Showing papers on "Measure (mathematics) published in 1968"

Christoffel's problem for general convex bodies

Abstract: Christoffel's problem, in its classical form, asks for the determination of necessary and sufficient conditions on a function φ, defined over the unit spherical surface Ω, in order that there exist a convex body K for which φ (u) is the sum of the principal radii of curvature at that boundary point of K where the outer unit normal is u. The figures Ω and K are in Euclidean n-dimensional space (n ≥ 3). It is assumed that φ is continuously differentiable and that K is of sufficient smoothness. A solution of Christoffe's problem was given in [6]. Yet that treatment is rather unsatisfactory in that the smoothness restrictions are set by the method rather than the problem, cf. [5; p. 60]. The present paper overcomes this defect. To do this it is first necessary to generalize the original problem so as to seek conditions on a measure M, defined over the Borel sets of Ω, in order that M be a first order area function for a convex body K. When K has sufficient smoothness, then φ is the Radon-Nikodym derivative of M with respect to surface area measure on Ω. It is this generalized Christoffel problem which is solved in what follows.

Statistical analysis of pair relationships: symmetry, subjective consistency and reciprocity.

Abstract: Pair data arise when, as in a sociomatrix, some relationship (e.g. "a choice") from person i to another person j is cross tabulated against the relationship from j to i. With a single such measure, the data may be analyzed in terms of tendencies toward symmetry. A coefficient of symmetry is discussed. With two items, four coefficients are shown to be possible: (1) a tendency toward symmetry in the first measure, (2) a tendency toward symmetry in the second measure, (3) subjective consistency, a tendency for i to give j similar choices on both measures, and (4) reciprocity, a tendency for i to give j a choice on one measure when j gives i a choice on the other measure. Lazarsfeld's analysis of spurious correlation is extended to give five rules that state the conditions under which one of these tendencies could be a spurious effect of the others. The results are used to codify possible outcomes in Tagiuri's "relational analysis," and to clarify some analytical issues in panel analysis of pair data. Tabulation procedures are explained.

Good lattice points in the sense of Hlawka and Monte-Carlo integration

Abstract: contained in Q~, of the absolute difference I p-1 v(F) /z (F) ], where #(P) is the s-dimensional measure of P , and v(I ~) is the number of points of X contained in it. Extending to an arbitrary number of dimensions [3] a theorem due to Koksrna [5], he showed that if /(x) (x = ) was of bounded variation in the sense of Hardy and Krause (for a precise definition of these concepts, see e.g. [3]), then

On the existence of consistent estimates and tests

Abstract: The paper concerns estimates of probability measures (=p-measures) determined from a countable number of independent realizations. The main results are: (2) For suitable topologies, the existence of a consistent sequence of estimates implies the existence of a sequence of estimates which is strongly consistent a. e. with respect to an arbitrary finite prior measure. In the case of a measurable parameter the existence of a consistent sequence of estimates implies strong consistency of the Bayes estimates for the quadratic loss function. (4) If a family ofp-measures is separable with respect to the supremum metric, there exists a sequence of strict estimates which is uniformly consistent with respect to the supremum metric on any totally bounded subset ofp-measures. (5) IfB is totally σ-bounded with respect to the supremum metric there exists a density-consistent sequence of estimates. IfB is separable with respect to the supremum metric there exists a sequence of estimates which is density consistent a. e. with respect to an arbitrary finite prior measure. (8) A sequence of uniformly consistent tests exists for all compact hypotheses in the case of an abstract sample space and for all weakly closed hypotheses in the case of a separable metric sample space.

Fourier analysis on wiener measure space

Abstract: The problem of representation of nonlinear systems on abstract spaces by a complete set of orthogonal functions defined on the same space was partly solved by Wiener, et al. (1–4) for nonlinear time invariant systems on the Wiener measure space (ΣI, BI, μ). This paper gives a simplified exposition of certain well-established results of Wiener and others (1, 6, 7, 8) in terms of non-rigorous concepts such as delta functions and white noise process in order to make the theory accessible to those knowing engineering mathematics. Proofs of Bessel's inequality and the Riesz-Fischer theorem which correspond directly to the modified Wiener's Orthogonal Set (9) are believed to be a contribution of this paper.

