Showing papers on "Metric (mathematics) published in 1984"

K-Means-Type Algorithms: A Generalized Convergence Theorem and Characterization of Local Optimality

Abstract: The K-means algorithm is a commonly used technique in cluster analysis. In this paper, several questions about the algorithm are addressed. The clustering problem is first cast as a nonconvex mathematical program. Then, a rigorous proof of the finite convergence of the K-means-type algorithm is given for any metric. It is shown that under certain conditions the algorithm may fail to converge to a local minimum, and that it converges under differentiability conditions to a Kuhn-Tucker point. Finally, a method for obtaining a local-minimum solution is given.

Metric bases in digital geometry

Abstract: Let S be a metric space under the distance function d. A metric basis is a subset B ⊆ S such that d(b, x) = d(b, y) for all b ϵ B implies x = y. It is shown that for Euclidean distance, the minimal metric bases for the digital plane are just the sets of three noncollinear points; but for city block or chessboard distance, the digital plane has no finite metric basis. The sizes of minimal metric bases for upright digital rectangles are also derived, and it is shown that there exist rectangles having minimal metric bases of any size ≥ 3.

An Optimal Global Nearest Neighbor Metric

Abstract: A quadratic metric dAO (X, Y) =[(X - Y)T AO(X - Y)]? is proposed which minimizes the mean-squared error between the nearest neighbor asymptotic risk and the finite sample risk. Under linearity assumptions, a heuristic argument is given which indicates that this metric produces lower mean-squared error than the Euclidean metric. A nonparametric estimate of Ao is developed. If samples appear to come from a Gaussian mixture, an alternative, parametrically directed distance measure is suggested for nearness decisions within a limited region of space. Examples of some two-class Gaussian mixture distributions are included.

On Quantum Field Theory in Gravitational Background

Abstract: We discuss quantum fields on Riemannian space-time. A principle of local definiteness is introduced which is needed beyond equations of motion and commutation relations to fix the theory uniquely. It also allows us to formulate local stability. In application to a region with a time-like Killing vector field and horizons it yields the value of the Hawking temperature. The concept of vacuum and particles in a non-stationary metric is treated in the example of the Robertson-Walker metric and some remarks on detectors in non-inertial motion are added.

Norms of random matrices and widths of finite-dimensional sets

Abstract: Precise orders are given for the Kolmogorov and linear widths of the unit ball of the space in the metric of for The determination of the upper estimates is based on approximation by random objects This method goes back to Kashin (Math USSR Izv 11 (1977), 317-333) The corresponding lower estimates were obtained in a previous article of the author (Vestnik Leningrad Univ Math 14 (1982), 163-170)Bibliography: 12 titles

A Unique Genetic Distance

Abstract: The problem of measuring distances between discrete frequency distributions is considered. Three conditions are stated, which are believed to reflect basic, intuitive requirements to be met by a distance measure of the above kind with particular reference to genetic frequency distributions. These conditions chiefly concern aspects of maximum distance and linearity. It is shown that exactly one function meets the conditions, and this function, having all properties of a metric, is explicitly given.

GENFOLD2. A Set of Models and Algorithms for the GENeral UnFOLDing Analysis of Preference/Dominance Data

Abstract: A general set of multidimensional unfolding models and algorithms is presented to analyze preference or dominance data This class of models termed GENFOLD2 (GENeral UnFOLDing Analysis-Version 2) allows one to perform internal or external analysis, constrained or unconstrained analysis, conditional or unconditional analysis, metric or nonmetric analysis, while providing the flexibility of specifying and/or testing a variety of different types of unfolding-type preference models mentioned in the literature including Caroll's (1972, 1980) simple, weighted, and general unfolding analysis An alternating weighted least-squares algorithm is utilized and discussed in terms of preventing degenerate solutions in the estimation of the specified parameters Finally, two applications of this new method are discussed concerning preference data for ten brands of pain relievers and twelve models of residential communication devices

The context dependent comparison of biological sequences

Abstract: A general method for comparing two macromolecules is developed. The method differs from more traditional procedures in that matches are evaluated dependent on sequence context. We first define a context dependent similarity score between sequences and give a dynamic programming algorithm for its calculation. Conditions are then described which allow the conversion of the similarity score to a metric distance. The class of metrics obtained in this manner includes the Sellers metric. An advantage of the method is the ability to make very rapid comparisons and to align long sequences.

