Showing papers on "Metric (mathematics) published in 1973"
TL;DR: The primary problem dealt with in this paper is the specification of a descriptive scheme, and a metric on which to base the decision of "goodness" of matching or detection.
Abstract: The primary problem dealt with in this paper is the following. Given some description of a visual object, find that object in an actual photograph. Part of the solution to this problem is the specification of a descriptive scheme, and a metric on which to base the decision of "goodness" of matching or detection.
1,536 citations
TL;DR: Three file structures are presented together with their corresponding search algorithms, which are intended to reduce the number of comparisons required to achieve the desired result.
Abstract: The problem of searching the set of keys in a file to find a key which is closest to a given query key is discussed. After “closest,” in terms of a metric on the the key space, is suitably defined, three file structures are presented together with their corresponding search algorithms, which are intended to reduce the number of comparisons required to achieve the desired result. These methods are derived using certain inequalities satisfied by metrics and by graph-theoretic concepts. Some empirical results are presented which compare the efficiency of the methods.
436 citations
01 Jan 1973
TL;DR: In this paper, a nonlinear problem in differential geometry is discussed: to characterize all smooth functions on a two-dimensional sphere which can be obtained in this manner from some riemannian metric.
Abstract: Publisher Summary This chapter discusses a nonlinear problem in differential geometry. Any metric ds 2 on a two-dimensional sphere S 2 determines a Gauss curvature function K satisfying the Gauss–Bonnet formula. This chapter discusses the converse question: to characterize all smooth functions on a two-dimensional sphere which can be obtained in this manner from some riemannian metric. To characterize all Gauss curvature functions belonging to metrics ds 2 which are conformally related to the standard metric ds 0 2 , so that ds 2 = λ ds 0 2 , where λ is a positive function on the sphere. This requires the determination of the single function λ in terms of the given function K .
190 citations
TL;DR: In this article, it was shown that for nonzero NUT parameter the fixed points of the bifurcate Killing horizons and the degeneracies at θ = 0,π cannot all be covered in one manifold.
Abstract: The Kerr‐Taub‐NUT metric is a local analytic solution of the vacuum Einstein‐Maxwell equations. When the metric is expressed in Schwarzschild‐like coordinates, two types of coordinate singularity are present. One occurs at certain values of the ``radial'' coordinate where grr becomes infinite and corresponds to bifurcate Killing horizons; the other occurs at θ=0,π, where the determinant of the components of the metric vanishes. It is shown that for nonzero NUT parameter the fixed points of the bifurcate Killing horizons and the degeneracies at θ=0,π cannot all be covered in one manifold. A maximal analytic manifold is constructed which covers the degeneracies at θ=0,π. It is non‐Hausdorff but contains maximal Hausdorff subspaces, topologically S3×R, which reduce to Taub‐NUT space for vanishing Kerr parameter. Kerr‐Taub space can be interpreted as a closed, inhomogeneous electromagnetic‐gravitational wave undergoing gravitational collapse. Another maximal analytic manifold is constructed which covers the f...
105 citations
83 citations
01 Jan 1973
TL;DR: The material in this chapter shows that the area between the MTF and the AIM curves, now called “TQF” or “MTFA” by different people, is a metric that is broadly applicable for general scenes but not necessarily a good metric for specific objects.
Abstract: In earlier material we addressed to the reader our misgivings about the parameters chosen by many systems evaluators. The material in this chapter shows that the area between the MTF and the AIM curves, now called “TQF” or “MTFA” by different people, is a metric that is broadly applicable for general scenes but not necessarily a good metric for specific objects.
46 citations
TL;DR: The integrability of a series in cosines under these conditions is equivalent to a theorem of Sidon as mentioned in this paper, which is the only known theorem for cosine series in trigonometric series.
