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Showing papers on "Metric (mathematics) published in 1977"


Journal ArticleDOI
TL;DR: In this paper, the authors define and study a parametric estimation procedure which is asymptotically efficient under a specified regular parametric family of densities and is minimax robust in a small Hellinger metric neighborhood of the given family.
Abstract: This paper defines and studies for independent identically distributed observations a new parametric estimation procedure which is asymptotically efficient under a specified regular parametric family of densities and is minimax robust in a small Hellinger metric neighborhood of the given family. Associated with the estimator is a goodness-of-fit statistic which assesses the adequacy of the chosen parametric model. The fitting of a normal location-scale model by the new procedure is exhibited numerically on clear and on contaminated data.

702 citations


Journal ArticleDOI
TL;DR: In this article, the Lattman metric is shown to be a scalar multiple of the metric tensor which arises naturally from the Jacobi formulation of the action principle for spherical tops.
Abstract: New sets of variables are studied which should lead to significantly improved numerical stability and efficiency for computer stimulation studies of rigid classical molecules. The search for these new variables is made by finding variables which lead to the simplest (i.e. euclidean) expressions for the metric tensor of orientation space. It is shown that the intuitively defined Lattman metric [1] is a scalar multiple of the metric tensor which arises naturally from the Jacobi formulation of the action principle for spherical tops. It is then shown that Euler's quaternion parameters lead to a euclidean form for the orientation metric. These parameters lead to many associated simplifications in the equations of motion of classical rigid bodies including the removal of singularities and spurious behaviour near θ = 0. It is felt that these benefits will translate into increased accuracy and efficiency both for numerical integration of the equations of motion and for performing Monte Carlo integrations of phas...

392 citations


Journal ArticleDOI
TL;DR: A one parameter family of variable metric updates is developed by considering a fundamental decomposition of the Hessian that underlies Variable Metric Algorithms and considers particular choices of the parameter.
Abstract: We develop a one parameter family of variable metric updates by considering a fundamental decomposition of the Hessian that underlies Variable Metric Algorithms. The relationship with other Variable Metric Updates is discussed. Considerations based on the condition of the Hessian inverse approximation indicate particular choices of the parameter and these are discussed in the second half of this paper.

333 citations


Journal ArticleDOI
TL;DR: In this paper, the exact, general relation between the center-of-mass 4-velocity and the 4-momentum is derived, and the physical meaning of quasirigidity is investigated by establishing an approximate connection with continuum mechanics.
Abstract: Dixon's approach to describe the dynamics of extended bodies in metric theories of gravity is elaborated. The exact, general relation between the center-of-mass 4-velocity and the 4-momentum is derived. Quasirigid bodies are defined, and their equations of motion are shown to be determinate for a given metric. Multipole approximations are considered, and the physical meaning of quasirigidity is investigated by establishing an approximate connection with continuum mechanics.

192 citations


Journal ArticleDOI
TL;DR: A simple algorithm is given to generate the metric tree for an additive Dissimilarity matrix and this algorithm is extended to non-additive dissimilarity matrices through the use of linear programming.

172 citations



Proceedings ArticleDOI
01 Dec 1977
TL;DR: It is shown that maximum likelihood and related Bayesian identification procedures converge to a model in the model set, which is closest to the actual system generating the observations in the information distance measure.
Abstract: The identification and modeling of dynamical systems in a practical situation, where the model set under consideration does not necessarily include the observed system, are treated A measure of the relevant information in a sequence of observations is shown to possess useful properties, such as the metric property on the parameter set It is then shown that maximum likelihood and related Bayesian identification procedures converge to a model in the model set, which is closest to the actual system generating the observations in the information distance measure The convergence analysis is restricted for simplicity to finite sets of models The analysis naturally suggests methods for approximating a high-order system by a low-order model and for selecting a representative model from a given model set, applicable to infinite and even noncompact model sets

118 citations


Journal ArticleDOI
TL;DR: It is shown how two independent gradients-soil base status and grazing intensity in grassland and tall-herb communities-can be satisfactorily reconstructed by a non-metric method.
Abstract: SUMMARY Two basic types of ordination, metric and non-metric, are distinguished and discussed. Non-metric ordination or non-metric scaling, unfamiliar in ecology, is considered to be more powerful in various respects than metric ordination (which includes principal components analysis). Its advantages are discussed with reference to the assumptions which have to be made if ecological inferences are to be drawn. The properties of two-dimensional metric and non-metric ordination are compared in two examples. The first illustrates a long plant succession, and contrasts the linear reconstruction by an appropriate non-metric method with the familiar arch produced by a principal components method. The second shows how two independent gradients-soil base status and grazing intensity in grassland and tall-herb communities-can be satisfactorily reconstructed by a non-metric method.

