From the Publisher:
Although it originated as a method to study general dynamical systems, symbolic dynamics is useful in coding for data storage and transmission as well as in linear algebra. Requiring only a undergraduate knowledge of linear algebra, this first general textbook includes over 500 exercises.

An Introduction to Symbolic Dynamics and Coding

Real and Complex Analysis

Most of the signal processing that we will study in this course involves local operations on a signal, namely transforming the signal by applying linear combinations of values in the neighborhood of each sample point. You are familiar with such operations from Calculus, namely, taking derivatives and you are also familiar with this from optics namely blurring a signal. We will be looking at sampled signals only. Let's start with a few basic examples. Local difference Suppose we have a 1D image and we take the local difference of intensities, DI(x) = 1 2 (I(x + 1) − I(x − 1)) which give a discrete approximation to a partial derivative. (We compute this for each x in the image.) What is the effect of such a transformation? One key idea is that such a derivative would be useful for marking positions where the intensity changes. Such a change is called an edge. It is important to detect edges in images because they often mark locations at which object properties change. These can include changes in illumination along a surface due to a shadow boundary, or a material (pigment) change, or a change in depth as when one object ends and another begins. The computational problem of finding intensity edges in images is called edge detection. We could look for positions at which DI(x) has a large negative or positive value. Large positive values indicate an edge that goes from low to high intensity, and large negative values indicate an edge that goes from high to low intensity. Example Suppose the image consists of a single (slightly sloped) edge:

http://ieeexplore.ieee.org/iel5/5/31307/01456462.pdf

Linear systems

Sphere Packings, Lattices and Groups

A challenging goal in materials chemistry and physics is spontaneously to form intended superstructures from designed building blocks. In fields such as crystal engineering and the design of porous materials, this typically involves building blocks of organic molecules, sometimes operating together with metallic ions or clusters. The translation of such ideas to nanoparticles and colloidal-sized building blocks would potentially open doors to new materials and new properties, but the pathways to achieve this goal are still undetermined. Here we show how colloidal spheres can be induced to self-assemble into a complex predetermined colloidal crystal-in this case a colloidal kagome lattice-through decoration of their surfaces with a simple pattern of hydrophobic domains. The building blocks are simple micrometre-sized spheres with interactions (electrostatic repulsion in the middle, hydrophobic attraction at the poles, which we call 'triblock Janus') that are also simple, but the self-assembly of the spheres into an open kagome structure contrasts with previously known close-packed periodic arrangements of spheres. This open network is of interest for several theoretical reasons. With a view to possible enhanced functionality, the resulting lattice structure possesses two families of pores, one that is hydrophobic on the rims of the pores and another that is hydrophilic. This strategy of 'convergent' self-assembly from easily fabricated colloidal building blocks encodes the target supracolloidal architecture, not in localized attractive spots but instead in large redundantly attractive regions, and can be extended to form other supracolloidal networks.

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Directed self-assembly of a colloidal kagome lattice

1.1. Description of results. A domino is a 1 x 2 (or 2 x 1) rectangle, and a tiling of a region by dominos is a way of covering that region with dominos so that there are no gaps or overlaps. In 1961, Kasteleyn [Kal] found a formula for the number of domino tilings of an m x n rectangle (with mn even), as shown in Figure 1 for rm = n = 68. Temperley and Fisher [TF] used a differenit method and arrived at the same result at almost exactly the same time. Both lines of calculation showed that the logarithm of number of tilings, divided by the number of dominos in a tiling (that is, mn/2), converges to 2G/7r 0.58 (here G is Catalan's constant). On the other hand, in 1992 Elkies et al. [EKLP] studied domino tilings of regions they called Aztec diamonds (Figure 2 shows an Aztec diamond of order 48), and showed that the logarithm of the number of tilings, divided by the number of dominos, converges to the smaller number (log 2)/2 0.35. Thus, even though the region in Figure 1 has slightly smaller area than the region in Figure 2, the former has far more domino tilings. For regions with other shapes, neither of these asymptotic formulas may apply. In the present paper we consider simply-connected regions of arbitrary shape. We give an exact formula for the limiting value of the logarithm of the number of tilings per unit area, as a function of the shape of the boundary of the region, as the size of the region goes to infinity. In particular, we show that computation of this limit is intimately linked with an understanding of long-range variations in the local statistics of random domino tilings. Such variations can be seen by comparing Figures 1 and 2. Each of the two tilings is random in the sense that the algorithm [PWI that was used to create it generates each of the possible tilings of the region being tiled with the same probability. Hence one can expect each tiling to be qualitatively typical of the overwhelming majority of tilings of the region in

/pdf/a-variational-principle-for-domino-tilings-33ch7ic1pe.pdf

A variational principle for domino tilings

We study configurations of points on the unit sphere that minimize potential energy for a broad class of potential functions (viewed as functions of the squared Euclidean distance between points). Call a configuration sharp if there are m distances between distinct points in it and it is a spherical (2m-1)-design. We prove that every sharp configuration minimizes potential energy for all completely monotonic potential functions. Examples include the minimal vectors of the E_8 and Leech lattices. We also prove the same result for the vertices of the 600-cell, which do not form a sharp configuration. For most known cases, we prove that they are the unique global minima for energy, as long as the potential function is strictly completely monotonic. For certain potential functions, some of these configurations were previously analyzed by Yudin, Kolushov, and Andreev; we build on their techniques. We also generalize our results to other compact two-point homogeneous spaces, and we conclude with an extension to Euclidean space.

/pdf/universally-optimal-distribution-of-points-on-spheres-292geyxxch.pdf

Universally optimal distribution of points on spheres

Building on Viazovska's recent solution of the sphere packing problem in eight dimensions, we prove that the Leech lattice is the densest packing of congruent spheres in twenty-four dimensions and that it is the unique optimal periodic packing. In particular, we find an optimal auxiliary function for the linear programming bounds, which is an analogue of Viazovska's function for the eight-dimensional case.

The sphere packing problem in dimension 24

We continue the study of the linear programming bounds for sphere packing introduced by Cohn and Elkies. We use theta series to give another proof of the principal theorem, and present some related results and conjectures. This article is in the arXiv as: arXiv:math.MG/0110010

/pdf/new-upper-bounds-on-sphere-packings-i-vp8oxu2km6.pdf

Henry Cohn

Papers

A variational principle for domino tilings

Universally optimal distribution of points on spheres

The sphere packing problem in dimension 24

New upper bounds on sphere packings I

Universally optimal distribution of points on spheres