Showing papers on "Euclidean distance published in 1981"
TL;DR: A local distance measure is shown to optimize the performance of the nearest neighbor two-class classifier for a finite number of samples using the difference between the finite sample error and the asymptotic error as the criterion of improvement.
Abstract: A local distance measure is shown to optimize the performance of the nearest neighbor two-class classifier for a finite number of samples. The difference between the finite sample error and the asymptotic error is used as the criterion of improvement. This new distance measure is compared to the well-known Euclidean distance. An algorithm for practical implementation is introduced. This algorithm is shown to be computationally competitive with the present nearest neighbor procedures and is illustrated experimentally. A closed form for the corresponding second-order moment of this criterion is found. Finally, the above results are extended to
251 citations
TL;DR: In this article, a constrained form of the Weber problem is formulated in which no path is permitted to enter a prespecified forbidden region R of the plane, and the shortest path between two points x, y ∉ R which does not intersect R is determined.
119 citations
01 Jan 1981
TL;DR: In this paper, the dimension of a distance matrix and its associated metric space is defined, and necessary and sufficient conditions for a metric space to be isometrically embeddable into suitable real inner product spaces and Euclidean spheres.
Abstract: We define the dimension of a distance matrix and its associated metric space, and use this to give necessary and sufficient conditions for a metric space to be isometrically embeddable into suitable real inner product spaces and Euclidean spheres. Also, for certain distance matrices C with irrational entries, we derive the bound w ≤ 2 f + 1 for the size w of C in terms of its dimension f . This result is applied to improve a bound by Larman, Rogers, and Seidel on two-distance sets in Euclidean space, and to characterize certain regular graphs as conference graphs.
48 citations
TL;DR: The performance analysis of CPM systems is extended to obtain bounds on the symbol error probability when the Viterbi algorithm is used, and it is concluded that the minimum Euclidean distance is a good performance measure for a broad class of C PM signals, even when the symbolerror probability is as large as 10-2.
Abstract: Recently the minimum Euclidean distance and bandwidth properties of continuous phase modulated (CPM) signals have been considered. It has also been shown that, for rational modulation indexes, a state description of these signals is possible and that the Viterbi algorithm (VA) can be used for demodulation. In this paper the performance analysis of CPM systems is extended to obtain bounds on the symbol error probability when the VA is used. The calculation of these bounds is based on the transfer function technique, which has been generalized. From numerical comparisons of the upper and lower bounds, it is concluded that the minimum Euclidean distance is a good performance measure for a broad class of CPM signals, even when the symbol error probability is as large as 10-2.
47 citations
TL;DR: Efficient algorithms, based on Goldman's concept of visibility, are proposed to provide the set of Pareto-optimal locations of the Weber–Rawls problem in which both efficiency and equity criteria are minimized.
Abstract: The Weber problem with maximum distance constraints is considered. Efficient algorithms, based on Goldman's concept of visibility, are proposed. Specifically, an O( m ) algorithm is given for the rectilinear norm case, where m is the number of users; an O( m 2 ) algorithm for determining the feasible region in the Euclidean norm case and an efficient algorithm for solving the corresponding location problem are presented. These algorithms are further used to provide the set of Pareto-optimal locations of the Weber–Rawls problem in which both efficiency and equity criteria are minimized.
28 citations
TL;DR: In this paper, the uniqueness of the topological placement of certain minimal surfaces in euclidean 3D space has been studied and the main result of Theorem 1 is proved.
26 citations
TL;DR: In this paper, a geometrical characterization of v4-submanifolds M of a Euclidean space E using the (p-2)th polar hypersurface KTM of the characteristic hypersurf ace K of the normal space T^M, m^M.
Abstract: §0. Introduction. In this paper we give a geometrical characterization of v4-submanifolds M of a Euclidean space E using the (p—2)th polar hypersurface KTM of the characteristic hypersurf ace K of the normal space T^M, m^M. This leads us to other characterizations involving Lipschitz-Killing curvature and second mean curvature. Several examples of ^4-submanifolds in E* are given. Finally we extend the notion of ^4-submanifold in a natural way to /U-submanifold according to the position of the mean curvature vector with respect to K%.
22 citations
01 Jan 1981
TL;DR: A scheme is proposed where both the Euclidean metric and a more simple city-block metric are utilized together to reduce overall classification time.
Abstract: Many pattern recognition computer programs use one of the clustering algorithm techniques. Often these algorithms use a Euclidean distance metric as a similarity measure. A scheme is proposed where both the Euclidean metric and a more simple city-block metric are utilized together to reduce overall classification time. The relation between the Euclidean and city-block distances is introduced as a scalar function. The bounds of the function are given and used to decide whether classification of each pattern vector is to be achieved by the computationally slow Euclidean distance or the faster city-block distance. The criteria is that the classification should be identical to the original Euclidean only scheme.
8 citations
TL;DR: In this article, it was shown that for every nxn doubly stochastic matrix A, for every entry aij, where is the Euclidean norm, the cases for which equality holds are discussed in detail.
Abstract: We show that for every nxn doubly stochastic matrix A, for every entry aij , where is the Euclidean norm. The cases for which equality holds are discussed in detail.
7 citations
01 Jan 1981
TL;DR: In this article, the Euclidean path-integral formula for the massive vector field is derived from a well-defined positive metric Hilbert space, and it is shown that the Minkowski and/or the Euclidan expres- sions can be obtained from this well defined pathintegral model by a simple change of the variable, and that Bo and B, are both real and run from -co to +co.
