Showing papers on "Metric (mathematics) published in 1972"

Variable-length codes and the Fano metric

Abstract: It is shown that the metric proposed originally by Fano for sequential decoding is precisely the required statistic for minimum-error-probability decoding of variable-length codes. The analysis shows further that the "natural" choice of bias in the metric is the code rate and gives insight into why the Fano metric has proved to be the best practical choice in sequential decoding. The recently devised Jelinek-Zigangirov "stack algorithm" is shown to be a natural consequence of this interpretation of the Fano metric. Finally, it is shown that the elimination of the bias in the "truncated" portion of the code tree gives a slight reduction in average computation at the sacrifice of increased error probability.

Variable metric algorithms: Necessary and sufficient conditions for identical behavior of nonquadratic functions

Abstract: Huang (Ref. 1) introduced a general family of variable metric updating formulas and showed that, for a convex quadratic function, all members of this family generate the same sequence of points and converge in at mostn steps. Huang and Levy (Ref. 2) published numerical data showing the behavior of this family for nonquadratic functions and concluded that this family could be divided into subsets that also generate sequences of identical points on more general functions. In this paper, the necessary and sufficient conditions for a group of algorithms to form part of one of these subsets are given.

Remarks on Kähler-Einstein manifolds

Abstract: The main purpose of this note is to characterize a compact Kahler-Einstein manifold in terms of curvature form The curvature form Q is an EndT valued differential form of type (1,1) which represents the curvature class of the manifold We shall prove that the curvature form of a Kahler metric is the harmonic representative of the curvature class if and only if the Kahler metric is an Einstein metric in the generalized sense (gs), that is, if the Ricci form of the metric is parallel It is well known that a Kahler metric is an Einstein metric in the g s if and only if it is locally product (globally, if the manifold is simply connected and complete) of Kahler-Einstein metrics We obtain an integral formula, involving the integral of the trace of some operators defined by the curvature tensor, which measures the deviation of a Kahler-Einstein metric from a Hermitian symmetric metric In the final section we shall prove the uniqueness up to equivalence of Kahler-Einstein metrics in a simply connected compact complex homogeneous space This result was proved by Berger in the case of a complex projective space and our proof is completely different from Berger’s

Set theory and metric spaces

Abstract: This book is based on notes from a course on set theory and metric spaces taught by Edwin Spanier, and also incorporates with his permission numerous exercises from those notes. The volume includes an Appendix that helps bridge the gap between metric and topological spaces, a Selected Bibliography, and an Index.

An RKHS approach to detection and estimation problems-- III: Generalized innovations representations and a likelihood-ratio formula

Abstract: The concept of a white Gaussian noise (WGN) innovations process has been used in a number of detection and estimation problems. However, there is fundamentally no special reason why WGN should be preferred over any other process, say, for example, an nth-order stationary autoregressive process. In this paper, we show that by working with the proper metric, any Gaussian process can be used as the innovations process. The proper metric is that of the associated reproducing kernel Hilbert space. This is not unexpected, but what is unexpected is that in this metric some basic concepts, like that of a causal operator and the distinction between a causal and a Volterra operator, have to be carefully reexamined and defined more precisely and more generally. It is shown that if the problem of deciding between two Gaussian processes is nonsingular, then there exists a causal (properly defined) and causally invertible transformation between them. Thus either process can be regarded as a generalized innovations process. As an application, it is shown that the likelihood ratio (LR) for two arbitrary Gaussian processes can, when it exists, be written in the same form as the LR for a known signal in colored Gaussian noise. This generalizes a similar result obtained earlier for white noise. The methods of Gohberg and Krein, as specialized to reproducing kernel spaces, are heavily used.

On Objective Functions of Phytosociological Resemblance

Abstract: Different objective functions are reviewed with regard to their intrinsic properties and generic relationshiDs as they relate to the measurement of phytosociological resemblance. These functions may take the form of a distance, probability, or information. Functions of these kinds are best conceived by the phytosociologist as abstractions which are meaningful only when placed within the bounds of a given sample space. In such a space the phytosociological objects, such as the individual stands of vegetation, are represented as points whose relative spatial placement is determined by the resemblance function. The spatial configuration of points, i.e., the manner of their placement relative to one another in sample space, is referred to as sample structure in the present paper. The first part of the paper includes a discussion of the sample space and sample structure, and it also deals with the concept of stochastic and deterministic resemblance functions. This is followed by the description of the different variants of distance, a probability-type coefficient, and several information theory functions. While the distance functions here represent metric divergences, which define the relative placement of objects in sample space, a probability-type coefficient, as a probability divergence, expresses the likelihood that given objects will be more dis-

Geometric Predictions of Scaling at Tricritical Points

Abstract: The shapes of the three critical lines meeting at a tricritical point are discussed in the light of the homogeneity hypothesis. We give a complete listing of possible shapes consistent with scaling. Certain possible geometries are thereby shown to be inconsistent with scaling. Application of these ideas to the crossover regions is shown to considerably simplify the "metric problem" posed by Riedel.

BEST APPROXIMATIONS OF FUNCTIONS IN THE Lp METRIC BY HAAR AND WALSH POLYNOMIALS

Abstract: In this work the modulus of continuity of functions in the Lp metric (1 ≤ p < ∞) is estimated through its best approximations in this metric by Haar and Walsh polynomials. Besides, estimates of best approximations of functions by Haar and Walsh polynomials in the Lq metric are obtained by the same approximations in the Lp metric (1 ≤ p ≤ ∞). In the last case, the results are analogous to those which were proved for approximations by trigonometric polynomials by P. L. Ul'janov and also by S. B. Steckin and A. A. Konjuskov. Bibliography: 26 items.

