Showing papers on "Euclidean distance published in 1979"

The Positive Action conjecture and asymptotically Euclidean metrics in quantum gravity

Abstract: The Positive Action conjecture requires that the action of any asymptotically Euclidean 4-dimensional Riemannian metric be positive, vanishing if and only if the space is flat. Because any Ricci flat, asymptotically Euclidean metric has zero action and is local extremum of the action which is a local minimum at flat space, the conjecture requires that there are no Ricci flat asymptotically Euclidean metrics other than flat space, which would establish that flat space is the only local minimum. We prove this for metrics onR 4 and a large class of more complicated topologies and for self-dual metrics. We show that ifR μ μ ≧0 there are no bound states of the Dirac equation and discuss the relevance to possible baryon non-conserving processes mediated by gravitational instantons. We conclude that these are forbidden in the lowest stationary phase approximation. We give a detailed discussion of instantons invariant under anSU(2) orSO(3) isometry group. We find all regular solutions, none of which is asymptotically Euclidean and all of which possess a further Killing vector. In an appendix we construct an approximate self-dual metric onK3 — the only simply connected compact manifold which admits a self-dual metric.

Mathematical Models of Road Travel Distances

Abstract: Some management science models require estimates of distances between points in a road network based on the point coordinates. This paper evaluates the accuracy of a range of reasonable forms for distance estimating functions using samples of urban and rural road distances. The intent is to derive better distance estimating functions for analyzing systems with distance related performance measures. Contrary to a standard assumption, the rectangular distance function is inferior to the simple Euclidean metric in the urban samples. More general functions provide still greater improvement over the rectangular metric. Statistical significance accompanies these conclusions. One of the more general functions appears particularly suited to rural distances.

Mixture Model Tests Of Hierarchical Clustering Algorithms: The Problem Of Classifying Everybody.

Abstract: Due to the effects of outliers, mixture model tests that require all objects to be classified can severely underestimate the accuracy of hierarchical clustering algorithms. More valid and relevant comparisons between algorithms can be made by calculating accuracy at several levels in the hierarchical tree and considering accuracy as a function of the coverage of the classification. Using this procedure, several algorithms were compared on their ability to resolve ten multivariate normal mixtures. All of the algorithms were significantly more accurate than a random linkage algorithm, and accuracy was inversely related to coverage. Algorithms using correlation as the similarity measure were significantly more accurate than those using Euclidean distance (p < .001). A subset of high accuracy algorithms, including single, average, and centroid linkage using correlation, and Ward's minimum variance technique, was identified.

Proof of the Positive-Action Conjecture in Quantum Relativity

Abstract: We extend our previous method of proving the positive-mass conjecture to prove the positive-action conjecture of Hawking for asymptotically Euclidean metric. This result is crucial in proving the path integral convergent in the Euclidean quantum gravity theory.

Generalization of the Euclidean algorithm for real numbers to all dimensions higher than two