GENFOLD2: A set of models and algorithms for the general UnFOLDing analysis of preference/dominance data

Abstract: A general set of multidimensional unfolding models and algorithms is presented to analyze preference or dominance data. This class of models termed GENFOLD2 (GENeral UnFOLDing Analysis-Version 2) allows one to perform internal or external analysis, constrained or unconstrained analysis, conditional or unconditional analysis, metric or nonmetric analysis, while providing the flexibility of specifying and/or testing a variety of different types of unfolding-type preference models mentioned in the literature including Caroll's (1972, 1980) simple, weighted, and general unfolding analysis. An alternating weighted least-squares algorithm is utilized and discussed in terms of preventing degenerate solutions in the estimation of the specified parameters. Finally, two applications of this new method are discussed concerning preference data for ten brands of pain relievers and twelve models of residential communication devices.

Electromagnetic mass models in general relativity

Abstract: The metric coefficients g/sub 00/ and g/sub 11/ of both the Schwarzschild and Reissner-Nordstroem metrics satisfy the relation g/sub 00/g/sub 11/ = -1. A coordinate-independent statement of this relation using the eigenvalues of the Einstein tensor is given. By considering the relation between the metric coefficients to be valid inside a charged perfect-fluid distribution, it is shown that the mass-energy density and the pressure of the distribution are of electromagnetic origin. In the absence of charge, however, there exists no interior solution. A particular solution which confirms the same and matches smoothly with the exterior Reissner-Nordstroem metric is obtained. This solution represents a charged particle whose mass is entirely of electromagnetic origin.

The Cauchy Problem for Einstein Equations

Abstract: A space-time, M = (V,g) is a differentiable C∞ manifold V of dimension 4 endowed with a lorenztian metric g, pseudo riemannian metric of signature (-,+,+,+).

Informative Geometry of Probability Spaces

Abstract: : This paper is concerned with the geometrical properties that are induced by the local information contents and structures of the parameter space of probability distributions. Of particular interest in this investigation is the Rao distance which is the geodesic distance induced by the differential metric associated with the Fisher information matrix of the parameter space. Moreover, following Efron, Dawid and Amari, some affine connections are introduced into the informative geometry of parameter space and thereby elucidating the role of the curvature in statistical studies. In addition, closed form expressions of the Rao distances for certain families of probability distributions are given and discussed.

Characterizations of the Kerr metric.

Abstract: For stationary, asymptotically flat solutions of Einstein's equations, covariant functionals of the metric variables are defined which characterize the Kerr metric uniquely. For instance, we obtain a generalization of the “Bach tensor” to stationary metrics, which vanishes if and only if the solution is Kerr. We also give a new interpretation of the “Schwarzschild-to-Kerr-transformation.” Our results might be applicable to simplify the proof of the uniqueness theorem for stationary black holes.

Configural processing of faces in the left and the right cerebral hemispheres.

Abstract: This study investigates the rules by which the component features of faces are combined when presented in the left or the right visual field, and it examines the validity of the analytic-holistic processing dichotomy, using concepts elaborated by Garner (1978, 1981) to specify stimulus properties and models of similarity relations as performance criteria. Latency measures of dissimilarity, obtained for the two visual fields, among a set of eight faces varying on three dimensions of two levels each, were fitted to the dominance metric model, the feature-matching model, the city-block distance metric model, and the Euclidean distance metric model. In addition to a right-visual-field superiority in different responses, a maximum likelihood estimation procedure showed that, for each subject and each visual field, the Euclidean model provided the best fit of the data, suggesting that the faces were compared in terms of their overall similarity. Moreover, the spatial representations of the results revealed interactions among the component facial features in the processing of faces. Taken together, these two findings indicate that faces initially projected to the right or to the left hemisphere were not processed analytically but in terms of their gestalt. Human information-processing capacities are the product of a highly adaptive and versatile nervous system that provides individuals with a large number of alternative means for achieving successful performance on any particular task. This versatility is partly attributed to the functional specialization of the two cerebral hemispheres whereby specific skills are alleged to be unilaterally represented, thus doubling the brain processing capacity while avoiding potential conflicts that would result from promiscuity. This specialization was initially characterized in terms of information that each hemisphere was better equipped to operate on (e.g., Milner, 1971). However, the diversity and heterogeneity of the type of information that each hemisphere could be shown to process, initially in experiments with commissurotomized patients, prompted researchers to inquire about the processes un

Differentiation of real-valued functions and continuity of metric projections

Abstract: We characterize the Freenet differentiability of real-valued func- tions on certain real Banach spaces in terms of a directional derivative being equal to a modified version of the local Lipschitz constant. This yields the continuity of metric projections onto closed sets whose distance functions have directional derivatives equal to 1, provided the Banach space and its dual have Frechet differentiable norms.

