Showing papers on "Metric (mathematics) published in 1986"

Metric and Euclidean properties of dissimilarity coefficients

Abstract: We assemble here properties of certain dissimilarity coefficients and are specially concerned with their metric and Euclidean status. No attempt is made to be exhaustive as far as coefficients are concerned, but certain mathematical results that we have found useful are presented and should help establish similar properties for other coefficients. The response to different types of data is investigated, leading to guidance on the choice of an appropriate coefficient.

Applying Metric and Nonmetric Multidimensional Scaling to Ecological Studies: Some New Results

Abstract: Metric (eigenanalysis) and nonmetric multidimensional scaling strategies for ecological ordination were compared. The results, based on simulated coenoplane data showing varying degrees of species turnover on two independent environmental axes, suggested some strong differences between metric and nonmetric scaling methods in their ability to recover underlying nonlinear data structures. Prior data standardization had important effects on the results of both metric and nonmetric scaling, though the effect varied with the ordination method used. Nonmetric multidimensional scaling based on Euclidean distance following stand norm standardization proved to be the best strategy for re- covering simulated coenoplane data. Of the metric strategies compared, correspondence analysis and the detrended form were the most successful. While detrending improved ordination configurations in some cases, in others it led to a distortion of results. It is suggested that none of the currently available ordination strategies is appropriate under all circumstances, and that future research in ordination methodology should emphasize a statistical rather than empirical approach.

The Fefferman metric and pseudo-Hermitian invariants

Abstract: C. Fefferman has shown that a real strictly pseudoconvex hypersurface in complex n-space carries a natural conformai Lorentz metric on a circle bundle over the manifold. This paper presents two intrinsic constructions of the metric, valid on an abstract CR manifold. One is in terms of tautologous differential forms on a natural circle bundle; the other is in terms of Webster's pseudohermitian invariants. These results are applied to compute the connection and curvature forms of the Fefferman metric explicitly.

Interpoint Distance Comparisons in Correspondence Analysis

Abstract: Correspondence anal/sis is a metric technique for finding a spatial representation of data that has particular applicability to the analysis of cross tabulations (or contingency tables). The author...

A pragmatic view of accuracy measurement in forecasting

Abstract: Accuracy measurement in forecasting is always a subject of debate because of its importance. An adequate metric is necessary to properly select a forecasting method for a specific application. Competitions to determine the best method have helped the practitioner. The criteria for selection have not received as much attention. Of the two kinds of measurement statistics—relative and absolute—the former may present problems for the user if zeros or near zero values appear. This is more a practitioner problem because artificially generated time series do not usually have zeros. The relative and absolute measures are discussed and a solution for the existence of zeros in the data is given. If symmetry of the errors is a problem solutions are discussed. Managers will select the metric depending on the application and their management style. Once the metric has been selected the decision as to which forecasting method to select in a given situation becomes a less difficult problem.

On type of metric spaces

Abstract: Families of finite metric spaces are investigated. A notion of metric type is introduced and it is shown that for Banach spaces it is consistent with the standard notion of type. A theorem parallel to the Maurey-Pisier Theorem in Local Theory is proved. Embeddings of Zp-cubes into the ¡i-cube (Hamming cube) are discussed.

An extended Čencov characterization of the information metric

Abstract: Cencov has shown that Riemannian metrics which are derived from the Fisher information matrix are the only metrics which preserve inner products under certain probabilistically important mappings. In (encov's theorem, the underlying differentiable manifold is the probability simplex E'xi = 1, xi > 0. For some purposes of using geometry to obtain insights about probability, it is more convenient to regard the simplex as a hypersurface in the positive cone. In the present paper Cencov's result is extended to the positive cone. The proof uses standard techniques of differential geometry but does not use the language of category theory.