114 citations


Journal ArticleDOI
TL;DR: A class of methods for minimizing a nondifferentiable function which is the maximum of a finite number of smooth functions is developed, which possesses many attractive features of variable metric methods and can be viewed as their natural extension to the nondifferentiability case.
Abstract: We develop a class of methods for minimizing a nondifferentiable function which is the maximum of a finite number of smooth functions. The methods proceed by solving iteratively qquadratic programming problems to generate search directions. For efficiency the matrices in the quadratic programming problems are suggested to be updated in a variable metric way. By doing so, the methods possess many attractive features of variable metric methods and can be viewed as their natural extension to the nondifferentiable case. To avoid the difficulties of an exact line search, a practical stepsize procedure is also introduced. Under mild asumptions the resulting method converge globally.

88 citations




Journal ArticleDOI
TL;DR: In this paper, the authors constructed measures of location differentiable at every density in the Hellinger metric and derived asymptotically optimal estimators for minimax robust location measures.
Abstract: Measures of location differentiable at every density in the Hellinger metric are constructed in this paper. Differentiability entitles these location functionals to the label "robust," even though their influence curves need not be bounded and continuous. The latter properties are, in fact, associated with functionals differentiable in the Prokhorov metric. A Hellinger metric concept of minimax robustness of a location measure at a density shape $f$ is developed. Asymptotically optimal estimators are found for minimax robust location measures. Since, at $f$, their asymptotic variance equals the reciprocal of Fisher information, asymptotic efficiency at $f$ and robustness near $f$ prove compatible.

R. Schattner1
01 Jan 1977
TL;DR: In this paper, the authors prove the existence and uniqueness of a zero-linear-momentum vector field and show the existence of a center-of-mass line which is a smooth timelike curve contained in a convex hull of the world tube of the body.
Abstract: In Dixon's theory of the dynamics of extended bodies in metric theories of gravity, a definition of a center-of-mass line is proposed. We prove the existence and uniqueness of a zero-linear-momentum vector field. Using this vector field we show the existence of a center-of-mass line which is a smooth timelike curve contained in a convex hull of the world-tube of the body.

Journal ArticleDOI
TL;DR: In this article, it was shown that the general form of the Robertson-Walker cosmological metric admits symmetry properties that are members of the symmetry family of contracted Ricci collineations.
Abstract: It is shown that the general form of the Robertson-Walker cosmological metric admits symmetry properties that are members of the symmetry family of contracted Ricci collineations. A particular form for the conservation law generator given by ▽ j [(−g)1/2(T −1/2δ )η i ] = 0 following in consequence of these symmetries is obtained and interpreted.



Journal ArticleDOI
01 Jan 1977-Topology
TL;DR: In this paper, the authors considered Hurewicz fiber maps with fibers which are ANRs and showed that all of the spaces (except function spaces) under consideration will be locally compact, separable and metric.

01 Jan 1977
TL;DR: In this article, the authors considered a set of denumerable stochastic matrices where the parameter set is a compact metric space and gave a number of simultaneous recurrence conditions on these matrices and established equivalences between these conditions.
Abstract: In this paper we consider a set of denumerable stochastic matrices where the parameter set is a compact metric space. We give a number of simultaneous recurrence conditions on the stochastic matrices and establish equivalences between these conditions. The results obtained generalize corresponding results in Markov chain theory to a considerable extent and have applications in stochastic control problems. COMPACT METRIC SET OF DENUMBERABLE STOCHASTIC MATRICES; SIMULTANEOUS RECURRENCE CONDITIONS; DOEBLIN CONDITION; SCRAMBLING CONDITION; OUASICOMPACTNESS CONDITION; EOUIVALENCES

Journal ArticleDOI
N.K. Nielsen, H. Römer1, Bert Schroer1
TL;DR: In this paper, a split point derivation of the m = 0 axial anomaly in an external Euclidean Riemann metric is given. And two new anomalous currents which lead to the Euler-density and the Hirzebruch L density are discussed.

Journal ArticleDOI
TL;DR: The self-scaling method, first introduced by Oren and Luenberger, and the method of Biggs are considered, and they are compared with methods which use correction formulae from the Broyden one-parameter family, in particular the BFGS formula and the Fletcher switching strategy.
Abstract: Two recent suggestions in the field of variable metric methods for function minimization are reviewed: the self-scaling method, first introduced by Oren and Luenberger, and the method of Biggs. The two proposals are considered both from a theoretical and computational aspect. They are compared with methods which use correction formulae from the Broyden one-parameter family, in particular the BFGS formula and the Fletcher switching strategy.

Journal ArticleDOI
TL;DR: In this paper, it was shown that the field equations for these spaces can be reduced to two ordinary differential equations, one of which is quasi-linear in one of the variables, and the metric is type D iff it possesses a two dimensional, abelian, orthogonally transitive symmetry group.
Abstract: This paper contains an investigation of algebraically special spaces with two commuting Killing vectors. It is shown that the field equations for these spaces can be reduced to two ordinary differential equations, one of which is quasi-linear in one of the variables. The metric is type D iff it possesses a two dimensional, abelian, orthogonally transitive symmetry group. Finally, the type D metrics of Kinnersley are expressed in various coordinates, including those of Plebanski and Demianski.

