Showing papers on "Euclidean distance published in 1975"

Preference Structures. II: Distances Between Asymmetric Relations

Abstract: In this paper the results of Preference structures I are extended to not necessarily transitive relations. The extension of the city block metric of [3] leads to a discussion of the relation of majority rule to the median of a collection of relations. The extension of the Euclidean metric of [3] leads to a discussion of the mean of a collection of relations and methods for finding means. Finally, the extension of the Euclidean metric leads to a discussion of statistical inference about random variables whose ranges are preference relations.

How to calculate shortest vectors in a lattice

Abstract: A method for calculating vectors of smallest norm in a given lattice is outlined. The norm is defined by means of a convex, compact, and symmetric subset of the given space. The main tool is the systematic use of the dual lattice. The method generalizes an algorithm presented by Coveyou and MacPherson, and improved by Knuth, for the determination of vectors of smallest Euclidean norm. 1. Formulation of the Problem. Let G be a lattice in the n-dimensional Euclidean space Rn, generated by n linearly independent vectors e (1) G = {x = zie z1 integers}. The norm in Rn is defined by a convex, compact set B which has positive measure and is symmetric about the origin: (2) lIxIl = min{X E R I x E XB}. Examples of these norms for x = (xl, . . ., xn) are (i) The Euclidean norm llxl = (X2 + ? ? x2)1/2. (ii) The Maximum norm lxii = max{ixi I I i = 1, ... , n}. Here Boo = { . . ., Xn)I Ixi? 1 for all i}. (iii) The norm lixil = lx11 + -'_ + lXn 1Here B1 = {(xl, . . . , Xn) IXI I + ?* + Xn I 1 The problem is to find a nonzero vector of shortest length (norm) in G. The main tool of the presented method is the use of the dual lattice, (3) G* ={x*= E z4e*Iz*integers), k=l where the e* are defined by eie* = 6ik; here 6ik is equal to 1 if i = k and equal to 0 if i # k, and e e* denotes the scalar product V7n= e.1e* . The polar of B, namely (4) B* = {b* E Rn I Ibb*l < 1, V bEB}, induces a length or norm in G* by (5) 1ix*i*1 min{X* E R Ix* G X*B*}. It should be noted that the Euclidean norm corresponds to itself, whereas the Maximum norm lxii = maxi IxiI corresponds to iix*ii* = Ixv + i ? + Ix*l and vice versa. Received May 16, 1974; revised August 6, 1974. AMS (MOS) subject classifications (1970). Primary 10E0S, 10E20, 10E25; Secondary 65C10.

Near-Optimal Solutions to a 2-Dimensional Placement Problem

Abstract: We consider the problem of placing records in a 2-dimensional storage array so that expected distance between consecutive references is minimized. A simple placement heuristic which uses only relative frequency of access for different records is shown to be within an additive constant of optimal when distance is measured by the Euclidean metric. For the rectilinear and maximum metrics, we show that there is no such heuristic. For the special case in which all access probabilities are equal, however, heuristics within an additive constant of optimal do exist, and their implementation requires solution of differential equations for which we give numerical solutions.

On isometries of euclidean planes

Abstract: In [1,5 ] the authors prove that any transformation of the euclidean plane,coordinatized by the real field, which preserves a single non-zero distance is necessarily an isometry. The purpose of this note is to give a synthetic proof of the same result forany euclidean plane. Hence, I do not assume the axiom of completeness.

Metric problems in elliptic geometry

Abstract: Publisher Summary This chapter discusses several problems that may be interpreted as metric problems in elliptic geometry. Any metric space M is isometrically imbeddable in the Euclidean plane whenever each 5 element subset of M is isometrically imbeddable in the Euclidean plane. Any M is isometrically imbeddable in Euclidean n-space whenever each (n + 3)-subset of M. The number n + 3 is the smallest number having this property and is called the congruence order of Euclidean n-space. For elliptic space, the situation is different. The congruence order of the elliptic plane turns out to be 7 and not 5, and that of elliptic (n − 1)- space, for n > 3, is unknown but much larger than n + 3.

Cepstral distance and the frequency domain

Abstract: Atal [J Acoust. Soc. Am. 55, 1304 (1974)] recently investigated weighted Euclidean distance measures for speaker identification and verification tasks. The cepstral coefficients of the filter A(z) determined through linear prediction analysis resulted in higher scores than other parameters such as predictor coefficients or area functions. The purpose of this paper is to discuss several properties of the cepstral coefficients obtained from A(z) and to suggest that the results illustrate the importance of the smoothed log spectrum as a basis for distance measurements in speech processing. From statistical experiments on speech, it is shown that for very small cepstral lengths (on the order of M, where M is the number of filter coefficients), there exists a very high correlation between the ceptral measure and the root‐mean‐square (rms) Euclidean distance measure between two log spectral models obtained using linear prediction analysis. Theoretically, an infinite number of cepstral coefficients are necessary...