Abstract: We consider problems concerning integrability and convergence in the metric L of trigonometric series, the coefficientsak of which satisfy the conditions:ak → 0, there exist numbers Ak such that Ak ↓ 0,\(\Sigma _{A_k } \) < ∞, and ¦Δ¦ ≤ Ak. The integrability of a series in cosines under these conditions is equivalent to a theorem of Sidon.
46 citations
TL;DR: In this paper, it was shown that Johnson's theory is valid if the vector field couples with a charged scalar field in the minimal interaction, provided that there are no other massless physical particles, under the assumption that the current j'\" is conserved.
Abstract: On the basis of the indefinite-metric vector field theory proposed previously, Johnson's proposition that the physical mass of a vector field tends to zero as its bare mass goes to zero, is shown to be valid if the vector field couples with a charged scalar field in the minimal interaction. In the case of the theory of spontaneously broken gauge invariance, the reason why the vector field acquires a non-zero mass in spite of the above theorem is clarified. The theory of a vector field which is massive owing to the spontaneous breakdown of gauge invariance is consistently formulated in the framework of the indefinite-metric quantum field theory. In this formalism, both renormalizability and the unitarity of the physical S-rnatrix are self-evident. § I. Introduction Recently, the present author 1 l'*l has proposed an indefinite-metric theroy 2 l of a massive vector field such that as its mass goes to zero the theory smoothly tends to the Landau-gauge quantum electrodynamics.3l' 4 l As one of important consequences of this theory, we can reasonably show the validity of Johnson's proposition 5 l'**l that if the bare mass of the vector field ufo goes to zero, its physical mass must also tend to zero, provided that there are no other massless physical particles, under the assumption that the current j'\" is conserved and does not explicitly depend on ufo. On the other hand, in connection with Weinberg's theory of leptons, BJ much attention has been paid to the spontaneous breakdown of gauge invariance in the massless vector field theories. Several years ago, Higgs and others 7 l noted that if gauge invariance of the theory is spontaneously broken, the massless vector field acquires a non-zero mass, but then Goldstone bosons do not appear in the Coulomb gauge because we do not have manifest covariance, which is necessary for the proof of the Goldstone theorem. 8 l If one reconsiders this situation in a covariant gauge, in which we have to introduce indefinite metric, Goldstone bosons appear but they become unphysical. This interesting phenomenon is now called the Higgs phenomenon. Recently, 't Hoofel has applied it to the Yang-Mills *> Unfortunately, the publication of this paper was much delayed. **> Johnson's reasoning was based on the conventional massive vector field theory whose mass-less limit is non-existent.
41 citations
TL;DR: A new class of quasi-Newton algorithms for uncon- strained minimization in which no line search is necessary and the inverse Hessian approxi- mations are positive definite are introduced.
Abstract: This paper introduces a new class of quasi-Newton algorithms for uncon- strained minimization in which no line search is necessary and the inverse Hessian approxi- mations are positive definite. These algorithms are based on a two-parameter family of rank two, updating formulae used earlier with line search in self-scaling variable metric algorithms. It is proved that, in a quadratic case, the new algorithms converge at least weak super- linearly. A special case of the above algorithms was implemented and tested numerically on several test functions. In this implementation, however, cubic interpolation was performed whenever the objective function was not satisfactorily decreased on the first "shot" (with unit step size), but this did not occur too often, except for very difficult functions. The numerical results indicate that the new algorithm is competitive and often superior to previous methods. 1. Introduction. This paper addresses the problem of minimizing a smooth real valued function f(x) depending on an n-dimensional vector x, assuming the avail- ability of the gradients Vf(x) = g(x) for any given x. An important class of algorithms for solving this problem is the quasi-Newton methods also known as variable metric algorithms. In these methods, the successive points are obtained by the equation (1) Xk+1 = Xk -aDkDkqk,
38 citations
01 Dec 1973
TL;DR: In this paper, a parallel variable metric algorithm was proposed to exploit parallel computing or vector streaming (pipeline) capabilities of computers, and the convergence of the iterates to the solution was proved for a quadratic functional on a real separable Hilbert space.
