Showing papers on "Measure (mathematics) published in 1976"

Poverty: An Ordinal Approach to Measurement

Abstract: The primary aim of this paper is to propose a new measure of poverty, which should avoid some of the shortcomings of the measures currently in use. An axiomatic approach is used to derive the measure. The conception of welfare in the axiom set is ordinal. The information requirement for the new measure is quite limited, permitting practical use.

On the Complexity of Finite Sequences

Abstract: A new approach to the problem of evaluating the complexity ("randomness") of finite sequences is presented. The proposed complexity measure is related to the number of steps in a self-delimiting production process by which a given sequence is presumed to be generated. It is further related to the number of distinct substrings and the rate of their occurrence along the sequence. The derived properties of the proposed measure are discussed and motivated in conjunction with other well-established complexity criteria.

A Unifying Tool for Linear Multivariate Statistical Methods: The RV-Coefficient

Abstract: Consider two data matrices on the same sample of n individuals, X(p x n), Y(q x n). From these matrices, geometrical representations of the sample are obtained as two configurations of n points, in Rp and Rq It is shown that the RV‐coefficient (Escoufier, 1970, 1973) can be used as a measure of similarity of the two configurations, taking into account the possibly distinct metrics to be used on them to measure the distances between points. The purpose of this paper is to show that most classical methods of linear multivariate statistical analysis can be interpreted as the search for optimal linear transformations or, equivalently, the search for optimal metrics to apply on two data matrices on the same sample; the optimality is defined in terms of the similarity of the corresponding configurations of points, which, in turn, calls for the maximization of the associated RV‐coefficient. The methods studied are principal components, principal components of instrumental variables, multivariate regression, canonical variables, discriminant analysis; they are differentiated by the possible relationships existing between the two data matrices involved and by additional constraints under which the maximum of RV is to be obtained. It is also shown that the RV‐coefficient can be used as a measure of goodness of a solution to the problem of discarding variables.

A measure associated with axiom-a attractors.

Abstract: The future orbits of a diffeomorphism near an Axiom-A attrac- tor are investigated. It is found that their asymptotic behavior is in general described by a fixed probability measure yt carried by the attractor. The measure It has an exponential cluster property, and satisfies a variational principle.

Exponential Models, Maximum Likelihood Estimation, and the Haar Condition

Abstract: Let f(x) be a probability density function with respect to the non-atomic measure μ, over the set χ. Suppose f(x) = exp [Σ m i = 1 τiϕi(x) − ψ m (τ)] for × ∈ χ, where ψ m (τ) = ψ m (τ1, τ2, …, τm ) is well-defined by exp [ψ m (τ)] = ∫ exp [Σ m i = 1, τiϕi,(x)] dμ(x) when the right side is finite, and the integration is over χ. Let ϕ0(x) ≡ 1 on χ and assume {ϕi(X)} m i = 0 is a collection of functions which satisfy the Haar Condition. Using convexity properties, we obtain some results on the almost sure existence or nonexistence of the MLE for τ.

A combinatorial property of p κλ

Abstract: In a paper on combinatorial properties and large cardinals [2], Jech extended several combinatorial properties of a cardinal κ to analogous properties of the set of all subsets of λ of cardinality less than κ, denoted by “pκλ”, where λ is any cardinal ≤κ. We shall consider in this paper one of these properties which is historically rooted in a theorem of Ramsey [10] and in work of Rowbottom [12].As in [2], define [pκλ]2 = {{x, y}: x, y ∈ pκλ and x ≠ y}. An unbounded subset A of pκλ is homogeneous for a function F: [pκλ]2 → 2 if there is a k < 2 so that for all x, y ∈ A with either x ⊊ y or y ⊊ x, F({x, y}) = k. A two-valued measure u on pκλ is fine if it is κ-complete and if for all α < λ, u({x ∈ pκλ: α ∈ x}) = 1, and u is normal if, in addition, for every function f: pκλ → λsuch that u({x ∈ pκλ: f(x) ∈ x}) = 1, there is an α < λ such that u({x ∈ pκλ: f(x) = α}) = 1. Finally, a fine measure on pκλ has the partition property if every F: [pκλ]2 → 2 has a homogeneous set of measure one.

Rotundity of Orlicz Spaces

Abstract: Rotund Orlicz spaces and Orlicz spaces that contain isomorphic copies of l ∘ and c o are characterized in the class of Orlicz spaces over measure spaces that are not purely atomic.

Time evolution of infinite classical systems with singular, long range, two body interactions

Abstract: Existence of dynamics for infinitely many hard-spheres inv dimensions is proven in a set of full equilibrium measure.

Similarity indices I: what do they measure.

