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

Similarity of automorphisms of the torus

Abstract: : The automorphisms of the Torus are rich mathematic objects. They possess interesting number theoretic, geometric, algebraic, topological, measure theoretic, probabilistic, and even information theoretic properties. Some of these are illustrated in this memoir whose main program is to prove that entropy is a complete invariant with respect to measurable changes of variable for continuous ergodic group automorphisms of the 2-dimensional Torus. A key tool in the proof is the use of symbolic dynamics. These automorphisms reveal a Markovian probabilistic behavior characteristic of a wide class of smooth mappings. (Author)

An Operator Theorem on $L_1$ Convergence to Zero with Applications to Markov Kernels

Abstract: A recent theorem of Orey [12] (see also [1], [6], [7], [13]) asserts that if $T$ is an $L_1$ operator induced on a discrete measure space by an irreducible recurrent aperiodic Markov matrix, then the condition (C) holds: $f \epsilon L_1, \int f = 0$ implies that $T^n f$ converges to zero in $L_1$. In an attempt to determine when (C) holds for more general operators, we at first prove the following (Theorem 1.1): Let $T$ be a positive linear contraction operator on $L_1$; if $T^nf$ and $T^{n+1}f$ intersect slightly, but uniformly in $f$ in the unit sphere of $L_1$, then $T^nf - T^{n+1}f$ converges to zero in norm. (C) follows if $T$ is conservative and ergodic (Corollary 1.3). In Section 2 we derive from this a simple proof of Orey's theorem. The main result of the paper is in Section 3 and could be called a "zero-two" theorem: Let $P(x, A)$ be a Markov kernel, and assume that there is a $\sigma$-finite measure $m$ such that for each $A, m(A) = 0$ implies $P(x, A) = 0$ a.e. and $m(A) > 0$ implies $\sum^\infty_{n=0} P^{(n)}(x, A) = \infty$ a.e. Then the total variation of the measure $P^{(n)}(x, \cdot) - P^{(n+1)}(x, \cdot)$ is either a.e. 2 for all $n$ or it converges a.e. to 0 as $n \rightarrow \infty$. In Section 4 it is shown that a version of the zero-two theorem essentially contains the Jamison-Orey generalization of Orey's theorem to Harris processes. Section 1 and Section 2 of this paper do not assume any knowledge of either operator ergodic theory or probability. Some known results in ergodic theory are applied in Section 3, but the proof of the main theorem does not depend on them.

Lattice dynamics of argon at 4 degrees K

Abstract: Neutron scattering techniques have been used to measure the phonon dispersion relations in a single crystal of 36Ar at 4 degrees K Data have been taken along the (110) and the (111) crystallographic directions The analysis is expressed in terms of a two-neighbour force-constant model with five disposable parameters; one linear combination of three of these parameters is not determined well by the data It is shown that the measurements are not inconsistent with a Mie-Lennard-Jones potential where the parameters are determined entirely from other macroscopic measurements on argon However, there is a discrepancy with recent measurements of the elastic constants

On the Notion of Randomness

Abstract: Publisher Summary This chapter focusses on the notion of randomness. The sequences satisfying the definition of randomness form a set of probability one with respect to the measure that makes all coordinates independently take the values 0 and 1 with probability ½. A recursive sequence is necessarily non random. It is proposed to avoid the paradox, born of the classical conception of the totality of all sets of probability one, to properties expressible in the constructive infinitary propositional calculus. The specific Borel sets considered are always obtained by applying the Borelian operations to recursive sequences of the defined sets, which means that they are hyperarithmetical. The chapter proves the theorem, which states that the intersection of all hyperarithmetical sets of measure one is a Σ 1 1 set of measure one.

Homeomorphic measures in metric spaces

Abstract: For any nonatomic, normalized Borel measure IA in a complete separable metric space X there exists a homeomorphism h: 9Y--X such that j = Xh-I on the domain of P, where 9Z is the set of irrational numbers in (0, 1) and X denotes Lebesgue-Borel measure in OZ. A Borel measure in 9Z is topologically equivalent to X if and only if it is nonatomic, normalized, and positive for relatively open subsets. 1. Defnitions and results. A topological measure space is a pair (X, ju), where X is a topological space and , is a measure on the class of Borel subsets of X. (X, ,u) is homeomorphic to (Y, z') if there exists a homeomorphism of X onto Y that makes v correspond to ,u, and then v is said to be topologically equivalent to,u. If B is a Borel subset of (X, ,u), then MB denotes the restriction of ji to the class of Borel subsets of B. A measure ,u is everywhere positive if ,u(G) >0 for every nonempty open set G, nonatomic if ,u({x 1) = 0 for each xEX, and normalized if ,u(X) = 1. Let 9t denote the set of irrational numbers in I= [0, 1], and let X denote the restriction of Lebesgue measure m to the Borel subsets of OZ. It is known [8, Theorem 2, p. 886] that a Borel measure in the n-dimensional cube In is topologically equivalent to n-dimensional Lebesgue-Borel measure in In if and only if it is everywhere positive, nonatomic, normalized, and vanishes on the boundary. A similar theorem will be shown to hold in Mt. THEOREM 1. A topological measure space (X, ,u) is homeomorphic to (s9, X) if and only if X is homeomorphic to K and IA is an everywhere positive, nonatomic, normalized Borel measure in X. In particular, any such measure in 9t is topologically equivalent to X. It is known [2, ?6, Exercise 8c, p. 84] that if X is a compact metric space, and , is a nonatomic, normalized Borel measure in X, then (X, ,u) is almost homeomorphic to (I, X), in the sense that there exist sets ACI and BCX such that X(I-A)=O, ,u(X-B)=O, and (B, JIB) is homeomorphic to (A, XA). We shall show that this conclusion still holds when X is a complete separable metric space, and that the set A can always be taken equal to 9l. Received by the editors July 3, 1969. A MS Subject Classifications. Primary 2813, 2810,-2870; Secondary 5435, 5460.

