Showing papers on "Metric (mathematics) published in 1979"
451 citations
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TL;DR: The dominant contribution to the path integral seems to come from metrics which have about one unit of topology per Planck volume as discussed by the authors, which can be described by introducing a Λ term as a Lagrange multiplier for the 4 volume.
361 citations
331 citations
297 citations
TL;DR: In this article, it was shown that if the constants an and bn are chosen in an optimal way, then the rate of convergence of (Yn − bn )/an to the extreme value distribution exp(e −x ), as measured by the supremum metric or the Levy metric, is proportional to 1/log n.
Abstract: Let Yn denote the largest of n independent N(0, 1) variables. It is shown that if the constants an and bn are chosen in an optimal way then the rate of convergence of (Yn – bn )/an to the extreme value distribution exp(–e–x ), as measured by the supremum metric or the Levy metric, is proportional to 1/log n.
236 citations
TL;DR: This paper evaluates the accuracy of a range of reasonable forms fordistance estimating functions using samples of urban and rural road distances to derive better distance estimating functions for analyzing systems with distance related performance measures.
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.
185 citations
TL;DR: In this article, a theory of the status attainment process is proposed and specified in a mathematical model, which justifies both a transformation of the conventional status scores to a metric producing an exponential distribution of attainments and a transition of educational attainments to a measure reflecting the competitive advantage conferred by education.
Abstract: This paper proposes a theory of the status attainment process and specifies it in a mathematical model. The theory justifies both a transformation of the conventional status scores to a metric producing an exponential distribution of attainments and a transformation of educational attainments to a metric reflecting the competitive advantage conferred by education. The new metric produces theoretically more meaningful results than the old one when used together with the proposed model in an analysis of change in status. The metric also performs better statistically with the proposed model as well as with conventional models for the level of status. An empirical analysis demonstrates that career opportunities are less favorable for women than for men and for blacks than for whites.
122 citations
01 Jul 1979
TL;DR: In this article, a method for solving the toroidal magnetohydrodynamic equilibrium equation in a coordinate system based on the magnetic field lines is presented for obtaining equilibria to use in tokamak stability and transport calculations.
Abstract: A method is presented for solving the toroidal magnetohydrodynamic equilibrium equation in a coordinate system based on the magnetic field lines. Both fixed boundary (conducting shell) and free boundary (external coil) boundary conditions are considered. A comparison with a special analytic solution is made. The method is useful for obtaining equilibria to use in tokamak stability and transport calculations.
121 citations
TL;DR: In this article, it is shown how this problem is similar to the renormalisation problem, and how the bifurcation velocity may be derived from a fixed-point theory.
Abstract: Endomorphisms of the real axis with one extremum have some universal metric properties which depend only on their analytic dependence near the extremum (bifurcation velocity, reduction parameter). It is shown how this problem is similar to the renormalisation problem, and how the bifurcation velocity may be derived from a fixed-point theory.
111 citations
TL;DR: In this paper, general results of the theory of separability for the geodesic equation in (Vn, g) were applied to deduce the canonical form of a separable metric withn- 2 Killing vectors.
Abstract: General results of the theory of separability for the geodesic equation in (V
n, g) are applied to deduce the canonical form of a separable metric withn- 2 Killing vectors. Applications to vacuum space-times with two Killing vectors are investigated.
TL;DR: In this article, it was shown that under fairly general conditions on a compact metric spaceY there are minimal homeomorphisms on Z×Y of the form(z,y)→(σz, hz(y)) where (Z, σ) is a arbitrary metric minimal flow andz→hz is a continuous map from Z to the space of homeomorphism of Y.
Abstract: It is shown that under fairly general conditions on a compact metric spaceY there are minimal homeomorphisms onZ×Y of the form(z,y)→(σz, hz(y)) where (Z, σ) is a arbitrary metric minimal flow andz→hz is a continuous map fromZ to the space of homeomorphisms ofY. Similar results are obtained for strict ergodicity, topolotical weak mixing and some relativized concepts.
Book•
01 May 1979
TL;DR: In this paper, the vector character of Euclidean n-space and familiar facts from linear algebra; concepts relevant to convex body theory; and the affine character of the space and the analogy between linear and affine concepts.