The quasihyperbolic metric and associated estimates on the hyperbolic metric

Abstract: In this paper we investigate the geometry of the quasihyperbolic metric of domains in R". This metric arises from the conformally flat generalized Riemannian metric d(x, aD)-'ldx I. Due to the fact that the density cl(x, aD) -t is not necessarily differentiable, the classical theories of Riemannian geometry do not apply to this metric. The quasihyperbolic metric has been found to have many interesting and varied applications in geometric function theory. In particular, quasiconformal mappings are quasi-isometries of this metric for sufficiently far-lying points, also bounds on the quasihyperbolic metric in terms of other metrics imply that a domain is uniform which then implies certain injectivity criteria for locally-Lipschitz mappings, amongst others. In fact there is quite a strong relationship between uniform domains and the quasihyperbolic metric. Most of these basic results on the quasihyperbolic metric can be found in [3], [2] and [5]. We note here that the quasihyperbolic metric is complete and generates the usual topology on a proper subdomain of R". Further, geodesics (length minimizing curves) always exist for this metric and these geodesics have Lipschitz continuous first derivatives, which is in fact best possible. We begin by calculating the quasihyperbolic metric, its curvature and geodesics for some nontrivial examples in R'. We also calculate the possible isometries and show that they are conformal mappings, and so when n > 2 are M6bius transformations. We then turn to the planar case for a more detailed investigation. Here it becomes possible to compute the Gaussian curvature of the quasihyperbolic metric in some basic examples and use these for comparison in more general cases. In particular, we show that the quasihyperbolic metric of a planar domain is an S-K metric, in the sense of Heins, if and only if the domain is convex. We denote euclidean n-space by R" and its one point compactification by R'.

Double Null Coordinates for the Vaidya Metric

Abstract: Einstein's equations with spherical symmetry are formulated in double-null coordinates, and the high-frequency approximation to a unidirectional radial flow of unpolarized radiation (the Vaidya metric) is studied in detail. For this case the Einstein equations reduce to a single first-order nonlinear partial differential equation. Integration of this equation introduces an arbitrary function (of one null variable) which must be chosen so as to regularize the metric across horizons. Although the problem is, in general, not analytically solvable, we are able to extend the class of known analytic solutions from the constant-mass case (Kruskal-Szekeres metric) to linear and exponential mass functions. In the linear case we give the first explicit regular covering of a spacetime with a naked shell-focusing singularity.

Topology change and monopole creation

Abstract: Lorentzian cobordism is considered as a mechanism for topology change in quantum gravity. It is shown that the condition for a topological cobordism to admit an appropriate metric is different in even and odd dimensions. In odd dimensions such a metric exists if and only if the initial and final manifolds have the same Euler characteristic. This means that pair creation of Kaluza-Klein monopoles cannot occur via the mechanism considered. Possible implications of this result are discussed. Lorentzian cobordism in two dimensions is also analyzed briefly.

An active set RQP algorithm for engineering design optimization

Abstract: Optimum design of complex engineering systems needs a globally and superlinearly convergent (robust and efficient) algorithm using active set strategy. Such an algorithm based on extensions of Pshenichny's linearization method is derived in the paper. In the original method, a linearized subproblem with a quadratic step-size constraint is used to compute a direction of design change. No second-order information is computed or approximated for use in the direction finding problem. Therefore, the rate of convergence is only linear. In the paper, we propose to incorporate a variable metric W which is a positive-definite approximation to the Hessian of the Lagrange function. This gives local superlinear rate of convergence to the algorithm. Some other computational improvements are discussed and incorporated. The proposed improvements appear to be quite simple. They are, however, quite significant for applications to engineering design problems. This is explained and several small-scale problems are solved using a program based on the modified algorithm. Results are compared to Pshenichny's original algorithm. The modified algorithm is considerably more efficient compared to the previous algorithm. It also appears to be quite robust, though more extensive testing is needed on a wider range of problems.