Abstract: A construction using integral matrices with determinant ± 1 is given which has as corollaries generalizations of classical theorems of Dirichlet and Kronecker. This construction yields a geometrically convergent algorithm successfully generalizing the Euclidean algorithm to finite sets of real numbers. Applied to such a set this algorithm terminates if and only if the set is integrally linearly dependent and the algorithm gives absolute simultaneous integral approximations if and only if the set is integrally linearly independent. This development applies to complex numbers, can be used to give proofs of irreducibility of polynomials and yields effective lower bounds on heights of integral relations. Let Z = rational integers, R = real numbers, Z\" = lattice points C R\" as row vectors, GLW (Z) C GLW (R) are n by n matrices with entries and invertible determinants in Z C R resp. For M = any matrix or vector, M = transpose, row,.M = ith row, coljM = ;th column, height (M) = max absolute values of entries of M. The entries ofxGR\" are Z-linearly dependent iff there exists 0 =£ m G Z such that xm = 0, m = Z-relation for x For 0 &x E R, x determines the line xR and orthogonal hyperplane x ~ {y &R: xy ==0}. A hyperplane matrix Q with respect to x is any matrix xQ = 0 such that the columns of Q transposed span r . The hyperplane matrix is a key idea here in three aspects: (I) it permits estimates of heights of relations (Theorem 1), (II) it measures how closely the rows of a GLW (Z) matrix are to the line xR (Lemma 1), (III) it underlies the definition of a crucial injection GLw(Z)c* GLW+1 (Z) (Lemma 2). We exploit the nonuniqueness of Q. THEOREM 1. Let 0 =£ x E R\". Then there exists a hyperplane matrix Q such that height m > 1/height ylô for m any Z-relation for x and any A G GL„(Z). SKETCH OF PROOF. The parallelotope /A/ = { Hf-coljA: \\fj\\ < 1 < ƒ < n} has easily characterized lattice points if A E GLW (Z). Let I = identity matrix and define Q to be the hyperplane matrix whose columns transposed are the vertices of the convex poly tope //ƒ O x. A GLW (Z)-algorithm is defined to be any construction (usually in response Received by the editors March 26, 1979. AMS (MOS) subject classifications (1970). Primary 10E45, 10F10, 10F20; Secondary 10F37, 12A10, 10H05, 02E10. 912 © 1 9 7 9 American Mathematical Society 0002-9904/79/0000-0505/$01-75 GENERALIZATION OF THE EUCLIDEAN ALGORITHM 913 to an x) of a sequence {M k} k > % ,M k^ GLW (Z). If xMk has a zero entry for some k then the entries ofx are Z-linearly dependent: the algorithm terminates. If the height of Mk Q approaches zero as k increases to infinity then the entries ofx are Z-linearly independent: the algorithm absolutely approximates x. A GLW (Z)-algorithm is split iff the algorithm terminates or absolutely approximates for every x G R\". LEMMA 1. If a GLn (Z) algorithm is split and does not terminate for some x then the distance of the rows ofMk l to the line xR approaches zero as k increases. SKETCH OF PROOF. For xx = 1, set Q = I xx to get a line-hyperplane decomposition of any matrix A = (Ax*)x -f AQ. Define the GL2 (Z) algorithm A2 by the following iteration. For x = (xl9 x2), let Xf be of largest and x* of next largest absolute value. Replace x( by the xt ± Xj of smaller absolute value. Then A2 splits. We will give an uncountable collection of GLW (Z) algorithms which split, An(b)9 n>2> l/b> 1. For brevity we describe them by induction on n by defining the injection/: GLW(Z) CL_» GL„ + ! (Z), / is also an integer ! < / < « «f 1. L e t x ^ G R \" bexGR\"\" \" with the /th entry deleted. For A G GLW (Z) set Ti = rowf.^l if i < ƒ or 7̂ = rowf_j A if i > / , c, = nearest integer to Tfc^yXj/x^x^y Define 7(4) G GLW + 1 (Z) to have a 1 in the (j, ƒ) position, zeros in that row, the c's in that column and A the minor matrix of the 1. Let Q^ be Q without the /th row and Tt = djtfy) + vi orthogonally. LEMMA 2. 77*e injection map J : GLW (Z) c-* GLW + j (Z) to rte property that row, (/(<4)0 ^ row;Qifi — j and S, row;. Ö + iQ(j) V ' ^ / w/*ere 16/1 < 1/2. To sketch the construction of An + l(b) from ^4„(è) and the proof that An splits, select the map / by choosing / to be the number of the row of Q with least height. By induction An splits; if An does not terminate then for 0 < e < (b — 1/2) height (row;. Q) there exists a finite number of iterations of An acting on Xfj} yielding XyyA~ for a certain A G GL„ (Z) with height (nt) < € I height Ö. By Lemma 2, height (row, (J(A)Q)) < b height (rowy Q) if i # / . Define one iteration of An+l by x *~*x(J(A)y~ . THEOREM 2. For every integer n and real number b, n> 2> l/b> I, the GLW (Z) algorithm An(b) splits, ie.9 for every nonzero x € R \" the sequence of matrices Mk, k> 1 is such that Termination) There exists a k such that a column of Mk is an integral relation among the entries ofx OR Absolute approx* imation) For every e > 0 there exists an integer K > 1 such that for each k > K the rows of Mk l give n linearly independent lattice points in Z, each within a distance e of the line determined by x. 914 H. R. P. FERGUSON AND R. W. FORCADE THEOREM 3. Let x = (xx, x2, . . • , xn) E R n where xx — 1, x2, . . . , xn are Z-linearly independent real numbers. Then for every e > 0 there exists an integral matrix P G GLW (Z) with first column N, N* E Z n such that height(Nx-P) 4. We have several examples that suggest Szekeres' [Ann. Univ. Sci. Budapest, Eötvös Sect. Math. 13 (1970), 113-140] algorithm does not split for n>5. DEPARTMENT OF MATHEMATICS, BRIGHAM YOUNG UNIVERSITY, PROVO, UTAH 84602