Abstract: A method for estimating the effects of environmental effusions on ecosystems is described. The characteristics of 25 similarity indices used in studies of ecological communities were investigated. The type of data structure, to which these indices are frequently applied, was described as consisting of vectors of measurements on attributes (species) observed in a set of samples. A general similarity index was characterized as the result of a two-step process defined on a pair of vectors. In the first step an attribute similarity score is obtained for each attribute by comparing the attribute values observed in the pair of vectors. The result is a vector of attribute similarity scores. These are combined in the second step to arrive at the similarity index. The operation in the first step was characterized as a function, g, defined on pairs of attribute values. The second operation was characterized as a function, F, defined on the vector of attribute similarity scores from the first step. Usually, F was a simple sum or weighted sum of the attribute similarity scores. It is concluded that similarity indices should not be used as the test statistic to discriminate between two ecological communities.

A Central Limit Theorem for the Subadditive Process and Its Application to Products of Random Matrices

Abstract: The purpose of the present paper is to give a central limit theorem for subadditive process in the sense of J. F. C. Kingman (cf. [3], [4]). Throughout this article (O5 , P) denotes a probability space on which all random variables are defined. Let T be a measure preserving transformation in what follows. According to Kingman, a family (xsy9 s

Composition operators induced by rational functions

Abstract: A necessary and sufficient condition for a rational function to define a composition operator on LP ( jt) is given in this paper, where yt is the Lebesgue measure on the Borel subsets of the real line. In particular, all polynomials inducing composition operators are completely determined. 1. Preliminaries. Let (X, 5, X) be a a-finite measure space and 4 be a measurable transformation on X into itself. Then we (roughly) define a linear transformation C. on the Banach space LP(X) (p > 1) into the space of all complex valued functions on X by C,1f = f o +. If C. is continuous with range in LP (X), then we call it a composition operator on LP (X) induced by 4. The following theorem will be needed. THEOREM 1. Let 4 be a measurable transformation on X into itself. Then C is a composition operator on LP (X) if and only if there exists an m > 0 such that A '(E) 0 such that X4 1(E) < mX(E) for every measurable set E. Then the measure X+1 is absolutely continuous with respect to X. Let fo denote the Radon-Nikodym derivative of X4 -. Then JE fdX= -'(E) S mX(E)= mdA. Therefore f0 < m (a.e.). Let f LP (X). Then fIf O (IP dX = JfIP dX<-1 = IIpfo dX < mIlAlp. This shows that C,, is bounded. This completes the proof of the theorem. In this paper we are interested in studying composition operators in a special case. We take X to be equal to R, the set of real numbers and we take yt to be the Lebesgue measure on the Borel subsets of R. Now 4 is a measurable (Borel) real valued function on R. The following theorem gives a Received by the editors September 19, 1975. AMS (MOS) subject classifications (1970). Primary 47B99; Secondary 47B99.

A simple proof of the maximal ergodic theorem

Abstract: Let X be a σ-finite measure space and let Tk, k any integer, be a group of positive linear transformations in Lp(X) such that with C independent of / and k. From now on / will be a positive function in Lp(X) and we will use the following notation:

The Structure of Imagery.

Abstract: Children's drawings of objects about to be moved in space (anticipatory images) were studied in relation to their judgments about Euclidean spatial relations. Tasks assessing Euclidean geometric operations were administered to 102 girsl between the ages of 4 and 13 years. 5 operative levels resulted: (a) failure on all operations tasks, (b) success on conservation of length, (c) success on conservation and 1-dimensional measurement, (d) success on the latter 2 tasks plus measurement in 2 dimensions, and (e) success on the latter 3 tasks and coordination of internal and external reference frames. The same children performed on 6 imaginal tasks involving movements of objects. Results confirmed several theoretical expectations: (a) children who were unable to measure had difficulty imaging successive positions for 1-dimensional movement; (b) children who were unable to coordinate a point in space by reference to 2 axes were unable to draw states of movement correctly for 2-dimensional movement; (c) children's errors in drawing successive positions of moving objects were related to the structure of operations at the lower levels; and (d) the coordination of object positions over successive states was dependent on the ability to measure from an external frame. The results were interpreted as support for Piaget and Inhelder's theory of imagery development.