Abstract: Euclidean path-integral formula for the massive vector field is derived from a well-defined positive metric Hilbert space. It is shown that the Minkowski and/or the Euclidean expres sions can be obtained from this well-defined path-integral formula by a simple change of the variable, and that Bo and B, are both real and run from -co to +co. It is also discussed that even though one employs the canonical formalism which contains the famous non-covariant Schwinger term in the vector case, one can get a covariant path-integral formula by starting from a Lagrangian with source terms.
6 citations
TL;DR: It is shown that at least one of the sums ±u1 ± … um has norm at most n12 given any family u1,…, um of vectors in Euclidean n-space of Euclidesan norm at least unity.
TL;DR: In this paper, the effects of uncertainty about impact multipliers on the optimal policy in a static linear-quadratic optimization model were examined, and it was shown that increasing uncertainty will act to reduce the vigour of policy if, in the optimal decision rule, the reaction matrices, corresponding to the cases before and after the change in uncertainty, commute.
TL;DR: In this paper, an iterative solution method is presented for solving the multifacility location problem with Euclidean distances under the minimax criterion, which is based on the transformation of the multi-acility minimax problem into a sequence of squared EuclIDEan minisum problems which have analytical solutions.
Abstract: An iterative solution method is presented for solving the multifacility location problem with Euclidean distances under the minimax criterion. The iterative procedure is based on the transformation of the multifacility minimax problem into a sequence of squared Euclidean minisum problems which have analytical solutions. Computational experience with the new method is also presented.
01 Jan 1981
TL;DR: The maximum area of a plane set S with property (2) is Ω( √ 2 )π, attained only when S is (essentially) a disc as discussed by the authors.
Abstract: Let us say that a set points in a Euclidean space has property (2) if no 2 points X, Y ∈ S have X Y > 1. Then an easy observation is the following: The maximum area ( Lebesgue measure ) of a plane set S with property (2) is ¼π, attained only when S is (essentially) a disc .
TL;DR: In this paper, a positive definite, asymptotically Euclidean metric was obtained on R4 by rotating the Levi-Civita metric in an appropriate coordinate range.
Abstract: By Wick rotating the Levi‐Civita metric in an appropriate coordinate range, a positive definite, asymptotically Euclidean metric is obtained on R4. This metric has the following properties: (a) it is smooth everywhere except on a 3‐sphere Σ, where it suffers (finite) discontinuities; (b) it is Ricci‐flat everywhere except on Σ; and (c) it is periodic in ’’imaginary time’’ both inside and outside Σ but with two distinct periods (reflecting the fact that the corresponding Lorentzian section covers the region bounded by two distinct horizons). In the Euclidean approach to quantum gravity, each region with a fixed period may be regarded as being at a fixed temperature. Therefore, in the semiclassical approximation, the metric represents an interesting extension of the familiar states of thermal equilibrium of the gravitational field.
05 Apr 1981
TL;DR: In this article, the authors used most likelihood, linear discriminant function, minimum Euclidean distance, and correlation coefficient methods to classify remotely sensed mu1 tispectral data.
Abstract: For classification of remotely sensed mu1 tispectral data the most likelihood method is employed very often. It, however, does not always give us the best results. Two reasons for that can be considered, that is, deviation from normal distribution of the data, and lack of generality of the training data used. one of them is dominant is shown. It use:; comparison of correct classification rates obtained by mutually related a1 gori thms. verification is given for the case of classification making land use map. Utilized algorithms were most likelihood, linear discriminant function, minimum Euclidean distance, and correlation coefficient methods. A way to determine which
TL;DR: In this article, the SU(2) Yang-Mills field equations in Minkowski space from Euclidean self-dual fields were solved by Bernreuther's method.
Abstract: It is examined, if it is possible, to obtain solutions of the SU(2) Yang-Mills field equations in Minkowski space from Euclidean self-dual Yang-Mills fields by method proposed by Bernreuther. It is shown that the conditions, which are imposed on the Euclidean self-dual fields by this method, make every Euclidean self-dual field and the corresponding Minkowski space field obtained from it, equivalent to a pure gauge field, Fab≡0.
Journal Article•
TL;DR: In this paper, it was shown that if A is n ×n doubly stochasticn > 2, and if per is the Euclidean norm, then Jn is the n×n matrix whose entries are all n − − 1.
Abstract: We show that if A is n ×n doubly stochasticn > 2, and if then per is the Euclidean norm; Jn is the n×n matrix whose entries are all n −
TL;DR: In this paper, the Gauss image does not move pointwise, and deformations where the tensor of deformation satisfies some differential equations, normal deformations and deformation of totally real submanifolds in a complex space are studied.
Abstract: Deformation of an m-dimensional submanifold immersed in a Euclidean n-space is studied when the Gauss image does not move pointwise. The present paper is the third one and contains deformations where the tensor of deformation satisfies some differential equations, normal deformations and deformations of totally-real submanifolds in a complex space.
TL;DR: In this paper, it was shown that if, then, a function defined in a subset of -dimensional Euclidean space has a total differential, i.e., the least uniform deviation of the function from the rational functions of degree.
Abstract: Denote by the least uniform deviation of the function , defined in a subset of -dimensional Euclidean space, from the rational functions of degree . It is shown that if , then, a.e. on , has a total differential. The case was previously treated by E. P. Dolženko. Bibliography: 9 titles.