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Showing papers on "Metric (mathematics) published in 1985"


Book
30 Jun 1985

570 citations


Journal ArticleDOI
TL;DR: A method to determine the three-dimensional structure of a protein molecule from such a set of distance constraints as can be determined by nuclear magnetic resonance studies, applicable to large molecules, with all atoms treated explicitly.

550 citations


Journal ArticleDOI
TL;DR: Here the authors present an elaboration and a quantitative example for a hypothetical neuronal process, implementing what they refer to as the metaorganization principle, which allows the internalization of external geometries into the central nervous system and a reciprocal and equally important action of the CNS geometry on the external (body) geometry.

314 citations


Book ChapterDOI
01 Jan 1985
TL;DR: In this article, the deRham cohomology class Ω of the Kahler metric is fixed, and the function space of all the kahler metrics in M in that class is considered.
Abstract: Given a compact, complex manifold M with a Kahler metric, we fix the deRham cohomology class Ω of the Kahler metric, and consider the function space ℊΩ of all Kahler metrics in M in that class. To each (g) ∈ GΩ we assign the non-negative real number \( \Phi (g) = \int\limits_{M} {R_{g}^{2}d{V_{g}}}\) (R g = scalar curvature, d V g = volume element).

266 citations


Proceedings ArticleDOI
01 Jun 1985
TL;DR: An asymptotically optimal algorithm for computing Voronoi diagrams based on convex distance functions that allows such diagrams to be defined for very general metrics and for distance measures that do not qualify as metrics.
Abstract: We present an “expanding waves” view of Voronoi diagrams that allows such diagrams to be defined for very general metrics and for distance measures that do not qualify as metrics. If a pebble is dropped into a still pond, circular waves move out from the point of impact. If n pebbles are dropped simultaneously, the places where wave fronts meet define the Voronoi diagram on the n points of impact.The Voronoi diagram for any normed metric, including the Lp metrics, can be obtained by changing the shape of the wave front from a circle to the shape of the “circle” in that metric. (For example, the “circle” in the L1 metric is diamond shaped.) For any convex wave shape there is a corresponding convex distance function. If the shape is not symmetric about its center (a triangle, for example) then the resulting distance function is not a metric, although it can still be used to define a Voronoi diagram.Like Voronoi diagrams based on the Euclidean metric, the Voronoi diagrams based on other normed metrics can be used to solve various closest-point problems (all-nearest-neighbors, minimum spanning trees, etc.). Some of these problems also make sense under convex distance functions which are not metrics. In particular, the “largest empty circle” problem becomes the “largest empty convex shape” problem, and “motion planning for a disc” becomes “motion planning for a convex shape”. These problems can both be solved quickly given the Voronoi diagram. We present an asymptotically optimal algorithm for computing Voronoi diagrams based on convex distance functions.

253 citations


Journal ArticleDOI
TL;DR: The process of finding the correspondence is formalized by defining a general relational distance measure that computes a numeric distance between any two relational descriptions-a model and an image description, two models, or two image descriptions.
Abstract: Relational models are frequently used in high-level computer vision. Finding a correspondence between a relational model and an image description is an important operation in the analysis of scenes. In this paper the process of finding the correspondence is formalized by defining a general relational distance measure that computes a numeric distance between any two relational descriptions-a model and an image description, two models, or two image descriptions. The distance measure is proved to be a metric, and is illustrated with examples of distance between object models. A variant measure used in our past studies is shown not to be a metric.

218 citations


Journal ArticleDOI
TL;DR: The problem of finding a translation to minimize the distance between point patterns is discussed and the sum of the distances in the minimal pairing is used as the “match distance” between the histograms.
Abstract: A metric is defined on the space of multidimensional histograms. Such histograms store in thexth location the number of events with feature vectorx; examples are gray level histograms and co-occurrence matrices of digital images. Given two multidimensional histograms, each is “unfolded” and a minimum distance pairing is performed using a distance metric on the feature vectorsx. The sum of the distances in the minimal pairing is used as the “match distance” between the histograms. This distance is shown to be a metric, and in the one-dimensional case is equal to the absolute difference of the two cumulative distribution functions. Among other applications, it facilitates direct computation of the distance between co-occurrence matrices or between point patterns. The problem of finding a translation to minimize the distance between point patterns is also discussed.