Some ergodic properties of multi-dimensional f-expansions

Abstract: where $, is an arbitrary function satisfying regularity conditions. He shows approaches (( 1 + x) log 2)', Gauss' measure. Ryll-Nardzewski [21] put this work in modern terminology by noting that T( ) was the shift on the digits of the continued fraction expansion. With his proof that T(.) was ergodic with respect to Lebesgue measure, the ergodic theory was completed by noting that Gauss' measure is an invariant measure for T(.). Renyi [ 181 generalized Ryll-Nardzewski's results to the f-expansions of Everett [6] and Bissinger [3]. Here the shift is T(x)=f-'(x)-[f-'(x)], x ~ ( 0 , l ) . To show Tergodic with respect to Lebesgue measure, Renyi imposed condition (C), a regularity condition. Rohlin [20] obtained some information theoretic results which were applied to Renyi's f-expansions. Recently Vinh-Hien [3 11 has extended Kuzmin's theorem to f-expansions and obtained a central limit theorem. Reznik [19] has used Vinh-Hien's work to obtain a law of the iterated logarithm. In 1869 Jacobi [lo] presented an extension of the continued fraction to two dimensions. Perron [ 171 extended Jacobi's work to n-dimensions. In 1964 Schweiger 1231 began an examination of the measure theoretic properties of Jacobi's algorithm (see [24 to 291). It was this work which motivated our paper. However Schweiger [30] has recently published some results which also concern general F-expansions for n-dimensions. The class of algorithms he considers does not include the Jacobi algorithm and is a natural generalization of Renyi [ls]. Our results generalize most of Schweiger's work and have the Jacobi algorithm

The Normal Range

Abstract: To the Editor.— The attempt by Dr. Elveback and her colleagues to exorcise the ghost of Gauss ( 211 :69-75, 1970), calls for some comment. As they state, Gauss was primarily concerned with the theory of errors, but the components in an error distribution are analogous to those in any other type of measurement. If I measure the length of a table with a ruler I should get a single answer; however, the lengths of the table and the ruler vary with temperature and humidity, and there are parallax errors in reading the ruler, so I do not get a constant reading. Since there are three definable sources of variation, the range of measurements should be predictable; unfortunately, the three primary causes of variation, temperature, humidity, and parallax are themselves resultants of more remote causes, so that ultimately there are not three, but an indefinitely large number of determinant factors; as

The Theory of Best Approximation in Normed Linear Spaces

Abstract: It is well known that the problem of best approximation of a function consists in the determination of a function belonging to a fixed family such that its deviation from the given function is a minimum, This problem was first formulated by P. L. Chebyshev, who investigated the approximation of continuous functions by algebraic polynomials of given degree and by rational fractions with numerators and denominators of fixed degree. As a measure of the deviation between two functions, Chebyshev used the maximum of the absolute value of their difference. Subsequently, a number of mathematicians have studied other specialized problems of best approximation whose content is defined by some choice of the measure of deviation and the function set used for the approximation. Among these, we should in the first place note A.A. Markov, Jackson, Bernshtein, de la Vallee-Poussin, Haar, and Kolmogorov. With the development of the theory of normed spaces it became clear that a wide range of problems of best approximation can be put into a general formulation in terms of normed spaces, if the norm of the space is taken as the measure of deviation.

Computation of optimal control for time-delay systems

Abstract: A numerical technique is presented for the computation of control for linear time-delay systems subject to a quadratic performance measure.

Note on tape reversal complexity of languages

Abstract: The number of tape reversals required for the recognition of a set of inputs by a 1-tape Turing machine (TM) has been proposed before as a measure of complexi It is shown that if a set is recognizable by a multi-tape TM with no more than R(n) tape reversals, where n is the input length, then the set is recog It is also shown that if a set is recognizable by a multi-tape TM within time T(n), then the set is recognizable by a 1-tape off-line TM with no more than Upper bounds on the number of reversals necessary for the recognition of a number of well-known classes of languages are also obtained.

