Showing papers on "Metric (mathematics) published in 1971"
TL;DR: It is proved that successful convergence is obtained provided that the objective function has a strictly positive definite second derivative matrix for all values of its variables.
Abstract: The variable metric algorithm is a frequently used method for calculating the least value of a function of several variables. However it has been proved only that the method is successful if the objective function is quadratic, although in practice it treats many types of objective functions successfully. This paper extends the theory, for it proves that successful convergence is obtained provided that the objective function has a strictly positive definite second derivative matrix for all values of its variables. Moreover it is shown that the rate of convergence is super-linear.
183 citations
146 citations
TL;DR: The development of algorithms and theory for the unconstrained optimization problem during the years 1967–1970 is surveyed and some numerical difficulties that can occur when using Davidon's (1959) variable metric algorithm are explained.
Abstract: We survey the development of algorithms and theory for the unconstrained optimization problem during the years 1967–1970. Therefore (except for one remark) the material is taken from papers that have already been published. This exception is an explanation of some numerical difficulties that can occur when using Davidon's (1959) variable metric algorithm.
144 citations
TL;DR: The search aspect of decision making is examined by construing the information environment in the form of a maze and measuring “transmitted” information according to the Shannon-Weaver logarithmic metric to assess the power of subjects' search heuristics.
Abstract: This paper examines the search aspect of decision making. An experiment was conducted to assess the amount of information utilized by subjects prior to committing themselves to a choice. By construing the information environment in the form of a maze and measuring “transmitted” information according to the Shannon-Weaver logarithmic metric, the power of subjects' search heuristics was assessed.
98 citations
01 Nov 1971
TL;DR: In this paper, the authors present metric theories of gravity, including the definition of metric theory, evidence for its existence, and response of matter to gravity with test body trajectories, gravitational red shift, and stressed matter responses.
Abstract: Metric theories of gravity are presented, including the definition of metric theory, evidence for its existence, and response of matter to gravity with test body trajectories, gravitational red shift, and stressed matter responses. Parametrized post-Newtonian framework and interpretations are reviewed. Gamma, beta and gamma, and varied other parameters were measured. Deflection of electromagnetic waves, radar time delay, geodetic gyroscope precession, perihelion shifts, and periodic effects in orbits are among various studies carried out for metric theory experimentation.
81 citations
TL;DR: All the discrete, memoryless, symmetric channels matched to the Lee metric are derived, and general properties of Lee metric block codes are investigated.
Abstract: Almost all nonbinary codes were designed for the Hamming metric. The Lee metric was defined by Lee in 1958 . Golomb and Welch (1968) and Berlekamp, 1968a , Berlekamp, 1968b have designed codes for the Lee metric. In this paper, we derive all the discrete, memoryless, symmetric channels matched to the Lee metric, and investigate general properties of Lee metric block codes. Finally, a class of cyclic Lee metric codes is defined and the number of information symbols is discussed.
42 citations
TL;DR: In this article, it was shown that if a metric projection P m (f) of f onto a closed linear subspace is nonempty and finite-dimensional, then if there is a continuous selection for P m it is unique.
30 citations
01 Mar 1971
TL;DR: In this article, it was shown that the space-time region inside an axially symmetric, infinite, rotating, cylindrical mass distribution is Minkowskian.
Abstract: It is shown that the space-time region inside an axially symmetric, infinite, rotating, cylindrical mass distribution is necessarily Minkowskian.
24 citations
TL;DR: In this article, the unconditional convergence of series in the Haar system in the metric of L(0, 1) was shown to be possible in the special case of series.
Abstract: We obtain some results concerning the unconditional convergence of series in the Haar system in the metric of L(0, 1).
TL;DR: Theorem A. as discussed by the authors states that a measure-preserving transformation T acts on a Lebesgue space (M, 8, ¡j) which is also a G-space for a compact separable group G.