180 citations


Journal ArticleDOI
TL;DR: In this article, the authors stress the concept of generators and that the existence of an expansive constant guarantees that a finite-time series would be sufficient for the calculation of the metric entropy, and propose optimal algorithms which are tested on a number of examples.
Abstract: The extraction of the Kolmogorov (metric) entropy from an experimental time signal is discussed Theoretically we stress the concept of generators and that the existence of an expansive constant guarantees that a finite-time series would be sufficient for the calculation of the metric entropy On the basis of the theory we attempt to propose optimal algorithms which are tested on a number of examples The approach is applicable to both dissipative and conservative dynamical systems

147 citations


Journal ArticleDOI
TL;DR: An example is given where truncation error, caused by finite computer arithmetic, leads to the BFGS variable-metric method becoming stuck, despite the approximated Hessian matrix, the gradient vector, and the search vector satisfying analytical conditions for convergence.
Abstract: An example is given where truncation error, caused by finite computer arithmetic, leads to the BFGS variable-metric method becoming stuck, despite the approximated Hessian matrix, the gradient vector, and the search vector satisfying analytical conditions for convergence. A restart criterion to eliminate the problem is suggested.

142 citations


Journal ArticleDOI
TL;DR: In this paper, the authors presented binary partitions of the plane which are generators under the action of the dissipative Henon map, at least to very good approximation, and the metric, topological, and correlation entropy all agree with previously conjectured formulas.

123 citations


Book
31 May 1985
TL;DR: In this paper, an introductory text on metric spaces that is written for students who are as interested in the applications as in the theory is presented, and the reader is expected to have had some exposure to elementary analysis.
Abstract: Here is an introductory text on metric spaces that is the first to be written for students who are as interested in the applications as in the theory. Knowledge of metric spaces is fundamental to understanding numerical methods (for example for solving differential equations) as well as analysis, yet most books at this level emphasise just the abstraction and theory. Dr Bryant uses applications to provide motivation and to sustain the development and discusses numerical procedures where appropriate. The reader is expected to have had some exposure to elementary analysis, but the author provides examples throughout to refresh the student's memory and to test and extend understanding. In short, this is an introductory textbook that will appeal to students of mathematics and engineering and will give them the required background for more advanced courses in both analysis and numerical analysis.

Journal ArticleDOI
TL;DR: In this paper, four types of metric scales are distinguished: the absolute scale, the ratio scale, difference scale and the interval scale, and a general coefficient of association for two variables of the same metric scale type is developed.
Abstract: Four types of metric scales are distinguished: the absolute scale, the ratio scale, the difference scale and the interval scale. A general coefficient of association for two variables of the same metric scale type is developed. Some properties of this general coefficient are discussed. It is shown that the matrix containing these coefficients between any number of variables is Gramian. The general coefficient reduces to specific coefficients of association for each of the four metric scales. Two of these coefficients are well known, the product-moment correlation and Tucker's congruence coefficient. Applications of the new coefficients are discussed.

Journal ArticleDOI
TL;DR: The geodesies of the quasihyperbolic meric metric have been shown to have Lipschitz continuous first derivatives as mentioned in this paper, which is the best possible.
Abstract: Let D be a proper subdomain of R" and kD the quasihyperbolic metric defined by the conformal metric tensor ds2 = dist(x, dD)~2ds2. The geodesies for this and related metrics are shown, by purely geometric methods, to exist and have Lipschitz continuous first derivatives. This is sharp for kD; we also obtain sharp estimates for the euclidean curvature of such geodesies. We then use these results to prove a general decomposition theorem for uniform domains in R", in terms of embeddings of bi-Lipschitz balls. We also construct a counterexample to the higher dimensional analogue of the decomposition theorem of Gehring and Osgood. the defining density dist(x, 3D)"1 need not be differentiable. We primarily are interested in studying the geometry of the quasihyperbolic meric and its geodesies. We show that the geodesies of this metric are C11, i.e. the arclength parametrisation has Lipschitz continuous derivatives. We show that this is best possible and obtain a sharp result on the euclidean curvature of such geodesies. These results are proved for a more general class of metrics, in particular metrics defined by locally Lipschitz densities. We then use these results to prove a gener- alisation of the decomposition theorem of Gehring and Osgood for uniform domains in ri-space. I wish to express my sincere thanks to F. W. Gehring for suggesting many of these problems, for many helpful ideas and simplifications throughout and for allowing me to present his proof of Theorem 3.7. I also wish to thank him and D. Herron for carefully reading the manuscript.

