The combinatorics of binary arrays

This paper gives an account of the combinatorics of binary arrays, mainly concerning their randomness properties. In many cases the problem reduces to the investigation on difference sets.postprin

Using alternating de Bruijn sequences to construct de Bruijn tori

A de Bruijn torus is the two dimensional generalization of a de Bruijn sequence. While some methods exist to generate these tori, only a few methods of construction are known. We present a novel method to generate de Bruijn tori with rectangular windows by combining two variants de Bruijn sequences called `Alternating de Bruijn sequences' and `De Bruijn families'.Comment: 21 pages, comments welcom

Growing perfect cubes

AbstractAn (n,a,b)-perfect double cube is a b×b×b sized n-ary periodic array containing all possible a×a×a sized n-ary array exactly once as subarray. A growing cube is an array whose cj×cj×cj sized prefix is an (nj,a,cj)-perfect double cube for j=1,2,…, where cj=njv/3,v=a3 and n1<n2<⋯. We construct the smallest possible perfect double cube (a 256×256×256 sized 8-ary array) and growing cubes for any a

On the Existence of de Bruijn Tori with Two by Two Windows

Necessary and sucient conditions for the existence of de Bruijn Tori (or Perfect Maps) with two by two windows over any alphabet are given. This is the rst two-dimensional window size for which the existence question has been completely answered for every alphabet. The techniques used to construct these arrays utilise existing results on Perfect Factors and Perfect Multi-Factors in one and two dimensions and involve new results on Perfect Factors with `puncturing capabilities&apos;. Finally, the existence question for two-dimensional Perfect Factors is considered and is settled for two by two windows and alphabets of prime-power size. 2

Exact sampling with Markov chains

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1996.Includes bibliographical references (p. 79-83).by David Bruce Wilson.Ph.D

Rigorous direct and inverse design of photonic-plasmonic nanostructures

Designing photonic-plasmonic nanostructures with desirable electromagnetic properties is a central problem in modern photonics engineering. As limited by available materials, engineering geometry of optical materials at both element and array levels becomes the key to solve this problem. In this thesis, I present my work on the development of novel methods and design strategies for photonic-plasmonic structures and metamaterials, including novel Green’s matrix-based spectral methods for predicting the optical properties of large-scale nanostructures of arbitrary geometry. From engineering elements to arrays, I begin my thesis addressing toroidal electrodynamics as an emerging approach to enhance light absorption in designed nanodisks by geometrically creating anapole configurations using high-index dielectric materials. This work demonstrates enhanced absorption rates driven by multipolar decomposition of current distributions involving toroidal multipole moments for the first time. I also present my work on designing helical nano-antennas using the rigorous Surface Integral Equations method. The helical nano-antennas feature unprecedented beam-forming and polarization tunability controlled by their geometrical parameters, and can be understood from the array perspective. In these projects, optimization of optical performances are translated into systematic study of identifiable geometric parameters. However, while array-geometry engineering presents multiple advantages, including physical intuition, versatility in design, and ease of fabrication, there is currently no rigorous and efficient solution for designing complex resonances in large-scale systems from an available set of geometrical parameters. In order to achieve this important goal, I developed an efficient numerical code based on the Green’s matrix method for modeling scattering by arbitrary arrays of coupled electric and magnetic dipoles, and show its relevance to the design of light localization and scattering resonances in deterministic aperiodic geometries. I will show how universal properties driven by the aperiodic geometries of the scattering arrays can be obtained by studying the spectral statistics of the corresponding Green’s matrices and how this approach leads to novel metamaterials for the visible and near-infrared spectral ranges. Within the thesis, I also present my collaborative works as examples of direct and inverse designs of nanostructures for photonics applications, including plasmonic sensing, optical antennas, and radiation shaping

Spatial discretizations of generic dynamical systems

How is it possible to read the dynamical properties (ie when the time goes to infinity) of a system on numerical simulations? To try to answer this question, we study in this manuscript a model reflecting what happens when the orbits of a discrete time system $f$ (for example an homeomorphism) are computed numerically . The computer working in finite numerical precision, it will replace $f$ by a spacial discretization of $f$, denoted by $f_N$ (where the order $N$ of discretization stands for the numerical accuracy). In particular, we will be interested in the dynamical behaviour of the finite maps $f_N$ for a generic system $f$ and $N$ going to infinity, where generic will be taken in the sense of Baire (mainly among sets of homeomorphisms or $C^1$-diffeomorphisms). The first part of this manuscript is devoted to the study of the dynamics of the discretizations $f_N$, when $f$ is a generic conservative/dissipative homeomorphism of a compact manifold. We show that it would be mistaken to try to recover the dynamics of $f$ from that of a single discretization $f_N$ : its dynamics strongly depends on the order $N$. To detect some dynamical features of $f$, we have to consider all the discretizations $f_N$ when $N$ goes through $\mathbf N$. The second part deals with the linear case, which plays an important role in the study of $C^1$-generic diffeomorphisms, discussed in the third part of this manuscript. Under these assumptions, we obtain results similar to those established in the first part, though weaker and harder to prove.Comment: 322 pages. This is an improved version of the thesis of the author (among others, the introduction and conclusion have been translated into English). In particular, it contains works already published on arXiv. Comments welcome