Abstract: The measure-preserving transformation T acts on a Lebesgue space (M, ¿8, ¡j) which is also a G-space for a compact separable group G. It is proved that if the factor-transformation on the space of G-orbits has completely positive entropy and a certain condition regarding the relations between the actions of G and T is satisfied, then T weakly mixing implies T has completely positive entropy. Introduction. It is very useful to know that a measure-preserving transformation Thas completely positive entropy; if this is the case, then Tis mixing of all orders ; if, in addition, T is invertible, then T is a Kolmogorov automorphism. An account of all this can be found in Rohlin's survey article [3]. The present paper considers completely positive entropy when the basic measure space (M, 38, p) is also a G-space for a compact separable group G. To be precise, the following theorem is proved : Theorem A. Let T be a measure-preserving transformation of a Lebesgue space (A7, 3$, p) which is also a G-space for a compact separable group G. If T satisfies the following conditions: (i) T is weakly mixing (has continuous spectrum, see [4, p. 39]) ; (ii) T o-commutes with G-action, i.e. Tg = ogT for all g in G, where a is a group endomorphism of G onto G (see §2) ; (iii) Ti(G) has completely positive entropy (see §1), then T has completely positive entropy. This theorem \"lifts\" the property of having completely positive entropy from the factor-transformation Taa) on the space of G-orbits to the transformation T itself. The concepts and notation used in stating the theorem will be considered in more detail in §§1 and 2; the proof is given in §§3 and 4 and §5 considers some applications. An immediate corollary of Theorem A is that an ergodic group endomorphism has completely positive entropy (see §5.1). This result was finally proved by Juzvinskiï [1] in 1965, the abelian case having been proved by Rohlin [2] the Received by the editors March 18, 1970 and, in revised form, October 27, 1970. AMS 1970 subject classifications. Primary 28A65, 22D40; Secondary 22D05, 22D10, 22E15.
TL;DR: In this article, the authors investigate criteria under which one may construct the energy tensor of a null radiation field from an algebraically special vacuum metric and show that the field bears the same relationship to the original metric as does Vaidya's to Schwarzschild's.
Abstract: We investigate criteria under which one may construct the energy tensor of a null radiation field from an algebraically special vacuum metric. The field bears the same relationship to the original metric as does Vaidya's to Schwarzschild's. As an example we generate a class of null radiation fields from a class of vacuum metrics without symmetry discovered by Robinson and Robinson.
TL;DR: In this article, a new method of approximation of nonperiodic functions by algebraic polynomials is introduced, and necessary and sufficient conditions for a function on the interval [-1, 1] to satisfy Holder's condition in the Lp metric are established.
Abstract: We introduce a new method of approximation of nonperiodic functions by algebraic polynomials. In particular, by this method we establish necessary and sufficient conditions for a function on the interval [-1,1] to satisfy Holder's condition in the Lp metric.
Patent•
22 Feb 1971
TL;DR: In this article, an arithmetic unit for a decoder for data encoded by convolution encoding is defined, which includes two channels, a main metric channel and a delta metric channel, where a metric is computed for a received symbol branch with respect to check bits from an encoder replica which is fed with a data bit, assumed to be a zero.
Abstract: An arithmetic unit for a decoder for data encoded by convolution encoding. The arithmetic unit includes two channels, a main metric channel and a delta metric channel. In the main metric channel a metric is computed for a received symbol branch with respect to check bits from an encoder replica which is fed with a data bit, assumed to be a zero. The delta metric channel computes a delta metric for the same branch. At the end of the computations the sign of the computed delta metric is used to control the changing of the data bit to a one and the adding of the computed delta metric to the metric in the main metric channel.
TL;DR: An unpublished algorithm due to Goode, although incomplete in its original form, turns out to be essentially equivalent to Stokes' method, but computationally very simple, the only (slight) disadvantage being that it may give superfluous solutions, which are easily excluded by a simple check.
