Euclidean spaces

A vector space (with multiplying real numbers a, (3) represents the Euclidean 

Inner product and distance. The concept of the inner product is used to introduce 

In this appendix we will discuss the mathematical tools necessary for developing the regularization theory of inverse problem solutions. They are based on the methods of functional analysis, which employ the ideas of functional spaces. Thus we should start our discussion by introducing the basic definitions and notations from functional analysis. Before doing so, I remind the reader of the basic properties of the simplest and, at the same time, the most fundamental mathematical space - Euclidean space. 

Conventional physical space has three dimensions. Any point in this space can represented by three Cartesian coordinates (.X], X2,. x.d. The natural generalization of three dimensional (3-D) physical space is the n dimensional Euclidean space (or Rn ), which can be described as the set of all possible vectors of order n  

By analogy with the length of the vector in 3-D physical space, we can introduce a norm of the vector a as 

It is easy to check that the norm introduced above satisfies the conditions 

Substituting equation (A.6) into (A.5), we obtain (A.4). In a general case of n dimensional Euclidean space, the triangular inequality comes from the Cauchy inequality, which we will discuss below. 

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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 ... 

Scalar Fields Consider a continuous field theory in Euclidean space with action... 

There are methods available to test whether or not two or more regression lines statistically differ from each other in the two major properties of lines in Euclidean space namely position (or elevation) and slope. This can be very useful in pharmacology. An example is given later in the chapter for the comparison of Schild regressions (see Chapter 6). 

We would like to discuss the questions raised above in more detail. Obviously, in numerical solution of mathematical problems it is unrealistic to reproduce a difference solution for all the values of the argument varying in a certain domain of a prescribed Euclidean space. The traditional way of covering this is to select some finite set of points in this domain and look for an approximate solution only at those points. Any such set of points is called a grid and the isolated points are termed the grid nodes. 

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. 

In Euclidean space we define squared distance from the origin of a point x by means of the scalar product of x with itself ... 

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. 

In many applications it is customary to define local coordinate systems indirectly by establishing their connection with the Cartesian coordinates in some underlying Euclidean space E if there is one. By labeling the points within each actual space (of local coordinate system) with the coordinate values in the underlying Euclidean space E there is a common reference for all local coordinate systems, and the compatibility conditions can be formulated within the Euclidean space E" of familiar and intuitively simple properties. 

The underlying Euclidean space E" also simplifies the definition of individual coordinate systems considerably. 

The role of a boundary in a manifold with boundary can be interpreted with reference to a hyperplane within a Euclidean space E using the concept of halfspace, where the hyperplane is in fact the boundary of the half-space. By appropriate reordering of the coordinates, a half-space Hn becomes the subset of a Euclidean space En containing all points of En with non-negative value for the last coordinate. 

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. 

When N > 4 there appears to be too many Zn, since N(N — l)/2 > 3N — 6. However, the Zn are not globally redundant. All Zn are needed for a global description of molecular shape, and no subset of ZN — 6 Zn will be adequate everywhere.49 The space of molecular coordinates which defines the shape of a molecule is not a rectilinear or Euclidean space, it is a curved manifold. It is well known in the mathematical literature that you cannot find a single global set of coordinates for such curved spaces. 

In contrast, SIMCA uses principal components analysis to model object classes in the reduced number of dimensions. It calculates multidimensional boxes of varying size and shape to represent the class categories. Unknown samples are classified according to their Euclidean space proximity to the nearest multidimensional box. Kansiz et al. used both KNN and SIMCA for classification of cyanobacteria based on Fourier transform infrared spectroscopy (FTIR).44... 

Fig. 12.2. IRAS differential source counts as a function of observed flux at 60 xm, normalized to a uniform population in Euclidean space. After Oliver, Rowan-Robinson and Saunders (1992).

A continuous connected group may be simply connected or multiply connected, depending on the topology of the parameter space. A subset of the euclidean space Sn is said to be k-fold connected if there are precisely k distinct paths connecting any two points of the subset which cannot be brought into each other by continuous deformation without going outside the subset. A schematic of four-fold connected space is shown in the lower diagram. 

