Non-Euclidean

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. ... 

The same idea can be developed in the case of a non-Euclidean metric such as the city-block metric or L,-norm (Section 31.6.1). Here we find that the trajectories, traced out by the variable coefficient kj are curvilinear, rather than linear. Markers between equidistant values on the original scales of the columns of X are usually not equidistant on the corresponding curvilinear trajectories of the nonlinear biplot (Fig. 31.17b). Although the curvilinear trajectories intersect at the origin of space, the latter does not necessarily coincide with the centroid of the row-points of X. We briefly describe here the basic steps of the algorithm and we refer to the original work of Gower [53,54] for a formal proof. 

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. 

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. 

An important question is whether the proximity measures are compatible with those of these references addresses the important issue of whether the proximity measure is compatible with embedding in a Euclidean space. For example, satisfying the distance axioms does not in itself guarantee that any distance matrix associated with a given set of molecules be compatible as the distance axioms are still satisfied in a non-Euclidean space. Gower has written extensively on this important issue, and his work should be consulted for details (89-91). Benigni (92), and Carbo (67) have also contributed interesting approaches in this area. 

The Fourth Dimension and Non-Euclidean Geometry in Modern Art. Princeton Univ. Press, 1983. 

Sommerville, D. M. Y. Bibliography of Non-Euclidean Geometry. Univ. of St. Andrews,... 

Cox98] H. S. M. Coxeter, Non-Euclidean Geometry, 6th edition, MAA Spectrum. Mathematical Association of America, 1998. 

Mag74] W. Magnus, Non-Euclidean Tesselations and Their Groups, Academic Press, 1974. 

The curve presented in the Fig. 6a should be considered periodic because it represents the vacuum properties of non-Euclidean spacetime. Here the axes of... 

All solutions of Einstein s equations are conditioned by the need of some ad hoc assumption about the geometry of space-time. The only indisputably valid assumption is that space-time is of absolute non-euclidean geometry. It is interesting to note that chiral space-time, probably demanded by the existence of antimatter and other chiral forms of matter, rules out the possibility of affine geometry, the standard assumption of modern TGR [7]. 

Application of Non-Euclidean Geometry Ideas in Polymer Topology... 

As was mentioned above, there are a lot of different ways of considering the Edwards-Frisch problem. However, from the methodological point of view and for the sake of a better clarification of non-euclidean geometry ideas for the description of topological constraints, we would like to present the method of conformal transformation. 

GEOMETRY OF COMPLEX NUMBERS, Hans Schwerdtfeger. Illuminating, widely praised book on analytic geometry of circles, the Moebius transformation, and two-dimensional non-Euclidean geometries. 200pp. 54 x 84. 

However, it is not quite clear whether this relation can be applied for non-Euclidean, ramified structures. Simulation results of carbon black formation under ballistic conditions by Meakin et al. [14] indicate that a scaling equation is fulfilled, approximately, between the number of particles Np in a primary aggregate and the relative cross section area A/Ap ... 

A more general formulation in terms of a non-euclidean manifold (curved space) has several advantages, the most important of which is a geometrical... 

Another method used for data analysis of non-Euclidean objects is fractals. A true fractal object is scale invariant (i.e., exhibits seU-similarity) thus a fractal dimension is obtained ITom the outline of an object by varying the scale of analysis. The ITactal dimension (FD) of an irregular geometry is a measure of the space-filling... 

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