Abstract: Helps students see mathematics as an organic whole by focusing on the geometric while presenting viewpoints and methods that require a general understanding and unification of previous mathematical backgrounds. Develops basic metric topological methods and algebraic needs. Reviews vector character of Euclidean n-space and familiar facts from linear algebra; concepts relevant to convex body theory; and the affine character of the space and the analogy between linear and affine concepts. Explains how n-dimensional convex bodies and surfaces of Euclidean n-spaces are identified and how basic properties are established, showing how to express fundamental concepts accurately and how to verify intuitive relations analytically in space of general dimensions. Also contains a selection of standard fundamental theorems. Excellent preparation for further study of convexity theory, optimization theory, or basic analysis, topology, and geometry.
TL;DR: In this article, the authors established a rather complete optimality theory for the average cost semi-Markov decision model with a denumerable state space, compact metric action sets and unbounded one-step costs for the case where the underlying Markov chains have a single ergotic set.
TL;DR: A survey of the theory of fixed points of mappings of metric spaces is presented in this article, containing some unpublished results of the author, including some unpublished conclusions of the authors of this paper.
Abstract: A survey of the theory of fixed points of mappings of metric spaces, containing some unpublished results of the author, is presented.
TL;DR: Two examples are presented to show that variable metric methods may cycle between two points instead of converging to the required solution.
Abstract: Although variable metric methods for constrained minimization generally give good numerical results, many of their convergence properties are still open. In this note two examples are presented to show that variable metric methods may cycle between two points instead of converging to the required solution.
TL;DR: In this article, a method for generating stationary, axisymmetric, asymptotically flat, vacuum solutions of the EKF with an arbitrary number of mass and angularmomentum parameters is presented.
Abstract: A method is presented for generating stationary, axisymmetric, asymptotically flat, vacuum solutions of Einstein's equation with an arbitrary number of mass and angularmomentum parameters. As an example, a two-parameter asympotically flat solution, generalizing extreme Kerr, is generated from flat space. It is conjectured that the method, applied to the general static metric, can be used to generate all stationary, axisymmetric, asymptotically flat metrics.
TL;DR: In this paper, the variability between different examples of the same syllable and adjusting the metric accordingly is analyzed and the sum obtained is assumed to be proportional to the log probability of the two patterns having the same identity.
Abstract: Time‐warping pattern‐comparison algorithms are widely used in speech recognition. Two words or syllables being compared are described by a series of time frames each containing values of a set of acoustic parameters. After time alignment, the squared distance between the patterns is summed over the parameters within a frame and then across frames. The sum obtained is assumed to be proportional to the log probability of the two patterns having the same identity. This assumption is generally invalid, but it may be made substantially true by analyzing the variability between different examples of the same syllable and adjusting the metric accordingly. Variability is estimated both as a function of frame position within the syllable as a function of the acoustic parameters. In the latter case, within‐ and between‐class covariance matrices can be estimated and standard linear discriminant analysis methods applied. This permits the combination of disparate acoustic parameters into a single distance measure. In particular, combining frame and frame‐difference parameters allows one to use time development information and to take inter‐frame correlations into account.
TL;DR: In this article, the authors characterize upper semicontinuity of multifunctions in terms of upper Hausdorff semicontuity, a measure of non compactness and active boundary, applicable in optimization, theory of best approximation and in metrizability theory.
Book•
01 Jan 1979
TL;DR: In this paper, the authors define a domain of plane geometry, defining not only concepts of incidence but also metric concepts such as orthogonality or reflection Verifies the interrelationships of three theories showing how they are different representations of a single unified theory.
