Euclidean

Partitional clustering using Euclidean distance as a measure of dissimilarity between pattern classes has been selected for the grouping of AE hits. 

To construct dissimilarity measures, one uses mismatches Here a + b is the Hamming (Manhattan, taxi-cab, city-block) distance, and a + h) is the Euclidean distance. 

This is done by calculating the Euclidean distance between the input data vector Xc and the weight vectors Wj of all neurons ... 

Euclidean and Hamming distance measures of torsional similarity. 

CONTOL Calculates ratio of the difference of the Euclidean norm (Lapidus and Pinder, 1982) between successive iterations to the nonn of the solution, as... 

For n = 2, this is the familiar -space Euclidean distance. Similarity values, are calculated as... 

In November 1919 Einstein became the mythical figure he is to this day. In May of that year two solar eclipse expeditions had (in the words of the astronomer Eddington) confirm[ed] Einstein s weird theory of non-Euclidean space. On November 6 the president of the Royal Society declared in London that this was the most remarkable scientific event since the discoveiy [in 18461 of the predicted existence of the planet Neptune. ... 

That Dfractai givcs the expected result for simple sets in Euclidean space is easy to see. If A consists of a single point, for example, we have N A, e) = 1, Ve, and thus that / fractal = 0. Similarly, if A is a line segment of length L, then N A,e) L/e so that Dfraciai = 1. In fact, for the usual n dinien.sional Euclidean sets, the fractal dimension equals the topological dimension. There are nrore complicated sets, however, for wliic h the two measures differ. 

Intuitively, a graph can be realized geometrically in a three-dimensional Euclidean space vertices arc represented by points and edges are represented either by lines (in the case of undirected graphs) or arrows (in the case of directed graphs). In this book, we will be concerned with both kinds of graphs multiple edges i.e. when vertices arc connected by more than one line or arrow), however, are not allowed. 

This Cantor set may be explicitly visualized in the n-dimensional euclidean space 7 " by defining a mapping xp F —> 7 " ([grass83] and [packl]). The coordinate of the resulting vector xpid) is given by ... 

The formalism for computing Lyajmnov exponents for continuous dynamical systems that was introduced in the last section can also be used, with only minor modifications, for determining exponents for CA as well. The major modification involves replacing the Euclidean norm, V t) - used for measuring the divergence of two nearby trajectories (see equation 4.60) - by the Cantor-set metric, d t) ... 

We first generalize the nomenclature. Consider a Euclidean d-dimensional lattice L, with translation group Gp. A frame, F, of L is defined to be a finite subset of (not necessarily contiguous) sites of L that is closed under (i) intersection, (ii) union, (iii) difference and (iv) operations g Gp. A block, Bp, is a specific assignment... 

Exact calculations have already been carried out for simple one and two dimensional Euclidean geometries by exploiting properties of polynomials (chapter 5.2.1) and circulant matrices (chapter 5.2.2) over the finite field J-[q, q p wherep is prime. We will here rely instead on the theory of input-free modular systems, which is more suitable for dealing with the dynamics of completely arbitrary lattices. 

The BBM gas consists of an arbitrary number of hard spheres (or balls) of finite diameter that collide elastically both among themselves and with any solid walls (or mirrors) that they may encounter during their motion. Starting out on some site of a two-dimensional Euclidean lattice, each ball is allowed to move only in one of four directions (see figure 6.10). The lattice spacing, d = l/ /2 (in arbitrary units), is chosen so that balls collide while occupying adjacent sites. Unit time is... 

The one dimensional rules given in equations 8.105 and 8.106 can be readily generalized to a d dimensional Euclidean lattice. Let T] r, t) be the d dimensional analogue of the one dimensional local slope at lattice point r at time Addition of sand is then generated by the rule... 

Fig. 8,20 First five iterations of an SDCA system starting from a 4-neighbor Euclidean lattice seeded with a single non-zero site at the center. The global transition rule F consists of totalistic-value and restricted totalistic topology rules C — (26,69648,32904)[3 3[ (see text for rule definitions). Solid sites have <7=1.

Vants represent the one of the simplest - and therefore, most persuasive - examples of emergence of high-level structures from low-level dynamics. Discovered by Langton [lang86], vants live on a two-dimensional Euclidean lattice and come in two flavors, red and bine. Each vant c an move in any of four directions (E,W,N,S). Each lattice site is either empty or contains one of two types of food, green food or yellow food. Vants arc fundamentally solitary creatures so that there is a strict conservation of the number of vants. 

To set up the problem and in order to appreciate more fully the difficulty in quantifying complexity, consider figure 12.1. The figure shows three patterns (a) an area of a regular two-dimensional Euclidean lattice, (b) a space-time view of the evolution of the nearest-neighbor one-dimensional cellular automata rule RllO, starting from a random initial state,f and (c) a completely random collection of dots. 

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