A Partial Vindication of Ergodic Theory

Abstract: Ergodic theory may well be a branch of physics which has received less attention from philosophers than any other shedding comparable light on philosophical problems. It is, therefore, heartening to note the care and attention bestowed upon it by Sklar in his "Statistical Explanation and Ergodic Theory" [11]. Sklar questions the relevance of ergodic theory as a basis for statistical thermodynamics on two grounds: first, it deals with infinite time averages, whereas all averages we measure in a mechanical system are taken over a finite interval of time; second, it summarily dismisses sets of Lebesgue measure zero on which pathological behavior (in the sense that the infinite time averages of functions on these sets may not exist or may differ from the phase averages of those functions over the rest of the phase space) occurs. In this reply I wish to defend ergodic theory against both of these criticisms and to justify the consideration of infinite time averages as well as the dismissal of sets of Lebesgue measure zero. This defense is not to be construed as an unqualified vindication of ergodic theory, however, for the relevance of ergodic theory can be questioned on a third ground, namely, its assumption (necessary to the derivation of Birkhoff's theorem) that the system is mechanically isolated from the rest of the universe. This assumption is not only not strictly true, but it is not even a good approximation.

11.—Equations d'Evolution du Second Ordre Associées à des Operateurs Maximaux Monotones

Abstract: This paper extends some recent results of V. Barbu and H. Brezis. It is concerned with bounded solutions of the problem pu″+qu′ ∈ Au, u(0) = a, where A is a maximal monotone operator in a real Hilbert space H and p and q are real functions. Existence and uniqueness theorems are proved, with results on integrability of solutions in various measure spaces on R+. T(t) denotes the family of contractions of D(A) generated by the equation and we obtain a regularising effect on the initial data. Some properties of this family of contractions are studied.

On the functional integral for generalized Wiener processes and nonequilibrium phenomena

Abstract: The attention will be focussed on a generalized Wiener diffusion process for which the macroscopic evolution y = c1(y) equals zero, of course, and where the variance of the process obeys gs2 = c2(y). The diffusion function c2(y) may be state dependent in an arbitrary way. We invoke our treatment of the general time-local Gaussian process as presented in a previous paper. This process will be seen to define a generalized functional Wiener measure. This measure has already been used implicitly in earlier work being concerned with nonlinear, nonequilibrium Markov processes. The sum of the generalized measure over the entire function space will be shown to be exactly related to the general Fokker-Planck equation for the driftless diffusion process. The relation between the well-defined functional sum and its corresponding functional integral will be studied in detail. The analysis demonstrates in clear fashion the origin of the deviations from other approaches, and provides an extension of our previous results on nonequilibrium, nonlinear phenomena to include generalized diffusion processes.

Cognitive Complexity and Social Prediction

Abstract: Problems in definition and measurement of ‘cognitive complexity’ are discussed and the importance of examining the predictive validity of the different measurement techniques is stressed. An experiment is described which examines the intercorrelations between several measures of ‘complexity’ and their relation to the subjects' predictions of how others' personal constructs are employed. The ‘hierarchical complexity’ measure of Smith & Leach (1972), Bannister's (1960, 1962) ‘Intensity’ score and Mehrabian & Komito's (1968) measure of factorial complexity share no variance with any scores derived from the prediction task developed in the reported experiment. The hypothesis that more ‘cognitively complex’ persons (Bieri's definition, 1966) are ‘set’ to seek less common features in their social environment receives strong support from the experiment. That this ‘set’ may lead to a significant, but spurious, association between ‘complexity’ and accuracy in social predictions is suggested by an analysis of the prediction tasks used in other studies. Bannister's Intensity is highly related to Mehrabian & Komito's measure, but contrary to expectation is unrelated to Bieri's complexity.

Interpolation of Lp-Spaces

Abstract: We investigate the real and complex interpolation of L p -spaces and Lorentz spaces over a measure space. In particular, we prove a generalized version of the Marcinkiewicz theorem (the Calderon-Marcinkiewicz theorem). We also investigate the real and the complex interpolation spaces between L p -spaces with different measures, thus extending a theorem by Stein and Weiss. In Section 6, we consider the interpolation of vector-valued L p -spaces of sequences, thus preparing for the interpolation of Besov spaces in the next chapter.

On the Asymptotic Behaviors of Transition Probability Densities of One-Dimensional Diffusion Processes

Abstract: problem of describing the asymptotic behavior of p(t, a, a) as f-»0[oo] in terms of the speed measure m. Concerning this problem, there have been several works when a e Q is the regular left end point except a trap: By completing I. S. Kac's result [3], Kasahara [4] showed that p(t, a, a) varies regularly in t if and only if m[0, x) does so in x. I. S. Kac [2] discussed the condition of the convergence of integrals related to the spectral function corresponding to the measure m(dx) and this result gives conditions for the convergence of the integrals \ cp(f)p(t9 Jo+ f°° a, a)dt and \ 0 or +00 using some nonincreasing functions Fa(t) defined in terms of the speed measure in. The proof of these inequalities is based on some inequalities similar to that of I. S. Kac [3] concerning

Estimation and decision for observations derived from martingales: Part I, Representations

Abstract: The observation process y considered is an additive composition of continuous and discontinuous components. The additive Gaussian, point, and jump process models, treated separately in the past, are all included here simultaneously. Representations for y in terms of its innovations and following a Girsanov-type measure transformation are derived. These are then used to develop a measure form of Bayes' rule that provides a convenient tool for the study of estimation and decision problems arising in a variety of applications including communication and control.