Analysis of an Algorithm for Finding Nearest Neighbors in Euclidean Space

Abstract: An analysis of Elias's algorithm for finding nearest nmghbors is made in n-dimensional Euclidean space. An expressmn for the executmn time is obtained when the data points being searched are grouped by arbitrary regular partitions. A lower bound corresponding to spherical partitions as obtained where the executmn tune increases exponentmlly with increasing dimensmn. The best known approach to th~s lower bound is the cubic par tmon whmh lies within 10 percent for dimensions 2 and 3 but whmh rapidly diverges for higher dimensions. The number of data points examined during a search is shown to be independent of the total number of data points.

A distance measure based on the derivative of linear prediction phase spectrum

Abstract: A new distance measure based on the derivative of linear prediction (LP) phase spectrum is proposed for comparison of speech spectra. Relationships among several distance measures based on the linear prediction coefficients (LPCs) are discussed. The advantages of the new measure and an efficient method of computing it are also discussed.

Bounds on the Optimal Location to the Weber Problem Under Conditions of Uncertainty

Abstract: Optimal and Heuristic bounds are given for the optimal location to the Weber problem when the locations of demand points are not deterministic but may be within given circles. Rectilinear, Euclidean and square Euclidean types of distance measure are discussed. The exact shape of all possible optimal points is given in the rectilinear and square Euclidean cases. A heuristic method for the computation of the region of possible optimal points is developed in the case of Euclidean distance problem. The maximal distance between a possible optimal point and the deterministic solution is also computed heuristically.

Strange complex structures on Euclidean space.

Abstract: On the topological space IR there are exactly two different complex structures, the canonical structure of C and that of the unit disc in C. In Opposition to this Situation one has on IR for n> l an infinite number of different complex structures. There are, for instance, äs extreme cases the famous examples of Calabi and Eckmann (see [2]) without any nonconstant holomorphic functions. On the other hand we remind the reader of the recent results of D. Burns, St. Shnider, R. Wells ([!]) and of S. Webster ([9]) both of which give large families of pairwise biholomorphically inequivalent strictly pseudoconvex domains in C\", #^2, all diffeomorphic to IR.

Evaluation of several schemes for classification of remotely sensed data: Their parameters and performance

Abstract: The author has identified the following significant results. Data sets for corn, soybeans, winter wheat, and spring wheat were used to evaluate the following schemes for crop identification: (1) per point Gaussian maximum classifier; (2) per point sum of normal densities classifiers; (3) per point linear classifier; (4) per point Gaussian maximum likelihood decision tree classifiers; and (5) texture sensitive per field Gaussian maximum likelihood classifier. Test site location and classifier both had significant effects on classification accuracy of small grains; classifiers did not differ significantly in overall accuracy, with the majority of the difference among classifiers being attributed to training method rather than to the classification algorithm applied. The complexity of use and computer costs for the classifiers varied significantly. A linear classification rule which assigns each pixel to the class whose mean is closest in Euclidean distance was the easiest for the analyst and cost the least per classification.

A characterization of isometries of rational Euclidean spaces

Abstract: We show that an injection of the rational Euclidean n-space, n≥5, which preserves the distances e, 1/2e, e an arbitrary non-zero rational number, is necessarily an isometry. Further, we show that the above characterization fails in case n=3 or 4.

On kinematics of the n-dimensional Euclidean space

Abstract: A one-parametric family of isometries of the n-dimensional Euclidean space En is called a motion of En. The kinematics of En deals with the geometry of the motions of En. In the cases of the dimensions n = 2,3 many results (excluding the recent ones) are scheduled in textbooks (for example: H.R. Muller: Kinematik, Berlin 1963; R. Beyer: Technische Raumkinematik, Berlin 1963; a.o.). The first important results in the case of arbitrary dimension n are due to H.R. Muller in [4].