On equations with sets as unknowns.

Abstract: We shall present here a number of results in set theory concerning the decomposition of a set E in various ways as sum (union) of its subsets . These results have connection with problems on countably additive measure functions in abstract sets, but they may also bear on the problems of the axiomatics of set theory and generally on foundations of set theory itself . Some of these results employ the continuum hypothesis or the generalized continuum hypothesis. The several _problems which will be presented also put these hypotheses in a certain limelight . The impossibility of defining a countably additive measure for all subsets of a set of power of continuum (a measure which would vanish for subsets consisting of any single point) was first established with the use of the continuum hypothesis by Banach and Kuratowski .l Very shortly afterwards, one of us showed the impossibility of such a measure for subsets of a set of power NI without the use of any hypothesis.' The same result was shown there to hold for sets of higher powers, in fact, for all the accessible alephs . More recently, these results have been extended to a large class of inaccessibles as well . These results show that this "problem of measure" is closely related to fundamental problems concerning the role of axioms of set theory . Recent developments have further clarified these relations . Important results have been obtained by Scott, Solovay, Martin, and others . The proofs of these relations make use of the methods introduced by Paul Cohen in proving the independence of the continuum hypothesis. Both the results of Banach and Kuratowski and the stronger result of Ulam are obtained by exhibiting purely combinatorial schemata of decomposition of abstract sets with certain properties : B • and K • show a countable sequence of decompositions of a set of power of the continuum, each into countably many disjoint subsets so that, no matter how one takes a finite number of sets from each of these decompositions, the intersection of all these finite unions contains, at most, countably many points . Sierpinski3 generalized the B • and K • schema in the following way. There exists a sequence of decompositions into aleph disjoint sets, each so that if one is selected from any countably many of these (not necessarily all), the union of the selected sets gives the whole of the space, except perhaps for countably many points . Decomposition given by U • show, without the use of the continuum hypothesis, the following phenomenon . A set E of power NI can be decomposed countably many times into NI disjoint sets in the following way : A "matrix" of sets cam be constructed such that we have countably many rows and noncountably many columns . Sets in each row are disjoint. The anion of sets in any column gives the whole set E except for possibly countably many points. As is easy to see, the existence of such a decomposition (a sequence

On the range of an unbounded vector-valued measure

Abstract: 1. In t roduc t ion . Consider a vector-valued measure (S, E, /~), that is, a space S, a o'-field E o f subsets o f S and a countably additive funct ion def ined on E and taking values f rom R n if finite, or infinity deno ted by o0. We assume that 0¢ + a = oo, E~=I a, = 0e if II~gol a,II as n ~ o0. T h e purpose o f this note is to describe the range o f such a measure. In the case that /z(S) is finite and the m e a s u r e / z is non-atomic, then a result due to A. A. Liapunov [4] says that the range o f / z is compact and convex. In the case we consider, the range remains convex, as can be easily seen f rom the Liapunov theorem, but need not be closed. However , we have the following result.

Extended Radiation Sources (Point Kernel Integrations)

Abstract: Radiation sources are considered here to be materials which emit either gamma photons or neutrons. The measure of the strength of a source is its rate of emission. For a source which is distributed in space, the source strength may be measured in terms of the emission rate for a limited quantity of the material, as, for example, neutrons per second per cubic centimeter.

Note on Discordant Observations

Abstract: SUMMARY Given a set of observations xl, ..., xn, a Bayes measure of discordance of an observation x is defined to be the distance between the posterior distributions of a parameter, in the presence or absence of x. A measure of dissimilarity between two observations is also proposed. For large numbers of observations, these two measures may be approximated by simple functions of the log likelihood, thereby avoiding dependence on prior distributions. The theory is applied to data in which 13 judges ranked 20 mothers; each judge is supposed to give an independent observation on the mothers, with the analysis showing which judges are discordant from the rest and which judges are similar to each other.