Abstract: An algorithm, designed to exploit the parallel computing or vector streaming (pipeline) capabilities of computers is presented. When p is the degree of parallelism, then one cycle of the parallel variable metric algorithm is defined as follows: first, the function and its gradient are computed in parallel at p different values of the independent variable; then the metric is modified by p rank-one corrections; and finally, a single univariant minimization is carried out in the Newton-like direction. Several properties of this algorithm are established. The convergence of the iterates to the solution is proved for a quadratic functional on a real separable Hilbert space. For a finite-dimensional space the convergence is in one cycle when p equals the dimension of the space. Results of numerical experiments indicate that the new algorithm will exploit parallel or pipeline computing capabilities to effect faster convergence than serial techniques.
TL;DR: The present study was undertaken to extend the earlier histoanatomical observations of Yakovlev and experimental studies of Caveness, with special attention to the development of the dendritic arbours and postsynaptic spines of the individual neurons of the motor cortex in relation to age.
Abstract: CLINICAL and electrocortical observations (Caveness, 1969; and Caveness et al., 1973) have indicated significant differences in the development and propagation of focal seizure activity induced by penicillin injection into the face-hand area of the motor cortex of the Macaca mulatto at birth and at 24 months of age. From histoanatomical study of the cerebra of the newborn and 24-month-old monkeys, Yakovlev (1962) concluded that these differences correlate with the early maturation of cortico-subcortical-cortical connexions and the later elaboration of the intracortical connexions. This interpretation has been further supported by recent electrophysiological experiments of Caveness et al. (1973) which showed that when the subcortical connexions of the face-hand area of the motor cortex of the newborn monkey were interrupted by undercutting the cortex prior to the injection of penicillin, there was no or only minimal spread of seizure activity. In contrast, subpial circumsection of the intracortico-cortical connexions of the face-hand area of the motor cortex without cortical undercutting did not prevent the spread of the penicillin-induced seizure activity to either the ipsilateral or contralateral hemisphere. The present study was undertaken to extend the earlier histoanatomical observations of Yakovlev and experimental studies of Caveness referred to, with special attention to the development of the dendritic arbours and postsynaptic spines of the individual neurons of the motor cortex in relation to age.
TL;DR: A Lagrangian-based metric theory of gravity is developed with three adjustable constants and two tensor fields, one of which is a nondynamic "flat space metric" eta as discussed by the authors.
Abstract: A Lagrangian-based metric theory of gravity is developed with three adjustable constants and two tensor fields, one of which is a nondynamic 'flat space metric' eta. With a suitable cosmological model and a particular choice of the constants, the 'post-Newtonian limit' of the theory agrees, in the current epoch, with that of general relativity theory (GRT); consequently the theory is consistent with current gravitation experiments. Because of the role of eta, the gravitational 'constant' G is time-dependent and gravitational waves travel null geodesics of eta rather than the physical metric g. Gravitational waves possess six degrees of freedom. The general exact static spherically-symmetric solution is a four-parameter family. Future experimental tests of the theory are discussed.
TL;DR: A general measure of retrieval effectiveness having full metric properties and treating the “retrieval system—arbiter of relevance” situation symmetrically, is the Marczewski-Steinhaus metric D, measuring the distance between the set of relevant documents.
01 Jan 1973
TL;DR: Several kinds of compact covering images of metric spaces are characterized in this paper, where the authors propose a compact cover image of the metric space and a compact covering image of metric space.
Abstract: Several kinds of compact-covering images of metric spaces are characterized.
TL;DR: In this paper, a procedure is proposed whereby questionnaire data, which is usually ordinal in nature and often error-ridden, may be transformed to reduce the error variance in the data and to improve the metric properties of the individual variables.
