Showing papers on "Euclidean distance published in 1969"

Goodness of fit for random rankings in kruskal's nonmetric scaling procedure *

Abstract: in order to establish null hypotheses for the goodness-of-fit measure. Various numbers of hypothetical stimuli were analyzed using the Euclidean distance metric in various dimensionalities, and the effects of tied ranks were studied. Kruskal (1964a, 1964b) presented a method of multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis. His method is an elaboration and refinement of the analysis of proximities proposed by Shepard (1962). In Kruskal's procedure, N points are positioned in an jw-dimensional space in such a way as to achieve the best possible approximation to a monotonic relationship be/N\ tween the ( . 1 interpoint distances and an experimentally obtained ranking of the dissimilarities among all pairs of N objects. For a given configuration of points, interpoint distance (d) is expressed as a function of ranked dissimilarity (8). The degree to which d is not a monotonic function of 8 for a given configuration is expressed by a measure called stress (S). Stress can be thought of as an

The embedding of nonlinear meson transformations in a Euclidean space

Abstract: The equivalence between the linear and nonlinear chiral transformations of the 0− mesons is demonstrated using a simple form of an embedding theorem of Palais.

On the (m+1)-terminal resistive-network problem

Abstract: It is shown that any (m+1)-terminal positive resistive network forms an mth order metric space Sm, whose characteristic is completely defined by the main theorem stated in this work. A metric transformation transforms Sm into the mth-order distance space Dm, which is congruently imbeddable in m-dimensional Euclidean space Em. As a consequence, any (m+1)-terminal positive resistive network may be considered to form an mth-order simplex or geometrical figure Pm in Em. This figure was shown to have all its first- and higher-order angles acute and was called hyperacute-angled. As a consequence, problems on resistive networks, particularly on resistive n-ports, are equivalent to geometric problems on acute-angled simplexes imbeddable in multidimensional Euclidean space. All the arsenal of knowledge, acquired over many years, which we posses on Euclidean spaces may therefore be utilised in the solution of the resistive n-port problem.

A weak projection of $C$ onto a Euclidean subspace

Abstract: In general there exists no projection of a space of continuous functions onto a subspace with the property that the distance between the projections of any two points does not exceed the distance between the two points. Nevertheless, we shall see that there is a projection of C(Sn-,) onto a suitable Euclidean n-dim subspace with the property that the volume of the projection of a k-dim box does not exceed the volume of the box if k > 1. The specific definitions are given later, but we can say here that volume, as defined in an n-dim Minkowski space, is equivalent to saying that the smallest box containing the unit sphere has volume 2n. We then prove a weak Kirzbraun type theorem which enables us to deduce that Kolmogorov's Principle, as applied to Lebesgue area, holds for surfaces in C. From this we conclude that a generalization of Lebesgue area is an extension. Previously it was known that the generalization agreed with Lebesgue area only on those surfaces for which a lower area agreed with Lebesgue area. The crucial result is, essentially, the following: Let