic integers behaves. An example of said ring are complex numbers of the form ℤ[ω] = {a+bω ∣ a, b ∈ ℤ} where ω2 + ω + 1 = 0. These are known as the Eisenstein integers, which form a triangular lattice in the complex plane, in contrast with the Gaussian integers, ℤ[i] = {a + bi ∣ a, b ∈ ℤ} which form a square lattice in the complex plane. The Gaussian moat problem, first posed by Basil Gordon in 1962 at the International Congress of Mathematicians in Stockholm [7], asks whether it is possible to walk from the origin to infinity using the Gaussian primes as stepping stones and taking steps of bounded length. Although it has been shown that one cannot walk to infinity on the real number line, taking steps of bounded length and stepping only on the primes, the moat problem for Gaussian and Eisenstein primes remains unsolved. We will provide the necessary background for the reader, then investigate the Eisenstein moat problem

Replica Field Theory for Deterministic Models: Binary Sequences with Low Autocorrelation

We study systems without quenched disorder with a complex landscape, and we use replica symmetry theory to describe them. We discuss the Golay-Bernasconi-Derrida approximation of the low autocorrelation model, and we reconstruct it by using replica calculations. Then we consider the full model, its low $T$ properties (with the help of number theory) and a Hartree-Fock resummation of the high-temperature series. We show that replica theory allows to solve the model in the high $T$ phase. Our solution is based on one-link integral techniques, and is based on substituting a Fourier transform with a generic unitary transformation. We discuss this approach as a powerful tool to describe systems with a complex landscape in the absence of quenched disorder.Comment: 42 pages, uufile with eps figures added in figures, ROM2F/94/1

A time-stepping dynamically-consistent spherical-shell dynamo code

A pseudo-spectral dynamo code, developed as a computational laboratory, is described. The magnetic, heat and Boussinesq Navier-Stokes equations, with inertia, non-linear advection, buoyancy with asymmetric gravity, Coriolis, viscous and Lorentz forces, are solved numerically in a rotating conducting fluid shell. The convection is thermally driven by prescribed boundary temperatures. The equations are discretised using toroidal-poloidal fields, Chebychev collocation in radius and spherical harmonic expansion in angles. Derivatives are performed spectrally. Products are evaluated in physical space for efficiency. Fields are transformed between physical and spectral spaces by fast Fourier and Gauss-Legendre methods. Linear terms are time-stepped implicitly and product terms explicitly using an Adams predictor/corrector. Results are presented for two benchmark models