Abstract: A procedure is proposed whereby questionnaire data, which is usually ordinal in nature and often error-ridden, may be transformed to reduce the error variance in the data and to improve the metric properties of the individual variables. The technique is suggested by a result of Eckart and Young. The properties of the method are investigated by means of a Monte Carlo study. Various matrices were generated representing the usual concept of “true scores”. These matrices were distorted using two levels of random errors and two kinds of categorization. The distorted matrices were in turn transformed by the proposed methods and compared to the “true scores”. In all cases an overall measure of similarity reveals the transformed matrices are better approximations to the “true scores” than the untransformed data. Some properties of the transformation are discussed and some possible applications of the general technique are suggested.
TL;DR: In this paper, it was proved that if a metric space is subjected to a mixing transformation, then there exists a positive numberx 0 such that the probability that any arbitrary set of positive measure is asymptotically mapped into a set of diameter less than 0 is zero.
Abstract: It is proved that if a metric space is subjected to a mixing transformation, then there exists a positive numberx
0 such that the probability that any arbitrary set of positive measure is asymptotically mapped into a set of diameter less thanx
0 is zero. Physical implications of this result, in particular the interpretation of Poincare recurrence, are discussed.
TL;DR: By examining the behaviour of geodesics approaching the singularity of the Curzon solution, it was shown that the metric is capable of being extended in such a way that almost all such geodesic are complete as discussed by the authors.
Abstract: By examining the behaviour of geodesics approaching the singularity of the Curzon solution, it is shown that the metric is capable of being extended in such a way that almost all such geodesics are complete. There are an infinite number of possible extensions. None are analytic, but all areC
∞.
01 Sep 1973
TL;DR: Precepts for use of the SI, conversion tables, and geophysical constants expressed in SI units are given.
Abstract: Two types of systems of units have been formed. The commercial is a system of weights and measures, such as the metric, defined by civil law. The scientific is a system of physical units generated through use of scientific laws which link corresponding quantities.
TL;DR: In this article, a natural metric, d, on the space of infinitely differentiable real valued functions defined on an open subset U of the real numbers, R, is defined, and the elements of C∞, which are analytic near at least one point of U comprise a first category subset of C ∞,.
Abstract: We define a natural metric, d, on the space, C∞, , of infinitely differentiable real valued functions defined on an open subset U of the real numbers, R, and show that C∞, is complete with respect to this metric. Then we show that the elements of C∞, which are analytic near at least one point of U comprise a first category subset of C∞, .
TL;DR: In this article, it was shown that for every number c ∈ (0, 1) there exists a metric p on X such that the metric space (X, p ) is separable and the mapping f is a.contraction with the Lipschitz constant c.
Abstract: The following statement is proved: Let X be a set having at most continuously many elements and f : X → X a mapping such that each iteration f n (n = 1, 2, …) has a unique fixed point. Then for every number c ∈ (0, 1) there exists a metric p on X such that the metric space ( X, p ) is separable and the mapping f is a.contraction with the Lipschitz constant c .
TL;DR: In this paper, a finite-dimensional, complete, metric AR (or ANR) is defined, and X×R∞ is R∞ (or an open subset of R ∞).
Abstract: If X is a finite-dimensional, complete, metric AR (or ANR), then X×R∞ is R∞ (or an open subset of R∞).
Patent•
07 May 1973
TL;DR: In this article, a pair of spaced, rigidly affixed members each provided with a two-cycle logarithmic scale and a slide plate disposed therebetween in a tight, slidable relationship, said slide plate being provided with plurality of conversion indicia on each side thereof.
Abstract: This device for converting British units to metric units and vice versa includes a pair of spaced, rigidly affixed members each provided with a two-cycle logarithmic scale and a slide plate disposed therebetween in a tight, slidable relationship, said slide plate being provided with a plurality of conversion indicia on each side thereof. Each conversion indicia includes a pair of spaced arrows pointing either toward one or the other of said pair of members with the distance between said pair of arrows representing a specific conversion factor which is determined by the distance between numeral ''''1'''' on logarithmic scale and the numeral corresponding to the conversion factor.