PatentDOI
TL;DR: In this article, a method and apparatus for noise suppression for speech recognition systems which employs the principle of a least-means square estimation which is implemented with conditional expected values is proposed.
Abstract: A method and apparatus for noise suppression for speech recognition systems which employs the principle of a least means square estimation which is implemented with conditional expected values Essentially, according to this method, one computes a series of optimal estimators which estimators and their variances are then employed to implement a noise immune metric This noise immune metric enables the system to substitute a noisy distance with an expected value which value is calculated according to combined speech and noise data which occurs in the bandpass filter domain Thus the system can be used with any set of speech parameters and is relatively independent of a specific speech recognition apparatus structure

Journal ArticleDOI
TL;DR: In this article, a brief survey on the theory of Riemannian submersions between almost contact metric manifolds is given, where the authors start with a brief introduction to the theory.
Abstract: In this paper we start with a brief survey on the theory of Riemannian submersions between almost contact metric manifolds.

Journal ArticleDOI
TL;DR: In this paper, the authors combine geostatistics and multicriterion decision making to design a regular observation network for several spatially correlated and anisotropic parameters, including network density, distance between observation points for the various parameters, and observation effort.
Abstract: Geostatistics and multicriterion decision making are combined to design a regular observation network for several spatially correlated and anisotropic parameters. The decision variables are network density, distance between observation points for the Various parameters, and observation effort. The estimation error is calculated as a function of the decision variables by use of a geostatistical model based on prior variograms. Composite programming, an extension of compromise programming with more than one value of p in the lp distance, makes it possible to account for the analytical characteristics of statistical criteria versus the economic value of observation effort. Thus an l1 metric is applied to statistical criteria and anl2 metric to observation effort criteria. The algorithm for solution uses gradient optimization. The approach is extended to the case when an existing network is to be augmented. The example of a two-layer aquifer system, where thickness and porosity are the parameters to be identified, illustrates the methodology. The composite solution appears to be quite robust with respect to parameter weight changes.

Journal ArticleDOI
TL;DR: In this article, a modification of the thermodynamic Weinhold metric is introduced by the statistical scheme of bit-number cumulants discussed in earlier papers, which is a metric in the space of intensive thermal variables.
Abstract: A modification of the thermodynamic Weinhold metric is introduced by the statistical scheme of bit-number cumulants discussed in earlier papers. It is a metric in the space of intensive thermal variables and becomes identical with the Weinhold metric if transformed into the space of extensive variables. The quadratic forms of the metric tensor are different for finite distances. The here presented metric is signified by quantities defined in general statistics, distinguished by important invariance properties and moreover it brings simplifications.

Journal ArticleDOI
01 Apr 1985
TL;DR: In this paper, the authors characterized the topology of uniform convergence in metric spaces whose hyperspace is at least as strong as the Vietoris topology, and showed that the Hausdorff metric topology on the hyperspace of a metric space can be obtained by a metric on the metric space C(X, Y) induced by a real-valued continuous function on X x Y compatible with product uniformity.
Abstract: Atsuji has internally characterized those metric spaces X for which each real-valued continuous function on X is uniformly continuous as follows: (1) the set X' of limit points of X is compact, and (2) for each £ > 0, the set of points in X whose distance from X' exceeds e is uniformly discrete. We obtain these new characterizations: (a) for each metric space V, the Hausdorff metric on C(X, Y), induced by a metric on X x Y compatible with the product uniformity, yields the topology of uniform convergence; (b) there exists a metric space Y containing an arc for which the Hausdorff metric on C(X, Y) yields the topology of uniform convergence; (c) the Hausdorff metric topology on CL(X) is at least as strong as the Vietoris topology. We also characterize those metric spaces whose hyperspace is such a space. Let {W, d) be a metric space. If K C W and e is positive, let SE (K) denote the union of all open e-balls whose centers run over K. If Ki and K2 are nonempty subsets of W, and for some e > 0 both S£(Ai) D K2 and Se(A2) D Kx, then the Hausdorff distance h? between them is given by the formula h?{Kx,K-i) = inf{e: Se(Ai) D K2 and Se(K2) D Kx}. Otherwise, we write h?(Kx,K2) = 00. If we restrict hd to the closed nonempty subsets CL(W) of W, then h

Journal ArticleDOI
TL;DR: It is shown that the Riemannian metric on the probability simplex ∑xi = 1 defined by (ds) 2 = ∑(dx i ) 2 x i has an invariance property under certain probabilistically natural mappings.

Journal ArticleDOI
D.R. Divgi1
TL;DR: In this article, the authors present a new procedure that combines the desirable features of Stocking and Lord's methods and is simpler than the one they recommended, which makes more complete use of available information.
Abstract: The t scale in item response theory has arbitrary unit and origin. When a group of items is calibrated twice, estimates from one calibration must be transformed to the metric of the other. A new method is presented for doing so. It is simpler than an earlier method based on test characteristic curves, and makes more complete use of available information. The metric of the 0 scale in item response theory (IRT) is arbitrary. In applications such as equating and bias analysis, a test or subtest is calibrated separately in two samples. It is necessary to transform one set of estimates to the metric of the other. Stocking and Lord (1983) have presented two methods for doing so. The objective of this article is to present a new procedure that combines the desirable features of their methods, and is simpler than the one they recommended.