Gini Ratios: Some Considerations Affecting Their Interpretation

Abstract: A CURRENTLY popular tool for studying concentration-the Gini ratio-is being used with increasing frequency [2, 3, 5, 7, 10, 11, 12]. The objective of this paper is to demonstrate the possible shortcomings of this tool as a measure of relative distributional inequality. It will be shown that the common practice of reporting Gini coefficients without explicitly describing the distribution from which they are derived may be analogous to reporting regression coefficients without their standard errors. More precisely, it will be demonstrated that in certain cases interpreting Gini coefficients on a cardinal basis may lead to erroneous conclusions and that even conclusions drawn from ordinal ranking of Gini coefficients can be misleading.

A “halo”-model for multidimensional ratio scaling

Abstract: A model for direct multidimensional ratio scaling is presented, based on the concepts “common” and “difference” of the “halos” of two percepts. Measures of halos and their differences are proportional to lengths of corresponding percept vectors and their distance in subjective space. Ekman type scaling judgements are assumed to reflect the ratio measure of the common/measure of the standard's halo. The model is supposed to yield results that are in line with the results of distance models of multidimensional ratio scaling since negative scalar products of percept vectors are admitted.

Integral inequalities for equimeasurable rearrangements

Abstract: For a real-valued function f on the domain [0,b], the equimeasurable decreasing rearrangement f* of f is defined as a function μ –1 inverse to μ, where μ(y) is the measure of the set {x|f(x) > y}. Inequalities connected with rearrangements of sequences as well as functions play a considerable part in various branches of analysis, and, for example, the concluding chapter of Hardy, Littlewood, and Polya [3] is devoted to rearrangement inequalities. Equimeasurable rearrangements of functions are also used by Zygmund [6, Vol. II, Chapters I and XII].

Comment on multiprogramming under a page on demand strategy

Abstract: It has been brought to my attention that Figure 5 in my paper [1] is not consistent with Figure 2 in the paper [2], as it was meant to be In the former the ordinate is defined as the incremental number of instructions executed between page demands, but the curve corresponds to the cumulative measure given in the latter

Generating Classification Rules FromDatabases

Abstract: Systems for inducing classification rules from databases are valuable tools for assisting in the task of knowledge acquisition for expert systems. In this paper, we introduce an approach for extracting knowledge from databases in the form of inductive rules. We develop an information theoretic measure which is used as a criteria for selecting the rules generated from databases. To reduce the complexity of rule generation, the boundary of the information measure is analyzed and used to prune the search space of hypothesis. The system is implemented and tested on some well known machine learning databases.

Translation in measure algebras and the correspondence to Fourier transforms vanishing at infinity.

Abstract: Let G denote a locally compact (not necessarily abelian) group and M(G) the collection of finite regular Borel measures on G. The set M(G) is a semisimple Banach algebra with identity under convolution *. It can be identified with the dml space of CO(G), the space of continuous complex-valued functions on G that vanish at infinity, with the sup-norm. The group G has a left-invariant regular Borel measure din(x) that is unique up to a constant and is called the left Haar measure of G. Let C ‘(G) denote the space of bounded continuous functions on G. For each x e G, we define on C ‘(G) the left-translation operator by the relation

A linear set of infinite measure with no two points having integral ratio

Abstract: It is not difficult to construct an unbounded set E on the positive real line such that, if x 1 , x 2 belong to E , then x 1 / x 2 is never equal to an integer. Our object is to show that it is possible to find such a set E which is measurable and of infinite Lebesgue measure. We were led to consider this problem through a study of those sets E , which are of infinite measure, yet, for each x > 0, nx є E for only a finite number of integers n . Sets of this type were first discovered by C. G. Lekkerkerker [2]. The set that we consider has both these properties. For another result on lattice points in sets of infinite measure see [1].

Some strengthenings of the Ulam nonmeasurability condition

Abstract: The purpose of this paper is to amplify the remarks made in the footnote on p 603 in [4] A complete account of the investigation initiated in [4] will be published in Fundamenta Mathematicae; in this paper we shall discuss various strengthenings of the Ulam nonmeasurability condition as well as their relative strength Some of the present results were announced in [6] We shall assume the familiarity with notations and terminology of [4] and [5]; we shall, however, review the more frequently used terms By an Ulam measure in a set X we shall mean a finitely additive 0-1 measure A whose domain is a field of subsets of X containing all the one-element subsets of X and such that ( { x )=0 for every xEX and ,u(X) = 1 A measure on X is a measure whose domain contains all subsets of X Given a cardinal m we will denote by Xm a set of cardinality m (in a topological context Xm will denote a discrete space of cardinality m) An infinite cardinal m is said to be Ulam nonmeasurable provided that no Ulam measure on Xm is countably additive An Ulam measure A is said to be rn-additive provided that ,u (U St) = sup { ,u (A): A E T } for every class J of subsets of the domain of ,u with card 9?m In [4] we have considered the following conditions on an infinite cardinal m mraEM: there is a collection of sequences A(t), A? **; of subsets of Xm such that card _ ? m and for every Ulam measure A on Xm the equality