Showing papers on "Metric (mathematics) published in 1970"

On the Nonorthogonality Problem

Abstract: Publisher Summary This chapter discusses three orthonormalization procedures, such as successive orthonormalization, symmetric orthonormalization, and canonical orthonormalization The simplest way of orthonormalizing a finite set of functions is by the classical Schmidt procedure, in which each member of the set in order is orthogonalized against all the previous members and subsequently normalized In solid-state theory, one could probably construct orthonormal combinations of the atomic orbitals of the system, which would still preserve the natural symmetry In such an approach, it would be necessary to treat the given functions ϕ = {ϕ 1 , ϕ 2 …, ϕ n } simultaneously, on an equivalent basis instead of successively as in the Schmidt procedure In molecular and solid-state theory, there are cases when also the symmetric orthonormalization procedure will break down, depending on the fact that, even if the basis ϕ = {ϕ 1 , ϕ 2 …, ϕ n } is linearly independent from the mathematical point of view, it may be approximately linearly dependent from the computational point of view This phenomenon causes a great many complications and may lead to very misleading results, since the associated secular equations may be almost identically vanishing Unfortunately, it seems as if many of the conventionally used basic systems are strongly affected by approximate linear dependencies In order to systematize this problem, it is convenient to study the metric matrix

Post-Newtonian metric for a general class of scalar-tensor gravitational theories and observational consequences

Abstract: Space-time Riemann metric for scalar-tensor gravitational theories with arbitrary omega parameter

Comparison of Gradient Methods for the Solution of Nonlinear Parameter Estimation Problems

Abstract: The performance of several of the best known gradient methods is compared in the solution of some least squares, maximum likelihood, and Bayesian estimation problems. Modifications of the Gauss method (including Marquardt’s) performed best, followed by variable metric rank one and Davidon–Fletcher–Powell methods, in that order. There appears to be no need to locate the optimum precisely in the one-dimensional searches. The matrix inversion method used with the Gauss algorithm must guarantee a positive definite inverse.

Source of the kerr metric

Abstract: Assuming that the Kerr-Newman metric is the field of a layer of mass and charge distributed over the equatorial disk spanning the ring singularity, the source distribution on the disk is computed explicitly. In the uncharged case, this interpretation automatically excises the noncausal parts of the manifold, so that one obtains the unique source of the causally maximal extension of the vacuum metric. A Newtonian field which gives the same source distribution is exhibited, and shown to be closely analogous to the relativistic case. In particular, the Newtonian particle orbits show the same avoidance of the ring singularity that is such a remarkable feature of geodesics in the Kerr geometry. In the charged case, we examine how the gyromagnetic moment (which is equal to that of the Dirac electron) is reflected in the character of the source distribution.

Nonmetric multidimensional scaling: Recovery of metric information

Abstract: The degree of metric determinancy afforded by nonmetric multidimensional scaling was investigated as a function of the number of points being scaled, the true dimensionality of the data being scaled, and the amount of error contained in the data being scaled. It was found 1) that if the ratio of the degrees of freedom of the data to that of the coordinates is sufficiently large then metric information is recovered even when random error is present; and 2) when the number of points being scaled increases the stress of the solution increases even though the degree of metric determinacy increases.

On metric multidimensional unfolding

Abstract: The problem of locating two sets of points in a joint space, given the Euclidean distances between elements from distinct sets, is solved algebraically. For error free data the solution is exact, for fallible data it has least squares properties.

About well-posed optimization problems for functionals in metric spaces

Abstract: A necessary and sufficient condition of correctness of extremal problems for lower semicontinuous functionals defined in metric spaces is given.

Metric properties of measure preserving homeomorphisms

Abstract: We study "typical" metric (ergodic) properties of measure preserving homeomorphisms of regularly connected cellular polyhedra and of some other spaces. In 1941 Oxtoby and Ulam proved (for a narrower class of spaces) that ergodicity is such a property. Using a modification of their construction and the method of approximating metric automorphisms by periodic ones, we prove in this paper that almost all properties that are "typical" for the metric automorphisms of the Lebesgue spaces are also "typical" for the situation under discussion.