Journal ArticleDOI
01 Jun 1985
TL;DR: A non Euclidean metric is defined on the space of polynucleotide concentrations with a Riemannian metric that follows the corresponding generalized gradient during the process of selection and, therefore, the rate of ascent is now maximum.
Abstract: General criteria of selection are derived from the kinetic equations of polynucleotide replication. As an illustrative example we discuss replication in the continuously stirred tank reactor (CSTR). The total rate of RNA synthesis is optimized during selection. The conjecture that the rate of approach towards the stable steady state is a maximum can be easily disproved. It is possible, nevertheless, to derive a potential function for polynucleotide replication in the CSTR. Following a method first introduced by Shahshahani we define a non Euclidean metric on the space of polynucleotide concentrations. In this space with a Riemannian metric the systems follows the corresponding generalized gradient during the process of selection and, therefore, the rate of ascent is now maximum. Potential functions can be derived also for some second order autocatalytic systems which are of interest in evolution, for a multidimensional Schloegl model in the CSTR and, as originally has been shown by Shahshahani, for the Fisher-Haldane-Wright equation of population genetics. In the general case, however, second order autocatalysis is not compatible with the existence of a potential. The elementary hypercycle is discussed as one simple example of a reaction network whose dynamics cannot be described by means of a generalized gradient system. Finite population size introduces a stochastic element into the selection process. Under certain conditions fluctuations in particle numbers become extremely important for the dynamics of selection. Two examples of this kind are: kinetic degeneracy of rate constants and low accuracy of replication.

Proceedings ArticleDOI
01 Jun 1985
TL;DR: Experiments show that with the quadratic metric used in this study, at least for homogenous interchangeable devices confined to a square grid, pairwise interchange suffices to move the placement very close to the global optimum over a range of 100 to 1600 devices.
Abstract: Placement algorithms for IC layout which are optimal are known to be NP-complete 5. As a result, heuristics such as pairwise-interchange techniques must be employed to generate satisfactory placements. Unfortunately, with these algorithms, there is generally no way of knowing just how far away the result is from optimum. With the quadratic metric used in this study, however, a useful absolute lower bound can be calculated for the score. Experiments show that with this metric, at least for homogenous interchangeable devices confined to a square grid, pairwise interchange suffices to move the placement very close to the global optimum over a range of 100 to 1600 devices; in particular, asymptotic approach to optimality is observed with increasing size. In addition, a theoretical model is developed which explains the observed deviation from optimality.

Journal ArticleDOI
TL;DR: Empirically comparing structural test coverage metrics reveals that test sets that satisfy one metric are likely to satisfy another metric as well as they can be distinguished.
Abstract: Empirically comparing structural test coverage metrics reveals that test sets that satisfy one metric are likely to satisfy metric as we.

Journal ArticleDOI
M. I. Wanas1
TL;DR: In this paper, three solutions with spherical symmetry are obtained for the field equations of the generalized field theory established recently by Mikhail and Wanas, which correspond to a field in a matter-free space.
Abstract: Three solutions with spherical symmetry are obtained for the field equations of the generalized field theory established recently by Mikhail and Wanas. The solutions found are in agreement with classical known results. The solution representing a generalized field, outside a spherical symmetric charged body, is found to have an extra term compared with the Reissner-Nordstrom metric. The space used for application is of type FIGI, so the solutions obtained correspond to a field in a matter-free space. A brief comparison between the solutions obtained and those given by other field theories is given. Two methods have been used to get physical results: the first is the type analysis, and the second is the comparison with classical known results by writing down the metric of the associated Riemannian space.