Least squares metric, unidimensional unfolding

Abstract: The partial derivatives of the squared error loss function for the metric unfolding problem have a unique geometry which can be exploited to produce unfolding methods with very desirable properties. This paper details a simple unidimensional unfolding method which uses the geometry of the partial derivatives to find conditional global minima; i.e., one set of points is held fixed and the global minimum is found for the other set. The two sets are then interchanged. The procedure is very robust. It converges to a minimum very quickly from a random or non-random starting configuration and is particularly useful for the analysis of large data sets with missing entries.

Liveness properties as convergence in metric spaces

Abstract: Four liveness properties of concurrent programs are characterized by the fact that their computations, represented as sequences of partial orderings of events, are convergent in suitable metric spaces. The corresponding topological completions do not therefore contain the infinite computations without the desired properties. The properties are: vitality (i.e. every running process will eventually produce an observable event), global and local fairness, and deadlock freedom. This approach proves fruitful since a universal scheduler is defined, which, when supplied with a particular metric, generates all and only convergent computations. Thus, this scheduler can be used to generate all and only vital, fair or deadlock free computations.

Approximation of infinite-dimensional Teichmüller spaces

Abstract: By means of an exhaustion process it is shown that Teichmuller's metric and Kobavashi's metric arc equal for infinite dimensional Tcichmuller spaces. By the same approximation method important estimates coming from the Reich·Strebel inequality are extended to the infinite dimensional cases. These estimates are used to ,how that Teichmuller's metric is the integral of its infinitesimal form. They are also used to give a sufficient condition for a sequence to be an absolute maximal sequence for the Hamilton functional. Finally, they are used to give a new sufficient condition for a sequence of Beltrami coefficients to converge in the Teichmuller metric. Introduction. The subject of this paper is Teichmiiller spaces of infinitely generated Fuchsian groups. By approximation techniques involving theta series and finitely generated subgroups of a given group, we extend certain important results already known in the finite case to the infinitely generated case. In § I we set up the approximation technique and cite the necessary theorems involving Poincare series and approximation by rational functions. In §2 we consider Kobayashi's extremal problem for Teichmiiller spaces with complex structure and prove the following new result. The theorem of Royden on the equality of the Kobayashi and Teichmiiller me tries remains true in the infinite cases. These cases include Teichmiiller spaces of groups of the first and second kind. In particular, the case of universal Teichmiiller space is included. In §3 we prove the important main inequality of Reich and Strebel [14). Its most significant consequences are upper and lower estimates for the extremal value of the dilatation in a given Teichmiiller class. The chief result of this section is that these upper and lower estimates hold even in the infinite dimensional cases. In §4 we derive the well-known [16) infinitesimal form of Teichmiiller's metric. We use this general form together with the Hamilton condition as developed by Reich and Strebel [14) to show that Teichmiiller's metric is equal to the integral of its infinitesimal form. O'Byrne already obtained this result in [6, 7, 13). The method used here is more direct and the theorem is proved in greater generality. In §5, we give a sufficient condition for a sequence C(Jn to be an absolute maximal sequence for the Hamilton functional H[p,). We do not know if this condition is also Received by the editors January 12, 19R2 and. in revised form. March 25. 1983. 19W Mathematics Suhject Classification. Primary 30C60; Secondary 30C70. Ker ..... ords and phrases. Teichmuller space. Teichmiillcr's metric, Kobayashi's metric. Hamilton functional. absolute maximal sequence. infinitesimal metric. I This work partially supported by the Research Foundation of CUNY. 367 t·1984 American Mathematical Society 0002-9947/84 $100 + $.25 per page License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

Geodesic-flow On So(4) and the Intersection of Quadrics

Abstract: This note studies the relationship between the complete integrability of geodesic flow on SO(4) with a left-invariant metric and the geometry of the intersection of four quadrics in P(6).

Digital data decoders

Abstract: A Viterbi decoder is provided with synchronisation error detecting means (58) which detects synchronisation errors by monitoring the growth of the path metric associated with a selected state A and to indicate synchronisation errors independently of the behavour of metrics associated with other states. The difference between the integral of the selected metric over a certain period of time and the sum of the absolute values of the input encoded data signal is determined and compared to a threshold to indicate whether or not a synchronisation error exists.

An algorithm for sizing software products

Abstract: This paper reports on efforts to develop a cost forecasting scheme based on a Function Metric called System BANG. A Function Metric is a quantifiable indication of system system size and complexity derived directly from a formal statement of system requirement. Conclusions from a small sample of projects are presented.