Showing papers on "Euclidean distance published in 1977"

Condition of self-duality for SU(2) gauge fields on Euclidean four-dimensional space

Abstract: The condition of self-duality for a SU(2) gauge field on Euclidean four-dimensional flat space is integrated once to obtain Laplace-like equations for three real variables or for variables one real and one complex. A special case of the latter reduces to the Corrigan-Fairlie-Wilczek-'t Hooft Ansatz.

Cluster Analysis Using Seed Points and Density-Determined Hyperspheres as an Aid to Global Optimization

Abstract: A model for finding the local optima of a multimodal function defined in a region A ? Rn is proposed. The method uses a local optimizer which is started from a number of points sampled in A. In order to reduce the number of function evaluations needed to reach the local optima, the parallel local search processes are stopped repeatedly, the working points clustered, and a reduced number of processes from each cluster resumed. A direct nonhierarchical cluster analysis technique is presented. The dissimilarity measure used is the Euclidean distance between points. Clusters are grown from seed points. The number of required distance evaluations is less than or equal to c(n-1), where n is the number of points and c is the number of clusters arrived at. Thresholds are determined by the point density in a body which in turn is determined by the given points. The covariance matrix is diagonalized, and a decision on the dimensionality of the space containing the points can be made. The volume of the body is proportional to the square root of the product of the corresponding eigenvalues. The performance of the clustering analysis technique is illustrated. It is demonstrated that there exist classes of global optimization problems for which the probability of obtaining a solution is greater for the proposed model than for multiple local optimizations. Some experiences gained from using the model are reported.

An Algorithm for Euclidean Sum of Squares Classification

Abstract: A ni analysis of surface pollen samples to discover iJ'they fall naturally in1to distinct groups oJ' simiiilar samiples is an example of a classification problem. In Euclidean classification, a set of n objects can be represented as n points in Euclidean space of p dimensions. The sum oJ'squares criterion defines the optimal partition of the points in1to g disjoint groups to be the partition which mninimzizes the total within-group sumn of squared distances about the g centroids. It is not usually Jeasible to examiine all possible partitions oJ'the objects into g groups. A critical review is mnade of algorithmiis which have been proposedfor seeking optimal partitions. The problem is reformulated itn non-linear programming terms, and a new algorithm for seeking the minimum sumi1 oJ'squares is described. The performance of this algorithm in analyzing the pollen data is Jound to compare vell vith the perJormance of three oJ the existing algorithms. An efficient hybrid algorithmi is introduced.

A Hölder estimate for quasiconformal maps between surfaces in Euclidean space

Abstract: In [2] C. B. Morrey proved a H61der estimate for quasiconformal mappings in the plane. Such a HSlder estimate was a fundamental development in the theory of quasiconformal mappings, and had very important applications to partial differential equations. L. Nirenberg in [3] made significant simplifications and improvements to Morrey's work (in particular, the restriction that the mappings involved be 1 1 was removed), and he was consequently able to develop a rather complete theory for second order elliptic equation with 2 independent variables. In Theorem (2.2) of the present paper we obtain a H61der estimate which is analogous to that obtained by Nirenberg in [3] but which is applicable to quasiconformal mappings between surface~ in Euclidean space. The methods used in the proof are quite analogous to those of [3], although there are of course some technical difficulties to be overcome because of the more general setting adopted here. In w 3 and w 4 we discuss applications to graphs with quasiconformal Gauss map. In this case Theorem (2.2) gives a H61der estimate for the unit normal of the graph. One rather striking consequence is given in Theorem (4.1), which establishes the linearity of any C2(R *) function having a graph with quasiconformal Gauss map. This result includes as a special case the classical theorem of Bernstein concerning C2(R 2) solutions of the minimal surface equation, and the analogous theorem of Jenkins [1] for a special class of variational equations. There are also in w 3 and w 4 a number of other results for graphs with quasiconformal Gauss map, including some gradient estimates and a global estimate of H61der continuity. w 4 concludes with an application to the minimal surface system. One of the main reasons for studying graphs satisfying the condition that the Gauss map is quasieonformal (or (A1, A2)-quasiconformal in the sense of (1.8) below) is tha t such

Large Sample Properties of Nearest Neighbor Density Function Estimators

Abstract: Publisher Summary This chapter presents the assumption that Let X1, X2,… are iid random variables having unknown density function f with respect to Lebesgue measure λ on Euclidean p-space RP. It presents an estimation of f(z) for a given z. A nearest neighbor estimator of f(z) is gn(z) = k(n)/n/λ{S(R(n))}. gn(z) is simply empiric measure divided by Lebesgue measure for the region S(R(n)). This estimator is essentially due to Fix and Hodges, and was explicitly introduced and studied by Loftsgaarden and Quesenberry. These and subsequent authors used the Euclidean norm, but for p > 1, other norms may be useful and proofs are unaffected by this generality.

The Equivalence of Three Statistical Packages for Performing Hierarchical Cluster Analysis.

Abstract: Three different software programs which contain hierarchical agglomerative cluster analysis procedures were shown to generate different solutions on the same data set using apparently the same options. The basis for the differences in the solutions was the formulae used to calculate Euclidean distance. Users of statistical software were warned of terminological confusion concerning cluster analysis.

Homological Methods for the Classification of Discrete Euclidean Structures

Abstract: Homology groups are important topological invariants of discrete structures. In this paper we present efficient algorithms for the computation of all the homology groups of a structure imbedded in a 3-dimensional Euclidean space, as well as an efficient algorithm for the computation of the $(n - 1)$st homology group of a structure imbedded in an n-dimensional Euclidean space.

Automated Routing of Large-Scale Circuits

Abstract: This paper presents a comprehensive state-of-the-art survey of the interconnection problem of large-scale circuits using design automation. The different types of routers in existence, as well as the implementation of some relevant algorithms are mentioned. The topological approach to layout design including applications of known results in Euclidean metric and some new results in rectilinear or Manhattan metric of topological invariants like thickness, are considered. The trend of future research and some of the important problems that demand attention, especially in the routing of multinets in multi-layers are cited. Attention is directed to the relatively new results as well as unreported results in progress, while the older results are adequately mentioned or referenced.

A Psychophysical Analysis of Complex Integrated Displays.

Abstract: : Five types of complex integrated displays were subjected to multidimensional scaling analyses. The display types were selected to be representative of a variety of characteristics that can result when dimensions are combined in an integrated fashion. These characteristics included perceptual separability, familiarity, emergent properties and perceptual interaction among dimensions. Of primary interest was the question of whether or not the Minkowski scaling metric would be diagnostic or predictive of any of these characteristics, as previous literature had indicated. The results showed that in virtually all cases the Euclidean metric produced better fits than the City-Block metric.

Error Bounds for Approximate Fixed Points

Abstract: The error bound between a given function and its approximate fixed-point, as computed through finite pivoting algorithms, is usually expressed in terms of max-norm. This is because the manner in which the integer labeling is defined lends itself to a simple and direct derivation of such a max-norm error bound. However, such norms are not invariant under rotation of coordinate axes. In this paper the best uniform error bound, expressed in euclidean norm, for a proper integer labeling of a triangulation of the standard n-simplex S n is derived.

Embedding Graphs in Euclidean 3-Space

Abstract: (1977). Embedding Graphs in Euclidean 3-Space. The American Mathematical Monthly: Vol. 84, No. 5, pp. 372-373.