A characterization of well-posed minimum problems in a complete metric space

Abstract: Given a functionalI:X →R, defined on a complete metric space (X,d), we give a necessary and sufficient condition for the minimum problem forI onX to be well posed or well posed in the generalized sense.

Metrics on test function spaces for canonical field operators

Abstract: In a canonical field theory, the field Φ(f) and momentum π(g) are assumed defined for test functionsf andg which are elements of linear vector spaces and , respectively. Generally, the continuity of the map onto the unitary Weyl operatorsU(f),V(g) is taken as ray continuity, the barest minimum to recover the field operators as their generators, i.e.,U(f)=e iΦ(f) ,V(g)=e iπ(g) . This leaves open the question of whether any wider continuity properties follow and what form they would take. We show that much richer continuity properties do follow in a natural fashion for every cyclic representation of the canonical commutation relations. In particular, we show that the test function space may be taken as a metric space, that the space may be uniquely completed in this topology, and that the map into the unitary Weyl operators is strongly continuous in this topology. The topology induced by this metric is minimal in the sense that it is the weakest vector topology for which the mapsf→U(f),g→V(g) are strongly continuous. An expression for a suitable metric can easily be given in terms of a simple integral over a state on the Weyl operators.

Psychological reality of physical concepts

Abstract: Fifty college physics students rated the similarity of pairs of concept words in analytical mechanics, and also provided 1 min of continued word associations to each individual concept word. The mean proportion of responses in common on the word association test was used as an index of associative similarity among concepts. Mean rating-scale judgments served as an index of perceived similarity. These two indices were interpreted as proximity measures and were scaled, using multidimensional scaling procedures with both a Euclidean and a city-block metric. Results suggest that either a two-dimensional configuration with a Euclidean metric or a three-dimensional configuration with a city-block metric describes the underlying structure of the similarity relations. The three-dimensional configuration correlated well with an hypothesized geometric model.

Note on metric dimension

Abstract: The metric dimension of a compact metric space is defined here as the order of growth of the exponential metric entropy of the space. The metric dimension depends on the metric, but is always bounded below by the topological dimension. More- over, there is always an equivalent metric in which the metric and topological dimensions agree. This result may be used to define the topological dimension in terms of the metric entropy as the in- fimum of the metric dimension taken over all metrics. Kolmogorov's concept of e-entropy, defined for compact metric spaces (3), has proved a useful measure of the "bulk" of such a space. In this note we relate this concept to the Hausdorff and topological dimensions of spaces for which these dimensions are defined. Our result appears as Theorem 2 below. Let X be a compact metric space, with distance function p. If U

Physical interpretation of complex-energy negative-metric theories

Abstract: It has long been held that the divergences of field theory could be removed if the theory was formulated on a Hilbert space with an indefinite metric. The difficulty of this approach was that the theory then possessed channels with negative probability and therefore violated unitarity. In a recent paper, Lee and Wick have observed that this difficulty could be eliminated if negative-metric states had complex energy. In this paper, we produce a model which shows several of the difficulties associated with the Lee-Wick proposal. The existence of complex energies leads to interpretation difficulties and, in higher sectors, does not completely solve the unitarity problem. We use the model to show how the Lee-Wick conjecture can be extended to overcome the difficulties that are produced. We describe the physical meaning of "good" theories with an indefinite metric.

Exploratory Investigation of Perceptual Color Scaling

Abstract: A viewing situation of moderate complexity was used in the present study. Two groups of color samples were used, one with constant hue and one with varied hue. Ratio judgments of color difference were obtained by having observers set the physical distance between pairs of color samples to represent the ratio of the size of visual differences relative to a standard difference created by setting two other color samples a fixed distance apart. The scaled color judgments were subjected to the Shepard–Kruskal nonmetric technique of multidimensional analysis. A comparison was made between the nonmetric analysis and more-conventional metric analyses where comparable scaling results were obtained. The outcome of this comparison was that with a nonmetric technique, and with an intrinsically imprecise scaling technique, meaningful metric visual-scaling information became available. Good agreement was found with Munsell scales and with Godlove and MacAdam predictions of visual distance.