Euclidean dimensionality

In order to determine the critical parameters of a magnetic, the functional dependence of any thermodynamic potential. should be deduced. Let us take advantage of the Ising lattice model which has turned out to be universal for many systems (see Figure 1.2.1). The Euclidean dimensionality of the lattice can be any—from 1 to oo. Its sites are assigned... 

In the case of no restrictions imposed on the diain flexibility, the polymer chain represents a system without any directional memory and evokes a path of Brownian diffusive motion [9]. A suitable description of this self-similar process can be the random walk model rwm) depending on Euclidean dimensionality, d. The rwm-hased distribution function determines the probability of the root mean square chain end-to-end separation,... 

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 ... 

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... 

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. 

Another simple example is the traiditional two-dimensional random-walk on a four-neighbor Euclidean lattice [toff89]. Despite the fact that the underlying lattice is symmetric only with respect to rotations that are multiples of 90 deg, the probability distribution p(s, y) for a particle that begins its random walk at the origin becomes circularly symmetric in the limit as time t —> oo p x,y,t) —> (see figure 12.12). 

Fig. 12.12. A circularly symmetric Gaussian probability distribution p x,y) describing a two-dimensional random walk emerges for large times on the macroscopic level, despite the fact that the underlying Euclidean lattice is anisotropic.

Jourjine [jour85] generalizes Euclidean lattice field theory on a d-dimensional lattice to a cell complex. He uses homology theory to replace points by cells of various dimensions and fields by functions on cells, the cochains, in hopes of developing a formalism that treats space-time as a dynamical variable and describes the change in the dimension of space-time as a phase transition (see figure 12.19). 

For a free noninteracting spinning particle, invariance with respect to translations and rotations in three dimensional space, i.e., invariance under the inhomogeneous euclidean group, requires that the momenta pl and the total angular momenta J1 obey the following commutation rules... 

Let CO be a finite set of nodes (a grid) in some bounded domain of the n-dimensional Euclidean space and let P G co be a point of the grid u>. Consider the equation... 

A locally one-dlinensional scheme (LOS) for the heat conduction equation in an arbitrary domain. The method of summarized approximation can find a wide range of application in designing economical additive schemes for parabolic equations in the domains of rather complicated configurations and shapes. More a detailed exploration is devoted to a locally one-dimensional problem for the heat conduction equation in a complex domain G = G -f F of the dimension p. Let x — (sj, 2,..., a- p) be a point in the Euclidean space R. ... 

The general theory of iterative methods is presented in the next sections with regard to an operator equation of the first kind Au — f, where T is a self-adjoint operator in a finite-dimensional Euclidean space. The applications of such theory to elliptic grid equations began to spread to more and more branches as they took on an important place in real-life situations. 

Vaisman, Foundations of Three-Dimensional Euclidean Geometry (1980)... 

A mathematically very simple classification procedure is the nearest neighbour method. In this method one computes the distance between an unknown object u and each of the objects of the training set. Usually one employs the Euclidean distance D (see Section 30.2.2.1) but for strongly correlated variables, one should prefer correlation based measures (Section 30.2.2.2). If the training set consists of n objects, then n distances are calculated and the lowest of these is selected. If this is where u represents the unknown and I an object from learning class L, then one classifies u in group L. A three-dimensional example is given in Fig. 33.11. Object u is closest to an object of the class L and is therefore considered to be a member of that class. 

Minimizing this function is equivalent to finding a low-dimensional configuration of points which has Euclidean object-to-object distances by as close as possible to some transformation/(.) of the original distances or dissimilarities, dy. Thus, the model distances by are not necessarily fitted to the original dy, as in classical MDS, but to some admissible transformation of the measured distances. For example, when the transformation is a general monotonic transformation it preserves the... 

The simplest formulation of the packing problem is to give some collection of distance constraints and to calculate these coordinates in ordinary three-dimensional Euclidean space for the atoms of a molecule. This embedding problem - the Fundamental Problem of Distance Geometry - has been proven to be NP-hard [116]. However, this does not mean that practical algorithms for its solution do not exist [117-119]. 

Three-dimensional electron densities have no boundaries they converge to zero exponentially with distance from the nuclei of the peripheral atoms in the molecule. Considering a single, isolated molecule, the exact quantum-mechanical electron density becomes zero in a strict sense only at infinite distance from the center of mass of the molecule. Consequently, the electron density is not a compact set, just as the embedding three-dimensional Euclidean space E3 is not compact either. However, the three-dimensional Euclidean space E3, as a subset of a four-dimensional Euclidean space E4, can be slightly extended (for example, by adding one point) and made compact by various compactification techniques. 

A space M where each point x e M has an open neighborhood homeomorphic to a set open within a Euclidean half-space Hn, is an K-dimensional manifold with boundary. 

It is possible, however, to avoid any violation of these fundamental properties, and derive a result on the local electron densities of non-zero volume subsystems of boundaryless electron densities of complete molecules [159-161]. A four-dimensional representation of molecular electron densities is constructed by taking the first three dimensions as those corresponding to the ordinary three-space E3 and the fourth dimension as that representing the electron density values p(r). Using a compactifi-cation method, all points of the ordinary three- dimensional space E3 can be mapped to a manifold S3 embedded in a four- dimensional Euclidean space E4, where the addition of a single point leads to a compact manifold representation of the entire, boundaryless molecular electron density. 

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