At any given instant the equation S(x, t) = const, defines a surface in Euclidean space. As t varies the surface traces out a volume. At each point of the moving surface the gradient, VS is orthogonal to the surface. In the case of an external scalar potential the particle trajectories associated with S are given by the solutions mx = VS. It follows that the mechanical paths of a moving point are perpendicular to the surface S = c for all x and t. A family of trajectories is therefore obtained by constructing the normals to a set of... 

In order to give a physical interpretation of special relativity it is necessary to understand the implications of the Lorentz rotation. Within Galilean relativity the three-dimensional line element of euclidean space (r2 = r r) is an invariant and the transformation corresponds to a rotation in three-dimensional space. The fact that this line element is not Lorentz invariant shows that world space has more dimensions than three. When rotated in four-dimensional space the physical invariance of the line element is either masked by the appearance of a fourth coordinate in its definition, or else destroyed if the four-space is not euclidean. An illustration of the second possibility is the geographical surface of the earth, which appears to be euclidean at short range, although on a larger scale it is known to curve out of the euclidean plane. 

The important result is the obvious symmetry between TM1/ and R u as shown in (42) and (43). Both of these tensors vanish in empty euclidean space and a reciprocal relationship between them is inferred The presence of matter causes space to curl up and curvature of space generates matter. 

In the real world the stress tensor never vanishes and so requires a nonvanishing curvature tensor under all circumstances. Alternatively, the concept of mass is strictly undefined in flat Minkowski space-time. Any mass point in Minkowski space disperses spontaneously, which means that it has a space-like rather than a time-like world line. In perfect analogy a mass point can be viewed as a local distortion of space-time. In euclidean space it can be smoothed away without leaving any trace, but not on a curved manifold. Mass generation therefore resembles distortion of a euclidean cover when spread across a non-euclidean surface. A given degree of curvature then corresponds to creation of a constant quantity of matter, or a constant measure of misfit between cover and surface, that cannot be smoothed away. Associated with the misfit (mass) a strain field appears in the curved surface. 

Interpreted, as it is, within the standard model, Higgs theory has little meaning in the real world, failing, as it does to relate the broken symmetry of the field to the chirality of space, time and matter. Only vindication of the conjecture is expected to be the heralded observation of the field bosons at stupendous temperatures in monstrous particle accelerators of the future. However, the mathematical model, without cosmological baggage, identifies important structural characteristics of any material universe. The most obvious stipulation is to confirm that inertial matter cannot survive in high-symmetry euclidean space. 

The quantity y/(a a) is known as the norm of a), analogous to the properties of vectors in Euclidean space. 

The phase space (r space) of the system is the Euclidean space spanned by the 2n rectangular Cartesian coordinates qL and pt. Every possible mechanical state of the system is represented by exactly one point in phase space (and conversely each point in phase space represents exactly one mechanical state). 

We now apply the generalized Matsubara formalism, discussed earlier, to a fermionic theory aiming to discuss effects of simultaneous spatial confinement and finite temperature. We consider the Wick-ordered massive Gross-Neveu model in a D-dimensional Euclidean space, described by the Lagrangian density (D.J. Gross et.al., 1974)... 

In some publications (e.g., [208]), the E parameter is considered as a topological dimension and E+1 as die dimension of the corresponding Euclidean space. 

Each object or data point is represented by a point in a multidimensional space. These plots or projected points are arranged in this space so that the distances between pairs of points have the strongest possible relation to the degree of similarity among the pairs of objects. That is, two similar objects are represented by two points that are close together, and two dissimilar objects are represented by a pair of points that are far apart. The space is usually a two- or three-dimensional Euclidean space, but may be non-Euclidean and may have more dimensions. 

This technique functions by taking observed measures of similarity or dissimilarity between every pair of M objects, then finding a representation of the objects as points in Euclidean space so that the interpoint distances in some sense match the observed similarities or dissimilarities by means of weighting constants. 

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