Book ChapterDOI
TL;DR: A sequence of minimum problems given on a metric space, together with a limit problem, is called equiwellset if every problem has exactly one solution, if the values converge, and if any asymptotically minimizing sequence converges to the solution of the limit problem as mentioned in this paper.
Abstract: A sequence of minimum problems given on a metric space, together with a limit problem, is called equiwellset if every problem has exactly one solution, if the values converge, and if any asymptotically minimizing sequence converges to the solution of the limit problem. When all problems are equal we get the classical definition of Tyhonov. A metric characterization of equiwellposedness is given that generalizes results of Vainberg for a single problem. Differential characterizations are also obtained extending a result of Asplund-Rockafellar. Applications are given to the epsilon method and to the perturbations of the linear quadratic problem.

Journal ArticleDOI
TL;DR: In this article, the authors defined the isotropy condition for spherical relativistic spheres of perfect fluid to be isotropic by Walker's (1935) condition, which permits the use of noncomoving coordinate systems, which are preferable to comoving systems in certain situations.
Abstract: Spherically symmetric relativistic spheres of perfect fluid are defined to be isotropic by Walker's (1935) isotropy condition. This condition permits the use of noncomoving coordinate systems, which, it is argued, are preferable to comoving systems in certain situations. It is assumed that these systems are such that the metric is orthogonal and involves three unknown functions. These functions are obtained by solving the equation expressing the isotropy condition in a number of cases defined by ancillary mathematical assumptions. Formulas are given for the pressure, density, and velocity components of the fluid, but the detailed physical analysis of the various cases found is reserved for a subsequent paper.


Patent
25 Apr 1977
TL;DR: In this article, a tree searching technique is used to determine the maximum metric for each of the states during each baud interval and generate a data bit stream therefrom, there being one for each state, which is updated by updating the bit stream corresponding to its associated survivor metric by deleting the oldest bit therefrom and adding as the newest bit, the bit value associated with the survivor metric path.
Abstract: Digital data which is correlatively encoded into discrete plural states and transmitted by modulating a carrier signal whose phase in each baud interval is a function of the data is detected through a tree searching technique which recursively determines the maximum metric for each of the states during each baud interval and generates a data bit stream therefrom. The maximum metric is determined by computing the transition metric for each possible path into a state from a comparison of the baseband signal with a reference signal generated therefrom and adding it to the state metric for the prior state from which the path originated to derive a path metric and then selecting the maximum path metric for each state designated survivor decision metric. The data bit stream, there being one for each state, is generated by updating the data bit stream corresponding to its associated survivor metric by deleting the oldest bit therefrom and adding as the newest bit, the bit value associated with the survivor metric path.

Journal ArticleDOI
TL;DR: The Wightman functions of a field theory with indefinite metric allow the reconstruction of a linear space with an indefinite sesquilinear form as mentioned in this paper, which is investigated when this form is given by a bounded operator on a Hilbert space.

Book ChapterDOI
01 Jan 1977
TL;DR: In this article, the canonical forms of the metric are established for all complex Einstein flat with the minimally (one-sided) algebraically degenerate -conformal curvature by applying the Plebanski-Hacyan theorem.
Abstract: By applying Plebanski-Hacyan theorem, the canonical forms of the metric are established for all complex Einstein flat with the minimally (one-sided) algebraically degenerate — conformal curvature. Then Einstein equations are integrated. The solution is expressed in the terms of only one fundamental key function which is determined by a differential equation of the second order and with quadratic non-linearity only, this equation being a generalization of the second heavenly equation.

Journal ArticleDOI
TL;DR: In this paper, necessary and sufficient conditions for the stability of the problem were presented for quasi-convex programming problems, where the upper and lower semiconformity of the function and the upper semicon form of the point-to-set mapping were taken into account.
Abstract: This paper presents necessary and sufficient conditions for the stability of the problem . Here M is a subset of a metric space X, λ is an element of some set ⋀ “with convergence” and f is a functional defined on the Cartesian product X×⋀. These conditions apply to the upper and lower semiconformity of the function and the upper semiconformity of the point-to-set mapping . The used set-convergence is less strong than the convergence induced by the Hausdorff metric. As conclusions theorems on the relationship between the f 0 and [fcirc] upper semiconformity and sufficient stability-conditions for some general problems (especially quasi convex programming) are received. The necessity of certain suppositions is illustrated by appropriate examples.


Journal ArticleDOI
TL;DR: In this article, a technique for determining an approximate simply-transitive three-parameter symmetry group of a three-dimensional positive-definite Riemannian metric is developed, which employs a variational principle to find a set of three orthonormal vectors whose commutation coefficients are as close as possible to a given set of structure constants.
Abstract: A useful step toward understanding inhomogeneous space–times would be to classify them, perhaps in a fashion analogous to that used for spatially homogeneous space–times. To that end, a technique for determining an approximate simply‐transitive three‐parameter symmetry group of a three‐dimensional positive‐definite Riemannian metric is developed. The technique employs a variational principle to find a set of three orthonormal vectors whose commutation coefficients are as close as possible to a set of structure constants. The Bianchi classification of the structure constants of three‐parameter groups is then used to classify these inhomogeneous metrics. Application of this technique to perturbed homogeneous metrics is discussed in detail. We find that only four types of symmetry groups can be considered generic in the space of all perturbed homogeneous metrics.