Abstract: A unidimensional ordered metric scaling (Coombs, 1964) of a set of objects is an ordinal scaling of them and also a (partial or complete) ordinal scaling of the distances between them. The problem of representing such a scaling reduces to that of finding the general solution of a system of strict homogeneous linear inequalities. There are two possible approaches to the latter problem.
The purely algebraic approach of Fourier (1890), which is a modification of the usual method (systematic elimination) for the solution of systems of linear equations, is computationally simple, but does not give a satisfactory representation of the general solution, only an inconvenient and non-unique set of back substitutions.
The coordinate-geometric approach of Minkowski (1896), upon which Stokes' (1931) method of solution is based, gives a matrix representation of the general solution, but is computationally very laborious. However, an unpublished algorithm due to Goode (Coombs, 1964), although incomplete in its original form, turns out, when completed by the application of a theorem of Farkas (1902), to be essentially equivalent to Stokes' method, but computationally very simple, the only (slight) disadvantage being that it may give superfluous solutions, which are easily excluded by a simple check.
TL;DR: In this article, a computational procedure for numerical evaluation of integrals having both gaussian and exponential functional dependence in the integrand is presented, and three ranges are distinguished and in each one, an appropriate evaluation method is suggested.
01 Feb 1971
TL;DR: Optimal metrization for metrizable topological spaces was introduced in this paper, where it is shown that the real line R is optimal and that the usual metric is optimal.
Abstract: The purpose of this note is to introduce the concept of "Optimal Metrization" for metrizable topological spaces. Let X be such a space, p a metric on X and K(p) the group of all those homeomorphisms of X onto itself which preserve p. The metric p is said to be "optimal" provided there is no p* with K(p*) properly containing K(p). A space having at least one optimal metric is called 'optimally metrizable." Examples of spaces which are and which are not optimally metrizable are given; it is shown that the real line R is, and that the usual metric is optimal.
TL;DR: A survey of modern dimension theory emphasizing the development since 1961 when the first Prague Symposium was held can be found in this article, where the authors discuss the dimension theory of metric spaces and non-metrizable spaces and compare the general imbedding theorem with the classical one for separable metric spaces.
Abstract: This chapter presents a survey of modern dimension theory emphasizing the development since 1961 when the first Prague Symposium was held. The chapter discusses the dimension theory of metric spaces and non-metrizable spaces. As M. Katetov and K. Morita extended principal results of the classical dimension theory like sum theorem, decomposition theorem and product theorem to general metric spaces and proved dim R = Ind R for every metric space R, there has been a remarkable progress in the theory for metric spaces. Comparing the general imbedding theorem with the classical one for separable metric spaces, it is noticed that P(A) has infinite dimension while every n -dimensional separable metric space is imbedded in the (2 n + l)-dimensional Euclidean cube I 2n + l . In contrast to the metric case, there are many fundamental problems to be solved in the nonmetrizable case. For example, it is not known whether ind R = Ind R for every compact space R though it is not true for normal spaces. The chapter presents a list of grade to show how good or bad the present status of dimension theory for nonmetrizable spaces is in comparison with the theory for separable and nonseparable metric spaces.
TL;DR: In this paper, the authors make use of tractable differentiable absolutely convergent Fourier series for the general problem to determine when a measure is a measure, and apply it to obtain new criteria that a Helson set be a set of spectral synthesis.
Abstract: LetE\( \subseteqq\)R/2πZ be closed with Lebesgue measure 0. We imbed the pseudo-measuresTsupported byE into a space of distributions on a specific compact connected group. The reason for this approach is to make use of the more tractable differentiable absolutely convergent Fourier series for the general problem to determine whenT is a measure. The specific results are outlined in the Introduction. Applications of the techniques presented here are used to obtain new criteria that a Helson set be a set of spectral synthesis in the author's forthcoming work (viz., “A Support Preserving Hahn-Banach Property and the Helson-S Set Problem” and “The Helson-S Set Problem and Discontinuous Homomorphisms on Metric Algebras”).