Response measure effects in children as a function of type and schedule of reward

Abstract: Kindergarten children were administered a series of lever-pulling trials subsequent to either 10 min or 0 min social isolation. Lever-pulling responses were reinforced on 0%, 50%. or 100% schedules; reinforcements were social (“Good”) or candy. Isolation had no effect on performance; partial-reward superiority was demonstrated on a starting-speed measure for social reward, but on a movement-speed measure for candy-reward. The results were interpreted in terms of an hypothesis of frustrative nonreward and S’s past history of reinforcement.

On a measure of complexity for stochastic sequential machines

Abstract: The need for a measure different from the number of states in analyzing stochastic sequential machines is pointed out. Using the decomposition previously demonstrated in association with actual physical realization of stochastic sequential machines4, a particular measure of complexity C(M) for a given machine M is introduced. The computational aspect of C(M) is discussed and an example exhibiting two state-equivalent machines M1 and M2 with #{S1} ≫ #{S2} (#{Si} ≡ number of states of machine Mi i=1, 2) but C(M1) ≪ C(M2) is given. Areas for future research are pointed out.

A computer program to generate and measure random forms

Abstract: A computer program is described that generates random form stimuli by a method parallel to that described by Attneave & Arnoult (1956) for construction forms with both angles and arcs in their perimeters. The program also performs a physical analysis of the forms by computing the values of a large number of physical variables describing each form. Already existing forms also may be analyzed by the program. Thus, it has the capability to perform physical analyses of approximations to natural forms.

Exponential approximation with piecewise linear error criteria

Abstract: The problem of approximating a continuous function on the interval [0, T] with a linear combination of real exponential functions is considered. Rather than restricting the analysis to mathematically well behaved error criteria, such as least-squares, the theory of distributions is used to analyze criteria based on piecewise linear weighting of the error. For such criteria, it is found that the first and second derivatives of the error measure E with respect to the parameters of the approximating function can be obtained from exact and easily evaluated formulae. A descent search to find the solution to the approximation problem by minimizing E is thus economically feasible, since no inaccurate and time-consuming numerical integrations are required to produce the needed derivatives. As an example of such a criterion, approximation in the L1 norm is treated.

An Alternative "Fundamental" Axiomatization of Multiplicative Power Relations among Three Variables

Abstract: Suppose that the axioms of conjoint measurement hold for quantities having two independent components and that the axioms of extensive measurement hold for each of these components separately. In a recent paper, Luce shows that if a certain axiom relates the two measurement systems, then the conjoint measure on each component is a power function of the extensive measure on that component. Luce supposes that each component set contains all "rational fractions" of each element in that set; in this note we present an alternative form of the axiom relating the measurement systems that enables us to prove Luce's result without requiring that such "rational fractions" exist.

A measure of efficiency for a class of decision feedback communication systems

Abstract: A measure of efficiency is introduced for a broad class of decision feedback communication systems in which the feedback channel is noiseless and both the source and the forward channel are discrete and memoryless. The proposed measure R, which can be interpreted as a transmission rate, is a non-negative number that never exceeds the transmission rate Ro that can be achieved with the source and forward channel without the aid of feedback. However, there does exist a necessary and sufficient condition for which R = Ro. Roughly speaking, this condition is that requests for retransmission are made only when signals received in the past fail to impart any information as to which message symbol has been selected at the source. The measure R also has the property that it vanishes if, and only if, the rate Ro vanishes. In other words, the measure R reflects the idea that it is possible to convey information from source to receiver using feedback if, and only if, the forward channel is capable of carrying information without the aid of feedback.

Digital simulation of continuous dynamic systems: an overview

Abstract: There has been a growing interest over the last decade, both in this country and abroad, in the use of the digital computer to model continuous dynamic systems. Prior to 1960, and still true in large measure today this application area had been the exclusive province of the analog computer. The initial motivation for integrating the ordinary differential equations characteristic of continuous system simulation models on the digital computer was one of accuracy; given enough time, solutions of essentially unlimited accuracy could be obtained from the digital computer. These accurate, but costly, solutions were then used as consistency checks on the many hundreds of solutions characteristic of a simulation study on a high speed electronic analog computer.