On the Dual of Besov Spaces

Abstract: This paper is a supplement to the author's paper [6]. Here we shall discuss the space Bpi00_($), a closed subspace of JB£>JO(,G), and determine the dual of Besov spaces ££ig(J2) . For a measure space (M, /*) and a Banach space X by L (M, p. ; X) we denote the space of all X-valued strongly measurable functions f(x) such that \f(x)lx 0 and as |y|-»oo. We shall make use of the following conventions: p , l/oo -=l/oo = 0. The space Bpt00_ (fl; X) is defined as follows:

A short proof of Schoenberg’s theorem

Abstract: Using positive semidefiniteness of Laplace transforms, we give a short and simple proof of Schoenberg's theorem characterising radially symmetric positive semidefinite functions on a Hilbert space. A slight generalisation of this theorem is also given. In his paper Metric spaces and completely monotone functions [3], I. J. Schoenberg raises the question about the connection between the class of Fourier transforms of (finite, nonnegative) measures in Euclidean spaces and the class of Laplace transforms of (finite, nonnegative) measures on the halfline R+ = [0, oo). He states: "In spite of the entirely different analytical character of these two classes, a certain kinship was to be expected for the following two reasons: 1. In both classes the defining kernel is the exponential function. 2. The less formal reason of the similarity of the closure properties of both classes, for both classes are convex, i.e. alf, + a2f2 (a, > 0,a2 > 0) belongs to the class if f1 and f2 belong to it, multiplicative, i.e., also f1 . f2 belongs to the class, and finally closed with respect to ordinary convergence to a continuous limit function." The answer Schoenberg could give to the above question was the remarkable result that to each continuous function f: R+ -C with the property that f o I ln is positive semidefinite on Rn for all n (I n denoting the Euclidean norm) there exists a finite nonnegative measure on R+ with Laplace transform f(V/t) [3, Theorem 2]. Positive semidefiniteness of a mapping g: Rn -, C has the meaning that the kernel K(x,y) = g(x y) is positive semidefinite. The proof of this theorem, even that given in the more recent book of Donoghue [1, pp. 201-206], however is rather complicated and technical in nature. But there is a further common feature of Fourier and Laplace transforms seemingly unknown until quite recently: Laplace transforms, too, are characterised essentially by positive semidefiniteness. More precisely, a functionf: RP -+ C is the Laplace transform of a finite nonnegative measure on the Borel sets of RP if and only if / is continuous, bounded and positive semidefinite in the sense that = Ej=1 ai ajf (ti + tj) > 0 for all (a1, . . .,ak) E Rk, (tl, , tk) E (RP))k, and k E N [2, Satz 1]. Using this we give a new proof of THEOREM 1 (SCHOENBERG). A continuous function f: R+ -C has the property that f o I In is positive semidefinite on Rn for all n E N if and only if there exists a finite nonnegative measure A on R+ such that Received by the editors July 17, 1975. AMS (MOS) subject classifications (1970). Primary 43A35; Secondary 44A 10. ? American Mathematical Society 1976

The uniform distribution in signal detection theory

Abstract: The robustness of signal detection theory (SDT) is investigated with respect to the form of the underlying distributions. Usually these distributions are taken to be normal; here an SDT model based on two overlapping uniform (rectangular) distributions is examined, for the Yes/No experiment and the rating-method experiment. In the Yes/No case the SDT measure (using uniform distributions) is found to be equivalent to a measure recently proposed by Hammerton & Altham (1971), and, from contingency-table considerations, it is a measure likely to give similar conclusions to the SDT measure using normal distributions. In the rating-method case it is surprising to find that the SDT model with uniform distributions yields non-significant goodness-of-fit statistics for many sets of data.

Made-to-measure potential functions in lattice dynamics—solid Cl2

Abstract: A simple method is outlined for identifying potential functions designed to reproduce given normal mode frequencies, equilibrium conditions and other constraints. It is applied to a model of solid Cl2 and a family of potentials found which satisfy a large number (14) of constraints but which have other, implausible features. The application is presented as a cautionary illustration of Szigeti's theorem [2]. A more useful area of application is seen to be in refining potentials whose principal features are already known.