Journal ArticleDOI
01 Jul 1985
TL;DR: In this article, the authors describe the use of metric techniques in an electronic support measures (ESM) data processing scheme developed at Smith Associates Consulting System Engineers Limited for the UK Ministry of Defence (Procurement Executive).
Abstract: The paper describes the use of metric techniques in an electronic support measures (ESM) data processing scheme developed at Smith Associates Consulting System Engineers Limited for the UK Ministry of Defence (Procurement Executive). The ESM data processing scheme takes in digital pulse data describing the radar pulses illuminating an ESM receiver. The output from the scheme is a continuously updated emitter table, which lists the emitters present in the environment and specifies the deduced parameters of these emitters. Metric techniques play an important role at two distinct stages of the scheme: firstly in direction-of-arrival (DOA) RF pulse filtering, and secondly in emitter table updating. In DOA RF filtering, the pulses are processed using two-dimensional cluster analysis. The method ensures that pulses from a single emitter are not separated into different batches, but that pulses from distinct emitters are separated, unless their parameters overlap in both DOA and RF. The same result cannot in general, be achieved using one-dimensional filtering. The scheme uses a computationally efficient method, in which the metric technique is applied to groups of pulses following sequential one-dimensional filterings in DOA and then RF. In emitter table updating, a metric technique is used to ensure that the pulses from an emitter already on the emitter table are correctly associated with that emitter table entry, and are not added as a new emitter seen for the first time. This overcomes the problem that, for some emitters, some of the parameters can change discontinuously when the emitter changes mode. The metric technique ensures that, provided some of the parameters remain approximately constant, the correct association is made.

Book ChapterDOI
01 Jan 1985
TL;DR: In this article, the tangent space, the Riemannian metric and the α-connections are introduced in a statistical manifold, and the present chapter is devoted to the introduction of fundamental differential geometrical structures of statistical models.
Abstract: The present chapter is devoted to the introduction of fundamental differential-geometrical structures of statistical models. The tangent space, the Riemannian metric and the α-connections are introduced in a statistical manifold. No differential-geometrical background is required for reading this monograph, because the present chapter provides a readable introduction to differential geometry.

Journal ArticleDOI
TL;DR: In this article, the problem of finding stationary axially symmetric solutions to the vacuum Einstein equations is reduced to a single quadrature when the seed solution is diagonal, and the possibility of having real odd number soliton solutions is investigated.
Abstract: The application of the Belinsky–Zakharov solution‐generating technique, i.e., the inverse scattering method, to generate stationary axially symmetric solutions to the vacuum Einstein equations is reduced to a single quadrature when the seed solution is diagonal. The possibility of having real odd‐number soliton solutions is investigated. These solutions represent solitonic perturbations of Euclidean metrics. The possibility of using instantons as seed solutions is also investigated. The one‐ and two‐soliton solutions generated from a diagonal seed solution are studied. As an application, a unified derivation of some well‐known static solutions, like the Schwarzschild metric and the Chazy–Curzon metric, as well as other new metrics is presented. By using these metrics as seed solutions, some known stationary solutions, like the Kerr‐NUT metric, the double Kerr metric, and the rotating Weyl C‐metric, as well as other new metrics are also derived in a unified way.

Journal ArticleDOI
TL;DR: Empirical evidence of loose satisfaction of these properties with real speech will be presented, allowing the assumption of a “loose metric space” structure in the set of parametric representations of words in a given vocabulary.

Journal ArticleDOI
TL;DR: Straeter's rank-one updating formula appears to be the only parallel extension within Huang's class with the property of quadratic termination, and a parallel extension of Broyden's (1965) rank- one updating formula is developed and proved.
Abstract: We classify Straeter's ideas for parallel unconstrained optimization and apply them to Huang's class of updating formulas. Straeter's rank-one updating formula appears to be the only parallel extension within Huang's class with the property of quadratic termination. We also develop a parallel extension of Broyden's (1965) rank-one updating formula and prove quadratic termination. Finally, we present numerical results, obtained by testing the algorithms on several standard example problems.

Proceedings Article
18 Aug 1985
TL;DR: A new measurement function is reported which permits the uncertainty over how to characterise the proximal relationship between rigid bodies in continum algorithms, and promises to significantly increase the capabilities of continuum path planning software.
Abstract: The problem of planning motions of robot manipulators and similar mechanical devices in the presence of obstacles is one of keen interest to the artificial intelligence community. Most of the algorithms previously reported for solving such problems have been combinatorial algorithms, which work by partitioning the problem domain continuum into a finite set of equivalence classes, and applying combinatorial search algorithms to plan transitions among them. However, the few continuum algorithms that have been reported, which do not rely on such a partitioning, have shown greater promise when applied to problems of complexity equivalent to that of planning a true manipulator motion. This is true even though the heuristics employed in these continuum algorithms have been extremely simple in nature. A significant barrier to the development of more refined heuristics for use in continum algorithms is the uncertainty over how to characterise the proximal relationship between rigid bodies. In this paper, a new measurement function is reported which permits such characterisation. An introduction is made to a new type of path planning algorithm which this function makes possible, which promises to significantly increase the capabilities of continuum path planning software.