Calculation of metric coefficients for streamline coordinates.

Abstract: A procedure is given for deriving the equations describing the surface streamline metric in a general form and in a form suitable for incorporation into computer codes to calculate an inviscid flow about three-dimensional bodies Sample results are included to show the application of the metric in conjunction with an axisymmetric analog to predict heat transfer to a typical space shuttle orbiter

Use of rank order data in functional measurement.

Abstract: Subjects estimated average value of angle pairs using magnitude estimations or graphic ratings. These numerical response data follow a simple averaging model. Functional scaling yields a linear relation between subjective and objective angle. The numerical data are then reduced to rank orders, and J. B. Kruskal's monotone analysis of variance (MONANOVA) procedure is applied. This allows a reconstruction of the original metric information from the strictly ordinal information, illustrating the power of MONANOVA in scaling. Limitations of MONANOVA in testing the underlying model are discussed.

A metric characterization of zero-dimensional spaces

Abstract: It is shown that a nonempty separable metrizable space X is zero-dimensional if and only if there exists a metric p on X, inducing the given topology of X and such that all nonzero distances p(x, y) are mutually different.

Metric considerations in cluster analysis

Abstract: : A variation of the 'k-means' method of cluster analysis is described which is designed to take into account and profit from the possibility that the separate clusters resemble samples from multivariate normal distributions with substantially different covariance structures. These covariance structures determine metrics which can be updated recursively. (Author)

Note on the Tightness of the Metric on the Set of Complete Sub $\sigma$-Algebras of a Probability Space

Abstract: The purpose of this note is to show that the usual metric on the set of complete $\operatorname{sub} \sigma$-algebras of a probability space is very tight; a recent result from E. B. Boylan on equi-convergence of martingales follows and is thereby, we believe, better understood.

On Canonical Commutation Relations and Infinite‐Dimensional Measures

Abstract: Recently, Araki has proved ray continuity for the measure‐space realization of the CCR's. In this paper, stronger continuity properties of this realization are derived, from which Araki's result follows as a corollary. It is shown that V(g) is continuous on complete metric subspaces of Uπ for any metric stronger than the weak topology. Surprisingly no further assumptions on the measure are needed. If Uπ is an F‐ or LF‐space, V(g) is continuous on the whole of Uπ. These results are equivalent to certain continuity properties of the Radon‐Nikodym derivative of the measure. Via these, it is then shown that every quasi‐invariant measure on a nuclear F‐ or LF‐space, such as S or D, is a superposition of ergodic measures. In the derivation, their close connection with irreducible representations of the CCR's is exploited.

Expected Utility and Continuity

Abstract: The theory of expected utility derives conditions in which an ordering > of a set P of probability distributions, all defined on some measurable space (X, 1), can be represented by the numerical order of the expected values J udP of some " utility " function u on X. The set P may comprise only the simple distributions usually considered in texts on utility theory, but may include all discrete distributions, or continuous distributions of various kinds, or perhaps all the probability measures which can be defined on S. The class U of permissible utility functions may also be defined in various ways: it may include all those for which the expected values are defined and finite, or may be restricted to functions having some prescribed properties, such as boundedness, continuity or differentiability. Professor K. J. Arrow [2] has recently proposed a condition of a type new to this theory, namely that the ordering be continuous with respect to a certain metric in the set P of distributions. Such an assumption reflects the intuitively appealing notion that distributions which are " close together " in their assignment of probabilities to events in X should also be neighbours in the order of preference. But there are various plausible ways in which " close together " can be defined mathematically, and the choice among them has radical implications-which can be far from intuitive-for the existence and properties of utility functions. The particular metric considered by Arrow turns out to be unsatisfactory in certain respects; but his proposal suggests an interesting general method, which it is the object of the present paper to explore systematically. The following general statement of the problem of the existence of utility functions is adequate for the present discussion; note that the structure defined in this statement, together with the stated properties of the symbols X, Y, P, 1, U, and Sy will be assumed without special mention throughout the paper:

Variable metric and conjugate direction methods in unconstrained optimization: recent developments

Abstract: This paper discusses and summarizes three topics that are of current interest in the theory of unconstrained optimization. It treats variable metric algorithms that minimize convex quadratics in a finite number of steps without performing linear searches, variable metric and conjugate direction algorithms that minimize nonquadratics in a finite number of steps, and numerically stable variable metric updating formulas.

Comment on "A Nonlinear Mapping for Data Structure Analysis

Abstract: An alternative metric for use with Sammon's nonlinear mapping is suggested. Rather than Euclidean, the Hamming metric is proposed as a means of reducing the iteration time.

Stationary Axially Symmetric Fields and the Kerr Metric

Abstract: In this note the axially symmetric metric for stationary gravitational field, in a slightly general form, is discussed. The vacuum field equations for this metric are given. Specialization of this metric leads to a different form of field equations previously discussed in literature. In particular, the Kerr metric is given in a new form. A justification for interpreting the Kerr metric as an exterior solution corresponding to a spinning rod or a rotating spherical body is given.

An Approximate Solution for the Static, Spherically Symmetric Metric Due to a Point Charged Mass in Brans‐Dicke Theory

Abstract: An approximate solution of the field equations of Brans‐Dicke theory is obtained for a static, spherically symmetric metric due to a charged point mass which can be considered to be an analog of the Reissner‐Nordstrom metric in Einstein's theory.