Intergroup Diversity and Concordance for Ranking Data: An Approach via Metrics for Permutations

Abstract: Motivated by the apportionment of diversity analysis due to C. R. Rao, a general approach to comparing populations of rankers is proposed. Each permutation metric corresponds to a particular population characteristic that forms the basis of the comparison. Tests of hypotheses concerning equality of characteristics are developed. Throughout, comparison is made with earlier work, most of which is based on the use of only the Spearman metric. Extension to tied rankings is discussed. Examples for two groups are presented which illustrate the computational feasibility as well as the value of the proposed procedures.

Metric Domains for Completeness

Abstract: Completeness is a semantic non-operational notion of program correctness suggested (but not pursued) by W.W.Wadge. Program verification can be simplified using completeness, firstly by removing the approximation relation from proofs, and secondly by removing partial objects from proofs. The dissertation proves the validity of this approach by demonstrating how it can work in the class of metric domains. We show how the use of Tarski's least fixed point theorem can be replaced by a non-operational unique fixed point theorem for many well behaved Programs. The proof of this theorem is also non-operational. After this we consider the problem of deciding what it means f or a function to be "complete". It is shown that combinators such as function composition are not complete, although they are traditionally assumed to be so. Complete versions for these combinators are given. Absolute functions are proposed as a general model for the notion of a complete function. The theory of mategories is introduced as a vehicle for studying absolute functions.

Variance of set-indexed sums of mixing random variables and weak convergence of set-indexed processes

Abstract: A uniform bound is found for the variance of a partial-sum set-indexed process under a mixing condition. Sufficient conditions are given for a sequence of partial-sum set-indexed processes to converge to Brownian motion. The requisite tightness follows from hypotheses on the metric entropy of the class of sets and moment and mixing conditions on the summands. The proof uses a construction of Bass [2]. Convergence of finite-dimensional laws in this context is studied in [16].

An optimal algorithm for the all-nearest-neighbors problem

Abstract: Given a set V of n points in k-dimensional space, and an Lq-metric (Minkowski metric), the All-Nearest-Neighbors problem is defined as follows: For each point p in V, find all those points in V-{p} that are closest to p under the distance metric Lq. We give an O(nlogn) algorithm for the All-Nearest-Neighbors problem, for fixed dimension k and fixed metric Lq. Since there is an Ω(n logn) lower bound, in the algebraic decision tree model of computation, on the time complexity of any algorithm that solves the All-Nearest-Neighbors problem (for k = 1), the running time of our algorithm is optimal upto a constant.

Uniqueness results for homeomorphism groups

Abstract: Let X be a separable metric manifold and let W(X) be the homeomorphism group of X. Then W(X) has a unique topology in which it is a complete separable metric group. Similar results are demonstrated for a much wider class of spaces, X, and for many subgroups of the homeomorphism group.

A family of chance-corrected association coefficients for metric scales

Abstract: A chance-corrected version of the family of association coefficients for metric scales proposed by Zegers and ten Berge is presented. It is shown that a matrix with chance-corrected coefficients between a number of variables is Gramian. The members of the chance-corrected family are shown to be partially ordered.

A note on the Berry-Meekings style metric

Abstract: A modification of the Berry-Meekings "style metric"—applied to software from the corporate environment—finds little relationship between this style metric and error proneness.

Teichmuller metric and length spectrums of riemann surfaces

Abstract: The purpose of this paper is to introduce a metric for Teichmuller spaces in terms of length spectrums of Riemann surfaces and prove the topological equivalence of this metric and the Teichmuller metric. The result of this paper solves a problem proposed by T. Sorvali in 1975.

Sequential decoding method and apparatus

Abstract: A sequential decoding of a convolutionally encoded sequence has been proposed. Decoding is carried out by extending paths along the branches of code-tree over plural tree levels at a time and calculating likelihood (metric) of the extended paths referring to the received sequence. Decoding is terminated when it proceeds to a node belonging to the last tree level of a data block. Then, one of the extended sequences associated with a node located at the last tree level which has the largest metric is output as a decoded sequence.