Abstract: Devoted to a domain of plane geometry, defining not only concepts of incidence but also metric concepts such as orthogonality or reflection Verifies the interrelationships of three theories showing how they are different representations of a single, unified theory These include a purely geometric theory based on the concept of incidence structures with orthogonality or with reflections, mainly as a treatment of Euclidean and non-Euclidean planes and certain subplanes of these planes; a theory of three-dimensional metric vector spaces with their natural geometric interpretation; and a theory of special types of S-groups and their group planes
01 Jan 1979
TL;DR: In this paper, the authors studied nonspacelike incompleteness using the Lorentzian distance d(p, q) and showed that no space-time can satisfy all of the following three conditions: (1) (M, g) is chronological, (2) each inextendible nonsPacelike geodesic contains a pair of conjugate points and (3) there exist two divergent sequences {pn} and {qn} which are causally disconnected by a compact set K. This particular result extends a theorem of Hawking and
Abstract: A space–time (M, g) is singular if it is inextendible and contains an inex-tendible nonspacelike geodesic which is incomplete. In this paper nonspacelike incompleteness is studied using the Lorentzian distance d(p, q). A compact subset Kof M causally disconnects two divergent sequences {pn} and {qn} if 0 < d(pn,qn) < ∞ for all n and all nonspacelike curves from pn to qn meet K. It is shown that no space–time (M, g) can satisfy all of the following three conditions: (1) (M, g) is chronological, (2) each inextendible nonspacelike geodesic contains a pair of conjugate points and (3) there exist two divergent sequences {pn} and {qn} which are causally disconnected by a compact set K. This particular result extends a theorem of Hawking and Penrose. It also implies that if (M, g) satisfies conditions (1) and (3), then there is a Co-neigh-bourhood of g in the space of metrics conformal to g such that any metric in this neighbourhood which satisfies the generic condition and the strong energy condition is nonspacelike incomplete.
TL;DR: In this paper, the junction conditions that these perturbations obey are expressed in terms of the perturbed first and second fundamental forms (intrinsic metric and extrinsic curvature) of the (perturbed) set of hypersurfaces one of which is the discontinuous one.
Abstract: Perturbations of a general background space-time with a hypersurface of discontinuity, such as the history of a collapsing star, are considered. The junction conditions that these perturbations obey are expressed in terms of the perturbed first and second fundamental forms (intrinsic metric and extrinsic curvature) of the (perturbed) set of hypersurfaces one of which is the discontinuous one. These junction conditions are applied to the odd-parity metric and hydrodynamical asymmetries of a slightly aspherical but otherwise general and realistic spherically collapsing star. The junction conditions are stated in terms of those metric and matter perturbational objects that are the most natural, economic, and versatile: gauge-invariant geometrical objects. For odd-parity perturbations these are scalars and vectors on the totally geodesic submanifold spanned by the time and radial coordinates. The end result is simple: The junction conditions amount to the continuity of the gradient of a master gauge-invariant scalar wave function from which all other perturbational quantities can be derived.
TL;DR: In this paper, a class of variable metric methods for unconstrained minimization is considered and it is shown that under appropriate assumptions on the function to be, minimized and stated conditions on the line search each algorithm in this class converges globally and superlinearly on convex functions.
Abstract: This paper considers a class of variable metric methods for unconstrained minimization. Without requiring exact line searches, it is shown that under appropriate assumptions on the function to be, minimized and stated conditions on the line search each algorithm in this class converges globally and superlinearly on convex functions.
TL;DR: In this article, the authors studied the time-dependent field equations for radiating fluid spheres, where they take the energy-momentum tensor as the sum of the EPT tensor for a perfect fluid and a radially expanding radiation and presented four exact analytic solutions with corresponding pressure, density, and luminosity distributions.
Abstract: We study the time-dependent field equations for radiating fluid spheres, where we take the energy-momentum tensor as the sum of the energy-momentum tensor for a perfect fluid and a radially expanding radiation. We present four new exact analytic solutions with the corresponding pressure, density, and luminosity distributions. We also present a case where two of the three metric coefficients are given as a function of the other, thus reducing the problem to choosing one of the metric elements such that distribution of physical quantities is reasonable. The solutions we have considered correspond to fluids at rest and have radius independent of time. Hence they represent fluid spheres in equilibrium.
TL;DR: In this article, it was shown that if an element y of Y belongs to T(D) if and only if there exists x 0ϵ D such that dist(T(x0), y) ⩽ dist(t(x, y), t(x), t) for all x ϵ X − D.
TL;DR: It is suggested that the operator calculus of Rota should be helpful in making explicit the isomorphism sought for the graph-like state, as well as the early programme culminating in the proof that the continum and discrete formulations of quantum theory are isomorphic.