Showing papers on "Euclidean distance published in 1970"

On metric multidimensional unfolding

Abstract: The problem of locating two sets of points in a joint space, given the Euclidean distances between elements from distinct sets, is solved algebraically. For error free data the solution is exact, for fallible data it has least squares properties.

Realization of all distances in a decomposition of the space R n into n+1 parts

Abstract: Let the sets A1, A2,...,An+1, form a covering of the n-dimensional euclidean space Rn (n>1). Then among these sets can be found a set Ai containing, for every d>0, a pair of points such that the distance between them is equal to d.

Psychological reality of physical concepts

Abstract: Fifty college physics students rated the similarity of pairs of concept words in analytical mechanics, and also provided 1 min of continued word associations to each individual concept word. The mean proportion of responses in common on the word association test was used as an index of associative similarity among concepts. Mean rating-scale judgments served as an index of perceived similarity. These two indices were interpreted as proximity measures and were scaled, using multidimensional scaling procedures with both a Euclidean and a city-block metric. Results suggest that either a two-dimensional configuration with a Euclidean metric or a three-dimensional configuration with a city-block metric describes the underlying structure of the similarity relations. The three-dimensional configuration correlated well with an hypothesized geometric model.

Sur la résolution de problèmes «inverses» en hydrogéologie

Abstract: We investigate the “inverse” problem consisting in the determination of the transmissivity and the storativity of an aquifer in every point of a net. The equations of the problem are generally undetermined, and one is led to introduce smoothing conditions. We also consider other two types of constraints: the knowledge of the transmissivity and the storativity in a certain number of points, and prescribed regions of equal values. Finally, we have several different solutions according to the relative importance of each constraint, and according to the selected norm. We explicit these solutions in the case of the Euclidean norm; they can be expressed very simply by means of pseudo-inverses.

Nonlinear regression methods for simultaneous property measurement

Abstract: A general procedure for the statistical estimation of thermal properties from experimental data is reviewed. The procedure, based on the techniques of nonlinear regression, includes all traditional methods of property determination and removes four specific restrictions frequently imposed on the design of property measurement experiments. This article surveys nonlinear-regression techniques and applications. A new convergence criterion for halting the iterative calculations is presented. The criterion Tnorm is based on the Euclidean norm of the scaled normal-equation residuals, equaling zero at a minimum of the root-mean-square deviation Trms and being scaled never to exceed Trms.

On a Simple Minkowski Metric Classifier

Abstract: A classifier which, in general, implements a nonlinear decision boundary is shown to be equivalent to a linear discriminant function when the measurements are binary valued; its relation to the Bayes classifier is derived. The classifier requires less computation than a similar one based on the Euclidean distance and can perform equally well.

Bounding immersions of codimension 1 in the euclidean space

Abstract: Let M be an (w + l)-dimensional differentiate manifold without boundary (compact or not) and ƒ : V—+M an immersion of the compact w-dimensional manifold without boundary V. We say that ƒ is a bounding immersion if there is a manifold W with boundary dW= V, and an immersion g\\W-+M such that f=g\\ V. If M and V are oriented, then V must be the oriented boundary of the oriented manifold Wy and g an oriented immersion of codimension 0. Using the classification of immersions (Smale [7], Hirsch [2]) and the work of Kervaire-Milnor [3], [4], we compute in this note the regular homotopy classes of all bounding immersions of the sphere S into the euclidean space jR and into the sphere S.

A Novel Firing Rule For Training Kohonen SelforganisingMaps

Abstract: Statistical Process Control Charts can exhibit six principal types of patterns: Normal, Cyclic, Increasing Trend, Decreasing Trend, Upward Shift and Downward Shift. All except Normal patterns indicate abnormalities in the process that must be corrected. Accurate and speedy detection of such patterns is important to achieving tight control of the process and ensuring good product quality. This paper describes an implementation of the Kohonen self-organising map which employs the Euclidean Distance as the firing rule for control chart pattern recognition. First, the structure of the network is outlined and the equations which govern its dynamics are given. Then the learning mechanism of the network is explained. The effects of different combinations of network parameters on classificatio n accuracy are discussed. A novel firing rule for the Kohonen self-organising map is proposed. This rule involves component-bycomponent comparison between the input pattern and the established class templates. When an input vector is presented, it is compared with the class templates in all the neurons in turn. The neuron containing the class template that best matches the input vector will subsequently fire. This approach is intended to enhance the generalisation capability and accuracy of the Kohonen selforganising map. The paper gives a comparison of the results obtained using the Euclidean Distance and the proposed firing rule.

Submanifolds of codimension 2 of a Euclidean space

Abstract: The main purpose of the present paper is to generalize, to the case of submanifolds of codimension 2, a famous theorem of Liebmann [1] and Sίiss [3]: The only convex hypersurface of a Euclidean space with constant mean curvature is a sphere. In § 1, we recall fundamental concepts and formulas for submanifolds of codimension 2 of a Euclidean space assuming that the mean curvature vector field never vanishes and taking it as the first normal to the submanifolds. In § 2, we prove integral formulas for general submanifolds of codimension 2 of a Euclidean space. § 3 is devoted to the study of submanifolds whose mean curvature vector field is parallel with respect to the connection induced in the normal bundle. In the last section 4, we study submanifolds which admit a normal vector field passing through a fixed point.